Beverage Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠑⢄⣀⣉⠀⠀⠀⠈⠉⣼⡈⡀⡄⠀⠋⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡅⣀⣤⣿⡟⣡⣵⢒⣶⣖⢂⣿⡀⠈⢁⣬⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢻⣿⣿⢿⣏⣤⡖⢉⣉⣺⢷⣅⢹⣇⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⢤⣉⠁⠇⠀⠋⠁⣀⣠⣾⣠⣿⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣠⣤⣦⠀⠀⠛⠛⠋⠉⠙⠉⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠈⠉⠘⠦⠂⠒⠋⠁⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡀⣤⢒⡾⡟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣴⡅⢾⣥⣤⣥⡤⣥⣤⣴⣶⣭⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⡀⠉⠉⠛⠛⠛⠛⠛⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣷⣶⣶⡶⠶⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⢰⣶⣆⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠠⣄⠀⠀⠀⢸⣻⢶⡠⡀⢹⠄⠀⠘⠇⠀⠀⠀⠀⠐⢆
⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠐⠀⠠⡟⠀⣤⠂⠈⡇⠀⠀⠘⣿⡗⠋⠡⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⣵⣄⣀⡀⠀⠀⡀⣀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⢨⣍⣀⣀⡀⠀⠀⠀⢈⣻⣍⣏⣐⠞⠁⠀⠀⠀⣿⣇⠶⢦⠠⠈⠳⠈⡇⠀⠀⠀⠀⠀⠀⠈⢻⣿⠖⠂⣤⠄⠀⢠⠋
⠀⠀⠀⠀⠀⠀⢸⣡⣍⠁⠿⢀⠀⠀⠈⠠⣿⠋⠁⠀⠀⠀⠀⠀⢺⣇⠳⣀⠶⠦⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣌⠁⠖⡔⠁
⠀⠀⠀⠀⠀⠀⠸⣿⢿⡷⠾⠽⠶⢶⠖⠈⣻⠀⠀⠀⠀⠀⠀⠀⣾⣄⢴⣟⡀⢠⡆⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠂⠈⠏
⠀⠀⠀⠀⠀⠀⠀⢿⡟⠓⠂⠀⢶⠀⢠⠋⡏⠀⠀⠀⠀⠀⠀⠀⡿⣟⣟⠋⠀⠻⠄⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⢸⡇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⠟⠳⢴⡀⡁⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢸⠸⡏⠀⠀⠈⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣠⣄⠀⠣⡖⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣋⠠⠃⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠤⠄⠤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⡛⠛⠓⠓⠛⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⣀⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣀⢀⠠⠐⠀⠀⠡⠄⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡄⠠⠠⠄⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡟⠛⠉⠀⠀⠈⢦⢀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣤⠀⠀⠠⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡆⠒⢀⡬⠷⣄⣻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣿⣀⠀⢀⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⡛⠳⠖⠉⠙⣣⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠿⠛⠋⠉⠁⠿⠀⠀⠀⠀⠀⠀⣄⢦⢀⣠⠀⠀⢠⣠⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⠯⢅⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠎⣤⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⣧⣄⡰⠂⠁⠈⠣⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⠀⡀⡀⠂⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠋⠹⡀⠶⠎⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⡀⠀⠈⢀⣤⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠾⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠶⠄⠸⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠚⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠤⠀⠐⠁⠀⠀⠀⠀⠀⠀⠙⠚⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠗⠀⠀⢀⣀⣀⠀⠀⠀⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣴⠞⣛⡤⢣⠤⠌⡭⣄⠀⣀⣀⣇⠀⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢰⣧⣀⠋⠰⠇⠤⠀⣁⠉⠀⠀⣠⠙⢠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣶⣶⣤⢸⣦⣤⡇⠀⠀⣠⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⢿⣿⣶⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⢿⣽⣾⣶⣾⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠟⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⣠⠈⠋⠈⠀⡿⣁⠓⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠠⣞⢻⡀⠀⠀⠀⣷⡀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢀⡿⠛⠃⠁⣀⠀⠀⠛⠃⠀⠀⠐⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⡎⠉⣀⣀⣠⡤⠴⣶⣶⡤⣤⣤⣀⣈⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠚⢻⡟⠀⠀⠀⣿⣟⣶⣄
⠀⠀⠀⠀⠻⠋⠉⠀⠀⢀⠀⣾⠟⠠⡄⠀⢽⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠⡔⠻⣄⠁⠘⣧
⠀⠀⠀⠀⠀⣷⠀⠛⠀⠀⢠⣿⣧⠀⠀⠀⠈⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣀⣀⣠⣾⡠⠶⡙⠀⣠⣿⣧
⠀⠀⠀⠀⠀⠹⡀⠄⠀⠀⠀⢈⡇⢁⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠘⢻⡿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠿⠀⠈⠁⠒⠻⢿⣥⣤⣶⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣎⣿⣿⡿⣃⡁⠁⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⢠⡆⠠⠄⠀⣀⡀⠀⠀⢩⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⡤⠿⡇⠀⠀⠀⠀⠌⢹⡇
⠀⠀⠀⠀⠀⠀⠘⢁⠀⢀⠀⠈⠱⠄⠠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠿⣿⣿⣿⣾⣷⡄⢲⠄⢀⣤⣿⠅
⠀⠀⠀⠀⠀⠀⠀⡾⣛⠉⠀⣀⡄⠀⢐⣬⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡖⠷⣽⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠾⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⢿⡿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢀⠶⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢾⡿⠋⠉⠙⠻⢄⠀⢸
⠀⠀⠀⠀⢀⡀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣟⢣⡉⣀⠀⠀⠠⣼⢤⡀
⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⣿⡟⣩⠉⡀⠀⠀⠀⠀⣈⠘⠃
⠀⠀⡄⠀⣀⠂⠐⢠⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠛⠀⠆⠀⠈⠀⠀⣤⣶⡧⣶⣤⣀
⠀⠀⠀⠀⠹⣿⡿⠿⠗⠐⠂⠀⠀⠀⠀⠀⠀⠄⠀⠀⡐⠀⠀⠀⠀⢠⣿⣿⡿⢧⣜⠀⠀⠀⠀⠀⠀⠸⠋⠀⣾⠀⣸
⠀⠀⠀⢀⣤⠟⣅⣀⡀⠀⠀⠀⠀⢀⣤⡀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠘⣿⡟⣿⡟⠻⠀⠀⠀⢀⡠⠶⡀⠄⢠⣿⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢆⠀⠀⡀⢀⣿⠃⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠸⣿⣯⠔⠀⣻⢦⣤⡈⠁⠈⠀⠄⣠⡍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⡋⠑⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⣹⡅⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢀⠀⠰⡗⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣁⠄⠀⣦⠛⠃⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⠀⣿⠛⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣐⢶⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣍⠉⠉⠉⣈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⡍⠀⢀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣭⣍⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠀⠀⠀⠁⠀⢀⡰⢎⡠⢈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠘⠁⠀⠢⠀⠀⡀⠐⠀⠔⠀⠀⠏⠐⠁⠠⠧⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠂⠀⢱⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⠌⡰⠀⠀⠀⠀⠀⠀⠀⠀⡆⠐⠒⠓⠃
⢀⠂⢀⠈⠉⠉⠉⠓⠲⡦⠄⠀⠀⠀⠑⢄⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⢀⢃⠠⠀⢀
⠀⠀⠉⠁⠒⠒⠒⠒⠚⠁⠃⠀⠀⡀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠈⠀⠀⠀⠁
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠐⠀⠀⠀⠈⠒⠤⠄⠀⣀⠀⠀⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠒⠈⢁⣀⣀⡂⢐⠠⣀⢄⠀⠀⢡⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠂⠁⠉⢉⣀⣠⣀⢍⠶⡐⠂⡄⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⡀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⡱⡀⠀⣸⠀⠀⠀⠈⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠁⠀⠀⠀⣓⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠩⠔⠁⢠⠁⠀⠄⠀⠠⠀⠐⡀
⠀⠰⢄⡀⠀⠀⠀⠀⢀⠠⠂⠀⠀⠀⠀⠘⠈⠐⠀⠙⠀⠠⡦⠀⢀⠐⠒⢄⠀⠀⠀⠀⢀⡠⠓⠒⠤⠀⠈⠀⠀⠀⠀⠂⠐⠀⠰
⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠐⠥⠃⠐⠄⣄⡀⠉⠓⠒⠚⠉⠀⡀⠼⠈⠀⠘⢌⠀⠀⠀⠀⠒⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⣘⡄⠀⠀⠀⠀⠀⢰⣊⣉⣀⣀⣉⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠤⠤⠈⢺⢦⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡀⢤⣤⡤⠄⠀⠥⣠
⠀⠀⠀⠀⣀⠀⠀⠀⣀⣀⠀⠀⠀⢸⢄⡀⠀⠀⠀⠀⣀⣠⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐⠁⠘⠛⣓⣚⣀⣐⠙
⠠⠀⢀⠀⠀⠀⠀⠀⠀⣀⠈⢲⠀⢸⠀⠀⠉⠉⠉⠉⠀⢸⠀⠀⠀⠸⠆⠀⠈⠈⠉⠁⢀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀
⠓⠂⠀⢀⣀⠀⢀⣀⣀⣀⢼⡾⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⡀⠀⠈⠀⠀⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠠⡅⠒⠀⠤⠤⠤⠀⠀⠒⢩⠇⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⣀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⢀⠌
⠀⠈⠲⠤⣀⠀⠀⢀⡠⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⠁⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠢⠖⠠⢄⠀⠀⠀⠀⠘⠂⢄⡀⠀⠀⣀⡠⠴⠃⠀⠀⠀⠀⢀⠀⡄⢦⠀⢀⠀⠀⠀⠀⠀⠡⡀⠀⢸
⠀⠀⠀⠄⢀⣀⡀⠀⠜⠀⠀⠀⠀⠀⠑⠦⠤⣀⣀⡠⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠃⠀⠀⠀⠀⠀⠙⠀⠘⠒⠐⡀⠠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠎⠛
⠀⠀⠀⠀⠈⠁⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠐⠀⣼⣿⣿⠀⠀⠀⠀⠀⢸⠏⠁⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⣂⣿⣶⣶⣨⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⡴⠋⠉⠹⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⢸⡿⠏⠙⡈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⢠⣿⡤⡀⠠⣿⡇⠀⠀⠀⠀⠀⠀⠣⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⡸⠁
⠀⠀⠀⠀⡜⠛⠿⠿⠿⢛⣥⠀⠀⠀⠀⠀⠀⠀⠲⣿⠀⠀⠀⣄⣀⣴⠀⠀⠀⠀⠀⠀⠀⠘⡦⢴⠁
⠀⠀⠀⠀⢿⣤⣤⣠⣤⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⣟⠉⠑⠛⠋⠉⣽⠄⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢸⠁⠹⣿⡿⠋⠉⡇⠀⠀⠀⠀⠀⠀⠔⠛⢢⣄⡐⠊⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠘⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠾⠀⠀⠀⠀⠀⣦⠀⠐⠒⠒⠒⠒⢒⣲⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠿⣿⡠⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⡟⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⢠⠀⠀⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⣛⠛⠉⠁
⠀⠀⠀⠀⠀⢸⣶⡂⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⣾⢿⠿⠴⢿⡇⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢻⠅⠀⠀⠀⠀⠀⠠⣀⣐⡲⠳⠏⢒⣀⡠
⠀⠀⠀⠀⠀⠙⠙⠛⠓⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⡀⠐⢦
⠀⠀⠀⠀⠀⠀⣠⢧⠀⢘⢏⣣⠀⠀⠀⠀⠀⡴⠒⢲⡄
⠀⠀⠀⠀⡴⣊⡀⢀⣉⡺⣈⠥⠶⠀⠀⠀⠀⡱⠒⠺⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢰⠮⠷⠆
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⢀⠃⠀⠀⡰⠂⠀⢀⠻⡄⠀⠀⠀⠀⠀⣟⣛⣻⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢸⠦⠶⠶⠤⠤⠤⡗⢒⣺⠀⠀⢰⠁⠀⠠⠈⠀⢹⠀⠀⠀⠀⢀⠃⠤⠚⡄⠀⠀⠀⠀⢸⠀⠀⡀
⠀⠀⠀⢸⠓⠛⠒⠚⠶⠾⡿⠛⢹⠀⠀⢸⠗⠶⠶⠶⠞⣛⠀⠀⠀⠀⡘⠢⣀⡤⠻⡄⠀⠀⠀⡸⠊⠉⢡⡀
⠀⠀⠀⢸⠀⢀⢤⢀⠀⠀⡇⠀⢾⠀⠀⢸⢠⠖⣴⣶⠍⠽⡆⠀⠀⠀⠁⠀⡀⠀⠀⣷⠀⠀⠘⠀⠀⠀⠀⠁
⠀⠀⠀⢸⠀⠈⡛⢊⠀⠀⡇⠀⢸⠀⠀⢸⡰⣤⠼⠗⠲⠆⡇⠀⠀⢰⠀⠀⠁⠠⠀⣿⠀⠀⡄⢀⢤⡀⠀⠀⡆
⠀⠀⠀⢸⡀⠀⠳⠚⠀⠀⣇⠀⢸⠀⠀⠘⣄⣀⣀⣠⣤⣤⠇⠀⠀⢸⠀⠀⢠⠀⠀⣿⠀⠀⢳⡅⠀⢸⠉⠉⡇
⠀⠀⠀⠀⡇⠀⢈⣁⠀⠀⣿⠂⢸⠀⠀⠀⡗⢀⡀⠰⠆⢸⡀⠀⠀⢸⠀⠀⠀⢀⠀⡿⠀⠀⢰⠈⢈⠅⠂⠉⠁
⠀⠀⠀⠀⡇⢀⣐⣃⣀⠀⣿⣾⢸⠀⠀⢠⠃⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⣧⠀⠀⢸⣜⠥⠤⠤⠤⡀
⠀⠀⠀⠀⡇⠸⠀⠀⣈⠇⢮⠛⢸⠀⠀⠘⡀⠀⠀⢀⠀⢰⡟⠀⠀⠸⣀⠀⠀⠀⢉⡿⠀⠀⠸⡀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⡇⠀⠀⠀⢀⣀⣸⡰⠋⠀⠀⠀⠙⠒⠚⠉⠙⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠀⠀⣀⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⠀⢀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠴⠦⠄⠀⠀⠀⠀⠁⠀⡟⠀⠀⠸⠀⠀⠀⠀⠒⠒⠒⠒⠒⠶⠶⠶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⢀⣠⣤⠶⠶⠛⠋⠁⠀⢀⣠⠴⠒⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠶⣤⣄⡀⠀⢀⣀⡀⠀⠉⠛⠻⣦⡀
⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⠀⠠⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⡀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠈⠙⣳⡄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⢶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡶⠶⠶⠞⠋⠀⠀⠀⢀⣀⣤⣶⠿⢃
⠀⢀⠙⠳⣶⣶⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⠶⠶⠾⣿⠟⠛⠉⠀⠀⠈
⠀⠈⠉⢉⠱⠿⢤⣭⣉⣉⣙⠛⢛⣛⢛⠓⠛⠛⠋⠀⠀⣓⠒⣛⠛⠛⠛⠛⠛⠋⣉⠉⠉⠉⠁⢀⡄⠀⠀⠀⠈⠀⠀⠀⢀⣀⡠⠄
⠀⠀⠀⣾⠀⣀⠀⠂⠈⠉⠉⠘⠛⠛⠛⠛⠋⠉⠁⢀⠛⠉⠉⠉⠀⠀⠀⠀⠔⠊⠉⠑⠀⠀⠀⠀⣀⣀⣤⣤⡤⠶⠚⠛⠋⢹
⠀⠀⠀⠸⠀⠈⠙⠛⠓⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠶⠶⠾⠟⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢠⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠘⣧⠀⠀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠶⢀⢸⡇
⠀⠀⠀⠀⠀⣿⣎⠀⠀⠀⠀⠀⢠⠖⠒⠒⠒⠀⠀⠀⠀⠀⣠⡴⠶⠾⠶⢤⣄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⢻⡏⠓⠤⠄⠀⢀⡎⠀⠀⠀⠀⠀⠀⠠⠐⠉⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠈⠀⠀⢐⡒⣤⠶⢷⣦⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⠖⠒⠒⠉⠉⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠠⠖⠋⠁⠀⠀⠀⠀⠈⠃⠌⠉⣉⠝⠋⠉⠉⠉⣉⡉⠉⠙⠛⠒⢤⡀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⡼⠃⠀⠀⠤⠀⠀⠈⠉⢄⡀⠀⠀⠀⠉⠆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⣷⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢻⡆⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡀⠀⠀⡀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠸⠀⠀⠀⣧⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⢤⣤⣤⡤⠐⠂⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠙⠀⠀⠀⢱⣀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⢀⡠⠤⠐⠛⠀⣀⣉⠤⠤⠤⠤⡴⠒⠓⠀⠀⠀⠠⠠⠀⠈⢁⡈⠁⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⠀⠀⠀⠀⢾⣋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠷⣶⣄⣀⠀⠈⠛⠂⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠛⠷⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠶⠛⠋⠉
⠀⠀⠀⠀⠘⠷⣤⣤⣀⣉⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠋⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠦⠤⢠⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡄⢰⣿⣦⠀⣰⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒⠚⠛⣿⣛⣿⣿⣿⡿⣿⣿⣿⣿⣿⡟⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠶⠤⠤⠂⠀⠲⣟⠻⣿⡿⠿⣉⠐⠒⢦⣝⢿⣿⣿⣿⢸⣿⠇⠳⡄
⠀⠀⠀⠀⠀⠀⠒⢲⠠⣀⣀⣀⠀⠀⠀⠈⠉⠉⠉⡁⠉⠀⢨⣉⠉⠉⠙⠟⠿⠏⢛⢣⣤⣠⣿
⠀⠀⠀⠀⠀⣿⠀⠸⣄⡙⠟⠛⢿⣧⣄⢠⣄⣠⣼⠇⠈⠁⠀⠈⠳⢦⣤⢽⣔⡄⠈⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠛⠶⠾⠻⠛⢋⠲⣌⠻⠿⢞⡉⠉⠀⠀⠀⠈⣿⠶⠖⠀⢸⣿⣷⠈⡽⣹⣿⠇
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠑⠌⠛⠛⢋⣠⣤⡀⠀⠀⠀⠈⢑⣠⡄⣼⣿⣿⡆⢸⣿⡟⢀⢠⡀
⠀⠀⠀⠀⠀⠀⠘⣤⠲⣦⠀⠀⠀⠀⠀⠘⠛⠁⠈⠉⠀⠀⠀⠉⢉⠉⡰⠃⣽⣿⠇⠸⣿⣧⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢯⣀⣤⡀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠻⠞⠧⡴⢛⣿⠈⠳⣌⣹⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡿⢱⣿⠀⠀⠀⠀⠀⠁⣿⡿⠆⠀⠀⠀⠀⠀⡟⠁⣿⣾⠇⠙⠒⠂
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⡽⠃⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠿⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠙⠆⠐⠆⠑⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣤⢾⣋
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠃⣀⣤⣤⣄⣀⣀⡤⠉⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠈⣧⡀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠘⠿⠿⠋⠀⠀⠀⢠⡔⠒⠉⠉⠉⠉⢉⣰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⣹⠉⠙⠛⠛⠛⠋⠉⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠠⠶⠶⣏⠘⠀⠀⠀⠀⠀⣀⠀⠀⠗⠚⣶⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠀⠈⢻⡅⣀⣙⠀⠏⣩⡟⣁⢀⡴⠋⠁⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⡆⠀⣖⡚⠇⢹⣿⣇⠀⠁⣛⡟⢃⡴⢊⡴⢧⠀⢀⣠⠞⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢡⡾⠁⠀⠀⠙⣿⣾⣿⣿⡆⠘⣻⣶⣫⡴⢋⣤⡾⠶⣫⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣦⡄⢀⡾⠁⣿⠀⠀⠀⠀⠀⢨⣿⣿⣭⠀⣀⠀⣻⣟⠶⠋⣡⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⡇⢹⡾⠁⠾⣧⣀⡀⠀⠀⠀⠈⣟⣛⣋⠀⡈⢙⡛⡋⢀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠞⠀⠀⠠⣬⣭⣿⡄⠀⠀⠈⣿⣿⠾⠿⠓⠾⢷⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠰⢤⣈⣹⠀⠀⠀⠙⠿⠶⠶⠶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⣀⣀⣀⣈⠉⣳
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⣰⡏⠀⢈⣥⠞⠋
⠀⠀⠀⠀⣿⠃⠀⠀⠀⢀⣴⠟⠀⠀⠈
⠀⠀⣴⢶⣯⠀⠀⠀⢶⡟⠁
⠀⠀⢰⡞⣻⣷⣀⠀⠘⠃⣶⡄
⠀⠀⠀⠉⣅⣻⡿⣿⢋⣼⣏⠁
⠀⠀⠀⠀⠙⠉⢠⣿⡆⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⠂⠀⢠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⢀⣴⣿⣿⣿⣿⣿⡀⠀⣹⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣆⠂⣼⣿⣿⣿⣿⣿⡯⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⡟⣡⣼⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠙⢿⡆
⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⣉⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⠋⣙⠻⣿⣿⣿⣿⠟⠁⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⠁⠉⠛⠋⠀⠀⠀⠀⠀⠈⡙⢂⠆
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣧⣈⡀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠈⠉⠉⠉⠀⠀⠀⣿⠿⠿⠋⠉⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠂⢐⡴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⣐⡞⢤⡤⠶⠀⠀⠀⠀⢸⢇⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠢⢄⡀⠀⣀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢱⣦⣄⣀⣠⡴⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠋⠉⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠧⠀⠀⠒⠂⠀⠒⠀⠄⡀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⡄⠀⢀⡀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⣠⣿⡿⠀⠀⠸⣿⣿⡇⠀⠀⠀⢤⢾⣧⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡅⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⢀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠿⣿⡿⠀⢀⣴⣦⡄⠀⣴⣶⡄⢠⣄⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡙⢣⣤⣄⡀⠀⠀⠀⠈⠻⠿⠟⠀⠻⡿⠇⠘⢿⡇⠀⠐⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣼⠿⠿⠃⠒⠈⠀⠈⠋⢠⡄⠀⠀⢠⣶⣄⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠋⣆⣰⡎⠀⠀⠀⠀⠀⠀⢸⣧⣄⠀⠸⣿⠟⠀⣾⣿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣴⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠀⢠⣴⣤⠀⠙⠁⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠻⠿⠟⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⢀⣤⡶⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠀⠀⣰⡄⠉⠁⠀⠈⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠿⠇⢀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣦⣶⣴⣶⣦⣴⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶⣶
⣿⣿⣏⣀⣁⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠂⠉⠁⠀⠀⠈⠁⠒⠤⡀
⠀⠀⠀⢀⣀⡀⠀⡠⠊⣁⣀⣀⠀⠐⠒⠤⠤⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⡀
⠀⠀⠀⡥⠄⡶⠉⠀⠢⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢉⠄⠀⠀⠀⠈⠑⢍⠁⠒⠠⣀
⠀⠀⠀⠀⠰⢡⠀⠀⠀⠙⡆⠀⡗⠒⠠⠴⠤⠤⠤⠤⠽⠉⠐⢶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢌⠖⠠⠃⠀⣴⠂⠀⠀⠀⠀⠑⡀⠀⠈⢳⣄⣀
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠙⠀⢰⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣂⢀⡁⢀⡌⡇⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠈⢯⠉⢆
⠀⠀⠀⠀⢠⠀⠁⠀⠀⠀⠀⠀⡌⠀⠀⠀⠈⠁⠀⢀⠖⠒⠒⠀⠓⠀⠉⡏⢑⡄⠀⠀⠀⠀⡍⠉⠀⠀⠀⠀⠈⠙⠃⠒⠀⢤⢀⣸⠀⠀⠀⠀⣼⠀⢨
⠀⠀⠀⠀⠈⠆⠀⠀⡜⠉⠱⠉⠀⠀⠀⠀⠀⠀⡀⢼⠀⠐⣆⠀⡤⠴⠾⣏⡁⠁⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢈⠰⢄⠀⠀⠀⠀⢠⠀⠈
⠀⠀⠀⠀⠀⠈⢆⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠣⣼⣆⠬⠁⣰⡇⠙⠢⠴⡖⠁⠀⠀⠀⠘⠈⠀⡄⠀⠀⠀⠓⠀⠀⠀⠀⠘⠤⠀⢄⠀⠀⠀⡌⢰
⠀⠀⠀⠀⠀⠀⠀⠑⠄⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢚⡩⠋⠉⡆⠀⠸⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢀⠜⠀⢸⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠂⡄⠀⠀⠲⡒⠉⡇⠀⢀⠜⠀⠀⠀⡎⠀⠀⠀⠀⠀⠘⢄⡠⠐⠈⠉⢘⠅⠀⠀⠀⠀⠀⠀⠀⢈⠦⠻⠀⠀⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠒⠀⢤⣿⡦⣤⠴⠋⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠎⠀⡠⠶⣶⢻⠀⠀⠀⣀⣀⠤⠔⠊⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠒⠢⠼⠛⠉⠱⡀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠈⠈⡎⠉⣿⠁⠀⠀⣸⠦⡀⠀⠀⠀⠸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢰⠁⠀⠀⠀⠀⠀⡸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⣇⠀⠘⠳⠶⠚⠋⠀⠈⢢⠀⠀⠀⠑⢄⡀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠳⠄⠀⠀⠒⠁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⣼⠀⠀⠀⠀⣀⣀⠀⠀⢡⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⣂⡀⢰⡀⣀⠤⠚⡅⠀⠀⡠⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣟⠀⠁⠀⠀⠀⠀⣀⠬⠦⠞⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡀⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡨⠋⠉⠘⠃⠉⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠲⠶⣐⣂⣙⢲⣖⣖⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠈⠀⠔⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄⠀⠒⠀⢠⢠⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⣉⣉⡉⠈⠉⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⠸⠖⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⢀⣀⢀⣀⣀⠀⢀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⣈⢉⣉⣀⡀⠬⢥⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠸⠀⠀⡇⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠀⠀⢆⡀⣙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠤⠴⠃⠀⠀⡗⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠄⠤⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⠜

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

⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠀⠀⠂⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠎⠀⠖⠉⠉⠀⢀⢔⠀⠔⢁⠀⠈⠦⣀⡀
⠀⠀⠀⡴⠂⢉⠄⠉⠔⠁⡔⠀⠀⠈⠀⣄⠈⠢⠀⠀⠀⢍⢢
⠀⠀⢸⠠⠊⡡⠂⠀⠖⠊⡤⠒⠁⠀⠀⠈⠙⠐⠢⡀⠂⡌⠀⡀
⠀⠀⡜⠁⠀⠀⠐⠒⠒⠊⠀⣠⠀⠀⠀⠀⠀⠀⠀⠈⠲⠐⠀⢠
⠀⠀⣤⡊⣇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠚⠀⠀⠀⠀⠣⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠠⠂⠤⣀⡀
⠀⠀⢠⠈⠋⠀⠀⠀⠈⡃⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⢂⢧⡺⣖⡇⣦⡄⡀⢔⠄
⠀⠀⠈⢂⠸⣄⠀⠀⠀⢣⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢧⡄
⠀⠀⡈⠀⠀⠈⠳⣄⠀⠀⠙⠒⠀⠀⠐⠁⠀⠀⠀⣠⢎⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠈⠀⠉⠉⠈⠀⠉⠒⠠⡤⡤⠤⡤⡠⠄⠒⢭⠤⣨⠡⠃⠀⠀⠀⠀⠀⠀⠀⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠰⠂⠀⠀⡠⠂⢰⠪⡕⣒⡥⠇⡄⠀⠀⢀⣄⠀⠀⡔⠉⠐⡄⠀⠀⠀⣄⠀⠀⡰⠂⠀⠀⠀⠀⢀⡳⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⡀⠤⠈⠀⠀⣌⢀⠀⠀⠀⠁⡆⠀⠀⢨⠻⠂⠀⠀⠀⠀⢸⠀⠀⠀⠈⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⣘⠘⠚⠀⠐⠂⢉⠘⠀⠀⠄⡆⢇⠀⢀⡎⠀⠀⠀⠈⠂⠀⠐⠀⠀⠀⠀⠀⠙⠢⢄⣀⣀⠀⣀⣀⣠⠴⠊⠁
⠀⠀⠀⠰⡅⠀⠁⠂⠀⠀⠃⠠⠤⣤⣀⡤⠬⠶⠎⠀⠀⠤⠤⠤⠤⡀⠀⢠⡄⠀⠀⠀⠀⣀⠀⣤⠏⠀⠋⠇⡠⠊⠉⡄
⠀⠀⠀⠀⠀⠬⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠈⠧⠈⣉⣉⡁⠀⡔⠀⢀⠀⠀⠘⢀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠠⠀⣀⠴⠀⠦⠜⢤⠷⠀⠀⠀⢀⠔⢂⠀⠰⠀⠘⠉⣑⠠⡐⠏⢀⠒⠂⠸⣅⠴⡋⢠⡤⣴⢏⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠠⠀⠪⠈⠁⠀⠀⠀⠘⡠⠃⠀⢹⡤⠭⠤⠤⠤⠤⠤⠤⠯⠤⢤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠉⠐⠄⠈⠀⠀⠀⠠⠄⠉⠀⠐⣔⠳⠀⠤⠠⠔⠒⠒⢒⠶⠶⢶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⢠⠀⠠⠊⠠⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣤⠃⡜⠁⠐⠈⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢒⠂⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣈⡂⣰⠁⠈⣉⠉⠉⢈⡇⠰
⠀⠀⠀⠀⠀⢐⠀⠴⠂⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⡆⠀⢤⠀⠀⠀⠖⠸⠁
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠸⠀⠀⠈⠀⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉⢀⢋⠀⠰⠄
⠀⠀⠀⠀⠀⠀⢀⡀⢤⠐⢒⠊⠁⠀⢡⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠴⠒⠴⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠀⠀⢁⡈⡒⠀⠠
⠀⠀⠀⢠⠊⢹⠀⠀⣈⠠⠼⠐⠲⠄⠤⠆⠀⠀⠀⠀⠀⠑⠀⢀⠀⠂⠀⠀⡀⠀⠀⠀⠙⠲⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠒⠛⠉⡟⠀⢀⠈
⠀⠀⠀⠈⠒⠊⠉⠁⠀⠀⠀⠀⠀⠧⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠈⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣖⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⢤⠅⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣩⠀⠙⡤⠤⡤⠤⠤⡤⠤⠤⠄⢀⡀⣀⠀⠀⠠⠠⠤⠄⠐⠒⢌⡀⠀⠀⡀⠄⠀⠀⠠⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⠨⠏⠌⠙⠒⠏⠉⢳⠉⠁⠈⠁⠉⠁⠐⠒⠒⠘⠋⠥⠹⠗⡠⢄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⣧⢝⡐⠒⠒⠂⢀⠼⠦⠴⠦⡤⠤⠤⡤⠤⠤⠤⠶⡂⠐⠒⠒⣂⠯⡼⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⠐⠢⣔⠁⠀⠀⠀⠀⣈⠗⢏⡀⠀⠀⠀⠀⠈⣢⠒⠉⠀⠀⠘⡇⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠄⡀⠀⠠⠊⠀⠉⠢⢄⡠⠊⠁⠀⠀⠉⠢⡀⡠⠔⠉⠀⠑⢄⠀⠀⣀⡆⠀⠀⠀⠀⢄⠀⠀⠀⡄⠀⢈⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡑⠣⣀⠀⠀⢀⠊⠑⠦⡀⠀⠀⢀⠴⠋⠢⡀⠀⠀⢀⡨⢶⠉⠀⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡜⠀⠀⠀⠑⣔⠁⠀⠀⠀⠈⢢⣔⠁⠀⠀⠀⢙⣤⡔⠁⠀⠀⢣⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠈⡶⢄⠀⠀⢀⠊⠀⠑⢄⡀⠀⡔⠁⠀⠱⡀⢀⠔⠁⠀⠑⡄⠀⠀⡠⢶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠉⢢⡃⠀⠀⠀⠀⡝⠪⡀⠀⠀⢀⠜⢣⠀⠀⠀⠀⢘⡔⠈⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠃⠈⠒⢄⢀⠌⠀⠀⠈⢢⡔⠁⠀⠀⠱⡀⣠⠞⠁⢡⢀⠜⢃⠀⡀⠀⠀⠀⠀⠀⠀⡀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⡳⡀⠀⠀⠀⠜⠢⠀⠀⠀⢀⣽⠀⠀⣀⠔⠁⠀⠈⠅⠂⠅⠀⠀⠀⠨⠅⠂⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠀⠬⠦⢀⣎⣀⣀⣱⡀⠠⠎⠄⠓⠉⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠐⠁


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⡁⠶⠂⠉⡑⡖⠒⠠⡉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡰⠋⠈⢄⢣⡀⠇⡇⢀⡠⢊⠦⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢰⣀⣀⣀⣈⣧⠙⠠⠖⣩⠖⠁⠀⢱⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠒⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡆⠐⠒⠀⢠⡄⢀⡀⢈⡉⣀⣐⣒⣚⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠓⠦⢤⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣂⣰⡀⣀⣴⢾⠶⠋⣽⡌⠣⡀⠉⠉⠹⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠍⠣⡉⠔⠀⠀⠀⣿⠈⠣⡹⢄⡰⢁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠈⠐⠀⠀⠄⣹⡀⡠⠜⢋⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢠⠶⢤⣤⣠⣤⠤⠶⠒⠲⢦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⠶⠦⣄⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢸⠀⠀⠀⠀⠀⢀⡠⠔⠈⠉⠑⡉⠒⠢⠤⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠈⣳⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠐⠁⠀⣠⠴⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡆⠀⠀⠀⠃⠀⠈⠀⠀⠘⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠐⣬⠒⠒⠒⠒⠒⠒⢦⡀⠀⠀⠀⠀⢸⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢳⠀⠀⠀⠈⢆⢄⡀⠀⢀⡠⠔⢸⠀⠀⢠⠀⠀⠀⠀⠘⢆⠀⠀⠀⣼⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠀⠀⠀⠀⠈⠢⠤⠚⠉⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⡟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡄⠀⠀⠀⠀⠀⠀⣠⠒⠢⢄⡘⢠⣀⠈⠀⠀⠀⠀⢀⢸⠀⠀⢠⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣷⠀⠀⠀⠀⠀⡰⠁⠀⢀⠀⠈⠙⠷⠄⠀⠀⠀⠀⠊⠜⠀⠀⢸⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢱⠀⠀⠀⠀⡐⠁⠀⡠⠉⠐⠢⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡆⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡇⠀⠀⠀⢃⠆⠀⠁⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⢀⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢇⠀⠀⠀⠈⢄⡀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢻⠀⠀⠀⠀⠀⠉⠂⠄⡠⢊⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣈⠓⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠖⠋⣠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠓⠦⣈⠉⠙⠛⠛⠛⠛⠛⠋⢉⣁⠴⠖⠋⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠦⠤⠤⠤⠤⠤⠤⠶⠖⠋⠁


⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣭⣭⣭⣭⣭⣭⣬⣭⣭⣉⣍⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣴⣦⣤⣿⣿⣿⣭⣿⣿⣽⣿⣯⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣊⡥⠐⢲⠄⠠⡐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠎⠀⢊⢂⡈⠀⡠⠊⠢⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠠⠄⠀⠀⠒⠒⠒⠒⠒⠒⠒⠂⠀⡇⣸⠀⢀⡀⠂⠁⠡⠎⠄⠀⣁⢰
⠀⠀⠀⠀⠀⠀⠀⡟⠢⢄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣁⣀⣠⣼⠵⢳⡜⢇⠀⠀⡀⠸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠁⠀⠀⢸⠈⠢⣢⠔⡡⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠂⣀⠥⠊
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢸⢰⠋⠒⠒⠒⠈⢁⠩⠳⡦⢄⡀⠀⣀⡠⠔⠂⠉⠉⠉⠲⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡘⡀⠀⠀⠠⠂⠁⣠⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⡄⠀⠁⠀⠡⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢺
⠀⠀⠀⠀⠀⠀⠀⠀⢁⢁⠀⠀⠠⠀⠀⠀⠀⢀⠸⠁⢀⠀⠀⠀⠀⠀⠀⠀⢘⡈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢘⠀⠀⠀⠁⠁⠄⠂⠁⢠⠀⢀⠁⠀⠁⠀⠀⠀⠀⡉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠠⢈⣀⠀⠀⠀⠀⢠⠀⠀⠀⢷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⡰⠀⢀⠆⠀⠈⠓⠀⠀⠀⠀⠂⠀⢰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢁⠀⠀⠀⡤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⠀⠀⠀⠰⡀⠀⠀⢀⢀⠂⠀⠀⠀⠀⠀⢀⢃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⢬⡑⠒⠶⠤⠤⠄⠴⠂⢒⣩⠔⡺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡈⠉⠁⠒⠒⠒⠊⠉⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠒⠒⠂⠉⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠚⠉⡀⢠⡤⠄⠀⠠⠀⠀⠉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡠⠀⡀⢄⠀⠀⠀⠀⡀⠀⠘⠀⠀⠢⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠘⢦
⠀⠀⠀⠀⢀⡀⠠⠄⠶⣒⣀⣠⣤⣬⢬⠭⠩⠉⠉⠉⠉⠉⠉⠁⠈⠉⠩⠭⠿⠿⣿⣦⣄⣠⣄⣰⣀⣀⣼⠤⡄⠀⢀⡀⠈⠁⠘⣦⠈⢧
⠀⠀⠀⠀⢸⠙⠋⠀⠐⡒⠶⣤⣤⣤⣤⠴⠶⠌⠉⠉⠀⠩⠬⣍⣁⣄⣥⠬⠥⠥⠴⠖⠒⠀⠀⢠⣤⣴⣶⣾⡇⠛⠉⠀⠀⠀⠀⠈⡁⢘⡄
⠀⠀⠀⠀⢸⣤⣦⠀⠀⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⣠⡽⣿⡇⠀⠀⠀⠀⠄⠀⠀⠤⠀⠇
⠀⠀⠀⠀⢸⠋⡇⠀⠀⠀⠀⠉⠉⠙⠛⠀⠂⠀⠀⠀⠀⠚⠷⠖⠒⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⡏⠉⠉⠉⠀⠀⠈⢭⡄⢸⠇
⠀⠀⠀⠀⢸⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⣦⡀⠀⠁⠀⠀⠰⠀⡼
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⣿⡖⢤⣀⠐⠁⣼⠃
⠀⠀⠀⠀⢸⢏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣢⣲⡲⠂⠀⠀⠀⣿⡄⠸⣿⡇⠀⠑⢤⡾⠁
⠀⠀⠀⠀⢸⠓⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⢰⠄⣿⠃⠀⣿⡇
⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠠⣄⣈⡁⣨⠤⠀⣿⡇
⠀⠀⠀⠀⢸⡂⡇⢐⡀⠀⠀⠀⣀⣤⠤⠄⠀⠀⠀⠀⠈⠉⠉⠁⠰⣦⣀⢀⡉⠉⠉⠉⠀⠉⠀⠈⠉⠈⠁⣿⣸⠀⣿⡇
⠀⠀⠀⠀⢸⡀⡇⠸⣷⣄⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⣠⠂⠀⠀⠀⠀⠀⠀⣐⣳⡀⠀⣿⣿⠀⢿⡇
⠀⠀⠀⠀⢸⡁⡇⠀⣿⣿⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢡⠁⠀⠀⠀⠀⢤⡀⠐⠮⠄⠀⢻⣿⠀⢹⡇
⠀⠀⠀⠀⢸⠁⡇⠀⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢽⡂⠀⠀⠀⠀⢒⠀⠀⠢⣀⠀⢸⣿⠀⢼⡁
⠀⠀⠀⠀⢸⠅⡇⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⠰⡗⠈⠀⠀⠀⠀⢀⣺⣗⠄⠀⠀⠀⢀⡀⠀⠠⢽⠀⠈⢸⠀⢻⡇
⠀⠀⠀⠀⢸⡇⡇⠀⢸⡿⠂⠤⠀⣀⠀⠀⠀⠀⠀⠌⠂⠀⠀⠀⠀⠀⢰⣯⣼⡧⠀⠀⠀⠲⠄⠈⢹⣯⠀⠀⢀⠀⢼⠂
⠀⠀⠀⠀⢸⡇⡁⠀⠀⠁⠨⢁⡀⣰⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⣠⣅⠀⠇⠀⠀⠀⣀⠀⠀⢈⣿⠀⢸⣿⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠰⠛⣉⣽⠀⢸⣶⣷⣤⣀⣀⠻⠿⠟⠁⠈⠛⠛⠁⠀⠂⠀⠀⠉⠠⠄⣿⣿⠀⢸⣿⠅⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣀⡀⠀⣸⠀⣈⣛⣆⠀⠀⠀⠛⠗⠤⠄⠀⠀⠖⠒⠀⠀⠤⠀⠀⠀⠀⣿⣿⡂⢸⣿⣑⣿⡄
⠀⠀⠀⠀⢸⡇⡀⠀⠉⠛⠿⠿⣿⣿⠀⣿⣿⣿⡾⣿⣥⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣿⣿⡇⢸⣿⢀⣿⠅
⠀⠀⠀⠀⢸⡿⣿⡀⠀⠀⠐⣂⠙⠿⠀⣾⣿⣿⣇⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠙⢻⣿⠃⠀⢽⠐⣿
⠀⠀⠀⠀⢸⡷⣾⠉⠀⠀⠒⢻⡿⠀⠀⠷⡿⣼⡁⠀⠀⣉⣉⣁⣀⣀⣀⣠⣀⣀⣀⣀⣤⠄⠄⠀⠈⢁⡀⠀⠸⢠⣿⡆
⠀⠀⠀⠀⢸⡗⢺⠃⠀⠀⣀⣬⣥⣤⢀⢤⣤⣾⣶⣶⣶⣦⣄⣀⣀⣀⡀⠂⠒⠒⠶⠶⢦⠤⠀⢀⣠⣼⡗⠀⢠⣼⣿⠃
⠀⠀⠀⠀⠀⠈⠀⠘⠡⠸⠿⠷⠿⣛⠿⣷⣿⣿⣿⣷⣿⣿⣶⣿⣏⣩⣭⣥⣤⣤⣴⣶⣤⣶⣾⠿⠿⠿⠷⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⠛⠛⠛⠛⠛⠛⠋⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣄⠀⠀⢠⡶⠀⠀⠀⠀⠶⡄⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠿⣿⡏⠀⠀⠀⠐⠋⠀⠀⢸⠈⠉⠀⠀⢸⠀⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢷⡙⠿⠀⠀⠀⢀⠀⠢⣄⣼⠀⠀⠀⠀⢸⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢷⠀⠀⠀⠴⢿⡀⠀⠀⠁⠀⣠⠀⠀⡾⠀⠀⠀⢻⡟
⠀⠀⠀⠀⠀⠀⠀⢠⣴⡓⠛⠲⠼⣧⡀⠀⠀⠀⠉⠓⠒⠒⠋⠁⠀⢠⠇⠀⠀⠀⢸⣗⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠛⠛⣶⡾⢻⠛⣦⡀⠀⠀⠠⠀⡆⠄⢀⣴⠋⠀⠀⠀⢔⢺⡿⢾⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠋⠉⢑⡾⠏⠀⢸⠀⠀⠙⣿⠲⡶⠶⡖⠛⢏⢻⠷⡄⠀⠀⢠⠺⣔⡈⠱⣣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⠦⠗⢶⣬⠀⠀⣼⠏⢠⠃⠀⢵⢀⠈⠈⢧⡙⣦⡀⠀⢱⠈⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡙⢸⠀⡇⣿⢘⠷⣼⠋⠀⠀⡤⡀⠀⢎⡷⠹⡎⢧⣧⠟⠀⢸⠀⣷
⠀⠀⠀⠀⠀⠀⡘⡟⢇⢸⠀⡇⣿⢸⠀⣿⡀⠀⠀⢳⠱⠞⠈⠁⣱⠥⡄⠱⡄⠀⡞⠀⢸
⠀⠀⠀⠀⠀⢠⡇⣗⢸⣮⠀⡃⣿⢨⣤⡾⠃⠀⣼⢈⡇⠀⣀⡤⠖⠋⠀⢠⣇⢰⠁⠀⢸
⠀⠀⠀⠀⠀⢸⡇⣿⢸⢻⣄⣥⡟⠋⠀⢳⣀⣠⡿⣶⠁⣼⣁⣠⢾⠻⣖⠋⠈⠛⠀⠀⣼
⠀⠀⠀⠀⠀⣿⡇⢿⢸⢸⡏⡟⢷⡀⠀⠘⣧⣁⡟⠁⡸⢹⢹⢱⡟⠀⠈⠳⡤⠔⠊⠈⠉⠓⠒⠠⣄
⠀⠀⠀⠀⠀⢻⣇⢸⢸⢸⠇⡇⠘⢷⡀⠀⠘⠋⠀⢰⡇⢸⢸⣼⠇⠀⠀⣰⢥⣉⣉⡉⠤⠬⠒⠈⢸⣿⣷⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⢸⡟⠛⠿⠾⠶⣦⣼⣬⣿⣄⠀⠀⢀⣯⣤⣼⣾⣿⣴⣾⣿⠋⢧⡀⠀⠀⠀⠀⢀⣠⠏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣴⣿⣷⣿⣿⣿⣿⣿⣿⣿⣦⣀⠉⠓⠒⠒⠊⠉⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣿
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠠⠖⠺⠷⠶⢤⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡟⠛⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠻⣦⠈⠙⣿⣿⣿⣿⣿⣿⣿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⢦⣤⣤⣤⣤⣤⣿⣧⣀⡀⠀⠀⠈⢳⡀⢹⣀⣨⣿⣿⠋⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠛⢦⡀⠀⠀⢻⣾⣿⣿⣿⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⢻⣄⣴⣿⣿⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢰⣿⣿⡿⢛⣛⠁⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠘⢿⣿⣶⣿⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⢶⣞⠛⠒⠒⠋⠀⣠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣼⡁⠀⠙⠦⡈⠳⣄⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠙⢦⡀⠀⠈⠢⡈⠳⣄⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⣦⠀⠀⠙⢦⡀⠀⠘⢦⡈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⢢⣠⠾⡅⠀⠀⠀⠀⠙⢦⡀⠀⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⡀⢀⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⠀⠀⠀⠀⢀⡼⠋⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⢈⡶⠋⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⢈⣶⠞⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠲⡴⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⣶⠟⠋




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⣀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⡄⢀⢄⢀⠀⠀⠀⠀⠀⠀⡖⠈⣀⢀⢰⡆⠀⠀⠀⠈⠁⠈⠉⠙⡇
⠀⠀⠀⠀⠀⠀⣟⣛⡟⠚⣿⡇⠀⠀⠀⠀⠀⡀⠰⠀⠌⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠈⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⡇⠀⠀⠀⠀⠀⡃⠀⠀⢀⠀⡇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⢾⠃⠀⠀⠀⠀⠀⠃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠓⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠐⠀⠑⠚⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠰⠆⠈⣴⠟⠀⠀⠀⠀⠀⠀⠳⠄⣀⣠⠾⠀⠀⠀⠀⠐⠦⠀⠐⠶⠃⠀⠀⠀⠀⠀⠠⠐⣠⣴⠇⠀⠀⠀⠀⠀⠸⠦⠀⣀⣀⣀⠶⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⢀⠀⣀⡀⠀⠀⠀⠀⢀⢀⣀⣀
⠀⠀⠀⠀⠰⣾⠀⠀⠨⢷⣩⡗⠀⠀⠀⠀⡋⠀⠀⠀⠁⠀⢹⠀⠀⠘⢆⠀⠀⠀⠀⣼⠃⠀⠀⢸⡛⠀⠒⠒⠒⠛⠻⠂
⠀⠀⠀⠀⠀⠈⢆⠀⠀⠠⠋⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠂⠀⠀⠀⠈⢢⠀⠀⡜⠁⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠰⠀⠀⢠⣷⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠋⠀⠀⠀⠀⠀⠀⠀⢃⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⣿⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣏⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠀⠀⠀⠀⠀⡸⣇⠀⠀⠀⠀⠀⠀⠀⠀⡀⢃⣀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠟⠁
⠀⠀⠀⠀⠀⠐⠄⠀⠈⠐⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠘⠤⠄⠀⠐⠀⠀⠀⠀⠀⠀⠢⠀⠠⠀⠂⠀⠀⠀⠀⠛⠀⠀⠀⠀⠼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢰⡶⠊⠉⠉⣀⣨⣷⠀⠀⠀⠀⢠⣇⣤⣾⣧⣤⡄⠀⠀⠀⢠⠀⠀⠄⠀⣄⠀⠀⠀⠀⢰⠏⠉⠉⢭⣿⠀⠀⠀⠀⢠⣄⣒⣒⢒⢲⣦⡄
⠀⠀⠀⠀⠈⡇⠠⠆⠤⠖⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠐⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⣁⠀⠀⠈⣿⠀⠀⠀⠀⢰⠂⣠⢿⡴⣣⢰⡇
⠀⠀⠀⠀⠀⢸⡀⠀⠤⠀⠀⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⣼⠀⠀⠀⠀⠈⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣗⠀⠀⠀⡟⠀⠀⠀⠀⢸⠢⠔⠪⠃⡼⠀⡇
⠀⠀⠀⠀⠀⠀⣇⠈⠀⠀⢰⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡟⠀⠀⠀⠀⠀⡀⠁⠀⢀⡅⠀⠀⠀⠀⠀⢹⠀⠀⠀⡇⠀⠀⠀⠀⢘⡀⠈⠉⠉⠁⣐⡀
⠀⠀⠀⠀⠀⠀⠙⠁⠀⠚⠃⠀⠀⠀⠀⠀⠀⠘⠂⠐⠚⠃⠀⠀⠀⠀⠀⠂⠀⠐⠛⠀⠀⠀⠀⠀⠀⠘⠀⠀⠚⠁⠀⠀⠀⠀⠈⠓⠂⠀⠈⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⣂⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡔⣥⠒⢺⠀⡄⠈⢢⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣤⠀⠙⠉⠀⢸⣼⠀⠒⢦⡗⢰⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⢿⡅⠀⠡⠶⠒⠒⣺⡈⣶⡎⢡⢰⡹⡁⠀⠜⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⢙⠣⠤⠄⠤⠤⠒⠓⠶⢿⣟⠘⢁⠐⠒⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⢀⣤⠀⠀⠙⠂⣀⣿⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡂⢀⠀⠠⠟⠗⠀⠤⣤⣤⣿⢟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢻⡈⠀⠀⠀⠐⠒⠒⠀⠈⡏⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣿⡞⠀⠒⠂⠐⠒⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⢸⡇⠀⠀⠀⠀⠀⠏⢰⡥⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠶⠾⠳⠦⠤⣤⣴⠤⠆⠘⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣤⣷⢶⣶⠀⠀⠀⠀⢀⢀⣤⣿⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠆⠀⠀⠰⠀⠀⡌⠉⠉⠉⠉⠉⠉⠉⠉⣱⠀⠁⠀⠀⠀⠀⢗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠚⠴⠿
⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢨⠁⠂⠀⢠⡇⠀⠀⠀⠀⠡⠀⠀⠲⢦⠄⠀⠀⠀⠀⠶⣾
⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠎⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢰⠀⠀⠘⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠦⡀⠀⠀⠂⠀⠤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠌⠀⠀⠀⠑⠀⢀⡀⠀⠁⠀⠀⠀⠀⠀⠀⢘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠄⠄⠐⠄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄
⠀⠀⠀⠀⠰⠀⣀⣄⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⡝⠃⠀⠀⡀⠰⠂⢀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠃⠄⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠸⠁⠆⠀⠀⠠⠄⠟⠀⠀⠀⠀⠀⠀⠀⢩⠄⠀⠠⡝⠓⡐⠀⠀⠀⠀⠀⠀⠖⠄⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠂⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⣤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠐⡦⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠔⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉
⣿⣿⣿⠿⣿⠿⢿⣿⣿⡿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣷⣮⣶⣷⣶⣿⣶⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠓⢤⡀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠙⢆⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠴⠖⠃⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢷⢄⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡷⠀⠈⠻⣆⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⡜⠀⡀
⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢺⡠⠞⠁⠀⡄⠠⡜⠀⠃⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢻⡀⠀⠀⠀⠀⢸⠀⠀⢀⣠⠁⠀⠃⠀⠈⣤⡁
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠎⠉⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢎⡰⠊⣉⠇⠀⠀⠀⢠⠇⣠⠜⠁⠸⡀⠀⠀⢀⠤⠒
⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⡆⠠⠊⢙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⡠⠛⠉⠀⠀⠀⠀⠉⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠙⢆⡀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠉⠛⠛⠋⠉⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠉⠀⠀⠀⠀⠀⠠⢄⢀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠉⠉⠉⡹⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⣀⠤⠊⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠉⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢄⠀⠠⢄⠀⡀⡀⠀⢀⡠⠤⠄⢈⠀⠀⠀⠀⠀⠀⢀⡱⠤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⠚⢒⡆⠈⠚⠁⠃⠊⠁⠀⠀⠀⠀⠈⠲⡄⠀⠀⠀⠀⠀⢠⠎⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠂⠦⡀⠀⡎⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⠀⠀⠀⢸⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⢱⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⡗⠊⠁⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⡸⠀⠀⠀⢸⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠛⠚⠛⢦⡀⠀⠀⠇⠀⠀⢀⠎⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⠤⠖⠁⠉⠲⢄⣀⢀⡠⠊



⣿⣿⡿⠻⠟⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣷⣶⣶⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣄⠉⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⣿⣧⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⣿⣿⣿⡿⠿⠛⠁


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

⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⡴⢋⣩⢭⡙⢦⠀⠀⢀⡴⠊⠉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠘⡆⢳⠦⣀
⠀⠀⢠⡶⣼⠃⡏⠀⠀⡽⢸⠇⠀⠀⣇⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⢹⠈⡆⠈⠑⢄
⠀⣠⠴⠷⠀⠀⠑⠒⣚⣡⠞⠀⠀⠀⠸⡄⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠿⡀⠀⠀⢇⢸⡀⠀⠀⠱⣄
⣼⢡⠞⠉⠙⡆⣤⡙⢯⡀⠀⠀⣀⣀⣀⣧⠀⠠⠬⠦⠤⡤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠠⣾⣿⣟⣦⣈⣳⣄⣸⣄⣧⣀⣀⣀⣈⢦
⢷⠸⣄⣀⡼⢡⠏⠉⠢⣭⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⠒⠦⢤⣀
⠈⠓⠶⠶⠞⠋⠀⠀⣴⣛⡏⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣓⣲⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⡾⠭⠭⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣤⣎⠍⠀⣀⣤⣴⣶⣶⣶⣦⣄⡀⢣⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠇⢀⣤⣴⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⢸⢸⡭⠭⠭⢽
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢠⣾⣿⡿⢛⣛⣙⣛⠻⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠟⣛⣉⣛⠻⢿⣿⣷⡄⠀⠀⠈⠒⠨⠭⠭⢽
⠀⠀⠀⠀⡎⠀⠈⠈⠃⢀⣿⣿⢋⠔⢁⠤⠤⢄⠑⠌⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⡐⠉⡠⠄⢄⡉⢦⢻⣿⣿⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠓⠤⠤⠤⠖⢺⣿⣿⢸⠀⡃⠀⠀⠀⡇⢸⢸⣿⣿⠒⠒⠒⠂⠀⠀⠀⠀⠈⠉⠙⣿⣿⠀⠁⢘⠀⠀⠀⡁⠀⠆⣿⣿⠓⠒⠒⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡌⢦⡘⠢⠤⠚⣠⠎⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠣⡈⠂⠤⠐⢁⠜⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣌⣐⣒⣈⣴⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣬⣑⣒⣊⣥⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠛⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠀⢹⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣻⣿⣶⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⠟⠉⠉⠘⠮⡝⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⢀⡴⠂⠈⠉⠉⠁⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠒⢦
⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠠⢗⠒⠀⠈⠓⠀⠀⠀⠀⠀⠀⣤⠊⠉⠁⠈⠉⠉⠀⠐⠢⡀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠈⢂⡤⢤⠀⠀⠀⠀⠀⠈⠀⠀⠛⠉⠓⠊⠉⠥⡄⠲⠊⠁
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇
⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠰⠆⢠⠟⠃⠈⢹⠏⠉⢹⠏⠉⠙⡖⣷⢀⡀⠀⠀⠀⢠⢻
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠠⢀⠀⢸⠀⠀⠀⢸⠀⠀⢸⠀⠀⢀⠀⣿⠀⠉⠛⠛⠒⠛⠈⢀⢀⠐⡠⠖⠒⠒⠂⠀⠀⠠⠤⠤⢠
⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠘⠆⠀⡄⠀⠀⠸⡀⠀⢸⠄⠀⠨⡄⣿⠀⠀⠀⠀⠀⠀⠀⠈⡖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⣀⣰⣀⣧⣤⠤⠴⠇⢠⡘⠷⠖⠶⠗⠿⣶⣦⣤⠤⠤⣤⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⠀⠀⠀⠀⣸⠏⠉⠀⠀⠀⠀⢠⠚⠀⠈⠑⣦⠀⢠⠊⠁⠀⠈⢳⡀⢸⡧⠴⠷⠤⠤⠤⠤⠀⢀⣀⣀⡀⠀⣘⠃
⠀⠀⠀⠀⠀⠉⠒⠤⡤⣤⠤⠤⠒⠉⠀⠀⠀⠀⡠⢊⠸⡆⠀⠀⠀⠀⠀⣯⠀⠫⠉⠁⣉⡁⢛⡀⠂⠀⠀⣠⡇⠈⠉⠙⠉⠉⠉⠉⠉⠁⠀⠀⠐⠀⠉⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠐⠰⠀⠂⠀⠀⠀⠀⠠⣥⢎⣴⡁⠦⠃⠀⠀⠀⠀⠀⠙⢦⣤⣤⣴⠟⠀⠈⠻⢦⣤⠶⠋⠀⠀⠰⢀⡠⠈⢄⣀⠄⠀⢄⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⢀⡴⢶⠉⠁⠀⠀⠈⠑⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⠶⠖⠛⠛⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠔⠁⢀⠼⠤⣄⡀⠀⠈⠑⠄⡀⠑⢄⠀⠀⠀⣠⠴⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⡾⡤⠰⢏⠀⠀⠀⠙⠦⡀⠀⠀⠈⠠⠈⢧⠀⠀⣿⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠷⠚⠉⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⢦⡀⠀⠀⢃⠘⡄⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰⠀⠀⠀⠀⠑⢄⠀⠀⠀⠹⣄⠀⠈⠀⠇⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢂⡀⠀⠀⠀⠈⠳⣄⠀⠀⠘⣆⠀⡸⠀⠀⠀⣧⠀⠀⢠⡄⠀⡀⠰⠂⠀⠀⠀⠀⠀⠀⢻⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠳⡀⠀⠀⠀⠀⠈⢦⠀⠀⢘⣾⠃⠀⠀⠀⢻⠀⠀⠀⠀⠀⢠⡶⠛⠳⡄⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠙⠢⢄⡀⠀⠀⠀⢑⣤⠞⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⠤⣉⡛⠳⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠘⣧⠀⠀⠿⠶⠶⠖⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠁⠀⠒⠙⠓⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣤⠤⠖⠚⠉⠀⣀⠤⣀⠠⠤⠤⣄⣠⠖⠢⡄
⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⡰⠁⠀⠀⠀⠈⠀⠙⣤⠇
⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⣄⠀⠈⢱⡀⠐⠛⠛⠃⠀⠀⢐⠀⡔⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠨⡄⠀⢀⠱⢦⣀⡀⠀⢈⣤⠊⠀⡇⠀⠒
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠤⣀⢈⠞⠁⠀⠀⠈⠁⠀⠀⠙⣦⠔⠀⢀⠜
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠐⠉
⠀⣀⠔⠂⠀⠀⠘⠆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠃⡀⠐⠲⠤⣀⠐⠀⠀⠀⠉⠛⠓⠀⠤⠤⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠈⢧⠀⠐⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠶⠶⠦⣤⡀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣺⠀⠀⠀⡀⠀⠀⠙⣄⠀⠀⠀⠊⠄⠐⠆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠰⠛⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠃
⠀⠀⠀⠀⠐⠒⠒⠊⠀⠀⠁⠠⡟⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢀⣠⠖⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⣀⡹⠉⠒⠒⠒⠲⡖⠒⠉⠀⢩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡌⠉⠉⠙⠻⠅⠀⣀⣀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⡀⠀⠀⠀⠀⠀⠑⠶⠴⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⣀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⣽⣆⣶⠀⡴⠀⢸⠀⢰⠀⢰⡄⠐⠇⠀⡧⠀⠰⠀⠸⠀⢰⡆⣼⣄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣀⣤⣀⣤⣄⣤⣄⣠⣄⣠⣤⣴⣎⣀⣴⢠⣶⣦⣴⣆⣰⣿⣿⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡏⢻⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠙⠢⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⠄⠀⠀⠀⠀⠀⢀⣤⠈⢕⠆⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⠆⡀⢠⣈⡷⠊⠑⠦⢄⣀⡘⠂⠈⠓⠁⢀⡤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⠧⠛⠊⠉⠀⠀⠀⠀⠀⠠⡛⢍⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢤⣀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢣⡀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⡀⠀⠀⠀⠹⡀⠀⡇
⠀⠀⠀⠀⢀⡀⠀⢀⠎⠀⠀⠀⡠⠒⠉⠉⠉⠑⢦⠀⠀⠀⠀⠀⠀⠀⠉⢱⠀⠀⠀⠀⡇⢠⠃
⠀⠀⢐⡮⠳⣄⢱⠃⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⢣⠃
⠀⠀⠆⠀⠀⠀⠘⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡟
⠀⠀⠱⡀⠀⠀⠀⠀⢠⠃⠀⣀⣠⠄⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠸⡄⠀⠀⢀⠔
⠀⠀⠀⠙⣄⠀⠀⣰⠁⢀⠺⣅⢸⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⢀⠌⠙⠢⠒⠁
⠀⠀⠀⠀⠈⠙⠒⠉⠙⠉⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⢀⡠⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⡠⠖⠉⣤⡺⠭⢝⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣁⣀⠤⠒⠉⠀⠀⠀⢸⣽⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⡟⠴⠞⠂⠀⠚⠒⢃⠞⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠇⠠⠆⠀⠀⠠⠀⠹⢋⠁
⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⢀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⢢⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠈⢄⠀⠉⠀⡨⠁⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠤⠀⡄⢄⢀⠀⠀⠐⠀⠀⠀⠀⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢱⡄⢀⠷⠀⠀⠀⠀⠀⢀⡜⡀⡸⢀⠀⠉⢀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢘⡾⠄⠀⡀⠀⠀⢸⠀⠠⠀⣠⣴⣶⣦⠀⠐⠀⢫⠉⠂⠂⠚⠓⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠂⡀⠀⢀⠬⡤⣞⣒⣒⡺⢮⣠⣿⣇⢈⠖⠁⠀⠀⡀⠀⠀⠀⠀⠀⠄⠄⠈⠁⠄
⠀⠀⠀⠴⢳⠀⠀⠀⠐⡁⢘⣶⡄⣶⣶⣶⡄⠉⠉⠉⠀⠀⠈⠀⠄⠀⠀⡀⠀⡀⣉⠀⠀⠄⠁⠀⠘⢦⡀
⠀⠀⠂⠒⠺⠷⠀⠤⠀⠰⠏⠻⣿⣽⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠒⠧⠉⠉⠀⠀⠀⠰⠃⠞
⠀⠀⠀⠀⠀⠀⠰⢴⢒⣲⡂⠀⠉⠛⠛⢁⣀⣀⣀⡤⡤⣤⣀⣀⣀⠠⠤
⠀⠀⠀⠀⠀⢰⣶⣿⠫⠈⢱⡆⠐⢸⣏⠉⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⡷⠁⠄⡀
⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠙⢦⡄⡘⠉⠉⠉⠋⠋⠋⠋⠉⠉⢉⣁⣠⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠃⠢⠬⠤⠥⠜⠉⠉⠛⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠄⠀⠀⠀⠘⢛⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠋⠀⠀⢀⠀⠀⠀⠾⠟⣿⡆⠀⠀⠀⠀⢣⡀⠀⠀⢀⣾⢯⡙⢦⠀⠀⠀⠀⠀⠀⢶⣄⣾⠃
⠀⠀⠀⢸⠿⢿⣿⣿⡆⢻⡏⠀⠀⠐⢀⣤⣤⣤⣤⣤⣤⣠⣀⠂⠀⠀⠀⢠⣤⣤⣤⣤⡄⠛⠁⠀⠀⠀⠀⢀⣅⣀⣀⣈⣉⣅⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿⣄
⠀⠀⠀⢰⣄⣰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⢿⡿⠈⢹⣿⠏⠀⠀⠀⠀⠀⣭⣭⣭⣭⣭⣭⡀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡈⠁
⠀⠀⠀⠸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡿⠃⠀⠀⠀⠀⠀⠸⣿⡿⠿⡏⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟⠀⠀⠀⠀⠀⠀⠀⢰⡛⣾⢭⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣶⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠸⣛⣛⢋⣙⠃
⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠛⠃
⠀⠀⠀⠑⠐⠛⠛⠂⠃⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠈⠒⠛⠛⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣦
⠀⠀⢀⠄⠀⢘⣿⣟⣻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⠀⠀⠀⠛⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠣
⠀⠀⢰⠀⠀⠈⠻⢿⡿⠏⠁⠀⠀⠀⠀⠀⠀⠀⢰⣮⣷⣛⡃⠀⠀⠈⠰⣶⣶⢮⣽⣾⠶⠉⠀⠀⠀⠀⠀⠀⢠⣴⣿⣷⣄⠀⠀⠀⠀⠀⠀⠁⠅⠀⠀⡈⠁
⠀⠀⠀⣧⣶⡷⢸⣾⠅⠀⠀⠀⠀⠀⡆⠀⠀⠀⠠⠏⠘⠟⠂⠀⠀⠀⠀⠈⢻⣟⣩⠌⠀⠀⠀⠀⠀⢀⣤⣤⣬⣍⡁⣿⣿⠀⠀⠀⠀⠀⢀⣰⣀⣰⣎⣀⡐
⠀⠀⠀⣿⣷⢰⣷⢸⡄⠀⠀⠀⠀⠀⠁⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⠀⠀⠀⠀⠀⣨⣩⣍⣩⣭⠀⠉⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣟⢵⡶⣻⣿⡅⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠿⠾⠵⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣻⣿⣿⡿⣿⡯
⠀⠀⢰⠈⠓⠘⠋⢸⡇⠀⠀⠀⠀⠀⠀⠿⢿⠿⠿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠸⣿⡮⣻⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡟⠉
⠀⠀⠐⠦⠶⠶⠦⠼⠁⠀⠀⠀⠀⠀⠐⠠⠴⠦⠄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠶⠾⠶⠄⠀⠀⠀⠀⠀⠂⠠⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠿⠷⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣄⠀⠀⠀⠀⣿⣷⡏
⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠄⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢈⠆⠀⠀⠀⠿⠟⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠿⠿⢷⡄⠀⠀⠀⢠⡄⠀⠀⠀⠠⡄
⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠂⣼⣁⣀⣀⣀⣀⣀⣀⠈⠂⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⠂⠿⠇⠀⠀⠀⢀⣿⣿⠻⣿⣿
⠀⠀⠀⠁⠁⠉⠁⠉⠁⠀⠀⠰⡇⢻⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣛⣛⣛⣻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡓⢿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠿⢿⣿⣿
⠀⠀⠀⢸⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣟⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣧⣤⣼⡟⡀
⠀⠀⠀⠘⠿⠿⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠉⠀⠇
⠀⠀⠀⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠒⠚⠛⠛⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠁

⣷⣶⣶⣶⣶⡾⢶⣾⣿⡿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⢶⡿⠷⢶⠶⣶⡾⠷⠶⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⢫⣭⣙⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣀⢁⣂⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠶⠶⣴⡏⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡶⠛⠿⠻⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠟⡇⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⢠⠀⠀⢣⡃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠆⡆⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡌⡈⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠊⠘⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠰⢠⠅⠀⠀⠀⠐⣷⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢠⢢⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⡄⡘⠀⠀⠀⠀⠀⠫⠆⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢡⠁⠀⠀⠀⠀⠀⠸⢁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠘
⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡆⡇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠸⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠇⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢰⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⢦⠦⠤⠐⠒⠀⠐⠂⠤⠤⠒⢒⡠⠃⠀⠀⡇⠈⠢⠤⠤⠤⠤⠔⠂⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠂⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠀⠚⠚⠀⠀⠁⠉⠀⠀⠀⠈⠀⠀⠋⠀⠈⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣐⠲⠂⢀⠐⠂⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠡⠁⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⡀⠀⠀⣀⣔⠈⠙⠀⠀⠀⠀⠀⠀⠀⠠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⠒⢄⠀⠀⢀⠀⢀⠀⠑⡀⠔⡟⠔⠆⠁⠘⠢⠀⠀⠀⠀⡀⢔⡡⠦⢋⠡⣐⠏⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠢⠄⠘⠀⠀⠈⠈⠠⠀⠠⠐⠔⡸⢀⠀⠐⠹⠀⠀⠉⢑⠀⠀⡸⡀⠀⢄⡴⣛⠜⡊⢧⠗⠉⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣈⠀⢀⣀⠀⣀⠘⠃⠀⠀⠱⠚⡙⣀⠥⠷⣵⢊⠖⠈⠚⠹⢂⠤⠽⡂⠁⠌⠾⠦⠵⠛⠈⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠺⣿⠟⠀⢀⠈⢢⠃⠀⠄⢖⠛⣹⡇⠰⢤⣠⠅⠎⠷⣰⣜⣠⠊⠉⠈⠥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠤⠠⠄⠀⠀⠀⠘⠀⠐⢆⠂⠩⡡⠾⠦⠈⠀⠀⠀⠜⠅⢈⠃⣀⡒⠀⣴⢄⠀⠉⠐⣠⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢉⠔⠀⠀⠀⠀⠀⢀⡀⠰⢦⣤⣥⣔⡀⡖⠀⠀⠀⠤⠀⠀⠈⠁⠈⢯⣿⠄⠙⢀⠀⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⡉⠁⠀⠉⠀⠠⠀⠐⠀⠰⠦⠀⢀⠀⣨⡀⠀⠈⡇⠀⡀⣀⠀⠤⠄⠀⠀⠐⠋⠁⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠁⢀⠊⠀⠀⠀⠠⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⡀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢀⢀⠀⠄⠀⠀⠀⠀⠁⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠘⢢⠀⠜⡐⠉⠁⠖⢂⠠⡀⢀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⢈⠀⡃⠅⠀⠠⢠⠀⡑⢄⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡀⠈⠁⠀⠀⠂⠁⠑⠂⠐⠥⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠒⠂⠤⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠆⠒⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠠⠤⠤⠀⠒⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣠⠤⣄⣀⣠⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⢠⣄⣀⣀⣀⣴⡀⠀⢐⠶⠭⠶⣒⠲⠭⠶⣚⠲⠅⠀⠐⢀⠀⠀⠀⢀⠐⢀⠀⠀⣠⡜⠻⢿⡿⠛⢢⣀⠀⠀⠀⡔⢢⣀⣀⡠⠲⡀
⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇⠀⢨⣝⣒⣫⣭⣝⣒⣫⣭⣝⡂⠀⠈⠁⡀⠀⠀⠀⠉⡀⠀⢸⣿⣧⣀⡔⢢⢠⣾⣿⡆⠀⠀⣿⣾⣿⣿⣿⣾⡄
⠀⠀⠀⢿⣿⣿⣿⣿⣿⠇⠀⠐⠮⣭⠵⠒⠮⣭⠵⠒⠦⡅⠀⠀⡀⠀⠀⠀⠀⠠⠁⠀⠘⢿⡏⠀⣸⡁⠀⢹⡿⠃⠀⠀⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⢨⣛⠒⣛⣭⣛⠒⣚⣭⣝⠂⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠉⠁⠀⠈⢱⣶⣶⠀⠀⢠⣶⣶⡄
⠀⠀⠀⠀⠀⢠⢳⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣀⣀⠀⠸⠿⠿⣤⣤⣼⠿⠿⣧⣤⣄⠀⢀⠤⢄⡀⢀⡠⠤⡀⠀⠀⠀⣤⢾⡛⠟⡳⣦⡀
⠀⠀⠠⡄⣤⣪⣿⣑⡄⡤⠄⠀⠀⣻⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⠀⠠⠀⠀⠀⠀⠀⠠⠇⠀⠀⣸⢣⠱⣪⣬⡪⠊⣷
⠀⠀⠀⠈⣋⣿⢿⣯⡎⠀⠀⠀⠀⢿⡿⠉⠭⠉⢿⡿⠀⢸⣿⣿⠀⠀⢸⣿⣿⡇⠀⠀⠀⢠⡀⠁⠀⠀⠀⢀⠄⠀⠀⢹⣞⣔⢝⡛⢝⢄⡿
⠀⠀⠀⢠⠧⠋⠉⠫⢧⠀⠀⠀⠀⠈⠛⠦⠠⠤⠛⠁⠀⠈⠉⠉⣷⣶⣾⠉⠉⣷⣶⣶⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠙⠿⣯⣬⡷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠛⠛⠛
⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡴⢖⣿⡲⣦⡀⠀⠀⠀⠀⢼⣿⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⡰⠆⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⡀
⠀⠀⠚⠛⣃⡀⠀⠀⠙⠛⠃⠀⢠⡟⠞⢕⣤⡪⡹⣻⡄⠀⠀⠀⢸⣯⣤⣬⣿⠀⠀⠀⠀⣤⢤⣴⣻⣟⣤⠤⣤⠀⠀⠘⠟⣉⠀⠀⣈⡻⠇
⠀⠀⠀⠀⠻⠋⡤⠀⠀⠀⠀⠀⠸⣇⢌⢜⣿⡣⡢⣹⠇⠀⠀⢠⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠙⣽⣿⣿⣻⠊⠀⠀⠀⠀⠘⠿⠣⡄⠻⠟
⠀⠀⠀⠈⠀⠠⠀⠄⠀⠀⠀⠀⠀⠙⢷⣵⣶⣮⡾⠋⠀⠀⠀⠸⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢸⠽⠓⠚⢭⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈
⠀⣠⣤⣄⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣉⡉⠉⠁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢿⣿⣿⣀⣀⣸⣿⣿⣀⣀⡀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠠⣭⣛⣫⠥⢭⣛⣫⡭⢭⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⡄
⠀⠀⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⢨⠀⠃⠛⠚⠁⡔⠀⢐⡲⠭⠴⣚⡲⠬⠵⣚⡲⠤⠀⣀⣴⠞⠿⠿⠳⣦⣀⠀⢀⣾⣧⠀⢈⡏⠀⣸⣷⡄
⠀⣶⣶⣾⠉⠉⢱⣶⣾⠉⠉⠁⠀⡏⠀⠀⢀⠀⠂⢹⠀⠨⣝⣓⣛⣭⣝⣓⣛⣭⣽⡓⠸⣿⣧⠀⠀⠀⢀⣼⡿⠀⠸⣿⡿⠃⠃⠜⠁⢻⣿⡟
⠀⠙⠛⠛⣤⣤⡜⠛⠛⣤⣤⡄⠀⠙⠀⠀⠀⠀⠀⠋⠀⠐⠶⣭⡵⢖⠶⣭⡵⠖⠲⣭⠀⠀⠙⠦⢤⠤⠾⠋⠀⠀⠀⠈⠣⣤⣾⣷⣦⡜⠋
⠀⠀⠀⠀⠿⠿⠇⠀⠀⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠚⠉⠙⠲⠚⠉⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠤⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠰⠆⣣⣤⡼⠃⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣀⠀⣱
⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠠⢀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠁⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢉⠒⠀⠀⠀⠀⠀⡐
⠀⠀⠀⡀⠀⠢⠀⠀⠐⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢁⣠⣤⣆⠏⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠈
⠀⠀⠁⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠈⠀⠀⠔⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠁⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠲⠀⠐⠄⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⣠⠤⠀⠒⠁⢀⡀⠠⠤⠤⠖⠈⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠆
⠀⠀⠀⠈⠂⠈⠉⣉⢀⣠⡤⠆⠂⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠐⠐⠀⠀⠀⠀⢀
⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠖⠀⠉⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⢀⠴⠂⢀⠀⠀⡆⠀⠀⠙⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠂
⠀⠈⠆⠁⠆⠀⠤⠄⡔⠂⠉⠁⡤⣀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠔⠁⢠⡄⠃⠀⠀⠅⠀⠀⡇⠀⠀⠒⠂⠙⠋⠘⠃⠁
⠀⠀⠈⠐⠓⠄⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠠⠐⠀⠔⠀⢐⠀⡇⠀⠀⠁⠀⢠⣴⠇⠀⠀⠀⠀⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠅⠌⠀⠀⠢⠀⠠⢍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠐⠀⠀⠤⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠆⠠⠀⠀⠘⠀⠐⠊⠉⠀⠀⠠⠤⠄⠐⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠂⠚⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠄⠀⠀⠀⡇⡁⢇⡡
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠘⠁⠀⠉⠁⠘⠓
⠀⠀⠀⡄⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠓
⠀⠀⢠⠦⠤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇
⠀⢠⠁⠀⠀⠈⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⡀⠠⠤⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⢀⠀⠀⠤⠤⠤⠀⠀⠀⠀⡀
⠀⠐⠀⠀⠁⠀⡀⠀⠀⠀⠈⠀⠀⠀⠳⠠⠄⠀⠀⠀⠀⠀⠀⠀⠐⠅⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠂⠀⠀⠀⠀⠀⠀⠀⠉
⠀⢀⠀⠤⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⡀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠈⢆⠀⠾⠡⠄⠆⡱⠀⠀⠀⠀⠀⠀⢀⡀⠘⠃⠤⠤⠛⠀⣀⣥⠀⠀⠀⠀⠀⠀⠐⠛⠐⠘⠉⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀
⠀⠀⠀⠐⠂⠀⠐⠒⠁⠀⠀⠀⠀⠀⢰⠀⢈⠈⠉⠉⣉⢉⡉⠀⢹⠀⠀⠀⠀⠀⠀⡀⢠⠀⠐⢬⡀⠀⠀⠘⠀⠀⠀⠀⢀⠚⠉⠀⠀⠆⠔⠀⠁⠓⣄
⠀⠀⠀⠀⠥⣬⠈⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠊⠜⠃⠨⠀⠄⠂⢼⠀⠀⠀⠀⠀⠀⠁⠋⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⠒⠂⠀⠆⠐⠔⠂⣸
⠀⢰⠀⠀⢀⠁⠉⢀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⣉⡁⠘⠋⠁⢀⣀⠔⠁
⠀⠈⢄⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⡀⠃⠀⢂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠈⠀⢀⠀⠀⠀⠀⠀⢰⣄⢀⡉⠉⠉⠉⠉⢁⣀⡴
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠜⠀⠀⠀⠀⠣⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠶⠶⠶⠶⠶⠶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠚⠒⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠐⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⢀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠐⡦⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢀⡦⠀⠀⠠⠄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁⠰⡈⠀⠀⠀⠀⠈⠀⠀⠐⠄
⠀⠀⠀⠀⠤⠀⡀⠀⠀⡀⡀⠠⠐⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠠⠄⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡃⠀⠀⡒⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡈⣀⣀⠤⠐⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠄⠠⢀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⢀⠀⠀⠀⡐⠐⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠋⢉⡀
⡄⠀⠐⠂⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠆⣈⠀⠀⢤⡒⠀⠀⠀⠀⢨⠀⠀⢰⠚⠓⢠⠀⠰⠃
⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠘⠧⠄⠇⠀⣈⠁⠀⠀⢀⠁⢀⠁⠚⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠙⠣⡀⠁⠀⠀⠁⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠦⠂⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣠⠤⠦⠆⢇
⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣬⠡⠤⠶⣖⠁⠀⠐⢄⠚⠏
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠚⠘⠅⢧⡘⠘
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠀⡄⠠⠀⣄⠀⠀⢴⡐⠀⢠⣄⠁⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠙⠀⠀⠈⠁⢀⠋⠁⠀⡜⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠈⠉⠀⠀⠀⠀⠀⠀⠁⠢⢄
⠀⠀⠀⠀⠂⠀⢰⠀⠀⠠⠄⠀⢀⡀⠀⠀⠌⠀⠀⢐⠀⠀⢀⠁⠀⠀⠂⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠙⠀⠀⠀⡅⠀⡐⠁⠀⠀⠆⠀⠀⡀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠓⠀⠂⢣⣀⡴⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠆⠀⠐⢐⡀⠃⠀⠑⠂⠈

⠀⠀⠀⠀⠀⢸⣿⣶⣶⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣶⣦⠄
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⣴⣦⣶⣦⡢⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣴⣦⣶⣦⡀⡀
⠀⠀⠀⠀⠘⠛⠛⠉⠀⠄⠀⠀⠀⠀⠀⠀⢤⣿⡿⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠈⠛⢛⠉⠐⠄⠀⠀⠀⠀⠀⠀⠼⣿⡿⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠺⣿⡿⣿⣿⣿⢿⣿⣿⣿⣿⢟⣾⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣼⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⢻⡟⢿⣿⣻⠟⠋⠁⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠹⣿⣿⣿⣿⡟⠋⠁⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⡛⠛⠫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢸⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠄⠀⠀⠀⠀⠰⠸⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠶⠶⠷⠿⠖⠄⠀⠀⠑⠜⠻⠿⠻⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠶⠶⠶⠾⠖⠄⠀⠀⠀⠘⠻⠿⠻⡛⠑

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⣀⠉⠑⣠
⠀⠀⠀⠀⠀⠀⠈⠒⠤⠀⣀⢠⠂⠁⠀⠀⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⠈⣄⠘⠛⠁⠀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠨⡇⠀⠰⢀⡐⠀⠘⠀⡰⡄⠂⠀⠀⠀⢊⠁⢀⠀⢀⣀⡆⠀⢠⠔⠉⠑⠂⠀⠺⣀⠀⠀⠀⠀⢠⠀⠀⠀⡀
⠀⠀⠀⠊⡜⠁⠀⣀⢀⠀⠧⢴⣀⡀⠈⠁⠀⠘⢀⡁⠀⠀⠀⢀⠀⡈⡔⣡⣶⠀⣴⡄⢆⠁⠀⠀⠀⠀⠀⠀⠘⠔⢈⡀⡔⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠉⠣⡀⠑⠂⢈⡠⢷⠚⠠⢄⠀⠀⠀⠀⠀⠈⢏⠐⠂⠀⠡⠈⡇⠈⠉⣂⡉⠁⢨⠀⠀⠀⠀⠀⠀⠀⠰⣀⠀⠙⢩⠦⠤⣧⣤⣆⡠
⠀⠀⠀⠀⢀⠌⠉⠉⠙⠦⠜⣤⡀⠀⠁⡄⠀⠀⡀⠀⠸⠄⢀⠀⠀⠀⠈⠂⠄⠉⠠⠔⠁⠡⠒⢀⠄⠒⢀⠀⠀⡼⠋⠁
⠀⠀⡀⠂⡌⠀⠀⠀⠀⠀⢦⠉⢈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠄⠀⠀⠸⠄⠀⠋⠀⠀⠀⠈⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈
⠀⠸⣄⣠⠅⠌⠀⠀⠈⠀⠸⡵⠶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠠⠔⠁⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⣀⠆⢴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠸⠀⠡⠀⠀⠈⠀⠀⠄⡀⠠⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣂⠀⠀⢠⠂⠩⠀⠁⠘⠂⠀⠀⡀⠀⠙⠢⠤⡀⣀⠀⠀⠀⣀⡀⠔⠁
⠀⠀⢔⠀⠁⢠⠤⠤⠴⠺⠄⠀⠀⠀⠀⠀⠀⠠⠄⠈⠀⠀⠀⠈⠀⠁⠈⠀⡃⠀⠠⠄⠀⠠⠀⠀⠄⠀⠀⠑⠤⣄⣀⠤⠂⠀⢀⠏⠁⠀⠄
⠀⠀⠀⠉⠉⠁⠠⣀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠠⠤⠸⠀⢐⡃⠀⠀⠀⠀⠂⡀⣀⣠⡄⣀⢠⠀⠀⠃⠀⠀⠔⠃⢀⠰⠆⠀⠀⠈⠳⠶⠆⠶⠎
⠀⡐⠁⠀⢀⡠⠐⠂⡀⡀⠀⠀⠀⠀⠁⠀⠣⢔⠃⠁⠀⠀⢰⠐⠒⠆⠀⠐⢃⡀⡀⢀⠀⡠⠀⠀⡅⠀⢘⡄⠰⠾⠀⢰⣷⠀⠠⠄⠠⠀⡀
⠰⠀⠀⡠⠋⠀⠀⠀⠀⠰⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠉⠀⠉⠁⠀⠀⠈⠣⣀⣴⣄⠨⡵⠤⠤⢴⣠⠄⠀⠀⠀⠄⡠
⠈⣄⡐⠀⡠⠂⠀⢀⠀⠘⠂⠀⠀⠀⠀⠀⡀⠀⠠⠠⠀⠰⠁⠀⠀⠠⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⠀⠒⢀⠀⠐⢢⡀
⠀⠘⢦⣀⠀⠀⠄⢂⠔⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠐⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠁⠀⠈⠎⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⡀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠖⠁⠰⠛⠉⠀⢁⠐⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⣠⠂⠀⠀⠀⠀⡀⠀⠁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡀⠀⢰⠃⠀⠀⡀⠀⠘⠿⠂⠌⠻⠃⢄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡇⠀⡇⠀⢠⠶⠦⠀⠀⠀⠀⠀⠀⢰⣿⣿⣯⣭⣉⣭⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠃⢸⠀⠀⡞⠰⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠸⡟⠛⠋⠀⢀⠎⠀⢰⣆⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠀⣯⠀⠀⠀⢿⣷⣦⣬⣂⠠⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠹⡀⠀⠀⠈⠁⠀⢉⠋⠁⠠⠀⢀⠀⠀⠀⢀⠈⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢾⠀⠀⠀⠀⠀⡤⠄⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡠⠀⠈⠀⠀⠈⠑⠒⠒⠂⠁⠐⡄⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⡀⠀⡀⠀⡠⠀⢄⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠐⢯⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡼
⠁⠀⠀⠨⣀⣤⠀⡀⠀⢇⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡸⠀⠀⠀⡏⠰⠖⠂⠛⠬⠉⠙⠦⠐⠂⠈
⠀⡄⠀⠀⠀⣄⠀⠁⠐⠈⠻⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠉⠀⠀⠀⢀⠔⠠⠤⠀⠀⠒⠒⠁
⠀⠀⠀⠀⢰⡀⠳⣀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠸⣿⣄⠀⠉⠁⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠹⡁⠀⠂⠄⢈⠉⠒⠢⠤⠤⣀⣀⣀⣀⣀⣀⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢧⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡌⠀⠀⠈⠀⠀⠀⢸⠀⠀⠀⠀⠈⠁⠙⡤⢀
⠀⠀⠀⢀⠀⠂⠉⠛⢄⠈⠑⠠⢀⠀⠀⠀⣸⠤⠄⠄⠐⠂⠀⠀⢀⠈⠁⠈⠔⢄⡀
⠀⠀⢠⠃⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⢀⣹⠄⢀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡱
⣀⣀⣈⣑⣂⣀⣤⣄⣄⣊⣁⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡏⠭⣟⣛⣛⣛⣛⢛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡇⠀⢠⣤⡆⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠘⣿⢱⠀⢸⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⢸⣷⣿⠀⠀⠀⠀⠀⡠⠀⠀⠀⣀⣀⠠⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⡄⠀⡀⢸⣿⣇⢸⣿⣿⠀⠀⠀⠐⠂⠀⠀⠀⠘⠉⠀⠀⠀⠰⠂⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠈⣿⣿⠸⣿⣿⡀⠀⠀⠀⠀⢀⠀⠠⠀⠀⢀⠀⢀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣀⠀⢸⣿⣇⣿⣿⡁⠀⠀⠀⡰⠁⠀⠀⠀⣐⡁⠀⠘⠒⠈⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣶⢾⣿⣿⣿⣿⣧⣤⣴⣤⣤⣶⣶⢶⠾⠛⠁⠀⠀⠀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢀⡛⠿⣿⣿⣧⠔⠈⠀⠀⢄⠔⠀⠘⠀⠀⠀⡄⢰⣇⣼⡇⢀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⢴⣶⢀⠀⢠⣧⣼⣿⣿⣴⣿⠟⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢀⠀⠂⣴⣤⠀⢀⠐⠀⠻⣆⠁⢸⣤⢸⣿⣿⣿⣿⣿⣷⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⣀⠀⣾⠰⢃⠀⠆⠀⠉⡀⠀⢀⡸⠂⠊⢸⡇⣸⣿⢸⣿⣿⣿⠟⣫⣥⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⡙⣿⣿⣿⡈⢻⣿⣿⣧⡄⠀⠀⠀⠐⣤⠂⠀⠀⠀⠀⠀⠀⠀⡟⢨⢻⡘⣇⢋⠴⣿⣿⡿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣧⣈⣿⣿⣧⡀⣿⣿⡿⠑⠘⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⠀⣼⠇⢨⠈⢇⣡⣴⣾⣿⡿⢛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⡐⠀⠀⢠⡀⠈⠀⠀⠀⠀⣀⡾⠃⠘⢰⡀⢂⣴⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠈⡇⠁⠀⠰⠘⣻⣧⣶⣶⣶⣛⠉⠀⢰⠀⠈⠔⣻⣿⠿⣛⣥⣶⣶⣶⣶⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣿⡿⢿⣿⣶⣾⣷⣾⣟⣿⠄⠀⣴⠟⠉⢉⠉⠙⠿⣿⣿⣆⠀⠀⢀⣾⢿⣷⣿⣿⣿⡿⠿⠿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠋⢀⣾⣿⣿⠉⣿⣿⣿⣿⢈⣆⠀⠀⠆⠈⠀⠀⠀⠈⢿⣿⡄⠀⠈⠛⢻⣦⣽⠟⢿⣷⣶⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣿⣿⣿⣿⠏⣠⣿⣿⠈⣿⡈⠉⢲⡔⠂⠈⣛⠆⠐⠂⠈⣿⡇⠀⣸⣿⢿⣿⣿⠳⣶⣮⣝⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⣵⣿⣿⣿⣿⣾⣿⣷⣄⣀⠈⠒⠂⠁⠸⠷⠀⢀⣁⡤⢶⣏⣿⣄⣿⣿⣷⣿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⢹⡀⠉⠙⡛⢻⠛⠛⠛⠋⠉⢹⢀⣸⠇⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠓⠚⠓⠒⠚⠒⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⣦⡀
⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠸⣿⣿⣿⠏⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢡⣶⣄⠀⠀⢀⣤⣶⣶⣶⣤⡀⠀⠀⢠⣴⣌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⡀⠀⠀⢉⣿⣥⣄
⠀⠀⠀⠀⠀⠈⠻⣿⣷⣶⣿⣿⠿⠋⣿⠿⣿⣶⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣄⣤⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⣧⡀⢀⣤⣿⡇⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⠀⠰⣶⣶⠆⠀⠀⠀⠀⠈⠻⡿⠟⣿⣿⣿⣿⡇⣤
⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⣿⣷⣶⣿⣿⡷⠶⠶⠶⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠷⠶⢶⡶⠾⠷⠶⠶⠶⠆⣠⣤⣤⣤⣤⣿⣿⣿⣿⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⣣⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇⣿⣽⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⣿⡇⢐⠟⠛⠛⠒⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⣿⡇⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠈⠿⠿⠿⠿⠿⠅⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣶⡤⠀⠀⠀⠈⠿⠿⠃⠥⠄


⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣈⣛⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣆⣀⣤⣾⣿⠇⠶⢄⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢀⡀⣿⣿⣿⣿⣿⣿⣿⣯⣤⣄⠔⠃⣾⣿⡷⠞⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠷⣾⣽⣿⣿⣶
⠀⠀⠀⠀⠀⠘⡟⣿⣿⣿⣿⣟⢛⢛⣛⣛⠛⣤⣤⣬⣿⣥⣤⣤⣤⣤⣧⣤⣤⡤⠀⠀⠀⠀⠀⠀⢠⣤⣤⣭⣽⣿⣿⣷⣬⣯⣭⣭⣤⡄
⠀⠀⠀⠀⠀⠀⣧⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠹⢷⣿⣿⣿⣿⡯⠉⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣻⣟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⣿⡿⠀⠀⠀⢀⣠⣾⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢻⣿⣿⣿⣿⣿⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠉⠉⠁⠀⠁

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠋⠑⣤⠂⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈⠀⠀⠉⠖⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣠⠊⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠧⠤⠤⢤⣤⡀
⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠖⠚⠋⠉⠀⢰⢯⡼⠛⠀⢀⣿⣧
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⢰⠋⠉⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⢀⣠⣶⣟⢾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠔⠋⠀⠀⣀⠀⠤⣄⠀⠀⣀⠠⠄⣾⢇⠉⢿⡿⣾⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠁⠙⠂⠺⠁⠀⡏⠀⠀⢸⠒⡉⠜⢆⣄⠙⠞⡾⡌⣵⢸⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡷⠶⣶⣿⠛⣇⠀⢁⠀⠀⣭⢁⠁⡸⣤⠯⡙⢶⠘⠀⣹⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⣟⢳⡐⡿⡷⣄⡀⢰⠂⠃⠀⠱⠘⢆⡙⢦⢡⠇⢸⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⢽⡢⠙⢦⠈⠈⠅⢫⠀⠠⠀⡧⣲⡌⢃⠀⠈⡞⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢔⠿⣿⠀⣧⠀⢲⡀⠣⠀⡀⠀⢌⢷⠈⢰⡇⣣⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠮⣷⣝⠃⢤⡀⠲⣄⣈⢦⠺⠦⠁⡆⣯⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣷⣶⣭⣄⣴⣦⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡿⠀⠈⠳⣄⣠⣤⣤⠤⠶⠶⠶⠦⢤⣤⣴⣋⠀⠀⢹⣼⡆
⠀⠀⠀⠀⣠⠶⠚⣛⣿⡟⣁⡴⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢤⣹⠿⢿⠓⠦⣄
⠀⠀⣠⣞⣡⠤⣶⣿⣠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡟⢿⣒⠺⢷⡄
⠀⢠⡟⣿⠒⠛⠁⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠈⢉⡿⢻⡀
⠀⢸⡇⠘⢷⣄⢀⡏⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⠟⠁⢸⡇
⠀⠘⣇⠀⠀⢉⡿⠀⠀⠀⠀⠀⢀⡏⠙⠦⣄⣀⣙⣷⣤⣀⡀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠈⢷⣄⡀⣼⠁
⠀⠀⢿⡀⣴⠞⠁⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⣠⡬⠁⠀⠠⣭⡉⠉⠉⠁⠉⠳⢦⣀⣀⠀⢀⣠⡼⢁⡟
⠀⠀⠘⣇⢹⣷⠶⠶⠶⠚⠋⠁⠀⠀⠀⠀⠀⠁⣴⠀⠀⢠⣦⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⡍⢧⣼⠃
⠀⠀⠀⢻⣯⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣼⡏
⠀⠀⠀⢸⡏⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⣤⣀⣤⠟⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣤⣤⡴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠳⣤⣀
⠘⢯⣄⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⣠⡿
⠀⠀⠉⠳⢤⣀⣀⣈⣳⣄⡀⠀⠀⠀⠀⢀⡴⠲⠤⢿⠷⢤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣃⣀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⢠⡞⠃⠀⠀⠀⠀⠀⠛⣦⠀⠀⣀⣠⡴⠚⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣽⣿⢿⣿⣶⣶⣶⣶⣶⣿⢻⣟⣻⣯⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⡞⢹⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⢹⡆⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⢸⠃⢸⠀⢀⣀⠀⠀⠀⠀⠀⣸⠇⠀⠀⣿⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠿⣤⠾⠛⠋⠹⣆⣀⣀⣀⣼⠋⠓⠶⣤⣽⠇⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠈⣩⣭⣿⡁⠀⠀⠀⠈⢷⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⢸⡀⠀⠀⠀⠀⠀⢠⣤⣤⠏⠀⠀⠀⠀⢸⠀⠀⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠛⠾⠷⣤⣄⡀⠀⠀⢤⣤⡽⠂⠀⢀⣀⣤⡿⠶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠈⠙⠛⣶⣴⣯⣤⠶⠛⠋⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠴⠿⠭⠤⠭⠶⠆⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠶⠦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⠴⠶⠞⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠉⠉⠉⢹⠉⠉⠉⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤⣴⣶⣤⣤⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡋⣿⣿⣿⣿⣿⣿⣿⠇⣿⡁⣿⣿⣿⣿⣿⣿⣿⡏⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢦⣤⣀⣀⣀⣠⣤⢞⣿⣟⢦⣄⣀⣀⣀⣀⣤⢴⣿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡠⡄⠀⠀⠀⠀⠀⠀⢰⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠶⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢡⣿⠻⡏⠀⠀⠀⠀⠀⠀⠸⣄⣿⣷⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⣸⣿⠐⣂⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⢻⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⢿⣿⣷⣶⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡍⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠁⢻⣿⠞⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠑⣿⣿⡏⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⣖⣡⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠀⠀⠚⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣲⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣯⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠈⠈⣿⡇⠈⠄⠀⠀⠀⠀⠀⠀⠀⠐⠛⢿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠉⢹⠉⠉⠻⡿⠀⠀⠀⠀⠀⠀⠔⠀⠀⠐⠂⠄⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠂⠀⡀⣴⡀
⠀⠀⠀⠴⢶⣦⣤⢶⣿⣿⣶⣶⡶⣶⣦⠀⠀⠀⠀⠈⠀⠤⠌⠀⠀⡔⠀⠀⠀⠀⠀⠀⣾⠆⠴⡤⠐⠀⣐⠀⠀⣏⠴⠚⠉⠁⠀⢰⡆⠘⡍⠠⠙
⠀⠀⠀⠀⠀⠉⠛⠂⠿⢿⣟⣛⣳⣿⠋⠀⠀⠀⠀⠀⢠⢄⣠⠾⡐⠁⠀⠀⠀⠀⠀⠀⠉⣀⠀⠐⠐⠃⠍⠀⠀⠘⢖⢉⡤⠶⠀⠀⠁⠰⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠋⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠞⠌⠝⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠂⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢟⡿⠛⢿⣿⣿⠀⠀⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⢸⣿⣿⣿⠟⠛⠋⣹⡿⠊⠉⠙⢿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⡎⠐⣆⣾⣿⣿⠀⠀⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⢸⣿⣿⣧⣦⡀⠀⣿⣇⠋⠀⢆⣼⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠷⡀⠀⢻⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣁⣛⣿⣿⣯⣿⣿⣿⠀⠸⣿⣿⡟⣿⣿⠀⣿⣿⣷⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣧⠄⠙⢆⣬⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣇⣿⣿⣟⣿⣿⡿⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣴⠀⢈⣾⣿⣿⣿⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⣿⣷⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⡿⢿⣷⠞⢻⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⠀⠀⣿⣿⣿⠀⠘⠟⠀⢸⣿⣿
⠀⠀⣿⣿⡿⣿⣿⣿⣾⣿⢿⠿⣿⣿⣿⣿⠀⠀⣿⣿⡏⢀⠊⠁⠀⣹⣿⣿⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣦⣤⡻⣿⠻⣿⣿⣿⣿⣟⢿⠯⣿⠀⠀⣿⣿⣧⣀⢀⣴⢰⣿⣿⣿⠀⠀⣿⣯⣼⣿⣯⣿⠞⣅⣽⣿
⠀⠀⣿⣿⣿⣿⣿⣶⣽⣟⣫⣥⣀⡤⣾⣿⠀⠀⣿⣿⣿⣟⣹⣿⣏⣿⣿⣿⠀⠀⣿⣿⣿⣙⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢢⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣬⠠⠀⠤⣷⢦⠌⢀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⣷⣀⡦⢽⢞⣬⡤⢚⡽⠁⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠹⣽⣿⣿⣇⣿⣽⣷⠾⠁⠀⠀⠀⠀⠀⠤⠤⢠⣿⡇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡼⠟⠉⠁⠀⡀⠀⠲⢄⡀⠀⢀⣴⢾⣿⠀⢀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢻⠳⠖⢶⠾⠶⢦⣤⡤⠿⠶⡶⠾⣿⣿⡟⢻⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣌⡇⠀⠀⠀⠀⠘⣯⠂⣤⠤⠀⠼⠿⣉⠀⣼⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠓⠲⠖⠒⠒⢶⣶⠶⠶⠾⣛⠛⠉⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠂⠀⠀⠘⢿⡟⠀⠀⢿⣧⡀⢠⡇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⣲⣤⣤⡟⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢀⣔⡾⠋⢠⣿⡀⠂⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠀⠀⠀⠀⣬⠋⠀⢠⡾⣿⣷⡃⢼⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⢸⡇⣀⡴⠋⣱⡏⢽⡁⣾⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠑⢄⡀⠀⠋⠁⢠⢾⠟⣽⣿⡁⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⣯⣶⣶⣾⠁⢀⣴⣿⡿⠟⣣⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣀⠈⠉⠉⠉⠉⠉⠉⠉⠁⢀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠒⠒⠒⠛⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⢽⣋⣛⣟⣙⣋⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⣴⣿⣿⣿⣿⣿⣿⠟⠀⠉⠈⠛⠿⠋⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⣼⣿⣿⠏⠉⠀⢉⣉⢀⣀⣶⣶⣦⣄⠀⢠⣿⣿⡟⠉⠉⠉⠙⠛⠛⣿
⢠⡿⠏⠋⠀⠀⠀⠈⢙⠋⢋⣀⣡⣬⣿⣷⣾⣿⠿⠁⠀⠀⠀⠀⠀⠀⡇
⣸⠃⠀⣀⣤⣤⣿⣦⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⣿⣤⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢀⠤⠄⠀⠀⠋⣧⠀⠀⡀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⣴⣿⠞⠋⠉⠳⣄
⠀⢻⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠚⠉⠛⠁⠀⠀⢠⠞⠁⠀⡠⠊⣹⠈⠓⢄
⠀⠀⠹⣿⣿⡏⢠⠖⠶⣦⠙⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢀⠎⠀⣘⡡⠄⠐⠢⣝⢦⣤⣄⣀
⠀⠀⠀⠹⣿⣇⠈⠀⠠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢠⠁⢠⠞⠁⢀⡤⠒⠊⢁⣴⠈⠳⣄⠉⠲⢄
⠀⠀⠀⠀⠈⠻⣦⡀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀⠸⠀⠉⠀⠰⠋⠀⡠⠖⠋⣠⠃⠀⠈⠓⢄⡈⢳⡀
⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠙⢆⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠻⠿⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⢶⡄
⠀⠀⠀⠀⠀⠀⢀⣠⠾⣥⣄⡀⠀⠈⠉⠛⠷⢦⣤⣄⣀⡀⠀⢀⣠⠎⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⠀⠀⡉⢢⡀
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠉⠛⢶⣄⡀⠀⠀⢀⣼⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⣤⡀⠀⠀⣸⠋⠀⠀⠀⠀⡰⠁⠀⢳⡀
⠀⠀⠀⢀⣾⡷⠞⠋⠉⠓⠂⠀⠀⠀⠈⢙⣶⣤⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠙⣿⣿⣿⣶⣄⠉⠀⠀⠀⣠⠞⠁⠀⠀⠀⢳⡀
⠀⠀⠀⣾⠋⢀⡠⠀⠀⠀⠀⢀⡠⠴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠉⠈⠿⠃⠀⢹⣿⣿⣿⣿⣿⣶⣶⣿⣿⠀⠀⠀⠀⠀⠀⢷
⠀⠀⢰⣷⠞⠋⠀⣠⡴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢻⣟⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⣿⣄⠀⠀⠀⠀⠀⢸
⠀⢀⡾⠋⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⣿⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡇⠘⢿⣦⠀⠀⠀⠀⢸
⢀⡾⠁⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⣤⣤⣽⣿⣶⣤⣶⠟
⣾⡇⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠷⠶⠶⠿⠟⠛⠋⠉⠉⠉⠉
⢹⣧⠀⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠙⣧⣿⣿⣿⣷⣤⣀⣀⠀⣠⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⣿⠿⠿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢀⡿⠀⠀⠀⠙⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃

⠀⠀⠀⠀⣀⠊⠗⠀⡀⠀⣴⣰⠛⠇⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⢠⠂⠻⠀⡀⠀⢰⣤
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠁⠀⠀⠀⠀⠣⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣥⠏⠀⠀⡀⠀⠀⠀⠀⠀⠆⠀⠀⠈
⠀⠰⠇⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣤⣤⠶⢄⠀⠀⠀⠱⠀⠀⠀⠀⠖⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⣄⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⠁⠂
⢿⠀⠀⠀⠀⠃⠀⠀⠃⢀⠀⠀⡀⠀⠂⠀⠈⠑⠠⠰⠀⠐⠈⠀⠀⠀⠀⡀⠀⢸⠆⠀⠀⠀⠁⠀⠁⠀⠀⡀
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡖⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀
⡀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⡀⠀⠀⠀⠨⣏⡤⠜⠃⠀⠀⠀⠀⢄⠀⠀⢸⡀⠀⠀⠀⠀⠀⢤⠄
⠈⠳⣶⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⢀⡼⠀⠰⠿⠀⠀⠳⣶⡾⠇
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀
⠀⠈⢀⡀⡀⡀⠀⠀⠐⠷⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠾
⠐⠀⠀⠀⠓⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠐⢒⠀⠀⠀⠀⠀⠀⢸⠁⡶⠲⠀⠀⠀⠛⠐⠓
⢀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣛⠤⠒⠩⠀⠀⠀⠀⠀⣴⡚⠿⠉⣀⡀
⠖⠠⠀⠀⠀⠀⠘⠆⠀⠴⠂⠀⠀⠀⠀⠙⠳⢤⡴⠇⠀⢀⣀⠀⠀⠀⠀⠀⢰⡷⠀⠆⠀⠀⠀⠀⠲⠀⡤⠞
⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠋⠀⠀⠀⠀⢮⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠴⠉⠀⠀⠀⢸⡀⠀⠀⠀⠀⠉⣄⣀⠃⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠏⠁⠀⠀⠘⣡⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠄⠁⠋⠀⠄⠀⠾⡼⠉⠃⠄⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠠⠁⠙⠀⠄⠀⠸⢶

⠀⠀⠀⠀⠀⠀⠀⡜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠒⠂⢤⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡤⠄⢀⢀⡤
⠀⠀⠀⠀⠀⠀⣀⠇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⣠⠴⠒⠚⠢⣀⡀⠀⠀⠀⠀⠁⠀⡼⠃⠀⠀⢸⠀⠀⢀⠆
⠀⠀⠀⠀⠀⡜⠀⡄⠀⢀⡡⠠⠉⠉⠀⠀⡀⠀⠀⠀⠀⢰⠃⠀⣠⠊⠁⠀⠀⠀⢀⠏⠐⡀⠀⠀⠀⠀⡜⠀⠀⣦⡐⠇⠀⠀⠐
⠀⠀⠀⠀⠎⠀⠀⡈⠀⠐⠟⠠⠤⠤⠀⠀⠄⢀⠄⠀⠀⢸⠀⣰⠃⠀⠀⠀⠀⢀⠞⠀⢀⡇⠀⠀⠀⠀⠁⠀⠁⠁⠔⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠈⡵⠉⠀⠀⡌⠠⠔⠒⠀⠔⠀⠐⠆⠀⠀⠀⠠⡇⠀⠀⢀⣠⠔⠉⠀⢀⡜⠀⠀⠀⠀⠀⣠⠖⠀⠀⣁⡈⠐⠚⠁⠠
⠀⠀⠀⠀⠀⡔⠀⠀⠏⢉⡁⠀⠀⠀⠤⠤⠤⢤⠀⠀⠀⠀⠀⠑⠐⠋⠉⠀⢀⣀⡴⠋⠀⠀⠀⠀⠀⠀⡇⠠⠴⠋⠀⠀⢠⠀⠀⠀⠐
⡔⠃⠀⢀⡜⠀⠀⠀⡠⠈⠀⠀⠀⠈⠁⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣠⠚⠉⠁⠐⠄⠀⣠⢆⣠⡴⠐⠂⣠⠆
⠀⠊⠀⠍⡇⠀⡾⠁⠀⠀⠀⠐⢀⣠⠍⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠇⠀⠀⠀⢀⠀⠀⢠⠎⠁⠀⠀⠀⢸⡆⢀
⠀⠏⡉⠙⠢⠈⣄⣀⠀⠀⠀⠀⠀⠀⠀⣰⣄⣤⠤⠶⢢⡀⠀⠀⠀⡰⠊⠁⠀⠀⠀⠀⠀⠀⠀⢀⡠⢊⡠⠔⠚⠃⠀⠀⠀⣠⡟⠀⠁
⠀⠀⠚⠁⠐⠠⠄⣀⢈⣉⣉⣀⣀⣈⣉⣉⣁⠤⠔⠒⠉⠀⠀⡠⠊⠁⠐⠒⠒⢁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⣀⣀⣀⡀⠈⠊⢀⡄⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠠⡀⢀⡠⠤⠖⠉⠀⠀⢏⡇⠀⢀⠀⠀⠀⡧⡰
⠀⢀⠄⠀⢀⠄⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⢀⠀⠎⠀⡌⠀⠀⣨⠁⠃
⠁⠈⠀⠈⠈⠐⠉⢀⡼⠀⠀⡤⠂⠀⢀⠀⢸⠀⠂⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠸⠀⠀⡀
⠀⠀⠀⠃⠐⠀⠀⡍⠀⠀⠀⢀⣀⡀⠀⠀⠉⠀⠁⠀⠂⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡏⠀⠀⢀⠤⠀⣠⠤⠲⡆⠈⠁
⠀⠀⠀⠀⢀⡨⠃⡇⠀⠀⠀⠎⠀⠀⣄⠀⠀⡔⠊⠀⠀⠀⠄⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡜⠁⢂⣸⣁⡴⠛⡄⡀⠀⢀⠔⠁
⠀⠀⢀⠋⠉⠀⠀⠇⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⢀⡴⠚⠉⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡽⣠⡾⠒⠲⡧⠀⠀⢰
⠀⠀⢸⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠍⢀⡇⠀⣠⠜⠀⠀⠐⠁
⠀⠀⢈⠀⠀⠀⡀⡀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠊⠀⣡⠀⠀⠒⡇⡠⠀⠀⠄⠀⠀⠀⢀⡔⠋⠀⠀⠁⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠄⢀⠀⠈⡀⠀⠐⣊⠀⠀⠊⢸⠀⠀⠀⡔⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡐⠓⠘⠷⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⡀⠀⡆⠀⠀⠀⠇⠀⠀⠑⠤⠀⠀⠀⠀⠀⠀⠀⠀⠤⠎⠐⠄⡀⠀⠀⢀⠄⠁⠤⠀⠠⠊⠀⣶⠁
⠀⠀⠀⠀⠀⠀⢂⠔⠂⠀⠀⠐⠁⠀⡅⠀⠀⠀⠀⠡⠐⠚⠀⠀⡰⠂⠒⠀⠀⠄⣀⡤⠖⠀⠉⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⢺⣅
⠀⠀⠀⠀⠄⠤⠊⠀⠀⠀⠀⢀⣀⠴⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⡀⠀⢋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⡴⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠿⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠈⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠀⠀⠀⠀⠙⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⣀⡀
⠀⠀⠀⠀⠀⠠⠄⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⣲⣤⣴⢖⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠀⠀⣶⣶⣶⣾⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠑⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇
⠀⠀⠀⠀⠀⠘⢻⡟⠋⣀⢿⡷⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠃
⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠐⠛⠂⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼
⠀⠀⠀⠀⠀⠤⠠⠤⠠⠀⠀⠀⠀⠀⠀⢿⢻⠷⠶⡶⠆⠀⠀⠀⠀⠀⠀⠩⣤⣤⠌⠀⠀⠀⠀⠀⠀⠀⠀⣂⣀⡲
⠀⠀⠀⠀⠀⡄⠂⢤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⠎⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣶⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣄⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠃
⠀⠀⠀⠀⠀⢳⣶⣶⡾⢀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠍⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠂
⠀⠀⠀⠀⠀⠀⢹⡏⢺⣿⣧⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄
⠀⠀⠀⠀⠀⠠⣾⣷⠄⠻⠿⠃⠀⠀⠀⠀⠀⢴⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠰⡿⠂⠀⠀⠀⠀⠀⠀⣬⣿⡄

⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⣰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤
⠀⠀⠀⠀⠀⢀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⢠
⠀⠀⠀⢀⣤⡈⢻⡟⣡⣶⣦⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⣤⡀⠀⠀⠀⠀⠀⠀⢸⣿⣠⣿⣾⣏
⠀⠀⠀⠀⠉⠃⠚⠓⠘⠛⠛⠀⠀⠀⠀⠀⠀⠘⠛⠃⠚⠀⠀⠀⠀⠀⠀⠐⠻⠓⠒⠐⠀⠀⠀⠀⠀⠀⠀⠚⠛⠋⠛⠛⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣒⣒⣯⣽⣵⣒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢹⣷⣾⣿⣿⣿⣿⣿⣿⠿⠗⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠉⠀⠈⠉⠉⠻⡶⣶⣦⣤⣤⣤⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠑⡎⠩⣿⣿⣿⡀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠆⠀⠰⠖⠂⠀⠀⠀⢞⣁⣘⣿⣿⣿⣷⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠃⠀⠀⠈⠋⠀⠀⠀⢸⣽⡿⣛⢻⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠁⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠈⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⠀⠈⣠⣶⣅⠀⠀⠀⠀⣠⠏⠀⠠⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⠀⠀⠂⠐⠂⡆⠀⠀⠀⠀⠀⡴⠚⠀⠐⠊⠁⠣⡰⠙⠃⡸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡡⣬⠃⠀⣀⠀⠀⠃⠀⠀⣀⠴⠞⠁⠀⢀⠐⠢⠀⢠⠇⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣀⢀⣹⣄⣈⣀⣵⠒⠔⠉⠀⠀⠀⠀⠀⠎⠀⢠⣣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠄⠀⠀⠀⠀⢙⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡉⢻⡏⣾⣿⡇⠊⠀⢠⠎⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢠⠊⠀⣀⣤⠴⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠀⠉⣻⣿⣼⣿⣿⠁⠀⠀⢸⡄⠀⠀⠀⣠⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠁⢠⠎⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠈⠀⠉⠉⠁⠀⠀⠀⠀⢧⠀⠀⢸⠁⠙⠀⠀⠀⠀⠠⢄⡀⠀⠀⠈⢢⠀⠈⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠁⠀⠀⡴⠁⠀⠀⠀⠀⣰⠏⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⠁⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠈⢣⡀⠀⠀⢀⠀⢧⠀⠀⢺⡁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣇⣠⣖⣁⣤⡜⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠄⢀⣧⡴⠖⠒⠒⠒⠢⠤⢀⡀⠀⢸
⠠⣄⣀⡀⠀⠀⠀⢀⣸⣦⣤⣤⡔⠒⠙⠛⠉⠉⠁⢸⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠸
⠀⠀⠀⠉⠙⣢⠞⠁⠀⠀⠀⠀⠉⠙⠓⠦⡀⢀⣀⣋⣀⣀⣘⠀⠀⠀⣀⡤⠤⠂⠀⠀⠑⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡄
⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠉⠀⠀⠀⠳⣤⠤⠞⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⣀⠞
⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠉⠳⢦⣤⣀⠀⡠⠤⠀⢸⣅⡧⠠⣤⣾⢿⣛⠛⠋⢉⡉⣉⢉⡟⠛⠛⠋⠉⠁
⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠈⠉⠛⠦⡀⣀⣠⠼⠒⠋⠁⠀⢀⣿⢠⣆⣼⣷⠿⣾⠃⢀⠂
⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢀⡠⠖⠋⠉⠀⢀⣀⠤⠶⡞⣹⠏⣾⣿⠻⠏⠀⠁⠀⢸
⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣠⠋⠀⢀⡤⠖⠋⠉⠀⢲⡄⣷⠏⠀⠀⠁⠀⠀⠀⠀⠀⣼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣛⠿⠛⠁⠸⣿⣿⡆⣴⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢠⣿⣟⢸⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡻⠿⣿⣿⠇⣿⣿⣿⣿⣝⢿⣿⣿⣦⣀⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡆⣿⣿⣿⣿⣿⡆⠹⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⡗⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⡆⠀⠀⠤⢤⡔⠒⡆⠀⠀⠀⡠⠐⠉⠣⢤
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⣀⡌⢰⣴⠾⡓⢛⠋⠀⠀⠀⠀⠀⠀⠀⠀⡸⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠙⣦⠀⢰⣃⠀⣀⠀⣰⠀⠀⢀⠸⠀⠀⠀⠺
⠀⠀⠀⠀⠀⠈⢁⠠⣤⠆⠐⠚⢂⡞⠊⣁⡈⣿⣆⠀⠀⠀⠀⠀⠀⢠⣅⠊⢛⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠉⢰⡒⠃⠀⠀⠘⠒⠀⠀⠃⠀⠀⠸⠤⠒⠤⠒⠊
⠀⠀⠀⠀⠀⠀⠈⠥⣄⠤⠒⠊⠣⡰⡎⠩⣆⢹⣿⡆⠀⠀⠀⠀⠀⡋⠒⡭⠟⠱⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⡴⠤⠝⠀⠀⠀⣀⢠⢶⡄⢀⣤⡀⠀⠰⡏⢧
⠀⠀⠀⠀⠀⠀⠀⡎⠀⢀⠀⠀⣀⠱⡐⠔⣋⣭⣽⣷⠀⠀⠀⠀⠀⢧⡀⡄⣀⠴⠀⢠⡄⠀⠀⠀⠀⠀⠙⠚⠁⠀⠀⠀⠀⣻⠹⡬⠇⠙⣩⡤⠟⡴⠓⠷⠘⣄⣀
⠀⠀⠀⠀⠀⠀⠘⢯⣷⣄⢴⣿⣂⣥⡵⠿⠿⠿⣿⣾⡆⠀⠀⠀⣰⠓⠚⠛⠓⠒⠘⠃⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣉⣠⠤⣆⢈⡧⠀⣤⠞⠀⠀⠀⣠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣿⣿⡿⠀⠀⠀⠀⠀⠈⠁⠀⢀⡔⠉⣢⠤⠔⢶⠒⠈⠡⡔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠀⠀⠘⠦⠶⣄⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠣⣀⠤⠚⠀⠀⠀⣂⠄⠀⢀⡴⠶⡛⢋⡿⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⢀⣀⡀⠀⣀⣀⣤⣄⣀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠐⠢⠴⡄⡇⠀⠀⡗⠀⠉⠁⠀⣴⡥⠤⢤⣼⠻⣤⠒⠉⠉⢦
⠀⠀⠀⣿⣛⡿⠋⠁⠀⠀⢀⡈⠻⣹⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⢸⠁⡤⠚⠉⡹⠻⣅⠉⠑⢦⣸
⠀⠀⠀⠙⢻⠁⠘⢟⡤⡤⢜⠃⠀⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⢠⠃⠀⠈⢣⠀⠀⢹⠀⠀⠀⠀⠀⠀⢀⡞⠋⡙⣶⠴⠶⣄
⠀⠀⠀⠀⠘⢧⡀⠸⣴⣷⡬⠃⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢦⡀⣀⣤⣄⠀⠀⠀⠀⠀⠘⢆⠀⣼⠀⠀⠀⠈⣇⣠⠃⠀⠀⠀⠀⣠⠖⠚⢇⡃⠀⡇⡀⠀⢨
⠀⠀⠀⠀⠀⣠⠟⠓⠒⠖⠒⠺⡅⠀⠀⠀⠀⠀⠀⣠⣤⣄⠹⠦⠈⣿⠋⢤⣼⠃⠀⠀⠀⠀⠀⠀⠙⠻⠤⠤⠤⠔⠋⠀⠀⠀⠀⠀⠀⢿⡀⠐⠛⢡⣤⠀⠩⠍⠙
⠀⠀⠀⠀⢰⡏⠀⣠⠶⠲⣤⣀⣽⡄⠀⠀⠀⠀⢰⡏⢀⣈⠳⣤⡞⣧⣠⠴⠞⠛⢦⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣻⠛⡆⢠⣁⢈⠙⢦⣠
⠀⠀⠀⠀⢸⡉⠉⣹⡇⢾⣉⣁⣠⡇⠀⠀⠀⠀⣴⠃⣼⠏⣷⡟⢹⠟⣡⡶⣷⣴⠋⠀⣠⠖⢉⣁⡤⠤⡤⠶⢶⠶⠶⢤⡀⣈⣉⣙⢶⡀⠹⣄⣀⣨⢯⡀⢀⣸⠁
⠀⠀⠀⠀⠈⠻⡿⠛⢷⠒⣻⠛⢻⡇⠀⠀⠀⠀⢻⣤⡽⠟⢻⢧⡟⢷⡙⢦⡗⢹⠀⠀⡇⢰⠉⢸⠀⠀⡇⠀⢸⠀⠀⠀⡇⡏⠉⢍⠓⠋⠀⠈⠉⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠙⠒⠚⠛⠉⠛⠉⠀⠀⠀⠀⠀⢠⡏⣴⣿⣸⠾⡇⣶⡙⡶⠶⠛⠀⡴⠃⠈⠒⠞⠶⠶⠓⠶⠞⠓⠒⠚⠁⠓⠒⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠼⠟⠁⠀⠳⣼⣧⠇⠀⠀⠀⠉⠒⠦⣴⡿⣷⡶⠶⠶⠶⠶⠶⣶⡶⣶⡔⠃
⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠁⠀⠀⠀⠀⠘⢿⣿⠿⠁
⠀⠀⢸⠁⢤⣬⣽⠳⢤⡴⠒⢦⡼⢳⣾⡯⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣛⣣⣀⣀⣚⠛⣛⡳⣦⣶⡲⣄
⠀⠀⠘⡇⠠⣿⡷⠶⢾⣇⣀⣠⣧⠤⢭⣿⣂⣼⠀⠀⠀⢀⡴⠚⠛⠛⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⢱⡿⢿⠁⠻⢀⠸⠋⡻⢧⡙⡟⡆
⠀⠀⠀⢹⣾⠁⣀⣤⡾⢩⣿⠈⠻⣟⠒⠶⠾⠃⠀⠀⠀⢸⡁⠀⣏⣹⠄⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⢁⣈⣧⣘⣊⣚⡰⠃⢸⣿⣇⣷⡀
⠀⠀⠀⠀⠙⠋⠉⣼⠃⢸⠻⡄⢀⣘⡆⠀⠀⠀⠀⠀⠀⠈⣷⢀⣤⣤⣀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣚⠥⣺⣧⠈⢳⠞⡋⢙⡧⣈⡛⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠿⠒⠾⠀⠙⠋⠉⠀⠀⠀⠀⢀⣠⣤⣼⡇⣾⠀⠀⢹⠎⠷⢛⡓⢦⡀⠀⠀⠀⠀⠀⠀⠈⠳⠞⢀⡏⠙⠰⠂⣁⣏⠑⠦⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢡⡴⣤⠀⠈⣛⠒⣋⡀⠀⢯⣹⠂⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣙⡃⠾⠆⠉⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠘⠳⠋⢀⡞⠉⠉⠉⠙⠦⣄⣤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢹⠛⢛⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣒⣛⡾⠀⢸⣚⣒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠁

⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⢸⣿⡿⣙⠋⡉⢿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⠟⠋⢻⣿⣿⣿⢿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣆
⠀⢸⣿⠇⠛⠆⠛⠘⣿⡿⣉⠉⣉⢹⣿⡇⢸⣿⣧⡀⠀⣤⣬⣛⣿⣿⣿⣿⣿⣿⠀⠀⢠⡖⠦⣄⣤⣤⣤⣄⣠⠤⣤⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⠀⠀⠀⠀⠀⣿⠇⠙⠲⠙⠀⣿⡇⢸⡿⠩⣧⡀⠈⣿⣿⣿⣾⣳⢹⣿⣿⠀⠀⠀⢧⣾⣿⣿⣿⣿⣿⣿⣤⡏⠀⠀⣿⣿⣿⣿⠟⢻⡟⠛⠿⢿⣿⣿⣿
⠀⢸⣿⣿⣧⣤⣼⣷⣿⠀⠀⠀⠀⠀⣼⡇⣼⣿⣤⠝⡇⢀⣾⣿⣿⠾⠿⣿⣿⣿⠀⠀⣀⣿⡟⢉⣻⣿⣿⣋⠛⣿⡆⠀⠀⣿⣿⠏⠀⣤⡄⢠⣶⡄⣾⣿⣿⣿
⠀⢸⣿⣿⣻⣿⣿⣿⡿⠟⣃⣀⣘⡛⠿⠇⣿⣿⣿⠈⣵⣿⣿⣿⣱⡿⢿⣮⣿⣿⠀⠀⠠⠼⢿⣶⣾⣟⣿⣧⣴⣿⠁⠀⠀⣿⡏⢸⡷⠈⢰⣦⠉⠀⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡆⣿⣿⣿⣦⣙⢿⣿⣿⣜⡿⠿⣫⣿⣿⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⣿⣾⣿
⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⢈⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠃⠀⠀⢀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟
⠀⢀⠀⠀⣤⣶⣤⠀⠀⢀⣴⣶⣄⠀⠀⠀⣾⣿⣿⣿⣿⡻⠽⣿⣿⣿⣿⣿⣿⣷⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡿⠻⢿⣿⣿⠟⢿⣿⣿⣷
⠀⢸⠀⠀⠘⣛⠃⠠⡴⠟⢻⣿⣿⣧⠀⠀⣿⣿⣿⡃⠈⠁⠞⠿⠻⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⡿⣷⣻⣧⢿⣿⢿⣿⣿⠀⣿⡿⣋⣀⠀⠀⠛⠁⠀⣀⣙⠿⣿
⠀⠈⠀⠰⣞⠙⡶⠂⠀⢠⣾⣿⢏⣿⡄⠀⣿⣿⡿⣍⣤⠀⣀⣷⣶⣿⣿⣿⣿⣿⠀⣿⡿⣾⣭⣽⡿⢿⢿⣿⡾⠿⣿⣿⠀⣿⣿⣿⠋⠀⣸⠃⢧⠀⠙⣿⣿⣿
⠀⢠⠀⠀⠛⣛⠓⠀⠀⠸⣿⡏⣼⢿⠇⠀⢿⣿⣿⣿⣿⠂⣻⣿⣿⡿⠛⠻⣿⣿⠀⣿⣿⣿⣞⣿⣿⣿⣿⣿⣟⣛⣿⣿⠀⣿⣿⣿⡀⢸⠇⣀⡸⡇⠀⣽⣿⣿
⠀⢰⠠⣤⣼⣿⣦⣤⠀⠀⠙⢷⣿⡄⠀⠀⣼⣿⣿⣿⣿⣾⣿⡇⠀⢱⠄⠀⢹⣿⠀⣿⣿⣿⣻⣟⣿⣿⢿⣟⣾⣭⣿⣿⠀⣿⣿⣿⣷⣌⠀⣿⠁⣡⣾⣿⣿⣿
⠀⢸⠀⢸⣿⣿⣿⠁⠀⣄⢀⣼⠟⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠒⢀⣿⣿⠀⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⡿⢋⣴⣿⣦⡙⢿⢻⣿⡿
⠀⠀⢀⣈⣁⣀⣈⣀⣀⣉⣉⣁⣀⣀⣀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠠⢤⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣄⣀⣀⣀⣠⡄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢇⣹⣿⡇⠀⠀⠀⠠⣿⣻⣿⣻⡇⠀⠀⠀⠀⠀⠀⣿⣿⣏⠞⢻⢏⣿⡙⡿⠻⣸⣿⣿⠀⠀⠀⠀⣸⢟⠻⣿⠟⠻⣧
⠀⠀⣿⣿⣿⡿⠿⠛⢛⡟⢀⣠⣾⣿⣿⣇⠀⠀⠀⠀⢈⣿⣿⡍⠀⠀⣀⡀⠀⠀⠀⣿⢯⣭⣰⣿⣿⠿⣿⣾⡆⣭⡽⣿⠀⠀⠀⢠⣿⣼⣧⠿⣼⣧⣿⡆
⠀⠀⣿⣿⣏⣀⣤⣶⣿⠟⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⣾⣿⣿⣷⠀⢸⠿⠇⠀⠀⢀⣿⣾⢿⡿⣩⣾⣿⣷⣍⢿⡿⢷⣿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷
⠀⢸⣿⣿⣿⣿⣿⡿⠃⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⢰⣿⣿⣿⣿⡇⢸⠀⠀⠀⠀⢸⣿⣿⣾⡇⣿⣿⣿⡿⣿⢸⣷⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢻⣿⣿⣿⣾⣿⣇⣤⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢼⣿⣿⣿⣿⡿⠎⠀⠀⠀⠀⢸⣿⣿⣿⣿⣮⣛⣿⣛⣵⣿⣿⣿⣿⠀⠀⠀⠘⠋⢻⣿⣿⣿⡟⠙⠃
⠀⠀⠛⠿⠿⠿⠿⠿⠿⠿⠿⠟⠻⠛⠛⠋⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢋⣩⡥⣬⣩⣝⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡁⢎⡉⠘⡀⢠⠇⠀⢹⣌⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣼⠁⠁⠋⣻⣿⣿⣦⡊⢁⣨⠄⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠛⠙⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣽⠲⠶⢾⣿⣦⣈⣿⡏⠁⠐⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠳⠦⣼⣷⣀⣠⣞⣛⣒⣤⣀⣀⠈⣷⡘⢣⣴⣤⣿⣿⡿⠿⡍⢨⡛⢡⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣴⡼⠋⠁⠀⠈⠉⠉⠉⢻⣿⣐⠀⣺⣿⣩⣿⣿⢳⡀⡜⠿⣣⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⣿⣋⡁⠀⠀⠀⠀⠀⠀⢀⡿⣻⣾⣖⡛⠛⢛⡿⣏⣈⣭⣵⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⣯⣐⠢⣈⠉⠉⠉⠛⠓⠛⢚⣿⡿⡿⣿⣿⣿⢀⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⠈⠙⠈⣝⣓⠀⢠⡾⡰⠋⠀⠀⠀⠀⣸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢻⣇⠀⠀⠈⣿⡔⢩⣾⠁⠀⠀⠀⠀⢀⣿⡷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⢹⣦⣀⣀⣘⣇⠈⠁⠀⠀⠀⠀⠀⣰⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⣻⡿⠯⠽⠾⣷⡾⠦⣲⣤⣶⣿⣿⣿⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⣙⣤⣤⣀⡙⣷⣀⣈⣺⡏⣸⣷⣿⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠿⣿⣏⣉⣉⡛⠛⣿⣙⣉⣵⣿⣿⠇⣸⣷⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣤⣤⣤⣄⣈⠉⠉⠉⢁⣤⣤⣤⣶⣤⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣶⣤⣴⣾⣶⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠁⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠁⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣁⣀⣀⣀⣀⣈⣧⣀⠀⠀⠀⠀⠀⠀⠀⢠⡐⠛⠛⠓⠀⠀⠚⠠
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣁⣀⣀⣀⣀⣀⣀⣈⣹⡇⠀⠀⠀⠀⠀⢠⣿⠿⠛⠃⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⠀⠀⠀⠀⠘⢷⠶⠶⠶⠶⠶⠶⠶⢶⠞
⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠟⠛⠛⠛⠛⠛⠛⠛⠛⠻⣇⠀⠀⠀⠀⠀⣴⠟⠉⠁⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⠔⠊⠉⢩⠁⠀⠀⠀⠀⢹⠉⠉⠒⢄⠀⠀⠀⠀⠀⢸⡆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⢯⠀⠀⠀⠘⠤⢀⣀⣀⡠⠜⠁⠀⠀⢠⠇⠀⠀⠀⠀⠈⣏⡀⠈⠀⠀⠀⠀⠀⠈⠀⠀⢸⠃⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠈⠻⣦⣄⡀⠀⠀⠀⠀⠀⢀⣀⣠⡾⠋⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⡇⢸⠉⠛⠙⠿⠞⠛⠋⢹⠇⡇⠀⠀⠀⠀⠀⠀⢸⡧⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠱⡈⢦⡀⠀⠀⠀⠀⣀⠞⣰⠃⠀⠀⠀⠀⠀⠀⢸⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠙⢦⣙⠒⠲⠒⢚⣡⠔⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢰⣟⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠘⣯⡉⠁⠀⠀⠀⣀⣀⣤⣴⣶⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠒⠒⠒⠒⠒⠚⠋⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣄⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡲⡽⣾⡽⠿⠚⠺⢽⣾⡖⣆⠀⠀⠀⠀⠀⠈⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡟⠞⠁⠀⡀⠀⠀⠀⠹⠉⠰⠆⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣊⠀⢃⠀⠁⠀⠀⠀⠀⢘⣡⠟⠀⠀⠀⠀⠀⡀⠤⠀⠈
⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠸⠀⠀⠀⢀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠂⣄⡐⠂⠈⠰⠏⠀⠀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠊⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡰⠆⠀⠀⠀⠂⠀⠀⠀⠨⠀⣀⣀⡠⠄
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠐⠋⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⢰⠀⠠⠀⠤
⠀⠰⠀⠠⠂⠀⠀⠁⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠎⠁⠀⠁
⠀⠀⠀⠀⢀⠀⣀⠤⠀⠀⠀⠀⠐⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠖⡀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠊⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠊⠁⠀⠀⠀⠀⢀⣋⣙⠒⠓⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠒⠶⠶⠤⠤⠤⠤⠤⠀⠀⠂⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⣠⣀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠄⠴⠀⠀⡜⠁⡀⣀⠠⠀⠀⠀⠋⠀⢀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⡄⠰⡆⠀⠐⠂⢘⠀⠐⠺⡀⠁⠘⠚⠂⠀⠀⠀⠘⡄⠀⠐⠷⠂
⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⡀⢀⠀⠄⠀⠀⠀⠀⠉⠁⠀⠀⠙⠒⠂⠂⠀⠀⠀⠀⠉⠚⠒⠀⠀⠀⠀⠀⠀⠈⠠⣄⣀⠀⠀⠀⠀⣀⠠⡀
⠀⠀⠀⠀⠐⢠⠀⡐⠀⠀⠸⡄⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢺⢡⠉⠋⠛⠊⠉⠠⠒⢼⡄
⠀⠀⠀⠀⠀⠀⠆⠀⢀⠀⠀⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠀⠀⠓⠦⠚⠀⠀⠘⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠂⠀⠀⢀⣀⣀⣀⡀⠀⠀⠜⠀⠀⠀⠀⠀⠀⡄⠀⠈⡄⢀⠀⠄⢀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢁⠐⠀⠀⠀⠳⡀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢠⠃⠀⠐⡀⠂⠐
⠀⠀⡠⢀⠀⠀⠀⠐⠄⠀⠁⠈⠀⠀⠀⠀⠀⡀⢀⡣⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠉
⠀⡠⠐⠈⠀⠀⠀⠀⢀⠒⠀⠂⠀⢀⠀⠀⠈⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⢌⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠠⣆⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⠐

⠀⠀⠀⠀⣀⠤⠒⠂⠉⠉⠉⠉⠉⠁⠒⠢⢄⡀
⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄
⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡠⠄⠀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠧⢼⡇⠙⠋⣿⢂⡥⡐⡄
⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣤⠖⣫⣽⠶⠖⠒⠈⣑⣳⣤⡇
⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⢀⣼⠋⣡⣴⣿⢧⠀⠀⠐⢮⣽⣿⣿⣿⡀
⠀⠀⠈⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠈⠛⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠐⡄⢰⠒⠒⠂⢙⠀⠀⠀⠀⠀⢀⡀⠀⠀⢸⠋⠀⠀⢙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⣄⠀⠀⢈⣙⠀⠀⠀⠊⠁⢀⠀⠀⠀⢶⣖⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠃⠙⠀⠀⠀⠈⠙⠃⠀⠀⠈⠛⠉⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢨⠤⠤⠤⣀⠀⠀⠀⠀⠀⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠷⠶⠞⠁⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡆⠀⠀⠈⠀⠀⣴⣄⣠⣶⠟⠁⠀⠘⠆
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠦⣥⠀⠀⠀⠀⢀⡿⠉⠉⠀⠀⠀⠀⢀⠀⠁⡀
⠀⠀⠀⠀⠀⠆⠂⠀⢀⣤⠀⠀⠀⠈⠁⣆⣀⡐⢏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⣄⣀
⠀⠀⠀⠀⢠⠀⠀⠀⢸⡇⠀⠀⣠⣶⣿⣿⣿⢡⣿⣧⠁⠀⢀⣠⣴⣶⣦⠀⠀⢸⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢠⣾⣿⣿⣿⣿⣷⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣧⡀⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠶⠒⠚⠲⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠻⣏⠁⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⣀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⢹⡇⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⠀⠀⣠⡞⠁⠀⠀⠀⣀⣀⣀⣀⠘⣧⡀⠀⢀⣸⠇⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠈⠉⠉⠉⣿⠀⠀⠀⢠⣎⠖⠉⠉⠑⢿⢎⠙⠛⠛⠁⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⣧⠉⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢀⣤⣤⣀⢀⡟⠀⠀⠀⠀⠡⣿⣄⠀⠀⢀⢴⡹⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⣰⠟⣰⠏⠉⠛⠁⢀⣀⣀⠀⠀⠈⠫⠶⠶⠍⠋⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⡼⠃⠀⣠⠞⠁⢰⢏⣤⣠⡴⠞⠋⠁⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁
⠀⠀⠀⢀⡾⠁⢠⡾⠃⠀⠀⣸⢿⣄⠙⢦⡀⠀⠀⣠⠇⠀⠀⢀⣀⣤⠴⢶⡾⠋
⠀⠀⠠⣟⣀⡴⠋⠀⠀⠀⣸⣇⠀⠙⢷⣄⠙⢦⣰⠏⠀⠰⣟⠉⠁⠀⠀⠀⣧
⠀⠀⠀⠈⠋⠀⠀⠀⠀⢸⣇⠈⠳⣄⠀⠙⠷⣄⣹⠆⣀⣠⢿⡄⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠳⣤⠈⢳⣤⡴⠛⠳⠛⢉⣤⠞⠁⠀⠀⠀⣀⡼⠆
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢀⣤⠞⠻⠋⠁⠀⠀⢀⡴⠋⠁⠀⠀⣠⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠰⣏⣠⠞⠋⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⣠⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣠⡾⠋⢀⣤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡴⠞⠋







⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⠿⠿⠿⠿⣿⣿⠿⠿⠿⠿⠿⢿⠿⠿⠿⠿⠿⢿⣿
⣿⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣷⣾⣶⣿⣿⣿

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠋⠁⠀⠐⠄⠀⠉⠙⠿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠉⠉⠉⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡈⣿⣷⠶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣹⣿⠀⠀⠈⠙⠻⢷⣦⡀
⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠸⣿⣦⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣸⣿⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣤⣙⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⡿⠇⠀⠀⠀⠀⠀⢠⣾⠿⣧
⠀⠀⠀⠀⢸⣿⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣶⣶⣶⣦⣤⣤⣤⣴⣶⣾⡿⠿⠛⠉⠀⠀⠀⠀⠀⣠⣴⣿⠏⠀⣿⡆
⠀⠀⠀⠀⢸⣿⡀⠀⠹⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠋⠀⢸⡀⣿⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⡆⠈⠙⠿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⠟⠋⠁⢸⡆⠀⢸⡇⣿⡇
⠀⠀⠀⠀⠀⣿⣇⠀⢸⡇⠀⠀⢀⠀⠈⠙⠻⡿⠿⢿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡿⠿⠿⠛⠋⠁⣸⡆⠀⠀⠈⣷⠀⠈⣷⣿⡇
⠀⠀⠀⠀⠀⠸⣿⡆⠘⡿⠀⠀⢸⡆⠀⠀⠀⡇⠀⠀⠀⡆⠀⠈⠉⢉⡋⠉⠉⢹⡇⠀⠀⠀⢸⠀⠀⠀⢹⡇⠀⠀⠀⣿⠀⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣶⣧⡀⠀⢸⡇⠀⠀⠀⡇⠀⠀⢰⡇⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⠘⣧⠀⠀⠀⣿⣤⣾⣿⠁
⠀⠀⠀⠀⠀⣰⡏⠉⠙⠿⣷⣦⣠⡇⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⢸⠁⠀⠀⢸⡇⠀⠀⠀⢻⡇⠀⠀⠀⣿⠀⣠⣴⣿⡿⠋⠉⢷⣄
⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣤⣅⣀⣀⢈⡇⠀⠀⠀⠘⠀⠀⠀⠘⠃⠀⠀⢀⣘⣧⣠⣤⣴⣿⡿⠛⠋⠁⠀⠀⠀⠀⠹⣷⡄
⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⢿⣿⣷⣶⣿⣿⣾⣿⣿⣿⡿⠿⠿⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡜⢿⣦
⠀⢰⣿⣧⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢻⣷
⢠⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⣼⣿⢸⣿⠀⠀⠀⠀⠀⢀⣀⣤⣶⠶⠖⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠶⢦⣀⡀⠀⠀⠀⠀⠀⢸⣿⡆
⣿⡇⢸⡇⠀⠀⠀⠀⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣿⡇
⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⡇⣼⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⣸⣿
⣿⡇⢸⡇⠀⠀⠈⠑⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠋⠁⠀⠀⠀⢘⡟⣿
⣿⡇⢺⡇⠀⠀⠀⠀⠀⠀⠙⠛⠷⣶⣶⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣶⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⠿⠿⣷⣶⣶⣶⣶⣶⣶⣶⣾⡿⠿⠿⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢹
⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸
⣿⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸
⠀⣿⡇⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⢹⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⢸⣷⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⢸⣿⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠸⡇⢸
⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢿⡇⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠈⣿⡄⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⠶⠿⠿⠛⠁⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠷⠶⠦⣤⣀⠀⠀⠀⣠⡤⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠘⣿⣄⠙⢧⡀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠈⢻⣧⡀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⡄⠀⠀⠀⣀⣤⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣦⣤⣛⣛⣓⠾⠿⠿⠿⣿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⣉⣉⣠⣤⣴⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⢿⣶⣾⣿⣷⣶⣶⣶⣶⣶⣾⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁