⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⢠⠀⣄⠠⣀⠄⣠⠀⣄⠠⣀⢄⡠⣀⢄⡠⣀⠄⣠⢀⠤⣀⠤⣀⢄⡠⣀⢄⡠⣀⢄⡠⣀⢄⡠⣀⢄⡠⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡘⣆⢣⠜⣡⣛⢤⡓⢬⢚⡤⢫⡔⣣⠜⣢⠱⡌⢦⠱⣌⡚⢤⢋⠖⣡⠞⡰⣊⠴⣡⠎⡴⣡⠎⣖⡱⢮⡕⣯⣞⡵⣉⠆⠄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⣱⢎⢇⡼⢧⣋⢶⣙⣎⡳⣜⢣⣞⡱⢮⡱⢳⡜⣣⢝⡢⡝⣎⢮⡙⢦⣍⠳⣌⡓⢦⠛⣔⢣⣛⡴⢭⣳⢿⣿⣿⡽⢦⡙⠀⢀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⡯⢋⣾⡝⣧⣿⣟⡾⢴⣛⣬⣓⢮⡝⣧⣛⢧⣞⡱⢮⡵⣹⡜⣲⢝⣣⢎⠷⣬⡙⣎⠽⣌⠳⣜⢺⢧⣿⣿⣿⣿⣟⢧⡝⢦⡈⠔⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣥⣿⣳⣿⣿⣿⡿⣽⣣⢟⡶⣹⢮⣻⣜⢿⣺⡼⣝⣳⣞⣵⣫⢗⡮⢧⣛⡞⡴⡹⣌⢳⢎⠷⣎⣯⢿⣿⣿⣿⣿⣞⣿⣺⠡⡘⠤⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣟⣿⣿⣿⣿⠷⣽⣻⣞⣷⣻⣾⢯⣟⡾⣯⢷⣻⢯⡽⣯⣟⣞⡷⣎⢿⡱⢧⡳⡝⣮⡜⡶⣽⣿⣾⣿⣿⣿⣿⡿⣽⣯⢀⠳⠠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣳⣿⣿⣿⣿⣏⣟⣳⣽⣿⣾⣿⣻⣿⣿⣿⣽⣯⣟⣯⣿⣾⡽⣾⣽⣻⢾⣹⢧⣛⡼⣵⢺⡵⣯⣿⣿⣿⣿⣿⣿⣿⣿⡇⠌⢎⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢳⡞⣼⢣⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⣽⣯⢿⣹⣳⢿⣼⣳⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣊⠆⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣟⣿⣚⣯⢾⡱⢋⡾⡛⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠐⡄⢊⠔⠡
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢰⡿⣯⢟⡽⠮⠝⠊⠁⠀⢐⢧⢻⠘⢿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡽⠀⠰⢈⠢⠘⢠
⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠴⣪⣟⡼⣇⠀⠙⠿⣿⡿⣟⣾⣻⣷⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠁⡈⠔⡁⠂⠉⠀⠀
⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⢀⡀⣤⡰⢮⣹⢯⣷⣻⡜⣯⢷⣤⠀⠀⠉⠙⠒⠙⠒⠉⠋⠛⠙⠯⠿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡱⠁⡐⢈⠐⠈⠀⡀⢂
⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠸⣝⡶⣻⢭⣳⣏⢾⣳⢿⣹⢮⡝⣟⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣻⣿⣿⣿⣿⣿⣿⡿⣿⢟⠝⠁⠠⠐⠀⠀⠄⣂⠁⠆⢠
⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠳⡹⣎⠗⢮⢏⠷⣫⣟⣯⠞⣽⣚⢾⣹⢳⢦⡔⣠⣄⡀⠀⠀⠀⠀⠈⡿⣿⣿⣿⣿⣿⣯⣽⡿⠋⠀⠀⠀⠀⡄⠑⠂⠄⠌⡐⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠀⠈⠘⠱⢾⣯⢝⡲⣍⢎⡳⠭⢎⡝⡳⡼⢭⡳⢦⣄⡀⠀⢘⣿⣿⣿⣿⡟⠖⠋⠀⠀⠀⠀⠠⠁⠄⡁⠊⠌⡐⢀⠂⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡂⠀⠀⠀⢺⣽⡘⢧⡹⠜⡃⠖⠨⠑⠣⡑⢮⢳⢮⡝⣧⢂⢈⣿⣿⣿⡟⠁⢀⣀⠀⠀⠀⠀⠂⠐⠠⢁⠂⠄⠡⠈⠄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠁⠀⠐⠃⢸⡷⣘⠧⡘⠁⢀⣤⠄⠀⠀⠀⠈⠣⢫⡜⣣⢏⣞⣿⣿⡟⠋⠠⠛⠁⠀⠀⠀⠀⠠⠁⠐⠀⡐⠈⡀⠡⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠛⠄⠀⠀⠀⣸⡷⣉⠖⠁⢠⣿⡇⠀⢀⡄⠀⠀⠀⠀⣝⢢⡝⣾⣿⠋⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⡀⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡤⣀⣀⣠⣴⣿⢧⡹⠌⠂⠸⣿⡀⠀⠀⠁⠀⢰⡇⠀⣾⣿⣿⣿⠃⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⣿⣟⣿⣿⣿⡿⣋⠦⡱⠄⠃⡀⠉⠛⠂⠁⠀⠐⣉⢤⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣼⣿⣿⡿⣱⢡⢚⡵⢫⣔⢢⢆⡁⣈⡐⣁⣩⣴⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣽⣥⣷⣫⣜⡳⣜⢫⡞⡵⢣⢏⡳⢽⡹⢻⠿⢻⠇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣭⠞⡼⡱⢫⢜⡡⢧⡝⣯⢏⡎⣤⠀⠀⠀⡀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠘⡴⢩⡓⢮⡱⢧⣿⣣⡞⠀⠨⠑⠊⡔⠠⠁⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⢥⡙⡰⢣⡝⣲⣽⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣯⣷⣮⣕⠣⢎⡵⣻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠿⢿⣿⠿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣎⢲⣹⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠎⠐⠀⡠⢌⣭⣦⣭⣉⠛⢿⣿⢰⣻⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠃⠀⠀⠀⢊⠬⣙⡛⣿⢿⡌⠣⢎⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀