𝕿𝖊𝖈𝖍𝖗𝖎𝖌𝖍𝖙𝖘 Bulletin for Monday, July 18, 2022
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⦿ 4 Years of Monopoly Tony (Mr. Campinos) Violating the Rights of Free Association/Expression/Speech as EPO President | Techrights

⦿ Debian Takes Money From Microsoft  -  an Epic Mistake | Techrights

⦿ [Meme] EPO Management Coming Out of the Closet | Techrights

⦿ Freedom of Speech Cannot Coexist With Crime at the EPO | Techrights

⦿ IRC Proceedings: Sunday, July 17, 2022 | Techrights

⦿ Razik Alex Menidjel: Software Patents Guru or Career-Climbing Poser Who Never Wrote Code? | Techrights

 ䷼ Bulletin articles (as HTML) to comment on (requires login):
		http://techrights.org/2022/07/18/battistelli-the-second-vs-speech/#comments
		http://techrights.org/2022/07/18/debconf22-welcomes-microsoft/#comments
		http://techrights.org/2022/07/18/epo-skeletons-in-the-closet/#comments
		http://techrights.org/2022/07/18/freedom-of-speech-epo/#comments
		http://techrights.org/2022/07/18/irc-log-170722/#comments
		http://techrights.org/2022/07/18/razik-alex-menidjel-swpats/#comments

 ䷞ Followed by Daily Links (assorted news picks curated and categorised):
		http://techrights.org/2022/07/18/alpine-3-16-1-released/#comments
		http://techrights.org/2022/07/18/assault-against-free-software/#comments
		http://techrights.org/2022/07/18/cutelyst-relicensed/#comments
		http://techrights.org/2022/07/18/steam-deck-at-4000-games/#comments

                    ䷩ 𝚕𝚒𝚗𝚎 65

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✐ 4_Years_of_Monopoly_Tony_(Mr._Campinos)_Violating_the_Rights_of_Free
Association/Expression/Speech_as_EPO_President⠀✐


Posted in Europe, Law, Patents at 9:30 am by Dr. Roy Schestowitz


The ‘anti-Jorge‘ Campinos or Jorge’s archenemy


          🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇ILOAT_Judgment_4551_“You’ve_Got_Mail”⦈_
               ILOAT Judgment 4551, as_explained_here_last_week


Summary: Communicating with colleagues is a right; what the EPO did for nearly
a decade was a violation of rights; Campinos violated these rights for 4 years
and even showed his teeth (along with Barking Bergot, the Battistelli Bulldog
he inherited and kept) just months after his term had started, so we know we’re
dealing with merely Battistelli the Second (he’s still not complying with the
ILOAT judgment and messages remain blocked) with a_second_term_one_year_away


The Central Staff Committee (CSC) of the EPO has circulated material which is
applicable to many institutions where management tries to interfere with staff
communications. What’s interesting is that after Benoît_Battistelli‘s term we
have seen his compatriot António_Campinos doing nothing at all to improve
things. As we showed years ago, he actively interfered with correspondence
between staff representatives and their colleagues. He had no intention of
fixing this.


“The Tribunal’s ruling is, as usual, coming far too late to have a truly
meaningful effect.”As it turns out, based on a ruling nearly a decade late,
this was all along unlawful.


The CSC notes that “[o]n 6 July 2022, the Tribunal ruled in Judgment 4551 that
the EPO Communiqué of 31 May 2013 restraining the use of email to no more than
50 recipients without prior authorisation breached the fundamental right to
freedom of association. The Tribunal considered that the true reason was to
exercise prior censorship on the content of communications (consideration 11,
p. 15, last §).”


“It also confirmed that the mass emails of the staff representation and unions
at the material time did not exceed the limits to freedom of opinion and
speech, and hence did not justify a mechanism of a prior authorisation
(consideration 11, p. 16, last § – p. 17, last §).”


“After 9 years of breach of freedom of communication (including 4 years during
the mandate of Mr Campinos), the Tribunal has declared the ban of mass emails
unlawful. Mr Campinos must now lift the technical restrictions imposed on staff
and their representation. It is yet another judgment confirming his lack of
willingness to restore fundamental rights at the EPO on his own initiative.”


Much of the same can be seen in the abstract below. “Since 2012,” the CSC notes
(yes, an entire decade!), “the EPO has been consistently testing the limits of
Employment Law and feels only governed by the Tribunal’s ruling.”


The Tribunal’s ruling is, as usual, coming far too late to have a truly
meaningful effect. This enormous delay is being exploited by gutsy abusers, who
treat laws, constitutions and treaties no better than toilet tissue.


For an HTML version (along with text and GemText later today) we’ve converted
the whole publication and left it intact, verbatim, with no annotation added:


     Zentraler Personalausschuss
     Central Staff Committee
     Le Comité Central du Personnel


     Munich, 14/07/2022
     sc22099cp – 0.2.1/0.3.2/5.1


     § ILOAT Judgment 4551⠀➾


     § “You’ve Got Mail”⠀➾


     On 6 July 2022, the Tribunal ruled in Judgment 4551 that the EPO
     Communiqué of 31 May 2013 restraining the use of email to no more
     than 50 recipients without prior authorisation breached the
     fundamental right to freedom of association. The Tribunal considered
     that the true reason was to exercise prior censorship on the content
     of communications1. It also confirmed that the mass emails of the
     staff representation and unions at the material time did not exceed
     the limits to freedom of opinion and speech, and hence did not
     justify a mechanism of a prior authorisation2.


     After 9 years of breach of freedom of communication (including 4
     years during the mandate of Mr Campinos), the Tribunal has declared
     the ban of mass emails unlawful. Mr Campinos must now lift the
     technical restrictions imposed on staff and their representation3. It
     is yet another judgment confirming his lack of willingness to restore
     fundamental rights at the EPO on his own initiative. Since 2012, the
     EPO has been consistently testing the limits of Employment Law and
     feels only governed by the Tribunal’s ruling.


     I. Changing his own rules during the game


     Beginning of 2013, the use of email communication was governed by
     Communiqué No. 10 of 29 March 2006 and the announcement
     “Communication means for CSC and LSC” of Mr Battistelli dated 28
     December 2011 allowing staff committee members to have recourse to
     mass emails. However, the social atmosphere at the EPO had started to
     deteriorate with the introduction in 2012 of the Investigation
     Guidelines severely criticised4 by the union as an authoritarian
     drift.


     On 30 January 2013, Mr Battistelli appointed5 as Principal Director
     Human Resources the wife of his closest advisor, thereby raising the
     suspicion of nepotism at the EPO. Management then started to express6
     disturbing intentions in the working groups on “performance
     management” and “well-being”. The deterioration continued with
     Communiqué No. 20 in which Mr Battistelli accused staff
     representatives of making unfounded personal attacks thereby
     triggering a rebuttal7 from those accused. During this period, staff
     representatives made use of mass-emails to timely inform all staff of
     the chain of events and the consequently planned industrial actions.


     On 13 May 2013, Mr Battistelli issued Communiqué No. 26 entitled
     “When enough is enough – The use of mass emails within the Office”
     announcing the setting of new rules on mass communication.


     ________
     1 Judgment 4551 cons. 11, p.15, last §
     2 Judgment 4551 cons. 11, p.16, last § – p.17, first §
     3 From a technical point of view this task should require only a few
     minutes from BIT services. Indeed, more than decade before the
     issuance of the President’s Communiqué of 2013, staff committees and
     unions were already sending mass emails to all EPO staff without any
     hindrance from the Administration.
     4 “Investigation Guidelines = EPO becoming a Police State”, SUEPO
     Munich paper of 15.11.2012 (su12112mp)
     5 “New Principal Director Human Resources as of 1 February 2013”, VP4
     Communiqué of 30.01.2013
     6 “Report on the CSC with the President on 20 November 2012 in
     Munich”, CSC paper of 26.11.2012 (sc12122cp)
     7 “Story-telling by Mr Battistelli”, SUEPO paper of 27.02.2013
     (su13024cp)


     =====================================================================
     Shortly after, on 15 May 2013, the union reacted8 with a “translation
     into plain English” containing a satirical cartoon.


     The next day, on 16 May 2013, a meeting9 took place between Mr
     Battistelli and the Central Staff Committee (CSC) during which Mr
     Battistelli informed of his intention to ban mass emails which had
     not obtained the requisite authorisation (see his Communiqué 27).


     On 31 May 2013, the Vice-President of DG4 (Mr Topic at the material
     time) issued a Communiqué entitled “Communication in the workplace”
     announcing that as from 3 June 2013 only authorised employees would
     be able to send emails to more than 50 recipients.


     II. Judgment 4551: the ban on mass emails of more than 50 recipients
     is unlawful


     On_receivability
     The complainants challenged the Communiqué No. 26 and the Communiqué
     of 31 May 2013. They were at the material time permanent EPO
     employees and also elected members of the staff committee and
     executive members of the SUEPO trade union.


     In their capacity as staff representatives, they argued that the ban
     on the use of mass emails affected them in their daily work since
     they could not “maintain suitable contacts with the staff” nor
     provide “a channel for the expression of opinion by staff” (Article
     34(1) ServRegs).


     In their capacity as individual employees they alleged an
     infringement of their rights to freedom of association, communication
     and speech which are granted to EPO employees.


     The Tribunal confirmed (cons. 3) that each staff member of an
     international organisation has a right to freely associate and the
     organisation has a corresponding duty to respect that right. This_is
     a_necessary_element_of_their_employment (see Judgment 4194, cons. 7;
     Judgment 911, cons. 3).


     According to the Tribunal’s case law, a general decision cannot be
     challenged by a staff member unless and until an individual decision
     is taken, “[h]owever there are exceptions where the genera decision
     does not require an implementing decision and immediately and
     adversely affects individual rights.” (Judgment 3761, cons. 14). This
     is equally true regarding the right to associate freely (Judgment
     496, cons. 6, and 3414, cons. 4).


     The Tribunal considered that the Communiqué of 31 May 2013
     immediately and directly affected the right of staff members to
     freely associate, by stating that as from 3 June 2013 emails sent to
     more than 50 recipients would be allowed only if authorised, and, if
     not, they would be automatically blocked and not dispatched. As to
     Communiqué No. 26 of 13 May 2013, it was the first step of the
     process; therefore, it was properly contested together with the final
     Communiqué of 31 May 2013.


     On_the_merits
     Freedom of association necessarily involves freedom of communication


     The Tribunal recalled (cons. 9) that there can be no doubt that
     freedom of association is a well-recognised and acknowledged
     universal right which all workers should enjoy. It is recognised as a
     right by the Tribunal as well as by a large number of international
     conventions and declarations, as enshrined in Article 30 ServRegs
     providing that “Permanent employees shall enjoy freedom of
     association; they may in particular be members of trade unions or
     staff associations of European civil servants”.


     The role of staff associations or unions is to represent the
     interests of members primarily in dealing


     ________
     8 “Communiqué No. 26 – Translation into plain English”, SUEPO paper
     of 15.05.2013 (su13068cp)
     9 “Note to staff on the Meeting with the President on 16 May 2013”,
     CSC paper of 22.05.2013 (sc13074cp)


     =====================================================================
     with their employing organisation on issues concerning the staff.
     Staff associations or unions should be able to do so unhindered or
     uninfluenced by the Administration of the employing organisation.
     Were it otherwise, the role would be compromised (see Judgment 4482,
     consideration 8).


     Freedom of association necessarily involves freedom of discussion and
     debate (cons. 9, p.12, last §), as well as freedom of communication,
     information and speech in all forms (cons. 10). Such rights are
     vested not only in their authors (usually the staff representatives),
     but also in the recipients. Every_limitation_to_the_right_of_staff
     representatives_to_send_mass_emails_to_the_staff_members,_is_also_a
     limitation_to_the_right_of_the_staff_members_to_receive_mass_emails.


     Free communication, information, and speech also imply:


     i. the right to the confidentiality of communication, information,
     and speech; and
     ii. the right to freely choose the means by which the communications
     are sent, information is provided, and speeches are given.


     Like any other freedom such freedom has its bounds


     In the proceedings, the Administration abundantly cited Judgment 3516
     in which the Tribunal held that, in specific cases, a prior
     authorisation for the dispatching of mass emails could be justified
     as “the right to freedom of speech does not encompass action that
     impairs the dignity of the international civil service, or gross
     abuse of this right and, in particular, damage to the individual
     interests of certain persons through allusions that are malicious,
     defamatory or which concern their private lives.”


     The Tribunal noted (cons. 9) that an organisation acts unlawfully
     only if the conditions for implementing a mechanism in practice leads
     to a breach of that right, for example by an unjustified refusal to
     circulate a particular message (see Judgment 3156, considerations 15
     and 16).


     When considering the Communiqué No. 10 of 29 March 2006 and the
     announcement of 28 December 2011, the Tribunal concluded that the EPO
     had actually already reached a reasonable balance in the use of mass
     emails (cons. 11).


     The Communiqué of 31 May 2013 is unlawful


     The Tribunal considered (cons. 11) that the Communiqué of 31 May 2013
     is unlawful because it sets out an indiscriminate limitation, without
     providing specific reasons for the prior authorisations, irrespective
     of technical difficulties for the dispatch, and moreover for an
     indefinite time. In the view of the Tribunal, the wording of the two
     impugned Communiqués revealed that the true reason for the
     requirement of the prior authorisation was to exercise a prior
     censorship on the content of the communications (cons. 11).


     Contrary to Judgment 3156 (cons. 16), the Tribunal held that in the
     case of the EPO, prior authorisation was imposed as a general
     preventive measure, independent of specific violations without any
     time limit.


     In the proceedings, the Administration filed as evidence several
     communications from the staff representation or the union, and
     focused in particular on the SUEPO paper10 of 15 May 2013 denouncing
     nepotism at the EPO with a satirical cartoon.


     The Tribunal considered the evidence at hand and was satisfied that
     it did not exceed the limits of freedom of opinion and speech, and
     therefore did not justify the mechanism of prior authorisation. Staff
     representatives have the right to criticise the employer. In the
     present case, the measures taken by the EPO were disproportionate
     (cons. 11). The EPO is not allowed (cons. 12) to impose certain
     communication means (such as, in the present


     ________
     10 “Communiqué No. 26 – Translation into plain English”, SUEPO paper
     of 15.05.2013 (su13068cp)


     =====================================================================
     case, the dedicated intranet webpage) rather than others (the mass
     emails). This is particularly true where the means offered (or
     imposed) are more complicated and less viable than the other ones
     technically available, or are under the supervision of the EPO
     itself.


     Finally (cons. 12), the ability of a body representing the staff to
     circulate emails to all staff members is not “a privilege”. Such body
     “has a legitimate right to avail itself of this facility, unless
     there is good cause for restricting it” (see Judgment 3156,
     consideration 14).


     Decision


     In view of the violations of the right to freedom of association,
     Communiqué of 31 May 2013 is set aside in part, and the former rules
     on mass emails in Communiqué No. 10 and in the announcement of 28
     December 2011 reinstated.


     III. Did Mr Campinos attempt to remedy the situation before the
     Judgment?


     After joining the EPO, on 5 November 2018, Mr Campinos announced11 a
     pilot allowing staff committees to send mass_email_invitations_for
     two_of_their_respective_general_assemblies_per_year. Such invitations
     could only contain the place, date and time of the assembly and the
     agenda items. No other content was allowed. Even these invitations
     required prior approval of VP4 and could only be sent by the EPO
     services and not directly by the respective staff committee.


     When the Covid-19 pandemic broke out in 2020, the Office premises
     were closed and teleworking became the rule. In order to maintain
     suitable contacts with staff in the new virtual environment, the CSC
     asked the ban on mass-emails to be lifted. Mr Campinos refused. Mr
     Campinos also turned down the requests for running Technologia
     surveys on @epo.org email addresses.


     On 18 November 2020, a meeting took place between Mr Campinos and
     SUEPO. Mr Campinos addressed12 the topic of mass emails as follows:
     “In any case, if SUEPO wants back access to mass-emails, it would be
     only in the framework of a Memorandum of Understanding. SUEPO will
     lose in Geneva anyway.” Since then SUEPO never met with Mr Campinos
     again.


     On 30 August 2021, Mr Campinos sent to the CSC a “Draft Agreement on
     the use of communication between the EPO and the CSC” and announced13
     its aim at “focusing on the safeguards to maintain professional
     decency according to ILOAT standards in all communication”.


     The proposal was actually below all standards and could not be
     considered as serious. Essentially, the document defined that mass
     emails would only be sent to staff members who had priorly opted in
     (like for RSS feeds). It also made reference to Judgment 3156, cons.
     15 to justify that the President would still be allowed to suspend
     the sending of mass emails temporarily or permanently if they are not
     to his taste.


     In the meantime, even the sending of Christmas wishes by the staff
     representation remained censored14 in December 2021, whereas the
     Administration did so.


     The Administration actually used the above proposal in legal
     proceedings before the Tribunal to argue that the matter was on the
     verge to be settled.


     ________
     11 “Communication means for Staff Representation Certain mass emails
     permitted”, President Communiqué of 05.11.2018
     12 “Report on the SUEPO meeting with the President on 18 November
     2020”, SUEPO paper of 01.12.2020(sc20037cp)
     13 “Report on CSC meeting: Update on strike regulations, the New
     Normal and Social Agenda 2021”,
     Communiqué of 17.09.2021
     14 “Censorship of Christmas wishes”, LSCMN paper of 12.01.2022
     (sc22001mp)


     =====================================================================
     IV. Conclusion
     After the illegal strike regulations15, the “Social Democracy”
     interference into staff representation elections16, the ruling of the
     Tribunal on the unlawful ban on mass emails17 is the third of the
     kind sanctioning an EPO breach of the fundamental right to freedom of
     association. The unlawful ban spanned over 9 years including 4 years
     during the mandate of Mr Campinos.


     Since 2012, the EPO has been consistently testing the limits of
     Employment Law and feels only governed by the Tribunal’s rulings. Mr
     Campinos will be President of the EPO until 2028. It is high time for
     him to reconsider his choice of advisors who have placed him in the
     untenable position of Head of an International Organisation who has
     repeatedly violated fundamental rights. We also question his decision
     to have the contracts of these advisors and high managers very
     recently renewed for another 5 years until 2027/2028.


     Following Judgment 4551 the CSC addressed Mr Campinos by letter on 7
     July 2022 stating that “we count on the immediate removal of all
     barriers to sending of mass emails by the CSC (and the LSCs) to the
     respective staff population”. At the time of drafting the present
     paper, the free sending of mass emails by staff committees is still
     impossible.


     We will keep you posted.


     The Central Staff Committee


     “A strange rumour has recently spread across Europe in some media,
     political circles and even with national judges: that the EPO has not
     been respecting fundamental human rights. What an accusation!”
     Management public statement18 of 18 March 2015


     ________
     15 Judgments 4430-4435
     16 Judgment 4482
     17 Judgment 4551
     18 https://www.epo.org/news-events/news/2015/20150318_de.html


Worse yet (than the above quote), the EPO_went_out_of_its_way_to_bully_bloggers
who_presented_evidence_of_the_abuse. In other words, the EPO’s dictators
resorted to yet more abuses to cover up evidence of their abuses. █


⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡇⣿⣿⣿⣿⣙⢮⢻⣿⣿⣿⡿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣮⣧⣤⣿⣾⣿⣿⣿⣮⣿⣯⣭⣤⣥⣬⣭⣴⣵⣵⣿⣽⣷⣷⣿⣿
⣿⡇⣿⣿⡿⣛⠎⡆⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣴⣼⣷⣿⣿⣵⣬⣤⣧⣿⣾⣾⣷⣿⣿⣦⣷⣿⣿
⣿⡇⣿⣿⡇⣋⢆⢇⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣯⣿⣼⣿⣿⣿⣿⣾⣿⣯⣿⣥⣽⣥⣯⣬⣽⣼⣿⣿⣽⣿⣿⣯⣽⣿⣿
⣿⡇⣿⣓⠮⢒⣡⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⢸⣿⡏⣶⡆⢻⢇⡜⣷⡆⣶⣿⣿⣷⢸⢹⡏⡿⢩⡁⡟⣽⢹⢙⡝⣏⢻⠫⠙⡏⣏⢻⢨⣿⡿⡁⢸⣐⡲⣇⡒⢿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣬⣭⣷⣭⣥⣯⣴⣶⣽⣧⣿⣿⣷⣭⣼⣦⣥⣷⣬⣤⡧⠭⣸⣼⣧⣿⣼⣬⣽⣧⣿⣼⣬⣿⣷⣦⣼⣭⣥⣯⣭⣾⣥⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡟⡟⢿⢟⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⢟⣛⢻⣿⣿⣿⢿⣿⡟⢻⡟⢻⣿⣿⣟⣻⢻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣾⠱⠗⡹⠸⢃⣿⣧⠉⣼⠰⠧⣿⣏⠸⠭⢹⠰⠇⡗⢰⣿⠇⡎⡐⣸⠲⢠⠇⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

                    ䷩ 𝚕𝚒𝚗𝚎 574

╒═══════════════════ 𝐀𝐑𝐓𝐈𝐂𝐋𝐄 ═════════════════════════════════════════════════╕

(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/07/18/debconf22-welcomes-microsoft/#comments
    Gemini version at gemini://gemini.techrights.org/2022/07/18/debconf22-welcomes-microsoft/

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 07.18.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Debian_Takes_Money_From_Microsoft_—_an_Epic_Mistake⠀✐


Posted in Debian, Microsoft at 2:33 pm by Dr. Roy Schestowitz


Very disappointing, especially as I use 2 Debian PCs and my wife also uses 2
Debian PCs. Last night we span 2 new machines for Tux Machines and Techrights
(new sites). They use Alpine.


             🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇DebConf22_welcomes_its_sponsors!⦈_
    As per today’s_announcement from The Debian Publicity Team, which wrote
 promotional fluff for the criminal_company that’s attacking us all. Will the
   money from Microsoft be used to pay lawyers to attack Debian volunteers?


Summary: Debian has resumed taking bribes from Microsoft after a year or two
without that; that does not inspire much confidence in the direction or the
judgment of the leaders, who are unable to recognise a company that attacks
GNU/Linux more viciously than any other company (right now it’s trying to
prevent laptops from even booting Debian)


⡟⠛⣿⣿⢟⣿⣿⡻⣿⣿⣟⣻⣿⠟⠻⣿⣿⠿⠿⣿⠿⣿⠛⠛⣿⡵⠶⠶⠶⡶⠶⠶⢶⠶⠶⠶⠶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠲⠶⡶⠶⠶⠶⠶⠶⢶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠧⠤⠿⠿⠮⠿⠿⠵⠿⠿⠦⠿⠿⠮⠵⠿⠿⠶⠶⠿⠾⠿⠦⠤⠿⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠐⠒⠒⠒⠒⠚⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⠒⠒⠒⠒⠒⠒⠚⠒⠒⠚⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⡿⣿⣿⣿⠿⣿⢿⣿⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⢿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣇⣄⣬⣸⣄⣨⣨⣢⣀⣚⣇⣀⣀⣀⣀⣀⣀⣗⣀⣀⣀⣀⣸⣐⣆⣀⣀⣰⣀⣀⣶⣅⣔⣐⣥⣸⣀⣀⣷⣂⣀⣸⣀⣠⣸⣕⣶⣇⣁⣸⣀⣀⣀⣇⣀⣀⣀⣦⣺⣒⣕⣜⣐⣼⣈⣈⣂⣇⣀⣸⣿⣿⣿⣿⣿⣿⣿
⣿⣇⣀⣈⣈⣀⡐⣁⣵⣔⣇⣑⡀⢀⣘⣇⣀⣸⣧⣎⣁⣀⣀⣡⣸⣀⣻⣂⣸⣀⣀⣾⣪⣢⣁⣀⣰⣇⣉⣄⣱⢠⡇⣈⣀⣆⣑⣰⣿⣈⣎⣺⣇⣉⣄⣡⢌⣇⣀⡖⣁⣁⣑⣈⣰⣂⣪⣆⣇⣀⣸⣀⣎⣸⣿⣿⣿⣿
⣿⡇⣀⣀⣀⣁⣔⣇⣈⣇⣉⣠⣁⣈⣡⣈⣇⣠⣁⣠⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

                    ䷩ 𝚕𝚒𝚗𝚎 620

╒═══════════════════ 𝐀𝐑𝐓𝐈𝐂𝐋𝐄 ═════════════════════════════════════════════════╕

(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/07/18/epo-skeletons-in-the-closet/#comments
    Gemini version at gemini://gemini.techrights.org/2022/07/18/epo-skeletons-in-the-closet/

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 07.18.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ [Meme]_EPO_Management_Coming_Out_of_the_Closet⠀✐


Posted in Europe, Patents at 2:52 pm by Dr. Roy Schestowitz


Skeletons in the closet, that is…


                      🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇Welcome_to_EPO⦈_


🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇Statement_from_the_management:_No,_the_EPO_is_not_violating
 fundamental_human_rights;_Yes,_you_do_in_countless_ways,_affirm_the_judges⦈_
                          They actually said that!!!


Summary: As rulings against the EPO pile up (violations of fundamental rights)
the EPO’s Web site, basically a barrage_of_lies, won’t age well… like a human
locked up inside a closet


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⡆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⣸⣿⣷⣦⣼⣿⣧⣤⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣜⠿⠷⠿⠋⠸⠿⠇⠿⠇⠿⠇⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢰⣿⡟⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⡇⠀⠘⣿⡟⠻⣿⣿⣿⠉⠁⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠹⣟⣿⠟⠁⠸⠿⠇⠈⢿⣿⣿⡟⠁⠀⠀⠿⠿⠿⢿⣿⠿⠀⠀⠈⠿⠷⠿⠟⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⡇⠀⠀⠀⠀⠀⠀⣼⠓⠛⣿⣄⡀⠀⠀⠀⢀⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣷⠀⠀⠀
⠿⠿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢧⠀⠀⠀⠀⠀⣸⠇⠐⠀⠘⣿⣿⣶⣶⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⠛⠛⠻⡛⣛⠀⠀⠀⠀⠀⠙⡇⠀⠀
⠀⠀⠀⠙⢿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⣄⣀⣴⣿⣿⣄⣀⣤⣡⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡋⠀⠀⠀⠀⢀⡏⠙⢷⣦⣄⡀⣀⣼⣧⠀⠀
⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⢀⣴⣿⡁⢀⣄⣿⣿⣿⣿⣏⠈⠀⠀
⣷⣶⣶⣶⣿⠁⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢯⠳⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀
⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠰⠟⠻⣿⣿⠿⠿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡧⣿⠾⠿⠿⠿⠋⠋⠉⢸⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡹⠛⠟⠋⠋⠉⠉⠉⠙⢿⠃⠀⠀
⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⢹⣷⡄⠀⠈⡟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⡀⡀⠀⢀⣀⣤⣴⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⣀⣀⠀⠀⢠⣼⡄⠀⠀
⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠹⠿⣿⣿⡶⣾⡁⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠏⣟⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢀⠖⠁⠀⣿⡇⠀⠀
⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠃⣿⣽⠋⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣿⠿⠛⣁⣴⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠠⢀⣀⣀⣤⣾⣿⣷⠀⠀
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⣏⣥⠀⠀⠀⠀⣠⣿⣿⣟⠿⣷⣤⣤⡀⠀⣀⣀⣀⡀⠀⠀⠚⠭⠿⠿⠛⢹⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣯⣿⣿⣿⠟⢻⡆⣀
⠈⠙⢷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢻⡶⢃⠀⠀⢰⣿⣿⣿⣿⣄⠈⢿⣿⣿⣿⠛⠉⣉⣉⡉⠒⠂⠀⠀⠀⠀⠟⡿⠨⠿⣿⣿⣶⣤⣶⣶⣿⣿⣿⣿⣿⣛⣿⣶⣦⠄⠀⠀⢀⣤⣿⠏⠉⣉⣁⠴⢿⣛⣿
⣤⣤⣾⣿⣿⣿⠿⠿⠿⠿⣿⣶⣦⣄⣴⣿⡿⠷⠀⢀⡀⠒⠀⠀⣿⣿⣿⣿⣧⠀⣸⣿⣿⣇⣤⣀⣀⣀⡈⠑⠀⣀⣀⣀⠀⢀⣠⣤⣶⡿⠟⠋⣉⣭⣵⣶⣾⣿⣿⣿⠀⢉⠀⢺⣿⣶⣶⣛⠋⠁⠀⠀⢢⣬⣾⣿⣿⠿
⠛⠛⢩⣿⣷⠀⠀⠀⠀⢀⣬⣽⣿⡅⠉⣩⣤⡀⣀⣀⣀⡤⠔⢻⣿⣿⡟⠙⠉⠀⣿⣾⡛⣩⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⠿⠛⠉⠁⠐⠚⣸⣿⣿⣿⣿⣿⣿⣿⡏⠀⠋⠀⠸⣿⣷⡟⡻⡇⣠⣤⣤⣦⡿⠉⠀⠀⢀
⣀⣴⣿⡿⠃⠀⠀⠀⢠⢾⣿⣿⣿⠇⠀⢿⡿⠗⠈⠉⠀⣴⣶⣿⣿⣧⠀⠀⡠⢊⠽⣿⡆⠻⣿⣿⣿⣿⣿⠿⠉⠉⠉⠀⠀⠀⠐⠊⢀⣠⣾⡿⢿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣼⣿⠹⣿⣿⢰⣿⣿⣿⣿⢀⣤⣤⣶⡿
⠙⠉⠙⣿⡄⠀⢀⡴⠞⡛⢸⣿⣿⠐⠂⣴⡄⠒⠒⠶⠒⢻⣿⣿⢻⣿⣷⣾⠁⠀⠀⠙⢿⡀⠹⢿⠛⠛⢁⣾⣷⣶⣶⣶⣶⣶⣶⠾⠿⠿⠛⠁⣸⣿⣿⣿⣿⣿⠟⣻⡆⠀⢀⣿⣯⡄⠈⠉⠁⠙⠛⠛⠁⠈⠋⠁⠀⠀
⣐⣦⣴⡿⠀⡴⠁⠀⢀⣿⣾⣿⣿⠀⠤⣤⡀⠠⢤⣤⣤⣿⣿⣯⡾⢻⣿⣿⣿⣤⣶⣶⣾⣷⣤⣄⡀⠐⠻⠛⠋⠉⠉⠀⠁⠀⠀⣀⡄⠚⠓⢲⣿⣿⣿⣿⡿⠋⠀⣏⣿⠀⠘⣿⡇⢷⣤⣤⣀⢿⣷⣿⣿⠂⠀⠀⠀⠀
⠛⠛⠻⣧⠞⠁⠀⠀⢸⣿⣧⣿⣿⠀⠀⠉⠏⠠⠀⠀⣾⣿⡟⠙⣿⣼⣿⣿⡻⣿⣿⣿⣿⢿⣿⣿⡛⠃⢸⣿⠀⠀⠀⠀⠀⠐⠈⠁⣀⣼⣶⣿⣿⣿⣿⣿⠃⠀⠀⢸⣿⡆⠀⢻⣿⠈⠛⠛⠻⢿⣿⣥⣤⠔⠾⠿⡿⠿
⠀⣦⣾⣷⠇⢶⠄⠰⣾⠷⢾⣿⣿⢀⠀⠉⠁⠀⠀⢸⣿⡿⠹⠿⡟⢸⣿⣿⣷⣿⣟⠛⢿⣈⠙⠛⠃⠀⢿⣿⣦⣤⣤⣤⣤⣶⣾⣿⣿⣿⠿⠟⠋⠈⣿⣿⡆⠀⠀⣿⣿⠃⠀⠀⣿⣦⣀⣀⡀⠘⠻⠿⠟⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⡆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣿⣿⡇⣿⣿⣸⣿⣷⣦⣼⣿⣧⣤⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣜⠿⠷⠿⠋⠸⠿⠇⠿⠇⠿⠇⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢰⣿⡟⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⡇⠀⠘⣿⡟⠻⣿⣿⣿⠉⠁⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⣟⣿⠟⠁⠸⠿⠇⠈⢿⣿⣿⡟⠁⠀⠀⠿⠿⠿⢿⣿⠿⠀⠀⠈⠿⢷⠿⠟⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⡇⠀⠀⠀⠀⠀⠀⣼⠓⠛⣷⣄⡀⠀⠀⠀⢀⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣷⠀⠀⠀
⠿⠿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣣⠀⠀⠀⠀⠀⣼⠃⠐⠀⠘⣿⣿⣶⣦⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⠛⠛⠻⡛⣛⠀⠀⠀⠀⠀⠙⡇⠀⠀
⠀⠀⠀⠙⢿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⣄⣀⣴⣿⣿⣄⣀⣤⣡⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡋⠀⠀⠀⠀⢀⡏⠙⢷⣦⣄⡀⣀⣼⣧⠀⠀
⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⢀⣴⣿⡁⢀⣄⣿⣿⣿⣿⣏⠈⠀⠀
⣷⣶⣶⣶⣿⠁⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢯⠳⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀
⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠰⠟⠻⣿⣿⠿⠿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⣿⠾⠿⠿⠿⠋⠋⠉⢸⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡹⠋⠟⠋⠋⠉⠉⠉⠉⢿⠃⠀⠀
⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⢹⣷⡄⠀⠈⡟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⡀⠀⢀⣀⣤⣴⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⣀⣀⠀⠀⢠⣼⡄⠀⠀
⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠹⢿⣿⣿⡶⣾⡁⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠏⣟⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠀⠀⠀⢀⠔⠁⠀⣿⡇⠀⠀
⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠃⣿⣽⠋⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣿⠿⠛⣁⣴⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠠⢀⣀⣀⣤⣾⣿⣷⠀⠀
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⣏⣥⠀⠀⠀⠀⣠⣿⣿⣟⠿⣷⣤⣤⡀⠀⣀⣀⣀⡀⠀⠀⠈⠭⠳⠿⠛⢹⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣦⣯⣿⣿⣿⠟⣻⡆⣀
⠈⠙⢷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢿⡶⢃⠀⠀⢰⣿⣿⣿⣿⣄⠈⢿⣿⣿⣿⠛⠉⣉⣉⡉⠒⠂⠀⠀⠀⠀⠝⣿⠨⠿⣿⣿⣶⣤⣶⣶⣿⣿⣿⣿⣿⣛⣿⣶⣦⠄⠀⠀⢀⣤⣿⠟⠋⢉⣁⠶⢿⣻⢿
⣤⣤⣾⣿⣷⣿⠿⠿⠿⠿⣿⣶⣦⣄⣴⣿⡿⠷⠀⢀⡀⠒⠀⠀⣿⣿⣿⣿⣧⠀⣸⣿⣿⣇⣤⣀⣀⣀⡈⠑⠀⣀⣀⣀⠀⢀⣠⣤⣶⡿⠟⠋⣉⣭⣵⣶⣾⣿⣿⣿⠀⢩⠀⢸⣿⣶⣶⣛⠋⠁⠀⠀⢠⣬⣾⣿⣿⠿
⠛⠛⢩⣿⡷⠀⠀⠀⠀⢀⣬⣭⣿⡅⠈⣩⣤⡀⣀⣀⣀⡤⠔⢻⣿⣿⡟⠙⠉⠀⣿⣿⡛⣩⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⠿⠛⠉⠁⠐⠚⣸⣿⣿⣿⣿⣿⣿⣿⡏⠀⠋⠀⠘⣿⣷⡝⡻⡇⣠⣤⣤⣦⡿⠉⠀⠀⢀
⣀⣴⣿⡿⠃⠀⠀⠀⢠⢾⣿⣿⣿⠇⠀⢿⡿⠗⠈⠉⠀⣴⣶⣿⣿⣣⡀⠀⡠⢊⠽⣿⡆⠻⣿⣿⣿⣿⣿⠿⠉⠉⠉⠀⠀⠀⠐⠊⢀⣠⣾⡿⢿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣼⣿⠹⣿⣯⢰⣿⣿⣿⣿⢀⣤⣤⣶⡿
⠙⠉⠙⣿⡄⠀⢀⡴⠞⡛⢸⣿⣿⠐⠀⣴⡄⠒⠒⠶⠒⢻⣿⣿⢻⣿⣷⣾⠁⠀⠀⠙⢿⡀⠹⢿⠛⠛⢁⣾⣷⣶⣶⣶⣶⣶⣶⠾⠿⠿⠛⠁⣸⣿⣿⣿⣿⣿⠟⣻⡆⠀⢀⣿⣧⡄⠈⠉⠁⠉⠙⠛⠁⠈⠋⠁⠀⠀
⣐⣦⣴⡿⠀⡴⠁⠀⠠⣿⣾⣿⣿⠀⠤⣤⡀⠤⢤⣤⣤⣿⣿⣯⡾⢻⣿⣿⣿⣤⣶⣶⣾⣷⣤⣄⡀⠐⠻⠛⠉⠉⠉⠀⠁⠀⠀⣀⡄⠚⠓⢲⣿⣿⣿⣿⡿⠋⠀⣍⣿⠀⠘⣿⡇⢷⣤⣤⣀⢿⣷⣿⣿⠂⠀⠀⠀⠀
⠛⠛⠻⣧⠞⠁⠀⠀⢸⣿⣧⣿⣿⠀⠀⠉⠏⠠⠀⠀⣾⣿⡟⠙⣿⣼⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⡛⠁⢸⣿⠀⠀⠀⠀⠀⠐⠈⠁⣀⣼⣶⣿⣿⣿⣿⣿⠃⠀⠀⢸⣿⡆⠀⢹⣿⠈⠛⠛⠻⢿⣿⣤⣤⠐⠾⠿⠿⡿
⠀⣦⣾⣷⠇⢶⠄⠰⣾⠷⢾⣿⣿⢀⠀⠉⠁⠀⠀⢨⣿⡿⠹⠿⡟⢸⣿⣿⣿⣾⣿⠻⣿⣈⠙⠛⠃⠀⢿⣿⣦⣤⣤⣤⣤⣶⣾⣿⣿⣿⠿⠟⠋⠈⣿⣿⡆⠀⠀⣿⣿⠃⠀⠀⣿⣦⣀⣀⡀⠘⠻⠿⠛⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⡆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣿⣿⡇⣿⣿⢸⣿⣧⣤⣼⣿⣧⣤⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⠿⠷⠿⠋⠸⠿⠇⠿⠇⠿⠇⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢰⣿⡟⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⡏⠉⠙⣿⡟⠻⢿⣿⣿⠉⠁⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢿⣻⡿⠛⠹⠿⠇⠀⢻⣿⣿⣿⠁⠀⠀⠿⠿⠿⠸⣿⡿⠀⠀⠘⠿⢿⠿⠟⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⠀⠀⠀⠀⠀⠀⢨⡟⠘⢿⣦⡀⠀⠀⠀⠀⢰⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀
⣿⠿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⢠⡿⠀⠂⠀⢿⣿⣶⣤⣤⣶⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⡿⠟⠛⠛⠟⢛⠋⠀⠀⠀⠀⠉⢻⠀⠀
⡁⠀⠀⠙⢿⣿⡿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⣀⣀⣶⣿⣧⠀⣀⣀⣼⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⣸⠛⢳⣶⣄⡀⠀⣠⣾⡀⠀
⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠾⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⣠⣾⡇⠀⣀⣹⣿⣿⣿⣿⠉⠁⠀
⣿⣦⣤⣤⣾⡟⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡳⢾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀
⣿⣿⣻⡿⠋⠁⠀⠀⠀⠀⠀⢀⠼⠿⣿⣿⡿⠿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢿⣹⣽⡿⣿⠿⠟⠛⠉⠁⣿⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⠟⢻⠛⠛⠉⠉⠉⠙⠿⡟⠀⠀
⢿⣽⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠈⢹⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⢀⠀⠀⡀⣀⣠⣴⣾⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠀⠀⢀⣀⠀⠀⠀⣤⣿⠀⠀
⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢈⠟⠙⢿⣿⣿⣦⣴⣎⢨⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠝⢹⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠉⠀⡠⠖⠀⢸⣿⠀⠀
⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠙⠿⢰⣯⡾⠁⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠿⠿⣿⣿⣿⣿⠿⠋⣡⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢀⡀⠀⣀⣀⣠⣿⣿⡆⠀
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣛⡅⠀⠀⠀⠀⣸⣿⣿⡻⣿⣦⣤⡀⠀⠀⣀⣀⠀⠀⠀⠘⠯⠽⠷⠶⠚⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠛⢷⣤⣽⣿⣿⣿⣿⠟⣷⠀
⠉⠙⢶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⢭⡕⠀⠀⠀⣼⣿⣿⣿⣧⡈⢻⣿⣿⣿⡟⠛⠋⠉⠋⠒⠲⠀⠀⠀⠀⠰⢿⡏⠻⣿⣿⣶⣤⣤⣶⣶⣿⣿⣿⣿⣿⢿⣶⣦⣤⠀⠀⠀⢀⣬⡿⠛⢛⣉⢡⠾⣟⠿
⣀⣤⣤⣿⣿⣶⣶⣶⠶⠿⣷⣶⣦⣤⣠⣶⣾⡿⠍⠁⠒⠁⠤⠀⢸⣿⣿⣿⣿⡅⢀⣿⣿⣿⣀⣀⡀⠉⠉⠙⠂⠀⣀⣀⠀⠀⢀⣀⣤⣶⡿⠟⠋⣉⣭⣽⣿⣾⣿⣿⡆⠈⡁⠀⣿⣦⣠⣴⠋⠋⠁⠀⠀⣤⣴⣾⣿⣿
⠿⠟⠛⣻⣿⡇⠀⠀⠀⠀⢨⣿⣿⣟⠈⣉⣩⣀⢀⡈⢉⣀⣠⠖⣾⣿⡿⠏⠛⠃⢸⣿⣟⠛⣽⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⠿⠛⠉⠁⠰⢆⣿⣿⣿⣿⣿⣿⣿⣿⠁⠸⠁⠀⢿⣿⣿⠛⢿⢀⣀⣀⣴⣼⠟⠉⠀⠀
⠀⣀⣴⣿⡟⠁⠀⠀⠀⣤⣿⣿⣿⡿⠀⠸⣿⡿⠊⠉⠉⢡⣶⣶⣿⣿⡁⠀⢀⡔⠉⢿⣧⠸⣿⣿⣿⣿⣿⣿⠟⠛⠋⠉⠀⠀⠀⠐⠊⢀⣤⣿⡿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢀⣿⡏⢿⣷⣇⣾⣿⣿⣿⡇⣀⣤⣤⣶
⠙⠛⠉⢿⣧⠀⠀⣠⡴⠿⠋⣿⣿⡗⠲⢤⣬⠠⠀⠤⠤⠾⣿⣿⡿⣿⣿⣦⡟⠀⠀⠘⢿⣇⠈⢿⡿⠿⠋⣡⣾⣶⣶⣶⣶⣶⣶⣶⣾⣿⠿⠋⠀⣿⣿⣿⣿⣿⣿⢟⣿⠀⠀⢸⣿⢡⠀⠉⠉⠘⠛⠛⠛⠀⠛⠋⠉⠁
⠀⣢⣠⣼⡏⢀⡜⠁⠀⣸⣷⣿⣿⡧⠤⣬⣭⠀⢤⣤⣤⣰⣿⣿⣣⡿⣿⣿⣷⣦⣤⣴⣶⣾⣤⣄⡀⠀⠰⠿⠛⠛⠉⠁⠉⠉⠁⠀⣠⠐⠒⠶⣼⣿⣿⣿⣿⠟⠁⢸⢹⡇⠀⢿⣿⢹⣦⣄⣀⡀⣷⣶⣿⡷⠀⠀⠀⠀
⠛⠻⠟⢿⣠⠞⠀⠀⠀⣿⡿⣽⣿⡇⡀⠈⠙⠂⠀⠈⢩⣿⣿⠟⢻⣧⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠀⣾⡆⠀⠀⠀⠀⠀⠐⠊⠁⣀⣶⣾⣿⣿⣿⣿⡟⠀⠀⠈⣿⣷⠀⠘⣿⡇⠙⠛⠿⠿⣿⣯⣭⡄⠶⠶⢶⣶
⠀⠆⢲⣾⠇⢶⠄⠰⢶⡿⠗⢿⣿⡇⢀⠀⠈⠀⠀⠈⣹⣿⡟⠶⠾⡃⣿⣿⣿⣶⣯⡻⢿⣇⡙⠻⠛⠀⢰⣿⣷⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⡿⠟⠉⢙⣿⣷⠀⠀⢰⣿⡿⠀⠀⢸⣷⣄⠀⠀⠀⠗⠿⠿⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣿⣿⡇⣿⣿⢸⣿⣧⣤⣸⣿⣇⣀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠿⠷⠿⠋⠸⠿⠇⠿⠇⠿⠇⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢰⣿⡟⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠙⣿⡿⠿⢿⣿⣿⠏⠁⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣷⣶⣤⡀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⣿⣿⡿⠿⠿⠇⠀⠻⣿⣿⣿⡃⠀⠀⠿⠿⠿⠸⢿⡿⠀⠀⠈⠿⢷⠿⠟⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⣿⠏⠀⠀⠀⠀⠀⠀⣿⠇⢿⣧⡀⠀⠀⠀⠀⠀⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡄⠀⠀⠀⠀⠀⣰⠇⠠⠀⢹⣿⣶⣤⣀⣤⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⣿⡿⠿⠿⠿⠿⠟⠉⠀⠉⠉⠉⠻⡄⠀
⡛⠁⠈⠙⢿⣿⣿⣽⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⠀⠀⣤⣴⣿⡀⠀⣐⢀⣿⣿⣿⣿⣿⣿⣯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠏⠁⠀⠀⠀⢀⡗⠻⣦⣄⡀⠀⠀⣰⡇⠀
⣿⠀⠀⠀⠀⢹⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣠⣾⠀⠀⣘⣿⣿⣷⣿⡟⠙⠀
⣿⣦⣀⣀⣠⣾⠇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣒⢴⣾⣿⣿⣶⣿⢿⣿⣿⣿⣿⣷⡄⠀
⣿⣿⠿⣿⡟⠛⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣛⣿⣽⣿⣿⣿⠻⠟⠛⢽⣦⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠌⠿⢻⡿⠟⠛⠛⠉⠛⢻⣿⠃⠀
⡿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⢿⣯⠀⠈⠙⡿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⠉⠈⠀⠀⣀⣠⣴⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀
⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⢶⣾⣿⣧⣀⣴⠃⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⡗⣷⣶⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠈⠉⢉⡤⠂⠈⣿⡇⠀
⢛⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠏⣿⣣⡏⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠴⣿⣿⣿⣿⣿⣿⠿⠋⣡⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠈⠀⣀⣴⣿⣧⠀
⣏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢟⠀⠀⠀⠀⢀⣾⣿⡿⣶⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢝⣛⣯⣷⠶⢺⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣤⣈⣟⣿⣿⣿⡿⢳⡄
⠒⠒⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣾⣿⣫⠄⠀⠀⢠⣿⣿⣿⣿⠈⠻⣿⣿⣶⣶⠾⠟⠛⠒⠢⠤⠄⠀⠀⠀⠀⣾⣿⢙⣿⣷⣶⣄⣀⣤⣤⣶⣾⣿⣿⣶⠶⣶⣤⣄⡀⠀⠀⠀⣉⣿⡟⠛⠛⠋⣠⣿⢷
⠀⢀⣀⣹⣿⣦⣤⣤⣤⣶⣶⣦⣤⣀⡀⣠⣤⣿⣏⠒⠡⠐⢀⡀⠘⣿⣿⣿⣿⣏⠀⢹⣿⣿⡏⠀⠀⠉⠉⠒⠢⠀⠀⠀⠀⠀⠀⠁⣀⣤⣿⣿⠿⠛⢛⣻⣭⣭⣿⣿⣿⠉⢛⠁⣽⣶⣤⢤⠾⠟⠋⠀⠀⢌⢁⣴⣾⣿
⣿⣿⠿⠿⣿⣿⠉⠁⠀⠀⠈⣹⣿⣿⠃⠛⢛⣁⠀⠀⠒⠀⠀⣠⢦⣿⣿⣟⠻⠻⠀⣿⣽⠿⢛⣿⣿⣷⣶⣶⣦⣴⣿⣿⣿⣶⣾⠿⠟⠋⠡⠴⢰⣿⣿⣿⣿⣿⣿⣿⡏⠀⡟⠀⢹⣿⣯⡿⢿⡆⢀⣀⣀⣬⣿⠟⠉⠁
⠀⠀⢀⣼⣿⠏⠀⠀⠀⢀⣼⣿⣿⣿⡇⠈⣿⣿⡇⠛⠛⠋⣩⣤⣾⣿⠏⠁⠀⢀⠔⣿⣿⡁⣿⣿⣿⣿⣿⣿⣿⠟⠟⠛⠉⠉⠀⠠⠴⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣸⣿⢻⣿⣦⢠⣿⣿⣿⣿⠀⣀⣀⣤
⠖⠾⠟⢻⣷⡀⠀⠀⣀⣼⠿⢹⣿⣿⠤⢄⣈⡋⠀⢀⣀⣤⢿⣿⣿⣿⣿⣦⣴⠃⠈⠀⢿⣷⠈⢻⣿⠿⠿⠋⣰⣦⣤⣤⣤⣤⣤⣤⣤⣶⣿⠿⠃⢸⣿⣿⣿⣿⣿⡿⢿⡄⠀⠀⣿⣏⡀⠙⠛⠊⠛⠿⠿⠋⠘⠟⠛⠉
⠀⠠⡀⣠⣿⠃⢠⠞⠉⠀⣷⣼⣿⣿⣀⣀⣛⠁⢀⣀⣀⣀⣼⣿⣟⣸⢿⣿⣿⣤⣀⣠⣤⣭⣧⣀⡉⠀⠠⢾⠿⠟⠛⠛⠙⠋⠉⠉⠁⡉⠤⢤⣤⣿⣿⣿⣿⣿⠟⠀⡿⣧⠀⢸⣿⡟⣧⣀⣀⠀⢴⣦⣶⣶⠄⠀⠀⠀
⠶⠾⠿⢿⡏⡠⠋⠀⠀⢸⣿⣟⣿⣿⠀⠈⠛⡇⠀⠉⠛⣻⣿⣿⠿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠆⢰⡶⠀⠀⠀⠀⠀⠠⠔⠊⠁⣰⣤⣾⣿⣿⣿⣿⠇⠀⠀⢱⣿⡄⠀⢿⣷⠘⠿⠿⢿⣾⣯⣉⣉⠠⣴⣶⣶
⠀⠆⠲⣶⡿⢷⠄⠰⠶⣾⠟⠸⣿⣿⠈⠀⠀⠂⠀⠀⢒⣿⣿⢣⣴⡿⢹⣿⣿⣧⣽⡿⢿⣿⠘⠻⠿⠇⠀⣼⣿⣄⣀⣀⣀⣀⣤⣤⣶⣿⣿⣿⣿⠟⠙⣿⣿⡄⠀⠀⣼⣿⡇⠀⠈⣿⣄⠀⠀⠀⠸⢿⣿⡿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣿⣿⡇⣿⣿⢸⣿⣧⣤⣸⣿⣇⣀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠿⠋⠸⠿⠇⠿⠇⠿⠇⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢰⣿⡟⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠙⣿⡿⠻⢿⣿⣿⡏⠁⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣾⣶⣤⣄⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⡿⠿⠿⠇⠀⠻⣿⣿⣿⡋⠀⠀⠿⠿⠿⠸⢿⣿⠃⠀⠈⠿⠷⠿⠟⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣿⡟⠁⠀⠀⠀⠀⠀⢸⡿⠿⣷⡀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣧⠀⠀⠀⠀⠀⢀⡿⠀⡀⠘⣿⣷⣤⣀⣠⣴⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⠿⠿⠿⠿⠿⠋⠀⠈⠉⠉⠙⢇⠀
⠛⠃⠈⠙⠻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⡄⠀⢠⣴⣾⡅⠀⢀⠂⣹⣿⣿⣿⣿⣿⣿⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡸⠉⠀⠀⠀⠀⡸⠛⢶⣤⣀⠀⠀⢀⣯⠀
⣿⡆⠀⠀⠀⠈⣿⡅⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢠⣾⡇⠀⣀⣻⣿⣿⣿⡿⠉⠃
⣿⣧⣄⣀⣀⣴⡟⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣗⠦⣶⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣆⠀
⣿⣿⡿⢿⣿⠟⠃⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢘⣟⣻⣻⣿⣿⣿⡟⠟⠛⢫⣷⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠹⠟⣿⠿⠛⠋⢋⠋⠛⣿⡇⠀
⣿⣿⣷⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠻⣿⡁⠀⠙⢿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠁⠀⢀⣀⣤⡼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀
⣶⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⢶⣶⣿⣷⣄⣠⡎⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⢾⢶⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⠀⠀⠉⠉⣡⠔⠀⢿⣷⠀
⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠙⠻⡟⢹⢷⣽⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢾⣿⣿⣿⣿⣿⡿⠛⣩⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠀⡀⠀⡁⢀⣠⣿⣿⡀
⣿⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⡃⠀⠀⠀⠀⣰⢿⡿⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢟⣻⣵⡴⠶⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣦⣥⣻⣻⣿⣿⣿⠿⣦
⠗⠒⠲⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣟⡥⠀⠀⠀⣼⣿⣿⣿⡆⠙⢿⣯⣷⣴⡶⠿⠛⠓⠒⠤⠤⠀⠀⠀⠀⢰⢿⡏⣻⣿⣶⣦⣀⣤⣤⣴⣶⣿⣿⣷⣶⢶⣦⣤⡀⠀⠀⠀⢈⣹⣿⠛⠛⠻⢡⣼⡿
⠀⠀⣀⣈⣿⣧⣤⣤⣤⣤⣶⣦⣤⣀⡀⢀⣤⣼⣿⡑⠊⠄⠀⣀⠀⢿⣿⣿⣿⣗⡀⢈⣿⣿⣿⠁⠀⠈⠉⠐⠲⠄⠀⠀⠀⠀⠀⠈⢀⣠⣼⣿⠿⠛⠛⣛⣯⣽⣽⣿⣿⡏⠘⠉⢩⣷⣦⣄⡴⠿⠛⠁⠀⠠⣉⣠⣶⣿
⣾⣿⡿⠿⢿⣿⡋⠉⠉⠀⠈⢙⣿⣿⡟⠙⠛⣋⠀⠀⠐⠂⠀⢀⡴⣼⣿⣿⠟⠿⠇⢸⣿⡿⠟⣻⣿⣷⣶⣶⣶⣤⣾⣿⣿⣶⣶⡿⠟⠛⠉⠤⠆⣾⣿⣿⣿⣿⣿⣿⣿⠁⠸⠃⠈⣿⣿⣽⠿⣶⠀⣀⢀⣠⣹⡟⠉⠉
⠁⠀⢀⣠⣿⡿⠀⠀⠀⠀⣀⣿⣿⣿⣿⠈⢹⣿⣷⠘⠒⠚⢉⣭⣴⣿⡿⠀⠀⠀⡠⠺⣿⡇⢸⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠁⠀⠤⠂⠀⣠⣾⢿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢀⣿⡿⣿⣾⡊⣾⣿⣿⣿⡇⢀⣀⣤
⠴⠶⠿⠛⣿⡆⠀⠀⢀⣠⡿⠟⣿⣿⠧⢀⢈⣛⢁⠀⣀⣀⣼⣿⣿⣿⣷⣷⣠⡞⠀⠁⠸⣿⡄⠹⣿⡿⠿⠟⢡⣶⣤⣤⣤⣤⣤⣤⣤⣴⣿⡿⠟⢈⣿⣿⣿⣿⣿⣿⠿⣧⠀⠀⢸⣿⡀⠈⠛⠓⠹⠻⠿⠟⠀⠿⠛⠉
⠀⠀⢄⢀⣼⡟⠀⡴⠋⠁⢸⣤⣿⣿⡄⠀⢘⡋⠀⣀⣀⣀⣠⣿⣿⣃⡿⣿⣿⣧⣄⣀⣠⣬⣵⣀⡈⠁⠀⢴⡿⠿⠛⠛⠋⠛⠉⠉⠉⢨⠡⠤⣤⣼⣿⣿⣿⣿⡿⠁⢸⢿⡄⠀⣿⣿⢻⣄⣀⡀⢰⣶⣴⣶⡦⠀⠀⠀
⠶⢶⣾⡿⣟⢁⡜⠁⠀⠀⣿⣿⣿⣿⡁⠀⠙⢳⠀⠈⠙⢻⣿⣿⠿⢿⡅⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠶⠀⣶⠆⠀⠀⠀⠀⠀⠤⠚⠉⢀⣤⣴⣿⣿⣿⣿⡟⠀⠀⠈⣾⣧⠀⠹⣿⡆⠛⠿⠿⣿⣿⣏⣉⣁⣤⣶⣶
⠀⠆⠲⣶⣿⢿⠄⠰⠶⢶⡿⠙⣿⣿⡇⠁⠀⠘⠀⠀⠐⣾⣿⡟⣤⣼⠇⣿⣿⣷⣯⡿⢿⣿⡏⠛⠿⠿⠀⢸⣿⣧⣀⣀⣀⣀⣀⣤⣴⣾⣿⣿⣿⠿⠋⢻⣿⣧⠀⠀⢰⣿⣿⠀⠀⢻⣧⡀⠀⠀⠀⠾⣿⣿⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⣸⣿⣇⣀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠿⠋⠸⠿⠇⠿⠇⠿⠇⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢰⣿⡟⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⡿⠿⢿⣿⣿⣿⠁⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣷⣦⣤⡀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣾⣿⡿⠿⠇⠀⠻⢿⣿⢿⣛⠁⠀⠿⠿⠿⠸⠿⣿⡏⠀⠈⠿⢷⠿⠟⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣽⣾⠋⠀⠀⠀⠀⠀⠈⣿⡾⣿⣆⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣽⡀⠀⠀⠀⠀⠀⣼⠃⠀⠈⢿⣷⣦⣄⣀⣤⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⡿⠿⠿⠿⠿⠏⠁⠀⠉⠉⠉⢻⡀
⠙⠛⠉⠉⠻⢿⣿⣿⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣷⡄⠀⠀⢤⣴⣏⠀⠀⡐⢸⣿⣿⣿⣿⣿⣿⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠋⠁⠀⠀⠀⢀⠞⢷⣦⣀⠀⠀⢀⣼⡅
⣿⣧⠀⠀⠀⠈⢻⣏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣽⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⣴⣿⠀⢀⣙⣿⣿⣶⣿⠏⠑
⣿⣿⣄⣀⢀⣠⣾⠇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡲⣴⣾⣿⣿⣶⣿⣿⣿⣿⣿⣿⣷⠀
⣿⣿⣿⠿⣿⡿⠛⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣻⣟⣿⣿⣿⣿⠿⠟⠛⣽⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠙⡿⢻⡿⠿⠛⠙⠙⠛⢻⣿⠀
⣿⣿⣿⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠸⣿⡏⠈⠉⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠉⠉⠀⠀⢀⣀⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀
⣤⣶⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠠⠤⠦⣶⣾⣿⣆⣀⣴⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡶⡶⣶⣾⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠄⠀⠈⠉⢁⡤⠀⠸⣿⡆
⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠭⠿⢿⠋⡿⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣶⣿⣿⣿⣿⣿⣿⠿⢛⣥⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠄⢀⠀⢈⢀⣀⣾⣿⡇
⣶⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠏⠀⠀⠀⠀⠠⣾⣿⣷⣶⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠩⠝⣛⣯⣥⠴⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢶⣬⣜⣿⣿⣿⣿⡿⣷
⡿⠓⠲⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⣾⣗⣫⠄⠀⠀⢠⣿⣿⣿⣷⠉⢻⣿⣽⣦⣶⠶⠶⠒⠲⠤⢤⡀⠀⠀⠀⠀⣿⡿⠻⣿⣶⣦⣄⣀⣤⣤⣶⣶⣿⣷⣶⡶⣦⣤⣀⠀⠀⠀⠀⣉⣿⡟⠛⠛⠏⣤⣿
⠀⠀⠀⠀⣻⣷⣄⣤⣤⣤⣤⣤⣤⣀⡀⠀⣠⣤⣿⣗⠐⠡⠄⣀⠀⢸⣿⣿⣿⣿⢀⠀⢻⣿⣿⡋⠀⠀⠒⠒⠒⠤⠀⠀⠀⠀⠀⠀⠁⣀⣨⣿⣿⠿⠛⢛⣻⣽⣭⣿⣿⣯⠙⠋⠉⣽⣦⣤⣤⠾⠛⠃⠀⠀⢍⢡⣴⣾
⣶⣾⣿⣿⠿⣿⣟⠛⠉⠉⠉⠉⠿⠿⣿⠃⠻⠟⠋⠀⠀⠒⠀⠀⣠⣤⣿⣿⡟⠻⠿⢠⣷⣿⡿⢛⣿⣷⣶⣶⣶⣆⣴⣿⣿⣷⣶⣾⠿⠟⠋⢡⡤⢰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠏⠀⢹⣿⣿⡿⢷⡆⢀⡀⣀⣤⣿⠋⠉
⠉⠁⠀⢀⣼⣿⠏⠀⠀⠀⢀⣸⣿⣿⣿⡇⠐⣿⣿⡖⠲⠒⠚⣩⣅⣾⣿⠏⠀⠀⢀⠜⢿⣿⠀⣿⣿⣿⣿⣿⣿⣿⠻⠟⠛⠋⠉⠀⠠⠤⠀⢀⣴⡿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣸⣿⢿⣷⡆⢡⣿⣿⣿⣿⠁⣀⣀
⠤⠤⣶⠿⢿⣯⠀⠀⠀⢀⣼⣟⢻⣿⡿⠀⣀⣙⠋⡀⣀⣀⣠⣿⣿⣿⣿⣾⣅⣴⠃⠈⠈⢿⣧⠈⢿⣿⣿⡿⠋⣵⣦⣤⣤⣤⣤⣤⣤⣤⣶⣿⠿⠃⣻⣿⣿⣿⣿⣿⡿⢿⡀⠀⢀⣿⣇⠀⠙⠛⠉⠟⠿⠿⠋⠸⠟⠋
⠀⠀⢀⡀⢨⣿⠃⢠⠞⠋⠁⣦⣸⣿⣇⠀⠀⢛⠃⠀⣀⠀⠀⣼⣿⡟⣸⢿⣿⣿⣤⢀⣀⣠⣽⣄⡈⠉⠀⢠⣾⠿⠿⠛⠛⠙⠛⠋⠉⠉⡉⠤⣤⣤⣿⣿⣿⣿⣿⠟⠀⣿⣧⠀⢸⣿⡟⣆⣀⣀⠀⣶⣤⣶⣶⠄⠀⠀
⠶⠶⣶⣿⣿⡋⣠⠏⠀⠀⢸⣿⣿⣿⣇⠀⠉⠟⡆⠉⠙⠛⣻⣿⡿⢿⣯⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠄⢠⡦⠀⠀⠀⠀⠀⠀⠤⠊⠁⣠⣤⣾⣿⣿⣿⣿⠃⠀⠀⢹⣿⡄⠈⢿⣷⠛⠿⠿⣷⣿⣿⣛⣉⣠⣴⣶
⠀⠆⠲⣶⢺⣿⠅⠰⠶⠶⣾⡟⢿⣿⣿⠈⠀⠠⠃⠀⠀⢠⣿⣿⣧⣤⡿⢸⣿⣿⣮⣿⢿⣿⣿⠙⠿⡿⠆⠀⣿⣿⣄⣀⢀⣀⣀⣠⣤⣶⣿⣿⣿⣿⠟⠙⣿⣿⡄⠀⠀⣾⣿⡇⠀⠘⣿⣄⠀⠀⠀⠸⢿⣿⡿⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣿⣿⡇⣿⣿⢸⣿⣧⣤⣸⣿⣇⣀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠿⠋⠸⠿⠇⠿⠇⠿⠇⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢰⣿⡟⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⡿⠿⢿⣿⣿⣿⠁⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣷⣦⣤⡀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣾⣿⡿⠿⠇⠀⠻⢿⣿⢿⣛⠁⠀⠿⠿⠿⠸⠿⣿⡏⠀⠘⠿⠷⠿⠟⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣼⣾⠋⠀⠀⠀⠀⠀⠈⣿⡾⣿⣆⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣽⡀⠀⠀⠀⠀⠀⣼⠃⠀⠈⢿⣷⣦⣀⣀⣤⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⡿⠿⠿⠿⠿⠟⠁⠀⠉⠉⠉⢻⠀
⠙⠟⠉⠉⠻⢿⣿⣿⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡄⠀⠀⢤⣴⣏⠀⠀⡐⢸⣿⣿⣿⣿⣿⣿⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠋⠁⠀⠀⠀⢀⠞⢷⣦⣀⠀⠀⢀⣼⡅
⣿⣧⠀⠀⠀⠈⢻⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣽⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⣴⣿⠀⢀⣙⣿⣿⣶⣿⠏⠱
⣿⣿⣄⣀⢀⣠⣾⠇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡦⣴⣾⣿⣿⣶⣿⣿⣿⣿⣿⣿⣷⠀
⣿⣿⣿⠿⣿⡿⠛⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣻⣛⣿⣿⣿⣿⠿⠟⠛⣿⣤⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠙⡿⢻⡿⠿⠛⠙⠙⠛⢻⣿⠀
⣿⣿⣟⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠸⣿⡏⠈⠉⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠉⠉⠁⠀⢀⣀⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣺⠀
⣤⣶⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠠⠤⠦⣶⣾⣿⣆⣀⣴⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡶⡶⣶⣾⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠈⠉⢁⡤⠀⠸⣿⡆
⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠭⠿⢿⠋⡿⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣶⣿⣿⣿⣿⣿⣿⠿⢛⣥⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠄⢀⠀⢈⢀⣀⣾⣿⡇
⣶⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠏⠀⠀⠀⠀⠠⣾⣿⣷⣶⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠨⠝⣛⣯⣥⠴⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣬⣘⣻⣿⣿⣿⡿⣷
⡿⠓⠲⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⣾⣗⣫⠄⠀⠀⢠⣿⣿⣿⣷⠉⢻⣿⣽⣦⣶⠶⠶⠓⠲⠤⢤⡀⠀⠀⠀⢀⣿⡿⢻⣿⣶⣦⣄⣀⣤⣤⣶⣶⣿⣷⣶⡶⣦⣤⣀⠀⠀⠀⠀⣉⣿⡛⠛⠛⠏⣠⣿
⠀⠀⠀⠀⣻⣷⣄⣤⣤⣤⣤⣤⣤⣀⡀⠀⣀⣤⣿⣗⠐⠡⠀⣀⠀⢸⣿⣿⣿⣿⢀⠀⢻⣿⣿⡋⠀⠀⠒⠒⠒⠤⠀⠀⠀⠀⠀⠀⠁⣀⣨⣿⣿⠿⠛⢛⣻⣽⣭⣿⣿⣯⠙⠋⠉⣽⣦⣤⡠⠾⠟⠃⠀⠀⠉⢁⣴⣾
⣶⣾⣿⣿⠿⣿⣛⠛⠉⠉⠉⠉⠻⠿⣿⠃⠛⠟⠋⠀⠀⠒⠀⠀⣠⣤⣿⣿⡟⠿⠿⢠⣧⣿⡿⢛⣿⣷⣶⣶⣶⣆⣴⣿⣿⣶⣶⣾⠿⠟⠋⢡⡴⢰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠏⠀⢹⣿⣿⡿⢷⡆⢀⡀⣀⣤⣿⠋⠉
⠉⠁⠀⢀⣼⣿⠏⠀⠀⠀⢀⣸⣿⣿⣿⡇⠀⣿⣿⡖⠲⠒⠚⣩⣅⣾⣿⠏⠀⠀⢀⠜⢿⣿⠠⣿⣿⣿⣿⣿⣿⣿⠻⠟⠛⠋⠉⠀⠠⠤⠀⢀⣴⡿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣸⣿⣿⣷⡖⠠⣿⣿⣿⣿⠁⣀⣀
⣤⠤⣶⠿⢿⣯⠀⠀⠀⢀⣼⣟⢻⣿⡿⠀⣀⣙⠋⣀⣀⣀⣠⣿⣿⣿⣿⣾⣅⣴⠃⠈⠈⢿⣧⠈⢿⣿⣿⡿⠋⣴⣦⣤⣤⣤⣤⣤⣤⣤⣶⣿⡿⠃⣻⣿⣿⣿⣿⣿⡿⣿⡀⠀⢀⣿⣇⠀⠙⠛⠈⠟⠿⠿⠋⠸⠟⠋
⠀⠀⢀⡀⢨⣿⠃⢠⠞⠋⠁⣦⣸⣿⣇⠀⠀⢛⠃⠀⣀⠀⠀⣼⣿⡟⣸⣿⣿⣿⣤⢀⣀⣤⣽⣄⡈⠉⠀⢠⣾⠿⠿⠛⠛⠙⠛⠛⠉⠉⡉⠤⣤⣤⣿⣿⣿⣿⣿⠟⠀⣿⣧⠀⢸⣿⡟⣆⣀⣀⠀⣶⣤⣶⣶⠄⠀⠀
⠶⢶⣶⣿⣿⡋⣠⠏⠀⠀⢸⣿⡿⣿⣇⠀⠉⠟⡆⠈⠙⠛⣿⣿⡿⢿⣯⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠄⢠⡦⠀⠀⠀⠀⠀⠀⠤⠊⠁⣠⣤⣾⣿⣿⣿⣿⠃⠀⠀⢱⣿⡄⠈⢿⣷⠙⠿⠿⣷⣿⣿⣛⣉⣠⣴⣶
⠀⠆⠲⣶⢹⣿⠅⠰⠶⠶⣾⡟⢿⣿⣿⠈⠀⠠⠃⠈⠀⢠⣿⣿⣧⣤⡿⢸⣿⣿⣮⣿⢿⣿⣿⠙⠿⡿⠆⠀⣿⣿⣄⣀⢀⢀⣀⣠⣤⣶⣿⣿⣿⣿⠟⠙⣿⣿⡄⠀⠀⣾⣿⠇⠀⠘⣷⣄⠀⠀⠀⠸⢿⣿⡯⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⣸⣿⣇⣀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⣏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣸⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠿⠋⠸⠿⠇⠿⠇⠿⠇⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢰⣿⡟⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⠿⢿⣿⣿⣿⡇⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣾⣶⣤⣀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⡿⠿⠏⠀⠻⠿⣿⡿⣛⡃⠀⠿⠿⠿⠸⠿⢿⣿⠀⠈⠿⠷⠿⠟⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢬⣷⡟⠀⠀⠀⠀⠀⠀⢹⣇⢿⣧⡀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣇⠀⠀⠀⠀⠀⢠⡟⠀⠀⠹⣿⣦⣄⣀⣀⣴⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⣿⣿⠿⠿⠿⠿⠿⠉⠀⠉⠉⠉⠙⡇
⠙⠿⠏⠉⠛⢿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣦⠀⠀⠀⣤⣾⡁⠀⠀⠂⣿⣿⣿⣿⣿⣿⣯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡘⠁⠀⠀⠀⠀⡐⠻⣦⣄⡀⠀⠀⣠⣧
⢰⣿⡄⠀⠀⠀⠙⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢯⣾⣾⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢠⣾⡇⠀⣈⣿⣿⣷⣾⡟⠉
⢸⣿⣇⡀⠀⢀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⢢⣶⣿⣿⣷⣾⣻⣿⣿⣿⣿⣿⡆
⣼⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⡟⣟⢻⣿⣿⣿⡿⠟⠛⢹⣧⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢻⠟⣿⠿⠟⠋⠋⠛⠛⣿⡇
⣿⣿⣿⢷⣿⠁⠀⠀⠀⠀⠀⠀⠀⠊⠉⠈⢿⣿⠉⠉⠛⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠈⠉⠁⠀⠀⣀⣠⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡇
⣤⣤⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢤⡴⢴⣤⣿⣧⣀⣀⡎⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⢶⣶⢶⣴⣶⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠉⠉⣠⠄⠀⢿⣷
⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠿⢿⡟⢹⢧⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣾⣿⣿⣿⣿⣿⡿⠟⣋⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠠⠀⠀⠀⡁⣀⣠⣿⣿
⣥⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠃⠀⠀⠀⠀⢴⣾⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠝⣛⣻⣭⡤⢶⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣦⣅⣛⣿⣿⣿⣿⠿
⣿⠿⠥⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣴⣿⣚⡁⠀⠀⠀⣴⣿⣿⣿⡎⠛⣿⣯⣶⣤⡶⠶⠖⠒⠦⠤⣄⠀⠀⠀⠀⢰⣿⠟⢿⣷⣶⣤⣀⣠⣤⣤⣶⣾⣿⣶⣶⢶⣤⣄⡀⠀⠀⠀⢈⣹⣿⠛⠛⠛⢁⣼
⡆⠀⠀⠀⠘⣿⣆⣀⣠⣤⣤⣤⣤⣀⣀⠀⢀⣠⣴⣿⡉⠛⠡⢀⡀⠀⣿⣿⣿⣿⡇⡀⠘⣿⣿⣟⠁⠀⠒⠒⠒⠢⠄⠀⠀⠀⠀⠀⠈⢀⣀⣭⣿⡿⠟⠛⣛⣿⣭⣽⣿⣿⡍⠛⠉⢩⣷⣤⡤⠴⠿⠛⠁⠀⠀⢉⣠⣶
⣴⣶⣾⣿⣿⣿⣿⡛⠋⠉⠉⠉⠹⠿⠿⡿⠺⠿⠟⠉⠁⠐⠒⠀⢀⣤⣼⣿⣿⢿⠿⠇⣸⣿⣿⡟⢻⣿⣶⣶⣶⣶⣠⣾⣿⣷⣶⣶⣾⠿⠛⠉⣥⠄⣾⣿⣿⣿⣿⣿⣿⣿⠁⠸⠀⠈⣿⣿⣿⠿⣶⠀⣀⢀⣠⣼⡟⠉
⠋⠉⠉⠀⢀⣾⣿⠁⠀⠀⠀⢀⣾⣿⣿⣷⠀⢺⣿⣷⠒⠶⠖⢚⣉⣠⣿⡿⠉⠀⠀⡠⠹⣿⡇⢸⣿⣿⣿⣿⣿⣿⡟⠿⠟⠛⠋⠉⠀⠠⠄⠀⣠⣾⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣿⡿⣿⣶⡈⣾⣿⣿⣿⡇⢀⣀
⣤⡤⢴⡾⠿⣿⡅⠀⠀⠀⣀⣿⡟⣿⣿⣇⣀⠈⡛⢁⣀⣀⣀⣸⣿⣿⣿⣷⣧⣀⡞⠀⠁⠹⣿⡄⠹⣿⣿⣿⠟⢉⣶⣤⣤⣤⣤⣤⣤⣤⣴⣾⣿⠟⢈⣿⣿⣿⣿⣿⣿⠿⣧⠀⠀⢸⣿⡀⠈⠛⠓⠹⠿⠿⠟⠁⠿⠛
⠀⠀⠀⡀⠀⣹⡿⠀⣠⠞⠉⢰⣀⣿⣿⡄⠀⠘⠛⠀⠀⠀⠀⢠⣿⣿⢃⣿⣿⣿⣧⡄⢀⣀⣭⣧⣀⠉⠁⠀⣠⣿⠿⠿⠛⠛⠛⠛⠋⠉⢉⠡⢤⣤⣸⣿⣿⣿⣿⡿⠃⢸⣿⡄⠀⣿⣿⢻⣀⣀⠀⢠⣦⣴⣶⣦⠀⠀
⠤⢤⣴⣾⣿⣿⠃⡴⠃⠀⠀⣼⣿⣿⣿⡁⠈⠻⢷⠀⠉⠛⠛⣿⣿⡿⣿⡅⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡦⠀⣴⠄⠀⠀⠀⠀⠀⠠⠔⠉⠀⣤⣴⣿⣿⣿⣿⡿⠀⠀⠀⣾⣧⠀⠹⣿⡆⠻⠿⢿⡷⣿⣟⣋⡁⣤⣶
⠀⠆⠲⣶⠁⣿⠎⠰⠶⠶⢶⣿⠋⣿⣿⡇⠀⠠⠬⠀⠁⠀⢼⣿⡿⣤⣼⠇⣿⣿⣧⣽⡿⣿⣿⡏⠻⣿⠷⠀⢸⣿⣆⡀⠀⠀⢀⣀⣠⣴⣶⣿⣿⣿⡿⠛⠻⣿⣧⠀⠀⢰⣿⡿⠀⠀⢻⣧⡀⠀⠀⠀⠿⣿⣿⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⣇⣀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⣏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣸⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠿⠋⠸⠿⠇⠿⠇⠿⠇⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣴⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢰⣿⡟⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⢿⣿⣿⣿⣿⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⣷⣦⣤⡀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⢿⣿⠿⠏⠁⠻⠿⣿⣿⢛⣛⠀⠿⠿⠿⠸⠿⠿⣿⡇⠈⠿⢷⠿⠟⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣼⣾⠃⠀⠀⠀⠀⠀⢘⣿⡸⣿⣄⠀⠀⠀⠀⠀⢀⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⣼⠃⠄⠈⣿⣷⣤⣀⣀⣠⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠟⠿⠿⠿⠏⠀⠀⠉⠉⠉⢻
⣘⠿⠿⠉⠛⠻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀⠀⠀⢠⣼⣏⠀⠀⠐⢸⣿⣿⣿⣿⣿⣿⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠋⠀⠀⠀⠀⢀⠛⢳⣦⣀⠀⠀⢀⣼
⠃⣿⣇⠀⠀⠀⠈⢿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣹⣷⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⣴⣿⠀⢀⣹⣿⣿⣿⣿⡋
⠀⣿⣿⣀⠀⠀⣀⣾⠇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡲⢴⣿⣿⣿⣶⣟⣿⣿⣿⣿⣿⣷
⣤⣿⣿⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢛⢛⣛⣿⣿⣿⣿⠿⠛⠋⣿⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠿⢻⡿⠛⠛⠙⡉⠙⢻⣿
⣿⣿⣿⡿⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣿⡏⠉⠙⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠉⠉⠀⠀⢀⣀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⡀⠀⠀⠀⢀⣻
⣤⣬⣵⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠠⢤⠦⣦⣼⣿⣄⣀⣰⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⣶⣦⣶⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠈⠉⢉⠤⠀⠸⣿
⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠽⠿⣿⠞⡿⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣶⣿⣿⣿⣿⣿⣿⠿⢛⣥⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠄⢀⠀⢀⢀⣀⣾⣿
⣭⣴⡞⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠏⠀⠀⠀⠀⠠⣾⣿⣷⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣛⣛⣯⣥⡴⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢶⣬⣜⣻⣿⣿⣿⡿
⣿⣿⠭⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⣾⣷⣋⠀⠀⠀⢠⣾⣿⣿⣷⠙⢿⣿⣵⣦⣴⠶⠶⠒⠶⠤⢤⡀⠀⠀⠀⢀⣿⡟⠻⣷⣶⣦⣄⣀⣤⣤⣶⣶⣿⣷⣶⡶⣦⣤⣀⠀⠀⠀⠀⣉⣿⡟⠛⠛⠍⣠
⣶⡀⠀⠀⠈⢻⣷⣀⣀⣀⣠⣤⣤⣄⣀⡀⠀⣀⣄⣾⣏⠙⠋⠄⣀⠀⢸⣿⣿⣿⣿⢂⠀⢻⣿⣿⡋⠀⠐⠒⠒⠒⠤⠀⠀⠀⠀⠀⠀⠁⣀⣨⣿⣿⠿⠛⢛⣻⣭⣭⣿⣿⣯⠙⠋⠉⣽⣦⣄⣤⠾⠟⠃⠀⠀⢈⢉⣴
⣡⣴⣶⣿⣿⣿⣿⣟⠛⠋⠉⠉⠙⠿⠿⣿⠗⠿⠿⠋⠉⠀⠒⠂⠀⣠⣤⣿⣿⡟⡿⠿⢀⣧⣿⡿⠻⣿⣷⣶⣶⣶⣆⣴⣿⣿⣶⣶⣶⡿⠟⠋⢩⡴⢰⣿⣿⣿⣿⣿⣿⣿⡏⠀⡟⠀⢹⣿⣿⡻⢷⡆⢀⣀⣀⣄⢿⠋
⠏⠛⠉⠁⠀⣰⣿⡟⠀⠀⠀⠀⢰⣿⣿⣿⡆⠐⣾⣿⣆⠶⠶⠒⣋⣁⣼⣿⠏⠀⠀⢀⠜⢿⣿⠠⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠉⠀⠀⠤⠀⢀⣴⡿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣸⣿⢿⣷⣆⢠⣿⣿⣿⣿⠀⣀
⣀⣤⡤⣴⡾⢿⣯⠀⠀⠀⢀⣸⣿⢿⣿⣿⣁⣀⠙⠋⣁⣀⣀⣀⣿⣿⣿⣿⣾⣄⣰⠋⠀⠈⢿⣇⠈⢿⣿⣿⡿⠋⣱⣦⣤⣤⣤⣤⣤⣤⣤⣶⣿⡿⠃⢸⣿⣿⣿⣿⣿⡿⢿⡀⠀⠀⣿⡇⠀⠙⠛⠋⠟⠻⠿⠃⠸⠟
⠀⠀⠀⢀⠀⢈⣿⠇⢀⡴⠋⠉⣦⣸⣿⣧⠀⠈⠿⠃⠀⠀⠉⠀⣼⣿⡟⣸⣿⣿⣿⣠⠀⣀⣨⣽⣆⡈⠉⠀⢀⣼⡿⠿⠟⠛⠛⠛⠛⠉⠉⡉⠤⢤⣄⣿⣿⣿⣿⣿⠟⠀⣿⣧⠀⢰⣿⡟⣧⣀⣀⠀⣴⣦⣶⣶⡄⠀
⣤⣤⣤⣴⣿⣿⠟⢀⠞⠀⠀⢠⣿⣟⣿⣿⠉⠙⠿⡆⠉⠛⠛⣻⣿⣿⢿⣯⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠄⢠⡤⠀⠀⠀⠀⠀⠀⠤⠚⠁⢠⣤⣾⣿⣿⣿⣿⠇⠀⠀⢱⣿⡄⠈⢿⣷⠙⠻⠿⢿⡾⣿⣛⣉⢠⣴
⠀⠆⠲⣾⠉⢶⣷⠻⠶⠶⠆⣸⡟⢹⣿⣿⠀⠀⠠⠅⠈⠀⠠⣿⣿⢧⣤⡿⢹⣿⣿⣮⣿⢿⣿⣿⠛⢿⣿⠆⠀⣾⣿⣄⠀⠀⠀⣀⣀⣤⣴⣾⣿⣿⣿⠟⠛⢿⣿⡄⠀⠀⣼⣿⡇⠀⠈⣿⣄⠀⠀⠀⠸⢿⣿⡿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⣀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠸⠿⠇⠿⠇⠿⠇⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣀⣠⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣰⣿⡟⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣾⣿⣷⣦⣄⡀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⠿⠋⠻⠿⠿⣿⡿⢟⡻⠿⠿⠿⠸⠿⠇⢹⣿⠋⠿⠷⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣾⣿⠁⠀⠀⠀⠀⠀⢸⣷⣿⣷⡀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣃⠀⠀⠀⠀⠀⢀⡾⠀⠀⠹⣿⣶⣄⣀⣀⣤⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⠿⠿⠿⠿⠿⠋⠀⠀⠉⠉⠉
⠀⠻⠿⡿⠛⠻⢿⣿⣿⡿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣦⡀⠀⠀⣠⣾⡇⠀⠀⠄⣹⣿⣿⣿⣿⣿⣿⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡸⠋⠀⠀⠀⠀⣸⠛⢷⣦⣀⠀⠀⢀
⠾⠇⣶⡆⠀⠀⠀⠙⣿⡏⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣵⣶⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⣴⡇⠀⣀⣹⣿⣿⣿⣿
⠀⠀⣿⣷⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡢⢴⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⣤⣠⣿⣿⣿⣷⣾⣿⠿⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣟⢟⣻⣿⣽⣿⡿⠿⠻⠋⣷⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠨⠙⡟⢻⠿⠛⠛⠙⡉⠙⢻
⣿⣿⣿⣿⣿⣼⡟⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⢻⣿⡏⠉⠻⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠉⠉⠉⠀⠀⢀⣀⣴⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⠀⣀
⣌⣉⣉⣽⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣤⣿⣷⡀⢀⡸⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⢴⣴⣴⣶⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠈⠉⢁⠴⠀⢸
⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢙⡛⢿⣿⢛⡷⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣾⣾⣿⣿⣿⣿⣿⠟⢋⣵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢄⡀⠀⢀⢀⣠⣾
⣛⣋⣥⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⠏⠀⠀⠀⠀⢠⣶⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⢛⣯⣤⡴⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢶⣤⣘⣻⣿⣿⣿
⣿⣿⣿⠥⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡶⢏⠀⠀⠀⢠⣾⣿⣿⡎⠙⢿⣿⣷⣤⣴⡶⠶⠒⠦⠤⣤⠀⠀⠀⠀⢠⣿⡟⠿⣶⣶⣤⣀⣀⣤⣤⣶⣶⣿⣷⣶⠶⣦⣤⣀⠀⠀⠀⠀⣉⣿⡟⠛⠛⠍
⣭⣭⠁⠀⠀⠉⣿⣦⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣀⣀⣿⡏⠽⡋⠀⠀⠀⢸⣿⣿⣿⣿⠀⠈⢿⣿⣿⠃⠀⠐⠒⠒⠢⠄⠀⠀⠀⠀⠀⠈⠁⢀⣨⣿⡿⠿⠛⢛⣻⣭⣽⣿⣿⣯⠙⠋⠉⣿⣦⣄⡤⠾⠟⠃⠈⠀⢉⢠
⣿⣩⣤⣴⣶⣾⣿⣿⠿⠟⠛⠛⠙⠛⢿⣿⣿⠶⠿⠿⠋⠁⠀⠤⠀⠉⣁⣨⣿⣿⡟⡿⠟⢠⣯⣿⡿⠻⣿⣶⣶⣶⣴⣄⣴⣾⣶⣶⣶⣶⡿⠟⠋⢩⡤⣰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠇⠀⢻⣿⣟⡿⢷⡆⣀⣀⣀⣄⣿
⣿⠟⠛⠋⠉⠁⣸⣿⡇⠀⠀⠀⠀⣰⣿⣾⣿⡆⠲⣾⣷⡤⠦⠤⠖⣚⣁⣾⣿⠏⠀⠀⢀⠼⣿⣿⠠⣿⣿⣿⣿⣿⣿⣿⠻⠿⠛⠛⠋⠁⢀⠤⠀⢀⣴⡿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣸⣿⢿⣷⡖⢸⣿⣿⣿⣿⠀
⣀⣀⣀⣤⣴⡾⣿⣟⠀⠀⠀⠀⣸⣿⢿⣿⣿⡀⠀⠻⠟⠁⠀⠀⣰⣿⣿⣿⣿⣶⡄⣰⠃⠈⠘⣿⣆⠙⣿⣿⣿⡿⠋⣱⣤⣠⣤⣀⣀⣀⣠⣤⣶⣿⡿⠃⣿⣿⣿⣿⣿⣿⡿⣟⡀⠀⢀⣿⣇⠀⠙⠛⠋⠟⠻⠿⠃⠸
⠉⠁⠀⠀⠈⠀⢘⣿⠄⢀⡴⠛⠉⣅⣸⣿⡇⠀⠘⠿⠀⠁⠈⠉⠉⣽⣿⡟⣹⣿⣿⣿⣀⠀⣀⣈⣻⣀⠈⠃⠀⢀⣼⡿⠿⠟⠛⠛⠛⠛⠋⠉⡩⠥⣤⣀⣿⣿⣿⣿⣿⠟⠀⢿⣇⠀⢸⣿⡟⣆⣀⣀⠀⢶⣦⣶⣶⠄
⣤⣤⣤⣤⣤⣷⣾⡟⢀⡞⠀⠀⢸⣿⣿⣿⡏⠈⠑⠿⡄⠘⠛⠿⣿⣿⣿⣿⣏⢹⣿⡿⣿⣿⣿⣿⣿⣿⣿⣶⣦⠄⢠⡤⠀⠀⠀⠀⠀⠀⠤⠊⠁⢠⣤⣾⣿⣿⣿⣿⠁⠀⠀⢹⣿⡄⠈⢿⣷⠙⠿⠿⢿⣾⣿⣋⣉⢠
⠀⠆⠲⣶⠋⢿⠽⡷⠿⠶⠆⠀⢸⡿⢿⣿⣿⠐⠀⠀⠇⠀⠀⠤⣿⣿⢇⣠⣿⢸⣿⣿⣝⣿⢿⣿⣿⠻⢿⣿⠆⠀⣿⣧⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⠟⠻⣿⣿⡀⠀⠀⣿⣿⡇⠀⠘⣯⣄⠀⠀⠀⠸⢿⣿⡿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⣇⣀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠸⠿⠇⠿⠇⠿⠇⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣀⣠⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣰⣿⡟⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣾⣿⣷⣦⣄⡀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣿⣿⠿⠋⠻⠿⠿⣿⠿⣟⡻⠿⠿⠿⠸⠿⠏⢹⣿⠋⠿⠷⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣶⣿⠁⠀⠀⠀⠀⠀⢸⣷⣿⣷⡀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣇⠀⠀⠀⠀⠀⢀⡾⠀⠀⠹⣿⣶⣄⣀⣀⣤⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⠿⠿⠿⠿⠿⠋⠀⠀⠉⠉⠉
⠀⠻⢿⡿⠛⠻⢿⣿⣿⡿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣦⡀⠀⠀⣠⣾⡇⠀⠀⠄⣹⣿⣿⣿⣿⣿⣿⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡸⠋⠀⠀⠀⠀⣸⠛⢷⣦⣀⠀⠀⢀
⡶⠇⣶⡆⠀⠀⠀⠙⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣵⣶⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⣴⡇⠀⣀⣹⣿⣿⣿⣿
⠀⠀⣿⣷⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡢⢴⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⣤⣠⣿⣿⣿⣷⣾⣿⠿⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣟⢛⣻⣿⣽⣿⡿⠿⠻⠋⣷⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠨⠙⡟⢻⠿⠛⠛⠙⡉⠙⢻
⣿⣿⣿⣿⣿⣼⡟⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⢻⣿⡏⠉⠻⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠉⠉⠉⠀⠀⢀⣀⣴⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⡀⠀⠀⠀⣀
⣄⣉⣉⣽⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣤⣿⣷⡀⢀⡸⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⢴⣴⣴⣶⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠈⠉⢁⠴⠀⢸
⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢙⡛⢿⣿⢛⡷⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣾⣾⣿⣿⣿⣿⣿⠟⢋⣵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢄⡀⠀⢀⢀⣠⣾
⣛⣋⣥⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⠏⠀⠀⠀⠀⢠⣶⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣻⣯⣤⠴⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢶⣤⣘⣻⣿⣿⣿
⣿⣿⣿⠥⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡶⢏⠀⠀⠀⢠⣾⣿⣿⡎⠙⢿⣿⣷⣤⣴⡶⠶⠒⠦⠤⣤⠀⠀⠀⠀⢠⣿⡟⠿⣷⣶⣤⣀⣀⣠⣤⣶⣶⣿⣷⣶⠶⣦⣤⣀⠀⠀⠀⠀⣉⣿⡟⠛⠛⠍
⣭⣭⠁⠀⠀⠉⢿⣦⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣀⣀⣿⡏⠽⡋⠀⠀⠀⢸⣿⣿⣿⣿⠀⠈⢿⣿⣿⠃⠀⠐⠒⠒⠢⠄⠀⠀⠀⠀⠀⠈⠁⢀⣨⣽⡿⠿⠛⢛⣻⣭⣭⣿⣿⣯⠙⡋⠉⣿⣦⣄⡤⠾⠟⠃⠀⠀⢉⢡
⣿⣩⣤⣴⣶⣾⣿⣿⠿⠟⠛⠛⠛⠛⢿⣿⣿⠖⠿⠿⠋⠁⠀⠤⠀⠉⣁⣬⣿⣿⡟⡿⠿⢠⣯⣿⡿⠻⣿⣶⣶⣶⣶⣄⣴⣾⣶⣶⣶⣶⡿⠟⠋⢩⡤⣰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠇⠀⢻⣿⣟⠻⢷⡆⣀⣀⣀⣄⢿
⣿⠟⠛⠋⠉⠉⣰⣿⡇⠀⠀⠀⠀⣰⣿⣾⣿⡆⠐⣾⣷⡤⠦⠤⠖⣚⣁⣾⣿⠏⠁⠀⢀⠼⣿⣿⠠⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠋⠁⢀⠤⠀⢀⣴⡿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣸⣿⢿⣷⡖⢸⣿⣿⣿⣿⠀
⣀⣀⣀⣤⣴⡾⣿⣟⠀⠀⠀⠀⣸⣿⢿⣿⣿⡀⠀⠻⠟⠁⠀⠀⣰⣿⣿⣿⣿⣶⡀⣰⠃⠈⠘⣿⣆⠙⣿⣿⣿⡿⠋⣱⣄⣠⣤⣀⣀⣀⣠⣤⣶⣿⡿⠃⣿⣿⣿⣿⣿⣿⡿⢟⡀⠀⢀⣿⣇⠀⠙⠛⠚⠟⠻⠿⠃⠸
⠉⠉⠀⠀⠈⠀⢈⣿⠀⢀⡴⠛⠉⣍⣸⣿⡇⠀⠘⠿⠀⠁⠈⠉⠉⣽⣿⡟⣹⣿⣿⣿⣀⠀⣀⣈⣻⣀⠈⠋⠀⢀⣼⡿⠿⠟⠛⠛⠛⠛⠉⠉⡩⠥⣤⣀⣿⣿⣿⣿⣿⠟⠀⢿⣇⠀⢸⣿⡿⣆⣀⣀⠀⣶⣦⣶⣶⠄
⣤⣤⣤⣤⣬⣷⣾⠟⢀⡞⠀⠀⢸⣿⣿⣿⡏⠁⠐⠿⡄⠘⠛⠿⣿⣿⣿⣿⣏⢹⣿⡿⣿⣿⣿⣿⣿⣿⣿⣶⣦⠄⢠⡤⠀⠀⠀⠀⠀⠀⠤⠊⠁⢠⣤⣾⣿⣿⣿⣿⠁⠀⠀⢹⣿⡄⠈⢿⣧⠙⠿⠿⢿⣾⣿⣋⣉⢠
⠀⠆⠲⣾⠋⢿⠽⡷⠿⠶⠆⠀⢸⡿⢿⣿⣿⠐⠀⠀⠅⠀⠀⢤⣿⣿⢇⣠⣿⢸⣿⣿⣝⡻⢿⣿⣿⠹⢿⣿⠆⠀⣿⣧⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⠟⠻⣿⣿⡀⠀⠀⣾⣿⡇⠀⠘⣯⣄⠀⠀⠀⠸⢿⣿⡿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡇⢸⣿⣯⣵⣾⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠸⠿⠇⠿⠇⠿⠇⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡟⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⣿⣿⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣿⣿⣶⣦⣄⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠻⠿⠿⠿⣿⠿⠿⠿⠿⠿⠸⠿⠇⠀⢻⣿⠿⠷⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢈⣧⡿⠁⠀⠀⠀⠀⠀⢹⣗⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⢀⡾⢀⠀⢹⣿⣦⣄⣀⣀⣴⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⠿⠿⠿⠿⠿⠉⠀⠀⠉⠉
⠀⠀⠿⢿⡿⠻⠿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣄⠀⠀⢀⣠⣾⡃⠀⠀⠄⣻⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⠉⠀⠀⠀⠀⡸⠛⢶⣤⣀⠀⠀
⣠⣼⢇⣶⡄⠀⠀⠈⠻⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⣷⣶⣾⣿⣿⣷⣶⣾⣾⣿⣿⣿⣿⣿⣿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⢠⣾⡇⠀⣀⣻⣿⣿⣿
⠋⠀⠀⣿⣇⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⡢⣶⣿⣿⣷⣾⣿⣿⣿⣿⣿
⣤⣄⣀⣿⣿⣷⣶⣶⣿⡏⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⡿⣿⢻⣿⣿⣿⡿⠿⠻⢻⣧⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠙⠟⢻⠿⠛⠛⢉⡉⠛
⣿⣿⣿⣿⣿⣯⣽⡟⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⢿⣿⠟⠛⠻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠙⠉⠉⠈⠀⢀⣠⣼⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⠀
⣈⣉⣉⣛⣻⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⣌⣿⣧⠀⠀⡸⢹⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⢴⣤⣴⣶⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠈⠉⣡⠔⠀
⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠈⢛⣿⣿⣿⢻⣧⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣾⣾⣿⣿⣿⣿⣿⠿⢋⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡐⠄⡀⠀⢀⢀⣠
⠿⠿⢛⣩⡴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣷⠏⠀⠀⠀⠀⢠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⡿⢿⣯⣥⠴⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣤⣘⣻⣿⣿
⣽⣿⣿⣿⣃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣿⡴⢆⠀⠀⠀⣠⣾⣿⣷⡍⠻⣿⣿⣦⣤⣤⣶⠶⠶⠤⣄⣀⠉⠈⠉⠁⢠⣿⡟⢿⣶⣶⣤⣀⣀⣠⣤⣴⣶⣿⣷⣶⡶⣦⣄⣀⠀⠀⠀⢀⣹⣿⠛⠛⠛
⣉⣉⣭⠁⠀⠈⠙⣷⣦⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣠⣿⡏⠽⢋⠄⠀⠀⢻⣿⣿⣿⣧⠂⠘⣿⣿⣿⠃⠀⠐⠒⠢⠤⡄⠀⠀⠀⠀⠀⠈⠁⢈⣩⣿⡿⠿⠛⢛⣛⣯⣯⣿⣿⡭⠘⠋⠉⣿⣦⣤⡴⠿⠛⠃⠀⠀⢉
⣿⣿⢋⣠⣤⣶⣶⣿⣿⡿⠿⠛⠛⠛⠻⣿⣿⣷⠴⣿⡿⠛⠃⠀⠤⠀⠉⣀⣸⣿⣿⡿⡿⡇⢰⣾⣿⣿⢶⣶⣶⣤⣤⣤⣀⣴⣶⣶⣶⣴⣶⡿⠟⠛⣩⡤⣴⣿⣿⣿⣿⣿⣿⣿⠇⠰⠇⠀⢻⣿⣟⡻⢷⡆⣀⣀⣀⣤
⣿⣿⡟⠟⠛⠋⠉⣹⣿⡇⠀⠀⠀⠀⣰⣶⣶⣷⠀⠠⣶⣶⡠⠤⠤⠴⢒⣉⣿⣿⠟⠀⠀⢀⠸⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠁⢀⠤⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣼⡿⣿⣷⡇⢰⣿⣿⣿⣿
⣉⠁⠀⣀⣀⣠⣶⣿⣟⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠿⠟⠁⠀⠀⢰⣿⣿⣿⣿⣦⡀⣰⠋⠀⠹⣿⡄⠙⣿⣿⣿⡿⠏⣥⣄⣀⣀⣀⣀⣀⣀⣤⣶⣿⡿⠁⣿⣿⣿⣿⣿⣿⡿⢿⡀⠀⢀⣿⢇⠀⠙⠛⠊⠛⠿⠿⠃
⠛⠛⠉⠉⠉⠉⠀⠘⣿⠀⠀⡠⠞⠋⡅⢻⣿⡇⠈⠸⠿⠀⠉⠉⠉⠉⣽⣿⡟⣹⣿⣿⣯⡀⠀⢀⣉⣻⡀⠉⠋⠀⢀⣼⣿⠿⠿⠟⠛⠛⠛⠋⠉⡩⠥⣤⣀⣿⣿⣿⣿⣿⠟⠀⣿⡇⠀⢸⣿⡾⣆⣀⣀⠀⣶⣦⣶⣶
⣀⣀⣀⣀⣀⣨⣶⣾⡿⢀⡜⠁⠀⢸⣿⣿⣿⡇⠀⢲⡾⡄⠒⠳⠶⣾⣿⣿⣶⣏⢹⣿⣻⣿⣿⣿⣿⣿⣿⣿⣶⣤⠀⣩⡄⠀⠀⠀⠀⠀⢀⠠⠚⠁⢤⣠⣾⣿⣿⣿⣿⠁⠀⠀⢸⣧⡄⠈⢿⣇⠙⠿⠿⢿⣞⣿⣋⣉
⠀⠏⠻⣿⠋⢿⠟⠹⣷⠾⠆⠀⠀⢸⡿⣗⣿⣷⠀⠀⣀⠃⠐⠀⢠⣿⣿⣇⣈⣿⢸⣿⣿⣝⡿⣿⣿⡿⠻⣿⣿⠄⠀⣿⣧⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⠟⠻⣿⣿⡀⠀⠀⣾⣿⠇⠀⠘⣿⣄⠀⠀⠀⠸⢿⡿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢹⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⣏⣥⣾⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⢇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠸⠿⠇⠿⠇⠿⠇⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢻⣿⣿⢿⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⣷⣶⣤⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠿⠿⠿⠟⢿⡿⠿⠿⠿⠿⠸⠿⠇⠀⠹⣿⡿⠷⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣹⣼⠋⠀⠀⠀⠀⠀⠘⣿⣺⣿⡀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⣸⠃⡀⠘⣿⣶⣄⣀⣀⣠⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⠿⠿⠿⠿⠿⠟⠁⠀⠈⠉
⠀⠀⠸⢿⣿⠿⠿⢿⣿⣿⣿⣻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣀⠀⠀⣀⣴⡏⠀⠀⠠⢸⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠋⠁⠀⠀⠀⢀⠗⠻⣦⣄⡀⠀
⢀⣠⣾⣰⣄⠀⠀⠀⠙⢿⡿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣳⣶⣶⣿⣿⣿⣦⣾⣷⣿⣿⣿⣿⣿⣿⡿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣴⣿⠀⢀⣘⣿⣿⣿
⡿⠋⠀⢹⣿⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣖⢴⣾⣿⣿⣶⣿⢿⣿⣿⣿
⣧⣀⡀⣸⣿⣷⣶⣶⣾⣿⠁⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⢿⡿⣿⣿⣿⣿⢿⠟⠟⣿⣬⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢉⠿⢻⡿⠟⠛⠉⣉⠉
⣿⣿⣿⣿⣿⣿⣭⣿⠋⠀⠀⠀⠀⠀⠀⠀⠜⠛⢻⣿⡿⠛⠻⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠋⠉⠉⠁⠀⣀⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⢀⠀⠀⠀
⠉⣉⡙⠛⣛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⣀⣹⣿⡄⠀⢀⠏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⡴⣦⣤⣶⣾⣿⣿⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠈⠉⢉⡤⠂
⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠙⣻⣿⣿⡞⣿⣔⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣵⣷⣿⣿⣿⣿⣿⡿⠟⣩⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠥⣀⠀⠀⡀⣀
⠿⠿⠿⢛⣥⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⡾⠁⠀⠀⠀⠀⣤⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣻⠿⣻⣭⣤⢾⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣦⣄⣟⣿⣿
⣭⣿⣿⣿⣟⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣸⣧⡴⡀⠀⠀⠀⣴⣿⣿⣿⠙⢿⣿⣷⣄⣤⣴⡶⠶⠤⢄⣀⡈⠉⠉⠉⠀⣼⣿⠻⣷⣶⣦⣄⣀⣀⣤⣤⣶⣾⣿⣶⡶⢶⣤⣄⡀⠀⠀⠀⣉⣿⡟⠛⠛
⣉⣉⣛⡟⠁⠀⠉⢻⣶⡀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣼⣿⠛⠟⠁⠀⠀⠘⣿⣿⣿⣿⡶⠀⢻⣿⣿⡟⠀⠀⠐⠒⠤⢄⠀⠀⠀⠀⠀⠁⠉⠀⣉⣽⣿⡿⠟⠛⣛⣻⣯⣽⣿⣯⠍⠛⠉⢹⣷⣤⣤⠼⠿⠛⠈⠁⠨
⣿⣿⡟⣃⣠⣤⣴⣾⣿⣿⠿⠿⠛⠛⠛⢿⣿⣷⡦⢾⣿⠿⠛⠀⠠⠄⠈⠁⣀⣿⣿⣿⢿⣷⠂⣾⣿⣿⡷⣶⣶⣤⣤⣤⣄⣠⣶⣶⣶⣤⣶⣾⠿⠛⢋⣥⢄⣾⣿⣿⣿⣿⣿⣿⣿⠀⠺⠀⢸⣿⣿⡛⠿⡶⢀⣀⣀⣤
⣿⣿⣿⠿⠿⠛⠛⠉⣿⣿⠀⠀⠀⠀⢀⣴⣾⣿⡧⠀⣴⣶⣄⠤⣤⠤⠖⡊⣹⣿⡿⠏⠀⠀⢀⢿⣿⡆⣼⣿⣿⣿⣿⣿⣿⡿⡿⠿⠛⠛⠉⠀⣠⠄⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢀⣿⡿⣿⣾⠈⣿⣿⣿⣿
⠛⠉⠁⠀⢀⣀⣤⣾⣿⡋⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠸⠿⠋⠀⠀⠀⣾⣿⣿⣿⣷⣔⢀⡔⠁⠈⢻⣧⠈⢻⣿⣿⣿⠿⢉⣄⣀⣀⣀⣀⣀⣀⣠⣴⣾⣿⠟⢸⣿⣿⣿⣿⣿⣿⠿⣇⠀⠀⣸⣿⡀⠉⠛⠋⠻⠻⠿⠟
⠟⠛⠛⠋⠉⠉⠉⠈⢻⡇⠀⢀⡴⠚⢫⠈⣿⣿⠉⠁⢶⠆⠈⠉⠙⠉⢉⣿⣿⠋⣿⣿⣿⣇⠀⠀⢈⣛⣇⠈⠙⠁⠀⣠⣿⡿⠿⠿⠛⠛⠛⠛⠉⢉⠭⣤⣄⣸⣿⣿⣿⣿⡿⠃⢸⣿⡀⠀⣿⣿⢳⣀⣀⡀⠰⣦⣴⣶
⣀⣀⣀⣀⣀⣀⣱⣴⣿⠇⢠⠏⠀⠀⣼⣿⢿⣿⠂⠐⢶⣆⠐⠲⠶⢶⣾⣿⣿⣾⡋⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⣦⡄⢈⣥⠀⠀⠀⠀⠀⠀⡀⠴⠋⠀⣄⣴⣿⣿⣿⣿⡏⠀⠀⠈⣿⣦⠀⠹⣿⡎⠻⠿⢿⣷⣻⣯⣉
⠀⠟⠺⣿⠋⢿⠟⠻⢿⡶⠇⠀⠀⠀⣿⢿⣾⣿⡆⠀⢀⣘⠀⠂⠀⣸⣿⣿⣀⣹⡇⣿⣿⣯⣻⢿⣿⣿⡟⢿⣿⡧⠀⢸⣿⣄⠀⠀⠀⠀⣀⣀⣤⣶⣿⣿⣿⣿⠟⢿⣿⣇⠀⠀⢰⣿⣿⠀⠀⢻⣧⡀⠀⠀⠀⠯⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢹⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⣏⣡⣼⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣇⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠸⠿⠇⠿⠇⠿⠇⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣴⣶⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⢿⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣾⣿⣿⣶⣦⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠿⠿⠿⠟⠻⣿⠿⠿⠿⠿⠸⠿⠇⠀⠈⣿⣿⠷⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣈⣿⡟⠁⠀⠀⠀⠀⠀⢿⡗⣿⣇⠀⠀⠀⠀⠀⠀⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⡇⠀⠀⠀⠀⠀⢠⡟⢀⠈⢻⣷⣤⣀⡀⣀⣴⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠿⠿⠿⠿⠁⠀⠀⠀
⠀⠀⠈⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⠀⠀⢀⣠⣾⠁⠀⠀⠄⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡙⠁⠀⠀⠀⠀⡘⠛⢶⣦⣀⠀
⠀⣀⣴⣆⣮⡀⠀⠀⠉⢿⣿⡟⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣶⣾⣿⣿⣷⣴⣿⣾⣿⣿⣿⣿⣿⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⣠⣾⡇⠀⣀⣻⣿⣿
⣾⠟⠁⠈⣿⡇⠀⠀⠀⢀⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣗⠢⣶⣿⣿⣷⣾⣿⣿⣿⣿
⣿⣀⣀⢀⣿⣿⣶⣦⣶⣾⡏⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡿⣿⢿⣿⣿⣿⡿⡿⠿⢻⣧⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠟⢿⠿⠛⠋⢉⡉
⣿⣿⣿⣿⣿⣿⣿⣽⡟⠁⠀⠀⠀⠀⠀⠀⠠⠛⠛⣿⣿⠟⠛⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠘⠉⠉⠉⠀⢀⣠⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀
⡈⢉⡙⠛⠛⣻⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣂⣈⣿⣆⠀⠀⡹⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢦⣦⣤⣴⣶⣿⣿⣿⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠈⠉⣡⠔
⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠻⣿⣿⣷⢿⣇⣺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣬⣼⣾⣿⣿⣿⣿⣿⠿⢋⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡐⢄⡀⠀⢀⢀
⠿⠿⠿⠟⣋⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣷⠏⠀⠀⠀⠀⢠⣴⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⢿⣯⣥⠴⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣤⣘⣻⣿
⣍⣽⣿⣿⣿⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⢆⠀⠀⠀⢠⣾⣿⣿⡍⠻⣷⣗⣦⣠⣤⣴⡶⠦⠤⣀⣀⠉⠉⠉⠀⢠⣿⡟⢿⣶⣦⣤⣀⣀⣠⣤⣶⣶⣿⣶⣶⠶⣦⣤⣀⠀⠀⠀⢀⣹⣿⠛⠛
⣉⣉⢙⣛⠋⠀⠈⠛⣶⣄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⣿⠟⠻⠋⠀⠀⠀⢿⣿⣿⣿⣷⠄⠘⣿⣿⣿⠃⠀⠀⠐⠦⠤⡈⠀⠀⠀⠀⠈⠈⠁⢈⣩⣿⣿⠿⠛⢛⣻⣿⣿⣿⣿⣭⠙⠋⠉⣿⣦⣠⡤⠿⠟⠁⠉⠀
⣿⣿⣿⢛⣀⣠⣤⣴⣿⣿⣿⠿⠿⠿⠻⠿⣿⣷⣶⢤⣿⣿⠟⠃⠀⠤⠀⠁⢀⣸⣿⣿⡿⣿⡇⢰⣿⣿⣿⢶⣶⣦⣤⣤⣤⣀⣴⣶⣶⣦⣤⣶⡿⠟⠛⣩⡤⣰⣿⣿⣿⣿⣿⣿⣿⡇⠐⠇⠀⢿⣿⣟⡻⢷⡆⣀⣀⣀
⣿⣿⣿⣿⠿⠿⠛⠋⢹⣿⡇⠀⠀⠀⠀⢠⣾⣿⣿⠀⢤⣶⣦⡠⣤⡤⠤⠖⠋⣿⣿⠟⠀⠀⠀⡸⣿⣷⢠⣿⣿⣿⣿⣿⣿⣿⢿⠿⠟⠛⠋⠁⢀⡤⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣸⣿⢿⣷⡇⢸⣿⣿⣿
⠛⠛⠉⠀⠀⢀⣠⣴⣿⡿⠁⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠿⠟⠁⠀⠀⢰⣿⣿⣿⣿⣦⡀⣠⠊⠐⠙⣿⡄⠹⣿⣿⣿⡿⠟⣡⣀⣀⣀⣀⣀⣀⣀⣤⣶⣿⡿⠃⣿⣿⣿⣿⣿⣿⡿⢿⡀⠀⠀⣿⣇⠀⠙⠛⠈⠟⠿⠿
⣿⠛⠛⠋⠋⠉⠙⠁⠙⣿⡄⠀⣠⠶⠛⡅⢹⣿⡏⠉⠀⣶⠀⠁⠙⠛⠉⣹⣿⡟⢻⣿⣿⣿⡀⠀⠀⣙⣻⡀⠙⠋⠈⢀⣼⣿⡿⠿⠿⠻⠛⠛⠋⠉⡉⢥⣄⣀⣿⣿⣿⣿⣿⠟⠀⣿⣇⠀⢸⣿⡟⣦⣀⣀⠀⢰⣦⣶
⢀⣀⣀⣀⣀⣀⣀⣦⣴⡿⠀⡼⠁⠀⢠⣿⡿⣿⡗⠐⠲⣶⡀⠒⠶⠶⣶⣿⣿⣷⣟⢹⣿⣿⣿⣾⣿⣿⣿⣿⣷⣦⣤⠈⣩⡄⠀⠀⠀⠀⠀⢀⠠⠞⠁⢠⣠⣾⣿⣿⣿⣿⠅⠀⠀⢘⣿⡄⠀⢿⣷⠝⠿⠿⢿⣞⣿⣍
⠀⠟⠻⣿⠛⢿⠟⠻⠿⣷⠞⠁⠀⠀⢸⣿⣷⣿⣿⠀⠀⢀⡃⠐⠀⢀⣿⣿⣇⣈⣿⢸⣿⣿⣹⣿⣿⣿⣿⠻⣿⣿⡅⠀⣿⣧⡀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⠟⠻⣿⣿⡀⠀⠀⣿⣿⡇⠀⠈⣿⣄⠀⠀⠀⠸⠽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢹⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⣁⣼⣿⣷⣿⣿⣿⣿⣿⣿⣾⣿⣇⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠸⠿⠇⠿⠇⠿⠇⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣠⣶⣾⣿⣿⣿⣿⣿⣷⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⣿⣿⠿⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣿⣷⣶⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠿⠿⠟⠛⣿⠿⠿⠿⠿⠸⠿⠇⠀⠀⢹⣿⡷⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⣾⣿⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⢏⠀⠀⠀⠀⠀⠀⣼⠁⡀⠹⣿⣦⣄⣀⣀⣠⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⠻⠿⠟⠋⠀⠀⠀
⠀⠀⠀⢹⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⣀⣼⡇⠀⠀⠄⢹⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠋⠀⠀⠀⠀⢸⠛⠳⣦⣄⡀
⠀⠀⣠⡶⣨⡅⠀⠀⠈⠻⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣴⣶⣿⣿⣿⣤⣾⣷⣿⣿⣿⣿⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢀⣴⣿⠀⢀⣘⣿⣿
⣶⡿⠋⠁⢿⣿⠀⠀⠀⠀⣸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡒⢴⣿⣿⣿⣶⣿⣿⣿⣿
⣿⣅⣀⠀⣸⣿⣶⣦⣤⣶⣿⠁⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⢿⠿⣿⣿⣿⣿⢿⠿⠟⣯⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠿⢻⡿⠛⠛⠉⣉
⣿⣿⣿⣿⣿⣿⣿⣫⣿⠋⠀⠀⠀⠀⠀⠀⠀⠼⠛⢻⣿⡿⠛⠻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠋⠋⠉⠉⠁⠀⣀⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⣀⠀⠀
⣈⠉⠛⠛⠻⢟⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣀⣻⣷⡀⠀⢈⠏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣴⣤⣤⣶⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢆⠀⠈⠉⢉⡤
⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢿⣿⣿⡾⣿⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣥⣷⣿⣿⣿⣿⣿⡿⠟⣡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠄⣀⠀⠀⡀
⡿⢿⣿⠿⢛⣩⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠀⣯⡿⠁⠀⠀⠀⠀⣤⣶⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡻⠿⣿⣭⡤⢾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣦⣄⣛⣿
⣌⣑⣶⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡵⠀⠀⠀⢀⣴⣿⣿⣯⠛⢿⣾⣲⣄⣤⣤⣤⠦⠤⢄⣀⡈⠉⠉⠁⠀⣼⣿⠻⣶⣶⣤⣄⣀⣀⣤⣴⣶⣾⣷⣶⣶⣶⣤⣄⡀⠀⠀⠀⣉⣿⡟⠛
⣈⣉⠙⢛⡛⠉⠉⠙⢶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢓⠞⢁⠀⠀⢸⣿⣿⣿⣿⡀⠀⢻⣿⣿⡟⠀⠀⠀⠢⠤⢄⠁⠀⠀⠀⠀⠁⠋⠈⣉⣽⣿⡿⠟⠛⣛⣻⣿⣿⣿⣯⠍⠛⠉⢹⣷⣤⣤⠼⠟⠋⠀⠁
⣿⣿⣿⡟⢃⣀⣤⣤⣼⣿⣷⣿⡿⠿⠿⠿⣿⣷⣶⡤⣼⣿⡿⠛⠀⢈⠄⠈⠀⣀⣿⣿⣿⣿⣿⠀⣾⣿⣿⡷⣶⣶⣤⣤⣤⣄⣠⣶⣶⣶⣤⣴⣾⠿⠛⢋⣥⢤⣾⣿⣿⣿⣿⣿⣿⡿⠀⡾⠀⠸⣿⣿⣛⠿⡦⢀⢀⣀
⣿⣿⣿⣿⠿⠿⠟⠛⠋⣿⣷⠀⠀⠀⠀⠀⣼⣯⣿⡥⠤⣤⣶⣄⣤⣤⡤⠴⠚⣹⣿⡿⠇⠈⠀⢀⢿⣿⠊⣴⣿⣿⣿⣿⣿⣿⡿⡿⠿⠛⠛⠉⠀⡠⠄⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢠⣿⠿⣿⣶⠀⣿⣿⣿
⡛⠛⠋⠁⠀⠀⢀⣠⣾⣿⠏⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠸⡿⠋⠀⠀⠀⣾⣿⣿⣿⣧⣀⢀⡔⠁⠈⣻⣧⠙⢿⣿⣿⣿⠿⢋⣄⣀⣀⣀⣀⣀⣀⣠⣴⣾⣿⠏⢸⣿⣿⣿⣿⣿⣿⠿⡇⠀⠀⣸⣿⡀⠈⠛⠃⠙⠻⠿
⣿⡟⠛⠛⠛⠋⠙⠋⠉⢿⡇⠀⢀⡴⠞⢛⠈⣿⣿⠉⠁⢶⡆⠈⠐⠒⠋⢉⣿⣿⠛⣿⣿⣿⡁⠀⠀⠈⣛⡆⠈⠛⠉⠁⣠⣿⣿⠿⠿⠟⠟⠛⠛⠉⢉⠭⣤⣀⣸⣿⣿⣿⣿⡿⠃⠘⣿⡀⠀⣿⣿⢷⣀⣀⡀⠰⣶⣴
⠁⣀⣀⠀⠀⣀⣀⣢⣤⣿⠇⢠⠏⠀⠀⣾⣾⣿⣿⠒⠒⢶⣆⠐⠲⠶⣶⣾⣿⣿⣾⠛⣿⣿⣿⣷⣿⣿⣿⣿⣿⣶⣤⡄⠉⣡⠀⠀⠀⠀⠀⠀⡀⠴⠋⠠⣄⣴⣿⣿⣿⣿⡯⠀⠀⠀⣼⣧⠀⠹⣿⡾⠻⠿⠿⣷⣻⣿
⠁⠟⠻⣿⠟⢿⠟⠻⠿⢿⡶⠋⠀⠀⠀⣿⣿⣾⣿⡆⠀⠀⢸⠀⠀⠀⣸⣿⣿⢈⣹⡇⣿⣿⣏⣿⣿⣿⣿⠟⢿⣿⣏⠀⢸⣿⣄⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⠟⢿⣿⣇⠀⠀⢰⣿⣿⠀⠀⢻⣧⡀⠀⠀⠀⠭⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢹⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⣸⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⠿⠇⠿⠇⠿⠇⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣤⣶⣾⣿⣿⣿⣷⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠿⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣿⣿⣿⣿⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣯⣿⣿⣿⠿⠟⠉⠻⣿⡿⠿⠿⠹⠿⠇⠀⠀⠈⢻⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣞⣿⠋⠀⠀⠀⠀⠀⢹⣧⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⣼⠉⠈⠹⣷⣦⣀⠀⢀⣠⣾⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⠿⠛⠻⠿⢟⠋⠀⠀
⠉⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣄⠀⠀⠀⣀⣼⡇⠀⠁⠄⢹⣿⣿⣿⣿⣿⣿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠫⠋⠀⠀⠀⠀⢨⠛⠻⣶⣤
⠀⠀⠀⢀⡄⣭⡍⠀⠀⠙⢻⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⣶⣿⣿⣷⣤⣶⣶⣿⣿⣿⣿⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢀⣴⣿⠀⣀⣘⣿
⣤⣤⣾⠟⠁⣿⣿⠀⠀⠀⠀⣹⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡲⢴⣿⣿⣿⣶⣿⣿⣿
⣿⣿⡇⠀⠀⢹⣿⣦⣤⣤⣴⣿⠁⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⣿⢿⠿⣿⣻⣿⣿⣿⠿⠟⣯⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠿⢻⡿⠛⠋⠉
⢻⣿⣿⣿⣷⣿⣿⣿⢛⣿⠏⠉⠀⠀⠀⠀⠀⠀⠼⠟⢿⣿⡿⠿⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠚⠛⠛⠛⠉⠁⠀⢀⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠀⠀⣀⠀
⢀⠉⠛⠛⠿⠿⠿⣯⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣈⣻⣧⡀⠀⢘⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠄⠀⠈⠉⢁
⣼⣿⣷⣿⣷⣶⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠿⢿⣿⣷⡶⣿⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣬⣷⣿⣿⣿⣿⣿⡿⠟⣡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠄⣀⠀⠀
⠻⢿⣿⣿⣿⠟⣋⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠁⣯⣿⠃⠀⠀⠀⠀⣤⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢛⢻⠿⣟⣩⣤⢾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣦⣤⣛
⣦⣉⢒⣶⣶⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣸⣯⡵⠀⠀⠀⠀⣴⣿⣿⣯⠻⣿⣶⣦⣄⣠⣤⣤⠤⢤⣀⣀⡈⠉⠉⠉⠀⣼⣿⠻⣶⣶⣤⣀⣀⣀⣤⣴⣶⣾⣷⣶⣶⣶⣤⣄⡀⠀⠀⠀⣈⣽⣿
⣈⠉⠛⠛⠿⠿⠉⠉⠓⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣒⡶⢁⠀⠀⢸⣿⣿⣿⣿⡀⠈⢿⣿⣿⡟⠁⠀⠤⠤⠤⣈⠁⠀⠀⠀⠀⠁⠋⠈⣈⣽⣿⡿⠟⠛⣛⣻⣿⣿⣿⣯⡍⢛⠉⢹⣷⣤⣠⠴⠟⠋⠀
⣿⣿⣿⣿⡿⠇⢀⣀⣠⣤⣿⣷⣶⣶⠾⠿⠿⣿⣶⣶⣄⣼⣿⡿⠓⠀⣈⡀⠐⠂⡀⣿⣿⣿⣿⣷⠀⣾⢿⣿⡷⣶⣦⣤⣤⣤⣄⣠⣴⣶⣦⣤⣴⣶⡿⠟⢋⣡⢄⣾⣿⣿⣿⣿⣿⣿⣿⠀⠸⠀⠸⣿⣿⣛⠿⣶⢀⣀
⣿⣿⣿⣿⣷⠿⡿⠿⠿⠛⣿⣿⠀⠀⠀⠀⠀⣼⣭⣿⡅⠀⣡⣦⡄⣀⣀⣠⠤⠞⢹⣿⣿⠛⠈⠀⢀⣿⣷⠏⣴⣾⣿⣿⣿⣿⣿⡿⡿⠿⠟⠛⠉⠀⣠⠄⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢀⣿⣿⣿⣶⡀⣾⣿
⣉⠻⠿⠛⠋⠀⠀⠀⣀⣴⣿⠟⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⠸⣿⠟⠀⠀⠀⣾⣿⣿⣿⣧⣤⢀⡴⠁⠊⣻⣧⠙⢿⣿⣿⣿⡿⢋⣄⣀⣀⣀⣀⣀⣀⣀⣤⣾⣿⠟⢸⣿⣿⣿⣿⣿⣿⠿⣇⠀⠀⢸⣿⡁⠈⠛⠋⠙⠻
⣿⣿⣷⠶⠶⠖⠒⠛⠛⠉⢿⣧⠀⠀⣠⠾⢛⠟⣿⣿⠛⠂⣲⡆⠐⠒⠒⠚⢛⣿⣿⠟⣿⣿⣾⡇⠀⠀⠈⠻⡆⠈⠛⠉⠉⣠⣿⣿⡿⠿⠟⠿⠛⠛⠋⢉⢩⣤⣀⣸⣿⣿⣿⣿⡿⠃⠸⣿⡀⠀⣿⣿⢻⣄⣀⡀⠀⣶
⠿⠋⠁⠀⠀⠀⠀⠀⢢⣠⣾⡏⢀⡞⠁⠀⣸⣶⣿⣿⠂⠠⢤⣆⠀⠶⣶⣦⣾⣿⣿⣾⠛⣿⣿⣿⣷⣾⣿⣿⣿⣷⣶⣤⣀⠉⣡⡀⠀⠀⠀⠀⠀⣀⠴⠋⠀⣄⣴⣿⣿⣿⣿⡿⠀⠀⠈⣟⣦⠀⠸⣿⣇⠛⠿⠿⣷⣻
⡿⠖⠻⣿⠟⢿⠟⠻⠿⠿⢿⣠⠎⠀⠀⠀⣿⣿⣿⣿⡆⠀⠈⠸⠀⠄⢀⣸⣿⣿⠉⢹⡇⣿⣿⣟⠻⣿⣿⣿⡿⣿⣿⣯⠁⢰⣿⡄⠀⠀⠀⠀⢀⣀⣠⣶⣾⣿⣿⣿⠟⢿⣿⣧⠀⠀⠀⣿⣿⠀⠀⢸⣧⡀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢹⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⣧⣶⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠿⠇⠿⠇⠿⠇⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣠⣴⣶⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⠿⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣾⣿⣿⣿⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣿⣿⣿⡿⠟⠉⠛⢿⣿⠿⠿⠿⠿⠇⠀⠀⠈⠻⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⢻⣿⠟⠁⠀⠀⠀⠀⠘⣿⢾⣿⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⢰⠏⡁⠙⣿⣦⣄⡀⠀⣀⣼⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⠿⠟⠛⠿⠿⡟⠁⠀
⠉⠉⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣇⠀⠀⠀⠀⣠⡟⠀⠀⠠⠸⣿⣿⣿⣿⣿⣿⣛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠏⠁⠀⠀⠀⠀⡟⠛⣶⣤
⠀⠀⠀⠀⣠⢉⣙⠁⠀⠉⠻⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣤⣴⣾⣿⣿⣦⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⣠⣿⡅⢀⣈⣿
⣤⣤⣴⡾⠋⢸⣿⡄⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣖⢦⣾⣿⣿⣷⣿⣿⣿
⣿⣿⣏⠀⠀⢸⣿⣧⣤⣤⣤⣾⡏⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⡿⣿⣿⣿⣿⣿⡿⠿⢻⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠌⠹⠟⣿⠟⠛⠉
⠛⣿⣿⣿⣷⣾⣿⣿⡟⣻⡿⠉⠁⠀⠀⠀⠀⠀⡠⠿⠿⣿⣿⠿⠿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠟⠛⠙⠋⠉⠀⢀⣠⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀
⠀⡉⠛⠻⠻⠿⠿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢙⣿⡄⠀⠈⡟⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣄⣤⣴⣴⣷⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⠀⠀⠉⠁
⣶⣿⣷⣶⣶⣶⣶⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⠹⠿⣿⣿⣶⢾⡇⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣣⣿⣿⣿⣿⣿⣿⣿⠿⢋⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⢀⡀⠀
⠛⢿⣿⣿⣿⡿⢟⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠃⣸⣽⠏⠀⠀⠀⠀⢠⣴⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⡻⠿⢿⣋⣥⣴⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣤⣘
⢠⣉⠐⣲⣶⣶⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣿⣩⠄⠀⠀⠀⢠⣾⣿⣿⡟⢿⣷⣶⣄⣀⣤⣤⡤⢤⣄⣀⡀⠉⠉⠉⠁⢨⣿⡿⢷⣶⣤⣄⢀⣀⣠⣤⣶⣶⣿⣶⣶⢶⣦⣄⣀⠀⠀⠀⢀⣩⣿
⡀⠉⠙⠛⠿⠿⠟⠉⠑⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⣷⠎⠀⠀⠀⣿⣿⣿⣿⣇⠀⠹⣿⣿⡿⠋⠀⠤⠤⠤⢄⡉⠀⠀⠀⠀⠈⠙⠁⢀⣹⣿⣿⠿⠛⢛⣛⣿⣿⣿⣿⡭⠙⡋⠉⣿⣦⣄⣤⠿⠛⠁
⣾⣿⣿⣿⣾⠿⠀⢀⣀⣠⣿⣿⣶⣶⡶⠾⠿⣿⣶⣶⣤⣠⣾⣿⠿⠂⢀⣉⠐⠒⠀⢹⣿⣿⣿⣷⡆⢰⣿⣿⡿⣶⣶⣤⣤⣤⣤⢀⣴⣶⣶⣤⣤⣶⡾⠿⠛⣉⡄⣴⣿⣿⣿⣿⣿⣿⣿⠇⠰⠇⠀⢿⣿⣟⠻⢷⠄⣀
⣿⣿⣿⣿⣿⣾⣿⡿⠿⠟⢻⣿⡏⠀⠀⠀⠀⢨⣭⣽⣯⠀⢉⣥⣄⢀⣀⣀⡤⠴⠋⣿⣿⢟⠀⠁⠁⣸⣿⡯⢡⣾⣿⣿⣿⣿⣿⣿⢿⠿⠿⠛⠋⠁⢀⡤⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣼⣿⣿⣷⡄⢸⣽
⡉⠻⠿⠿⠛⠁⠀⠀⢀⣠⣾⡿⠃⠀⠀⠀⢠⣿⣿⣿⣿⠀⠈⣿⡿⠃⠈⠀⢰⣾⣿⣿⣿⣄⠀⣠⠎⠐⢻⣿⡈⠻⣿⣿⣿⣿⠟⣡⣀⣀⣀⣀⣀⣀⣀⣠⣴⣿⡿⠁⣿⣿⣿⣿⣿⣿⡿⢿⠀⠀⠀⣿⣇⠀⠙⠛⠉⠛
⣿⣿⣷⠶⠶⠶⠒⠚⠛⠋⠻⣷⠀⠀⢀⡴⠞⠋⢻⣿⡟⠒⢰⣴⠀⠒⠒⠒⠚⣻⣿⡟⢻⣿⣷⣏⠀⠀⠀⠙⢷⠀⠛⠋⠉⢀⣼⣿⣿⠿⠿⠿⠟⠛⠛⠉⡋⣥⣀⣀⣿⣿⣿⣿⣿⠟⠀⢻⣇⠀⢸⣿⡟⣆⣀⣀⠀⢰
⠿⠟⠉⠀⠀⠀⠀⠀⠀⣄⣰⣿⠀⣰⠋⠀⢠⣷⣾⣿⡇⠄⢤⣥⠀⠤⣤⣤⣴⣿⣿⣷⡟⢻⣿⣿⣷⣶⣿⣿⣿⣷⣶⣤⣄⠈⣉⡀⠀⠀⠀⠀⠀⢀⡠⠞⠁⢀⣠⣾⣿⣿⣿⣿⠁⠀⠀⢸⣽⡄⠈⢿⣷⡙⠻⠿⠿⣞
⣶⠷⠾⣿⠿⢿⠟⠻⠿⠿⠿⣇⡴⠁⠀⠀⢸⣿⣿⣿⣧⠀⠀⠉⠇⠠⠀⡀⣿⣿⡏⠙⣿⢸⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⠍⠀⣿⣇⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⡿⠻⣿⣿⡀⠀⠀⣿⣿⡇⠀⠈⣿⣄⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢹⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⠇⠿⠇⠿⠇⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣤⣴⣶⣾⣿⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣼⣿⣿⡿⠟⠋⠙⢻⣿⠿⠿⠿⠿⠇⠀⠀⠈⠻⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⣯⡿⠋⠀⠀⠀⠀⠀⢿⡗⣿⣇⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⢀⡾⢁⠈⢻⣷⣤⣀⠀⢀⣠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡿⠿⠛⠻⠿⢛⠃⠀
⠛⠛⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⡀⠀⠀⠀⢀⣼⠁⠀⠀⠄⢿⣿⣿⣻⣿⣿⣟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡸⠁⠀⠀⠀⠀⣸⠛⢻⣶
⠀⠀⠀⠀⢀⡬⣛⡋⠀⠈⠙⢿⣿⡿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⣤⣶⣿⣿⣧⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⢠⣾⡇⠀⣀⣹
⣤⣄⣤⣶⠟⠁⣿⣧⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡲⣴⣿⣿⣿⣾⣿⣿
⢩⣿⣿⡁⠀⠀⢿⣿⣤⣀⣀⣴⣿⠁⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⡿⣿⢿⣿⣿⣿⣿⣿⠿⢟⣧⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⢉⡟⢻⠿⠛⠋
⠙⣻⣿⣿⣿⣶⣿⣿⣿⢟⣿⠟⠉⠀⠀⠀⠀⠀⢀⡼⠿⢿⣿⡿⠿⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠻⠛⠋⠛⠉⠁⠀⢀⣴⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⣀
⠀⠈⠙⠛⠿⠿⠿⠿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣿⣇⠀⠀⢹⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣤⣠⣤⣦⣶⣾⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠄⠀⠈⠁
⣿⣾⣿⣶⣶⣶⣶⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠿⢿⣿⣷⡶⣿⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣜⣼⣿⣿⣿⣿⣿⣿⡿⠟⣡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠄⣀⠀
⠛⠿⣿⣿⣿⣿⠿⢛⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢐⣯⡿⠁⠀⠀⠀⠀⣠⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢛⠿⠿⣟⣩⡤⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣤
⠀⣌⡀⠺⣭⣶⣾⣯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣸⣏⡥⠀⠀⠀⠀⣼⣿⣿⣽⠻⣷⣶⣦⣀⣀⣤⣤⠤⣄⣀⣀⠀⠉⠉⠉⠀⣿⡿⠿⣶⣦⣤⡀⣀⣀⣤⣤⣶⣾⣷⣶⣶⣶⣤⣄⡀⠀⠀⠀⣈⣽
⠀⠉⠙⠛⠿⠿⠿⠛⠓⠲⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢿⡶⠁⠀⠀⢸⣿⣿⣿⣿⠀⠘⢿⣿⣿⡟⠁⠠⠤⠤⠤⣈⠁⠀⠀⠀⠀⠉⠃⠉⣈⣿⣿⡿⠟⠛⣛⣻⣿⣿⣿⣯⠉⢛⠉⢹⣷⣤⣤⡞⠟⠋
⣸⣿⣿⣿⣶⣿⠇⠀⣀⣀⣸⣿⣧⣶⣶⣶⠾⢿⣶⣶⣤⣄⣴⣾⡿⠷⠀⣀⡁⠒⠀⠈⣿⣿⣿⣿⣴⠀⣞⣿⣿⢷⣶⣤⣤⣤⣤⡄⣠⣴⣶⣦⣤⣴⣶⡿⠟⢋⣡⢠⣾⣿⣿⣿⣿⣿⣿⡿⠀⠾⠀⠸⣿⣿⣟⠿⡶⢀
⣿⣿⣿⣿⣿⣷⣾⣿⡿⠿⠟⣿⣯⠁⠀⠀⠀⠀⣩⣿⣿⡅⠈⣩⣤⡀⣀⣀⣀⡤⠜⢻⣿⡿⠃⠈⠉⢀⣿⣿⠍⣴⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠀⣀⠄⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢠⣿⡿⣿⣾⡀⣾
⠉⠛⠿⠿⠟⠋⠀⠀⠀⢀⣴⣿⡟⠀⠀⠀⠀⣴⣿⣿⣿⡇⠀⢹⣿⠟⠈⠉⠀⣴⣷⣿⣿⣧⡀⢀⡴⠁⠚⣿⣇⠙⢿⣿⣿⣿⡿⢋⣄⣀⣀⣀⣀⣀⣀⣀⣤⣾⣿⠟⢸⣿⣿⣿⣿⣿⣿⠿⡇⠀⠀⣸⣿⡄⠈⠛⠋⠙
⣿⣿⣷⣶⠶⠶⠒⠲⠻⠟⠛⢿⣆⠀⠀⣠⡴⠟⠙⣿⣿⠓⠒⣦⡄⠐⠒⠒⠒⢛⣿⣿⠟⣿⣿⣾⡁⠀⠀⠈⠻⡆⠘⠛⠉⠁⣠⣿⣿⡿⠿⠿⠿⠛⠛⠋⢙⢩⣤⣀⣸⣿⣿⣿⣿⡿⠁⢸⣿⡀⠀⣿⣿⢳⣄⣀⣀⠀
⠻⠿⠟⠁⠀⠀⠀⠀⠀⢠⣀⣼⡏⢀⡞⠁⠀⢸⣶⣿⣿⠤⠤⣬⡅⠠⢤⣤⣤⣾⣿⣿⣼⠛⣿⣿⣿⣷⣶⣿⣿⣿⣶⣦⣤⣀⠉⣁⡀⠀⠀⠀⠀⠀⣀⠴⠋⠀⣀⣴⣿⣿⣿⣿⡯⠀⠀⠘⣿⣧⠀⠸⣿⣯⠛⠿⠿⣷
⣶⣶⠶⣾⠿⢿⠶⠿⠿⠿⠿⢿⣁⠞⠀⠀⠀⣿⣿⣿⣿⡄⠀⠈⠹⠀⠄⠀⢸⣿⣿⠉⢻⡇⣿⣿⣟⢿⣿⣿⣿⡿⣿⣿⣯⠁⢸⣿⡄⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⠟⢿⣿⣧⠀⠀⠀⣿⣿⠀⠀⢸⣧⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢹⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⣧⣤⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠿⠇⠿⠇⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿
⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣯⣿⣿⣿⠿⠛⠙⠻⣿⣿⠿⠿⠿⠇⠀⠀⠈⠻⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿
⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢻⣿⠟⠁⠀⠀⠀⠀⢸⣷⢾⣷⡀⠀⠀⠀⠀⠀⢠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢿⠀⠀⠀⠀⠀⠀⣸⠏⠈⠹⣿⣦⣀⠀⠀⣀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⠿⠟⠛⠿⠟⡋⠁
⠞⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⣠⡟⠀⠁⠠⢹⣿⣿⣿⣿⣿⣿⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠋⠀⠀⠀⠀⢀⡏⠻⣶
⠀⠀⠀⠀⠀⢨⢝⡛⠁⠈⠙⢿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣤⣴⣿⣿⣷⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⣴⣿⠁⢀⣈
⣆⣀⣠⣤⡾⠛⢸⣿⡄⠀⠀⠀⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣖⢴⣾⣿⣿⣷⣿⣿
⢈⣿⣿⡟⠀⠀⢸⣿⣧⣀⣀⣠⣾⡏⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⢿⣿⠿⣿⣿⣿⣿⣿⡿⠟⣻⣄⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠌⣻⠛⡿⠟⠛
⠈⠻⣿⣿⣶⣷⣾⣿⣿⡿⣿⡿⠋⠁⠀⠀⠀⠀⠀⣠⠾⢿⣿⣿⠿⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠟⠟⠛⠛⠋⠉⠀⠀⣀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⢀
⠀⠀⠉⠻⠿⠿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⣷⡀⠀⠈⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣄⣤⣤⣴⣶⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⠀⠀⠉
⣿⣶⣽⣧⣶⣶⣶⣶⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠻⣿⣿⣷⣤⣾⡅⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣣⣿⣿⣿⣿⣿⣿⣿⠿⢋⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⢀⡀
⠙⠿⣿⣿⣿⣿⣿⠟⣋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⠇⣾⣽⠏⠀⠀⠀⠀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⡻⡿⢿⣋⣥⣴⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢶⣤
⠀⢠⣙⠛⢭⣭⣶⣾⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣫⠄⠀⠀⠀⢠⣽⣿⣿⠛⢿⣶⣦⣄⣀⣠⣤⣤⢤⣀⣀⡀⠈⠉⠙⠁⢸⣿⠿⢷⣶⣤⣄⢀⣀⣠⣤⣴⣶⣿⣶⣶⣶⣦⣤⣀⠀⠀⠀⢀⣩
⠀⠈⠙⠻⠿⠿⢿⠟⠒⠒⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⣿⠆⠀⠀⠀⣿⣿⣿⣿⡇⠀⢻⣿⣿⣿⠋⠀⠤⠤⠤⣈⡉⠀⠀⠀⠀⠉⠙⠈⢁⣹⣿⣿⠿⠛⢛⣛⣿⣿⣿⣿⡍⠙⠋⠩⣷⣦⣄⡴⠻⠛
⢀⣿⣷⣲⣶⣶⣿⠀⠀⣀⣀⣿⣷⣤⣶⣶⣶⣶⣶⣶⣤⣄⣀⣶⣾⠿⠌⠀⣈⠐⠂⠀⢻⣿⣿⣿⣧⡆⢰⣻⣿⡷⣶⣤⣤⣤⣤⣀⢀⣤⣶⣦⣤⣤⣴⣾⠿⠛⣉⡄⣴⣿⣿⣿⣿⣿⣿⣿⠃⢰⠇⠀⢿⣿⣿⠻⢷⠆
⣿⣿⣿⣿⣿⣿⣶⣾⣿⡿⠿⢿⣿⡏⠉⠀⠀⠀⢈⣿⣿⣟⡈⣉⣥⣄⢀⣀⣀⣀⠤⠛⣿⣿⡿⠑⠉⠁⣸⣿⡏⢡⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁⢀⡠⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣼⡿⢿⣷⣇⢸
⠉⠛⠻⣿⠿⠟⠋⠁⠀⠀⣀⣾⣿⠃⠀⠀⠀⢠⣾⣿⣿⣿⠀⠘⣿⡿⠃⠉⠁⢠⣾⣾⣿⣿⣄⠀⣠⠎⠐⢻⣷⡘⢻⣿⣿⣿⣿⠟⣡⣀⣀⣀⣀⣀⣀⣀⣠⣴⣿⡿⠁⣿⣿⣿⣿⣿⣿⡿⢿⠀⠀⢀⣿⣧⠀⠙⠛⠉
⣿⣿⣷⣶⡶⠦⠤⠴⠶⠾⠟⢻⣷⡀⠀⢀⣠⠾⡓⢿⣿⡟⠒⢠⣬⠀⠂⠲⠶⠚⣻⣿⡟⢿⣿⣷⡏⠀⠀⠈⠛⢷⠀⠛⠏⠉⢁⣼⣿⣿⡿⠿⠿⠿⠛⠛⠙⡋⣩⣀⣀⣿⣿⣿⣿⣿⠟⠀⣿⣇⠀⢸⣿⡟⣦⣀⣀⠀
⠙⠿⠿⠋⠀⠀⠀⠀⠀⠀⣀⣠⣿⠁⢠⠋⠀⢠⣷⣾⣿⡧⠄⢤⣭⠀⠤⣤⣤⣴⣿⣿⣧⡿⢿⣿⣿⣷⣶⣾⣿⣿⣷⣶⣤⣄⠈⣉⣁⠀⠀⠀⠀⠀⢀⡠⠞⠁⢀⣠⣿⣿⣿⣿⣿⠇⠀⠀⢻⣿⡄⠀⢿⣿⠟⠻⠿⢿
⣶⣶⣶⣶⠿⢷⠶⠶⠶⠿⠿⠿⡏⡰⠃⠀⠀⢸⣿⣿⣿⣧⠀⠀⠉⠇⠀⠀⠀⣿⣿⡏⠙⣿⢸⣿⣿⢻⣿⣿⣿⣿⢿⣿⣿⠉⠀⣿⣇⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⡿⠻⣿⣿⡀⠀⠀⣼⣿⡇⠀⠀⣿⣄⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢹⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⣇⣤⣿⣿⣶⣿⣿⣿⣿⣷⣿⣿⣸⣿⡟⣿⡏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠿⠇⠿⠇⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⡿⠛⠋⠻⢿⣿⡿⠿⠿⠇⠀⠀⠈⠻⠿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿
⡾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠹⣿⡿⠋⠀⠀⠀⠀⠈⣿⣾⣿⡇⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡃⠀⠀⠀⠀⠀⢀⡟⠁⠉⢿⣶⣄⡀⠀⣀⣴⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⡿⠟⠛⠻⢟⢛⠁
⡴⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⢀⣾⠁⠈⠀⠈⣿⣿⣿⣿⣿⣿⣟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⠁⠀⠀⠀⠀⣸⠙⠻
⠁⠀⠀⠀⠀⠀⡝⣛⠛⠉⠉⠻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣤⣾⣿⣿⣦⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⣠⣾⡇⠀⣀
⢷⣀⣀⣠⣴⠟⠃⣿⡇⠀⠀⠀⠈⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡲⢶⣿⣿⣿⣾⣿
⡀⣹⣿⣿⠋⠀⠀⣿⣿⣄⣀⣀⣴⣿⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢻⣿⡿⢿⣿⣿⣿⣿⣿⠿⢟⣧⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⢙⡿⢻⠿⠛
⠁⠙⢿⣿⣷⣶⣴⣿⣿⣿⢿⣿⠟⠋⠀⠀⠀⠀⠀⢀⡴⠿⣿⣿⡿⠿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠿⠛⠛⠛⠉⠁⠀⢀⣸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡑⠀⠀⠀
⡆⠀⠙⠻⠿⠿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣿⣆⠀⠀⢹⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣤⣠⣠⣤⣶⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠄⠀⠈
⣿⣷⣼⣶⣴⣶⣶⣴⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠛⢿⣿⣿⣦⣴⣏⢨⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣞⣼⣿⣿⣿⣿⣿⣿⡿⠟⣡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠄⣀
⠈⠻⢿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠙⠻⢰⣯⡿⠁⠀⠀⠀⢀⣠⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣛⢿⢿⣟⣉⡤⣾⣯⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦
⠀⠐⣄⠛⠭⣭⣵⣶⣏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⡥⠀⠀⠀⠀⣬⣿⣿⣟⠻⣷⣶⣤⣀⣀⣤⣤⠤⣄⣀⢀⠀⠉⠉⠉⠀⣿⡿⢿⣶⣦⣤⡀⣀⣀⣤⣴⣶⣿⣷⣶⡶⣶⣤⣤⡀⠀⠀⠀⣈
⠀⠀⠙⠛⠾⢿⣿⡿⠒⠒⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠿⡷⠀⠀⠀⢸⣿⣿⣿⣷⠀⠘⣿⣿⣿⡟⠁⠠⠤⠤⢄⡉⠀⠀⠀⠀⠈⠙⠃⠉⣉⣿⣿⡿⠟⠟⣛⣻⣿⣿⣿⣿⠉⢛⠉⢹⣷⣄⣠⡾⠏
⡀⣸⣿⣶⣶⣶⣶⡇⠀⢀⣀⣸⣿⣦⣤⣶⣶⣶⣶⣶⣦⣤⣀⣴⣶⡿⠧⠁⢀⡀⠒⠀⠘⣿⣿⣿⣿⣴⠀⣾⣿⣿⣶⣦⣤⣤⣤⣤⡈⣠⣴⣶⣤⣤⣴⣶⠿⠛⢋⣡⢰⣾⣿⣿⣿⣿⣿⣿⡿⠀⡾⠀⠸⣿⣿⣟⠿⣷
⣸⣿⣿⣿⣿⣿⣧⣶⣿⣿⡿⠿⣿⣿⠉⠁⠀⠀⠀⣹⣛⣻⡃⣉⣩⣤⡀⣀⣁⣀⡠⠜⢻⣿⡿⠏⠋⠉⢀⣿⣿⠉⣴⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠀⣀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢠⣿⡿⣿⣿⡄
⡟⠛⠛⢿⣿⠿⠟⠈⠁⠀⢀⣰⣿⡿⠀⠀⠀⠀⣴⣿⣿⣿⡇⠀⢻⣿⠟⠉⠉⠁⣴⣶⣿⣿⣧⣀⢀⡴⠁⠚⣿⣆⠙⣿⣿⣿⣿⡿⢋⣄⣀⣀⣀⣀⣀⣀⣀⣤⣾⣿⠏⢸⣿⣿⣿⣿⣿⣿⠿⣇⠀⠀⣸⣿⡆⠈⠛⠛
⣿⣿⣿⣶⣦⡤⣤⠤⠶⠶⠿⠛⣿⡆⠀⠀⣀⣴⠿⠙⣿⣿⠓⠀⣤⡅⠐⠐⠶⠖⢛⣿⣿⠟⣿⣿⣾⠁⠀⠀⠙⠻⡄⠘⠻⠉⠉⣠⣿⣿⣿⠿⠿⠿⠟⠛⠋⢛⢩⣅⣀⣸⣿⣿⣿⣿⡿⠃⠸⣿⡀⠀⣿⣿⢻⣄⣀⣀
⠉⠻⣿⠟⠋⠀⠀⠀⠀⠀⢄⣀⣼⡟⠀⡜⠉⠀⣸⣶⣿⣿⠤⠤⣬⡅⠠⢤⣤⣤⣾⣿⣿⣼⠿⣿⣿⣿⣶⣶⣿⣿⣿⣶⣦⣤⣀⠉⣉⡀⠀⠀⠀⠀⠀⣀⡴⠋⠀⣀⣼⣿⣿⣿⣿⡿⠀⠀⠈⣿⣧⠀⠸⣿⣯⠛⠿⠿
⣶⣶⣶⣶⠿⢷⠶⠶⠶⠶⠿⠿⢿⢁⡞⠁⠀⠀⣿⣿⣿⣿⣄⡀⠈⠹⠀⠀⠀⢸⣿⣿⠉⢻⡇⣿⣿⡟⢿⣿⣿⣿⡿⢿⣿⣏⠁⢸⣿⡄⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⠟⢿⣿⣧⠀⠀⠘⣿⣿⠀⠀⢸⣧⣄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢹⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⣇⣠⣿⣿⣶⣶⣿⣿⣿⣷⣿⣿⣸⣿⡟⣿⡏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⢿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠿⠇⠿⠇⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿
⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⠿⠛⠻⠿⣿⣿⡿⡿⠇⠀⠀⠈⠻⠿⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿
⢰⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⣿⣿⠟⠁⠀⠀⠀⠀⢻⣶⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠀⠀⠀⠀⠀⠀⣼⠋⠉⢻⣷⣤⣀⠀⠀⣠⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠛⠛⠟⠛⡋
⢀⠾⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡄⠀⠀⠀⢀⣰⡏⠀⠁⡀⢹⣿⣿⣿⣿⣿⣿⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠋⠀⠀⠀⠀⢀⡏⠛
⠈⠀⠀⠀⠀⠀⣘⡛⠿⠋⠉⠛⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢦⣤⣼⣿⣿⣷⣤⣴⣶⣾⣿⣿⣿⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⢀⣴⣿⠀⢀
⠯⣄⣀⣀⣠⡾⠟⢼⣷⠀⠀⠀⠀⢻⣯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣓⠶⣾⣿⣿⣷⣿
⠀⢈⣿⣿⡟⠀⠀⢸⣿⣄⣀⣀⣠⣾⠇⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⣿⣿⡿⣿⣿⣿⣿⣿⡿⠟⣿⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠻⠋⡿⠟
⡇⠘⠿⣿⣷⣶⣦⣼⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⢠⡾⢿⣿⣿⠿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⠟⠛⠛⠋⠉⠀⠀⣀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠂⠀⠀
⣧⠀⠈⠛⠿⢿⢿⣿⣿⣿⣷⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢿⣧⠀⠀⠙⡟⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣤⣄⣄⣤⣴⣶⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⠀⠀
⢹⣷⣦⣴⣦⣤⣤⣤⣤⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠲⠞⢻⣿⣿⣷⣤⣶⡁⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣣⣿⣿⣿⣿⣿⣿⣿⠿⢋⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠠⢀
⠀⠛⠿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠒⠛⠻⠇⣾⣽⠏⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⣿⣿⣿⣋⣡⣴⣿⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶
⠀⠀⢀⠚⠭⣭⣭⣵⣾⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣫⠄⠀⠀⠀⢠⣿⣿⣿⠻⣿⣦⣦⣀⣀⣠⣤⣤⣄⣀⡀⡁⠊⠉⠙⠁⢸⣿⠿⣷⣦⣤⣄⢀⣀⣠⣤⣶⣶⣿⣶⣶⢶⣦⣤⣄⠀⠀⠀⢀
⠀⠀⠈⠛⠶⠿⣿⣿⡗⠒⠢⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⢭⠆⠀⠀⠀⣿⣿⣿⣿⣆⠈⢻⣿⣿⡿⠋⠁⠤⠤⠤⣉⠁⠀⠀⠀⠀⠉⠙⠀⢉⣻⣿⣿⠿⠻⢛⣛⣿⣿⣿⣿⡍⠛⠋⢨⣷⣦⣤⣴⠿
⣄⢀⣶⣶⣶⣶⣴⣶⠀⠀⠀⣈⣿⣷⣤⣤⣤⣶⣶⣶⣤⣤⣄⢠⣶⣾⡿⠍⠑⠂⠐⠄⠀⢿⣿⣿⣿⣧⡆⢠⣿⣿⣷⣤⣤⣤⣤⣀⣀⢁⣤⣶⣦⣤⣤⣴⣾⠿⠛⣉⡄⣴⣿⣿⣿⣿⣿⣿⣿⠃⢰⠇⠀⢿⣿⣟⠻⠿
⠡⣿⣿⣿⣿⣿⣿⣴⣶⣿⣿⠿⢿⣿⡏⠉⠀⠀⠀⢉⣛⣛⣟⠉⢙⣭⡀⢀⡈⢁⣀⡤⠛⣿⣿⠿⠙⠉⠁⣿⣿⡍⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⢀⡀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣼⣿⢿⣷⡆
⣮⠛⠛⢻⣿⡿⠿⠃⠉⠁⠀⢀⣾⣿⠇⠀⠀⠀⢀⣿⣿⣿⣿⠀⠸⣿⡿⠋⠉⠉⢠⣶⣾⣿⣿⡄⠀⣠⠎⠈⢿⣷⡘⢿⣿⣿⣿⣿⠟⣡⠀⠀⣀⣀⣀⣀⣀⣠⣴⣿⡿⠁⣿⣿⣿⣿⣿⣿⡿⣿⡀⠀⢀⣿⣗⠀⠙⠛
⢻⣾⣷⣶⣦⣤⣤⡤⠤⠴⠾⠟⢿⣧⠀⠀⠀⣠⡾⠏⢿⣿⡟⠂⢠⣭⠀⠀⠤⠴⠞⣿⣿⡿⢿⣿⣷⡏⠀⠀⠈⠻⢧⠀⠛⠟⠉⢁⣼⣿⣿⡿⠿⠿⠿⠟⠛⠙⣋⣉⣀⣀⣿⣿⣿⣿⣿⠟⠀⢿⣇⠀⢸⣿⡟⣧⣀⣀
⠉⠛⣿⣿⠛⠁⠀⠀⠀⠀⠠⡀⣠⣿⠀⢠⠎⠁⢀⣧⣾⣿⡧⠀⣈⣭⠀⢠⣤⣤⣤⣿⣿⣧⡿⣿⣿⣿⣷⣴⣶⣿⣿⣶⣶⣤⣀⠈⢋⡁⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⣨⣿⣿⣿⣿⣿⠇⠀⠀⢪⣿⡄⠀⢿⣷⡝⠻⠿
⣶⣶⣶⣶⠾⢶⠶⠶⠶⠶⣶⣿⣿⡏⣠⠋⠀⠀⢸⣿⣿⣿⣇⡀⠐⠋⡇⠀⠈⠉⣿⣿⡏⠛⣷⢸⣿⣿⢿⣿⣿⣿⣿⢿⣿⣿⠉⠀⣿⣇⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⡿⠻⣿⣿⡄⠀⠀⢸⣿⡏⠀⠀⣿⣤⡀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⣀⣿⣿⣶⣶⣶⣾⣿⣷⣿⣿⣼⣿⡟⣿⡏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡿⠇⠿⠇⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⢧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⣿⣿⣿⣿⣿⣿⠿⠻⠿⢿⣿⣿⣿⡿⠀⠀⠈⠻⠷⠿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿
⠀⢰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠹⣟⣿⠋⠁⠀⠀⠀⠀⢿⣆⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⡇⠀⠀⠀⠀⠀⠀⣼⠉⠉⢻⣷⣄⡀⠀⢀⣠⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡿⠟⠛⠛⠟⠛
⠀⢀⡾⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢳⡀⠀⠀⠀⠀⣼⠇⠀⠀⡄⢻⣿⣿⣻⣿⣿⣟⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⠋⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⢘⠻⠿⠋⠙⠻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⣠⣤⣿⣿⣧⣤⣴⣦⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢀⣴⣿⠀
⠰⢵⡀⠀⢀⣠⡶⠿⣼⣧⠀⠀⠀⠈⢻⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢡⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣲⠶⣿⣿⣿⣷
⠀⠀⢙⣿⣿⡟⠁⠀⢸⣿⣀⠀⠀⣀⣾⠇⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣿⣿⡿⣿⣿⣿⣿⣿⠿⠟⣯⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠻⠛⡿
⢀⡇⠘⢿⣿⣷⣴⣤⣼⣿⣿⡿⣿⡿⠟⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⠼⠟⠛⠛⠋⠉⠀⠀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠀
⢸⣧⠀⠘⠛⠿⣿⣿⣿⣿⣿⣶⡿⠁⠀⠀⠀⠀⠀⠀⠀⠁⢀⠈⢿⣇⠀⠈⠙⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣠⣄⣀⣤⣴⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀
⠀⢹⣦⣤⣴⣄⣤⣤⣬⣥⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠗⢶⣾⣿⣧⣤⣶⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣭⣿⣿⣿⣿⣿⣿⣿⠿⢋⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠡
⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠚⠛⠿⠃⣿⣷⠏⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⢿⡿⣿⣋⣡⣴⣿⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⢀⠹⢟⣛⣋⣭⣶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣫⠀⠀⠀⠀⢠⣿⣿⣟⠿⣷⣤⣤⣀⢀⣠⣤⣤⣄⣀⠀⠁⠈⠉⠋⠁⢸⣿⠿⣷⣦⣤⣄⢀⣀⣠⣤⣶⣶⣿⣶⣶⢶⣦⣤⣀⠀⠀⠀
⠀⠀⠀⠘⠳⠤⣿⣿⣿⡿⠥⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣾⠿⢭⠆⠀⠀⠀⣿⣿⣿⣿⣄⠈⢿⣿⣿⡿⠉⠉⠠⠤⢀⣉⠁⠀⠀⠀⠀⠉⠛⠀⢈⣻⣿⣿⠿⠻⢛⣛⣿⣿⣿⣿⡏⠛⠋⢨⣷⣦⣠⣤
⣤⣀⠀⣰⡦⣤⣤⣤⣆⠀⠀⠀⠈⣿⣧⣠⣤⣤⣤⣶⣦⣤⣄⣀⢠⣴⣾⣿⠌⠁⠂⠠⠄⠀⢿⣿⣿⣿⣧⡄⢨⢿⣿⣧⣤⣤⣤⣀⣀⣀⢁⣤⣶⣤⣤⣤⣴⣾⠿⠛⣉⡄⣶⣿⣿⣿⣿⣿⣿⣿⠃⢰⠃⠀⢿⣿⣟⣛
⠈⠉⣾⣿⣿⣿⣿⣿⣥⣶⣾⣿⡿⢿⣿⡛⠋⠁⠀⠀⢙⣻⣿⡟⠙⢛⣉⡀⠀⠈⢁⣀⡠⠞⣿⣿⠿⠙⠉⠁⣿⣿⡝⢡⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠋⠁⢀⡀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣼⡿⢿⣷
⣦⣬⠛⠿⢿⣿⣿⡿⠛⠉⠉⠀⢀⣼⣿⠇⠀⠀⠀⢀⣾⣿⣿⣿⠀⢸⣿⡿⠉⠉⠉⢡⣶⣾⣿⣟⡄⠀⢠⠎⠈⢿⣧⠘⢿⣿⣿⣿⣿⠟⣁⠀⠀⢀⡀⠀⢀⣀⣠⣴⣿⡿⠉⣿⣿⣿⣿⣿⣿⡿⢿⠀⠀⢠⣿⣗⠀⠉
⠙⠻⣶⣶⣦⣭⣤⣤⣤⣤⡤⣶⠿⢿⣧⠀⠀⠀⣠⣽⢟⢿⣿⡷⠤⢤⣭⠠⠤⠤⠤⠾⣿⣿⡿⢿⣿⣶⡏⠀⠀⠈⠻⣧⠈⠻⠏⠋⢁⣼⣿⣿⡿⠿⠿⠿⠟⠛⠙⣋⣉⣀⣀⣿⣿⣿⣿⣿⠏⠀⢿⣇⠀⢸⣿⡟⣧⣀
⠀⠘⠻⣿⣿⠟⠁⠀⠀⠀⠀⠀⡀⢈⣿⠀⢠⠞⠉⢀⣦⣼⣿⡇⠀⣈⣩⠀⢠⣤⣤⣤⣿⣿⣥⡿⣿⣿⣿⣧⣴⣶⣿⣿⣶⣦⣄⣀⠈⢋⡁⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⣩⣿⣿⣿⣿⣿⠁⠀⠀⣳⣿⡄⠀⢿⣷⡜⠻
⠣⢶⣶⣾⣧⢶⠶⢴⠶⠶⢶⣶⣿⣿⡏⢠⠏⠀⠀⢸⣿⣿⣿⣇⠀⠘⠛⡆⠀⠈⠉⣿⣿⡟⠻⣷⢸⣿⣿⢿⣿⣿⣿⡿⣿⣿⣿⠋⠀⣿⣇⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⡿⠻⣿⣿⡀⠀⠀⢺⣿⡇⠀⠀⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢹⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⢀⣿⣿⣴⣶⣶⣾⣿⣷⣿⣿⣼⣿⡟⣿⡏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠇⠿⠇⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⢸⣿⣿⣿⣿⣿⡿⠻⠿⠿⣿⣿⣿⣿⠀⠀⠈⠻⠷⠿⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿
⠀⠀⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢿⣻⡟⠉⠀⠀⠀⠀⢸⣷⢾⣿⡀⠀⠀⠀⠀⠀⠀⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⣿⣿⣿⣿⣿⣿
⠀⠐⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣿⠀⠀⠀⠀⠀⠀⢰⠏⠈⠹⣿⣦⣀⠀⠀⣀⣼⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⠿⠛⠛⠻⠛
⠀⠀⣠⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⣆⠀⠀⠀⠀⣠⡿⠀⠁⢠⠸⣿⣿⣿⣿⣿⣿⣛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠏⠀⠀⠀⠀⢀⡞
⠀⠀⠁⠀⠀⠀⠀⠀⡿⠿⠟⠙⠻⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣄⣤⣿⣿⣿⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⣰⣿⡇
⠀⢾⣄⠀⠀⢀⣠⡾⢧⣿⡄⠀⠀⠀⠹⣿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣗⡲⣾⣿⣿⣿
⠀⠀⠙⢻⣿⣿⠋⠀⠀⣿⣇⡀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⢿⣿⣿⣿⣿⡿⠿⢻⣅⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢙⡟⢻
⠀⢰⠀⢻⣿⣿⣦⣤⣠⣿⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠷⠿⠛⠛⠛⠉⠁⠀⢀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡐⠀
⠀⣿⡄⠀⠺⢻⢿⣿⣿⣿⣿⣷⣾⠏⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠻⣿⡀⠀⠉⢿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⣄⣀⣠⣤⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠄
⠀⠈⣷⣄⣤⣤⣠⣤⣅⣭⣽⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⠶⣶⣿⣿⣤⣤⡆⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣩⣿⣿⣿⣿⣿⣿⣿⡿⠟⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣎
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠟⢸⣟⡾⠁⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢻⣿⣿⣟⣉⣤⣾⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠨⠿⣛⣛⣫⣥⡖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢛⡁⠀⠀⠀⠀⣸⣿⣿⡿⢿⣦⣤⣄⠀⣀⣤⣤⣤⣀⡀⢈⠀⠉⠛⠉⢀⣿⡿⢿⣶⣤⣄⡀⣀⣀⣤⣴⣶⣿⣷⣶⡶⣶⣤⣤⡀⠀⠀
⠀⠀⠀⠀⠳⠦⣴⣿⣿⣿⠭⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⡿⢯⡵⠀⠀⠀⢼⣿⣿⣿⣧⡄⠹⣿⣿⣿⡟⠉⠁⠤⢄⣈⡉⠀⠀⠀⠀⠈⠙⠃⠈⣙⣿⣿⡿⠟⠟⣛⣻⣿⣿⣿⣿⠙⢛⠉⢹⣶⣤⣤
⣤⣄⡀⢰⣶⣤⣤⣬⣼⡅⠀⠀⠀⢹⣷⣄⣤⣤⣤⣤⣤⣤⣤⣀⡀⣠⣴⣿⡧⠈⠐⠀⠤⠀⠸⣿⣿⣿⣿⣤⠀⣽⣿⣿⣤⣤⣤⣀⣀⣀⡈⣠⣴⣦⣤⣤⣤⣶⡿⠟⢋⣡⢰⣾⣿⣿⣿⣿⣿⣿⡟⠀⡾⠀⠸⣿⣿⣟
⠈⠉⢱⣿⣿⣿⣿⣿⣯⣥⣶⣾⣿⡿⣿⣟⠛⠉⠉⠉⠉⣻⣿⣿⠋⠛⢛⣁⠀⠀⠉⢀⣀⡴⢻⣿⣿⠏⠋⠉⢸⣿⣯⠋⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠉⠁⣀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢠⣿⡿⣿
⣶⣤⡿⠿⠿⣿⣿⣿⠿⠛⠉⠉⠀⣠⣿⡿⠀⠀⠀⠀⣰⣿⣿⣿⡇⠀⢿⣿⡿⠙⠋⠉⣴⣶⣿⣿⣧⠀⠀⡴⢃⠹⣿⡄⠻⣿⣿⣿⣿⡿⢋⡀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⠏⢸⣿⣿⣿⣿⣿⣿⠿⣇⠀⠀⣸⣿⣆⠈
⠉⠻⣷⣶⣶⣬⣭⣤⣀⣠⣤⣤⣾⠿⣿⡅⠀⠀⢀⣠⡿⣻⣿⣿⠧⠤⣈⡏⠠⠤⠤⠴⢿⣿⣿⠿⣿⣿⣾⠁⠀⠀⠙⢿⡄⠘⠿⠛⠋⣠⣿⣿⣿⠿⠿⠿⠿⠛⠋⢛⣉⣁⣀⣸⣿⣿⣿⣿⡿⠃⢸⣿⡄⠀⣿⣿⢻⣤
⣀⠀⠚⢿⣿⡿⠛⠉⠀⠀⠀⠀⡀⠀⣹⡗⠀⡴⠋⠁⣰⣤⣿⣿⣀⢀⣙⡁⢀⣀⣀⣀⣼⣿⣿⣴⢿⣿⣿⣷⣤⣶⣶⣿⣷⣶⣤⣀⡀⠙⣉⠁⠀⠀⠀⠀⠀⢀⡤⠋⠀⣀⣽⣿⣿⣿⣿⣿⠀⠀⠘⣾⣧⠀⠘⣿⣧⠙
⠛⠷⣶⣶⣯⣶⠴⠶⠶⢶⣶⣴⣾⣿⣿⠃⡴⠁⠀⠀⣿⣿⣿⣿⡄⠀⠙⢳⠀⡈⠉⢹⣿⣿⠛⢿⡆⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣟⠁⢸⣿⡀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⠟⢿⣿⣧⠀⠀⠀⣿⣿⠁⠀⢸⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⣴⣶⣶⣾⣿⣷⣿⣿⣼⣿⡟⣿⡏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠿⠇⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⡏⠀⠈⠻⠷⠿⠟⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢹⣿⣿⠏⠁⠀⠀⠀⠈⣷⡶⣶⡇⠀⠀⠀⠀⠀⠀⣸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⣿⣿⣿⣿⣿⣿
⠀⠀⠋⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⡇⠀⠀⠀⠀⠀⠀⡿⠉⠉⢿⣶⣄⡀⠀⢀⣠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣺⣿⡿⠟⠛⠛⠟
⠀⠀⢀⣼⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⢀⣼⠇⠈⠀⡌⣿⣿⣿⣿⣿⣿⣟⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠋⠀⠀⠀⠀⣠
⠀⠀⠈⠀⠀⠀⠀⠀⠸⠿⡿⠛⠛⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣠⣴⣿⣿⣦⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢀⣼⣿
⠀⠰⣧⡀⠀⠀⣀⣴⠿⣾⣦⠀⠀⠀⠙⢿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢡⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣒⢶⣿⣿⣿
⠀⠀⠀⢻⣿⣿⡿⠁⠀⢹⣿⡀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠟⣧⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠏⣻⠛
⠀⢀⡇⠸⣿⣿⣧⣤⣄⣼⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠿⠟⠟⠛⠛⠉⠀⠀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀
⠀⢸⣧⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠉⢀⠘⣿⣇⠀⠈⠻⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣠⣀⣀⣤⣴⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧
⠀⠀⢻⣦⣀⣠⣄⣀⣉⣉⣩⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢤⠶⣶⣾⣿⣦⣠⣴⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣿⣿⣿⣿⣿⣿⣿⣿⠿⢋⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠇⣿⣳⠏⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⣿⣿⣿⣋⣡⣴⣿⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠽⣛⣛⣛⣩⣴⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢋⠀⠀⠀⠀⢀⣿⣿⣿⠿⣷⣤⣤⡀⢀⣀⣀⣤⣀⣀⠀⠙⠊⠙⠋⠉⣸⣿⠿⣷⣦⣤⣄⠀⣀⣠⣤⣶⣾⣿⣶⣶⢶⣦⣤⣄⠀⠀
⠀⠀⠀⠀⠸⢷⣤⣿⣿⣿⡯⠥⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣾⣿⣯⠆⠀⠀⠠⣿⣿⣿⣿⣅⠉⢿⣿⣿⣿⠋⠉⢀⢤⣈⣉⠁⠀⠀⠀⠀⠉⠛⠀⢉⣻⣿⣿⠿⠻⢛⣛⣿⣿⣿⣿⡍⠛⡋⠈⣷⣦⣤
⣤⣄⡀⠀⣤⣤⣤⣤⣭⣍⠀⠀⠀⠙⣿⣦⣀⣀⣤⣤⣤⣤⣤⣀⣀⢀⣤⣼⣿⠀⠁⠐⠠⠄⠀⢿⣿⣿⣿⣧⡄⢨⣿⣿⣧⣤⣤⣀⣀⣀⣈⢁⣤⣤⣤⣤⣤⣴⣾⠿⠛⣉⡄⣴⣿⣿⣿⣿⣿⣿⣿⠇⠰⠇⠀⢿⣿⣿
⠈⠉⠙⣾⣿⣿⣿⣿⣿⣩⣤⣶⣾⣿⣿⣿⡟⠛⠉⠉⠉⠙⣿⣿⡟⠙⠛⣋⡀⠀⠈⠉⢀⣠⠶⣾⣿⡿⠙⠋⠁⣿⣿⡟⢡⣽⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉⢀⡀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣼⣿⢿
⣶⣦⣴⡿⠿⢿⣿⣿⣿⠟⠛⠋⠉⠀⣼⣿⠇⠀⠀⠀⢀⣾⣿⣿⣿⠀⢹⣿⣿⠞⠛⠋⢩⣶⣼⣿⣿⡄⠀⢀⠔⡹⢿⣧⠘⢿⣿⣿⣿⣿⠟⣁⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⡿⠃⣿⣿⣿⣿⣿⣿⡿⣿⡀⠀⠀⣿⡿⡀
⠈⠻⢿⣶⣦⣬⣍⣉⣀⣀⣀⣤⣴⡾⢿⣯⠀⠀⠀⢀⣼⣿⣿⣿⡿⠤⢄⣛⠁⠤⠤⠤⠼⣿⣿⡿⣿⣿⣶⡏⠀⠀⠈⠻⣧⠀⠻⠟⠛⠁⣼⣿⣿⣿⡿⠿⠿⠿⠛⠛⣛⣉⣀⣀⣿⣿⣿⣿⣿⠟⠀⣿⣧⠀⢸⣿⡇⣧
⣀⠀⠘⠿⣿⣿⡿⠋⠉⠉⠁⠀⠀⠀⢈⣿⠆⢀⠜⠉⠀⣮⣼⣿⣇⣀⣈⣛⠀⣀⣀⣀⣠⣿⣿⣣⡾⣿⣿⣿⣦⣤⣶⣾⣿⣶⣦⣄⣀⠘⢋⡉⠁⠀⠀⠀⠀⠀⣀⠔⠃⢀⣈⣽⣿⣿⣿⣿⡇⠀⠀⢿⣿⣄⠀⢻⣿⡯
⠛⠷⢶⣶⣯⣿⠤⣴⣶⣶⣦⣤⣤⣷⣿⡟⢀⠞⠀⠀⢸⣿⣿⣿⣇⠀⠈⠛⡆⠀⠉⠉⣿⣿⡟⠻⣷⢸⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡋⠀⣿⣧⠀⠀⠀⠀⠀⠀⣀⣠⣶⣾⣿⣿⡿⠻⣿⣿⡄⠀⠀⢸⣿⡟⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⣰⣶⣶⣾⣿⣷⣿⣿⣾⣿⣿⣿⡏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠿⠇⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠻⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⡇⠈⠻⠷⠿⠏⠙⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣿⡿⠋⠀⠀⠀⠀⠘⣷⣶⣶⡄⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⣿⠃⠀⠀⠀⠀⠀⢀⡿⠉⠙⣿⣦⣄⠀⠀⢀⣰⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⡿⠟⠛⠛
⠀⠀⠀⢀⣼⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⠀⠀⠀⠀⢀⣾⠃⠈⠀⡘⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡸⠁⠀⠀⠀⠀
⠀⠀⠀⡋⠀⠀⠀⠀⠀⢹⣿⣿⠛⠿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⣠⣴⣿⣿⣦⣠⣶⣴⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⢀⣾
⠀⠀⢠⣧⠀⠀⠀⢀⣴⠿⣶⡆⠀⠀⠀⠙⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡲⢶⣿⣿
⠀⠀⠀⠉⠷⣶⣾⡟⠁⠀⣿⣿⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣻⣿⣿⣿⡿⣿⣿⣿⣿⢿⢟⣧⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠈⡿
⠀⠀⠀⡆⢸⣿⣿⣧⣠⣀⣼⣿⣿⣷⣶⣿⠿⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠾⠟⠛⠛⠁⠀⠀⣰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣅
⠀⠀⢸⣇⠀⠹⠿⣿⣿⣿⣿⣿⣿⣽⡿⠁⠀⠀⠀⠀⠀⠀⠐⠉⢁⠹⣿⡇⠀⠉⢻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣠⣀⣀⣤⣶⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠈⢻⣄⣠⣠⣀⣉⣉⣙⣫⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠠⢤⡦⣶⣾⣿⣆⣠⣼⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣸⣿⣿⣿⣿⣿⣿⣿⠿⢃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠭⠟⢿⠋⣿⣾⠇⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⣿⣿⣿⣋⣠⣴⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢟⣿⠿⢟⣋⣵⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢋⠀⠀⠀⠀⢀⣿⣿⣿⢿⣶⣤⣄⡀⢀⣀⣀⣀⣀⡀⠈⠑⠚⠛⠋⠁⣸⣿⠿⣶⣦⣤⣀⠀⣀⣠⣤⣶⣾⣿⣶⡶⢶⣦⣤⡄⠀
⠀⠀⠀⠀⠀⠰⣶⣬⣿⣿⣿⣿⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣾⣿⣯⠆⠀⠀⢠⣿⣿⣿⣿⣀⠙⣿⣿⣿⡿⠋⠉⣀⣌⣈⡉⠁⠀⠀⠀⠀⠉⠛⠀⢉⣻⣿⣿⠿⠻⢛⣛⣿⣿⣿⣿⡏⠛⠋⢨⣷⣦
⢦⣄⣀⠀⠀⣤⣤⣈⣉⣉⡍⠁⠀⠀⠙⣿⣆⣀⣀⣀⣤⣤⣤⣀⣀⡀⢀⣤⣾⣿⠐⠡⠐⠠⠀⠈⣿⣿⣿⣿⣧⡄⣸⣿⣿⣧⣤⣤⣀⣀⣀⣈⢀⣤⣤⣤⣤⣤⣴⣾⠿⠛⣉⡄⣶⣿⣿⣿⣿⣿⣿⣿⠃⢸⠃⠀⢿⣿
⡀⠉⠛⠛⣶⣿⣿⣿⣿⣿⣋⣤⣴⣶⣾⣿⣿⡿⠛⠋⠉⠉⠻⡿⢿⡟⠛⠟⣋⠀⠀⠐⠁⢀⣠⠶⣿⣿⡿⠙⠙⠁⣿⣿⡟⢡⣽⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉⢀⡀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣼⣿
⢿⣦⣤⣰⠿⣿⣿⣿⣿⣿⠟⠛⠛⠉⠁⣼⣿⡇⠀⠀⠀⢀⣾⣿⣿⣿⠈⢸⣿⣷⠞⠛⠋⢩⣴⣼⣿⣟⣀⠀⢀⠔⡙⢿⣇⠹⢿⣿⣿⣿⣿⠟⣃⠁⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠉⣿⣿⣿⣿⣿⣿⡿⣿⡀⠀⢀⣿⡟
⠀⠉⠻⣿⣤⣤⣬⣙⣉⣁⣀⣀⣠⣴⣾⣿⣏⠀⠀⠀⢀⣾⣟⣿⣿⡿⢄⣈⣛⢁⠄⣄⣤⠼⢿⣿⡿⣿⣿⣶⡏⠀⠀⠘⠿⣇⠈⠿⠟⠛⠁⣴⣿⣿⣿⡿⠿⠿⠿⠛⠛⣛⣉⣀⣀⣿⣿⣿⣿⣿⠟⠀⣿⣧⠀⢸⣿⡏
⣄⣀⠀⠺⢿⣿⣿⡿⠏⠉⠉⠉⠉⠉⠀⠈⣿⠆⢀⡴⠋⢁⣎⣼⣿⣇⠀⠘⣛⠀⣀⣀⣀⣰⣿⣿⣣⡿⣿⣿⣷⣦⣤⣶⣶⣿⣶⣤⣄⡀⠘⢋⡉⠁⠀⠀⠀⠀⠀⣀⠔⠁⢀⣉⣽⣿⣿⣿⣿⠇⠀⠀⢹⣿⡄⠀⠿⣿
⠙⠿⠷⣶⣯⣿⣏⢰⣶⣶⣦⣤⣤⣤⣷⣾⡿⢀⡞⠀⠀⢸⣿⣿⣿⡇⠀⠩⠛⡆⠀⠉⠉⣿⣿⡟⠻⣷⢸⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠛⠀⣿⣇⠀⠀⠀⠀⠀⠀⣀⣠⣶⣾⣿⣿⡿⠻⣿⣿⡄⠀⠀⢸⣿⡟⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⣰⣶⣶⣾⣿⣷⣿⣿⣾⣿⣿⣿⡏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡿⠇⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠻⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⡟⠈⠻⠷⠿⠏⠉⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⠟⠁⠀⠀⠀⠀⢿⣦⣶⣆⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠸⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣿⡟⠀⠀⠀⠀⠀⠀⣸⠋⠉⢻⣷⣄⡀⠀⠀⣠⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⡿⠟⠛⠛
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⡀⠀⠀⠀⠀⣰⡏⠀⠂⡀⢻⣿⣿⣿⣿⣿⣿⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠋⠀⠀⠀⠀
⠀⠀⠀⢰⠉⠀⠀⠀⠀⠘⣿⣿⡟⠻⢿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣀⣤⣿⣿⣧⣀⣴⣦⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⢀⣴
⠀⠀⠀⣴⡀⠀⠀⠀⣠⣾⢷⣦⠀⠀⠀⠘⢿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⠶⣿⣿
⠀⠀⠀⠈⠳⣶⣶⣾⠟⠁⢸⣿⡇⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⣿⣿⣿⣿⢿⣿⣿⣿⡿⡿⣿⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⢰⠰⣿⣿⣿⣀⣄⣀⣿⣿⣷⣶⣾⡿⠇⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠾⠷⠿⠛⠛⠋⠀⠀⣀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⣟⡀⠈⢿⢿⣿⣿⣿⣿⣿⣯⣿⠏⠀⠀⠀⠀⠀⠀⠀⠊⠉⡈⢿⣿⠁⠉⠛⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣄⣀⣠⣴⣶⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠹⣧⡀⢄⣈⢈⣉⣋⣛⣹⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠤⡼⢴⣶⣿⣧⣀⣤⡇⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣇⣿⣿⣿⣿⣿⣿⣿⡿⢛⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠽⠿⡟⢹⣳⡿⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠿⣿⣿⣟⣁⣤⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠙⣿⠿⠿⢛⣩⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⡁⠀⠀⠀⠀⣺⣿⣟⡿⣷⣦⣤⣀⠀⣀⣀⣀⣀⣀⠀⠉⠒⠙⠛⠉⢀⣿⡿⢷⣶⣤⣄⡀⣀⣀⣤⣴⣶⣿⣷⣶⠶⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⢶⣤⣽⣿⣿⣿⣓⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣰⣿⣟⡥⠀⠀⠀⣾⣿⣿⣿⣇⡈⢻⣿⣿⣿⠟⠉⢁⣀⣉⣉⠉⠀⠀⠀⠀⠈⠙⠃⠀⣙⣿⣿⡿⠿⠟⣛⣻⣿⣿⣿⣿⠉⢛⠁⢹⣷
⠷⣤⣀⡀⠀⢠⣤⣀⣉⣉⣹⠋⠀⠀⠈⢻⣶⣀⣀⣀⣀⣤⣤⣄⣀⣀⠀⣤⣴⣿⡗⠊⠄⠂⠄⠀⢹⣿⣿⣿⣿⣤⢀⣿⣿⣿⣤⣤⣄⣀⣀⣀⠁⣠⣤⣤⣤⣤⣤⣶⡿⠟⢋⣡⢰⣾⣿⣿⣿⣿⣿⣿⡟⠀⠞⠀⠸⣿
⣆⠈⠛⠟⢣⣾⣿⣿⣿⣿⣟⣡⣤⣶⣶⣿⣿⣿⠟⠛⠋⠉⠛⠿⠿⣿⠛⠻⢟⡁⠀⠀⠊⠀⣀⡴⢾⣿⣿⠏⠏⠋⢸⣷⣿⠋⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁⣀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢠⣿
⠻⣷⣤⣀⡾⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⢁⣿⣿⠀⠀⠀⠀⣰⣿⣿⣿⡏⠁⣿⣿⡶⠛⠛⠉⣥⣤⣿⣿⣃⡄⠀⡰⢊⠻⣿⡈⠿⣿⣿⣿⣿⡿⢛⡈⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠏⢹⣿⣿⣿⣿⣿⣿⢿⣇⠀⠀⢸⣿
⠀⠈⠙⢿⣷⣤⣤⣉⣛⣉⢀⣀⣀⣤⣴⣿⣿⡁⠀⠀⠀⣰⣿⣿⣿⣿⠧⡄⣘⡋⡀⢀⣠⡤⢿⣿⣿⢿⣿⣦⣼⠁⠀⠀⠻⢿⡀⠹⠿⠛⠋⢠⣾⣿⣿⣿⠿⠿⠿⠟⠛⢛⣋⣁⣀⣸⣿⣿⣿⣿⡿⠃⢘⣿⡄⠀⢿⣿
⣤⣀⠀⠀⠿⣿⣿⣿⠿⠛⠉⠉⠉⠉⠁⠀⢹⡇⠀⣠⠞⠉⢱⣭⣿⣿⠀⠀⢙⡃⢀⣀⣀⣀⣼⣿⣟⣸⢿⣿⣿⣶⣤⣴⣶⣾⣷⣦⣄⣀⠐⠛⣉⠉⠀⠀⠀⠀⠀⢀⡴⠊⠀⣈⣩⣿⣿⣿⣿⣿⠀⠀⠘⣿⣧⠀⠘⣿
⠙⠟⠿⣶⣯⣶⣟⡱⣶⣶⣦⣀⣠⣤⣼⣶⣿⠇⣰⠃⠀⠀⣿⣿⣿⣿⠀⠈⠽⢳⠀⠈⠉⢻⣿⣿⠻⢿⡆⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡟⠃⢸⣿⡀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣿⣿⣿⠟⢿⣿⣧⠀⠀⠀⣿⣿⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣿⣿⡇⣿⣿⢸⣿⣷⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⣰⣶⣶⣾⣿⣷⣿⣿⣾⣿⣿⣿⡏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⣧⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠇⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠻⢿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⠉⠻⠷⠿⠏⠈⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣿⡿⠋⠀⠀⠀⠀⠸⣿⣶⣶⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣿⠁⠀⠀⠀⠀⠀⢠⡟⠋⠙⣷⣦⣀⠀⠀⢀⣴⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠛⠛
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣄⠀⠀⠀⠀⢀⣾⠀⠐⢀⠘⣿⣿⣿⣿⣿⣿⣟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⠁⠀⠀⠀
⠀⠀⠀⠀⠞⠉⠁⠀⠀⠈⢿⣿⣿⠿⠿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⣄⣠⣼⣿⣿⣤⣠⣶⣴⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⣠
⠀⠀⠀⢠⣄⠀⠀⠀⢀⣴⡶⣾⡅⠀⠀⠈⠻⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡳⢾⣿
⠀⠀⠀⠈⠙⢶⣶⣶⡿⠉⠀⣿⣷⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣇⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠈
⠀⠀⠀⠀⡆⢾⣿⣿⣇⣀⠀⣹⣿⣷⣶⣶⣿⡿⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠺⠟⠛⠛⠁⠀⢀⣸⣯⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣸⣇⠀⢹⣿⣿⣿⣿⣿⣿⣿⣻⡿⠁⠀⠀⠀⠀⠀⠀⠐⠉⢉⠻⣿⡋⠈⠙⢿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣠⣀⣠⣤⣶⣾⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⣿⣄⢠⣀⡉⣉⡛⠛⣛⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠠⣤⣧⣦⣿⣿⣄⣀⣼⢘⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣸⣸⣿⣿⣿⣿⣿⣿⡿⠟⣣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠩⠽⠿⣿⠉⡟⣾⠇⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⣿⣿⣿⣋⣠⣼⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠙⣻⡿⠿⠿⣛⣥⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢏⠀⠀⠀⠀⢐⣿⣿⡿⢿⣦⣤⣄⠀⢀⣀⣀⣀⣀⡀⠈⠑⠊⠛⠋⠁⣸⣿⢿⣶⣤⣤⣀⢀⣀⣤⣤⣶⣾⣿⣶⡶⢶⣦⣤⡄
⢀⠀⠀⠀⠀⠀⢶⣤⣭⣿⣿⣿⣟⣂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣾⣿⣫⠄⠀⠀⢰⣿⣿⣿⣿⠀⠙⣿⣿⣿⡿⠋⠉⣀⣈⣉⡉⠁⠀⠀⠀⠀⠉⠛⠀⢉⣻⣿⡿⠿⠿⢛⣻⣿⣿⣿⣿⡏⠙⡁⠠⣿
⠻⣦⣀⡀⠀⠀⣤⣀⣉⣉⣙⡟⠁⠀⠈⠹⣷⣄⢀⣀⣀⣀⣤⣄⣀⣀⠀⢀⣤⣾⣟⠒⠡⠐⢠⠀⠈⣿⣿⣿⣿⣧⡀⣸⣿⣿⣧⣤⣄⣀⣀⣀⡈⢀⣤⣤⣤⣤⣤⣴⣾⠿⠛⣉⡄⣶⣿⣿⣿⣿⣿⣿⣿⠁⠸⠃⠀⢿
⣷⡀⠙⠻⠿⣴⣿⣿⣿⣿⣿⣛⣠⣤⣶⣶⣿⣿⠿⠟⠛⠛⠙⠻⠿⣿⡟⠻⠿⣋⠀⠀⠒⠂⢀⣠⢶⣿⣿⡿⠛⠛⠀⣿⣿⡟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉⢀⡀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣼
⠻⣷⣤⣄⣠⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⣼⣿⡇⠀⠀⠀⢀⣾⣿⣿⣷⠈⢸⣿⣷⠞⠒⠛⢩⣤⣼⣿⣟⣅⠀⢀⠔⡙⢿⣇⠻⣿⣿⣿⣿⣿⠟⣃⠁⠀⠀⠀⠀⠀⣀⣠⣴⣿⡿⠈⣿⣿⣿⣿⣿⣿⡿⣿⡀⠀⢀⣿
⠀⠀⠙⢻⣿⣤⣤⣌⣛⣋⠁⠀⣀⣀⣤⣾⣿⣟⠀⠀⠀⢀⣺⣿⣿⣿⡏⣀⢈⣛⢁⣀⣀⣠⡼⣿⣿⡿⣿⣿⣶⡏⠀⠀⠘⢿⣆⠈⠿⡟⠛⠁⣴⣿⣿⣿⡿⠿⠿⠿⠛⠛⣛⣉⣀⣀⣿⣿⣿⣿⣿⠟⠀⡿⣧⠀⠸⣿
⣦⣄⡀⠀⠸⣿⣿⣿⣿⠟⠙⠉⠉⠉⠉⠀⠘⣿⠀⢀⡴⠛⢉⣷⣹⣿⡇⠀⠘⣛⠀⣀⣀⣀⣠⣿⣿⣃⡿⣿⣿⣷⣦⣤⣶⣶⣿⣶⣤⣀⡀⠚⢛⡉⠁⠀⠀⠀⠀⠀⣠⠞⠁⣀⣩⣿⣿⣿⣿⣿⠇⠀⠀⢳⣿⣄⠀⢻
⠙⠟⠿⣷⣯⣶⣝⣻⠷⣶⣆⣀⣀⣀⣤⣷⣾⡿⢀⡞⠀⠀⢸⣿⣿⣿⣇⠀⠹⠟⡆⠈⠙⠛⣿⣿⡟⠿⣷⢹⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡛⠀⣿⣇⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⡿⠻⣿⣿⡄⠀⠀⢸⣿⡟⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⣶⣼⣿⣷⣿⣿⣾⣿⣿⣿⡏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠻⠿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⡟⠻⠷⠿⠏⠀⠉⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣟⣿⠟⠁⠀⠀⠀⠈⣿⡶⣶⣆⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠸⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⣼⠛⠉⢻⣶⣄⡀⠀⠀⣠⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⡿⠟⠛
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⣰⡇⠀⠂⡀⢻⣿⣿⣿⣿⣿⣿⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠋⠀⠀⠀
⠀⠀⠀⠀⢰⠋⠉⠉⠀⠀⠙⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⣀⣤⣿⣿⣧⣄⣴⣦⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⢀
⠀⠀⠀⠀⣠⡀⠀⠀⠀⢀⣴⢾⣯⠀⠀⠀⠙⢿⡿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠲⢿
⠀⠀⠀⠀⠙⠳⣶⣶⣾⠟⠁⢸⣿⡆⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠀⣾⣿⣿⣀⡀⢈⣿⣿⣶⣶⣾⣿⠇⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠷⠿⠛⠛⠋⠀⠀⣀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣾⠀⠈⣿⣿⣿⣿⣿⣿⣿⣟⣿⠟⠀⠀⠀⠀⠀⠀⠀⠚⠉⠙⢿⣿⠉⠉⠻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣄⣀⣠⣴⣶⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠹⣧⡀⡄⣈⠙⠛⠛⠛⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢤⣼⣿⣧⣀⣠⡇⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣃⣿⣿⣿⣿⣿⣿⣿⡿⢟⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠿⢿⡟⢹⢷⡿⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠿⣿⣿⣟⣁⣤⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠉⢙⣿⠿⠿⢟⣋⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⡃⠀⠀⠀⠀⣸⣿⣿⠿⣷⣦⣤⣀⠀⣀⣀⣀⣀⣀⠀⠉⠒⠛⠛⠉⢀⣿⡿⣿⣶⣤⣄⡀⣀⣠⣤⣶⣶⣿⣿⣷⠶⣶⣦⣤
⡀⣀⠀⠀⠀⠀⠈⢦⣬⣹⣿⣿⣿⣓⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣰⣿⣞⡡⠀⠀⠀⣼⣿⣿⣿⣇⡈⢻⣿⣿⣿⡟⠉⢁⣀⣉⣉⠉⠀⠀⠀⠀⠈⠙⠃⠀⣙⣿⣿⡿⠛⢛⣛⣿⣿⣿⣿⣷⠈⢉⠁⢸
⠛⢶⣄⡀⠀⠀⢀⣀⣈⣉⣙⣛⡋⠀⠀⠉⢻⣦⡀⢀⣀⣀⣀⣄⣀⣀⠀⠀⣀⣤⣿⡇⠚⠥⠂⣄⠀⢹⣿⣿⣿⣿⣄⢀⣿⣿⣿⣤⣤⣀⣀⣀⣀⠁⣠⣤⣤⣤⣤⣤⣶⡿⠟⢋⣡⢰⣾⣿⣿⣿⣿⣿⣿⡿⠀⠻⠀⠈
⣿⡆⠉⠛⠿⢧⣴⣿⣿⣿⣿⡟⣃⣠⣴⣶⣾⣿⣿⠿⠟⠛⠛⠻⠿⣿⣿⠟⠿⠟⡃⠀⠀⠒⠀⢀⡴⢾⣿⣿⠟⠟⠋⢸⣿⣿⠋⣬⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁⣀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢀
⠘⢿⣦⣄⣀⣸⣿⣿⣿⣿⣿⣿⠛⠻⠛⠋⢉⣿⣿⠀⠀⠀⠀⢠⣿⣿⣿⡆⠀⣿⣿⡶⠓⠒⠋⣩⣤⣾⣿⢇⠀⠀⡠⢊⠻⣿⡈⠿⣿⣿⣿⣿⣿⢛⡈⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⠏⢸⣿⣿⣿⣿⣿⣿⠿⣧⠀⠀⢸
⠀⠀⠉⠛⣿⣿⣠⣌⣙⡛⠋⠁⠀⢀⣀⣴⣾⣿⡋⠀⠀⠀⢀⣿⣿⣿⣿⢇⣀⢘⡋⢀⣀⣀⣤⠿⣿⣿⢿⣿⣷⣼⠃⠀⠀⠻⢿⡄⠸⠿⠛⠋⢀⣾⣿⣿⣿⠿⠿⠿⠟⠛⢛⣋⣁⣀⣸⣿⣿⣿⣿⡿⠃⢸⢹⡆⠀⢸
⣶⣤⣀⠀⠀⠿⣿⣿⣿⡿⠛⠛⠋⠉⠉⠉⠀⢹⣷⠀⢠⠞⠋⢩⣉⣿⣿⠀⠀⠙⡃⠀⣀⣀⣀⣼⣿⣿⣠⠿⣿⣿⣶⣤⣤⣶⣶⣷⣦⣤⣀⡀⠛⣋⠉⠀⠀⠀⠀⠀⢀⡠⠚⠀⣀⣩⣿⣿⣿⣿⣿⠀⠀⠈⣾⣿⡄⠘
⡉⠟⠿⣿⠿⣶⣌⢻⡿⢶⣆⣀⣀⣀⣀⣸⣶⣾⠟⢠⠏⠀⠀⣸⣿⣿⣿⡍⠈⠻⢳⠈⠉⠛⢻⣿⣿⠿⢿⡆⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣟⠋⢠⣿⡅⠀⠀⠀⠀⠀⢁⣀⣤⣶⣿⣿⣿⠟⠻⣿⣧⡀⠀⠀⣿⣿⠇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⣶⣼⣿⣷⣿⣿⣾⣿⣿⣿⣏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠻⠿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⡿⠷⠿⠏⠀⠀⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢻⣿⣿⠟⠁⠀⠀⠀⠈⣿⡶⣶⡆⠀⠀⠀⠀⠀⠀⣸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢺⣿⣿⣿
⠀⠀⠀⠀⠀⡸⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣿⡇⠀⠀⠀⠀⠀⠀⡿⠛⠉⢿⣦⣄⡀⠀⠀⣠⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⠀⠀⠀⠀⠀⣼⠇⠈⠀⡈⢿⣿⣿⣿⣿⣿⣿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠁⠀⠀
⠀⠀⠀⠀⠀⢰⠛⠉⠉⠉⠈⢹⣿⣿⡿⢿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣆⣀⣤⣿⣿⣦⣀⣴⣤⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀
⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢀⣴⢹⣍⠀⠀⠈⠙⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠲
⠀⠀⠀⠀⠈⠛⢷⣤⣤⣶⠟⠉⢹⣿⡀⠀⠀⠀⣸⡗⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⣿⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⡀⠀⠀⠀⠀⢠⢠⣿⣿⣿⠀⡀⢸⣿⣷⣶⣤⣶⣿⠅⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠖⠟⠛⠛⠋⠀⠀⣀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢷⠀⠀⠀⠀⣿⠀⠹⣿⣿⣿⣿⣿⣿⣿⣟⣿⠟⠀⠀⠀⠀⠀⠀⠀⠚⠉⠙⣿⣟⠉⠙⠻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣀⣀⣤⣴⣶⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣴⠀⠀⠀⠀⢻⣇⡀⡀⡙⠙⠛⠛⠿⣻⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣤⣼⣿⣆⣀⣠⠃⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣃⣿⣿⣿⣿⣿⣿⣿⠿⢛⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠻⠻⢿⡏⣻⢧⡿⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⢿⣿⣿⣛⣁⣤⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⠿⢛⣥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠁⠀⠀⠀⠀⣺⣿⣿⢿⣷⣤⣤⡀⠀⣀⣀⣀⣀⡀⠀⠉⠒⠛⠛⠉⢀⣿⡿⢷⣶⣤⣄⡀⣀⣠⣤⣶⣾⣿⣿⣷⢶⣶⣦
⣄⢀⢀⠀⠀⠀⠀⠨⣦⣉⣲⣶⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣞⡡⠀⠀⠀⣼⣿⣿⣿⡇⠈⢿⣿⣿⣿⡟⠉⢁⣀⣉⣉⠉⠀⠀⠀⠀⠈⠙⠃⠀⣙⣿⣿⠿⠛⢛⣛⣿⣿⣿⣿⣶⠉⢋⠁
⠉⠳⢧⣀⠀⠀⠀⢀⣈⡉⠛⠛⢛⡛⠉⠉⠙⢷⣦⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⣀⣴⣿⡇⠛⠡⠂⣀⠀⢻⣿⣿⣿⣿⡄⢀⣿⣿⣿⣤⣤⣀⣀⣀⣀⠁⣠⣤⣤⣤⣤⣤⣶⡿⠟⢋⣡⢰⣾⣿⣿⣿⣿⣿⣿⡿⠀⠞⠀
⠀⣶⡄⠙⠻⠳⢆⣼⣿⣿⣿⣿⡿⢃⣀⣤⣴⣾⣿⣿⠿⠟⠛⠛⠻⢿⣿⣿⠾⠿⠿⠋⠁⠀⠒⠀⢀⡴⣾⣿⣿⠏⠟⠋⢸⣿⣿⠛⣭⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁⣀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀
⠀⠘⢿⣦⣀⡀⣸⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⢉⣿⣿⠀⠀⠀⠀⢠⣾⣿⣿⡆⠐⣾⣿⣶⠒⠒⠛⣩⣤⣿⣿⢇⠀⠀⡠⢊⢻⣿⡘⢿⣿⣿⣿⣿⡿⢛⡈⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⠏⢸⣿⣿⣿⣿⣿⣿⠿⣧⠀⠀
⠀⠀⠀⠙⠻⣿⣿⣈⣉⡛⠟⠛⠉⠀⠀⣀⣤⣾⣿⡋⠀⠀⠀⢀⣿⣿⣿⣿⣇⣀⠙⡛⢁⣀⣀⣠⣿⣿⣿⣿⣿⣦⣼⠃⠀⠀⠻⣷⡀⠹⢿⠛⠋⢠⣾⣿⣿⣿⠿⠿⠿⠟⠛⢛⣋⣁⣀⣸⣿⣿⣿⣿⡿⠃⢸⢻⡆⠀
⣷⣦⣄⡀⠀⠈⣿⣿⣿⣿⣿⠟⠓⠛⠛⠙⠉⠀⢻⣷⠀⢀⡴⠋⢩⣍⣿⣿⠀⠈⠻⠃⠀⠀⠀⠀⣼⣿⣿⣰⣿⣿⣿⣶⣤⣤⣶⣶⣷⣤⣄⣀⠀⠛⢋⠉⠀⠀⠀⠀⠀⢀⡠⠚⠀⣀⣩⣿⣿⣿⣿⣿⠀⠀⠈⣾⣿⡄
⢀⡟⠻⣿⢿⣶⣄⡹⠿⠿⢆⣀⣀⣀⣀⣀⣱⣤⣾⡏⢠⠏⠀⠀⣸⣿⣿⣿⡉⠉⠻⢧⠈⠙⠛⢻⣿⣿⠿⢿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃⢰⣿⡁⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⠟⠻⣿⣧⡀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⣶⣼⣿⣷⣿⣿⣾⣿⣿⣿⣏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠻⠿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⡿⠷⠿⠏⠀⠀⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢻⣿⣿⠟⠁⠀⠀⠀⠈⣿⣶⣶⡆⠀⠀⠀⠀⠀⠀⣸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⣿⣿⣿
⠀⠀⠀⠀⠀⡸⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣿⡇⠀⠀⠀⠀⠀⠀⡿⠋⠉⢿⣦⣄⡀⠀⠀⣠⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⠀⠀⠀⠀⠀⣼⠇⠈⠀⡈⢿⣿⣿⣿⣿⣿⣿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠁⠀⠀
⠀⠀⠀⠀⠀⢰⠛⠉⠉⠉⠉⢹⣿⣿⡿⢿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣆⣀⣤⣿⣿⣦⣀⣴⣦⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀
⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢀⣴⢩⣍⠀⠀⠈⠙⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠲
⠀⠀⠀⠀⠀⠙⢷⣤⣤⣶⠟⠉⢹⣿⡀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⣿⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡀⠀⠀⠀⠀⢰⢠⣿⣿⣯⡀⡀⢸⣿⣷⣦⣤⣶⣿⠅⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⡷⠿⠛⠛⠋⠀⠀⣀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢷⠀⠀⠀⠀⣿⠀⠙⣿⣿⣿⣿⣿⣿⣿⣟⣿⠟⠁⠀⠀⠀⠀⠀⠀⠚⠉⠙⣿⣟⠉⠙⠻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣀⣀⣤⣴⣶⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣴⠀⠀⠀⠀⢻⣇⡀⡀⡙⠙⠛⠛⠿⣻⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣤⣼⣿⣦⣀⣠⠃⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣇⣿⣿⣿⣿⣿⣿⣿⠿⢋⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠻⠻⢿⡏⣻⢧⡿⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⢿⣿⣿⣛⣁⣤⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⠿⢛⣥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠁⠀⠀⠀⠀⣺⣿⣿⢿⣷⣤⣤⡀⠀⣀⣀⣀⣀⡀⠀⠉⠓⠛⠛⠉⢀⣿⡟⢷⣶⣤⣄⡀⣀⣠⣤⣶⣾⣿⣿⣷⠶⣶⣦
⣄⢀⢀⠀⠀⠀⠀⠀⣦⣉⣳⣶⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣞⡡⠀⠀⠀⣼⣿⣿⣿⣇⠈⢿⣿⣿⣿⡟⠉⣁⣀⣉⣉⠙⠀⠀⠀⠀⠈⠙⠃⠀⣙⣿⣿⠿⠛⢛⣛⣿⣿⣿⣿⣶⠉⢛⠁
⠉⠻⢧⣀⠀⠀⠀⢀⣈⡉⠛⠛⢛⡛⠉⠉⠙⢷⣦⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⣀⣴⣿⡇⠚⠡⠂⣀⠀⢻⣿⣿⣿⣿⡄⢀⣿⣿⣿⣤⣤⣀⣀⣀⣀⠁⣠⣤⣤⣤⣤⣤⣶⡿⠟⢋⣡⢰⣾⣿⣿⣿⣿⣿⣿⡿⠀⠞⠀
⠀⣶⡄⠙⠻⠷⢆⣼⣿⣿⣿⣿⡿⢃⣀⣤⣴⣾⣿⣿⠿⠟⠛⠛⠻⢿⣿⣿⠖⠿⠿⠋⠁⠀⠒⠀⢀⡴⣾⣿⣿⠏⠟⠋⢸⣿⣿⠋⣭⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁⣀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀
⠀⠘⢿⣦⣀⡀⣸⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⢉⣿⣿⠀⠀⠀⠀⢠⣾⣿⣿⡆⠐⣾⣿⡶⠲⠒⠛⣩⣤⣿⣿⢋⠀⠀⡠⢊⢻⣿⡘⢿⣿⣿⣿⣿⡿⢛⡈⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⠏⢸⣿⣿⣿⣿⣿⣿⠿⣧⠀⠀
⠀⠀⠀⠙⠻⣿⣿⣈⣉⡛⠟⠛⠁⠀⠀⣀⣤⣾⣿⡋⠀⠀⠀⢀⣿⣿⣿⣿⣇⣀⠙⡛⢁⣀⣀⣠⣿⣿⣿⣿⣿⣦⣼⠃⠀⠀⠻⣷⡀⠹⢿⠛⠋⢠⣾⣿⣿⣿⠿⠿⠿⠟⠛⢛⣋⣁⣀⣸⣿⣿⣿⣿⡿⠃⢸⢻⡆⠀
⣷⣦⣄⡀⠀⠈⡽⣿⣿⣿⣿⠟⠛⠛⠋⠙⠉⠀⢻⣷⠀⢀⡴⠋⢩⣈⣿⣿⠀⠀⠻⠃⠀⢀⠀⠀⣼⣿⣿⣸⢿⣿⣿⣶⣤⣤⣶⣶⣷⣤⣄⣀⠀⠛⢋⠉⠀⠀⠀⠀⠀⢀⡠⠚⠀⣀⣩⣿⣿⣿⣿⣿⠀⠀⠈⣾⣿⡄
⢀⡟⠻⣿⢿⢶⣄⡹⠿⠿⢇⣀⣀⣀⣀⣀⣱⣤⣾⡏⢠⠏⠀⠀⣸⣿⣿⣿⡉⠉⠻⢧⠈⠙⠛⢻⣿⣿⠿⢿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⢰⣿⡁⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⠟⠻⣿⣧⡀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣾⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⣶⣸⣿⣷⣿⣿⣾⣿⣿⣿⣿⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣷⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⠿⠿⠏⠀⠀⠈⠙⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⠛⠁⠀⠀⠀⠸⣿⣶⣶⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⣿⣿
⠀⠀⠀⠀⠀⠀⡼⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠃⠀⠀⠀⠀⠀⢀⡿⠋⠙⣿⣦⣀⠀⠀⢀⣰⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⣿⠿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⢀⣼⠃⠘⠀⡈⣿⣿⣿⣿⣿⣿⣟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠁⠀
⠀⠀⠀⠀⠀⠀⠠⠛⠛⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣀⣠⣤⣿⣿⣤⣠⣴⣴⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣬⢻⡏⠀⠀⠉⠻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿
⠀⠀⠀⠀⠀⠈⠛⣦⣤⣤⣴⠟⠉⢻⣿⠀⠀⠀⠀⣹⡗⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠟⣧⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣄⠀⠀⠀⠀⠀⢠⢨⣿⣿⣏⠀⠀⢸⣿⣶⣤⣤⣴⣿⠁⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⠾⠟⠛⠛⠋⠀⠀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠿⣇⠀⠀⠀⠀⣇⠈⢻⣿⣿⣿⣷⣿⣿⣿⣟⣿⠟⠉⠀⠀⠀⠀⠀⠀⠚⠉⠻⣿⡟⠉⠛⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣄⣀⣀⣀⣤⣴⣶⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣀⡜⠀⠀⠀⠈⢷⣄⠀⡉⠛⠛⠻⠿⠿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣤⣽⣿⣄⢀⣠⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣏⣿⣿⣿⣿⣿⣿⣿⠿⢋⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⡇⠀⠀⠀⠀⠘⣿⣷⣿⣷⣾⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠈⣛⠿⣿⡟⣿⢦⡟⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⢿⣿⣿⣛⣁⣴⣿⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡘⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⡿⢛⣥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠁⠀⠀⠀⠀⣾⣿⣿⢶⣶⣤⣄⡀⠀⣀⣀⣀⣀⡀⠈⠉⠒⠛⠛⠉⣸⣿⡿⣷⣶⣤⣄⣀⣀⣠⣤⣶⣾⣿⣿⡿⢿⣶
⣦⣀⡀⠀⠀⠀⠀⠀⢰⣬⣙⣲⣶⣾⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣞⡅⠀⠀⠀⣾⣿⣿⣿⡄⠉⢿⣿⣿⣿⠟⠉⣁⣈⣉⣉⠁⠀⠀⠀⠀⠊⠙⠁⠈⣹⣿⣿⠿⠛⢛⣻⣿⣿⣿⣷⣶⠉⢉
⠉⠙⠻⣧⡀⠀⠀⠀⢀⡉⠙⠛⠛⢻⠋⠉⠉⠛⣷⣄⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⣀⣰⣿⡋⠛⠡⢂⣀⠀⢿⣿⣿⣿⣿⡄⢨⣿⣿⣿⣤⣄⣀⣀⣀⣈⠁⣠⣤⣤⣤⣠⣴⣶⠿⠟⣋⣡⢰⣿⣿⣿⣿⣿⣿⣿⡟⠀⠟
⠀⠀⣶⡌⠛⠿⣶⣄⣼⣿⣿⣿⣿⡿⢆⣀⣤⣤⣾⣿⣿⡿⠿⠟⠛⠻⣿⣿⣿⠶⠿⠿⠋⠁⠀⠒⠀⢀⡴⣼⣿⣿⠏⠟⠃⢸⣿⣟⠋⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠉⠀⡀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀
⠀⠀⠹⣿⣄⣀⠀⣸⣿⣿⣿⣿⣿⣿⢿⠿⠟⠛⢋⣿⣷⠀⠀⠀⠀⢠⣶⣿⣿⡖⠒⣾⣷⡴⠶⠖⠚⣩⣤⣿⣿⢋⡀⠀⡠⢊⢻⣿⡘⢿⣿⣿⣿⣿⡿⢋⠈⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⠇⢹⣿⣿⣿⣿⣿⣿⢿⣇⠀
⣆⠀⠀⠈⠛⠿⣿⣧⣉⣉⠻⠿⠛⠉⠀⠀⢀⣠⣾⣿⠋⠀⠀⠀⢰⣿⣿⣿⣿⣁⣀⠙⠟⣁⣀⣀⣠⣿⣿⣿⣿⣿⣦⣼⠁⠀⠀⠻⣷⡀⠻⢿⠛⠋⢠⣾⣿⣿⣿⠿⠿⠿⠟⠛⢛⣉⣀⣀⣸⣿⣿⣿⣿⡿⠃⢸⣻⡆
⠻⣷⣤⣀⠀⠀⠉⣽⣿⣿⣿⣿⠷⠒⠛⠛⠛⠋⠉⢿⣇⠀⢀⡴⠛⣩⣉⣿⣿⠀⠉⠿⠃⠁⠀⠀⠀⣸⣿⡟⣸⢿⣿⣿⣶⣠⣤⣶⣾⣷⣤⣄⣀⠐⠛⠋⠉⠀⠀⠀⠀⠀⢀⡴⠋⠀⣈⣽⣿⣿⣿⣿⣿⠀⠀⠈⣿⣿
⣀⣞⠻⣿⠿⣷⠴⡰⠿⠿⠟⠁⡀⠀⠀⢀⣀⣰⣤⣾⡇⢠⠏⠀⠀⣿⣿⣿⣿⠉⠘⡻⢧⠈⠙⠛⢻⣿⣿⠿⣿⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃⢸⣿⡀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⠟⠻⣿⣷⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣾⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⣶⣸⣿⣧⣿⣿⣾⣿⣿⣿⣯⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠻⠿⠿⠟⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠉⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⣿⣿⡿⠉⠀⠀⠀⠀⢿⣷⣶⣆⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣿⣿
⠀⠀⠀⠀⠀⠀⢠⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣿⡟⠀⠀⠀⠀⠀⠀⣸⠛⠙⢻⣶⣄⡀⠀⠀⣀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⡀⠀⠀⠀⠀⣰⡟⠀⠂⡀⢻⣿⣿⣿⣿⣿⣿⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠋⠀
⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠛⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣆⣀⣰⣾⣿⣧⣀⣤⣦⣾⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⣿⣿⠃⠀⠈⠙⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠘⢷⣄⣠⣤⣾⠛⠹⣿⡆⠀⠀⠀⢈⣿⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⡿⢻⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣄⡀⠀⠀⠀⠀⠀⡄⣽⣿⣿⠁⠀⠀⣿⣷⣤⣤⣤⣾⡏⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⡷⠿⠻⠛⠛⠁⠀⢀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠿⣷⡀⠀⠀⠀⢰⠃⠙⣿⣿⣿⣿⣶⣿⣿⣿⢻⡿⠋⠁⠀⠀⠀⠀⠀⠰⠛⠛⢿⣿⠋⠙⠻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⣀⣀⣀⣠⣤⣶⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣄⣠⠃⠀⠀⠀⠻⣧⡀⢈⠙⠛⠿⠿⠿⢿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣠⣢⣬⣿⣧⡀⣀⡜⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣹⣹⣿⣿⣿⣿⣿⣿⡿⠟⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢿⣿⠁⠀⠀⠀⠀⢻⣿⣾⣿⣶⣶⣶⣶⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠊⢛⣻⣿⣿⢻⡷⣼⠇⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠿⣿⣿⣟⣋⣠⣾⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠃⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⠿⣋⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⠇⠀⠀⠀⠀⢰⣿⣿⣷⣶⣦⣤⡀⠀⢀⣀⣀⣀⣀⠀⠉⠑⠚⠛⠋⠁⣿⣿⢿⣶⣦⣤⣀⣀⣀⣤⣴⣶⣿⣿⣿⡿⣷
⣮⣄⣀⠀⠀⠀⠀⠀⠀⣦⣍⠓⣶⣶⣾⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶⣋⠀⠀⠀⢰⣿⣿⣿⣿⠈⠻⣿⣿⣿⣿⠋⢉⣀⣉⣉⠉⠀⠀⠀⠀⠀⠉⠋⠀⢉⣻⣿⡿⠟⠛⣛⣻⣭⣿⣿⣷⡎⠉
⠈⠉⠻⣦⣁⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠋⠉⠑⢶⣦⡀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⣀⣾⡿⠙⠋⠄⣀⡀⠸⣿⣿⣿⣿⣧⠀⣹⣿⣿⣇⣀⣀⣀⣀⣀⡉⢀⣤⣤⣤⣄⣤⣶⣾⠿⠛⣉⡄⣶⣿⣿⣿⣿⣿⣿⣿⠃⠰
⠀⠀⢠⣤⠙⠻⢶⣦⣠⣿⣿⣿⣿⣿⠷⢀⣀⣤⣤⣿⣿⣿⡿⠿⠟⠛⢿⣿⣿⡷⠾⠿⠟⠉⠀⠐⠂⠀⢠⣤⣿⣿⡿⠻⠛⠀⣿⣿⡛⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠁⢀⡀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀
⡄⠀⠈⢿⣧⣀⠀⢀⣿⣿⣿⣿⣿⣿⣾⣿⠿⠟⠛⢻⣿⡆⠀⠀⠀⠀⣴⣾⣿⣷⠒⢰⣿⣦⠶⠶⠖⢋⣥⣼⣿⡿⡃⠀⢀⠔⡙⣿⣇⠻⣿⣿⣿⣿⣿⠟⣃⠁⠀⠀⠀⠀⠀⢁⣠⣴⣿⡿⠀⣿⣿⣿⣿⣿⣿⡿⢿⡄
⣷⡀⠀⠀⠙⠻⣿⣿⡍⠉⠛⠿⠿⠛⠁⠀⠀⣀⣴⣿⡿⠁⠀⠀⠀⣺⣿⣿⣿⣿⡀⠈⠿⠋⣀⣀⣀⣼⣿⣿⣿⣿⣷⣤⡏⠀⠀⠘⢿⣆⠘⠿⡟⠛⠁⣴⣿⣿⣿⡿⠿⠿⠿⠛⠛⣛⣉⣀⣀⣿⣿⣿⣿⣿⠟⠀⣿⣿
⠻⢿⣦⣀⠀⠀⠈⢹⣿⣿⣿⣿⡶⠶⠶⠒⠛⠛⠋⠹⣿⡄⠀⣠⠞⠛⣍⢻⣿⡇⠈⠹⠿⠈⠀⠈⠁⢀⣿⣿⢇⣿⣿⣿⣷⣆⣠⣤⣶⣿⣦⣤⣀⡀⠚⠛⠉⠁⠀⠀⠀⠀⠀⣀⠔⠃⢀⣈⣽⣿⣿⣿⣿⡇⠀⠀⢸⣿
⣀⣞⠻⣿⠿⣷⣦⠸⠿⠿⠿⠋⠀⠀⠀⠀⠀⢀⣦⣴⣿⠁⡸⠁⠀⢀⣿⣿⣿⡏⠀⢒⡿⡄⠉⠛⠛⣿⣿⡿⢿⣧⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠀⣿⣏⠀⠀⠀⠀⠀⠈⣀⣠⣴⣾⣿⣿⡿⠛⢿⣿⣆⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠟⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⣶⢸⣿⣧⣿⣿⣾⣿⣿⣿⣯⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠻⠿⠿⠟⢹⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⠿⠏⠀⠀⠀⠈⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⣿⣿⡿⠋⠀⠀⠀⠀⢹⣿⣶⣶⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣽⣿⠁⠀⠀⠀⠀⠀⢠⡟⠋⠻⣷⣤⣀⠀⠀⢀⣴⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⡄⠀⠀⠀⠀⢀⣾⠁⠀⢀⠹⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠟⠛⠛⠋⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣀⣀⣶⣿⣿⣄⣠⣶⣴⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣬⢿⡋⠀⠈⠙⢿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠘⠿⣄⣀⣠⣴⠿⠉⢿⣷⠀⠀⠀⠀⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡿⣿⣿⣿⣿⢿⠿⣧⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣀⠀⠀⠀⠀⠀⢀⢩⣿⣿⡏⠀⠀⢸⣿⣦⣤⣤⣴⣿⠁⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⢿⠾⠟⠟⠛⠃⠁⠀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡿⢿⡆⠀⠀⠀⠀⡇⠈⢻⣿⣿⣿⣷⣾⣿⣿⡟⣿⠏⠉⠀⠀⠀⠀⠀⠀⠞⠛⢻⣿⡟⠙⠛⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⣤⣴⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣄⣀⡜⠀⠀⠀⠘⣷⡄⠀⠉⠛⠿⠿⠿⠿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣤⣽⣷⣄⢀⣨⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣛⣿⣿⣿⣿⣿⣿⣿⠿⢋⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠿⣿⠃⠀⠀⠀⠀⠸⣿⣶⣿⣷⣶⣶⣶⣶⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠜⠑⠙⣻⣿⣿⡟⣿⢬⡟⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⢿⣿⣿⣛⣁⣴⣿⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡘⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⡿⢟⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⡿⠀⠀⠀⠀⠀⣾⣿⡿⣶⣦⣀⣄⠀⠀⢀⣀⣀⣀⠀⠈⠙⠚⠛⠛⠁⣸⣿⠿⣷⣶⣤⣄⢀⣀⣠⣤⣶⣾⣿⣿⡿⢿
⣶⣅⣀⠀⠀⠀⠀⠀⠀⢠⣌⡛⢻⣷⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣞⡄⠀⠀⠀⣾⣿⣿⣿⡀⠙⢿⣿⣿⣿⠟⠉⣁⣉⣉⡉⠃⠀⠀⠀⠀⠈⠛⠀⠉⣻⣿⣿⠿⠛⢛⣛⣯⣿⣿⣷⣶⠉
⠈⠉⠛⢿⣄⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠟⠉⠉⠲⣶⣄⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣀⣰⣿⠉⢛⠠⢀⡀⠀⣿⣿⣿⣿⣿⡄⢈⣿⣿⣯⣀⣀⣀⣀⣀⣈⠁⣠⣤⣤⣄⣠⣴⣶⠿⠟⣋⣤⢰⣿⣿⣿⣿⣿⣿⣿⡏⠀
⠀⠀⠀⣦⡉⠻⢶⣦⣄⣸⣿⣿⣿⣶⡿⠆⢀⣀⣤⣼⣿⣶⣿⠿⠿⠟⠿⣿⣿⣿⠴⢿⠿⠋⠁⠀⠒⠀⠀⣤⣼⣿⣿⠏⠟⠃⢸⣿⣟⢋⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠉⠀⡀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀
⣠⠀⠀⢹⣷⣄⠀⠀⣹⣿⣿⣿⣿⣿⣷⣿⡿⠿⠛⢛⣿⣷⠀⠀⠀⠀⢠⣶⣿⣿⡄⠢⣾⣷⡴⠶⠶⠚⣩⣤⣿⣿⢉⡀⠀⡠⢚⢿⣷⡘⢿⣿⣿⣿⣿⡿⢋⠈⠀⠀⠀⠀⠀⢈⣀⣤⣾⣿⠇⢹⣿⣿⣿⣿⣿⣿⢿⣧
⣿⣄⠀⠀⠈⠻⢿⣷⣯⠙⠛⠿⠿⠟⠋⠀⠀⠀⣠⣾⣿⠏⠀⠀⠀⢰⣿⣿⣿⣿⣃⠀⠻⠟⠁⣀⣀⣰⣿⣿⣿⣿⣿⣧⣼⠁⠀⠀⠻⣷⠀⠻⢿⠛⠋⢠⣿⣿⣿⣿⢿⠿⠿⠟⠛⢛⣉⣁⣀⣸⣿⣿⣿⣿⡿⠃⢸⢹
⠙⢿⣧⣄⡀⠀⠀⠉⣿⣿⣿⣿⣶⠶⠶⠖⠚⠛⠛⠉⢿⡇⠀⢀⡴⠞⢫⡝⣿⣿⠈⠉⠿⠇⠁⠈⠉⠀⣼⣿⡟⣸⢿⣿⣿⣴⣀⣤⣴⣾⣧⣤⣄⡀⠐⠛⠋⠉⠀⠀⠀⠀⠀⢀⡤⠛⠀⣈⣽⣿⣿⣿⣿⣿⠀⠀⠈⣽
⣀⣆⡻⣿⠿⣷⣶⠰⠿⠿⠿⠟⠁⠀⠀⠀⠀⠀⢢⣤⣾⡇⢠⠏⠀⠀⣿⣿⣿⣿⠀⠐⣺⢧⠈⠙⠛⢻⣿⣿⠿⣿⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⢸⣿⠀⠀⠀⠀⠀⠀⢁⣀⣤⣾⣿⣿⣿⠟⠻⣿⣷⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠻⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⣶⢸⣿⣧⣿⣿⣾⣿⣿⣿⣿⣿⣇⢸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠻⠿⠿⠟⠁⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣿⣿⠟⠁⠀⠀⠀⠈⣿⣷⣶⡆⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⢠⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢪⣿⡏⠀⠀⠀⠀⠀⠀⣿⠛⠙⢿⣦⣄⠀⠀⠀⣠⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢧⠀⠀⠀⠀⠀⣸⡇⠈⠀⡈⢿⣿⣿⣿⣿⣿⣿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⣀⣰⣾⣿⣧⣀⣴⣤⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣠⣿⣛⠁⠈⠉⠻⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠻⢦⣀⣀⣠⣾⠟⢹⣿⡄⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⡿⣿⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣄⣀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⠁⠀⠈⣿⣧⣤⣤⣤⣾⡇⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⡿⡶⠿⠻⠛⠛⠈⠀⢀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣷⠀⠀⠀⠀⣸⠀⠛⣿⣿⣷⣷⣶⣿⣿⣿⢻⡿⠋⠁⠀⠀⠀⠀⠀⠰⠛⠛⣿⣿⠋⠛⠿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣀⣠⣤⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢢⡀⣨⠃⠀⠀⠀⢻⣆⠀⠉⠙⠻⠿⠿⢿⢿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⡀⣠⣠⣨⣿⣆⠀⣀⡍⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣋⣿⣿⣿⣿⣿⣿⣿⡿⠟⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠿⢿⡟⠀⠀⠀⠀⠀⢿⣷⣾⣿⣴⣶⣶⣶⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠚⢛⣿⣿⣿⢻⣧⣿⠃⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠿⣿⣿⣟⣋⣠⣾⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠃⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⠿⢋⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⠇⠀⠀⠀⠀⢸⣿⣿⣷⣶⣄⣄⡀⠀⢀⣀⣀⣀⡀⠀⠉⠓⠛⠛⠋⢀⣿⡿⢿⣶⣦⣤⡀⣀⣠⣤⣶⣾⣿⣿⣿⡿
⣶⣮⣀⠀⠀⠀⠀⠀⠀⠀⣄⡙⠻⣯⣶⣾⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣾⡶⣣⠀⠀⠀⢰⣿⣿⣿⣇⠈⠻⣿⣿⣿⡿⠋⢉⣁⣉⣉⠙⠀⠀⠀⠀⠘⠙⠃⠀⢙⣿⣿⡿⠟⠛⣛⣯⣯⣿⣿⣶⡎
⠀⠉⠛⠷⣷⡁⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠋⠉⠓⢶⣤⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⣾⡯⠽⡋⠄⣀⠀⢸⣿⣿⣿⣿⣇⠀⣹⣿⣿⣅⣀⣀⣀⣀⣀⡉⢀⣤⣤⣤⣀⣤⣶⡾⠟⢋⣡⡄⣾⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢠⣌⠙⢷⣶⣤⢀⣿⣿⣾⣷⣾⠧⠀⣀⣀⣤⣿⣷⣶⡿⠿⠿⠿⣿⣿⣾⡶⢾⡿⠟⠉⠀⠐⠂⠀⢠⣤⣿⣿⡿⠿⠛⠀⣿⣿⡛⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠋⠁⢀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⡄⠀⠘⣿⣦⡀⠈⠉⣿⣿⣿⣿⣿⣿⣾⣿⡿⠿⠛⣻⣿⡆⠀⠀⠀⠀⣴⣿⣿⣧⠒⢶⣷⣦⠶⠶⠖⢋⣥⣼⣿⣟⡁⠀⢀⠔⡻⣿⡆⠿⣿⣿⣿⣿⣿⠟⡁⠁⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠀⣿⣿⣿⣿⣿⣿⡿⢻
⣼⣧⡀⠀⠀⠙⠿⣿⣾⡟⠛⠻⠿⠿⠛⠉⠀⠀⢀⣤⣿⡿⠀⠀⠀⠀⣾⣿⣿⣿⣟⡀⠘⠿⠋⠀⡀⣀⣾⣿⣿⣿⣿⣯⣠⡏⠀⠀⠸⣿⡄⠘⠿⡟⠛⠁⣴⣿⣿⣿⡿⡿⠿⠿⠛⠛⣛⣉⣀⣀⣿⣿⣿⣿⣿⠟⠀⠟
⠘⠻⣷⣤⣀⠀⠀⠈⢻⣿⣿⣿⣶⡶⠶⠶⠖⠛⠛⠋⠹⣿⠀⠀⣠⠶⢛⣟⣻⣿⡇⠉⠻⠿⠈⠀⠉⠁⢀⣿⣿⢃⡿⣿⣿⣧⣆⣠⣤⣶⣿⣤⣤⣀⡀⠚⠛⠉⠁⠀⠀⠀⠀⠀⣀⠔⠃⢀⣈⣿⣿⣿⣿⣿⡇⠀⠀⢸
⣀⣆⡺⣿⠿⢿⣶⡴⠾⠿⠿⠿⠋⠀⠀⠀⠀⠀⠐⣤⣴⣿⠀⡸⠁⠀⢸⣿⣿⣿⡇⠀⢲⡿⡄⠙⠛⠛⣿⣿⡿⢿⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⣿⣇⠀⠀⠀⠀⠀⠈⣀⣠⣴⣿⣿⣿⡿⠛⢿⣿⣆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⣧⣿⣿⣾⣿⣿⣿⣿⣿⣧⣸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⢿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣿⡄⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠻⠿⠿⠟⠁⢨⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠙⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⠟⠁⠀⠀⠀⠸⣿⣴⣶⡆⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡇⠀⠀⠀⠀⠀⠀⡿⠋⠙⣿⣦⣀⠀⠀⢀⣠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⣼⠇⠘⠀⡈⣿⣿⣿⣿⣿⣿⣟⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠿⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣀⣀⣤⣿⣿⣤⣀⣴⣤⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⣨⣟⡻⠃⠉⠙⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠿⣆⣀⣀⣤⣶⠟⢸⣿⠀⠀⠀⠀⢹⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⣿⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣀⣀⡀⠀⠀⠀⠀⠀⠀⢉⣽⣿⣿⠁⠀⠘⣿⣧⣄⣀⣠⣾⡇⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⡽⡶⠿⠿⠛⠋⠈⠀⣀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣧⠀⠀⠀⠀⣸⠀⠻⣿⣿⣶⣶⣴⣿⣿⡿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠰⠛⠻⣿⡿⠛⠛⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡀⣀⣠⣴⣶⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⢥⠀⢉⠆⠀⠀⠀⢿⡆⠀⠙⠻⠿⠿⡿⣿⣿⣻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣹⣿⣄⠀⢀⡏⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⢏⣿⣿⣿⣿⣿⣿⣿⡿⠟⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠶⠾⣿⡟⠀⠀⠀⠀⠈⣷⣦⣾⣶⣴⣶⣶⣶⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢩⠋⠙⢻⣿⣿⡿⢻⣧⣿⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⢿⣿⣿⣟⣉⣠⣾⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠀⠈⠃⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⡿⢋⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣾⠇⠀⠀⠀⠀⢰⣿⣿⣶⣶⣄⣄⡀⠀⠀⣀⣀⣀⡀⠀⠉⠓⠛⠛⠋⢁⣿⡿⢿⣶⣦⣤⡀⣀⣠⣤⣶⣾⣿⣿⣿
⣶⣶⣎⡀⠀⠀⠀⠀⠀⠀⠐⣀⡚⠭⣭⣵⣾⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡶⢃⠀⠀⠀⣰⣿⣿⣿⣇⠉⢿⣿⣿⣾⡟⠋⢉⣉⣉⣉⠙⠀⠀⠀⠀⠈⠙⠃⠀⢙⣿⣿⡿⠟⠛⣻⣿⣿⣿⣿⣶
⠈⠉⠙⠛⢿⣔⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠛⠓⠲⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡏⠽⣋⠄⡀⠀⢸⣿⣿⣿⣿⣁⠀⣻⣿⣿⣄⣀⣀⣀⣀⣀⡉⢀⣤⣤⣤⣀⣤⣶⡿⠟⢋⣡⠄⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣍⠻⢶⣤⣄⢀⣿⣶⣶⣶⣶⡧⠀⢀⣀⣬⣿⣷⣶⣶⠿⠿⠿⣿⣿⣶⡶⣾⣿⠟⠋⠀⠤⠂⠀⢠⣤⣿⣿⡟⠿⠟⠠⣿⣿⠟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠋⠁⢀⠄⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡄⠀⠘⣿⣄⠀⠉⠑⣿⣿⣿⣿⣿⣷⣶⣿⣿⠿⠟⣻⣿⡄⠀⠀⠀⠀⣼⣿⣿⣧⠀⣴⣷⣦⠴⠴⠖⢚⣁⣼⣿⡏⠁⠀⢀⠔⡻⣿⡄⢿⣿⣿⣿⣿⣿⠟⡁⠁⠀⠀⠀⠀⠀⣀⣠⣶⣿⡿⠀⣿⣿⣿⣿⣿⣿⡿
⠀⢸⣧⠀⠀⠈⠻⢿⣷⣾⡟⠛⠻⣿⡿⠟⠉⠀⠀⢀⣠⣿⡿⠁⠀⠀⠀⣶⣿⣿⣿⣟⠀⠘⠿⠋⠀⠀⣀⣿⣿⣿⣿⣿⣮⣰⡏⠀⠀⠺⣿⡄⠘⢿⡟⠛⠁⣼⣿⣿⣿⡿⡿⠿⠿⠛⠙⣋⣉⣀⣀⣿⣿⣿⣿⣿⠟⠀
⠀⠘⠻⣷⣄⡀⠀⠀⠈⢻⣿⣿⣷⣶⣶⡶⠶⠶⠞⠿⠋⠻⣷⠀⠀⣠⠶⢛⡫⢿⣿⡏⠉⠻⠿⠈⠉⠉⠉⢠⣿⣿⢃⣿⣿⣿⣧⣆⣠⣤⣶⣿⣤⣤⣀⠀⠚⠛⠉⠁⠀⠀⠀⠀⠀⣀⠞⠁⢀⣨⣿⣿⣿⣿⣿⡇⠀⠀
⣀⠆⠲⣾⠿⢿⣷⣶⡶⠾⠟⢻⡿⠏⠁⠀⠀⠀⠀⠀⣄⣰⣿⠀⡰⠁⠀⢸⣿⣿⣿⡇⠂⢲⡿⡀⠘⠛⠿⣿⣿⡿⢿⣇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⣿⣇⠀⠀⠀⠀⠀⠁⣀⣠⣴⣿⣿⣿⡿⠛⢿⣿⣆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⣇⣿⣿⣼⣿⣿⣿⣿⣿⣧⣸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⣣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠻⠿⠿⠟⠁⠀⣼⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠈⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⡿⠋⠀⠀⠀⠀⢿⣧⣶⣆⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡿⠀⠀⠀⠀⠀⠀⢸⠟⠉⢻⣷⣄⡀⠀⠀⣀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⢠⡿⠀⠃⢀⢹⣿⣿⣿⣿⣿⣿⣟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠿⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⣀⣠⣾⣿⣷⣄⣠⣦⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⣻⣿⠟⠉⠉⠻⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⡀⢀⣀⣤⡾⠃⣿⡇⠀⠀⠀⠈⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣇⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠀⠀⢿⣿⣄⣀⣀⣴⣿⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢯⣷⠾⠿⠟⠛⠁⠁⢀⣸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡄⠀⠀⠀⠀⡇⠙⢿⣿⣷⣶⣦⣾⣿⣿⢿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠞⠛⢿⣿⠟⠛⠻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⣤⣴⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠩⡁⠈⡱⠀⠀⠀⠸⣷⠀⠈⠛⠿⠿⢿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣈⣿⣧⡀⠀⣸⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⣹⣿⣿⣿⣿⣿⣿⣿⠿⢃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣷⡶⣿⣿⠇⠀⠀⠀⠀⢹⣷⣦⣶⣦⣤⣦⣴⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⡙⠉⠛⣿⣿⣿⠟⣿⣼⡏⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⢿⣿⣿⣋⣁⣴⣿⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡀⠀⠹⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⠟⢋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣷⡿⠀⠀⠀⠀⠀⣾⣿⡷⣶⣦⣠⣀⠀⠀⢀⣀⣀⣀⠀⠈⠉⠚⠛⠛⠉⣸⣿⠿⣷⣶⣤⣄⣀⣀⣤⣴⣶⣿⣿⣿
⣶⣦⣤⡃⠀⠀⠀⠀⠀⠀⠀⢀⣘⠫⢭⣭⣶⣾⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣸⣷⠾⡀⠀⠀⠀⣾⣿⣿⣿⡈⠙⣿⣿⣷⣿⠛⠉⣁⣉⣉⡉⠃⠀⠀⠀⠀⠋⠛⠀⠈⣻⣿⣿⠿⠛⢛⣿⣭⣿⣿⣿
⠉⠉⠉⠛⠿⣧⡒⠀⠀⠀⠀⠈⠛⠻⠿⢿⣿⠟⠒⠒⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⠩⢟⡁⢀⡀⠀⣿⣿⣿⣿⣏⡀⢈⣿⣿⣯⣀⣀⣀⣀⣀⣈⠁⣠⣤⣤⣤⣠⣴⣾⠿⠛⣉⡤⢰⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣬⡙⠷⣦⣤⡀⣰⣷⣶⣶⣶⣾⡄⠀⣀⣀⣽⣿⣶⣶⡶⠿⠿⢿⣿⣶⣶⢴⣿⡿⠛⠁⠠⠔⠀⠀⣠⣽⣿⣿⢿⠿⠇⢸⣿⣟⢏⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠉⠀⡀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⠀⠀⢻⣧⡀⠈⠉⢹⣿⣿⣿⣿⣿⣷⣾⣿⠿⠿⠛⣿⣿⠀⠀⠀⠀⢠⣯⣿⣿⡤⢠⣶⣶⡤⠤⠴⠒⣋⣡⣿⣿⢋⠀⠀⡠⢚⢿⣷⠸⢿⣿⣿⣿⣿⡿⢋⠈⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⠇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⣾⡄⠀⠀⠙⠿⣷⣶⣾⠛⠛⢿⣿⠿⠛⠉⠀⠀⢀⣼⣿⠏⠀⠀⠀⢠⣾⣿⣿⣿⡃⠀⠻⠟⠁⠀⠀⣰⣿⣿⣿⣿⣿⣇⣼⠁⠀⠀⢿⣧⠀⠻⣿⠛⠋⢠⣿⣿⣿⣿⢿⠿⠿⠟⠛⢛⣉⣁⣀⣸⣿⣿⣿⣿⡿⠃
⠀⠀⠘⢿⣦⣀⠀⠀⠀⠙⣿⣿⣿⣶⣦⣶⣤⠶⠶⠾⠟⠙⢿⣆⠀⠀⣤⠞⢛⠝⣿⣿⠉⠙⣾⠇⠉⠉⠉⠁⣼⣿⡟⣸⣿⣿⣿⣴⣀⣤⣴⣾⣧⣤⣄⡀⠐⠛⠋⠉⠁⠀⠀⠀⠀⢀⡠⠚⠀⣀⣭⣿⣿⣿⣿⣿⠀⠀
⣾⠆⠲⣾⠟⢿⣿⣶⣶⠶⠟⠛⣿⡿⠋⠀⠀⠀⠀⠀⠠⣀⣼⡟⢀⡞⠀⠀⣿⣿⣿⣿⠒⠐⣶⢇⠀⠛⠿⢿⣿⣿⠿⣿⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃⢸⣿⡁⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⠟⠻⣿⣷⡀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣾⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⣶⢸⣿⣇⣿⣿⣼⣿⣿⣿⣿⣿⣧⣼⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⢃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠻⠿⠿⠟⠁⠀⢸⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠈⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⣿⣿⠟⠁⠀⠀⠀⠹⣷⣴⣶⡀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⣿⠃⠀⠀⠀⠀⠀⢀⡿⠋⠛⣷⣦⣀⠀⠀⢀⣴⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠊⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⡆⠀⠀⠀⠀⠀⣾⠃⠐⠀⠘⣿⣿⣿⣿⣿⣿⣟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠿⠿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⣀⣶⣿⣿⣄⣀⣴⣤⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣸⣛⠿⠉⠉⠛⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢡⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣄⠀⢀⣠⣶⠟⢺⣷⠀⠀⠀⠀⢹⡯⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⡿⠁⠀⢸⣿⣦⣀⣀⣠⣾⠃⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣵⡶⠿⠿⠛⠋⠈⠀⣀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣆⠀⠀⠀⠀⣸⠘⠿⣿⣿⣶⣦⣴⣿⣿⡿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠼⠛⢻⣿⡿⠛⠻⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⡀⣀⣤⣴⣶⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠉⣍⠀⢙⠄⠀⠀⠀⣿⡆⠀⠛⠻⢿⣿⣿⣿⣿⣻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣹⣷⡄⠀⢀⠏⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⢏⣿⣿⣿⣿⣿⣿⣿⡿⢛⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣶⣾⣷⡿⠀⠀⠀⠀⠈⣷⣦⣴⣶⣤⣤⣤⣤⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠹⠉⠙⢻⣿⣿⡿⣿⣆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠿⣿⣿⣟⣁⣤⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢈⠇⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣹⣾⠃⠀⠀⠀⠀⢴⣿⣿⣶⣶⣄⣄⡀⠀⠀⣀⣀⣀⡀⠀⠉⠓⠛⠛⠋⢀⣿⡿⢿⣶⣦⣤⣀⣀⣠⣤⣶⣿⣿⣿
⣶⣦⣤⣜⠀⠀⠀⠀⠀⠀⠀⠀⣐⠻⠭⣭⣭⣶⣏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡴⢃⠀⠀⠀⣰⣿⣿⣿⣇⠙⢿⣿⣿⣾⡟⠋⢉⣈⣉⣉⠙⠀⠀⠀⠀⠈⠙⠃⠀⣙⣿⣿⡿⠛⠛⣻⣭⣿⣿⣿
⠉⠉⠉⠛⠻⢧⣰⠂⠀⠀⠀⠀⠙⠻⠶⠿⣿⣿⠒⠒⠲⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡏⠽⣋⠄⠀⠀⢹⣿⣿⣿⣿⣃⠀⣻⣿⣿⣀⣀⣀⣀⣀⣈⠁⢀⣤⣤⣤⣀⣤⣶⡿⠟⢋⣡⢄⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⡙⠻⣶⣤⣄⢀⣶⣶⣶⣶⣶⣇⠀⢀⣀⣈⣿⣷⣶⣶⡶⠿⠿⣿⣷⣶⡦⣾⣿⠟⠋⠀⠤⠀⠉⢠⣬⣿⣿⡿⠿⠟⠠⣿⣿⠟⣩⣿⣿⣿⣿⣿⣷⣿⣿⣿⠿⠿⠋⠁⣀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⡀⠀⠘⣿⣄⠀⠉⠉⣿⣿⣿⣿⣿⣿⣶⣾⣿⠿⠿⢻⣿⡍⠀⠀⠀⠀⣼⣽⣿⡧⠄⣴⣶⣤⠤⠤⠖⢚⣁⣼⣿⡟⠁⠀⢀⠔⡻⣿⡄⢿⣿⣿⣿⣿⣿⠟⡁⠁⠀⠀⠀⠀⠀⣀⣠⣶⣿⡿⠈⣿⣿⣿⣿⣿⣿
⠀⠀⢰⣧⠀⠀⠈⠻⣿⣶⣾⡟⠻⢿⣿⡿⠿⠋⠁⠀⠀⣠⣿⡿⠁⠀⠀⠀⣴⣿⣿⣿⡇⠀⠸⠿⠋⠀⠀⢀⣾⣿⣿⣿⣿⣷⣠⡏⠀⠀⠺⣿⡄⠘⢿⡟⠛⠁⣼⣿⣿⣿⡿⡿⠿⠿⠛⠛⣋⣁⣀⣀⣿⣿⣿⣿⣿⠟
⠀⠀⠘⠿⣷⣄⠀⠀⠀⠙⢛⣷⣶⣶⣦⣴⣤⣤⠴⠶⠿⠛⢻⣷⠀⠀⢠⡴⢟⡻⣿⣿⡏⠉⢳⡷⠈⠉⠉⠉⢡⣿⣿⢃⣿⣿⣿⣧⣆⣠⣤⣶⣷⣤⣄⣀⠀⠚⠛⠉⠁⠀⠀⠀⠀⠀⣠⠞⠁⢀⣨⣾⣿⣿⣿⣿⡇⠀
⢸⡗⠲⣶⠟⢿⢿⣶⣶⡶⠞⠛⢿⣿⠟⠁⠀⠀⠀⠀⠀⣄⣰⣿⠀⣰⠃⠀⢸⣿⣿⣿⡗⠒⢲⣶⡀⠚⠿⠿⣿⣿⡿⢿⣅⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⣿⣇⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⡿⠛⢿⣿⣇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⣶⢸⣿⡇⣿⣿⣼⣿⣿⣿⣿⣿⣷⣼⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠻⠿⠿⠟⠁⠀⠸⣿⣿⣿⣿⣿⣿⡿⠟⠿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⣿⣿⡟⠉⠀⠀⠀⠈⢿⣧⣶⡆⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣿⡏⠀⠀⠀⠀⠀⠀⣸⠛⠙⢻⣶⣄⡀⠀⠀⣠⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠘⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⠀⠀⠀⠀⠀⢰⡟⠀⠂⡀⢻⣿⣿⣿⣿⣿⣿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠿⠿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣀⣀⣰⣾⣿⣧⣀⣠⣦⣾⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣛⠿⠏⠉⠙⠻⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⢀⣤⡾⠻⣿⡆⠀⠀⠀⠘⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⣿⠏⠀⠀⣿⣧⣀⣀⣀⣴⡟⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠿⣶⠾⠿⠟⠛⠁⠀⢀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⡇⠻⣿⣿⣷⣴⣤⣾⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠾⠛⢿⣿⠟⠛⠿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣀⢀⣠⣤⣶⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⠉⢩⡉⠉⠣⠀⠀⠀⢸⣧⠀⠘⠛⠿⣿⣿⣿⣿⣿⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣈⣿⣦⠀⠀⡸⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡹⣹⣿⣿⣿⣿⣿⣿⣿⠟⣣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣷⣦⣿⣶⠇⠀⠀⠀⠀⢻⣧⣥⣴⣦⣤⣤⣤⣴⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠟⠉⠿⠿⣿⣿⢻⣷⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⣿⣿⣿⣋⣡⣼⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⣷⡟⠀⠀⠀⠀⠠⣾⣿⢷⣶⣦⣀⡀⠀⠀⢀⣀⣀⣀⠀⠈⠙⠚⠛⠛⠉⣸⣿⠿⣷⣶⣤⣄⢀⣠⣤⣴⣾⣿⣿
⣤⣤⣤⣤⡂⠀⠀⠀⠀⠀⠀⠀⢄⡩⠽⣟⣭⣵⣾⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠶⡀⠀⠀⠀⣾⣿⣿⣿⡈⠻⣿⣿⣷⣿⠛⠉⣉⣉⣉⡉⠃⠀⠀⠀⠀⠋⠛⠀⢈⣹⣿⡿⠟⠛⢛⣯⣽⣿⣿
⠉⠉⠉⠛⠻⢿⣆⡄⠀⠀⠀⠀⠈⠛⠶⠿⣿⣿⡿⠒⠲⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠩⠟⡁⠀⠀⠀⣿⣿⣿⣿⣟⡀⢘⣿⣿⣏⣀⣀⣀⣀⣀⣈⠀⣠⣤⣤⣄⣠⣴⣾⠿⠛⣉⡤⢰⣿⣿⣿⣿⣿⣿
⠄⠀⠀⠀⠀⠀⣈⠻⢶⣤⣄⡀⢰⣶⣶⣶⣤⣶⡀⠀⢀⣀⣻⣿⣴⣶⣶⠾⠿⢿⣷⣶⣶⢴⣿⡿⠛⠁⠠⠄⠈⠁⣠⣽⣿⣿⢿⠿⠃⢼⣿⡿⢋⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠉⠀⡀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⠀⠀⢻⣧⡀⠈⠛⢻⣿⣿⣿⣿⣿⣧⣶⣾⣿⠿⠟⣿⣭⠁⠀⠀⠀⢨⣭⣽⣿⡤⢤⣶⣦⡠⠤⠴⠖⣋⣡⣿⣿⠁⠀⠀⡠⢚⢿⣧⠸⣿⣿⣿⣿⣿⡿⢋⠈⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⠇⢸⣿⣿⣿⣿⣿
⠀⠀⠀⣾⡀⠀⠈⠙⢿⣶⣦⣾⠻⠿⣿⣿⡿⠟⠉⠁⠀⢀⣼⣿⠏⠀⠀⠀⢠⣾⣿⣿⣿⠀⠀⠿⠟⠁⠀⠀⣰⣿⣿⣿⣿⣿⣅⣼⠁⠀⠐⢿⣧⠀⠻⣿⠛⠋⢠⣾⣿⣿⣿⢿⠿⠿⠟⠛⢛⣩⣀⣀⣸⣿⣿⣿⣿⡿
⠀⠀⠀⠻⣿⣄⡀⠀⠀⠈⠛⣻⣶⣶⣶⣤⣤⣤⣤⠴⠾⠟⠛⣿⡆⠀⠀⣠⠾⢛⢽⣿⣿⠉⠋⣶⠆⠉⠉⠉⠉⣸⣿⡟⣸⣿⣿⣿⣤⣀⣠⣴⣾⣧⣤⣄⡀⠐⠛⠋⠉⠁⠀⠀⠀⠀⢀⡠⠚⠀⣀⣠⣿⣿⣿⣿⣿⠀
⠀⢿⠲⣶⠋⢿⠿⣷⣶⣶⠶⠛⠻⣿⡿⠋⠀⠀⠀⠀⠀⢠⣀⣼⡟⢀⡞⠁⠀⣿⣿⣿⣿⠒⠒⢶⣆⠐⠻⠿⢷⣿⣿⠿⣿⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃⢸⣿⡀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⠟⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠿⠟⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⣶⢸⣿⡇⣿⣿⣼⣿⣿⣿⣿⣿⣷⣾⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠻⠿⠿⠟⠁⠀⠀⠿⣿⣿⢿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣿⣿⠟⠉⠀⠀⠀⠈⣿⣶⣶⡆⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣿⡏⠀⠀⠀⠀⠀⠀⣿⠛⠙⢿⣶⣄⡀⠀⢀⣠⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣇⠀⠀⠀⠀⠀⣰⡇⠀⠄⡀⢿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣷⣀⣀⣰⣾⣿⣧⣀⣴⣤⣾⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⡻⢿⠏⠉⠛⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣆⠀⠀⢀⣤⡾⢻⣿⡄⠀⠀⠀⠹⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⢻⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⠏⠀⠀⣿⣧⣀⠀⣀⣼⡟⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⡿⡷⠿⠻⠟⠛⠁⠀⢀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⣾⣿⣿⣿⣦⡀⠀⠀⠀⢀⡇⠻⣿⣿⣦⣴⣤⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⢠⠾⠻⣿⣿⠟⠛⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣀⣠⣤⣶⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠋⠉⠉⢛⠉⠙⠣⠀⠀⠀⢸⣇⠀⠘⠿⡿⣿⣿⣿⣿⣿⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣘⣿⣆⠀⠀⡹⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣹⣹⣿⣿⣿⣿⣿⣿⡿⠟⣡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⣴⣦⣤⣷⣶⡇⠀⠀⠀⠀⢻⣦⣤⣤⣄⣤⣤⣬⣵⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠟⠉⠿⢿⣿⣷⢻⣧⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⣿⣿⣟⣋⣠⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠀⠉⠁⠈⠻⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⢸⣷⡟⠀⠀⠀⠀⠠⣾⣿⣷⣶⣤⣀⡀⠀⠀⣀⣀⣀⣀⠀⠈⠙⠚⠛⠋⠁⣸⣿⠿⣷⣶⣤⣄⣀⣠⣤⣶⣾⣿
⣤⣤⣤⣄⣀⠇⠀⠀⠀⠀⠀⠀⠀⢄⠩⢟⣛⣫⣥⣾⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠖⡀⠀⠀⢀⣾⣿⣿⣿⡈⠻⣿⣿⣷⣿⠛⠉⣉⣉⣉⡉⠃⠀⠀⠀⠀⠉⠛⠈⢈⣩⣿⡿⠟⠛⣛⣭⣽⣿
⡙⠉⠉⠛⠛⠿⢶⣆⠤⠀⠀⠀⠀⠘⠻⢶⣿⣿⣿⡿⠒⠢⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠻⠟⡡⠀⠀⠘⣿⣿⣿⣿⣏⡀⢸⣿⣿⣇⣀⣀⣀⣀⣀⡉⠀⣠⣤⣤⣄⣤⣴⣾⠿⠛⣩⡤⣰⣿⣿⣿⣿⣿
⣿⠀⠀⠀⠀⠀⠀⣉⠳⣦⣤⣀⡀⢰⣶⣤⣤⣤⣶⡀⠀⠀⣀⣻⣷⣤⣶⣶⣶⠿⣿⣶⣶⣦⣴⣿⡿⠛⠀⠠⠄⠈⠉⣠⣽⣿⣿⢻⠿⠃⣾⣿⡿⢋⣽⣿⣿⣿⣿⣿⣾⣿⣿⡿⠿⠛⠉⢀⡠⠀⢀⣰⣿⣿⣿⣿⣿⣿
⠟⠀⠀⠀⢀⠀⠀⢿⣇⠀⠉⠛⢳⣿⣿⣿⣿⣿⣧⣶⣾⣿⡿⠿⣿⣯⠉⠀⠀⠀⢨⣭⣽⣯⠤⢤⣶⣦⡤⠤⠤⠖⣋⣠⣿⣿⠁⠀⠀⡠⢚⣿⣧⠸⣿⣿⣿⣿⣿⡿⢋⠈⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⠃⢸⣿⣿⣿⣿
⠄⠀⠀⠀⣼⡀⠀⠈⠛⢿⣶⣤⣼⠿⠿⣿⣿⡿⠟⠈⠁⠀⢀⣼⣿⠏⠀⠀⠀⢠⣾⣿⣿⣿⠀⠈⠿⠟⠁⠀⠀⣸⣿⣿⣿⣿⣷⣆⣼⠁⠁⠐⢿⣧⠀⠿⡿⠛⠋⢠⣿⣿⣿⣿⠿⠿⠿⠛⠋⢛⢩⣤⣀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠉⠻⣿⣶⣶⣦⣥⣤⣤⣤⠴⠶⠿⠛⣿⡆⠀⠀⣠⠾⢻⢿⣿⣿⠉⠘⣶⠆⠉⠉⠉⠉⣹⣿⡟⣸⣿⣿⣿⣤⣀⣤⣴⣾⣦⣤⣄⡀⠐⠛⠋⠉⠀⠀⠀⠀⠀⢀⡴⠚⠀⣀⣤⣿⣿⣿⣿⣿
⠐⠶⢾⣶⠉⢿⠿⢿⣶⣶⣆⠀⠛⠿⣿⣿⠛⠈⠀⠀⠀⠀⢄⣀⣼⡇⢀⡞⠁⠀⢾⣷⣿⣿⠒⠲⢶⣆⠐⠺⠾⢶⣿⣿⣿⣯⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⢸⣿⡀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⠟⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠿⠟⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⣼⣿⣿⣿⣿⣿⣷⣾⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠻⠿⠿⠟⠁⠀⠀⠿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣿⣿⠟⠉⠀⠀⠀⠈⣿⣶⣶⡆⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⣿⡏⠀⠀⠀⠀⠀⠀⣿⠛⠙⢿⣦⣄⡀⠀⢀⣠⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣇⠀⠀⠀⠀⠀⣰⡇⠀⠄⡀⢿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣷⣀⣀⣰⣾⣿⣧⣀⣴⣤⣾⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡿⠿⠏⠉⠛⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣆⠀⠀⢀⣠⡾⢯⣿⡄⠀⠀⠀⠹⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⢻⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⠏⠀⠀⣿⣧⣀⠀⣀⣼⡟⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⡿⡷⠿⠻⠟⠛⠁⠀⢀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣾⣿⣿⣿⣦⡀⠀⠀⠀⠀⡆⠻⣿⣿⣦⣴⣤⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⢠⠾⠻⣿⣿⠟⠛⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣀⣀⣀⣠⣤⣶⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠋⠉⠉⢛⠉⠙⠣⠀⠀⠀⢸⣧⠀⠘⠿⡿⣿⣿⣿⣿⣿⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣘⣿⣆⠀⠀⡹⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣹⣹⣿⣿⣿⣿⣿⣿⡿⠟⣡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠠⣴⣦⣤⣷⣶⠇⠀⠀⠀⠀⢻⣦⣤⣤⣄⣤⣤⣬⣵⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠟⠉⠿⢿⣿⣷⢻⣧⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⣿⣿⣟⣋⣠⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠁⠉⠁⠈⠻⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⢸⣷⡟⠀⠀⠀⠀⠠⣾⣿⣷⣶⣤⣀⡀⠀⠀⣀⣀⣀⣀⠀⠈⠙⠚⠛⠋⠁⣸⣿⠿⣶⣶⣤⣄⣀⣠⣤⣶⣾⣿
⣤⣤⣤⣄⣠⡇⠀⠀⠀⠀⠀⠀⠀⢄⠩⠟⣛⣫⣥⣾⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠖⡀⠀⠀⢀⣾⣿⣿⣿⡈⠻⣿⣿⣷⣿⠛⠉⣉⣉⣉⡉⠃⠀⠀⠀⠀⠋⠛⠈⢉⣩⣿⡿⠟⠛⣛⣭⣽⣿
⡛⠉⠉⠛⠛⠿⢶⣄⡄⠀⠀⠀⠀⠘⠻⢶⣾⣿⣿⡿⠒⠢⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠛⠟⡡⠀⠀⠘⣿⣿⣿⣿⣏⡀⢘⣿⣿⣇⣀⣀⣀⣀⣀⡉⠀⣠⣤⣤⣄⣤⣴⣾⠿⠛⣩⡤⣰⣿⣿⣿⣿⣿
⣿⠀⠀⠀⠀⠀⠀⣈⠳⣦⣤⣀⡀⢰⣶⣤⣤⣤⣶⡀⠀⠀⣀⣻⣷⣤⣶⣶⣾⠿⣿⣶⣶⣦⢴⣿⡿⠛⠀⠠⠄⠈⠁⣠⣽⣿⣿⠻⠿⠃⣾⣿⡿⢋⣽⣿⣿⣿⣿⣿⣾⣿⣿⡿⠿⠛⠉⢀⡠⠀⢀⣰⣿⣿⣿⣿⣿⣿
⠟⠀⠀⠀⢀⠀⠀⢿⣇⠀⠉⠛⢳⣿⣿⣿⣿⣿⣧⣶⣾⣿⠿⠿⣿⣯⠁⠀⠀⠀⢨⣭⣽⣯⠤⢤⣶⣦⡠⠤⠤⠖⣋⣠⣿⣿⠁⠀⠀⡠⢚⣿⣧⠸⣿⣿⣿⣿⣿⡿⢋⠈⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⠃⢸⣿⣿⣿⣿
⠄⠀⠀⠀⣾⡀⠀⠈⠛⢿⣶⣤⣼⠿⠿⣿⣿⡿⠟⠈⠁⠀⢀⣼⣿⠏⠀⠀⠀⢠⣾⣿⣿⣿⠀⠈⠿⠟⠁⠀⠀⣸⣿⣿⣿⣿⣷⣅⣼⠁⠀⠐⢿⣧⠀⠿⣿⠛⠋⢠⣿⣿⣿⣿⠿⠿⠿⠛⠋⢛⢩⣤⣀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠉⠻⢿⣶⣶⣦⣥⣤⣤⣤⠴⠶⠿⠛⣿⡆⠀⠀⣠⠾⢟⢿⣿⣿⠉⠈⣶⠆⠉⠉⠉⠉⣹⣿⡟⣸⣿⣿⣿⣤⣀⣠⣴⣾⣦⣤⣀⡀⠐⠛⠋⠉⠀⠀⠀⠀⠀⢀⡤⠚⠀⣀⣤⣿⣿⣿⣿⣿
⠐⠶⠾⣶⠉⢿⠿⢿⣶⣶⣆⠀⠛⠿⣿⣿⠛⠉⠀⠀⠀⠀⢄⣀⣼⡗⢀⡞⠁⠀⣺⣷⣿⣿⠒⠰⢶⣆⠐⠺⠾⢶⣿⣿⣿⣯⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⢸⣿⡀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⠟⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠿⠟⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⣸⣿⣿⣿⣿⣿⣿⣾⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠻⠿⠿⠟⠁⠀⠰⠿⢿⣿⡿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣿⣿⣿⠟⠁⠀⠀⠀⠸⣿⣶⣶⡀⠀⠀⠀⠀⠀⠀⣹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢼⣿⠇⠀⠀⠀⠀⠀⠀⡿⠉⠙⢿⣦⣄⡀⠀⢀⣤⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⣼⡇⠈⠀⡈⢿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⣀⣰⣾⣿⣦⣀⣴⣤⣿⣿⣿⣿⣿⣿⡿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠈⠿⢿⠏⠛⠻⢿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⠀⠀⠀⣠⡾⢿⣷⡄⠀⠀⠀⢹⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢻⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣿⣿⠟⠀⠀⣿⣇⡀⠀⢀⣼⡟⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⡽⡶⠿⠻⠛⠛⠀⠀⢀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣶⣾⣿⣶⣦⠀⠀⠀⠀⠀⡇⢿⣿⣿⣦⣤⣀⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⢠⠿⠿⣿⣿⠟⠻⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣀⣠⣤⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠘⠛⠛⢛⠛⠛⢇⠀⠀⠀⢸⣇⠀⠹⠿⣿⣿⣿⣿⣿⣿⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣙⣿⣆⠀⠀⡙⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢙⣻⣿⣿⣿⣿⣿⣿⡿⠟⣡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⣤⣮⣬⣷⣴⡇⠀⠀⠀⠈⢻⣤⣤⣠⣄⣤⣭⣩⣵⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢈⠟⠙⠿⢿⣿⣷⢿⣧⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠿⣿⣿⣟⣋⣤⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠘⠁⠉⠉⠉⠻⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢸⣷⡏⠀⠀⠀⠀⠠⣾⣿⢷⣶⣤⣀⡀⠀⠀⣀⣀⣀⣀⠀⠈⠙⠊⠉⠋⠀⣼⣿⠻⣷⣶⣶⣤⣀⣤⣴⣶⣿
⢀⣠⣤⣄⣀⣀⠇⠀⠀⠀⠀⠀⠀⠀⠈⠽⢟⣛⣛⣭⣶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠖⡀⠀⠀⢠⣾⣿⣿⣷⠈⠻⣿⣿⣷⣿⠛⠉⣉⣈⣉⡉⠃⠀⠀⠀⠀⠉⠛⠈⢉⣭⣿⡿⠟⠛⣫⣭⣿
⣟⡛⠛⠛⠛⠻⠷⣶⣀⡀⠀⠀⠀⠀⠘⠿⣶⣿⣿⣿⡿⠷⠤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠛⠟⡡⠀⠀⠘⣿⣿⣿⣿⣇⡀⢘⣿⣿⣇⣀⣀⣀⣀⣀⣈⠀⣠⣤⣤⣄⣤⣶⣾⠟⠋⣩⡤⣸⣿⣿⣿⣿
⣿⣿⡄⠀⠀⠀⠀⠀⠙⠳⣦⣄⣀⠀⢰⣤⣤⣤⣤⣶⡀⠀⠀⢈⣻⣷⣤⣴⣶⣶⡿⣿⣶⣶⣦⣼⣿⡿⠛⠀⠠⠄⠈⠉⣠⣿⣿⣿⢻⠿⠇⣾⣿⡿⢋⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⢀⡠⠀⢀⣴⣿⣿⣿⣿⣿
⣿⠿⠁⠀⠀⠀⠀⠀⢻⣇⠈⠙⠛⢷⣿⣿⣿⣿⣿⣯⣴⣾⣿⡿⠿⣿⣯⠉⠀⠀⠀⢨⣽⣿⣯⠤⢤⣶⣤⡤⣤⠤⠖⣚⣡⣿⣿⠋⠀⠀⡠⢚⣿⣧⠸⣿⣿⣿⣿⣿⡿⢋⠈⠀⠀⠀⠀⢀⣀⣠⣴⣿⣿⠃⢸⣿⣿⣿
⡀⠀⠀⠀⠀⣼⡀⠀⠈⠛⣷⣦⣤⣼⠿⠿⣿⣿⣿⠿⠋⠉⠀⢀⣼⣿⡏⠀⠀⠀⢠⣾⣿⣿⣿⠀⠈⢿⡟⠁⠀⠀⢸⣿⣿⣿⣿⣿⣅⣼⠁⠀⠐⢿⣧⠀⠿⢿⠛⠋⢠⣿⣿⣿⣿⠿⠿⠿⠛⠋⢉⢩⣤⣄⣸⣿⣿⣿
⠠⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠉⠻⢿⣶⣶⣬⣭⣤⣤⣤⡴⠶⠿⠛⣿⡆⠀⠀⣠⡾⢏⢻⣿⣿⠉⠰⣶⠆⠉⠉⠉⠉⣽⣿⡟⢹⣿⣿⣿⣤⣀⣠⣤⣾⣦⣤⣄⡀⠐⠛⠋⠉⠀⠀⠀⠀⠀⢀⡴⠛⠀⣄⣤⣿⣿⣿⣿
⠀⠶⠶⣶⠏⢶⠝⠻⢿⣶⣦⣀⠀⠛⠿⣿⣿⠟⠉⠁⠀⠀⠀⢀⣀⣸⡿⠀⡞⠁⠀⣾⣿⣿⣿⠒⠲⢶⣆⠐⠲⠶⢶⣿⣿⣿⣯⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃⢰⣿⡄⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿⠟⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠿⠟⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⣸⣿⣿⣿⣿⣿⣷⣾⣿⣯⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⢸⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⢰⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠸⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠻⠿⠿⠟⠁⠀⠰⠿⠿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢿⣿⣿⡿⠋⠀⠀⠀⠀⢿⣷⣶⣆⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣿⡿⠀⠀⠀⠀⠀⠀⢸⠟⠉⠻⣷⣤⣀⠀⠀⣠⣾⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⢀⡿⠀⠁⢀⠹⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⣀⣀⣶⣿⣷⣄⣠⣦⣼⣿⣿⣿⣿⣿⣿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠹⠿⡿⠛⠛⠿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⢀⣴⠿⣵⣆⠀⠀⠀⠙⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣧⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣶⣾⡿⠃⠀⢸⣿⡀⠀⠀⣠⣿⠃⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢿⢿⠾⠟⠟⠛⠁⠀⠀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣶⣶⣶⣦⡄⠀⠀⠀⠀⢰⠸⣿⣿⣷⣤⣄⣸⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⡴⠿⢿⣿⡿⠛⠿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⣤⣴⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠛⠛⠻⡛⠛⠻⡀⠀⠀⠀⣟⡀⠈⠿⢿⣿⣿⣿⣿⣿⣧⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⢈⢻⣧⡀⠀⢈⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣏⣿⣿⣿⣿⣿⣿⣿⠿⢋⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⣴⣤⣹⣦⣴⠀⠀⠀⠀⠹⣷⣄⣄⣤⣈⣈⣉⣭⣽⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠻⢿⣿⣿⡾⣿⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⢿⣿⣿⣛⣡⣴⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠋⠉⠉⠉⠛⡇⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠃⣿⣾⠃⠀⠀⠀⠀⢴⣿⣿⣶⣦⣄⣀⠀⠀⢀⣀⣀⣀⡀⠀⠉⠒⠉⠙⠃⢀⣿⡟⠿⣶⣶⣦⣄⣠⣤⣶⣾
⠀⢀⣀⣀⣀⣀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠩⠿⣛⣛⣋⣥⡖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣴⢂⠀⠀⠀⣴⣿⣿⣿⣇⠙⢿⣿⣿⣾⡟⠋⢉⣀⣉⣉⠙⠀⠀⠀⠀⠈⠙⠃⠉⣉⣽⣿⠿⠛⣛⣭⣽
⣐⣛⠛⠛⠛⠻⠿⣶⣦⢀⡀⠀⠀⠀⠀⠻⢶⣴⣿⣿⣿⠯⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠻⢋⠄⠀⠀⢻⣿⣿⣿⣿⣃⠀⣻⣿⣿⣀⣀⣀⣀⣀⣀⡉⢀⣤⣤⣤⣤⣤⣶⡿⠟⠉⣥⠄⣿⣿⣿⣿
⣿⣿⣧⠀⠀⠀⠀⠀⠈⠻⢦⣄⣀⡀⠀⣤⣤⣤⣤⣥⣇⠀⠀⠀⢙⣿⣦⣤⣴⣶⣶⣾⣶⣶⣶⣤⣾⣿⠟⠓⠀⠤⠄⠉⢁⣬⣿⣿⡿⡿⠿⠠⣿⣿⠟⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⣀⠄⠀⣀⣾⣿⣿⣿⣿
⣿⣿⠟⠀⠀⠀⠀⠀⠐⣿⡀⠉⠹⠛⣾⣿⣿⣿⣿⣿⣭⣶⣾⣿⡿⢿⣿⡏⠉⠀⠀⠀⣩⣿⣿⡥⠤⣵⣶⣄⣤⣤⠴⢒⣋⣸⣿⡏⡁⠀⢀⠔⣻⣿⡄⢿⣿⣿⣿⣿⣿⠟⣁⠁⠀⠀⠀⠀⣀⣀⣠⣾⣿⡿⠁⣿⣿⣿
⠃⡀⠀⠀⠀⢀⣞⠀⠀⠙⢿⣶⣤⣤⡿⠿⢿⣿⣿⣿⠟⠉⠉⠀⢀⣾⣿⠃⠀⠀⠀⣴⣿⣿⣿⡿⠀⠸⣿⠛⠀⠀⠀⣾⣿⣿⣿⣿⣯⣠⡏⠀⠀⠺⣿⡄⠘⠿⡟⠛⠁⣴⣿⣿⣿⡿⠿⠿⠟⠛⠉⡉⢥⣤⣀⣿⣿⣿
⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠉⠙⢿⣧⣤⣤⣭⣩⣄⣀⣤⣤⣶⠟⢿⣧⡀⠀⢀⣤⠿⡛⢻⣿⡏⠑⢴⣶⠘⠈⠙⠉⢉⣿⣿⠇⣿⣿⣿⣧⣄⣀⣤⣴⣿⣤⣤⣀⠀⠚⠛⠉⠁⠀⠀⠀⠀⠀⣀⠔⠃⠠⣠⣼⣿⣿⣿
⠀⠶⠶⣶⡿⢷⠄⠻⢿⣶⣶⣄⡀⠸⠿⢿⣿⡿⠏⠉⠁⠀⠀⠀⡀⢀⣿⡇⢠⠏⠀⠀⣿⣾⣿⡗⠀⠤⣶⡀⠒⠶⠶⣾⣿⣿⣿⣇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠀⣸⣯⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣿⣿⡿⠋⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠿⠟⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⣶⢸⣿⡇⣿⣿⣸⣿⣿⣿⣿⣿⣷⣾⣿⣯⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠨⣿⡿⠿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠠⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠘⣿⡏⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠻⠿⠿⠟⠁⠀⠠⠿⠿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠉⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⣿⣿⣿⠟⠁⠀⠀⠀⢸⣿⣶⣾⡀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⣿⠃⠀⠀⠀⠀⠀⢀⡟⠋⠙⣿⣦⣄⡀⠀⢀⣴⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⠀⣼⠃⠘⠀⡈⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⣀⣴⣾⣿⣤⣀⣴⣤⣿⣿⣿⣿⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠘⣿⣿⠟⠛⠿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⣠⡾⢿⣶⠀⠀⠀⠈⢻⣟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣻⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣶⣿⠟⠀⠘⣿⣇⠀⠀⢀⣼⡏⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⡼⡷⠿⠻⠛⠋⠀⠀⣀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣴⣶⣶⣶⣄⠀⠀⠀⠀⢠⠆⢿⣿⣿⣄⣤⣀⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⢠⠿⠿⣿⣿⠟⠿⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣀⣠⣤⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠚⠛⠛⠿⠛⠻⠆⠀⠀⠀⢼⣁⠀⠹⢿⣿⣿⣿⣿⣿⣿⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⢙⣿⣄⠀⠀⡛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢉⣻⣿⣿⣿⣿⣿⣿⡿⠟⣡⠄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣠⣆⣘⣦⣤⡆⠀⠀⠀⠈⢿⣄⣠⣤⣀⣉⣉⣉⣭⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠙⠿⣿⣿⣶⢿⣇⣺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠿⣿⣿⣟⣉⣤⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠘⠉⠋⠉⠙⢿⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢸⣷⡏⠀⠀⠀⠀⢠⣾⣿⣶⣶⣄⣀⡀⠀⠀⣀⣀⣀⣀⠀⠈⠙⠛⠙⠉⠀⣽⣿⠻⣶⣶⣶⣤⣀⣤⣴⣶
⠀⠀⣀⣀⣀⣀⣀⠇⠀⠀⠀⠀⠀⠀⠀⡈⠿⣿⣟⣛⣩⣴⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣼⣭⠖⡀⠀⠀⢠⣾⣿⣿⣇⠈⠻⣿⣿⣷⣿⠛⢉⣁⣉⣉⡉⠃⠀⠀⠀⠀⠉⠛⠈⢉⣭⣿⡿⠟⢛⣫⣭
⣀⣻⠟⠛⠻⠛⠿⢷⣦⡄⡠⠀⠀⠀⠀⠸⠶⣤⣿⣿⣿⡯⠥⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠛⠟⠠⠀⠀⠸⣿⣿⣿⣿⣇⡀⢹⣿⣿⣇⣀⣀⣀⣀⣀⡈⢀⣠⣤⣤⣤⣤⣶⡾⠟⠋⣡⡤⣸⣿⣿⣿
⣿⣿⣷⡀⠀⠀⠀⠀⠈⠙⢷⣤⣀⡀⠀⢠⣤⣤⣤⣩⣽⠀⠀⠀⠈⣿⣷⣤⣤⣴⣶⣶⣶⣶⣦⣤⣴⣿⡿⠛⠀⠠⠄⠈⠁⣀⣿⣿⣿⢻⠿⠃⣿⣿⡿⢋⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁⢀⡠⠀⢀⣴⣿⣿⣿⣿
⣿⣿⡿⠁⠀⠀⠀⠀⠀⢿⣆⠈⠙⠿⢳⣿⣿⣿⣿⣿⣭⣥⣶⣿⣿⢿⣿⣟⠉⠁⠀⠀⢈⣿⣿⣯⠨⢩⣶⣤⡤⣤⡤⠖⣚⣡⣿⣿⠃⠀⠀⡠⢚⣿⣧⠸⣿⣿⣿⣿⣿⡿⢋⠈⠀⠀⠀⠀⣀⣀⣠⣴⣿⡿⠇⢸⣿⣿
⡏⠀⠀⠀⠀⠀⣸⡀⠀⠘⠻⣷⣤⣄⡼⠿⣿⣿⣿⣿⡿⠉⠉⠁⠀⣸⣿⡇⠀⠀⠀⢠⣼⣿⣿⣿⠀⠈⢿⡟⠁⠀⠀⢸⣿⣿⣿⣿⣷⣅⣼⠁⠀⠘⢿⣧⠀⠿⣿⠛⠋⢠⣿⣿⣿⣿⠿⠿⠿⠛⠋⢉⢩⣤⣄⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠈⠙⠿⣿⣤⣤⣭⣉⣅⣀⣠⡤⣴⡿⠿⣿⡄⠀⠀⣠⡼⢿⢻⣿⣿⠓⠲⣶⡆⠃⠙⠋⠉⣹⣿⡟⢸⣿⣿⣿⣤⣀⣠⣤⣾⣦⣤⣀⡀⠐⠛⠋⠉⠀⠀⠀⠀⠀⢀⡴⠋⠀⣄⣴⣿⣿⣿
⠀⠶⠶⣶⠿⢷⠄⠹⠿⣷⣶⣄⣀⠀⠻⢿⣿⣿⠿⠉⠉⠁⠀⠀⢀⠀⣸⡗⠀⡴⠁⠀⣸⣷⣿⣿⠀⠠⢴⣆⠐⠲⠶⢶⣿⣿⣿⣯⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃⢸⣿⡀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠿⠟⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡇⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⣶⢸⣿⡇⣿⣿⢸⣿⣿⣿⣿⣿⣷⣾⣿⣿⣥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⢐⣿⡿⠿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⢀⣿⣷⣤⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠙⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠸⠿⠇⠀⠻⠿⠿⠟⠁⠀⠠⠿⠿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⣿⣿⡿⠉⠀⠀⠀⠈⢿⣷⣶⣇⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣿⡏⠀⠀⠀⠀⠀⠀⣸⠛⠉⢻⣷⣤⣀⠀⠀⣠⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⠀⠀⠀⠀⠀⢰⡏⠀⠂⢀⢹⣿⣿⣿⣿⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⣀⣠⣶⣿⣧⣄⣠⣦⣾⣿⣿⣿⣿⣿⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⢿⢿⡿⠛⠻⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠉⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⢀⣴⡟⣶⡆⠀⠀⠀⠙⣿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⣧⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀
⣀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣶⣶⡿⠃⠀⢻⣿⡀⠀⠀⣠⣿⠁⠀⠀⠀⠀⠀⠀⣀⣶⣾⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢿⢿⠾⠟⠛⠛⠁⠀⢀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠀⠀⢠⣶⣶⣶⣦⡀⠀⠀⠀⠀⡰⠰⣿⣿⣧⣠⣄⣸⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⡼⠿⣿⣿⡿⠻⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⣤⣴⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠛⠛⠿⠿⠛⠷⠀⠀⠀⠀⣇⠀⠨⢿⣿⣿⣿⣿⣿⣿⣧⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⢻⣧⠀⠀⢸⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣙⣿⣿⣿⣿⣿⣿⣿⠿⢋⡤⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣀⣤⣀⣳⣤⣴⠀⠀⠀⠀⠹⣧⣀⣄⣌⣉⣉⣉⣉⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠲⠛⠻⢿⣿⣷⡾⣿⣐⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⢿⣿⣿⣋⣡⣴⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠉⠙⠛⠉⠻⡇⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠃⣿⣾⠁⠀⠀⠀⠀⣴⣿⣷⣶⣦⣀⣀⠀⠀⢀⣀⣀⣀⡀⠀⠉⠛⠉⠉⠁⢠⣿⡟⠷⣶⣶⣦⣄⣠⣤⣶
⠀⠀⠀⣀⣀⣀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢈⠙⢿⡿⢿⣛⣥⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣿⣴⢂⠀⠀⠀⣴⣿⣿⣿⡀⠙⢿⣿⣿⣾⡟⠋⢉⣈⣉⣉⠙⠀⠀⠀⠀⠈⠙⠃⠀⣩⣽⣿⠿⠋⣛⣭
⣀⣰⡿⠿⠻⠿⠿⠿⣦⣤⣀⠀⠀⠀⠀⠀⠶⣦⣼⣿⣿⣿⠭⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⡻⠋⠀⠀⠀⢻⣿⣿⣿⣿⡁⠈⣿⣿⣿⣀⣀⣀⣀⣀⣀⡁⢀⣤⣤⣤⣤⣴⣶⠿⠛⠉⣤⢄⣿⣿⣿
⣿⣿⣿⣆⠀⠀⠀⠀⠀⠉⠻⣦⣄⡀⠀⠀⣤⣄⣨⣍⣩⡏⠀⠀⠀⢹⣿⣦⣤⣤⣶⣶⣶⣶⣶⣤⣤⣾⣿⠟⠃⠀⠤⠀⠉⢀⣸⣿⣿⡟⡿⠟⢸⣿⣿⠟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⣀⠄⠀⣠⣾⣿⣿⣿
⣿⣿⣿⡟⠀⠀⠀⠀⠀⠸⣷⡀⠙⠻⠿⣴⣿⣿⣿⣿⣿⣩⣴⣶⣿⣿⢿⣟⡋⠉⠀⠀⠀⣹⣿⣿⡅⠍⣵⣦⣄⣤⣤⠤⠖⣋⣼⣿⡟⠀⠀⢀⠔⣻⣿⡄⢿⣿⣿⣿⣿⣿⠟⣁⠁⠀⠀⠀⣀⣀⣀⣤⣾⣿⠿⠁⣿⣿
⣿⠃⠀⠀⠀⠀⢀⡄⠀⠀⠛⢿⣦⣤⣀⠿⣿⣿⣿⣿⣿⠏⠋⠉⠁⢀⣾⣿⠃⠀⠀⠀⣤⣿⣿⣿⡇⠀⠹⣿⠛⠀⠀⠀⣾⣿⣿⣿⣿⣧⣠⡏⠀⠀⠺⣿⡄⠸⠿⡟⠛⠁⣼⣿⣿⣿⡿⠿⠿⠟⠛⠉⡉⢥⣤⣀⣿⣿
⠇⠈⠀⠀⠀⠀⠀⢽⣦⡀⠀⠀⠉⠻⣿⣧⣤⣤⣍⣉⣀⣀⣀⣤⣶⡿⢿⣧⠀⠀⢀⣤⠿⡟⢿⣿⡟⠒⢲⣶⠐⠈⠛⠉⢉⣿⣿⠇⣿⣿⣿⣧⣄⣀⣤⣴⣿⣤⣤⣀⠀⠚⠛⠉⠁⠀⠀⠀⠀⠀⣀⠔⠁⠠⣤⣼⣿⣿
⠀⠗⡶⣶⠿⢷⠆⠸⠿⢿⣶⣤⣀⠀⠸⠿⣿⣿⣿⠏⠉⠉⠁⠀⠀⡀⢀⣿⠇⢠⠎⠀⠀⣿⣾⣿⡇⠀⠤⣶⡀⠒⠶⠶⣾⣿⣿⣿⣅⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠀⣼⣯⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠿⠟⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⣶⢸⣿⡇⣿⣿⢸⣿⣿⣿⣿⣿⣷⣾⣿⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⢀⣿⡏⠉⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠸⠿⠇⠀⠻⠿⠿⠟⠁⠀⠈⠿⠿⠿⠹⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⡟⠁⠀⠀⠀⠘⣿⣶⣶⡆⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⣿⡏⠀⠀⠀⠀⠀⠀⣾⠃⠉⢻⣶⣤⣀⠀⢀⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢧⠀⠀⠀⠀⠀⣰⡇⠀⠂⡀⢻⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣷⣄⣀⣰⣾⣿⣧⣀⣴⣦⣾⣿⣿⣿⣿⣿⣿⢷⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⠀⠀⠈⣿⣿⡟⠛⠿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣼⡆⠀⠀⠀⠀⠀⠀⠀
⠀⠙⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀⣴⡏⣾⡄⠀⠀⠈⠹⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢻⣧⣿⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⣶⡿⠃⠀⣿⣧⠀⠀⠀⣠⣿⠁⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢯⣷⠾⠟⠛⠉⠁⠀⢀⣼⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀
⣾⠃⠀⠀⢠⣤⣶⣦⣤⡀⠀⠀⠀⠀⡠⢰⣿⣿⣇⣀⣀⣸⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⡼⠿⣿⣿⡿⠿⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣠⣤⣶⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠉⠀⠀⠀⠿⠿⠿⠿⠿⢧⠀⠀⠀⢀⣇⠀⠩⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣿⣦⠀⠀⢹⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⣹⣿⣿⣿⣿⣿⣿⣿⠟⣁⡦⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣀⣠⣀⣳⣄⣴⠀⠀⠀⠀⢻⣇⠀⣄⡉⢉⣉⣙⣋⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠲⠛⠿⣿⣿⣷⡶⣿⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⢿⣿⣿⣋⣤⢴⣿⣷⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠟⠛⠛⠛⠻⡏⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠙⠃⣯⣿⠁⠀⠀⠀⠀⣶⣿⣷⣶⣦⣀⣀⠀⠀⢀⣀⣀⣀⡀⠀⠉⠛⠉⠉⠁⢠⣿⡟⢷⣶⣶⣦⣄⣤⣴
⠀⠀⠀⠀⠀⡀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠉⢙⣿⠿⠿⣛⣡⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣯⣵⠂⠀⠀⠀⣴⣿⣿⣿⡈⠙⢿⣿⣿⣾⡟⠉⣉⣈⣉⣉⠙⠀⠀⠀⠀⠈⠙⠁⢀⣩⣽⣿⠟⠋⣉
⠀⠀⣰⠿⠿⠿⠿⠿⣷⣤⡄⡀⡀⠀⠀⠀⠀⢶⣤⣭⣿⣿⣿⠭⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⡻⠃⠀⠀⠀⢿⣿⣿⣿⣿⡀⠈⣿⣿⣿⣀⣀⣀⣀⣀⣀⡁⣀⣤⣤⣤⣤⣴⣾⠿⠛⠉⡤⢄⣿⣿
⣾⣿⣿⣿⣆⠀⠀⠀⠀⠈⠙⠿⣤⣀⠀⠀⠀⣤⣀⣉⣉⣩⡏⠀⠀⠀⢹⣷⣄⣤⣤⣴⣶⣶⣶⣤⣤⣤⣾⣿⠟⠃⠀⠤⠀⠉⢀⣸⣿⣿⡿⡿⠟⢸⣿⣟⠟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⣠⠄⠀⣠⣾⣿⣿
⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠰⣷⡈⠙⠻⠷⣴⣿⣿⣿⣿⣿⣩⣤⣶⣿⣿⣿⣟⡛⠉⠉⠀⠈⣙⣻⣿⡁⠉⣥⣦⣀⣤⣤⠤⢖⣋⣼⣿⡏⡀⠀⢀⠔⣻⣿⡄⢿⣿⣿⣿⣿⣿⠟⣁⠁⠀⠀⣀⣀⣀⣀⣤⣾⣿⠟⠀⣿
⣿⣿⠏⠉⠀⠀⠀⠀⡄⠀⠀⠻⢷⣤⣄⣠⣿⣿⣿⣿⣿⣿⠯⠛⠉⠁⢀⣾⣿⠁⠀⠀⠀⣤⣿⣿⣿⡇⠀⠹⣿⠟⠀⠀⠀⣾⣿⣿⣿⣿⣧⣠⡏⠀⠀⠻⣿⡄⠘⢿⡟⠛⠁⣼⣿⣿⣿⠿⠿⠿⠛⠛⠉⡉⠥⣤⣄⣿
⣿⡏⠀⠁⠀⠀⠀⠸⢿⣆⠀⠀⠀⠙⠻⣿⣧⣤⣬⣙⣉⣁⣀⣀⣤⣴⡿⢿⣧⠀⠀⠀⣠⡿⣟⢿⣿⡟⠒⢶⣶⠐⠈⠛⠋⢉⣿⣿⠏⣿⣿⣿⣧⣄⣀⣤⣴⣷⣤⣤⣀⠀⠚⠛⠉⠁⠀⠀⠀⠀⠀⡠⠞⠁⠠⣤⣾⣿
⡿⠗⠷⣶⠯⢷⡶⠸⠾⠿⣷⣦⣄⣀⠀⠸⢿⣿⣿⣿⠏⠉⠉⠉⠀⠀⡀⢈⣿⠆⢠⠞⠁⠀⣷⣾⣿⡧⠀⠤⣶⡀⠲⠶⠶⣾⣿⣿⣾⣅⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠀⢸⣯⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⡹⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠿⠟⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡄⢸⣷⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡇⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⣶⢸⣿⡇⣿⣿⢸⣿⣿⣿⣿⣿⣷⣾⣿⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢀⣁⣽⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⠃⠈⠉⠉⠿⠛⢿⣿⣿⡿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠙⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀
⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠸⠿⠇⠀⠻⠿⠿⠟⠁⠀⠀⠿⠿⠿⠸⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢷⣿⡿⠋⠀⠀⠀⠀⢻⣧⣶⣶⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣾⣿⠁⠀⠀⠀⠀⠀⢸⡟⠈⠹⣷⣦⣄⡀⠀⣠⣼⣿⣿⠇⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⡀⠀⠀⠀⠀⢀⣾⠀⠁⢀⠘⣿⣿⣿⣿⣿⣿⣯⠉⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣄⣀⣀⣶⣿⣿⣄⣀⣶⣴⣿⣿⣿⣿⡿⣿⡿⡆⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠉⠀⠀⢹⣿⣿⠟⠿⢿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⣷⠀⠀⠀⠀⠀⠀⠀
⠀⠉⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣀⣶⢠⣦⠀⠀⠀⠙⢿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⠟⣻⣬⡏⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢱⣤⣤⣾⠟⠁⢸⣿⡄⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠼⡶⠟⠛⠋⠉⠀⠀⣠⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀
⣶⡞⠁⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⢀⠀⣼⣿⣿⣀⣀⢀⣿⣿⣷⣶⣿⠿⠂⠀⠀⠀⠀⠀⢠⠿⢿⣿⣿⠿⠿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣀⣀⣠⣴⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀
⠋⠁⠀⠀⠸⠿⠿⠿⠿⠿⡄⠀⠀⠀⢸⡀⠈⢽⣿⣿⣿⣿⣿⣿⣯⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠻⣷⡀⠀⠈⡏⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢉⣻⣿⣿⣿⣿⣿⣿⠿⢋⣰⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡀⢀⣄⠐⣄⣠⡆⠀⠀⠀⠘⣷⡀⢠⣈⠉⣉⣛⣛⣻⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠻⢿⣿⣿⣶⢾⣇⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⣿⣿⣟⣡⡤⣾⣿⣆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⠙⠛⠛⠻⢿⠁⠀⠀⠀⠀⠸⣿⣾⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠙⠛⢸⣿⡏⠀⠀⠀⠀⢰⣾⣿⣶⣶⣄⣀⡀⠀⠀⣀⣀⣀⣀⠀⠈⠙⠋⠉⠉⠀⣽⣿⠻⢶⣶⣶⣤⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠈⠙⣻⡿⠿⠿⣋⣵⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣼⣭⠖⠀⠀⠀⢠⣾⣿⣿⣇⠉⠻⣿⣿⣷⣿⠛⢉⣀⣉⣉⡉⠃⠀⠀⠀⠀⠁⠋⠀⣈⣽⣿⡿⠛⣉
⠀⠀⠀⠶⠿⠿⠿⠿⣿⣦⣄⡀⢀⠀⠀⠀⠀⠰⣤⣈⣽⣿⣿⣿⠧⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣓⠞⠀⠀⠀⠸⣿⣿⣿⣿⣇⠀⢹⣿⣿⣇⣀⣀⣀⣀⣀⣈⢀⣠⣤⣤⣤⣤⣶⡿⠟⠋⢡⠴⣸⣿
⣠⣿⣿⣿⣶⡀⠀⠀⠀⠀⠙⠳⢧⣀⡀⠀⠀⢠⣀⣈⣉⣉⣹⠁⠀⠀⠉⣿⣦⣠⣤⣤⣴⣶⣶⣦⣤⣤⣴⣾⡿⠓⠀⢀⠄⠈⠁⣀⣿⣿⣿⠿⠿⠃⣿⣿⡻⢋⣽⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⢀⡤⠀⢀⣴⣿⢿
⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢴⡆⠉⠻⠿⢦⣾⣿⣿⣿⣿⣋⣥⣴⣶⣿⣿⣿⣛⠋⠉⠁⠀⢙⣻⣿⣏⠈⢉⣤⣄⣠⣤⡤⠴⢚⣡⣿⣿⠁⠀⠀⡠⢚⣿⣧⠸⢿⣿⣿⣿⣿⡿⢋⡀⠀⠀⢀⣀⣀⣀⣠⣴⣿⡿⠃⢸
⣿⣿⡿⠉⠁⠀⠀⠀⢠⠀⠀⠘⠿⣦⣄⣀⡸⣿⣿⣿⣿⣿⡿⠟⠋⠉⠁⣰⣿⡟⠀⠀⠀⢀⣼⣿⣿⣿⠀⠈⢿⡿⠃⠀⠀⢰⣿⣿⣿⣿⣦⣄⣼⠁⠀⠐⢿⣧⠀⠻⢿⠛⠋⢠⣿⣿⣿⡿⠿⠿⠟⠛⠉⢉⠩⠤⣤⣸
⣿⣿⠁⠈⠀⠀⠀⠀⢘⣷⡀⠀⠀⠉⠛⢿⣿⣠⣤⣙⣛⣉⣀⣀⣠⣤⣾⠿⣿⡄⠀⠀⢀⣼⠿⡿⣿⣿⠒⠂⣶⡆⠂⠘⠚⠉⣹⣿⡿⢹⣿⣿⣿⣤⣀⣠⣤⣾⣧⣤⣄⡀⠐⠛⠋⠉⠀⠀⠀⠀⠀⢀⠠⠊⠀⢤⣴⣿
⣿⡟⠲⣶⠯⢷⠴⠴⠶⠿⢿⣶⣤⣀⠀⠀⠾⣿⣿⣿⡿⠋⠈⠉⠀⠀⢀⠀⢹⣿⠀⡴⠋⠀⢸⣷⣿⣿⠄⠠⢤⣆⠐⠲⠶⢶⣿⣿⣿⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⣿⡅⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⡹⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠿⠟⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡆⢸⣷⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⢸⣿⣿⣿⣿⣿⣷⣾⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡅⢀⣁⣽⣿⣿⣿⣿⣿⣿⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⢋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⠃⠈⠉⠉⠿⠻⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠛⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠸⠿⠇⠀⠻⠿⠿⠟⠁⠀⠘⠿⠿⠿⠸⢿⣿⣟⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⡿⣿⣿⠟⠁⠀⠀⠀⠘⣿⣴⣶⡄⠀⠀⠀⠀⠀⠀⢸⣯⡀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢼⣼⡇⠀⠀⠀⠀⠀⠀⡿⠁⠉⢿⣶⣤⣀⢀⣀⣴⣿⣿⡿⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢠⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⣸⠇⠈⠀⡀⢻⣿⣿⣿⣿⣿⣿⡍⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⣀⣀⣰⣾⣿⣦⣀⣴⣦⣾⣿⣿⣿⣿⣿⣿⢷⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠋⠉⠉⠀⠈⣿⣿⡿⠻⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⡆⠀⠀⠀⠀⠀⠀
⠉⠉⠛⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀⣰⡇⣾⡀⠀⠀⠈⠻⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⠿⢻⣧⣿⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠸⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⣤⣴⡿⠋⠀⣿⣇⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⣷⠾⠛⠛⠉⠁⠀⣀⣼⣷⣇⠀⠀⠀⠀⠀⠀⠀
⣶⣶⠋⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⠀⠀⡀⢠⣿⣿⣇⣀⠀⣹⣿⣿⣶⣾⡿⠇⠀⠀⠀⠀⠀⠀⡼⠿⣿⣿⡿⠿⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣠⣤⣶⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠛⠋⠀⠀⠀⠿⠿⢿⡿⠿⣧⠀⠀⠀⠀⡇⠀⠹⣿⣿⣿⣿⣿⣿⣿⣭⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣿⣆⠀⠀⢹⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⣙⣿⣿⣿⣿⣿⣿⡿⠟⣡⡆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⠀⢢⣀⣤⠀⠀⠀⠀⠻⣆⠀⣄⡉⢙⣙⡛⣛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⡺⠛⠟⣿⣿⣷⡶⣧⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⢿⣿⣟⣋⣤⠴⣿⣷⡄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡟⠻⠛⠛⢿⡏⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠛⠁⣯⣿⠁⠀⠀⠀⠀⣶⣿⣷⣶⣦⣀⣀⠀⠀⢀⣀⣀⣀⡀⠀⠉⠊⠉⠉⠁⢠⣿⡟⠻⣿⣶⣶⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠉⢛⣿⠿⠿⢟⣩⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣯⣵⠀⠀⠀⠀⣴⣿⣿⣿⡈⠙⢿⣿⣿⣾⡟⠋⣁⣈⣉⣉⠉⠀⠀⠀⠀⠈⠘⠁⢀⣨⣿⡿⠟⠉
⠀⠀⠀⠠⣾⡿⠿⠿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⢦⣄⣩⣿⣿⣿⡷⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⡲⠃⠀⠀⠀⢿⣿⣿⣿⣯⡀⠈⣿⣿⣿⣀⣀⣀⣀⣀⣀⡁⣀⣴⣦⣤⣤⣴⣾⠿⠛⠡⠤⢆⣿
⢀⣴⣿⣿⣷⣄⠀⠀⠀⠀⠉⠛⠾⣄⡀⠀⠀⠀⣀⣈⣉⣉⣉⡏⠀⠀⠈⠻⣷⣄⣤⣤⣤⣶⣶⣶⣤⣤⣤⣶⣿⡟⠂⠀⡠⠀⠉⢀⣸⣿⣿⡟⠿⠟⢸⣿⣿⠟⢩⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⣠⠄⠀⣠⣾⡿
⢸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠠⣦⠈⠛⠿⠶⣴⣿⣿⣿⣿⣟⣋⣤⣶⣶⣿⣿⣿⡛⠉⠉⠀⠈⣛⣻⣿⣁⠉⣡⣤⣀⣤⣤⠤⠖⣋⣼⣿⡏⠀⠀⢀⠔⣻⣿⡄⠿⣿⣿⣿⣿⣿⠟⣁⠀⠀⢀⣀⣀⣀⣀⣤⣾⣿⠟⠀
⢸⣿⣿⠏⠉⠁⠀⠀⠀⡀⠀⠀⠻⣷⣤⣀⣀⢿⣿⣿⣿⣿⣿⡿⠛⠋⠉⢀⣾⣿⠃⠀⠀⠀⣠⣿⣿⣿⡇⠀⠹⣿⠟⠀⠀⠀⣾⣿⣿⣿⣿⣄⣠⡏⠀⠀⠺⣿⡄⠘⠿⡟⠛⠁⣼⣿⣿⡿⠿⠿⠿⠛⠋⠉⡉⠤⢤⣤
⢸⣿⡟⠀⠀⠀⠀⠀⠐⣿⣄⠀⠀⠈⠙⠻⣿⣇⣀⣈⣛⢋⣁⢀⣀⣠⣴⡿⢿⣧⠀⠀⠀⣠⡿⠿⢿⣿⡗⠂⢲⣶⠐⠐⠒⠋⢉⣿⣿⠏⣿⣿⣿⣧⣄⣀⣤⣴⣿⣤⣤⣀⠀⠚⠛⠉⠁⠀⠀⠀⠀⠀⡠⠖⠁⠠⣤⣶
⢼⣿⠷⣶⠯⢷⠴⠴⠶⠿⠿⣷⣤⣀⡀⠀⠀⣿⣿⣿⣿⠟⠃⠉⠉⠀⠀⠀⠈⣿⡇⢠⠞⠁⠀⣷⣾⣿⡧⠀⢬⣴⡀⠲⠶⠶⣾⣿⣿⣾⣇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⢸⣯⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⡹⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠟⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣷⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⢸⣿⣟⣿⣿⣿⣷⣾⣿⣿⣷⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢀⡁⣹⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⢋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⠃⠈⠉⠉⠿⠛⢿⣿⣿⡿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠙⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠸⠿⠇⠀⠻⠿⠿⠟⠁⠀⠀⠿⠿⠿⠸⠿⣿⣿⢿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢹⣿⠇⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢻⡿⣿⡿⠋⠀⠀⠀⠀⢿⣧⣶⣆⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣧⡿⠀⠀⠀⠀⠀⠀⢸⠏⠈⠹⣷⣦⣀⡀⣀⣠⣾⣿⣿⠃⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡾⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⢠⡿⠀⠁⢀⠹⣿⣿⣿⣿⣿⣿⣯⠁⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⣀⣀⣶⣿⣷⣄⣠⣶⣼⣿⣿⣿⣿⣿⣿⢿⡄⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠉⠉⠁⠀⢻⣿⣿⠻⠿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⡇⠀⠀⠀⠀⠀⠀
⠋⠉⠙⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⣾⣰⣅⠀⠀⠀⠙⣿⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣻⣿⣿⣿⡿⣿⣿⣿⣿⠿⠟⣷⣼⡇⠀⠀⠀⠀⠀⠀
⡀⠀⠀⠀⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣤⣤⣾⠟⠁⢹⣿⡀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢷⡾⠟⠛⠋⠉⠀⢀⣤⣿⣿⡀⠀⠀⠀⠀⠀⠀
⣷⣶⡞⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⢠⠠⣾⣿⣯⣀⡀⢘⣿⣿⣶⣶⣿⠿⠀⠀⠀⠀⠀⠀⣠⠿⢿⣿⡿⠿⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⣀⣤⣴⣶⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀
⠟⠛⠁⠀⠀⠼⠿⢿⣿⠿⢷⡀⠀⠀⠀⣸⠀⠈⢿⣿⣿⣿⣿⣿⣿⣯⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⣧⡀⠀⠈⠏⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠋⣻⣿⣿⣿⣿⣿⣿⠿⢋⣴⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠄⠰⣄⣠⡅⠀⠀⠀⠙⣆⠀⢀⣈⠉⠋⠛⢛⣻⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⢲⠛⠻⢿⣿⣿⣶⢾⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠻⣿⣿⣛⣡⡤⣿⣷⣆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠸⠛⠿⠛⠻⣿⠁⠀⠀⠀⠀⠹⣷⣾⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠃⢸⣼⠏⠀⠀⠀⠀⢰⣾⣿⣶⣦⣄⣀⡀⠀⠀⣀⣀⣀⣀⠀⠈⠑⠉⠉⠉⠀⣽⣿⠛⢿⣶⣶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡆⠀⠀⠀⠀⠀⠀⠈⠛⣿⠿⠿⠿⣋⣵⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣼⣭⠆⠀⠀⠀⢠⣿⣿⣿⣇⠉⢻⣿⣻⣾⣿⠛⢉⣀⣉⣉⡉⠀⠀⠀⠀⠀⠁⠋⠀⣀⣽⣿⠿⠋
⠀⠀⠀⠀⣴⣾⠿⢿⣷⣾⣦⣄⡀⠀⠀⠀⠀⠀⠱⣤⣉⣽⣿⣿⣟⣂⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣓⠞⠀⠀⠀⢸⣿⣿⣿⣿⣅⠀⣹⣿⣿⣇⣀⣀⣀⣀⣀⡈⢀⣤⣶⣦⣤⣤⣶⡿⠟⠉⠤⠶⣸
⠀⣠⣶⣿⣿⣦⡀⠀⠀⠀⠈⠙⠳⣧⣀⠀⠀⠀⢀⣀⣉⣉⣉⣹⠁⠀⠀⠙⣿⣦⣀⣤⣤⣤⣶⣶⣤⣤⣄⣰⣾⡿⠓⠀⢈⠄⠉⠁⣀⣿⣿⣿⢻⠻⠀⣿⣿⡻⢋⣽⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⢀⡤⠀⢀⣼⣿
⠀⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣶⡄⠙⠻⠷⢄⣾⣿⣿⣿⣿⣛⣠⣤⣶⣾⣿⣿⡟⠛⠉⠁⠀⢙⣛⣻⣏⡈⢉⣤⣄⣠⣤⡤⠴⢚⣡⣿⣿⠁⠀⠀⡠⠛⣿⣇⠸⣿⣿⣿⣿⣿⡿⢋⠀⠀⢀⣀⣀⣀⣀⣠⣴⣿⡿⠃
⠀⣿⣿⡿⠋⠉⠀⠀⠀⢀⠀⠀⠘⢿⣦⣀⡀⢸⣿⣿⣿⣿⣿⡿⠟⠛⠉⠁⣰⣿⡇⠀⠀⠀⢀⣼⣿⣿⣿⠀⠈⢿⡿⠃⠀⠀⢰⣿⣿⣿⣯⣯⣄⣼⠁⠁⠐⢿⣧⠀⠻⡿⠛⠋⢠⣿⣿⣿⠿⠿⠿⠛⠛⠉⢉⠡⠤⣤
⠀⣿⣿⠃⠀⠀⠀⠀⠀⢟⣧⡀⠀⠀⠙⠻⣿⣿⣀⣀⣙⡛⣉⢀⣀⣀⣤⣾⠿⣿⡀⠀⠀⢀⣼⠟⡻⣿⣿⠒⠰⣴⡄⠂⠒⠚⠉⣹⣿⡿⢹⣿⣿⣿⣤⣀⣠⣤⣾⣦⣤⣄⣀⠐⠛⠋⠉⠀⠀⠀⠀⠀⢀⠴⠋⠀⢤⣴
⢠⣿⡿⣶⠏⢿⠅⠰⠶⠾⠿⢿⣦⣄⡀⠀⠀⢻⣿⣿⣿⣿⠙⠉⠉⠁⠈⠀⠀⢹⡷⠀⡴⠋⠀⢸⣷⣿⣿⠤⠠⣥⡄⠐⠶⠶⢶⣿⣿⣷⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⣿⡅⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⠀⠠⡹⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⢸⣿⡟⣿⣯⣿⣷⣾⣿⣿⣷⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢀⣁⣽⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⠋⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⠃⠈⠉⠉⠿⠛⢿⣿⣿⡿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠙⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⠿⠿⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠸⠿⠇⠀⠻⠿⠿⠟⠁⠀⠀⠿⠿⠿⠸⠿⠿⣿⣟⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣿⣿⡟⠁⠀⠀⠀⠈⣿⣦⣶⣆⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢌⣿⡏⠀⠀⠀⠀⠀⠀⣼⠃⠉⢻⣷⣦⣀⡀⣀⣠⣿⣿⡿⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡾⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢇⠀⠀⠀⠀⠀⣰⡏⠀⠂⠀⢹⣿⣿⣿⣿⣿⣿⣭⠁⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣀⣀⣠⣾⣿⣧⣀⣤⣦⣾⣿⣿⣿⣿⣿⣿⢻⡀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠛⠉⠉⠉⢿⣿⡿⠿⢿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⡇⠀⠀⠀⠀⠀
⣿⠋⠉⠛⢿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣀⡞⣬⡅⠀⠀⠈⠻⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣧⣼⠇⠀⠀⠀⠀⠀
⢻⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⣤⣤⣾⠟⠁⢻⣷⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⢿⠾⠟⠛⠉⠁⠀⢀⣤⣿⣿⠀⠀⠀⠀⠀⠀
⣿⣷⣤⡖⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⢠⢠⣽⣿⣇⠀⡀⢸⣿⣷⣶⣶⣿⠿⠀⠀⠀⠀⠀⠀⣰⡿⣿⣿⡿⠿⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⣀⣤⣴⣾⣿⣿⣾⣿⠀⠀⠀⠀⠀⠀
⡿⠿⠛⠁⠀⠀⣼⣿⣿⣿⠿⣷⠀⠀⠀⠀⣼⠀⠙⣿⣿⣿⣿⣿⣿⣿⣯⣿⠃⠀⠀⠀⠀⠀⠀⠀⠁⢀⠈⢿⣧⠀⠀⢘⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⢋⣿⣿⣿⣿⣿⣿⣿⠟⢋⣴⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣀⣠⠃⠀⠀⠀⠻⣇⠀⢈⡘⠙⠛⠛⠛⣻⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⢲⠟⢻⣿⣿⣷⡶⣾⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿⣿⣿⣛⣥⡴⣿⣷⣆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⠙⠿⠟⢿⣿⠁⠀⠀⠀⠀⢻⣷⣶⣿⣾⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠙⠃⣿⣼⠃⠀⠀⠀⠀⢴⣿⣷⣶⣦⣄⣀⠀⠀⠀⣀⣀⣀⣀⠀⠈⠐⠉⠉⠁⠀⣿⣿⠻⣿⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⠿⢋⣥⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣩⠆⠀⠀⠀⢰⣿⣿⣿⡆⠙⢿⣿⣿⣾⡟⠋⢉⣀⣉⣉⡉⠀⠀⠀⠀⠈⠁⠉⠀⣨⣽⣿⠟
⠀⠀⠀⠀⠀⣴⣾⣿⣷⣶⣾⣄⣀⠀⢀⠀⠀⠀⠀⢠⣄⣉⣶⣿⣿⣟⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣒⠞⠀⠀⠀⢹⣿⣿⣿⣿⢁⠀⣻⣿⣿⣅⣀⣀⣀⣀⣀⡈⢀⣤⣶⣤⣤⣴⣶⡿⠛⠉⠤⠖
⠀⠀⣠⣶⣾⣶⣤⠀⠀⠀⠀⠉⠛⠷⣇⡀⠀⠀⠀⢀⣈⢉⣙⢛⣛⠁⠀⠈⠙⣷⣄⣀⣠⣤⣤⣶⣦⣤⣤⣄⣰⣾⡿⠗⠀⣈⠄⠉⠀⣈⣿⣿⣿⢻⠿⠀⣿⣟⡻⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⢀⠄⠀⢀⣼
⠀⢐⣟⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣶⡈⠛⠷⣶⢄⣴⣿⣿⣿⣿⢛⣠⣤⣶⣾⣿⣿⡟⠛⠉⠉⠈⢙⣛⣿⣏⡉⣩⣤⣄⣀⣀⡤⠴⢚⢹⣿⡿⠃⠀⠀⡠⠻⣿⣇⠹⣿⣿⣿⣿⣿⡟⢃⠀⠀⢀⣀⣀⣀⣀⣠⣶⣿⡿
⠀⠀⣿⣿⡿⠛⠋⠀⠀⠀⢠⠀⠀⠹⣷⣤⣀⠀⢸⣿⣿⣿⣿⣿⡟⠟⠛⠋⠉⣸⣿⡇⠀⠀⠀⢀⣾⣿⣿⣿⠀⠘⣿⡟⠃⠀⠀⢸⣿⣿⣿⣷⣧⣄⡼⠀⠁⠘⢿⣆⠈⠿⡿⠛⠃⢠⣿⣿⣿⠿⠿⠿⠛⠛⠉⢉⠥⠤
⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠞⣆⠀⠀⠀⠙⠿⣿⣿⣉⣉⣛⠛⠉⠁⠀⣀⣠⣾⢿⣿⡀⠀⠀⢀⣼⠿⢿⣿⡟⠒⠲⣴⡀⠒⠒⠚⠋⣹⣿⡟⢹⣿⣿⣷⣤⣀⣠⣤⣾⣦⣤⣀⡀⠐⠛⠉⠁⠀⠀⠀⠀⠀⢀⠴⠋⠀⢤
⡆⢶⣿⣿⠿⢿⠝⠰⠶⠶⠞⠻⣷⣦⣀⡀⠀⠈⣽⣿⣿⣿⣯⠈⠁⠉⠉⠉⠁⠀⢻⡇⠀⡴⠋⠀⣶⣶⣿⣿⠄⢤⣭⡄⠀⠶⣶⣶⣿⣿⣷⣯⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃⢀⣿⡅⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⡹⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡆⢸⣷⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⢸⣿⡟⣿⣯⣿⣷⣾⣿⣿⣷⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢀⣁⣽⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⠋⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⠃⠈⠉⠉⠿⠛⢿⣿⣿⡿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠙⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⠿⠿⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠸⠿⠇⠀⠻⠿⠿⠟⠁⠀⠀⠿⠿⠿⠸⠿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⢿⣿⡟⠁⠀⠀⠀⠈⣿⣶⣶⣇⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⣿⡏⠀⠀⠀⠀⠀⠀⣼⠃⠉⠻⣷⣦⣀⡀⣀⣠⣿⣿⡿⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡾⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⣰⡏⠀⠂⠀⢹⣿⣿⣿⣿⣿⣿⣭⠁⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣀⣀⣠⣾⣿⣧⣀⣴⣦⣾⣿⣿⣿⣿⣿⣿⢻⡀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠛⠉⠉⠉⢿⣿⣿⠿⢿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠽⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⡇⠀⠀⠀⠀⠀
⣿⠋⠉⠛⢿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣠⡎⣬⡅⠀⠀⠈⠻⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣧⣼⠇⠀⠀⠀⠀⠀
⢻⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣦⣤⣤⣾⠏⠁⢻⣷⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢶⢿⠾⠟⠛⠉⠁⠀⢀⣤⣿⣿⠀⠀⠀⠀⠀⠀
⣿⣧⣤⡖⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⢠⢠⣽⣿⣇⠀⡀⢸⣿⣷⣶⣶⣿⠿⠀⠀⠀⠀⠀⠀⣰⡿⣿⣿⡿⠿⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⣤⣴⣾⣿⣿⣾⣿⠀⠀⠀⠀⠀⠀
⡿⠿⠛⠁⠀⠀⣼⣿⣿⣿⠿⣷⠀⠀⠀⠀⣼⠀⠙⣿⣿⣿⣿⣿⣿⣿⣯⣿⠃⠀⠀⠀⠀⠀⠀⠀⠁⢀⠈⢿⣧⠀⠀⢙⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠋⣿⣿⣿⣿⣿⣿⣿⠟⢋⣴⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣀⣀⠃⠀⠀⠀⠻⣆⠀⢈⡘⠙⠛⠛⠛⣻⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⢲⠟⢻⣿⣿⣷⡶⣾⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿⣿⣿⣋⣥⡴⣿⣷⣆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠸⠛⠿⠟⢿⣿⠁⠀⠀⠀⠀⢻⣿⣾⣿⣾⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠙⠃⣿⣼⠃⠀⠀⠀⠀⢴⣿⣷⣶⣦⣄⣀⠀⠀⠀⣀⣀⣀⡀⠀⠈⠑⠉⠉⠁⠠⣿⣿⠻⣿⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⠿⢋⣥⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣩⠆⠀⠀⠀⢰⣿⣿⣿⡆⠙⢿⣿⣿⣾⡟⠋⢉⣀⣁⣉⡉⠀⠀⠀⠀⠈⠁⠉⠀⣠⣽⣿⠟
⠀⠀⠀⠀⠀⣴⣶⣿⣷⣶⣾⣄⣀⠀⠀⠀⠀⠀⠀⢥⣌⣉⣶⣿⣿⣟⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣲⠞⠀⠀⠀⢹⣿⣿⣿⣿⢁⠀⣹⣿⣿⣅⣀⣀⣀⣀⣀⡉⢀⣤⣶⣤⣤⣤⣾⡿⠛⠉⠤⠖
⠀⠀⣠⣶⣾⣶⣤⠀⠀⠀⠀⠉⠛⠶⣅⡀⠀⠀⠀⢀⡈⣉⣙⢛⣛⠁⠀⠈⠙⣷⣄⣀⣠⣤⣤⣶⣦⣤⣤⣄⣴⣾⡿⠗⠀⣈⠄⠉⠀⣈⣿⣿⣿⢻⠿⠀⣿⣟⡻⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⢀⠄⠀⢀⣼
⠀⢠⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣶⡈⠛⠷⣶⢄⣾⣿⣿⣿⣿⢛⣠⣤⣶⣾⣿⣿⡟⠛⠉⠉⠈⢙⣛⣿⣏⡉⣩⣤⣄⣀⣀⡤⠴⢚⢹⣿⡿⠃⠀⠀⡠⠻⣿⣇⠹⣿⣿⣿⣿⣿⡟⢃⠀⠀⢀⣀⣀⣀⣀⣠⣶⣿⡿
⠀⠀⣿⣿⡿⠛⠋⠀⠀⠀⢠⠀⠀⠹⣷⣤⣀⠀⢸⣿⣿⣿⣿⣿⡟⠝⠛⠋⠉⣸⣿⡇⠀⠀⠀⢀⣾⣿⣿⣿⠀⠘⣿⡿⠁⠀⠀⢸⣿⣿⣿⣷⣧⣄⡼⠀⠁⠘⢿⣆⠈⠿⡿⠛⠋⢠⣿⣿⣿⠿⠿⠿⠛⠛⠉⢉⠥⠤
⠀⠀⣹⣿⠃⠐⠀⠀⠀⠀⠿⣆⠀⠀⠀⠙⠿⣿⣯⣉⣉⣛⠛⠉⠁⠀⣀⣠⣾⢿⣿⡀⠀⠀⢀⣼⠿⢻⣿⡟⠒⠲⣴⡀⠒⠒⠚⠉⣹⣿⡟⢹⢿⣿⣷⣤⣀⣠⣤⣿⣦⣤⣀⡀⠐⠛⠉⠁⠀⠀⠀⠀⠀⢀⠴⠋⠀⣤
⡆⢶⣿⣿⠿⢷⠝⠰⠶⠶⠞⠻⣷⣦⣀⠀⠀⠈⣽⣿⣿⣿⣯⠛⠉⠉⠉⠉⠁⠀⢻⡇⠀⡴⠋⠀⣶⣾⣿⣿⠄⢤⣭⡄⠀⠶⣶⣶⣿⣿⣷⣯⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃⢀⣿⡅⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⡹⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡆⢸⣷⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⣶⢸⣿⡇⣿⣿⢸⣿⡟⣿⣏⣿⣷⣾⣿⣿⣷⣶⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡅⢀⡁⣽⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⠋⠸⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡸⠃⠈⠉⠉⠿⠛⢿⣿⣿⡿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠙⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⡏⢩⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠿⠿⠿⣿⣿⣿⠀⠀⠀⠀⠀
⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣶⣶⣤⣀⠀⠀⠸⠿⠇⠀⠻⠿⠿⠟⠁⠀⠀⠿⠿⠿⠸⠿⠇⢸⣿⡿⣿⣿⣿⣿⠿⠟⠿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⡟⣿⣿⠟⠁⠀⠀⠀⠘⣿⡶⣶⣆⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣼⡇⠀⠀⠀⠀⠀⠀⣼⠉⠉⢻⣷⣦⣀⣀⣀⣴⣿⣿⡿⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⢇⠀⠀⠀⠀⠀⣰⡇⠀⠂⡀⢻⣿⣿⣿⣿⣿⣿⣭⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣦⣀⣀⣠⣾⣿⣧⣄⣴⣦⣾⣿⣿⣿⣿⣿⣿⢻⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠛⠋⠉⠙⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠾⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⡇⠀⠀⠀⠀
⣾⣿⠟⠛⠻⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡏⣭⡁⠀⠀⠉⠻⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣿⣿⣿⣿⣿⣻⣿⣿⡿⠻⠋⣷⣼⠇⠀⠀⠀⠀
⠛⢻⠀⠀⠀⠈⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣄⣀⣤⣾⠟⠁⢿⣷⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠮⢷⠾⠟⠛⠉⠁⠀⣀⣤⣿⣿⠀⠀⠀⠀⠀
⠀⣼⣧⣤⡴⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⢠⢠⣿⣿⣇⠀⠀⢸⣿⣶⣶⣶⣿⡿⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠿⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⣤⣴⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀
⣿⣿⡿⠟⠁⠀⠀⣼⣿⣿⣿⣿⣧⠀⠀⠀⠀⡼⠈⠙⣿⣿⣿⣷⣿⣿⣿⣋⣿⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⢿⣇⠀⠀⢙⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⢏⣿⣿⣿⣿⣿⣿⣿⠟⢋⡴⠀⠀⠀⠀
⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⢡⡀⢈⠃⠀⠀⠀⢻⡆⠀⠈⠙⠛⠛⠛⠛⣻⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢰⠗⢿⣿⣿⣷⡶⣾⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿⣿⢿⣋⣥⠴⣿⣷⣆⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠠⠞⡷⠾⢿⣿⠁⠀⠀⠀⠀⢻⣧⣾⣿⣾⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠙⠃⣿⣽⠃⠀⠀⠀⠀⢴⣿⣷⣶⣦⣄⣀⠀⠀⠀⣀⣀⣀⣀⠀⠈⠈⠉⠉⠁⠀⣽⣿⠻⢿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⡿⢛⣥⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣩⠆⠀⠀⠀⢰⣿⣿⣿⡆⠙⢿⣿⣿⣾⡟⠋⢉⣀⣁⣉⡉⠀⠀⠀⠀⠀⠁⠉⢀⣠⣽⣿
⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣅⡀⠀⠀⠀⠀⠀⠀⢠⣍⡑⣶⣾⣿⣟⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢓⠞⠀⠀⠀⢸⣿⣿⣿⣿⢁⠀⣹⣿⣿⣅⣀⣀⣀⣀⣀⡈⢀⣤⣶⣦⣤⣴⣾⠿⠛⠉⠤
⣷⡀⠀⢀⣴⣶⣶⣄⠀⠀⠀⠈⠙⠻⢧⣆⠀⠀⠀⠀⢀⡉⠙⠛⠛⣛⠁⠀⠈⠻⣷⣄⣀⣀⣤⣤⣴⣤⣤⣄⣄⣰⣶⡿⠗⠀⣉⠀⠉⠀⣈⣿⣿⣿⡻⠿⠀⣿⣟⡿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠀⢀⠄⠀⢀
⣿⠃⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣤⡉⠛⢷⣶⣄⣼⣿⣿⣿⣿⠛⣀⣤⣶⣶⣿⡿⡟⠛⠋⠉⠉⢙⣟⣻⣋⣉⣉⣤⣄⣀⣀⡤⠴⢚⢩⣿⣿⠁⠀⠀⡠⠻⣿⣇⠸⣿⣿⣿⣿⣿⡟⢃⠀⠀⣀⣀⣀⣀⣀⣤⣶⣿
⣿⠀⠀⣿⣿⣿⠛⠋⠀⠀⠀⢀⠀⠀⠹⣷⣄⡀⠀⢹⣿⣿⣿⣿⣿⣟⠻⠟⠋⠉⢹⣿⡇⠀⠀⠀⢀⣾⣿⣿⣿⠀⠘⣿⡿⠀⠀⠀⢸⣿⣿⣿⣿⣇⣄⣼⠁⠁⠘⢿⣇⠀⠿⢿⠛⠋⢠⣿⣿⣿⠿⠿⠿⠛⠛⠉⢉⠡
⣿⠀⠀⣽⣿⠇⠀⠀⠀⠀⠀⢾⣆⠀⠀⠀⠛⠿⣿⣿⣉⣉⡛⠛⠋⠁⠀⢀⣠⣴⣿⣿⡀⠀⠀⢀⣼⡿⢻⣿⣿⠒⢤⣤⡄⠂⠒⠚⠋⣹⣿⡟⢹⣿⣿⣿⣤⢀⣠⣤⣾⣦⣤⣀⣀⠀⠛⢋⠁⠀⠀⠀⠀⠀⢀⠴⠋⠀
⣿⡿⠶⣿⡿⢶⠖⠸⠶⠶⠶⠀⠻⣷⣤⣀⠀⠀⠈⣽⣿⣿⣿⣿⠛⠛⠛⠉⠉⠁⠀⢹⡇⠀⡴⠋⠁⣴⣼⣿⣿⠄⠠⣭⡄⠠⠶⣶⢶⣿⣿⣷⣿⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⠀⣿⡅⠀⠀⠀⠀⠀⢀⣠⣴⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣷⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⠀⠠⡹⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠟⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡆⢸⣷⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⢸⣿⡟⣿⣏⣿⣷⣾⣿⣿⣿⣶⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡅⢀⣁⣹⣿⣿⣿⣿⣿⣿⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⠋⠸⢿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⠃⠈⠉⠉⠿⠛⢿⣿⣿⡿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⡿⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠙⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⡏⠉⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠿⠿⢿⣿⣿⡇⠀⠀⠀⠀
⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣶⣶⣦⣄⡀⠀⠸⠿⠇⠀⠻⠿⠿⠟⠁⠀⠀⠿⠿⠿⠸⠿⠇⠀⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢿⣿⡿⠋⠀⠀⠀⠀⢻⣶⢶⣷⡀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣿⣿⠁⠀⠀⠀⠀⠀⢸⡏⠈⠙⣿⣶⣄⣀⣀⣠⣾⣿⣿⠇⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⡀⠀⠀⠀⠀⢀⣿⠀⠐⢀⠈⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣆⣀⣀⣴⣿⣿⣦⣠⣶⣶⣿⣿⣿⣿⣿⣿⡟⣇⠀⠀⠀⠀
⢿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠛⠛⠉⠉⢻⣿⣿⡿⢿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⠀⠀⠀⠀
⣠⣿⡿⠛⠛⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣨⢍⣍⠀⠀⠈⠙⢿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠟⢹⣦⡿⠀⠀⠀⠀
⠛⠛⡇⠀⠀⠀⢹⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⣀⣀⣴⡾⠋⠸⣿⡆⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⠶⠿⠟⠛⠋⠉⠀⢀⣠⣾⣿⡇⠀⠀⠀⠀
⠀⢠⣿⣤⣤⠆⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡀⣽⣿⣿⠀⠀⠀⣿⣷⣶⣶⣶⣿⠇⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠿⠿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣄⣄⣀⣠⣴⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀
⣿⣿⣿⠿⠟⠀⠀⢠⣾⣿⣿⣿⣿⡄⠀⠀⠀⢀⡇⠙⢻⣿⣿⣿⣾⣿⣿⣟⣹⡟⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠻⣿⡀⠀⠈⡻⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡙⣹⣿⣿⣿⣿⣿⣿⡿⠋⣡⠆⠀⠀⠀
⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠉⣅⠀⡙⠀⠀⠀⠘⢷⠀⠀⠉⠛⠛⠛⠻⢛⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡾⠺⣿⣿⣿⣶⣶⣇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠿⠿⣟⣡⡤⢾⣿⣾⡄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⡖⢻⠶⢾⣿⡟⠀⠀⠀⠀⠘⣷⣶⣾⣷⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠙⠛⢸⣯⡟⠀⠀⠀⠀⠀⣾⣿⣶⣶⣤⣀⡀⠀⠀⢀⣀⣀⣀⡀⠀⡉⠉⠉⠉⠀⢸⣿⡟⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⠿⣋⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡵⠀⠀⠀⠀⣴⣿⣿⣿⠈⠙⣿⣿⣷⣿⠟⠉⣁⣀⣈⣉⠁⠀⠀⠀⠀⠁⠈⠁⣀⣤⣾
⣦⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣴⣮⡀⠀⠀⠀⠀⠀⠀⠀⣬⣉⢲⣶⣿⣿⣃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠛⡲⠃⠀⠀⠀⣿⣿⣿⣿⣇⡀⢈⢿⣿⣯⣀⣀⣀⣀⣀⣀⡁⣠⣶⣶⣦⣤⣶⡿⠟⠋⠁
⣸⣆⠀⠀⣠⣶⣶⣮⡀⠀⠀⠈⠉⠛⠿⣶⡁⠀⠀⠀⠀⣈⠉⠛⠛⢛⡋⠀⠀⠙⢿⣦⣀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣶⣾⡿⠆⢀⡁⠈⠁⢀⣸⣿⣿⣟⡿⠇⢸⣿⣻⠟⣩⣿⣿⣿⣿⣿⣿⡿⣿⠿⠟⠛⠁⠀⡠⠆⠀
⣿⡟⠀⢐⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⣌⠙⠳⣶⣦⣠⣿⣿⣿⣿⡟⢃⣠⣤⣶⣾⣿⣿⠛⠛⠉⠉⠉⣛⣿⣿⡉⣉⣩⣤⣀⣀⣠⡤⠖⡋⣽⣿⡏⠀⠀⢀⠜⣻⣿⡀⠿⣿⣿⣿⣿⣿⠛⣁⠀⣀⣀⣀⣀⣀⣠⣴⣾
⣿⡇⠀⢸⣿⣿⡟⠛⠃⠀⠀⠀⣐⠀⠈⢿⣦⣀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠉⣿⣿⠀⠀⠀⠀⣠⣿⣿⣿⡇⠀⢻⣿⠟⠀⠀⠀⣿⣿⣿⣿⣿⣥⣠⡏⠀⠀⠺⣿⡄⠘⠿⡟⠛⠁⣼⣿⣿⡿⠿⠿⠟⠛⠋⠁⣉
⣿⡇⠀⢸⣿⣿⠁⠀⠀⠀⠀⠐⣷⡀⠀⠀⠘⠻⢿⣿⣎⣉⣙⠻⠛⠉⠀⠀⣀⣤⣾⣿⣏⠀⠀⠀⣀⣿⠿⣿⣿⡗⠲⢤⣤⠐⠒⠒⠛⢉⣿⣿⠋⣿⣿⣿⣧⣄⣀⣤⣴⣿⣤⣤⣀⡀⠘⠛⣉⠀⠀⠀⠀⠀⢀⠠⠞⠁
⣿⣿⠷⣾⣿⢷⠖⠲⠶⠶⠶⠀⠘⢿⣦⣄⡀⠀⠀⢉⣿⣿⣿⣿⡽⠛⠛⠋⠉⠉⠁⠈⣿⡆⢀⡞⠉⠀⣤⣾⣿⡇⠀⢬⣥⠀⠴⢶⣶⣾⣿⣿⣾⣇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠁⢸⣯⠀⠀⠀⠀⠀⠀⣀⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣷⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⡹⣿⣿⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠻⠟⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣷⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡇⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⣷⣼⣿⣿⣿⣷⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢀⣁⣽⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⠋⠸⠿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⠃⠈⠉⠉⠿⠛⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⡿⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠙⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⡏⠉⢧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿⠿⠿⣿⣿⣿⠀⠀⠀⠀
⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣷⣶⣦⣤⡀⠀⠸⠿⠇⠀⠻⠿⠿⠟⠁⠀⠀⠿⠿⠿⠸⠿⠇⠀⢸⣿⡿⣿⣿⣿⣿⠿⠛⠿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⣷⣿⣿⠟⠁⠀⠀⠀⠘⣿⣾⣾⡄⠀⠀⠀⠀⠀⠀⣼⣷⣄⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⣾⠃⠀⠀⠀⠀⠀⠀⡿⠁⠉⢿⣷⣦⣀⣀⣠⣴⣿⣿⡟⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⣼⡇⠈⠀⡄⢻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⣀⣠⣾⣿⣦⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⢻⠀⠀⠀⠀
⠿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠛⠋⠉⠙⣿⣿⡿⢿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠾⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢼⡇⠀⠀⠀
⢀⣿⣿⠟⠛⠿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⢠⡍⣭⡁⠀⠀⠙⢿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿⣿⣿⣿⣿⣿⣿⡿⡿⠻⠩⣷⣼⠃⠀⠀⠀
⠟⠛⢯⠀⠀⠀⠈⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⣀⣤⣾⠏⠉⣿⣇⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠶⠿⠿⠛⠛⠉⠁⠀⣀⣼⣿⣿⠀⠀⠀⠀
⠀⠀⣿⣦⣀⡤⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢀⢉⣿⣿⡇⠀⠀⢻⣿⣶⣶⣶⣿⡏⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⠿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣠⣀⣀⣤⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀
⣿⣾⣿⣿⠿⠁⠀⠀⣼⣿⣿⣿⣿⣦⠀⠀⠀⠀⡾⠈⠻⣿⣿⣿⣷⣿⣿⣿⣛⣿⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⢿⣄⠀⠀⢹⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣉⣿⣿⣿⣿⣿⣿⡿⠟⣡⡶⠀⠀⠀
⠻⠿⠁⠀⠀⠀⠀⠀⠁⠀⡈⢥⡀⢉⠂⠀⠀⠀⢿⡆⠀⠉⠙⠛⠻⠛⠟⣻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢴⢗⢿⣿⣿⣷⣴⣿⡠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠻⡿⢿⣋⣤⠴⣿⣷⣆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠲⢾⡶⠾⣿⣿⠀⠀⠀⠀⠀⢻⣶⣷⣿⣾⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠒⠋⠛⠃⣿⣽⠃⠀⠀⠀⠀⣴⣿⣷⣶⣦⣀⣀⠀⠀⢀⣀⣀⣀⣀⠀⢈⠉⠉⠉⠁⢀⣿⣿⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡏⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⡿⢛⣥⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣡⠆⠀⠀⠀⣰⣿⣿⣿⡆⠙⢿⣿⣿⣾⡟⠋⢉⣀⣁⣉⡉⠀⠀⠀⠀⠈⠀⠉⢀⣠⣴
⣷⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣴⣁⠀⠀⠀⠀⠀⠀⠀⣠⣅⢐⣶⣾⣿⣗⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣳⠞⡀⠀⠀⢹⣿⣿⣿⣿⢁⠀⣻⣿⣿⣄⣀⣀⣀⣀⣀⡈⢀⣴⣶⣶⣤⣴⣾⠿⠋⠉
⣀⣦⠀⠀⢀⣤⣶⣯⣍⠀⠉⠉⠉⠛⠻⣷⣂⠀⠀⠀⠀⠈⠉⠛⠛⠛⡛⠁⠀⠈⠻⣷⣄⣀⣀⣤⣤⣤⣤⣤⣄⣀⣰⣾⡿⠗⠀⣉⠀⠉⢀⣈⣿⣿⣿⡿⠿⠀⣿⣿⡿⢩⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠠⠔⠀
⣾⣿⠀⠀⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣤⡙⠻⣶⣶⣀⣼⣿⣿⣿⣿⠛⣀⣤⣴⣶⣿⡿⠟⠛⠋⠉⠉⢹⣻⣿⣋⢉⣉⣤⣄⣀⣀⣤⠴⢊⢹⣿⡿⠁⠀⠀⡠⠻⣿⣇⠹⣿⣿⣿⣿⣿⠟⢃⡀⣀⣀⣀⣀⣀⣀⣤⣶
⣿⡟⠀⠀⣿⣿⣿⠛⠛⠀⠀⠀⢀⠀⠀⠹⣷⣄⡀⠀⢹⣿⣿⣿⣿⣿⣿⠻⠟⠛⠉⣹⣿⡇⠀⠀⠀⢀⣾⣿⣿⣿⠀⠘⣿⡿⠂⠀⠀⢸⣿⣿⣿⣷⣧⣀⡾⠀⠁⠘⢿⣇⠀⠿⠟⠛⠉⢠⣿⣿⡿⠿⠿⠟⠛⠛⠉⢉
⣿⣿⠀⠀⣿⣿⠇⠄⠀⠀⠀⠀⣶⣆⠀⠀⠈⠻⢿⣿⣿⢉⣉⠻⠟⠋⠁⠀⢀⣠⣴⣿⣿⠀⠀⠀⢀⣼⡿⢻⣿⡿⠒⢢⣬⡀⠒⠒⠚⠋⣹⣿⡟⢹⣿⣿⣷⣤⢀⣠⣴⣾⣦⣤⣄⣀⠐⠛⢉⠁⠀⠀⠀⠀⠀⣀⠴⠋
⢿⣿⡿⣶⣿⣿⠴⠲⠶⠶⠶⠀⠀⢻⣷⣄⡀⠀⠀⠉⢽⣿⣿⣿⣿⠏⠉⠙⠉⠉⠁⠀⢻⡇⠀⡰⠋⠁⣴⣾⣿⣿⠀⢠⣭⡄⠠⠶⣶⣶⣿⣿⣷⣿⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⣿⡅⠀⠀⠀⠀⠀⣀⣠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⠀⢀⡹⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡆⢸⣷⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡇⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⣧⣼⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢀⣁⣹⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⠋⠸⠿⢇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡸⠃⠈⠉⠉⠿⠛⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⡿⢿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠙⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣧⣼⣿⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⡏⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠻⠿⣿⣿⡇⠀⠀⠀
⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣷⣶⣶⣤⣄⠀⠸⠿⠇⠀⠻⠿⠿⠟⠁⠀⠀⠿⠿⠿⠸⠿⠇⠀⠀⣿⣿⣿⣿⣿⣿⡿⠟⠛⢿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⡾⣿⡿⠋⠀⠀⠀⠀⢻⣾⣿⣧⠀⠀⠀⠀⠀⠀⢠⣿⣦⡀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣷⡟⠀⠀⠀⠀⠀⠀⣸⠋⠈⠹⣿⣶⣤⣀⣀⣴⣾⣿⣿⠃⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⢠⡿⠀⠁⢠⠘⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣆⣀⣠⣼⣿⣷⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⡟⡆⠀⠀⠀
⠿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠛⠛⠋⠉⢻⣿⣿⡿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿⠀⠀⠀
⠀⢸⣿⡿⠛⠻⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⣨⢉⡋⠀⠀⠈⠻⣿⡯⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣿⣻⣿⣿⠿⠟⠏⢽⣶⡟⠀⠀⠀
⡿⠛⠿⡇⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣀⣴⡾⠉⢸⣿⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠾⠟⠟⠛⠋⠉⠀⣀⣠⣾⣿⡇⠀⠀⠀
⠀⠀⢸⣧⣀⣠⠄⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⡈⣹⣿⣿⠀⠀⠈⣿⣷⣶⣴⣾⣿⠇⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠿⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣄⣀⣴⣶⣿⣿⣿⣿⣿⡇⠀⠀⠀
⣿⣷⣾⣿⡿⠏⠀⠀⢠⣿⣿⣿⣿⣷⡀⠀⠀⠀⢰⠁⠙⣿⣿⣿⣿⣾⣿⣿⣟⣹⡟⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢻⣧⡀⠀⠈⡟⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡙⣹⣿⣿⣿⣿⣿⣿⠟⢋⣴⠆⠀⠀
⠿⠿⠟⠀⠀⠀⠀⠀⠈⠁⢈⠩⡁⠈⡱⠀⠀⠀⠘⡦⠀⠈⠉⠛⠛⠿⠻⢟⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠠⠾⣺⣿⣿⣷⣶⣾⣇⢼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⡿⠿⣛⣥⡴⢾⣿⣿⡄⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢶⡶⢿⣿⡇⠀⠀⠀⠀⠘⣷⣶⣾⣷⣶⣷⣾⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠚⠙⠛⢸⣯⡟⠀⠀⠀⠀⢠⣾⣿⣶⣶⣤⣀⡀⠀⠀⣀⣀⣀⣀⣀⢀⡈⠉⠉⠉⠀⢸⣿⡏⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⠿⣋⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡴⠀⠀⠀⢀⣾⣿⣿⣷⠈⠻⣿⣿⣷⣿⠋⠉⣀⣠⣀⣉⡁⠀⠀⠀⠀⠀⠈⠁⣀⣤
⣶⡆⠀⠀⠀⠀⠀⢀⣠⣶⣶⣦⣤⣎⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⢲⣶⣿⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢛⡶⢃⠀⠀⠈⣿⣿⣿⣿⡏⡀⢈⣿⣿⣧⣀⣀⣀⣀⣀⣀⠁⣠⣶⣶⣶⣴⣶⡿⠛⠉
⣁⣰⡄⠀⠀⣠⣤⣾⣭⡉⠉⠉⠉⠙⠻⢿⣶⠁⠀⠀⠀⠀⠉⠙⠛⠛⢛⠋⠀⠈⠙⢿⣦⣀⣀⣀⣤⣤⣤⣤⣤⣀⣀⣶⣾⠿⠆⢈⡁⠈⠁⣀⣹⣿⣿⠟⠿⠇⢸⣿⣻⠟⣩⣿⣿⣿⣿⣿⣿⡿⡿⠿⠛⠋⠁⠀⠤⠂
⣾⣿⡟⠀⢰⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢠⣌⠛⢶⣶⣄⣠⣿⣿⣿⣿⡿⢃⣠⣤⣶⣾⣿⠿⠛⠛⠉⠉⠙⣻⣿⣛⡉⣉⣡⣤⣀⣀⣠⡤⠖⣋⣿⣿⡏⠀⠀⢀⠜⢻⣿⡀⠿⣿⣿⣿⣿⣿⠋⣄⣀⣀⣀⣀⣀⣠⣤⣴
⣿⣿⡇⠀⢸⣿⣿⡿⠟⠃⠀⠀⠀⠐⠀⠈⢿⣦⡀⠈⠈⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⣿⣿⠀⠀⠀⠀⣰⣿⣿⣿⡇⠀⢻⣿⠗⠀⠀⠀⣿⣿⣿⣿⣿⣤⣠⡇⠈⠀⠻⣿⡄⠘⠿⠟⠋⢁⣾⣿⣿⠿⠿⠻⠛⠛⠉⠀
⢿⣿⡇⠀⢸⣿⣿⠁⠄⠀⠀⠀⢠⣷⡀⠀⠀⠙⠿⣿⣿⡏⢉⠙⠿⠛⠋⠀⠀⣀⣠⣾⣿⣏⠀⠀⠀⣀⣿⣟⣿⣿⠗⠲⣤⣥⠐⠒⠒⠛⢉⣿⣿⠋⣿⣿⣿⣧⣄⣀⣤⣶⣷⣤⣤⣀⣀⠘⢋⣁⠀⠀⠀⠀⠀⠀⠠⠞
⠸⣿⣿⣷⢿⣿⡧⠴⠶⠶⠶⠀⠀⠘⣿⣦⣀⠀⠀⠈⢹⣿⣿⣿⣿⡿⠛⠛⠋⠉⠉⠀⠈⣿⠄⢀⡞⠉⢀⣶⣿⣿⡇⠀⣬⣥⠀⠴⢶⣶⣾⣿⣿⣾⣇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠁⢸⣿⠀⠀⠀⠀⠀⢀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⠀⢀⡹⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡆⢸⣷⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡅⢀⣁⣹⣿⣿⣿⣿⣿⣿⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⠋⠸⠿⠇⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡸⠃⠈⠉⠉⠿⠛⢿⣿⣿⡿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⡿⢿⣷⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠙⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣧⣼⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⡏⠉⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠿⣿⣿⠃⠀⠀
⣶⣶⣶⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣷⣶⣶⣦⣤⣀⠸⠿⠇⠀⠻⠿⠿⠟⠁⠀⠀⠿⠿⠿⠸⠿⠇⠀⠀⠘⣿⣿⣿⣿⣿⣿⠿⠛⠛⢿⣿⡿⠿⠇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣾⣿⡟⠁⠀⠀⠀⠈⢿⣖⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⣦⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣿⡏⠀⠀⠀⠀⠀⠀⣸⠋⠈⠹⣿⣶⣤⣀⣠⣴⣿⣿⡿⠁⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠀⠀⢠⡏⠀⠁⠠⠘⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⣀⣠⣼⣿⣷⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⡟⡆⠀⠀
⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠟⠛⠛⠛⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⠀⠀
⠀⠀⢸⣿⣿⠿⠿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣬⢉⡋⠀⠀⠉⠻⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⣿⣿⡿⣿⣻⣿⣿⠿⠟⠋⣵⣾⡟⠀⠀
⣶⣶⠿⢿⡅⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⣀⣴⡾⠋⢻⣿⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠾⠟⠛⠛⠉⠁⠀⣀⣴⣿⣿⡇⠀⠀
⡏⠀⠀⢸⣧⣀⣠⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣹⣿⣿⠀⠀⠘⣿⣷⣦⣤⣾⣿⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠿⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣄⣤⣴⣴⣿⣿⣿⣿⣿⡇⠀⠀
⣿⣿⣶⣾⣿⣿⠇⠀⠀⢠⣾⣿⣿⣿⣷⡀⠀⠀⠀⢰⡆⠙⢿⣿⣿⣷⣾⣿⣿⣟⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢻⣧⡀⠀⠈⡟⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⣽⣿⣿⣿⣿⣿⣿⠟⢋⣴⠃⠀
⠻⡿⠿⠟⠁⠀⠀⠀⠀⠈⠉⠉⠩⡉⠉⡣⠀⠀⠀⠸⣤⠀⠈⠙⠛⠟⠿⠿⢟⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠠⡾⢺⣿⣿⣷⣶⢾⣇⢼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⡿⢿⣫⣥⠴⢻⣿⣿⡄
⡨⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⣶⣦⣿⣾⡇⠀⠀⠀⠀⠹⣶⣾⣾⣶⣶⣶⣶⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠚⠙⠋⢸⣯⡟⠀⠀⠀⠀⢠⣾⡿⣶⣶⣤⣀⡀⠀⠀⣀⣀⣀⣀⣀⣀⡈⠉⠉⠁⠀⢸⣿⡟⠛
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⠿⣋⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡔⠀⠀⠀⢀⣾⣿⣿⣷⠈⠻⣿⣾⣷⣿⠋⠉⣀⡠⢄⣈⡁⠀⠀⠀⠀⠀⠈⠁⣀
⣤⣶⣆⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣞⠀⠀⠀⠀⠀⠀⠀⠀⣈⠙⢲⣶⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢓⡾⢁⠀⠀⠘⣿⣿⣿⣿⡏⡀⢈⣿⣿⣧⣠⣤⣀⣀⣀⣀⡀⣠⣶⣶⣶⣴⣾⡿⠛
⠻⣉⢹⡄⠀⠀⢀⣤⣼⣯⡉⠉⠉⠉⠛⠻⢷⣴⠁⠀⠀⠀⠀⠉⠙⠛⠛⢻⠋⠁⠉⠙⢷⣦⢀⣀⣀⣤⣤⣤⣤⣄⣀⣠⣶⣾⠿⠆⢈⡁⠈⠁⣀⣹⣿⣿⠟⠿⠇⢸⣿⣻⠏⣩⣿⣿⣿⣿⣿⣿⡿⡿⠿⠛⠋⠁⠀⠤
⢴⣿⣿⡿⠀⢠⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢠⣍⠻⢶⣦⣄⣠⣿⣿⣿⣿⡿⢆⣀⣤⣴⣾⣿⡿⠟⠛⠉⠉⠙⣻⣿⣟⡉⣉⣡⣄⣀⣀⣠⡤⠖⠉⣿⣿⠋⠀⠀⢀⠜⢻⣿⡀⠻⣿⣿⣿⣿⣿⠋⣄⣀⣀⣀⣀⣀⣠⣤
⠀⢻⣿⡇⠀⢸⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠈⢿⣦⡀⠈⠁⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⣿⣿⠀⠀⠀⠀⣰⣿⣿⣿⡇⠀⢻⣿⠟⠀⠀⠀⣿⣿⣿⣿⣾⣀⣠⡇⠈⠀⠺⣿⡄⠘⠿⠟⠋⠁⣼⣿⣿⠿⠟⠛⠛⠛⠉
⣶⣿⣿⡇⠀⢸⣿⣿⠃⠤⠀⠀⠀⠠⣿⡀⠀⠈⠙⠿⣿⣶⡏⠉⠛⠿⠟⠋⠀⠀⢀⣠⣾⣿⡏⠀⠀⠀⣀⣿⡟⣿⣿⠗⠲⣤⣅⠐⠒⠒⠛⢉⣿⣿⠃⣿⣿⣿⣧⣄⣀⣤⣶⣷⣦⣤⣀⣀⠈⢋⣁⠀⠀⠀⠀⠀⠀⠠
⠑⠶⣿⣿⡿⢾⣿⡿⠶⠶⠶⠀⠀⠀⠙⣿⣦⣀⠀⠀⠈⢙⣿⣿⣿⣿⡾⠖⠚⠛⠉⠉⠁⠘⣿⠄⢀⡞⠉⢀⣦⣼⣿⡇⠀⣨⣥⠀⠴⢶⣶⣾⣿⣿⣾⣇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠁⢸⣿⡀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣠⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⠀⠠⡹⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡆⢸⣷⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡇⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⣧⣸⣿⣿⣷⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡅⢀⣁⣹⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⠋⠸⠿⠇⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡸⠃⠈⠉⠉⠿⠛⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⡿⢿⣷⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠙⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣧⣼⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⡏⠉⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠻⢿⣿⡇⠀⠀
⣶⣶⣶⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣶⣶⣦⣤⣀⠸⠿⠇⠀⠻⠿⠿⠟⠁⠀⠀⠿⠿⠿⠸⠿⠇⠀⠀⠈⢿⣿⣿⣿⣿⣿⡿⠟⠛⠻⣿⣿⠿⠿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢳⣿⡿⠋⠀⠀⠀⠀⠹⣷⢻⣿⡀⠀⠀⠀⠀⠀⠀⣿⣷⡄⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣼⣿⠁⠀⠀⠀⠀⠀⢠⡏⠉⠉⢿⣷⣦⣄⣀⣤⣾⣿⣿⠇⠀
⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⡄⠀⠀⠀⠀⢀⣾⠁⠈⠀⠄⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⣀⣤⣿⣿⣦⣤⣴⣶⣿⣿⣿⣿⣿⣿⡿⢳⠀⠀
⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠿⠟⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣾⡆⠀
⠀⠀⠀⣿⣿⡿⠿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡍⣙⠁⠀⠈⠙⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⣿⣿⣿⢿⣿⣿⣿⡿⠿⠛⢩⣶⣿⠂⠀
⣤⣶⡾⠿⣏⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⢀⣀⣶⠟⠙⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠿⠛⠛⠋⠉⠀⢀⣠⣾⣿⣿⠀⠀
⣿⠁⠀⠀⣿⣀⣀⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣿⣿⡃⠀⠀⢿⣿⣦⣤⣴⣿⡇⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⠿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣠⣠⣤⣶⣾⣿⣿⣿⣿⣿⠀⠀
⣿⣷⣶⣶⣿⣿⣿⠁⠀⠀⣰⣿⣿⣿⣷⣄⠀⠀⠀⠀⡄⠙⠻⣿⣿⣾⣶⣿⣿⣿⣏⣿⠃⠀⠀⠀⠀⠀⠀⠈⠁⠀⠹⣿⣄⠀⠀⢹⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣨⣿⣿⣿⣿⣿⣿⡿⠛⣡⡞⠀
⠛⢿⠿⡿⠋⠁⠀⠀⠀⠀⠉⠉⠙⣋⠉⠙⠄⠀⠀⠀⢧⡄⠈⠛⠻⠻⠿⠿⠿⣯⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⢤⠷⡷⣿⣿⣦⡴⣯⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠛⠻⠿⣟⣭⣤⠞⣿⣿⣧
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⢴⣧⣼⣷⣾⠀⠀⠀⠀⠈⣵⣤⣧⣶⣶⣶⣶⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠒⠛⠛⠁⣿⣽⠃⠀⠀⠀⠀⣴⣿⣷⣶⣦⣀⣀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠉⠉⠁⠀⠀⡿⣟⠙
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⠟⣡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣡⠆⠀⠀⠀⣰⣿⣿⣿⡆⠉⢿⣿⣿⣾⡟⠉⢁⣠⠤⣀⣉⠀⠀⠀⠀⠀⠀⠁⣀
⣠⣴⣶⡀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣴⠃⠀⠀⠀⠀⠀⠀⠀⢀⡘⠛⣵⣶⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣾⡿⢳⠎⡀⠀⠀⢻⣿⣿⣿⣿⢁⠀⡹⣿⣿⣤⣤⣀⣀⣀⣀⡀⢠⣴⣶⣶⣦⣶⡿⠟
⠟⢟⠉⣧⠀⠀⠀⣀⣤⣽⡉⠉⠉⠉⠛⠻⢷⣤⡂⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⠉⠉⠙⠻⣶⡄⢀⣀⣀⣤⣤⣤⣄⣀⡀⣰⣶⡿⠧⠀⣈⠀⠁⢀⣈⣿⣿⡿⠻⠿⠀⣿⣿⡿⢩⣽⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠀⠠
⠤⣮⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣌⠙⠶⣦⣤⡀⣼⣿⣿⣿⣿⠷⢀⣠⣤⣶⣿⣿⠿⠛⠋⠉⠉⢛⣿⣿⣏⢉⣋⣤⣀⣀⣀⣠⠴⠚⢹⣿⡿⠀⠀⠀⡠⠛⣿⣇⠸⢿⣿⣿⣿⣿⠟⢡⣀⣀⣀⣀⣀⣀⣤
⠀⠘⣿⣿⠀⠀⢞⣿⣿⡿⠟⠁⠀⠀⠀⡀⠀⢻⣷⣀⠀⠁⢸⣿⣿⣿⣿⣿⡜⡿⠟⠛⠋⢹⣿⡇⠀⠀⠀⢀⣾⣿⣿⣿⠀⠘⣿⡿⠁⠀⠀⢸⣿⣿⣿⣷⣇⣄⣼⠀⠀⠐⢿⣧⠀⠻⠿⠙⠁⢠⣿⣿⡿⠿⠛⠛⠛⠋
⣶⣾⣿⣿⠀⠀⢹⣿⡟⠀⠄⠀⠀⠀⣀⣄⠀⠀⠉⠻⢿⣶⣾⠙⠛⠻⠿⠛⠁⠀⠀⣀⣴⣿⣿⠁⠀⠀⠀⣸⣿⣿⣿⡿⠆⢤⣬⡀⠒⠒⠒⠋⣹⣿⡟⢹⣿⣿⣿⣤⣀⣠⣴⣾⣧⣤⣄⣀⡀⠙⣉⡀⠀⠀⠀⠀⠀⠀
⠈⠆⢿⣿⣿⢷⣿⣿⠷⠶⠶⠀⠀⠀⠈⢿⣷⣄⡀⠀⠀⠙⢻⣿⣿⣿⣶⠶⠂⠋⠋⠉⠉⠈⢻⣧⠀⣠⠟⠁⣴⣥⣿⣿⠀⢀⣩⡄⠠⠶⣶⣶⣿⣿⣷⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⣿⣇⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡀⠠⡹⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡆⢸⣷⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⣶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⣇⣸⣿⣿⣷⣾⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢀⣁⣹⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⠋⠸⠿⠇⠿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡸⠃⠈⠉⠉⠿⠻⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⡿⢿⣷⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠙⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣧⣼⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⡏⠉⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠛⠿⣿⣿⠀⠀
⣶⣶⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣶⣤⣄⡸⠿⠇⠀⠻⠿⠿⠟⠁⠀⠀⠿⠿⠿⠸⠿⠇⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⠛⠻⢿⣿⡿⠿⠇⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⣿⣿⠟⠁⠀⠀⠀⠈⣿⡿⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⣦⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢼⣿⡇⠀⠀⠀⠀⠀⠀⣾⠁⠈⠹⣿⣶⣤⣀⣠⣶⣿⣿⡿⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢧⠀⠀⠀⠀⠀⣰⡏⠀⠁⠠⢸⣿⣿⣿⣿⣿⣿⣷⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⣀⣠⣾⣿⣷⣤⣴⣷⣾⣿⣿⣿⣿⣿⣿⠻⡆⠀
⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢸⣷⠀
⠀⠀⠀⢹⣿⣿⠿⢿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡍⢙⡃⠀⠀⠙⢿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣿⣿⣿⡿⣿⣻⣿⣿⠿⠛⠋⣿⣾⡇⠀
⣤⣴⣶⠿⢿⡁⠀⠀⠉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⠀⣀⣤⡾⠃⢻⣷⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠾⠟⠛⠛⠉⠁⠀⣀⣴⣿⣿⡇⠀
⣿⡟⠁⠀⢸⣇⡀⣀⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⣿⠁⠀⠸⣿⣦⣤⣤⣾⣿⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠿⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣄⣤⣴⣾⣿⣿⣿⣿⣿⡇⠀
⣿⣷⣶⣦⣾⣿⣿⡏⠀⠀⢀⣶⣿⣿⣿⣦⠀⠀⠀⠀⢸⠈⠛⢿⣿⣶⣷⣾⣿⣿⣟⣿⡏⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⢿⣧⡀⠀⠈⡟⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⣽⣿⣿⣿⣿⣿⣿⠟⢁⣴⠃
⠻⠿⡿⣿⡟⠉⠁⠀⠀⠀⠘⠉⠙⢛⡉⠉⠣⠀⠀⠀⢾⣄⠀⠙⠛⠿⠿⠿⠿⢿⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠤⡾⢾⣾⣿⣷⣴⣾⣄⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⡿⢿⣯⣥⠴⢻⣿⣿
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⣮⣤⣷⣶⠇⠀⠀⠀⠀⢹⣶⣼⣶⣶⣶⣶⣶⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠚⠛⠏⢸⣽⡏⠀⠀⠀⠀⢠⣾⣿⣶⣶⣤⣀⡀⠀⠀⣀⣀⣀⣀⣀⣀⡈⠁⠀⠀⠀⣸⣿⡋
⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢹⠁⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡿⢋⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣋⡔⠀⠀⠀⢀⣾⣿⣿⣧⠈⠻⣿⣿⣿⣿⠋⠉⣀⠤⠤⣈⡁⠀⠀⠀⠀⠀⠈⢀
⣀⣠⣴⡄⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡎⠀⠀⠀⠀⠀⠀⠀⢀⣀⠛⢿⣴⣾⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⣿⢗⡾⢁⠀⠀⠘⣿⣿⣿⣿⡟⡀⢈⣿⣿⣧⣤⣤⣀⣀⣀⣀⠀⣤⣶⣶⣶⣶⣾⠿
⡿⠻⡋⢹⡄⠀⠀⢀⣠⣼⣟⠉⠉⠉⠙⠛⠿⣶⣴⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠏⠉⠉⠛⢷⣆⠀⣀⣀⣠⣤⣤⣤⣄⣀⣀⣶⣾⠿⠀⢀⡁⠈⠀⣀⣹⣿⣿⡟⠿⠇⢸⣿⣟⠋⣭⣿⣿⣿⣿⣿⣿⡿⡿⠟⠛⠋⠀⠀
⡤⢴⣿⣿⡿⠀⢀⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢀⡙⠳⣶⣤⣄⢠⣿⣿⣿⣾⡿⠆⣀⣤⣴⣾⣿⡿⠟⠛⠋⠉⠙⣻⣿⣿⡉⣙⣡⣄⢀⣀⣠⡤⠖⢉⣿⣿⠋⠀⠀⢀⠎⢻⣿⡈⠿⣿⣿⣿⣿⡿⠋⣄⣀⣀⣀⣀⣀⣤
⠀⠀⢻⣿⡇⠀⢘⣿⣿⡿⠿⠏⠀⠀⠀⠠⠀⠘⣿⣄⠀⠉⠁⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⣿⣿⠀⠀⠀⠀⣰⣿⣿⣿⡇⠀⢿⣿⠟⠀⠀⠀⣿⣿⣿⣿⣾⣠⣠⡇⠈⠀⠺⣿⡄⠘⠿⡟⠋⢀⣾⣿⣿⠿⠟⠛⠛⠛
⢶⣶⣿⣿⡇⠀⢸⣿⣿⢃⡀⠀⠀⠀⢀⣿⡀⠀⠈⠙⢿⣷⣶⡟⠛⠻⠿⠟⠋⠀⠀⢀⣠⣾⣿⣏⠀⠀⠀⢀⣿⣿⣿⣿⠗⠤⣤⣅⠐⠒⠒⠚⢉⣿⣿⠋⣿⣿⣿⣧⣄⣀⣤⣶⣶⣦⣤⣀⣀⠈⢋⣁⠀⠀⠀⠀⠀⠀
⠀⠿⠶⣿⣿⣷⢴⣿⡿⠶⠶⠀⠀⠀⠀⠻⣷⣦⡀⠀⠀⠈⠛⣿⣿⣿⣷⡶⠶⠲⠛⠋⠉⠁⠘⣿⡄⢀⡴⠋⠀⣮⣼⣿⡇⠀⣈⣥⠀⠴⢶⣶⣼⣿⣿⣾⣇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⢸⣿⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⠀⠠⡹⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡆⢸⣷⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡇⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⣇⣸⣿⣿⣷⣾⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡅⢀⣁⣹⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⠋⠸⠿⠇⠿⢇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀
⠀⠀⠀⠀⠀⠀⡸⠃⠈⠉⠉⠿⠛⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⡿⢿⣷⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠙⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣧⣼⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⡏⠉⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠻⢿⣿⡇⠀
⣤⣶⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣶⣶⣤⣤⣸⠿⠇⠀⠻⠿⠿⠟⠁⠀⠀⠿⠿⠿⠸⠿⠇⠀⠀⠈⠻⣿⣿⢿⣿⣿⣿⡿⠟⠛⠻⣿⡿⠿⠏⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢿⠘⣿⡿⠉⠀⠀⠀⠀⢻⣿⢿⣷⡀⠀⠀⠀⠀⠀⢀⣿⣷⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣶⡿⠀⠀⠀⠀⠀⠀⢸⠏⠀⠉⢿⣷⣦⣄⣠⣴⣾⣿⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡽⡀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡆⠀
⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⣠⣼⣿⣿⣦⣤⣾⣾⣿⣿⣿⣿⣿⣿⡟⣷⠀
⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠿⠟⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⠄
⠀⠀⠀⠈⣿⣿⡿⠿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢨⠍⣝⠀⠀⠉⠻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⣿⣿⡿⣿⣻⣿⣿⡿⠟⠛⢹⣷⡿⠀
⣀⣤⣴⡾⢿⡏⠀⠀⠈⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⡀⢀⣠⣶⠟⠹⣿⡆⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠾⠿⠛⠛⠉⠀⠀⢀⣴⣾⣿⣷⠀
⣿⣿⠋⠀⠀⣿⣀⠀⣀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣿⣿⠃⠀⠀⣿⣷⣤⣤⣴⣿⡇⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⣤⣠⣤⣶⣿⣿⣿⣿⣿⡿⠀
⣿⣿⣶⣦⣴⣿⣿⣿⠁⠀⠀⣰⣾⣿⣿⣷⡄⠀⠀⠀⠀⡆⠙⠿⣿⣷⣾⣶⣿⣿⣿⢻⡿⠁⠀⠀⠀⠀⠀⠀⠈⠁⡀⠻⣿⣄⠀⠀⢹⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣨⣿⣿⣿⣿⣿⣿⡿⠋⣠⡞
⠛⠿⢿⣿⣿⠋⠉⠀⠀⠀⠀⠋⠉⠙⡋⠉⠛⠄⠀⠀⠰⣧⠀⠈⠛⠻⠿⠿⠿⠿⣯⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⠴⠷⣷⣿⣿⣶⣶⣧⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⣛⠿⣿⣥⡴⠞⣿⣿
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⢴⣤⣼⣶⡾⠀⠀⠀⠀⠘⣷⣦⣷⣦⣶⣶⣶⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠒⠛⠻⠁⣯⣿⠁⠀⠀⠀⠀⣶⣿⣷⣶⣦⣄⣀⡀⠀⣀⣀⣤⣤⣀⣀⣀⠉⠀⠀⠀⠀⡿⣿
⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⡇⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⠟⣁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣡⠀⠀⠀⠀⣰⣿⣿⣿⡆⠙⢿⣿⣿⣿⡟⠉⠁⡠⠤⠤⣈⠀⠀⠀⠀⠀⠀⠁
⣄⣀⣤⣤⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣅⣴⠁⠀⠀⠀⠀⠀⠀⠀⣀⡘⢩⣭⣶⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣾⡿⣲⠏⡀⠀⠀⢻⣿⣿⣿⣿⢁⠀⡹⣿⣿⣤⣤⣄⣀⣀⣠⣄⢠⣶⣶⣶⣶⣶⡿
⣿⠟⢟⠋⣧⠀⠀⠀⣀⣠⣿⡛⠉⠉⠙⠛⠻⢶⣦⡎⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠉⠉⠙⠻⣦⡀⢀⣀⣀⣤⣤⣤⣄⣀⡀⣴⣶⡿⠧⠀⣉⠀⠉⢀⣈⣿⣿⡟⠻⠿⠀⣿⣿⡟⢩⣽⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁⠀
⣯⣀⣼⣾⣿⡇⠀⣴⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣈⠛⢶⣦⣤⡀⣸⣿⣾⣷⣾⠷⢀⣠⣤⣶⣿⣿⠿⠛⠛⠉⠉⢻⣿⣿⣏⢉⣉⣤⣀⣀⣀⣠⠴⠚⢹⣿⡿⠀⠀⠀⡰⠙⣿⣇⠘⢿⣿⣿⣿⣿⠟⢡⣀⣀⣀⣀⣠⣤
⠁⠀⠘⣿⣿⠀⠀⣻⣿⣿⡿⠿⠁⠀⠀⠀⠀⠀⢹⣧⡀⠈⠉⢸⣿⣿⣿⣿⣿⣾⡿⠿⠛⠋⢹⣿⡇⠀⠀⠀⢀⣾⣿⣿⣿⠀⠸⣿⡿⠁⠀⠀⢸⣿⣿⣿⣷⣇⣄⡼⠀⠁⠐⠿⣧⠀⠻⠿⠙⠁⣰⣿⣿⡿⠿⠛⠛⠋
⠓⣶⣶⣿⣿⠀⠀⢻⣿⡿⢀⠀⠀⠀⠀⢸⣆⠀⠀⠉⠿⣷⣶⣾⠛⠛⠿⠿⠟⠃⠀⠀⣀⣴⣿⣿⠁⠀⠀⠀⣸⣿⣿⣿⡿⠢⠤⣬⡄⠒⠒⠒⠋⣹⣿⡟⢹⣿⣿⣿⣤⣀⣠⣴⣶⣷⣤⣄⣀⡀⠙⣁⡀⠀⠀⠀⠀⠀
⠀⠞⢷⣾⣿⣷⠆⣾⣿⠷⠶⠀⠄⠀⠀⠈⢽⣦⣀⠀⠀⠀⠙⢻⣿⣿⣿⣶⠲⠂⠞⠛⠉⠉⠈⢻⣧⠀⣠⠞⠁⢰⣦⣿⣿⠀⢀⣩⡄⠠⠶⣶⣶⣿⣿⣷⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⣿⣇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⠀⢀⡹⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡆⢸⣷⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡇⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⣯⣷⣶⣿⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡅⢀⡁⣹⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⠋⠸⠿⠇⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀
⠀⠀⠀⠀⠀⠀⡸⠃⠈⠉⠉⠿⠛⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⡿⢿⣷⢠⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠙⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣧⣼⣿⢸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⡏⠉⠁⢸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠙⠛⠛⠛⠻⢿⣿⡇
⣠⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣶⣦⣤⣼⡿⠇⠀⠻⠿⠿⠟⠁⠀⠀⠿⠿⠿⠸⠿⠇⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⠿⠛⠛⢿⣿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠘⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢻⡾⣿⡟⠁⠀⠀⠀⠀⢻⣷⢿⣧⡀⠀⠀⠀⠀⠀⢀⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣷⡟⠀⠀⠀⠀⠀⠀⣸⠋⡀⠙⢿⣷⣦⣤⣤⣴⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡵⡀⠀⠀⠀⠀⣠⣿⠀⠀⠠⠈⣿⣿⣿⣿⣿⣿⣿⡆
⠿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠋⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣶⣀⣠⣼⣿⣿⣦⣤⣿⣾⣿⣿⣿⣿⣿⣿⡏⣷
⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠉⣿⣿⣿⢿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢨⢛⡛⠀⠀⠉⠻⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢿⣿⠿⣿⣻⣿⣿⠿⠟⠛⢸⣷⡿
⣀⣀⣠⣤⣶⣿⡏⠀⠀⠈⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣆⠀⠀⣠⣶⠟⢸⣿⡄⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠾⠿⠛⠉⠉⠀⠀⢀⣴⣾⣿⣧
⣿⣿⣿⠋⠀⠀⣿⡀⠀⣀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⠃⠀⠀⣿⣷⣤⣤⣴⣿⡇⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠿⢿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⡿
⢻⣿⣿⣶⣤⣤⣿⣿⣿⠁⠀⠀⣰⣶⣿⣿⣶⡀⠀⠀⠀⠀⡇⠙⢿⣿⣷⣶⣶⣿⣿⣿⢻⡿⠁⠀⠀⠀⠀⠀⠀⠈⠀⡀⠻⣿⡀⠀⠈⢻⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣬⣿⣿⣿⣿⣿⣿⠟⠋⣴
⡘⠛⠿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠛⠛⠛⡋⠉⠳⠀⠀⠀⠰⣧⠀⠈⠛⠿⠿⠿⠿⠿⣯⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⡴⠶⣶⣿⣿⣦⢶⣧⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⣛⣿⣿⣥⡶⠚⣿
⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣤⣽⣶⡾⠀⠀⠀⠀⠘⣶⣤⣷⣦⣶⣶⣶⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠚⠛⠻⠀⣯⣿⠁⠀⠀⠀⠀⣶⣿⣷⣶⣦⣄⣄⠀⠀⣀⣠⣤⣤⣀⣀⣀⠉⠀⠀⠀⢠⠿
⣷⡶⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠙⠇⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⠟⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣵⠀⠀⠀⠀⣴⣿⣿⣿⡀⠙⢿⣿⣿⣿⡏⠉⠠⠤⠤⠤⣈⠀⠀⠀⠀⠀⠀
⠿⣄⢀⣠⣤⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣁⣰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠩⣭⣶⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣿⡿⣶⠃⡀⠀⠀⢿⣿⣿⣿⣿⢀⠀⣹⣿⣿⣤⣤⣄⣀⣠⣤⣄⣠⣶⣶⣶⣶⣶
⣰⣾⡿⣟⠋⣄⠀⠀⠀⢀⣠⣿⡛⠛⠛⠛⠛⠷⣶⣆⠄⠀⠀⠀⠀⠙⠛⠾⠿⠿⠿⠉⠉⠙⢷⣦⡀⢀⣀⣀⣤⣤⣤⣄⣀⡀⣴⣾⡿⠇⠀⣉⠀⠉⢀⣈⣿⣿⡟⡿⠿⠀⣿⣿⡛⢩⣽⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁
⣿⣏⣀⣼⣼⣿⡆⠀⣴⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣉⠛⢶⣤⣤⠀⣼⣶⣶⣶⣾⠷⢀⣠⣤⣴⣿⣿⠿⠟⠛⠉⠉⢻⣿⣿⡋⢉⣋⣤⡀⣀⣀⡤⠴⠊⣸⣿⡏⠁⠀⠀⡔⠙⣿⣇⠙⢿⣿⣿⣿⣿⠟⢡⣀⣀⣀⣤⣤
⣿⠁⠀⠘⣿⣿⠀⠀⣿⣿⣿⣿⠿⠁⠀⠀⠀⠤⠀⢻⣧⡀⠉⠋⢻⣿⣿⣿⣿⣿⡾⣿⠿⠛⠋⢹⣿⡇⠀⠀⠀⢀⣾⣿⣿⣿⠀⠸⣿⡿⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⣾⠀⠁⠐⠿⣧⠀⠻⠿⠉⠁⣠⣿⣿⠿⠟⠛⠛
⣿⠷⣶⣶⣿⣿⠀⠀⣿⣿⡟⢀⠀⠀⠀⠀⢸⡆⠀⠀⠉⠿⣷⣶⣾⠛⠛⠿⡿⠟⠃⠀⠀⢀⣴⣿⣿⠁⠀⠀⠀⣸⣿⢿⣿⡿⠲⢤⣬⡀⠒⠒⠒⠋⣹⣿⡟⢹⣿⣿⣿⣤⣀⣤⣴⣶⣷⣤⣤⣀⡀⠉⣁⡀⠀⠀⠀⠀
⣿⠆⠺⣿⢿⣿⣷⠶⢿⣿⠇⠀⠠⠀⠀⠀⠙⢻⣦⣀⠀⠀⠀⠙⢻⣿⣿⣷⣶⠶⠶⠶⠛⠙⠉⠈⢻⡧⠀⣠⠞⠁⣶⣼⣿⣿⠀⢀⣩⡄⠠⠴⣶⣶⣿⣿⣷⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⣿⣧⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣾⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⠀⠠⡹⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣷⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡇⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⣶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⣯⣷⣶⣿⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢀⣁⣹⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⠋⠸⠿⠇⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀
⠀⠀⠀⠀⠀⠀⡸⠃⠈⠉⠉⠿⠻⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⡿⢿⣷⢠⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠛⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣧⣼⣿⢸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⡏⠉⠁⢸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠙⠛⠛⠛⠻⢿⣿⡇
⣠⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣶⣦⣤⣼⡿⠇⠀⠻⠿⠿⠟⠁⠀⠀⠿⠿⠿⠸⠿⠇⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⠿⠛⠛⢿⣿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠘⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⡿⣿⡟⠁⠀⠀⠀⠀⢻⣷⢿⣧⡀⠀⠀⠀⠀⠀⢀⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣷⡟⠀⠀⠀⠀⠀⠀⣸⠏⡀⠙⢿⣷⣦⣤⣤⣴⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣵⡀⠀⠀⠀⠀⣠⣿⠀⠀⠠⠈⣿⣿⣿⣿⣿⣿⣿⡆
⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠋⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣶⣀⣠⣼⣿⣿⣦⣤⣾⣾⣿⣿⣿⣿⣿⣿⡟⣷
⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠉⣿⣿⣿⢿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸⢛⡛⠀⠀⠉⠻⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡻⣿⣿⡿⣿⣻⣿⣿⠿⠟⠛⢸⣷⡿
⣀⣀⣠⣤⣶⣿⡏⠀⠀⠉⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣆⠀⠀⣠⣶⠟⢸⣿⡄⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠾⠿⠛⠉⠉⠀⠀⢀⣴⣾⣿⣷
⣿⣿⣿⠋⠀⠀⣿⡀⠀⣀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⠃⠀⠀⣿⣷⣤⣤⣴⣿⡇⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠿⢿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣤⣤⣠⣴⣶⣿⣿⣿⣿⣿⡿
⢻⣿⣿⣶⣤⣤⣿⣿⣿⠁⠀⠀⣴⣶⣿⣿⣶⡀⠀⠀⠀⠀⡇⠙⢿⣿⣷⣾⣶⣿⣿⣿⢻⡿⠁⠀⠀⠀⠀⠀⠀⠈⠀⡀⠻⣿⡀⠀⠈⢻⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣨⣿⣿⣿⣿⣿⣿⠟⠋⣴
⡘⠛⠿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠛⠛⠛⡋⠉⠳⠀⠀⠀⠸⣧⠀⠈⠛⠿⠿⠿⠿⠿⣯⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⡴⠶⣶⣿⣿⣦⣶⣧⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⣻⣿⣿⣥⡶⠚⣿
⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣤⣽⣶⡾⠀⠀⠀⠀⠘⣦⣤⣷⣦⣶⣶⣶⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠚⠛⠻⠀⣯⣿⠁⠀⠀⠀⠀⣾⣿⣷⣶⣦⣄⣄⠀⠀⣀⣠⣤⣤⣀⣀⣀⠉⠀⠀⠀⢠⠿
⣷⡶⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠙⠇⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⠟⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣱⠀⠀⠀⠀⣴⣿⣿⣿⡄⠙⢿⣿⣿⣿⡏⠉⠠⠤⠤⠤⣈⠀⠀⠀⠀⠀⠀
⠿⣄⢀⣠⣤⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣁⣰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠩⣭⣶⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣿⡿⣷⠇⡀⠀⠀⢿⣿⣿⣿⣿⢀⠀⣻⣿⣿⣤⣤⣄⣀⣠⣤⣄⢠⣶⣶⣶⣶⣶
⣰⣿⡿⣟⠋⣄⠀⠀⠀⢀⣀⣟⡛⠛⠛⠛⠛⠿⣶⣦⠄⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠉⠉⠙⢷⣦⡀⢀⣀⣀⣤⣤⣤⣄⣀⡀⣴⣾⡿⠇⠀⣉⠀⠁⢀⣈⣿⣿⡟⡿⠿⠀⣿⣿⡛⢨⣽⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁
⣿⣯⣀⣼⣼⣿⡆⠀⣼⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣉⠻⢶⣤⣤⡀⣼⣶⣶⣶⣾⠷⢀⣠⣤⣴⣿⣿⠿⠟⠛⠉⠉⢻⣿⣿⡋⢉⣋⣤⡀⣀⣀⡤⠴⠊⣸⣿⡏⠁⠀⠀⡔⠙⣿⣇⠹⢿⣿⣿⣿⣿⠟⢡⣀⣀⣀⣤⣤
⣿⠁⠀⠘⣿⣿⠀⠀⣿⣿⣿⣿⠿⠁⠀⠀⠀⡤⠀⢻⣧⡀⠉⠋⢻⣿⣿⣿⣿⣿⣾⣿⠿⠛⠋⢹⣿⡇⠀⠀⠀⢀⣾⣿⣿⣿⠀⠸⣿⡿⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⣾⠀⠁⠐⠿⣧⠀⠻⠿⠙⠁⣠⣿⣿⠿⠟⠛⠛
⣿⠷⣶⣶⣿⣿⠀⠀⣿⣿⡟⢀⠀⠀⠀⠀⣼⡇⠀⠀⠉⠿⣷⣶⣾⠛⠛⠿⡿⠟⠃⠀⠀⣀⣴⣿⣿⠁⠀⠀⠀⣸⡿⢿⣿⡿⠲⢤⣬⡀⠒⠒⠒⠋⣹⣿⡟⢹⣿⣿⣿⣤⣀⣤⣴⣶⣷⣤⣤⣀⡀⠉⣁⡀⠀⠀⠀⠀
⣿⠆⠺⣿⢿⣿⣷⠶⣿⣿⠇⠀⠠⠠⠀⢀⠙⣻⣦⣀⠀⠀⠀⠙⢻⣿⣿⣷⣶⠶⠶⠶⠛⠙⠉⠈⢻⡧⠀⣠⠞⠁⣶⣼⣿⣿⠀⢀⣩⡄⠠⠴⣶⣶⣿⣿⣷⣿⡈⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠋⠀⣿⣧⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣠⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣾⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⡹⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡄⢸⣷⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡇⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⣯⣵⣶⣾⣿⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢀⡁⣹⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠈⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⠋⠸⠿⠇⠿⠇⠿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀
⠀⠀⠀⠀⠀⠀⡸⠃⠈⠉⠉⠿⠛⢿⣿⣿⡿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⡿⢿⣷⢠⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠙⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣧⣼⣿⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⡏⠉⠁⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠛⠛⠛⠛⢿⣿
⢀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣶⣶⣤⣼⣿⡇⠀⠻⠿⠿⠟⠁⠀⠀⠿⠿⠿⠸⠿⠇⠀⠀⠈⠻⠷⢿⣿⣿⣿⣿⣿⣿⠟⠛⠛⢿⣿⠿⠿⠇⠀⠀⠀⠀⠀⠀⢸⣿
⡿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⣾⣿⠟⠁⠀⠀⠀⠀⢿⣿⣿⣧⡀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣿⡇⠀⠀⠀⠀⠀⠀⣼⠁⡀⠙⢿⣷⣦⣤⣤⣶⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⣰⡏⠀⠀⠠⠈⣿⣿⣿⣿⣿⣿⣿
⡻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠋⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣀⣠⣾⣿⣿⣦⣴⣿⣾⣿⣿⣿⣿⣿⣿⡏
⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀
⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⢘⢛⡛⠀⠈⠙⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠽⣿⡿⠿⣟⣻⣿⣿⠻⠋⠋⢸⣷
⣀⡀⣀⣀⣤⣴⣿⡏⠀⠀⠙⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣄⠀⠀⣀⣴⠟⢺⣿⠀⠀⠀⠀⢹⡗⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠞⠟⠛⠉⠁⠀⢀⣠⣴⣾⣿
⣿⣿⣿⣿⠋⠀⠀⣷⠀⠀⣀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠃⠀⠈⣿⣧⣤⣤⣴⣿⡇⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠿⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿
⠈⢿⣿⣿⣤⣄⣠⣿⣿⣿⠁⠀⠀⣰⣶⣶⣶⣦⡀⠀⠀⠀⠀⡇⠛⢿⣿⣷⣶⣴⣿⣿⣿⢻⡿⠁⠀⠀⠀⠀⠀⠀⠊⠀⡀⠻⣿⡀⠀⠈⡛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣎⣼⣿⣿⣿⣿⣿⣿⠟⠉
⠂⡘⠻⣿⣿⣿⣿⠟⠋⠁⠀⠀⠈⠛⠛⠛⡛⠙⢣⠀⠀⠀⠰⣧⠀⠘⠛⠿⠿⠿⠿⠿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠠⡼⠲⣶⣿⣿⣦⢶⣇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⣻⢟⣯⣥⠶⠛
⣀⠀⠀⠀⠊⠈⠁⠀⠀⠀⠀⠀⠀⣀⣴⣤⣹⣦⣾⠀⠀⠀⠀⠸⣶⣤⣶⣦⣴⣶⣶⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠻⠐⣯⣿⠀⠀⠀⠀⠀⣾⣿⢷⣶⣦⣠⣄⠀⠀⣀⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⢀
⣿⣷⡶⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠙⠇⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⠟⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣵⠀⠀⠀⠀⣴⣿⣿⣿⡀⠙⢿⣿⣿⣿⡏⠉⠠⠤⠤⠤⣀⠀⠀⠀⠀⠀
⠿⢿⣁⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡐⠋⠩⣭⣶⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣿⡷⣲⠃⠀⠀⠀⢿⣿⣿⣿⣿⣀⠀⣻⣿⣿⣤⣤⣄⣠⣤⣤⣄⣠⣶⣿⣷⣶
⢀⣰⣿⡿⡟⠟⣄⠀⠀⠀⢀⣀⣛⠛⠛⠛⠛⠻⠷⣶⣀⠄⠀⠀⠀⠀⠙⠛⠾⠿⠿⠿⠉⠉⠙⢷⣦⡀⠀⣀⣀⣤⣤⣤⣄⣀⡀⣴⣾⡿⠇⠀⣁⠀⠁⢀⣘⣿⣿⡿⡿⠟⠀⣿⣿⡛⣨⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠉
⣿⣿⣟⣀⣼⣴⣿⡆⠀⢴⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⣙⠳⣦⣤⣄⠀⣶⣶⣶⣶⣾⠧⢀⣀⣤⣴⣿⣿⠿⠟⠛⠉⠙⢻⣿⣿⡋⢙⣋⣤⡀⣀⣀⡤⠴⣊⣸⣿⡏⠀⠀⠀⡔⠙⣿⣇⠹⢿⣿⣿⣿⣿⠏⣡⣄⣀⣠⣤
⣿⣟⠁⠀⠙⣿⣿⠁⠀⣹⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢻⣦⡀⠙⠋⢸⣿⣿⣿⣿⣿⡾⡿⠿⠟⠋⢹⣿⡇⠀⠀⠀⢀⣿⣿⣿⣿⠀⠹⣿⡿⠀⠀⠀⢸⣿⣿⣿⣷⣯⣀⣾⠀⠁⠘⠿⣧⠀⠙⠽⠉⠁⣰⣿⣿⠿⠛⠛
⣿⣿⠷⣤⣴⣿⣿⠀⠀⢽⣿⡿⢀⡀⠀⠀⠀⣸⡄⠀⠈⠙⢿⣶⣦⣜⠛⠛⠿⡿⠟⠋⠀⠀⢀⣴⣿⣿⠁⠀⠀⠀⣸⣿⣿⣿⡿⠢⢤⣬⡀⠒⠒⠚⠋⣹⣿⡟⢹⣿⣿⣿⣤⣀⣤⣶⣶⣷⣤⣤⣀⣀⠉⣁⡀⠀⠀⠀
⣿⣿⠲⣾⠏⢿⣿⣷⠶⣿⣿⡇⡀⠄⠀⠀⠀⠘⣿⣦⡀⠀⠀⠈⠛⠻⣾⣿⣶⣶⠶⠶⠖⠛⠛⠋⠈⢻⡇⠀⣠⠞⠉⣤⣴⣿⣿⠀⢀⣩⡄⠠⠴⣶⣶⣿⣿⣷⣿⡈⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠉⠀⣿⣧⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡀⠀⡹⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣄⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⣶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⣏⣥⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡅⢀⣁⣽⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⠋⠸⠿⠇⠿⠇⠿⢧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⡸⠃⠈⠉⠉⠿⠛⢿⣿⣿⡿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⡿⢿⣷⢰⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠙⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣧⣼⣿⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⡏⠉⠁⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠙⠛⠛⠻⣿
⠀⢀⣀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣤⣼⣿⡇⠀⠻⠿⠿⠟⠁⠀⠀⠿⠿⠿⠸⠿⠇⠀⠀⠈⠻⠷⠿⢿⣿⣿⣿⣿⣿⡿⠛⠛⠻⣿⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⣿
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣿⣿⡿⠋⠀⠀⠀⠀⠘⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣽⣿⠀⠀⠀⠀⠀⠀⢠⡏⢀⠈⠻⣿⣶⣦⣤⣴⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⢀⣾⠁⠀⠀⠆⣹⣿⣿⣿⣿⣿⣿
⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⣠⣴⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⡅⠀⠀⠀⠈⠉⢻⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⡛⣻⠃⠀⠉⠻⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢻⡿⢿⢿⣿⣽⣿⡿⠛⠙⠁⣿
⠃⠀⠀⢀⣠⣤⣾⣿⠋⠀⠉⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⢀⣠⡾⠓⣿⡆⠀⠀⠀⠈⣿⠂⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠻⠛⠋⠉⠀⠀⣀⣤⣶⣿
⢻⣿⣿⣿⡿⠋⠀⢸⡆⠀⠀⡀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⡟⠀⠀⢻⣿⣤⣤⣤⣾⣿⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣤⣤⣤⣶⣾⣿⣿⣿⣿
⠀⢹⣿⣿⣧⣤⣀⣼⣿⣷⡟⠀⠀⢀⣴⣶⣶⣶⣄⠀⠀⠀⠀⢸⠘⢿⣿⣷⣶⣦⣾⣿⣿⡟⣿⠏⠀⠀⠀⠀⠀⠀⠐⠉⢀⠘⢿⣇⠀⠀⢙⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣡⣿⣿⣿⣿⣿⣿⡿⠋
⠐⠀⠛⢿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠚⠛⠛⠟⠛⠛⠄⠀⠀⠀⣿⡄⠀⠛⠻⠿⠿⡿⣿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢤⠗⢶⣾⣿⣷⣴⣾⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⣻⣻⣯⣶⠞
⣄⡀⠀⠀⠐⠉⠉⠁⠀⠀⠀⠀⠀⢀⣠⣬⣈⣶⣴⡇⠀⠀⠀⠈⢿⣦⣴⣶⣤⣶⣶⣶⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠛⠛⠇⢺⣽⡏⠀⠀⠀⠀⢰⣿⣿⣶⣶⣤⣤⣀⠀⢀⣠⣤⣤⢤⣀⣀⡀⠀⠀⠀⠀
⣿⣿⣶⡶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢻⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⡿⢋⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣩⡆⠀⠀⠀⢠⣾⣿⣿⣧⠈⠻⣿⣿⣿⣿⠉⠁⠤⠤⠤⢤⡀⠀⠀⠀⠀
⠿⠿⣏⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠚⠫⢭⣭⣶⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣸⣿⢓⡞⢀⠀⠀⠘⣿⣿⣿⣿⣇⡀⢈⣿⣿⣧⣤⣤⣤⣤⣤⣤⣀⣴⣿⣿⣿
⠀⢀⣼⣿⢿⡛⢻⡀⠀⠀⠀⣀⣸⡿⠛⠛⠛⠻⠿⢶⣦⡠⠀⠀⠀⠀⠈⠛⠷⠿⠿⠿⠟⠉⠙⠳⣶⣄⠀⢀⣀⣀⣤⣤⣤⣀⣀⢠⣶⣿⠿⠀⢈⡁⠈⠁⣀⣹⣿⣿⣿⠿⠇⢸⣿⣟⠃⣼⣿⣿⣿⣿⣿⣿⡿⠿⠟⠉
⣿⣿⣿⣇⣀⣷⣾⣷⠀⢀⣾⣿⣿⣿⣦⠀⠀⠀⠀⠀⠈⡙⢶⣤⣤⡀⢀⣶⣶⣶⣶⡿⠄⢀⣠⣤⣾⣿⣿⠿⠛⠋⠉⠛⢿⣿⣿⠉⣛⣡⣄⣀⣀⣠⠤⠖⣁⣿⣿⠏⠀⠀⢠⠊⢻⣿⡄⠻⢿⣿⣿⣿⡿⠋⣤⣀⣀⣤
⠿⣿⡟⠉⠈⢻⣿⡟⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⢀⠀⠘⣷⣄⠈⠛⠻⣿⣿⣿⣿⣿⣷⣾⡿⠿⠛⠋⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⡇⠈⢿⣿⠇⠀⠀⠀⣿⣿⣿⣿⣾⣥⣰⡇⠈⠀⠚⢿⣄⠈⠛⠏⠉⢀⣼⣿⡿⠟⠛
⣿⣿⣿⢤⣤⣼⣿⡇⠀⠈⣿⣿⡏⢀⡀⠀⠀⢀⣶⠀⠀⠈⠻⣷⣶⣶⡟⠛⠻⣿⠿⠟⠁⠀⠀⣠⣼⣿⡏⠀⠀⠀⢀⣿⣿⣿⣿⠷⠤⣤⣥⠐⠒⠒⠚⠉⣿⣿⠋⣿⣿⣿⣧⣄⣠⣴⣶⣾⣶⣤⣤⣀⡈⢉⣠⠀⠀⠀
⣿⣿⡷⣾⠿⣿⣿⣿⡶⠶⣿⣿⢀⠁⠀⠀⠀⠀⢿⣷⣄⠀⠀⠀⠉⠛⣷⣿⣷⣶⣶⠶⠶⠞⠛⠛⠉⠙⣿⡄⢀⡴⠋⢁⣦⣿⣿⣇⠀⣈⣥⠀⠤⢶⣶⣾⣿⣿⣾⣧⢹⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡏⠁⢸⣿⣄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⡹⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡆⢸⣷⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡇⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⣏⣥⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡅⢀⣁⣹⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⠋⠸⠿⠇⠿⠇⠿⢇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⡸⠃⠈⠉⠉⠿⠛⢿⣿⣿⡿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⡿⢿⣷⢰⣿⡟⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠙⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣧⣼⣿⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⡏⠉⠁⢸⣿⡇⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠉⠉⠙⠛⣿
⠀⠀⣀⣀⣠⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣤⣼⣿⣇⠀⠻⠿⠿⠟⠁⠀⠀⠿⠿⠿⠸⠿⠇⠀⠀⠈⠻⠷⠿⢿⣿⣟⣿⣿⣿⡿⠟⠛⠛⢿⡿⠿⠟⡁⠀⠀⠀⠀⠀⠀⢸
⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣿⢻⣿⠏⠁⠀⠀⠀⠈⢿⡟⣿⣧⡀⠀⠀⠀⠀⠀⣸
⣟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣾⡇⠀⠀⠀⠀⠀⠀⣾⠁⡀⠙⣿⣷⣦⣤⣤⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⣰⡇⠀⠀⠰⢘⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣆⣀⣠⣽⣿⣿⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡷⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠀⠀⠀⠈⠉⠛⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⣸⢛⠟⠀⠉⠙⢿⣿⡏⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠿⣿⠿⡿⣿⣯⣿⣿⠻⠋⠋⢸
⠣⠀⠀⠀⣀⣀⣴⣿⡟⠁⠈⠙⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⣠⣼⠟⢾⣷⠀⠀⠀⠀⣻⡇⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠚⠟⠋⠉⠁⠀⢀⣠⣴⣿
⠛⣿⣿⣿⣿⠟⠁⠉⣧⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⠁⠀⠘⣿⣦⣤⣤⣴⣿⠁⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠿⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⣤⣤⣴⣶⣿⣿⣿⣿
⠀⢛⢿⣿⣿⣤⣄⣠⣿⣷⣾⠁⠀⠀⣠⣶⣶⣶⣤⡀⠀⠀⠀⢀⡇⠻⣿⣿⣦⣶⣴⣿⣿⡿⣻⡿⠁⠀⠀⠀⠀⠀⠀⠈⠀⠈⢿⣯⡀⠀⠈⡟⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⣸⣿⣿⣿⣿⣿⣿⠟
⠀⠄⠘⠿⣿⣿⣿⣿⠟⠛⠉⠀⠀⠐⠛⠛⠿⠛⠛⢧⠀⠀⠀⢸⣧⠀⠘⠛⠿⠿⣿⢿⡿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠠⡼⠶⣶⣿⣿⣦⣶⣇⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⢟⣋⣴⠶
⣤⡀⠀⠀⠀⠋⠉⠉⠀⠀⠀⠀⠀⠀⣀⣤⣀⣱⣤⣾⠀⠀⠀⠀⠹⣶⣤⣲⣦⣤⣦⣶⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠻⢸⣯⡟⠀⠀⠀⠀⠀⣾⣿⢷⣶⣦⣠⣄⡀⠀⣀⣤⣤⢤⣄⣀⣀⠁⠀⠀⠀
⣿⣿⣧⣶⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠛⡇⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⠟⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣵⠀⠀⠀⠀⣼⣿⣿⣿⡀⠙⢿⣿⣿⣿⠏⠉⠠⠤⠤⠤⣀⠀⠀⠀⠀
⠿⠿⣟⡁⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠠⡈⠛⠭⣭⣴⣾⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⣶⠃⠀⠀⠀⢿⣿⣿⣿⣷⣄⠀⣻⣿⣿⣤⣤⣤⣤⣤⣤⣄⣠⣾⣿⣿
⠀⠀⣠⣽⡿⡟⠛⣄⠀⠀⠀⠀⣀⣿⠿⠛⠛⠛⠿⢶⣦⣀⠄⠀⠀⠀⠀⠙⠳⠾⠿⠿⠿⠋⠉⠛⢶⣦⡀⠀⣀⣀⣠⣤⣤⣀⣀⠀⣴⣾⡿⠆⠀⣁⠀⠁⢀⣨⣿⣿⡟⡿⠛⠀⣿⣿⡛⢠⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⣿⣿⣿⣿⠀⣹⣴⣿⡆⠀⣰⣿⣿⣿⣷⡄⠀⠀⠀⠀⠈⡙⠷⣦⣤⣀⠀⣶⣶⣶⣶⣶⠧⠀⣀⣤⣴⣿⣿⠿⠟⠛⠋⠙⢻⣿⣿⡏⢛⢋⣤⡀⣀⣀⡤⠴⢊⣸⣿⡿⠀⠀⠀⡔⠙⣿⣇⠙⠿⣿⣿⣿⣿⠟⣡⣄⣠⣤
⠻⣿⣿⠋⠉⠙⣿⣿⠁⠀⣽⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢿⣆⠈⠙⠛⢿⣿⣿⣿⣿⣿⣶⣿⠿⠟⠛⢹⣿⡇⠀⠀⠀⠀⣾⣿⣿⣿⠀⠹⣿⡿⠁⠀⠀⢸⣿⣿⣿⣷⣯⣄⣾⠀⠁⠘⠻⣧⠀⠙⠿⠉⠁⣰⣿⡿⠿⠛
⣾⣿⣿⣧⣤⣤⣿⣿⠀⠀⢻⣿⣿⠁⢀⠀⠀⠀⣸⡇⠀⠈⠛⢿⣶⣤⣿⠛⠛⣿⣿⠟⠋⠀⠀⢀⣠⣿⣿⠁⠀⠀⠀⣸⡟⢿⣿⡿⠦⢤⣭⡀⠒⠒⠚⠋⣹⣿⡟⢹⣿⣿⣿⣤⣀⣤⣶⣶⣷⣦⣤⣀⣀⠉⣁⡀⠀⠀
⣽⣿⣿⣾⠛⢿⢽⣿⣷⠶⣿⣿⠇⢈⣀⣀⠀⠀⢹⢿⣦⡀⠀⠀⠈⠛⢿⣶⣶⣶⣶⢶⠶⠶⠚⠛⠋⠉⢿⣇⠀⣠⠞⠉⢴⣭⣿⣿⡀⢀⣩⡄⠠⠴⣶⣶⣿⣿⣷⣿⡈⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀⣿⣧⡀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡏⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⠀⢠⡹⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⣶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⣏⣡⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢀⣁⣽⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⠋⠸⠿⠇⠿⠇⠿⠇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣸⠃⠈⠉⠉⠿⠛⢿⣿⣿⡿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⢀⣿⡿⠿⢸⣿⡿⢿⣷⢠⣿⡟⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠛⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣧⣼⣿⢸⣿⡇⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⡏⠉⠁⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⠉⠙⢻
⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣦⣾⣿⣇⡀⠻⠿⠿⠟⠁⠀⠀⠿⠿⠿⠸⠿⠇⠀⠀⠈⠻⠷⠿⠟⣿⣿⣻⣿⣿⣿⠿⠛⠛⠻⣿⠿⢟⡋⠀⠀⠀⠀⠀⠀⠀
⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢿⡏⣿⡟⠁⠀⠀⠀⠀⢹⣿⣿⣿⣄⡀⠀⠀⠀⠀⢀
⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣷⣿⠀⠀⠀⠀⠀⠀⢸⠇⢀⠈⢻⣿⣷⣦⣤⣶⣿
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⣀⣿⠀⠀⠀⠂⣻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢀⣠⣼⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡷⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡇⠀⠀⠀⠉⠙⢻⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢈⡛⠻⠁⠈⠉⠻⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⢿⣿⣿⢿⣫⣿⣿⡟⠛⠉⠁
⠟⠄⠀⠀⢀⣀⣠⣿⣿⠋⠀⠙⠻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⢀⣤⡿⠳⣿⡆⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠙⠉⠉⠀⢀⣀⣤⣿
⠛⢿⣿⣿⣿⡿⠋⠉⢹⡄⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⡟⠀⠀⢿⣷⣤⣤⣤⣾⡏⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢴⣴⣤⣤⣾⣿⣿⣿⣿
⠀⠨⠻⣿⣿⣧⣤⣀⣼⣿⣶⡏⠀⠀⢀⣤⣶⣶⣦⡄⠀⠀⠀⠀⣸⠘⢿⣿⣷⣴⣤⣾⣿⣿⢛⣿⠏⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⣿⣄⠀⠀⢹⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣠⣿⣿⣿⣿⣿⣿⠿
⠀⠠⠀⠻⢿⣿⣿⣿⡿⠛⠋⠁⠀⠀⠞⠛⠛⠿⠛⠻⡄⠀⠀⠀⣿⡄⠀⠛⠻⢿⢿⣿⣿⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢤⠷⢶⣿⣿⣶⣴⣿⡠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠿⣻⣥⡶
⢦⣄⠈⠀⠀⠸⠉⠉⠁⠀⠀⠀⠀⠀⢀⣠⣄⣘⣦⣴⡅⠀⠀⠀⠈⣷⣦⣔⣶⣤⣶⣴⣶⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠚⠛⠛⠇⣿⣽⠃⠀⠀⠀⠀⢴⣿⣿⣶⣶⣤⣤⣀⠀⣀⣤⣤⠤⠤⢄⣠⡀⠀⠀⠀
⣿⣿⣿⣴⡤⠇⠀⠀⠀⠀⠀⠀⠀⠀⢈⠋⠛⠉⠙⢿⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⡿⢋⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣩⡆⠀⠀⠀⢰⣿⣿⣿⣇⠈⠻⣿⣿⣿⡿⠉⠀⠤⠤⠤⠤⡀⠀⠀⠀
⠿⠿⣿⣏⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⢄⠋⠯⢭⣵⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣸⣿⢗⠞⠀⠀⠀⠸⣿⣿⣿⣿⣤⡀⢸⣿⣿⣧⣤⣤⣤⣤⣤⣤⣀⣴⣿⣿
⠀⠀⠀⣬⣿⣿⠿⢧⠀⠀⠀⠀⢀⣼⡿⠟⠛⠛⠻⠷⣶⣄⡠⠀⠀⠀⠀⠈⠛⠷⠿⠿⠿⠏⠉⠙⠳⣦⣄⠀⢀⣀⣀⣤⣤⣄⣀⣀⢠⣶⣿⠷⠀⢈⡀⠈⠁⣀⣽⣿⣿⢻⠟⠃⢸⣿⣟⢃⣼⣿⣿⣿⣿⣿⣿⡿⠿⠛
⣿⣿⣿⣿⡏⢈⣇⣾⣷⠀⢠⣾⣿⣿⣿⣆⠀⠀⠀⠀⠀⢉⠻⢶⣤⣄⡀⢰⣶⣶⣶⣶⡾⠄⢀⣠⣤⣾⣿⡿⠿⠟⠛⠉⠛⢿⢿⣿⠉⣛⣡⣄⢀⣀⣠⠤⢚⣁⣿⣿⠃⠀⠀⣠⠊⢻⣿⡄⠻⢿⣿⣿⣿⠿⢋⣤⣠⣤
⠘⢿⣿⡟⠉⠉⢻⣿⡟⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠸⣷⡄⠉⠛⢛⣿⣿⣿⣿⣿⣷⣾⣿⠿⠛⠛⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⡇⠈⢿⣿⠇⠀⠀⠀⣿⣿⣿⣿⣾⣤⣰⡇⠀⠀⠙⢿⣆⠈⠛⠏⠉⢀⣾⣿⠿⠟
⢠⣿⣿⣿⣤⣤⣼⣿⡇⠀⠘⣿⣿⡏⠀⡀⠀⠀⢀⣿⠀⠀⠙⠻⣷⣦⣬⡟⠛⢻⣿⡿⠟⠁⠀⠀⣀⣼⣿⡏⠀⠀⠀⢀⣿⣿⣿⣿⠷⠤⣬⣅⠐⠒⠒⠛⢉⣿⣿⠃⣿⣿⣿⣧⣄⣠⣴⣶⣾⣶⣶⣤⣄⡈⢉⣤⠀⠀
⢻⣿⣿⣿⠟⢿⡟⣿⣿⣶⠾⣿⣿⠁⣁⣀⡀⠀⠈⣿⣷⣄⠀⠀⠀⠙⠻⣷⣾⣶⣶⣶⣶⠶⠖⠛⠛⠉⠙⣿⡄⢀⡴⠋⢁⣦⣾⣿⣇⠀⣈⣥⠀⠴⢶⡶⣾⣿⣿⣾⣧⢸⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣏⠁⢸⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡏⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡀⠠⡹⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣷⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⡏⣁⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢀⣁⣽⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⠋⠸⠿⠇⠿⠇⠿⠇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣸⠃⠈⠉⠉⠿⠻⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⡿⢿⣷⢰⣿⡟⣿⣷⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠛⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣧⣼⣿⢸⣿⡇⣿⣿⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⡏⠉⠁⢸⣿⡇⣿⣿⢘⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⠉⠉⠙
⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣦⣾⣿⣇⣀⠻⠿⠿⠟⠁⠀⠀⠿⠿⠿⠸⠿⠇⠀⠀⠈⠻⠷⠿⠏⠀⣿⣿⢻⣿⣿⣿⠟⠋⠙⠻⣿⡟⣛⡃⠀⠀⠀⠀⠀⠀
⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢻⡟⣿⠟⠁⠀⠀⠀⠀⢹⣿⢻⣿⣄⡀⠀⠀⠀⠀
⣾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡟⠀⠀⠀⠀⠀⠀⣸⠃⠀⠈⢻⣿⣷⣦⣴⣾
⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄⠀⠀⠀⠀⣠⣿⠀⠀⠀⠂⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⣤⣼⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡇⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢘⡻⠻⠉⠉⠙⠻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⠿⣿⢿⣯⣿⣿⠟⠛⠉
⢻⡿⠀⠀⠀⠀⣀⣠⣿⣿⠋⠉⠙⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⢀⣴⡿⢻⣿⡄⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠉⠁⠀⢀⣠⣴
⣿⠾⢿⣷⣾⣿⡿⠋⠉⢹⡀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠻⣿⣿⡏⠀⠀⣿⣧⣤⣀⣤⣾⡇⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡴⣦⣶⣿⣿⣿⣿
⠁⠀⠘⠻⣿⣿⣇⣀⣀⣼⣷⣶⡞⠀⠀⢀⣤⣶⣶⣦⡀⠀⠀⠀⠀⡔⠸⢿⣿⣷⣴⣤⣾⣿⣿⢻⣿⠋⠀⠀⠀⠀⠀⠀⠈⠁⠀⠹⣿⣄⠀⠀⢹⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣤⣿⣿⣿⣿⣿⣿
⠀⠀⠠⠁⠿⣿⣿⣿⣿⡿⠟⠛⠁⠀⠀⠾⠟⠻⠿⠛⠻⡀⠀⠀⠀⣿⡀⠀⠛⠿⢿⢿⣿⣿⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠠⢤⢷⢶⣿⣿⣦⣴⣿⡰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⢟⠻⠵
⣀⢦⡄⠈⠀⠀⠨⠉⠉⠁⠀⠀⠀⠀⠀⠀⣠⣄⣘⣦⣴⡅⠀⠀⠀⠘⣷⣤⣒⣦⣤⣴⣤⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠚⠛⠛⠇⣿⣽⠃⠀⠀⠀⠀⢴⣿⡿⢶⣶⣤⣤⣀⠀⣀⣤⣤⠤⠤⠤⣤⡀⠀⠀
⣿⣿⣿⣷⣤⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⢨⠋⠛⠋⠛⢿⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⡿⢋⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣩⠆⠀⠀⠀⢰⣿⣿⣿⣇⠈⠻⣿⣿⣿⡿⠁⠀⠤⠴⠤⠤⡀⠀⠀
⠿⠿⢿⣿⡟⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢄⠙⠿⢭⣭⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣸⣿⢓⠞⠀⠀⠀⠸⣿⣿⣿⣿⢤⡀⢨⣿⣿⣧⣤⣤⣤⣤⣤⣤⣄⣶⣿
⣀⠀⠀⠀⣹⣷⣿⡿⢷⠀⠀⠀⠀⠀⣴⡿⠿⠿⠿⠿⢷⣶⣄⢠⠀⠀⠀⠀⠘⠻⠶⠿⠿⠿⠟⠉⠙⠶⣦⣄⠀⢀⣀⣀⣤⣤⣄⣀⡀⢠⣶⣿⠷⠀⢈⡀⠉⠁⣀⣽⣿⣿⠻⠟⠃⢸⣿⣝⢃⣼⣿⣿⣿⣿⣿⣿⡿⠻
⣿⣿⣿⣿⣿⡏⢈⣧⣼⣷⠀⢀⣴⣿⣿⣿⣆⠀⠀⠀⠀⠀⢙⠻⢦⣤⣀⡀⢰⣶⣶⣶⣶⣾⠄⢀⣠⣤⣿⣿⡿⠿⠟⠛⠉⠻⢿⣿⣟⢉⣛⣡⣄⢀⣀⣠⠴⢚⣁⣿⣿⠁⠀⠀⣠⢊⢻⣿⡄⠻⢿⣿⣿⣿⡿⢉⣤⣤
⠀⠈⢿⣿⡟⠋⠉⢿⣿⡟⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⢀⠀⠸⣷⡀⠙⠛⢳⣿⣿⣿⣿⣿⣷⣾⣿⠿⠛⠛⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⡇⠈⢿⣿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣥⣰⡇⠀⠀⠙⢿⣆⠈⠛⠏⠉⢀⣾⣿⠿
⣤⢀⣾⣿⣿⣤⣄⣼⣿⡇⠀⠸⣿⣿⡏⠁⡀⠀⠀⠠⢷⠀⠀⠙⠻⣶⣦⣼⡿⠿⢿⣿⡿⠟⠁⠀⠀⣀⣼⣿⡏⠀⠀⠀⢀⣿⣿⣿⣿⠷⠤⣤⣅⠐⠒⠒⠋⢉⣿⣿⠃⣿⣿⣿⣧⣆⣠⣴⣶⣾⣶⣶⣤⣤⡈⢉⣤⠀
⢰⣿⣿⣿⡟⢿⠿⠻⣿⣿⣾⠀⣿⣿⠉⣉⣀⡀⠀⠈⣿⣷⣄⠀⠀⠈⠙⢿⣷⣶⣶⣶⣶⣶⠶⠶⠛⠛⠉⠹⣿⡄⢀⡴⠋⢡⣦⣿⣿⣇⡀⣈⣥⠀⠴⢶⡶⣾⣿⣿⣿⣧⢹⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣯⠁⢸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⣧⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡀⢀⡹⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡆⢸⣷⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡟⣿⡇⢸⣿⡏⢁⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢀⡁⣹⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⠋⠸⠿⠇⠿⠇⠿⠇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡸⠃⠈⠉⠉⠿⠻⢿⣿⣿⡿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⡿⢿⣷⢰⣿⡟⣿⣷⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠙⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣧⣼⣿⢸⣿⡇⣿⣿⠀⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⡏⠉⠁⢸⣿⡇⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠈⠉⠉⠉
⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣾⣿⣧⣀⡻⠿⠿⠟⠁⠀⠀⠿⠿⠿⠸⠿⠇⠀⠀⠈⠻⠷⠿⠏⠀⢹⣿⣿⣿⣿⣿⡿⠛⠉⠛⢿⣿⢛⣛⠀⠀⠀⠀⠀⠀
⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⣿⢻⡿⠋⠀⠀⠀⠀⠈⣿⡟⢿⣦⣀⠀⠀⠀⠀
⣴⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠁⠀⠀⠀⠀⠀⢀⡟⠀⡀⠙⣿⣿⣶⣶⣶
⣿⢏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⢀⣼⡇⠀⠀⡐⣸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⣠⣴⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠀⠀⠈⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣛⠟⠏⠉⠉⠛⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠽⡿⢟⣿⣿⣽⣿⠿⠛⠉
⣛⢿⠇⠀⠀⠀⠀⣀⣼⣿⡟⠉⠉⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⣠⣾⠟⣾⣧⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠉⠈⠀⠀⣀⣤
⣿⣷⠿⣷⣶⣿⣿⠟⠉⠉⣇⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⣿⣿⣿⠁⠀⢸⣿⣤⣄⣠⣴⣿⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠿⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⢶⣴⣾⣿⣿⣿
⡟⠁⠀⠛⢿⣿⣿⣀⡀⣀⣿⣶⣶⠃⠀⠀⣠⣴⣶⣦⣄⠀⠀⠀⠀⢠⡆⢿⣿⣿⣦⣦⣴⣿⣿⡟⣻⡟⠁⠀⠀⠀⠀⠀⠀⠉⠀⠘⢿⣧⡀⠀⠈⡏⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠌⠸⢿⣿⣿⣿⣿⡿⠛⠋⠀⠀⠰⠿⠿⠿⠟⠻⢇⠀⠀⠀⢸⣄⠀⠙⠿⡿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠤⡼⡶⣾⣿⣷⣦⣾⣇⢼⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠽⡛⠿
⡀⢠⣤⠀⠀⠀⠉⠝⠙⠋⠀⠀⠀⠀⠀⠀⢀⣠⣀⣳⣤⣮⠀⠀⠀⠈⢻⣤⣤⣲⣄⣤⣤⣴⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠛⠛⠻⢸⣯⡟⠀⠀⠀⠀⠠⣾⣿⣷⣶⣦⣤⣤⡀⢀⣠⣤⡤⠤⠤⠤⣤⠀⠀
⣿⣿⣿⣿⣦⣤⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠛⠛⠙⠻⡇⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠟⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣵⠀⠀⠀⠀⣾⣿⣿⣿⡀⠉⢿⣿⣿⣿⠏⠀⠠⠴⠒⠤⠤⠀⠀
⠾⠿⠿⣿⣿⠃⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠠⡈⠹⠿⣭⣵⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣿⡟⡲⠃⠀⠀⠀⢿⣿⣿⣿⣧⣄⠀⡽⣿⣿⣤⣤⣤⣤⣤⣤⣤⣰⣾
⣀⡀⠀⠀⢈⣿⣶⣿⠿⡆⠀⠀⠀⠀⢀⣾⠿⠿⠿⠿⠿⣦⣤⣀⡄⠀⠀⠀⠀⠛⠷⠶⣿⡿⡿⠋⠉⠳⢶⣦⡀⠀⣀⣀⣠⣤⣤⣀⣀⣀⣴⣾⡿⠆⠀⣁⠀⠉⢀⣬⣿⣿⡟⠿⠛⠀⣿⣯⡟⢠⣿⣿⣿⣿⣿⣿⡿⠻
⣿⣿⣷⣶⣿⣿⠁⢹⣠⣿⡆⠀⢠⣾⣿⣿⣷⡀⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⠀⣶⣶⣶⣶⣶⡧⠀⣀⣠⣼⣿⣿⠿⠿⠛⠋⠉⠻⣿⣿⡏⢙⣋⣤⣀⣀⣀⡤⠖⣋⣸⣿⡏⠁⠀⢀⡔⠙⢿⣧⡘⠻⢿⣿⣿⣿⠟⣠⣦
⠀⠀⠹⣿⣿⠟⠋⠹⣿⣿⠇⠀⣿⣿⣿⣿⣿⠇⠀⠀⠀⣀⠀⢿⣆⠈⠛⠋⢺⣿⣿⣿⣿⣿⣶⣿⡿⠟⠛⢻⣿⡇⠀⠀⠀⢀⣾⣿⣿⣿⠀⠹⣿⡿⠀⠀⠀⢸⣿⣿⣿⣷⣧⣄⣾⠀⠁⠈⠻⢷⡀⠘⠻⠀⠁⣠⣿⡿
⣤⣄⣰⣿⣿⣧⣄⣠⣿⣿⠀⠀⢹⣿⣿⠋⠁⠀⠀⠀⣸⡇⠀⠈⠛⢿⣶⣤⣿⠿⠿⣿⣿⠿⠃⠀⠀⢀⣠⣿⣿⠁⠀⠀⠀⣸⣿⢿⣿⡿⠦⢤⣬⡄⠒⠒⠚⠋⢹⣿⡟⢸⣿⣿⣿⣶⣀⣤⣶⣾⣷⣶⣶⣤⣄⠉⢡⡤
⠀⣿⣿⣿⣿⢿⠿⠿⢿⣿⣷⡄⢸⣿⡏⠈⣁⣀⡀⠀⢹⣿⣦⡀⠀⠀⠉⠻⢿⣶⣶⣶⣶⣶⣶⠶⠖⠛⠋⠉⢻⣧⠀⢠⠞⠉⢴⣴⣿⣿⣀⢀⣩⡄⠠⠴⣶⢶⣿⣿⣷⣿⣄⢿⣿⡿⣿⣿⣿⣿⣿⢿⣿⣿⡍⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⣿⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⠀⠠⡹⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡿⢸⣿⣧⡄⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡆⢸⣷⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡇⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⡏⢁⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⢸⡇⢀⣁⣹⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠈⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⠋⠸⠿⠇⠿⠇⠿⠇⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡸⠃⠈⠉⠉⠿⠛⢿⣿⣿⡿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⡿⢿⣷⢰⣿⡟⣿⣷⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠙⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣧⣼⣿⢸⣿⡇⣿⣿⠀⢘⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⡏⠉⠁⢸⣿⡇⣿⣿⠀⢳⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣾⣿⣧⣄⣻⠿⠿⠟⠁⠀⠀⠿⠿⠿⠸⠿⠇⠀⠀⠈⠻⠷⠿⠋⠀⠘⣿⣟⢿⣿⣿⡿⠛⠉⠉⢻⣿⢟⣛⡃⠀⠀⠀⠀⠀
⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⣿⣿⠋⠀⠀⠀⠀⠀⢻⣿⠿⣷⣄⡀⠀⠀⠀
⣠⣾⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣿⡇⠀⠀⠀⠀⠀⠀⣼⠁⠄⠈⢻⣿⣷⣶⣶
⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⣰⣿⠀⠀⢐⢀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣄⣤⣾⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣿⣴⠇⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⡻⠿⠉⠉⠙⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⢿⠿⣟⣻⣽⣿⠿⠟⠋
⣙⢿⡿⡀⠀⠀⠀⢀⣠⣿⣿⠋⠉⠛⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⢀⣴⡿⢸⣿⡀⠀⠀⠀⢹⡷⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠁⠀⠀⢀⣠
⢻⣇⠟⢿⣶⣶⣿⡿⠋⠉⢻⠀⠀⠀⠨⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠳⣿⣿⠏⠀⠀⣿⣧⣄⣀⣤⣾⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣴⣶⣶⣿⣿⣿
⣸⠋⠀⠘⠿⣿⣿⣅⣀⢀⣸⣷⣶⡞⠀⠀⢀⣤⣴⣶⣤⡀⠀⠀⠀⠀⡆⠹⣿⣿⣦⣴⣤⣿⣿⣿⢻⣿⠋⠀⠀⠀⠀⠀⠀⠈⠁⠀⠻⣿⣄⠀⠈⢹⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣠⣿⣿⣿⣿⣿
⡇⠀⠀⢀⠁⢿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠾⠿⠿⠿⠛⠷⡀⠀⠀⠀⣧⡀⠘⠿⢿⢿⣿⣿⣿⣯⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⢴⢷⣶⣿⣿⣦⡴⣧⣰⡟⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠟⠛
⡃⠀⣄⡀⠀⠀⠈⠩⠋⠛⠁⠀⠀⠀⠀⠀⠀⢀⣄⣱⣦⣴⡁⠀⠀⠀⠙⣧⣄⣂⣤⣤⣤⣤⣴⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠁⣿⣽⠃⠀⠀⠀⠀⢾⣿⡿⢶⣶⣤⣤⣀⣀⣠⣤⣤⠤⠤⠤⠤⡄⠀
⢳⣷⣿⣿⣷⣤⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠛⠋⠛⢿⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⡿⢋⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣡⠆⠀⠀⠀⢰⣿⣿⣿⣇⠈⠹⣿⣿⣿⡿⠁⠀⠤⠒⠒⠤⠄⠀
⠶⢿⡿⣿⣿⡟⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠙⠿⣯⣭⣶⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣸⡿⢓⠞⠀⠀⠀⢸⣿⣿⣿⣿⢤⡀⢨⣿⣿⣧⣤⣤⣤⣤⣤⣤⣄⣶
⣄⡀⠀⠀⠀⣙⣷⣾⢿⢇⠀⠀⠀⠀⠀⣰⡽⠿⠿⠿⠿⢷⣤⣄⣠⠀⠀⠀⠀⠈⠛⠶⠿⢿⣿⠟⠋⠓⠶⣦⡀⠀⢀⣀⣀⣤⣤⣄⣀⡀⣰⣶⣿⠷⠀⢈⡀⠉⠁⣀⣽⣿⡿⠻⠟⠃⢸⣿⣟⢃⣼⣿⣿⣿⣿⣿⣿⠟
⣿⣿⣷⣶⣾⣿⡏⠈⣇⣼⣧⠀⢀⣴⣿⣿⣷⣆⠀⠀⠀⠀⠈⠙⠳⢦⣄⣀⡀⢰⣶⣶⣶⣶⣾⠄⢀⣀⣤⣿⣿⡿⠿⠛⠛⠉⠙⢿⣿⣟⢙⣋⣥⣄⢀⣀⣤⠴⢚⣁⣿⣿⠁⠀⠀⣠⢊⠻⣿⣄⠛⢿⢿⣿⣿⠿⢁⣶
⠀⠁⠈⢿⣿⡿⠛⠙⢿⣿⡟⠀⢸⣿⣿⣿⣿⡿⠀⠀⠀⢀⡀⠸⣷⡀⠙⠛⢳⣿⣿⣿⣿⣿⣷⣾⣿⠿⠛⠛⣿⣿⠀⠀⠀⠀⣸⣿⣿⣿⡆⠈⢿⣿⠃⠀⠀⠀⣿⣿⣿⣿⣾⣤⣰⡇⠀⠀⠙⠿⣆⠈⠛⠏⠁⢀⣾⡿
⣤⣤⠀⣾⣿⣷⣀⣀⣸⣿⡇⠀⠘⣿⣿⡏⠉⠀⠀⠀⢀⣿⠀⠀⠙⢻⣶⣤⣼⡿⠿⢿⣿⡿⠟⠈⠀⠀⣀⣼⣿⡏⠀⠀⠀⢀⣿⣿⣿⣿⠶⠄⣤⡅⠐⠒⠒⠋⢉⣿⣿⠃⣿⣿⣿⣧⣆⣠⣴⣶⣿⣶⣶⣦⣤⡀⠉⣤
⠁⢶⣿⣿⣿⣿⠟⠿⠿⣿⣿⡀⠀⣿⣿⠁⢁⣀⡀⠀⠈⣿⣧⡀⠀⠀⠈⠙⢿⣷⣶⣶⣶⣤⣶⠶⠖⠛⠛⠉⠙⣿⡄⠀⡴⠋⢁⣦⣼⣿⡇⠀⣨⣥⠀⠴⢶⠶⡶⣿⣿⣾⣧⢸⣿⣿⢿⣿⣿⣿⣿⠿⣿⣿⣯⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣾⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⡆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⣿⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⡹⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡟⢸⣿⣧⡄⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡇⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⡏⠁⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⢸⡇⢀⡁⣽⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠈⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠾⠿⠃⠘⠿⠿⠿⠋⠸⠿⠇⠿⠇⠿⠇⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡸⠃⠈⠉⠉⠿⠻⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⡿⢿⣷⢰⣿⡟⣿⣷⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠙⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣧⣼⣿⢸⣿⡇⣿⣿⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⡏⠉⠁⢸⣿⡇⣿⣿⠀⠘⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣾⣿⣧⣄⣻⡿⠿⠟⠁⠀⠀⠿⠿⠿⠸⠿⠇⠀⠀⠈⠻⠷⠿⠏⠀⠀⣿⣿⣿⣿⣿⣿⠟⠋⠉⠙⣿⣿⢛⣛⠀⠀⠀⠀⠀
⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢿⣿⡟⠁⠀⠀⠀⠀⠘⣿⡽⢿⣦⣀⠀⠀⠀
⢀⣴⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⣿⠀⠀⠀⠀⠀⠀⢠⡇⠠⠀⠙⣿⣿⣶⣶
⣾⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⣀⣾⡃⠀⢀⡂⣸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣤⣤⣴⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣽⣯⡏⠁⠀⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁⠀⠀⠀⠀⣿⠿⠏⠉⠉⠻⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠽⡿⣟⣛⣯⣿⡿⠿⠛
⢉⡻⣿⠇⠀⠀⠀⠀⢀⣼⣿⠟⠉⠙⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⣠⣾⠇⣿⣇⠀⠀⠀⠈⣿⠄⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠀⠀⠀⣀
⠈⢿⡼⢷⣶⣶⣾⣿⠟⠋⠛⡆⠀⠀⠀⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⢶⣿⡿⠁⠀⢸⣿⣤⣀⣠⣼⣿⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠿⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⢶⣶⣾⣿⣿
⢀⡟⠁⠀⠻⣿⣿⣧⣀⡀⢰⣿⣦⣴⠃⠀⠀⣠⣤⣤⣤⣄⠀⠀⠀⠀⢰⠸⢿⣿⣷⣤⣦⣼⣿⣿⡟⣿⡟⠁⠀⠀⠀⠀⠀⠀⠉⠀⠘⢿⣧⡀⠀⠉⡏⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣾⣿⣿⣿⣿
⣿⠀⠀⠀⡈⠸⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠰⠿⠿⠿⠟⠿⢆⠀⠀⠀⢸⣄⠀⠻⠿⡿⡿⣿⣿⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠤⡾⣶⣾⣿⣷⣔⢾⣄⣾⠃⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠝⡛
⡟⡀⢠⣀⠀⠀⠀⠉⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⣠⣈⣦⣤⡎⠀⠀⠀⠈⢻⣤⣰⣠⣤⣤⣤⣤⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠛⠛⠋⢸⣿⡟⠀⠀⠀⠀⠠⣿⣿⡷⢶⣦⣤⣤⣀⣀⣤⣤⠴⠦⠤⠤⠤⠀
⠙⣽⣷⣿⣷⣆⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠙⠛⠙⠻⡇⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠟⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡴⡀⠀⠀⠀⣾⣿⣿⣿⡀⠉⢿⣿⣿⣿⠏⠀⠀⠒⠒⠒⠤⠀
⠶⠾⣿⣿⣿⣿⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡈⠻⢯⣭⣵⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣿⠟⡲⠃⠀⠀⠀⢿⣿⣿⣿⡧⣄⠀⣽⣿⣿⣴⣶⣦⣤⣤⣶⣦⣰
⣄⣀⠀⠀⠀⢉⣳⣶⣿⡿⡄⠀⠀⠀⠀⢀⣾⠿⠿⠿⠿⢿⣦⣤⡄⡄⠀⠀⠀⠀⠛⠷⠾⢿⣿⡿⠋⠙⠲⢶⣄⠀⠀⣀⣀⣠⣤⣤⣀⣀⠀⣴⣾⡿⠆⠀⡡⠈⠉⢀⣬⣿⣿⡟⠻⠛⠀⣿⣮⡍⢰⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣿⣶⣶⣿⣿⠃⢹⣠⣿⡄⠀⣠⣾⣿⣿⣶⡀⠀⠀⠀⠀⠉⠛⠦⣤⣀⣀⠀⣶⣶⣶⣶⣶⡧⠀⣀⣤⣼⣿⣿⠿⠟⠛⠋⠙⠻⣿⣿⡋⣙⣩⣤⣀⣀⣠⡤⠖⣋⣸⣿⡏⠁⠀⢀⡔⡙⠿⣧⡘⠻⣿⣿⣿⡿⠋⣰
⠀⠉⠁⢹⣿⣿⠟⠋⠻⣿⣿⠇⠀⣿⣿⣿⣿⣿⠇⠀⠀⠀⣀⠀⢿⣆⠈⠛⠉⢾⣿⣿⣿⣿⣿⣶⣿⡿⠿⠛⢻⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⠀⠹⣿⡟⠀⠀⠀⢸⣿⣿⣿⣷⣯⣄⣾⠀⠀⠈⠻⢷⡄⠈⠛⠀⠀⣰⣿
⣀⣤⣄⢠⣾⣿⣆⣀⣀⣿⣿⠀⠀⢻⣿⣿⠋⠉⠀⠀⠀⢰⡇⠀⠈⠛⣷⣦⣤⣿⠿⠿⣿⣿⠿⠃⠀⠀⠀⣠⣿⣿⠁⠀⠀⠀⣸⣿⣿⣿⡿⠦⢤⣬⡄⠒⠒⠚⠉⣹⣿⡟⢸⣿⣿⣿⣶⣤⣴⣶⣿⣿⣶⣶⣤⣤⠈⢡
⠉⠟⣿⣿⣿⣿⠟⠻⡿⢿⣿⣇⡀⢸⣿⡏⠈⣁⣀⠀⠀⢘⣿⣄⠀⠀⠀⠉⠻⢿⣶⣶⣶⣦⣴⣶⠶⠞⠛⠋⠉⢻⣧⠀⢠⠞⠉⣰⣼⣿⣿⡀⢀⣩⡄⠠⠶⡶⢶⢿⣿⣿⣿⣄⣿⣿⡿⣿⣿⣿⣿⡿⢿⣿⣿⡅⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⣯⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡀⠠⡹⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡏⢸⣿⣧⡄⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣄⣀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⡏⠁⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⢸⡇⢀⣁⣽⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⠋⠸⠿⠇⠿⠇⠿⠇⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡸⠃⠈⠉⠉⠿⠛⢿⣿⣿⡿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⢀⣿⡿⠿⢸⣿⡿⢿⣷⢰⣿⡟⣿⣷⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠛⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣧⣼⣿⢸⣿⡇⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⡏⠉⠁⢸⣿⡇⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣿⣧⣤⣿⣿⡿⠟⠁⠀⠀⠿⠿⠿⠸⠿⠇⠀⠀⠈⠻⠷⠿⠏⠀⠀⠀⣿⣿⣻⣿⣿⣿⠟⠉⠉⠙⣿⣟⣛⣥⠀⠀⠀⠀
⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⡏⣿⠟⠁⠀⠀⠀⠀⠘⣿⠽⢿⣦⣄⡀⠀
⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇⠠⠀⠙⣿⣿⣷
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⢀⣠⣿⡀⠀⢀⣃⣸⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠸⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣤⣤⣶⣿⣿⣿⣶⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡟⠉⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⠀⢈⡿⠿⠏⠉⠙⠻⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠹⣿⣿⣻⣯⣿⡿⠟
⠉⢛⠿⣿⡇⠀⠀⠀⠀⠀⣸⣿⠟⠋⠛⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⣠⡾⠇⣿⡆⠀⠀⠀⠈⣿⠂⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠁⠀⠀⢀
⠀⢸⣿⡿⣾⣶⣶⣶⣿⡿⠋⠛⡆⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣶⣿⡟⠁⠀⢻⣿⣄⣀⣠⣼⣿⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠿⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⡶⢶⣶⣾⣿
⠀⢠⡿⠁⢈⠿⣿⣿⣯⣀⠀⢸⣿⣤⣴⠂⠀⠀⣀⣤⣤⣤⣄⠀⠀⠀⠀⢰⠘⢿⣿⣷⣤⣦⣼⣿⣿⡟⣿⡟⠉⠀⠀⠀⠀⠀⠀⠉⢀⠘⢿⣇⠀⠀⠙⠏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢼⣾⣿⣿⣿
⢀⣾⠀⠀⠀⡘⢸⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⢰⠿⠿⡿⠿⠿⣆⠀⠀⠀⢼⣄⠀⠻⢿⣿⣿⣿⣿⣿⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⢤⡾⢶⣾⣿⣷⣴⢾⣄⣿⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⢨⣿⠀⢠⣀⠀⠂⠈⠉⡟⠛⠋⠀⠀⠀⠀⠀⠀⠀⣠⡀⣦⣠⣎⠀⠀⠀⠈⢿⣤⣰⣠⣤⣤⣤⣤⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠛⠛⠋⢸⣽⡏⠀⠀⠀⠀⢰⣿⣿⡿⢿⣶⣤⣤⣀⣠⣤⣴⠶⠲⠦⠤⠤
⠀⠻⣿⣿⣿⣷⣆⣀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠻⠛⠛⠿⡇⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠟⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⡶⡀⠀⠀⢀⣾⣿⣿⣿⡄⠉⢿⣿⣿⣿⠃⠀⠐⠒⠒⠒⠠
⠐⣶⣾⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡈⠿⣿⣭⣴⣾⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣿⣛⡺⠁⠀⠀⠀⢿⣿⣿⣿⣧⣄⠀⣽⣿⣿⢴⣶⣦⣶⣶⣶⣦
⣦⣄⣀⠀⠀⠈⢉⣳⣶⣿⡿⡀⠀⠀⠀⠀⢠⣶⠿⠿⠿⠿⣿⣦⣤⣀⡄⠀⠀⠀⠀⠻⠷⢾⣿⣿⡿⠛⠛⠲⢶⣄⠀⠀⣀⣀⣠⣤⣤⣀⣀⣀⣶⣾⡿⠆⠈⡡⠈⠉⢠⣬⣿⣿⡯⠟⠛⠀⣿⣿⡍⣴⣿⣿⣿⣿⣿⣿
⢿⣿⣿⣷⣶⣶⣾⣿⠋⢹⣠⣿⡄⠀⣠⣾⣿⣿⣦⡀⠀⠀⠀⠀⠉⠻⢦⣤⣀⣀⠀⣶⣶⣶⣦⣶⡧⠀⣀⣠⣼⣿⣿⠿⠿⠛⠋⠙⢻⣿⣿⡋⣙⣩⣤⣀⣠⣤⠤⠖⣋⣸⣿⡏⡁⠀⢀⠜⡙⠿⣧⡘⠻⣿⣿⣿⡿⠛
⠀⠈⠉⠉⢻⣿⣿⠟⠒⠻⣿⣿⠇⠀⣿⣿⣿⣿⣿⠇⠀⠀⠀⣀⠀⢿⣆⠉⠛⠛⣾⣿⣿⣿⣿⣿⣶⣿⡿⠿⠛⢻⣿⡇⠀⠀⠀⢀⣿⣿⣿⣷⠀⠹⣿⡟⠀⠀⠀⢸⣿⣿⣿⣷⣯⣤⣾⠀⠀⠈⠛⠷⡄⠈⠛⠀⠀⣰
⣀⣀⣀⡄⢠⣿⣿⣆⣀⣀⣿⣿⠀⠀⢿⣿⣿⠋⠉⠀⠀⠀⣰⡇⠀⠘⠻⣷⣦⣤⣿⠿⠿⣿⣿⠿⠋⠁⠀⢀⣠⣿⣿⠁⠀⠀⠀⣸⣿⣿⣿⡷⠶⢤⣬⡀⠒⠒⠛⠉⣹⣿⡟⣸⣿⣿⣿⣶⣤⣴⣾⣿⣿⣷⣶⣤⣤⠀
⠉⠟⠻⣿⣿⣿⣿⠻⠿⡿⢿⣿⣇⡄⢸⣿⡏⠀⢈⡀⠀⠀⢻⣷⣄⠀⠀⠀⠉⠻⣿⣶⣶⣶⣤⣶⣶⠶⠞⠛⠛⠉⢿⣧⠀⢠⠞⠉⣼⣼⣿⣿⢀⢠⣭⣄⠀⠶⠶⢶⣿⣿⡿⣿⣄⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⣯⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⠀⠠⡹⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣿⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⣀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡆⢸⣷⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡇⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⡏⠁⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢸⡇⢀⣁⣽⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⠋⠸⠿⠇⠿⠇⠿⠇⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡸⠃⠈⠉⠉⠿⠛⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⡿⢿⣷⢠⣿⡟⣿⣷⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠙⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣧⣼⣿⢸⣿⡇⣿⣿⠀⠀⢸⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⡏⠉⠁⢸⣿⡇⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣿⣧⣤⣿⣿⡿⠟⠁⠀⠀⠿⠿⠿⠸⠿⠇⠀⠀⠈⠻⠷⠿⠏⠀⠀⠀⣿⣿⢻⣿⣿⣿⠟⠉⠉⠙⣿⣛⣛⣥⠀⠀⠀⠀
⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠻⡏⣿⠟⠁⠀⠀⠀⠀⠘⣿⠽⢿⣦⣀⡀⠀
⠀⠀⣴⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣿⣿⠀⠀⠀⠀⠀⠀⢰⡇⠠⠀⠙⣿⣿⣷
⢀⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⣠⣿⡀⠀⢀⣃⣸⣿⣿
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠸⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣤⣤⣶⣿⣿⣿⣶⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣻⣿⣿⡏⠉⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⠀⢈⡟⠿⠏⠉⠙⠻⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣟⣻⣯⣾⡿⠟
⠉⠛⡻⣿⡇⠀⠀⠀⠀⠀⣸⣿⠟⠉⠛⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⣠⡾⠇⣿⡆⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠁⠀⠀⢀
⠀⠘⣿⡽⣾⣶⣶⣶⣿⡿⠋⠛⡆⠀⠀⠀⠟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣶⣿⡿⠁⠀⢻⣿⣄⣀⣠⣼⣿⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠿⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⢶⣶⣾⣿
⠀⠀⡟⠁⠀⠻⣿⣿⣏⣀⠀⢸⣿⣤⣴⠂⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⢰⠈⢿⣿⣷⣤⣦⣼⣿⣿⠟⣿⠟⠉⠀⠀⠀⠀⠀⠀⠉⢀⠘⢿⣇⠀⠀⠙⠏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢴⣾⣿⣿⣿
⢀⣺⠀⠀⠀⡘⢸⣿⣿⣿⣿⣿⡿⠟⠃⠀⠀⢰⠿⠿⡿⠿⠿⡆⠀⠀⠀⢸⡄⠀⠻⢿⣿⣿⣿⣿⣿⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⢤⠶⢶⣾⣿⣷⣴⢾⣄⣿⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠸⣻⠀⢠⣀⠀⠂⠈⠉⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⣠⡀⣦⣠⡎⠀⠀⠀⠈⢿⣤⣰⣠⣤⣤⣤⣤⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠚⠛⠋⢸⣽⡏⠀⠀⠀⠀⢰⢿⣿⡿⢿⣶⣤⣤⣀⣀⣤⣴⠶⠲⠦⠤⠤
⠀⠻⢿⣶⣿⣷⣆⣀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠹⠛⠛⠿⡇⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠟⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡖⡀⠀⠀⢀⣾⣿⣿⣿⡀⠉⢿⣿⣿⣿⠃⠀⠐⠒⠒⠒⠠
⠐⣶⣾⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡈⠻⣿⣩⣴⣾⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣿⣛⡺⠁⠀⠀⠀⢿⣿⣿⣿⣧⣄⠀⡽⣿⣿⢴⣶⣦⣴⣶⣶⣦
⣦⣄⣀⠀⠀⠈⢉⣳⣶⣿⡿⡀⠀⠀⠀⠀⢀⣴⠿⠿⠿⠿⢿⣦⣤⣀⡀⠀⠀⠀⠀⠻⠶⠾⣿⣿⡿⠛⠙⠲⢶⣄⠀⠀⣀⣀⣠⣤⣤⣀⣀⠀⣶⣾⡿⠆⠀⡡⠈⠉⢀⣬⣿⣿⠛⠟⠛⠀⣿⣿⡍⢰⣿⣿⣿⣿⣿⣿
⢿⣿⣿⣷⣶⣶⣾⣿⠋⢹⢀⣿⡄⠀⣠⣶⣿⣿⣦⡀⠀⠀⠀⠀⠉⠻⢦⣤⣀⡀⠀⣶⣶⣶⣦⣶⡧⠀⣀⣠⣼⣿⣿⠿⠿⠛⠋⠙⢻⣿⣿⡋⣙⣩⣤⣀⣠⣤⠤⠖⣋⣸⣿⡏⠁⠀⢀⠜⡉⠿⣧⡘⠻⣿⣿⣿⡿⠛
⠀⠈⠉⠁⢻⣿⣿⠟⠒⠻⣿⣿⠇⠀⣿⣿⣿⣿⣿⠇⠀⠀⠀⣀⠀⢿⣄⠉⠛⠛⣾⣿⣿⣿⣿⣿⣶⣿⡿⠿⠛⢻⣿⡆⠀⠀⠀⢀⣿⣿⣿⣿⠀⠹⣿⡟⠀⠀⠀⢸⣿⣿⣿⣷⣯⣄⣾⠀⠀⠈⠛⠷⡄⠈⠛⠀⠀⣰
⣀⣀⣀⡄⢠⣿⣿⣆⣀⣀⣿⣿⠀⠀⢿⣿⣿⠋⠉⠀⠀⠀⣰⡇⠀⠈⠻⣷⣦⣤⣿⠿⠿⣿⣿⠿⠋⠀⠀⢀⣠⣿⣿⠁⠀⠀⠀⣸⣿⣿⣿⡗⠂⢠⣬⡀⠒⠒⠚⠉⣹⣿⡟⣸⣿⣿⣿⣶⣤⣴⣾⣿⣿⣷⣶⣤⣤⠀
⠉⠟⠻⣿⣿⣿⣿⠻⠿⡿⢿⣿⣇⡄⢸⣿⡏⠀⢈⡀⠀⠀⢹⣷⣄⠀⠀⠀⠉⠻⣿⣶⣶⣶⣤⣤⠴⠶⠞⠛⠋⠉⢻⣧⠀⢠⠞⠉⣼⣼⣿⣿⠀⢀⣭⡄⠀⠶⠶⢶⣿⣿⡿⣿⣄⢿⣿⣿⢿⣿⣿⣿⣿⢿⣿⣿⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⣿⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡄⢠⡹⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⣟⢸⣿⣧⡄⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣄⣀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⡏⠁⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⢀⣁⣽⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⠋⠸⠿⠇⠿⠇⠿⠇⠸⠿⠿⢿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡸⠃⠈⠉⠉⠿⠛⢿⣿⣿⡿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⡿⢿⣷⢰⣿⡟⣿⣷⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠙⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣧⣼⣿⢸⣿⡇⣿⣿⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⡏⠉⠁⢸⣿⡇⣿⣿⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣧⣤⣿⣿⣿⠟⠁⠀⠠⠿⠿⠿⠸⠿⠇⠀⠀⠈⠻⠷⠿⠏⠀⠀⠀⠈⣿⣿⣻⣿⣿⡿⠛⠉⠉⠙⣿⣟⣫⣥⠀⠀⠀
⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣷⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠻⡏⣿⠏⠀⠀⠀⠀⠀⠸⡿⠿⢿⣦⣄⡀
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠀⠀⠀⠀⠀⠀⢸⠇⠠⠀⠙⣿⣿
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⢀⣠⣿⡀⠀⢀⣃⣼⣿
⠀⣾⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣠⣤⣾⣿⣿⣿⣷⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⡟⠉⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠉⠀⠀⠀⠈⣿⢿⠏⠉⠙⠻⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣟⣿⣻⣿⣾⠿
⠀⠙⠛⠿⣿⡇⠀⠀⠀⠀⠀⣸⣿⠟⠛⠻⢿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⣠⣾⠧⣿⡆⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠁⠀⢀
⠀⠀⢸⣿⡻⣿⣦⣤⣶⣾⡿⠛⢻⡄⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠶⣶⣾⡿⠁⠀⢿⣷⣀⣀⣠⣼⣏⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠿⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⣷⣾⣿
⠂⠀⢀⡿⠃⠈⢿⣿⣿⣏⠀⠀⢸⣷⣤⡴⠂⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⠀⢰⠸⢿⣿⣷⣤⣤⣼⣿⣿⡟⣿⠟⠁⠀⠀⠀⠀⠀⠀⠉⢀⠙⢿⣇⠀⠀⢙⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢴⣾⣿⣿
⠀⠀⣾⠁⠀⠀⠐⢹⣿⣿⣿⣷⣿⣿⠿⠃⠀⠀⢰⡿⢿⣿⠿⠿⡆⠀⠀⠀⣼⡀⠀⠻⢿⣿⣿⣿⣿⣿⣺⣿⠀⠀⠀⠀⠀⠀⠀⠀⢤⡶⢶⣾⣿⣷⣴⢾⣄⣿⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⣿⠀⢀⢀⠁⠀⠉⠛⡟⠛⠛⠀⠀⠀⠀⠀⠀⠀⡠⠐⣦⣀⡆⠀⠀⠀⠈⢿⣤⣰⣠⣤⣤⣤⣤⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠚⠙⠋⢸⣽⡏⠀⠀⠀⠀⢰⣿⣿⡿⢿⣶⣴⣤⣀⣀⣤⡶⠖⠒⠒⠤
⠒⠊⢻⣿⣿⣿⣶⣄⣀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠻⠟⠛⢿⡇⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠟⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⡖⡀⠀⠀⢀⣿⣿⣿⣿⡀⠈⢻⣿⣿⣿⠃⠀⠐⠒⠒⠒
⠀⠰⣦⣽⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢈⠩⠻⣛⣫⣵⣾⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣿⣛⡺⠁⠀⠀⠀⢿⣿⣿⣿⣧⣄⠀⣽⣿⣿⢶⣶⣶⣶⣶⣶
⣶⣤⣀⡀⠈⠀⠉⠙⣶⣴⣾⣿⡄⠀⠀⠀⠀⢀⣶⣿⠿⠿⠿⣿⣤⣤⠀⡀⠀⠀⠀⠀⠻⠶⣾⣿⣿⡿⠚⠒⠲⣶⣄⠀⠀⣀⣀⣠⣤⣤⣀⣀⣀⣶⣾⡿⠆⠀⡡⠈⠉⢀⣼⣿⣿⡏⠟⠛⠀⣿⣿⡍⢰⣿⣿⣿⣿⣿
⠻⢿⣿⣿⣷⣦⣤⣾⣿⠋⢹⣡⣷⡄⠀⢀⣴⣾⣿⣦⠀⠀⠀⠀⠈⠉⠻⢦⣄⣀⡀⠀⣶⣤⣴⣦⣶⡧⠀⢀⣠⣼⣿⣿⠿⠿⠛⠋⠙⠻⣿⣿⡋⣙⣩⣤⣀⣠⣤⠤⠖⣋⣸⣿⣿⠀⠀⢀⠌⡈⠿⣷⡈⠛⣿⣿⣿⡿
⠀⠀⠉⠉⠉⢻⣿⣿⡿⠒⠺⣿⣿⡇⠀⣿⣿⣿⣿⣿⠇⠀⠀⠀⢀⠀⢿⣄⠉⠻⠟⣶⣿⣿⣿⣿⣿⣴⣿⡿⠿⠛⢻⣿⡆⠀⠀⠀⢀⣿⣿⣿⣷⠀⠹⣿⠟⠀⠀⠀⢸⣿⣿⣿⣷⣧⣆⣾⠀⠀⠈⠛⠿⡄⠈⠙⠀⠀
⠀⢀⣀⣀⡀⢠⣽⣿⣆⣀⡀⣿⣿⠀⠀⢿⣿⣿⠋⠉⠀⠀⠀⣰⡇⠀⠈⠿⣷⣤⣤⣿⠿⠿⣿⣿⡿⠃⠁⠀⠀⣠⣿⣿⠁⠀⠀⠀⣸⡿⣿⣿⡿⠦⢦⣬⡀⠂⠒⠛⠉⣹⣿⡟⣸⡿⣿⣿⣶⣤⣴⣾⣿⣿⣷⣶⣦⣤
⠙⠟⠿⣿⢿⣿⣿⣿⠿⠿⡿⢿⣿⣇⡄⢸⣿⡟⠐⠈⠀⠀⠀⢸⣷⣄⠀⠀⠈⠉⠻⣿⣴⣶⣶⣤⣤⣤⡶⠞⠛⠛⠉⢻⣧⠀⢠⠞⠉⣼⣽⣿⣿⣀⢠⣭⡄⠐⠶⠶⢶⣿⣿⡿⣿⣄⢿⣿⣿⢿⣿⣿⣿⣿⢿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡿⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡄⠀⡹⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠈⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡟⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡟⣿⡇⢸⣿⡏⠁⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⢀⣁⣽⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⠋⠸⠿⠇⠿⠇⠿⠇⠸⠿⠿⢿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡸⠃⠈⠉⠉⠿⠻⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⡿⢿⣷⢠⣿⡟⣿⣷⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠙⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣧⣼⣿⢸⣿⡇⣿⣿⠀⠀⠀⢀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⢀⣿⡏⠉⢸⣿⡏⠉⠁⢸⣿⡇⣿⣿⠀⠀⠀⠘⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣧⣤⣿⣿⣿⡟⠁⠀⠨⠿⠿⠿⠸⠿⠇⠀⠀⠈⠻⠷⠿⠏⠀⠀⠀⠀⢻⣿⣟⣿⣿⣿⠟⠉⠈⠉⢻⣿⣩⣭⡄⠀⠀
⠀⠀⠀⠀⠀⢀⣠⣶⣶⣿⣿⣿⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⢿⢹⡟⠁⠀⠀⠀⠀⠀⣿⠯⠿⢷⣤⣀
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡃⠀⠀⠀⠀⠀⢀⡟⠀⠄⠈⢻⣿
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⣀⣼⣇⠀⠀⣈⣠⣿
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⠟⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠁⠀⠀⠀⢹⡿⡟⠉⠉⠛⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣻⣿⣿⣿⡿
⠀⠈⠛⠿⣿⣿⠀⠀⠀⠀⠀⢀⣿⡿⠛⠛⠿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⠀⢀⣴⡿⢼⣧⠀⠀⠀⠀⣻⡇⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡉⠉⠀⠀⠀
⠀⠀⠀⢻⣏⣿⣦⣤⣴⣶⣿⠟⠛⣧⠀⠀⠀⢸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⣶⣿⠋⠀⠸⣿⣦⣀⣀⣴⣿⡁⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠿⠿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠾⣾⣿
⠐⠀⠀⢸⠟⠀⠻⣿⣿⣿⡃⠀⠀⣿⣦⣤⡖⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⡆⢾⣿⣿⣦⣤⣤⣿⣿⡿⢻⡿⠉⠀⠀⠀⠀⠀⠀⠊⠀⡈⠻⣿⡀⠀⠈⡛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠿⠿⣿
⠀⠀⢰⡏⠀⠀⠀⠊⣿⣿⣿⣿⣾⣿⡿⠿⠀⠀⠀⣾⠿⣿⡿⠿⢧⠀⠀⠀⢰⣇⠀⠙⠿⣿⣿⣿⣿⣿⣟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⣼⠶⣶⣿⣿⣶⠾⣧⣸⡇⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣿⡇⠀⡀⠈⠂⠈⠙⢻⠛⠿⠃⠀⠀⠀⠀⠀⠀⢀⡄⢲⣄⣰⠁⠀⠀⠀⠻⣧⣄⣦⣄⣤⣤⣤⣴⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠒⠋⠛⠀⣯⣿⠁⠀⠀⠀⠀⣾⣿⣿⡿⣷⣦⣶⣤⣀⣤⣶⠶⠒⠒⠒
⠒⠒⠙⢿⣷⣿⣷⣦⣀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠿⠛⠿⣿⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⢋⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣵⢂⠀⠀⠀⣸⣿⣿⣿⣧⠀⠙⢿⣽⣿⡟⠀⠀⠒⠒⠒
⠀⠀⠶⣬⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡩⠝⢿⣛⣭⣴⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣸⣟⣳⠏⡀⠀⠀⠸⣿⣿⣿⣿⣦⡄⢸⣿⣿⡷⢶⣶⣶⣶⣶
⣷⣦⣄⡀⠀⠁⠈⠉⢳⣦⣴⣿⣧⠀⠀⠀⠀⠀⢠⣾⣿⠿⡿⢿⣧⣤⣄⣠⠀⠀⠀⠀⠸⠷⣶⣿⣿⣿⠗⠚⠓⢶⣦⡀⠀⢀⣀⣠⣤⣤⣄⣀⣀⣰⣶⣿⠗⠀⠠⠄⠈⢁⣤⣽⣿⣿⠿⠛⠁⢸⣿⣯⠁⣾⣿⣿⣿⣿
⠻⢿⣿⣿⣿⣦⣤⣴⣿⡿⠉⡏⣼⣦⠀⠀⣠⣶⣿⣷⡄⠀⠀⠀⠀⠉⠙⢷⣤⣀⣀⠀⢰⣦⣤⣦⣴⣾⠄⠀⣀⣤⣿⣿⡿⠿⠟⠛⠉⠛⣿⣿⣟⣉⣋⣥⣄⣠⣤⡤⠖⣊⣀⣿⣿⢉⡀⠀⡰⠃⠙⢿⣆⠙⠻⣿⣿⣿
⠀⠀⠈⠙⠛⠙⣿⣿⣿⠗⠒⢿⣿⣿⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠸⣧⡈⠙⠿⢳⣾⣿⣿⣿⣿⣷⣾⣿⠿⠟⠛⣿⣷⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⢿⣿⠃⠀⠀⠀⣿⣿⣿⣿⣾⣶⣴⡇⠀⠀⠈⠻⢧⠀⠉⠋⠀
⠀⠀⢀⣀⣀⠀⣬⣿⣷⣀⡀⢸⣿⡏⠀⠸⣿⣿⡿⠙⠁⠀⠀⢀⣾⠀⠀⠙⢿⣶⣤⣼⡿⠿⢿⣿⣿⠟⠉⠀⠀⢀⣼⣿⡏⠀⠀⠀⢀⣿⣿⣿⣿⡗⠲⣤⣥⠐⠐⠛⠉⢉⣿⣿⠃⣿⢿⣿⣷⣦⣤⣶⣿⣿⣿⣷⣶⣶
⠙⠟⠿⣿⠟⣿⣿⡿⣿⠿⢿⠿⣿⣿⣀⠀⣿⣿⠃⠂⠉⠀⠀⠀⣿⣦⡀⠀⠀⠉⠙⢿⣧⣶⣶⣦⣤⣤⣤⠶⠛⠛⠋⠙⣿⡄⠀⡴⠋⠁⣧⣿⣿⣧⡀⢬⣥⡀⠲⠶⠶⢾⣿⣿⠿⣷⡘⣿⣿⡿⣿⣿⣿⣿⡟⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⣏⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡄⢀⡹⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡿⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡟⣿⡇⢸⣿⡏⠁⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⢀⣁⣽⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠷⠿⠋⠸⠿⠇⠿⠇⠿⠇⠸⠿⠿⠿⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡸⠃⠈⠉⠉⠿⠻⢿⣿⣿⡿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⡿⢿⣷⢰⣿⡟⣿⣷⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠙⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣧⣼⣿⢸⣿⡇⣿⣿⠀⠀⠀⠀⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⡏⠉⠁⢸⣿⡇⣿⣿⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣧⣤⣿⣿⣿⡟⠁⠀⠰⠿⠿⠿⠸⠿⠇⠀⠀⠈⠻⠷⠿⠏⠀⠀⠀⠀⠸⣿⣯⣿⣿⣿⡿⠋⠁⠉⠙⣿⣏⣭⣥⠀⠀
⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣿⣷⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢻⡏⣿⠋⠀⠀⠀⠀⠀⢸⡿⠟⠿⣦⣤
⠀⠀⠀⠀⣠⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡇⠀⠀⠀⠀⠀⠀⣼⠃⠠⠀⠙⣿
⠀⠀⢀⣾⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⣀⣴⣿⡀⠀⣠⣁⣾
⠀⠀⣞⣿⣯⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡼⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡿⣿⣿⣿⡟⠋⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠘⡿⠿⠋⠉⠙⠿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠻⣟⣿⣽⣷⡿
⠀⠀⠙⠻⢿⣿⡇⠀⠀⠀⠀⠀⣸⣿⠟⠛⠻⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⣠⡾⢷⣿⡄⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠉⠁⠀⠀
⠀⠀⠂⠸⣷⣹⣷⣤⣤⣴⣶⡿⠛⢿⡄⠀⠀⠀⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠶⣶⣾⡟⠁⠀⢿⣷⣀⣀⣠⣾⡏⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠿⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢝⠗⢷⣿
⠀⠂⠀⢀⡿⠃⠙⢿⣿⣿⣏⠀⠀⢸⣧⣤⡴⠂⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⢸⠸⣿⣿⣧⣤⣤⣼⣿⣿⡟⣿⠏⠁⠀⠀⠀⠀⠀⠐⠁⢀⠙⢿⣆⠀⠀⢙⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠾⠿⣿
⠄⠀⠀⣾⠁⠀⠀⠐⢹⣿⣿⣿⣷⣿⣿⠿⠇⠀⠀⢸⡿⢿⣿⠿⢿⡄⠀⠀⠀⣾⡀⠈⠿⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢤⡷⢶⣿⣿⣷⡶⢿⣄⣿⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢸⣿⠀⢀⠀⠁⠀⠉⠛⡿⠻⠟⠀⠀⠀⠀⠀⠀⠀⣠⠐⣆⣀⡄⠀⠀⠀⠘⣿⣄⣰⣤⣠⣤⣤⣤⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠚⠙⠃⢸⣽⡏⠀⠀⠀⠀⢰⣿⣟⣿⢿⣶⣶⣦⣄⣠⣴⠶⠖⠒⠒
⠐⠢⠀⠻⣶⣮⣿⣶⡄⣀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠻⠟⠻⢿⡇⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠟⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⡖⡀⠀⠀⢀⣿⣿⣿⣿⡄⠈⢻⣿⣿⣿⠃⠀⠐⠂⠉
⠀⠀⠠⢤⣽⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢈⠝⢿⣛⣫⣥⣾⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣿⣛⡾⢁⠀⠀⠀⢿⣿⣿⣿⣧⣤⠀⣿⣿⣿⠶⣶⣶⣶⣶
⣿⣶⣤⣁⠀⠉⠉⠉⠛⢦⣤⣶⣷⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⢿⣿⣤⣄⣀⡀⠀⠀⠀⠀⠿⢶⣾⣿⣿⡿⠒⠓⠲⣶⣄⠀⠀⣀⣀⣤⣤⣤⣀⣀⣀⣶⣾⡿⠂⠀⠤⠈⠉⣠⣼⣿⣿⡟⠟⠋⠀⣿⣿⡍⢰⣿⣿⣿⣿
⠛⠿⣿⣿⣿⣶⣤⣤⣾⣿⠏⢻⣁⣷⡄⠀⢀⣴⣾⣿⣦⠀⠀⠀⠀⠈⠙⠻⣦⣄⣀⡀⠀⣶⣤⣤⣤⣶⡧⠀⣀⣠⣼⣿⣿⠿⠿⠛⠉⠉⢻⣿⣿⡋⣉⣩⣤⣀⣤⣤⠴⢒⣁⣼⣿⡏⣁⠀⢀⠎⡈⠻⣷⡌⠛⢿⣿⣿
⠀⠀⠀⠉⠛⠛⢻⣿⣿⡿⠖⢺⣿⣿⡇⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⠀⣷⣄⠙⠻⠟⣶⣿⣿⣿⣿⣿⣶⣿⡿⠿⠛⢻⣿⡆⠀⠀⠀⢀⣿⣿⣿⣿⠀⠸⣿⠟⠀⠀⠀⢸⣿⣿⣿⣿⣷⣄⣾⠀⠀⠀⠙⠻⡄⠈⠙⠀
⠀⠀⠀⠀⣀⡀⢀⣽⣿⣆⠀⠀⣿⣿⠁⠀⣿⣿⣿⠟⠋⠀⠀⠀⣰⡇⠀⠘⠻⣶⣦⣤⣿⠿⠿⣿⣿⡿⠋⠁⠀⠀⣠⣿⣿⠁⠀⠀⠀⣸⣿⣿⣿⡿⠒⢦⣬⡀⠂⠚⠋⠉⣹⣿⡟⣸⣿⣿⣿⣶⣤⣶⣾⣿⣿⣿⣷⣶
⠙⠟⠿⣿⠟⢿⣿⣿⣿⡿⠿⣿⢿⣿⣇⡀⢸⣿⡟⠐⠊⠀⠀⠀⢹⣷⣄⠀⠀⠈⠉⠻⣿⣴⣶⣶⣤⣤⣤⠶⠞⠛⠛⠉⢻⣧⠀⢠⠞⠉⢰⣴⣿⣿⣄⠠⣭⣄⠐⠲⠶⠶⣿⣿⡿⢿⣆⢻⣿⣿⢿⣿⣿⣿⣿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⣯⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡄⠠⡹⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡟⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡇⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⡏⠁⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⢀⣁⣽⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⠋⠸⠿⠇⠿⠇⠿⠇⠸⠿⠿⠿⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡸⠃⠈⠉⠉⠿⠛⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⡿⢿⣷⢰⣿⡟⣿⣷⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠛⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣧⣼⣿⢸⣿⡇⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⡏⠉⠁⢸⣿⡇⣿⣿⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⣤⣿⣿⣿⣟⠁⠀⠸⠿⠿⠿⠸⠿⠇⠀⠀⠈⠻⠷⠿⠏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠛⠉⠀⠉⢿⣿⣭⣽⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⣻⣿⡟⠁⠀⠀⠀⠀⠈⣿⠟⠻⣷⣤
⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠈⢻
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⣠⣾⣇⠀⢀⣌⣼
⠀⠀⣰⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⠏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣤⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⣻⣿⣿⣿⣿⣿⣿
⠀⠀⢿⣿⣿⣿⣿⠛⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⣻⠿⡏⠉⠙⠻⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⣿⣫⣯⣿⢾
⡇⠀⠘⠛⠿⣿⣿⠀⠀⠀⠀⠀⢀⣿⣿⠛⠻⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢀⣴⠟⣿⣧⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡉⠉⠀⠀
⠀⠀⠀⠀⣷⡞⣯⣤⣤⣤⣶⣾⠟⠻⣇⠀⠀⠀⢹⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣾⣿⠃⠀⢸⣿⣄⣀⣀⣴⣿⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠿⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡻⢻⣾
⠀⠐⠀⠀⣾⠗⠈⠻⣿⣿⣿⡁⠀⠀⣿⣤⣤⠖⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⡆⢿⣿⣿⣤⣤⣤⣿⣿⡿⣻⡿⠉⠀⠀⠀⠀⠀⠀⠋⠀⠈⠿⣧⡀⠀⠈⡏⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠿⠿
⠠⠀⠀⠰⡇⠀⠀⠁⠪⣿⣿⣿⣿⣾⣿⡿⠿⠀⠀⢀⣾⣿⣿⡿⠿⣆⠀⠀⠀⢸⡂⠀⠹⠿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠠⡼⡲⣾⣿⣿⣶⠾⣧⣸⡇⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣿⡇⠀⡀⠈⠠⠈⠛⢻⠿⠿⠃⠀⠀⠀⠀⠀⠀⢀⠄⢢⣀⣰⠀⠀⠀⠈⢻⣦⣄⣤⣄⣤⣤⣤⣴⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠒⠋⠛⠀⣧⡿⠀⠀⠀⠀⠀⣿⣿⣿⠿⣷⣶⣶⣤⣀⣤⡶⠶⠒⠒
⠀⠢⠄⠙⣷⣶⣽⣷⣤⢀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢘⠻⠿⠛⠿⣯⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣵⢂⠀⠀⠀⣼⣿⣿⣿⣧⠀⠙⣿⣿⣿⡟⠀⠐⠒⠉
⡀⠀⠀⢤⣬⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⡉⠿⣛⣛⣭⣶⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⡳⠏⠀⠀⠀⠸⣿⣿⣿⣿⣤⡀⣼⣿⣿⡷⢶⣶⣶⣶
⣻⣷⣤⣈⠀⠈⠉⠉⠙⠳⣤⣴⣿⡆⠀⠀⠀⠀⠀⣰⣾⣿⣿⡿⣿⣧⣤⡀⣀⠀⠀⠀⠀⠺⠷⣶⣿⣿⣿⠗⠒⠒⢶⣤⡀⠀⢀⣀⣠⣤⣤⣄⣀⣀⣰⣶⣿⠗⠀⠠⠄⠉⢁⣤⣿⣿⣿⠱⠛⠁⢻⣿⣯⠁⣾⣿⣿⣿
⠙⠻⢿⣿⣿⣷⣦⣤⣴⣿⡟⠙⣏⣸⣆⠀⠀⣰⣶⣿⣷⡄⠀⠀⠀⠈⠉⠻⢧⣄⣀⡀⠀⢰⣤⣤⣤⣴⣾⠀⢀⣀⣤⣿⣿⡿⠿⠟⠋⠉⠛⣿⣿⣟⢉⢉⣥⣄⣠⣤⡤⠖⣋⣠⣿⣿⢀⡀⠀⡰⢁⠹⣿⣆⠙⢻⣿⣿
⠀⠀⠀⠉⠛⠛⠛⣿⣿⣿⠷⠲⣿⣿⣿⠀⢸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸⣦⠈⠛⠿⢳⣾⣿⣿⣿⣿⣶⣾⣿⠿⠟⠛⣿⣷⠀⠀⠀⠀⣸⣿⣿⣿⡆⠈⢿⡿⠃⠀⠀⢀⣿⣿⣿⣿⣾⣶⣼⡇⠀⠀⠈⠛⢧⠀⠉⠃
⠀⠀⠀⠀⠀⣀⠂⣹⣿⣷⡀⠀⢸⣿⡇⠀⢸⣿⣿⡟⠛⠁⠀⠀⢀⣾⠁⠀⠛⢷⣦⣤⣼⡿⠿⢿⣿⣿⠟⠁⠀⠀⢀⣼⣿⡏⠀⠀⠀⢀⣿⣿⣿⣿⠓⠰⣤⣄⠐⠘⠛⠉⢉⣿⣿⢃⣿⢿⣿⣷⣦⣴⣾⣿⣿⣿⣿⣶
⠙⠟⠿⣿⠿⢿⣿⣿⣿⣿⠿⢿⡿⣿⣿⣠⠀⣿⣿⠃⠒⠁⠀⠀⠘⣿⣦⡀⠀⠀⠉⠹⢿⣷⣶⣶⣤⣤⣤⡰⠶⠛⠛⠋⠹⣿⡄⠀⡴⠋⢡⣦⣿⣿⣧⠀⢬⣥⡀⠒⠶⠶⢾⣿⣿⠿⣧⡘⣿⣿⡿⣿⣿⣿⣿⡟⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⣟⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡄⠠⡹⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⣿⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡟⣿⡟⢸⣿⡏⠁⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⢀⣁⣽⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠷⠿⠋⠸⠿⠇⠿⠇⠿⠇⠸⠿⠿⠿⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡸⠃⠈⠉⠉⠿⠻⢿⣿⣿⡿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⡿⢿⣷⢠⣿⡟⣿⣷⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠙⠃⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣧⣼⣿⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⢸⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⡏⠉⠁⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣤⣿⣿⣿⣟⣁⠀⠸⠿⠿⠿⠸⠿⠇⠀⠀⠈⠻⠷⠿⠏⠀⠀⠀⠀⠀⠀⣿⣷⣽⣿⣿⡿⠋⠁⠀⠉⢿⣯⣭⣥⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠻⣯⣿⠏⠀⠀⠀⠀⠀⠈⡿⠟⠛⣷
⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⢰⡇⠐⠀⠈
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⣤⣿⣇⡀⢠⣤
⠀⠀⠀⢸⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⣶⣾⣿⣿⣿⣿⣿⣿
⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠹⣿⣿⣿⣿⣿⣿
⣷⠀⠀⢿⣿⣿⣿⣿⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⡚⠉⠁⠀⠀⠈⣿⠿⠏⠉⠙⠻⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⣿⣫⣭⣿
⣿⡆⠀⠘⠶⣿⣿⣿⠀⠀⠀⠀⠀⢈⣿⡿⠟⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⢀⣼⠇⣿⣆⠀⠀⠀⠈⣿⠄⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡉⠁⠀
⣿⠀⠀⠀⠀⣶⣞⣿⣤⣤⣤⣴⣶⠟⠿⡇⠀⠀⠀⢹⠗⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠵⣶⣶⣿⠃⠀⢹⣿⣄⣀⣀⣴⣿⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠿⠿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡻⠻
⠏⠀⠰⠀⠀⣿⠗⠩⢿⣿⣿⣿⠁⠀⠀⣿⣄⣠⠔⠀⠀⢀⣀⣠⣄⡀⠀⠀⠀⠀⢀⡦⢿⣿⣿⣤⣤⣼⣿⣿⡿⢿⡿⠉⠀⠀⠀⠀⠀⠀⠉⠀⠈⠿⣧⡀⠀⠈⡏⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠿
⠀⠠⠀⠀⢰⡏⠀⠀⠁⢛⣿⣿⣿⣷⣾⣿⣿⠿⠀⠀⢀⣾⣿⣿⡿⢿⣆⠀⠀⠀⢸⡇⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠠⡼⡾⣾⣿⣿⣶⠾⣧⣺⠇⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠒⠀⠀⠀⣿⡇⠀⠀⠁⠤⠉⠛⢻⠿⠿⠃⠀⠀⠀⠀⠀⠀⢀⠄⢦⣀⣸⠀⠀⠀⠈⢻⣤⣀⣤⣄⣤⣤⣤⣴⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠚⠉⠛⠈⣷⡿⠀⠀⠀⠀⠀⣿⣿⣟⡿⣿⣶⣶⣤⣠⣴⡶⠖⠛
⠀⠐⠠⠄⠻⣷⣴⣿⣧⣤⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⢺⠻⠿⠟⢿⡏⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⡵⢀⠀⠀⠀⣼⣿⣿⣿⣇⠀⠙⣿⣿⣿⡏⠀⠀⠊
⣀⠀⠀⠀⢤⣬⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⡩⠿⣛⣛⣭⣾⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⡻⢋⠀⠀⠀⠹⣿⣿⣿⣿⣴⡀⣼⣿⣿⡷⢾⣷⣶
⣿⣿⣶⣤⣈⠉⠉⠉⠉⠛⠳⣤⣴⣶⡆⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⣾⣧⣠⡀⣀⠀⠀⠀⠀⠺⠶⣶⣿⣿⣿⠗⠒⠒⢶⣤⡀⠀⢀⣀⣠⣤⣤⣤⣀⣀⣰⣾⡿⠓⠀⠠⠄⠈⢀⣤⣿⣿⡿⠻⠋⠁⢹⣿⣯⠡⣾⣿⣿
⠉⠙⠻⣿⣿⣿⣷⣤⣤⣤⣿⡿⠛⣏⣸⣆⠀⠀⣠⣶⣿⣧⡀⠀⠀⠀⠈⠉⠳⢧⣄⣀⡀⠀⢰⣤⣤⣤⣴⣶⠀⢀⣀⣤⣿⣿⡿⠿⠛⠋⠉⠻⣿⣿⣏⣉⣉⣥⣄⣠⣤⡤⠖⣋⣠⣿⣿⢀⡀⠀⡴⠡⠙⣿⣦⠉⢻⣿
⡄⠀⠀⠀⠉⠛⠛⠻⣿⣿⣿⠧⠴⣿⣿⣿⠀⢸⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢸⣦⠉⠛⠿⢳⣾⣿⣿⣿⣿⣷⣾⣿⠿⠟⠛⣿⣷⠀⠀⠀⠀⣸⣿⣿⣿⡄⠈⢿⡿⠃⠀⠀⢀⣿⣿⣿⡿⣾⣶⣼⡇⠀⠀⠈⠛⣧⠀⠉
⠀⠀⠀⠀⠀⠀⢀⠀⣹⣿⣷⠀⠀⢸⣿⡇⠀⢸⣿⣿⡿⠛⠁⠀⠀⢀⣮⠀⠀⠻⢷⣦⣤⣼⡿⠿⢿⣿⣿⠟⠁⠀⠀⣀⣾⣿⡏⠀⠀⠀⢀⣿⣿⣿⣿⠒⠲⣤⡆⠐⠈⠋⠉⢁⣿⣿⢃⣿⢿⣿⣷⣶⣴⣾⣿⣿⣿⣿
⠙⠟⠿⣿⠿⢿⠿⣷⣿⣿⣿⠻⢿⡿⣿⣷⣠⠀⣿⣿⠃⠒⠀⠀⠀⠘⣿⣦⡀⠀⠀⠉⠻⢿⣧⣶⣶⣭⣤⣤⡴⠶⠛⠛⠉⠹⣿⡀⠀⡴⠋⢱⣮⣽⣿⡧⠀⢬⣥⡀⠒⠶⠶⣾⣿⣿⠿⣷⡘⣿⣿⡿⢿⣿⣿⣿⡟⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⣿⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡄⢠⡹⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⣿⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡟⣿⡏⢸⣿⡏⠁⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⢀⣁⣽⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⠋⠸⠿⠇⠿⠇⠿⠇⠸⠿⠿⠿⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡸⠃⠈⠉⠉⠿⠛⢿⣿⣿⡿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⡿⢿⣷⢰⣿⡟⣿⣷⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠛⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣧⣼⣿⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠠⣿⡏⠉⢸⣿⡏⠉⠁⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣇⣤⣿⣿⣿⣿⣁⡀⠸⠿⠿⠿⠸⠿⠇⠀⠀⠈⠻⠷⠿⠏⠀⠀⠀⠀⠀⠀⢸⣿⣞⣿⣿⣿⠟⠉⠀⠈⠹⣿⣭⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⢿⣼⡟⠁⠀⠀⠀⠀⠀⢹⠟⠛⠻
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⠀⠀⠀⠀⠀⠀⣾⠀⠂⠀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⡀⠀⠀⢠⣼⣿⣀⢀⣬
⠀⠀⠀⠀⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣴⣾⣿⣿⣿⣿⣿⣿
⣄⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢿⣿⣿⣿⣿⣿
⣿⡆⠀⠸⣿⣿⣿⣿⡟⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠀⠀⠀⢹⡿⡿⠉⠉⠛⢿⣿⡿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢻⣿⣭⣽
⣿⣷⠀⠀⠻⢿⣿⣿⡇⠀⠀⠀⠀⠀⣹⣿⡿⠛⢿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣠⡿⢸⣷⡀⠀⠀⠀⢹⡧⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣾⣷⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠈⠀
⣿⡏⠀⠀⠀⢰⣾⣿⣧⣄⣠⣤⣶⡾⠿⢿⠀⠀⠀⠈⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠶⣶⣾⡟⠀⠈⣿⣧⣀⣀⣠⣾⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠿⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠟
⠹⠃⠀⠄⠀⢘⡿⠏⠹⣿⣿⣿⡏⠀⠀⢸⣧⣀⡤⠂⠀⠀⣀⣀⣤⣀⠀⠀⠀⠀⠀⣠⠼⣿⣿⣦⣤⣤⣿⣿⣿⢟⣿⠏⠁⠀⠀⠀⠀⠀⠈⠁⠠⠹⣿⣄⠀⠀⢹⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠾
⠂⠀⠀⠀⠀⣼⠁⠀⠈⠸⣻⣿⣿⣿⣶⣾⣿⡿⠇⠀⠀⣰⣿⣿⣿⠿⣷⠀⠀⠀⠀⣿⡀⠘⠿⢿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢴⢷⢷⣿⣿⣷⡶⢿⣔⣿⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠂⠀⠀⢸⣿⠀⠀⠈⠠⠌⠙⠻⡿⠿⠟⠀⠀⠀⠀⠀⠀⠀⡀⠰⣄⣀⡇⠀⠀⠀⠙⣷⣄⣠⣤⣠⣤⣤⣤⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠙⠃⢹⣾⠇⠀⠀⠀⠀⢸⣿⣿⣛⠿⣷⣶⣦⣤⣤⡶⠿⠛
⠀⠀⠂⠤⠌⢿⣷⣮⣿⣤⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠿⠿⠻⣿⡅⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠟⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣼⣥⠆⡀⠀⠀⢠⣿⣿⣿⣿⡄⠈⠻⣿⣿⣿⠁⠀⠀
⣄⡀⠀⠀⠠⣤⣽⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢉⠽⢿⣛⣫⣵⣾⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣛⠟⡁⢀⡀⠀⢿⣿⣿⣿⣷⣦⠀⣿⣿⣿⠾⣿⣷
⣿⣿⣷⣦⣀⠉⠉⠉⠉⠙⠻⢦⣠⣴⣶⡀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣶⣿⣤⣄⢀⡀⠀⠀⠀⠐⠿⢶⣾⣿⣿⡿⠒⠒⠲⣶⣄⠀⠀⣀⣀⣤⣤⣤⣤⣀⣀⣾⣿⠟⠂⠀⠤⠀⠀⣠⢼⣿⣿⡿⠛⠉⠀⣿⣿⡇⢰⣿⣿
⠉⠉⠛⢿⣿⣿⣿⣦⣤⣤⣴⣿⠟⢻⣁⣷⡀⠀⢀⣴⣾⣿⣆⠀⠀⠀⠀⠉⠙⠾⣤⣀⣀⠀⠀⣦⣤⣤⣤⣶⡇⠀⣀⣠⣼⣿⣿⡿⠿⠛⠉⠙⢻⣿⣿⣉⠉⣡⣤⣄⣤⡤⠴⢚⣁⣼⣿⡿⣀⠀⢀⠎⠘⢛⣿⣄⠙⢿
⢄⠀⠀⠀⠈⠛⠛⠛⢻⣿⣿⡿⠤⢾⣾⣿⡇⠀⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣷⡌⠙⠻⠿⣶⣿⣿⣿⣿⣿⣴⣿⡿⠿⠛⢻⣿⡆⠀⠀⠀⢀⣿⣿⣿⣿⠀⠹⣿⠟⠀⠀⠀⣸⣿⣿⣿⢿⣿⣦⣿⠀⠀⠀⠉⢛⣆⠀
⡀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣿⣇⠀⠀⣿⣿⠃⠀⢻⣿⣿⠟⠋⠀⠀⠀⢰⡏⠀⠘⠿⣶⣤⣄⣿⠿⠿⣿⣿⡿⠋⠁⠀⠀⣠⣿⣿⠁⠀⠀⠀⣸⣿⣿⣿⡟⠒⢲⣶⡄⠊⠉⠉⠁⣸⣿⡿⣸⡿⣿⣿⣷⣦⣶⣿⣿⣿⣿
⠙⠟⠿⣷⠿⢿⠿⢿⣾⣿⣿⡟⢿⣿⠿⣿⣇⡀⢸⣿⡿⠐⠒⠁⠀⠀⢻⣷⣄⠀⠀⠈⠙⠿⣿⣶⣶⣮⣥⣤⣤⡶⠞⠛⠛⠉⢻⣧⠀⢠⠞⠉⢸⣿⣿⣿⠤⠤⣭⣆⠐⠲⠶⠶⣿⣿⡿⢿⣦⢻⣿⣿⠿⣿⣿⣿⣿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡗⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡄⠠⡹⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⣏⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡇⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡟⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⢀⣁⣽⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⠋⠸⠿⠇⠿⠇⠿⠇⠸⠿⠿⠿⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣸⠃⠈⠉⠉⠿⠻⢿⣿⣿⡿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⡿⢿⣷⢰⣿⡟⣿⣷⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠙⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣧⣼⣿⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠸⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⡏⠉⠁⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣇⣠⣿⣿⣿⣿⣁⡀⠰⠿⠿⠿⠸⠿⠇⠀⠀⠈⠻⠷⠿⠏⠀⠀⠀⠀⠀⠀⠈⣿⣿⣻⣿⣿⡿⠉⠀⠀⠈⢿⣯⣵⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠹⣇⣿⠋⠀⠀⠀⠀⠀⠈⡿⠊⠛
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⠀⠀⠀⠀⠀⣸⡇⠐⠀
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⣴⣿⣇⡀⣠
⠀⠀⠀⠀⢰⣿⣿⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣶⣶⣿⣿⣿⣿⣿⣿
⣄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠹⣿⣿⣿⣿⣿
⣿⣧⠀⠀⣿⣿⣿⣿⣿⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠁⠀⠀⠈⣿⠿⠋⠉⠙⠻⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⣿⣿⣭
⣿⣿⡆⠀⠸⠾⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⠟⠿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⢀⣼⠇⣿⡆⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠁
⣿⣿⠀⠀⠀⠀⣶⣽⣿⣄⣀⣠⣴⣶⠿⠿⡇⠀⠀⠀⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣶⣶⣿⠃⠀⢹⣿⣄⣀⣀⣴⣿⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⡿⠿⠿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡻
⡛⠏⠀⠠⠀⠀⣿⠿⠉⢿⣿⣿⣿⠉⠀⠀⣿⣀⣠⠄⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢰⠰⣿⣿⣷⣤⣄⣼⣿⣿⡿⣻⡟⠉⠀⠀⠀⠀⠀⠀⠉⢀⠈⢿⣧⡀⠀⠈⡏⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠇⠀⠀⠀⠀⢰⡏⠀⠀⠂⢿⣿⣿⣷⣶⣶⣿⣿⠿⠀⠀⢀⣾⣿⣿⡿⣿⡆⠀⠀⠀⢸⡆⠀⠻⠿⣿⣿⣿⣿⣿⣽⣿⠁⠀⠀⠀⠀⠀⠀⠀⠤⡾⡾⣾⣿⣿⣶⠿⣧⣺⠇⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠤⠤⠀⠀⠀⣿⡇⠀⠀⠑⠌⠙⠻⢿⠿⠿⠋⠀⠀⠀⠀⠀⠀⢀⠀⢦⣀⣸⠀⠀⠀⠈⢿⣤⣀⣤⣄⣤⣥⣬⣴⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠛⠘⣷⡿⠀⠀⠀⠀⠀⣿⣿⣟⡻⢿⣶⣶⣤⣤⣴⠾⠟
⠀⠀⠀⠤⠄⠿⣷⣶⣷⣦⣄⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⢲⠿⠿⠾⢿⡏⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⡿⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⣯⡴⢀⠀⠀⠀⣼⣿⣿⣿⣧⠀⠙⢿⣿⣿⡏⠀⠀
⣤⣀⠀⠀⠀⢤⣙⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢈⡩⠿⣛⣛⣭⣾⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⡻⢃⠀⣀⠀⠹⣿⣿⣿⣿⣲⡀⣼⣿⣿⡷⢿⣿
⣿⣿⣿⣦⣄⡈⠉⠉⠋⠙⠛⠧⣄⣤⣶⡆⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣾⣧⣀⡀⣀⠀⠀⠀⠀⠻⠶⣶⣿⣿⣿⠗⠒⠒⢶⣤⡀⠀⢀⣀⣠⣤⣤⣤⣄⣀⣰⣾⡿⠓⠀⠠⠄⠀⢀⡤⣿⣿⣿⠛⠉⠁⢹⣿⣿⠠⣿⣿
⠉⠉⠙⠿⣿⣿⣷⣶⣤⣤⣤⣿⡿⠛⡏⣹⣆⠀⠀⣠⣴⣿⣧⡀⠀⠀⠀⠈⠙⠳⢧⣄⣀⠀⠀⢠⣤⣤⣤⣴⣶⠀⢀⣀⣤⣿⣿⡿⠿⠟⠋⠉⠻⣿⣿⣏⡉⢉⣤⣤⣠⣤⠤⠖⣉⣠⣿⣿⢁⡀⠀⡴⠁⠙⣻⣧⡈⠻
⢠⣀⠀⠀⠈⠙⠛⠛⠻⣿⣿⣿⡧⠴⣷⣿⣿⠀⢸⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢸⣦⠉⠛⠿⢷⣾⣿⣿⣿⣿⣧⣾⣿⡿⠟⠛⣿⣷⠀⠀⠀⠀⣸⣿⣿⣿⠆⠈⢿⡿⠃⠀⠀⢀⣿⣿⣿⡿⣿⣶⣼⡇⠀⠀⠈⢙⣳⣀
⣀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣯⠀⠀⢸⣿⡏⠀⢸⣿⣿⡿⠛⠁⠀⠀⢀⣾⠀⠀⠻⢷⣦⣤⣸⡿⠿⢿⣿⣿⠟⠉⠀⠀⢀⣼⣿⡏⠀⠀⠀⢀⣿⣿⣿⣿⠓⠲⣦⣆⠐⠉⠋⠉⢀⣿⣿⢃⣿⢿⣿⣿⣷⣴⣾⣿⣿⣿
⠙⠟⠶⣶⠿⢿⠿⠿⣷⣿⣿⣿⠻⣿⡿⣿⣷⣀⠀⣿⣿⠃⠒⠀⠀⠀⠸⣿⣦⡀⠀⠀⠉⠻⢿⣧⣶⣶⣭⣤⣤⣴⠶⠛⠛⠋⠹⣿⡄⠀⡴⠋⢠⣷⣿⣿⡧⠄⢬⣵⡀⠒⠶⠾⢾⣿⣿⠿⣷⡜⣿⣿⡟⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡄⠀⡹⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡏⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⢀⣁⣽⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⠋⠸⠿⠇⠿⠇⠿⠇⠸⠿⠿⠿⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣸⠃⠈⠉⠉⠿⠛⢿⣿⣿⡿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⡿⢿⣷⢠⣿⡟⣿⣷⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠛⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣧⣼⣿⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⡏⠉⠁⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢇⣀⣿⣿⣿⣿⣁⣀⠰⠿⠿⠿⠸⠿⠇⠀⠀⠈⠻⠷⠿⠏⠀⠀⠀⠀⠀⠀⠀⢿⣿⣞⣿⣿⡿⠋⠁⠀⠀⠹⣯⣭⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⢻⢹⡟⠀⠀⠀⠀⠀⠀⣿⠓⠙
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡀⠀⠀⠀⠀⠀⢠⡿⠀⠂
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣄⠀⠀⠀⢤⣾⣧⣀⢀
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣶⣾⣿⣿⣿⣿⣿
⣤⡀⠀⠀⢸⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢽⣿⣿⣿⣿
⣿⣿⡄⠀⢸⣿⣿⣿⣿⡿⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠀⠀⠀⢹⠿⡟⠉⠉⠻⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢻⣿⣯
⣿⣿⣇⠀⠈⠷⣿⣿⣿⠁⠀⠀⠀⠀⠈⣿⣿⡿⠿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⣴⡿⣽⣧⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠈
⣿⣿⡏⠀⠀⠀⢰⡿⣿⣦⣀⣀⣤⣴⡾⠿⢿⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣶⣿⠏⠀⠘⣿⣦⣀⣀⣠⣿⡇⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠿⠿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝
⡟⠻⠁⠀⡄⠀⢸⣷⠯⢻⣿⣿⣿⡟⠀⠀⢸⣆⣀⡠⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⡆⢼⣿⣿⣦⣤⣠⣿⣿⣿⢟⣿⠋⠁⠀⠀⠀⠀⠀⠈⠀⡀⠻⣿⣄⠀⠀⣹⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⠀⠀⡀⠀⠀⣾⠁⠀⠀⠼⣿⣿⣿⣶⣶⣾⣿⡿⠇⠀⠀⣼⣿⣿⣿⣿⣧⠀⠀⠀⢀⣗⠀⠹⠿⣿⣿⣿⣿⣿⣯⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⢴⢷⣿⣿⣿⣷⡾⣿⣔⡟⠀⠀⠀⠀⠀⣠⣄⡀⡀⠀⠀⠀⠀⠀⠀⠀
⠥⠤⠄⠀⠀⢼⣿⠀⠀⠈⠢⠉⠻⠿⡿⠿⠟⠁⠀⠀⠀⠀⠁⠀⡈⠠⣄⣠⡃⠀⠀⠀⠻⣇⣀⣄⣤⣠⣤⣤⣤⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠒⠉⠛⠃⣿⣾⠁⠀⠀⠀⠀⣸⣿⣿⡛⠿⣷⣶⣲⣤⣴⡶⠟
⠀⠀⠀⠠⣄⠸⣿⣦⣿⣶⣤⡀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⢐⠾⠿⠶⠿⣿⠁⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠟⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣤⢆⡀⠀⠀⢰⣿⣿⣿⣿⡄⠈⠻⣿⣿⣿⠁⠀
⣤⣀⠀⠀⠀⢀⣈⣻⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣉⠿⢟⣛⣫⣴⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠛⠟⡠⢀⡀⠈⢿⣿⣿⣿⣇⣆⢰⣿⣿⣿⠻⣿
⣿⣿⣿⣶⣤⣀⠉⠉⠙⠙⠛⠻⣄⣠⣴⣶⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣧⣄⣀⠀⡀⠀⠀⠀⠘⠷⢶⣾⣿⣿⡗⠒⠒⠶⣦⣄⠀⠀⣀⣀⣤⣤⣤⣄⣀⣀⣾⣿⠟⠂⠠⠤⠀⠀⣠⢼⣿⣿⡏⠋⠁⠈⣿⣿⡆⢼⣿
⠉⠉⠙⠻⣿⣿⣿⣶⣤⣄⣀⣴⣿⠟⢿⣉⣦⡀⠀⢀⣤⣾⣿⣅⠀⠀⠀⠈⠉⠛⠾⣤⣀⡀⠀⠀⣤⣤⣤⣤⣶⡆⠀⣀⣠⣼⣿⣿⠿⠟⠛⠉⠙⢿⣿⣿⣉⠉⣡⣤⣠⣤⡤⠖⢊⣅⣼⣿⡏⣀⠀⢠⠎⠌⢛⣷⣄⠙
⠀⣀⡀⠀⠀⠉⠛⠻⠟⢿⣿⣿⣿⠤⢾⣿⣿⡇⠀⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⣶⡈⠛⠿⠿⣾⣿⣿⣿⣿⣧⣶⣿⡿⠿⠛⣻⣿⠄⠀⠀⠀⢀⣿⣿⣿⡇⠀⠹⣿⠟⠀⠀⠀⣺⣿⣿⣿⢿⣷⣦⣿⠀⠀⠀⠉⣛⣆
⣀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡅⠀⠀⣿⣿⠀⠀⣿⣿⣿⠟⠋⠀⠀⠀⢰⡍⠀⠙⠿⣦⣤⣀⣿⠿⠿⣿⣿⡿⠊⠁⠀⢀⣰⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡟⠒⢲⣴⠀⠈⠉⠉⠁⣼⣿⡟⣸⡿⣿⣿⣿⣶⣶⣿⣿⣿
⠉⠟⠷⣶⠷⢶⠿⠿⢷⣾⣿⣿⡿⢿⣿⣿⣿⡆⡀⢸⣿⡟⠐⠒⠀⠀⠀⢻⣷⡀⠀⠀⠉⠛⠿⣿⣴⣶⣬⣥⣤⣤⡶⠞⠛⠋⠉⢿⣇⠀⣠⠞⠁⣾⣼⣿⣿⠤⠤⣥⣆⠐⠲⠶⢷⣿⣿⡿⢿⣦⢻⣿⣿⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⡇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡄⠀⡹⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡧⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣿
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡟⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⢀⣁⣽⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠷⠿⠋⠸⠿⠇⠿⠇⠿⠇⠸⠿⠿⠿⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣸⠃⠈⠉⠉⠿⠛⢿⣿⣿⡿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⡿⢿⣷⢰⣿⡟⣿⣷⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠛⠃⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣧⣼⣿⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⡏⠉⢸⣿⡏⠉⠁⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⢀⣻⣿⣿⣿⣥⣀⣸⠿⠿⠿⠸⠿⠇⠀⠀⠈⠻⠷⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣟⣿⣿⡿⠋⠀⠀⠀⢹⣯⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⢹⣻⠏⠀⠀⠀⠀⠀⠀⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⡀⠀⠀⠀⠀⠀⢠⡟⠀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣄⠀⠀⠀⣴⣿⣧⣄
⠀⠀⠀⠀⠀⢀⣾⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢰⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣶⣿⣿⣿⣿⣿
⣦⣤⡀⠀⠀⢸⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣻⣿⣿
⣿⣿⣷⡀⠀⢸⣿⣿⣿⣿⡿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⠀⠀⠀⢻⢿⡟⠉⠙⠻⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⢹⣯
⣿⣿⣿⡇⠀⠘⠷⣿⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⡿⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⢀⣴⡿⣿⣧⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣿⣿⣿⡇⠀⠀⠀⢠⣽⣟⣆⣀⣀⣤⣴⡾⠿⣏⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⣶⣶⣿⠏⠀⠸⣿⣦⣀⣀⣴⣿⡁⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠿⠿⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⡟⢿⠁⠀⡄⠀⢸⣷⠝⢻⣿⣿⣿⡏⠁⠀⢸⣄⣀⡤⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⡆⢾⣿⣿⣤⣤⣠⣿⣿⡿⣻⡿⠋⠀⠀⠀⠀⠀⠀⠈⠀⡀⠻⣿⣄⠀⠀⡹⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠙⣿⠀⠀⡀⠀⠀⣼⠁⠀⠀⠝⣿⣿⣷⣶⣴⣾⣿⡿⠇⠀⠀⣼⣿⣿⣿⣿⣧⠀⠀⠀⢀⡇⠀⠹⠿⣿⣿⣿⣿⣿⣏⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⡼⠖⡿⣿⣿⣷⡾⣷⣼⡏⠀⠀⠀⠀⢀⣠⣄⣀⣀⠀⠀⠀⠀⠀⠀
⡾⠡⠤⠄⠀⠀⢸⣿⠀⠀⠘⠀⠉⠻⠿⣿⣿⠟⠁⠀⠀⠀⠀⠁⠀⡈⢠⣀⣠⠃⠀⠀⠀⠻⣧⣀⣄⣤⣠⣤⣤⣤⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠂⠉⠛⠃⣿⣾⠁⠀⠀⠀⠀⣸⣿⣿⡟⠻⣷⣾⣶⣤⣴⡶
⠁⠀⠀⠀⠤⣀⠸⣿⣴⣿⣶⣄⡀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⢰⠾⡿⠾⠿⣿⠁⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠟⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣴⢆⠀⠀⠀⢰⣿⣿⣿⣷⡄⠈⠻⣿⣿⣿⠁
⣶⣤⣀⠀⠀⠀⢀⣈⣻⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣉⠿⣛⣛⣫⣴⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠛⢟⡠⢀⡀⠈⢿⣿⣿⣿⣧⣆⢰⣿⣿⣿⠛
⢿⣿⣿⣿⣶⣄⡀⠚⠉⠛⠛⠛⠯⣄⣀⣴⣦⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣧⣀⣀⠀⠀⠀⠀⠀⠘⠷⢶⣾⣿⣿⡗⠒⠒⠶⣦⣄⠀⠀⣀⣠⣤⣤⣤⣤⣠⣀⣾⣿⠟⠂⠠⠔⠀⠀⣴⢾⣿⣿⡏⠋⠁⠈⢿⣿⡆⢸
⠀⠉⠛⠛⠿⣿⣿⣷⣶⣤⣀⣀⣴⣿⠟⢿⢉⣦⠀⠀⢀⣤⣾⣿⣄⠀⠀⠀⠈⠉⠛⠾⣤⣀⡀⠀⠀⣤⣤⣤⣤⣶⡄⠀⣀⣠⣼⣿⣿⠿⠟⠛⠉⠙⢻⣿⣿⣈⠉⣥⣦⣄⠤⠤⠖⢋⣥⣾⣿⡇⡠⠀⢠⠎⠐⠛⣿⣆
⠀⠀⣀⡀⠀⠀⠉⠛⠿⠿⢿⣿⣿⣿⡤⢼⣾⣿⡇⠀⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣶⡈⠛⠿⠟⣾⣿⣿⣿⣿⣷⣶⣿⡿⠿⠛⣻⣿⡄⠀⠀⠀⢀⣿⣿⣿⡇⠀⠹⣿⠟⠀⠀⢀⣸⣿⣿⣿⢿⣿⣧⣿⠀⠀⠀⢈⣛
⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⡅⠀⠈⣿⣿⠀⠀⣿⣿⣿⠟⠛⠀⠀⠀⣰⡏⠀⠹⠿⣦⣤⣀⣿⠿⠿⣿⣿⡿⠋⠀⠀⢀⣰⣿⣿⠀⠀⠀⠀⣼⣿⣿⣿⡟⠒⢲⣶⡄⠉⠉⠉⠁⣸⣿⣟⣸⡟⣿⣿⣿⣶⣾⣿⣿
⠉⠟⠿⣶⠶⢶⠶⠶⠿⢷⣾⣿⣿⡿⣷⣶⣿⣿⣆⠀⢸⣿⡿⠠⠒⠀⠀⠀⢿⣷⣄⠀⠀⠈⠛⠿⣿⣴⣶⣬⣥⣤⣤⡶⠞⠛⠋⠉⢿⣇⠀⣠⠟⠁⣼⣾⣿⣿⠤⠤⢴⣆⠐⠚⠿⠿⣿⣿⡿⠿⣦⢹⣿⣿⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣤⣤⣾⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡿⠿⢸⣿⡇⠀⣼⣿⢻⣿⣆⢰⣿⡟⣿⣧⢸⣿⣿⡇⣿⣿⣇⢸⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡄⠀⡹⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⡇⣿⣿⢸⣿⣧⣤⢸⣿⡇⠀⣿⣿⠘⠛⠛⢸⣿⡇⣿⣿⢸⣿⣿⣷⣿⣿⡇⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣿
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡏⢸⣿⡏⠁⢸⣿⡇⠀⣿⣿⢰⣶⡶⢸⣿⡇⣿⣿⢸⣿⡟⣿⡟⣿⡇⢸⣿⡏⠁⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⢀⣁⣽⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠘⠿⠿⠇⠸⠿⠿⠿⠸⠿⠿⠇⠙⠿⠿⠿⠃⠘⠿⠿⠿⠋⠸⠿⠇⠿⠇⠿⠇⠸⠿⠿⠿⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣸⠃⠈⠉⠉⠿⠛⢿⣿⣿⡿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠇⣾⣿⢻⣷⡄⠀⠀⣿⡿⠿⢸⣿⡿⢿⣷⢰⣿⡟⣿⣷⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠛⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣷⣤⢸⣿⣧⣼⣿⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢸⣿⡇⠀⢀⣿⡏⠉⢸⣿⡏⠉⠁⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⢀⣻⣿⣿⣿⣥⣀⣸⠿⠿⠿⠸⠿⠇⠀⠀⠈⠻⠷⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣟⣿⣿⡿⠋⠀⠀⠀⢹⣯⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⢹⣹⠏⠀⠀⠀⠀⠀⠀⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡀⠀⠀⠀⠀⠀⢠⡟⠀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣄⠀⠀⠀⣴⣿⣧⣄
⠀⠀⠀⠀⠀⢀⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢰⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣶⣿⣿⣿⣿⣿
⣦⣤⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿
⣿⣿⣷⡀⠀⢸⣿⣿⣿⣿⡷⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⠀⠀⠀⢻⠿⡟⠉⠙⠻⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⢹⣯
⣿⣿⣿⡇⠀⠘⠷⣿⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⡿⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⢀⣴⠿⣿⣧⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣿⣿⣿⡇⠀⠀⠀⢰⣿⣿⣆⣀⣀⣤⣴⣾⠿⣏⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⣶⣶⣿⠏⠀⠸⣿⣦⣀⣀⣴⣿⡁⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠿⠿⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⡟⢿⠁⠀⡄⠀⢸⡿⠝⢻⣿⣿⣿⡏⠁⠀⢸⣄⣀⡤⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⡆⢾⣿⣿⣤⣤⣠⣿⣿⡿⢿⡿⠋⠀⠀⠀⠀⠀⠀⠈⠀⡀⠻⣿⣄⠀⠀⡹⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠙⣿⠀⠀⡀⠀⠀⣼⠁⠀⠈⠝⣿⣿⣷⣶⣶⣾⣿⡿⠇⠀⠀⣼⣿⣿⣿⣿⣧⠀⠀⠀⢀⡇⠀⠹⠿⣿⣿⣿⣿⣿⣗⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⡼⢗⡿⣿⣿⣷⡾⣷⣼⡇⠀⠀⠀⠀⠀⣠⣄⣀⣀⠀⠀⠀⠀⠀⠀
⡾⠡⠤⠄⠀⠀⢸⣿⠀⠀⠘⠀⠉⠻⠿⣿⣿⠟⠉⠀⠀⠀⠀⠁⠀⡈⢡⣀⣠⠃⠀⠀⠀⠻⣧⣀⣄⣤⣠⣤⣤⣤⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠂⠉⠙⠃⣿⣾⠁⠀⠀⠀⠀⣸⣿⣿⡛⠻⣷⣾⣶⣤⣴⡶
⠁⠀⠀⠀⠤⣀⠸⣿⣴⣿⣶⣄⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠾⡿⠾⠿⣿⠁⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠟⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣴⢆⠀⠀⠀⢰⣿⣿⣿⣷⡄⠈⠻⣿⣿⣿⠁
⣶⣤⡀⠀⠀⠀⢀⣈⣻⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⣉⠿⣛⣛⣫⣴⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠛⠟⡠⢀⣀⠈⢿⣿⣿⣿⣧⣆⢰⣿⣿⣿⠛
⢿⣿⣿⣿⣶⣄⡀⠚⠉⠛⠛⠛⠯⣄⣀⣴⣦⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣧⣀⣀⠀⠀⠀⠀⠀⠘⠷⢶⣾⣿⣿⡗⠒⠒⠶⣦⣄⠀⢀⣀⣠⣤⣤⣤⣤⣀⣠⣾⣿⠟⠂⠠⠔⠀⠀⣴⢾⣿⣿⠏⠋⠁⠈⢿⣿⡆⢸
⠀⠉⠛⠛⠿⣿⣿⣷⣶⣤⣀⣀⣴⣿⠟⢿⢉⣦⠀⠀⢀⣤⣾⣿⣅⠀⠀⠀⠈⠉⠛⠾⣤⣀⡀⠀⠀⣤⣤⣤⣤⣶⡄⠀⣀⣠⣼⣿⣿⠿⠟⠛⠉⠋⢻⣿⣿⣈⠉⣥⣦⣄⠤⠤⠖⢋⣥⣾⣿⡗⡠⠀⢠⠎⠐⠛⣿⣆
⠀⠀⣀⡀⠀⠀⠉⠛⠿⠿⢿⣿⣿⣿⡤⢼⣿⣿⡇⠀⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣶⡈⠛⠿⠟⣾⣿⣿⣿⣿⣷⣶⣿⡿⠿⠛⣻⣿⡄⠀⠀⠀⢀⣿⣿⣿⡇⠀⠹⣿⠟⠀⠀⢀⣸⣿⣿⣿⢿⣿⣧⣿⠀⠀⠀⢈⣛
⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⡅⠀⠈⣿⣿⠀⠀⣿⣿⣿⠟⠛⠀⠀⠀⣰⡏⠀⠹⠿⣦⣤⣀⣿⠿⠿⣿⣿⡿⠋⠀⠀⢀⣰⣿⣟⠀⠀⠀⠀⣼⣿⣿⣿⡟⠒⠶⣶⡄⠉⠉⠉⠁⣸⣿⣟⣸⡟⣿⣿⣿⣶⣾⣿⣿
⠉⠟⠿⣶⠶⢶⠶⠶⠿⢷⣾⣿⣿⡿⣷⣶⣿⣿⣆⡀⢸⣿⡿⠠⠐⠀⠀⠀⢿⣷⣄⠀⠀⠉⠛⠿⣿⣴⣶⣬⣥⣤⣤⡴⠞⠛⠋⠉⢿⣇⠀⣠⠟⠁⢴⣾⣿⣿⠤⠤⢴⣆⠐⠚⠿⠿⣿⣿⡿⠿⣦⢹⣿⣿⡻⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣇⡐⠂⡅⡍⠉⢩⢨⠋⠍⡏⣩⢩⡉⠋⠍⠏⣩⢹⢨⣿⡅⡌⣩⢩⡍⡏⣭⢩⢹⣿⡅⡅⣩⢹⠩⠍⣿⡏⣭⢩⠙⠩⠉⡏⣩⢹⠩⠉⠏⣍⠙⠩⠙⠉⡍⣩⢹⠩⠙⠉⡍⡏⣽⣹⣿⡇⡘⠇⡏⣭⢹⣿⣿⣿⣿⣿
⣿⣧⣭⣥⣧⣥⣭⣬⣬⣦⣥⣧⣿⣼⣧⣦⣥⣧⣿⣼⣬⣿⣧⣧⣿⣬⣥⣧⣿⣼⣼⣿⣧⣥⣿⣼⣬⣥⣿⣧⣿⣼⣤⣬⣥⣥⣿⣼⣬⣤⡇⠅⣰⣬⣴⣤⣧⣿⣼⣬⣥⣤⣧⣷⣽⣼⣿⣧⣷⣤⣧⣭⣼⣸⣿⣿⣿⣿
⣿⡿⡟⠻⣿⣿⢿⣿⣿⠛⠛⡟⠛⠻⡿⠛⠛⢿⣿⢻⡿⣿⣿⣿⡿⣿⡿⣿⡿⣿⣿⣿⣿⢻⡿⣿⢻⣿⢿⡿⣿⣻⡿⣿⣿⣿⣿⣿⢛⣿⣿⣿⡿⣿⡿⠛⣿⢿⣿⡿⣿⣿⣿⢿⣿⡿⣿⢿⣿⢿⡟⣿⣿⣿⣿⣿⣿⣿
⣿⡆⡆⢰⢸⠀⡂⣿⣿⠨⠭⡇⣉⣡⡇⠿⠿⢸⣿⢸⢀⠂⣿⡇⣶⢈⢰⠆⡆⣾⣇⠙⣴⠈⢰⠌⢸⠀⠀⡇⢾⠀⢰⢸⠰⠆⣿⣷⢰⢸⠇⡇⣶⢈⢴⠀⢀⠀⡇⣶⢰⠀⠁⡂⡃⣴⢸⢰⠂⠀⡇⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣶⣾⣾⣷⣶⣿⣿⣶⣶⣷⣿⣿⣿⣶⣶⣿⣿⣾⣶⣶⣿⣷⣿⣾⣶⣾⣷⣾⣿⣶⣿⣾⣶⣾⣾⣶⣶⣶⣶⣶⣾⣾⣅⣀⣿⣿⣾⣶⣶⣷⣿⣾⣶⣶⣶⣶⣶⣿⣾⣶⣷⣶⣷⣿⣾⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿
⣿⡏⠛⢿⢻⡟⡟⠛⠟⢿⠟⠻⡟⠛⢿⣿⠛⡏⡟⠻⠋⠙⠻⠛⡟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡇⣿⢸⠘⠃⡇⡇⣿⢸⠀⡀⠃⣿⢸⣿⢸⡇⡇⠛⠀⢸⠀⡘⡃⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⢡⠋⠙⠋⢹⣿⣿⢻⢻⢋⡙⢻⡏⣿⡟⣉⢸⢋⡹⣿⢺⢙⡙⣿⡟⣉⢏⡙⡏⣿⢹⢉⡹⡁⡁⡟⠉⠛⠉⡟⠉⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣬⣽⣭⣼⣸⡿⢆⣾⣬⣵⣮⣥⣿⣧⣭⣼⣬⣴⣿⣼⣼⣧⣿⣧⣭⣦⣥⣧⣭⣼⣼⣧⣧⣥⣧⣭⣭⣥⣯⣤⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣿⢟⡛⡛⣿⡿⣿⢿⣿⣿⣿⢿⢻⢿⣿⢿⣿⡟⣿⣿⣿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡘⡄⢱⠃⠈⡜⢱⠄⢺⢿⣿⠂⢸⢰⡆⡆⡇⣶⣷⡆⣶⢸⣷⠰⢰⡆⡇⠀⣿⡇⡇⠿⢸⠰⠆⠱⠆⡇⠀⠡⠐⣿⣿⣿⣿⣿⣿
⣿⡯⠏⢹⠉⠙⠉⠉⠙⠉⡯⠉⠍⠽⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣥⣿⣷⣿⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣥⣿⣾⣿⣷⣿⣯⣤⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿
⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⢸⣿⣿⣿⡇⢏⢉⡩⡉⠉⣿⠉⡩⡯⠀⠉⠉⠉⣿⠀⡉⣉⠍⣹⠁⠉⢩⠩⠉⠀⡏⡍⡩⣩⠩⡇⡍⡏⢉⢉⠋⠈⠉⠁⣍⢉⣹⠀⡊⠉⡉⠉⡁⠉⢹⡝⡉⣉⠀⠊⠍⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣃⣸⣁⣀⣄⣀⣌⠀⣈⣇⣘⣡⣀⣁⣙⣡⣇⣜⣈⣘⣇⣁⣁⣁⣡⣀⡀⣸⣀⢀⣡⣀⣀⣁⣿⣈⣀⣈⣈⣈⣘⡇⠀⠃⠈⠀⠀⢀⣀⣀⣎⣈⣈⣄⣀⣸⣧⣌⣀⣀⣈⡘⣿⢈⣀⣀⣀⣀⣀⣻⣁⣁⣄⣁⣈⣀⣿
⣿⡿⠿⠿⢿⡿⠿⠿⠿⢿⠿⠿⠛⠿⡿⠿⠿⠿⠿⠿⠿⢻⠻⡿⠿⠿⠿⠿⢿⠿⠿⠿⢻⠟⠿⠿⡟⠋⠈⠟⠋⠁⠀⠀⢀⠤⠶⠞⡟⠿⠿⠿⠿⣿⠿⠿⠿⠿⠿⠿⢿⠿⠿⡿⢿⠿⠿⠻⠿⠿⠿⠿⠟⠿⡟⣿⣿⣿
⣿⣧⣤⣤⣼⣧⣦⣧⣤⣼⣧⣼⣤⣧⣧⣤⣬⣵⣵⣧⣤⣼⣤⣴⣴⣤⣤⣤⣼⣦⣧⣤⣼⣦⣧⠄⠁⠀⠀⠀⣀⣤⣤⣤⣼⣼⣤⣤⣧⣴⣴⣤⣼⣿⣧⣤⣤⣴⣴⣴⣼⣧⣤⣿⣮⣼⣤⣤⣤⣿⣤⣤⣷⣥⣴⣿⣿⣿
⣿⣯⠉⠉⢩⠉⠏⢹⢩⠩⢩⠁⠉⠿⠌⠀⠨⠉⠩⠏⠉⠽⠏⠉⠉⠉⠉⠉⠩⠍⠍⠉⠸⠟⠁⣀⣀⣀⣀⣀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

                    ䷩ 𝚕𝚒𝚗𝚎 3005

╒═══════════════════ 𝐀𝐑𝐓𝐈𝐂𝐋𝐄 ═════════════════════════════════════════════════╕

(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/07/18/freedom-of-speech-epo/#comments
    Gemini version at gemini://gemini.techrights.org/2022/07/18/freedom-of-speech-epo/

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 07.18.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Freedom_of_Speech_Cannot_Coexist_With_Crime_at_the_EPO⠀✐


Posted in Europe, Patents at 10:50 am by Dr. Roy Schestowitz


Video_download_link | md5sum 0cd6204c8f5ec04fa0f5e613e5b4dd6a
Free Speech Issues at EPO
Creative Commons Attribution-No Derivative Works 4.0


http://techrights.org/videos/epo-free-speech-ruling.webm


Summary: The corruption in Europe’s second-largest institution and largest
patent office has become so profound that the only way to sustain this
corruption is to silence and/or intimidate those who speak about it


THE post_we've_just_published shares a 5-page document about how Monopoly Tony
(António_Campinos) and his EPO appointer Benoît_Battistelli (Campinos lacks
prior history at the EPO) illegally suppressed communication at the Office for
nearly 10 years.


“Freedom of expression scares tyrants the most because they tend to have the
most to hide.”This suppression is perhaps expected. When you routinely break
the law you don’t want your staff and your clients/customers (that’s how
today’s EPO views applicants) to find out. The video above points out that an
honest, law-abiding patent office does not need to worry about people speaking
to one another openly; moreover, the importance of circulation information
becomes ever more apparent when management goes rogue and does everything
wrong. When there’s diplomatic immunity it is essential.


Freedom of expression scares tyrants the most because they tend to have the
most to hide. █



                    ䷩ 𝚕𝚒𝚗𝚎 3058

╒═══════════════════ 𝐀𝐑𝐓𝐈𝐂𝐋𝐄 ═════════════════════════════════════════════════╕

(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/07/18/irc-log-170722/#comments
    Gemini version at gemini://gemini.techrights.org/2022/07/18/irc-log-170722/

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 07.18.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ IRC_Proceedings:_Sunday,_July_17,_2022⠀✐


Posted in IRC_Logs at 2:12 am by Needs Sunlight


Also available via the Gemini protocol at:


    * gemini://gemini.techrights.org/irc-gmi/irc-log-techrights-170722.gmi
    * gemini://gemini.techrights.org/irc-gmi/irc-log-170722.gmi
    * gemini://gemini.techrights.org/irc-gmi/irc-log-social-170722.gmi
    * gemini://gemini.techrights.org/irc-gmi/irc-log-techbytes-170722.gmi
Over HTTP:


                                      🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇H                                     🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇HTML5_logs⦈_


                                         #techrights_log_as_HTML5                                                                                  #boycottnovell_log_as_HTML5


                                      🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇H                                     🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇HTML5_logs⦈_


                                    #boycottnovell-social_log_as_HTML5                                                                               #techbytes_log_as_HTML5


                                      🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇t                                      🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇text_logs⦈_


                                          #techrights_log_as_text                                                                                   #boycottnovell_log_as_text


                                      🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇t                                      🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇text_logs⦈_


                                     #boycottnovell-social_log_as_text                                                                                #techbytes_log_as_text


                          Enter_the_IRC_channels_now


===============================================================================
§ IPFS Mirrors⠀➾


CID                                              Description     Object type


                                                 IRC log for
 QmbEkm4fYuQbd27yM5RthP7ocvNCxoHNsoBFP7k7KZ8KMk #boycottnovell  🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
                                                 (full IRC log
                                                 as HTML)


                                                 IRC log for
                                                 #boycottnovell
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                                                 as plain/ASCII
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                                                 IRC log for
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                                                 as HTML)


                                                 IRC log for
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 QmYhZ4eCjJgftGYfcLFLa1PFWSHUUgvmXi4t4LQPcKi6yB social          🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇text logs⦈
                                                 (full IRC log
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                                                 IRC log for
 QmZqAPo5gxNPfDz8TngPEyBtQXGQtdzqbEZXoXL16jw8Qm #techbytes      🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
                                                 (full IRC log
                                                 as HTML)


                                                 IRC log for
                                                 #techbytes
 QmNvP1cH3kgU9upsB6g7Kfzw1h5ojMdzbGcQLqdKqeVDfb (full IRC log   🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇text logs⦈
                                                 as plain/ASCII
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                                                 IRC log for
 QmShERFSe96vjQ3moRgPEqMXAFwHMB3rENr7sfYYM9k5Qr #techrights     🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
                                                 (full IRC log
                                                 as HTML)


                                                 IRC log for
                                                 #techrights
 QmfR8y3FMGGyTfAeWGqmKtSKuwr6zK7dRQsrFuA2MZC1aT (full IRC log   🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇text logs⦈
                                                 as plain/ASCII
                                                 text)


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇IPFS logo⦈


§ Bulletin for Yesterday⠀➾


Local_copy | CID (IPFS): QmXptQogxzP3vUxmf2ttL7Vyb5aAwUT9Ux3nmXxPkuNRNp



                    ䷩ 𝚕𝚒𝚗𝚎 3185

╒═══════════════════ 𝐀𝐑𝐓𝐈𝐂𝐋𝐄 ═════════════════════════════════════════════════╕

(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/07/18/razik-alex-menidjel-swpats/#comments
    Gemini version at gemini://gemini.techrights.org/2022/07/18/razik-alex-menidjel-swpats/

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 07.18.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Razik_Alex_Menidjel:_Software_Patents_Guru_or_Career-Climbing_Poser_Who_Never
Wrote_Code?⠀✐


Posted in Europe, Patents at 11:11 am by Dr. Roy Schestowitz


Video_download_link | md5sum 8a56aacc67638b3def912d524e30efbf
The COO Coup
Creative Commons Attribution-No Derivative Works 4.0


http://techrights.org/videos/razik-patent-office.webm


Summary: The EPO‘s most_notorious_French_dictator promoted a compatriot, Razik
Menidjel, who was later promoted by another Frenchman, António_Campinos, the
“f***ing president” (in his own words) who expects to be in the Office for a
decade, promoting European_software_patents in violation of the EPC


THE ongoing series shows how the EPO’s management plots to slow down the
bursting of a patennt bubble long in the making. In Part_V and Part_VI we
explained the role of Razik Menidjel, who was put in charge of disciplines he
does not understand and has zero experience in.


The EPO is out of control. With the pseudoscience of ‘economics’ on its side
(another Frenchman put there by Battistelli to just mindlessly parrot buzzwords
like “4IR”– a codeword/buzzword/acronym for software patents) we can expect
more lobbying in the direction of unprecedented divergence from the EPC (i.e.
breaking the law) and the granting of hundreds of thousands of Invalid Patents
(IPs) instead of European Patents (EPs). █



                    ䷩ 𝚕𝚒𝚗𝚎 3234

╒═══════════════════ 𝐃𝐀𝐈𝐋𝐘 𝐋𝐈𝐍𝐊𝐒 ═════════════════════════════════════════════╕

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 07.18.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Links_18/07/2022:_Alpine_3.16.1_Released⠀✐


Posted in News_Roundup at 2:59 pm by Dr. Roy Schestowitz


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇GNOME bluefish⦈


§ Contents⠀➾


    * GNU/Linux
          o Audiocasts/Shows
          o Applications
          o Instructionals/Technical
          o Games
    * Distributions_and_Operating_Systems
          o New_Releases
          o Fedora_Family_/_IBM
          o Canonical/Ubuntu_Family
          o Devices/Embedded
          o Mobile_Systems/Mobile_Applications
    * Free,_Libre,_and_Open_Source_Software
          o Web_Browsers
                # Mozilla
          o GNU_Projects
          o Programming/Development
                # Perl_/_Raku
    * Leftovers
          o Hardware
          o Security
                # Privacy/Surveillance
          o Civil_Rights/Policing
          o Internet_Policy/Net_Neutrality
    * Gemini*_and_Gopher
          o Personal
          o Technical
                # Internet/Gemini
    * § GNU/Linux⠀➾


          o § Audiocasts/Shows⠀➾


                # ⚓ Video ☛ Is_Ubuntu_a_bad_Linux_Distro?_–_Invidious⠀⇛


                       Ubuntu – the distribution that many members of the
                       Linux community love to hate. But why? Is the hate
                       warranted, or are people overreacting? In this
                       video, I’ll give you my thoughts on some of the
                       criticisms that Ubuntu faces nowadays.


                # ⚓ Video ☛ Display_file_contents_in_columns_with_the_“column”
                  command_–_Invidious⠀⇛


                       The ‘column’ command is a standard command line
                       utility that is part of the util-linux package.
                       What ‘column’ does is takes input and adds columns
                       to the output. This is especially useful for files
                       that have field separator characters like CSV
                       files.


                # ⚓ Linux_User_Space:_Episode_3:02:_The_Utopic_Tale_of_Ubuntu⠀⇛


                       Linux User Space
                       A podcast focused on connecting user space and
                       community


                # ⚓ Video ☛ AYANEO_just_declared_WAR_on_SteamOS_(Plus_other
                  Deck_news)_–_Invidious⠀⇛


          o § Applications⠀➾


                # ⚓ DebugPoint ☛ Top_10_Free_and_Professional_Video_Editors_in
                  2022⠀⇛


                       We list the top 10 best and most free video editors
                       for Linux and Windows. Have a look.


                       Video editors are costly software, especially those
                       that are more advanced such as Adobe Premiere Pro.
                       However, there are plenty of known/unknown Video
                       Editors available, which are free of cost and open
                       source. Here we list ten free video editors which
                       might be helpful for you and your use case.


          o § Instructionals/Technical⠀➾


                # ⚓ HowTo Geek ☛ How_to_List_the_Installed_Packages_on_Linux⠀⇛


                       With thousands of free Linux applications, it’s
                       easy to lose track of what you once installed but
                       no longer use. Here’s how to list the installed
                       applications on the major Linux families.


                # ⚓ MakeTech Easier ☛ How_to_Set_Up_WordPress_Using_Docker_in
                  Ubuntu_–_Make_Tech_Easier⠀⇛


                       Save your time and maintenance effort by running
                       WordPress in Docker container.


                # ⚓ HowTo Forge ☛ How_to_Install_Joomla_with_Apache_and_Let’s
                  Encrypt_SSL_on_Ubuntu_22.04⠀⇛


                       Joomla is a free, open-source, and award-winning
                       content management system that allows you to build
                       websites and online web applications. It is written
                       in PHP and uses MySQL as a database backend. In
                       this post, we will show you how to install Joomla
                       CMS with Apache and Let’s Encrypt SSL on Ubuntu
                       22.04.


                # ⚓ H2S Media ☛ How_to_Install_DaVinci_Resolve_on_Ubuntu_22.04
                  LTS_Jammy⠀⇛


                       Get the steps to install DaVinci Resolve video
                       editor on Ubuntu 22.04 LTS Jammy JellyFish Linux
                       using the command terminal.


                       Professional software and special editing station
                       hardware manufacturer for color correction and non-
                       destructive video editing with an extensive
                       powerful tool range Blackmagic Design offers a
                       freemium tool known as DaVinci Resolve. The
                       company’s solutions have been used in many film and
                       television productions such as Solo Star Wars,
                       Jurassic World, The Walking Dead…


                       Although the offerings of Blackmagic design are
                       paid, a few years ago, they started offering the
                       basic version of the software for free. With each
                       version number, the number of features increased,
                       and mainly the editing functions, but also e.g. the
                       audio and effect editing, grew in scope.


                       The free edition is available for Mac OS X,
                       Windows, and Linux. Even if the origin of the
                       program lies in color correction, many use it
                       mainly as a non-destructive video editing program.
                       This means that you can use it to cut and edit
                       videos without changing the source material.


                # ⚓ ZDNet ☛ How_I_revived_three_ancient_computers_with_ChromeOS
                  Flex_|_ZDNet⠀⇛


                       Linux is a great operating system. I’ve expounded
                       its virtues for over 20 years. I was the founding
                       Sr. Technology Editor of the now defunct Linux
                       Magazine in 1999. I was a principal consultant for
                       open source data center technology at Unisys and
                       IBM.


                       Today I work for the Linux Foundation as its
                       Editorial Director of Research. So yes, I get
                       Linux.


                # ⚓ Vitux ☛ How_to_Install_KDE_Plasma_Desktop_on_Debian⠀⇛


                       KDE Plasma is the graphical workspace environment
                       that is created by KDE for Linux systems. It is
                       based on the Plasma framework and is open source.
                       This contemporary desktop environment is highly
                       customizable making it users’ top choice. In this


                # ⚓ OSNote ☛ How_to_Install_and_Use_NetworkManager_(NMCLI)_on
                  Ubuntu_–_OSNote⠀⇛


                       NetworkManager is a software application that
                       manages network connections. It is included in the
                       default installation of most Linux operating
                       systems. An instance of NetworkManager runs for
                       each network connection managed, which includes Wi-
                       Fi and Ethernet connections, mobile broadband (GSM/
                       3G), and others; NetworkManager will attempt to
                       always maintain at least one active network
                       connection available.


                       If you are a system administrator and manage Ubuntu
                       servers, you should hear about NetworkManager. In
                       this tutorial, we will show you how to install and
                       use the basic NMCLI commands on Ubuntu 22.04 and
                       Ubuntu 20.04.


                # ⚓ Linux Handbook ☛ Find_Exec_Command_in_Linux:_9_Useful
                  Examples⠀⇛


                       The find command in Linux is an excellent tool to
                       find files and directories based on given criteria.


                       You can take your findings to the next level by
                       actually doing specific operations on the found
                       files.


                       For example, you found all the files with .jpeg
                       extension. How about renaming them with .jpg
                       extension?


                       You cannot just pipe redirect the find command
                       output to another command. It won’t work that
                       simply.


                # ⚓ Red Hat Official ☛ How_to_encrypt_sensitive_data_in
                  playbooks_with_Ansible_Vault_|_Enable_Sysadmin⠀⇛


                       Ansible Vault lets you keep sensitive data, such as
                       passwords and keys, in encrypted files. Here’s how
                       to use it in playbooks to improve automation
                       workflow safety.


          o § Games⠀➾


                # ⚓ Steam_is_no_longer_just_for_Windows,_Mac,_and_Linux_users
                  [Ed: What a dumb headline. Like ChromeOS, this one too uses
                  Linux.]⠀⇛


    * § Distributions and Operating Systems⠀➾


          o ⚓ Linux_distro_for_Apple_silicon_Macs_is_already_up_and_running_on
            the_brand-new_M2⠀⇛


                 Unlike Intel Macs, Apple silicon Macs were designed to
                 run only Apple’s software. But the developers on the
                 Asahi Linux team have been working to change that,
                 painstakingly reverse-engineering support for Apple’s
                 processors and other Mac hardware and releasing it as a
                 work-in-progress distro that can actually boot up and run
                 on bare metal, no virtualization required.


                 The Asahi Linux team put out a new release today with
                 plenty of additions and improvements. Most notably, the
                 distro now supports the M1 Ultra and the Mac Studio and
                 has added preliminary support for the M2 MacBook Pro
                 (which has been tested firsthand by the team) and the M2
                 MacBook Air (which hasn’t been tested but ought to work).


          o ⚓ Liliputing ☛ Asahi_Linux_adds_support_for_Macs_with_M2_chips⠀⇛


                 
                 Apple began shipping Macs with Apple Silicon processors
                 in late 2020 and while those laptop and desktop computer
                 ship with macOS, it didn’t take longer for developers to
                 begin porting Linux to work with Apple’s new hardware.
                 Earlier this year the Asahi Linux project released the
                 first public beta of a GNU/Linux distribution designed
                 for Macs with M1 series chips.


                 But M1 was just the beginning. The first Macs with M2
                 chips began shipping this summer, and the folks at the
                 Asahi project have already managed to port their Linux-
                 based operating system to work on computers powered by
                 Apple’s new processor.


          o ⚓ The Register UK ☛ Disentangling_the_Debian_derivatives:_Which
            should_you_use?⠀⇛


                 There are probably more variants of Debian than any other
                 Linux distro, which can make it confusing. To provide
                 some clarity,The Reg has lined up the main suspects.


                 Toy Story, the movie that saved Pixar (and Steve Jobs’
                 fortune), was released in 1995. Although Debian 0.01
                 appeared in 1993, you can date Debian from Toy Story,
                 because all of its stable releases are codenamed after
                 characters from that film.


                 The first named release, “Buzz”, came out in 1996. Debian
                 2.1, “Slink”, appeared in March 1999, and delivered the
                 Advanced Package Tool (APT), which brought automatic
                 dependency resolution to Linux software management.
                 (Arguably, FreeBSD’s ports system had something
                 comparable earlier.)


          o § New Releases⠀➾


                # ⚓ Alpine_3.16.1_released⠀⇛


                       
                       The Alpine Linux project is pleased to announce the
                       immediate availability of version 3.16.1 of its
                       Alpine Linux operating system.


                       This release includes various security fixes,
                       including…


          o § Fedora Family / IBM⠀➾


                # ⚓ The Register UK ☛ Rocky_Linux_9_and_its_new_build_service
                  enter_the_ring⠀⇛


                       The Rocky Linux Project has released version 9 of
                       its RHEL-compatible distro and debuted its new
                       build service.


                       Rocky Linux 9, codenamed “Blue Onyx”, is here at
                       last, some two months after the upstream distro on
                       whose source code it is based. Its progenitor, Red
                       Hat Enterprise Linux 9, was announced on May 10.
                       The best-known other modern CentOS Linux
                       replacement, AlmaLinux 9, went into beta in April,
                       and shipped just over a fortnight after Red Hat.


          o § Canonical/Ubuntu Family⠀➾


                # ⚓ CNX Software ☛ Libre_Computer_ROC-RK3328-CC_now_supports
                  Ubuntu_22.04_with_Linux_5.18.2⠀⇛


                       
                       Libre Computer’s latest Ubuntu 22.04 desktop and
                       server images for ROC-RK3328-CC board are based on
                       Linux mainline, namely Linux 5.18.2, and other
                       boards from the company, based on Allwinner H2+/H5
                       processor or Amlogic S905X, have gotten the same
                       treatment.


                       Some companies will churn out boards regularly but
                       provide limited software support. Libre Computer
                       seems to have taken a different approach, as they
                       released most of their board in 2017 and 2018,
                       starting with AML-S905X-CC “Le Potato” board based
                       on the Amlogic S905X processor, followed by ALL-H3-
                       CC “Tritium” SBC with Allwinner H2+, H3, or H5
                       processor, and finally ROC-RK3328-CC “Renegade”
                       board, but still release updated OS images in 2022.


          o § Devices/Embedded⠀➾


                # ⚓ The DIY Life ☛ The_reTerminal_E10-1,_the_First_Expansion
                  Board_for_the_reTerminal_–_The_DIY_Life⠀⇛


                       Last year Seeed Studios launched the reTerminal, a
                       Raspberry Pi Compute Module 4 based touch display
                       terminal with a pretty good list of features. One
                       of the features that looked promising was their
                       high-speed expansion interface on the back, which
                       they said would be used to add plug-in modules to
                       expand on the reTerminal’s functionality and IO.


                # ⚓ Geeky Gadgets ☛ Mixtile_Edge_2_Kit_IoT_edge_computer
                  supports_Linux_&_Android⠀⇛


                       
                       IoT hardware solution provider Mixtile has created
                       a new IoT edge computer supporting both Linux and
                       Android based embedded systems in the former the
                       Edge 2 Kit. Designed to provide users with a ready
                       to run IoT edge computer the compact computer
                       features a 3.5-inch form factor, 6 Antenna ports,
                       mini-PCIe and M.2 modules and comes preloaded with
                       an Android 11 and Linux container.


                       Features of the IoT edge computer include support
                       for dual hard drives, dual network ports or Hi-Fi
                       music decoding and other functions, meets the needs
                       of a variety of application scenarios at a low
                       cost. Reserved CAN, I2C, and UART internal
                       interfaces for developers with hardware capability
                       or customization needs together with an industry-
                       standard interface RS-485 to meet the compatibility
                       needs of other peripheral expansion. As well as
                       Gigabit Ethernet ports and supports Wi-Fi 6 for
                       faster IoT and wider accessibility.


          o § Mobile Systems/Mobile Applications⠀➾


                # ⚓ Gadgets Now ☛ 8_‘dangerous’_apps_Android_smartphone_users
                  must_delete_right_now_|_Gadgets_Now⠀⇛


                # ⚓ TechRadar ☛ Android_apps_can_now_potentially_lie_about_the
                  data_they_collect_|_TechRadar⠀⇛


                # ⚓ Express ☛ Samsung_just_made_owning_its_popular_Galaxy
                  phones_way_more_affordable_|_Express.co.uk⠀⇛


                # ⚓ India Times ☛ Oppo_launches_its_first_Android_tablet_in
                  India:_Price,_features_and_more_–_Times_of_India⠀⇛


                # ⚓ Make Use Of ☛ Android_13_Is_Almost_Here!_Here’s_What_You
                  Can_Expect⠀⇛


    * § Free, Libre, and Open Source Software⠀➾


          o ⚓ Its FOSS ☛ AppFlowy:_An_Open-Source_Alternative_to_Notion⠀⇛


                 
                 So, what about an alternative that is more transparent,
                 private, and available for Linux?


                 That’s where AppFlowy shines!


                 Build with Rust and Flutter, AppFlowy follows a minimal
                 approach to simplify things yet with enough room for
                 tweaks.


                 AppFlowy is fairly new. We reported its development
                 status last year after its initial launch.


                 It is an open-source project that aims to overcome some
                 limitations of Notion in terms of security and privacy.
                 It helps you manage tasks, add to-do lists, add due
                 dates, track the events, add pages, and format text for
                 your notes/tasks.


          o § Web Browsers⠀➾


                # § Mozilla⠀➾


                      # ⚓ Its FOSS ☛ Tor_Browser_11.5_Adds_Automatic_Censorship
                        Detection_With_HTTPS-Only_Mode_as_Default⠀⇛


                             Tor Browser is the de-facto standard for
                             users looking to utilize Tor network, and
                             want strict privacy settings on their web
                             browsers.


                             With the latest 11.5 update, Tor Browser made
                             some major improvements to secure the
                             experience for users.


                             Let me give you a quick overview of the
                             features introduced with this release.


          o § GNU Projects⠀➾


                # ⚓ GNU ☛ GNU_Health_–_News:_GNU_Health_project_to_implement
                  REUSE_compliance_[Savannah]⠀⇛


                       The REUSE initiative[1] is a Free Software
                       Foundation Europe program that facilitates the
                       documentation of licenses of Libre projects like
                       GNU Health.


          o § Programming/Development⠀➾


                # ⚓ Ubuntu Pit ☛ 15_Free_Coding_Games_To_Learn_Programming_for
                  Beginners⠀⇛


                       Have you ever read sci-fi books that envisioned the
                       world becoming a digital utopia around this decade?
                       The reality isn’t quite as glamourized – however,
                       we are getting there as machine learning and AI
                       technology is advancing like no other. In
                       retrospect, learning to program is the trend now,
                       and there are tons of free coding games that make
                       it a fun experience.


                       Let’s be honest – programming as a beginner can be
                       daunting. While it’s thrilling to see your first
                       program run successfully, many people struggle to
                       apply programming concepts to real problems. That
                       said, there’s an abundance of resources to learn
                       programming on your own from scratch and challenge
                       yourself in several ways to grow skills.


                # ⚓ gem-compare’s_new_contributors⠀⇛


                       gem-compare v1.2.0, new contributors, and 200+
                       GitHub starts.


                       My little RubyGems plugin got some new
                       contributions. Read the v1.0 announcement to know
                       what’s the gem about.


                # ⚓ LWN ☛ Conill:_How_efficient_can_cat(1)_be?⠀⇛


                       Ariadne Conill explores ways to make the Unix cat
                       utility more efficient on Linux.


                # § Perl / Raku⠀➾


                      # ⚓ Rakulang ☛ 2022.29_Hot_–_Rakudo_Weekly_News⠀⇛


                             It is hot. It will keep getting hotter in the
                             coming years. Enjoy the coolest summer you’ll
                             ever have. Not much else to say.


    * § Leftovers⠀➾


          o § Hardware⠀➾


                # ⚓ The Next Platform ☛ KAIST_Shows_Off_DirectCXL_Disaggregated
                  Memory_Prototype⠀⇛


                       The hyperscalers and cloud builders are not the
                       only ones having fun with the CXL protocol and its
                       ability to create tiered, disaggregated, and
                       composable main memory for systems. HPC centers are
                       getting in on the action, too, and in this case, we
                       are specifically talking about the Korea Advanced
                       Institute of Science and Technology.


                       Researchers at KAIST’s CAMELab have joined the
                       ranks of Meta Platforms (Facebook), with its
                       Transparent Page Placement protocol and Chameleon
                       memory tracking, and Microsoft with its zNUMA
                       memory project, is creating actual hardware and
                       software to do memory disaggregation and
                       composition using the CXL 2.0 protocol atop the
                       PCI-Express bus and a PCI-Express switching complex
                       in what amounts to a memory server that it calls
                       DirectCXL. The DirectCXL proof of concept was
                       talked about it in a paper that was presented at
                       the USENIX Annual Technical Conference last week,
                       in a brochure that you can browse through here, and
                       in a short video you can watch here. (This sure
                       looks like startup potential to us.)


          o § Security⠀➾


                # ⚓ Hacker News ☛ Juniper_Releases_Patches_for_Critical_Flaws
                  in_Junos_OS_and_Contrail_Networking [Ed: As per revelations
                  amid Snowden's NSA leaks, security is not their goal]⠀⇛


                       Juniper Networks has pushed security updates to
                       address several vulnerabilities affecting multiple
                       products, some of which could be exploited to seize
                       control of affected systems.


                       The most critical of the flaws affect Junos Space
                       and Contrail Networking, with the tech company
                       urging customers to release versions 22.1R1 and
                       21.4.0, respectively.


                # ⚓ LWN ☛ Security_updates_for_Monday_[LWN.net]⠀⇛


                       Security updates have been issued by Debian (mat2
                       and xen), Fedora (butane, caddy, clash, direnv,
                       geoipupdate, gitjacker, golang-bug-serial-1,
                       golang-github-a8m-envsubst, golang-github-apache-
                       beam-2, golang-github-aws-lambda, golang-github-
                       cespare-xxhash, golang-github-chromedp, golang-
                       github-cloudflare, golang-github-cloudflare-
                       redoctober, golang-github-cockroachdb-pebble,
                       golang-github-cucumber-godog, golang-github-
                       dreamacro-shadowsocks2, golang-github-
                       dustinkirkland-petname, golang-github-etcd-io-
                       gofail, golang-github-facebookincubator-contest,
                       golang-github-facebookincubator-dhcplb, golang-
                       github-facebookincubator-go2chef, golang-github-
                       facebookincubator-ntp, golang-github-
                       facebookincubator-nvdtools, golang-github-goccy-
                       yaml, golang-github-gojuno-minimock, golang-github-
                       google-wire, golang-github-hexdigest-gowrap,
                       golang-github-intel-goresctrl, golang-github-j-
                       keck-arping, golang-github-jamesclonk-vultr,
                       golang-github-liamg-scout, golang-github-liamg-tml,
                       golang-github-mattn-colorable, golang-github-
                       mdlayher-ethernet, golang-github-moby-buildkit,
                       golang-github-mock, golang-github-niklasfasching-
                       org, golang-github-nxadm-tail, golang-github-path-
                       network-mmproxy, golang-github-rakyll-statik,
                       golang-github-shopify-toxiproxy, golang-github-
                       shulhan-bindata, golang-github-skynetservices-
                       skydns, golang-github-sophaskins-efs2tar, golang-
                       github-spf13-cobra, golang-github-spyzhov-ajson,
                       golang-github-task, golang-github-temoto-robotstxt,
                       golang-github-theoapp-theo-agent, golang-github-
                       tinylib-msgp, golang-github-tklauser-numcpus,
                       golang-github-valyala-fasthttp, golang-google-
                       protobuf, golang-honnef-tools, golang-k8s-kube-
                       openapi, golang-k8s-pod-security-admission, golang-
                       k8s-sample-cli-plugin, golang-mvdan-sh-3, golang-
                       storj-drpc, golang-x-tools, gopass, harfbuzz,
                       hcloud, manifest-tool, moby-engine, mqttcli, nex,
                       php-laminas-diactoros2, podman-tui, seamonkey,
                       snapd, tinygo, vgrep, vultr, vultr-cli, weldr-
                       client, xen, and yubihsm-connector), Mageia (golang
                       and java), Oracle (grub2, kernel, kernel-container,
                       and squid), and SUSE (crash, kernel, nodejs12,
                       nodejs14, and nodejs16).


                # ⚓ MakeTech Easier ☛ Internal_Roblox_Documents_Hacked_and
                  Posted_Online⠀⇛


                       It seems like if you’re a company with a popular
                       online presence that you
                       should expect to be hacked at some point. The
                       latest to be hacked is the gaming platform Roblox.
                       A cybercriminal connected with the right person in
                       Roblox and posted internal employee documents
                       online in what appears to be an extortion attempt.


                # ⚓ LinuxSecurity ☛ Hacker’s_Corner:_Complete_Guide_to_Anti-
                  Debugging_in_Linux_–_Part_3⠀⇛


                       A breakpoint is intentional “pause” in normal
                       execution of a program, generally used to inspect
                       the internals of said process in more detail. This
                       is the *most* used feature of any debugger.


                       On x86 CPUs, there are two types of breakpoints:
                       hardware breakpoints and software breakpoints.
                       While they overlap to a certain degree they are not
                       exactly the same.


                       In most of debugging cases, you will be using
                       software breakpoints, which do not need any special
                       hardware support. These are implemented using same
                       interrupt mechanism which is used by pretty much
                       everything else. On x86, 3rd interrupt is used to
                       implement a breakpoint. When you set a breakpoint,
                       your debugger overwrites target address (where you
                       want to put the breakpoint) with INT 3 (0xCC in
                       hex). When this instruction gets executed, debugger
                       gets the control back from target process, and can
                       inspect its state (registers, memory etc). To
                       resume the execution, debugger will silently remove
                       breakpoint, execute the instruction, and set the
                       breakpoint again before letting the process resume
                       (until it terminates, or breaks). Features like
                       step over, step out are also implemented using
                       “transparent” software breakpoints, which are set
                       and removed automatically by debugger. Generally,
                       you can set any number of software breakpoints;
                       however these cannot be set on non-code address
                       (i.e. these can break the program only when target
                       address content is executed; but not if the address
                       is read from or write to).


                       Hardware breakpoints, on the other hand, are much
                       more powerful and flexible than software
                       breakpoints. These can be set to break not only on
                       execution, but also on memory access (read and
                       write both), I/O port access etc. These debuggers
                       are set by writing into special “debug registers”
                       which are largely platform specific (and not all
                       platforms will have support for hardware
                       breakpoints). On x86, registers DR0-3 and DR6-7 are
                       used to set these breakpoints (DR4-5 are reserved
                       as of now). If you have ever used “watchpoints”
                       which let you break when certain memory address is
                       accessed, you have used hardware breakpoints.


                # ⚓ Implementing_Zero-Trust_on_Kubernetes_–_Container_Journal⠀⇛


                       As a cornerstone of the cloud-native community,
                       Kubernetes empowers enterprises to deploy and
                       manage containers in production environments more
                       efficiently. Although Kubernetes was initially
                       designed with basic security capabilities, broad
                       and rapid adoption and an increasingly
                       sophisticated threat landscape have made Kubernetes
                       more vulnerable to attacks. Developers and security
                       experts are now tasked with extending Kubernetes’
                       built-in security to effectively protect against
                       cyberattacks that are more complex, volatile and
                       frequent.


                       The previous “trust but verify” approach has proven
                       ineffective for the complex distributed nature of
                       cloud computing, so Kubernetes must move to the
                       “never trust, always verify” ideology of the zero-
                       trust model to provide greater protection to
                       businesses.


                # § Privacy/Surveillance⠀➾


                      # ⚓ Bruce Schneier ☛ Facebook_Is_Now_Encrypting_Links_to
                        Prevent_URL_Stripping⠀⇛


                             Some sites, including Facebook, add
                             parameters to the web address for tracking
                             purposes. These parameters have no
                             functionality that is relevant to the user,
                             but sites rely on them to track users across
                             pages and properties.


                      # ⚓ Ghacks ☛ Facebook_has_started_to_encrypt_links_to
                        counter_privacy-improving_URL_Stripping_–_gHacks_Tech
                        News⠀⇛


                             Facebook has started to use a different URL
                             scheme for site links to combat URL stripping
                             technologies that browsers such as Firefox or
                             Brave use to improve privacy and prevent user
                             tracking.


          o § Civil Rights/Policing⠀➾


                # ⚓ Haaretz ☛ Israeli_Pegasus_Spyware_Was_Used_Against_Thai
                  Pro-democracy_Activists,_Report_Says⠀⇛


                       After years of concerns that Thailand was
                       purchasing Israeli-made cyberoffense surveillance
                       technologies, rights groups have found the first
                       forensic evidence linking the NSO Group’s infamous
                       Pegasus spyware to the Southeast Asian kingdom.


                       A new report by Citizen Lab, published Monday
                       together with two Thai civil society groups, says
                       digital forensic evidence indicates that at least
                       30 cellphones belonging to activists in Thailand
                       were targeted with Pegasus.


                       “We can now officially add Thailand to the growing
                       list of countries where people peacefully calling
                       for change, expressing an opinion or discussing
                       government policies may trigger invasive
                       surveillance with a profound toll on an
                       individual’s freedom of expression, privacy and
                       sense of security,” said Etienne Maynier, a
                       technologist at Amnesty International, who peer
                       reviewed the report and its methodologies.


          o § Internet Policy/Net Neutrality⠀➾


                # ⚓ Public Knowledge ☛ Public_Knowledge_Urges_Congress_To
                  Swiftly_Move_New_Net_Neutrality_Bill_Forward_–_Public
                  Knowledge⠀⇛


                       Today, reports indicate that Senators Ed Markey (D-
                       MA) and Ron Wyden (D-OR), along with Rep. Matsui
                       (D-CA) and other co-sponsors, will soon introduce a
                       bill to reinstate net neutrality and the Federal
                       Communications Commission’s authority over
                       broadband service. The “Net Neutrality and
                       Broadband Justice Act” would codify the FCC’s
                       authority to classify broadband as a “Title II”
                       communications service and regulate it as such.


                       The bill effectively reverses the unprecedented
                       abdication of authority over broadband that FCC
                       Chairman Pai instigated in the agency’s 2017 repeal
                       of its 2015 Open Internet Order that established
                       the net neutrality rules under Chairman Tom
                       Wheeler. Those rules prevented broadband providers
                       from blocking websites, throttling web traffic, or
                       creating “fast lanes” only for those able to pay
                       for prioritization. Prior to Chairman Pai’s
                       abdication, Republican and Democratic Commissions
                       had agreed the FCC had authority over broadband –
                       the question was just under which title of its
                       statute. The bill also prevents a future-FCC from
                       backtracking on other broadband consumer
                       protections – fundamental in an increasingly online
                       world. Public Knowledge urges Congress to swiftly
                       pass this bill once introduced.


    * § Gemini* and Gopher⠀➾


          o § Personal⠀➾


                # ⚓ SpellBinding:_CELNTXY_Wordo:_IMPEL⠀⇛


                # ⚓ a_game_in_my_head⠀⇛


                       One thing that I’ve never told anyone is that I
                       freated a game that I’ve been playing for more than
                       10 years.


                       The concept of the game is really weird, as it’s
                       about floor patterns, the position of the feet, and
                       breathing. There are floor patterns that requires
                       me to put my left foot and right foot in a certain
                       position, some that may require me to stop
                       breathing once I stepped on it, some are non
                       rewaeding, some are very rewarding. It’s quite a
                       mess, it’s also quite dangerous as I have to look
                       at thr floor for most of the time, but it’s fun for
                       me, and I somehow survived playing it.


                # ⚓ Deus_Ex⠀⇛


                       I’ve been a pseudo conspiracy junkie since I was
                       very young. Not in that I believe in any, but that
                       I’ve always found them interesting. Belief in such
                       phenomena is very quaint; not in a negative way,
                       but in that it harkens back to a very “early human”
                       mindset. Which makes it ironic then, that I became
                       such a big Deus Ex fan without knowing how central
                       it played a part in the originals. I got enthralled
                       by HR, but only ever really gave IW a decent try;
                       always failing to struggle through the OG.


                # ⚓ Spooky_Anime_#1⠀⇛


                       A while back I stumbled across a well-put-together
                       thread on /x/ about weird/spooky anime. The OP of
                       that thread created these beautiful collages for
                       each one and really piqued my interest in checking
                       them out. I finally got around to watching some of
                       them and decided to finally make use of my page
                       here to talk about them as I go through them. My
                       main motivation for this right now is to have a
                       personal record of these shows and movies that I
                       can look back on. But I also hope that I can spread
                       some enjoyment to others out there should any of
                       these interest you 🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇:D⦈


                # ⚓ Kink_aesthetics⠀⇛


                       i recently had an excellent conversation with a
                       fellow kinkster about kink aesthetics, during which
                       we discussed the pervasive idea that a kink event
                       must (apparently) have a medieval-industrial vibe:
                       a ‘dungeon’ flooded with industrial music[a]. Not
                       only that, but there is a relative homogeneity to
                       the sort of outfits seen as appropriate for such an
                       event: leather, for example, is expected in a way
                       that brightly-coloured drag gear is …. not. Still,
                       and much to my delight, they were planning to
                       attend just such a kink event in just such gear.


          o § Technical⠀➾


                # § Internet/Gemini⠀➾


                      # ⚓ Gemini_Radio:_Episode_43⠀⇛


===============================================================================
* Gemini_(Primer) links can be opened using Gemini_software. It’s like the
World Wide Web but a lot lighter.



                    ䷩ 𝚕𝚒𝚗𝚎 4212

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Gemini_version_available_♊︎


✐ Links_18/07/2022:_More_on_Microsoft’s_Assault_Against_Free_Software⠀✐


Posted in News_Roundup at 5:18 am by Dr. Roy Schestowitz


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇GNOME bluefish⦈


§ Contents⠀➾


    * GNU/Linux
          o Kernel_Space
          o Instructionals/Technical
    * Distributions_and_Operating_Systems
          o Red_Hat_/_IBM
    * Free,_Libre,_and_Open_Source_Software
          o Licensing_/_Legal
    * Leftovers
          o Education
          o Hardware
          o Health/Nutrition/Agriculture
          o Privatisation/Privateering
          o Defence/Aggression
          o Environment
                # Energy
          o Finance
          o AstroTurf/Lobbying/Politics
          o Civil_Rights/Policing
          o Internet_Policy/Net_Neutrality
          o Digital_Restrictions_(DRM)
          o Monopolies
                # Patents
    * Gemini*_and_Gopher
          o Technical
                # Science
    * § GNU/Linux⠀➾


          o § Kernel Space⠀➾


                # ⚓ The Register UK ☛ Improve_Linux_performance_with_this_one
                  weird_trick_•_The_Register⠀⇛


                       A cryptic website with a single line of text
                       promises to make your Linux box more responsive –
                       if you are willing to accept some risk.


                       Another day, another bleed-ing vulnerability. New
                       speculative-execution attacks keep being
                       discovered, and OS kernel developers keep finding
                       ways to block them – at the cost of some CPU
                       performance for each mitigation. But what if you
                       run an isolated, standalone box? What if you just …
                       turned off all the mitigations?


                       Many years ago my then-lodger – a Perl and Linux
                       guru – attempted to show me a quick way to move
                       some stuff from one Linux box to another. He was
                       very surprised to discover that I didn’t have SSH
                       enabled, and that there was no way to access any of
                       my Linux boxes from any of the others. This was and
                       is the default setting for the desktop versions of
                       Ubuntu and Mint, and for a lot of people, that’s
                       what you want: a computer that is sealed off to the
                       outside world.


                       If that describes your needs, then you might find
                       Jean-Michaël Celerier’s admirably terse make-linux-
                       fast-again.com useful. The site is a little dated –
                       you only need all of the switches if you’re running
                       a kernel older than version 5.1.13. These days,
                       just the last one is enough. Add it to the end of
                       the kernel line in /etc/default/grub, run update-
                       grub and reboot.


          o § Instructionals/Technical⠀➾


                # ⚓ Linux Capable ☛ How_to_Install_Brave_Browser_on_Rocky_Linux
                  9⠀⇛


                # ⚓ Linux Capable ☛ How_to_Install_OnlyOffice_on_Rocky_Linux
                  9⠀⇛


                # ⚓ Query_cheat.sh_from_the_termianl._A_quick_access_cheat
                  sheet_for_a_range_of_linux_commands!⠀⇛


                # ⚓ Quickly_run_any_gif_as_a_ASCII_animation_in_a_terminal
                  window⠀⇛


    * § Distributions and Operating Systems⠀➾


          o § Red Hat / IBM⠀➾


                # ⚓ Fedora Magazaine ☛ Community_container_images_available_for
                  applications_development⠀⇛


                       Fedora containers are based on the latest stable
                       Fedora content, and CentOS-7 containers are based
                       on components from the CentOS-7 and related SCLo
                       SIG components. And finally, CentOS Stream
                       containers are based on either CentOS Stream 8 or
                       CentOS Stream 9.


                       Each container, e.g. s2i-php-container or s2i-perl-
                       container, contain the same packages which are
                       available for a given operating system. It means,
                       that from a functionality point of view these
                       example containers provides the PHP interpreter or
                       Perl interpreter,respectively.


                # ⚓ Red Hat ☛ Simplify_client_connection_configurations_with
                  service_contexts_|_Red_Hat_Developer⠀⇛


                       The latest release of rhoas, the command-line
                       interface (CLI) for Red Hat OpenShift application
                       services, adds a powerful and flexible feature
                       called service contexts that makes it easier than
                       ever to connect clients to your instances of
                       OpenShift application services. This article
                       illustrates this new feature and shows how it can
                       accelerate your development workflows for stream-
                       based applications.


                # ⚓ Enterprisers Project ☛ IT_talent:_How_upskilling_can_help
                  boost_digital_transformation⠀⇛


                       “Many organizations are struggling with the digital
                       transformation efforts that they have begun,” says
                       David Rogers, author of The Digital Transformation
                       Playbook and professor at Columbia Business School,
                       in the latest Harvard Business Review Analytic
                       Services report sponsored by Red Hat.


                # ⚓ Enterprisers Project ☛ 5_ways_fear_can_derail_your_digital
                  transformation_strategy⠀⇛


                       What could go wrong with digital transformation?


                       You might think the main problems would spring from
                       something technical such as a bug, a virus, or
                       maybe incompatibility. Seldom do we think about the
                       people who are expected to use these tools – but
                       they may be the most dangerous part of your new
                       digital infrastructure. That’s because, with
                       humans, fear is in control.


    * § Free, Libre, and Open Source Software⠀➾


          o § Licensing / Legal⠀➾


                # ⚓ TechCrunch ☛ Dissecting_Microsoft’s_proposed_policy_to_ban
                  commercial_open_source_apps_–_TechCrunch⠀⇛


                       Bradley M. Kuhn is a free and open source software
                       (FOSS) activist who serves as “hacker-in-residence”
                       at Software Freedom Conservancy, a not-for-profit
                       organization that provides support and legal
                       services for open source software projects. In the
                       aftermath of Microsoft’s policy change, Kuhn and
                       his colleague Denver Gingerich penned an in-depth
                       blog post outlining their issues with the new
                       policy (as well as more long-standing gripes with
                       Microsoft’s attitude to open source), and the main
                       thrust of their argument was that the very nature
                       of open source software is that it’s free of
                       restrictions, with no favoritism over who can and
                       can’t monetize it.


                       “We believe that the rights ensured by FOSS, as is
                       well-enshrined in the licenses themselves, allow
                       everyone to monetize FOSS,” Kuhn told TechCrunch.
                       “FOSS licenses have always treated all commercial
                       and non-commercial actively equally. It’s free as
                       in freedom, but also it’s free as in market.”


                       While there are many different kinds of open source
                       licenses available (whether they are all truly
                       “open source” is a debate for another day), Kuhn is
                       referring specifically to so-called “copyleft”
                       licenses. Such licenses have few restrictions, but
                       they do mandate that any software derived from the
                       original open source project must be released under
                       a similar open source license. This runs in
                       contrast to more “permissive” licenses that don’t
                       impose such restrictions (meaning that private
                       companies can easily adopt an open source project
                       as part of a proprietary product).


                       Put simply, the spirit of open source is all about
                       the freedoms it permits.


                       “Copyleft licenses require that you provide
                       correct, complete, and corresponding source code to
                       all customers, and have various rules about
                       patents, but there are otherwise generally not
                       intended to be serious restrictions on the ability
                       monetize FOSS,” Kuhn said.


                       A recurring discussion point around Microsoft’s
                       proposed policy change is how it may protect
                       trademarks or avert “brand confusion.” But Kuhn
                       argues that this is already well-provisioned for in
                       existing laws, and it’s not specifically a FOSS
                       problem — it’s a problem relating to software in
                       general.


                       “Trademark rules control the rights to name and
                       market a product under a particular name,” Kuhn
                       said. “Trademark restrictions on using a name are
                       completely compatible with FOSS and have long been
                       encouraged. Now, this is not a FOSS-specific
                       problem, but cloned software by fly-by-night
                       entities and malware on app stores is a broader
                       problem.”


                       While all of this is probably true, independent
                       open source developers generally don’t have the
                       resources to pursue what are often faceless
                       entities over trademark violations (assuming that
                       they actually own the trademark at all). And that
                       is precisely why a policy that deters “fly-by-
                       night” developers from capitalizing on the hard
                       work of others will likely be well received
                       whenever it’s finally introduced.


    * § Leftovers⠀➾


          o ⚓ Hackaday ☛ It_Turns_Out_You_Can’t_Just_Fly_A_Drone_Under_Water⠀⇛


                 The differences between a drone and an underwater remote-
                 operated vehicle (ROV) aren’t actually that large. Both
                 have powerful motors that move large volumes of fluid
                 (yes, air is a fluid), a camera, a remote, and an onboard
                 battery. So when [RCLifeOn] got his hands on a cheap used
                 drone, he reckoned that it could fly underwater just as
                 well as it did in the air.


          o ⚓ Hackaday ☛ Observe_Airflow_Using_Smartphone_And_Background-
            Oriented_Schlieren⠀⇛


                 Multiple people have recently shared this exciting
                 demonstration (nitter) with us – visualizing airflow
                 using a smartphone, called ‘background-oriented
                 schlieren’. On a hot summer day, you might see waves in
                 the air – caused by air changing density as it warms up,
                 and therefore refracting the light differently. Schlieren
                 photography is an general set of techniques for
                 visualizing fluid flow, but of course, it can also be
                 applied to airflow. In this case, using some clever
                 optical recognition tricks, this schlieren method lets
                 you visualize flow of air using only your Android
                 smartphone’s high resolution camera and a known-pattern
                 printed background!


          o § Education⠀➾


                # ⚓ Counter Punch ☛ Students,_Campuses_and_Dominant_Corporate
                  Power⠀⇛


                       Much of campus activity these days focuses on
                       diversity, tuition, student loans, “politically
                       correct” speech demands and conforming conduct.


                       This campus environment is strangely oblivious to
                       the corporate abuses of our economy, culture and
                       government. This indifference extends to the
                       endless grip of corporate power over the
                       educational institutions that the students attend.


          o § Hardware⠀➾


                # ⚓ Hackaday ☛ A_Home_Made_Sewing_Machine_May_Be_The_Only_One⠀⇛


                       The sewing machine is a tool that many of us will
                       have somewhere around our workshop. Concealed
                       within it lies an intricate and fascinating
                       mechanism. Some of us may have peered inside, but
                       very few indeed of us will have gone to the effort
                       of building our own. In case you had ever wondered
                       whether it was possible, [Fraens] has done just
                       that, with what he claims may be the only entirely
                       homemade sewing machine on the Internet.


                # ⚓ Hackaday ☛ Beautiful_Inductors,_Now_Not_Such_A_Lost_Art⠀⇛


                       As ferrite technology has progressed into a mastery
                       of magnetic permeability, the size of inductors has
                       gone down to the point at which they are now fairly
                       nondescript components. There was a time though
                       when inductors could be beautiful creations of
                       interleaving layers of copper wire in large air-
                       cored inductors, achieved through clever winding
                       techniques. It’s something that’s attracted the
                       attention of [Brett], who’s produced a machine
                       capable of producing something close to the
                       originals.


                # ⚓ Hackaday ☛ Robot_Arm_Has_The_Touch⠀⇛


                       [Maurizio] built a robot arm, which is always a
                       great accomplishment. But his project includes a
                       very cool touch interface for an Android device
                       that sets it apart from many other similar
                       projects.You can see a very fast summary of the
                       construction in the video below.


          o § Health/Nutrition/Agriculture⠀➾


                # ⚓ Orac ☛ Can_you_get_Omicron_every_three_weeks?⠀⇛


                       I have been very critical of the Great Barrington
                       Declaration (GBD) ever since it was first published
                       in October 2020. The GBD, as you might recall, was
                       a document published as a result of a meeting held
                       in Great Barrington, MA at the headquarters of the
                       right wing libertarian free market think
                       tank American Institute for Economic Research 
                       (AIER). The reason that I’ve been so critical of
                       the GBD—calling it right from the
                       beginning “magnified minority” and eugenics—is
                       because it advocated a “natural herd immunity”
                       approach to the COVID-19 pandemic; i.e., what we
                       referred to as as a “let ‘er rip” approach in which
                       SARS-CoV-2, the coronavirus that causes COVID-19,
                       should be allowed to spread through the “low risk”
                       young and “healthy” population, in order to produce
                       “natural immunity” in enough people to reach herd
                       immunity, meanwhile using “focused protection”
                       (never well defined) to keep the elderly and those
                       with chronic health conditions that rendered them
                       susceptible to death and the worst complications of
                       the virus. (Remember, this was a couple of months
                       before the mRNA-based COVID-19 vaccines received
                       emergency use authorization.) It’s an approach
                       that never would have worked, given the practical
                       difficulties and the observation (predicted even in
                       October 2020) that “natural” postinfection immunity
                       was not likely to be durable, given the rise of
                       immune-evading variants like Omicron. Even given
                       the history of the last two years, GBD
                       adherents still claim that “herd immunity” would be
                       a mere 3-6 months away if we just followed their
                       plan. Worse, GBD authors like Martin Kulldorff and
                       Jay Bhattacharya have pivoted to help
                       spread antivaccine disinformation, along with the
                       “spiritual child” of the GBD, the Brownstone
                       Institute (for which Kulldorff is the scientific
                       advisor), which has apparently never seen a public
                       health intervention that it liked—although it does
                       very much like comparing public health
                       to fascism and Communism.


          o § Privatisation/Privateering⠀➾


                # ⚓ Scheerpost ☛ The_Corporatization_of_Just_About_Everything_—
                  Including_the_Climate_Crisis_and_Outer_Space⠀⇛


                       Not only is there more corporate control in more
                       areas of life but (worse) there has been
                       consolidation of corporate power into monopolies
                       that are so powerful that they seem unstoppable.


          o § Defence/Aggression⠀➾


                # ⚓ Common Dreams ☛ Opinion_|_The_FBI_Is_Still_Targeting_Black
                  People_for_Entrapment⠀⇛


                       Romeo Langhorne is the latest victim of an FBI
                       phony terror entrapment scheme. On July 7, 2022 he
                       was sentenced to 20 years in prison for uploading a
                       bomb-making video. Langhorne didn’t make a bomb. He
                       uploaded a video while under the direction of an
                       FBI informant. The video had in fact been produced
                       by the government.


                # ⚓ Counter Punch ☛ Get_Gota:_Holding_a_War_Criminal
                  Accountable⠀⇛


                       There have been various efforts in train regarding
                       a man who ruthlessly concluded his country’s civil
                       war in an orgy of mass killing. The war itself,
                       waged between the forces of Sinhalese Buddhist
                       nationalism and the minority Tamils seeking
                       independence, was the rotten fruit of
                       discrimination, exclusion and ethnocratic politics
                       heralded by the passage of the Sinhala Only Act in
                       1956. That legislative instrument, implemented by
                       Prime Minister S.W.R.D Bandaranaike, made Sinhalese
                       the country’s official language while banishing
                       Tamils from important positions of employment.


                       Gotabaya’s entry into Sri Lankan politics was a
                       fraternal affair. His brother Mahinda, on becoming
                       president in 2005, picked him as defence secretary.
                       Prior to that, “Gota” worked as a computer systems
                       administrator at Loyola School in Los Angeles,
                       during which time he became a US citizen.


                # ⚓ Counter Punch ☛ Maasai_Evictions_Trigger_Frankfurters’_Fake
                  Condemnation⠀⇛


                       It’s also the first time – and the two “firsts” are
                       connected – that the violence inherent in a
                       conservation land grab has been broadcast around
                       the world in real time. Within a few minutes of
                       Maasai uploading mobile phone footage it was in the
                       public domain, with its unarguable drama: the thuds
                       of the bullets; the Maasai fleeing in their red
                       robes, overtaking others who hadn’t yet seen the
                       danger; the shakiness of a cameraman close to the
                       line of fire.[3] This was cinéma-vérité on a level
                       previously unimaginable in the history of
                       conservation.


                       People like me, who have been campaigning against
                       similar crimes for decades, were able to assess the
                       footage, appreciate its genuineness and relay it on
                       in just a couple of minutes. By the time the
                       Tanzanian authorities realised the scale of the
                       exposure, and were making a feeble effort to deny
                       it had happened, the horse had bolted.


                # ⚓ Telex (Hungary) ☛ Western_allies_puzzled_by_Hungary’_mild
                  reaction_to_Russia’s_hacking⠀⇛


                # ⚓ Common Dreams ☛ ‘Remember_Abu_Ghraib,’_MBS_Tells_Biden_When
                  Pressed_on_Khashoggi:_Report⠀⇛


                       Saudi Crown Prince Mohammed bin Salman reportedly
                       told President Joe Biden during their meeting in
                       Jeddah Friday that while the murder of journalist
                       Jamal Khashoggi is “regrettable,” U.S. hands are
                       not clean and other journalists are killed with
                       impunity.


                       “Bin Salman’s smarmy reply underlines the way in
                       which the U.S. government’s lawlessness… and its
                       weird commitment to Israeli apartheid… undermine
                       its moral standing.”


                # ⚓ Common Dreams ☛ Opinion_|_Plotting_War_on_Iran⠀⇛


                       The United States, Israel and Saudi Arabia are
                       plotting a war with Iran. The 2015 Iranian nuclear
                       arms accord, or Joint Comprehensive Plan of Action
                       (JCPOA), which Donald Trump sabotaged, does not
                       look like it will be revived.  U.S. Central Command
                       (CENTCOM) is reviewing options to attack if Teheran
                       looks poised to obtain a nuclear weapon and Israel,
                       which opposes U.S.-Iran nuclear negotiations,
                       carries out military strikes.


                # ⚓ Counter Punch ☛ Palestinian_Universities_are_Once_Again
                  Under_Attack⠀⇛


                       Later this month, the Israeli authorities are
                       expected to put into effect a 97-page ordinance,
                       called Procedure for Entry and Residence for
                       Foreigners in Judea and Samaria Area (PDF), which
                       would grant the Israeli Ministry of Defence and
                       thus, the military, absolute power to determine how
                       many and which foreign academics and students can
                       visit, study or work at all 15 Palestinian
                       universities and colleges in the West Bank.


                       The “procedure” limits the number of staff allowed
                       to work for any of these 15 universities and
                       colleges to no more than 100 “distinguished
                       lecturers and researchers,” noting that
                       “applications for a permit under this section will
                       be approved if it is demonstrated, to the
                       satisfaction of the authorised [military] official,
                       that the lecturer contributes significantly to
                       academic learning, to the area’s economy, or to
                       advancing regional cooperation and peace”.


                # ⚓ Scheerpost ☛ “The_American_Production_System”:_The_Rise_and
                  Fragmentation_of_a_Manufacturing_Empire⠀⇛


                       It can be said that since the 1960s, the U.S.
                       government and elites have had a persistent
                       manufacturing anxiety.


                # ⚓ Scheerpost ☛ Chris_Hedges_on_Trauma_and_Teaching_Writing_in
                  Prison⠀⇛


                       Joining Mansa Musa on Rattling the Bars, Hedges
                       speaks about his book and the transformations he
                       witnessed among the men he taught behind prison
                       walls.


                # ⚓ Scheerpost ☛ Patrick_Lawrence:_The_Imaginary_War⠀⇛


                       It began when the Biden regime and the press
                       misrepresented Russian aims in Ukraine. All else
                       has flowed from it.


                # ⚓ TruthOut ☛ Let’s_Replicate_Chicago’s_Antiwar_Organizing
                  Victory_Against_Boeing⠀⇛


                # ⚓ Counter Punch ☛ The_Second_Amendment_and_Gun_Violence_in
                  the_U.S.⠀⇛


                       The Second Amendment to the United States
                       Constitution, ratified on December 15, 1791, reads:
                        A well regulated militia, being necessary to the
                       security of a free State, the right of the people
                       to keep and bear arms, shall not be infringed. This
                       amendment was inspired by the right to keep and
                       bear arms recognized in the English common law, and
                       by the English Bill of Rights of 1689 that
                       prevailed in the American colonies. The scope and
                       prerogatives arising under the Second Amendment
                       have been the subject of considerable controversy,
                       and its vague interpretation has had serious legal
                       consequences.


                       Opponents of gun control emphasize the last part of
                       the sentence, “…the right of the people to keep and
                       bear arms, shall not be infringed,” neglecting that
                       this constitutional entitlement is centered on a
                       “well regulated militia,” which at the time was
                       deemed “necessary to the security of a free
                       state.” As noted by Alexander Hamilton in the
                       Federalist Papers, a well regulated militia is “the
                       most natural defense of a free country.”


                # ⚓ Counter Punch ☛ The_West_Is_Experiencing_a_Contraction_of
                  Its_Power,_Not_Necessarily_Its_Decline⠀⇛


                       Meanwhile, what Westerners understood by the West
                       was changing. It began as Christianity and
                       colonialism, then changed to capitalism and
                       imperialism, and then metamorphosed into democracy,
                       human rights, decolonization, self-determination,
                       and “rules-based international relations”—it was
                       made clear that the rules would be established by
                       the West and would only be followed when they
                       served its interests—and finally into
                       globalization.


                       By the middle of the last century, the West had
                       shrunk so much that several newly independent
                       countries made the decision to align themselves
                       neither with the West nor with the bloc that had
                       emerged as its rival, the Soviet bloc. This led to
                       the emergence, from 1955-1961, of the Non-Aligned
                       Movement (NAM). With the collapse of the Soviet
                       bloc in 1991, the West seemed to go through a time
                       of enthusiastic expansion. It was around this time
                       that former Russian President Mikhail Gorbachev
                       expressed his desire for Russia to join the “common
                       home” of Europe, with the support of then-U.S.
                       President George H. W. Bush, a desire reaffirmed by
                       Vladimir Putin when he took power in 2000. It was a
                       short historical period, and recent events show
                       that the “size” of the West has since shrunk
                       drastically. In the wake of the Ukraine war, the
                       West decided, on its own initiative, that only
                       those countries applying sanctions against Russia
                       would be considered part of the pro-Western camp.
                       These countries comprise about 21 percent of the UN
                       member countries, which constitute only 16
                       percent of the world’s population.


          o § Environment⠀➾


                # ⚓ Common Dreams ☛ Over_1,000_Die_as_Record-Breaking_Heat,
                  Wildfires_Scorch_Europe⠀⇛


                       Record-breaking heat has killed over 1,000 people
                       in Western Europe over the past week, while
                       firefighters battle to contain blazes scorching
                       swathes of three countries amid a worsening climate
                       emergency, officials said this weekend.


                       El País reports heat killed 360 people in Spain
                       between July 10 and July 15. This follows the heat-
                       related deaths of more than 800 people last month,
                       according to the Spanish government’s Carlos III
                       Health Institute. Madrid-Barajas International
                       Airport recorded an all-time high temperature of
                       108°F Thursday, while some Spanish municipalities
                       registered highs of 110°F to 113°F.


                # ⚓ Counter Punch ☛ Nuclear_Power_is_Not_the_Solution_to
                  Climate_Change⠀⇛


                       Although reducing carbon emissions is critical to
                       slowing the pace of climate change, it must not be
                       our only litmus test for moving toward a “clean”
                       energy future, similarly to how our overall health
                       cannot be measured solely by our blood pressure or
                       weight.


                       In the case of nuclear power, we must consider its
                       high cost compared to renewable energy sources,
                       such as wind and solar. According to Climate Nexus,
                       the minimum cost per megawatt hour to build a new
                       nuclear plant is almost 3 times higher than
                       utility-scale solar ($112 vs. $46, respectively)
                       and almost 4 times higher than wind power ($122 vs.
                       $30, respectively). That’s like paying $70,000 for
                       a car when you could purchase an equivalent car, in
                       terms of its overall value, for one-third or one-
                       quarter of the cost.


                # ⚓ TruthOut ☛ The_National_Weather_Service_Is_Unwittingly
                  Obscuring_Reality_of_Global_Warming⠀⇛


                # § Energy⠀➾


                      # ⚓ Common Dreams ☛ Opinion_|_Petro,_Lula,_and_the_2024
                        Venezuelan_Election⠀⇛


                             Last month I followed Colombia’s elections
                             very closely. Unlike many of my countrymen,
                             I’m very passionate about the history and
                             affairs of our closest neighbor. To be
                             completely honest, I thought Petro was not
                             going to make it.


          o § Finance⠀➾


                # ⚓ Counter Punch ☛ Does_the_Future_Belong_to_People_Who_Profit
                  Off_Our_‘Excessive_Wealth_Disorder’?⠀⇛


                # ⚓ Counter Punch ☛ James_Webb_Space_Telescope_and_Feeding_the
                  Hungry⠀⇛


                       But if we saw Putin and Trump each misusing power
                       that unleashed unnecessary death and havoc, on that
                       same ordinary day we also saw humans at their
                       extraordinary best. Approximately 20,000 scientists
                       from all over the world celebrated as they shared
                       with us some of the first images downloaded from
                       the Webb telescope, or, as they would like it
                       called, either the JWST (James Webb Space
                       Telescope) or the Webb Observatory.


                       It took a quarter of a century of highly technical
                       creative work, involving 14 countries, $9.7
                       billion, and most of all a scientific spirit of
                       cooperation that sadly feels far rarer than it
                       ought to be on our small planet, to put those
                       microscopically aligned mirrors millions of miles
                       out beyond the distortions of our atmosphere. Three
                       hundred forty-four procedures, “points of failure,”
                       any one of which gone wrong would have completely
                       halted the mission, had to go exactly right over
                       the deployment period of 30 days. And every one of
                       them did, including the launch of the reliable
                       Arianne rocket, the intricate unfurling of the
                       layers of protective foil that keep the instruments
                       from overheating, and the mind-bogglingly complex
                       imaging technology that has now begun to send back
                       crystal-clear images of the early universe, a gift
                       to all of us on earth.


          o § AstroTurf/Lobbying/Politics⠀➾


                # ⚓ Counter Punch ☛ The_Age_of_Weird_Men:_Gonzo_Politicians
                  From_Berlusconi_to_BoJo⠀⇛


                       These scholars of the future will have the great
                       advantage of knowing how the story ended and what
                       long term scars were left by “Johnsonism”, a
                       curious coalition of plutocrats, media magnates,
                       traditional conservatives and populist English
                       nationalists. This combination may fragment or may
                       live on like “Trumpism”, which survived Donald
                       Trump’s loss of the White House, and is poised to
                       win back Congress this year and the presidency in
                       2024.


                       People in England prefer to compare their politics
                       to America rather than Italy, but it was Silvio
                       Berlusconi – dominant in Italian politics from 1994
                       to 2011 – who first used the political skill set
                       later adopted by Trump and Johnson. I do not mean
                       that they consciously imitated him, but they chose
                       the same political formula to win power. Its
                       ingredients are not easy to identify because to use
                       them effectively, a leader has to be a chameleon,
                       making false and contradictory promises to
                       different parts of his electoral support.


          o § Civil Rights/Policing⠀➾


                # ⚓ Common Dreams ☛ Opinion_|_House_Panel_Told_Pregnant
                  Patients_Without_Abortion_Access_Face_‘Death_Sentence’⠀⇛


                       Witnesses told a U.S. House committee this week
                       that pregnant patients who can’t obtain abortions
                       will face higher mortality rates if they are forced
                       to carry their pregnancies to term.


                # ⚓ Counter Punch ☛ Beyond_Crisis:_The_Possibilities_and
                  Growing_Challenges_in_Building_a_Left_Asian_American
                  Politics⠀⇛


                       The two white men who attacked Chin, while Chin was
                       out celebrating at a bar with friends, did so
                       believing he was Japanese. At the time, much like
                       China now, Japan was viewed as the major economic
                       competitor of the U.S., and therefore, the main
                       reason for the economic downturn in cities like
                       Detroit. Aside from the fact that Japan wasn’t the
                       main reason for why there was another U.S. economic
                       recession, Chin had as much control over the
                       machinations of global capitalist order as any
                       individual working person, which was none.


                       “They made him a scapegoat,” said Annie Tan, Chin’s
                       cousin.


                # ⚓ Counter Punch ☛ LGBT_Adults_Are_More_Likely_to_Experience
                  Mental_Health_Hardships,_But_Less_Likely_to_Get_Needed_Help⠀⇛


                       To inform policy and other discussions around
                       LGBTQIA+ mental health, this article uses the
                       Census Bureau’s Household Pulse Survey (HPS) to
                       document mental health disparities by gender
                       identity and sexual orientation. Consistent with
                       prior research, it documents that LGBT adults are
                       more likely to report experiencing mental health
                       hardships than non-LGBT respondents (the analysis
                       is limited to LGBT adults because the HPS does not
                       include sufficient questions to cover the full
                       range of LGBTQIA+ identities). Yet at the same
                       time, LGBT respondents were less likely to get the
                       counseling and therapy they needed.


                       About one in two LGBT adults (48 percent) reported
                       feelings of anxiety for more than half the days in
                       the two weeks prior to responding to the survey,
                       compared to less than one-quarter (23.5 percent) of
                       non-LGBT adults.


                # ⚓ TruthOut ☛ NLRB_Is_Understaffed_and_Underfunded_at_Critical
                  Moment_for_Labor_Movement⠀⇛


          o § Internet Policy/Net Neutrality⠀➾


                # ⚓ Why_Internet_Access_Needs_to_Be_Considered_a_Basic_Human
                  Right⠀⇛


                       When United Nations Secretary-General António
                       Guterres highlighted the global effort required to
                       get back on track towards achieving the sustainable
                       development goals in his report in 2021, he
                       specifically called out the need for universal
                       access to the internet, among the other issue
                       areas.


                       “Now is the time to renew the social contract
                       between governments and their people and within
                       societies,” he said. “[The contract should] include
                       updated governance arrangements to deliver better
                       public goods and usher in a new era of universal
                       social protection, health coverage, education,
                       skills, decent work and housing, as well as
                       universal access to the internet by 2030 as a basic
                       human right.”


                       Acknowledging internet access as a human right
                       follows an announcement by the UN in 2016 that
                       “measures to intentionally prevent or disrupt
                       access to or dissemination of information online
                       (is) in violation of international human rights
                       law.”


                # ⚓ Resilience_Amid_Oppression_in_Myanmar’s_‘Digital_Coup’⠀⇛


                       Over a year since the coup that brought Myanmar
                       back under military dictatorship, the country
                       continues to face serious digital threats and
                       numerous human rights violations. The ongoing push
                       to enact a draconian cybersecurity bill and the
                       sale of Telenor’s Myanmar telecom unit to a
                       military-linked company are just some of the
                       developments fuelling fears of further digital
                       oppression in the country.


                       In this episode of Pretty Good Podcast, Asia-
                       Pacific analyst and Access Now’s Myanmar lead Wai
                       Phyo Myint elaborates on how these developments are
                       affecting the efforts of Myanmar’s resilient
                       digital rights actors, individuals, and
                       organisations to restore democracy in the country.
                       She also shares concrete ways that the
                       international community can continue to or update
                       their support for changemakers.


          o § Digital Restrictions (DRM)⠀➾


                # ⚓ Game Rant ☛ Analyst_Thinks_Microsoft_Could_Acquire
                  Netflix⠀⇛


                       Microsoft is currently aggressively acquiring video
                       game studios for its Xbox subsidiary. The pace of
                       acquisitions has slowed somewhat due to the ongoing
                       acquisition of Activision Blizzard, in part due to
                       the publisher’s size and the amount of
                       international oversight involved. But one analyst
                       believes that Microsoft could have an even bigger
                       acquisition in mind. They believe that once
                       Activision Blizzard closes, Microsoft may consider
                       exploring the possibility of acquiring video
                       streaming service Netflix.


          o § Monopolies⠀➾


                # § Patents⠀➾


                      # ⚓ Counter Punch ☛ Public_University_Patents_are_a
                        Racket⠀⇛


                             But if I gave your neighbor Bob a million
                             dollars of YOUR money to invent that
                             mousetrap, on the same conditions, you’d
                             probably take issue with the idea.


                             Even if you were interested in investing in
                             mousetrap innovation, you’d probably want a
                             return on your investment.


    * § Gemini* and Gopher⠀➾


          o § Technical⠀➾


                # § Science⠀➾


                      # ⚓ Hackaday ☛ Peer_Into_Space_Through_This_James_Webb-
                        Style_Hexagonal_Mirror⠀⇛


                             The James Webb Space Telescope (JWST)
                             generated considerable excitement when its
                             first test images were released earlier this
                             year: they proved that the instrument was
                             working and helped its engineers to set up
                             all systems for maximum performance. But the
                             real proof of the pudding came last week,
                             when the first batch of beautiful full-scale
                             pictures was unveiled. If you thought those
                             pictures were pretty enough to hang on your
                             wall, you’re not the only one: [Fredrik], als
                             known as [Cellar Nerd], built a wall-mounted
                             display, shaped like the JWST’s main mirror,
                             that cycles through images taken by the space
                             telescope.


===============================================================================
* Gemini_(Primer) links can be opened using Gemini_software. It’s like the
World Wide Web but a lot lighter.



                    ䷩ 𝚕𝚒𝚗𝚎 5220

╒═══════════════════ 𝐃𝐀𝐈𝐋𝐘 𝐋𝐈𝐍𝐊𝐒 ═════════════════════════════════════════════╕

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 07.18.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Links_18/07/2022:_Archcraft_July_2022_Release_and_Cutelyst_Relicensed⠀✐


Posted in News_Roundup at 5:49 pm by Dr. Roy Schestowitz


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇GNOME bluefish⦈


§ Contents⠀➾


    * GNU/Linux
          o Desktop/Laptop
          o Audiocasts/Shows
          o Applications
          o Instructionals/Technical
          o Desktop_Environments/WMs
                # K_Desktop_Environment/KDE_SC/Qt
                # GNOME_Desktop/GTK
    * Distributions_and_Operating_Systems
          o New_Releases
          o Rocky_Linux_/_Fedora_/_Red_Hat_/_IBM
          o Canonical/Ubuntu_Family
          o Open_Hardware/Modding
    * Free,_Libre,_and_Open_Source_Software
          o Licensing_/_Legal
          o Programming/Development
                # Perl_/_Raku
    * Leftovers
          o Health/Nutrition/Agriculture
          o Security
          o Finance
    * Gemini*_and_Gopher
          o Personal
    * § GNU/Linux⠀➾


          o § Desktop/Laptop⠀➾


                # ⚓ FOSSLife ☛ TUXEDO_Computers_Releases_Water_Cooling_System
                  for_Linux_Laptops⠀⇛


                       “Aquaris supports the conventional air cooling of
                       the notebook, resulting in significantly reduced
                       surface and CPU and GPU temperatures and overall
                       lower system fan noise. Alternatively, the
                       additional cooling capacity can be invested in
                       maximum CPU and GPU performance for rendering at
                       record speeds,” the announcement states.


                       The unit weighs 1.39 kg with dimensions of
                       203x75x186 mm and is designed mainly for stationary
                       use at the desk. The Aquaris additionally offers
                       three fan speeds with control through a graphical
                       user interface.


          o § Audiocasts/Shows⠀➾


                # ⚓ Late_Night_Linux_–_Episode_186_–_Late_Night_Linux⠀⇛


                       Thinkpads that won’t boot Linux by default, Lennart
                       moves to Microsoft, the Firefox Snap is finally a
                       lot faster, Reddit shows its true colours, KDE
                       Korner, and more.


                # ⚓ Video ☛ The_Linux_Desktop_Wouldn’t_Exist_Without_This_Group
                  –_Invidious⠀⇛


                       Xorg, Wayland, Pulseaudio, pipewire, dbus, mesa and
                       many many more incredibly important linux desktop
                       projects are developed under the banner of a single
                       group, that group is Freedesktop


          o § Applications⠀➾


                # ⚓ GstVA_H.264_encoder,_compositor_and_JPEG_decoder_–
                  Herostratus’_legacy⠀⇛


                       There are, right now, three new GstVA elements
                       merged in main: vah264enc, vacompositor and
                       vajpegdec.


                       Just to recap, GstVA is a GStreamer plugin in gst-
                       plugins-bad (yes, we agree it’s not a great name
                       anymore), to differentiate it from gstreamer-vaapi.
                       Both plugins use libva to access stateless video
                       processing operations; the main difference is,
                       precisely, how stream’s state is handled: while
                       GstVA uses GStreamer libraries shared by other
                       hardware accelerated plugins (such as d3d11 and
                       v4l2codecs), gstreamer-vaapi uses an internal,
                       tightly coupled and convoluted library.


                       Also, note that right now (release 1.20) GstVA
                       elements are ranked NONE, while gstreamer-vaapi
                       ones are mostly PRIMARY+1.


                       Back to the three new elements in GstVA, the most
                       complex one is vah264enc wrote almost completely by
                       He Junyan, from Intel. For it, He had to write a
                       H.264 bitwriter which is, until certain extend, the
                       opposite for H.264 parser: construct the bitstream
                       buffer from H.264 structures such as PPS, SPS,
                       slice header, etc. This API is part of
                       libgstcodecparsers, ready to be reused by other
                       plugins or applications. Currently vah264enc is
                       fairly complete and functional, dealing with
                       profiles and rate controls, among other parameters.
                       It still have rough spots, but we’re working on
                       them. But He Junyan is restless and he already has
                       in the pipeline an encoder common class along with
                       a HEVC and AV1 encoders.


          o § Instructionals/Technical⠀➾


                # ⚓ RoseHosting ☛ How_to_Install_ownCloud_on_Ubuntu_22.04_–
                  RoseHosting⠀⇛


                       ownCloud is free and open-source software written
                       in PHP that’s used for data synchronization and
                       file sharing. ownCloud is very similar to DropBox
                       and other cloud storage services such OneDrive,
                       iCloud, and Google Drive.


                       The main difference is that this is self-hosted,
                       which means that your data really only belongs to
                       you. This of course is preferable if you want your
                       data to be more secure.


                # ⚓ ZDNet ☛ Linux_101:_How_to_create_a_zip_file_in_Linux_|
                  ZDNet⠀⇛


                       When was the last time you had to create a zip
                       file? Personally, I have to interact with those
                       types of files all the time (either receiving or
                       sending them to various clients, family, and
                       friends). For most people, the creation of zip
                       files is second nature. And even on the Linux
                       operating system, working with such files isn’t a
                       challenge… once you know what you’re doing.


                # ⚓ Pragmatic Linux ☛ Single_window_mode_for_the_Lazarus_IDE_–
                  PragmaticLinux⠀⇛


                       Does the Lazarus IDE look a bit too old fashioned
                       for your liking? Multiple windows all over your
                       desktop. Don’t let this put you off. With just a
                       few clicks you can reconfigure the Lazarus IDE for
                       single window mode. Resulting in a sleek and modern
                       looking developing environment for cross-platform
                       rapid application development.


                # ⚓ AddictiveTips ☛ How_to_download_wallpapers_to_the_Linux
                  desktop_with_Wallpaper_Downloader⠀⇛


                       Wallpaper Downloader is a slick Linux wallpaper
                       downloader and manager. It supports most Linux
                       desktop environments and is very easy to use. This
                       guide will show you how to download wallpapers to
                       the Linux desktop with Wallpaper Downloader.


                # ⚓ Video ☛ How_to_install_Audacity_on_Pop!_OS_22.04_–
                  Invidious⠀⇛


                       In this video, we are looking at how to install
                       Audacity on Pop!_OS 22.04. Enjoy!


                # ⚓ Linux Made Simple ☛ How_to_install_Dolphin_Emulator_on_a
                  Chromebook⠀⇛


                       Today we are looking at how to install Dolphin
                       Emulator on a Chromebook. Please follow the video/
                       audio guide as a tutorial where we explain the
                       process step by step and use the commands below.


                # ⚓ Linuxiac ☛ How_to_Install_Pi-hole_on_a_Raspberry_Pi_to
                  Block_Ads_Network-Wide⠀⇛


                       With Pi-hole, your Raspberry Pi can filter ads
                       before they reach any of your devices across your
                       network. Here’s how to do it!


                       Ads on websites and applications are a common way
                       of generating revenue. However, the truth is that
                       many websites, including this one, would not exist
                       without the income earned.


                       But the big truth is that ads have always been an
                       unwanted “addition” for end users and their user
                       experience.


                       Unfortunately, advertising is a necessary evil for
                       the internet to exist in its current form. First,
                       to be able to use all the resources it offers for
                       free. To even be able to read this article.


          o § Desktop Environments/WMs⠀➾


                # § K Desktop Environment/KDE SC/Qt⠀➾


                      # ⚓ Preprocessing_for_improving_the_quality_of_the
                        output⠀⇛


                             Based on the results of previous research, I
                             realize that Tesseract does various
                             processing internally before doing the actual
                             OCR. However, some instances still exist
                             where Tesseract can result in a signification
                             reduction in accuracy.


                             So in this post, I would like to introduce
                             some pre-processing methods that apply
                             directly before passing in Tesseract.


                      # ⚓ GSOC_Update_1⠀⇛


                             GSOC is in full swing and here is my first
                             progress update! I’ve been spending time
                             getting familiar with the Krita code base.
                             The first step in my project was making SVG
                             appear as an export option to test start
                             testing the export code. While this may seem
                             straight forward (I certainly thought it
                             would be) there are a few things that we’ll
                             need to do.


                             First, how does Krita know what files it can
                             import/export as? Well that is easy enough to
                             answer, in a database. Specifically Krita has
                             a class KisMimeDatabase that stores all
                             available file formats Krita supports. Adding
                             a new option to this database is fairly easy
                             as there are plenty of examples in the
                             KisMimeDatabase.cpp. We can mostly copy/paste
                             how other options are added but replace that
                             file name with svg. Neat :).


                # § GNOME Desktop/GTK⠀➾


                      # ⚓ Status_update_18/07/2022_–_Sam_Thursfield⠀⇛


                             Summer is here!


                             All my creative energy has gone into wrapping
                             up a difficult project at Codethink, and the
                             rest of the time I’ve been enjoying sunshine
                             and festivals. I was able to dedicate some
                             time to learning the basics of async Rust but
                             I don’t have much to share from the last
                             month. Instead, let me focus on some projects
                             I’m keeping an eye on.


                             Firstly, in the Tracker search engine, Carlos
                             Garnacho has landed some important features
                             and refactors. The main one being stream-
                             based serializers and deserializers.


                             This allows more easily backing up and
                             importing data in and out the tracker-store,
                             and cleaning up some cruft like multiple
                             different implementations of Turtle. It seems
                             ideal having a totally stream-based codec so
                             you can process an effectively infinite
                             amount of data, but there is a tradeoff if
                             you serialize data triple-by-triple – the
                             serialized output is much less human-readable
                             and in some cases larger than if you do some
                             buffering and group related statements
                             together. For this reason we didn’t yet land
                             the JSON-LD support.


                      # ⚓ Pitivi_GSoC:_2nd_Update⠀⇛


                             This is my second GSoC update blog, in the 2
                             weeks since my last update, and we have
                             reached further in the port progress.


                             [...]


                             In my last update, I said that we will move
                             on to changing the version to GTK4 once the
                             event controllers are implemented, but it
                             didn’t go well, not all event controllers
                             were backported and one of the most used ones
                             had a different name in GTK3+ thus, for the
                             time being, we have delayed event controller
                             implementations and moved onto changing the
                             dependencies to GTK4, officially starting the
                             Breaking Phase.


    * § Distributions and Operating Systems⠀➾


          o § New Releases⠀➾


                # ⚓ Archcraft_July_2022_Release_Available⠀⇛


                       Just a regular ISO update


                       July release fixes a bunch of issue, updated
                       packages etc.


          o § Rocky Linux / Fedora / Red Hat / IBM⠀➾


                # ⚓ Talospace ☛ Rocky_Linux_9.0⠀⇛


                       With new version 9.0 Rocky Linux joins the list of
                       ppc64le-compatible CentOS clones, along with the
                       already extant AlmaLinux 9 and Circle Linux based
                       on Red Hat Enterprise Linux 9 (itself based on
                       Fedora 34). Rocky Linux explicitly requires a
                       POWER9 CPU. Other than that, the big difference is
                       the branding and the governance, but more choice is
                       always good. Download ISOs are available.


                # ⚓ Video ☛ Rocky_Linux_9.0_overview_|_Enterprise_Linux,_the
                  community_way._–_Invidious⠀⇛


                       In this video, I am going to show an overview of
                       Rocky Linux 9.0 and some of the applications pre-
                       installed.


                # ⚓ ZDNet ☛ Rocky_Linux_optimized_for_Google_Cloud_arrives_|
                  ZDNet⠀⇛


                       Rocky Linux is on a roll. A few days after Rocky
                       Linux 9 arrived, Google Cloud recommended Rocky
                       Linux as a replacement for CentOS 7.


                       Now, CentOS 7, unlike CentOS 8, which got an abrupt
                       end of life in December 2021, will still be
                       supported until June 30, 2024. But that’s soon
                       enough for big companies like Facebook, Disney,
                       GoDaddy, RackSpace, Toyota, and Verizon, which had
                       depended on CentOS, the Red Hat Enterprise Linux
                       (RHEL) clone, to look at alternatives. And Rocky
                       Linux is 100% compatible with RHEL.


                # ⚓ CentOS ☛ CentOS_Community_Newsletter,_July_2022⠀⇛


                       CentOS is hosting an in-person Dojo at DevConf.US.
                       Dojos are free mini-conferences with sessions on a
                       range of topics in the Enterprise Linux ecosystem.
                       This Dojo takes place on August 17 at Boston
                       University. Registration is free but strongly
                       recommended. We also have a room block at the
                       nearby Residence Inn. See the event page for
                       details.


                # ⚓ NeuroFedora ☛ Next_Open_NeuroFedora_meeting:_18_July_1300
                  UTC_|_NeuroFedora:_Blog⠀⇛


                       Please join us at the next regular Open NeuroFedora
                       team meeting on Monday 18 July at 1300 UTC The
                       meeting is a public meeting, and open for everyone
                       to attend.


                # ⚓ Red Hat Official ☛ AnsibleFest_2022_registration_is_now
                  open⠀⇛


                       After two years of virtual gatherings, we’re
                       excited to announce that AnsibleFest 2022 is
                       returning to an in-person gathering, happening
                       October 18 and 19 in Chicago, Illinois.


                       AnsibleFest is one of our favorite events because
                       it brings together our customers, partners,
                       community members and Red Hatters to talk about the
                       latest happenings and about what’s possible for the
                       future of automation and open source technologies.


                       This year’s event will have multiple content tracks
                       covering topics for everyone from Ansible beginners
                       to automation experts. It will also be a great
                       opportunity to network with industry pros and to
                       get expert answers for all of your automation
                       questions.


          o § Canonical/Ubuntu Family⠀➾


                # ⚓ Video ☛ PopOS_is_SPECIAL,_and_I’m_moving_to_it,_here’s_why!
                  –_Invidious⠀⇛


                # ⚓ ZDNet ☛ Pop!_OS_might_have_a_complicated_name_but_it_makes
                  using_Linux_so_easy_|_ZDNet⠀⇛


                       In the world of operating systems, the strangest
                       name would have to go to System76′s in-house
                       version of Linux, called Pop!_OS. This is the OS
                       that ships with their desktop and laptop systems
                       and make for remarkably seamless integration. In
                       fact, I would go so far as to say that with
                       Pop!_OS, System76 has done for Linux what Apple did
                       for its operating system, only with the addition of
                       making everything open-source. With Pop!_OS running
                       on System76 hardware (such as my Thelio desktop
                       machine — which is, hands down, my favorite desktop
                       PC I’ve ever owned), it goes well beyond the
                       “everything just works” mantra and into the realm
                       of “everything works to perfection.”


          o § Open Hardware/Modding⠀➾


                # ⚓ Arduino ☛ An_LED_cube_circuit_sculpture_shield_for_the_UNO
                  Mini_LE_|_Arduino_Blog⠀⇛


                       Few devices are more iconic in the maker community
                       than the Arduino Uno board. To celebrate the Uno’s
                       history and beloved status, we released the UNO
                       Mini Limited Edition in late 2021. This little
                       board is a tiny replica of the standard Uno, but
                       with a special black and gold color scheme. While
                       the UNO Mini LE is collectible, it is also a fully
                       functional development board. Hari Wiguna took
                       advantage of that fact to create this LED cube
                       circuit sculpture shield for it.


                       The UNO Mini LE’s specs are almost identical to the
                       standard Uno Rev3, since they share the same
                       Microchip ATmega328P microcontroller (just in a
                       different package). But the small size of the UNO
                       Mini LE means that it isn’t compatible with normal
                       Uno shields. Wiguna’s shield fits on the UNO Mini
                       LE and provides a flashing circuit sculpture cube
                       of LED goodness.


    * § Free, Libre, and Open Source Software⠀➾


          o ⚓ Apache Blog ☛ The_Apache_News_Round-up:_week_ending_15_July
            2022⠀⇛


                 Happy Monday, everyone — let’s review the Apache
                 community’s activities from the past week…


          o § Licensing / Legal⠀➾


                # ⚓ Cutelyst_v3.5_relicensed_as_BSD-3-Clause_–_Dantti’s_Blog⠀⇛


                       Cutelyst the Qt Web framework just got a new
                       license, the more permissive BSD-3-Clause. Back in
                       2013 when I started the project the LGPL was a
                       perfectly fine license as software on servers can
                       be closed as long they are not AGPL, thus it was
                       permissive enough for the web and REST backends
                       use-cases I had in mind.


                       Fast-forward almost 10 years and I have used for a
                       few of projects where it was embedded into another
                       application, and I realized that there might be
                       users with commercial Qt license that would like to
                       use it but can’t due the current license.


                # ⚓ Permission_to_Sell_FOSS_in_the_Microsoft_App_Store_Restored
                  –_Software_Freedom_Conservancy⠀⇛


                       Software Freedom Conservancy congratulates
                       Microsoft for changing the terms of their app store
                       to again allow commercial distribution of Free and
                       Open Source Software (FOSS). Microsoft deleted
                       their previously proposed text (originally slated
                       to go into effect over the weekend) which would
                       have prohibited “profit[ing] from open-source …
                       [when that] software … is otherwise generally
                       available for free”.


                       Last week, we explained the disaster that would
                       ensue if the policy was enacted. While some argued
                       that the policy was essential to avoid the problem
                       of “clones” of FOSS programs on app stores, our
                       Policy Fellow also explained how the FOSS community
                       has long used trademark policies and enforcement to
                       mitigate this problem in “app stores”. We remain
                       open to discussing this in detail with app store
                       policy-makers, including Microsoft.


          o § Programming/Development⠀➾


                # ⚓ Drew DeVault ☛ Status_update,_July_2022⠀⇛


                       Hello there! It’s been a hot July week in
                       Amsterdam, and I expect hotter days are still to
                       come. I wish air conditioning was more popular in
                       Europe, but alas. This month of FOSS development
                       enjoyed a lot of small improvements in a lot of
                       different projects.


                       For Hare, I have introduced a number of
                       improvements. I wrote a new standard library module
                       for string templates, strings::template, and a new
                       third-party library for working with pixel buffers,
                       pixbuf. The templating is pretty simple — as is
                       typical for the standard library — but allows a
                       fairly wide range of formatting options. We’ll be
                       extending this a little bit more in the future, but
                       it will not be a complete solution like you see in
                       things like Jinja2. Nevertheless, it makes some
                       use-cases, like code generation, a lot cleaner,
                       without introducing a weighty or complex
                       dependency.


                       pixbuf is pretty neat, and is the first in a line
                       of work I have planned for graphics on Hare. It’s
                       similar to pixman, but with a much smaller scope —
                       it only deals with pixel buffers, handling pixel
                       format conversions and doing small operations like
                       fill and copy. In the future I will add simple
                       buffer compositing as well, and extending modules
                       like hare-png to support loading data into these
                       buffers. Later, I plan on writing a simple vector
                       graphics library, capable at least of rendering
                       TinyVG images and perhaps later TinySVG as well.
                       I’m also working on hare-wayland again, to provide
                       a place to display these buffers.


                # § Perl / Raku⠀➾


                      # ⚓ Perl ☛ Ideas_from_TPRC2022:_Bug/Task/Issue
                        independent_interface_and_a_cli_tool_|_dean_
                        [blogs.perl.org]⠀⇛


                             We have a Database Independent Interface aka
                             DBI and a Unified Cache Handling Interface
                             aka CHI which both provide a generalized
                             interface to similar backend services.
                             Similarly we have AnyEvent – the DBI of event
                             loop programming and Log::Any. With the
                             Nopaste cli provides an agnostic tool to send
                             data to pastebin like services.


    * § Leftovers⠀➾


          o ⚓ Jim Nielsen ☛ Writing_and_Waiting_–_Jim_Nielsen’s_Blog⠀⇛


                 This is a reflection on my inner desire for a “links”
                 section on my blog. However, it applies more broadly to
                 any of my writing.


                 I go back and forth on having a “links” section on my
                 blog separate from my “posts”. Instead of my monthly
                 readingNotes, each of which is one blog post with a
                 collection of links in it, I want split those out into
                 individual links I can post at any moment.


                 However, I enjoy my current process where I note stuff
                 down and then come back to it at the end of the month and
                 read it with fresh eyes. It requires an editing process
                 to the whole thing. Sometimes I cut links out completely.
                 Sometimes I end up commenting on the excerpt so much, I
                 pull it out into its own blog post with additional
                 commentary.


                 What’s one to do? Write about it, of course! It helps me
                 think.


                 [...]


                 This results in a queue of thoughts, many of which never
                 get published.


                 The technologist in me wants to find a way to automate
                 this.


          o § Health/Nutrition/Agriculture⠀➾


                # ⚓ ‘My_troubles_begin_after_I_get_a_good_harvest’⠀⇛


                       Farmers like Rajiv Kumar Ojha in Bihar face high
                       volatility in crop prices, made worse by the
                       absence of APMCs in the state. Their situation
                       predicts the future when the new farm laws take
                       full effect across India


          o § Security⠀➾


                # ⚓ USCERT ☛ CISA_Updates_Advisory_on_Cyber_Actors_Continued
                  Exploitation_of_Log4Shell_in_VMware_Horizon_Systems [Ed: When
                  it is Windows CISA does not even name Windows or Microsoft.
                  This is selective attention.]⠀⇛


                       CISA has updated the joint CISA-United States Coast
                       Guard Cyber Command (CGCYBER) Cybersecurity
                       Advisory AA22-174A: Malicious Cyber Actors Continue
                       to Exploit Log4Shell in VMware Horizon, originally
                       released June 23, 2022. The advisory now includes
                       IOCs provided in Malware Analysis Report (MAR)-
                       10382580-2.


                # ⚓ Ubuntu Handbook ☛ CPU-X_–_Gather_PC_Hardware_Specs_in_Linux
                  with_CPU-Z_Style_User_Interface_|_UbuntuHandbook⠀⇛


                       For those want to check PC hardware specs in Linux,
                       CPU-X is a good choice for user switching from
                       Microsoft Windows.


                       It’s a free open-source system profiling and
                       monitoring application, that looks quite similar to
                       CPU-Z for Windows. With it, you can check your CPU
                       specification, vendor, codename, clocks, and L1/L2/
                       L3 caches.


                # ⚓ MITRE_Engenuity_ATT&CK:_What_endpoint_security_vendors
                  learned_after_Evals_|_SC_Media⠀⇛


                       The Evals results can tell you, for example,
                       whether a particular endpoint security product
                       succeeded in blocking privilege escalation or
                       failed to block password harvesting. Highly
                       successful threat actors such as those emulated by
                       the Engenuity tests use attacks involving dozens of
                       steps, and the Evals results track the outcome of
                       each step.


                       Because the results and methodologies of the
                       evaluations are freely posted on the MITRE website,
                       organizations that are considering new endpoint
                       security products and are also familiar with the
                       MITRE ATT&CK framework can pore over the results to
                       see how well specific offerings did, and which
                       might be the best fit for their individual security
                       postures.


          o § Finance⠀➾


                # ⚓ DaemonFC (Ryan Farmer) ☛ GM_is_in_very_serious_financial
                  trouble_and_thinks_that_the_solution_is_to_force_every_new
                  Buick_and_GMC_owner_into_a_$1500_three_year_OnStar
                  subscription_–_BaronHK’s_Rants⠀⇛


                       GM is in very serious financial trouble, and thinks
                       that the solution is to force every new Buick and
                       GMC owner into a $1500 three year OnStar
                       subscription.


                       Buick and GMC fans are furious. General Motors is
                       in bad fiscal trouble again after the stock
                       collapsed over the past six months and analysts are
                       warning about “earnings misses” and “recessions”
                       this morning related to GM and large investors are
                       dumping the stock.


                       GM learned nothing the last time it went bankrupt
                       and re-emerged. Some structural problems were
                       fixed, such as having too many brands that were
                       basically all the same car, and some were not.


                       The biggest problems “New GM” faced were that the
                       Obama Administration managed the bankruptcy and
                       they had to agree to carry over many of their
                       legacy problems into the new company, which
                       continued to drain their coffers.


                       While nobody should want retirees/pensioners and
                       current workers to “take a bath” on their pay and
                       benefits, at the same time, none of GM’s main
                       rivals in the auto industry have to deal with
                       anything like it.


                       Then came the government mandates to build
                       compliance and electric cars that nobody (at the
                       time) wanted, because gas was cheap, so GM killed
                       them off as soon as Obama was gone. Then gas got
                       very expensive again, and it forced them to stop
                       what they were doing and run a crash program to get
                       efficient and electric vehicles to market.


                       Obama allowed the GM bankruptcy to go through while
                       allowing all of the people who ran them into
                       bankruptcy to remain at their posts without
                       punishment.


                # ⚓ Michael West Media ☛ Punching_above_our_state:_foreign
                  missions_abound,_but_where’s_the_agency_for_taxpayers?_–
                  Michael_West⠀⇛


                       The appointment of a former NSW deputy premier over
                       a more qualified candidate to a plum overseas
                       posting is rightly being lashed. But the John
                       Barilaro saga is just one example of the energetic
                       outreach to the world by our state governments.
                       Public good or puffed-up parochialism? Mark Sawyer
                       examines the reach of the program.


                       For generations of Australians in London and
                       missing home, it was a nice little perk. If it had
                       been too long since you had heard about the goings-
                       on back home, there was a place to go. Lob into the
                       magnificent lobby of Australia House in The Strand,
                       flash the passport and perhaps work that accent,
                       and next you know, you’re in a comfortable armchair
                       with an Australian newspaper on your lap. News from
                       home, Aussie news, and only a week old!


                       Simpler times – and maybe you had to be a news
                       junkie. Anyway, in the past three decades, the need
                       for such a service has, not surprisingly, subsided.


    * § Gemini* and Gopher⠀➾


          o § Personal⠀➾


                # ⚓ Week_26/27/28:_Photos⠀⇛


                       Our hedgehog, Kino, was initially quite scared of
                       us. He has grown more friendly and curious in
                       recent months, but he still does not like to be
                       touched. We take hin outside using a fabric bag
                       called a “snuggle sack”, and a few nights ago he
                       emerged from the sack into the darkness of the
                       backyard.


                # ⚓ Re:_The_Dutch_Baby_Incident⠀⇛


                       I didn’t get what this was until I looked at the
                       picture. In England, that would be called Yorkshire
                       pudding, and it’s a savoury thing that you serve
                       with a roast dinner. You’d normally make it in a
                       baking tray instead of a pan with a handle, which
                       might save burns.


===============================================================================
* Gemini_(Primer) links can be opened using Gemini_software. It’s like the
World Wide Web but a lot lighter.



                    ䷩ 𝚕𝚒𝚗𝚎 6094

╒═══════════════════ 𝐃𝐀𝐈𝐋𝐘 𝐋𝐈𝐍𝐊𝐒 ═════════════════════════════════════════════╕

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 07.18.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Links_18/07/2022:_EasyOS_Dunfell-series_4.2.7,_EasyOS_4.2_Reviewed,_Steam
Deck_at_4,000_Games⠀✐


Posted in News_Roundup at 3:57 am by Dr. Roy Schestowitz


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇GNOME bluefish⦈


§ Contents⠀➾


    * GNU/Linux
          o Audiocasts/Shows
          o Kernel_Space
          o Applications
          o Instructionals/Technical
          o Games
    * Distributions_and_Operating_Systems
          o Reviews
          o New_Releases
          o Fedora_Family_/_IBM
          o Devuan_Family
          o Open_Hardware/Modding
          o Mobile_Systems/Mobile_Applications
    * Leftovers
          o Hardware
          o Security
                # Privacy/Surveillance
          o Monopolies
                # Patents
                # Copyrights
    * Gemini*_and_Gopher
          o Personal
          o Politics
          o Technical
    * § GNU/Linux⠀➾


          o ⚓ 9to5Linux ☛ 9to5Linux_Weekly_Roundup:_July_17th,_2022⠀⇛


                 This week we had lots of great news, starting with the
                 release of the Linux Mint 21 beta operating system and
                 Lubuntu’s new Backports PPA to get the latest LXQt
                 desktop releases, and continuing with lots of goodies for
                 both KDE Plasma and GNOME desktop users, as well as some
                 goodies for Linux gamers.


                 On the hardware side of things, System76 launched their
                 Launch Lite configurable keyboard and TUXEDO Computers
                 announced Linux support for their external water cooling
                 system for Linux laptops. On top of that, I give you the
                 heads up on Ubuntu 21.10′s end of life and the SCaLE 19x
                 conference in late July.


          o § Audiocasts/Shows⠀➾


                # ⚓ Jupiter Broadcasting ☛ All_Hands_on_Deck_|_LINUX_Unplugged
                  467_|_Jupiter_Broadcasting⠀⇛


                       We try and bust a common Linux distro myth. Then
                       what surprised Chris about his new Steam Deck.


                # ⚓ Open Source Security (Audio Show) ☛ Episode_332_–_PyPI:_2FA
                  or_not_2FA,_that_is_the_question_–_Open_Source_Security⠀⇛


                       Josh and Kurt talk about PyPI mandating two factor
                       authentication for the top 1% of projects. It feels
                       like a simple idea, but it’s not when you start to
                       think about it. What problems does 2FA solve? How
                       common are these attacks? What are the second and
                       third order effects of mandating 2FA? This episode
                       should have something for everyone on all sides of
                       this discussion to violently disagree with.


                # ⚓ GNU World Order (Audio Show) ☛ GNU_World_Order_469⠀⇛


                       **kdev-php** , **kdev-python** , **the kdevelop
                       IDE** , **kdewebkit** , **kdf** , **kdiagram** ,
                       **kdialog** from Slackware set **kde**.


                # ⚓ IRL:_Online_Life_is_Real_Life⠀⇛


                       Where should tech builders draw the line on AI for
                       military or surveillance? Just because it can be
                       built, doesn’t mean it should be. At what point do
                       we blow the whistle, call out the boss, and tell
                       the world? Find out what it’s like to sound the
                       alarm from inside a big tech company.


                       Laura Nolan shares the story behind her decision to
                       leave Google in 2018 over their involvement in
                       Project Maven, a Pentagon project which used AI by
                       Google.


                       Yves Moreau explains why he is calling on academic
                       journals and international publishers to retract
                       papers that use facial recognition and DNA
                       profiling of minority groups.


          o § Kernel Space⠀➾


                # ⚓ LWN ☛ Kernel_prepatch_5.19-rc7_[LWN.net]⠀⇛


                       The 5.19-rc7 kernel prepatch is out for testing.


                # ⚓ The Register UK ☛ Linus_Torvalds_says_Linux_kernel_has
                  addressed_‘Retbleed’_•_The_Register⠀⇛


                       Linux kernel developers have addressed the Retbleed
                       speculative execution bug in older Intel and AMD
                       silicon, but the fix wasn’t straightforward, so
                       emperor penguin Linus Torvalds has delayed delivery
                       of the next version by a week.


                       “When we’ve had one of those embargoed [hardware]
                       issues pending, the patches didn’t get the open
                       development, and then as a result missed all the
                       usual sanity checking by all the automation build
                       and test infrastructure we have,” Torvalds wrote in
                       a post announcing the delivery of release candidate
                       seven for version 5.19 of the kernel.


                       “So no surprise – there’s been various small fixup
                       patches afterwards too for some corner cases.”


                       Retbleed is not the only reason for the delay.


          o § Applications⠀➾


                # ⚓ OMG Ubuntu ☛ ‘Cider’_is_an_Open_Source_Apple_Music_Client
                  for_Linux_Desktops_–_OMG!_Ubuntu!⠀⇛


                       If you’re an Apple Music subscriber wanting to use
                       the service on Linux, you have to check out Cider.


                       Cider (inspired name, eh?) is an “open-source,
                       community-oriented Apple Music client” for Windows,
                       macOS (not that they need it), and Linux. It is
                       Electron-based but the developers behind it say it
                       was written ‘from scratch with performance in
                       mind’.


                       As well as letting you browse songs, artists, and
                       albums in the Apple Music catalog the app also
                       gives you access to Apple Podcasts too.


                # ⚓ Linux Links ☛ Best_Free_and_Open_Source_Alternatives_to
                  Apple_Notes⠀⇛


                       
                       There are many things to admire about Apple’s
                       hardware and software. Apple make great looking
                       (albeit expensive) hardware. Over the years key
                       successes include the iPhone, iPad, iPod, and the
                       MacBook Air. The company designs its own hardware
                       and software. This gives them the power to make an
                       operating system and suite of apps that are tailor-
                       made and optimized for their hardware. Apple also
                       operates the Apple Music and Apple TV media
                       distribution platforms.


                       Mac OS X is Apple’s proprietary operating system
                       for its line of Macintosh computers. Its interface,
                       known as Aqua, is highly polished and built on top
                       of a BSD derivative (Darwin). There’s a whole raft
                       of proprietary applications that are developed by
                       Apple for their operating software. This software
                       is not available for Linux and there’s no prospect
                       of that position changing.


          o § Instructionals/Technical⠀➾


                # ⚓ TecMint ☛ How_to_Mount_and_Unmount_an_ISO_Image_in_Linux⠀⇛


                       An ISO image or .iso (International Organization
                       for Standardization) file is an archive file that
                       contains a disk image called ISO 9660 file system
                       format.


                       Every ISO file has .iso an extension has a defined
                       format name taken from the ISO 9660 file system and
                       is specially used with CD/DVD ROMs. In simple
                       words, an iso file is a disk image.


                # ⚓ TecMint ☛ PlayOnLinux_–_Run_Windows_Software_and_Games_in
                  Linux⠀⇛


                       In our earlier articles on this blog, we used the
                       Wine program to install and run windows based
                       applications on Debian-based and and other Red Hat-
                       based Linux distributions.


                       There is another open source software available
                       called PlayOnLinux that uses Wine as its base and
                       gives feature-rich functions and a user-friendly
                       interface to install and run windows applications
                       on Linux.


                # ⚓ OpenSource.com ☛ How_I_configure_a_DHCP_server_on_my
                  personal_network⠀⇛


                       The Dynamic Host Configuration Protocol (DHCP)
                       provides a centralized and automated method for
                       configuring the network attributes of hosts when
                       they connect to the network. The DHCP server
                       assigns IP addresses to hosts, along with
                       configuration information such as DNS servers, the
                       domain name used for DNS searches, the default
                       gateway, an NTP (Network Time Protocol) server, a
                       server from which a network boot can be performed
                       if necessary, and more. DHCP eliminates the need to
                       configure each network host individually.


                       DHCP is also useful for configuring laptops, mobile
                       phones, tablets, and other devices which might
                       connect as unknown guests. This configuration is
                       typical for WiFi access in public places. However,
                       DHCP offers even more advantages when used in a
                       closed, private network to manage static IP address
                       assignments for known hosts using the central DHCP
                       database.


                       The DHCP server uses a database of information
                       created by the sysadmin. This database is entirely
                       contained in the /etc/dhcp/dhcpd.conf configuration
                       file. DHCPD stands for DHCP Daemon, which is the
                       background server process. Like all well-designed
                       Linux configuration files, it is a simple ASCII
                       plain text file. This structure means that it is
                       open and knowable. It can be examined by standard,
                       simple text manipulation tools like cat and grep,
                       and be modified by any text editor such as EMACS or
                       Vim, or a stream editor such as sed.


                # ⚓ OpenSource.com ☛ Create_a_JavaScript_API_in_6_minutes⠀⇛


                       This article demonstrates creating a base API with
                       Express and JavaScript. Express is a NodeJS
                       minimalist web framework. This combination allows
                       for minimal effort to get an API up and running at
                       the speed of light. If you have six minutes of free
                       time, you can get this API working to do something
                       useful.


                # ⚓ OpenSource.com ☛ Monitor_your_Linux_firewall_with
                  nftwatch⠀⇛


                       Netfilter tables (nftables) is the default firewall
                       shipped with modern Linux distros. It’s available
                       on Fedora and RHEL 8, the latest Debian, and many
                       others. It replaces the older iptables that was
                       bundled in earlier distro releases. It’s a powerful
                       and worthy replacement for iptables, and as someone
                       who uses it extensively, I appreciate its power and
                       functionality.


                       One of the features of nftables is the ability to
                       add counters to many elements, such as rules. These
                       are enabled on demand. You need to explicitly ask
                       for it on a per line basis using the “counter”
                       argument. I have them enabled for specific rules in
                       my firewall, which gives me visibility into those
                       rules.


                       This got me thinking. How can I look at these
                       counters in real time? At first I tried “watch”
                       which allows things like refresh rate, but I didn’t
                       like the default format and it wasn’t scrollable. I
                       found using head and tail and awk less than ideal.
                       A user-friendly solution didn’t exist. So I wrote
                       my own, which I’d like to share with the open
                       source community.


                # ⚓ Linux Shell Tips ☛ How_to_Clean_or_Kill_a_Linux_Zombie
                  Process⠀⇛


                       Exposure to the Linux operating system ecosystem
                       introduces its users to an in-depth understanding
                       of Linux process management footprints. By
                       definition, a process is a program that is
                       continually executing.


                       A program that is not executing does not qualify as
                       a process since it is a passive entity. Its
                       executing state makes it an active entity hence a
                       process. It is also worth mentioning that a single
                       program in execution can be associated with
                       multiple processes.


                # ⚓ Linux Shell Tips ☛ How_to_Compare_Two_Directories_on_Linux
                  System⠀⇛


                       Comparing directories in Linux might seem like a
                       light and dismissive task for a user just starting
                       to explore the milestones associated with using
                       Linux as a primary operating system.


                       Directory comparison is everything for matured
                       users who have thoroughly explored the Linux file
                       management spectrum. For instance, you might have
                       two directories that are very similar in terms of
                       the content stored in them.


                       In this case, you might want to get rid of one of
                       these directories but you are unsure if they are
                       perfect duplicates. Such a scenario calls for the
                       Linux directory comparison expertise.


                # ⚓ Trend Oceans ☛ How_to_Install_Fedora_36_on_VirtualBox_–
                  TREND_OCEANS⠀⇛


                       If you want to use an operating system that is
                       close to RedHat, then Fedora should be on top of
                       your list because it has the same package manager
                       that RedHat does use, and apart from that, you will
                       get the same experience.


                       So let’s see how you can install Fedora 36 on your
                       VirtualBox with the following steps, and after
                       that, you can enhance your skills with an
                       enterprise-grade OS.


                # ⚓ Linux Capable ☛ How_to_Install_Nextcloud_Client_on_Ubuntu
                  22.04_LTS⠀⇛


                # ⚓ Linux Capable ☛ How_to_Install_Quod_Libet_on_Ubuntu_22.04
                  LTS⠀⇛


                # ⚓ LinuxTechi ☛ How_to_Install_Rocky_Linux_9_Step_by_Step_with
                  Screenshots⠀⇛


                       In this guide, we will cover how to install Rocky
                       Linux 9 step by step with screenshots.


                       Rocky Enterprise Software Foundation has released
                       its latest operating system ‘Rocky Linux 9’. Rocky
                       Linux is free and open-source operating system for
                       workstation and servers. It is considered as drop-
                       in replacement for CentOS Linux.


                       ‘Blue Onyx’ is the Code name for Rocky Linux 9, it
                       is the clone of RHEL 9. The main difference between
                       Rocky Linux and RHEL is that it has its own open-
                       source build system called ‘Peridot’.


                # ⚓ Trend Oceans ☛ How_to_Install_ZSH_Shell_along_with_OhMyZsh
                  and_Powerlevel10k_on_Ubuntu_22.04_–_TREND_OCEANS⠀⇛


                       As a starter, you might think that Ba(sh) is the
                       only shell available to use in UNIX-like
                       distributions, but that’s not the case in modern
                       Linux distributions.


                       There are several other shell interpreters
                       available to help you out, like KornShell (ksh), C
                       shell (csh), Z shell (zsh), and fish. From this
                       list, you can use the zsh shell, which is highly
                       customizable with more than 100+ themes and plugins
                       that can help you to perform your work swiftly with
                       an aesthetic look.


                       Z Shell, or zsh, is a shell interpreter that
                       inherits features from bash, ksh, and tcsh, which
                       can be used as an interactive shell, utilities, and
                       basic programming language functionality.


                # ⚓ Linux Capable ☛ How_to_Install_mpv_Media_Player_on_Ubuntu
                  22.04_LTS⠀⇛


                # ⚓ Linux Shell Tips ☛ How_to_Keeping_SSH_Session_Alive_in
                  Linux⠀⇛


                       SSH or Secure Shell is an invaluable utility when
                       it comes to undertaking remote login objectives on
                       remote machines. For Linux users, the SSH utility
                       offers more than just remote login solutions.


                       The mentioned users are also able to effortlessly
                       accomplish Linux administrative tasks. Regardless
                       of whether the targeted remote machine is
                       untrustworthy or whether the network that
                       facilitates the communication between the two
                       machines is insecure, SSH ensures that these
                       communications are secure and encrypted.


                       It is sometimes frustrating when an SSH session
                       ends too soon while we are multitasking on the
                       Linux terminal environment. Finding a way of
                       keeping the SSH session alive for as long as
                       possible can be a game changer for Linux users who
                       don’t want to keep on re-initiating new SSH
                       sessions.


                # ⚓ Linux Shell Tips ☛ Differences_Between_PID,_TID_and_PPID_in
                  Linux⠀⇛


                       A Linux operating system provides the perfect
                       environmental exposure for users to understand the
                       Ins and Outs of process management. The simplest
                       definition of a process is any program in execution
                       (running).


                       For instance, the web browser you are using to read
                       this article piece only becomes a process once that
                       web browser is up and running. Before your web
                       browser was installed and launched, it only existed
                       as a program (a process in waiting).


                       When you dive deeper into the concept of process
                       management, you will get to acknowledge the various
                       properties associated with processes. This article
                       will walk us through differentiating three
                       processes’ properties namely: PID, TID, and PPID.


          o § Games⠀➾


                # ⚓ GamingOnLinux ☛ Steam_Deck_hits_over_4,000_titles_marked
                  either_Verified_or_Playable⠀⇛


                       As more developers work to optimize their games for
                       the Steam Deck, a bunch of Native Linux games
                       release and Valve continue improving Proton – the
                       Steam Deck continues to see more games go through
                       Deck Verified and there’s now over 4,000 listed
                       officially.


    * § Distributions and Operating Systems⠀➾


          o § Reviews⠀➾


                # ⚓ Distro Watch ☛ Review:_EasyOS_4.2⠀⇛


                       Puppy Linux has offered a minimal and portable
                       Linux system for almost 20 years, making it not
                       only one of the oldest continuous Linux distros,
                       but a legend among anyone who ever used a desktop
                       in a hotel business center. Insert a USB with Puppy
                       on it, answer e-mail, print a boarding pass, and
                       never once worry about malware, spyware, or
                       navigating Internet Explorer.


                       Puppy impresario Barry Kauler started development
                       on EasyOS in 2017. The goal was to take what he had
                       with Puppy and Puppy successor Quirky and turn it
                       into a cutting-edge, but still minimal, Linux
                       distro. Hence, EasyOS runs off a USB stick, takes
                       up only 641MB and needs just 2GB of memory (though
                       8GB is recommended). The Linux kernel (currently,
                       the 5.15 series) loads into RAM at the first boot.
                       Then it can save itself to the USB at the end of
                       the session. In addition, EasyOS uses some of
                       Linux’s most modern technology, including
                       containers, and its EasyShare app makes finding
                       network printers and shares almost impossibly
                       simple.


                       In this, EasyOS is fast, efficient, effective, and
                       impressive. And, if the desktop is a bit unsightly
                       – “retro,” says Kauler – it doesn’t get in the way
                       of the work. (Yes, we’re talking about you, GNOME.)


                       Still, know that EasyOS – to be polite – has its
                       idiosyncrasies. I downloaded EasyOS Dunfell 4.1 for
                       this review in the middle of June; two weeks later,
                       the OS had been updated at least three times, and
                       the update mechanism was broken in the process. In
                       addition, the download comes as an IMG file and not
                       an ISO, and Kauler is adamant about that
                       difference. Finally, EasyOS is a work in progress,
                       and not everything always happens as it is supposed
                       to happen. As such, it’s not for everyone, as well-
                       intentioned as it is.


          o § New Releases⠀➾


                # ⚓ Barry Kauler ☛ EasyOS_Dunfell-series_4.2.7_released⠀⇛


                       Of course it features the small “difference file”
                       update, by clicking on the “update” icon on the
                       desktop. By sure to have the current image file,
                       for example easy-4.2.1-amd64.img’, at /mnt/wkg (/
                       mnt/${WKG_DEV}/${WKG_DIR}), so that a small
                       difference-file may be downloaded and merged with
                       it, to create ‘easy-4.2.7-amd64.img’.


          o § Fedora Family / IBM⠀➾


                # ⚓ Power10_Entry_Machines:_The_Power_S1014⠀⇛


                # ⚓ What_Happens_To_IBM_i_In_A_“Zero_Datacenter,_Zero
                  Mainframe”_FedEx?⠀⇛


                # ⚓ Guru:_IBM_i_Experience_Sharing,_Case_5_–_Using_IBM_i_PDI
                  Charts_To_Answer_Performance_Questions⠀⇛


                # ⚓ Support_For_Java_7_Ending_WAS_8.5_On_IBM_i⠀⇛


                # ⚓ Now_Is_Your_Chance_To_Ask_IBM_i_CTO_Steve_Will_Some
                  Questions⠀⇛


          o § Devuan Family⠀➾


                # ⚓ [Old] Devuan ☛ We_are_making_progress⠀⇛


                       We are finalizing the branding for the installers.
                       Thank you so much for letting us tag your logo in
                       our spin. It will be a water mark for slides during
                       the install (calamares). As soon as I get a final I
                       will post a screen here of the installer.


          o § Open Hardware/Modding⠀➾


                # ⚓ CNX Software ☛ DEBIX_Model_A_–_A_Raspberry_Pi-inspired_NXP
                  i.MX_8M_Plus_SBC_for_AI_applications_–_CNX_Software⠀⇛


                       DEBIX Model A is a single board computer powered by
                       an NXP i.MX 8M Plus quad-core Cortex-A53 processor
                       with a 2.3 TOPS AI accelerator in a form factor
                       that combines Raspberry Pi 4 and 3 Model B designs.


                       The SBC comes with up to 6GB RAM, a microSD card
                       slot, an optional eMMC flash of up to 128GB
                       capacity, HDMI 2.0a, LVDS, and MIPI DSI display
                       interfaces, a MIPI CSI camera interface, Gigabit
                       Ethernet and WiFi 5 connectivity, plus a 40-pin
                       GPIO expansion header.


                # ⚓ CNX Software ☛ Geniatech_XPI-3566_is_a_business-card_sized
                  Rockchip_R3566_SBC⠀⇛


                # ⚓ Linux Gizmos ☛ RISC-V_based_ICE-V_Wireless_board_includes_a
                  Lattice_ICE40_UltraPlus_FPGA⠀⇛


                       The ICE-V Wireless from QWERTY Embedded Design is a
                       tiny board that combines a Lattice iCE40 UltraPlus
                       FPGA and a ESP32-C3-MINI-1 Wi-Fi/BLE module from
                       Espressif Systems. This device supports popular
                       programming platforms such as Arduino and
                       MicroPython for easy software development.


                       The FPGA featured on this board is the low-cost
                       ICE40UP5K-SG48 from Lattice Semi which provides up
                       to 120 Kbits of EBR RAM, 1024 Kbits of SPRAM and
                       5280 LUTs. According to Lattice, this FPGA has been
                       optimized for Machine Learning and AI
                       implementations.


          o § Mobile Systems/Mobile Applications⠀➾


                # ⚓ SlashGear ☛ Why_You_Should_Never_Let_Your_Android_Battery
                  Hit_Zero⠀⇛


                # ⚓ 9to5Google ☛ Google_advertises_Android_apps_in_curious
                  Amazon_storefront_–_9to5Google⠀⇛


                # ⚓ Express ☛ Android_alert:_Don’t_make_this_error_today_–_it
                  WILL_break_your_phone_|_Express.co.uk⠀⇛


                # ⚓ Android Central ☛ Mobvoi_teases_its_next_TicWatch_flagship
                  smartwatch_running_Wear_OS_|_Android_Central⠀⇛


    * § Leftovers⠀➾


          o § Hardware⠀➾


                # ⚓ As_I_See_It:_Pass_The_Chips_–_IT_Jungle⠀⇛


                       Decades ago, I was sent to Taiwan and met a
                       colleague there to install a computer system in a
                       silicon wafer manufacturing facility and provide
                       necessary training to the staff. I took a cab from
                       the airport to my hotel in Taipei and recall
                       marveling at the extraordinarily generous freeways
                       that served surprisingly light traffic flows. In
                       many places, six to eight lanes stretched for miles
                       in both directions supporting traffic that could
                       have been accommodated by no more than two.


                       I became curious and finally asked my driver if he
                       knew why the roadways were so incongruously wide?
                       It was, he explained, a precaution against a
                       preemptive Chinese invasion. “If they bomb our
                       military airfields,” he explained, “our pilots will
                       have somewhere to land.”


          o § Security⠀➾


                # § Privacy/Surveillance⠀➾


                      # ⚓ Citizen Lab ☛ GeckoSpy:_Pegasus_Spyware_Used_Against
                        Thailand’s_Pro-Democracy_Movement_–_The_Citizen_Lab⠀⇛


                             The Kingdom of Thailand is a constitutional
                             monarchy with a parliamentary-style
                             government divided into executive,
                             legislative, and judiciary branches. The
                             country has been beset by intense political
                             conflict since 2005, during the government of
                             former Prime Minister Thaksin Shinawatra.
                             Corruption allegations against the regime
                             culminated in a military coup on September
                             19, 2006 that ousted Thaksin. The military
                             launched another coup on May 22, 2014 and
                             seized power following mass protests against
                             the civilian government led by Thaksin’s
                             sister, Yingluck Shinawatra. The junta
                             claimed that the 2014 coup was needed to
                             restore order and called itself the National
                             Council for Peace and Order (NCPO).


          o § Monopolies⠀➾


                # § Patents⠀➾


                      # ⚓ IT Wire ☛ Apple_5G_device_sales_blocked_in_Colombia
                        due_to_patent_issue⠀⇛


                             A court in Colombia has banned Apple from
                             selling 5G iPhones and iPads in the country
                             due to the infringement of a patent owned by
                             Swedish telecom giant Ericsson.


                             The affected devices are iPhone 12 and 13
                             models and also the newer iPads.


                             The court order, issued in the first week of
                             July, concerned a patent issued to Ericsson
                             in 2019 and valid until the end of 2037,
                             according to the FOSS Patents blog which is
                             published by long-time IP activist Florian
                             Mueller.


                             Apple’s infringement of the patent was
                             determined in April and the following month
                             Ericsson posted a bond of US$50,000
                             (A$73,496), a prerequisite for enforcement.


                # § Copyrights⠀➾


                      # ⚓ Torrent Freak ☛ Bungie_Sues_Twitch-Streaming_Destiny
                        Cheater_‘MiffysWorld’_For_Being_Awful⠀⇛


                             After targeting cheat sellers, game developer
                             Bungie has now sued a US-based gamer who uses
                             cheats. At least initially, Bungie believed
                             it was dealing with just another ban-evading
                             cheater but the investigation into Twitch-
                             streamer ‘MiffysWorld’ turned up a lot more.
                             As it turns out, expressing a desire to burn
                             down Bungie’s offices wasn’t the worst thing
                             discovered.


                      # ⚓ Torrent Freak ☛ Research_Suggests_That_Software
                        Piracy_Lowers_Poverty⠀⇛


                             Piracy is a controversial topic and research
                             into the effects of copyright infringement is
                             widely debated as well. A new academic study
                             adds an interesting angle by suggesting that
                             software piracy might lower poverty. While
                             this is a plausible conclusion, there could
                             actually be more factors at play.


    * § Gemini* and Gopher⠀➾


          o § Personal⠀➾


                # ⚓ Kink_contracts⠀⇛


                       Alexis’ ownership of X is based on ongoing mutual
                       informed consent, in the context of Risk-Aware
                       Consensual Kink. Nothing in this Title Deed should
                       be interpreted otherwise.


                # ⚓ A_surplus_of_empathy⠀⇛


                       I feel too much, sometimes. Right now my wife and I
                       are both away from home, and while I know our cat
                       is well-cared-for, she misses us and she doesn’t
                       know why we’re gone. I just wish I could explain it
                       to her.


                # ⚓ That_Time_I_Helped_a_Sad_Drunkard⠀⇛


                       I was visiting a friend down south many years ago.
                       I believe I was still in upper secondary school
                       (high school, basically) so it must have been
                       sometime 2001-2004.


                       My friend and I went to a city nearby. It was
                       before lunch. Soon after getting there we saw a
                       very inebriated miss a step as he was about to
                       cross a road. His foot came over the edge of the
                       pavement without him being ready for it and he fell
                       head first on the asphalt. Although he got up by
                       himself we felt that a fall that hard may have
                       consequences.


                       We went over to see how he was doing. Of course
                       with him being under the influence it was hard to
                       tell. The safest thing seemed to be to have medical
                       personnel have a look. We must have left our phones
                       at home because I remember that we to a hotel.
                       Maybe it was the one he was staying at. We sat down
                       in the lobby together.


                # ⚓ Acceptable⠀⇛


                       They say speech is acceptable
                       When it fits into norms
                       They say it’s reprehensible
                       If it strays in its forms


                       They want us to believe their lies
                       That truth always prevails
                       We all should simply close our eyes
                       Ignoring all details


          o § Politics⠀➾


                # ⚓ The_US_Is_Not_Ok⠀⇛


                       The US is not ok, and it hasn’t been for a long,
                       long time. Given its history, it has never been ok.
                       But this is different. Now it is fracturing, and
                       the seismic disturbances continue a-pace.


                       [...]


                       Two years ago, I lived through a coup in Sri Lanka.
                       It was stupid. The minority party threw chili
                       powder at everyone in Parliament and took over by
                       farce. Math, however, requires a majority and the
                       courts kicked them out. They gave in. We’d been
                       protesting for weeks and yay, we won.


                       No.


                       I didn’t know it at the time, but we had already
                       lost. No one knew — but oh my God, what we lost.
                       The legitimate government came back but it was
                       divided and weak. We were divided and weak. We were
                       vulnerable.


                       Four months later, on Easter Sunday, some assholes
                       attacked multiple churches and hotels, killing 269
                       of us. My wife and kids were at church, I had to
                       frantically call them back. Our nation was
                       shattered. Mobs began attacking innocent Muslims.
                       It was out of control. The coup broke our
                       government, and four months later, that broke us.


                # ⚓ Re:_Unconditional_Basic_Income⠀⇛


                       I grew up in your typical midwestern US household.
                       My father worked for 30+ years in a factory and my
                       mom was stay at home until I (being the youngest
                       child) was old enough to not need constant
                       supervision. I was taught all of my life about the
                       value of hard work. As such, for much of my life I
                       very much subscribed to the idea that the world
                       doesn’t owe you anything; that everything you get
                       should be something that you worked for. This is a
                       very difficult notion to get rid of. It’s only now,
                       as I settle into middle age, that I’m seeing
                       another truth. That truth is that long before I was
                       born, all of the valuable land was already carved
                       up amongst the wealthy. All of the resources are
                       spoken for, by and large. In fact, our society is
                       borrowing against our future generations’ resources
                       and potential. In short, we were all bought and
                       sold down the river from day one. So yeah, maybe
                       we’re owed something for that.


          o § Technical⠀➾


                # ⚓ Ocarina_of_Time_Feels_Like_a_New_Game⠀⇛


                       I’ve been playing it the past week and it’s the
                       most fun I’ve had with OOT. I’ve started OOT over
                       multiple times over the years through emulators and
                       various Nintendo releases, but this might be the
                       time I actually finish it.


                       The best feature is the frame interpolation which
                       lets you play far beyond the game’s native frame
                       rate of 20 or so. It’s almost perfect and makes the
                       game feel like it’s actually running at a higher
                       frame rate despite the extra frames only being
                       interpolated. The controls even feel more
                       responsive and everything just feels tighter, which
                       is nice even if it’s placebo.


                # ⚓ Keyboarding_Education⠀⇛


                       Preface: I’m not in the hobby – I just like to ooh
                       and ahh over pretty arrangements of keycaps and
                       cases.


                       I subscribe to the mystique of the HHKB. Despite
                       the humorous saddle metaphor, the HHKB is a
                       utilitarian workhorse. It is not without its many
                       detractors who think it’s a 60% toy nevertheless.


                       Whenever I hear someone boast about their 100+ wpm
                       prowess with Qwerty, I cringe because even if
                       they’re really good with it an inferior layout has
                       worked its way into their muscle memory.


                       [...]


                       I think the standard mouse is harmful excepting
                       trackballs (often placed in the middle between two
                       keyboard halves). That being said, I think a one-
                       handed keyboard and mousing with the other hand
                       makes a lot of sense. I didn’t forget about
                       trackpads and pointing sticks, but that’s a story
                       for another day.


                       Keyboards still have a lot of room to become more
                       personalized. What do I mean by that? For example,
                       the Esrille Nisse comes in two sizes to accommodate
                       for Japanese people’s smaller hands. The keycaps on
                       the Medium model are milled to be thinner from
                       regular keycaps. There should be some sort of
                       program to figure out from people’s hands and a
                       heat map of how they type what keyboard is best for
                       them. From there, it’s off to the text editor
                       races.


                # ⚓ Self_Hosting_Feed_Reader⠀⇛


                       I installed FreshRSS last night (it wasn’t entirely
                       straightforward for someone who hasn’t dealt with
                       php installs in ages) and I gotta say it already
                       blows Feedly out of the water.


                       Feedly was great when I started using it a bunch of
                       years ago. It’s been the only feed reader I’ve used
                       and it really changed how I interact with a lot of
                       websites. But feedly has constantly gotten worse
                       over the years in an attempt to be more feature-
                       full.


                       [...]


                       None of these are really big changes in and of
                       themselves, but they do clutter up the interface a
                       bit and especially the first two are truly annoying
                       to me. What I dislike the most I guess is the
                       general trend towards a heavier and slower website
                       made so by features that aren’t for me.


                       FreshRSS let’s me just read my feeds. I installed
                       it on my Raspberry Pi 2B, where I host this website
                       and capsule along with thelounge.chat and Antenna
                       (by far my most used service). Despite the low
                       powered and much utilised server the FreshRSS
                       interface is a lot faster than Feedly. And I really
                       do mean a lot.


===============================================================================
* Gemini_(Primer) links can be opened using Gemini_software. It’s like the
World Wide Web but a lot lighter.


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