Bunny rabbit Emojis & Symbols

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠸⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⡌⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠛⠁⠀⠀⠈⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋⠁⠀⠀⠀⠀⣸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢻⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠈⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠘⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣾⡿⡇⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⢹⣿⣿⣿⡇⠀⠀⠀⠀⣸⣿⣿⣿⣾⣧⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⢸⣿⣿⣿⣇⢀⣄⢀⣠⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⢻⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⢿⣿⠿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣞⣿⣿⣿⣿⣿⣷⣮⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⢸⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠿⠿⠟⠁⠀⠀⠀⠉⠉⠛⠋⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠈⢿⣿⣿⡄⠀⠀⠀⣠⣴⠟⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⢸⣿⡏⠃⠀⠀⣴⣿⠇⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⣿⡇⣷⠀⣸⣿⣿⠘⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⣾⣿⣤⡿⢿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣟⠶⠒⣾⣷⡮⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣸⠎⢩⡇⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣇⣨⣴⣾⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⢯⡇⠁⢹⡆
⠀⠀⠀⠀⠀⠠⡬⠁⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣧⣤⣾⠁
⠀⠀⠀⠀⠀⠀⢹⣾⣿⣿⣿⣿⣿⣿⣿⣿⣽⡿⢿⣿⣿⣿⣿⣿⣿⣿⠅
⠀⠀⠀⠀⠀⢠⣾⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⡟⠟⠛⠛⠛⢁
⠀⠀⠀⠈⢻⣿⡿⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⠁⠁⠀⠀⠀⠀⠀⠈⠀⢀⠁
⠀⠀⠀⠀⢨⣿⣿⠀⠀⠀⣸⣿⣿⣿⣿⣿⣾⠟⣧⠂⡀⣀⡀⡀⠀⠀⠀⠟
⠀⠀⠀⠐⠨⣿⣿⣟⣂⣼⣿⣿⣿⣿⣿⣿⣿⣾⣵⣿⣷⣿⣿⣿⣿⠗⢿⡀⢀⡠
⠀⠀⠀⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠙⢿⣿⣿⣟⣁⠀⣿⣿⣿⣿⣶⢿⠂
⠀⠀⠀⠀⠉⠙⠿⣿⣧⣈⣋⣿⣿⣿⣿⣿⣿⣿⣆⠀⡘⢿⣿⣿⣿⣤⣻⣿⣿⣿⣯⣶⠿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠿⠟⠛⠉⠛⠛⠉⠀⠩⠙⠏⠛⠛⠛⠋⠋⠙⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡄⠀⢀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠙⣿⣄⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠘⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆⠀⣸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣤⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⡀⠀⢀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝⠈⠿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⠟⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠐⢶⡶⠀⠆⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢀⠀⣸⠿⠿⠻⡇⢀⣠⣆⣀⣨⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠳⡄⠀⠀⠠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣤⠀⠀⠐⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⢱⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡛⠀⢿⣯⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡟⢻⢶⢮⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣮⣲⣾⣏⡀
⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⣀⠀⠃⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⡋⠁⠐⣶
⠀⠀⠠⠀⠐⠀⠁⠀⠀⠀⢰⠀⠀⠀⠋⠚⠋⣠⣤⣤⣶⣶⣶⣾⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠟⠋
⠀⠀⠀⠀⠂⠂⢁⠀⠘⠀⠀⢀⢂⣀⣠⣠⣾⣿⣿⣿⣿⠿⡿⢍⠙⢷⡤⣤⣬⣝⠻⢿⣿⠁⠘⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠄⠀⠀⠠⠘⠀⠚⠀⡰⠓⠛⠋⣸⣿⣿⣿⣿⣯⠧⠾⢷⣖⠀⠀⢳⣼⣿⣿⣷⠈⡇⠀⠀⡀⡤⠀⠀⠀⠀⠍⠀⠠⠀⠠⠴⠶
⠀⠀⠀⠀⠄⠐⠀⢧⠀⠀⡇⠀⠀⢸⣿⣿⣿⣿⣷⣿⣧⡀⣼⠋⠀⠀⡸⣀⢡⣿⡟⢰⠁⠀⡔⠘⡤⠋⠀⠠⠤⠄⢀⠂⠙⠒⠒
⠀⠀⠀⠀⡀⠀⠃⠈⢂⠀⢣⠀⠀⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣇⣴⡎⠜⣿⣿⡼⠁⡏⡠⠊⢀⠀⢠⠴⠒⠁⠀⢠⠈⠀⠐⠂⠴⠒
⠀⠀⠀⠄⠀⠑⢄⡀⠀⡆⠈⠆⢸⣿⣿⡿⠟⢁⣠⣶⣦⡀⠙⣿⢟⡀⢉⠈⠉⠀⣼⠟⠁⡐⠃⠀⣡⠖⣁⡠⠒⠁⠀⠐⠁⢀
⠀⠀⠀⠀⠀⡀⠀⠤⠔⣣⣤⡀⣾⣿⢟⣴⣾⣿⣿⣿⣿⣿⠀⣿⣿⣾⡃⠀⠀⢸⣿⠀⠀⣁⠈⠁⠔⠋⠁⠀⠉⠀⠀⠀⠈⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣿⣷⣿⣿⣿⣿⣷⠟⢻⣿⣿⡿⢰⣿⣿⣿⡟⠀⠀⣿⡿⠀⠐⠁⠀⠂⣀⣀⠢⢀⠀⢀⣄⢠⢀
⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⣽⣿⡿⠿⠟⠁⠀⠄⠈⡺⠿⢡⣿⣿⣿⠏⢹⠀⠀⣿⠇⠀⠀⠄⠀⢲⡀⢤⡄⠀⠑⠠⠀⠁⠠⠄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⡀⠔⠃⢄⠀⠉⠀⣼⣿⡿⢁⠀⠘⡄⠀⢹⠠⠀⠀⠀⠀⠀⠙⠂⠉⠐⠂⠀⠑⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣉⡏⣄⢳⣷⣾⣿⣿⠀⠀⣰⣿⣿⡾⠟⠛⢋⣻⠀⠀⢢⠀⠀⠠⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⡀⣀⣠⣼⣶⣿⣿⣿⣿⣿⣿⣿⠋⠈⠀⠛⠛⠛⠛⠛⠋⠉⠈⠣⣀⠀⢀⡈⣶⣿⣿⣿⣾⣿⣷⡠⢠⣄⡀
⠀⠀⠀⠀⠀⢿⣿⢿⣿⢿⠟⠿⢿⣿⣿⣿⣇⣀⣀⣤⡤⠤⠤⠤⠤⠴⠦⠤⠶⠛⠛⠈⠁⠙⠋⠋⠉⠋⠀⠀⠀⠈⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣄⡀⠀⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠈⠉⠙⢦⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣂⠀⠀⠀⠀⠙⢷⣿⣧⣠⡰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⢿⣿⣿⣿⣽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⣿⣷⣻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢶⣤⣼⣿⠿⠟⠉⠛⠿⣦⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⠀⣴⣩⣟⢦⢈⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠂⠒⠒⠒⠤⢤⣀⣀⣤⣄⣿⣿⣿⠃⠀⠙⠻⠟⠻⠘⣿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠉⡿⢿⣿⣛⡓⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠶⣆⣁⠀⡀⠀⣰⡦⠔⢀⠌⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⡿⠛⠒⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⢔⡮⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠈⢻⣞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⡁⠢⣀⣀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣿⡞
⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣟⡅⠀⠀⠉⣶⣥⠃⠀⠀⢀⡴⠃⠀⠀⠀⠀⣰⣾⡄⠀⠀⠟⡡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣀⠀⠀⢹⢷⣶⡞⢺⠋⠀⠀⠀⠀⢀⣼⣭⣐⣟⣡⣴⣾⡇
⠀⠀⠀⠀⠀⢠⢴⡀⠀⠻⣿⡿⣡⠀⢀⣼⣾⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠙⠒⣁⣠⠵⠛⠀⠴⠾⠛⠛⢿⣿⣿⣷⡀⠀⠀⠠⣿⡿⠿⠋⠙⢿⣿⡿⠣
⠀⠀⠀⠀⠀⢅⠀⠃⠀⠄⠀⡀⢀⣀⡀⢰⡄⠦⠛⠿⠛⠃⠀⠀⠘⡄⢆⠀⠀⠀⠀⠉⡁⠀⡀⡴⣦⡀⠀⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠉⠁⠈⠈⠉⠁⠠⠠⠀⠁⠊⠑⠊⠀⠀⠀⣀



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠉⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡘⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢸⠙⡄⠀⠀⠀⠀⠀⠀⠀⢠⠇⡇⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠘⡀⢳⠀⠀⠀⠀⠀⠀⢀⡞⢰⠃⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡇⠘⡄⠀⠀⠀⠀⠀⡼⠁⣸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢳⠀⢳⠀⠀⠀⠀⢠⠇⠀⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡀⠀⠀⠀⠀⠘⡇⢸⡀⠀⠀⠀⣸⠀⠀⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡇⠀⠀⠀⠀⠀⢻⢸⡇⠀⠀⠀⣟⠀⠀⣇⠀⠀⠀⠀⡔⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⠀⠀⠀⠀⠀⢸⠨⡇⠀⠀⠀⡟⠀⠀⢿⠀⠀⠀⢰⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣃⠀⠀⠀⠀⢸⠄⡇⠀⠀⠀⣿⡀⠂⢸⠀⠀⢠⠃⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⡀⣀⡀⢸⣆⣇⠀⠀⠀⣿⢧⣆⣼⢠⣦⢟⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣵⣮⣻⣻⣉⡉⠘⠳⠾⣏⣻⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⢛⠩⡑⠙⠫⠄⠀⠉⠁⠑⣌⠹⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⡜⠁⠀⠀⢡⠀⠀⠀⠀⠀⠀⠆⣱⣄⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠘⠛⠋⠁⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⣿⣿⠣⡀⠀⠀⠠⠀⠀⠀⠀⠀⡼⣿⣿⢹⡸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⠻⣿⣿⠔⠉⠀⠀⠀⠶⠀⠀⠀⠀⠑⠿⢿⠿⠁⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠓⠦⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡀⠀⠀⢀⡀⠀⠘⡁⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⡰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠾⠔⠀⠀⡬⢀⡀⠘⢿⣕⡒⣨⡴⠂⠀⠀⢱⠀⠀⢀⣈⣁⣦⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⢷⣶⣶⣐⣣⢷⠤⠤⣀⠀⠈⠹⠉⠀⠀⡀⡀⣼⡙⣃⠀⡐⣶⡗⠀⠤⠀⠤⠄⢀⡈⠉⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠤⠀⠒⠈⠻⣿⣷⣿⣿⣆⣀⠀⢀⡤⠚⠲⢤⣀⣠⠄⣠⢤⣀⣆⣥⠟⠉⠀⠂⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⢷⣍⣁⣀⣀⢀⣀⣠⣤⣾⠿⠷⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠞⠁⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢻⣇⣤⠀⠀⠀⠀⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⢿⠷⣤⣴⡞⢿⣿⣿⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⡿⢁⠀⣼⣿⣄⢋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⡿⢿⣿⣟⣿⠃⠁⣀⠙⣿⣿⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠛⠙⠉⠁⠈⢹⣿⡏⢀⣾⣿⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠸⣿⣿⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡄⠀⠀⠀⠀⠺⣿⠁⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠘⣝⡈⠋⣾⣷⡆⠀⠀⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢶⡈⠻⠟⣁⣼⣿⡟⠶⢤⣤⣤⡁⠀⠀⠰⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡸⢿⣿⣿⡿⠿⠓⠀⠀⠀⠺⡿⠟⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠍⠥⠀⠀⠂⠀⢀⣀⣀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⠲⣆⣽⣿⣿⣿⣿⣧⠠⠈⠁⠀⠐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡇⠀⠹⠛⢛⣙⣽⣧⣿⡽⣻⠿⠟⠂⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠟⠿⠻⣿⠛⠏⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠈⣿⣮⢻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠃⠀⠀⠀⠀⠘⢻⡏⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⣄

⣤⣄⣀⡀
⣽⡻⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣤⣶⣲⣆⣤⣦⣤⣄⣀⣀
⠻⢟⡿⣿⣏⣻⣿⣿⣿⣿⣶⣦⣤⣀⣀⠀⣀⣠⣤⣶⣞⣛⡻⢯⡟⣽⣷⣷⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⡀
⣿⢲⣿⣌⣿⡇⠡⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣫⣿⣟⣿⣿⣟⣿⣷⠿⠿⠿⠿⣿⣿⣿⣿⣦⣄⡀
⣿⣿⣿⣾⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡿⢿⣿⣿⣿⢛⣭⣌⠛⡋⢠⣤⣤⣶⣦⣤⣤⣈⣿⣿⣿⣿⣿⣦⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠷⣬⣭⠉⠅⠓⠴⠶⣬⣀⣿⡿⣿⣿⣿⣿⠻⣿⣿⣿⣷⠙⡿⣿⣿⣿⣿⣶⣤⡀
⣦⣬⣝⣉⣻⣟⣷⡛⢩⣛⣹⣿⣿⣿⣿⣿⣿⡧⣚⡛⠯⢤⣵⡾⠅⣶⣚⣞⡛⠙⢛⣛⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣁⠡⠞⢿⣿⣿⣿⣿⣿⣦⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣵⠿⣽⢏⣿⣿⣿⢫⣿⣄⠙⠩⢥⢏⡝⡫⢂⡿⣿⡦⣾⣶⡌⠻⣿⣮⣝⠿⠿⣭⠾⢋⡉⠙⣣⣤⣦⣿⣿⣿⣿⣿⣿⣿⣻⣦⣄
⠿⠙⠛⣉⣷⣪⣭⡽⣛⣘⣃⣽⣿⣾⣿⣿⣶⣰⣾⣿⡵⣲⣿⡿⣿⣿⡿⢿⣿⣿⣿⠠⣈⠛⠛⢿⡖⣖⠉⠀⣠⡆⣦⣶⣿⣿⣽⣻⣟⣿⣿⣿⣿⣿⣿⣷⡄
⠞⠛⠛⠉⠉⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⠶⣿⣶⣿⣿⣦⣿⣯⣿⣿⣿⣿⣾⣷⡶⣫⣖⣥⣶⣷⣮⣾⣿⣾⣿⣿⣿⣶⣷⣞⡚⠻⠥⣾⣿⣉⣴⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣟⡟⣻⣫⢿⠿⢺⣿⣿⣿⣽⠿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⡻⠀⡜⠈⠙⢿⣿⡿⠁
⣀⣀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⢿⣏⣻⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢡⡞⢀⠀⣢⠀⠀⠀⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣲⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡻⠳⠶⠟⠛⠱⢱⣿⣟⣿⣫⠟⣽⠏⢀⠃⢠⠃⡌⠀⠵⠱
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣻⣭⡽⡟⠛⢿⣿⢿⣿⣵⠿⠁⠀⣀⣤⠄⣰⡖⣻⣃⣿⣿⣿⢦⣿⢢⡟⢶⢃⣮⣵⠗
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⡟⢯⢍⣶⡋⠐⠀⠀⢀⠈⠙⠻⡖⠋⣙⣃⢪⠏⠙⠃⠻⢋⡿⣿⣿⣿⣿⣿⡿⣿⣿⠇⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣭⣤⢶⣆⣄⡀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡼⠋⢹⠋⢉⠋⡙⠘⠀⠡
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣸⢿⣿⣿⣾⣧⣉⣹⡎⣀⡊⣀⣀⠀⣀⠀⣀⣤⣤⣺⡿⠃⡆⠀⡌⠀⠸⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣏⣳⣿⣿⣿⣿⣿⣽⣿⣯⣟⣵⢫⣿⡿⢷⣿⣿⣿⡟⠿⠁⠘⠀⢠⠁⠀⠇⠀⠀⠀⠆
⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⢆⠙⡿⣽⣿⣿⢿⡽⣯⡟⣀⡼⠟⠳⡿⣏⠙⡟⠠⠐⠀⠇⠀⡄
⣻⣿⣷⣟⣿⣯⣟⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣛⠴⠄⣮⠻⢹⣯⠙⠣⣌⣈⣭⡄⠰⠊⠀⠘⣤⡗
⣿⣿⣿⣞⠟⣻⠿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢦⣜⣷⣶⣦⣛⡄⠀⢄⠈⠒⠀⠀⠰⣦⢰⡶⠀⠀⠘⠀⠀⠀⢀⠀⠀⠐
⣶⣞⣿⣿⡾⣷⣾⣿⡷⣽⣿⢿⣽⣿⣿⣿⣿⣿⣿⣽⣿⣿⡞⣹⣿⠷⢀⠀⠀⠵⠖⠀⠠⣀⣥⣿⠁
⣿⣿⣟⣿⡿⠷⢿⣭⡿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷⡄⠬⢧⠈⠀⠀⠀⠀⠐⠠⡀⢿⡛
⣿⣿⣛⡿⢿⣿⣽⢻⢿⣿⣿⣷⣿⣿⣿⡿⣯⣿⣿⣿⣿⣿⣿⣧⣈⡲⡀⠀⠀⠀⠀⢀⣮⣭⣾⠁
⣿⣿⣿⡿⠠⣽⣍⣺⡹⣃⣊⡍⣚⢙⠿⠹⢦⡛⡻⢿⣿⣿⣷⣍⠛⢷⠍⢶⠀⠀⠀⣨⣽⡟⠁
⣿⣿⣿⣍⣝⡃⣩⠿⢳⡤⣤⣤⣀⣤⠑⢄⣼⣥⠙⠾⣿⣿⣿⣿⣂⠀⡶⡤⠀⠅⣶⣿⣿
⣿⣿⣿⣿⡿⡏⠧⣦⡠⢁⡉⡤⠈⠷⣰⣌⠸⠺⡄⠈⣾⣿⣿⣿⠷⣷⠦⠁⠀⢀⣿⣿⠁
⣿⣿⣿⣿⢸⠧⣗⣹⡟⣪⡇⡇⠠⣾⣤⡿⡄⢀⠁⠀⢹⣿⣿⣿⠰⠀⢰⣀⣾⣿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⡄⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠄⠁⠀⢀⠠⠒⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠂⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠐⠀⠆⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐
⠀⠀⠀⠈⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⢰⣶⠆⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠑⠄⠀⢀
⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⠀⠀⠀⠀⠁⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣈⣑⣀⣀⣀⣀⣀⣀⣀⣀⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡈⡿⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⢹⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣾⡿⢿⣿⣿⣯⡍⠉⠢⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠈⠋⠁⠉⠀⠀⠀⢹⣿⣷⣦⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⢠⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⢤⡤⠄⠀⢀⠛⠻⠍⠀⠉⠛⠻⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⣤⡠⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣤⣀⠀⣸⣄⠀⡀⠀⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣯⣽⣿⣿⣿⣿⣯⣾⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⠍⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡏⠀⠋⢀⢩⡄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣶⣾⢀⠁⠀⠀⠀⠒⠷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣾⣆⡠⢠⡶⠀⠀⠀⠀⠀⠀⠐⠀⠤
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣾⣿⣿⣿⣿⣿⣿⠋⠀⠀⠹⣿⣿⣶⣾⠀⠀⠀⠀⠀⢠⣧⣤⣄⡀⠂
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣽⣿⣧⠀⠀⠀⢻⣿⣿⣟⠀⠀⠀⠻⣶⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢹⣿⣿⣧⣀⡀⠀⡸⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠚⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠉⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡠⠷⠖⠒⠒⠒⠒⠒⠲⠼⢅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⡼⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠸
⡇⠀⠀⠀⠀⠀⠀⢀⡴⡇⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡇⠙⢄
⠹⢄⣀⣀⣀⡠⠔⠋⠀⡇⠀⠀⠀⠀⣼⣏⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣹⣦⠀⠀⠀⠀⢧⠀⠀⠙⠦⣄⣀⣀⣀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠰⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠆⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠉⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢁⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢢⠀⠀⠀⡔⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠈⡆⠀⢠⠃⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢣⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⡼⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⡤⠤⠤⡑⠤⣀⠀⣠⠃⠀⠘⣄⠀⣀⡠⢊⡤⠤⢤⡀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠰⠃⠀⠀⠀⠈⢦⠀⠈⠀⠀⠀⠀⠀⠉⠀⡴⠁⠀⠀⠀⠈⢆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡆⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡇⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢾⣿⡄⠀⠀⠀⠀⢀⣀⣠⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠚⠁⠀⠀⢀⡤⡺⠛⠀⠨⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠢⠁⢂⣼⡍⡞⠀⠀⢀⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠉⠄⠄⠀⣾⣿⢀⣿⣆⣀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠤⣴⣹⣿⡈⢿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣦⠬⣅⠽⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⣛⣋⠔⠋⢬⣔⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢨⢁⢌⣁⣴⠮⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠀⠀⠀⠀⠀⠀⢸⣏⠃⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠑⠀⠀⠀⠀⠀⠀⡠⡿⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠡⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⠀⠀⠀⢀⣠⣾⠛⠓⢀⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠀⠀⠀⢀⣿⡋⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠀⡠⠀⠀⠀⠸⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠺⠖⠂⠂⠀⠻⠦⠴⠤⠆⠀⠀⠀⠀⠀⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣻⡟⡛⢻⡛⣻⢿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⠀⠀⠢⡄⠀⠀⠀⠀⠀⢀⠀⠈⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⠀⠁⠐⡀⠀⠀⢌⡎⢸⢐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⡄⠀⡇⠈⣇⠀⢀⣼⠁⢸⢞⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⢿⠇⡼⠟⠀⠛⠟⢄⠈⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡟⠆⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⣩⠿⠛⢀⣷⠐⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣠⠐⠁⢀⠆⠘⠿⠀⢀⣤⣦⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠁⠀⠀⣬⠀⢀⠆⠀⠤⠁⠀⠋⡀⢬⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠸⠀⡄⠀⠀⠀⠻⣦⣌⠄⠀⠀⠲⠗⠀⠆⣸⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠘⠁⠀⡰⠻⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠛⠙⠋⠁⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠒⠳⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⠀⠀⠀⣃⠀⠀⠀⣼⠿⣿⣿⡀⠀⢸⣿⣤⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠛⣯⡀⠀⣀⢹⠀⠘⠻⠦⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⡄⠀⠀⠀⠀⠀⢀⣴⡶⢶⣄⠀⠀⠀⠀⠀⣠⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠸⠫⠀⠌⠛⠀⠀⠀⠀⠀⣞⠡⠀⠄⢻⡆⠀⠀⠀⢰⡟⠩⠉⠙⢻⠀⠀⠀⠀⠀⠀⠟⡛⠉⠿
⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀⣴⡄⠀⠀⠀⠀⠀⠘⠂⠀⠀⠋⠀⠀⠀⠀⠀⢛⣦⣠⣴⡋⠀⠀⠀⠀⠀⠀⢠⡀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣹⡏⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠐⠞⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠠⣤⠀⠀⠀⠀⠀⠀⠈⢋⡥⠀⠀⠀⠀⠀⠀⠀⢠⡄⠿⠀⠀⠀⠀⠀⠀⠋⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠿⢷⡄⠀⠀⠀⠀⠀⣴⣿⠷⣶⡀⠀⠀⠀⠀⠀⣾⣿⡿⣿⡄⠀⠀⠀⠀⢠⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠈⠁⠀⠀⠀⠀⠀⡍⠀⠐⢈⡅⠀⠀⠀⠀⠀⠉⠀⠐⠈⠁⠀⠀⠀⠀⣟⠡⠀⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠸⣷⣴⣦⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⢰⣶⣴⣦⠀⠀⠀⠀⠀⠙⠂⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠐⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⣴⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠙⠿⠖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠈⠁⠀⠀⠀⠀⠀⠠⠖⠰⠆⠀⠀⠀⠀⠀⠀⠈⠃⠶⠀⠀⠀⠀⠀⠲⠖⠁⠀⠀⠀⠺⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠛⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠈⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⣀⡈⡸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⡛⣵⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⢟⠀⢘⠧⠤⠀⠀⠀⠀⠀⠀⠀⢀⠄⠖⡇⢹⠀⢈⠑⠒⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠠⢰⠈⠀⠀⠀⠀⠀⠀⠄⠀⢲⡧⠬⠈⠁⠀⢰⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡀⠀⠀⠀⠀⣵⠀⠀⠀⠀⠀⠀⡈⡀⠐⢉⠧⠰⠀⠀⠀⠈⠧⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⢠⠁⠀⡠⢃⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠅⠀⣀⡀⠠⡆⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡰⡀⢠⠀⡀⣘⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢈⠿⣁⠘⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢠⠀⠀⠀⣰⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⠜⠒⠒⠴⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢰⣿⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⡀⠀⠀⠀⢠⠟⣽⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠐⠀⠀⢧⠂⠀⣿⠁⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠓⠀⠀⠀⠐⠇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡁⠀⠐⠉⠓⠉⠈⡈⠉⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠰⠀⢀⠀⠀⠀⠀⠀⠘⠀⠠⠀⠀⠀⠀⠀⢀⠀⢸⡥⠤⠥⡴⣀⡀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠃⠁⠃⠀⠀⠀⠀⠀⠸⣇⠀⠆⠀⠀⠆⡉⠉⠉⠉⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⢸⠀⠀⠈⠉⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠘⢩⠁⠀⠀⠀⠀⠀⢸⡾⠃⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⠀⠀⢈⣷⠋⠁⠀⠀⠀⠑⠄⠀⠀⠀⡍⢶⡇
⠀⠀⠀⠀⠀⠔⠋⢡⠀⠀⡼⢹⠑⠠⠆⡱⠀⣠⣛⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⣆⢀⡜⠛⢀⡀⠀⠀⠀⠀⠈⠢⠀⠀⡁⠘⠙⣄
⠀⠀⢀⠔⠁⠀⠁⠁⠀⠀⠈⠈⠀⠈⠐⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠈⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠑⠀⠉⠁⠀⠉
⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠆⠶⠤⠀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠍⠋⢁⠀⠀⢠⡄⠀⠈⠉⠉⠱⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠈⠀⢀⠈⠀⡀⠀⠚⠐⠗⠀⠋⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡛⠃⠀⠀⠁⠛⠲⠀⡈⠠⠄⠁⡀⠠⡄⠀⢀⡀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡇⠀⠀⠠⠄⢀⠰⢀⣆⡦⡈⠅⠂⢠⠀⠀⠀⠁⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠈⠁⢀⡀⠈⠰⠠⢙⠿⣗⡥⢀⠂⠀⡐⠀⠀⠔⠀⠼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡄⠀⠀⢀⠀⠀⠈⠘⠀⠀⠀⠹⠗⠔⠂⠁⡀⠐⠂⣀⡀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⠁⠀⠉⠀⠀⠀⢃⠩⠈⡀⠡⠉⡈⠠⡄⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣦⠀⠀⠠⠀⠐⠃⠀⠈⣀⠐⠀⠘⠀⢣⡀⣀⢀⡀⠀⠹⡡⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⣄⠀⠀⠀⢀⠰⠆⠉⠀⠀⠈⠀⡀⠀⠀⠀⠀⣠⠞⠡
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠋⠙⠷⣤⣠⡌⢀⠀⠀⠘⠂⠀⠀⠉⠀⢀⡨⠚⡉⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠃⠁⠀⠀⡧⠀⠀⠈⠉⠳⠭⠦⣴⣶⣤⣤⣶⠶⠚⠋⢀⠀⠅⠀⠀⠀⠀⡼⠇
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣀⣀⠡⠀⠆⠀⠐⠀⠀⠀⣠⠀⠂⠀⠑⠀⠀⢸⠷⠄⢀⠀⠠
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣈⣤⠐⠶⠄⠀⠀⠀⠀⠀⡚⠀⠀⠀⢬⠀⠀⠀⠀⠀⠛⢭⡄
⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⡟⡁⠀⠀⠀⠀⠀⠐⢃⣠⢁⠐⠆⠀⠀⠄⠀⠁⠀⠀⠀⢀⠁
⠀⠀⠀⠀⡠⠞⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⢸⡟⠃⠿⣶⡄⠀⡰⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠁⠀⠀⠁⠘⠁⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠂⠀⠤⠀⠀⠸⡀⢰⠑⡀⠀⠀⠀⠹⠀⠀⠁⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠖⠀⠄⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⡀⠀⢀⡀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢤⣼⣧⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⣾⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡃
⠀⠀⠀⠀⠀⠀⠘⣽⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠛⡵⠃
⠀⠀⠀⠀⠀⠀⢀⡟⢿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣋⡉⠉⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠱⡏⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡆⠀⢠⡟⠿⠂⠀⠀⠀⠀⠀⠀⢸⠃⢻⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠀⢧⠘⣆⣀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠚⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⡦⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢇⡀
⠀⠀⠀⠀⠀⠠⣤⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣇⢹⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⢿⣿⡀
⠀⠀⠀⠀⠀⠀⢹⣾⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢷⡀⠈⣿⣷
⠀⠀⠀⠀⠀⢀⣼⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⡀⠀⠀⠀⠀⠀⠀⢰⡋⠀⣰⡋⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠙⣦⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠑⢾⣷⠄⠀⠀⠀⠀⠀⠀⠀⠘⠻⡄
⠀⠀⠀⠀⠀⠀⣀⣼⣿⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣿⠉⠁⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣿⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠋⢛⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠛⠂⠀⠀⠀⠀⠀⠀⠘⠟⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠙⠿⠟⠃⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠄

⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣆
⠀⠀⠀⠀⠙⢣⣁⣸⣇⣹⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠾⢿⡿⠷⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠉⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⠏⡇
⠀⠀⠀⠀⠀⠀⠀⡏⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⢷⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠒

⠀⠀⠀⠀⠀⢰⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡄
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⡀⠀⠀⡄⠐⣆⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢿⣷⣴⠀⠀⣼⣷⣶⣶⣯⣀⣄⣴⣦⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢦⡖⠋⠀⠀⠀⢈⡁⢻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠠⡎⠀⠀⠀⠀⣵⠄⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠘⣷⠀⠀⠀⠀⠀⠃⣿⠀⢇⠀⠀⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠉⠀⠈⠀⠀⠃⠀⠉⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⡰⠀⠀⠀⢼⡧⠀⠀⠀⡄⠀⣿⡂⠀⠀⠀⠀⠀⠀⠈⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣟⣽⡄⢂⠀⢀⡐⣼⡷⣂⡀⢀⠃⣰⣿⣷⠀⠀⠀⠖⠀⠀⠀⠘⡉⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠝⡶⢆⣀⡀⠉⠛⠛⠉⢀⣄⡴⣿⣿⡿⠀⠀⠀⠆⠀⠀⠀⠀⢣⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣳⡇⠀⠀⠀⠣⠀⠀⠀⠀⢸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣿⡃⠀⢀⣰⣴⡃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⣷⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⠁⠀⣾⣿⣷⡧⠄⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⠿⣄⡀⠀⣀⣀⣠⠶⠟⢋⣶⣩⣧⣾⣿⡯⠬⠁⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣀⠀⣼⡿⢻⠏⠉⠁⠀⣰⡿⠿⠿⠿⢏⠏⣰⠂⠰⢆⣠⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠮⠴⠤⠴⠋⠀⠦⠴⠤⠼⠴⠋⠀⠀⠀⠀⠛⠒⠛⠓⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣦⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠉⠁⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⠀⣼⣿⣿⣿⣿⡿⠛⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠒⢸⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⣿⣿⢻⡿⠋⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠘⠊⠀⠈⠙⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢶⣶⣿⣿⠁⠀⠀⠀⠹⡟⠂⠀⠀⠀⣀⡸⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⡇⠀⠀⠀⠐⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⡙⠁⠀⠀⠀⠀⠀⢀⣴⠅⠂⠐⠶⣄⠀⠀⠋⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠉⠀⠈⣿⡦⢀⠀⡃⠀⠀⠀⠀⢀⡠⠐⠈⠀⠀⠀⠀⢀⠆⠰⠀⠙⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠈⠀⠀⠀⠀⠀⢠⡛⢱⠀⠈⢱⡶⠢⡀⠀⢁⠀⢀⠤⡄⠀⢡⠄⠊⠀⠀⠑⠤⠄⠈⠈⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡤⠀⠂⠈⣥⣴⠗⠀⠀⠓⢀⡸⠃⠁⠠⠀⠀⠀⠀⠆⠐⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠰⠀⠀⠀⠀⢰⠊⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢃⠀⠀⠀⠏⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⠀⠡⡀⠀⢠⠊⠀⠑⠄⠀⠀⠀⠀⠀⠀⡰⠀⠃⠀⠀⢀⠜⢢⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠸⠀⠀⠀⢰⢱⠀⠀⠀⠀⠀⠀⠀⢀⠆⠘⠀⠀⠱⣀⡬⢠⣤⠊⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⣀⠔⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⡆⠀⠀⣼⡎⠀⠀⠀⠀⠀⠀⠔⡁⠀⠀⠃⢀⡴⠋⢠⡎⠈⠁⠒⠀⠄⠀⡤⠁⠀⠁⠀⠀⠁⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⠰⠙⠂⠤⠀⣀⡀⢀⠌⠀⠐⡀⠀⠈⠀⢀⠴⢤⣿⣀⡀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢘⡂⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠈⠢⣀⠔⠁⠰⠋⠀⠀⠈⠈⠓⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠟⡒⠂⠠⡤⠄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡌⠀⠀⠁⠀⠀⠈⠒⠣⠀⢀⠀⠀⠀⠀⠀⠀⢀⠂⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠐⡀⠀⢅⡂⠤⢀⠀⠀⠀⠈⠁⠁⠂⠄⣀⢀⠎⠀⠀⠀⢀⠎⠈⠉⠀⠒⠒⠒⠒⠂⠄⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡰⡙⠃⠂⢌⣀⡀⠀⠁⠀⢧⠀⠀⠀⠀⠀⠉⠂⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⠕⠀⠀⡠⠃⠀⠀⠀⠉⠉⠉⠉⠁⡐⠀⠀⠀⡆⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠛⠤⣀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠂
⠀⠀⠀⠀⠀⢀⠊⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⢀⠔⠃⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠷⠺⡀
⠀⠀⠀⠀⢀⠂⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠹⠶⠁⢠⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⣀⠀⠈⢄
⠀⠀⠀⢀⠂⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡠⡀⠀⣠⢾⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠭⠂
⠀⢀⣔⣇⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢦⡍⠀⠀⠈⢄
⠴⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠫⢐⡤⣀⠑⢄
⠈⠃⢑⡄⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋
⣀⣀⣀⣙⣲⣌⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⠀⢀⣀⡀
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⡀
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠿⠿⠿⠿⠿⠟⠁⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣤⣀⣬⣶⣿⣷⣄
⣸⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠇⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⢻⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠈⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠸⣿⣿⣿⣿⣿⣿⣦⣄⠀⠸⠀⠀⢴⡦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠀⣀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠈⠉⠻⣿⡏⠉⠁⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡏⠘⣿⣿⣿⠿⠃⠐⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠘⠋⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠋⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣀⣀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠀⣹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣦⣤⣾⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣠⣴⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡒⢢⣤⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⣠⣄⡤⣤⣠⣤⣤⣠⣤⣤⣄⣄⣠⣤⣄⡤⣤⣠⣤⣤⣤⢤⣄⣤⣀⣤⣄⣄⣀⣀⣀⣀⣀⣠⣀⣀⣤⣤⡄⠀⢀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⣹⢿⡷⣷⢯⣿⣿⠷⢿⣿⡽⣿⣿⣯⣿⢶⣿⠭⢧⡿⣿⣿⢿⣿⠾⣻⣮⡿⣻⡛⢽⡿⣿⣿⣾⣷⣿⣞⣼⣿⣿⣿⣷⣄⡠
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡷⡴⣿⣿⣿⡿⡿⢳⢛⢛⣟⡛⣿⣿⣭⡿⢿⣹⣿⣹⣾⢳⣾⡖⣧⠻⢿⣷⣿⡿⢶⣿⡻⢝⠿⠋⢽⣦⡤⣿⣿⣟⣻⡟⠃
⠀⠀⠀⠀⠀⠀⠀⣤⣌⣷⣶⣟⣯⣿⣿⣣⡏⣼⣷⣼⣵⡿⣾⣿⣿⣿⡽⢻⠗⣿⢼⣿⡿⣷⣿⣨⠟⣁⡘⣿⣷⣤⣾⣩⣴⡂⣼⠻⣻⣽⡿⢟
⠀⠀⠀⠀⠀⠀⠀⣃⡽⡮⢁⣹⣻⣟⣿⣿⣀⣿⣿⣿⡿⢷⣚⠊⣽⡿⣿⣿⠷⢸⣿⣿⣿⢟⣭⡟⣡⣜⣝⠻⣽⣗⡸⣻⢾⡟⣻⣗⣿⡿⢷⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⣿⡛⢿⣿⠌⢫⣿⠗⢙⣟⣿⣯⠹⠓⢈⣥⠞⣛⡟⣿⣿⣁⢘⣻⣭⣴⣛⡯⣽⡻⣶⠭⣠⠶⢡⣤⠞⣻⣧⣿⡞⣷⣞⡷⢞⣉⡃
⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⡿⢡⣾⡛⣾⣿⡻⠙⢀⣤⣶⣿⣶⣶⣿⣧⣿⣿⣿⣿⣻⣏⣿⣿⣿⣐⣷⣿⣿⣾⣦⡾⣧⡴⣟⣿⢻⣅⣿⡟⣃⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠻⡎⢿⠈⣿⠻⣿⣿⠏⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡿⣶⡿⡿⣿⡬⣵⣲⣿⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢩⡄⠀⢹⡧⠙⣯⣼⣿⣿⣿⢥⣄⣷⣸⣿⣿⣿⣿⣿⣿⣯⢸⣿⣸⣿⣿⣿⣿⣿⡿⢿⢿⣿⣷⣉⣿⡾⢡⢶⣷⣿⠻⠄
⠀⠀⠀⠀⠀⠀⠀⢂⠀⡄⠀⢸⡇⠸⠇⣿⣿⣿⣿⠀⠀⠈⠉⣿⣿⣿⣿⣿⡏⠁⠉⠀⣽⡆⣿⣿⣿⣿⣿⣧⣜⣿⣿⣿⣿⣿⣾⠡⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣯⡐⢣⠀⠀⢁⠀⠰⣿⣿⣿⡿⢦⡐⠀⠴⣿⣿⣿⣿⣿⣷⣄⠀⠀⠺⣷⡿⢿⣿⡿⠿⣿⣿⣿⣽⣿⡿⢟⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠘⠐⠚⠀⠀⠀⠁⢸⠿⠛⠁⠀⠀⠀⠣⠄⠀⠈⢻⣿⡿⠛⠍⠛⣶⠤⠀⠀⠈⠿⠀⢀⣿⡿⢁⣭⣶⣷⠉⠁⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⣤⠤⢀⡒⠀⠄⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡆⠀⠀⠀⠀⠀⣨⠄⠀⢠⣀⢻⡇⣼⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠉⠂⠹⣾⠿⡡⢤⣄⡀⠀⣀⡀⢀⡄⠘⣿⣿⡏⠀⠀⠀⠀⠀⢻⠀⠀⢘⣿⣿⣧⢹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡷⢼⣿⠋⠛⠉⠉⠉⠉⠁⢀⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⡆⢁⣀⣄⠀⢄⡈⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠄⠀⢀⣧⠀⠀⢰⣈⣰⡶⢀⣀⣄⡀⢀⠀⡀⣃⡀⠶⠛⠛⣿⡇⢸⢿⠭⠀⠦⡱⠂⠿⠂⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⠀⠀⣿⣿⣿⣶⡶⠀⠸⣷⠀⠀⢿⣿⣷⣄⠀⠈⠀⠠⡏⠀⠀⠀⠁⠙⡧⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠀⠀⣀⠀⠁⠀⠈⠉⠙⢷⣳⠦⠀⠘⢻⣄⣠⣀⣰⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠱⠄⠘⠻⣯⣿⣿⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⢠⠀⠀⢸⣯⡟⢆⠸⡏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣵⣿⠀⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢸⡟⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⠀⠀⢸⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⢼⣿⣾⣳⠀⣠⠧⣘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠢⡀⠼⢺⣿⢿⣿⣏⣀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⡿⢳⣾⡟⣶⢻⡟⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠉⠿⣷⣾⡿⣿⢿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢀⣼⡿⠃⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠊

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂

⠀⠀⠀⢀⠄
⠀⠀⠀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠉⠕⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣗⣿⢆⢠⣿⣶⣇⡀⠂⡀⢠⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠃
⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣄⡀⢈⠐⢿⣿⣿⣿⣷⡀⠀⡀⢤⢄⠀⢀⣠⣾⣷⣄⠀⠀⠐⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠃⠀⠀⡀⠀⠀⣎⣠⣤⣢⣽⣿⣿⣿⣿⣿⣼⣿⣞⣿⣰⣶⣿⣿⣿⣿⣿⣿⡟⡀⠀⠀⠀⠀⢀⠀⢢⠀⢀⡊
⠀⣠⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡆⠀⠐⠀⠀⠀⢳⣄⣸⣷⣻⣿⢋⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⡄⠀⡀⠀⣆⡘⠂⡆⣷⡾
⠀⠧⡞⠀⠀⠀⠀⠀⠀⢢⠈⢷⣅⠀⣛⢌⢢⣧⣌⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣿⣷⣼⡇⣦⣸⣇⣰⣿⣿⣧⣶
⠀⠸⡧⠀⣄⠰⡀⠑⢠⢸⣾⣼⣿⣿⣿⡿⣿⡿⣿⣯⣿⣿⣿⣿⣿⢿⣿⣿⡟⣿⣾⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡿⣿⣿⣿⢿⣏⡆
⠀⠀⠉⠀⠀⠀⠉⠀⠀⠙⠃⠋⠻⠛⠘⠙⠁⠉⠁⠃⠈⠙⠋⠃⠙⠛⠙⠛⡟⠏⠉⠁⢻⡟⠘⠛⠏⠙⠉⠙⠛⠈⠛⠋⠙⠁⠉⠉⠙⠁⠁⠈⠛⠈⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣀⣀⡀⡀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢶⡄⠀⢿⡴⠛⠡⠟⠩⠿⠶⠚⠁⠀⣀⠈⢦⡀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢸⠈⡆⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣿⢳⡀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢻⡄⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠹⣆⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠹⣆⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⠶⢤⣀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⣻⡟⣦⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠿⢦⣄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢠⣿⡌⢷⡀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣦⣄⠀⠀⠀⠀⠀⠀⠀⠹⠶⠞⠁⠀⠀⠀⠀⠀⣀⠀⠀⠙⠓⠛⠀⠀⠀⢀⣴⣇⠀⠹⢦⡀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣄⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⣀⡽⠀⠀⠀⠀⠀⠀⠈⠁⢀⡼⠁⠀⠈⠛⢦⣄⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡏⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡤⠖⠊⠉⠉⠉⠒⢦⣼⠁⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡆⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢠⠇⠀⣰⠀⠀⠀⣅
⠀⠀⠀⢠⡶⠶⠶⣤⡀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⡸⢇⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⡼⠀⢀⣿⡄⠀⠀⢿
⠀⠀⠀⣿⠀⠀⠀⠀⠙⠷⣄⢰⠇⠀⠀⠀⠀⠀⠀⠀⠇⠘⣧⠀⠀⠈⣇⠀⠀⠀⠀⢸⠇⢀⡾⠀⡇⠀⠀⢸⡇
⠀⢀⣀⣽⡄⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⢷⡀⠀⢻⠀⠀⠀⠀⣼⠀⡞⠀⠀⢱⠀⢀⣸⡇
⡼⠋⠀⠀⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⠶⠟⠀⠀⠀⠀⠈⠋⠀⠀⠀⢼⠀⠀⢻⡇
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⢷
⠈⢷⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢸⡆
⠀⠀⣼⠋⠉⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣼⠁
⠀⠐⢧⣀⣀⣀⡄⠀⠀⠀⠀⣠⢿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢⣤⣤⣤⣀⡀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢠⣾⣁
⠀⠀⠀⠉⠉⠙⢧⣀⣀⡴⠞⠁⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⠠⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠾⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠛⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠹⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣴⠛⠚⠿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣆⠀⢻⡅⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠈⠀⠈⡇⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⡀⢸⡀⠁⠀⠀⡼⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠱⢸⡇⡀⠀⠀⣥⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⢼⠀⡁⠈⢿⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⣀⡀⠹⣆⠀⠩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡰⢻⣷⣦⣠⡌⢿⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣴⡇⠿⣿⣟⠘⢓⣾⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠱⠟⠋⠛⠻⣏⠁⠀⠉⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣠⣠⠶⠀⠀⢰⡤⠿⠶⠤⢤⠤⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢾⣿⣯⣾⣿⠃⢘⣠⡶⣄⢀⠤⠖⠋⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠩⣥⣀⡛⠛⠋⠁⠀⢠⠀⠀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠋⣘⠯⠁⡰⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢻⠀⠈⠙⠇⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⢄⠀⠂⠠⠄⠘⠂⠲⠀⠀⢀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⣸⣇⢈⠁⠀⠀⠰⡆⠀⠀⠀⠂⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢻⣿⡈⠀⠀⠁⠠⣿⠀⠀⢀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡼⣻⢷⡄⢿⡁⠀⠠⠀⠀⣿⣿⣁⠀⠀⠀⠘⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠈⣥⠋⠀⢷⡈⢷⠔⠀⠀⠀⣿⣿⡟⠉⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⢈⣶⡀⠀⠁⠘⣤⠴⠂⠀⣿⡿⢳⢠⡜⠁⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠹⢻⡄⠀⠀⢹⡁⠀⠀⡋⠀⠀⡇⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠉⣀⣤⠌⢀⡀⠀⠁⠀⠀⠀⠀⠀⢏⣄⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡵⣶⠇⡠⡴⠞⠑⣰⠈⡀⢠⠀⠀⠀⠀⠀⠀⠀⣠⣾⡴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠙⠃⠉⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢲⠢⡀⠀⠀⠀⠀⡰⠚⡳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠈⣧⠏⠉⢢⠀⣀⡇⠀⡇⠀⠈⡆
⠀⠀⠀⠀⢀⣴⠒⠒⠒⠲⢤⡀⡴⠛⡄⠀⢸⠀⠀⣼⠋⠀⠸⣀⣇⣀⣼⣁
⠀⠀⠀⢠⠃⠀⠙⠢⣄⠀⠀⢹⡀⠀⠘⢦⠀⡇⡰⠉⡇⢀⠴⠋⠁⠀⠀⢸⡆
⠀⠀⠀⢸⡀⠀⠀⠀⠈⠑⣄⠀⡏⠑⠲⣄⠳⣿⠁⢠⠗⠁⠀⠀⠀⠀⢀⡜⠀⠀⣀⣀⣀⣠⣤⣀⣀⡀
⠀⠀⠀⠀⠙⠢⢤⠤⠖⠒⠺⢷⡇⠀⠀⠘⡆⣅⡴⠃⠀⠀⠀⠀⢀⣠⠾⠒⠊⠉⠉⠀⠀⠀⠀⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠙⡄⠀⠀⣹⠟⠀⠀⠀⣀⠴⠚⠉⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⡀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢩⠋⠉⠙⠺⣄⣴⠏⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠚⠻⠿⠿⠿⠿⠿⠟⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣤⠤⠤⢿⡼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠤⠶⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠀⠀⠀⠀⠀⠀⠘⣆⢀⡠⠤⠤⠤⠤⠤⣀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣔⠁⠀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢰⠁⠀⢀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⣀⠼⠁⠀⠀⠀⢰⠃⣀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⢀⡞⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⠻⣅⣸⠆⠀⠀⠀⠀⠊⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⣣⡀⠀⠈⠓⠒⠊⠀⠀⠀⣲⣯⠤⠤⠤⠐⠂⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⣠⠭⠷⠤⣄⣀⠀⠀⢠⠎⠁⠀⠀⠀⠀⠀⣀⣤⢀⣤⣤⡴⠿⠂⠀⠀⠀⣠⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⢉⡝⠓⠤⠤⠤⠤⠶⣎⠁⣀⠜⠁⢮⣠⠊⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢸⣄⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠿⠃⢻⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢚⡁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣀⠀⠀⠀⢀⡎⣧⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠋⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠘⠃⠀⠀⢸⡇⣿⡄⠀⠀⠀⢀⠔⠻⢡⡏⣤⠔⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⠀⠀⢄⢀⢸⠀⣿⡇⠀⠀⣴⠟⠀⢀⣾⢡⡏⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠻⣿⠏⢠⣿⡇⠀⣼⣿⠂⠄⣸⣿⠟⢱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⠴⠇⠰⠻⠿⠧⠼⠿⣶⣿⠀⠿⢣⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢀⠐⠋⠀⠄⠲⠄⠀⠀⠉⢶⣡⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⢡⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠈⢼⡟⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠃⠀⠀⠠⡁⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⡿⢿⠀⡀⠀⢀⡡⠀⠠⠀⡁⢀⠀⣆⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠓⠺⡷⠋⢠⡁⠰⣿⠋⢀⡼⠇⢁⣴⡄⠈⠂⠈⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠅⠀⠀⠤⠄⠀⣷⣤⣹⡀⠈⠀⠀⠀⡏⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠁⠀⠀⠀⠉⠋⠙⠛⠲⣤⣶⣜⣃⣤⣴⠞⡽⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠄⠀⠀⠀⠀⢿⡶⠀⠀⢀⠀⠀⠐⠋⠋⠉⠉⠄⠀⢁⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⢢⠀⠀⢴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡘⡿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡟⠀⢠⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠃⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠂⣴⣿⣧⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡏⠀⣀⡾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠁⠀⠋⠈⠛⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠂⠀⣿⡟⠐⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠁⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡄⢰⣿⠀⠀⢿⠠⡀⠼⠀⠀⢀⠀⠀⠀⢀⠈⠃⠠⢀⣾⠃⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣾⣿⠀⠀⠀⠀⠁⣤⣤⠾⣿⣀⠀⠀⠁⠀⠀⠀⣿⡧⠀⠀⠀⠀⢧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⣿⠡⠀⠀⠀⣸⣿⢏⣵⣿⣿⡇⠀⠀⠀⠀⢰⣿⠃⠀⣀⣀⡀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⡭⠉⢻⡿⠀⠀⢀⠀⣿⣯⣈⡙⢉⡻⣿⠃⠀⠠⢠⣿⣿⡃⠀⢉⣉⠙⠛⠒⣄
⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣁⣼⣶⡔⣿⣷⠶⡤⠀⢀⣾⡿⠿⠿⢿⣷⣾⡄⢀⡄⢄⣿⣿⣷⣾⡟⠉⢧⠀⢀⢸⢆
⠢⠁⠄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠛⣺⣿⣤⣠⣤⣠⣾⠃⠀⠀⠀⠀⠀⠹⣷⣤⣥⠈⠫⠿⠄⠈⠛⠛⠛⠓⠛⠓⠛⠃
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⢢⢀⡀⠀⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡄

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠘⠃⢀⣒⠲⠶⢦⣕⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⢀⠔⠻⠶⠞⢛⠟⠀⠁⠀⠀⠈⠢⣀⡠⠔⠂⠉⠁⠀⠀⠈⠉⠢⡄⠀⣀
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠈⠊⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠠⠄⠀⠀⡂⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⣀⠐⠁
⠀⠀⠀⠀⠐⣲⣅⠀⠄⠁⠠⠃⠄⣀⠖⠒⠀⠈⠀⠀⠀⠀⢀⡈⠦⠄⠀⠀⠀⡴⠃⠈⠀⠀⢀⠴⢦⣄
⠀⠀⠀⠀⠜⠛⠀⠀⠀⠀⠘⠒⠒⠓⠠⠤⠤⠤⠤⠴⠋⠹⣀⣀⣀⡀⠀⠤⠊⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⢀⣀⣤⣴⣤⠔⠒⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⢀⣀⡀⠀⠀⡀⠠⢄⣤⣤⣄⠄
⠀⠀⠀⠀⠈⠉⡼⠉⠢⠠⠀⠤⠤⠭⠑⠀⠀⣀⠠⠄⠊⠁⠁⡾⣿⠿⠂⣀⠔⠈⠀⠀⠠⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠀⠠⠤⠤⠲⠚

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣟⣴⣿⣿⣿⣿⡿⣿⣿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣀⣀⣀⣀⣤⣤⣤⣴⣦⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⠿⡿⠿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⣟⠋⠙⠉⠉⠁⠀⠀⠀⠀⠀⠛⠟⠻⢷⣶⣿⢶⡤⠴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠿⠿⣿⣿⣿⣿⣿⠿⠿⠋⠁
⡯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠀⠀⠀⠙⠟⠋⠉⠉⠉⠊⠁⠀⠁⠘⠃⠀⠀⢀⣀⠊⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣴⡿⡄⠀⠀⠿⢶⡶
⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀
⣶⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠢⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠛⠉⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢘⠙⡄⠀⢸⣄⠀⡰⠃⢀⡤⡄⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠘⠤⠇⠤⠼⠈⠷⠃⡴⠃⡠⠃⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⡝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠊⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡤⣼⠈⣆⡤⠴⠒⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡈⠀⠀⠀⠀⠯⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⢾⡠⠊⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠶⢤⣼⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉
⡞⠳⣄⠀⠀⠀⠀⣠⠇⠐⠒⠒⠃⡠⠤⣀⠀⠀⠀⠀⠠⠤⠄⢲⡖⠋⠁
⣇⠀⠈⢧⠀⢀⡞⠁⠀⠀⠀⠀⠀⢣⠀⠀⠑⠂⠄⠀⠀⠀⠀⢸⡇
⢻⡀⠀⠈⢧⡼⠀⠀⠀⠀⠀⠀⠀⠀⡵⠒⠛⢢⠀⠀⠀⠀⠀⠈⡇
⠈⢧⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠈⡗⠂⠀⠀⠀⠀⡇
⠀⠈⢧⠀⠀⠘⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢰⠓⠦⡄
⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⡠⠃⠀⠐⣇
⠀⠀⠀⠀⠉⠙⠲⢤⣄⣀⠀⠃⠀⠀⠀⠀⣰⠃⢀⣀⣤⡊⠀⠀⣰⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢦⣀⣀⡴⠋⠉⠉⠁⠀⠉⠛⠋⠁
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⢀⠀⡠⠤⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠈⠀⡀⠀⡇⠀⠀⠀⢀⠠⠠⠀⠀⠀⠀⠀⠀⡤⠀⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⢸⠀⢢⢄
⠀⡄⠁⠀⡇⠀⡇⠐⠂⠰⡃⠀⢀⡘⢂⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠘⠒⠀⠀⠀⠰⠁⣀⠀⠀⠡⠄⢸⠀⢸⠸
⠀⠀⠁⠀⡇⠀⡇⢸⠀⠀⠹⡁⠀⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠎⠀⠧⠤⠡⠆⠈⠀⠸⠀⣩
⠀⠀⠘⠀⠃⠀⡇⢸⠀⠀⠀⡁⠀⡜⠀⠀⠀⠀⠀⣀⠀⠀⠢⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⠀⠀⡇⠀⠀⠀⡆⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠐⢤⠀⠐⠀⠄⠈⠁⠀⡄⠄⠀⠀⠁⠀⠀⠀⠁⠀⢠⣠⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠱⡀⠀⠁⠀⠀⠀⢆⠀⠀⠐⠀⠀⡰⠀⠀⠤⠀⠀⠠⠁⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠉⠐⠂⠀⢠⠤⡕⠢⠀⠠⡊⠀⡤⡀⡀⠄⠂⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠄⣀⢀⠎⠀⡇⠀⠀⠀⠂⠈⡅⢩⠀⠀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠰⢄⡀⠀⡀⠎⢠⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠀⠀⡆⠀⠀⠈⡀⠠⠀⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣤⣶⣶⣷⣆⠀⠀⠃⠀⣃⣉⢡⠀⢃⢀⣠⣼⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠈⢻⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⠋⠁⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠖⢪⠉⠻⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⠇⠀⡠⠀⢄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠑⢄⣹⡿⠿⠟⠛⠛⠛⠛⠿⠿⠿⣿⡿⠦⠊⠀⠀⢠⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡤⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠠⠤⡌
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡯⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⢶⡀
⠀⠀⠀⠀⠀⢀⠊⠁⠈⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣀⠈⠈⠉⠒⠄
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠂⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣶⣦⣤⣤⣀⣀⣠⣤⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⠿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠐⠒⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠁⢀⠙⠿⠋⠀⠀⠀⠀⠀⣤⣦⠀⡀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠡⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⡐⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠐⠠⠄⠄⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣆⠹⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣌⡛⢿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⣿⣿⠁⢠⣤⣀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣬⣙⠻⣿⣿⣿⣿⣿⣧
⠀⠀⣿⠃⠀⠘⣿⣿⣿⣶⣶⣶⣶⣦⣾⣿⣿⣿⣿⣿⣿⣿⠿⣷⣦⡙⢿⣿⣿⣿⣇
⠀⢸⣿⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⣿⣿⡄⢿⣿⣿⣿
⠀⢸⡿⠀⠀⢄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸⣿⣷⠘⣿⣿⣿⡆
⠀⢸⡇⠀⠈⠊⠉⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠈⣿⣿⡆⢹⣿⣿⣷
⢂⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠘⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⣿⣿⣿⡿⠟⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠷⠘⠋⠁⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⡶⠟⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣰⣦⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢀⣄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣿⣿⣇⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⣴⣿⡿⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⣿⣿⡄⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣸⣿⣿⠇⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⣿⣿⣷⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢠⣿⣿⡏⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠘⣿⣿⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣼⣿⡿⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠹⣿⡇⠸⣧⠤⠖⠒⠚⡒⠒⠒⠦⢄⣿⠀⢀⣿⡿⠁⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠙⠃⠀⠀⠀⠀⠙⠈⠁⠃⠀⠀⠀⠀⠀⠘⠛⠁⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠙⠲⠶⠋⠀⠀⠀⠀⠀⠀⠈⠣⣤⠼⠁⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⠤⢼⠦⠤⠤⣤⠀⠀⠀⠀⠀⠈⠻⣿⠟⠁⠀⠀⠀⠀⠠⠒⠒⠚⠋⠉⣻⠉⠉⠉⠉⠉⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣸⣄⡤⠤⠄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠒⠒⡶⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠈⠙⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠤⣄⣀⣀⣀⣀⣀⣀⣀⣠⡤⠤⠖⠛⠉




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠐⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠃⠀⠀⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠂⠕⠄⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⡆⠈⠀⠀⠀⠀⠀⠀⢀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠲⠀⠀⠀⠁⠀⠀⠀⠀⠇⠀⠀⡀⠂⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆⠊⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠈⠀⠠⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠰⠄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⡀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢡
⠀⠀⠉⠐⣆⠄⠀⠀⠀⠀⠀⠂⠂⠂⠀⠈⠐⠄⠂⠀⠀⠀⠀⠀⠀⠀⠂⠐⢃
⠀⠀⠀⢠⠇⠤⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⡐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⠀⠖
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠈⠀⢀⠜⠁⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡠⠊⠁⢀⡀⠤⠞⠓⠂⠉⢠⣇⠤⠤⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢂⠠⡆⢹⣀⠄⠊⠁⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠈⠑⢄⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠣⠀⠘⠀⠀⠀⠀⣀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⢱
⠀⠀⠀⠀⠀⡠⠚⠁⠀⠁⠀⠀⠀⠙⠫⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠶⠶⠡⠔⠂
⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠠⠊⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠒⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⡤⠚⠒⠤⠤⠤⠴⠒
⢀⣔⠁⡀⣒⢂⠀⠀⠀⠀⡄⠒⠪⠀⠀⠀⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⡍⠀⠩⠿⠏⠀⠀⠀⠀⢝⣿⡶⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠘⢢⠀⠀⠀⠈⠢⡤⠔⠊⠀⠀⠀⠀⠀⢀⠓⠁⠀⠀⠀⠀⣠⠔⠁
⠀⢀⠇⢄⡀⠀⠒⠱⢄⣀⠄⠀⠀⢀⠴⠃⠀⠀⢀⡠⠔⠋
⠀⠸⠀⠀⠀⢩⠜⠓⠁⠒⠲⠒⠊⠁⠀⢀⡖⠉⠁
⢠⠁⠀⠀⡠⠃⠀⠀⠀⠀⡜⠀⠀⠀⢠⠎
⠀⠀⠀⠔⠁⠀⠀⠀⠀⡊⠀⠀⢀⠔⠁
⣼⣴⣮⣤⣤⣤⣤⣤⣤⣙⣈⣉⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⢀⣼⣿⡟⢀⡤⠂⠀⠀⠀⣠⠄⠀⠀⠀⠀⣠⡤⠖⠀⠀⠀⠀⢐⣿⢂⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⣀⣴⡿⠟⠋⠀⠈⠛⢉⠗⠀⢰⣿⠀⠀⠀⢀⣾⠋⠀⠀⣶⡀⠀⠀⡼⠋⣿⠃⢀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠈⠛⠂⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⠘⠁⠀⠈
⠀⠀⠀⠈⠉⠛⠋⠉




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣄⣠⣀⣀⣄⣤⣤⣀⣠⣄⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣩⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⠛⠿⣿⣿⣿⣿⣿⣿⠃⢸⣿⣿⣿⣗⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠢⠀⠀⠈⠉⠻⢿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠉⠂⢄⠀⠀⠇⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠐⡀⢀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠰⡀⠀⠀⠀⣅⠐⠂⠀⠂⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣦⣀⣀⣀⡀⢁⣤⣤⣂⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠁⠀⠛⠟⠒⠠⠤⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠑⠉⠀⠀⠀⣀⠀⠀⠀⠀⠋⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣮⠆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠜⢜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢀⣠⡾⠿⢶⡄⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡧⠀⠂⠀⠀⠈⢻⣄⣀⣼⠇⠀⠀⠄⠀⠘⡇⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣤⡶⠂⠉⢹⠥⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠰⠁⠀⠀⠙⣗⢆
⠀⠀⠀⠀⠀⠀⠴⠟⠁⠀⠀⠀⢡⣲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣅⠀⠀⠀⠀⠈⠛⠦
⠀⠀⠀⠀⠀⠀⣶⣤⡤⠊⠀⠀⢱⠋⠺⢂⠠⡀⠀⠀⡀⠀⠀⣠⠠⠀⠀⢹⡄⠀⠀⠰⢦⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡺⣏⠀⠀⠀⠋⠉⠚⡯⢽⠉⠁⠈⣆⠀⢠⡾⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢠⠇⠀⢰⣀⠀⡠⠆⠙⠓⠠⡄⠀⣿⠄⠈⢻⡞⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣾⡇⠀⠈⠉⢬⠀⢰⠊⢱⠀⢸⡌⠃⠀⠀⠀⢻⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠗⠀⠀⠀⠀⣤⠀⠉⠁⢠⡄⠀⠀⠀⡇⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣤⣥⠀⢹⡁⠀⢈⣄⡀⠀⠙⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢠⣤⣤⢊⣱⢯⠴⠾⣧⣼⡏⢬⣿⠀⠈⠉⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠙⠋⠁⣿⢀⡠⡼⢻⠆⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⣸⠁⠘⢲⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⣀⠿⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⡾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢃⠑⡀
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢄⠀⠀⠀⢠⣾⣿⣿⣿⠾⡄⠰
⠀⠀⠀⠘⢄⣼⡂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡞⠀⠀⠀⣿⣿⠟⠉⠁⠀⣷⠒⠓⡄
⠀⠀⠀⣠⣶⣿⠿⠢⢄⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⡄⠀⠀⠹⣿⣄⡀⠀⢀⡍⠀⠀⢰
⠀⠀⠀⣿⣿⣇⠀⠀⠀⠱⡄⠉⠒⣄⠀⠀⠀⠀⠀⠀⡆⠰⡀⣀⣀⠈⢿⣿⠏⠀⣉⠏⠒⣼
⠀⠀⠀⠈⠻⣷⣄⣀⡀⠤⠦⠔⠒⠈⠧⣀⠀⠀⠀⠀⠰⡰⠉⠀⠀⠀⢀⠎⠉⠉⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⠠⠃⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⠣⠤⠄⠒⢩⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⢀⠜⠁⠡⡀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠣⠀⠀⡠⠄⠐⢲
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠱⢮⠀⠤⠤⠤⢄
⠀⠀⠀⠀⠀⠀⢹⠤⡤⠔⠓⣄⠀⠀⠀⠀⠀⠀⣀⠤⠊⠀⠡⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠸⠀⠃⠀⠀⠀⠑⠄⣀⠄⠊⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⡠⢦⠀⠀⠀⢰⠀⠀⢠⠃⠀⠀⢀⠔⠲⢄⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠈⠄⡇⠀⠀⠀⠀⡜⠀⠀⡠⠊⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠑⢄⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠊⢉⠂⠀⠀⡘⠀⢠⠞⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠃⡔⠁⠀⠀⠀⠀⠀⠀⠀⠱⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡰⢁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠩⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠲


⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣾⠁⠀⡀⠀⣼⣤⡄⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⠶⢟⣁⢂⣉⣁⣀⣀⡀⠀⠀⠈⣹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡉⡈⢀⣴⣿⣿⠿⠿⠻⢿⠿⢿⣿⣷⣶⣬⣥⠿⠷⠶⣤⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣿⡿⠛⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣷⣄⠘⢯⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⡀⠀⡄⠀⠀⠉⠀⠀⠀⣤⠀⠀⠘⠀⠀⠀⠛⠀⠀⡀⠀⠀⠹⣿⣧⡀⠁⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢙⣾⡷⠲⣶⠁⣠⠀⠀⠀⠀⢠⡄⠀⣤⠀⣀⠀⠀⠋⠀⠀⠀⠀⢻⣿⣤⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡟⠁⢀⡇⠀⠀⠁⠀⠸⠖⢀⣁⣤⣠⡄⠉⠀⣤⠀⠀⠶⠂⠀⠀⢿⣿⡙⣿⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣶⣶⣦⣶⣦⣷⠖⠋⠰⠆⠀⠀⠚⠂⢰⣺⠟⠛⠁⠿⣿⡆⠀⣤⠀⠀⠀⠀⠀⠈⣿⣿⢻⣷⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡿⢏⣽⠉⣿⡷⠀⠀⠀⠀⠀⠰⠆⢰⣋⡍⠀⠀⠀⠀⢹⣿⠀⢠⡀⠀⠈⡁⠀⠀⠈⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⣿⠇⡀⢻⣷⠀⠀⠒⠀⠀⠠⡤⠈⢿⡆⠀⠀⠀⣀⣶⡟⠀⣀⠀⠀⢤⡉⠻⣿⣦⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠁⢸⣿⣦⠀⠀⢀⡀⠀⠀⣤⠀⠙⠺⠯⠿⠛⠃⢠⡄⠉⠀⠀⠀⢻⡀⠈⠙⣦⣈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣴⠀⠈⢿⣿⡄⠀⠈⠁⠀⠀⠀⠀⠶⠀⢰⡄⠰⠇⠀⠁⠀⠀⠓⠀⣀⣶⣶⡆⠀⠉⡄⠙⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣸⡆⠹⣿⣦⡄⠀⠀⠒⠀⠀⢀⠀⠀⠀⠀⠀⠀⠰⠆⢀⡴⢊⣵⣿⠛⣧⠀⣾⣿⡙⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣉⣹⠆⠸⣿⣷⣤⡀⠀⠀⠀⠉⠀⠀⠀⠙⠀⠀⠀⠀⣈⣰⣿⣿⠁⣾⠛⣆⢿⣽⣟⢿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃⠀⠀⡇⠉⠻⢿⣿⣷⣤⣀⣀⣤⣄⣀⣀⣤⣞⣾⡯⠛⠉⠇⠀⠙⡟⠁⠈⠻⠏⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⢷⡀⠀⠀⠉⣿⡿⠛⠛⣿⡿⠛⠛⢻⡽⠋⠀⠀⢀⣶⡀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠘⣿⠁⠀⠀⢀⡜⠀⠀⠸⣅⡽⠀⠀⢰⠀⠀⠀⠀⠀⢻⠀⢸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠃⠀⢠⣤⣿⡏⠀⠀⠰⣏⡛⠀⠀⢸⠀⠀⠀⠀⠀⠸⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⢦⡀⣿⣿⣿⠃⠀⠀⠀⣸⡁⠀⠀⠸⠀⠀⠀⠀⠀⢰⢿⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣿⣽⣿⡿⢣⣿⡿⠃⠀⠀⠀⢠⡋⣷⠀⠀⠀⢧⡀⠀⠀⠀⠸⣾⢻⣾⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣟⠀⠈⠉⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⢻⣿⣷⡀⠀⠀⢀⣰⣿⣿⣶⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣻⡄⠀⠀⠀⠀⠀⠦⢴⣶⣶⣿⣆⠀⠀⢸⣿⣿⣧⠀⠀⠀⢸⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠐⣻⣿⣿⣿⣿⠁⠀⠀⠈⠻⣿⠀⠀⠀⠈⢉⣿⡟⣻⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢠⣾⣻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠹⠃⠀⠀⠀⠸⣿⣷⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣫⣿⣿⣿⣿⣿⣿⣽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣿⡗⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣏⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣤⡒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⢀⡶⠈⠀⠀⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⡏⣧⡼⠀⡼⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⡏⠀⠀⠐⣤⣼⣿⣿⡿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣟⣻⣿⣷⠀⠀⣰⠉⠘⠋⠉⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣰⠋⠀⣠⣶⣲⣿⣿⣿⣿⣿⣾⠋⣿⠀⠀⡄⠃⠀⠀⠀⡄⠀⢀⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣷⣇⣼⣿⢀⣾⣿⣿⣿⡯⠿⣀⣨⣿⣧⣾⡿⡇⠘⡇⠀⣀⢢⡠⢀⣤⢿⣿⡹⡄
⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⠠⠀⠀⠉⠉⠁⠀⠀⡀⠀⠙⣄⠈⣆⠃⡾⠁⠀⠉⣷⠇
⠀⠀⠀⠀⣠⡾⣿⡿⢿⣿⣿⣿⣿⣴⣷⣾⣿⢻⣿⡗⣿⠀⣆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢶⣯⠴⠃⠀⠀⠀⣹
⠀⠀⠀⢰⣟⠀⠙⢿⣿⣿⣿⣿⠿⢿⠋⣴⠟⢸⣿⣿⠍⠳⡌⢦⣴⣤⣶⣤⣤⡀⠀⢨⠀⠀⢇⠀⠀⠖⠀⠁⢸
⠀⠀⠀⢸⣷⣀⣴⣾⣿⣿⣀⣀⣹⣾⠓⠀⢠⠃⢿⣿⡤⠶⠟⠘⠡⣀⡀⠸⡁⠀⠀⣸⡄⠤⠀⣷⣶⡇⠄⢀⣼
⠀⠀⠀⠀⡛⠛⣿⣿⣿⣉⣿⣭⣩⣤⣾⡜⠛⠷⠤⠼⠿⠿⠧⠤⠄⠉⠉⠉⠛⠒⠒⠛⠓⠂⠚⠋⠛⠓⠂⠚⠛⠁
⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠂⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣤⠄⠊
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣾⣿⣶⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⣨⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠇⡀⢠⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⡀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡇⠀⠁⢠⠀⠀⠀⠀⠀⠀⢀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢉⠂⠠⢬⠀⠀⢠⡾⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣿⠀⠀⠀⠘⠣⡀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⢠⢀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠄⢠⠀⠂⢳⣄⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡔⠐⣿⣷⣤⣿⣏⣤⣤⣌⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠙⢿⣿⣿⣿⣿⣿⣿⠈⠀⠈⠳⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠙⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣄⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣯⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠁⠀⠈⠉⠛⠛⠛⠛⠛⠻⠿⠿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠴⠢⣄⡀⠈⠀⠀⠀⢀⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠿⠿⣿⣿⣿⣷⣶⣀⠢⠄⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⠒⠂⠸⠂

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⢸⣿⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢼⣿⠀⣴⡆⠀⠀⢀⣴⠒⠀⣠⢄⢀⠀⢀⡀⠰⢹⣧⡐⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢸⣿⠀⣿⡇⣠⠀⣾⡏⠀⣼⡇⢸⡿⠀⠍⣿⠃⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⣿⠀⠈⢹⣿⠀⢿⡇⡼⣿⣇⡼⣇⡘⠀⣿⠀⢸⡏⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠘⣿⢀⠌⠈⠿⠜⠀⠉⠀⠈⠉⠀⠉⠀⠀⠉⠀⠈⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⣠⣤⠴⠖⠒⠋⠉⠉⠉⠉⠀⠉⠉⠉⠉⠐⢺⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠃⠀⠀⢀⣀⠀⠀⠉⠉⠀⣀⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⣴⠏⠘⠀⠀⠀⠀⣼⠏⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⢸⡿⠀⠀⠀⠀⠀⣸⡟⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢘⠃⠀⣠⠔⢤⠀⠀⠀⡠⠔⠐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⢰⣿⢀⠔⠉⣷⠀⣰⡟⠀⣼⡏⠀⢺⡇⡠⠊⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⣿⠆⣿⠇⢀⣿⠁⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢠⢺⣿⠀⠻⣄⠎⠹⠶⢁⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣁⡾⠃⠀⠀⠀⠀⠀⢠⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢀⡼⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠉⠉⠁⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⣀⣱⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣀⠔⠉⠀⢸⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⣀⠸⠎⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠐⣶⠏⠒⠀⠀⠀⠀⣄⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⡡⠔⠒⠀⢲⠲⡀⠀⠀⠀⠸⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠊⠁⠀⠀⠀⠀⢈⠀⠱⡦⣀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⡜⠀⠀⠲⡀⠀⠀⡍⠙⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢀⠤⠴⠒⠈⠁⡧⠤⠊⠀⠀⠀⠀⠑⠤⠤⠁⠀⢡⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠘⡀⠈⢆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢰⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠇⠀⠀⠑⠢⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣂⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⢰⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠘⡀⠀⠀⠀⠀⣠⠎⢀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡉⠀⠀⠀⠃⠀⢀⡠⠊⠁⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠓⢄⠀⢰⠤⠄⢀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠱⡀⠀⠀⠸⠖⠁⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢈⡅⡆⠀⠀⠈⠁⠐⠒⠂⠀⡴⠋⠀⠀⠀⠱⡀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⢀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠈⠢⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠋⠀⠈⠛⠓⠀⠀⠠⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠦⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠠⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠉⠒⢠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⢠⠎⠉⠁⠀⠒⠒⠒⠂⠀⠀⠀⠤⠤⡄⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⠉⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢀⠀⠀⠀⢀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠈⠳⣀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠋
⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡌
⠀⠀⠀⠀⠀⡰⠁⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠀⡸
⠀⠀⠀⢀⡰⠁⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠘⠛⠹⡀
⠀⠀⣴⢉⣧⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⡀⠀⠀⠀⠘⢆
⡴⣏⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠖⢋⡀⢂⣀⠀⠑⢦
⠑⠪⡓⢄⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠛⠳⠶⠋
⠀⠀⠈⠲⣷⣄⠈⢦
⣀⣀⣀⣀⣀⣉⣳⣦⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠻⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣤⡤⢴⣶⡞⠁⣹⡟⢀⣶⡤⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠰⢿⣌⣿⢷⣎⠴⠿⣼⡿⠿⢷⠾⠿⢿⣶⣼⣿⡀⠿⠆⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⣹⣶⡿⠛⠋⠀⠛⣤⣶⠛⢷⣤⣀⠀⢠⣦⣀⣈⠙⠋⠿⣾⣿⣥⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠿⣇⣴⡟⢻⣄⢠⣾⠛⢿⣿⡯⠁⠀⠀⠘⠿⣿⣿⠛⠻⣿⣇⠀⣠⡜⠛⣿⣿⣇⣴⣦
⠀⠀⠀⠀⠀⠀⣀⠛⣿⣴⣿⣿⠟⠛⠻⠿⠿⡓⠛⠛⢀⣤⣶⠷⢤⡀⠙⠉⠳⣴⡿⠿⠟⠿⠛⠾⢿⡉⠹⣿⣶⣶⡆
⠀⠀⠀⠀⢀⣠⠛⣶⡿⠁⢨⡏⠀⠀⠀⠀⣀⠀⣶⣆⡀⠋⠀⠄⠀⠳⢀⣴⡆⠀⢶⣤⡀⠀⠀⠀⣼⡄⠀⠈⠿⣿⢷⣶
⠀⠀⠀⠀⣈⠛⣱⡿⠁⠀⣼⣿⠀⢀⣰⠟⠈⠦⠘⣿⣇⠀⢠⣤⡄⢀⣿⣿⠅⠀⠈⠡⠟⠁⠀⢀⣿⣥⣀⠀⠀⢿⣿⣵⣦
⠀⠀⠀⣘⡛⣳⣿⠇⢱⣶⢿⣿⠠⠸⠟⠀⢠⣴⣦⣼⡿⠷⠟⠿⣾⣿⣟⠿⣠⣶⡔⠀⠀⠛⠀⠛⠛⠛⣿⣷⡀⠀⣿⣦⣴⣦
⠀⠀⢀⡟⣿⣿⠋⠀⠸⣿⣤⡀⢠⣤⣤⣄⣸⣿⠿⠃⠁⠀⠀⠀⠈⢀⠉⠀⢿⣮⡁⣤⣶⣶⡶⠂⠀⢠⣿⡟⠁⠀⠈⣿⣟⣁
⠀⠀⠘⢿⣿⡟⠀⣤⣶⡿⠛⠁⣄⡉⠿⢿⠿⠉⠂⠀⠀⠀⠷⡀⢰⠃⠀⡀⠀⠈⣥⢿⠿⢋⠀⣀⡈⠀⠀⠿⢶⣴⠚⢷⣮⠻⠃
⠀⠘⡿⠻⣿⣇⣼⡿⠋⠀⠀⠶⠉⠀⠀⣤⠋⠑⠄⠘⠠⣤⣴⣿⣾⡗⠉⠁⠀⠀⢠⣶⣤⡄⠀⠘⠻⣄⠀⠀⠀⣽⠷⣤⣅⣿⡆
⠀⠸⠿⢶⣿⠿⢿⣄⠀⠀⣰⡆⠀⠀⠀⢻⡾⢁⠀⣀⠤⠄⢻⡿⢿⡇⠀⠶⡄⠀⣼⣋⠉⠃⠀⠰⣦⠀⠀⣀⣴⠏⢀⣽⢟⣶⡄
⠀⠰⠿⣿⣿⡄⠈⠻⠷⣄⣀⣴⠀⢠⣼⣾⣶⣾⣶⠉⠀⠀⠘⠃⠀⢠⡀⠀⠀⢰⣿⣿⣿⣷⣤⡀⠀⠿⢿⣿⡋⡀⠘⣿⣦⣭⠁
⠀⠀⢻⡟⣿⡆⠀⠀⢠⣾⠛⣀⠾⠛⠋⠉⢿⣟⣻⠀⢀⠀⠀⠀⠀⠈⢀⠠⣔⣿⡟⠉⡉⠉⠙⠛⢀⡀⢦⣿⡿⠁⣼⣿⣽⡏
⠀⠀⠘⠟⠻⢿⣄⠀⠸⢿⣄⣹⡷⠀⢳⣀⠼⠛⠛⣿⣿⣤⣶⣶⣿⣶⢾⣶⣏⠛⠃⠀⠈⣶⡖⠒⣿⣿⣶⠟⢃⣠⣿⣟⡛⠃
⠀⠀⠀⠸⠷⠸⣿⡆⠀⠀⢉⣿⠇⠀⠈⠻⢿⣶⠀⢹⣿⡟⠁⠘⠋⠁⠘⢿⣿⡆⠠⣌⠿⠃⠀⠀⢨⣿⡀⠀⢀⣿⣿⡿⠃
⠀⠀⠀⠀⢾⣗⣸⢿⣄⠀⠀⣿⡀⠀⠀⠀⠀⢸⡆⠘⠛⡀⠷⠀⠈⣠⣤⠀⠙⠁⠀⣡⣄⠀⠀⠀⠈⣿⠁⢰⣿⣿⣼⠟
⠀⠀⠀⠀⢴⠟⢻⣼⣿⣧⣀⠻⢷⡶⠦⢶⡞⢻⣿⠂⣾⣿⣄⠈⠻⠇⠀⢠⣆⡀⠛⣿⡟⢳⣿⡶⢾⣿⣷⣿⣿⠛⠻⣶
⠀⠀⠀⠀⢰⡄⠀⠙⢿⣿⢿⣷⣄⣀⠀⠀⠀⠈⠿⠾⠟⠛⣿⠇⠀⠀⣰⣿⠙⠿⡿⠏⠀⠀⣀⣰⣾⣿⣯⡝⠋
⠀⠀⠀⠀⢨⡀⠀⠀⠀⠉⠿⢧⣭⠙⡿⢦⣤⣤⣄⡀⠀⠀⠙⢷⣤⡾⠋⠀⠀⠀⣤⣤⣶⡶⠛⢹⣧⡌⠛⠁⠀⠀⠀⣦
⠀⠀⠀⠀⠘⠁⠀⠀⠸⡇⠀⠈⠉⠹⣿⠈⣿⡍⢻⣿⠲⣶⣴⣶⡟⣷⡆⠸⠿⣛⡋⣽⣦⠴⠿⠿⠍⠀⠸⠇
⠀⠀⠀⠀⠹⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣋⠁⠘⠿⠠⢾⠿⢻⡷⠄⠾⠷⠀⠛⠃⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠆
⠀⠀⠀⠀⠸⠂⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⣸⣧⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⢀⣿⣟⣷⡀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣸⣿⢿⣯⣿⡿⢷⣄
⠀⠀⠀⠀⠀⢀⣜⢻⣿⣇⠀⠀⠀⠀⠘⡟⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⢠⣿⢿⣆⣩⣤⣄⠀⠈⠁
⠀⠀⠀⠀⣼⡟⢿⣿⣿⣿⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢸⡿⢤⣾⠿⢿⣿⣶⣶
⠀⠀⠀⢰⡏⢁⠈⡉⢻⣿⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡘⢧⠀⣾⠋⣩⣿⣷⡀⢀
⠀⠀⠀⣤⣿⣿⡇⠀⠾⢻⡇⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⢀⠀⠋⢀⣧⡤⣿⡟⠋
⠀⠀⢸⣿⣿⣿⣤⡋⠀⢿⠃⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠛⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠻⣄⣰⡿⠋⣀⣨⣧
⠀⢀⣾⢿⣿⣟⠉⢿⡄⢺⡀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⡴⢀⣈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠸⠃⠈⠋⣁⡖⢉⣍⣩⡆
⠀⢸⣿⣿⣿⣿⣶⣤⣿⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠚⣿⠾⣯⣽⣄⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣴⡏⢱⣿⡉⢿⡇
⠀⢨⣿⣿⡟⠁⡉⠉⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⠠⠞⣧⡌⠯⣉⢻⣷⡀⠀⠀⠀⠀⠀⠀⣡⠀⣾⠉⣷⣶⣿⣿⣆⡤
⣀⠸⣿⣿⠇⠀⢿⣦⡀⠸⠟⠳⠟⠀⠀⠀⠀⠀⠀⢸⠁⣼⠁⣠⣿⡿⠶⡉⠢⣜⢳⡀⠀⠀⠀⠀⠀⢹⣆⠙⣿⡏⢀⣿⢯⣿⠇
⠉⠀⣽⠁⠰⠶⢿⣿⣧⣤⣴⣦⠀⠀⠀⠀⠀⠀⠀⣾⠆⢁⣴⠟⠉⣿⣶⡟⢷⣉⢏⠀⠀⠀⠀⠀⠀⠠⡍⢿⣯⣴⣿⣏⢸⣿⡆
⠀⠀⠠⠄⠀⠀⠘⣿⣧⣄⣿⠏⠀⠀⠀⠀⠀⢰⣿⠋⢠⣾⠋⠀⠀⢿⣿⣿⣿⣧⡌⣡⡄⠀⠀⠀⠀⠀⠀⣶⣾⡿⠋⣹⠟⣿⡇
⠀⢰⡀⠀⠉⣱⣾⣿⣿⣾⠃⠀⠀⠀⠀⠀⠀⡾⠋⣠⣿⠃⠀⠀⠁⠄⠀⠙⢿⣿⣾⣏⢠⠀⠀⠀⠀⢀⡀⢿⡇⠴⡛⢃⡀⠻
⠀⠈⣥⠔⢠⡙⣧⠘⣿⣿⡇⠀⠀⠀⠀⠀⠘⠃⣴⡿⠃⠀⢶⠀⢠⡀⢀⣶⡄⠹⣿⣿⡃⠠⠄⠀⠀⠀⠁⠀⣠⣾⣿⡄⠁
⠀⠀⠍⢐⣾⣿⣤⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⢠⠔⣻⣿⡿⠛⠀⠀⠈⢿⣷⠀⠀⠀⠀⠀⠀⠀⠛⢁⣶⠁
⠀⠀⠰⠞⠿⣿⡧⠉⢉⣽⡇⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⣡⣾⡿⠉⠀⠀⠀⣰⡷⠈⣿⡆⠀⠀⠀⠀⠀⠀⣠⣾⠃
⠀⠀⠀⠀⠀⠉⠀⠈⣿⡿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣼⣿⣿⣿⣄⠰⣦⠙⠁⠀⠀⢀⡷⠀⠀⠀⠀⠀⠀⠹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⡜⠁⠀⣠⣿⡿⣿⣿⣿⣇⠘⠁⠀⠀⡀⠲⣷⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⢁⠀⢀⣿⣿⣿⡿⢋⣸⠋⢠⠀⣶⣦⠴⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣾⣿⣿⠟⢀⣼⣿⣶⣶⣄⠉⠁⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢠⣿⣿⠏⠀⠿⣽⠏⢰⣤⠻⣿⢷⡮⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢀⣼⢇⣀⠛⢧⠈⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⣾⠇⣸⣿⢀⣀⡸⠧⣰⡟⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⣿⠠⡿⠹⣇⠙⢷⣤⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣶⣁⣶⣾⣧⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠉⣹⣯⡛⠉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡘⠿⣿⡿⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⣨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣀
⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣷⣶⣶⣶⣾⣶⣶⣶⣷⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⡿⣿⠿⢿⣿⣿⠿⠛⢿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣇⣈⣤⣀⣸⣧⣤⣠⣾⣀⣄⣰⣷⣸⣀⣶⣿⣋⣰⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⠀⠀⣾⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⡇⢰⣿⣿⠿⠛⠁⠀⠀⠀⠀⠈⠉⠻⢿⣧⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣡⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⡀⢀⣀⣀⣀⢠⣤⠀⠀⠀⣰⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣼⣿⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠹⢿⣿⡿⢛⠿⠛⢿⣿⣿⡟⠁⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⢸⣿⣿⣦⣄⣶⣿⣿⣿⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⠛⠻⢿⣾⣷⡿⠋⠉⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣧⡀⠀⠀⠀⠀⠀⠀⣼⣇⠀⠀⠀⠀⠀⠀⣴⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣦⣤⣀⣠⣤⣾⣿⣿⣦⣤⣤⣤⣴⣿⣥⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⢿⠿⠛⠋⠙⠈⠙⠛⠿⣿⣿⣿⣿⠿⠏⠉⠁⠀⠀⠈⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⢀⡀⠄⠠⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠉⣑⣾⣿⣛⣳⣦⠑⠤⢀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⣔⡩⡟⣿⢭⡝⠉⠀⠀⠀⠀⠑⢤⠔⠊⠁⠀⠀⡀⠀⠀⠈⠑⣄⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢸⡀⡽
⠀⠀⠀⠀⢀⢀⣴⣶⣶⠟⢳⠀⠀⠔⣂⣀⣠⠇⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⣸⣛⣁⣤⣄⡀
⠀⠀⠀⠀⣩⣽⠟⠋⠀⠀⣯⣙⣲⣯⠅⠀⠀⠀⢀⡴⢧⠖⠛⠛⠂⠀⢀⡔⠉⠉⠛⠻⠿⠛⡟⠣
⠀⠀⠀⠀⠁⠀⡀⠀⣀⣀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠘⠒⠒⡚⠒⠊⠉
⠀⠀⠀⠐⠺⠟⡻⢾⣿⣿⣿⣷⣦⣤⡀⠀⠀⠀⢀⣠⣤⣄⣸⣡⣶⠒⠀⣠⣴⣾⣿⡾⢟⠓⠂
⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠉⠉⠉⠀⢀⣤⣶⣾⣿⣿⣿⡿⠋⠛⠀⠐⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣴⡿⢿⣿⠛⠛⢿⣿⣿⠛⠻⣷⠶⣧⣤⣄
⠀⠀⠀⠀⠀⠀⢀⣰⣿⣾⠟⢉⣿⠋⠙⠳⢾⣇⠀⠈⠷⣦⠉⠻⣮⡙⠿⣶⠂⡀
⠀⠀⠀⠀⢀⡐⣷⣿⣴⠟⠃⣾⠿⠂⣴⡇⠀⠙⢿⡁⠀⢹⣷⠀⠻⣿⠀⢺⡿⣟⣤⡀
⠀⠀⠀⠀⢾⣿⣏⠁⣿⣅⣀⣿⣀⠀⡆⠙⢦⡗⠀⣣⡴⠉⣷⣀⠀⣿⡄⠀⣦⠙⢿⣱⡄
⠀⠀⢀⣸⣿⣡⠿⠋⣋⣁⣬⣿⣹⣶⣧⣤⣼⣿⣴⣿⣿⣤⣿⠉⣈⡏⠀⢈⡏⠀⣼⣻⣿⡀
⠀⠀⣼⣿⡿⢻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⢂⣾⠇⠀⣼⠹⣿⣇
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣹⣿⣿⣿⣿⣿⡿⠿⣿⣿⣇⠀⣹⠿⢀⣿⣿⣄
⠀⣿⣿⣿⣹⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠈⠙⠿⣧⡀⣹⡏⣿⣟
⠀⣿⣿⣿⣿⠙⣿⣿⣿⣧⣄⠀⠘⣿⣿⠀⠀⠉⠁⣸⣿⣿⠁⠀⠀⠀⠀⢀⣀⣀⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣶⡇⠈⢰⡟⠻⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠼⣶⠒⠛⢻⡉⠉⢩⣿⣿⣿⣿⠇
⠀⢻⡟⣿⣿⣿⣿⢤⣾⡁⢀⣾⡇⢨⡟⠉⢹⡏⠙⣿⡉⠙⡧⠄⣈⣦⣀⢘⣷⠔⣿⣿⣿⢿⡇
⠀⠀⡷⠙⣿⡿⣿⣀⣿⡁⠈⣿⠀⠀⣷⠄⠛⣧⣀⠛⠻⣾⡇⠀⠈⣉⡀⢸⣇⣠⣿⣿⠏⣸⡇
⠘⠂⢇⠀⠈⢻⣾⡿⣿⣿⠀⢿⣗⠂⢹⣦⠀⡸⢿⣤⣀⡼⠛⠛⢿⠟⠛⣿⣿⣿⡿⠃⠀⣿⡆
⠀⠀⠈⠀⠄⠀⠙⢿⣾⡿⣷⣤⣿⣷⣤⠿⢿⣄⠀⣹⠟⠻⢦⣾⣿⣶⣾⣽⡿⠋⠀⠀⠀⡿
⠀⠀⠘⠇⠀⠀⠀⠀⠈⠻⢷⣿⣟⣻⣿⣶⣶⣿⡿⣻⡿⣷⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠉⠀⠀⠹⢆⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⡀⠀⣀⣀⣀
⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠑⣶⣾⣿⡛⠉⠉⠛⢳⣄⠠⠀⠘⢿⣿⣿⣿⣦⣄
⠀⣰⣿⡙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⣯⡝⠉⠐⣀⡀⠀⠀⠙⡆⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄
⢠⣿⣿⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠆⠀⢰⣿⡇⠀⠀⠀⠴⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤
⣿⣿⣿⡇⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠂⠀⠀⠀⠀⣤⠀⠀⡀⠉⠀⠀⠀⣰⡆⠀⠀⠘⣿⣿⣿⣏⢻⣌⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢷⣼⣵⣜⢦⣤⣼⣿⠁⠀⠀⠀⢻⣿⣿⣿⡀⠻⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠉⠉⠉⠉⠀⠈⠁⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⠈⠿
⢻⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣠⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠈⢻⡇
⠀⢿⣷⣿⠃
⠀⠀⠹⣿
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣇⢻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣫⣿⣿⣿⣿⣿⡾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣽⣿⢿⣿⣿⣿⣿⣿⡿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣡⣾⠿⠛⢿⣿⣿⣷⡋⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣫⠞⠋⠀⠀⠀⢛⣿⣿⣿⣿⣶⡈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠁⠀⠀⠀⠀⠀⠈⢻⣿⣝⣿⣿⣿⣏⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠙⣿⣿⣷⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⡋⢻⣿⣧⣹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⡄⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⡿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠁
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡏⠀⠈⢀⠀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢀⠄⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠋⢀⠂⠀⢀⠄⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⡞⢠⠃⠀⢀⠆⢀⠜⠀⠀⠀⠀⠀⠀⠀⣀⡄⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⡘⢁⠘⠀⠀⠀⠀⢠⢀⡆⠀⠠⠃⡠⢋⡠⠤⠀⠀⠀⠀⠠⢼⣅⣀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⢠⠁⢸⠠⠀⠀⠀⠀⠆⢠⠀⢠⠁⡸⠋⠁⠀⠀⠀⠀⠀⢀⣀⡀⠀⠉⠪⣐⠠⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢀⢀⡠⢣⠀⠈⡇⠀⠀⠀⢀⠀⡆⢀⠋⠌⠀⢀⣏⢹⠄⠀⠀⠀⠮⣠⠗⠀⠀⠀⠀⠳⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢃⠀⡇⠅⣠⡂⠹⠰⠀⠆⡘⠂⠤⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡬⠆⠒⠛⠋⠉⠚⠓⠰⠤⣁⠀⠀⠈⠢⡀⠀⠠⠄⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠙⢄⠘⠢⠞⠁⠀⠀⢰⠒⢣⠀⠀⠀⠀⠆⢸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀⠈⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠸⡀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⣀⢠⡀⠀⠀⠀⠀⠀⠤⠰⡀⠀⠐⢄⡀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⢀⢼⠁⠀⠀⠀⠐⠤⠠⠔⠀⠀⠀⠀⠠⣀⣀⠄⠀⠀⠀⠀⠘⡆⠀⠀⠣⣤⠃⠀⠀⠀⠀⠈⠃⠔⠁⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⢠⠖⢁⠞⠀⠀⢠⠊⠑⡄⠀⠀⣠⠀⢠⠀⠀⠀⡠⠀⢄⠀⠀⠀⢀⡇⠀⠀⠠⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠈⡙⠀⠑⢄⠀⠑⠊⠀⠀⠢⠠⠛⠡⠤⠂⠀⠣⠤⠌⠀⠀⣀⠜⠇⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⣀⠠⠄⢒⠜⠁
⠀⠀⠀⠀⠀⠀⢸⠱⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⠈⠀⠀⠈⢄⡀⠀⠀⠀⠀⢀⡠⠒⠈⠀⡀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠈⠢⣑⠀⠤⠛⠢⠄⣀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⠬⠒⠀⠀⢒⡁⠠⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⠠⠤⠉⢁⣐⣒⣒⣚⣄⠀⠀⠀⢀⣠⣤⠄⠊⠥⠄⠀⠐⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠶⠶⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⣄⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⠟⠳⢦⣄⣀⠀⠀⠈⠳⣌⡻⣦⡀⠀⠀⠀⣶⣈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣿⠀⠀⢀⣀⠉⠙⠛⠒⠶⣾⣿⡾⠳⣄⠀⢰⡏⠙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠙⢷⡇⠀⠚⠋⠉⠙⠂⠀⠀⠀⠉⠉⠙⠦⠙⣦⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠳⣶⠞⠛⣧⠀⠀⠀⢸⡇⠀⠀⠸⡿⠂⠀⠀⠀⠀⠸⡿⠆⠀⠀⢸⡟⣇
⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⣿⣤⡀⠻⣷⡶⠶⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⠶⠃⠈⠙⢦⡈⠙⠶⣄⡈⠳⣄⠀⠀⠀⠳⠶⠶⠞⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠙⣶⣌⡛⠶⢦⣤⣤⣤⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣞⠁⠈⠛⠛⢻⡇⠀⠀⣺⠛⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣰⠟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⣄⠈⠳⣄⠀⠀⣀⡤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⢷⣄⠘⢧⣰⡏⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇⣴⠖⠓⠿⣇⠀⠙⣧⣀⢀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣀⣀⣀⠀⠀⢸⣿⡇⠀⠀⠀⣹⠶⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣰⡏⠉⠉⢹⣆⣀⣸⡏⠳⣤⣤⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⢹⡇⠀⠀⢸⡏⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠷⡤⣼⡇⠀⠀⢸⡧⣤⣼⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠶⠶⠾⠇⠀⠀⠸⠷⠶⠶⠴⠾⠷

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⡄⠀⠀⠀⠘⣽⣄⣤⣤⡄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣨⣼⠤⠀⠋⡟⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⢀⣄⣀⠀⠀⠀⢦⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⡿⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠐⣴⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣆⢀⣰⣧⣤⣤⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠈⠁⠀⠤⠀⠄⠀⠀⠇⠀⢠⡄⠀⠂⠀⠀⠀⢀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠐⠀⠀⠀⠀⠀⠀⣠⠀⠈⠀⠄⠀⠠⠄⢠⠊⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⣦⣌⣀⣀⣀⣀⣥⢆⣷⡀⢀⣀⣀⣠⡇⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⢉⠂⣹⠋⠈⠁⠈⠻⢫⠉⠉⠉⠉⣽⢿⠟⠛⠂⠀⢀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠈⠁⢠⠋⣴⢀⢠⠀⠀⠀⠀⠀⠛⠣⠾⠤⠠⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁



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