Joint Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠃⠀⠀⠀⠈⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠈⣇
⠀⠀⠀⠀⡞⠉⡉⠙⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢉⡉⠙⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠘⣆
⠀⠀⠀⠀⠉⠲⣌⡓⢶⣋⠀⠉⠙⠒⢦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢬⡑⢶⣏⡀⠈⠉⠒⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢬⡛⠦⣄⡀⠀⠀⠈⠉⠉⠓⠦⢤⣀⠀⠀⠀⠀⠀⠉⠛⢬⡙⠷⣦⣄⡀⠀⠀⠉⠉⠒⠢⢤⣀⠀⠀⠀⠀⠈⠲⠄⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢬⣙⠷⢶⣤⡀⠀⠀⠀⠀⠈⠙⠳⠦⢤⣀⡀⠀⠀⠈⠓⢬⣙⠻⢶⣤⣄⠀⠀⠀⠀⠈⠙⠳⠦⢤⣀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⡐⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⡀⠀⠈⠉⠳⢤⡐⠶⣦⣆⣄⠀⠀⠀⠀⢀⣤⣿⣻⣶⣤⣄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠮⣙⠻⢶⣴⣴⣀⡀⠀⠀⠠⠀⣀⠉⢷⡀⠀⠀⠀⠈⠓⠦⣍⡛⢿⣶⣴⣴⠻⣿⣿⣟⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢬⣙⠿⢅⣀⠉⠍⢈⠉⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠉⠓⠬⣙⣿⠷⣿⣿⣿⣾⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣍⣛⡒⣀⣀⣀⣀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⠉⠉⠙⠲⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠤⢄⣀⣀⠀⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡂⠁⠀⠁⠀⢀⠀⠤⠤⠤⠤⠤⠤⠤⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡵⠀⠐⠧⣀⡀⠀⠤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣖⣈⣉⣉⠉⠉⠀⠀⠀⠀⠀⢠⠤⠤⠤⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠠⠤⠃⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠄⠀⠐⠐⠒⠀⠉⠁⢒⠦⠀⠀⠀⠀⠀⠀⠀⣠⠤⠄⠀⠐⠒⠖⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣉⣉⡉⠉⠉⣙⠆⠀⠑⠢⠤⣀⡀
⠀⠀⢀⣤⠤⠤⠤⠤⢤⠤⠤⠄⠶⠶⠀⠀⠀⠀⠀⠲⡒⠒⠒⠒⠀⠉⠉⠉⠉⡽⠒⢄⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀
⠀⠀⡇⡇⠀⠀⠀⢀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⢀⠗⠘⠂⣇⠀⠀⠀⢀⣀⡀⠀⠤⠤⠤⠤⠚⠁
⠀⠀⠈⠑⠒⠒⠀⠀⠀⠼⠦⠀⠤⠤⠤⠀⣀⣀⣀⠀⠀⠈⢦⠀⠀⠀⠀⢠⣃⣈⠀⢠⠇⠀⢀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠓⠒⠒⠒⠂⠒⠚⠓⠉⠉⠉







⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡴⠂⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠈⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠉⠀⠑⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⣢⣄⠀⠀⠀⠀⠉⠷⠶⠦⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣙⣛⠲⠤⣔⣒⡒⠤⠤⠤⠤⠬⢭⣭⣛⣒⠦⢤⣄⣀⡠⠺⠃⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠙⠛⢛⣛⠛⠻⢷⣆⠈⠛⢿⠗⠒⡖⠒⠒⢶⡶⢭⡙⠒⢠⡙⠓⢤⡀⠀⠀⠀⠀⠀⣠⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠐⢄⠱⠑⣄⣸⠀⠀⢿⠓⠀⡻⣭⠊⠙⠻⠦⣝⢧⡈⠉⠒⠀⠠⠔⠈⠁⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠽⣇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⡇⠀⠀⠈⠓⠀⢀⣺⠀⠀⠀⠀⠈⠑⠨⠖⠂⢀⠀⠀⠠⠮⡄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡈⠀⠀⢕⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⠀⠀⠀⢤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⣼⠉⠓⠯⠁⠀⠀⡀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠸⡇⢋⣠⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢦⡈⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢷⠀⠀⠀⣀⠀⠀⠈⠲⣤⣾⡇⠀⠀⠀⠀⠀⠀⢰⡃⢾⢽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢸⣿⣶⣶⣿⠀⠀⠀⠀⠀⡏⠻⣶⣶⣄⡀⠀⠀⠀⣿⡎⠀⠑⡀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⡟⠖⣀⣽⠀⠀⠀⠀⣸⠁⠀⠀⠈⠻⣿⣤⠀⠀⢷⠻⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠈⠈⣻⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⢻⡇⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠂⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠢⡀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠐⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣲⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠀⠀⠀⢸⢽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⡆⠀⠐⠦⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⠀⠀⠀⣸⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠇⠐⠦⣀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣂⣁⣌⣹⡒⢀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣯⡘⣡⢣⢲⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠁⠀⠀⠀⣯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣣⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠸⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⣀⣒⣀⣀⡤⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⢰⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣻⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⣶⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠘⡟⢷⡀⠀⠀⠀⠀⠀⣠⠾⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⣁⣩⣴⣷⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⣧⠀⠀⠀⢠⠾⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⢀⣜⣰⣿⣿⡿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠟⠁⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣷⡄
⠀⠀⠀⠀⠰⠀⠀⠀⢠⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⢀⣤⡶⠛⠉⠀⠀⠀⠀⢹⣿⣿⣿⢿⡟⣹⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣙⣷⡄⠀⠀⠉⣠⣤⣴⣤⣤⡀⠀⠀⠀⠈⠙⢒⡈⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣤⠿⠟⡗⠀⠀⠀⢯⣀⡽⠿⠏⠀⠀⠀⠀⠀⠀⠚⠁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠜⠀⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⢩⡛⠻⠛⠉⠀⠀⠀⠀⠀⠀⣀⣴⠏⠀⠀⣿⡿⠿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⣴⣾⠿⠀⠀⠀⠘⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠺⣄⠀⠀⠠⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⣤⣶⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⠈⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠉⠀⢀⠀⠀⠀⠀⠀⠀⢰⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠐⠁⠐⣀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠀⠀⠀⠀⠀⠀⢹⣧⡀⠀⠀⢿⣿⣿⠀⠀⣠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠾⠿⠛⠉⠀⠀⠀⠀⢠⣀⠀⠀⠀⣿⣷⡄⠀⣿⣿⣿⠀⣰⣿⡟⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣦⣄⠘⣿⣿⣄⣿⣿⣿⣼⣿⡟⠀⣀⣴⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠿⠿⣯⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠞⠟⠋⠁⠀⠀⠀⠈⠙⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢠⡤⣄⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢸⠅⠀⣼⢾⡖⡂⢸⣇⠼⠃⡇⢈⡱⠀⠀⣰⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠢⡀⠀⢿⡶⠾⢻⡄⠈⡇⠈⣇⠀⢸⡇⠀⠀⡟⠉⣇⡙⢣⡏⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣰⠇⠀⢹⡇⠀⢸⡄⠘⠂⠀⠀⠀⠀⠉⠀⠰⠇⠀⠀⠀⠈⡇⠀⢀⠤⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠾⠇⠀⠈⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠈⢀⣀⢨⠂⠀⢀
⠀⠀⠀⠀⠀⢠⠂⠀⠁⠀⡠⠏⢹⣇⠀⠀⠀⠀⠀⢀⠴⠴⠋⠐⠀⠈⣿⣄⡀⠀⠀⢀⣤⣄⣀⣀⠀⡀⠀⣰⠖⠉⠛⠹⡄
⠀⠀⠀⠀⠀⠀⠈⠈⢀⠼⠀⠀⣀⣼⠃⣠⠖⢲⡄⠘⡈⠀⠀⡀⣠⠀⠀⠀⢻⡆⢠⡏⠂⠘⠁⠘⠏⠙⣷⣛⠴⡆⢀⠦⠂
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣀⣸⡗⢠⠛⠀⣸⡇⢼⠀⢀⠖⠋⢹⠗⠀⠰⢿⡇⡜⠁⢀⠀⠀⢠⡄⠀⠉⣿⡆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠈⢿⣀⡜⠀⠀⢽⡆⠈⠙⠁⢀⡴⠞⠀⡀⣤⣿⠛⣧⠀⢀⣿⠒⠋⢳⠆⠀⠻⣧⠀⠀⠠⠀⢀⡠
⠀⠀⠀⢀⣀⣀⡠⢼⣧⣠⠆⠀⠈⠈⠀⠀⠀⣾⠃⠀⠀⣠⢼⡌⠀⢈⡿⠛⠁⢠⠏⠀⠈⣿⡄⠀⢸⡄⠀⠔⣿⠀⠀⠀⠀⠀⠀⠀⠚⠠
⠀⠀⢰⠉⠀⠈⠀⠀⢇⡀⠈⠒⢯⠀⠀⠀⠘⣿⡄⠀⢸⠱⠀⢠⣤⡾⠃⠀⠀⢿⡀⠀⠀⢿⣇⣄⡏⠀⠘⢺⡏⠀⠀⠀⠠⡀⣠⡀⢰⠂
⠀⠀⠈⠳⣄⣠⠀⢠⠴⠃⠀⢰⠃⠀⠀⠠⣾⠟⠀⢰⡏⠀⠀⡼⢯⠔⡶⣤⠴⣤⣏⠀⠀⠐⠃⠂⠁⠀⣔⣸⠇⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠉⠋⣡⣤⣤⣄⣘⡂⠀⢀⣀⡿⠀⠀⣏⡀⣀⠀⠀⠀⠀⣀⠀⣠⣿⠏⢻⠀⠀⠀⠀⠀⠠⠼⢟⣀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣯⠙⠛⠛⢾⣥⣄⡀⠈⠓⠶⠾⠓⠬⠖⠋⠛⠛⠉⠀⠘⠷⢖⣪⣤⡶⠛⠛⠋⣽⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠈⠉⠛⠓⣦⣄⡀⠀⠀⠀⠀⢀⣠⡴⠚⠛⠉⠁⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢯⣷⣄⣰⣄⣰⡀⠀⠀⠠⣌⣉⣛⠻⠟⠛⢶⣭⣭⣉⡀⠀⠀⢀⡇⣠⣂⣠⣾⡽⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣻⠿⠶⣶⣤⣀⣀⠀⠀⣀⡉⠉⠁⠉⠙⠛⠛⠶⣬⣛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⠉⠀⣀⣀⣩⣽⠟⠛⠛⠷⠶⠶⣤⣄⣀⣀⠀⠈⠉⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⣋⣁⣀⣤⡴⠶⠟⠋⠉⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠉⠙⠻⠶⢦⣤⣄⣌⣙⡓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣦⣿⣶⣋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡽⠻⢯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠇⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢸⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡄⠀⠀⠙⢦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠙⠦⣤⣤⣤⣤⣍⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣤⣤⣤⠤⣤⣄⠈⠻⡄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⣷⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠞⠉⠀⠀⠀⣀⣼⢷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⢀⣀⡤⣖⣴⡴⠯⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⠋⠁⣀⡠⣄⣰⡾⠿⠓⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⠁⣀⡠⣄⣲⠭⠗⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⣎⠉⣀⣤⣴⡴⠞⠓⠉⠁
⠀⠀⠀⢀⡠⠔⠒⣉⣠⣴⡶⠿⠛⠋⠉
⠀⢰⣋⣡⣴⠾⠟⠛⠉
⠀⠀⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠈⢀⡱⠆⢠⣄⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⢀⠉⠉⠉⠉⠻⣷⡀⠀⠄⠀⠘⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⡾⠃⠐⠀⠀⠈⠀⠉⠐⠦⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⡤⠶⠤⠀⠀⠀⠀⠀⢀⣴⠧⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⢀⡀⠀⠀⠀⠀⡀⠾⠟⠛⠛⣧⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡶⠖⠂⠀⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠈⠀⠙⠛⠠⡄⠈⠙⠁⠶⠶⠤⠄⠀⠀⠳⠌⠁⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⣀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⣄⢀⠠⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠂⠼⠛⠀⠀⢤⡄⠀⠀⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢋⡁⠀⠀⠉⠀⣠⡄⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣼⣅⡀⠲⣤⢾⣿⣿⣷⣦⣤⡓⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠁⠀⢰⣄⣉⠛⢻⣿⣿⣷⠀⠀⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⢸⣿⠟⠻⢿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠘⠁⠐⠋⠁⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣦⣤⣄⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣾⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠛⠛⣿⡆⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣷⠀⠀⠀⠀⣾
⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⡀⠀⠀⠀⠐⣀⠀⠀⠀⢀⣸⣿⣿⡿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠈⠉⠻⣿⣷⣤⣤⣀⡀⠀⠀⠘⢿⣿⣦⡀⠀⠀⣿⡀⠀⠀⣞⣿⣫⣿⡇⠀⠀⠀⢰⣏⠀⢀⣠⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡿⢿⣷⣶⣄⣀⠿⣿⣿⣦⠀⣿⣧⡄⣾⣿⣿⢻⣿⠃⠀⠀⢀⣼⡿⣆⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣏⡻⠿⣿⣿⣿⣿⣷⣾⣿⣷⣟⢻⣿⣾⣿⠀⠀⡴⢌⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣯⣿⣿⣿⣿⣿⣿⣿⣼⢛⣿⣿⣤⢛⣤⣿⡿⠋⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣺⣿⣯⣴⣿⣿⣿⣄⣠⣶⣶⣶⣿⣿⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠾⠿⠶⠿⠿⠿⠿⠿⠛⣻⣿⣿⡿⠋⠀⠘⢿⣿⣿⣿⣿⣿⣿⣭⣿⣯⡟⠷⠆⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⡿⠛⠁⠀⠀⠀⠀⠛⢻⣿⣿⣿⣍⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣯⣵⣶⣾⢿⣶⣶⣿⣿⣿⣇⠛⠹⠿⣿⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠿⣿⣿⡿⠿⠛⠉⢉⣠⣾⣿⡿⣿⣿⣿⣿⣷⣄⠀⠀⠈⠙⠛⠻⠷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠁⠀⠀⠀⠀⣠⣴⣿⡿⣷⣿⣷⠻⣿⣿⠿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠟⠀⣿⡿⠇⠀⢿⣿⡧⠀⠘⢿⡟⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠛⠁⠀⣿⣿⡿⠀⠀⠘⣿⣿⠀⠀⠀⠘⠿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠃⠀⠀⢸⣿⣿⠃⠀⠀⠀⢿⣿⡷⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠁⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠃⠀⠀⠀⠀⠀⠀⠘⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠁⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣼⣿⣿⣿⣿⣆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠐⢦⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣠⠄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣙⣿⣿⡿⢡⣾⣿⣿⣿⣿⣿⣿⣦⠹⣿⡿⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠰⠤⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣀⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣛⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⡿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣀⠀⠀⢰⣿⣶⠶⣾⣿⣿⣷⣦⣤⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⣿⣿⣿⠀⠉⣉⡉⠀⠀⠈⠙⠛⠋⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠁⣠⠀⠈⣿⠁⠀⠀⠈⠉⠀⠀⠈⣿⠏⠀⡀⠉⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⣿⣿⡄⢾⣿⣿⣿⣿⣿⣿⣿⡆⢠⣿⣿⣷⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠁⠀⢺⡍⢷⣄⠀⠀⠀⠀⢀⡼⠋⠀⠀⠙⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣿⣏⠰⣌⠻⣿⣿⣿⣿⣿⠟⣰⣿⣿⣷⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⢻⡄⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡏⢠⢹⣿⣿⣿⠏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠈⣇⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇⢸⡀⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡿⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣇⣸⣇⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣀⣤⠤⢤⡄
⠀⠀⠀⢸⠉⠁⠀⠀⢸
⠀⠀⠀⣼⠀⠂⠀⠀⡄
⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⠀⣸
⠀⠀⢸⠀⠇⠀⠀⡿
⠀⠀⣼⠀⠀⠀⢸⡇
⠀⢠⡇⢸⠀⠀⢸⠃
⠀⣸⡇⢸⡀⠀⠘⡆
⣴⠃⡟⢸⡀⠀⠀⠈⡐⠤⡀
⢻⣹⡇⠀⣧⣄⠠⢄⣀⣅⠈⢢
⠀⣿⣷⣧⡙⠀⠀⠀⠉⠈⠳⣼⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⢠⡏⣼⠿⣧⠀⠀⠀⢀⡗⠀⢹⡧⠙⠦⣤⣤⡤⠤⢶⠒⠋⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠶⠶⠶⠶⠶⠶⢶
⣾⣸⠇⠀⠈⢄⠀⠈⠁⠀⣀⣾⡆⠀⡄⠂⠂⠈⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣀⣀⣸
⡇⢻⡀⠀⠀⠀⠉⠙⠉⠉⠉⢿⡁⣠⣁⣀⣀⣀⣀⡀⠀⠈⠈⠉⠉⠁⠈⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠚⠻⠷⠭⠭⠭⠿⠍⠛⠒⠒⠶⠶⠤⠤⠤⠤⢤⣤⡤⠤⠤⢤⡄
⠙⠲⠧⠤⠤⠴⠶⠿⠟⠻⠿⠶⢶⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⡀⠀⢸⣿⣿⡇⠀⢀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦⢸⣿⣿⡇⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠙⢶⣦⣤⣈⡻⣿⣷⣿⣿⢰⣿⡿⣋⣤⣴⣶⠞⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣾⣿⣿⣷⣤⡀⠀⠀⠀⢀⣠⣴⣾⣿⡿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⣴⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠘⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⡿⢷⡀⠀⠀⠉⠉⠉⠀⠛⠁⠀⠙⠃⠈⠉⠙⠀⠀⢀⡾⢿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠃
⠀⠀⠀⠀⠀⠘⢷⣿⣿⣿⣿⣿⣿⡏⠀⠙⢾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠷⠋⠀⢹⣿⣿⣿⣿⣿⣿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣟⠛⠈⡁⠀⠀⠀⠘⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⡁⠀⠀⠀⠈⠁⠛⣻⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⠀⠀⠀⠀⠀⠌⢂⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡏⠰⠡⠀⠀⠀⠀⠀⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⠀⢿⢄⠀⠀⠀⢐⡏⠻⣷⣄⠀⠀⠀⠀⣠⣾⠟⠙⡄⠀⠀⠀⡠⡷⠀⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠈⠀⠃⡂⠀⠘⠀⠀⠈⠻⣷⣀⣠⣾⠟⠁⠀⠀⠃⠀⢰⠘⠀⢁⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣔⣤⠐⢄⠀⠀⠀⠀⠀⠹⣿⣿⡟⠒⠀⠀⠀⢀⡠⠂⣤⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣌⠳⡀⠀⠀⠀⠀⢻⠻⢿⣦⡀⠀⢠⠞⣡⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⡉⠢⠀⠀⠀⠈⡄⠀⣱⣝⢶⣁⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⠙⠿⣷⣄⠐⢄⠀⠀⠀⠀⠀⠈⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠀⠀⠀⠀⠀⠀⢀⣼⣿⣶⣌⠑⢄⠀⠀⠀⠀⠀⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠟⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠻⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⡄⢀⣀⣠⣶⣾⡿⠛⠁⠀⠀⠀⠀⠈⠛⢿⣷⣶⣄⣀⡀⢀⠀⠘⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠟⠁⣀⠤⠿⣿⢿⣽⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣭⡻⣿⠯⠤⣄⠈⠻⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠊⣡⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣌⠓⠀⠀⠀⠀⠘⠻⣦
⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⡀⣀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⢀⠀⠀⠀⠉⢷⣄
⠀⠀⠀⠀⢀⡞⠁⠀⠀⣀⣤⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣶⣤⣀⠀⠀⠈⢳⡀
⠀⠀⠀⣠⣿⠀⠀⠀⡈⢡⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡏⠁⠀⠀⠀⣿⣆
⠀⢀⡼⠋⠀⠀⣀⣤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣀⠀⠀⠙⢧⡀
⣰⠏⠀⢠⣶⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⡄⠀⠹
⠹⢄⡴⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠦⣠

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡟⠛⢋⠙⠿⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠉⠀⠀⠉⠀⠀⠀⠀⠉⠋⠳⠤⢀⣀⣀⡀⠠⢤⡔⠛⠃⠀⠉⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠀⠀⠈⠉⠃⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣄⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠠⡀⠀⠀⠀⠻⣧⡀
⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⢠⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡱⣀⡀⠀⣼⠗⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠈⣿⡷⣤⠄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢳⠄⢻⣷⣬⡸⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢹⣿⣿⡟⢁⣄⣿⣵⡄
⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⢩⣿⣿⣤⣿⣯⡍⡣⠬
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠠⢭⣿⣿⣿⣯⣈⣥⣶⠆
⠀⢸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠸⣿⣿⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠹⣿⣿⣿⣿⣿⣷⡈⠳⡆
⠀⠀⣿⣿⣿⣧⣄⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠉⢿⣟⠀⠀⠀⠻⣿⣿⣿⣿⣿⡃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠹⣿⡿⣯⣽⣻⡝⢷⠄⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣦⣌⣿⣇⠀⠾⠀⢹⣿⣿⣿⣿⣷⣄⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠂⣠⡏
⠀⠀⠀⢸⣷⣬⣙⠻⣿⣄⡤⢦⣦⡀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢹⢿⣿⣿⠿⣿⣷⣤⣈⠛⠀⠀⠀⢀⠀⢀⣼⡶⣏
⠀⠀⠀⢸⡟⢿⣟⣿⢷⣶⢶⡶⢶⣶⣾⠿⠿⠿⠿⠛⢉⡉⠛⢿⣿⣿⠀⠀⠀⠀⢚⣿⣿⡘⠁⠝⠛⠛⠿⠿⠷⠶⠿⢿⣏⡨⠆⠸⣧
⠀⠀⠀⢸⣧⡈⠫⡀⠈⠁⠂⠀⠀⠀⠀⠃⠠⠀⠠⠀⠀⠀⢠⢼⣿⣿⠀⠀⠀⠀⠀⠻⢿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢸
⠀⠀⠀⢸⣿⠿⣆⠀⠠⠀⠀⠀⠀⠐⠛⠗⠀⠀⠀⠀⠀⠀⠀⠘⠛⡿⠆⠘⠀⠀⠀⠑⠀⠉⠙⠛⠛⠒⠲⣦⣒⣲⡶⠟⡠⠄
⠀⠀⠀⠈⣿⣦⣌⠐⠀⠀⠀⠒⠒⠢⠾⣤⣤⠤⠤⢤⣤⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠴⠶⣶⣤⠖⠋⠀⠀⠉⠁⠀⠚⣧⠀⠀⢠⡄
⠀⠀⠀⠀⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡖⠁⠀⣀⣶⣶⠆⠰⢆⠴⠶⠒⠶⠦⣤⣤⡤
⠀⠀⠀⠀⣿⣿⣿⣿⢿⣟⣿⣧⣷⣶⣤⣦⣴⠶⡦⠀⠠⣶⣶⢶⡞⠉⣣⣶⠿⠛⠁⠀⣠⡰⠶⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⢸⣿⣍⣿⠀⠛⠛⣿⡏⠉⠫⠀⢀⡀⠀⠀⠀⡀⠉⣈⡴⠛⠋⠀⠀⢀
⠀⠀⠀⠀⢸⣿⡇⢿⠀⠓⠀⠛⠀⠀⠀⠀⣟⡁⠐⠲⠀⠀⢨⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⢀⠀⠀⢨⣿⠋⠀⠰⠀⠀⠀⠠⠀⠀⠀⣿⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠒⠛⠀⠀⣈⠁⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⣿⣶⡀⠀⠀⠀⠀⠀⣠⣄⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠻⣿⡉⠉⠀⠀⢠⡞⠁⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠋
⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⣿⡇⠐⡇⠀⡾⠁⠀⢠⣷⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠞⠁
⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⡤⢤⡞⠀⠀⠀⣸⣿⠀⢠⡖⠀⠂⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠰⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⣿⡇⠀⠀⠀⣿⣿⡇⠀⠛⠀⢸⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠸⣶⣷⠀⠀⠀⠀⠀⠀⠀⣤⠙⢿⣿⣿⣿⣿⡿⣷⡀⠀⠀⣿⣿⡇⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⢸⠇⠀⠈⢿⣿⣿⣿⡇⠈⠷⡀⠈⣿⣿⡇⠀⠀⠀⢉⣿⠂⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣿⣷⠀⣄⠀⠀⠀⢀⠀⠀⠀⠈⣿⣿⣿⣿⡀⠀⢱⡀⠻⠿⠁⢠⡎
⠀⠀⠀⠀⠀⠸⣿⣸⣿⠀⠀⠀⢸⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⢀⠻⣤⣤⠀⠠⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢹⣿⣯⠀⠀⠀⠟⠀⠀⠀⠀⠀⢻⣿⣿⣧⠆⠈⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠸⣿⠙⠀⠀⠀⠀⠀⠐⠀⠀⠀⢸⣿⣿⣿⣤⣶⠶⠶⢤⠄⣠⡄⣀
⠀⠀⠀⠀⠀⠀⠀⣧⢸⠀⠀⠀⠀⢀⠀⡠⠀⠀⢘⣿⣿⣿⢿⡇⠀⠀⠘⠀⠃⠀⠻⠃⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⣿⣼⠀⠀⠀⠀⢎⠀⠀⠀⠘⠆⣿⣿⣿⢤⠡⠒⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣿⣿⡷⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣾⠿⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠘⠀⠃⠀⠀⣿⣿⣿⠋⠉⠉⠛⠛⠛⠋⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣋⣙⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⠿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣀⣀⣀⣀⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠘⢶⣄⠀⠀⠸⣇⠀⠀⠀⡔⠀⠀⡠⠀⢀⣠⣴⡶⠋⠠⡔
⠀⠀⠀⠀⠀⠀⠀⠻⣶⣄⠀⣿⠀⠀⣾⠀⠠⠊⢀⣴⡿⠁⢨⡿⠂⠀⠆
⠀⠀⠀⠀⠀⣀⣀⣀⡈⠻⣧⣈⠂⣸⠇⡰⠁⢀⣿⠁⢀⣼⠋⠀⠀⢠⡁
⠀⠀⠀⠀⠀⠀⠉⠉⢛⣛⣻⣿⣾⣧⡴⠁⠀⣼⠁⢠⣾⠇⠀⠄⡰⠿⣄
⠀⠀⠀⠀⠀⠐⠒⠛⠋⠁⣰⡟⠀⡼⠁⠐⠛⢉⣹⡿⠋⠀⠀⣸⣷⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠄⠀⡄⠀⠀⠀⠺⠋⠀⣀⣤⡄⣿⠙⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠁⠀⠀⠀⠀⠀⠀⠁⣽⣇⣿⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⣿⡼⡀⠀⠀⠀⠀⠀⠀⢊⣧⣿⠀⣰⣿⠏⠀⠀⠀⣠⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⢿⣿⣧⠀⠀⠀⠀⠀⢀⣿⣿⠹⣶⣿⡟⠀⢀⢐⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣄⠘⣿⣿⣆⠀⠀⠀⠀⢸⣿⣿⢠⣿⡟⢀⠔⣩⠞⠁
⠀⠀⠀⣀⣀⣀⡀⠀⠀⠈⠻⣿⣷⣜⣿⣿⡆⠀⠀⠀⠀⠻⢫⣾⡿⠀⣼⣾⠃
⠀⠀⠀⠉⠙⠿⣿⣿⣶⣤⣀⠈⠻⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣥⣿⡿⠁⢀⣠⣤⣤⣤⣦⠶⠶⠂⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣶⣾⣟⣿⣿⡆⠀⠀⠀⠀⣼⣯⣴⣯⣴⣿⡿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣉⣿⡿⢭⣿⡇⠀⠀⠀⠀⠀⠙⢿⣽⣯⣥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠿⠉⣡⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⣿⣷⣤⡀
⠀⢀⠀⠀⠀⠀⠀⠐⠛⠛⠉⠀⣠⣴⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠦⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠁
⠀⣉

⣿⣷⣦⣀
⣿⣿⣿⣿⣷⣦⣄
⢿⣿⣿⣿⣿⡿⠻⣷⡢⣀
⢈⡋⠿⠝⠯⠀⠃⠀⠀⠙⠷⣦⣄
⠀⠀⠀⠐⠃⠀⠀⣀⣣⣶⣤⣽⣿⣿⣶⣄⠀⠀⠀⠀⣠⠤⠤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠠⠀⠀⠀⠀⠚⣹⣿⣿⣿⣿⡾⣿⡿⠁⠀⠀⠀⠇⣤⣴⣀⣤⣀⠈⠑⢦⣀⡀⠀⣀⣴⠾⠿⠦⡀
⠀⢀⠃⣀⠠⠤⠐⠒⡿⠟⠛⣋⣉⣽⠟⠀⠀⠀⠀⠀⠾⠿⠋⠈⠉⠉⠓⢶⡏⠛⠻⣿⣿⡏⠀⠀⠀⣹⣦⡀
⣼⣿⣿⣾⣿⣷⣶⣦⣶⣾⣿⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣠⣿⣿⣿⣦⣤⣾⡿⣿⣿⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠈⣿⣿⣿⡿⡟⠉⠀⠘⠃
⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⡟⠁⣠⠶⢄⣠⠒⢁⡼⠒⣆⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠋⢀⡾⠛⠁⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠙⠛⠿⣿⡿⠀⠀⠻⠞⠋⢿⣷⡏⢁⣴⢋⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡟⢁⣼⠃⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠛⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⡏⠀⠀⠀⣿⠶⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠈⢻⡇⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠁⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠙⢷⡀⣿⠀⠀⠈⠃⠀⠀⠀⠉⠀⠀⠀⠿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠘⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡴⠖⠲⡄⠀⠀⠀⠀⠀⢻⡆⠀⠀⢹⣿⠀⢠⡶⠛⠛⠛⢶⣴⠞⠛⠛⠳⣦⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠳⣆⣀⡀
⠀⠀⠀⠀⣠⡴⣴⡇⠀⠀⣿⠛⢻⡆⠀⠀⠘⣷⠀⠀⠈⠟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣿⠀⠀⠀⠀⠀⢠⡶⢶⣿⠀⠀⣿⠋⠙⣧
⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⣿⠀⢸⣇⣀⡀⠀⢻⡆⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⣿⣀⣀
⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⣿⠀⢸⡏⠉⢻⡆⠘⣷⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⢀⣠⠾⠁⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⣿⠉⢹
⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⣿⠀⢸⡇⠀⢸⡇⠀⢹⣆⠀⠀⠀⠀⠀⠀⠙⢷⣄⢀⣴⠟⠁⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠸
⢀⣤⡀⠀⣿⠀⢸⡇⠀⠀⣿⠀⢸⡇⠀⢸⡇⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⣼⠃⠀⢀⣤⣄⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⣿
⣿⠁⠹⣦⣿⠀⠈⠃⠀⠀⠁⠀⠘⠇⠀⠈⣧⠀⠀⠀⠙⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⢿⡅⠙⣧⣿⠀⠀⠉⠀⠀⠀⠀⠀⠛
⢸⡇⠀⢻⣿⠀⢀⣠⣤⣀⣀⣤⣤⡀⠀⠀⣿⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢸⣇⠀⠸⣿⠀⢀⣠⣤⣄⣀⣤⣤⣄
⠈⣷⠀⠈⠿⢀⡿⠁⠀⠙⠋⠁⠈⢹⡆⠀⣿⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⠀⠀⠻⠀⣿⠁⠀⠈⠋⠀⠀⢹⡇
⠀⢸⡄⠀⠀⠈⢻⣄⠀⠀⠀⠀⢀⡾⠃⢀⡿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸⣆⠀⠀⠀⠻⣦⡀⠀⠀⠀⢠⡾⠃
⠀⠈⣷⡀⠀⠀⠀⠙⠷⣄⣠⡾⠋⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠈⠻⣦⣠⡾⠋⠀⠀⢰
⠀⠀⠘⣷⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣸
⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢹⠇
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣾⣷
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠉⠙⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠛⠋⠉⢿⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⣽⣇⡀⠀⠀⠀⠀⢀⣴⠶⠶⠶⠶⠶⢶⣤⣤⣄⣀⣤⣿⡄⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⣼⠏⠉⠛⠻⠿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⣷⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⢠⡾⠋⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢸⡇⠀⠀⢰⡟⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠸⣧
⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢸⡇⠀⠀⣾⠃⠀⠀⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠘⠳⢦⣄⡀⢀⣼⡃⠀⠀⠀⠀⠀⠀⠸⣧⣤⡾⠃⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢀⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠻⣦⣠⡶⢶⣄⣀⣀⡀⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠙⢷⣄⠀⠀⢀⣴⠟⠻⠶⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡋⠀⣠⡿⠋⠉⣻⡄⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⠀⠀⠉⠻⣾⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢾⡟⠁⢀⣴⠟⠛⣷⠀⠀⠀⠻⣶⣄⠀⠀⠙⢷⣦⡀⠀⢘⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣴⡿⠁⠀⣰⣿⣀⢶⣄⠀⠀⠙⢷⣄⡀⠀⢹⡿⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣾⠋⠈⣹⡆⠙⢷⣤⡀⠀⢹⡿⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠿⣦⣀⣀⣿⠛⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠀⠀⠀⢀⣤⣾⣿⣿⣧⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⢿⣿⣿⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣤⣤⠺⣭⣝⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠑⠒⠀⢀⡀⢀⠄⠠⠀⠀⠀⠈⢿⣾⢷⠀⠀⠀⢠⣴⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣴⠀⠈⡷⣬⡅⠀⠀⠀⠀⠀⠀⠀⠒⠂⣀⠙⠒⠂⢀⣠⠤⠟⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⣦⣴⣦⣾⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠐⠁⢀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢘⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠹⠆⠀⢸⣶⠎⠁⠙⣦⠀⡀⠀⢀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⡤⠖⠋⠘⠀⠀⠙⠋⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠀⠀⠶⠈⠳⠿⠚⠋⠀⣠⠀⠀⠀⠀⠀⠀⠀⢘⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⡤⠴⠖⠛⠁⢀⡤⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠛⠉⠀⢀⣀⣤⡴⠀⣰⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠠⡀⠀⠀⠀⠀⠀⠠⠀⢀⢀⡀⠖⠲⠟⠗⠀⠀⠀⣀⠟⠁⢡⠀⠀⠀⠀⠀⢀⣿⣿⣷⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢯⢹⡄⠳⣁⢠⣄⠰⠂⠁⠀⠣⠀⠀⠈⠒⢤⣀⠀⠀⠰⠛⠁⠀⠀⠈⣆⠀⠀⠀⠀⣼⣟⢉⣥⣀⡍⢿⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠸⣎⢣⣙⢶⣛⠿⠃⠀⠀⠠⠄⠀⠰⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠉⠀⣀⣼⣿⣿⣿⣿⣿⣿⣦⢧⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠵⡘⢢⠘⠿⡹⠆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⡙⢣⡈⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢰⡾⢷⠀⠀⠉⠛⠿⢿⣿⣿⣿⢶⡝⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠉⠉⠿⣿⣾⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣉⠉⠉⠉⠛⠛⠲⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣀⡈⠉⠉⠙⠒⠀⠀⠀⠀⠘⢧⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⡈⠉⠓⠲⠤⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣍⡉⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠉⠛⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢻⡟⢹⡷⠶⣄
⠀⠀⠀⠀⢦⣀⣠⡴⠟⠁⠀⠀⠀⠀⠀⢀⣰⣟⠀⠀⣀⡠⠤⠀⠤⣤⡀⠀⠀⠀⣴⠋⢀⡞⠀⡼⠁⣸⠏⣷⡀
⠀⠀⠀⢠⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠈⠉⠉⠀⠀⠀⠀⢠⡟⠛⢦⣀⡜⠁⠀⡜⠀⡸⠁⢀⠏⢀⡏⠙
⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⣸⠃
⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠶⣴⠇⠀⡆⠀⣸⠁⠀⠀⠀⣠⠤⣄⡀⢸⠇⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⢸⠀⣸⠀⢠⠃⢀⠆⠀⡐⠁⠀⠈⠻⣧⣀⣤⡤⠶⠶⠒⢦⣴⠋⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠛⠳
⠀⠘⢴⡇⢀⡎⢀⡾⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⢀⣴⠟⠉
⠀⠀⠀⠙⠛⠷⠾⠷⠛⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⢀⡀⠀⠒⠦⣤⣍⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢿⡆⠀⠀⠀⠀⠀⠉⠙⠒⠤⣄⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⣄⣀⠀⠈⠒⠒⠤⢤⣀⣈⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⡶⠦⠤⠤⠌⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⢿⠟⢿⠟⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣭⣽⣤⣽⣳⢿⣷⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⣿⡿⣿⡿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣤⡀⠀⠀⢿⣿⣤⣿⣦⣾⣦⣾⣿⠇⠈⠉⢀⣤⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣆⠀⠈⣿⣿⡿⢿⠿⠿⠛⠁⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⠀⢀⣀⣀⢀⣀⣀⢀⠀⣀⢀⣀⡀⢀⣀⡀⠀⢀⡀⢀⣀⡀⢀⣠⣀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠟⣰⠿⣇⢸⡶⣟⠀⣿⠀⣿⠳⣿⢸⣗⡂⢸⠶⡟⠀⢸⡇⢯⣀⡀⢿⣀⣽⢸⡟⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠈⠀⠈⠀⠈⠀⠈⠀⠀⠀⠉⠁⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⢁⡈⠉⠁⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣴⠛⠛⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠈⠛⢻⡆⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢻⣄⠀⠚⢻⠟⠁⠀⣠⡟⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠙⢷⡀⠀⠀⠙⠀⢤⣤⣤⡴⠞⠋⠀⠀⢀⡾⠋⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡄⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢠⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣸⣀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣀⣇⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣯⢴⣰⢰⣇⢴⢰⢲⢴⣶⠔⣦⢰⣆⠀⣶⣲⢠⣶⢰⡆⣟⢰⢲⣤⣶⠆⡖⢲⡆⡟⣦⣾⢰⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠀⠁⠀⠀⠀⠀⠁⠀⠁⠈⠀⠀⠉⠁⠀⠈⠀⠀⠈⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠘⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠈⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢠⣤⣤⣤⣴⡀⠀⠀⠀⠠⠃⠀⠀⠀⠀⡐
⡀⠀⠀⠀⠀⠈⠀⢄⠀⠀⠀⠀⠀⠀⠢⠀⣶⣫⠭⠾⡧⠀⠠⠋⠁⠒⠠⣀⠀⠔
⠀⠁⠒⡀⡀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠁⠻⠀⠀⠀⠐⡀⢈⡒⠒⠄⡀⠀⡋
⠀⠀⠀⠇⠀⢈⠐⡤⣀⠀⠀⠑⢄⠀⠀⠀⠐⣆⢀⣄⡀⣱⠋⠀⠀⢀⠞⠊
⠀⠀⢀⠂⠈⠀⠀⠟⠾⢉⠖⠤⠄⠈⠢⢀⣄⠭⢟⡾⠞⠉⠀⠀⢠⡃
⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣨⠡⡴⠋⠀⠀⠀⠀⢠⣗⣅
⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⡿⢋⣤⠂⢀⠀⠀⢀⢷⡥⣏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⣶⡿⢃⣴⠏⣰⢃⠎⣜⢽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠒⠱⢵⣯⣋⣼⣇⠎⡜⠉⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠎⠀⠙⣆⠀⠘⡿⠯⣧⠉⠀⣠⣾⠟⠙⢆
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠈⢄⠀⠀⠸⠒⠈⠀⢠⣽⣶⢶⢿⣧⡀⠀⠀⠑⢄
⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢈⡀⠙⠉⠀⠀⠀⠈⠘⠉⠁⠸⠢⡳⣦
⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢌⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠟⠁⠀⠀⠀⠀⠀⠀⠀⠘⢠⠀⠀⠀⠁⠀⠀⠑⢍⠣⣠⡀
⠀⠀⠀⠀⢀⠄⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡀⠀⠀⠀⠑⠈⠻⡆
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⠒⠀⠠⠤⠀⣀⠠⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢀⡉⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⡀⠀⣀⠔⠃⠀⠤⣄⠀⠀⠀⠀⠀⠀⠈⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⢀⡠⠤⠒
⠀⠀⠀⠀⠀⠀⠈⠀⠎⠈⠠⣀⠀⠀⠀⠂⠁⠢⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠐⢄⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⡜⠀⠀⢠⠦⡄⠀⠀⠀⠠⡀⠀⠀⠑⠀⠀⠀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡃⠀⠀⡆⠀⢨⠀⢄⠀⠀⠘⢄⠀⠀⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⡀⠀⠠⡀⠀⠑⢄⠀⠀⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣦⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⣿⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⣿⠀⣿⡇⣿⣿⣿⣿⣿⢠⣶⡆⢀⣀⣴⣾⣿⣿⣯⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⢸⣿⣾⣿⣶⣿⣶⣿⡇⣿⣿⣿⣿⣿⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣭⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣼⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣭⣍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⢸⣿⡇⠻⠿⢿⣿⣿⣿⣿⣿⣿⣩⣩⣭⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠏⠀⠋⠉⠉⠉⠋⠀⠀⠀⠀⠀⢀⣍⣉⣉⣉⣉⣉⣉⣉⡙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣌⣉⣉⣉⣉⣉⣉⣉⣙⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⡀⣰⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣛⣙⣛⣿⣿⣿⣿⣿⣿⣷⣶⣦⢸⣿⡇⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡟⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⣿⣿⣿⣿⣿⢸⣿⠿⣿⠿⣿⡿⣿⡇⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⣻⣿⣿⡿⠟⠉⠁⠸⠿⠃⣿⣿⣿⣿⣿⢸⣿⠀⣿⠀⣿⡇⣿⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⣿⠀⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠻⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢴⡟⢻⡶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⠙⣿⢸⡇⢸⡇⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠾⠷⠶⠶⠶⠶⠶⠾⠷⢦⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠋⠈⠁⠈⠁⢸⡇⢀⣤⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠾⢷⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⢸⡿⠋⢉⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠾⢷⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⠀⠀⠀⠀⠀⠀⢀⣴⠟⠗⠲⠶⠶⠶⠶⠶⠿⠶⢶⣄⠀⠀⠲⣶⠶⢶⡶⠲⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠻⢦⣄⠈⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠛⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣤⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⡀⠙⠳⣦⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⣷
⠀⠀⠀⠀⠀⠀⣠⡴⠶⠾⠷⠶⠿⠦⠀⠀⠙⠷⢶⣶⠶⠶⠶⠶⠶⠶⢶⣴⠟⠁⠀⠀⠀⠀⠀⠀⣿⣶⡄
⠀⠀⠀⠀⠀⢀⡿⠶⠶⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠘⢷⡶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣁⣠⣾⡇⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠘⢷⡶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠛⠉⢸⡇⢀⡀⢀⡀⣠⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠳⢶⡶⠶⠶⠶⠶⠶⢶⡶⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇⢸⡇⢸⡇⣿⣄⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⣧⣼⠗⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠛⣆⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢻⡀⢸⡇⢸⡴⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⡇⠸⡇⠘⣇⠀⣷⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢷⠀⣷⠀⢿⠀⢻⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡁⠀⠰⡏⠳⣸⡀⠘⠀⠈⠀⠈⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣄⠻⠃⠀⠀⢻⡀⢹⡇⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠳⣌⢧⡀⠀⠀⠀⢳⡀⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠈⢧⡙⢦⠀⠀⢀⣷⡀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣮⠷⠖⠉⠀⠙⢲⣀⣀⣀⣤⠤⠤⠿⣄
⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⣀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⣼⠭⠤⠶⠒⠒⠛⣉⡉⡇
⠀⠀⠀⠀⠀⠀⠰⣇⣤⠞⠁⣠⠶⠀⠀⠀⠀⢀⡀⠀⡴⠳⢤⣀⡀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠛⠃⢿
⠀⠀⠀⠀⠀⠀⢠⡞⢁⣴⠞⠁⣠⠞⠃⣠⢶⠏⢀⡞⠁⠀⠀⣸⢳⣄⡤⠶⠟⠶⠶⠤⠤⠤⣤⣄⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠈⡽⠋⢁⣴⠟⢁⡤⠚⢱⠇⣰⠋⠀⠀⠀⢰⢃⡞⠀⠀⠀⠀⠀⠰⠶⣤⣤⠤⠞⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠳⠞⢿⣁⡴⢯⣀⡀⠈⠉⠁⠀⣀⡀⢀⡏⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣈⠙⠓⠦⢤⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣄⣛⠃⣼⣰⣃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠙⠲⠶⣤⣌⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠁⠈⠳⢤⣀⠀⠀⠀⠀⢀⡈⠙⠓⠶⣤⣄⣈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⢤⣀⡉⠛⠶⢦⣤⣈⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠴⠃


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⣹⣏⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⡇⠀⢸⡗⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠈⡇⠀⢸⡇⠀⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⡇⠀⢸⡇⠀⣿⠁⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⢸⡇⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢤⡀⢻⠀⠀⡇⠀⠸⠇⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢳⣼⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠲⣄⣀⠀⠀⢻⠀⠀⣿⢀⡶⠋⠙⠲⠞⠋⠛⢦⠀⣿⠀⠀⠀⠀⡀⡶⠛⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢹⡇⠀⡇⢸⣇⠀⠸⡇⠀⠈⠸⡄⠀⠀⠀⠀⠀⠀⣸⠀⣿⠀⠀⠀⣾⠉⡇⠀⣯⠀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⡇⢸⡏⢹⡄⢷⠀⠀⠀⠙⢦⣀⠀⠀⣠⠞⠁⠀⡏⠀⠀⠀⣿⠀⡇⠀⣿⠀⡏⠙⡇
⠀⠀⠀⠀⠀⠀⢀⣀⠀⡇⢸⡇⠀⡇⢸⡇⢸⡇⠈⢧⠀⠀⠀⠀⠈⠳⠞⠁⠀⠀⢰⠇⠀⣀⠀⣿⠀⡇⠀⣿⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⢺⠉⢷⡇⠈⠁⠀⠁⠘⠃⠀⡇⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢸⡍⢳⣿⠀⠁⠀⠀⠀⠃⠀⣷
⠀⠀⠀⠀⠀⠀⠘⡇⠘⡇⣴⠚⠳⠖⠛⢦⠀⡇⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣧⠀⠿⢰⠚⠛⠞⠛⢳⡀⣿
⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠹⣄⠀⠀⣠⠟⢀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡄⠀⠘⢦⡀⠀⣠⠞⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⠓⠞⠁⠀⢸⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢳⡀⠀⠀⠙⠚⠁⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢀⡤⠤⠤⣀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣠⣶⣶⣦⡀⠊
⠀⠀⠀⠀⠀⠀⣀⣀⢠⡏⠀⠀⠀⠈⡆⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣼⣿⣿⣿⣿⣿⡄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠼⢧⠀⠀⠀⣠⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠹⣿⣿⣿⣿⡿⠡⣄
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢳⣀⣸⠁⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⡠⠂⠘⣿⣿⡟⠀⠲⡌⢳⡀
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠐⣶⡖⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢒⣒⠂⠀⠀⠀⠀⢷
⠀⠀⠀⢀⠖⠛⠓⢆⠀⢀⡤⠤⠤⡀⠀⣰⠋⠉⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣤⡀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣴⣶⣦
⠀⠀⠀⢸⡀⠀⠀⣸⠀⡟⠀⠀⠀⢸⠀⣳⣀⠀⣠⣏⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡿⠀⠀⣼⣿⣿⣿⡄⠀⠸⣿⣿⣿⠃
⠀⠀⡴⠋⠉⠛⠋⠉⣳⠿⢤⣀⡠⠾⢏⠀⠈⠉⠁⠀⢳⠀⠀⠀⠀⠀⠀⣠⣦⣭⣯⣵⣦⠀⡻⣿⣿⠿⡃⢰⣾⣶⣿⣶⣷⡄
⠀⠀⡇⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢣⣾⣿⣷⣶⣾⣿⣦⢻⣿⣿⣿⣿⣿
⠀⠀⠧⣤⣀⣀⣀⣼⠀⠀⠀⠀⠀⠀⠀⢸⠤⠤⠤⠤⠼⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣺⣿⣿⣿⣿⣿⣿⣿⡸⣿⣿⣿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠸⢤⣤⣀⣀⣀⣠⣤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠈⠙⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⣞⡽⠓⠲⠤⣤⠞⠉⠉⠉⠉⠙⠛⠒⠒⠒⠚⢯⢳⡀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⢠⢎⡞⠀⠀⠀⡰⠃⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡳⡀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⣠⢫⠏⠀⠀⠀⡞⠁⢀⣠⠞⠉⠉⠓⢤⡀⠀⠀⠀⠀⠀⠀⢳⡹⡄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⣰⣣⠋⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⡹⣽⣄⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢳⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⡜⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡴⠁⠀⣳⠊⠉⢢⠀⠀⠀⠀⠀⠓⢤⡀⠀⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⡜⠁⠀⡰⠋⠙⡆⠀⠓⢦⡀⠀⠉⢳⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣠⡞⠁⠀⡴⠛⠲⣄⠀⠈⢳⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⣞⠁⠀⡴⠋⣳⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠊⠙⠒⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⠤⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠙⢹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣤⢴⠻⠻⠃⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡔⠀⠀⠀⠈⠻⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⢀⠀⠙⠆⠀⠐⢄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢳⡀⠀⠀⠀⣀⣤⡴⠆⠒⠒⠻⡆⠈⡆⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣤⠔⠛⠋⠁⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⡀⠈⠁⠀⠀⠠⠖⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⠞⠁⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠠⡀⠀⠀⢀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠖⠁⣠⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠀⠱⣄⠀⠈⠂⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡠⠊⠁⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠒⠂⠀⢀⡠⠄⠀⠉⠀⠀⠐⢮⡀⠀⠀⠀⠀⠀⠙⣤⡀⠀⠀⣀⠀⠀⠒⠁⠀⢀⡠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⡐⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠃⠀⠀⠀⠀⠀⠀⢀⠀⠈⠻⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠊⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠱⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⢀⡴⠋⠀⠈⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢀⣀⣤⠔⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣆⠀⢷⣠⠤⠤⠤⠞⠛⠉⠀⠀⠀⠈⢳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢦⠀⠀⢀⠀⠘⡄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣣⡀⠀⠂⠀⢈⡷⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⢠⢠⣠⠦⠗⠋⠁⠈⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁


⣿⠿⠿⠿⢿⡛⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣛⣛⣻⣛⣿⣟⣻⣛⣿⣛⣻⣟⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣷⣄⠀⠀⣼⣿⠛⠛⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠛⠛⠛⠛⢿⣷
⠀⠀⠀⣿⡏⠀⠀⠀⠀⣀⣤⣿⣿⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⣼⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣿⡏⠉⣿⣿⣿⣶⣄⠀⢸⣿
⠀⠀⠀⣿⡇⠀⢀⣴⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠙⣿⡇⠀⣿⣿⠀⣿⣿⣤⣼⣿
⠀⠀⠀⣿⡇⠀⢸⣿⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⡇⠀⣿⣏⠀⣿⣿⠛⢿⣿
⠀⠀⠀⣿⡇⠀⠀⠙⢻⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⡇⠀⣿⣿⠀⣿⣿⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠸⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⢠⣾⡿⠿⣿⣿⠀⠀⠀⠀⣀⡀⠀⢸⣿⡇⠀⣿⡇⠀⣿⣿⠀⣿⣿⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⣩⣽⣿⡿⠟⠀⠀⠀⠀⣠⣾⡿⠁⠀⣿⣿⠀⠀⢰⣿⡿⢿⣷⣸⣿⡇⠀⣿⡇⠀⣿⣿⠀⣿⣿⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⢁⣀⣤⣤⣶⣾⣿⠿⠛⠁⠀⠀⠘⢿⣿⣾⣿⣿⡆⠈⢿⣿⣿⡇⠀⠉⠁⠀⠉⠁⠀⠈⠁⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠈⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟
⠀⠀⠀⠀⣠⣾⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠈⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⢀⡀⠀⠀⡀⠀⠀⡀⠀⢸⣿⣿⣷⡀⠸⣿⣿⡿⣿⣷⣄⠀⠀⠀⣠⣶⣿⡿⠿⠛⠛⠉⢁⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⣿⣿⠀⣾⣿⠀⢸⣿⠀⢸⣿⡏⢿⣷⣶⣿⠇⠀⠀⣿⣿⠀⢀⣾⡿⠋⠀⠀⠀⠀⣴⣾⣿⣟⣋⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⣿⣿⠀⣿⣿⠀⢸⣿⠀⢸⣿⡇⠀⠉⠉⠁⠀⠀⠀⣿⣿⣶⣾⣿⠃⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡆⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⣿⣿⠀⣽⣿⠀⢸⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣷⣄⠀⠀⢸⣿
⠀⠀⠀⠀⣿⣷⣤⣿⣿⠀⢾⣿⠀⢸⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢙⣿⡇⠀⢸⣿
⠀⠀⠀⠀⣿⡿⠛⣿⣿⠀⣹⣿⠀⢸⣿⡀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠹⢿⣿⣿⣿⡀⣸⣿⡿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⡿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⣿⣿⠛⠋⠁⠀⠀⠀⣸⣿
⠀⠀⠀⠀⢿⣷⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣿⡟⠀⠀⠹⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣿⠟
⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⢠⠔⠰⢠⡆⢊⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢠⣤⣮⣭⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠑⢈⡊⠒⠘⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣤⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠀⠛⠛⠃
⠀⠀⠀⢲⢤⡠⠆⠀⢦⢠⡄⠆⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⡰⢸⠰⠲⠆⠞⠈⠀⠆⠳
⠀⠀⠀⠛⠈⠃⠘⠁⠈⠉⣂⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣄⣠⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠀⢀⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⣿⢻⡏⣿⢩⡏⡻⡟⣻⡏⣿⢽⢿⣿⡯⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣵⣷⣾⣵⣽⣼⣧⣧⣷⣿⣷⣿⣾⣶⣿⣧⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⢰⣶⣶⡶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠛⠛⠋⠛⠿⠿⠛⠋⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⠦⣄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣽⣿⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⡼⣠⠤⡀
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⢛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠗⢦⡣⠤⠏
⠀⠀⠀⠀⠀⠀⢠⠀⠀⡄⢠⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠒⠚
⠀⠀⠀⠀⠀⠀⠈⠘⠐⠑⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⢸⡆⡷⠴⡄⣾⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠙⠋⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠘⠀⠁⠈⠀⠁⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⢤⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠙⠂⢙⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣇⢰⡘⢱⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠁⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠐⢲⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢠⠈⡌⢫⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠁⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢀⣖⣒⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣡⢔⡳⢸⡇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠅⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⣄⠀⢘⡙⠛⡛⡋⡋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⣀⢍⠉⣡⢠⠄⡄
⠀⠀⠀⠀⠀⡯⢸⠇⠣⠅⠹⠻⠃⢱⢘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠟⠡⡘⠄⠏⠀⡇⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⣄⣤⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣿⣿⣿⡿⣿⢻⡿⣿⢻⡿⣿⣻⣿⢿⡟⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣿⣿⣽⣽⣷⣿⣸⣿⣶⣾⣷⣿⣾⣷⣽⣾⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠻⠟⠛⠻⠿⠿⠟⠋⠈⠉⠉
⠀⠀⠀⠀⠀⣿⣿⣿⣥⣽⣿⡇⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣿⣯⣷
⠀⠀⠀⠀⠀⣀⠈⠀⡀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢿⣷⣿⣾⣭⠟
⠀⠀⠀⠀⠀⠎⠰⠴⠳⠎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⢶⠖⠓⣦⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⡉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⣀⣶⣙⠀⠀⠀⠀⠀⢸⡎⣿⡀⣼⣀⢺⢣⠹⡄⢹⢔⡦⣿⡀⣼⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡍⠿⠉⠀⠀⠀⠀⠀⠘⠁⠁⠉⠀⠉⠈⠈⠀⠁⠈⠀⠀⠁⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠒⢰⡄⣀⢠⡔⠐⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠘⠢⠻⠃⠸⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣾⣴⣶⣦⠀⠀⠀⠀⠀⠀⣴⣿⣅⣼⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣉⠉⠁⠀⢀⣠⣀⠀⠀⠉⢉⡹⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⢠⣤⠏⠉⠉⣷⣄⠾⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠘⠛⣧⠀⢰⡟⠋⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠛⠛⡆⢀⡀⢿⣦⠿⣶⢀⠀⠘⠛⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⠀⢼⡇⠁⠁⠀⠀⠀⠀⠈⢐⣀⢀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠶⠿⠿⠇⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠿





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

⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠸⠿⠟⠀⠀⠀⠀⠘⠿⠿⠀⠀⠀⠀⠀⠿⡿⠃⠀⠀⠀⠀⠀⠘⠿⠿⠃⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠻⠿⠇
⠀⠀⣴⣿⣼⣼⣷⡄⠀⣰⣿⣼⣼⣷⡄⠀⢠⣾⣧⣧⣿⣆⠀⠀⠀⢠⣾⣶⣶⣷⡄⠀⢠⣾⣧⣧⣿⣦⠀⢀⣾⣧⣧⣿⣧
⠀⢀⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣧⠀⣸⣿⣿⣿⡿⢿⣶⣤⣶⡿⢿⣿⣿⣿⣷⠀⣸⣿⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⡄
⠀⢸⡟⣿⣿⣿⡟⣿⠰⡿⣿⣿⣿⡏⣿⠀⣿⢻⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⡏⢿⠀⣿⢿⣿⣿⣿⢻⡇⢾⢿⣿⣿⣿⢹⡇
⠀⠀⠀⣿⡟⣿⡇⠀⠀⠀⣿⡟⣿⡇⠀⠀⠀⢸⣿⢿⣷⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡆
⠀⠀⠀⣿⡇⣿⡇⠀⠀⠀⣿⡇⣿⡇⠀⠀⠀⢸⣿⢸⣿⠀⠀⠀⠀⠀⣸⡏⢸⣯⠀⠀⠀⢸⣽⢨⣯⠀⠀⠀⢸⣿⢹⣿⡇
⣤⣤⣤⣿⣧⣿⣧⣤⣤⣤⣿⣧⣼⣿⣤⣤⣤⣼⣧⣼⣿⣤⣤⣤⣤⣤⣿⣧⣼⣿⣤⣤⣤⣼⣿⣼⣿⣤⣤⣤⣼⣿⣤⣿⣧⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⣿⠿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⣦⣤⣄⣀⡀⠀⠀⠀⣀⡀⢀⠀⠀⠀⣼⡟⠁⡸⠀⢸⠈⠹⠛⣼⣿⣿⣿⣿⣿⣿⡀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⢿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⡇⠀⠃⠀⠌⠀⡤⠀⢨⣿⣿⣷⣤⣤⠈⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⣿⣿⣿⡄⢰⣆⠀⠀⠀⠃⢠⣾⣿⣿⣿⣿⣿⠄⠀⠙⠚⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⡿⣴⣇⡀⠀⢸⣿⣿⣿⣿⣿⣿⣄⠀⠀⠐⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⢿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠇⠀⡻⣿⣿⣿⣿⣿⣏⠀⠀⠀⠰⠾⠛⠋⠉⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡇⠋⠉⠀⠀⠁⠈⢻⣿⣿⣿⡿⠿⠶⠤⠴⠂⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣄⣀⣤⠀⠀⠀⣀⡀⠀⠀⢸⣿⢹⡴⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⡇⠀⠀⠀⠀⠉⢻⣦⣼⣏⡼⣣⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣬⣽⡟⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⣻⣆⠀⢀⠀⠠⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣉⣳⠀⠀⠀⠀⠀⣼⣿⡿⢸⣿⢹⣿⠀⣾⠂⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠳⣶⣶⣤⣾⣿⣿⣷⣾⡿⢸⣿⢰⡇⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠿⠿⣿⣿⣿⡇⠀⠀⠐⠿⠿⢿⡿⠿⠿⠿⠫⠛⠛⠧⠺⠷⠀⠉⠉⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠷⣤⡤⠞⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠤⠤⠤⠀⠀⠤⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠶⣄⠀⠀⠀⢠⠀⠀⠀⢀⡴⠖⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⣹⠀⠀⢀⣼⣄⠀⠀⢸⡁⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠶⠶⠞⠛⠳⣄⠉⠀⠈⢁⡴⠛⠛⠶⠶⠞⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠶⠶⠶⠶⠶⠟⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢔⢪⡢⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠢⡑⠝⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠑⢄⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠐⠃⠬⠤⠄⠔⡡⠞⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠯⢍⣒⡶⠀⠀⠸⢀⠔⡱⠀⣑⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈⢆⠠⢤⢤⠊⠀⠉⠁⣒⡒⠩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡌⡄⡀⡀⡖⣌⡆⠸⠘⡀⠀⠀⢀⢐⡂⠼⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡰⠰⢰⢹⢀⠤⠤⠐⠋⠉⠉⠉⠁⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠑⠛⠁




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

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣀⠀⢀⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⠟⢁⣴⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣡⣶⣿⣿⣿⣿⣿⠿⢋⡙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠻⣿⣿⣿⡿⠛⣡⣾⣿⣿⣦⣀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⡿⢛⡛⠿⣿⣿⣿⣦⣤⣭⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⢿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠈⠻⠿⢋⣴⣿⣿⡇⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⢿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⣡⣾⣿⣦⠹⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⣡⣾⣿⣿⡿⣣⣶⣿⣷⡈⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠿⢋⣴⣿⣿⣿⢟⣴⣶⣶⡌⣿⣿⣿⣿⣿⣿⡿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠟⣵⣿⣿⣿⡿⢁⣿⣿⣿⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠿⢋⣴⡿⠋⠉

⠀⠀⠀⠀⠀⣀⡤⠖⠒⠒⠒⠒⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠒⠒⠒⠲⠤⣀
⠀⠀⠀⢠⠞⠁⢀⣀⣤⣤⣀⣀⡀⠀⣀⡩⠷⠶⠶⠖⠲⠶⠦⠤⣄⡀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠖⠲⢶⡤⠞⠉⠀⠀⢀⣀⣀⣤⣤⣀⡀⠈⠳⡄
⠀⠀⢠⠏⠀⢰⠁⠀⠀⠀⣀⠴⠛⠉⠀⣀⣀⠤⠤⢤⡤⠤⣄⣀⠀⠉⠳⡄⣠⠞⠁⠀⣀⣠⠤⢤⡴⠋⠀⢀⣠⠴⠚⠛⠦⣀⠀⠀⠀⠈⡆⠀⠸⡄
⠀⠀⢸⠀⠀⣿⠀⠀⣠⠞⠁⢀⡤⠒⠉⠁⠙⢦⡀⠀⢳⡀⠀⠀⢙⡆⠀⣸⣇⠀⢰⡋⠀⠀⢀⡞⠀⢀⡴⠋⠀⠉⠒⢦⡀⠈⠳⣄⠀⠀⣿⠀⠀⡇
⠀⠀⢸⠀⠀⢸⣀⡜⠁⢀⠞⠁⠀⠀⠀⠀⠀⠀⠹⡄⠀⢇⠀⡰⠋⠀⡰⠃⠘⢦⠀⠙⢄⠀⡼⠀⢠⠏⠀⠀⠀⠀⠀⠀⠈⠳⡀⠈⢣⣀⡇⠀⠀⡇
⠀⠀⠘⣇⠀⠈⢿⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠸⠎⠀⣠⠊⠀⠀⠀⠀⠑⢄⠀⠳⣇⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⢻⠁⠀⣸⠃
⠀⠀⠀⠹⡄⠀⠘⢧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣇⠞⠁⠀⠀⠀⠀⠀⠀⠈⠳⡄⠈⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⣇⢠⠏
⠀⠀⠀⠀⠙⣆⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠘⡀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢆⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢣⠀⢸⡏
⠀⠀⠀⠀⢸⠏⠳⣄⠀⠈⠓⠦⣄⣀⠀⢀⡠⠞⠁⢀⠜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠣⡀⠈⠳⢄⠀⠀⣀⣠⠴⠚⠁⠀⣸⠀⠸⡇
⠀⠀⠀⠀⢸⡆⠀⡏⠙⠲⠤⣄⣀⣠⠝⠋⢀⡠⠶⠉⡀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⡈⠶⠐⠒⠉⠉⢀⣀⣠⠤⠖⠋⢹⠀⢰⡇
⠀⠀⠀⠀⠈⡇⠀⢳⠀⠀⢀⡠⠊⠁⣠⠔⠉⠒⠀⠀⠀⠁⠊⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠓⠈⠀⠀⠀⠒⠘⠫⣍⠈⠑⢄⠀⠀⠀⡞⠀⢸⠁
⠀⠀⠀⠀⠀⠹⡄⠀⠳⠴⠋⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠙⠦⠜⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠙⢦⣀⣠⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⣠⠞⠉⠙⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠉⠲⣄
⠀⠀⠀⠀⠀⣰⠃⠀⡴⠲⣄⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⣠⠖⢦⠀⠘⣆
⠀⠀⠀⠀⢀⡇⠀⡸⠀⠀⠀⠑⢄⡀⣙⣦⡀⠤⠀⠀⠀⡀⢤⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⡠⢀⠀⠀⠀⠤⣀⠔⠋⢀⡤⠊⠀⠀⠀⢧⠀⢸⡀
⠀⠀⠀⠀⢸⠇⠀⣇⣠⠴⠒⠋⠉⠁⣀⣀⠤⠐⠶⡈⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⣁⠶⠊⠁⣠⣔⠋⠉⠙⠒⠦⣄⣸⠀⠸⡇
⠀⠀⠀⠀⢸⡆⠀⡏⠀⢀⡤⠖⠋⠉⠀⠀⠑⢦⡀⠈⢣⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡜⠁⢀⡴⠊⠁⠀⠉⠙⠲⢤⡀⠀⠙⢦⣠⡇
⠀⠀⠀⠀⣸⡇⠀⢣⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠱⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⡄⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠹⣄
⠀⠀⠀⣰⠃⢱⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⡀⠈⢦⡀⠀⠀⠀⠀⠀⠀⢀⡴⢻⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢳⡄⠀⠘⣆
⠀⠀⢠⡏⠀⠀⣧⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢹⢆⠀⠙⢄⠀⠀⠀⠀⡠⠋⠀⡴⡆⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⣷⡀⠀⢹⡄
⠀⠀⢸⠀⠀⢸⠉⢣⡀⠈⢦⡀⠀⠀⠀⠀⠀⠀⣰⠃⠀⡞⠀⠑⣄⠀⠱⡄⢠⠞⠀⣠⠊⠀⢳⠀⠘⣆⠀⠀⠀⠀⠀⠀⢀⡴⠁⢀⡜⠁⡇⠀⠀⡇
⠀⠀⢸⠀⠀⣿⠀⠀⠙⢦⡀⠈⠓⠤⣀⠀⣠⠞⠁⠀⡼⠁⠀⠀⣈⠇⠀⢹⡇⠀⠸⣅⠀⠀⠈⢧⠀⠈⠳⣄⢀⣀⠤⠚⠁⢀⡴⠋⠀⠀⣿⠀⠀⡇
⠀⠀⠘⣆⠀⠸⡀⠀⠀⠀⠉⠲⣤⡤⠖⠋⠁⠀⣠⠞⠓⠒⠚⠉⠀⣀⡴⠃⠙⢦⣀⠀⠉⠛⠒⠚⠓⠒⠒⠉⠉⠀⣀⣤⡖⠉⠀⠀⠀⢀⠇⠀⣰⠃
⠀⠄⠀⠘⢦⡀⠈⠉⠛⠛⠉⠉⠁⠀⠀⣀⡴⠞⠷⠤⠤⠶⠖⠒⠋⠁⠀⠀⠀⠀⠈⠙⠒⠲⠶⠦⠴⠶⠶⢶⣛⠉⠀⠈⠉⠉⠛⠛⠉⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠉⠓⠦⠤⠤⠤⠤⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⠤⠤⠤⠴⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢋⡑⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢚⡙⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣟⣴⣶⣿⢟⣀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡻⣿⣷⣦⣻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡿⣡⣾⣿⣿⡿⣫⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⢿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⢟⣴⣿⣿⣿⢋⣾⣿⣿⡟⣛⣛⢿⣿⣿⣿⣿⣿⣿⣿⣦⡹⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢣⣾⣿⣿⣿⡇⣾⣿⣿⢏⣼⣿⣿⣷⡙⢿⣿⣿⣿⣿⣿⣿⣷⣌⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣧⣝⣋⣵⣾⣿⣿⣿⣿⣿⣦⡙⠿⠿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣥⣬⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠳⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⢟⣤⣭⡻⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣾⣷⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠇⣙⠻⣿⣿⣿⣯⣴⣿⣯⡙⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣣⣾⣿⣿⡌⢿⣿⣯⡻⣿⣿⣿⣦⡙⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢩⣾⣿⣿⡿⢋⣴⣷⡎⣿⣿⣮⡻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠋⣴⣿⣿⢟⣅⢻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠿⠿







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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣁⣠⣤⣴⣶⣷⡤⣄⢀⡄⠀⠀⠀⢀⣤⣄⢺⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠋⡟⠻⢫⣅⡛⠿⢉⢻⣿⣿⡷⣄⣀⢸⣿⣿⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⣾⣴⣧⣤⣀⣠⣤⡶⢃⢼⡍⣹⡾⣷⣿⣶⣬⣥⣯⣀⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⣿⣿⣿⣿⣿⣿⣇⣆⡞⢸⣷⣿⣿⣧⣿⣿⣿⣯⣾⢩⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡞⣻⣿⣿⣿⣾⣿⣿⣿⣯⣴⣿⣿⣽⣽⣿⣿⡿⠿⠿⢋⣴⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿⣯⣼⠉⢻⣇⣀⣬⡼⣿⣀⣽⣿⣿⣷⣎⣸⣤⡟⠻⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡷⠻⡇⢙⡛⣹⣟⢻⣿⣿⣿⣿⣿⣿⣯⣭⣛⠛
⠀⠀⠀⠀⠀⠀⣀⡀⣿⣿⣿⣿⣿⣿⡿⢿⠷⠒⠟⢻⠋⢻⡿⠿⠿⣿⣿⠿⣿⠿⢿⣿⣷⣄
⠀⠀⠀⠀⠀⢘⡿⢳⣿⣿⣿⣿⣿⠋⠀⠂⢀⣼⠀⠈⠀⣼⣷⡀⢠⣿⣿⣦⠘⢶⠀⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣦⡔⢠⣿⣟⢀⡄⣨⣿⣿⣿⣿⣿⢿⣿⣶⣼⣷⣾⡿⢿⠇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣦⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⢻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢻⣿⣿⠿⡿⣿⣿⡿⢿⣿⠿⠟⠿⠿⢿⡏⠘⠿⠿⠿⠿⠡⢼⢿⣿⣿⡿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⣡⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣦⡙⢿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⢾⣿⣿⣿⣿⣿⣿⣿⣿⣯⡉⠛⠿⡿⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣂⡀⣴⣮⢛⡛⢿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⢸⣿⠇⣿⡇⣾⣿⢘⣛⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠟⣋⡻⡏⣼⣿⢸⣿⠃⣿⡇⣾⣿⢸⣿⠟⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣿⣇⠁⣿⣿⣼⣿⣸⣿⣃⣿⡇⠚⠁⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣭⣭⣍⣙⣛⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠿⠿⠿⢿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠛⠋⠉⠉⠉⠉⠉⠉⠛⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢰⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠋⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠙⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⣀⣀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡏⠀⣰⣿⣿⣿⣷⡘⢿⣿⢃⣼⣿⣿⣿⣆⠀⢹⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⢻⣿⣿⠁⢰⣿⣿⣿⣿⣿⣷⡌⢁⣾⣿⣿⣿⣿⣿⡆⠀⢿⣿⡟⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢻⡏⠀⣸⣿⡟⢹⣿⣿⣿⣿⣿⣿⣿⣿⡏⢻⣿⣇⠀⠘⡟⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠁⠀⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠈⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠻⣿⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠛⢶⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠛⠶⣄⠀⠀⣸⠃⠀⠀⡿⠘⣧⠀⠀⣠⡶⠚⠓⠶⣄⠀⠀⢰⡏⢿⠀⠀⠘⣇⠀⠀⣠⠶⠛⠛⠳⣄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹⡆⢀⡟⠀⠀⢸⠇⠀⢻⡄⢰⡏⠀⠀⠀⠀⢹⡇⠀⣿⠀⠸⡇⠀⠀⢻⡀⢰⡏⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⢀⡼⠁⣸⠃⠀⠀⣿⡀⠀⠘⣇⠘⣧⡀⠀⠀⢀⣼⠃⢸⠇⠀⠀⣿⠀⠀⠘⣇⠈⢧⡀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣙⣛⣛⣋⣠⡴⠋⠀⠀⣸⠟⣇⠀⠀⠙⢦⡈⠙⠓⠚⠛⢁⣴⠟⠀⠀⢰⡿⣇⠀⠀⠙⢦⣄⣙⣛⣛⣋⣁⡀
⠀⠀⠀⠀⣴⠞⠋⠉⠉⠉⠉⠉⠁⠀⠀⢀⡴⠋⠀⠙⢦⡀⠀⠀⠉⠙⠛⠛⠋⠉⠁⠀⢀⣴⠏⠀⠙⢦⡀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠳⣦
⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⡿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿
⠀⣸⠇⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠘⣇
⢠⡟⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⢻
⣼⠃⠀⢀⡟⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⢻⡀⠀⠘
⢿⣄⣀⣼⠃⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠘⣧⣀⣠
⠀⠉⠉⠀⠀⣿⠀⠀⠀⣤⡄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⣤⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⣤⠀⠀⠀⣿⠀⠀⠉⠉
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣹⣿⡄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⢰⡏⣷⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⢻⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⣾⢹⡆⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⡿⠀⠀⢸⡇⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⢸⡇⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⢸⡇⠀⠀⢿
⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⢸⡇⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠠⡇⠀⠀⢸⡇⣿⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠻⣦⣴⡟⠀⠙⣦⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⣴⠋⢀⢻⣦⣴⠟⠀⠀⠀⠀⠀⠀⢀⠘⢷⣤⣴⠋⠀⢻⣦⣴⠟
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠤⠀⠀⠀⠀⣀⠂⠀⠀⣴⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣛⡂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡿⢻⠈⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠼⢁⠀⠀⠀⢲⡄⠊⠀⠀⠀⠀⠀⠹⣿⡑⡀⣀⢀⣿⣾⠆⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠠⠋⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠓⠄⠀⠀⠒⠋⠁⠀⠀⠀⠀⠀⢀⠀⠀⢹⣷⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢸⡄⠀⣠⣤⡜⠛⢧⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠶⠠⠀⠀⠾⢏⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠁⠀⠉⠋⠀⠀⡌⠱⣤⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠒⠉⡀⠀⠀⠀⠀⠀⠀⣼⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⡠⠴⠖⠊⠁⠀⣀⣀⡄⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠴⠲⠿⠮⠁⠀⠀⡨⠟⠈⡆⠀⠀⠀⠀⠀⣾⣷⣲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢺⣹⣤⠓⣁⢤⣠⠔⠈⠀⠘⠀⠀⠈⠑⠤⡀⠀⠀⠐⠋⠀⠀⠀⢱⡀⠀⠀⠀⣸⣿⣈⣤⣀⡙⠣⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⢆⠱⣌⢶⣙⠟⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠄⠀⠀⠀⠘⡍⠁⣀⣴⣿⣿⣿⣿⣿⣿⢴⣥⠃
⠀⠀⠀⠀⠀⠀⠀⠐⢆⡱⠌⠻⠌⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠉⣄⠙⠻⣿⣿⣿⣿⣿⣿⣼⢗⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢞⠛⡆⠀⠀⠉⠛⠿⣿⣿⣿⡿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠻⢿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢉⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⣤⣤⣤⡾⠃⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢰⣏⣀⣠⡴⠋⠀⢠⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⢀⣀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠋⠉⢳⣤⠶⢦⡀⠀⠀⠀⠀⠀⠀⠙⢷⣄⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⣠⠟⠁⠀⢠⡇⠀⠀⠀⠀⠰⢦⡀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡞⠁⠀⢀⡴⠛⠙⣦⠀⠰⣄⠀⠈⠛⢦⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⠋⠀⠀⣠⣿⡀⠀⠈⠳⣄⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⠞⠁⠀⣻⣦⡀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠿⠛⠛⠛⠋⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⣁⢖⣃⠲⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⡗⢊⣏⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⠛⡴⢦⢡⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⢸⣿⠹⣿⢇⣯⠀⠀⠀⠀⠀⢀⢀⡗⡈⢉⣁⠶⠌⣿⣿⠫⠀⢄⠀⠀⠀⠀⠀⠒⣼⣦⣭⣍⣮⠁⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⢣⣙⡚⠷⣖⠁⠀⠀⠀⠀⣞⡿⢃⠀⣱⠋⠀⠀⠀⠉⠙⠡⢈⣀⢠⡀⠀⠀⠀⠐⢏⡽⠛⣩⠏⡃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠉⠁⠀⠀⣴⢌⢦⣜⣐⠰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠻⠞⠃⢠⠁⢶⣀⡀⠀⠁⠚⠞
⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠄⠀⠀⡂⠀⠀⡼⠙⣿⣶⡿⠟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢲⣖⢨⡝⣷⠆⠀⠀⠄⠀⢠⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⣎⢹⡢⢼⡄⠀⣡⠀⡸⣵⡖⡆⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⡥⣖⡄⠀⠰⠆⠀⠤⣌⢉⠠⠄
⠀⠀⠀⠀⠀⠀⣠⡮⢿⣿⣿⢦⢀⣬⡀⣩⣧⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⢈⡀⠸⢇⣠⣾⣿⢃⠀⣸
⠀⠀⠀⠀⠀⠸⡿⡿⣶⣶⣶⣉⣩⣋⡹⠟⠃⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣒⠠⣿⡖⠠⠛⡫⠡⣟⣷⣃
⠀⠀⠀⠀⠀⣼⠃⣃⠐⡠⠌⣻⣀⡽⢧⣼⡅⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠼⣵⡋⣼⣹⡲⠛⢁⡿⢟⣃⠻
⠀⠀⠀⠀⠰⣽⠆⣈⠀⢼⠻⡎⢉⢱⣒⠛⣶⡫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢨⢰⢿⣠⡔⣡⠄⠀⡿⡇⢯⠇
⠀⠀⠀⠀⢰⢩⠁⠘⠀⢾⣯⠅⡞⣨⣿⣴⡮⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣃⡍⠈⣼⠉⣉⢿⣑⠀⠰⡶⠷⠂
⠀⠀⠀⠀⠸⠒⠺⠀⠀⢘⠛⣆⠸⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠢⢄⣁⢀⠀⠄⣛⡁⠀⢿⠀⡰⠂
⠀⠀⠀⠀⠐⠸⠆⠀⠀⣭⠑⠘⣥⣟⣩⡙⠟⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣳⣁⡰⠗⠤⠠⠜⠛⠃⠀⠘⠀⣨⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢰⠇⢸⢹⣃⢿⢌⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣤⠄⡆⢸⣿⡇⣞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⡉⠀⣨⢈⣶⢲⣮⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣤⡉⢌⡇⢼⡧⣳⣼⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣧⢠⠁⠀⢻⣤⣏⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⣏⠠⠇⢘⣾⣯⢰⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣈⠛⡀⠀⢠⠟⢀⠘⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣧⣣⣴⠀⠈⠇⡭⠸⢏
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣶⣶⡤⠀⠀⢘⠶⣰⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣍⠸⠓⠀⠀⣛⣟⢱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠟⣻⠃⠀⠀⠀⣒⡰⢶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠚⣃⠃⠀⠀⢹⠫⠉⣩⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠇⣯⠸⡃⠀⠀⠀⡷⠿⡾⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣤⢮⠃⠀⠀⠈⢾⠃⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠇⣨⡁⠀⠀⠀⠐⠠⣆⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢨⢹⠃⠀⠀⢨⣶⡼⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠉⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠃⠀⠀⠀⠀⠙⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⣠⣼⣷⣄⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣿⡍⠉⠋⠀⣽⣯⠀⠙⠉⢹⣿⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣴⡖⠒⠋⠀⢸⣿⣦⣀⣤⣯⣼⣤⣀⣰⣿⡇⠀⠛⠒⣲⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⢿⡶⣤⡠⠞⠉⠁⠀⡐⠀⠀⠀⠀⠈⠉⠳⣄⣠⢶⡟⠋⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⠃⠀⠈⡷⠋⠀⠀⠁⠀⠰⠀⠠⠄⠐⠆⠀⠉⠀⠀⠑⣞⠀⠀⠙⠲⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠹⡿⢷⣶⣶⡾⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢦⣶⣶⡾⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⠻⠀⠀⠀⠀⠀⠀⠒⠀⠀⠰⠀⠀⠀⠒⠂⠀⠈⠀⠀⠈⡟⠁⠀⢼
⠀⠀⠀⠀⠀⠀⣀⣾⣁⣤⠴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢳⢤⣄⣀⣵⣄
⠀⠀⠀⠀⠀⠈⠙⢟⠋⠛⠓⡏⠀⠈⠁⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⣹⠶⠚⠛⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⡗⠀⢀⣴⠀⠀⠀⠀⠀⠀⠤⠀⠀⠰⠀⠀⠀⠤⠀⠀⠀⠀⠀⢀⣧⡀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⣸⣷⡾⠿⠟⢳⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠾⠿⠿⣶⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠂⣤⠀⢀⡷⢄⠀⠀⣀⠀⠰⠀⠐⠂⠀⠆⠀⣀⠀⠀⡠⢮⡀⠀⢠⠔⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⣾⠖⠋⠑⢢⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡔⠋⠙⠲⣷⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠧⠤⣄⠀⢸⣿⠏⠁⠙⡟⣿⠉⠉⠻⣿⡇⠀⢠⠤⠼⠟⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣾⣃⣠⣄⠀⣿⡇⢀⣠⣀⣹⣿⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⠀⠙⢻⡟⠋⠀⠀⠉⠙⠃