Borel-Cantelli Emojis & Text

Copy & Paste Borel-Cantelli Emojis & Symbols ℛAₙ)∈{0️⃣1️⃣} | ∑=∞+🎲↔️🎲⇒P(🎯∞)=1 | ∑<∞⇒P(🎯∞)=0

ℛAₙ)∈{0️⃣1️⃣}
new borel-cantelli measure theory zero-one law limsup measure probability space infinite events
∑=∞+🎲↔️🎲⇒P(🎯∞)=1
new borel-cantelli probability sum divergence infinitely many events independent events measure theory converse borel-cantelli
∑<∞⇒P(🎯∞)=0
new borel-cantelli probability sum convergence finitely many events measure theory sequence of events
⊥+∑📈⇒✨∞=1️⃣
new borel-cantelli second borel-cantelli probability theory sum of probabilities infinitely often almost surely limsup events
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∑📉⇒✨∞=0️⃣
new borel-cantelli first borel-cantelli probability theory sum of probabilities infinitely often almost surely limsup events

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⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⡀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠈⠀⠀⠀⡀⠀⠀⠀⠀⠨⢀⣀⠁⠀⠘⠇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⡀⠀⢈⠠⢀⢀⠠⣤⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠈⠁⣰⣟⣐⣂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠠⠤⡀⣨⠃⠔⠀⢠⢄⡑⠮⣐⢋⠎⣓⡞⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠐⠔⠂⢐⢢⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢱⣀⡴⠋⢀⡿⡜⠻⡌⢉⡰⢀⠈⢡⡒⢢⣤⣘⠮⡆⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠷⠀⠀⠀⠀⢀⡔⢐⠬⠃⡔⣚⡘⡒⢩⠖⣩⡃⠀⠀⠀⠀⠉⠀⠒⠀⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⣰⠃⣠⣿⣟⡿⣺⣄⣤⠖⣢⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⣼⡯⢠⣴⡃⢔⠱⡈⣥⣴⣿⣽⣿⢈⣵⣷⠇⣿⣷⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠞⠀⠀⠀⠀⠜⣁⡞⠜⢡⣛⡜⡥⢆⡍⡤⠌⣇⠀⠐⠊⠠⠔⢀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂⣤⢎⠃⠀⠀⣰⠃⡼⡟⢿⡾⣽⣳⣟⣿⠁⠀⠑⠈⠁⠈⠢⣠⣤⣠⡀⠀⠀ ⠀⠀⠀⠀⢼⣿⡿⣰⠻⣍⢲⠖⢠⡽⠴⠬⢙⣛⣛⡀⢀⢻⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣬⢀⢀⡄⠀⢾⣫⢐⡪⣓⠏⣻⡬⡑⣼⣸⠓⠚⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠚⠠⠴⢂⡠⠄⣀⠈⢻⡤⣠⢴⣧⢽⡰⣌⣣⢿⣷⣻⣾⡿⠁⠀⠀⠀⠀⠂⠄⠀⣄⣌⡁⠀⠂ ⠀⠀⠀⠀⠈⠹⣷⡯⡷⠤⣅⡐⢦⣶⣔⡖⣿⢢⡡⠭⣹⣫⠶⡚⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣵⡃⡀⣤⢒⣵⠚⢮⡥⠐⡭⡔⣛⣟⡅⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⣲⠋⠀⠀⡌⠠⠩⡙⣍⠻⣽⢶⣯⣿⠁⠀⠀⠀⠑⠒⠂⠀⢰⡿⣎⣿⠛⠳ ⣆⣀⠀⠀⠀⢠⡿⠷⣟⣶⠿⣛⣿⢣⡛⢫⠞⣧⠟⣯⠹⣥⠃⠘⢻⣖⠇⠀⠀⠀⠀⣾⡖⠀⢀⡷⣿⣰⠁⠈⢃⡔⡫⠈⣜⠁⢀⡿⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠄⡑⠲⣄⣣⣿⣟⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠝⠊⣁⠄ ⡟⠈⠀⠀⠉⠀⠙⢶⣷⣴⣞⡿⣽⣶⠹⢿⣶⣼⡞⡅⢲⣜⠫⣑⣼⠏⠀⢴⣦⡆⠠⢾⠃⠀⢠⣿⣿⡎⠀⠰⠎⢓⠵⢯⣁⣶⠚⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠐⠸⢑⠦⣿⣿⢯⣿⠇⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠀⠁⠂⣈ ⠀⠀⠐⠀⢄⡀⠀⠀⢾⣓⡾⠿⣿⣟⣯⣴⡺⣷⣫⣟⣵⣲⡳⣥⣺⣳⣀⡀⠀⠀⠀⣘⡝⠔⢲⣿⢿⠅⠀⠠⣬⠋⢤⢽⡺⡻⣽⣏⠃⠐⠒⠠⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⠤⢀⣀⠦⢀⠀⠀⠀⠀⠉⠑⠻⠿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⡛⠊⠉⠉⠂⢻⡿ ⠀⠀⠀⠀⠄⡀⣡⣶⣾⣿⣿⣿⣶⣾⣭⣭⣛⣓⠿⠮⣷⣏⣷⡡⢧⣹⣹⣉⠀⠀⢬⣿⣽⠶⣸⢿⢷⣿⣆⠀⠈⢫⢵⢤⡖⣁⣏⡼⣎⠀⠒⢀⠤⠀⢀⠉⠁⠒⠂⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢣⠒⠂⠤⣄⣀⡀⠀⠁⢺⣿⠁⠀⠀⠀⠀⠀⠀⢀⣿⡧⠀⠀⠀⢀⣼⣿ ⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣿⣿⣿⣻⣝⡻⣿⣿⣿⣶⣶⣯⣵⣙⣻⡿⠛⡜⢊⣿⣋⡐⢴⡏⣿⢸⡙⣯⢶⣂⠀⢹⣢⣞⣧⡝⠾⡹⠐⠂⠠⠄⣀⡀⠀⠀⠐⠀⠠⠄⠀⡀⠉⠐⠒⠠⠤⠀⣀⠀⠀⠀⠘⠀⠀⠀⠀⠐⠋⠽⠓⣶⡏⠀⠀⠀⠀⠀⠀⠀⣸⣿⠥⠄⠀⠸⣿⣟⣾ ⠀⠀⠀⠀⠀⠀⢨⡿⢿⣿⣿⣿⣷⡿⣿⣞⡿⣷⣻⢾⣭⣻⣭⣟⠻⢿⡿⣿⣿⢿⣭⣙⣿⡿⢣⡴⢧⡗⣤⢙⣶⣃⠨⢣⢛⠘⠉⢀⣘⠀⠀⠀⠀⠀⠀⠉⠁⠒⠠⠤⠀⣀⠀⠀⠁⠀⠂⠠⠄⢀⡈⠉⠐⠢⠤⢀⣀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠠⣿⣿⡽⣞ ⠀⠀⠀⠀⠀⠀⢀⠇⠈⠛⠛⠿⣾⣷⢿⣿⣽⣿⣿⣿⣷⣻⢷⢭⣟⣺⢡⠻⣋⢠⡼⣿⢿⠛⣾⡻⡽⣜⠣⣊⣕⢻⠢⢿⠦⠤⢾⣩⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠠⠤⠀⣀⠀⠀⠀⠐⠀⠀⠀⠉⠉⠳⢶⡴⠃⠀⠀⠀⠀⠀⠀⠀⢺⡿⠀⠀⠀⣸⡿⠞⠃⠉ ⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡽⠟⠛⠟⢻⣿⢻⣷⣿⣏⣟⡲⡾⣸⡔⠺⣹⣷⣿⢿⣻⣷⣢⢽⡾⢡⠙⠼⣼⣻⣞⣷⣌⣀⢳⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠢⠤⢀⣀⠀⠈⠁⢰⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠀⠀⣶⣿⡶⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣺⠞⠐⢨⣋⠬⣙⠼⡅⣏⣇⣰⡇⢪⣽⣿⣷⣿⣿⣍⢾⡱⣎⣒⡼⣿⠹⣞⡦⣽⠇⣤⣮⣴⣢⣴⣾⣿⣿⣿⣽⣊⣤⣶⣥⣤⣄⠠⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢰⡶⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢰⣿⣿⡇⠀⠀⠀ ⠀⠀⠀⠀⣂⠀⡇⢀⡀⠀⠀⠀⢸⠀⠀⠀⠀⣟⡀⠉⠚⣙⣒⣓⢾⢹⡑⡌⢿⢿⡗⣾⣿⣾⠟⣿⣋⣧⢟⡧⣣⡟⣧⣻⣽⣾⣿⣟⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣯⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣶⡇⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⣾⢿⣿⠇⠀⠀⠀ ⠀⠀⠀⠀⠀⢻⣷⣶⣼⣤⣀⣀⣞⠀⠀⠀⢰⡇⠠⠀⢷⣽⣿⡷⣋⡰⠬⠫⠭⢳⣅⡫⠿⡏⡻⢿⣇⣉⠙⠚⠿⣽⣷⣿⣿⣿⣻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣄⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⣸⣿⣿⡟⠀⠀⠀⠀ ⠀⠀⠀⠈⢡⣄⣉⣛⢋⠿⣿⣿⣿⣿⣿⣷⣾⡤⢄⢀⡾⠿⠿⣟⣭⡱⢥⡀⠐⠀⢀⠒⠿⣛⠟⠏⠋⠩⠀⣄⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣒⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⢠⣿⢻⣿⠁⠀⠀⠀⠀ ⣀⡀⣀⣀⣀⣈⣓⣪⣗⣚⠘⠀⠉⠛⠟⠻⠿⠿⠓⠛⠶⠘⠧⣠⠌⠙⠲⠂⣁⣍⠉⠔⡀⠰⣅⠓⠀⠠⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡟⠁⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⣮⣯⣿⠇⠀⠀⠀⠀⠘ ⢷⣹⢶⡻⣖⣳⢬⢓⡰⢢⡒⢉⠞⡤⣰⢁⣀⠂⣀⠆⢭⠷⠮⠪⢐⠪⡜⣌⢏⢆⣡⡵⠝⢦⢜⠣⡌⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⣸⡼⣾⡟⠀⠀⠀⠀⢀⠀ ⣿⣹⡞⣝⢯⣷⣺⢏⡯⢧⣝⡱⢊⡱⢀⡎⢔⣒⣠⡿⣛⢩⣆⡨⢖⣒⣎⣁⡠⣄⣄⠩⢽⣬⠛⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠄⠀⠀⠀⠀⣼⠟⠀⠀⠠⣷⣻⣿⠁⠀⠀⠀⠀⠀⠀ ⣯⢷⣻⣭⢟⣞⣳⢯⣟⡿⡜⢇⣝⣣⠃⢔⡤⣚⣝⣛⠯⡟⠾⢇⣼⣲⣗⠟⡲⣤⢝⡿⣌⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⡿⠀⠀⠀⢰⣿⣿⠏⠀⢀⠀⠀⡀⠀⢸ ⣿⣯⣖⣯⣞⡿⣽⣻⢮⡽⣯⣶⣟⣲⣸⣾⣥⣫⠾⠿⢧⠷⠽⢾⣤⣳⡎⣨⣗⡻⢿⣦⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⣸⠃⢀⡀⢀⣿⣿⣟⣹⢻⣮⢤⣼⣁⠀⢸ ⣿⠾⡍⢮⣹⣞⣳⣫⣷⣻⡱⣭⡶⢿⣥⣶⢷⣶⣲⢞⣛⣿⣯⡽⣿⣿⣫⣷⠿⣷⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡛⣟⠾⡝⢯⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⣿⠤⠟⠛⣿⣿⡿⣿⣿⣿⣿⣿⣧⠿⣷⣾ ⠷⠛⠚⠉⠓⠛⠫⠭⠍⠉⢉⡵⣡⣺⣹⣦⣬⣿⣽⡻⢏⣸⡿⣾⣾⣿⢟⣷⣻⣮⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠍⡘⠡⡙⠤⠣⢌⠃⡌⠓⡌⠻⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡼⠵⠒⣂⣸⣟⡝⠳⣿⢻⠯⣶⣿⣿⣿⣿⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣴⣾⡽⣆⣻⣿⣟⣷⣿⢛⣷⣫⣿⣿⣾⣟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⡀⠄⠀⠂⠁⠂⠀⠀⠁⠈⠅⠊⢍⡻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⠳⢶⢴⣚⣧⠛⣾⡽⠿⣿⢿⣿⣿⣿⣿⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢟⣷⣿⣾⣋⠕⣻⣿⣻⣿⣮⢯⡛⠭⣼⣿⡖⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠡⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠌⠛⠛⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠷⠬⢭⡊⣯⣟⣻⣽⣆⢻⣿⣿⣿⣿⣿⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡺⡿⣿⣻⣿⣾⣿⣿⣿⣯⡷⣿⡿⣿⣿⣿⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⠏⠈⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢤⣐⠺⡇⣭⣳⣯⣟⣿⣿⣾⡿⣾⣿⣿⣟ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢶⣟⣹⣿⣟⢿⣿⣿⣿⢿⣿⣿⣿⣳⣿⣷⡿⣿⢯⣿⣿⣿⣿⣿⣿⣿⡟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢃⠎⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠻⣟⢾⣙⣾⠁⣟⣯⣿⢻⣷⣿⣿⣿⣿⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⢨⣿⡶⣿⣟⣿⣿⣤⣿⣽⣻⣿⣿⣍⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣏⢔⣠⣶⣶⡷⢂⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠌⠢⡙⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣿⣧⢺⣴⡚⣾⢿⣡⣿⡿⣿⣿⣿⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢐⢿⡿⣿⣿⣿⣿⣿⡇⢛⣿⣽⢻⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢅⠺⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⣏⣳⣼⣿⣧⡝⣟⣘⡿⢟⣿⣿⢿⣾⣿⣽ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⣶⡷⣿⣿⣿⣿⣿⣴⣿⣻⢹⣮⣽⣿⣿⣿⣿⣿⣿⣫⣿⣏⣫⠙⡛⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⡀⠀⣀⢠⠐⣦⣦⣨⣄⣀⣀⣀⠀⠈⠐⡨⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣟⢿⡟⣞⡯⣍⢳⠝⣮⣇⣲⢌⠫⣔⢫⡙ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠡⠋⡹⢛⣹⣿⣿⣿⣿⣿⣧⣼⣿⣟⡿⣿⣿⣿⣿⣿⢿⠛⣻⠿⠟⠻⢿⣿⣷⣿⣿⣿⣿⣿⣟⡛⠿⡿⣶⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣄⠐⠄⠡⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣉⣷⣷⣻⣻⣯⡼⣘⢦⠲⣄⢋⡳⠎⠗⣮ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠉⠉⠹⠿⢿⣿⣿⣿⣯⣭⣷⡽⢿⣿⣿⣿⡏⠘⣬⣿⣾⡿⢟⣿⣿⣿⣷⣯⡻⢿⣿⣶⣤⣀⠈⢇⣈⣛⣿⣿⣿⣿⣿⣿⣿⣿⣻⠟⡿⣿⣿⣿⣿⣿⡌⠂⠁⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⢿⣻⢿⣿⣿⣾⠤⣟⡞⢮⢷⡜⣢⠵⣍⢎⡤ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢟⣾⣷⣾⣿⣿⣿⠀⠐⢠⡙⢿⠿⣼⣿⣿⠿⡿⠿⣿⢇⣿⣿⣿⣿⣷⣾⣿⡻⢻⠿⣿⡟⠿⠟⡟⠻⠿⣿⣷⣶⣤⡉⠿⣿⣿⠠⠁⠼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣛⣽⣿⣿⣻⢀⣇⣞⠣⢷⣜⣢⡙⣬⢚⠶ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠂⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⢄⠀⠀⠘⢦⣙⡴⣩⢜⡱⣍⡶⠃⢾⣿⠋⠀⠀⠘⢿⡇⡇⡍⢎⠴⣿⣟⠿⣿⣿⣶⣦⣌⠙⠿⣿⣷⣾⣿⣇⣈⣘⣿⣿⣿⣿⣿⣿⣿⣽⡫⡿⣿⣟⣉⢻⣐⣍⢧⢯⡟⢿⣲⠦⣭⡐⣯⢞ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⠤⠀⣀⠀⠀⣿⠙⠿⣵⣆⡄⠀⠈⠉⠉⠉⠉⠀⢀⣼⡿⠁⠀⠀⠀⠀⢸⣇⡇⠈⡘⢸⣦⡍⢳⢿⣿⡿⣃⣽⣿⡆⢰⣿⡿⢛⠻⠿⣻⣿⣿⣿⣿⣿⣿⣯⠻⣷⣿⣿⡍⢿⣷⡿⡗⢿⣽⣙⡻⠎⠷⣥⣛⡴⢋ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⡟⡔⠂⠀⠈⠉⠛⠛⠲⠶⢶⣶⣶⡾⠟⠁⠀⠀⠀⠀⠀⠐⣿⡰⡀⠀⠀⠘⠛⠷⣦⣍⡃⣍⠘⠤⠐⢸⡟⠀⠂⠁⠂⣽⣿⣿⣿⢛⣏⣟⠻⣿⡾⢿⣟⣿⣾⣿⣧⡁⠈⣞⣳⠽⣛⡟⡲⢧⢭⢻ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠐⡀⠀⠀⡀⠀⢀⠐⡤⣿⠋⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠘⢷⣽⣦⡄⡀⠀⠀⠀⠈⠉⠉⠈⠀⢀⡿⠁⠀⠀⢀⡐⣿⣿⠏⡔⣻⣯⣾⣾⣷⢻⣦⣾⢾⣯⣫⠯⠈⣁⣯⣽⣛⣻⡶⢻⣬⢳⣌ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠱⣀⠃⡄⠠⢈⠢⢱⠱⢻⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⠋⠛⠛⠶⠶⠶⢦⡴⡶⠶⠟⠁⠀⠀⠀⠄⣼⣿⠃⠀⢰⣿⣿⣿⣿⣿⣿⠿⣷⣞⠈⡿⢠⣾⣷⣣⣞⡱⢧⡛⢯⣒⡦⢽ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠱⣌⠲⣈⠕⡢⢎⡥⣉⣆⣿⣿⣿⣷⣡⣄⠀⣀⣀⣠⣀⠀⠀⢀⡟⠐⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⢥⣿⡏⡖⢀⣾⣿⣡⣼⣾⣿⣿⣧⣿⠻⡴⠀⣞⡶⢮⣭⣝⣋⡹⠿⢲⣵⢮⢧ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢎⡲⡱⢥⣚⣵⣾⣶⣿⣿⣿⣿⣿⣿⣿⣙⣟⣻⣛⡿⣿⡿⡶⠚⠁⡁⠆⠠⠀⠀⠀⠀⠀⠀⠀⠠⠠⡈⠵⣈⣾⣿⡱⢌⡎⠉⠉⠙⠛⢿⣿⣿⣿⣿⡤⠁⢰⣿⣼⣳⣖⣣⠞⡍⢟⡓⡲⢬⢬ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣢⠱⡑⢾⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣵⡀⠀⠀⡍⡐⠠⡁⠄⠠⢀⠀⡀⠄⢢⠑⢲⣿⣿⠏⢴⣞⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧⣀⠀⢈⢯⢭⣑⣚⣻⠿⢿⣾⡶⢽⣘⣦ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⡱⢌⢿⣿⣿⠿⣿⣿⣿⣾⣿⣷⣶⣤⣍⣿⡿⣿⣿⡿⣿⣿⣷⣿⣧⡐⣀⠓⡌⢆⡡⢂⠜⣠⡙⡰⢬⣿⣿⠇⢠⠏⠈⠙⠓⠶⠤⠴⣋⣻⣟⢿⡉⠀⠹⣟⣎⢞⡰⠩⣍⢻⡒⢶⠮⣭⣓⣚ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣓⠎⣼⣿⢣⢇⠹⡜⢿⠷⣰⠀⠋⠉⢹⠛⠛⡿⢭⣶⡳⣏⡽⣿⣿⣿⣤⢃⢍⠢⣑⠣⢎⡡⢚⣴⣿⣿⡿⣄⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠚⠿⢯⣽⣿⣷⠾⣦⡼⣦⣓⢆⠳⢌ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⣜⣾⣿⢏⢎⢆⡩⢄⠂⠀⠉⠒⠶⠤⠤⠤⠤⡤⢇⠽⢿⣿⣿⣿⣿⣿⠎⡄⢢⢁⢳⠬⡸⣼⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⢿⣻⣛⣲ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣫⣜⢎⣶⣙⣷⡄⠀⠀⣀⡀⠀⠀⠀⠀⣈⠜⡃⠥⠾⣿⣿⠟⡣⢄⢣⡚⣜⡲⣵⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠬⡔⠉⢿⣿⣿⣿⣿⣶⣶⣬⡔⡴⢢⡓⡌⢦⡉⠓⡌⡽⣟⡅⠰⣈⢶⣹⣷⣽⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣖⣈⡀⢬⣽⣿⡿⣿⡿⣿⢿⣏⢣⠙⡸⢐⡸⢥⡙⢿⣿⣞⡰⣐⣮⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣻⡾⣫⡿⢋⢝⡽⡔⢣⠂⠀⠂⡘⣄⡑⡢⢼⣙⣶⣿⣿⣶⣿⣿⣿⣿⣿⠟⢹⣿⣏⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡠⣚⣽⣿⣿⣿⣿⣿⣿⣿⣵⡾⣾⣽⣽⡼⣧⡶⣅⣼⣮⣷⣷⣬⣿⣿⣿⣿⣿⣿⣿⣿⡿⠍⣈⢸⡿⠘⠆⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠂⡄⠜⠂⢀⣭⣾⢟⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣮⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠿⡒⠬⣐⣰⣿⢁⡑⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⣨⡼⠋⠦⢀⢔⡯⠞⠁⣾⢧⢿⣿⡸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⡟⣡⠣⡑⠄⣶⣿⢃⡳⠁⢄⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ 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⠀⠀⠀⠀⠀⠀⠀⠑⢌⣿⢼⢻⡀⠀⠀⠀⠈⡀⠀⢆⡏⢦⢣⡘⠐⢕⢅⠢⡑⠮⣰⠇⢂⡲⣰⣱⠣⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠡⡐⡰⡱⡑⡙⡀⠁⠂⠅⡒⠨⣙⣲⡂⠸⠽⢸⣛⡟⠰⢌⣈⣓⠒⠶⢤⣍⣀⡒⠨⡐⠄⡘⡆⠀⠀⠀⣀⠤⡐⠌⠑⣀⢩⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣼⣃⠞⠳⡶⠀⢠⠁⠘⡼⡘⡎⢢⠱⠀⠀⠑⢕⠨⣒⠟⠀⣜⢣⣯⢣⣱⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠡⡑⡰⡱⠡⡑⡐⠀⠀⠀⠀⠀⠁⠂⠀⠀⠏⠃⡿⣸⠣⡟⣒⠀⠬⢉⡒⠲⠆⢍⣙⠓⢤⡕⢀⣡⠄⣂⠩⣠⠥⠖⢚⣉⣥⠠⠖⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⡎⠹⣽⡿⡀⠀⠁⢀⡆⠀⢳⢣⢳⠘⠆⠆⠀⠀⠀⠀⠉⡌⠐⢱⣜⣜⢯⣾⠣⣅⠨⢐⠄⡀⠀⠀⠀⠀⠀⢠⠡⡑⡰⣱⢁⠣⢡⠁⠀⠀⠀⠀⠀⠀⠀⡘⡸⠘⢸⢣⢇⡙⠀⠀⠈⠁⠂⠠⠅⣐⠂⠌⡱⣾⢻⡄⢖⡣⢔⣋⡄⠖⠛⡉⠠⠀⣒⠈⠤⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⣿⢸⢆⠰⡂⡘⠀⠈⡞⡌⡇⠻⡘⠀⠀⠀⠀⠀⢰⠀⡐⡼⡘⡝⢤⠂⡑⢬⡙⠦⣁⠨⡐⢄⠀⠀⠠⢡⠱⣰⢣⠇⢂⢃⠃⠀⠀⠀⠀⠀⠀⠀⢀⢃⢃⠃⣏⠎⢦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡼⡘⣧⣷⣺⠷⠈⡁⠄⣊⠡⠐⠈⠁⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⣤⢀⠀⢸⠀⢸⣾⠋⠢⢕⡪⣇⠀⢸⣇⢻⢹⡄⠇⠀⠀⠀⠀⠀⡇⠀⢱⢧⡻⠑⢆⠈⠢⡐⠌⡓⢬⡓⢦⡑⠈⡢⢀⠣⢡⢇⡟⡈⡌⡈⠀⠀⠀⠀⠀⠀⠀⠀⡘⠘⡈⡼⡜⡘⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢷⢣⣟⢻⡈⢎⠇⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠘⢗⡭⡎⠀⠰⠃⠀⠀⠀⢨⢕⢝⣾⢸⡸⡈⢇⠀⠀⣾⡇⠀⡼⠀⣠⡟⢮⢣⡙⠌⠀⠀⠀⠑⠨⡂⢍⠳⣍⠳⢌⡆⠇⡾⣼⢡⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢣⢱⢳⢡⢸⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠞⢥⢹⡎⢧⠰⠸⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠘⣁⢰⣦⠀⠀⠀⠀⢸⠀⠁⣿⢸⡧⣷⢬⢣⡘⠿⠃⢀⠇⢀⡚⡜⡎⢦⠑⠀⠀⠀⠀⠀⠀⠈⠐⠅⡢⠙⡺⣌⣺⣭⢃⡦⠣⢀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡘⡄⡏⡟⠈⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⡣⢊⢣⢹⡸⡆⠇⠆⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⢰⠁⣌⠋⠀⠀⠀⠀⡆⠀⡀⡿⡜⢇⡏⢃⠢⡙⡆⠀⣼⠀⡘⢱⢹⢘⠆⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠰⢵⡰⡻⣷⡋⠦⣌⣒⣂⣉⣉⣛⡉⠭⢐⠠⢠⠱⢸⢳⡇⡇⠇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠊⠀⠆⡎⡌⣇⢷⠸⠸⡈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⢠⠇⣸⣆⠀⠀⠀⠀⢀⠁⢰⢘⢳⡙⡞⡌⠢⠣⡙⢌⣺⣏⠀⠟⡎⣇⡛⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢢⡗⣱⢟⡟⡠⠍⣛⣒⣒⣒⣒⡒⠮⣭⣛⠢⡌⡚⡍⣾⣺⢅⣱⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠰⠸⢰⢸⡼⣆⢣⢡⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ 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