Plant stem Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⢸⣏⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠘⣿⡇⠀⠀⠀⠀⠀⣠⠀⠀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠲⢆⡙⠳⠦⣄⠀⠀⠀⠘⠿⠀⣻⡇⠀⢠⡞⠀⡼⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⣷⢖⡄⠒⢂⣻⣶⣄⠀⢿⢘⣿⠇⠀⣘⢀⣼⡥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠛⠉⠉⠉⠻⣷⣿⡀⣿⡇⢀⠟⠋⣀⠀⢀⡤⠤⠴⠦⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⢸⣷⠃⣠⡾⠵⠾⢧⣦⣤⡔⠲⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⣿⣼⠃⠀⠀⠀⠀⠙⠻⢬⣙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢸⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢿⣷
⠀⠀⠀⣀⣀⡀⠀⠀⢀⣀⡀⠀⠀⠀⣄⣀⡀⠸⠘⢦⡀⠀⠀⠀⠀⠀⠀⠘⣿⡘⣿⣣
⠀⠀⠀⠑⢌⠉⡛⠳⠴⡶⠉⡓⠶⣄⠀⡈⠙⠳⠄⠀⠹⣄⠀⠀⠀⠀⠀⠀⢈⣧⢹⣷⡀
⠀⠀⠀⠀⠀⠑⢮⡀⠀⡀⠀⠈⠢⢈⠙⢾⣄⠀⠘⢦⠀⠈⢷⡄⠀⠀⠀⠀⠈⢻⡄⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢈⠶⠈⠓⠦⣄⣀⠑⢄⡈⠃⠀⠀⠱⡄⠀⢻⡄⠀⠀⠀⠀⢿⣧⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡒⠒⠒⠚⠛⠲⠬⣦⣄⡀⠀⠐⡀⠈⣧⠀⠀⠀⠀⠸⣷⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠤⡄⠀⠀⠀⠈⠉⠓⠢⣰⣄⣿⠀⠀⠀⠀⠀⣿⠀⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠉⠉⠉⠉⣉⡥⠉⠩⣙⢦⡀⠀⠀⣽⡈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⡶⠒⠒⠉⡁⢼⠀⠀⠀⠘⢷⠝⢦⠀⢹⠀⣿⡉⠀⢠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠚⢀⣼⠀⠀⠀⠀⠀⠘⠿⣷⣼⣀⣿⣀⡴⠋⠉⠁⣀⣠⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⢺⠀⠀⠀⠀⠀⠀⠀⠚⣿⠋⢿⣿⣗⣛⠿⡛⠧⣄⣠⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡴⡀⠀⠁⢿⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣿⡇⠀⠈⠳⡜⣆⠀⠉⠛⠲⠶⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢧⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢘⣷⠀⠀⠀⠸⡄⠉⠛⠿⣦⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣉⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢸⣿⡀⠀⠀⠀⠛⠀⠀⠀⠈⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠀⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⡇⢘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠼⠿⣿⡀

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⠀⠀⣤⣶⣶⠇
⠀⠀⠀⠀⢀⡀⠀⠀⢿⣏⣀⠤⣾⣿⣛⠥⠤⣴⣾⣿⣿⣶⠦⠀⠀⠀⠀⠀⠀⠐⠢⢼⠀⠀⠐
⠀⠀⠀⠀⢼⡷⠀⢀⠼⣿⣤⣄⠈⠻⢿⣷⣄⠀⠉⠉⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠈⠻⣤⡎⠀⠈⠻⠿⢧⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠀⠀⠀⠁
⠀⠀⠈⢿⣶⡀⢸⠀⣠⣾⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥
⠀⠀⠀⠈⠛⠓⠺⣿⠉⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠠
⠀⠀⠀⠀⠀⣀⣀⣈⢄⠀⣰⠿⠋⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀
⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠧⣀⡀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⡌⠀⠂⠀⠀⠀⠀⠀⠀⠸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢲⡆⠀⠀⠀⠀⠀⠑⠰⣽⠀⠑⠀⠀⠀⠀⠀⢦⡀⢹⡇⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⡾⣆⠀⢰⠁⢀⡜⠀⠀⠀⠀⠀⡀⠀⠐⠾⠀⠄⠄⠀⠀⠀⠀⠈⢻⡆⡇⢨⡟
⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⠀⣮⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⠐⠐⠀⠀⠀⠀⠀⣄⡀⠙⣧⠞
⠀⠀⠀⠀⠀⣀⠀⠈⢇⣸⡔⣁⠤⠖⢒⡶⠀⠀⠀⠠⠒⠂⠐⠟⠀⠠⠀⠄⠀⠀⠀⠘⢷⡀⢸⠀⣰⡿
⠀⠀⠀⠀⢰⠉⠳⡄⢈⡿⠾⠥⠤⠔⠋⠀⠀⠀⠀⠀⠀⠀⠈⣿⡁⠀⠐⠂⠀⠀⠀⠀⠀⠱⣼⢠⠏
⠀⠀⠀⠀⠘⢆⠀⣹⠎⣀⠤⣀⣀⡀⠀⠀⠀⠀⠀⠄⠉⠀⢀⣽⠀⠀⠀⠄⠀⠀⠀⢸⣦⠀⢹⠋⠀⢀⡀
⠀⠀⠀⠀⠀⠈⢳⡿⢯⣤⣠⠤⠊⠀⠀⠀⠀⠀⠀⠀⠉⠀⢄⣀⡀⠀⠀⡀⠀⠀⠀⠀⠈⢧⢸⠀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠊
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⢘⡆⠰⡌⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡷⣾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⢔⠟⣤⡄⡁⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⣷⣯⢴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠈⠀⠆⠀⠀⠀⠀⠀⠀⠀⢤⡄⢻⣟⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣼⡆⢀⠀⠀⠀⠀⠀⠀⠀⣶⣨⢷⣼⣾⣯⣻⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠤⠀⠀⠀⠀⢠⡾⠟⣏⢘⣃⠄⠀⠀⠀⠀⠀⢀⠿⡟⢺⠟⠛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠀⠀⠨⣷⠆⠀⣠⠄⠀⠀⠀⠸⣷⢤⠖⠀⠀⠀⠀⣄⠮⠄⣹⣿⠻⡿⡿⠄⠰⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢂⡌⠀⣩⣄⢠⡙⢤⠀⠀⠀⠀⠙⢬⠀⠀⠀⠀⠀⢸⠒⢠⣿⠏⢠⣌⡀⢚⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⢾⠇⠀⠀⠀⣸⡶⢂⠈⢇⣀⣠⡄⠻⡿⢁⣰⡋⡐⠰⠂⠀⠄⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠀⡀⢦⠀⠀⠀⠀⠈⠡⠀⠀⠈⣯⡯⢥⠄⠁⣸⠟⠀⠙⠛⠒⢶⠌⠑⠂
⠀⣰⢆⡀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠐⠆⢓⠘⢠⠀⠀⢈⡫⠺⡀⠀⠀⠀⠀⠀⠀⠀⡘⢈⣿⠀⢠⣌⡱⠏⡀⠀⢠⠄⠄
⠀⠬⢸⣵⠿⠯⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠒⠀⠀⣑⠶⣧⠀⠀⠀⠐⠆⠀⠤⢓⡺⣿⢘⣿⡣⢂⢠
⢠⣆⣄⣣⡀⣬⢷⠆⣤⢂⢀⡀⠀⠀⠰⠀⠉⣶⣇⠀⢀⣻⢂⡽⣆⠀⠀⠀⠀⢀⣤⣬⠁⣿⣾⣰⡍⠸⠀⠉
⠀⠀⢀⣽⠷⣴⠶⣿⣿⠀⢸⣧⠆⠀⠀⠠⠝⠹⣷⡄⢠⣤⠋⠀⢹⡆⠀⠀⠀⠀⣥⡵⣬⣿⡀⠁⠘⣧⠀⣿⣤⣼⠀⣦⠄
⠀⠣⠘⠳⠇⣨⣻⡾⣍⣠⠚⠿⠛⠀⠀⠀⠀⠀⠘⢧⠀⢼⣄⣄⣸⡧⣤⠀⠀⠀⢠⡴⢿⠏⡁⠡⣖⠉⠀⠉⠉⠙⠙⠁
⠀⠀⠀⠀⠈⠩⠇⠿⠈⠉⢳⣧⣴⣶⡀⢠⣿⡦⠀⢈⣿⣏⡌⡹⣿⣷⡍⠠⣿⣿⡾⢱⣿⣖⢈⣄⡨⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⣶⣽⡆⢈⡹⣿⣿⣷⣿⣮⣴⡼⢠⣽⣿⣾⢛⢼⣷⣯⢀⣸⣿⣷⡟⠛⠛⠀⢣⡙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠏⠀⠁⠘⠛⣿⣿⣷⣽⣿⣯⣾⡷⢸⣿⣦⣿⣿⡿⠟⠛⠛⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣶⢯⣍⣻⠿⠿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠋⠀⠀⠀⠀⠀⠉⢻⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧⣤⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⢻⣧⠘⣿⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⠀⠰⣇⠀⠀⢰⣿⠟⠀⢺⣿⣶⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡄⠀⣿⠀⠀⠀⠁⠀⡠⣾⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⡀⠀⠀⠀⣼⣿⣿⣿⡇⠘⢿⠀⢴⣧⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡆⠀⠀⢹⣿⣿⣿⠀⣴⣾⡿⠿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣄⣠⡄⠘⣿⣿⣿⠀⠟⠁⣠⡄⠛⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⣇⡀⢿⣿⡿⢆⣠⣜⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣠⣾⣿⣿⡟⣿⣟⣅⣸⣷⣾⠻⠟⢉⣿⣿⣿⠃⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡟⠻⣿⣿⢹⣯⠀⣴⡶⠈⠉⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣯⠙⢷⣷⣦⣹⡇⠀⠀⢦⡀⢠⣤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡇⠈⢻⣿⡿⠳⠶⠶⡄⠙⣎⣵⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣀⣼⡏⣠⣤⣤⠀⣾⠃⣤⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠉⠉⠛⠻⠶⠾⠿⠿⠿⠟⠛⠋⠉
⠀⢲⣶⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⢸⡆⠀⢠⡟
⠀⣀⠉⠀⢰⣶⠄⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣤⠀⠀⢸⣧⣐⣿⡟⠀⠀⠀⠀⢀
⠀⠉⣷⡶⢸⣿⡆⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣦⠘⢿⡿⠋⢀⣠⠶⢀⣴⠟
⠀⠀⢿⠇⠀⢻⣧⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣷⣾⣷⠞⠋⠁⣠⢎⠛
⠀⠀⠀⠠⠾⠁⠀⠀⠀⢄⠀⠰⢾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⠙⢻⣿⣿⣿⣏⢠⣄⣴⣛⡴⠊
⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⣷⠀⣼⠏⠀⠄⠀⠀⠀⠀⠀⠀⢠⣤⡈⢻⣷⡀⠙⣿⣿⣽⣷⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠯⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⡀⣿⡇⠀⣧⠘⠋⠁⠀⠀⠀⢤⠄⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⡀⠉⠀⠀⣸⠇⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣎⢻⣠⡿⠀⡀⢠⡾⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣾⣿⢀⣾⡟⠀⠀⣀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠄⠀⠀⠀⠀⢀⡰⣦⣀⠀⢿⣿⡿⠈⢿⣿⣷⠌⠛⣣⣶⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⢠⡿⢧⡜⢿⣷⣼⣿⠀⠀⠀⠸⠀⣠⡿⢋⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣿⣿⣧⣾⡆⠀⠘⠿⠈⢻⣿⡇⠀⣇⠀⠀⣴⣿⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣸⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣵⣆⡀⢠⣤⡀⢿⣧⠀⢻⡆⠀⣼⣿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠈⢿⣧⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⢋⠈⠉⠉⠁⠀⠻⣿⠾⣿⣆⢈⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡄⣶⣦⣴⣅⠀⠈⠀⣸⣿⣴⣿⠇⠀⠀⠀⣠⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⠀⠀⣦⢁⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠙⢿⣿⣿⣶⣄⡀⠈⢻⣿⠃⢀⡀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣼⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣦⣼⠀⣼⡿⢿⣿⣿⡏⡆⢸⡏⢀⡜⢻⣴⠟⠁⠀⠀⠀⠀⠀⣀⡀⠀⠀⡄⠀⠀⢾⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢠⠛⠀⠀⠹⣿⠁⢵⡀⠹⣿⣧⡾⠋⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠃⠀⢠⡜⠀⠁⣠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣶⣄⡀⠀⠀⣠⠀⠈⢀⣤⠁⠀⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠙⢿⡀⠀⡌⣳⣤⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣆⢀⣿⠀⠀⠉⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣦⣄⣿⡶⠻⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣾⠏⠀⠀⠰⢼⣿⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣟⠉⢻⣿⣿⠀⣀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⡀⠈⢛⡀⠀⠀⣤⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⡀⠾⢙⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡘⢇⠀⢸⣿⠀⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠘⣯⣿⡴⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣤⣀⠀⠀⠀⠙⠻⣼⣧⠘⣿⠀⢹⡟⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⢠⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣷⣦⣀⡘⢷⣄⣁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡤⠚⠀⣼⣆⠀⡌⢀⣴⣶⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣌⣿⣇⣀⠀⢠⡀⠀⠀⠀⢀⣠⣴⣶⠄⢻⣿⣤⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣥⠉⢿⣿⣿⣿⣿⠀⢸⡇⢀⡄⢀⣼⠿⠛⣿⡆⠈⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠈⠻⣿⣿⠀⢸⠁⠘⣡⡾⠋⣠⣤⣿⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⠀⠈⠻⣿⣿⠃⠼⠋⠀⠀⣿⡟⠛⠇⠀⣠⣤⣶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣆⠀⠠⠀⠈⣿⠀⠀⣾⡇⢠⣿⡇⠰⠆⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⢸⡄⢰⣿⡇⠐⣿⣷⡦⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣆⠀⠀⢸⣷⠈⢻⡇⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣧⡄⢸⣿⠀⠘⠃⠀⢋⣤⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠛⣿⣿⣿⠿⠿⣸⣿⠀⢀⠀⠀⠘⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⠀⠙⠋⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⠙⣿⣆⠀⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢸⣿⣄⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣤⣤⠀⠀⠠⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡄⢀⡀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠓⠀⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⢤⣶⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣔⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣦⡀⠀⠈⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠉⠰⣾⠀⢀⣀⠀⠃⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⡤⠀⠀⠀⠀⠀⠀⠘⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣤⣄⠀⠀⠀⠀⡀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠻⣿⡄⢰⠾⠁⢀⡄⢂⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠰⠟⢶⡿⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⢴⠇⠀⠠⠟⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠿⣇⡼⠇⣠⡶⠆⠀⠀⢰⡆⠶⡄⠀⠀⠀⠶⠴⠖⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⣠⠀⠈⣣⡀⣹⠗⠀⣠⠀⠀⠀⠀⠀⠀⠐⠂⠀⠸⠒⠂⠀⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠾⠉⠉⠻⣄⠀⠉⠀⠀⠀⠀⠀⠀⢠⣤⡀⣿⣟⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣀⠴⠀⠀⠀⠀⠀⢀⠀⠀⠀⠲⠄⣠⡄⠈⠀⣤⠀⠀⠀⠀⠀⢶⡏⠙⣿⣦⣭⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄⠀⠰⡇⠀⠀⢿⡄⠀⣤⡄⠀⠀⢀⡀⠂⠈⡇⠀⠀⣴⠀⠀⠻⡇⠹⠏⠉⠉⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢠⡌⠁⠠⠄⠀⠀⣀⠀⠈⢷⣄⠀⠀⢨⡀⠀⠀⣿⣠⠀⢺⣷⣀⣠⣿⣇⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⢠⣦⠲⠂⠀⠀⢀⣀⠀⣾⠉⠂⠀⠈⠳⠀⠀⢸⡏⠀⠀⠉⠙⢿⡝⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⡛⠀⠀⠀⢰⠞⠉⠒⠻⠏⠑⢀⣶⠀⠛⠦⠘⠤⠀⢸⣧⢀⠀⠀⠀⣠⣄⣤⣴⡄⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⢉⡄⠀⠀⠀⢀⠀⣀⡀⢀⡀⠀⢀⣈⠳⠄⣡⡀⠀⢠⡏⠀⠛⢷⡄⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⠆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⡄⠀⠀⠈⠀⠉⠀⡄⠑⣆⢀⣍⠀⠀⠉⠁⠶⣾⣇⠀⠀⣈⠁⣄⣀⡀⠀⠀⢎⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⢻⣤⣄⣡⣾⡂⢀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⢸⣿⣤⣀⠙⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠉⠘⢿⣏⢁⣀⣀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢹⡟⢀⠤⠀⠀⠀⠀⠀⠀⠰⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⣀⣤⡀⣸⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⢸⣇⠸⠇⠈⠳⡄⠀⠀⠀⣤⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⢰⡏⠀⣶⡿⠃⠀⠀⢀⠀⠀⠠⠤⠀⠀⠀⠈⣿⠿⠋⢀⠤⡀⠰⠀⠀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢤⣀⠙⢣⣴⣆⣹⡆⠀⢀⣀⣿⠇⠀⡀⠄⠈⠀⠀⢸⡇⠀⠀⠙⢆⡀⢶⡀⠀⢀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠹⣿⠠⠶⠋⢻⡇⠀⠀⠀⠀⠈⠀⢀⣬⡇⠀⠀⠀⠀⠁⠠⠃⠲⣾⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣐⠀⣠⣄⠀⣤⡄⠀⡀⢈⣷⠀⠀⠀⠀⢀⡀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⣰⣶⠟⠉⠁
⠀⠀⢠⣤⣤⣤⠀⠀⠀⠀⠀⠉⠀⠉⠉⠓⡈⠰⠎⠁⡞⣁⠀⠀⠀⣀⡘⠁⠀⠀⢹⣷⠀⢀⠀⠀⠀⣀⡘⢳⣤⣴⠆
⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠟⠀⠈⠀⠀⠀⠋⠀⠀⠀⠀⠀⠋⠺⠟⠗⠀⣠⣯⣴⠞⠁
⠀⠀⠀⢴⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣃⡀⠀⣆⠸⠆⠀⠀⠈⣾⠃⠀⠀⢠⠀⠘⠇⠀⠀⠀⠶⠀⠈⡁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧
⠀⠀⠀⠀⠙⠿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠶⢀⣀⠸⢦⡈⡀⠉⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠⣤⣤⡄⢹⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠘⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠀⠀⠙⣧⠀⠀⣀⡀⠀⠀⠰⣟⠀⠀⠠⣽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠋
⠀⠀⠀⠀⠠⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠠⣾⡆⠠⣦⡈⠃⠸⠆⠀⣶⠞⠀⠀⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡶⠀⢰⣶⡄⠀⢠⣦⡈⢋⣤⡿⠃⣀⡜⠋⣀⠀⠀⠀⡟⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⠁⠀⠀⠙⠻⢿⣿⣧⣿⠋⠔⠃⢈⠀⠀⣠⠃⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⣤⣶⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⡀⠀⠀⠀⠈⢻⣿⡏⠀⠴⠂⠈⠁⢀⣿⠀⢠⡄⠀⠀⠀⠀⠀⢀⠀⢀⣴⣦⠄⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣴⣶⡄⢀⣿⠁⠀⠀⠀⠀⠀⠘⠇⠀⠆⠀⠀⠀⢀⣠⣤⣼⠷⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⣻⣿⡂⠀⠀⠀⠀⠀⢘⣷⡄⠀⠤⠘⠛⠛⠛⠋⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⡶⠀⠀⠀⠀⠘⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣷⣶⡁⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣦⡖⠀⠉⠧⠄⠀⣿⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠘⣹⣶⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢻⡇⠀⢰⣶⡦⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⢴⣄⠀⠀⠘⢷⣿⣧⠐⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡄⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣚⣭⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡥⠞⣹⢫⠏⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣡⠋⣠⠞⡱⢋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡳⢁⡼⣣⢞⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡞⣹⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣵⣿⠿⠛⢁⣀⣀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣸⠁⣿⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠷⠶⠶⠾⣿⣛⠓⠒⠲⠾⢿⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡿⠀⢹⠀⡇⠀⠀⠀⠀⠀⠀⣠⡴⠟⣋⣁⣀⡀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣷⠀⢸⡆⡇⠀⠀⠀⢀⣤⣾⣵⣺⣭⣭⣭⣉⣉⡙⠳⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢹⡄⢸⢃⡇⠀⢀⣴⠟⠛⠛⠻⢭⣙⠓⠶⣤⣉⡉⠛⠓⠶⢿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⢳⣸⣸⠃⣠⠞⠁⠀⠀⠀⠀⠀⠈⠙⠲⢤⣌⣉⣛⣛⣩⡽⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡏⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢀⣠⡴⠖⠚⠛⢉⣉⣉⣭⣿⣿⣷⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢀⣠⠖⠋⠀⢀⣠⡴⠞⣫⡽⠖⢋⡭⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⣠⠴⠋⣀⣤⠶⢚⣉⣥⠶⠛⠁⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣤⣶⣿⣵⣒⣫⠥⠶⠚⠋⠁⣀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠈⠉⠉⠛⠛⠛⠒⠒⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⢀⣠⣴⠟⠁
⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⡂⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡦⢤⣤⣄⣧⣽⡈⢿⡇⠀⠀⠀⣀⣠⠤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠥⠶⠦⠶⠽⠿⣾⡇⠀⡰⠋⠁⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠙⢦⡃⡀⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⣿⣭⣹⣭⣿⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⢣⠀⠀⢀⣤⢶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⡆⢠⣏⡔⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣮⡶⣄⣸⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠛⠻⣇⠀⢀⣠⣤⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣼⣿⢟⢱⠏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡵⠗⣁⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⡰⠋⢁⣴⠛⠵⣳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣄⠀⠀⠀⣀⠂⣼⣡⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣧⠀⠀⠀⠀⢀⡄⠀⠀⠀⣄⠀⠀⠀⠀⠈⣿⢮⣳⠀⢀⠟⠚⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠁⠘⢦⠀⠀⠀⢸⣧⠀⠀⠀⣿⢶⣄⠀⠀⠀⡟⢘⡟⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⠠⠀⠀⢦⠀⢸⠘⠘⡆⠀⢸⣍⠀⠈⢳⡀⠀⢷⣿⣟⣥⣤⣤⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠂⠀⡀⣼⠀⠈⢦⠠⣱⠀⠀⠻⢦⣤⡼⢃⣠⠞⠛⠛⠫⢄⢀⣀⣀⡤⠟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣠⣿⠇⠀⠀⠀⠁⣿⠀⠀⠀⢀⣽⣛⣉⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠲⠦⠤⢤⣿⣯⣄⣀⣠⣤⣤⣽⡤⠒⠈⠀⠀⠉⠻⢿⣿⣿⣿⣶⡶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢷⡄⠀⠀⠀⢀⣟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢄⢿⡄⠀⢀⣾⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⣮⠇⠈⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣣⠽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣣⠟⢠⠟⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡡⢼⡈⠎⣂⡰⣏⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠇⡑⠚⣧⠀⠄⣸⢯⣀⣏⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡗⠉⠑⢈⠤⠿⠈⢉⠵⡂⠠⣹⡟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡧⡋⡲⠎⣼⡁⡑⠎⢻⣉⣅⣁⢜⡑⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡱⣨⠁⢀⣌⠉⣱⠓⠟⠡⠀⡷⢰⠧⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢌⡓⠂⢐⠟⢋⡶⠛⠛⠾⠥⡵⠖⠋⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠯⠤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⡴⢻⠝⣋⡭⠭⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡆⠊⡡⠈⢲⣉⠮⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣉⣥⡀⣽⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⣻⡓⢙⣶⣎⣃⠀⣸⠨⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣽⣫⢼⣀⡵⠔⠳⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢈⡳⢜⣃⠜⡑⠰⣀⡓⢊⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢡⡁⡤⢸⢇⡀⠒⢹⣦⣀⣀⠀⠀⠀⠀⠀⠀⡤⣞⠑⠈⡗⢦⣗⡈⡟⣉⡉⢉⣛⢲⡹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠼⣦⢱⣏⣣⡴⠋⣱⣤⣤⣧⠌⣹⠶⣄⠀⠀⢸⠗⢇⠗⢶⠋⡻⠼⢎⣷⡥⢞⠐⠒⡖⢙⡞⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⢳⢤⡾⠶⢎⢁⡋⡎⠰⣶⠀⣜⡙⢌⠢⣈⣇⠀⠘⢦⣜⠳⣙⣦⣒⣡⠛⣀⣯⡽⠶⠚⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠺⣄⡱⢀⡟⠣⣑⡴⠿⡃⠈⠃⣸⠀⣇⡟⠀⠀⠀⠉⠙⠛⠚⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣼⣧⡨⢇⠆⢀⣾⠋⠂⡔⠁⠑⡟⢲⢊⡝⣰⠟⠁
⠀⠀⠀⠀⠀⠀⣰⠧⣠⣧⠮⡖⠻⠤⡟⢩⠑⢢⣴⣫⣾⠿⠚⠁
⠀⠀⠀⠀⢀⣴⡷⠾⠗⠒⠛⠛⠛⠛⠋⠉⠉⠉⠉
⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡃
⠀⠀⠀⡀⠀⠀⢄⠈⠃⠀⠀⣾⣶⠄⠀⠀⣾⡁⠂⠀⠀⠐
⠀⠈⠀⠒⠈⠀⠀⢀⠀⠀⠀⠈⠉⠂⠀⢰⠟⠁⣀⣀⡈⠈
⠀⠀⠀⠀⠀⢀⣿⠆⠠⡄⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⠆
⠀⠀⠀⠀⠀⠀⠈⠀⠠⠇⠀⠀⣠⣷⡤⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠋⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠉⠀⠁⠀⣀⣤⠀⢠⣆⠀⠀⠀⠀⢐⣾⣷⡆⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⡃⠀⠨⠀⠀⠀⠀⠈⠁⠁⠁
⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⢧⠀⠀⡈⠁⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠁⠀⠚⠁⠀⠀⠀⠀⠛

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⡄⣠⣶⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣠⣾⣫⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡏⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⡟⢀⣾⣿⣿⣽⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠛⢉⣽⣿⠏⢀⣾⣿⣿⢷⣿⣟⣿⣿⢡⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⣾⣿⣿⣿⢸⡟⣼⣿⢃⣿⣿⣿⣿⣿⣿⣿⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⣿⣿⣿⣿⣿⢱⣿⠃⣾⣿⣿⡟⣿⣿⣿⡿⠀⣾
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⢀⠀⠀⣧⢹⣿⣿⣿⣿⣿⠏⣼⣿⣿⢋⣾⣿⣿⡿⢡⠀⣿⡆
⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⣾⠀⠀⣿⣆⢿⣿⣿⣿⡟⣼⣿⢟⣵⣿⣿⡿⢋⣴⡟⠀⣿⡇
⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⣿⠀⠀⢿⣿⣎⢿⣿⣿⢰⣿⣷⣿⠿⢛⣡⣶⡿⢋⡀⠀⣿⡗
⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⢿⣷⡀⢻⡄⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣵⣾⣿⠟⣋⣴⠟⠀⣰⣿⡇
⠀⠀⠀⣄⠸⣿⣿⣆⠀⠀⠀⠀⠀⢻⣿⣜⣷⡀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣟⣭⣶⣿⠟⠁⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⣠⡶⠁
⠀⠀⠀⠸⣆⢹⣿⣿⣷⡀⠀⠀⠀⠀⠙⣿⣿⣷⡀⠀⠈⢻⣿⣿⣿⣿⣿⣿⠿⠋⠁⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⢀⣴⣾⠟
⠀⠀⠀⠀⣿⡄⢿⣿⣿⣿⡄⣄⠀⠀⠀⠈⢻⣿⣧⠠⡀⠀⢻⣿⣿⠿⠋⠁⠀⢀⣴⣿⠟⠁⠀⢀⣀⣤⠄⣠⣾⣿⣿⠏⣠
⠀⠀⠀⠀⢻⣿⡜⣿⣿⣿⣿⡌⢧⡀⠀⠀⠀⠻⣿⣧⠹⣦⣼⡟⠁⠀⠀⠀⢠⣾⠟⣡⣴⣶⠿⠛⠉⣠⣾⣿⣿⣿⠃⣼⠃
⠀⠀⠀⠀⢸⣿⣿⡜⣿⣿⡏⠛⠈⢷⡄⠀⠀⠀⢹⣿⣧⠘⣿⡇⠀⠀⠀⢀⣾⣣⣾⡿⠋⠁⣠⠞⣴⣿⣿⣿⣿⢏⣾⡇
⠀⠀⠀⢠⠀⢿⣿⣿⣎⢿⣿⣄⠀⠈⢿⣄⠀⠀⠀⢻⣿⣆⢻⡇⠀⠀⠀⣸⣿⡿⠋⠀⢀⡾⠋⠀⠁⣿⣿⡿⢣⣾⣿⠁
⠀⠀⠀⠘⣷⣜⣿⣿⣿⣷⡙⢿⣦⡀⠘⣿⡄⠀⠀⠈⣿⣿⣿⡇⠀⠀⢀⣿⡟⠀⠀⣰⡿⠁⠀⢀⣾⣿⢟⣵⣿⣿⡿
⠀⠀⠀⠀⠹⣿⣿⣿⣿⠿⢿⣮⠻⣿⣄⢹⣿⡀⠀⠀⢹⣿⣿⡇⠀⠀⢸⡿⠀⠀⣰⡿⠁⢀⣴⡿⢟⣵⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣦⣀⠀⠘⢿⣦⣿⣇⠀⠀⠈⣿⣿⡇⠀⠀⣾⡇⠀⢰⣿⢃⣴⡿⠋⠐⠛⣛⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣷⡄⠈⢿⣿⣿⡀⠀⠀⢿⣿⡇⠀⢀⣿⠀⢀⣿⣿⣿⠟⠁⣠⣴⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠆⠘⣿⣿⡇⠀⠀⢸⣿⣿⠀⢸⣿⠀⣼⣿⣿⠃⢀⣾⠿⠛⠋⠉⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠘⠛⠶⣶⣦⣤⣄⣀⠤⣤⡀⠀⠀⠀⢶⣌⠻⣿⠀⠀⢸⣿⣿⠀⣾⡇⢠⣿⣿⠇⠰⠋⠀⣀⣤⣶⣾⣿⣿⣿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⢭⣿⣿⣿⣿⣶⣧⣄⡐⢤⡈⢻⣷⣌⠀⠀⢸⣿⣿⠀⣿⡇⣸⣿⡿⠀⣠⣴⣿⣿⠿⣿⣿⡿⠋
⠐⢦⣤⣤⣤⣤⣤⣽⣿⣿⣿⣟⢿⣿⣷⣌⠢⣿⣿⣆⠀⠘⣿⣿⢸⣿⠃⣿⠟⣱⣾⣿⣿⠟⣡⣾⣿⠟⣠⠞
⠀⠀⠈⠙⠿⢿⣿⣿⣿⠟⠻⣿⣷⣍⠻⣿⣷⣼⣿⣿⣧⠀⣿⣿⢸⣿⠐⢃⣾⣿⣿⠟⣡⣾⣿⠟⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣧⡈⠻⣿⣿⣿⣿⣆⣿⣿⣿⣿⣰⣿⣿⡿⢁⣾⣿⢟⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⡿⢋⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⢻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣿

⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠸⠀⢠⠄
⠀⠀⠀⠀⠀⠐⢄⠀⠀⠄⠀⢠⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠤⠀⠀⠀⣸
⠀⠀⠀⠀⢀⡀⠀⢠⠁⠀⠀⠀⢉⡀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡴⠑⠂⠁⠀⣀⠐⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⢀⠐⠀⢄⠀⠀⠀⠀⠈⡀⠀⢠⠀⠀⠀⡀⠘⡃⠀⠀⠀⠰⠄⠄⢀⡤⠪⠁
⠀⠀⠀⠀⠀⠈⠀⠀⡈⠁⠉⠀⢠⡀⠈⠁⠀⠀⠀⠀⠁⠀⠈⢷⢄⠀⠈⠀⡀⠀⠀⠀⠀⠅⡠⢊⠂⠀⠀⠸⠀⠀⠈
⠀⠀⠀⠀⠀⡀⠀⠘⠁⠀⠸⠀⠀⠁⠀⠀⠀⠐⡂⢄⠀⠀⠀⠈⠳⢐⠄⠀⠇⠀⠀⠀⠀⢸⢀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠌⠖⠀⠐⠄⠀⠀⠀⠀⢀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠡⠄⠓⠢⡖⢄⠀⠀⠀⡀⠈⠁
⠀⠀⠀⠈⠙⠲⠄⡄⠀⠁⠀⠀⠀⡐⠁⠀⠀⠀⠀⠄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠂⢅⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠂⠂⠀⠈⠈⠀⠀⠀⠀⠈⠀⣀⠤⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠢⠈⠸⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠂⠄
⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠘⠐⠙⠀⠀⢲⢊⣀⠀⠀⠀⠀⠀⠐⠄⠀⠈⠀⠀⠀⠀⠇⡆⠀⠀⠀⠀⠠⡀⠀⡀⠙⢆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⡌⡌⠀⠀⠉⠙⠁⠈⠀⡀⠀⠀⠠⠂⠒⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠣⠄⠈
⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠒⠀⠠⢄⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠠⠀⠰⠀⠠
⠀⠀⠀⠀⠀⢀⠊⠀⢀⠄⠀⠠⡀⠀⠄⠀⠂⠀⠊⠀⢀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⡀
⠀⠀⠀⠀⢀⡀⡠⠢⠈⠢⠀⠀⠀⡁⠀⠀⠄⠠⠀⠂⢁⠀⡀⠀⠉⠀⠀⠀⠀⠠⡀⠀⢃⠀⠀⠀⡄⠂⠀⠀⠀⠁⠐⢀
⠀⠀⠀⠀⢰⠀⠃⠀⠋⢀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⡁⠀⠈⠀⠀⠰⠀⠀⠀⠀⠐⡀⠀⠁⠀⠀⠀⢀⠀⠀⠉⠐⠀⠔
⠀⠀⢀⠀⠀⢧⡀⠀⣠⣂⡀⠃⠀⠀⠖⠈⠀⠀⠀⠀⠁⠀⠀⢀⠤⠂⠠⠂⠀⠀⠀⠐⠄⠀⠀⠀⠀⠂⠀⠀⠒⠐⠀⢄
⠀⠀⠀⠠⠀⠐⠬⢉⠀⠀⠀⠀⠄⠊⠀⠀⠀⠀⠀⠀⡆⠠⡀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠈⠄⢀⠀⠀⢂⠀⠈⠀⠂
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢄⠀⠈⠂⠀⠀⡠⠔⠒⠒⠒⢄⠂⠀⠀⠁⠀⠀⠁⠀⠘⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⠀⠀⠀⠂⠀⠀⣀⣈⠀⢀⣀⡀⠀⠀⠁⠀⢡⠔⠒⠀⠤⠤⠤⠄⠀⣑⣌⠂⠠⠀⠀⠀⠊⡀
⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠂⠐⡁⢄⠨⠁⠩⠀⠈⢙⠀⢣⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠠⡀⢱⠘⢀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠈⠀⡐⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠡⣀⣆⡂⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠯⠀⠀⠀⠀⠀⠤⠔⠀⠀⡠⡀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠤⢀⣒⣀⣀⠖
⠀⠀⠀⠀⡰⠊⠄⡁⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠊⠀⠀⢀⠘⠂⠀⠁⠀⠀⠈⠊⢔⡤⠠⡀⠀⠀⠀⠈⠒⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⣀⠀⣿⣿⣇⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⣴⠂⠀⠠⠈⠀⣼⣿⡟⢋⣁⡈⠻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠙⣏⠀⠀⠆⠀⠀⢈⣭⡖⠚⠿⠃⢨⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣄⠀⠀⢸⣿⠃⢸⡿⠀⠀⠃⠀⠀⠙⢿⡿⢣⣶⣾⡝⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣋⠉⠙⣆⠀⠈⢿⠂⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⡂⠜⠀⠀⠀⠀⠘⠄⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢺⣿⣿⣿⣠⣶⣦⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠛⢄⣁⠈⠹⢿⣿⠇⠀⠀⠀⠀⢸⠀⠀⠀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⡧⠀⠾⣿⡗⠀⣻⣦⣄⠀⠀⠀⠀⡍⠀⠀⠀⠋⠙⢋⣈⣹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣆⡔⢀⣀⣤⣿⣿⡟⠀⠀⠀⢠⠉⠀⠀⠚⠀⠲⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⢿⣿⣿⡷⠀⠉⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⡜⠀⢠⠆⠀⠀⣼⣷⣿⣷⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡠⠃⣴⣶⣤⣹⠿⠿⢻⡥⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⠄⠀⢾⣿⣿⡿⠈⢣⣄⣌⠀⠙⣻⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠐⠀⠀⢉⣩⡖⠒⠺⠿⣷⠀⢉⣽⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠰⣿⣿⣿⣆⣴⢀⣀⣰⢼⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡀⠀⠀⠀⡀⠂⠘⠿⠟⠉⢼⣿⣿⣿⡇⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠔⠀⢀⠄⠈⠀⠀⠀⠀⠀⠀⠈⠛⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⢿⣶⣄⠀⡘⡏⠀⠀⠁⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣄⣀⣈⡀⠙⠛⢠⡿⠁⠀⠀⢀⠖⠉⠛⠛⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠛⠿⠷⠒⠐⢮⠃⡄⠀⠀⠉⠀⠐⢶⣶⡄⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡥⠀⠀⠀⠀⠀⠀⠈⠻⣆⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢆⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢳⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠳⣄⠀⠙⠳⠤⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡟⣿⠿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠛⣁⣀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠠⢬⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠉⡋⠀⠈⠑⠓⠄⠀⠀⠀⠀⠰
⠈⠋⠉⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⡔⠀⡄⠀⠀⠀⠣⠄⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠳⠀⠀⠐⡄⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠠⠀⠀⠐⣆⠀⠀⠀⠀⢀⡀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡁⠠⠤⠄⠐⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⢄⡀⠀⠀⠀⠂⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣵⣤⢄⠐⠂
⠀⠀⠁⠀⠀⢐⣀⣠⡀⠐⠲⠶⣶⡶⠀⠈⠙⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠒⠠⢄⠀⠀⠀⠀⠀⠀⠀⠋⠁⢠⠆⠀⠀⠀⢘⠀⠀⠀⢠⣤⣴
⠀⠐⠁⠀⠀⠀⠉⠁⣀⠀⠀⡄⠁⢐⡆⠀⠀⠀⠀⠀⠐⠈⠆⠶⣄⣀⡀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠈⠉⠑⠦⠀⢀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠀⠀⠀⠀⠀⠀⠈⡀⠐⠀⠀⠀⠀⠀⠀⣴⢀⠀⢰⠁
⠀⠀⢀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠧⠠⠀⠃⠀⠀⠠⢄⡂⠐⢦⠘⠱⡀⢧⠇⢀⡠⠤
⠀⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢀⡓⠦⢙⣦⣁⣆⢀⡡⠀⡨
⠀⠀⠈⢀⠀⠠⡄⠁⠈⠀⠀⠀⠀⠀⡘⠖⠉⣠⡼⢡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠚⠛⠙⠻⢿⣿⣿⡶⠒⠛⠁
⠀⠀⠀⠈⠷⠢⣤⠀⡀⠀⠀⠀⢀⠐⢁⣦⠟⡉⠈⠁⠁⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⢰⣥⠴⡺⠋⢁⠞⡿⢹⢟⢣⣀⠀⠁⠁⠀⠐
⠀⠀⠀⠀⠤⠒⠄⠁⠀⠀⠀⠀⠀⣴⠏⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠄⠤⠂⠁⠐⠀⣰⠈⣥⠜⢆⠉⠁
⢄⠀⡄⠂⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⡀⠀⠀⠀⢠⡄
⠀⠈⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠐⢆⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⠎⡇⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⢐⡀⠀⠀⠀⠴⠀⠀⠀⢀⡠⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠄⠀⠃⠀⠈⠸⠀⣴⡃⡀⠄⡠⠀⠀⠀⠀⠐⣡⠾⠀⠀⠀⠀⠐⠀⠂⡨⠿⢧⡀
⠠⠄⠀⠀⡀⠀⡐⠀⠀⠃⠀⠈⠁⠀⠊⠀⠀⠀⠀⢸⣿⣁⡀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡇⢸⠀⠀⠁⠈⠁⠀⠀⡠⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⡀⠀⠀⠀⠀⠀⢀
⠅⠊⡀⠌⠀⠀⠠⠄⢀⠠⣈⣉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⡠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠈⠁⣀⣒⠒⠂⠢⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⣈⠄⡤⠈⠓⠒⠂⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡤⠊⠉⠄⠘⠉⠉⠐⡄⠀⠀⠀⠤⢀⡐⠠⠀⠠⣄⠉⣠
⠀⠀⠒⠀⠂⠉⠀⠀⠀⣀⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠰⢄⡙⣳⣄⡀⠁⠘⠛⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠱⣤⠉⠛⠶⣶⣸⠀⠋⠃⡀
⣉⣯⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡠⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠐⢒⡀⠙⢛⢃⣀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡆⠀⠀⠀⢀⣼⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⠀⠀⢠⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⢠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⣿⣿⣿⡟⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⢰⣿⣿⡿⣸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⣿⣶⣤⣀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⡇⢸⣿⡿⢡⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠃⣿⣿⣿⠃⣼⣿⢃⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⡀⠀⣷⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⣿⣏⣾⣿⠟⠁⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡀⢿⡆⠀⣼⣷⡀⠀⠘⣿⣿⢠⣿⣿⡿⠋⠀⠀⠀⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣇⢸⣿⡀⣿⣿⣷⡀⠀⣿⡇⣸⣿⠟⠀⠀⠀⣴⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣆⢿⣧⠹⢿⣿⡇⠀⣿⣧⣿⠏⠀⠀⢠⣾⣿⣿⣿⣿⣿⠃⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⡈⠙⢿⣿⣿⣧⢸⣿⡇⠀⣿⣿⡟⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣼⣿⣷⡀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣦⡄⠈⢿⣿⣇⠹⡿⠀⣿⣿⠃⠀⣸⣿⣿⡿⣹⣿⣿⣿⠇⠀⠀⠀⠀⣿⣿⣿⣧⠀⡄⠀⠀⢀⣴⣿⣿⣿⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⣤⡀⠙⣿⣆⠀⠀⣿⡿⠀⢰⣿⣿⣿⢡⣿⣿⣿⠏⠀⠀⠀⠀⠀⢻⣿⣿⡏⢰⡇⠀⢠⣾⣿⣿⣿⣿⠇
⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⣦⡘⢿⣆⠀⣿⡇⠀⣿⣿⣿⢃⣿⣿⣿⠏⠀⢰⣄⠀⠀⠀⠘⣿⣿⣇⣿⠁⣰⣿⣿⣿⣿⡿⠃
⢸⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣆⣿⡇⢀⣿⣿⠇⣼⣿⡿⠃⠀⠀⣼⣿⣧⠀⠀⠀⠘⣿⡿⠃⣰⣿⣿⣿⠿⠋
⠈⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠐⣶⣶⣤⣤⣄⣀⠀⠀⠀⠹⣿⣿⡇⢸⣿⡟⣸⣿⡟⠁⠀⠀⠀⣿⣿⣿⠀⠀⠀⣴⠟⣡⣾⣿⠿⠋⠁⠀⠀⠀⣀⣠⡀
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠈⢿⣿⣿⣿⣿⣷⣦⡀⠀⢻⣿⡇⢸⣿⣷⣿⠏⠀⠀⣤⡀⠀⣿⡿⠃⠀⣠⣾⣷⡾⠟⢉⣀⣤⣴⣶⣾⣿⣿⣿⠏
⠀⠀⠘⢿⣿⣿⣿⣿⢿⣿⣿⣷⡄⠀⠻⣿⣿⣿⣿⣿⣷⣄⠘⣿⡇⢸⣿⣿⠃⠀⠀⠀⢿⣿⡔⠟⠁⢀⣾⡿⠟⣉⣴⣾⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠈⠻⣿⣿⣿⣷⣍⡻⢿⣿⣦⠀⠙⣿⣿⣿⣿⣿⣿⡆⢻⡇⢸⣿⡇⣀⠀⠀⠀⠘⣿⠇⢀⣴⡿⠋⣠⣾⠿⣛⣯⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣦⣝⢻⣷⣄⠈⠻⣿⣿⣿⣿⣿⣼⡇⢸⣿⡆⣿⣷⡀⠀⠀⣟⣰⡿⠋⣤⠞⣩⣴⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣷⣄⠈⢿⣿⣮⡻⣿⣿⠈⣿⡇⠘⢿⣷⠀⣰⣿⠏⠀⢪⣴⣿⣿⠟⠛⠉
⢤⣴⣶⣶⣾⣷⣶⣶⣶⣶⣤⣤⣀⡉⠙⠻⢿⣿⣷⣄⠙⢿⣿⣿⣿⡇⢻⣿⠀⠈⠻⢠⡿⠁⢀⣴⣿⠟⠉
⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡉⠻⣿⣷⣄⠙⢿⣿⣿⡘⣿⡆⠀⠀⣿⠁⣰⣿⠟⠁
⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣭⣝⣋⣛⠛⠻⠿⢶⣌⠹⣿⣧⡀⢻⣿⣧⢹⣷⠀⢸⠃⣴⡿⠁⢀⣤⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿
⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡶⠀⠀⠈⠻⣿⣄⢻⣿⡄⢿⡀⡏⣰⡟⠀⠴⠿⠿⠛⠁⢀⣴⣶⣿⠟⠀⣀⠔⢀⣴⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠹⣿⡆⢿⣷⠸⣷⣱⣏⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣁⠴⠚⠁⠶⠿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣤⡀⠹⣿⡘⣿⠀⣿⣿⠋⠀⠀⠀⣀⣤⣴⣿⡿⠭⠤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣷⣿⣇⢻⣶⣿⠃⠀⣠⣴⠿⠛⠉⠁⠀⠀⠀⠀⢺⣿⣿⣄⠙⠓⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⢸⣿⠇⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⣿⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢷⡄⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⡄⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢳⠂⠀⠀⠀⠀⠀⠀⠡⡀⢲⡀⣰⢀⡅⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣙⠢⣑⣘⣜⡐⠉⠴⠂
⠀⠀⠀⠀⠠⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢒⣽⢿⢿⠒⣒⠒⠂
⠀⠀⠀⠀⠀⠢⠆⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⢻⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⡰⢃⠂⡇⠑⡌⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⣠⡄⠀⠀⣏⠰⢿⣇⠀⠀⠠⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠂⠉⢓⠋⠁⠐⠀⠀⠀⣿⣿⠏⠀⣀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠨⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠂⠀⠈⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠠⠀⠉⠢⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠄⠀⠀⠀⠛⠀⠀⠀⢀⡍⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠃⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠈⠒⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠍⠀⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⢸⡀⠀⠀⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠐⠄⠀⡀⠀⠀⠀⠀⠀⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡇⣰⠀⠀⢸⡇⠀⠀⠈⢸
⠀⠀⠀⠀⢶⣦⡂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢧⠆⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢪⠉⠀⠄⢸⡇⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠇⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⣀⣭⡆⠀⡄⠂⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠰⠂⠈⣄⡀⠀⢸⡇⠀⠀⢀⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⣶⠆⠀⠀⠀⠠⡝⢻⡇⠀⠀⠆⠣⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⣁⠀⢸⡇⠀⢀⠄⢠⣤
⠀⠀⠠⠀⠐⠠⠀⢀⡀⠀⠀⠀⠘⠀⠰⡿⠂⠀⠀⠀⠀⠀⠀⠀⠁⠉⠃⠀⠀⣆⡄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⠀⠀⠹⣄⣿⡇⢀⡎⠀⠀⠙
⠀⠀⠀⠂⠀⠀⠁⢸⠃⠀⠒⡠⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠻⣿⣇⡞
⠀⠀⠀⠀⠀⠀⠐⠚⠐⠂⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠸⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠀⠂⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢣⠀⠀⢀⣠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠸⠀⢈⠀⡔⠁⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠈⢢⠀⠀⣀⡄⠻⡲⣕⣾⡰⠧⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⢸⢠⠋⠀⠀⠀⠈⡩⠯⡖⠉⠁⠉⠑⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠐⠒⢤⠙⣤⣀⠴⢁⡤⠊⣀⠠⠬⢕⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠄⠠⠷⣜⠀⡰⠻⠛⠭⡑⠄⠀⢀⠱⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠐⠳⠀⣸⠊⠀⠸⡀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠈⣆⠀⢀⡐⢤⣴⠗⠉⢉⠇⠈⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣈⠢⣗⡔⠁⡄⡸⠋⢰⣌⢡⠊⠉⠀⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠣⢄⣀⣱⠾⣧⡀⠃⠃⢠⣜⢼⡶⠤⡴⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣍⣿⡟⠰⠀⡇⠘⣄⡥⢎⠀⢀⡹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠉⡟⠲⡄⠡⣀⡿⢀⠄⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⢒⡩⠤⣀⢱⠀⣹⡗⠁⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣎⣀⡀⠴⣋⠻⡀⠈⠉⡩⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠉⠆⠙⢄⠀⡁⢘⣠⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠉⠑⠪⣢⡄⣴⣭⣛⣺⣾⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣒⣂⣉⠤⣏⠁⠀⡠⠤⠤⢬⠖⠒⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡠⠁⠰⠁⠱⠀⣇⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠊⠀⠀⠈⢢⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣿⣿⣿⣻⣟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠈⠢⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢁⣀⠄
⠀⠀⠀⠀⠈⠢⡀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡤⠞⠪⢊⠛⠒⣂⠄⡆
⠀⠀⠀⠀⠀⠀⠈⠲⣴⡧⠤⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⠙⠌⣀⢸⣷⡬⠼⠃⠀⠀⠀⠂⢾⣙⢠⡏⢸⢸⠇⠘
⠀⠀⠀⠀⠄⠀⢀⡀⢻⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⠉⠻⠥⣢⡈⠛⡩⢁⠀⢀⠀⠀⠀⠀⠢⠬⠿⢹⠼⠬⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠀⠀⠀⠀⠀⣀⢠⣤⡀⠉⢱⠛⠋⠅⠭⠅⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣙⣔⡀⡙⠛⢙⠢⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⡙⢷⡀⠉⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠠⠺⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠋⠛⠂⠀⠀⠀⠀⠠⠀⠀⠄⠑⠠⢌⡞⢁⡄⠁⠀⡀
⠀⠀⠀⠀⠀⠠⠢⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣄⡀⢀⡀⠀⠀⠀⠀⠀⠒⢤⣀⢀⢰⠉⡠⢁⣤⠘⠉
⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⣠⣰⠲⠆⠬⢶⡾⣿⠀⠀⠀⠀⠀⠀⢀⡐⠉⠑⣯⣦⡷⠃⠄⠁⠁
⠀⠀⠀⠨⢄⠶⢐⠀⠉⠁⠆⠀⠀⠀⠀⠀⠀⣠⠤⠲⢌⣳⣶⢞⣻⣿⣝⠄⠀⠀⠀⠠⠐⢄⠀⣶⡼⣿⢿⠓
⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣈⡺⢿⠋⠤⡟⡀⠿⠀⠀⠀⠀⠀⠀⠐⠡⠈⡡⠼⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠐⠄⠋⡬⠹⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣭⣴⣭⣯⣽⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡄⢀⠀⣠⣀⡆⠀⠀⠀⠀⠰⠶⣶⣄⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣄⠈⠠⣌⠉⠀⠀⠀⠀⠀⠀⠀⠸⢿⡏⠓⢐⠀⠹⠃⠀⠀⠀⠒⠀⣠⣯⣀⣤⣄⠀⠀⠀⠀⠀⠀⠈⠉⠉⢢
⠀⠀⠀⠀⢀⡀⠹⣦⠺⠟⠀⠀⠀⠀⠀⠴⢲⡆⠀⠈⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⢠⡿⡿⠟⢿⣿⠇⠀⠀⠀⠰⠃⠀⠀⠀⠀⢠⠐⠄
⠀⠀⠠⣾⡟⠻⡷⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⢀⣾⣿⡄⠀⠀⠀⠀⠡⠆⡀⠐⠂⠀⠀⠄
⠀⠀⠈⠣⡁⠴⠁⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠈⠁⠀⠈⢳⠀⠀⠀⠀⠀⣠⣶⣾⣿⠿⠿⠿⠃⠀⠀⠂⠀⢠⣌⡤⠀⠀⠀⢠
⠀⠀⠀⢀⡀⠀⣰⣿⠿⢿⡷⣄⠀⠀⠀⠀⠊⠁⠀⣸⣀⠰⠴⠞⠀⠀⠀⠀⢰⢟⣵⡿⠷⡄⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠠⠾⠀⢀⡆
⠀⠀⠀⠀⠁⢸⡟⣥⠀⠸⢃⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠘⢸⣿⠃⠀⠁⠀⠀⠀⠀⠀⠀⢸⣿⡧⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠈⠀⠈⠓⠒⠛⠁⢀⡀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇
⠀⠀⠀⢀⠠⠤⠄⠀⠀⠀⠀⡠⠄⣽⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢲⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⢦⣤⡀
⠀⠀⠀⠃⣶⣦⠀⠀⠀⠀⠸⣴⣦⡾⠃⠀⠀⠀⠀⠀⠀⠀⢰⣯⠤⡀⠻⠉⠙⣷⠀⠀⠀⠀⢤⡤⢄⠀⠀⠀⠀⢸⡀⠀⠀⠉⠙⠂
⠀⠀⠀⠀⠈⠘⠀⠤⠀⠀⠀⠀⠀⠀⡇⠀⣴⣄⠀⠀⠀⠀⠏⣠⠄⠰⠞⠀⠀⠿⠀⠀⠀⠃⠻⠷⠞⠀⠀⠀⠀⠀⣩⠤⠴⠉⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⢳⡅⠀⣤⣄⠀⠀⠀⠀⠀⠀⠃⠀⠸⠻⠂⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣠⠄⢠⣄⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠸⠃⠀⠐⠋⠛⠀⠀⢀⡀⠤⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⠀⠀⣤⠀⠀⠀⣠⠶⠟⠛⠛⣿⡿⠶⡀
⠀⠀⠀⠀⠀⠒⠚⠁⠀⠀⢀⣼⣿⠷⠀⠀⠀⠀⠀⠘⠁⠀⠀⢠⠀⠀⠀⠻⡀⠀⠀⠀⠀⠡⡴⠂⠀⠀⠠⠚⠁⠀⠀⠀⠀⠚⠁⠀⠉⡄
⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⣼⠟⠁⢀⣤⡄⠀⠀⠀⣰⠶⣄⢀⣼⠀⣀⠀⠀⠈⠢⣀⣀⣤⠶⠃⢀⡠⠀⠀⢲⣶⣄⢠⡾⠛⢷⠀⠀⠀⠁
⠀⠀⠀⠂⠀⠀⠀⢀⣴⠟⠁⣀⠈⡾⠍⠡⠀⠀⠀⠃⣀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠙⠿⠈⢄⣀⠀⠀⣀⠴
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠂⠁
⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⢠⣤⢀⣴⣿⢋⡉⠀⠀⠉⠀⠀⠀⠀⢀⣤⠶⠄⠀⠀⠀⠀⠀⢘⣧⣄⡀⠀⠀⠀⠀⠀⣠⢴⣦⣤⡀
⠀⠀⠀⠠⡄⠀⠘⠈⢿⣼⣿⠂⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⡍⠀⠀⠀⠀⠀⠀⠠⠿⠿⣿⠠⡀⠠⠀⢀⡴⠀⠀⢀
⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣤⡀⢳⣶⡶⠂⠀⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠿⠲⣄⣉⣁⣀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠒⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠉⠠⡀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠸⣆⣀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠇⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠘⡻⠋⡀
⠀⠀⠀⢀⡀⠀⠈⢀⠙⢤⠘⠿⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⡦⠀⠀⠀⢀⡀⠀⠀⢦⡀⠻⠃⠀⠀⠀⠀⠀⠀⣀⣁⣹⣦⣄⠈⠁
⠀⠀⠀⠉⠉⠀⠀⠈⢡⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠈⣡⣴⣶⣤⡙⡄⠀⢐⣦⡀⠀⢠⠞⠉⠉⠉⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⣼⠀⠀⠀⠀⠀⠀⠀⢀⠔⣡⠊⠀⠀⠀⠀⢸⣿⣥⣌⠻⣿⣿⠀⠉⠋⠀⠀⠇⠠⣾⣶⠀⡀⠄⠀⠁⡄
⠀⠀⠀⠀⢀⣀⠀⠀⣼⣴⢃⠀⠀⢀⠀⠀⠀⢠⢋⡴⢁⠀⢄⡀⠀⠀⠀⠛⢛⡃⠀⣿⣏⡀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠢⡀⣠⣇
⠀⠀⠀⠀⠘⠛⣠⣼⢿⡋⠀⠀⢠⡄⠀⠀⠀⣿⡟⠏⠀⢠⡀⣸⠀⣠⡤⢄⠈⢁⣶⡟⣉⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⢧⠠⠄
⠀⠀⠀⠀⠀⣰⡟⢡⠁⢹⡄⠀⠀⠁⠀⠀⢠⣿⡁⠀⠀⠀⠀⠁⠀⠻⠶⠀⣰⣿⣿⡿⠿⣿⣷⣞⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠐⣠⣄
⠀⠀⠀⠀⠀⣿⡇⣮⠀⢸⡷⠀⠀⠀⠀⠀⢸⣿⣿⣄⠀⠀⠀⠀⠀⠀⣀⢀⣿⣿⠋⢠⣾⣿⣿⣿⠀⠀⠀⠀⠀⣿⡇⠀⠐⠆⡐⠹⠿⠁
⠀⠀⠀⠀⠀⢿⣧⣻⣀⣾⠃⠀⠀⠀⠀⠀⠈⢿⣷⡙⢷⣦⣀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠻⠿⠿⠋⠀⠀⠀⠀⠀⠹⣷⣄⡠
⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣽⣿⣿⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡿⢿⡿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠐⢀⡀⠀⠀⢠⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⣀⣠⣤⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠐⠀⠞⠁⠈⠀⣀⠀⠀⡴⣌⠂⢄⡈⡙⠛⠋⠉⠉⠀⠀⠠⣄⣀⠈⠀⠈⠉⠒⠒⠀⠀⣔⠒⢄⠀⡀⠀⢀
⠀⠀⢠⠀⠳⣬⡛⠻⢋⢙⡁⣌⠀⠀⠀⢰⠀⢂⣠⠭⠤⠆⢀⠐⠉⢠⠀⣀⠩⢀⢀⠀⠄⠀⠁⠀⢠⡀⠀⡀
⢀⠀⠺⠷⣥⣤⠔⠊⢱⡟⠀⢸⡌⠲⢤⣴⣾⢯⣤⣦⡤⠄⠐⠒⠒⢄⠱⡝⠳⣤⣒⡉⠀⢀⠈⠀⠀⡁⠐⡄
⠸⣮⣧⣀⠋⢡⠊⠀⢸⢰⡇⠀⢿⡄⠀⠲⣭⣇⡘⢯⠙⣷⣦⠠⣀⠰⡦⣬⣀⠀⠉⣉⣃⣠⠤⠖⠖
⠀⢀⡵⡣⠠⢷⠆⢠⠾⢸⣷⡀⠈⢻⡄⠀⠀⢉⠻⣮⠁⠘⠇⢷⠄⠑⡞⣦⠉⣙⠳⢌⠓⠦⣤⣉⠉⠉⠉⠆
⠀⠋⢠⡇⠀⣿⡄⢸⡄⢸⡯⣷⠀⢠⣷⠈⢀⠀⠁⠏⡄⢀⠀⠸⠠⠀⠛⠈⢧⡈⠁⠂⠄⠀⠀⠨⠛⢦⡀
⠐⠀⠘⡇⠙⢸⡇⠸⣷⡈⣧⢻⠀⢘⡍⠀⠠⠀⠀⠠⢧⡀⠈⢄⡀⠀⢠⣌⣀⠹⠦⠀⠘⠀⠀⠜⠀⠀⠑⡄
⠀⢠⠃⣇⡜⢸⠃⡄⠌⡅⢻⠀⠀⢈⡇⠀⠀⠀⠀⠂⣸⠃⢤⠀⠀⣄⠚⢮⡙⠓⢦⡀⠐⠤⠤⣤⡀⠐⠀⠁
⠀⠀⠀⠻⡀⠀⠐⠘⠀⠈⠞⠘⠀⢸⢣⡄⠀⠂⠀⠀⢻⡐⠈⠂⠀⡟⠓⢤⡛⣦⠀⠁⢀⡂⠐⠀⠙⢆
⠀⠀⠂⠄⠀⢠⠘⡆⢸⠆⠘⡄⡼⠀⡜⣧⢀⠀⠀⠀⠠⠀⠄⢡⠸⡧⡄⠈⠉⠈⢳⡀⢠⡀⠀⠘⠀⠁
⠀⠀⠁⠀⠀⠀⣼⠃⠈⡀⢀⣼⢃⡜⢅⢹⡄⠉⠩⣭⣶⣀⡒⠈⠀⠁⢷⠄⠘⠆⠢⠹⠀⠝⢧⡀⡀
⠀⠁⢀⣀⣀⡴⡿⢠⠀⠁⡄⠈⢈⢀⡏⢸⡇⣶⣄⠀⠤⡈⠙⠀⢀⠀⠈⠂⠀⠀⠀⠀⠀⠐⡁⢷
⠀⠒⠀⣈⣉⠔⠁⢸⡄⠄⢁⡠⡺⣸⠃⢸⣧⡇⢙⣦⠀⠀⢀⠀⡈⢷⣤⡀⠀⠀⡀⢠⠇⠀⠺⡌
⠀⡀⠀⠁⠀⠀⡴⢸⠁⢘⡿⢠⠃⢰⠇⡿⠛⠀⢈⢎⠀⠠⡄⠀⠙⢸⣇⠙⡆⠀⢀⡾⢰⡀⢀⠈
⠀⠁⠄⠀⠀⢸⠃⠂⠰⠛⢠⠃⢠⣼⠘⢡⠶⡄⠀⠸⡄⠄⠀⠀⠀⣾⡙⡇⠀⠀⣼⢇
⠀⠀⠠⠀⠀⠏⡂⠀⣂⠔⡁⣠⢘⠘⢠⡏⠠⢿⠀⠠⢠⡠⡄⠀⠀⠹⣇⠀⠠⠀⣿⠀⠂
⠀⠀⠀⠀⣢⢞⠀⠀⢡⠊⡇⠸⠃⠄⠘⣇⠐⠈⠀⡀⢸⣇⠈⠈⠀⠀⠉⠀⠀⠀⢿⡀
⠀⠐⣠⣶⠏⢈⠀⠠⢡⣰⣿⠀⠀⠐⣀⠘⢆⠐⣼⠃⢦⡘⣧⡑⢶⠀⠀⠀⠀⠀⠀⠁
⠀⠀⡿⡏⠀⢞⠀⠀⡀⢹⡌⠇⢄⣴⣧⢠⠀⣼⢃⠀⠀⢭⣿⡝⠒⠆⢀
⠀⠀⠀⢷⠀⠘⠀⠀⠃⢸⠁⢠⡞⢹⢹⣆⠠⡟⢸⣷⡀⠀⠬⠙⣧
⠀⠀⠀⠈⠃⠀⠰⠀⠀⠀⠂⡏⠂⢘⠀⠹⠀⠇⠈⣯⣷⠠⠈⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠔⠆⠀⠀⠀⣿⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢢⠀⠀⢢⠘⠀⠃⠿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣋⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠐⠐⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠤⠀⠀⠒⢀⠀⠀⠒⠂⠀⠠
⠀⠀⠀⠀⠁⠀⠈⠀⢀⡀⠐⠀⡀⠀⣀⠀⠀⠈⠉⡀
⠀⠀⠀⠀⠠⠀⠀⠀⡀⠀⠠⠀⠁⡀⠉⡄⠀⠀⠀⢅⠀⠁⠀⠄⠀⠀⠠⠊⢀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠠⠄⣀⣁⡀⠀⠀⠀⠀⠀⠀⠈⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶
⠘⢀⡀⠀⢀⠀⢀⠀⠙⠉⠁⠀⠠⠛⠀⠀⠀⠀⠘⠣⠄⠀⠀⠄⠀⠄⠀⡀
⠀⠀⠈⠀⠁⠀⠀⠁⠁⠤⠤⠐⡀⠡⡔⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⠁⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠠⠌⠀⠒⠀⠀⡀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠠⢆⠀⡀⠐⠀⠠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⢠⠎⠀⠁⢈⠒⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⠄⠄⢗⡌⠀⠐⠀⠰⠂⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣡⠀⠊⠀⢉⡄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢃⠈⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠘⡀⠀⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠁⠐⡀⠤⢀⠀⠴⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⡀⠀⠀⠀⠀⠀⠀⢠⠤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⡀⣀⠔⠓⠀⠐⠁
⣶⣶⣶⣶⠶⣶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⡶⠶⠶⣶⣶⠶⢶⡶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⢀⣤⡀
⠀⠀⢠⡆⠀⢠⡶⠀⠀⠀⣦⡄⠀⢀⣀⠀⠀⠸⡄⡏⡄⠀⠀⠀⠿⣿⣤⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠘⢷⡀⣼⠟
⠀⢀⠘⡇⡴⢋⡀⠀⣀⡀⠹⠿⣾⡿⠟⠀⢀⠳⣜⢊⡴⢀⠀⢀⣠⡛⠁⠀⠀⠀⣴⣿⣷⣦⡄⠀⠀⠀⠀⠈⠳⢁⣴
⠀⠘⣧⡈⢠⠾⠁⡄⠻⣷⣄⠀⣠⣴⡦⠀⣬⢷⡈⡡⠴⢋⡄⠘⠿⣿⣿⠀⢠⣷⣹⣯⣿⣾⣽⣷⡄⠀⠀⣈⢣⡿⠋
⠀⠀⢨⠛⠆⣠⡾⠃⠀⣨⡍⡀⠛⢋⠀⢀⡙⢇⢹⣠⠶⢟⠀⠀⢀⠀⡀⠀⣼⡛⣿⣿⣿⣿⣿⣻⣥⠀⠈⣿⢿⠁
⠀⠀⠸⣦⡀⠋⠀⠀⠀⠘⠿⣧⣾⡿⠃⠀⡳⡌⣯⣶⠞⠁⠀⢾⡿⣿⣿⠀⣻⠿⣯⣿⣿⣿⣿⢿⣿⢃⣀⠈⢿⢋⣤⡄
⠀⠀⠀⠈⠳⣠⡄⠀⠀⢿⣦⣏⣴⣶⠀⠀⠳⣘⣡⠆⠀⠀⠀⢀⣀⠀⣀⠀⢿⣋⣿⡟⢻⣏⣹⣿⣦⠈⠿⡀⡸⠚⡋
⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠙⠟⠛⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠸⢿⠿⠿⠂⠀⠉⠉⠀⠀⠻⠿⠿⠃⠀⠀⠰⢡⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢏
⠀⠀⠀⣶⣄⠀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣿⣆⢀⣤⣤⠀⠀⠀⠀⠀⠀⣿⢀⡄⠀⢸⣧⠀⠀⡀
⠀⠀⠀⢻⣿⠆⠃⣀⡀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠈⢁⣈⠛⠋⠀⠀⠀⠺⣦⡀⢹⡿⠃⠀⠀⠙⠀⣾⠇
⠀⠀⠀⠀⠉⣥⣾⣿⠇⠀⠀⠀⠘⢿⡇⣠⣴⣶⠀⣤⡄⠸⡟⢷⡶⠀⢀⣤⣤⡬⢥⡄⣀⣀⣄⠻⣶⡤⡄⠀⣰⣿⠂
⠀⠀⣀⣤⣀⠘⠋⠀⣀⣀⠀⢀⣄⠀⡀⣿⡿⠃⠀⢙⣥⡄⠀⢰⣶⡆⠈⠛⣡⡀⠀⠱⠀⢉⡋⠀⠀⠀⠬⣀⠛⢁
⠀⠀⠻⣿⡿⠿⡇⣾⣿⡿⠂⠸⣿⣷⠸⡇⢀⣴⡆⠈⠻⠓⢤⠘⠛⡁⠀⠘⢿⣧⠀⢆⣾⡿⠃⠀⢀⣤⣀⢸⠐⠁
⠀⠀⠀⠀⠀⣀⣸⠈⠉⠀⠀⠀⠉⠻⡄⣶⣿⣿⠟⠀⠀⠐⣄⢀⠞⠀⠀⠀⠀⠉⢠⠋⠁⠀⠀⠀⠀⠙⢻⠘
⠀⠀⠀⠰⣾⣿⡿⣠⣴⣆⢸⣿⣷⡄⢸⠛⠋⠁⠀⠀⠀⠀⠈⡟⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⡄⡞
⠀⠀⠀⠀⠈⠁⠀⠟⠛⠁⠀⠙⠛⠓⠷⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐⢀⠏⠀⠀⠀⠀⠀⢠⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠠⠴⠶⢶⣶⣶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠙⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠈⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣷⣄⠀⠈⢿⣿⣿⣄⠀⠀⠀⠀⢀⣴⣿⣿⡟⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣷⡀⠘⣿⣿⣿⡄⠀⠀⢀⣾⣿⣿⡟⠀⠀⠀⠀⣠⣴⣾⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⡄⠈⣿⣿⣷⠀⢀⣾⣿⣿⡟⠀⠀⠀⣠⣾⣿⣿⣿⠏
⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣷⣶⣄⡀⠹⣿⣿⣿⣷⠀⠸⣿⣿⡄⣼⣿⣿⡟⠀⠀⢠⣾⣿⣿⣿⣿⣿
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠘⣿⣿⣿⡆⠀⢿⣿⣧⣿⣿⡟⠀⠀⣠⣿⣿⣿⣿⣿⣿⠇⠀⡄
⠀⠀⢸⣿⣿⡿⠋⠁⠀⠈⠉⠛⢿⣿⣿⣆⠘⣿⣿⡇⠀⢸⣿⣿⣿⡟⠀⠀⢠⣿⣿⣿⣿⣿⣿⠋⠀⣰⡇
⠀⠀⠸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆⠸⣿⣧⠀⠈⣿⣿⣿⠃⠀⠀⣿⣿⣿⣿⣿⠟⠁⢀⣴⣿⡇
⠀⠀⠀⠹⡇⠀⠐⠒⠲⠶⠶⣶⣶⣤⣀⠙⢿⡄⠹⣿⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⣿⡿⠋⣠⡄⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢷⣌⠻⡄⢻⡇⠀⣿⣿⡇⢰⡄⢸⣿⣿⡟⢠⣾⡟⢡⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀⠙⠳⣼⡄⢻⡄⢸⣿⣧⢸⡇⠈⣿⣿⢠⣿⠋⣠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⣄⢳⠸⣿⣿⢸⣇⠀⢹⣿⢸⠇⣴⣿⡿⠃⢀⣴⣶⣶⣶⣄
⠀⠀⠀⠀⠀⢠⣿⡿⠟⠋⠉⠉⠉⠉⠉⠙⠻⢿⣿⣦⡙⢧⡀⢿⣿⡀⣿⡀⠀⢿⠀⣸⣿⡟⠁⠀⠸⠟⠉⠈⠉⢻⣧
⠀⠀⠀⠀⢀⡿⠋⠀⠀⠀⢀⣤⣶⣶⣶⣶⣦⣄⡈⠛⢿⣎⢳⡘⣿⡇⠸⣧⠀⠀⠂⣿⡟⠀⣦⡀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠘⠁⠀⠀⠀⣴⣿⣿⡿⠿⠿⠿⣿⣿⣿⣷⣄⠙⣷⡅⢹⣿⡀⢿⡆⠀⠀⣿⠁⠀⣿⣷⣤⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣌⢻⣄⢻⣧⠘⣿⡄⢤⡈⠀⠀⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣄⣀⣾⣿⣷⡄⠀⠀⠈⢻⣿⣿⣧⡹⡎⢿⡆⢹⣷⠸⣇⠀⠀⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠙⢿⣿⣷⣿⠘⣿⡀⢿⡇⢿⡄⠀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡇⢻⣇⢸⣿⢸⣇⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣶⣶⣤⣀⠈⢻⣿⡘⣿⠀⣿⠘⣿⠀⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠋⢁⣉⠉⠻⢷⣄⠹⣧⣿⡀⣿⡄⣿⠀⠀⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢰⣿⣿⣿⣶⡄⠻⣷⢹⣿⡇⣿⠀⣿⠀⢠⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⣀⠉⢉⣿⣿⡿⠀⢹⣧⣿⡇⣿⢸⡿⠀⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠿⠁⠀⠀⣿⣿⣷⣿⣾⠃⣼⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠁⠀⠀⠀⠀⢹⣿⣿⣿⡏⣼⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⡜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡔⣀⣀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠁⠁⢀⠀⠀⠀⢀⡠⢊⣀⡠⠂⡀⠀⠀⣚⡫⣇⣤⣠⣄⠨⠄⠀⣃⡀⠀⢀⣀⣀⣀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⢦⡐⠖⡉⠛⣷⠀⠀⠻⣅⠈⠩⢅⠈⠓⠲⠌⠉⠀⠈⠙⠀⣞⠁⠀⠂⠀⢍⠲⠭⠉⠉⠉⠩⢏⠂⠋⡇
⠀⠀⠀⠀⠀⢀⠉⠀⠉⠐⠒⠁⡀⠀⠘⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠠⠃⣀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢰⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠠
⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡅
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠨⠻⠿⠂
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠄⢁⠀⢘⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡀⠈⠀⠠⣀⠀⠀⢢⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠬⢥⠴⠤⠤⠤⠮⠤⠤⠴⡿⠄⠀⣶⣾⡆⠀⠀⠤⠤⢒⠒⡶⢐⠿⣄⠀⠀⠙⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠉⠉⠀⡐⠀⠀⠀⠠⠔⠉⠐⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣂⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣬⠇⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⢠⡄⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐
⠀⠀⣻⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠙⠁⠀⠀⠀⠀⠋⠠⡥⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⡤⠄
⠀⠀⠻⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣄⠀⠀⠀⢀⣀⣀⠀⡀⠓⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⣀⠀⠈⠰⠦
⠀⠀⠰⡀⠀⢠⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠨⣿⣿⣿⡿⠿⠟⠻⢿⣿⣦⡄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⠃
⣦⠀⠴⠄⠀⠈⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠙⠀⢀⣄⣀⠄⠀⠙⠛⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁
⣿⠙⠒⣦⠀⡘⠀⠀⣬⠀⠈⠆⠀⠀⠠⠀⠀⠀⢀⡀⠘⠀⠀⠀⠀⠉⠁⠀⠀⠀⠜⠢⠀⠰⠀⠀⠀⠐⠀⢠⠀⠀⠐⠀⠀⠤⢀⠆⠠
⢻⢀⠀⠈⠀⠃⠀⠀⣽⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸⣷⣺⣶⠃⠀⠀⠀⠀⠀⢀⠄⠀⠀⠄⠀⠇⠀⠀⠀⠄⠀⢀⠀⠀⠰⠤⠇⡉
⠀⢯⣠⡀⠈⠁⠀⠀⣽⠀⠈⣠⡀⠀⠀⠀⠀⠀⠀⠘⣿⡽⢀⡠⡶⢀⠀⠊⠍⠀⠀⠀⣔⠀⢠⣰⣿⡀⠀⠀⠀⠠⣄⠠⠇⣀⣄⠘⠃
⠃⠉⢀⠙⠄⠰⡆⠀⢺⣁⣼⠀⠀⠀⠀⠀⠀⠀⠠⠄⠙⣇⠬⠴⠃⠸⢂⠀⠀⢀⡞⢡⠘⠚⠀⣏⡣⠀⠀⠀⠁⠀⠈⠶⣌⠉⠀⠀⢴
⠀⣰⠏⢳⡀⠀⠆⠀⣽⠛⣿⣧⠀⡀⠀⠀⠀⠀⠀⡘⢀⠃⠒⠚⢃⣸⠏⣰⣤⣻⠂⣻⠀⢦⢠⣿⠃⠀⣄⠀⠀⠀⠀⠀⠈⠚⢁⠊
⠀⠁⠀⠀⣷⢀⣶⠄⣿⠀⡟⢹⣏⠁⠀⠀⢀⠄⣾⠇⣼⠆⠀⠀⢎⢈⠓⠈⠠⣨⡁⠚⣥⠜⠮⠝⠂⠀⠈⢢⠄⡀⠀⣀⠀⣀⠈⠐⠆
⡴⡄⠀⠓⠮⠋⠦⠀⣼⠰⠁⠙⣿⠳⡄⢀⠌⢠⡟⠀⠀⠓⢦⡇⡠⠈⡀⠀⠀⠀⠁⣞⣾⣷⣄⣠⠀⠐⢀⡰⢠⢴⠐⠣⠠⠘
⠀⢸⣦⣆⣠⡠⡾⠿⣿⠦⠃⢨⣟⠀⢶⠀⢰⡾⡶⠄⡄⠤⣭⢜⣓⢺⡏⠉⢀⡠⠀⠘⣺⣏⡀⡦⠀⢀⣉⡀⣼⡜⡙⠃⠀⠁⠀⠶⡀
⠆⠈⣥⣄⠐⢛⣿⣀⣿⡀⠐⠢⡌⠀⡈⠰⢦⣞⠁⢠⠓⠂⠀⠀⠀⢾⣁⣠⡜⠱⢾⡠⠝⠛⠛⣰⠆⡙⠛⠁⠛⠁⠒⣤⡳⢄⣤⣤⠙
⠻⢎⣿⣿⣡⣬⣭⡙⣿⢛⠀⠀⠸⠀⣷⣶⣾⣿⣷⣷⣶⠌⢀⠀⢴⡟⡱⠾⢆⣟⣻⣀⣗⠘⠋⠝⢨⡅⠀⠀⢀⣄⣀⠬⠁⣹⠘⠓⠶
⡖⠺⣟⡽⠿⢋⣙⣫⠁⡀⠀⢤⡄⢨⣯⣾⠽⠿⢿⡿⠀⠒⠤⠀⠘⣝⠿⢛⠈⢻⠿⢿⠏⠉⢰⠓⠃⢩⠑⢂⣼⠾⢻⣤⣺⣭⠶⣄
⠉⠴⣿⣂⢶⡄⢀⣤⢞⣿⣿⠶⣲⢹⣏⡈⢥⣣⠛⣟⣀⣤⡗⢠⣦⣯⡟⣁⢀⠘⠸⡏⠃⢀⠘⡁⠻⢻⠄⠛⢣⢛⣜⣷⢷⢿⡙⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡌⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⢀⡹⠁⠄⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠠⠀⠈⡟⠁⡀⠐⡤⠀⠠⠀⠈⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⣉⠀⡌⠁⠀⢳⡥⡞⠀⡥⢠⠆⠀⠈⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⢤⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠠⡱⣇⣅⣂⢴⠡⣀⢄⣠⠄⠐⢈⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠘⠀⠀⠀⠀⠅⣠⠖⠉⠀⠀⢹⠟⠉⠉⠙⡄⢈⠋⠠⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⠁⡀⢀⣡⡄⠀⠀⡞⠁⠀⠀⠀⢀⠃⠀⠀⠀⠀⠙⡅⠰⠂⠀⠠⠀⠀⢤⢀⠀⠄⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡲⠛⠁⠀⠀⣰⠷⠓⠂⢄⡀⠀⢸⠀⠀⡀⢠⣀⠴⠁⠀⠁⢂⠀⣰⠰⠟⠓⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠃⡠⠤⠀⡜⠁⠀⠀⠀⠀⣤⢖⠟⠙⠛⠫⡙⢥⣀⠀⠀⠀⠘⡌⠉⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢀⡄⠄⠀⠐⢉⠁⠀⡀⠀⠀⢸⣷⠁⡠⠀⡀⠀⠀⢂⢳⠠⣀⣠⠴⢧⠀⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠒⠘⠓⠚⠀⠀⠀⢼⣀⠀⠐⠂⠈⠐⢀⣼⣿⠚⠁⠀⠀⡼⠀⡐⠈⠐⠲⢤⣂⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⢀⡲⢆⠀⠀⠀⠎⢿⣶⡤⣄⢠⣤⡾⠏⠀⠈⠐⠒⣹⠰⣂⡀⢀⠂⠀⢢⠀⠙⠉⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⣤⣄⣆⢴⠁⠀⢀⡥⣖⠀⠀⠀⠀⠩⠋⠙⠈⠀⠀⠀⢹⠒⠊⠪⠼⣆⣲⠋⣘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠠⠜⠃⠚⢆⠀⠈⠀⠈⢄⡀⢀⣠⠂⠀⠀⠀⢀⣀⣠⣼⠳⣷⣞⠁⠈⠇⠋⠱⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠈⡄⠀⠁⠀⠄⠐⢩⡆⠠⠀⠐⠂⠀⠈⠁⠁⢠⡖⠒⢲⠉⢽⠅⡛⠀⣿⢫⠩⠁⠶⠀⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠀⠀⡤⠈⠀⢀⠣⣌⡾⢀⠤⢀⡠⡀⠀⠑⣷⡖⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠐⠀⢀⡀⠀⠁⠀⠛⢻⠀⠀⠀⠀⠇⠒⠀⠀⣳⣀⠠⠀⠀⠀⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠈⠛⠂⠀⠀⠀⠐⠀⠀⠀⣆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⣲⡦⠀⠀⠀⠀⠀⢀⣀⣠⠈⣐⠖⠀⠀⠀⠀⠀⠀⠀⣀⣔⣾⢿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⡿⠃⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣋⣿⣥⡿⠯⠀⠐⣀⣰⣦⣤⣈⣿⡍⠈⠀⠀⠀⠀⠀⠀⠀⠘⠋⢀⢠⣼⣿⣶⣶⣶⡆⠀⠀⠾⠿⠿⠋⠋⣰⣤⣿⣿⣧⣤⣀
⠀⠀⠀⠀⠀⣀⡄⣠⣀⡉⠛⠋⡟⠉⠉⠀⠀⣺⣿⣿⡻⣟⣷⣻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡤⣮⣿⣿⣿⣿⣾⣿⣷⡀⠀⠀⠀⠀⣲⣿⣿⢿⣷⣿⣿⣿⣿⡁
⠀⠀⠀⠐⣰⣿⣛⣷⣽⣿⡆⠀⠃⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣷⣿⣿⠷⠀⠀⠀⠀⠀⢀⣾⣵⣿⣿⢿⣿⣿⣿⣿⣿⣿⠂⠀⠀⣩⣿⣯⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⢀⢿⣿⣶⣿⣷⣿⣿⣶⡄⠀⠀⠰⢹⣻⣿⡏⢈⣿⣯⣽⣿⣿⣿⣿⠀⠀⠀⠀⠀⠰⣿⣿⡿⠷⣟⡽⠛⣯⣿⡏⠋⠀⠀⢸⡣⢿⣿⣿⣿⠿⣿⣻⣿⣷⡆
⠀⠀⠀⡾⣿⣯⣝⣏⣯⣿⡿⠀⡇⠀⠀⠉⠛⣿⡆⡆⢀⣽⣷⣾⠿⠟⠉⠀⠀⠀⠀⠀⠀⢀⡹⣀⠀⢸⠀⠀⠿⣿⡇⠀⠀⠀⢸⠈⠀⠙⢹⡇⠀⠈⠉⠋⠉
⠀⠀⠀⠙⢿⠟⢸⡁⠀⠉⡿⢿⣿⣶⠀⠀⠀⠉⠁⡇⢸⠙⣿⡇⠀⠀⠀⣀⣀⡀⠀⠀⠀⠻⢹⣿⡇⢸⠀⠀⠀⠀⠀⠀⣀⣤⣼⣄⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡅⢩⣉⣿⠀⠀⠀⠀⠀⡛⠈⠓⠛⠓⢠⣶⣿⣿⣿⣿⣷⣶⡀⠛⠹⠿⠦⢾⠤⠤⠤⠤⠤⠠⠨⠹⣿⣿⡇⠀⢨
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠓⠛⠿⠷⣾⣿⣿⢷⣶⢿⣤⣀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢀⡤⣿⣶⣿⣷⣿⣦⠀⢰⡴⣿⣿⣀⣀⣸⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣸⣇⣀⣀⣀⢀⡀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⡿⣃⣶⣶⣽⣲⣶⣿⣯⡿⣟⡡⠊⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⡟⠁⡰⠄⣿⣿⣿⣿⣿⣦⣽⣿⣿⣍⣉⣿⣾⡯⣥⣴⣶⡶⣶⣶⣶⣿⣿⣷⣾⣶⣄
⠀⠀⠀⠙⣿⣿⣿⣿⣿⡿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠳⣀⠉⢻⣩⠽⣉⡿⠬⠏⣀⠇⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣷
⠀⠀⠀⠀⣿⣟⣿⣿⣿⠟⠛⠛⠛⢛⣛⣩⣭⣵⣾⣿⣿⣿⣿⣿⣿⣷⣷⣶⣿⣶⣒⣻⣥⣾⣿⣿⣿⣿⣿⣷⣿⣿⢥⣟⣛⣻⣿⢿⣏⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠸⠯⠿⠿⠿⠤⢤⣴⣾⣿⣿⣿⣿⡿⠿⣿⣿⣿⡟⣻⢿⣛⣿⣹⡽⣯⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡟⣿⣶⣦⣌⠉⠙⢻⡇⣹⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡛⢚⣿⣿⣿⣿⣷⣿⣯⣿⣿⢿⣿⣿⣿⣿⣟⣿⡄⢹⣿⣿⣿⣟⣿⣿⣿⢿⡆⠀⠀⠁⠙⠛⠛⠛⠛⠛⠛⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⡆⢹⣿⣿⣿⠷⠮⣿⣽⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠿⢿⣿⣯⣽⣿⣿⡽⠟⠋⠈⠓⠛⠛⠛⠛⠛⠒⠒⠒⠚⠛⠒⠒⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⢀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⢀⡴⠞⢁⣠⠖⠋⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⢧⣀⣠⠞⣉⣤⠶⠋⣡⡶⢋⠭⠥⠤⠤⣄⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣉⡥⠴⠚⠉⠉⠉⠉⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⡀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠺⡂⢀⠀⠀⠊⠉⠀⠉⠉⠓⠦⣄⣉⡳⢤⣚⣩⠥⠒⣶⠠⠀⠀⢀⡤⢲⠆⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠃⠀⠀⠀⠐⠀⢀⣀⠤⠒⢉⣻⠶⣍⡙⠒⠋⠁⢀⡀⠀⠈⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠤⠴⠒⠉⠀⠀⠀⠈⠉⠓⠚⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⣤⡂⠀⠀⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡞⠀⠉⠑⠂⠠⡴⠐⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠠⡤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠲⢤⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠁⠀⠀⠀⢙⡶⣁⠠⠂⠉⠁⠒⠦⣀
⡠⠄⠀⠀⠀⠀⡔⠀⡀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣼⠀⠈⠙⠀⠀⠀⠀⢀⠈⠳⣅⠀⠀⠀⠀⠀⠀⠀⢀
⠃⠀⠀⠀⠀⠀⠑⠈⠀⠀⠀⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⣠⠎
⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣀⣀⣀⣀⣀⣀⣉
⣿⣿⡋⣋⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣌⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣌⠻⣿⣧⡀⠀⠀⠀⠀⢀⣾⡿⢱⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧⡙⣿⣷⡀⠀⠀⠀⣾⣿⢃⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣌⢿⣇⠀⠀⢰⣿⡟⣸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣶⣶⣶⣦⣤⣀⡀⠀⠈⠻⣿⣦⢿⡄⠀⢸⣿⠇⣿⣿⠏⠀⠀⠀⠀⠀⠀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣶⣮⣭⣛⠻⠿⣿⣷⣦⡀⠀⠙⠿⣧⠀⠘⣿⢸⣿⡏⠀⠀⠀⠀⣠⣴⡿⣻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣷⣦⣍⡛⢿⣦⡀⠀⠘⣷⡀⡿⣾⠟⠀⠀⠀⣠⣾⡿⢋⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⢿⣷⣮⣿⣦⡀⠘⣧⣼⠋⠀⠀⠀⣴⣿⠟⣰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⣀⡀⠀⠀⠀⠈⠉⠉⠉⠓⢾⡏⠀⠀⠀⣼⣿⠏⣼⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣝⡻⢿⣷⣦⣄⠀⠀⠀⠀⠀⠀⢿⡀⠀⢠⣿⢏⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣌⡻⢿⣿⣦⡀⠀⠀⠀⢸⡇⠀⢸⣏⣾⡿⠋⠀⠀⠀⢀⣀⣤⣴⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣦⡙⢿⣷⡄⠀⠀⠀⣿⠀⣬⡿⠋⠀⠀⢀⣠⣶⡿⢛⣵⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣦⣙⢿⡄⠀⠀⢻⡾⠋⠀⠀⠀⣴⣿⡿⢋⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣀⡀⠀⠀⠀⠉⠛⠿⣦⣿⡄⠀⢸⡇⠀⠀⢠⣾⣿⢋⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣯⣙⠿⣿⣦⣄⠀⠀⠀⠀⠉⠛⢦⣸⡇⠀⢀⣿⡟⣡⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣮⡛⢿⣷⣦⠀⠀⠀⠀⠀⢹⡇⠀⣸⢟⣼⣿⠟⠋⠀⢀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡙⢿⣷⡄⠀⠀⠀⢸⡇⢠⣿⡾⠟⠁⣀⣴⣾⢟⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣦⡻⣿⡀⠀⠀⢸⣷⠟⠁⠀⣠⣾⡿⢋⣵⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⢦⣄⡀⠀⠈⠛⠿⣷⣜⢷⡀⠀⢸⡇⠀⢀⣾⣿⠟⣵⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡝⢿⣶⣄⠀⠀⠈⠙⠻⢧⡀⢸⡇⢀⣾⡿⣡⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⡹⣿⣷⡄⠀⠀⠀⠀⠙⣿⠁⣸⡟⣴⣿⡿⠋⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡌⢿⣿⡄⠀⠀⠀⠀⣿⢀⣿⣾⠟⠋⢀⣤⣶⣾⠿⣻⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣎⢻⣿⡄⠀⠀⢠⣿⠞⠋⠁⣠⣶⣿⠿⣋⣵⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆⢻⡇⠀⠀⢸⡏⠀⢀⣼⣿⠟⣡⣾⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣎⣿⠀⠀⣿⠀⢀⣾⢟⣥⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄⢠⡏⢀⣾⣣⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣡⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃

⡀⠀⠀⡠⠔⠒⠒⠒⠢⢤⡀⠀⠀⠀⢹⠉⢃⠒⠦⣀⠀⠀⠀⠀⢠⡧⣀⡀
⠙⠦⡍⠐⠂⠶⠄⣌⣂⡐⠌⢢⠀⠀⡞⢀⠀⡆⠀⠈⠳⡀⠀⡰⠁⣄⠄⡈⢂
⠀⠀⠈⢦⡀⠀⠀⠄⠐⠙⡛⢮⣇⠀⢣⠀⠀⠹⣆⡠⠀⢧⠀⠃⠐⢧⠀⠀⠀⠃
⠀⠀⠀⠀⠈⠓⠤⠤⠬⠤⠒⠉⠈⠙⠲⢕⣦⣘⠛⢦⡀⠇⠀⠡⣀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠀⢒⠀⢄⣤⠞⠓⢬⣍⣻⡄⠀⠀⠈⠂⠤⠻⣄⠁
⠀⠀⠀⠀⠀⠀⢀⠜⢀⢀⣈⠖⠒⠋⠉⡸⠀⠀⠀⠀⠈⠉⠓⠒⠦⢄⣀⡀⠈⣧⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢸⠀⠠⢞⠀⠀⠂⢀⡴⠁⠀⠀⡀⠔⠊⠉⠁⣒⢺⠉⠈⠉⠉⠉⠉⠀⠀⠉⠉⠉⠓⠦⣤
⠀⠀⠀⠀⠀⠀⠈⢄⠐⢀⡠⠤⠔⠉⠀⠀⢀⡎⠀⠀⠀⢒⠴⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⡸⠁⠤⠤⠛⠋⢘⡴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠞⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⡹⣮⢦⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡈⢧⢳⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣄⠘⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣆⡀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⢻⡿⠷⠾⣿⣿⣿⣷⡄⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠦⠤⢤⣄
⠀⠀⠀⠀⠀⠀⣠⠴⠶⣿⠁⠀⣠⠚⠉⠀⠀⠀⠈⠻⣿⣿⣷⣆⢠⢬⢆⠀⠀⠀⠀⢀⠔⣫⠔⡈⠀⢂⣀⠤⠤⠼⠗⠒⠲⢷⣀⣠⡄⢀⣀⣠⢔⡿
⠀⠀⠀⠐⢶⡀⠁⠀⣰⠇⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣇⠂⠆⢣⡀⣀⣠⣿⢾⡷⠦⣶⣭⣥⣀⣀⡤⠀⠀⠂⠒⠀⢀⣸⠟⠉⣉⣕⣹⡇
⠀⠀⠀⠈⠒⠿⠷⢶⡿⠄⠴⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣶⣖⣬⣭⣥⣾⣇⠈⢢⡀⠈⠳⡀⠈⠳⣌⠉⢻⡛⠲⢟⡓⠒⠒⠚⠛⣺⠟
⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⣀⠀⠀⠉⠓⠦⣤⡀⠄⣤⣤⣤⣠⣿⣿⣿⣈⢦⣀⠢⠀⠉⢳⣦⣤⣀⣄⡉⠢⡀⠀⠙⠢⢽⠶⠤⠈⠑⣒⣠⡼⠛
⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠤⠟⠒⠒⠒⠈⠉⠛⣛⡿⠛⠛⠿⣿⣿⣿⣿⣿⣿⣷⣡⡀⢿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣷⣶⣶⣶⣖⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣏⠁⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⢚⣩⠿⠶⠶⢶⣶⣶⣶⣾⡿⠛⠁⠀⠀⠈⢿⣿⣿⣿⣦⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠙⠛⠶⣬⣙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣆⠰⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠉⠛⠶⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣟⡅⢡⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣾⡀⣇⠀⠀⠀⠀⠉⠈⠙⠿⠛⢿⣿⣟⣻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣯⡐⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣽⣿⣧⣠⡘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣯⣿⣿⣷⣝⢄⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠛⠉⠀⢻⣿⢻⣿⣦⡱⡈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⡶⣪⠏⠀⠀⠀⠀⠀⠙⣷⣿⣿⣷⣌⢂⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⡤⠴⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⡁⢄⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣦⣕⡀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⣐⣦⡤⠴⠖⠒⠛⠛⢉⣰⣶⣦⣽⣿⣿⣿⣷⣦⡀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⡶⠟⠋⠀⠀⠀⠈⠙⠻⣿⣿⣿⣷⣼⡆⠪⡓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⣭⣥⣤⡤⠿⠉⠀⠀⡠⠞⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⡛⢿⣿⣿⣿⣶⣟⡗⢝⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠤⠤⠤⠤⠠⠖⠒⣻⡟⠉⠉⠉⠉⠉⠛⠻⣿⣿⣿⣷⣿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠃

⠀⠀⠀⠀⠀⠀⣳⠘⠁⠀⠀⠠⠂⢠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⠀⠀⠀⠑⠀⠀⠀⣷⣶⣾⡓⠀⠀⢡
⠀⠀⡀⠄⠀⠀⠛⢀⡌⠀⡴⢁⠔⠉⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠴⢂⡀⠀⠴⠖⠈⠈⢹⡿⠏
⠀⠀⠀⠀⣘⣿⣀⠀⠀⠈⠐⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⣸⠃
⠀⠀⠀⠀⣿⠟⠻⡁⠀⠀⠈⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠈⡇⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⣐⣿⣧
⡀⠀⢀⣌⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠙
⠿⣿⣿⣍⠀⠀⠘
⠀⢨⡿⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠆⠤⣄⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠑⠲⣀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣈⡂⢇
⠀⣴⣄⢀⡠⠄⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈
⠉⢹⣿⣿⠉⠀⢤⠀⠀⠀⣀⠔⢁⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂⠀⠀⠀⣶⣤⣴⠯
⣠⡾⠋⠁⠀⠀⠐⠁⢀⡔⠡⠠⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠙⣿⣿⠆
⠁⠀⠀⠀⠀⠂⣺⣶⣾⡋⠁⠉⢖⠒⠂⠐⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠉⠁⠀⢹⡟⠁⠀⠀⡀⠀⢀⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⣶⣶⡆⢀⡀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣀⣀⣀⡈⠛⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣰⣿⣿⣿⣿⣿⡷⡀⠙⢛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢿⠟⠉⣀⣀⠀⠉⠻⡆⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⠀⠀⠀⣰⢿⣻⣻⡿⣆⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⡶⠷⣿⣷⣴⣝⣳⣦⣄⠀⠀⠀⠀⠙⠣⠿⠆⠀⠻⠽⠇⠈⣮⣽⡆⢸⠀⢠⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢀⠀⠙⢿⣿⣿⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⢃⡟⢀⣾⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋⢀⠀⠈⢿⣿⣿⣿⡇⠀⠀⢀⣤⣶⣶⡶⢦⣾⣿⣿⣿⠟⣡⠞⣠⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⠸⣿⣿⣿⡇⠀⢠⣾⣿⣿⣿⣗⣾⣿⡿⠛⣡⢞⣡⣾⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀⣿⣿⣿⠃⣴⢟⣿⢟⣿⠿⠚⠋⠁⣀⣭⣶⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁⣿⣿⡟⢰⣿⠨⠟⠋⠁⣀⢰⣶⣿⣿⣿⡿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢰⣿⡿⢀⡿⠋⢀⡤⣖⣻⡿⣾⣿⣿⣿⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠏⣼⡿⠁⣨⢤⢾⣽⣿⣿⣛⣳⣿⣟⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣤⡀⠙⢷⡀⢰⡟⣡⠾⠛⠉⠉⠁⠀⠀⣀⣬⣍⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⠻⠟⠀⠈⠃⣿⠜⠁⠀⠀⠀⠤⠒⠚⠳⠿⠟⠵⣿⣿⣻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⢀⣀⣠⣤⣶⣶⣦⣤⡀⠀⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠉⠉⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢀⣿⠃⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⡀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⣾⠇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡟⠀⠀⠀⠀⠀⠀⠀⠿⣿⣛⣻⢿⣿⣿⣟⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡇⠀⠀⣠⣤⠤⢤⣄⠀⠙⠿⠿⣬⣿⣿⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢱⠀⣾⠋⠀⡠⣤⣉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢺⡆⠀⠈⢾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣮⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠐⠀⠀⢀⣆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢠⠄⠀⠀⠀⣾⠆⠀⠀⠀⠀⠄⠈⡀⠴⠘⡄⠁⠄⠀⠀⠀⠂⣰⠈
⠀⠀⠀⠀⠀⣠⣤⣾⢿⣧⣤⠜⠀⠀⠀⠀⠀⠠⠀⠀⠀⢘⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠄⠀⣴⠁⢌⡱⢐⠀⠀⢀⢈⠇⠈
⠀⠀⠀⠐⠀⢙⣿⣅⣏⣿⡟⠀⠀⠀⠀⠀⠀⡜⠀⠀⢠⠈⠀⠀⢄⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡉⢀⠒⠢⢀⠀⠀⠀⢀⠚⡆
⠀⠀⠀⠐⠀⠈⠟⠛⠛⠟⠇⠄⠂⠀⠀⢀⡨⠇⠂⢠⡈⢄⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⠀⠀⠈⠀⠠⠈⠃⠀⠀⠀⠀⠀⠐⢧⠀⠂
⠀⠀⠀⠀⠀⠀⠠⠀⠠⠐⠀⠀⠀⠀⣤⠼⠀⠀⠀⠈⢁⠀⠀⢰⣿⠛⠁⠀⠁⠀⠀⢄⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢻⡄
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⣾⠁⠀⠀⠀⢀⣰⢠⡁⠠⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠠⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠀⠀⠁⠦⠘⠘⠸⠁⠀⠀⡀⠀⠀⠀⠀⢀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠀⢄⠂⠐⣀⢆⠄⠀⠀⠀⣀⣸⣄⠂⠀⠠⠐⣾⡂⠀⠀⠀⠀⠁⠰⡶⠀⠀⠀⠀⠀⡀⠐
⠀⠀⠀⡀⢀⠀⠀⠀⠀⣰⠇⠂⠀⢌⢄⠂⠤⡩⣚⠌⡄⠀⠈⠔⡷⠾⠉⠂⠀⠀⢸⡇⠁⠀⠀⠀⠄⠀⢰⠀⠀⠀⠀⣠⠁⠀⠀⠀⢀⡴⠂
⠀⠀⠄⢢⠼⢦⠂⠀⠀⠅⠀⠀⠠⠂⢌⠦⠐⡖⠀⢂⡱⡁⠀⠀⠀⠀⠀⠀⠀⠐⠸⣇⣀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⡏⠀⢀⠀⠁
⠀⠀⠨⠜⠓⠋⠁⠀⠀⠀⠀⠀⠠⠎⠎⡜⠨⡰⢁⠃⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⢏⠀⠀⠀⠀⢠⡆⠀⠈⠁⠘⠁⠀⠄⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⡀⢠⡄⠀⠀⠀⠀⠈⠚⠱⠱⠑⠂⠉⠀⠀⠀⠀⠀⡐⠈⡄⠀⡀⠀⡀⠈⠆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠙⠁
⠀⠀⢀⣀⠀⠈⠋⠈⢠⡦⠀⠀⠀⣤⡀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⢥⢀⡖⠀⠀⡀⠁⠀⢀⡀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠗⣀⠀⢂⠠⠀⢀
⠀⣠⡌⠋⢶⢆⡄⣁⢠⡀⠀⣠⡀⠉⢀⠀⠀⣁⠀⡄⠀⠀⠈⠢⠄⡠⢿⣿⠗⠃⢀⠀⠀⠚⠋⠀⠀⠀⠀⡀⠀⠀⢀⣄⠀⠊⠓⡌⠀⡄⠙
⠀⠀⠁⠨⢄⠈⠁⡄⠈⠳⡶⠙⢁⡀⠀⡔⢆⠉⠀⠁⠜⠠⠀⠀⢠⠀⢀⠋⠠⠀⠀⠄⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠠⣄⠀⠈⡗⠋
⠀⠀⠘⠓⠀⢒⠀⢼⠀⠸⠆⠇⠘⠁⠀⠀⢀⣀⠤⠤⠀⠀⠀⠀⠘⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡰⠖⠀⠀⠀⠀⠈⢈⠑⡼⠂⣀⢰⠆
⠀⠀⠀⣀⡀⣘⠃⠐⢦⠀⠋⠘⠓⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠴⠏⠀⠀⠄⠀⠟⢠⠀⠀⢹⠄⠒⠉⠀⠀⡀
⠀⠀⢠⡝⠃⡛⠀⠛⢀⢡⠀⠿⠀⠀⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⣸⠛⠛⠅⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⣇⠄⠀⠀⠀⢀⠇
⠀⠀⠈⠁⠀⣤⠁⠉⠉⠀⢳⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⢀⠀⢸⡀⠀⠀⠄⠀⢀⠋⠉⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡸⠧⠀⠐⠀⠀⡘⠀⠀⠀⠈⠸⡟⠋⠀⠂⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⢴⡗⠃⠀⠀⠀⠀⠀⢰⣶⡴⣾⡆⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⢸⠀⠀⠀⠀⠘⣤⢧⡤⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⣾⣯⣝⣿⡷⠤⠂⠀⠸⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠚⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠈⠻⢅⠀⠘⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠂⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⢓⣄⡀⠀⠀⠀⠀⢀⣔⠀⡀⠀⠀⠀⡐⠁⣼⢡⣯⠤⠄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⣷⠀⣟⠀⢁⠀⠀⠀⠀⠀⠀⢀⠷⠀⠀⠀⠀⠀⠀⠀⠹⣦⠄⠠⠾⡿⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⢲⠃⠁⠀⠁⠀⢸⡄⠙⣴⠋⠉⠁⠀⡎⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠐
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠷⣤⡙⠖⠂⠈⠀⠀⢙⠁⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠂⠀⢿⣿⡿⠷⠶⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠶⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠗⠀⠀⠀⠀⠀⠐
⡐⠒⠲⠄⣀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⡰⠊⠀⠀⠀⠀⢠⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠸⡄⠳⠊⠀⠀⠀⠀⠀⠀⠀⠙⠏⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⡄⣠⠜⠀⠀⠰⠀⠀⠀⢠⣿⠆⢀⠈⠀⠀⠀⠈⢦⡑⠙⢁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⣄⣠⡄⠀⠀⢃⢀⠀⣧⡍⠀⠀⠀⠀⠀⣀⣾⣿⣿⠆⠀⠀⠀⠀⠀⡈⠄⢿⣧⠴⢊⠀⠀⠀⠀⠀⠀⢹⡔⠀⠀⢀⡀
⠀⠀⠀⠀⠠⠤⠀⠬⠹⡄⠀⠀⠀⣾⢆⠋⠀⠀⠀⠀⠀⠀⠈⡻⠛⠁⠀⠀⠀⠀⠀⠀⢛⣄⣞⠠⢀⡀⠀⠀⠀⠀⠀⠀⠘⠳⡄⠊⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠏⠉
⠀⠀⠀⠀⠐⣷⠀⢂⡀⢀⠉⠀⠀⡋⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠹⠈⢻⡇⢈⡇⠀⢠⠁⢀⡀⠀⠀⢀⣀⡤⠀⡰⠀⠀⡠⢾⠟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⣿⣟⡴⡾⠿⡏⠀⠔⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡸⢁⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠃⢀⠀⠈⠄⠀⠀⢀⡤⠂
⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠈⠛⡁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠛⠷⣬⠀⡄⡰⠋⢀⠀⠁
⠀⠙⢛⢷⡶⠂⠀⢀⠀⠀⠀⠀⠀⡏⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⣶⢠⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⢄⠠⠀⢂⠙⠛⠂⠂
⠀⠀⠀⠉⠋⠐⠂⢀⡻⢦⡀⠀⠀⠄⠐⣾⡋⢻⡄⠀⠀⠀⠀⠀⠹⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠂⣀⣠⠀⠉⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡀⠙⡆⠀⠙⢛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣁⣈⣠⣉⡛⠍⠉⡋⠉⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠇⠸⠀⠀⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⡉⠳⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⢀⣀

⡄⠀⠀⣌⠀⢘⠉⠒⠋⠒⠊⢓⠀⢀⡄⠀⢀⣄⠀⠀⢠⠁⠐⠋⠑⠊⠑⠊⠙⠂⠈⡄⠀⠀⣠⡀⠀⢠⡀⠠⡚⠑⠒⠙⠒⠉⡣⠀⣠
⡿⡧⠀⠌⠲⠞⠀⢠⣄⡤⠀⠐⠶⠉⠄⠨⢿⣿⠯⠀⡀⠑⠶⠁⠀⣄⣠⠀⠈⠲⠎⠀⠀⡽⣿⡻⡅⠠⠉⠶⠂⠀⢤⣠⡄⠀⠳⠖⠡⡀⠼
⠿⠀⠀⣠⡀⠠⠨⣭⣿⣫⠅⠠⠀⣠⡀⠀⠽⠻⠇⠀⣀⣄⠠⠀⢭⣻⣟⡭⠐⠄⣀⣀⠀⠸⠟⠯⠀⢀⣄⠀⡄⠩⡝⣿⣬⠅⠄⢀⣄
⢀⡀⢐⡀⠘⠦⠀⠛⠿⠋⠂⠠⠎⠀⡂⠀⣀⠀⣀⠀⡃⠀⠳⠀⠈⠻⠟⠓⠀⠞⠀⢘⠀⣀⠀⣀⠀⢐⠀⠱⠄⠐⠙⠿⠛⠀⠴⠃⢠⡂⢀
⢇⠸⠻⠄⠐⠀⠀⠀⢀⠀⠀⠀⠒⠀⡽⠫⡠⠯⡠⠛⠅⠀⠂⠀⠀⢀⡀⠀⠀⠐⠀⠨⠟⢄⠿⢄⠝⢯⠀⠒⠀⠀⠀⡀⠀⠀⠀⠂⢠⠝⠇
⢀⠀⠀⣁⡔⠋⠀⣲⣿⣷⡂⢈⠳⣄⡅⠀⢀⠀⠀⠀⢎⡤⢋⠀⢰⣾⣷⡆⠀⠙⢨⡱⠀⠀⠀⡀⠀⢠⣠⠚⡁⢐⡾⣿⣔⠀⠙⢂⣈⠄
⡿⣒⠀⠀⠀⠈⠀⢱⣿⣞⠁⠈⠀⠀⡀⢐⣺⣿⢗⠂⡀⠀⠀⠀⠉⣾⣷⠉⠀⠀⠀⢀⠐⡺⣿⣗⡂⢀⠀⠀⠁⠈⣹⣿⡊⠀⠁⠀⠀⠀⣒
⡿⠧⠄⡰⠈⠱⠀⠀⠀⠁⠀⢘⠙⢤⡀⠬⠿⣿⠯⠄⣠⠌⠉⠆⠀⠁⠈⠀⠰⠉⠡⣄⠤⠝⣿⠿⠥⢀⡤⠋⠃⠀⠈⠀⠀⠀⠎⠁⢎⠀⠼
⠁⠀⠀⠠⠀⢙⡦⠊⣶⠋⡢⣞⠀⠄⠀⠀⠀⠁⠀⠀⠀⠄⢘⣢⡆⢱⡎⢱⣔⠃⠠⠀⠀⠀⠈⠀⠀⠀⠠⠀⣳⢔⠑⣖⠑⣴⡃⠠⠄
⣦⡤⠄⢢⠀⢨⠀⠈⠀⠉⠀⠈⠂⠴⠂⠤⣤⣷⣤⠄⠲⡄⠚⡅⠈⠁⠈⠁⢀⠂⢠⠖⠠⣤⣾⣄⠤⠐⢦⠐⠁⠀⠁⠀⠁⠀⡕⠀⡔⠠⢤
⣯⡓⠀⠀⠉⠁⠀⢰⣶⣖⠀⠈⠉⠀⠂⢐⣫⣿⣟⠀⠂⠈⠉⠀⢀⣶⣶⠀⡀⠉⠁⠐⢀⣻⣿⣝⡂⠐⠀⠉⠁⠀⣲⣶⡆⠀⠉⠉⠐⠀⢚
⠙⠀⢀⠶⢆⠈⠀⣳⣿⣵⡃⠐⣀⠖⠄⠀⠚⠙⠃⠀⠔⠦⡐⠁⣺⣾⣷⣗⠈⢂⠴⢢⠀⠘⠋⠓⠀⠠⠲⣀⠃⢘⣮⣿⣞⠀⠁⡰⠖⠄
⡤⢄⣠⠅⢈⠁⠀⠀⠙⠁⠀⠘⡁⠀⢥⡠⠤⣠⠤⣀⠅⠀⡙⠀⠀⠘⠃⠀⠀⢋⠀⠨⣀⠄⢤⠤⢄⡬⠀⢈⠃⠀⠈⠋⠀⠀⠘⡀⠨⣄⡠
⠑⠊⠙⠂⢀⡤⠀⡀⣠⡀⡀⢠⡀⠀⡚⠑⠒⠙⠒⠉⡃⠀⣠⠀⣀⣠⣄⣀⠀⣄⠀⢘⠉⠒⠊⠒⠊⢓⠀⢀⡄⢀⢀⣤⢀⠀⢤⡀⠐⠋⠑
⣤⠀⠈⠲⠊⠠⠠⠽⣿⡻⠅⠠⠉⠶⠂⠀⢤⣠⠄⠀⠳⠖⠡⡀⠼⢿⡿⠧⢀⠌⠲⠜⠀⢠⣄⡤⠀⠐⠶⠉⠄⠨⢟⣿⠯⠀⡀⠑⠖
⣿⡥⠀⠄⣠⣄⠈⠹⠟⠯⠁⢀⣄⠀⠄⠡⣝⣿⡥⠀⠀⢀⣄⠈⠉⠿⠿⠉⠁⣠⡀⠀⠀⣭⣿⣫⠌⠠⠀⣤⡀⠀⠴⠻⠏⠁⣠⣄⠠⠈⢬
⠏⠑⠀⠞⠀⢜⠀⣀⠀⣀⡀⣐⠀⠹⠄⠐⠙⠿⠉⠂⠴⠃⢠⡃⢀⡀⢀⡀⢘⡄⠘⠦⠐⠋⠿⠋⠒⠠⠏⠠⡂⠀⣀⠀⣀⠀⡣⠀⠳⠀⠚
⡀⠀⠀⡐⠀⠨⠛⠄⠽⠤⠝⠯⠀⠊⡀⠀⠀⡀⠀⠀⠀⠂⢠⠟⠣⠼⠧⠜⠻⡄⠐⠀⠀⠀⢀⠀⠀⢀⠐⠀⡽⠫⠠⠿⠠⠛⠅⠀⢂
⣷⣖⠀⡹⢠⡸⠀⢀⡀⡀⠀⠪⣠⠚⠁⢐⡾⣿⣖⠂⠈⢢⣀⠆⠀⡀⢀⠀⠰⣠⡔⠁⠀⣲⣿⢷⡂⠈⠓⣄⠅⠀⢀⢀⡀⠀⢆⡤⢋⠐⣲
⣷⠭⠀⠀⠀⢀⢀⣺⣿⢏⡂⢀⠀⠀⠀⠨⣾⣿⡮⠀⠁⠀⠀⠀⣐⢿⡿⣐⠀⠀⠀⠀⠠⢵⣿⣿⠅⠈⠀⠀⡀⢐⡹⣿⣗⡀⡀⠀⠀⠀⠭
⠀⠀⠰⣉⠱⣄⠀⠝⣿⡿⠄⢈⡴⢉⠁⠀⠀⠀⠀⠀⢌⠉⢎⠀⠸⢿⡿⠧⠀⡰⠉⡱⠀⠀⠀⠀⠀⠈⡉⢦⡁⠠⠻⣿⠫⠀⣠⠎⣉⠂
⡎⢱⢔⡃⠠⠀⠀⠀⠈⠀⠀⠀⠤⠀⣳⢔⠁⣖⢑⣴⡃⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⢘⡶⡊⣲⠉⡢⣞⠀⠤⠀⠀⠀⠁⠀⠀⠀⠄⢐⡢⡎
⠈⠁⠠⠁⢠⠖⠀⣤⣾⣤⠄⠐⠦⠀⡅⠀⠉⠀⠉⠀⡅⠀⠔⠀⢤⣴⣦⡤⠀⠢⠀⢨⠀⠈⠀⠉⠀⠨⠀⠴⠂⠠⣤⣷⣤⠀⠲⡄⠘⠀⠈
⣶⠠⠀⠉⠁⠐⢐⣻⣿⣜⡂⠜⠀⠉⠁⠠⢲⣶⡢⠄⠉⠉⠀⠀⣚⣽⣯⣓⠀⠀⠉⠉⠠⢴⣶⡖⠄⠈⠉⠀⠣⢐⣳⣿⢟⡂⠂⠈⠉⠀⠄
⣷⡟⠀⢂⠴⢢⠀⠘⠋⠋⠀⠠⠲⣀⠁⢘⣾⣿⣞⠀⠀⡰⠖⡄⠀⠋⠙⠀⢠⠲⢆⠈⠀⣳⣿⣷⡃⠈⣀⠖⠄⠀⠙⠙⠃⠀⡔⠆⡐⠀⣻
⠃⠀⠀⢋⠀⠨⣀⠔⢤⠔⢄⡬⠀⠈⠁⠀⠈⠋⠀⠀⠈⡀⠠⣄⡠⢢⡴⢄⣠⠄⢀⠁⠀⠀⠙⠁⠀⠘⢁⠀⢥⡠⠢⡤⠢⣀⠅⠀⡙
⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠁⠀⠀⠀⠈⠙⠒⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡠⠴⠋⣠⣤⠀⠀⠀⠀⠀⠀⠀⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠁⠀⠈⠻⢄⣀⠘⠿⠏⣠⣶⣦⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠈⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⣤⣽⣷⣄⡀⠛⠛⠁⠀⢺⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠐⠡⠀⠨⠭⡽⠁⠈⠉⠉⡏⠻⣿⡷⡀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠋⣱⠈⠉⠁⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠘⡍⡀⢀⣀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠍⠉⡂⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢀⣀⠤⣄⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠎⠁⠀⠉⠉⠉⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠄⠀⠉⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄
⠀⠀⠀⠀⠀⠀⠀⡠⠤⠄⢀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠉⣳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⡠⠤⠄⣀⣽⠀⠀⠀⠈⠑⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢠⡴⠡⠒⠀⢄⣀⡖⠀⢄⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠑⠢⣀⠀⢹⡄⠈⠙⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠐⠲⠤⠤⠤⠶⠖⠚⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣭⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣭⣭⣿⣯⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀⢸⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⢀⣴⣿⣿⣿⣿⣷⣦⡈⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⣾⣿⡿⠟⠛⠛⠻⢿⣿⣆⠈⢿⡇
⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠻⢋⣴⣾⣿⣿⣶⣄⠹⣿⡆⠈⣧
⢠⣾⣿⣿⣿⣿⣿⣦⣠⣴⣦⣤⡀⠀⠀⠀⠀⠘⢿⣿⣶⣄⠀⣾⣿⣿⡿⣿⣿⣿⡆⢹⡇⠀⣿
⣿⣿⣿⡿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠙⠛⠛⠀⠘⠛⠛⠁⣸⣿⣿⡇⣸⠇⢀⡏⠀⣴⡀⠠
⠻⣿⣿⣆⣀⡤⠀⠈⠻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣶⣿⣿⡟⢡⠟⢀⣾⣶⣾⣿⠇
⠀⠈⠛⠛⠋⢁⣤⣀⠀⢻⣿⣿⣿⣿⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡴⢋⣴⣿⣿⣿⣿⡟⠀⠠
⠀⠀⠀⠀⠀⢿⣿⣿⣧⠈⣿⣿⣿⣿⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣉⠴⣋⣴⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠈⢿⣿⣿⠀⣿⣿⣿⡏⢀⣿⣿⣿⣿⣿⣿⠿⠛⠁⢀⣨⣴⣾⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⣿⣿⣿⠁⣼⣿⣿⡿⠟⠉⣀⣤⣶⣾⣿⣿⣿⠿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠈⣿⢀⣿⣿⠃⣸⡿⠛⣡⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡟⣸⣿⠏⠐⣫⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⢀⡶⠋⠉⠙⢶⣄⠀⢀⣿⢏⣤⣾⣿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⢸⣤⣿⣿⡆⠀⢹⡄⣼⣷⡿⠛⠁⠀⠀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠙⠛⠋⠀⠀⢸⣿⡿⠋⠀⠀⠀⠀⠉⠉⢉⣉⡉⠉⠙⠛⠿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⣠⣴⡾⠿⠿⠿⠿⠿⣿⣷⣦⣄⠀⠈⠉⠁
⠀⠀⠀⠀⠀⣴⠃⠀⢀⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣄
⠀⠀⠀⠀⣸⠏⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢠⣿⠀⢰⡿⠁⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡄⠀⢈⣿⣿⡄
⠀⠀⠀⢸⡇⢀⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⠃
⠀⠀⠀⣾⡇⢸⡿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⢻⡇⢸⡇⠀⠀⢀⣠⣤⣤⣄⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⢸⡇⠘⣿⠀⣴⠟⠁⣠⣄⡙⣷⡀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣿⡀⢹⣿⡏⠀⠸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠘⣷⡀⢻⣧⠀⠀⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠘⢿⣦⡹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⢀⣤⡀
⠀⠀⢠⡆⠀⢠⡶⠀⠀⠀⣦⡄⠀⢀⣀⠀⠀⠸⡄⡏⡄⠀⠀⠀⠿⣿⣤⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠘⢷⡀⣼⠟
⠀⢀⠘⡇⡴⢋⡀⠀⣀⡀⠹⠿⣾⡿⠟⠀⢀⠳⣜⢊⡴⢀⠀⢀⣠⡛⠁⠀⠀⠀⣴⣿⣷⣦⡄⠀⠀⠀⠀⠈⠳⢁⣴
⠀⠘⣧⡈⢠⠾⠁⡄⠻⣷⣄⠀⣠⣴⡦⠀⣬⢷⡈⡡⠴⢋⡄⠘⠿⣿⣿⠀⢠⣷⣹⣯⣿⣾⣽⣷⡄⠀⠀⣈⢣⡿⠋
⠀⠀⢨⠛⠆⣠⡾⠃⠀⣨⡍⡀⠛⢋⠀⢀⡙⢇⢹⣠⠶⢟⠀⠀⢀⠀⡀⠀⣼⡛⣿⣿⣿⣿⣿⣻⣥⠀⠈⣿⢿⠁
⠀⠀⠸⣦⡀⠋⠀⠀⠀⠘⠿⣧⣾⡿⠃⠀⡳⡌⣯⣶⠞⠁⠀⢾⡿⣿⣿⠀⣻⠿⣯⣿⣿⣿⣿⢿⣿⢃⣀⠈⢿⢋⣤⡄
⠀⠀⠀⠈⠳⣠⡄⠀⠀⢿⣦⣏⣴⣶⠀⠀⠳⣘⣡⠆⠀⠀⠀⢀⣀⠀⣀⠀⢿⣋⣿⡟⢻⣏⣹⣿⣦⠈⠿⡀⡸⠚⡋
⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠙⠟⠛⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠸⢿⠿⠿⠂⠀⠉⠉⠀⠀⠻⠿⠿⠃⠀⠀⠰⢡⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢏
⠀⠀⠀⣶⣄⠀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣿⣆⢀⣤⣤⠀⠀⠀⠀⠀⠀⣿⢀⡄⠀⢸⣧⠀⠀⡀
⠀⠀⠀⢻⣿⠆⠃⣀⡀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠈⢁⣈⠛⠋⠀⠀⠀⠺⣦⡀⢹⡿⠃⠀⠀⠙⠀⣾⠇
⠀⠀⠀⠀⠉⣥⣾⣿⠇⠀⠀⠀⠘⢿⡇⣠⣴⣶⠀⣤⡄⠸⡟⢷⡶⠀⢀⣤⣤⡬⢥⡄⣀⣀⣄⠻⣶⡤⡄⠀⣰⣿⠂
⠀⠀⣀⣤⣀⠘⠋⠀⣀⣀⠀⢀⣄⠀⡀⣿⡿⠃⠀⢙⣥⡄⠀⢰⣶⡆⠈⠛⣡⡀⠀⠱⠀⢉⡋⠀⠀⠀⠬⣀⠛⢁
⠀⠀⠻⣿⡿⠿⡇⣾⣿⡿⠂⠸⣿⣷⠸⡇⢀⣴⡆⠈⠻⠓⢤⠘⠛⡁⠀⠘⢿⣧⠀⢆⣾⡿⠃⠀⢀⣤⣀⢸⠐⠁
⠀⠀⠀⠀⠀⣀⣸⠈⠉⠀⠀⠀⠉⠻⡄⣶⣿⣿⠟⠀⠀⠐⣄⢀⠞⠀⠀⠀⠀⠉⢠⠋⠁⠀⠀⠀⠀⠙⢻⠘
⠀⠀⠀⠰⣾⣿⡿⣠⣴⣆⢸⣿⣷⡄⢸⠛⠋⠁⠀⠀⠀⠀⠈⡟⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⡄⡞
⠀⠀⠀⠀⠈⠁⠀⠟⠛⠁⠀⠙⠛⠓⠷⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐⢀⠏⠀⠀⠀⠀⠀⢠⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⠈⠛⠻⠿⠷⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣆⡀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣷⣀⡀⠀⠀⠀⠀⠀⠈⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⢿⠿⣿⠋⠈⠁⠀⠀⠀⠀⠀⠀⣄⣠⣤⣤⠶⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⠋⡀⠁⠀⠀⡟⢤⠀⠀⢀⠀⠀⠀⢠⣤⠾⢛⠏⠀⠑⠁⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠀⠹⣿⣿⣿⡿⣯⣶⣶⣿⣇⣀⡤⠊⠀⠀⠀⠀⠀⠈⠁⢀⡞⠁⣴⠁⠓⢄⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠉⠻⣿⠿⠿⢷⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠁⣀⣼⣿⣥⡀⠩⡀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠓⢄⠀⠬⠷⡆⣅⣹⣵⠤⠤⣶⣶⣤⠤⠤⠖⠛⠻⠉⠿⢝⠋⠢⠅⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠃⣴⠔⠛⠀⠀⠀⠠⢴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣤⠃⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⣛⣭⣿⣿⣿⠏
⠀⠀⠀⢠⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠿⠛⠛⠛⠛⠛⠛⠻⠷⣶⣤⣤⣤⣤⣠⣤⣤⣴⡶⠿⠛⣩⢖⣴⠞⣹⡿⠋⠉
⠀⠀⠀⢸⣿⣝⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⢉⡴⠂⠀⠀⠀⠀⠀⠀⣀⣠⠤⠂⠀⡉⠉⠉⠉⢉⡭⠀⠀⠀⣼⡵⠋⠀⣰⡿⠁
⠀⠀⠀⠀⠈⠙⠻⣿⣷⣄⡀⠀⠀⠀⠀⠀⣠⣴⡿⠋⣠⠞⠁⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⣠⡾⠋⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣶⣶⣴⣶⠿⠛⣁⠴⠊⠁⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠀⣀⡴⠿⠧⠦⠔⠒⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣟⠳⠶⢶⣮⣅⣀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⣀⠴⠋⢀⣀⣤⡴⠞⠉⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠈⢫⡙⠓⠶⠾⣭⣤⣤⣤⡤⠴⠶⠖⠚⠛⠛⠛⠉⠙⠓⠦⣄⡀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠑⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠠⡀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠙⠲⢄⡀⠀⠀⠀⠀⠀⢀⣀⣠⣭⣦⣤⣤⣤⣴⡾⠟⠋
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣀⡀⠀⠀⠀⠀⠉⠓⢀⣠⣶⠾⠟⠛⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠶⠶⠶⠿⠿⠛⠉


⠀⠄

⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠊⠀⢴⣾⣿⣿⡄⠈⠂⠙⠛⠀⠀⠐⠀⠀⠀⠀⠎⠀⠀⠈
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠋⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣐⣀⣀⠀⠀⠀⠀⠀⠀⠋⡿⠃⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⣼
⠀⡠⠒⠃⠀⠒⢀⠀⠀⠀⠀⠀⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢄⢀⣾⣟
⠞⠀⠀⠀⠀⠀⠈⠁⡄⠀⠀⠀⠉⠉⠤⠤⠤⠄⢀⡴⠒⠒⠒⠠⡀⠀⠀⠈⠃⡀⠀⠀⠀⠄⠠⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢀⣀⣶⣿⡟⢉
⠂⠒⠂⠐⠄⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⣄⡀⢈⢢⠤⢤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣷⣷⣿⣿⣷⣾⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⠐⠒⠄⣀⠀⠀⠀⠀⠀⠀⠀⠉⠩⢯⣿⣿⢿⣿⣿⣯
⠷⠄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠘⠈⠃⠀⠀⠀⡠⠒⠀⠀⠐⢉⣿⡿⠄⠉⠏⠛
⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢵⣤⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠰⢄⡀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣿⣆⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣶⠇
⠀⠀⠀⠀⠈⡟⠒⠠⠀⠀⠀⠣⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠉⠉⠙⠧⠀⢀⠀⠀⠀⠒⠤⢤⣿⣿⣿⡇⠀⣿⣇⣠⣴⢅⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠀⠠⣤⡼⣿⣿⣿⣸⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⢀⢈⡄⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡿⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠠⠀⠀⠀⠑⠀⢀⡎⠀⠄⠺⠃
⠀⠚⠀⢤⡀⠀⠀⠀⠀⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⢔⣾⠿⠛⢿⡇⠐⠦⠀⠀⢀⣀⡀⠀⠀⠀⠃⠀⠀⠤⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀
⠀⠈⠐⠛⢧⡀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠄⣀⣀⣀⡀⠤⢰⠞⠋⡠⠀⡀⠀⠙⣄⠀⠀⠀⠀⠀⠉⠁⠀⠊⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⠁⠀⠀⠀⠀⠀⠀⠀⠸⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠖⠒⠻⣷⢶⣷⠀⡀⠀⠀⠀⠀⠀⠀⠄
⠔⠋⠀⠀⠀⠀⠀⠀⠀⠂⠣⣄⣀⡄⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠄⠀⠀⠀⠀⠀⠁⠉⠉⠉⠚⢆⠀⠀⠀⠀⢀⠀⠀⠀⣠
⠀⠀⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⠀⠉⠘⠈⠁⠁⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠀⢸⣇⣸⣀⣠⣾⠿
⠀⠀⠀⢀⡇⠉⠉⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⢱⣄⣆⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⣽⣧⣾⣿⣿⡿⠿⠃
⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠀⠀⢿⣿⣆⡀⠀⡜⠁⠀⠀⢀⣴⣞⡥⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢿⡏⡁⠀⠀⠀⠐
⠀⠀⠀⡀⠠⡈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠀⠀⢤⣀⠀⠀⠀⠀⢻⣿⣷⣀⣠⡄⠀⣤⣿⣿⠟⢀⡀⠀⠀⠀⠀⠢⡄⢀⠍⠀⠂⠉⠷⣁⠂⠤
⠒⠀⡉⠁⠀⠈⠀⠀⠐⠄⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠈⠙⢿⣿⡛⠃⢰⣿⣿⣩⠆⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠉⠓⠒
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣿⣿⣶⣶⣉⡄⠓⢦⣹⣿⣡⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠁⠀⠀⠀⠒⠋⠉⠉⠈⠞⠛⠙⠋⢻⡿⢿⠅⣠⣆⠋⣼⣿⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⣾⣿⡿⢋⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠤⣄⡀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⣴⣿⣿⡟⣰⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⡀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⣼⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄⠀⢿⣿⡀⠀⠀⠀⢀⣿⣿⠀⠀⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⠈⢻⣇⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⡿⠛⠉⠀⠀⠀⠀⣀⣀⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⠀⠀⠀⠀⢳⡄⠀⠀⠀⢿⣿⣧⠀⠹⠆⠀⠀⣾⣿⣿⡀⠀⣿⣿⡇⢻⣿⣾⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⣿⡄⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⣿⣿⣿⣇⠀⣿⣿⡇⠸⣿⣿⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣇⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⣿⣿⣿⣿⡀⣿⣿⡇⠀⣿⣿⣿⠀⠀⠀⠀⣀⣤⣶⠶⠒⠒
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⣿⣿⣿⣿⡇⢹⣿⡇⠀⣿⣿⣿⠀⠀⢠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⢠⣿⡟⠀⠀⣾⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⡇⣼⣿⡇⢀⣿⣿⡏⠀⢠⣿⡟
⠀⠀⠀⠀⢀⣴⣿⣿⡟⣱⣿⣿⣿⠃⠀⠀⠀⢠⣾⣿⡇⠀⣼⣿⣿⣿⣿⣿⠀⠀⣾⣿⣿⣿⣿⠀⣿⣿⠁⢸⣿⣿⠃⠀⣾⣿⠇
⠀⠀⠀⢠⣾⣿⣿⢏⣼⢟⣿⣿⠇⠀⠀⠀⣴⣿⣿⠟⢀⣾⣿⣻⣿⣿⣿⠃⠀⣰⣿⣿⣿⣿⠇⣸⣿⡏⠀⣿⣿⠇⠀⣼⣿⣿
⠀⠀⣰⣿⣿⣿⢏⣾⠏⣾⣿⡏⠀⠀⢀⣾⣿⣿⠋⣰⣿⡟⣱⣿⣿⡿⠁⢀⣴⣿⣿⣧⣿⡟⢠⣿⡿⠀⣸⣿⠋⠀⣼⣿⣿⠇
⠀⢰⣿⣿⣿⣯⣿⠏⣼⣿⡿⠀⠀⢠⣾⣿⡟⢡⣾⣿⢋⣼⣿⡿⠋⠀⣠⣾⣿⣿⣿⣾⡟⢠⣿⡿⠁⢀⡟⠁⢠⣾⣿⣿⡏
⢀⣿⣿⣿⣿⣿⡿⢸⣿⣿⡇⠀⠀⣾⣿⡟⢠⣿⡿⢡⣾⣿⠏⠀⣠⣾⣿⣿⣿⣿⣿⣟⣴⣿⡟⠁⠀⠈⢀⣴⣿⣿⣿⡟
⣸⣿⣿⣿⣿⣿⡇⣾⣿⣿⠀⠀⢸⣿⣿⢣⣿⡿⣱⣿⡿⠁⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⢀⣴⣿⣿⣿⣿⡟
⣿⣿⣿⠀⣿⣿⣇⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣱⣿⡟⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⢀⣴⣿⣿⣿⣿⣿⡟⠀⠀⣷⡀
⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡀⠀⢸⣿⣿⣿⣿⣿⡿⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⣠⣶⣿⠿⣻⣿⣿⣿⠏⠀⣰⠀⣿⣧
⣿⣿⣿⡀⢿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⠇⠀⣸⣿⣿⣿⣿⣿⣿⡿⠋⠀⣠⣾⣿⠟⣡⣾⣿⣿⡿⠃⢀⣼⣿⠀⣿⣿⡇
⢹⣿⣿⣇⠈⢿⣿⣿⣿⣿⣿⡀⠈⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⠟⠀⣠⣾⣿⠟⣡⣾⣿⣿⣿⠏⠀⣠⣿⣿⣿⠀⣿⣿⣿
⠈⣿⣿⣿⣆⠈⢿⣿⣿⣿⣿⣧⡀⢹⣿⣿⣿⣿⠀⢠⣿⣿⣿⣿⣿⠋⠀⣴⣿⠟⣡⣾⣿⣿⣿⠟⠁⢠⣾⣿⣿⣿⡏⠀⣿⣿⣿⡇
⠀⠘⢿⣿⣿⣧⡀⠙⢿⣿⣿⣿⣷⡀⢻⣿⣿⣿⠀⢸⣿⣿⣿⣿⠇⠀⣼⣿⢋⣾⣿⣿⣿⠟⠁⢀⣴⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⡇
⠀⠀⠈⢻⣿⣿⣿⣶⣄⠙⢿⣿⣿⣿⣦⣿⣿⣿⠀⠀⣿⣿⣿⡏⠀⢰⣿⣧⣿⣿⡿⠟⠁⣀⣴⡿⢟⣵⣿⣿⣿⡟⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣦⣙⠻⣿⣿⣿⣿⣿⡄⠀⣿⣿⣿⠁⠀⣿⣿⣿⡿⠋⠀⣠⣾⠿⣫⣶⣿⣿⣿⣿⠟⠀⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡇⠀⢻⣿⡿⠀⢠⣿⡿⠋⢀⣠⣾⣿⣿⣿⣿⣿⣿⡿⠟⠁⢀⣴⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⠀⠸⣿⡇⠀⠸⠏⠀⣰⣾⠟⣩⡾⠟⠛⠋⠉⢁⣀⣠⣴⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡆⠀⣿⡇⠀⠀⠀⣼⠟⠁⠈⢁⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⠀⢹⡇⠀⠀⣼⠃⠀⣠⣾⣿⣛⣭⣭⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠘⡇⠀⠀⠃⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⡇⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⣼⣿⡿⠛⠉⠁⠀⠀⠀⠈⠉⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣊⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣼⣃⠖⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡿⠩⠂⠀⠀⠀⠀⠀⠀⢠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⡗⠵⠀⠀⠀⠀⣄⡀⠀⢾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣹⡄⡠⠀⠀⠀⠀⡄⠈⠆⢸⠛⡡⠄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣼⠥⡄⠀⠀⣷⣤⠐⢄⣰⢃⡼⠀⠔⠃⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡨⡄⠀⠀⠻⣿⡇⠀⡹⢁⣤⣤⠀⠀⡀⠄⠚⠅⣀⠄⠒⡩⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠣⢄⡀⠛⡄⠀⠀⠀⢠⢿⡂⢅⠀⠀⠠⢌⠣⣀⢣⣿⣿⠏⠀⠌⢡⠀⢰⢊⡀⠤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠠⠌⣦⡘⡄⢰⣶⠀⣐⣗⠧⠀⠀⠐⡄⢱⣸⠼⠛⠁⠀⡐⠀⢀⡴⠋⠁⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠈⠑⣆⠀⠁⠀⠃⣿⢎⠄⠀⢀⠜⠁⠀⠀⠉⠐⠒⠇⣀⣼⠤⠐⢒⠞⠀⢀⠤⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠤⠀⣈⣤⡀⠘⡟⠪⣷⣂⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⡔⠋⠀⢠⠁⢀⠌⢀⡠⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠈⢧⠀⡷⠊⠀⠇⠀⠀⠀⠀⠀⢀⠄⠒⠒⠄⡀⠀⠀⠈⠐⠤⣴⠖⠃⠒⠮⣀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠠⠄⡉⠒⠤⠄⡤⠋⠀⠀⠀⡀⠤⠤⠤⠄⣀⡎⠀⠀⠀⠀⠈⢊⠀⠀⠀⠀⠈⢋⢣⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠥⣦⠊⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⢀⣠⠊⠀⠈⠢⠀⠀⡤⢀⡀⠀⠀⠸⠄⣉⣁⡉
⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠄⣀⢰⠖⢹⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡔⠉⠉⠑⠄⠀⠀⡄⢀⡅⠀⠈⠢⡀⣰⠢⡘⠦⠈⠆
⠀⠀⢀⡤⠒⠫⡀⠀⠀⠱⡄⠀⠈⠑⠺⡀⠀⠀⠀⢨⠀⠀⠀⠀⠀⢣⠀⠀⠶⠂⠀⢠⠗⠈⢡⠀⠀⠀⠘⡅⠣⣈⠄
⠀⡴⢋⡠⠄⠒⠒⠓⣒⣦⠬⠶⣄⣀⡀⢡⡀⠀⠀⠈⠄⠀⠀⠀⠀⡠⠥⡀⠀⣀⠔⠁⠀⠀⡘⠀⠀⠀⠀⢰
⣰⣀⠁⠀⢀⡔⠊⠉⠀⠀⢀⠠⠂⠉⠉⠀⡱⣄⠀⠀⠈⡶⠠⠄⠚⢄⠀⠈⠈⠀⠀⠀⠀⡠⠁⠀⠀⠀⢀⠃⢀⣠⡟
⠁⠀⠀⠀⠀⠀⠁⠒⠂⠴⠥⠄⠤⠄⠒⡨⢤⣫⠶⢆⠈⠀⠀⠀⠀⠀⠑⠤⣀⣀⣀⡤⠊⠁⠀⠀⠀⢀⠌⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣬⢛⡡⠑⠉⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠤⠀⠐⠒⠂⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡾⢏⠢⠀⠀⢀⠔⠊⠱⢄⠀⠀⠀⢀⠠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡽⡟⠌⠊⠀⢀⠜⢁⣀⠤⣴⠋⠉⠐⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡸⡗⡌⠀⠀⢀⠊⠉⠀⢀⠔⡇⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⣔⣾⠣⠈⠁⠀⠀⣈⠀⡀⢴⠏⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⣆⣽⢣⠉⠀⠀⠀⠀⡏⠁⠀⠎⠀⠀⡄⠔⠁
⠀⠀⠀⠀⠀⠀⠮⣽⣉⡍⠀⠀⠀⠀⠀⢀⠀⠐⠀⢀⡤⠋
⠀⠀⠀⠀⠀⠀⠰⡰⠀⠀⠀⠀⠀⠀⠀⠈⠆⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣀⠀⠀⠀⢠⡤⠀⡀⠀⠀⠀⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠐⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢟⡀⠁⠀⠀⠤⠄⠀⠀⡴⠊⢠⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⡀⠠⠒⠠⠀⠀⠤⠤⠀⠀⠈⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⢙⣀⣀⡀⠀⣀⠀⠀⠀⠀⠰⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠋⠠⠀⠶⠈⢉⡅⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⣍⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⠑⢆⠉⠉⠀⠀⠀⡤⠂⠈⢠⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠂⠤⡀⠀⠀⠀⡀⠀⠀⠇⠀⠈⠒⠒⠀⠀⠀⢸⠄⢀⢀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⡏⡤⠤⠀⠀⠐⠀⠀⠀⠁⠀⠀⠁⠀⠈⠀⠁⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁⠄⠐⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⠤⢀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠑⡀⠸⠀⡄⠀⠀⠀⠀⠀⠀⠈⢢⡈⠦⡀⠀⠠⠀⠀⠀⠈⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂⠐⠌⠈⠁⠀⠀⠰⠤⠀⠀⠀⠀⠃⠀⡙⠀⢀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⡄⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢈⣉⡀⢀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⢠⠆⢀⠀⠑⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠤⡄⠀⠘⠛⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢈⠁⠠⠠⢤⠀⡀⠀⠈⠀⠈⠰⠛⠀⠉⠁⠘⣆⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠃⠀⠛⠁⠀⠀⠀⠀⠀⠠⡉⠀⠦⠐⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢨⠐⠀⠘⠦⢤⠄⠠⡌⠀⢠⠀⢄⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⠃⠀⡔⠣⠈⣰⠃⢀⠋⡀⠠⠀⠠⠔⠠⢄⡀⠀⠀⠀⠀⠀⠑⠀⠀⠁
⠀⠀⠀⣀⣠⠀⠀⡰⠁⠠⠀⢠⠀⠄⢀⡔⠀⢐⡏⣄⠚⡁⢤⣚⡡⣀⠛⠯⡐⠥⣸⡄⠒⠀⠀⢄⡉⠲⢄⡀⠠⡀
⢀⣜⠳⣽⠧⢤⡾⢃⠤⣠⠄⠫⡥⠐⠪⠈⠁⢌⠉⠙⢅⠀⠃⠙⠶⢉⠣⡄⠀⢀⡔⠈⠑⠒⢄⠀⠀⠀⠀⠙⠆
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⣾⣿⣿⣿⡗⠒⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣠⢿⣿⣿⣿⠆⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣷⡄⠀⠀⠀⠀⢀⣾⣷⡆⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠴⠙⣿⠏
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢁⣤⠀⠀⠀⢾⣿⢿⣄⠀⠙⠇⡀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠾⠟⢀⠀⠀⠈⠁⠈⢙⢄⠀⠀⠁⠀⠀⠄⠱⢤⡀⠀⣤⡀⠀⢸⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢡⣀⠀⠀⢸⡆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⣆⠈⢀⣾⣿⣿⠀⢸⠀⠀⢀⡟
⠀⣀⡀⣶⣾⣿⠷⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠘⡄⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣨⡿⢿⣿⣿⠃⢸⠀⠀⠀⠁
⠀⠀⠂⣿⣿⣿⡟⠋⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⢱⡹⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⡇⠐⠈⠙⡏⠀⢸
⠀⠀⠀⠹⡟⠑⠀⠀⠀⠀⠀⣾⠆⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠀⣷⠀⠰⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⢀⠀⠀⡇⠀⣀⠀⠀⠰⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⠀⠀⢸⡆⠀⠀⣙⣷⠀⠀⠀⠀⠀⠰⠀⠀⢸⡇⢆⠀⡃⠀⢾⣀⠀⠀⠀⠀⢀⠾⠁⣹⢿⡆
⠀⢸⡄⠀⣇⣰⣿⡆⠀⠀⠈⢉⡛⠆⠀⠀⠀⠀⠀⣀⣀⣴⣧⣾⣿⠁⣷⠀⢀⠿⠀⣴⠀⢾⠀⠀⠀⠀⠀⠈⣧⠘⡆⢿⠀⢼⠈⠳⡄⠀⠀⠀⠀⠀⡉
⠀⠈⣧⣸⡿⣿⣿⣇⣠⣴⣤⡾⠃⠀⠀⠀⠐⠴⣾⣿⢿⠟⣿⡛⠉⣧⠸⣆⡡⠐⢠⡇⠀⠈⠀⢀⣶⡚⠀⠀⣿⠀⢹⣼⡀⣺⠐⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢸⢿⡇⢤⣿⡿⣿⣭⢿⡇⠀⠀⠀⡆⠀⠀⠈⢳⣄⠀⠘⡇⠀⠸⡆⣿⠅⠀⣼⠃⠀⢐⠀⠈⠀⠁⠀⠀⢹⡇⠈⣿⡇⣿⠀⡇⢀⠻⡆⠀⠀⠀⠀⠀⣰⠃
⠀⠀⢸⠀⣇⢸⣯⣤⣴⣾⣿⣧⣤⣤⢦⣇⢀⣠⠀⠘⣿⣦⡀⣿⠀⠀⢿⢸⡄⠀⡟⠀⠀⢸⠀⠀⢠⠆⠀⠐⣼⡷⣤⣼⣿⣿⠀⣿⡎⠀⢻⡀⠀⠀⣠⣾⠇
⠀⠀⣾⡄⣿⡾⢻⣿⡛⢿⣿⣿⣦⡁⣾⣿⣿⣿⣿⡀⢿⣿⣷⣿⠀⠀⠸⡿⣇⢸⡇⠀⠀⡼⠀⠀⡼⠀⣠⠀⢸⣷⡀⠈⣿⣿⠀⢻⠋⡄⠈⣧⡀⠀⢻⣿
⠀⠀⡟⡇⡏⡇⠈⡏⣇⠘⣍⠙⢿⣛⠹⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⠀⣷⣿⢸⠇⠀⢠⢑⠰⣾⣧⡾⠋⠀⣼⡏⢳⢰⣿⣿⢀⣸⡀⢣⠀⢻⠀⠀⣼⡏⠄
⠀⢰⡧⣇⣷⣿⣠⣷⢸⡆⠘⣧⠸⡇⠀⣿⠿⣿⣿⠇⢸⣿⡿⢿⡇⢰⠀⣿⣼⣾⠀⣀⣼⣸⣶⣿⣉⢷⡀⡼⢸⡇⢨⣿⠀⣿⢻⣿⣆⡸⡆⢸⡇⣷⣿⠀⠸
⠀⣸⠀⣯⡿⣿⡉⣿⡀⣷⣿⣿⣧⣿⣴⣿⣷⡏⣿⠀⠸⣯⡇⠘⡇⢸⠀⢸⣿⣿⣿⣿⡟⣿⣿⠛⣻⠛⢰⠃⠘⡇⢀⣿⡄⣿⢸⡇⡇⠙⣧⡈⣿⡇⣿⠀⠀⡇
⠀⣿⠀⢿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⢿⡀⠀⣿⡇⠀⡇⢸⠀⢸⣿⣿⣿⠈⣷⢸⡏⣰⠃⠀⡿⠀⢀⣧⠦⣿⣇⣿⣾⡇⡇⠀⢹⢻⣿⡗⡏⡁⠀⢹
⠀⣿⠀⢼⣧⣾⡟⢻⣿⣿⣿⣿⣿⡇⣿⣿⣿⡇⢸⡇⢰⡿⡇⠀⡇⢸⠂⢸⡿⣟⣷⣤⣇⣸⢧⡏⢀⢰⣇⡴⢛⡟⠀⣿⣿⣿⣿⣧⡇⠀⢸⣈⣿⣧⡏⣿⠀⢸⡀
⠀⣿⠀⣾⣿⠨⡇⢸⡿⣿⣿⣿⣿⣿⣿⣿⡇⢣⣸⡇⡼⣷⣷⠀⣧⣼⡇⢸⠆⣿⣿⣿⢻⣿⣾⣷⣾⣾⠟⠀⢸⡇⠀⣿⣿⣿⣿⣿⣿⠴⢻⡅⣿⣿⡇⢹⠀⠸⡇
⠀⣿⣴⣿⣿⠀⡇⢸⡇⢻⣿⣿⠟⢹⡿⣿⡷⣿⣿⣿⣷⣿⣿⣷⣿⣿⣿⢸⠀⡿⠹⣿⢸⣿⣿⣿⣸⡿⠶⡀⢸⠃⢀⠹⣿⣿⣿⡿⣷⡄⢸⡇⣿⣿⡇⢸⡇⠀⡇
⠀⢿⡟⢹⣿⣆⡇⢸⣷⣾⣿⣿⣧⣾⣧⣿⠇⢸⢻⣿⡏⡟⠀⡏⢿⢻⠈⣿⠀⡇⠀⣿⣼⣿⣿⣿⣿⡇⠀⠀⣿⠀⢸⢀⣿⣿⣿⡇⢹⡇⢸⡇⣿⣿⡇⠘⡇⠀⡇
⠀⣿⠃⠈⣿⣟⣷⣿⣿⠿⣿⠙⡟⢿⡏⣿⠀⢸⣾⣿⠀⡇⡆⣇⣿⢸⠀⣿⡆⡇⠀⣿⡿⣿⠏⡟⢸⣇⣀⣀⣿⠀⣼⡞⢿⣿⣯⡇⠀⠀⠈⡇⣿⣿⣧⠀⡇⠀⣿
⠀⣿⠀⢸⣿⡿⡏⠈⡿⣧⣸⠀⡇⢸⡇⣿⠀⢸⣿⣿⠀⢧⠃⢻⣿⠸⠀⣻⣿⡇⠀⢻⡇⡇⢸⣿⣿⣿⡉⢩⣿⠀⣿⠇⢸⣿⡏⠁⠀⠀⠀⡇⣿⣿⢹⠀⡇⠀⣿
⠀⢸⠀⢸⣿⡇⡇⣦⡇⠀⢹⠀⡇⠀⡇⣿⠀⢸⡟⣿⠀⢸⡀⢸⣿⠀⠀⣿⣿⡇⠀⢸⠁⡇⣼⠋⢸⠈⠛⢸⣿⠀⣿⠀⢸⣿⠀⠀⠀⠀⠀⡇⠛⣿⢸⠀⣷⠀⣿

⢲⣤⣤⣤⣤⠠⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢠⣀⣄⡀⠀⠀⠐⡶⣶⣶⡶⠶⣶⣶⣤
⠀⢹⣧⡉⠙⠷⣷⣷⡘⡦⣇⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣈⠩⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠿⡟⠛⡄⠀⠀⢘⣶⢮⡱⣞⢀⠓⠀⠁⢂
⠀⠀⠍⠿⢗⣀⠀⠈⠉⠃⠁⣿⣦⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢲⡆⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠴⢒⠓⠲⠸⠄⠀⠀⢿⣿⢿⣦⣈⢣⠀⠀⣼
⠀⠀⠘⢿⣭⡽⠞⣣⡄⡀⠀⠙⠃⠀⠀⢀⣔⠀⢸⡁⣤⠀⢠⡀⠀⠀⠀⠀⠀⠀⣠⠀⢀⠀⠀⠀⢀⠀⠀⠂⡄⠀⠀⠨⠿⣷⣎⡈⠁
⠀⠀⠀⠀⠑⠂⠁⠀⠈⠺⠷⠦⣀⠠⠀⠀⠙⠛⠎⠓⠚⠓⠲⠇⠀⠀⠀⠀⠀⠉⠀⣤⣀⢻⣄⣄⣬⡿⣿⣷⠿⠆⠀⠀⠀⠻⠯⠴⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣅⢹⣶⣽⣟⣿⠿⠛⠀⢀⢀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠔⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠰⠀⠀⠀⡄⡏⠉⠠⠀⠢⡀
⠀⡀⠀⠖⠀⠀⠀⠀⠀⠨⢻⣤⠀⠀⠀⠀⠀⡐⢿⠇⢰⢃⠀⠀⢀⢠⢤⡄⢀⢀⠀⣤⡤⢀⡀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢩⠀⠸⠀⠀⠰⠆
⢸⣷⠋⢰⠆⠀⠰⡄⠀⠀⠂⢽⡇⠀⠀⠀⠀⡝⣿⠀⢨⡟⠀⣰⢻⠿⣼⢣⡤⠾⠋⣁⢴⠖⠀⠀⠀⠀⠀⠀⠈⣑⠦⠀⠠⡀⠀⠀⠀⠀⠀⠀⣆
⣸⣿⡿⠁⠀⠀⣤⠘⣆⠈⣻⣲⡇⠀⠈⡐⢀⣤⡌⠀⣼⡧⣠⣿⠂⠴⠋⠀⠀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠄⣀⠐⠀⠀⠀⠀⠀⠀⠀⣸
⢹⣛⡄⠂⠄⠀⠀⢀⠀⢠⣜⡟⠀⠀⠀⠁⠀⠙⠁⠀⠻⠁⠛⠁⠀⠠⠤⣼⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡤⡄
⠀⢿⠡⢎⡄⠀⠀⣄⢁⡰⡎⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⡐⠷⠶⢶⣔⡶⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠴⣬⣌⡀
⠀⠘⢷⣎⠄⠀⢦⢰⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠛⠿⠳⠦⢤⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀
⠀⠀⠀⢻⡆⢼⡼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡀⢛⠑⠀⠀⠀⢀⡀⡐⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠈⣧⣄⣀
⠀⠀⠀⠀⠁⠀⠀⠀⣀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢳⠘⡅⠘⢧⢉⠀⠙⠟⠟⠈⠉⠀⠀⠀⠀⠀⢫⣄⢀⠀⠘⠀⠀⠸⣯⠿⡿⣷⣶⡆⢠⣄
⠀⠀⠀⠀⢠⣤⣄⡀⠙⢿⣷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⣀⣤⣤⡶⣾⣾⣷⣦⣄⠀⠀⠙⢿⡄⡀⠀⢠⣶⣿⣄⠸⠇⢸⡟⢿⡿⣠
⠀⠀⠀⠀⠈⠳⣯⣟⣷⢮⡙⠆⠀⢀⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⠈⢻⣿⣟⠋⠻⣮⠻⣿⠝⣻⠆⠀⠀⠀⠙⠁⠀⢘⡻⢿⡖⠂⠀⠘⠀⠞⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠟⠂⢤⣄⣉⡉⠉⠁⠀⠀⠀⠀⠀⠀⣠⣤⣽⡷⠖⣚⣶⣦⡆⠁⠉⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣯⣙⡀⠀⠈⠐⠋
⠀⠀⣴⣶⡶⠶⢷⠖⠂⠀⠀⠘⠳⣶⣿⣛⠲⡄⠀⠀⠀⢀⠴⢿⣿⢯⣭⠄⠀⠈⠙⠉⠄⠀⢀⠀⠀⠀⠀⠁⠀⠈⠻⢿⣿⣽⣥⡄⠀⠰⡀
⠀⠀⢈⣙⣛⣛⣛⡛⠃⠀⢤⡙⢷⣤⡀⠉⠙⠉⠀⠀⣠⣾⡛⣛⣫⣤⠴⢀⣀⣴⣤⡾⠂⣀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠉
⠀⣾⣿⣿⣯⣿⣯⡅⢀⡐⢦⡙⠳⣽⣻⣆⠀⠀⠀⠀⠉⠛⠟⠿⣿⣿⣿⠟⣻⣿⡏⡅⣴⣟⣰⣿⠇⠀⠀⠀⠀⠀⠀⢦⣶⡆⠀⠀⣠⡄
⠀⣈⣭⣭⣽⡶⠄⠠⣌⣳⣦⡛⢷⡌⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣾⢳⣿⣿⠿⠏⠀⠀⠀⠀⠀⢰⠤⢹⣙⣤⡞⣽⢟⡏
⢼⣯⣿⣿⡿⠟⠀⠸⣿⡿⣿⣟⠿⠿⠀⠀⠀⢀⣠⣤⠤⢦⠀⠠⣤⣤⣀⠀⠈⠉⠁⠁⠁⠀⠀⠀⠀⢠⣶⠀⢀⡶⣢⣾⡿⠋⣻⣿⣆⣋⡜⢳⡀
⠀⣤⡶⠟⢛⡯⢠⣷⣬⡻⢿⣿⡆⠀⢀⣤⡾⠿⡛⠣⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢸⣤⡌⠀⠛⠙⠳⢚⢉⣭⡸⣟⣿⡯
⠘⠻⠟⠛⣩⣧⡌⢻⣿⢿⡄⠀⠀⢠⡾⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠁⠀⠀⠀⠀⣀⠄⠤⠚⠀⠉⢀⡈⢠⣥⣈⡘⠻⣷⣾⣥⣿⣧
⠀⠀⠀⢸⣿⣿⡇⠀⠙⠻⠃⠀⣠⣏⠀⠀⠀⠀⠀⠀⢠⢂⢂⣾⣶⡾⠁⠀⠀⠀⠀⠀⠊⠁⣀⣠⠴⢤⣯⠉⠙⠿⣿⣿⡛⢦⣬⣽⣣⡌⠉
⠀⠀⠀⠘⢿⠟⠁⠀⠀⠀⠀⠈⠉⠉⠙⠓⠀⢠⣤⣤⣷⠶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠘⠛⠛⠂⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⠀⡀⣉⠀⠀⣠⠀⠂⠀⠀⠀⠐⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡈⠀⠁⠉⢠⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠠⠄⡄⠀⡩⠄⠁⠐⠂⠀⠀⠀⠀⠠⠇⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⠈⠳⣄⠘⠀⣠⠄⢒⠀⠁⠀⠀⠀⠀⢀⠄⠄⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⡈⠀⠈⡁⠤
⠀⠀⠀⠀⠀⠻⠂⠀⡀⠀⠀⠀⠀⠀⠈⠀⠁⠐⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠉⢃⠀⠭⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⠋⠀⠀⠀⢠⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠓⠈⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⠠⣀⠰⡌⠁⣀⡇
⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡅⠊⠤⡶⡱⢄
⠀⠀⠀⠀⠀⠁⠀⢰⠄⠀⠀⠀⠀⠀⠐⠀⠈⠂⠀⠀⠂⠀⠀⠀⠀⠀⢠⠀⠀⠐⡈⠛⠁⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠤⠄⠣⠀⠀⠉⠁
⠀⢀⠤⢄⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠢⠀⢠⡀⠀⣀⡀⠤⡀
⠀⡇⠲⣦⡷⠒⡒⢶⠒⡒⡗⣾⠒⡒⢢⠖⡒⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⠆⠘⠂⠁⠀⠀⠒⢀⡇
⠀⡞⢶⠄⣷⠸⠜⢸⢸⠓⡇⣿⢸⢹⢸⡘⠟⢸
⠀⠈⠒⠚⢃⢰⠚⠉⠛⠀⠙⠋⠊⠙⠺⡍⠃⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⡄
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠄⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠴⠒⠊⠉⠉⠉⠍⣉⠉⠉⠂⠒⠈⠉⡀⠀⠀⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠚⠉⠅⠒⠒⠒⠒⠒⠲⠤⢀⡀⠀⠀⠀⠈⠑⠦⡀⠀⠀⠈⠂⠒⢄⣀⣀⠑⢂⡀⠮⠭⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠠⠤⠤⠤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉⠒⠄⡀⠀⠀⠈⢓⡤⢤⠖⠊⡩⠋⠁⡜⠁
⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠢⣀⠀⠀⠀⠀⢀⡨⠕⡲⠉⠁⢀⠎⠀⢠⠁⠀⡼⠁
⠀⠀⠀⠀⠀⡔⣁⡤⠤⠤⠤⠤⠤⠤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣩⠝⡶⠊⠁⠀⢀⠁⠀⢀⠎⠀⢠⠃⢀⠞
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡠⣴⠚⠉⠀⢠⠁⠀⠀⠀⡎⠀⠀⡎⠀⠀⠁⡠⠋
⠀⠀⢠⢃⠀⠠⠄⠀⠠⠤⠤⢀⡀⠀⠀⣀⡤⠚⠁⢠⠁⠀⠀⠀⡆⠀⠀⠀⢸⠀⠀⡰⠁⠀⢡⠎
⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠝⠋⡝⠀⠀⠀⡘⠀⠀⠀⢰⠃⠀⠀⢀⠇⠀⠠⠁⠀⡰⠁
⠀⡜⠀⠀⠀⠀⠀⠄⠠⣤⢴⠋⠀⠀⢸⠁⠀⠀⢠⠃⠀⠀⠀⡎⠀⠀⠀⠎⠀⠠⠃⢠⠞⠁
⡜⠀⠀⠀⠀⠀⢀⠴⠊⢠⠇⠀⠀⠀⡌⠀⠀⠀⡎⠀⠀⠀⡜⠀⠀⢀⠎⠀⠐⣀⠔⠁
⡇⠒⠉⢉⠩⠟⠁⠀⠀⠎⠀⠀⡠⠊⠀⠀⠀⡼⠀⠀⢀⠎⠀⠀⠐⢁⡤⠖⠉
⡇⡠⠂⠁⡄⠀⠀⣠⠎⠀⠀⡔⠁⠀⢀⠤⠊⢀⡠⢖⣁⣠⠤⠖⠊⠁
⠛⠒⠲⠤⠤⠤⠬⣅⣀⣀⣈⣀⣀⣠⠥⠤⠖⠊⠉⠉⠁

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀
⠀⠄⠀⠘⠱⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠶⠀⠀⠀⠓⠛⢁
⠀⠀⠀⠀⠁⠓⡀⠀⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⢀⡚⢠
⠀⠠⠀⠈⠀⢠⢁⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢌⡉⡸⠀⡀
⠀⠀⠀⠀⠀⡤⠀⢤⠀⡻⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠐⠲⠤⡀⢀⠜
⠀⠂⠀⠀⠈⠀⠠⣀⠆⡴⢎⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠐⠆⠄⢀⠀⠠⠒⠁
⠀⠀⠀⠀⠨⠀⠀⢠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⡠⠀⠁
⠀⠈⢥⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠁⡀⠀⠀⢨⠟⢃⠀⠂⠤⠄⠀⠊⠐⡀⠀⠀⡤⢄⠀⠀⠀⠀⠀⡄
⠀⠀⠸⡁⠀⠀⠀⠀⣴⠂⠀⠀⠀⣉⣁⠤⠀⠀⠀⠀⠘⢅⢁⠂⢀⠄⢄⠀⣤⠘⠂⠀⠀⠀⠀⠀⢀⠢⠤
⠀⠀⢘⡇⠀⠀⠀⡶⢗⠀⠀⠀⣀⣀⠀⠀⠄⠀⠀⠀⠖⠂⠠⢤⠞⠀⢀⠈⠁⠀⣀⠄⠀⠀⠀⢤⠀⠑
⠀⠀⢸⠁⠀⠀⠢⣭⠵⠀⡀⠀⠀⠀⠁⠀⢀⠄⠀⠀⠀⠐⢃⣬⣄⠛⢆⠲⠷⠢⠀⠀⠀⠀⠠⠀⠀⢈⣤⡀
⠀⠀⡎⠀⠀⠙⠦⠶⠍⡉⠀⠀⠲⠄⢀⠠⠡⠀⠀⠀⠀⠀⠘⠀⠀⠫⠀⠆⠠⡀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠉⠲⢶⠾⠛⠁⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁
⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣶⣄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣰⣿⠛⢿⣇⠈⣿⡿⠛⣿⡆⠀⣴⣿⣟⡛⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢻⣯⡉⠉⠙⠛⣳⣶⣿⣤⣟⣡⣶⣿⣁⣤⣬⣿⡿⠟⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣤⣶⠿⠋⠉⠛⠿⣶⣤⣀⣠⣴⣿⡿⠟⢛⠛⣉⠙⠋⡉⠙⠋⠉⠉⠉⠉⠁⢀⣠⣴⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⡛⠉⢍⠉⠛⠛⠛⠛⠶⢦⣴⣤⣀⣀⣽⠿⠛⢉⠔⠀⠀⠠⠄⠔⠁⡀⢤⠀⠀⢠⣥⣷⣶⠟⢻⣿⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⢶⣾⡿⠁⠀⠣⠔⠀⠀⢎⣥⣤⣤⣤⣀⠈⣉⠛⠋⠉⠀⠈⠁⡀⠤⡀⠢⠆⠀⠀⠉⠁⠀⢰⣿⠋⢩⡿⢠⡾⠋⣀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⣰⡟⣡⣶⠿⠋⠀⠀⠀⠢⠅⠀⣰⠟⢁⢄⠄⠀⠉⢷⡈⢧⠀⠀⠠⢠⠀⠣⠐⠁⠀⠀⠀⠀⠀⣀⠀⠀⠻⠿⠟⠃⠉⠾⠿⠟⠛⢿⣦
⠀⠀⠀⠀⠀⠀⣿⣴⠟⠁⠀⠀⡎⢀⠁⠀⠀⠀⡏⠀⠠⢸⣷⠁⠀⠈⡇⢸⠀⠀⠀⢡⠠⡀⠀⠀⠠⠤⠤⣤⣾⡿⣷⣤⣶⡶⠾⣷⡜⢿⣿⣶⣶⡾⠏
⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠫⠆⠀⠀⡀⡀⠀⠀⢻⡀⠀⠁⣙⠌⠀⣰⠇⠈⠀⢰⠀⠈⠀⠀⣀⣀⣀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠸⣿⡀⠙⣿
⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠂⠊⠀⠀⠀⠙⠶⢤⣤⠴⠞⠁⠀⠀⠀⣿⣤⠶⠞⠛⠛⢛⣿⠃⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠙⠿⠾⠟
⠀⠀⢀⣿⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⣼⡏⠀⣸⡟
⠀⠀⢸⣿⠀⠀⠛⠁⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡗⠀⠀⠀⣰⣿⣷⠶⠶⣶⣶⠾⠿⠀⠀⣿⡁
⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⣀⡴⠛⣛⡇⠀⢀⣴⣿⣿⣀⣀⣠⣤⣶⣶⡆⠀⢀⡘⢿⣦⡀
⠀⠀⠀⠀⠉⠻⢷⣦⣤⣄⣀⣀⣀⣀⣀⣀⣤⣴⣾⣿⣶⣿⣭⣤⣶⡿⠟⠀⠈⠉⠉⠉⢁⣴⡿⣰⣿⡌⣿⣦⡙⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠀⠉⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⣿⡿⣿⠘⣿⣧⡀⠈⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⣰⡿⠐⣿⠀⢸⣿⠻⢶⠾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠻⣷⣾⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠙⢦⡘⢶⣦⡀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠲⣄⠀⠀⠀⢳⡀⢻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⣠⡿⠀⠀⠀⠘⡇⢸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠀⠀⠀⠀⢀⡇⢈⣴⡾⢿⣿⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⡀⠰⡄⠀⢀⣧⠀⠀⣸⠃⣨⠴⠶⠾⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇⠀⠤⠋⢀⣿⢀⣿⡟⠀⣰⢃⡠⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⢠⡀⠀⠀⠀⣠⣿⣿⣿⠃⠀⠀⢀⣿⠏⡸⠛⣠⠞⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢤⣀⠀⠀⠀⠀⢰⣌⠉⠁⣼⡇⠀⠀⣰⣿⢟⣿⠏⠀⠀⢀⠾⢋⣠⠴⢛⣡⣶⣿⣿⣿⡶⠖⠈
⠀⠀⠀⠀⣿⣷⡄⠀⠀⣼⣿⡀⠀⢸⠇⠀⣰⠟⠵⠋⠁⠠⠴⠖⠒⠊⠩⠤⠶⢛⣋⣽⣿⡿⠋
⠀⠀⠀⠀⢿⣿⣿⡆⠀⣿⣿⠁⠀⢋⣀⣤⣴⣶⣾⡿⠿⠛⠒⠶⢤⣀⠀⠀⠉⠉⠉⠉⠉
⢀⣇⠀⠀⠸⣿⡿⣷⠀⢹⢋⣠⣾⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠈⠳⣄⠹⣿⣶⣶⣶⣦⣄⡀⣀
⢸⣿⣷⣄⡀⢹⣧⡇⠀⣠⣿⡿⠟⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠈⠙⠛⠛⠿⠟⠋⠁
⠸⣿⣿⣿⣷⠀⡟⢠⣾⢟⣥⣶⣿⠟⠓⠶⣄⠀⠀⠀⠀⠀⠀⢠⠄⠄⠀⠀⠈⡇⠲⣄⣀⣀⣀
⠀⠙⣿⣎⢿⡇⢀⣿⡿⠛⠛⠀⠀⠀⠄⠄⠈⡀⠀⠀⣾⡟⠁⠀⠀⠀⠀⠀⠀⡇⠀⠈⠙⠻⠿⠷⠤⠄
⠀⠀⠈⢿⡜⡇⣿⠃⠀⠀⠀⠀⠀⠁⣀⣀⡰⠁⠀⠈⣿⡇⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠈⠇⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⣤⣤⠶⠋
⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⢀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⡤⣡⢅⠀⢀⠀⠀⠀⠀⠀⠀⢠⢁⠀⠀⠀⠀⠀⢰⣦⣀⡰⣞⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠠⠤⠀⠈⠀⠀⠺⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠱⠌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠂⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠙⣯⣿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠴⠰⠆⠀⠠⠤⠀⠠⠄⠰⠄⠀⠀⡀⠀⠀⠄⠀⠠⠀⠀⢼⡴⣦⡄⢄⡰⠀⠀⠀⠱⠷⠀⠀⡐⠀⠀⠠⡄⣶⣄⠉⠁
⠀⠀⠀⠀⠀⠀⣀⢒⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⡄⠒⣠⠐⢦⡤⠀⠀⠀⠉⠁⠀⠀⡄⠀⠀⠀⠀⢠⢞⣁⠑⠄⠀⠠⠖⠈⠀⠲⠶
⠀⠀⠀⠀⠀⢄⡩⡄⡍⢰⣿⠇⠀⢠⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⢀⠰⠠⠄
⠀⠀⢠⣶⡄⠈⠙⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠁⠀⠁⠠⠀⢀⠀⡀⢀⠈⠀⠀⣤⠀⠀⡀⢀⠀⢀⡐⣨
⠀⠀⠀⢂⡀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⢀⠆⠀⠀⡤⣶⠆⢃⠀⣀⡀⠀⠀⠀⠀⠘⠿⠀⠸⡿⠁⠀⠈⠀⠀⠙⠎⠀⠀⢱⠇
⠀⠰⠶⣷⠶⠄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⣬⡄⠂⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⡀⢠⣤⡴⠀⠀⠀⡀⢀⡀⠀⠀⠀⣤
⠀⣄⠀⠀⢀⡀⠈⠻⠷⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡠⡊⠉⢠⣀⡦⣴⡂⡄⠀⢠⠀⢠⠴⣄⠀⢾⢟⠀⠈⠁⠀⠀⠻⠂⠈⠀⠀⠀⠀⠀⠏
⠀⣽⣷⣴⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⠀⠀⠈⠀⠃⠙⠁⠙⠉⠉⠋⠉⠉⠁⠀⠀⠀⠂⠀⠀⢤⣆⠒⢸⠴⣪⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠈⠈⠀⠉⠀⠀⠤⠀⠀⠀⡶⠶⠎⠀⠀⠀⠀⠢⠀⠀⢀⣺⣭⢸⠷⢺⢶⣂⡆⡶⢤⡦⣖⣡⠀⠐⠂⠀⠀⠀⠠⡐⢾⠀⠓⠃⠄⠀⠀⠀⠋⠀⠀⠁
⠀⢠⡈⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⣹⣉⡱⠊⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣀⠀⠀⢰⣿⣿⠂⠀⠀⠀⠀⢠⡄⠀⣠⠂⠀⠀⠱⡀⠀⠀⠀⠾⠂
⠀⢀⣟⣛⣧⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠿⢀⣤⣄⠉⠉⠀⠀⠦⠀⠀⡾⠀⠀⠐⠰⣿⣾⢾⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢷⣄⡄
⠀⠀⠿⠿⠏⠀⢠⣠⠤⣄⠀⢀⣀⡀⠀⢠⣾⣿⣿⣶⡀⠘⣾⣏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣌⣠⣦⠄⠀⢠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⣀⠀⡄⠀⠀⢿⣿⣿⣿⣿⠇⠀⠀⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢛⠄⠀⠀⠘⠘⠚⠙⠉⠀⠀⢰⣦⣀
⠀⣰⠀⡠⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⣤⣄⠉⠛⠟⠃⠀⠀⠀⠐⠂⠀⠆⠀⡿⣿⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠸⡟⠀⠃⠚⠒⠂⠀⠙⠑⠂⠈⠛⠉
⠀⣿⢰⠃⠐⢃⠌⠀⠀⠠⣿⣿⡇⠀⠛⠋⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧⠀⣤⣤⠄⠀⣾⣿

⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢀⡀⣀
⠀⠀⠀⠈⠊⠀⠀⠂⠀⠐⠀⠐⠠⠔⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⠊⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠰⡁⠀⠀⠀⠃⠰⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⠲⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠆⠸⠁⠐⠀⠸⠀⠆⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⢀⡠
⠀⠀⠀⠈
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⢀⠠⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠐⠂⠀⠂⠉⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠄⠀⠀⠣⠀⠀⠜⠓⠒⠃⠀⠀⠀⠀⠑⠒⠒⠈
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠘⠤⠤⠀⠂⠁⠐⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠠⡀⢨⠀⠀⠀⠀⢠⠆⠒

⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠄⠐⠒⠀⠀⠀⠀⠀⠀⠠⠀⠂⠉⠙⠲⠀⣄⠀⠀⠀⠀⢀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠀⠀⠀⠐⠌⠓⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠐⠀⠀⠑⠦⠀⠀⠀⠔⠩⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠠⠄⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢈⠀⠀⠑⠀⠀⠀⠀⠀⠀⢈⡖⠐⡂⠐⠄⠀⠀⠀⠀⠀⠀⢀⡀⢐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣄⣹⣠⡞⠉⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣷⣄⣤⣤⣴⣶⣶⣶⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠰⣆⠀⣰⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢻⡇⢠⠟
⠀⠀⠀⠀⣴⣤⣤⣼⣷⣧⣄⡀⠀⠀⣾⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⣾⣿⣷⣦⣤⣘⡛⣦⣷⡏⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⡼⢿⠛⠛⠿⣷⣦⣼⣿⡇⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⢠⣿⣿⣿⠉⠛⠿⢿⣿⣿⡷⠒⠚⠻⠃⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠰⠿⠀⠾⠆⠀⠀⠈⠙⠻⠿⠃⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣸⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡟⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢻⣿⣷⣤⣤⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣻⢿⡿⣿⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣡⠏⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠈⠋

⠀⠀⢦⠀⠲⡀⠀⢀⡇⠀⠀⠐⠃⣠⠜⠁⠀⠀⠀⡀⠀⠈⠀⠀⠀⢆⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠅⠘⢎⠀⠉⠓⡞⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠈⡍⠀⠀⠀⠀⠀⠀⠳⡀⠀⢠⠇
⢀⠀⠀⠰⡄⠀⣠⡇⣀⠀⠀⢐⠁⠀⠀⠀⠐⣲⠀⠈⢆⠀⠀⠈⠰⠀⠘⢄⠀⣀⣀⠸⡄⠀⠈⠓⡞
⢪⠀⠀⠀⡧⠀⠆⠐⢆⠀⢒⡟⠀⠀⠀⠀⠀⠁⠀⠀⠈⢧⡀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠇⠀⣀⣸⡇⣀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠹⣎⠀⠀⠀⠁⠀⡠⠀⠀⠀⠀⠀⠱⣄⠀⠀⢀⡄⠐⠌⠁⠀⢣⠀⠘⡇⠁
⠀⢀⠀⡀⠸⡆⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢷⣠⠏⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⡀
⠠⠀⠈⠈⠙⢇⠀⠠⠀⠀⠀⠀⠙⣄⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠆⠀⠀⢸⠁⠉⠁
⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠠⠀⠀⠈⢷⡋⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠘⢦⡀⠀⣠⠀⠀⠀⡈
⠀⠠⡀⠀⠀⢸⡄⠀⠀⠀⠀⠆⠂⡶⠀⠳⡄⠀⠀⠀⠀⠆⠀⠈⠲⡀⠀⠀⠹⣶⠅⡀⠀⢠⠃
⠀⠀⡸⠓⠂⠘⡇⠀⠄⢦⡀⢀⠎⢀⠀⠀⠹⣄⠀⠀⠀⠰⡤⠂⠀⠀⢄⠀⠀⡈⠀⠈⠂⡌⠀⠀⠀⢀⢀
⢄⡖⠀⠑⠀⠒⡇⠁⠀⠀⢰⠃⠀⠀⠢⠐⠈⠙⣆⠀⢀⣤⡄⠀⠀⠀⠀⠱⡀⠀⠀⠀⣸⠂⠀⠀⣀⠀⠈
⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⢀⡸⠞⠉⠀⠀⢤⠀⠀⠀⠀⠘⢦⡀⣰⠏⠉⠋⠉⠁
⠀⠀⠀⡀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠁⠀⢀⡴⠃⠤⠄⠀⢠⠀⠀⢣⡀⠀⠀⠀⠘⣦⠛⠀⠀⠀⠀⠀⠀⢰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢁⠀⠀⠀⠀⡈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣀⡀⢠⠀⠘⡀⠀⠀⢠⠁⠀⡀⢀⡀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠈⠁⠀⠀⠱⡀⢀⠆⠀⠀⠈⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡀⠀⠀⠀⠙⠃⠀⠀⠀⢀⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠴⠠⠄⡀⡀⠀⠀⡌⠑⣄⠀⠀⠀⠀⠀⢸⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⡇⠀⠀⠀⠀⠀⡠⠊⢁⠀⠀⢀⢀⠄⠄⠦
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠠⢄⡉⠑⠘⠀⠀⡀⠑⢄⡀⠀⠀⢸⠀⠀⠀⠑⡄⠀⠀⠀⠀⢠⠃⠀⠀⠀⡇⠀⠀⢀⡤⠊⢀⠀⠈⠓⠃⢁⡠⠄⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠉⠢⣀⠀⠀⠰⠀⠀⠈⠐⠂⠊⠀⠀⠀⠀⠘⡄⠀⠀⡠⠃⠀⠀⠀⠀⠑⠐⠂⠁⠀⠀⠂⠀⠀⡠⠔⠉⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠈⠓⢄⠀⡆⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈⢦⡔⠁⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⢐⠀⡠⠊⠁⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠃⠀⠀⠀⠀⠈⠱⡅⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢀⠀⠀⠀⢰⠎⠀⠀⠀⠀⠀⠘⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠐⠠⢀⠀⠀⠀⠀⠀⠘⢆⠀⠀⢡⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⡌⠀⠀⡔⠁⠀⠀⠀⠀⢀⡀⠔⠂⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠈⠉⠐⠂⠤⠄⣀⣱⡄⠘⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠃⢠⣎⣀⠠⠤⠐⠂⠉⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠑⡄⠀⠀⢠⠃⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠈⢦⡔⠁⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠨⠆⠀⠉⠀⠒⠒⠒⠀⠠⠤⠵⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠧⠤⠄⠀⠒⠒⠒⠈⠁⠀⠰⠅⢀
⠀⠀⠀⠀⠀⢀⡀⠤⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⢀⡀
⠀⠀⠀⠀⠻⢄⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢆⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⡠⠛
⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠉⠒⠒⠂⠹⢤⡤⠤⠄⠀⠀⢀⣀⣀⣀⣀⣀⣑⣤⣤⣊⣀⣀⣀⣀⣀⡀⠀⠀⠠⠤⢤⡴⠃⠒⠒⠂⠉⠀⠀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢉⡉⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄⠀⠀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⠔⠋⡟⢢⡀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢀⡴⠋⣀⠔⠁⠀⠀⡇⠀⠉⠢⣀⠑⢤⡀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠔⡡⠔⠊⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠑⠢⢍⢢⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃
⢀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡞⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⢀
⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⣦⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⡀
⠀⠀⠀⠀⠀⠀⡶⠀⠈⠄⣠⣤⣀⣀⠀⠀⠀⠀⠀⠀⣠⣾⣆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣶⠇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣯⡵⣤⣴⡾⡅
⠀⠀⢀⠀⠀⠀⠀⠠⠂⣰⠿⣿⣿⣿⣿⢷⣶⣤⣤⣀⡣⣍⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢹⡟⠀⠀⢀⠀⠀⠀⠀⢼⣧⠀⠀⠋⠀⢉⠙⠀⡅
⠀⠀⢈⠀⠀⣼⠀⡆⣼⣇⢾⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⠀⡄⠻⡇⠀⠀⠀⠀⢰⠀⣀⣿⣿⣦⠀⠀⠀⡏⣠⠈⡇
⠀⠀⠀⢳⠠⢿⣆⣴⠿⠻⣿⣿⡿⠿⣿⣿⢟⠾⠛⣿⣿⣿⣿⡄⠀⠀⠘⠀⢀⡀⠀⠀⣦⠀⠀⠀⠀⡀⡇⢘⡇⠀⠀⠀⣠⣿⣦⣿⣿⡛⠿⣷⡆⠀⠈⠀⠀⡁
⠀⠀⢀⡸⣷⣸⣿⣧⢿⣾⣿⣿⣧⣤⣽⣿⡟⠿⢿⣿⣿⡿⢿⣷⡄⠀⢨⠀⡌⠀⠀⠀⣿⠀⢀⢄⠀⢻⠁⣯⡇⠀⠀⣲⣿⡟⣅⢱⣀⣠⡀⢈⠻⠀⠈⠿⠁⠑
⠀⠀⠀⢢⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣯⡤⠀⢀⡏⠉⢉⣻⣿⣶⣶⣤⣄⡀⠀⠊⠉⣿⡄⠫⠘⢨⠋⣷⣏⣧⡼⣴⣿⣟⡙⢻⣶⣾⣻⣆⣿⣾⣂⠠⠄⠀⠁
⠀⠀⠈⡈⣿⣻⣿⣿⣿⣿⣿⣿⣿⠿⢹⣿⣟⣻⣦⣶⠐⢲⣾⣿⣿⣿⣯⣿⣿⣷⣶⣶⣿⣺⣦⡶⣾⢻⣿⣿⡿⣇⣦⣼⣇⠀⢀⣹⡉⣿⣿⣿⡿⣿⠀⡀
⠀⠀⠀⠁⢿⣾⣿⣿⠛⣿⣿⣿⣿⣶⣿⣯⡽⣝⣿⣿⣄⡈⣿⣿⣿⣿⡿⢻⣿⣿⢿⣿⣿⢿⢿⡿⠿⢾⣿⣻⣿⣿⠿⣿⡿⣴⣾⡿⢃⡷⡯⣝⣯⣝⣻⣿
⠀⠀⠀⠒⣼⡿⣟⣼⣷⣿⣿⡥⠸⣏⠉⢙⣿⢾⣯⡇⢸⣈⡿⢿⣷⣶⡟⣾⣷⠷⣶⣿⣿⣾⣟⣿⣅⣸⣿⠟⠻⣾⠞⣣⣿⠟⣻⣿⠿⢷⣾⣭⣽⣾⣟⣿⠀⠈⠃
⢠⡦⠀⠀⠀⠿⣡⣾⡟⣛⣻⡵⢶⣿⣿⣿⠿⡷⡼⣻⣿⣱⣆⣲⢿⣿⣻⣿⠛⢹⣿⣿⣿⣿⣧⣼⣎⣿⣿⣽⣇⢷⡼⣯⣧⠂⠉⢘⣿⣏⣣⣿⣿⡿⢷⣟⠀⢷⠂
⠈⠡⣀⡆⠘⣰⡧⢿⠛⣡⡿⣏⢛⡟⣇⣁⡃⣠⠜⢡⡀⣽⣩⢟⣘⠻⣿⣏⢈⣈⣻⣿⣿⣿⣿⣯⣿⢿⡿⣿⣏⠻⣿⡛⣿⡄⠀⣀⣿⣿⣿⢛⡿⣧⣾⣿⠀⠨
⠀⠙⢻⡁⡼⠁⠀⣂⡼⣿⣤⢫⢿⣿⣈⡯⢷⣅⣮⢠⡿⠇⣹⣤⣾⢕⣿⣿⣳⣿⡟⣿⣻⣿⣿⣿⣿⣿⣿⣿⣭⣹⣿⣿⣟⣻⣿⣿⣿⠻⣟⣹⡿⢿⣽⡿⡇⢸
⠀⠀⠀⢣⣠⡆⣼⢿⣷⣟⣿⠰⢚⡽⢿⡿⢦⡋⡵⠾⠂⢶⣧⢹⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣿⣼⣷⣿⡙⠿⣿⢷⣾⣷⣿⣿⢻⣧⣼⣻⣿⣫⣠⣿⡇⢠
⠀⠀⠀⠀⠧⢸⢧⣸⣿⣾⣧⣷⣤⣾⡏⣳⢼⣿⣷⣿⣿⣿⣿⣸⣿⠟⡻⠛⢿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⠒⠤⣤⡌⢶⣿⣿⣯⣻⣏⣈⣇⣹⣿⣿⣿⣿⣧⠘
⠀⠀⠀⠀⠘⣾⣾⢿⣿⣏⣉⣉⢻⣟⣿⣿⣿⣿⣿⣿⣿⣿⡿⡼⣀⣰⣿⡿⢻⣿⠿⣼⠁⠉⢊⡁⠀⠻⠿⠷⡶⣿⣿⡏⢹⣿⣧⣿⢟⡿⠏⡹⣷⣶⡇⠉⡹⠘
⠀⡞⣁⡉⡑⣾⣇⣘⣿⣿⣿⣿⣷⣽⠹⣿⣿⣿⣿⣿⣿⣿⡗⣣⡿⢻⣿⣷⣾⣿⣦⡼⣀⣸⠋⣁⣸⣠⣄⣿⠀⣨⣾⣿⣾⣿⣿⠿⣏⡁⠘⣤⣌⣷⡏⢿⡦⠖⠋
⠈⠋⠀⠈⠹⣿⣦⡉⣿⣿⣿⣿⣟⠿⣦⡼⣿⣿⣿⣿⣟⣿⣾⣿⣦⣿⣿⣟⣻⣿⣿⣿⣿⣿⢿⡟⡄⢸⣿⣿⣿⣿⣿⣿⠿⣿⣿⠀⠀⢍⣢⠈⠁⣿⢷⣷⣤
⠀⠀⠀⠀⠀⢸⣿⣹⣿⣿⣿⠋⠀⢠⣿⣿⢶⣽⣿⣿⣿⣿⣿⣿⢡⣟⣽⣿⣻⣏⣭⣿⡗⢻⣾⡦⢶⣿⡻⡏⢛⠋⣿⢙⠀⣿⡏⣲⣾⣿⣿⣿⢶⣟⢌⣿⠋
⠀⠀⠀⠀⠀⢸⣿⣻⡟⣿⢣⡄⣰⣿⣿⣿⣞⣿⣿⣿⣿⣯⣻⣿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣼⣿⣵⣾⣿⡅⢠⡊⠀⣿⣭⣿⣿⣽⡁⠩⠏⢙⠀⠀⢹⣿⣿⡗
⠀⠀⠀⠀⠀⠀⢻⡇⠹⣿⠸⡰⠟⠃⠟⠻⡟⢿⣿⡿⡿⠟⣿⡿⣿⠻⠿⢿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡿⣷⣤⣶⣾⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠻⠈⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠉⠉⠀⠙⠁⠉⠉⠈⠉⠈⠉⠀⠀⠀⠈⠃⠋⠙⠙⠙⠉⠉

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣶⣆⠀⣀
⠀⠀⠀⠀⠀⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⣠⠞⢀⠀⠀⠀⠄⢁⠀⠀⠀⠀⠀⠀⣬⣿⣼⡿⣃⡀
⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⣴⣦⣸⣋⣀⠀⠀⠀⢢⠀⢸⠃⣠⠏⠀⠀⠀⠀⠉⢣⡶⠀⠀⠀⠀⠙⣿⣭⣾⡿⢃
⠀⠀⢶⣦⣤⡈⢻⢠⣶⣿⠇⠀⠀⢀⣬⡉⡟⣻⡟⠀⠀⠀⢸⡆⡴⠞⢃⡄⠀⠀⠀⠠⠀⣄⣇⠀⠀⠀⠀⠀⢉⣭⣭⣶⡿⠃
⠀⠀⠀⠙⠻⠿⠾⠟⠛⣁⡄⠀⠀⢈⣛⢻⡟⠻⠛⠀⠀⢸⡄⠙⠂⣠⡞⠀⠀⠀⠀⠀⠀⠀⢹⣠⠄⠀⠀⠀⠈⣭⣍⣧⣶⡷
⠀⠀⠻⣿⣶⣄⢀⣴⣿⡿⠁⠀⠀⠻⠿⢿⠿⢿⡿⠀⠐⣄⢿⣄⠋⠁⣴⠃⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣹⣛⣋⣡⣤⡀
⠀⠀⣀⣈⣙⠛⢻⠉⣁⣠⣤⡄⠀⣾⣷⣾⣥⣶⣶⡄⠀⠸⣆⠈⡤⠞⣡⠎⠀⠀⠀⠀⠀⠉⢹⠀⠀⠀⠀⠀⠀⣙⣛⣛⣩⣭⡀
⠀⠀⠙⠿⠿⠷⠾⠾⠿⠟⠋⠀⠀⠉⠉⠁⡏⠙⠋⠀⠀⠀⠈⠓⣀⡴⠋⠀⠀⠀⠀⠀⠀⣀⠸⠛⠀⠀⠀⠀⠈⣛⡛⣟⣻⣭
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⠀⠀⠀⠀⠀⠈⠻⠟⡟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⡆⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⢠⡀⠀⡀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⢀⠀⣇⣿⣰⠏⠀⠀⠀⠀⠀⡀⠀⠠⢎⢞⡇⠀⠀⠀⠀⢸⢀⠞⠉⠀⠀⠀⠈⣿⣷⡀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⡠
⠀⠀⠀⠈⣧⢿⣿⣫⡶⠟⠁⠀⠀⠀⢝⢯⣢⡿⠚⣤⡄⠀⢸⠃⢠⢃⡰⠇⠀⠀⠀⢠⡙⢿⢃⣴⡗⠀⠀⢀⡀⠀⣀⠀⠀⢀
⠀⠀⢀⢳⣼⣿⣯⣴⣾⣛⡁⠀⠀⣤⡤⢅⣸⡫⠍⣊⠀⠀⠘⢄⡞⠉⢀⡀⠀⠀⠀⠸⣿⣦⠿⠟⣁⠀⠀⠀⠁⠂⣬⡄⠀⠉
⠀⠀⣈⢷⣝⢿⣛⣿⣯⠵⠂⠀⢀⣈⡊⠽⡿⢒⠭⠋⠀⣀⠀⢨⠃⡠⠊⠁⠀⠀⠸⣦⣌⢹⣶⣿⠏⠀⠀⠐⠆⠀⣈⡀⠀⠞
⠀⠀⣌⡳⣝⠢⠥⢶⣞⣛⡁⠀⠙⣕⠮⢵⡖⡭⢽⡿⠀⠙⣄⢸⠉⠀⠀⠀⠀⠀⢠⣙⠻⢟⣩⣶⡏⠀⠀⢠⡀⠀⠙⠃⢀⡄
⠀⠐⢮⣝⡾⣝⣛⣻⣭⡥⠄⠀⢶⢖⡋⢭⡭⢛⣃⣤⠠⣦⠀⢹⣀⡼⠋⠀⠀⠀⢈⠿⣷⡞⣛⣭⣴⠀⠀⠀⠀⠀⠲⠗⠁
⠀⠀⠀⠉⠛⠺⠵⠶⠟⠛⠁⠀⠈⠑⠎⠹⠋⠍⠚⠁⠀⠈⠢⣼⠁⠀⠀⠀⠀⠀⠘⢿⣶⣼⠛⠛⠁⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠣⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⣿⣆⠀⢸⡄⠀⢀⣴⡟⠁⣼⡏⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⣿⣿⣿⣆⠈⣷⣠⣿⠋⠀⣸⣿⠁⠀⠀⠀⢀⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀⢿⣿⣿⣿⠀⢿⡿⠁⠀⣰⣿⡏⠀⠀⣠⣴⡿⠋⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⢸⣿⣿⣷⣌⢿⣿⣿⠀⢸⡇⠀⢠⣿⣿⠁⣰⣾⡿⠋⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢿⣿⣿⣿⡎⠿⠋⠀⣼⡇⢀⣿⣿⠇⣴⣿⠟⣠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡟⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠓⠈⢻⣿⣿⣿⡀⠀⢠⣿⠀⣾⣿⣿⣾⡿⢃⣾⣿⠋⢀
⠀⠀⠀⠀⠀⠀⢠⣿⠇⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠈⢦⡀⠀⠀⠈⠻⣿⡇⠀⣼⠃⣼⣿⣿⣿⢟⣴⣿⡿⠁⣠⠏
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⢻⣦⠀⠀⠀⣿⡇⢰⠏⣰⣿⣿⣿⣿⣿⣿⠟⢀⡴⠃
⠀⠀⠀⠀⠀⠀⣿⣿⡀⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⢻⣷⡀⠀⣿⡇⠏⣰⣿⣿⣿⣿⣿⡿⢃⣴⠟⠁
⠀⠀⠀⠀⠀⠀⢿⣿⡇⠘⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡌⢿⣷⣠⡿⠀⣰⣿⣿⣿⣿⠟⢁⣴⠟⠁
⡆⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠙⠛⠀⠀⠀⠀⠀⣄⠀⠀⠀⣸⣿⣷⠘⣿⣿⠇⣰⣿⣿⠟⢋⣠⠾⢋⡀
⣇⠀⠀⠀⠀⠀⠘⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⣿⠀⣿⡟⢠⣿⡿⢁⡴⠛⣡⣾⠏
⣿⡀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣄⠀⠀⠀⠀⠀⢠⣿⣧⠀⢸⣿⡟⢻⠀⣿⠃⣾⠟⠠⠋⣠⣾⣿⠟
⢹⣧⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣦⠀⠀⠀⠀⠸⣿⣿⡇⣸⣿⡇⠘⢠⡟⢸⡏⠀⢀⣾⣿⡿⠋⣠⡆
⠈⣿⣷⣄⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⡄⠀⠀⠀⢿⣿⣿⣿⣿⠁⠀⢸⡇⡟⠀⠀⣸⠿⣫⣴⣿⡿⠁⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠈⠻⣿⣿⣶⣤⣀⠀⠀⠘⢿⣿⣿⣿⣦⠀⠀⠀⠻⣿⣿⣿⠀⠀⣿⠀⠀⠀⡴⠃⣾⣿⠿⢋⣠⡴⠖⠛⠋⠉⠉⠉⠉⠉⠙⡛⠶⣄
⠀⠀⠀⠈⠙⠻⣿⣿⣿⣦⣄⠀⠻⣿⣿⡻⣷⡀⠀⠀⠘⣿⣿⠀⠀⡿⠀⢀⡞⠁⠀⢀⣤⠞⠋⠁⣀⣀⣀⣀⣀⣀⣀⣠⣴⡾⠃⠀⢨⡙⣦
⠀⠀⠀⠀⢠⣄⠈⠻⣿⣿⣿⣷⣄⠈⠻⣿⣝⢿⣄⠀⠀⣿⣿⠀⢸⡇⠀⡼⠀⢀⡴⠋⣀⡴⠞⠛⣉⣩⣤⣿⣿⣿⣿⣿⠉⠀⠀⠀⠘⣿⡌⢳⡀
⠀⠐⣦⡀⠀⢻⣷⣦⣈⠻⣿⣿⣿⣷⣄⠙⢿⣦⡹⣆⠀⢸⣿⡀⢸⠃⢰⠁⣰⠋⣠⠞⠉⣠⣶⣿⣿⠿⠛⠉⠀⠀⠈⢿⣷⡀⠀⠀⠀⠙⡇⠀⢧
⠀⠀⠘⣿⣷⣦⣍⡻⢿⣧⡘⢿⣿⣿⣿⣷⡄⠙⢷⡈⠀⢸⣿⡇⢸⠀⡇⡼⢁⡾⢁⣴⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⢶⡈⣿⣇⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠈⠻⣿⣿⣿⣷⣯⡁⠀⠻⣿⣿⣿⣿⣦⡈⠻⡄⠀⣿⣇⢸⢸⡼⢡⠟⣠⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢸⣿
⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣦⡀⠈⢿⣿⣿⣿⣷⡄⠁⠀⢻⣿⣾⣾⢃⡏⣰⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣾⡿⢸⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠙⣿⣿⣿⣿⣆⠀⠸⣿⣿⡿⣸⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⣰⣿⢇⡿⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣧⠀⣿⣿⣷⣧⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⣡⣾⣿⠏⣼⠃⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣿⣇⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⠁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡛⠙⠻⣧⣄⣤⣀⠤⠂⠘⢇⠀⠀⠀⢿⣶⣦⠀⢀⣼⣾⡇⢁⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠈⠁
⠀⠀⠀⠀⠀⣠⡌⡵⠇⠀⠘⣿⣿⣮⣿⡦⢻⣁⡘⢷⣦⣠⣤⠟⠃⣼⠟⠛⠛⠁⠘⠀⢴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⠖⠃
⠀⠀⠀⣰⣰⣿⠟⠃⠀⠀⠀⠉⠀⠈⡀⠀⠚⠋⠁⠈⢋⡫⡀⠀⠾⣝⣴⣾⠿⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠨⠉⠁⠀⠀⠀⣠⠀⠀⠀⠀⢠⣶⡄⠀⠀⣸⡁⢀⣷⣦⣶⣿⣿⣿⣿⣀⠀⢿⣷⣶⡦⠀⢰⣁⠆⠀⢀⠰⡲⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠄⢂⢀⡚⠮⢿⣦⣀⠀⠨⠉⠛⣷⣶⣿⠃⡄⢀⣽⣥⣿⣿⡿⣻⣷⣄⣼⣿⢀⣬⡻⠽⢯⠄⡀⣀⠉⠀⠤⣄⠀⠄⠀⡀
⠀⠀⠀⢀⡀⢠⠼⠮⠛⠀⢨⣧⢈⣠⢀⣀⡁⠨⢁⣍⢓⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣷⣿⠋⢀⣟⢁⣤⣙⣿⣿⣧⣖⣶⣫⢀⡐⢠
⠀⠀⠀⠀⣽⠀⡐⠒⠶⠀⠀⣿⡽⢿⣿⣿⡦⣠⣾⣀⣷⣿⡿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢿⡿⠻⣛⣻⣿⣿⣾⣷⣮⣠⢡⡏⠄⠁
⠀⠀⡄⠀⡉⠀⡇⠀⠀⠀⢀⣿⠀⢸⣿⠿⠇⢁⠛⠛⣿⣿⣧⣾⣿⣿⣿⣿⣿⣷⣽⣿⣿⡟⠀⣿⡏⠘⠙⣿⣿⣿⠛⣻⣉⠿⠦⡤⠄
⠀⢀⡇⢀⣡⢨⡇⠀⠀⠀⢸⣿⡇⣸⡏⣠⣶⣾⣤⣀⣽⣿⣿⣿⣿⡿⢿⣿⣿⣿⣻⢿⣿⣷⣶⣿⣇⢀⣴⣿⣿⣿⣾⣿⣷⣿⡟⠐⠀⣀
⠀⠈⡇⠠⡦⢸⣷⣴⣾⢳⣿⣿⣇⣸⣷⣿⣿⣧⡿⢯⣽⣿⣿⣿⣾⣽⡿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⣿⣿⣷⣾⡿⠿⠿⠋
⠀⢘⣇⣀⠀⢸⡇⠉⠁⠶⠖⢺⣇⢸⡷⣾⣛⣗⣬⣵⣚⣿⣿⣿⡿⢿⣿⡏⠀⠀⢻⣿⣿⣿⣿⣿⣿⣧⣿⣿⢋⣼⡥⠚⠲⠁
⠀⠺⣿⣿⢀⠘⡏⠀⠀⠀⢠⣻⣗⣸⣇⢉⣉⣿⠛⠉⠙⢛⣿⣿⣿⣛⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠄
⠐⢶⣝⡧⠊⠈⠍⠀⠀⠶⡈⢻⣿⣽⣇⠀⠀⣟⢀⣠⣤⠄⠩⠤⣽⣿⣿⣄⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⡘⢴⠞⠀⠀⠀⠀⢀⣘⣾⡿⠿⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠐⣉⣩⣽⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡄
⠀⠷⣿⠇⠀⠀⠀⠀⠀⠈⠹⠿⣶⠂⡐⠀⠀⠿⠃⠀⠀⠀⢀⣀⣶⠤⠭⢿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡟⠁⠁⠀⠀⠀⣀⣀⣀⡀
⠀⠚⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠀⠀⠢⠲⠂⠀⠀⠀⠘⠋⠃⠀⠀⠀⠁⠤⠈⠉⠉⠈⠉⠹⠤⠀⠉⠙⠿⠦⠁⠀⠀⠀⠈⠀⠉⠉

⠀⠐⠷⠀⠀⢶⠄⢴⠀⠀⠀⠀⠀⠀⠀⠀⣄⠄⠀⠀⠀⠀⠦⠶⠢⡀⣰⢰⣠⠖⣣⣴⠐⢀⠀⢀⣦⠛⠀⠀⠀⠀⠐⣄⠀⣀⣀
⠨⠤⢀⡂⠀⣠⡄⣔⠂⡈⠑⡄⠈⠐⠋⠀⠀⢦⠀⠶⢃⢄⡃⣎⡐⣢⣿⡿⠁⠉⠰⠇⡘⢋⠞⣁⣒⠀⠀⠀⠀⠀⠀⣾⣿⠿⢿
⠀⠠⠈⢃⢈⠛⠃⢬⠑⢱⠊⠈⣀⠂⠄⠔⠄⠞⠬⡋⢽⢀⠂⠜⠀⠀⠈⠁⠀⠠⡙⠓⣕⠡⠔⡛⠇⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠃⠀⣀⠀⠘⠆⠁⠀⠀⠈⠈⠀⢢⡴⣒⠎⣠⡂⢌⠂⠟⠛⠛⠛⠹⠗⠶⣶⠤⠄⣑⡀⠘⡟⣧⠄⠀⠀⠀⠀⠀⠿⢿⣶⡾
⠶⢂⣠⡄⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠁⡆⠀⠛⠀⠘⠃⠀⠘⠁⠐⠇⠈⣀⣰⣗⠵⣀⠡⠠⠀⠀⠀⠀⠈⠀⠀⠘
⡦⢤⡷⣾⡆⠐⣄⠀⣀⢀⡀⠀⢀⠀⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠛⠟⠛⠀⠛⡽⣦⡐⠰⠂
⠃⣸⠍⠈⠅⠀⣼⣿⠿⣿⠳⠆⠀⠸⠆⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⢀⠀⠀⠙⠉⣸⠠⢲⣀⣀⠀⡀⢀
⠀⠈⠀⠀⠀⢀⣿⣇⢀⣠⣵⠶⠆⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⠄⠌⠀⣀⡃⢈⣘⣭⢰⠂⠡
⠄⠠⠀⠀⠀⢀⠉⠩⠖⠠⠃⠀⣤⠨⡄⠈⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠃⡀⠉⡃⢉⣂⠘⠤⣀⣀
⠀⠀⠀⠀⠈⠂⡀⠻⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡅⣧⠀⠁⠀⠀⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⢤⠎⢯⠛⡟⡄⢚⠏
⠀⢀⠂⠀⢀⠀⠀⢀⡣⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢲⣇⢀⡀⣄⠀⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠁⠊⠉⠖⠃⢈⠒
⠀⠀⢐⠐⠂⠀⣶⣾⢦⠙⡡⠀⠀⠀⠀⠀⢠⠂⠀⡀⠀⢀⠀⠀⣀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡈⠻⠀⠀⠘⠇⠲⡉⢀
⠐⢿⢿⠾⠀⠀⠊⠠⢰⣾⣿⡄⠀⠀⠀⠀⠈⢆⠀⠀⡅⢐⡀⠀⠀⠀⠈⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠒⢨⣄⢀⣠⡗⡀⠈⠄
⣴⡡⠀⠀⠀⠁⠀⣄⣶⡊⢢⠠⠀⠀⠀⠀⠀⠈⠀⡏⠀⠈⠀⠀⢀⠀⠀⠘⢀⠀⡇⠐⠀⠀⠀⠀⠀⠀⠠⠆⠩⢀⠜⢛⠁
⠢⢉⡇⠀⠀⢂⠈⠹⠛⢃⡅⠀⠀⠀⠀⠀⠀⠐⠀⠃⠀⠘⠐⠀⠰⠀⠀⠀⠘⠀⠱⠔⠀⠀⠀⠀⠀⠀⠀⠈⠄⢠⡄⠈⠢⡄⠁
⠾⠀⠀⠠⠴⠆⠀⠐⡼⡾⣵⢠⠙⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣤⢄⠉⡠⠠⠎⠈⡀
⢀⡀⠀⠀⠙⠂⠀⠀⠈⠟⠢⠁⣀⣧⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣗⢾⢲⣿⠓⠀⢺⣶⠄⠁⠈
⠈⠉⠀⠠⡦⠴⠿⠙⢈⠡⠤⢂⡚⠙⠙⠩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢭⡀⠳⠲⠟⠘⠋⠠⠀⠻⠻⢀⡜⠃
⠀⡀⠀⣤⡄⠠⠄⠶⠀⡔⡰⢓⠂⣠⣦⣯⡶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡆⠢⡊⢧⠙⡤⠄⢀⣤⣮⡱⠀⢀⣡⡀⠉
⢀⣄⠀⠈⠀⠀⠘⠦⠠⣾⠕⠉⢳⠪⠊⠉⠀⠥⠈⠍⠙⠛⠶⠷⠶⠾⠚⠗⡛⠉⠁⠐⠂⠁⠀⠀⠀⢊⠊⠟⠙⠌⠀⢸⠿⡇⠤
⠈⢋⠀⢠⢤⠔⣀⢀⣁⠁⢀⠾⠑⣆⠁⠈⠓⠀⠀⠀⠀⠄⠐⡀⠠⠂⠠⢂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄⠄⠀⠀⠐
⠀⢀⣤⡄⠂⠠⠈⠉⠈⠁⢸⡏⡀⠉⠀⣰⠆⠀⠀⠺⠂⠀⠸⣶⣦⡂⠂⠈⠀⠀⠀⠀⠱⣄⣤⣤⣄⠁⠀⠀⠀⠀⠀⠀⠆⠠⠈
⠀⠀⠁⠀⠀⠂⠐⢀⣬⢀⠉⢠⡟⣿⠁⠈⢀⠀⠄⢜⠀⠈⠀⠀⠀⠑⠀⢀⠀⠀⠂⠀⣼⣿⠛⠛⢿⣿⡀⠀⠀⠀⠀⠠⠀⠀⢄
⠀⠀⠨⣶⠒⠀⠀⠀⠩⠀⠀⢠⠛⠇⠀⠂⠘⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠯⠀⠀⡼⡿⠅⠄⠀⠀⠀⠀⡜⠇

⠙⢹⡏⠉⠏⠀⠀⠀⠀⠀⠙⠁⢰⣬⣿⣿⣿⣿⢿⣿⣻⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣿⠃⠊⠉
⠀⣼⠁⠀⢀⡀⠀⠀⠀⠀⠀⢠⡼⠗⣹⣿⣿⣿⣿⣿⣻⣻⣧⣾⣿⣷⣿⣿⣿⣿⣾⣿⣿⣿⣯⣹⣿⣏⠳⠰⠆
⠀⢹⡀⠀⠘⠙⢿⡶⠄⣴⢰⣯⣦⣿⣋⡼⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⠟⢶⣀⠁
⠀⠈⠁⢀⣶⣶⡏⢿⣻⢋⣉⣙⣿⣯⣿⣠⣾⣿⣿⣿⣷⣿⣇⣀⠙⣿⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠂⣿⡍⡆
⠀⠀⠀⠙⢣⡙⠗⢽⡉⠇⠈⢻⠟⠛⣟⡿⡟⠛⠛⠛⠻⣿⢿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠑⠉⠈⠹⣀⠑
⢀⠀⣍⣽⡇⠀⠀⢠⣉⣓⠶⣶⡶⣟⡽⡀⢀⠀⠀⠀⣤⣧⣀⣉⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠦⣶⣾⡍⠤
⠀⠛⣿⣿⣿⣿⣧⡶⢦⠋⠹⣿⣿⣿⢀⣁⠀⠒⠂⠙⠛⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣿⣿⣿⢉⣸⡿⠉⠐⡀
⠀⠀⣇⢹⣷⣿⣿⣷⡆⢳⣿⣿⣷⣿⢟⡉⠤⣿⣖⣀⣀⣀⣬⣷⣿⣿⣿⢿⠚⣻⣿⣾⣿⣿⣯⣼⡷⡆⠀⠀⠁
⠀⡄⢽⢹⡇⢻⣿⣿⡑⠀⠑⡻⣿⣿⡟⢱⢴⢾⠿⠧⠀⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠷⠃⡅
⠀⠻⣿⢺⡧⣞⣿⣿⣅⣤⣶⣦⣿⣿⣻⣿⣻⣤⣴⣶⣶⣿⣿⣿⡿⢧⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⡀⡇
⠀⠀⢿⣿⣇⠈⠋⣿⡛⢋⣀⢀⡩⣍⠛⢟⣛⣿⣿⣥⣿⣿⣿⣋⣵⠾⠟⠛⡉⣛⣽⣿⣿⣿⣿⣽⠃
⠀⠀⠂⣼⣿⣿⢇⣾⣇⠀⠙⠻⠻⠛⠾⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠁⠲⠭⢍⠁⠌⠉⢉⣻⣭⣀⠄
⠀⠀⠀⠈⠋⠁⠀⠁⠈⠀⠄⠀⠀⠀⠀⠁⢠⣿⣿⣿⡿⠀⠀⢀⠀⠀⢀⡈⡉⠛⣏⣠⣄⣻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢿⣿⣿⣿⣇⣀⠀⠀⠀⠀⠙⠷⠧⢾⣯⣻⣿⡟⣷⢶⡄⡴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠊⠛⢋⣛⡯⣓⣹⣦⣄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⠀⠉⠛⠛⠻⣯⠁⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠠⠄⠀⠀⠛⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⡀⠀⠢⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠄⠀⠀⠐⠒⠒⣀⠀⣀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⡀⣠⠉⠀⠀⠱⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⢐⠈⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠒⠋⠀⠀⠀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠢⠄⠀⠐⠄⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠔⠀⠁
⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠕⠀⡀⠁⠢⠄⢀⡀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⠒⠀⠀⠢⠀⠀⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈
⠀⢀⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠰⠀⠁⠀⢨⠈⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠁⠤⠈⠀⠘⠒⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⠠⢄⡀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⢂⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠀⠀⠀⠀⠀⠱⠈⠀⠙⢢⠀⠀⠀⠈
⠀⠀⠀⣠⠀⢄⠀⠐⠀⠐⠂⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠓⠂⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠈⠃
⠀⠀⠀⠁⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠂⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⢸
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠠⡀⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢜⠀⠀⠀⠀⠀⠀⠁⠀⠀⠡⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠠⠠⡀⠀⠁
⠀⠀⠀⠀⠀⠈⠂⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⢀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠄⠈⡀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠨⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠈
⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠈⠀⠀⠀⠀⠈⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠋
⠀⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠑⡄⢃⠒⠂⠒⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠶⠀⠀⠀⠀⠐
⠀⠀⠀⠀⡰⠴⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠁⠀⠀⠂⠌⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⣄⠡⣀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⠑⢄⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠉⠁⠀⠀⠀⠈⠂⠀⠈⠁⠐⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠂
⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡧⠴⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⡈⠈⠁⠀⠀⠁⢠⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⡀
⠀⠐⣶⡄⠀⠀⠀⠀⣴⣏⡼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⣘⡒⠓⢲⠔⢢⠆
⠀⢠⣿⡇⠀⠀⠀⡀⠀⠉⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⢶⣛⡭⠥⠟⠛⠛⣡⡀
⠀⠸⣿⡇⢀⣴⣿⡟⠀⠀⠃⠀⠀⠠⢈⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠐⠠⠁⠰⠒⠃⣀⣊⡠⢔⠂
⠀⠀⠙⠿⠎⠉⠙⠃⠀⠉⠉⠈⠀⠚⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⠇⠀⠀⠈⠆⠠⠄⢠⠞⠁⢀⣹⣿⣶⣾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠄⠀⢂⣤⢀⢐⡠⢀⣠⣿⣿⣿⣿⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡤⠜⠁⠀⠀⠙⠛⠋⠉⠉⠀⠉
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢩⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⣂⠐⢁⣰⠐⡐⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡮⠿⣧⠟⡈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠲⠊⠁⠘⠠⠢⠄⡖⠒⠆⢹⠊⠁⠈
⠀⠀⠰⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢨⣄⠘⠓⢂⠀⠀⢀⡀⣾⣗⠦⢀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⢀⡴⠈⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠐⠀⠀⠀⠀⠘⣏⠑⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣰⡦⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠁⠉
⠀⠀⠀⠀⠀⣀⠩⠉⠀⣀⡿⡄⠀⠠⣤⡄⠀⠀⢀⣄⠀⠀⣀⠀⠀⠀⠀⢠⣄⠄⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠤⠛⣿⡆⠀⣤⣾⣷⠟⡴⢾⣿⢀⣠⠾⠋⣠⠁⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⢈⣀⠀⠀⢀⡀⠂⠺⢃⡈⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠑⠊⠁⠀⠘⡂⠈⠡⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⡀⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⢠⡆⣤⠈⡄⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⡛⢀⠔⠀⠀⠀⠀⢤⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢄⠀⠐⠁⠀⠀⠘⠁⡀⠀⠒⠲⢄⢀⠠⠀⠀⡥
⠀⠀⠀⠀⠀⠀⠣⣀⠂⠠⠆⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠡⠈⣿⣵⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠳⢄⠤⣤⣀⣁⠐⠤⠀⠐⢞⣻⠎⡁⠀⢭⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠀⠠⠅⠀⣼⣿⡿⠀⠒⢄⠀⠀⠉⡆⠈⠀⠀⠉⠸
⠀⠀⠀⠀⢀⠄⠀⠀⠼⠀⠀⠀⠐⢇⠀⠋⠹⠁⠀⠀⠀⠑⠀⠁
⠀⠀⠀⠀⠂⠀⠀⢀⡜⠒⢀⣤⡂⠀⠀⠀⢰⡀
⠀⠀⠀⠁⠀⠀⠁⢁⣤⣴⠊⢨⠀⠀⠀⠀⡨
⠀⠀⠀⠀⠀⠀⠀⠁⡾⠆⢀⠀⡆⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠲⠀⠈⠀⠰⠀⠀⠀⠀⢰⠄⡃⠰
⠀⠀⠀⠀⠀⠀⠀⢰⠇⢀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠈⠂⠤⣤⣷⣠⠀⠀⠀⢐⠂⡀⠐⠁⠁
⠀⠀⠀⠀⠀⠀⠃⠀⠈⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂⢠⠰⠆⠀⠀⢀⢿⣾⡯⠓⠒⢀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡐⠀⡀⢈⠠⠁⠀⡏⡐⠒⠠⢀⣰⠿⡐⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠈⠆⢘⠀⠳⡀⠀⣰⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⢢⠀⠀⠀⠴⠐⠀⢠⠃⠀⡬⣿⣹⠶⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠈⠀⠁⠢⣤⡷⠛⣮⣀⠤⠚⡁⢀⡄⠀⠀⠻⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠰⠄⠱⠤⢴⣀⡀⠠⢽⣷⣤⡿⣇⠀⠀⠀⢨⠃⢶⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢁⡄⠒⠁⣘⢹⡀⢀⠀⠀⡘⠙⠀⠘⢆⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠆⣀⠠⢶⠻⠠⠂⣡⠀⡛⠰⠀⠈⢃⡀⠀⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⢠⡏⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠀⠀⠸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠢⠐⠀⠀⠀⠀⢀⡀⠐
⡀⠀⠀⠀⠀⣀⠀⠀⢀⠀⢨⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡤⣀⣀⣀⡀⠀⠀⢀⡐⠠⠒⠚⠉⠀⠀⠈⠉⠙⢃⠀⠿⠂⠠⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠴⠤⠤⠤⠆⠀⠈⠑⢢⣤⣠⡄⣀⡀⠀⠀⠀⠠⡀⢀⠘⣧⡀⠀⠘⠐⠉
⠀⠀⠀⠀⠤⡄⠀⠚⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠁⠈⠛⠚⠋⠁⠀⠀⠀⠛⠉⠑
⣀⣀⠐⠃⠉⣀⣤⣠⣤⣰⣤⣴⠜⠂⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠂
⣄⠴⠿⠒⠛⠋⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢄⠤⠰⣄⠀⣤⡴⠤
⠀⠀⠀⠀⠀⣀⡄⣠⠀⠘⠃⢀⡀⠀⡀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠙⠠⠦⠄⠈⠉⠁
⠀⠀⠀⠘⢳⠀⣤⣀⠠⠄⠀⠀⠀⠀⠀⠀⠁⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠄
⠀⠀⠀⡿⠿⡶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠀⢠⡄
⠂⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⠀⠀⠀⠀⡠⠶⠶⣶⣦⡀⠀⢠⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶
⠀⠀⠀⠀⠀⠈⠁⣠⣴⠞⠋⡋⠀⠀⠀⠀⠷⣤⡀⠛⣇⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣠⠮
⠀⠉⠀⠹⣆⠀⠀⠙⠏⢠⡜⢀⡄⠀⠀⣵⡀⢙⣿⣆⠸⠋⠓⣤⢻⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠌⠚
⡄⢠⣤⠈⣿⣄⠐⣷⣦⣼⠀⣼⡇⠀⠀⠻⣖⠄⢚⣿⠡⡎⠇⢠⣾⣧⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣷⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣻⣿⣿⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⣳⣿⣿⣷⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⡿⣱⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣱⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⣿⣳⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣿⢯⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⣄⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣹⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⢐
⠀⠀⠀⠀⠀⠀⠀⠀⣌⣿⣿⣽⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣐⣒⣒⡒⠒⠒⠒⠒⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡉
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠰⠯⠭⠭⠭⠭⠭⣉
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠶⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠠⡌⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠒⠢⠄
⠀⠀⠀⠀⠀⠀⠀⢼⣷⡿⠋⠀⠀⠀⠀⠀⣀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠸⠀⡠⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠟⠈
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣻⣿⣿⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣨⢐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠀⠀⠀⠴⠄⠀⢻⣿⡟
⠀⠀⠀⠀⠀⢀⢢⡂⠀⠀⠨⡀⠀⠀⠈⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠁⠚⠋⠀⣥⣄⠀⠈⠀⠀⠇⠀⢀⣄⠀⠚⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠈⠱⢄⠀⠀⠘⡂⠀⠀⠀⠰⣼⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⠀⡠⠂⠀⠑⣄⠀⡇⠀⠀⠀⠀⠙⢛⠇⠀⠀⠀⠀⠄⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡠⢤⠀⠁⠈⠶⠁⣀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠘⣦⣿⣴⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⢀⣀⡴⠋⠀⠠⠀⠀⠀⠀⠀⠙⢆⠀⠀⡞⠀⠄⠀⠀⠰⡆⢙⠟⠁⣀⡀⢀⠀⠈⠉⠠⡀
⠀⠀⠀⠀⠀⠀⠐⠉⡿⠿⠋⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⢣⡀⢿⡀⠀⢀⣀⠠⠥⠊⠀⠀⠉⠀⠀⢁⢀⢀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⡠⠴⠶⠦⠻⠆⣷⠊⠁⠀⠀⠀⠀⠀⠀⠈⠔⢂⠛⠳⣾⣽⣷⡄⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠈⢢⡄⠈⠀⠀⠀⠀⠀⠀⠘⣦⣀⣤⡈⠀⠀⠀⡑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄⠠⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠟⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠈⡇⠐⢁⠴⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⡀⢐⡀⠤⢀⣟⣴⡁⢠⣀⠀⠀⠀⠀⠀⠱⠀⢀⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢀⢀⡸⠋⠀⠰⠂⠉⢣⡀⠀⠈⠑⢦⡀⠀⠁⠀⢠⣿⣾⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣱⣼⣿⣷⡀⠀⡀⠀⠀⢸⡇⠀⠀⠀⠀⠙⠦⢀⡀⠜⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠋⠘⠿⠁⠀⠱⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⡀⠜⠆⠶⠛⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠚⠀⠀⠀⠀⠀⠀⠈⢻⣄⠄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⣄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⢀⠃⠀⠉⠛⠛⠻⠷⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠂⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠰⠦⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡄⣉⣄⣠⡀⠁⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣼⡏⠉⠉⠛⣷⡄⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣀⢻⣦⣤⠄⠀⠘⣧⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⢀⡿⢀⣴⠾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢧⣟⠐⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠾⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣶⢶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢰⡿⠋⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢷⣄⠀⣿⠁⠀⠘⠦⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⡁⢙⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢠⣶⠿⠿⢾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⢻⣏⠀⠀⢀⠹⣷⠀⠀⠀⠀⠀⠀⢀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡐⣈⠙⠛⠁⡰⠂⣿⠀⠀⠀⣠⡤⠞⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠟⢰⡇⠓⠀⢀⡿⠀⣠⠟⣩⠖⠛⣇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠁⠂⠀⣾⠃⣼⣣⠞⡕⢺⠇⣧⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⢀⠖⠁⠀⠈⠛⠀⠀⠈⣸⣾⣏
⠀⠀⠀⠀⣀⣀⣤⠤⠤⠶⠦⠶⣤⣄⡀⠀⠀⠀⠙⣶⣦⣴⡿⠀⠀⠀⠀⠘⢃⣼⠟⠉⠀⠈⠳⢄
⠀⠀⠀⠀⠙⢿⣄⠰⣖⢛⡿⣶⢶⣬⣇⠀⠀⢀⣀⣽⣿⣿⡷⢦⣤⠶⠖⠋⠛⢿⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠛⣷⡈⠫⡻⣄⡈⠉⢛⡷⠀⠉⠁⠉⢻⡏⠁⣾⠁⠀⠀⠐⠀⢼⣿⣆⣤⣤⣶⠷⠚⠓⠓⢆
⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⣌⠊⠗⣦⣌⠀⠀⠀⠀⠀⣈⡳⠂⠉⠀⠂⣤⣤⣴⣾⣿⣏⣹⣿⡿⠂⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠆⠉⠘⢦⡀⠀⡀⣤⡿⠁⠀⠀⠀⠀⠁⠈⠙⠛⣻⣿⣿⣿⣷⣄⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠟⢿⣿⣂⠠⠤⠤⢤⡁⢀⠊⠉⠁⠀⠙⢯⡀⠙⢣⣤⣜⣓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢚⣿⠶⣿⣏⡿⠀⠀⠀⠀⠙⢬⡄⠀⠀⠀⠀⠀⠉⠀⠀⠛⢉⠻⢿⠦⡙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⢻⣌⡇⠀⠀⠀⠈⠀⣸⣷⠀⠀⠀⣀⡀⠂⠠⣶⣁⠦⠤⠴⠴⠙⣣⡘⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⣿⠗⣀⡀⣤⣤⣤⣼⣿⡷⢿⣾⠉⠀⠀⠀⠀⢩⣤⣥⣉⣉⣉⣉⣥⣼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⢻⡄⠀⠀⠀⣠⣳⠞⠉⠁⠀⠀⢀⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢡⠉⠈⠉⠁⠉⠁⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢁⡼⣻⠟⠁⡋⢃⡤⡆⠀⠀⠀⠀⠀⠀⢸⡿⠁⢻⠿⣦⡀⠀⠀⠀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢠⡿⢱⣯⣴⠶⠟⡫⠜⣣⡀⠀⠀⠀⠀⣠⠏⠀⠀⠈⠀⠈⠟⡹⢿⣿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣿⠧⠖⠒⠒⣉⣡⣴⠞⠛⠁⠉⠒⠀⠂⠹⠀⠀⠀⠀⠀⠀⠈⡅⠉⢋⢷⡹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣠⣤⣴⡶⠾⠿⠋⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⣀⠀⢀⣤⣾⡓⢦⣈⡌⣧⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠋⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠙⢷⣝⠦⣍⡃⣿⢸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣮⣙⠛⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⢤⣤⣀⡀⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢿⡿⢿⣿⡿⣷⣤⣀⡀⠂⠁⠀⠀⠀⠀⠀⠉⠻⣦⡄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡋⣴⣿⣾⠷⣦⣷⣶⠿⠏⣀⡴⠿⠿⢿⣷⣶⣄⠀⠀⠈⢻⡆⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⢠⣿⠋⠀⠀⡈⢿⡇⠀⠀⠀⠀⠀⣴⠀⢀⡙⢿⣧⠀⠀⢸⣿⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠉⠘⢿⡄⠀⣀⣀⠜⠁⠀⠀⣄⠀⠀⣿⣓⣛⣿⢾⠏⠀⠀⣼⡏⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢀⡔⠂⣨⣿⣷⢚⠟⠋⠀⠀⠀⣰⣿⠀⠀⠙⠛⠛⢡⡀⠀⠀⣼⡟⠀⠀⡸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠈⣿⣿⣛⣿⠟⠁⠀⠀⠀⢀⠖⠁⢿⣧⡀⠐⠄⠀⠘⣳⣶⣦⠙⣷⡀⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠈⣩⠟⠁⠀⠀⠀⢀⠒⠀⠀⠀⠈⠛⠿⠿⢶⣦⣤⣤⣄⡉⠲⣿⣗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠀⠈⠛⣻⣿⣿⡆⠀⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠃⠀⠀⠀⠐⠈⢀⣈⣉⡐⠀⠀⠈⣵⣶⡿⢷⡦⠀⣧⣿⣿⡇⠀⠀⠙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢔⡺⣟⣻⣿⡛⠂⠀⠀⠁⠙⠛⠋⠀⢰⢿⣿⠟⣵⠃⠀⠀⢀⣜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠁⣄⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⠃⣾⡇⠀⠀⠀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠈⠷⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣧⣄⠙⣷⣦⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢀⠂⠀⠀⣠⣶⠶⡒⠀⠀⠀⠙⣯⣿⠃⣾⣿⡇⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣠⠞⠁⠀⠀⠀⣠⢟⣤⣄⠀⠈⠛⠀⠀⠀⠀⠀⣠⡿⢃⣻⣿⡟⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⣠⡾⢫⡾⣽⣿⡄⠀⠀⠀⠀⠀⢀⡶⢋⣴⣿⣿⠏⠀⠀⠘⡷⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⢀⣴⡟⣱⡟⣸⣿⣿⡇⠀⠀⠀⠀⠀⣙⠀⢈⣉⣿⣿⣄⠀⠀⠀⠀⠚⠣⣀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣿⡏⠀⣿⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠈⠛⠶⢦⡙⠟⣙⢷⡄⠀⠀⠀⠀⠁⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣄⠀⠀⠀⢿⣧⡀⠙⢶⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡿⠫⢻⡇⠀⠀⢀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⠶⠉⢙⣿⣿⡶⣦⡀⠈⠙⠆⠀⠉⢄⠀⠀⠀⠀⠀⢀⠀⡀⠀⠠⠾⣥⠦⣠⠟⠀⠀⠀⠀⠛⠛⠲
⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠐⠀⠀⠀⢻⣿⣿⣿⣜⠃⠀⠀⡐⠀⠀⠈⣦⠀⠀⠀⠀⢈⠋⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡀⠀⣾⣿⠿⡟⠁⣀⣴⡎⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠁⠿⠶⢤⠀⠚⢆⡰⠀⠀⠀⠀⠀⠀⠰⠇⣼⡇⢀
⠀⠀⠀⠀⠘⣦⢶⣶⣶⠀⠀⠀⠀⠀⠀⠁⣾⢡⡞⣡⣼⡇⠀⠀⠀⠀⢸⡇⠀⠁⠀⡠⠀⠀⠰⠏⠀⢀⠸⣇⣀⠀⠀⠀⡠⣀⡀⢠⡈⠁
⠀⠀⠀⠀⠀⠐⢻⡟⠁⠀⠀⠀⣼⣿⣤⣶⠉⡾⠿⣫⡿⠁⠀⠀⠀⣠⠟⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⠆⠠⣿⣿⡙⡀⠀⠀⠀⣞⠱⠌⠃⠀⠁
⠀⠀⠀⠀⠀⠐⠾⣷⣄⣴⣶⣾⠿⣿⣏⡀⠀⣀⡾⠋⠀⠀⠀⣠⣾⡏⠀⠀⡀⠈⠀⠀⠀⠀⡀⠀⢰⣦⠀⠀⢠⡞⠆⠀⣾⣿⣿⣼⡟⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⢀⢼⡿⣿⣈⣁⢀⡘⠻⣿⣴⠏⠀⠀⢀⣤⠞⠁⣿⢆⠀⠲⠉⠀⠀⠀⠀⠂⣠⣴⠾⡿⢦⡀⠸⣇⠀⠀⠸⣿⣏⠀⠈⠀⠀⠐
⠀⠀⠀⠀⠀⠠⠀⢸⠇⣿⡏⠹⢛⣦⣠⣿⡇⠀⠀⢠⡟⠁⠀⠸⣄⣠⣉⣴⠢⢄⠀⠀⢀⣾⡉⡀⢀⡓⣾⠁⠀⡽⠀⠀⣸⣿⠿⣡
⠀⠀⠀⠀⠀⠀⠃⣾⢤⣼⣥⣶⢛⡷⢦⡿⠿⠀⠀⢿⣦⠀⠀⠀⠀⢩⠁⠠⡄⠀⠀⠀⠈⠛⠛⠓⠞⠳⠁⢀⠞⠀⢀⣴⠿⣿⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠨⠬⡿⠇⠸⠇⠈⠀⠀⠁⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⠰⠂⡀⠰⠀⠀⠀⠑⠂⠀⠉⠀⠐⠛⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠠⢀⠂⠐⣶⠏⠀⠉⠔⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⣆⡴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⣇⣩⠗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢠⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⣀⠔⠂⠀⠀⠀⠀⠀⠀⣀⣄⠀⠲⠐⠂
⠀⠀⠀⠀⠀⢀⡀⣐⡻⠿⡄⣥⠃⠀⠀⠀⠠⠄⠐⠀⠀⣀⣠⠼⠶⠖⠒⠒⠐⣺⢿⠉⠁⠀⠀⢀⣀⡄
⠀⠀⠀⠀⢠⣽⢣⣛⠙⠲⠖⠊⠁⢀⠀⡄⣀⠄⠁⠄⠀⠀⣀⠀⡀⠀⠀⠈⠀⠈⠓⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠉⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⢠⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠀⠀⡀⠀⠀⠉⠀⠀⠀⠀⠀⠠⣞⠇
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠡⠀⠁⠀⡀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂
⠀⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈
⠀⠉⠃⠀⠀⠐⠈⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢄⢸⣿⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠚⢤⠄⠀⢠⠀⡀⡀⠠⠀⠀⠀⠀⠔⠀⠀⠀⠀⠁⠀⠀⠂⠰⠍
⠀⠀⠀⠐⠀⠀⠄⠀⠀⠆⠀⠠⠀⠠⠀⠈⡐⠉⠉⠑⡃⠀⠁⠀⠀⠠⠤⠀⠀⢰⣶⡄⠀⠀⢰⣖⡆
⠀⠀⠀⠀⠠⠏⠀⢀⠀⠀⡀⠤⠼⠂⠀⠀⢶⣄⠂⠀⠜⠀⠀⡀⠀⠀⠀⠀⠀⠠⡀⢪⠐⠀⠀⢀⠀⡄
⠆⠀⠀⠀⠀⠀⢀⡛⠛⠁⠚⠒⠀⣤⣠⡄⠀⠹⣤⡠⢀⠄⠀⠛⠀⠀⠀⠀⠀⠋⠜⡄⠁⠀⢐⢠⠙
⠀⠀⠀⠀⠀⠀⠙⠁⠐⠒⠀⢀⢋⠁⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠲⠗⠐⠀⠈⠀⡛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠄⠀⢐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢠⠇⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠐⠤⠐⠓⠒⡉⠁⠐⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠁
⠀⠀⢘⠃⢀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠠⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋
⠀⠀⡬⠂⠁⠃⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⢀⠤⠔⠛⠁⠀⠀⠀⠠
⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡐⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠁
⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⢀⡷⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⡇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢀⠀⠠⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠁⢠⡄
⠀⠀⠀⠀⠀⠀⣀⠀⢔⠀⠀⠁⠀⠀⠀⠀⡴⠀⠰⠀⠀⢁⡀⢠⡄⠰⠦⡚⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⡀⠂⠀⠀⠀⢠
⠀⠀⠀⠤⠈⠓⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣀⠜⠂⢈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⣠⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠀⣀⠵⠿⠁⡁⠉⠐⠂⠈⠁⡄⠀⠀⠀⠀⠀⠁⠆⢡⢘⣁⠁⠰⢀⡀⠰⠆⠸⠇⠀⠀⠁⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠀⠁⠀⠀⠀⠀⠆⠀⠀⠀⠁⠁⠀⠀⠀⠠⠀⠀⠈⠉⠀⠀⠀⠁⠈⢣⠀⠃⠀⠀⠁⠀⠀⡀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⢀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⡂⠐⠈
⠀⠀⠀⠈⠀⠀⠀⠂⣀⠠⢌⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠢⠂⠐⠢⠀⠀⠀⠀⠀⠀⢀⠒⠀⠀⠐⠀⠁⠀⠀⠀⠀⠤⠀⠀⠠⡀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢤⡀⠀⠀⠀⠀⡀⠀⠀⠀⠢⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠐⠀⠓⠈⠂⠈⠀⡀⠀⠀⠔⠀⠁⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠈⠈⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⡠⠀⠁⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠄⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠃⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠰⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠈⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁