Scrawny Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠊⠉⢉⡧⢦⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⣋⠁⠀⢠⠋⠀⠘⡇⠀⠀⠀⠀⣠⠖⠛⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠁⠿⢚⢀⢤⠑⡄⠀⡇⣀⠤⢀⡜⠁⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢋⡠⠤⣼⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠈⡤⠀⠈⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⣒⠐⠂⠘⠀⠒⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣉⣉⣁⡄⠀⢹⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠑⢤⡂⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⡜⠈⠑⠒⠤⠤⠤⠤⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡂⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠰⣰⠫⠦⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠨⡄⠀⠀⠀⠈⠋⠀⠛⠁⠀⠀⠀⢀⠆⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠏⢿⠤⠂⡏⠀⠀⠀⠀⠀⡇⠈⠉⠉⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠂⠀⠀⠇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠀⠤⠤⠄⠀⠠⢷⠄⠲⠶⠶⠂⠶⢯⣤⠀⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢈⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠦⡤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠤⢼⣴⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣄⣀⣀⣀⣀⣀⣀⣸⣄⡀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠥⠤⠚⠉⠀⠀⠀⠀⠉⠃⠤⢬⠏⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣰⠇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣧
⠀⠀⠀⠀⢛⣻⣶⣾⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⣠⠹⣜⣧⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡘⣷⣿⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⡟⣷
⠀⠀⠀⠀⠀⠤⠞⢿⣼⣿⡿⣷⠀⠀⠀⠀⠀⠀⠙⢶⣿⢻⢻⣧⣴⠶⠂⠀⠀⠀⠀⠀⠀⠙⢿⣿⣟⣳⡞⠂⠀⠀⠀⠀⠀⠀⠀⣦⠀⣼⡟⣿⣯⠆
⠀⠀⠀⠀⠀⠀⠒⠋⠉⠙⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠘⢷⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣆⣀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣦⣄⣀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣦⣄⣀
⠀⠀⠀⠀⠀⠚⢿⡿⣿⣿⣿⣿⡿⠗⠀⠀⠀⠀⠀⠿⢿⡿⠿⠿⢿⣿⠋⠁⠀⠀⠀⠀⠛⠛⢿⣿⣿⡿⠟⠛⠋⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⣿⠟⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢷⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡁⣾⢿⡆⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⡆⣀
⠀⠀⠀⠀⠀⠀⠈⢧⣘⣧⣼⡅⠀⣀⠀⠀⠀⠀⠈⠓⢮⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣸⣿⣿⣾⣧⠟⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⢠⡜⣲⣼⡷⠋
⠀⠀⠀⠀⠀⠀⠈⠲⣿⣿⣼⡗⣶⠟⠀⠀⠀⠀⠙⠲⢾⣿⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣼⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠛⠛⠛⠻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣀
⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⣦⠄⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⡦
⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠙⠛⠛⠛⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠉⠛⠉⠙⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠖⠋⠉⠁⠀⠀⠀⠀⠉⠉⠙⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠦⣄⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⣠⠶⠦⠤⠴⠞⠉⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢀⡇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣄⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢀⡜⠀⣀⡴⠛⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢦⡀⠀⠈⠙⢦⡀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠤⣴⠋⢠⠾⠥⠤⠤⠤⢭⡛⢒⣒⢲⡶⠤⣄⡀⠀⠀⢳⡀⠀⠀⠀⠹⡄⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⠀⠹⣄⢿⠀⠀⠀⠀⠀⠀⣿⠽⠾⢿⡀⠀⠀⠙⣄⠀⠀⢹⡄⠀⠀⠀⢹⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⡷⣎⢻⠳⠶⠶⠶⠶⢶⠃⠀⡄⠀⣉⡉⠉⠉⢹⡆⠀⠀⣷⠀⠀⠀⣼⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠹⡄⢸⡀⠀⠀⠀⠾⠛⠀⠀⡇⠀⠙⠛⠀⠀⡾⢹⠀⠀⡟⠀⠀⢠⠇⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠉⠉⠱⣄⠀⠀⠀⠀⠀⠀⣧⡀⠀⢀⡀⢰⠇⢸⢀⡼⠁⠀⣠⠏⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠈⢧⠘⢦⠀⠀⠀⠛⠁⠀⢸⠁⡾⠀⠟⠋⢀⡤⠞⠁⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⣇⠘⠀⢀⣄⣀⣄⡀⠈⢰⡇⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣽⠀⢶⡿⠤⣭⠤⣷⠦⢸⠁⠀⠀⡇⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⠳⠤⣤⠴⠋⠀⣼⠀⠀⢠⡇⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⢐⣶⣂⣠⣾⠁⠀⠀⣼⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⡉⣀⢹⠀⣸⠀⣠⠜⢁⣠⠖⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⢀⡼⠁⠉⠇⠈⠷⣟⣩⡥⠖⠋⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠉⢛⣒⣿⠯⠤⠤⠀⠀⠰⠶⠶⠶⠒⠚⠉⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠚⠛⠋⠉⠉⣹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⢠⡏⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⣁⣠⡄⠀⠀⠀⠀⠀⢀⡴⢾⠀⠠⠾⢋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⣘⣓⣉⡉⠀⠾⠒⠒⠉⣹⣧⠶⠛⡇⢸⡆⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠛⠉⠉⠉⠙⠷⣄⠀⠀⠀⠉⠀⠀⠀⡇⠈⣇⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⣄⠀⡇⠀⢺⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⠦⣤⠈⡼⠀⠀⢸⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣸⠟⠁⠀⣾⠟⠁⠀⠀⠈⣯⡇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⢠⣄⣼⠃⠀⠀⣼⠋⠀⠀⠀⠀⠀⡟⣷⡴⠞⠋⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠾⣿⡷⠂⣠⠞⠁⠀⠀⠀⠀⠀⢀⠟⣉⢴⡶⢆⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠈⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠉⠀⠀⠀⠂⡽⠃⠀⠀⠀⠀⠀⠀⢠⡶⠶⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠁⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⠀⡤⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⠀⠀⢀⣩⠴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⡀⠒⠲⣤⣤⡀⠀⠀⠀⠀⢀⣴⣏
⢰⣏⠉⠉⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠈⠻⠿⠆⢀⣴⡟⠀⠙⢷⣄
⠀⠈⠳⢤⡀⠙⢆⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢛⣶⠶⠞⢋⠏⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠙⣦⣼⡟⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⢰⠛⠁⢹⣅⣠⡆⢰⠀⠀⠀⠀⢀⣠⠶⠋⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠈⡟⠉⣁⡽⣧⠂⣼⡤⠤⠤⠒⠋⠁⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠹⣟⣉⡴⠿⣧⠛⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠸⣀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣸⣄⠨⡄⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠒⠀⠀⠠⠤⠤⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠾⢏⣹⢻⠀⢸⣤⣼⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⡀⣀⣀⠉⠐⢦⣀
⠀⠀⠀⠀⠀⢀⡔⠁⠀⠈⢧⣜⣳⡟⠠⠞⣹⠗⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣅⣀⣈⣀⣀⡼⠓⠢⠔⠋
⠀⠀⠀⠀⣰⠏⠀⠀⠀⢀⠃⣼⠋⠀⠀⠀⠀⠙⣶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⢠⡏⠀⠀⢀⣠⡎⢰⠃⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⢸⠀⠀⠀⠄⠙⣿⠿⡶⠄⠀⠀⣠⡏
⠀⠀⠀⠘⡄⠀⠀⠀⢰⡏⠀⠀⠁⠀⠀⢩⣏
⡠⠤⠤⠤⣿⡄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠹⡄
⠓⠆⠠⠼⠧⣤⠛⠒⠺⡿⠦⢤⠀⠀⠀⠀⠀⠹⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠳⣄⠀⠀⠀⠀⢀⣤⡶⣔⣪⠤⠤⠤⠄⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠛⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠠⣄⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣂⣁⡀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⠀⠀⠀⠒⠚⠍⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠘⣡⠋⠀⢀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣼⣿⡿⢶⣀⠠⣄⠀⣄⡀⠀⢀⡀⠀⠀⠀⠀⠀⡀⡄⠃
⠀⠀⠀⠀⠀⢀⡌⣰⣟⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⣀⣠⡤⠑⠛⠁⠊⠀⠀⠈⠁⠉⠓⠀⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⢮⣄
⠀⠀⠀⠀⠀⢾⣱⣟⣿⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⣠⡤⠀⠀⠄⢠⡙⠆
⠀⠀⠀⠀⠀⡜⠿⣾⡿⢀⠀⠀⠀⢀⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢤⣿⣾⡀⠀⠀⢀⠹
⠀⠀⠀⠀⠀⠇⠃⡟⡽⢢⠀⡘⠧⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢸⣶⣿⣿⣷⡀⢦⢸
⠀⠀⠀⠀⠀⠀⡆⢀⣿⠋⡄⠀⢠⠀⠀⠀⠤⠒⠶⠖⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⡁⠀⠀⠸⠺⣹⣿⡟⢣⣼⠆
⠀⠀⠀⠀⠀⠀⠈⠈⠋⠂⠀⠀⢸⡀⠀⠀⠠⠴⢶⠖⠒⠀⠀⠀⠀⠀⠀⠐⠁⠊⠉⠛⠁⠉⠂⠀⠀⢈⣯⡌⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡁⡇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢴⣼⣿⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⢀⣠⠀⠀⠀⠀⠀⠀⡠⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣂⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠉⠉⠉⠙⠛⠉⠁⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢧⡰⡀⠀⠀⠀⠈⠛⠛⠃⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡠⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣄⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⡶⠶⠶⠖⠋⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⡿⠿⠿⣿⡻⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣛⣿⣻⣿⣛⣛⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡔⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⣼⠋⠈⠳⣀⢹⡄⣴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠳⡄⠀⢰⡏⠀⢀⡴⢫⣄⠛⢣⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⣠⠖⠋⠁⠀⠀⠹⡄⠀⠀⡾⠁⣠⠞⠀⠸⡏⠳⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⣷⢀⠞⠀⣴⢣⠞⠟⢣⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⢤⣄⡀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠀⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⣀⣀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⡿⣿⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠸⣿⣥⣴⡇⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢸⠟⠉⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠖⠛⠋⠁⢷⣴⣶⡀⠀⠀⠀⣠⡟⠀⠀⠰⠃⣠⣞⡿⢓⡾⠁
⣴⠚⠛⠛⠛⠒⠶⠶⠦⠤⣤⣤⣤⣤⡤⠤⠤⠴⠒⠒⠉⢿⣟⣁⣠⣴⠾⠋⠀⠀⠀⠀⠀⢀⣠⠴⠋
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡽⠃⠀⠀⠙⠢⢤⠤⠀⠀⠀⠀⠀⢠⠟⠁
⠹⣆⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠙⠳⢤⣅⣀⣉⣁⣈⣡⣤⠴⠖⠛⠉⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡘⠫⡑⢦⡀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⣉⠳⡙⣆⣷⠀⠈⣧⢰⣄⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡉⠒⡿⠋⢉⡗⠀⠸⡆⠈⢻⣦⣀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠆⡽⠋⠁⠀⠀⠀⢹⡄⠈⢻⡽⠆⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠰⠀⠀⠀⠀⠐⠀⠀⣻⡀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠲⠤⠔⠒⠒⢚⣋⣉⡟⠁⠀⠀⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠇⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡇⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⠀⠀⠀⠀⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣇⠘⠀⠀⠀⠀⡇⡇⢸⡤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⠙⠛⠒⠒⢺⡇⠟⠁⠤⠛⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⢿⣄⡀⠀⠒⣺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⢤⣤⣤⣀⠀⠀⠀⠉⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⣤⣿⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢻⡇⠚⢶⣶⡖⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⠇⠀⣾⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡇⢸⣄⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣛⢶⡒⠉⠉⠁⠀⠀⠀⣀⣈⣷⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⠿⣦⣤⠄⠒⠊⠉⠁⠀⠀⠀⠀⠉⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⠃⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⢶⣀⡶⠒⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⠀⣠⣤⡀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡿⢹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⣼⣤⣾⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡤⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣇⣿⣿⣿⣿⣷⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⣠⣾⣿⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣎⠤⣀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢟⣧⣠⣤⣿⢻⣯⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡞⠁⠀⠀⠀⢨⣽⣿⣿⣿⣿⣾⣿⣼⠟⣻⠿⣿⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡙⠛⣷⣿⡆⠀⢰⡃
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣾⣿⣶⣷⣾⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣷⣿⣦⣤⣀⡀
⠀⠀⠀⢤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛
⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⣤⡶⠿⢛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⠿⢿⣿⣿⣿⡿⠟⣠⣿⠃⣴⣶⠿⠛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⣶⠈⣿⠟⢁⣴⡄⢻⡟⢡⣴⡾⠟⢃⣼⡟⢠⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠛⣡⣾⡿⠛⣠⣿⣴⣿⠟⣡⡿⢡⣿⠃⣴⡾⠛⣋⣠⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⢋⣤⣾⣿⠏⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⣿⠛⠛⠛⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠛⣉⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⢽⣿⣿⣿⠇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠈⣷⣤⣤⣤⡌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⣴⣟⣉⣉⡉⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣁⣿⠛⠛⠛⢃⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣍⡙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠂⣿⠿⠿⠿⠇⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢀⡆
⠀⠀⠐⢲⣶⣦⣽⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⣯⣻⣤⣟⠀⠀⠀⠀⠀⠀⠀⠀⣶⡘⣧⣾⣥⠞⠃
⠀⠀⠀⠈⣩⠽⣿⣽⣿⣿⣦⠀⠀⠀⠀⠀⠘⢷⣼⣿⡿⣿⣀⣤⠦⠀⠀⠀⠀⠀⠛⢷⣿⣿⢿⡶⠆
⠀⠀⠀⠀⠐⠶⠛⠛⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿
⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣦⣀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣦⣄⣀
⠀⠀⠀⠀⠺⢿⠿⣿⣿⣿⣿⠿⠆⠀⠀⠀⠀⠘⠿⢿⠿⠿⠿⣿⡟⠉⠀⠀⠀⠛⠛⢿⣿⣿⣿⢟⠛⠋
⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠩⢀⠒⠠⢄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢒⣬⣍⠑⠂⠀⠈⠪⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⣄⡀⠀⠀⠑⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠠⡀⠀⠀⠀⠀⢰⠀⠈⠑⢄⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⠧⢤⡤⠥⣀⠀⠀⣠⠃⠀⠀⠀⠸⠀⢘⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠂⠰⠆⠂⠫⢁⢂⣥⠊⠁⠀⠀⠀⣼⠄⠠⠜⠃
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡤⠀⣀⠀⠀⠼⠧⠀⠠
⠀⠀⠰⠠⠁⠀⣤⣖⢐⢒⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠚⢅⠀⠐⠒⠆⠭⠥⠴⠰⠿⠗⢀⠤⡈⠉⣁⡄
⠀⠀⠀⠃⠁⠀⢄⠉⠉⠉⠉⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠢⠤⣀⢀⠀⠀⢀⠔⠁⠀⠀⠠⠀⠐⢠⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠔⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠃⠀⠀⠀⠀⠈⠆⡀⠶⣤⡡
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠄⠐⠀⠀⠤⠤⢀⠀⢀⠀⠉⠚
⠀⠀⠀⠀⠀⠁⠘⠀⠀⠡⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⠔⠀⠈⠁⢨⠁⠀⢀⠋⠉⠀⠐⠊⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⠢⡀⢠⠀⠀⠀⠀⠀⠀⠀⣀⠰⠡⠁⡀⠀⣀⠔⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢰⠃⠄⠀⣀⡀⣬⢾⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⣥⠊⠁⠀⠀⠀⠀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠎⡄⠀⠀⠀⠀⠀⠘⣀⠄⣀⣠⠟⡓⠒⠒⡒⠂⠉⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢶⠉⣻⡀⢰⡇⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⣇⣀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠘⡈⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠄⣾⡅⢸⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠘⡈⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠁⢠⠂⠐⠀⠤⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡀⠀⠀⢃⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⠀⠄⠡⡀⠀⠘⡈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠓⠂⣀⠠⠤⠤⠤⠤⠊⠀⢀⠃⠑⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢄⠀⠀⢁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣄⠈⠛⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⣠⣾⠿⣿⣦⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄⠀⠙⢿⣿⠟⠁⣀⠀⠙⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠟⠁⠀⠈⠻⣿⣦⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠁⣠⣾⣿⣷⣄⠀⠙⢹⣿⣿⣿
⠀⠀⠀⠀⠀⣤⢀⣤⡀⠈⠁⠀⢠⣄⣤⡀⠚⠛⠃⠀⠀⠀⢸⣿⣿⣿⣿⣿⠛⡿⠛⢿⣷⣾⣿⡟⡿⠛⢿⣥⣤⣼⣿⣿⣿
⠀⠀⠀⠀⢀⡿⢟⣥⡙⢦⡀⢀⣼⢟⣥⣝⢶⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢀⡴⠛⢦⡙⢿⡿⢁⡴⠛⢦⡙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⢛⣵⣿⠿⢿⣷⣝⢛⣵⣿⠿⢿⣷⣝⠃⠀⠀⠀⢸⣿⣿⣿⣿⡴⠋⢀⣤⡀⠙⢦⡴⠋⢀⣤⡀⠙⢦⣽⣿⣿⣿
⠀⠀⠀⠀⣼⣿⡯⠀⢸⣿⣿⣼⣿⡿⠀⢸⣿⣿⡄⠀⠀⠀⢸⣿⣿⣿⣿⠃⠀⢸⣿⡇⠀⠘⠇⠀⢸⣿⡇⠀⠐⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢠⡟⠿⢻
⠀⠀⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⢸⡇⠀⢸⡀
⠀⠀⠀⠀⢸⡇⣶⠿⠯⠭⠭⠽⠿⣆
⠀⠀⠀⠀⢸⡇⢻⣤⣤⣤⣤⣤⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⣠⣿⣿⠟⠻⠛⠛⠻⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⣼⣿⡿⠋⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠁⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⡴⠒⢺⢛⣡⢄⠀⠈⠀⠀⠀⠈⠁⠀⣠⠈⡛⣗⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⠀⠀⠀⠀⠉⠙⠻⣿⣿⠟⠿⢿⣿⣿⡄
⢧⡀⢻⠀⠉⠉⠀⠀⠢⠄⠠⠔⠁⠀⠈⠉⠁⣽⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣇⠀⠰⠀⠀⠀⠰⠆⠀⠘⠷⢀⣀⣿⠿⣿⠃
⠀⠉⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢻⡁⠀⠐⠤⠔⠀⠀⠘⠑⠀⠀⠁⠀⠀⢸⠆
⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣤⣾⣄
⠀⠀⠀⠀⠐⡿⢻⠛⡖⠒⠒⣲⠛⡟⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣛⡖⠒⠒⠒⢛⡏⠛⣿⣿⡿⠟⠛
⠀⠀⠀⠀⢰⠃⡞⠀⠘⠒⠒⠁⠀⢻⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠈⢳⡙⣟⠿⢿⠋⠙⠦⡀⢸⡿⠿⠄
⠀⠀⠀⠀⡏⢸⠁⠀⠀⠀⠀⠀⠀⠀⡇⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⠀⢘⡇⢸⠀⠀⠀⠀⠀⠸⡆⢧
⠀⠀⠀⣸⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⢀⡼⢁⠞⠀⠀⠀⠀⠀⠀⣇⠘⡄
⠀⠀⣠⠃⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢶⠿⡖⠉⠀⠀⠀⠀⠀⠀⠀⢹⠀⠳⡀
⠀⠀⠙⠒⢺⡗⡶⢶⡦⠤⠴⢶⡶⢶⡞⠓⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢺⠀⣷⡶⣶⠤⠤⠴⢶⡶⢶⠛⠒⠚⠁
⠀⠀⠀⠀⢸⡇⢧⠀⠈⡗⢺⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⡇⠀⢹⡒⢺⡅⠀⢸
⠀⠀⠀⠀⢸⡇⢸⡄⠀⡇⢸⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢻⠀⢸⡇⠀⡇⠀⢸
⠀⠀⠀⠀⢸⡇⢈⣇⠀⣇⣸⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠘⡆⠀⡇⠀⣧⠀⣸
⠀⠀⠀⣴⠚⠉⠉⣽⣀⡿⢹⣀⣺⠉⠉⠛⣢⠀⠀⠀⠀⠐⠒⠀⠒⠘⠛⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⡗⠛⣇⣀⡏⠉⣿⣀⡸⠓⠒⠢⣄
⠀⠀⠀⡏⠉⠙⠒⠒⠒⠒⠒⠒⠒⠒⠋⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡗⠒⠒⠲⠶⠶⠶⠒⠒⠒⠚⠉⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠹⡀
⠀⠀⠀⢾⠓⠦⣄⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⢷⠀⠀⡠⠊⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⢆⠀⠈⠳⡀⠀⢀⡎⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡼⠀⡜⠁⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢳⠀⠀⢀⡴⠒⠉⣹⠃
⠀⠀⠀⠀⠈⠳⣄⡀⢳⠀⢸⠀⢀⡇⠀⠀⠀⠀⠀⠀⢀⡤⣀⠀⠀⢹⣸⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢸⠀⡰⠋⠀⢀⡴⠁
⠀⠀⢠⡤⠖⠒⠲⠭⢟⣦⣿⠀⣸⠁⡠⢦⠀⠀⠀⠀⠀⢇⠀⠑⢆⢸⡇⣠⠋⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢳⠀⢸⣰⠧⠴⠚⠁
⠀⠀⠀⠉⠓⠦⠤⠤⠤⠬⢿⡰⠃⡼⠁⢸⠀⠀⠀⠀⠀⠈⢦⡀⠈⢿⡛⠁⠀⠀⠀⠀⠀⠀⠀⢠⠇⢱⡀⠘⣆⣾⠓⠚⠉⠉⠙⠒⣢⠄
⠀⠀⠀⠀⠀⢞⡉⠉⠙⠒⢬⣇⢸⠁⢀⡞⠀⠀⠀⠀⠀⠀⣀⡈⠓⠺⡇⣠⠖⢻⡆⠀⠀⠀⠀⢸⡀⠀⣇⢀⠞⣀⣉⣓⣒⡒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠉⠓⠒⠦⠶⠺⣟⡠⠊⠀⠀⠀⠀⠀⠀⠀⡇⠙⢆⠀⡷⠁⠀⡼⠀⠀⠀⠀⠀⠀⢣⠀⢸⣾⣉⣀⣀⣀⣀⠬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⣦⡇⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢱⡏⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣸⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠤⠤⠶⠶⠶⠶⠶⠦⢤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⣚⣻⡍⠉⠁⠀⠲⠖⠒⠛⠓⠀⠀⠀⠀⠀⠲⠶⠦⣭⡉⠙⠓⠦⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⠐⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠺⠛⠻⢯⣍⠉⠉⠉⠛⠛⠶⠶⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣤⣤⣤⡤⠴⠶⠶⠖⠛⠛⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣧⡆⠀⠰⠖⠛⠛⢻⡆⠀⠀⠀⠀⠀⢀⡏⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣻⣏⣀⠀⠀⠀⠀⠀⠀⠀⠉⠇⠀⠀⠀⠀⣀⣸⡇⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠈⠉⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡇⠀⠀⠀⠀⠀⣸⠃⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠐⣶⠀⢀⣄⣀⣤⣤⣼⡇⠀⠀⠀⠀⢀⡟⢠⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⣌⣽⠇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠇⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠐⠛⣻⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣼⠁⠀⠀⠀⠀⣼⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡟⠀⠘⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠐⣿⠄⢠⣤⣤⣤⡤⣿⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠰⠶⣿⠒⠀⠀⠀⠀⠀⠾⠧⠄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢘⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⣿⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣿⣁⡀⠀⠀⠀⠀⠀⣼⡇⠀⠴⠶⠶⠚⠋⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⣀⣤⣴⡶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠒⠒⠲⠶⠶⠶⠶⠶⠖⠛⠛⠉⠉⠁

⠀⠀⠀⣀⣀
⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢿⣿⣧⡀⠀⠀⠀⠀⠀⢀⣴⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⣠⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣾⣿⠟⠁⢻⣿⣇⠀⠀⠀⠀⠀⣰⣿⣿⣿⣧⡀
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠃⠀⠀⠈⢿⣿⣆⠀⠀⠀⣰⣿⡿⠁⠻⣿⣷⡀
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⣰⣿⡿⠁⠀⠀⠹⣿⣷⡀
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣠⣿⣿⠃⠀⠀⠀⠀⠹⣿⣷⡄
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⣀⣤
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿
⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠙⢿⣦⡀⠀⢀⣴⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⡈⠙⣿⣿⣿⠟⢁⠈⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⢿⣶⣿⠟⠁⠈⠻⣷⣄⠀⣠⡄⠀⠀⣿⣿⣿⣿⣿⣿⣦⡀⠛⢁⣴⣿⣷⣄⠈⠻⣿⠟⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠈⣻⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣷⠄⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠃⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣤⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛



⠀⠀⢤⡀
⠀⠀⠀⠑⢄⢀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠑⢷⣄⠀⠀⠀⠈⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⢿⣟⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠙⠳⠆⠀⠈⠀⠀⢉⣻⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠷⠄⠀⠀⠀⣀⣴⠿⠛⣋⣉⣉⠛⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣀⠀⠀⠀⢀⣼⠟⣡⡾⠟⢛⣙⠛⠿⣶⡌⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠙⠀⠀⢀⣾⠃⣾⠏⣴⡾⠛⠛⢿⣦⠘⣿⡀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠲⠀⠀⢸⡏⢸⡏⢸⡏⠀⠀⠀⠀⢻⡇⣹⡇⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⡶⠶⢶⣶⠶⠶⣶⣶⣤⣤⣄⣀⡀⠀⠀⠀⠙⢷⣦⡀⢸⡇⢸⣇⠸⣷⣄⠀⠀⢀⣾⢃⣿⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠛⢿⡄⠀⠉⠳⠀⠀⠙⠃⠀⠈⠙⠂⠈⠻⠟⠛⢿⠶⣶⣤⣤⣭⣿⣿⣿⡄⠻⣷⣬⣉⣠⣴⠟⣡⡾⢃⡾⠃
⠀⠀⠀⠀⠀⣀⣤⣶⡟⠋⠉⠙⠷⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣶⣤⣤⣤⣄⣀⡙⠊⠉⠻⠍⠙⠿⣶⣬⣉⣉⣩⣥⡾⢋⣴⠟⠁
⠀⠀⠀⠀⠺⣿⡁⠘⠿⠆⠀⠀⠀⠀⣀⣤⣴⡶⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⢷⣶⣤⣤⣄⣀⣉⣉⣉⣩⣥⡶⠟⠁
⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⢀⣠⣶⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣷⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣌⣛⣿⣷⣾⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡙⢻⣿⣿⣿⣿⡿⠛⢉⣠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠋⣠⠾⠋⣁⣠⣄⣀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⣿⣿⣿⡟⢀⡾⠃⣰⠞⢋⣉⡉⠙⢻⣄⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⣿⣿⣿⠁⣼⠃⣼⠃⣰⣿⣿⣿⣷⠀⢻⡄⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⣿⠀⣿⠀⢻⣿⣿⣿⣿⠃⣼⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⠐⢶⣶⣦⡐⣶⣶⣄⠠⣤⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⢻⡄⠘⢷⣄⣸⣿⡿⠋⣰⠏⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠲⣿⣿⣦⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣷⣾⣿⣦⣶⣦⣀⢤⣀⠀⣀⣀⠈⠻⣦⣄⣉⣉⣁⣤⠞⢁⡴⠋
⠀⠀⠀⠀⢀⣠⡀⠘⢿⣿⣿⣶⣿⣿⣿⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⢿⣾⣿⣷⣦⣀⣉⣉⣉⣉⣤⡶⠛⠁
⠀⠀⠀⠀⠈⢻⣿⣶⣾⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠙⣿⡿⠛⠉

⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⡠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⢃⣤⡀
⠀⠀⠀⠀⠀⢸⢰⠞⠁⠁⠀⠀⠀⠉⠄⠀⠀⠀⠀⠀⠀⠀⡠⡄⠀⠀⠀⠀⠈⣣⣼⡉⣠⠄⠄⠀⣀⠀⠠⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⢸⡤
⠀⠀⠀⠘⠦⠀⠿⠂⠘⠃⠀⠀⠀⢆⠰⠀⠀⢀⢰⡆⠀⠀⣩⣷⠀⠀⠀⠀⣤⡀⣄⢃⣨⡶⠗⠀⠉⠁⠀⢀⠈⠳⠄⢀⠀⣀⠀⠀⠠⢀⠀⠘⠟⠁
⠀⠀⠄⡀⠠⡿⠤⠤⠔⠂⠀⢶⠄⠈⢶⡀⠐⠁⡘⠀⠀⠀⠉⢹⠀⠀⠀⠀⠈⠛⣤⣏⡏⠒⣧⠔⠀⠀⠀⠀⠁⠤⢮⢠⣤⡄⠀⠈⠦⠐⢂⡴⠆
⣀⠀⠑⠼⣶⠆⠀⣀⡰⠀⠀⠀⠀⡀⠘⣧⡔⠀⠀⠀⠀⠀⠘⠝⡇⠴⠀⠀⠀⠀⠀⢸⣽⠻⠃⣤⠀⠀⠀⠙⢦⡀⠈⣥⡅⠀⠀⠐⠶⠂⠾⠆⠀⢀⡀
⠀⠀⢠⣤⠅⠀⢀⡴⠀⠀⠀⠀⠠⡈⣤⡼⠀⠐⠒⢰⠇⠀⢀⢠⣧⠤⡄⠀⠀⢰⡟⠂⢸⠀⣀⡰⠆⠀⠀⠀⠀⠐⠄⠘⡁⠀⠀⡀⢠⡀⢀⠀⠀⠘⠊
⠀⠀⢰⠉⢁⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⣸⣀⣁⠠⠂⠀⠀⠘⠈⡁⠀⠀⠀⠀⠀⠘⠟⣛⣯⡌⠁⠀⠀⡶⠀⠀⠀⠈⣿⠀⠀⠀⠙⠛⣁⢨⠦⡤⢀⠬
⠀⠀⠀⠀⠻⠀⣀⠀⠀⠀⠁⠀⢢⠀⠀⠉⠀⠀⠤⢺⠃⠀⠀⠻⠟⠀⠃⠀⢀⠀⠀⣀⠈⢃⠀⡠⢪⡞⠁⠀⠀⠀⠀⠘⠀⠀⠠⣠⠀⠻⠂⣤⠀⠾⠆
⠀⠀⠃⣀⠀⠀⠀⠑⢄⠀⠈⣦⠀⠁⢀⡄⣃⠠⢀⠂⠀⠀⠸⠽⣷⡰⠀⠀⠈⠙⠦⠬⠽⢶⣔⡤⠊⠀⠀⢠⠎⡆⠀⠰⠀⠀⢠⠼⠦⠀⠀⠉⣀⡀
⠀⠀⠀⡌⠀⠂⠀⠀⠀⢢⠀⠈⠳⣇⣸⠁⠉⢀⠁⠀⠀⠀⡀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠁⢃⡅⣀⠀⠀⠀⠈⠚⠁⡤⣄⢠⠉⠋
⠀⠀⠀⠆⠀⠩⡀⠀⠀⠀⡆⠀⠀⠈⠏⠁⠀⠁⠀⠀⠀⠀⠑⠊⠋⠀⠀⠀⠀⡤⠄⠀⠀⠘⠀⠀⠈⢇⠀⡆⢸⠃⡋⢀⠀⠀⠀⠐⣀⣿⠁⠀⡤⢌⡆
⠀⠀⠀⠘⡀⠀⢧⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⢐⠀⠀⠐⠀⡀⠀⠀⢰⠀⠀⠀⡀⠀⠲⣆⢠⠀⠉⠂⠀⠀⠀⠠⠈⠀⢹⠁⡤⡑
⠀⠀⠀⠀⠀⠦⠘⠦⠀⢀⠃⠀⠀⠀⡆⠀⠀⠀⢈⣋⠀⠀⠀⠀⠀⠈⠄⢁⢰⠀⠀⠀⠀⠈⠀⠀⠈⠙⠁⠀⡹⠊⢀⠀⠁⢠⠞⡇⠀⠀⠀⠌⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠁⠀⢀⠀⠀⠐⠀⢀⡀⠉⠉⠰⢄⠀⠰⡄⣄⠈⢠⡌⠀⢸⠇⠀⠠⠀⠀⠀⠑⢵⣀⢻⠀⢂⠀⠠⠋⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢑⠀⠀⠀⠈⠳⠀⠀⠑⢬⡃⠀⠀⠫⡁⡀⢣⡀⠂⠃⠀⠀⠀⠀⠰⣄⠀⠙⢢⡦⠂⠀⡡⠀⢠⠊⠇
⠀⠀⠀⠀⠀⡀⠰⣆⠀⢀⠀⠀⠘⢆⣴⡄⠀⠳⠇⠀⠀⠀⠀⠀⢀⡀⠀⠳⣘⢁⣅⡀⠐⡦⠀⠀⠰⡈⢆⢂⠀⠡⡀⣸⠃⠀⠃⠀⠂
⠀⠀⠀⠀⠀⠐⡀⢈⣆⠈⠀⠀⠰⠻⢀⡀⢠⠶⢀⠀⠀⠀⠀⡀⠀⠆⠐⠦⢜⡆⣑⣀⠄⠀⠀⠀⠀⠈⠒⠶⣄⠀⠡⠀⠀⢠⡖
⠀⠀⠀⠀⠀⠀⠀⠉⠻⠀⠀⠀⠀⠀⠋⠀⠈⢆⠱⣅⠀⠀⡸⠁⠀⠀⠄⠠⠘⡗⠁⠀⠀⠀⠀⢀⠀⠀⢀⡀⠈⠀⠢⣇⠀⠋⢀⡶
⠀⠀⠀⠀⠀⠀⠃⠈⡒⢷⠀⠀⠾⠀⠀⠀⠦⠀⠑⢾⠇⠈⢀⡀⠀⠀⢆⢀⠁⠀⠰⣠⠁⠠⠀⠉⠀⠀⠈⠫⣆⡃⠀⠘⡀⢀⣨⠌
⠀⠀⠀⠀⠀⠀⣤⠀⡀⠹⡇⣀⠆⠀⠀⠀⠈⢌⠄⢸⡆⠀⠘⠁⠀⠀⢀⠑⣤⡔⡐⢡⡎⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠀⡟⠉⠁⠀⣤
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⡟⠡⢂⠏⠀⠀⠀⠀⠙⢾⡇⡴⠰⡄⠀⠇⠈⠃⢸⢪⡴⠊⠁⢀⡄⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⡄⠀⡧
⠀⠀⠀⠀⠀⠀⠀⣤⠈⡘⢿⠔⣃⠀⠀⠀⢂⠀⠀⢸⡟⢁⡠⠁⠀⠛⠷⣷⠀⡼⠀⠀⡄⠘⠀⠀⠀⠀⢊⡁⠀⠀⠀⠀⠀⠀⠐⠠⠀⢱⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠺⡇⠀⠁⠀⠀⠀⠈⠢⡀⢄⣧⠈⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⢰⠀⡠⠁⠀⠠⠀⡤⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠀⠘⢢⡀
⠀⠀⠀⠀⠀⠈⠣⡀⠀⣀⡅⢐⠂⠀⠀⠀⠀⠄⢈⡀⡟⠠⠐⠹⠁⠀⠀⢰⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⡥⠀⠀⠀⠀⠀⢈⠁⠀⠀⠐⠃⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⢈⠓⠻⠃⠍⣀⠀⠀⠀⠀⠀⠀⠈⢟⠐⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⢠⠃⠀⠀⠀⡤⡄⡠⠀⠨⠀⠀⠀⠤⠀⠀⠀⠀⠒⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢇⢰⣿⣠⠍⠀⠀⠠⢀⠤⠀⠀⡏⠀⠀⠀⠴⡄⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠐⠼⠧⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠀⠀⡠⢀⢍⠜⠀⠀⠀⠀⠀⠓⠀⠀⠀⡇⠀⠀⠀⠠⠀⠀⠀⠀⠀⢶⣄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠶⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄

⠀⠀⠀⠀⣴⣶⣄
⠀⠀⠀⠀⠙⠛⠛⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⢀
⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣦⠀⠀⠀⠁⠂⠄⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠑⠠⣀⣀⣄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⢇⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡔⠒⠒⠒⠒⠒⠒⠢⠀⠀⠀⠈⠐⠠⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠁⠢⢄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢶⣿⣷
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠒⠀⠀⠀⠀⠀⠂⢲
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⢘⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡄⠀⠀⠀⠀⠀⣠⠊⠉⠒⠉⢱⣀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡁⢘⣆⠀⠀⠀⠀⠀⠀⠀⢶⣤⡀⢀⣠⡤⣤⡀
⠀⠀⠀⢀⡴⠊⠉⢠⡰⢏⠃⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⣘⣁⠙⣉⣏⠀⠀⠀⠀⠀⠀⠀⣀⣽⣷⣾⣿⢶⣤⠁
⠀⠀⠀⢺⢀⡷⠚⠃⠀⡜⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢀⣍⢤⣩⣀⣯⠀⠀⠀⠀⠀⠀⢸⠟⠈⢲⡷⠊⠀⠘⠃
⠀⠀⠀⢸⠻⢥⣤⠤⠊⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⢲⠓⠂⠚⠀⠀⠀⠀⠀⠀⠼⠥⣌⣡⣨⢽⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚

⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⢀⠀⠀⡄⡀⠀⢰⣀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⢀⣠⡴
⠀⠀⢀⢤⠀⡊⠀⢈⡀⡏⣷⠀⠀⠀⠀⠀⠀⢠⡀⠀⡰⠟⢠⠖⠁⠀⠀⢈⠙⣶⠂⣿⡃⢀⣞⣥⠾⠤⠀⠀⠀⠀⢹⣧⠀⢀⢔⡿⠉
⠀⠀⢸⣸⣦⣰⠀⢸⣾⢟⠟⠀⠀⠀⠀⣀⣈⢳⣿⣠⢶⣟⣿⣀⠀⠀⠀⠋⠉⠙⠫⡹⡴⠟⠛⠻⢯⠀⠀⠀⠀⠀⠀⣯⣇⢎⡞
⠀⠀⠈⠳⠥⡟⠀⢰⠏⠁⠀⠀⠀⠀⠀⠈⢎⠳⡽⢡⢻⡟⣥⠴⠀⠀⠀⠀⠀⠀⠀⢧⠡⡄⠀⠀⠀⠀⠀⠀⠀⠀⢤⡏⣿⣼⡀
⠀⠀⠀⠀⠀⠱⠤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣇⣬⣬⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣇⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣆⣸⣿⢳⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⢤⠤⡄
⠀⠀⠠⠶⢤⡄⡾⢁⣤⡆⠠⠀⠀⠀⠀⠀⣴⣸⣿⣷⢟⣾⣶⡁⠀⠀⠀⠀⠀⢀⠮⠭⠥⠬⠭⢷⠀⠀⠀⢀⡴⠅⣤⢤⠒⠶⠻⣄⣄
⠀⠀⠀⠑⠂⠉⣧⠟⠭⠔⠀⠀⠀⠀⠀⠀⣿⣿⣾⡏⣼⣿⣹⣃⠀⠀⠀⠠⣦⠞⠀⠀⠀⢀⠀⢈⢣⣄⠀⢨⠧⡶⠿⢀⡀⢐⠶⠀⠘⢶⡄
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⢬⡟⣾⡛⣿⢗⡿⡏⠁⠀⠀⠘⠓⠚⠙⢦⡜⠉⠑⠊⠀⠀⠀⠀⢱⠃⣠⠋⠉⠙⠋⠙⣦⠀⢣
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠟⠚⠃⠛⠂⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀
⠀⠀⠀⠀⡤⠢⡔⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢟⣻⠒⢤⠀⡴⢫⠝⠋⠉⠀⠀⠀⠀⠀⠀⣀⠤⢤⡀
⠀⠀⣀⣀⣃⢀⣀⡸⠊⠉⢿⠀⠀⠀⠀⠀⠠⡀⠀⡆⢀⡠⠔⠂⠀⠀⠀⠀⣀⠤⡤⡭⣿⣿⣧⡀⣲⢦⡀⠀⠀⠀⢀⠞⠁⠀⠀⠙⡄
⠀⠘⠢⡤⠈⣿⢉⡷⠠⢤⠚⠀⠀⠀⢀⣀⣀⡈⢦⣷⢞⣡⡤⠤⠀⠀⢠⣮⢴⠎⣰⠇⠈⠘⡆⢷⠈⠉⠁⠀⠀⠀⠸⠀⣄⠀⠀⢀⢱
⠀⠀⠘⠤⠚⢈⡉⡧⣸⠞⠀⠀⠀⠀⠉⠀⠀⠉⠻⣷⠋⠁⠀⠀⠀⠀⠈⠀⠈⡼⠀⠀⡇⡆⠙⣾⠀⠀⠀⠀⠀⠀⠘⠚⢻⠒⢲⠛⠋
⠀⠀⠀⠀⠑⠊⠳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢠⢖⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⢿⠸⣠⠇⠀⠀⠸⣆⠈⣩⡆⠀⠀⠀⠀⢀⠄⠀⠀⠀⣿⠀⢛⡛⠁⠀⠀⠀⠀⠀⠀⢀⡤⠘⢁⡀
⠀⠀⢈⠐⢋⡶⠀⠀⠀⠈⢈⡵⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡈⠚⡏⠀⠀⠀⠀⡀⠀⠀⠀⠈⢿⠴⠛⠁
⠀⠀⠸⣦⡈⢠⢴⡆⠛⣩⠃⣞⣴⠃⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⠇⠀⠀⠀⠀⠁⠀⠀⠸⣷⡄⣰⡦
⠀⠀⠰⣄⠙⣠⠶⠀⢤⣘⠧⢁⣴⠇⠀⠀⠐⠀⠀⠀⠀⣄⡻⠄⣀⡶⠀⠀⠁⠀⠀⠀⠀⣤⣽⣡⠄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠁⠋⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠚⠉⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⢰⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⡶⢀⣤⣄⠀⠀⠀⠀⠀⠀⡀⠀⡀⢠⠆⠀⠀⠀⡈⠗⠾⠇
⠀⠀⠰⡆⢚⡋⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⠰⠈⠛⠋⠀⠀⠀⠀⠀⠊⠸⣀⠁⢋⣠⠄⠀⠀⣻⢇⢴⣦
⠀⠀⢀⠐⢋⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣂⣠⡄⠀⠀⠀⠀⠀⢀⠀⠈⠁⠋⢡⠆⠀⢀⡹⠀⢰⠇
⠀⠀⠘⠇⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠻⡾⠍⡉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣗⡏⢠⡶⠘⣛⠁⢉⡙⠁
⠀⠀⠰⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠋⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⣈⣹⣏⣤⠀⠙⠳⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠘⣿⡄⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠈⢿⣆⠘⢿⣦⡀⣿⡇⣠⣾⠟
⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠙⢿⣿⣿⠟⠁
⠀⠀⠀⢠⣿⠃⢠⠞⠙⣦⠀⣰⠂⠀⠈⢿⡆⠀⠀⠉⠁
⠀⠀⠀⣾⡇⠀⠈⠳⠶⠋⡼⠃⠀⠀⠀⠸⣿
⠀⠀⠀⢿⡇⠀⠀⠀⢀⡾⢁⡶⠒⢦⠀⢠⣿⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠘⣿⣄⠀⠀⠞⠀⠈⠳⠤⠞⢠⣾⠏⠀⠀⠀⠀⣼⡟⠙⣿⡄⠀⠀⠀⢀⣴⡿⣿⡿⣷⣄
⠀⠀⠀⠀⠈⠻⣷⣤⣀⣀⣀⣀⣠⣶⠿⠃⠀⠀⠀⢀⣾⡟⠀⠀⠘⣿⡄⠀⠠⡿⠋⠀⣿⡇⠈⠻⡷
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⢀⡴⠲⣆⠀⢠⡆⠀⠈⢿⣆⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠘⠧⡤⠟⣠⠏⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⣰⠃⣤⠶⢤⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⡼⠁⠀⢧⣀⡼⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣄⡀⠀⠀⠀⣀⣤⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⠂⣀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀⠀⠀⠀⣿⣿⠋⣿⣆⡹⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣇⠀⢸⡄⠀⠀⠟⠁⠀⣽⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢸⣿⣿⣿⡆⢸⡇⢀⡆⠀⣠⡄⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣦⠀⠀⢻⣿⣿⣟⢀⣬⣱⣶⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣏⣀⡀⠈⣿⣿⡿⠸⠏⠁⣤⠛⠳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣼⣿⣿⣇⡀⢿⣿⣿⣄⣠⣏⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣠⣾⣿⣿⣹⡟⣋⣠⣶⡿⠻⠋⣹⣿⣿⠋⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣋⡻⠿⣹⡧⠀⠶⠖⠉⠀⢀⣠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣇⠙⣿⣷⣬⣇⠀⡀⢢⡀⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡄⠈⣿⠟⠉⠉⠙⣄⠰⢶⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣾⠿⠶⢷⣆⣨⣧⣼⣿⣿⡿⠿⠿⠛⠛
⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⡀⠀⣸⠇⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠘⠋⠉⣤⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⣀⠀⠀⢸⡅⢰⣿⠄
⠀⠓⣤⡄⣿⣧⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣄⠸⣿⡟⠁⣀⡴⠀⣠⡞
⠀⠀⣿⠁⠙⣯⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣴⣾⣶⠚⠉⠠⢞⠛
⠀⠀⠀⠰⠃⠀⠀⠀⢀⡀⠐⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡌⠻⣿⣿⣿⣁⠤⣠⣾⡵⠊
⠀⠀⠀⠀⠀⠀⢤⣴⣀⡷⢠⡾⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠙⣿⡄⠘⣿⣿⣿⡿⠟⠋⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⡟⠁⠀⣀⠀⠀⠀⠀⠀⢠⣿⣿⣧⡸⣇⠀⣾⠉⠁⠀⡀⢀⠐⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⠀⠀⣠⡏⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣸⣶⠏⣠⡄⠟⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⠀⠛⠀⠀⠀⢀⡀⠀⠘⣿⣿⡟⣿⣿⣟⠰⣿⣀⣠⣴⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⣴⡎⡹⣶⣄⢿⡟⠀⠈⢻⠋⢁⣴⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣾⣷⣠⡆⠉⠘⣧⠘⣿⣿⠁⢠⠀⠀⣠⣟⣱⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⢸⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣣⣄⠀⢠⣄⠸⣿⡄⠸⡆⠀⣹⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⢿⣏⠉⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⡟⡋⠉⠙⠁⠈⢿⣧⢿⣷⠀⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡇⢰⣦⣴⣄⠀⠉⠠⣿⣧⣿⠇⠀⠀⢀⣴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⢀⣾⣀⣠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠡⡌⢻⣿⣿⣷⣤⣀⠙⣿⠇⢀⡄⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣼⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⡃⣼⠏⠻⣿⣿⠱⠀⢾⣠⡎⢡⡾⠃⠀⠀⠀⠀⠀⢦⣄⠀⢀⡁⠀⠙⡏⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠑⠀⠀⠀⠙⠟⠀⠗⠀⢿⣷⠋⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⡄⠀⠀⢶⡀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣦⡀⠀⣾⠀⠀⢴⣧⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠙⣄⣸⣶⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣴⡟⠀⠀⢈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡿⠿⣿⣿⡅⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠙⠟⠀⠀⠀⡀⢻⡷⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⡟⢻⣷⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠐⡇⠀⣾⡇⢸⣿⡀⠀⠀⠀⠀⠀⠀⢠⡾⠀⠘⣟⣧⡴⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣄⡀⠀⠀⠉⠻⢼⡆⠻⡇⠈⣿⠁⠀⠀⠀⣴⠀⠀⠈⠁⠀⢠⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣤⣈⠳⣎⣁⠀⠀⠀⡿⠀⠀⠀⠀⠀⠄⠃⢰⣧⠀⠆⣠⣴⡶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣮⣿⣦⡄⠀⣦⠀⠀⠀⢀⣴⣷⡆⠘⣿⣦⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠙⢿⣿⣿⡇⠀⡷⢀⡄⢠⡾⠋⢹⣿⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠙⢿⣧⣤⣥⢀⣴⠟⢀⣶⣾⣿⠀⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⡀⠀⠀⠉⢿⡇⠈⢡⡄⢸⣿⠀⣈⢠⣶⣿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡀⠀⠀⠈⡇⢀⣾⡇⢸⣿⣦⣈⡵⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⡀⠀⠀⣿⠈⢿⡇⠀⡻⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⣿⣿⣷⣄⠀⣿⡆⠘⠃⠀⢃⣤⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⢿⣿⣿⠿⢇⣿⠀⢀⠀⠀⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢿⣿⡆⠈⠛⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠛⣿⡄⢰⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣧⣼⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣧⡈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠦⠾⣽⣷⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠃⠠⠄⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠶⣤⠁⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⣺⣿⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠖⠒⠋⢋⠉⠄⠰⠿⣻⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⣉⢠⡤⠀⠆⣒⣂⣩⡤⠴⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡫⣶⢀⣃⡴⠴⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣾⣟⣷⣶⣶⣶⣶⡶⡶⡖⣖⣒⣚⣛⣟⣋⣭⡩⡴⠦⠶⠒⡛⢋⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠿⠿⠟⠛⢛⡛⢉⡉⢉⡉⢩⡅⢡⡅⠠⣄⠀⡆⠰⠇⠘⣃⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢩⡄⠰⠆⠘⣁⣘⣃⣀⣥⣬⣥⣤⠥⠬⠥⠶⠶⠖⠒⠚⠋⠉⠉⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣬⣶⣾⣿⣿⣿⣯⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠤⠖⡒⠋⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⡿⠿⠿⠛⠛⢛⡋⠉⡭⠉⢩⠉⠁⠄⠀⢰⠀⠀⠇⠀⣓⣀⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢩⠄⠀⡆⠀⠩⢀⢀⡃⣀⣁⣀⣨⣡⣤⣤⠤⠴⠴⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠻⣂⣠⠤⠶⠒⠛⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣿⣷⣤⣤⣤⣤⣤⣤⡤⡤⢤⠤⠤⠤⠴⢶⢶⣶⣶⡶⢶⣶⣶⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣿⣿⣿⣿⣷⡿⣶⣺⢹⣦⣞⣼⣣⣷⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠓⠛⠛⠛⠚⠓⠓⠚⠛⠛⠛⣛⣛⣉⣩⡭⠿⠿⣛⠻⢴⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⠶⢖⠒⢋⢉⣩⠉⠤⡀⡶⠀⠿⢀⣚⡤⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢐⠹⢂⣚⣡⣭⠴⠴⠴⠒⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⢛⢻⡛⡟⣻⠟⣿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣄⣀
⢀⣾⣷⣀⢠⣶⠖⡀⠀⠀⠀⠀⠀⣬⣣⣒⣻⠀⠀⠀⠀⠀⠀⠀⠲⡄⠀⢀⠰⠞⠁⢀⠀⠀⠀⠀⠀⠀⣿⣭⣧⠼⠿⢟⠃
⢹⢿⡿⣿⣵⣥⡩⠒⠀⠀⠀⠀⢓⣑⣢⣶⠗⠆⠀⠀⠀⠀⠶⠀⣀⠈⡲⢤⡆⠒⡀⠀⡤⠆⠀⠀⢠⣶⣿⡿⣿⣶⣷⣿⣿⣄
⠠⠉⢲⣿⣿⢬⡟⠛⠃⠀⠀⠀⣷⣾⣶⣶⣾⠆⠀⠀⠀⠀⠀⠀⠖⣠⣄⣁⣴⠖⠒⢤⣀⠀⠀⠀⠸⠟⢠⣾⣿⣿⡿⠉⠛⠁
⠀⢂⡾⠜⣿⣷⡽⠲⠢⠀⠀⠀⠻⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿
⠀⠸⠁⢈⡿⠹⡗⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡟
⠀⠀⢀⣶⠅⢺⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⡆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⡀⠀⠀⠀⠀⠀⣀⣤⣴⣴⣦⣤
⠀⠀⠚⣯⠀⣿⠇⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣸⣾⠟⠘⠁⠀⠀⠀⠀⠀⠀⠀⣦⡀⣦⢀⣴⠟⠀⠀⠀⠀⠀⣠⢿⣿⣿⣷⣽⣖
⠀⠀⠀⢀⣸⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⣇⣿⣿⣿⣥⣾⠃⠀⠀⠀⠀⠞⠋⣿⢻⡉⠙
⠀⠀⠀⠀⠱⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣼⣿⣿⡧⢾⣒⠀⠀⠀⠀⠀⠀⠀⣿⡄⠁
⠀⠀⠀⠀⠀⠃⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠼⢻⣽⣿⣿⣱⣆⠈⠁⠀⠀⠀⠀⠠⠶⠿⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠈⠈

⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⠀⠀⠓⠈⠅⠀⠀⢰⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠀⠀⠍⣾⠈⠙⠒⠒⠒⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠠⡀⠀⠀⠈⠁⠀⠀⡀⠀⢀⡄⠀⡥⠀⠀⠀⣰⠊⠁⠈⠻⣆
⠀⠀⠀⠀⠤⠤⠤⣤⡤⠴⣦⣤⡤⠤⢆⠄⡀⠀⠰⣄⣀⣀⡈⡟⠉⣍⠂⡺⢀⡀⠀⠀⢠⣴⠇⠀⠀⠀⠀⠈⠉⠳⢦
⠀⠀⠀⣀⣀⢀⣌⠁⠀⠘⢻⣿⡿⠆⠈⠃⣄⣀⡤⠻⣿⣿⡐⡟⠀⡈⣠⠇⠀⢉⡆⠀⠈⢿⣆⠀⠀⡀
⠀⠀⠀⠀⠀⢺⣤⡀⣶⣿⣿⠆⠠⢀⠀⠀⠸⠀⠀⠇⢸⢸⣽⣷⣰⡷⢺⠀⠀⢸⢁⠀⠀⠀⠉⠳⠮⠤⠖⢂⣢⡀
⠀⠀⠀⠀⠀⠘⢿⣿⠟⠸⠉⠀⠀⡀⠀⠀⠘⠢⣂⡀⠀⢘⢻⣇⣻⡷⠞⠅⠀⠈⠈⠑⠠⡀⢀⣀⣣⣼⡿⠿⣯⣙⠓
⠀⠀⠀⠀⠀⠀⠘⢯⠀⠀⢀⡀⠀⡠⠀⠀⠀⠀⠀⠉⠓⠺⠞⠋⠘⣏⠔⠁⠀⢂⡀⠀⠀⠀⢸⡄⠁⠀⠀⠀⠈⠛⠕
⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠚⣮⡉⠉⠁⠀⠀⢸⠁
⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠘⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡹⣧⠀⠀⠶⠀⢺⣦⠤⢤⠤⠶⠶⠛⠛
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡄⠀⠉⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣝⢧⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢰⠀⠀⢀⠀⡅⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣆⢡⠀⠀⠀⠀⠀⠠⣠⡴⠞
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡙⠳⣥⣀⣀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠄⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⢷⡙⢏⠛⠛⠯⣉⠁
⠀⠀⠀⠀⠀⠀⢁⠀⡀⠀⠀⠁⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣽⣌⣳⣄⢀⠄⠉
⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⢤⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠟⣯⠁⠠
⠀⠀⠀⠀⠀⠀⢀⡯⡁⠀⠈⠈⠉⠉⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡻⡇⣿⡀⠁
⠀⠀⠀⠀⠀⠀⣟⠁⠚⡇⠀⠀⠆⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡕⢄⠨⠫⢸⠄
⠀⠀⠀⠀⠀⠰⣧⠀⠨⠇⠀⠀⢈⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠳⡄⠈
⣶⣶⠶⠶⠶⣶⣿⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣾⣶⣷⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠨⢴⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣆⠀⠴⠀⠳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠓⠀⢤⣤⠌⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠄⢉⣀⠁⠰⠋⠁⠀⠠⠜⢳⡁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⣴⠀⢆⡀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠨⠋⠀⠛⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⡁⣀⠀⠀⢲⣶⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠉⠈⠰⠟⢀⡀⠙⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠛⠡⠞⢁⠉⠹⠂⡀⠘⢻⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠾⠟⠉⠷⠀⠻⠂⣤⣈⠦⠀⠘⣡⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⢫⡾⠂⢶⡄⣦⠠⣤⠀⣁⠀⢀⡤⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⣚⠛⠧⢀⠀⢶⡄⠁⠈⠂⠈⠀⠁⠺⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢴⣈⠉⣀⠈⠓⠀⠈⣡⡄⠰⠟⠒⠀⠻⣟⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠲⣃⠛⠁⠶⠉⢤⣍⠀⢦⣀⠘⢉⠁⠶⠌⠷⠸⢆⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠛⠒⢈⠀⠋⠰⢆⡐⠴⠄⣉⠛⠀⠋⠺⠋⠲⠆⠐⠂⠀⣘⢦
⠀⠀⠀⠀⠀⠀⡐⠃⢘⢡⡄⠉⡴⠀⠈⠳⠆⢤⠹⠇⡈⠰⣆⠛⢀⠀⢾⡶⠄⡙⢈⠃
⠀⠀⠀⠀⠀⢰⢁⢄⣉⠠⢙⡀⠀⢲⡿⠀⠓⠀⢶⡀⢱⡄⠛⠲⠈⠀⢀⣀⠛⠁⡌
⠀⠀⠀⠀⠀⢀⡟⠀⠁⠠⡌⠃⠾⠆⠀⢰⣆⠻⡀⠁⣤⠀⣦⠐⠲⣈⠈⣩⠶⠷
⠀⠀⠀⠀⢀⣾⣄⠁⠀⣀⠰⠐⢶⣴⡉⠀⠉⢀⡁⣠⡄⠀⠈⢁⠀⡌⢸⡃⠀⠁
⠀⠀⠀⢠⡞⠁⠈⠄⠠⠉⣃⠐⠀⠉⠉⠉⠃⠉⠀⠀⠀⠀⠀⠀⠀⠻⠎⡁⢀⠀⠠⡀
⠀⠀⢀⣿⡄⠀⢠⠀⠀⠀⣩⡄⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠐⠫⡄
⠀⠀⣼⣿⡇⠀⡜⠀⠀⠰⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠂⠀⠀⠀⠀⠄⠀⠁⣄
⠀⣴⣿⠟⢀⠬⠁⠀⡄⠀⠀⠀⠀⢆⠀⠀⡆⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑
⠀⠻⠋⠀⡃⠀⠀⡙⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠱
⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠃⠀⠀⠀⠀⠀⠀⠠⠀⡆
⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡸
⠀⠀⠀⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁
⠀⠀⠀⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⣰⢀⡆
⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢅⠁⢀⠋⢰⠀⠀⠀⠀⢀⠆⠈
⠀⠀⢣⡐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠌⢡⡎⢣⡀⠀⠀⠻⣿⣦
⠀⠀⠀⣵⣾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠳⠿⠛⠀⠀⠀⠀⠉
⣿⣿⣿⡿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣴⣤⣷⣤⣦⣤⣶⣥⣶⣭⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⠤⠤⢤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠤⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⢸⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⡇⠀⠀⢀⣀⠀⠄⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⢸⣿⣿⡇⠀⡟⢻⣿⣿⡟⢿⣿⣿⡇⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⡇⠀⠀⢸⣿⢰⣦⠈⠐⢠⡄⠀⠀⢸⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⢸⣿⣿⡇⠀⡇⢸⡇⢸⠟⢻⣿⣿⡇⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⡇⠀⠀⢸⣿⢸⡇⢸⡇⣠⡄⠀⠀⢸⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⠸⠿⠿⠷⠶⠷⠾⠷⠾⠷⠾⠿⠿⠇⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣇⣀⣀⣈⣋⣈⣁⣈⣁⣙⣁⣀⣀⣸⢸⣿⣿⣿⣿
⠀⠀⠀⠀⣠⠞⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠳⣄⠀⠀⠀⠀⣿⣿⣿⣿⠟⣵⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣮⢻⣿⣿⣿
⠀⠀⠀⠸⢥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡬⠇⠀⠀⠀⣿⣿⣿⣏⣚⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣓⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿




⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⡇⢠⡄⠀⠀⢀
⠀⠀⠀⠀⢨⡃⠀⠀⠀⠀⠀⠘⠃⢸⡇⢠⡀
⠀⠀⠀⠀⢘⡅⣀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⠀⠀⢀⠀⠀⠀⢲⣤⡄⠀⠀⢀⠀⠀⡸⣆⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⣿⣶⠘⣧⣿⡟⠀⠀⠈⠿⣿⣄⢐⣾⡷⢳⣽⡍⠀⠀⠀⠀⠀⢻⣖⠀⠀⣤⣶⠧⠀⠀⠀⢀⠀⢀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣽⡤⣤⣠⡿⣌⢡⡄⢶⣷⣤⢨⣉⡱⣎⣠⡔⠊⠀⠀⠀⠀⠻⣤⣀⠘⣿⡦⠴⠛⣧⣴⠀⠀⠙⣾⣇⠈⣯⣵⡄
⠀⠀⠀⠀⠀⠀⣀⣀⣴⡓⢿⡀⠘⡯⣞⠛⢿⠃⠠⣍⣡⣔⠛⣵⠋⠸⣿⡍⠀⠀⣰⡦⡿⠶⣏⢠⡟⢀⠀⠀⠸⠻⠄⢠⣄⡄⢉⣌⣰⡞⠟⠃⢰⣠⣆⡤
⠀⠀⠀⠀⠀⠀⡼⠟⠋⠀⢉⣭⡿⡿⢟⢧⣈⣯⣟⣯⠃⠀⠙⢻⡄⢀⠙⠠⣄⣤⠀⣀⣼⣄⡎⣻⠹⡿⠶⠂⠀⢀⡰⠀⠽⠻⡟⢣⠶⣀⢀⣤⣊⠃⠛⠛⠃
⠀⠀⠀⠀⠀⠀⢠⣄⣰⠀⠀⠀⠀⠀⡿⡿⠙⣷⢼⡟⠀⠀⠀⠀⢻⣜⣿⣤⡽⢿⠆⡁⢀⠈⠁⢿⡘⣶⡂⣀⡴⣮⡛⢃⣀⣺⣡⡴⠞⣉⠁⠀⠨⠗⠢⣴⣧⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣽⠿⡃⢦⣾⣧⠄⠀⢀⠙⣲⣆⢺⣴⣧⢀⡄⠹⣿⢿⣟⢪⡴⣷⠿⣄⠀⡀⢹⡝⠁⣼⣟⣋⡉⢚⡿⠋⠀⢙⡘⠿⢾⡀⠀⢸⣷⠃⠉⠟⠷
⠀⠀⠀⠀⠀⠀⡼⠟⠋⢉⣿⢶⣿⡁⠀⠒⠻⠟⡿⠛⢿⡁⠀⢠⣿⣶⣿⠉⣽⢦⡶⣤⣤⠀⠀⠽⣾⢀⣔⠉⣀⠿⢳⢎⣴⠃⠀⢿⡿⠁⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⠻⠃⠈⠙⣦⡷⣏⠀⢠⠄⠈⣷⠀⠀⠀⠀⠹⣦⡆⠈⠞⢳⢾⡇⠀⢀⡟⠋⠈⠽⣋⣠⢸⠋⠉⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠟⣁⣾⢷⣄⣀⣬⣷⡀⣹⣿⣶⠀⢿⡄⠀⠀⡽⡏⢻⡇⢀⠴⢶⡍⢉⡝⢺⡇⠀⢠⣴⣍⣇⢸⠀⠀⠀⠀⢀⢠⣤⠄
⠀⣀⣶⣼⣦⢤⣤⣄⡤⢿⣿⡿⠟⠋⠀⠈⠻⢶⡶⠿⠶⢾⣷⣦⣤⣅⣄⣼⡇⣸⠻⠎⢀⣻⡛⠀⣷⢢⣴⠚⠟⠙⣿⠀⠀⣳⢆⢸⠟⠁⢀⣦⡞⠀⢠⡀⣴⣀⡀
⠘⠉⠋⣸⠗⠋⢀⣹⠳⠈⠸⠆⠀⠀⠀⠀⢻⡻⠀⢀⢀⣤⣤⡋⠉⠉⠉⠛⢿⣷⡄⢤⡬⠉⣥⣠⣟⠁⠉⣉⣢⠀⡏⠰⣾⣈⣋⡇⠀⣀⣼⢷⠶⢖⣉⡛⠉⠈⠁
⢀⣀⡞⣋⣓⠀⠈⠏⠃⠰⣆⣀⡀⢲⡤⣶⣫⡝⢳⣿⠀⡴⣿⠏⠀⠀⣄⣲⠺⣉⣿⣎⣹⠏⣙⠈⣧⢼⡔⣊⢉⡾⢡⣸⡏⣙⣫⣷⡋⠀⠛⠻⠦⠈⢿⡦⠄
⣘⣽⠻⠁⣯⠀⠀⠀⢀⣏⣹⣻⣥⣿⠾⠧⣿⣛⣹⣏⣀⡀⠘⠂⠀⠀⠚⣫⡾⢹⠇⢻⣇⠰⣿⢷⡇⣰⣽⡛⢻⣇⣹⡿⠛⣉⣡⣸⣷⢀⣠⣦⣶⡀⠀⠛
⠉⡎⠃⠀⠀⠀⠀⠀⢶⣾⣣⡝⠁⣞⡆⠀⠀⣉⢹⡭⠛⠛⠻⠶⢦⣴⣿⣤⠈⣏⣻⣾⢻⣧⢀⣾⣥⡿⢟⡏⢹⡟⠻⢞⣰⡜⢻⣽⠖⢺⣿⢩⢧⣇⡀
⠀⠁⠀⠀⠀⠀⠀⠀⠘⠁⠾⡿⠀⠙⠇⠀⠈⡽⠟⠃⠀⠀⢠⣠⡄⠈⢛⣿⣶⠋⠓⠀⠀⢻⣿⠋⣷⡤⠘⣿⣿⠘⢶⣸⣻⣶⠻⡄⢐⣤⠖⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣝⢫⣽⣍⠛⣦⡀⠀⠀⢰⣿⣆⠘⠿⠀⢠⡿⠤⠚⣹⡦⠈⢿⡉⠉⢩⢾⠃⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣯⠐⢻⡏⠀⠈⠻⣦⣀⡿⠀⣿⣄⣀⣀⣿⣁⣰⡧⠄⢹⢢⣼⣧⣗⢀⣼⣤⠤⠴⣚⣟⠻⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⢸⣿⣉⣿⣿⣽⣫⡴⠶⠂⡉⢸⣯⢀⡟⠀⠀⡀⠀⠋⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡷⣰⠄⠻⣷⣄⣼⣿⣿⡇⢡⡈⢯⣥⡄⠙⠛⠛⢻⣾⠖⣰⡮⠷⠍⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⣼⣿⣱⣦⣿⣿⣿⣿⣿⡇⠺⣟⡄⡿⠀⠲⣿⠋⣸⣿⣉⠀⣻⣆⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⣤⣸⣿⡎⢸⣿⣿⣿⡇⠈⠉⢙⠇⠀⢸⣠⡾⣅⠀⠞⣗⠈⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢀⣹⣾⣿⣿⣿⣿⣶⣶⣾⠶⣤⣾⠻⣟⠟⠃⠈⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡝⠀⢹⣿⣿⣿⣿⡏⢠⢷⡂⠸⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⣿⣿⣿⣿⡇⠀⢫⡟⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢄⣰⣿⣿⣿⣿⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⣷⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣾⣿⡄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠋⠀⠀⠈⠉⠀⠈⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⣿⠃⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⠀⠈⣿⣿⣿⣿⣿⣿⣿⠃⠀⠙⢻⡿⠛⠛⠛⠛⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⢠⣾⣿⠀⢀⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⣿⣿⣿⣧
⠀⢀⡞⣻⡿⠀⢸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⠻⣿⣿⣿⡄
⢀⠘⣹⣿⠁⢠⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠹⣿⣿⣷⡀
⡜⢠⡿⠏⢰⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠹⣿⣿⣷⡀
⠀⠀⠘⠀⠘⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢹⣿⣿⣷
⠀⠀⠀⠀⠀⣿⣿⡟⠈⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠁⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⢠⣿⡏⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⠇⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆
⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠙⠿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠄⢠⢠⡄⠀⠀⢠⠀⡀⢀⠀⢠⡀⠀⠀⠀⢀⢀⡄⠀⠀⠀⠀⠆⠀⠀⡴⢀⡄⡀⠀⠂⡴⠐⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠀⠀⠀⢀⠜⣽⠋⠁⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠜⠁⠠⠒⠉⠉⠁⠒⠤⠀⢀⣀⣤⡶⠏⠀⠀⠀⠒⡲⠖⠐⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⢀⢀⡴⠅⠀⠀⠠⢤⣤⣶⣶⢿⢿⠭⠭⠓⠒⣚⠡⠄⠊⠁⠀⠀⠁⠀⠄⠀⠀⠀⢠⣤⣦⡾
⠀⠀⠀⢀⣠⣴⡏⠀⠀⠀⣀⣤⠶⠚⠉⠀⠀⠀⠀⢀⠔⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢝⣿⠁
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⡀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⡴⠒⢒⣲⡖⠉⢉⣈⠉⠉⠋⠀⠀⡄⠐⠀⠆⠀⠀⡍
⠀⠀⠀⠀⠀⡸⣋⠄⠀⠀⠁⠀⠀⠀⠀⠀⠁⢐⠀⠀⠠⢀⡂⠠
⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⠄⠀⠀⠀⠘⠀⠀⢀⠄⡉⢆⠀⢐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠁⠔⠁⠠⠁⠀⢀⡀⠀⠀⢠⣄⡀⠀⠀⠀⠀⣼⠆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⢾⣦⡀⣀⡜⠊⠅⢰⠂⠰⠀⠀⠀⠀⠚⠋⠁⠀⡀⠠⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢹⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣦⣿⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠂⠤⠀⢀⠘⢿⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠈⠠⠀⠐⢂⡀⢬⣿⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡞⠀⠀⢠⢃⠄⣀
⠀⠀⠀⠀⠀⠀⠀⡁⢠⠊⠁⠀⠀⢈⠡⠴⢒⠂⠒⠀⠐⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⡾⠉⡿⠿⠿⠄
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠈⢛⣶⣧⡀⠂⢀⠀⢀⠀⠀⠄⡄⠀⠀⠀⠀⠀⠓⣄⢸⣶⢸⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⢉⡿⠿⡇⠄⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣆⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠞⠀⠐⢠⠄⠀⠀⠀⠀⠀⠀⠀⠹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄
⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅
⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⡀⠀⢻⣿⣿⠛⢡⣿⣿⣿⡄⠀⠘⣿⣿⡿⠁⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣾⣿⣶⣄⠉⢁⣶⣿⣿⣿⣿⣿⣶⣤⡀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⣿⣿⣿⠿⠿⠛⠉⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡏⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠁⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠁⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠖⠀⠉⣉⣶⣄⣀⣨⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠃⠀⠀⠀⠈⠉⠉⠙⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠠⠎⠀⠀⠀⠀⢀⡰⠂⠀⠀⠀⠀⠀⠈⠶⠄⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠘⠿⠏⠀⠀⠸⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠀⠀⡌⠁
⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠙⠷⣤⣤⣀⠀⠀⠀⠀⣀⣤⡄⠲⠞⠁⠀⠀⠀⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠸⢧⡀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠷⢦⣤⣤⣄⣀⣠⣤⣤⡄⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⠀⠀⠀⠈⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢧
⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⢠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⡸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⣼⡇⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣧⡀⢀
⠀⣿⡄⢸⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠉⢁⠉⠉⣇⠈⠀⢸
⠀⠛⠛⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠻⠆⠀⢸
⠀⢸⡇⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠈
⠀⠈⡇⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⢸⡂
⠀⠀⢁⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢿⠀⣀⢀⠀⢀⣀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠒⠒⠒⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠠⠤⠤⢤⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⣀⡀
⣤⣀⣀⣠⣼⣷⣦⣤⣤⣤⣬⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣀⣀⣀⣀⣄⣠⣿⣿⣿⣷⣶⣿⣿⣅
⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣾⣷⣶⣾⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣦⣀⣠⣴⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣾⣿⣿⡟⠉⠙⠛⠋⠉⠉⠛⠿⠿⠿⠧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⣀⠀⣤⣤⣀⣀⣀⣀⠶⢦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⡛⠁⠈⠛⠛⠛⠛⠛⣠⣼⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠘⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠛⢻⠿⠿⠿⢿⣿⣿⣿⣿⣷⣤⣼⡿⣿⣷⠶⠟⠛⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠶⠟⠛⠀⠀⠀⠀⠀⢀⡀⠈⢹⣿⡟⠉⣿⣡⣤⣤⣿⣿⣶⣶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠋⠀⠀⣤⣤⣄⠀⠀⠀⣠⣾⣿⡆⠀⠀⠀⢸⡿⠻⣿⣿⣿⠿⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣀⣀⠀⠀⠀⠲⣤⣤⣤⣤⣄⠀⣤⣤⣽⣿⣦⣶⠿⠏⠀⠀⠉⠀⢠⣾⠟⢠⣾⠟⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⠿⠟⠿⡟⠋⠁⠀⠀⠀⠀⣀⡰⣿⡇⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣠⣼⠟⠁⠈⣤⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢠⠚⢻⣧⣤⡾⠿⠉⠉⢡⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠠⠶⠈⢠⣾⡛⠉⠀⠀⢀⣴⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣦⠀⣴⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡟⠉⠻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠟⠁⠘⠇⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠉⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠘⠂⠀⠀⠀⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⢦⠀⠀⠀⠀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡄⠀⠀⠠⠆⢸⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣄⡀⠄⠀⠈⢀⣀⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠋⣁⠀⠀⠤⠈⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣧⠀⠛⠛⠀⢀⡀⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣯⡀⠀⣠⡆⠀⣀⠈⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⠛⠋⠉⠋⠀⠀⠉⣀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢺⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠰⠆⠐⠷⠀⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠠⠀⢀⣾⣿⡿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠛⠀⠀⣀⣴⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠈⠀⣤⠀⣠⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠃⠀⠀⠊⠁⠀⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠠⠆⢀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣤⣀⣀⣠⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⣸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣏⡀⢀⣀⣼⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⠛⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣤⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⠋⢹⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣾⣿⡛⣻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠛⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣪⣵⡾⠷⢶⡭⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣹⡟⠁⠀⠀⠀⠙⡌⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣏⣓⢦⡀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣪⡷⠿⣻⣿⣸⠛⣇⠀⢀⣟⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣽⠞⠉⠀⠀⢈⠝⣡⣾⣿⡤⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣪⠖⠋⠀⠀⢀⡤⣚⡵⠞⣽⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⣭⠞⠁⠀⢀⡠⢖⣽⠞⠉⢀⢾⠏⢸⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠵⠋⠀⠀⣠⣖⣭⠞⣻⠏⠀⢀⣾⡟⠀⣸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁⠀⣠⣔⡿⠞⠉⠀⣸⠏⠀⢀⣮⡟⡇⠀⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠋⠀⣠⣴⡿⠞⠉⠀⠀⠀⣴⠏⠀⢀⢮⠟⠘⡇⠀⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣾⠟⠁⣠⣴⡿⠟⠉⠀⠀⠀⠀⠀⣸⠏⠀⢠⢫⠏⠀⡆⡇⠀⡇⡇
⠀⠀⠀⠀⠀⠀⣀⣴⣾⡯⠵⠒⠲⣼⡿⣅⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⣰⢣⠏⠀⠀⢱⡇⠀⣧⡇
⠀⠀⠀⠀⣠⣞⡵⠛⠁⠀⠀⠀⠀⡸⢠⢟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⣰⣣⠏⠀⠀⠀⢸⡇⢠⢻⠃
⠀⠀⢀⢞⡵⠋⠀⠀⠀⠀⠀⠀⢰⠃⡞⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⣰⣳⠏⠀⠀⠀⠀⢸⡇⢸⢸
⠀⢀⣏⡞⠀⠀⠀⠀⠀⠀⠀⣰⢃⡜⢁⣿⡟⠀⠀⠀⠀⠀⠀⢀⣠⠷⢷⣴⣿⠏⠀⠀⠀⠀⠀⢸⡇⢸⢸
⠀⢰⡿⠀⠀⠀⠀⠀⠀⢀⠔⡡⠋⡠⣿⡟⠀⠀⠀⠀⠀⢀⡴⣫⠴⠚⢯⢹⣿⣆⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠈⢿⣄⠀⠀⣀⡠⢖⣵⣪⠴⣫⡾⠋⠀⠀⠀⠀⠀⢀⣞⡼⠁⠀⠀⢸⢀⡇⢻⡆⠀⠀⠀⠀⢸⡇⠸⣿
⠀⠀⠀⠙⠛⠿⠿⠿⠭⠥⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⡼⣸⠁⠘⡇⠀⠀⠀⢠⣾⠽⢿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⣰⣷⠇⠀⣸⠃⠀⠀⢠⣿⠃⠀⠈⣿⡜⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⡆⠀⢀⡴⡿⠋⢀⣴⠃⠀⠀⠀⢸⣿⠀⠀⠀⢹⡇⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣽⠮⢴⣺⠟⠁⠀⠀⠀⠀⢸⣿⡀⠀⠀⣸⠃⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢧⣣⣀⣴⣟⣴⠟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣛⣉⣁⣀⣀⣀⣀⣀
⣿⣿⠋⢛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣀⣀⡿⣿⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠿⠿⣀⠀⡟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⡞⢉⡖⠦⠙⢦⢸⣿⡇⣤⣤⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣡⣾⣫⣙⣶⡹⡇⠀⢸⠟⠛⣿⣿⣿⣿⣿⣿
⠀⠀⢸⡀⢘⡗⠀⠀⢸⢸⣿⡇⣿⣿⢀⣤⣄⠀⠀⠀⠀⣿⣿⡇⣿⣧⢨⣿⣿⡇⡇⠀⢸⠀⠀⣿⠛⢻⣿⣿⣿
⠀⠀⠀⠳⣌⣉⣉⡴⣋⢸⣿⡇⣿⣿⢸⣿⣿⠀⠀⠀⠀⣿⣿⣿⣌⡳⠶⠶⣛⠴⡇⠀⢸⠀⠀⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀⠈⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛


⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠉⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠠⢀⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠸⠀⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠠⠖⠑⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢀⠴⠚⠤⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠙⠋⠲⠶⣦⣄⣀
⠀⡠⠠⣭⡀⠀⠀⠀⠀⠄⠠⣀⠀⠈⠀⠛⠁⠠⠠⣦⣤⡀⠀⠉⣡⣶⡀
⠀⢀⣤⣴⣶⣶⣶⣤⣤⣤⣀⣲⡤⢌⣉⡐⠒⠢⢄⠀⡌⣫⡔⠛⠋⠹⣿⡄
⠀⢸⣿⣿⠟⠛⠻⠿⠿⠿⢿⡿⠿⣿⣷⣾⣭⣓⢶⣴⡜⠇⠀⠀⠆⠀⠹⣿
⠂⣾⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠿⠿⣿⣧⡹⣿⣎⠀⠀⠀⢀⣀⠀⠠
⡀⡇⠀⠀⠀⢀⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⢹⡟⠀⠀⣴⢻⡿⠀⠀⠄
⡀⡇⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢿⣦⡀⢹⣆⣧⢀⢠⡄
⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⡎⡿⠇⢀⡿⠈⠰⡌⡇
⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢁⢻⠀⢸⡗⡀⠀⢹⡟
⡀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠘⠛⡄⡜⡇⠇⠀⣼⠁
⠁⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⣠⠙⢱⡇⣆⠔⠁
⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠁⠀⠀⠀⠀⠀⠀⡅⠈⣸⣿⣿⣆
⠀⠠⢌⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣠⡶⠟⢝⡯
⠀⠀⠀⠀⠉⠒⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣦⠔⠁⠀⠈⠁⠒⠠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⢙⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⢄⣀⠀⠀⠸⣿⣿⠗⠲⠤⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠋⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣄⣀⠶⠀⠀⠀⢁⣒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣄⣀⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠂⠀⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠂⠰⣤⣀
⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠄
⠀⠀⡀⠊⠀⠀⠀⠀⠠⠔⠋⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⡀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⡔⠀⢀⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠆⠰⢶⢤⣤⠴⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⡌⢠⠏⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢿
⡇⡎⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠺⠇⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⡀
⢁⠀⠀⠀⠴⣀⢤⡆⣰⡆⠰⠂⣠⡶⠀⠀⠀⠀⠀⣤⠣⠁⠀⠀⠀⣠⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⢸⠃⠀⠀⠛⠃⠋⠈⠟⢡⠑⠀⠛⠁⠃⠀⠀⠀⣼⠃⠀⠀⠀⣠⠞⠁⠀⠀⠈⠛⠷⢶⣤⣀⠀⠀⠀⠀⠀⠛⢷⡀
⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⣥
⠀⢹⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡦⠁⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠻⡷
⠀⠀⠻⢮⣁⡒⠀⠒⠚⠛⠛⠛⠁⠀⠐⢛⣡⠄⠰⠞⠺⠿⠿⠿⢿⣿⠿⠶⣶⣶⣤⣄⣀⣼⠃⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠈⠛⠻⠷⡶⣦⣄⣀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠸⡟⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠀⠀⠀⠀⠈⢳⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠙⣮⡉⠛⠶⢦⣤⣴⣦⣴⣄⣠⣄⣀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠉⠉⠉⠁⠀⢠⣬⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⢀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⢸⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢷⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠈⢳⡀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠳⣴⣿⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⢹⣇⠀⠀⠀⠔⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣦⡀⠉⠠⣤⣀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠀⠙⣿⣦⠀⠈⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠈⠻⣦⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⢀⠔⠁⠀⠈⢣⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⢣⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣠⠤⠒⠊⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠚⠉⠉⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠘⣇
⠀⠀⠀⠀⠀⡤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠹⡆
⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⡄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦

⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣆
⠀⠀⠀⠀⠀⠀⠀⣿⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠹⣆
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⡎⣧⠀⠀⠘⡇⣾⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⢻⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢰⢇⡟⠀⠀⠀⢿⡜⣧⡀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠘⣧⡙⢦⡄⠀⢠⡞⠀⠀⠀⠀⠈⠻⣤⡀⠀⠀⠴⣫⠞⠁⠀⠀⠀⠈⠻⢦⣝⣛⣃⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢧
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠈⢧
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⣸⠁⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⡟⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⢠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠈⣟⣇⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢠⣸⡆⠀⠀⠀⣿⢿⡀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠹⡜⣆⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠈⢷⣄⡀⠀⠀⠜⣸⠃⠀⠀⠀⠹⣌⠷⣄⡀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠙⢮⣝⣲⠶⣂⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠉⠓⠶⠶⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⣿⣿⠛⠿⢻⠿⡿⢻⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠒⠒⠛⠒⠺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⣀⣀⣀⣀⣀⣤⣀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⣁⣀⡈⠁⣀⣀⢸⡆⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢳⣼⠀⠤⠀⡆⠠⠄⢸⡗⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⠀⠀⠀⢼⣁⠀⠀⠈⣁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠠⣖⣲⡆⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠲⢤⡥⠴⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⣶⠛⠁⠀⠀⠀⠀⠈⠙⡶⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠈⣇⠸⡄⠀⠀⠀⠀⠀⠀⣰⠃⡼⠁⠈⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢹⠀⠈⠲⠤⠤⠤⠴⠚⠁⠀⡇⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢀⣴⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣧⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢀⡞⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢳⡀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣏⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣹⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢸⡄⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢠⡇⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢸⠇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⡇⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢸⠀⢰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⠀⠀⡇⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⣙⣾⠉⠉⠉⠉⠉⠛⠉⠉⠉⠉⠁⢹⣴⡚⢁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⡧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠶⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣟⡉⠉⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡷⠂⠙⢿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⢈⣽⣦⡄⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣰⣿⣿⣿⣿⡟⠁⠀⠀⠀⠹⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠀⠀⠀⠀⢠⣿⣿⣿⣿⣆⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢹⣿⣿⣿⣿
⠀⠀⠀⠸⠿⠿⠿⠿⠏⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣿⣿⣿⣧⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣡⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠋⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⡀
⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⢽⣇⣀⣀⡀⠀⠀⠀⠀⠀⠀⢴⣄
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠲⣄⣠⠖⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢀⡸⠁⠀⠸⡄⠀⠀⠀⠀⣀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣸⠃⢀⣄⠙⢦⡀⠀⠳⡀⠀⠀⢀⣠⣤⣝⠀⣠⣤⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠾⣍⢠⠏⢸⢀⣀⠳⡄⠀⠱⡄⢠⠏⠀⠀⠈⣿⠁⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘⠋⠈⢳⣘⣆⠀⠙⣾⠀⠀⠀⠀⣿⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣼⣦⣤⠾⣆⠀⢀⡴⠋⠓⠤⠶⠥⣤⣤⣤⠤⠤⠤⠶⠒⠒⠚⠛⠉⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⣿⣇⠀⠈⠁⠉⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⢀⣀⣹⣟⡀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢄⣀⣀⣀⣀⣀⣀⣐⣀⣐⣦⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣰⠋⢠⣲⠮⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠤⠶⢾⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⡴⠋⢩⠇⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢧⣠⣸⣀⣄⣴⣀⠞⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠊⠉⠀⠀⠀⡰⠋⢾⣝⠉⢤⡉⠓⠢⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢀⠞⠁⠀⠀⠯⠷⠚⠳⠒⡆⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⡂⠉⢈⣓⣒⡒⠒⠀⣀⣠⠤⠤⠤⠜⢱⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠔⠋⢉⡇⠀⢸⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡴⠁⠀⠀⣸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡎⡸⠁⠀⠀⢠⠇⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠈⠀⠛⢦⣀⣠⠎⠀⠀⠀⢰⠈⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣠⡀⠀⡾⠒⢫⡛⠦⠤⠤⠤⠤⠐⠛⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠼⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⣖⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢀⣻⣤⣤⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⣿⣿⠀⡘⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡎⢀⠇⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠼⢰⣡⣊⠐⠓⠤⠠⠤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠀⠀⠈⠁⠀⠀⢁⠉⡉⢳⣌⣹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠂⠲⠼⠦⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⢒⡛⠭⠉⢂⣀⠒⠠⠤⠶⣤⡖⣂⡀⠉⠭⠝⣒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢋⠩⢔⡂⠭⣐⣂⣭⣑⣦⣶⣶⣗⣚⣛⣛⣓⣚⡾⢿⣷⣥⣶⢬⣕⣪⢙⠲⣄
⠀⠀⠀⠀⠀⠀⢀⡴⢋⠕⡪⢔⡨⣐⡬⣟⣿⠿⠛⡋⠋⠙⠉⠙⡏⠛⠻⠛⠿⣯⣭⣛⠲⢭⣛⠦⣍⠳⣝⠢⡙⢦
⠀⠀⠀⠀⠀⡴⢋⠚⡡⡪⢊⢵⠟⣡⠞⡏⠀⠀⠀⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠃⠀⠋⡹⢶⠈⠳⣌⢧⡈⢷⡌⢎⢧
⠀⠀⠀⠀⣼⢡⠃⡔⡱⠱⢱⠁⣼⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢁⣟⠀⠀⠸⣾⠡⠈⠘⠀⡊⡇
⠀⠀⠀⠀⣿⢀⠘⠀⡇⠀⢸⢠⡏⠀⠀⠀⠀⣀⣠⠤⠤⠴⠶⠶⠦⠤⣤⣀⡀⠀⠀⠇⣼⠉⠳⡄⣰⢃⠎⡜⢠⢀⣵⡇
⠀⠀⠀⠀⣿⡜⡀⢄⢇⠰⢰⠈⣧⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⣤⠇⠀⣠⣿⠗⡡⢊⠔⡡⣫⠏⡇
⠀⠀⠀⠀⡇⣿⡐⡌⢎⢢⠑⡕⢌⠳⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡖⢟⡫⢔⡨⢒⡥⣪⠞⠁⠀⡇
⠀⠀⠀⠀⣧⠉⠻⣌⠢⡱⢕⢬⡑⠬⢚⣽⣓⣶⢤⣤⣤⣤⣤⣤⣴⠶⠶⠞⢛⡋⠭⢐⠪⠅⡒⠥⣢⣵⣯⡥⠴⢶⠖⠟⠒⠶⠒⠖⠲⠶⠒⠲⢦⢤
⠀⠀⠀⠀⣿⠀⠸⠋⢻⢮⣑⠩⣙⠲⠦⣤⣉⣉⣀⣀⣈⣉⣉⠀⠀⠴⠆⠑⣂⠠⣉⣂⣬⣵⠞⢻⠉⠉⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠇⣼
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⢹⠛⠶⢬⣍⣙⡛⠒⠖⠒⠒⠒⠀⢒⣈⣃⣽⣾⠿⠻⠋⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢠⠇
⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠈⠀⠀⠀⠁⢹⠉⠙⠛⠛⠛⡏⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⡞
⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠖⠒⠛⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠴⠖⠚⠋⠉
⠀⠁
⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠤⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠜⠉⠀⠀⠈⠀⠀⠈⠀⠐⢠⡀⠑⡀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡜⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠹⣆⢱⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡦⠀⢠⢡⠀⠀⢻⣰⠀⡇
⣤⣄⡪⠎⠉⠑⢒⠒⠒⠠⠤⠄⢀⣀⡀⠀⠂⠀⠀⠀⠀⠀⠛⠁⠀⠁⡼⠀⠀⣽⢸⢀⡇
⠘⠋⠉⠋⠙⠓⠒⠶⠦⠤⢬⣄⣐⠀⠀⠉⠡⡀⡀⠀⠀⠙⠒⠒⠒⠊⢠⠔⠾⠗⠯⠴⠦⠀⠀⡀⠀⠀⢀⢄⣤⢄⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⠒⠆⠈⡢⣄⣀⡀⠀⢀⣀⣬⡴⠒⣀⣞⣛⢳⡶⠴⡖⠀⠠⠮⠾⣫⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠓⠀⠀⠀⠰⠃⠘⠛⡏⣟⠐⠛⠖⣺⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢞⣡⠗⠖⠖⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⢯⣿⣗⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡣⡤⠜⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⡂⢡⡗⢄⡄⡔⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡈⢷⣄⡋⣿⣿⣧⡗⢀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣪⡿⡠⠂⡇⠶⠒⣙⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⢩⣿⣦⣱⣨⠷⡽⣿⡶⡚⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢂⠀⠀⢀⣀⣀⠀⠈⠘⢿⣷⣿⡿⣀⣤⠿⢏⡰⢒⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠒⠻⠿⠃⠀⢀⡜⠃⠚⠋⠁⠀⠀⠈⡓⠀⠀⠀⠀⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⡄⣤⡀⠀⠀⡀⢺⣤⣲⡇⠀⢀⠀⢀⣺⡥⠀⢀⣀⣤⣤⣶⡿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣍⡉⠻⢿⠀⠀⠙⢿⣾⠏⡏⣀⢸⠀⢸⡯⠐⠂⠉⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣼⣷⡀⠀⢨⠉⠘⡇⡎⡄⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠹⣆⢦⣿⡰⠀⡴⠁⠀⠀⠐⢱⣿⠿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠈⠢⡸⣾⣽⢃⡜⣠⠖⢲⠦⠾⢯⡢⣶⡼⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡾⠎⠉⠙⠢⣄⠀⢹⣿⣧⡎⣴⣥⡾⠋⠀⠀⠀⠀⠈⠉⠈⠛⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠂⣠⣔⣶⣿⣿⠃⠀⠀⠀⢀⣠⣵⣾⣿⣿⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢀⣴⠥⠞⠃⠀⠀⠀⢰⣿⠋⣀⣽⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣤⠐⠋⠀⠀⠀⠀⣀⣤⢤⣼⡷⣻⣷⣿⣿⣿⣿⣯⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠚⠉⠓⠟⠛⠛⠿⠿⠛⠻⢿⣻⡿⠿⠿⠿⠿⠇

⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠉⠉⠉⠉⠛⠛⠻⠭⣛⡳⢶⣤⡀
⠀⠀⠀⠀⠀⣠⠾⠛⠓⠉⠉⠉⠉⠉⠒⠲⠤⣄⡀⠀⠀⠉⠳⣌⠛⢶⣄
⠀⠀⠀⢠⡞⠁⠀⠀⣀⣠⣤⣤⣄⣀⡀⠀⠀⠀⠉⠲⣄⡀⠀⣸⠇⠀⠹⣧⡀
⠀⠀⢠⠏⠀⣠⡶⠛⠉⠉⠀⠈⠉⠉⠛⠷⣦⣄⠀⠀⠀⠉⠛⠁⠀⠀⢀⡼⣷⡀
⠀⢀⡟⢀⡼⠋⢀⣤⠶⠖⠒⠶⠦⣤⣀⠀⠀⠙⢷⣄⠀⠀⠀⢀⣠⠞⣡⠴⠜⣷
⠀⢸⠇⢸⠁⢰⠋⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⠀⠀⠙⢷⡀⠠⡏⢠⠊⠀⠀⠘⣼⡇
⠀⢸⡀⢿⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠈⢿⡀⠹⡌⡄⠀⠀⠀⣐⣷
⠀⠘⡇⠸⣀⡈⠉⠓⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⣷⠀⢹⡐⣀⡴⠚⠉⢿
⠀⠀⡵⠦⢤⣉⠙⠲⢤⣀⠉⠓⢤⡀⠀⠀⠀⠀⠀⢱⠀⠀⢸⡀⠀⣇⡅⠀⠀⠀⠘⡄
⠀⢀⡇⠀⠀⠈⠙⠓⠤⣈⠛⢦⣀⠈⠲⣄⠀⠀⠀⢸⠀⠀⢸⡇⠀⣿⡀⠑⠲⠴⡾⠁
⠀⠈⣷⠀⠀⠀⠀⠀⠀⠈⠑⠢⡈⠳⢤⡀⠑⢄⣠⠏⠀⠀⡾⠀⢠⡟⣁⣀⢤⣾⡁
⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⢦⡀⠁⠀⢀⠞⠁⠀⢼⣠⠤⠚⠉⠀⢹
⠀⠀⠀⠈⠛⠷⠶⠶⠶⣶⣤⣤⣀⡀⠀⠀⠀⠙⠶⠚⠁⠀⠀⠀⠀⠀⠀⣀⠴⠛⠻⢤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠲⠶⠤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠸⠿⠶⣲⡴⢤⣄⣀⣀⠀⠀⠀⢀⡖⠒⡶⠤⢤⣤⣈⣉⡭⠿⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠦⠶⠚⠁⠀⠀⠀⠀⠉⠉⠛⠓⠾⠿⣿⣶⣤⣤⣂⡀⠠⡠⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠷⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⢀⠠⣤⠮⠁⢤⣾⣷⣿⠁⠀⠐⠋⡿⠉
⠀⢀⣠⡏⠸⠀⣀⠈⠁⠀⢠⠞⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⢈⣉⡈⠀⠀⠀⠀⢢⣠⣠⣿⣾⠟
⠀⠙⠡⠀⠀⠀⠛⠾⢻⣿⣧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠈⢛⠙⠁⠜⡅⠋⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠺⣿⠛⢹⡇
⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢱⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢸⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘
⠀⠀⠀⠀⠀⠀⠘⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠃⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡀⠛⠁⠀⠙⠁⣤⠀⠀⠀⠀⠠⡆⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢸⠁⠀⢰⠃⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⣷⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡔⠀⠀⢿⡀⠀⠸⠀⢠⢦⠀⠀⣷⡀⠀⠀⡆⢃⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠐⠃⠀⠀⠀⢸⠀⡆⠸⡏⠀⠀⠀⠃⠀⠀⠀⠀⠀⠃
⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠶⠂⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⢸⣀⠰⠀⠀⡀⠀⠀⠀⢧⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠈⠀⠀⠀⣶⠀⠀⠀⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢈⡟⠀⠆⠐⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⢀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡇⠃⠀⠀⠱⣶⢸⡇⠘⡇⠀⠀⠀⠀⠀⢠⡿⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⣷⡇⢀⠀⢰⣿⢿⠁⢀⡇⡀⠀⡇⠀⠀⠘⡃⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⡿⣧⣌⠀⠈⡟⠘⠃⠘⣷⢁⢸⠇⠀⡐⠀⠁⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠃⣿⠀⠉⣷⡇⣦⠀⠀⡿⣼⠈⠀⠼⠀⠀⡜⢳⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡆⠃⠀⠀⢿⠀⣿⠀⢀⢧⡋⠀⠀⡆⠀⠀⠃⢸⡄⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠃⠀⠀⠀⡸⠀⣿⠀⡸⠸⡇⠀⠀⡇⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⠀⠀⠀⡇⠀⣿⣸⡇⠀⣇⠀⢸⠇⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢇⠀⡟⣿⠀⠀⣿⠀⣾⠀⠀⠀⠀⠀⡇
⣀⣀⣀⣀⣀⣀⣀⣤⣀⣀⣀⣀⣀⣤⣀⣀⣀⣀⣀⣠⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄⣽⣿⣀⣤⣠⣼⣠⣧⣿⣀⣠⣿⣤⣿⣀⣀⣠⣄⣀⣥⣠⣤⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠻⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠁⠀⠀⠀⠈⠉⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣠⠟⢹⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠋⠉⠀⢸⣿⣿⣿⣿⣿⣿⣷
⠀⠀⢰⣯⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⡇⠈⠻⣿⣿⣧
⠀⢀⣿⡿⠀⣼⣿⣿⠟⠁⠹⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠙⢿⣿⣷⡀
⢀⣿⡿⢁⣼⣿⣿⠏⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠙⣿⣿
⠈⠉⠀⢸⣿⣿⠏⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠻⣿⣿⡀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣸⡿
⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣿⠇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⣿⡇⠀⠀⠀⠀⣸⣿
⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣤⣿⣧⠀⠀⠀⣠⣿⡏
⠀⠈⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣦⡀⠀⠙⢿⡇
⠀⠀⠘⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠄⠀⡀⠀⠀⠀⡀⢠⠀⠀⠀⢀⠀⠀⠀⠀⠄⢠⡀⠀⢠⡀⡀⡀⠠⢠⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⠀⠀⠀⠰⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠈⢿⣿⣿⣆⠀⠀⠀⠀⣴⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⡀⢀⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡿⠁⠈⢻⣿⣿⣦⠀⠀⢠⣾⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠟⠁⠀⠀⠀⠙⢿⣿⣷⣴⣿⣿⠟⠻⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠋⠀⠀⠈⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠙⣿⣿⣷⣠⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠄⢀⣂⣰⣶⠶⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠄⠒⣂⣉⣭⡤⠤⠶⠦⠘⠂⠐⠒⠐⠒⠐⠒⣦⣴⡿
⠀⠀⠀⠀⠀⠀⠀⢠⠶⣲⡾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠒⠒⠒⠒⠒⢒⣒⣒⡒⢒⠲⣽⡿⠋
⠀⠀⠀⠀⡀⠤⠤⢼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣿
⠀⢀⠔⡡⠐⠂⠀⠺⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠒⡺⡇
⠀⡎⠎⢀⢔⣮⠿⣶⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣒⣂⣀⣀⢠⣷⠇
⢠⢸⠀⢸⡎⠇⠀⠙⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⠀⠘⢸
⠈⡎⡄⢸⢧⠇⠀⠀⢹⠼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⡞⡇
⠀⢱⢣⠈⡾⡸⡀⠀⢸⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⢤⣧⡇
⠀⠀⢫⢆⠸⣷⣣⠀⠀⡇⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢹
⠀⠀⠀⢫⢦⠘⢷⣷⡴⣿⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⡾⡞
⠀⠀⠀⠀⠳⣕⢄⠈⠉⠁⣦⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢠⣥⠇
⠀⠀⠀⠀⠀⠈⠑⠯⣒⣦⡇⡎⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠺⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠐⠚⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠠⠴⠤⢼⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⢧⡿⢭⣀⣒⣒⣒⣒⣒⡒⢒⠒⠒⢒⣊⣉⣽⣴⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣒⣊⣒⣒⣒⣂⣀⣀⣨⣍⣭⣉⣉⣙⣐⣒⣚⣁⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡫⠟⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣷⡎⠹⠇⣤⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣓⣮⠂⠚⢟⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⠀⠀⠀⠀⢀⡀⠀⠀⢥⣾⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠓⠰⠦⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣿⠕⠂⣤⢈⠉
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣠⡭⠀⠘⢦⡀
⠀⠀⠀⠀⠀⢠⣿⣿⡿⠛⣦⡀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⣠⡾⠋⠀⠀⠐⠄⠹⡄
⠀⠀⠀⠀⢀⣿⣿⣿⠃⠀⠈⠻⣦⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⡾⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠡⠹⡄
⠀⠀⠀⠀⣼⣿⣿⡏⠀⠀⠀⠀⠙⢷⣄⢻⡀⠀⠀⠀⢨⣬⠀⣶⣆⠀⢸⠇⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⢠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⠀⠀⠀⢸⣿⠀⣿⣿⠀⣿⣠⡾⢋⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣞⣷
⠀⠀⠀⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠶⠶⠶⠿⠿⣿⢿⣿⣿⣿⣿⠿⠿⠿⠗⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⡿⠟
⠀⠀⠀⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⠶⢶⣶⣶⣿⣻⣶⡆
⠀⠀⠀⠈⣿⣿⣿⢿⣿⣶⣶⣶⣶⣿⣿⡿⠿⠿⠿⠿⠿⣶⡆⢀⡇⠀⠀⠀⠹⣏⢉⣉⣉⣤⣤⠴⠶⠖⠛⢻⣿⣿⣿⣿⣿⣿⠷
⠀⠀⠀⠀⠉⠉⠉⠋⠉⠉⠙⠂⠀⠉⠉⠛⠿⢶⣦⣤⣼⡟⠀⠨⠇⠀⠀⠠⣶⠟⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠃⠀⠀⠀⠀⠀⢠⡟⠋⠻⣶⡤⠀⠀⠀⠀⠐⣡⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠂⠀⠀⠀⠀⠀⢠⡟⠁⢀⣴⡟⢱⠀⠀⠀⠀⠀⠹⣶⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⢀⣾⠏⠀⢰⠀⠀⠀⠀⠀⠀⠹⣧⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣰⡿⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⣴⠟⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣰⣿⢃⣼⠏⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣾⣿⣧⣾⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⣹⣿⠿⢶⡆⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⣠⣿⣷⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣿⣿⠟⠉⠉⠀⠀⠰⡆
⠀⠀⠀⠀⠀⢀⣤⣦⣤⣄⢰⣿⣀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠠⠤⠴⠛⠁⣸⠁⠈⠻⢯⠙⠛⠛⠛⠓⠀⠂⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠰⠀⠀⠄⠙⠛⠒⢠⣿⡧⠀⠀⠀⣀⣠⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠶⠿⠏⠀⠠⠤⠞⠉⠸⠃⠈⠑⠒


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⢠⡀⠀⠀⣄⣀⢰⢂⣠⢠⠂⡄⢀⣢⡆⣤⠀⡄⣸⢀⣀⣄⢠⡄⣠⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁