Christmastide Emojis & Symbols

⠀⠀⠀⠀⠀⣠⡄⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⢤⣀⡀
⠀⠀⠘⠟⠀⢾⣶⣿⣿⣿⣿⠖⠀⠀⠀⠀⠀⠀⠀⣠⡾⠷⢶⣶⣶⣤⣄⡙⠳⢤⣄
⠀⠀⠀⢰⠠⣿⠉⠟⠛⠛⢁⣄⡀⠀⢀⣠⠤⠂⠾⠷⠾⠿⠛⠛⠿⠿⣿⣿⣷⣤⡙⢿⣶⣤⣀⡀
⠀⠀⠀⡿⣄⣀⣌⠀⠠⢹⣷⣬⣿⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣍⠛⢿⣿⣦⠹⣿⣿⣿⣦⡀
⠀⠀⠘⠛⣟⢻⣿⣿⣮⣌⡀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⡆⠀⢿⣿⣧⠝⣿⣷⠈⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠸⠛⠻⡾⠉⠀⢠⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡏⠺⣮⣽⠟⢣⣿⣿⣇⢸⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠒⠲⢶⣶⣦⣤⣤⣄⡀⠀⢈⣯⣤⣶⣦⣼⡿⡿⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡻⢀⣀⣀⣀⠲⢤⣈⠀⠀⠀⠉⠙⠻⣿⣆⠀⠀⠛⢛⣫⣶⠃⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⢉⣡⣤⣽⣷⣿⠠⣤⣤⢀⣀⣥⣝⢻⡄⠀⢠⣿⣿⠇⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⡄⠘⠃⣀⡾⠛⣭⠃⢆⠉⢉⣛⣿⡿⣥⠅⠀⣼⣿⣿⡄⢸⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣼⡟⣀⣀⣠⡴⢀⡌⠁⠀⢄⠀⠈⠛⣿⡷⣿⣄⡀⢻⣿⣿⣇⠈⢿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⣀⡘⠁⣠⠅⠀⣻⣇⡉⠉⠉⠹⢦⠿⠶⠶⠞⠱⢦⣴⡿⠟⢂⡟⠶⠜⣧⠀⠉⢀⣼⣿⣿⣿⠟
⠀⠀⠀⠈⢁⠤⠤⠒⠒⠀⠀⠛⠉⠒⠶⠶⣤⣀⣀⣄⡀⠀⠀⠀⠀⢀⣼⣧⠌⢱⣮⣻⣾⠿⣿⠿⢿⣟
⠀⠀⠒⠊⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠨⠽⠿⣦⣉⠒⠂⠉⠛⠋⠉⠙⢷⣾⠿⠋⠣⠉⠀⠀⠠⣹⣿⠆
⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠹⠻⠗⠀⢀⠀⠀⠀⠈⢮⣿⡀⠀⠀⠀⠀⢀⣴⣼⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⢄⠀⠀⠀⡄⠀⠀⠀⠀⠰⢀⡂⠀⡄⢸⣿⡇⠀⠀⠀⠀⠘⢻⣿⣿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠈⠄⠡⠇⠀⠀⠀⠀⡀⠀⠀⠃⠘⡃⠘⣰⣿⣤⣷⣬⣳⣶⣾⢻⣟⢤⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠐⡼⠀⣸⡀⠀⠀⢸⠀⣾⠁⣴⠀⣇⢰⣿⣿⣿⣿⣿⣿⣿⣷⠀⣿⣮⠟
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠘⠃⡴⠀⠠⠀⣠⣇⢶⣌⢿⣿⣃⡏⢸⣿⣯⣿⣿⣿⡿⣿⡿⢰⣿⣿⠿
⠌⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⢀⡞⠠⠀⠈⠠⠞⠁⢩⣿⣿⣿⣿⡟⢀⣼⣿⣿⣿⣿⣿⣶⠉⣧⡿⢫⣤⡟
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠺⢀⠄⠀⠀⠀⢀⣼⠏⣸⣿⣿⣿⢅⣾⣿⣿⣿⣿⡿⠋⢁⠒⠉⢀⣾⣿⣅
⠀⠢⠁⠀⠀⣀⡤⠤⠀⠀⠀⠀⠀⠀⠈⢀⣀⡤⣠⣾⠋⣴⣿⡿⢙⣷⡿⣿⣻⠿⠺⡻⣖⡷⢶⡶⢈⣿⢻⠾⠁
⠀⢠⡀⠀⠈⠉⣠⣤⣤⣴⣶⠖⢀⣴⡾⠋⣡⣾⣯⣴⣿⣿⣿⣝⣯⡗⠋⠈⠀⠀⡠⠤⠤⠤⠭⠁⡉⠁⠈
⠀⠀⠁⠀⠀⠈⠹⣾⣿⣿⡁⠀⢹⣯⢀⣼⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠐⠀⠀⠀⠀⠈⠉⠛⠃⠀⠛⠃⢨⣵⡾⠿⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⢉⡉⢉⠉⠍⡍⡉⡉⡉⠍⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⠛⠁⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠛⠃⠀⠂⠃⠛⠓⠃⠄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠨⠕⡇⡇⠉⢹⣿⠈⠄⢉⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣴⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠀⠀⠁⠁⠀⠀⠈⠈⠀⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⢼⣯⣀⣽⣷⡄⣰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡴⠄⣠⠀⢶⠀⡄⣄⡄⣤⠤⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠀⠀⠈⠿⠃⠀⠀⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣧⠀⣟⡆⢸⠀⡇⡿⡇⣿⢲⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣙⣉⣁⣁⣘⣀⣁⣁⣁⣙⣋⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠦⣄⣠⣄⣠⡴⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠬⣿⣾⠥⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣤⡤⠶⢿⣿⠲⢤⣤⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣿⣿⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢽⣿⣿⡦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡾⣟⣹⣏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡃⠏⠃⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇

⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣠⡶⢷⡄⠀⠀⠀⠀⢀⠠⣀⣀⠀⠀⣀⣶⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡔⠒⢂⠀⠀⠀⠀⡇
⠀⢰⠀⠀⠀⠀⠉⢀⠀⠁⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠠⠋⠀⠀⠀⢠⡀⠃⠀⠀⠀⠀⠀⠀⠈⠁⣀⠈⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⣴⡂⣠⠔⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢸⣟⠁⠁⠒⠈⠁⠀⢈⣭⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⣿⠅⠀⢈⣠⡠⡂⠀⢸⣿⡆⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢰⣷⠀⢀⡞⠛⠃⠸⡆⠀⣾⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⠁⠀⢀⡙⠀⠀⢖⡀⠀⠋⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠙⠶⠶⠋⠀⠀⠀⢀⣼⡇
⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⠀⠀⠚⠷⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣿⠀⢀⡀⠀⠀⠀⠀⠰⡆⠐⣟⠊⠃
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡉⠀⠈⠀⠀⠀⠀⠀⠀⢳⠀⠻⢠⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠃⢀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢺⠘⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠀⠀⠄⠀⠀⠀⠀⠤⠀⢠⠀⢀⡆
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠒⠀⠀⠓⠀⠀⠀⠀⠘⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⠄⠀⠀⠀⠀⠰⠀⠀⠀⠀⠂⠈⠇
⠀⢸⠀⠀⠀⠀⢐⠲⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂
⠀⢸⠀⠀⠀⠀⢺⣖⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⢰⡀
⠀⢸⠀⠀⠀⠀⣾⣮⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⢾⠃
⠀⢸⠀⠀⠀⠀⣾⡷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⣺⡃
⠀⢸⠀⠀⠀⠀⠬⢽⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠯⠔⠀⠀⠀⠀⡄
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠋⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⢸⣛⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣾⡿⠿⠿⢿⣿⣿⣿⣷⠿⠿⠾⣞⣟⣷⣟⣿⣻⣾⣾⣿⣳⣷⣻⣾⣾⣷⣿
⠀⢸⣽⣿⣿⣿⢛⠙⠻⢿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠙⣿⠟⠁⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠟⠉⠈⠙⣿⣿⣿⢿
⠀⢸⢿⣿⣿⣧⡼⠂⠀⣾⣿⣿⣿⣿⣿⠁⢀⠀⠀⠀⠀⠀⠈⠀⠠⠖⠠⢤⡀⠀⢹⣿⣿⣿⣿⣿⣦⠐⠂⠁⣿⣿⣟⣻
⠀⢸⣽⣿⠟⠋⠀⢤⠸⣿⣿⡟⣿⣿⣿⣇⠀⣁⠀⢀⠖⢀⣾⡄⢹⣀⠁⣤⣄⢠⣿⣿⣿⣿⣿⣿⣿⣧⠀⠆⣻⣿⣯⣽
⠀⢸⢾⠇⠀⠀⠀⠁⢙⠂⢹⣿⡇⡟⠃⠀⠀⢠⢼⠋⠲⣿⣿⣿⠗⠒⠶⣃⢲⠀⠙⣿⣾⠟⠛⠻⣠⠡⡤⢿⣶⢀⣿⣺
⠀⢸⣛⠆⠀⢰⠄⠱⣞⢁⡀⠀⢱⣄⡀⠀⡀⡸⢿⢀⡀⠛⠋⠃⠂⢰⠀⠀⡀⠄⠀⠙⣿⣠⡴⠀⢹⣧⣀⢠⣾⣿⣿⣽
⠀⢸⣽⡆⠀⠉⠀⢀⠽⠟⠻⠁⡸⣇⡝⠃⠙⠢⣀⠘⠀⠀⠀⠀⠀⠸⢳⡿⠟⠂⢄⣠⠻⣿⠛⠟⢻⠟⠛⠢⣙⣽⣿⢾
⠀⢸⣾⣧⡷⠂⠁⠀⠀⠀⠻⡮⢳⡿⢷⣶⠷⢩⣿⣧⠀⠀⠀⠀⠀⠃⣼⣥⡤⠴⠞⣗⠶⡏⣹⣶⠚⠀⣤⢄⠈⢻⣿⣻
⠀⢸⣽⣿⡅⠀⠀⠀⠀⠀⠀⢼⠀⠘⠈⣠⣄⣬⣝⡻⡏⣀⡄⠈⣀⣟⡻⣷⠀⠐⡤⠀⠀⠀⠈⠣⡄⡈⠈⢁⢣⡞⢿⣽
⠀⢸⢶⡟⠀⠀⠀⠀⣤⡜⠀⠀⠐⠁⠀⠜⣄⡿⠏⣰⠉⠁⠃⠀⣿⠩⣼⡿⠇⢰⣶⣄⠁⠉⠀⠀⡈⢁⠀⢀⠈⠀⣿⣿
⠀⢸⣿⡏⠠⠃⠀⢀⣿⡇⠀⠀⠄⡸⠀⠀⠈⠀⠞⠉⣘⡁⠀⣀⣽⢧⣁⡙⢧⠆⠙⠙⢷⡀⠀⠀⢁⡘⣧⣶⣿⣴⣿⣿
⠀⢸⣽⣷⣾⣆⣂⣀⣿⣷⣄⣀⣒⣷⡄⠀⣀⣄⣀⣀⣇⣿⣿⣿⣟⢀⣿⡁⣄⢀⢀⡀⣸⡁⠀⠀⠀⠁⢹⣁⣁⣛⣷⣼
⠀⠸⠿⠿⠿⠽⠿⠯⠿⠿⠿⠽⠿⠿⠯⠿⠿⠽⠽⠯⠯⠿⠽⠽⠭⠯⠯⠿⠽⠽⠿⠯⠯⠿⠽⠽⠯⠯⠿⠽⠽⠽⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⣤⣾⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠖⠀⠀⠀⠀⠀⠐⠉⠐⠂⠄⠀⢠⣤⠔⠻⣟⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠼⠟⠉⣤⠰⠆⡤⠀⠀⠁⠀⠤⠀⠀⠀⠀⠉⠀⠘⢿⢿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣈⠋⡀⠀⠀⣌⢁⡀⢰⡔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⣠⠯⣼⢣⡾⣃⡀⠀⠉⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠈⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢾⡝⠓⠉⠁⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⢠⡆⢀⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣲⣳⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣡⡄⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢰⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⣷⠈⠉
⠀⠀⠀⠀⠀⢀⠀⠀⢸⠀⠃⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡐⢳⡀⠀⡠⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⡄⠙⠀⢀
⠀⠀⠀⠀⠀⠈⡄⠀⠀⡂⠠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⣇⢸⣧⡆⢻⡇⣘⡷⣴⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⣠⡄⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠸⡇⢿⢸⡿⠧⠈⠋⢙⠲⣥⠟⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡮⠿⠆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠸⣧⡟⢀⡄⢸⠃⠿⠀⠀⢤⢀⠀⠰⣾⣵⡋⠀⠀⠀⢐⣻⠶⠂⠀⠀⠀⢀⣼⣿⣟⠀⢀⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣄⡀⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠀⢸⣄⠐⢿⡆⡄⢠⠍⣹⠏⣀⣉⠉⠉⠉⢁⡀⠀⠀⠀⠀⠀⠨⡿⡿⠉⠀⠘⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢿⣷⣶⠀⠀⠀⣤⣤⠀⠀⠀⠀⣿⠀⠀⠀⢸⡏⣇⢾⣻⣧⣬⠟⣿⣼⢿⢹⣶⢲⠆⣸⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣷⠀⢾⣿⡿⠂⠀⠀⠀⠸⣆⠀⠀⢸⣇⠙⠀⠁⠀⠀⠀⠀⠀⠀⠈⠙⢛⡫⣿⠛⠛⠀⠀⠀⠀⠀⠀⠻⠿⠇⠀⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣝⠛⠃⠀⠀⠀⠀⠀⠘⠳⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣯⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣬⡃⠸⢃⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣠⠀⢄⢦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡁⡘⠏⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⡀⣀⠀⠰⢦⢸⣦⢲⣜⣻⢶⡏⠹⠆⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢁⣠⣤⡄⠙⠀⠀⠀⠰⠷⠀⠀⠰⠆⠙⠃⠘⢁⡻⠇⠀⠰⠟⢡⡟⠰⠟⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢀⣀⣤⣤⣶⣤⠄⠘⠃⠦⠐⠛⢀⣤⣦⠆⠠⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠋⠁⠀⠀⠀⠀⠀⠘⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣌⢤
⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⢠⣴⠎⠀⠀⠄⠀⠀⠐⣁⠀⠀⠀⢠⢀⠀⣷⠀⠀⣸⡿⢛⣦
⠀⠀⠀⠀⠀⣇⡆⠀⠀⠀⢀⣮⠄⠀⠀⠘⣴⠀⠀⣛⠀⡄⠀⠀⢸⣩⡆⠀⠀⢠⣿⠿⠁⢈⡅
⠀⠀⢀⢀⣶⣃⡀⢠⡀⠀⣊⠀⠀⠰⡄⢰⢲⡇⣴⠉⢀⡀⢸⡤⠾⠉⠷⢄⠀⠘⣋⡴⠚⠉⠹⠀⣧
⠀⠀⢂⡿⢟⡡⠀⡐⠲⠀⠟⢠⣄⠿⠇⡼⠛⡆⠀⢰⢾⠋⡿⠒⠖⠒⠒⠚⣫⡀⠋⣷⡠⠞⠀⠰⠆⢷
⠀⠈⡎⣷⣈⠇⠐⠀⡈⠈⠀⠈⠉⠀⠀⠀⠀⢹⡀⠀⠀⠘⣿⣳⣞⣶⣻⢾⣿⠇⠈⠋⡀⠂⠀⡜⠀⠘⡆
⠀⠀⠈⠻⢿⠟⠉⠈⠊⠀⠀⣠⣧⣀⣀⠀⠀⠀⡟⢀⠀⡀⠈⠳⢶⠒⢒⠶⠋⠀⠀⠀⠁⢀⣼⣯⣥⣬⣽⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠿⢯⠀⢆⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢃⠐⠰⠈⠆⠀⠆⠀⠀⠐⠀⠂⠀⠀⠈⠖⠒⠀⠆⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣶⡶⠆⣀⡐⠛⠛⠉⠉⢉⣴⠶⢛⣥⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢸⣿⣿⣿⣿⣿⣿⣿⡿⢋⣥⣾⣿⣿⣿⣷⣦⡀⠀⣾⣯⠸⠿⢛⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⣫⣥⣶⠘⣿⣿⣿⠟⣋⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣶⡄⣷⣬⣍⣍⣉⣉⣍⡻⣿⣿⠟⣋⣴⣾⠿⢛⣩⣥⣦⡙⢿⣿⣘⠛⣋⡥⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡾⢟⣩⣶⣾⣿⣿⣿⣿⣿⣶⣬⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢛⣛⠛⢛⣛⣃⣛⡛⣛⠛⡛⠛⣛⣧⡙⠿⠷⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣷⣶⣶⣶⣶⡆⣶⣶⣶⣶⣶⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⡄⣤⣤⣤⣤⣤⣤⣤⣤⡄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇⠿⢟⣫⣥⠤⢀⣤⣤⣤⢠⣤⣬⣙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟⠿⠿⠿⠿⠃⠻⠿⠿⠟⠿⠿⠿⠟⣣⣶⣿⠟⡁⢾⣿⣿⣿⣿⠸⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⡟⣰⣿⣿⠏⣼⣿⣦⡙⢋⣥⣶⣶⣦⣍⡻⢋⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⢰⣿⣿⡟⢸⣿⣿⡿⢱⣿⣿⣿⣿⣿⣿⣷⡜⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣭⣭⣥⣬⣭⣭⣍⢩⣭⣭⣭⣥⣭⢸⣿⣿⡇⠛⠛⠛⠃⣸⣿⣿⣿⣿⣿⣿⣿⡇⠿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⢸⣿⣿⡇⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠟⠸⠿⠿⠿⠿⠿⢸⣿⣿⡇⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⡇⣬⣤⣬⡄⢺⣿⣿⣿⣿⣿⣿⣿⡇⣩⣭⣭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡇⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠁⠈⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉





⣿⣾⣿⣾⣶⡿⢶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⠶⠶⠶⠶⠿⣶⣶⡶⠶⠶⠶⠶⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⢴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⢸⠃⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡋⣴⣾⠀⢀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⠘⣿⡆⢿⡼⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢻⢿⣤⡄⢠⠤⠀⡀⣀⢤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⢣⠘⠀⢦⠐⠻⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠃⠀⠀⢠⣴⣶⣶⣦⣴⡆⠈⠀⠈⢧⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⣿⣿⣿⣿⣽⡇
⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⢀⣼⠎⠀⣷⠀⠀⠻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⡾⠟⢀⣀⢀⣴⠀⢲⣿⣿⡏⠽⡅⠀⠀⠀⠀⠀⠠⣀⠀⠠
⠀⠀⠀⠀⠀⠀⡀⠁⠄⠀⠀⣸⣃⣰⡆⢩⠟⠙⣇⡁⠻⣯⡄⢤⠑⠢⠀⠀⠀⡀⣨⣹⣧⠀⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⡇⣸⠅⡀⡷⣿⠏⠃⢠⣏⣤⣄⠶⠆⣰⡤⠀⠘⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⢻⡄⠀⠀⠾⣷⣷⡶⠀⠀⣾⣿⡉⠀⠀⠀⠧⠙⠛⡿⠀⣾⠀⠀⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⣧⣸⣷⣤⢐⣿⣿⣇⡀⢸⡟⣳⡄⠸⣿⠀⠄⠄⢨⡍⠃⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣼⡦⠉⠏⠀⠀⠘⣿⣛⣷⣴⣹⠟⠳⡀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠉⠃⠀⠀⠀⠀⣀⣿⣟⣿⣿⣀⡋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠾⠻⠿⡿⠛⠛⠿⠿⠟⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⢀⠀⢓⡆⢀⠀⠀⠀⠠⣀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠉⠁⠠⠀⠈⢀⡿⢧⡈⠳⠀⠀⠀⠈⠁⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⣿⡿⣿⣿⠿⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⢀⠖⠀⠀⠀⣠⡔⠇⠄⠀⣠⡴⢿⣷⣿⣿⠠⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⠸⡷⠀⡀⢸⣿⠀⢰⡾⠁⠋⠁⠘⠀⠛⠿⠂⣾⠄⠹⠃⠀⣤⣴⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣳⣷⢸⣇⣼⣿⣿⠃⣰⡆⣇⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⣀⠛⣼⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠘⠛⢾⣿⡛⢹⣿⣶⣿⡧⠀⠀⣀⢀⣀⡤⠏⠀⢠⣤⡀⢹⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⢙⠋⠉⢡⡄⢀⠈⠉⠙⠀⠀⢀⡈⠉⠉⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣿⠅⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⣿⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣩⣽⡶⠒⣀⠀⠀⠀⠀⠀⠀⠀⠐⢛⠛⣧⣀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡉⢽⠼⣦⠅⠀⠀⠀⠀⠀⠀⢀⢀⣈⢿⣿⡯⣿⡟⠀⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⠒⢻⡍⢠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⠞⣀⠙⣶⠋⢴⡌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⠀⠀⠀⠚⠂⠀⠈⠄⠀⠀⠘⡇⢰⣿⡀⡏
⠀⠀⠀⠀⠀⠀⢀⡶⣄⡀⠀⠀⣴⣲⣌⠇⡸⢄⣀⢠⣷⣿⡆⣖⠀⠀⠀⠈⡟⢿⣿⣤⡇
⠀⠀⠀⠀⠀⠀⢸⣷⡿⢷⡄⠀⣿⣿⣿⣇⠳⠚⢿⣿⣿⣿⣿⢧⡀⠀⠀⢰⣾⣿⣿⣿⡯⣶
⠀⠀⠀⠀⠀⠀⢸⠏⣱⣾⣷⣴⣿⡿⣿⣏⣵⢶⣾⣿⣿⣿⢏⢘⣧⣤⣤⣿⣿⣿⣿⣿⡟⠀⠁
⠀⠀⠀⠀⠀⠀⠈⣤⡿⠿⣿⣣⢸⡇⣿⣏⣿⡿⣿⣎⣫⣻⣷⣟⣿⣻⡿⢿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠰⠆⠽⠿⠿⠿⠷⠷⠤⠭⠤⠻⠵⠏⠉⠙⣻⣿⢿⣷⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⣀⣤⣴⣿⡿⡱⠶⠶⢶⢶⡒⡖⢢⡀⠴⠔⠒⡴⡶⡾⠿⢿⠿⡾⡿⠿⣿⣿⠿⣻⣿⣿⣷⣦⡀
⠀⠀⣼⣿⣿⣿⡏⣌⠀⠁⢰⢃⢂⠀⢀⠁⢀⢸⢸⠇⡇⡓⢣⢴⠷⢿⣷⠱⢳⢱⣷⡕⡹⣹⣿⣿⣿⣿⡄
⠀⢸⣿⣿⣿⣿⢹⢸⠈⠀⠀⠀⠈⠀⢸⠘⠈⢸⠸⠀⠇⠘⢸⠠⢸⠈⡇⣦⣼⠈⣹⣧⣵⡧⣿⣿⣿⣿⣷
⠀⢸⣿⣿⢻⡿⡘⠘⠀⠀⡜⠊⡈⡰⠘⠀⡈⢰⢰⡄⡇⣶⣸⣰⣾⣴⣷⣯⣧⣼⣿⣾⣿⣿⣿⣿⣿⣿⣇
⠀⠀⣿⣿⣏⣿⡀⠀⠃⠀⠰⢓⢁⠁⠆⠆⡇⢸⡾⡇⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⡟⠐
⠀⠀⠈⢿⣿⣿⣇⢃⠀⠀⠀⡜⡌⡸⠰⠰⢡⢸⡇⡗⣷⣿⡇⣿⣏⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣟⠛
⠀⠀⠀⠈⢻⣿⣟⣭⢂⠀⠀⢠⢁⠀⡁⠀⢸⠸⡀⡇⢰⢸⠙⠛⠟⣻⢹⠓⠋⣼⣿⣿⣿⣿⣿⣿⣿⣬⣄⠀⢀
⠀⠀⠀⠀⠈⣿⣿⣷⠂⡌⠀⠈⡎⠀⠇⠆⠀⠀⡇⠇⢸⢸⣇⠄⡦⡏⡾⡘⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⢀
⠀⠀⠀⠀⠀⠹⢿⣿⡇⠀⠀⠁⡁⠀⠀⠁⠀⡅⡇⠃⢸⣹⡟⢰⣧⠃⡅⠀⣽⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢀⠰⣄⠘⠿⠃⠲⢦⡀⠤⠘⠘⠀⠀⠇⠇⠀⠸⠉⠶⠸⠋⣃⣦⣦⣿⣷⡿⢷⣺⣿⣽⡿⣿⡿⣝⣿⢇
⠀⠀⠖⠲⣾⣧⣄⣉⠛⠐⠚⠀⠀⠀⠁⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠘⠸⠷⠳⢿⠟⠯⣉⣭⣵⣾⣿⣟⣿⡟⠋
⠀⠀⠀⠀⡷⢿⣿⡏⠙⠘⠛⠒⠦⠖⠦⠤⠤⢤⡤⡤⠤⢠⠄⠤⠒⠶⠶⠻⢷⣶⣿⣿⣿⣿⣾⣿⣿⣿⣛⣹⣶
⠀⠀⠀⣶⣵⣤⣙⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄⠀⠠⠀⠀⠂⠀⠄⠀⠂⠀⠒⠀⣿⣿⣇⣭⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠨⠼⠾⠿⠷⠶⠷⣤⣤⣬⣤⣤⣤⣠⣀⣤⣠⣠⣄⣠⣀⣀⣤⣤⣾⡖⣶⡉⢿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠉⠉⠉⣫⡍⠉⠉⣉⠉⢩⠉⠉⡉⠉⠘⠛⠉⠀⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⢲⣠⣄⠀⠀⣠⠔⠚⠉⠉⠉⠉⠉⠉⠓⠢⣄⡴⠚⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⣀⣀⣇⠀⠸⠃⡼⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢸⣃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢳⠘⣇⠈⠻⠀⠠⣞⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⠃⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⡼⠀⠈⠑⠦⢄⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⣀⣹⣾⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠉⠉⠙⡖⢒⣲⣤⣳⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⣶⣽⡃⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣤⣤⣤⣤⣏⣻⣬⣿⣿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⢿⣿⣿⣶⣋⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⢋⣉⣿⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠛⢋⣿⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠿⠛⣿⠀⠀⠀⠈⡇⠀⠀⠀⠘⠉⠓⣀⣤⣄⣀⠘⠉⠃⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⠶⠟⣿⠀⠀⠀⠀⢻⡀⠀⠀⠀⣠⠋⢵⠉⠉⣻⠑⣄⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣤⠶⠺⡿⠀⠀⠀⠀⠀⡽⢲⡦⠤⢧⢤⣬⣙⣫⣥⡤⠽⠤⠴⣖⠚⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣄⡴⠳⢤⣀⣀⣀⣸⣇⡹⠟⠰⠂⠾⠗⠰⠆⠺⠷⠐⠂⠘⢛⣀⡽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠉⢹⠋⣉⠙⡟⠛⠛⠋⠉⠉⠉⠉⠉⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⡶⠀⢠⠏⣤⡄⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡴⠁⣄⠉⣱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⣿⣄⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⠚⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⡠⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢸⡍⠉⠙⠛⠉⠉⢹⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠒⠺⡇⠀⠀⠀⠀⠀⡼⠒⠒⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡤⣄⡽⠀⠀⠀⠀⠰⣇⣠⣄⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠾⠻⠋⠙⠟⠋⠿⣳⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⢾⣿⡿⢷⣘⣡⣦⢁⣦⣁⡒⠆⠈⢍⣹⠛⣻⢿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⢀⠀⢒⣴⢖⣢⣖⢺⣉⠀⠀⣿⡟⠜⠁⢀⠀⠀⠞⣳⣾⣿⣿⣾⣍⠛⣻⣿⣦⡀
⠀⠀⠀⠀⢀⣴⣿⠀⣸⠗⠋⠁⠀⠉⠛⠒⠚⠀⡃⠀⠀⠀⣾⠀⣦⣾⣿⣿⣿⣿⡤⠛⣬⣝⢿⣿⣦⡀
⠀⠀⠀⣠⣿⡿⠋⠈⠑⣆⠀⠀⠀⢠⡀⠀⠀⠠⣤⣄⢀⣴⡿⠰⢏⣻⣯⣍⣿⣿⣿⣾⣿⣿⣷⣽⣿⣿⣄
⠀⠀⣰⠟⠁⠀⠀⠀⠀⠉⠆⠀⠠⠸⠅⠀⠀⠀⡘⣿⡿⢋⣴⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣮⢿⣿⣆
⠀⣰⡿⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⠀⠀⠀⠀⢤⢸⣿⡃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⢿⣆⢮⡻⣿⣆
⠀⠋⠀⠀⠀⠀⠀⠀⢀⣈⠁⢈⣁⠀⠀⠀⠀⠀⠘⢿⣷⡌⢹⣿⣿⣡⣹⣿⣿⣿⣿⣿⡟⠉⣝⣿⣿⠘⢳⣎⢻
⣀⠀⠀⠀⠀⠀⠀⢰⡇⠈⠀⠀⠁⠀⣦⠀⠀⡄⠀⠈⠛⠁⠈⠛⠛⠻⠶⠉⠉⢿⡿⠛⠇⠀⠟⠛⠿⢀⠀⠹⣇
⣿⣷⣌⣣⡀⠀⡀⠘⠇⢠⡆⠀⠀⠀⠘⠠⢄⡟⠁⠀⠀⠀⠠⠤⠤⠤⠴⠶⠟⣦⡤⠠⠌⠶⠜⠀⠀⠘⣢⠀⢻
⣿⢿⢽⣿⣿⣶⣧⠀⠀⣸⣇⠀⠀⠀⠀⠤⣬⡄⠈⠑⠄⠀⠀⢤⣤⡄⠀⠆⠆⣹⣲⣊⢀⣠⠟⠁⠀⠀⣩
⢿⠿⠟⡿⠻⠛⠈⢿⣿⣟⣹⣷⣤⡘⢆⠐⣋⠀⡀⠀⡄⠀⠀⠀⠈⡍⠒⠂⠀⠀⠘⠛⠉⢀⣀⡀⠀⢀⡬
⠰⠊⠹⠐⢦⡚⢀⣰⠍⠂⠙⠠⠄⡀⢌⠁⢉⠁⣄⣈⠁⠐⠦⠐⣿⡿⠒⠂⠀⠉⠈⠀⣰⣶⡿⣵⠖⠖⠣⢄⣸
⠀⠀⠀⢣⠉⠁⡄⠀⠀⢀⡀⣾⣯⡏⢁⠰⣔⣦⠀⢈⣼⠻⠷⡆⠀⢃⠀⠀⢀⡀⠀⠀⣃⣛⣸⡏⣶⣿⣿⣿⣿
⠀⠀⠀⠈⢆⠀⠃⡰⠞⡟⢦⣀⠀⠀⠀⠠⣧⡿⢀⣨⠭⣅⣶⠐⠧⢾⣴⢶⡟⡆⢠⣂⣈⣁⣩⣃⢻⢏⣿⣿⠇
⠀⠀⠀⠀⠈⢣⠀⢇⡰⠊⠦⡀⠀⠀⠀⠓⠾⠀⣾⣿⡀⣿⠏⠀⠀⠀⠀⣀⠁⢮⣻⢉⣩⣭⣟⣰⡌⢈⢿⠏
⠀⠀⠀⠀⠀⠀⠹⢼⠇⣲⠄⣨⠳⣄⠈⠑⠲⠴⢹⡗⡁⣿⠀⡀⠀⢀⠸⠉⠁⠀⠉⠩⠬⠞⣿⣥⠵⠊
⠀⠀⠀⠀⠀⠀⠀⠘⠰⡶⣟⠻⣀⠈⠢⠀⢤⠀⠙⠀⢃⡸⠀⠀⠀⣎⡤⣤⠶⠛⠐⣢⣄⣠⣶⠒⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠺⠗⠧⣀⠀⠀⠀⠀⠀⣻⠿⠄⠀⠀⠚⠒⠒⠉⣀⠤⠒⠉⠓⠗⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠿⢸⡗⣦⢰⡤⢸⠀⡖⠆⢺⠂⡶⢲⠰⡄⠴⢢⠀⡖⠆⢸⠂⡇⢰⠖⡇⣰⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡶⢸⡇⣿⢸⠇⢸⠀⣍⡷⢸⡀⡇⢸⠀⡇⢾⣹⠀⣍⡧⢸⡀⡷⢸⣀⡇⢻⣡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣶⣄⡄⠀⠀⢀⣤⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠜⠳⠀⠀⠀⠀⠀⣠⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⣀⡃⠰⡉⡇⠀⢠⡿⠿⠿⢿⣽⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠰⠞⢽⡷⣕⠝⠀⠀⠸⣗⠡⠌⠩⠉⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⠀⠀⠀⠀⠀⠙⢿⣿⣿⠿⠁⠀⠀⠀⠀⠰⠞⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠬⠤⢤⡄⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡇⠀⠒⠀⠐⠒⠀⡇⠀⠀⠀⠶⠀⠐⢶⠀⠀⠀⠀⢀⠓⢹⣟⠙⢁⠀⠀⠀⠀⠀⣰⣶⡀
⠀⠀⠀⠀⠀⠀⡇⠈⠐⠀⠈⠁⠀⡇⠀⠀⣄⠠⣟⣿⣄⣄⠀⠀⠀⠈⠉⢹⣗⠉⠁⠀⠀⠀⠀⢠⣿⣿⣧
⠀⠀⠀⠀⠀⠀⡇⡃⡆⡇⢰⡆⠅⡇⠀⠀⠉⠀⣨⡅⠀⠉⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠾⢿⣿⠿⠆
⠀⠀⠀⠀⠀⠀⠳⠭⠤⠤⠤⠤⠴⠇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠅⠈⣿⠀⠈⢨⠀⠉⠀⠀⠀⠩⠀⠁⢸⡇⠉⠀⠁⠀⡅⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣶⣶⣿⣶⣶⣶⣶⣾⣤⣶⣶⣶⣶⣶⣾⣷⣷⣶⣷⣶⣷⣾⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡖⠲⠄⠀⠀⠀⠀⠀⠢⣤⡾⠒⣇⠒⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠛⠀⠶⠆⠆⠀⠀⠀⠀⠀⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⠁⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡶⢀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⢀⡀⠤⠚⠀⠣⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⣤⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣧⣹⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠸⣹⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣄⡀⠀⠀⠀⠉⢹⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠺⠛⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠿⠓⠲⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠨⢨⢤⠁⠀⠀⠀⠠⠾⠃⠀⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢸⣅⠔⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣾⣥⣀⣀⣿⡇⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣿⣿⣿⠻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⡀⠀⠀⢸⠀⠀⠘⢿⣦⡤⠂⠀⠀⢀⠀⠀⠸⠃⡟⡟⢸⠀⠛⠀⠀⠀⠀⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢙⢙⡉⡋⠀⠀⢸⡀⠀⠀⠈⠉⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰⡖⠁⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢴⡆⠃⠀⠀⢸⠀⠀⠀⠀⢤⠀⠀⠀⢰⣿⠀⠀⣠⣤⠀⠀⠀⠀⠘⠋⠉⢹⡇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢸⠀⠀⠀⢀⣸⡄⢀⣴⣿⣿⡆⢠⣿⡄⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⠀⠀⠀⢸⡀⠀⠹⣿⣿⠏⢸⣿⣿⣿⡿⣿⣿⣿⠀⠀⠀⠸⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⢾⡷⠀⠀⠀⠀⠀⠀⣟⠋⢻⠀⠉⠻⠁⠀⠀⠀⠀⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⢀⣶⣶⣄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⣻⡿⠀⠀⠀⢠⠄⢀⡄⢸⡋⣴⢰⣯⣤⣄⢠⡟⠉⠙⠿⣿⣷⡶⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢀⣀⣼⣤⣼⣶⣾⣷⣿⣾⣷⠙⣿⣿⡇⠀⠀⠞⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡧⠀⠀⠀⠀⠀⠈⠛⠻⡿⠟⠛⠛⠉⠙⠛⠅⠙⠛⠏

⠀⠀⠀⠀⣯⠂⠀⠀⠀⢀⠀⢀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠰⠀⢀
⠀⠀⠀⢐⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠂⠀⠀⠀⣦⣼⣤⡄⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣔⠀⠤⠀⢀⢁⠋⠀⠌⠊
⠀⠀⢀⠀⠀⠀⢀⠀⢀⠀⠀⡀⠀⠀⠀⠃⠀⠀⠐⠀⠀⠰⠐⢶⠄⣇⠀⠀⠀⠄⢨⡑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⢀⠀⠀⠈⣀⠄⢀⠀⢂
⠀⠀⠀⠁⠀⠀⠁⠀⠀⠐⠀⠀⢆⣄⠁⠀⠂⠀⠀⠀⠀⠀⡀⢈⡁⠀⠀⠐⠁⠀⠀⢁⠠⠀⠀⠀⠀⡗⠦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⡀⡀⠀⠁⠀⠀⠁⠈⠐⠛⠤⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⠊⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢰
⠀⠀⢀⠀⡀⠀⠀⠀⠠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠀⠀⠱⡄⠀⠀⠸⣉⡇⠀⠀⠀⠀⠱⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠘⠀⠁⠠⠀⠐⠀⠌⠁⠁⠀⠀⡠⡆⡀⠀⠀⠈⠇⠀⠀⠀⠃⠀⠠⠀⠀⠀⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢬⠓⣥⣧⡕⢫⠀⠀⠀⢀⠀⠀⢠⠖⠁⠑⢆⠀⠀⠊⠉⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⠺⣧⣤⡟⢐⠛⠀⠀⠉⠀⠀⢇⠀⠀⠀⢨⠀⠀⠀⠀⠀⢀⠼⠀⠀⠀⣀⠀⠀⠠⠇⠀⠀⣀⠀⠈⠠⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠄⡇⠌⠳⠀⠀⡀⠀⠀⠀⠈⠳⠀⠔⠋⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⠁⡄⣆⣆⢣⡈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠠⠤⡄⠭⢰⡏⣛⣗⠬⠀⡦⠄⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⡟⡛⡱⠍⠀⠀⠀⢀
⠀⠀⠀⠔⢻⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⡾⠁⠀⠀⠀⠀⠐⠂⠀⠀⠘⠃⠈⢠⡅⠁⠈⠳⠀⠀⠜⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⢀⠀⠀⡀⠀⠀⢄⠀⠁⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⣀⠀⠰⠀⢐⠀⠀⠀⠀⠀⠀⡈⢇⣀⠀⠀⠀⠀⠀⣦⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⢀⡨⣩⠕⢭⠭⡀⠀⠀⠈⠁⠀⠀⠈⠓⢂⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠆⠰⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠈⢐⣙⡦⣚⣒⠉⠀⠀⢀⣠⣥⣀⠀⠀⠀⠈⠁⠀⠀⠀⠸⠖⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠉⠀⠰⠀⠈⠀⠀⠀⠈⣿⠜⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡞⠀⠀⢈⡟⠒⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠈⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⣄⠀⠀⢀⡄⠀⠀⠀⠀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⣯⠀⣿⠸⣍⢸⠀⠀⠀⢸⣤⣦⣼⢡⢿⣆⢸⣧⣷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠷⠃⠻⠶⠳⠳⠞⠹⠀⠀⠀⠀⠟⠙⠟⠊⠉⠻⠸⠘⠇⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⠖⣀⣀⠀⠀⢠⡶⢶⡄⣀⣀⡀⡂⣀⢀⢰⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢼⠁⢸⠇⠀⢸⣇⣠⠟⡷⣶⠋⡇⡿⣾⢸⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⠉⠀⠀⠈⠉⠁⠀⠁⠈⠁⠉⠁⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⡇⡴⠶⣍⡏⠁⠀⢠⡞⢃⡄⢠⢠⠟⠙⡇⣤⠆⡼⠉⢳⣤⠀⠀⢀⣤⡉⣿⣵⠶
⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⠿⣅⣠⠏⡇⠀⠀⠘⣷⡮⡟⢻⠸⣄⣴⠃⣿⡤⢧⣠⡞⢸⣄⡀⠞⠺⣷⣿⣿⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⢀⣀⣀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⣆⢸⣷⡧⣶⠙⢷⠀⠀⠸⣆⣦⣸⢃⣾⣦⢸⣰⡟⢻⡤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠻⠸⠘⠇⠿⠶⠋⠀⠀⠀⠿⠙⠿⠘⠉⠻⠼⠙⠷⠺⠀⠛
⠀⠀⠀⠀⠀⢠⡤⢀⠀⡀⡤⠶⣤⠀⢀⡶⡶⣆⢀⡀⠀⣴⡀⠀⣀⠀⠀⢠⣄⣴⠀⣀⣀⢰⡀⠀⡄⢠⠶⠆⢀⡀⣴
⠀⠀⠀⠀⠀⣿⣀⣿⠴⡇⣷⣞⠁⣆⠳⢄⡇⡿⣾⡇⣸⠽⣧⡘⠤⡀⠀⢸⠿⢿⣼⠁⢨⡟⣧⡼⢰⣾⠟⠃⡳⢄⡿
⠀⠀⠀⠀⠀⠘⠋⠙⠀⠁⠉⠈⠁⠋⠙⠋⠉⠃⠀⠃⠀⠀⠙⠙⠛⠁⠀⠘⠀⠘⠈⠓⠋⠀⠙⠁⠈⠉⠛⠃⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⢀⣘⣛⣂⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢁⡿⠄⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⣰⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀
⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣹⢸⡟⣇⠀⠀⠀⠀⠀⠀⡀⠀⠉⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⡍⠰⠀⠀⠀⠀⠀⠴⣠⠄⡧⣠⠀⠀⠀⠀⣰⢻⡇⣿⢸⡇⢹⡄⠀⠀⠀⠐⠀⠀⠀⠆⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣬⠭⠵⣫⡴⣿⡂⠀⠀⣠⠇⡾⢸⣿⢸⣿⠈⢷⡀⠀⠀⠐⠀⠁⠀⠖⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⠥⠞⠑⠃⠀⠀⠉⠈⠁⠉⠉⠈⠉⠀⠈⠁
⠀⠀⠀⢰
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠸
⠀⠀⠀⢸
⠀⠀⠀⠈⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⠇⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⢽⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠆⠃⠐⢩⢔⠀⠀⠀⠀⠀⠚⠥⠀⠀⠀⠀⠀⠀⢠⢋⡂⠀⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⡔⠃⢒⡀⠀⠀⠀⠀⡈⠉⠭⠂⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠐⠠⠠
⠀⠀⠀⠀⠄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠃⠀⡈⠀⠰⠣⠀⠀⡀⠭⠭⠭⠙⠀⠀⠀⠀⢸⠗⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠄⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠋⠉⠌⠄⡠⠀⠀⢑⣒⣒⢒⣒⣲⠄⠀⠀⠍⠀⠀⠀⠀⡄⠀⠀⡄⠀⡀
⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠌⠌⠀⠀⠀⠀⠀⠀⠸⠡⠤⠬⡭⠥⠤⠄⠀⠐⠦⠀⢀⠀⠀⠀⠀⠀⠀⠁⣀⡐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣴⡅⠐⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⢾⣇⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣠⠈⡄⠘⠂⠀⣀⡤⠒⠊⠉⠉⠀⠀⠀⠀⢘⣟⠉⠉⠑⠒⢴⢅⠀⠈⠻⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡄⠀⠀⠀⢀⡠⠒⠉⢹⣵⣦⡀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠇⠀⠉⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡀⣀⠔⠉⠀⠀⠀⠘⣿⣿⣷⠀⠀⠀⠀⠀⠀⢿⡿⠃⠀⠀⠸⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠽⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⢲⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠃⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡶⢮⠤⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢠⡄⠙⠀⠀⠀⡴⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡏⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠅
⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣧⣤⣷⣤⢠⡶⢉⣥⠀⢸⣧⠁⢸⣇⠜⠀⣾⡆⢰⣦⠀⣶⡄⢀⣴⠋⣡⠀⢠⠓⣶⠀⢸⡿⣂⡷
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⡏⠉⡏⠁⠻⠷⠊⠻⠦⠜⠻⠦⠔⠻⠶⠊⠘⠿⠃⠻⠟⠙⠷⠚⠿⠖⠻⠷⠊⠀⠻⠦⢺⡿⣅⣀⠀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡇⠀⠀⢠⡄⢰⡦⢠⡄⣴⠤⡀⢠⢀⡤⣄⢤⣤⠤⢴⣦⠀⣴⡀⢀⣦⠀⣤⢤⡀⠀⠁⠈⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣿⠀⢸⡇⢸⡇⣿⡞⠁⢸⠘⣇⠹⠀⣿⠀⢸⡏⠛⢸⡇⣼⢿⡀⣿⠸⠇⠀⠀⠀⠰⣦⣤⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⠀⣿⠹⣄⣸⠀⢹⡀⠀⣿⠀⢸⡇⠀⢸⡇⡷⢺⡇⠘⡇⠀⠀⠀⣤⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⣀⠁⠤⠀⠈⠐⠘⠇⠈⠀⠛⠀⠀⠘⠀⠻⠃⠀⠟⠀⠸⠃⠀⠘⠙⠃⠘⠇⠳⢏⣤⣤⡀⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡛⡁⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣤⣾⣷⣄⠀⠀⣸⣿⢿⣃⡉⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡟⠉⠸⣿⡿⠺⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⠏⠉⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣼⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣨⣿⠋⣥⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⣿⣿⣿⣷⣫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡈⣻⣿⡟⠛⠫⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡾⢿⡏⢙⣧⣀⣴⠛⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡀⢀⣿⠟⢿⡟⠛⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⡿⢿⣿⣾⣿⡿⠿⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣏⠀⠀⣿⣄⣼⣇⠀⣀⡟⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠉⢹⣿⡿⣿⡿⠛⠻⣿⣿⠿⢿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣦⣤⣼⣷⣴⣿⣇⠀⣀⣿⣧⣤⣾⣭⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡻⠛⣿⣿⠛⠛⢻⣿⠟⠿⣿⣧⣽⣿⡏⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⣴⣿⣿⣄⣀⣸⣿⣤⣤⣿⡟⠉⠹⣿⡿⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡿⠛⢿⣿⡿⢻⣿⣿⣿⣧⣤⣶⠿⠿⢿⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣼⣷⣤⣼⣿⣿⣿⣿⣧⣀⣿⣿⣏⡀⠀⣼⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣟⠋⢻⣿⢉⣻⡏⠀⠈⢻⣿⣿⠋⢹⣿⣿⣾⣿⠟⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⡿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣾⣿⣿⣿⣿⣿⣤⣾⣿⣦⣶⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣷⣍⣀⣸⣿⠁⠀⢹⣿⠛⠛⣿⣿⣥⣿⡟⠁⠉⢿⣏⠁⠹⣟⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠭⣿⣿⣿⣿⣦⣴⡿⢻⣶⣾⣿⠋⢹⣿⣧⣄⣠⣿⣿⣿⣿⣿⣿⣿⣟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠛⣛⣿⣿⠿⢿⣶⣾⠿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠁⠀⠀⠉⠁⠀⠈⠉⠉⠉⠉⠁⠉⢫⠝⠛⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⠀⠀⠀⠀⠄⠀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠐⠤⠀⠤⣄⠠⠀⠀⠉⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁

⣿⢿⠿⠿⡿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⣴⠋⠀⠀⠀⣀⠀⠈⢲⡀⠈⣧⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⢸⠀⠀⡈⠉⠛⠻⡿⣶⣿⣌⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⣸⡄⠀⠳⣄⠀⣸⠇⢸⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢶⣦⡼⣿⣯⣴⣟⣡⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡐⡆⢀⣆⢀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣉⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣄⣛⣿⣿⣿⣾⣦⣶⣶⣾⣿⠇⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⠟⠋⠉⢿⣿⣿⣿⠃⠀⠀⠀⠀⣾⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠻⣿⣿⣷⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠸⣶⣶⣤⣤⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⡇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣷⡆⠀⠈⣿⡿⠻⢿⣿⠟⠛⠛⠛⢻⣿⡟⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠉⠛⠛⠉⢹⣿⡏⣠⣾⡿⠃⠀⠀⣿⠃⠀⢸⣿⠀⠀⠀⠀⠀⢻⣿⠀⠻⣿⠄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢿⣇⠀⠀⠀⠀⢸⣿⡿⠋⠁⠀⠀⠀⠀⣿⠀⠀⠘⣿⠀⠀⠀⠀⢀⡾⠁⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠈⢿⣄⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⣿⠀⠀⠀⢀⡾⠁⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠘⠿⠆⠀⠀⠰⣿⡀⠀⠀⠀⠀⠀⣼⠇⠀⠀⢠⣿⠀⠀⢠⣿⠃⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡿⣋⣴⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣾⣿⣿⣿⡿⠋⣁⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⢟⣩⣶⣿⠟⣉⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⣫⣾⣿⠞⣫⠿⣫⣴⡿⢛⣡⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⣉⣴⣿⣿⡿⢟⣭⣶⡿⠋⣀⣴⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⢉⣴⣾⣿⣿⣿⡿⢞⡿⠟⣩⣴⣿⠿⠋⣀⣴⣶⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠾⢛⣽⣾⠿⢛⣥⢞⣫⣶⣿⠿⢋⣡⣶⡿⠿⠿⠿⡿⠛⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢋⣤⣾⣿⡿⣿⣿⣿⣵⣾⠿⣋⣥⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠿⣋⣵⣾⠟⣫⣿⣿⣵⡾⠛⠉⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠟⢉⣴⡾⠟⣩⣴⣿⣿⠿⠛⠁⣀⡴⠞⢋⣡⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠋⢁⣤⡾⠛⣉⣴⣾⣿⠿⠋⢁⣠⡶⠟⣉⣤⣾⡿⠟⣁⣤⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⢁⣴⣾⣿⠿⠋⣁⣴⠾⢛⣡⡴⢛⣽⢞⣫⣶⣿⣿⣿⣿⢛⣿⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⣤⣾⣿⠿⢋⣡⣶⠟⣋⣥⠾⣋⣵⣾⣿⣿⣿⣿⣿⣿⣿⡟⠋⡛⠛⠃⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣋⣥⡾⠟⣫⡴⠞⣉⣴⣾⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣾⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣵⡾⢛⣭⠶⠛⣡⣴⣿⣿⣿⣿⣿⡿⣟⣯⣷⣿⣿⣿⣿⣿⡿⣟⣽⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⣛⣵⠞⢋⣠⣶⣿⣿⣿⣿⣿⢟⣫⣵⣾⣿⣿⣿⣿⣿⢟⣫⣵⣿⣿⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠛⢉⣠⣶⣿⣿⣿⣿⣿⢟⣭⣶⣿⣿⣿⣿⣿⡿⢛⣩⣶⣿⣿⣿⡿⣿⣿⣿⣿⠿⣿⡇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⢁⣠⣶⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣿⣿⣿⠟⣋⣵⣾⣿⣿⣿⠿⣛⣭⣾⣿⠿⢛⣥⢞⣭⠶⣿⣅
⠀⠀⠀⠀⠀⠀⠸⠟⠉⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣾⣿⡿⢛⣩⣴⣿⣿⣿⣿⠟⣫⣵⣾⣿⠟⢋⣡⣾⣿⣿⣿⣿⣿⡿⠛⠿⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣟⣯⣷⣿⣿⠿⣋⣥⣶⣿⣿⣿⡿⢟⣫⣶⣿⡿⠟⢋⣤⣾⣿⣿⣿⣿⣿⣿⠿⠋⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣫⣴⣾⣿⣿⣿⡿⣛⣽⣾⣿⡿⠛⣉⣴⣾⣿⣿⣿⣿⣿⣿⡿⣛⣥⣾⣿⣿⣿⣯⡀
⠀⠀⠀⢀⣠⡶⣛⣿⣿⣿⣿⣿⣿⣿⡿⢟⣩⣶⣿⣿⣿⣿⡿⣻⣵⣾⣿⠿⢋⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⢋⣤⠀⣠⡄
⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣷⣿⡿⠟⢋⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⣿⣿⣾⣯⣵⡾⣋⣤⡶
⣸⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⣿⣯⣷⣿⣿⣿⣿⣿⣿⠿⠛⣁⣤⣾⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⣵⣾⣿⣿⠟⣫⣿⣿⣿⣿⣻⡷⠻⣆⣠
⠉⠛⠉⠛⠁⠉⠁⠀⠀⠘⠋⢩⣽⣿⣿⣿⡿⠟⢋⣡⣴⣿⣿⣿⣿⣿⣿⠿⢛⣩⣾⣿⣿⣿⣿⣿⣿⣿⠻⣿⣛⣠⣾⠟⣉⣴⣾⣿⣿⣿⢛⡿⠛⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠟⠉⠀⠘⢋⡽⢿⣿⡿⠿⠛⢋⣡⣶⣿⣿⣿⠟⠉⠉⠋⠙⠛⠉⠃⠺⠏⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⣿⣿⠿⣿⠟⠁⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣡⣞⣥⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢋⡵⢋⣤⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠞⠋⣼⣷⡄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣻⣿⣿⡿⣿⣿⢻⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡶⠿⠿⠿⠿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣵⣶⣶⣾⣿⣿⣶⣶⣶⣭⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣻⣭⣭⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣭⣭⣟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠿⠿⠿⠛⠛⠛⠛⠉⠉⠉⢉⣿⣿⡉⠉⠉⠉⠛⠛⠛⠛⠿⠿⠿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠤⢄⣀⣀⣀⣀⣀⣀⣈⣻⣟⣁⣀⣀⣀⣀⣀⣀⡠⠤⠤⠖⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢀⡀⣿⣿⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢿⡿⣿⣿⣿⣟⡳⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⣯⣿⣿⣿⣿⣋⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣶⣶⣿⣿⣿⣿⣿⠿⠛⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠛⠻⣿⡿⠻⢿⣿⣅⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡿⠿⡀⣴⣿⣿⣿⣿⡟⢿⣿⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣰⣆⣡⣾⣿⣿⣿⣿⣿⣿⡟⠦⠙⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣮⣿⣿⣿⣿⣿⣿⣧⣽⣿⡛⢿⡿⣷⣶⣶⣦⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢩⣍⡉⠉⢠⣷⣬⣾⣿⣾⣿⣤⣦⣿⣿⣿⣯⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣽⢿⢷⣈⣤⣽⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⣶⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣾⣿⣿⢿⠿⡟⢷⣾⢿⣿⣿⣿⢿⣿⣷⣛⠿⢿⣾⣅⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⣿⢟⡀⢀⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣈⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣜⠛⠉⣱⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⡿⡏⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣝⣿⣿⣶⣿⣯⣽⣿⣿⣿⣿⠟⢿⣿⣏⣽⣿⣬⣛⣻⣿⡿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣟⣻⣿⠏⠛⠻⠇⠙⠃
⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣿⣿⠿⠻⣿⣿⣩⣿⣿⣿⣿⣿⣿⣿⡟⠫⠈⠁⣿⣿⣿⣿⣶⣿⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠛⣿⣷⣤⣼⣾⣿⣿⡿⠋⣻⣿⣿⣿⣿⣇⣰⠀⢀⣿⣿⣿⣽⠙⠿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣦⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢿⣿⡻⣿⣿⠿⢿⣿⠏⠹⣿⣿⣿⣏⠛⠻⣿⡷⠉⠙⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠙⠋⠀⠀⠉⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⠀⡀⠀⡀⠀⢀⠀⠀⠀⢠⠀⡄⢠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠁⠀⠀⠀⠁⠁⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣾⡀⣀⡖⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢿⡏⠓⢿⣀⣠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡁⣾⠛⠻⠿⣟⢆⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣁⡴⠶⠦⢭⣿⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠈⢦⣿⡿⡄⠉⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⡆⠀⠀⢯⣷⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⡇⠀⠀⢰⡏⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣴⠀⠀⠀⠀⠀⠀⠀⢈⣾⣧⡤⣠⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠻⢿⣦⣀⡀⠀⠀⢠⣺⡾⢋⣩⡾⠟⠋⠉⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣵⣶⣌⣉⣙⣓⣪⡴⣫⣶⠟⠋⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣤⣀⣤⣤⣤⣴⣶⣶⠶⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡐⠐⠟⢨⣿⣿⣿⣧⣿⢗⣤⣀⢀⢀⣄⣠⣤⣶⣿⣿⣿⡿⠿⢛⣻⣿⣿⣿⣿⠏⠁⠂⠐⠲⠝⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠿⠛⣟⢛⣏⣸⣿⣈⠉⠛⠛⠿⠻⢿⡟⣿⠟⠙⠉⠘⡃⠸⣿⣿⣿⣿⣿⣷⡄⠀⠀⢰⣌⢸⡿⢂⣀
⠀⠀⠀⢠⡄⡀⢀⠀⠀⠀⠀⠀⠐⠒⠒⠒⡻⡻⠻⠟⢻⣿⣿⣿⣿⣶⣤⣤⠇⠀⠀⠀⣧⢀⠀⠀⣥⣾⣿⣿⣿⣿⣿⣿⣷⣔⢧⣾⣿⣿⡷⠿⠿⠿
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⣀⡀⣠⣴⣿⣿⡿⢿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣿⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⠿⠛⠉⠋⣧⡤⠆⠀⠰⣦⠀⠋⣹⡿⢿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠁⠀⠀⠀⠀⢶⣤⠿⠟⠛⠛⠋⢻⣿⣿⡿⣿⣿⠟⠛⠛⠛⠛⣿⣿⡆⠀⢻⣿⣇
⠀⠀⠀⠀⠀⢠⠀⠀⠴⠦⠤⣤⣤⣤⣤⣄⠀⠈⠂⣠⠸⡧⠤⠶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⡏⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠓⢄⡀⠀⠀⠀⠀⠀⠉⠀⢹⠿⣦⣤⡶⣾⡑⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⡇⠀⢀⠀⠄⠀⠀⢸⣿⠃⠀⠸⣿⠃
⠀⠀⠀⠀⠀⠰⣄⣷⡄⠀⢀⣀⠀⠀⢀⡏⠀⣀⣼⠑⠿⣇⠀⠀⠀⠀⠀⢃⡠⠠⣤⣬⣳⠟⢿⣿⡇⢸⡃⣼⣴⣷⣶⠧⠂⢸⡏⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠠⢸⠟⠉⠙⠛⠺⢲⣿⣿⡿⣿⡿⠛⠳⣦⣨⠀⠀⠀⠛⠞⣷⣥⣾⣿⣿⣿⣤⣸⣿⢠⣼⠛⠉⠀⠀⠀⠀⠀⢻⠁⠀⠀⢀⣟⠠⠄⠀⢠⠄
⠀⠀⠀⠀⢠⠆⡇⠀⠀⠀⠀⠀⠌⠀⣿⡇⠈⠁⠀⠈⠀⠀⠄⣐⣥⣟⣆⣻⣿⠿⠛⠋⠉⠉⠈⣉⠀⢸⠀⠀⠀⢀⣀⣀⠀⣼⣤⣶⣶⠜⣿⢶⣶⣶⡆
⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠘⠇⠀⠸⡇⢲⠐⣄⡤⣀⣀⣤⠘⠫⠙⠉⠁⠀⠀⠀⠀⠀⠀⣀⢻⠀⠀⣄⣀⣤⡾⣿⣥⣍⣵⣿⣿⣯⣍⣹⡿⠟⠋
⠀⠀⠀⢀⡃⡀⠀⣀⣤⣤⣄⠀⠰⢤⣤⠳⣼⣿⠿⠷⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠶⠿⠛⠓⠲⢯⣏⣉⣁⣀⣩⠟⠂⠉⠉⠈⠉⠁
⠀⠀⠀⠀⠀⠈⠁⠉⠁⠈⠈⠀⠐⠂⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⡄⠀⢈⡔⠀⠀⢀⠀⠰⠀⠀⠐⠻⠿⠟⠋⠋⠉⠉⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠱⣿⣄⣤⣀⡀⡀⠀⠖⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠾⠚⠛⠛⠻⠿⣿⣿⣦⣦⣤⣠⣶⣶⣄⣀⣠⣄⣀⡛⢮⣀⡀⣟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⢉⡀⠀⠀⠀⠉⠉⠛⠻⣿⣿⡿⠯⠟⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠈⠁⠈⠋⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠛⠛⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠋⠛⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⣀⣠⣤⣤⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠁⠀⠈⣇⠀⠀⠀⡀⠠⠐⠂⢐⠄⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠪⣁⣀⣉⠉⠁⠀⠀⠀⢸⡀⠀⠀⢀⡟⠀⠀⠀⠀⠐⠒⢊⣁⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣤⣤⣤⣤⣤⣤⡤⠶⠚⠋⣹⠟⠛⠻⢿⡛⠛⠛⠻⢿⣉⠻⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⣠⠞⠁⠀⠀⣠⠞⠁⠀⠀⠀⠀⠉⠳⣄⠀⠀⠈⠳⣌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢠⡞⠁⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢀⣙⣎⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣯⡤⢤⠀⢠⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⣿⠀⠁⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠈⣧⣼⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⡁⢀⠀⣀⠀⠀⠀⠘⠇⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠻⣆
⠀⢀⣀⣠⡶⠶⠛⠋⠁⠀⠀⠰⠶⠀⠂⠀⠀⠠⢀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠁⠀⠀⢠⣤⠁⠒⠀⠹⣦
⢀⡾⠉⠉⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠁⠀⠀⠀⠊⠁⠀⣸⡆⠀⠀⣤⠀⡀⠀⠀⠀⠁⡀⠈⠁⠂⢠⡌⠀⠀⠂⠙⢷⣄⡀
⣼⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠮⡀⠀⢈⠀⡽⣷⠀⠀⠀⢴⠄⡉⠀⠀⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⣦⣤⣀⡀
⠈⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢹⠀⠈⠀⠂⠰⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠄⠒⠂⠈⠉⠉⠉⠉⠉⠉⠉⢻⡄
⠀⠀⠀⠈⠙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⠔⠐⠒⠒⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠒⠊⠉⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣤⣀⡀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⢀⡠⠔⠂⠉⠀⠀⠀⠀⢢⡀⠀⠀⠀⡠⠃⠀⠀⠀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⣶⠟⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠉⠀⠀⢀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣯⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣀⣀⣀⣀⣀⣀⣤⣤⡴⠶⠞⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠀⠈⠁⢀⠶⠂⠀⠀⠀⠀⠀⠀⠀⢸⣫⡆⢐⢮⡏⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠺⠂⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣤⣄⠁⠀⣠⡄⠀⠀⠀⠀⠀⣀⣤⣠⡋⠈⢩⣤⣶⣦⠄⠀⠀⠀⠀⠀⠐⠛⠢⣤⡤⠒⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⠈⠛⠚⢈⡶⠭⢶⣁⠉⠀⠀⠀⠀⠀⠀⠀⣔⡶⠊⠉⠈⠐⡶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⣠⡄⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⢼⡝⠂⠙⠾⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣺⡃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣠⢀⡘⠓⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣀⠁⡀⠀⣤⠀⠀⠀⠀⠀⠀⠰⣖⡴⠈⠪⠞⢱⢰⣲⠂
⠀⠀⠀⠀⠀⠀⠐⠷⡵⠄⡦⢱⡎⣥⢀⡴⣠⡀⠀⠀⠀⠀⠀⠀⠬⣖⣭⡲⠅⠀⠀⠀⠀⠀⠀⠀⠀⣓⠊⢑⡊⢰⠉⠐⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⠇⠻⡂⠀⠘⠈⠀⠀⠀⠀⠀⡀⠶⠀⠔⠙⠂⠀⠶⠀⠀⠀⠀⠀⠀⣠⣖⢭⠃⣨⡁⢹⣱⣲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠑⠀⠠⢻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢔⡮⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢦⣧⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢋⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡡
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠇⡰⠲⠔⢦⢸⣷⠂⠀⠀⠀⠀⠀⠛⣨⣄⠀⠭⢀⣸⡀⠓⠀⠀⠀⠀⠀⠘⣂⣅⡀⠸⠇⣀⣦⡘⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠺⣁⠀⠀⣁⡶⠈⠁⠀⠀⠀⠀⠀⢈⡀⢈⡹⣿⢎⠀⣀⡁⠀⠀⠀⠀⠀⠀⡀⢈⣉⠾⠦⣉⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠰⢿⡇⠳⢞⠦⠆⢰⠯⡄⠀⠀⠀⠀⠀⡤⢘⠇⢀⣖⠀⢸⠑⡴⠀⠀⠀⠀⠀⠠⠌⠋⠁⢸⡆⠉⠏⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⠓⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣤⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣌⣱⠎⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢤⣿⣷⡶⢀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠞⠛⢁⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣶⣾⢹⡅⣠⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢡⡿⣧⣾⣷⣤⠆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣄⡋⣀⡿⠿⠛⢠⣼⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢉⣁⣅⣠⠄⢾⠿⠾⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠏⠈⢻⣄⣘⣷⠈⠑⠲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠠⣤⠟⠁⡈⣠⡴⠋⠉⡾⠆⡷⠂⣈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢂⣾⣿⠋⡪⣴⣀⢱⡤⠶⢂⣤⣩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⣄⠴⠲⠾⠿⢃⢠⣄⣼⠛⠋⡀⠐⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄⢠⡶⢠⣀⣈⣉⠀⠴⠶⡅⠀⣀⡤⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⠘⡻⠊⣸⡿⠃⠀⠘⠋⠀⠀⡈⠁⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⡀⣤⡂⠀⠈⠀⢸⡿⢁⠠⣀⠖⠀⠀⠀⠈⠓⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠐⣿⣾⠋⠀⠉⠀⢠⠀⠀⠈⠰⠄⣠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⣄⠀⢀⠀⠄⢻⣿⣶⡀⠀⡰⢿⠀⢀⢀⠄⠂⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⠀⠈⠁⠀⠀⡰⠃⠀⠀⢀⣧⡀⠀⠀⠦⠈⡀⠰⠊⠀⠀⡀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠉⠒⠥⠰⠚⠃⠐⠒⠀⠚⠛⠟⢉⣠⠤⠊⡑⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠂⠐⠂⠀⠂⠠⣤⣦⠔⠀⠀⠹⠷⡰⠚⠓⠋⠀⠀⠀⠀⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⣤⣄⣀⠆⠚⢸⣿⠏⠀⠀⠀⣶⡆⠀⢈⡀⠀⠀⠒⢀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠄⣨⡼⠏⠀⠀⡸⢡⠀⣀⣠⠁⠀⠀⠀⠀⠘⢆⠀⢀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠁⣿⣿⣿⡇⠀⢠⡈⠀⢀⠀⠈⠡⠀⠈⠀⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠻⠝⣿⠁⠐⠈⠁⠀⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠘⠀⢀⡀⠰⠠⠀⠛⠀⣈⠀⠀⣤⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠀⠈⠀⠃⠈⢈⡁⠀⠤⠤⢠⠀⠈⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠖⣠⡄⠈⢸⠐⢈⣉⡀⡀⠠⠤⠀⠠⠀⢈⠁⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢸⠅⠀⠀⠀⠀⠀⠀⢀⡠⠀⠈⠘⠃⣠⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢨⡄⠐⣶⠠⠀⠀⢀⣸⣤⣀⡀⢽⠶⠾⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡮⠃⠀⠀⢠⡀⢰⣿⡙⢿⣿⣿⣿⣀⣼⣿⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣿⡇⠈⠛⢄⠼⢻⣿⣿⣿⠻⣜⡄⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠷⣿⠗⣼⣿⣿⣿⣿⣿⣀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠈⠀⣿⣿⣿⣿⢿⡿⣉⠗⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡀⠀⠀⢠⣷⡧⢉⡁⠀⠀⣇⣈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣜⡅⡌⢣⠀⠀⠀⠡⠝⠰⣹⣿⡿⠷⣖⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠉⣼⡆⠀⠀⠀⠀⠐⢿⡟⠀⠀⣁⡀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⢴⣤⠆⠈⢽⠁⠀⠀⠀⠓⠈⣩⠉⠀⡀⠈⠀⢸⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠶⣾⣧⣷⡄⠈⠀⣀⣀⣤⡆⠀⠈⠀⡽⢅⠀⠀⠚⡂
⠀⠀⠀⠀⠀⠀⠀⠋⠉⠒⠾⠃⠀⣿⣿⣏⠋⡀⠸⣿⣿⣾⡿⠂⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣏⡟⠠⠀⠀⠿⢿⣿⠇⠀⠀⠉⠟⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠑⠠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⠄⠄⡴⠀⠀⠀⠲⡒⠈⠀⠀⠀⠀⢤⠀⠀⠀⠀⠜⠄⡠⡄⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⡀⠀⠀⢀⡀⡊⡀⠀⠀⠀⡀⠀⣐⠆⠀⠀⠂⠠⡀⠀⠀⠘⠁⢀⠃⠉⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠎⢀⠁⠄⠰⠀⠐⠦⢰⠤⠀⠀⠀⠀⠑⠄⠀⠐⢠⣰⡄⠂⠉⠀⠘⠟⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⡂⠀⡀⢁⠈⣉⣆⣣⠄⠁⠀⠀⠀⠀⠀⠀⢀⡀⢄⡈⠱⠘⠀⠀⠀⠀⠈⢤⠀⢀⠠⠐
⠀⠀⠀⠀⠀⠀⢀⣀⣲⠠⡀⠑⠒⢈⠤⠨⠅⠒⠈⠉⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠉⡋⠙⠓⠀⠀⠀⠂⠀⠀⠀⠐⠊⠀⠀⠀⢀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠚⠉⠁⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⢠⢀⠐⡖⠐⠃⠀⠀⠀⠀⢀⣀⠀⠀⢠⠐⠁⠀⠐⠈⠀⠀⠀⠌
⠀⢀⠄⠒⠈⠉⠀⠂⢄⠀⠉⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠂⠦⡅⠀⠀⢀⡒⠀⠀⢸⣄⡄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⢀⠀⠌⢁⠁⠀⠀⠀⠑⠄⡆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡭⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀
⠠⠒⠉⠉⢉⠢⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡂⠤⢄⠀⠄
⡆⠀⠂⠀⠰⠐⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠉⠀⠘⠀⠈
⠇⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢒⠀⠁⠀⠀⠀⠀⡀⠔⠁⠀⡐
⠀⠄⡀⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⢀⡀⡀⠄⠊
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠰⢁⡄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⡀⢰⠁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠂⣀
⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠁⠉⠉
⠁⠒⠂⠁⠀⠃⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⣂⠥⣌⠂⠀⠤
⠀⠀⠀⠐⠀⠀⠊⠐⠄⣁⢀⠀⠀⠀⠒⠁⠀⠀⠀⢀⠔⠂⢀⠀⠀⣀⠤⠄⣀⡀⠤⠔⠊⠀⠀⠈⠓⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢢⠆⠀⠀⠈⠈⠣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢇⠀⠀⡤⠀⡀⠀⠀⣆⠀⠉⠉⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠘⡇⠁⠘⠳⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⣀⠀⠀⣄⠀⠸⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠐⠀⠀⠀⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡶⠁⠀⠀⡌⠀⠀⠀⠀⠀⠘⡳⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠁⠸⠞⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⢠⡇⠀⠀⠀⠀⠀⣀⢀⡁⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⡄⠣⡀⠀⢠⠔⠋⠉⠀⠉⠑⢊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⢃⠀⠙⠊⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⠀⠀⠀⠀⠀⡀⠀⢀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⠢⣀⠀⠀⢠⠁⠀⣸⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠈⢳⠒⢁⠀⠀⠟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⡄⠀⠀⢠⠀⠇⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢁⠀⠀⢸⠀⢸⠀⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠀⢸⡄⠀⠈⠀⢸⡄⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡇⠀⠘⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡧⠀⠀⠆⣀⠆⡀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠃⠀⠁⠘⣅⣰⠀⠈⠁⢸⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠋⠉⠉⠛⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠛⠉⠉⠀⠀⠀⠉⢉⡱⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠀⣤⢀⡤⠴⠛⠓⢓⠖
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠆⠙⠁⠀⠀⠁⠀⠀⠀⠀⠀⠉⠲⡄
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠤⢀⡀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠎⠀⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣄⠀⠀⠉⢶⡤⠦⠤⡿
⠀⠀⠀⠀⠀⣸⡇⠀⠀⢀⡤⢄⣀⡠⡀⠀⠸⢀⠖⠀⣐⠁⠀⠀⠀⠀⠀⢀⣠⡤⠴⠒⠛⠉⠀⠀⠙⠓⠰⢦⣽⡄
⠀⠀⠀⠀⡸⠉⢧⠀⣴⠋⠀⠠⠤⠤⡜⣉⡿⠛⢶⠋⠁⠀⠀⢀⡠⢔⣛⣩⡤⠀⡀⠀⠀⠀⢀⠔⠊⠘⠲⡀⡏
⠀⠀⠀⢰⠃⠀⠀⢳⠈⢳⠤⠤⣀⡴⠋⠀⠈⡦⠚⠀⢀⡤⠖⣡⣎⣁⡀⠀⠀⠀⢀⡷⠀⠀⢸⠀⢀⠀⠀⢰⢣⣀⣀⣀⣀
⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠚⢁⡠⠞⣯⠀⠀⠀⠀⠈⠁⠈⠛⠀⠛⠀⠀⢀⢠⡎⢁⠑⠄⢸⣀⡀⠀⠀⠉⠙⢦⡠⣄
⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⠔⠋⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⡐⢤⣀⣀⡀⠘⣆⣻⣾⣷⣷⠞⠉⠉⢢⠀⠀⠀⠈⠳⡟⢦⡤⣤⣀
⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⣠⠿⠄⠀⠀⠉⠀⠀⡀⢤⡟⠉⠈⠉⢷⠋⠉⠉⠁⠙⢦⣀⣠⡞⠀⠀⠀⠀⣠⠇⠀⢧⠀⠉⠓⢄
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢀⣠⠞⠀⠀⠀⠀⠀⢸⠏⠉⠈⠉⠓⠶⠊⠁⠀⠀⡼⠀⠀⠀⠺⣷
⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⠇⠀⠀⠀⠀⠀⠀⣠⠴⠒⠂⠀⠙⠻⣏⠔⠀⡀⠀⠀⢀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡁⠀⠀⠀⠀⠈⢷
⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡆⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢯⠳⣄⣀⣀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⣶⠖⠉
⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠳⣄⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣹⠓⠒⠐⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⢀⣤⡄⣴⣤⡴⢎⠁⣀⠨⠓⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡟
⠀⠀⡠⠒⠚⠙⠓⠦⢤⣀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠁
⠀⡼⠇⠀⠀⠀⠀⠀⠀⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⣶⡇⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠈⠲⠤⣤⣀⣠⠔⠷⠞⠀⠀⠀⠈⠙⠦⢄⣀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠺⠛⠛⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⠛⠲⢤⣀⣀⠀⠀⠀⠀⠀⠀⢾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠖⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⠀⠰⣄⡙⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⣀⣀⣀⣀⣉⣉⣛⠺⢿⣶⣤⣉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠶⠚⠋⠉⠉⠉⠉⠉⠉⠉⠛⠻⢿⣿⣿⡑⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠻⣿⣦⣀⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⡀⠀⠀⣀⣠⣤⠤⠶⠶⠶⠦⢤⣄⣀⠈⠲⠄⠘⣿⠋⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣴⠞⡉⢓⡲⡄⠀⠀⠀⢀⣔⣋⠛⠻⣦⣀⠀⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣼⢹⠀⠋⢁⣭⡙⠀⠀⠀⠘⣩⡤⡉⠃⠹⡿⣾⡙⢦⣰⣄⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⡸⡆⠠⠿⠿⡇⡀⠀⠀⠀⡟⠛⠧⡀⠀⡇⡿⢹⡄⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠽⣶⡟⠁⢀⣠⣤⡞⠀⠀⠀⢘⣦⣤⣄⡀⠈⣿⣷⠾⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⡷⠲⣏⠓⠚⠉⠀⠂⠻⢦⣤⣤⠞⠉⠐⠀⠉⠛⠋⣸⠆⠙⢯
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠹⣦⣄⠀⣀⣀⣤⣾⣋⣻⣶⣤⣀⣀⢀⣒⣾⠟⠀⠠⣌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡀⠀⠀⠉⠉⠉⠉⠀⠹⣛⣛⡭⠈⠉⠉⠉⠉⠀⠀⠀⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⢰⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣸⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡞⢻⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⢠⣆⢀⣀⣤⣦⠴⠚⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠈⠋⠁⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⢀⡼⠃⠤⣤⣙⠉⠀⡴⠆⠀⠀⠀⠀⢠⡀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⡴⠛⣀⠀⠀⠨⣥⡀⣆⠛⠀⠀⠀⢶⣦⢸⣇⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠀⡷⢛⡁⠀⠀⠁⠲⣄⣧⢚⣥⠄⠀⢢⡝⠞⣥⠆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣝⣷⠋⢐⣷⠘⣶⣤⠘⡇⣚⠅⢀⠸⣦⠙⣏⢉⣥⣴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡈⠉⠙⠳⠾⣇⡠⣤⣹⣴⡟⣡⠀⣾⠀⡀⢠⠷⠿⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⣧⡘⠖⠈⢛⣶⣭⣽⣦⣁⠀⢿⡄⢹⡿⠖⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡄⢩⣟⠛⣩⢅⡟⠁⢶⣭⣝⣿⣷⣾⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣼⠿⠃⠾⠋⠸⠁⠀⠈⠛⠻⣿⣿⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢤⣤⣶⠿⡿⢀⡿⢛⣿⣿
⠀⠀⠀⠀⠀⠀⣠⣴⢾⢟⣿⡛⢿⣿⡿⣿⢻⣶⣶⢶⠿⣻⢿⠇⣾⡽⠿⣾⣷⠿⠶⢿⣿⡟
⠀⠀⣴⣾⣿⣿⣟⣓⣛⠻⢭⡿⢿⡦⣧⣿⠧⠾⢛⣡⠾⣶⠏⡼⢡⢖⣛⢲⢬⣙⣛⣛⣿⠃
⠀⠸⠋⠉⢸⡿⢉⡵⣛⡓⢦⡹⣯⣿⠾⢷⣔⡛⠛⠒⣋⣵⣚⣷⡐⢜⡽⢢⡇⡌⣭⣿⠃
⠀⠀⠀⠀⢸⡿⡏⠸⡩⠿⢢⠇⣿⣿⠻⣳⣴⣶⡦⣤⣤⣔⠿⣿⡷⣔⢶⡟⣇⣿⣼⡇
⠀⠀⠀⠀⠘⣇⡇⢸⣬⢍⣭⣾⣿⣿⣿⣿⣼⣿⣿⡌⣧⡙⣷⣬⣿⢾⣇⣦⣿⣿⣿⣅
⠀⠀⠀⠀⢰⣿⢃⣇⣹⣶⣿⣿⣿⠃⠉⠙⠻⠿⠿⠿⠿⠿⠿⠛⢿⣿⣼⣿⣿⢿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠈⠉⠛⠛⢿⣿⡇
⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⡿⠃⠸⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⢸⡟⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⣿⠃⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⣶⡿⠃
⢸⣿⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠉
⠸⣿⡆⠀⠀⠀⠀⠀⠹⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠛⣗⢲⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠻⢇⡿⠉⠑⠝⣆
⠀⠀⢤⣀⡀⠀⠀⠤⢤⣄⣀⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⢰⠆⠀⡗⠠⡀⠀⠈⠑⢤⡀
⠀⠀⠀⢻⣯⠳⣄⠀⠀⢀⠀⠈⢿⣿⣿⣿⣿⣄⠀⠀⠀⠀⢸⠀⠀⡇⠀⠈⢷⢄⠀⠀⢉⢢
⠀⠀⠀⠀⠹⣷⣼⣧⡀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣦⡀⠀⠀⡄⢸⢠⡇⣠⠄⠈⠈⢣⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠈⠛⠿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡇⠀⣼⣰⣟⡀⢀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠏⠉⠛⠋⠹⡟⠛⠛⠛⠛⢛⠿⣿⡹⠋⠉⢈⡤⠂⠀⢀⡗⠂⢀⣀⣀⣀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠀⠀⠈⠁⠠⡇⠀⠀⠀⠀⠀⠀⢧⠀⠉⠁⠀⠀⠠⠴⢰⠃⠀⠀⣠⠞⠉⠉
⠀⠀⢀⣀⣀⣀⣀⣰⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣀⠔⠁⢀⡠
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠓⠒⠒⠒⠓⠒⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢙⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠙⠀⢈⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡋⣳⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠰⡦⠹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢈⣟⠛⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠆⣤⡄⠀⡀⠀⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡏⢹⣧⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⡷⠁⠤⠀⠀⠀⢰⣿⠎⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠉⠉⠉⣿⡏⠉⠉⠉⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠰⣦⠀⠀⢺⣇⠀⠀⡰⠆⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠛⣤⢘⣇⡤⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣳⣆⠀⡐⡿⠁
⠀⠀⠀⠀⠰⠾⠾⠒⡒⣇⣪⡓⠓⠛⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⣄⣟⢆⢸⢃⣤⡼⠞⠃
⠀⠀⠀⠀⠀⠀⣐⡶⠋⢐⣇⠙⢦⡄⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⣈⣨⡭⠝⡿⡗⠤⣶⣴⡄
⠀⠀⠀⠀⠀⠐⠋⠂⠀⢐⡗⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⠉⠁⣺⡇⠈⢶⣀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁⠀⠈⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠜⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⢀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠
⠀⠀⠀⠀⠀⠀⢬⢵⠷⣛⠀⠀⠀⠀⠀⠀⠀⠠⡄⣆⡄⠀⠀⠀⠀⠀⠀⣀⡇⠀⢠⡥⠀⣀⠜⠄
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠁⠀⠀⠀⠀⠀⠀⠀⠒⡺⡿⡖⠀⠀⠀⠀⢀⠀⠀⡘⠣⣀⣀⡔⠓⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠼⠉⠉⠉⢉⡽⠯⣉⠉⠋⠉⠙
⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠰⡆⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⠋⢀⡆⠘⢣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣥⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠑⠀⢨⣇⠀⠈
⠀⠀⠀⠀⠀⢀⣄⠀⠀⠘⠉⡴⢰⡈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲

⣿⣟⣛⣟⣛⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠠⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⡆⠀⠀⠀⠀⠀⣧⠀⢰⠀⡇⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⡀⠀⠘⡄⠀⠀⠀⢠⠈⣇⢸⢀⡇⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡤⠙⠀⠀⡀⠀⠠⠀⠀⠄⠰⠀⠀⡀⠀⡞
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠦⠤⢤⣀⣀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠈⠀⠀⠀⣠⠴⠊⣁⡤
⠀⠀⠀⠀⠀⠀⠀⠠⡈⠡⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡇⠀⠀⠀⠀⠀⠀⠁⠌⢡⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⢹⢗⡧⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠧⣩⡟⠟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⡠⠴⠶⠒⣛⢀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⠈⣦⡎⠉⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⣋⣀⣉⣉⣉
⠀⠀⠤⠤⠤⠄⠤⠀⠀⠀⠀⠀⣔⠺⣌⣥⡴⠾⠷⠉⠋⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣂⡤⠤⠐⢹⣃⢯⣰⠆⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢀⡀⠤⢒⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⢉⠥⠒⢈⠠⠒⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠄⠁⠉⠒⠤⣀
⠀⠀⠀⠀⠐⠊⠁⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠠⠀⠀⠈⠓⠂⠤⣀⠉
⠀⠀⠀⠀⠀⠁⠔⠁⠀⠀⠀⠀⣠⢀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠘⠂⠀⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢡⠊⢀⡄⠀⠁⠂⢀⠀⠀⠀⠐⠀⠁⠀⠀⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⣠⠃⠀⢸⠀⠀⠀⠀⢸⠀⠰⡆⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠇⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠃⠈⠃



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣍⣬⣯⣠⣍⣽⣍⣽⣫⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣾⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⣿⣿⡿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠎⠁⠘⠛⠳⠀⠀⠀⠉⠈⠻⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⢈⣴⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠻⠿⠾⢃⣦⣤⣼⣄⣷⢦⣤⣀⠀⠀⠀⠈⢿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⣤⣾⡿⢻⣯⠴⠟⣷⡄⠀⢀⣉⣻⣧⣄⣿⣿⣿⠌
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣶⣶⣤⣄⠈⠛⢻⣿⠇⢀⡾⠉⠿⢛⠻⠃⣴⣿⣯⣃⢸⣿⣿⣁⣈⣴⣶⣿⣿⣷⣆⣀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣸⠇⠀⠺⣇⣤⣠⡤⠶⣤⣨⡋⠈⠁⢸⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢉⣿⠁⠙⠟⢻⣿⣿⣿⣿⣿⠃⣾⣷⡾⣿⣿⣽⣦⣾⣯⣭⣽⠷⠶⣿⠟⣿⣿⣿⠋⠀⣔⣻⣿⣿⣿⣿⣿⡁
⠀⠀⢀⣀⠀⠀⢿⣿⠆⠀⠸⠄⠈⣽⣿⣿⣿⣆⣼⣿⣤⣾⣿⣶⣷⣭⣭⡿⢿⠷⢊⡟⢻⣿⣿⣯⠶⠆⠈⠉⠻⣿⣿⣿⣿⡿⠂⠀⣀
⠀⠀⡾⣭⣷⣄⠀⠉⠀⠲⠀⠀⢠⣠⡼⠛⢉⣽⣿⢻⡹⣧⡸⠋⠁⠈⣷⡄⢸⢧⡿⢧⣸⣿⣿⣿⡀⠰⠿⠗⢀⣽⣿⣿⣿⣿⣶⠞⢿⠂
⠀⠀⣁⣙⣷⣽⠿⠖⠻⣤⣤⣤⣾⠛⠀⣰⡟⠉⣿⡟⠀⠞⠁⢾⡆⢀⣿⡇⠙⢈⠁⣄⣿⣿⡛⠻⢿⣦⣤⡶⠟⠉⠀⢀⣨⣼⣿⣆⠙⢻⡧
⠀⠀⠈⠙⠉⠙⢀⠀⡀⠀⠈⠙⠛⢳⣾⠋⠀⢸⣿⣿⠆⣤⣤⡞⢗⠘⠳⣄⠀⣼⣿⣿⣿⣿⣁⠲⣠⣿⣟⢁⣤⣴⠟⠻⣿⣧⣈⣙⣷⡏
⠀⠀⠀⠀⠀⠠⠾⢿⣇⡦⣤⣆⣀⣤⡄⣰⣿⣿⠿⣿⣧⠽⡏⣁⢨⡖⠛⣷⢄⣼⡟⢿⠿⠛⠛⠛⠉⠉⣭⢉⣉⡧⢸⣶⠀⣉⣿⣿⣿⣷
⠀⠀⠰⠁⠀⠀⠀⢈⣽⠿⠿⠷⠟⢻⣿⠿⣿⣿⡇⣿⣿⣿⣷⣟⣿⣿⣵⣧⡾⠋⣡⡶⠶⠶⣦⣠⣶⣿⣿⣧⡙⠇⢸⣿⢿⣿⣿⣿⣿⡿⡇
⠀⠐⠃⠀⠀⠀⢀⡿⠀⠀⠀⠚⠛⠛⠉⠲⣯⠿⠷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡇⠀⢉⠿⠋⠗⠘⣿⡮⡻⣷⣿⣿⣀⣿⣷⣿⣩⣷⣇
⠀⠀⠀⢀⠀⣄⢸⣯⠀⠀⠀⠀⣙⣛⣿⣿⣠⣀⣠⣿⣿⣿⣿⣤⠙⠿⢟⣿⢿⣿⠿⣷⠶⠈⢀⡶⢋⠀⣼⣿⣴⡘⣿⢻⣿⡷⠀⠚⢿⣥⠋
⠀⠀⡂⠀⠀⠾⡇⢹⡿⢶⣦⣤⣾⣿⠋⠉⠉⠛⢛⣽⣿⣿⣿⡄⣶⣾⣿⣷⢿⠧⠉⠀⠀⠀⠁⢀⡏⣠⡿⢿⡟⣣⠘⠻⢿⡏⣤⠀⠸⣿⣀
⠀⢰⣿⣿⣥⣤⣴⣾⠁⠈⢫⡛⣹⣿⠀⠀⠀⠀⣿⡿⠛⢛⣿⣿⣷⣾⣟⣷⣾⣦⣄⠺⡿⢒⡖⠈⣿⣯⣴⡶⠛⢛⣾⠟⠸⢿⣿⣿⣿⣿⣿
⠀⠈⢿⣿⣿⣿⣿⣷⣶⣦⣼⣽⣿⣿⡇⠀⠀⢸⣿⠃⣿⣿⢛⣿⠟⢉⣿⣿⣿⣿⣟⣿⣧⣉⣿⢿⠿⠟⠋⣠⣾⣋⣥⣶⣿⣦⣽⣿⣿⣿⣿
⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⣷⢶⣿⣿⣤⣿⣿⢸⣿⣄⣹⡿⢏⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⣾⠿⣿⣿⣿⣦⣤⣼⣿⡿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠈⣽⡏⣿⠃⣼⣿⣿⣿⣿⡽⣿⣿⣿⣿⣿⣿⣿⣿⠿⣄⠈⠉⡝⠻⣿⣿⣿⣿⢿⣯⣽⣸⣿⡿⣿⣷⣮⡻⣷⣜⠇
⠀⠀⠀⠀⠀⢀⣀⣿⢀⢻⢰⣿⣿⣿⡿⣿⣧⢿⣿⣿⣿⣯⣌⡙⠶⠛⠿⣿⣾⣷⣶⣤⣌⠙⡄⢻⣿⣟⡴⠟⠀⣿⢿⡿⣛⣶⠟
⠀⠀⠀⠀⢸⣿⣿⣦⣀⠉⢿⣿⣿⣿⣷⡄⢹⡿⠻⢿⣿⣿⣿⣿⣶⣤⣴⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠴⠆⠉⠛⠋
⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣾⣿⣿⣿⣿⣷⣾⠇⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣻⡟⣭⣶⣶⣿⣿⣿⡧⠚⠁⢹⣧
⠀⠀⠀⠀⠀⣠⡶⢿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⡿⠂⢈⣤⣾⠿⣿⣷⣷
⠀⠀⠀⠀⣰⠏⠐⠀⠰⠉⠉⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣟⣻⣿⡿⠿⠛⠋⠉⠛⠁⠀⠀⠈⠙⣷⡄
⠀⠀⠀⠀⢯⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢉⣯⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢠⣀⡄⠀⠀⠀⠀⠀⣴⣺⣿⡏
⠀⠀⠀⠀⢸⣞⠃⠀⠀⠀⠀⣀⣷⠀⠀⠀⠀⢸⡃⠖⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠈⠄⢀⠀⡀⢠⡄⢰⣿⠟⠁
⠀⠀⠀⠀⠈⢿⣦⣘⠆⠀⠀⠈⠀⠀⢠⡄⠄⢸⡇⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⠒⠀⠀⠉⢰⢀⣿⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠓⠶⠶⣶⣶⣮⣬⣵⣄⠈⣿⡏⠀⠀⠀⣀⠂⠀⠃⠀⠂⠀⢀⣁⣀⣀⣀⣀⣀⡴⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡟⠳⣶⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠙⠻⠿⠿⠿⠛⠙⣿⣿⣿⠀⠸⣿⣏⠙⠿⠿⠿⠿⠟⠋⠀⢹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣆⡀⠀⠀⢤⣤⡴⠀⣘⣿⠇⠀⠀⠻⣿⣶⣆⣀⡀⢀⣀⣠⣤⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣶⣶⣴⣶⣾⣿⣿⣿⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣧⠀⠁⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⡀⠂⠀⠀⠀⡀⠀⠀⣸⣿⣿⣷⣾⣿⣧⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡶⠒⠃⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⠛⠋⠉⠉⠀⠀⠀⢀⣴⣿⣩⣙⣿⣿⣿⣿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⣀⠀⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⣀⣼⡖⢩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡿⣯⣏⣽⣶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣇⡘⢑⡙⠎⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡤⠠⡤⠄⠀⠀⢀⡀⠀⣀⣀⣀⣀⣠⢀⣀⣤⣭⡍⠛⠊⠻⣿⣿
⠀⠀⠀⠀⠀⡀⠀⠀⢹⠄⠉⡀⢉⡖⠶⡈⠈⣃⢰⣿⠂⢹⠠⣿⣺⣇⣽⣿⣷⣤⣤⣴⣿⣿
⠀⠀⠀⠀⣾⢛⡁⠈⠀⠀⢂⠀⢆⠓⢠⡀⠠⠰⠀⢙⡯⢼⡛⠉⣗⡑⠘⠿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⢸⣯⠘⣷⠀⠄⢀⠀⠀⠈⠃⠐⠃⠘⠃⢀⠉⠀⡉⠉⠀⠀⠃⢷⡀⠈⢻⣟⣿⡟
⠀⠀⠀⢸⡏⢠⠈⢷⡄⠀⠀⠀⠂⣸⣟⠂⠰⣯⣤⣍⣻⣧⣾⣶⣸⣷⣴⣷⡔⣻⣿⡿
⠀⠀⠀⠈⠓⠉⠀⠁⠈⠆⠀⢠⡿⠛⣿⣿⣾⡛⣷⡞⠓⢽⣿⣯⣷⠛⢳⣏⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⡴⢃⠀⢠⢒⠎⠀⠀⠉⠉⠛⠛⠓⠒⠒⠒⠛⠛⠙⣦⢻⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠂⡀⣠⠀⠐⢫⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠿⡆⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⣈⡼⠁⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⣧⢳⠀⠀⠀⠠⠖⠂⠀⠀⣰⠖
⠀⠀⠀⠐⢠⠃⠀⠀⠀⠀⠓⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠃⠈⠁⠀⠉⠀⠀⠀⠀⣅⡈
⠀⠀⠀⠘⠎⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡖⠐⠐⠋⠀⠉
⠀⠀⠠⠖⠀⠀⠀⠀⠠⠄⠀⠈⠁⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠿⠄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⠇⠀⠜⠃⠀⠀⠀⠀⠀⠘⠲⠓⠉⠁⠀⠀⠁
⣶⣶⠶⠶⢶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⣴⠂⠀⣲⠿⠃
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠆
⠀⠀⢰⣶⣿⣷⣽⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡄
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣦⣤⣤⣤⣾⣿⣿⣿⡍⠁
⠀⠀⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⢿⣷⣦⡀⣀⣄⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣇⣽⣿⣿⣯⢙⣓⣒⣀⣀⣸⣿⣿⡿⢿⣿⢿⣿⡿⣿⣦
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠙⣆⠀⠀⠀⠀⢸⡿⣿⠁⠀⠀⠀⣿⡷⠾⠋
⠀⠀⣴⠂⠀⣻⡿⠿⠿⠿⠿⠿⠿⣿⡋⢀⣠⠏⠀⠀⠀⠀⠀⣿⡼⣧⠀⠀⠀⣿⡄
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡈⠀⠀⠀⠀⡠⠔⣒⢄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⣴⠇⠀⡤⡤⠈⠃⠄⠈⡈⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢵⠏⠀⠁⢧⠼⣧⠤⠀⠀⡤⠀⠀⡤⠘⢄⢼
⠀⠀⠀⠀⠀⠀⢀⡤⠦⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠄⢀
⠀⠀⠀⠀⠀⢠⡟⠀⠆⠔⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⠈⠒⠂⢧⡴⠁
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢸⢧⡄⢠⠀⠄⠀⠀⢰⠄⠐⠁⠀⢠⠔⣤⢲⠀⠠⢦⠄⠀⡆
⠀⠀⠀⠀⠀⢿⣆⡀⢀⣰⠃⢠⣀⠈⢀⡀⢺⡀⢈⠂⠀⣦⡤⡟⠀⡇⢸⣠⣀⡞⠀⠀⡵
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠀⠈⠁⠀⠀⠁⠈⠁⠈⠁⠀⠉⠀⠀⠀⠀⠈⠁⠉⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⢁⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠊⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⣢⣀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⠙⢮⢠⠃⢀⡠⠄⠀⠒⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⡼⢸⠖⠁⠀⠀⢀⠀⢀⣀⣀⡀⠺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠢⡀
⠀⠀⠀⡔⠉⠉⠑⠄⢀⠤⡀⠀⠉⠀⠀⠀⣏⠀⠀⡠⠖⠡⠀⢀⣁⠈⣘⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⢢⠀⠀⠀⠀⢸
⠀⠀⠀⡀⠀⠀⠀⠈⠃⠀⠘⠀⠀⠀⠀⢀⠜⠒⠒⢄⢠⡀⠀⢀⣀⠉⣴⡄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣄⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⢣⡀⠀⠀⠀⠀⡠⠖⠒⠦⡀⠀⣇⠀⠲⠦⣀⠈⣄⡐⠌⠋⠀⠀⠀⠀⠈⢳⠠⠀⠀⠤⡀⠀⠀⠀⡞⠁⠀⠀⠱⡄⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠑⢤⡀⠀⡜⠁⠀⠀⣤⡸⠀⠈⠒⡔⠁⠀⠠⠤⠈⠑⠢⡀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠈⢆⠀⠀⢹⠛⢆⠀⠀⠘⢦⠔⠉
⠀⠀⠀⠀⠀⠀⢉⡏⠀⠀⢠⠎⠈⢧⠀⠀⠜⢀⠀⠀⠘⠟⠀⠀⠀⠉⠒⠂⠒⠈⡀⠀⠀⠀⠀⠀⣼⠀⡴⠃⠀⠈⢣⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠇⠀⠀⠀⠑⢄⠃⡍⠀⠀⠀⣄⠀⡉⠐⠒⠂⠒⡆⠀⠀⠀⠀⠀⠀⠀⠳⠊⠀⠀⠀⠀⠈⡇⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠈⠢⣄⣸⡄⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠘⢿⣿⣿⣿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⠞⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⠙⠠⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⢠⠀⠈⠦⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠜⠋⠀⠐⣄⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠙⡇⠀⠀⠀⠈⠉⠒⠤⢀⡀⢀⡀⠀⠀⣠⠒⠁⠀⠀⠀⠀⠀⠸⡄⠀⢀⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠉⠓⢦⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠑⠶⠁⠂⠀⠀⠀⠀⠀⠀⣰⡷⠊⠁⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣤⣀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠤⡀⠀⠀⢀⣴⣶⣿⣗⣀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣀
⠰⠂⠀⠠⠤⢤⣀⠀⠀⠀⠀⠀⢠⠞⠁⠈⠛⢿⣿⣿⣿⣶⣦⣼⢰⣶⣶⣶⣶⣶⣶⠈⣶⣾⣿⣿⡿⠋⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⣀⠈⠳
⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⢀⣤⣿⣄⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⢸⣿⣿⣿⣿⣤⣤⢘⡿⠟⠋⠁⠀⠀⠀⢀⣾⡀⠀⠀⢀⠤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠙⠛⠛⠓⠒⠠⠤⠤⠤⠤⠤⠤⠬⣤⣭⣭⣭⣭⣭⣥⣜⣀⣀⣀⣀⣀⣀⣠⣾⣿⣿⣷⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢼⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⢼⠀⠀⠀⢰⠄⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⡿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢃⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠘⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⠐⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⢠⠃⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⡞⠀⠀⠜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠷⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣷⠀⠇⡁⠀⠈⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⢰⠱⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠐⣴⢃⠀⠸⡃⢄⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠈⠓⡀⠀⠀⠀⠀⠀⠀⠀⡠⢘⠀⠀⠜⣶⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠛⢄⠘⡿⡄⠀⠙⢆⢦⡀⠀⠀⠀⠀⠀⠜⡄⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢀⡼⡰⠃⢀⣾⣶⠃⡰
⠀⠀⠀⠀⠀⠠⠄⠀⠠⠀⠀⠤⠀⠑⢤⡌⠰⠄⠀⢠⡙⠆⠀⠀⠀⠀⠄⢧⠀⠀⠀⠀⠀⠀⠰⠘⠀⠀⢀⠀⠴⢋⠔⠀⣴⢟⣿⣿⡎
⠀⠀⠀⣀⠀⠀⡀⠀⢀⠀⠀⣀⠀⠀⡀⠙⢢⡴⢒⣒⠛⠓⠦⠌⠐⣤⣸⠀⠂⠀⠀⠀⠀⠠⠀⣃⣤⠂⢡⠴⠛⠛⢂⣲⣾⣿⣯⣾⣿⣄
⠀⠀⠀⠁⠀⠀⠁⠀⠈⠀⠀⠉⠀⠀⠁⠀⠈⠀⠀⠈⣌⣂⣀⡀⠢⣄⠈⢑⣦⠀⠀⠀⢀⣆⡈⢁⣠⠔⢀⣀⣂⣡⢰⣻⣿⣿⣿⣿⣯⣿⣆
⠀⠀⠀⠓⠀⠀⠃⠀⠘⠀⠀⠚⠀⠀⠃⠀⠐⠂⠀⠒⠠⠀⠀⠉⠳⠠⡙⡀⡀⢡⠀⣀⡄⠀⢀⠉⠠⠞⠉⠀⢀⡿⣹⣷⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⢁⡜⢅⠀⠼⢠⡅⠦⢀⡀⢃⠌⠀⠀⢀⡾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢑⡉⠀⠁⠀⠀⢰⠀⠁⣊⣦⠐⢀⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠳⣐⠆⠚⠓⠰⣀⠞⢠⠂⢀⣾⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠈⢃⠀⠀⠌⠹⠀⡏⠀⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘⠆⠈⠛⠛⠂⠸⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡆⠘⠃⠀⠀⠶⠃⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠜⢂⠠⠴⠦⠄⠺⠣⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⣒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠀⠑⠐⠂⢊⠀⡠⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⠭⠭⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠛⠠⣀⣀⠄⠛⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣒⣒⣒⣒⣒⣒⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠠⠬⠭⠭⠭⠭⠭⠭⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠸⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⣀⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡄⢠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠌⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠳⣦⣄⠀⣿⠀⣠⣴⠞⠃⠀⠀⠀⣰⢤⡹⣽⢃⣤⡆⠈⠀⠀⠀⠀⠀⠀⠶⠀⠰⠇
⠀⠀⠀⠀⠀⠰⣶⠆⠀⠀⠀⠀⠀⠀⠂⠀⢉⣓⢾⣞⡉⠀⢐⠀⠀⠀⠀⢈⣛⣻⣿⣟⣋⡁⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠠⡷
⠀⠀⢂⠨⠀⠀⠀⠀⠀⠀⡐⠀⠀⢀⣤⠾⠋⠁⣾⠀⠙⠻⢦⡀⠀⠀⠀⠻⠞⢱⣻⡜⠾⠇⠀⠀⠀⠀⠀⠀⢐⡴⠀⠰⡄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠤⣤⢠⣬⣼⣥⡤⢤⡤⠀⠀⠀⠀⠠⠄⠈⠀⠉⠀⠄⠠⠀⠀⠀⢀⡀⢀⢀⠀⠄⠀⢀⠀⠀⠀⠀⢀⠀⠀⠻⡦⠀⣴⠦⠄
⠀⠀⠘⢣⣿⣯⣝⣿⣿⡜⠁⠀⠀⠀⠀⠂⠀⠀⣤⡀⠀⠐⠀⠀⠀⠀⠀⠀⢀⡉⠀⢉⡤⠀⠀⠀⠀⠀⠘⠶⠖⠀⠀⠀⠉
⠀⠀⢰⣮⢟⣿⣽⣿⡿⢴⡆⠀⠀⠀⠀⠂⠀⠈⠉⠀⠀⠐⠀⠀⠀⠀⢠⡄⠡⢔⠀⠤⠀⢠⠀⠀⠀⠀⠀⠀⣰⠄⢀⡀⠘⠛⢧
⠀⠀⠉⠉⠀⠡⣭⡍⠀⠉⠉⠀⠀⠀⠀⠈⠁⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⡿
⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠃⠭⠘⢠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⡄⢓⣿⢿⢠⡄⣠⡄
⠀⠀⠀⠀⠠⣀⣀⣀⠀⠀⠀⠀⠀⠀⢬⠑⢣⣄⣻⣠⡄⠪⡥⠀⠀⠀⠀⠀⠠⠠⠡⠀⠀⠀⠀⠀⠀⠀⢩⢻⡶⣍⣹⣩⡶⣟⣯⠁
⠀⠀⠀⠀⠠⠘⠛⠣⠀⠀⠀⠀⠀⠀⠴⢄⡠⠜⢻⠳⢆⡠⠦⠀⠀⠀⠀⠀⠀⠡⠡⠌⠀⠀⠀⠀⠀⠀⠰⣰⣥⠞⣿⠳⣮⣇⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠡⠆⡀⢭⢀⠰⠮⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠫⠆⡹⣿⢾⠱⠏⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠾⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⠍
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣰⠛⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢳
⠀⠰⣶⢫⠧⣩⠉⣏⠴⡜⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⠢⣤⣾⢾⣆⣤⠴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⢙⡛⢴⢽⣿⡯⡦⢚⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣧⣽⣻⣯⣿⠋⠀⠀⠀⠀⠀⡀
⠀⢀⣮⣱⡄⠾⠭⠗⣀⢟⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣫⠷⢾⣹⡷⢿⣑⣄
⠀⠀⠉⠉⠁⢏⣒⡟⠈⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣹⠁
⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⡟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⡶⣤⣤⣴⣤⡾⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⠀⠀⠀⠀⣼⣋⣻⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣯⡤⠉⣗⡆⠀⠀⠈⠉⠉⠀⠀⢤⣵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⡶⢠⡾⠿⣷⣀⡀⠀⠀⠀⠀⢞⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠛⠛⠉⠛⢷⡤⡄⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⢦⣴⣄⡱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠏⠀⣽⠿⢿⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣶⡧⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⡤⢸⣶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣖⡿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⣀⡀⣀⡤⣤⣴⣲⠖⠻⣾⠿⢯⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠭⠗⣾⣽⣶⣿⠽⠛⠓⠋⠉⢹⣯⣿⡆⠀⠀⠙⠛⠋⠀⣀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣎⣥⣆⣴⢞⣁⣻⣭⣽⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⣀⣈⠳⣤⣶⣌⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⣩⡿⣅⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣉⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠉⠉⠛⣽⣶⣖⣦⣠⢤⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠻⢿⠇⠀⠀⠀⠘⣿⣭⡏⠀⢈⣿⣻⡷⠛⠻⣽⣶⣟⠦⢴⣲⠤⣖⠶⠾⡿⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⢲⣾⣵⣧⡶⠃⠀⠀⠀⠀⠀⠀⠀⠈⠳⠾⠃⠀⠘⣯⣭⠇⠀⠀⠀⠀⠙⣦⣴⣾⣷⠚⠳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡤⣀⣴⣺⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢟⣶⣶⣖⣦⣤⢲⣤⣴⣢⣤⣖⣦⠴⣲⠶⠾⣽⡿⣷⠛⠉⠛⠁⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠗⢲⠿⣿⣪⣿⣾⠇⠉⠀⠨⣿⣙⡧⠀⠀⠀⠀⠀⠻⣶⠿⠀⠀⠀⠀⠐⢿⡾⢿⡖⠽⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣯⣿⢿⣲⣤⣠⢤⡙⣋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠁⠉⠀⠋⠀⣽⡯⣿⠊⠉⠓⠛⠛⢯⡿⣾⡽⠶⢾⡷⠶⢏⣶⣶⡭⠖⠺⠽⠓⢦⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣊⢥⡶⠃⠀⠀⢀⡴⣤⢈⡟⣻⠀⣠⡄⢠⠿⣸⣯⣿⡁⣠⣄⢰⡄⣴⣯⣬⣧⡀⠀⠀⠙⢶⠬⣳⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠰⠞⠒⢏⡤⠞⠹⠋⠱⠊⠷⠛⠋⠁⠈⡇⠀⠀⠀⢸⠋⠁⠈⠛⠙⠞⠻⠏⠻⠏⠲⢄⡟⠚⠳⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⢠⠉⣤
⠀⠀⠀⠀⠐⠆⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢆⠀⠀⢀⠿
⠀⠀⠀⠀⠇⢀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠑⠪⠆⠒⢸⡀⠀⡀⠤⢄
⠀⠀⠀⠀⠐⢄⡡⢀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠀⠡⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣁⡠⠆
⠀⠀⠀⠄⣀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡐⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡡⠞⠁⠀⠉⠉
⠀⠀⠦⠀⠤⠄⠁⠈⠢⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣋⡀⠓⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠈⢀⠶⡈⠑⠆⡀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⣉⡤⠚⠁⠈⠶⣄⠀⢀⡙
⠀⠀⠀⠀⠀⠀⠈⠀⠐⠁⠀⠈⠉⠒⠒⠀⠈⠉⠉⠀⠀⠤⣄⠀⠀⠀⠀⠄⠀⢀⠐⢄⠀⠀⡤⠒⠀⠒⠒⠒⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠞⠁⠀⠀⠀⠀⠈⠲⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⢀⠀⠀⡠⠄⠀⠀⠀⠀⢀⡀⠒⠒⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⣤⡀⠁⠀⣾⠆⠀⠀⠈⠥⠄⠀⠀⠀⢀⡀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠇⠀⠀⠀⢠⡀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢠⠀⠀⠀⢆⠀⠀⠋⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠑⠀⠀⠀⠉⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠠⠴⡿⠦⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⣻⢢⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢻⡌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⣀⡤⠄⠠⠦⡄⡀⠀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠈⠣⡀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠉⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⡄⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⢑⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣠⠏⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠰⣄⡀⠀⠀⢠⠁⠀⠀⣸⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠛⠦⠒⢛⠀⠀⠀⠻⡄⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢰⠀⢸⠀⠀⢰⠀⢡⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠘⠀⢸⠀⠀⢸⠃⢸⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢢⠸⠀⠀⠈⠀⠈⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢸⣄⠀⠀⠀⠀⠀⣦⠀⠆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⠀⠀⠀⠀⠀⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠘⠀⠀⢀⡇⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠂⠀⡿⠀⣤⡄⢠⡠⠚⢁⠖⠀⠤⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⠀⡇⠀⠀⠀⠀⢣⣀⡎⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠤⠘⢷⣄⣀⣀⣀⣼⠟⠙⠢⠤⠤⠴⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡿⣙⣿⠛⣿⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣶⠂⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣿⣗⠀⠻⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠿⠁⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣵⣶⣾⣶⠤⠦⠤⣤⣤⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣣⣿⣿⠿⠟⣉⣉⣉⠉⠛⢒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢭⣿⣿⣿⣿⣿⡋⠀⠀⠀⢦⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠰⢿⣟⣛⣫⣭⣭⣭⣥⠤⠤⣄⣘⣃⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⡀⠒⠂⠈⢽⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⢿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⣐⣿⣿⣌⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣟⣽⣶⣿⡿⠋⠉⠈⠻⣿⣿⣿⣿⣦⡄⠄⠈⢹⣿⣿⣿⣟⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣾⣿⣿⣿⣿⣧⡀⠀⠀⢀⣿⣿⣿⣿⣿⠶⠀⢰⣿⣿⠿⣟⡥⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠗⣴⣶⣿⣿⣿⣿⣿⣿⣛⣭⣿⠿⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣴⠏⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⢤⣾⠿⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⢉⣤⣀⣠⡾⠟⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣏⣁⣠⣤⠔⠾⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠲⠾⣿⣿⠟⢫⡉⣁⢀⣠⡞
⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⢀⣿⣿⠿⠋⢙⠋⠉⢁
⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⣸⣿⣷⠾⠿⠛⠟⠉⢁
⠀⠀⠀⡤⠚⠟⠀⠀⢀⣤⣾⣿⣿⠿⣿⣾⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠘⢿⢿⣿⣽⣥⣿⡿⠛⢻⡗
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠟⠋⠀⠀⠀⣷⠆
⠀⠀⠀⠀⠀⣠⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⢠⣯⠁
⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡫⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣝⣦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢋⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁

⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠹⡉⠓⠲⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠖⠚⢉⠏
⠀⠀⠱⡈⢻⠛⠻⠿⢿⣯⣉⠙⢲⣶⣶⣦⡤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⣴⣶⣶⡖⠋⣉⣿⡿⠿⠗⠒⣟⣀⠎
⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠓⠶⠤⠼⠿⠿⠿⠧⠤⠬⠿⠿⠿⠷⠶⠒⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⣼⣿⡟
⠀⠀⠀⠈⡟⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡤⣄⢀⣤⡄⢠⣤⡄⢠⣤⡄⠠⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⢿⠁
⠀⠀⠀⠀⢷⣤⣸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣈⣁⣩⣬⡿⠥⠬⠭⠧⢬⣭⣇⣀⣛⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡼
⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢀⣤⠤⣄⠀⣠⠴⡄⠀⠀⠀⢀⡔⡒⠒⠦⢄⣰⠂⠀⠀⢴⢲⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⢸⠛⢻⡇⠀⣰⠏⠸⠥⠞⢠⣟⡼⠁⡀⢀⣄⢀⡏⢁⣤⠀⢠⠏⢉⣓⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⡇
⠀⠀⠀⠀⣟⠀⢸⠀⠀⣿⠀⠀⠀⣀⣼⠋⣹⣀⠙⣟⣁⢸⣃⡼⢷⡄⡟⠀⡾⢩⡏⣟⢰⠏⣽⢁⣸⣇⣠⠀⠀⣇⣀⣳
⠀⠀⠀⢀⣿⣿⡏⠀⠀⠈⠓⠒⠋⠁⠉⠀⠈⠁⠀⠈⠀⠈⠁⠀⠉⠁⠉⠁⠁⠈⠀⠉⠈⠉⠉⠉⠀⠿⠀⠀⠀⢹⣿⣿⡄
⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⣀⣀⣠⠤⠤⣤⣶⣶⡖⠒⢲⣶⣶⣖⠛⠒⢶⣶⣶⣶⠒⠶⢤⣤⣤⣀⣀⣀⠀⠀⠀⠀⢻⣿⣧
⠀⠀⣰⠁⣿⣤⣶⣶⡋⣉⣿⣿⠿⠤⠤⠾⠛⠛⠛⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠾⠿⠿⠿⣦⣈⣉⣿⣶⣶⡤⣷⠈⢆
⠀⣰⣁⡤⠴⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠦⢤⣈⣆
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⢤⣤⣤⣤⣤⣤⡄⠛⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠠⡟⠁⠀⠈⠳⣄⢠⠏⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠛⢣⡤⢤⣀⠀⠀⠙⠻⣿⣿⡀⠀⣠⣤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⠋⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⠟⢉⡁⠈⠀⠀⢰⣿⡙⣆⠀⠘⢿⡇⠀⠿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢀⡟⠓⢦⡀⠀⠀⠀⠀⠀⣠⠖⠈⡏⠀⠸⣦⣬⡗⠂⠀⠀⢠⠄⠀⠛⡄⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠶⠒⠉⠀⠀⠀⠙⣦⠀⠀⣠⠎⠁⠀⠀⢷⣄⠀⠈⠉⠉⠀⠰⢶⣷⣶⡄⠀⠸⣄⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⣽⣄⣴⠁⠀⠀⠀⠀⠀⠙⠛⠻⣽⡿⣦⣀⣀⣀⣠⠔⠀⠀⠈⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⣠⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⢶⡚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢶⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠿⠿⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⢶⠀⠀⣼⣿⣿⣦⣤⣤⣀⣀⡀⢀⠀⢀⣀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⡞⠀⡜⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⣿⣿⡟⠛⠻⢿⠛⠉⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠛⠀⣸⣧⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡇⠀⠀⢳⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀⡤⠞⠀⠀⠀⢀⡇⠀⠀⠀⠀⢠⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠛⠉⠉⠉⠉⠙⠛⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣄⠀⠀⣀⣤⣾⣁⣀⣀⣀⣤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⠰⠶⠶⠶⠶⣶⣶⣶⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢤⣄⡈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⡀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠿⠛⢿⣿⣿⣿⣿⡿⠟⠛⠉⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣀⣤⣽⣿⡟⣧⣠⣤⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣻⣿⣿⡿⢿⣿⡇⣰⣿⣿⣿⠿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠁⠀⠈⠉⠰⣷⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠂⣐⠀⠂⣆⠀⡴⠖⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠉⠀⠀⠀⠀⠀⠁⢀⠉⠉⠌⠀⠀⢤⡀⠀⠀⠈⣴⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠚⠀⠀⠀⠀⢀⠀⠀⢀⣤⠜⠁⠀⠰⠏⠀⠄⡀⠉⢢⠀⠠⠃⠀⠀⠀⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢅⣠⣤⠶⠞⠛⠉⠙⠛⠛⠻⠷⠶⢮⣔⡀⢀⠀⠀⠀⠄⠀⠜⠢⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡼⠂⢘⣭⣶⠛⠁⠀⠀⠀⠀⠐⠶⠄⠀⠈⠀⠀⠀⠙⠿⣷⣦⡀⢀⠀⠀⠀⠀⠈⢠
⠀⠀⠀⠀⠐⠀⠉⠨⠉⠀⢄⣵⣿⣿⠃⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⡐⡿⠂⠐⢂⢠⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣢⣾⣿⣿⢱⣾⠿⠃⢐⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⣿⣿⣿⣎⠛⠀⡀⠈⠀⠱
⠀⠀⠐⠁⠀⠀⠀⠠⣰⣿⣿⣿⣿⣿⣷⡀⣀⠀⠁⣄⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⡘⣇⠀⢻⣿⣿⣿⣧⢠⣤⠀⠀⠀⢣
⠀⠀⠄⠀⠀⡀⠀⢥⣿⣿⣿⡏⢿⣿⣿⣷⣿⣄⣴⣿⢀⣠⣔⡋⠉⠀⠀⠈⣉⣈⠻⣿⣦⣼⣿⣿⣿⣿⣇⢭⡂⠀⠀⠀⡀
⠀⠠⠀⠀⠐⠁⡀⣾⣿⣿⣿⣯⡝⢿⣿⣿⣿⣿⣿⠟⢿⡏⠉⣀⣠⡄⠸⣏⣉⠁⠀⠛⠖⠿⣿⣿⣿⣿⣿⠘⠄⠘⠢⡀⢀
⠀⠸⠀⢠⠀⠀⡇⣿⣿⣿⣿⣿⣷⡾⣽⣿⣿⣿⣿⣆⠘⠞⠊⠁⠘⢿⡆⠀⢠⡄⠀⠉⠀⠄⠀⠉⠻⣿⣿⡇⡆⠈⣃⠈⠹
⠀⢀⠀⠀⠀⡄⠁⣿⣿⣿⠛⠟⠹⠂⠂⢹⣿⣿⠙⠻⣿⡗⠰⢀⡤⠎⠄⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⡅⠀⠀⠀⢘
⠀⠐⢰⠀⠈⠀⠀⣿⣿⣿⢳⡄⠀⢀⣾⣿⣿⡿⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⢀⣀⣴⣤⡶⠀⠀⠀⠀⣿⠣⠀⠀⠹⣤⠘
⠀⠀⠈⢂⠀⠀⠀⢸⣿⡿⠊⠀⠀⠙⠛⠻⣿⣿⣄⠀⠀⠀⢀⣠⣤⣴⣴⡾⠟⠁⠁⠀⠀⠀⠀⠀⠀⠀⢫⣜⡇⠀⢠⡘⠃
⠀⠀⢡⠈⠣⣀⣴⣿⣿⠁⠀⠀⠆⠀⠐⠒⢿⡟⠻⣦⡄⠐⠘⠉⡋⠓⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⡿
⠀⠀⢀⣤⣾⠿⠉⠛⠃⠀⠀⠀⠀⠀⠀⠐⢾⣷⠁⠘⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣤⡁
⠀⢀⣾⡿⣡⠎⠀⠠⠂⠀⠀⠀⡀⠀⣘⣠⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣷
⢀⣿⠛⠁⢉⠀⠀⠁⠀⠀⠀⠀⠂⣴⣿⣿⣿⣧⣾⠇⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃
⢸⠟⡇⠈⠈⠀⠀⠀⠐⡞⠁⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣯⡉
⠘⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣾⣿⣧⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⡟⠛⠛⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡿⠿⠿⢿⢿⣿⣿⡀⢠⣿⣈⢀⠀⠀⠀⠀⣤⣠⣶⣿⡿⠋⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠟⠆⠀⢿⠂⢰⢆⣤⣾⡿⠿⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠞⣿⠟⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠶⠿⠟⠛⢻⡟⠛⠿⠷⠶⢶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠉⠁⠀⠀⠀⠀⢀⡄⠈⣡⢤⡀⠀⠀⠀⠀⠈⠉⠛⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠟⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠐⢿⠀⠐⢺⠀⠀⠀⠀⠀⠐⠓⠂⠚⠓⠂⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⢀⣹⡀⢀⣸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢧⠀⠀⠀⠀⠀⠀⠙⣷⣄
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⢠⣾⠋⢀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⣷⡀
⠀⠀⢠⣿⠋⠓⢩⡇⢰⡇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⠀⠋⠉⣿⡄
⠀⢀⣿⠃⠀⠀⠼⠧⠈⠣⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣥⠀⠀⠀⠘⣿⡀
⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠇⠀⣠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠘
⣿⠶⠆⠳⠤⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡃⠰⠶
⣿⡄⠀⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠴⠃⠀⢠
⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢰⡟
⠀⠈⣿⡄⠀⠀⠀⣏⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣹⠀⠀⠀⢀⣿⠁
⠀⠀⠘⣿⣀⣠⠐⣇⣨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⠁⢤⣠⣿⠃
⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠐⢒⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢤⡀⠀⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠃⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠈⢓⡚⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⠶⢶⣶⣤⣼⣧⣤⣴⡶⠶⠾⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⢹⣿⠁⠀⠀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣶⣄⢸⣿⢀⣴⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⠋⠀⠀⠀⣀⠀⠀⠀⠀⠰⣿⡀⠀⠀⣠⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣄⠀⠀⢸⣿⠀⠀⠀⢴⡆⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⣿⡇⠀⣸⣿⠁⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⣨⣿⣆⡀⠀⣿⡇⠀⠀⠀⢠⣤⠀⠀⠀⢸⣿⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⣸⣿⣴⡾⠟⠛⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠋⠙⠿⣷⣿⣧⠀⠀⠀⠀⠀⣤⡀⠀⢸⣿⠀⠀⣰⡶⠀⠀⠀⢀⣠⣶⡿⠿⠿⢷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠿⠿⢿⣦⣄⡀⠀⠙⣿⣄⢸⣿⠀⣰⡿⠁⣀⣤⣾⠿⠋⠁⠀⠀⠀⠀⠀⠉⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠉⠀⠀⠀⠀⠀⠈⠙⠻⣷⣦⣜⣿⣾⣿⣴⣿⣷⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀⢀⣀⣤⣤⣭⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⠶⠀⠰⠆⠀⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠛⠃⠈⠉⠉⢉⣩⣿⢿⣿⣿⣿⡿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠋⢁⣾⠟⣿⡿⣿⡄⠉⠛⢿⣶⣄⡀⠀⠀⠀⠀⢀⣠⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣶⣶⣶⡿⠟⠉⠀⠀⢠⣾⠏⠀⣿⡇⠘⢿⣆⠀⠀⠈⠙⠻⢷⣶⣶⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⣴⣾⠟⣿⡿⠁⠀⠀⠀⠀⠘⠋⠀⠀⣿⡇⠀⠈⠉⣀⠀⠀⠀⠀⠀⢹⣿⠿⣷⣦⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠉⠉⣿⡏⠀⢠⣿⠃⠀⠀⠀⠀⠸⠃⠀⠀⠀⣿⡇⠀⠀⠈⠛⠀⠀⠀⠀⠀⢸⣿⡄⠀⠉⢿⣏⠉⣹⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣶⡾⠟⠀⠀⢸⣿⠀⠀⠀⠀⠰⠆⠀⠀⠀⣰⣿⣧⡀⠀⠀⠀⠺⠇⠀⠀⠀⠀⣿⡇⠀⠀⠈⠛⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⡿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⣿⡇⠈⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⣿⣇⡀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⣠⣶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⡄⠀⡜⠁⡏⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠸⡈⠳⡀⢀⠎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⠲⠁⠸⡇⠀⡜⠁⢡⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠁⠳⡀⠀⡇⠀⠳⠋⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠙⠞⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠓⠚⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀⠀⠀⠓⢄⡀⠀⠹⡦⢄⣀⣰⣄⡄⠀⠀⢀⡴⠃⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠈⠓⢤⡀⠀⠙⣄⠤⠷⣄⠀⠈⠉⡿⠤⣤⡏⠀⢀⠴⠊⠁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⡤⠖⠒⠋⠉⠁⠀⡀⣉⠑⠢⣌⡶⠋⠀⠀⠀⠀⠉⠒⠊⠁⠀⠀⠉⢶⣡⠔⠊⡉⣀⠀⠈⠉⠙⠒⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠀⠀⠁⠀⠉⠙⠂⡞⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢳⡐⠛⠉⠀⠈⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣀⣠⠴⠚⢹⠁⠔⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠣⠀⡏⠓⠦⣄⣀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣰⣿⣛⣷⡀⠀⠀⠀⠀⠀⣾⣿⣹⣧⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠹⡻⠟⡢⠁⠀⣀⡀⠀⠀⢿⠿⢟⡜⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡈⠉⠀⣠⣤⣶⣶⢦⣄⠀⠉⢁⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⣨⣇⡀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⠦⢤⣀⣑⣊⣀⡠⠴⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢹⠀⠀⠀⠀⠀⠀⠀⠀⢫⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢹⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⣆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢀⠏⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡈⢦⡇⠀⠀⡆⠀⠀⠀⠀⠀⢸⣧⠎⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣤⡇⠀⠀⣇⣀⣀⣸⠀⠀⢸⣷⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡿⠃⠀⠀⠈⠿⣶⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⠿⠿⠿⠿⠿⠿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠛⠉⠁⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠉⠉⠛⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⠰⠀⢀⠀⠀⠀⠀⠀⢳⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠆⠙⠻⣷⣦⡀
⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠦⣄⠈⢧⡀⠀⠀⠀⠘⣇⢸⢸⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠈⠙⣿⣦⡀
⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠈⠳⣄⠳⠀⠀⠀⠀⠈⠈⠈⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⣰⣿⢋⡀⠀⠀⠀⠀⠀⠀⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⣿⣆
⠀⣰⣿⠃⠀⠠⠤⢤⣄⣀⡀⠀⠈⠛⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣚⡥⠀⠀⠘⣿⡆
⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠵⠊⠁⠀⠀⠀⠀⢸⣿
⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠠⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⢸
⣿⡇⠆⠀⠖⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿⠒⠒⠒⠒⠓⠚⠿⠟⠛⠉⠀⠰⠿⠿⠿⠽⠀⠶⢸
⣿⡇⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⢸
⢹⣿⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠘⣿⡆⠀⠀⠴⣚⠵⣚⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣝⡲⢤⡀⠀⠀⢰⣿
⠀⠸⣿⡄⠀⠈⠵⠊⠁⣤⠆⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⣝⠲⣍⡲⠁⠀⢠⣿⠇
⠀⠀⠹⣿⡌⠁⠀⡤⠚⠁⠀⠀⣰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠁⠈⢡⣿⠏
⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠀⣰⢣⠏⢠⡆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠘⠡⠋⠀⡼⠀⠀⠀⠀⢸⠀⢿⠀⠀⠀⠀⠀⠀⠈⠳⠄⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣄⠰⠀⠀⠃⠀⠀⠀⠀⢸⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠆⣀⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣤⣀⡀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⢀⣀⣤⣶⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⢿⣶⣶⣶⣶⣶⣶⡿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠶⠾⠿⠿⠛⠛⠿⠿⢶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⣠⠴⠊⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠈⠻⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⣠⣞⣁⣤⠴⠖⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠿⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⣠⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢤⣄⣀⠀⠀⠀⢠⡀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠞⠛⠛⠛⠛⠛⠛⠿⠶⠶⣭⣝⡲⢦⣼⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⣧⢀⠀⢀⣤⠞⠋⠉⠤⠶⠦⢴⠶⣄⠀⠀⢀⡴⢶⠤⠶⠦⠉⠛⠷⣦⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢠⡇⢸⡿⣫⠀⠀⠀⠐⢀⣤⣶⢤⠠⠟⠀⠀⠘⠧⢠⢴⣶⣆⠐⠀⠀⠈⠹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⣸⢃⡌⢱⠛⠀⠀⣤⣞⠉⣿⣿⠀⠀⡇⠀⠀⠰⡂⠀⢸⣿⡏⣹⣦⡄⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢰⢋⣾⡇⠀⠀⢀⠀⠀⠠⠖⠒⠶⠉⠀⠀⠀⠀⠀⠀⠈⠱⠖⠒⠦⠀⠀⢀⠀⠐⣾⡳⡀
⠀⠀⠀⠀⠀⠴⢶⣦⡀⠀⠁⡴⢋⡴⠀⠀⡽⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠱⣅
⠀⣠⠆⠀⠀⠀⠀⠙⢷⣄⣾⠀⣸⠏⠀⠺⢷⠀⠀⠀⠀⠀⠀⣀⣧⠶⠄⡀⠴⠦⣆⡀⠀⠀⠀⠀⠀⢸⠟⠂⠘⢿⡀
⣼⣿⠀⠀⠀⠀⠀⠀⣸⣯⠁⣰⠋⠀⠀⠀⠘⣶⣀⠀⠀⡴⠋⠁⠀⣀⣠⣤⣀⠀⠀⠉⢳⡀⠀⢀⣰⠎⠀⠀⠀⠈⢳
⠀⠿⢧⣤⡀⢀⣀⣴⣿⡏⢰⣧⡆⡀⠀⠀⠀⠛⠻⢀⣼⡇⢀⣠⠞⠛⠿⠿⠟⠛⢦⣀⠘⣿⣄⠸⠛⠀⠀⠀⠀⡀⣦
⠀⠀⠉⠙⢿⣿⡿⠿⠛⠃⠘⠹⣿⡇⠀⠀⠀⠀⠰⠟⠛⠛⠛⠁⠀⠙⠒⠶⠒⠋⠀⠙⠛⠛⠛⠷⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⡴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣀⣠⡄⠀⣤⣀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢿⣷⢰⣿⢿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟

⠀⢀⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣴⠶
⠰⣿⡏⠀⠀⢀⣀⣤⣀⣤⣄⣀⣠⣦⡀⠘⠟⠀⠀⠀⠀⠀⠀⣴⣿⣷⣶⣶⣶⣶⣶⣶⣦⡄⠀⠀⠀⠻⢿⣷⠀⠀⠀⠀⠀⣠⣿⡿
⠀⣿⣇⠀⠀⢾⣿⣿⣿⣿⣉⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠟⢿⣿⣿⣿⣿⣿⣿⡇⠀⣀⣀⣤⣤⣁⡀⠀⢰⣷⠸⠿⠇
⠀⠘⠟⠂⢀⣬⣿⡿⣿⣿⣿⣏⠙⠛⢻⣶⣶⣾⣷⣤⠀⠀⢠⡞⠉⠀⣀⣀⠀⠉⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣶⡞⠋⣀⣀⡀⠻⠃
⣀⣀⠀⢀⣾⣿⡿⠃⠀⠈⠻⣿⣦⣴⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⡤⢾⣿⣿⠷⣀⣉⣙⣿⣿⣿⣿⠿⠿⠟⢿⣿⣿⣧⣼⣿⣿⣿⠀⠀⢀⣄
⠿⠋⣶⣿⣿⡟⠀⠀⠀⣀⣴⡿⢿⡿⠟⠛⠛⠿⢿⣿⣿⣿⣶⠖⠓⠼⣿⣿⣾⣿⡿⢿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⣿⡿⠛⢿⣿⣧⡀⠈⠛⢶⠄⠀⡀
⠀⠈⠛⠻⣿⠃⠐⠀⠻⢿⣿⣤⣼⣧⠀⠀⠀⢀⡀⠹⢿⡿⠋⠀⣀⡀⠘⢿⣿⣷⣆⠀⠹⣿⣿⣄⠀⠀⠘⠋⢹⣿⣿⡟⠀⠀⠀⠙⢿⣷⠀⣀⣀⣾⡿
⠀⠀⠀⣠⣤⣤⣄⣠⣶⣾⣿⣿⣿⣿⠀⠀⠀⠸⣿⣆⠀⠀⠀⠈⠻⠿⠆⠀⠙⠻⠋⠀⠀⠋⠉⠁⠀⠀⣀⠀⠸⣿⣿⣿⣦⠀⠀⠀⠀⠙⠿⣏⣠⣀
⠀⠀⠀⠉⠙⠛⠛⠛⢿⣿⣿⣿⣿⣷⡀⠀⠀⣹⣿⣿⡆⠀⠀⠀⢀⣤⠀⠀⠀⣠⣤⡀⠀⠀⣠⣿⣶⣼⠋⠀⠀⠀⣠⡄⠉⣶⡄⠀⠀⠀⠀⢿⣿⠏⠘
⠀⠀⠀⠰⢷⣦⡄⠀⢸⣿⣿⣿⣿⣿⡇⣰⣿⣿⣿⣿⣿⣶⡀⠀⠈⠉⠀⢀⠘⠋⣁⡅⠀⢰⣿⣿⣿⣿⠚⠛⣾⣿⣿⣿⣶⣍⣤⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⣶⣶⣿⣧⣴⣿⣿⣦⣼⣿⠛⠻⣿⣆⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠠⠄
⠀⠀⣠⣄⠀⠀⠀⠚⠋⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⠟⢻⣿⠟⠛⠋
⠀⠸⠿⠿⠿⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠟⠉⠀⠉⠉⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠙⠿⠿⠿⠿⠿⠿⠿⠷⠾⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡀⠀⢀⣿⣿⣿⠿⠛⠻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢁⡝⠛⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠛⠻⣿⣿⡇⠀⠈⢿⣏⠀⠀⠀⢰⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⣾⣿⣶⣄⡸⠿⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠀⠀⣤⡿⠋⠀⠀⠀⠀⠙⠓⠀⣤⠀⠘⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠿⢿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠁⠀⢠⣿⣿⡄⠀⠀⠀⠀⢀⣤⣤⣀⣀⣤⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣿⠟⠀⠈⠋⢻⣿⣿⣿⡌⠹⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣰⣾⣿⣿⣷⣄⠀⠀⢀⣾⣿⣿⡟⠛⠛⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠟⠻⢿⣿⠃⠀⡙⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣼⣿⣿⣿⣿⣆⣴⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⣀⣁⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠐⣿⠟⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠻⠿⣿⣿⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⢿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣴⣷⣦⣴⣿⣤⣿⣿⣷⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠉⠉⠁⠀⠀⠀⠉⠙⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⢠⣴⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡌⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣷⠀⠀⠀⣿⣿⣶⣿⠁⠀⠀⠀⢀⡀⡀⠀⢸⣿⣿⣷⣿⡏⠁
⠀⠀⠀⠀⣠⣶⣦⡀⢻⣿⣟⠀⠀⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣿⡏⠀⠀⠀⠈⠙⠻⠟⠀⠀⠀⠀⣈⡉⢙⣤⣼⢿⣿⣿⠋
⠀⢠⣴⣿⣿⣿⣿⣿⣿⣽⣿⠃⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣾⣷⡀⠀⠀⣿⣿⡿⣿⣿⣀⣿⣷⣤⡀
⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠰⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣦⣴⣿⣿⡆⠉⢹⣿⣿⣩⣿⣿⠆
⠀⠻⢿⣿⠿⠛⠁⣽⣿⣿⣿⣿⣿⣿⣿⠏⠀⠈⢻⡶⠤⣤⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⣶⣿⣿⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠐⠂⡀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⠟⠉⠿⠟⠀⠰⠖⣤⣄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠻⣿⣿⣿
⠀⠀⠀⠀⣄⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣤⡄⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠻⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠈⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⠿⠿⠛⠿⢿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠁⠀⣠⡀⠀⠀⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⡇⠀⡀⠀⣀⣤⣾⡿⠛⣿⣿⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠐⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⣿⡟⠁⠸⣿⠟⠿⣿⣿⣧⣾⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣄⡀⠀⣷⣄⣼⣿⣿⣿⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣷⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠉⠉⠙⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠹⠿⠛⠛⠋⠱

⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⣾⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⣀⣀⣀⣀⣄⣹⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⡿⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠉⠀⢀⣄⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⣀⣀⣀⠀⠀⣤⡖⠀⠀⠀⠠⣾⣿⣽⣷⡀⣀⠀⢠⣄⠈⢹⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣄⢀⣤⠖
⣿⣿⣿⣿⣾⠟⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⣿⠟⠃⢀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⣿⠟⠃⠀⠀⠠
⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠰⠋⠉⠉⠀⠈⠉⠀⠀⢸⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣄
⣿⣿⣿⡿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⠋⠠⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⣶⡿⠷⠀⢠⣾⠿⠏⣉⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢉⣹⣿⣷⣦⡀⠀⠀⠀⠀⠀⢁⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⠀⠆
⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⡖⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⢳⣦⠉⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣷⡀⠀⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⣿⣿⣶⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣤⣤⡄⠀⠀⠙⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏⠉⠙⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠁⢰⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⣠⣤⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⣀⠀⠉⠀⠸⠿⠟⠀⠀⢀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣧⣤⣀⠀⠀⠀⣴⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡟⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣟⡛⢛⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⠀⠃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠿⠋⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠀⠈⢟⠀⠀⣿⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⢰⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣿⡇⠀⠈⠁⠘⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⣤⣴⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠉⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠉⠻⠿⠋⠙⠛⠛⠛⠛⠛⠃

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠖⠀⣀⣤⡶⠶⠾⠿⠿⠛⠛⠓⠶⣤⣀⡀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣴⣾⠯⠅⠀⠀⠀⠀⡀⢀⣤⣤⣄⡀⠉⣿⣷⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣴⣿⣿⣿⡿⠿⢷⣶⣶⠀⣿⣿⣿⣙⣳⡄⠀⢸⢇⡛⠳⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠿⠻⣿⣯⠉⠉⠠⠄⢺⣷⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠘⣿⣿⣿⡿⠟⠛⠛⠻⣿⣿⠀⠀⢀⡀⠀⠀⣠⠞⠀⠈⣿⠃⠀⠙⠛⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⡤⠄⢸⣿⣿⡆⠀⢀⣙⣿⣯⡀⠀⠀⠀⠘⢧⣀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡶⠿⠛⠛⠿⠒⠉⠀⠀⠀⠉⠳⢤⡀⠀⠀⠙⢛⣣⣤⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡥⢀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠿⠿⠛⠋⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣦⣤⣀⠀⠀⠀⠀⠀⢀⣽⡟⠀⠘⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣈⠒⠀⠀⠀⠀⠀⢠⣞⣴⠿⠋⠉⠉⠉⠁⠀⠉⠛⠻⠷⠦⠴⠖⠋⠁⠀⠀⣲⠆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣹⣷⣶⣶⣶⣶⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠟⠃⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡴⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠵⠃⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢐⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠐⣤⠀⠀⠀⠀⠀⠰⣄⡀⠀⠀⠀⠐⣤⢹⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⢀⠀⣧⠀⡆⠸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠠⡰⣄⡈⢳⡄⣄⣀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⢧⣑⣻⡆⣧⢠⠀⠀⠀⣾⡀⠘⠇⠀⣠⣠⣧⣿⣿⣄⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠈⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠶⠶⠿⠶⠶⠶⣶⣤⣄⡀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⢀⣴⠟⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠤⠤⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠐⣿⠖⠒⠺⢿⣓⠒⠒⠶⠶⠶⠤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣀⡀⠀⢹⣧
⠀⠀⠀⠀⠙⠿⣦⣄⡀⠈⠉⠑⠲⠤⢄⣀⠀⠀⠀⠉⠉⠉⠛⠓⠲⠶⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢽⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣦⣤⣀⡀⠀⠉⠙⠒⠲⠤⠤⣄⣀⡀⠀⠀⣾⠀⣿⠙⠓⠶⠦⣤⣀⡀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠷⣶⣤⣤⣄⣀⡀⠀⠉⣽⠟⠋⠉⠛⠲⣤⣤⣤⣀⣈⣉⣛⣲⣤⣀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⣿⠁⠀⠀⠀⠀⠀⠘⣧⣀⠀⠀⠀⠀⠁⠀⠈⢙⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⡿⠛⠛⠛⠛⠿⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉
⠀⡀⣀

⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⣰⣣⡄⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⠘⣣⠒⠀⠀⠀⠀⢾⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄⠀⠀⠀⠀⣴⣶
⠀⠀⠀⢠⡯⠏⠁⡀⠀⠀⠸⢿⡿⠇⠀⠀⠀⠠⠰⠀⠀⠀⠀⠴⣿⣿⡷⠄⠀⠀⠀⠀⠀⠀⡂⠀⠀⢀⣾⣿⣿⣧⡀⠀⠀⠀⣷⣿⣆
⠀⠀⠀⠚⠉⠙⠛⠛⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠘⠉⠉⠉⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢀⣴⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⣸⡀
⠀⠀⠀⢀⣼⣿⣧⠀⠀⠀⢀⠡⠀⠀⠀⠀⠀⣼⣿⡄⠀⠀⠀⢠⣾⣿⣦⠀⠀⠀⠀⢾⣿⣿⡂⠀⠀⠀⢀⣽⣿⣄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣴⣿⣿⡄
⠀⠀⠀⣠⣾⣿⣷⣄⠀⠀⠈⠐⠀⠀⠀⠀⢠⣾⣿⣦⠀⠀⠀⣴⣿⣿⣷⡄⠀⠀⠰⣿⣿⣿⡦⠀⠀⠀⣠⣾⣿⣧⠀⠀⠀⠀⠄⠀⠀⠀⠀⠘⣿⣿⣿⡷
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠙⠙⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣉⣈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣀⣴⡟⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠙⢳⡀
⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⣰⡏⠀⢸⣿⣸⣆⠘⣿⡇⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⡸⠃
⠀⠀⠀⠀⠀⢸⣿⣏⠀⠀⠹⣿⣤⣈⣿⣿⣟⡸⠟⠋⠉⢹⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣷⣾⡁
⠀⠀⠀⠀⠀⠈⠻⠿⣷⡶⠶⠶⣻⣿⠙⠛⣋⡶⠀⡀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⠿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⣰⡾⢻⡟⣥⠀⠀⠹⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣶⣶⠀⢰⡅
⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⢀⣾⣿⠃⣾⡟⢿⣷⡀⠀⣠⣵⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠈⠛⠤⡀⢿⣿⣇⣸⣿⠃⢸⣟⣵⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣇⠈⣧⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⠿⣿⣿⠀⢘⣾⣿⠋⠙⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣛⡛⠛⣋⣉⣉⡉⠙⠛⣋⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠛⠛⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠾⠹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⢀⠀⠀⣼⡆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⣟⡀⣠⣾⣧⢀⣿⡇⢀⣶⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢀⡙⠻⠷⣯⣿⣿⠿⢷⣿⣿⣷⣶⣶⠶
⠀⠀⠀⠀⠀⠀⢸⡇⠠⣤⣤⣾⡿⢿⡾⢿⣿⣿⡿⠂⣠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡅
⠀⠀⠀⠀⠀⠀⠈⣿⣇⠻⠟⠋⣴⡿⢿⡄⠈⠙⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣷⣶⣦⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⢸⣧⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠸⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢙⣻⣿⣿⣿⣿⣿⣋⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣲⣾⣿⡂⠀⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠦⣤⣤⣽⣯⣭⠯⠼⠿⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⢿⣿⣷⣶⣿⣿⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡿⠿⢿⣤⣤⣤⣤⣶⣶⠾⠿⢿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉


⣿⠿⠿⠿⢿⡿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⢿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠠⠦⢄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⡀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⢶⣀⠐⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⠈⠀⠀⠀⠀⠀⠀⠐⡺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⢋⠃⠀⠀⠀⠀⠀⢠⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠁⠀⠀⢀⢔⡗⣉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣦⡤⠆⣀⠀⠀⠀⢀⣲⠿⠛⠁⠀⣀⡠⢤⢎⣼⣶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⣀⣴⠀⠉⠙⠛⠓⠛⠛⠛⢿⣷⣄⢀⣷⣐⠥⢆⢀⡾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⣛⢿⣿⣿⣿⣋⢵⣿⡟
⠀⠀⠀⠀⠀⠀⡹⠀⠀⠀⠀⠀⠀⠀⠦⣽⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣾⡆⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠟⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀⠘⢃⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡶⠿⠿⣿⣰⡿⢿⠃
⠀⠀⠀⠀⠀⣾⣿⣡⣴⠀⠤⠄⠐⠰⠤⡁⠀⠰⠶⢦⣀⣀⠀⠀⠀⠀⠀⠋⠹⠉⠁⠀⠀⠈⠛⠙⠋
⠀⠀⠀⢀⢸⣿⣿⠟⡽⣶⣮⣰⠷⠶⠶⠾⠀⠀⣰⣋⣩⣿⣆
⠀⠀⠀⠊⣼⡿⢁⣾⣿⣿⡟⠁⠀⠀⠀⠀⠀⣽⡏⠁⠀⠀⠉⠢⠄⠀⠀⠀⢀
⠀⠀⠀⡟⡟⢰⣋⣿⡿⠋⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⣿⡂
⠀⠀⠀⢹⡇⠀⢶⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢺⡄
⠀⠀⠀⢺⡇⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠼⣧
⠀⠀⠀⠸⣇⢼⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠐⣿
⠀⠀⠀⢶⣿⠿⠀⢸⡻⠀⠀⠀⢰⡇⠀⠀⣸⣧⠀⠀⠀⠀⣄⣠⠀⠀⢸⡆
⠀⠀⠐⠿⠟⠀⠀⠸⣿⣷⡀⠀⠈⠉⠂⠰⢿⣿⠜⠓⠒⠂⠁⠀⠀⠀⣾⣿⢆⡀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠨⠶⠿⠗⠀⠀⠀⠘⠻⡷⠶⠂⠀⠀⠀⢠⣐⠠⣶⣩⣤⡥⠀⢈⡁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⠒⠒⠛⠓⠛⠓⠒⠲⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⢚⣉⣥⠴⠒⡖⠋⢙⡏⡀⠀⠀⡈⣟⠉⠓⡖⠲⢤⣍⣙⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢚⣡⠴⠚⣏⣀⡼⠶⠖⠛⠋⠉⠉⠉⠒⠊⠉⠉⠉⠛⠓⠶⠾⣄⣈⡟⠲⢤⣙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⣡⣖⠋⣹⡶⠞⠉⠀⠀⠀⠀⠀⠀⠐⣆⡐⠀⡗⢰⡆⠀⠀⠀⠀⠀⠀⠈⠑⠢⣾⡉⢓⣦⡙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⢏⢁⡬⠋⠁⠀⠀⠐⢆⠀⢀⡀⠀⠀⠀⢹⣆⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠉⠪⢄⢉⠶⣌⠳⣄
⠀⠀⠀⠀⠀⠀⣰⠋⡠⢎⣀⠜⠁⠀⠀⠀⠙⢦⡀⠈⣆⠸⣗⠀⠀⠀⠄⠹⠴⠇⠾⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠙⢄⣈⠦⡈⢳⡀
⠀⠀⠀⠀⢀⡞⢁⠎⣀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠙⢢⠘⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⡨⢆⠙⣆
⠀⠀⠀⢠⠟⢠⡃⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣱⣤⡤⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠈⢂⠀⣡⡘⢦
⠀⠀⢠⡏⣰⠁⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣿⠈⣷⠀⠀⢻⠃⠀⠀⠀⠀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠀⠀⢣⠀⢱⡈⢧
⠀⢀⡞⢠⠋⢱⠃⠀⠀⠙⠛⠳⠶⢦⠄⠀⠀⠀⣀⠐⠠⠀⠀⡝⣧⣿⡇⠀⠈⠀⠀⢀⡆⠀⡅⠀⠠⠐⠀⠀⠀⠀⠀⢤⠞⣋⣤⠖⠃⠀⠀⢳⠉⢣⠘⣆
⠀⣼⠁⠞⠒⠆⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠂⢀⣰⠈⠉⡉⣿⣇⢀⣠⣄⠀⢈⣡⡈⠉⢱⣀⠀⠂⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠶⠚⠆⢹
⢠⡏⢸⠤⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡨⠤⠤⠴⠆⠀⠀⠀⠀⣈⢿⣴⠃⢻⠤⢄⡀⠀⠀⠀⠀⠶⠤⠤⠭⡰⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠨⡤⢼⠈
⢸⠃⣏⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠠⠤⢤⡄⣪⠁⢠⠾⣅⠈⣿⣄⣼⡀⢀⠟⢦⠀⢩⡀⣤⠤⠠⠠⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⡇
⣾⠐⠋⠉⠁⠀⠀⢀⣀⣠⠤⠆⠀⠀⠀⠀⢁⠄⠴⠀⠀⠃⢀⡯⣀⡠⠃⢹⣿⣯⠈⢳⣀⠨⣧⠀⠃⠀⠰⠄⢄⠁⠀⠀⠀⠀⡤⠤⠤⠤⠴⠀⠀⠋⠈⠓
⣾⠀⠀⠀⡄⠀⠈⠉⢀⣀⣀⡀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠘⡁⣀⡅⠀⢹⣯⣿⠆⠀⣂⠀⣙⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠀⠀⡀
⢻⠐⣦⣼⡇⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠉⢀⠀⠀⠀⢋⠀⢈⠦⣀⣈⣁⡠⢎⠀⢈⠇⠀⠀⢀⠈⠀⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠀⠀⣧⣤⡆
⢸⡆⢧⣀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠓⢅⣀⡜⠉⢹⠀⣈⠶⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣀⠇
⠈⣧⢸⡉⢹⡀⠀⠀⣀⣠⡤⠖⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠉⠙⠛⠛⠉⢁⣀⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣸⠉⡹⢠
⠀⢹⡄⢷⠚⢧⠀⠈⣉⣤⠶⢛⣩⡤⠀⠀⠀⠀⢀⠂⠀⠘⢀⡄⠀⣼⠁⠀⠀⠀⠀⢹⡌⠠⡄⠙⠀⠀⢂⠀⠀⠀⠀⠀⠳⢤⣀⠀⠀⠀⠀⢠⠟⢲⠇⡼
⠀⠀⢳⡘⣦⠼⢇⠀⠉⡤⠞⠋⠁⣤⠂⠀⠀⠀⠀⠐⠈⢀⠀⡅⠀⠋⠐⠒⠀⠀⠒⠀⠃⠈⠍⠀⠈⠐⠀⠀⠀⠀⠀⠐⠦⣄⡈⠙⠶⠀⢀⠾⢤⡞⢰⠃
⠀⠀⠀⢧⠘⣄⡨⢦⠀⠀⠂⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡡⢠⠀⠀⠑⠀⠐⠁⡀⠀⠠⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠀⢠⠿⣀⠞⢠⠇
⠀⠀⠀⠀⢣⡘⢦⢈⣧⡀⠈⠁⠀⠐⠀⠀⢀⡼⢢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣏⣠⠎⣰⠃
⠀⠀⠀⠀⠀⠱⣄⠻⡄⢙⢦⡀⠀⠀⠀⣠⠋⣰⠋⢀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠀⠀⠀⠀⢀⢞⠀⡼⢋⡔⠁
⠀⠀⠀⠀⠀⠀⠈⢢⡈⢣⡀⣱⢤⡀⠈⠁⠼⠃⠀⣸⠁⠀⠀⠀⠀⠀⠀⡏⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣄⠀⠀⠀⣠⢴⡁⣠⠟⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⠣⣄⣹⠶⣄⡀⠈⠀⠛⠀⠀⠀⠁⠀⠀⠀⡇⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⡴⢾⣁⡤⠛⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣌⡑⠦⣴⠋⠓⣤⣄⡀⠀⠀⠀⠀⠀⠘⠃⠠⠀⠇⠀⠀⠀⠀⠀⠀⣀⣤⡔⠋⢳⡤⠖⣉⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢬⡙⠲⠧⣄⣹⠛⠒⡶⠦⢤⡤⡤⠒⠠⡤⣤⠤⠶⡖⠛⢻⣁⡠⠷⠚⣩⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣭⣙⡒⠓⠶⠶⠧⠀⠀⠀⠠⠷⠴⠖⢓⣋⣩⣥⠴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠒⠒⠒⠒⠒⠒⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠃⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣧⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⡿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⣄⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢯⣄⠈⢷⣜⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢫⣿⣾⠁⢳⣄⠻⡷⣝⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣱⣿⣿⡟⢲⡀⠻⣧⡈⢮⡳⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⢃⣼⣻⣿⡟⣇⠀⠹⣦⡈⢯⡢⣙⢦⡑⢽⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢟⡽⢋⣴⡟⣱⣿⠟⣴⡏⢷⣄⠈⠻⣦⡙⢮⡓⢭⣢⣌⠛⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⠖⢋⣴⣿⢋⣼⣿⢋⣾⠳⡇⠀⠹⣷⢄⠙⢿⡳⣽⣦⡈⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠵⠚⣩⠞⣡⣾⠟⣡⣿⠃⢠⡟⢦⡀⠈⠳⡙⢦⣙⢦⣉⠻⢶⣄⡀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⢾⡿⣫⡾⡿⠃⣠⢿⣦⡀⠻⣦⡀⠙⢦⡈⠓⢬⡳⢤⡈⠛⠳⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠿⠛⣱⣟⡴⣫⠞⢁⣼⠟⢸⡏⠻⣦⡈⠻⣷⣄⡙⠦⣀⠈⠑⠻⢷⣤⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢋⡴⠃⣴⣿⠋⣰⢻⠷⣄⠙⢿⣷⣌⠻⣝⠲⣬⡳⢤⡀⠀⠀⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢁⡴⠋⣠⣾⠟⢡⣾⠏⣸⠧⣜⢷⣄⡙⠻⣟⢮⣝⣦⣍⠓⠿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⢀⡴⣋⡴⣚⡿⢁⡴⣫⢏⡼⢻⠀⠈⠳⣝⢿⡦⣌⠙⠮⣙⠻⢿⣶⣤⣉⡙⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠛⣡⠖⢋⡴⢋⡜⣡⠞⠁⢸⠀⠀⠀⠈⠓⢿⣮⣝⠲⠮⣽⣶⣄⡉⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢚⣡⠖⢋⡴⢋⡾⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⡈⠉⠙⠛⠶⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡾⠿⠚⠉⢀⡴⢋⡴⠋⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣫⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢬⣵⣦⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣷⢶⡀⠀⠰⠿⠁⠀⡠⣤⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢇⡀⣐⣤⠐⣿⠉⠉⠧⠀⠀⢸⣞⡁⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⢇⣱⠤⢄⣖⠰⠆⢴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡴⢷⣠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⢶⣶⡛⣠⢤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠙⣿⣿⡿
⠀⠀⠀⠀⢀⣤⣶⣾⣯⣻⣵⣾⠩⢉⡁⠊⠸⣿⣿⣴⣾⣯⣉⣻⣚⣇⠜⢶⣟⣉⣟⡛⢹⣏⣽⣿⡃⠀⠠
⠀⠀⢀⠐⣾⠃⠞⣿⣿⠀⠀⠁⠀⠀⠀⠛⣶⢞⣿⡿⠟⣛⠘⠃⠿⠿⡟⣠⣤⡿⢿⣿⠛⣹⡾⢻⣿
⠀⠀⠻⢸⣿⠠⠀⢸⣿⠀⠀⠀⠀⢀⡀⠀⢰⠘⣷⣻⡟⠈⠙⠀⠂⠀⠘⠛⠻⣧⠿⣿⣽⢿⡛⣿⣿⠄⠀⠀⣠⣤⡆
⠀⠀⠀⢟⣿⡀⡀⢸⡟⠀⠀⠀⠀⠈⠉⠀⢀⠽⣏⣿⣌⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⢛⣀⠠⣛⣵⣟⣠⠐⣾⣿⡟
⠀⠀⠀⠈⢻⣇⠡⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡍⣩⣿⠀⠀⠀⠀⠀⠀⠀⣰⣦⡾⠩⠄⢺⣏⣻⠿⣿⡟⡹⠋
⠀⠀⠀⠀⠀⣿⡄⠉⠀⡄⠀⠀⠀⠀⠀⠀⢀⡀⢥⡄⠀⠀⠀⠀⠀⠀⠀⠀⡽⠟⠀⠀⠀⠉⠉⠏⠀⠙⡟⠁
⠀⠀⠀⠀⠀⠸⣿⣆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠙⠁⠤⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡄⠐⠀⠀⠀⠀⠀⠀⠘⠛⠂⠀⠠⠄⠀⠀⠀⠀⠒⠒⢷
⠀⠀⠀⠀⠀⠀⢠⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⣀
⠀⠀⠀⠀⠀⠀⣤⡀⠸⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⣰⣄
⠀⠀⠀⠀⠀⠀⠀⠰⣄⣼⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠟⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣀⣀⢄⣀⠤⠒⣐⣂⠈⠈⠈⣐⣒⡒⠤⣀⡀⢀⣀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠔⢤⡎⡡⢊⡴⠟⡩⠔⠊⠉⠀⠀⠀⢀⠀⠀⠨⠉⠑⠦⢍⡳⢧⡔⠪⠱⣀⠤⣄⣀
⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠐⠚⡃⢴⠂⢨⣵⠏⡤⡊⠀⠠⢴⠒⡒⢄⢀⠞⢶⣶⡶⠒⠒⠲⢶⣿⣦⡙⣶⡙⢁⢴⠄⠞⠶⠀⠀⠀⠀⠀⠀⠀⠠
⢀⠖⢂⡑⡇⠀⢠⠀⢀⠀⢀⢀⡀⠀⠛⢹⠀⡴⢡⣊⠋⠀⠀⠠⢘⠁⣡⠋⠈⣶⡶⢀⢘⠀⠠⠀⠀⡀⠀⠿⣌⢣⡀⢸⠀⢀⡄⠀⠀⠀⢀⠄⠀⠘⠃⠦⢂⡤⡀
⠉⠲⠤⢄⡇⠀⠉⠁⠀⠁⠀⠀⠀⠀⠁⢸⡾⢡⡏⢅⠀⠀⠀⣠⣴⣿⠁⠈⡬⡬⡀⠆⠉⠀⠀⠀⠀⡇⠀⠀⠘⡄⢷⢿⠀⠀⠀⠀⠈⠀⠀⠉⠓⠒⢠⠋⡡⠠⠜
⠀⠀⠀⡔⡧⠄⢠⠀⠤⠄⣄⠀⠀⠀⠀⢾⡇⡎⠑⠒⣶⣶⣿⣿⣿⣿⣤⡀⠀⠀⠈⠖⠃⠀⠀⠐⠂⠃⠀⠀⠀⠸⠘⡽⣤⠤⠀⠄⠠⡤⠤⡄⠤⠤⠴⠃
⠀⠀⠀⠈⠁⢸⣁⣁⠤⠎⠙⠒⢄⠉⠂⢿⠀⠇⠀⠀⣻⣿⣯⣿⡿⡿⠁⠀⢀⡤⡆⠀⠠⠀⠀⠰⠄⠀⠀⠀⠀⠀⡇⣿⠇⠊⢀⠔⠊⠑⢤⣈⣂⣘⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣻⠀⡇⠀⠀⠈⢡⠈⠁⠀⠀⠂⠠⡎⠀⡇⡲⢖⢀⢦⠐⠀⣆⠀⠀⠀⠀⡇⡗⡙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⣧⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠰⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡘⣄⠀⢀⠎⠀⠐⠀⠀⠀⠀⣯⡆⠀⠀⠸⠂⠐⠀⠘⠀⠀⠀⢠⠃⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⢶⡞⠀⠀⠀⠀⠀⠀⡆⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢃⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡑⢤⠀⠀⠄⠀⢈⠅⠈⢿⡆⠀⢠⠀⠀⠀⢠⡠⠊⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣍⡚⠥⣤⣴⣶⣾⣿⡁⠀⣰⣀⡤⠖⣉⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠬⢭⠍⠉⠩⠭⠩⠄⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠴⠦⠤⠶⢶⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⣤⠀⠀⠠⡄⠀⠀⠀⠘⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣴⠣⠀⠀⠀⢀⠀⡀⠂⡀⠀⠈⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠉⠠⠀⠀⠀⠀⡌⠛⠀⢠⠀⠀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠠⠀⠀⠈⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡖⠄⠀⠀⠠⠠⠀⠄⠀⠀⠀⠙⠃⠀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⢃⡀⣀⣀⠀⠀⠀⠀⣼⠏⠁⠁⠄⠀⡀⢴⣦⣀⣀⠀⠀⠠⠇⠀⢀⣹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠈⡟⠀⠉⠳⡀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠨⠁⠈⠉⠀⠀⠀⠀⠀⠈⠛⠹⣿⣿⣿⡧⡤⣄
⠀⠀⠀⠀⠀⠀⠑⠤⣀⠀⠀⢀⡠⠁⠀⠀⠘⣤⡾⠀⠀⠀⣀⠠⠤⠴⠚⠒⠀⠀⠀⠒⠠⡀⠀⠀⠀⠈⠂⣬⠛⠿⠇⠀⠈⡄
⠀⢀⣀⣀⣀⠀⠀⢐⡏⠘⠃⠈⠀⠀⠀⢀⣴⠿⠷⠀⠒⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠇
⢀⡈⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⢤⠤⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢹⣷⣦⣤⠴⠞
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⣄
⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⢀⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣆
⠀⠈⠶⠤⠴⡦⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣯⣿⣿
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢀⡄⠂⠉⠉⠛⠻⢿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠁⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠃⠀⠤⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠙⠄
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠋⠀⠀⢠
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⠀⠀⠀⠀⢰⣆⣤⣶⣿⣿⠿⠛⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣴⣿⠿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠏
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠖⠚⠉
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣤⣤⣶⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⠛⠛⠛⠛⠛⠉⠈⢡
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⣀⣠⠞⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣾⣶⣿⣿⡇
⠀⠀⣀⠠⢤⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⣿⠁
⠀⣸⠀⠀⠀⠀⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣴⡿
⠀⣇⠀⠀⠀⠀⠀⠠⢤⣤⣴⡶⠦⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠿⠋⠁
⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠏
⠀⠀⠙⠒⠶⢤⣤⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡀⠀⡰⠃⠀⠀⠀⠀⠀⢀⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⢤⣄⣀⡀⠀⠀⠀⠀⠀⠈⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠢⢤⣤⠀⣀⣀⣄⣀⡀⢀⣀⣠⣤⠖⠋

⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⣀⣀⣀
⣿⡇⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⣞⣁⠈⣿⡏⠀⣿⣷⠀⡔⢉⠀⣿⣿⠀⠀⣽⣿⠀⠹⠇⠀⠀⠀⠐⠿
⣿⣧⠚⣻⡷⠈⣿⡇⠀⣿⣷⠚⢿⣷⠀⣴⣿⠉⣿⣏⠀⠀⠀⠙⠋⠀⣿⡇⠰⣿⣁⠀⠛⠛⠀⣿⣿⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⠘⣿
⣿⡇⣼⣟⠁⠀⣿⡇⠀⣿⣿⠀⢸⣿⠀⢻⣧⠀⣿⡟⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠈⠻⣿⣦⠀⠀⠀⠀⣿
⣿⠇⠙⣿⣆⠀⢿⡇⠀⢿⡿⠀⣼⡟⠀⢰⣯⣭⣁⣀⠀⠀⠀⠀⠀⠀⠿⣧⣠⡼⠟⠁⠀⠀⠀⢿⡿⠀⢠⣦⠀⠀⣿⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠈⢻⣷⡤⠀⠀⠀⠀⠈⠉⠀⠀⣼⠉⠉⢙⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠐⠀⠁⠄⠀⠀⠖⠀⠀⠂⠀⢀⠀⠀⠐⠀⢘⠀⠸⠁⢰⠀⠀⠐⠀⠀⡃⠀⠀⠀⠐⠀⣀⢠⠀⡀⠀⠌⠀⠀⠀⠀⠀⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢠⣤⠤⣄⠀⠀⣠⠴⠦⣄⠀⢠⣤⣤⣤⡄⢀⣤⣤⣤⡀⠀⣴⣦⣰⣶⠀⢶⡄⣤⣤⣤⣤⡄⠀⠀⠀⣤⠀⠀⠀⠀⡀⠀⠀⡄
⢸⣿⠀⠈⣇⢰⠃⠀⠀⠸⡆⢸⠉⠉⠉⠁⢸⣿⣤⣈⠁⠀⣿⣿⣿⣿⠀⠘⠁⠛⢻⡟⠛⠃⠀⠀⠀⣿⡆⠀⠀⣼⡇⠀⣸⢻⡄
⢸⣿⠀⠀⡏⢸⡀⠀⠀⢠⡇⢸⠿⠿⠇⠀⠀⠈⠻⣿⣧⠀⣿⠹⣿⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⢻⡀⡼⢹⡇⢠⡿⠾⢷⡀
⠸⠿⠀⠊⠀⠀⠑⠤⠴⠚⠀⠸⠿⠿⠿⠂⠘⠶⠶⠿⠋⠀⠿⠀⠹⠿⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠇⠈⠿⠁⠸⠇⠼⠀⠀⠀⠷
⠀⠀⠀⠀⢠⡀⠀⠀⠀⣤⣄⢠⣤⠀⣤⡀⠀⣠⠀⠀⠀⠀⣀⡤⣄⡀⢠⣤⣤⣤⡀⢠⣤⡄⠀⢠⡄⠀⣀⣤⣀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⡾⢳⡀⠀⠀⣿⣿⣿⣿⠀⠸⣧⣰⡏⠀⠀⠀⠰⣧⡀⠀⠀⢸⣀⣀⣀⠀⢸⣿⠹⡄⢸⡇⢸⣇⠀⠀⠀⡟⠉⠉⠉
⠀⠀⠀⢰⡷⠾⢧⠀⠘⣿⢻⣿⣿⠀⠀⢻⡿⠀⠀⠀⠀⠀⠙⠻⢷⡄⢾⠉⠉⠉⠀⣼⣿⠀⠹⣸⡇⠀⠙⠿⣶⠀⣿⠶⠶
⠀⠀⠀⠞⠀⠀⠈⠧⠀⠿⠀⠹⠿⠀⠀⠿⠁⠀⠀⠀⠀⠀⠤⣤⠾⠃⠸⠤⠤⠤⠀⢿⣿⠀⠀⠻⠇⠠⣤⣤⠟⠀⠷⠶⠶⡶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣙⣿⣡⠄⢄⢸⣇⡤⠀⠀⠀⡆⣶⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠑⢄⠀⢹⡏⢀⡴⠃⢀⡿⠯⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⡆⠀⢰⠈⠀⣤⠑⢼⡷⢋⣄⠀⠈⡆⠀⢰⢀⡴⠋
⠀⠀⠀⠀⠀⠀⢤⣠⠀⡄⠠⠤⠷⢇⠀⢸⠀⠘⣿⠇⣸⣇⠸⡿⠀⠀⡇⢀⡼⠯⠤⠄
⠀⠀⠀⠀⠀⠀⢘⣛⣷⣇⢀⣀⣀⣠⣵⣾⠀⠀⡠⠊⢸⡏⠳⣄⠀⢀⣷⣋⣀⣀⣀⡀⠀⢀⡔⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠁⠀⠀⠀⠀⠀⡱⢎⠀⢠⢾⣧⡀⢈⡵⢏⠀⠀⠀⣀⠀⠐⠚⡻⣟⠻⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣄⠈⠳⣄⠺⠿⠗⣠⠞⠁⢀⡝⢇⢸⣇⡽⠫⡀⠀⠑⢄⠻⠿⠛⣠⠊⠀⡨
⠀⠀⠀⠀⠀⠰⠶⠾⡷⠶⠾⡷⠶⢾⠷⠶⢶⡷⠶⢶⣿⣿⠶⠶⣾⠷⠶⢾⡷⠶⢾⡶⠶⢾⡶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⡊⠀⡠⠋⣤⣶⣦⠑⢄⠀⢉⡶⣋⢸⡇⢱⣞⠁⢀⡴⠋⢴⣾⡦⠙⢦⡀⠙
⠀⠀⠀⠀⢀⣀⣦⣽⣮⡤⠀⠀⠉⠀⠀⢀⡵⢋⡀⠈⢻⡷⠃⠀⡱⢎⠀⠀⠀⠀⠀⢀⡀⠉
⠀⠀⠀⠀⠀⠀⠋⠜⠁⠀⠈⠉⠉⢉⡽⣿⠀⠀⠙⢦⣸⣇⡠⠊⠀⠀⡟⢏⠉⠉⠉⠀⢹⢿⣭⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⡿⡏⠀⢹⠀⠀⣾⡆⢹⡏⢰⣿⡄⠀⡇⠀⣷⢖⠒⠂⠘⠀⠋⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠇⠀⠘⡀⠀⠙⣡⢾⡗⣄⠛⠀⠀⠇⠀⠸⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣲⣾⠁⢠⠞⠁⣸⣇⠀⠑⢄⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠿⠸⠀⠀⠀⠚⢹⡇⠑⠐⢋⣿⣍⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠠⠀⠀⠀⠀⠀⠠⣆⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⢀⡀⠀⠀⣰⣶
⠀⠀⢻⣤⣴⣆⠀⠀⢠⠿⠛⠻⢦⡀⠀⢰⠿⠛⠻⣦⠀⠀⢠⠿⠟⢷⡀⠀⠀⠀⣴⣼⣻⣿⣿⣾⠋⠁
⠀⠀⠺⠋⠀⠉⠑⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⣘⣿⠿⠿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠒⠚

⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀
⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢀⣴⣿⣿⣿⣯⣿⣿⣷⣄
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⡆
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠈⠻⠿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠙⠻⠿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠈⠻⠿⣿⣾⣿⠿⠋⠁
⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀
⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣠⣾⣿⣿⣿⣹⣿⣿⣿⣦⡀⠀⠀⢀⣴⣿⣯⣿⣏⣿⣿⣿⣿⣆
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⠀⠀⣿⣯⣽⣿⣿⣿⣿⣿⣿⣭⣿⡇
⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠘⣿⣯⣾⢿⣿⣿⣿⣿⣮⣿⡟⠀⠀⢻⣿⣽⣿⣿⣿⣿⣿⣿⣽⣿⠃
⠀⠀⠈⠻⢿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⠟⠁

⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⣹⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡄
⠀⠀⠀⠀⢀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⢀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣿⣟⠦
⠀⠀⠀⠀⣸⣯⣿⣧⠀⠀⠀⠀⠀⠀⢴⣶⣾⣶⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⣠⠐⠂⢁⠀⠀⠀⠀⠀⠀⢾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⣿⣟⣆⡀
⠀⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⠀⠀⢠⣶⣷⣶⣷⣶⣿⣿⡆⠀⠀⠀⠀⠀⠂⠠⢀⡀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠈⠉⣽⣿⣿⣿⣿⠛⠄
⠀⠀⢀⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣬⣟⣛⣛⣛⣫⣭⣾⣿⡆⠀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣰⣿⣿⣿⣿⡆⠀⠀⠀⠀⢀⣤⣿⣾⣿⣿⣿⣿⣇⡀
⠀⠀⠈⠉⠉⠹⠿⠉⠛⠃⠀⠀⠀⠉⠈⠛⠋⠿⠋⠈⠉⠈⠁⠀⠀⠈⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⠏⠉⠉⠀⠀⠀⠀⠈⠉⠈⠉⠹⠟⠛⠿⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠐⠚⠛⠿⠿⣿⣿⣿⣻⢿⣿⣟⡲⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⡠⠆⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣯⡻⣿⣌⡻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⡴⠋⠀⣠⡤⠀⢀⡀⠀⠀⠀⠀⠀⠠⠌⠑⠌⢻⣿⡹⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠟⣉⣤⣶⣿⣯⡽⠿⠒⠩⢭⣥⣖⣀⣠⠀⠈⢢⡀⠈⢻⡇⢸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡟⢀⣿⣿⠦⠄⠉⠁⠈⠩⠉⠛⠻⢿⣦⡀⠀⢻⡌⢊⠃⣾⡿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠈⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠱⠀⢺⢟⣠⠍⣛⣻⣿⣿⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⣴⣆⣠⣶⣷⣄⢀⠈⣿⡆⢠⡀⠀⢽⡆⠀⢀⡄⠈⠐⠋⠉⠉⣩⣿⠋⠉⠻⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠒⠫⠵⡻⠟⠋⠙⠆⣐⠐⠚⠉⠁⠀⠈⢧⡀⠂⠀⠀⠠⠀⠤⠾⠋⠁⠀⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠀⢿⢸⣆⣤⠴⠛⠁⠈⠀⠀⠁⠉⠉⠢⣤⣤⡄⡆⠀⢈⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⠓⠌⣼⣿⠃⢠⠀⡁⣴⣿⣿⣦⡀⢄⡀⠈⠙⠓⠛⠂⠀⢃⡀⢶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠃⠀⢠⠀⠀⣤⠻⠀⠁⠙⢿⡿⠋⠈⠀⠙⠒⠒⠂⠀⠀⠀⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠑⠀⠀⢀⠀⠊⣄⠀⠀⠦⢂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠉⠑⠦⠤⢤⣄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣨⣦⣀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢿⢰⣾⣶⣤⡆⣷⣼⡖⠄
⠀⠀⠀⠀⢀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣾⣿⣿⣿⣿⣿⢛⣤⣶
⠀⠀⠀⠀⠘⣿⣿⣷⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⣾⣟⠁
⠀⠀⠀⠀⠀⣽⣿⣿⠟⠂⢄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠢⠛⠛⣽⣿⣿⣿⣿⣿⣿⡶
⠀⠀⠀⠀⠀⠙⢫⠃⠀⢀⣾⠓⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠳⣌⣉⣋⣤⣿⣿⣿⣿⣫⡼⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⡋⢠⡄⢲⡖⢠⣾⣦⣹⣿⣿⣶⣶⣶⣾⣧⣴⣿⣿⣿⣿⣿⣿⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠋⠀⠜⠠⢄⣠⠴⠿⠿⠈⠛⠻⠿⠿⠿⠿⠿⠟⠛⠉⠙⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣿⡇⠀⠀⢸⣿⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡃⠀⠙⢿⡇⠀⠀⢸⡿⠋⠀⢘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⢻⣷⡄⠀⠈⠃⠀⠀⠘⠁⠀⢠⣾⠟⠀⠀⠀⢀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⣿⠟⠉⠻⣿⡄⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⢠⣿⠟⠉⠻⣿⠀⣀⣤⣤⣀
⠀⠀⠀⠀⣼⡿⠋⠙⠻⢿⣿⣧⠀⠀⠹⣿⡄⠀⠀⠀⠈⢿⣧⡀⠀⠀⢀⣼⡿⠁⠀⠀⠀⢠⣿⠏⠀⠀⣼⣿⡿⠟⠋⠙⢿⣧
⠀⠀⠀⠀⠹⣷⣤⡀⠀⠀⠈⠛⠦⠀⠀⠹⣷⡄⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⢠⣿⠏⠀⠀⠴⠛⠁⠀⠀⢀⣤⣾⠏
⠀⠀⠀⠀⠀⠈⣹⣿⣷⣦⣄⠀⠀⠀⠀⠀⠹⣷⣄⡀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⢀⣠⣾⠏⠀⠀⠀⠀⠀⣠⣴⣾⣿⣏⠁
⠀⠀⠀⠀⠀⣼⡿⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣶⣤⣿⡇⠀⠀⢸⣿⣤⣶⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢿⣇
⠀⠀⠀⠀⠀⠸⣿⣦⣤⣤⣤⣶⣶⣶⣶⣶⣦⣀⠀⠀⠈⠙⠻⡇⠀⠀⢸⠟⠋⠁⠀⠀⣀⣴⣶⣶⣶⣶⣶⣤⣤⣤⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠻⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣄
⠀⠀⠀⠀⠀⢠⣶⡿⠿⠿⠿⣶⣶⣶⣶⣶⡿⠛⠉⠀⠀⢀⣠⡆⠀⠀⢰⣄⡀⠀⠀⠉⠛⢿⣶⣶⣶⣶⣶⠿⠿⠿⢿⣶⡄
⠀⠀⠀⠀⠀⢺⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⣿⡇⠀⠀⢸⣿⠿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡗
⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠟⠀⠀⠀⠀⠀⢰⣿⠟⠋⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠙⠻⣿⡄⠀⠀⠀⠀⠀⠻⢿⣿⣿⡟⠁
⠀⠀⠀⠀⢠⣾⠿⠋⠁⠀⠀⣀⡄⠀⠀⢠⣿⠇⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠸⣿⡄⠀⠀⢤⣀⠀⠀⠈⠙⠿⣷⡄
⠀⠀⠀⠀⢿⣇⡀⢀⣠⣴⣿⡿⠀⠀⢠⣿⠏⠀⠀⠀⠀⣴⡿⠃⠀⠀⠘⢿⣦⠀⠀⠀⠀⠹⣿⡄⠀⠀⢿⣷⣦⣄⡀⢀⣼⡿
⠀⠀⠀⠀⠈⠛⠿⠿⠛⠉⣿⣇⠀⣠⣿⠏⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠈⢻⣷⡀⠀⠀⠀⠹⣿⣄⠀⣨⣿⠉⠛⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠋⠀⠀⠀⣰⣿⠋⠀⠀⡀⠀⠀⢀⠀⠀⠹⣿⣆⠀⠀⠀⠙⠻⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡁⠀⢀⣼⡇⠀⠀⢸⣧⡀⠀⢘⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣶⡿⣿⡇⠀⠀⢸⣿⢿⣶⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⣾⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⡀⣆⠀⢰⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠜⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠤⠃
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠂
⠀⠀⠀⢐⠀⠀⠈⡈⠉⠈⡀
⠀⠀⠀⢸⠀⠀⠀⡁⠀⠀⡁⠀⠛⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢈
⠀⠀⠀⢸⡥⠤⠤⡿⢶⣏⡇⢠⢴⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⣸⡀⢀⡦⣤⡀⢀⣯⠒⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠘⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⡠⢖⡥⠁
⠀⠀⢠⠤⡬⡬⠴⠰⡠⡇⢸⠉⠀⠀⡍⡇⠀⠀⠀⠀⠀⢀⡠⠔⠈⡅⠐⠙⠿⠿⠯⠿⠛⠀⢀⡠⠥
⠀⠀⠠⢠⡜⠀⠪⠀⠄⡜⢼⠀⠂⠀⡃⠁⠀⠀⣀⠔⠊⠁⠀⠀⠈⠀⠀⠀⠋⠀⠀⠀⠈⠀⠀⠀⢀⡀
⠀⠀⢠⣨⣟⢄⣩⣨⣠⣌⣸⣀⣀⣀⣆⣇⠔⠊⠠⠇⡄⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠒⡆⠂⠀⠀⠀⠀⠙⡆⠀⠈⢤⡀⠀⠀⠁
⠀⠀⠆⠀⠀⠊⠁⠀⠀⠀⠐⠒⠀⠓⠂⠀⠒⠁

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡀⣀⣜⣿⣟⣄⡀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣖⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢦⣿⡴⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡂
⠀⠀⠀⣽⢿⣿⡈⢹⠋⣸⣿⢿⡅⠀⠀⠀⢻⣿⣶⣨⣿⣅⣰⣿⣿⠂⠀⠀⠀⢳⣾⣄⠀⣿⢀⣠⣷⡚⠀⠀⠀⠈⡿⣷⣆⢉⣏⢁⣶⡿⡟
⠀⠀⠀⠰⣏⣁⣹⣿⣿⣁⣉⡷⠀⠀⠀⠀⠀⣈⣙⣿⣿⣿⣏⣉⠀⠀⠀⠀⠀⣍⠀⣈⣽⣿⣯⡁⢀⣉⠀⠀⠀⠀⠀⣍⣈⣿⣩⣏⣀⡅
⠀⠀⠀⠽⣿⣿⢁⣼⣄⢸⣿⡾⠅⠀⠀⠀⡼⣿⠿⢙⣿⣍⠹⢿⢿⠄⠀⠀⠀⠼⢿⠉⠀⣿⠀⢙⡿⠧⠀⠀⠀⠠⢿⡟⠏⣀⣇⡀⠿⣷⠧
⠀⠀⠀⠀⠁⠈⠩⢾⠮⠋⠁⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠛⣿⠓⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠆

⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆
⠀⠀⠀⢤⡀⣦⠙⢾⠞⢡⣆⣠⠄⠀⠀⠀⣴⡆⢿⠶⣿⡴⣿⢀⣦⠄⠀⠀⠠⣤⣴⠀⢙⣿⠋⢀⣆⣤⠀⠀⠀⠠⣤⣄⠀⣸⣿⡁⢀⣤⣤
⠀⠀⠀⢠⠽⠛⣦⠞⢦⡞⠻⢥⠀⠀⠀⠀⣉⡽⠿⢦⣿⡤⠿⢯⣉⠁⠀⠀⠀⠹⠛⣷⢯⣿⡽⡾⠛⠏⠀⠀⠀⠀⠙⠛⡿⣬⣯⡼⢿⠛⠁
⠀⠀⠀⠘⣳⣶⠛⢶⠞⢳⣶⡚⠀⠀⠀⠀⣍⣹⣶⠚⣿⠙⣶⣟⣉⡀⠀⠀⠀⣰⣤⡿⣟⣿⣻⢷⣴⣆⠀⠀⠀⠀⣠⣼⢷⢛⣟⢳⢿⣤⡄
⠀⠀⠀⠉⠁⠟⡤⢾⠶⡌⠇⠉⠁⠀⠀⠀⠛⠃⣾⠞⣿⠳⢿⠈⠟⠁⠀⠀⠈⠛⠛⠀⣠⣿⣄⠀⠋⠛⠀⠀⠀⠈⠉⠉⠀⢩⣿⡁⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠷⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡆⠀⠀⠀⢸⣿⣿⣿⣄⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⣷⡆⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣤⣤⣤⣤⣀⣸⣿⣿⣿⣿⣿⣿⣿⣟⠀⢠⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣾⣿⣿⣿⣿⣿⣿⡏⠻⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢷⣿⢿⣿⣿⣿⣿⣿⣿⣅⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⣾⣟⣴⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣴⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠈⠻⢿⣿⠟⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⠉⢈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠒⢶⣶⣶⠚⠋⠉⣠⠀⠀⠀⢀⡀⢸⣿⠛⡿⠉⣹⣟⣿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠸⠟⠁⠀⠀⠀⠀⣠⠀⣿⠀⠐⣿⡁⠙⠷⠀⠉⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠏⠀⠀⠛⠇⠀⠀⠀⠀⠀⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡇⢀⠄⠀⠀⠀⠀⠀⠀⠀⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠊⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⢳⠄⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠒⠙⠞⠰⠛⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⢀⠔⠃⠤⡄⠀⠀⠀⠀⠀⠀⢀⡖⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠐⠋⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠈⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⣾⡆⢠⣿⠃⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣧⣸⣿⣠⡴⢏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣦⣀⠀⢸⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠉⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠈⠻⣷⣼⡇⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⣸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣤⢿⡄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⢠⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠌⢿⣶⣶⣤⣤⣀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⢀⣀⣴⣶⣿⣿⣾⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠘⠛⠻⠿⢾⣷⣄⠀⠀⠀⠀⠀⠀⢠⣿⣼⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⣿⣆⠀⠀⠀⠀⢐⣼⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⡀⢰⣞⠋⢹⣇⠀⠠⠊⠛⣿⣁⡀⠴⠶⠆⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣶⣮⡿⠏⠁⠈⠉⠀⠈⢠⣿⡟⢡⣴⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠈⢿⡟⠁⠀⠀⠀⣄⣠⣄⠀⠘⠋⢀⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠃⠀⠀⠀⠐⠛⠛⠛⠁⠀⠀⠚⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⣈⣿⣿⣿⣿⡿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣭⣄⣀⣀⡀⠀⠀⣀⣶⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠟⠉⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡏⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠻⣿⣇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡏⠀⢻⣿⣷⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⣿⣿⡋⠀⢙⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⣻⡯⠀⠀⠈⣿⣿⣿⠿⣻⣿⠃⣾⣿⡻⣿⣿⡿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡋⠀⠘⠿⠀⠀⠀⢿⣿⣥⣖⣺⣿⣵⣿⣿⣷⣮⣿⣇⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡃⠀⠈⡓⠀⠀⠀⠚⢻⣿⣿⣿⠟⢳⣿⣿⣿⡿⢻⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣯⣁⡀⣀⣹⣿⢰⣿⣿⣿⣿⣿⣿⣶⣿⣿⢿⣿⠇⠈⡟⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⡿⡟⣿⢻⣿⣿⡿⠀⢟⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡏⣿⣿⡿⠸⢿⠁⠀⠁⠀⠈⠁⣿⠇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣛⣿⣿⠿⠰⢷⣶⣦⣤⡀⠀⣤⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢿⣿⣿⣷⣶⣾⣮⣿⣿⣿⣶⣿⣿⣷⣶⣯⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢿⣿⣿⣿⣿⣾⣝⣛⡛⠿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠿⠋⣁⠠⠀⠀⠀⠀⠠⢀⠉⠻⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡿⢁⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡟⠠⠂⠀⠀⠀⠀⠀⢺⠂⠀⢀⡀⢀⡀⣴⠣⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠇⠂⠀⡀⠄⠀⠀⠁⠀⠀⠙⢀⣀⣀⢠⠉⠀⠁⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡆⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠂⡀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠁⠀⢹⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠘⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⠋⠉⢻⡆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣀⣰⣧⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢸⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣽⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠈⠉⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⢀⡀⠀⠀⠀⢸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣶⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⣀⣠⠓⠀⠤⠀⠀⠀⠀⠀⡠⠊⠐⢸⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡉⢀⢀⠀⠄⠀⠀⠀⠈⠂⢤⡾⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣇⠀⢀⣨⣦⣤⣶⣶⣶⡶⠶⠶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⣿⠋⠀⠀⠈⢿⠉⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣿⣀⣀⠀⠀⢸⡇⢠⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⣭⣭⡽⠇⠀⠘⢷⣤⣭⣭⡿⠇

⠀⠀⠀⠀⠀⢀⡀⠀⣀⣠⢤⣄⠀⠀⠀⠀⠀⢀⣀⡀⡤⠤⠶⠒⣶⡶⠶⠶⠤⢤⣀⣀⠀⠀⠀⠀⠀⣠⣤⣤⣄⡀⢀⡀
⠀⠀⠀⠀⠀⠈⠃⠀⢬⡙⠀⠙⠀⠀⠠⠖⠀⠉⠀⠀⠁⠀⣄⠀⠀⣤⢨⣍⠀⠀⠀⠉⠙⡲⢦⣄⠈⠋⠀⠒⣩⠭⡌
⠀⠀⠀⠘⠂⡀⠒⠀⠀⠀⢀⡀⠒⠾⠂⣤⡀⠀⠀⠀⠀⠀⢻⡄⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠐⠻⠟⠦⣄⠀⠁⠀⠘⠃⠌⢉
⠀⠀⠀⣰⣇⢳⠄⠀⢀⠔⠁⠀⣀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠘⣧⠀⣿⢸⡇⠀⠀⠀⠀⠀⢀⡼⠂⠀⠀⠈⠳⢤⠀⠀⠰⠃⣾⡄
⠀⠀⠀⢷⡈⠀⠀⠀⠉⠀⠀⠀⠈⠻⢦⡀⠈⢳⡀⠀⠀⠀⠀⢹⣤⣿⣸⡇⠀⠀⠀⠀⢠⡞⠁⠀⠀⡠⠂⠀⠀⠻⣄⠀⠀⣼⠃
⠀⠀⠀⠈⠛⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡻⠄⢀⣨⣥⣤⣤⣤⣤⣄⣀⠀⠀⣰⠏⠀⠀⣠⠊⠀⠀⠀⠀⠐⠘⢧⡀
⠀⠀⠀⠀⢠⠾⠏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⣾⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣶⣄⡀⣳⣾⠁⠀⠀⠀⣀⣤⠶⡀⠿⢷⡀
⠀⠀⠀⢀⡎⠀⠐⠛⠓⠶⠶⠶⠶⠦⠄⢀⣾⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⣿⣿⣷⡎⢀⣠⡶⣛⣭⠶⠛⠉⠀⠀⢷
⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠋⠀⠀⠀⣠⣴⣶⣾⣷⣶⣦⣄⣸⣥⣿⠿⣿⣷⡀⠳⠛⠋⠁⠀⠀⠀⠀⠀⠘⡆
⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⢠⣾⣿⣿⣏⣿⣿⣎⣻⣿⣿⣟⠋⠀⠘⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⢰⡇⠀⠀⠀⠀⢀⣀⣠⡤⠆⢸⣿⡗⠀⠀⠀⣿⣿⣿⣏⣼⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢹⣿⡄⣤⣤⣤⣤⣤⣤⣤⡄⠀⢸⡄
⠀⠀⠸⣷⠀⠶⠖⠛⠉⠉⠀⠀⡀⢿⣿⣧⠀⠀⠈⣿⣧⣶⣾⣿⣿⣿⣿⣤⣤⣼⣿⠇⠀⠀⢸⣿⡇⠲⠶⠶⠶⠶⠶⠶⠆⠸⢿⡇
⠀⠀⠀⡇⠀⠛⠛⠛⠛⠋⠉⠉⠁⢸⣿⣿⡀⠀⠀⢻⣿⣿⠟⢻⠟⢻⣿⣿⣿⣿⡟⠀⠀⠀⣾⣿⠀⠛⠛⠛⠛⠛⠛⠛⠃⠠⢸⠂
⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⣀⣤⢴⡀⢿⣿⣷⠀⠀⠀⠙⢿⣴⣿⣾⣿⣿⣿⡿⠋⠀⠀⠀⣰⣿⠇⢠⣀⠀⠀⠀⠀⠀⠀⠀⡀⣾
⠀⠀⠀⠘⡄⠀⢠⠶⣚⣯⣵⣾⣿⠟⠂⠻⣿⣷⣄⠀⠀⠀⠈⠉⠉⠉⠉⢿⠓⣆⡀⣠⣾⣿⠃⣠⣓⢯⣟⡶⣤⣀⡀⠀⠀⢠⠏
⠀⠀⠀⠀⢳⡀⠈⢿⡽⠞⠋⠁⣰⠞⠁⠀⠈⠻⣿⣿⣦⣄⡀⠀⠀⠀⠀⣯⣿⣿⣿⡿⠋⠁⠈⠳⢮⡓⢮⣝⠶⣭⡻⠁⢠⡞
⠀⠀⠀⠀⠀⢳⣶⡈⠀⠀⣠⠞⠁⠀⠀⠀⣠⢶⡌⠙⠻⠿⣿⣿⣿⣿⣿⠿⠿⢿⡇⢀⠀⠀⠀⠀⠀⠙⠷⣬⡛⠶⠁⢿⡞
⠀⠀⠀⠀⠀⠀⠹⣄⠂⠙⠁⠀⠀⠀⣠⡾⣣⠟⣠⡆⠀⠀⠀⢠⡄⠀⡄⠀⠀⠀⠰⡈⣷⣄⠀⠀⠀⠀⠀⠈⠋⠐⠠⠌⠀⢀
⠀⠀⠀⣼⠃⠀⠀⠈⠳⡐⢄⠀⢀⣴⠟⣴⠏⢀⡿⠀⠀⠀⠀⢸⡇⢠⣧⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⡀⠈⠰⠈⠀⠀⠈⢳
⠀⠀⠀⢻⠇⡴⠀⠀⠀⠉⠲⣄⠀⠉⠺⠇⠀⣸⠃⠀⠀⠀⠀⢸⡇⠀⢿⡀⠀⠀⠀⠀⠀⠀⠈⠛⡁⠀⠀⠀⠋⠀⠀⠀⢦⠐⡞
⠀⠀⠀⢈⣀⠃⣀⠀⠀⠀⠀⠈⠑⠺⣷⠀⢀⠙⠀⠀⠀⠀⠀⢸⡇⠀⢸⣇⠀⠀⠀⠀⣀⠠⠀⠺⠗⠐⠈⠀⠀⠀⠀⣀⠀⣀⣄
⠀⠀⠀⠈⠁⠀⡍⠐⢚⣉⠀⠀⣀⠀⠈⠙⠳⠦⣤⣐⡂⠤⠄⠉⣁⣀⠈⠉⠀⠂⣀⡈⠤⠔⠋⠁⠀⠀⠀⠀⡐⠢⠀⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠃⠈⠙⠛⠚⠛⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠛⠒⠙⠛⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡄⠀⠀⠀⠀⠀⠀⢹⡿⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠲⡄⣸⢃⠜⠡⣄⡀⢸⠃⣾⠇⣿⠁⡀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⠎⢀⣤⠄⢠⠇⣿⡠⣦⠀⢹⣇⣿⡴⢻⡴⢧⢼⡿⣻⡷⢋⡿
⠀⠀⠀⠀⠀⠀⡿⠀⠻⠤⠔⠋⠀⢸⣧⣿⣸⠀⠛⠈⠱⡽⠃⠀⠈⠁⠛⢡⡟⣠⠴⣛⣣⡄
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⡜⠀⠈⠁⠀⢀⣀⢠⣤⢤⣶⠲⠶⠽⠂⠉⠀⠀⠸⠫⢴⡾⠷
⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⠄⠐⣈⠀⠀⢰⣼⣿⢼⣿⠺⣭⣰⣶⣰⣿⣼⡇⠀⠀⠀⢰⣓⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢨⣷⠆⠀⣾⣿⣾⢿⠟⣿⢻⣿⡛⡏⣿⣧⠀⠠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⡿⡿⠂⡸⠦⠀⣿⡸⣿⢸⢸⡿⡎⣿⡇⣧⢹⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⢸⡇⣭⣜⣎⣣⣿⣬⡿⠼⠾⠟⠀⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢉⡍⠳⡌⢠⠀⠈⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⡔⢊⠸⡄⠘⡥⢾⣿⡀⣿⠀⡇⡾⡆⣾⢢⢀⣤⠀⣀⠀⢽⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠈⢟⡄⠺⣿⣧⠈⢿⣧⠸⠚⠿⠇⠇⢷⢋⡞⡅⢠⡟⢁⡆⣙⢿⣿⣿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠙⣿⣧⠈⣿⣧⠀⠀⠀⠀⡄⢠⡄⠁⠚⠠⠟⣴⣫⠖⡏⡻⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡈⢿⣧⠼⠛⠁⠀⠀⠰⠿⢷⡇⠀⠀⠀⠀⠁⠧⣺⠟⣡⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠉⠀⠀⣀⠀⢠⠀⠀⠀⣀⢠⡀⠀⢀⡄⣠⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠀⠀⢨⣏⡼⢸⣷⡻⢾⣽⠀⡧⡟⢸⡇⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⡟⠀⠀⠀⢠⠋⡹⠃⠀⠁⠀⠫⠛⠟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡨⠒⠈⠐⠢⢄⣉⠀⠀⠀⠀⠀⠀⠀⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⢤⠀⠀⠀⠈⠙⠒⠲⠶⠶⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠌⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣦⣤⣤⡀⠀⠀⠁⠂⠤⣀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠘⠉⠉⠉⢿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠨⠐⠦⡄⠀⠀⠘⠋⠙⢿⣿⣿⣿⣷⡑⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣷⣶⡄⠀⠀⠪⠙⣦⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⣸⣇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣷⣞⣿⣿⢿⣿⣷⣤⣴⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣽⣿⣿⣿⣿⣟⣉⣚⠢⢍⢿⣿⣿⣿⣦⣤⣤⣤⣤⣴⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡛⠿⣿⣿⣿⣿⣿⣿⣿⡿⡟⠋⠁⠀⠀⠈⠑⣄⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡤
⠀⠀⣀⡀⠀⠀⠒⢒⣲⣶⡿⠦⡀⠀⠀⢣⠜⠁⠉⢛⠛⠻⡁⠀⠀⠐⠄⠀⠀⠀⠀⠀⢸⡴⢫⠋⠙⠛⠻⠿⢿⡟⠛⢹⠁⢸⡇
⢀⢜⡻⡍⢻⡉⠁⠀⢀⠀⠄⢀⠛⢢⣀⣜⢀⣀⡤⠂⠀⠀⠘⣆⠀⠀⠈⠢⣀⣀⣀⣀⣸⠅⠃⠀⠀⣀⣀⣠⡂⣵⠀⠀⢀⣼⠁
⠸⣗⡠⠃⣠⣋⢢⣀⣀⣀⣀⣀⣀⣠⠞⠁⠀⠀⣀⣀⠀⠀⢀⣈⣳⣄⣀⣀⣀⣉⣡⡜⢁⠀⠀⠈⠉⣀⣀⣀⡴⣃⣠⣴⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣠⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⢀⣁⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠳⣤⣴⢿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢧⠃⠂⠀⠀⠀⠀⠀⠀⠈⠘⠁⠀⢹⡷⡆⠄⣀⣀⠀⠀⢀⡻⠎⢀⠈
⠀⠀⠀⠠⢆⠀⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠃⠰⠻⠁⠀⠀⠘⠛⠻⠺⢙
⠀⠀⠀⠀⠈⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣄⣀⣀⣀⣀⣲⠄
⠀⠀⠀⠀⠀⠀⠀⢀⢠⡀⠀⢈⠛⠀⢀⣤⣤⢀⣀⣀⣠⣄⣴⡆⠀⠀⣀⣶⣿⣿⣿⣿⣿⣾⠟⠁⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠘⣠⠀⠶⣬⣽⠿⠀⢻⣿⣿⣿⣿⢿⣿⣿⣷⢀⡄⡹⣿⣿⣿⡼⢿⣿⠋
⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⠀⠀⠀⠘⠙⡿⠁⠈⠀⡀
⠀⠀⠀⠀⠈⠞⠀⠀⠀⠀⠀⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠀⠠⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣈⢢⣄⠀⠀⠀⢄⠆⣀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⢨⡆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠲⣈⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠰⠀⠀⢠⡄⠀⠀⠀⠀⠰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄
⠀⠀⠀⠀⠀⠠⠋⠁⠀⠀⠀⠀⠞⣿⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⢠⣿⣄⠀⠀⠀⠀⠀⠀⠰⠆⠀⢀⣦⡀
⠀⠀⠀⠀⠀⠉⣶⠶⠁⠀⠀⢀⠾⣿⠣⡀⠀⠀⠀⢀⣾⡄⠀⠀⠀⣹⡅⠀⠀⢩⣿⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⣹⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡼⣿⢷⡈⠀⠀⢀⣾⣿⣿⡀⠀⢈⠼⢌⠀⠐⢛⣿⡟⠀⠀⠀⣼⣧⠀⠀⠀⠉⣹⣿⣏⣉⡀
⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⢈⡴⣿⣧⡙⠂⠀⣾⣿⣿⣿⣷⠀⡨⢘⠢⡁⢠⣾⣿⣿⡆⠀⢠⡿⢼⡄⠀⠀⢙⣿⣿⣏⣉⣀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢉⡴⢿⢦⠑⠆⣸⣿⣿⣿⣿⣿⣧⠠⠞⠦⠀⠈⣸⣿⣆⠀⠀⣾⡿⢹⣷⠀⠲⠟⣻⣿⣿⣭⣄⡀
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠐⠋⠀⢸⠀⠑⠀⠟⠿⠿⠿⠻⠟⠛⠂⢸⠀⠀⠼⠿⠿⠿⠇⣸⣿⣧⣽⣿⡆⠠⡾⠿⣿⢿⣧⣀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣀⣠⣿⠁⠀⠀⠀⠀⣤⣤⣤⣀⣤⣿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⢺⣿⡿⠿⠿⠿⢷⡂⠤⠔⣲⠿⠛⠛⠛⢿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⢠⣴⣶⣾⠿⡷⢤⠐⣻⠋⠀⠀⠀⠀⠀⠉⠀⠚⠁⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣿⢄⠀⡰⡿⠁⠀⠀⠃⠀⠙⠁
⠀⠀⠀⢤⣼⣿⣷⣴⣧⣆⡀⠀⢀⠜⡿⠁⠀⠖⠁⠀⠛⠤⠁
⠀⠀⠀⢘⠙⣿⣿⣿⣿⣿⠿⠀⠀⠊
⠀⠀⠀⠀⠀⠘⠛⠛⠉

⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠓⠁⠈⢘⣶⣶⣶⣶⣶⣤⣄⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠙⠷⠐⠈⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠛⠛⠛⠛⠉⠉⠰⠆⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠦⢤⣀⣀⣀⣠⣤⡶⡾⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⡥⢨⣀⣀⠂⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠠⠤⠄⠀⠩⠃⣁⣠⠟⡀⠉⢳⠀⠀⠀⠀⠀⠀⠀⢀⠌⡔⢠⡤⢀⠉⢛⡤⠔⠂⠘⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⡀⠀⢠⠞⠢⠀⠀⠀⢄⠃⣀⠎⠀⠘⠛⢧⠀⠀⢀⡴⣒⡶⠁⠠⢄⣀⠀⣠⠾⠽⠄⠐⠒⢲⣤⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⠐⠈⠣⠤⣇⡀⠀⠀⠉⠀⠀⠀⠋⢷⠋⢀⠔⠚⣩⣈⡵⠌⠁⠀⢀⠎⠑⠀⠀⠀⢀⠀⢺⠾⢃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠹⣄⣶⣂⣼⣀⣸⠦⡴⠋⠀⡇⠀⣰⠁⠠⠈⠀⠂⢀⡟⠀⠀⠀⠀⠀⠀⠀⠉⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠑⢤⠘⢷⣯⣹⡶⣫⡍⠓⠐⢢⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣷⣶⣶⣶⣦⣤⣤⣄⣀⠀⠀⠙⢆⠻⣿⠁⠣⣇⠈⢻⡟⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠋⠠⠀⠀⢈⣭⣦⡼⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠢⢰⠟⣇⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⡠⠏⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢷⡀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡏⠉⠛⠛⢻⣿⠛⠛⠛⠛⢻⣿⠛⠉⠉⠉⢇⢛⣷⡄⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣶⣶⣾⣿⣦⣤⣤⣤⣼⣿⣧⣤⣤⣶⣾⣤⣴⣾⣦⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⢳⡀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠄⠠⠀⣀⠴⠚⡇⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣀⣠⣥⣤⣶⣾⣯⣄⠀⢠⣇⣀⣀⠀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⡶⠶⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣫⠭⠒⠛⠉⢉⠉
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠈⢉⢛⡻⢿⣿⡿⢟⣵⣾⠟⠛⣆⠀⠀⣠⠋⢻
⠀⠀⠀⠀⠀⢀⠞⠛⠿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⣀⡀⠠⠐⠈⠀⠀⠈⢒⡾⠻⢿⡟⠁⠀⠀⠈⠳⠚⠉⢰⡞
⠀⠀⠀⠀⢀⠊⠀⠀⠀⢀⡼⠟⠿⠿⠛⠀⠀⣠⠊⠉⢱⡄⠀⠀⠀⠀⡠⠊⠀⠀⠀⠙⢦⡀⠀⠀⠀⢀⣰⠟
⠀⠀⠀⠀⡜⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⢭⢰⣀⠥⠛⠢⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠙⠲⢦⣀⣈⠟
⠀⠀⠀⠰⢢⠟⠳⣄⠘⠓⢤⡀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⣀⠀⣀⠤⠒⠊⠉⠉⠙⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠁⠀⠀⠀⠠⠄⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣤⣤⡀
⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠚⠁⠈⠁⠈⠀⠀⠘⢧⡀
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠰⢄⡀⣀⢰⡄
⠀⠀⠀⠀⣿⠀⠀⢀⣀⣀⣀⠀⠸⠁⠄⢀⠈⠀⠀⠀⠀⣀⣠⠤⠔⠒⠉⠐⠦⢤⣹⡉⠉⠁
⠀⠀⠀⡜⠙⡄⢔⠁⠀⠤⢤⡋⣟⠓⡎⠁⠀⢀⣠⣶⠉⠥⠀⠀⠀⠀⠠⠂⠀⠂⣿⠁
⠀⠀⡸⠀⠀⠙⠀⠓⠀⠲⠻⣀⣠⠋⢀⡤⠊⠑⠃⠲⠀⢤⠀⡜⠀⠀⢃⠤⠄⠀⡝⠒⠒⠦⢄⡀
⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡄⡠⠚⠁⢸⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡸⣼⣦⣤⠗⠒⡄⠀⠀⠹⣽⢦⣀⡀
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣠⠾⠀⠀⠈⠀⠀⢰⠏⠉⠉⣶⠉⠉⠉⠹⣄⣤⣏⡀⠀⢀⠞⠀⡇⠈⠑⢄
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⡞⠀⠀⠀⠀⠀⢀⡀⠘⢧⣤⠔⠃⠀⠀⠀⢠⠇⠀⠀⠈⠉⠁⠀⢀⠇⠀⠀⠙⢇
⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠈⠁⠀⠒⠠⠒⠉⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠈
⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⣸⢻⠣⢀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢀⠴⠂⠁
⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⢠⠃⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠼⡉⡄⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠘⡄⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠙⠒⠤⠤⠤⠄⠂⠉⠉⠉⠫⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢸⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣼
⠀⣠⠊⠉⠉⠓⠲⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃
⡰⠉⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⡏⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠙⢆⠀⠀⠀⠀⡀⡰⠃⠘⢦⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠈⠉⠛⠒⠉⠉⠀⠀⠀⠀⠈⠑⠒⢧⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠟⠓⢤⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠢⠤⢄⣀⣀⣀⠀⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⠶⡄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣙⠻⠀⠟⢋⡽⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⢰⣋⠳⠿⠸⠷⠊⣹⢀⣤⡀
⠀⠀⠀⠀⣀⠀⡏⢹⢀⣠⡖⠮⣷⠀⣾⠵⢲⣄⡀⡏⢹⠀⣀⠀⠀⠀⠀⢠⠶⢤⡇⢸⢠⠬⣕⣦⢠⣖⡫⢤⢸⡀⣧⡴⢦
⠀⠀⠀⠘⢏⡙⠃⢸⣸⠀⡟⠲⣤⠀⣠⠖⢻⠀⣇⡇⠘⢋⡹⠇⠀⠀⠀⣈⡷⠄⠀⢿⡞⢠⣄⡉⠈⣉⣤⠘⣯⠇⠀⠴⣏⡀
⠀⠀⠀⣴⠚⣡⠴⣦⡉⠀⢧⣄⣸⠀⣇⣠⠼⠀⠉⣴⠦⣌⠓⣦⠀⠀⠘⠥⠴⣚⡽⠆⠀⠸⡤⠿⠘⢧⡼⠀⠀⠾⣝⡲⠤⠽
⠀⠀⠀⠈⠉⠱⣏⣡⠖⠛⠲⣌⡙⠀⢋⣡⠖⠛⠲⣌⣹⠎⠉⠁⠀⠀⣞⠳⣄⢧⡀⢾⣉⡗⠀⠀⠀⠀⠐⣏⣹⠆⣠⢇⡴⢛⡆
⠀⠀⠀⣠⢤⣘⢧⣉⠓⠶⠚⣉⣤⠀⣤⣉⠓⠶⠚⣉⡼⣃⡤⣄⠀⢀⡬⠷⠈⠛⣋⠀⢩⣤⠀⠀⠀⠀⢠⣬⠁⢈⠓⠋⠰⠯⣄
⠀⠀⠀⠙⢦⣌⠙⢋⣡⠀⡟⠁⢸⠀⡇⠈⢹⠀⣌⡙⠋⢡⡴⠋⠀⠘⠲⢲⠂⡾⡽⠀⠷⠼⠃⡴⠶⡄⠻⠤⠇⠸⡽⡆⢲⠲⠚
⠀⠀⠀⠰⣏⡤⡆⢸⢹⣀⡷⠞⣉⠀⣉⠓⢾⣀⡏⡇⢰⢦⣹⠆⠀⠀⠀⠻⠴⢃⣙⣒⣲⠂⣤⠙⠞⢡⡄⢲⣒⣚⣁⠳⡼⠃
⠀⠀⠀⠀⠀⠀⢧⡼⠀⠉⢓⣫⠽⠀⠿⣝⡚⠉⠀⢧⡼⠀⠀⠀⠀⠀⠀⠀⠀⢯⣥⣤⠀⡈⢯⡳⠴⣫⠏⠀⢠⣬⣭⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⡤⢶⠀⡶⢤⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡼⠳⣄⣳⢰⣃⡴⠳⣌⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡶⢼⠉⣧⠶⡆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣙⠾⠹⢚⣩⠇
⠀⠀⠀⠀⠀⢠⡶⣴⢋⣇⣠⢽⣶⠀⣶⡯⣅⣸⡝⣧⢶⡄⠀⠀⠀⠀⠀⠀⡴⣄⢠⣄⠰⣟⠻⢿⢰⠿⢚⡽⠀⣀⢠⡤⡄
⠀⠀⠀⠀⢠⡴⢛⣀⡸⡿⣸⠓⢮⠀⡵⠚⣇⢻⠇⣀⡛⢦⡄⠀⠀⠀⠀⡴⠇⣿⣿⢸⣀⡬⠟⣂⢀⡚⠯⣄⣸⢹⣿⡆⠷⣄
⠀⠀⠀⠀⣀⠉⣹⣛⣋⣤⡻⣄⢸⠀⡇⣠⠞⣠⣘⣛⣯⠉⣁⠀⠀⢀⡴⢋⣤⣬⠋⠘⣩⡖⠋⢹⢸⡏⠓⣮⡙⠈⢫⣤⣌⡓⣦
⠀⠀⠀⣘⣏⢳⣧⡹⣄⣀⣙⣮⡻⠀⢟⣡⣊⣀⣠⠏⣼⡾⣹⣃⠀⠀⠉⢹⣋⡵⣶⢰⠦⣝⠲⢾⢸⠷⢚⡥⢶⢰⡶⢍⣳⡍⠁
⠀⠀⠘⠯⡭⢠⣭⢄⡬⠭⠭⣭⠅⠀⡨⣭⠭⠭⢥⡠⣭⡄⢭⠽⠃⠀⠀⣠⢤⣀⣿⢸⣀⡤⠟⣂⢀⡚⠯⣄⣸⢸⣇⣠⣥
⠀⠀⠀⠸⠥⠋⣯⠾⣤⣠⢞⡵⢻⠀⡟⢮⡳⣄⣤⠷⣽⠙⢮⠇⠀⢠⡶⢼⣶⣬⠋⠘⣭⡖⠋⢹⢸⡍⠓⣮⡝⠈⢫⣴⣾⠖⡦
⠀⠀⠀⠀⢠⣖⠾⠿⢖⡀⢿⠀⣸⠀⣇⠀⡿⢀⡲⡾⠷⢲⡄⠀⠀⠀⠉⣗⡈⣱⣶⢰⠦⣝⡲⢾⢸⠷⢚⡥⢶⢰⣶⡍⣐⡏⠁
⠀⠀⠀⠀⠀⢩⣟⣤⢸⡿⣼⣋⣥⠀⣬⣙⣧⢿⡇⣤⣻⡍⠀⠀⠀⠀⠀⠉⣇⣹⠻⠼⢀⡤⠟⣂⢀⡘⠧⣄⠸⣼⢿⣇⡏⠁
⠀⠀⠀⠀⠀⠀⠉⠘⠷⠇⢠⢟⣫⠀⣝⡻⡄⠸⠶⠃⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⢒⡿⠟⠘⠻⣷⡋⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠶⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠖⠛⠞⠛⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠿⠛⠛⠛⠿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⢠⠟⢁⣠⠴⠶⠶⠶⠦⣄⡈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠟⠋⠉⠁⠀⠀⠀⠀⠈⠉⠛⠿⣿⣆⠀⠀⠀⠀⠀⠀⠘⣴⣿⠉⠘⡅⠀⢠⠓⠉⢹⡖⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠀⢀⣤⡶⠚⠛⠿⠿⠟⠳⢶⣤⡀⠈⠻⣧⠀⠀⠀⠀⢀⠠⡏⠱⣄⡠⠄⠀⠠⢤⣠⡎⠱⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣅⣠⣾⠛⠛⠴⠶⠂⠀⠀⠐⠶⠦⠙⠛⣦⣀⡽⡀⠀⠀⣴⣿⡓⢤⣄⡀⣀⢤⣦⡄⡀⣀⣠⠔⠈⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢰⣿⣿⡈⠀⠐⠘⠇⠀⢀⠸⠗⠂⠀⠁⣿⡇⢀⠘⠀⠀⠈⢻⡁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠐⣴⠋⠀⣧⣀⣀⣤⢤⣀⠀⣤⣤⣀⣀⣼⠀⠙⠆⢰⡀⠀⠀⠀⣏⡀⢦⠀⠀⠀⠀⠀⠀⠀⠔⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣧⡈⠁⠒⠀⠉⠁⠀⠀⣀⠀⠀⠀⠉⠙⠒⠀⠀⣠⠛⣿⣦⡀⠀⠈⢧⡌⣧⣀⠀⠀⠀⢠⡌⢁⡬⠃
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡄⠙⠲⠦⣤⣄⡤⠔⠻⣭⡽⠳⠤⣤⠤⠰⠶⠋⠀⠰⣿⣿⣿⣄⠀⠀⠓⢤⠈⠉⠛⠉⢁⠤⠚
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣦⡀⠀⠉⠳⠴⠋⠈
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣶⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡅⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⢸⣿⣿⣿⡿⠿⠛⠛⠿⢿⣿⣿⣄⡈⠛⣇⢀⡀⠀⠀⠀⠀⢀⣀⣩⠉⢁⣠⣿⣿⣿⠿⠛⠋⠉⠙⠻⢿⣿⣿⡄
⣿⣿⣿⠏⠀⠀⢀⣠⣤⣄⣈⣻⣿⣇⠀⠙⠹⣷⡶⣆⣴⢦⡾⠉⠁⢀⣼⡿⠿⣿⣥⣤⣶⣦⣤⡀⠀⠈⢿⣿⡇
⢹⣿⡟⠀⠀⣴⣿⣿⠟⠋⠉⠉⠁⠈⣿⣧⠀⠀⠀⠈⠁⠀⠀⢸⣿⣿⣿⣇⠀⠀⠀⠀⠉⠻⣿⣿⡄⠀⢸⣿⡇
⠈⢿⣧⠀⢰⣿⣿⠁⠀⠀⠀⠀⠀⠘⣿⣿⣷⣿⢧⣠⣴⣿⣶⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⣸⡟
⠀⠈⠻⣄⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠉⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣴⠋
⠀⠀⠀⠘⢷⣽⣧⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣇⣀⣄⣀⣻⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢀⣼⠟⠋⠁
⠀⠀⠀⠀⠀⣿⣿⣷⣤⣀⣀⣀⣠⣾⣿⣿⡏⢰⣶⣶⣆⢸⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣴⣾⠁
⠀⠀⠀⠀⢸⡇⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠬⢍⣉⠭⠼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣿⠆
⠀⠀⠀⠀⠈⢷⡀⠀⠉⠛⠿⢿⣿⣿⣿⣿⠇⠀⠀⣿⠀⠀⠻⣿⣿⣿⣿⠿⠟⠋⠁⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣄⡀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⡏⢻⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡟⠀⠀⢸⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣧⠀⠀⣼⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡆
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⠀⡀⠀⠀⠀⠘⠀⠃⠀⠀⠀⢀⠀⢔⡄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⢀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣭⡛⠀⢀⠀⠀⠅⠀⠀⢚⣥⠒
⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠊⡄
⠀⠀⠀⠀⠤⠀⢀⠄⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⠁⠐⠀⠢⠄⡀⢁⣄
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠈⠤⠛⠀⠀⠈⠀⠠⠔⠀⠳⠠⢁⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠸⡀⡇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠜⠋⠡⠄⢀⠀⠀⠀⠀⠂
⠀⠀⢀⣢⣀⠀⠀⠀⢀⢰⡀⠀⠀⠀⣀⣐⠄⠂⠀⠀⠀⠀⠠⠠⢤⡂⠆⠂⠈⠁⢠⠀⠀⠀⠋⠐⠤⠤⢣⠄
⠀⠀⠀⠀⠉⠐⣰⠄⣀⠈⣀⠰⣂⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠁⢠⡀⠀⢾⡀⣰⠶⠂⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠘⠓⢒⣉⠡⢴⣄⠀⠰⠦⠄
⠀⠀⠀⠀⣀⠠⠻⠀⠉⡀⠁⠰⠏⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠂⠀⠠⠶⠋⠀⣷⠀⠈⠁⠀⣆
⠀⠀⠀⠍⠁⠀⠀⠀⠈⠏⠁⠀⠀⠀⠉⠹⠄⠀⠀⠀⠀⠀⠀⠀⢪⠔⠃⠆⢠⠀⠀⠀⠃⠀⣀⠰⠰⠎⣟⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⢠⡔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿