Tidy Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⠆⠙⠦⣄⡴⢛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢻⡀⢀⣴⣶⣌⠻⣏⠙⠲⣽⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠟⠁⣠⣬⡳⣄⠈⠻⣿⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⣾⣿⡿⠃⠈⠳⣄⢹⡆⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠙⢿⡟⠀⠀⣠⣦⡈⠻⡵⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⠡⠀⠀⠀⠀⠀⠀⠀⠈⠻⣴⣟⣿⡿⠋⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⡀⠀⣠⣤⣘⣧⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠅⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⠤⠤⠤⠤⣤⣤⣤⣤⣀⡙⠾⣿⣿⡿⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠛⠛⠋⠉⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠉⠙⡇⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢋⠭⢄⣀⣠⣤⣶⠶⠾⠿⠛⠛⠛⠛⠛⠋⠛⠛⠛⠛⢛⡿⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠛⣿⠉⢉⣀⡀⠀⠀⠀⣿⠛⢻⣿⣿⠂⢀⣶⠶⣿⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⢻⡟⣷⠀⠀⠀⡟⠀⠈⣽⣿⠀⢸⡇⢸⡟⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠈⢟⣿⡄⠀⠰⡇⠀⠀⣿⡟⠀⢸⠀⡨⢱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠈⢸⡇⠀⠀⡇⠀⢸⣿⡇⠀⣾⠀⠘⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢼⡇⠀⠀⡇⠀⢛⣿⡇⠀⡿⠀⢄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⡇⠀⠀⡇⠀⢸⣿⡇⠀⡇⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⡇⠀⠀⡇⠀⣼⢻⡇⠀⡇⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢸⡇⠀⠀⣇⠀⣿⢸⡇⢀⡇⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⣷⠀⠀⣿⠀⠐⢿⡇⢸⣇⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣾⣃⡀⠀⠿⣤⣤⡼⠇⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⠿⠿⠛⠋

⠀⠀⠀⠀⠐⠶⠒⠒⠒⠒⠲⠂⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣌⣈⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⡀⠀⢀
⠀⠐⠀⠀⠀⠀⠙⠃⠛⠛⠓⠐⠒⠷⠀⠀⠆⠀⠀⠉⠀⠀⠀⢷⡇⠀⠀⠀⠀⠀⡀⡀⢀⡀⣁⢂⢄⠀⡐⠹
⠀⠀⠐⠀⠀⠀⠀⠀⠋⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣱⠀⠀⢸⣀⠀⠀⠨⡀
⠀⠀⡀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⠀⡄⠀⠘⠂⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢀⡋⠀⠀⠀⠸⠀⠀⠛⠛⠃⠀⠀⠀⠁⠃⠀⢐⢢⠀⠀⠀⠄
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠄⠀⠀⠂⠈
⠀⠀⡄⡆⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠨⠅⠅⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠄
⢀⠀⢠⡄⠀⠀⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⢀⠀⠀⢸⠀⢨⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢥⠳⠸⠘⠏⠏⠏
⠀⠠⠀⠀⠀⠀⠀⣀⡒⢃⢁⣰⢀⠛⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⡈⠉⠙⠛⠉⠉⣘⢛⡂
⠀⠀⠀⠀⠀⠀⠤⣄⠈⢽⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀⡀⡀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠐⣀⠀⠁⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⡀⠀⠲⢈⡙⠗⡦⣐⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣜⢦⣀⠑⠠⠉⠒⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠈⠛⢿⣷⡤⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠋⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡀⣠⣄⢀⡀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣠⠚⠉⠁⠀⠉⠈⠙⠙⠢⠤⠤⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⣻⠠⠄⠀⡤⠀⠀⠰⠀⠀⠀⣠⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠚⠋⠉⠛⢦⠘⢆⡀⠐⠒⠒⠃⠀⣀⠤⠚⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢹⠃⠀⡀⠠⣌⡀⠀⣇⡠⠜⠋⢚⣊⠭⠟⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠁⢀⡰⣇⣀⡤⠒⠉⠀⠀⠀⢀⣀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⢤⡔⠒⠉⢰⡍⠀⢣⣀⣠⣤⠒⠋⠉⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢆⢠⠖⢲⠀⠀⡜⠁⣰⠃⠀⠀⠀⠷⢤⠊⠁⠀⠘⡄⠀⠀⠀⣀⣀⣠⠤⢼⠀⠀⠀⠀⠀⠀⢀⣤⡖⢒⡶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡼⠀⣏⣠⣀⣧⣀⠟⡆⠀⠀⠀⠀⢸⠀⠀⠀⠀⡟⠉⠉⠉⠁⠀⠀⠀⠘⡆⠀⠀⠀⠀⢠⠏⠀⡏⠉⠙⡄⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠠⠀⠀⠄⠈⢦⣠⣿⡛⠋⣳⢄⣀⣤⠔⣯⠽⡇⠀⢠⠇⠀⠀⡠⢄⠀⠀⠀⠀⣇⠀⠀⠀⠀⢸⡀⠀⡇⠀⠀⢸⠀⡸
⠀⠀⠀⠀⠀⠀⣴⣯⣇⣀⣀⣙⣊⣀⣀⣸⣀⣀⣈⣶⣁⣀⣮⠼⠊⣀⠀⡇⠀⢸⠀⠀⢠⠃⠈⣆⠀⢀⣀⡼⠀⠀⠀⠀⠀⠑⢤⣧⣀⣀⣸⠟⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢸⠀⡇⠀⣸⣉⣙⠏⠀⠀⠈⣯⠥⠴⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠔⠋⡠⠣⣖⣁⣀⡼⠀⠀⠀⢴⣃⣠⠴⠃⠀⢀⠴⣒⣒⠛⣒⠢⡀
⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠐⠃⠚⠘⠒⠃⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠯⠗⠒⠒⠫⠝⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⢀⠤⣄⠀⠑⠠⡀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⡀⢀⠀⠀⣀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠃⠁⠄⢀⠌⠀⠄⠀⠀⠀⠑⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠈⡀⠈⢢⡀⠀⠀⠀⣤⣨⣥⣤⣬⣿⣇⠑⠾⢁⣷⣤⣄⡀⠀⠒⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠄⠇⠀⠁⢁⠀⠸⢃⡁⠀⣠⠟⠛⢿⣿⠏⣍⣿⣿⣾⣿⡈⠉⣿⣷⠀⢀⠀⠆⠀⠔⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠒⠀⠀⠀⠂⠀⠀⠀⠐⠛⠃⢃⡘⠀⠏⠼⢫⣿⠿⠿⠒⠌⣙⠋⠀⠀⠸⠁⢀⠊⠀⠀⠀⠀⠀⠠⢂⡀⠀⠠⠎⠀⠃
⠀⠀⠀⠀⠀⠤⠀⡀⠑⠀⠀⠀⡰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⢒⢃⡴⠋⠁⠐⠀⠀⠀⠈⢢⠄⠀⣾⣄⠠⡁⠢⡀⢂⢀⠌⡁⠀⢀⡀⡤
⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣶⡾⠁⠀⠀⠀⠒⠀⠠⠀⣀⠄⢸⠋⡼⠁⠐⢀⠀⢀⠀⠀⠀⠀⣱⢴⣇⢹⣷⡤⢀⣀⣁⠉⠀⠀⡀⠈⡧⠉⠀⠀⠁
⠀⠀⠀⠈⠀⢈⡴⠶⣄⠀⡹⠁⠀⡄⡈⠁⠀⠈⢦⠦⡉⢲⠂⡰⠁⠀⠀⠀⠃⠐⠀⣶⠄⠀⢰⠆⠈⠀⠋⠀⢀⢡⠝⠄⠀⠒⠉⠉⠁⠀⠀⡀⠈⠉
⠀⠀⠀⠀⠈⠘⠷⠆⣼⡷⠁⠀⣀⡀⣉⠇⠀⠤⡄⢰⠏⠋⠐⢛⡀⠀⠀⠀⠀⠀⣼⠃⡐⠀⣼⣷⡀⠀⠀⣄⣈⣙⠁⠀⠀⠀⠀⠀⠀⠀⢈⣟
⠀⠀⠀⢀⣠⡿⠟⠉⠙⢁⠀⠀⠈⠛⠉⠀⠤⡀⠁⢨⡳⠀⠀⠱⢄⠀⠀⠀⠀⣸⣯⣴⠁⣤⠸⠛⠇⠀⣀⢹⣿⣽⣿⣤⠀⠀⠀⠀⠀⠀⣻⣿⣇⡀
⠀⠀⣸⣿⠃⠠⠆⡀⠀⠀⠠⣀⣀⠀⠀⠀⠈⠃⠀⡾⡃⠀⠀⠀⠀⡇⠀⠀⠀⢿⣿⠃⡸⠃⡀⠀⠔⡡⢺⡄⢿⡀⢹⣿⡏⠀⠂⠐⠀⠀⠉⢿⣿⣿
⠀⠀⠈⠙⢣⠀⠀⠄⠀⠀⠀⠀⡉⠁⠀⣤⣀⠀⡎⡈⡑⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⡼⠀⣰⢋⢉⠐⠆⡜⠁⠀⠀⠈⠭⢄⠀⠀⢀⣈⡇⠀⢾⡿⠟
⠀⠀⠀⠐⠄⠑⠤⠀⠀⠀⢠⡜⢡⠴⣿⣿⣿⠛⢴⣦⠃⠀⠀⠀⠙⠀⠁⣤⢠⢤⠀⠠⠤⠄⡘⠀⠀⢠⡦⠀⠀⢀⡀⠰⠋⠀⠤⠜⠉⠢⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢀⣀⡀⠀⢩⠃⠀⠰⠛⠦⡄⣉⠓⠠⢾⡏⠀⠀⠀⠀⠀⢀⠀⢙⣀⠀⣃⠰⠒⠂⠀⠀⢠⠌⠓⡁⠞⢟⢻⡀⢷⠿⡀⢈⠀⢀⣂⣀⠀⠀⠀⡀
⠀⠀⠀⠀⢿⠏⢠⣾⣿⠀⠁⠀⠀⠀⠀⠉⠉⠒⠠⢹⡀⢣⠀⠀⠀⠘⠀⠁⡠⠌⠁⠌⠀⠀⠒⢄⠠⠦⠐⢁⠊⠛⣽⡧⠈⡄⣀⠀⠡⡈⠛⢿⡷⠀⡆
⠀⠀⠀⠀⠀⠀⢾⣟⡛⠄⠉⡙⠠⠄⡀⠀⠀⠀⠀⠀⠓⣤⠠⠀⠀⠀⠘⢢⠍⢀⡀⠠⠄⠀⠃⠺⡅⠀⣠⠂⠀⣼⣿⡇⢣⡅⠉⢢⣀⣼⣶⡾⠁
⠀⠀⠀⠠⠀⠀⠈⠙⠁⠈⠂⠈⠂⢤⣨⠜⠒⠂⢤⣄⠀⣿⣦⣤⣠⣄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⠃⢀⡚⠉⠀⡆⠀⠀⡀⠀⠹⣿⡿⡁
⠀⠀⠀⠀⠉⠀⠀⠀⢀⠤⠐⠚⡟⠉⣀⠧⠄⠤⠤⠀⣀⠈⠉⠙⢹⠆⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⢚⠃⢀⠎⠀⠄⠀⠱⠀⠀⠀⠀⠀⡋⢀⠃⠔
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠠⡏⠈⠀⠀⠀⢠⡒⠒⠠⠈⠦⠀⠒⠀⣎⠀⠀⠀⠀⠀⠉⠀⠀⣀⠎⡰⠏⢂⠠⠦⢀⠀⠒⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⢿⡇⠀⠀⠀⠀⠠⡁⣀⠀⠈⢂⠠⠀⠊⠉⡕⠒⠄⣀⢀⢀⡠⠿⡏⡴⢠⠞⠣⠀⢡⠰⠤⠴⠓⣀⠀⠀⠀⠂⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢂⠀⠻⠀⠀⠿⠀⡔⠒⠀⠀⠀⢈⠡⠄⢠⠆⣅⠀⠤⠸⣄⡀⠀⠙⠳⢌⡲⠌⢹⠀⢀⠀⢰⣾⡷⠆⠒⠫⢵⠄⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠢⡈⠁⠀⠀⢉⣴⡆⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠐⣠⣤⣼⣶⣤⡀⠠⢈⠐⠸⢀⡈⢱⣾⡏⠀⠀⠀⠀⠀⠑⢄⠀⠠⠀⡂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠋⠙⢿⡀⠄⠀⠀⠀⠀⠀⠀⠀⢠⠞⠫⠞⠛⠛⠉⢉⣉⢈⠁⡶⢄⡀⠀⡁⠘⣿⡇⠀⠀⠀⠀⠀⠀⢂⢂
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⠡⡈⠄⠀⠀⠀⠀⢀⣀⣼⠐⠀⠋⠁⠀⠉⠀⠀⠀⠀⢀⣀⣿⣷⡈⠋⠙⠇⠀⢠⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠀⠀⠀⡙⠒⠒⠀⠀⠀⠙⠛⠻⠦⢤⣤⣦⣴⡶⠲⠶⠶⠉⠉⡉⢉⢉⢉⢋⠀⠘⠨⠃⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢘⡠⠸⠲⡶⡄⠀⠀⠀⠀⠀⢀⠀⡀⠄⠀⠈⠁⠄⠀⠤⠀⠀⠀⡤
⠀⠀⠀⠡⡈⠺⡃⢾⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠈⠲⢶⠖⢃⣤⠤⢀⣖⠶⠶⣶⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠙⢏⣀⣶⠆⣭⡕⠂⠽⣤⠇⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠔⠢⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠙⠲⢿⡁⣰⡷⠀⢑⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠃
⠀⠀⠀⠀⠐⠂⠀⠀⣀⣀⣠⠤⠤⠤⠬⣍⣓⠂⠿⠏
⠀⠀⠀⠀⢀⠖⢚⣉⡡⠤⠴⠖⠒⠒⠒⠶⡤⠇⠀⠀⠘⠛⠦⠰⢤
⠀⠀⠀⠀⠀⠉⢫⠴⢦⠀⢸⠉⣿⠀⠏⢭⠇⠀⠀⠐⠒⢠⠀⢐⡩⠐
⠀⠀⠀⠀⠀⠀⠸⡀⢸⠀⢸⠀⣿⢸⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠸⡀⢸⠀⡇⢸⢀⡇⢠⠒⠒⠶⣤⢤⡀⠀⠀⠀⠈⢔⠀⠒⠼⠁⠈⡁⠡⢊⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠘⢴⣁⣸⣤⣗⣨⠞⠀⠀⡽⣆⠀⡤⠬⠽⠆⠀⠀⠀⢀⠐⠀⡰⠋⡊⢸⣞⣩⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠁⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⣶⠟⠛⠛⠦⣄⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣰⠊⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⢴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣤⠤⠀⠈⢀⡞⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡠⢦⡀⠀⢸⣤⣴⠟⠓⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠿⠞⠀⠀⠀⠈⠁⢀⡄⠀⠒⠺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⣀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⡆⠀⠀⠀⠀⠀⠀⡟⢛⢫⣥⣀⠀⠀⠀⠀⠀⠀⢀⠞⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⣀⣠⡴⣦⠀⢸⠀⠀⢀⣽⣤⣶⣤⣀⣠⣤⢋⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡕⠂⠘⠓⠒⠁⠀⣻⣿⣿⣿⡿⣟⣝⢚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⣤⣶⣿⣇⣠⣴⣿⣿⣿⣿⡷⣌⢀⣾⡀⠀⠀⠀⠀⢀⡀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⠟⡵⠫⣄⠀⠉⠲⣤⠶⠋⠉⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⢋⠞⢁⣠⡽⣦⡀⠀⠈⣦⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢯⡙⢿⠿⠿⠿⢿⠟⡡⠛⣾⣋⣁⣀⣈⡙⠲⠾⠋⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢀⡼⠃⠀⠀⣰⢋⠞⢵⠋⠙⠿⣶⣄⠀⠀⠈⣹⡟⠑⠒⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣾⠀⠀⢀⡜⡵⠃⠀⠀⡇⠀⠀⠀⣻⣷⣄⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣀⡤⠤⠀⠀⠀⠀⠈⢷⡄⠘⡶⢊⢫⠞⠁⠀⠀⣰⠃⠀⣴⣿⣿⣿⢯⣼⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⢠⢂⡾⠆⠀⠀⡴⠃⠀⣀⣸⣿⡿⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⣮⠞⠁⠀⢀⡞⠁⠀⢸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣡⠋⠀⠀⠀⢠⠎⠀⠀⣀⣤⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢋⠜⠁⠀⠀⠀⣰⣧⣤⣀⣀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡷⠋⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠸⡇⠀⠀⠀⠀⠀⣠⠶⠚⠒⠲⣿⡞⠁⠀⠀⠀⠀⠘⠋⠉⠉⣇⠀⠀⣿⢆⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠹⠀⠀⠀⢀⠜⢡⡀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣿⠘⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⡀⠈⠢⡀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⢻⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠀⡠⠊⡀⠀⠀⠈⢢⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⡟⠀⠘⣇⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠰⠞⢋⣁⡤⠂⣠⠔⠊⢀⡠⢂⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠠⡇⠀⠀⢹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠩⢭⣴⠲⠭⢐⣠⠔⣋⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠇⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠬⢭⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⠀⠀⠀⠈⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⢸⣿⣿⠁⠀⠈⣿⣿⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⢸⣿⣿⣷⣤⣾⣿⣿⡇⢠⣤⣶⣿⣿⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣤⣤⣤⡎
⠀⠀⠀⠀⠀⠀⣠⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣆⡤⣶⣦⣀
⠀⠀⠀⣶⣶⣶⣶⣶⠄⠀⠀⠀⠀⠀⠀⠀⢀⣠⢖⣯⣷⣿⣿⣿⣾⣽⡲⢤⡀
⠀⠀⠀⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⣿⣿⣿⢿⢿⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⣤⣤⣤⣤⣄⠠
⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⢹⣿⣿⣀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠛⠛⠛⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢰⡇⠀⠀⠀⠈⠁⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣀
⠀⠀⠀⠀⠀⠀⢀⣿⣇⡀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣦⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠈
⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⡇⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢠⠀⠸⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠎⢱⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⠀⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⡀⠀⠀⠀⠀⡂⠈⢦⣤⣶⣶⡄⠀⢸⣿⣿⣿⢿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣷⣦⠀⠀⠀⠇⠀⢸⣿⣿⣿⡇⠀⠀⠉⠉⠁⠘⠧⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⢀⣤⣧⡄⣼⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⣀⠀⣀⢀⡀⢀⣀⠀⠀⢀⣀⡉⠙⠋⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣶⣶⣶⣶⡶⠂⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⡿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣉⣉⣉⣁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠉⠉⠈⣯⠉⠉⠈⡏⠉⠉⠉⡏⠉⠉⢩⡏⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⠶⠶⣿⠶⠶⠶⡷⠶⠶⢶⡷⠶⠶⢾⠇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⣾⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⠛⠛⢻⠛⠛⠛⡟⠛⠛⢻⠟⠛⠛⡟⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣸⣆⣀⣀⣇⣀⣀⣸⣀⣀⣸⡇⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠃⠀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍⠉⢹⡏⠉⠉⡏⠉⠉⣿⠉⠉⣹⠁⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁⠀⠀⠈⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⣼⣧⣤⣤⣧⣤⣤⣿⣤⣤⣿⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣇⠀⠀⡇⠀⠀⡟⠀⢠⡏⠀⠀⠀⠀⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠶⠶⣿⠶⠶⡷⠶⢶⡷⠶⢾⡇⠀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⡇⠀⢸⡇⠀⣾⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠛⠒⠚⠛⠒⠚⠓⠒⠛⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠁⣀⠀⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡠⢠⣾⡿⠇⠀⠀⠀⠀⠀⠀⢁⠄⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠘⡀⠈⠒⢀⠀⢀⠉⠁⠀⠀⢰⡆⠢⣌⠙⠃⢀⠀⠀⡄⠀⠰⡏⠀⠀⠀⠀⠀⠀⣤⣀⠠⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠑⣾⠀⠀⠠⠀⠘⣇⡐⣶⠇⠰⠘⠁⠀⠀⠀⢰⠂⠀⠁⠀⠀⠠⢀⢹⡟⣡⣄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠁⠀⠁⠀⢠⠀⢠⣼⠿⠄⠀⠀⠀⢑⠂⡾⠇⠈⠁⠈⠀⠀⠀⠠
⠀⠀⠀⢠⠊⠀⠀⡐⠇⠀⠀⠀⠀⠀⣀⠀⡀⠀⠄⠀⠀⠐⠀⢠⠀⠆⠀⢐⡀⠀⠘⡤⡁⠐⠦⠀⠀⠀⠄
⠀⠀⠀⠈⠃⠀⡀⠀⡂⠀⠀⠀⠀⠀⠀⢠⡿⠁⢨⢂⠂⠦⡤⢆⠤⡋⠁⢦⣤⣀⡜⠀⠁⠀⠷⠀⠂⣴⣶
⠀⠀⠀⣀⡤⠐⠠⣁⣩⠦⢠⡀⠤⠚⠉⡀⠀⢂⣸⠂⡔⠤⡐⡆⠜⠀⠀⠀⠉⠈⠉⠁⠒⡶⣌⠐⠢⠈⢻⡧⠄
⠀⠀⠀⣿⣇⠀⠄⠯⠀⠄⠀⠀⠀⠀⠀⢄⠠⠄⡑⣀⠱⠟⢋⠀⡀⠀⠀⢀⡁⠀⠀⠀⡀⠑⠘⣧⡀⠑⡀⠓
⠀⠀⠀⡟⠉⡉⢃⠄⡀⠐⠄⠀⠀⠀⠀⠈⡉⢀⣁⠘⠁⠈⠀⠀⠐⢀⢠⢸⣇⠀⢒⡔⠀⠤⢊⡻⠗⣊⠽⠀⠦
⠀⠀⠀⠃⠀⢀⡌⠁⠹⠄⢈⡄⠀⠮⢰⢀⡀⠀⠂⠱⡀⠀⠀⠀⠄⠀⠁⢄⡛⠾⡿⠠⡔⣋⠤⠒⠊⠀⣀⠤⠒
⠀⠀⠀⠀⢀⣌⠀⠀⠀⢀⠆⡔⠂⠀⠥⠭⡀⠀⠐⠲⢈⠠⠂⠀⠀⠀⠀⣄⣱⡷⠉⠉⠉⠉⠡⠂⠎⠁
⠀⠀⠀⠁⣿⣿⣤⠄⢖⣡⣭⡧⠀⡀⠀⠀⡀⠴⠤⡀⣻⣶⡆⠀⠁⠀⠀⠁⠥⣾⡖⠀⢰⣦⣛⡄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡉⠉⡥⢸⡟⢉⡹⢧⠁⠁⠀⠀⠱⠀⢀⢁⡿⠻⠁⠂⠆⠀⠆⡂⠀⣽⠇⠀⠈⢥⡀⢁⠀⠀⡀⠄⠀⠂⡀
⠀⠀⠀⠀⠱⠀⠠⠘⠧⠀⠀⣦⣥⠀⠈⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠠⠄⢀⣸⡇⠀⠀⠀⠀⠈⠞⠡⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⣙⠃⠀⠛⠂⠤⣾⠃⠀⠀⠀⠀⠐⣬⠦⠚⠙⠃⠀⠀⠒⠀⠐⡇⠀⠤⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢘⠁⠺⠃⠸⠀⠀⠀⢙⣤⠄⠐⠂⠁⠀⠀⠀⠀⢀⠄⡀⠂⠀⠀⡇⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠃⡀⠀⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⢀⠐⠋⠀⠁⠲⣄⠀⢀⠠⠋⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠀⠀⠀⠅⠀⠀⠀⢀⣠⡔⠁⠀⠀⠀⠀⢀⣼⣶⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠄⠀⠀⠈⢹⠛⣿⣿⣶⠒⠒⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣚⣭⡙⢦⣀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⠿⠃⠀⠈⠳⣤⡞⣩⣉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⢀⣴⣿⣶⡄⠙⢧⡈⠙⢦⣟⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡿⠋⠀⣠⣤⡙⢦⡀⠈⢿⡯⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⣠⣾⣿⣿⠇⠀⠙⢦⡀⢳⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠟⠁⠀⠀⣠⡀⠙⣞⣠⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⣠⠾⢿⣿⠆⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡼⠋⠁⠀⠀⠀⢻⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⢼⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⢀⡴⣷⣦⡿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠶⠶⠶⠶⠒⠒⠛⠛⠛⠷⠶⠶⠶⣦⣤⣙⠻⣿⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠿⠛⠋⠉⠉⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠈⢹⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⢁⠤⢀⣀⣠⣤⣤⠶⠶⠟⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⣿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠶⢿⠋⠉⠁⠀⠀⠀⠀⠀⢰⡶⠖⢷⢿⢿⡆⠀⢠⣤⠶⠶⠆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⡴⠾⠿⣆⠀⠀⠀⢸⡇⠀⠀⣛⣽⠇⠀⣼⠃⠰⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠘⠂⢻⡀⠀⠀⢸⡇⠀⠀⡾⣿⠀⠀⡿⠀⠘⠁⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢸⡇⠀⠀⢸⡇⠀⢠⠁⣿⠀⠀⡇⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⡇⠀⣸⢻⡟⠀⢀⡇⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢸⡇⠀⠀⢸⡇⠀⢁⡼⡇⠀⢸⠇⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢸⡇⠀⠀⢸⡇⠀⢸⡇⡇⠀⢸⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⢸⡇⠀⢸⠀⡇⠀⢸⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣷⠀⠀⢸⡇⠀⡌⢀⡇⠀⣸⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣿⠀⠀⢸⡇⠀⠁⢸⡇⠀⣿⠀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣸⡟⠀⠀⢸⡇⠀⠀⢐⡇⠀⠛⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣤⣉⣛⣿⣋⣡⣤⡾⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣛⣛⣋⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡿⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣶⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠈⠀⠠⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠠⢳⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠀⠀⠀⠁⠀⠀⢀⠶⡇⠀⠀⠚⠂⠈⠀⣀⡠⠔⣠⣿⢯⠠⣾⠛⠆⠂⠀⠀⠀⠀⠄⣀⠠⢠⠄⡰⢿⣿⠻⢆⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠡⡄⠈⠂⠀⡀⠀⠀⠀⣴⣷⡀⠀⠀⠀⠀⠀⠀⠋⢉⢄⡔⠀⢰⣿⡃⠚⠁⡔⠀⠠⠀⠀⢡⣴⣶⠀⠗⢈⠱⠄⠠⠛⠀⠁
⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠉⢻⡇⠀⠟⠁⠱⡀⠀⠀⠄⠂⠈⠁⠠⠉⠁⠉⠈⣙⠻⠶⠊⠀⠁⠐⠀⢀⠼⣿⠋⠤⠂⠂⠀⠀⠀⡀⠀⠢
⠀⠀⠀⠠⠤⠔⠀⡀⢁⠀⠀⠴⠌⠈⠻⠀⠈⠤⢆⣴⣄⡚⠐⠂⠈⠄⠀⠀⠀⠀⠀⢸⠀⠀⠠⠀⠁⡴⠊⢀⡈⠻⣄⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠠⡄⠀⢠⣆⢐⠀⠀⠠⠁⠈⠦⠀⠒⠄⠂⠍⢉⣥⠀⠀⠀⠀⠀⢠⠀⠈⡇⠀⠀⠀⡸⠁⡜⢿⡏⠐⡈⠛⡃⠂⡨⢻⡀⠁
⠀⠀⠀⠀⠆⠀⠀⠀⣸⡿⣿⣉⠀⠈⠀⠀⢏⡁⠀⡷⠾⢦⡝⢿⡆⠀⠀⠀⠀⠈⡄⣰⣳⡄⠚⠛⠒⠚⠻⠦⣈⣉⣀⣴⡀⠄⠀⠀⠹⣶⡿⠂
⠀⠀⠠⢀⠀⠑⠀⣴⠉⠒⠀⠈⠉⠂⠥⠀⠈⠢⣴⣄⡤⠠⠠⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠄⠂⣒⠒⢂⠀⢸⠟⠁⠀⣷⠤⠤⠤⠀⣿⣿⡀⢀
⠀⠀⠀⠀⠈⠀⠀⠈⠃⠘⠅⢆⠀⠀⠀⠀⣀⠀⣈⣉⠉⠉⠍⢉⠉⠉⠍⠉⠉⠉⠁⠤⣄⣑⠆⠀⠐⢠⠀⠰⠏⠀⠀⠀⠰⠆⠐⠀⠀⣿⣟⠓⠄
⠀⠀⠀⠀⠀⡀⠀⢼⡀⠀⠀⠄⠢⠀⠀⢠⠋⠁⠄⠀⠀⠒⣲⠀⠓⠀⡖⠳⣆⡀⠀⠀⠦⡀⣄⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠰⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠁⠈⠣⠀⠀⠀⠀⠠⡀⡳⢀⢀⠈⠀⠀⠀⢭⡒⣄⠀⡡⠀⠁⠈⠈⣉⠠⢙⠦⢭⠀⣓⣀⠠⠜⢆⢀⠔⠂⠒⠈⣶⠿⣶
⠀⠀⠀⠀⠀⠀⣄⠀⠀⢢⡀⠀⠀⢄⠹⡇⠀⢐⠉⠁⠀⠀⠀⠀⢸⠆⠀⠐⠛⠁⠁⠉⠀⠈⣱⣄⣹⠆⠉⠉⠉⠉⠲⠒⠶⠖⠂⠰⠶⠋
⠀⠀⠀⠠⠀⢸⣿⣇⠀⢀⠜⠋⠁⠙⠂⠀⡀⠠⠂⠀⠀⠀⠀⠀⢸⢀⣶⠄⠀⢶⠰⠠⠶⠃⠙⠻⡅⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠀⠰⠆⠀⡀
⠀⠀⠀⠁⢂⣸⣿⣿⡿⠁⠀⠐⣐⠀⠀⣤⠘⠔⠂⠀⢀⡀⢄⠀⢬⠸⣿⣼⣷⢠⠖⠠⠓⢂⠑⠿⡸⣦⡃⠀⠀⠀⠀⢀⠏⠀⢀⠀⠀⢀⠀⠃⠊
⠀⠀⠀⠀⠘⠿⠿⠿⠇⠀⢀⠟⠋⠙⠒⡄⢢⠘⢦⠀⠈⠐⢀⡠⢺⣦⠜⣄⠊⠀⠀⠠⠀⠀⠀⡄⠻⢾⣄⠀⠀⠀⣰⠎⣴⣄⠀⠠⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠪⡿⠖⣼⠈⢧⠈⢣⣀⠠⠙⠦⡀⠀⣀⣇⠀⠀⠀⠀⠀⠀⢂⠉⠂⢀⢿⡶⠖⠈⠁⢿⠟⣡⡴⠀⣶⠀⠀⠁
⠀⠀⠀⠀⠀⠄⠀⠀⢀⢰⣾⣷⢀⣶⣾⠿⠉⠶⡆⠀⠸⠀⠀⠠⢚⣻⡿⠟⠊⠁⢀⡀⠀⠀⠀⠁⠀⢈⠘⠀⠀⠀⠀⣴⡿⢟⣥⣾⡿⠧⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⡛⣾⠋⣠⣾⠁⠀⠀⠀⠀⣀⣴⡆⠉⠀⠀⠀⠀⠀⠈⠈⠰⣄⠀⢀⠄⠃⠀⠀⠀⢠⡾⢋⣴⣿⠟⣡⡄⠙⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠑⠃⠀⠛⠋⡆⢀⢤⣶⣷⡈⢛⠀⠀⢀⣀⡀⠀⠂⠈⠀⢐⣅⠀⠀⠀⠀⢀⡀⢤⡜⢴⡿⠋⠡⠾⠋⠐⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠀⠀⠈⠀⠒⠸⢙⣷⣦⡽⠟⠃⠈⠀⡀⢀⡁⠀⢀⡀⢠⣾⣿⠇⢠⣦⣴⣤⠊⠀⠀⠣⠤⠃⠀⠀⠀⠀⠀⠀⠷
⠀⠀⠀⢀⠈⠢⠐⠀⠀⠀⠑⠂⠀⠐⠨⠐⠘⢿⣿⡛⠀⠀⢀⠀⠈⠐⠑⢈⠉⠘⠛⠉⠀⠐⠳⠌⠙⠋⠀⢀⠀⠀⠀⠠⠀⠃⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠈⠀⢾⣿⠿⢄⠀⡈⠁⠀⢆⣠⠄⠈⡀⠀⠀⠑⣏⠓⠀⠀⠀⠀⠀⠙⠢⣄⠀⠉⠋⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣆⣤⣶⠿⠋⠁⢀⡀⠑⠀⠀⣀⣠⣿⡒⠀⠀⠀⠀⠀⠀⠀⠙⠢⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢠⠄⠀⠂⠀⠀⠠⠘⠻⣿⣿⣿⣷⣶⠄⠄⠀⠀⠀⡀⠀⠀⠄⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠲⠤⢴⣞⣉⣷⣤⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢥⣀⣀⣀⣀⣠⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣭⣉⣁⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠿⠿⣿⠿⠿⣿⡿⠿⢿⣿⠿⠿⣿⠖⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⣀⣷⣀⣀⣸⣀⣀⣰⣇⣀⣀⡟⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⣿⠉⠉⢹⠉⠉⢹⡏⠉⢩⡇⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠶⠶⢿⠶⠶⢾⠶⠶⢾⠷⠶⢾⠁⠀⠀⠀⠀⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣸⣀⣀⣸⣀⣀⣸⣀⣀⣿⠀⠀⠀⠀⢰⣿⣿⣿⡿⠁⠘⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⢹⡏⠉⢹⠉⠉⣿⠉⠉⡇⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠶⠾⡷⠶⢾⠶⠶⣿⠶⢾⡇⠀⠀⠀⠀⣾⣿⡿⠁⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣇⣀⣸⣀⣀⣇⣀⣸⠀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠉⣯⠉⢹⠉⢉⡏⠉⡿⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠤⠿⠤⠾⠦⠼⠧⠴⠇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⢻⢰⠉⢹⡄⠀⡷⣼⢷⡋⢙⡆⢹⠋⠀⢸⠀⠀⡇⢹⡏⠈⣿⠁⣿⠭⢸⡯⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠈⠙⠋⠀⠀⠁⠙⠁⠙⠋⠀⠈⠁⠀⠈⠛⠀⠁⠈⠁⠀⠉⠀⠙⠛⠈⠁⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠒⠊⠉⠉⠉⠉⠉⠉⠓⠒⠲⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠉⣀⣠⣶⢀⡴⣛⣥⠾⠋⠉⠉⢻⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠻⠟⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⢻⣾⣿⣏⣡⢼⡟⠉⠳⡄⠀⠀⠀⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡃⠀⠀⠘⠛⠻⢿⣿⡧⣟⠀⢰⠇⠀⠀⢀⡾⠀⠀⠀⠀⠀⢠⡶⠶⠶⢤⡴⠋⠙⠓⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢃⡟⠀⠀⠀⢀⡴⠛⢉⢿⣰⠋⠀⡏⠀⡴⠞⠋⣤⣤⠀⠀⠀⠀⣸⠃⣴⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡾⠀⠀⠀⠀⢾⡀⣴⠋⠦⡧⠀⠈⠧⠞⠀⠀⠀⠉⠉⠀⠀⠀⢿⣁⠐⡏⢳⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣼⠁⠀⠀⠀⠀⠀⣿⠃⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡀⢉⣙⣧⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⢀⡴⠒⢦⡀⠀⠀⢠⠃⢠⡏⠀⠀⠀⠀⠀⠀⠀⠠⢶⡞⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢯⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢀⡴⠶⢦
⡞⠀⠀⠀⢹⡄⠀⣼⠀⢸⠃⠀⠀⠀⠀⠀⠀⣠⠴⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣪⠀⠀⣀⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⢠⡏
⡇⠀⠀⠀⠀⣇⠀⡇⠀⢸⣆⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣰⠎⣁⡈⢿⠷⣮⠓⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸
⡇⠀⠀⠀⠀⢸⠀⡇⠀⠘⣿⠀⡴⠒⢦⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠛⠉⠁⠙⢦⣠⠶⠲⠖⣁⠀⢰⡛⣦⡴⢦⠀⣾⣿⠀⡇
⡇⠀⠀⠀⠀⢸⡀⡇⠀⠀⠸⣆⣧⣤⢶⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣝⢇⠀⠉⢩⠇⠸⣿⣿⣿⠀⡇
⡇⠀⠀⠀⠀⠈⡇⢳⠀⠀⠀⠈⠳⣼⡆⠙⠷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡀⣠⠞⠀⠀⣿⢹⡏⢸⠁
⡇⠀⠀⠀⠀⠀⡇⠘⡀⢠⠔⠢⣄⠈⠻⣄⣴⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣇⣠⠖⢦⣄⣼⠁⢸
⡇⠀⠀⠀⠀⠀⢳⠀⢻⡏⠀⠀⠈⢣⠀⣸⠇⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⢹⡟⠀⡾
⡇⠀⠀⠀⠀⠀⢸⠀⠸⡇⠀⠀⠀⠈⡆⢧⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⠀⠀⠀⠀⠀⢀⡞⣻⠀⠀⠀⠀⢸⠇⠀⡇
⢳⠀⠀⠀⠀⠀⢸⠀⠀⢷⠀⠀⠀⠀⠸⡌⢧⡀⠀⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣼⣡⠇⠀⠀⠀⠀⡿⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠘⡆⠀⠘⡇⠀⠀⠀⠀⠹⣆⠙⢦⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢀⡜⣿⠏⠀⠀⠀⠀⣸⠃⠀⢰⠃
⢸⠀⠀⠀⠀⠀⠀⢧⡀⠀⢻⡀⠀⠀⠀⠀⠙⢦⡈⢇⠀⠀⠀⣴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡴⢋⡼⠃⠀⠀⠀⠀⠀⡟⠀⢀⡼⠀⠀⠀⠀⠀⢀
⠈⡇⠀⠀⠀⠀⠀⠀⠉⠲⣜⣇⠀⠀⠀⠀⠀⠀⠙⢾⣆⢀⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢋⡴⠋⠀⠀⠀⠀⠀⠀⣸⣣⠖⠉⠀⠀⠀⠀⠀⠀⢸
⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⠤⣤⣤⣄⣀⣀⣠⣤⣤⠤⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⠈⣉⣁⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣴⠶⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠛⠶⣦⡀
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⡿⠛⢻⡟⠛⠛⠛⢿⠟⠛⠛⠻⣿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠰⣿⣿⠀⢸⠶⢿⡇⠀⡽⠶⣿⠀⢈⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣷⣤⣼⣧⣤⣤⣤⣿⣦⣤⣤⣴⣿⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠛⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠰⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏⢻⣿⡏⢻⣿⡟⢹⣿⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⡇⢸⣿⡇⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠘⣿⡇⢸⣿⠇⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⡇⢸⣿⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢻⡇⢸⣿⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣾⣷⣾⣷⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠉⣩⣉⣹⡏⠉⣹⣿⣯⠉⢩⣉⣹⣿⠋⠉⠹⣿⡏⠉⠙⢯⠉⣽⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⡀⠚⠉⣹⡇⠀⠛⢻⡿⠀⠐⠋⢹⡏⠀⡄⠀⢻⡇⠀⣦⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⣺⠄⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠇
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠐⠐⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠂⠀⠀⠀⠉⠂⢄
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡰⠀⠈⠈⠀⠀⠀⢀⣀⠀⠢⠈⠀⠀⠀⠠⠄⠉⠑⠀⠁⠈⠄⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠠⠊⡐⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⠂⠐⠳⡄⠀⡀⠘⡀
⠀⠀⠀⠀⠠⠁⠌⠠⠂⠀⠀⠀⠊⠀⠐⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠱⠀⠀⢙⣈⠁⠀⠀⢡
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢹⣃⠀⠸
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠄⠁⠀⠀⡤⠂⠀⠀⠀⠀⠈⠀⠀⢀⣤⠦⣬⠀⠀⠀⢸⡆⠐⡈⣀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠁⠀⠀⠀⢠⠣⠀⠀⠈⠂⠂⠀⠁⠀⠀⠈⠑⠈⡑⢫⠋⠩⠂⠈⠙⣗⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠀⠀⠀⠀⠀⠁⠈⠈⠙⠉⡊⠈⠀⠀⠀⠀⠀⡆⠀⠀⠀⡀⠀⠀⠀⠉⢪
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⡠⡀⣀⠂⠀⡁⠀⠀⠁⡀⠀⠁⠉⠡⡞
⠀⠀⠀⠀⡂⠀⠀⠀⠈⠀⠀⠂⡀⠂⠀⠀⠐⠀⠀⠀⠀⠎⠁⠉⠈⠉⠉⠠⠀⠀⠈⠑⡦⠂⠀⢠⡁
⠀⠀⠀⠀⠅⠀⠀⠀⠀⠐⢠⠐⠈⠳⠤⠜⠱⢃⢑⣦⡌⢀⠐⠀⠱⠰⠄⠖⠄⠤⡤⠄⠀⠀⠘⣠⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⢀⠀⠀⠉⢏⠀⠈⡛⠋⠋⢻⣀⠀⠠⠂⠐⢂⠀⠐⠀⠐⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⡜⠃⠠⠔⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⡀⢀⠀⢀⢀⠄⠀⠀⠠⠄⡀⢡⡄⠀⡀⠀⠀⠀⠀⠀⠀⠂⠄⡀
⠀⠀⠀⠀⡄⠀⠂⠀⠂⠀⡄⠆⡬⢡⢀⣶⠄⠀⠀⠀⠀⠈⠀⠀⣀⠙⠦⡀⠸⠘⠀⠠⡀⠂⡦⡤⢀⠀⢠⡀
⠀⠀⠀⠀⡇⠀⠀⠳⠖⢂⡤⠤⠄⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣁⣀⣁⣀⡀⠠⠤⣤⡰⠶⠾⢊⡇
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠀⡄⠈⡧
⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡄⠀⡧⠀⡠⢀⠄
⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡠⢁⠌⡀⠀⠂⠀⡀⠁⠀⠀⠄⠀⠀⠀⡠⠊⠐⡁⠐⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⣤⢄⣀⡀⠀⠀⠀⠀⠄⠡⠊⠔⠁⠈⠀⠡⠂⠈⠀⢀⣀⡄⡤⡶⢚⠁⢂⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠒⠒⠒⠲⠒⠐⠒⠒⠒⠃⠙⠩⠀⠉⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⡿⡿⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⡦⣜⢷⣶⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣿⣿⣭⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠚⠉⠁⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠴⠖⠚⠉⢁⣰⡏⢳⡴⢦⡀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⡀⠸⡄⠀⠀⠀⠀⣯⠈⣇⠈⡇⠘⡗⠓⡆⠘⡆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢰⠇⢹⢀⢷⠀⠀⠀⠀⢸⡀⢹⠀⢻⠀⢿⠀⢷⠀⠹⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣸⠀⢸⠟⠈⣇⠀⣠⢤⡈⡇⠘⠇⠘⠃⠸⠆⢸⡄⠀⢻⡀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⡏⠀⠘⡇⠀⠘⣆⢳⡀⠹⣿⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⣇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢻⡀⠀⠁⠀⠀⠹⡄⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⡿⠤⠴⠿⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢷⠀⠙⢦⣀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⡴⠚⠸⠶⠖⠒⠈⢹⡀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣟⠙⠋⠙⠋⠙⠋⠙
⠀⠀⠀⠀⠁⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⣏⠀⢀⠀⠄⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠈⠰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⠶⠓⠒⠒⠊⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠰⠃⡀⡄⠀⠀⠀⠀⠀⠈⠓⠒⢦⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢀⠀⠈⠀⢄⠃⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠁⢠⠡⠙⢆⠀⠀⠘⠂⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠢⠗⠀⠀⠀⠀⠹⡖⠼⡶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠠⠀⠀⢀⠀⠓⠒⡤⠂⠈⠀⠀⠹⢶⠼⡦⡴⠀⠀⠠⢂⠆
⠀⠀⠀⠀⠀⠀⠀⢦⠀⢰⡀⣠⢀⠀⢤⠀⣠⠀⠁⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⡰⣂⣽⢦⡀⠀⣡⣶⡄⡀⢀⣠⣌⠀⠉⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠁⣃⣠⡈⠉⠁⠀⡐⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡁⠀⠂⠂⣀⠙⠋⠙⣀⠈⠛⠋⠀⠠⠃⢦⡦⠤⡀
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⡀⡁⠀⢀⢀⡄⠈⢀⡄⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⢠⠂⠀⠈⢻⣷⣠⠀⠀⢀⡀⠀⠀⡄⠐⡌⢢⡃
⠀⠀⠀⠀⠀⠀⠘⠀⢀⢤⣳⠥⠬⠦⠥⠴⣫⢖⡄⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢀⢄⡣⡧⠬⠴⠌⣴⡥⣶⡀⠀⡸⠁⠀⠈
⠀⠀⠀⡀⠀⠀⠀⢀⠮⣗⠯⠭⠭⠭⠭⠽⠉⡷⢖⡁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠫⡳⢻⠭⠭⢽⣿⣿⡟⢔⡻⠄⠈
⠀⠀⠀⠀⠆⠀⠀⠶⡽⠃⣏⣉⣉⣋⣉⣙⠛⢭⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠳⠝⠀⣬⣭⣭⣭⣭⣭⣇⠈⢪⠽
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣯⠉⢉⣩⢉⣤⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢀⠀⠓⠶⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠤⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠄⠀⡄⠂⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠰⠟⠀⠨⠅⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄⠰⠠⠳⠜⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠅⠽⠈⠁⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠉⠛⢶⣤⣀⣀⣀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⣼⠂⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣿⣶⣦⣤⣤⣤⣤⣤⣤⣼⣷⡄⠀⠀⠹⣧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⣾⣯⣤⣤⣤⣤⣤⣴⣶⣾⣧⣤⣤⣤⣤⣤⣤⣤⣄
⢀⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⢻⣆⠀⠀⠀⣾⠃⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣄
⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠷⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠶⠶⠶⠶⣶⣶⣶⠶⠶⠶⠶⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⣠⣄⠀⠈⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⢀⣾⠋⠙⣷⡀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣾⠃⠀⠀⠘⣷⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⣿⡀⠀⠀⢀⡿⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠈⠛⠷⠞⠛⠁⣀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⢶⣤⣤⡶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠙⠷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡶⠟
⣶⣶⡶⠶⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣀⣠⠤⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⡞⣩⣶⣾⠿⠿⠿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣬⣿⣾⣿⣷⣾⣿⣿⡿⠶⣦⣭⣭⠵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⢿⣿⢿⣿⡻⣿⣿⣿⣿⣿⣶⣌⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣿⡿⣾⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣿⡆⡀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⢀⣿⣿⡷⣿⣏⣿⣿⣿⢿⣿⣿⡈⠻⣿⣿⣿⡎⢻⣿⣿⣿⣿⣿⣿⢠⠞⢓⣿⢹
⠀⠀⠀⠀⠀⢰⢮⣗⡛⢦⠀⠸⣿⣽⣧⣿⣿⣇⣙⣿⣧⡙⢿⣷⣄⠈⠙⠿⠓⠆⢿⣿⣿⣿⣿⣿⣾⠂⠈⣩⠟
⠀⠀⠀⠀⠀⠘⢏⠉⠀⠸⣄⠀⣿⣿⣿⣿⡟⠋⢀⣤⡀⠈⠀⠈⠉⠀⢀⣴⣦⡀⢸⣿⣿⣿⣿⡿⠃⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠘⢦⣽⣟⡿⢿⣷⠀⠚⠉⠛⠀⠀⢀⠀⠀⠈⠁⠀⠁⠸⠏⣭⢳⠟⠀⢀⣼⣯⣑⣒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠙⢿⡘⠃⠙⠀⠀⠀⠀⢠⡀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣠⠞⠀⣠⣾⣯⣛⣋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣦⡀⠀⢹⣦⣴⣆⠀⠀⠀⠘⣿⣿⣿⣿⠇⠀⠀⢀⣾⣿⣿⣤⣾⣿⡛⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠶⣿⣿⣿⣿⣷⣤⣀⣀⠈⠛⠛⣋⣀⣤⣶⣿⣿⣿⠟⠁⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⠉⠉⢉⠉⠉⢀⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦⣤⣾⣷⣶⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠉⢛⠋⠉⠉⢉⡉⠀⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢠⣏⣀⣀⠀⠀⢻⡄⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⣾⠁⢹⣷⣿⡇⠀⠘⣇⠀⣀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣸⢿⣿⡇⠀⠘⣯⣿⠿⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢻⣿⡇⠀⠀⠘⣧⠤⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣦⡙⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡎⢻⣿⣿⣿⣿⣿⣿⣿⣿⣆⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⠟⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣀⣀⣀⣀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠒⠛⠛⠛⠛⠛⠛⠃⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡟⠙⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠰⣌⡻⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠘⠿⠿⠿⠿⠟⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠤⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⠐⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⠁⠒⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠁⠀⠠⣈⡠⠂⠀⠀⠀⠀⠀⠀⢀⠤⠤⠂⠀⠀⣀⣀⠀⡠⢀⡀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠔⢀⠠⠃⠀⠀⠀⠀⠀⣀⠤⠔⠈⠐⠁⠀⠀⠀⠀⠊⠀⠈⡀⠀⠅⣂⠁⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⠔⢁⠖⠁⠀⢀⠄⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠁⠀⣈⣗⡀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆⠀⠃⠀⠀⢠⠆⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⠀⠈⣥⣂⠡⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⡈⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⡀⠀⠠⠂⠀⠀⠀⠄⠔⢀⡆⠀⠀⠈⢏⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠁⠀⠀⢲⠂⠀⠀⠀⠀⠀⢈⠀⢀⣠⣴⢢⢤⠁⢊⡂⣈⠇⠈⡭⣅⠮⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡐⠃⠀⠀⠀⠀⠀⠜⠢⡈⡀⢀⡐⢁⣀⠀⠀⠀⠀⢉⠐⠉⠩⠉⠁⠒⠈⠉⣳⠭⡂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠐⠄⠀⠀⠀⠀⠀⠈⠀⠁⠩⠍⡉⣁⡂⡠⠀⠀⢀⡀⠀⠀⠂⠁⠀⠀⠀⣈⠻⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣌⡤⣨⣌⠀⣀⡀⠀⠀⠁⡀⠀⠒⠉⠁⢾⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠐⢄⠐⢂⠁⠠⠊⠀⠐⠀⠀⠌⠊⠁⠈⠉⠉⠄⢄⠀⡄⠑⢦⠐⠐⠠⡎⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡕⠊⢛⠦⡴⠋⠛⣘⣹⣭⡁⢨⠂⠀⠖⡲⢀⠖⠔⠤⣤⠄⠠⠊⠚⢀⢗⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠍⠊⠀⠈⠳⣅⠀⢉⠚⠉⠩⣒⣀⠁⠨⠊⠐⢂⡈⠀⠊⠈⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠈⢐⠛⠂⠈⠓⠐⠀⠀⠁⠀⠀⠐⠂⡀⠠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⠐⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠒⣻⣿⣶⠃⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠌⣣⠀⠀⠀⠀⡋⠀⠀⠀⠐⠛⠛⠛⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠆⠀⠀⠄⣄⣲⢠⢟⡀⢣⣄⠐⠁⠀⠀⠈⢒⠀⠁⠈⠻⣍⠙⢻⠻⠛⢁⠃⡆⣶⢡⡉⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠤⣀⡤⠀⠀⣈⣈⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠈⠉⠀⠃⠃⣂⣁⡀⠉⢠⣄⣡⡤⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠐⠚⠛⠋⠉⠉⠁⠀⠨⠭⠭⠍⠉⠉⠉⠐⠒⠒⠂⠈⠉⠉⡁⠀⢠⢁⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄⠀⠀⠀⠀⠀⠀⢆⢀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⡐⢁⠈⡐⡠⢂⠈⡐⠀⠠⠁⢀⠀⠊⡡⢋⡠⢊⠄⡀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⣀⡀⠀⠀⠀⠠⠐⠀⠈⠀⡠⢊⠔⡠⠊⠐⠡⢂⠔⡠⠁⠌⠐⠐⢀⣁⡤⠖⡈⠐⡁⢀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠛⠒⠦⠦⠤⣤⣬⣄⠁⣈⡀⢈⣤⢤⣤⠄⡠⠶⠂⠜⠛⠉⠊⠔⠈⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠒⠂⠀⠐⠢⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⠴⡄⢠⠎⠁⠈⠉⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⠁⠀⠧⠊⠀⣈⡉⠁⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⡠⠃⠀⠀⠀⠀⠉⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣤⠊⢁⠀⠀⠀⢿⣄⠤⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⢳⣄⣠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠢⢄⡀⣪⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⡀⠔⠠⢍⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡵⠋⢽⢶⡒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⢀⣀⣀⠔⠋⠀⠀⣱⡀⠨⠷⢬⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠐⡀⠀⠀⠀⠀⠀⠈⢣⡀⠀⢀⠾⠂⠁⢀⡀⢀⡸⠁⠀⠀⠀⠀⠑⠌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠘⠦⡀⠀⠀⠀⢀⠌⠑⠊⠁⠀⢀⠴⣞⠿⢃⣐⢐⣂⣀⣀⣶⣶⣶⣿⣬⡢⡀
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⣠⣏⠀⠀⣀⣤⠒⠁⠀⠈⠻⡥⢯⣩⣭⠭⠥⠤⠤⢤⣟⡛⣙⢮⡢⡀
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠑⠊⠁⢸⡀⠀⡠⢤⠀⠈⢻⡶⠒⠒⠒⠂⠋⠭⠿⣇⣰⣃⣙⢮⡢⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠱⣄⠑⠘⠀⠀⠀⠙⣭⠿⠷⠿⠷⠦⠶⠉⠩⠭⣿⣄⣱⣜⢄
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠣⡑⢄
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠘⢦⠳⢄
⠀⠀⠀⠀⠀⣧⠤⠤⠤⠤⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣄⣀⡀⠀⠤⠤⠤⠤⠤⠤⠤⠤⢤⣴⡇⠀⠀⠑⢄⠳⣄
⠀⠀⠀⠀⢀⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠠⡂⡢⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠳⡌⠂
⠀⠀⠀⠀⢸⠇⠀⠀⡌⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠤⣀⠀⠀⠈⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⡸⠀⠀⣰⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢠⡃⠀⢀⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠉⠂⢄⡀⠀⡇
⠀⠀⠀⠈⠉⢹⠻⠆⠶⠄⠀⠀⠠⢤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⡀⠀⠀⠈⠙⠇
⠀⠀⠀⠀⠀⠘⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⡀⠀⠀⠑⠒⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡎⠀⠀⢸⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠁⠂⠀⣀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠃⠀⠀⢸⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠉⠐⠢⢄⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢀
⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠊⠁⠉⠉⠉⠉⠉⠉⠉⠉⠈⠁⠀⠀⠀⠉⠈⠉⢱⠀⡖⠒⠒⠒⠋⠉⠉⠈⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢢
⠀⠀⠀⠀⠉⡏⠀⠀⠀⠀⠘⠛⠀⠀⠈⠛⠂⠀⠀⠀⠀⠛⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠂⠀⡀⠐⢀⠀⢀⠀⠀⠀⠀⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠂⠀⠂⠘⠃⠠⠛⠀⠀⠀⠀⠀⢀⣁⠤⣺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠘⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠉⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⠔⡇⠀⠀⠀⠢⠀⠠⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢀⢄⣨⠐⢁⠃⠐⠒⠀⠐⠒⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠑⠒⠀⠠⠤⡄⠀⠀⡅⠀⠈⣠⢤⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢠⢼⡒⠉⠀⠀⢰⣏⠀⣲⠄⠀⠀⠀⢰⠤⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⡜⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⠀⡘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠁⢀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣤⣤⣤⠀⡤⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠿⠹⠿⠀⠇⠷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠰⣆⢰⡆⣶⠶⢠⡶⢦⡰⣶⡶⢤⡶⣦⠐⣶⢢⡄⣶⠀⠶⣶⢰⡶⠆⠀⢠⡶⣄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣾⠁⣿⠤⢸⡇⠈⠁⣿⡇⢻⡇⣿⡗⣿⣼⠇⣿⠀⣼⠇⢸⡧⠄⠀⢈⣿⢃⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢹⡟⠀⣿⣀⠘⣧⣼⠃⢻⠇⠸⣇⣿⠡⣿⢸⡆⢿⠆⣿⣀⢸⣇⡀⠀⠸⣯⡿⡁⠀⠀⢸⣿⣿⣿⣉⠁⢉⢏⠉⡉⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣤⣀⢠⡄⠀⣤⣤⡄⢠⡄⠀⣄⠀⡄⠀⢠⡄⢠⡄⣤⣤⣀⠀⠀⠀⠀⠀⠀⢸⣏⢹⠙⠉⡉⣉⢉⠉⠉⡉⢉⣙⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠛⢺⡇⠀⣿⣇⠀⣾⣿⠀⣿⣦⡇⠀⢸⡇⢸⡇⣾⣇⡿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⣴⢎⣇⠀⣾⡅⢀⡿⢾⡆⣿⢹⡇⠀⠘⣧⢸⡇⢿⠅⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠷⠾⠯⠭⢿⡃
⠀⠀⠀⠀⠀⠀⠀⡉⠉⣀⠉⢉⠈⣉⢉⡁⢈⠁⡉⣀⡁⢀⣀⣈⡉⣀⣉⡀
⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⣿⣸⣷⣏⣽⡇⢸⣧⣿⣿⡷⢸⣿⢿⣿⡇⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠓
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⣀⡀⠀⣀⡀⢀⡀⣀⢀⡀⣀⢀⡀⣀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⡏⢸⡇⣿⣀⡿⢀⣿⡇⢸⣷⣿⣼⠁⣿⢸⣷⣼⠀⡏⠘⠇
⠀⠀⠀⠀⠀⠀⢸⣇⣸⡇⣿⠙⣇⢸⡷⣿⠈⣿⢫⣿⠀⣿⢸⡟⣿⠀⣇⢹⡇⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢀⡇⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠙⠀⠋⠙⠁⠙⠀⠉⠈⠋⠀⠋⠈⠁⠘⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⡀⣀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢸⣿⡿⠹⠁⠘⡿⢀⣿⠉⣿⣿⣿⣿
⠀⠀⠀⠀⠘⣿⡇⠈⠁⠀⠀⢀⡀⠀⠈⠻⢻⠏
⠀⠀⠀⠀⠀⠘⢟⠠⠛⠐⠀⠈⠁⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⢈⡁⣠⣔⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠁⠀⠈⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⢡⠀⠀⠀⠀⣰⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣊⠑⣾⠀⠀⣠⡎⠣⣀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣤⡠⠤⠲⢾⣟⣶⣤⣀⡀
⠀⠀⠀⢰⢁⠿⢷⠂⣯⢜⠒⠊⡠⠜⠶⠋⢸⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣷⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⣀⠭⠷⡶⠧⢤⣎⣀⣿⠀⢀⣀⣠⣾⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠙⠉⠀⠀⠀⠀⠙⣻⣿⣿⡇⠀⠀⠀⠀⢹⡿⣿⣿⣯⡟⠒⠠⢤⠴⣰⡆
⠀⠀⢸⣀⡠⠤⠿⠾⠤⠤⣀⣨⡿⣭⣿⣿⣷⣿⣴⢶⣲⡀⠀⠀⠀⢹⣿⠀⠰⠆⠘⢃⠀⠀⠀⠀⢉⡹⠁⠀⠀⠀⣀⠴⠋⠛⠛⠋⠉⠀⠒⢚⡶⣸⠃
⠀⠀⢹⡀⠈⠉⠁⠀⠉⢉⠔⢹⣟⣿⣿⣿⣿⣾⣧⣻⣿⠇⠀⠀⠀⠀⢋⡂⠰⠆⠀⠘⠁⠁⠀⣠⠎⠀⠀⣠⠴⣊⢃⡆⠀⠀⠀⠀⠀⠀⠀⠘⠟⠋
⠀⠀⠀⢳⠶⠶⠦⠔⠊⠁⠀⡿⠽⠾⠳⠿⠷⠛⠉⠙⠁⠀⠀⢀⠤⠠⡴⢍⣄⣀⢀⣀⠡⢴⠊⠁⠀⡠⠋⠀⡴⣻⠞⠁
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⣕⡂⠉⠀⡎⠢⢄⠀⠀⡐⠣⡄⣠⠞⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠑⠂⠤⠤⠤⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠻⣧⣠⠂⡤⢀⡀⢁⡶⠉⡦⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠈⠀⠀⠀⡨⠂⠈⠀⡎⠑⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀⠈⠹⠵⣢⡠⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢠⣾⣿⠁⠀⠀⠀⢰⣷⡄⠀⠀⠐⠦⢄⠴⠖⠀⠀⣀⣶⣷⡆⡀⠀⢀⡀⠘⠿⠀⠀⠠⠚⠁⠈⠁⡆
⠀⠀⠘⠸⠟⠀⠀⠀⡀⣈⢛⡁⡀⠀⠀⠀⢸⠀⠀⠀⠀⠨⠻⡿⠣⠀⠀⠸⠋⠠⣴⡤⠀⠀⠈⠉⢩⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⢀⣴⠙⢁⣦⠀⠀⣿⣿⡇⠀⠀⠀⠀⣠⡞⢶⡆⠀⠀⣀⣀⢀⢀⡀⠀⠀⠀⣤⡅⠂⠀⢠⣤⣤
⠀⠘⡩⡄⠘⣏⠃⠀⣻⣿⣻⣿⡇⠀⠈⣿⠀⢀⡟⠀⠰⣏⣿⣸⣮⡇⠀⠈⠈⣿⡃⠌⠀⠀⣾⣿⠟⠻⠆
⠀⠀⠈⠑⠀⠁⠀⠀⠛⠀⠛⠛⠃⠀⠀⠛⠉⠛⠃⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠉⠉⠀⠀⠃
⠀⠀⠀⡐⢂⡀⠀⠀⠀⢀⡗⢠⠀⠀⠀⠀⣠⠀⠀⠀⠀⢀⡴⢰⣀⠀⠀⠀⡤⠀⢄⡀⠀⠀⢀
⠀⠀⢀⡁⣉⠀⠀⠀⠀⣾⡉⠛⠀⠀⢀⣴⣧⣀⠡⠀⠀⢹⢐⡍⠔⠂⠀⠘⠌⣮⢚⠃⠀⢠⣾⡂⢐⣿⠂
⠀⠀⠈⠻⠟⠀⠀⠀⠰⠃⠃⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠻⠛⠛⠉
⠀⠀⡀⠀⠀⣀⠀⠀⠀⢠⡤⣄⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⡀
⠀⠘⠿⣀⣘⠿⠃⠀⢰⣿⠟⠈⠀⠀⠀⣤⣿⣿⡇⠀⠀⣔⡀⠰⣄⠀⠀⣶⠒⢲⡄⠀⠀⠀⣿⢰⡆⣤
⠀⠀⠀⠧⠟⠀⠀⠀⠸⠏⠀⠁⠃⠀⠀⠸⠦⠤⠃⠀⠀⠻⠛⠛⠿⠃⠀⠻⠗⠚⠛⠿⠂⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⣄⣤⣠
⠀⠀⢠⢖⡢⡄⠀⠀⠀⢸⡆⢹⠀⠀⠈⠃⠸⠀⠛⠀⠀⠒⣿⠷⠄⠀⠀⠀⢠⡾⠅⠀⠀⠀⠈⣿⢁⣀
⠀⠀⠣⣘⣃⠎⠀⠀⠀⢼⡧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡄⣶⠂⠀⠀⠀⠙⠀⠀⠀⠀⢰⢻⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⡴⢣⡐⢦⠀⠀⣴⣧⢀⣿⡆⠀⠀⢰⣶⡇⠀⠀⠀⡔⢉⣥⠛⡄⠀⠀⣠⣼⣶⡄⠀⢀⣾⡏⢹⣷⡄
⠀⠀⢣⡉⢃⠞⠀⠀⢣⣨⣶⣀⠇⠀⠀⠲⠌⠶⠃⠀⠀⢧⡘⢛⣠⠃⠀⠀⠟⠁⡼⠃⠀⠈⢿⣿⣺⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀
⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂
⠀⠀⠀⠒⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢤⠤
⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠠⠄⠀⠀⠀⡠⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠁⠀⠀⠀⠀⠀⠀⠁⠀⢂⡤⠆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⣴⣒⠁⠈⢃⠶⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠂⠘⠂⢠⠄⠀⠉⠁⠀⢈⠆⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠘⡤⠤⠀⠒⠀⠉⠸⣄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠀⠀⠀⢱⡄⠀⠔⡈⠠⠄⠀⣆⠀⡀⠀⠁⡐⠂⠈⠁⠐⠒⢄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠂⠄⠐⠢⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⢠⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠆⠠⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠀⠀⠂⠐⠆⠀⠀⠸⠠⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣶⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣥⠀⢀⡠⠰⠏⣈⣄⡀⠀⠀⠀⠄⡄⠀⠒⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⣠⠴⠚⠁⠀⡤⣡⠄⣠⡁⡖⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠈⢰⠂⠀⠉⠙⠋⠉⠓⢂⡪⠤⢤⣀⣀⠀⢉⠀⠂⠱⣀⡠⠖⠋⠁⣠⡀⠀⡴⢁⣤⣶⣦⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⠀⠀⠉⠀⠛⠊⠻⠋⠈⠀⠀⠀⠀⠉⠀⠀⡇⠀⠃⠠⠄⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠀⠀⠀⠉⠄⠀⠀⢸⡂⠚⠃⠀⠀⠈⠀⠠⠀⠀⠀⠀⡇⠀⢐⠆⡰
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⢛⠃⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠎⠘⠛⠳⠀⠂
⠀⠀⠀⠀⠀⠀⠊⠈⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢃
⠀⠀⠀⠀⠀⠀⠋⠀⠈⠃⠀⢠⡁⠘⠠⠆⠁⡀⠂⡠⠄⢀⠀⠀⠀⠀⠀⠈⢀⠘⠀⠀⡀⢄⠢⠁⠁⠈⠢⢶⡷⠴⠀⣤⣉⣠⡄
⠀⠀⠀⠀⠀⠄⣀⠉⡦⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠖⠀⠀⠀⠀⠀⠈⢀⠀⠁⠈⠀⢀⠠⠖⠠⠆⠀⠀⠀⣀⡤⣭⣠⠄
⠀⠀⠀⠀⠀⠄⣄⠀⠶⠀⢠⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠴⠤⠀⣀⡀⢄⡀⠤⠼⠇⠀⠀⠀⠺⠄⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢀
⠀⠀⠀⠀⣀⡄⢄⣀⡤⠀⢀⠀⡤⠀⠐⠂⠀⠀⠀⠈⠀⢀⠄⠀⠈⠁⠍⠀⠀⠀⡀⠀⠀⠀⡴⠋⠀⠠⡄⠀⠀⢀⠀⠐⣲⠆
⠀⠀⠀⠀⠁⠀⠀⠀⠀⡄⠉⠀⠶⠀⠀⡤⠒⠂⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢁⡌⠁⠀⠀⠘⠈⠉⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⢨⡄⠘⢠⡀⠂⠤⢄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠉⠒⠄⠠⢀⠐⢈⠀⠀⠀⠀⠀⠄⠰⠨
⠀⠀⠀⠀⠀⣥⠀⡀⠀⡄⡈⠀⡌⠀⠀⠀⢠⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠒⠲⡄
⠀⠀⠀⠀⠀⠀⢀⡴⢦⡀⠀⠀⠀⢪⠐⠇⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠛⣠⠞
⠀⠀⠀⠀⠀⠀⡞⠓⢤⡜⠀⠀⠀⠸⡆⠀⠈⡇⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⣀⠴⠋
⠀⠀⠀⠀⠀⢸⠃⠀⠘⠦⣄⡀⠀⠀⢳⠀⠀⢻⠀⡏⠀⠈⠉⠉⠙⠛⠲⣤⠞⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠙⡆⠀⠈⣇⠀⠸⡆⢹⠀⣰⢲⠶⠤⣤⠟⠁⣀⡴⠋
⠀⠀⠀⠀⠀⡇⠀⠀⡠⠐⠢⠄⢸⣀⡠⠞⠉⠙⢧⡏⢠⡏⢛⡶⠞⢁⡴⠮⡉⠳⡀
⠀⠀⠀⠀⠀⢹⠀⠀⡇⠀⠀⠈⠄⢫⠀⢠⠖⠉⠁⠀⠈⠳⣏⡠⠴⠿⠓⠒⠛⠉⠉⠱⡄
⠀⠀⠀⠀⠀⠈⢧⠀⢇⠀⠀⠀⢸⠈⢆⡜⡔⠉⠉⠒⠢⠀⢸⠀⣀⠤⠴⠒⠒⠒⢄⠀⣧⡤⠖⠲⢤⣀
⠀⠀⠀⠀⠀⢀⣸⡄⠈⢦⠀⠀⣸⠀⠘⣤⠁⠀⠀⠀⢠⠀⢸⠀⠇⠀⠀⠀⠀⠀⠈⡄⢏⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠰⣅⠀⡱⠀⠀⠉⠉⠁⠀⠀⠈⡀⠀⠀⠀⠸⢀⡏⡀⢀⠀⠀⠀⠀⠀⠀⠇⢸⠀⡠⠖⢆⠐⡆⢳
⠀⠀⠀⠀⠀⢹⠂⠉⡑⠒⠒⠒⠒⠒⠠⠤⠼⠤⠤⠤⠧⠼⠽⠧⠼⠤⠤⠤⠤⠤⠤⠼⠤⠯⠔⢀⡸⠀⠇⠈⣇
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠉⠉⠙⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⢸⠀⠸⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣧⠀⢸⢦⣠⠇
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠶⢻⡀⡸⠀⠁
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠖⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡰⠀⠉⠁⠐⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⡈⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣥⢹⡃⠀⠀⠀⠀⣛⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠘⠃⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⣦⣈⣤⣾⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⡗⠾⠿⠿⠟⠛⢋⣿⣿⡙⢢⡤⠤⠤⠤⢤⡞⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠰
⠀⡐⢦⣶⡶⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠘⢄⠀⠀⠀⡀⠠⢠⠎⠀⢸⣇⠀⠀⣀⣀⣤⣬⣽⣿⣡⡾⣹⠆⠀⠀⠛⠢⠀⠀⠀⠀⠀⡀⠀⡎⣌⢢⠇
⠀⠀⠘⠛⠉⠀⠀⣀⠀⠀⠀⠀⠀⠱⡀⠀⠈⢆⠉⠀⠀⣠⢷⣤⣀⢸⣿⣾⣿⣿⣿⣿⣿⣿⠿⠇⠉⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⡸⠁⡿⣏
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠱⡀⠀⡈⢆⠀⡰⠛⠳⣬⣍⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⣷⡼⠀⠈⠉⠙⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠱⡐⠂⠈⠋⠀⡙⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠹⡸⣿⣷⣧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⣌⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣦⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⠃⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠐⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡀
⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣷⡀
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣙⠿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡿⠟⡋
⠀⠘⢷⣦⣭⣭⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣴⣿⣄⣀⣀⣀⣀⣀⣈⣉⣉⣉⣭⣤⣶⡾⠁
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠹⣿⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹

⠋⠁⢀⡞⢄⠢⡑⢿⣿⣿⣶⣿⠿⡿⠀⠀⠀⠀⢠⢞⠁⠀⠀⠀⠀⠀⣾⡀⠀⠀⢰⣿⠿⡏⠀⠀⠀⠀⢠⢞⠀⠀⠀⠀⣀⠀⠀⠀⠂⠀⠀⠀⠐
⣄⣰⠋⠀⣤⡀⠑⠌⠛⠛⠆⠈⠀⠀⠀⠀⠀⠀⠀⠼⠲⠆⠀⢰⣶⡿⠿⢷⣦⡄⠆⠁⠀⠀⠀⠀⠀⠀⠀⠾⠲⠆⠀⠰⠿⠄⠀⠀⣠⡀
⣟⢻⡄⠆⠁⢩⣰⢀⠑⠘⠃⠀⠀⠀⠀⠀⠸⠖⠀⢀⠔⠄⢠⣾⣿⣷⣶⣿⣿⣟⠃⠀⢀⠀⠀⢀⣸⠖⠀⠀⠔⠄⠀⠀⠀⠀⠀⠺⠿⠅
⣿⣿⣯⠅⠔⢢⡄⡘⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠀⣼⣿⣿⣿⣽⣿⣿⡿⢊⣐⣇⠀⠫⠁⠀⠀⠀⠙⠒⢆⠀⠀⠘⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⢀
⠋⠀⠀⠰⣿⠏⣅⣐⣤⣶⣿⣿⣶⣶⣆⠀⠀⠀⠄⠀⠀⠀⠘⣿⣿⣿⣿⣿⡟⠀⡾⠛⠿⣷⣿⣷⣶⣦⣄⡀⠀⠀⠐⠃⠀⠀⡈⠀⠀⠀⠨⠃⠁⠀⡀
⣀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⡰⣦⡴⢂⡆⠀⢹⣿⣿⣿⣿⣿⣤⣁⡀⠀⠀⠀⠉⠛⠿⠛⢿⣦⡀⠀⠀⡀⠠⠄⠀⢀⡄⠀⠀⠐⣀⣀⣀⡀
⡿⠀⠀⠀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠟⠂⣸⠑⠂⠈⣿⣿⣿⣿⣿⠿⠿⠿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠉⠀⣟⠁⠀⣾⣿⣿⠐⡀⢀⣞⣿⣿⣿⢧
⠁⠰⠀⢻⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣤⣷⠒⠀⣠⣿⡏⣿⠿⢷⣤⣀⣤⠍⠛⠛⠻⢿⣿⢶⣤⣄⣀⣠⡃⠀⢂⠙⣩⣤⣦⠀⢈⡻⢿⣿⣿⢟
⠠⠄⠉⠀⠈⠛⣻⣿⣿⣿⡿⠿⡿⠛⠿⢿⣿⠿⠿⡿⠇⣺⣾⣿⣿⣿⠇⠀⡠⠀⠋⠹⣀⠀⠀⠀⠀⠁⠀⠹⣿⣿⠏⠿⠒⠈⢦⡈⢻⡏⡄⣼⣿⣮⡀⠀⠈
⠀⢨⡄⢠⣾⣿⡿⠟⣻⣿⡧⠀⠷⠂⠐⣂⣤⣄⣶⣶⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠓⠻⠷⠆⠀⠘⠛⠶⢴⡿⠃⣸⡆⡓⢄⣘⣻⣦⣽⣿⣛⣩⡄⠀⠀⠉
⣶⠀⣀⠘⠛⠋⠀⡈⠀⠉⠀⣀⣀⣀⣾⢫⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠠⠀⠀⡀⣰⡿⠗⣠⣴⣾⣿⢟⣽⢿⣿⣮⠥⠥⠄⠉
⠃⠀⠩⠀⠀⢀⠁⣀⠀⣰⡿⠫⢍⠹⠟⢸⡿⢿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⢸⣿⣿⢂⣾⣿⠿⠛⡱⢋⡿⠌⢿⣿⣧⡀⢠⣤
⣀⠠⠄⠉⠀⠙⣀⣹⣿⣿⣧⡀⢀⣷⡰⠚⠀⠀⠀⢢⠂⠙⣿⣿⣿⠏⠉⡍⠙⠻⠿⢿⣿⣿⢿⣿⣿⣷⣶⣿⣿⣿⣦⣴⡄⢀⣈⠀⣼⣿⣦⣆⠻⣿⣧⠸⣿
⠿⠆⠘⠀⡄⠰⠈⢿⣿⣿⡿⠛⣉⣀⣁⡈⠛⠀⠀⠀⠀⠀⢠⣿⣿⣶⣄⣀⡀⠀⠀⠀⠉⠁⢰⣿⣄⣼⣿⣿⣿⣿⣿⡀⠑⠀⢱⡾⢿⣿⡿⢃⣄⠤⢍⣀⣻
⠀⠀⠀⠸⣿⡄⠀⢠⠉⣿⣴⣿⣿⡧⠈⠀⢠⡤⠀⢠⡔⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⠞⠄⠢⠐⣽⣾⣿⠤⣺⢿⡏
⠀⠀⠀⠀⢻⢇⠀⠀⠁⠀⢻⣿⠀⠀⣀⣤⠘⠂⠠⠞⠉⢲⣿⣿⣿⣿⣿⣿⣷⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣰⠃⠀⣭⠀⠑⠌⠛⠲
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠉⣀⣀⠀⠉⠉⠁⠀⠀⠀⠀⡉⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠒⢤⡑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⣠⠊⠀⠀⡠⠐⠀⠀⠀⠀⠀⠢⢄⠀⠈⠉⠒⠢⢄⡈⠢⡙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡰⠁⠀⡴⠋⠀⠀⠀⠀⠀⣀⠀⠂⢄⡈⠐⠢⠄⣀⣀⠈⠢⢌⠈⠆
⠀⠀⠀⠀⣀⢀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠰⠃⠀⠊⢀⠄⢁⠔⠁⡠⠊⠀⠑⠢⣀⠈⠐⠢⠄⣀⠀⠈⠁⠀⠂⠸
⠀⠀⢠⠤⣣⠹⡄⢃⡐⠂⠀⠀⠀⠀⠀⠀⢸⠀⠀⠤⠐⢊⡡⠔⢁⡤⠊⠀⠀⠀⠀⠀⠀⠉⠒⠠⠤⢀⣉⠉⠉⠁⠀⢀
⠀⠀⢀⡵⡌⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⣊⡩⠤⠊⠁⢠⡄⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠈⡏⣙⡏⡀
⠀⠀⠀⠑⠤⣀⠀⣀⠔⠉⠙⢆⠀⠀⠀⠀⢰⠹⡩⢡⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⠴⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⠧⠊⢀⡇
⠀⠀⠀⠀⠀⠀⢩⠀⠀⠀⠀⠀⠑⢄⠀⠀⢈⣄⠈⠚⢆⠀⠀⡔⠒⡄⠀⠀⠐⠤⡤⠆⠀⠀⠀⡆⢐⠄⠀⠀⡜⢀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠑⠢⣜⠀⠐⣠⡨⠢⡀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣠⠎⠂⡅⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠑⠲⠅⣀⠀⠈⡦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠚⠁⠀⠀⠘⢎⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡄⠀⠈⠉⠒⠒⠀⠀⠴⠖⠂⠉⠀⢠⠕⢦⡀⠀⠀⠸⣤⣠⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢬⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⢀⣀⠠⢤⠤⡄⠀⠀⠠⠤⠋⠀⠀⠹⢄⠀⠀⢹⣖⣫⣽⡦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢉⠏⡿⡲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⡟⡏⢇⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠑⠠⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠣⡀⠀⠁⠈
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣇⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡇⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠈⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⣷⠀⢀⡠⠔⠊⠉⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⢽⣠⣇⣀⣀⣀⣀⣀⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⡿⠊⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠉⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠉⠀⠀⣰⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡛⠛⠦⣠⡴⠀⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢉⣙⣿⡟⠁⠀⢈⠀⠀⢻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⢸⡾⢿⢏⣿⣿⣶⣟⢀⣷⣤⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠢⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡏⠀⠀⠐⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠛⠛⠉⠋⠉⢀⡠⠤⠖⠒⢢
⠀⠀⠀⠀⠀⠀⠀⢀⠞⡝⣿⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⢀⡎⠰⠀⣿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⣾⠀⠃⠀⡏⢖⠈⢀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⣘⠸⡘⠀⠀⢰⠈⢦⡀⠀⠀⠀⠑⠀⠀⠀⠀⠀⣀⠔⠁⢀⡠⠊
⠀⠀⠀⠀⢠⡏⠀⣷⠀⠀⠘⡄⠈⢯⡢⢄⡀⠀⠀⠀⠀⠀⠈⡠⠔⠒⠁
⠀⠀⠀⠀⠚⣷⠀⢜⢢⡀⠀⠸⡄⠀⠙⠦⣈⠉⠁⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀⣀⡠⠋
⠀⠀⠀⠀⠠⢿⣄⢱⠀⠙⢤⡀⠘⢄⠀⠀⠈⠉⠒⠢⠤⠤⠤⠤⠤⠄⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠈⠲⢬⣳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠉⠓⠶⡦⠤⠤⠤⠤⠤⠤⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⠤⣀⣀⣀⣠⠴⠀⠀⠀⠀⠀⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠤⢤⡤⠤⠤⠤⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⡂⠀⠀⠀⠀⠉⠠⣤⠀⠈⠁⢀⠄⢒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠡⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⠐⠼⠁⠂⢄⠑⢄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣀⠀⠀⠀⢈⠀⠀⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠊⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠑⠂⠁⠀⠀⠀⠀⠀⣇⠂⠒⠐⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠉⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠆⣁⡀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢹⠅⠡⠄⠤⠠⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠂⢀⠇⠀⠀⠔⠀⠀⢄⠀⠀⠀⠀⠀⢠⠨⢄⠀⠀⢌⠀⡀⠀⠀⡠⠤⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⢁⡀⠀⡀⠀⠔⠀⠀⠨⠀⠀⠃⠀⢈⠀⠀⠀⢐⠂⢁⠀⠀⠀⡇⠀⠈⠀⠂⠀⠀⠀⠉⠀⠀⠧⠀⠀⠠⠌⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡡⠀⡐⠀⠀⠀⠀⠀⢰⠈⡀⠀⠀⠠⠄⠀⠀⢈⡀⠸⠀⠀⠀⠁⠀⠀⡇⠀⠀⠀⠠⠀⠀⠀⠈⠂⠀⠒⠁⠀⠀⠢⠙⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⡂⠈⢀⢐⠀⣀⣀⢀⡀⢀⡀⠔⠁⠀⢀⣉⠔⠀⠀⠃⢊⠂⢀⠐⠀⡄⠀⠰⠄⠀⠠⠀⠀⡀⡀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠂⠀⠂⠀⠀⠈⠀⢀⠐⠄⠀⠀⠀⠀⠀⠀⡐⠁⢀⠀⠀⢀⠀⠀⡉⠀⠀⡈⠉⡉⠀⠀⠀⠀⠀⢠⠁⠀⠘⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⡆⢰⠂⠈⡄⠀⢁⠤⠄⠀⡄⠀⠀⠁⠀⠀⡆⠐⠀⠀⠀⠎⢀⠀⠄⠀⠐⠀⠤⠤⠴⠤⠄⠀⠀⠃⠄
⠀⠀⠀⠀⠀⠦⠤⠶⠀⠆⠀⠀⠀⢴⠂⡀⠀⠅⠘⠂⡐⠀⠀⠈⠀⡀⠀⡇⠀⠘⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⢠
⠀⠀⠀⠀⠀⠀⠚⠒⠀⢄⠀⠀⠀⣸⡀⠂⠀⡅⠸⠁⠰⠀⠀⠀⠡⢀⠀⠡⠈⢁⠔⠀⠈⠐⠔⢀⠁⠀⡀⠘⠀⢠⠀⠀⠀⠀⠀⠄⡀⠀⡠⠠⠀⢀⠀⠄
⠀⠀⠀⠀⠈⠁⠁⠉⠢⠀⠀⠀⠀⡨⢄⠀⠀⠐⠂⠈⠈⠠⡈⠀⠈⠁⠀⠀⠀⢀⡀⠀⠀⠒⠂⠁⠰⠀⠈⠐⠀⠁⠀⠀⠀⠀⠀⠡⢈⣎⠠⠁⠀⢸
⠀⠀⠀⠀⠠⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠂⠀⡐⡀⢀⠬⡀⠀⡀⠀⡀⢠⠀⠀⡀⠠⢀⠀⠀⡆⠀⠀⡠⠐⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢢⡄⠄⠁⠀⠘⠀⠀⠀⣀⠀⠀⡆⠂⠄⠁⢢⡠⢃⢴⠀⠑⣠⠆⢀⠈⠀⠰⠀⠁⣸⠀⣀⠋⡹⡀⠁⡤⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠄⠀⠀⡌⠂⠀⠀⠈⠛⠀⠀⠦⠁⢀⠨⠃⠍⠀⠉⠊⠁⠘⠀⠱⢗⠋⠓⠀⠊⠀⠒⠀⠃⠈⠀⠉⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠠⠀⠁⠀⠀⢠⢀⠀⠚⠂⠀⣀⡀⠀⠀⡀⠀⠀⠀⠀⢠⠊⠙⠇⠀⡄⠀⠀⠀⠀⠀⠀⡐⠀⠠⢂⠄⢀⠤⠀⠀⠰⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡑⢄⠔⡁⠔⠀⠀⣀⠀⠀⠀⠀⠀⠘⠐⠊⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠔⠁⠀⠊⠀⠂⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠹⠀⠀⠀⠀⠀⠈⠆⠀⠀⠠⣀⠢⠀⠀⠀⠄⠃⠀⠀⠀⠀⠀⠠⣂⡁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠐⢄⠀⠐⠀⢈⠄⠀⠀⠀⢈⠄⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠰⣘⡀⢀⡀⠀⡄⢱⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠁⠀⠀⠀⠀⠀⣀⠐⡰⢅⠈⠀⠀⠔⢡⢳⣤⠀⡀⢄⠀⠀⡀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠁⠁⠀⡄⡄⢰⡴⠁⠦⠄⠐⠂⠀⠀⠂⠸⣆⡈⠄⡀⠀⠉⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠎⠠⠄⠀⠀⠀⠀⡀⠀⢐⠬⠥⢈⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠂⠄⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⠿⣷⣄⣩⣥⣤⣤⣤⣤⣬⣍⣠⣾⠿⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣟⢙⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡿⠟⣋⣉⣉⣙⡛⢿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠏⣴⣿⢿⣿⣿⡿⣿⣦⠹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⣿⣿⣴⣿⣿⣧⣾⣿⢀⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣷⣌⡛⠿⠿⠿⠿⢛⣡⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠁⠉⠉⠉⠉⠀⢿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢰⡆⠀⢀⡀
⠀⠀⠀⠀⠀⠰⠶⠶⣶⡾⠿⠿⣿⡶⣶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠴⠂⠐⠤⡉
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⡀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢇⣄⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡸⣿⣷⣶⣿⢡⠂
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡁⠸⡟⠃⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⠉⠉⢱⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⠠⠜
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠾⠀⠄⠾⠸⠲

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣳⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠙⠟⡇⣠⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠐⡂⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡷⣄⣿⠻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⣤⣀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣴⣌⡃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣀⠀⠀⠀⢀⣀⣾⠃
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⣶⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⡿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⠀⢸⡿⠛⠁
⠀⠀⠀⠀⠀⠀⡀⣠⢠⠄⡀⢀⠄⣠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⡉⢀⠈⠁⣀
⠀⠀⠀⠀⠀⠀⠇⠧⠸⠜⠧⠸⠄⠿⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⠄⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠓⠀⠛⠛⠀⠒⠛⠚⠂⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠶⡄⣴⠄⣠⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠒⠖⣡⣾⢟⠕⢮⡻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⠶⠦⠴⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠾⢿⠗⢡⢀⢠⠘⢮⠻⠷⠀⠀⠀⠀⠀⢀⡴⠚⢻⣿⡑⠤⡀⢀⢃⣽⡞⠛⢶⡄⠀⠀⠀⠀⠀⣿⣿⣿⣏⠈⠙⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⡠⠄⠚⠀⠀⡀⠀⠀⠀⠀⢸⡅⠀⠋⢀⡗⠀⠈⠛⢺⡗⠛⠂⠀⣿⠀⠀⠀⠀⠀⣿⣿⠏⢹⣿⡶⠛⢿⣿⡇
⠀⠀⠀⠀⠀⠁⠶⢀⠀⣀⢈⠀⢀⣤⢄⠲⠈⠀⠀⠀⠀⠈⠻⠶⠶⠟⠁⠀⠀⠀⠀⠛⠶⠶⠞⠁⠀⠀⠀⠀⠀⣿⣿⣦⣤⣿⣤⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⡿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡏⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⢠⠞⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣠⣦⡈⠻⠁⠀⠀⠀⢀⡴⠁⢀⠞⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⣠⠤⠒⠒⠠⠤⢄⡀⠀⠀⡇⢸⠀⠈⠟⠁⠀⠀⠀⠀⡠⠊⠀⡴⠃⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡇
⠀⠀⠀⠀⢀⠞⣡⢢⠖⠀⠀⢠⣀⠙⢆⠀⣷⣾⠀⡤⠤⠄⠐⠒⠀⠾⡅⣠⠊⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⢠⠀⡇
⠀⠀⠀⠀⡜⠠⢣⠃⠀⠀⢀⠎⢀⡩⢾⠀⣿⣿⠀⢷⡏⠀⢀⢀⠈⠀⠟⠁⠀⠀⠀⠀⣴⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢀⡔⠁⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⡖⠒⠁⠀⠀⠐⠸⡄⡟⢻⠀⢻⣇⣄⡿⣿⣷⡄⢰⠀⠀⠀⠀⠀⠘⡇⢸⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⡠⠋⠀⠀⠀⡇
⠀⠀⠀⠀⠸⡀⠀⠀⢱⠀⠀⠀⠀⣀⣰⠁⡇⢸⠀⣹⡇⢎⠀⢀⠝⠀⢸⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⢀⠔⠁⢠⠞⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢀⠎⢹⠒⠢⡇⠀⢀⣀⣀⡠⠃⠀⡇⢸⠀⣟⣣⠳⢴⠁⠀⠀⡄⠀⡀⠠⣿⠆⠀⡇⢸⠀⠀⠀⠀⡰⠋⢀⡔⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⢸⠀⢀⡇⣀⡧⠤⢄⣀⠀⠀⡇⢸⠀⣸⠃⠀⡎⠉⠉⠉⠀⠲⣿⠆⠈⠀⠀⡇⢸⠀⠀⣠⠎⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡸⢀⣞⢞⢱⠈⢱⢂⠇⠀⠈⠉⠒⠳⠼⡤⠃⠀⡜⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡇⢸⠀⠐⠁⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠛⠒⠉⡜⡘⡜⠀⡇⡜⠀⠀⠤⣀⡀⠀⠀⠀⠀⡰⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠧⠤⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠤⠄
⠀⠀⠀⠀⢀⠃⠃⡇⢰⠀⡇⠀⠀⣠⢿⠈⠉⠒⠂⠔⠁
⠀⠀⠀⠀⢸⠀⠸⡇⢸⠸⢀⣀⠔⠁⠈⡆
⠀⠀⠀⠀⠸⡀⢸⢉⠊⠉⠁⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢣⢸⢸⡇⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢿⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⢸⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠈⠈⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⢿⣿⣿⡷⠖⠂⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣀⣀⣀⣀⠤⠚⠁⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠀⠀⢠⣿⡆⢀⣿⣿⠀⣰⣦⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⣰⣆⠀⠀⠀⢀⣀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠈⠙⢿⡟⠋⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿⣸⣿⡏⠀⣤⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⣿⣿⡇⣿⣿⣧⣿⣿⠃⣸⣿⠇⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⣿⣿⣷⣿⣿⣿⣿⡟⢰⣿⡟⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣿⣿⣷⠀⢀⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⣿⣿⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢀⣀⡤⠤⠶⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⢀⣠⠼⢿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠚⠉⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉⠁


⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠰⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⡄⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠰⠀⠀⠀⠒⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒⢒⠀⠉⠁⠈⠸⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡿⠁⠀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠁⠠⠀⠀⠠⠀⢄⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⣾⠇⠀⠀⣷⡆⠀⠆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠸⡇⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⢄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⢀⡐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⠰⠀⠀⠀⠀⢀⠶⠂⠀⠲⠄⢠⣄⠀⢀⡀⠴⠞⠌⡁⢋⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢀⣵⢀⡃⠀⠀⢨⣧⣷⠉⠠⣀⢀⡀⢞⣀⠑⢄
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠂⠀⠀⠣⢠⣿⠃⠐⠧⠠⠄⣾⡿⠉⠢⠄⠀⠂⠁⠉⠁⠀⠀⠇
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣄⣁⣤⣴⣷⠀⢸⠄⠀⠀⠀⢰⣿⠁⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠈⣽⠁⠀⣾⠀⢻⠀⠀⠀⠀⠘⡅⠐⣤⣀⠀⠤⠀⠀⠀⣀⣤⡔⢫⣄⠀⢀
⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢸⠁⠘⠃⠀⢠⠏⢠⡞⣧⠀⠀⠀⠀⠳⠾⠖⠀⠛⢒⣁⣶⠛⠉⠹⣁⣼⡋⠀⠀⢁
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠒⠖⠺⣄⠲⠓⠲⠊⠀⢾⡽⠾⠃⠱⠤⠤⠚⣥⠀⠀⠀⠀⠉⠀⢸⡀⡱⠛⣟⠛⢻⠆⠘
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠅⢘⣤⣄⡰⠋⠀⠠⠀⠀⣀⣿⠻⡆⠃⠁⠀⠀⠀⠋⠠⠀⠀⠀⠀⠀⢸⠛⠁⠀⠈⢷⣤⣤⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣐⣁⣀⣸⠁⠠⢀⠄⢀⠶⠄⠀⠳⣉⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⢀⡠⠤⠤⠘⡅⠀⠠⢀⠀⠀⠈⢉
⠀⠀⡇⠀⠀⠀⠀⢀⣾⢿⡿⢏⡄⣠⡽⣤⠠⠁⠀⠰⠄⠀⠀⠀⠙⠦⣀⠀⠀⡠⠀⠀⢀⡄⠀⠀⠀⢰⠛⠁⠀⠀⠀⠈⠢⠤⢈⡀⠀⠀⠃
⠀⠀⣷⠀⠀⠀⠰⣿⠈⡇⠈⠸⠯⣶⠀⠀⠐⠲⠁⠀⠲⠰⠀⠀⠀⠀⠀⠀⠉⠁⠀⠐⠊⠳⠄⠠⡴⠃⠀⠀⠀⠀⠀⠀⠰⠒⠉⠀⠰⠂
⠀⠀⢹⡀⣀⠀⠀⠙⣶⣷⣆⣡⡖⠛⢷⠖⠻⡷⠁⠠⡌⠀⠀⢈⡳⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶
⠀⠀⠈⠃⠈⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠃⠀⠀⠰⠄⠀⠀⠀⠁⢦⣀⣀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠰⠒⠃⠠⠂⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡀⠀⣀⠤⣠⠀⠆⠀⠀⠀⠀⠀⠀⢰⢆⡀⠀⠀⠀⢠⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢣⠀⡠⠂⠀⠀⠀⢦⣦⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠪⠤⠀⠒⠃⡅⡈⢳⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠴⠦⣴⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠁⡸⠀⠠⡖⠈⠛⠃⠧⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣈⡀⠈⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⠀⠀⠞⡀⠀⣤⠂⠉⠁⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠋⠀⠚⠂⠀⠤⠆⠑
⠀⠀⠀⠀⠀⠀⠀⠠⠉⢅⠀⠠⠤⡄⣄⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠐⢓⠀⠐⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠗⠀⠀⠀⠀⠀⠰⢀⠆
⠀⠀⠀⢀⠀⠀⠀⠐⠃⠺⣔⠨⠄⡙⠀⠨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡴⠁⠀⠀⠀⠁⠀⠀⠚
⠀⠀⠀⢸⠀⠀⠀⠀⠸⢀⡘⠐⠈⢸⢈⠃⠜⠂⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⣀
⠀⠀⠀⠈⠀⠀⠀⠘⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠃⠰⠀⠒⠀⠄⠀⠉⠀⠀⠀⠀⠀⠀⢤⠀⠈⠤⠇⠸⢦⡀
⠀⠀⠀⠀⠠⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢶⠆⠀⣠⣤⡙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⡶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣿⣧⣤⣤⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣤⣤⣼⣿⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠠⡄⠸⣧
⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠲⠀⢻⡆
⠀⠀⠀⠀⠀⠀⢰⡟⠀⢘⠀⠀⠀⠀⠹⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⡤⠤⢤⣤⣤⡤⣤⠇⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⣿⠁⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⣴⣿⣤⣤⣤⣤⣤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⢷⡄
⠀⠀⠀⠀⠀⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣄⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⢾⡇
⠀⠀⠀⠀⠀⢻⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣼⠇
⠀⠀⠀⠀⠀⠈⠉⠉⠙⣿⠉⠉⠙⣿⠉⠉⣿⠏⠉⠉⠉⠉⠉⠉⠉⢹⡏⠀⢈⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣼⠇⢀⡀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢿⣤⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣼⠏⠀⠀⠀⠀⠀⠀⠀⠹⠷⠶⠟⠀⠸⠃⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⢻⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⣀⣀⣀⣀⣻⡆⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠛⠷⠶⠶⠶⠶⠶⠆⠀⠀⠀⠘⢿⡏⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⣤⡄
⠀⠈⠀⠀⠀⠀⠀⠀⢸⡏⢠⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⣿
⠀⠐⠀⠀⠀⠀⠀⠀⢸⡇⢐⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡅⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⠿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⣼⡷⡂⢸⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣉⣀⣿⣯⣇⣸⡟
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⠛⠉
⠀⢠

⠀⠈

⠀⠀⠀⠀⠀⠲⠖⠶⠶⠶⠖⠶⠆⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠸⣵⠄
⠀⢀⠀⠀⠀⠀⠐⣿⢟⣽⣶⣄⢀⢀⣐⡀⠀⠀⠀⠓⠺⠿⠒⠀⠀⢀⣢⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠠⢀⠂⠐⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⡀⠤⢀⠀⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠐⠀⠆⡠⡆⠐⠀⡅⠂⠐⠀⡁
⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠀⠀⠀⠀⠀⠀⡀⠀⠢⠇⠀⠀⠧⠄⠀⠀⠒⠔
⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⢀⢀⣀⠀⠒⠒⢂⡀⡇⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⢨⠅⠀⠀⠀⠀⠀⠀⠘⠉⠉⠀⠀⠀⠀⠀⠃⠀⢸⣉⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⠐⠐⠒⠃⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠁⠠⠀⠐⠀⠁
⠀⠀⡤⢲⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠈⡬⠨⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⠀⠀⠀⠄
⢀⠀⢀⣄⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡅⢰⠀⠀⠀⡇⠀⣉⠀⢠
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡈⠀⠀⠂⠁⠀⠉⡀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⢐⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡔⠒⠀⠀⣲⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⢀⠀⠀⠀⢀⠚⠢⣦⠴⠞⠀⠁⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠇⠐
⠀⠀⠀⠀⠀⠀⠀⢹⠖⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠒⠒⠐⠀⠀⠀⠀⡀⠈
⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢀⡀⠠⡈⠃⢤⡁⠫⠲⣬⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣬⣳⣤⡀⠑⠠⠈⠓⠊⠁
⠈⠙⠷⣿⡦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡆⣿⣷⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⡋⠋⠛⠛⠛⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⣿⣿⣿⣿⡇⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣤⣤⣤⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⣿⣿⣿⡿⠇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣤⣤⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⡉
⠀⠀⠀⠀⠀⢠⣾⣷⣆⠸⡆⠀⡿⠁⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⡏⣿⣿⡏⠀⣷⠀⡇⣀⣤⣤⣶⠀⠰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠒⣶⣿⣶⣧⡘⣧⡀
⠀⠀⠀⠀⠀⠸⣻⣿⡟⢀⡇⠀⣿⢻⣿⣟⣻⣄⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡘⣷⡄⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⡶
⠀⠀⠀⠀⠀⠀⠈⠁⣠⠞⠀⠀⣿⣸⣗⣷⣹⣽⣿⣿⠁⠀⠀⠀⢰⡆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡘⣿⣆⠀⠀⠀⠀⠀⠀⠀⣇⢠⡇
⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡘⣿⡄⠀⠀⠀⠀⠀⠀⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠙⠛⠛⠉⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⡘⣷⠀⠀⠀⠀⠀⠀⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⡘⡇⠀⠀⠀⠀⠀⠙⠙
⠀⠀⠀⠀⠓⠒⠶⠶⠶⢤⣤⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⡈
⠀⠀⠀⠀⠤⠤⠶⠶⠶⠟⠛⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡏⣾⠂⠀⠈⡇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⡏⠀⠀⠀⢃⠀⠀⠁
⠀⠀⢀⣀⣤⣤⣤⣤⣴⣾⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣴⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠚⢃⠀⠀⠀⠀⠀⠁⠀⠀⡀
⠀⠀⠸⠟⠿⠟⣿⣿⠿⠿⠿⠟⠻⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠀⠀⠀⠀⣀⣾⠁⠀⠀⠀⠈⠛⠿⢿⠿⠿⣿⠟⠛⠀⠀⠀⠀⠈⠀⢄⠀⠀⠀⠀⠀⠠⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠒⢁⠀⠛⠋⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢀⡀⠀⠀⠀⠀⣿⣟⠀⠐
⠀⠀⠀⠀⠠⣤⣤⣼⣶⣿⣿⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⡄⠀⠀⠀⠀⠀⣒⣶⠶⠶⢶⣖⠀⢠⠀⠀⢸⡏⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠴⠶⠾⣿⣿⣿⣿⣶⣤⣤⣶⣴⣧⣤⠾⠯⠍⠀⠀⠀⠀⠀⠀⠀⡪⠭⠭⠭⠭⠵⠀⠈⢧⡀⠀⠻⠶⠂⠀⠀⠹⣿⣄⣠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠿⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠭⣽⣯⡭⠭⠳⠶⠞⠓⣄⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠋⠉⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣀⣠⣼⠽⠿⠦⠼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠛⡍⠀⢠⠭⠁⠀⢐⢢⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠬⠶⡄⠀⠈⠚⠁⣀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣞⠢⣄⣀⣀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠛⠉⠉⠈⠊⠓⢾⣆⡀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣖⠽⠃⠀⠀⠀⠀⠀⢀⡀⠁⠀⠙⣆⠀⠀⠀⠀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⡞⠁⠀⡤⣶⠻⠀⠀⡔⠁⠈⡆⢠⠀⠸⡄⠀⠀⠀⠸⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠳⡀⠀⠳⡇⢀⣀⣼⠔⠂⠀⡇⡾⡁⠀⠽⠦⠄⣀⣠⣤⡦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠹⣄⣀⠷⢚⡄⠸⢄⣀⡴⠁⡿⣜⡂⠤⢀⡀⢡⠊⣄⣛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢴⡋⣁⣤⢴⠟⠲⣄⣀⣀⣀⢧⠀⠉⠙⠲⢆⡽⠤⢿⣤⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡝⠋⠉⢸⠀⢰⡏⢁⡄⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠇⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠁⠘⠁⠀⠀⠈⠳⡈⢣⣾⠆⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⢰⢹
⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣹⣶⣄⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠏⡄
⠀⠀⠀⠀⠀⣠⠞⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⢳⠈⡓⠀⠀⠀⣸⠀⠀⠀⠀⠀⢓⢡
⠀⠀⠀⢀⡞⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣇⠀⢀⡌⢠⠁⠀⠀⠀⡏⠀⠀⠀⠀⠀⠘⣸⠀⢀⠀⣄
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠀⣾⢀⡏⣀⣀⣀⣐⣳⠐⢯⠍⣰⠖⠀⢱⠀⠈⢟⣉⡓⢲⠇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣛⣳⣜⠯⣝⣓⠠⠤⠬⣓⡦⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠧⠴⠋⠀⠀⢈⡉⣉⠍⢥⡀⠀⠀⠀⡤⠶⠋⠛⠒⢲⡄
⠀⠀⠀⠀⠈⠙⠲⢤⣤⣤⣀⣀⠀⠀⠀⠀⢀⣀⡴⠋⠀⠀⠀⠀⠀⡅⠙⠄⠀⣀⠅⠀⠀⠘⠓⠒⠢⣤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠐⠒⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⠚⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢴⠀⠀⠀⠀⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⢠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⢀⡄⠀⠀⠀⠀⠔⠀⠀⠀⠀⡀⠀⣘⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣗⠀⠤⠙⠛⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⡀⠀⠘⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣧⣬⡓⠀⠀⠀⠀⠀⠀⠤⠂⠂⠀⠀⠀⠄⠀⠀⠀⠒⠀⠀⠒⠂⠤⠤
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠸⡕⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⠀⠀⠧⢄⣠⡼⠋⠈⠀⠂⠀⠂⠀⠐⠅⠀⢠⡁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠄⠈
⠀⠀⠀⠀⠄⢨⠀⠀⠈⠉⠀⠀⠀⠘⠒⠀⠄⠀⠀⡰⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠠⡀⠀⠀⠁⠂⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠻⠿⠛⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⠟⠛⠋⠉⠉⠉⠉⠛⠛⠿⣿⣿⣿⣷⣶⣦⣼⣿⣿⣿⣿⡿⠟⢻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⢀⣴⣿⣿⠿⠋⠑⠂⠐⠂⠢⡠⠀⢤⠀⠀⠀⠀⡤⠝⠻⢿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⡟⢉⠢⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠈⠁⠲⠀⠻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⢻⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣀⡘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿
⣿⣿⣿⣿⡛⠉⠁⠀⠀⠀⠀⠀⠈⣿⣿⡟⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠠⠌⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⣿⣿⣿⣿⣿⣶⠄⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠘⢄⠀⠀⢠⠀⠂⠐⢊⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⢄⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣤⣤⣀⣻⣇⠀⠀⠀⠀⠈⠑⠒⠚⢼⡅⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢢⠀⢠⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠉⠂⠤⠀⠀⠀⠀⠀⡀⢀⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢿
⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠈⠢⡄⠀⠀⠀⢀⠎⠀⠀⠀⠀⢰⣡⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢀⡠⠼
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠈⡄⢀⠔⠁⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⠤⠒⠉
⣟⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣿⣿⣶⣤⣄⣀⣀⣣⣃⣀⣠⣤⣴⣾⣿⣿⣿⠟⠁⠸⣿⣿⡿⠟⠛⠉
⣿⣷⣦⣄⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⠁
⣿⣿⣿⣿⣿⣶⣄⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠁⣠⠬⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣙⣺⣤⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠷⢴⣏⣁⣀⣀⣠⣬⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠃⠀⠀⠈⠉⠉⠉⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡷⠀⠀⠰⣧⣀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣇⠀⠀⣤⣀⣉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠉⣇⡇⠹⣄⡀⠀⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⠁⠀⢀⣤⡆⢠⢻⡇⠀⠀⡽⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣴⡋⣽⢃⡏⠀⠉⠛⠚⢡⡟⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⣷⡿⠛⠀⠀⠀⠀⠀⢹⡷⡀⣙⡳⠶⠶⢶⣶⣶⣒⣒⣒⣒⣲⠶⠶⠶⠶⠶⠶⠤⠤⠤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⢻⡇⠀⠀⠀⠀⣀⣠⡼⠟⠛⠉⣉⡩⠿⣟⠒⠒⠒⠒⠮⠭⠭⠭⠭⢭⣿⣿⣓⣲⡶⣾⠋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡖⠛⠉⢙⡶⣺⣿⡿⠷⠶⠤⠴⣄⣀⣠⠴⣾⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣤⠖⠋⢹⡇⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠸⠾⣅⢸⡇⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⢻⡟⠀⠀⢸⡇⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⣦⡴⠞⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⣿⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢹⣿⣀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢸⡇⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠾⣻⠟⣿⠙⢻⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⡟⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣴⣟⣡⣶⠋⠁⠀⣿⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢸⡇⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣐⣢⠼⣿⠳⣬⡙⣦⢠⡟⠀⢸⡇⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠃⣇⣴⡿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⠋⣿⠉⠁⠀⠀⣿⣦⠞⣹⠟⣸⠃⠀⣼⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣸⣶⠟⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⢠⡟⣡⠞⠁⣰⠇⢀⣴⠋⠀⣸⡷⠶⢤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⣠⣴⠟⠋⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⢀⣿⠖⠛⢂⣚⣛⡛⠛⢧⡀⣾⡟⢦⣤⡤⣀⣀⠈⠉⠉⠉⠛⠛⠓⠶⠶⠦⣤⣼⣿⡾⠛⠁⢀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠸⡆⣹⣄⣀⣸⣟⣓⠲⠶⠶⠾⢶⣚⣩⣇⠹⠿⢿⣥⣀⣀⣀⠈⠉⠁⠒⠒⠤⠤⣤⣀⣀⣶⣿⠉⠀⣠⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠈⠉⢹⣷⢭⣍⣉⣉⣉⠭⠥⣶⡏⠀⠀⠀⠀⠀⠉⠉⠉⠙⠒⠒⠦⠤⢤⣀⣀⣈⣿⣿⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠒⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠄⠀⠀⡠⠀⠠⡔⠀⠀⣾⠃⠁⡀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⢐⣷⣦⠀⠏⠀⠀⡧⠃⠀⠃
⠀⠀⠀⠀⠀⠞⠁⠀⠀⠒⣺⣿⡿⠀⠀⠀⠸⠁⠞⢀⡅
⠀⠀⠀⠀⡔⠀⠀⠀⣤⣾⠋⣡⣦⣤⡶⠴⠛⠀⣐⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡄
⠀⠀⠀⡀⠀⠀⠀⠼⠛⠻⣿⣿⣷⠶⠤⡤⡦⠀⡤⣀⠈⠂⢠⣤⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠘⠇⠘⡠⠶
⠀⠀⠀⢇⠀⠀⠀⠀⠀⠰⢡⣿⠻⠓⠀⠀⠀⠀⠀⣚⠗⣼⠆⠈⣠⣴⡶⠶⠶⠛⠛⢩⣍⣤⣴⡶⢸⡟⢻⡇
⠀⠀⠀⠀⠉⠲⡖⠚⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡉⠈⣁⡄⢨⡆⢰⡶⠟⠛⣿⠀⣿⡇⣿⣄⡄⢸⣇⣼⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⣸⢿⣋⣀⣀⣀⣀⡠⠔⠚⠃⠀⣿⣷⣸⡇⢸⣧⣤⠀⢿⣧⣿⠀⣿⣋⣁⢸⡏⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⠋⠉⠉⠉⠀⠀⠀⢀⡄⠀⣿⠸⣿⡇⢸⣧⣤⡄⠸⣿⣇⠀⢻⡟⢩⣌⣷⡟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⠤⠤⠀⠐⠒⠂⠉⢽⡇⠀⢿⣀⡙⣣⣼⣏⢁⣶⢺⡟⠋⠀⢸⡇⢸⣿⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡟⢉⣋⣿⠙⣿⣸⡇⢸⡿⠇⠀⢸⣇⣸⡟⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⣀⣀⣀⣀⣀⣴⠶⣟⢸⣧⢈⣿⣿⡆⢻⣿⠁⠸⠟⠛⠀⣀⠩⠭⠔⠒⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠛⢛⣋⣉⣥⠤⠶⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠒⠒⠒⠒⠒⠋⠉⠘⡟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠂⠀⠀⠀⠈⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠⠤⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡠⠤⠤⢤⡼⣅⡀⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠉⠉⠉⠻⣦⡀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠹⡱⡀⠀⠀⢠⣤⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡀⠤⠄⠀⠀⠀⡴⠁⠱⡀⠀⡾⠈⠃⠀⠀⠀⢱⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⡠⠀⣀⠀⢞⠀⠀⠀⠱⣀⠇⠀⠈⠀⠀⠀⠀⢉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠢⢶⣷⣠⡄⠁⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠠⠤⠬⢿⠀⣀⠈⡀⠉⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⢉⡖⠀⠀⠀⠀⠄⠈⡆⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠣⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢠⠦⣵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠐⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠠⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠈
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠐⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠉⣰⠔⠀⠀⠀⠀⠑⡄⠀⠀⠀⠲⠏⠀⡀
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⢎⣠⡤⠤⢠⣤⡄⢀⠐⠀⢀⠀⠀⢀⡜⠹⣇⣉⣀⣁⣀⣀⣈⣂⣤⠴⠦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢣⠀⢀⢮⠊⠀⠀⠀⠀⠀⠀⢸⠀⠀⠆⠠⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣜⠈⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⡀⠀⠀⡀⠀⠄⠀⢠⠀⠀⠀⠈⢳⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣧⡤⠀⠀⠀⠀⠀⠀⣢⠄⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠑⠀⠉⠲⠀⠐⠚⠀⠀⠈⠀⣄⠀⠀⠀⠙⢄⠀⣿⣀⣀⡀⣤⠤⠖⣋⣵⢊⢿⣶⠀⠀⢀⠄⡑⢉⣄⣰⣈⠱
⣘⠀⠐⣈⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⡤⠀⠈⠢⢀⠀⠀⠀⠳⡇⠀⠀⠀⢨⠄⢚⣿⣿⡷⣔⣊⢉⠒⠀⠚⡋⢟⣿⣿⡥⠄
⠙⠐⠂⠪⠛⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠏⠁⠀⠀⠈⠉⠂⠂⠈⠀⠀⠀⠀⠀⠓⠥⣨⡀⠤⠋⠀⠈⠄⠀⠀⠙⢬⣌⣁⣩⠞
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠚⠋⢵⡉⢥⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠂⠀⠀⠘⠁⠂⠀⠸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠆⠀⠀⠀⠀⠀⢀⡍⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠂⠀⠀⠠⢂⡀⠀⢀⠈⠐⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣂⠀⠀⠀⠀⠀⠩⠁⠰⣦⡄⠀⡀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠚⠿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠑⠂⠃⣤⠁⠓⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⠙⣧⢀⢀⡇⡐⠂⠀⢀⣍⣸⠆⠀⡤⠴⡊⠑⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⡈⠛⠀⠘⡭⢇⢈⠀⡇⢰⠆⢉⣴⡟⣀⣠⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠆⠀⠀⡄⢀⠤⠈⣅⠄⠠⢶⣊⡍⠜⠡⡏⢠⣖⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒⠀⠀⠈⣐⠀⠁⠀⠀⠀⠀⠉⠙⠒⠤⢌⣻⣛⠀⠀⠀⠤⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃⠀⢀⣿⡂⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢶⡈⠄⡀⠆⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠄⠒⠋⠙⡆⠉⠉⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠄⠀⠃⠁⣀⠀⠠⠅⠀⠄
⠀⠀⠀⠀⠀⠀⣶⣛⠍⠉⠀⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠐⢠⠌⠑⠌⠡⠆
⠀⠀⠀⠀⠀⡤⠥⠍⡳⠢⠄⠱⠶⣀⡀⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣠⡀⡀⢀⡀⢈⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⣀⡃⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⣀⣑⡐⠿⣄⡁⢁⠚⠃
⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⢀⠊⠹⡀⢩⡟⠃⠀⣀⣀⣀⠀⠀⡀⠀⠀⠀⠀⠀⣠⡴⠋⣵⢀⡉⠛⢀⠉⠉⢂⠀⠈⠁⠀⠀⢀
⠀⢠⣁⡀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠙⡓⠻⠺⠄⠈⢀⠀⠀⠀⠀⠀⠀⢀⢤⣚⢽⡓⡇⡌⠀⠀⣀⡄⠀⠀⠀⢀⡀⠳⠀⠶⠈⡐
⠀⠁⠀⠔⠀⠀⠁⠐⠀⠀⠀⠀⠀⣸⢭⠔⠂⠘⠀⠉⠀⠁⠀⠀⠀⣴⠉⠀⠁⠀⢈⣧⠀⢀⠀⠤⢤⡺⡅⠀⡈⣠⠤⠁⠀⡈
⢠⠀⠠⡀⠀⠠⢄⠐⢃⠀⠠⠤⣿⡜⢂⢀⠀⠈⠁⠰⢀⡀⢀⡀⠀⠈⠲⣄⣀⠀⣴⠊⠂⢀⠀⠀⠀⡀⠠⠀⡀⣭⠄
⠀⠀⠀⠁⠰⠐⠦⢀⡒⠓⠀⠲⠀⠀⢉⠈⠀⠊⠀⠔⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⡀⠉⠀⡆⠁⠥⠀⠉⠀⠀⢣⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠰⠆⠀⠉⠂⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡄⠀⠀⠀⠀⠀⠐⠁⠈⠋⠄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠖⠛⠛⠛⠛⠛⠓⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠳⣮⡟⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⣿⢉⣤⣤⡀⠀⠀⠀⠀⠀⠀⢘⣻⡆⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠙⠋⣴⢮⣷⠀⠀⠀⠀⠀⢠⣾⣏⣤⣀⣀⣀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⣉⠉⢹⡇⠀⠀⠹⡆
⠀⢀⡶⠛⢷⠀⠀⢸⣇⢰⡞⠛⠉⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠉⠉⠻⣎⠛⣷⣠⣄⣿⠀⠀⢰⠟⠳⣆
⠀⣼⠁⠀⢸⡇⠀⣾⠻⣮⣻⣦⣀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢿⣤⡈⠀⠈⢹⡆⠀⣿⠀⠀⢹⡆
⠀⡟⠀⠀⢸⡇⠀⢹⡄⠀⠈⢻⡏⠛⠛⢦⡄⠀⠀⠀⠈⠛⠶⠟⣦⠀⠀⠀⣠⡞⠁⠀⠀⣸⠁⠀⣿⠀⠀⠘⡇
⢸⡇⠀⢀⡸⠇⠀⠘⣇⠀⠀⠸⣇⠀⠀⠀⠙⠛⣻⠄⠀⠀⠀⠀⢸⡇⠀⢀⡿⠀⠀⠀⢀⡟⠀⠀⠟⣀⠀⠀⣿
⢸⡇⢠⡟⠙⢷⡀⠀⢹⣆⠀⠀⠙⠳⣆⠀⠀⣀⡿⠀⠀⠀⠀⠀⢸⠀⢠⡟⠁⠀⠀⢀⣾⠁⠀⣰⠟⠙⣧⠀⣿
⣼⠀⠈⣧⠀⠀⠻⣆⠀⠹⣦⠀⠀⠀⠹⣦⣸⠋⠀⠀⠀⠀⠀⠀⠿⠶⠋⠀⠀⠀⢠⡾⠁⢀⡾⠋⠀⢠⡏⠀⢸
⣿⠀⠀⠘⣦⠀⠀⠈⠳⣄⡈⠳⣄⡀⠀⠘⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⣀⡴⠋⠀⠀⢠⡟⠀⠀⢸
⢿⡀⠀⠀⠈⢷⣄⠀⠀⠈⠻⣦⡈⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⣠⡾⠋⠀⠀⢀⣴⠏⠀⠀⠀⣸
⠈⠻⣦⡀⠀⠀⠙⣷⠀⠀⠀⠈⠻⣆⠀⠀⠉⠉⠛⠛⠛⠛⠛⠋⠉⠁⠀⢀⡾⠋⠀⠀⠀⢰⡟⠁⠀⠀⣠⡾⠋
⠀⠀⠈⠻⣆⠀⠀⠘⠁⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠙⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠻⠀⠀⣀⣀⣀⣀⣠⣬⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣤⣀⣀⣀⣀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⢸⡛⠛⠋⠉⠉⠉⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠉⠉⠉⠛⠛⢻
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣀⣀⣀⣀⣤⣬⡇⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣄⣀⣀⣀⣀⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠟⣋⠽⢛⣻⣷⣦⣤⣤⣤⡶⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⢛⡥⠖⣋⣥⣾⠛⣿⢻⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣸⠗⢊⡡⠖⠋⠁⣸⠁⠀⢻⡈⢻⣤⣀⣠⣤⣤⣤⣤⣤⣄⠀⠀⢹⡆⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⢯⠴⠚⠻⣶⣶⣶⠶⢿⣆⠀⠀⢷⡀⠈⠉⠁⠀⠀⠀⠀⠀⣼⠃⠀⢸⡇⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠆⠀⣿⣶⣄⠀⠙⢲⡦⠀⠀⠀⠀⠀⢸⡟⠀⠀⣾⡇⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠀⠀⢿⡻⣿⡳⠦⣼⠁⠀⠀⠀⢀⣀⣼⠃⠀⢀⣿⠁⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠈⠻⣷⠛⠛⠛⢛⣿⡿⠋⣿⢀⣶⠞⠃⠀⠀⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⡿⠀⠀⣰⠟⠋⣠⡾⠃⣼⠃⠀⣀⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⣸⠁⢀⡼⠃⢠⡾⠋⠀⣼⠏⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⡇⠀⢸⠃⠀⣾⠀⠀⣾⠃⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢶⣤⣀⣼⠃⠀⢸⣇⠀⣿⠀⢠⡇⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞⠁⠀⠀⠀⠀⠈⣿⡄⠘⢷⣼⠃⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⣸⠃⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠈⣿⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣠⣾⡇⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⢠⡟⣱⠶⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⠀⢷⣀⣀⣀⣀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⣾⣷⠋⠀⢷⡟⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠁⠀⠈⢻⠻⣿⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠉⣁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣾⣋⣀⣤⣤⡶⠾⠛⠛⣀⣿⠀⠈⣷
⠀⠀⠀⢀⣤⡴⠾⠛⠋⠉⠉⠻⣦⡀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⢀⣠⣴⠞⠋⠁⠀⠀⢹⡇
⠀⠀⠀⠀⠻⣦⣀⠀⣀⣠⣤⣶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣉⣀⣀⣠⣤⡤⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⢸⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠀⠀⠀⠈⠈⠀⡀⠀⠈⠛
⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⡇⠀⠇⠀⠀⡿
⠀⠀⠀⠀⠸⠀⣀⣴⣿⣶⣶⣛⠛⠳⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀
⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡴⠏⠁⠂⠀⠀⢀⣀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⡝⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⠇
⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠶⠂⠀⠀⠀⢠⣤⣄⡲⡦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣉⡉⠉⠛⠛⠦⠉⠁⠀⠀⢀⠁⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠐⠀⠸⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣷⣶⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣷⠘⠁⣄⠀⣤⠀⠃
⣿⣿⣿⣿⣿⣿⣿⠟⢿⣶⣄⡰⠎⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢈⠀⣾⠃
⣿⣿⣿⣿⣿⣿⣿⣇⠀⠙⠛⠛⠂⠀⠀⠀⠠⠎⠁⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠘⣿⣿⣿⣿⣿⣿⣿⡆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡇
⠀⠈⠻⣿⣿⣿⠉⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠷⢠
⠀⠀⠀⠈⢻⣿⣷⣦⣤⣾⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⠀⠀⠀⠀⠀⠀⠒⠀⠈⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⢸⠈⠉⠛⣿⣿⣿⣿⡀⢠⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣶⣿⣄⢠⠤⣀⣀⠀⠀⠀⠀⠀⠰⠚⠉⢉⣧
⠀⠀⠀⠀⢸⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿⣿⣶⣿⠿⠆
⠀⠀⠀⠀⢸⠀⠀⠀⠀⢙⡿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣯⠓⢀⡀
⠀⠀⠀⠀⢸⠀⠀⠀⢀⣼⣿⣿⣸⣿⡻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣅⡼⠇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠾⠋⠁⠈⠻⢾⣿⣦⣌⣻⢷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⣼⠀⠀⢠⠄⠀⢀⠀⠀⠀⠙⢿⣿⣷⣿⣿⡟⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⡀⠀⠀⠀⠂⠀⢰⡇
⠀⠀⠀⠀⢸⠀⠀⠚⠀⠀⠀⠀⠀⠀⠠⠀⢈⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⢷⣆⠀⠈⠁⠀⢀
⠀⠀⠀⠀⢸⠀⠀⠸⡇⠀⠄⠀⠀⠀⠀⠀⠀⠀⠙⠁⠙⢿⣿⣷⣤⡤⠀⠀⠀⠀⠀⠀⠀⠙⠦⣈⠻⣷⣤⠀⠀⠙
⠀⠀⠀⠀⢸⠀⠀⠀⣻⠒⠂⠀⠒⠂⠄⠀⠀⢀⣀⠀⠀⠈⠉⠙⠻⠷⣄⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠙⢷⣄⠀⠀⡂
⠀⠀⠀⠀⢸⠀⠀⠀⢈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⣄⡀⠀⠀⠙⢷⡌⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠹⠌⠀⠋⠀⠘⠛⠞⠀⠀⢠⠤⠄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⣤⠋⠀⠀⠘⠂⠄
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣶⣶⣶⣆⣀⣀⣀⣀⣀⡀⠁⠀⠀⠀⠀⠀⠂⡀
⠀⠀⠀⠀⢸⠀⠀⠀⣀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠟⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⢸⢀⣴⣾⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⠆⠀⠛⠋⠉⠁⠀⠀⠉⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢰⢸⣿⠟⠋⢉⣙⣫⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠳
⠀⠀⠀⠀⢸⣿⣯⣶⣹⣾⢿⣿⡿⢦⠀⠀⠀⠀⠀⠀⠀⢸⡶⠏⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠖
⠀⠀⠀⠀⢸⣿⣿⣿⣫⡷⣾⡿⣁⣿⣿⣦⠀⠀⠀⠀⠀⠈⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣤⣤⣴⣶⣿⣿⠃⠀⠃
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣧⣿⡏⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⡿⢯⡿⣾⠏
⠀⠀⠀⠀⠸⢻⣿⣿⣿⡿⢡⣼⣿⣧⣿⣿⣿⣷⣦⣀⡀⠀⠀⠀⠀⣀⣤⣄⣴⣾⣿⡿⠏⠀⢸⡟⣱⣟⣿⠟⠀⠀⠀⠁⡄
⠀⠀⠀⠀⢴⢸⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣟⣯⣭⢻⣿⣶⣶⡶⠟⢋⣤⣾⣿⣿⠉⠀⠀⠀⠀⢸⣿⣽⠇⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⢰⠈⣿⣿⣿⡟⢿⣿⣿⣯⣾⢿⣿⣻⣵⣿⡝⣧⣍⢀⣴⢟⣷⡿⠃⠀⠀⠀⠀⠀⠀⡰⣾⡟⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⢸⠀⠻⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⠿⠋⢀⣿⣿⠿⠏⠹⠋⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿
⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⠿⣿⣿⣿⣿⣿⠛⠉⠀⣴⡟⣷⣾⣯⣶⣧⡀⠀⠀⠀⠀⠀⠀⣠⣻⡿⠁⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢻⠀⠀⠘⣿⣿⣾⣿⠿⠋⠀⠀⢀⣤⢾⣤⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⣰⣿⣿⠁⠀⠀⡆⠀⠀⠀⠀⢀⠀⡆
⠀⠀⠀⠀⢸⠀⠀⠀⠹⡿⠋⠀⠀⠀⢀⣴⣿⣿⣾⣿⠿⠉⠀⠋⣩⣽⣿⣿⠃⠀⠀⢠⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⢸⠀⠀⠀⠀⡄⠀⠀⢀⣴⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠉⠀⠀⠛⠆⠀⠐⢿⣿⡟⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢠⣴⣿⣿⣹⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢌
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⣿⠟⣺⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢻⣷⡿⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⠿⠿⠿⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⡆

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣾⠟⠉⠉⠉⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠉⠉⠉⠻⣷⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣿⠀⠀⣠⣤⣤⣄⠀⠀⡆⠀⠀⠀⠀⠀⠀⢰⠀⠀⣠⣤⣤⣄⠀⠀⣿⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠸⡄⢰⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣰⣿⣿⣿⣿⡆⢠⠏⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣿⣿⣿⣿⣿⠀⡧⢀⣀⣀⠀⠀⠀⢻⠀⣿⣿⣿⣿⣿⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡙⢿⣿⣿⠟⣰⣿⣫⣮⣝⣷⡀⠀⠸⣄⠹⣿⣿⡿⢋⡄⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠻⢶⣤⣴⣾⡟⠋⠈⣍⣙⢿⠃⠦⠀⣻⣷⣦⣤⡶⠟⠁⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣀⣀⣤⡄⠀⠀⠀⣿⣧⠀⠀⠀⠈⠉⠀⠀⢀⣾⣾⣾⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣹⡇⢠⣤⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⣭⠟⠁⠀⠀⠀⠀⣿⣿⣧⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⠀⠈⠻⢯⡛⠿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠟⠉⢀⣴⣿⡄⠀⠀⠀⠀⠀⠀⠉⣹⡟⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⣿⠀⠀⠀⢠⣿⣦⡀⠉⠛⢿⣿⣿⣿
⣿⣿⡿⠋⠀⠀⢠⣾⣿⣿⣷⡀⠀⠀⠀⠀⠀⠸⢿⠇⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⡿⠿⠟⣻⣿⣦⠀⠀⠀⠏⠀⠀⢀⣾⣿⣿⣷⡄⠀⠀⠙⢿⣿
⡿⠋⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⡾⠁⣀⣄⠀⠀⠀⠉⢩⣉⠈⢷⡶⣄⡀⠹⠿⠟⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡄⠀⠀⠀⠙
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣧⠀⠀⢀⣀⣤⡤⠖⠚⢉⣴⡯⠶⣻⠟⠁⠀⠀⠀⠀⠈⠻⣟⠶⢽⣦⡉⠛⠶⢦⣤⣀⡀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⠟⠓⠛⠉⠉⠀⢀⣤⠶⠋⢁⣠⠞⢁⣀⡀⠀⠀⠀⠀⢀⣀⡈⠳⣤⡈⠙⠳⣤⡀⠀⠉⠉⠛⠚⠻⠿⠿⠿⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠞⠋⠀⢀⣴⠿⠛⢛⣽⠟⠁⠀⠀⠀⠀⠀⠙⢯⡛⠛⠿⢦⡀⠀⠉⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⣤⣀⣀⣀⣤⡤⠴⠚⠋⠁⠀⢀⣠⠶⠋⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠙⠲⣄⡀⠀⠈⠙⠓⠶⢤⣤⣄⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠋⠁⣀⣴⣚⣥⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣬⣓⣦⣀⠀⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠴⠶⠖⠚⠋⠁⢀⣠⡴⠾⠛⢋⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣙⠛⠿⢦⣄⡀⠈⠙⠓⠲⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠶⠖⠛⠉⠁⣀⡤⠞⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠳⢦⡀⠈⠉⠛⠲⠶⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡛⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣌⢻⣿⣿⣿⣿⣿⣄⣠⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠈⠙⣦⠀⠀⠀⠀⠀⠹⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠋⠀⠻⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⣰⣿⣧⡀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡿⠁⠀⠀⠰⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⢀⣼⣿⣿⣿⣁⣀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢤⣤⣤⣤⣤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣟⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠁⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠹⣿⣿⣿⣿
⠀⠀⠀⠀⢻⣿⣿⣿⡿⠋⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⠀⠀⠀⠀⣴⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⡛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⣀⣀⣈⣛⣩⡔⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⣿⣿⣿⣤⡴⢏⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⣴⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠀⠈⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣶⣿⣆⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠈⢻⡟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣆⢠⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣾⣿⣤⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡀⠀⢀⢀⣠⡴⣂⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠭⠿⠚⠊⠒⠛⠙⠿⠿⢭⣥⣄⡀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⢿⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣶⠄⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣢⠀⠀⠀⠀⠀⠀⠸⢿⣿⠐⡀⠀⠀⠀⠺⠉⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠰⠚⣻⢷⣦⣤⠶⠖⠗⠂⠺⠶⠦⢤⣄⠀⢳⡇⠈⠀⠀⠀⠀⠀⠀⠀⠙⣿⣊⣍⡆⠀⣎⡉⡆⠈⣿⡿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠚⢿⠃⠈⣩⣀⠀⠀⠀⢀⠐⢙⣳⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡇⠈⡋⢁⣀⠙⣫⠁⠀⠩⢖⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣬⣆⠀⠸⠶⠶⠃⠀⠰⣭⣉⠾⣛⡦⠤⠤⢤⣀⣠⠤⠒⢒⡢⣄⢻⠈⠻⣿⣓⣸⠏⠀⢀⡜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠈⢆⠙⣦⣴⣄⣀⣠⡯⢅⠞⠋⠀⠀⢠⢋⡼⠶⠮⡙⢢⢹⣘⣷⢑⣦⣌⡻⠊⣀⡴⢻⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⢀⠴⠃⢀⠀⠀⠀⠱⡮⠿⢟⣩⠥⠚⢋⠄⠀⠀⢠⡇⠞⠀⠀⠀⣸⡎⠭⠝⠉⠙⡶⡶⠈⠛⠭⠔⢫⠃⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⢠⠃⠀⠀⣼⠀⠀⠀⠀⢧⠀⠀⠀⠀⢠⠛⠒⠐⠂⣠⠀⠀⢀⡀⠀⠀⠀⢀⣀⡴⠞⠀⢧⣀⡤⠤⠤⠿⠚⠓⢦⡀⣸⠃
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢹⠀⠀⠀⠀⠸⡀⠀⠀⣤⠀⣀⠀⢀⠘⠛⠒⠒⠛⢣⣞⣿⢺⠧⠶⢆⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⢼⡷⢋
⠀⠀⠀⠀⢠⠇⠀⠀⠀⠘⣦⡤⠤⠴⡶⠃⢀⣼⡄⠀⢈⠟⠀⠉⠉⠉⣉⠽⠃⠯⣅⠘⠦⠤⠔⠋⣿⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠸⣄⠀⠀⠀⣀⣸⣇⠀⠀⢧⡜⠁⠀⠀⠂⡿⠀⣤⡀⣤⣾⡟⠀⣠⣦⣼⣆⣀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⢸⠈⠙⣦⡾⠻⡓⠻⠭⠟⣿⣷⣦⣤⣀⡂⠃⣸⠿⢤⣠⠉⠀⠀⠛⠿⢻⣯⣽⡿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡇
⠀⠀⠀⠀⢸⣀⡴⠋⢀⠜⠉⠙⠒⠦⠤⢀⣈⡉⠙⠛⠻⢿⢶⣾⠷⠚⠛⠛⠛⠉⠉⠀⠀⠀⠀⢸⡸⠛⠛⠛⠛⠛⠉⠉⠉⠉⠀⠀⠇
⠀⠀⠀⢀⣼⠟⠁⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣲⡾⢃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠛⢳⡄⠀⠀⠀⠀⠀⢸
⠀⠀⣠⡾⠃⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⣠⠋⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠸
⢀⣼⠋⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠕⢁⡔⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠒⢶⣶⣤⠤⢼⣀⣀⣀⣀⣀⡠⠔
⢻⣧⣴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡿⠋⡴⠋⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⣿⠒⢺⣄⠀⠀⣯⣉⣈⣇
⠀⠑⠨⢝⡳⢦⣤⣀⠀⠀⠀⠀⠀⢠⣾⠏⡠⠊⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⡧⠿⠷⠴⠋⣆⠀⣸⣤⣀⣼⡀
⠀⠀⠀⢰⢏⡛⠲⢭⣟⡲⣤⣄⣴⠕⣡⠞⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣶⡶⠛⠛⠳⠷⠛⠁⢸⡀
⠀⠀⠀⠀⠑⠪⠭⠭⠭⠟⠃⠛⢯⡴⢥⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡏⠀⠀⠀⢸⡦⠤⣤⣤⣤⣖⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣀⡀⠀⢸⠀⠀⠀⣀⡀⠤⠤⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⡙⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣦⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣇⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣴⣾⣏⠈⠉⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⣿⠿⢟⣛⣿⣮⣛⠻⠿⠟⣿⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣤⣾⣿⡟⢡⡾⣫⡿⣿⣿⢿⡿⣿⡿⣯⣻⣿⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣭⣳⣤⣽⣿⡿⢱⣿⣿⡟⣰⢏⣿⣹⣗⣿⢷⡙⢿⣮⣛⠶⣿⣤⡀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣩⣽⣿⣿⣿⠃⣼⡿⣹⡇⣿⣿⢿⠙⣿⣽⣆⣿⢦⡙⢿⣷⣮⠻⣿⣷⡶⣆⠀⠘⠻⠶⠶⠾⠟
⠀⠀⠀⠀⠀⣠⣤⠖⠛⠉⢱⠞⢻⣿⡽⢟⣡⠞⣽⣿⣿⡇⣿⣿⣿⡆⣿⣿⢿⣧⣾⠻⣆⠹⣎⢷⠘⣯⠻⣟⠛⠛⢶⡄⠀⣤⠶⠖⠛⠻⢦⣄
⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠘⠻⣯⣭⠶⣻⣿⠾⢋⡾⣡⣿⣿⣏⣼⠇⣿⣿⡘⣏⢹⣧⣹⠌⣿⢿⡆⢿⡛⠋⠀⣴⣟⣁⠀⠙⠛⠛⠛⠛⠛⠋
⠀⠀⢿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣏⣰⣟⣫⡾⢿⡏⣿⣿⡟⣰⡏⣿⣇⢿⡿⣿⣿⠀⣿⣌⢳⣼⣿⡆⠀⠀⠀⠉⢻⡆⠀⠰⣾⣿⠆
⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⢻⡆⠀⢾⣿⣧⠾⠷⠞⠻⣷⢿⣟⣱⢿⡇⢿⣿⣎⢻⣿⢿⡶⣷⠾⠀⠉⠋⠀⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢀⣀⣀⣀⣹⣾⠟⠋⠀⣿⣼⠁⣿⢻⡇⠘⠷⠛⢀⣀⣠⣤⡴⠶⠛⢉⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠈⠛⠓⠒⠛⠛⠛⣋⣉⣩⣉⠛⠦⣤⣤⣤⣬⣅⡀⢙⣋⣡⡤⠶⠛⠛⠉⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣤⠴⠟⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢉⡆⠀⠀⣎⣑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡷⠀⠀⢾⢷⢦
⠀⠀⠀⠀⠀⠀⠀⠸⠓⢒⣁⡀⠀⠀⡠⠔⠚⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⢛⣁⡀⠀⠀⢠⣤⣶⣇
⠀⠀⠀⠀⠀⠀⠀⢣⠀⢇⡀⠉⢦⠀⠘⢶⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢺⣿⣿⣧⠀⠈⠻⣿⡟
⠀⠀⠀⠀⢸⠹⡀⠈⢆⠀⢹⡰⠃⠀⠀⢀⡷⠁⢰⢉⡆⠀⠀⠀⠀⠀⣷⠀⠈⢦⡀⢸⣿⠃⠀⠀⠀⡹⠁⠀⣾⠇
⠀⠀⠀⠀⠘⡄⢡⠤⡀⠑⠢⠤⠤⠤⠒⠉⣠⠤⡌⢸⠀⠀⠀⠀⠀⠀⣿⣦⣤⡀⠉⠂⠤⠤⠤⠔⠊⢀⣠⣤⣿
⠀⠀⠀⠀⠀⢣⠀⠱⡌⠓⢄⠀⠀⠀⡴⠊⢡⠊⢀⠎⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶⣄⠀⠀⠀⢠⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠙⠢⡄⠀⢸⡀⠀⢀⠁⠀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⠀⠀⠀⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣎⣉⡱⠀⢏⣉⣚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡶⠀⢴⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈

⠀⠀⠁⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢸⠀⠀⡏⡅⠀⠀⠇⢤⣤⣤⠤⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠈
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡅⠀⢸⠀⠀⡇⡇⠀⠀⡅⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠠⠀⠀⠀⠸
⠀⠀⠀⠰⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⢸⠇⠀⠀⠄⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠘⠀⠀⢄⡀⠠⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⡄
⠀⠀⠀⠀⡄⠀⠘⠀⠀⠈⠈⠀⠀⢀⡃⠀⠀⠀⠀⡇⠀⠀⠀⡇⢸⠀⠀⠀⠀⢀⡼⠤⠀⠀⠀⠀⠀⢀⠀⠃
⠀⠀⠀⠀⠐⠂⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡇⢸⠀⠀⠀⠀⠑⢲⠉⠀⠀⢀⠀⠀⠘⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⠀⠀⠀⠀⡌⠀⠀⠀⠈⠀⠀⡆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⣰⢶⡖⠓⠒⠧⣄⠀⠀⡇⢘⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡠⢠⠢⠄⣈⠔⠉⡀⠟⠁⠀⠀⠀⠂⠠⠤⢙⢅⡉⠹⠦⣄⣀⣀⣠⡇⡄
⠀⠀⠀⠀⠀⣷⢀⠠⠒⠋⠔⠈⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⡇⠂⢄⠈⡅⠀⠀⠀⠉⠉⠉⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠸⠉⠀⣀⣼⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⣿⡀⠀⠛⠀⠠⠆⠀⠁⠶
⠀⠀⠀⠀⠀⠀⠀⠨⣁⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡹⠀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⢠⠀⡇⡖⠅⡀⠒⠒⠐⠂⡀
⠀⠀⠀⠀⠀⠀⢇⠄⠀⠀⠀⢑⢤⠀⠤⠤⠠⠤⠀⣀⠐⠁⠀⠀⠈⠀⠁⢣⢀⣹⠭⠏⣀⣀⡈⣄
⠀⠀⠀⠀⠀⠀⢸⠖⠀⡐⠊⠀⣠⡠⠀⠤⠠⢀⡀⢠⡈⠋⢍⠉⠀⢯⣴⣆⣀⣀⣀⡈⣀⣀⣀⣦⣠⣀⣀⣀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⡼⠀⢮⡀⣀⠜⠛⡠⠀⠀⠀⠀⢤⠛⠺⣀⣈⠀⠀⠀⢌⡁⠀⠀⠉⠙⠀⠉⣿⡧⣤⣄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠐⠆⠐⠊⠙⢉⡔⠀⠀⡀⠀⡀⢀⠀⠀⠊⠉⣩⣉⣉⡁⠘⣫⣀⣀⣈⣀⣒⣶⣟⣀⣀⣘⣇⠀⠀⢀⡇
⠀⠀⠀⠀⢀⡇⠂⠔⠀⠀⡨⠄⡄⠀⠀⠀⠉⠠⠁⢤⠤⢆⠑⠈⠅⢂⠀⠈⡄⠀⠀⡀⠠⠤⠀⡀⠀⠀⠰⠀⠀⠀⠁
⠀⠀⠀⠀⡜⠤⠡⠔⠒⠸⣀⡠⠉⠉⠒⠒⠒⠒⠈⠉⢄⢀⠃⠐⠂⠀⠀⠀⠱⡄⠈⠀⠤⠤⠂⠅⡀⠀⠀⢁
⠀⠀⢀⠐⠂⠂⠀⠀⠀⠈⠉⠁⠉⠁⠀⠀⠈⠂⠀⠀⠈⠈⠀⠉⠉⠉⠁⠀⠀⠘⠂⠘⠀⠀⠀⠀⠁⠀⠀⠀⡄
⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠐
⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⣄⣀⣀⣂⡀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣼⠃⠀⣄⠀⠀⠀⠀⠀⠀⠀⠒⠚⠀⠐⢠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⢰⣿⠗⢸⣟⡄⠐⣬⡖⠀⠀⠀⠀⢸⣤⣤⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠿⢿⣼⣿⣼⡸⣿⢃⠂⠼⠇⠀⢀⢀⠀⠀⠙⣯⣟⣻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣼⣧⣹⣏⣭⣤⣋⣽⠖⠋⡉⣉⠛⠦⠀⠀⢀⣿⣾⣿⣿⣿⣗⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠙⣿⣿⣾⡟⠛⢿⣜⣿⣿⣍⣉⡙⡇⠠⠙⡻⠟⣿⠟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠰⠂⣠⡇⣻⣿⣽⢿⣷⣴⡞⠉⠀⠀⢳⠈⠀⠉⠝⠉⠁⠏⢉⡴⢻⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡞⢋⡉⢓⣟⣫⣴⢿⡿⡁⡠⠴⠿⣿⡿⣿⣶⡶⠒⠒⠊⠙⣷⣤⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⠸⡿⢤⡴⠶⢺⡉⠀⣰⡟⠋⢀⡄⠀⠈⣧⠉⡿⠃⠀⠀⣠⠔⠻⣟⡛⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⣿⣾⠓⠛⠋⠈⢠⣿⣇⢠⣼⡇⠀⠀⢸⣯⠷⠟⠛⠙⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⠧⠾⠛⠍⠂⠀⠲⠋⢸⡿⢨⡿⡇⠀⠀⢸⠁⠀⠀⠀⠀⠀⠈⠻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢧⡆⢷⠃⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⣫⠞⣇⠀⠻⠛⠤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠋⢰⡹

⠀⠀⠀⠀⠀⣀⠤⠒⠂⠈⠉⠉⠉⠑⠒⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠀⠉⠉⠉⠉⠒⠦⣀
⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠒⢄⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⡯⠉⢳⡀⣱⣄⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⡌⠀⠀⠀⠀⠀⠀⠀⢰⠉⠹⣶⡄⠀⢸⡏⠀⠸⣿⣿⣿⣧⣶⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢹⣾⣿⣇⠀⢹⣿⠛⣿⣿⣿⣿⣷⣤
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢸⣿⣿⣿⣦⣘⣿⡄⢿⣿⢋⣶⣿⠿⣷⣶⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣷⣿⢩⢻⡿⣿⣿⡧⠌⣟⣿⡟⢁⣤⠾⢻⣿⣦⣤
⢡⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣾⣿⡏⣧⢸⠏⣷⡌⢹⢇⣶⡿⢸⡼⣿⣋⣴⣿⣿⠿⣿⣧⣄
⠈⡄⠀⠀⠀⠀⠀⠀⣠⢿⣿⡟⢸⡇⠇⠟⡦⡿⣸⣦⠸⢹⢳⡿⢰⣿⢹⣿⠋⢠⣾⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠘⡄⠀⠀⠀⠀⣼⢃⣵⣾⠘⣼⣷⣶⣮⠁⣵⠛⡾⡇⡞⢸⢃⣿⠃⣸⡏⢹⣿⣿⠿⢛⣹⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠙⡄⠀⢀⡼⠡⣪⢟⢕⠟⢉⠛⢛⣣⣴⡇⢸⡇⡇⠁⣿⢸⡟⣰⡿⠁⣸⡟⢳⣤⣿⣿⠿⠛⢿⣷⣄⠀⠀⠀⢀⠄
⠀⠀⠀⠘⢄⢞⡵⠉⠑⢁⡵⠾⠛⠛⢋⡽⢹⣧⢸⡇⠃⠀⡇⣿⢱⡿⠁⣴⣿⠇⣼⣿⣟⠁⢀⣤⣶⣿⣿⡿⠂⢀⠎
⠀⠀⠀⣠⣿⣫⣀⣠⣞⣥⣶⣶⣶⠿⠋⠀⣾⣿⡄⣷⠀⣸⢃⣿⣿⠄⣼⣟⡞⢸⣿⠇⢘⣿⣿⣿⣿⠟⠋⠀⡰⠃
⠀⠀⣼⣿⣿⣿⣿⣫⠥⠤⣀⡉⠛⣿⡧⠀⣿⡇⢣⣹⠀⣿⣿⣿⠏⣼⠏⠊⣰⣿⢣⠀⣨⣿⣿⠟⠁⠀⠀⠔⠁
⢀⡼⣼⣿⣿⣿⣿⣿⡿⠟⢾⣿⡆⢸⡇⠀⣿⣇⣆⣿⣰⢏⣾⠏⣼⣳⠃⣰⣿⣿⡆⢰⣿⣿⠏⠀⠀⠀⠊
⣾⣿⣿⣻⣿⣿⣿⠿⢶⣶⣤⢸⡇⡼⡰⢠⣿⣿⣿⣿⢃⡾⠃⣼⣿⠃⡰⣿⣿⡟⢀⣿⡟⡏⠀⢀⠔⠁
⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢼⡼⠑⠋⠁⠀⠀⠹⣿⠟⠁⣼⣿⠃⡔⣼⣿⡟⢀⣾⣿⢱⠁⡠⠃
⠀⠙⠿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠘⢄⠀⠀⠀⠀⠀⣿⠀⣸⣿⠃⣘⣾⣿⡟⠁⣼⣿⣏⡯⠊
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠘⣶⡟⢁⣴⣿⠫⠋⢀⣾⣿⣿⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠀⠉⠉⠛⠓⠒⠛⢛⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⢀⠔⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣱⣔⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣯⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣭⣿⣯⣽⣯⣭⣭⣭

⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠒⢤⡀⢀⡠⠐⠒⠒⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⣇⠀⣠⠀⠀⠀⢠⣤⠀⠀⠀⠔⠀⠉⠁⠢
⠀⠀⠓⠃⠀⠀⢰⠁⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⢁⣀⡀⠀⠉⠉⠀⠀⢀⡀⠈⠀⠀⢘⢀⡄⠀⠀⠀⠀⡔⠁⠉
⠀⠀⠀⠀⢠⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⢀⠀⠀⠀⠀⢀⡜⠢⡠⠚⣄⠀⠀⢀⡴⠥⠄⠠⠤⢄⡀
⠀⠀⠀⠀⣈⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠥⢀⠀⠀⠀⠀⠀⠃⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠙⢦⠀⠠
⠀⠀⠀⠁⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠒⠈⠀⠀⠀⠀⠒⠎⠀⠀⣠⣒⡒⠦⠔⠒⠢⠤⡄⠀⠈⠑⢄⠒
⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⡇⠰⡞⡵⣶⡦⠀⠀⣬⣭⣁⠁⠀⠀⠀⣸
⠀⡏⢉⠉⡉⠉⡉⢉⠉⣀⣀⡄⠬⠤⠄⢀⡀⠀⠀⠁⠀⠇⠀⠄⡄⠀⠀⠀⠀⠀⠀⠀⢹⢶⠀⢏⡻⡇⠀⢸⡸⣿⠁⢃⠒⣄⡴⠃⡖
⠀⡇⠀⠀⢀⢸⡀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠛⢭⠭⠍⠀⡥⠤⠤⠤⠤⠤⠤⠤⠤⢤⠀⢸⠈⢆⠀⢰⡥⠤⣴⠉⠁⢀⡌⢴⠉⠀⠆⡗
⠀⡇⢀⠀⡰⠊⠀⠀⠤⠤⢤⣄⠀⠐⠒⣠⢄⡀⢣⡠⠀⡇⠰⠐⠒⠐⠂⢲⠒⠂⠀⠀⢸⠀⠨⠷⡦⢭⣨⡥⡴⢚⠉⡹⢋⢧⣁⣤⣷
⠉⠀⠉⠀⡇⠀⠀⠀⠉⠐⣚⣧⠀⠀⢴⣻⡄⢃⡿⢤⠀⡇⢀⠀⠀⠀⠀⢸⠀⢀⠀⠖⠁⠁⠐⡞⠀⢰⠀⠅⢀⢸⠊⠐⠁⠈⡇⡐⡒
⣀⡄⠉⠁⠳⢦⠃⡵⠇⠀⣌⢫⠇⠀⣩⡛⡣⢸⠃⠜⠀⠃⢸⠀⠀⠀⠀⢸⠀⠈⠀⠐⠂⠈⠀⠳⠄⣨⡀⢘⢇⣀⠉⠒⠒⠘⣧⠆⢇
⢀⣠⣤⠦⠄⠠⠛⢦⠀⠀⠈⡵⢦⣤⣷⠍⠀⠌⠀⣁⠀⡁⢸⠀⠀⠀⠀⢸⡔⠁⠠⠔⠀⣰⢯⢻⡄⢙⡷⢡⠞⡀⠈⠑⠒⠒⠙⡀⢸
⠈⡆⠀⡬⢤⢰⣪⠞⠓⢤⣀⠘⠂⢐⣋⣤⠞⢋⡡⠜⡇⡀⢸⠀⠀⠀⠀⢸⠐⠤⢄⣀⣄⣀⡄⠀⣠⢈⣱⡁⠀⣣⣀⣀⣀⠠⠔⡇⠸
⣀⣿⠀⠇⠲⡼⡅⠀⠀⡇⠉⢛⠫⠚⡩⠔⠊⠁⠀⠰⠁⡀⠗⠠⠦⠠⠄⣸⠄⠂⠌⠣⠸⠄⢤⠔⡹⡏⢸⠁⠀⠀⢳⠀⠀⠀⠀⣭⢉
⠤⠘⠀⡇⡘⢰⣀⡀⠠⢼⢠⠋⡝⠁⠀⠀⠀⢀⠠⠺⣒⣊⡉⠔⠓⡄⠢⠔⠢⡀⠈⡉⠤⠶⣭⠊⠀⡆⢸⠀⠀⠀⢸⡇⠀⠀⠀⢸⠄
⠒⠉⠀⢹⠁⠈⠲⠒⠛⢋⡇⠸⠀⢀⡠⠔⢊⡡⢐⠩⠐⠉⡠⠔⠊⢳⠴⠒⠛⠩⠀⠀⠀⠀⣿⠈⠀⠉⢸⠀⠀⠀⡞⠹⡄⠀⠀⢸
⠀⠶⠄⠱⢤⣀⡀⠑⠢⢄⡑⠦⣮⣅⠴⠮⡔⠊⣀⠤⠂⠁⠀⠀⠀⠸⡄⢀⣀⠀⠀⠀⡐⠀⢸⠠⠶⠄⠸⣽⣖⣱⠇⠀⢳⡁⢮⡼⢀
⠀⠀⠀⡐⠁⠀⠈⠉⢲⠚⠻⣄⣠⣏⠀⠂⠓⢎⠀⠀⢠⠊⣿⠽⠂⣄⢗⠝⠁⠀⠀⠠⠀⠀⣴⠀⢀⣄⢰⠃⠄⢇⠀⠒⣂⢇⡀⣇⠈
⠀⠀⠀⡇⠀⠀⠀⢰⡟⠈⠀⠘⢟⠋⠀⠀⠀⠈⠄⠀⠠⢡⠀⢀⡶⣼⠚⡤⠖⠤⠒⣦⣐⠊⡿⠴⣥⠤⠥⠤⠤⢬⡖⠛⠚⠀⠀⢈⡆
⠀⠀⠀⡇⠀⠀⠀⠘⢹⠀⠀⠀⠘⡣⡀⠀⠀⠀⠈⠀⠘⢯⣉⡜⠂⠁⠀⡛⢄⢠⡾⠜⣹⡦⣤⡔⠉⠉⠈⠈⠁⠉⠓⠀⠈⠉⠓⠒⣁
⠀⠤⡄⡇⠀⠀⠀⡇⠈⡆⠀⠀⠀⡗⠓⢄⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢀⠜⠉⠛⠉⠙⢎⠀⠀⢙⢆⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⡧⠤⠤⢤⡇⠀⢸⡄⠀⠠⡇⠀⠈⢢⡀⣀⠀⠈⡄⠀⠀⠀⠀⠎⠀⡘⠀⠀⠈⡈⢣⠀⠀⠁⢻⠉⠙
⠀⠀⠀⢹⣁⣹⣉⡀⠀⠀⠹⠤⠽⡥⢄⠀⠋⣕⢄⠱⡀⠘⣄⣀⠤⢸⠀⢠⠀⠀⠀⠀⠰⠈⡆⢀⠈⠈⡆⠀⠀⠀⠀⠀⠐⠓⠶⠳
⠀⠤⠤⢾⣀⠀⠁⣈⣧⠒⢷⢄⣀⣀⠠⡧⠡⣑⣀⡤⠋⠀⠉⠀⠀⢸⠀⠘⠄⠀⠀⠀⠀⠀⢡⣀⣙⣀⣱⣄⡀
⣀⣀⣀⣀⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣐⣲⣆⣀⣀⣀⣀⣀⣘⣙⣁⣘⣒⣒⣔⣒⣒⣚⣒⣒⣚⣃⣀⣀⣈⣁⣉⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣋⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠉⠉⠉⢛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⢆⠺⣷⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⣴⠒⠒⠀⢸⣁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠈⠳⣄⣸⠀⠀⠀⣀⣤⠀⠀⠀⠀⢀⣠⣼⡀⢸⠀⠀⠀⢸⢁⣧⡤
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠻⡟⠣⣀⣠⠞⠋⣁⡈⣇⠸⢒⣒⣒⢾⢸⢁⣇
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⣠⡶⠛⠒⠦⣄⡀⠀⢠⠃⠀⠀⣻⡴⠚⠁⢷⢹⠀⢒⣿⡤⣿⢸⠘⣸⠆
⠀⠀⠀⠀⣀⡤⠖⠋⠑⢦⣤⣴⣋⣠⠁⠀⠀⠀⠀⠙⠆⣇⠀⠀⠛⢅⠳⡀⠀⣸⢼⡆⢀⡀⠀⣿⠘⡰⠁
⠀⠀⠀⠞⠁⠀⠀⠀⢀⠞⠀⠀⠉⠛⢶⣄⣠⡤⣤⣀⣓⡙⣆⠀⠀⣸⢣⣱⡞⠁⢸⡇⠙⠓⠀⣿⠰⠥⠤⢤⣀
⠀⠀⠀⠈⠢⢀⣄⢀⠎⠀⠀⠀⠀⢀⡜⢩⣴⢦⠇⠙⣄⠀⠘⡀⠀⡈⢣⠳⣷⠀⢸⡇⠀⠀⠀⢻⣷⠄⠀⢀⠴
⠀⠀⠀⠀⢀⡔⠁⡌⠀⠀⠀⠀⠀⠸⠤⠴⠒⠉⠀⠀⠈⣆⡠⠷⣾⡿⠀⠡⠸⡇⢸⠃⠀⢀⡠⠞⢁⠤⠊⠁
⠇⠀⠀⠤⠤⣍⣉⣓⡒⠂⠤⠤⠤⠴⣧⣀⣀⣀⣀⡀⠀⠀⠀⠀⣮⠀⠀⠀⢁⣿⣼⠴⢚⡡⠔⠊⠀⠀⡠⠔⡎
⠀⢤⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠀⠐⠒⠒⠂⠀⠤⠌⠭⠭⠭⠭⠭⠭⠭⠍⠉⢡⡔⠊⠁⠀⣀⠤⠒⠉⠀⠀⡇
⠀⢸⡀⠀⠀⠀⠀⠀⠈⠁⠁⠈⣷⠒⠒⠒⠒⠶⢦⠤⠤⠤⠤⢤⣤⣄⣀⣀⣀⣀⡁⠤⠒⢉⡠⡀⠀⠀⠀⠀⡇
⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣄⢊⠔⠀⠤⢄⠀⢠⠁
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡀⠨⠂⠀⠤⠺⡀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠐⡜⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢀⣀⡸⠐⢢⠀⠁⢔⠇
⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠤⠄⠊⠓⠊⣀⠴⠋
⠀⠀⢠⠎⠁⠀⠀⠀⠀⠉⠉⠉⡓⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⢀⠠⠐⠊⠁
⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⣼⢰⣶⣹⠤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠄⠒⠈
⠀⠀⠸⣄⣀⡀⠀⠀⠀⠀⠀⢿⠈⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣀⣀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠽⠦⠀⠀⠀⠀⠀⠀⠙⠖⠚⠉⠉⠙⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⣀
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⢀⣶⣿⡿⢿⣷⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢿⣿⠏⠀⣼⣿⣿⣧⠃⠀⠀⠀⠀⠀⠀⠀⢸⣷⣀⣤⣶
⠀⠀⠀⠀⠀⢀⡼⠂⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢰⣿⣿⣿⣿⡀⠀⠀⠐⣆⠀⠀⠀⣼⣿⠿⠛⠁
⠀⠀⠘⠻⠿⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣾⣿⣿⣿⣿⣿⣷⣦⠀⣿⠀⠀⢰⣿⣛⠢⢄⡀
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢀⣆⣠⣾⡟⣸⣿⣿⣿⣿⣿⣿⣿⣯⣠⣿⣤⣴⣿⣿⣿⣿⣶⣍⡲⢄⡀⠀⠀⠀⣀⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢰⣿⣩⣿⣯⣴⣿⣿⣿⠟⠻⣿⣻⣿⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠙⠶⠒⠋⡴⠊⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢰⡿⠛⡿⣿⣽⣿⣿⣿⣿⣿⠻⠿⠿⠛⠋⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠘⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣶⡇⣸⣿⡷⠶⠿⠿⢋⣿⣿⣿⣿⠷⠒⣄⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠙⢤⠀⣠
⠀⠀⠀⠀⠀⠀⠠⠞⢩⣿⣿⣿⣿⡆⠀⣠⣾⣿⡯⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣏⠙⣿⣛⣁⣀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡞⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠄⠿⠋⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣉⠻⣿⣿⣿⡃⠀⠀⢠⠊⠑⢢⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣸⣿⣿⣳⣄⠀⠏⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠟⢡⣿⣿⣿⠉⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣟⠻⢿⡎⠙⢦⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣦⣼⣿⣿⣿⣷⡄⠀⣤⣿⣿⣿⣿⡿⠿⠿⢟⣷⣦⡉⠳⣶⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣠⣤⣀⣀⣀⣀⣀⣀⡀⠀⣰⣿⣿⣿⣿⣿⠀⠀⠘⢦⡈⣿⣿⣿⡿⠦⢭⣭⣶⣾⣿⡿⢠⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣴⣿⣿⠟⣹⣿⣿⣏⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢨⡇⣼⣿⣿⣧⢸⢸⣿⣿⣿⣿⣅⣾⣿⣿⣿⣿⣿

⠀⠀⠶⢤⢤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠤⠶⡦⠠⠶⠀⠀⠄⣀⣀⡀⣲⢐⠀⠀⢰
⠀⠀⠴⠚⠉⠁⠸⠀⠀⠀⠀⠈⠑⢄⡀⠀⠀⢀⠠⢤⠄⠤⠰⠒⠒⠤⠠⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⣦⣀⠠⠤⡆⠀⠀⠰⢂⠁⠀⠸⢉⠦⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡆⠀⠀⢀⣀⢠⡿⠂⠂
⠀⠀⣀⣂⣤⣴⠁⠀⠀⠀⠠⠐⠃⢦⣸⠄⠀⠀⠾⡿⠇⢺⢿⣷⠞⠀⠀⠀⠀⢀⣇⡠⠂⠉⠁
⠀⠀⠿⠿⠿⣿⠀⠰⢄⡁⠀⠀⠀⠀⠙⠤⢤⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡼⢫⣤⡄
⠀⠀⠠⠀⣀⡄⠀⠀⡄⠀⢀⡆⠀⠀⠀⠀⠀⠉⠉⠢⠀⠓⣤⠔⠢⡘⠉⠉⡔⠀⠈⡟⠀⠀⠉
⠀⠀⠀⠀⢀⠃⠀⣦⠂⡐⠈⡇⢠⠀⠶⠆⠀⠀⠀⢸⡶⣄⠀⠀⠒⠂⠰⡴⣤⢠⡖⢶⣄⠀⢤⡤⣤⣤⣤
⠀⠀⠠⠄⡾⠀⠀⣀⣀⠀⢰⡇⢌⠆⠄⡠⠀⣐⠁⢰⠀⢸⠀⡦⠀⠀⢀⠃⠀⢸⣷⣬⣿⠆⠄⠉⠉⠉⠉
⠀⠀⠀⠀⡆⠀⠈⢠⠌⠂⣸⠇⠊⢀⡶⠇⢀⣿⢈⢸⠀⢸⢈⣿⠀⠀⠘⠀⠀⣴⠲⠌⢀⡢⠀⢀⠂⡀
⠀⠀⠀⠀⠀⠀⠈⠚⠀⠂⢻⠶⢡⣌⠝⠀⠁⠹⠠⢸⠀⢸⠀⣿⠀⡆⠀⠀⢘⣮⠆⠐⡏⡀⡲⠎⠐⣠⡄
⠀⠀⠀⢸⠀⠀⡏⠉⠉⠉⡟⢁⣀⡔⣠⣁⣀⠠⠀⠰⠀⢄⠀⣽⣀⠓⠂⠀⠀⠇⠐⠁⣴⠁⠀⢰⢰⠙⠹
⠀⠀⠀⡄⠀⣸⣧⡠⣤⡖⠿⢷⣤⣿⡅⢴⡃⢦⣭⢹⠀⠈⠂⠀⠈⠃⠀⠀⠀⣏⠀⡄⠩
⠀⠀⢀⠁⠀⡭⠭⠅⠾⠁⠐⠚⠉⠁⠃⢜⡁⠔⢢⠀⠀⠀⠀⠀⢀⡷⠋⠁⠀⢻⠀⠛⡀⠐⠀⠘⠊⠁
⠀⠀⡌⠀⠀⠀⠀⠀⠀⢀⠀⢀⢠⡀⢠⡾⡇⠸⡧⢸⠯⠸⠤⠒⠁⣀⠄⠀⠀⠀⠆⠀⠀⢶⠄⠀⠀⠠
⠀⠀⡆⠀⢀⣀⣠⡄⠀⠸⠟⠋⠉⡅⢠⣠⣤⠄⢠⠘⠀⠀⠀⠀⠔⠀⠀⠀⠄⠀⠀⢡
⠀⠀⠃⠀⠭⠑⠈⠁⠀⠀⢀⣠⡤⡛⢀⣉⢥⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⢀⠽⠀⠀⠲⢀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⣠⣇⡘⢀⣠⣄⡤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠦⠀⠀⠀⠀⠾⠷
⠀⠀⠀⠤⡶⠛⠋⠁⠀⠠⠐⠘⠛⠙⠉⠙⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠰⠀⠀⠀⠒⠒⠂⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠑⠂⠆



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠂⠐⠂⠀⠒⠀⠀⠀⠒⠀⠐⠂⠀⠐⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢀⣀⠀⢀⡀⠀⠀⡄⣀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⣠⣤⣤⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣫⣿⠁⠈⠛⣆⢸⠁⠀⣸⢻⠀⠟⢹⠂⡀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢟⣿⡿⠀⠈⠛⢹⡿⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠺⡶⠏⠈⠛⠚⠁⠚⠀⠛⠛⠀⠛⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⢡⣿⠏⠀⠀⠀⢠⣿⢷⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠚⠛⠛⠓⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⢿⣿⢀⣬⡁⣀⣈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣸⣿⣥⣼⣿⣾⣿⢿⣿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣻⣿⣯⣿⣿⢈⣤⣼⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣹⡴⠂⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟⠋⠉⢹⣿⠈⠉⢹⣿⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⣸⣿⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢸⡇⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢰⠇⠀⠀⠹⡄⠀⠀⠀⠀⣠⡄⠀⠀⣠⡀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢏⣾⡗⠀⠀⠙⢦⡀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⢀⣼⣃⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⡄⠀⢀⣀⣿⣧⣤⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⡴⠋⠉⠙⠛⠲⣤⣉⣈⡉⢉⣉⣉⡤⠞⠉⠉⠉⠉⠉⢳⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠘⠛⠋⢻⡿⠁⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠂⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⡆⠀⢀⠀⠛⠛⠛⣁⣀⣠⣤⣿⣷⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⣼⡿⠛⠉⢉⣿⣿⡀
⠀⠀⠀⠀⠀⠀⡎⠀⡞⠀⠀⠀⢸⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢸⡇⠀⠀⣧⠀⢸⠀⠀⢠⣿⠃⠀⢀⣾⠏⠹⣷⣄
⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⣿⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⠀⠀⠀⣿⠀⣼⠀⢀⣾⡏⣤⣴⡿⠋⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⢧⠀⡇⠀⠀⠀⣿⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⢸⠀⠀⠀⠿⢠⠇⠘⠻⠟⠁⠉⠁⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⡟⠀⠀⢰⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡇⠀⠀⢸⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⡤⢆⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⠀⢠⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠛
⠛⠋⠙⠛⠉⠙⠉⠉⠉⠉⠉⠙⡇⠀⠘⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠶⠶⠶⠾⠶⢿⠀⠀⡷⠶⠶⠶⠶⠶⠶⠶⠶⢶⡦⠶⠷⠶⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠛⠃⠠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠞⠀⠀⡳⣄⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⢶⡖⣶⣲⠮⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠯⠶⡆⣴⢰⡌⢷⡈⢳⡀⠀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠾⠿⠷⠿⠇⠀⠀⣀⡤⠖⢋⣭⡍⣉⡉⠉⠁⠒⢠⠀⢧⡿⢦⡟⠾⣋⣶⡷⠶⠒⠦⢤⣄⣀⣀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣖⠋⠁⠀⠀⠉⠰⠷⣿⠃⣀⡤⣾⠟⠀⠀⠀⣠⡴⠋⠭⢩⡽⠀⣠⣄⣄⠀⠀⠈⢩⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⡭⠭⣭⣍⣛⣛⣓⣒⣲⡿⠷⠛⠁⠀⣠⠴⠊⠁⠀⠀⠀⠀⠓⠘⠫⠿⠋⠀⣠⠞⣣⢞⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣍⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣷⣮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣠⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠿⠟⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⢈⡙⠿⢿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⢟⣡⠆
⠀⠀⠀⠀⠀⠈⢻⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠄⠀⠀⠀⠀⢀⣼⣿⣀⣀⣀⣀⣀⣀⣀⣛⣭⣥⣶⣿⠏
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠂⠀⠀⠀⠀⠙⣿⣿⠿⠿⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⢀⠀⠀⠀⢀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⢀

⠀⠀⠀⠀⢀⠤⠒⠀⠐⠢⡀
⠀⠀⠀⠰⠁⠀⠀⡠⠄⡠⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠲⠒⠒⠒⠒⠒⠒⠒⠒⢒⣒⡒⣒⣒⣂⣉⣉⠉⢢
⠀⠀⠀⠃⠀⣴⠘⠃⠨⠄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠉⠈⠉⠉⠉⠉⠀⠀⣀⣀⠠⢄⠀⣀⣤⠄⠤⢬⡆⢸
⠀⠀⠐⡆⠀⡸⠀⠀⢀⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⡇⡇⠀⠀⠀⢸⠅⢸
⠀⠀⠀⠈⠒⡃⠀⡆⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⡇⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⡇⠀⠀⠀⢸⡇⢸
⠀⠀⠀⢀⠎⡉⠚⣣⡀⠀⢰⢖⣒⣀⠠⠴⢄⠀⠀⠀⡇⠸⢄⠤⠤⠤⠤⠤⠤⠇⠀⠐⠒⠒⠓⠓⠒⠒⠒⢚⠃⠈
⠀⠀⠀⠸⠰⠀⠀⠈⠳⡀⢸⢸⠀⠀⠀⠀⢸⠀⠀⠀⡇⢠⠖⠒⠒⠒⠀⠀⠀⠈⠁⠉⠉⠉⠉⠉⠉⠉⠁⠈⡇
⠀⠀⠀⡇⠀⢣⡀⠀⠀⡨⢾⠸⠀⠀⠀⠀⢸⡤⡄⠀⡃⠘⠀⣀⣀⣀⣀⢀⡀⠤⠠⡀⢀⠤⠤⠤⡀⠀⠀⠀⡇
⠀⠀⢰⠀⠀⠀⠑⠤⢾⢀⠀⠙⠂⠐⠒⠒⠊⡠⠃⠀⡇⠀⡆⡇⠀⠀⢸⠸⠀⠀⠀⠃⠀⠀⠀⠀⢱⠀⠀⠀⠅⡆
⠀⠀⠸⠀⠀⠀⠀⠀⢸⠀⠁⠒⠀⠀⠀⠈⠉⠀⠀⠀⡆⢀⣁⡇⠀⣀⣸⣠⠄⠀⠀⠤⠤⠤⠤⠤⠼⠤⠤⠤⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡤⠤⠄⠀⠀⠐⠒⠒⠂⠀⠀⠒⠒⠒⠒⠒⠒⠲⡀
⠀⠀⠰⡤⠠⠤⠤⠤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⢀⣀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡠⠤⡄⡇
⠀⠀⠀⢷⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡇⠏⠁⢸⠁⠉⡏⠉⠘⠀⠀⢸⠀⠀⣿⠀⠀⣷⡇
⠀⠀⠀⢈⠱⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣇⠀⠀⣸⠀⠀⡇⠀⢠⠀⠀⣸⡀⢀⣿⠀⠀⣸⠇
⠀⠀⠀⢠⠀⡄⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣉⣁⣀⣉⣈⣀⡀⠀⡀⠀⠀⠀⢀⣀⣀⣀⣀⢉
⠀⠀⠀⡆⠀⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⣀⡠⠤⠤⠤⠤⠤⡀⢠⠤⠤⠤⠤⠤⠤⢤⠈⡄
⠀⠀⠀⠁⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠅⠀⠀⠀⠀⠀⠀⠃⢸⠀⠀⠀⠀⠀⠀⠸⠀⡇
⠀⠀⢘⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣄⡀⠀⡀⠀⠀⠀⠀⣤⣼⣀⠀⠀⠀⠀⠀⠸⠤⠃⢰
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣉⠡⠤⠤⠄⠀⠀⠀⠀⠒⠒⠒⠒⠒⠂⠒⠒⢲⠀⢸
⠀⠀⢧⠤⠤⢦⡤⠤⠤⠤⠾⠤⡀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠈⠒⠒⠚⠦⠠⠤⠤⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠟⠉⣻⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⠦⠖⠦⡀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣷⠀⠀⡷⡇⠀⠀⠀⠀⠀⣠⡞⠉⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⢀⣼⡀⠀⠀⠀⠀⠀⠀⠀⡯⡀⠄⣐⡑⡘⢦⡀⠀⠀⠀⠀⡼⢡⣤⡌⣧⠀⡇⡇⠀⠀⠀⠀⠀⣿⣿⣶⣄⣠⢤⣼⡟
⠀⠀⠀⢀⣰⠛⠉⠙⢦⡄⠀⠀⠀⠀⠀⠙⢶⣄⠉⣡⠽⠊⠀⠀⠀⠀⠀⡇⠈⢭⠀⣿⠀⣱⡇⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⡿⠋
⠀⠀⠀⠀⠙⠓⠴⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⣟⠓⠛⠚⣿⣤⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢉⠉⠉⠛⠋
⠀⠀⣠⠔⢒⡒⠒⢤⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⢶⣶⣤⣤⡄
⠀⢸⡇⡈⠀⠒⡃⠴⠃⠀⢸⠀⠀⠀⠀⣀⠴⢊⠅⣢⢟⠎⠳⡄⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⡛⢑⡀⢃⢲⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢐⡧⢀⠋⠾⠓⡁⠔⣨⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣻⠈⠀⢸⢸
⠀⠸⡇⠀⠀⠀⠀⢠⣄⣀⡟⠀⠀⠀⠀⡷⣄⠑⣖⣡⠴⠋⠀⠀⠀⠀⠀⢀⣠⠔⣒⣿⡟⠉⢳⡆⠀⠀⠀⠀⠀⣷⣽⢀⠀⠀⣿
⠀⠀⠈⠒⠦⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠩⠭⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⣳⣄⣤⣤⣤⡶⠟⠁⠀⠀⠀⠀⠀⠙⠾⠶⠶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⢠⣴⡳⣷⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢴⠞⠓⡆
⠀⠀⠘⣎⢳⠘⠉⠸⣿⡀⠀⠀⠀⢀⡤⠴⠉⠀⠀⠀⠀⠈⢓⣢⠀⠀⠀⠀⡶⣯⣶⡯⠕⠒⠢⣄⠀⠀⠀⠀⠀⠀⡨⠷⠾⣄
⠀⠀⢴⡚⠂⠀⠀⠀⢻⠃⠀⠀⠀⢸⡕⠀⢄⡀⠀⢀⣠⠒⢁⣼⠀⠀⢀⡴⢇⣻⠀⠀⣀⣴⣾⣿⠆⠀⠀⠀⡴⠋⠀⠀⠀⠀⠳⡀
⠀⠀⠀⢹⠓⢄⠀⠀⠈⢳⡀⠀⠀⠸⣧⣄⠀⠈⢙⠁⣀⣴⠟⠋⠀⠀⣸⣻⣦⣤⣴⡞⣣⠟⠉⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⢼⣀⣈⣆⣀⣤⠾⠁⠀⠀⠀⠀⠙⠻⢶⣶⣶⠞⠁⠀⠀⠀⠀⠈⠙⠾⣽⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⢇⡀⠀⠀⠀⢀⢀⡞
⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⣠⣤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠘⣇⣀⡼⣏⠀⠀⠀⠀⢰⠛⢦⣀⣀⡀⠀⠀⠀⣠⢶⡄⠀⠀⢀⡤⠖⠒⠒⠒⠦⣄⠀⠀⠀⠀⠀⢀⡿⠛⣧⣞⣲⣦
⠀⠀⠀⠀⠀⡿⠙⡇⠈⠀⠀⠀⠀⠈⣧⡾⠀⠈⠉⠉⠉⠉⢧⢸⠇⠀⠀⠙⡟⠒⠶⠶⠶⠒⡿⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠙⡄
⠀⠀⠀⢠⣎⠀⠀⢙⡆⠀⠀⠀⠀⠀⠈⠣⣄⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⢻⠢⠤⠤⠴⢺⡇⠀⠀⠀⠀⠀⢸⡏⠈⠉⠉⠁⠹⡇
⠀⠀⠀⢸⣇⣀⣀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠈⠳⠤⢤⡤⠞⠀⠀⠀⠀⠀⠀⢫⠢⠤⠤⠤⠤⠒⡇
⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⢠⡿⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⣠⣴⢶⣤⣤⣤⣄⣀⠀⠀⠀⡾⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⢫⣾⡿⢣⣶⡙⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡟⣱⣿⣿⢡⣿⣿⣷⠸⣿⣿⣿⣿⣿⣿⣿⣷⣄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡟⣼⣿⣿⣿⡜⢿⣿⡟⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⣠⣴⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣦⣭⣙⡛⠟⠼⢿⡿⠛⠿⣿⣶⢠⣾⣿⠟⠛⣿⡟⡛⠿⢛⣻⣿⣿⣿⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⢰⣿⣶⡆⢹⣷⣦⡙⠘⣫⣶⣿⡇⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⡀⠀⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⡘⢿⣿⣿⠰⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣶⣭⣥⢠⣬⣥⣾⣿⣿⣿⣿⣿⣟⡻⠟⣛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣥⣤⣤⣤⣤⣤⢠⣤⣤⣤⣤⣤⣤⣤⣤⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⠀⠀⠛⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⠏⠹⣿⣿⣿⠟⡹⣿⣿⣿⠟⡙⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⢠⣶⣿⣿⣶⣬⣴⣾⣿⣶⣤⣴⣾⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⠸⠿⠟⣛⣛⣛⣉⣉⢸⣿⠟⢿⣿⣿⡿⠻⣿⣿⣿⠟⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣛⣩⣥⣶⣦⠹⣿⣿⣿⣿⣿⣿⢈⣵⣾⣦⣭⣭⣶⣷⣬⣭⣵⣾⣦⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⡹⣿⣿⣿⣿⣿⢸⣿⣿⢿⣿⣿⣿⡿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣌⢿⣿⣿⣿⢘⣋⣵⣦⣙⣛⣫⣴⣬⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢷⡙⢿⣿⢸⣿⣿⣿⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⢂⣲⡤⣿⣾⢤⣤⣤⣀⠀⡇⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢹⡀⣾⡍⣰⣀⣿⣿⣸⣴⣻⣈⣽⣿⣦⣿⣷⢄⡴⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⢘⣯⠊⢷⣝⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣩⠖⢤⡾⠁⡀
⠀⠀⠀⠀⠀⠠⣈⡷⣿⣬⣿⣻⣿⡿⢿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣷⣖⣹⣾⣏⡤⡾⢁⠀⡀
⠀⠀⠀⠀⢰⡠⢋⡷⣿⣿⣿⠿⠁⣴⡿⠛⠁⠀⣀⣤⣤⣀⡀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣽⣭⡴⢃⡌
⠀⠀⠀⠨⣔⣽⢿⣿⢻⣿⠃⠀⣼⣿⠁⠀⣰⣿⠟⠛⠛⠛⢿⣷⡄⠀⠘⣿⣯⡻⣿⣿⣿⣿⣟⣿⣷⣫⡴
⠀⠀⠀⣤⣿⣿⠟⠁⢸⣿⠀⠀⢿⣿⠀⠀⢻⣿⣤⣶⣦⠀⠀⢹⣿⡀⠀⢸⣿⢀⠀⠛⢿⣿⣿⣿⣟⣋⣄
⠀⢀⣼⣿⡿⣱⠁⠀⠘⣿⡄⠀⠘⢿⣧⣀⠀⠈⢀⣼⣿⠁⠀⢸⣿⠃⠀⣸⣿⠃⠀⢠⡿⣝⢿⠛⣛⠋⠁
⠀⠛⠋⡜⠀⠈⡀⠀⠀⠙⣿⣦⡀⠀⠙⠻⠿⠿⠿⠛⠁⠀⣠⣿⠟⠀⣠⡟⠁⠀⣠⣿⣝⠛⡓⣾⠏
⠀⠀⠀⠠⠠⠄⢤⣌⡒⠠⠬⣻⣿⣶⣤⣤⣄⣀⣤⣤⣴⣾⠟⠃⡠⠖⠉⢀⣤⡟⢝⢿⠻⣦⢃⡟
⠀⠀⠀⠈⠙⣒⠦⡠⠋⠑⡶⠶⣤⣍⣉⣙⣛⣛⡛⠉⡉⡀⣀⣀⡴⠶⢞⠛⣎⠃⠀⠀⢀⠏⡼⠁
⠀⠀⠀⠀⡴⠲⢿⣖⠄⡠⠀⠀⡜⠀⠉⠋⠉⡟⠉⠉⢩⠹⡍⠉⢢⡈⠃⠑⠘⠃⢀⠀⢁⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠌⠑⠢⢄⡀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠈⠃⠀⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⠄⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠐⠒⠂⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⠄⡀
⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⡀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠄⠀⠀⠀⠀⠀⠈⠠⢤⡠⠤⠄⠤⠠⠤⠄⠤⠠⢴
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢠⠒⠄⠈⠢⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠋⣠⣠⢈⠑⠖⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡅⢀⠀⡉⢻⠎⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠊⠈⢀⠈⣐⠔⠁⠀⠀⠀⠀⠀⠀⠀⣬
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⡀⠔⡌⣙⠢⡀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠃⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠄⠀⠈⠢⢌⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣌
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠢⡀⠀⠀⠀⠨⠃⢲⢄⣀⡀⠀⠀⠀⢉⣀
⠀⠀⠀⢀⠘⠐⠀⠒⠁⠀⠀⠀⣤⠤⠳⠄⢀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⢸⣀⣠⠶⣤⡆⠀⠑⢌⣂⣀⣀⣀⡍
⠀⠀⠀⢠⠇⣀⠀⣀⣉⣑⠦⢈⣳⡆⠀⢀⠬⢧⡀⠀⠀⠀⡬⠈⠀⢚⠙⠛⠛⢻⠈⢃⠛⠀⠉⠉⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢴⣀⠉⢏⢀⣀⣀⠴⠉⠀⠤⠭⠭⠭⠭⠀⠀⠠⠛⠁⠈⡇⠤⠒⠁⠀⠀⠀⠈⢋⣿⡸⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠐⠐⡒⢶⣐⢐⣰⣆⣰⣂⡂⠠⣝⣹⣽⠏⠁⠀⠀⠀⡇⠀⠀⠀⠀⡞⠀⠀⠈⠟⢁⠆⠀⣀⣀⣀⠀⠀⡄
⠀⠀⠀⠀⢯⠵⣭⣺⣽⣯⡽⣽⠂⢨⣿⢿⢿⡀⠀⠀⠀⠀⡅⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢠⡖⡀⣀⣀⣀⢀⡀⣀⢀⡀⣆
⠀⠀⠀⠀⠸⣾⣷⣯⢺⣗⣮⢾⡀⠜⠁⢕⣢⠉⠀⠈⠛⠂⠃⠀⠀⠀⠐⠚⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠂⠉⢉⠁⣡⠩⠈⠀⡅⠉⢈⠁⡽
⠀⠀⠀⢈⠆⠀⠀⠀⠀⢠⠔⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⡄⠀⠀⠀⠅⠀⠸⠀⠃
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠒⠉⠉⠉⠙⠂⠀⠒⠀⠀⠐⠀⠀⠐⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠈⠄⠀⠓
⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠈⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠒⠂⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠐⠒⠒⠒⠒⠒⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠠⠀⣀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠒⠂⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠄⠁⠀⠈⠑⠠⣀⠀⠀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⢀⠔⠁⠀⠀⠀⠁⠉⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⡐⠠⢤⣀⠀⠀⠈⠑⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠀⢀⠀⠈⠁⠂⠠⠱⠄⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⠀⢀⡾⠃⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢀⣸⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢥⡄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢄⢀⠀⠀⠘⠃⠀⠀⠀⠀⠰⣷⠀⠀⠴⢦⡀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠁⠀⠐⠶
⠀⠀⠀⠀⠀⠀⡀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠘⠀⠁⠀⠀⠀⠀⢀⡤⣀⠀⢡⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡌
⠀⠀⠀⠀⠐⣇⡀⠠⠀⣀⠁⠠⣀⡀⠀⠀⠀⣀⣠⠐⠁⢀⡀⠀⠀⠀⣀⡀⠀⢘⣦⠀⠀⢀⡛⢄⣂⡤⠀⣀⣀⣀⠠⠐⢻⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠂⠠⠄⠈⠁⠙⠂⠈⠓⠂⠀⠤⠀⠐⢈⣈⠂⠀⠀⢸⡆⠀⡌⠁⡑⠀⣀⠱⠈⠀⠀⠀⠑⠢⢀⠀⠀⠀⠁⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠰⠀⠀⠀⠀⠀⢀⡀⣰⠀⠲⠯⠤⠴⠄⠴⠾⠷⠤⠖⠲⠆⠀⠀⠉⠛⠀⠀⠀⠀⡀⠈⠓⠀⠠⠄⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠈⠉⠀⠒⠂⠈⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠐⠀⠀⡀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⢀⡀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡀⠀⠈⠆⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡘⠐⠰⠂⠀⠆⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⡀⣴⠀⠐⡄⠒⢴⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⡐⠀⡠⠀⠀⠀⠈⢢⠀⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡠⠤⠤⠤⠤⢤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⡀⠀⣤⠀⠀⠠⠀⠀⢀⠀⠀⠀⠐⠒⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⠀⣠⠴⠟⢲⠗⢛⠛⠀⢹⣛⡀⡀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢰⠁⡌⡀⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢰⡟⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠀⣀⡂⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣾⠃⠀⠀⠀⠀⢠⣾⣿⣿⣿⣧⠀⠀⠀⢰⣿⣿⣿⣷⡌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣸⢡⣼⠀⠀⠀⠀⡀⠀⣿⣿⣿⣿⣿⠇⠀⠀⠸⣿⣿⣿⣿⡇⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⡆⢻⠄⠀⠀⠀⠓⢀⠙⠛⠟⠛⠋⠀⠀⠀⡀⠈⠙⠛⠛⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢷⣸⠀⠀⠀⠀⠘⠀⠀⠀⠀⠇⠀⠀⠀⣸⣇⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢿⢰⡟⣟⢟⠳⡄⠀⠀⠐⠀⠈⢆⠐⠀⠐⠛⠛⠆⠀⠀⣠⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⢰⠛⢻⡬⢧⠙⣦⡤⣤⣴⣟⣀⣧⣦⡤⡶⡶⠒⣞⣻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⡈⡗⠶⢿⡼⢷⣤⣿⣻⣿⣿⡿⢿⠿⢿⠛⠛⠛⠛⠛⠉⡃⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⣀⣀⠀⣇⡀⡀⣼⣻⣹⣟⣶⣲⣶⣾⣷⣿⣖⣲⡶⣶⣶⠴⠶⠄⠐⠒⠁
⢀⠀⠀⠀⠀⠀⠀⢀⡾⣭⣤⣔⠒⠾⢿⢿⣿⠭⠽⣿⣿⣿⣟⣻⣷⠟⣻⠛⢦⠘⣌⠂
⡇⠀⠀⠀⠀⠀⠀⣿⠀⣹⠺⢝⡻⢷⣾⣷⣶⣗⣿⣿⣿⣿⣿⣯⣥⣿⣇⠀⢀⡇⢿⡠
⠀⠀⠀⠀⠀⠀⠰⢹⡰⠫⣶⡲⢽⣶⣬⣭⣭⣭⣶⣿⣿⣿⣿⣯⣥⡞⣠⠆⠀⢸⡌⣷⠐
⠀⠀⠀⠀⠀⠀⢂⡟⡇⠈⣀⣉⣓⣿⣿⣿⣿⣽⣶⣿⣿⣿⣿⡿⠟⣯⡏⠈⠈⠙⣷⡜⠧⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⡀⠀⠀⠀⢰⡟⣼⠀⠈⢷⠤⢬⣭⣿⣛⣿⣿⡿⠿⣿⣿⣿⣿⣿⡿⠆⠀⠀⠀⠈⣷⡘⡄⠑
⠰⠁⠀⠀⣠⣟⣰⠇⠀⠀⠀⢶⣖⣏⣯⣽⣵⣶⣲⡿⠛⠛⠿⠶⠾⠟⠀⠀⠠⢀⠀⠘⣇⠹⡄
⠀⠀⠀⣰⣟⣾⡟⠁⠀⠀⠀⠀⢠⣯⣯⡽⠟⠚⠛⠽⣂⣀⢘⣀⡄⠀⠀⠀⠀⠀⠢⠀⢻⡄⢹⠀⠠
⠀⢀⡴⣡⢺⡟⠀⠀⠀⠀⠀⠀⠦⣖⣸⣶⣶⣯⣿⣿⣿⣿⣷⣿⣷⡶⣦⣄⡀⣄⠀⠀⢢⣇⠘⢇⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠕⠆⠦
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⡠⢀⣀
⠀⠀⠀⠀⠀⠀⠀⡘⠉⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⡏⢡⡀⠑
⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠉⠒⠉⠉⡠⠂⠀⠀⢀
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⢐⠆⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀
⠀⠀⠀⠀⢠⠂⠂⠒⠀⠀⠀⠀⠀⢤⢀⡗⠈⠢⠤⠄⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⠉⢀⠽⡽⠋⠒⠂⠀⠀⠀⠀⠤⠐⡇⠀⠀⠀⠀⠀⠀⠈⠆⢄⠀⢄⢀⡀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠲⠛⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠⢠⡃⠀⢱⠂
⠀⠀⠀⠀⢬⠇⠀⠀⠀⠀⠀⢸⠌⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡶⢫⣜⠒⠊
⠀⠀⠀⠀⠂⠲⢁⠀⠀⠀⠀⠇⠀⠠⠒⠎⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡰⠡⠃⠀⠉⡆
⠀⠀⠀⠀⠀⠃⠈⡄⠲⠀⠴⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠰⠖⡐⠠⠁⠀⠀⠠⠁
⠀⠀⠀⠰⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠜⡰⠁⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠏⡐⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠁⠀⠀⠀⡠⠄⠒⠀⠈⠀⠈⢸⠄⠀⡠⠂⢸
⠀⠀⠀⠓⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠔⠀⢀⠼⠠⣀⣀⡀⠀⢠⡁⠊⠆⠀⠠⠀⠀⣉⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⢈⡏⠈⢻⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⠇⣌⣴⡶⠀⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣆⠘⢿⣿⠀⠀⣸⡿⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢯⡙⢄⡹⣿⠀⠐⠇⢀⠞⠋⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⢿⣦⢻⡀⡁⠀⠀⣀⣴⡏⠀⣠⣿⢋⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢻⢹⢸⡇⣿⢸⡇⣟⣧⠀⣾⢻⠁⣿⠟⠙⢻⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⢨⣰⠃⡿⢸⡇⠿⢸⣇⣯⠟⣰⡌⢸⢰⣿⠀⣴⠏⢉⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣚⡽⣡⡾⠃⣼⣿⣶⣦⣨⢎⡼⢋⡇⣿⢸⡇⢸⣿⠰⠏⠈⣽⣿⣤⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣋⡾⠋⣁⣈⡙⠻⠾⢫⠌⠈⣇⢾⢰⡟⢸⡇⣼⡿⢰⣦⡾⠛⢻⣿⣧⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⣩⡶⠟⠋⠉⠙⠷⣦⡄⠘⡆⢻⠀⡿⢰⢸⣀⡿⢁⣾⡏⡀⣠⡿⢻⣿⣗⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢻⣯⣿⡿⠗⠛⠛⠿⣿⣦⡀⠀⢻⡄⣧⢸⡄⢁⠈⢸⡿⢡⣾⡿⠁⣿⡏⠀⣻⣿⣿⡀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣿⡿⣵⠾⠿⣷⣦⣄⠀⠹⣷⡄⠸⡇⢸⠈⣧⡘⠆⣾⢣⣿⡟⠁⣼⣿⡇⡞⠉⢼⣿⠁⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡗⣥⣦⣄⠀⠉⠛⢷⡄⢈⡿⠀⡇⢸⡆⡎⣧⢸⠏⣾⡟⠀⣼⣿⡿⠁⣷⡤⢿⡇⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢿⣿⣿⣧⠉⠛⠻⠿⣶⣤⠈⠇⣼⠃⣸⠇⣸⠀⣷⢨⠞⣼⣿⠁⣼⣿⡿⠁⣸⣿⣷⣿⡃⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣠⣾⡇⢹⣿⠳⢒⠲⢤⣄⠸⠀⣼⣣⢴⡟⢠⡟⢠⡏⠀⣰⣿⠃⣸⣿⣿⠃⣸⣿⡇⢸⠿⣇⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣠⣿⣛⣠⣿⣿⣿⣿⣷⣇⣿⣼⣸⡷⠿⠏⢰⡋⠶⠟⣠⣼⡟⠁⣰⣿⣿⠏⢀⣿⣿⡇⣾⢸⡏⣷⣀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠁⠀⠀⠀⢀⠻⣶⠿⠟⠋⢠⣾⢟⣿⠇⢀⣾⣿⣿⠁⣿⣾⣷⡿⣧⢹⣦⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⣉⣴⡿⠃⢀⣼⣿⣿⡇⢰⣿⣿⠃⣿⠸⡆⢯⠙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⢰⣾⣿⠟⠀⢀⣾⣿⣿⠟⠀⣼⣿⡇⢰⣿⠀⢿⢸⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟⠁⠀⢠⣿⣿⡿⠋⠀⣴⣿⣿⠇⣾⣿⠀⢸⡆⣿⣿⡏⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⢠⣿⣿⠟⠁⢀⣾⣿⣿⡏⢰⣿⣿⠀⢸⣧⢹⣿⡇⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠻⣆⠀⠀⠀⣿⣿⠋⠀⣴⣿⣿⣿⡿⠀⣼⣿⣿⠀⠀⣿⠀⣿⡷⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠙⢿⣶⣾⡟⠁⣠⣾⣿⠟⠋⠀⢀⣴⣿⣿⣿⠀⠀⣿⡀⣿⡗⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⣄⠉⠉⠀⠠⠛⠉⠀⣀⣤⣴⣿⣿⣿⣿⡟⠀⢸⣿⡇⣿⡷⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠉⠉⠛⢦⣤⣤⣀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⣾⣿⠀⣿⣧⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⡀⠀⠀⠀⠀⠙⠻⠿⠟⠛⠉⠙⠛⠛⠋⠉⠀⠀⢀⣼⣿⡏⢠⣿⡇⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠀⣾⣿⡧⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⠟⠁⢀⣼⠿⣿⡇⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⠀⠀⢻⣷⢦⣤⣤⣤⣤⣴⣾⣿⡿⠟⠛⠁⠀⣠⡾⠋⠀⣿⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠹⢦⣽⠿⠛⠋⠉⠉⠉⠀⣀⣀⣀⡾⠋⠀⢀⣴⡇

⡴⠒⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠓
⡇
⡇
⠻⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠
⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠺⣅⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣨⠗⢤⡀⠀⠳⠴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣠⠞⠁⠀⠴⣏⣀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⣠⠞⠁⠀⠀⠀⠀⠀⠈⠉⢉⡤⠚⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⢢⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣸⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⡟⠲⠤⢄⣀⡀⠀⠀⠀⠀⠀⣠⢦⣀⡼⠖⠊⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣴⡁⠀⠀⠀⠀⠉⠉⢲⠀⢀⠜⠁⠈⠁⠀⠀⢀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⢤⣀⣸⠀⠈⠑⢦⣰⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠈⠢⠤⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠤⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠒⠒⠒⠤⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⡀⠀⡀⣸⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⡯⣁⠂⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠐⢁⠄⠀⠀⠤⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡞⠣⠤⠔⢳⠀⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠁⠀⠴⠀⠀⠆⢇⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡐⠓⢒⡲⠂⢈⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡏⠃⠀⠀⠀⠐⠁⠀⡘⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠎⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡦⡀⠀⠐⠄⠔⠠⠃⠀⠀⠀⠀⠀⠀⡇⠀⡔⠁⠀⡰⠢⠄⠀⢤⣾⠤⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡖⠀⠠⡖⠁⠀⠀⠀⠀⠀⠀⢀⡳⠊⠀⣠⠾⠀⠀⠀⠀⠘⠢⢈⡐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⠀⠀⠳⣤⠤⠤⠤⠖⠚⡎⠁⠀⡠⠚⠁⢀⣀⣀⣀⣀⣀⣀⣀⣈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢤⣀⠑⠂⠊⠀⢂⠀⠀⠀⠀⢧⠔⠁⢠⠴⠮⠥⠧⢄⢠⡠⠀⠤⡎⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠣⡀⠀⠀⠘⣄⣀⡠⠔⠋⠀⠀⠘⠀⠀⠀⠀⠀⢀⣤⡀⠀⡇⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠡⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢾⠀⢀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡀⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡜⠘⡄⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢡⠀⢀⠠⠖⠋⠀⠀⠀⡆⠀⠀⢀⠴⠒⢾⠀⠀⠀⠀⠌⠀⠀⢸⣸⠀⡧⠤⠀⠒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠡⠀⠀⠈⠀⠀⠀⣇⠀⠊⠁⠀⠀⠟⠉⠉⠳⠀⠃⠀⡴⠋⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠓⠒⠾⠥⠤⠤⠤⠔⠚⠧⠤⠤⣴⠦⢆⣀⣧⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠡⠤⠤⠤⠤⠤⠤⠤⠄⡀⠀⠠⣭⣭⣿⠠⠂⠛⠙⣇⣀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠛⠛⠁⠐⠤⠤⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢘⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠁⠀⠈⠉⡉⠁⠀⣈⣉⣉⢉⣉⠉⣀⣠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠚⠛⡏⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠇

⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣾⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⡿⠇⠙⠉⢻⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠃⠀⠁⠀⠀⢀⠠⣌⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⡀⠀⠀⢠⣿⣧⢘⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⠷⣄⠀⠘⠀⠀⣘⠿⣿⣷⣆⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣷⢄⠈⠑⠀⠀⠃⢺⣾⠿⣾⡿⢀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣭⡀⢡⠦⠄⠀⠂⢈⡙⠄⣿⣏⠘⠀⠐⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⡼⣿⣇⢸⠯⣿⣶⣴⡁⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⠟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠼⣞⡆⠹⢷⣿⣿⡻⡶⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠈⠀⠀⠀⠈⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠨⣿⡶⠠⣛⣿⣱⣿⣧⠀⢻⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⢻⠟⣺⣿⡟⠁⢀⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠘⠿⠟⠠⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠟⠀⠀⠀⠰⢿⡄⣿⣷⣿⡆⠀⣸⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⢀⡄⣰⣾⣿⠟⠋⢠⣾⡿⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠗⠀⠀⠀⠐⠆⠨⣷⣿⡿⠗⠐⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⣠⣿⣶⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢤⠰⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⠇⠀⠀⠀⢀⣾⣷⣾⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠉⢛⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢻⣿⣿⣿⠄⠀⠀⠀⠠⢾⡿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⡀⠀⠀⢠⣷⢾⡿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣇⣀⣤⣏⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⡁⠀⠀⠀⠉⠻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣷⠀⠀⠀⠀⠘⢶⣽⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣟⣁⠀⠀⠀⠐⠻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣾⣿⡀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣧⠀⠀⠀⠀⠈⠻⢻⣿⣿⣿⣿⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠲⢿⣿⣿⣿⣿⣀⠀⠀⡀⠀⠀⠠⡀⠀⣠⠾⠋⢀⣀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠈⡩⢿⣽⣿⠋⠳⢂⠱⢄⡞⠀⠰⠟⠁⠀⠰⠿⠿⣶⣼⡦⠤⠼⠛⠋⠉⠉⠉⠉⠉⢻⠶⠀⠘⠙⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⡀⠀⡀⠀⠀⠀⠠⠆⢈⠀⠛⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠑⢦⣄⠀⠸⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡄⠁⠠⠀⠰⣦⠀⢠⠶⠘⣾⣗⣀⡄⣀⠀⠀⠈⠙⠉⢙⠋⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⢽⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⣘⢐⣍⠁⠀⠉⠀⠀⠰⡿⣀⣠⠴⠂⠐⠀⠴⣶⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⢸⣿⢿⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⠧⠀⠀⠀⠀⠁⠀⠰⠶⠒⠛⠀⠐⠲⠤⠄⠁⠀⠦⠞⠀⠀⠀⠀⠀⠉⠉⠐⠒⠦⠀⠒⠒⠀⠀⠠⠀⠐⠛⠿⠿⠿⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⠒⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⣀⠤⠄⣀⡀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⡞⠁⣀⠀⠀⠈⠙⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠎⠑⠃⠀⣤⠀⠀⣀⢣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢦⠀⠀⠀⠀⠤⠌⠉⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢧⡀⠐⢄⡠⠄⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠒⠀⣖⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⣍⠢⣄⣠⠟⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢢⠀⠀⠀⠀⡆⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⡄⠀⠀⠀⣧⠤⠒⣇⠀⠀⠀⠀⠀⠀⠀⢠⠴⢖⠒⠒⠒⠒⠀⠒⠒⠒⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡤⢶⠊⠙⡄⢀⡀⠠⠿⠒⠊⠉⠉⠉⠑⠒⠤⣀⠀⠈⣦⢀⣉⠒⠠⣄⣀⢀⣀⣀⣀⣀⣀⣈⣉⣒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡀⠀⢹⠉⠓⠢⠄⢀⡀⠠⠤⠐⠒⡯⠀⡸⠆⠀⠀⠀⡈⠉⠒⢺⡦⠤⠀⠄⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠱⣄⣈⠲⠤⢤⣀⣀⣀⢰⡶⠊⠑⠒⠃⠐⠁⠀⡄⠀⠤⠆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠈⠉⠑⠒⡶⠲⠖⠚⠀⠀⠀⠀⠀⠀⢀⣀⣙⣦⣄⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠒⠖⠒⠒⠒⠒⠢⢴⠃⠀⠀⠀⠀⠀⢸⠻⢭⡉⠀⠀⠀⠀⠀⠈⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡞⢶⡀⠈⡳⠒⠒⠈⠉⠉⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠹⣉⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠑⠤⣇⣀⣀⡀⢀⣀⢀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⢀⢠⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⡌⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠥⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢠⠁⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡄⠀⢸⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠑⠒⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠃⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠴⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠈⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠉⠒⠤⢄⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢀⡼⠴⠖⠦⢤⡀⢀⠀⠀⣡⠖⠉⠀⠀⠉⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣹⠢⣄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠛⠠⠤⠤⠤⠤⠞⠉⠀⠀⠙⠒⠂⠀⠀⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠑⢮⡤⢤⣤⠤⠤⠤⠤⠤⠤⣤⠤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠇⠀⠀⠀⠀⠀⠀⠸⠤⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⠶⡶⠶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⣴⠋⠀⠀⣷⠀⠈⠹⣄⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠳⣤⣼⠁⠀⠀⠀⣿⠀⠀⠀⠘⣦⡤⠞⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⢸⠋⠛⠒⠶⠶⣿⠴⠶⠒⠋⠹⡆⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠒⠒⠒⠺⡟⠒⠒⠒⠒⠚⣿⠒⠒⠒⠒⠒⢻⠓⠒⠒⠒⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢻⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢀⡏⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⣸⣷⠖⠛⠉⠉⣍⠉⠉⠛⠲⣾⣅⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣏⠁⠸⣆⠀⠀⠀⣿⠀⠀⠀⣴⠃⠈⣻⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣌⣷⣄⡀⣿⢀⣠⣾⣥⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠑⠢⠀⢀⠀⢀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠒⠒⠒⠠⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢠⠴⠒⠚⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⣱⠤⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠦⠤⠤⠤⠤⠤⠤⠤⠔⠀⠒⠀⠀⠀⠈


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⢠⠂⠀⠉⠉⠉⠉⠉⠉⠉⠀⠐⠄⠀⠀⠀⠀⠀⠢⠀⠈⠀⠂
⠀⢀⠄⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⡀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⣀⡀⢀⡀⣀⢠⢀⣀⠐⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠰⠀⢠⢠⠟⡇⡸⣇⣣⢸⢸⣇⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠤⠤⠄⠀⠀⠀⢀⡠⠄
⠀⠀⠀⠀⠀⠒⠀⠁⠀⠀⠁⠀⠀⠀⠊⠀⠁⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⢀⠄
⠀⠀⠀⣃⠀⠀⠀⢠⣺⠉⠹⣒⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠈⢀⠈⣀⠔⠁⠀⠀⠰⢤⠴
⠀⠀⡏⠉⢹⠀⠀⢾⠁⠀⠀⢸⡆⠀⢘⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡇⠀⢸⢄⡀⠈⠛⠊⠝⠋⠀⡀⣤⠀⢸⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠉⠉⠉⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⡠⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣴⣿⡀⢀⡖⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢀⣴⣾⡿⣽⣿⣿⣿⣿⣿⣿⡿⣷⡞⠫⡏⠉⠐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢰⣿⣇⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⠥⠀⡀⠄⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⢸⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠳⠶⠚⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠉⠀⠀⠠⡿⠿⠿⠟⠛⠋⠉⠉⠀⠀⢀⣀⣤⣴⡖⡀⠀⠀⠀⡄⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣀⣤⣤⣶⣶⣿⣿⣿⣿⠟⢩⣗⠠⠀⢰⣷⠈⢠⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⢠⣀⣿⠀⠀⠋⢀⣤⠀⣀⣍⠴⣍⡛⢿⣿⣿⡿⡟⢿⠿⢿⣿⠛⠍⠀⣼⣿⡀⠒⣸⣿⠀⣤⠆⢀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⢸⣼⡿⠞⠀⢈⠗⠲⣂⠄⢈⡅⠉⢁⡀⠆⣉⡩⠛⠉⠁⢀⠐⠂⡄⠑⠐⠭⣢⡀⡀⢨⠑⠣⠛⠻⢇⠁⣀
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⠊⠀⠀⠠⠖⠧⢀⡄⠀⠀⡀⠀⠁⠘⠋⠈⠁⠀⡀⠄⠂⠙⡀⢰⣷⠀⠠⡀⠀⠀⠈⠻⠗⡤⠀⠀⠈⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠔⠃⠀⠀⢀⠂⠠⢸⣇⣾⠀⠈⠁⠀⠀⠀⢠⡆⠈⢀⠀⠂⢠⣇⣸⣿⡀⣀⣀⠄⢀⠅⠀⠁⢠⡀⠀⢀
⠀⠀⡆⠀⠐⠀⠀⠀⠤⣀⣀⠀⢀⣠⣤⡤⣬⣍⠀⠀⠴⠄⠀⣠⡄⣺⣇⢒⠶⠖⠠⡚⠛⠙⠉⠉⢀⠄⠊⠀⠀⠀⠁⠀⠀⠀⠈
⠀⠀⠁⠨⠑⠠⠄⠉⠀⢀⠀⠀⠈⠙⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠐⠆⠨⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠂⠈⠘⢿⣿⣶⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⡰⢠⢤⣄⡄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⠓⠦⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⣝⠻⠖⣲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠞⠀⠀⠀⣠⢾⡷⡾⢀⡜⠙⣆⠀⠉⢫⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣃⠀⢀⡠⢞⣁⠀⠀⠷⠋⢀⣀⠘⢆⣦⢀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡽⡹⡴⠋⠀⠈⠈⠁⠀⠀⠀⠈⠈⠀⠈⠙⡶⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡇⠀⠀⠀⢠⠄⠀⠀⠀⠰⡦⠀⠀⠀⡟⠱⡄⠀⢀⠜⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣇⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡷⠚⢁⡴⣫⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠢⣤⠔⠀⠀⠀⢀⡴⠓⠋⠻⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣤⣀⣀⣀⣀⣠⣤⡖⠻⡇⠀⠀⢀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢙⠲⠋⠀⠳⠋⢘⡶⡹⠓⠒⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⢸⡀⠀⡠⠔⠶⣯⢾⣁⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠙⠲⣇⠀⠀⣸⠘⡍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡏⠓⢤⣀⠜⡽⠋⠉⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⣄⣀⠀⠐⣏⠙⠒⢢⠀⠀⠀⠀⠀⣿⣷⡶⣫⣾⣶⣶⣶⣶⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠉⠰⢧⡐⢥⡄⣰⠇⢀⣤⢖⠒⠒⢿⣏⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡀⠀⢀⣠⠇⠀⠉⠁⡴⠋⠘⠦⣳⢄⠀⠹⡟⢻⠛⠉⠛⢿⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠱⠤⠞⠁⠀⢀⡤⢊⠀⠀⢀⠀⠉⠻⣷⣤⠇⢸⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣔⠋⠀⠀⠀⡸⡏⠉⢹⡀⠀⠀⣸⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⠴⣗⣋⣉⣹⠇⠀⠀⠿⢝⣛⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⣠⣶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣤⡤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠛⠷⠀⠟⠋⠀⠀⠀⠀⠀⠀⠀⠠⡿⣱⡿⠃⠀⠈⢷⡝⠷⠀⠀⠀⠀⠀⣼⣿⣿⡇⠈⢻⣿⣆⠀⠀⠀⠀⠀⢸⠀⣘⠀⣁⠀⡃⢈
⠀⠀⠀⢀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠄⢤⣤⣤⠀⡐⠀⠀⠀⠀⠀⢸⡟⠙⢿⠁⠀⣸⣿⣿⠀⠀⠀⠀⠀⠠⠠⠤⠠⠤⠠⠤⠠⠄
⠀⠀⠘⠿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠘⠦⣈⣙⣻⠔⠁⠀⠀⠀⠀⠀⠀⢻⣦⣬⢧⡾⢋⣼⠏⠀⠀⠀⠀⠀⢈⠬⠭⠬⠏⢈⡉⠈⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠋


⠀⠀⠀⣄⡞⣉⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠒⢦⡄
⠀⠀⠠⠠⠄⢤⠀⠤⢰⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠙⢆⠂⠁⠔⠀⠀⠀⠀⠀⠀⣀⣀⠀⢸⠿⠋⠀⠀⠀⠀⠀⠀⠀⣰⠃⣀⠀⣠⣶⠶⠹⡆
⠀⠀⢀⣔⡆⣰⣠⣶⣆⣀⡀⠀⠀⠀⠀⠀⢡⣄⣀⠀⡈⣧⠴⡞⡃⠀⠀⠀⠀⠀⠈⠻⠷⠀⢠⡶⠆⠀⠀⠀⠀⠀⠀⢻⡀⠉⠁⠉⠁⠀⢠⡇
⠀⠀⠀⠙⠻⠿⠿⠿⠿⣿⣿⠀⠀⠀⠀⠀⠀⠉⢿⡜⠛⣻⡴⠟⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⣿⣶⣶⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⡀
⠀⠀⢠⣾⠏⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⡧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⢀⣀⡴⠉⠉⠉⠉⢳⣀⡀
⠀⠀⢸⠫⠠⣤⣤⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⢬⣿⣻⣖⢄⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⣽⣛⣿⣿⣻⣿⣛⣻⠁
⠀⠀⠰⣦⣤⣽⣿⣷⣷⣦⣄⠀⠀⠀⠀⠀⠀⡆⠞⠁⠆⠰⠠⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠹⠿⠋⠉⠏⠉⠻⠟
⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⢿⠀⠀⠀⠀⠀⠀⠐⢿⣀⣃⡜⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠛⠂


⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⡀⢈⣤⠶⣦⣌⢀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠾⢿⣿⣭⡇
⠀⠀⠀⠁⢾⡁⣾⠍⣹⠈⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⣤⡿⠀⠀⠀⠀⠀⠀⠀⣤⣾⣇⡸⡿⢠⡿⠟
⠀⠀⠀⠁⢈⠻⠷⠾⢋⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⠖⠉
⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⣟⢧⠻⣿⡿⣿⢻⣿⢿⣿⣟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠿⠘⠳⣿⡟⠃⠀⠀⠀⢴⡟⠾⢿⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢀⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠠⠂⠀⠂⠂⠀⠂⠍⠍⠐⠒⠠⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠒⠢⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⡤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂⠤⠤⣜⣠⠞⠉⠙⠚⢦⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡻⢟⡲⢯⣒⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⡉⠐⠒⠺⠔⠟⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠓⣁⠜⣙⢛⣀⣁⣩⣫⣋⣝⣙⣉⣋⣝⣹⣰⣒⣒⣖⣶⣄⣠⣤⣤⢤⡤⠤⠤⠴⠁⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⢇⠀⠐⢒⣂⡒⠒⠚⠛⠋⠙⠓⠛⠛⠚⠒⠓⠛⠓⠓⠚⠃⠛⠛⠚⠛⠋⠉⣟⠫⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⡡⣴⣿⢸⠀⠀⠄⠀⠀⠀⠀⠂⠂⠃⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂⢦⡄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣨⡟⠁⠈⡘⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢸⡶⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠰⡇⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⠤⡄⠀⠀⠀⠀⠀⠀⠠⢡⢞⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢨⣇⠀⠀⠁⠇⠀⠀⠀⠀⠀⠀⢀⠔⠊⣁⠠⠅⠀⠰⡉⠀⣘⠢⢄⡀⠀⠀⠀⠀⠀⡦⣲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⠈⢓⠢⣼⢀⠀⠀⠀⠀⠀⢰⠃⠔⠉⠀⠀⠀⠀⠀⠈⠶⠀⠑⢦⠈⢆⠀⠀⠀⢸⠅⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣣⣹⢸⠀⠀⠀⠸⡍⠁⠘⢒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡀⠀⠀⡼⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡟⠀⠀⠀⠀⠈⢦⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃⠀⠀⣇⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡘⠀⠀⢀⣓⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⠀⠀⠀⠀⠈⢦⡙⢄⡀⠀⠀⢠⣀⡀⠀⡤⢊⣰⠞⠀⠀⠀⢸⣲⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠀⠀⠀⠀⠀⠀⠑⠢⢌⣁⡒⠊⠀⠈⡅⠉⠉⠀⠀⠀⠀⠀⣘⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠎⠀⠀⠀⠀⠀⠀⠀⠀⣗⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠯⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠔⡀⢸⠍⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢹⡄⠀⠠⠀⡠⢀⠀⣀⠔⢉⠔⣊⠔⢁⠔⣡⠔⡡⠊⡀⢸⣑⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⠙⠇⠀⠐⡡⢔⠡⢊⡠⢊⠥⠊⡠⠐⡡⠊⡠⢊⡠⢊⡀⢸⣒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⣤⣠⡉⠒⢄⠀⠐⠁⠔⡡⠔⢁⠤⢊⡠⠊⡠⢊⠔⢉⠔⢃⠄⠸⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢴⣯⠒⠙⠙⢧⠀⢧⠀⠀⠈⡠⠒⢁⠔⢁⠔⢋⠔⢁⠔⢁⠔⢁⠄⢨⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠽⢎⠓⠓⣐⠎⠀⣺⠥⠀⠀⠠⠔⡡⠚⣡⠖⡡⠜⡡⠚⡡⠜⡡⠂⢸⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⡉⡉⠀⣐⣴⣿⡕⠀⠀⠀⠊⠀⠊⠀⠊⠀⠈⠀⠘⠀⠊⠒⠁⡸⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠡⠒⠠⠔⢊⠽⢓⠿⢋⠥⣀⠝⡻⠟⣲⠾⢻⠯⣿⠿⡿⢿⡿⢿⠿⢹⡯⣥⠌⡠⢼⠄⡠⢄⡀⡀⢀⡀⡀⢀⡀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠁⠈⠀⠈⠀⠁⠈⠀⠁⠈⠀⠁⠈⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡀⠀⠀⡐⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⠷⠀⠀⠀⠀⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠊⠀⠀⠀⠄⠀⠁⠀⠀⠀⠈⠁⠀⠐⣆⢤⣼⣜⠀⠀⠠⠀⠀⠀⠒⠀⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠛⠫⠭⠁⠙⣮⡇⠀⠀⢱⡏⠀⠀⠀⠲⣤⣴⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠲⡜⠓⠀⠈
⠀⢠⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢨⠁
⣆⠁⠙⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⢤⡀⠀⠀⠄⠀⠀⠀⠀⠐⢦⡀⠀⠀⡜⠙⠊⠀⠀⣌⣉⣀
⢋⣀⠃⠀⠀⠀⠀⠀⠀⠊⢡⠀⠀⠀⡠⢎⡂⠀⠀⠀⠀⠀⠀⠘⠙⠹⢶⡞⠀⠀⠀⠀⠘⠈⡌⠹⠷⡆
⢸⣽⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠄⠀⠲⠀⠀⠰⠷⠄⠰⠀⠠⢘⠠⠀⠀⠀⠣⢑⢠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠶⣢⠀⠀⠀⠀⠄⢀⠝⠀⠀⣄⠀⠤⣄⠈⠀⠀⠀⠀⠀⠀⠀⡠⢤⡀
⠀⠀⠀⠀⠀⠠⣄⣒⣤⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠑⠀⢁⡩⠤⠔⠊⠉⠀⠀⠀⠿⡭⠄
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠓⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠠⠌⢠⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⣀
⠀⠀⠀⠀⠀⠀⣀⣸⢁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣄⣠⡨⣆⡀
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⡴⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢦⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⠴⢦
⠀⠀⠀⠳⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣀⣀⣀⣀⣀⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠏⠀⠀⠀⠀⢿⡀⢸⠀⡴⠃⡠⠊
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠐⢋⠉⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⡝⠀⠉⠀⠘⠓⠉⠉⣱
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠈⢦⠔⠁⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⢀⣘⠛⠯⡍
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⡇⠀⢀⣲⡀⢸⢀⣦⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⣠⠴⣊⡤⠚⠉⠉⠉⠈⠓⠤⠋
⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⡤⠀⠀⢡⡀⠈⠛⠁⡸⡈⠛⠀⢀⠜⠀⠀⠀⠀⠀⠀⢸⡥⠴⣚⡭⠖⠋⠁
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠱⡀⠀⠀⠙⠂⠀⠚⠁⠈⠐⠂⠁⠀⠀⡰⠁⠀⠀⠀⣸⠈⠉
⠀⠀⢀⡠⢖⡩⢿⠀⠀⠀⠀⠘⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⠀⠀⠀⠀⡏
⠀⢴⠁⢶⡉⠀⢸⡄⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠈⠁⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠑⠦⢌⡑⠺⡇⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠄⠓⡤⣄⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⢠⠖⠒⢼⠁⣧⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢈⡵⠒⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠘⠦⡤⠒⢀⣾⡄⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠳⠖⣫⣬⡇⠀⠀⠀⠀⠀⠀⢰⡶⠀⠀⠐⢾⡆⠀⠀⠀⠀⠀⠀⢸⣧⣄
⠀⠀⠀⠀⠀⢸⠁⠀⣇⠀⠀⠀⠀⠀⠀⠀⢿⣿⠆⣶⡾⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡆
⠀⠀⠀⠀⠀⢸⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⡿⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⢿⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠣⣤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣤⠜