Ele Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⡤⠤⢤⠔⠂⠉⠉⠉⠉⠑⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⡠⠔⠂⠉⠩⢭⣴⡖⠚⢤⠤⢄⣼⣄⣀⣀⣄⣀⣀⠀⢠⡅⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⢡⡀
⠀⠀⢠⠎⠀⠀⠀⠀⠀⠠⠉⠟⣀⡘⠀⠀⠀⢀⡤⠂⢀⠠⣌⠉⠉⣀⡤⢚⣛⡻⣏⡉⠉⠉⠉⣷⠉⢨⣟⠛⠉⠉⠒⠦⣄
⠀⢠⠇⠀⠀⠀⠀⠀⠀⢀⠂⣾⠉⠇⠀⠀⠐⡏⢸⣿⣏⣁⠈⠀⠀⡛⢰⣿⣏⣷⠈⡇⠀⠀⠀⣾⠻⡏⠹⣧⠀⠀⠀⠀⠈⠳⡄
⠀⡄⠀⠀⠀⠀⠀⠀⢀⡎⠀⠙⠄⣴⠀⠀⠀⢡⡘⠻⠿⠋⡰⣀⣀⠳⣈⠛⠿⢋⡴⠃⠀⠀⠀⣟⣰⠁⠀⢹⡄⠀⠀⠀⠀⠀⠹⡄
⠀⡇⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠈⡄⠀⠀⠀⠙⠒⠒⠋⠀⠉⠉⠂⠈⠙⠚⠉⠀⠀⠀⠀⣸⠋⠁⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢻⡀
⢰⡃⠀⣀⠤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⢸⠀⠠⠀⠀⠀⠀⠈⢷⣀
⠸⠷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⠓⠒⠛⠁⠀⠀⠀⣀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠈⠀⠐⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣀⣉⣁⣀⣠⡤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠄⢠⣤⣤⣒⡛⠀⣸⣖⣦⡤⠤⢄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⣀⡴⠎⠉⠀⠀⣨⠷⠛⠢⡀⠀⠙⠓⢤⠀⠀⠀⠈⠙⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠃⠀⠀⠀⠃⢸⠇⠀⠀⠀⠹⢆⠘⢠⠀⠇⠀⠀⠀⠀⠀⡇⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠾⠀⠀⠀⠀⠀⠷⢃⠀⠀⠈⠃⠘⠆⢸⠀⣤⡆⠀⠀⠀⠀⡇⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⠂⠒⠀⠀⠀⠀⠈⡄⠀⠛⠀⠀⢠⠈⠀⠉⠱⡀⠀⠀⠀⡇⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡆⠀⠀⠀⢱⠀⠀⠀⠀⠘⢀⠃⠀⠀⢸⠂⠀⠀⢀⡇⠀⠀⠀⣷⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⢃⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⣤⡀⣠⠜⠇⠀⠀⠀⡿⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆⠈⢳⠖⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⠹⣅⠀⢰⠀⠀⢀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠳⠚⠁⠀⢙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⠓⠚⠀⠀⢀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡼⠧⢤⣀⣀⣀⣀⣀⣀⣀⡤⠞⡄⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣒⢠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣀⣀⡤⠎⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣏⡉⠉⠑⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠮⠥⠄⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢨⡟⠛⠛⠶⡶⠦⢤⣤⣤⡤⠦⠴⡶⠶⠖⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣋⢀⡀⠈⢸⠇⠀⠀⠀⡶⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠳⡤⠀⣱⠇⠀⠀⠀⢠⠇⠰⣄⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⡞⠀⠀⠈⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⣰⠞⠁⠀⠀⠀⠀⠀⡏⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⣀⠜⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠰⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣼
⠀⠀⠀⠀⠀⠀⣠⠴⠊⠀⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡇
⠀⠀⠀⣠⠴⠋⠁⠀⠰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⢸⡇
⢀⠶⠋⠀⠠⠀⢠⠜⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠙⢦⡀
⡎⠀⠙⠄⣀⠖⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢎⠡⠀⠀⠙⠢⣄
⢣⣄⣀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡑⠀⠄⠖⠉⢱
⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⢀⡴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠐⠒⠒⠒⠒⠒⠒⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⣜⠁⠀⠀⠀⠐⠀⠀⡀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢠⠃⠀⣶⠆⠴⠶⣶⡶⢿⣿⣧⣤⣶⣶⣶⣴⣦⣤⣤⣤⣤⣶⣤⣤⣼⣿⣶⣤⣴⣶⣶⣶⢤⣦⠀⠀⠈⡇
⠀⠀⠀⣿⠀⠀⣽⠀⠀⠀⠀⣤⣤⣤⣤⣀⡰⣄⠀⠀⠀⠀⠀⢀⡀⢀⣀⣉⣀⠠⣄⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠇
⠀⠀⠀⠋⠀⠀⠘⡄⠀⠀⠀⢙⠷⣾⣿⡿⠃⢹⠀⠀⠀⠀⠀⣾⣏⣹⣿⣿⡇⠀⠘⠃⠀⠀⠀⠀⣾
⠀⠀⠀⣧⠀⠀⠀⢿⡀⠀⠀⠀⠒⠈⣁⡀⠐⠁⢀⡴⠲⣄⠀⠘⢫⡙⠛⠉⠀⣤⠂⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⡐
⠀⠀⠀⣿⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠹⣦⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠘⠳⢦⣀⣀⣀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠋⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⢀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠻⣦⠘⣃⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⢀⣠⡴⠛⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⣸⡛⠳⠶⠦⢤⣤⣄⣤⣤⡤⠶⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⢹⠂⠀⠀⠀⣿⠁⠈⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠘⠁⠀⠀⣭⡝⢢⡄⢻⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠⣛⡾⠇⠰⢶⡉⠉⠙⠈⠉⠃⢀⠈⠇⠙⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠰⣦⠉⠀⠀⠀⠈⠉⠲⠒⠒⠖⠒⠈⠁⠀⠀⢀⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⢸⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⡇⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠸⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⣀⣀⣀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠛⠛⠛⠛⠓⠶⢦⣤⣤⣄⣀⣀⣀⣀⣠⣤⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠶⣄⣀⣀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠸⠀⠀⠀⡂⢀⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠛⠳⠶⣦⣤⣤⣴⣤⣤⣴⡦⠶⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡿⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⡇⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢠⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠠⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠾⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠈⡀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠚⠃⠀⠀⠀⠀⠀⠉⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⠀⠀⠀⢀⣀⡠⣾⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠃⠀⠀⠈⢇⠀⠀⠀⠀⠀⣙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣠⠶⠟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣏⡶⠚⢿⣏⣁⣈⣉⣉⣉⣉⣀⣀⣉⣀⣈⣁⣈⣁⣈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠃⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⡀⠀⠀⠀⠀⠀⢀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣝⣛⣻⣻⣿⣿⣿⣿⣏⠀⠈⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠘⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢰⡇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⢀⣦⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⠃⢀⣀⣤⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⡿⠀⠀⠻⣿⣿⣿⣿⣿⠁⠀⠸⣿⣿⣿⠀⠘⠻⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⠈⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢻⠙⠃




⠀⡁⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠖⠁⠀⠀⠀⠀⠀⢾⠂⠄⢙⡾⣿⣿⣵⠟⡧⠌⠉⠁⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢈⣾⣦⣼⣻⡋⠀⠈⠀⠀⠀⠀⠀⠀⠐⠓
⠀⠀⠀⠀⠀⡃⠀⠃⠀⠀⠀⠂⠀⣴⣿⠿⠙⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⢀⡀⠨
⠀⠀⠀⠀⠀⠉⡀⠀⠀⠤⡀⢀⠃⠸⢉⢀⣠⣿⠇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠃⠠⠤⢅⡀⠀⠀⢠⣿⢛⣿⡟⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠁⢐⡔⠁⠀⠁
⠀⠀⠀⠀⠀⠈⣂⣉⠓⢲⣿⠄⠀⣸⣧⢸⡇⠅⠀⠀⠀⠀⠀⠀⠑⠀⠀⣂⡤⠺⠀⢡⡀⠄
⠀⠀⠀⠀⣀⣘⠁⠤⣌⣦⣿⠀⢠⣿⢿⢿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⠋⠁⠀⠀⠈⣐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡠⠀⠀⠀⢀⠀⠀⣄
⠀⠀⠀⠀⢠⠤⠰⠀⣼⡿⠏⠀⣼⠻⠂⢹⠉⠀⠀⢀⠀⣴⣶⣶⣿⣿⠇⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠀⠀⠀⠀⠀⣠⡾⢧
⠦⠤⠤⠒⠐⡳⠮⣭⣉⡤⠀⠊⠁⠀⠀⠈⢤⣤⣤⣴⣿⣿⣻⣿⣏⠃⠀⠀⠀⠀⠀⠀⠀⢻⠠⡀⠀⠀⠀⠀⠀⠀⠀⠾⠁⠀⠀⠀⠀⠀⢐⣿⠃⠈⢡
⠀⠀⠀⠀⠀⠠⠁⣸⡏⠀⠀⠀⠀⠀⠀⠀⢸⠻⠹⣿⣿⣿⣽⣿⣮⣆⠀⠀⠀⠀⠀⡀⠀⠸⣇⠄⠀⠀⠀⠠⠀⠐⢠⠐⠀⠀⠀⠀⠀⢀⢺⠏⠀⢠⠻
⠀⠀⠀⠀⠀⠑⣀⢼⡇⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢿⣿⣿⣿⣷⠀⠉⡂⠀⠀⠀⠀⠁⠀⠀⢦⠂⠀⠀⠀⠀⠀⡄⡼⠀⠀⠀⠀⠀⢀⢤⠟⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⢼⡟⣿⣿⣿⠀⠀⠑⠀⠀⠀⠀⠀⠀⢄⠆⠀⠓⠀⠂⠀⡤⣦⢷⠀⠀⠀⠁⠀⢂⡞⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠘⡷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⠠⠀⠀⢀⡞⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⢀⠀⢬⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⢽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠂⠀⠀⠙⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡀⢠⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢰⡞⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠁
⠀⣠⡠⠜⣀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⡐⣻
⠀⠉⠙⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠏⠃⠀⠀⠀⠀⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢀⡀⠀⢾⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⠀⠀⢸⠃⠀⠀⢸⡄⠀⠀⠀⢀⣤⡀⠀⠂⠀⠀⠙⠮⡆⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠈⠀⠀⠀⠀⠀⠂⡀⢀⢀⢀⠼⠁⠁⠀⠀⢀⡀⣀⠀⠜⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠉⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡈⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠒⢄⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⣠⠊⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠈⠐⠢⢄⠉⠢⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡰⠁⢀⠔⠁⠀⠀⡀⠀⢀⠤⡀⠀⠀⡀⠁⠒⠠⠀⠀⡈⠢⡀⠡
⠀⠀⠀⠀⠄⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠈⡠⠀⡠⠊⢀⠔⠁⠀⠈⠂⠄⡀⠁⠂⠠⢄⣀⠀⠀⠈⠀⡄
⠀⠀⠰⡂⢧⠹⠀⠢⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⢈⡁⠔⠊⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠤⠀⠀⣀⣀⣠
⠀⠀⠠⡚⠂⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⢦⡀⣀⠄⠀⠂⠁⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠄⠁⠆
⠀⠀⠀⠈⠐⠄⡠⠐⠁⠀⠈⠢⡀⠀⠀⠀⠀⠂⡐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠀⣀⠀⠀⠀⠀⠖⠈⢀⡄
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠈⠢⡀⠀⡀⠀⡀⠑⡀⠀⠊⡈⠂⠀⠀⠈⠓⠚⠁⠀⠀⠘⠄⠄⠀⠀⡔⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠈⠃⢄⡀⡠⠐⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠁⣀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠠⢀⠉⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠣⡀⠀⠀⠀⠈⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠢⡀⠀⠀⠡⣀⡠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡩⢿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠁⠐⠐⠂⠈⠀⠀⠀⠀⠈⢄⠀⠀⢣⡶⡟⢫⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠊⠘⠁⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢢⠀⠈⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠲⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠓⠀⠀⠒⢄⠀⠀⠀⡀⠸⠀⠠⠔⠂⠐⠒⠒⠊⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠀⠉⠉⠉⠉⠉⠉⠉⠠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⣴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢴⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⡇⠀⢶⡶⠒⠾⢢⣴⠾⢿⣏⣁⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣷⣦⣤⣤⣤⣤⣤⣤⡴⠆⠀⠘
⣶⠁⠀⠸⡆⠀⠀⠀⢠⣄⢉⣉⣉⡉⢤⡀⠀⠀⠀⠀⠀⠀⠄⣉⣉⣉⡁⢠⡀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⣧⠀⠀⠀⠉⣹⣤⣿⣿⡟⠀⡧⠀⠀⠀⠀⢰⣿⠀⣻⣿⣿⠇⠀⢙⠀⠀⠀⠀⠀⡸⠃
⣿⠀⠀⠀⢻⡄⠀⠀⠈⠡⠌⠉⡁⠀⠖⠀⣠⠶⢄⡀⠀⠰⠙⠛⠛⠋⢀⡠⠈⠀⠀⠀⠀⣸⡇
⣿⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠈⠃⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⢀
⢸⣆⠀⠀⠀⠈⠻⣄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⢀⡼
⠀⠙⢷⣄⣀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢦⣀⡼⠃⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠈⠉⠛⠒⠓⠚⠛⠷⣤⡀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⢀⣤⠾⠿⠶⠶⠶⠶⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⢁⣼⠷⣦⣄⣀⣀⣀⣀⣀⣀⣀⣠⡶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠀⠈⠉⣤⠀⠈⠉⢙⡟⠉⠙⡋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠷⠀⠀⢀⡸⠷⢀⠸⢟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢠⣦⡼⣤⣤⡀⠘⠛⠀⠛⠃⠊⠉⠲⠆⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⠉⠁⠀⠈⠳⠆⣀⣀⣀⡀⠴⠆⠀⠀⠀⡀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣾⠁⢠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢡⡀⠋⡀
⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⣿⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⣸⠁⠐⠚⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢁⡀
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣿⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠛⡀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⡀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠉⠉⠉⠉⠉⠓⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠛⠢⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠛⠳⠷⢶⣤⣠⣶⠴⠶⣶⠶⠞⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠃⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⢸⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠘⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⡆⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⡇⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⡇⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⡄⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠞⠁⠀⠀⠉⠁⠉⢫⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠉⠀⠀⠀⠀⣀⢰⡇⠀⠀⠀⠙⠣⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠶⠖⠉⠉⠀⢻⡄⠀⠀⠀⠀⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⣀⣀⣀⡴⠟
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠛⠉⠙⠛⠋⠙⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢉⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠻⢿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⡈⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⠀⠀⠈⣻⣿⣿⣿⣿⣽⡿⣻⣿⣯⣭⣯⣹⣻⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣤⣀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢮⣭⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠻⠁
⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⢲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣷⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣿⡷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣩⣝⣉⡃⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⢷⣿⣿⣿⣿⣟⣿⣭⣽⣲⡷⣿⡿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠿⢿⣿⣿⣿⣿⣿⣿⣷⣿⣟⣿⣿⣿⣿⣷⡀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡏⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠉⠛⠛⠛⠛⠛⠛⠋⠁⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠀⠤⣤⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⢀⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀
⠀⠀⠀⠀⠀⡰⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡆
⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠶⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣶⣄⠀⠀⠀⠈⢠⡀⠀⠀⠀⠀⠀⠀⣶
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠰⠀⠀⢀⠀⠀⠀⠉⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠻⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⢤⣤⣤⠀⠀⠀⠀⠀⠀⠀⢱⣦⠀⡀⠀⠀⠀⠀⢰⠀⢸⠀⠀⠀⠀⣿⠀⠀⠃⠀⠀⠀⠀⠀⠀⠸⠀⢠⠄⠙⠂
⠀⠀⠷⠀⠀⠀⠀⠘⠃⠀⠠⠊⠁⢰⣿⣇⡀⠀⠀⠀⠀⠀⠀⢿⣯⣼⠇⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⡸⠇⠀⠄⠀⠀⠀⣤
⠀⠀⠀⠆⠀⠀⢠⡆⠀⠀⠀⠶⠖⠀⠙⠟⠁⠀⠀⠀⠀⠴⠢⠞⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⣿⠀⠀⠘⠀⠀⢀⡸⠃⠀⠀⠀⠀⢠⠀⠘
⠀⠀⠀⡀⠶⠆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⣿⠀⣀⠀⠀⠀⠞⠁⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⢻⡇⠀⠀⠀⠀⠀⠠⠀⠀⠉⠁
⠀⠀⠀⠆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠻⠿⠿⠗⠀⠀⠀⠀⠀⡠⠆⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⠒
⠀⠀⠀⠀⠀⣤⡸⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠚⢣⣤⣾⡿⠟⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠋⠐⣆⠀⠦⠀⠀⠀⡀
⠀⠀⠗⠀⠀⢸⡇⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠙⠃⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣤⣤⡾⠛⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠟⠀⠀⣇⠀⠁⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠎⠀⠀⣿⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛
⣆⠀⠀⡄⠀⠂⠀⠀⠀⠀⠙⢷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠻⡄⠀⡞⠑⣤⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢘⠛⠳⢶⣦⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣴⣶⠞⠛⠁⠀⢀⠘⠀⣼⡀⠀⠀⠇⠀⠈⠃⡀
⣿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠉⠉⠉⠉⠻⠟⠹⡿⠋⠉⠀⠀⠀⠀⠀⢠⠋⠀⣠⠈⠿⠀⠀⠂⠀⠀⢸⡇⠀⠀⠀⢸
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢎⠁⠀⠀⠀⠀⢰⡎⠀⢠⡀⠛⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠂
⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠰⠀⠈⠁⠀⠀⠀⢠⡞⠀⢀⠈⠁⠀⠀⠀⠀⠇⠀⠀⢸⡇⠀⠀⠀⠘
⠀⢿⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠃⠀⠀⠀⠀⢸⡇⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⣷⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠘⠀⠀⠀⢰
⠀⠀⠀⢷⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⡟⠘⠓⠒⠀⢤⣤⣤⠟
⠀⠀⠀⠀⠛⣦⡀⠀⢰⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠁⡄⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠀⢀⠀⠀⠀⢰⠀⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣀⣤⡀⠀⠀⠄⢀⡈⠈⠉
⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠻⠶⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⣀⣀⣀⡤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠂⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠰⠶⠶⠶⠶⠤⣀⣀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠃⠀⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠉⠙⠉⠛⠛⠁⠀⠀⣀⠀⠀⠁⠀⠀⠈⠛⠠⡄⠠⠤⢤⡤⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠤⠦⠀⠀⠀⠀⠀⠀⠀⠠⡶⠤⢤⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⢱
⠀⠀⠀⠀⣀⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⣀⣀⡀⢀⣠⣤⣀
⠀⠀⠀⠀⣏⠀⠀⠀⠓⢤⣀⠀⢀⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠉⣴
⠀⠀⠀⠀⠹⣤⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠙⢃⣤⣤⣤⣔⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⢀⣀⣀⣀⡘⠛⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠛⠋⠘⠛⠛⠛⠙⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⣀⣀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣾⣿⣶⣿⣷⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠛
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⢀⣀⣴⣿⣿⠃
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⠇⠀⠀⠀⠈⠛⠛⢿⣿⣉⠉⠉⠁
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠸⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡽⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣜⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠙⢻⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⠈⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠈⣿⣿⣿⣿⠃⢸⣿⣿⣿⣿⠀⠀⠙⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⠀⢸⣿⣿⣿⡿⠀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⣿⣿⣿⣧⠀⢸⣿⣿⣿⡇⠀⠀⠀⢀⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣧⠀⠀⣿⣿⣿⣿⡀⣸⣿⣿⣿⡇⠀⠀⣤⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣧⣿⣿⣿⣿⣇⠀⠀⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠹⣿⣿⣿⣿⣿⡄⠀⠈⠛⠻⠿⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠿⠻⡿⢛⠿⢿⣿⣻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⢀⡀⠤⠐⠒⠒⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠁⡠⠄⠀⠀⠀⠀⠀⠀⡀⠀⠈⠐⠢⢌⠲⢄
⠀⠀⠀⠀⠀⢀⠔⠁⠀⡔⠉⢀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠄⡀⠑⠈⠣
⠀⠀⠀⠀⢠⠃⠀⢠⠊⠀⡔⠁⠀⡐⠀⠀⠀⠀⡀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠆⠀⠰⠁⠀⠜⠀⠀⠔⠀⠀⡔⢄⠀⠈⠂⠄⣀⠈⠉⠒⠒⠤⠄⠀⠘⠢⢄⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢠⠊⠀⣠⠊⠀⠀⠉⠢⢄⡀⠀⠉⠐⠒⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⢠⠉⡎⣀⠔⠆
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠁⣠⠾⣁⠤⡀⠀⠀⠀⢠⢚⠍⠡⢔⢖⠶⡲⠚⠉⠇⠀⠀⠀⠀⠀⠘⡍⠚⠀⠁⠉⠼⡲
⠀⠀⠀⠀⠈⠹⡑⠢⠤⡤⢮⢱⠡⣤⠈⠎⠯⠌⠁⢇⢇⠺⠇⢈⢸⠉⠀⠸⠋⠀⠀⠀⠀⠀⠀⢀⠥⡀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠈⠂⡇⠈⠉⠊⡢⠤⢔⡡⠁⠐⠊⠈⠚⠥⠠⢑⡁⠀⠀⡘⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠉⢐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⡔⠒⡄⠀⠀⠀⠲⠔⠃⠀⠀⠀⠧⠼⠀⡴⠁⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠦⡈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⣊⠀⠠⠔⠂⠉⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠘⢗⠂⠠⢤⠤⠤⠤⡄⠐⠲⠋⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠜⠒⠤⣀⠀⠀⠑⢄⠈⢤⠤⢤⠃⡰⠁⠀⡠⢖⠁⠀⠀⠀⠀⠀⢀⡠⠒⠁
⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠁⠒⠊⡂⠳⡄⠀⠀⡖⠀⡏⠉⠀⠈⡄⠀⢀⡠⠔⠂⠁
⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠠⠊⠀⠀⠈⠢⢌⢢⠌⡠⠊⠀⠀⠀⠀⢱⠉
⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠁⠈⠈⠉⠉⠉⠙⣞⣉⣥⡭⠷⠦⠤⠤⠺⢤⣄⡀⠀⠀⠀⠀⣀⣠⠤⠤⠦⠤⠦⠤⣤⣄⣀⡀
⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠼⡏⠀⠀⢀⡀⣀⣀⢀⣀⣀⣀⠉⠳⡄⡰⠊⠁⣀⣀⣀⡀⣀⡀⣀⠀⠀⠈⠇
⠀⠀⠀⠘⠤⢀⣀⠠⠤⠤⠒⠈⠉⠀⠀⠀⠠⠇⠬⠥⠤⠤⠐⠒⠒⠒⠖⠀⠬⠑⠾⠖⠩⠤⠀⠒⠒⠒⠒⠀⠀⠤⠭⠄⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠹⡆⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢀⡤⢤⣀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢹⡤⢤⡀⢠⠞⠁⠀⣿⠶⠶⠦⣤⡀
⠀⠀⠀⠀⢠⠋⠀⠀⠈⠓⢤⣤⣤⠴⠋⠀⠀⠈⠙⠶⣄⣀⣀⣠⣤⣀⠀⢺⣇⣀⡴⢾⠀⠀⣿⠀⠀⠀⢰⡟⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢌⡀⠀⢛⣉⡀⠈⠓⠚⠛⠄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣷
⠀⢀⠔⠈⠀⠀⢀⣀⣀⠀⢀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⠈⠙⠛⠉⠉⠙⢷⠀⠀⠀⠘⢦⣀⣼⠃⠀⠀⠀⠀⠀⠀⢸⠇
⠰⠃⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠀⡠⢆⠀⠀⠴⡄⠀⠀⠀⢸⠃⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢰⠀⠀⡾
⣇⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠞⠉⠀⠀⠁⢀⣀⡀⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠰⡇
⠸⣆⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⡆⢠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡠⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡀⠙⠶⠤⢤⣤⣀
⠀⣿⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⢀⡼⠁⠀⠳⢤⣀⠀⠀⠀⢀⣀⡤⠞⠁⠀⠀⣹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⡀⠀⠀⠙⢷
⣴⠃⠀⡠⠃⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⣠⠏⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⢸
⢧⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⣤⠟
⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣏⡠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡴⠚⠉
⠀⠀⠈⠐⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠓⠶⣤⣤⣄⣀⣀⣤⣤⠶⠚⢉⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⢀⣼⠃⣠⠴⠄⠀⣰⠶⡦⢤⡨⠏⠁⠙⠦⢤⣾⠛⣆⡀⠀⠀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠂
⠀⠀⠀⠀⠀⠀⠙⠦⢤⣀⣤⠴⠛⠁⠘⠁⠀⡠⢊⡙⠀⠇⠀⢦⡀⠀⠀⢀⡼⠃⠀⡯⠙⠛⠛⠛⡁⠀⠀⠉⠠⢄⣀⣀⣀⣀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡠⠊⠔⢋⠅⢠⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠄⢆⠑⢄⠰⠅⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡡⢀⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠈⡖⠠⢠⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠷⣀⣁⠬⢄⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠛⠒⢠⠄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⠃⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⣀⠀⠀⠀⠐⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠋⢻⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⠆⠤⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠇⢀⣀⣀⣀⣀⣀⣀⣀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣾⠀⡆⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⡇⠀⢹⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠁⠀⠘⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⡇⠀⠀⠀⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣇⠀⠀⠀⡇⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡋⢙⠉⠀⠀⠀⠁⡟⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⠀⠀⠀⠀⠀⡇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠚⠉⠁⡜⠀⠀⠀⠀⢠⣇⠀⠈⠐⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⢀⠀⢀⡶⠋⠀⠀⠀⠀⠈⢢⣁⠂⠤⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠈⠑⣀⣭⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⢄⣀⡠⠆
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣀⣈⣀⣀⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⣀⠀⠀⢀⣠⣤⣶⣶⣶⣿⣶⣶⣤⣤⡀⠀⠀⢀⣀⣤⠤
⠀⠀⠀⠀⠀⠀⣠⠖⢋⣀⣤⣴⣤⠀⠀⠀⠀⠈⠙⣿⣷⡿⠛⠉⠉⠙⠛⠛⠛⠉⠉⠙⠿⣷⣿⡏⠁⠀⠀⠀⠀⣠⣦⣤⣄⡉⠲⣄
⠀⠀⠀⠀⣠⠞⠁⠘⠋⠉⠉⢻⣿⣷⣤⠀⠀⣠⣾⣿⠋⣀⣠⠆⠀⠀⠀⠀⠀⠀⠀⣦⣀⠘⢿⣿⣦⡀⠀⣴⣾⣿⣟⠉⠙⠛⠳⣌⠳⣄
⠀⠀⠀⣴⠏⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⡿⢃⠄⠀⠀⠀⠀⠀⠀⠀⡌⢻⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠘⣧
⠀⠀⢸⡟⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿⡿⠟⢋⣡⡞⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡙⠛⠿⣿⣿⣿⣿⣿⣿⣿⡉⠉⠀⠀⠀⠀⠀⢹⣇
⠀⠀⣾⣧⣴⡟⠁⣠⠄⠀⠈⠋⠙⣿⠿⢿⣿⡏⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⢸⣿⡿⠻⣿⠋⠙⠻⣄⠀⣄⡈⠻⣦⣬⣿
⠀⠀⠛⢿⣟⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⢸⡿⣦⡀⠀⠀⠀⠠⠤⠤⠤⠀⠀⠀⠀⣠⡾⣿⠀⣿⠁⠀⠀⠀⠀⠀⠈⠀⠈⢿⣦⣹⡿⠟
⠀⠀⠀⠀⠻⢿⣿⣶⡇⠀⠀⠀⠀⠀⠀⠀⣿⣄⠈⢻⡟⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⣿⠋⢀⣿⠀⠀⠀⣀⡀⠀⠀⢰⣷⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠻⣿⣷⣶⣤⣤⣀⠀⠀⠀⣾⣿⡆⠀⢠⡀⠀⠀⠠⠤⠴⠦⠤⠤⠀⠀⠀⠀⡀⠀⣻⣿⣷⣶⣾⣿⣿⣿⣶⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣷⡀⠀⣿⣿⣷⡀⠈⠇⠀⠀⠠⠴⠶⠶⠦⠤⠀⠀⠀⣾⡀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣴⣿⣿⣿⣿⡆⠀⡇⠀⠀⠒⠚⠛⠒⠒⠀⠀⣆⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠷⣦⣤⡇⠀⠐⠒⠒⠛⠒⠲⠶⢾⣧⣤⠶⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠛⠁⠀⣼⢿⡇⠀⠒⠛⠛⠛⠛⠛⠚⢻⣿⣿⡀⠈⠙⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⣼⠃⠀⢧⠀⠀⠈⠉⠉⠉⠁⠒⣿⠃⠈⢷⡀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣼⠃⠀⠀⠈⠀⠀⠉⠉⠉⠉⠉⣹⡏⠀⠀⠈⢷⡀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⡇⠀⠀⠀⠀⠀⠀⢀⠈⠉⠉⣈⡿⠀⠀⠀⠀⠘⣇⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢸⡀⠀⠀⠀⠀⠀⡀⠀⠀⢤⡤⢴⡇⠀⠀⠀⠀⠀⡿⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⣇⠀⠀⠀⠀⠀⡇⠀⠀⠶⠖⣺⡇⠀⠀⠀⠀⢠⠃⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠌⠓⠀⠀⠀⠀⣇⠀⢉⣚⢩⣿⠀⠀⠀⠀⠐⠁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠠⠤⢼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⢚⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣿⣧⣤⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠛⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣍⣬⣻⣤⣉⣽⣋⣽⣏⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡴⠚⠉⠁⠀⠀⠀⠈⠉⠙⠒⣤⠤⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠋⠉⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣤⠄⠘⠠⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡅⠀⠀⠀⡁⠀⡆
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠉⢲⠶⠚⢃⠈⠱⡀
⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠒⠉⢷⡀⢹⡄
⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠻⡀
⠀⠀⠀⣿⠶⠶⠞⢻⠛⠛⠛⠛⢙⣿⣿⠛⣿⠛⠛⠛⡻⠓⣶⣶⣤⣤⣄⣤⣀⡀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⢸⡇⠀⠀⢠⡌⠀⠀⠈⢦⡘⢿⣿⣿⡟⠀⠀⠀⡇⢸⣿⣿⣥⣸⡏⠀⠈⡟⠛⡶⠶⢾⠀⠀⢸
⠀⠀⣾⠁⠀⠀⠀⣧⠀⠀⠀⠀⠉⠀⠉⠉⢀⣤⠀⠀⠐⠂⠹⢿⠿⠋⠀⠀⠀⣷⠐⠁⠀⠀⠀⠀⠘⡄
⠀⠀⣿⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠡⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⣿⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣀⠀⣀⡀⠔⢻⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⠊⡠⠁⠠⢤
⠀⢰⣿⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣀⣠⣤⢾⡟⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠐⢂⡈⣄⠰⠁⠶⠈⣠⡴⠎⣀
⠀⠈⣯⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⡀⠖⣡⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡁⠘⢦⡀⠀⠈⡀⠒⠒⠉
⠀⠀⣿⠀⠀⠀⠀⠀⢀⣀⣀⣠⡽⠶⠤⢤⣀⣀⣀⣀⣀⣀⡤⠔⠚⠛⠒⠒⠒⠒⠠⠤⠤⣄⣀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠁⠀⢀⠤⠌⠰
⠀⠀⢹⡤⠤⠔⠚⠛⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⣿⠉⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⣾⠀⠀⡀⠃⣀⣀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠴⠶⠀⠉⠀⠈⠳⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠰⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠉⠉⠉⠉⠁⠠⠦⠀⣤⡄⣀⡔⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⠀⣌⠀⠀⠀⠀⠈⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⡠⠙⠢⡀⠀⢰⠀⠀⠀⠀⠀⠁⠠⠀⠈⠁⠀⠀⠀⠀⠛⠂⡀⠀⠋⠛⠓⠒⠶⠶⠶⠛⠁
⠀⠀⠀⠀⠀⡠⠈⠀⢀⡴⠛⠲⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⢈⠀⠀⠰⢋⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠙⠢⠤⣀⠈⠑⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡆⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣀⠀⠤⠤⠤⠤⡶⠲⢤⡤⠖⠛⠦⣄⣀⣀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢶⠀⠀⠈⡄⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢰⡇⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠏⠀⠀⠀⠀⢱⠀⠀⡠⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡎⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠠⠐⠉⠀⠀⠀⡠⡾⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠈⠢⣀
⠀⠀⠀⢠⡐⠀⠀⠀⣀⡀⠤⠐⣋⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⢁⢆⡀⠀⠀⠀⢈⠗⡀
⠀⠠⡄⠀⠈⠙⢉⣀⡠⠴⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡙⠠⠆⠚⠃⢀⡵
⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠶⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠔⠒⠛⠛⠓⠒⠦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⡀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡈⣧
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢿
⠀⠀⠀⠀⠀⢀⡿⠀⠀⠠⠄⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⡄⣷
⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠁⣿
⠀⠀⠀⠀⠀⢿⠶⠶⣶⠿⢶⠞⢺⣿⣷⡶⠶⣶⠶⠶⠶⢶⡶⠶⢤⣤⣄⣀⣀⣀⣀⢀⡇⠀⠀⢸⡄⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠸⣄⠘⣿⣿⣧⣴⠟⠀⠀⠀⠸⡆⠀⢰⣿⣿⣿⠏⠉⣿⠟⠛⢿⡟⢻⡀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠈⠑⠛⠛⠋⢁⠒⠊⠀⠀⠀⠙⢦⣼⣿⣿⣷⣿⠿⠋⠀⢠⡟⢃⡼⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣠⡟⠉⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠰⣄⣀⡰⠃⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣀⣴⠟⠉⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣤⣄⡀⣀⠀⢀⣀⣠⣤⣶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠛⠋⢻⣏⡀⠀⠛⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠺⢤⡼⠋⢁⣰⠀⣀⣤⣤⠴⠷⠾⠷⢤⣤⣄⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠒⡴⠋⠀⠀⠀⣤⣤⡄⠈⠻⠟⠻⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠞⠉⠀⢀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⡀
⠀⠀⠀⠀⠀⢀⡴⠋⠁⢀⣠⠤⢚⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠙⠣⣄
⠀⠀⠀⠀⠀⠛⡀⠀⠐⢏⠁⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢦⣄⠀⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠹⢆⠀⠈⢣⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠈⠳⢦⡄⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠸⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⡠⠊⠈⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⠏⢀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⢀⠝⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠆⠀⠈⠁⠒⠒⠒⠒⠒⠒⠀⠈⠉⠀⠀⢡⡄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠈⠉⠉⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⡀
⠀⠀⠀⠿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⡶
⠀⠀⠀⠀⠈⠉⠑⠲⣤⡀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⣀⣤⠶⠶⠶⠶⠶⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢙⡟⠙⠶⠶⠶⡾⢛⠋⠀⠈⠉⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⣧⢸⡇⣀⡀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢿⣸⡇⠈⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠈⠻⠷⣤⡄⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⠘⠂⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢦⣶⣀⡀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⠒⢦⣤⠀⠀⠀⠈⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⠁⠙⠳⠶⣀⡀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⡇⠀⠀⠀⠀⠀⢹⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⢀⡿⠁⠀⠀⠀⠀⣀⠼⠆⠸⠃⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠀⢀⡰⣬⠃⠀⠀⠀⠀⠶⢄⣀⣀⣀⡀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠑⠂⠒⠒⠋⡰⠊⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⢀⣀⣠⡴⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠛⠂⡀⠈⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠀⠀⡀⠀⢀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢠⣶⣶⡟⠁⠀⠀⠀⠀⠀⠀⢸⡏⢹⡏⠉⠉⠉⠉⢹⡏⢹⡇⠀⠀⠀⠀⠀⠀⠈⠲⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠘⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⢜⣃⣘⣃⣀⣀⣀⣀⣘⣃⣘⡣⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣬⡵⠤⠶⠴⠶⠶⠷⠶⠴⢮⣽⡴⠀⠀⣤⣤⣤⣤⣤⣤⣤⡤⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠁⠀⣾⡏⠉⡍⠉⢹⡇⠀⠁⠁⠀⠀⡿⢻⡇⠛⠋⢻⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣇⣀⣼⡇⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡇⠀⣿⡇⠀⠀⠀⠀⠀⠿⠼⠧⠤⠤⠼⠧⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⡏⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⢿⣗⣒⣓⣒⡺⢇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠻⣏⠉⠉⠉⠉⠉⠉⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⡴⠓⣌⠳⡄⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣡⠎⠀⠀⠈⠳⡘⢯⣉⣉⣉⣉⣉⣉⣉⣉⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢃⠞⠁⠀⠀⠀⠀⠀⠘⢦⡑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣊⣴⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣌⣻⣉⣉⣉⣉⣉⣉⣉⣉⣉⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡞⡉⠉⠉⠉⠉⢳⠀⠀⢸⠀⠀⡏⠉⢙⠉⠉⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⡇⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⡧⠀⠴⠄⠠⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⡇⠀⠀⣀⠀⢸⠀⠀⢸⠀⠀⣇⣀⣨⣀⣀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⡇⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣇⡇⠀⠀⠀⠀⣸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣧⣤⣤⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⣶⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢍⠙⡶⠊⣀⡤⠬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣦⣶⣿⣷⡟⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠉⠉⠘⢶⣟⣺⣿⢦⣴⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠀⢉⣿⠞⣹⠮⠿⠟⠂⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⣠⢽⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢈⣷⠞⢉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠂⠖⠻⠗⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠌⣿⠄⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠺⠻⠿⣯⣟⢻⣿⣾⣿⣤⣊⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠀⠀⠀⠀⠀⠘⠛⠽⣿⢿⣽⣗⣖⣠⢄⣀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣄⣀⣀⣀⣀⣀⣰⣧⣤⣿⣥⣤⣭⣭⣋⣉⣀⣀⣰⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣝⣛⣩⣛⣻⣟⣽⣟⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⣀⡴⠋⠓⢦⡀⠀⠈⠛⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⢀⡴⠞⠁⠀⠀⠀⠀⠉⠳⣤⡀⠀⠈⠻⢿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⢀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠙⠻⣄⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⠙⠦⣄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⣀⡴⠚⠁⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠈⠛⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠿⠋⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣤⣤⣤⣤⣼⣤⣤⣤⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠙⠻⣿⣿⣿⣿
⣿⣿⡿⠛⠁⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠈⠙⠿⣿
⣿⣏⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠤⠤⠤⠼⠤⠤⠤⠤⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⣸
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠈⡇⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⢰⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⣧⣤⣤⣤⣼⣤⣤⣤⣤⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⣤⣤⣤⣼⣧⣤⣤⣤⣿⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠈⡇⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⢧⡤⣤⢤⠼⢤⣤⣤⢤⠇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠿⠤⠤⠤⠼⠧⠤⠤⠤⠿⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠤⣄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠶⠊⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠀⠂⢄⡀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠨⠓⠁⠀⠉⢀⠀⠙⠴⡀⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠂⢀⣂⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠸⠀⠀⠘⡖⡇⠀⠀⠀⢠⠀⠀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠠⠀⢀⠐⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠁⠂⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⣀⡀⣀⣀⠄⢁⣀⣀⢀⠄⣀
⠀⠀⠀⠀⠀⢁⠄⠀⠀⠀⠐⠤⠀⢤⠀⠀⠀⠀⠀⠉⠉⠃⠀⠀⠀⠀⡄⠈⣁⠀⠢⠀⠾⠙⡮⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠈⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢂⠴⠒⠄⠀⢰⡀⠂⠀⠄⠄⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡀⠀⠀⠀⣀⠐⢀⠀⠀⠀⠀⠠⠀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⡄⠀⢀⣤⣾⠿⠋⠀⠀⠀⠈⠛⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⠉⣿⣇⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣠⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣿⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⡏⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢰⣿⡿⠿⠿⠿⠿⠿⢿⣿⡇⠀⠀⠀⠀⣿⣿⠿⠿⠿⠿⠿⠿⣿⣷⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣧⣤⣤⣤⣤⣤⣤⣿⣿⣤⣤⣤⣤⣿⣧⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⡀⢀⠀⠀⠄⢀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠀⠁⠀⠈⠁⠁⠀⠀⠂⠀⠁⠈⠁⠁⠈⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠉⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠘⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡗⠛⠛⢻⡟⠀⠀⠀⠀⢀⣴⠟⠻⣦⡀
⠀⠀⠀⣰⡶⠶⠶⠶⠶⢾⡇⠀⠀⠸⠷⠶⠶⢶⡾⠋⣀⡤⣄⠀⠙⢷⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡄
⠀⠀⠀⡿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⠖⠉⣠⠞⠁⠀⠈⠑⢤⡀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⢠⡏⠀⠀⠀⠀⠀⠘⠃⠀⢀⡤⠚⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⣀⠴⠋⣠⠞⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠈⠓⢤⡀⠙⠦⣀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⣿⠀⠀⠀⠀⠀⣠⠞⢁⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠈⠳⣄⠀⠀⠀⠀⠘⣧
⠀⢀⡟⠀⠀⢀⡴⠋⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠙⢦⡀⠀⠀⢿
⠀⢸⡇⣠⠔⢉⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠉⠢⣄⢸⡇
⣠⡾⠛⣡⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣤⡈⠛⢷
⠙⠛⠛⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠣⠬⠤⠤⠥⠤⠤⠤⠬⠤⠤⠤⠤⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⢀⢀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⢀⣀⣀⣀⡀⣀⣀⣀⡀⠀⠀⣿
⣶⣶⡶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶
⣿⣿⣴⣤⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡤⠤⡤⠀⠀⠀⠀⠀⢀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣦⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⣸⠀⠀⣳⠀⠀⠀⠀⣰⢋⣴⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⢨⡷⡇⡤⡄⠀⠀⠀⠀⠀⣀⣀⣀⡀⡾⠁
⠀⠀⠀⢹⠁⢐⡷⠀⠀⠀⠀⠘⢿⡟⣾⢂⣶⢐⣶⢰⣆⠀⠀⠀⠀⠈⠀⠈⡯⡇⠀⠁⠀⠀⠀⠀⠀⠨⠉⠉⠁⡾⣆
⠀⠀⠀⢩⡣⠊⠀⠀⠀⠀⠀⠀⠀⢬⡯⢨⣯⠨⣿⠬⣿⡄⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠙⠄
⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⢙⣿⠉⠋⠉⠁⠀⠀⢀⣀⣀⣤⣋⣼⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠐⠠
⢤⡀⢀⣴⣦⣾⣷⣤⣄⠀⠀⠀⠀⠀⢀⡀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⢳⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⢸⡇⣸⡇⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠿⠉⠀⠹⠇⠀⠀⠀⠈⡿⠿⣿⢸⣿⣿⡏
⠀⠘⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢼⣹⣏⣿⣻⣻⡇⠀⠀⠀⠀⣁⠀⢰⣶⠆⠀⠀⠀⠀⠀⠀⢰⡉⡎⣹⢸⣿⣿⣷
⠀⠀⠉⠾⠎⠛⠱⠿⠊⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠠⠿⠀⠾⠀⠧⠀⠠⠄⠀⠀⠀⠾⠿⠿⠿⠯⠿⠿⠿⠆
⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠰⠛⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠛⠛⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄
⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠞⠳⢴⠀⠀⠀⠀⠀⠀⢰⣷⠖⣲⠒⡶⢲⢲⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠒⣇⠘⠇⣸⠒⠀⠀⠀⠀⠀⣼⠏⠀⠁⠈⠀⡿⣼⡇⠀⠀⠀⢀⣀⣰⣺⣲⣀⣀
⠀⠀⠀⢀⣻⣟⡅⠀⠀⠀⠀⠀⠀⢀⡀⣋⢙⡋⣙⣀⡀⠀⠀⠀⢀⣿⢀⡀⠠⠀⢸⣇⣿⡇⠀⠀⠀⢈⣿⣉⢹⠉⣹⣏
⠀⠀⠀⠚⠓⠓⠓⠂⠀⠀⠀⠀⠀⠐⠂⠒⠐⠂⠒⠐⠂⠀⠀⠀⠈⠀⠀⠁⠀⠀⠉⠙⠛⠃⠀⠀⠀⠐⠂⠒⠐⠒⠂⠒
⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣀⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠀⠀⠀⠀⠀⠀⠀⡎⣭⣭⡝⠀⢀⣀⠀⠀⠀⠀⠀⠀⣠⡀⡴⠋
⠀⠀⢰⣿⠀⠀⢰⡇⠀⠀⠀⠀⣀⣤⣄⢴⠟⠻⣇⠀⠀⠀⠀⠀⠀⠀⡇⠶⠶⢮⠀⣼⣿⡀⠀⠀⠀⢀⣴⣿⣿⣦⡀⢀⡴
⠀⠀⠘⣿⠀⢠⣿⡇⠀⠀⠀⠀⢿⣿⣿⣦⣄⢀⣭⣦⣄⠀⠀⠀⠀⠀⣏⣛⣛⣉⠀⠛⣿⠃⠀⠀⠀⢀⣿⣿⣿⣿⣿⣮
⠀⠀⠀⠻⡄⣾⡿⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⢺⣿⣿⣿⡇⠀⠀⠀⠀⡧⠷⠿⢼⠀⠀⠁⠀⠀⠀⢀⣼⣿⠿⠻⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⡾⡧⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⡷⣥⡀
⠀⠙⠋⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⣽⣯⡧⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢀⠠⡷⠷⢚⠆⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣟⠐⠀⠀⠀⣀⠀⠀⠀⣴⣖⣉⡁
⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡷⡿⠉⠀⠠⠤⠴⠦⠄⠀⠗⡆⣖⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠤⠰⠀⠀⠀⢛⣻⣿⠒⠒⠒⠒⠶⠒⠂⠈⢷⡥⢶⠛
⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⡤⠄⠀⠀⢀⠀⠀⠀⡼⠓⢘⡆⠀⠀⠀⣶⣾⠄⠼⢾⡗⢀⣸⡄
⠀⠀⠀⠀⠀⠀⢀⠀⠰⡶⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠨⠇⠀⠀⠈⢉⠀⠀⠰⠛⠃⡠⠷⠀⠀⠀⣟⠻⠀⡈⢿⡅⠸⠁⣇
⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠘⢶⡀⠀⠀⢀⡤⢾⠭⣧⢀⠀⣂⣀⡀⡰⠛⢠⣤⣼⣉⢛⣿⢻⠄⠀⢀⡇⠀⣧⠁⡞⣇⠐⡟⣿⠒
⠀⠀⠀⡇⠀⣼⣠⣸⣧⣷⣶⣶⣿⢶⠶⠿⠿⢷⣛⣛⣛⡙⠯⠭⢿⣷⢰⣾⠿⠻⣿⡿⣿⡟⣷⣾⣿⣴⣦⣻⣶⣯⣿⣅⣿⣿⣷⡆
⠀⠀⣠⣧⣸⡭⣟⣹⣿⣻⠟⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠛⠓⠦⠬⢭⣍⣀⣚⠛⠾⠿⢿⣷⣿⣿⣿⣿⠍⣿⣿⣿⣷⡽⠿⣷⠃
⠀⢠⠤⣗⣨⠶⣫⠽⠋⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⢠⡀⠀⠀⠀⠀⠉⠉⠙⠓⠒⢶⣤⣤⣥⣉⡛⠺⠿⠿⣿⣿⣷⣶⣾
⠀⠀⢘⣩⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⢯⣷⣦⣤⣄⡉⠁
⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣄⡀⠸⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⣋⣙⢿⣿⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⢛⣵⣾⣿⣿⣷⣮⡛⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⢟⣥⣾⣿⣿⣁⣀⣀⣈⣿⣿⣷⣬⡻⢿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢟⣩⣾⣿⣿⣿⣿⣿⣄⣸⣇⣠⣿⣿⣿⣿⣿⣷⣍⡻⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢸⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⠦⠤⠤⢼⠦⠤⠤⢼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢸⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣾⣶⣶⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠓⠐⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠈⠙⣶
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠇⢐⠀⠀⡀⡀⠀⠀⠀⠀⢀⣴⠾⠿⣶⣆⡀⠀⠀⠠⠅⠠⠅⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠄⠀⠙⣿⡇⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠠⣒⠒⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⢁⡿⠋⠀⠀⠀⠐⠒⠃
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⣛⠀⣀⣀⠀⠀⠀⠀⠘⠓⠀⠶⠛⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠸⣶⣴⡟⠀⠀⠄⠡⠄⠀⠀⠀⠈⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢘⣵⡿⠛⢿⣧⠀⠀⠀⠀⠉⠁⠀⢀⡄⡀⠴⡀
⠀⠀⠀⠀⠀⠀⠹⠟⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠁⠀⠀⡀⠀⣀⡀⠀⠀⠀⠀⠀⢃⠁⠈⠀⠠⢹⢃⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠀⠀⠀⢀⣿⣧⣀⣀⣠⠁⠀⠀⢻⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠁⠀⠈⠀⠴⠆⠤⢀⠁⠞⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠤⢀⡶⢾⣷⣦⠀⠀⠀⠀⠀⠀⢀⠀⠀⠓⠒⠀⠀⠈⠀⠀⠀⠀⡴⠖⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⡇⣄⠀⠀⠀⠒⠁⠀⠀⠀⠀⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠂⢀⡀⠈⠀⠀⠀⠀⠀⠀⢠⣿⣿⣶⡗⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠘⠀⠉⠁⠐⠀⠀⢀⠀⠀⠀⠠⠁⣠⡄
⠀⠀⠀⠀⠀⠰⠁⠈⠀⠀⠀⠀⠀⠃⠠⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⢂⠈⠁⠉⠀⠀⠀⠀⠀⠐⣆⠂⠘⠀⠀⠀⠀⡀⣠⣶⣷⣾⣧⣄⢰⣿⣿⡇
⠀⠀⠀⠀⠂⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⢠⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠨⠂⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠊⠀⡄⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠈⢽⣿⣿⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠂⠀⠈⣿⣿⣿⣿⠀⠀⡈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣿⣿⣿⣿⣿⡇⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠃⠛⠛⠋
⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠋⠉⠳⣄⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⡇⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡷⢤⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⡠⠚⠛⠛⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⠉⠉⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢾⠀⠀⠀⠀⠀⢀⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠘⢿⡿⠃⠀⠀⢀⣀⣼⣄⣀⣀⣧⣀⣀⣸⣇⣀⣀⣼⣀⡀⠀⠀⠀⠈⠻⡿⠃⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⢸⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⡇⠀⠐⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠁⠀⠈⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⡇⠁⠉⠉⠁⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣅⣀⣀⣀⣀⣀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡷⠤⠤⠤⠤⠤⠤⠤⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡧⠤⠤⠤⠤⠤⢴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⣉⣉⣉⣉⣉⣉⣡⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠒⠒⠒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠒⠒⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣯⡀⠀⠀⠀⠀⠙⠿⢿⠿⠿⠿⡿⠿⠿⢿⡿⠿⠿⢿⠿⠟⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⣤⣶⣄⠀⠀⠈⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⣠⠞⠀⠀⣤⣷⣄⠀⢸⡇⠀⠀⠈⠉⠉⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠗⠒⠚⠁⠀⠀⠀⠿⣿⠟⠀⠘⢧⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢉⠀⠀⠄⠙⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠛⠛⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠉⠀⠁⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣤⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⣤⣀⣠⡴⣿⡀
⠀⠀⠀⠀⠀⠀⠀⡴⣆⠀⠀⢸⡇⣿⠀⠀⠀⣀⡴⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠉⠁⣿⣿⣿⣷⡀⠀⠀⢀⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣼⣃⣿⠤⠒⠉⠁⠀⣠⣤⡾⠿⣦⠀⠀⠀⢀⣤⡴⣤⡀⠀⠀⠀⠉⠲⢄⣸⣿⣿⣿⣿⣷⣄⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠸⣿⣇⠀⢀⢿⡄⠀⠀⢸⡏⠀⠈⡇⠀⠀⠀⠀⠀⢀⡿⠿⣿⡿⠿⡿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢀⡀⠸⠋⣿⣿⡄⣠⡄⢠⣄⡙⣶⣾⣇⣠⣶⠀⠀⢀⣾⣼⣿⣤⣾⣭⣽⣥⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠈⠉⠒⣾⣿⣿⣿⠀⠀⠀⠀⢙⣿⣿⣿⠀⠀⠀⠀⠸⠿⠿⠿⡿⠿⢿⠿⠿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡇⠀⠀⠀⠀⠀⠾⠿⡿⡿⠿⠃⠀⠀⠀⠀⠻⡿⡟⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⣜⣈⣧⣀⣀⣀⣀⣀⣀⣠⣇⣣⣄⣀⣀⣀⣀⣀⣤⣇⣧⣀⣀⣀⣀⣀⣀⣀⣤⣇⣀⣈⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀








⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⣠⢞⡥⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⢠⡖⢋⣩⡼⠷⡄⠀⠀⠀⠀⠀⠀⢠⢞⡭⠂⠀⠀⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⡞⢡⠘⣿⡆
⠀⠀⠀⠀⣏⢸⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⣼⠛⣉⣥⠴⠾⡇⠀⠀⠀⠀⠀⠀⡏⡞⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣿⡇
⠀⠀⠀⠀⠹⣌⠆⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢼⣋⠉⣤⠶⡟⠁⠀⠀⠀⠀⠀⠀⣧⢣⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⡧⢸⠀⣿⡇
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣤⣿⣤⣧⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⠀⠀⠀⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⣯⣼⣤⣿⡇
⠀⠀⠀⠀⠀⠈⣳⣶⣶⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣶⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠐⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡿

⠀⠀⠀⠀⠀⣀⢀⣤⢠⣄⢀⡀⠀⠀⠀⠀⠀⠀⢠⠖⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣠⣤⣶⣤⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⣰⣞⠛⣶⡀
⠀⠀⠀⢀⣘⠋⢾⡏⠉⡽⠎⢛⣀⠀⠀⠀⠀⠀⠙⠦⣀⣬⣬⡃⣤⠀⠀⠀⠀⠀⠀⠀⡇⣼⣿⡿⠉⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡙⠿⠾⠏⡁
⠀⠀⠀⠾⠿⠸⡿⠀⠀⢛⡿⠸⠿⠆⠀⠀⠀⠀⠀⢠⣿⣿⡸⢞⣵⢟⣀⠀⠀⠀⠀⠀⣧⣿⣿⡇⠀⠘⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⡸⠁
⠀⠀⠀⢶⣶⠰⣿⡶⢠⣾⡷⢰⣶⠆⠀⠀⠀⠀⠀⠘⣿⣿⣧⣌⠡⠿⡋⠀⠀⠀⠀⠀⠹⣿⣿⣷⡗⣰⣿⣿⠇⠀⠀⠀⠀⠀⣠⠀⣾⣮⣩⣶⡆⢠⡀
⠀⠀⠀⠈⢩⣄⢶⣷⢻⣶⢆⣬⠉⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⢻⡂⢯⡀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣿⡿⠋⠀⠀⠀⠀⠀⢸⣿⠀⠿⠿⠿⠿⠇⢸⣿
⠀⠀⠀⠀⠀⠈⠀⠛⠘⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠈⠑⠻⢦⡀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠘⠛

⠀⠀⠀⣀⠀⢀⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⡀
⠀⠀⠀⣿⣿⣷⣿⠉⠉⠉⠉⢹⣿⠀⠀⠀⠀⠀⠀⢸⠟⢉⣉⠛⣿⠀⢠⢀⠀⠀⠀⠀⠀⣿⣿⠟⠛⠻⣿⣿⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠉⠳⣄
⠀⢀⡄⣿⣿⣿⣿⢰⣀⣠⣶⢾⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⣴⣴⠀⠀⠀⠀⠀⣿⣿⡈⠀⢁⣿⣿⠀⠀⠀⠀⠀⡞⠀⢠⡀⢙⣶⣄⠀⠀⠸⡆
⠀⠘⠇⣿⣿⣿⣿⢸⣿⠶⠶⢸⣿⠀⠀⠀⠀⠀⠀⣿⣶⣷⣾⣷⣿⠀⢿⢿⠀⠀⠀⠀⠀⣿⣿⠟⠛⠛⣿⣿⠀⠀⠀⠀⠀⡇⠀⠀⠸⣿⣿⣿⠀⠀⢀⡇
⠀⠀⠀⣿⣿⣿⣿⠸⣿⠛⢻⢸⣿⠀⠀⠀⠀⠀⠀⠙⠿⣿⢿⠿⠟⠀⢸⢸⠄⠀⠀⠀⠀⣿⣿⣈⣠⣀⣿⣿⠀⠀⠀⠀⠀⠹⣄⠀⠀⠈⠉⠉⡇⢀⡼⠁
⠀⠀⠀⠛⠋⠩⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣼⣄⣀⣠⣎⡞⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⣤⠼⠃⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⡄⠀⠀⠀⠀⢼⠄
⠀⠀⠀⠀⠀⠀⣀⣶⣶⣶⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⠀⠀⠀⢫⣻⠇⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⢀⠰⠚⠋⠁⠸⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠟⠀⠀⠀⣾⣭⣶⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⣤⡤⢸⠩⣭⣭⡭⢻⠠⣤⡀
⠀⠀⠀⠉⠛⠗⣶⣶⣤⣀⡀⡶⠒⢦⠀⠀⠀⠀⠀⠀⢨⡿⢿⡅⠀⣴⡟⠉⢻⣆⠀⠀⠀⣶⡆⠀⢬⡥⠀⢠⣥⠀⠀⠀⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡆
⠀⠀⢠⣶⣶⣴⣬⣿⣿⣿⣿⣿⣾⣿⣷⣦⣄⠀⠀⠀⣿⠀⠀⣿⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⣾⣧⠀⢸⣷⠀⠀⠀⠿⠟⣵⡎⠿⠿⠿⠿⠏⣶⠹⠿⠃
⠀⠀⠘⠿⣟⢙⡿⠿⠿⠿⠿⠿⢿⡟⢙⡿⠿⠀⠀⠀⣿⠀⠀⣿⠀⣿⠀⠀⠀⡟⠀⠀⠀⣿⡇⠀⣿⣿⠀⢸⣿⠀⠀⠀⢸⣿⣮⣴⣿⣿⣿⣿⣷⣭⣾⣿
⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠉⠀⠀⠉⠀⠉⠀⠀⠀⠃⠀⠀⠀⠙⠁⠀⠙⠋⠀⠈⠋⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⢀⡀⠀⣿⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠈⠉⣤⣶⣦⡉⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⣷⣿⣷⡆
⠀⢰⣶⣶⣶⣷⣿⣿⣿⣿⡇⣈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠹⣿⡿⠁
⠀⠀⢻⣿⣿⣻⣿⣻⣿⡁⢤⠈⠁⠀⠀⠀⠀⠀⠀⠐⢤⠤⣿⡧⢤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠿⠿⠿⠿⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠇⠻⠿⠇

⠀⣀⣠⣴⣦⣤⣤⣴⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⡀⠠
⠀⠛⢛⣉⣉⣉⢩⣭⠡⣤⡄⣶⡄⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⢸⣿⢸⣿⠀⣿⡇⣿⡇⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠛⠢⠄
⠀⠀⢸⣿⢸⣿⠀⣿⡇⣿⣇⣿⣇⡀⠀⠀⠀⠀⢸⡇⠀⠀⡀⡄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⢰⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⡇⠠⣾⣿⣿⡆⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠘⠿⣿⣿⡿⠿⢿⣿⣿⡿⠿⠟⠁⠀⠀⣿⣻⣿⣿⣛⣿⣻⡿⢿⣛⢿⣿⡟⠿⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⠿⠶⠿⠿⠿⠷⠾⠿⠛⠃⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣀⠀⣠⣄
⠀⠀⠀⢀⡒⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠲⠛⠓⠂
⠀⠀⠀⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣤⣤⣀⣀⡀
⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡆⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠙
⠀⠀⢰⣿⢻⡖⣶⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⢀⣀⣀⠘⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐
⠀⢀⣸⣿⣼⣧⣿⣼⣇⣴⣿⣦⣤⣀⠀⠀⠀⣀⣿⣿⣿⣿⣴⣿⣿⣿⣿⣙⣀⡀⠀⠀⠀⠒⣁⡐⠀⠀⠀⠀⠀⠀⠼⠾⠆
⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠰⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⡋⠈⡻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡧⠀⢀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣤⣀⣤⣀⣴⡿⣫⣾⡇⠀⣿⣮⡻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡿⠋⣾⣿⣿⡇⠀⣿⣿⣿⡎⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⣫⣷⠀⣿⣿⣿⡇⠀⣿⣿⣿⡇⠐⣮⡻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢫⣾⣿⣿⠀⣿⣿⣿⡇⠀⣿⣿⣿⡇⠀⣿⣿⣮⠻⣷⣄
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⠀⢸⣿⣿⣿⠀⣿⣿⣿⡇⠀⣿⣿⣿⡇⠀⣿⣿⣿⡂⢨⡻⣷⣄
⠀⠀⠀⠀⠴⠿⠿⠿⣿⣿⣿⢿⣿⣿⣿⣿⢿⣿⡿⣫⣴⣿⣶⣾⣿⣿⣷⣶⣾⣿⣿⣷⣶⣾⣿⣿⣶⣶⣿⣿⣿⣶⣶⣷⢬⡻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣷⡾⣫⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣮⣾⣿⡧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⢹⠀⢸⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⡉⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣀⣼⣀⣸⣿⣿⡯⣿⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣄⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⢹⠀⢸⣿⣿⣶⣶⣶⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⣾⣤⣼⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣟⣿⣯⣽⣿⣿⣿⣯⣿⣟⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⣿⣿⣿⣿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⡟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡇⣿⣿⣿⣿⡏⣾⣿⣿⣷⣶⣾⣿⣿⣿⣷⡮⣿⣿⣿⣿⡇⣿⣿⣿⣿





⡿⠿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣻⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠋⠀⠀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡇⠀⠀⣾⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠀⠀⠈⠛⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀
⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄
⠀⠀⠀⠀⢀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄
⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷
⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿
⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠀⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟
⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁
⠀⠀⠀⠀⠀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣤⡀⠀⠀⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡿⠀⠀⢸⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠀⠀⣠⣾⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠀⠀⠀⠀⠐⠓⠶⢦⣤⣀⡀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⢀⣀⣤⡴⠶⠒⠂⠀⠀⠀⠀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⣠⠶⢋⣠⣤⣶⣶⣦⡀⠀⠀⠀⠀⠀⠙⢻⣿⣿⠿⠋⠉⠉⠉⠙⠛⠛⠋⠉⠉⠉⠙⠿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⣤⣶⣶⣤⣀⠉⠳⣤⡀
⠀⠀⠀⣰⡟⠁⠀⠋⠉⠉⠉⢻⣿⣿⣦⡄⠀⠀⣀⣴⣿⡿⠃⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠈⠻⣿⣷⣄⠀⠀⢀⣴⣾⣿⡟⠋⠉⠛⠻⢦⡈⠻⣦
⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⡀⣼⣿⣿⣿⣷⣶⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣿⣿⣿⣷⢀⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠹⣷⡄
⠀⣾⠏⠀⠀⠀⠀⠀⠀⠚⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠳⠀⠀⠀⠀⠀⠀⠹⣿
⢸⡟⠀⠀⣀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢀⠀⠀⢿
⣾⣷⣴⡿⠁⢀⣤⠄⠀⠀⠀⠀⠈⠿⠛⠙⢻⣿⡏⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢸⣿⡟⠉⠙⠏⠀⠈⠙⠣⠀⠀⢤⡀⠀⢻⣶⣾
⠉⠛⢿⣁⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁⢸⣛⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⡃⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⣈⣿⠟
⠀⠀⠘⠿⣿⣧⣤⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠈⠛⢿⡟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⣷⡛⠁⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣾⣿⣿⠟
⠀⠀⠀⠀⠈⣿⣿⣷⣄⣀⣀⡀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⣄⣤⣶⣿⣷⣶⣤⣤⣼⣿⣿⠋
⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣦⠀⠀⠀⣿⣿⣇⠀⠀⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣧⠀⢰⣿⣿⣿⣦⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⣧⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣷⠶⣄⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢾⣇⣀⡤⢼⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⠿⠛⠁⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⡀⠀⠙⠿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢰⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⣼⠃⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢈⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⣨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⣿⣄⣀⣠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⣿⣿⡏⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠰⠄⠀⠀⠀⠁⠀⣤⠀⠀⠀⠁
⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⡌⡇
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣹⢰⠀⠃⡆⠀⢠⣤⣤⡀⠆⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠑⠂⠀⠴⣤⣄⠀⠀⢀⣴⣶⣶⣦⠀⠀⣤⡀⠒⠀⠀⠀⠀⠐⠓⠂⠀⠐⠀⠛⠛⠀⢰⡆⣿⡉⢸⣿⣿⣡⡆⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠈⠙⢯⣽⣽⣿⣛⣁⣀⣉⠋⠐⢲⣶⣶⡀⢀⣀⣀⣀⠀⠀⠀⠈⠉⠙⠉⣩⠄⣾⣿⣿⡦⠀⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠊⠀⠀⠀⠒⠀⠀⠺⠿⠿⠿⠿⠿⠶⠶⠶⠛⠛⠙⠉⠉⠉⠁⠀⠀⠀⠀⠁⠒⠲⠬⢴⣿⣏⣹⣷⣠⣶⣿
⠀⠀⠀⢠⠀⢠⠀⠀⠇⠀⠉⣀⣌⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⣏⣩⣉⢀⡒⠂⣰⠄⠐⠢⠀⠠⠤⠄⢀⢩⣙⡛⠿⣿⣿
⠀⠀⠀⠈⠠⣤⡰⣶⡎⠿⣷⢹⣿⡜⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢐⠒⠂⡮⠤⢴⣀⣠⡂⡀⣼⠃⠉⡸⠁⠑⡒⠚⠀⡤⠆⠀⣀⠇⠀⠀⠉
⠀⠀⠀⠀⠀⠙⡇⠀⢱⠀⠸⣆⠈⢷⡀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⡠⠀⢀⠌⠀⢠⠃⠀⣰⠏⠈⠉⠋⠑⠓⡖⠐⠥⡤⢰⣀⡀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⢿⠀⠘⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠤⢤⣆⣀⣠⠁⠀⣰⡟⠀⠀⡌⠀⠀⡌⠀⠀⢰⠁⠀⠀⡇⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⠁⠀⠀⠀⠀⠈⠻⠿⣶⣶⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⢿⣶⠛⠠⠤⠼⣀⣀⠰⠁⠀⢀⡇⠀⠀⢸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⡖⣾⠖⠤⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡀
⣿⣿⡟⠛⠿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣶⣶⣶⣶⡾⠿⠿⢿⡟⠛⠛⠛⣟⠙⠉⠛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⠿⠿⠿⠛⠛⢿⠛⠉⠉⠉⠉⢳⡀⠀⠀⠙⣄⠀⠀⠹⣄⠀⠀⠘⢦⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠻⣿⡄⠀⠈⢆⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⢳⡀⢀⣀⣈⣦⠤⣤⠼⠶⠶⠒⣞⠛⠉⠉⢯⡻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⣸⣿⡄⠀⠈⢧⣀⣀⣀⣤⠤⠷⠴⠒⢲⠊⠉⠉⠁⠹⡄⠀⠀⠈⢣⡀⠀⠀⠈⢧⡀⠀⠀⠱⡝⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⢀⡼⠃⠸⣿⡖⠋⠉⠉⠀⠈⢧⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠙⣄⠀⠀⠀⠱⣄⣀⣀⣤⠵⠶⠖⢖⠛⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⢠⡾⠁⠀⢀⣾⣿⡄⠀⠀⠀⠀⠀⢳⡀⠀⠀⢀⣈⣧⣠⣠⡤⠤⠞⠒⠛⣏⠉⠉⠀⠱⣄⠀⠀⠀⠈⠣⡀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⣰⠏⠀⠀⢠⣾⣉⣙⣿⣄⣀⣀⣤⡤⠖⠛⠚⠙⢯⠉⠀⠀⠀⠱⡄⠀⠀⠀⠈⢦⡀⠀⠀⠈⢦⠀⠀⢀⣀⣹⠦⠜⢻⣧
⠀⠀⠀⠀⢀⣾⠟⠀⠀⣼⠋⠀⠀⢠⡿⠲⠤⠬⢽⣿⡅⠀⠀⠹⡀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠘⢆⠀⣀⣀⡤⠵⠴⠒⢶⡚⠉⠉⠉⠳⡄⠀⠀⠀⠻⣷⡀
⠀⠀⠀⣠⣿⠋⠀⢀⡼⠃⠀⠀⠀⠈⠉⠙⠛⠶⠶⠾⣿⡆⠀⠀⠹⡄⠀⣀⣀⣤⠷⠴⠖⢲⡚⠉⠉⠀⠈⢣⡀⠀⠀⠀⠳⡄⠀⠀⠀⣘⣦⣤⣴⣶⡿⠇
⠀⠀⢰⣿⣥⣤⣤⡾⠷⠶⠦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠘⣿⡶⠖⠒⠉⠉⠀⠱⡄⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠱⣄⣀⣤⣤⣾⡶⠿⠟⠛⣿⡏⠁
⠀⠀⠈⠉⠉⢹⣿⠛⠛⠓⠶⠶⠦⣬⣍⣉⣛⠳⠶⠶⢤⣼⣿⡄⠀⠀⠀⠀⠀⠙⣆⠀⠀⢀⣀⣨⣧⣤⣶⡶⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠶⠶⢤⣼⣿⣆⠀⣀⣠⣤⣴⣾⠿⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡴⠶⠺⡟⠋⢹⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡏⠉⠛⠛⠿⠷⣶⣤⣄⡀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣤⡴⢶⠞⠛⠛⣷⠀⠀⢸⡇⠀⠀⡇⠀⢸⡇⠀⢠⣿
⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠉⢹⡇⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⢈⣇⣤⣼⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⠀⢀⣠⡟⠀⠀⢸⣿⣿⡿⠿⠛⠋⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⣿⣠⣤⠾⣟⣫⡽⠇⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠘⠃⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⢿⠷⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢤⣚⣯⣿⡧
⠀⠀⠀⠀⠀⠀⣿⡗⠲⢤⣨⣿⣄⣀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⣚⣭⣵⡾⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⢿⣟⡾⢯⣟⡳⢦⣤⣀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⣋⣭⣴⡾⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡛⠶⣬⣙⡳⠮⣭⣛⠶⣭⣿⢿⣿⣀⡀⠀⠀⣿⠀⢀⣠⠤⠖⣋⣩⣴⡶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣬⣙⠳⢶⣽⡷⠿⢀⣸⣿⣤⣭⣉⡓⣿⣊⣩⣴⣶⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢷⣶⣷⣶⡾⠟⠋⠉⠈⠉⠛⠻⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣆⠀⠀⠶⢿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠿⠛⠉⠁⠀⠀⠀⠀⠈⠙⠃⠀⠀⠈⠉⠙⠻⠿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠙⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⡀
⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀
⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷
⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠇
⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⢠⣤⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷
⠀⠀⠀⠹⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⢛⣿⣿⣿⣿⣶⣶⣶⣶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⡀
⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⢻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣦⣄⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣶⣤⣤⣀⣀⣰⣦⣄⡀⠀⠀⠀⣠⣴⣶⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⢿⣿⣿⠿⠛⠀⠀⠘⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⢶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠋⠀⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⡿⠿⠛⠉⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⡟⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠛⠉⠛⠁⠀⠀⠈⠀⠀⢠⣿⣿⠏⠀⠈⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣀⢀⣀⣤⣄⠀⠀⠀⠸⠟⠁⠀⠀⠀⠀⠈⠛⠳
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡈⣿⠘⣿⣿⣿⣷⡀⠀⣇⠀⠀⠀⠀⣄⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⡇⢠⣿⣄⠀⠈⠃⠀⠈⠁⠀⣼⣷⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⡄⠄⠀⠀⢠⣾⣿⣿⠇
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠻⣿⡿⢻⣿⣿⣿⣿⡄⠈⠙⠛⣑⠦⠤⠞⣂⠉⠉⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣧⡀⠀⢠⣿⣷⣶⣾⣿⡆⠀⠀⠀⢸⣿⣿⠀⣀⡀
⠀⠀⠀⠀⢰⣿⡿⢿⣿⣿⣿⣿⢿⣿⡿⠋⠀⠀⠉⡛⠿⢾⣿⢿⣿⣿⣿⠉⣿⡀⣠⣶⣿⡟⢿⣿⣭⠁
⠀⠀⠀⠀⡀⢈⠁⠀⢸⣿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣿⣿⣿⠀⠘⡿⠛⠛⠁⠀⠈⠻⢿⣷
⠀⠀⠀⠀⠉⠁⠀⠀⢸⣿⣿⣿⠀⡀⠚⠀⠀⠁⠀⠛⠀⠈⠀⣴⣿⣿⣿⣆⠀⠁⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⣿⣦⣄⣀⣀⣠⡄⠀⠀⣸⣿⣿⣿⣿⣿⣧⠀⠀⠙⠀⠀⠘⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣿⣦⣠⣿⣿⣿⣿⣿⣿⣿⣆⢠⣾⣶⣴⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⠟⢿⣿⣿⣿⣿⡜⢋⠝⠛⠛⠛⠛⠛⠫⠤⠀⣿⣿⣿⣿⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⡇⠘⣷⣿⣿⣻⡟⣛⠁⠀⠀⠀⢀⡀⢀⣀⠀⠀⠈⢛⡋⢙⣛
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠛⠛⠃⠀⠛⠙⠿⠟⠁⠿⠦⠀⠐⠛⠉⠀⠀⠉⠁⠀⠀⠚⠛⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢗⣻⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢵⢟⣩⢿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣿⠏⣿⣳⣿⣿⣯⣞⣯⣿⢿⡿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠜⠁⠙⠙⠉⣾⠋⡩⣻⣟⢕⠿⣻⢻⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠺⠿⣷⠿⣻⣷⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢊⣠⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣫⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣻⣿⣿⣠⡔⠁⠀⠀⠀⠀⠀⠀⣠⣾⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢿⣿⣻⣿⣾⣿⣟⠕⠀⠀⠀⠀⢀⠀⡺⣫⣽⢟⢽⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⢛⣿⠟⠁⠀⠀⠀⠀⠁⠀⠈⢫⣴⣷⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⢇⣬⢠⡔⠀⠀⠀⠀⠀⠀⢀⣤⠚⠁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣧⣾⣿⣿⣻⣷⠛⠔⠀⠀⠀⠠⢛⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⠿⢿⣿⡿⠃⠀⠀⠀⠀⠺⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣡⣴⡷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠃
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⠀⠀⠀⠀⢰⣶⣶⠀⠀⡀⠤⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡒⠀⣒⠐⡂⢐⡂⢐⠂⢸⣿⣿⠀⠀⠁⠀⢀⠀⡀⢀⠀⡀⢀⠀⡀⢀⠀⡀⢀⠀⡀⢈
⠀⠀⠀⠀⠀⠀⠤⠀⠤⠠⠄⠠⠄⠠⠄⢸⣿⣿⠀⠀⠀⠀⡀⠀⡀⠀⠀⣀⡀⠀⢀⣀⣀⣀⣀⣀⠀⣀⡀
⠀⠀⠀⠀⠠⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠘⠉⠙⠀⠀⠀⠀⠿⠿⠇⠀⢸⡉⢹⠆⠈⠀⠀⠀⠀⠀⠀⠄⠈
⠀⠀⠀⠀⢄⠀⠀⢀⠀⠀⠀⡀⠀⡔⠈⠁⠀⠈⠁⠈⠑⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠈⠈⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠰⠄⠀⠠⠤⠀⠀⠦⠀⠤⠀⠀⠀⠀⠀⠀⠀⠄⠤⠠⠄⠤⠰⠀⠁⠄⠠⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠉⠉⢲⡤⠒⠊⠉⠉⠓⠲⢤⠒⠉⠉⠉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠊⠉⠉⢲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠱⡖⠉⠉⠓⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢸⠀⠀⠘⠃⠀⠀⠀⠀⠛⠀⠀⢸⠂⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠣⡀⠀⢰⢠⠤⡄⡄⠀⠀⢀⠞⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠞⠒⠼⡀⢂⡂⢧⠴⠚⠣⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣈⣉⡉⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢰⡆⠀⠀⢸⣿⢸⣿⠀⠀⠀⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⣿⣴⡆⣿⣿⢸⣿⣿⣶⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠷⣄⣴⣿⣿⣸⣿⣿⡷⠶⠾⢸⣿⣿⣿⣿⣿⣿⣿⣶⣤⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡟⢿⣏⣽⣿⣯⠿⠟⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣰⣿⣼⡟⣿⣩⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣷⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠿⠿⠿⣺⣧⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣹⣿⣻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣯⣷⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠰⣷⡖⣻⣿⡇⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⡇⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢿⡇⣿⣹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⡖⣶⣼⡏⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠹⣿⣶⣯⣿⣙⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡿⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶⣶⢸⣏⣻⣿⣟⣶⣦⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⡶⠛⠸⣿⣿⢹⣿⣿⡷⠶⢶⢸⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⣿⠻⠇⣿⣿⢸⣿⣿⡿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠸⠃⠀⠀⢸⣿⢸⣿⠀⠀⠈⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠒⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠰⠀⠀⢀⡠⠔⡂⠀⠈⠉⠀⠙⠁⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠤⡀⠁⡄
⠀⠀⠀⠀⠀⠀⣀⠀⠈⠁⠀⢀⡀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠆⠄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠈⠁⠀⢄⠀⠀⢀⡠⢴⡏⠀⠀⠀⡖⠁⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠐⠁⠈
⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠠⡔⢊⣩⣴⡾⠀⠀⢀⡜⠀⠀⠀⠀⠀⢀⡴⠊⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⢻⣏⣉⠝⠁⣀⠴⠋⠀⠀⠀⠀⡠⠔⠁⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⠤⠤⠤⣤⠤⠖⠋⠁⠀⠀⠀⡠⠔⠉⢀⡠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠒⠀⢀⠒⠂⠐⠒⠁⣠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠁⢒⠀⠄⠀⠀⠀⢐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⡔⢁⠔⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢎⠐⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠸⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠀⠀⠠⠤⠤⠴⠖⠒⠒⢲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠇⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡎⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈⡆⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰
⠀⠀⠀⠀⠀⠀⢀⠠⠐⠉⠒⢃⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠴⠆
⠀⡠⠖⠒⠒⠶⣅⠀⠀⠀⡠⢼⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⣠⡤⠤⢤⣀
⢸⠀⠀⠀⠀⠀⠈⠖⣈⠥⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠪⢁⠢⢄⡞⠁⠀⠀⠀⠈⢣
⠈⠒⠒⠒⠒⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠤⠤⠤⠤⠤⠜

⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⣾⣿⣿⠃⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠋⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣿⡟⠀⠀⠀⠀⢀⣾⣿⡶
⠀⠀⠀⠀⣿⣿⣿⣿⠋⠀⠀⠀⢠⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠈⠉⠉⣿⡿⠃⠀⠀⠀⠀⠘⠛⣿⣿⠟⠀⠀⠀⠀⠀⠸⢿⣿⠋
⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⡿⠃
⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⢀⣿⣿⠏⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⢀⣼⡿⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⣴⡿
⠀⠀⠀⠀⢠⣾⣿⡇⠀⠀⠀⠀⠀⢀⣼⣿⣿⣀⡀⠀⠀⠀⢰⣿⣿⡗⠀⠀⠀⠀⢀⣾⣿⣇⣀⣀
⠀⠀⠀⠀⠛⢿⣿⣷⣦⠀⠀⠀⠀⠘⣻⣿⣿⣟⡀⠀⠀⠀⢀⣿⣿⡀⠀⠀⠀⣠⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⠀⠀⠘⢻⣿⣿⠟⠁⠀⠀⠀⠈⢹⡟⠁⠀⠀⠀⠉⠁⣸⣿⡟⠁
⠀⠀⠀⠀⠀⣾⠟⠀⠀⠀⠀⠀⠀⠀⣸⡿⠋⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⣿⠏
⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⣤⠄⠀⢀⡴⠋⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⠖⠶⢶⡞⠃⣴⠟⠁⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⣛⣻⣧⡞⠁⠀⠀⠀⣀⣀⣀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⢀⡽⠋⠀⠀⠀⠀⡾⠉⣿⠉⢳⡄⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠏⠀⠀⠀⠀⠀⠸⡟⠛⣿⠛⢻⡇⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠘⡇⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣽⡦
⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⣀⣀⣿⠀⠀⠀⠀⢠⡶⠶⢶⡶⠶⢶⡆⠀⠀⠀⠀⠀⠀⣶⠶⠶⡶⠶⢶⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢀⣾⠉⠙⢷⣰⣏⠀⠙⣿⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⣏⠀⢀⣇⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⢰⠏⠉⢁⣀⣼⣯⣉⡙⠂⣿⠀⠀⠀⠀⢠⡟⠛⢻⡟⠛⢻⡇⠀⠀⠀⠀⠀⠀⣿⠛⠛⡟⠛⢻⡇⠀⠀⠀⠀⣿⠻⢦⣀
⠀⠀⠀⠘⢳⡞⠋⠉⠀⠀⠉⠉⠀⣿⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⢾⡀⠀⠀⢀⣤⠀⠀⠀⣿⠀⠀⠀⠀⠘⠷⠶⠾⠷⠶⠾⠃⠀⠀⠀⠀⠀⠀⠿⠶⠾⠿⠶⠞⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠙⠂
⠀⠀⠀⠀⠈⠛⠶⠚⢻⣇⣠⡆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤⣤
⠀⠀⠀⠀⠀⠀⠀⢰⠟⠉⣩⠻⠦⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠀⠛⠲⣶
⠀⠀⠀⠀⠀⠀⠀⠻⣤⡴⣯⣀⢀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⣉⣙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⣿⣿⠿⠻⡿⠛⠿⢿⡿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⢀⣤⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⠀⣾⣿⣿⣿⣿⡟⣠⣾⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡇⣼⣿⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⢠⣿⡿⠿⠛⠛⠿⢃⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⠈⣿⠿⠟⢋⣉⣤⣤⣶⣾⣿⣷⡀⢺⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⣿⣿⣿⣿⠇⠹⣿⣿⣿⣿⣿⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣈⣉⣙⢋⣠⣄⡙⠛⠻⠿⠟⠀⠹⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣦⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣧⠀⣉⣉⣉⣙⣛⠁⣼⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⠛⠛⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⠿⠿⠿⠻⠿⢿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡛⣛⣛⡛⢛⣻⣿⣿⠿⠿⢿⡿⠿⣿⣿⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡠⠶⠒⠛⠉⠉⠉⠉⠛⠒⠶⢤⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⠋
⠀⠀⠀⠀⣰⠋⡠⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠟⠀⢴⡆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⠆
⠀⠀⠀⣰⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢀⡄⠀⠔⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉
⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⣠⠉⢠⡀⠀⠀⠠⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⢂⡀⡀
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⢠⡀⡀⠀⠀⡰⡛⠁⠀⠀⠀⠀⠀⠀⢘⡄⣀⣡
⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⣀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢆⠀⢁⠜⡝⠁⠀⠀⠀⠀⠀⢀⠤⠚⠛⠛⠋⠀⠀⠀⠀⡀⠀⠀⠱
⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⣿⢠⠃⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠑⠚⠋⠀⠀⠀⠐⠀⠀⠂⠁⠀⠀⠀⡀⠠⠀⠀⠒⠂⠀⠀⠀⡔
⠀⠀⠀⠀⠈⢦⠀⠀⠀⢀⡇⡞⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⢀⣤⣴⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⣜⠣⢄⡀⣼⣸⠁⠀⠁⣀⡖⠋⠁⠀⠀⠀⡴⠋⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⢀⠤⠂⠈⠁⠀⠁
⠀⠀⠀⠀⠀⣍⠳⠄⢀⠀⠈⠉⠉⠉⢑⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠚⡡⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⢸
⠀⠀⠀⠀⠈⠆⠉⠒⠢⠄⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⣀⡤⠔⠂⠀⡄⢪⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠑⠢⠤⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢢⣄⣀⣀⣀⠀⢀⠔⠒⠀⠀⠀⠀⠀⢐⣉⣶⠆⠀⣀⢬⠇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠁⠀⠀⠀⠀⠀⣠⠴⠛⠉⢀⡤⣚⡭⠟⠀⠀⠀⣰⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⡠⢚⡥⠋⠁⠀⠀⠀⠀⠐⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢢⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⠛⣓⡶⠀⠀⠈⠑⣣⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠐⢁⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣸⠞⠁⠀⠀⢀⠔⠋⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡰⠂⠀⠰⠤⠀⠀⠠⠂⠀⠀⠀⣠⠛⡀
⠀⠀⠀⠀⠚⠢⠀⠈⠁⠀⢀⡴⠊⠁⠀⠀⡠⠚⢁⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠁⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠯⣤⡄⠀⠀⢯⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠙⠲⠶⠶⠖⠒⠋⠁⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⡴⠋⠀⠀⠀⣀⠬⠆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⡠⠔⠊⠁⢰⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⣀⠀⠀⡀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⢀⠞⠁⡠⠔⠉⠀⢀⡴⠒⠁⠀⠀⠀⠀⠀⠀⢰⠀⠈⠂⠐⠋⢹⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⣰⣫⠖⠉⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⣀⣀⠀⢨⠀⠀⠠⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⢀⡾⠋⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⢆⢀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠉⡄⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⢲⡀⠀⠂⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁
⠀⠀⠀⠀⠱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠤⠤⠤⠴⠖⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠂⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠈⠀⠀⠀⠀⢀⣰⡿⠙⠲⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠚⠆⠀⠀⠄⠤⠶⣬⣭⠗⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠉⠁⠈⢙⣷⡏⠀⣐⣿⠌⡖⠀⠀⡆⠠⠀⠀⠀⠀⠀⠀⠀⠀⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠛⠧⣈⣸⡟⢠⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠆⠀⠀⠀⠀⠀⠀⠈⠛⠛⠆⠀⣴⠟⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠮⠀⠀⠀⠁⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⢄⠀⠀⣰⠞⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⡀⠁⠠⠆⣠⠀⠄⠀⠀⠀⠀⠀⠀⠀⡜⠀⣾⠟⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⢰⣲⣴⣶⣤⣘⠄⠀⠀⠀⠀⠀⢡⣷⢸⡿⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣾⡵⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡥⠄⠀⠈⠀⠀⠉⠽⣛⣫⡅⠀⡀⠀⠀⢠⠈⠀⠘⣸⣡⢏⡀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⣼⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠐⠢⠑⠀⠀⣀⠨⠁⠀⠀⠀⠀⣌⠍⠀⠀⠀⢾⣯⡿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⣸⣿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⣥⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠂⣿⠟⠁⠊⠁⡀⠄⠀⠀⠀⠀⠀⣠⣿⣿⣷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠓⠲⢄⠈⠓⠀⠀⠀⠀⠀⠐⠂⢢⠀⠀⠀⠀⢠⡕⣳⠏⠀⠀⣠⡰⠁⢇⠀⢀⡴⢠⣾⣿⣿⣿⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⡀⠀⠉⠒⣠⢞⣀⠘⣆⣣⣷⡼⢄⠄⠀⠄⢡⣶⣿⠀⠀⠀⠁⠀⠀⠈⠇⠈⣰⣿⣿⣾⡿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠐⠀⠭⢉⣰⠟⠋⣹⣷⣿⣿⣿⣿⣡⣤⣴⠾⠻⣿⡟⠀⠈⠀⠀⠀⢀⣤⣾⣿⣿⣿⢁⠇⢆⠃
⠀⠀⠀⠀⠀⢀⠠⠐⣀⣛⠉⠉⠀⠀⠁⣠⣾⣿⣿⡿⠟⠋⠉⠉⠀⠀⠀⠀⠈⠓⠀⠀⢀⣠⣾⡿⢰⡿⣿⣿⠟⠈⢰
⢠⡄⢀⣀⡤⠤⠖⠚⠉⢩⠤⠀⣁⣴⢾⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⣶⣻⣿⣿⡇⣿⡳⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠄⣂⣥⣶⠛⣩⣥⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣿⡇⣷⠀⠈⠀⠀⠀⠘
⠀⠀⣀⣄⣤⡤⠶⠛⠉⠐⡄⢀⠀⡩⢽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇⣿⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠸⠭⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠔⢀⠂⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠀⣤⠀⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃⠀⠀⠀⠘⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢿⣿⣿⣆⠀⠀⠀⠀⠈⠛⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠻⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠉⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⣿⠀⠀⠀⠀⠹⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⢿⡁⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠈⣿⠀⠀⠀⠀⢀⣿⡇⠹⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢸⡏⠹⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⡤⠞⠁⠀⠀⢀⣴⡿⠟⠀⠀⠘⣿⣿⣧⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢹⣄⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⣿⠁⢀⠜⠀⢀⣤⣤⣾⡟⠉⠀⠀⠀⠀⠀⠈⣿⣿⣷⡀⠀⠀⢠⠞⠋⠀⠀⠀⠀⠀⠀⠘⣷⣄⡀
⠀⠀⠀⣠⣤⣤⣤⣴⣶⡾⠟⠋⠁⠀⢸⡟⠀⠁⠀⠚⠁⠀⠀⢸⣧⠀⠀⠀⠀⠀⢀⣾⣿⡿⣿⣧⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⣇⣀⣀⡀
⠀⠀⢰⠟⠀⠀⠀⣠⡟⠁⠀⠀⠀⠠⣿⠁⠀⠀⠀⠀⠀⣠⣴⡿⠋⠀⠀⠀⠀⢠⣾⣿⡟⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⠈⠁⠈⢳
⠀⠀⡻⠀⠀⠀⣾⠁⠀⠀⠀⠀⣠⡤⡼⣷⣴⣶⡶⠿⠛⠋⠁⠀⠀⠀⠀⠀⣠⣿⣿⠏⠀⠀⠀⢻⣷⣴⣶⣶⠶⠶⠶⣦⣀⠀⠀⠀⠀⠈⠀⠀⠀⢸⠃
⡠⠞⠁⠀⠀⠀⢻⠀⠀⠀⢀⠞⠁⢀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣿⣿⠃⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠈⠛⣷⠀⠀⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠘⠀⠀⠎⠀⠀⢀⣀⣠⣤⡶⠶⠟⣿⡟⠋⠉⠙⣿⣿⣆⠀⠀⠀⠀⠙⠷⠶⣤⡀⠀⠀⠀⠀⢈⡿
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⣴⠏⠀⠀⢠⣿⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠘⢿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠈⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠈⣿⣿⣿⣷⣦⣄⣀⣀⣀⣀⣤⣤⠤⠤⢄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢹⣿⣇⠈⠉⠛⠿⢿⣧⣄⠀⠀⠓⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⠀⠀⠀⠀⠀⠀⣰⠚⠁⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠉⠀⠀⠀⠀⠀⢀⡸⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⣇⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⠀⠀⠀⠈⠿⢶⣶⣤⣤⣤⣄⣀⣤⣤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠉⠉⢠⡞⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⠀⢻⣆⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠟⣫⣿⣿⠟⠀⠀⣿⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠋⠁⣠⣾⣿⣿⠁⠀⠀⠀⢹⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⣾⣿⠟⠁⢻⣇⠀⠀⠀⠈⠻⣆⣠⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡼⠁⠀⠀⣴⣿⠟⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠉⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠘⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⣸⣿⠇⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠻⣆⠀⠀⠀⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠙⠦⣀⠀⠀⠀⠀⠻⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⠞⢻⣿⠀⠀⠀⠀⠀⢽⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢻⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠷⠴⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃

⠀⠀⠀⠀⠀⠀⣀⢠⠔⠒⠢⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡠⠊⠉⠑⢦⢀⡀
⠀⠀⠀⡀⠔⠉⠀⠁⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠁⠀⠂⠤⠤⢄⠀⠀⠀⠀⠀⠀⠘⠂⠁⠢⠄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠢⠤⠤⡀⠀⠀⠀⠀⠀⠀⠑⠀⠐⠤⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠁⠑⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⠁⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⡆⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⢀⣀⣀⣀⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⡖⠒⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣒⣒⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⣀⣠
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⢀⣀⠀⣀⡀⢀⣀⠀⣀⣀⠀⣀⡀⢠⣶⣶⣶⠀⠀⠀⠀⠀⣃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡆
⠀⠀⠀⢈⣉⠀⡉⡁⢈⣉⡁⣉⣉⢀⣉⡁⢸⣿⣿⣿⠀⡇⠀⡇⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠃
⠀⠀⠀⢨⢨⠀⣍⡅⢠⣩⡄⣨⣥⠀⡍⡄⢸⣷⣶⣾⢀⠀⠀⡇⠀⠀⠀⣒⣒⡆⠀⣖⣒⡂⠀⢰⣒⣲⠀⢰⣒⣒⠀⠀⣒⣒⡆⠀⡇⠀⠀⠀⠂
⠀⠀⠀⠤⠠⠀⠄⠄⠠⠠⠄⠤⠤⠠⠄⠄⢸⣗⣒⣺⠸⠀⠀⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⡀⢀⡇⠀⠀⠀⢰
⠀⠀⠀⠐⠰⠀⠖⠆⠰⠶⠂⠲⠶⠐⠂⠆⠸⠗⠒⠺⣄⠀⠀⡇⠀⠀⣗⣒⣚⡇⠀⠀⣼⠩⠩⢳⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠸
⡔⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⣑⣄⣀⣀⣀⣄⣀⣀⣇⣀⠀⠁⠀⠀⠁⠀⠀⠹⡆⠀⠟⠀⠀⡇⠐⡁⢸⢐⡉⢹⢸⡉⢱
⠑⣤⠤⠤⠤⠤⣤⠤⠤⠤⢤⣤⠤⢼⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⠀⣀⣀⣀⣀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠁⠁⠀⠁⠁⠀⠁⠁
⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⢸⠀⢈⠈⡇⣈⢈⡇⣈⢈⡇⢸⠀⠀⠀⢀⡀⠀⠀⠀⡄⠀⠀⠀⠀⡇
⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠙⠋⠀⠀⢀⣇⣠⣀⠀⠀⡇
⣴⣿⣷⣶⣶⣶⣿⣶⣶⣶⣾⣿⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣾⣶⣶⣶⣶⣶⣷⣶⣶⣾⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⢿⡿⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢻⠳⢤⣖⠉⠀⠠⡇⠀⠉⣲⠤⠚⡟⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠀⠀⢸⠀⠀⠈⢱⠀⠈⡇⠀⡏⠀⠀⠀⡇⠀⣤⣀⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⡇⠀⣿⣿⣿⣦⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⣇⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠛⠢⣄⡀⢸⠀⢀⡇⠀⡇⠀⣀⠼⠿⠧⡈⠙⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠔⠊⠀⠀⠀⠀⠀⢉⠿⠀⣨⣏⠰⣷⣉⠀⠀⠀⠀⠀⠑⠠⢄⠁⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⢁⠀⠀⠀⠀⢀⡠⠔⠊⠀⣠⠔⠻⠟⠢⣄⠈⠑⠢⣄⡀⠀⠀⠀⠀⡨⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠂⠤⠒⠁⢀⡠⠖⠉⠀⣀⢴⣦⣀⠀⠉⠲⢤⣀⠉⠲⠤⠔⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠄⠊⠁⢀⡠⠔⠈⠀⠐⡇⠈⠙⠢⢄⡀⠈⠑⠦⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⡉⠀⠀⢠⠂⠁⠀⠀⠀⠀⣴⣷⡀⠀⠀⠀⠉⠲⡄⠀⠀⢙⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢼⠀⠀⠀⡠⠔⠊⠉⠉⠙⠢⢄⡀⠀⠀⡧⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⢀⠠⠐⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⣠⣶⣾⣿⠿⠛⠉⠀⠀⠀⠀⠉⢲⣦⣤⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⢀⡴⠋⠁⠀⠀⢀⣉⣙⣻⣷⣶⣦⣤⠀⠀⠀⠀⠀⠀⠙⢢⡹⣄⡀
⠀⢀⣤⠤⠖⠋⠉⠉⠉⢉⡟⠋⠀⠀⠀⠀⠀⠈⠙⠋⢻⣿⣿⣿⣿⡿⠇⠀⡐⢀⠀⠀⠀⠀⠀⠙⢲⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢻⡿⠛⠛⠋⠐⠛⠛⠷⢿⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣷⡀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠀⠀⠀⠀⠀⠀⢣⠀⠀⡠⢣⠁
⠀⠀⠀⠀⠀⠀⠘⣇⠐⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣴⣄⡀⠀⠀⠀⠉⣾⣿⣷⣶⣶⣆⠀⠀⢀⡡⠊⡰⠁⠀⠀⠀⠀⡼⠇
⠀⠀⠀⠀⠀⠀⠀⣸⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡀⢀⡆⠁⠒⠿⠿⠿⠿⠿⠷⠚⠁⡠⠊⠀⠀⠀⠀⢀⠜⡜
⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣤⣥⣤⣤⣀⣀⣀⣀⣤⣤⡖⠉⢀⡀⠀⠀⠠⠒⣁⠞
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⣀⠂⠀⠀⠀⠀⢀⠀⢸⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⣿⣿⣿⣾⡿⣻⣭⣠⡤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣷⡄⠀⠀⠀⠀⠀⣾⣿⣿⠟⠉⠀⠈⠉⠉⠉⠀⠀⠀⢠⣿⣿⡿⠛⠁⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇⠀⠀⠀⣠⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠁⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠋⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⠽⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠂⠁⠀⠀⠀⠀⠀⡄⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣍⠀⠀⠀⠀⠀⠀⠀⢷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠅⠀⠀⠀⠀⢀⢜⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣦⣄⣐⣉⣥⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⠿⠿⠟⠋

⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⣿⣿⣿⡀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠂
⢹⣿⣿⣷⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⠏
⠘⣿⣿⣿⣷⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠛⠁⠀⢀⣠⣴⣾⣿⣿⣿⣿⡿⠃
⠀⠹⣿⣿⣿⣷⣄⠀⠙⢿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠹⣿⣿⣿⣿⣷⣤⣀⠈⠉⢁⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣶⣶⣭⣭⣙⣛⣛⣛⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⣿⣿⣿⣿⣿⢡⣶⣬⡻⣿⠀⠀⣴⣿⣿⣿⣷⡄⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠉⠀⣿⣿⣿⣿⣷⠘⣿⣿⣷⡜⠀⠸⣿⣿⣿⣿⣿⡟⠀⠀⣠⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⠹⣿⣿⣿⣦⣄⠙⠿⣿⡿⠟⠁⣠⣾⣿⣿⣿⠃
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⢿⣿⣿⣷⣌⠻⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣦⡹⣿⣿⣿⣷⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡇⣿⣿⠟⣋⡙⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣶⣤⣄⡀⠘⠿⣿⣿⣿⠟⠁⣫⣴⣿⣿⠇⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⡿⣫⣾⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⡙⠿⣿⣿⣿⣿⣿⣿⣟⢡⣾⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠙⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡟⠀⢸⣿⣿⣿⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡇⠀⢸⣿⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠈⢿⣿⡇⠀⠀⠀⠀⢹⣿⣿⡏⣿⣿⡏⠀⣿⣿⣿⠇⠀⠀⠿⣿⠇⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⡇⠀⢿⡿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⢶⡋⠉⠳⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠲⠶⠃⠀⠀⡀⢀⢄⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠀⠀⠀⠸⡉⠊⠀⠀⠀⠀⠀⠎⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢠⢒⣒⣦⠀⠀⠘⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠭⠍⢶⠀⠀⠀⠂⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⣧⣀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⢀⣼⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⣧⣄⡀⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⣀⣸⡀⣄⣴⡇⠀⠀⠀⣧⢠⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢸⠋⠁⠀⠙⠶⠶⠚⠉⠉⠙⠓⡧⠥⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠷⡿⣄⣀⣀⡼⠀⠀⠀⠀⠀⠀⢀⠀⠄⢒⡥⠶⠊⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠉⠓⠲⠬⣅⣒⠀⠄⢤⠰⣂⣩⠴⠒⠋⡡⡀⢐⢰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠨⢉⢁⠒⠶⠤⢄⣉⠙⢺⠊⢁⡀⢰⢒⠀⡇⡇⠨⠘⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⣉⠑⠒⠰⠬⠇⢸⠀⠡⠄⠸⠈⠀⠁⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⡿⠿⠛⠉⠁⠙⠲⢄⡀⠀⢰⠀⠀⠀⠀⢀⠀⠄⠂⢀⢄⢸
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡿⠻⠛⠋⠉⠁⠄⠀⠀⠀⠀⠀⠀⠀⠈⢒⣼⡀⠠⠐⠀⣀⠀⡆⠂⠐⠀⢸
⠀⠀⠀⠀⠀⡆⠉⠉⢡⡄⣀⡐⠢⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠊⠁⢸⠀⡤⢠⢸⠰⠀⠅⠀⠀⠀⣸
⠀⠀⠀⠀⠀⡇⠀⠀⠸⢵⣒⣸⡇⠀⠑⠦⣀⡠⠴⠒⠉⠁⠀⠀⠀⢠⡦⢸⠀⠋⠙⠀⠀⠀⢀⣠⠔⠋⠁
⠀⠀⠀⠀⠀⡇⢠⠤⢀⡀⠀⠈⠀⠀⠀⠀⡍⠀⠀⠀⢠⡄⠀⠯⠀⠈⠀⢸⠀⠀⠀⢀⡤⠖⠉
⠀⠀⠀⠀⠀⡇⠀⠂⠠⠄⡉⠐⠀⠠⣄⠀⡗⠰⡬⠀⠈⠁⠀⠀⠀⠀⠀⢸⣠⠴⠚⠁
⠀⠀⠀⠀⠀⢧⣈⠀⠀⠠⢀⠁⠀⠀⢼⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⢄⡈⠀⠀⢼⠀⡇⠀⠀⠀⢀⣠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢬⡀⣃⣀⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣦⣷⣶⣤⣶⣴⣯⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⠳⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠐⢀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠈⠀⠀⠀⢕⢆⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠣⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⢔⠁⠀⠀⠀⣀⡀⠀⠀⠑⢄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠡⠀⠠⡀⢀⠔⠁⠀⠀⣴⡿⠛⠛⢷⡄⠀⠀⠀⠌⠂
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠈⡣⡀⠀⠀⠀⣿⡀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠨⡢⢀
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠔⠁⠀⠀⠀⠀⠀⠀⠈⠢⡠⠀⠀⠙⠿⠶⠾⠟⠁⠀⠀⠀⠀⠀⠀⠈⠪⠅

⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⠛⠛⠛⢻⠀⠻⣿⣿⣿⣿⣿⣿⡇⠀⠀⢽⠛⠛⠛⠛⠛⠛⠛⠛⠛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠒⠒⠒⠒⠒⠒⠒⢿⠀⠀⣿⣟⣟⣻⣳⢹⡇⠀⠀⢸⠒⠒⠒⠒⠒⠒⠒⠒⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣿⡧⡧⢼⢼⣻⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢠⠄⠐⠘⠓⣦⣤⣤⣤⣤⣤⣤⣿⠀⠀⣿⣗⣗⣺⣺⣿⡇⠀⢀⣼⣦⣤⣤⣤⣤⣤⣤⣤⣴⣧⠄⠂⠤⠠⡀
⠀⠀⠀⠀⠀⢸⡀⢀⠀⢀⣸⠔⠂⠂⡀⠀⠀⠈⠀⠀⣗⠈⠉⠉⠉⣿⡇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠉⠀⠀⠀⠀⣸⠂
⠀⠀⠀⠀⠀⢰⢾⠭⠭⢽⠶⢶⠶⣶⠳⠶⠶⠴⠀⣀⣧⣀⣀⣀⣀⣿⣇⠀⠶⠶⠶⠶⠦⠶⣤⣤⡶⠶⠶⠶⢿⠭⠭⢽⠶⠆
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠈⠉⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠈⠓⠚⠀⠀⠀⠀⠈⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⠄⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⡶⠒⠛⠛⠙⠉⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢫⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢣⠏
⠀⠀⠀⢠⡇⡰⣊⠩⢀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⢀⡟
⠀⠀⠀⢸⡇⠁⠁⠀⠙⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢸⡇⠀⢀⣀⣀⡀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠘⠒⠒⠲⢦⡄
⠀⠀⠀⢸⠁⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠘⠛⠛⠉⠉⣹⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠉⣹⠇⠀⠀⠀⡜⠀⣀⣀⡀⠀⠀⠀⣴⠋
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠛⢉⡟⠀⠀⠀⠀⣼⠃⠀⠀⢀⡀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣯⣤⣄⠀⠀⠀⢲⡏⠀⠀⠀⠀⠉⠉⢉⡽⠃⢀⢀⡼⠁
⠀⠀⠀⣿⣀⣠⣤⠀⠀⠀⢀⡞⠀⠀⠀⠀⢠⣏⣴⡶⠟⢋⡏⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣢⡟
⠀⠀⠀⠀⠀⠀⡿⠀⠀⢀⡾⠁⠀⠀⠀⠀⠘⠋⠁⠀⠀⢸⡇⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣾⣁⣀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠟⢛⡿
⠀⠀⠀⠀⠀⠰⡇⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⢠⠏⠀⠀⠀⠀⠿⠛⠒⡖⠀⢠⡟
⠀⠀⠀⠀⠀⢸⡧⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⡤⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢸⡇⣰⠟
⠀⠀⠀⠀⠀⢸⡇⢡⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⠋
⠀⠀⠀⠀⠀⢸⡇⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡞⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠃
⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡶⢶⡶⠶⠶⣦⠀⠀⠀⠀⠀⠀⠐⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠠⡇⢷⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⡴⠶⠖⠒⡒⣲⠆
⠀⠀⠀⠀⠀⢰⣶⠛⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⢀⡇⠈⢣⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⡶⠛⠁⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⢣⠀⠀⠀⠀⠀⡾⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⣠⡏⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⢠⡶⠞⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⡞⠉⠀⠀⢀⢂⡴⠛⠀⠀⠀⠀⠀⡇⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣰⢏⠀⠀⠀⠀⢠⣿⣃⣀⣀⡀
⠀⠀⠀⠀⣿⡄⠀⠀⠀⣠⣭⠇⠀⠀⠀⠘⣇⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠐⠉⠉⠉⣩⠞⠁
⠀⠀⠀⠀⠸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣧⣤⡄⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠠⣞⠃⠀⠀⣈⣷⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠓⢲⠀⠀⠀⠀⠀⣸⠀⢀⡀⠀⢠⠞⠀⠀⠀⢰⡯⠶⠖⠚⣿⠀⠀⠁⣴⠋
⠀⠀⠀⠀⠹⣆⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡜⡀⢸⠀⠀⠀⠀⠀⠙⠋⢹⡟⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠹⣆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢸⡄⠀⠀⠀⠀⠀⢀⡟⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠹⣆⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣽⡇⠀⠀⠀⠀⠀⢸⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠳⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⣀⡀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠞⠉⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣠⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⢀⡄⣀⡽⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡴⠋⠀⠀⠀⠠⣀⣤⡌⠄⠀⠀⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢹⡇⠶⣶⠀⣿⠀⠀⠀⠀⣿⠰⡶⡦⢸⡏⠀⠀⠀⢐⡂⢰⢀⠠⢠⣤⣐⡂⠀⠀⠀⡏⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢨⣮⠀⣿⠀⠀⠀⠀⣿⠀⡇⣥⢸⡇⠀⠀⠀⣈⣈⠸⠘⠙⠸⠙⣁⣂⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠒⠒⠀⠿⠀⠀⠀⠀⠿⠐⠒⠒⠸⠇⠀⠀⠀⠨⠁⢰⠠⢆⢴⢤⠍⠅⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢦⡀⠀⠀⠀⠀⠀⠀⠘⠐⠘⠈⠛⠙⠉⠁⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠁⠀⠙⢯⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⡇⠀⠀⠀⢠⠀⢹⡏⠈⠁⠀⠀⠀⣤⣤⣤⡄⠈⠉⠉⠀⣿⡇⠀⡄⡄⡄⠀⣿⡇⠀⠀⠀⢸⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⠜⠘⢨⠃⢸⡇⠀⠀⠘⠛⠀⣿⣿⣿⡇⠀⠀⠀⠀⣿⡇⠃⡂⡆⡃⠃⣿⡇⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠌⢈⠩⠁⢸⠇⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⢿⡇⡁⠍⠌⡅⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠈⠈⠰⠁⢸⣦⣀⣀⡀⠀⠀⣿⣿⣿⡇⠀⣀⣀⣀⣼⡇⠁⠍⠏⠇⡁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠼⠿⠉⠉⠀⠀⠠⠿⠿⠿⠇⠈⠉⠉⠁⠿⠇⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠐








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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠐⠫⠊
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡤⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠚⠋
⠀⠀⠀⠀⠀⠀⠀⣰⡟⢀⣀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠆⠀⠀⠀⠀⣰⣿⣿⡶⠛⠀⠀⢀⡄⣀
⠀⠀⠀⠀⠀⠀⠼⢿⣿⡿⠋⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⢈⡽⠟⠁⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠠⠎
⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⢀⠐⡁⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠁⠌⠁⠀⠀⠀⠀⠀⠐⠁⢠⢌⠁⠀⠀⠀⠀⠀⠀⠐⠐⠒⠀⠀⠀⠀⠀⢃⠔⠃
⠀⠀⠀⠀⢀⢀⡘⠂⡰⠀⠀⠀⠀⠈⠀⡪⠐⠁⠀⠀⠀⠀⠈⣱⠃⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⢀⠔⠁⠈⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⢴⡩⠇⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⡠⠏⠀⠀⠀⠀⠀⠀⠀⠚⠋
⠀⠀⠀⠀⠀⠠⣸⣁⣀⡀⠀⠀⠀⡠⠋⢠⣎⠁⠀⠀⠀⣠⠊⢘⡂⠖⠀⠀⠀⠀⢠⠎⣠⠄
⠀⠀⠀⠀⠰⢓⢖⡴⠋⠀⠀⠀⠈⣩⡿⠒⠁⠀⠀⠀⠀⣩⠏⠋⠀⠀⠀⠀⠀⠀⡠⠂⠁
⠀⠀⠀⠀⣰⠟⠉⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠰⠎
⠀⠀⠀⠈

⠀⠀⠀⠀⠬⠀⠀⠀⣀⡀
⠀⠀⢠⣄⣾⣦⡀⣠⠤⡹⠻⣆
⠀⢀⣠⣼⣿⢿⠄⠂⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁
⠖⠓⠘⠛⣿⢰⡄⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢈⣧⣿⣞⣋⠉⠈⠉⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠐⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣤⠾⢻⡙⣿⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣽⣿⡃⠉⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸
⠁⠀⠸⢳⣿⢳⡦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠚⢙⢹⣧⡄⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠑⠒⢰⣾⣖
⠀⣀⣴⣶⣿⠻⠇⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢾⠯⠅⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢈⣽⡟⠙⠢
⠛⠁⢘⠹⣿⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠰⢸⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠘⠀⠉⠨⣿⣿⣤⡀
⠀⠀⡝⢂⣿⠄⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⢴⣾⣟⠇⠀⠙⠆⠀⠀⡠⠖⡀⣠⣤⠀⠀⠀⠀⢀⣤⢤⣾⡶⠀⠈
⠀⠀⡄⠘⣿⠈⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢄⢹⡏⠃⠀⠀⠀⠀⠀⠀⣀⡀⣹⡿⠲⢄⡀⠀⠁⠀⢘⣽⡿⠖⡤
⠀⠰⠈⢤⣿⠃⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⡇⠃⠀⠀⠀⠀⠀⠘⠉⠉⢿⣿⣤⡀⠀⠀⠀⠀⢀⢸⡇⠂
⠀⠐⠀⢸⣿⡂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⡇⠤⠀⠀⠀⠀⠀⠀⡠⠀⢺⡏⠀⠈⠀⠀⠀⠀⡀⢼⡇
⠀⠁⠈⣺⢿⠉⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣸⡇⢈⡀⠀⠀⠀⠀⠀⠀⠀⢾⡟⡉⠂⠀⠀⠀⠀⠁⣸⡇⠰
⢸⣳⣄⢼⣾⡧⣀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡀⣭⣧⡀⠂⠀⠀⠀⠀⠀⠀⠀⢊⡇⠀⠀⠀⠀⠀⠠⠁⣼⡇⢀⡄
⣻⣀⣈⣻⣻⣃⣉⣹⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣏⣈⣿⣗⣉⣵⣀⣀⣀⣀⣀⣀⣀⣠⣇⣀⣀⣀⣀⣀⣸⣉⣻⣟⣉⣇
⣿⣿⠋⠛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣽⣯⣿⢷⡄⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣄⢀⣀⣴⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⣿⣿⡏⢘⣿⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣿⣿⣧⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣻⣿⣋⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⢿⡿⢿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⣾⣿⣿⣿⡆⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠾⡿⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠙⠿⡿⠟⠁⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠒⡗⠂⣠⣾⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣦⣼⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠀⠀⠰⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣽⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠴⠖⠓⠛⠛⢿⣿⣿⣿⣿⣷⠶⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⢰⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⡴⠶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⢠⠀⢠⠀⠀⠀⢸⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⢸⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣤⢠⣇⢀⣸⣀⣸⣀⣤⣆⣸⣇⣰⣇⡀⢸
⠀⠀⠀⣤⣷⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡆
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠉⠉⠉⠉⠉⠁
⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣼⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣶⣦⡀⢸⡟⣁⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣦⡀
⠀⠀⠀⠈⠙⠻⠌⠜⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⢠⡀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⠿⠛⠙⢏⡇
⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣾⡆⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⠞⢛⣿⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢸⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣷⣿⣷⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠸⡆
⠀⠀⠀⠀⠀⣿⡥⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢾⡧⣿⠴⡧⢼⡧⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣸⠓⠒⢻⡀⠀⠀⠀⠤
⠀⠀⠀⠀⣸⡏⢉⡺⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢺⡗⣿⠒⡗⢺⡗⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣷⡖⠒⠒⢧⠀⠀⠀⠶
⠀⠀⠀⠠⣿⣿⣧⣤⣾⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡇
⠀⣀⣀⣀⣀⣤⣤⣴⣶⡞⢠⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⢹⠋⠉
⠀⠻⣿⣿⣿⣿⣿⣿⡟⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⢠⣤⣄⣼⣿⡁⢹⣿⠏⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣽⣄⣀⢀⣀⡀
⠀⠀⠈⢻⣿⠿⠟⠋⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣭⢨⣭⣯⣿⣭⡅⠘⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠊⢡⣏⣺⣯⠽⠛⠉⠁
⠀⢰⡟⢳⡴⠖⠻⣭⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣬⣿⣼⣿⣯⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⢳⣿
⠀⠈⠓⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠖⠓⢾⡇


⠀⠀⠀⢀⣀⣤⣄⣀
⠀⣠⡾⠋⣤⣶⣶⠍⠛⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠺⠓⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣯⣭⣤⣦⡔⣠⡀
⢠⡟⠀⠀⠘⢿⠏⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣀⣆⣀⣀⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣵⣿⣷⡀
⢸⡇⣶⣶⣶⠎⢶⣶⣶⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⣽⢹⣿⣿⡗⡀
⠀⢻⣜⠿⠋⠀⠈⠻⢟⣼⠃⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣉⣿⠻⠟⠃⢻⣧⣀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣇⣺⣿⣿⡇⡇
⠀⠀⠉⠳⠶⠴⠴⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠿⠿⠷⠝⠿⣿⡯⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠒⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⡀⠀⠀⠀⠰⡆⠀⠀⠀⢀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢏⠀⡗⠀⠀⡇⠀⢸⠂⢸⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠳⡈⢢⠀⡇⡴⠋⡔⠋⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠘⠉⠉⠉⠳⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⡔⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⣀⣀⣀⣀⣀⣿⣿⣟⡍⣭⢹⣭⡍⣭⣩⣩⡝⣿⣿⣀⣀⣀⣀⣀⡤⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣄⣀⣀⣀⣀⣿⣿⣯⣥⣭⣬⣭⣥⣭⣭⣬⡅⣿⣿⣀⣀⣀⣀⣀⠔⠉⠉
⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠉⠉⠉⠉⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⠉⠉⠉⠉⠉⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⠉⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⠔⢁⠜⠀⡇⠳⣄⠣⡀⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣇⠀⡧⠀⠀⡇⠀⢸⠄⢹⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠁⠀⠀⠀⠰⠇⠀⠀⠀⠈⠀⠸

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡄⢀⣧⠀⠀⢀⡇⠀⠀⠀⠀⢠⠀⡄⠀⠀⠀⠀⠀⡆⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⡄⡄⣤⢠⣤⡄⠀⠀⠀⠀⠀⠀⢸⡄⡇
⠀⣠⣀⡇⢸⣿⣟⡃⠸⣷⣦⣦⣶⠀⢸⣾⣇⣀⣧⠀⠀⢠⣧⣀⣠⣇⠀⡀⠀⣠⣶⣴⣦⣶⣄⣯⠀⣠⣇⣧⣿⣼⣿⣧⠀⣤⠤⢤⣶⣾⣼⡧⣷
⠀⠋⠹⠿⠿⠿⠿⠇⠠⠿⠿⠿⠿⠀⠸⠿⠿⠿⠿⠿⠀⠸⠿⠿⠿⠿⠿⠷⠀⠿⠶⠷⠶⠶⠿⠿⠀⠘⠿⠿⠿⠸⠿⠇⠐⠶⠰⠰⠿⠿⠻⠗⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣤
⠀⢀⣲⡆⡄⠀⠀⡀⠀⠀⢀⣀⣇⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⣤⢸⡇⠀⢀⣤⢤⡀⡇⠀⠀⠀⣾⣿⣷⠤⠤⡤⠤⡇⠀⢸⣿⢤⣤⡄⠀⡂
⠀⣿⣿⣿⣧⣷⣧⡇⢠⣶⣎⠉⣻⣿⠀⣤⣿⣷⣸⣾⣷⠀⢸⣿⣿⣿⣭⡅⢸⣿⣿⣿⣿⣿⣄⠀⢰⣿⣿⣿⡇⣸⣷⢸⣿⠀⢰⣿⣿⣿⣧⣤⣆
⠀⠈⠉⠁⠀⠉⠁⠀⠈⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠁⠈⡁⢈⠉⠀⠈⠉⠉⠈⠀⠉⠁
⠀⠀⠀⠀⠀⠀⢠⠀⢀⢀⢰⠀⠀⠀⢀⣀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⡇⢰⣦⡇
⠀⢠⣄⣠⣄⣀⣸⠀⢸⣿⢸⡆⠀⠀⢸⣿⠇⠀⠀⡇⠀⢀⣄⢀⣤⡀⢸⡇⠀⢸⣿⡆⠀⣾⣿⢀⠀⠀⠀⣴⣶⣀⣿⣿⠀⠀⠀⠒⡟⢋⣸⣧⡇
⠀⢿⢿⡿⢿⡇⣿⠀⣼⢿⣾⠷⠶⠤⠨⠿⣷⣿⣿⡇⠀⠸⣿⠿⡯⠽⣿⡷⠀⣿⣿⠿⠿⠿⠿⠿⢷⠄⢾⣿⣿⠩⣿⣿⢶⠆⠸⣿⣿⡿⠉⠉⡇

⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠦⣤⡴⠛⠛⠛⠶⣤⡀
⠀⠀⣠⡴⠛⠉⠉⠉⠛⠶⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⢰⡏⠀⠀⠀⠀⣀⣤⠶⢦⡀⠀⠀⠀⠀⠀⢹
⣸⠇⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢸⡇⢀⣠⠶⠛⠉⠀⠀⠀⠙⠳⣤⣀⡀
⣿⠀⠀⠀⠀⠀⣀⡤⠞⠋⠈⠳⣦⣀⡀⠀⠀⠀⢸⡇⠀⠀⢸⣧⡟⠁⠀⠤⠄⠀⠀⠀⠀⠠⠤⠍⠙⠛⣧
⣿⠀⠀⣾⠛⠉⠑⠲⠀⠀⠀⠀⠖⠋⠙⢛⣿⣀⠈⣇⣀⣀⢨⣟⣡⣀⠀⠰⠆⠀⠀⡀⠀⠀⠶⠀⠀⠀⢙
⣿⠀⠀⢹⡄⠀⠘⠃⠀⠀⡄⠀⠘⠃⣰⠏⣠⡿⠛⠉⠉⠉⠙⠻⣧⠉⢷⠀⠀⢀⠀⠋⢀⠀⠀⠀⢀⡾⠛
⢻⡀⠀⠀⢻⣄⠀⠀⠠⢈⡀⠄⠀⠀⢻⣶⠏⣀⣤⠖⠛⢦⣄⡀⠈⣷⣾⡁⠀⠈⠉⠉⠁⠀⠀⣠⠟⠁
⠘⣧⡀⠀⠀⠙⢦⣄⠀⠀⠀⠀⣀⡴⠋⣿⡟⠭⡆⠀⠀⠀⠈⡿⠛⣿⡋⠛⢶⣄⣀⣀⣠⡴⠟⠁
⠀⠈⠛⠶⠦⣤⣤⣽⣿⠲⠖⣿⢯⣤⣤⣯⣀⠀⠁⠀⠰⠀⠀⠁⢀⣹⡇⠀⢠⣿⣉⣉⣹⣇
⠀⠀⠀⠀⠀⠀⣀⡼⣿⣿⣿⣿⣧⣀⠀⠈⠹⣦⡀⠀⠉⠁⠀⣠⡞⠉⣠⡴⠟⠻⠿⠿⠿⠛⢶⣄⡀
⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠈⠙⣷⣶⣶⡛⠁⣰⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⡀
⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣴⠞⢉⠀⠈⠛⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣇⣀⣿⣛⣻⣇⣀⣷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹
⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⡤⠔⠚⠙⠒⠒⠒⠒⠒⠒⠒⠲⣦⠀⠀⠀⠀⠀⠀⢠⡶⠒⣒⣂⣀⣒⣒⣒⠠⠍⠻⠀⠳⣄
⠀⠀⣀⣴⠏⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠛⠛⠛⠛⠛⠛⠛⠛⠛⠦⣄⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⢠⣾⢛⣥⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⣾⣇⠻⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠙⠻⣷⡄⣿⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⣀
⠀⠀⣿⢱⠃⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠈⠙⢷⣦
⠀⠀⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢿⡄
⠀⠀⠘⠳⣦⣄⠀⠀⠀⠀⠀⠀⠹⣧⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢔⣼⠟⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⢻⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⣀⡤⠄⣂⣤⠖⠋⠀⠀⠀⠀⠀⢦⢹
⠀⠀⠀⠀⠀⠘⢿⣦⣤⣀⠀⠀⣠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡜
⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⠷⣶⣿⡟⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠳⣄⠀⠀⣀⡴⢚⡵⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢦⣬⣡⣿⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⣧⢹⣶⡾⣫⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣈⠳⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⡾⣿⣶⣾⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢈⡇⢰⡀⠀⠀⠀⠀⠀⠀⣠⣾⠀⠘⣿⡿⠋⠀⠈⠁⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣼⣿⣆⢳⡀⠀⠀⠀⠀⠀⣿⣿⣧⡀⠘⣧⠀⠀⠀⢀⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢰⡿⣿⣿⡘⣧⠀⠀⠀⠀⠀⣿⣿⡿⢳⡄⠘⣆⠀⠀⢸⣧⢙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⣿⢁⣿⣿⣇⢿⡀⠀⠀⠀⠀⣿⡏⠀⠀⠙⢦⡘⣆⠀⢸⣿⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣾⢇⣼⣿⣿⣿⡸⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠙⢿⣦⣿⣿⣦⣍⣁⡀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⣿⣿⣿⣧⢻⡀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡉⠉⠙⣿⣿⣿⣿⣯⢇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠉⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⠃⠀⠀⢸⣿⣿⠋⢻⡎⡆⠀⠀⠸⡇⠀⠀⠀⢸⡇⠈⣿⣿⣿⣿⣿⡇⠀⡏⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠈⢿⠃⠀⠀⢻⡘⡀⠀⠀⣧⠀⠀⠀⣸⠁⠀⣿⢉⠁⠀⢹⡇⠀⣧⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡇⠀⠀⠀⣠⠏⠀⠀⠀⠈⣧⠀⠀⠀⢹⠀⠀⠀⣿⠀⠀⣿⣸⠀⠀⠈⡇⠀⠸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⡄⠀⠀⣰⠏⠀⠀⠀⠀⠀⠸⣇⠀⠀⠈⣇⠀⠀⣿⠀⠀⣿⣿⠀⠀⢸⠇⠀⠀⢻⡀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢹⡀⠀⡏⠀⠀⡏⡇⠀⠀⠸⣆⠀⠀⢸⡧⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠈⣧⠀⡇⠀⣰⠇⠀⠀⠀⠀⣽⠀⠀⢸⡇⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⢸⡄⣇⣰⡇⠀⢤⡀⠀⢠⡇⠀⠀⢸⡇⢀⣄⢀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡃⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢧⣤⣸⡇⣿⠻⠿⠶⠶⠶⠶⠟⠃⠀⠀⠈⠳⠶⣬⣥⡶⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠹⠗⠄⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⣀⣈⠛⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠒⠒⠒⠚⠛⠃⠀⠀⠀⠀⠀⠀⠀⠘⠻⢶⣥⣼⠶⠾⠛
⣿⣿⡿⠛⠿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢰⣶⣿⡯⠍⠀⡭⣙⠙⠀⠀⠢⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡽⠾⢿⣿⠛⠁⠀⠀⠴⠀⠈⠁⠲⣄⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⢿⠅⢀⣴⣿⣃⣠⣄⣀⡀⠀⠔⠀⠀⠀⠈⠛⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠁⡤⢫⣿⣿⠏⠀⠀⣉⠙⠳⢦⡀⠀⠈⠀⠀⠀⠳⣤⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠫⠐⡴⠃⢠⣿⡏⠀⠀⠘⠛⣆⠀⠈⠻⢶⡀⠀⠀⠀⠀⠈⠹⡄⠀⠀⠠⣀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢟⠀⠀⣸⡟⠁⠀⣿⣿⣄⢘⠂⠀⢠⠃⠀⠀⠃⠌⡿⣶⡇⠂⠐⠲⠀⢷⡀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢻⣆⠀⡼⢻⣿⣿⠁⠀⡴⢦⡞⢉⣾⠄⠀⠒⠈⠻⣶⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣗⠀⣿⠀⢸⣿⠀⣦⠀⣾⣥⣾⣿⢿⣶⣴⢀⣴⣻⠿⣾⣶⡆⠀⣻⣦⠀⢀⠈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⢏⡀⣿⠀⠸⣿⣿⣇⠀⣇⠈⢉⡡⠾⣿⣿⣧⣸⠁⠸⠟⣿⠃⠠⠃⡆⠀⡏⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠸⠐⣿⣄⡀⠈⠛⢿⣷⣿⣧⣾⣷⣤⣿⣿⣿⣿⣤⣤⣴⣟⣤⣖⣨⠃⣘⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡄⠀⠀⠀⠀⠀⢻⣟⣫⣿⣦⡈⣠⣭⣿⣽⣿⣿⣉⢻⢿⣿⣿⣿⣿⣿⣿⡏⣈⢧⡀⠟⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠻⠋⠉⠉⠛⢻⣾⡿⣛⢻⣿⡿⠟⠻⢿⣿⣿⣿⣿⣿⣧⣥⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠈⠛⠿⠿⠟⠷⠷⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⠅⠀⠰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠄⠀⠀⢘⠟⠋⣿⣧⣿⣿⡿⢽⡿⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢵⡄⠈⠀⠘⣭⡾⢶⣿⡻⠟⠋⣰⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣉⣁⣄⣠⣬⠋⠀⠀⣽⣿⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣟⠋⠁⠀⠀⢀⣀⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣎⠁
⠀⠀⠀⠀⠀⡇⠀⠀⣀⣀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀
⠀⠀⠀⠀⠐⠛⠛⠋⠉⠉⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣤⣶⡶⠾⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣀⣀⣀⡄⠀⠀⠀⣀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠋⠉⠉⢀⡇⠀⠀⠈⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣿
⠀⡀⠀⣠⡶⢻⡄⠀⠀⢀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠙⣦⠄⠀⠀⠀⠀⠀⠘⣿⠙⠛⠳⠿
⠀⠱⠞⠁⠀⠈⢷⣀⣴⠟⠉⣷⠀⠀⣠⠔⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠁⠀⠀⢠⣤⣄⣀⡀⣿
⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠘⣧⠞⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡇
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡉⠲⣤⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣙⣷⡄
⠀⠀⠀⠀⠀⢀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡟⢿⣿⣿⠿⠟⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠛⠋⠁⠀⠈⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠰⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣼⣿⣿⡟⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠧⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⢠⠞⠉⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣄⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠀⣤⠟⠀⠀⠀⢲⡄⠀⠀⠀⢀⡾⠋⢁⣠⣌⠙⢷⠀⠀⠀⣸⣿⣿⣿⡿⠿⢤⡄
⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⢼⠁⠀⣼⣿⠛⢻⣶⠀⠀⠀⢾⠀⢠⣿⣿⠉⣰⣾⠁⠀⠀⢿⣿⣿⣿⠃⠀⠈⢡
⠀⠀⠀⡘⠃⣼⣿⠟⠛⠀⠀⠸⣦⣀⢻⣿⣿⣿⠏⠀⠀⠀⠈⣷⡄⣿⣿⣿⣿⠃⠀⠀⠀⠀⠈⠙⡿⠀⠀⢠⡦
⠀⠀⠀⢳⣄⠀⣸⡄⠀⠀⠀⠀⠈⠛⠉⠉⠁⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⢰⡿⢶⠜⠋
⠀⠀⠀⠀⠉⠁⠿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⣄⡀⣀⠀⠀⠀⣈⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠺⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⣉⣠⡴⠾⠋⣹⣿⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣉⣁⣀⣤⣾⣿⡿⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡟⠉⣡⡝⠁⠀⠀⠀⣤⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢶⣶⣤⣀⣀⣀⣀⣀⡤⢶⣶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣻⠁⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠶⠋⠀⠀⠀⠉⠲⠶⠶⣦⣄⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⣠⠔⠀⢰⡏⠀⠀⠉⠿⠆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⣠⡞⠋⠀⠀⠀⠀⠀⠀⠈⠻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠉⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⢷⡆
⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠠⠎⣡⡎⠁
⠀⠀⠀⠀⠀⠀⠰⠋⢡⡀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⡄⢃⡴⠋⠱⣆
⠀⠀⠀⠀⠀⡘⠁⠀⠸⢁⡄⠀⣠⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣴⠂⣴⠋⠀⠀⠀⠸⡆
⠀⠀⠀⠀⡘⠃⠀⠀⢠⣿⣁⣿⠃⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠿⠉⢆⡀⠀⠀⠀⢸⡆
⠀⠀⠀⢸⡇⠀⠀⠰⣇⠀⠛⠃⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⣱
⠀⠀⠀⠘⢧⡀⠀⠀⠀⢀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠖⠀⠀⠀⠀⢀⣰⠏
⠀⠀⠀⠀⠈⠻⣆⠀⠀⠈⣠⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣠⠟⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠉⢦⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⠋⠀⠀⢀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣤
⠀⠀⠀⠀⠀⠀⢠⠀⣻⡃⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠹⣾⡋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣉⣁⣀⣀⣀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠟⠀⠐⠀⠐⠒⠂⠐⠒⠒⠒⠒⠒⠀⠀⠀⠒⠀⠀⠀⠀⠈⠙⠻⣶⠶⠶⠖⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠶⠤⣄⣀⣀⡀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣀⣄⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠈⠉⠉⡿⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⢻⡇⠀⠀⠛⠉⠉⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠤⠀⠀⣀⣸⡆⠀⠀⠀⠀⠀⣿⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⣶⣄⣀⡉⠙⠋⠀⠀⠀⠀⠀⠀⠙⡀⢸⣷⠀⠀⠀⠘⠆⠠⠤⠤⣶⡶⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢷⣶⠶⠶⣦⣄⣤⣠⣼⡇⠈⢿⣄⣀⣀⣀⣀⣀⣀⣀⣠⡴⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡇⠀⠀⠀⠈⣿⠋⠉⠀⠀⠀⠈⠙⠋⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠸⡏⠉⠙⠛⠂⣤⣿⠀⠀⠀⠀⠀⠀⠠⣴⠀⠀⠀⠀⢠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢤⡀⠀⢠⡄⠀⠀⠀⠀⠈⠿⠀⠀⠀⣤⣤⠀⠈⠃⠀⠀⠀⠀⠀⢠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠸⡃⠀⠀⠰⠀⢿⠀⠀⠀⠀⠁⠀⣀⣀⠴⠿⠷⢦⣄⣠⡏
⠀⠀⠀⠀⠀⢀⡠⠶⠶⢤⣀⣀⣀⡀⢀⣸⠇⠀⠁⠀⠀⠲⠶⠾⣄⠀⠀⠀⢠⡆⠉⠁⠀⠀⠀⠀⠈⠉⠻⢦⡀
⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠛⠁⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⣄⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠒⠒⠚⣧⠀⠆⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣄⠀⠘⢇
⠀⠀⠀⠰⣧⠀⣀⣀⣀⣀⣀⠀⠈⠀⠀⠀⣤⣤⣤⣤⠴⠚⠛⠛⠛⠋⠀⢰⣄⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠙⠛⠛⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⢡⣶⠠⣾⣟⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⢉⠉⠋⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣥⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣴⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣫⠞⢮⡻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⣠⢞⠜⠁⠿⠇⠙⢌⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢄⠀⢀⠀⢸⡀⠀⠀⠀⠀⢀⠔⡵⠃⠀⢠⠤⢤⠀⠀⠱⣝⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⡳⢋⡴⠀⡇⠀⠀⠀⡴⣫⠊⠀⠀⠀⢸⢺⣺⠀⠀⠀⠈⢣⡱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠑⢶⠊⠀⢀⡇⠀⠀⣪⠞⣹⠀⠀⢀⣀⣀⣉⣁⣀⣀⠀⠀⢸⢹⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⢄⢸⡴⢃⡞⣠⢔⡫⠖⢹⢽⢀⣀⡛⠒⣒⣒⣒⡒⠚⣃⣀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣽⢲⠋⠋⡔⡯⠒⠢⢸⢽⡧⠤⡇⠀⣷⠉⢱⡇⠀⡧⠾⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⡇⡗⠒⠒⢸⢸⠈⠉⠁⠀⣿⠀⢸⡇⠀⠉⠉⢹⢸⢀⢄
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣤⣼⣼⣾⣭⣷⣧⣤⣤⣼⣾⣤⣤⣴⣵⣿⣷⣿⣿⣬⣦⣤⣼⣼⣤⣮⣤⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⠵⣝⢦⡀⢀
⠀⠀⠀⠀⠀⠀⠰⡿⢿⠆⢀⡴⣫⠎⠙⢮⡣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠔⠁⠀⠈⠳⢝⠯⡏
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡴⣫⠞⠁⡤⠤⡀⠙⢌⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣊⠔⠁⠀⢠⢤⢤⠀⠀⠑⣌⠢
⠀⠀⠀⠀⠀⠀⢀⡷⣫⠞⠁⠀⠀⣟⣛⠇⠀⠀⠱⣝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠞⢹⠀⠀⠀⢸⠲⢾⠀⠀⠀⡏⠙
⠀⠀⠀⠀⠀⡠⣫⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢆⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠛⠛⠀⠀⠀⡇
⠀⠀⠀⠀⣪⡞⠁⡇⢠⢤⠤⡀⢰⡶⠶⢶⠄⢠⢤⢤⠀⡏⠣⠑⠀⠀⠀⠀⠀⣰⣼⠀⠀⠀⢠⠤⢤⠀⠀⠀⣇⣦
⠀⠀⠀⢈⣁⡇⠀⡇⢘⠚⠒⡇⢸⡾⢉⣾⠀⠟⢺⢚⠀⡇⣠⢄⠀⠀⠀⠀⢀⡇⢿⠀⠀⠀⢸⠚⢻⠀⠀⠀⣿⢘⡄
⠀⠀⢸⡁⣀⣙⡆⣇⢺⡍⢉⠓⢸⣯⣠⣿⡀⠋⠋⣽⢂⡇⢧⡴⠃⠀⠀⠀⣸⠁⣸⡀⠀⠀⠈⠋⠋⠀⠀⠀⣟⠀⣧
⠀⠀⠒⣿⣿⣶⣷⣿⣶⣶⣶⣶⣿⣷⣷⣿⣷⣶⣶⣶⣶⣷⣶⣷⡖⠀⠀⠐⢺⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⠟⠒
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠄⠠⠄⠠⠤⠤⠤⠤⠄⠀⠈⠓⠊⠉⠉⠀⠀⠀⠀⠀⠙⠋⠛⠛⠋⠙⠋⠋⠉⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠑⠒⠲⠤⣀⣀
⠀⠀⠀⠀⠀⡰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⡀
⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⢰⠇⠀⠈⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠑⠤⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⡇⠀⠀⠀⢀⠴⠒⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⣇⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⣧
⢹⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠘⢧
⠀⢣⠀⠀⣇⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠓⠦⣄⠀⠀⠀⣇
⠀⠈⡆⠀⢹⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠸⡄
⠀⠀⠘⠀⠈⠀⠀⠀⠀⢸⠁⠀⠀⡊⠀⠀⠀⠀⠈⣇⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢰⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠹⡄
⠀⠀⠀⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢣⠀⠀⠀⠀⣠⠇⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⢸⠀⠀⠀⣼⠁⠀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠚⠁⠀⠀⠀⠀⠀⣞⠁⠀⠀⣀⡼⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⡤⠤⢤⠔⠂⠉⠉⠉⠉⠁⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⡠⠔⠂⠉⠩⢭⣤⡆⠚⢤⠤⢄⣼⣄⣀⣀⣄⣀⡀⠀⢰⡆⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⢱⡀
⠀⠀⢠⠊⠀⠀⠀⠀⠀⠐⠉⠖⣀⡘⠀⠀⠀⢀⡤⠂⢀⠠⣄⠉⠉⣀⡤⢈⣛⡻⣍⡉⠉⠉⠙⢷⠉⢠⣛⠘⠋⠉⠒⠦⣄
⠀⢠⠇⠀⠀⠀⠀⠀⠀⢀⠃⣾⠉⠇⠀⠀⠐⡏⢸⣿⣏⣡⠈⠀⠀⠛⢰⣿⣍⣷⠈⡇⠀⠀⠀⣼⠻⡏⠙⣇⠀⠀⠀⠀⠈⠳⡄
⠀⣆⠀⠀⠀⠀⠀⠀⢀⡎⠀⠙⢤⣴⠀⠀⠀⢡⡘⠻⠿⠋⡰⣀⣀⠳⣈⠻⠟⢋⣴⠁⠀⠀⠀⣟⣰⠃⠀⢹⡄⠀⠀⠀⠀⠀⠹⡄
⠀⡇⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠈⡄⠀⠀⠀⠙⠓⠚⠋⠀⠉⠉⠂⠈⠙⠓⠉⠀⠀⠀⠀⣸⠋⠁⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢳⡀
⢰⠁⠀⣀⠤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⢲⠀⠠⠀⠀⠀⠀⠈⢷⣄
⠸⠷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⠓⠒⠛⠁⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠐⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣀⣉⣁⣀⣠⠤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠠⣤⣤⡒⠟⠀⢸⡆⢤⡤⠤⠄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⣠⠴⠎⠁⠁⠀⣨⠷⠙⢢⡀⠀⠁⠒⣄⠀⠀⠀⠈⠙⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠃⠘⠀⠀⠂⠸⠇⠀⠀⠀⠹⣆⠀⢠⠀⠇⠀⠀⠀⠀⠀⡇⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠲⠀⠀⠀⠀⠀⠷⠀⠀⠀⠈⠃⠘⠆⢸⠐⣤⡄⠀⠀⠀⠀⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠐⠂⠒⠀⠀⠀⠀⠈⠄⠀⠘⠀⠀⢠⠉⠈⠈⠑⡀⠀⠀⠀⡤⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡆⠀⠀⠀⢱⠀⠀⠀⠀⠐⢁⠃⠀⠀⢸⠂⠀⠀⢀⡇⠀⠀⠀⣻⡀⢰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⢃⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠀⠀⠀⢘⣄⡀⢀⠜⠇⠀⠀⠀⡟⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄⠈⢱⠖⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⠹⣅⠀⢰⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠳⠚⠁⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⠳⠞⠀⠀⢀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡼⠧⢤⣀⣀⣀⣀⣀⣀⣀⡤⠞⡄⠀⠀⠀⠀⠀⠀⢰⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠥⡒⢠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣀⣀⣀⣀⠠⠎⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣏⡉⠁⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠮⠥⠄⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢒⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡟⠛⠓⠶⡴⠦⢤⣤⣤⡤⠦⠴⡶⠶⠖⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡋⢀⡀⠈⢸⠆⠀⠀⠀⣦⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠳⡤⠀⣱⠏⠀⠀⠀⢠⠃⠠⡄⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⡴⠃⠀⠀⠀⠀⡘⠀⠀⠈⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⡏⠀⠀⠀⢶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⣀⠜⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⣀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⣼
⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡅
⠀⠀⠀⣠⠔⠋⠁⠀⠠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇
⢀⠶⠋⠀⠀⠀⢠⠔⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⡀⠙⢦⡀
⡎⠀⠑⠄⣀⠖⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢌⠤⠀⠀⠙⠢⣄
⢣⣄⣀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡑⠠⠄⠒⠉⢱
⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⢀⡴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠀⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠉⠉⠉⠀⠒⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⢀⡀⠠⠤⠤⠬⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠀⠀⠀⠀⢀⠃⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⡄⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠈⠸⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠾⠰⡦⡀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣧⡆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡜⠀⠑⢧⠈⠉⠒⠒⠀⠒⢄⠀⡇⠀⢀⢼⠀⠀⠀⣴⠄⠀⠀⠀⠀⣰⡆⠀⠘⡀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢰⡷⠀⠀⠘⡀⠉⢑⢈⠸⠀⠀⠀⠉⠀⢰⠀⠀⠀⠉⠀⠀⠀⠑⠁⠀⢑
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⡏⠀⠀⣷⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢡⡥⠊⠈⠒⠇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡰⢆⢰⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣼⠃⠀⠀⠀⠘⡀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢧⢂⠀⠀⠀⠀⠘⢄⡀⠄⠀⠀⢀⠎⠈⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⢀⢦⡠⡄
⠀⠀⠀⠀⣤⢄⣀⣄⠀⠀⠀⠙⢷⠢⢀⣀⠀⠀⣀⡀⠤⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢒⣤⠤⠤⠐⣆⠁⠀⠀⠸⠙⠂⠃⠖⠱
⠀⠀⠬⣓⠀⠀⠁⡎⠀⠀⠀⠀⠀⠁⠀⢀⣠⢠⠇⠀⠀⠘⠂⠠⡤⢀⡀⠀⠀⠀⠀⡀⠴⠙⠦⡀⠀⡀⠟⠢⢄⡠⡒⠃⠀⠀⢀⠇
⠀⠀⠀⠉⠂⠢⡀⠈⠢⡀⠀⠀⠀⠀⢰⣫⠶⡁⠀⠀⠀⠀⠀⡠⠁⠀⠈⠑⠢⢔⠉⠀⠀⢀⠀⢉⡞⠀⠀⠀⠀⠈⠢⡉⣖⠂⠉⣆
⠀⠀⠀⠀⠀⠀⠑⡄⠀⣈⠦⠀⠒⠚⠣⢄⡀⠑⠢⠀⠠⠄⠊⠀⠀⠀⣀⡀⠀⡠⠓⠢⡀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⢀⠜⠉⣆⡠⠄⠀⠒⠒⠠⣈⠢⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⡲⠅⠀⠒⠘⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠈⡄
⠀⠀⠀⠀⠀⢀⠊⢀⠔⠁⠀⠀⠀⠀⠀⠀⠈⢧⠵⢌⠑⡆⠀⠀⠀⠀⣷⣊⡀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⠀⠀⣴⢤⡀⠨⠇⠀⠀⣴
⠀⠀⠀⠀⠀⠘⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⢀⣈⢣⠀⠀⠀⣽⠁⠀⠑⠂⠿⠲⠆⠀⠀⠀⠀⠇⠀⠀⠀⣿⠀⠙⠳⠦⣤⣤⠟
⠀⠀⠀⠀⠀⢰⢸⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠇⢏⠈⡎⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠁⢸⠀⠀⠀⡇⠀⠀⠀⠙
⠀⠀⠀⠀⢠⠚⢪⡖⠀⠀⠀⠀⠤⠀⠁⠀⠀⠀⡼⢄⡞⠀⡦⢤⠤⠒⡀⠈⠁⠀⠀⠀⠖⠀⠀⢄⠄⣀⠃
⠀⠀⠀⠀⠸⡀⢇⢀⡠⡀⠀⠀⠀⠐⠈⣀⠤⣎⡠⠚⠣⡜⠀⠓⡄⢇⠑⢄⠀⠙⠀⠂⠀⠀⠀⠀⣠⡮⢀
⠀⠀⠀⠀⠀⢄⠀⠀⢀⠌⠡⠒⠀⢂⡽⠀⠠⠦⠄⠀⣠⠃⠀⠀⠱⡈⢳⠆⠷⢰⠂⠠⣤⠴⡖⠋⡩⠄⠚⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠐⠂⢻⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⣀⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠁⠉⡀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠟⠀⠘⠉⠉⠁⠉⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣠⣇⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠴⠈⠉⠑⠒⠁⠒⠤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠒⢋⣀⠀⠀⢀⢄⡠⢤⠒⠂⠀⠀⠀⠀⠈⠂⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⢰⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠀⠐⠤⡀
⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⡔⠉⠁⠀⠠⠢⡀⠀⠀⠀⠀⠀⣅⡀⢀⠀⠈⡆⠀⠀⢄⡀
⠀⠀⠀⠀⠀⢰⠁⡀⠀⠀⠀⠈⠂⠀⠀⠀⠀⢣⠀⠰⣁⡀⠀⠀⠀⠀⠀⠀⠉⠇⠀⠀⢇
⠀⠀⠀⠀⠀⠸⠀⢦⠀⠀⠀⡀⠀⠀⡀⠀⠈⠁⣄⠀⠀⠑⠲⣄⠀⠀⠀⠀⠀⡀⠀⢀⢨⠀⠀⠀⠀⠀⢀⢒⡀⠢⢀⡀
⠀⠀⠀⠀⠀⢰⠀⠘⠤⠄⠀⠉⠉⠉⣀⠔⠉⠉⠀⠉⠈⠦⣀⡀⠠⡄⠀⠀⠀⠑⠒⠊⡠⠁⠀⠀⠰⠢⠜⠘⠃⠗⣕⡁
⠀⠀⠀⠀⠀⠐⣄⣀⣠⡠⠖⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠉⠉⠒⠊⠀⡤⡴⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⡡⠃
⠀⠀⠀⠀⠀⠀⠣⡐⠈⡀⠀⠀⠀⠀⢴⠆⠀⠀⠀⠀⠀⠀⠾⠀⠈⠂⠤⠐⢹⠉⢂⠄⠀⠀⠀⡰⠉⠉⠐⠠⡔⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠣⠀⠀⡠⢄⠀⠀⠀⠀⠐⠂⠀⠀⠀⢀⠤⡀⠀⠀⡜⠈⠀⠀⢀⠔⠊⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠑⠒⠀⠀⠀⠀⠦⠞⠀⠀⠀⠑⠒⠁⢀⠜⠀⢀⠠⠒⠁⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢉⠖⠈⠁⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠉⠒⠤⢀⡉⡀⠒⠒⠒⢂⣈⡡⠄⠊⠁⠀⠀⠀⠀⠀⠀⢀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡠⠒⠁
⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠈⠁
⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠰⠁⣀⣀⠀⠀⠈⠪⣢⣀⣀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠭⠭⠶⠭⠤⠴⢝⣆⠀⠀⠀⠀⣠⠦⠖⠒⠛⠛⠛⠛⠒⠒⠲⢤
⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⣇⡀⠤⠤⠀⠐⠒⠒⠒⠀⠠⠍⣢⣠⢊⠡⠤⠀⠐⠒⠒⠂⠀⠠⠠⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⣤⣴⣶⡟⠉⠹⣿⠀⢨⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⢊⣉⣩⡍⠙⠛⠻⣆⠸⣿⣧⣼⣿⣗⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⢁⡒⠛⠛⠿⢿⣧⢠⣆⣿⡛⣯⡭⠏⠛⠉⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⡈⣽⣵⣁⡃⢀⣸⣿⣦⡿⣿⣿⣷⣶⣶⣶⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣿⣿⣿⣿⢿⢺⡇⣿⣯⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⣸⣘⣿⣿⣿⣿⣿⣿⣿⣿⠻⠀⠉⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣟⠈⢁⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡟⡕⡻⢻⣿⣿⣿⣿⠙⢿⣍⠢⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣹⢏⠌⠀⠉⣟⢅⠙⣆⠻⣢⣽⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡎⣀⡀⢀⣬⣺⣶⣌⣷⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣷⣶⣾⣿⣿⣿⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⡿⣿⢸⣿⡟⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣷⣿⣽⡿⣧⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠉⠹⡇⣿⣿⢹⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢿⣿⠀⢘⡇⠻⠋⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⣾⡇⠄⢸⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⠇⠀⢸⡄⡄⠀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡷⢸⠀⠀⠈⡇⡇⠀⣿⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⣷⡇⠀⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⡇⠀⢸⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡇⠀⠀⠀⠀⠘⣿⠀⠘⣷⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⢹⠀⠀⢸⡼⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣃⣀⣀⣀⣀⣀⣈⣀⣀⣀⣓⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⣶⣤⡀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠿⣿⠿⠿⠆⠀⠀⣾⡿⠉⠉⠉⠉⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠛⠀⠰⠀⠰⠆⠀⠀⠀⠉⠁⠀⠘⠃⠀⠛
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣶⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⢸⡿⠿⠿⠛⠛⠛⠀⠀⠈⠁⠀⣤⠀⠀⣠⠀⠀⠀⠀⠀⣴⣷⣀⣭⣄⣾⣦⠀⠀⠀⠀⣠⣤⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠖⠀⠘⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⡆⠀⠀⠀⣸⠟⣿⣿⣿⠟⢿⠻⣇⠀⠀⣾⡟⠻⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⣤⣄⡀⠐⠒⢀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠀⣿⠀⣿⣿⣿⣷⣡⠀⣿⠀⠀⣍⠀⠘⠃⠈⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⣶⣿⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠐⠋⠀⣿⣿⣿⣿⣿⠀⠘⠀⠀⠾⠆⡀⠉⠁⠚⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⣿⣿⠉⣿⣿⡇⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⣿⣿⠇⠀⠀⠀⠀⠸⠅
⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⣤⣤⣄⠀⠀⠀⠀⢠⣀⡀⢠⣀⡀⠀⠀⠀⢀⣴⣿⣿⣀⣿⣿⣦⡀⠀⠀⢀⣤⣤⢠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠉⠉⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠀⠀⠀⠉⠉⠉⠀⠉⠉⠉⠀⠀⠀⠀⠉⠉⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠣⠤⢴⠃⠀⠀⠀⠀⠐⠃⠀⠂⢀⣑⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢢⠁⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢻⠗⣋⠔⠋⠀⠀⠑⢌⡐⢀⠈⠀⠀⠀⠀⠀⡀⢀⡀⠐⡆⠊⠀⡖⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢂⡤⠊⠁⠀⠀⠀⠀⠀⠀⠉⠢⢍⡃⢄⠀⠀⠠⢀⠉⠀⠀⠊⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⡄⣂⡠⠒⠁⠀⠀⢀⣀⡀⠀⠀⠀⢀⣀⠀⣀⣉⡒⠪⢔⡀⢤⡢⣤⣀⣉⡾
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⢸⡉⠈⠈⠸⠀⠀⠀⠸⠀⠀⠁⠀⠃⠀⠀⠁⠉⠀⠐⠳⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠮⠤⠦⠀⠀⠀⠸⠦⠦⠤⠤⠇⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠀⡀⠀⢀⡀⠀⠀⣀⡀⠀⠀⠀⡀⠀⣀⢸⢀⠀⠀⡀
⠀⠀⠀⠀⠁⠠⠐⠀⠁⠄⠀⠀⠀⠘⠉⠋⠉⠙⠀⠀⢠⡀⢠⡀⢌⠀⠀⢠⡄⢠⢤⠈⣯⠇⢠⡄
⠀⠀⠀⠀⠀⠛⠀⠛⠀⠛⠀⠀⠀⢨⠄⠀⠐⠀⠀⠀⢸⡤⠊⣤⣼⠀⠀⠈⠁⠈⠀⠀⡏⠁⠈
⠀⠀⠀⠀⠀⠿⠀⡻⠀⠟⡄⠀⣠⢸⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠸⠍⠀⣯⠀⠰⠆
⠀⠀⠀⠐⠒⠓⠓⠚⠓⠛⠓⠒⠒⠚⠒⠒⠒⠒⠒⠒⠒⠐⠀⠀⠂⠒⠒⠛⠖⠗⠲⠲⠓⠒⠒⠒⠖⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣶⣄⣀⣀⠀⢀⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢈⡿⣯⡶⠿⢾⣝⢿⠠⣼⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢀
⠀⠀⠀⠀⠀⠀⢰⣾⣿⡇⠰⣷⠀⣿⢾⣷⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢈⣧⡿⢦⣤⡼⣿⣾⠁⡀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠻⣿⠛⠉⣋⡼⣿⡽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢠⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⡃⢶⠄⣄⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢯⣷⣶⣾⡿⠂⢄⠀⠀⠙⣄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠁⠀⠀⠁⠄⠀⠘⢦⠀⠀⠀⢸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠘⡇⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠱⠄⠐⢃⣀⠭⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠔⠚⡈⠁⠀⠐⠢⣶⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⠀⠀⠀⠀⠄⠁⠘⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⢰⣿⣶⡆
⠀⠀⠀⠀⠀⠀⣼⣿⡟⢁⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⣾⣿⠟
⠀⠀⠀⠀⠀⣼⣿⣿⣾⠏⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⣼⣏⣀⡀⠀⠀⠀⢰⣿⣯⣤⡄
⠀⠀⠀⠀⠰⢫⣿⣿⢏⣠⠄⠀⠀⠀⠀⢰⡿⢀⣠⡄⠀⠀⠀⢰⣿⣿⣶⠀⠀⠀⠐⠛⣻⡟⠀⠀⠀⠀⠞⢻⣿⡿
⠀⠀⠀⠀⢀⣾⣿⣿⣿⠏⠀⠀⠀⠀⢠⣿⣿⣿⡟⠀⠀⠀⠀⢛⣿⣿⠃⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠚⢁⣿⣿⠃⠀⠀⠀⠀⠀⠟⠋⣽⡟⠀⠀⠀⠀⠀⣸⣿⣿⡶⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⣸⣿⣃⡀
⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠟⣿⡿⠁⠀⠀⠀⠼⠿⣿⠁⠀⠀⠀⢀⣿⣿⡟⠁
⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠘⣹⡟
⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⢰⣶⣶⡶⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣶⣶⣶⡶
⠀⠀⠀⠀⠀⢀⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠋⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⣰⣿⣿⣿⠃⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⣼⣿⣿⣿⣶⡦
⠀⠀⠀⠀⣰⣿⡟⠀⠀⣀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⢠⣿⣿⣿⠃⠀⠀⠀⠀⠀⢠⣿⣿⣀⠀⠀⠘⠛⢻⣿⣿⣿⠁
⠀⠀⠀⢠⣿⣿⣿⣿⣿⠏⠀⠀⠀⢠⣿⣿⣤⡄⠀⠀⢀⣿⣿⣿⣿⡶⠀⠀⠀⠀⠀⠛⣿⣿⡟⠀⠀⠀⠀⣾⣿⣿⣷⣶⠂
⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠾⢿⣿⡿⠁⠀⠀⠀⠀⣿⣿⡿⠁⠀⠀⠀⠀⠀⣼⣿⣿⡶⠀⠀⠀⠀⠉⢹⣿⣿⠃
⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⠀⠀⠀⠀⢉⣿⡿⠁⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⢀⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣍⡉⠀⢀⣀⣀⣤⣤⣶⣾⣿⣿⣿⡿⠿⠛⠛⠉⠉⠉⠉⠉⠉⠻⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⠿⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⣶⣶⡄⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⡄⠀⠀⠀⢒⣒⡒⠀⠀⠀⣠⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣤⣤⠀⠀⣠⣶⣿⣿⣿⣿⣷⣦⠀⠀⣿⣿⣿⢀⣴⣿⠟⣁⡈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣿⣿⣿⣿⠟⣡⣾⣿⣿⣦⡈⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⣾⣿⣿⣿⣿⣿⣿⣦⡈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣽⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣷⣄
⠀⠀⠀⠀⢀⣄⠀⣀⣼⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⣠⣾⡇⢠⣤⢠⣤⡄⣿⣿⣿⣿⣿⡇⣤⣤⢠⣤⠈⣿⣷⡌⠻⣷
⠀⠀⢀⣰⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢈⣉⢀⣉⡀⣿⣿⣿⣿⣿⡇⣉⣉⢈⣉⠀⣿⣿⡇⠀⠈
⠀⣴⣾⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣿⣿⡇⠘⠛⠘⠛⠃⣿⣿⣿⣿⣿⡇⠛⠛⠘⠛⠀⣿⣿⡇
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠸⠿⠘⠿⠃⣿⣿⣿⣿⣿⠀⠿⠿⠸⠿⠀⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣇⣀⣿⣀⣸⣿⣀⣸⣿⠀⠀⠀⣀⣀⣀⣀⡀⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⡇⢠⣶⣄
⠿⠿⠟⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⡏⠉⣿⠉⢹⣿⠉⢹⣿⠀⠀⢸⣿⣿⣿⣿⡇⠈⠉⠉⣉⣉⣉⣉⡉⠉⠉⠉⠉⠉⠉⣉⣉⣉⣉⠉⠉⣻⣿⣾⣿⣯⢀⣀
⠀⠀⠀⢠⣿⣿⣿⠿⣿⣿⣿⣷⣦⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⢸⣏⠉⣿⠉⣿⠀⠀⠀⠀⠀⢸⡏⢹⡏⢹⡇⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠐⠻⡿⠷⠀⠠⣿⣿⣿⢿⣷⣴⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣽⡇⠀⠀⢸⣏⢉⣿⠉⣿⠀⠀⠀⠀⠀⢸⡏⢹⡏⢹⡇⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠶⠶⠶⠶⠶⠶⠶⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣭⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⠶⠶⠿⠃⠹⠿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢠⣞⢉⣿⢮⡓⣤⣤⢤⣀⡀⣀⣠⢤⣄⡔⣫⢾⣏⢙⣦⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢠⣯⣿⢹⢘⣶⣋⡌⣼⡾⢿⢤⢼⠿⢾⡌⣌⣳⣞⢸⢹⣯⣧⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠻⡴⣯⠸⣻⣸⢨⣽⢏⣿⣙⣒⣿⣏⣿⣓⣚⣹⣏⢿⣭⢰⣸⡻⢈⡳⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣞⣿⣾⣞⠿⣿⠶⢮⣻⢣⢻⣫⠶⢾⡿⢟⣾⣾⣟⡶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠭⢁⠡⡢⣿⡯⢶⣬⣯⣯⣴⠮⣿⡯⡣⢁⠩⠕⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⣻⣏⢿⡕⣯⡗⣽⢏⣿⣧⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⢸⠻⣗⡱⣓⡿⢻⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢣⡏⣴⢿⣌⣾⢷⡌⣧⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⣿⣾⢽⣾⡿⣟⣾⢵⣾⣧⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢿⣟⠾⣿⣿⡇⢹⢞⣿⠇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡰⠸⣎⣳⣿⡿⣦⣇⣋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣷⣤⢿⣿⣿⡇⣿⣿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⣿⣋⣽⠇⢿⣝⣻⡧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠁⠀⠈⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠐⢀⠄⠂⠈⠁⠈⠉⠀⠢⢄⠀⠈⠂⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⢠⠀⣆⣔⠀⠀⠀⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠎⠁⠀⢈⠆⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠄⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠴⠠⠀⠀⠀⠊⠄⠀⠀⠀⠐⣀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠆⠂⠀⢸⠀⠠⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⣽⠟⢻⣏⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⢁⢠⢴⠊⠐⢀⣨⠉⠕⡢⢦⢬⡉⠁⠀⢁⠅⢤⣀⠀⢂⢤⣄
⠀⠀⠀⠀⠀⢀⠌⢠⠔⠚⣉⡀⠀⢠⡖⠗⢒⣦⣐⣂⣁⢄⣩⡆⣤⢷⠖⠢⡆⣐⣀⢌⠑⢿⣆
⠀⠀⠀⠀⠀⡈⠀⣎⠀⠞⣁⡅⢀⠏⠘⢠⠋⠀⠀⠀⠀⠀⠀⠀⡉⠉⢹⡇⢡⠀⢈⠽⠂⢈⠈
⠀⠀⠀⠀⠀⠠⡀⠈⠓⠤⠤⣉⣀⢤⣀⡆⡄⠉⠙⠀⠀⠀⠐⢡⡴⠾⢼⡧⣤⣭⢥⣤⣴⣶⣷
⠀⠀⠀⠀⠀⠀⠈⠐⠀⠂⠠⠀⠼⠯⣈⡻⠓⠲⣒⠲⠶⠴⠲⢾⠷⠶⠊⡀⠤⠄⠰⠼⠿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⠐⠉⠠⠾⠅⠠⠀⠀⠢⠸⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠄⠂⠐⢀⠀⢠⠀⠀⡐⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠁⠀⠀⠈⠀⠌⡂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠈⠀⢁⣀⡔⢐⠨⡀⠐⠒⠑⠀⢸⠀⣀⡄⠄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⠉⠚⠛⠉⠄⠂⠔⠨⢘⠳⡦⠄⡠⠐⠜⠡⢐⠠⠀⠁⠈⠪⠐⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠶⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠖⠋⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠴⠚⢲⠁⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠚⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠏⠀⠀⠀⠀⠘⡆⣀⣀⣴⡶⠋⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢀⡀⣀⠀⢰⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠠⠖⠚⠁⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣭⣍⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠖⠛⠶⢠⣿⣿⣿⡀⠀⣿⣿⣿⡇
⠀⠀⢸⡿⠿⠿⠿⠿⢿⡇⠀⠀⠀⠀⠀⣿⣿⣷⠀⣿⠿⠿⣿⠿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡆⢸⣿⣿⣿⡇⢰⣿⣿⣿⡇
⠀⠀⢸⣇⣀⣀⣀⣀⣸⡇⢸⣶⣤⣄⣸⢿⣿⣿⠀⣿⣤⣤⣿⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⣸⣿⣿⣿⣇⢸⣿⣿⣿⣷
⠀⠀⢸⡏⠉⠉⠉⠉⢹⡇⢸⡿⠿⠿⡿⠿⠾⣯⠀⣿⣀⣀⣿⣀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⣿⠿⣿⡿⢿⣿⠿⣿⡿⢿⡿⠿⣿⠿⢿⣿
⠀⠀⢸⡟⠛⠛⠛⠛⢻⡇⢸⣇⣀⣀⣇⣀⣀⣿⠀⣿⡀⢀⣿⠀⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣴⣿⣤⣾⣧⣼⣿⣤⣿⣦⣼⣷⣤⣿⣦⣼⣿
⠀⠀⢸⣧⣤⣤⣤⣤⣼⡇⢸⣇⣀⣀⣇⣀⣀⣿⠀⣿⠉⠉⣿⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣉⣿⣉⣹⣏⣹⣿⣉⣿⣏⣹⣏⣉⣿⣉⣹⣿
⠀⠀⢸⡇⢀⣀⣀⡀⢸⡇⢸⣿⡏⠉⠉⠉⣿⣿⠀⣿⠛⠛⣿⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⣿⠛⢿⡏⠉⠉⠉⠉⠉⢹⡿⠛⣿⠟⢻⣿
⠀⣤⣾⣿⣿⣿⣿⣿⣿⣧⣾⣿⣧⣤⣤⣤⣿⣿⣦⣿⣶⣶⣿⣶⣶⣿⣤⡄⠀⠀⠀⠀⠀⠀⢸⣷⣶⣿⣶⣿⣇⣀⣀⣀⣀⣀⣸⣿⣶⣿⣶⣾⣿
⠀⠁
⠀⡂

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢤⣀⣀⠀⣠⠤⠶⠤⣄
⠀⠀⠀⠀⣠⠚⠉⠉⠉⠙⠢⠤⢤⡞⠑⠞⠁⠀⢸⠁⠀⠉⡡⠤⢤⡀⠈⢳⠲⠤⣄
⠀⠀⠀⢰⠁⢀⡤⠒⠦⣀⣄⡀⠀⠓⢤⡀⣠⠒⢉⣰⠉⠉⠀⠀⣀⣱⠀⠸⣄⠀⠈⠣⢤⡀
⠀⠀⠀⢸⡀⢺⠀⠀⠀⠉⠀⠃⠀⠀⠀⢱⣇⠀⠏⠀⠀⠀⠀⠘⠁⠀⣳⠀⠈⠳⠤⡀⠀⠙⠦⠲⢄
⠀⠀⠀⠀⢳⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠾⣌⡶⠒⠲⠤⠒⠢⢤⡀⠀⠁⠀⠀⠀⠀⢸⡤⠔⠒⠤⡼
⠀⠀⠀⠀⡾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡔⠙⠚⠉⢦⡀⠙⣄⡀⠀⠀⠀⡴⠲⠔⠉⠉⠳⡘⡆
⠀⠀⠀⠈⣇⠈⠉⠉⠱⣄⠀⠀⣀⢔⣒⣦⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣹⣄⣀⠀⠀⠀⠀⠀⠀⠰⢣⡃
⠀⠀⠀⠀⠈⠑⠒⢤⣀⠴⣍⠉⢁⡜⢲⠚⡖⢻⠒⠒⠒⣿⠒⠒⢺⠓⡞⢲⣾⣦⣀⣤⣄⣀⣀⣀⣀⣱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠘⡏⣏⢹⠉⠉⠉⣿⠉⠉⢹⠉⡏⣟⣏⣻⣛⣛⣻⣛⣛⣿⣹⣛⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣲⣀⣀⣖⣆⣯⢿⢹⠉⠉⠉⣿⠉⠉⢹⢹⢋⣧⣯⢼⠤⠤⣼⠤⠤⣯⣼⡼
⠀⠀⠀⠀⠀⢀⣀⣀⣇⣀⣀⣀⣀⣀⣀⣹⢸⢸⡇⠀⢠⡛⡄⠀⢸⢸⢸⢹⣷⢸⠀⠀⢸⠀⠀⡇⣸⡇
⠀⠀⠀⠀⠀⠘⢦⢤⣤⣤⣤⣤⣤⣤⣤⣼⣼⣸⣇⣠⡻⠋⢞⣆⣼⣸⣼⢸⣿⣸⠀⠀⣸⠀⠀⣇⣻⡇
⠀⣀⣀⣀⣀⣀⣸⢸⣽⢼⢽⣭⡯⡯⣯⡽⡿⢸⡵⣿⣧⠀⣼⣯⢿⠘⡿⣼⡏⢹⠉⠉⢹⠉⠉⡏⢹⣇⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣸⣸⣺⢺⢺⣒⡗⡗⣗⡇⡇⣼⡝⠉⢉⣿⡉⠁⢻⣣⡇⡟⡇⢸⠀⠀⢸⠀⠀⣇⢸⢻
⠀⠀⠀⡏⢉⣉⠉⣉⠉⣉⠉⣉⠉⣉⣹⢡⠿⣼⠷⠶⠾⣿⠷⠶⢾⡧⢧⢹⠃⡎⠀⠀⢸⠀⠀⢸⠘⡜⡄
⠀⠀⠀⡇⢸⣿⠀⣿⠀⣿⠀⣿⠀⣿⡏⡸⠀⡇⠀⠀⠀⣿⠀⠀⠈⡇⠸⡈⣦⣇⣀⣀⣸⣀⣀⣸⣄⣧⣷
⠀⠀⠀⡇⢸⣿⠀⣿⠀⣿⠀⣿⠀⡿⢳⠗⢒⠗⠒⠒⠒⣿⠒⠒⠒⢷⠒⢷⠻⣄⣀⣀⣸⣀⣀⣀⣇⣘⣆⣧
⠀⠀⠀⢧⣸⣿⣤⣿⣤⣿⣤⣿⣴⠧⡼⠤⣼⠤⠤⠤⠤⣿⠤⠤⠤⠼⡤⠼⡦⢵⡤⠤⢼⠤⠤⠤⢼⡤⠽⡼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡾⠒⢲⠗⠒⠒⠒⠒⣿⠒⠒⠒⠒⢷⠒⠺⡖⢻⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⡀⣦⠀⠠⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠫⣅⣽⣿⣷⣴⣶⢀⢋⣭⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⢠⡄⢾⣿⣿⣿⠟⣿⣥⣼⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣀⡀⣸⣧⣶⣶⣾⡟⠁⠀⢀⠀⠀⠘⠟⢩⡷⢸⣿⢿⡁⠐⣷⣿⠾⢯⣿⠆
⠀⠀⠀⠀⠀⠀⠀⢠⣷⣠⣤⣤⢾⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣾⣷⣤⣤⡄⢾⣇⣰⣿⣽⣷⣾⣿⠀⣠⣀⣤
⠀⠀⠀⠀⠀⠀⠤⢽⣾⣿⣛⠁⡭⢄⠀⠻⠛⠋⠉⠛⠛⡗⠰⡶⡄⣼⣿⣿⣿⡇⣸⣿⣿⣿⣿⣭⣙⢳⣾⣾⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣴⣿⡇⠐⣰⠀⠰⣶⠂⠀⠀⠑⠀⠀⣿⣸⣧⢇⣼⡿⠋⠉⢿⣅⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿
⠀⠂⠀⠀⠢⣶⣿⣿⣻⡣⠀⠀⠀⠒⠀⣿⡖⢒⣒⠒⠚⣿⡏⢫⢾⢿⣶⠖⠀⣿⣿⠙⣿⣿⡯⡌⡿⠁⠈⠀⠈⣿
⠀⠀⠀⠀⠈⠉⢹⡅⠘⢃⣠⣶⠀⠀⠀⣼⣷⣌⣉⠁⢀⣿⣿⣴⣾⣀⠀⠁⠀⣿⣿⣠⣿⣿⡇⠃⣴⠀⠀⠀⠀⣯
⠀⠁⣸⡄⠀⢒⣉⡋⣉⡉⠉⠙⠀⢰⣶⣿⡟⢻⣿⣿⡟⢿⢿⡿⠿⠿⠶⠶⠒⠚⣩⣉⣉⣉⣁⣤⣤⡔⠂⠀⠉⠁
⠀⠈⠉⣉⣙⠛⠛⠛⠻⠷⠬⠷⠤⠜⠟⠛⠛⠋⠉⠉⢁⡀⠀⢦⡄⠀⣤⣀⣴⣽⣿⣯
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠤⢤⣤⣀⣀⡉⠉⠉⠉⠙⠛⠚⠻⠻⠿⠿⠿⠿⠶⠤⠄

⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⡉⠑⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠶⠒⠂⠀⠀⠙⠒⠀⠄⡀
⠀⠀⠀⢀⡠⠴⠛⠛⠉⠉⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀
⢀⡴⠊⠁⠀⠀⠀⠀⢀⣀⣀⣠⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⡏⠀⠀⠀⣀⠔⢲⣿⠭⠥⠔⠓⠒⠀⠀⠀⠉⠑⡄⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⢳⣀⡤⠖⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠃⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⣰⠋⠀⠀⠀⣀⣠⡤⠶⠖⣒⣈⣉⣩⠭⠉⠒⠲⣄⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⢿⡀⠀⣀⡻⠮⠟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠏⠀⠁⠂⠆⠀⠀⠋⠈⠀⠇
⢈⠟⠋⠁⠀⠀⠀⠀⢀⣀⣠⣤⠤⠤⠰⠶⠶⢋⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀
⣿⠀⠀⣀⠤⠶⣚⣛⣯⠥⠶⠒⠚⠃⠉⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢱⠀⡄⠁
⠙⢦⣀⣀⡴⠋⠉⠀⠀⠀⠀⠀⣀⣀⣀⣀⠤⠴⠋⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⢀⡞⢡⡞⠀⠀⢀⡤⠶⢶⣽⡋⣩⠟⠉⠋⠛⠢⡄⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⢸⠀⠸⡀⠀⢀⡿⢱⠀⠀⠀⠙⡏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠁
⢸⠀⣀⣈⣩⣭⠤⠼⠤⠤⠤⠤⠧⠤⢤⣤⣀⣀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⢳⠀⠀⠀⠀⠀⠀⠠⠁
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⡀
⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠈⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠱⡀⠀⠀⠀⠀⢰
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣼
⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠼⠀⠀⠀⠀⣸⣄⣀⣀⣀⣴⠁
⠀⠀⠘⢦⠤⠄⠲⠶⠒⠒⣶⣒⣒⣲⡒⠲⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠴⠤⢀⠃
⠀⠀⠀⢸⠀⠀⠀⣀⣀⣀⣀⣀⡀⢀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠃⠀⢀⣀⣀⣀⣀⣂⣉⣭⣽⡇⢈⠆
⠀⠀⠀⢼⠃⠀⢤⣤⣤⡤⠤⠤⠴⠶⡒⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠃⠀⠀⠀⠀⣀⣀⣈⣭⣤⣵⡶⢈⠆
⠀⠀⠀⣼⡃⠤⠤⠤⠤⠴⠶⠿⠿⠝⡓⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⣐⣢⣭⣭⣥⢈⠆
⠀⠀⠀⢼⡃⠤⠤⠤⠤⠴⠶⠿⠿⠿⡖⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠉⠉⠉⢉⣩⣭⣥⣶⡎⣈⠂
⠀⠀⠀⠘⢧⣀⠀⠐⠒⠒⠒⠺⣋⣗⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠉⠉⡭⢛⠜⠁
⠀⠀⠀⠀⠀⠉⢻⠛⠛⠛⠛⡿⢹⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⡛⣛⣷⠞
⣤⣤⣤⣤⣤⣤⣤⣽⣷⣶⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡿⠿⠿⠿⠿⠿⠿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠋⠁⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠉⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⣤⣤⡄⠀⣿⣿⠀⣤⣤⡄⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿⠀⣿⣿⡇⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠿⠿⠇⠀⠿⠟⠀⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡆⣸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣠⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣶⠾⠿⠟⠋⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠚⠉⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠂⠀⠀⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⡀⠄⠒⠀⠐⠂⠄⠀⠀⠈⠖⠀⠀⠀⠀⠀⠂⠀⠀⡆
⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠠⠒⠀⢀⠑⠄⢀⡏⠀⠁⠀⠀⠀⠀⠀⠐⠇⠠⠈⢀⠤⠀⠀⠀⠀⠀⠀⠄⠀⡅⠠⢤⠀⠠⠈⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠂⢁⠐⠘⠆⠀⠀⠀⡂⠀⠀⠀⠀⠄⠀⠀⠁⠀⠑⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠒⣠⠁
⠀⠀⠀⠀⠀⢀⠀⢀⠔⠡⠀⠀⠈⠀⠈⡤⠀⠀⠄⠀⠀⠀⠀⠀⢀⠌⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⣆⠀⠇⠀⠀⠀⠠⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠒⠙⡀⠀⠀⣀⡅⠀⠀⠆⠀⠀⡀⠒⢤⠤⠤⡰⠋⠄⠀⠀⠀⠀⠀⢁⣀⠀⠀⠠⠌⠐⠀⠈⢐⢦⠀⠀⡤⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣢⠤⠤⡠⠐⣇⠀⠀⠌⠀⡄⠀⠀⠀⢀⡴⠒⠒⠂⢄⠑⠤⣄⡁⡤⠴⡇⠈⡆⠀⠀⠠⠀⠀⠁⠀⡀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠑⠀⠘⠊⢀⠀⠀⠀⢀⡁⠀⠀⢀⠃⠔⣮⠉⠨⠄⣏⠄⠠⠜⢀⠀⠘⠄⡅⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡀⠀⠀⠀⢲⠩⠘⠂⠠⣠⠔⠀⠐⠀⡀⠀⢡⠆⠀⢀⠈⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣃⢀⢘⠀⢂⠰⠂⠒⡒⠊⡅⠀⠰⠀⠐⢀⡄⢴⠐⡡⠂⡅⠀⠀⠀⠀⢐⠋⠁⠀⠀⠀⢋⢊
⠀⠀⠀⠀⠀⠀⠀⠀⢢⠭⠉⠈⠉⠀⠈⡚⢸⢴⡀⠁⣀⠀⢠⠅⠸⠁⠀⠈⠉⢧⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠁⠀⠀⠀⠈⠀⠀⠈⠝⡂⠀⢀⠑⣀⢡⡅⠀⠀⢸⢄⡇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠸⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢰⠀⠀⠀⢾⡇⠀⢰⣷⠋⠇⡏⠉⡉⠙⢸⠂⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠨⣄⣀⣠⣭⡄⡄⠰⠰⠗⠸⠀⠁⠒⠰⠐⠆⠈⠀⠀⠀⠀⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢘⣇⠡⠀⠀⠀⠀⣾⠀⢹⡞⠀⠊⠀⣬⣨⣄⠀⠀⢰⠴⠤⠆⠀⠀⠀⠀⠀⠀⢨⢃⡓⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠧⠷⠾⠼⠶⠧⠿⠼⠼⠧⠧⠾⠼⠶⠧⠷⠾⠼⠶⠧⠷⠼⠼⠧⠷⠾⠼⠼⠧⠷⠾⠼⠶⠧⠷⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⢻⡹⡏⠉⠉⠋⠙⢟⢻⠉⠉⠉⠉⠛⡟⡟⠉⠉⠉⠉⢻⠻⡏⠉⠉⠉⠉⠛⡟⡟⠉⠛⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⣸⢸⠀⠀⠀⠀⢀⡇⡇⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⣠⡇⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⢠⠞⣽⢸⠀⠀⠀⡠⢋⡇⡇⠀⠀⢀⡔⣹⡇⡇⠀⠀⢠⢞⡵⡇⣿⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⣰⠣⠎⣸⢸⠀⠀⡼⠱⠃⡇⡇⠀⢠⠎⠜⣹⡇⡇⠀⣰⠣⢎⠐⡇⣿⠀⢠⠎⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⢰⠃⢂⠎⢾⢸⠀⡼⠐⣰⠃⡇⡇⢀⠏⠐⡼⢹⡇⡇⣰⠃⢃⠎⠞⡇⣿⢀⠎⠐⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣧⠇⢀⡄⠈⣼⢸⣼⠁⢠⠀⢁⡇⣧⡞⠀⡤⠀⣹⡇⣧⠇⢀⡄⠈⡜⡇⣿⡞⠀⡤⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠏⠀⠀⠀⠘⢹⣸⠃⠀⠀⠀⠊⣇⡟⠀⠀⠀⠀⢻⣇⠏⠀⠀⠀⠘⠁⡇⡿⠀⠀⠀⢸
⠀⠀⠀⡠⢤⣤⢞⡩⣍⠲⡤⣬⢏⡔⣒⢢⣀⡁⡸⠃⢂⣀⡄⠈⢀⡟⠐⠂⢀⠀⠉⢰⠏⠐⠀⡀⠀⠁⣀⡟⠐⠂⢀⠀⢹
⠀⣠⠞⠀⠀⢯⣏⢀⡀⠀⠀⠈⠛⠙⠀⠁⠉⠉⣧⡯⡾⠶⣄⣢⠎⡰⣣⠃⠋⢐⡷⢃⡜⡼⡐⡁⠘⣠⠟⡰⣣⠃⠋⠀⢻
⠀⢧⠀⠀⡔⠋⠁⢏⠙⠋⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⢳⣴⣵⣳⣳⡷⠋⣠⢞⣜⣝⣝⣽⠟⢁⣞⣵⡵⣱⣳⢣⣾
⠀⠀⠭⠭⠭⠭⠉⠉⠉⣭⣭⡍⠉⢉⣉⠉⢉⡉⠉⣉⠩⠭⡉⠉⠹⠿⠏⠉⠉⠉⠩⠭⠭⠭⠉⠭⠉⠩⠭⠭⠭⠉⡩⠭⠉
⠀⠀⠠⠬⠭⠉⠉⠥⠤⠀⠰⠆⠀⠐⠒⠒⠒⠀⠀⠐⠲⠈⠉⠉⠀⠉⠉⠩⠍⠉⠉⠡⠄⠀⠀⠒⠒⠂⠘⠛⠦⠌⠵⠶
⠀⠀⠀⠀⠚⠛⠓⠀⠀⠀⠉⠁⠀⠀⠉⠉⠉⠉⠀⠀⠉⠉⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠂⠀⠀⠀⠉⠉⠉