Apricot Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⡞⣾⡟⢏⢱⣾⣿⣉⣉⣉⣶⣤⣄⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⣭⡭⢄⣈⡱⢄⡀
⠀⠀⠀⠀⠀⠀⢸⢣⡿⣷⠈⣿⠋⢻⡿⣿⣛⣯⠤⠈⢳⣄⠀⠀⢀⣼⠟⠁⠀⣀⣀⣀⣀⣀⠠⠤⠈⠰⢀
⠀⠀⠀⠀⠀⠀⠈⢼⣟⣿⡆⠹⣆⠈⢷⠈⡿⢿⣶⣖⣈⢿⣫⡉⠉⠻⣿⣟⡉⠉⠉⠉⣉⣛⣲⣶⢶⣶⠚⢦
⠀⠀⠀⢀⠀⠐⠒⠚⠧⣽⣿⣦⣽⣆⠘⠀⣟⠀⠙⠻⠷⣦⣹⢿⣦⠄⡬⣿⣟⡛⠶⢯⡅⠀⠀⠀⠀⠀⢉⠛⡆
⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⢦⣿⡀⢀⠐⣷⢈⡝⠋⡧⠀⠀⠀⠙⢷⣄⠀⠙⣷⣟⠙⠓⠶⠦⠤⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠙⠲⣦⣈⡿⠀⠘⠁⠀⠀⠀⠀⠀⠈⠳⣄⡈⠷⢾⣟⡛⡒⠖⢚⡇
⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠠⠾⣫⣅⣉⠉⠁⠒⢧⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⢿⠙⠒⠮⠿⣧⣭⣂⣳
⠈⢢⣀⠀⠀⠀⠀⠀⢀⡠⠔⠋⡟⠀⣠⣼⣷⠶⣬⣓⢤⡀⠙⢆⠰⡀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠈⠙⠻⣆
⠀⠘⢄⠀⠀⠈⠉⠉⠁⠀⠀⢰⠁⠀⢸⣿⣿⠀⠈⠉⠓⠜⢆⠈⣆⠳⡀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠁⠂⠤⢀⣀⣀⣠⠴⠾⡀⡀⠀⣿⡹⣦⡀⠀⠀⢸⣿⡀⠘⡄⠙⢄⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠠⡀⠈⢻⣭⣿⣦⣀⣲⣿⣷⠀⣇⠀⠀⠑⠠⣀⣀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡑⢄⠀⠙⠿⠮⠟⠿⣿⠂⣰⠀⠑⠒⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣙⠢⠄⣀⣈⡉⣡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠁



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣟⣿⣟⣻⣿⣿⣻

⠀⠀⠀⠀⠀⠀⠀⣠⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⣴⠁⣸⣠⣤⢦⠄⡄⠀⡤⢀⡄⡀⡼⠁
⠀⠀⠀⠀⠀⠀⠃⠀⡟⠏⠀⠘⠀⠃⠀⠁⠈⠉⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠉⠉⠉⠉⠉⠉⠑⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠇⢰⡄⠀⠀⠴⠒⠒⠒⠲⢤⡀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⣄⣀⣠⠎⠀⠀⠀⠀⠉⠑⠣⣀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⢸⡀
⠀⠀⢀⠀⠠⠄⠒⠒⠐⠒⢢⣾⠁⠀⠀⠀⠀⠀⠈⠭⠀⠀⠀⠀⠠⣀⠀⠀⠀⠙⢧⣀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠀⠀⢀⠔⡇
⠀⠀⠈⠀⠀⠒⠠⠌⣢⡐⡆⢻⠀⠀⠀⠀⠀⠀⢀⠘⢄⡀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠙⣏⠛⠓⠒⠒⠒⠂⠁⠀⠀⠀⡘⠰
⠀⠀⠀⠀⠀⠂⠀⠀⠀⡉⡇⠈⢧⠀⠀⠀⠀⠀⠈⠺⣷⡉⢲⠤⢀⠀⢈⡇⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⢀⡐⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠠⡤⠋⠱⡀⠀⠳⢄⡀⠀⠀⠀⠀⠀⠉⠛⠻⠦⠽⠞⠁⠀⠀⠀⠀⠀⣸⠤⠤⠤⠤⠀⠒⠒⡋⠔⠁
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢳⡀⠀⠀⠉⠓⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⢒⠿⢤⡠⠄⠐⠐⠂⠁
⠀⠀⠀⠀⠀⢀⠂⠀⠈⠑⣢⣬⡷⡀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠶⠶⠒⠋⠁⠀⠀⣠⠚⣀⡄⠈⠒⢄⡀
⠀⠀⠀⢀⠔⠃⢀⠠⠕⠻⠃⠀⠀⠟⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠹⠉⠙⠆⠲⠒⠀⠙⢆
⠀⠀⠀⠪⢀⣈⠀⠀⠀⣀⡀⠤⠐⠒⠀⠉⠑⠢⠤⣄⣀⣀⣀⣠⠤⠥⠒⠚⠒⠀⠤⠤⠄⠀⠀⠤⠤⠴⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⣀⠤⠶⠖⠒⠒⠒⠒⠒⠛⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⡰⠋⠀⢀⡴⣄⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⡜⠁⠀⡴⣫⠞⠁⠀⠀⠀⠀⠀⢀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⡅⢀⡞⡵⠃⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡟⠻⢳⠎⣜⠁⢀⣀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣇⠎⡜⠁⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠖⠂⠘⠛⠛⠒⠲⠦⣼⢠⡟⡼⠖⠒⠛⠛⠛⠀⠲⢤⣄⡀
⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⢱⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⡇
⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⣀⡤⠖⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣲⣶⣤⣤⣤⣶⣶⣚⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣟⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠛⠱⠦⠤⣀⡀⠑⠈⢢⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⠀⠄⠈⠉⠲⡇⠀⢉⢢⡀⡼⡁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠘⠦⣸⡀⣻⠁⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⣀⣀⣴⣡⠼⠋⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠵⠶⠶⢤⣤⣼⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⣠⡄⠀⠀⠀⠀⠀⠈⠓⢦⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠋⠉⠙⠛⠒⠒⠛⠉⠁

⣿⣿⠻⠿⢻⠿⡿⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠉⠉⠉⠑⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠾⢿⣿⠿⠿⠿⠿⠟⠛⡛⠖⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⢀⣡⠄⠀⠀⠀⠐⠶⢤⡀⠐⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠤⠖⠃⠀⠀⠀⠀⠀⠀⠈⠀⠙⢦⡀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢁⣠⣠⡀⠀⠀⠀⠀⠀⢠⠻⡀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⣶⡦⣀⡶⠿⠒⠀⡀⠉⠛⠛⠻⠷⣦⡄⠀⠀⠌⠰⠞⣦⢳⡀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠀⠀⠀⠀⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣙⢷⣶⣶⣿⣿⢿⣿⣫⣴⡭⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠡⠝⢍⠉⠙⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢢⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠘⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⢱⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⡀⣺⡄⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣝⡃⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢀⠀⠹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⣼⣿⠉⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡷⢶⣿⣿⢻⡄⢀⠀⠀⠀⠀⠀⠀⢀⣤⢾⣿⣠⠏⣡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢙⡛⣙⣧⣾⠠⠢⠤⠠⠖⠾⠞⣃⡠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠂⠈⠉




⣀⣀⣀⠀⣀⣀⣀⣀⢀⣀⣀⣀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⡠⠀⠀⠀⠐⠠⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⠀⠰⠶⣄⡀⠀⢹⣄⡀⠀⠀⠀⢀⣠⣶⣶⣶⣴⣦⣤⣶⡔⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⡀⠀⠀⠠⠊⠀⠀⠉⠉⠀⠀⠀⠀⠀⠹⠷⠾⠿⠿⠖⠁⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡧⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢶⢀⠀⠀⠀⠀⠸⠿⠿⣿⣿⣿⣿⣿⣿⣿⣟⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡎⠀⠀⠀⠀⠄⠀⠀⠉⠉⠉⠉⠉⠉⠀⠂
⠀⠀⠀⠀⠀⠀⠓⠠⣀⡀⠀⠀⠀⠀⢀⢴⠟⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠩⠝⠨⠙⠟⠃⠁⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡠⠄⠉⠁⠀⠉⢗⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣺⣧⠀⠀⠀⠀⠀⣀⣦⡾⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋⠀⠀⠀⠀⢀⠀⠉⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⢦⡄⠀⠲⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⢀⡀⠀⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣰⣿⣷⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣠⠷⠀⠀⢀⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⢀⣠⠀⠀⢀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁
⠀⠀⠀⢀⠀⠀⠀⢝⣀⣰⣶⣿⠟⠁⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⠁⠁⣀⡴⠞⢁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠈⣦⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⡜⠀⠀⠀⠀⠀⠀⠴⠒⠉⠁⠀⣠⡟⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠘⢮⣀⠀⠀⠀⠀⠀⡀⢀⣊⡼⠋⠀⠀⠠⠃⠀⠂⠀⠀⠀⠀⣠⠶⠊⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡅⠆
⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⠶⠶⠒⠛⠉⠀⠀⠀⢀⠜⠂⠀⠤⠤⠴⠶⠛⠉⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣋⡽⠃
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣄⢀⣤⣠⣤⢠⣄⣶⠼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠂⠀⠀⠀⠀⠀⠀⡆⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠶⠛⠛⠋⠉⠀⢠⠔
⠀⠀⠀⠀⠀⠀⠱⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⣀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣄⣀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠠⠄⠀⠄⠉⠉⠛⠖⠒⠀⠀⢈⣠⠴⠒⠉⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⣀⢠⡤⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⣠⠾⢣⠊⠁⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣼⠆⠿⠧⠿⣥⣀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠒⣱⠛⠁⠀⠀⠀⠀⠀⠈⠙⢤⡀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡀⡠⠄⢄⣩⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠂⢄⢀⡼⠛⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢡⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⡹⡄⠁⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠤⠒⠳⣭⣈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢀⡆⠀⠀⡜⠀⠀⢳⡴⢀⠀⠈⢣⡀⠀⠀⠀⠀⣴⠿⣾⣛⣦⠈⠙⢿⠱⣎⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠈⠇⣠⠞⠅⢠⠀⠀⠙⣮⣄⢞⠀⡈⠒⢦⡤⠚⠁⠀⠁⠀⠙⢦⠀⠈⡇⠏⠛⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠜⠁⠂⠌⢬⠀⠀⠀⠀⠙⠛⣳⣷⡟⠟⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡘⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠈⠜⡙⠂⠀⣀⣴⡻⠍⡿⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣀⠂⠀⢋⡀⠔⠀⢿⣽⠀⠉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⠥⠠⢿⡀⠀⠀⠘⣿⣡⣀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠒⠊⠆⠹⠀⠀⠈⡏⠻⡿⢦⣄⣀⡠⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠊⠀⠀⢀⠀⠀⣰⡧⠀⣸⡒⠠⢄⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢀⠄⠀⢰⠂⢀⡄⠀⠀⠀⢀⠙⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡠⠀⠈⠀⠀⠀⠁⠀⠆⠈⢀⡾⠛⠛⠻⢦⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡤⠀⣤⢾⢐⠀⣠⣴⠯⡟⠀⠀⠀⠀⠈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠜⣿⢛⣸⠉⠢⣀⠀⠀⠀⠀⠔⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣎⣧⠄⡿⠀⠹⣞⣰⣧⠀⠈⠉⠒⣲⠾⣎⣀⡀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠁⠀⠀⢈⡟⠳⢲⡆⠊⠉⠇⠡⠀⠙⠡⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⢀⡐⢵⣃⣀⢤⠤⣾⠀⠰⠇⠁⠀⠀⠀⡀⠀⠀⠐⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡴⠞⣸⠈⠛⠨⢄⠀⠀⡛⡆⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⢊⢓⢋⠀⢠⡜⠆⠀⠀⠈⢃⡊⠙⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⢸⡆⠀⢀⠏⠀⠀⠀⠀⠀⠈⠃⡈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢐⢿⠀⡘⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢞⠄⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠂⠐⠐⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⠀⠀⠐⠒⠤⣀⠀⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠙⢦⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢡⣤⣦⣀⣀⠀⠀⠠⢳⣛⣮⣢⢄⡐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡾⠒⠀⠀⠀⠀⠈⠉⢛⡶⣶⣾⣿⣿⠷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⠘⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢊⠀⣀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⣤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣗⡘⡄⠀⠀⠀⠀⠀⠀⠀⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢷⠠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⣄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⠁⠀⠀⠀⢀⢀⣶⡀⠀⠀⠀⠀⠀⢡⠀⠀⢰⣯⣜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⢠⠀⠀⠀⠀⠀⢰⡇⡉⠀⠀⠄⣠⣾⡿⡿⠁⠀⠀⠀⠀⠀⢸⠀⢀⣦⢶⠎
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣧⡀⡄⠀⠀⠀⠀⢠⣉⣄⣐⢺⣟⣯⡽⠟⠀⠀⠀⠀⠀⠀⠀⣹⠐⣳⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⡀⠀⠀⠀⠀⠶⡦⣼⣿⣿⣿⣆⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⡔⣴⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢻⡓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⣫⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣅⠀⠄⠀⠀⢀⣀⡤⢤⠤⢒⣩⣴⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣦⣶⣤⣦⡤⠶⠿⠛⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠛⠛⠛⠛⠛⠛⠉⠉⠩⠉⠁


⣀⡀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠩⣄⠐⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠑⢤⡀⠱⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⠱⣄⠀⠀⠀⠀⠀⢠⡾⠟⠙⣛⠽⡏
⠀⠀⠀⠀⠀⣀⠀⠀⢀⡴⠋⠀⢹⣧⣀⣠⠤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠑⠒⠂⠉⢂⠀⠀⠀⢠⣯⣦⣾⣈⠴⠋
⠀⠀⠀⠀⠀⠁⠀⡴⠋⠀⠀⢀⠔⠿⠟⠒⠤⣀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⣱⠃⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠈⠣⣠⡀
⠀⠀⠀⠀⡇⢸⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠠⣽⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣺⠿⠟⠛⠀⠉⠀⠉⠙⠒⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢠⣴⠾⠉⠉⠉⠉⠒⢄⠀⠀⠘⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣌⠳⢄⠀⠀⠀⠀⠀⢸⡁⠀⠀⢠⣶⣿⠁⠀⠀⠀⠀⠀⢠⣿⠿⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣭⣔⣤⣤⣴⣼⣇⠀⠀⠘⠻⢿⣷⣤⣀⣀⡠⠴⠛⠁⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⣦⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡶⠤⣤⡀⣤⣤⣤⣤⠤⢶⣿⣿⣿⠋
⠀⠀⠀⠀⣰⣷⠏⠚⠁⠉⡹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣆⠀⠀⠀⠀⠀⠀⣀⣾⠿⠋
⠀⠀⠀⢠⣿⠋⠀⠀⢀⣼⠅⠀⠀⠀⠀⢠⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠋⠉
⠀⠀⠀⠘⠿⠶⠶⠚⠫⠊⠀⠀⠀⠀⠀⣸⣅⣢⣰⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣭⠍⠁⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣶⣶⣤⡘⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣻⣿⣿⣷⣿⣿⣬⣦⠙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⠿⠀⢠⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠟⠁⠀⠉⠉⠡⠥⠶⠒⠢⠀⠤⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠋⠀⠀⢀⡀⠊⠀⠀⠀⠀⠀⢠⣤⣶⣅⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⡿⠃⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠰⣶⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠉⠻⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣤⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣿⡀⢨⢹⣿⣿⣆⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢔⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⢿⣿⡷⣿⣿⣿⣿⣿⠀⠀⠀⡠⠄⢀⡞⠀⠀⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⠀⠀⠀⣼⣾⠏⠀⣠⣦⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣾⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣉⣲⣠⠤⠤⣠⣀⣤⠤⠖⣉⠉⠉⠉⠁⠀⠀⡀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢉⠁⠁⠈⠉⠀⡀⠠⠎⣯⢐⠰⢈⠀⠃⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⣦⠂⡄⠆⠠⠀⡋⠈⠁⠉⡈⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣴⣧⣿⣿⣷⣶⣦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⡿⠟⠛⠛⠛⠿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣀⣀⣔⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠭⠿⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⢀⣼⣿⢟⣷⣿⣿⣿⠟⠛⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠿⡿⠄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣦⣦⣄⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣃⡀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣤⡶⠶⠶⠶⠤⣄⡀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⡴⡀⠀⠀⠀⠀⠀⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡅⠀⠀⠀⠀⠈⠹⣷⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡜⡄⠀⠀⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⠹⣦⣉⠲⡀⠀⠀⠀⢀⣿⣿⡀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡗⣽⡀⠀⠀⠀⠀⢸⣿⣿⡇⠀⢀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠹⣿⣧⣿⣷⣷⠌⡲⣶⣬⣿⣿⣿⡇⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣽⠁⠀⠀⠀⠀⢸⣿⣿⢧⡀⣼
⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⡆⠀⠀⠀⣸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⠀⢠⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⢻⣷⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣧⣤⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠻⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠛⠛⠿⢿⣿⣿⣿⣿⣿⡿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣦⣄⣀⠀⠀⣀⠦⢀⣶⣶⣶⣶⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠑⢆⠀⠀⠀⢀⣠⠤⠶⠒⠒⠓⠒⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣤⠴⠒⠒⠒⠛⠲⢤⣀⡞⠁⠀⠀⠀⠈⢦⣠⠞⠩⠐⠀⠀⠀⠀⠀⠀⢀⢻⡌⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠘⢦⠀⠀⣼⠁⠀⠀⢧⢈⡴⠋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠘⠓⠳⠄⣽⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠸⣷⠈⠄⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⡀⠀⣹⣆⣀⣿⠁⠀⢀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠠⠀⡿⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⣄⡴⠚⠉⠉⠉⠉⢙⣛⣉⣀⠀⠀⠀⠈⠙⠲⣄⠀⠀⠀⠠⡜⢀⠌⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⣰⠏⠀⠐⠊⢉⠲⢄⠀⠀⠀⠈⠻⣦⡀⡰⠁⠚⠈⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡖⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠀⠢⠑⢄⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢧⠀⠀⠀⠀⠀⠻⡄⠠⡀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠈⢧⠀⢄⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢳⡄⠀⠀⠀⠀⠻⠶⣬⣑⠢⢦⣄⠀⠀⡇⠀⠀⠀⠀⠀⠸⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠏⠳⣄⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠲⠇⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠀⠀⠙⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣅⡀⠀⠀⠈⠙⠒⠒⠦⠤⢤⣤⣤⣤⠴⣲⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣄⣀⣀⣀⡀⢀⣴⣒⣻⠿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣩⣍⣫⣭⣉⣭⣩⣟⣹⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡤
⠀⠀⠀⠀⠀⠀⠀⢢⠠⡄⠀⠀⠀⢾⣿⣶⣴⡃⢀⣎⡀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠤⢤⡬⣄⡹⠦⢀⡚⣿⣿⠟⠝⣲⣿⣝⢢⠀⠀⠀⠀⣠⠦⢀⡀⢴⣦⣹⣇⠼⠇
⠀⠀⠀⠀⠀⣰⡾⢉⣽⣿⣷⣶⡾⣽⣴⡶⣿⣿⣧⣿⠿⠋⠀⠀⠀⣿⣶⣾⣿⡀⡛⠛⠿⣀
⠀⠀⠀⠀⠇⠙⠃⡐⠻⣿⠛⢻⣿⣏⡉⣥⢁⡐⣴⣅⠸⣧⣤⡤⠀⠙⣻⠿⠿⠷⠿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣳⣿⣿⣷⣥⣿⣿⣿⣿⣿⣿⣿⣕⣌⠙⠓⡠⣤⡀⠀⢠⡀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢯⣭⣥⣿⢿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣷⡿⣻⣿⣷⣦⢇⠀⠘⠓
⠀⠀⠀⠀⠀⠀⠀⢠⡼⡯⣝⡿⠛⠉⣯⣿⣿⣿⣾⣽⡶⢳⣾⣿⢝⣿⣿⣿⣻⣻⠿⣷⡀⣀⣈⢠
⠀⠀⠀⠀⠀⠀⠀⡀⡤⢶⢠⣀⣤⣮⢰⣿⣿⣿⣿⣿⣿⣶⣿⣿⡂⣿⣿⣟⣟⣈⡁⠙⢿⣿⡽⠡⡀
⠀⠀⠀⠒⣫⡷⣾⢸⣋⣴⠾⢿⣿⠿⠛⢉⠛⣿⣿⣿⣿⣿⣿⡷⢦⣛⣯⠜⣿⠛⣽⣷⣀⣉⠁
⠀⠀⠀⠀⠴⠞⢳⠞⢛⡷⠗⠈⡟⠀⢁⣤⠉⠟⢿⣿⣿⣿⣟⠁⡈⠉⠀⠀⠋⠈⡿⠀⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⡤⠀⠀⠀⠘⠀⠄⠄⠀⣿⣿⡿⠃⠀⠀⠀⠀⠀⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣾⣿⣿⣿⣷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡲⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡷⡜⢀⠀⠀⠶⠃⠁⢠⠀⠀⠻⠆⠀⢀⠘⡴⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠸⠆⢀⠛⠄⠷⣠⠐⠀⢸⢀⠐⢠⡰⠇⡜⢃⠀⠾⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠤⡷⠀⠛⣢⡘⡂⠀⣌⠃⠀⡀⡀⣰⢼⡌⠔⡀⠀⠀⢋⡄⠀⡚⣠⡚⠃⠐⣦⠄
⠀⠀⠀⠀⠀⠀⠈⠛⣠⠄⢀⠡⣀⠀⡀⢙⣆⠀⠘⡃⠁⣸⠃⡀⡛⠀⢀⣞⠁⡀⢀⡠⢁⠀⢤⡘⠋
⠀⠀⠀⠀⠀⠀⣠⡄⢠⡤⠘⠛⠉⢇⠈⢲⡁⠹⢶⡇⠀⣿⠀⢠⣷⠾⠁⣱⠊⢀⠏⠙⠛⠠⣤⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠅⠀⠠⠀⢶⡖⠾⢧⣘⠋⠛⠢⢽⣆⣿⣠⣾⠥⠚⠋⢛⡠⠿⠖⣶⠆⠠⠀⠀⠍
⠀⠀⠀⠀⢀⡆⠀⠚⠢⡠⢔⡒⠳⠦⣤⢬⣽⣿⣶⣶⣿⣿⣿⣷⣶⣾⣿⣭⢤⡤⠶⠓⣒⠤⠠⠚⠂⠀⣄
⠀⠀⠀⠀⠈⠱⠶⡄⠀⠀⠈⢹⡄⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣼⠉⠀⠀⠀⡴⠶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⠴⠶⠾⠿⠿⠿⠿⠿⠷⠶⠦⣤⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠀⠈⣇⠀⠀⡅⠠⠄⠀⠆⠀⠀⠀⠀⢆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⣆⡴⠃⠀⠀⠎⠀⠀⠀⠠⠀⠀⡏⠀⡠⢇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⡸⠀⠁⠀⠀⠀⠈⠒⢇⠀⠀⠈⠢⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣈⡇⠀⠀⠀⠀⠀⠀⢄⢸⡇⢸⣰⠃⢃⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⡀⠀⠀⠀⢸⡇⠀⣼⠅⠀⠀⠀⢀⠀⠀⠀⢸⡇⣸⠏⣀⡆⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠛⢻⣧⠞⠁⠀⠀⠀⠀⠀⢣⡀⠀⢸⣧⠟⠉⠀⣈⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢾⣿⠀⠀⠈⠠⣀⠀⠀⣼⡇⠀⣼⡟⠛⠛⠫⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⡀⠀⠀⠀⢀⠀⠀⢸⣿⠀⠄⠀⠀⣸⠀⣠⣛⣇⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⠈⠙⠓⣶⠶⠶⣷⣄⣸⣿⠀⠈⢢⡴⠛⠛⠁⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠑⠦⣀⡨⠶⠋⠁⢀⠠⠈⠉⠻⣿⣆⢠⡿⠀⠀⠀⣠⣿⣿⣶⢶⣄⣀⠠⠄⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠈⣷⣿⠏⠀⠀⠀⠉⠛⠛⢶⣒⣒⢛⠓⠒⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡄⣼⣿⠏⠀⠀⢠⠀⠤⠀⠀⠀⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⠀⢀⡴⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡥⠖⠋⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇
⢀⣀⣸⣄⠀⠀⠀⢀⣤⣤⣴⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⡆⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⣄⢀⡄⠀⠀⠀⠀⠀⠀⢀⣠⣀⣠
⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣶⣶⣶⣶⣷⣾⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⣤⡀⢀⣤⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣀⠸⠿⠀⢿⣿⠺⢿⡶⢀⡄⣿⠇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⣿⣾⢛⠹⠷⠞⠀⢰⣦⡀⣴⡇⢿⠇⠀⢸⠟⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⠉⣤⡉⣿⠇⠀⣷⣦⢠⣝⠃⡙⠁⣠⣴⠆⣀⢼⡿⠇⢀⡄⢠⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣜⠃⣶⡠⢰⣬⣄⠈⣷⠄⡙⢀⡟⢣⣾⣶⠙⠋⣾⡿⢀⣴⣇⢻⠟⠛⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣤⡉⠉⠀⠉⠁⠳⣍⠙⣇⣠⠊⣴⡄⠶⣶⢧⣸⣴⣦⠉⠀⠘⢿⡣⠔⠀⠿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣃⣙⣿⠗⢾⡛⠋⠻⢾⣿⡄⠛⠀⠀⣤⣤⣹⣇⣹⣴⠗⠫⣿⣶⠆⠀⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣦⡜⠛⣋⡀⢈⣭⣝⠲⣾⣿⣋⢻⣇⠙⠛⠢⣉⣹⡿⢛⣭⣽⣓⠒⢰⡾⠀⣶⠶⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣭⢀⡘⢿⣆⡭⠴⣾⣿⣍⠽⠿⣿⣷⣤⣤⣾⣿⣷⣮⣭⡭⠁⠀⣶⡿⠃⠀⠘⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠏⠠⡶⠀⢀⡀⢺⣷⠀⠀⠈⢿⣿⣿⠏⠀⠀⠀⠙⠍⠛⣻⣯⣅⠀⠘⠿⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠈⠁⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢰⣌⠙⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠘⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⢻⣿⢿⣿⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠒⠖⣛⠿⢛⢿⡥⡴⠋⣹⡈⢯⡛⢤⣘⠿⣛⡛⠛⠛⠒⠲⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠸⡀⠀⠃⠈⠂⠉⠑⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣤⢸⣷⠀⠀⠀⣤⠀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣬⣿⣤⠉⣾⢰⣆⠻⣚⣛⠁⠀⢴⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⣀⣻⣿⡆⣶⠉⠓⣶⢣⡘⢿⢢⡭⢁⡀⣿⣌⢱⣿⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡘⣿⡇⢉⠩⣤⣤⡈⠀⣧⢉⡋⢻⡞⣰⣾⡿⢀⢿⣿⢈⡁⣰⡄⣶⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣧⣬⡀⠟⣶⠀⢭⣬⡶⣤⡸⢳⡎⣷⢛⡋⠀⢸⣷⠀⢛⡁⣿⣇⡉⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢤⣤⣄⡀⣦⡀⠠⣭⣤⡤⡉⣾⠀⢹⡇⠨⣤⣪⡁⣿⠛⢡⣦⠷⢈⣀⡍⣁⣼⣋⣛⠛⠛⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡗⠙⢹⡇⡀⠉⠁⢘⢦⡼⢻⣰⣷⠨⣥⡙⣿⢠⡞⢿⣴⣣⠴⢺⡅⠈⠛⠿⠓⠀⢸⣿⣀⣴⠂
⠀⠀⠀⠀⠀⣀⠀⠀⠈⠛⠻⠃⠠⡼⠇⠸⠆⣿⡆⡙⢾⣜⠁⢶⣌⠁⣿⢈⣷⠿⣷⣤⡛⢂⠀⣜⢣⡄⢰⣦⡘⠟⣩⣵⠆
⠀⠀⠀⠀⠀⠸⣷⡄⣦⠀⣌⠛⠃⢹⡍⣛⣇⡪⣹⠇⣶⡙⣷⡜⠿⢶⣿⣿⠃⢠⡌⢉⠉⢸⣷⣉⠴⢛⢈⠿⡷⠘⣛⡋
⠀⠀⠀⠀⠀⠰⢾⡄⢉⠰⣿⡆⣷⣄⢷⡈⠉⢤⡬⠓⠾⣤⣘⣿⣆⣼⣿⡏⣀⠳⠕⣫⡄⣦⠙⡿⢚⣋⡘⢇⡜⢙⣋⠁
⠀⠀⠀⠀⠀⠰⠾⠿⠗⠑⢮⣄⣙⠿⠈⢳⣄⣀⣚⣛⡛⠀⠹⣿⣿⣿⣿⢣⣼⠷⡞⣩⡤⣉⡾⢳⡿⢟⣥⠟⣴⣿⢏⣾⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⣃⣤⣭⡝⢛⣛⣛⢋⣹⠿⠿⠿⣶⣼⣿⣿⣿⠈⠿⣴⣧⣽⣿⣿⠷⠶⠶⠿⠧⠤⠽⢥⢠⣅⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠛⣠⣤⣉⠝⣭⠸⡆⠀⠀⠘⣿⣿⣿⣿⠀⣴⠟⠛⠉⢩⡍⣙⠲⣟⣧⡄⢾⣷⡌⠓⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣶⡆⢿⡇⠁⠀⠀⠀⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠛⢸⣿⢀⡓⠈⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠇⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⡿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣠⣶⣶⣶⣖⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⢤⣀⣴⣿⣿⣿⣿⣿⣯⣿⣿⣿⣻⣿⣆⣄⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣀⡄⣤⠼⣿⣾⣿⣟⣿⣾⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀
⠀⠀⠀⠀⢀⣠⣠⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣿⣶
⠀⠀⠀⢰⣾⠿⣿⢿⣿⣿⣞⣳⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣷⣛⠿⠛
⠀⠀⠀⠸⢿⣿⣿⣿⣿⣽⣿⢯⣯⣯⣿⣿⣩⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⣙⣛⣿⣿⣿⣶⣽⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣻⣿⣯⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡿⠟⠉
⠀⠀⠀⠀⢹⣾⣿⣿⢿⣿⣿⣿⡿⠿⣧⣿⡏⢻⣽⣿⣹⣿⡿⠋⢉⡽⣿⣟⣿⣿⣿⣾⣿⢿⡆
⠀⠀⠀⠀⠀⠀⠻⣿⣾⣷⣿⣿⠝⠒⠚⠛⢷⣾⣻⡿⣹⣿⣽⠟⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠀⠀⠀⠀⠀⠈⠏⠫⣷⣿⣿⡟⠀⠀⠀⠀⠀⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢰⣟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡀⣿⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠿⠿⠿⠧⠿⠿⠿⠿⠿⠶⠶⠦
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣥⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑
⠀⢀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⠠⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠘⠣⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠘⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠄⠐⠦⣀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠰⠀⠰⢆⡀⠀⠀⠀⠀⠀⠐⠂⡀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⡘⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣄⣀
⠀⠁⠀⠀⢀⠀⠀⠀⠘⠃⠈⠉⠓⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⠀⠈⠙⠳⡄
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠑⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⡀⢠⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⢠⡀⠀⠀⠀⠀⠀⠄⢠⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠
⠀⠀⠉⠳⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠉⠦⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⢀⠋
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣾⣿⣶⡀
⠀⢰⠆⠀⠀⠀⠀⠀⠹⣿⠀⠀⠄⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⠀⠠⠀⢰⠆⠀⠀⠀⠀⠁⠠⣶⡿⠁
⠀⠘⣦⡀⠀⠀⠀⠀⣠⠟⠀⠀⠈⢃⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⢀⠉⡼⠁⠀⠘⣦⡀⠀⠀⠀⠀⣠⠟
⠀⠀⠈⠙⠺⡤⡴⠚⠁⠀⠀⠀⠀⠈⢳⣴⡿⠁⠀⠀⠀⠠⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠘⢾⢿⡞⠀⠀⠀⠀⠈⠙⠺⠤⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⠀⠀⠀⠀⡉⠁⣀⡀⢊⠀⠰⠉⠀⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸⢿⡟⠚⠖⠁⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠁⠘⠀⠀⡃
⠀⠀⠀⠀⠀⠤⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣤⣾⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠆
⠀⠀⠈⠐⠒⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠈⠐⠒⠒⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⣠⣶⣿⣿⣿⣝⢷⣄
⠀⠀⠀⣴⢿⣿⣿⣿⣿⣿⣧⠹⣷
⢀⠀⣰⣿⣿⣿⢿⣿⣿⣿⣿⡄⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡳⣦⡀
⠙⠒⢤⣬⣭⡅⢀⢈⣿⣿⣿⡇⢸⡿⠉⢢⢤⣤⣤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣠⣯⣿⣿⣿⣿⣷⡘⣷
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⡇⠀⠇⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠿⠷⢾⠇⠙⡿⠛⢻⣿⣿⡇⣿⣆⡀
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⢏⣼⡟⠑⠀⠰⡀⠀⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⠈⢷⠀⢹⣶⣾⣿⣿⣇⣿⠛⠛
⠀⠀⠀⠈⠻⢿⣿⣿⣛⣵⡿⠋⠀⠀⡆⠀⠱⡄⠀⠀⠀⣼⣿⡿⢋⡙⢇⠀⠀⠀⢸⠆⠀⣿⣿⣿⢟⣾⠏
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⢹⡄⠀⠹⣄⠀⠀⣿⠛⠀⠈⠀⢸⠀⠀⠀⣎⡀⠀⡿⠿⠷⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠱⣄⡙⠂⢀⣀⠀⢸⣄⡤⠋⠀⠁⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⣠⣀⠉⠑⠢⣌⠙⠻⢦⡀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠙⠉⠂⠀⠀⢹⣧⡀⠀⢳⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠠⢄⣸⣿⣿⣷⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⣼⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣠⣼⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠟⠛⠋⠷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⠉⠁⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠁⠀⠀⠀⠀⡤⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣰⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⣠⣿⡇⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡆⢢⡄⣀⣀⠀⢠⠋⠙⠊⠉⠁⠐⡇⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣦⣻⡀⠀⠈⠓⠁⠀⠀⠀⢀⡠⠚⢏⣁⣩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣿⣾⣁⣀⣉⣳⣤⣀⠀⠀⣠⣴⣋⣀⣀⣼⣿⣿⣷⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠁⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷

⢿⣍⣉⣉⣉⣉⣉⣉⣹⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⠋⢹⠉⢹⡟⢾⣝⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣸⣇⣞⣠⠟⢉⣷⣻
⠀⠀⠀⠀⠀⠀⠀⠘⢯⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⣛⣉⣨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣠⣾⡿⣭⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⠜⠁⠈⢿⢦⣈⡽⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⡤⠉⠀⠀⠀⠈⠳⣟⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣠⠎⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡈⢻⣿⣿⣿⣿⣿⣿⣿⡿⢁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠙⢿⣿⣿⣿⡿⠋⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠛⠛⠛⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠂⢲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣴⠀⢸⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠚⠋⠉⠉⠁⠀⢹⣀⣸⠀⠈⠉⠉⠉⠓⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⠿⠃
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣇⡴⠪⡀⣨⡶⢋⡵⠊
⠀⠀⠀⠀⠀⣿⠗⣠⢟⡡⢞⠍⢀⠂⠀⠀⢈⠤⠀⢀⣤⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢺⣷⢷⣷
⠀⠀⠀⠀⠀⣿⣾⡫⠋⠈⣀⣤⡶⠞⢁⠆⢀⣤⠔⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢾⣿
⠀⠀⠀⠀⠀⣿⣿⣴⠞⢛⣉⣵⣶⣤⠾⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢸⣿⣺⣿
⠀⠀⠀⠀⠀⣿⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⣿
⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣄⠀⠀⠀⠀⠀⢻⣿⣼⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠃⠀⠀⠀⠀⢹⣿⢺⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⢸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣿⣿
⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡏⢣⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣤⠀⢰⣦⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡙⠛⠃⠘⡿⠀⣾⣿⠃⠀⢀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⠛⠁⠀⣶⡄⠈⢢⡉⠀⠀⠀⣟⡰⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣄⡀⠀⠸⡇⠀⠀⠀⠁⢢⠀⣧⡴⠚⠋⠀⠀⠰⣷⣞⠿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⠐⢤⡃⠀⣴⡄⠀⠀⣷⠏⢠⡶⠀⠀⢀⣀⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⢧⡀⢿⠇⠀⢸⡿⠊⠀⠀⢀⠰⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠗⠐⠊⣩⡍⠉⠛⣶⣤⡀⢸⡇⠀⢀⣠⠾⠒⢒⣀⠀⠴⣿⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠂⣀⣉⣀⣀⠞⠁⠈⢻⣾⣿⣴⣯⠥⣤⠄⠈⠻⠷⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠉⠁⠀⠀⢠⣄⠀⠀⢻⣿⣿⢀⣀⠀⠀⠀⣤⠲⠦⠈⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠈⠀⠀⢸⣿⡇⠈⠉⣷⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷

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

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠈⢣⡄⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⡰⠁⠠
⠀⠀⠀⠀⠀⠀⠑⢤⠀⠀⠀⠄⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠈⠀⠀⡂⢀⠢⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠂⠀⠀⠁⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠑⠠⢐⠠⠂⢁⠠⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡤⠀⠠⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠔⡀⠀⠈⠀⠀⠀⡐⠀⠀⢀⠔⠁
⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠂⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠠⠀⠀⡀⠐⠀⠀⢀⠔⠁
⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠌⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠐⠠⠀⠑⠈⠀⢀⠔⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢎⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢩⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⢒⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠐⠀⠈⠀⡠⠊⠀⠀⠀⠀⠀⠁⠐⠀⢀⠠⠐⠈
⠀⠀⠁⠂⢀⠀⠀⠀⠠⠀⠀⠁⢄⠀⠀⠀⠀⠡⡂⠀⠁⠀⠀⠁⠀⠘⠀⡄⠀⡈⠀⠀⢀⠇⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠘⢆⠠⠄⠀⠄⠠⠠⠄⠁⠠⠀⠀⣠⠃⠀⠀⠀⡠⠈⠀⠀⠀⠀⠂⢀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⢠⠄⣑⢄⠠⠀⠈⠁⠀⢀⠀⠁⠀⠀⠉⠀⠒⠤⢄⠀⠜⡢⠈⠀⠀⠠⠀⡀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈
⠐⠀⠀⠤⢀⠀⠀⠂⠀⠀⠀⠀⠂⢄⠈⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡢⡀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⠀⠤
⠀⠄⠀⠀⠀⠀⠈⠐⠠⠄⣈⠀⠠⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⢀⣀⠀⠀⠀⠐
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠂⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡄
⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠠⠠⠤⠤⠠⠠⠤⠄⢄⢤⢠⢀⢠⢠⠠⣠⠀⡄⣤⣦⠧⠤⠤⠤⠤⠤⠤⠸⠠⠠⠤⠤⠤⠦⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⠒⠒⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠈⠁
⢀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀
⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠛⠋⠉⠀⠀⠀⠀⠀⢀⣀⠀⠉⠙⠛⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠉⠀⠀⣤⣶⣶⣿⣟⣶⡀⣰⣿⣿⣿⣶⣦⣤⡀⠀⠉⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⢁⣤⣤⣤⡈⢿⣿⣯⣿⡿⣿⣧⣿⣿⣿⢿⣿⣯⣿⠇⣠⣤⣀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⣠⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⢋⣼⣿⣿⣿⣷⣦⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⢰⣿⣿⣮⣿⣿⣿⣿⣧⠻⣿⣿⣿⣿⢸⣯⣿⣿⣿⢣⣾⣿⡿⣿⣿⣿⣿⣧⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠈⠿⣿⣿⣿⣟⣿⣿⣿⣧⡹⣿⣿⣿⢸⣿⣿⣿⢣⣿⣾⢿⣿⣿⣿⣽⡿⢋⣀⠀⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢀⣶⣿⣿⣶⣤⣙⠻⢿⣿⣽⣿⣿⣷⡜⣿⣿⢸⣿⣿⣣⣿⣿⣿⡿⢟⣻⣭⣶⣿⣿⣿⡇⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⣸⣿⣛⣿⣷⣿⡿⢿⣷⣮⣭⣟⣿⣿⣿⣎⠿⠟⠿⢳⣿⠿⣻⣵⣾⣿⣻⣿⣿⣿⣿⣿⣷⡀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡍⠀⠀⠀⠈⢽⣿⣿⣿⣻⣿⣿⢿⣷⣿⣿⣿⣿⣷⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢻⣿⣿⡿⢿⣿⣿⣷⣿⣿⡿⠾⢛⣩⣥⣶⢀⣀⢀⣀⢶⣿⣿⣯⣭⣟⣛⣿⣿⣿⣿⣿⣿⡟⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠛⠿⠿⠿⠿⣛⣋⣭⣤⣶⣿⣿⣿⣿⢣⣿⣿⡿⣿⣷⡻⣿⣿⣿⢿⣯⣿⣿⣿⣯⣭⡉⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⣿⣯⣾⠃⣾⣿⣿⣿⢿⣿⣿⣮⡻⣿⣿⣿⣿⣿⢿⣿⣿⣿⠄⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⢸⣿⣿⣿⣿⣿⣻⣿⣿⣿⡿⢃⣼⣿⣿⣿⣿⡾⣿⣿⣿⣿⣌⠻⣿⣿⣿⣿⣿⣿⠏⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⠈⢿⣿⣿⣻⣿⣿⣿⣿⡿⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣿⣿⡿⠋⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣌⠻⣿⣿⣿⣿⣿⠟⣴⣿⣿⣿⣿⣿⣿⣾⣿⡾⣿⣿⣿⣿⣿⣿⣷⡈⠉⢠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠙⠛⠛⠁⢸⣿⣷⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⡿⣣⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣦⣀⠀⠘⠿⣿⣿⣿⣾⣿⣿⣿⡿⠃⠈⠛⠛⠛⣉⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣶⣦⣤⣌⣉⣉⣉⣉⣀⣀⣠⣤⣴⣶⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠠⣤⢀⡀⠉⠤⠈⡿⠴⢤⠀⠀⢄⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⢐⣛⡣⠀⢋⠛⠈⣡⡄⡔⠒⠊⢠⣐⢎⡉⢁⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣭⠀⣭⣉⢀⠣⠆⢀⣰⠀⠒⠁⠂⣦⡀⣚⣁⠜⠃⠾⠃⣦⣀
⠀⠀⠀⠀⠀⠀⢠⣤⣤⠆⡂⠩⢗⡉⠛⢠⡌⠻⢼⡩⠧⣽⠀⠘⠀⠿⠀⢤⡉⠈⡆⣙⠁⢷⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠹⢢⣅⢠⢨⠛⠖⡄⠰⠆⢺⠛⣶⣭⡓⣤⡃⣦⣻⠞⣁⠨⠶⢈⠷⠀⣄⣇⠤
⠀⠀⠀⠀⠀⠀⣤⡴⠘⢂⣑⠚⣄⠥⣤⢀⣤⢖⡻⣟⠰⡀⠀⣶⢶⡞⢱⣄⣿⣘⣦⣊⣒⠇⢠⡌⠔⠃
⠀⠀⠀⠀⠀⠠⠾⣤⠘⠿⢧⠇⢈⡱⢦⣌⡬⢻⡁⢻⣄⣴⠊⠷⣼⢠⡞⢭⢀⠖⣤⠐⢀⣵⡊⠁⡆⡒⠶
⠀⠀⠀⠀⠀⠀⢀⣀⠋⠴⠀⣂⡘⠁⣛⣉⢷⡖⠘⡙⣿⡑⠄⡽⣣⡟⡰⠦⣼⠳⠀⣶⢀⡘⣥⠰⠇
⠀⠀⠀⠀⠀⠀⠠⠝⢡⠤⣃⠤⣠⠴⠶⢤⣩⡟⢾⣅⢸⣧⣾⣵⠟⢸⣷⣎⡬⣕⠂⣽⣦⠉⡀⠠⠜⣤⡀
⠀⠀⠀⠀⠀⠀⠸⠇⣲⣀⠘⠂⣈⠋⣰⢿⣮⠳⣦⡙⣿⣿⣿⣷⡿⣿⡉⣱⡦⠝⢃⡄⢍⡚⢋⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠨⠌⢁⠱⠀⣧⠡⠀⣀⠈⢻⣾⣿⣿⡏⠀⠈⠃⢀⡰⠒⠀⠠⡌⠑⢇⡐⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠃⠀⣅⠷⠀⠰⡶⡉⠀⠀⢻⣿⣿⠀⠀⠀⠀⠨⠤⢀⣄⠃⠀⠠⠾⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠉⠀⠀⠘⠁⠙⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁

⠀⠀⠀⣀⡠⠄⡀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠤⢍⣩⠩⣽⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣣⡄⠀⠀⠀⠀⣾⣷⣾⣂⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣦⡀
⠀⠀⠀⠨⠿⠞⠛⣟⣻⣿⣿⣧⡤⣀⡀⠀⠀⠒⣻⣿⡿⠅⠀⠀⣰⣾⣿⣿⣿⣿⣶⣦⠀⠀⠼⣿⣿⣟⣻⡿⣿⣿⡉⠃
⠀⠀⠀⢀⣀⣤⣴⣿⣿⣟⢿⣿⣿⣟⡿⢷⣤⣀⠈⠇⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠈⠁⢸⡏⠁⠉
⠀⠀⠐⠋⠉⠉⡬⣭⣿⠿⣟⡯⢿⣿⣷⣾⠛⠁⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠘⢃⣀⡀⣐⣦⠤
⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠈⠻⠄⠀⠀⠀⠀⠀⢠⡄⢀⣀⠀⠉⠛⠉⠉⢹⠏⠉⠙⠋⠁⠀⠀⠀⠀⢠⣿⣿⣿⣧⣿⣯⣵⡆⡻⢶⡀
⠀⠰⣿⣿⣿⣿⣤⡄⠀⠀⠀⠀⠀⣠⣤⣶⣿⢐⣬⠶⢉⡇⠀⠀⠀⠀⠀⠸⠇⠀⠀⣀⣴⣶⣦⡄⠀⠀⠋⠛⠛⢿⡻⡿⠿⠿⠃
⠀⠘⡿⣽⣿⣿⡿⠄⠀⠀⠀⢠⣿⣾⣿⣿⣏⣿⣿⣶⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣜⢻⣿⣿⡟⠛⡃⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠘⠉⣉⣉⠁⠀⠀⠀⠀⠾⠘⠋⠿⢿⣿⣿⣿⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⢯⣈⣻⣿⣿⣭⣤⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠠⣽⠦⠀⠀⠀⣠⣤⣤⣤⣤⣠⣀⡀⠈⠉⠱⡿⠉⠀⠑⠀⠠⣤⣶⣦⣤
⠀⣠⣴⣿⣿⣿⣿⣿⣋⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣏⣻⣇⠤⠄⠀⠀⠀⠀⠀⣾⡿⢿⣿⣿⣿⣷⣦⡄
⠀⣿⣯⠽⠿⠿⠾⣿⣿⣿⣿⡟⠿⣿⣿⣿⣀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣄⣀⠀⣴⢶⡾⣿⣧⣾⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠈⠀⠀⠀⠀⠿⠿⠯⠟⣿⠧⢴⣿⣿⣿⣿⣦⠀⠀⠉⠈⠉⠻⣻⣧⡿⠋⠉⠈⠉⠉⠁⠀⠀⠀⠈⠁⠈⢻⡿⢩⠋⠻⠟⠻⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⢻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠁

⠀⠀⠀⢀⡀⠀⠄⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠻⠀⠀⠀⠀⠀⣀⡩⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠉⠁⠟⠀⠀⠀⠘⠠⢤⣤⢤⠜⠃⠀⠀⠙⠃⠀⢠⠴⣯⠈
⠀⠀⠀⠀⠀⠈⢆⠀⡬⠁⠀⠀⠀⢰⡦⠤⠀⠀⠄⣴⠀⠀⢀⠀⠀⠀⠘⡆⠀⠀⠀⢻⡾⠿⣀⠀⠀⠀⠀⠀⠈⠂⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠈⠀⠀⠠⠄⠀⠀⠀⠀⠘⠇⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠼⠀⠀⠀⠀⠐⠺⠉⠀⠀⠀⠀⠧⡀⣈
⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⠖⠀⠀⠀⠀⠀⣓⡀⠀⠀⠀⠀⠷⠆⢀⠀⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⢩⠙⢢
⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠊⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⢏⡇
⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠶⠿⠶⠦⣴⡀⠀⠀⠀⠀⠀⠠⠀⠀⢠⠀⠀⠈⢄⠀⠀⠀⢠⣴⠃
⠀⠀⠀⠀⠀⢀⣞⢑⡀⠀⠀⣈⣱⠀⠀⠀⠀⠀⠐⠧⠀⠀⠀⢻⠃⠀⠀⠈⠀⠄⢀⢠⠀⢸⠀⠀⠀⠀⠁⠲⡄⠊⠁
⠀⠀⠀⠀⠀⣨⡉⠲⠅⠠⠀⢨⣟⠀⠀⠆⠀⠀⠀⠐⣴⠀⡜⠀⠀⠀⠀⠈⠁⢰⡀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⢹⣇⣀⣀⡀⠀⢸⣿⠀⠄⠘⠀⠀⠀⠀⠛⠀⠃⠀⠀⠀⠀⠀⠰⠐⠤⠚⠀⠊
⠀⠀⠀⠀⠀⠀⠙⠦⠧⠤⠤⠼⠋⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠚⠀⠀⠀⠀⠀⠲⠤⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⠀⠀⠂⠚⠉⠘⠢⠔⠃⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠊⠀⠀⠀⠀⠀⠀⣀⣴⣄⣀⡀⠀⣀⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣠⡀⠀⠀⠀⠀⢀⣸⣯⢉⡙⠁⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠂
⠀⠀⠀⢀⡀⠤⠀⠒⠂⠚⠁⠹⠛⠆⢴⣏⣲⣿⣿⣿⣯⣧⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣲⣊⣄⢰⣿⡛⠁⠈⠉⠙⠐⢣⣤⡀⣄⢚⡃⠀⢠⠌⡧⠴⠛⠀⠀⠀⣤⠀⠀⡄
⠀⢀⣀⠀⣠⣀⣠⡀⢀⠀⠲⣼⣿⣿⣿⣿⡿⢆⣙⢶⣤⣆⡸⣷⣿⣯⣿⣦⠿⠷⣦⡄⣶⣬⣥⣀⣀⣾⣷⣊⡀⠂
⠀⠀⠁⠚⠒⠛⠿⣿⣿⣷⣿⣿⣿⣿⣿⣷⡶⠿⠿⣿⣿⣿⡿⢻⣿⣽⣥⣔⣒⡋⠁⠐⠋⠉⠋⠁⢛⣿⣿⠛⠁
⠀⠀⠀⠄⠠⠀⠂⠀⠒⠝⠛⢀⠛⠀⢜⢻⣅⢿⣯⣼⣿⣻⣷⣸⣿⢿⣷⣶⣷⣶⡆⢠⣄⣤⣄⣼⠛⠉⠁⠀⢀⠉⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⢓⣦⣀⡚⠓⠙⠛⢛⣿⡿⢷⣇⡿⠋⠈⢿⡿⣿⠿⣿⡷⡿⠀⠉⠀⠀⠀⠀⣀⣈⡀⠀⠀⢀
⠀⢐⣂⡄⠀⢀⣀⡀⠀⠠⠉⢩⡭⢉⢽⡶⢦⣼⣿⡟⢁⡿⢋⡀⣸⣠⡏⢀⣤⣬⣟⣂⣄⠂⠀⠀⢠⣬⠣⠘⣫⠄⣤⣇⣩⡁
⠀⠈⢻⣶⢆⣍⠿⠖⣛⡉⣀⣮⡟⢥⣿⣧⣦⣿⣿⣧⣼⡧⣿⠛⢻⣿⢻⡏⠀⠸⠆⢴⠀⠀⠴⢡⣶⣷⠮⣿⣿⣾⣿⡶⣸⠿
⠀⠀⠀⠀⠉⠐⡿⣾⣿⣷⣶⣿⣿⣷⣾⣿⣿⣎⣿⣿⣿⣇⣿⣴⣿⣧⣾⣯⣤⣾⣧⣶⡭⠾⢗⣋⣿⣿⣿⣿⡿⣛⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠿⠛⣫⠿⣿⣿⣿⣿⠟⢿⣿⣿⣾⣿⠻⣿⢟⣿⣿⣿⣿⣷⣎⢻⡁⢠⡾⠉⠘⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⣿⣿⡿⠙⡃⠐⠀⠉⠛⠻⠟⠛⠛⠛⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⠀⢀⣀⣤⣤⣴⠶⠿⠿⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⢦⣶⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣰⡯⠙⠛⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣸⣿⣿⣿⣏⣤⣿⣆⠀⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣏⢷⡿⠏⠲⣤⠙⣆⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⢺⠿⠛⣿⣿⣿⣿⣿⡷⠀⠀⠀⠈⠃⢹⠂⠄⢠⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄⠀⠀⣀⣴⠶⠋⠀⠀⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠸⢿⢸⠀⣼⣿⡟⠁⠙⠿⢆⣄
⠀⠀⠀⠀⠀⢰⣿⣿⠛⢿⣿⠻⣿⠋⠀⠀⠀⠀⠠⠿⠿⢿⣟⣛⣂⣠⣤⡤⠤⠴⣾⢸⡆⢻⣿⡇⠘⠟⣿⡟⠋
⠀⠀⠀⡄⠀⢸⣿⡇⢰⣂⢥⣼⠷⠶⠒⠒⠚⢛⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠈⣧⣷⡜⣿⣿⣦⣴⣿⠇
⠀⠀⠀⢣⢠⠸⣿⣿⣄⣴⣿⣿⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠣⣧⣙⢿⣿⣿⡿⠃⠀⠀⠀⠀⠐⢿⣿⣿⡿⣟⠛⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢠⠘⠁⣿⠇⢻⡄⠘⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⣸⡏⠀⠀⣇⡰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢹⣷⠀⠀⠈⠃⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⢄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⣇⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣭⣾⣿⣇⠀⢛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠉⠉⠁⠀⠓⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠈⠂⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠞⠀⠶⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠈⢣⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠰⠀⠀⠇⠀⠠⠘⠒⠒⠂⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡔⠀⠀⢀⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠀⢀⡄⢚⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢄⡀⠀⠀⠀⠀⣼⠊⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⡇⠈⡀⢉⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠐⠁⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢣⠀⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠚⠦⠐⢀
⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢃⡁⠀⠁⠘⠁⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠈⡀⠀⠆⡀⠆⠡⠁
⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠆⠀⠑⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⡴⢀⡀⠀⠀⠉⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⡤⢀⣀⡀⣀⣈⡄⠰⠆⠀⠀⠓⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃⠀⠀⠀⠀⢁⠈⠁⢃⠐⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠰⠄⠠⠀⠘⠄⠘⡆⠀⠀⢀⡀⠀⠶⠀⠀⣑⡤⠀⠤⠄⡀⠀⠀⠀⢀⠠⠜⣶⣠⠀⠠⠆⠦⠠⡖⡠⠃⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡐⠀⠐⠀⠒⡀⡇⢀⠀⢀⡀⠀⢀⡤⠊⠀⡤⠀⠄⠀⡈⠑⢄⡰⡫⠊⢹⠛⠲⣎⠱⣞⢀⡊⣠⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⣀⠉⣄⢡⡽⢌⡁⣀⡠⠔⠉⡅⠚⡀⡠⠀⠆⢠⡄⠀⣼⠭⣀⠁⢀⡡⣄⠀⠓⣌⣏⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠥⠄⠆⠺⡤⠧⠩⠄⠚⡀⠴⠈⡁⠠⠠⠄⣄⠀⠘⠉⡐⠊⢠⠌⠐⠮⠕⠢⣈⡿⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠚⣇⣹⣆⣂⢙⡀⠴⠘⣃⠥⠔⠒⠰⢀⡈⠀⠄⢀⣀⢉⠜⢁⢂⣜⡵⠋⠉⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢭⠱⣿⣶⠄⠲⠀⢃⢠⠴⠚⠓⠃⠦⢀⠚⠀⠖⡰⢊⢬⡥⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢯⣃⠩⡄⠆⢋⣽⣍⡚⠠⠂⣄⣉⠘⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠛⠺⠼⠿⠾⠄⠡⠶⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠈⠀⠀⠀⠀⠀⠀⠀⠐⢨⡐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠉⢃⣱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠒⠀⠉⠈⠉⠉⠒⠠⣄⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣎⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢢⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣿⡷⣰⠃⠀⢀⣀⣀⣀⣀⣀⠀⠀⣾⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠷⠒⠩⣥⡄⠒⠂⣰⢬⠉⠛⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣵⡁⠀⠚⠈⠂⠉⠉⠀⠂⢀⡀⢲⡙⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⡴⢁⡀⠀⢠⠀⣠⠀⠐⠆⡠⠶⠀⠀⣨⢡⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⠋⠠⣌⡁⠀⠰⣆⡀⠀⢀⡄⠀⠀⠐⠢⠆⢀⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠤⠀⠀⠃⠀⣀⠀⠀⠀⠉⠀⠀⢠⠀⠶⢠⠌⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠟⠛⠿⠫⢿⣧⣦⣤⣀⣄⣰⣠⣴⣾⣇⢸⠂⠁⠁⠀⣀⠘⢥⠀⠘⡂⠘⠂⠈⠀⡰⢋⡼⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣐⠶⠤⠤⢀⣀⠉⠛⠛⣿⠿⠿⠿⠿⠛⢻⣆⡱⢤⡄⢀⡀⠀⠀⠦⠴⠷⢀⣠⠖⢊⣴⢞⣵⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠤⣀⠘⠋⣹⠋⠀⠀⠀⠀⢄⣰⠋⢘⣿⡦⣌⣉⣓⠒⠛⠒⢓⣀⣩⡴⣞⠋⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠝⠋⠀⠀⠀⠀⢀⡠⠞⠁⢠⣼⠏⠉⠢⢬⣉⡉⠉⠉⢉⡉⢀⣤⣼⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠤⡶⠶⠤⠤⠖⠶⠊⢁⣤⣦⡴⠗⠁⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣦⣤⣤⣤⣤⣤⡤⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠻⠟⠛⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠷⠾⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠶⠆
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠹⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⢿⣿⣿⣦⡀⠀⠀⠀⠀⣴⣿⣿⣦⠀⠀⠀⠀⢀⣠⣿⣿⡿⠏⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⣀⡀⢸⣿⣿⣿⣿⠀⠀⢀⣴⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣶⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠁⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠈⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠘⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠦⣄⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠘⢦⣄⡀⠀⠀⠀⠀⣻⠀⠀⠀⠀⣸⠇⠀⠀⠀⢠⡴⠚⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠠⠔⠃⠀⠀⢀⠴⠋⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡆⠀⠀⠀⠉⠉⠛⠛⠛⠶⠀⠀⠈⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠚⠛⠛⠛⠳⠀⢀⡤⠶⠛⠋⠉⠉⠉⠉⠙⠛⠷⣦⣀⠀⠘⠉⠀⢀⡴⠋⠀⠙⠓⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⠀⠀⢠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠰⠋⢀⣠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⣠⢤⣄⠀⠀⣠⠞⠁⠀⠰⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⡴⠋⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠁⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⠒⠒⠒⠒⠒⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠓⠒⠛⠒⠒⠒⠲⠶⠤⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⡤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠖⠚⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣀⣀⣤⣤⠤⠶⠶⠖⠒⠛⠛⠛⠛⠛⠛⠛⠛⢛⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠦⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠶⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢢


⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢐⣶⣾⣷⣿⣿⠶⠁⡺⣿⣿⢿⣿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣹⣿⠿⢉⣉⣿⣿⣤⣬⢻⣿⡭⢄⣻⣿⣿⣿⣁⣤⡖⠆
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣿⣿⣥⣼⣿⣯⣷⣦⣾⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣽⣿⣿⣏⣩⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⠠
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣼⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣭⣿⣯⣛⣛⣩⣴⣤⣀⡀
⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⢿⣿⣿⣧
⠀⠀⠀⠈⣹⠟⠚⠛⣿⣟⢻⣲⣷⣿⣿⣿⣷⣿⣿⣿⣿⠆⣙⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣻⣦
⠀⠀⠀⠀⠀⡀⣾⣿⣿⡟⠛⢉⠺⢛⡛⢻⣿⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⢷⣼⡿
⠀⠀⠀⠀⠀⠚⠫⠋⣀⣠⣄⠴⣾⣿⣿⢀⡘⢯⣻⡟⢿⣿⣿⣿⣿⡿⢿⣿⡟⡫⣿⣿⡽⣿⣯⡳⢿⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠁⢠⣽⣿⣿⣤⠀⠀⠙⢻⣦⣿⣿⡿⠉⠃⠀⠛⠀⠀⡓⠻⡆⠈⣿⠁⠹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠉⠈⢻⠇⠀⠀⠀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣴⣀⣀⣠⣠⣤⣤⣴⣶⣿⣿⣿⣿⣶⣿⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣤⣤⡶
⠀⠀⠀⠀⠂⡞⠿⡿⢿⠟⠛⢿⣿⡿⡿⡿⠿⠋⠉⠁⠠⠬⠭⠿⠿⡿⠿⠿⢿⡿⠻⠟⠻⠻⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠴⠤⠤⠤⠤⠄⠤⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣼⣆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⡏⢹⣥⡿⢾⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠂⠁⠀⠀⣧⠉⠀⠟⣠⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠹⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣇⠀⠀⠀⠀⠀⡀⠴⠂⢀⠖⣍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠚⠛⢢⡴⢦⣶⣤⣠⣄⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠀⢀⣀⡶⣞⡳⣿⣭⢞⣾⣷⢻⣅⢘⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠂⠀⠀⠀⢻⣧⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣼⢮⠏⠀⡐⡖⡯⢟⡧⢟⣻⢺⣺⠼⣻⣻⠝⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢟⠳⣄⠀⠀⠁⠀⠻⣿⣤⠀⠀⠀⠀⠀⠀⡀⣲⣽⣽⡃⠀⠶⣟⡯⣽⣿⣝⡿⣾⣻⣽⣻⠽⠊⣠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠺⠽⣷⡶⣶⣶⣾⣿⣿⣿⣤⣤⣀⣀⣳⡐⢻⠷⠃⣠⣽⢬⣻⡭⢯⣯⢿⣽⣾⡾⠿⠀⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠞⠛⠋⣩⠝⡋⠀⠉⠙⠛⠛⠛⠛⠓⠄⣀⠈⠳⢞⣾⣳⣯⡿⣾⡽⠚⢋⣠⢴⣻⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡴⢛⣡⠤⠶⠚⠋⠁⠔⠀⢀⠀⣤⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣤⣄⣠⣬⣤⣴⣾⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠞⠷⠁⠤⣔⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠜⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⣿⣿⣿⣿⣿⣿⠿⢿⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⠺⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⣡⣷⠟⣻⡖⢹⣮⡙⢷⣌⡻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⣼⣿⠀⣴⣿⠀⢘⣿⣧⠀⢻⣿⣺⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⣼⣿⠃⢀⣿⡇⠀⢘⣿⣿⡀⠈⢻⣿⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢠⣿⣏⠀⠸⣿⡇⠀⢸⣿⣿⡿⠀⢸⣿⡇⣿⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⠀⠂⣈⠛⢿⠀⠀⣿⡷⠀⠘⣈⣿⠓⠀⢽⣿⣿⡟⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠐⢆⣈⡁⢀⣐⠢⢀⡐⠛⠁⠀⠉⠻⣀⢀⣟⠉⠁⠂⠀⠈⠉⠁⠀⠀⠀⠀⠀⡧
⠀⠀⠀⢠⠄⡀⠀⠰⠀⠢⠒⠂⠐⠧⡴⠲⠞⠀⠀⠀⠀⠀⠀⢘⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠘⠢⣑⣒⡲⢣⠤⢯⠤⢿⣒⡻⠻⠷⣄⠀⠀⠀⠀⡴⠉⠁⠀⠈⠋⣍⠀⠀⠀⠀⠀⢀⣤⣴⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠒⠛⠎⠀⠀⠀⠘⠋⠉⠉⠻⡄⠀⠀⠀⠀⡎⠀⠀⠀⢰⣤⣾⣿⠿⠿⠿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠶⠶⢇⠀⠀⢀⢸⣷⣦⣤⠤⠾⠷⣤⣀⣰⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠛⠛⠉⠀⠀⠀⠀⠀⠀⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⡇⠀⠀⢠⡄⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴
⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠘⣆⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠐⢆⠀⠈⠳⡀⠀⠀⠘⣆⠀⡄⢸⡇⡆⢰⠇⡤⠀⣠⠋⠀⠀⢀⡴⠆⠀⠀⢀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⠀⠀⠀⠑⢤⡀⠙⢄⠘⣆⠘⡄⡇⢸⠀⠃⡏⡼⢁⡴⢁⡔⢀⡴⠋⠀⢀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⢠⡙⢮⠢⠣⡈⠂⠁⠀⢸⠀⠀⠐⢡⠎⡠⢋⠔⠋⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⠀⠀⢉⠲⢌⠢⠀⠀⢈⠀⠂⣀⣈⣈⠀⠐⢅⠈⠀⢁⡤⠞⢁⡠⠀⠀⣀⣠⠴⠖
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠈⠉⠙⠲⠤⣉⠂⠍⢢⠔⢁⠔⠉⠀⠀⠀⠈⠒⢄⠡⡒⠁⠐⢊⠡⠤⠒⠉⠁⠀⠀⠀⠀⠀⢀⣀⣄⣀⡀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠓⠒⠶⠦⠬⠥⣀⣀⡀⠂⢠⠇⠀⠀⠀⠀⠀⠀⠀⠈⡆⠠⠀⢈⣀⠀⠤⠴⠖⣒⠿⠥⠖⠚⠓⠋⠀⠀⠉⠉⠉⠁⠀⠈⠉⠁
⠀⣀⣀⣀⣀⣄⣤⠴⠒⠒⠒⠒⠒⠶⢤⡀⠀⠀⠒⠀⠀⠀⠉⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣀⣀⣀⡒⢀⣠⠼⠃
⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⡀⠀⠀⠈⠀⢣⣀⣤⣤⣄⣀⡤⠴⠚⠉⠉⠉⠉⠉⢠⡞⠁⣀⠴⠀⠐⠂⠖⠂⠀⠀⠀⠀⠀⢠⡄⠠⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠙⠚⠫⣉⢀⣀⠀⠀⠀⠀⠀⠀⡀⢀⣠⠤⠴⠚⢉⠰⠲⠁⢀⣀⠀⢀⡤⠴⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠒⠂⠠⠤⠄⢈⡓⠁⠐⠀⠓⡻⣲⣭⣀⠉⠉⠉⠉⠉⢩⠟⢁⠀⠀⠀⠀⠉⠀⠀⠈⠓⡂⢮⠖⠲⣤⠤⢤
⠀⠀⠐⠒⠀⠛⠃⠈⠃⠀⠐⠂⠈⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣰⡟⠉⡴⡀⢰⣐⣋⡂⠆⠀⠀⠐⢉⠀⠜⠅⠀⠐⠳⠶⠎⠡⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠚⣁⣀⣜⣫⡶⠛⠉⠙⠋⠁⠉⠉⠳⠶⠶⠶⠦⠖⠛⠒⠲⠦⠶⠶⠶⠶⠧⠿⢦
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⢀⡠⠤⠴⠶⠶⠚⠭⠍⠭⠽⡟⠉⠉⢰⡄⠒⠐⠒⠒⠋⠉⠒⠒⠊⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠰⠾⠋⠉⠉⠁⠀⠀⠀⠀⠉⠹⠦⢤⠀⠀⠀⠀⠐⠶⠞⠻⠭⠥⠥⠄⠠⠀⠀⠀⠀⠀⠀⠘⠛⠓⠒⠒⠒⠛⠋⠀⠀⠙⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠒⠉⠉⠉⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠒⢤⡀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⢀⣿⠀⠈⣷⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⡄⠘⠿⠋⠀⢠⡿⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⢦⣄⣠⣴⠟⠁⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣰⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⡀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⠋⢿⡄⠈⠻⣿⣦⡀⠀⠀⢀⣠⣤⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣶⣿⣿⣿⠿⠛⠉⢿⣿⠀⠈⠛⠀⠀⠀⠙⢿⣶⣜⠉⠀⠀⠀⠀⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣆⠀⠀⢠⠀⢀⣴⣶⣿⡟⠉⠛⠳⡄⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡏⠀⠀⠀⢱⣾⣿⣿⣿⣿⠀⢀⣄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢹⣿⣿⣿⠈⠉⠀⠀⠁⢀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⠀⠀⠉⣯⣈⡅⡀⠀⠀⠠⠆⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡍⠉⣀⣤⠀⠀⠉⢤⣷⣶⡶⠶⠀⢀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠸⣿⠀⠀⢸⡆⢹⣿⡿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣿⠀⠀⠀⠃⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⢀⠁⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⢀⠔⠀⠀⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠿⣷⣦⣤⠴⠖⠁⠀⠀⠀⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣀⣀⣀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣶⣿⠟⠋⠉⠁⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄⠀⠀⠐⠢⣄⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠟⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠂⠀⠀⠀⠀⢈⣷⣄⠀⠀⠀⠀⣸⣿⠋⠁⠀⠀⠀⢀⣠⣴⡿⠁⠀⠀⡠⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⣠⣿⣷⣦⣤⣶⣶⣿⣿⣿⣿⡇⠀⠀⢰⢣⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢀⠀⠀⠀⢀⣼⣿⣿⠃⣀⣠⣴⡿⠿⠿⠟⠛⠛⠋⠉⠀⢸⣿⡇⠀⠀⠉⠆
⣠⣴⣦⡤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠊⠀⢠⣿⣾⣿⠟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠸
⣿⣿⣿⠀⣤⣀⣀⠀⠈⠑⣦⣤⡤⢾⡇⢀⡅⠀⣰⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠇
⢹⣿⣿⣷⠟⠛⠿⣷⣤⣤⡇⠀⠀⠈⢻⣏⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠃⠀⠺⠇
⢻⣿⣿⡇⠀⠀⠀⠀⠙⢿⡇⣀⠀⠀⠀⠙⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣠⡄
⠘⠿⢿⣷⠄⠀⠀⠀⠀⠀⠻⠼⠗⠢⠴⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣦⣄⣀⣀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⣦⡌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⢄⡀⠀⠀⠀⢸⢻⣴⢈⠈⢻⡄⠀⠐⡇⠀⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠤⠤⠤⠤⠄⣴⠇⢠⡠⠀⠹⠀⠀⠀⢸⣿⠆⠗⢃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢑⣄⠀⠀⠀⠀⠀⣀⢀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⢘⡛⠀⠀⠀⠀⠀⠀⠀⠺⣶⣺⡛⠤⡼⠀⠀⠀⠠⣿⡇⣤⣄⢀⠐⢺⠈⡇⠀⠀⠀⠀⠀⠀⠈⠻⣿⣯⠤⢀⣈⠈⠁⢹⠋
⠀⠀⠀⠀⠀⠀⢸⡟⡛⠋⢻⠆⠀⠈⠀⠛⣹⢿⠛⠉⠀⠀⠀⠀⠀⣿⡏⣦⠈⣤⣄⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣆⠘⢩⣄⡴⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⣠⣖⣧⣤⣀⡄⠺⡏⠀⠀⠀⠀⠀⠀⠀⣽⡉⣄⡿⠀⢀⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠁⢂⠎
⠀⠀⠀⠀⠀⠀⠀⢻⣯⡭⠤⠈⢠⡈⠉⢰⢢⠃⠀⠀⠀⠀⠀⠀⠀⣿⢷⡴⠟⠩⢾⡃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⠀⠀⠈⣧⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⣭⣿⡿⣤⣀⠀⠶⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⠀⠀⠀⠿⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⠯⠀⠀⠀⢖⣠⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⠀⢀⣀⣀⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡷⢀⡌⠁⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠢⣠⡀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣌⣄⣴⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠒⠂⣀⠶⠶⠔⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⡆⠀⠀⠀⠀⠘⢛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣤⠄⠉⠉⢀⠀⠉⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⢿⡄⠐⠐⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡀⢀⡀⠀⣄⠈⠇⣐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠟⢿⡆⠀⠀⠰⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⣀⣶⠒⣳⣦⣿⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⢾⣷⣄⣤⣼⢘⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣯⣩⠋⠁⠀⢨⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢟⡉⠁⠀⠀⠈⠃⠀⠀⠀⠀⠀⢰⡧⠳⠢⠔⡀⢀⡸⠰⠆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⡛⢁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⢰⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⠜⠁⠀⠀⠙⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣰⣀⣄⠄⠈⢀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡋⠁⠘⠈⠺⠋⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⠀⠀⢀⣴⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠚⠇⠀⠐⢢⡀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠒⠒⠒⠛⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⡁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠲⠒⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠉⠚⠂⠀⠀⠐⠀⠈

⠀⠀⣀⠄⠂⠀⠀⡐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣐⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣉⣐⡤
⠀⠜⠁⠀⠀⠀⠐⢿⠇⠐⡀⠀⠀⢸⠃⠢⢀⠀⢀⣀⡄⣻⢻⣇⠀⠀⠀⠀⠀⠀⠀⠠⢀⡰⢄⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡗⠤⠄⣀⠩⠥⠊⡡⡺⠹⠀⠀⠀⢀⡀⢔⡩⠂⠐⡍⠉⢱⠺
⠀⡄⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢱⠚⣚⠒⠒⢒⠩⠊⢀⠂⠀⠀⠀⢡⠓⢮⣈⣖⡚⠓⢪⠰⣋⠆
⠀⠈⠢⠄⣀⢀⡀⠤⠊⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠈⠉⠚⠒⠛⠒⠉⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠑⠄⡀
⠀⠀⠀⠀⠠⢨⠢⣴⡄⠀⠀⠀⠀⢀⡤⠂⠙⠋⣏⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠀⠈⠲⡄
⠀⠀⡀⠀⠔⡳⣄⠊⡀⢀⠄⠀⢠⠊⠀⠀⠀⠀⠀⡱⢈⠕⣀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢄⠀⠘⡀
⠀⡌⡁⠂⠀⡅⢈⠳⡄⡵⠄⠀⠑⠤⡀⠀⠀⠀⠀⣱⢿⣃⣀⠕⣙⢦⠀⠀⠇⠦⡀⠀⠀⠀⢸⠀⢀⡇
⠸⠀⠄⠘⠀⠄⠈⡀⡇⠹⠀⠀⠀⠀⠈⠐⠂⠒⠊⢸⣠⣯⣝⠤⠜⠇⢣⠀⠈⠑⠨⠥⠤⠒⠓⠒⠋
⡗⢐⠀⡈⢀⣘⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣴⣠⣼⣊⠔⠊
⠈⠙⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⢻⣰⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⡀⢖⠺⠶⢶⠂⠀⠀⠀⠀⠘⠳⣲⡏⡿⣷⣫⢟⣉
⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⢆⡧⣀⣀⣠⡃⠀⠀⠀⠀⠀⠀⢰⣥⣥⣮⢤⣖⣫⣉
⠀⢀⠔⡪⠙⠌⠝⠌⣛⣩⡑⢢⠀⠀⠀⢰⢑⡀⠐⠚⠃⠀⠀⠀⣧⠀⠀⠀⢀⡔⠻⡉⢇⢻⣳⡂⡟⠃
⠀⡇⠈⠁⠹⠚⡞⠑⠳⠀⠀⠀⡇⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠰⡟⠀⠀⣰⠁⣠⠀⠤⠠⠤⣄⠇⠁
⠀⢿⢄⢄⣀⣀⡀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣱⠁⠀⠀⡏⡀⢠⠀⡀⢠⠄⡞
⠀⠀⠙⠒⠂⠀⠒⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⠣⠔⠒⠂⠤⠔⠁⠀⠀⠀⠳⣆⠰⢠⢇⡴⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣤⡖⢢⠤⢀⡠⣴⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⢡⡿⣇⠛⣛⣿⢿⣿⣿⠏⠻⢻⢷⣤⣀⣵⡀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣬⡫⠩⠝⠂⠍⣄⠘⣛⣦⣶⢶⣾⣿⣿⢻⠿⠿⢿⣾⣷⣄⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠐⠘⠀⠒⣶⠤⠤⣹⣾⣷⢾⡉⣾⡵⠻⢿⣤⢠⡄⢉⣽⣿⣿⣿⣯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡀⣾⣿⠩⠉⢀⠀⠁⠐⠂⠳⡄⠠⠙⡃⠹⣿⡴⣦⡟⠻⡆⢳⣌⢽⣿⣿⣿⠥⠠⡄
⠀⠀⠀⠀⠀⠀⠀⢴⣼⣤⠄⢿⣇⣆⢠⢈⠀⠃⠀⡀⣀⠀⡀⠤⠈⣧⠀⠀⢁⢰⣆⠞⣼⠼⣽⣿⣿⣷⣿⣗⡃
⠀⠀⠀⠀⠀⠀⠀⠰⢇⣻⣶⠾⠛⠻⡶⠂⠀⠤⠀⣠⣙⡷⢂⠀⡠⢄⡐⣄⣸⣛⡿⡻⠁⠐⠢⢻⣿⢿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠓⡂⠀⠀⣀⡆⠄⣒⣠⣾⣿⣿⣿⣴⣶⣿⣿⣿⣿⣿⣟⠳⢈⣷⣾⣿⡛⠿⣟⣻⣿⡇⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣾⣯⣠⣾⣷⣿⣯⢿⠿⠿⣿⣿⣿⣿⣿⣿⢿⣿⢻⣿⢿⢿⣿⣿⡿⠿⠷⣾⣽⣿⣿⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣿⣿⣿⣿⣿⣿⣿⣛⠛⠻⣿⣄⢨⣿⣿⣥⣴⠟⣻⣿⣾⣿⣿⣿⣦⣧⣼⣿⣿⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣾⢰⠾⡟⠛⠿⠟⠛⠛⠛⠻⣿⣾⠋⣿⡏⢁⣴⣿⠟⢩⣽⣿⣿⣿⣿⣿⣿⣿⣷⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢿⠟⣹⡿⠋⠀⢾⢿⣿⣿⣿⡿⣯⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⢀⠀⢰⣿⠁⠀⠀⠀⠀⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢸⢀⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡸⢺⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⣷⡜⠿⢿⢷⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣦⡘⢃⣤⠄⠀⢀⡄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⣶⣄⠿⢃⡀⠀⠈⠛⠘⠉⢠⡄⠸⢟⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⣙⣃⠀⡈⠉⠐⠛⠋⠀⠀⢠⠀⠀⣿⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⠁⠀⠀⠙⠲⡀⠀⠀⢀⣿⠖⠉⠀⠀⠀⠀⠀⢠⣀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡻⠂⠀⠀⠀⢠⡼⣦⠀⣼⠁⠀⣿⣆⣴⠀⠀⠀⠈⠋⠒⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠸⠃⠀⠀⠸⠛⠀⠘⣷⡿⠘⠿⠮⠙⠋⠀⠀⠀⠀⠀⠀⢠⣶⠟
⠀⠀⠀⠀⠀⢀⣤⣀⠿⡆⠀⢸⡆⠀⠀⠀⢰⣿⢀⣀⠀⠀⣿⡇⠀⠀⢀⣀⠤⠒⠂⠈⠉⠁⠀⢴⡜⠿⠆
⠀⠀⠀⠀⡀⠈⠉⣽⠶⠂⠀⠀⠁⢀⡀⠻⢷⠅⠿⠃⠀⠀⣿⣷⠀⣠⠞⠁⠀⠀⣠⡄⠀⠀⠀⠈⢁⠀⣠⡄
⠀⢀⣤⣀⠿⡆⠀⣀⠀⠐⠈⠉⠉⠉⠛⠓⠶⢦⣤⡀⠀⠀⢻⣿⣼⠃⠀⠀⠀⢷⡝⡷⠆⠀⠀⠀⠻⢎⣯⡀⠀⠰⠿⣡⣤⡀
⠀⠀⠉⣩⡶⠀⣼⠟⠇⠀⠀⣀⣀⠀⠀⡀⠤⠶⠾⢿⣷⣄⠸⣿⡿⠀⠀⠀⠀⣠⣤⣤⣤⣤⡤⠤⠤⠬⠅⠀⠀⠀⢶⡌⠉⠁
⠀⠀⠀⠉⠀⠀⠈⠀⠀⠺⢶⡌⠉⠃⣼⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠀⢀⣴⡿⠋⠉⠀⠀⠀⠀⠀⣀⢾⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣆⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡲⠖⠒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣤⡿⠤⢄⣀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠞⠀⢇⠀⠀⠀⠉⠲⢄⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣀⣘⣆⣀⣀⠀⠀⠀⠑⢄⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠒⠢⣌⢢⣤⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡜⡇⠀⢀⡠⠤⠐⠒⠒⠒⠒⠠⠤⣀⡀⡰⢫⠃⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢑⣶⡊⠁⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣼⣡⠃⢀⠀⢱⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡃⢆⠈⠑⠲⠤⢄⣀⣀⡀⢸⠣⡀⠐⠄⠉⠀⠈⠀⠘⣉⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠳⢜⣄⠀⠀⠀⠀⠀⠀⠀⠈⢣⡙⢢⠄⠀⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⠄⣀⣀⣀⣀⣴⢡⠋⠀⠆⢀⡆⠀⠂⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⡀⠀⣠⠎⠘⢆⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠉⠀⠀⠀⠀⠉⠓⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⡀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⣀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣄⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠈⠟⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠟⠛⠋⠁


⡶⠶⢶⢶⢶⡶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⡶⡶⣶⣶⣶⣶⡶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⣴⢲⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⣬⣦⣤⢤⣍⣻⣿⣿⡶⣿⠾⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣀⠰⠦⣿⣉⠀⠐⣶⣶⣿⠳⠞⣿⣿⠿⠋⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡦⣻⣿⣧⡳⣾⣿⡞⢲⠛⡻⣿⣤⣶⡾⣤⣤⠖⠺⣿⣯⣣⣤⢠⠒⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠿⣿⣿⣿⣿⣿⣿⣶⣬⠵⢿⣿⣧⣠⣃⣽⣶⠀⠀⢘⠛⠛⢲⠿⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⢿⠻⠟⠁⡟⠻⣿⣿⣽⣷⣿⣿⣿⣿⣿⣷⣴⣾⣦⡄⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠀⠠⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⡀⣤⣤⣥⣥⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⣷⣿⣿⠛⣿⣿⠿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⣤⢀⣀⣤⡟⢿⡿⠷⡌⢤⡛⠻⢿⢿⣿⣿⣿⣿⣻⣿⣿⡿⢝⠻⠿⢣⣖⢾⢾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠛⣻⣿⣟⣿⣿⣷⣦⣾⣿⣶⣦⣼⣿⣿⡿⣷⣿⣿⠐⣀⣉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠇⠈⢋⣻⣿⠇⢿⡻⣿⣿⣿⢫⣾⣿⣿⣿⡟⠻⠉⠀⢴⣴⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⣬⣿⣿⡿⣧⢹⣿⣿⣿⣿⣵⡆⢀⣷⣼⣿⣶⣤⡐⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣄⡀⢀⢀⡛⣿⣿⣿⠹⣿⢠⣿⢿⣿⣿⣿⣇⠀⢿⣯⡍⠍⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⢿⣿⣿⣮⣭⣯⣥⡉⠀⣿⣧⣶⣄⢋⣭⣿⣍⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠹⠷⠇⢡⣿⠉⠹⡟⠀⣿⣷⠛⠷⠻⣼⣿⣻⣿⣃⣤⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⡉⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⠒⠢⠀⡈⠙⠟⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣭⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠺⠛⠃⠃
⣿⣿⠿⠿⡿⢻⠿⣿⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⢠⠟⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⣿⠀⢁⣤⣆⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢿⡆⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⠜⣡⠞⠋⡛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠰⢶⣶⣤⡳⣌⠳⠀⢴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⣼⡇⡴⣿⠷⠾⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣥⣌⡻⣆⢸⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⣿⠀⣨⣞⣤⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠚⣧⡓⢸⡇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠿⣷⣫⣥⣤⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣮⣿⣜⣿⡀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣼⠛⢉⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣄⠹⣏⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢱⠃⠴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡃⢻⡄⠇⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⣤⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⢻⡄⠸⢁⡄
⠀⠀⠀⠀⠀⠀⣠⣶⣦⣴⣿⡿⠛⠛⠛⣿⣧⠀⠀⠀⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⢿⡄⣿⠁
⠀⠀⠀⠀⣠⣾⣿⣿⢿⣿⡏⠀⠀⠀⠀⣿⡏⠀⠀⣼⣿⠏⠀⣾⣷⠆⠀⠀⠀⠒⢦⡀⠀⠀⢰⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠁
⠀⠀⠀⣰⣿⣿⠟⠁⠀⢻⣧⠀⠀⠀⣰⡿⠁⢀⣾⣿⠋⠀⠀⠻⠃⠀⡀⠀⠀⠀⢸⣿⡄⠀⠘⠟⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠙⣀
⠀⠀⢠⣿⣿⠏⠀⠀⠀⠀⠙⠛⠛⣻⡿⠁⢀⣾⣿⠃⠀⣸⣿⠀⠀⢠⣿⡇⠀⢠⣿⡟⢠⣿⡇⠀⠀⣠⣾⣿⠿⠀⠀⠀⠀⠀⠾⠃⢠⣷⣄
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⢀⣾⡿⠁⢀⣼⣿⠋⠀⣰⣿⡿⠁⣴⡿⠋⢠⣿⡟⠁⢠⣾⣿⠟⣡⣦⠀⢀⡜⠁⠀⠀⣠⢿⣿⠏⠀⠀⡀
⠀⠀⠈⣿⣿⡀⠀⠀⠀⢀⣠⠾⠃⠀⠀⣾⡿⢁⡴⢻⣿⠃⣠⠞⣿⡟⣠⠾⠋⠀⠀⣿⡟⢀⣴⣿⡿⣡⣾⣿⠏⡴⠋⠀⠀⠀⠀⠁⢸⣿⣀⡤⠞⠁
⠀⠀⠀⠈⠻⠷⠶⠶⠶⠛⠁⠀⠀⠀⠀⠙⠟⠋⠀⠈⠻⠚⠁⠀⠛⠟⠁⠀⠀⠀⠀⠛⠟⠋⠀⠛⠛⠁⠘⠻⠋⠀⠀⠀⠀⠀⣠⠖⣻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡿⣠⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⢰⡿⠀⠀⡛⠲⠀⢉⢸⡄⡇⡼⠰⠀⡇⠀⠸⠀⠘⣇⠀⣿⢸⡇⣏⢸⢀⡇⢘⠀⠖⢛⡀⠀⢶⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠘⡃⠲⠆⠀⠀⢀⡈⠀⠁⠁⠀⠈⠀⠁⠀⠈⠀⠀⠀⠀⠈⠀⠀⠁⠈⠀⠁⠀⠀⠀⢀⡀⠀⠸⡇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻⣄⠰⠆⠠⣌⡛⠦⡜⢦⣄⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⣀⣀⠀⣀⠀⠀⣿⠁⣴⡦⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣽⣧⡀⠠⣤⣍⠳⢦⣤⣙⡛⠈⠻⣄⠲⣤⣤⡀⠲⢶⣶⣀⣬⡽⠷⢛⡉⢠⡇⢿⠀⠈⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⡘⢿⣄⠠⣙⠗⠀⠈⣉⠛⠛⠛⠻⠷⠾⠿⠛⠛⢛⣉⣉⣀⠶⠰⠟⢀⣼⠇⠘⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⣦⣹⠷⢲⡈⠋⠀⢙⣷⡄⠀⠀⠶⠶⢦⣈⣩⣭⠿⠶⠛⠁⠈⣉⣤⡾⠋⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣿⠟⠶⣿⣰⡇⠀⠈⣉⡿⠛⣛⣭⠿⠋⠀⣰⡾⠋⠀⠚⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠋⠉⠀⢼⡿⠁⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⣷⣶⣿⣯⣽⣛⣛⣓⣖⣲⢤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠋⣤⠖⠈⠉⠉⠉⠉⠉⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡞⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢋⠏⠀⠀⢸⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠤⠤⢄⡀⠀⢀⣠⣀⣀⣀⠞⣱⠋⠀⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⠀⣤⠀⠀⣠⣾⡿⠟⠛⠛⢑⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⡡⠂⢀⣤⢨⣬⠀⣸⣿⣫⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢿⢷⠁⠀⢾⣿⠸⣷⣾⡿⠛⠉⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢩⡏⡇⠀⠀⠀⠈⠻⢿⡛⠳⠀⢀⠀⠀⣿⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⢧⡇⠀⠀⠀⠀⠀⠘⣿⣄⣤⠛⠁⣼⡏⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡰⠷⠀⠀⠀⠀⠀⠰⠞⠿⠃⠀⢀⣿⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡈⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠘⣥⠞⠀⠀⠀⣿⡄⠀⠀⠈⠉⢉⣉⡙⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣀⠀⠀⠀⠲⠾⠟⠃⠰⣻⠁⠀⠀⢀⣐⡻⣧⢀⡤⣲⠽⠋⠉⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣤⣥⣤⢀⡴⠀⢀⣀⣀⣀⣿⣤⣶⣿⠿⢿⠿⣿⠷⡟⣠⡄⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⣠⡴⢋⡀⠀⠀⢀⣹⣿⣿⣻⣿⡏⠉⠙⣿⠋⢹⠧⢂⣴⢿⣿⡴⣟⠼⣝⢦⡀⠀⠀⠀⢠⣆⢹⡄
⠀⠀⠀⠀⠀⢠⣲⢏⣴⡟⠀⠀⠀⠀⠙⡾⠟⠋⠽⠁⠀⠀⣿⠀⣞⣼⡿⢃⡾⠁⢱⣝⠀⠈⠙⠳⠀⠀⠀⡠⣿⢸⡇
⠀⠀⠀⠀⢀⣿⠃⢸⡿⠀⠀⠀⠀⠀⠀⠤⣤⠤⡞⠀⠈⠀⢹⡀⣿⣿⠀⣸⠁⠀⠈⡟⡄⠀⠀⠀⠀⠀⢀⣷⣯⣾⠇
⠀⠀⠀⠀⢸⡏⠀⣿⡇⠀⠀⠀⠀⠀⠀⠐⠒⠈⠀⠀⠀⣿⢸⡇⢻⣿⡘⠁⠀⠀⠀⢳⡇⠀⠀⠀⠀⢈⣾⡿⣹⡟
⠀⠀⠀⠀⢸⣷⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⣧⠀⠛⠃⠀⠀⠀⠀⠀⠃⠀⣠⣶⢫⣾⢏⣴⡟
⠀⠀⠀⠀⠈⣿⣏⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⡆⠛⠉⠀⠀⠀⠈⠂⠠⠤⣴⡶⣿⣿⠷⣫⣴⡿⠏
⠀⠀⠀⠀⠀⠘⣽⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢃⠞⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⣻⠷⠶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠈⢻⣎⠀⠀⠀⠀⠀⠀⣤⠾⢛⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣻⣿⣷⣦⣄⣀⡀
⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⡿⣶⣶⣿⣿⣿⠿⠿⠿⠿⣯⣍⠫⠭⠍⠹⠯⠽⠿⠿⠿⠿⠛⠟⠶⠶⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢢⣿⣿⣿⣦⡠⠤⠒⠀⠀⢀⣀⣀
⠰⠶⠴⠤⠤⠤⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠨⣿⢿⣿⣶⡖⠒⠒⠚⠂⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢐⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠺⢿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⠂⠀⠀⠉⠁⠀⠀⠨⠥⠀⠀⢀⢀⣈⣉
⠐⠛⠋⠉⠉⠉⠀⠀⠀⠀⠉⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠈⠉⠀⠀⠀⠀⠐⠃⠙⠛⠛⠛
⢶⣶⡶⠶⢶⠦⢤⣤⣤⡤⠤⠤⠤⠤⠤⢤⣤⣀⣀⠀⠀⠀⢀⣀⣠⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⢠⣤⣤⣄⡀⢄⣀⣐⣲⢶⣶⣖⡀⢰⣶⣶⣶⣶⡶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⢀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⣿⠛⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⠿⠛⣛⣛⠻⠿⢿⣿⣽⣿⣿⣭⣿⡂⠀⠈⠉⠻⢿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠘⠿⠿⠿⠿⠻⠟⠛⠋⠁⠙⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠂⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣟⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣀⠀⠙⠋⡿⠟⠿⡿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣤⠴⢶⣦⣶⣄⣠⣤⣝⣻⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣯⣥⣤⣀⠘⠾⣽⣬⣤⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⢿⣿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠁⠀⠈⠐⠉⠙⠛⣿⢿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣷⣾⣯⣭⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠒⠒⠶⣶⣭⣭⣟⣻⣯⣵⣶⣤⣻⣝⢿⣿⣿⣿⣿⡻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣻⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿
⠐⠒⠒⠒⠚⠲⠚⣿⣶⣦⣽⠷⢶⣤⡈⠙⢽⣿⣿⣿⣿⣿⢟⣉⠉⠀⢈⡛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣾⣋⣉⣡⣾⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣚⣁⣀⡀⣾⣿⣦⣌⣉⣤⣤⣤⣀⣀⡉⢿⣿⣿⣿⣿⣷⣦⠤⠭⠹⢿⣿⣿⣯⣤⣤⣌⡛⠿⢿⣿⣾⣿⣿⣥⣤⣶⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠛⠛⠛⠋⠋⠀⠙⠛⠉⠻⡿⢿⣿⣿⣿⣿⣶⠤⠤⠀⠀⡤⠿⠿⠿⣿⣿⣿⣿⣷⣶⣶⣾⣭⣟⣛⣛⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠶⢦⣶⣶⡿⣿⣿⣶⣤⣮⣥⣤⣤⣀⠀⠀⠀⢠⣤⣶⣶⣦⣬⣤⣤⣤⣭⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤
⠀⠀⠀⠀⠀⢠⣤⣾⠛⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣶⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢛⣷⣤⡄
⠀⠀⠀⠀⠀⢸⡇⢸⠀⣿⢸⡇⠀⠀⢠⣤⣤⣤⣤⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠻⣮⠙⠷⣤⠀⠀⠀⠀⢀⣤⣤⣤⣤⡄⠀⠀⢸⡇⣿⢸⡇⢸⡇
⠀⠀⠀⠀⢸⣿⡇⢸⠀⣿⢸⣷⣶⣶⣾⣇⡀⢀⣼⣷⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣿⣶⣶⣶⣶⣾⣧⡀⢀⣸⣷⣶⣶⣾⡇⣿⢸⡇⢸⣿⡇
⠀⠀⠀⠀⠀⢹⡇⢸⠀⣿⢸⡇⠀⠀⠀⠉⢻⣏⠙⢧⡀⠀⠀⠀⢹⣣⡟⠳⠶⣤⣤⣤⣤⣤⠀⣿⠀⠀⠀⢠⡾⠉⣹⡟⠁⠀⠀⠀⢸⡇⣿⢸⡇⢸⡏
⠀⠀⠀⠀⠀⠸⠷⣾⣀⣿⢸⡇⠀⠀⠀⠀⠀⢹⣄⠈⢻⣄⠀⠀⢹⣿⡃⠲⠶⠀⠀⠾⠆⢸⡆⣿⠀⠀⣰⠟⠀⣰⠏⠀⠀⠀⠀⠀⢸⡇⣿⣨⡷⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⠾⠇⠀⠀⠀⠀⠀⠀⠻⣆⠀⠹⣦⣄⠘⢿⣇⠀⢀⣀⣀⠀⢀⣾⣿⠃⣠⡼⠋⠀⣰⠏⠀⠀⠀⠀⠀⠀⠸⠷⠿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠙⠳⣌⠙⢦⣀⣉⣉⣠⠾⢋⣵⠞⠉⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠘⢷⣴⣟⠉⠉⣿⣤⡾⠁⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⣠⡟⠁⠙⠛⠛⠋⠈⢷⣄⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢀⣤⡄⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢹⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠉⠉⠉⠉⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣤⣤⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⡇⢹⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢦⣾⠀⠈⣷⠴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⣿⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢰⡏⠀⠀⢻⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⢸⡇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣤⣼⠁⠀⠀⠘⣧⣤⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠁

⠀⢀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⡟⣉⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣠⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣾⣿⣿⣿⣿⣟⠻⡿⣿⣿⣿⣿⡿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⣿⣿⣿⢾⣿⣿⢿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⢀⣀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢽⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢀⣀⣤⣀
⠀⠘⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠋⠁⣼⠁⠀⠙⣩⠿⠛⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣛⣁⣶⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⠽⠛⠚⠹⡇⠀⠰⣇⠀⠀⢰⠇⠀⠀⣤⣾⣿⣷⣝⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭
⠀⠀⢸⡿⢿⡿⠋⢻⢻⠿⢟⡟⣿⣿⣿⠿⠿⠿⣿⣉⡀⠀⢰⠏⠀⠀⠀⠀⡇⠀⠀⠙⢦⠀⡜⠀⠀⢸⡇⠀⠀⠀⠉⠻⠶⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣓
⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠙⠿⣷⣾⠀⠀⠀⠀⠀⣧⠀⠀⠀⣸⡗⠁⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠛⠛⠿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠈⠳⣤⠀⣿⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⣄⡀⠀⢀⣿⡿⠋⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⡀⠀⣰⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠻⠿⠷⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⣤⣶⣚⣋⣭⣬⠹⣆⡀
⠀⠀⠀⠀⠀⣿⠏⠉⠡⠤⠬⣿⡏⣿
⠀⠀⠀⠀⢸⡇⢋⣀⠠⠤⠤⣼⣷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠖⠒⠒⠒⠀⢀⣠⠄
⠀⠀⠀⠀⢸⠀⠈⠀⣐⣶⠤⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠒⠶⠶⠶⠞⠋⠁
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⡶⢋⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⢿⠀⣯⣀⣀⣠⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⣀⣀⣀⡠⣄
⠀⠀⠀⠀⠘⣆⠀⠉⠭⡵⡞⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⠀⠰⣡⠚⠷⠚
⠀⠀⠀⠀⠀⢻⡀⢄⠀⠈⠙⣆⠹⣆⠀⠀⠀⠀⠀⣠⡶⠛⠉⠉⠉⠉⠙⠛⢿⣿⢶⡄⠠⣤⡀
⠀⠀⠀⠀⠀⢸⡇⠘⣆⠀⠀⠈⠃⠙⣦⡀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠹⣆⠈⠻⣆
⠀⠀⠀⠀⠀⢸⡇⠀⠘⣆⠀⠀⠀⠀⠈⠛⢦⣼⢇⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⠙⢷⣤⣀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠻⣮⣆⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⢀⠉⠙⢦⡈⠙⠛⠳⠶⠶⠶
⠀⠀⠀⠀⠀⠸⡇⢄⠀⢸⡳⣄⠀⠀⠀⠀⠀⠀⠘⣟⢻⡄⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⢸⠀⡆⠀⠇
⠀⠀⠀⠀⠀⠀⢿⡨⣂⠸⡟⢾⣶⣲⠤⠀⠀⠀⠀⣻⣤⡇⣀⣀⡀⢀⣀⣴⠟⠁⣠⠄⠀⢸⣤⡇
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣷⡀⠀⠀⠀⠀⢀⣀⣠⠟⢿⠑⠀⠉⠉⠉⠉⡁⠀⣞⠟⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⠻⠶⢤⣤⣤⣤⠅⠀⠀⠸⠀⣤⠀⠀⠀⠐⢬⣊⣸⠄⠀⠀⣾⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠘⢦⡀⠀⠀⠀⠀⠀⣠⠤⠖⠃⠀⠀⠀⠀⠀⠀⠀⠁⣠⡾⠃⠈⠻⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⡙⠲⠤⠴⠀⠈⢀⣀⣠⣤⣀⣀⣤⣤⣤⡴⠾⠛⠉⠀⠀⠀⠀⠀⠉⠛⠛⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣤⡴⠞⠛⠉⠉⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⢀⣀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢲⣶
⠀⠀⠀⢠⣴⣿⣿⠋⣀⣀⣾⠛⣷⠂⠀⠀⠀⠀⢰⣶⣶⣿⣷⠀⠀⠀⠀⠀⠀⠀⢠⡼⢼⣿⣤
⠀⠀⠀⠀⠀⠀⢠⣦⣤⣤⣤⠘⠏⠀⠀⠀⠀⠀⠘⡇⠉⢉⡇⠀⠀⠀⠀⠀⠀⠀⣿⠶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⣿⡆⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠹⡳⡄⠀⡜⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣯⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢽⠟⠀⠀⠀⠀⢠⠀⠀⢀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⡶⣶⣶⣶⠛⠁⠀⠀⠀⠀⢰⣶⣲⣶⣾⣿⣆
⠀⠀⠀⠀⠀⡠⠤⠤⠄⠴⠠⠄⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠉⠀⠀⠀⠀⠀⠀⢿⣍⠁⠉⠉⠉⡹
⠀⠀⠀⠀⠀⢸⣧⠴⢿⣦⢴⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠶⠒⠉
⠀⠀⠀⠀⠀⢈⣧⣤⣼⣄⣨⣏⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠠⠶⠿⠷⠦⠀⠀⠀⠀⠀⠀⠀⠰⠶⠿⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡜⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢹⣆⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⡏⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⡿⠟⢁⣴⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣤⣌⣉⣉⣉⣀⣡⣴⣾⣿⣿⣿⣿⡿⠋⢻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⢀⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣄⡀⠀⢉⣿⣿⠋⠉⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⣀⣀⣀⣀⣀⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠙⣶⣄⡀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠂⠀⠀⠀⠙⠿⢿⣷⣶⠶⠶⠶⢦⣀⡀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠁⣼
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⡀⠃⠰⡀
⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠀⡃⠀⣷⡀⣀
⠀⠀⠀⠀⠀⠙⠲⣦⣄⣄⣀⣄⣤⣶⠿⠋⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⢰⠆⠀⣧⠘⢦⡑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉⠉⠁⠀⠀⠀⠘⢦⡀⠀⠀⢐⣆⠀⠀⠈⢯⡳⢬⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⢈⠀⡀⢹⣆⠀⠹⣆⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⢀⠀⠀⠉⠊⠉⣽⠀⠀⣶⡀⠙⠶⣤⣻⣶⡀⠀⣀⠀⠀⣀⡠⠞⠉⠀⠀⠀⡇⠁⠂⢿⣦⡄⢸⡄⣿
⠀⠀⠀⠀⠀⠰⡴⠀⠀⠀⢀⡌⢸⠇⠀⠀⠀⠀⠑⠢⠴⣴⠋⠉⠠⠒⢹⠇⠀⠀⠀⠈⠉⠙⠫⢿⠷⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡇⠂⣇⠇
⠀⠀⠀⠀⠀⠀⠐⢢⠆⠞⣉⠑⣿⣆⠀⠀⡄⠀⠀⠀⠀⠀⠁⠒⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⣡⢼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠈⢽⡀
⠀⠀⠀⠀⠀⠄⢈⠢⡄⠠⠛⠐⠯⢾⡋⠀⢀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡠⠚⠁⡜⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⢁⢹⠠⣶⣾⣇⠀⠈⡀⠠⢠⣼⡇⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡠⠊⣀⣤⡾⠇⢸⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⠃
⠀⠀⣄⠀⠀⠀⣾⢿⡞⠁⠙⢡⠃⢠⡄⢁⢀⡞⢸⠃⠀⠀⢀⠀⠀⠀⠀⠀⣠⠤⠤⠴⠞⠛⠁⠀⠀⢸⢈⡗⠰⣦⢀⣀⣀⣀⣀⣴⠟⠁
⠀⠀⢀⠹⣷⠀⢨⠙⠙⠛⣿⢰⠀⣞⣀⢬⡿⢁⡟⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠁⠀⠀⠉⠉⠛⠉⠉
⠀⠀⠀⢢⠹⠾⣟⢦⡀⠆⠘⣤⣸⢏⣽⠟⣰⠟⠀⠀⠀⠀⠀⠀⠓⠂⠀⠀⢀⠀⣠⣤⣀⠀⣀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖
⠀⠀⠀⠀⠁⠀⠉⠳⢝⡢⡀⠘⠛⣿⣣⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠉⢉⣲⠦⣤⣀⣈⣙⣲⣄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠤⠀⠉⠛⢶⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠳⣄⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⡀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣱⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⣀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠙⠦⣤⣤⣄⣀⣀⣠⣴⡾⠛⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⣀⡴⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠶⠶⠶⠺⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠲⠶⠖⠛⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⢀⠄⠀⢀⡀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢻⣷⢸⣿⡇⠀⠀⠁⠀⠀⠈⠀⠀⠀⠁⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⡀⠘⣿⣾⣿⡇⠀⠀⠠⠀⠈⠀⡀⠁⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡶⣿⣿⣿⡧⠴⡶⠶⠶⠶⠚⠋⠀⠀⣀⢀⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠊⠙⠛⠓⠀⠀⠀⠀⢀⣀⠀⠀⡆⣿⣾⣠⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢀⠀⣠⡀⢁⠀⠰⢎⠁⣦⢸⣿⣿⣿⣿⣥⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡶⣤⣤⡀⠇⠃⠐⠀⠁⠀⠀⠜⠀⠸⡇⢿⣿⣿⡿⣿⣿⣖⠋
⠀⠀⠀⠀⠀⠀⣶⣶⣿⣿⣏⢻⣿⣇⠹⣿⣿⠄⠀⠀⠐⠀⠐⠀⠀⠄⢠⡇⠈⠇⢃⣨⣿⣿⠅
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣿⣿⣿⣿⣿⣿⣧⣀⠂⠀⣄⠀⠀⠀⢀⣠⠏⠐⡄⢈⣴⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠈⠻⠿⢛⣻⣿⣿⣟⣿⣻⣤⡆⠀⢈⣭⠿⣿⣿⣯⣄⠰⠀⢈⣽⢿⣯⣷⣿⣿⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠟⢋⣿⣿⢻⣿⣷⡇⣀⠋⠀⠐⠀⠈⠛⣿⡆⠀⠸⢿⣟⡿⢿⣯⣥⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⣿⣿⢏⣼⣿⢩⣏⠉⠢⡀⠀⡳⠀⠂⢸⡇⢀⣿⠾⣿⡗⢶⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⢿⣿⣿⣿⣿⡗⠤⢬⢅⣤⣽⡤⠤⠖⢻⠿⠷⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠴⠰⠄⠠⠦⠐⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠄⠀⠰⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠈⠐⠀⠐⠈⠀⣀⠀⠰⠀⠁⠈⠀⠀⠀⠀⠀⠠⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠀⠀⠉⠀⡄⢠⠀⠠⡄⠈⠈⠂⠀⠚⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠊⠩⠀⠘⠰⠀⠄⠈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠉⠈⠒⠢⣄⣀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⠁⠀⢀⠀⠈⡐⠀⠀⠀⠀⠀⠀⠀⠈⠠⢄⠀⠀⢐⣤⡜⠂⢎⡀⠀⠈⠁⠉⡒⠰⠖⢢
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⡀⡀⠂⠀⠈⡂⠀⠀⠈⠒⡇⡇⠲⠐⠛⠓⢒⡒⡎⠬⠁⠀⠀⠀⠣
⠀⠀⠀⠀⠀⢀⢀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠤⡄⠐⠒⠀⠀⠀⠀⠀⠀⠀⠐⢄⢠⠤⡇⠁⠈⠀⠀⠀⠈⢂⠀⡇
⠀⠀⠀⠀⠀⡖⠀⠀⢀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠠⠀⠀⠆⠤⠠⠁⠀⠒⢢
⠀⠀⠀⠀⠀⠈⠀⠃⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠀⠀⠂⠄⠂⠀⠀⠀⠀⢀⢁⠀⠀⠠⠤⠀⢠⠘⠁
⠀⠀⠀⠀⠀⠀⢀⠊⡀⠀⠀⢀⠀⢠⠀⠀⠀⠀⠀⠆⠈⠀⠂⠐⡀⠠⣌⣀⣀⣀⡀⠀⠀⠀⠐⠀⠈⠠⢄⠄⠀⠱
⠀⠀⠀⠀⠀⡈⠀⠀⠀⠁⠄⠐⠀⠁⠄⠀⠀⠀⣀⢀⠤⠤⣤⠈⠑⠤⢄⠠⡅⠠⠋⢏⠈⢃⣀⣀⡀⠀⠀⠀⡀⡠⠃⠂
⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠁⠀⠀⠈⠀⢄⣠⢿⠊⣈⠤⠀⠀⠀⠀⠀⠀⠠⠘⠄⣀⣈⣇⡀⠹⢂⠉⠉⠉⠀⠀⡀⠡⠨
⠀⠀⠀⠀⠀⠀⠀⢀⢈⠒⠈⠆⢀⠀⢀⢀⠠⠊⠀⠁⡄⢀⡀⡀⠀⡀⡄⡀⡀⡄⠠⢠⠓⠆⠐⠁⢀⠒⠤⠤⠰⠄
⠀⠀⠀⠀⠀⠂⠠⢀⠀⠀⢀⠈⠉⠀⡀⠀⠀⠀⠀⢠⢈⣀⡀⢈⠉⠈⢀⠁⠯⠙⣻⢀⢀⡀⢀⣠⠀⠢⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡌⠑⠒⢀⢂⠓⡲⠀⢄⠚⠈⠁⠃⡐⠂⢅⢀⣠⠀⢌⢠⡐⠃⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡌⠀⠂⠀⠀⠀⠐⠒⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠤⠤⠴⠀⡄⠄⢄⣄⡴⠄⣄⠤⠴⢰⣆⣠⣀⡀⢠⣴⣦⡄⣆⢤⣤⢠⣤⢀⣤⣄⣄⢠⣤⣤⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠉⠈⠀⠀⠀⠀⠁⠀⠁⠀⠀⠈⠀⠀⠁⠀⠈⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣦⢱⡀
⠀⠀⠀⠀⢀⣤⣸⣿⣿⣿⣿⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⢢
⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⢸⡏⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣿⣿⣿⣿⡎⡇
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢹⣿⣿⣿⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠈⠻⣿⣆⡀⠀⠀⠀⢠⣿⣿⣷⡄⠰⠃⡤⢿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣜⠦⢀⣀⣸⣿⣿⣿⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣾⣯⣿⣿⣿⣧⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢿⣿⣿⣿⣿⣇⣀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣡⣿⣿⣿⣿⣿⣿⡿⣻⣽⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠋⠉⠉⠉⢉⣼⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⢰⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⠀⠀⠀⠀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣤⣤
⠀⢀⣸⣿⣿⣿⣷⣀⣄⣀⣀⣀⣀⣠⣤⣤⣤⣤⣀⣠⣄⣰⣿⣿⣿⣿⣀⡀⢀⣰⣿⣿⣿⣿⣠⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⢤⣿⣿⣿⣿⣄⡀
⠀⠈⠹⣿⣿⣿⣿⠈⢿⣤⡀⠀⠀⣀⣀⠀⠀⢀⣤⡾⠃⠸⣿⣿⣿⣿⠀⠀⠀⠈⣿⣿⣿⣿⠀⠻⣦⣄⠀⠀⣠⣄⠀⠀⢀⣴⡾⠃⠀⣿⣿⣿⣿
⠀⠀⠀⠉⠉⠉⠁⠀⠀⠙⣿⣦⣄⣿⣿⣀⣴⣿⠋⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠈⢿⣷⣤⣿⣿⣤⣴⣿⠏⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠇⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⠁⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠅⠀⠀⠀⠀⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠴⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠈⠉⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⠶⠒⠂⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⡀⠀⠀⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠴⠒⠈⠀⠀⠀⠀⠠⠔⢁⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⣿
⠀⠀⢀⡴⠶⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿
⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣿
⠀⠀⠀⡀⠂⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠜⠀⠀⠀⣬⡿
⠀⠀⠀⢇⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⣾⠁
⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠘⣆⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠁⠀⠀⢠⣼⠟
⠀⠀⠀⠀⠀⠈⢦⡀⠀⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠛⣆⠀⠚⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣈⠛⠳⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣥⣽⣓⡲⠤⣤⣀⢀⣀⣀⢀⣀⠀⣀⣀⣤⣤⡶⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠙⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁




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

⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣂⣠⣄⠀⠀⠀⠀⠄⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⢀⣴⣀⡔⣄⣄⠀⠀⠀⠀⠀⠇⠀⠤⠄⢿⠀⠀⠀⠀⠀⠀⠉⠀⠻⠀⠀⠀⠀⢀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠘⠯⠝⠉⢹⣿⠀⠀⠀⠀⠀⡄⠐⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⠻⡇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡟⠀⠀⠀⠀⠀⡄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⡆⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⡁⢠⣀⣸⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣁⠀⠄⣸⠃⠀⠀⠀⠀⠀⣄⠀⠀⢀⣸⠀⠀⠀⠀⣀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⡀⠀⣠⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠀⠀⠀⠀⠀⠈⠀⠐⠋⠁⠀⠀⠀⠀⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠐⠛⠋⠀⠀⠀⠀⠀⠀⠉⠀⠒⠒⠂⠉⠁

⠀⠀⠀⠀⢠⡄⠀⠀⣲⡕⣠⠀⠀⠀⠀⠔⠂⠀⠐⠂⠐⡆⠀⠀⢎⠁⠈⠉⠁⢹⡖⠀⠀⢰⠖⠂⣀⢐⡶⠶⡦
⠀⠀⠀⠀⠈⠳⠀⠀⠀⡿⠃⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠃⠀⠀⠀⢣⠀⠀⢠⠋⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢣⡀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠰⠆⠀⠀⠀⠀⠈⠀⠀⠀⠈⠙⣷⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣋⣀⠃
⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠟⠁
⠀⠀⠀⠀⠀⠢⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠌⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠘⠀⠀⠀⠀⠼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⢸⠚⠀⠀⠤⠤⢽⡆⠀⠀⠀⠠⠧⠴⠿⠷⣶⡄⠀⠀⠀⠄⠀⠂⠀⣤⠀⠀⠀⠀⢸⠋⠀⠀⢾⡇⠀⠀⠀⠀⣖⠀⣀⠨⡲⢶⡄
⠀⠀⠀⠀⠈⡄⠘⠀⠚⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⠀⠀⢸⡇⠀⠀⠀⠀⡍⠴⢻⡖⣵⠸⡇
⠀⠀⠀⠀⠀⢱⠄⠐⠂⠁⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡟⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠁⠀⠀⠀⠀⠯⠂⣑⣁⠞⠐⠇
⠀⠀⠀⠀⠀⠈⣄⠀⠀⣠⠁⠀⠀⠀⠀⠀⢦⠀⢀⣠⡇⠀⠀⠀⠀⠦⠠⠀⣠⡆⠀⠀⠀⠀⠀⠃⠀⠀⣸⠀⠀⠀⠀⠀⣤⠀⠀⠀⣠⣬⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠉⠒⡄⠀⠀⠀⠀⠀⣴⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣬⣅
⠀⠀⠀⣴⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣠⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢻⠿⠿⠿⠇
⠀⠀⠀⢿⣿⣿⣿⣥⣤⣤⣤⣤⣠⣄⣄⠀⠀⠀⠀⠀⣿⣾⣷⣿⣿⣿⡏⠛⠷⠀⠀⠀⠀⠀⢩⢿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠉⠉⠻⢿⣿⣿⣿⣿⡿⠿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣍⣉⣭⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣿⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠂⠤
⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣶⣶⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⠤⠤⠤⠤⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠉⠟⠁⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣛⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠿⠿⠶⠆⠀⠀⠀⠀⠀⠀⠀⠰⠶⠿⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠿⠷⠦
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⣦⠀⣠⠞⣽⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣻⡿⠁⠀⡸⣿⣷
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣆⠠⣿⣿⣿⣄
⠀⠀⠀⠀⠀⢀⡟⣻⣿⣿⣶⣿⣿⣿⣿⣅⡀⠀⢸⡄
⠀⠀⠀⠀⠀⣼⣡⣿⣿⣿⣿⣿⣿⣿⣿⣯⡁⢠⣿⣇
⠀⠀⠀⠀⢀⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦
⠀⠀⠀⠀⢸⣯⣽⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣾⣿⣿
⠀⠀⠀⠀⠈⣿⣦⣿⣿⣿⣿⣍⠉⢉⣙⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣷⣦⡀⢀⡀⠀⠀⠀⣀⣤⣠⣴⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⢻⣁⣤⡈⢙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣎⣉⣙⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠿⠛⠛⠛⠙⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠑⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠿⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⠀⠀⠀⠀⢀⣈⣽⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣆⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⡃⣿⣠⣤⣶⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣶⣦⣶⣿⣾⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⠀⠀⢻⡆⠀⠀⢻⠀⠀⠸⠇⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠓⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⡶⠿⠿⠿⠿⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⢀⣤⣶⠿⠛⠋⠉⠀⠀⠀⢀⣀⣀⣀⣲⣄⣈⡉⠛⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢶⡄⠀⠀⠀⠀⣠⣾⠿⠋⠁⠀⠀⠀⠀⠀⠂⠉⠁⠁⠀⠀⠀⢣⠁⠀⠉⠑⠢⢙⠻⣷⣄
⠀⠀⠀⠀⢀⢀⠀⠀⠈⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀
⠀⠀⠀⠀⡅⠀⢣⠀⠈⡁⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⢸⠀⢠⠇⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷
⠀⠀⠀⢀⠀⠀⠈⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣴⣤⣀⡀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢰⡏⠀⠉⠙⠛⠾⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⣸⠃⠀⠒⠀⠀⠀⠀⠀⠉⠉⠳⠴⠤⢄⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⣿⡅⠀⠃⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏
⠀⠀⢸⣇⠰⠖⢋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠠⢈⠉⢹⠛⠲⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠠⣤⡄
⠀⠀⠸⣿⡀⢱⠉⢃⡴⠊⠀⠀⠠⠀⠀⠀⡀⠀⠀⠄⠈⠓⢤⠁⣰⠏⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢆⣰⡿⠋⠀⢀
⠀⠀⠀⢻⣧⠀⠱⡋⠀⠀⠄⠀⡆⠀⠀⠀⠸⡄⠀⢀⠞⠀⠀⡰⠃⠀⢠⣿⠀⠀⠀⠀⠀⠀⠰⣧⣼⡾⠋⠀⠀⠀⠈⠻⣶⣤
⠀⠀⠀⠈⢿⣆⠀⠙⣜⢮⠀⢠⠃⠀⣀⠀⠀⠹⡄⠀⠈⣠⠞⠁⠀⣰⢯⠇⠀⡀⢀⣀⣤⣴⡾⠟⠉⠀⠀⠀⠠⣄⠀⠀⠀⠉
⠀⠀⠀⠀⠈⢿⣧⡀⠈⠳⣄⡚⠀⠀⠀⠀⠀⢀⣀⡤⠞⠁⠀⣠⠞⢡⣿⠶⠷⠿⠛⠛⠉⠁⠀⠀⠀⠀⣄⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠉⠙⠓⠒⠚⠋⠉⠁⢀⣠⡴⢚⣡⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⣧⡀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⠶⣦⣤⣤⣶⣶⣾⣿⡿⠿⠟⠋⠁⠀⠀⠀⠀⢠⡀⠀⢰⡀⠀⠘⣦⡀⠀⠘⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠖⠒⠛⠉⠉⢉⠙⠛⠒⠢⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠈⠉⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁⠀⠀⠀⠀⠉⠉⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢦⡀⠀⠙⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠑⠢⣀⠙⢀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠈⢦⠘⢷⡄⠀⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠆⠌⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⡀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠈⠀⠀⡁⠆⠀⠀⠀⠀⡐⠒⠀⠀⠀⠀⠀⠀⡐⠁⢀⠘⡇⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⡄⢹⠀⠀⠀⠀⠀⢨⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠈⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠈⠀⠈⠀⠀⠠⠴⣦⡄⠀⠀⠀⠳⠀⠀⠀⢀⡔⠀⠰⠁⠀⠀⠐⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⢐⠒⠊⠉⠴⡆⠀⢸⠃⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⢠⠄⠛⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡾⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⠀⣀⠔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡆⠀⠈⠓⠂⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⢀⣉⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⢈⠋⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⣄⠀⡀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢼⠰⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣇⠀⠑⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣤⡀⠀⠠⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣍⠀⠀⠈⠙⠶⠤⠴⠋⠐⠀⠤⠄⠀⠀⠀⣀⣠⠶⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣬⣀⠐⠲⠤⠤⠴⠄⠐⠒⠲⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣾⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⣶⣶⣷⣾⣷⣾⣿⠿⠟⠿⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣿⣿⣯⣽⡏⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⡿⢀⣄
⠀⠀⠀⠀⠀⠀⣠⡿⠋⠉⠳⡀⠀⠀⠀⣻⣿⣿⡆⠀⢸⡀⠀⠀⢀⡀⣀⣤⠛⠙⠛⢶⣿⣿⢸⣧⣸⣿⣿⠆
⠀⠀⠀⠀⠀⣰⣿⡇⢀⣤⣠⣷⣶⣶⣭⣽⢀⠀⢠⣄⣸⠛⠛⠛⠛⠋⠉⠉⠛⣿⡆⢠⠈⢿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠘⠛⣿⣿⣾⠏⢨⡇⢀⣴⠋⠁⠀⠀⠀⠹⠧⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⡇⠀⠈⠻⠟⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⠛⣴⡿⠃⢀⡖⠀⠀⠀⣀⠀⠀⠀⠀⢦⣀⡀⢀⣼⣷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢠⡁⠀⠀⠀⠉⠉⠃⠀⠀⠈⠣⠄⠙⠃⠈⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⢤⠀⠀⠀⠀⠰⣶⣶⣄⣀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠰⣿⠋⣸⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⡄⠀⠀⠀⢀⡼⠋⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⣀⠀⠉⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣤⣌⣁⣠⣤⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠛⢿⣿⡿⠟⠛⠛⠛⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠹⡇⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠇⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⠀⠀⢠⡇⠀⠀⠀⠀⣈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⣠⠋⢹⣄⡀⠀⢀⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣀⠁⠀⠀⣿⡧⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠏⠀⠀⠀⠇⠀⠀⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⣀⠀⠀⠀⡘⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢰⠇⠀⠀⠀⢷⡔⠆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡾⠀⠀⠀⠀⠈⢿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣵⣄⡀⠀⠀⠀⠸⡇⠀⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠟⠛⠁⠀⠀⠀⠀⣿⣤⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢀⢄⣤⡴⡦⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⡶⢲⣄⢞⠘⢺⠔⢏⣢⣷⠗⢿⠂⠀⣏⣧⢤⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⡦⡄⣥⢘⣺⠼⡲⣠⣑⡶⢤⡙⡆⢩⣺⢅⢛⠛⣘⣣⢤⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣾⣹⣿⣷⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⢯⣗⣻⡿⠰⣀⢝⣹⣿⠜⡽⣴⢳⣩⠿⠽⢟⢊⣈⡭⣷⠂⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⢿⣏⠿⣫⣿⢿⣧⣿⣿⡌⢟⣛⣛⣻⣷⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡼⣅⢙⣦⣼⢼⡴⡇⢸⡋⢡⢟⣬⡇⣤⢺⣈⣱⢹⡽⢲⠰⡄⠀⠀⣿⣿⣿⣿⣿⣿⣷⣝⠃⣿⣿⢸⣇⢾⡟⣱⣿⣿⡿⣟⣽⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢖⠢⡝⣹⠛⡾⠊⢯⡁⠑⠦⣽⡿⢠⡾⠋⢡⡗⠆⣉⣄⢁⡤⡟⠁⠀⠀⣿⣿⣿⣿⣿⡿⣿⣿⣷⡘⢿⣮⣟⠂⢠⡿⢩⣵⣾⣿⣿⡿⣻⣿⣿⣿⣿⣿
⠀⠀⠸⣖⣶⢈⠟⢺⠹⡍⠛⠷⣦⣸⣷⡾⠓⢁⣏⡙⣴⣁⣿⣻⣾⡥⠀⠀⠀⣿⣿⣿⣿⣿⣷⣯⣿⣶⣶⣦⣌⡻⣷⠸⢃⣽⣿⣿⣶⠏⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠮⣝⣶⠦⠛⠛⠒⠻⣿⣿⣧⡴⠾⠷⠚⠁⠀⠀⠈⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣮⣜⣃⣤⣤⣽⣅⡀⠀⠘⣛⣫⣭⣭⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣭⣥⣿⣿⣿⣽⡁⠀⣴⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠿⠿⠷⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣀⣀⣈⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠤⠀⠀⠀⠠⠤⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣄⠀⠀⣀⣀⡠⠀⠁⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠈⠙⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠈⠉⠀⠀⠙⣷⡀⠈⢋⡀⠴⢚⣀⣈⡀⠀⠀⠀⠀⠉⠛⠒⠢⢀⡀⠀⠀⠀⠈⠑⠦⣄⡀⠀⠀⠀⢀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⣀⡀⠀⢀⡀⣤⢸⢷⠒⢁⣀⡀⠀⠀⠈⠉⠛⠶⣄⠁⠢⡀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠘⣱
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣀⡊⠀⣞⡛⢛⣀⠀⠀⠀⠀⠀⠀⠈⠓⣦⡈⠒⢄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠙⠛⠻⣷⣤⣂⠉⠉⠙⠒⠲⠤⣀⠀⠀⠀⠉⠲⣤⡑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠄⡤⠒
⠀⠀⠀⠀⠀⢠⡞⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠻⣌⠁⠀⠀⠀⠀⠙⢦⣀⠀⠀⠹⣆⠁⠙⠛⠒⠒⠲⠶⠗⠊
⠀⠀⠀⠀⢰⠏⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣀⠄⠀⠀⠀⠀⠙⣷⡀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⣇⠀⠀⠀⠀⠁⠀⠘⣿⡄⠀⣿⠀⠀⠀⣠⡴⣊⠀⠈⢿⣦
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⡆⠀⠄⠀⠀⠀⠀⠘⣿⠀⢸⣦⠖⢋⠁⠀⠬⣹⡆⢸⠻⣷⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢣⠠⠀⠀⠀⠀⠀⣀⡽⠗⠋⡰⠛⠀⠀⠒⠀⢸⣷⠀⡇⢻⢷⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡼⠂⠀⠀⣀⣴⠾⠋⠀⠀⠀⠀⠀⠀⡌⠉⡤⠈⠟⠀⢃⠈⠋⣷
⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣇⡴⠛⠉⢤⡤⠜⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⡇
⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡿⠐⠐⠂⠈⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡸⠀⠀⠀⢻
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠁⠀⠸⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⡘⠀⠸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠇⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢠⠁⢀⠆⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠂⢀⠊⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡠⠃⢠⠂⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢀⣦⠓⠀⠀⠀⠀⠀⠈⠀⢠⠀⠃⠀⠀⢀⡠⠊⢀⡴⠃⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣤⡀⠒⠂⠀⠀⠀⠀⠀⠈⠀⠀⠛⣵⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠁⢀⡴⠋⠀⠀⠀⠀⠀⢔⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣆⣀⠀⠐⠂⢀⣀⣠⣴⣾⣤⡈⢉⠓⠒⠛⠂⠀⠀⠀⠀⠈⠀⠀⠀⠚⠉⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠷⠦⢤⣾⣿⡧⠬⠿⠷⢾⣥⣤⡤⠄⠀⠀⠀⠀⠀⡀⠀⠀⣀⣠⣤⡴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠶⠾⠿⠶⠞⠛⠛⠉⠁

⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣶⣶⣶⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⡁⣠⣾
⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣝⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣾⣿⣿
⣴⣄⠀⠉⠉⠻⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⣿
⠿⠟⠛⠀⠄⠀⠘⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢽⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣖⣧⣶⣿
⠈⠁⠀⠀⠀⠀⠰⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣋⠙⠛
⣿⣾⣿⣿⣿⣶⣶⡿⠛⠛⠋⠉⠙⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣧⡤⠉⠙⢫⣭⣤⣤⣤⣤⣤⣄
⣿⣿⣿⣿⣿⣿⣿⣥⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡁⠉⣀⣰⣾⣿⣿⣿⡟⣿⣿⣿⡿⠿⠿⣿⣿⣿
⣛⣿⣿⣿⣿⣿⣿⡯⠀⡀⠐⢀⣤⣶⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⠂⢘⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⢙⣿⣿⣿⣿⣿⣿⣿⣿⡄⢰⣿⣿⣿⣿⣿⣿⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣼⣿⣿⣿⣿⣿⣿⣿⣟⡃⠠⢤⣤⣄
⣨⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣷⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣼⣿⣿⠿⣿⣿⣯⣭⣉⠉⠀⠀⠁
⠈⠈⠀⢻⣹⣯⣍⢻⣷⡞⠻⠿⠛⠉⠙⢻⠟⠋⢶⠒⠦⠄⠀⠀⠀⠀⠀⢠⠀⠀⠠⠿⠛⠁⠀⢹⠃⠹⠟⠻⢷⡀⠀⠀⠀⠀⠚
⣠⣀⣀⣴⣾⣿⣿⣿⣿⣿⣷⣄⠀⠠⣾⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⣈⡀⣿⣄⣁⡄⠀⠀⠀⠀⢈⡁⠀⠀⣀⣀⣙⣷⣇⣀⣠⣿
⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⢻⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⡟⠀⠀⠀⠀⠀⠐⠺⣇⠛⠻⠿⠿⠿⣿⣿⢿⣿⣿
⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⠶⢶⡿⠿⠿⠗⠂⠀⠐⠑⠒⠂⠐⠒⠒⠾⠟⠓⠒⠒⠒⠒⠒⠒⠚⠉⠁⠀⠀⠀⠀⠋⠙⢛⣛⣛
⣿⣿⣿⣿⣿⣿⡿⠁⠠⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣽⣿
⠉⠩⢷⡨⠭⠁⣀⣠⣤⣤⠤⠶⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠙⣆⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠲⡄⠀⠀⠀⠀⠀⠀⠙⣆⠈⢧⠀⣀⡤⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⢹⡄⠀⠀⠀⠀⠀⠀⠸⡄⢸⠟⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⠇⠀⠀⣠⠴⠖⠒⠶⢿⡎⠀⢀⠎⠀⣿⣠⣤⠤⠤⣤⣄⡀
⠀⠀⢠⡞⠛⠛⠶⣄⠀⠀⠀⠀⠈⠀⠀⠘⣷⡄⠀⠀⠀⠀⢣⡔⠁⣀⡜⠉⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠘⢧⣄⣀⣀⣸⡶⠀⠀⠀⠀⠀⠀⠀⣿⠘⡆⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢿⠛⠒⠒⠛⠁⠀⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⢶⡒⠦⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠟⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡿⢍⡁⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠲⠶⠶⠶⠶⣤⡴⣤⠶⠶⢤⣤⣤⣤⠴⠖⠋⠁


⣿⡯⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⢂⣀⣀⣀⣈⡀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣌⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠤⢄⣀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⢀⡀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⡠⠤⠤⠄⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠂


⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣶⣵⣶⣤⣾⣵⣿⣽⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠘⣦⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠌⠤⠀⢧⠼⡷⢧⡤⠁⠀⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠀⣬⡘⣧⡼⣐⣫⣩⠽⢛⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣶⣾⣤⠏⣼⠿⣟⣁⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⢸⣿⣿⣿⠿⠛⠀⠈⠉⠉⠀⠀⠀⠉⠉⠉⠐⢺⣿⡗⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⣇⡾⠛⠋⠀⠀⠀⣀⠔⠊⠓⢤⣀⠀⠀⠀⣠⡿⢾⣿⠁⠀⢀⣨⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⣷⢶⣶⠶⣴⣾⣤⣤⣤⣤⣤⣽⣶⣤⣶⣴⣾⣾⣿⣾⠿⢿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⢻⡀⠀⢀⠁⠀⠀⠉⣿⡯⣉⠉⢉⠀⠍⠤⣾⣿⠟⠃⠀⣾⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣇⡀⠀⠀⠀⠀⠀⠘⠷⠤⠛⠛⠋⣁⣉⣉⣥⣤⠄⢰⡟⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠉⠉⠛⠛⠛⠉⠉⠉⣿⣿⠟⠛⠻⠟⢋⡉⠉⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠂⠀⠀⠀⠻⣿⣿⠆⠀⢀⢲⣿⣆⠀⠀⣸⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⡀⢈⢹⡿⠋⠀⣿⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⢻⣿⠁⡀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⡹⠋⠀⢠⣾⣷⠀⡃⢀⣿⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⡆⠀⠄⠀⠀⠀⠀⠀⣌⠏⠀⠀⣠⣾⣿⢿⣴⠇⢸⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣇⠈⠀⠀⠀⠀⠀⢸⡏⠀⢀⣴⡟⠃⣼⠋⣿⠃⢸⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠠⡀⠀⠀⠀⢸⡧⠾⠋⠁⣠⣾⠿⣌⣽⠆⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⣇⠀⠈⠳⣄⡀⠀⠀⢀⣠⡾⠻⠋⣸⣿⣿⣄⢸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⢦⣄⣐⡮⠽⡶⢶⣿⡇⠀⢀⣴⣿⣿⣿⡿⢋⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠈⠉⠛⠛⠛⠛⠛⠓⠛⠛⠛⠋⢉⠀⠐⣩⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⣤⣤⣀⣀⣀⣀⣀⣤⣤⠴⠖⠋⠁



⣿⡿⠛⣻⡿⣿⣿⣿⢿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣥⣽⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⠶⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠀⠀⠉⢯⢢
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠙⡜⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣘⠀⢀⡴⣦⡀⠈⡆⢃
⠀⠀⠀⠀⠀⢀⣆⣀⢔⠺⡆⠀⠀⢁⠸⣖⣾⡋⢶⠤⠭⠭⠭⠭⠭⠑⠺⣯⢽⡥⠤⠨⣗⣼⠇⠀⡇⢸⠗
⠀⠀⠀⠀⠀⠙⡖⠂⠻⠶⠃⠀⠀⡈⢠⠀⠀⢣⠈⢆⠀⢀⣠⢤⡴⣤⡠⠋⠻⠘⣄⠀⠀⠀⠀⣰⢁⠆
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⣠⢃⠎⠀⠀⠘⡄⠈⣴⠋⢁⣀⣀⣸⠃⢠⠃⠀⠈⠢⠤⠤⠴⠥⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠴⠞⠓⠉⠀⠀⠀⠀⠘⡄⠹⣦⡏⢙⢷⣿⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢹⢿⡻⠿⣇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠁⠀⠙⢒⡃⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⣀⢀⡁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡤⢺⡎⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⠺⢣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⢭⡥⢒⡚⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠘⠀⠀⡨⡓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣘⣤⣸⠈⡒⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢻⠑⢺⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣅⣀⠔⠛⢩⠄⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠟⠢⡄⠸⠄⠉⢉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠢⡂⠂⢎⢦⠂⠀⡀⡀⠐⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠤⡀⠑⠐⢄⠀⠀⠀⠣⠣⣠⣱⣀⠄⠀⠀⠈⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣐⡈⠐⠠⢈⡐⢤⡠⡷⠒⢉⣙⣃⣭⣟⣿⡼⡆⣄⣀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠬⠁⡒⠺⢷⡾⢿⡶⡾⢭⠷⣿⢿⠾⢿⢿⣿⣽⡏⣰⠀⠀⠀⠀⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠂⠤⢄⣀⠀⠀⠀⠡⢀⡀⠀⠀⠀⡀⢀⢀⢿⣿⣿⣥⡇⠀⠀⣴⡮⠞⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠌⠀⠀⠀⣀⠀⠁⠁⢐⡦⠤⣀⣭⣖⣲⣬⢿⡿⠯⠟⠛⠻⣿⣶⣿⣭⠖⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠬⠩⠅⠀⠀⠀⠀⣀⠀⠀⢀⡍⣤⡭⠠⠬⠵⠿⢿⣿⣾⡟⠀⠀⠀⠀⠈⣿⣿⣷⠒
⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⣀⣀⡐⢹⣩⣍⣙⡢⠒⠂⣔⡘⣛⣿⣗⣀⠀⠀⢀⣜⡿⣿⣛⠷⣂⣠⣎⡭
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠝⢉⠢⢘⡴⠞⢾⡿⠻⢯⠿⢋⣻⣦⣬⠿⣽⣿⢿⣿⣷⣿⣿⣯⡿⣦⣿⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠔⢲⣋⣾⣿⣿⣿⣤⣔⣦⣴⣦⣦⣠⣼⣷⣶⣤⣄⡿⢛⣿⣺⠻⣿⣿⣿⣿⣿⣕⢢⡉⠪⡂
⠀⠀⠀⠀⠀⠀⠀⢀⠨⠛⡛⠽⡛⢟⡻⠛⢛⠋⠽⠻⣿⣽⣻⣿⣿⠿⢻⠏⡝⠁⠀⢹⣻⢋⠯⢫⡿⣣⡑⠀⠨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠊⢁⡀⠔⠊⠂⢅⡤⠀⡀⣴⣮⠿⡻⢷⠟⢡⡲⠣⠀⡄⠀⠀⠈⠛⠃⠈⠀⠙⢎⢽⡔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢕⠮⢒⡵⢌⣡⡴⣖⣾⣿⣿⣪⡔⡱⡠⡳⡱⠡⡰⡈⡐⠀⣤⣠⣀⣈⣀⣀⣪⣣⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣽⣾⣿⣿⣿⣿⣿⣿⡿⠛⢻⡟⠻⠷⣷⣷⢧⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⠿⠙⠛⠀⡀⠉⡀⠄⢢⣭⣟⡿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣶⠀⠮⣥⣄⢀⡀⠤⢊⠔⡉⢍⠟⠕⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠛⠻⠧⢖⠪⠭⡐⠠⠐⠌⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠃⠀⠈⠛⣿⠏⠀⠹⣦
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠀⠀⠀⠀⠸⡆⠀⠀⠙⢷⡄⠀⠀⠀⠀⣀⣤⡖⠚⠉⢉⣉⣉⡐⠲⢄⡀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⠿⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢻⣆⠀⣠⣾⣿⣍⣷⢷⣆⠉⠛⠛⠛⠀⠀⢹⣦
⠀⠀⠀⠀⠀⢠⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣴⣿⣿⣿⣿⣷⣿⠿⠟⠛⠛⠛⠛⠛⠛⠻⠇
⠀⠀⠀⠀⠀⣸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢀⠀⡄⣀⣀⡀⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣷⡆⠀⠀⠀⣤⣤⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⡄⣐⣲⣦⠾⠷⠛⠧⠿⠿⠿⠿⣷⣧⢀
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣦⣶⣿⣿⣿⣿⡏⠀⠀⠀⠀⡴⣩⡧⠾⠶⠛⠛⠋⠉⠉⢁⣀⣀⣀⣠⣀⣄⣀⣀⣀⣀⣏⡏⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣜⣿⣇⠀⢀⣀⣀⡤⠤⣶⣶⢏⡵⠒⠐⠖⠒⠓⠐⠦⠶⠲⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⠿⠿⣿⣿⡁⡄⠀⠀⠈⠋⢾⡿⠭⠵⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣤⠝⣈⣿⣯⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⢤⣤⣤⣤⣤⡶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⢠⣃⡤⠚⢙⡠⠐⠁⠀⠈⠀⣿⣷⣦⣤⣀⣀⣀⣀⡀⠠⠀⠀⠒⢂⣨⣥⣴⣶⣶⠶⠾⠿⠿⠍⠁
⠀⠀⠀⠀⠀⠀⠐⠋⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣯⣧⣤⣤⣤⣤⣶⡶⠟⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉⠉⠁

⢠⠂⠉⠙⠆
⡇⠀⠀⠠⡂⠀⠀⣀⢀⡀
⠑⡄⠀⠀⠱⣤⠋⠀⠀⠈⠓⡄
⠀⠈⠐⠄⡀⠈⠀⠀⠁⠀⠀⠸
⠀⠀⠀⠀⠀⠉⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⠀⢠⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠠⠤⣄⠀⡇⠈⣇⡠⠐⠤⡀
⠀⠀⠀⠀⠀⠀⢡⠀⠮⠤⠃⠀⠈⠂⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠑⢢⠄⠀⣁⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⢀⠗⠤⠤⠤⣀⡀⢱⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠀⠀⠙⢆⢧⠀⢀⡴⠊⠉⠉⠉⠒⠤⣠⠴⠀⠀⠠⢀⡀⠈⠙⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⠈⠶⠘⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣆⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⡰⠂⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠁⠁⠂⠇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠐⡀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠈⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⣀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠙⠲⠦⣄⣀⣀⣤⡤⠤⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠒⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⢙⠲⣄⡀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⠀⢀⣴⣾⣿⣿⣿⢿⣿⣶⣤⠐⠉⠙⠓⠿⠿⠶⣤⣤⣴⣺⣭⣭⣽⣶⣦⣶⣶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⣿⣰⣿⣿⣿⣿⣿⡿⠟⣿⣿⣿⣷⡀⠀⣀⠤⠔⠊⠉⠀⠉⠉⠉⠉⠉⢉⣹⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢨⣄⡛⠿⣿⣿⣿⣿⣦⡹⣿⣿⣿⣏⠉⠀⠀⠀⠀⠀⠀⢠⣤⣠⣴⣶⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⢸⣿⡿⠦⢀⣀⣉⣉⣉⣉⠘⠻⢿⡿⠀⠀⢀⣀⣤⣴⣶⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢀⠤⠒⢵⣾⣿⣿⣿⣿⣿⣿⣿⡇⣷⠄⣴⣾⡿⠿⣿⡿⠋⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣊⠁⠀⠀⠀⠻⢿⣿⡿⠛⣁⣤⣤⡄⣠⡾⠉⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠤⠶⣦⢶⡷⠒⠺⢟⣻⠋⠉⣿⠿⠶⢶⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⢋⡵⠊⠀⠀⠀⠀⣧⣄⣰⡿⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⣠⣿⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠛⠊⢉⣩⠟⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡽⠋⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠒⠒⢒⣒⣂⣀⣀⣛⣛⣛⣛⣿⣟⣛⣓⣀⣒⣶⠶⠾⠿⠿⠿⠿⠿⢿⣿⡿⠿⠻⠿⣿⣿⣷⣂⣀⣀⣀
⠀⠀⠀⠀⠀⠙⠛⠛⠋⢁⣉⣉⡛⢛⣛⣛⣛⣛⣋⣀⣀⣀⣒⣒⠒⣛⣛⣛⣓⣒⣒⣂⣀⣤⣤⣤⣄⠠⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⣤⣤⣤⡤⠴⠶⠶⠶⠒⠊⣉⣭⣿⠿⣿⣿⣿⣿⣷⡦⠄⢠⣤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣩⣭⡁⠤⢤⣭⣭⣭⣉⡉⣉⣭⣡⠴⣤⣶⣿⣷⡶⠶⠾⠟⠛⠋⠉⠁⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠶⠮⠍⠉⠉⠥⢶⣶⣾⣿⣯⣭⣭⣉⣉⣱⣶⣶⣤⡤⠖⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠒⠲⠶⠖⠂

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

⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣄⡀
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⢸⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠲⡄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠉⠀⠀⠀⠀⣹⡀
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠄⣧⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⡿⠀⠀⠀⠀⠀⠈⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠚⠋⠁⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⡠⠴⠞⠋⠉⠀⠀⠑⣶⣆⡛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⢸⣅⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣤⣄⣀⣀⣀⣀⣠⣤⠶⠚⠋⠂⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠙⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠻⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣷⣶⣤⣦⣤⣄⡐⠀⣀⡀⢀⣴⣶⠦⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢚⠀⣌⡈⠀⣴⣀⠄⠐⠀⠆⠀⠀⠀⠀⠀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣦⣠⣴⣆⣜⢟⣤⢌⢓⣚⣡⠄⠀⡔⠤⠥⢘⢿⣅⡀⠀⢸⢮⠁
⠀⠀⠀⠀⠀⠀⠂⣨⡙⢳⣨⢹⣷⡻⣽⠏⣿⡎⣮⠁⣹⡃⠑⡓⣇⣧⣴⣏⢣⡉⣄⡠⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢒⣾⡿⣾⢿⣿⣛⣻⣿⠿⣷⣭⣜⣿⣿⣥⠈⠬⢉⡽⣿⡾⠁⣞⣩⣴⣦⢷⡮⠁
⠀⠀⠀⠀⠀⢀⡀⢽⠍⠠⠁⣾⣔⡿⣽⣿⣿⣿⣽⣿⣟⣻⡵⣹⠾⣿⣇⣿⣿⣾⢟⣟⡭⠼⢅⠈
⠀⠀⠀⠀⢠⠀⣙⣋⣶⡅⠈⡥⣻⢆⣴⡿⣿⣿⣗⣝⣻⣹⡿⣱⣼⣝⡏⢿⡿⢋⣼⣁⢩⡄⠘⢹⢲⣄⠀⡤
⠀⠀⠀⠀⠨⡔⣉⡫⠂⠌⠢⣽⣿⣤⢉⢃⣿⣿⡻⣳⣦⢾⢝⡥⡟⣿⠇⣠⣴⡯⢿⠃⣼⢟⢴⢹⠒⢉⢀⢡⠔⢠
⠀⠀⢠⠀⣀⠫⠚⢽⣧⣳⣂⡠⣿⣿⣋⢧⣧⣽⢽⣮⣿⢸⣨⣾⡟⢁⣾⣻⣾⣾⣧⣾⠶⠬⠙⣁⣯⡶⠾⢋⡰⠘
⠀⠈⢋⢰⣎⣮⣰⣱⡄⢗⡾⣿⣟⢶⢪⣧⠹⣙⣷⡙⣾⡆⠻⡿⢠⣺⣽⣿⡟⣽⣋⡰⣌⣶⣼⣿⡧⢢⡋⣤
⠀⠀⠀⠀⠙⡿⣸⡿⡟⢶⣶⣦⣈⣉⢁⣤⠄⡈⠈⠁⡉⠂⡐⡄⠛⠻⠒⢉⣀⣥⡶⢾⡭⣴⡏⠖⢹⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠍⠗⠛⡟⢟⠀⣹⢟⢛⣿⡠⠀⠙⢡⣞⡿⠻⣿⣿⣷⢿⢻⡠⠲⠤⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠁⠀⠀⠈⠀⣳⡧⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠐⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣠⣾⡖⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣿⣾⣿⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠉⠒⢛⣾⠷⣟⣴⣿⣷⣶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠈⠛⠀⠋⠁⠠⠔⠃⠀⠈⠀⠾⠷⠏⠾⠯⠈⠛⠶⠦⠙⠿⠿⠶⠛⠲⠦⠤⠠⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⣀⣀⡀⠀⠀⠀⣀⡀⢀⡔⡩⠰⠖⠎⣴⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠱⣌⣠⣤⡀⠀⠀⠀⠀⠀⠀⣿⡄⢀⠢⡄⠀⠊⢰⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⣀⣰⣿⡿⠃⣈⣶⠶⢶⣶⠶⢄⣷⠀⠈⠉⢁⣈⠁⠀⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣚⢿⣿⣿⣿⣾⣿⡟⣈⣉⡭⠓⠚⡹⣯⡄⠠⢿⡏⠀⠠⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⡀⠻⢿⡿⣫⠿⠶⣿⠷⠀⠒⠒⣺⡏⢿⣆⣼⣥⡧⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⠻⠦⣤⠀⠛⠀⠀⠀⢀⣠⣤⣾⣿⣵⡿⣯⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣠⣤⣶⡀⠉⠉⠉⠙⠉⠉⠉⠘⠘⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠉⠉⠉⠁⠧⡤⠤⠤⠚⠒⠉⣀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠄⠀⠀⠀⠀⠀⠀⠀⢀⡀⣼⣿⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⠠⣀⣀⡀⠤⠤⠾⠥⠔⠛⠃⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣖⣿⠈⢿⣶⣶⣶⣶⣒⣲⣶⣶⣿⣿⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣧⠀⠈⠈⠉⠛⠛⠛⠛⠛⠛⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣷⣶⣶⣤⣤⣤⣤⣤⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠋⠉⠁