Bugs Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠠⣀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⢀⡀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢶⡄⠀⠀⠀⠀⠀⠀⠀⢉⡝⠻⣍⠀⠀⠀⠀⠀⠀⠀⠀⣠⢪⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⢷⠀⠀⠀⢀⡠⠤⠠⣯⣤⣤⣼⠧⠄⠤⢄⠀⠀⠀⢠⠇⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡘⣆⠀⢠⠏⢀⣤⣤⣤⣄⠀⣠⣤⣤⣤⡀⢱⠀⠀⠎⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣝⣄⣸⡀⢿⣿⣿⣿⡿⠀⣿⣿⣿⣿⡇⢸⣤⣢⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣬⣧⡈⠻⠿⣿⣷⠀⣿⣿⣿⠟⢁⣞⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦⠀⣿⣿⠀⣽⣿⠀⣰⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠚⣩⡥⠽⢿⣶⡿⠿⣍⡉⠈⠙⠒⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⢁⣤⣶⣾⣯⠀⠀⠀⣿⠀⠀⢈⣿⣿⣶⣦⠀⠸⡖⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡥⢼⠁⣼⣿⣿⣿⣿⡇⠀⠀⣿⠀⠀⢿⣿⣿⣿⣿⡇⠀⢹⠘⢻⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠀⢀⡎⠀⠘⢿⣿⣿⡿⠁⣀⣤⣿⣦⣄⡈⠻⠿⠿⠟⠀⠀⠈⡆⠀⢎⢧
⠀⠀⠀⠀⠀⠀⠀⢠⠂⡞⠀⣼⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢹⠀⠈⠄⢧
⠀⠀⠀⠀⠀⠀⠀⡞⡼⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠘⡇⠀⠱⡀⢣
⠀⠀⠀⠀⠀⠀⢸⡼⠁⠀⢸⠁⠀⣠⣤⣤⣄⣀⠀⠙⢿⣿⣿⣿⣿⠟⠀⢀⣠⣤⣤⣤⡀⠀⢳⠀⠀⠐⠾⡄
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⡟⠀⣾⣿⣿⣿⣿⣿⣿⣶⡀⠉⢹⠉⠁⣠⣾⣿⣿⣿⣿⣿⣧⠀⠸⡄⠀⠀⠀⠙⢦⣀
⠀⠀⢀⣤⠟⠁⠀⠀⠀⢸⡇⠀⢿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⠀⠀⢿⣿⣿⣿⣿⣿⣿⠟⠀⠀⡇⠀⠀⠀⠀⠀⠹⡍⠂
⠀⠀⠋⠏⠀⠀⠀⠀⠀⢸⡀⠀⠀⠙⠿⣿⣿⣿⣿⠏⠀⢀⣼⣆⠀⠈⠿⣿⣿⠿⠛⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠈⢉⣤⣶⣿⣿⣿⣿⣿⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⢸⢱⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠎⣹⡂⠀⣀⣀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢀⠀⠀⢸⠲⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠖⠹⣶⣿⣿⣿⡇⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⣾⣿⣿⣦⡌⠀⡎⣇
⠀⠀⠀⠀⠀⠀⠀⡟⡇⠀⢿⣿⣿⣿⡿⠀⠀⠀⠀⠉⠛⠛⢿⡿⠟⠛⠉⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢰⠸⡄
⠀⠀⠀⠀⠀⠀⢸⢩⠃⠀⠘⣿⣿⣿⠃⠀⢀⣤⣶⣶⣄⠀⢸⡇⠀⣠⣶⣶⣤⠀⠀⠹⣿⣿⡿⠀⠀⠸⡄⢱
⠀⠀⠀⠀⠀⠀⡟⢸⠀⠀⠀⠹⣿⠏⠀⠀⣼⣿⣿⣿⣿⠆⢸⡇⠀⣿⣿⣿⣿⣇⠀⠀⠈⣿⠃⠀⠀⠀⢣⠀⡇
⠀⠀⠀⠀⠀⢀⠃⣾⠀⠀⠀⠀⠈⢧⡀⠀⣿⣿⣿⣿⣿⠀⢸⡇⠀⣿⣿⣿⣿⣿⠀⠀⡴⠋⠀⠀⠀⠀⠈⢷⣟
⠀⠀⠀⠀⠀⣨⠾⠃⠀⠀⠀⠀⠀⠀⠙⢦⡙⠿⣿⠿⠃⠀⢸⡇⠀⠈⠻⠿⠟⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⣤⣤⣼⣧⣤⣤⡤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡷⢤
⣠⢖⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢠⣯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⢀⡀⡠⠂
⠀⠀⠀⠀⠀⠀⢘⣿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣿
⠀⠀⠀⠀⠀⠀⠘⠛⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢹⣇⡿⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠉⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⣶⡦⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣤⡆⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⢢⣾⡯⢷⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⡿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠂
⠀⠀⠀⢀⡴⠛⣽⡇⢳⡙⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⠒⣟⣿⣿⠂⡀⠀⠀⠀⠀⠀⠀⠄⢹⣿⣿⣏⠡⡀
⠀⠀⠴⠋⠀⡼⢹⡇⡎⢷⠀⠉⠣⠀⠀⠀⠀⠀⠐⠈⠐⢿⣿⡿⠂⠁⠀⠀⠀⠀⠀⠀⠐⠙⣿⣿⠏⠁
⠀⠀⠀⠀⠐⠁⠈⣇⠂⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠉⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⣠⣶⡄⠀⠀⢈⠉⠀⠀⠀⠀⠀⠈⢛⣲⣦⣾⣿⣶⣖⣛⠁⠀⠀⠀⠀⠀⠉⡁⠀⠀⢠⣶⣄
⠀⠀⠀⠀⣼⣿⣿⣷⡀⢈⠀⠀⢀⠀⠠⠒⡀⠈⡡⢺⣿⣿⣿⣿⡗⢌⠁⢀⠒⠄⠀⣀⠀⠀⡑⢠⣾⣿⣿⣧
⠀⠀⠀⠀⠈⠻⣿⣿⣿⠀⠀⠀⠻⠿⠀⠈⠀⠈⢐⡞⣿⣿⣿⣿⢫⡂⠁⠀⠁⠀⠿⠟⠀⠀⠀⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣧⡄⠀⠀⠀⠀⣰⡶⣴⠋⠈⢿⣿⣿⡿⠁⠙⣦⢶⣆⠀⠀⠀⠀⢠⣼⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠁⠄⠁⡴⠁⠀⢘⢸⣿⣻⡇⠃⠀⠈⣦⠈⠢⠀⠂⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣆⠴⢊⢃⠔⠁⠘⠸⣿⣿⠇⠋⠘⠢⡘⡑⠦⣰⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠛⠁⠀⠀⠀⢠⢿⡿⡌⠀⠀⠀⠈⠛⡄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⠂⠀⠀⠁⠀⠀⠀⡌⢣⠀⠀⠀⠀⠀⠀⠐⣀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠆⠀⢠⠠⠔⠁⠈⠢⠄⡄⠀⠰⠄⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡄⠀⡀
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣹⡀⠀⠀⠀⠁⠈⢷
⠀⠀⠀⠀⠡⣤⢆⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⡀⠈⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⢄⣀⣤⣾⣶⣾⠟⠁⠀⢸⣿⣿⣤⡄⠀⠀⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣧⠤⣤⣴⡿⠻⡿⢛⣿⣶⣆⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⢨⠹⢶⣶⣿⡤⢴⠓⡞⣻⣿⣁⣴⣤⣤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢔⣯⣀⠀⠀⠘⢿⣯⡷⡆⣠⡾⢋⠜⣡⡞⠀⣐⠀⠀⠉⠑⠶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⡀⠀⣹⣿⣿⡿⢚⣵⣿⣶⣷⡤⠃⠀⠠⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠛⠁⣻⣶⡿⣈⣸⣿⣿⣦⡄⠚⠀⠈⠛⠙⡄⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠹⣿⣯⣃⣠⠿⣿⣧⣄⢴⠳⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⠻⣿⣯⣠⣬⠿⣿⣦⣤⠒⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢰⡋⠙⢿⣶⣤⣾⡻⣿⡀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠙⢦⡀⠀⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⣠⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣄⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣇⠀⠀⣠⣿⣽⡆⢰⣯⣿⣄⡀⠀⣸⣽⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡾⠿⠋⣿⠁⠀⠀⠠⣾⠙⠿⢷⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣏⡀⢀⣠⣷⢀⣼⣧⠀⣿⣆⡀⢀⣹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠻⢿⣿⣿⡿⡞⢿⡿⢳⢿⣿⣿⡿⠟⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⣀⣀⣠⠄⢼⣿⣿⡿⢿⣿⣿⣇⣤⣄⣀⣀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢸⡛⠿⣿⣿⠿⠿⠟⠁⠘⠻⠿⣿⢿⣿⠿⢛⡇⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢸⣇⠀⣧⣭⣄⣄⣀⡀⢀⣀⣠⣤⣴⣿⠀⢸⡇⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⣿⣆⢿⡿⠛⠛⡛⠃⠙⠟⠻⠟⣿⡿⣰⣿⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⣸⠇⢼⣷⣤⣤⣤⡇⢠⣴⣤⣴⣿⣏⠸⣇⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢀⣜⡿⠀⣸⠿⢿⡿⢿⡄⢰⡿⢿⣿⣿⣇⠀⢿⣣⡀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠸⠏⡀⣾⣿⣤⣠⠀⠀⠀⡀⠀⠀⣄⣦⣿⣶⣄⠩⠅⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣦⡼⠋⢹⣿⣿⣾⣿⣾⣷⣿⣿⣿⣿⡏⠙⢿⣶⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⣧⠀⠀⠘⣇⠀⠀⠈⢸⡇⠁⠀⢀⣼⠃⠀⠀⣼⠃⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠈⢷⡀⠀⢿⣿⣿⣷⢾⣷⣾⣿⣿⡿⠀⢀⡾⠁⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠈⣿⡄⠘⣧⣀⣉⣼⣇⣁⣀⣼⠃⢠⣿⠁⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠈⣷⠀⠘⢿⣿⣿⣿⠿⡿⠃⠀⣾⠁⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡄⠀⠀⠀⠰⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠂⠀⠀⠀⠠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡇

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡀⠀⢠⣾⣷⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⣨⡇⠀⢀
⠀⠀⠀⠀⠀⠹⣤⠿⠋⠿⢷⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⡿⢾⣏⡠⠋
⠀⠀⠀⠀⠀⡠⢤⣛⢻⠛⡃⡦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠒⣾⣿⡀
⠀⠀⠀⠠⠚⣠⣾⣿⣾⣇⣷⣧⡈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠀⣿⡇⢡
⠀⠀⠀⠀⢠⡇⠈⠿⡿⠿⠟⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⣶⣿⠟⠆
⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠓⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⡂⡄⣸⣀⠀⡁⡒
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⣀⠈⢦⡿⠿⣿⣿⣿⡥⠃⣀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠐⠊⠉⠀⠈⠛⠛⠥⠹⢶⣤⣤⣽⣶⣿⣿⣿⣿⣿⣷⣯⣤⣠⡤⠶⠭⠍⠛⠋⠁⠀⠉⠀⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠈⠉⢉⣉⠏⠉⢡⡟⢻⣿⣿⢿⠛⣿⡟⣏⡉⠐⠂⠚⠂
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡠⠒⠉⡝⣠⠀⠊⣱⣿⣿⣿⣮⠀⠃⢃⢺⠉⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣤⢾⣿⣿⣿⣿⣿⣷⣄⣸⡿⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢸⠀⠻⣿⣿⣿⠟⠁⡏⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠉⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⣆⢀⣀⡀⣴
⠀⠀⠀⠀⠀⠀⠀⣠⡿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢧⣼⣩⣧⠍⠠⡀
⠀⠀⠀⠀⠀⢀⡄⠀⠚⠃⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢠⠴⠾⣿⠷⠦⡄⢡
⠀⠀⠀⠀⠀⠁⠁⠶⠀⠰⠇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⣰⣄⠋⣠⣆⣆
⠀⠀⠀⠀⠀⢀⠲⡆⠀⠀⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡉⣟⠀⡻⢈⠈
⠀⠀⠀⠀⠀⠊⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠛⠁⢾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⡔⠊⠀⠀⠀⠀⠀⠀⠇⢀⡀⠀⠀⠀⠀⠀⠀⢂⠀⢀⠀⠂
⠀⠀⠀⠀⢐⢀⠂⢐⠀⠀⠀⠀⠀⢉⡜⠐⣗⠀⠂⠀⠀⠀⠀⠐⡗⠀⠀⠀⠀⠀⠀⠀⠖⠁⢠⡄
⠀⠀⠀⠀⢀⡙⠀⠠⠂⠀⠀⠀⠀⠊⡇⢰⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⡁
⠀⠀⠀⠀⠃⠀⠀⢀⠓⠀⠀⠀⠈⠀⢇⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠘⠀⠀⠈⠃⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠠⣀⠀⡔
⠀⠀⠀⠀⠌⠀⠐⢁⠁⠀⠀⠀⠀⠀⠀⢀⠨⠦⠐⠶⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⡄⠠⣬⡂⣡⣤⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⣀⣀⣀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⡠⢊⣠⠞⢈⢡⠂
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠐⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠳⠿⠥⠒⢡⠻⡄
⠀⠀⠀⠠⠖⣋⣽⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠿⠛⠋⠙⠳⠴⠤⠄⠀⠀⠢⡀⠀⠄⣢⡮⠈⠤⠀⠤⠂⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠘⠋⠋⠀⠘⠀⠀⠀⠀⠀⠀⠄⠔⡒⣂⣀⠂⣠⠤⠤⠈⠁⠀⠀⠀⢀⢈⡔⡀⢲⠟⠁⢈⣢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⣂⣀⠂⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠁⠁⠀⠡⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⢳⠀⠀⠀⣎⣃⣀⠀⠀⠀⡀⠠⠦⠀⠀⠀⢨⢿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⣆⠀⠀⣈⣟⠉⠀⠀⠀⠹⠖⠶⠀⠀⢠⠃⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⣱⠊⠀⠉⠉⠈⠉⠉⠉⠉⠑⣄⡤⣮⠴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠠⡄⠒⢚⡩⠝⠻⡴⠋⠀⠀⠈⠙⡗⠒⠒⡖⠀⠉⠉⠻⣄⠠⡤⣀⡀
⠀⠀⣀⣀⣤⣶⡶⡧⠚⠁⠀⠀⢀⠆⠀⠀⠀⠀⠀⠙⢲⠚⠀⠀⠀⠀⠀⠘⡟⠁⠢⢌⠙⠤⣀
⡠⡮⠘⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠑⣶⠹⣦⣄⡀
⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠈⠀⠀⠙⠛⣕⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢛⡶⡇⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⣼⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⡜⠀⢣⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡼⠻⡟⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢸⠁⠀⠀⠙⢀⣀⣀⠀⢰⠀⢀⠀⢀⣠⠞⠁⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣶⡌⠀⠀⠀⠀⠀⠙⠻⠬⣿⣨⡭⠗⠋⠁⠀⠀⠀⢸⡀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⣌⠆⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠻⡅
⠀⠀⠀⠀⠀⢠⢺⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⡵⡄
⠀⠀⠀⠀⣰⣃⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢳⡀
⠀⠀⠀⠠⠇⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢸⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠹⣷⡀⠀⠀⣀⣿⣇⡀⠀⠀⣰⡿⠀⠀⠀⠀⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡀⠀⠀⠹⣿⣦⣼⣿⣿⣿⣿⣤⣾⡟⠁⠀⠀⣠⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⡄⠀⠀⢹⣿⣯⣿⣿⣿⣯⣿⣿⠁⠀⠀⣴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣽⣿⣿⣿⣿⢿⣿⣥⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠙⢻⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⠛⠉⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡇⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠁⠀⠀⠀⠀⠀⢠⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣧⠀⠀⠀⠀⠀⠀⠱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠻⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠈⠛⠿⠟⠿⠟⠋⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⢛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⠀⠀⠀⠀⠀⢀⡠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠄
⠀⠀⠀⠀⠀⢀⣾⣇⡀⠀⠀⠈⢻⡄⢾⡛⢆⠀⢀⠔⣻⡇⣼⠋⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢧⡾
⠀⠀⠀⠀⠀⡴⣿⣿⢄⠀⠀⠀⠀⠹⢤⠭⠿⣿⡿⠯⢥⠴⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣾⣶⡇
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢆⠀⠀⢻⠇⠀⢀⠆⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⢘⣿⣿⡁
⠀⠀⠀⠀⠀⠀⣇⣹⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣠⣀⣀⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⣰⣶⠀⠀⠀⠛⠿⣿⣿⣶⣶⣞⣳⣶⣶⣿⣿⠿⠛
⠀⠀⠙⢿⣿⣷⣤⣴⣿⣿⡿⠃⠀⠀⠀⠀⣿⣻⡅⠀⠀⠀⢀⣠⣴⣾⡿⢻⡟⢿⣷⣦⣄⡀
⠀⠀⠀⠀⣿⣿⡟⢿⣿⣿⠀⠀⠀⠀⠀⠀⢿⣿⠃⠀⠀⠀⠈⠉⠉⠀⠀⢸⡇⠀⠀⠉⠉⠁⠀⣰⣿⣷⡄
⠀⠀⠀⠀⠈⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⢿⣦⡀⠀⠀⠀⠀⠀⠀⣾⣇⠀⠀⠘⢿⣿⣷⣦⣔⣦⣴⣾⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠬⠕⠀⠀⠀⠀⠀⠾⠟⠋⢀⣉⠈⠙⠻⠆⠀⠀⠀⠀⠀⢿⡷⠀⠀⠀⠀⠐⠿⠿⢻⡟⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣻⣷⢆⠀⠀⢀⠴⣿⣻⡟
⠀⠀⠀⠀⠀⢀⣾⣇⡀⠀⠀⠀⠸⣿⣇⠀⣁⣤⣁⢀⣿⡿⠀⠀⠀⠀⢀⡜⢣⡄⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⢀⣾⢿⢿⣷⠀⠀⠀⠀⣼⡟⠋⠩⠟⠉⠛⢿⡆⠀⠀⠀⠀⢸⣷⣾⡇⠀⠀⠀⠀⢠⣾⣿⣿⣧⡀
⠀⠀⠀⠀⣼⡟⡲⡚⢿⡇⠀⠀⠀⢹⣷⣤⠄⠀⢠⣴⣾⠁⠀⠀⠀⠀⠘⣿⣿⠃⠀⠀⠀⠠⠿⠛⢻⠟⠛⠷
⠀⠀⠀⠀⠙⠁⠉⠁⠈⠃⠀⠀⠀⡜⠉⠁⠀⠀⠀⠉⠉⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡀⠀⠀⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯⣤⡶⠶⠒⠻⣽⣦⡿⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣿⣿⣯⣀⠀⠀⠀⠀⠀⢹⣿⣾⣿⡿⢢⡀⠀⠀⠀⣀⠠⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡈⠛⠻⠿⣷⣶⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⢀⣀⣠⣴⣿⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠋⠀⠀⠀⠀⠈⠉⠙⠲⣤⣈⠙⠿⣿⣿⣿⣿⣿⡇⠉⠛⠾⡆⠀⠀⢰⡿⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⣿⠿⠷⠦⢌⡙⠻⣿⣿⡇⠀⠀⠀⣷⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣫⣾⣿⡆⠀⠀⠀⠉⠲⣌⠙⢿⣦⡀⠀⢿⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢫⣾⣿⣿⡿⠃⠀⠀⠀⠀⠀⠈⠳⣄⠻⣿⠶⢸⣶⠟⠁
⠀⠀⠀⠀⠀⠀⢀⣠⡾⣿⠆⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠴⠋
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠙⠻⠿⠋⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⣰⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣦⠀⠀⠀⠀⠀⢀⣴⣦⣄⠀⠀⢰⣷
⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠇⠀⠀⠀⠀⠹⢿⣿⠟⠀⠀⣸⡿
⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃
⠀⠀⠀⠀⠀⣿⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀
⠀⣀⣠⣴⡾⠃⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠙⠟⠛⠗
⣠⡿⣿⠁⠀⠀⠀⠀⠀⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢹⣆

⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢤⠀⠉⠛⠷⠾⠛⠉⠀⡤⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣀⠀⠀⠀⠀⠀⠈⠢⣘⡌⠀⠀⢩⣃⠜⠁⠀⠀⠀⠀⠀⣈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⠀⠀⠀⢽⣟⣠⣀⢻⡇⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣇⠀⠀⢀⣠⣴⣭⣽⣿⣿⣭⣹⣦⣄⠀⠀⠀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⣰⠋⢮⣾⣯⣿⡇⢹⣿⣹⣿⡳⠹⣄⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⠀⢸⠋⣿⣿⡇⢸⣿⣿⠙⡇⢀⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠿⣇⢿⠀⢿⣿⡇⣸⣿⡟⢠⡷⣸⠿⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⢁⡴⠚⠻⠿⣷⣌⡛⠃⠛⢛⣡⣾⠿⠛⠳⢶⡌⠻⣦⣄⡀
⠀⠀⠀⠀⢀⡴⠖⠒⠚⠋⠙⠋⠀⠀⢸⢼⣿⣆⠀⢀⠙⣟⣏⣹⣫⠋⣀⣀⣀⣷⡷⡇⠀⠘⠋⠉⠉⠓⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⣟⡼⢿⡃⢟⠜⠣⢻⢾⢋⣟⢹⡟⢱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣷⣶⣆⣿⡦⢰⡇⣤⣿⣫⡼⣿⢋⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⠙⡼⣯⠾⣷⣸⡇⣾⠿⢽⣸⠋⠃⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⣤⡤⠟⣦⣽⢼⡟⢧⡿⣭⣴⠶⢢⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢀⣌⣿⡓⡿⢸⡇⢻⣙⣯⢲⠀⣾⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣉⣗⠑⠃⢸⡇⠉⠙⣼⣉⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠛⠓⠛⠛⠛⠒⠚⠛⠋⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⣠⣤⣿⣤⣅⣀⡀⢀⣀⠀⠀⠀⢠⠄⠀⠀⢈⠤⢌⡁⠀⠀⠠
⠀⠀⠁⠀⠀⠈⠀⠀⢠⡼⠓⢧⡄⠀⠀⠀⠀⢀⡜⠀⠀⠀⡠⠗⣠⠀⠀⠀⠀⠈⠀⠀⢀⡠⠷⣾⠿⣉⠿⣷⠯⠀⠀⠀⠀⠁⠀⠀⠠⠀⡀⠀⠀⠀⠀⣄⠀⠆
⠀⠀⠀⠀⠀⠤⠠⠀⠛⣧⠀⣾⠟⠀⠀⠰⡄⠁⠀⠀⠀⢘⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠓⢶⡶⠛⠃⡀
⠀⠀⠀⠀⠀⠀⠒⠚⠓⠈⠙⠉⠚⠓⠒⠀⠀⠀⠀⠀⠀⠐⡀⠘⢆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠘⠀⠀⠘⠀⠀⢀⠉⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠼⠏⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠂⡀⠀⠀⡀⢒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⢉⡁⠊

⠀⠀⠀⠀⠀⠀⡀⠠⣀⠀⠀⣀⠄⢀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠰⠢⠀⡠⠂⠀⠀⠀⢀⡀⠤⠤⠀⠀⠤⣀⡘⢀⡤⠄⠀⠄⠠⠄⣀⡀⠀⠀⠀⢠⣤⣆
⠀⠀⠀⠀⠀⠉⢠⠀⢨⠗⢸⡅⠀⠜⠉⠀⠀⠀⠀⠀⠢⡀⠃⢠⣤⠘⠁⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⡀
⠀⠀⠀⠀⢠⠀⣤⠓⢺⡄⢠⡷⠚⣤⠀⡄⠀⠀⠀⠀⠀⠈⢃⣀⠀⡈⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠸⣧⡼⠀⢠⣄⣀⡠⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠘⣄⡈⠛⠈⠉⠈⠁⠚⢁⡴⠃⠀⠀⠀⠀⠀⠆⣀⠉⠃⣁⡀⠢⡀⠀⠀⠀⠀⠀⠀⠿⡜⣦⣤⣘⣡⣤⣜⡶⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠄⠀⠁⠉⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠁⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠋⠽⠥⠓⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠐⠈⠒⠢⠄⢢⣰⣤⣆⡔⠠⠔⠒⠁⠀⠀⠀⠀⠀⢰⠀⠀⢠⡄⠀⢆⠀⠀⠀⠀⠀⠁⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢘⠀⠀⠀⠀⢰⣖⢉⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣟⠀⣻⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⢘⠀⡠⠚⠓⢄⣈⠀⠀⠀⠀⠀⠁⠀⢿⠀⠀⠀⢠⡤⡄⠀⠀⠀⡿⠀⠈⠀⠀⠀⠀⢈⣿⡀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠐⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⡤⣤⢠⠃⠀⠀⠀⠀⠀⠀⠀⠸⣇⢀⡰⢾⣿⡷⢆⡀⣼⠣⠀⠀⠀⠀⠀⠀⢠⣿⡇⡿
⠀⠀⠀⠀⠀⡠⠁⠾⡀⠀⢀⠷⠈⢆⠀⠀⠀⠀⠀⠀⠀⡀⢂⠀⠀⠀⠀⠘⠠⡀⠀⠀⠀⠀⢠⠢⣿⠀⠙⢿⣿⡿⠋⠀⣿⠔⡀⠀⠀⠀⠀⠀⠈⣷⢴⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠡⠀⠌⠀⡄⠀⠀⠀⠀⠀⠀⠀⠰⠀⣰⠀⠀⠀⠀⢤⠀⠘⠀⠀⠀⠀⠠⠀⠈⢧⡀⠀⠈⠀⢀⡼⠁⠀⠅⠀⠀⠀⠀⠀⠀⣷⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠑⠀⠀⠊⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣂
⠀⠀⠀⠀⠀⢣⣀⠀⠀⠀⠀⣀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⢀⢀⡔⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢠⡔⢠⠘⢢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⢤⣀⡤⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠁⢉⠀⠇⡄⠀⠀⠀⠀⡀⢀⠀⢀⣐⡀
⠀⠀⠀⠀⠀⢀⠜⠑⠚⠒⠋⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣮⠉⡵⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡈⠀⢈⠅⠃⠀⠀⠀⠀⠈⠣⢀⣠⠤⣄⡀⠴⠁
⠀⠀⠀⠀⢀⢾⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⡀⠘⠛⠙⢀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠒⠑⠀⠀⠀⢨⠀⠡⠀⠃⠀⡀⠀⠈⠈
⠀⠀⠀⠀⠂⠘⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⢻⡠⠈⡔⠛⠀⠛⡀⠁⠄⠛⠀⠀⠀⠀⠀⠀⠀⢨⠀⠰⠀⣤⠀⠃⠀⠀⠀⠀⠴⡀⠀⠴⠁⢀⠆
⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⢀⢜⠀⠀⠀⠀⢰⣄⠠⠀⠀⠃⠀⠘⠁⠀⠈⠃⠀⠐⠀⠀⠤⣰⡆⠀⠀⠀⢈⠂⠀⠀⡰⠀⠀⠀⠀⠀⠈⢄⠐⠆⠀⠑⠎⠠⠁
⠀⠀⠀⠀⠀⠀⠐⠍⠋⠛⢁⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣿⣿⡿⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢤⣴⣤⣤⡀⠰⣿⣿⡿⣽⣿⣿⠇⢀⣠⣤⣦⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣾⣿⣿⠟⠶⣤⣰⠏⠀⠀⠀⠀⠀⢀⣀⣀⢠⠤⠄⢀⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣿⣿⣿⡀⠀⠀⢸⣧⡠⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠝
⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⣽⣿⣿⣿⣶⣶⣿⣯⢵⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠾⣿⣿⣿⣿⣿⡊⠱⠤⠀⠀⠀⢀⡀⠀⠀⠀⠀⠠⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠀⠀⠀⢤⣶⣶⣾⣿⣿⣿⣿⣶⣶⣦⣤⡄⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣾⠁⢰⣿⣿⣿⣿⣏⣃⣀⣭⡆⠀⣧⠀⠀⠈⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⡟⠀⢸⣿⣿⣿⣿⣿⣿⠏⣿⣿⠀⣿⡇⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡝⠀⢸⣿⣿⣿⣿⣿⣛⠓⣛⡟⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠻⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠄⢠⣤⣤⡤⣀⡀
⠀⠀⣴⡿⠃⢹⣿⣇⢀⠛⠂⠉⠉⣙⣒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢒⣋⡉⠁⠐⠀⠀⠈⠿⣿⣁⠙⣿⡆
⠀⠀⢿⣿⡂⠿⠥⠄⠀⠀⠀⠀⣤⡿⠏⠉⠓⠪⠑⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠙⠈⠉⠘⠿⣶⣆⠀⠀⠀⠐⢒⡸⠇⣼⣿⠃
⠀⠀⠈⠫⢷⣾⡍⢀⣶⡀⠀⠀⠀⠀⢦⣤⣠⣶⠾⣿⡊⣳⢄⠀⠀⠀⠀⠀⣠⢞⠝⣻⠿⠻⣷⣤⣴⠊⠄⣀⣀⡀⢭⣏⢐⡾⢿⡷⠁
⠀⠀⠀⠀⠉⠲⣛⣿⡷⢐⣛⡅⠀⢀⡻⣿⣇⠚⠒⠶⢤⣈⡓⢷⡔⠆⣆⣴⢿⣳⡬⠔⠒⠛⢐⣿⡿⢃⣀⠀⢬⣉⠐⠾⣯⠝⠋
⠀⠀⠀⠀⠀⠀⠉⠶⣛⣷⡄⠒⣩⡅⠰⢿⣟⣛⠛⠁⠰⠭⡿⠟⣿⣧⣿⡟⠋⣩⠤⠌⠉⠩⢙⣿⠿⠂⢬⡉⠒⣶⣎⡛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⢿⢾⣥⣔⠊⠭⠠⠐⠒⠉⠁⠀⢁⠔⣡⢻⠧⠑⢄⠀⡀⠁⠠⠔⠂⠠⢍⠵⣶⣿⠻⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣿⠧⠍⢠⠆⠐⠀⠀⡠⠀⠀⡰⠛⠨⠐⡀⡀⡑⠢⡁⠀⠀⠀⠢⠈⠐⠿⣍⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣔⡦⠔⢣⠀⡠⠔⠊⠀⠄⡰⠁⠀⠘⠀⢡⠐⡘⡄⠈⡂⠠⡌⢠⡃⠲⢥⣾⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⣾⣡⢄⡔⠁⢠⠀⢀⠁⡘⢠⠃⠀⠀⠈⠀⠀⠂⠠⠘⡄⠁⠀⠰⡀⠙⢄⠢⣸⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡞⣳⡋⣰⢃⠇⡄⢸⢠⠃⠆⡌⠀⠀⠀⠀⠀⠀⠄⠂⢱⠘⡄⣆⢳⠹⡆⣻⡟⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢏⣿⢳⣾⣰⣃⣇⣾⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢸⣧⣧⣸⡤⢿⠛⣧⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠿⠘⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠀⠛⠚⠉

⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢰⡿⠛⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣶⡀
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠁⠀⣹⣤⣼⡇
⠀⠀⠀⠀⢹⣷⣴⣿⣿⠟⠁⠀⢴⡆⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡠⠊⣤⡀⠀⠀⠀⠚⣿⣯⣽⠇
⠀⠀⠀⠀⠸⣿⣤⣿⡿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠐⣪⣬⣭⣿
⠀⠀⠀⠀⠀⠻⣿⣽⣯⣅⣀⠀⠀⠀⠀⠀⠓⢴⣶⣤⣤⣾⣿⣁⡤⠴⠂⠀⠀⠀⠈⠙⢙⣋⠏
⠀⠀⠀⠀⠀⠀⠙⠻⠿⠖⠛⠛⠒⠒⠒⠒⠲⣿⣿⣿⣿⣿⣿⡷⠤⠤⠀⠀⠀⠀⠀⠒⠊
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣴⣧⡴⠋⠀⠀⠀⠀⢰⡿⣷⣿⢿⢻⣿⢻⠘⠿⣿⡀⠀⠀⠀⠀⠀⠈⢑⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣧⡦⡘⠀⠐⠋⢀⡵⠋⠀⠸⠟⠀⠀⠀⠉⠀⠀⠀⠀⣈⠲⢦⣬⣿⡄
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡃⠀⠀⠈⠀⠀⠀⠀⡆⠀⠀⠀⡀⠀⠀⠀⠀⠈⠳⣶⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣠⣄⣠⣤⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉

⠀⠀⠀⠀⠀⠀⣴⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣶⣤⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⣦⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣯⠛⠁⠀⠀⠈⠀⠦⡽⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⢸⣿⣶⡄⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣆⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⣠⠒⠉⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿
⠀⠀⠀⠀⠀⣼⣟⡻⠟⠀⠀⠀⠀⠀⠀⠘⣿⣯⡉⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⣠⠖⠀⠀⠘⠉⠉⠉⣡⣿
⠀⠀⠀⠀⠀⢻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠉⢽⡷⢌⡛⢷⡀⠀⠀⠀⢀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣻⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣝⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠙⠳⣭⣬⣿⣦⣀⢀⣿⣿⣿⣣⣀⠀⠀⠐⠀⠀⠀⠀⠀⢴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠙⣛⡛⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠒⠿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠁⠨⠀⠀⠀⠀⠀⣀⣹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠀⠀⣠⣀⣠⣤⢠⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⢶⣦⣤⣀⠀⠀⣸⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠁⠀⠀⠈⠁⣨⣿⣿⣿⣿⠟⢛⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⡿⠟⠠⣌⢱⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⠀⢠⠀⠋⠉⠁⠀⠀⠀⠀⣩⣿⣿⣿⡟⠂⠀⠀⠉⠉⠉⠉⠀⠀⠀⢸⣾⣄⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣯⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠏⣽⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⢀⣀⡀⠀⠀⠀⠀⠠⠂⢀⠄⠀⠀⠀⣿⡇⢻⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠘⠿⠟⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⣿⡇⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣤⣄⡁⣸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠈⠙⢄⠀⠀⠀⠀⠀⠛⠋⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⡀⠘⠿⢂⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣚⣻⣿⠷⢿⣿⠿⠛⠛⠛⠛⠛⠛⠓⠶⠦⣤⣼⣿⣿⣤⡤⠶⠖⠛⠛⠛⠙⠛⠛⠻⢿⣿⠾⢿⣿⣛⣲⣦⣤⣀⡀
⠀⠀⢀⣠⢶⣿⠿⠟⡫⠀⠉⡁⠴⢒⣿⡁⠀⣀⣀⡀⢀⡀⠀⠀⠀⠀⣤⣿⣿⠘⣧⠀⠀⠂⠀⠀⣀⠀⣀⣀⡀⠀⣻⣗⠢⢄⡉⠁⠬⡛⠿⢿⡷⢦⡀
⠀⢠⣾⣁⡟⠁⠀⠀⢀⠔⠉⡠⠞⠉⠀⢀⠔⠋⠀⢀⠌⠀⠀⠀⠀⣰⣿⣿⣿⠀⢹⣷⡀⠀⠀⠀⠐⠄⠀⠈⠲⣄⠀⠈⠓⢄⠈⠐⢄⠀⠀⡀⠹⣏⣹⣦
⠀⢿⣹⣅⣻⣎⢀⡔⠁⢀⡄⠀⠀⣠⠞⠁⠀⠀⡠⠂⠀⠀⢀⣴⣾⡿⠁⣿⣿⠃⡀⢻⣿⣦⣄⠀⠀⠈⠢⡀⠀⠀⠑⢄⠀⠀⠑⣄⠀⠱⡀⢸⣾⣋⣿⣺
⠀⠈⠷⢿⣷⣾⡏⡿⢤⣏⣹⣟⣻⣇⠀⠀⡠⠊⠀⠀⢀⣴⣿⠟⢁⠂⠀⠹⣿⠀⠁⠀⠈⠙⢿⣷⣄⠀⠀⠈⢦⡀⠀⢀⣿⣛⣿⣉⣧⠿⡩⣟⣶⣯⠾⠋
⠀⠀⠀⠀⠉⠉⠉⠙⠛⠳⠿⣿⣦⣿⣻⣿⡁⠀⢀⣾⣿⠟⠁⡠⠘⠀⠀⢰⣿⢀⠀⠀⠈⠠⡀⠙⢿⣷⣄⠀⠀⣿⣿⣻⣦⣿⡿⠗⠛⠋⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣇⣼⢛⣶⣿⠟⠁⠂⠂⠀⢰⠀⠀⣾⣿⠀⡇⠀⣶⠀⠀⠂⠂⠙⢿⣷⡞⢿⣨⣷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡾⣽⢯⠉⠁⠠⠀⠀⠀⠀⠸⠀⠀⢹⣿⠀⠇⠀⠙⠀⠀⠀⠀⢀⠀⠉⠭⣯⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣟⣠⡠⡆⠀⠀⡇⠀⠀⣀⠀⢸⠛⡋⠀⢀⡀⠀⠀⠀⠀⢀⡄⣤⢘⣭⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣾⣹⡯⢽⡯⣟⣲⣞⣫⣿⣿⠀⢻⣿⣿⣛⣶⣚⡯⣿⢭⣟⢹⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠓⠿⠻⠗⠟⠛⠋⠀⠘⠛⠻⠷⠿⠿⠓⠛⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⢠⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠒⢐⣶⣶⡿⣷⣒⡆⠐⠂⠀⠀⠐⠀⠀⠐⠒⠂
⠀⠀⠑⢠⡄⠀⠼⡇⢸⡷⠀⠀⡄⠈⠀⠀⢀⠀⡼⠀⠀⠀⠀⠀⠐⠀⠘⠉⢿⠈⠉⠀⠂⠀⠀⠀⠀⠀⡸⠦⣤⠴⢇
⠀⠀⠀⠀⠐⠒⠓⠉⠋⠚⠃⠂⠀⠀⠀⠀⠘⣄⢣⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠈⠂⠀⠈⠀⠐⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠁⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⢀⠤⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡀⠀⢀⠀⢀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠐⣀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠁⢂⣰⠉⢧⣀⠈⠀⠀⠀⠀⠐⢄⡁⠶⢈⡠⠄⠀⠀⠠⠀⠀⠀⠀⢸⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⢠⠘⠧⡙⠂⢛⡠⠃⡠⠀⠀⠀⠀⡀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠰⣞⣆⡈⢿⠁⣰⣳⠆⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠙⠒⠀⠈⠀⠐⠊⠁⠀⠀⠀⠀⠀⠑⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠂⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⣀⡀⢀⢠⢠⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⢠⠠⢤
⠀⠀⠀⠀⠈⠀⡿⠩⡇⠈⠀⠀⠀⠀⠀⠀⠐⢀⡠⠧⠀⠆⠀⠀⠀⠀⠈⡇⠀⠀⣀⣀⠀⠀⢸⠃⠀⠀⠀⠀⢸⡆⢸
⠀⠀⠀⠀⠀⢠⡚⠚⢢⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠤⠬⠁⠀⠀⠀⠀⠀⢸⡄⣐⣿⣿⢂⢀⡟⠀⠀⠀⠀⠀⢸⡧⡞
⠀⠀⠀⠀⠀⠘⠄⠀⠜⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⢂⠀⠀⠀⠈⠸⣇⠈⠛⠟⠁⣰⠗⠃⠀⠀⠀⠀⢸⡓⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⡇⠀⠀⠀⠀⠈⠀⠘⠂⠀⠀⠐⠃⠀⠁⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠂
⠀⠀⠀⠀⠢⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢰⠄⠄⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄
⠀⠀⠀⠀⢀⠯⠼⠤⢇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⠒⣋⠉⠀⠀⠀⠀⠀⠀⠀⠡⠀⠂⠊⠇⠀⠀⠀⢤⡀⠐⠒⠀⣠⠄
⠀⠀⠀⠀⠎⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠠⠀⢀⠘⠶⠃⡀⠀⡄⠀⠀⠀⠀⠀⠚⠑⠂⠱⠂⠀⠀⠐⠐⠀⠋⢈⠁⠀⠐
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠁⣈⠁⢁⠈⢠⠇⠀⠀⠀⠀⠀⢠⠀⠀⠰⠈⠀⠀⠀⡀⢂⠀⠊⠀⠂
⠀⠀⠀⠀⠀⠦⢤⠤⢚⠀⠀⠀⠀⠇⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠰⠆⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡁
⠀⠀⠀⠀⠀⠈⠂⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡀⠀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡀⠀⠀⠀⠀⠀⠀⠐⣦⠀⢀⢈⠙⠛⢉⢀⠀⢰⡞⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣇⣀⠀⠀⠀⠀⠀⠀⠈⠢⣘⡎⠀⠀⠈⣞⡠⠎⠀⠀⠀⠀⠀⠀⠀⣀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢼⡟⣀⣄⡘⣿⠅⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⣠⣴⣉⣩⣿⢸⣏⣉⣳⣤⣀⠀⠀⠀⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⢀⡴⠫⢏⣾⢟⣿⡿⠸⣿⣟⣿⣝⢏⠳⣄⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⠀⣿⢟⣿⣿⡧⢰⣿⣿⡟⢿⡆⠀⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣭⣇⢀⣏⠘⣿⣿⡇⠀⣿⣿⡟⢈⣅⢀⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⣋⣿⣮⣿⡀⢻⣿⣷⢸⣿⣿⠃⣸⣯⣾⣍⡉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⠟⠁⢠⢞⣉⡉⠙⠛⢿⣶⣍⣉⣈⣉⣵⣾⠟⠛⠉⣉⢙⢷⠈⠻⣿⣶⣤⣀⣀⡀
⠀⠀⠀⠀⠀⢠⠚⠋⠉⠉⠉⠈⠉⠁⠀⠀⢸⡸⣿⣿⡀⣀⢠⠈⣧⢳⢰⢯⡎⢀⡆⣠⣤⣿⡿⢻⠀⠀⠈⠉⠀⠀⠉⠉⠉⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢹⣿⢡⠿⣿⡋⢏⠏⢇⠋⡖⡻⣁⣗⢹⣿⠁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢨⣿⣾⣶⣎⡿⣦⡘⡬⢠⣸⡟⣡⣼⣿⢟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢘⠉⢯⡽⣟⣼⣷⣀⡗⢰⣛⣏⣿⢐⡟⠺⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⣀⣚⣘⡯⣤⢾⣉⡷⢾⣱⡖⣟⣨⣤⡄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠉⠉⠣⡤⢭⣟⢀⡯⠨⡿⢧⡇⡫⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⠼⢮⣛⢎⡏⠠⡿⠰⠷⡞⣷⠼⢄⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⣾⣧⣀⣀⣠⣷⣀⠀⣬⣿⣶⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠉⠉⠁⠀⠁⠀⠉⠉⠁⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⡿⠿⠿⠿⠿⡻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡃
⠀⠀⠀⠢⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣸⠀⣇⠄
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⡀⠀⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣒⠄⢠⡀⠄⠀⠀⢀⠙⠀⡋⡀⠀⠀⠀⢀⡀⠠⣚⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠅⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣉⠻⢀⠂⡀⠀⠀⡇⠀⠐⠀⢚⣩⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⡀⠀⠉⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠉⠠⠀⠀⠀⣠⠒⡀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢁⠀⡀⠀⠀⡀⠇
⠀⣀⠀⠀⠀⠉⠁⠠⠔⠁⠀⠀⠄⠀⠀⠈⠉⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠖⣀⠠⠄⠀⢠⠀⠁⠓⠀⠄⡐⠶⣄⡀
⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠃⡀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⡄⠱⠞⠋⠀⠁⠀⠀⠀⠅⠖⠉⠀⠹⠒⠦⣀⡆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⢾⠀⡷⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣍


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⢰⠭⠶⠀⠠⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⠂⢈⡄⠈⠁⠀⠈⠉⢠⡔⠀⠂⠀⢀⡀
⠀⠀⠀⠬⠳⣤⠀⠀⠀⠀⠠⡄⣧⠀⠀⠀⠀⢠⣀⡨⠆⠉⠀⠀⠀⠀⠀⠀⠆⠀⠶⢭⡚⠒⠃⠀⠀⠀⢼⠀⠀⠀⠈⠀⡈⠰⢗⡊⠵⠂⠠
⠀⠀⠀⠀⠀⠁⠀⢃⡶⢄⡀⠄⢃⡀⠰⣂⠐⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡭⠀⢄⡀⠀⠐⢿⠀⣰⡶⠞⠉⠀⠀⢤⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡌⠁⠈⢀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢤⠄⠀⠙⢷⠤⠌⠁⢫⣀⣄⣀⡀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠄⡰⢖⠁⠐⠄⠰⠀⢀⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢀⣠⡞⠘⣧⠀⠉⠳⠄⠀⢠
⠀⠀⠀⣀⡠⢷⠈⠁⠈⠀⢠⣦⢠⠀⠈⠉⠉⢠⣦⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠲⢆⡀⠠⠒⠋⠉⠀⠀⡟⠀⠀⠀⠀⠀⡳⢆⡀⠀⢀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠘⠒⠄⠀⠀⠀⠀⠀⠀⠀⠈⠡⠞⠉⠈⠃⢠⡆⠀⠀⠀⠇⠀⢠⡄⠜⠈⠛⠰⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠁⠀⠚⣠⢰⡖⠒⢀⠂⠀⠁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠁⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣠⣀⣀⠀⠀⠀⠀⠀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠖⠲⣤⣀⣄⣠⡴⠒⠦⠠⣤
⠀⠀⠹⣯⠿⣱⣆⣀⣠⣮⣽⣮⡟⠀⢰⡔⣠⣄⡀⠀⢀⣠⣄⢲⡆⠀⠙⠲⣄⡀⠈⢉⣿⠉⠀⣀⡴⠚⠁⠀⢶⡶⢢⣤⣤⣠⣠⣤⡄⠲⠆
⠀⠀⠀⢉⣟⣿⡽⣿⢻⣽⣫⡍⠀⠀⠀⠻⠟⣿⣿⣿⣿⣿⠻⠟⠀⠀⠀⠀⠈⢿⣶⡾⠙⣶⣶⡏⠀⠀⠀⠀⠈⠠⣻⣿⣋⢻⣋⣻⣟⡦⠉
⠀⠀⠀⠘⣿⣿⠇⠛⠰⣯⣿⠃⠀⠀⠀⠀⢢⣌⣙⠙⣋⣠⡴⠀⠀⠀⠀⠀⠀⠸⠉⠀⠀⠈⠙⠄⠀⠀⠀⠀⠀⠀⠙⠿⠍⠀⠨⠽⠛⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣀⣀⢀⣀⠀⠀⠀⣀⡀⣀⣀⡀⠀⠀⢰⣶⣶⡶⠤⢀⡀⠴⢶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠂⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣦⣀⠀⠘⠿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣽⣿⠿⠃⠀⠀⠹⢿⣧⠀⢸⡇⠀⣜⡿⠋
⠀⠀⠀⠀⠀⠀⠜⠿⠇⠀⠀⠀⠀⢠⣶⣿⣼⣿⣿⣿⣿⣿⣾⣿⣦⡀⠀⠈⢹⡟⢛⠛⣟⠛⡿⠁⠀⠀⠀⠀⠀⠀⠘⣷⣶⡎⣴⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠉⠛⠻⠿⠾⠿⠶⠿⠋⠙⠁⠀⠀⠀⠀⠈⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠉⠁⠈⠉⠣

⠀⠀⢠⠀⠤⠀⡀⡀⡤⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣾⣿⣿⡟
⠀⠀⠈⠓⠀⠀⢻⠁⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠈⢷⣨⣿⣿⣰⠏⠀⠈
⠀⠀⠀⠀⠘⠙⠀⠙⠁⠀⠀⠈⠗⢠⣄⠺⠁⠀⠀⠘⠃⠀⠋⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⠈⠿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠷⣄⠛⢢⢤⣤⣠⠴⠲⢃⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠑⠀⠀⠀⠀⠀⠐⠋⣼⠁
⠀⠀⠙⢷⡄⠸⢻⠙⠄⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡠⠀⠀⠀⠀⠧⠀⡀⠀⢐⠀⠀⢀⠀⠝
⠀⠀⠀⠈⢧⣄⠀⢀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⢀⠸⠇⠀⠄⠀⠀⠀⠀⠀⢤⠀⠀⠈⢈⠀⠁⠀⠀⡆
⠀⠀⠀⠀⠂⠈⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⠀⠀⠀⢀⣤⠜
⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤
⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⠙⢷⣟⠛⠉⠂⠀⢀⠀⠐⠋⠙⢻⣷⠟
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢷⣀⣠⣤⠴⢿⠶⣤⣤⣀⣸⠃⠀⠑⢄⠀⠀⠀⢰⡄⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⢸⠀⠀⠈⢷⡄⠀⠀⠀⠀⠃⡀⠀⢸⠇⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣀⠘⣄⣀⣠⡜⠀⠀⠀⠀⠀⠀⠐⠤⠔⠦⠴⠃
⠀⢶⣶⠒⠤⠀⠀⠀⠀⠠⠔⢲⣶⠂⠀⠀⠀⠉⠁⠀⠀⠉⠁⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⢻⣆⠀⠀⠀⡄⠀⠀⢀⣾⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣏⠫⣖⣤⡀⠀⠀⣠⣴⡾⢫⣿
⠀⠀⠈⢩⠀⠀⠀⠇⠀⠀⢈⠁⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠈⠿⣦⡈⠹⢞⣾⡞⠾⢁⣠⡿⠇
⠀⠀⠀⠸⣆⣠⠀⠀⣄⣀⡾⠀⠀⠀⠀⠳⣤⠀⠀⠠⠀⠀⢠⡤⠇⠀⠀⠀⢰⣿⠀⢸⠸⢸⠀⣸⡇
⠀⠀⠀⢀⠏⠉⠀⠀⠈⠉⢇⠀⠀⠀⠀⠸⡁⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠻⣷⡎⠀⠀⣶⡿⠃
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠁
⠀⠀⢀⠁⠀⢀⠀⠀⠀⡀⠀⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⢀⣴⣶⡖⣴⣶⣤⣦⣶⢶⡶⣦⣦⡀⠀⠀⣿⠒⠀⠂⠂⠀⢠
⠀⠀⠀⠀⠂⠀⠀⣁⠀⠀⠀⠀⠀⠀⠙⠻⣷⣶⣌⢹⡟⣡⣶⣼⠟⠋⠁⠀⠀⢧⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠁⢀⡅⡀⠀⠁⠀⠀⠀⠀⠀⠈⢯⣟⡿⢻⣛⡽⠃⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣴⢂⣼⡄⢀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡿⣿⣿⣿⠂⠉⣽⡦⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢰⣄⠀⠻⡿⣷⠞⠻⠋⠁⡥⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠷⣬⣉⣽⣦⣤⣽⣿⣿⡿⢳⣤⣴⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠤⢤⣶⣿⡛⣿⣤⡽⣿⣿⣷⣿⡿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠇⢸⣿⣿⣿⣾⠟⠃⠀⠙⢻⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣭⣷⠀⠠⠾⠀⠸⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣧⡿⠆⠀⠀⣀⡀⢸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣿⣷⠃⠀⠀⠀⣠⣿⡧
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⣿⣋⢙⢻⢿⣉⠉
⠀⠀⠀⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣶⣶⡔⣉⣹⠋⠉⠉⢹⠈⠛⢛⠀⣤⠶⣶⡀
⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠸⢂⠐⡻⠖⡏⠀⠀⠀⢸⡗⠦⠈⠉⢙⠧⣯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠹⣾⢠⣶⠃⠀⠀⠀⢸⡗⠀⢛⡼⢋⣡⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⣼⣷⠘⠈⢻⠁⠀⠀⠀⠀⠗⠀⠈⣴⣟⠁⠀⢸⡀
⠀⠀⠀⠃⢀⡀⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡌⢃⠐⡟⠀⠀⠀⠀⠀⣧⢃⣴⣿⡟⠀⠀⠈⡇⠀⠀⢀⣦
⠀⠀⠀⢀⠞⡉⠀⠢⠀⠀⠀⢀⣿⣿⣿⣿⡇⠈⡎⠇⠀⠀⠀⠀⠀⠿⣼⠋⡉⢡⡀⠀⠀⠀⢠⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣰⣿⣿⣿⣿⣿⡇⠀⣻⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢼⣿⡆⠀⠀⢸⣿⣿⣧
⠀⠀⠠⠀⠇⢀⡁⢤⣀⣾⣿⣭⣉⣿⣿⣿⡇⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠚⠀⠈⠳⣬⣿⣿⣿⣿⣿⣿⣿⣷⣶⣴⣄⣀⣀⢀⣀⣀⣄⣠⣶⣾⣿⣿⡇⠀⠀⢸⣿⣿⣿⡀
⠀⠀⠰⠆⠀⣤⣀⣸⣿⣿⣿⢿⣿⣿⣿⣿⡟⠻⢿⣽⣿⡿⢻⣿⠿⠟⡻⢿⠫⠙⢿⠁⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠿⣿⣿⣿⣷⣾⣧⣴⣿⣧⣿⡿⠀⠀⠀⠀⠈⠁⠈⠀⠨⠁⠀⠀⠀⠀⢈⣀⠴⡴⣿⣿⣿⣿⡇
⠀⠀⠀⣄⡉⠹⣻⣿⣿⣧⣼⣿⣿⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⣠⣘⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢲⠿⣿⣿⣿⣿⣿⣧⣾⣧⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢤⣌⡛⠛⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠁⠸⣻⣮⠿⣿⣿⣧⣹⡟⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠚⠈⠁⠀⠘⣿⣿⣿⣿
⠀⠀⠀⠀⠰⢶⣥⣿⣄⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠱⠆⠀⠀⠀⢘⣿⣿⡇
⠀⠀⠀⠀⠀⢚⡿⢾⣿⣿⡟⢿⣇⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣠⡄⢣⠙⣿⡃
⠀⠀⠀⠀⠀⠈⠶⣿⣿⣿⣵⡿⠉⠁⣴⠄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⢺⣯⠸⠋⠁⠀⣸⡇
⠀⠀⠀⠀⠀⢀⣶⡴⢻⣿⣿⣷⡀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠉⠿⠂⠉⠀⢀⣀⣼⡿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⠿⣿⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣙⡆⠙⠀⢡⡘⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠀⢷⣁⠌⣾⢿⣿⣿⣿⣿⣿⢽⡾⡏⢻⣿⣿⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠳⠈⠻⣿⣿⣿⣿⣿⠈⠁⠃⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⣻⡿⠃⠀⠀⠀⠈⢿⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⡀⠀⠀⠀⠀⠀⢀⣀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡄
⠀⠀⠀⠀⠀⢀⢸⣷⢀⠀⠀⠀⠙⣧⠰⣟⠳⡄⠀⢀⠴⢻⡇⣰⠟⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠸⣧⠇
⠀⠀⠀⠀⠀⢀⣿⣿⣇⠀⠀⠀⠀⠘⢧⣊⣭⢾⣶⡾⢮⣕⣰⠋⠀⠀⠀⠀⠀⡯⠀⠀⠀⠀⠀⠀⠀⣦⣼⣴⠆
⠀⠀⠀⠀⠀⠁⣿⣿⡇⠁⠀⠀⠀⠀⠀⡄⠀⠀⣿⠀⠀⠀⡄⠀⠀⠀⠀⠀⠰⣷⠀⠀⠀⠀⠀⠀⠀⢺⣿⣟
⠀⠀⠀⠀⠀⠀⡟⠹⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟
⠀⠀⠀⠀⠀⠀⠋⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢴⣶⣶⣤⣤⣤⣤⢷⣤⣤⣤⣤⣶⣶⠆
⠀⠀⠺⣽⣷⢦⣄⣀⣤⣶⣿⡽⠂⠀⠀⠀⠰⣿⣿⠀⠀⠀⠀⠈⠉⣛⣿⣿⡿⣾⣿⣿⣟⣋⠉
⠀⠀⠀⠙⣻⣿⣿⣿⣿⣿⡘⠁⠀⠀⠀⠐⢸⣷⣿⠁⠀⠀⠀⠐⠟⠛⠛⠉⠀⡇⠈⠙⠛⠛⠷⠀⠀⢠⣴⣤
⠀⠀⠀⠀⠙⠿⠋⠈⠻⠛⠁⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣯⣄⡀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⣄⣤⣄⣀⠀⣀⠀⢀⣠⣤⣄⡄
⠀⠀⠀⠀⠀⠀⡸⡿⡀⠀⠀⠀⠀⠀⣠⣾⡿⠡⠭⠻⣿⣦⡀⠀⠀⠀⠀⠀⢺⣿⠀⠀⠀⠈⠛⢻⣿⣿⣧⣿⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠈⠉⠁⠀⠰⠇⠀⠀⠉⠁⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠈⠉⠉⣿⠈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠘⣿⣿⡟⠢⠀⠀⡰⠛⣿⣿⠏⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢠⣼⣿⣤⠀⠀⠀⠀⠹⢟⣇⣀⣠⣦⣀⣠⣟⠟⠀⠀⠀⠀⢀⣮⠍⣦⠀⠀⠀⠀⠀⠀⢀⣶⣆
⠀⠀⠀⠀⢀⣾⡿⠿⣿⡆⠀⠀⠀⢐⣿⠋⠀⠘⠇⠀⠙⢻⡆⠀⠀⠀⠀⢸⣿⣾⣿⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢸⡿⢙⣚⠹⣿⠀⠀⠀⠀⢻⣷⡴⠀⠀⠢⣶⣿⠁⠀⠀⠀⠀⠀⢻⣿⠏⠂⠀⠀⠀⠸⠛⠉⠿⠋⠙⠿
⠀⠀⠀⠀⠈⠁⠈⠉⠀⠉⠀⠀⠀⠰⠃⠈⠀⠀⠀⠀⠉⠈⠦

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠄⠂⠐⠦⠀⢠⡆⠀⠀⠀⠀⠀⢀⡜⢈⢉⢣⠀⠀⡈⠀⢰⣦⠀⠀⠀⠂⠉⢾⣧⡀
⠀⢀⠀⠠⣤⠀⠲⣤⠷⠶⠉⠀⠀⠀⠀⠀⠘⠃⢠⣤⣽⣆⠀⣏⡈⢀⣿⣄⠀⠐⠶⠀⠙⢿⣿⡀
⠀⠈⠀⢰⠳⢦⡜⠙⣦⠀⠀⢠⡇⠀⠀⠀⠀⠀⠠⠋⠀⠘⠂⢻⠃⠪⠈⠌⠦⠀⠘⠀⠀⡀⠀⠻⢧
⠀⠀⠀⣤⣄⣤⠗⠴⠛⢀⠀⢸⠁⠀⠀⠀⠀⣆⡰⠀⠀⠀⠀⡆⠢⠀⠀⠈⠺⢦⡀⠐⡿⠀⢰⡗⠈⣷⡀
⠀⠀⠰⢤⠈⣁⣘⠒⠴⡁⢠⠞⠀⠀⠀⠀⠀⢣⡆⠀⠀⠀⠀⠘⢶⣄⠀⠃⠀⢤⠅⡎⠁⣀⣼⠇⠀⢹⣿⡄
⠀⠀⢤⣀⡉⠀⢀⡄⣰⡗⠋⠀⠀⠀⠀⣠⠏⠉⠑⣚⠿⣿⡛⠛⡛⢌⠉⠛⠀⣿⣶⠿⠛⠉⠀⠀⠀⠘⣸⡟
⠀⠀⠀⠀⠇⠀⢾⠁⢙⣇⠀⠀⠀⠠⠤⠀⠀⠀⠀⢤⢀⠏⠀⠀⠀⠀⠀⣠⠞⠀⢸⡆⠀⠀⢀⡤⢶⢿⣿⡇
⠀⠀⠠⠀⠀⠀⢸⠀⠿⣿⡄⠀⠀⠐⠀⠀⠀⠀⠀⢠⠀⡄⠀⠀⠀⠀⠀⠘⢦⠀⠀⢳⠀⢹⠋⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡄⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣼⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⣀⠀⠀⠱⢸⣿⡇⠀⠐⣙⠉⠁⠈⢉⡿⠋⢩⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠞⠁⠀⠙⠃⢠⣼⠇⠀⠘⠈⠀⠀⢀⡞⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⢤⣤⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⢀⠄⢟⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠁⢹⡇
⠀⠘⢠⣄⡀⠀⠀⠐⠿⣻⠀⠀⠀⠀⠈⡹⠚⠀⠈⠄⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⣿⡄⠀⠀⠈⡇
⠀⠀⠑⠒⠻⡳⡒⢾⣉⡀⠀⠀⠀⠑⠊⠁⠀⠀⠀⠀⠀⠀⠀⠤⠤⢤⣤⣠⣤⣤⣴⠿⠛⠁⠀⢀⡀⣇
⠀⠀⠀⠀⠀⠀⠅⠚⠛⢢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠒⠒⠐⢒⣃⣙⡆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠿⠛⠛⠛⠱⣼⠀⠀⠀⠀⠀⠈⠱⣦⡀⠀⣠⣤⣤⣄⠀⢀⣤⠏⠁⠀⠀⠀⠀⢠⣇⡎⠛⠛⠛⠿⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠟⣻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣸⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⡴⠫⠈⣩⡿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠱⣄⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣤⣿⣥⣤⣾⠿⠅⠹⠇⠻⠏⣋⠁⡿⣘⣻⣿⣦⠶⣿⣿⣤⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢯⣶⣾⣧⣦⣿⣴⣷⣀⣴⣿⣶⣿⣿⣿⣳⣿⣷⣬⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡖⣉⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠪⠉⡉⠉⠉⠉⠉⠉⠛⣛⣿⣿⣟⡛⠛⠉⡉⢩⠁⣤⠀⠀⠢⢦⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⡄⠀⠀⠀⢂⠐⣰⣾⣿⣿⣿⣿⣿⣶⣄⠡⡜⠇⠀⠀⠀⠀⠀⠛⡜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠊⠀⠄⠐⠀⠀⠀⠀⠁⠘⣿⣿⣿⣿⣿⣿⣿⠉⠇⠁⠀⠀⠀⠀⡀⠀⠀⡀⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣶⠁⠀⠈⠔⠀⠀⠂⠀⠀⠀⠈⠻⢿⡏⠛⡍⠥⠡⠈⠀⠀⢆⠀⠄⠠⠀⠀⡀⠄⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⣯⠀⠀⠈⠎⡀⠀⠀⠀⠀⠀⣀⡙⠾⠁⠤⢱⡚⡆⢣⠀⠀⠈⡄⠠⠀⠂⠀⠀⣠⡎⣿⡆
⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⣿⣿⢸⣿⡇⠂⠈⠰⢀⠀⠀⠀⡠⠳⣲⢳⣻⠠⠘⡀⢣⢭⠀⠀⠀⠀⠱⠀⠀⠠⠀⢸⣿⡧⢹⣿⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠋⠉⠉⠙⠿⣦⣸⣿⣧⠜⠟⠁⡢⠈⠃⣌⠠⠀⠠⣻⢧⣟⢾⣿⠀⠠⣹⡌⡂⠂⠈⠀⢀⡀⠠⠆⠀⠄⠄⠛⠃⢸⣿⣇⣶⡿⠋⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⡐⠁⠋⣠⣽⣿⣿⣿⣧⡺⡿⣿⣾⠀⠱⣼⣧⣀⣴⣶⣿⣿⣿⣷⣦⣇⠘⡌⠀⠁⢸⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢁⠂⣾⣿⣿⣿⣿⣿⣿⣿⣿⢷⡛⢸⢀⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⠰⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠌⡰⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣦⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⠣⡈⡆⠁⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠁⠍⡙⠛⠿⠟⠻⣿⠟⠠⠷⣿⠸⣌⣟⢼⠋⡟⠛⠋⣩⢩⠉⡔⣆⢩⠐⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠈⠀⠀⠂⡖⡲⠆⠀⠴⣗⡳⢽⢪⠐⢩⣾⡿⡇⢩⠀⢣⠤⠄⢤⠤⡸⡀⠆⠰⢰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⡄⣄⠀⢀⠁⠄⠂⢠⡄⢨⢐⢨⠰⠀⢺⡇⠀⢁⠀⡀⡰⢦⢱⠐⡀⢀⣱⣘⢂⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠻⣿⣧⣌⠀⡃⠍⢉⡴⢸⠚⢰⢸⢩⣿⢁⠀⠈⣀⠳⢨⣏⡐⣦⣿⣿⣿⣯⢺⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠙⣿⣿⣷⡀⠂⣡⠜⢹⢾⠨⣽⠓⡏⢸⠀⡇⢯⠸⠄⡄⢡⣿⣿⣿⡿⢃⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⠈⠻⣋⠡⡄⡁⠈⢹⢘⠀⡇⡀⡇⡘⠄⠹⢨⠙⠆⣡⢨⠝⣣⠞⠁⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⠀⠀⠈⠳⢍⡁⠠⣿⢸⡀⣧⡅⠂⡋⣆⢸⠄⠲⣷⢪⡠⠞⠁⠀⠸⣿⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠋⠁⠀⠀⠀⠀⠀⠉⠒⠿⣼⣃⡘⡇⢁⡁⠥⠘⣁⠤⠚⠉⠀⠀⠀⠀⠀⠈⠛⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣶⣶⣾⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡿⣻⣿⣿⡿⣟⣿⣿⣿⢋⣉⣻⠟⠉⠛⢛⡟⠉⠙⢛⠛⠻⣟⠛⠿⢿⣟⠛⠻⣷⣄⡀
⠀⠀⠀⠀⠀⢾⠿⡿⠿⣿⠿⠿⢿⣿⣷⡿⠿⣿⣷⣶⣷⣿⣶⣶⣶⣿⣽⣶⣶⣬⣦⣤⣤⣵⣤⣤⣽⣦⣤⣤⣬⣿⣦
⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠑⣄⢀⠀⠹⣄⠀⠀⠹⡅⠈⠈⠀⠀⡇⢸⢹⠀⠠⢀⠋⠀⢀⡼⣋⠵⣊⠝⠋⠁⠀⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠠⡉⠻⠿⣆⠀⠀⢱⠄⠀⠂⠀⡇⠀⡈⠀⠀⡰⠢⢐⣟⡪⠕⠊⠁⢀⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⢄⠀⠈⠛⢦⡌⠂⠀⡃⠀⣧⠀⡆⣠⣾⢱⢾⠟⠉⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⣿⢤⣿⣾⡿⠟⠋⠀⠀⠀⠀⠀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠀⠀⠀⢨⡝⡿⣍⠀⠀⢀⠂⠀⣠⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠠⡀⡾⢲⠷⣿⡀⠠⠁⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣹⣿⣾⣳⣎⣿⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⡩⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣖⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠒⠢⢤⡀⠀⠀⠀⢀⣠⠤⢄⡀
⠀⠀⠀⠀⠀⣠⠒⢛⣛⡒⠢⢄⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠹⡄⠀⡴⠋⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⢄⡀
⠀⠀⠀⠀⠀⣇⠀⡏⠀⠉⠑⠦⣈⠳⢄⠀⠀⠀⠀⠣⡀⢀⡆⠀⠀⢱⣸⠁⠀⢰⠀⠀⢠⠇⠀⠀⠀⡠⠖⣋⠥⠒⠒⢢⠀⢘⡄
⠀⠀⠀⠀⠀⠘⢦⡙⢄⡀⠀⠀⠈⠑⠦⡙⢦⡀⠀⠀⠈⠉⠀⠀⢀⣸⣇⠀⠀⠈⠓⠒⠋⠀⣀⠔⣫⠔⠋⠀⠀⠀⢀⠞⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠙⠢⣍⠒⠤⣀⠀⠀⠈⠲⣝⠦⡀⠀⠀⠀⡴⠋⠆⠰⠑⡄⠀⠀⠀⣠⢞⡥⠊⠁⠀⠀⣀⡤⠒⣁⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢬⡙⠢⢄⡀⠈⠑⣍⢦⡀⠀⣇⠀⠑⠃⢀⡇⠀⡠⢊⠔⠉⠀⢀⡠⠔⢋⡡⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣌⡓⠢⢌⣷⠙⢦⡈⡷⠠⢶⢋⡤⠊⡔⣁⣠⠴⢊⣡⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⣤⠀⠀⢹⠓⠒⢾⠋⠀⢀⣩⠥⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⣋⡭⠤⡄⢀⣧⣤⣀⣸⡀⠀⣀⣉⡙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠊⣁⠴⠚⠉⢀⡴⠊⢀⣼⠀⠀⠀⠈⡟⢆⠑⢄⡈⠑⠲⢤⡉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⡠⠔⠋⠀⠀⣠⠔⠉⢀⠔⠁⡏⠉⠉⠉⠙⡇⠀⠱⣄⠙⠦⡀⠀⠈⠑⠦⣈⠙⠢⣄
⠀⠀⠀⠀⠀⣠⠞⠁⣀⠔⠋⠀⠀⢀⡤⠊⠁⡠⠞⠁⠀⢸⠷⠤⠤⠤⢤⡇⠀⠀⠈⠑⢄⠈⠓⢄⡀⠀⠀⠑⠦⡀⠙⢦⡀
⠀⠀⠀⠀⡴⠁⢀⠞⠁⠀⣀⠤⠚⢁⡠⠔⠋⠀⠀⠀⠀⢸⣄⣀⣀⣀⣰⠁⠀⠀⠀⠀⠀⠙⠦⣀⠉⠲⢄⡀⠀⠈⠢⡀⠙⢦
⠀⠀⠀⢰⡁⠀⠘⠒⠒⣉⣡⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⡈⠓⠢⠤⠜⠀⠈⡆
⠀⠀⠀⠀⠑⠒⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠉⠛⠛⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣄⣀⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠦⠤⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡗⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡢⢤⡤⣒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⢀⠀⠀⡠⣀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠀⠀⠨⣘⡯⠀⠀⠀⢠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀⠀⢇⡀⠁⣀⠔⠛⠭⠀⠀⠀⠀⠀⠀⠀⡲⢤⣄⡄⠀⡀⠇⢠⠀⣸⣴⠾⡓⢄⡀
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠈⠓⣄⣾⠕⣫⠀⡤⡀⠀⠀⠀⠀⠀⠀⠀⠊⠐⠊⠔⠋⢑⣚⣿⣷⣊⣀⠀⠁⠈⠁⠈
⠀⠀⠀⠀⠀⠐⢻⠟⠉⠉⠩⣭⢛⠟⣍⠓⠬⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠔⢹⠙⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠁⡜⠀⠘⠁⠀⠀⠉⠲⣶⡄⠀⠀⠀⠀⠀⠀⠤⣐⣒⠾⠻⠀⠈⠀⢋⢷⣒⡒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠛⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠄⠀⠀⠀⠀⠈⡌⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⣙⣇⠄⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡯⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡀⠀⠀⡀⠡⠀⣤⠀⣂⠗⠠⠀⠀⠀⠀⠀⠀⢴⡰⡀⠠⡀⡀⠀⣇⢘⠀⠀⢠⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠫⠽⠓⠤⠬⣙⣬⣠⡧⠊⣩⡅⠀⠠⠤⠀⠀⠀⠀⠩⠭⠛⠢⢔⡳⣄⡿⢁⠤⠒⠛⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠓⣂⢟⠿⡦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⡻⡿⣓⠒⠣⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠊⠠⠎⠰⠈⠖⠈⢹⠳⡶⠀⠀⠀⠀⠀⠀⠀⠉⠀⢁⡔⢡⢇⠈⠑⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠙⠀⠀⡀⠀⠀⣃⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠿⠀⠀⠄⠀⠀⠸⢱⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠐⡹⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢻⡇⠀⠀⠀⡾⣿⠿⠿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠃⠀⡻⠀⠀⠀⠀⡅⠀⠀⠈⢿⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠄⠀⠀⠀⣸⠋⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⣠⠴⠋⠁⠻⡆⠀⢣⢱⠀⠀⠀⠴⠛⠉⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⣀
⠀⠀⠀⠀⠀⢀⠔⠊⢉⠟⢦⡄⠀⠀⣰⠏⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⢀⣶⣄⢀⣤⡈⠙⠛⠋⢰⡀⠀⠀⡰⠒⠋⠉⠉
⠀⠀⠀⠀⣴⡿⠄⠀⡼⠂⠀⠀⡀⠉⠁⠀⠀⠀⠀⠉⠙⠓⠀⠠⢠⣄⣼⠃⠀⠀⣾⡟⢹⡿⠟⣷⡀⠀⠀⠀⠙⠲⠚⠁
⠀⠀⠀⠸⠿⠶⠴⠄⢣⣀⣀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⢱⡟⠱⠀⠘⣻⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⣠⣄⠀⠟⠛⠳⢦⡀⠀⠀⠀⠀⢹⣸⠀⠀⣠⠞⠁⠛⢷⡄⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣾⠃⠀⠀⡀⠀⠀⠀⠙⠲⢤⡀⠀⠀⢿⣤⡎⠉⡀⠀⠀⣰⡇⠀⢠⡿⠋⠻⠋⠣
⠀⠀⣀⣤⠤⠀⢀⠀⠀⢀⠔⠁⠈⣿⠃⠀⠸⡇⠀⠀⠀⢻⡄⠀⠀⠀⠀⠈⠻⠂⠀⠀⠀⠁⠀⠀⠀⠒⠋⠀⢀⡾⠁
⠀⠀⠙⣿⡀⠀⠈⠑⢤⡏⠀⠀⠠⡿⠀⠀⣼⠁⠀⠀⠀⡀⠋⠀⠀⠀⠀⠀⢰⡶⢤⡀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠇⠀⠀⠀⠀⡀
⠀⠀⠀⠘⢷⡤⠄⠀⢸⡇⠀⠈⠉⠳⡄⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠩⠞⣣⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⢀⣏
⠀⠀⠀⠀⢞⠀⠀⠀⠀⠁⢀⠀⠀⣼⠃⠀⠈⠀⢀⡀⠀⢀⣀⠀⠀⠀⠀⠀⣼⡇⠀⢼⣿⣧⡀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠰⠋⠁
⠀⠀⠀⠀⠈⠷⣤⣠⠇⠀⠈⠓⠚⠃⡀⠀⠀⠀⡿⠁⢰⠟⠁⠀⠀⠀⠀⢸⣿⠀⣠⠀⣿⣿⣷⣦⣄⡀⠀⠙⠓
⠀⠀⠀⠀⠀⠠⠅⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⡞⠁⠀⠀⣤⠀⡀⠀⠀⠀⠀⠛⠇⠀⠘⣩⠉⠙⠛⢡⡞⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⣄⠀⠀⣦⣥⣀⣤⡦⣦⣼⣽⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠹⣄⠀⠈⠉⠋⠉⠀⠀⠀⠀⠀⠀⢀⣤⣠⣶⡄
⠀⠀⠀⠀⠀⠀⣰⣛⣿⢶⣿⡄⠀⠀⠛⠦⠌⠀⠀⢠⠶⠟⠂⠀⠀⠀⠀⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠳⣿⢻⡇
⠀⠀⠀⠀⠀⢰⣯⣽⣙⡓⢺⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⢀⣀⡾⠁
⠀⠀⠀⠀⠀⢸⣴⣥⣌⠃⠀⢿⣿⣤⠀⠀⠀⠀⠀⠀⢴⣿⡃⠀⠀⠀⠀⠙⠆⠀⠀⠰⠟⠓⠂⠀⢀⣴⣿⠽⠃⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠈⣿⣿⣿⠇⠀⠀⠙⣿⡅⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢻⣿⠃⠀⠀⠀⢀⣠⡧
⠀⠀⠀⠀⠀⠀⠾⠿⢿⠂⠰⠦⠀⣿⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⡿⠀⠀⠀⠀⠀⠠⠤⠾⢷⡆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⡀⠀⠀⠀⠀⢀⣀⠀⢰⣿⡆⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⢶⣢⣶⣄⣀⡀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣖⣶⠄⡀
⠀⠠⢀⡐⣒⠚⠍⠁⠀⠁⢳⣿⡞⠀⠀⠉⠹⠓⣀⢀⡀⠄⠀⠈⢧⣕⣿⣯⣃⣹⢁⣸⠗⠠⢶⠂⠚⣏⡀⣏⣘⣽⣿⣚⣵
⠀⠀⣀⣀⠈⠁⠐⠄⠀⠂⢾⢛⡇⠐⠀⠠⠂⠉⠁⣀⡀⠀⠀⠀⠀⠀⣿⣿⣖⡠⣶⣪⣿⣷⣈⣼⣿⣯⣲⢄⣰⣿⣿⠃
⠘⠓⠒⠂⠀⠀⠀⣀⣁⡤⠜⣍⠢⢤⣈⣀⠀⠀⠀⠐⠲⠚⠀⠀⠀⠀⠉⠻⢿⢷⠏⢡⡟⣯⣟⣯⡟⡄⠙⣶⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⡄⠑⠊⢀⡟⡄⠙⠒⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠘⠠⢶⢄⠃⠀⠀⠈⠻⡶⣶⠞⠁⠻⣶⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠀⠤⠄⢀⣑⣾⣭⣿⣊⡀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢹⢿⣤⡿⠏⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣷⠇⢨⣿⡇⠘⣾⢃⠀⠀⠀⠀⠀⠸⣄⡀
⠀⠀⠀⠀⠀⢠⠀⡏⠀⠀⠀⢸⠀⣆⠀⠀⠀⠀⠀⠀⠁⠃⠃⢸⣿⣇⠈⠈⠈⠀⢸⠀⠀⢠⣄⠀⡇⠀⣰⠏⠙⠒⠂⢄
⠀⠀⠀⢀⣠⡤⠑⢮⡀⣶⣀⠥⠊⠠⣄⡀⠀⠀⠀⠀⠀⢀⡜⢻⣿⡟⢢⡀⠀⠀⠸⡗⠀⣼⣿⣿⣷⣶⣇⣠⠴⠓⠒⢤⡀
⠀⠀⠀⠀⠤⡤⠀⣀⣻⠭⣿⣁⠀⠤⠤⠀⠀⠀⠀⠈⠓⠋⠀⠀⠉⠀⠀⠙⠚⠁⠀⣈⡛⢻⣿⣟⢟⣿⣽⡷⠚⢧⡀⠀⠈
⠀⠀⠀⠀⢠⠘⠋⣩⣤⠛⣡⣍⠙⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⠈⢉⣻⡿⣿⠿⠇⠀⠀⢱
⠀⠀⠀⠀⠘⠰⡄⠙⢿⣶⠿⠋⠀⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⣰⡏⠘⠁⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣿⣿⣿⣿⣛⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣿⣻⣿⣛

⠀⣶⣦⡀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠀⠀⣶⡶⠄⠀⠀⠀⠀⣀⡴⣶⡆⠀⠀⠀⣠⡶⣶⡀
⠈⣿⣭⣷⣄⠀⠀⠀⠘⣈⠑⠶⠀⠼⡤⠠⢟⣀⠃⠀⠀⠸⣇⠀⠐⣶⣶⡞⠋⣁⡿⠁⠀⠀⠰⣅⢰⡹⣧
⠀⣽⠿⣿⣿⣆⠀⠀⠀⠻⠶⠂⠀⠀⠀⠐⠺⠟⠁⠀⠀⠀⢾⣿⣆⣿⡏⠹⠜⣫⠇⠀⠀⠸⣷⡈⣷⢻⣧⡀
⠀⠹⣄⠈⠙⣿⣦⠲⡷⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠁⠀⠈⠁⠀⢸⢿⣧⣼⣿⣿⡇⣻⣷
⠀⠀⠈⠁⡼⣽⣧⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⡄⠐⣶⠦⠤⠄⠀⣀⠀⠲⡄⠙⣼⣿⣶⣿⣿⣟⠟⢠⠿⠃
⠀⠀⠀⣾⣾⠟⠋⣁⠬⠻⣿⣿⡿⢲⣦⡀⠀⠀⠀⠀⠏⠀⠀⠀⠸⣦⣄⢢⣯⣟⠻⣿⡿⠿⠈⠛⠂
⠀⠀⠀⡉⠀⠀⠄⡍⠓⠲⠾⠛⠛⠛⠛⠋⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠉⠉⠉⠻⣷⣶⣶⠾⠃
⠀⠰⢩⠀⠀⠛⡰⠁⠀⢀⣀⠀⢿⡇⠒⢤⡀⠀⠉⠉⠍⠉⠉⠀⢀⡤⣒⣼⡷⢀⣤⡀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠘⠷⠿⠂⠀⠀⡰⣿⣿⡆⠈⢿⣷⢊⡻⠦⡀⢀⠀⠔⢀⠴⢷⠛⢲⣿⠁⢸⣿⣿⡄⠀⠀⠀⢀⡰⠉
⠀⠀⠀⡀⠀⠀⠀⣸⣿⣧⣿⣧⡀⠐⢻⣧⣅⡈⠝⡆⢸⣶⣌⡂⢌⣼⣿⠇⠀⣸⡿⢹⣿⠀⠀⠀⠈
⠀⠀⠀⠁⠀⠀⢸⣿⣿⣿⣿⡿⠁⠀⢴⣖⠠⠀⢀⡟⢻⡿⠁⢀⡀⢠⡦⠀⠀⣻⠗⣿⣿⡇⠀⠀⡤⢀
⠀⠰⣶⣄⡀⠀⣸⡛⠏⠉⠁⡐⡄⠀⠈⢻⣆⡿⡁⠈⠛⠉⠄⢯⡱⡟⠁⠀⠰⣿⠂⣿⣿⣿⠀⠀⠀⠈⠁
⠀⠀⠘⢿⣿⣆⠸⣟⣾⠀⠀⠁⡗⠰⠀⢠⣿⠙⠀⠀⠀⠀⠈⠻⣿⡀⠀⠠⢽⣿⣸⣿⠟⠁⠀⠀⠳⣿⣆
⠀⠀⣀⣤⠀⠀⠀⠈⠙⠿⠶⠤⠅⢀⠠⠊⠁⠀⠀⢀⣤⡀⠀⠀⠉⠱⣄⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢻⣿⣿⣤⡀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠺⢿⣿⣦⣄⡀⠀⠀⢠⣾⣶⣆⣀⠀⠀⠀⠀⣤⣛⣿⣿
⠀⠀⣼⣿⣿⣿⣿⡄⣀⣠⣶⣷⣿⣿⠿⠂⠀⠀⠀⠒⠤⢤⣤⡈⠹⣷⠀⠀⣿⢙⣿⣿⣷⣄⣤⣾⣿⣿⣣⣿⠂
⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠁⠋⠀⠀⠙⣻⢪⣿⣿⣿⣿⢿⣿⣿⢿⣧
⠀⠀⢿⣿⣿⣿⡟⣿⢻⣿⣿⣿⣿⠉⠀⣀⡀⠀⠀⠑⠐⠂⠒⠉⠀⠀⠀⠀⠀⢿⣿⣿⣿⡏⠛⠹⢿⣷⣿⡿
⠀⠀⠀⠙⠛⠉⠀⠁⠘⢿⣿⡿⠏⠀⠀⡿⢣⣶⣄⡀⠀⠀⠀⠀⣀⢠⣤⠀⠀⠉⠛⠿⠏⠀⠀⠀⠀⠛⠋
⠀⠀⠀⠀⡀⠀⠀⠠⣀⠈⠁⠀⣀⡀⠀⢨⣷⢯⣯⣵⣶⣦⣘⣞⣿⣋⡟⠀⢰⣶⣀⡀⠐⠆⠀⠐⠂
⠀⠀⠀⠀⢘⣦⣴⠿⠛⠋⠀⣠⣿⣿⡄⠀⠛⠿⠿⠙⣾⠏⣿⣿⣿⡗⢀⣠⣿⣿⣿⠿
⠀⠀⢠⡶⠂⠉⠀⠀⠀⠀⠀⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⣾⣿⠿⡍⢾⡏⠀⠶⠀⠀⠀⡔⣷⣷
⠀⠨⠊⠀⢀⣠⡶⣤⠀⠀⠀⢘⡿⣿⣿⣿⣄⣀⠀⠀⠀⠀⢀⣴⣶⡾⢻⣵⣾⣷⣿⠁⢀⠖⠄⠀⠀⠄⠹⣿⡀
⠀⠀⢰⣦⣻⣿⠟⠉⠀⠀⠀⢸⣷⣄⣿⣿⣿⣿⡇⠀⠀⢰⣿⡿⢡⡌⣻⢿⣿⣸⡇⠀⠉⠀⠀⠀⢠⣵⣤⣿
⠀⠀⠘⠿⠏⠁⠀⠀⠀⠀⣰⣿⣿⣿⠿⢿⣿⣿⠇⠀⠀⢸⣯⣴⣿⠿⠟⠙⢟⣋⣤⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⡀⢤⣶⡶⣦⣟⣿⣿⣿⡗⠏⢧⡀⠀⠁⠀⠀⡠⢤⣤⣀⣀⡀⠀⠛⠉⡿⣿⣭⣛⡶⠦⣤⣄⡀⠈⠁
⠀⠀⠐⢽⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⣠⠔⣓⣦⣴⠞⠛⠋⠉⣠⣴⣿⣏⠙⡿⠿⠟⣛⣳⣬⣥⣾⡗
⠀⠀⠀⠀⠈⠉⠉⠉⠙⢿⣯⣿⣿⠟⢁⣼⣿⡿⠋⠉⠉⠀⠀⠀⠀⢯⣥⣴⣮⣴⣶⠟⠻⠿⠿⠛⠛⠻⠧⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⡄⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡙⢴⡧⢊⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡀⣸⣇⢀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣤⢸⡇⣤⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠙⡇⣬⡀⠆⠸⠀⣷⠀⠀⠓⢦⢣⣄⠰⣿⣺⣷⣿⠆⣠⡜⡴⠞⠀⠀⣾⠀⠗⠰⢀⣭⠸⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠝⠿⣶⣦⡁⣤⣤⣄⢸⡯⣟⣷⡖⣡⣌⢲⣾⣻⢽⡇⢀⣤⣄⢈⣤⣴⠿⠫⠅⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢁⡤⠉⣛⣿⣾⠯⠼⠻⡟⠉⡅⢿⣿⢸⠉⢻⠿⠧⢽⢷⣷⣛⠉⢤⡀⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠙⠾⠛⡆⢿⣷⣮⣓⣼⡼⢧⣷⣒⣵⣾⡷⢠⠻⠷⠋⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢦⠴⠾⣟⣿⠂⠪⢭⣿⠉⢄⡠⠉⣿⡭⠕⠐⣿⣻⠷⠦⡴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠙⠻⢿⡿⢂⣚⣭⣿⣄⡈⢉⣠⣿⣭⣓⡐⢿⡿⠟⠋⠙⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⣴⣿⣶⠃⠿⠿⢋⠔⡻⢻⡟⡟⠢⡙⠿⠿⠘⣴⡿⣦⠀⢀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠄⣉⣤⣵⢿⣿⣯⢱⡾⣧⣠⠇⣿⣿⢸⣀⣼⢷⡎⣽⣾⡿⣮⣤⡉⠡⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣈⣱⣾⠛⠋⣄⠙⠋⠁⢼⢟⣯⡿⠳⣌⣡⠞⢿⣽⡻⡇⠈⠙⠋⣠⠛⠛⣶⣆⡁⢄
⠀⠀⠀⠀⠀⠀⢀⡴⠦⠩⠁⠂⠼⠀⠟⠀⠀⠶⠋⡎⠁⠰⣿⢿⡿⣿⠆⠈⢱⠗⠦⠀⠀⠿⠀⡦⠐⠨⢍⠰⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣈⡉⣹⡏⢉⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⡁⢹⠏⢈⡉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⢹⡏⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢙⡣⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠐⠒⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠠⣤⣀⣙⣿⣿⣿⣶⠦⣠⣾⣯⢿⣿⣷⡖⢲⡶⢶⣦⣤⣤⡀⡀
⠀⠀⠀⠀⠀⢀⣀⣈⣙⣿⣿⡿⢋⣵⣾⣿⣿⣿⢸⣿⣿⡇⢸⡸⢇⡿⠉⢉⠿⠋
⠀⠀⠀⠀⠀⣀⣭⣭⣽⡿⢃⣴⣿⣿⣿⣿⣿⠏⡾⠟⠀⠁⢄⠉⠉⠀⠀⠛⠒⠶
⠀⠀⠀⠀⣀⢠⣴⣾⠋⣰⣿⣿⣿⣿⣿⣿⣿⣦⠈⠃⠀⠀⠀⢹⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⣀⣬⣿⢃⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⢄⠀⠁⠠⠄⢦⣑⣲⠖⠁
⠀⠀⠀⠈⢀⣹⠃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠘⠓⠦⠀⠀⠀⠉
⠀⠀⠀⠀⢙⡏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶
⠀⠀⠀⠐⢻⡃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⠀⣀⣶⣶⠊⠉
⠀⠀⠀⠰⢿⡃⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⡟⢀⣿⣾⡿⠟⠁
⠀⠀⠀⠀⠞⡇⣿⣿⣿⣿⣿⣿⣗⠫⠌⠿⢶⣾⠟⠋
⠀⠀⠀⠀⠠⢿⢸⣿⣿⣿⣿⣿⡷⠁⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⢁⡘⡃⢿⣿⣿⣿⣿⣮⡖⠀⠀⠀⠀⠀⡰⠛⠉⠁⠉⠉⠙⠻⡿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠈⣷⡈⠞⣿⣿⣿⣿⣿⣾⠆⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠙⡌⠘⢿⣦⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠈⢻⢸⣞⠻⣿⣿⣿⣿⣟⡄⠠⣴⣿⣋⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⢀⡞⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠇⢻⣜⢿⣿⣿⣿⣿⣄⠈⠁⠀⠙⠓⠂⠀⠀⠀⠀⠀⠀⠀⣡⣽⣶⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠠⢋⡑⣾⠛⢿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠠⢤⣌⣉⡻⠿⡿⣿⣿⣷⣶⣶⣾⣿⡾⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠰⣷⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⠀⠀⣤⠀⠀⠀⠀⠀⢠⡄⠀⠀⠰⢢⡶⠀⠀⠀⠀⠀⠀⠰⣆⠶⠀⣶⠒⠂⢀⠀⠀⠀⠠⡐⢰⡶
⠀⠀⠀⠈⢧⣀⡀⢰⡆⢀⣀⡼⠁⠀⠀⢰⡇⡶⢡⢀⢠⡱⡆⣷⠀⠀⠀⠈⡷⠀⠀⢀⡀⠀⠀⢸⡇⠀⠀⠘⡆⠀⠀⢥⣤⠄⠀⠀⣾⠁
⠀⠀⠀⠀⢰⠀⠀⢸⡇⠀⠀⡆⠀⠀⠀⠈⣡⡇⠻⣼⣼⠃⣧⡉⠀⠀⠀⠀⢀⡄⠀⠈⡇⠀⠀⣄⠀⠀⠀⠀⠙⡶⢠⡴⡿⣦⡐⢶⠃
⠀⠀⠀⠀⠘⢷⣤⡤⢤⣤⡾⠃⠀⠀⠀⠀⠹⣿⡄⠸⠁⣼⡿⠃⠀⠀⠀⠀⠙⢷⣤⣄⢁⣤⣼⠟⠀⠀⠀⠀⠘⢧⡛⡧⠁⣜⢃⡼
⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠈⠃⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠰⠋⠁⠀⠉⠧⠀⠀⠀⠀⠀⡀⠈⠛⠁⠀⠉⠋⠀⢀
⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⣠⠠⡄⡀⠀⠀⠀⣀⠤⠄⡄⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢧⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠘⣇⠀⠀⠀⢠⡀⠀⠀⠀⡼⠁⠀⠙⣆⠦⠏⠶⣤⠞⠻⠶⣴⠃⠀⠸⡇⠀⠀⠀⣀⠀⠀⠀⢠⡗⠀⠀⠈⣷⢦⠀⣤⠠⢾⡾
⠀⠀⠀⠙⣷⠀⠀⢸⠇⠀⠠⡾⠃⠀⠀⠀⢻⢤⡀⠦⡿⠞⢀⠴⡟⠀⠀⠀⢷⡄⠀⠀⢿⠀⠀⢀⡼⠃⠀⠀⢠⠀⠀⠀⠟⠀⠀⢠⡀
⠀⠀⠀⠀⠹⣷⣄⡈⢠⣰⣾⠇⠀⠀⠀⠀⠈⠳⣙⡆⠀⢒⣫⠞⠁⠀⠀⠀⠀⠳⣀⡀⠈⢀⣀⡼⠁⠀⠀⠀⠈⢧⣀⡀⠀⣀⣠⡞
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠈⠁⠀⠀⣀⠀⠀⠀⠀⠉⠁⠀⠈⠉
⠀⠀⠘⣧⠐⡢⠀⠀⠀⠠⠔⢂⡼⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣄⡠⠀⠀⢄⢠⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀
⠀⠀⠀⠈⢧⡀⠀⢸⡆⠀⢠⠞⠁⠀⠀⢰⣆⠀⠀⠘⣿⠁⠀⠀⣰⡆⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠁⠀⠀⠀⠀⠀⠴⣬⣿⣱⠆
⠀⠀⠀⠀⢺⣄⣀⠸⢁⣀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠈⠁⠀⣰⠃⠀⠀⠀⢠⣘⠷⠶⢢⠴⢾⣣
⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⠁⠀⠈⠛⠒⠂⠀⠀⠀⠀⠦⠀⠀⠀⠀⠐⠒⠛⠀⠀⠈⠧⠀⠀⠼⠁⠀⠀⠀⠀⠿⠆⠽⠵⠿⠮⠏⠸⠗
⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⢿⠃⣼⡗⠀⠀⠀⠀⢻⡆⠹⡟⠀⠸⡟⠳⠗⣦⣀⣠⣔⠻⠛⡿⠀⠀⢿⠛⠓⢄⠀⠀⠚⠛⣿⠃⠀⠛⣿⠂⠀⠀⠀⠀⠀⢺⡿⠋
⠀⠀⠈⠀⢸⣆⠀⢸⡇⠀⣸⠇⠀⠀⠀⠀⢻⡀⣰⡆⢸⠁⢼⡀⣰⠇⠀⠀⢸⡁⠀⠀⣶⠃⠀⠀⡿⠀⠀⠀⠹⣇⡀⠀⣿⠀⢀⡼⠃
⠀⠀⠀⠀⣄⠙⠶⠜⠷⠾⢋⡄⠀⠀⠀⠀⠀⣧⠈⠀⠈⠀⠈⢀⡇⠀⠀⠀⠀⡏⠉⠀⠹⠀⠈⢉⡇⠀⠀⠀⠀⢼⡃⢀⣧⡀⢸⡇
⠀⠀⠀⠀⠘⠿⠶⠀⠐⠶⠟⠁⠀⠀⠀⠀⠀⠸⠛⠃⠀⠀⠛⠻⠁⠀⠀⠀⠀⠹⢾⠃⠀⠀⢷⠞⠁⠀⠀⠀⠀⠈⠻⠞⠈⠷⠛

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⠀⠀⠀⠈⠋⠀⠀⠀⢸⣿⡇⠀⠀⠀⠘⠋⠀⠀⠀⣿⣿⠀⠀⠀⠈⠋⠀⠀⠀⢸⣿⡇⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⣤⠀⠀⢤⣄⣹⣆⠀⠀⠠⣤⠄⠀⠀⣰⣏⣠⡤⠀⠀⢤⣤⠀⠀⢤⣄⣹⣆⠀⠀⠠⣤⠄⠀⠀⣰⣏⣠⡤
⠁⠀⠀⠀⠈⠙⠋⠀⠀⠀⠉⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠁⠀⠀⠀⠈⠙⠋⠀⠀⠀⠉⠀⠀⠀⠙⠋⠁
⢀⡆⠀⠀⠀⠀⠀⠀⠀⣆⠀⣰⠀⠀⠀⠀⠀⠀⠀⢰⡀⢀⡆⠀⠀⠀⠀⠀⠀⠀⣆⠀⣰⠀⠀⠀⠀⠀⠀⠀⢰
⣾⠁⠀⠀⠰⠶⠀⠀⠀⢹⣶⡏⠀⠀⠀⠲⠖⠀⠀⠈⣷⣾⠁⠀⠀⠰⠖⠀⠀⠀⢹⣶⡏⠀⠀⠀⠲⠖⠀⠀⠈
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⡀⠀⠀⣠⡴⢾⡷⠀⠀⠀⣀⠀⠀⠀⢾⡷⢦⣄⠀⠀⢀⡀⠀⠀⣠⡴⢾⡷⠀⠀⠀⣀⠀⠀⠀⢾⡷⢦⣄
⠋⠀⠀⠀⠀⡾⠁⠀⠀⠈⠛⠁⠀⠀⠈⢷⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⡾⠁⠀⠀⠈⠛⠁⠀⠀⠈⢷
⠀⡄⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠉⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢀⠈⠒⢄⠀⡀⠤⠤⠀⠀⠰⢀⠎⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠈⡦⠒⠁⠀⠀⠀⠀⠀⠈⠙⠢⡀⠀⠆⠈⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣀⣮⠴⠀⠀⠀⠚⠛⠒⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⡡⠠⡀⠀⠀⠖⣠⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢗⣿⠏⠀⠀⠦⠿⠻⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⡨⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡘⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢛⠁⠉⡉⢢⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⡇⠀⠀⠀⡀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠄⠰⣜⠜⠀⢀⠔⢁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⠎⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠁⠺⣁⣤⣒⠡⣖⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠘⠀⠀⠀⠀⠈⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⢢⡀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠠⢧⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠋⢂⠀⠀⠀⠑⡀⠀⠀⠀⢀⠆⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠐⠶⠀⡠⠃⠀⠘⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢣⠀⠀⠀⠀⠋⠀⠀⠀⡔⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣤⠀⠀⠀⠀⠀⠀⢾⠆⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⢳⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⠢⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠐
⠀⠤⣄⡀⠷⠄⣷⣀⠃⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⢀⢠⠀⠀⠀⠀⠀⠀⠀⠰⠀⠃⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠃
⠴⠒⠃⠀⡠⠀⠤⣈⠑⢀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠈⡌⠦⠀⠀⠀⠀⠀⠀⠇⡐⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⣰⠄⣀⡀
⠀⠀⢂⠐⡀⠀⠀⠀⠑⢄⠑⡄⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠰⡀⡁⣄⡀⠀⣠⠛⣠⠁⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⡀⠄⠚⢉⠈⡈⠙⣏
⠀⠀⠀⠣⡑⠤⡀⠀⠀⠈⡂⠘⡀⠀⣀⠤⠀⠀⠚⠄⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⡐⠃⠄⡀⠀⠀⡀⠊⡠⠐⠈⠀⠀⡁⠠⠁
⠀⠀⠀⠀⠈⠂⢌⡁⠒⠒⠁⢀⠗⠈⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⣀⠱⠀⠀⠀⠋⡀⠀⠀⠀⡀⢠⣀⠄⠀⠈⠢⡌⠀⠘⠀⠀⠀⠀⢐⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠐⢰⡇⢸⡄⢸⡆⠻⡅⠀⠀⠀⠑⡿⠨⡧⠠⣷⠐⡇⠀⠀⠀⠀⠐⠄⣈⣉⡉⠤⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠄⠀⠠⠤⠀⠀⠀⠀⠉⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⣆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠛⠿⠿⠷⠤⣤⣦⣤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣉⢻⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢄⡐⠀⠀⠀⠀⠀⡀⠀⠲⣴⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠖⠛⠀⠀⠀⢸⡿⠈⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠀⣀⠀⢠⠀⠀⠀⣰⡿⠃⣠⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡿⣃⣞⣠⡤⠾⢋⣠⣾⠿⠉⠇⠀⠀⠀⠀⠀⠀⠒⢟⠀⠀⠀⠀⠀⠀⢠⠝⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣉⢉⣉⣩⣶⣾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣿⡜⠀⠀⠀⠀⠀⢀⢢⠀⠀⠀⠀⠀⠀⡰⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠿⣿⣿⡟⢻⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⠀⠀⠀⢤⡿⠁⠀⠀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢾⡿⣿⣷⡀⠘⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣽⠀⠀⠀⣾⣟⡆⠀⠀⠀⠀⠀⢪⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠹⢏⣿⣆⠀⠩⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⡐⡀⣰⣿⠗⠀⠀⠀⠀⠠⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠙⣿⣿⣆⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣯⣿⠇⠀⠀⠀⠀⢀⢸⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡟⠙⠻⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢘⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢹⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡿⣿⣿⠿⡇⠀⠀⠀⠀⠀⠺⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⠁⠀⠀⠀⠆⠈⠀⠀⠀⠀⠀⣠⣿⢯⠀⣿⡟⢸⣷⠀⠀⠀⠠⡀⢴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⡀⢠⡵⡿⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣟⡁⠞⠁⠀⠀⠀⠀⠁⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠛⠒⠒⠛⠁⠀⣤⢀⣄⣠⢀⣀⣴⣦⠼⠋⠉⠉⠉⠁⠾⠳⠆⠲⠖⠐⠾⠓⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⢀⣀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂⢄⡀
⠀⡠⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠢⡐⢦⡙⠓⠢⠰⣎⠀⠀⠀⠀⠀⠠⠑⢂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⡀⠀⠀⢁⡍⠑⣦⡄⢠⠀⠀⣀⣀⣤⣄⣀
⡈⣼⣿⡎⣿⡅⠀⠀⠀⠀⠀⡀⠀⢠⠈⣿⡄⠀⡁⢻⡀⠀⠀⠀⠀⠀⣷⠈⣗⡢⣬⡀⡀⣀⣿⣀⣄⣀⢀⣿⣆⡀⢘⣿⠄⢻⡇⠀⢣⡐⣿⡄⠀⠉⣿⡷⡄
⠇⣿⣿⡇⣽⣧⣿⣠⣤⣣⣸⣷⣆⣿⣷⣿⠇⠀⣆⣼⣅⠀⡀⢀⠠⣦⣿⡀⣾⣵⡿⠷⢛⣉⠩⠍⠁⠀⠈⠉⠉⠛⠻⣿⣧⣿⣷⣶⣺⡇⣣⡇⣷⣤⣾⣷
⠈⠪⠿⠕⠋⠉⠉⠉⠉⠛⠻⠿⠿⣿⣿⣏⣄⣼⣿⣿⣿⣖⣠⡾⠾⣛⣛⣭⣿⢛⠖⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠛⢿⣿⣧⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠛⣿⣿⣿⣿⢏⣡⠖⠋⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢷⣄⠊⢻⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠙⠿⣿⢸⡃⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⢛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠋⢸⠀⠀⡴⣾⠟⣩⢉⡀⢠⣉⠙⣦⡀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠛⠉⠉⠉⠃⠲⢤⠀⠸⠒⠖⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡓⢤⡀⢀⡁⠸⠀⡾⠐⠀⢠⡎⢐⣄⡄⢈⠵⡄⠹⣎⠋⣁⣀⣤⣾⡟⠁⠀⡠⢖⣾⢰⣶⢀⠀⠑⡄⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢲⠌⡑⡆⢸⣼⣷⡂⠀⡿⠤⣾⣥⣻⡤⠤⢭⠀⣿⣿⣿⣿⣿⡿⠀⠀⠼⠛⢛⡟⢻⣄⠴⠆⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠸⣆⣤⠑⠂⢞⣿⣇⠀⢒⡒⠘⢿⠛⠝⢢⣿⠀⡟⠀⠀⠀⠸⣷⠀⠀⢲⠂⠈⣿⠿⠟⠲⠂⠀⠷⠤⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠌⠉⠛⠛⠛⣛⣿⣆⠈⠙⠦⠿⠤⡿⠟⢁⣔⣀⣀⣀⣤⣤⣿⣦⡀⠈⠲⠆⣀⡄⠔⠁⢰⡖⠀⠀⣀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠶⠦⠤⠤⠤⠶⠟⠻⠿⠿⢿⠿⠿⠟⠛⠛⠶⣄⡠⢄⣀⣠⣴⣾⣿⣿⣿⣿⠟⠋⠁

⣀⡀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⣀⡀⠴
⢩⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣫⣁
⠈⠉⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢳⣿⠞⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡙⠈⠉⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⢤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣸⣀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⢀⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠾⣿⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⡿⠇
⠀⢀⠠⠒⣀⡰⡖⠀⠀⠘⡀⠀⠰⢦⣀⠐⠢⠀⠀⠀⠀⠀⠊⣌⠋⣬⠁⠀⠀⠀⢀⠠⠒⣀⡐⡖⠀⠀⠈⠀⠀⠲⢦
⠀⠁⠀⠀⠀⠀⠀⠐⡇⠀⡰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠦⠠⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⡰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂
⠀⣀⣤⣤⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⣄⡀⠀⢀⠀⡀⠀⠀⣀⣤⣤⣦⡄
⣩⡽⣻⣿⠟⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡽⣦⣍⡓⢄⢤⡤⣒⣩⡽⣻⣿⠟⠀⠀⠀⠀⡀⠄⢀
⣦⣼⡿⠃⠀⠀⠀⠀⠀⣤⡀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣶⠛⣩⢤⡙⠳⣦⣼⡿⠃⠀⠀⠀⠀⠀⣤⡀⢠⣄
⣤⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⡞⠹⡜⡌⣦⣮⣿⠀⠀⠀⠀⠀⠀⠀⣿⠃⠸⣟
⠾⠋⠀⠀⠀⠀⠀⠀⠀⠠⠃⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠃⠀⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠩⠃⠘⠏

⣀⡀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣀⡀⠘
⢵⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⢧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣰⢵⣥
⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠿⠋⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡌⠀⠈⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠢⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⡿⠏⠋⠻⣏⡀⠀⠀⠀⠀⠀⠀⢘⠿⢙⡙⢿⣁⠀⠀⠀⠀⠀⠀⠲⣾⣶⣿⣿⣮⠴⠀⠀⠀⠀⠀⠀⠑⠄⣒⠤⠁
⠀⠀⠀⠀⠀⠀⠙⣿⡄⡉⢠⡟⠉⠀⠀⠀⠀⠀⠈⢻⣧⢙⠃⣼⠋⠁⠀⠀⠀⠀⠀⠤⢽⣿⣿⣿⡿⡤⠀⠀⠀⠀⠀⠉⠀⡡⢛⠪⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠋⠝⠟⠈⠙⠀⠀⠀⠀⠀⠀⠀⠘⠩⠻⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠠⠎⠋⠮⠑⠀⠀⠀⠀⠀⠀⠀⠚⠀⠰⠀⠈


⠀⠀⠀⠀⠀⠀⢀⡄⢐⢂⣔⡄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠰⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣶⡴⡴⡠⠀⠀⠀⠀⠀⠀⠀⣄⠦⣶⢖⡄
⠀⠀⠀⠀⠀⠀⣄⠐⣾⣿⣷⠢⠆⠀⠀⠀⠀⠀⢀⡀⠐⢬⡌⠄⠤⠀⠀⠀⠀⠀⠀⠈⣟⢲⣿⣷⢺⢒⠀⠀⠀⠀⠀⢘⡊⢡⣷⡎⠬⠅
⠀⠀⠀⠀⠀⠀⠀⣬⠹⠛⠏⡅⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠘⡹⣤⣉⣧⢟⠂⠀⠀⠀⠀⠀⠐⢢⢄⢸⢴⡎⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠖⠞⢶⣀⠀⠀⠀⠀⠀⠀⠀⡰⢈⠬⢤⡔⠄⠀⠀⠀⠀⠀⠀⠀⠢⣴⠛⠶⠶⠂⠀⠀⠀⠀⠀⠐⢶⣶⣿⣶⣗⡄
⠀⠀⠀⠀⠀⠀⠂⣿⡸⠽⢀⡧⡂⠀⠀⠀⠀⠀⠐⠂⣗⡺⢇⡿⡘⠀⠀⠀⠀⠀⠀⠰⠈⣿⠶⡤⠤⠠⠀⠀⠀⠀⠀⠾⢿⠿⢿⠿⣿⡛
⠀⠀⠀⠀⠀⠀⠀⠎⠳⡓⠋⠒⠀⠀⠀⠀⠀⠀⠀⠠⠊⠍⡍⠈⠂⠀⠀⠀⠀⠀⠀⠀⠰⠋⢢⡖⠲⠀⠀⠀⠀⠀⠀⠀⠴⠰⢰⠦⠟⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⣤⡎⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⢪⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠲⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣠⢀⣸⣇⡀⣄⣤⠆⠀⠀⠠⣼⡆⢹⡟⢁⣿⡤⠀⠀⠰⣤⣆⢶⣼⣇⡶⢠⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⠋⠙⣿⡍⠀⠀⠀⠀⣛⡋⣙⣿⣿⣟⢉⣛⡀⠀⠀⠨⣭⣿⣿⣿⣿⣭⡍
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⠻⠷⡾⠟⡶⣤⡀⠀⠀⠸⢿⡏⢹⣇⠉⣿⠏⠀⠀⢀⣴⡶⣻⡿⡿⣟⢲⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠐⠛⠏⠓⠈⠀⠀⠀⠀⠀⠀⠉⢼⡷⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣦⣀⠦⣽⠴⣀⣴⡶⠀⠀⠲⣤⣀⠀⣹⡁⠀⣠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣝⣳⣾⣞⣛⠌⠀⠀⠀⠀⠀⠉⣛⣾⣶⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠛⡇⣿⢬⡛⣶⣦⠀⠀⣠⡶⠞⠉⢺⠒⠉⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠀⢀⣀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣤⠀⠘⡏⠀⣠⣄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣶⣷⡚⠉⠁
⠀⠚⠒⠂⠒⠂⠒⠒⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠋⠉⡇⠉⢻⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠄⠀⠀⠀⠀⠀⠠⠞⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡖⠀⠀⠀⠀⠀⠀⠀⠐⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢠⠄⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⢻⣷⣤⣀⡀⡄⡀⢀⣠⣴⣿⠃⠀⠀⢀⠈⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡌⠉⠀⠀⠈⣻⣿⣿⣿⣷⣾⣿⣿⣿⠋⠀⠀⠀⠀⣠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⡴⢚⣿⣗⣻⣶⣶⣦⣤⣌⣹⣿⣿⠿⢿⣿⣿⣉⣠⣤⠶⠶⠻⢒⣶⡶⠖⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣯⣬⣤⣄⣀⠀⠈⣨⣿⣿⡿⢻⣿⣿⣿⣰⣾⣿⣿⣿⣿⣿⣿⣿⣷⠿⠛⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⢀⣠⡤⠜⢻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣂⣀⣀⣤⣤⠶⣮⣍⣀⡀
⠀⠀⠀⢀⣴⣾⣿⣷⣦⣰⣞⣂⣀⣀⣈⣉⣛⠛⠻⢿⣿⣶⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⢿⣛⣛⣉⢉⣙⣿⣇⣤⣭⣭⣿⣷⣤⡀
⠀⠀⠰⠛⠉⠈⢉⠽⠛⢿⣿⣿⣿⣿⣿⣥⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡻⢿⡻⣿⣛⠉⠙⠻⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠛⡿⣿⣿⣿⣿⣿⢿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⡟⠻⠿⣿⣻⣯⡻⠙⠢⠁⠀⠙⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠁⠀⢠⣿⠋⠀⢻⣿⣻⣿⣿⠛⢻⣿⣿⣿⣧⡿⠀⠘⢿⡀⠀⠈⠙⠗⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⡴⠟⢻⣼⣿⣿⡁⢹⣿⣿⣿⠉⠻⡆⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠈⣾⣿⣿⡇⣸⣿⣿⠏⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⡀⠀⠈⠻⣿⣷⣿⡿⠋⠀⠀⢸⠇⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣿⣧⣿⣿⢿⣿⣿⠓⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡶⢶⣶⣿⢿⣿⣯⣽⡆
⠀⠀⠀⠀⠀⠀⣿⣾⣿⣷⠘⢿⡛⠉⢹⣿⡿⠉⠙⢿⣿⠶⣤⣀⠀⠀⠀⠀⣰⣧⡀⠀⠀⠀⣀⣤⢶⣾⠟⠛⢻⣿⣿⠛⠻⢿⡿⢸⣿⣿⣽⡇
⠀⠀⠀⠀⠀⠀⢿⣶⡿⣿⣏⡋⢁⣠⡾⡿⢦⣀⠀⠈⠻⠀⠈⠙⠿⣶⣄⣠⣿⣿⣇⣠⣶⠟⠋⠁⢸⠏⠀⠀⣀⣽⣿⣄⠀⠙⢒⣿⣿⣧⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⣟⠉⠁⠀⢀⡀⠤⠘⢛⣓⡶⠖⠛⢛⣻⣿⣿⣿⣿⣿⣿⣿⣷⠶⠶⠶⢶⣖⠒⠾⢍⣀⠀⠈⠉⠙⠛⣿⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡶⠒⠉⢀⣠⠴⠒⠉⣁⡤⠖⠛⢉⣩⣤⢾⣿⣿⣿⣿⣥⣉⠙⠒⠦⣤⣈⠙⠒⠤⣄⡀⠁⠒⢤⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⡚⠉⢀⡠⠖⠋⣁⣤⣶⡯⢭⣤⠶⢺⣷⣿⣿⣿⠦⣭⣍⢓⣦⣄⡈⠙⠦⢤⡈⠉⠓⢦⣼⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣾⣧⢴⣶⡛⠉⢻⣿⣿⠏⣠⡾⡿⣿⣿⣿⡿⡷⣄⡙⣾⣿⣿⠛⠓⢶⣦⣬⣷⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣴⣿⣿⣿⣿⣁⠞⣿⠏⡝⣹⠁⣿⣿⣿⡇⢳⠹⡙⢾⡿⢌⣳⣾⣿⣿⣦⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣿⣿⣿⣿⣿⢼⠋⡜⢰⠇⢸⣿⣶⣾⣷⠈⢧⠱⡈⢷⣼⣿⣿⣿⣿⡏⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣍⣻⢿⡿⢿⠃⡼⠁⡼⠀⠘⠹⣿⡿⠙⠀⢸⡆⢱⡈⢺⢿⣿⣿⢟⣹⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⢳⠇⣠⠇⠀⡇⠀⣠⢰⡟⣧⢀⡀⠀⡇⠀⣧⠈⣷⠷⣼⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⣤⣾⢧⣼⣷⣥⣿⣾⡆⢿⣼⣯⣢⣿⣀⣾⣇⣸⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠉⠙⠛⠉⠀⠀⠀⠀⠉⠈⠉⠉⠻⠟⠛⠛⠳⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣁⡀⢌⡢⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡐⠊⠀⢤⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠁⠻⠯⠷⡀⢀⣀⡀
⠀⠀⠀⢀⠖⡂⠉⠉⠉⢰⣀⡄⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠐⠻⢷
⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⢿⡄⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠶⠿⢖⢀⠸⠷
⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡋⠀⠀⠀⡐⢠⡄⡁
⠀⠀⠀⠘⡈⠀⠀⠀⡖⠁⠀⠀⠀⢀⠀⡀⠀⠀⢀⠀⠀⠐⠢⠀⠏⣷⠀⠀⢑⠀⠉⢁⠔⢆
⠀⠀⠀⡆⣇⠀⠀⢀⣷⠀⠾⢂⡨⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢩⢡⠀⠈⠇⠐⠀⠁⠈⠆
⠀⠀⢀⠵⢸⡸⠜⠡⠜⠄⠐⠿⣷⠀⠳⣶⠀⠀⠀⠀⠀⣠⠀⠀⠅⠈⢸⣰⠷⣕
⠀⠀⠄⠈⠈⢿⣦⣄⡀⢀⢰⠒⠲⣷⠶⠚⢃⡀⣀⣠⠾⠀⠀⠀⢀⠀⢨⡇⢸⡀⠀⠀⠀⠀⠠⢄
⠀⠀⠀⠃⠀⣿⣽⠜⠉⢸⠀⡇⠀⠀⠀⠁⠀⠉⠘⡈⠀⠀⠀⠀⠀⠀⠀⣁⠀⡄⠠⣤⠀⠀⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⡟⡀⠀⠀⠸⠀⢡⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⡧⢴⢁⠀⡇
⠀⠀⠀⠀⠀⠂⢰⠃⠰⠸⡀⢨⠁⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠁⢀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐
⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⢀⡠⣌⡄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢁⣼⣅⡀⠀⠀⢀⣠⣤⣤⣾⡶⣼⣷⡤⢠⣌⠉⠈⠁⢰⡾⠃
⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠈⠉⠀⠈⠉⠉⡙⠓⢶⠟⠉⠀⠄⠄⠍⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣶⣿⣿⣻⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣭⣿⣿⣿⣿⡿⠿⠛⢿⣿⣿⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⣿⣿⣿⣿⡿⠀⠀⠀⢀⡾⣿⡿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣯⣶⣿⣿⣿⡿⠿⠋⠀⠀⠀⣠⡟⣱⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣭⣿⣿⣿⡿⠛⠋⠀⠀⠀⠀⠀⣼⢏⣼⣿⣿⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣛⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⣾⣯⣾⣿⡿⢫⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣾⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⢃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣵⣶⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡏⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣫⣼⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⢿⣧⡿⡿⣿⢱⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣻⡿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⣿⠃⣾⡟⣸⠇⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠸⣿⠿⠷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠁⣸⡏⢠⣿⢁⡿⠀⢻⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⣿⢠⣿⡿⢸⡇⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢰⣿⡿⢹⡇⢸⡇⠀⠀⢀⣿⡆
⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢨⣿⠁⠈⣷⢸⣇⠀⢠⡾⠉⣿
⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⡄⠀⣿⠈⢿⡄⠻⠃⠀⠿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⣿⠀⠀⠛⠒
⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣄⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉

⠀⠀⠀⠀⠀⡤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⢠
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⡈⡡⠀⠧⠤⢀⢀⠀⠀⠀⠀⠀⠀⠠⠂⠤⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢢⡄⠀⡐⠉⠑⠢⢢⢐⠠⠠⡂⠀⠀⠀⠀⠀⠀⠂⠀⡠⠀⡠⠤⠀⠀⠀⠀⠀⠑⡄⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡄⠴⠁⠁⠀⠀⠀⠀⠈⡌⢢⡀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠘⠄⢆⠀⠀⠀⠀⠀⠘⡑⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠎⠸⠀⠀⠀⠀⠀⠀⠀⢀⡆⠐⡀⠀⠀⠀⢀⢈⠰⠀⠀⠀⠀⠘⢁⠢⠀⠀⠀⠀⡰⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡄⢂⠰⡀⠀⠀⠀⡠⠊⢅⠀⠂⠈⢂⠐⡀⠁⣈⢺⠀⠀⠀⠀⣸⠈⠀⠑⠐⢣⢸⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠁⢠⠁⠆⠉⠀⠈⠀⠀⠄⢀⠀⠄⠂⢰⠐⠨⠐⢒⠐⠈⠀⠀⠀⠀⠀⠀⡆⣧⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢝⠁⠂⢱⢰⠈⠁⠀⠘⠐⣦⠀⠀⠀⠀⠀⠀⢠⢸⠀⠀⠀⣤⣸
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢃⠸⠐⡶⡆⠀⠀⠀⢸⠀⠀⣸⢸⠀⢀⠄⠌⢠⢰⡄⠀⠀⠀⠀⠀⠈⠀⡇⡓⠸⠁⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠀⡌⡆⢸⠀⠀⠘⢰⠁⠀⠘⢸⠀⠜⡀⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠇⠇⡱⠁⡜⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢀⡇⢰⠀⠀⠂⡙⣄⠀⠀⡇⡄⠀⠃⡄⠀⠀⡀⠀⠀⠀⠀⠀⠘⡄⠐⠁⢀⠂⢸
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠘⠈⠏⠀⡄⠀⠐⠀⢱⠀⠀⡇⠀⠁⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⡎⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠢⠈⠀⠀⠈⡀⠀⡇⠈⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⡇⠀⢠⢀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠇⠂⠀⠀⢀⠑⢀⡇⠰⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⠁⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⠁⠂⠀⠀⠀⠰⠋⡇⠀⠀⠼⡀⠀⡇⢀⠀⠀⠀⠀⠀⢸⠀⡰⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡗⠁⠀⠀⠀⠀⠐⠆⠃⠀⠀⠀⠛⠸⠁⠠⠀⠀⠀⠀⠀⢪⠀⠇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠆⠀⠀⠀⠀⠀⠀⢈⣘⡀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣦⠀⠀⣿⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⣿⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⣤⡈⢻⣿⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢶⣄⡀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣿⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⣸⣇⠀⣶⠀⠀⠀⠀⠀⣤⣘⣿⣁⣤⠀⠀⠀⠀⠀⣴⠆⣰⡟⢀⣠⣴⠾⠛
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣾⠿⢿⣦⣿⡀⢴⡀⢠⣤⡈⢻⡿⠋⣡⣄⠀⣠⡀⢰⡟⣰⣿⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡶⠶⠿⠛⢷⣾⣇⠘⠻⣇⢸⡇⢀⡿⠛⢀⣿⣤⡿⣿⣉⡉⠛⠛⠳⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣉⠙⠳⣦⣽⣾⣷⣟⣠⡶⠟⠛⠿⢦⠄⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠗⠲⢶⣿⣿⣿⣿⠷⠶⠶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠐⠻⣶⣤⣴⠞⢋⣽⢿⡿⣟⠻⢦⣄⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠐⠷⠶⢦⣤⣤⣈⣉⣿⣾⠛⣿⠁⣴⣾⠁⢸⡇⢹⣦⡄⢹⡿⢷⣤⣶⠶⠾⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⣿⠏⣼⠇⠈⠋⠀⠉⢉⣠⣾⣧⡈⠛⠃⠸⠷⠨⣿⠻⣷⣶⣿⠿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⣤⡶⠟⠋⠁⣼⠏⠰⠟⠀⠀⠀⠀⠀⠛⢉⣿⡍⠛⠀⠀⠀⠀⠀⠶⠀⢻⡏⠛⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⣿⠻⣦⡀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠈⠙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⣧⡈⠛⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⣿⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⣿⠀⠀⠻⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿

⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠰⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⡀⠀⠀⠀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⡝⢦⡀⣼⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠛⢛⣛⣒⣶⣂⠀⠀⠀⠀⢀⠬⠁⡇⢱⡀⠈⢿⣿⣿⠟
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣶⣤⠔⠁⠀⠀⡇⠀⢣⡀⠀⠈⠁⠀⣀⣒⣀
⠀⠀⠀⠐⠛⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠰⠿⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⡇⠀⠀⠻⠂⠤⠀⣼⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢠⡄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠇⠀⠀⠀⠙⣄⠀⠈⠻⠿⠟
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⠀⢹⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡍⠑⢄⣀⣤⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠆⣠⡀⠀⢰⡦⠀⠀⠀⠀⠀⠰⡶⠀⠀⠀⠀⢉⣠⠴⢄⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⡇⠀⠠⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⢸⣾⣿⣿⡆⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⢠⠀⠀⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⠈⠻⣿⠿⠃⠒⠀⠖⠀⠙⠀⠀⠀⠀⢠⡶⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⣋⣉⠐⠠⠀⠀⠀⠀⠀⢠⡔⠂⠀⠀⡈⠀⠰⡀⠀⠀⠘⢧⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡤⠄⠙⠛⠂⠀⠃⠀⠀⠀⡐⠀⠀⠀⠑⡀⠀⠀⠀⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⣯⠅⠀⠀⢀⣴⣾⣶⣄⠀⡐⠀⠀⠀⠀⠀⣸⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⣿⣿⣿⣯⡶⠁⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠱⠋⢠⠀⠀⠀⠀⢠⠸⣿⣿⣿⣿⣿⣿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣮⣽⣂⡄⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⢸⠀⠀⠀⠀⢸⡄⢹⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣆⣀⣀⣀⣀⣀⡠⠂⠀⠀⠠⠚⡄⠀⠀⣠⡶⢦⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⡠⠈⠀⠀⠀⢠⠀⠀⢠⠀⢴⣯⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠿⠋⠀⠀⢀⡄⠚⠀⠀⠀⣀⣄⣤⡀⠀⠀⢂⢀⡙⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⡄⠀⢘⣳⣿⣿⣿⡆⣠⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣤⣤⣀⠠⠊⠀⠀⠀⠀⢠⠀⠀⠈⠙⠛⠛⠛⠀⠷⣟⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣾⣿⣷⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣄⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢰⣿⣿⣿⣿⣿⡄⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⡝⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⡄⠀⠀⠀⠀⠀⢰⠎⠙⠻⣿⣿⣿⠟⠁⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⡿⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⠀⠀⠀⠀⠀⣾⠁⠀⠀⠘⠟⠁⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⣀⡀⠀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⠃⠀⠈⠀⠀⠀⠀⠀⠻⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠸⠿⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠘⠁⠀⢀⡀⠀⠀⠀⢀⣀⠙⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢠⣤⠀⠀⠛⠃⢀⣤⡀⣀⡬⣈⠀⠀⠀⠀⠚⠋⠀⠀⠀⠈⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠅⢀⠈⠀⠀⠀⠉⠀⠁⠀⠈⠀⠀⠀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠆⠀⠈⠋

⣶⣶⣆⣀⣤⣴⣦⣴⣀⣶⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⢲⣖⠀⠐⠀⠀⠀⢲
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⣤⠀⢀⣀⣀⣉⡀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣦⡀⠀⠀⠀⠀⢸⣿⣾⣿⣿⣧⣤⣤⢀⣿⣿⣿⣿⣇⡀⠀⡄⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠶⠖⠷⠄⣴⣿⣿⣿⣿⣿⣿⣟⣋⣏⣿⣿⣿⣿⠁⢶⣿⣷⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣇⣐⣀⣀⣀⣀⣀⣤⣈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠿⠛⠿⢿⠿⠻⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡏⠠⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣤⣴⣶⣤⢄⠤⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡿⠛⠛⢻⣿⣿⡿⠟⠛⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠉⣻⠀⠀⠀⠀⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣕⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣟⣾⡟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⣨⠟⣩⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢻⡿⢻⡿⢛⡿⣯⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣤⣿⣤⣿⣴⣾⣿⡟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⠟⣹⠋⠁⢀⡿⣾⣻⣟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⣘⣰⠃⣠⠔⠋⠀⣸⣿⣻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠉⡷⠊⠁⢀⡴⠚⠁⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⢸⣿⠇⠀⢰⢃⣠⠖⠉⠀⢀⣤⣾⡿⠋
⠀⠀⠀⠀⣶⣶⣶⣦⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣸⡟⠀⠀⡾⠉⠀⠀⢀⣴⣿⡿⠋⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠻⣿⢿⡛⢦⣿⠿⢭⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⢰⠃⠀⢀⣴⣿⣿⣿⠿⠛⠙⢻⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠈⠻⣟⣾⣿⠲⢶⣿⠿⠝⡛⣿⣿⣷⣦⣄⡓⠀⠀⠀⠀⠀⠀⠀⡄⠀⣿⠀⢠⠇⢀⣴⣿⣿⡿⠋⢁⡠⠖⠊⢁⣀⡬⠷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢯⣿⣿⡉⠉⠒⠺⣇⠈⠉⠛⠿⢿⣷⣦⣄⠀⠀⡀⠀⡀⠀⡟⠀⣾⣶⡿⢟⣡⠿⣴⣊⣁⣤⠖⠊⠉⣀⣀⠠⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣽⡶⠖⠒⠚⠛⠲⢦⣄⣀⡀⠉⠛⠿⢶⣄⣹⣿⣿⣷⠿⠛⣉⠴⠋⠀⠀⠀⠀⢀⣈⠽⠿⣉⡀⠀⠀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⠖⠒⠚⠉⠁⠀⠈⠉⠙⠛⣛⣿⣿⣿⣿⡻⣷⣿⠷⠒⠒⠒⠒⠒⠛⠭⢄⣀⠀⠀⠉⠙⡺⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣤⣤⣤⣤⣴⣶⣾⣿⡿⠟⣉⠴⣻⣿⣾⣿⣝⡓⠦⢤⣀⡀⠀⠀⠀⠈⠙⠲⣤⣴⡷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣿⠿⠿⢯⣥⠖⠊⠁⢠⠃⢙⢿⢿⣷⣭⣒⢤⡀⠉⠙⠒⠦⢤⣀⣠⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⡤⠊⢹⡄⠀⢀⡏⠀⢸⠈⢆⢻⣷⡀⠉⠙⠳⢤⣀⣀⣠⣾⣯⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⣠⠟⢣⣀⡾⡄⠀⠘⡇⠘⡆⣿⣿⣄⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⢸⡉⢆⠘⣄⠀⢱⠀⠘⣿⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣬⣧⣈⣣⡀⣃⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠟

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣛⣻⣿⣫⣻⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣭⣿⣿⣿⣯⣿⣽⣯⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⢀⢨⢿⠠⠶⣾⣁⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⣼⣿⡎⠀⠀⠀⠐⣿⣿⡄⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢆⠀⣿⣿⣷⢀⢀⠀⣨⣿⣿⡇⠀⠀⢀⡴⣡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣸⡆⠈⢛⣾⣾⣾⣷⣿⣭⣤⣁⡀⢀⢾⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠟⠙⠛⠉⠁⠉⠉⠙⠹⣿⣻⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢊⣶⠔⢀⡀⠀⠀⠀⠀⠀⣀⢠⣿⣤⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⣤⡴⠊⡐⡂⣺⢿⣿⣾⣤⣴⣆⡄⣄⣮⣷⣿⣭⠭⢳⠀⠈⠳⣤⣄⣄⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠚⠓⢚⠍⠁⠀⡰⠃⠉⢉⣹⣿⠻⠿⡏⡿⣟⠿⢻⣙⣓⠚⠃⢣⠀⠀⠈⠛⠒⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢠⢁⣞⣯⣽⡃⠀⠀⠁⠀⠈⠁⣀⣠⡏⠿⣭⣛⠀⠃⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠈⢠⡙⣆⠋⠹⠓⠶⠠⠖⠒⠚⠛⠉⠁⠘⠄⢀⡀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⢠⠁⠀⠈⣇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠇⠀⠰⠀⠘⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢠⠞⣽⠟⠒⠀⠄⠄⠀⠀⠤⠠⠐⠊⠈⠙⠀⢲⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠠⢇⡾⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⣧⠀⠀⠻⣝⡂⠃⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⠀⠀⠿⠾⠧⡀⠀⠀⠀⠀⣀⣀⡥⠘⠏⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⢀⡰⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢻⠶⠤⢄⡀⣄⠤⡤⠾⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠀⢀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⢀⡴⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠹⣟⢀⠈⣻⠛⠁⣩⡤⠀⠒⠒⡆⠀⣀⣄⠄⠈⢀⣀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⠒⠚⠣⠂⠀⡾⠓⡌⠁⣀⠀⡟⠡⣄⣀⠰⠑⠰⣿⣿⢷⣠⣿⣦⡀⠁⢠⣤⢸⠋⠉⠃
⠀⠀⠀⠀⠀⡄⠀⠀⣤⡴⠛⡀⠄⠀⠀⠀⠛⠃⠠⠃⠈⠀⠀⠓⠐⡏⣀⠀⢳⠂⣹⡇⠀⠈⠛⠛⠡⠂⠀⢃⠈⠀⠳⡇
⠀⠀⠀⠀⠀⡀⠀⣀⡻⠃⠂⠀⠾⠁⢈⡿⠤⠀⠠⢠⠀⠀⠃⠀⠀⠃⠌⠂⠈⠆⠸⠗⠀⢺⡇⢈⡴⢤⣀⣐⣦⣁⣤⣴⠀⢀⡀
⠀⠀⠀⠀⠇⠀⠈⠀⡇⠀⠈⠀⠀⠀⡾⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣀⠊⠀⠀⠀⠀⠄⢈⡛⠝⢀⢸⣶⣿⠁
⠀⠀⠀⠀⠲⡀⠀⠀⣟⢣⡗⠀⢠⡆⠰⣗⢲⠀⠒⠇⠲⠘⠇⢰⠂⡇⢴⠂⣲⡄⠉⠘⠄⠀⣄⠀⡀⢠⠄⠸⠀⠦⠀⢼⢟⡊
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠊⠀⢿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣹⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠄⠀⢀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⣻⣤⡀⠀⡀⢀⢺⣗⠀⠀⡀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠁⢻⠄⡄⠢⡝⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢝⡳⣦⢰⠟⠎⣀⣲⠵⣛⠫⠁⠀⠀⠀⠀⡀⡀⠠⣦⡀⢫⢛⠌⣠⡞⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⣰⢒⢿⠐⣏⠂⠁⠀⠀⠀⠀⠀⠀⢐⠭⡻⠓⠂⠨⣷⣿⡯⠤⠄⢶⢶⠆
⠀⠀⠀⠀⠀⠀⢀⣠⡵⢿⠛⠸⠹⡾⠅⠋⠛⢥⡢⠄⠀⠀⠀⠀⠀⠀⠀⠐⠛⣡⢂⣝⡌⠱⠶⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⢭⠏⡂⠀⠀⠀⠸⠱⠆⠀⠀⠀⠀⠀⠠⢐⣸⡆⠀⠃⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠹⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢊⣟⡠⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠹⡅⠂⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⢤⠀⠀⠁⢀⠀⣨⠾⠡
⠀⠀⠀⠀⠀⠀⠠⣁⡀⠀⠀⠙⡇⢀⠀⣆⢠⠶⡀⠀⠀⠀⠀⠈⠟⡺⠤⢀⡀⠀⠣⣸⣀⡪⠀⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠊⠔⠩⠛⠒⢴⣶⣅⣊⠉⠑⠉⠂⠁⠀⠀⠀⠐⠋⠀⠀⢀⣀⠉⢶⡉⢹⢐⠊⠛⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢉⠔⢹⠱⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠔⡱⢻⠙⢆⠈⠹⡳⡣
⠀⠀⠀⠀⠀⠀⠀⠤⢄⡶⠉⠀⠀⠐⡙⣦⠴⠤⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠘⢁⡌⠀⠀⠃⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠡⠁⠀⠀⠀⠱⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢱⢯⢢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣶⣶⣿⣿⣷⣦⣄⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⡟⠿⠿⠿⢿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⢸⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⡟⠻⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠀⠙⢿⣧⠀⠀⠀⠈⢿⣿⣿⡿⠿⣿⣷
⠀⠀⠀⠀⣿⣿⠟⠁⠀⠀⢹⣿⣿⣿⣿⡅⠀⠀⢨⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠋⠀⠀⠀⠈⡿⠀⠀⠀⢀⣠⣿⣿⠃⠀⠈⣿
⠀⠀⠀⠀⠻⢿⠦⠀⠀⠀⠀⠛⠛⠛⠿⠿⠂⠀⠀⠉⠀⠀⠀⠀⢀⣦⣤⣤⣤⣦⠀⠈⠉⠉⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠉⠉⠁⠀⣀⠀⢀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠠⣤⣾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⡀⣶⣄⠀⠀⠀⢀⣉⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣀⣘⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠸⠋⠛⣻⣻⡟⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣻⣿⡀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⢠⣶⣦⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⡀⠀⠀⢿⣿⣿⠟⠻⣿⣿⡿⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢿⣿⣿⠿⠿⣿⣿⣿
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⢿⣿⣿⣿⣿⡿⠋⠀⠀⣘⣿⣧⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⢙⣿⣟⣿⡄⠀⠀⠀⠀⢘⣿⣯⠀⠀⢹⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣷⣶⣦⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠏⠁⣤⣈⡙⢆⣈⣙⣿⠘⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢁⣰⣾⣿⡿⠟⠂⠉⢿⣿⡏⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⢻⣫⣾⠟⣋⣠⣴⣿⣦⡀⣹⡟⢑⣥⣼⣿⣯⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢷⣨⣿⣿⠋⠉⠙⠚⠿⠕⠺⠿⢷⣦⢭⣵⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣽⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣿⣿⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣟⣿⡇⠰⠛⣛⡓⠀⠀⢀⢘⣯⣉⡓⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣙⣿⠀⠀⠺⠻⠟⠊⠀⢸⠛⠛⠋⠁⠸⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢿⣿⡄⠀⠀⠀⠀⠁⣀⢸⣄⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢷⠀⢰⠐⠀⣀⡍⢭⣀⠌⠀⢀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⡀⠀⠀⠈⢍⡭⠇⠀⢀⡾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢿⡉⠳⢦⣀⡀⣀⣀⣴⣿⠇⢹⣦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⣾⠉⠛⠈⠷⣤⡀⠉⠙⠛⢉⡽⠋⠀⠸⠿⣿⠉⣿⣷⢦⣄⣀
⠀⠀⠀⠀⢀⣀⡴⣿⡉⡇⣧⠸⣆⠀⠀⠀⠈⠙⢳⢖⠶⣾⠁⠀⠀⠀⠀⣿⠀⢳⣿⣸⣿⣿⣭⠷⣄
⠀⠀⢠⣾⣋⣵⣶⡏⡇⢿⢹⡀⢹⣆⠀⠀⢀⣤⣾⣜⣄⣵⣷⣦⣄⡀⢀⡟⠀⢸⢸⣿⣿⣿⣿⣿⡟⣷⡀
⠀⣰⣿⡄⡇⣿⣿⣿⢡⠀⠀⠃⠀⢹⣷⣾⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠈⠘⣿⣿⣿⣿⣿⡇⣿⣿
⠀⣿⣿⣷⢹⢹⣿⣿⡞⡆⠀⠀⠀⠀⠹⣿⣿⣿⠉⣿⣿⡟⣿⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣿⣻
⠀⣹⣿⣿⣏⣏⣿⣿⣷⢳⠀⠀⠀⠀⠀⠙⣿⣿⠀⣿⣿⡇⣿⣿⣿⠟⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣿⣿
⢠⡿⠸⣿⣿⣸⡼⣿⣿⣏⣇⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠓⠛⠋⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⣿⢧⡿⢻
⢸⡿⠶⣦⣿⡇⡇⣿⣿⣿⡜⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⣿⣿⣧⣿⠼⠁⢿
⣿⣷⣌⣶⣿⣿⢱⢹⣿⣿⣿⣝⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣳⣿⣿⡇⣿⣿⢀⣴⣿

⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢸⡅⠀⠀⢀⡗⠀⠀⠀⣠⡤⢠⢀⢄⣠⣀⢄⢠⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⢢⠀⢻⢛⡛⢿⠃⡀⠀⠀⠙⢦⡀⢢⣾⡷⣿⣦⠂⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏
⠀⠀⠣⢼⣯⣽⡯⠔⠁⠀⠀⠀⠀⠹⣦⣀⣹⣋⣠⡼⠁⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢎⠉⠁⠀⠂⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠀⠀⠀⠀⢀⣺⣿⣷⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠸⣶⣧⡶⠄⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡠⣼⠿⢿⢤⡀⠀⠀⠀⠇
⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⢰⣯⡧⠀⠀⠀⠀⠀⠈⡜⢹⣄⣸⢱⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⠀⠀⠀⠀⠈⠁⠈⠻⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣽⣷⣇⠀⠀⠀⠀⠀⢐⣛⡂
⠀⠀⠸⣿⠇⠀⠀⠀⢀⣿⣷⣾⣧⠀⠀⢀⡜⠻⡿⠛⡀⠀⠀⠀⣿⣽⣿⣯⡟⠀⠀⠀⠀⠹⣿⣿⡐⠄
⠀⠀⠀⣉⠀⠀⠀⠀⢸⢀⢠⢇⣭⡇⠀⠀⣵⣦⣶⣶⡉⠀⠀⠪⡇⠈⠛⢀⣶⡅⠀⠀⠈⠁⢸⡏
⠀⠀⠀⠉⠀⠀⠀⠀⠰⠻⠾⠾⠣⠀⠀⠀⠙⠻⠿⠛⠁⠀⠀⠰⡹⣴⣶⣿⢿⠃⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣟⣛⣛⣛⣛⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣯⣿⡄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡸⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠶⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣋⠄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡖⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢔⡴⣪⠆⠀⠀⠀⠀⠀⠀⣨⣍⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⣈⠁⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⣁⠬⣴⢁⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⣾⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠅⠰⡾⠓⢢⣴⢏⡆⡦⡄⢠⡟⣭⠌⠀⠀⠀⢀⣀⣀⠤⠶⠟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠈⡀⡐⠠⠎⢹⣧⣣⣔⣯⣵⣿⣧⣵⡿⣿⣿⣶⣶⣶⡶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣌⣿⣿⣦⢰⡿⣩⣵⣿⣟⡿⢿⠻⣧⣿⣿⣿⡿⢛⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢛⣭⣿⣿⣿⣧⡘⠿⢿⢫⡖⡿⣿⣿⣿⣿⡟⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢨⢁⢟⣿⣿⣿⣿⣶⣦⣹⣧⣳⣿⣿⠛⠋⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠼⢛⣿⣿⣿⣿⣿⣿⣿⡿⠋⣡⣶⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢻⣿⣿⣯⣿⡟⠐⡄⠾⠋⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣌⢿⡿⣡⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣀⠀⠙⢿⡿⠋⠀⣀⣤⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢻⣿⣿⡄⣿⢸⡇⣿⢠⣿⣿⡟⠀⠀⠀⠀⢠⡄
⣀⣀⡀⠀⠀⠀⠀⠀⣸⣿⣦⠀⠀⠀⠀⣬⣭⣿⣿⣸⣇⣿⣿⣭⣥⠀⠀⠀⠀⣴⣿⣇⠀⠀⠀⠀⠀⢀⣀
⠈⢿⣿⣿⣦⢹⡇⠀⠹⣿⣿⢠⡦⢸⣷⣄⡀⠀⠈⢻⡟⠁⠀⢀⣠⣶⡇⢴⡀⣿⣿⠏⠀⢸⡇⣴⣿⣿⡿
⠀⠀⠙⣛⣛⣿⣿⣜⡻⢾⣿⡿⠁⠘⣿⣿⣇⢠⡆⢸⡇⢰⡄⣸⣿⣿⠃⠈⢿⣿⡷⢟⣣⣿⣿⣛⣛⠉
⠀⠀⠀⠉⠉⠀⢶⣭⣛⢶⣽⣇⠀⠠⣬⣝⣿⣿⠁⢸⡇⠈⣿⣿⣫⣥⠄⠀⣸⣯⡶⣛⣭⠶⠀⠉⠉
⠀⠀⠀⣠⣾⣿⣶⡾⣿⠟⠛⠙⠻⣦⣄⡈⠉⢿⡄⢸⡇⢠⡿⠉⢉⣠⣴⠞⠋⠙⠻⣿⢷⣶⣿⣷⣄
⠀⠀⠉⠛⠛⠛⠋⢼⠏⢀⣀⠐⢶⣄⠉⠛⢶⣤⣻⣾⣷⣟⣤⡶⠛⠉⣠⡶⠂⣀⡀⠹⡧⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣿⣿⡷⣿⡷⠶⠶⢾⣿⣿⣿⣿⡷⠶⠶⢾⣿⢾⣿⣿⣿⡷⠆
⠀⠀⣀⣤⣤⣤⣄⢺⣆⠈⠉⠡⠾⠋⣀⣤⡶⠛⣿⢿⡿⣯⠛⢶⣤⣀⠙⢷⠌⠉⠁⣰⡗⣠⣤⣤⣄⣀
⠀⠀⠈⠙⢿⣿⡿⢷⣿⣦⣄⣠⣴⠞⠋⢁⣀⣾⠃⢸⡇⠘⣷⣀⡈⠙⠳⣦⣄⣠⣴⣿⡾⢿⣿⡿⠋
⠀⠀⠀⣀⣀⠀⠶⢛⣭⡾⣿⡏⠀⠐⠛⣫⣿⣿⡀⢸⡇⢀⣿⣿⣝⠛⠂⠀⢹⣿⢷⣭⡛⠶⠀⣀⣀
⠀⠀⣀⣭⣽⣿⣿⢋⣵⢾⣿⣷⡀⢰⣿⣿⡟⠸⠇⢸⡇⠸⠇⢻⣿⣿⡄⢀⣾⣿⡷⣮⡙⣿⣿⣯⣭⣀
⢀⣼⣿⣿⠟⣹⡇⠀⣠⣿⣿⠘⠗⢸⡿⠋⠁⠀⢀⣼⣧⡀⠀⠈⠙⢿⡇⠺⠃⣿⣿⣄⠀⢸⣏⠻⣿⣿⣧
⠉⠉⠉⠁⠀⠈⠀⠀⢹⣿⠟⠁⠀⠀⠀⢛⣛⣿⣿⢹⡏⣿⣿⣛⡛⠀⠀⠀⠈⠻⣿⡏⠀⠀⠁⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⣼⣿⣿⠇⣿⢸⡇⣿⠸⣿⣿⣷⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠉⠀⣠⣾⣷⣄⠀⠉⠛⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢋⣾⣷⡙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣠⣶⣿⠿⠿⣷⣦⣆⠀⠀⠀⢀⠔⠀⣠⡶⠋⠀⠉⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠶⠛⠁⢠⣾⡟⠉⠰⣤⣤⣄⡉⠻⢷⣶⣶⣷⣤⣿⣏⡶⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠋⢁⣠⣴⣶⣿⡿⠀⢈⢳⣿⣿⣿⣿⣦⣀⠀⠀⠉⠉⠉⠛⠿⣶⣤⣀⣄⡀⠀⠀⠀⣤⣤⡆
⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⢁⣠⣶⣿⣿⣿⠟⣩⣿⡀⢨⣿⣿⣿⣿⠿⠟⠛⠛⠒⠒⠀⠀⠀⠀⠀⠉⠻⡛⠷⢶⡾⠿⠟⠉⣿⡆⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠰⠟⠁⢀⡴⣿⣿⣿⣿⠟⠛⠉⣉⠽⠛⠀⠹⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⣀⣾⣿⠄⡀⠄
⠀⠀⠀⠀⠀⠀⣠⠞⣫⣾⣿⡿⠋⠀⠀⠀⣈⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡇⠀⠀⡀⢝⢿⣷⠊
⠀⠀⠀⠀⠀⠚⣡⣾⣿⣿⡟⠀⠀⣠⡿⠛⠁⠀⣠⢏⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡀⠘⣿⠀⠀⠃⠈⢿⣷
⠀⠀⠀⠀⢠⣾⣿⡿⣻⡟⠀⢀⣼⣿⡶⠂⠀⢸⠟⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣦⡀⠀⠀⠀⠀⠀⠘⢆⠃⣠⠂⠀⠀⢸⣯
⠀⣴⠊⣴⣿⠟⢈⣿⣿⡇⠀⢞⣿⠏⠀⠀⠀⢀⡾⢁⡄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠙⠻⣦⡀⠀⠀⠀⡀⠘⢀⠃⠀⠀⠀⠀⣿
⠀⢀⣾⠟⠁⢩⣿⣿⣿⡇⢠⡾⠟⠀⠀⠀⢀⡿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣳⣦⠽⠿⠶⠶⣦⣷⣴⣿⣤⡀⣀⡴⠛⢻⣇
⠠⠋⠀⠀⠂⠞⣼⣿⣿⣇⣾⠃⠀⠀⠀⣠⠀⠀⣿⣠⠎⠀⠀⠀⠀⣠⠶⢛⣽⡿⠏⠀⠀⠀⠀⠀⠀⠬⣽⣿⣛⡛⢿⡄⢀⢹⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⣴⢀⣼⣇⡤⠀⣿⢿⠀⠀⠀⠀⠈⠁⣰⠟⢁⣼⣿⣷⣦⡀⠀⠀⣯⣶⣶⣭⣭⣭⣄⠹⣇⢸⡿
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣸⣿⣿⣿⣿⠁⠀⠘⠘⠀⠀⠀⢠⡀⣸⠃⢀⣾⣿⣿⣿⣿⢷⣄⠀⣿⣿⣿⣿⣿⣿⡟⠀⣧⣿⠃
⠀⠀⠀⠀⠀⠁⠀⢿⢿⣿⣿⣿⣿⣿⣿⡇⢠⡆⠀⣀⠀⠀⠀⢸⡇⠃⠀⣼⣿⣿⣿⣿⣿⠀⢹⡗⢮⣿⣿⣿⣿⣛⣼⣰⣿⠃
⠀⠀⠀⠀⣀⠀⢠⡌⠘⠏⢿⣿⣿⣿⣿⣿⣼⣧⢰⣿⠀⡀⠀⠸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣠⣿⣶⣷⣿⣾⠍⡽⣭⣼⡏⠃
⠀⠀⢀⣼⢃⠜⠃⠀⠀⠀⠀⠻⡜⢿⣿⣿⣿⣿⣿⣿⣆⣧⠀⠀⠁⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠟⣻⣿⣯⣾⣷⣿⠟
⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠌⢿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣼⡟⣿⣿⣿⣿⣿⣿⡯⢰⠟⣛⢁⡟⠁⠁⢀⠀⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡀⡆⠀⣿⡷⣿⣿⣿⣿⢟⣿⢇⡴⠋⠀⠀⢀⣤⠞⣥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠟⣻⣿⣿⣿⣿⣿⣧⣿⠈⡟⣿⠯⣽⣿⣯⣼⡿⠋⠀⢀⣤⠾⠋⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣠⣾⡿⠟⣡⡾⠟⡍⢿⣿⠀⢳⣌⠙⠛⠛⠚⣹⢧⣾⡷⠟⠁⠀⠐⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠉⠀⠋⠉⠀⡠⠂⠘⢿⣴⣄⡉⠙⠛⠋⣡⣿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⡾⠛⢿⣿⣿⣷⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠸⠉⠁

⠀⠀⠄⠀⠀⠀⣤⡤⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠤⣶⠄⠀⠀⢀⠊
⠀⠀⠘⡂⠀⠀⠈⠀⠀⠀⠀⠀⠉⠁⠀⠀⠄⣀⠀⠀⠀⠀⣠⡴⣦⡄⠀⠀⠀⣀⡤⠔⠂⢀⣯⡄⡀⠀⠀⠠⠦⠀⠀⠀⡠⠃
⠀⠀⠀⠈⢆⡀⠀⠀⠐⠀⠀⠻⠱⡂⣄⡀⠂⠨⢉⣓⣼⣏⣥⣼⣧⣉⣾⣴⡫⢛⠀⠀⠰⠬⠛⠊⠀⠀⠠⠀⠁⠀⠠⠔
⠀⠀⠀⠀⠀⠱⠄⢆⠀⠀⠀⠀⣠⣀⣀⣀⣀⣈⡠⠾⣤⣘⣿⣷⣿⣟⣃⣽⠷⣤⢤⢄⣠⣤⣄⠀⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠀⠀⠀⠀⠙⢢⣿⣿⣿⡛⢻⣿⣿⣿⠂⠁⠉⠁⠀⠀⠀⠈⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣿⡿⣿⣿⣟⢿⢿⡋⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⢣⣿⠿⣿⡿⢿⡗⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣿⠛⠂⣽⡀⠙⡑⠏⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡉⡿⠿⢿⣋⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠑⣒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦⣬⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣆⠰⣶⡿⢧⣷⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣶⡶
⠀⠀⠀⠀⠀⠀⠀⢰⣏⣐⣿⣿⣿⣷⣾⣿⣿⡄⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠹⢿⣏⠉⢽⣿⠟⠁
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠙⢂⣠⣿⣿⣧⣀⠀⠀⠀⠀⠀⢀⣄⠹⣮⣽⣿⣷⣦⣄
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⡇⠀⠀⣠⣆⣇⠀⠻⡿⢋⣼⣿⣿⣶⣤⡄⣴⣿⣿⣷⣾⣹⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣇⣸⣿⣿⣿⣿⣿⡇⠀⣰⣿⣿⣿⣷⡜⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡁⣹⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣭⣻⢿⣿⣿⠀⣿⣿⣿⢟⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⡆
⠀⠀⠀⠀⣼⣿⣿⣿⣟⣷⣿⣿⣿⡏⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡟⠩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⠛⣻⣿⣿⣄
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣠⣹⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠈⠉⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠻⡏⠀⣠⣼⣯⠹⣿⠟⠛⠛⠛⠿⡟⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⣀⡟⠛⠛⠻⠿⡿⠿⠿⢿⣣⡿⠿⠛⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠻⠶⣷⣤⣭⡟⠻⠴⠟⠀⠀⠀⠀⠀⠈⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⢻⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⢣⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡙⠛⣿⣿⣿⣅⡅⠀⠀⠀⠀⠀⠀⡾⣹⣿⣿⣿⠟⣵⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣤⣌⢿⣿⣷⣷⣤⣶⣶⣶⣤⣴⣥⣿⣿⣿⣣⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣷⣿⣿⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣿⣿⣿⣿⣿⡿⠟⠿⢿⣿⣿⣿⣿⣿⣿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⣽⣿⣿⣿⣏⠀⠀⠀⠀⣹⣿⣿⣿⣿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⣿⠛⠛⠻⢿⣿⡟⣻⣿⣿⡿⠟⠛⠛⡞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣦⣀⠀⠀⢻⣿⣿⣿⠋⠀⠀⣠⣾⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡏⠛⠛⢿⣿⣶⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠉⠛⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠀⠀⠀⠀⠆⠠⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⡆⠀⠀⠀⠄

⣶⣾⣿⣿⠲⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⢒⣿⣿⣿⡿⢶
⢿⣸⣿⣿⣿⠳⠆⢹⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢻⣿⣷⠀⢘⣿⣿⣿⣧⡿
⠈⢳⣼⣿⣿⣷⣄⢾⣿⡇⠈⢻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠀⠀⣿⡿⢴⣺⣿⣿⣧⡟⠁
⠀⠀⢳⣹⣿⣿⣤⣀⠺⠧⠀⢸⣿⡟⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⢹⣿⣿⣄⡤⠘⠛⢻⣿⣿⣯⡾⠁
⠀⠀⠀⢻⣙⣿⣿⡙⠣⠀⣀⠘⣿⡏⠀⣸⣿⣿⣦⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣾⣿⣿⠀⠀⢿⣿⡁⢊⣤⣼⣿⣿⣿⣼⠇
⠀⠀⠀⠀⢿⣸⣿⣿⡟⠆⠀⠀⢹⣧⠀⣿⣿⡟⠙⢿⣦⣄⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⣠⡾⠋⠉⢿⣿⡄⢠⠞⠉⠀⠤⢌⣽⣿⣿⣥⡟
⠀⠀⠀⠀⠘⣏⢹⣿⣿⣦⠶⣬⠉⠉⠑⢿⣿⠁⠀⢰⣿⣿⣦⡘⠢⢌⠢⡀⠀⠀⡠⠞⣁⠔⣢⣾⣿⣶⠀⠀⠈⣿⡏⠀⠀⠀⡐⠶⣿⣿⣿⣇⣾⠁
⠀⠀⠀⠀⠀⠹⣞⠻⣿⣿⣇⣤⣒⣒⠈⣾⠗⠐⠀⣿⣿⣿⠙⣿⣦⡈⡆⢰⣿⣿⣄⡰⣡⣾⠟⢛⣿⣿⡂⠄⡀⡝⠁⠀⠀⢀⣠⣴⣿⣿⣿⣽⠃
⠀⠀⠀⠀⠀⠀⠻⣦⠿⣿⣿⣍⣉⠀⠀⠊⠁⠀⠀⣻⣿⠇⢐⣨⣿⣿⣮⣪⣽⣯⣿⣿⣿⣗⠀⠀⢹⣿⣇⠀⠈⠁⠀⣀⡐⢈⣹⣿⣿⢋⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣌⣿⣿⣿⠟⣃⠀⠤⠄⣰⣿⠟⠀⢀⡘⠿⢿⣿⣿⣿⣿⣽⣿⡛⠢⣤⣦⡾⠻⢿⣦⣤⣀⡀⢘⣻⣿⠿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠓⣒⣂⠈⠀⢀⢹⣧⠐⠂⢻⠿⣿⣿⣿⣿⣿⠃⠀⢸⣿⠇⢀⡀⢸⣷⣤⡉⠉⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡟⠠⠒⠃⠀⢻⡄⠀⡀⢠⢙⣿⣿⡿⠏⠂⠀⢸⡇⠀⠈⡝⠶⣿⣅⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⢛⣿⣏⡀⠀⢠⠀⠀⠈⣓⢀⠀⡆⣿⢿⣿⠐⣷⠈⠄⢸⠃⠀⠀⣴⠤⣾⣿⢿⡃⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⡟⡔⠉⢧⠀⠀⢾⠃⠀⠁⡇⣼⣿⠏⡙⡆⠀⣞⠀⠀⣠⡟⠐⢿⡇⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠈⣿⣿⣧⣶⡀⠠⠠⠀⢸⡇⢰⠀⡇⠹⡿⠀⡷⡄⠀⡇⠀⠺⠋⢤⣄⣼⡷⣾⠇⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣴⣿⣿⣿⣿⢀⣤⡀⡀⢸⡇⠸⠂⡆⠀⠀⢠⠟⡇⣸⡁⠀⣇⡐⠈⢹⡟⣴⣿⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⢹⣿⣿⣿⡿⠁⡀⢈⣇⠰⢓⠃⠀⠀⠘⢀⠁⣿⣧⡀⠈⢹⣿⣿⣧⠏⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⡿⣿⢻⣷⡿⢿⡿⢿⡿⣛⡃⠀⠀⢘⣛⣰⣿⣿⣿⣶⣟⣵⢾⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣌⢳⣟⣁⣾⣇⣰⣿⠉⠁⠀⠀⠈⢹⠋⡹⣿⠇⠸⣿⣠⣴⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠩⣿⡿⠛⠻⡟⣋⣿⠀⠀⠀⠀⠀⠸⣿⡻⣿⠟⢿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣶⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠙⢽⣿⣶⣦⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡤⣀⠀⣩⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠐⡦⢈⢰⢰⣴⣆⣬⠾⢯⡀⡠⠆⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣧⣄⠠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⡄⣿⣧⠀⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⡷⠀⠀⠠⠚⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢸⣿⣶⣿⣯⣅⣿⣿⣿⣿⣿⣿⡷⡾⡅⠀⣤⣄⡀⣼⠆⠀⠀⠀⠀⢯⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⢴⣿⣝⢸⣿⣯⣿⣇⠀⣿⣿⣿⣿⣿⣯⣄⣿⣭⡃⠟⠚⡇⠀⠀⠀⢀⠀⠀⢸⡅⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡘⠀⣿⠁⢸⣿⣧⣿⣿⣥⣿⣿⣿⣧⣟⡐⢤⠘⠿⢨⢉⡀⠀⡀⠆⡰⠃⠀⢀⣿⠀⠀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣀⣸⡇⣀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠉⢃⠛⠈⠉⠏⠀⠃⢸⣁⣰⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡁⠉⢹⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⣰⣶⣶⣄⠐⠀⠀⠀⢸⠀⢸⣿⣿⣿⡀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⠁⠀⣸⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡿⠀⣠⣿⣿⣿⣇⡐⡀⠐⢠⣾⠤⢼⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠈⡧⣿⣤⠀⢹⠂⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣤⣿⣿⣿⡿⡾⣤⣄⠈⠁⣿⣤⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡄⠀⠙⣿⣿⣀⣨⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⠘⡇⡿⣻⣾⣿⣬⣿⣿⣿⣿⠿⠷⠿⠿⠆⢐⣾⣿⣿⣿⣿⣷⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⠇⢹⣿⣿⣿⢿⣿⣿⣻⢿⡅⢼⣻⡿⣿⣟⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣟⣷⡆
⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣼⣟⣿⢸⣾⠿⠛⣿⣣⣻⠾⣿⣿⣿⣷⣼⣾⣤⣵⣥⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠁⠒⣼⣿⣿⣿⡈⣿⠉⠉⠛⠛⠛⠂⠻⠿⠚⣿⣿⡿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠁⠠⣸⣯⣿⣿⡿⣡⣿⣷⡀⠐⢶⡶⠖⢠⢤⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⣶⣿⣿⣿⣷⡼⡇
⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⠃⢠⢳⣿⡿⣿⣿⣿⣿⣿⢿⣿⣄⠀⠻⣿⣾⡀⢻⣄⠱⡀⠀⠀⠀⠀⠀⠀⠀⠡⣬⣿⣿⣿⣿⣿⡄
⠀⢽⣾⣿⣿⣿⣿⣿⣿⣿⢏⣀⣴⣿⣿⣇⣿⡿⢿⠛⣿⣾⣿⣿⣦⠀⠹⣿⢷⠈⣻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣄
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣏⣴⣿⣿⣿⣿⣿⣷⡀⢈⣿⣿⣿⣾⡏⣿⣧⡀⠈⢾⣵⣿⣿⣷⣌⠡⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣦⣄⡀
⠈⠿⠿⠿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⢻⣿⣿⣾⡶⠏⣿⣷⡄⠀⠉⠉⠉⠉⠙⠂⠈⢆⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠾⠟⠿⠿⣋⢳⣶⡟⣿⣿⣶⣶⣾⣾⣿⣶⣶⣶⣾⣷⣶⣶⣾⣿⣿⣿⣿⣿⠿⠻⠿⠛⠛⠛⠒
⠀⠀⠀⠀⠲⠤⠤⣀⡀⠀⠉⠉⠉⠙⠛⠛⠿⠟⠲⢿⣷⡀⢡⣶⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⡿⠿⠿⣿⣿⡿⠿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡉⢙⣫⣭⣍⣉⠀⠀⣴⣦⣤⡀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠶⠶⠦⣤⣤⣤⣔⠛⠾⣿⣿⣿⣿⣿⣷⠰⠿⢿⠿⠇⠀⠀⠀⠀⠀⠀⣀⡤⢤⠴⠖⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⡿⠭⠕

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⣶⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣄⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣧⡀⣶⣶⣄⠀⠀⠀⠀⠀⠀⢠⣶⣤⣶
⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿
⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠈⠁⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣶⣷⣦⣠⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠋⠙⠛⠿⠟⠛⠛⠛⠿⠿⢿⣿⠿⠋⠀⠀⠹⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⢹⣿⣿⣿⠏⠀⠈⢻⣿⣿⣿⠇⠈⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⢿⣿⡀⠀⠀⠀⢀⣻⣿⣷⡀⠀⠀⠀⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⠟⠷⠿⠿⠇⠀⠀⠀⠛⠛⠛⠛⠁⠀⠀⠀⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣠⣤⡀
⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⢿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢠⣿⣿⣿⣿⣶⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠰⣾⣿
⠉⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠈⠉
⠀⠀⠀⠀⣼⣿⣿⡟⠻⣿⣿⣧⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⣦⡀⢸⣿⣿⣿⡿⠃⠀⠙⢿⣿⣿⣿⣿⣿⡇⠀⠉⠛⠿⣿⣿⣧
⠀⠀⠀⣸⣿⣿⠏⠀⠀⠈⠻⣿⡇⠀⠀⠀⠀⠘⣿⣿⣿⠉⠙⢻⣿⡇⠈⣿⣿⡟⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣇⠀⠀⠀⠸⣿⠿⠁
⠀⣤⣶⣿⡿⠁⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠠⣴⣿⣿⠏⠀⠀⠀⠻⠃⠀⠙⠿⠿⠂⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⣴⣦⣀⣤⣤⣠⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠢⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⣲⡾⣷⡐⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⣴⡷⣆⡠⣯⣠⣿⡿⢀⢠⡄⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣝⣿⣿⣷⣤⠉⢭⣿⣿⣿⣷⣾⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⢈⣼⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⣽⣷⠽⣿⣿⣯⣥⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣂⣼⠟⠉⠈⠈⠁⣲⣷⠊⠈⠈⠩⠿⢷⣙⠷⠽⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⣼⢿⠻⣷⣦⣤⢽⣿⣷⡤⢠⣴⡾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⢀⢴⣷⣶⣤⣌⣥⣙⡋⣹⣿⣿⡇⣀⣈⣢⣽⣽⣷⢺⣓⠄⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠐⠋⢘⣋⣁⠉⠉⠋⠉⣿⣿⣿⡟⠻⠛⠛⢙⣿⣿⡇⠈⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠉⣋⣛⣛⣻⣿⣿⣿⣿⣟⣛⢛⣛⠿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⢿⠉⠉⢫⣟⣿⣿⣿⣿⣿⣟⣉⢛⣿⣿⣿⠅⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠸⣿⡟⠙⠉⢿⣿⣿⣿⣿⣛⡻⢻⣿⣿⠋⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢺⣿⠛⣻⣿⣿⣿⣋⣝⣛⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢯⣽⡿⣷⣶⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠅⢀⠔⠁
⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠠⡀⠀⢐⢐⠀⠀⡠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣄⢠⠂
⠀⠀⠐⡖⢢⠀⠀⠀⠰⡀⠀⠈⠀⠀⠀⠀⠊⠀⠀⠃⠀⢀⡰⠀⠄⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠠⠀⢀⡀⠉⡃⠁⣀⠀⠀⠀⡐⠀⠀⠄
⠀⠀⠀⣀⣀⡉⠤⡠⠀⠁⠀⠀⠀⠨⠀⠀⠀⠀⢘⠀⠤⠁⠀⠠⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠚⠬⡀⠀⢁⠐⠒⢈⠀⠀⢀⠀⢅⢋⣀
⠀⠀⠀⠀⠀⣀⣀⣀⠀⠒⠄⠀⡀⠸⠐⠀⣄⠠⠉⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠗⢄⡐⢖⣀⠀⠳⢌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡐⠂⢨⢣⠣⣉⠐⠒⠀⠁
⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⢀⢰⠆⠀⢀⠐⠍⢀⠄⡀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⢄⡠⠌⠂⡠⠈⠁⢈⠑⠶⡀
⠀⠀⠀⢀⣀⠀⠀⢀⡀⠘⠉⠀⠈⠀⢨⠀⠂⠀⠀⠀⢠⣦⠄⠤⠄⠀⠀⠀⠀⠀⠀⣀⠄⠘⢲⠃⠀⠀⠀⢘⡄⠄⡘⠀⠠⠂⠤⠁
⠀⠀⠀⠀⢀⣠⠄⠉⠁⢠⠀⠀⠀⠀⠰⠀⠀⠀⠀⢆⠀⠠⠈⢃⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⢔⡢⡀⠀⠀⠀⠀⠀⠈⠒
⠀⠀⠀⠔⠀⠁⢠⠃⠀⠂⢠⠔⠁⠀⢠⠀⠐⢄⠀⠘⠀⠀⠳⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⣉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢀⠁⡀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢾⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠙⠀⠰⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣄⠀⠐⠀⠐⣦⡀⠀⢹⠀⠀⠀⣐⠂⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠠⡀⠠⠊⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⢦⣀⠀⢠⠈⠲⡟⢄⡀⢀⠃⠀⢀⣈⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠈⢆⠈⠐⢀⠴⠁⠀⠈⢀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠙⡛⠈⠁⠀⠁⠘⠀⢚⠶⡾⠋⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⢀⡆⠙⠰⠁⡈⠀⠶⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠖⡊⠉⠄⠈⠠⠂⠄⠀⡠⠒⠀⢀⣀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⡈⠉⠀⠀⠈⠁⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣀⠀⠀⠀⣠⣭⡄⢂⠀⠀⠁⠀⠀⠃⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠐⠂⠀⢀⡀⠀⠏⠀⢀⠀⠫⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠚⠉⠁⠎⢈⠇⠀⢠⠁⠂⠞⢁⣄⠀⠀⠤⠅
⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢠⠔⠉⣈⠀⢤⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠚⠊⠀⡀⠀⠀⡸⠂⠁⠠⣦⠀⠀⢬⠀⠀⠀⠃⠤
⠀⠀⠀⠀⠀⠉⠈⠀⠀⠆⠀⢁⡀⠀⠸⠀⡀⠀⠰⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⡟⢀⠀⠀⠁⠀⠘⡄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣤⠄⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠞⠀⠼⠈⢆⠒⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⠛⠛⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⢼⡿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠻⠃⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠿⢿⣶⣀⣿⡀⠀⢀⣀⣠⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠉⠉⠙⠐⢿⣿⠷⡶⠟⠋⣸⡇⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣠⣾⠃⠀⡤⣀⠀⠀⠀⠘⣿⣥⣤⣴⣾⡿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠤⠰⢤⣼⠂⠀⠀⠀⠀⠙⣏⢀⠘⠃⢼⠀⠀⠀⠀⡿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠐⠀⠀⠀⠀⠠⠖⣤⠄⠀⢰⠃⠙⣍⣻⠷⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢠⣤⣤⣀⡀⠀⠀⠀⠈⡇⠀⠀⠰⣦⡄⠠⡶⠿⠿⢷⣤⡁⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠈⠛⢿⣿⣻⣷⣤⡀⠀⢿⡄⠀⠀⠀⠀⢀⣠⣶⣾⣿⡿⠁⢠⣾⣽⢿⣄⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⣾⣿⣿⣿⠂⠀⠁⠀⡀⠔⠖⠸⣿⣿⣿⣿⠶⣾⣉⠙⠡⢆⣿⣆⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠈⢿⣿⣿⡏⠁⠐⠀⠀⠀⢀⡄⠀⠀⠀⢿⣿⠛⢦⣹⣿⡇⠲⠚⠁⠁⠀⠀⠀⠀⠀⠠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠈⠈⣩⣤⣤⠶⠶⠃⣼⣿⡶⠦⣄⣨⣙⢿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⣼⠛⠉⠁⠀⠀⠀⠜⣿⡇⠀⠈⠙⢿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⢀⣿⡆⠠⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⠠⠀⠉⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠿⣷⣶⣄⠀⢀⣠⣿⣿⡷⣼⣆⣀⣠⣿⣿⡿⣆⠀⠀⢘⡂⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⠟⠻⣽⠀⠑⣤⡝⠛⠙⠛⠋⣿⣿⣷⣤⡤⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⣄⣸⣷⡶⣀⣤⠀⢻⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⣐⣦⠀⠀⠀⠀⠀⠠⣦⣾⣿⣿⡿⣿⣷⠙⠋⠀⣸⣿⣿⣣⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢨⣿⣿⣟⣛⣳⣿⣿⠀⠀⠀⢹⡿⣮⠍⠀⠀⠀⠢⠄⢀⠄⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢤⡤⠀⠀⠞⠉⠉⢿⠿⠿⠩⠁⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠠⠄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣦⣤⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⠁⠀⠀⠀⠀⠀⠀⣠⣷⡶⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠚⢆⢀⠘⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢀⣶⣖⣲⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣟⣿⣿⣶⣾⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⢻⡿⠉⠉⠛⠿⣿⣄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠘⠷⠶⠀⠛⠀⢹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⣔⣶⡀⠀⠀⠀⣸⡟⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢶⣄⡄⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣦⣀⠄⠀⢀⡀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠂⣠⠉⠤⠠⢾⡟⠋⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡅⣠⠂⠀⠠⣄⠀⢸⣿⠀⠀⡙⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠂⠘⢧⡀⠀⠉⠄⠈⣿⡇⠘⠟⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡷⠀⠀⠘⢿⣄⡀⠈⠀⣿⣿⢠⡀⠀⢻⡄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢠⠞⠓⢶⣬⣿⣿⣿⣷⣿⣿⣿⡟⣀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⠃⠀⠀⠁⠀⠀⣀⠉⣹⠟⠛⠋⠘⠙⠃⢹⣇⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⠀⢀⡞⠁⢰⠃⢤⠀⠀⠁⠀⣀⡀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⣦⠀⣀⡴⠀⡀⠀⠀⡀⠀⢠⣿⡄⢀⡀⣿⣿⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⣿⠛⠀⠞⠋⠀⣀⠁⠀⢸⠛⢲⣿⣿⡟⣿⡁⠘⠟⢠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣻⣶⠁⢀⣴⠋⠀⠈⣿⠃⢀⣿⠀⢀⠸⣿⡆⢿⣇⢀⣀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠿⠋⣠⡾⠁⠀⢀⣸⣿⠀⣼⣷⠀⠨⢀⣿⡇⠼⣿⡌⠻⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣟⣋⣠⣴⣾⠟⠁⠀⣀⣦⡀⠄⢠⣿⣿⠀⠀⠃⣿⣇⠀⣻⣿⠰⡧⠄⠈⣿⣧
⠀⠂⠀⠀⠀⠀⢀⡀⠀⢰⡀⣰⣿⣿⣯⣉⣽⣿⣿⡥⠀⣛⣨⢠⣸⠄⢂⣿⣯⣿⡇⣧⠸⢿⣿⠀⡻⣿⣆⠀⠘⢀⣿⣿⡆⠀⢄⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣼⡟⣿⣯⣭⣤⣉⣿⠃⠀⠀⣤⠘⠷⠀⠀⣸⣿⣽⣿⠀⠀⡀⠸⣿⡀⠙⡩⣿⣆⢠⣿⡔⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡼⣰⣿⣿⢯⣴⣦⣼⠟⠃⠀⠀⢀⡀⣤⠀⠀⢴⡟⠉⠝⠁⠀⠀⠤⠀⣿⡇⠀⢤⠹⣿⡄⠘⢧⣈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⣿⣿⣟⣷⠟⢁⣀⡀⠀⠈⣮⣷⠏⠀⢰⣦⣤⢠⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠓⠹⣿⡄⠈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⠟⣡⣶⠿⢁⡶⠾⣛⠋⠁⠀⢠⣿⣿⠃⢸⠀⠀⠀⢠⠀⠀⣿⣧⠀⠰⠆⠀⢸⣿⣆⠈⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⣡⡾⠋⠁⠀⠈⢦⣴⢿⣀⣠⣾⡿⣿⣿⡄⠈⠀⠀⠀⠀⣶⠀⣿⣿⠀⣧⠀⠀⠸⣿⣿⣦⠸⣿⣿⡄
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⢋⡀⠠⠀⠀⢠⡾⠛⠇⣈⣽⣿⠇⡿⠉⣧⡆⠀⠀⠀⠈⣙⠀⣿⣿⠀⣿⢿⣄⠀⢹⣿⣿⣷⡝⣿⣇
⠀⠀⠀⠀⠘⣷⣿⣛⡟⣋⢶⣿⢧⠀⠸⠟⠉⢁⡴⣾⣿⠟⣿⠀⣧⠀⢹⡇⠀⠀⠀⠀⠃⠀⣼⣿⡀⣿⠚⢿⡶⠘⢛⣻⡙⢿⣿⣿⡄
⠀⠀⠀⠀⢀⣼⣿⡟⣧⣿⠂⣝⠂⢶⠀⠂⠀⣠⣼⠟⣹⣧⠁⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⣿⣿⡇⢩⠀⠀⠀⠀⣙⣿⣇⠈⣿⣿⣷
⠀⠀⠀⠀⢸⣿⣟⠾⣿⣿⢛⣿⡀⠃⢀⣴⣿⢋⠀⣾⡇⢻⣦⢘⣴⣄⣉⣄⠀⠀⠁⠀⠀⠀⣿⢿⡇⢸⣀⠀⠐⠈⢿⣿⡯⢶⡌⢻⣿
⠀⠀⠀⢰⣯⡙⣿⣶⠌⠹⣿⣥⡾⣣⣿⣿⡇⠸⣧⠘⠷⢀⣿⣿⠙⠣⠉⡙⠇⡀⢀⠆⠀⠀⣿⢸⡇⠛⣿⣄⠀⢼⡆⠈⣵⣿⡇⠈⣩⡾
⠀⠀⠀⣾⣿⣷⣄⣉⣀⣘⣍⠁⣼⣟⣿⣿⡅⢦⣙⢷⣄⣸⣿⣿⣷⣀⠀⠰⠶⠟⢻⠿⠦⠀⣿⠀⡇⠀⢾⣿⣧⠄⠀⡐⠋⢉⣅⠀⢹⣧⡀
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣟⠹⣧⠈⢳⣿⣿⠃⠘⢿⣾⣿⣿⣿⣿⠇⣿⣦⣀⠀⠀⠛⠂⣾⠀⣿⠀⣷⠀⠛⠾⠁⣠⠞⠁⠀⢸⣷⠀⠸⣧⣁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠻⠂⣼⣿⣿⡆⠀⠀⠙⣿⣿⣿⡿⢠⣿⡟⠙⢷⣦⣄⣀⠀⠀⣿⠀⣿⠀⣀⣤⣾⠃⠀⢀⡠⢿⣿⣤⠀⣿⣿
⠀⠀⣾⣿⣿⣿⣿⠿⠋⠈⠳⠀⠐⣿⣿⣿⣿⡄⢀⠀⠹⣿⣿⡇⣸⡟⠁⠀⠉⠉⠉⠙⠛⠶⠿⠀⠹⠟⠋⢀⣾⡇⠐⣃⠀⢸⣿⡃⠀⢿⣿⠄
⠀⠀⠙⠛⠛⠻⣧⣤⣤⣄⣀⣀⣴⣿⣿⣿⣿⣧⡀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠐⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⡇⢠⠍⠀⢸⣿⠀⠀⢸⣿⢦
⠀⠀⠀⠀⠀⠀⠘⠛⠻⠿⣛⣿⣿⣿⣯⣿⣭⣿⣟⣀⣤⣾⣿⣿⣶⣶⣦⣤⣀⣈⡁⢀⣀⣀⡀⠀⠀⣀⣀⣶⣿⣧⣀⣀⣠⣼⣿⡤⠴⠛⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣬⣿⣟⣻⡿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠉⠉⠉⠙⠛⠛⢶⣾⣿⣿⡿⠿⣻⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠉⠀⠀⠀⠠⠤⠄⠀⠀⠀⠉⠀⠛⠛⠛⠛⠛⠛⠻⠶⠿⠓

⠀⠀⣀⢀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠈⠏⠀⠑⠀⢀⠀⠔⠁⠀⠟⠀⠀⠘⣇⢾⣶⢀⠀⠀⢠⢰⣾⢸⡏⠀⠀⠨⣯⢥⠆⠀⢀⠀⠠⣮⣹⠅
⠀⠀⠲⠦⠤⠤⣽⡧⠤⠤⠴⠃⠀⠀⠀⠘⢦⡜⠛⢲⣷⠚⠳⣴⠟⠀⠀⠀⠀⠸⡮⠂⢒⣿⡖⠐⢳⠏
⠀⠀⠀⠀⠠⠀⠿⠇⠠⠄⡀⠀⠀⠀⠀⠀⣿⣇⠓⠏⠛⠗⢢⣿⠇⠀⠀⠀⠀⡖⣾⠣⠀⠘⠀⠜⢿⡶
⠀⠀⠀⠨⠤⠀⠀⠀⠠⠤⡁⠀⠀⠀⠀⠀⢩⠿⠛⠁⠀⠙⠿⢏⠀⠀⠀⠀⠀⠤⠵⠎⠀⠀⠀⠸⠮⠠⡀
⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⣡⣤⡤⠀⢀⠀⢨⣀⡀⠀⠀⠀⣀⡌⠀⢠
⠀⢹⣯⠁⠑⢄⠀⢀⠔⠁⠩⣿⠁⠀⠀⢳⣦⠁⠈⣲⣶⠈⠀⣡⡾⠃⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⢿⡤⠬⠤⣿⡧⠤⠥⢼⡏⠀⠀⠀⠀⢹⠋⠩⠸⡿⠍⠙⣿⠁⠀⠀⠈⠲⡀⠀⠀⠀⣿⡇⢀⠀⠀⠖
⠀⠀⢽⣀⠀⠂⠾⠇⠒⠀⣸⡇⠀⠀⠀⠀⠈⢧⣤⠆⠀⢦⣴⠃⠀⠀⠀⠀⠀⠀⠊⠀⠀⣛⠀⠀⠉
⠀⠀⠈⠛⠛⠂⠀⠀⠚⠛⠋⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⡀⠀⠀⠀⡀⠀⢀⡄⠀⠀⡿⣖⣤⠀⠀⠀⠀⠀⢤⣲⢿
⠀⣇⡈⠒⠒⠀⢸⠆⠐⠒⠊⣀⡵⠀⠘⢀⣀⣀⡈⣰⣷⡈⣀⣀⣀⠈⠀⠀⢻⠉⠈⠈⠣⣀⠜⠁⠁⠉⡝
⠀⠈⠒⠀⠊⠁⢸⡀⠉⠒⠐⠚⠀⠀⠀⠠⠊⢀⠀⣸⣿⡀⢀⠈⠂⠀⠀⠀⠀⡜⠉⠋⠙⣿⠋⠙⠉⢳⡀
⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠐⠀⠰⠀⠀⠀⠀⠀⢧⣀⣀⠘⠙⠁⣀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⣀⣀⠉⠉⠀⠀⠀⠈⠉⢀⣀
⠰⡇⠈⠢⡀⠀⠀⠀⠀⡠⠊⠁⡿⠀⠀⣿⡏⠈⠀⠀⠀⠀⠈⠘⣿⠇⠀⠀⢿⣿⢯⡦⡀⠀⢀⠴⡽⢿⡿
⠀⠘⠂⠀⠈⠀⢠⠀⠊⠀⠀⠒⠀⠀⠀⠘⢷⠤⠀⢰⣶⠀⠠⢴⠏⠀⠀⠀⠈⢿⣦⠹⢿⣶⡿⡏⣵⡿⠃
⠀⠀⠀⡠⠂⠀⢾⡗⠀⠰⠀⠀⠀⠀⠀⠠⡇⠀⠀⡘⠏⡀⠀⢀⡇⠀⠀⠀⠀⠀⣿⣄⣰⢿⣧⣠⣿
⠀⠀⠈⠇⣀⠀⠘⠁⠀⡀⠘⠀⠀⠀⠀⠀⠙⠻⠋⠀⠀⠙⠻⠛⠁⠀⠀⠀⠀⠀⢙⠿⠓⠀⠚⠿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠃

⠀⠀⡠⠀⠠⠄⠀⠀⠐⠀⠁⠀⠉⠉⠉⠝⣂⡀⠠⠶⡒⢢⣤⣀⣀⣀⢀⡀⠀⠀⢀⣠⣄⠀⣀⣀⣀⣀
⠀⠈⠀⠚⣦⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⣶⠈⠆⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⢳⡦⢄⡀
⠀⠀⣠⠐⠅⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⢠⡀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠸⠀⠀⠀⠀⠀⠶⠄⠀⠀⠀⠰⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠰⡟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙
⠀⠀⠰⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠱⣀⠀⠀⠀⢠⠀⣶⣶⡄⢠⠀⠀⠀⢃⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠘
⠀⠀⠀⠀⠈⠃⠀⠠⠈⠀⠈⠀⠁⠘⠂⠠⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⡷⣀⠳⠀⠘⠛⢷⡃⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠇⣝⠀⣆⠀⠱⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠁⢨⣄⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠠⣼⠿⣦⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⠀⠀⠄⠀⠀⠀⠀⠀⠀⢙
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⠀⠀⠁⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⢘⠀⠸⡕⢀⣦⢄⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠐⠀⣀⡄
⠀⠀⠀⠀⠀⠀⢀⢠⠀⠛⢹⣿⡷⡟⠙⠋⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⠊⢀⡀⠀⣆⡜⠉⠙⠓⠦⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠸⠀⠀⡘⢣⣯⢛⠇⠀⠀⠀⠀⢓⡤⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⢠⡄⠤⠀⢀⠎⡾⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠈⠒⣼⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢀⠀⢀⠃⠄⠁⠀⠀⠀⠁⢸⠇⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡀⡐⠇⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠆⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡠⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢠⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠁⠊⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⡘⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⠞⠀⢠⣤⡀⣆⠀⠀⠀⠀⢰⠃⠀⠀⠀⢀⡞⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑
⠀⠀⢀⡠⠖⠋⠀⡔⢸⠼⠃⠀⠀⠀⠀⡰⠛⠀⠀⠀⢠⠊⠀⠀⠀⣀⡤⠊⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡎⢡⠀⠀⠀⠀⡀
⠀⢰⢣⢄⠦⢠⢰⡆⠰⣘⣼⣧⡤⠖⠋⠀⡀⠀⠀⠀⡎⠀⠀⡤⠚⠉⠑⠤⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡁⠀⠀⠀⠀⠀⠸⡇
⠀⠈⠀⠈⠀⠒⠀⠒⠉⢉⠋⠟⠈⡥⠂⠀⠀⠀⠀⠀⡇⠀⠒⠐⠀⠄⠴⢦⡄⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⡎⠤⠀⡆⠐⢠⣤⣤⡴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⣿⣦⡤⠬⣉⠛⠷⢦⣄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣤⠶⠟⢋⠩⠤⣤⣾⡇
⠀⠀⠀⠀⠀⠹⣿⣿⣦⣤⠀⠁⠀⢬⡙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⣩⠄⠂⠁⢠⣤⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⢻⣿⣶⣽⣷⣤⣀⣀⡀⠁⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠁⣀⣀⣠⣄⣿⣵⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠈⣿⣧⣴⡿⣷⣬⡹⢟⡶⣄⡀⠉⠊⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠊⠚⠁⣀⣴⣞⠿⢩⣔⡿⣷⣤⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠸⣎⠣⣉⣉⣹⣿⣤⣭⡻⢿⣷⢤⡈⠘⢷⡀⠀⠀⣀⣄⠀⠀⡰⠞⠂⣠⢴⡿⠿⣫⣤⣜⣿⣉⣉⡡⢋⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠉⢻⣙⣭⡵⢾⠟⠷⠮⣝⣿⣷⣤⡙⢷⣤⣿⣿⣧⠟⣡⣴⣿⣟⡫⠶⠟⢿⠶⣬⣝⣹⠋⢡⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠃⣀⠐⡐⢒⣿⣿⡏⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣏⡉⣿⣿⣟⠒⡐⢀⡀⠃⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⡀⠀⠈⠉⠁⣀⣀⣉⡉⠤⠼⠵⣿⣿⣿⣿⣿⡧⠹⠤⠌⣉⣁⣀⡀⠉⠉⠀⠀⣀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⣩⣾⠋⢁⡀⠄⠒⠊⢁⣴⣿⣿⣿⣷⣄⠉⠐⠢⠄⡀⠉⢻⢦⡉⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠶⠠⠊⠀⠀⢀⠔⣿⣿⢱⠁⢹⢻⣿⡗⢄⠀⠀⠈⠢⡰⠎⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡰⠆⠁⠀⠀⡠⠊⣰⣿⣿⢸⠉⢹⢸⣿⣿⡀⠢⡀⠀⠀⠁⠶⡈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠰⠂⢱⠀⠘⠁⣰⣿⣿⠏⠾⠒⢺⠈⢿⣿⣷⡀⢹⠀⢰⠁⠲⠈⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡴⠉⠁⠀⠀⢠⣿⣿⠏⠀⡶⠤⢾⠀⠈⢿⣿⣧⠀⠀⠀⠙⠱⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⡀⠀⣢⡿⠛⠁⠀⠀⢷⠲⡎⠀⠀⠀⠙⠻⣧⡁⠀⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡒⢆⣤⣐⢒⠂⣠⣺⣟⢙⣷⣷⣶⢶⣤⠀⠀⠀⡀⣤⣴⣶⣞⡟⡟⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠲⠋⣻⡉⠁⠚⢀⠛⠀⠀⢦⣁⠛⢓⣼⣿⣯⣿⣷⡿⢿⣻⡍⠂⠀⣰⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⠐⠛⢻⠶⣯⣯⡀⣼⡤⣶⣽⡿⣿⣷⡝⠻⣉⣵⣿⣟⢹⣧⡗⢡⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠃⠈⠓⠛⠛⠛⠀⢒⣾⠓⠖⠚⢿⠝⣪⠟⠛⠘⠷⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠠⠄⠂⠛⠀⠀⠀⠀⠀⠀⢀⡴⠋⣇⠀⠀⠠⣿⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠺⣄⠀⠈⠙⠆⠘⠉⢿⣿⠚⣖⣖
⠀⠀⠀⠀⠀⠀⠀⢉⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃
⠀⠀⠀⠀⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣸⣶⣷⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⢀⠀⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⢀⠀⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡄⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠴⡈⠘⠉⠁⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠀⢀⡁⠀⡀⠀⠀⠀⠀⠀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠙⠋⢁⣰⠇⣀⠀⠀⢀⡄⢰⡇
⠀⠀⠀⠠⠀⠠⠤⡀⢀⡘⠒⠛⠋⠁⠀⠙⣿⠳⠁⠀⢸⡇
⠀⠀⠀⠀⢀⠀⢶⣛⣿⠁⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⢸⡷⡄
⠀⠀⠀⠀⠀⠘⠛⢛⡉⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⢸⡗⢃⠀⢀⠔⡦⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠘⣿⣿⠇⠀⠀⠀⣴⣿⡿⠀⠀⠀⠀⠀⣿⡄⢹⡠⠊⢃⡇⠀⠀⠘⢿⣷
⠀⠀⠀⠀⢀⣤⣿⣿⠀⠀⢠⣾⣫⡿⠁⠀⠀⠀⠀⠀⠧⢻⣿⣶⠶⠟⠀⢠⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠸⣿⣿⠇⠀⠀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⠀⠀⠠⡀⠀⠀⢹⡆
⢀⠀⠀⠀⢀⣿⣯⠀⠀⣠⡿⣽⠃⠀⠀⠀⡀⠀⠀⠀⣦⡤⠀⠀⠀⠀⢀⠀⠄⠀⠀⢀⠀⢣⡀
⠀⠀⠀⠀⣸⣿⠃⠃⣴⠛⢠⡏⠃⠸⠀⠀⠓⠀⠃⠨⠉⠘⠂⠀⠀⠓⠀⠂⠘⡄⠀⠊⠢⠀⢷
⠀⠀⠀⠀⣿⡿⠀⣼⠃⠀⡞⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠈⠆
⠀⠀⠀⢰⣿⠇⣰⡿⡎⣼⢁⠃⢠⠇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠘
⠀⠀⠀⢸⣿⣄⣿⢡⢠⡿⣤⠀⢸⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢸⡀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⢀
⠀⠀⠀⢸⣿⣿⣯⡏⡿⠀⠜⢠⡆⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢇⠀⠀⠈
⠀⠀⠀⡈⣫⣽⡭⢀⠇⠀⠀⣼⠀⠀⡠⠀⠀⠀⠀⠀⢸⠰⣀⠀⠀⠃⠀⢠⠶⢄⡀⠈⠀⠀⠀⢠⡀
⠀⠀⢠⢸⣿⢿⣧⣾⣶⡄⠀⣇⣀⣴⠛⣶⢄⡀⠀⠀⣿⡆⢋⠳⠢⣴⠠⡆⢃⠀⠀⠚⡶⠂⠃⠀⠀⢄
⠀⠀⢸⣎⠻⣷⣿⣿⣿⢏⣴⣾⠁⣿⠀⠇⠈⣿⠙⠚⣻⡇⢸⠀⠇⢛⠳⣹⡜⡆⠀⠀⡀
⠀⠀⢸⣿⣷⣦⣽⣿⡶⢸⡏⣿⠀⣿⠀⠀⠀⢿⠀⠀⢹⢻⠀⡇⠀⠈⡄⢸⣧⢰⠀⠀⢄⠀⡀
⠀⠀⣸⣿⣿⢿⡏⢸⠇⣼⠷⡏⢐⡇⠀⢠⠀⢸⠀⠀⠈⢸⡄⣿⡆⠇⠀⠀⢿⠀⠂⠀⠈⠀⠁
⠀⠀⢿⣿⣄⡀⠀⡟⠠⠋⠸⠃⢸⠷⠀⠿⠂⠈⠁⠰⠀⢸⠄⢸⠀⣀⠀⣀⣼⣦⣤⣄⣤⣤⣤⣴⣶⠖
⠀⠀⢻⣿⣿⣿⣿⣿⣷⣾⣶⣶⣷⣴⣴⣶⣴⣾⣿⣷⣶⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠁
⠀⠀⠀⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯
⠀⠀⠀⠀⠹⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠉⠉⠉⠉⠁⠉⠁

⠀⠀⠀⠀⠀⣠⣤⣠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⣠⡿⠿⢿⣆⠀⠀⠀⠀⢀⠀⠀⠀⠀⣴⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣴⣆
⠀⠀⠀⠀⠘⠙⠛⠋⠃⠀⠀⠀⢰⡟⣇⠀⠀⠠⣿⢿⣿⣇⠀⠀⠀⣴⡀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠻⠟⠀⠀⠘⠍⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⠇⠀⠀⠂⠘⠺⠛⠁⠂⠀⢀⣿⡇⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⣀
⠀⠀⠀⢀⣾⣿⣖⣀⠀⠰⣦⣀⡀⡈⠀⣀⠀⠀⣄⢀⣀⡀⠀⠀⠎⣿⠡⠀⠀⢠⣀⡄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢀⣽⣽⣯⠀⠀⠀⠀⢙⡟⡡⠀⠈⠀⠀⡏⠀⠀⠁⠀⠀⠀⠉⣀⣀⠤⣰⠚⣦⣄⢄⡀⠀⢸⣿⠀⠀⠀⠀⠷
⠀⠀⠀⠀⠸⣿⠃⠁⠀⠀⠀⣾⣾⣄⡾⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢁⣰⣛⣶⣻⣷⡪⠎⠀⠀⠿⠀⠀⠀⠀⠖
⠀⢀⣤⣀⠀⠀⠀⠀⣀⣤⡄⠘⠾⠟⠃⠀⠀⠀⠀⠀⠀⢀⢶⠀⠀⠀⠀⠘⠧⠵⠻⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⢻⣜⣇⣤⣤⣮⣛⡿⠁⠀⠀⠀⢠⣴⡄⠀⠀⠀⠀⠸⡘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢳⡄
⠀⠀⠀⣟⠏⡽⠿⡩⢝⡇⠀⠀⠀⠀⡧⢻⢜⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⢰⠀⠤⠀⠀⠀⠀⣙⣛⡁
⠀⠀⠀⠉⠋⠀⠀⠘⠋⠀⠀⠀⠀⠀⠙⠻⠊⠀⠀⢀⣠⡀⠀⠀⠀⠀⠁⢹⠑⠀⠀⠀⠀⠀⠲⠷
⠀⠀⠀⢒⣤⡀⢤⣤⣤⣤⣴⣿⣤⣤⣤⡤⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⣾⡂⠀⠀⠀⢀⣀⠀⠀⠀⢀⠀⠈⠓⠒⠿⠒⠈
⠀⠀⠀⠀⠉⠇⠐⠒⠿⠿⠟⠟⠿⠿⠟⠒⠀⠀⠀⣿⣿⢿⡆⠀⠀⠀⠀⡟⡇⠁⠀⠀⣸⣿⠆⠀⠀⣿
⠀⠀⠀⠀⢠⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡾⠂⠀⠀⠀⠀⢧⠇⠀⠀⢀⢻⣿⠇⠀⠀⠘
⠀⠀⠀⠀⠀⠉⠚⠃⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋⠈
⠀⠀⠀⣠⣤⣄⠀⡀⢀⠀⢀⠀⢀⠀⠀⣀⢀⠀⠀⡀⣀⢀⣀⠀⢀⡀⢀⣀⢀⢀⡀⡀⠀⢀⣀⠀⡀⣀⢀⡀⠀⢀
⠀⠀⠀⢿⣿⡿⠀⠇⠸⠀⠞⠆⠨⠳⠀⠷⠸⠀⠀⠧⠜⠸⠿⠄⠜⠧⠘⠾⠂⠠⠙⠆⠀⠸⠽⠀⠧⠀⠘⠤⠆⠬⠆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠿⠾⠻⠇⠀⠸⠻⠿⠺⠇⠀⡼⠟⠻⠿⠷⠀⠸⠏⠞⠿⠆⠀⠟⠻⠷⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠂⠀⠀⡠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠐⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠤⠠⠴⢃⡞⠀⠀⠐⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⣀⢒⢲⠌⠃⠀⠀⠀⠀⠀⠀⠀⠘⢀⣤⡀⠀⢀⠀⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠁⠈⠀⣁⣀⣤⣤⣤⣤⣤⣶⣶⣾⡟⠁⠹⣷⣶⣶⣦⣤⣤⣤⣤⣄⣀⡖⠚⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⣀⣶⣟⡿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣧⠀⢠⣿⣿⣿⣿⣿⣯⣿⣿⣟⢿⣻⣶⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⣼⣿⡿⣟⣻⣯⢿⢿⡿⣿⣷⣿⣿⣿⣿⣆⣼⣿⣿⣿⣿⣿⣿⡿⢿⣽⣟⣿⣿⣿⣧⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⢻⣿⣿⣿⣾⣿⣿⣫⣪⣾⣿⣿⣿⢿⣿⣿⣿⣿⢿⣿⣿⣷⣗⣝⣿⣿⣿⣿⣿⣿⡟⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠙⠻⣿⣿⣿⣾⣿⣿⣿⡿⠛⠉⠛⣿⣿⣿⡿⠋⠉⠙⣿⣿⣿⣷⣿⣿⣿⠿⠋⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⢀⣠⣠⠉⠉⠩⠹⣿⣿⣇⠀⠑⠀⣽⣿⣿⣇⠀⠁⠀⣽⣿⠟⠋⠉⠉⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⢀⡰⠶⢓⡤⠀⠀⠈⠛⢿⣷⣶⣾⣿⡟⠉⠙⢷⣶⡿⠟⠋⠀⠀⡩⣍⡛⡄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠈⠉⠙⠁⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠊⠆⡉⠈⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⣀⠤⠤⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤
⠀⢣⠀⠀⠀⠀⠈⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠁⠀⠀⠀⠀⡀
⠀⠀⢑⡆⠀⠀⠀⠠⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⢐⡊
⠀⠀⠈⢷⡀⠀⠉⠀⠈⠦⣀⣙⢦⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣶⠆⠀⠀⠀⠀⠰⣾⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⡴⣋⢀⠴⠁⠀⠉⠀⢀⡾⠁
⠀⠀⠀⠈⢿⠤⠀⠲⠦⣄⣘⠛⢣⡷⣄⠀⠀⠀⢠⠞⠀⠈⠉⠑⣤⣶⣶⣶⣶⡴⠊⠉⠁⠀⠳⡄⠀⠀⠀⣠⢾⡜⠻⣓⣠⠴⠖⠀⠠⡿⠁
⠀⠀⠀⠀⠘⣀⣀⠀⠀⠀⠀⠤⡆⠐⢌⢣⡀⠀⠸⣄⡀⠀⠀⣠⡿⣿⣿⣿⣿⢿⣄⠀⠀⢀⣠⠇⠀⢀⠜⡩⠂⢰⠤⠀⠀⠀⠀⣀⣄⠃
⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠐⠋⠃⠀⠀⡀⠙⢦⡀⢸⣿⢀⣾⢽⣾⣿⣿⣾⣿⣷⡯⣷⡀⣿⡇⢀⡴⠋⢀⠀⠀⠈⠙⠂⠀⠀⣀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠙⣦⡹⢿⣿⣤⣿⣿⣿⣿⣿⣿⣤⣿⡿⣏⣴⠋⣠⠖⠁⠀⠁⠀⠀⠀⠀⡀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠆⠀⠀⠀⠴⠶⠿⢿⣷⣾⣿⡛⡿⠟⢛⣿⣿⣿⣿⡛⠻⠿⢛⣿⣷⣾⡿⠟⠶⠦⠀⠀⠀⠰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠀⠀⠔⠉⠛⣿⣿⡿⡀⠀⢘⣷⣿⣿⣾⡃⠀⢀⢿⣿⣿⠛⠉⠢⠄⠀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⢀⠈⣸⣿⡇⡀⠀⡚⣿⢿⡷⣿⢃⠀⢈⢸⣿⣎⠁⡀⠀⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡀⠈⣸⢻⡇⠀⠠⠋⢻⣼⣇⡗⠙⠄⠀⢸⡟⣇⠑⢀⠠⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⣼⣿⣿⡇⠀⠀⠼⢻⣻⣟⡿⠇⠀⠀⢸⣿⣿⣧⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣷⡇⣿⣇⠂⢀⠀⡐⣿⣿⢂⠀⠀⠈⣸⣿⢸⣾⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢻⣿⣾⣄⠓⢿⣿⣿⡿⠚⣠⣷⣿⡟⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠉⠻⠶⠶⠟⠉⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⣠⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠲⣴⠒⡤⠤⣤⣀⣀⣀⣮⣾⡏⣽⣧⣷⣀⣀⣀⣤⠤⢤⠒⣦⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠈⠀⠘⣦⣿⣻⣿⣿⣭⢹⠅⢸⡇⡬⣉⢙⠏⢋⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠴⠀⠀⡎
⠐⠣⠋⣍⠁⠈⡣⢘⡾⠫⠘⠂⠘⠢⠉⣵⡁⢘⢁⠉⣩⠛⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠀⠀⢰⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠑⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠤⠐⡖⠀⠀⠀⣖⠠⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠨⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠁⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠁⠀⢠⠈⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢢
⠀⠀⡀⠢⡤⠄⠀⠀⠈⢠⡅⠄⠠⢨⡀⠁⠀⠀⠠⢤⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡄⠂⠀⠒⣄⠌
⠀⠀⠀⢚⡌⣀⡈⢄⣌⣼⠀⠀⠀⢈⣇⣥⡠⢁⣀⢁⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢈⡀⠀⣨⠀⠰
⠀⠀⠀⠀⠘⠂⠉⠃⠰⢨⡆⠀⠀⢠⡁⠆⠘⠁⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠒⠈⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠴⠂⠐⣟⠀⠀⠀⠀⣳⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠐⠀⠀⠀⠀⠐⠠⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠹⡀⠀⠀⢀⠃⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠐⠀⠀⠂⠀⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⡌⠀⠀⠀⠀⠀⡌⠂⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠠⠁⠀⠀⡀⠀⠀⠠⢠
⠀⠀⠀⠀⠀⠀⠀⢡⠀⠉⢢⡕⢢⡔⠉⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠄⠸⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⢆⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠀⠀⠀⠓⠀⠃⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⢠⠒⠘⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠀⠀⣄⠔⠂⠀⠒⢦⡀⠈⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠁⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠐⠁⣰⠂⠀⠀⠀⢲⠀⠑⠀⠌⠁
⠀⠀⠀⠀⠀⠀⠀⡸⢀⡆⠀⠀⠀⠀⢰⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠊⢂
⠀⠀⠀⠀⠀⠀⠜⠁⠄⠐⠀⠀⠀⠀⠂⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⢦⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢩⠀⠀⢠⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠼⠀⠀⠀⠀⠀⠀⠨⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⢀⠐⠁⠀⠃