Cowardly Emojis & Symbols

⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣶⣶⣶⣦⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣠⠞⠠⠤⠤⢰⡀⣀⠀⣴⠟⠁⠀⠉⠙⠛⠛⠿⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠀⠈⡆⠀⡠⠂⠙⠦⣈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠒⠀⠀⠀⠀⢠⠇⠐⠏⠀⡃⢠⠃⢠⠀⠀⢠⠀⠀⠱⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠄⣀⣀⠤⠼⡀⠀⠀⠀⠇⠘⠀⠉⠀⠀⢰⠀⠀⠀⠤⡀⠀⠸⠁⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠊⠁⠀⠀⠀⠳⣄⠠⠎⠀⢸⠀⠀⠀⠀⡌⠀⠤⠀⠀⠘⡆⠀⠐
⠀⠀⢀⣉⠢⠀⢸⣿⣿⣿⣿⣶⣶⣤⣄⡈⠁⠀⠀⠀⠀⢴⢤⡤⠞⠀⢀⠎⠐⡀⠀⢃⠇
⠀⠀⠀⠐⣄⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠁⠀⠀⣠⠏⠀⠀⢣⠀⠈⠈
⠀⠀⠀⠀⠈⠓⠂⠀⠻⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⢀⣀⡤⠞⠁⠀⠀⠀⠈⡆⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠢⡀⠀⠄⠀⠀⠠⡉⠉⠉⠁⠀⠑⣄⠀⠀⠀⠀⡇⠀⠀⢀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠴⠚⠈⡇⠀⠀⠀⠀⠈⢆⠀⠀⠀⠁⠀⠀⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠀⠠⠊⠀⠀⠀⠀⠀⠉⢻⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠉⠀⢀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣄⣀⣀⠀⢀⣀⡀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⡀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂

⠀⠀⢀⣀⣀
⠀⠀⠘⠻⣶⣭⣙⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠉⠛⠿⢷⣶⣬⣵⣦⣤⣀⠀⣠⡆⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⣀⣠⣴⣶⣿⣖⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣭⠿⠛⠉⢳⡤⠤⠤⣴⠃⠈⠛⠲⢶⡿⠛⠉⠁⠀⠙⢿⣷⣌⡲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⣧⠀⢀⠟⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣬⣵⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢠⡀⠀⢸⠀⣸⠀⣰⣃⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣿⡖⠀⢸⠀⡏⠀⠘⠟⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠈⠀⠀⢸⠀⡗⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⠀⢧⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠰⢀⡇⠀⢸⡀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⢧⠀⠐⠀⠀⠀⢠⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⣄⠀⠀⣠⠞⠁⠀⠀⠀⠈⢷⡀⠀⠀⣠⠏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠨⠽⣛⣡⡤⠤⠤⠤⠤⠤⠤⢽⣿⡏⠁⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢰⣧⣤⣤⣤⣤⣤⠄⠀⠀⠀⢀⣠⡼⠀⡀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⢹⣉⠀⠀⠀⠀⠙⢿⣿⣯⣥⣤⣴⣶⣶⣿⣿⠟⠀⡀⠀⠀⠀⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⢀⣀⠽⣟⠋⠿⣿⡟⠃⢉⠷⣀⠀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢫⠉⠀⠀⡼⠞⠘⠉⠉⠉⡄⡆⠀⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠰⠙⠀⠀⠀⠀⠀⠸⢳⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠙⠒⠒⠒⠒⠒⠒⠒⠉⢹⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢣⡤⡄⠀⠀⠀⠀⠀⢠⠖⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡄⡖⠃⠀⠀⠀⠀⠀⠀⡇⡰⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡀⢸⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣁⠼⠟⣿⠿⠿⠟⠻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⣠⣤⠀⠄⠈⠀⠸⣿⣿⠴⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣿⠄⠀⠀⠊⢀⠀⠀⠀⠀⠘⠓⢻⠇⢰⣸⡀
⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⡎⠏⠀⠀⠰⢲⡆⢸⡆⠈⡸
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠿⣫⣿⣿⣿⠀⠀⠀⠐⢋⣠⣤⡀⡀⠉⢀⣸⣷⣚
⠀⠀⣠⣿⣿⣿⣿⡿⠋⣥⣾⣿⣿⣿⣿⡀⠀⠀⠀⠘⢻⣿⡿⠃⡀⠸⣿⣿⣿⣿⣿⣦⡀
⠀⣾⣿⣿⣿⡿⠉⢀⣾⣿⣿⣿⣿⣿⣿⣿⡴⠂⠀⠀⠀⠅⠀⠀⠹⣀⣿⣿⣿⣻⢿⣿⣿
⢸⣿⣿⡿⣽⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⡱⠄⢀⡾⢦⠀⢀⠀⢸⣿⣿⣎⢿⣧⠿⠋
⠘⣿⣿⡁⣧⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣰⠊⠀⠘⣄⠀⣡⣾⣿⣿⣿⣮⠃
⠀⠈⠙⠁⡏⡆⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⡇⢳⡀⠀⠀⠙⠿⡿⢽⡆⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠙⣿⣿⠏
⠀⠀⠀⡼⠀⠀⠑⢄⠀⠀⠀⠀⠀⢂⠀⠀⠈⢹⠿⣿⣿⣿⣿⠟⠛⠀⠀⠛⠁
⠀⠀⡼⠁⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠑⠦⢠⡋⢀⠈⠿⠻⡅
⠀⢰⠃⠀⠀⠀⠀⠀⠀⠄⠑⡄⠀⠀⠀⠀⠀⡇⠀⠑⠀⠀⠻⣄
⠀⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⣘⠤⠼⠦⠀⠱⠧⣀⣰⠦⡀⠀⡈⠉⢢
⢸⠄⠀⠀⠀⠀⠀⢸⣢⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⠚⠳⢤⠏
⠈⡆⠀⠀⠀⠀⠀⠺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⡇
⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠄
⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⣠⠎⠀⢀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣿⣿
⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡋⡀⠀⣠⠾⠁⢂⡨⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠿⠛⣁⣀
⠀⠀⠀⠀⠀⣩⠶⠀⠀⠀⠈⠁⠀⠀⠀⠀⡈⢣⠤⠒⠈⠁⢀⠂⠈⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠾⠷⠦⠴⠞⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠛⠳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⡔⣐⠆⣠⡴⠶⠟⠛⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⣀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠘⢧⣠⠶⠶⠀⠀⠀⠈⠉⠙⠂⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠙⢷⣄⠀⠀⠈⣷⣄⠀⠀⠀⠀⠈⢳⣄⣤⣄⣀⣀⠤⠶⠲⣤⡴⠞⠋⢻⣄⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠙⠳⣤⣀⠣⣀⣱⣄⡀⢀⡔⠉⠀⠀⠀⠈⠳⡴⣦⠞⠋⠀⠀⠀⠀⠈⠛⠳⠶⣤⣤⣤⣾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⣌⣙⣿⠀⠀⣶⠀⠀⠀⠀⣻⠁⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⡔⠒⡦⠀⠀⠉⢻⡀⠀⠀⠀⠀⠀⢀⠇⠀⠀⡀⢀⠆⠀⠀⠙⠒⠉⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⠶⠿⠿⠶⠶⠶⢶⡾⠛⠦⣄⣀⡤⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⢀⡑⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⢀⡴⢻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⢠⣾⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁
⠀⠀⠀⠀⣰⠟⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⡷⠲⣶⠶⠶⠖⠛⠋⠉
⠀⠀⠀⢠⡏⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠛⠉⠉⢿⣿⣿⠁⢸⡇
⠀⠀⠀⢸⡇⠀⢹⡄⠀⠀⠀⢀⣴⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠉⠉⠙⠛⠛⢀⡟
⠀⠀⠀⠀⢷⡀⠀⠙⠶⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠙⢦⣀⢠⡟⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠋⠛⠳⢦⣄⠀⢀⡾⠻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠈⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡞⠁⠀⠈⠙⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣤⣀⡀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣆⢘⣄⡿
⠀⢀⣠⠶⠒⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⠀⠀⠀⠀⠀⠀⣀⣀⣤⡶⠶⠶⠶⠶⠶⠶⣆⠀⠀⠠⡀⣿⠞⠋
⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠿⣶⠶⠶⠖⠚⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣤⡷⠋
⢿⣀⡠⠆⢀⣤⣤⣤⣤⣤⣤⣤⣴⣶⠶⠛⠋⢁⣠⠞⠁
⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡄⣼⠋⠁
⣠⣀⣄⣠⣠⣄⣠⣄⣄⣀⣀⣀⣌⣿⣾⣿⣯⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣠⣤⣀⣠⣀⣄
⣿⣿⣏⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣈⠙⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠶⣄⣶⢽⣦⡿⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡃⠀⢿⡇⠐⢧⢺⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣄⢀⣯⣧⣀⣸⠝⢦⡼⠛⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⢻⡷⡄⢰⣿⣿⣿⣿⠓⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⡾⢏⡣⡤⠽⡟⠛⣗⣒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡙⣼⠿⣆⣀⡄⠱⠤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⡏⠀⠈⢦⣿⡀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⣇⠀⠀⠀⠈⠙⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠢⢀⣀⣀⡠⠚⠛⢶⡀

⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣦⣄⡀⣀⠔⠠⢺⢕⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢿⣿⣿⣿⣿⣿⣿⡆⠘⠂
⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⢻⣿⣿⣽⠟⠿⣿⣟⢀⠀⢁⣀⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠆⠀⠀⠀⠉⢹⣿⣯⣞⣡⣄⣉⣟⣛⣋⣀⣤⣤⣿⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⠴⠂⠁⠡⠉⠛⠿⣿⡿⠟⠛⠉⠙⠉⠉⠛⠍⠉⠉⠉⠉⠈⠉⠓⠶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠐⠁⡀⠀⠀⠀⡀⠀⢠⠟⡑⠄⠀⠀⡎⠀⠀⣴⠁
⡀⠀⠀⠀⠀⠀⢀⠔⠂⠀⢠⠊⠀⠀⠐⠋⠀⠀⠛⣆⢉⣤⠀⢀⡘⠂⢀⡀⢱
⣿⡄⠀⠤⢀⠀⠸⡀⠀⡀⠈⠉⠉⠠⣤⣄⠀⠀⣰⡾⢧⣿⢀⣿⣿⡦⠢⢌⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴
⣿⣷⣦⣄⣀⣖⠢⣍⠳⣷⡄⠀⠀⠀⠋⠛⢀⢖⡿⠀⠈⢻⣸⣿⣿⡟⠢⡀⠀⠀⢀⣀⣀⣶⣍⣰⣾⣿⣿
⣿⣿⣷⣿⣿⣿⣦⣠⠞⡡⢛⢄⣀⣀⣠⣴⠃⣸⡇⠀⠀⢸⣿⣙⠃⢀⣀⠤⠖⠚⣉⣭⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠁⠜⣰⣧⣄⠀⠉⢉⠁⢈⣿⠥⠄⠤⢼⣇⣼⣯⣭⣴⣶⣾⣿⣿⣋⣉⣿⣿⣿⣿⣾⣿
⣿⣿⣿⣿⣿⠏⢀⠈⣠⣿⡿⠘⢧⠒⢃⣰⣡⣷⣦⣄⡀⠂⣿⣭⢹⣟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣂⠾⢋⣵⠟⠀⣠⠞⠛⠉⠉⢁⣽⡍⠉⣿⣾⣿⣿⡆⠹⣇⠘⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿
⣿⣿⠿⠛⠉⣀⣴⡿⠋⣠⠞⠁⠀⢀⣀⣴⡿⡟⠁⠀⠙⠛⠿⠿⣿⡖⠢⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣶⣶⣾⣿⡿⣋⣀⣼⣿⠀⠀⠈⠛⠉⠀⠀⠛⠀⠀⠀⠛⠀⠀⠈⣿⣦⣈⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣿⣶⡿⢿⡄⠀⠀⡄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢸⣜⢉⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⣿
⣾⣿⣿⣿⣿⣿⣿⡉⠀⠺⣿⠧⠀⢁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⠿⣷⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠟⣋⣴⣾⣾⣿⣇⠀⠀⠢⠀⠀⣀⢀⡌⠀⠱⣄⡀⢀⠄⠀⠈⠑⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠟⣡⣾⣿⠟⠛⠉⠀⠀⠀⠀⠀⠈⠙⢛⠟⠁⠀⠀⠈⠛⠃⠀⠀⠀⠀⠚⠉⢉⡽⠿⢿⣿⣿⣉⣿
⣿⣿⠋⣼⣿⡟⣡⠖⠈⠻⣿⡟⠙⠒⢦⣀⠀⠀⠀⠀⠀⠀⢀⣾⣶⣶⣶⣾⣯⣴⣾⡿⠁⠀⠀⠙⢿⣿⣿
⣿⡇⣼⣿⠏⣾⡇⣾⣷⠀⢹⡄⠀⠀⢸⡏⠙⣿⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⢸⣿⣿
⣿⡇⢻⡿⢸⣿⡃⣿⡿⠀⠈⣇⠀⠀⢸⣇⢀⣿⣧⣀⡀⠀⠀⠀⠈⣹⣿⠿⢿⣿⣿⣿⣶⣤⣤⣤⣿⣿⣿
⣿⣿⣬⣧⠘⢿⣇⣈⣀⣠⡾⠿⠶⣶⠛⠋⠙⠟⠻⡿⠿⣿⣶⣾⠿⠟⣉⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⡀⠉⠉⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢲⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿
⢁⣼⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡟⠻⣿⡿⢿⣿⠛⠛⠃⣸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣦⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⢻⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠿⣿⣿⣿
⣿⠿⠉⢙⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣦⡞⢫⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠖⢋⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⡀⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⣤⣤⣄⡀⠀⠀⠀⠈⠁⠈⠳⠾⢿⣿⣿⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣌⠛⠀⠀⠀⠀⠈⠽⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣸⣿⣯⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣿⡝⢦⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⠓⠒⣢⠀⠀⠀⣿⡿⣿⡇⠸⡆
⠀⠀⠀⠀⠀⠀⢤⣤⣴⣷⡀⠀⠀⠀⠀⠀⡸⢱⣿⣿⣿⡈⢳⡄⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⡿⠟⢻⡿⠋⠉⠛⢷⣦⠈⠛⠃⠀⢰⣿⠁⠹⣿⠀⠁
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟⠁⠀⠀⠀⠈⢡⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠈⢻⣿⠀⠰⠿⠃⠀⠀⠀⠘⠋⠀⠀⠀⠀⣸⡇⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡆⠀⠀⠀⠀⠐⠛⠛⣿⠿⣿⡛⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠘⣿
⠀⠀⠀⠠⡤⠶⠛⣿⣿⣿⡿⣷⡀⠀⠀⠀⠀⠀⢹⡄⠈⠙⠳⢦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣧⡀⠁⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⢻⠿⠛⢿⠿⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⣤⣿⡆⠀⠀⠲⣦
⠀⠀⠀⠀⠀⠐⠂⠀⠀⣨⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⣻⣿⣶⣶⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠿⠉⠿⠻⠇

⠀⠀⠀⢰⡆
⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠠⠄⠠⡖
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⣿⣷⣿⣶⣧⡀⡀⣄
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⣠⣭⠍⠉⠀⠈⠉⢻⣿⣿⣿⣷⣄
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢡⣼⣿⡅⠀⠀⠀⠀⠀⡄⠈⠉⣉⠉⠿⣦⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣼⡿⡿⠋⠰⠄⠀⠀⠀⠀⠀⠘⠙⢿⣿⣿⣷
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⢀⣤⠀⠀⠈⠁⠀⣀⠈⣿⣿⣿⣿⣧⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡂⠀⠀⠀⢀⣀⡀⠀⢸⣦⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⢿⣆⠀⠘⠿⠟⢁⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠙⠾⣿⣦⢰⣖⣋⣭⠏⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣥⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣿⣿⣾⣿⣉⣤⣶⣿⣿⣿⣿⣿⣿⣿⡏⠈⠈⠛⢿⣷⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠀⢀⣠⡀⢸⡍⠑⠂
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣎⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠟⢹⠇⠸⣷⣀⣷⡘⢷⡄
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⣿⣿⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠘⣷⠀⢻⣿⢳⡷⣦⡑
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡠⣾⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠟⢿⡏⠀⠀⠀⢠⣿⣤⢸⣿⡜⣿⣿⡁
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣟⣿⣿⣿⣿⡿⣿⣿⠿⠛⠋⠙⠛⠋⠙⠛⢙⣷⣤⣤⣦⠀⠀⣸⣿⣿⣼⣿⣧⢹⣿⣷
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⠿⠁⣿⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⢻⣿⣇⢀⣿⣿⣿⣻⣿⣯⢺⣿⣿⡄
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⢿⠟⣽⣿⡋⣰⠾⣟⣁⠐⢿⡆⠀⠀⠀⠀⠀⠈⡁⢿⣿⠀⢠⡏⠛⢸⣿⣿⣿⣿⡿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢰⣿⣇⢹⣿⡄⠉⢙⣛⣼⡀⠀⢀⣀⣀⣐⡦⢰⣼⣿⢠⡏⢯⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣶⠁⡀⣀⣈⣹⣿⠗⣤⣿⣿⣿⣿⣿⡏⠩⣿⡿⠁⠈⢻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠹⣿⣵⣿⣿⣿⣿⣦⣈⣥⣶⣿⣿⣿⣿⣷⣤⣀⠀⠷⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⠻⢾⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣼⣿⣿⣿⣿⣿⠟⡛⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣭⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⢀⣼⠭⣺⡿⡏⢹⠛⠋⠙⣹⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣯⡅⢀⣾⡆⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡟⠋⣩⣿⣿⣿⠃⢀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⣿⡿⢫⣿⣿⢄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡟⣴⣿⣷⣿⣤⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⢊⣿⣿⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⢸⡿⣦⠻⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⣿⣿⣇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣿⣿⣻⡿⣠⣿⣴⠉⢶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠸⡄⣿⣿⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⢿⣿⣿⣿⣟⣿⢇⣠⠻⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠘⢿⣿⡇
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣿⣿⣿⣿⣾⣿⡟⢽⣿⡇⢸⣯⣈⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢠⣧⡇
⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠼⠿⠿⠿⠛⠛⠒⠛⠓⠚⠛⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠂⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⣀⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢣⠞⠂⠉⠉⠁
⠀⠀⠀⡒⠒⠒⠒⠂⠈⠉⠀⠀⠀⢀⣀⠑⢦⡀⠀⠀⠀⠀⡤⣄⢀⣔⡱⠑⢆
⠀⠀⠀⠀⠁⠒⠀⠠⠤⠤⠄⠒⠈⠁⠀⠉⠢⣝⣦⠒⢈⢟⠔⠧⡈⠉⠉⡟⠐⠵⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠚⠋⠁⠀⠀⠙⡄⠀⡇⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠀⠄⠀⠀⡇⠀⠀⠀⠀⢸⠀⣇⢀⣄⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⣠⣄⠀⠀⡇⠰⣿⠄⠀⢸⠀⠘⡜⠋⠀⠀⢸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠸⡇⠀⠀⠘⡄⠉⠀⠀⢸⠀⠀⠈⠦⣀⣠⡎⠀⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠄⢸⠸⡀⠀⠀⠈⠢⣀⣠⠇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢈⣳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⢸⡀⡝⢄⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣠⡤⠴⠒⠚⠋⠉⠉⠉⠉⢹⠦⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠈⡙⢦⡀⢹⠢⠤⣀⠀⠀⠀⠀⣞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢳⠢⢌⠓⠤⢄⡇⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠱⢄⣙⣆⠀⠻⣄⠀⠀⠀⠀⠀⢀⣩⠗⠲⠤⠤⠞⠁⠉⠓⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⠒⡲⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢼⡀⠀⠀⠀⠀⠀⠀⢀⠄⠡⣤⠂⣳⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠈⠛⠜⠀⠑⠂⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⠓⠒⠠⠴⠤⠐⠒⠊⠁⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢷⡀⠀⠀⠀⠀⠀⠀⢀⠤⠆⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡰⠊⠀⠀⠀⠀⠀⠀⠀⠙⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠸⣿⣿⣷⣶⣶⣤⣤⣀⣀
⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠈⠙⣦⠀⠠⠸⣿⡄⠀⠀⡀⢀⣤⣾⡿⠛⠙⢿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⢆⠀⠀⠀⠀⢠⠈⠛⠶⢤⣄⠙⢏⠁⠀⠀⠀⠀⠈⠛⠿⠿⢿⡿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠞⠉⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠘⠀⠀⠈⠀⠀⠀⣤⡀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⡆⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⢡⡀⠀⠀⣠⠆⠀⠀⠀⠀⠱⡀⠀⠀⡠⠂⠀⠀⠀⠔⠰⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠈⠒⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠉⠐⠀⠀⠀⠀⡠⠁⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⢀⠬⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⢆⡰⠀⠐⠇
⠀⠀⠀⠀⣀⠰⢿⣷⣶⣶⣦⣤⣤⣴⠶⠖⠂⠀⠀⢀⣠⣾⠃⠀⠀⠀⠠⠔⠈⢀⠀⢀⠎⠀⠀⠀⠘
⠀⠀⢀⡊⠀⠀⠀⠙⢿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⠟⠁⠀⠀⡀⠀⠀⠀⠀⠀⠋⠁⠀⠈⠀⠀⢀⠁
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠈⠀⠀⠀⠀⣀⡄⠀⡠⠐⠈⠀⠃⠀⠀⠀⠉⠢⢄
⠀⠊⠀⠑⠤⢀⡀⡀⣀⠀⠔⠉⠛⠿⠿⠛⠏⠁⠒⠠⠄⠀⠀⠀⡦⠀⠉⠈⠟⠀⠀⢠⠂⠀⠀⠀⠀⠐⠂⠄⡁⠢⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠈⠀⠣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠔⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠡⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡀⠄⢊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠐⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣀⠁

⠀⠀⣀⡤⠴⠶⠒⠶⠦⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⣄⣀⡀⠀⠀⢀⣀⣤⡶⠃
⠀⠘⠿⣶⣶⣶⣶⣶⣶⣶⣶⡿⢿⣶⣤⣀⠀⢠⠄⠀⢄⠀⠀⠀⠀⠀⣰⡿⠛⠁⠈⠙⠻⠿⠿⠿⠿⠿⠛⠋
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢉⡹⠟⢃⠀⠀⢺⢧⡉⠉⠐⢚⠯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⡀⠀⠀⠀⡜⠀⢉⠉⠉⡁⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⣧⠀⠀⠀⡇⢰⣯⠀⠀⡇⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠁⣸⠀⠀⣇⠈⠉⠀⣸⠃⠀⠀⠀⠀⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠘⣄⣀⡴⠃⠀⠀⠘⣆⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠤⠊⠀⠀⠀⠀⠀⠑⠄⠄⠀⠀⠀⢀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⣀⢠⡶⠓⠒⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣄⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⢰⠁⠸⣿⣷⣾⣷⣶⣾⣷⡄⠀⠀⢀⠀⠀⠀⣀⣴⣾⣿⣿⣿⡄⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠑⠂⢿⣿⣿⣿⣿⠿⠋⠀⢀⣠⠊⠁⠈⠻⣿⣿⣿⣿⣿⣿⣇⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠖⠻⡄⠀⠀⠀⠀⠈⢿⣿⠏⠁⠀⡌⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠸⠇⠀⢀⡴⠁⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠠⠠⠒⠉⠀⠀⠀⠀⣰⣏⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡆⠸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠐⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠟⢦⣀⠀⠀⠀⠀⠀⢀⣀⣴⠃⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠀⠀⠈⠉⠛⠛⠛⠛⠉⠁⠘⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢓⡦⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⣻⣶⡹⣆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣏⢰⢏⣼⣿⡙⣯⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢸⠆⠋⠀⠉⠉⠈⠀⠋⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠆⢠⡈⠀⣄⠀⠀⠀⢀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠠⠀⡀⢀⣀⠈⠂⠀⠈⣆⠘⡆
⠀⠀⠀⠀⠀⡶⣧⠀⡀⢀⡤⠀⠘⠀⣁⠉⢶⣄⠀⠀⠀⣹⢏
⠀⠀⠀⠀⠀⢷⡞⠻⣄⠈⠊⠆⠐⠀⠀⡀⠀⡀⠀⠀⡀⠱⢾⣆
⠀⠀⠀⠀⠀⡈⠑⠒⠚⣧⠶⠴⣿⣿⣿⣿⠆⢰⡀⠀⠀⢂⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠰⠀⠙⠿⠋⠁⠀⢆⠁⠀⠀⢸⡿⣏⣷
⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠊⠀⠀⣠⡀⠀⠀⠸⡧⠀⠀⣈⣿⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠀⣴⣿⣿⣿⣷⡄⠀⣷⡠⢀⡿⠃⠋
⠀⠀⠀⠀⠂⠀⡀⢩⣿⣿⢻⣿⠿⣿⣿⢿⣿⠘⣿⣧⠀⡀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠺⠄⣻⠂⠛⠶⠤⠔⠊⠋⠀⣗⠸⠀⢀⢞⣿
⠀⠀⠀⠀⠘⠀⠀⠀⠀⢸⣴⣆⠀⠀⠀⠀⠀⡀⢿⣆⠀⠀⣼⠃⠀⠀⠀⠂
⠀⠀⠀⡀⠀⠐⠀⠀⠀⠘⡝⣿⡀⡄⠀⢀⢔⡔⠻⠈⠀⠘⠟⢅⠀⠀⠀⠠⠐
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⢿⢢⢸⠷⠋⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣸
⠀⠀⠀⡀⠀⠀⠀⡑⠀⠀⠀⡆⠈⠳⠂⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢉⡹
⠀⠀⠀⠇⠀⠀⠀⢧⡆⠈⠀⢹⡀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠆⠀⠀⠀⡆⠀⠀⠀⠀⠇⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈⠈⢹⡶⡀⢰⠄⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠘⠀⠀⢠⡇⠘⡖⡀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡇⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠘⠢⡈⠀⠸⢇⠀⣷⡁⠀⠈⢧⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠗⠀⡈
⠀⠀⠀⠀⡀⢤⡀⠀⠄⡀⣷⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠜⠀⠀⠀⠀⣾⡿⠸⠁⠁⠀⠃⠀⠈⢸
⠀⠀⠀⢀⣆⠴⢙⢶⣤⠊⡉⠉⠁⠂⠀⠀⡆⠀⠀⠀⠘⠂⠀⠂⠠⠚⠘⠁⠀⠀⠀⠀⡄⠀⡾
⠀⠀⠀⠘⠀⠀⠋⠘⠣⠈⠀⠀⡀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⠀⡠⠔⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⢀⡴⠊⠀⠀⢠⡴⠋⠁⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣿⠀⠀⠀⠀⣠⢾⡿⠃⠀⢠⣾⠏⠀⠀⠀⠐⠁⣀⣴⠿⠋⣁⠀⠀⣀⣤⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⣿⠀⠀⢀⡼⣣⡟⠁⢀⡴⣳⢋⣀⣠⣠⣂⠀⢨⡿⢡⡞⢉⢄⡴⠟⠉⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⢹⠀⢠⡟⣱⡟⣀⣴⣿⠿⠛⢋⣭⣾⡿⠁⢠⣟⣠⣿⣟⣡⣿⣶⣋⣭⣋⡉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⢠⠀⠀⣿⠀⣿⢸⣰⣿⣶⡿⠛⢉⡟⡡⣢⠾⠛⣩⠿⠛⡊⠭⠍⣉⢥⣶⣞⣛⠛⢻⣿⣿⣗⣒⣒⣤⢀⣀⡀
⠀⠀⠀⠀⠀⠘⣆⣠⣿⠷⢿⠿⠛⠿⣥⢶⠀⠘⠻⠜⠁⢀⡞⣡⠖⡡⠐⢀⠴⠛⠂⠉⢉⣛⣭⡭⠽⠿⠿⣿⡥⠄⡀
⠀⢣⠀⠀⢰⡀⣿⠋⡀⣤⣈⡐⠀⠀⠈⠂⠣⠀⠀⠀⠀⠉⠛⠷⠊⠀⢀⡈⠈⠒⠒⠋⠉⢃⠠⠒⢉⣉⣉⣉⣙⠲⠒⠀⠲⢤⣀
⠀⠈⢷⡀⠘⣧⣿⢸⡏⠙⠦⣅⠀⠀⠀⠀⠀⢧⣄⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠈⠉⠈⠁⢀⣀⣀⡿⣿⣿⣷⣄⣀⣈⣁
⠀⠀⠈⢷⣄⠹⣿⣿⠀⠀⠀⠀⢳⣄⣀⠀⣠⠆⠀⠓⣄⠀⠀⠀⠀⣸⣇⣀⣠⣤⣴⣶⣶⣶⣦⣶⣤⡤⠶⠶⠖⢻⣿⣄⠉⠙⠦
⠀⠠⡀⠈⢮⠳⣽⢻⡄⠀⠀⠀⠀⠈⣿⡿⠃⠀⠀⠀⠈⠀⠀⠀⣰⣿⣶⣶⣶⠾⢿⣿⠟⠚⠋⠁⠀⠀⠀⠀⠀⠈⣿⠙⠳⣄
⠀⠀⠘⠢⣀⣑⣿⣾⣧⣤⡀⠀⣀⠽⣷⡁⠀⠀⠀⠀⠀⠀⠀⡰⠷⠶⠶⠶⠶⠿⠿⢿⡾⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⢄⠀⠈
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣝⣻⣷⡄⠀⠀⠀⠀⢀⣾⣥⣿⣶⣴⣖⡂⠉⣉⡤⠴⠄⣰⣶⠆⠀⠀⠀⠀⠀⣿⢠⣝⡢⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢽⣿⣿⣿⣿⣿⣿⣿⣷⣶⣮⣴⣿⣓⣒⣒⣓⣀⣀⣄⣠⣴⣶⣿⣿⣿⣶⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⠶⠍⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠒⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠶⠶⢦⣤⡀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⡀⢀⣠⠴⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⢶⡀⠀⠀⢸⡇
⠀⠀⣴⠟⠋⠁⠀⠈⢻⡄⠀⠀⠀⠀⣠⣤⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣷⡄⠀⢻⡄⠀⢸⡇
⠀⣾⠃⠀⠀⠀⠀⠀⠀⣷⣀⣠⠴⠛⠁⠀⠀⠀⠈⠙⠓⠶⠶⢤⣤⣤⡤⠤⠤⠴⠶⠋⢀⣠⣤⣶⣶⣶⣶⣶⣤⣄⡀⠀⠈⢻⣷⣶⠘⢿⠇⡀⠈⣿⠀⢸⡇
⢰⠇⠀⢠⡄⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠉⣁⠀⠀⢰⡇⠀⢻⡄⢸⡇
⣼⠀⠀⢸⣷⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠻⣦⣾⣇⠀⢸⡇⢸⡇
⢹⠀⠀⠀⡿⠁⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡿⢟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈⢻⣿⠀⢸⡇⢸⡇
⠸⡄⠀⠀⠀⣤⠈⠀⠀⣤⣶⣷⣄⠀⣠⣾⣿⣿⡿⠿⢛⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣴⣄⠀⠻⠀⢸⠀⢸⡇
⠀⣧⠀⠀⣸⠃⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣭⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣇⠀⠀⡾⠀⢸⡇
⠀⢸⡆⠀⣿⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢻⡀⠀⠀⣠⠟⢿⣌⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇⠀⢸⡇
⠀⠀⢷⠀⠛⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⡟⠋⢀⣤⠀⡄⠀⢰⢁⡤⠀⠙⠷⣤⠼⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡇⠀⢸⡇
⠀⠀⠈⢅⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⢸⡿⠀⠃⠀⠇⢸⡿⠀⠀⠀⢰⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⠀⢸⡇
⠀⠀⠀⠀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠈⠁⢀⠃⢰⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢸⡀⢸⡃
⠀⠀⠀⣸⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⡼⠀⣿⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⢠⡟⠀⢸⡇
⠀⠀⢰⠇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢿⠀⠀⠀⠀⡴⠁⠀⢸⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⠀⢠⠋⠀⠀⢸⡇
⠀⠀⠹⢶⡆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⡘⢷⣤⡴⠞⠀⠀⠀⠘⣇⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⡀⠀⠀⢸⡇
⠀⠀⢀⡿⠀⠀⢸⣿⣿⣿⣿⣿⠿⠿⠿⠧⠶⠤⠤⠤⣄⣀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢹⣦⣤⣤⡾⠃⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⢹⣶⠀⢸⡇
⠀⠀⢸⠃⠀⠀⣸⣿⣿⣿⣿⣧⣤⣀⣀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣤⠀⠀⠀⠀⠀⠀⠀⠳⠄⠀⣀⣀⣀⣀⡀⠀⠹⣿⣿⣿⣿⣿⣇⠀⠀⠀⢀⡾⠁⠀⢸⡇
⠀⠀⠻⣦⠀⢀⣿⣿⣿⠿⠛⣿⣿⣿⡛⠻⠛⠛⠛⠀⠀⠀⢀⣠⣼⠃⠀⣠⡀⢀⣀⣠⠴⠺⠋⠉⠉⡉⠉⠙⠳⣦⠸⣿⣿⣿⣿⣿⣆⠀⠀⣾⠁⠀⠀⢸⡇
⠀⠀⢀⠎⢀⣾⣿⢿⠋⠀⠀⠘⢿⣿⣿⣷⣶⣶⣶⣶⣶⣾⣿⠟⠁⣀⠀⣇⠙⢋⡿⠙⠒⠷⣄⡀⢘⠃⠀⠀⡇⠘⡇⢿⣿⣿⣿⣿⣿⣧⡀⠙⢿⠀⠀⢸⡇
⠀⠀⡌⠀⢘⣭⣽⣿⣃⠀⠉⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿⠟⡁⢰⠛⣆⣺⢤⡄⢋⠀⠀⠀⠀⠀⠉⠻⣄⡀⠀⣇⡼⢃⣾⣿⣿⣿⣿⣿⣿⣷⡀⢻⠀⠀⢸⡇
⠀⢰⠀⠀⠻⠿⠫⢶⣿⣶⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣄⠀⢉⠴⠀⢠⢿⡞⢡⢿⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⢧⠀⢸⡇
⠀⣾⠀⠀⠀⠀⠀⢀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣶⡿⡸⠋⠀⠈⡇⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠘⣧⢸⡇
⠀⣿⠀⠀⠀⠀⠀⣸⣇⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⡆⠀⠀⠱⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠟⢁⡀⠀⠀⠀⠀⢹⣿⡇
⠀⢸⡆⠀⠳⣶⡆⠙⠿⠀⡄⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠃⠀⣿⡇⠀⣠⡄⠀⢸⣿⡄
⠀⠀⢿⡀⠀⠘⠋⠀⠀⠀⣧⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣿⣿⣿⣿⠁⠀⠀⠿⠃⣾⣿⠁⠀⣿⢻⡇
⠀⠀⠈⢳⡀⠀⢠⣀⠀⠀⢹⡄⠈⢿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣿⣿⣿⠇⠀⡄⠀⠀⡀⠉⠁⠀⣼⠏⢸⡇
⠀⠀⠀⠀⠙⢦⡀⠙⠷⣶⣾⣿⡀⠘⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⢻⣿⣿⠏⠀⣼⣷⡶⠋⠀⠀⢀⣼⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠙⢦⣀⠈⠙⠛⠃⠀⠘⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠁⠀⢸⡿⠋⠀⠀⢀⣴⠟⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠶⠶⢦⣀⣠⡴⠛⠳⣤⣀⣠⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣤⣤⣤⣶⠿⠟⠋⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠐⠒⠒⠒⠲⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⠉⠉⠉⢉⡷
⠀⠀⠀⠀⢠⠖⠛⠋⠉⠉⠉⠉⠀⠀⣀⣌⠑⢄⠀⠀⠀⣄⠀⠀⢠⣦⢠⢖⡿⠤⠄⠀⠤⠤⠒⠊⠁
⠀⠀⠀⠀⠀⠉⠒⠒⠒⠒⠒⠒⠒⠉⠁⠀⠑⢮⡣⠴⣾⡟⠛⠓⠒⠟⠿⠯⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⢤⠞⠉⠀⠀⡀⠐⠀⣤⠀⢰⠀⠀⠀⠀⣠⣄⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡤⣄⠀⠀⡞⠀⠀⢰⠀⠀⢿⠀⠀⠀⠀⠈⠀⢘⡆⠀⠀⣠⢗⣸⣸⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣙⡷⡃⢠⠞⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠉⠁⠚⠑⠯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢣⡏⠀⢉⠛⠉⠀⠀⠀⠀⣠⠇⠀⠀⠁⠠⠂⠀⠀⠀⠀⠀⠀⠉⠱⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⣇⠀⠎⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠁⠀⠀⡐⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣤⣀⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⡇⠀⠀⠀⠀⡠⠂⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⣰⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠂⡇⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠣⣀⣀⣀⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣄⡼⠀⠊⠀⠀⠀⠀⠀⠀⣤⠀⠄⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠴⡞⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠗⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠥⢔⡣⣌⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠊⠁⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢠⠄⢀⠀⣀⡠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡎⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠱⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠒⠒⠒⣻⣿⣿⣿⣿⣦⣽⣶⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣙⣻⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢤⡻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣾⡿⠁⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠐⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣒⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⣠⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢲⣤⣹⣿⣿⣿⣿⣿⣿⣿⣷⣯⣖⣦⣀⣠⠴⠿⠦⣤⣧⣦⣤⣤⣈⢻⣧⠶⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⣠⣴⣾⣿⣿⣿⣿⣟⡏⠀⢀⣨⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠘⠛⣫⣿⣿⣿⣿⣿⣿⣤⣀⣙⣫⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⢓⠒⠛⠁⠇⠀⠉⠉⠿⢿⡄⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡹⣿⣿⡏⠀⠁⠉⠉⠀⢀⣴⣿⠁⠀⠀⠀⠀⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣿⣿⡇⡀⠀⠀⠀⢠⣾⠋⠘⠻⣶⣴⣶⣿⣿⡄⣆⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣙⣿⣶⣸⣰⢀⣾⡏⣠⣤⣤⣿⣿⣿⣟⣻⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠙⣿⣿⣿⡏⣿⣏⣉⡉⢨⠉⡏⣟⣿⣇⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⣿⣿⣿⠿⠛⣿⣟⠻⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣏⠉⢿⣤⣀⣈⣉⣀⠈⠻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣆⠈⠘⠛⠙⣁⣾⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⠛⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠶⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⢀⠒⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣤⣶⣶⣿⣷⣶⣶⣶⣶⣶⣿⣿⢖⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⠛⠛⠛⠛⠛⠿⠿⠿⠿⠋⠀⠀⠀⠀⠒⠁⠀⠁
⠀⠀⠀⠀⠀⠐⠀⠠⠔⠂⠀⠀⢠⣿⣿⣿⠋⠁⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⠤⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠘⠛⠀⠀⠀⠀⠀⠀⠘⠡⠀⠀⠛⠸⠀⡃
⠀⠀⠀⠀⠀⠀⢄⡠⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠊⠀⠀⠐⠒⠊⡁⠀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⢀⠀⢠⠄⠁⠀⠈⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡀⢀⠄⠀⠀⠸⠀⠀⠀⠊⢀⠀⠀⡀⠔⠂⠐⠂⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠡⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⠤⠄⠒⠉⢀⠠⢠⠒⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠢⠀⠂⠀⠔⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⢀⠀⠆⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠈⢀⠕⠀⠀⠀⠀⠀⠀⠀⠂⠀⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠒⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠒⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⠀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢉⠑⢄⠔⠁⠀⠈⠓⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡠⠔⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡠⠤⢄⡀⠀⠀⠀⢀⡞⠁⢀⠋⠒⢤⡀⠀⠀⢀⡜⠉⠀⠂⢤⡀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠹⣄⠀⠀⠈⠀⠀⠌⠀⠀⠀⠙⢦⢀⡾⠀⠀⠀⠀⠀⡇⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⡜⠀⠀⠀⠀⠀⠈⢻⡅⣀⣄⡀⠀⢰⡇⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠈⠱⡀⠀⠀⠀⠀⠀⠀⢿⣤⣿⣿⣠⠟⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠙⣄⠀⠀⠀⠠⢶⣼⡟⠛⠉⠁⠀⠀⠀⢤⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⣾⠀⠀⠀⠀⠀⠈⠳⢦⡀⠘⠛⣿⠇⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢉⣤⣤⣈⠙⠣⠀⠀⠀⠀⠀⠀⠀⠀⢉⣓⡾⠋⠀⠀⠀⠉⠓⠚⠋⠁⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠉⠉⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠤⠀⠄⠂⠀⢀⣀⡀⣤⡤⠤⠤⣠⠶⠶⠖⠚⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠈⣀⠴⠛⠁⣀⢔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⢉⣀⡤⠚⠁⢱⡀⠈⠒⠄⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠉⠁⠀⠀⠀⠘⡏⠀⣐⣀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠔⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⢹⣀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠈⠉⠀⢖⡽⣿⡀⠀⠀⠀⠀⢰⣀⣨⠿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣆⡴⠃⠊⣿⠒⠉⠛⠓⠚⠚⠉⠉⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡤⠆
⢠⣤⣤⡄⠀⠐⠒⠒⠈⠉⠉⠁⠀⠐⠒⠀⠀⠤⠀⠀⠄⢀⡀⢸⣤⣤⣤⣤⣄⣤⣴⠷⠚⢻⡀⣀
⠀⠲⡖⠠⠤⠄⣀⣀⠀⠐⠂⠤⠤⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠉⢀⣈⠀⣄⢠⡄⠘⡾⠦⠤⠄⠀⣀⣀⣀⡀
⠀⠀⠱⡀⠀⠀⠀⠀⠀⠉⠁⠐⣲⣦⣤⣤⡌⠉⠀⠐⠒⠀⠤⠤⢘⠶⢶⡶⠖⠶⠟⠋⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠒⠒⠒⠂⠠⠤⠤⣤
⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⠀⠀⠀⠀⠈⠁⠐⠒⠀⠤⠄⣀⣁⠉⠉⠐⠒⠂⠤⠤⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⣊⡩⠄
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣼⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠑⠒⠒⠠⠤⠤⣀⣈⠉⠉⠁⠀⠐⡦⠀⢀⡩⠤⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠉⠀⠒⠒⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠒⠓⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠂⠒⠂⠐⠀⠀⠆⠂⠒⠀⠀⠂⠀⠐⠀⠐⠐⠂⠀⠀⠀⠀⠐⠀⠃⠁⠘⠀⠀⠘

⠀⠀⠀⣀⣤⣤⣤⣤⣀⠀⠀⠀⠀⣠⣤⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣤⣀⠀⠀⠀⢀⣠⣴⣶⣶⣦⡀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣦⣠⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣸⣿⡿⣿⣿⣿⣿⣷⣆⣰⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⣠⣶⣿⣷⣦⣄⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⡄⠀⠀⠀⣴⣾⣿⣿⣷⣶⣤⡀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣶⡄
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣄⢀⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣯⣿⣿⡿⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠘⣿⣿⣿⣿⣿⣻⣿⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⢏⣴⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣶⣌⠻⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣧⡘⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⡾⢰⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⡿⢂⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⠃⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣦⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⠻⠄
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠄

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠒⠒⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⡉⠀⠀⠀⠀⠀⠀⠠⢝⠢⣄
⠀⠀⠀⠀⠀⠀⢀⠔⠩⠐⠊⠁⠁⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠄⠃⠀⠀⠀⠀⠀⠀⠀⡐⠦⡙⢎⠣⡀
⠀⠀⠀⠀⢀⠔⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢠⠎⠤⠉⠀⠀⠀⠀⠀⠀⠀⠐⠀⢄⠱⡙⡄⢣⠱⡄
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡄⠀⠀⣠⡀⢠⠏⠌⠁⠀⠀⠀⠀⠀⠀⢀⠐⠁⠐⡄⢢⠡⠘⡀⡆⢧
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠳⣌⠋⠀⠀⠀⠀⠀⠀⠀⢀⠐⠄⠁⠀⠱⠰⠀⡆⠃⢁⢰⢸
⠀⠀⠀⡆⡡⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣳⠃⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⢌⢂⠱⠀⢠⠘⡀⠃⡆⢡⢰⠈⢸⢨
⠀⠀⠀⡇⠔⡡⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠈⠱⡢⣄⠀⠄⡐⠘⠈⡌⠀⠈⡴⠀⡇⠀⠁⢸⢸⠀⠼⡀
⠀⠀⠀⠀⠐⡡⢂⠀⠉⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡆⠀⠑⠈⠀⣹⡆⢀⠀⠃⡀⢀⠁⡇⠀⠇⠀⠀⢸⠘⢠⢀⠃
⠀⠀⠀⠰⠐⡱⢁⠔⡀⠈⠀⠊⡔⢁⠔⣠⢂⠔⡠⢀⠆⡄⠀⠀⢞⡟⠁⠀⠐⢣⡾⠋⠀⠀⠀⣄⠣⣽⠀⠟⠀⠆⡄⡆⡈⠀⡈⡌
⠀⠀⠀⠀⠡⠀⢁⠎⢠⠂⡠⠀⡀⠀⠀⠀⢀⠄⢀⠀⡀⢠⢀⢂⠌⣱⡄⠀⠀⠀⢱⣄⠱⡱⠰⠘⢠⠇⡘⡏⠰⢠⢀⠁⠁⡄⠷
⠀⠀⠀⠀⠀⢣⠀⠔⢁⠎⠠⠊⡠⠊⡰⢣⠊⠔⢁⠌⡐⡡⢣⠊⡴⣿⢃⡄⠀⣰⡟⠁⡰⢡⢃⢆⣾⢀⠿⢀⠃⡆⠘⠰⡰⢠⠃
⠀⠀⠀⠀⠀⠀⠡⡀⠈⠰⢁⠜⢀⠌⡐⢁⠊⠠⢃⠌⡐⠠⣡⢞⠞⠁⠈⠁⣴⡟⠀⡴⢡⢊⠎⣼⢃⠎⠇⡜⢠⢠⢃⢃⢃⠂
⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠎⢠⠊⡜⠠⠃⡐⢡⠂⠌⠠⢁⠻⣏⠲⠀⠶⠀⢻⡄⠊⣠⢃⠎⡐⡌⡘⡜⢰⢁⠇⡆⡜⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠁⡌⠰⠁⡔⠠⢁⠎⠠⢡⠃⣮⡿⠀⠀⠀⠠⡟⠁⡔⢁⠎⡰⡱⡑⢰⢀⠇⡜⡰⡰⡡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⡔⠠⢁⠊⢠⢃⠂⡄⡈⠑⣦⡀⠙⠀⠉⣢⡀⠃⠴⠵⠳⢡⠇⠞⡴⣰⢣⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢁⠃⡌⠀⠂⡌⠠⢠⢁⠮⡻⠡⠰⠀⣼⠟⢁⠞⡴⠱⢃⠏⠞⠰⠑⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠑⠤⡈⢰⢠⠁⠃⢸⣾⠁⠀⠀⠀⢹⠰⢃⠞⡰⢁⠏⡜⢸⡡⠋⠔⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠬⣐⠰⢀⢿⡆⣠⡀⠀⠀⢇⠉⡴⡡⠊⢎⠔⠋⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠼⠇⠀⠀⠀⠀⠘⣌⣐⡥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠀⠀⢜⠆⠀⣬⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠀⢀⣀⠀⠰⡦⠈
⠀⠀⠀⠀⢀⠄⡠⠀⡠⢀⠄⡠⢀⠄⢀⠄⡀⣀⠀⡀⠀⡠⢀⠀⡀⠈⡉⠀⠀⡀⢀⠀⢠⠀⡀⢀⠀⡀⢀⢀⠀⢀⠀⡀⢀⠀⡀
⠀⠀⠀⠀⠀⠉⠐⠉⠒⠑⠊⠐⠁⠒⠁⠊⠐⠡⠊⠴⠋⠔⠡⠊⠠⠊⠔⠠⠊⠔⠡⠊⠠⠊⠔⠁⠊⠐⠑⠃⠂⠁⠊⠐⠁⠊⠐⠁⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⢀⡀⠀⠀⠀⠀⠠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠸⡉⠈⠀⠐⢣⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠒⢄⠀⠀⠈⢑⠆⡠⠄⠀⢜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⡀⠘⠄⠈⢐⢀⠀⠡⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⠀⣀⡀⠀⠣⡀⠳⠎⠁⣈⠆⢀⡷⠁⠀⠀⠀⠀⠀⠀⠤⣐⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡈⠧⠤⠚⠀⠘⣓⢈⠑⡆⠀⠀⠉⠈⠐⠂⠒⠒⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⣀⠤⠊⠀⠀⠀⠀⢠⠀⠤⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⡖⠀⠉⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠘⢢⡄⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢄⠈⡌⠀⠁⠢⠜⡄⠐
⠀⠀⠀⠀⠀⠀⢀⠠⠆⠀⠄⣈⠢⡁⠺⡄⠀⠀⠀⢁⢧⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠻⠄⠐⠂⢈⢠⣈⠆⠈⠐⠂⠉⠁⡀⠀⠀⡞⡩⠃
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⠝⠁⠀⠀⠀⠀⠀⢀⠃⠮⠄⠁⢅⢄⠒⡄
⠀⠀⣾⠉⠑⣄⠃⠈⠐⠄⣴⠁⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠈⠣⠕⠂
⠀⠀⠈⠛⢶⣿⡣⡀⠠⠑⠨⢃⣀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠈⠉⠛⠃⠀⠀⠀⠙⠣⣒⠒⠤⠴⢏⡰⠳⠤⣦⣄⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⡀⠙⢄⠀⣿⡥⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠦⠼⠀⠁⠒⠊
⠀⠀⢀⣀⡀⣀⠀⣀⠀⠀⠀⣀⡀⠀⢀⡀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⡇⢱⡀⠀⣴⡄
⢀⢸⠹⠀⢀⠀⠀⠀⠀⠀⢀⣶⠀⣿⠘⡄⣇⠀⢳⡀⡇⠹⡀
⢸⢺⡄⢳⠟⠀⠀⠀⠀⠀⢸⠁⠆⣿⠀⠘⢿⠀⠀⢻⡇⠀⠱⡀⡷
⢀⢧⠁⠀⠀⢀⠀⠀⠀⡷⣸⡀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠡⢠
⠈⢿⣆⠀⠀⠿⡇⠀⢸⡇⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠹⣦⡀⣷⡄
⠀⢈⢆⠀⠀⠀⢹⣠⡘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠉⢹⣄
⠀⠀⠻⡷⠀⠀⠀⠿⠙⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⡙⠃
⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⢻⡄⠀⣱⣆⣤
⠀⠀⠀⠈⠻⡷⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⡠⠙⠛⠁⠀⠀⣷⡄
⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⣰⡆⠀⢠⠀⢸⡀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠺⣧⣬⣀⣀⣈⣀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⠖⠋⠁⣷⠀⣾⣇⢸⡇⠀⠀⠀⠀⠀⠘⣆⠀⣿⡄⠀⡠⢨⡏⠉⠾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⢸⡀⡇⠀⠀⠀⠀⠀⠀⢹⡀⣿⢱⡈⣷⡷⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⡇⠀⣇⢷⠀⠀⠀⠀⠀⠀⠀⣧⣿⠀⢣⢹⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠸⣼⡀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⢇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⡀⠘⣇⢀⢀⠀⠀⠀⠀⠀⢸⣧⢀⢀⠀⢻⡄⢀⢀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠴⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣦⡀⠀⠀⠀⠁⢀⠀⡈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⣽⣹⣿⡁⠀⠀⠀⠃⢝⡐⠀⢈⠁⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢉⣽⣿⢤⣶⣄⡓⠮⠐⡀⠐⠈⠀⠀⠀⢠⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣟⣿⣮⣮⡟⡍⠜⢁⠢⠄⠉⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣳⣾⣿⣿⣿⣗⣿⣗⣮⠑⢞⠃⡀⠒⠈⡈⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣷⣶⣭⡛⠋⠁⠀⣻⣹⣶⡤⠊⢈⢣⡓⠀⠄⠐⡀⠀⢀⠀⠈
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣟⢉⢉⣿⣿⣿⣦⣆⣽⣿⣿⡿⠡⠁⡀⠂⠉⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢹⣤⠶⣶⡶⠶⣾⣿⣿⣿⣿⠿⠿⠿⠿⣿⡿⣿⣿⣿⡿⠆⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⢀⡀⠈
⠀⠀⠀⠀⠀⠀⠠⠤⢤⣤⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⣿⣷⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⡰⠋⣩⠆
⠀⠀⠀⠀⠀⠀⣾⡿⢋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⡇⠀⠘⠓⠉⠁
⠀⠀⠀⠀⠀⣠⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠙⢿⣿⣷⡀⠀⠠⣖⠒⠒⣦
⠀⠀⠀⠀⢀⣿⣿⣾⣿⣿⣿⣿⣿⠿⠿⠛⠻⠿⠿⠿⠟⠛⠁⠀⠀⠀⠀⣀⠀⢻⣿⣇⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢻⣿⣤⣠⣤⣄
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢦⡀⠻⣿⣿⣿⣿⣿⣦
⠀⣀⠀⠀⠀⢿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⡇⠀⢉⣸⣿⣿⣿⣯⣄⣀⡀
⢎⣉⣩⠷⠀⠸⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢠⡟⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠋⠙⣦⠶⣄
⠀⠀⠀⣀⢠⡀⠹⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠻⣾⠇⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠈⣿⡟⠉⢻⡇⢀⡜⠁⢀⡼⣄
⠀⠀⡞⢁⡞⠀⣀⣹⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⣁⣀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠙⠟⢦⡴⠋⢀⣸⠇
⠀⠀⠉⠁⠀⣸⣿⣿⣿⣿⠟⡛⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⡄⠀⠀⠀⣸⠿⣿⠄⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠐⣿⣿⣿⣿⡇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⢀⣿⠞⠁⠀⢀⡴⠚⠋
⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣦⣤⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡉⣩⠖⠋⠉⠀⢠⣞⠁⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⠿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢀⣠⡶⠋⠙⢧⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⢻⠟⠻⠿⣯⠀⠘⣿⠉⠳⢦⣄⣀⡀⠀⠀⠀⣀⣀⣠⡴⢖⣫⠟⠁⠈⠲⣾⠉
⠀⠀⠀⠀⠀⠀⢀⣽⠦⠤⢄⣸⢇⠀⠘⠷⠶⠶⠞⠛⣿⠉⠉⠉⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠸⣇⣀⣀⣀⣵⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⣀⣠⣤⣤⠞⠛⠛⠛⠛⠋⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠞⠛⠛⠛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣀⣤⡴⠖⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣍⡀⠀⠀⠀⠀⢀⣀⣤⠶⠚⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⣀⣀⡈⠙⣿⠓⠚⠛⣿⠁⠀⠀⠀⢻⣀⣀⣠⡿⣤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⣹⠉⠉⠉⠉⠀⠀⢀⡿⠀⠀⠀⠀⣸⠉⠉⠈⠓⠓⢺⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠁⡠⢿⣄⣀⣀⡀⣀⣀⡼⠃⠀⠀⠀⠀⢾⣤⣭⣭⣭⣭⡭⠤⠤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣤⣍⣲⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠐⢀⣽⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠧
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⣀⣴⣾⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⢀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⢟⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠿⠿⠟⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠂⠒⠐⠐⠀⠀⠠⠀⠐⠈⠈⠈⠀⠀⠀⠀⠀⠈⠀⠒⢜⡄⠀⠀⠀⠀⠰⠠
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⢀⠀⡄⣀⠄⠀⠈⠉⠍⠻⠥⢐⢄⠀⠀⠀⠀⠁⠉⠀⠀⠶⠞⢗⣦⠬⠓⠂⡀
⠀⠇⠀⠀⣆⡀⢰⠚⡏⠀⣀⠠⠴⡔⠠⢄⠀⠸⡠⢤⣼⠀⢀⡤⠤⢤⠤⢠⠀⢸⠿⠀⣰⠠
⡀⠀⠠⠀⠯⠓⠂⠠⠯⠄⠃⠀⠈⠻⠀⠀⠃⠨⠂⠀⠿⠠⠉⠀⠠⠛⠄⠀⠇⠔⠁⠀⠓⠂⠀⠀⠀⠠⠐
⠀⠀⠀⠀⠄⠤⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠂⠙⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠉⠠⣀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠤⠤⠤⣀⠀⠀⠐⠦⡀⠈⠁⡀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⢒⣂⠤⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠄⠋⠉⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠠⠀⠀⠀⠀⠘⠀⢡⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⣠⡴⠚⠉⠉⠀⠈⡁⢀⣀⠂⠠⠖⢀⠀⠀⠀⠀⠀⢤⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⢺⠁⠴⠋⠀⢠⠂⠀⠀⠉⠉⠑⠀⠀⠀⠱⠀⠀⠀⢀⡈⠄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠴⠃⠀⠀⡆⠠⠐⠒⠀⠀⠀⠈⠉⠑⠄⠀⠀⣤⠣⠀⠀⡴⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⡾⢠⠀⢰⣇⡀⠀⠀⠀⠀⣠⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠜⠉⡘⠘⠿⢠⠀⠒⠀⠘⠃⠤⠄⠴⠀⠀⠀⡐⢀⡄⠴⣼⠀⡠⠤⢄⢀⡔⠈⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡘⠁⠀⠀⠀⠀⠂⠀⠀⡤⡀⠀⠀⠀⠠⠞⠟⠉⠀⠀⢀⠋⠀⣠⠋⠀⢀⣗⣈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠈⠁⣨⠀⢠⠇⢀⡔⠋⠀⢀⠗⠂⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⢀⠀⠀⢈⠐⠀⠀⡀⠀⠀⠀⣠⢊⠁⠉⠉⢡⠀⠀⠀⠈⠀⠴⠚⠡⡤⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⠜⠉⠉⠉⠙⢷⠀⢿⡀⠁⡒⣉⣋⡁⣸⠓⠂⠀⠙⠀⠀⠀⠀⡠⠀⣤⡂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠴⠒⠉
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⠒⢴⠒⠊⠉
⠀⠀⠀⠀⡀⠄⢈⠀⠀⠀⠀⠾⠰⡀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠢⢤⠀⣤⠃
⠀⠀⡠⠊⠀⠐⠠⠀⠀⡀⡠⠂⠰⡇⠀⠀⠀⠀⡀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠌⠀⠀⠀⠀⠀⠘⢠⠞⠁⠀⠀⠈⢢⡀⢰⠈⢀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠊
⡎⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠈⡱⠇⠀⡠⠦⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢀⣀
⠳⠤⣄⣀⡠⠔⠥⡎⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠐⠂⠤⠤⠄⡀⠀⠀⠀⡜⠀⠀⠀⠉⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠉⠐⠒⠶⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⡀⠄⠒⠒⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣀⣀⣀⠠⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠱⣁⣔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⢀⡤⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⣀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡄⠀⠀⠀⠀⠀⡔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠂⣀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠄⠀⠀⠀⠠⠄⠦⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⢀⠠⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠶⠞⠛⠛⠛⠛⠛⠛⠛⠛⠉⠲⠶⠦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⡀⠀⠀⠉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⣄⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢂⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⢀⠖⠈⠑⣤⠀⠀⠀⠀⠈⠓⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⡀⠀⠀⢀⡘⠈⠙⢦⣀⠀⠀⠀⠀⠈⠃⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⣩⡷⣦⣀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠⡾⠋⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠆⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠈⠛⠶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠈⡇
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠂⠀⠀⠀⣠⡟⠀⠀⣠⡶⠿⠄⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣤⣀⣀⠀⠰⠿⢶⡄⠀⢀⣠⣾⠋⠀⠀⣿
⡄⠒⢤⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⠶⠶⠷⠶⠋⠁⢸⡇⠀⠀⣿⠀⢀⠐⠒⡀
⠁⠰⠆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢸⠇⠀⠀⣿⠀⠘⠃⠐⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣴⠶⠶⣦⣸⡟⠀⠀⠀⢰⠏⢀⡀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⣀⠀⠿⠄⠀⠀⡟⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢠⠟⠁⣤⠀⠀⠛⠁⠀⠀⠀⠘⣄⠈⠁⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠉⢀⡶⠀⠀⢸⣿⡄⠀⠀⡟
⡏⢀⠜⠀⠀⠀⢸⡄⢸⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⠀⢿⣀⡿⠀⢸⡇⠀⠘⢃⠈⡁
⠀⠀⠀⠀⠀⠀⢸⡇⠈⠻⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣶⣶⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣄⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠄⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠙⠻⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣼⠇⢠⡝⠃
⠀⠀⠀⢀⣀⣰⠳⢆⠀⣠⢄⠙⢶⡠⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⣠⡾⠁⠘⣾⠁⠀⠀⡠⣄⠀⣠⠞⣦⣀
⠀⠀⠀⠻⣍⡙⠳⠈⠛⠁⢸⠁⠈⠻⣆⠀⠀⠙⠻⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠞⠋⠀⢂⡿⠁⠀⠀⠈⠣⠈⠟⠁⠾⢛⣉
⠀⠀⠀⠰⣯⣁⡀⠀⠀⠀⠘⠧⣤⣄⠉⢳⣄⠀⠀⠀⠈⠉⠙⠓⠒⠶⠒⢶⡤⠀⠠⡤⠶⠶⠶⠞⠋⠉⠀⠀⠰⣹⠞⠁⠀⣀⣤⡸⠇⠀⠀⠀⠀⣉⣽
⠀⠀⠀⠀⠀⠀⠈⠛⠃⠚⠲⠦⣄⡀⠙⠋⠻⠷⠌⣡⣄⣀⣠⣶⡶⠶⠶⠾⣄⠀⢀⡴⢶⠶⠶⢶⣆⣀⣀⣀⣴⠿⠔⠛⠋⢀⣀⠴⠖⠁⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⢤⡄⣀⡀⠀⠁⣹⡇⠀⠀⠀⠂⠀⠤⠌⠀⠀⠀⠀⢸⡏⠈⠀⢀⣀⣠⡤⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠻⢷⣴⡆⠀⠀⠀⠀⠀⠀⠀⢸⣦⣼⡿⠟⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠛⠛⠛⠛⠛⠛⠛⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣄⡀⠀⡀⠀⣀⣠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠋⠻⠟⠻⠟⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⣿⡇⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢰⣿⡇⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠹⡇⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣾⠀⣧⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⣿⠀⣿⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣿⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⢿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⣿⠀⢸⡃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠾⣟⠛⣿⠀⣸⡛⣿⠿⠶⣦⣀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣷⣶⣶⣦⣤⣟⣀⣻⣤⣤⣶⣶⣾⣿⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡧⣖⣰⣶⣶⢰⡶⣶⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣿⣿⣿⣿⢿⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⡉⠋⠉⠉⠁⠠⢙⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟⡛⠿⡏⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠃⠀⣼⡿⠿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢝⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠤⣄⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⠀⠀⢸⠀⠀⠀⠈⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠘⢄⡀⠀⠀⠚⠋⠉⠉⠉⠙⠛⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠉⢒⡶⠒⣉⣽⣃⣀⣀⡠⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡴⠋⠐⣋⠭⠤⠶⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠤⠖⢛⣻⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣋⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢸⡃
⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⢈⣷
⠀⠀⠀⠀⠀⠀⣾⢀⠀⣀⠿⠟⠛⣶⡀
⠀⠀⠀⠀⠀⢸⣧⢸⣺⡏⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠤⠞⣿⠈⢻⣿⣶⣶⣤⠾⠃⢠⡄
⠀⠀⢈⡇⠀⠀⢿⣦⡀⢰⣷⣄⡄⠀⢀⣾⠃⠀⣶⠆
⠀⠀⠘⠓⠀⠀⠀⠙⠻⣾⣿⣻⣷⢀⡸⡎⠀⣼⠋⢀⡀
⠀⠀⠂⠀⣀⢀⠐⠀⠚⠉⢉⡉⠙⠉⠈⠿⣿⣧⣀⣼⠓⠀⠀⢀⣀⣤⣄⣀
⠀⠀⠀⠀⡘⣇⡀⠀⢀⡤⠛⠓⠒⠀⠀⢀⣽⣿⣿⣷⣠⣶⠎⠀⠀⠀⠀⠙⡧
⠀⠀⠀⢸⣿⣻⣷⡞⠉⠀⠀⠀⠀⠀⠀⠿⡁⣿⣿⡟⢠⡄⠀⠀⠐⠸⣟⡶⠁⠀⣆
⠀⠀⠀⣿⣿⣿⣿⣁⣤⣶⡄⠀⠀⠀⠀⢲⡟⣸⣿⡿⢿⡹⣆⠀⢰⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡀⣄
⠀⠀⠀⢿⣿⣿⣿⣼⣿⡉⢿⡄⠀⠀⠀⢸⣧⣿⣿⣿⠈⢁⢹⡄⠀⠂⠀⠀⠀⢰⣏⢀⡶⠀⠀⠀⠈⢷⡀
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠹⠀⢿⡆⢠⣴⣿⣿⣿⣿⣿⣷⣴⣿⣇⠀⠀⠈⠀⢠⡿⢁⣾⠁⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣷⣶⣶⣾⡟⢨⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣆⡀⠀⠀⢹⡇⣼⠃⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣬⣿⣿⣿⣿⣿⡿⣻⣿⣴⠆⠀⣸⠇⢿⡄⠀⠀⠀⠀⠀⠀⢸⣿⠂⣀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⢧⣈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣥⣤⣾⣃⡼⢾⣧⡀⠀⠀⠁⠀⠀⣼⡟⡶⠃⢀⠐⠁
⢀⡀⠀⠀⠘⣷⡟⠀⠀⠀⣾⣩⣿⣿⣿⣿⣩⣽⣿⢉⣿⠛⣿⣿⣿⣿⣿⣿⣛⣧⡄⠙⣿⣶⣤⣤⣤⣴⡿⠏⠠⠊
⠈⣿⡀⠀⠰⡟⣻⣦⡑⣶⣾⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣠⡇⢻⣧⢈⡙⠓⠀⠀⠀⠀⣠⠾⠃
⠀⠈⠿⣦⣄⣸⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠛⠟⠛⠛⠛⣿⡿⣿⣿⣿⣿⣿⣯⣿⣦⣿⢧⡄⠀⠀⠀⢀⠀⠊⠁
⠀⠀⠀⠠⢙⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⣿⣿⣿⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⣾⣯⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⣿
⠀⠀⠁⣸⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⡀
⠀⠐⠿⠿⣿⣿⣧⣿⡇⠀⢀⣤⣶⣶⣶⣶⣦⣶⡀⠀⢠⣾⣿⠿⠛⢿⣿⣶⣤⣿⣿⣿⣧⣼⣷⠛
⠀⠀⠀⠀⠈⢹⣿⣿⣿⣾⣿⠿⠋⠀⠀⠀⠙⣿⡟⠀⢸⣿⠉⠀⢀⡐⣋⣿⣿⣿⣿⣿⣿⣿⠯
⠀⠀⠀⠀⠀⣸⠁⣿⣿⣿⣿⣤⣤⣤⣤⣶⣾⣿⡄⠀⠸⠿⠿⠿⠿⠿⠟⠛⠋⠈⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠇⢰⣿⣿⠇⠙⠻⠋⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠈⠿⠀⠀⠀⢃⡀⠀⠀⠀⠀⠀⠀⠚⠳⠚⠁⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠠
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⢀⣤⣤⡤⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⢀⣰⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⣷⣶⡋⢻⡿⠛⠋⠉⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡆⠸⣿⣿⣿⣧⡀⠀⣀⡴⠂⠼⠃⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⣵⣾⣯⣥⣭⡿⠛⢻⠟⠋⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⣤⠤⠄⠀⠀⠈⣷⠂⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠘⠃⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠲⠌⠘⠃⠉⣩⢛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡃⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠪⠅⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠉⠑⣄⠀⠀⠀⠀⢸⠋⠁⠒⠀⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠈⠇⠀⠀⠀⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣠⠜⠃⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠐⠛⠓⠒⠒⠦⠀⠀⠘⢦⡀⣇⠀⠀⠀⠀⠀⢀⣂⣀⣀⣀⠀⠀⠀⢀⣤⡀⠀⠀⠐⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠀⠀⠀⠀⠀⠀⢤⣒⠄⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⡀⡀⠀⠀⠀⣀⠤⠊⠀⠀⠀⠀⠀⠀⡠⠠⠤⠤⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠤⢤⡤⠔⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢀⣀⣠⠁⡹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢄⡀⠹⡈⠀⠈⠑⠣⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⠉⠢⠱⠀⠀⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠁⠊⠀⠀⠀⡈⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⢈⣁⣰⢤⡀
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠘⠤⠔⠀⠢⡀⢄⡘⢲⡄
⠀⣴⡋⠙⠲⡀⠘⠀⠉⠂⢄⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠈⢦⣱⠤⠃
⠀⠈⢷⠀⢀⣰⣧⡀⠀⠀⡦⠘⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠐⠝⢣⣕⣄⡰⠁⠀⠙⠂⢠⢤⡀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠙⠂⢈⠑⠒⠀⠚⠇⠰⠛⠛⠀⢵⡆⢀⠂⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⡀⠀⠈⢦⠀⠀⣿⣓⡰⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⡑⣄⠃⠙⠤⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋
⠀⢀⣤⣤⡤⠤⢀⣰⣤⡀⠠⠄⢀⣤⡤⠄⣀⣤⣄⡀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⢠⣊⡱⢀⣤⠶⠞⠛⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣤⣀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠹⣦⡴⠶⠆⠀⠀⠀⠉⠉⠛⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠙⢦⡀⠀⠀⠘⣧⣄⠀⠀⠀⠀⠘⣷⣤⠤⣄⣀⣠⠴⠒⢲⣤⠶⠛⠙⢧⣀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⠳⣤⣈⠢⣈⣳⣄⠀⢠⠞⠁⠀⠀⠀⠈⢣⣰⡶⠛⠁⠀⠀⠀⠀⠉⠛⠶⠦⣤⣤⣴⡟⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⣌⣙⡏⠀⢰⡦⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⠀⠀⠀⣧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⢠⡒⣒⡆⠀⠀⠉⢧⠀⠀⠀⠀⠀⠀⡰⠁⠀⠠⣀⡰⠀⠀⠀⠑⠊⠁⠀⠀⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠒⠶⠿⠷⠶⠶⠶⢶⡟⠓⠦⣤⣤⠴⠚⠁⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣀⣠⠇⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⣠⡾⣻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⣴⣿⣿⣷⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋
⠀⠀⠀⠀⣰⠏⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⢰⡿⣿⣿⣿⣿⣿⣿⣿⠟⢳⡶⠶⠚⠛⠋⠉
⠀⠀⠀⢰⡏⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⢋⣀⡙⣿⣿⡟⠀⣾⠁
⠀⠀⠀⢸⡇⠀⢹⣄⠀⠀⠀⣠⡶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉⠉⠉⠉⠙⠃⣸⠇
⠀⠀⠀⠀⢻⡄⠀⠙⠳⣶⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠙⢶⣄⣰⡟⠀⠀⠀⠀⠀⠀⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠉⠉⠉⠻⢦⣄⠀⣰⠟⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⡋⠀⠀⠈⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢦⣄⡀⠀⡀⢹⡆
⠀⠀⠀⠀⠀⠀⢀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⣹⣼⠃
⠀⢀⡴⠟⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⣀⣀⣠⣤⠴⠟⠒⠒⠖⠛⠛⢾⣅⠀⠀⢠⢸⡟⠋⠁
⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⢻⡿⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⠾⠟⠁
⠻⣤⣴⢃⣠⡶⠶⠦⠤⠴⠶⠶⢶⡾⠛⠋⠁⣠⡴⠋
⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⡄⢠⢀⡿⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⢁⠀⠀⠀⡀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⡄⠀⠀⠀⠰⠄⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣶⣄⠀⠀⢻⣾⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢠⡄⠈⠿⠛⠀⠀⢻⡿⠀⠀⡀⠇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⣯⠀⡀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡜⢳⠷⠀⠀⠀⡴⠂⠀⢀⡼⠁⣼⠃⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣳⣄⠀⠀⠀⠈⠀⠀⠐⠁⠀⠊⣠⠖⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⣦⠀⠀⢀⠶⢶⣶⠆⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃⢠⠋⠀⣸⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡋⣴⣦⠀⠀⣿⠀⠸⠦⢀⣀⣠⣄⠰⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢢⡶⠻⣷⠘⠏⣠⣼⠋⠀⠀⠀⠘⡿⠋⢻⠀⠀⠈⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠋⠉⠀⢾⠀⠀⣿⠇⠘⠉⣼⠀⠀⠀⠀⡾⢣⠀⡌⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⠖⠀⠀⠀⠀⠀⠈⠃⠀⠹⣷⡆⣴⠇⠀⠀⠀⢸⠃⠀⠀⠇⠀⠀⡆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠈⢢⠀⠀⠀⠀⣿⣤⡁⢀⣀⣀⣀⠘⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠃⠀⠢⠀⠐⠀⠻⠿⠿⠘⠛⠉⢁⣠⡴⣦⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⢠⡷⠀⡶⠀⠀⢠⠞⠋⡄⠉⠀⢀⠀⠀⢱⡆⠀⣠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣤⣤⣤⢿⡄⠀⠀⠈⢳⣤⠀⢀⣴⠏⢠⡜⠀⠀⠆⠀⠀⠰⣄⢀⡀⠙⠎⣵
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠉⠀⠀⠀⠰⣿⣄⣼⠃⠀⡌⠀⠀⡰⠀⠀⠀⠘⠛⢿⠉⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠈⣧⠀⠀⢠⡏⠻⣷⠀⡜⠀⠀⠈⣡⣂⣠⡄⠀⡱⠈⣷⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠈⠀⣰⠟⣱⠈⣿⠆⠀⠀⡾⠺⠇⢹⡟⠀⠀⡁⠈⢸⣧⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠃⢀⣿⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⠃⣠⣾⣿⠏⣀⣠⣤⡄⣤⣄⣀⣀⣀⣠⣤⡀⠀⠀⠈⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣰⣿⣾⡿⢻⠟⢪⡍⠁⠉⢦⣬⣯⠉⠉⠉⠉⠙⣄⡀⠀⠀⠀⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⢙⣹⠟⠉⠀⠙⣿⡗⠲⠆⠀⠀⠀⠀⠀⠀⣀⣽⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⢸⠃⢧⣀⣠⣤⣤⣤⣤⣴⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⠟⠻⢶⣤⣄⣘⢃⣌⠀⠀⠀⢙⡏⠙⠛⠟⢻⡟⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡐⣀⣀⡀⢩⡻⠛⠛⠻⣿⠀⣼⡇⢀⣀⠀⠈⠛⢉⠉⠻⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⣼⠟⠛⠁⠀⠈⠉⣿⠃⠘⢶⡦⡏⠀⢻⣷⡏⠉⣉⠹⠟⠉⢠⣤⠹⣧
⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣄⣼⣫⠶⠊⠉⠀⠀⠀⠀⠉⡄⠈⣇
⠀⠀⠀⠀⠀⣴⣿⠇⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠛⠻⣿⣸⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⣿
⠀⠀⠀⠀⣾⣿⣿⠀⠀⣿⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⢻⣿⡀⠀⣍⢿⠀⣶⡄⠀⠀⠀⠀⠀⠠⠀⣿
⠀⠀⠀⠀⠹⣿⡟⠀⢸⡏⠀⠀⠀⠀⠀⠈⠃⠀⠀⢀⣿⢹⠁⠀⣿⣾⠀⠘⣷⠀⠀⢠⡄⠀⣦⠀⣿⡄
⠀⠀⠀⠀⠀⠹⣿⣀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠻⠀⠀⢻⣧⠀⠀⢻⡇⠀⠈⠁⢠⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠓⠾⠟⢲⣤⣤⣀⡀⠀⢠⣿⣿⡿⠀⠀⠀⠈⢿⡆⠀⠈⠙⢷⣤⠀⠄⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠿⣿⣿⡿⠃⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠐⠒⠒⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠭⠛⠉⠀⠰⠊⢁⡎⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢂⡇⢠⠀⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣠⠃⠀⣈⠀⢀⣀⣀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠨⣉⢢⣴⡷⠁⠀⣠⡞⠉⠁⠀⣸⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⣢⠌⠁⠁⠀⢠⠋⠈⠋⠀⢀⠴⣻⡆⠈⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡯⢆⡤⠤⠶⠶⠶⠯⠅⠀⠀⠀⠀⠈⠑⠒⠚⢋⠉⠤⣉⣁⡠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠇⠀⣇⣾⠀⢴⣊⣛⠺⢏⣖⣦⣀⠀⠀⠀⠀⠀⣀⠼⢄⠀⠀⠉⠉⠒⠢⠤⢤⣀⡀
⠄⠀⠤⠤⠀⡀⠀⠀⠀⠀⠀⠀⣀⠔⠖⠊⠁⡸⠀⠀⡇⠀⠁⠀⠙⠂⠭⣱⡊⠉⠉⠳⣄⡤⠒⠉⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠉
⠀⠈⢻⠃⠉⠉⠑⠄⠀⢀⣏⠁⠀⠀⠀⠀⠀⠓⠒⠄⡇⠀⠀⠀⣀⣀⡤⠤⠌⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣂⣀⣀⣠⠔
⠀⠀⠈⡄⠀⡠⠶⠓⠊⠁⠈⠣⡀⠀⠀⠀⠀⠤⠀⠈⠑⠒⠿⣏⠉⠀⢤⡄⣀⡠⢀
⠀⠀⠀⠑⢄⢰⠀⠀⠀⠀⠀⠀⠈⠑⢤⣀⠀⠀⠑⣀⡠⢶⣿⣿⠤⢾⡶⠼⡀⠵⡴⢳⠀⢀⠼⣥⣼
⠀⠀⠀⠀⠀⠉⠒⠂⠠⠤⠄⣀⣀⠀⠀⠈⢉⠒⠒⠩⠽⣁⠀⠛⠙⠒⠂⠭⣁⠸⡱⢳⠆⠇⠔⡍⢠⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠚⠥⡀⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠙⠒⠉⠈⠂⡸⠙⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠑⢄⡠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⣀⡴⣾⡤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠉⠉⠉⠉⠋⠙⠓⠛⠛⠛⠒⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡅⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⡇⢀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠁⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢸⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⠋⠐⠒⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠙⠁⠉⠻⠟⠀⠙⠉⠉⠉⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⡆⠦⠀⠎⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣾⡇⠰⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠴⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡃⢸⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠑⠼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠡⣆⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⢸⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⢀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠸⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡸⠃⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢶⠀⠀⠠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠐⠀⠀⠀⢰⢲⡌⠁⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡁⠈⢆⠀⠀⠀⣀⠀⠀⠀⡘⣸⠃⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠀⠀⠀⢻⠀⡄⢰⢣⠏⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⣎⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠈⠁⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢐⡆⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠃⠀⠀⠀⠀⠀⠀⠀⡤⢲
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠒⠋
⠀⠀⠀⢀⠀⠀⢧⠉⠢⡀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠁⠀⠀⠱⢤⠞⠀⠚⠀⠀⠀⣀⣠⡾⠉⠉⡓⢦⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⡓⢌⢦⡤⠋⡀⢈⢳⡄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢀⡀
⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⣇⠐⠒⢶⠋⠀⢁⠀⢹⡶⠞⣿⠩⠹⣦⠀⠀⠀⠀⠀⠀⢀⠔⢉⠇
⠀⠀⠀⠀⠰⡓⢮⡂⠀⠀⠀⠀⢠⡶⠛⠓⢄⠀⠑⠂⠀⠀⠀⣿⣤⣼⡷⠄⠘⡇⠀⠀⠀⢀⠜⢁⡴⠋
⠀⠀⠰⠦⠀⠈⠀⠝⣦⡀⠀⠀⢸⡀⠀⠀⠚⠀⠀⣠⠊⠀⠀⠀⠀⠄⢀⣀⡼⠃⠀⠀⠠⠊⡰⠋⠀⠀⠀⠀⠀⠀⠀⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠙⠳⠶⣄⠐⠀⠁⠀⠀⠀⠀⠀⣰⠟⠉⠀⠀⠀⠀⡧⠞⠁⠀⠀⣠⠔⠊⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢧⣄⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠉⢳⡎⠀⠀⠁⢸⡀⠀⠈⠀⠀⠀⠞⠀⠀⠀⠠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⣸⣧⠀⠀⠀⢸⣇⢀⠄⠀⣠⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⢢⡀⡟⠀⠀⠀⠀⣇⠀⠴⠶⠃⠄⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠲⠤⢤⠀⠳⣷⠀⠀⢠⢀⢻⠄⢀⠀⠤⠔⠂⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠈⠛⠀⠀⠤⢽⣿⠀⠀⢈⣈⣾⣇⣤⣷⡀⠈⠀⠺⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠩⣭⣵⣒⣲⢿⡿⠿⠿⠿⠿⠛⠛⠿⠉⠙⠂⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⠤⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠘⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡠⡀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⢠⡆⠀⠀⠊⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡸⠊⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⡀⠀⠀⠀⠀⠀⢀⡀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⣰⠀⠀⠀⠀⣠⡊⠭⠊⠀⡆
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡌⠀⡰⠉⢆⠈⢦⠊⠁⠀⠀⠀⢠⠃
⠀⠀⠀⠪⡐⡡⠀⢆⠀⠀⠀⠈⠈⠀⠀⠈⠢⡀⠀⠀⠀⣀⢴⣉⣈⡈⠐⠂⢄⡀
⠀⠀⠀⠡⠱⢴⡖⣸⡑⠒⠀⠰⠄⠀⡀⢀⡰⠚⠉⠉⠁⠀⠀⢃⠀⠀⠈⠑⠠⣈⠢⡀
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠉⠁⠀⡄⠉⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠑⢤⠃⠀⠀⠀⢄⠙⠄⠀⠀⠀⢀⣀⠠⠄⠂⠒⠒⠂⢄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⢸⠐⢍⢧⡈⡢⠝⠀⠀⠀⠉⠉
⠦⡀⠀⠀⠀⢸⠁⢠⠀⠑⠤⣀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠨⣀⡄⠀⠹⡄
⠐⡌⠠⡀⠀⡜⠀⠘⠣⡀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢄⠀⠸⠤⠤⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠈⠢⣈⠲⠀⢀⠃⠀⠈⠑⠄⣀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢣⠠⡇⠀⠀⠀⠈⠢⢄⡀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠁
⠀⠀⠀⠀⢑⠑⡼⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠠⠀⢀⢀⣀⣀⣀⠔⠠⡀⠀⠀⠄⡈⢳⡈⠢⡀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠀⠉
⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⠢⣤⢀⠀⠀⢣⡀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠤⠊⠀⠀⠀⠱⣀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⡴⠤⣀⠀⢀⣠⠛⠧⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠉⠐⢢⡴⠋⠀⠀⠈⠑⠟⠁⠀⠀⠈⠳⣤⠞⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠶⣄⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢼⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢏⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠰⣄⠸⡇⠀⠀⠀⢀⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⡶⠘⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠢⡄⠀⠀⠀⢳⠀⠀⢰⡎⠁⠀⠀⠙⣷⡀
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⡀⠰⠂⠀⠀⢠⣤⡀⢨⣄⠀⠀⠾⠷⠶⠞⠀⠀⢾⡇⠀⣿⡇
⠀⡞⠃⣰⠏⢨⠆⣀⠤⣄⠀⣿⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢈⡁⠀⠀⣀⣠⣤⣤⡀⠀⠀⢀⣰⠟⠀⠀⠰⠆
⣀⢰⣆⠘⠷⠈⠾⠋⠰⠋⠀⠻⢆⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⢀⡉⠀⠀⠘⢣⠀⠀⠀⠀⣤⠋⠉⠀⠀⣿⠃⠀⠈⠙⠶⠖⠉⠁
⠿⣀⣁⠀⠀⠀⠀⠀⠘⠦⠴⣄⢸⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠄⠀⡿⠀⠀⠀⠀⠀⠀⣀⣀⢠⡞⢷⣄⣀⡀
⠀⠀⠙⣀⠀⠀⠀⠀⢀⡒⠚⠁⢸⡀⠀⢤⡀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢀⣤⡘⢏⡉⢻⡗⢀⣿⠏⢱
⠀⢠⡏⠉⠀⠀⠀⠀⢹⠇⠀⠀⠸⠷⠶⣼⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡷⠀⠀⠀⠀⠺⣄⡙⠾⠁⠀⠀⠈⢀⣤⣾
⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠋⠻⡄⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠛⠣⠜⠛⠀⠈⠰⠒⢀⡿⠠⠄⠀⠀⠀⠀⢈⣹⠃⠀⠁⠀⠀⣠⡭⣤
⠀⢸⡇⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⣿⠀⣀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣠⡟
⠀⢸⡇⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⡀⠀⠹⠦⠀⠀⡄⠠⣤⢀⡤⠤⠄⣴⠏⠀⠀⠟⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⣽⠂
⠀⠘⡇⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⠀⠉⠉⠀⠀⠀⠀⠀⠐⣾⠏⠀⠀⠀⠀⠀⠀⠀⢀⡘⠀⠀⠀⠀⠀⠀⢠⠋
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠙⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢶⣄⣀⣠⠖⠲⠶⠋⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⡾
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠈⠉⢹⡀⢀⣄⠀⠀⠀⠀⠀⠀⢀⣤⠟⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⣀⠀⠀⠀⠀⠀⢀⣾⠁⠈⢻⡄⠀⢀⡸⠋⠁⠸⠷⠄⠴⠶⠶⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠩⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢢⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⢦⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠴⠶⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⢆⡀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠸⠷⠀⠀⠀⠀⠘⡟⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠘⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⣸⣀⣀⠀⠀⠀⠀⠀⣨⠆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣿⣶⣶⣦⣶⣿⣦⣴⣿⣀⣀⣀⣀⣀⣘⣉⣻⣤⣤⣤⣤⣤⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⣄⡀⠀⢀⣆⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢲⡆
⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⣠⣾⠟⠀⠀⠀⢀⡀⠀⠀⠁⠢⡄⠀⠀⠀⠀⠀⣀⠤⠐⠚⠻⣇⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠁⣀⡤⠤⢶⠟⠃⠘⣷⠠⠀⢸⡇⠀⠉⠁⠚⠛⣱⣄⠤⠒⠉⠀⠀⠀⢀⡤⠛⠀⠀⣴⡾⠛⠛⠋⠂
⠀⠀⠀⠀⡐⢀⣸⡇⠀⡀⢘⡀⠀⠀⣹⡇⠀⠈⠁⠀⠀⠀⣠⡾⠋⠀⠠⣄⡀⢀⣤⠔⠉⠀⣤⠤⠤⢿⠇⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠘⡻⠋⣿⠡⣾⢀⣠⣴⢾⣿⡿⠁⠀⠀⠀⠀⢠⡾⠋⠀⠀⣐⡀⠀⠉⠉⠀⠀⢀⣄⣸⣧⠀⠀⠀⢠⣆⠀⢻⠶⣴
⠀⠀⠀⢰⠁⠈⠁⠐⢻⡿⠋⠀⣾⡿⠁⠀⠀⢠⣾⣴⡟⠀⠀⠀⢀⡠⠘⠀⠀⢠⢀⡴⠋⠉⣿⠇⠀⣀⣠⣾⣿⣶⠟⢀⠞
⠀⠀⠀⢸⡲⠁⠀⡴⠋⢀⣀⠠⠿⢟⣿⣿⣿⠏⠀⢻⠀⠀⣠⡖⠁⠀⠀⠀⠀⠘⠉⠀⡀⣸⣿⣶⣿⠿⠟⠛⠁⠉⢠⠎
⠀⠀⠀⢸⡀⠀⠚⢶⣴⡏⠀⠀⠀⣿⣿⡏⠀⠀⠀⠈⠇⠊⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⣙⣿⡍⠉⠀⠀⠀⠀⢠⡴⠋
⠀⠀⠀⠀⠳⡄⠀⠀⣿⣏⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣀⣴⡿⠉⠷⣄⡀⠀⠀⢀⣸⣿⣿⡉⠀⠀⠀⠑⠒⣤⠞⠁
⠀⠀⠀⠀⠀⠹⣆⢲⣿⡏⠀⢠⣴⠿⠋⠀⡠⠒⠶⠞⢱⣿⣇⠀⠀⠈⠻⠿⠿⠟⠋⠉⠻⢿⣦⡀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠘⢾⡿⡅⡀⠘⣿⡄⢀⡜⠁⠀⠀⠀⣴⠋⠘⣷⣄⠀⠀⠀⠀⠀⠀⢀⣰⠞⠛⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣧⠀⠈⢻⣾⠀⠀⠀⢠⡤⠽⠆⠀⣼⡿⠀⠀⣦⠀⣠⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢐⡿⢸⡀⠀⣾⠁⢀⣀⣠⠟⠀⠀⠀⢰⣿⡇⢀⣠⡿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠈⠷⡾⠛⠀⣿⠈⠉⠀⢿⣇⢀⣸⠿⠿⠛⠋⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⣆⣀⡤⠞⠉⠛⠁⠀⠀⠀⠴⠟⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠾⠋⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣺⡿⣿⣿⢿⣕⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡢⡀⠀⣀⣀⣠⣯⡟⠁⢀⣸⣇⡀⠈⢻⣽⣄⣀⣀⠀⢀⢤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣉⠁⠀⠉⠙⣗⠕⠩⠝⠪⣺⠋⠉⠀⠈⣉⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠘⠿⣧⢀⣠⣜⣯⠴⠞⠳⠦⣽⣓⣄⡀⣼⠿⠃⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡤⣾⡄⠀⠁⣰⢯⣳⣾⢡⢾⠽⠯⣗⡌⣷⣞⣽⣆⠀⠀⢠⣷⢤⡀
⠀⠀⠀⠀⠀⠀⣰⣻⠟⠉⢷⢤⡾⣼⠾⣛⡛⠈⡚⠎⠱⠃⠉⢛⣛⠳⣧⢶⡤⡾⠉⠻⣞⣆
⠀⠀⠀⣀⣀⣰⣿⣯⣄⣰⡕⡁⣰⠇⣞⠖⡺⠬⢾⣷⣾⡿⠤⢗⢲⣳⠸⣆⢈⣪⣆⡠⣽⣿⣆⣀⣀
⠀⠀⠀⠉⠉⠹⣿⣟⠉⠹⡳⡁⠹⣆⢯⠧⢵⠂⣾⡿⢿⣷⠒⡾⡼⡽⢰⠏⢈⢜⠏⠑⣻⣿⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠹⡿⣦⣀⡾⠚⠷⢿⢦⣭⣥⢀⢤⣇⣸⡬⣀⣼⣭⡶⡟⠾⠓⢷⣀⣴⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢿⠃⠀⠀⠹⣞⣽⢿⡘⢼⣲⣖⡯⢣⡿⢯⣷⠏⠀⠀⠘⡿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢠⣶⡟⠈⠙⢫⣟⠲⢦⡴⠖⣻⡍⠋⠁⢻⣶⡄⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣉⡀⠀⣀⣠⡯⡣⣔⣆⢔⢽⣄⣀⠀⢀⣉⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠕⠁⠀⠉⠉⠙⣟⣧⡀⠈⠹⢏⠁⢀⣼⣻⠋⠉⠉⠀⠈⠪⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢽⣷⣿⣿⣾⡿⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⢠⡶⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⣄⡴⠋⠉⠉⠛⢶⣄
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠂⠉⠁⠀⣠⡟⠁⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡤⠖⠛⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⢿⡄⣀⠔⠊⠁⠀⠀⠀⠀⣀⣠⡴⠞⠉⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣿
⠀⢀⣀⣀⣸⣿⣁⢀⣀⣀⣤⠴⠖⠛⠉⠁⣀⣤⠴⠖⠚⠛⠋⠉⠉⠉⠁⠀⠀⠀⣴⠏
⢰⠟⠁⠀⠈⠉⠉⠉⠉⠀⠀⢀⣠⡴⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⣿⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋
⣿⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⣀
⢹⡆⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠶⡟⠋⠁⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠞⠛⠉⠙⠈⠁⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢀⠂⡡
⠀⠀⠀⠉⠛⠛⠓⠚⠛⠛⠛⠉⠁⠀⠀⠀⢀⣀⣀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⠃⢸⣿⠈⠉⠙⠻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⢀⣿⣿⣿⣿⠟⠁⣼⠿⠟⣫⡿⠿⠛⠁⠀⠰⡏⠀⠀⠀⠀⠈⠻⣿⣿⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠶⠂⠀⠀⠀⠀⠀⠀⠈⠛⠄⠀⠀⢹⣿⠇⠀⠱⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣈⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣠⠀⢠⡀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠏⠀⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣯⠈⢣⠀⠀⠀⠀⢀⣏⠈⠇⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠠⡄⠀⠀⠼⠀⢀⣾⣿⣿⣿⣿⣿⠋⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠣⠤⠃⠀⠀⠀⠀⠀⠣⠤⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠙⢶⣄⣀⣤⣾⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠟⠛⠋⠉⠀⠀⠤⠭⠋⠀⠀⠀⢻⣇⠀⣀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⡆⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⢦⡈⠂⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⠀⢀⡼⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⢰⡟⠋⠉⠉⠀⢙⠁⠀⠀⢀⣴⠏⠀⢻⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠉⠻⣦⣀⠀⠀⠀⠀⠦⠄⠒⠀⠀⠈⠀⢀⣴⠟⠁⠀⠀⢀⡙⠓⠰⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠁⠀⠀⠉⠒⢤⡙⠻⠶⣤⣤⣀⣀⣀⣀⣠⣴⠾⢿⠁⠀⠀⠀⠀⣸⠋⠁⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⢯⡈⢢⡤⠚⠃⠄⠀⠀⠈⠲⣄⣀⣈⣉⣉⣉⠟⠛⣤⠠⣄⠁⠀⠀⠀⠀⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⢹⠃⠀⠀⠀⠀⠈⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠠⠴⠦⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠤⠞⠚⠀⠀⠀⠀⠀⢸⡇⠀⠀⡀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣠⡶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⢦⠀⠀⠀⠐⣿⠙⠛⠿⠿⠷⣶⣶⡶⢿⡀⠀⢨⠛⣿⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠈⢳⡄⠀⢠⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⢈⣠⡾⠃⠀⠈⠉⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⣄⢀⣤⠿⡶⠋⠀⠀⣠⠶⠷⠶⠶⠶⠶⠶⠟⠛⠉⠙⠛⣶⣖⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠘⢳⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⣴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢤⣸⡇
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣀⣀⣀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⢠⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢸⡇⠀⠚⠀⠀⠀⠀⣀⣼⢁⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣰⡇⠀⠀⠀⠁⠀⠀⢀⠆⠀⠀⠀⡀⠈⠃
⠀⣸⠃⠀⠀⠀⢠⠀⡀⠀⣁⠟⢓⠀⠠⣄⠠⠂⢀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣧⢾⣷⡆⠀⠀⠀⠀⠀⠀⡄⠀⠀⠘⠇⠀⡄
⠀⠙⠀⠀⠀⠂⠁⠀⡿⢦⡝⠰⠋⠁⠚⠋⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⡆⠠⣤⠀⢠⡿⡧⠇⢸⠏⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⢐⠅⣿⡃⠃⠀⠀⠀⠀⠀⢀⠀⠠⣀⠀⠀⠀⠀⠀⢸⣷⠀⠁⠀⠀⠀⠡⡦⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠸⣤⣤⣤⣆⠀⠀⠀⡄⠀⠈⣀⣄⣼⣀⡠⢠⡄⣠⣘⣏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠈⠀⠀⠀⠀⠀⣀⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠃⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⣴⢿⢹⣿⡇⠀⠀⠀⠀⠀⠀⢀⣸⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡟⠁
⠀⠀⠀⠀⠀⠀⠼⣿⠞⣸⠇⠀⠀⠀⠀⠀⡀⠀⠀⢻⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡲⠶⠀⠀⢀⠀⠀⠀⠠⠴⡂⠂⠀⠒⠒⠛⠛⠡⣤⣰⠴
⠀⠀⠀⠀⠀⠜⠁⠉⠀⠻⠒⠀⠀⠀⠀⣼⠀⢠⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣼⡁⣀⢸⡀⢸⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠅⢈⠛⡏⠷⠌⠃⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣿⠃
⠀⠀⠀⣀⣀⡀⠀⡀⠠⠀⠀⠀⢰⠅⢀⣿⠀⠀⢀⠀⠀⠀⠀⢰⣿⠀⠘⠀⠀⠀⠀⠺⠻⠭⡇⠀⠀⠀⠀⠀⠀⠀⢈⠀⠤⠀⠀⠀⢀⠶⣆⣠⣿⡟
⠀⠀⢠⠈⠀⠉⠀⠀⠐⠀⠀⠀⣡⣀⠌⣅⣠⠀⠸⠀⠂⠀⠀⣿⡏⠀⠀⠀⠀⡠⢤⠚⢿⡽⠄⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⡌⠔⠁⠈⠛
⠀⢀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠁⠀⡇⠀⠀⠀⠀⠀⠀⠠⣿⡧⡀⢀⣠⣤⣴⣼⣷⣼⡇⠀⠀⢀⣀⣀⠀⠀⠀⠌⠰⠀⠀⠀⠀⢰⢤⠀⠀⠒⡇
⠀⠈⠊⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠰⠿⠷⠿⢿⣽⣣⣻⣿⣞⣛⣳⣤⣬⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠆⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠉⠉⠉⠉⠀⠉⠉⠀⠈⠱⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢠⢄⠒⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡀⠀⠀⣠⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠻⢿⣿⣿⡟⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡏⠀⣰⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⣸⣿⠏⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣀⣀⠀⢠⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠠⣿⢿⣿⡟⠁⠀⠀⠀⠀⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠃⣰⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠃⠘⠛⢻⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⣼⣯⣤⡶⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣡⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠈⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣶⣶⣶⢶⣶⠶⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣿⣿⣿⣿⣿⣷⣾⣶⣷⣾⣿⣾⣶⣾⣷⣷⣾⣶⣶
⣿⣿⣶⣷⣷⣾⣶⣿⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⣀⣠⣤⣴⣶⣣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⡷⠞⠀⠀⠠⠄⠤⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⣾⣿⣿⡿⠋⠁⠀⠄⢄⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⢠⡄⠁⠀⢠⠈⠀⠘⠀⠀⠀⠀⠐⠒⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⠀⠂⡀
⠀⠀⠀⠀⠐⠁⠐⠒⠉⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠀⡀⡠⠀⠀⠀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⠀⠈⠀⠀⡠⠀⢀⡀⠄⠀⠀⠉⠀⠀⠀⠀⠠⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠤⠀⠰⠂⠐⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠄⣀⠄⠀⠀⠀⢡⠀⠀⠈⠣⢠⠀⠀⡤⠂⠁⠉⠉⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠈⣁⠀⠔⠀⡎⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠂⠐⠂⠀⠒⠆⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡀⠀⢠⢀⠂⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠁⠀⢀⠕⠀⠀⠀⠀⠀⠁⠀⠒⠠⠤⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠠⠤⠀⠒⠁

⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⣠⠤⠖⠦⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⢳⡀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣣⠆⠀⠀⠀⠀⢰⡷⣌⡳⢤⣀⠀⠀⠀⠀⠀⠀⠀⢠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⣰⡿⠠⠎⠉⠛⠛⠛⢛⣛⠛⢻⡿⣅⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣠⣾⠟⢀⡤⠞⠛⡷⣄⢀⡞⠉⠙⠢⣷⣌⠳⠀⠘⢦⡀
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠰⠖⣫⣿⠃⠀⣾⠀⠀⠺⠿⢺⢿⣇⢿⠗⠀⣹⣿⢧⡀⠀⠀⠙⢦
⠈⠉⠁⠀⣰⠃⠀⠀⠀⠀⠀⢸⠟⠻⠀⠐⠮⣷⣶⠶⠞⠉⠀⠙⢷⣶⣶⡯⢾⣄⣳⡀⠀⠀⠈⣧⠀⠀⣀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⣷⣀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣤⡼⠃⠀⠀⠀⣫⡏⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠘⢦⣌⠳⣄⡀⠀⢻⡘⠿⣧⡤⠀⢀⣤⣴⠶⢶⠶⠶⡶⠶⣶⢦⣄⠙⣄⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢓⣒⠒⠛⡟⢠⣾⢻⡀⣸⣄⣸⣄⣀⣇⣠⣿⠤⣿⣇⣿⠀⠀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠙⣾⣇⢘⣿⢟⡿⠋⡿⢸⠄⣿⠙⢧⡙⢦⠘⡏⢿⠻⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢹⡇⠸⠁⠰⠃⠙⠀⢿⢧⡈⠁⠘⠃⠛⢸⡇⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠈⣟⣎⢧⠀⠀⠀⠀⠀⠀⣸⠀⣿⠀⠀⠀⠀⡾⠁⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢻⡏⣨⡷⠂⠀⠀⠀⢀⡟⠀⢿⡀⠀⠀⠘⢷⡖⠒⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠴⠒⠛⠛⣻⠟⠁⠀⠀⠀⢠⡶⠁⠀⠀⠀⠙⣆⠀⠀⠀⠙⢿⡲⣬⣧
⠀⠀⠀⠀⠀⠀⠀⣰⠟⢁⡴⢖⣻⠟⠁⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠙⠘⣯
⠀⠀⠀⠀⠀⠀⠀⢿⣴⣿⠀⠈⠁⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢷⣦⣀⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣤⠴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⢾⣿⡏⣤⠈⣿⣿⠿⠋⠀⠀⠀⠐⠂⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣗⣂⣽⣿⡶⠞⠋⠁⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠉⠀⠀⠀⠀⠀⢀⣶⡆⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡄⠘⣿⣿⡀⡀⠤⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣶⣶⡶⠿⠟⠛⠙⣿⣷⠠⠙⠉⠁⠀⠀⠐⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⣀⣀⣀⣤⣤⣤⣤⣴⣆⠀⠆⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⣶⣾⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠆⠀⠀⠀⠀⠀⢡⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠈⠉⢙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡍⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⢃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠂⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⠀⠑⠠⠀⣀⠀⠀⠀⠀⠀⣀⠠⠤⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀⠤⠚⠁⠀⠀⠀⠀⠀⠀⢀⡠⠄⠂⠈⠡⠀⠀⠴⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠀⠀⠀⢀⠀⠒⡉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠂⠉⠀⠀⠀⠀⠀⠀⠡⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⢡⠀⠀⠀⠀⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠁⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⡠⠔⠊⠐⠒⠀⠠⠃
⠀⠀⠀⠀⣀⠊⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠉⠀⠀⠂⠊⠐⠀⠂

⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⢀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⠿⠿⠟⠿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡟⣿⣿⢿⣿⠿⣿⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣤⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⡀⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠁⠂⢠⣤⡄⠤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⢹⠉⠈⠁⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⢸⠉
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠈⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡇⠀⠀⠀⠀⠀⠀⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣷⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢻⣿⣿⣿⣟⣁⣼⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⡿⢻⣤⣾⣿⣁⣤⣿⣿⣿⣿⠿⣿⣿⣿⡁⢀
⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣏⣻⣿⣿⣿⣟⢡⣼⣏⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⠿⣉⣹⣿⣿⣿⠿⣿⣿⣿⣶⡿⠛⠻⣿⣿
⢙⣿⣿⣿⣟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⣿⣿⣿⣷⣾⣿⣿⡿⣿⣿⣿⣿⣞⢉⣟⣻⣿⣿⣶⣿⢿⣿⣿⣧⡦⣸⣿⣿
⣿⣿⠿⣿⣿⣿⣿⣿⣿⣏⣹⣿⣯⣿⣿⣿⣿⣼⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡿⢾⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣿⣻⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠟⠻⠟⠃⠉⢡⣝⠿⣿⣿⣿⣿⣙⣷⣦⣯⣿⠿⣿⣾⣿⣿⣿⣿⣯⣴⣏⣰⠿
⣿⣿⣯⣿⣷⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣼⣿⣿⡿⠟⠉⠉⢉⠁⠈⠁⠀⠈⠉⠉⠁⠠⠄⠈⠏⠿⣿⣧⣾⣿⣿⣿⣿⠉⣹⣿⣟⢻⣿⡿⢉⣿⣆
⣿⣿⣿⣿⣿⣿⣿⣏⢸⣿⣾⣿⣿⣿⣿⡿⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⢿⣿⣿⣋⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⡟⢻
⠿⣟⣻⣿⡿⣾⣻⣿⣿⣿⣿⣿⣿⣿⣿⡯⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⠠⠀⠀⠀⠄⠀⠀⣸⣿⣽⢣⣾⣿⣿⣻⡿⣿⣿⣿⣿⢷⣾
⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣟⣿⣿⣽⣿⣶⣄⠀⠀⠐⣄⢻⣿⣿⣠⣾⣿⣿⣿⣥⣿⣿⣿⣿⣿⣿
⢿⣿⣿⣿⣿⣿⠿⣿⣿⢻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣟⣿⣽⣿⢿⣧⣿⠀⠉⠘⠏⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿
⣶⣿⣿⣿⣿⡏⢀⣠⣿⣿⣿⣿⣿⣿⣿⢿⣦⣯⠀⠀⠀⠀⠐⣿⣿⣿⣮⣩⣽⣿⣿⣿⣿⣿⡄⠀⠀⠀⣰⣿⣿⣿⣟⣿⢿⣿⣿⢿⣿⣿⡏⠉⢻
⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⡆⠀⠀⠀⠀⠘⢻⣿⣿⣿⢹⢽⣿⣿⣿⣿⣿⡀⠀⠘⣿⣿⣿⣿⢿⣷⣾⣿⣿⣿⣿⣿⣿⣷⣾
⣟⠻⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⡇⢀⠀⠀⠀⠀⢈⢽⠛⠷⢾⣿⣿⣿⣿⣿⣧⣀⠀⣿⣿⣿⣿⣿⣿⣟⣿⣿⣯⣿⣿⢿⣿⣿
⣿⣾⣿⣿⣿⠷⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡀⠀⠀⠀⣿⣏⣾⣷⠄⠀⠉⠻⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⣿⣅⣀⣉⣻
⠈⠿⣿⣿⣿⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⠋⠋⠛⠻⡿⣿⣿⣶⣤⣠⡄⢍⣩⣻⠟⠀⠀⠀⠀⠉⢿⣋⣿⠿⢿⣾⡿⢾⣿⣿⣿⣿⣦⣾⣿⣿⠿⣿
⠀⠀⠘⠛⣿⣿⣿⣿⣿⣯⣿⡿⠿⠋⠁⠀⠀⠀⠀⠀⠈⠻⠿⠿⣿⣿⡄⠈⣽⣤⣀⠀⠀⠀⠀⠈⠠⣼⣿⣿⣿⣶⣿⣿⡿⢿⣿⣿⣿⣿⡏⠀⠉
⢻⣆⠀⠀⢸⣯⣽⣿⣿⡿⡇⠁⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠍⠉⠈⠉⠹⣿⣷⡀⠀⠀⠀⢾⢿⣿⣿⣿⣟⣿⣿⣿⣾⣿⣿⠟⠻⠋
⠀⠙⠧⠀⠀⢯⣻⣿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠖⠈⠹⠿⠃⠀⠀⠀⠠⣼⣿⣿⣿⣿⣯⣿⣿⣿⣿⠃⢰⡇
⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⣠⡀⣠⣼⣿⣿⣷⣶⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⠟⠛⢻⣏⠀⠈⠁⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⢈⣿⣢⡄⢀⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇⠀⠀⠸⠟⠀⢠⠶⣶⣏⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣴⡟⠫⠃⢸⣶⣦⣭⣧⣤⣤⣾⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢠⡇⠙⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⡿⠋⢠⣾⣿⡇⠀⢰⣾⣏⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠠⣤⣴⢃⡄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠂⢸⣿⡉⢟⣿⣭⣿⣿⣿⣿⠿⣿⢻⡟⠋⠁⠠⢶⡀⢦⢧⣜⠃⠄
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣿⠿⢿⠟⢻⡇⠀⠠⠟⠀⠀⢀⣤⡟⢠⠀⠀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⣘⢀⠀⠀⠀⠀⢤⣄⡀⠐⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⢀⣸⠟⠁⠀⠉⡈
⠀⠀⠀⠀⠀⠀⠀⠉⢀⣀⢀⣴⣦⡀⠉⠉⠻⠷⠺⠶⣤⠀⠀⠀⡶⢀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢈⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠈⢣⣈⠁⠉⣀⢀⡐⠂⠀⠀⠀⠆⠁⠀⠀⠀⢾⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⡄⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠹⣷⣤⣈⡿⢿⣫⣄⣀⠀⠀⠀⠀⠀⠀⠸⠗⠁⠀⣰⠀⣀⠁⠀⠀⠀⢰⠦⠀⠳⠚⠋⠻⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣙⣧⠉⠐⠒⠀⠀⠈⠑⡀⠀⠀⠀⠀⠀⢦⣿⠄⠀⢀⣙⠔⠛⠒⠂⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⠦⣴⣶⣦⡀⠈⠙⢻⣿⠋⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⣤⠛⣛⣦⡐⣛⢙⠂⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⣿⡁⢀⣘⡉⠈⠡⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡿⣭⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠹⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣭⣍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⠀⣉⣛⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣿⣷⣶⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢋⣿⣿⣿⣿⣧⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⣹⣿⣿⣿⣋⣀⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠐⢿⠟⠁⢸⣿⣿⣿⣿⠿⠿⠟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⠁⣰⣿⣿⣿⡟⠿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⡏⠈⠻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣿⣿⠛⠛⣿⣿⡅
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⢹⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠏⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⠀⠈⠙⠢⡀⠀⠀⣠⠖⠉⠁⠀⠀⠀⠈⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠂⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢣⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⢀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠒⠒⠲⠶⢦⣄⣀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣍⠀⣀⠀⠀⠀⠀⠀⢤⣤⣁⡡⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⡛⢿⠷⣖⣂⣀⠀⠀⠈⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠚⢃⣈⡇⠀⠱⡤⠤⡤⠀⣤⣭⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣇⡀⠉⠀⠀⠰⢛⣷⣶⣶⠒⣭⣥⡄
⠀⢺⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⡄⠀⠀⢿⣷⣶⣶⣮⣭⣭⡝⠋
⢀⣀⣿⣷⣶⣄⢀⣀⠀⠀⠀⢀⣀⣀⣀⣀⠀⢤⡿⠀⣠⡾⠟⡟⠛⠛⠋⠉
⠙⣿⣿⠧⣿⣮⠻⢽⣏⣉⣉⣙⣻⣿⣶⣿⠿⠿⠿⠟⠛⠛⠛⠳⣄
⠘⠻⣿⢶⣶⡮⠁⠀⠉⠉⠉⠉⠉⠉⠹⡟⣦⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠘⠟⠛⣿⡿⠛⠛⠛⠛⠛⠉⠉⠉⠹⡿⠷⣤⣤⣄⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⣀⣠⣽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⢦⣤⣤⣽⣷⣶⠖⠋⠉⠁⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣹⡄
⠀⠀⠀⠀⠸⡟⣦⡀⠀⠀⣾⠁⠀⠀⢀⣴⠞⠙⠛⠲⠶⠶⠤⠤⠴⠶⠶⡖⠃⠀⠀⣿⠀⠀⠀⠀⠀⣀⡴⠚⣏⣀⣀⢹⣇
⠀⠀⠀⠀⠀⠙⣿⣗⠶⠞⢻⣤⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠈⢧⣀⣠⠴⠚⠁⠀⢀⣿⡁⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠘⢿⣷⣤⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠠⣿⠋⠀⠀⣀⡴⠞⠋⠉⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣠⡤⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⣯⣭⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣟⡛⠛⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡆⠈⣿⡇⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠿⠋⠁⢿⡏⠀⣻⡇⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⣠⣼⠇⠘⠁⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣿⣿⣿⣿⣿⡏⣿⡇⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⣿⣿⣿⣿⣿⣿⣿⡇⠀⡄⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⡄⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣧⣸⡇⢠⣿⡏⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠹⣦⣹⡄⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣧⣞⣥⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣜⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⢋⣥⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢾⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢶⠀⢀⣠⠤⠴⠒⠒⠒⠒⠦⠤⣄⡀⠠⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⣤⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⠀⠀⣀⠀⠀⣀⠀⢸⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠖⠂⠉⠱⠐⠙⠲⠶⡉⠒⠚⠉⠓⠋⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡋⠄⠉⠀⠀⠀⢀⣠⢐⡂⠉⠀⠈⠒⠢⡀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⢄⠀⢰⠉⠀⠀⠈⠉⠢⡀⠀⠀⡄⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠛⠒⠶⣤⡤⢀⡀⠀⢸⣐⡪⢴⣾⡧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢀⠈⠙⠪⠭⠏⠳⢄⣠⠿⠬⣳⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣊⠉⢲⠊⠉⡏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠋⠀⢠⠇⠀⠀⠀⠀⠀⡾⠤⢤⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠢⢤⠤⠤⠞⠑⠒⠤⠤⣄⣀⠟⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠴⠓⡏⠀⠀⠀⠀⢀⠴⠒⠒⠦⣄⠀⠀⢀⡿⠴⠊⡹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠉⠁⠀⠀⠀⠀⠀⠱⡤⠋⠑⠦⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣖⣠⣶⢃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢉⡤⠕⠋⠁⠀⢀⠤⢒⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⡞⠀⠀⠀⠀⠒⠋⠉⠐⠲⠯⠉⡹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⡇⣠⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣼⠁⠀⣠⡿⣫⠔⠛⢋⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⢠⣔⠚⢭⡻⢦⠀⠉⢸⣃⡴⠚⠉⠀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⠀⠀⠸⢄⣉⡹⠇⠀⠀⠀⠘⢯⣁⣀⣤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠓⠈⠁⠀⠀⠀⠀⢸⠀⠀⠀⢠⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠁⠀⠀⠀⠀⢀⡤⠄⢤⠀⠀⣼⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⣿⢹⠏⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠛⢒⡥⠎⠀⠀⢿⠴⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣴⡋⠀⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⣀⣀⣀⣀⣤⣤⠶⠚⠉⠀⠀⠀⠀⠐⠣⠾⠷⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠿⣿⣧⣀⣠⠗⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⢇⠀⢀⡠⣄⠲⠬⠓⠟⠬⣅⡲⣄⠈⢳⡀⠀⠀⢀⣀⡠⠊⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⢯⣴⢶⠶⣞⡻⣶⣿⣿⣼⡆⠀⡿⣶⠞⠋⠁⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠦⡤⠠⡄⠸⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢶⣒⠿⣿⣿⡇⣇⣼⠃⡟⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢂⣀⠀⠀⠉⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⣯⣍⣙⠻⠦⡼⠟⡽⡿⠃⡸⠁⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⣴⠋⢙⣦⠀⠰⠚⠋⢀⠀⢰⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡄⠀⠀⠀⠀⠀⠙⠓⣶⡊⠉⠒⢒⣥⠞⢀⣴⠁⠀⠀⠀⠀⠘⢧⡄⠀⠀⠀⢀⣠⣿⣧⣌⡙⠳⣄⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡙⡏⣏⢏⡀⠀⣀⣀⡀⠀⢀⣨⠷⣴⠞⠋⠁⣠⡾⠃⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⣶⣿⣿⠟⠁⠀⠙⢷⣘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠛⠀⡇⠀⣠⢞⡟⠉⠀⠈⠉⠋⣸⡷⠊⢸⡇⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠥⠔⠛⠻⣇⠀⠀⠀⠀⠀⢉⡀⠊⠸⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⠊⠀⠀⣠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⣠⣾⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣁⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣷⣶⣶⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⡿⠟⠋⠁
⢀⣄⡀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠁
⡇⠹⣎⢳⡄⢀⣾⣿⣿⣿⣿⣿⣿⠟⠉⠉⠙⠛⠉⠉⠉⡄⢠
⣇⠀⢻⡀⢿⣾⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈
⢹⣄⣸⠇⣸⢿⠿⠿⠟⠋⠁
⢆⡘⣽⠴⠃
⣾⣿⣷⠶⡶⢶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠾⣏⠐⠀⠀⢀⣴⣶⠒⠂⠀⣤⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠐⣤⣽⣦⣄⡼⢁⣹⣆⠐⠂⠀⢀⡀⠈⠑⣢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠐⢊⣤⠤⠀⣨⣿⣿⡇⠸⠟⣹⠿⣧⡀⢨⡇⠀⠀⠉⣄⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠋⣁⠀⠀⠲⠀⠈⢹⣿⣄⡸⠏⡀⠀⢽⠛⣦⢤⣀⣄⠈⠁⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠸⢃⠔⠋⠀⠀⠀⠀⠀⣴⣏⣾⢋⣿⢟⣿⣶⣦⡸⢿⡼⢻⣽⢾⣆⠀⠀⠔⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣠⠃⠀⠀⠀⣠⠤⠔⠚⠉⢿⠏⠘⠛⠟⣣⣾⣋⣸⠿⣷⣸⣟⠛⠹⣞⠃⢈⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣠⠀⠀⡀⠀⣠⣄⣼⣿⡿⢛⠻⢾⣿⣼⢿⣄⣰⢟⣿⠻⣃⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠘⠛⣁⠀⠰⠀⣤⢤⣿⡟⠉⠐⢥⡴⠆⢙⠱⡟⢻⣯⡿⢋⡇⠙⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠋⢀⣀⡀⠉⢀⠀⢠⠄⡼⠟⡀⠀⠀⠐⠃⠤⠙⠀⠸⢿⣿⡄⡾⢿⠀⠉⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣾⣿⣿⣿⣿⣿⣿⣧⠔⠀⠠⠂⣠⣀⣤⣤⣤⣤⣄⣙⣇⠸⣿⡿⠁⠈⣧⢀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣿⣿⣿⣿⣿⣿⣿⣷⣮⡄⠁⣴⣿⣿⣿⣿⣿⣿⣷⣿⣟⡀⠻⣟⠀⠓⡿⢠⣀⠈⡇
⠀⠀⠀⠀⢀⣀⠠⠖⣐⣼⠟⣡⣿⣿⣿⣿⠿⣿⣿⣿⣿⣆⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣯⢰⣦⡇⠬⠋⠀⡇
⠀⢀⡴⢈⣩⣵⣶⣿⣿⡟⠀⠈⠿⠛⡉⢁⣾⣿⣿⣿⣿⣿⠄⣈⡙⢿⣿⣿⣿⣿⣿⣿⣿⡟⢹⣿⠘⢛⣴⡄⣠⢰⣄⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣶⣿⣧⣾⣿⣿⣿⣿⣿⣿⣀⣩⣽⣦⣤⣍⣉⣉⣉⣉⣡⣀⣼⣿⣿⣿⣿⣿⣷⣿⣶⣾⡆
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⢉⣙⣢⡖⣉⣁⣉⣑⣶⣉⣡⡍⠻⢦⣥⣤⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣹
⠀⠸⣿⢿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⠟⡉⣨⣛⣭⣿⣿⣷⣿⣿⣿⡟⣩⠦⠿⠏⠉⢠⡟⢥⣿⣷⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡇⠘⠻⣿⣿⠛⠿⣿⣿⢿⣿⡇⠉⠐⣶⣠⣼⣿⢋⠳⣤⡴⠋⠋⠙⢋⣴⠆⠀⢀⡎⢀⡿⢏⡱⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢱⠴⠁⣿⣿⡦⣄⣿⣿⣾⡟⢹⠀⠀⠀⢩⣽⣮⠀⢿⠛⣿⠃⠀⣰⢿⡇⢀⠀⢸⠃⢨⠴⠋⡑⣯⣿⢾⣿⣿⣿⣿
⠀⠀⠘⣇⣌⡻⣿⡄⡏⣿⣿⣿⡇⠀⣇⢈⡀⠠⣏⠳⡀⠀⢱⡇⠀⣰⣿⣿⠄⢸⠀⣾⠀⢸⣣⠆⢀⠊⢉⠣⣿⡟⣿⠃
⠀⠀⠀⢹⡎⣷⣿⡟⢃⣽⣿⣿⠇⠀⢿⠀⡇⠀⠟⢿⡅⢶⢈⠁⡄⣿⣩⢿⠀⠘⠠⡇⠀⠈⣥⠊⠿⢠⣎⠐⣻⣡⡏
⠀⠀⠀⠀⢳⡌⢿⠟⠛⡼⢟⠼⠀⠀⠞⠀⠁⢠⣇⣰⡄⠀⠀⠀⡆⣿⣽⣾⠂⠀⢸⡇⠀⢰⡆⣠⡇⡀⠼⢸⣿⠏
⠀⠀⠀⠀⠀⠹⣄⠀⠤⣑⣄⠘⡄⠀⣷⠀⠀⢈⡟⢹⡧⢶⠀⠀⠀⢿⣉⢻⡁⠀⣼⢀⡄⠰⠃⢼⠇⢻⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣖⡈⢱⣏⠳⡀⠨⠀⠠⠄⢿⡎⠓⠀⠘⠡⠠⡞⢛⡺⠁⢸⠟⡞⢠⣧⢴⣏⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣦⢒⣤⠹⢆⠀⠈⢣⡶⠇⠰⠆⢾⡟⣾⠃⢠⡆⠀⠀⣺⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠓⠚⠷⠒⠾⣿⠿⠿⣾⠿⣿⡶⠲⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠆⢀⠁⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⡴⠶⣿⢿⢃⣹⠷⢆⣙⣿⢦⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⢀⣤⡰⣾⣿⣿⡻⣏⠿⢿⣿⡿⣷⣬⣧⣰⣿⣄⣠⣸⣥⣥⣤⣾⢿⡟⢿⡹⢿⣭⢿⡗⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠸⣾⣿⣿⣿⣾⣿⣯⣤⣄⣸⣥⣮⣿⣿⣾⣿⣺⣷⣾⣷⠖⢿⣤⢻⣝⠒⢃⣀⣥⣈⣹⣿⣷⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠫⠿⣿⣿⣏⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⢀⠀⡀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠃⠚⠒⠢⢄⠀⠀⠘⠂⡀
⡀⣀⡀⠀⠀⠀⠀⡠⠄⢤⠀⡀⠀⠤⡀⢀⡧⣄⠀⠀⠙⠪⣃⠀⠀⠈⠉
⠑⠄⣈⠉⠐⠂⠤⢧⣀⣼⡉⠀⢀⡴⠓⢉⡤⠚⠁⠀⠀⢀⠈⢧⠀⠒
⠀⠀⠀⠹⡋⠀⠁⠀⠀⠣⣅⠐⠉⠀⠢⢔⡇⠀⠀⠀⠀⠈⢏⠢⡇⠀⠀⠰⠀⡀
⠀⠀⠀⠀⠈⠒⠐⠴⠴⠴⢤⣄⠀⠀⠎⠉⢢⣤⡀⢠⡀⢰⡈⡀⢄⢢⡀⠤⣢⢪⡄
⠀⠀⠀⢀⣀⠠⡀⠀⣀⠀⡀⠴⠽⣦⡀⠄⠀⠁⠱⡘⢳⢸⠘⠀⠀⡹⠀⠄⠓⠋
⠀⠀⠀⢠⣠⢙⡖⠦⡈⠘⠒⠒⠈⠻⠑⡄⠢⡀⠀⠀⢈⠁⠀⢠⠊⠈
⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠐⠀⠈⠃⣍⢀⠀⠈⠢⡐⠊⠳⣖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠣⠀⠈⠰⠶⡁⠀⠈⠋⠀⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⢢⠀⠀⠀⠀⠐⣠⡀⠀⠈⠅⢬⢇⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠏⠛⠷⢮⣉⡓⠀⠉⠀⠀⠈⠓⠤⣀⢊⠂⠈⠈⠁⠂⠀⠀⠀⢀⠀⢠⠀⠀⠠⠙⠛⠝⠶⠑⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢀⠀⠤⠔⡀⠀⠀⠀⠀⠀⠒⠠⠤⠤⠔⠊⠛⠓⠀⠂⠀⠀⠀⠀⠀⠀⠀⢙⠫⠀⠠⠴⡄⢀⠠⠚⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡨⠏⠀⢤⠆⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⡄⠶⠇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠤⢤⣤⠤⠀⣤⢀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠟⠀⠀⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣴⣶⣿⣿⠿⠛⠁⠀⠀⠀⠈⠻⢿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⠟⠋⠉⠉⠉⠁⠀⠀⠑⢄⠀⠀⠀⠀⠀⡠⠒⠉⠉⠙⠛⠛⠿⡶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣴⣿⠗⠆⠀⠀⠀⠈⡄⠀⡎⠀⠀⠀⠀⣠⡶⠄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠙⠿⠷⠃⠀⠀⠀⢰⠀⠀⢁⠀⠀⠀⠀⠙⠿⠊⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠛⠛⠻⠶⠶⠶⠴⠮⠀⠀⠀⠀⠀⠀⠉⠦⠤⠶⠶⠶⠶⠮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠁⠀⠀⠀⠀⠉⠉⠉⠛⠓⠲⢶⣦⣤⣤⣤⣼⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡤⠖⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠛⠻⠿⠧⠤⠤⠤⠄⠒⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⢤⣤⡀




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣙⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣶⣶⣶⣦⣤⣤⣤
⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣶⣿⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⢈⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠃⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⠀⢉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⢈⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⠻⣿⣿⣿⡿⠋⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⡀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢹⣿⡷⠀⢀⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⢀⣴⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⣿⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣿⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣟⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣷⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣇⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⣿⣿⣿⣿⣿⣿⣿⣼⡟⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠈⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⠀⣾⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦⠀⡟⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⠀⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢹⣿⣿⣿⡷⢀⠁⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⡁⠀⠘⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣡⣼⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣸⣿⣿⣿⡀⡇⠀⠀⠀⠀⠀⠀⣿⣿⡿⠛⠛⠉⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⡅⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⡆⠀⣿⠛⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡄⡇⠀⠀⠀⠀⠀⠀⣼⡇⠀⢸⡇⢰⣿⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠸⡇⣾⣿⡀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠙⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠏⢠⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⢹⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⡈⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⢸⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠸⣿⣿⣧⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⢸⣿⣿⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⣀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠠⣤⣤⣤⣤⣤⣼⣧⡘⣿⣭⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣹⣯⣤⣛⣛⣛⣛⣑⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⢹⣿⣿⣯⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣼⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠻⠃
⣶⣶⣶⣦⣴⣶⣶⣶⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠻⠛⠛⠿⠋⠛⠛⢛⣛⠟⣻⡿⢟⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣿⣷⣾⣿⣿⣾⣿⣿⣶⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠙⣆⠀⠀⠀⠀⠀⠀⠀⡔⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠑⠁⢀⢄⠀⣀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⡀⠀⠀⢻⡀⠀⠀⠀⢳⣠⠴⠄⠀⠀⠛⠓⠀⠉⢈⡾⠉⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠑⢤⡀⠀⡷⢄⠀⠀⠀⡹⠦⠢⢤⡴⠒⣒⡶⠚⠉⠳⣄⣀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠙⠲⢬⣖⡓⢦⡎⠀⣀⠀⠀⠘⡟⠁⠀⠀⠀⠀⠀⠉⠉⠛⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⡔⢢⠄⠉⠙⣇⠀⠁⠀⠀⡰⠁⠀⠀⠀⠀⠈⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⣻⠋⠓⠢⠔⠊⠁⠀⠀⠀⡀⠠⡀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⡴⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⣰⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⣀⡤⠞⠁
⠀⠀⠀⡾⢡⡏⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠉⠀⠀⠀⠀⠀⡟⢿⠿⢿⣿⣿⠋⣿⠛⠋⠉⠁
⠀⠀⢸⡇⠸⡄⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠖⠒⠒⠻⠿⢰⠇
⠀⠀⠀⢧⡀⠙⠶⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠙⠦⣼⠁⠀⠀⠀⠀⠒⢤⡀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠉⠓⢦⣠⠟⠹⠦⣄⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⣀⡀⠀⠙⣧
⠀⠀⠀⣀⣀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⣠⡾
⢀⡴⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣁⣀⣤⣤⠤⠴⠖⠋⠉⠉⠉⠉⠻⣄⠀⣢⡟⠁
⢯⣀⠄⢀⣤⣄⣀⣠⣤⣤⡤⠖⠚⣁⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠉⠉⠉⠀⠀⠀⠀⠀⢻⣀⣄⡟⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠾⠿⠛⠛⠻⠿⠶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠞⠛⠛⠛⠿⠶⣶⣤⣄⡀
⠀⠀⠀⠀⢀⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⡀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⢀⣴⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣟⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠂⠀⣨⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟
⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋
⠀⠀⠀⠀⠙⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠸⣯⡀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠙⢿⣄⠀⠙⢿⣄⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⢹⣷⠀⠀⣿⠃⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⠀⣾⠇⠀⣼⣏⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡏⠀⢰⣿⠟⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡄⠀⡞⡹⠃⠀⠀⠀⣰⡩⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠜⠁⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣄⡀⠀⠀⠀⠀⠀⠀⡔⢲⠄⠀⠀⠀⣠⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣫⡁⠀⠈⠉⠙⠢⡀⠀⠘⠊⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠤⣽⢒⣦⣄⠀⠀⠀⢱⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⡜⢀⣇⣘⡉⠠⢖⢲⠄⡼
⠀⠀⠀⠀⠀⢠⠎⢠⠊⢯⡙⠁⠀⠀⣫⠞⠁⠀⠀⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠓⠬⣓⣴⣮⣭⡔⢙⠀⠀⠀⣀⣀⣀⣀⡤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢜⠀⠈⡞⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⢖⠚⡅⠸⡄⠀⢀⣀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⡄⠈⣦⢱⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡿⠧⠴⠠⠤⠯⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡸⠤⠬⠭⠥⠞⢐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠘⣂⠀⠀⠀⠀⠀⡱
⠀⠀⠀⠀⠀⠀⢀⣏⠀⠀⠀⢀⣀⢴⠊⠈⢲⡆⠀⠀⠈⠉⠁⠀⠀⠀⠀⠘⢤⠾⠛⠀⢀⣀⣤⣾⡁⠀⠉
⠀⠀⠀⠀⠀⠀⡇⠈⠲⠤⣊⠁⠀⠀⠉⠒⠓⠬⣑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠉⡯⢫⡏⠀⠐⠋⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠉⠒⠢⠤⣠⠏⡇⠀⠀⠀⠀⠀⠈⣇⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡠⠃⠀⠀⠀⣀⣀⣀⠤⠤⠤⠤⠤⠤⠖⠒⠒⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⢀⣀⡤⠤⢀⣀⣎
⠀⠀⠀⡔⠁⠀⠁⠀⠀⠀⠁⠉⠢⡀
⠀⠀⢸⠠⠒⠒⠄⡠⢊⣉⠑⢆⠀⢡
⠀⠀⠸⡇⢉⠁⠈⠁⡠⢄⠁⢸⠀⢸
⠀⠀⠀⢧⠁⣰⠀⠈⠀⣾⠀⡎⠀⡎
⠀⠀⡸⠘⠠⠟⠀⡔⠈⠉⠀⠘⢪
⠠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠑⠈⣂⡄⢄⣈⣀⣀⢤⠒⠒⠊⠉⢁
⢸⠁⠀⠷⠈⠢⢀⡠⠊⠀⡊⠉⠂⢆
⠀⠢⠄⢺⢀⡀⣀⡀⢀⠸⢅⠀⢀⣴⣦⠄
⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠉⠋⠎⠒⢄
⠀⠀⣀⠼⢄⡀⠒⠀⠙⣷⢄⣀⣠⠃⠀⡼
⠀⡌⠀⠈⠢⡐⣧⠀⠀⠁⠀⠀⠀⠉⠉
⠀⠑⢄⠀⣀⡸⢠⠃
⣤⣤⣤⣤⣬⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣧⣾⣭⣭⣭⣭⣽⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣷⣦⣀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣷⣦⡀⣰⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠶⠶⡾⠛⠛⠛⠛⢿⢷⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⡈⢠⡄⠒⠤⠙⠂⠀⠀⠙⠛⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⢰⡀⢰⠀⠩⠄⠈⠃⠀⠀⢀⠀⠀⠀⠀⠘⢍⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣡⡄⡏⠈⠁⠈⣁⠀⠘⠄⠀⠀⠀⠀⠰⠀⠐⡀⠀⡈⢊⢿⡄⠀⣀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⢠⣿⠛⠉⠀⠁⠀⠀⠀⠈⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⡀⢰⡰⠟⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢲⠄⠘⠀⠀⠀⠈⢿⣄⠈⠀⠀⠀⠀⠀⠲⠂⠀⡀
⠀⠀⠀⠀⠀⠀⣿⠂⢀⠀⠀⠀⠄⠀⠈⠀⠀⢈⡀⠀⠐⣎⠀⠚⠀⠀⢢⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⣿⡰⠈⡄⠠⠀⠈⠄⠀⠰⣀⠁⠑⠀⠀⠘⡀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠸⣷⡡⠐⣀⢰⡀⠸⠄⠀⠙⠀⠀⠀⠀⠀⠑⠀⠀⠡⠀⠀⢃⠀⠀⠀⠀⠀⠈⣿⡆⠀⡀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⢹⣷⠧⠻⠀⠡⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠂⠀⠀⠀⢠⠰⣼⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠄⢡⣄⢵⠀⠰⠄⠀⠲⠄⠀⢲⡐⣀⠠⠀⠀⠁⡀⠀⢦⠀⠡⠈⢃⠐⡹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣎⠀⠻⠉⢣⠀⠙⠀⠀⠁⠀⠀⠁⠀⠀⠠⠂⠀⠥⠀⠈⠁⠀⢡⠀⢢⠘⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠄⠡⠀⠡⠀⠀⠀⠀⢠⠀⠀⢡⠀⠀⠡⠀⠀⠁⠀⠀⠄⠀⢣⠀⢆⠐⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣎⡄⠡⠀⢠⠀⢛⡀⠀⠓⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠄⠀⢣⠀⠂⢈⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣮⣆⠧⠀⠃⠀⠡⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣠⣶⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣎⠄⠁⠀⢀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣎⠄⠂⠈⠄⠀⠡⠀⠀⠀⣀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠈⠄⢀⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣈⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣯⣤⣤⣶⣦⣴⣤⣤⣄⣀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠋⠉


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣏⣍⢿⣯⣿⣽⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢦⢰⠋⣆
⠀⠀⠀⠀⠀⡠⠒⠉⠉⠀⠀⠀⢱⣄⠀⠀⠀⠀⠀⠀⡖⠲⣌⢇⠘⣾⠀⣿⠀⠀⠀⠀⠀⡸⢣⡰⢱⢠⢦
⠀⠀⠀⠀⡜⠀⠀⠀⢀⠀⠀⠂⠈⠯⡀⠀⡀⠀⠀⣀⣹⣄⠈⢻⡆⢿⠀⡏⠀⠀⢀⡀⠀⣄⠘⡇⣸⢇⡎
⠀⠀⠀⢸⠃⠀⠀⠀⠃⠀⠒⢄⠀⠀⠚⡼⢹⠀⠀⢧⣈⠙⠿⣦⡙⠈⠠⠗⠒⠋⣁⠜⡀⢹⠀⡇⣟⡼⠒⡩⠃
⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⠀⠃⠀⠀⠚⠁⠈⠑⠢⣄⠈⠑⠂⣬⠀⠀⠀⡴⠒⠉⠀⢫⡍⠉⠀⠀⢀⡠⠚⠁
⠀⠀⠀⠈⣇⠀⠀⡀⠀⠀⠀⠀⠀⠀⡂⢄⠀⠀⠀⢈⠷⠒⠉⢀⡠⠒⠈⣀⡤⠖⠊⢁⣠⠤⠒⠒⠁
⠀⠀⠀⠀⠈⢲⠀⠁⠄⠀⠀⠀⡾⠀⠂⠈⠀⠐⠀⠀⣀⡤⠞⣉⡤⠖⠋⣁⡤⠔⠊⠉
⠀⠀⠀⠀⠀⠈⠒⠒⠉⠳⡄⠀⠀⠀⠀⠁⠀⣀⡴⣞⡭⠖⠋⣁⠤⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠁⠀⢀⡤⠾⠟⠋⣡⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠉⠐⠀⠰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⢄⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⠀⠀⠹⡄
⡴⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡄⠀⠀⠀⠀⠀⠀⠘⢆
⠇⠀⠀⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⡰⢦⡀⠀⠈⠳⡀
⠘⢄⡀⠀⠀⠈⠙⠶⣄⠀⠀⠀⠀⡰⠁⠀⠀⢀⡞⠁⠀⠙⢦⡀⠀⠙⢆
⠀⠀⠈⠉⠙⡆⠀⢠⠎⠀⠀⠀⡼⠁⠀⠀⣠⠏⠀⠀⠀⠀⠀⠑⢄⠀⠀⠳⣄
⠀⠀⠀⠀⢀⡇⠀⢸⠀⠀⠀⡼⠁⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠳⣄
⠀⠀⠀⠀⢸⠁⠀⢸⠀⢠⠞⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠈⠑⢤⡀⠀⠀⠀⠀⠀⠀⢀⡠⠤
⠀⠀⠀⠀⢸⡀⠀⠘⠖⠁⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠙⠢⣄⠀⠀⠀⡼⠁⠀⡠
⠀⠀⠀⠀⠈⣇⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠙⠒⠺⠁⢀⡜⠁
⠀⠀⠀⠀⠀⠈⠑⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣤⡁⢀⠎
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶
⣿⣿⣤⣤⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⣤⡀⠀⠀⠀⠀⠈⣿⠟⠀⠀⠀⠀⠀⠀⢀⡀⣾⠇⠀⠀⠀⠀⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⡀⠀⠀⠀⠀⢿⡄⠛⣳⣄⣀⣀⠀⠀⡿⢠⠀⠀⠀⣀⢀⣤⣻⢁⡼⠀⠀⠀⠀⠀⢿⣴⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠁⠀⠀⠀⠀⠈⢻⣆⠻⠙⠛⢉⣶⣤⡇⢸⣤⣴⠻⠟⠉⣿⠁⣼⢃⣷⡀⠀⠀⠀⣴⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⠀⠀⠀⠀⢤⠀⢿⣦⡀⠀⠀⠿⡙⠁⣿⡏⣹⡄⢀⣀⣻⠾⠁⠈⣭⠀⢲⣄⡸⢈⡇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣷⠆⠀⠀⠙⠀⠀⠈⠛⠿⣷⣤⣿⠀⠀⣧⣿⣷⠞⠋⠁⠀⠀⠀⠁⠀⠀⣼⠃⣾⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣸⡇⢻⣶⢰⠄⣴⠀⠀⠀⠀⠀⠈⠉⢹⠀⠺⡏⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣴⠇⢰⡏⠐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣷⠈⢿⡎⠀⠘⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⡘⠃⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⠸⣇⠀⣀⣤⡴⠶⠶⠟⠛⢻⡿⢢⡾⢾⣿⠋⡉⠉⠙⠛⠲⠶⣴⡟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣀⣿⣿⣋⠀⢀⡀⠀⠀⡔⢠⣄⡀⣤⠈⣥⠀⣴⡎⣠⠀⠐⠆⠘⠛⢶⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠛⠡⠄⠻⡄⠋⠀⠀⠀⢸⣆⠉⣩⡀⠀⢿⡀⢈⡄⠉⠣⢁⠀⠈⠑⠢⡀⠘⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⢀⡀⠀⣀⣀⣄⠀⣤⣄⣄⡘⠆⠀⠀⡿⠃⠀⠼⠟⢀⣤⡶⠤⢄⠵⣦⣤⣤⡈⠳⠀⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢀⣻⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣦⣶⣶⣶⡠⣿⣵⣷⣿⣶⣿⣷⣾⣷⣿⣁⢀⠀⢀⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣼⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢷⡈⢇
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠘⢿⡛⣇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠀⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠋⢁⣿⣿⣤⣿⣿⣽⣿⣿⣿⠈⢛⠛⠛⠻⠻⠿⢿⣿⡿⣿⠿⠘⠀⠈⡿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠹⡀⠀⠹⠛⠛⠉⠉⠀⠀⠀⡰⣾⣦⡄⢾⣷⢛⣏⢸⡿⢿⣿⠛⢀⡄⠹⣶⣶⠤⠀⢀⣁⠀⠀⢶⠟⢀⣰⡿⢸⡆
⠀⠀⠀⠀⠀⠀⠀⣿⢀⣸⣷⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⣾⠓⣾⠏⣿⠀⢸⡇⠀⣿⠁⢘⣿⠀⢾⡇⣀⠀⠉⠱⢶⡄⠛⠰⠘⢿⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣠⡿⠋⠀⠻⠶⠀⢢⠐⠇⢀⣀⡾⢷⡀⠸⣇⣸⣧⣸⡇⢀⣿⣤⣼⡇⢠⡾⠁⣿⡄⣼⢣⡌⠀⢉⡟⡀⠈⠛⢿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠋⠰⠞⣠⣶⣤⡀⢸⠀⣄⣌⠛⢇⣼⣿⣶⣿⣿⣿⣿⡀⢨⣿⣿⣿⣷⣾⣿⣾⡁⣼⣇⠘⠀⢤⣿⣟⣹⣿⠀⢠⡆
⠀⠀⠀⠀⠀⠀⢸⡁⠐⣧⢀⣿⣧⣿⣿⣾⢀⣏⠹⣷⣿⣿⣿⣿⣿⣏⡙⣿⡇⢸⣿⠋⣹⣿⣿⣿⣿⣿⡁⣿⢇⣼⣿⣿⣿⣿⣿⡆⢰⣿⢀⡀
⠀⠀⠀⠀⠀⢤⡀⣿⣤⣨⣿⣿⣿⣿⣿⣿⣦⣙⣴⣿⣿⣿⣿⣿⠋⠀⢉⣿⠇⠈⣿⣾⡉⠈⠹⣿⣿⢿⣿⣧⣼⣿⣿⣿⣿⣿⣿⡧⢜⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠙⢿⡛⢿⣿⣿⣿⡿⢿⣿⣿⡟⣼⣿⡅⠈⢛⣥⣶⣿⣟⠁⣶⣾⣧⢸⣿⣿⣟⠉⢁⣼⣿⣻⣿⣿⡿⢻⣿⣟⣫⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠿⢦⣌⠙⣿⡅⠸⣿⣿⣿⣿⣇⣿⣷⠾⠛⠉⣿⣿⣿⢫⣽⢿⣿⣿⣯⣉⡛⣿⣿⣿⣿⣿⣿⡇⢰⣿⠿⠿⠇⢾⣀⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠚⠛⠛⣻⣻⣧⠀⢿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣡⣿⣿⣀⣿⣿⣏⣻⣿⣿⣿⣿⣿⣿⣿⣿⡀⣼⣿⣶⣶⢤⡴⠍⣀⡙⢦⡀
⠀⠀⠀⠀⣠⣿⠷⠶⠂⡿⠛⢻⣿⣿⡆⠀⣿⣿⣻⣿⣿⣿⣿⣿⢿⠉⢻⠀⢠⠄⢈⡏⠉⡿⠋⢿⣿⣿⣿⣿⣿⠋⠄⣿⣿⢿⡟⢨⡄⠀⠀⠁⠻⣷
⠀⠀⠀⠀⠋⠁⠀⠀⠀⠐⣶⠟⣡⣿⣿⡀⠹⣿⣯⣿⠛⠋⠙⣇⣸⣧⡼⢦⣼⠧⣾⣧⣼⡇⢰⣏⣿⠛⣿⠻⡿⠀⣼⢿⣿⠈⠱⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠐⠊⠳⣈⠿⣧⠀⠹⡁⠙⣦⣼⠿⠋⠆⠁⠀⠀⢻⡀⢿⠀⢸⠿⣟⢻⡿⠿⣧⡴⢡⠞⠹⠀⣟⡜⠃⠲⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⢻⠂⢹⡇⠀⠉⡄⠉⠃⠀⠀⡐⠀⡄⠀⢼⡇⠀⢀⠃⠀⠃⠀⢠⡇⢸⡇⠀⣤⣤⠀⢻⡇⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠘⠣⠔⠀⢠⣦⠀⢠⣼⠛⠀⠃⠀⠀⠀⣼⠈⢣⡀⣠⣧⡞⢠⡿⠷⠀⠹⠿⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⠀⠸⠋⠀⠀⠀⣠⣤⡄⠄⠐⠀⠹⠀⠀⠀⠛⠹⡿⠀⣠⡄⢰⡗⠠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠈⠀⠀⠐⠛⠗⠀⢁⣀⣠⣤⡤⠴⠶⣶⣦⣤⣤⣤⠀⠸⠉⠀⠀⡠⢾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⣀⣀⣤⠶⠛⠉⢿⣿⣷⠀⣾⣿⣿⡟⠉⢹⡇⠶⢶⣿⣁⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠰⠟⠋⠀⠀⣠⡶⢛⡿⣿⡿⠂⢻⣿⠉⡏⠑⢦⡀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠉⢀⣶⠐⠉⢻⠃⣿⠁⠸⣷⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠈⠁⠀⢀⣿⠀⣿⣄⠀⠀⠀⠀⠀⠘⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠐⠛⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡖⣶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠂

⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠈⠀⠀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡀⠂⢀⣀⣀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⠄
⠀⠀⠀⠀⢠⣾⣿⣿⡆⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠂⠀⠐
⠀⠀⠀⠀⣚⣹⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⣩⣿⣿⡿⠁
⠀⠀⠀⠘⠉⢩⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡀⢀⡁⢸⡿⠏⣀⣴⣶⡄
⠀⠀⠀⠒⠠⢾⡿⠀⠀⢀⣠⠴⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣲⡏⠐⠈
⠐⠷⠷⠀⠀⠀⠻⣶⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⣹⣿⠛⠁
⠀⠀⠀⠀⠀⣀⣠⡼⢿⣿⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠻⠾⠿⠟⢛⡿⠁
⠀⠀⢠⣤⡴⠁⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⡀⠀⠀⠀⠀⠀⠀⠘⠲⠧⠀⠀⠀⠀⢀⣮⣤⠄

⠀⠀⠀⠀⠀⠄⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠶⢟⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡲⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⢀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⠀⠀⢀⠒
⠀⠀⠀⠀⢶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠐⠂
⠀⠀⠀⠀⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉