Petite Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡤⠤⠤⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣀⠀⠀⠀⡄⠀⠀⠰⡆⠀⠀⣰⠆⠀⢠⡄⠀⠀⡄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠻⣄⣀⡴⠛⠶⠶⠖⠋⠛⠋⠉⠓⠒⠋⠳⠶⡞⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣸⠁⢴⣿⡆⠀⠀⠀⠀⠀⠀⢰⣶⡆⠀⠀⠈⠓⣫⢭⢳⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢠⡏⠀⠀⠁⠀⣀⠀⠀⢀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣍⣡⠞⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⢸⠭⠯⡽⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠘⠢⣄⡀⠀⠀⠉⠊⠁⠀⠀⠀⠀⠀⣀⣠⡖⠃⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⢹⡟⡖⠒⠒⠒⠒⢒⡟⡟⠉⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣧⠿⠶⢤⣤⣤⠴⠇⣧⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⡴⠀⠀⠀⢸⠃⠀⠘⠛⠓⢦⣠⡞⠁⠉⠙⠓⠾⠀⠀⠀⠈⡇⠀⢰⡄⠀⢰⣀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⣄⡀⣄⠘⢧⣀⠀⠀⠀⣀⣈⣽⠦⣄⠀⠀⠀⠀⠀⠀⣼⣁⣠⠜⠳⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠛⡟⠉⡏⠉⡇⢸⡇⢸⠙⣷⠲⡦⠶⢾⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢀⡇⢠⡇⢸⡇⢸⠀⢻⠀⣷⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢸⠀⢸⠁⢸⠀⢸⠀⢸⠀⢹⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⡿⠶⠾⠤⠼⣤⡼⠤⠾⠶⢺⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⣰⢣⠇⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠙⠢⢤⣀⣰⢃⣾⣀⣀⣀⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⠞⠁⠺⢤⣬⣉⣩⡽⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣖⣲⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⣿⣿⣿⣿⣟⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣷⡤⠞⠙⠻⣭⣿⣽⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣯⢿⡭⠟⢢⣴⣾⡿⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠙⢷⣽⣷⣦⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣿⣻⣟⡷⠋⠀⠐⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣻⣯⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣜⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢏⡹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢙⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠠⠴⠶⠿⠿⣷⣶⣦⣄⡀⠀⠀⠀⠀⢠⣴⣶⡿⠿⠛⠛⠛⠒⠂⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⢀⣀⣀⣈⠉⠛⠃⠀⠀⠀⠀⠉⠉⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠠⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⠀⠀⠀⠀⠈⠀⠰⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⠓⠀⠁⠀⠀⠀⠀⠀⡆⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠈⠁

⣶⣶⣶⢷⢿⠷⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⠷⣶⣶⣶⣶⣶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⠶⠾⠿⠟⠛⠛⠛⠛⠿⠷⠶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠦⣄⡀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠻⣧⡀
⠀⠀⠀⠀⠀⣸⡟⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠙⣷⣄
⠀⠀⠀⠀⢠⣿⠁⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠈⢿⣧⣀
⠀⠀⠀⠀⣼⡇⠀⢰⠏⠀⠀⡀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡙⢿⣷⣄⡀
⠀⠀⠀⠀⣿⠁⢠⡏⠀⣠⠞⠁⠀⠀⣀⣴⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⣿⠀⣾⢁⡼⠋⣀⣤⣶⠿⠛⣉⡼⠿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⢤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠈⠉⠛⠻⠶⠶⣶⣤⣤⣤⣤⣤⣤⣿⣿⡿⠏
⠀⠀⠀⠀⢻⣷⣷⣾⡷⠟⠋⠁⠀⢠⠞⢡⣾⣿⡿⠿⣿⠿⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠻⠟⢶⣶⣶⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⣨⣿⣟⠁⠀⠀⠀⠀⠀⣿⠀⢿⣿⣿⣿⣴⣿⠀⢸⡏⠉⠛⠛⣿⢿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⠿⢿⣿⠿⠟⠛⠛⠋⠉⠁
⠀⠀⠀⢸⡏⠀⠈⠀⠀⠀⠀⠀⠀⢻⡀⠈⠻⠿⠿⠿⠃⢀⣾⠃⠀⠀⠀⢿⠘⣿⣿⣾⣤⡿⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠈⢿⣤⣴⠂⠀⠀⠀⠀⠀⠀⠙⢦⣤⣀⣀⣤⡶⠟⠁⠀⠀⡀⠀⠈⢷⣌⣙⠋⢉⣀⣤⠾⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠙⢳⡆⠀⠈⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⡶⣄⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠠⡖⠤⡑⡌⢣⡠⢶⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⣀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠈⠲⣀⠛⢀⣧⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠁
⠓⠀⡬⢁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⠙⠳⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡿⠟⠉
⠀⠂⠐⠈⠙⠚⢃⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢶⣶⣶⣶⣶⣶⣶⠿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠘⣆⠳⣄⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⡇⢀⢀⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⠲⠤⢰⡏⠉⠉⠉⠉⠉⠉⢷⣌⣁⣈⡿⠀⠀⠀⡀⠘⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠸⣇⠀⠀⣠⠆⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣷⡀⠀⠈⢛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣦⣤⠟⠤⣉⡳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢁⡰⠞⣋⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡎⠡⠒⠉⡵⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⠈⠛⠧⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡌⠳⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣁⣀⣠⣤⣶⠶⠟⠛⠛⠛⠛⠛⠛⠛⠛⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠻⡿⠟⠛⠛⠋⠉⡛⢻⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣸⣧⠀⠀⠀⠀⣀⣷⠾⡶⠶⠶⡶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠉⠙⢦⣀⣤⡤⠶⠋⠀⢸⣇⠀⢠⠞⡙⠀⠀⠀⠀⠀⠁⠈⠙⢷
⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⡀⢀⢈⣆⣀⣀⣀⣀⣀⣉⢼⡀⣿⠀⣃⠀⠀⠀⠀⠀⠀⠀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⣏⣀⣀⣁⣀⣈⣁⣀⣀⠤⠤⠤⠶⠤⠋⠙⠤⠤⢀⣈⣁⣀⣈⣁⠠⠭⠟⠁

⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⡀⠠⠀⢤⠀⠀⠀⢀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠄⡤⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⣥⠥⠰⠊⠁⠤⣀⠬⠭⠥⠃⠀⠀⠀⠀⠀⠀⠀⡴⡳⣎⠉⠳⠻⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⡖⠀⠀⠀⠀⠀⠀⠀⢀⣀⢹⣦⠀⢀⡀⢀⣿⠀⠀⠴⣠⠴⠀⠀⠀⠀⠀⠐⣉⣅⡉⠉⢉⠉⡇
⣿⠀⠀⠀⠀⠀⠀⠀⣀⡀⠘⣏⠀⠀⠀⠀⠀⠀⠀⠉⠺⣁⠘⠀⢘⣩⠎⠈⠒⣒⣲⠟⠀⠀⠀⠀⠀⠀⠀⠉⠂⠠⣴⠾⠋⠁
⣿⠀⠀⠀⠀⠀⠀⠘⢺⣿⡿⠋⠀⠀⠀⠀⠀⠀⣤⣠⢀⣈⡶⠀⡠⢿⡔⡓⠺⠁⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⣿⠀⠀⠀⠀⠀⠀⢀⢸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠉⠱⣏⠹⡃⡾⠁⡆⠈⣩⣟⣉⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅
⣿⠀⠀⠀⠀⠀⠀⠀⢰⢣⡆⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⢰⡁⠀⣤⠞⠁⢏⣡⡅⠒⠞⠀⠀⠀⠀⠀⠀⠀⠀⢈⡅⢠⠴⢄
⣿⠀⠀⠀⠀⠀⠹⠶⠿⢸⣤⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠈⠁⠞⠈⡄⠶⣶⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡴⠒⠊
⣿⠀⠀⠀⠀⠀⠀⠀⡘⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠁⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣰⡆⠀⠳
⣿⠀⠀⠀⠀⠀⠀⠀⠘⠻⡶⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⢠⠀⠀⠀⠀⠈⢦⡜⢀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠈⢧⠸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⠀⠀⠀⠀⠐
⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤
⣿⠀⠀⠀⠀⠀⠀⠀⢻⡄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢨⠆⠄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠤⣉⡽⠆
⣿⠀⠀⠀⠀⠀⠀⣐⣉⡓⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢹⠤⠈⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⣰⠏⠀⡎⠀⠀⠈⢄
⣿⠀⠀⠀⠀⠀⠸⠿⠼⠟⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠠⠂⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⣿⠀⡴⠀⠘⠀⠄⢀⡄
⣿⠀⠀⠀⠀⠀⠠⠀⢄⠃⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠤⢴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡕⣄⠠⠨⠀⠀⠠
⣿⠀⠀⠀⠀⠀⠈⠿⢀⡀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠠⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⠢⠄⠠⠈
⣿⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⣿⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡕
⣿⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⣶⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣛⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⢸⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣿⣿⠟⢿⣻⣿⣻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⢸⣿⡟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⡟⠛⢧⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿
⠀⠀⠀⠀⣿⡇⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠘⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡆
⠀⠀⠀⠀⢹⠀⢸⣿⣿⡹⡿⣿⡟⡿⠟⣿⣿⣿⣿⣶⣿⣿⣿⣿⣟⣿⣿⡿⠛⠻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠘⠀⢸⣿⣿⣇⣾⡟⢈⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠁⢀⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⡆⠀⠸⣿⣿⡏⢻⣦⣴⣷⡍⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⣮⣽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠁⠀⠈⣿⣿⡇⣤⠏⣿⣷⣍⣧⣤⣼⢿⣿⣿⣿⣿⣿⣿⣦⢔⠷⢫⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣻⣿⣷⢸⡇⢿⣿⣿⣏⠉⢿⣼⣿⣿⣿⣿⣿⣿⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢰⠀⠀⠀⢿⣿⣿⠈⣧⡈⢿⣿⣿⠀⣾⣿⣿⠃⠹⣿⣿⣿⣷⣾⣿⣿⡻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣆⡘⢷⡌⢻⣟⣾⣿⣿⠿⠀⠀⢹⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⣿⣷⣤⡻⠆⡛⣿⣿⡏⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣬⣿⡿⢿⣯⣉⣉⣁⣥⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡏⠒⠄⠍⠙⣿⣿⡿⠋⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣥⡀⠀⠀⠀⢠⡟⠀⠀⢀⡶⣅⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⢶⣶⣺⡟⡟⠳⠶⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠘⠈⠂⢟⣿⣿⣿⣷⣄⡈⠀⢀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢙⢿⣿⡻⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠄⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠈⠁⠀⢤⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠣⡄⠀⠐⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⢀⢠⢤⡔
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠊⠙⡈⠹⠟⠦⢄⢀⡄⢻⠀⢰⠂⠎⠀⠈
⠀⠠⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⢤⣄⣀⣀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠬⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡠⠒⠈⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠰⠀⠄⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠂⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠐⠀⠀⠀⠘⠲⠑⠶⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡁⠂⢠⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠂
⠀⠀⠀⠀⠀⠤⠀⠑⡀⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⢀⠀⠀⢀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠘⠘⠀⠘⠀⠀⠒⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⣶⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠋⠙⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠟⠋⡠⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠋⠉⠙⠛⠻⠷⣦⠀⠀⠀⠠⠾⠟⠛⠉⠉⠉⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⣷⠖⠀⠀⠀⠀⠀⠀⠀⠲⣿⠗⠂⠀⠀⠀⠀⠈⠀⡂
⠀⠀⠀⠀⠀⠀⠈⠄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠐⡈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⠋⠉⠉⠉⠛⠟⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⣠⠐⠀⠐
⣿⠀⠀⠀⠀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠣⠀⣶⣿⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢀⠀⠀⠀⠀⠀⠀⡀⢹⣬⣶⡾⣽⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠠
⣿⠀⠀⠀⠀⠀⠘⠃⠀⠂⡞⠁⢋⢸⠀⠀⠀⠀⠀⠀⠈⠀⠛⠛⠓⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠍⢷⣄⣠⡄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡶⠜⠁⠀⠀⠒⠀⠀⠀⠀⢠⡀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠈
⣿⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⣤⣀⣀⠒⠂⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⣀⠼⠅⠄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢤⡀⠀⠠⡀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠠
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢨⠇⠀⢂
⣿
⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⣿⠀⠀⠀⢀⠘⢢⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⡠⠀⠀⣄⡀⠈⠁⠀⠀⠀⢀⠀⠁⠀⠀⠐⠐⠀⢤⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠁⠀⠈⠃
⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈
⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⣠⡄⠀⠀⠐⠁
⣿⠀⠀⠀⠀⠁⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢳⡀⠚⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠈⠻⢿⣇⢀⠄⠀⠀⠈
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⠀⢀⡶⣿⣷⠀⠀⠀⠀⠀⠀⢀⡈⢻⡜
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⡉⠚⡟⠀⠀⠀⠀⠀⠀⠀⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠀⠀⠓⠈⠀⠀⠀⠀⠀⠀⠀⣀⠂⠀⠀⠀⠀⠀⠠
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠉
⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⣶⣶⣶⣶⣶⣶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣿⣿⣿⣿⣿⣿⣏⢸⣿⡋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠑⣿⣿⣿⣿⣿⣿⣿⣿⡀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣿⣿⢿⣿⢿⠟⣿⣿⣷⣶⣾⣿⣿⣿⣿⠟⠛⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢼⡇⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣏⣠⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢀⣿⣿⣟⣀⣩⣿⣿⣿⣿⣿⣏⢡⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡞⡇⢿⣿⣏⠙⣿⣿⢿⣿⣿⡇⢀⣴⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⢻⡌⢿⡧⣼⣿⣯⠀⢻⣿⣿⣿⣿⣯⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣽⠦⢻⣿⡿⠁⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣷⡿⠟⠿⣬⣈⣤⡼⡿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠈⣻⠋⠀⣠⠀⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣶⡶⣶⠿⠲⠤⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⣤⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉



⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠉⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣴⣾⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡉⠉⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡟⠋⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⡏⠀⠀⠀⢈⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠟⠁⢀⣀⣄⠈⠛⠛⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠏⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣤⣤⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣽⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡇⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣧⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣥⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠁⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠋⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠈⠙⠛⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⡀
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣿
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉
⠀⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⡤⠄⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⢠⡾⠇⢲⠀⠀⣰⣄⣄⡦⠦⠀⠀⡀⠘⠂⠰⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣶⣉⡱⠌⠀⣄⣵⣦⣾⡾⣾⢋⠉⣭⣷⢠⠀⠘⣁⠈⠀⡀⠁⠀⠶⠦
⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢠⠖⠮⠥⠘⢣⠀⡀⢸⣿⠟⣋⣤⢻⣯⡙⢈⡟⣠⢄⣧⢉⣰⣞⡋⠁⠀⠁⠀⠀⢀
⠤⠤⠴⠶⠋⠉⠉⠉⠁⢀⣤⣾⠿⡶⠲⠆⠛⡉⠣⠴⢣⣤⠛⠋⠁⢨⠈⢷⣿⠀⠘⠾⠔⠙⣹⠫⡄⠢⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⢸⣿⢾⠢⣼⢇⠘⣂⡀⠘⣅⠂⣀⠁⡄⢸⡏⣤⠐⠷⣼⣠⢿⡠⣬⡄⠁⣈⠀⣀⠀⠬⠄⢢
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣉⠶⣿⢿⣇⡀⠀⠘⡃⠈⠀⡀⠘⣱⠏⠈⠀⣿⠙⣡⣌⣿⠁⠌⢁⣼⠏⠀⠀⢀⠃⠁⠐⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡐⢯⡾⣿⣿⡇⠙⠻⣇⡙⣶⣤⠀⠀⡿⠀⠀⠀⣾⠌⠙⣞⢳⡌⣦⣾⣃⠀⣸⠃⠀⠀⠀⢀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⣤⣼⣯⡞⣿⢈⡁⣈⢧⡄⠹⣦⠼⠃⣿⣄⢴⣿⡀⠸⠗⠿⢁⣿⡷⢉⣡⡶⠎⠀⢀⠀⡀⠀⠄⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠋⡸⣿⣿⣿⣶⡃⡉⢄⢠⠔⢻⡗⢀⠈⣿⠀⣧⡄⠀⣀⡠⠛⢁⡤⢚⠉⠀⣸⣾⡗⠂⡄⢲⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣼⠳⢇⢁⡸⢻⣷⣟⣷⣼⣄⠀⠻⣧⡀⣿⠀⣿⠀⣼⣯⡤⠈⠀⡁⠀⣻⣿⠟⠛⠀⡆⡔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣾⣮⣕⢻⣛⣿⣷⣮⣿⣈⠂⡿⣧⡘⡆⢻⡼⠿⠉⣰⠗⣪⣵⠖⠛⣀⠀⠋⢼⣁⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣭⣿⣿⣿⣿⣿⣿⣿⣿⣏⡀⢹⣿⠁⣼⠃⢀⣡⣼⡾⠻⠿⠟⢛⣥⣤⣄⡄⠃
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢻⣽⠛⡿⣿⣿⣿⣿⡁⢼⣿⣿⡿⢫⣅⣤⣤⣤⣾⣾⡧⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢿⣿⣿⣆⠻⠏⢠⠘⠉⠉⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⣾⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⣄⣤⣤⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⢠⠀⠐⠂⠰⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠓⠚⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇
⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⡀⠀⠀⠀⠀⠀⣠⠃⡀⠀⠀⢀⡀⠐⠀⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣼⣃⣠⣿⢸⢧⠘⡇⢰⡀⢨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠒
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣠⠞⢉⣽⣿⣿⢸⡄⠄⠁⠸⣷⠈⠟⠀⠀⠐⠛⠛⠛⠛
⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⢀⡾⠁⢠⣾⣿⣿⣿⡦⠞⢰⢄⠀⡏⢧⡀⣤⠀⠀⠀⡒⣰⣒⠂⠰⠄⢠⣤⣤⣤⠄
⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠇⡄⠀⠀⠀⠀⣾⣿⣿⠟⢸⠿⣥⡈⠀⠀⣷⡼⣷⡹⣷⡀⠀⠀⡀⠈⢠⡄⣠⡀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⡟⣻⠛⠳⠐⢇⣠⡄⣦⣿⣿⡿⠫⢠⠐⢆⠸⠀⠀⠀⢿⣴⣌⣷⣌⣿⣦⣾⣄⠁⢰⣾⣥⣾⣥⣴⣤⣤⣀⣙⣁⡀
⡿⣿⣿⣿⣿⣿⣟⡋⠰⠧⠀⠀⠐⡞⢯⣷⣾⣿⠟⠃⠡⡄⢱⣾⣶⣶⡶⢶⣽⣏⣿⣏⣥⣦⣽⢳⣌⣀⠛⢻⣿⣿⣿⣿⣧⡛⠋⠉⠙
⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣄⣀⣢⣿⣿⣿⣿⣿⣧⣤⣤⣦⣾⣟⣙⡉⠴⠂⠀⣈⣿⣿⣧⣿⣽⣿⣿⣿⣤⢀⣉⠹⢻⣿⣿⣷⣶⣦⠿
⢻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣼⣿⣿⣿⡿⠿⢿⡿⣿⣿⣿⡆⠉⢰⣺⣿⣿⣿⣿⢿⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⣦⣰⣻⡟⢿⡿⢿⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⣀⡀
⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⠿⠛⠛⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣿⠿⠿⠿⠿⠿⣿⡿⠿⠿⠿⠿⠿⠏⠙⠛⠛⠀⠀⠈⠉⠛⠻⠷⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠁⠀⠙⢦⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢠⠶⣄⢀⣄⣳⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣠⠶⣤⠞⣶⡋⠀⣈⣉⠀⠙⣆⠀⠀⠀⠘⢒⠖⢀⣠⠔⠒⠒⠒⠒⠒⠒⠚
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠶⠦⠤⢥⡎⣤⢹⣼⢱⡎⡶⠒⠻⢳⠀⡠⣚⠥⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠧⣄⠀⠀⠀⣄⠳⢤⠞⠸⠦⠜⡡⠀⣸⠛⠞⠋⢧⡖
⠀⠀⠀⠤⡼⠀⠀⠀⠘⢆⠀⠀⠈⠑⠒⠶⠖⠒⠉⠀⡴⠃⠀⣀⠴⠃⠉⠁
⠀⠀⠀⠉⠁⠙⠳⢦⣀⠈⢑⣦⣄⣀⡀⢀⣀⣠⢴⣾⣃⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣿⠙⠺⢭⣽⣹⠶⠋⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⡏⡇⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠉⠁⠀⠀⢘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⡟⢳⠒⡖⢶⠲⡛⢏⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⡇⡜⢠⠃⠈⢆⠱⡈⢆⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢸⠀⡇⢸⠀⣶⡈⢧⠱⣈⣦⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢺⠶⠶⠼⠋⠳⠚⠋⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⣄⣨⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⢀⣿⠏⢀⣤⡴⠛⠛⠛⠛⠛⠛⠿⠷⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣾⣿⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠶⠶⠾⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⠀⠀⠀⠀⠘⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠘⢣⡀
⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣦⢀⣼⣶⠟⢹⣿⢷⠿⣆⠘⣷⡄⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⢰⣿⣾⣿⡿⠁⠉⠀⠘⠃⠀⠀⢻⣾⢷⣾⠇⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢀⠟⢠⣿⠇⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣴⠀⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⡟⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡟⠀⢸⡇
⠀⠀⠀⠀⠀⢀⣿⢇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠛⠁⣇⠀⢺⠇
⠀⠀⠀⠀⠀⠸⡏⢸⡇⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⢸⡟⢰⡇
⠀⠀⠀⠀⠀⠀⢻⡎⢻⡄⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠈⢷⡄⠻⠆⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠈⢫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⣄⣀⡀⢤⡤⠛⠃⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋⠙⠛⢳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣴⠦⣀⣀⣀⠀⢀⣀⣀⣠⡶⠞⠛⣿⡇⠀⠀⠀⠀⠈⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠿⠋⠹⣷⣄⠈⠉⠉⠛⠋⠉⠉⠁⠀⠀⣠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠈⠛⠿⢷⣶⣤⣀⣀⣤⣤⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⢀⣴⠛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣴⣿⣿⡿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠶⣄⣀⣠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠋⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠶⣬⣭⣽⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣻⣧⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⣿⣿⣶⠶⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⣶⣾⡿⠿⠟⠙⣿⡠⢄⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣴⣆⠀⣿⠁⠈⠉⠛⠛⠛⠿⠿⣷⠶⠶⢾⣿⣿⣿⠿⠿⠿⠿⠟⠛⠛⠛⠉⠉⠀⠀⠀⠀⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⢡⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⡾⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⡿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⢸⣇⣤⡾⠛⠛⠳⣦⣄⡀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⢀⣠⠶⠖⠉⠙⠛⠛⠷⢦⣼⠟⠁⠀⠀⠀⠀⠀⠈⢳⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣧⣄⠀⠀⠀⠀⠀⠀⢸⣿⣦⣄⣠⣤⣤⣤⣤⡴⠛⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣦⣴⣾⡻⠾⠋⠈⠛⠛⠛⠶⠶⠶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡟⠛⡟⠛⠛⠟⠛⠛⠛⣧⠈⠛⠛⢿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣏⣹⣿⣉⣉⣉⣉⣉⣩⣉⣉⣽⣿⣿⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣹⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠀⠉⠉⠉⠉⠉⠐⠢⠄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⡄
⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⣀⣴⣀⣀⣀⣀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢀⠃⣄⡠⠔⠂⠽⠃⢀⡀⠀⠀⠀⠀⠉⠉⡀⢒⡠⠃⣀⠀⠀⠀⠀⢠
⠀⠀⣠⡄⣰⠏⠀⠀⢀⠖⣡⣶⣀⡁⢀⠀⠀⢠⣃⣶⣧⣌⣱⡀⠉⢲⡄⠀⢸
⠀⢸⠀⠀⣿⠀⠀⠀⠰⠒⠋⠉⠉⠉⠉⠀⠀⠈⡉⠀⠀⠀⠈⠁⠀⠈⣇
⠀⠈⢳⠒⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢨⠀⠀⠀⠀⠀⠀⠀⡿⢤⡀
⠀⠀⠘⡄⠈⢧⠀⠀⠀⠀⠀⠀⢦⢀⡀⠀⠀⠀⣀⠤⠆⠀⠀⠀⠀⢸⠇⠀⠹⠂
⠀⠀⠀⠙⢆⡈⠳⣄⠀⠀⠀⠀⠀⠳⣌⣉⠉⣁⣠⠞⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⡀⢀⡀⠀⢀⠠⠐⡲⠀⡀
⠀⠀⠀⠀⠀⠙⠲⠾⠷⢤⣀⡀⠀⠀⠀⠉⢩⠉⠀⠀⠀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠂⠀⢑⣀⠂⠀⠮⠔⠉⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⢶⠶⠤⡶⠶⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠑⠀⠀⠤⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢤⡤⠜⠀⠀⠳⢤⡤⠤⠄⣀⣀⠀⠀⠀⠀⠀⢀⡠⠜⠂⠀⠀⠀⠀⠀⢠⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠑⠢⠤⠤⠔⠋⠀⠀⠀⠀⠀⢹⡇⠈⠉⠉⠁⠀⠀⠀⣀⠀⠀⠀⣠⢄⣉⡰
⠀⠀⠀⠀⠀⠀⠀⣼⣁⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢈⣇⣀⣀⣀⣠⡤⠴⠚⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⠂⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠈⠉⠁
⠀⠀⠀⠀⠀⠀⡆⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣘⠀⠀⠰⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠁⠀⠀⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⡘⠁⠀⠀⢄⢸⣤⣤⣤⣤⡤⠤⠤⠤⠤⠤⠤⠤⢼⠇
⠀⠀⠀⢰⠀⠀⠀⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⢀⠀⠀⠀⠀⢠⠀⠆⠀⠀⠀⢀⠞⢳⡀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠈⠓⠦⡸⣄⠤⠛⠊⠀⠀⠀⠘⠀⠈⣇⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠄⠀⠀⠀⠇⠀⠀⢹⠀⠀⠠⠤⠆⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢰⠃⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡇
⠀⣀⠴⠒⠭⡁⠀⠉⠉⠉⠉⢩⠛⢧⠀⠀⠀⠀⠀⡟⠁⣉⡉⠭⠥⠈⢱⡄
⡜⠁⠀⠀⠀⠈⢠⡀⠀⠀⠀⠈⣲⠾⡇⠀⠀⠀⠀⡗⠉⠀⠀⠀⠀⠀⠀⠈⠲⡄
⠳⠶⠤⠤⠤⠤⠤⠧⠤⠤⠤⠬⢥⠤⠇⠀⠀⠀⠀⠳⢤⣄⣀⣀⣀⣀⣀⣀⣤⠟
⣿⣿⡿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⣰⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤
⠀⠀⠴⢿⣿⡟⠋⠀⠀⠀⠀⠈⠻⣿⣿⠀⠀⠀⠀⠀⠀⢀⣽⣶⡶⠆⠀⠀⠀⠀⠐⠞⣿⣿⠿⠂
⠀⠀⠀⢨⡟⠁⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡁⠀⠀⠀⠀⠀⠀⠀⢈⣿⣄⡀
⠀⣼⣿⣿⣿⣷⡆⠀⠀⠀⢠⣶⣾⣿⣷⣶⡄⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣦⠀⠀⠀⢰⣿⣿⣿⣿
⢠⡟⢿⣿⣿⣿⡇⠀⠀⠀⣾⠛⣿⣿⣿⠙⣧⠀⠀⠀⢀⣿⣿⣿⣿⡟⠀⠙⢷⡄⠀⠈⢿⣿⣿⣿
⢸⡇⠘⣿⣿⣿⡇⠀⠀⠀⣿⠀⣻⣿⢛⡀⣿⠀⠀⠀⣸⡇⢹⣟⣛⠁⣠⡴⠛⠁⠀⢠⣾⣿⡿⢿⡄
⠀⢻⡀⢼⣿⣿⣧⠀⠀⠀⠘⠷⢀⠉⠐⠄⠇⠀⠀⢰⡟⠀⠀⠈⠉⠘⠉⠁⠀⠀⠀⣸⣿⣿⣷⠈⣷⡀
⠀⢸⡇⢨⣿⣿⣿⠀⠀⠀⠠⠂⠀⢄⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⣷⡈⢠⣶⠀⢀⣿⣿⣿⣿⡄⠙⣇
⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠑⡆⠀⠀⠁⠀⠀⠘⠇⠀⠀⠀⠀⠠⣿⣷⣸⠃⠀⣼⣿⣿⣿⣿⣿⠀⠙⠁
⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠛⢻⣿⣿⣿⠿
⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿
⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣼⣿
⠀⠀⣸⠃⡿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⣿⢹⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⣴⠇⣼⡇⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡇⠀⠀⠀⠀⠀⠀⠀⠉⢹⡄
⢸⠟⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠋⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠁⠀⠀⢀⠀⠀⠀⠄⢀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢨⠃⠀⢠⡄⠀⠀⠐⠂⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⡘⢿⣦⡀⠠⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠐⠈⠟⠋⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣤⣀⣤⠤⠖⠉
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⠰
⠀⠀⠀⢠⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⠊⢐⠀⠀⠀⠀⠀⡀⡀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠠
⠀⠥⠤⢼⠀⠂⠀⢀⠴⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠰
⠀⠀⢀⡠⠀⡤⠀⠀⠀⠀⠠⠀⠂⡩⠀⠀⠀⠀⠀⠝⠢⠤⣀⣀⣄⠀⠑⠀⠴⠇⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠋⠁⢦⣄⠀⣀⡴⠠⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠈⠉⠉⠁⡀⠀⠀⠀⠈⡀⠈⠢⠠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡜⢃⠀⠀⠀⠀⠃⠀⢠⠂⠀⢀⢀⠀⢀⠀⢀⠀⠀⠠⢀⠈⠌⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠠⠀⠀⠄⠘⡀⠀⠒⠂⢰⠀⠁⠀⠀⠈⢂⠀⢀⠀⡰⠒⠠⠂⡄⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣀⡀⢀⣀⠀⠀⢧⣀⣀⣀⠨⢤⠀⠀⠀⠀⠀⠀⠈⠒⠳⢀⠤⠐⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⡄⠀⠦⢀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠁⠀⠀⠀⠀⡳⠤⠈⠒⠛⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⢎⣰⣀⣸⠂⠈⠉⠓⠒⠲⠖⠒⠲⢍⡀⠀⡀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠐⠓⠋

⠀⠀⣄⣺⠗⠁⠀⠀⠠⠀⠉⠁⠉⠉⠀⠷⡿⠀⠁⠀⠁⠀⠀⠉
⢰⠿⠉⠁⠀⣀⠀⠀⠂⠠⠀⠀⠐⠋⠀⠋⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠺⠇⠀⣤
⠀⠀⠀⠀⢈⠻⠈⠀⠀⢀⡄⢰⡄⠐⠂⠀⠀⠀⠀⠀⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠁⠰⡇⢰⡾
⠀⠀⠀⠀⡎⠀⡄⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠂⠀⠀⠐⣃⠠⡆
⠀⠀⠀⠀⠩⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢀⡨⠖⠺⠛⢀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡠⢄⣀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠆⣀⡄⠐⠋⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡿⠿⡇⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠐⠃⢀⣠
⡀⠀⡄⠀⢐⠂⠀⠀⠀⠀⠀⠀⠀⢿⣯⡀⢠⣟⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠄⠐⠋⢀
⠀⠀⠄⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠐⠃⢈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡙⡁⠀⠀⡄⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠒⠃⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⡷⠰⠂⣇⣸⣿⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠢⠀⠀⠀⠀⠤⠆⠐⠃⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣀⠀⠀⢘⣿⣿⣿⣿⡇⠀⣳⠿⣿⣿⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢤⠀⢠⣶⣶⠀⠀⠀⢀⣖⣂⠸⠯⠌⡒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢜⣭⣽⣙⠃⠠⠘⠀⡿⠻⠄⠀⠀⠀⠀⠐⠙⠀⠀⠀⠀⠈⠀⠈⣸⣿⣠⠴⠉⠀⣿⣇⠐⢃⡘⠉⣀⣀
⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠈⠘⠙⣬⠂⡶⠀⣿⡻⠃⠀⠀⠀⠀⠀⠐⠁⠀⠀⡀⠀⠀⠀⠀⢸⣿⣴⡖⢻⡟⣯⡍⢨⣯⢨⣿⡇⠉
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡶⠛⣧⡿⠺⣧⡤⢾⠳⠄⢀⡆⠀⠀⠐⡆⠀⠀⢤⡾⢶⡿⠿⡿⡆⠇⢹⠇⠘⡇⠸⠟⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠊⠣⠀⠻⠇⠀⣸⡗⣾⡧⠒⠀⠐⠀⠀⠐⠐⠒⠀⠈⠀⢈⣡⣤⠐⠢⡄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⠫⡹⣆⠔⡀⠳⡄⠀⠀⠊⠸⠃⠱⠠⡤⣤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣃⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣽⡔⢱⡷⠀⠙⢜⣌⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⠁⣽⣇⠴⠆⠊⡹⠅⡀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢠⣃⠞⣹⠾⠆⠀⠀⢡⣡⣜
⠀⠀⠀⠀⠀⠀⠀⢰⠇⣦⢻⢃⠘⠃⠀⠀⠀⠀⠀⠣⣿⣣⣎⡀
⠀⠀⠀⠀⠀⠀⢠⡿⢚⣀⢃⣾⠄⠀⠀⠀⠀⠀⠀⢰⡀⣳⣵⠰⡄
⠀⠀⠀⠀⠀⠀⠀⡼⢡⡟⣨⣿⠀⠀⠀⠀⠀⠀⠀⠀⣱⢎⡁⢤⡟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠳⡟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⡙⢼⠖⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⡴⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⡶⠀⠀⠀⣀⣠⣤⠶⠶⠚⠛⠛⠛⠛⢛⣓⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⣠⢞⣠⡴⠞⠛⠉⠁⠀⠀⠀⠀⣠⡤⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⣛⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢶⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢞⣋⠁⠀⠀⠈⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⡴⣫⡴⠒⠛⠛⠓⠶⣄⠈⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣳⡶⠆⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠉⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠋⠁⣀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠠⣌⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⠀⠀⠀⠰⣞⠉⠀⠀⠀⢠⠞⠉⠁⠀⠀⢸⡀⠀⣠⣤⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠻⣦⠀⢠⡴⠶⢦⡀⠀⠨⡅⠉
⠀⠀⠀⠀⠀⡞⣩⣷⠀⠀⠀⠀⠀⢶⡞⠂⠀⠀⠀⠟⠀⠀⣠⡶⠚⠛⠛⠲⣥⣽⡿⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣀⣀⠈⠳⡜⠦⣤⠞⠉⠙⢳⣠⣤⡀
⠀⠀⠀⠀⠀⠟⢹⠃⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⣹⡆⠀⠀⠀⠀⢀⡴⠃⠀⠰⠊⠁⠀⣽⠛⠓⢿⣠⠏⠉⠉⠡⠋⠀⢀⡽⠀⡶
⠀⠀⠀⠀⠀⠀⣿⠀⣠⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⣷⡀⠀⠀⠀⠐⢻⡷⠶⠶⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⣸⠗⠲⣌⠉⠀⠀⢀⡤⠞⠋⠉⠀⠘⠡
⠀⠀⠀⠀⠀⠀⠸⣶⢻⡀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠈⠳⢦⣤⣤⠴⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⢁⡀⠀⣸⡇⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠳⣤⡤⠴⠶⠒⠲⠦⠾⡄⠀⣠⣴⠂⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢎⡁⠀⣰⢻⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⣠⡶⠒⢲⡦⡄⠀⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⢾⠉⠉⠳⣄⠉⠻⣥⠞⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠈⠁⠉⠉⠉⣀⣴⠛⠦⣄⡀⠀⠀⠀⠉⠙⠛⠒⠒⠒⠚⠛⠉⠁⠀⠀⣈⣷⡄⠀⠈⠓⠀⠈⢳⡄⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣶⢶⡶⠖⠛⠉⠀⠀⠀⠈⠙⠳⠦⣄⣀⣀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠛⠉⢰⠟⣆⠀⠀⠀⠀⣸⠷⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠱⣏⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⡽⠿⠶⠶⠤⢤⣄⣼⠓⠛⠳⠶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⣀⣾⡋⢳⡄⣠⣿⣿⣦⣄⡀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠸⡅⠈⠳⠜⠟⠁⢈⡹⠋⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⡾⠓⠒⠀⢀⣠⠞⠓⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠾⠃
⠀⠀⠀⠀⢀⡼⠛⠓⠒⠚⠋⠁⠀⠀⠀⠀⣨⠟⣩⡟⠛⠛⠋⠉⣩⠞⠋
⠀⠀⠀⣀⣞⠁⠀⠀⠀⠀⠀⣀⠀⢀⣠⠞⣡⠞⠁⠀⠀⢀⡴⠛⠁
⣰⠚⠉⣩⣿⠳⣦⣄⣀⠀⠀⠈⠉⣹⠛⠛⠛⢒⣂⣤⢾⠯⣤⡀
⣿⣀⣀⣏⣈⣛⣉⣀⣭⡽⠷⢶⡾⣷⣶⣚⣛⡙⠿⠖⣹⣃⣠⡿
⠻⠤⠤⠬⠥⠤⠴⠶⠖⠒⠛⠉⠉⠓⠲⠭⢭⣍⣉⣉⣩⣭⠽⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠛⠲⠤⠚⠉⠁⠀⠀⠸⠉⢳⡀
⠀⠀⠀⠀⠀⠀⢠⡖⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⢺⡳⢤⣼⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣹⠗⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣤⣬⡭⠟⠁
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢠⡀⠀⠀⠈⠃⠀⠀⠀⣀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⣦⠀⠘⡗⢤⣀⠀⠀⠙⠢⣄⠈⢻⣦⠈⣷⣄⠀⠀⠀⡀⢀⡀⠀⢀⢳⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢿⢳⡀⣜⠦⠈⠉⠉⠐⠒⠌⢳⡄⣿⣿⠇⡙⢿⡓⠲⢌⢳⣷⡀⣾⡌⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠉⣻⣭⡤⠤⢤⡄⠀⠀⠀⣿⠿⠋⢀⣈⣁⣙⣀⠀⠹⣿⣷⣿⣷⠇
⠀⠀⠀⠀⢠⣞⠀⠀⠀⠹⡆⠈⠁⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠑⠀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠈⢳⡤⣄⠀⢱⠀⠐⠋⢹⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣏⠑⠂⠀⣼⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠈⡇⠈⢶⡎⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⣿⠋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⢡⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠁⠀⠀⠀⢰⠁⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠙⢦⣤⢄⣀⡀⢤⣤⠔⠁⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠉⠓⠊⠉⠁⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⣄⡀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⢹⠀⠉⠓⠦⠤⠤⠖⠋⡟⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠹⡝⠀⠀⠀⠀⠀⠀⠀⠀⢉⡆⠀⢸⣅⡀
⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀⠏⡜⠉⠛⠲⢤⣄⡀
⠀⠀⠀⠀⡠⠞⠉⠀⠀⠀⠀⠀⢄⠀⠀⠀⠈⢧⡀⠀⠀⠀⢀⡞⠀⠀⢈⠞⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠱⡀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠄⠒⠒⠈⠉⠉⠉⠉⠉⠐⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣐⣶⣤⣀⡀⠀⠀⣷⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣠⣿⡴⠞⠛⠉⠉⣠⣤⣌⠻⠉⠛⠶⣾⣦⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡶⠂⠙⢋⡤⠒⠲⣄⠀⡞⠁⠀⠈⢧⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣸⠃⠀⠀⡜⠀⣠⡶⢾⢸⠁⢀⣾⠋⣿⠀⠀⠀⠀⠀⠘⡇⠀⠀⢰⣧
⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⣧⡟⠀⠀⠀⢧⡀⣿⣷⡿⠈⣦⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⠠⢤⡞⠁
⢠⠖⡇⠈⠟⢢⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠈⢙⣾⣉⠴⠀⠀⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢹⠄
⡼⡆⢻⡀⠸⠀⠇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠟⠋
⡇⢡⡀⠳⠀⠁⢸⣀⣀⡀⠀⠀⠹⡆⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠸⡀⠁⠀⣀⣀⣸⠃⢀⠠⠇⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠶⣤⣤⡾⠃⠀⠀⠀⠀⠀⢀⣼⠏
⠀⢳⡀⠚⠉⠀⠉⠀⢹⡀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠻⣄⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠉⠳⢦⣤⣀⣀⣀⣀⣀⣀⣠⣴⡾⠋⠙⣶⣤⡀
⠀⠀⠀⠈⠳⢶⡀⠘⣇⠀⠀⠀⠀⠀⠀⢀⣠⠴⢚⠋⠉⢫⠉⠉⠉⠉⢀⡼⠃⠀⠀⠘⣧⠉⠳⣄⡀
⠀⠀⠀⠀⠀⠘⣇⠀⠘⡆⠀⠀⠀⠀⡰⠋⠁⢀⡟⠀⠀⠈⠛⠶⠶⠖⠋⠁⠀⠀⠀⠀⢸⡆⠀⠈⢹⡀
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠸⡀⣠⠖⢛⡇⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⢸⡟⢦⡀
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠋⠀⣠⢼⣇⣠⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⠳⢾⣇⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠷⠤⠖⠋⠁⠀⠉⠁⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠈⢦⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣈⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢀⡜⠁⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠎⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣤⠿⠀⢡⣴⠖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⠒⠒⠒⠒⠒⠛⠛⠛⠉⠉⠉⠉⠉⠁⠀⠀⠸⣇⠀⠀⠘⠋⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠓⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢈⡶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣸⠃⠈⣧⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠄⠀⠀⠀⠀⣀⣀⣀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⣶⠒⠒⢶⠶⠶⠟⠀⠀⠘⠛⣿⠛⠛⢋⡟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠈⣧⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢸⡆⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣸⠁⠀⠀⠀⠀⠀⣿⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢀⡏⠀⠀⠀⠀⠀⢠⡏⠀⠀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣼⠃⠀⠸⡇⠀⠀⠀⠀⠀⢸⠷⣄⣀⣨⠏⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⣰⠶⣏⠉⠁⠀⠀⠀⠴⠖⠛⠳⢤⣤⣷⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠓⠀⠀⢈⣙⣢⣀
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣌⣙⠒⠤⠄⠀⢀⣀⢀⣀⡄⠠⡾⠀⠀⠀⠀⠀⠀⢿⡂⠒⠒⠛⠛⠉⢉⣉⣩⣽⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠲⠃⠀⠀⠀⠀⠉⠀⠀⠀⢢⢬⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⠄⢀⠔⠙⢌⠢⣀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠳⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⡡⠖⠁⠀⠀⠀⠑⢬⡀⢄⡄⠀⠀⠀⠀⢄⡢⠀⢈⢄⠀⠘⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢬⣐⠄⡀⠉⠢⠀⠀⠀⡠⠠⠉
⠀⠀⠀⠀⠀⠀⠀⠅⠶⠊⠀⠀⡤⠄⠤⠄⠀⠀⠀⢠⠄⠄⠠⠤⡄⠉⠒⠬⠴⠀⢒⡆⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢀⡄⢀⠀⠀⠀⠘⡄⢀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠈⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠄⠀⠄⠀⠄⠀⠀⠀⠐⡤⠠⠤⠤⢔⠀⠀⠂⠀⠀⡰⠆⠀⠀⡀⠂⢀⠢⣼⡔⠀⠐
⠀⠀⠀⠀⠀⡄⠀⡄⠀⡆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡇⠠⠂⣱⠀⠀⠘⠃⠘⠁⠀⢿⠀⠘⠁
⠀⠀⠀⠀⠀⣄⠀⣦⠠⡄⠀⠀⠀⡇⠀⠀⠀⠸⠀⠈⠉⠉⠉⠉⠀⠀⢰⠀⢠⠄⢸⡾⠀⣤⠀⠂
⠀⠀⠀⠠⠤⠥⠤⠴⠬⠾⠤⠤⠥⠧⠤⠤⠤⣼⣄⣄⣀⢀⠀⡀⣀⢀⠧⣄⡤⣤⣼⠭⣤⣭⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠉⠉⠉⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠉⢛⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠂⠀⠀⠀⣰⠏⠉⠛⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⡹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡰⠁⠀⣰⠃⠀⠀⠀⠀⠀⠈⠙⢦⣐⢦⣠⣀⠀⠀⡀⠀⠐⡀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣁⣠⠞⣁⣤⠶⠶⠶⠂⠀⠀⠀⠀⠈⠛⠿⣦⣵⣦⣐⡄⠀⢱⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡁⠈⠁⠀⠐⢋⣠⣤⣤⣤⡀⠀⠀⠀⠀⢀⣀⣀⠈⠁⠀⠈⠳⡀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢙⠃⠀⠀⡔⢡⣿⣿⣯⣀⡟⢂⠀⡠⢺⣿⣯⡟⢻⣆⠀⠀⠀⠙⠋⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠰⡈⠻⣿⡿⠟⡡⠃⠀⢣⡈⢿⣿⣿⡿⠈⡆⠀⠀⢲⡤⠞
⠀⠀⠀⡤⠄⠀⠀⠀⠀⢷⡀⠀⠀⠉⠓⠒⠒⢉⡤⠄⠀⠀⠑⢦⣭⣭⡴⠊⠁⠀⠀⣸⠁
⠀⢯⣑⡤⢐⣠⠖⡢⠀⠈⢷⡀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠑⠠⡏⠀⡏⠀⡎⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⡆⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠐⠒⠒⠄⣠⣐⢥⡀⠀⠀⠀⠀⠈⠙⠦⣄⣀⠙⠒⠚⠁⠀⠀⠀⣀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢠⠋⢀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠉⢛⣻⢻⣞⡛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣃⡴⣐⠁
⠀⠀⠀⠀⠈⠓⢧⡀⠀⠀⠈⠉⠛⠒⠒⠒⠒⡏⡸⢯⠀⢸⠆⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠃⠙⡄⠘⡏
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⡀⠀⣠⠀⠀⠀⠱⠀⠈⠷⠋⠀⠀⠈⠙⠒⠒⠒⠚⠋⠫⠈⢯⠶⣄⡡⠖⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢄⣀⣀⣀⣀⣠⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠐⠂⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⠤⣤⣤⣤⣼⣄⣀⣀⣀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡆⠀⢻⡀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⣷⠀⢸⡇⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⢸⡆⠠⣇⡀⠀⠀⢀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⠐⠒⠉⠉⠙⠛⢻⠁⢰⠋⠉⠉⠉⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠤⠤⢤⣤⣴⣶⠒⠒⠿⠀⠸⣓⠦⠤⠤⠴⣚⡇

⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣠⣾⣿⣿⣶⠤⠀⠀⠀⢀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠐⣛⣉⣤⡀
⠀⠀⠀⠀⠋⢻⣿⡿⠀⠀⠀⠀⠀⣾⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠐⠻⣿⣿⠋⠁
⠀⠀⠀⢀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠉⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⣻⣯⣀⣀⡀
⠀⠀⢠⣿⣿⣿⣿⣿⣷⠄⠀⠀⠀⣴⣾⣿⠆⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⡆⠀⠀⢀⣶⣾⣿⣿⣿⣿⣿⣷⡀
⠀⠀⣾⡿⣿⣿⣿⣿⣿⠀⠀⠀⢠⠛⠿⠿⠰⣿⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣿⣿⣿⣿⣿⣧⠀⠀⢸⣿⣿⣿⣿⣿⡿⠀⠙⢿⣦
⠀⢀⣿⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡜⢿⡇⠀⠀⠀⠀⠀⣠⣿⣃⣈⣿⣿⣿⡿⠁⣿⡀⠀⣸⡏⢹⣿⣿⣿⠁⠀⢀⣴⡿
⠀⠈⢿⡄⠈⢛⣿⣯⣿⠄⠀⠀⢀⡴⢀⠘⠄⣸⡇⠀⠀⠀⠀⠐⠛⠛⠉⠙⣿⣿⡿⠁⢀⣼⠇⢠⣿⠁⠸⠛⠷⣖⣤⡶⠛⠁
⠀⠀⠈⣷⠀⢾⣿⣿⣿⡇⠀⠀⠈⠀⠀⠀⠀⣿⠀⠀⠐⠤⣀⠀⠀⠀⣠⣿⣿⣿⣧⡾⠛⠁⢀⡿⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⣿⡄⢀⣿⣿⣿⣿⠀⠀⠘⠘⠀⠀⠀⣿⡀⠄⠀⠀⠈⠙⣷⢰⣿⣿⣿⣿⣿⠇⠀⢀⣾⠁
⠀⠀⠀⠈⠁⠘⣿⣿⣿⡿⠀⠀⠀⠀⠀⢀⠀⠈⠙⠀⠈⠢⣄⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠈⠃⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⡀⢠⣤⣿⡀⠀⣀⡤⠂⠀⠀⠀⠘⣿⣿⣿⠃⠀⠀⠀⢠⣿⡟⠀⢤⣴⣤⠄
⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⠀⢰⣿⣏⠙⠀⠉⠁⠀⠀⠀⠀⠀⠀⢻⣿⡏⠀⠀⠀⠀⠀⠻⠅⠀⢸⣿⣿
⠀⠀⠀⠀⠀⣰⣿⣿⠇⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⢀⣿⣿⡿⠀⠀⠀⠀⠀⣿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⣸⠏⣿⠇⠀⠀⠀⠀⢸⠏⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⡿
⠀⠀⠀⣠⡏⣠⣿⠀⠀⠀⠀⢀⣿⠀⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣿
⠀⠀⣰⡿⠠⢻⡿⠀⠀⠀⠀⢸⡏⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⡄
⠀⠘⠟⠁⠀⠸⣇⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠚⠋⠉⠉⠉⠉⠉⠛⠛⠒⠲⠶⠤⠤⠤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⢀⣠⠤⠶⠒⠒⠒⠒⠒⠲⠤⣄⡀⠀⠀⠀⠀⠳⢤⣀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⡴⠋⠓⠒⠋⣀⣀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣯⠲⣄⡀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⠀⡜⠀⠀⠀⠰⠛⣉⡉⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⠁⠀⠉⢳⡒⣶⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⡔⠉⢀⣀⠑⡄⠀⠀⡰⠈⢀⣀⠉⢢⠀⠀⠀⠀⣧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡆⠀⠀⠀⠀⣇⠀⢿⣧⣥⠇⠀⠀⢇⠀⣿⣧⡆⢸⠀⠀⠀⠀⡿⡸
⠀⠀⠀⢀⣠⠔⠒⡒⠒⠲⢤⣀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠉⠓⠖⠛⠉⡠⠴⠢⣌⠓⠤⠥⠒⠃⠀⠀⠀⢰⢣⠇
⠀⢠⠞⠉⠉⣉⠟⠣⣤⡒⠉⠉⠑⢄⠀⢠⠖⠒⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠻⡉⠀⠀⢀⣿⡯⠤⢄⡀
⢰⠗⠚⠉⠉⠀⡠⠊⠀⡹⢲⡉⠐⠺⣇⢻⡀⠈⠈⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠘⡡⠀⠀⠀⡇
⡏⠀⠀⠀⢀⠜⠁⠀⢀⠀⠀⠹⡄⠀⢸⡄⠉⠋⠙⠢⠔⠣⣄⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠤⠚⠁⠀⣠⠶⣄⠀⣠⠖⠚⠁
⣇⠀⠀⢀⠎⠀⠀⠀⠸⠀⠀⠀⢱⠀⢸⠃⠀⠀⠀⠀⠀⠀⠈⠑⠦⢄⣀⡀⠀⠀⠀⠀⢴⣒⡤⠔⠊⠁⠀⠈⠉⠁
⠸⡄⠀⡎⠀⠀⠀⠀⡄⠀⠀⠀⠈⣅⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⡏⠉⠁
⠀⠙⢶⡀⠀⠀⠀⠀⡇⠀⠀⢀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢢⢶⠊⠀⠙⡶⠶⠒⠢⡀
⠀⠀⠀⠉⠓⠲⠤⠼⠤⠔⡾⠋⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⠒⣬⠈⠦⠄⠔⠁⠅⡌⢆⠘⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡑⣤⡀⠤⡄⠀⠀⠀⠀⡜⢡⠃⢠⠃⠀⠀⠀⠀⠀⠈⢣⠀⣑⠆⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠡⠵⣤⠂⠳⠀⠒⠉⢉⠤⠃⠀⠸⠀⠀⠀⠀⠀⠀⠀⠸⢪⡔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⡐⢘⢢⠀⡆⠐⠊⠁⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢸⡀⠈⠧⢂⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠤⠤⠜⠁⠀⠀⠀⠀⠀⠀⢰⠉⠉⠀⠉⠉⠉⠉⢡⠈⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡀⠀⣿⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠬⡇⠀⠀⠉⠹⡉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠠⡇⠀⠀⠀⠀⡵⠚⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠼⠦⠤⠇⠀⠀⠀⠀⠓⠒⠻⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠃⠤⠤⠠⢄⡞⠀⠀⠀⢠⠛⠚⠓⠒⠚⢳⡀

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠂⠀⡄⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⣠⣤⣀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⠂⠀⠀⠀⠹⠉⠛⠋⢁⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡞⠻⣤⣶⣿⣁⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢀⣤⠀⠀⡄⣄⡝⢃⡼⠷⠃⠈⠤
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⣧⠄⠠⠀⠹⣶⠁⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠘⠡⣀⠀⢠⣿⣟⡀⠠⠀⠀⠠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣷⠂⠀⠀⢸⣏⢷⠈⢢⣜⣳⢆⣠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⡀⠀⢰⣿⡇⠀⠀⠈⣿⢸⡇⡄⠙⢣⣴⠋⠑⢆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢤⣾⠿⠃⠀⠀⠀⠈⢸⠇⡇⠀⢈⠁⢀⣉⡋⠓⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠙⢷⣼⣥⠀⠀⠀⠀⠀⠀⣸⠀⡇⢀⣼⠟⢩⣤⣿⠓⠀⠀⠀⠑
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⠌⢻⡀⠀⠀⠀⠀⢀⡇⠀⠁⢹⠿⠉⠉⠛⠋⢦⠀⠀⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⡘⠀⡀⢠⠌⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⣜
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠰⣤⠠⠉⢹⣰⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠘⠧⠤⣀⣀⣀⠴⠇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢰⠆⠀⠄⠀⡤⠤
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⡁⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠀⠸⢇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⠆⠹⠇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⡜
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⢸⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠉⠑⠲⠤⣀⣠⠤⠖⠘⠉⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠲⣤⠶⣇⠀⠀⠀⢠⠖⠂⠒⠒⠋⠀⠀⠀⠀⠙⠂⠒⠒⠢⡄⠀⠀⠀⡸⠶⣤⠲⠤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠘⡄⠘⣆⠀⣠⠋⠀⡤⠖⠂⢤⠀⠀⠀⢀⠤⠒⠲⣄⠀⠹⣄⠀⡰⠁⢠⠃⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠘⣄⣼⠁⠀⠀⢸⠁⢰⣿⡆⠁⠀⠀⠀⢰⣿⡆⢘⡇⠀⠀⠈⣧⡠⠃⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢸⠈⡆⠀⠀⠈⠳⠤⠠⠔⢁⣠⣄⠈⠦⢤⡤⠞⠁⠀⠀⢰⠁⡇⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⣤⠶⠚⠉⠀⠀⠀⠀⠀⢀⡰⠃⠀⠘⣄⠀⠀⠀⠀⠀⣀⡈⣁⠈⠁⢀⠀⠀⠀⠀⠀⣠⠃⠀⠘⣄⠀⠀⠀⠀⠀⠀⠉⠓⠶⣤
⠀⠀⠀⠀⠙⠲⠤⣀⣀⣀⠠⠔⠋⠀⠀⠀⠀⠈⠣⣀⠀⠀⠈⢯⡉⠉⠉⢉⡝⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠙⠢⠄⣀⣀⣀⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠓⠢⠤⣀⣉⡒⣒⣉⣀⠠⠔⠚⢥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠦⣀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠔⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠈⠓⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠚⠁⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠈⠙⠛⠒⠒⠶⠶⠒⠂⠛⠋⠉⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⢤⣄⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡤⠰⠶⢇⠀⠀⠀⡙⠋⠙⢋⠁⠀⠀⡸⠒⠢⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⢀⡤⢦⡶⢿⠀⠀⠀⣇⠀⠀⣸⠀⠀⠀⡿⢶⡤⢤⡀⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⡜⠁⠠⡋⠀⠈⠀⠀⢀⠟⡃⢸⢿⡀⠀⠀⠀⠀⢹⠄⠈⢢⠀⠀⠀⠉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⣄⠀⠀⠉⠒⠒⠒⠊⣡⠜⠁⠈⠣⣉⠒⠒⠒⠒⠉⠀⠀⣰⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠈⠙⠲⠶⠲⠶⠒⠋⠁⠀⠀⠀⠀⠈⠙⠒⠶⠶⠶⠖⠋⠁⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠒⠒⠒⠀⠈⠉⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣛⣛⣻⣛⣛⣫⣘⡛⣿⣛⡛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣹⣏⣿⣭⣉⣹⣿⣿⣍⣛⣛⣛⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⢀⣠⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⣼⠏⣠⠶⠛⠉⠀⠀⠀⠀⠈⠙⠛⠳⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣹⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠸⡇⠀⠀⠀⠀⠸⣅
⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣤⠀⣴⣴⠟⣿⣶⣶⣦⠀⢳⣄⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢀⣿⣶⣿⡿⠋⠉⠀⠛⠁⠀⠈⣷⣾⣿⡆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢸⠉⣷⡞⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢡⠀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⢰⡆⠀⠀⠀⠀⠀⠀⢸⣇⣰⠇⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡏⠀⢨⠇
⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠈⠁⣌⠀⣿
⠀⠀⠀⠀⠀⢸⡏⣷⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠐⣿⠆⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⢏⡸⠁
⠀⠀⠀⠀⠀⠈⢷⡘⢷⡀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠘⢷⡈⠁⠠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠤⣷⣄⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠓⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣳⣦⣤⣀⣀⠀⠀⠀⠀⣀⣠⣤⠖⢻⣷⠀⠀⠀⠀⠙⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠙⢿⣄⠈⠉⠛⠓⠒⠉⠉⠁⠀⢀⣼⡟⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠙⠻⠶⢶⣤⣀⣀⣠⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⣠⣜⣹⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣾⣿⣿⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⠳⠦⣤⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠳⠶⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣻⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢀⣿⣷⣶⣆⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⣶⣿⠿⠟⢹⣃⣦⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠴⣶⠀⣿⠁⠀⠙⠛⠛⠿⠿⠿⠟⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⠛⠋⠉⠀⠀⠀⠀⠸⣿⢃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⢳⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⠀⣀⣀⣀⣠⣤⠞⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣀⡴⠶⠶⠶⢦⣀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⣀⡴⠞⠛⠛⠛⠛⠷⣤⡾⠋⠀⠀⠀⠀⠀⠉⢻⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢺⣯⣤⢀⡀⠀⠀⠀⡀⣼⣿⣤⣤⣤⣤⣤⡤⠶⢛⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⢿⣶⣶⣿⡿⠟⠀⠈⠉⠉⠐⠶⠶⠶⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠉⠉⠉⠉⠉⠉⠉⠑⠪⢧⣼⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡠⠀⠀⢀⠠⠔⠂⢩⣿⡧⢦⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⢸⠁⡠⠊⠁⠀⠀⠀⠉⠁⠀⣴⡯⢲⠂⠀⠀⡖⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⢸⡐⠁⣀⣀⠀⠀⠀⠀⠀⠀⠉⠀⡇⢀⡄⢸⠇⡀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⢀⡠⠔⠒⠂⣒⣒⢄⠀⠀⠀⠸⠘⡿⠁⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⢴⠓⠀⡒⢄⠸⣱⣎⠇⠑⡄⠀⢠⠃⣃⠞⢉⣳⡄⠀⠀⠀⠀⠀⠀⠀⠛⣷⣸⡇⠘⠆⣧⠀⠀⠀⠀⢠⠀⠀⢠
⠀⠀⠀⡎⢧⠐⡺⡢⣸⣀⠬⠐⠊⢉⠝⠀⡆⢸⠀⢰⣾⣿⡇⠠⡄⠀⠀⠀⢀⣀⣀⠸⠉⣿⡀⠀⢿⡀⠀⠀⠀⢸⠀⠀⠸
⠀⠀⠠⢀⠏⠑⠒⡶⢋⣀⡤⠔⠚⠓⡆⠀⢰⠆⠀⠘⠛⠛⠁⠘⠁⠀⠀⠀⢉⣩⣿⣷⡵⡈⢷⡀⠘⣇⠀⠀⠀⡟⢠⢃⠇
⠀⠀⢸⠉⠀⠀⠘⠞⣭⡤⠧⣤⠴⠊⠀⠀⠘⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣾⣿⣿⣿⠇⠘⢌⠳⠀⠘⠂⠀⠘⣠⠋⠎
⠀⠀⠀⡀⠀⠀⣀⣀⣀⠠⠏⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢠⣤⢀⡉⢀⠀⠀⠈⠙⢋⠁⠀⠀⠀⡱⠀⢀⡀⠤⠤⠥⡎
⠀⠀⠀⣏⠉⣁⣀⡀⠤⠐⢺⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠘⢟⡒⠐⣿⠇⠀⠀⠀⠀⠀⠀⠀⡜⠀⡔⢁⣴⡛⠙⠲⠱
⠀⠀⠀⢱⠤⠀⠒⠒⠈⢉⡠⢇⠀⠀⠀⣀⣀⡠⠤⠬⠷⣄⡀⠬⠬⠅⠀⠀⠀⠀⠀⠀⠀⠀⠑⠎⡀⠈⠀⠟⢀⡴⠃
⠀⠀⠀⠘⠂⠒⠂⠈⠉⠀⣀⣼⣤⢊⣥⡈⠉⠀⠸⣿⠞⢩⢟⠑⠲⠤⢄⣀⣀⣀⡀⣤⠤⢒⡚⠉⠀⠉⣡⠍⢹
⠀⠀⠀⠀⢷⣲⣲⣲⣺⠽⠜⠺⣹⠈⠛⠁⠺⣿⠞⠁⠀⠘⢌⡒⠤⠤⠔⡫⠔⠁⣩⣮⣦⡈⠉⠻⣦⣼⠃⢀⠃
⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠈⡤⣿⠆⣰⠋⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠀⣔⠉⠛⠉⢻⣖⢤⠈⠃⣀⠧
⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠱⡀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠿⠃⢠⣤⠘⠛⠳⣌⠻⡏⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣀⡀⠘⠛⠀⣴⣄⢹⣇⠀⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢲⠇⠀⠀⠀⠀⠀⡤⢄⡀⠀⠀⠀⠀⢸⠀⠻⠃⠀⣠⡀⠉⢁⡸⠫⣝⡧⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣠⠤⣙⢦⠙⠶⣄⠀⠀⢸⣰⣶⡄⠀⣛⠧⠞⡏⠀⠀⠈⠻⠑⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⢉⣒⣤⡉⠁⠸⡤⣠⣤⠴⠛⠛⠛⠛⠶⠀⠀⡇⠀⠀⠀⠐⢆⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠓⠠⢤⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠥⠤⠤⠤⠒⠤⠤⠤⠤⠤⠤⠤⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣠⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠛⠋⠛⠋⢉⣿⣿⠿⠟⡋⠉⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢖⣋⠀⠀⠘⠉⠀⢀⣨⣉⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⣏⣤⣷⠀⠀⠀⠀⣟⣠⣬⡇⢸⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠻⣿⣧⠻⠿⠟⠐⠋⣉⡷⠈⠛⠛⠁⠈⢋⡅⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠗⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⢰⠄⠀⠀⣌⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠛⠦⣄⡀⠀⠈⠁⠒⠒⠈⠁⣀⣤⣞⣀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢲⠒⠒⠲⡚⠋⠁⣘⡅⠋⠀⡝⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢟⣤⣀⣤⣿⠦⡸⡅⠘⠆⡄⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠉⠉⠉⠁⠀⠑⡝⣦⠈⣁⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢰⢠⠴⠒⠒⠂⠑⡆⡠⡻⠁⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠳⡎⣼⠀⠀⠀⠀⠀⡇⣷⠁⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡰⠁⠈⠛⠛⠛⠒⠚⠁⠘⡆⢀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⠀⣀⣠⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠙⠒⣶⠿⠋⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣇⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⠀⢸⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣤⣿⣄⣀⣸⣀⣀⣼⣥⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠉⠁⠈⣹⠉⠐⠋⠁⠀⠈⡧
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣒⣒⠒⣒⣋⣉⣓⢢⣤⣤⡴⣚⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠐⠂⠀⡓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢓⠲⠲⠆⠀⠀⠀⠉⢀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⡀⠀⠀⠀⠎⠁⠑⢄⠡⠵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠬⢒⠏⠁⠀⠒⠀⠀⠀⠀⠂⠡⡀⢨⡄⠀⠀⠠⢊⡉⡑⡄⢄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠆⠈⠀⡀⠄⠤⣴⠀⠀⡕⣐⠁⠀⠔⡐⠃⠀⡴⠚
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⠐⠤⣀⠈⠀⠀⠀⣀⠀⠀⡄⠘⣦⢊⠔⡀⠔⠈⠢⡤⠄⠀⠀⠀⠁
⠀⠀⠀⠀⢀⡀⡈⠀⡠⢬⠅⣂⠘⠶⠀⠀⠀⠁⠀⠀⠠⡀⡠⠋⠀⠀⠀⠀⠀⢈⠒⠠⠒⠙⠀⢀
⠀⠀⠀⠈⠀⠀⠛⠒⠂⢤⠀⠄⠨⠁⠒⣤⡃⠀⠀⠀⠀⣼⣿⡆⠀⠀⠀⠀⠀⡏⠖⠃⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡤⡎⠀⠀⠀⠀⠐⣿⣷⣤⣤⣴⠾⠛⠛⠣⡀⠀⠀⠀⢀⠡⠔⠁⠤⢘⠊⢠⠃
⠀⠀⠀⠀⢰⠀⣀⡐⠓⢬⣆⠀⠀⠀⠀⠙⠏⠁⠀⠐⠀⠀⠀⢀⠀⠀⠀⢀⠌⡠⡀⢀⡠⢂⠔⠁
⠀⠀⠀⠀⠈⡄⢡⠀⠢⠠⠣⠑⢄⠀⠀⠀⣄⣀⡤⠊⠉⠁⠚⠫⠭⠈⠥⠋⡐⡡⠊⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠈⢆⡎⢡⡉⠅⢀⡑⠪⠥⠴⢊⠤⢀⢀⣀⢠⠘⠁⢒⠒⠚⢃⡠⠊
⠀⠀⠀⠀⠀⠀⠁⠄⢘⠀⡀⢀⣄⡈⠁⠀⠀⡁⣀⡀⣀⢥⠀⠫⠙⠁⠀⠀⢀⡂⠀⠀⡠⠢⡄⠀⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡚⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠐⠇⡀⠀⠰⠁⠀⢸⣷⡋⠦⢀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢲⠃⠀⠀⢰⣿⠉⠀⡈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢀⣸⡿⠀⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⢀⡀⠤⠒⠅⠀⠀⢀⠌⠱⡘⣿⠇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠉⠀⠀⠀⠀⠡⣈⡤⠐⠁⠀⠀⠫⣞⠔⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠀⠀⠀⠈⠉⢱⡀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⣤⣤⣤⣴⠦⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⢱⠨⠟⠓⠛⠛⠿⡿⢿⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠈⠀⠀⠄⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠚⠛⠉⠉⠉⠙⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⢀⠟⠁⡀⢀⠀⠀⠀⠀⠀⢀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢀⣠⠤⠶⠶⢤⣄⡀
⠀⠀⠀⣰⠃⠀⡜⢠⠃⠀⠀⠀⠀⠀⢸⠀⣿⠦⣀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣴⠟⠂⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⢰⠇⣀⣀⣇⣸⠀⠀⠀⠀⠀⠀⠈⢇⢘⣦⣈⠑⠢⢤⣀⣀⠀⢸⡗⢺⣥⡖⠂⠈⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠈⠉⠉⣹⠹⣿⣿⣹⣿⢲⠒⠒⣶⣿⣿⣙⡏⡦⠀⠀⠀⠉⠉⡙⠋⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⢨⠦⣌⡭⠵⠃⠈⢧⡀⠙⠿⠿⢟⡕⠁⠀⠀⠀⠀⠀⢳⡶⠃⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢶⡂⠀⠀⠉⠙⠲⠒⠃⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠙⢧⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⡆⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠑⠤⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡇⢘⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢸⠁⡜⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠬⡉⠟⠳⣻⠛⠻⠭⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠂⡰⠁⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠵⡤⢀⢀⠤⣀⠐⠴⢅⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢠⠀⠀⢠⢸
⠀⠀⠀⠀⠀⠀⠀⣀⢼⣄⣸⠀⠀⠀⠀⠀⠀⢧⢀⣩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣴⡄⠘⢌⢣⡀
⡄⣀⠒⠀⢠⠠⠄⠂⠁⢈⠇⠀⠀⠀⠀⠀⠀⢸⡍⠐⠮⣢⢄⡀⢀⡠⠡⠀⠤⠀⠀⠀⠀⠀⠉⠛⠶⠤⠽⠿⠛
⠐⠆⢒⠀⠁⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠈⠉⠚⠫⡀⠁⡩⡖
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠈⠀⠒
⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⣄⣀⣀⣀⣤⣤⣀⣀⣀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⡝⠸⠀⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠋⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠸⠀⣴⠧⡸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⠻⠿⢿⣿⣿⠿⠿⠛⠉⠀⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠐⠉⠉⠀⠀⠀⠀⠀⠉⠙⠓⠀⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⠿⠀⠀⠀⢨⣦⡀⠀⠀⠀⠈⣀⣿⡆⢈⣫⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠛⠀⠀⠰⣿⠿⠃⠀⠀⠀⠀⢿⠿⠃⢸⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠭⠶⠂⠤⠄⠲⠬⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠥⠆⠐⠫⠒⠤⠄⣪⠓⠈⢲⠴⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠉⠀⠀⡂⠀⠀⠀⠀⠀⠁⠠⠄⢸⠀⠀⠘⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠃⠀⠀⢀⠔⡇⠀⠀⠀⠀⠀⠀⠘⠁⠙⢆⠀⠀⠀⠉⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠴⠣⠾⣉⣀⠓⢲⡀⠀⠀⠀⠀⠠⠖⠒⡌⣿⠠⠮⠷⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢨⣀⡠⠀⠀⠀⠀⠀⠐⢄⣠⠗⢤⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡠⠊⡐⠀⠀⠀⠀⠀⠀⠀⠀⢄⠢⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⡌⠀⢠⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠖⠂⠀⠀⠀⠑⠰⠖⠒⠈⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⢺⠀⠀⠀⠀⠀⠀⠀⢸⠀⢐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡍⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠡⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢀⠀⣀⡼⠃⠀⠀⠀⠀⠀⠀⠳⣄⡀⢀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠤⠃

⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣶⣶⣶⣶⣶⣶⣷⣶⣿⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣷⣶⣿⣷⣿⣿⣷⣾⣶⣿⣶⣿

⠀⠀⠀⠀⠀⠀⢀⡀⠒⠀⠐⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⡀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋⠉⠓⠂⠠⠀⡀
⠀⠀⠀⠀⠀⣀⠉⠀⠀⠐⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢈⠂⠒⠒⠒⠒⢀⠈⡆
⠀⠀⠀⠀⠀⠙⠇⠈⠇⠀⠠⠠⠒⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⡰⠄⡔⢀⡀⠀⠀⡀⠀⡘⠀⠀⠀⠀⠀⠀⠀⡌⠘⠀⠀⠀⡤⠀⢀⠃
⠀⠀⠀⠀⠀⠀⣆⡀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⠃⠰⠁⠦⠈⠈⠁⠚⢠⠴⠀⠇⠀⠀⠀⠀⠀⠀⢰⠀⠀⣀⣀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣠⡞⠀⠘⣀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠁⡄⠀⡀⠌⣀⠀⠉⢀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠄⠀⠰⠊
⠀⠀⠀⠀⠀⠀⠈⠀⠐⠊⠿⠁⠀⡠⠁⠀⠢⡀⠀⠀⠀⠀⠀⠈⠚⠉⠢⢌⣐⠒⠂⣠⠆⢀⡃⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⡄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠐⠒⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠇⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⣀⠤⠐⠒⠂⢄
⠀⠀⠀⠀⠀⠀⢀⠔⠒⠉⠀⠀⠀⠈⠳⢤⠀⠀⠀⠀⠀⠀⠀⣀⡰⢤⠴⠒⠛⠉⠉⠉⠒⠲⢤⣀⠀⠀⠀⠀⠀⢖⠶⠈⠉⠁⠀⠀⠀⠀⠀⠀⠉⠲⡀
⠀⠀⠀⠀⠀⢰⢁⡀⠀⡀⢀⡀⣠⣤⡄⠈⡇⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⢀⢀⡀⢀⣀⠀⢀⣀⠀⠩⡇⠀⠀⠀⠘⠥⣀⠀⠀⠤⠴⠒⢹⢶⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠧⠀⠀⠈⣉⣉⠀⠸⠀⡒⢣⠀⠀⠀⠀⠀⢸⠁⠀⠀⡜⢁⠤⠍⠉⠀⢉⣥⣀⠙⡖⠃⠀⠀⠀⠀⠀⡇⠀⢠⣦⠀⣤⡄⠈⣶⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠘⣯⡅⠀⠰⠖⠀⠀⠀⠁⠘⠀⠀⠀⠀⢠⠞⡲⡆⠀⡇⠀⢀⣤⠀⠀⢀⣠⠀⢠⠇⠀⠀⠀⠀⠀⠀⡇⠀⢺⣯⠀⣿⡁⠀⠸⠔⣚⢢
⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⢀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠸⣄⡁⠈⠛⠃⠀⠘⠋⠀⠠⡈⠛⠀⢸⠀⠀⠀⠀⠀⠀⠀⠇⢀⣈⠉⠀⠉⢁⡀⠀⠀⠈⡰
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠤⠵⠂⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠈⠙⢄⠀⠀⠀⠀⠠⣄⣐⣋⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠘⠣⢏⣞⣺⡹⠟⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⢀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⠤⣄⣀⣀⣠⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠠⢄⡀
⠀⠀⠀⠀⠀⡠⠖⠈⠉⠉⠁⠀⠒⠒⠒⠂⣙⡠⠊⠁⠀⠀⠀⠀⢀⠴⠁⠀⠀⠀⠀⠀⠈⠁⠂⡀⠀⠀⠀⠀⠀⢀⡤⠖⢻⠏⠉⠛⠲⢤⡀
⠀⠀⠀⢠⡏⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠅⠀⠀⠀⡇⠀⠀⡘⠋⠉⠉⠒⠀⠀⡀⢰⠀⠀⠀⠀⢠⡏⠀⢀⡏⠀⠀⢀⣀⣀⣹⣆⣀
⠀⠀⠀⠀⢉⠀⠐⠁⠀⠀⠉⠉⠉⠒⠒⠒⠂⢹⣄⡴⠃⠀⠀⣠⠤⣀⡎⢀⣀⣀⠀⠀⠠⠤⢸⠎⠀⠀⠀⠀⣿⠀⣀⣸⣴⠞⢫⣍⠉⣵⡚⢻⣿⣿⣷⡄
⠀⠀⠀⢀⠘⠄⡀⠀⠀⠀⢽⣷⡄⢀⠂⠹⢿⣦⡇⠀⠀⠀⠀⠇⠀⠉⠀⠀⠐⠋⠀⠐⡄⠁⢸⠀⠀⠀⢠⠖⣛⠻⣩⣤⠟⠀⢈⣿⠂⣻⡷⠘⣟⠛⠉
⠀⠀⠀⢠⡈⢇⡀⠀⠀⠀⣰⡿⠃⡈⠢⢰⣿⠟⡅⠀⠀⠀⠀⠈⠐⡇⠀⡀⠀⠀⠄⠀⡳⠀⠘⠀⠀⠀⠸⣄⠀⣀⠀⠀⠀⠀⠈⠋⠀⡙⠃⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⡉⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠠⠤⠤⠀⠀⡘⠀⠀⠀⠀⠈⠙⢧⡄⠀⠂⢀⠀⠠⠤⠄⠀⠄⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⡀⠀⠀⠁⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠀⠀⢀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠉⠳⠷⣤⣤⣦⣤⣶⡾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠖⠒⠒⠒⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣛⣈⣁⣀⣂⡥⠺⣄⢄⠀⠰⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⢹⠀⠀⢤⡄⠀⠀⢈⠓⢤⣀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣦⣤⣼⠀⠀⠀⢀⡄⠀⠸⠗⢀⡷⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⡄⠀⠐⠤⠀⠀⢀⡴⠏⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠢⠤⠤⣴⠚⠁⠘⠛⠶⠷⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠂⠻⢍⡀⠀⢀⡹⠋⢣⡴⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠉⠢⣀⠠⠛⠻⡉⢀⡠⠋⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⢀⡠⠊⡴⡅
⠀⠀⠀⠀⠀⠀⠀⢀⠟⢎⡷⢄⣀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡧⡜⠉⠀⠘⡀
⠀⠀⠀⠀⠀⢀⠖⠁⠀⢀⡝⢋⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠘⢆⠀⠀⠳⡀
⠀⠀⠀⢀⠔⠁⠀⠀⡠⠊⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠈⢆⠀⠀⠙⢄
⠀⠀⡰⠁⠀⠠⣔⠈⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⢧⡀⠀⠈⣇⠀⠀⠀⡴⢒⣒⣄
⠀⠀⠃⠀⠀⢠⡎⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠉⢢⡀⠈⠓⢤⡀⢧⣏⠤⠛⠂⠒⢤⡀
⠀⠀⠳⠶⠴⠋⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⣴⠊⢋⠀⠄⡀⡇⠌⠀⠰⡦⠀⠀⠀⠙⣆⣀⣀
⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠻⣄⠈⠙⠶⠶⣹⢀⠖⢻⣶⣄⢠⣤⠀⢸⢓⡌⡇
⠀⠀⠀⠀⠀⢘⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⠳⢄⣀⡤⠛⡘⣶⠚⡍⠈⠇⠁⠀⡸⠿⠚
⠀⠀⠀⠀⠀⠀⠙⠐⢟⣰⡤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣿⠂⠀⢨⠏⠀⠀⠘⠪⣙⡛⠂⢁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⣤⣏⡹⠒⣶⠂⣤⢤⡤⢤⠤⡔⢲⣊⡻⠜⠉⠁⠀⢠⠏⠀⠀⠀⠀⠀⠀⠈⡝⠙⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⣿⢛⠀⠊⠁⠉⡏⠈⠁⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡇⢸⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢀⡼⡇⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠸⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠎⠀⢙⣄⣀⠀⢀⣀⡠⠞⠉⣆⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡜⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⡞⠀⠀⡞⠀⠉⠙⡍⠁⢡⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢰⠁⠀⠀⡇⠀⠀⡎⠀⠀⠀⠀⠀⢰⠁⠀⢰⡇⠀⠀⣰⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡟⠁⠀⣀⢾⠀⠀⣸⢧⠀⠀⠘⣶⣀⠀⠀⠀⠈⠦⠤⠟⠀⠀⣴⠃⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠙⠒⠊⣡⠊⠀⠀⠘⠦⡉⠒⠚⠁⠀⠉⢢⠀⠀⠀⠀⠀⠀⠀⠙⠤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣠⠔⠁⠀⠀⠀⠀⠀⠈⠲⣄⡀⣀⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣿⣽⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣟⠁⠲⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⠁⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣆⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣷⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠿⡿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠿⠿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠉⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣶⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⢿⣿⣿⣿⣟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣮⣿⣿⣿⣿⣿⣿⡿⣿⣿⣷⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣋⣱⣾⣿⡿⠏⠙⠛⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣠⣾⡿⢻⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢀⣾⣟⣿⡟⠙⣻⣿⣏⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣭⢠⣿⣿⣿⣟⡏
⠀⠀⠀⠀⢠⣾⣿⣾⣿⢱⣿⣿⣿⣿⣿⣿⠟⠉⣁⣿⣿⣷⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⠏⣿⠉⠉
⠀⠀⠀⠀⠈⠻⣿⣿⣿⡜⣿⣿⣿⣿⣿⣴⡶⣾⡿⣡⣿⣿⡿⠿⢿⡿⠸⣿⣹⣿⢍⣀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⢸⠿⣿⢷⣼⣿⡟⠻⢷⣿⣄⣿⣿⠿⠛⣁⣀⣀⣉⣀⣤⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠻⠄⠀⢠⣿⣿⡿⠶⣾⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡿⠿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣇⠀⠙⢉⣿⠛⢿⠿⢿⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠿⢿⣿⣦⠴⠟⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⣷⣦⣤⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠭⠉⠉⠀⠀⠀⠀⠀⠀⠀⠚⠿⠿⠿⠏⠉⠋⠹⠿⠿⠶⠆
⠀⠀⠀⠀⠀⠀⠿⠋⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⣀⠀⠙
⠀⠀⠀⠀⠀⠈⠹⣶⣄⣀⣀⣠⣾⡿⠃⠀⠀⠀⠀⠀⠀⣴⣷⣄⣀⣀⣤⣶⠟⠀⠀⠀⠀⠀⠀⢾⣿⣷⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠃⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠉⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁⢠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡄⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣿⣿⡇⠀⠀⠀⠀⠸⣿⣧⡤⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠃
⡿⢿⢿⠿⠿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠀⢉⠉⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠉⠁⠀⢀⡴⢏⡘⠙⡆⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⡀⠀⢀⠴⠯⡀⠈⢧⣀⠼⡀⠀⠀⠀⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠏⠉⡰⢋⣠⣍⠁⠀⠀⠈⣉⣷⡀⢄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⢆⠀⠀⠀⠀⢿⣀⡸⠇⠈⠯⠸⠃⣀⣀⠸⠿⠥⠳⣄⣱⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢳⡈⠧⢸⠀⠀⠀⠀⠀⠈⢦⣀⠀⠀⢠⣀⣁⣈⢀⡄⠀⠀⢸⡧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠩⡙⠂⠀⠀⠈⠦⢀⠦⠀⠀⠀⠀⠙⠦⣀⠀⠁⠀⠀⠚⠁⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢈⣹⠶⣶⣶⠖⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠇⠀⠀⠀⣠⠴⠊⠉⠻⣧⣌⣀⣀⣉⣹⠶⠶⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣀⣤⡞⠁⠀⡀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠄⠙⣿⡀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢀⣀⡠⠼⢷⡞⠈⠉⠉⠉⠉⠛⠒⠒⠒⠒⢲⡆⣀⢤⣼⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⡸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣯⠖⠉⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡆⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢹⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣤⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⡞⠀⢸⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠁⠀⠀⢸⠇⡇⠀⠀⠳⣄⢸⠀⢳⡤⠚⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠈⢻⠠⢸⡆⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠈⠆⢹⡗⠤⠖⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡟⡇⠀⠀⠀⠀⠘⡌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⣄⣀⣀⣀⣠⠃⢻⣀⣀⣀⣀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸⠁⠀⠀⠀⠘⡄⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡼⠀⠀⠀⠀⠀⢣⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡦⠤⠴⠇⠀⠀⠀⠀⠀⠸⡦⠔⠚⣹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⠚⠛⢤⡑⠒⢒⣃⠀⠀⠀⠀⠀⠀⣩⠐⠚⡡⡞⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠁⠈⠂⠓⠒⣉⣀⣼⣏⣿⠀⠀⠀⠀⠀⠀⢹⣵⣙⠌⠑⠶⠜⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠭⠭⠭⠭⠍⠐⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠦⣒⠀⠤⢚⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⢢⣴⠇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢠⢰⠶⣿⠘⣷⡎⠸⣧⡾⣫⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠙⡈⣮⡤⢛⡿⠛⠛⠓⢾⣿⢋⣠⠀⠀⠀⠀⠀⠀⣠⠀⢰⠆
⠀⠀⠀⠀⠀⠀⢰⡀⠀⣴⠀⠀⠀⣤⣈⠑⣷⣯⡿⠓⠀⠀⠀⢠⠴⠙⢿⣶⣬⣄⠀⠀⠀⠀⢸⣇⡟⣠⠴⠂
⠀⠀⠀⠰⢦⣀⠈⢷⣼⣇⠀⠀⠸⠒⠹⢎⣾⠋⣴⣄⠀⠀⠀⣠⠞⠓⠈⣿⡏⠁⠀⠀⠐⠢⣼⠟⣿⣡⣀⡀
⠀⠀⠀⠀⢀⣈⣽⣾⣇⣽⡆⠀⠈⠉⢰⣾⠃⠘⠿⠟⠀⠀⠀⢃⠺⢇⠀⣸⣇⠀⠀⠀⠀⢀⡿⠟⠁⠀⠈⠁
⠀⠀⠀⠀⠈⠉⠁⢸⠟⠋⢿⠳⡀⠀⢸⣿⠀⠈⠠⠐⠀⣉⣁⠀⠉⠁⠀⢸⣿⠆⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠱⡙⢦⡈⢧⢣⡀⠀⢠⣾⡟⠻⣿⡆⠀⣠⣿⠟⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⠛⢮⡣⡙⢦⣜⠳⠾⠿⠛⣀⡴⠏⢁⣤⣾⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣇⠀⠻⠏⢿⡿⢭⣶⣦⢾⡋⣡⡾⠉⢙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠻⣄⣰⠃⠀⠙⠟⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣃⠀⠀⠈⠿⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠖⠶⠶⠶⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠻⣖⣀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⢀⣼⢻⣏⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣷⣦⡾⠃⢸⣿⣄⣠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠉⢹⣯⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⢦⣀⣀⣴⡿⠃⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣶⣦
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡴⠖⠛⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡴⠶⠚⠉⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⢦⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⢠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠁⠀⠀⠠⣄⣀⠀⠀⠀⠁⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⢠⡄⠀⠀⣀⠀⠀⠙⢦⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⠀⠀⠀⠀⠀⠠⠀⠐⠈⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⢀⠀⣿⢠⣶⡐⠀⠀⠀⠀⠀⠀⠹⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⡁⠀⠀⠈⠉⠀⠈⠀⠀⢠⠄⠀⠀⠀⠀⠓⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠈⠙⠒⠒⠋⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⢰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⠀⠀⠈⠂⠂⢀⡠⠴⠋
⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡋⡋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠑⠂⠀⠢
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠐⠄⡀⠀⠀⠂⠄⡔⠀⠀⠁⠀⠀⠀⠀⠀⠠⠉⠀⢠⠠⡠⠐⠃⠀⠀⠀⠀⠀⠀⠀⠠⠢
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠁⠂⠐⠉⡁⠀⠰⠈⠂⢧⡀⠀⠀⠀⠀⢀⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠀⠀⠤⠔⠁⠀⠀⢠⠀⠀⠀⠙⢔⠂⠐⠊⠁⠀⠀⠀⠠⠀⠀⠀⠀⠃⢠⠤⠀⠂⠁
⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠂⢎⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡆⠀⡀
⠀⠈⠢⡀⠀⠀⣀⡀⠀⠀⢀⡀⠀⠀⠀⠈⠀⠀⠀⡆⡆⠘⡄⢂⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠄⠀⠀⡏⠂⠐⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⠙⢆⣑⣄⠀⠀⠀⠈⠧⣄⣱⣀⣀⡀⠉⢹⠓⠃
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠠⠝⠋⠑⢄⡀⠀⠀⠀⢀⡔⠁⠈⠳⠀⡄
⠀⠀⠀⡇⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀⡤⠎⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠉⠉⡋⢹⠀⠀⠀⢀⠀⢇
⠀⠀⠀⠸⡄⠀⠀⢀⡧⠠⢤⠀⠀⠀⠀⢸⠸⠀⠠⠀⠀⠐⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡎
⠀⠀⠀⠀⠉⢀⠤⢊⠀⢀⠆⠀⠀⠀⠀⢸⠀⠀⢈⠀⢀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⡇
⠀⠀⠀⠪⡀⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⢸⢰⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⢐⠙⡾⠀⠀⠀⠳⢀⠃⠀⠀⠀⢤
⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠜⡀⠀⠀⢑⠘⡀⠂⠀⠀⢀⡀⠐⠤⣀⣀⣠⣀⣀⠤⠐⠈⠀⠀⠀⠀⠉⠂⠥⣀⡀⢀⣀⣀⠴⠂
⠀⠀⠀⠀⠀⠰⠀⠀⠀⣤⠃⠀⠈⠁⠉⠀⢧⣀⣀⣀⡤⠃
⣶⣶⡶⠶⠶⢶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⠀⠀⠀⠀⢠⡔⠒⠛⣁⣈⠷⠒⠦⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⣠⠤⡀⠚⠃⠈⠉⠉⠀⠀⠀⠀⠀⠠⢤⣀⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡠⢊⡥⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⠀⠀⠄⣰⡊⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡆⠀⠀⠠⠱⠀⠐⠀⠀⠀⠀⠀⢀⡤⠧⢂⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⠀⠁⠀⡇⠀⢀⠄⡠⠐⡠⠐⡏⠀⢣⡈⠧⡈⢢⡘⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⡀⠀⢹⣀⡀⡀⠐⠁⢀⠀⠀⠀⠀⠹⠀⠉⠓⣽⠦⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠃⠀⠘⠙⠋⠀⠀⠀⣤⣀⣀⠀⠀⠀⠀⠀⡀⠈⡁⠀⠀⠀⠀⠀⠀⢠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠀⠀⠀⠀⠀⠻⠿⠋⠀⠐⡀⠀⠀⠀⠆⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡇⣤⠀⢈⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡃⢀⠎
⠀⠀⠀⠀⠀⡀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣸⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠺⢀⠃⠏
⠀⠀⠀⠀⡘⠀⠀⢀⠔⠁⠀⠀⠀⢠⠀⠐⢸⡷⠃⠂⠒⠀⠘⠑⠀⠀⠀⠀⠰⡐⠊⠀⠀⠘⠜
⠀⠀⠀⠀⠁⠀⠀⠀⢀⣤⠖⠀⠀⠀⠀⠀⠀⠓⣀⠰⡶⢋⠀⠂⠀⠀⠀⠀⠀⡠⢰⡴⠀⠀⠈⢦⠰
⠀⠀⠀⠀⠀⠀⠀⠠⠋⠁⠠⠊⠀⠀⠀⢀⠀⠐⠁⠀⠁⠀⠀⠂⠤⠀⠀⠀⠡⠴⡎⠀⠀⠀⠁⡒⢀⡀
⠀⠀⠀⠀⢢⡀⠀⠁⠀⠀⠀⠄⠀⢀⠀⡘⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠙⠄⠀⢠⠖⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠰⢦⡀
⠀⠀⠀⠀⠀⣀⣀⡴⠁⠀⠀⠠⠋⠀⠒⠦⣄⠀⠀⠜⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⡤⠊
⠀⠀⠀⠀⠏⠈⠀⡀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠒⠋⠀⠀⠀⢘⡀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠠⠄⠈⠉⠁⠀⠀⠉⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠰⣀⠀⠙⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠠⠰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠒⠶⠖⠠⠤⣀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠛⠛⠃⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠘⠀⡆⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢣
⠀⠀⠀⠀⠀⡁⠓⠒⠚⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢰⠇⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠔⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠐⠚⠉⠀⠀⢀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠹⡄
⡀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣀⣂⣐⣒⣂⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣀⣠⣄⣀⣀⣀⣀⣀⣠⣤⣤⣄⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠛⠛⠛⠛⠛⠁⠠⢤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠶⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⡀⠤⠤⠤⠤⠤⠤⠄⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣠⠶⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠉⠉⠶⣆⠀⠀⢾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢸⡇⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢘⡅⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⢀⠀⣀⡀⠃⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢨⡇⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢸⡇⢴⠋⣻⣿⣿⡟⢣⠀⠀⠀⠘⡟⣸⣿⣿⠋⣷⠀⢸⡇⢸⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⣧⣼⡇⠀⠻⢿⣿⣿⡿⠃⠀⠀⠀⠀⠻⣿⣿⣿⠿⠉⠀⢸⡇⢸⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢀⡏⠀⠀⠀⠀⠀⠈⠁⠀⢠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠛⢿⠃⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢦⣄⡀⠀⠀⠀⠀⣀⣀⠰⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡎⠉⠉⠀⠀⠀⠀⢈⡟⠓⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠃⠀⠸⣆⠀⠀⠀⠀⠀⡸⠃⠀⠀⠈⠛⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠁⠀⠀⠀⠀⠀⠈⠳⠶⠶⠶⠎⠁⠀⠀⠀⠀⠀⠀⠈⠉⠶⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠇⠀⠀⠀⣷⠀⠀⢀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⡾⠁⠀⠀⠸⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⢀⡷⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠒⣾⡂⠀⠀⠀⠀⠘⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠁⠀⠀⠀⠀⣴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠛⣦⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⣾⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⣧
⠀⠀⠀⠀⢸⠇⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠸⡇
⠀⠀⢀⣰⠇⠀⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠰⣆
⠀⠀⠸⠇⠀⠀⠀⠈⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣄⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠰⠶⠚⢋⡀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⠸⠇
⠀⣠⣇⠀⠀⠀⠀⣠⠶⠀⠽⠃⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠀⠘⢯⡀⠶⣀⠀⠀⠀⠀⣸⡆
⣄⢋⣾⣇⡀⣄⡀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢀⣠⢄⣸⣧⡜
⠉⢹⣧⣸⡇⣿⠇⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⣾⠸⣿⢸⣇⣹⡇
⠀⠀⠉⠙⠿⠁⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠈⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠄⠀⠠⣤⣤⣤⣤⣤⣤⣦⠀⠈⢳⣤⣤⣤⢤⣤⣤⣤⢤⣤⣤⣤⣴⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⠀⠀⠀⠹⡁⠀⠀⠀⠀⠀⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠁⠈⠉⠉⠛⠒⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠔⠛⠁⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠢⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠖⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠺⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠰⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠒⠲⠶⠶⠶⠶⠶⠶⠶⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣿⣾⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⠿⠿⢿⣿⡿⠿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡗⠀⠀⠂⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠒⠲⠒⠒⠙⠋⠒⠒⠶⠒⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣤⠦⠾⠤⠒⠒⠤⠷⠴⣤⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⢞⠵⢒⠦⡀⠀⠀⢀⡴⢒⠪⡷⡆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡒⢄⡗⠃⢺⣾⣿⣥⡇⠀⠀⢸⣼⣿⣥⡿⠘⢾⡠⢖⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣁⡀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⢀⣘⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠢⠔⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠢⣄⣀⣀⡠⠔⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠒⢤⠤⢤⡤⢤⡤⠖⢮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠲⠤⡷⠤⠤⢾⠤⠖⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡄⠀⡇⠀⠀⢸⠀⢠⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣸⠀⠀⡇⠀⠀⢸⠀⠀⣇⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢲⢿⠴⠟⣇⣀⣀⣸⠻⠦⣿⡖⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⡾⠛⣶⠶⠋⠀⠀⠙⠶⣶⠛⢶⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣇⣀⣀⣸⣇⣀⣀⣸⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠉⡏⢹⠉⠉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠤⠤⡇⢸⠤⠤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⣖⡲⠿⠻⢖⣲⠾⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠉⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣙⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣀⠠⡌⣿⡿⠛⣻⠻⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⢿⣦⣤⣀⣤⣀⣀⣀⣤⣴⣤⡄⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⢸⣿⣉⣛⣾⣿⣿⣿⣿⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡟⠀⢿⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣙⡻⣿⣿⡇⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡈⢻⣦⠀⠀⠀⠀⠀⠀⠀⣠⡙⠦⣬⣿⣿⣿⡇⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡙⢧⡀⠀⠀⠀⣠⣾⠋⠙⠳⣾⣿⣿⣿⣿⣆⢸⣧⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⢻⣦⡀⣰⣿⣿⣧⣀⠀⠈⠁⠎⣙⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣯⣿⣿⣶⣶⣤⣀⡛⠲⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣻⡟⣿⠟⣿⡋⢿⣿⣿⣿⣿⣷⣦⣙⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢡⣿⢸⡟⠀⠻⠇⣆⠀⠈⠉⣿⣯⡙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣼⢿⡿⠀⢸⠀⠀⠏⠀⠆⢰⠁⠈⠹⣦⡙⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣼⠁⣾⠃⠀⣀⣀⣠⣤⣀⣀⣵⠀⠀⠂⠛⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣼⠇⣸⠇⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣆⠘⠀⠀⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⡐⠁⠀⠇⢀⣾⣿⠉⣿⣿⣿⠉⠉⢻⣿⣿⣧⣳⡄⢻⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃⢀⡤⠚⠁⢀⡼⠀⢸⣿⣿⡀⠹⣿⣿⠀⠀⣾⣿⣿⣿⣷⠹⡈⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⡴⠋⠀⠀⠀⡈⠁⠀⣾⣿⣿⣇⠀⢻⣿⡇⢠⣿⡟⣿⣿⣿⡆⡾⢻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣾⠁⠀⠀⠀⢀⠃⠀⢀⣿⣿⣿⣿⡄⠀⢿⡇⢸⣿⣇⡿⢸⣿⣿⡆⠈⡿⣿⣿⣧⠹⣷⡈⠙⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢉⠀⠀⠀⢀⡾⠀⠀⣸⣿⣿⣿⣿⣷⠀⢸⡇⣸⣿⢻⠇⡜⣿⣿⡇⠀⠀⢛⣿⣿⣧⣹⣷⡄⠀⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣾⠃⠀⢀⡞⠁⠀⣴⢿⣿⣿⣿⣿⡿⠗⣾⣿⣿⣿⢸⡀⠇⣿⠹⣷⡄⠀⠀⣿⣿⠿⡿⢿⣿⣆⠀⠸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠿⠀⠀⠎⠀⢠⡾⢁⣿⣿⣿⣿⣿⠁⠀⣿⣷⣿⣿⢸⡇⠐⢻⡇⣿⣧⡑⠀⣻⣟⠀⠹⣄⠙⣿⣷⡀⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠁⠀⠀⣼⣇⣾⣿⣿⣿⣿⡿⠀⠀⣿⣧⣿⣿⢸⡇⡆⢸⡇⣿⡟⣿⣄⠹⣿⡀⠀⠈⢧⡘⣿⣿⣬⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⡄⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⠋⣦⣿⠀⢸⡇⣿⡇⠉⠻⣦⣿⣿⡒⠢⡈⠳⡉⢻⣿⡟⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢠⠁⠀⠀⠀⢠⣾⢯⣿⣿⣿⣿⣿⣿⣇⠀⠀⣿⣿⡟⠀⠁⣿⠀⢸⡇⢿⡇⢃⠀⢿⠉⠻⣿⣦⡈⢢⠳⣼⣿⣇⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⣾⣿⣿⣿⣿⣿⣿⣿⡀⠀⣿⣿⠀⠀⠀⣿⠀⣼⡟⠘⣷⠀⠡⠸⠀⠀⠈⠻⣿⣄⠀⠘⣿⣿⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠸⢿⣆⠀⠀⠀⠀⠀⠀⢀⣾⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⣿⠀⠀⢠⡟⠀⣿⡿⠀⠹⣧⠀⠀⠸⠀⠀⠀⠈⢿⣆⠀⢹⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣀⠀⠀⢀⣶⠟⢋⣴⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⠀⢀⣿⠀⠀⣿⡇⠇⠀⣿⡄⠇⠀⡇⠀⠀⠀⠸⣿⡆⠘⣿⢃⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⣿⣷⣤⣿⣿⣿⣿⠇⣹⣿⣿⣿⣿⣿⡟⣷⣼⡏⠀⠈⢻⠀⠀⢸⠇⠸⠆⣿⣧⠀⠀⢰⣦⣠⣀⣀⣽⠃⣠⣾⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⡟⢰⡟⠘⣷⡆⠀⠄⠀⠀⣼⣷⣶⣾⣿⣿⣦⠀⠘⣿⠟⠛⠋⠁⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠁⠙⢿⣿⣿⡿⢿⡇⢸⣧⠀⢹⣷⣦⣦⣴⣦⣾⡿⠛⠛⠻⠿⣿⡿⠶⠿⠀⠀⠀⠀⠠⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⣿⡿⠁⢀⣨⣏⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⠛⠛
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡛⠋⡛⠻⢋⡉⡛⠛⢻⡛⠋⣛⡿⢛⡏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣾⣿⣷⣿⣿⣷⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⡾⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣌⢲⣿⣿⢃⣾⢋⣦⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢶⡘⡗⡏⢋⣿⣤⣯⣯⣌⣙⣿⣷⠾⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⢦⣄⣇⢹⡾⣏⡴⠊⠁⠀⠀⠙⠿⣯⡴⠖⢀⡀⠀⠀⠀⠀⠀⣧⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⢾⡆⠀⣾⠃⠀⠀⢠⠶⣤⢈⢿⣿⡾⠋⠙⠀⠀⠀⠠⠞⠃⠘⢿⣿⠶⣆⠀⠀⠀⠀⠀⢹⣴⣏⣴⠞⠃
⠀⠀⠀⠀⠲⢦⣄⠀⢻⣼⣧⡀⠀⠀⠛⣓⠺⣏⣾⠏⢰⣧⡌⠂⠀⠀⢠⠞⡉⠓⠘⣿⡉⠁⠀⠀⠀⠙⢲⣿⠛⣿⣤⣄⣀⡀
⠀⠀⠀⠀⠀⢀⣨⣽⣾⡇⣸⣧⠀⠀⠉⠉⢨⣾⡏⠀⠻⢿⠇⠀⠀⠀⢨⡸⠿⡠⠀⣿⡇⠀⠀⠀⠀⠀⢘⡿⠟⠁
⠀⠀⠀⠀⠀⠉⠙⠁⢸⡟⠛⢻⡿⣄⠀⠀⢿⣿⠀⠀⠢⠤⠞⠀⠛⠁⠀⠉⠊⠁⠀⣩⣿⡆⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠙⣌⢳⡄⠘⣧⢣⠀⠀⠀⢠⣶⡿⢿⣷⣦⠀⠀⣠⣿⡿⠀⢀⣤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣽⠷⣘⢦⠳⣄⡀⢾⣽⣵⣶⣾⠏⢀⣴⡿⠋⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣷⡄⠘⢦⣷⢦⣻⣶⠤⣍⡀⢀⣴⠟⠋⣠⠾⢿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄⠈⠁⠈⢻⡏⠛⠚⡿⠙⢷⣴⡾⠃⠀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⣀⠀⠀⠙⢦⣼⠁⠀⠀⠉⠁⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡀⠀⠀⠈⠛⠀⠀⠀⠀⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣷⠤⣤⣤⣤⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡤⠀⠀⠀⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠈⣱⣷⡦⠤⢴⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣀⢀⣼⠏⢸⣿⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⠿⠿⠋⠀⠸⡿⣷⣶⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡟⠳⢤⣤⣤⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⢰⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠖⠊⠙⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣧⠶⠞⠋⠉⣿⣟
⠀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠻⣦⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃
⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⣀⡠⠤⠖⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠙⢦⠐⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠁⠀⠃⠀⠀⡀⠀⠀⠀⠃⠀⠀⠀⠈⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠂⠀⠀⠀⠀⣀⡤⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢰⠁⡔⢸⠁⡔⠉⢇⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠠⠀⢀⡼⠓⠛⠈⠣⠷⠄⠈⠑⠉⠁⠳⡄⠘⠂⠀⠀⠀⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠒⡋⣤⠖⠦⢄⡀⠀⠀⢀⡠⠤⠰⠦⣜⢦⡤⠈⠁⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠒⠂⠀⠀⣿⣏⡠⠤⠤⠝⠀⠰⠭⠤⠤⠤⠤⣼⢸⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⡀⢨⠀⠀⢰⡶⠀⠀⠀⠀⠰⡶⠆⠀⠀⢸⣰⣀⣀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⢹⢺⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣼⡇⢸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠛⡄⠀⠀⠤⠤⢀⣀⠤⠤⠃⠀⠀⠀⡏⢁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢇⠀⠤⠤⠠⠤⠤⠤⠤⠤⠤⠄⠀⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⣄⡀⠀⠀⢀⣀⠤⣞⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⢉⣇⠀⠈⠉⠉⠁⠀⢀⡧⠤⠞⠉⣓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠒⠒⠲⢭⣓⢤⡀⠀⣠⣺⠥⠖⠒⠋⠉⠁⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠒⠂⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⣀⣀⣀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣉⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠔⠋⠀⠀⠀⠉⠑⠶⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡰⠁⠀⠀⠄⠀⠀⠀⠀⠀⠠⡀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⢠⣀⠀⠀⠀⢠⣄⠀⠀⠀⠘⢄⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠙⠋⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⡁⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢃⠀⠀⢐⠂⠤⢀⠐⠤⠚⠀⠀⠀⠀⠀⠀⠰⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠘⢿⠀⠀⣤⣤⣤⡿⠃⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠐⠀⠈⠛⠉⠀⠀⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⣤⠠⠤⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠢⠀⠀⠠⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢠⢄⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠒⠁⠂⣀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠂⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠣⠀⠀⠐
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠒⠈⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⡀⠀⠀⠀⠀⡄
⠀⢀⡀⠀⠑⠄⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁⡀
⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⢄⡀⠀⠀⠀⢠⠖⠛⡂⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣀⣀⣈⠂⠀⠀⡈⠀⠀⡇⠀⠀⠀⡄⠤⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⡆⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠈⠀⠀⠀⡀⠀⠀⡇⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠐⠚⠀⠀⠀⠃⠀⠀⡇⠀⠀⠀⡄⢀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⡄⠀⠀⡇⠀⠀⠀⠀⠈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⢤⡀⣀⣀⣀⠧⠀⠀⠧⣀⣀⣀⣀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠃⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⠤⠴⠀⠀⠀⠀⢰⠤⢤⡽⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠈⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⠀⠀⠂⠀⠤⠤⠤⢳⠀⠀⢀⠆⣀⣀⣀⣀⠀⠄⠛⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠶⠂⠀⠀⠤⠤⠞⠀⠀⠺⠤⣤⡀⠀⣠⡄⠠⠰⠊⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣽⣏⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡠⠞⢠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡎⡇⠀⠀⠀⠀⠀⠀⠠⡠⠢⠄⣰⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⣠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢔⣀⠈⠀⠀⠡⠀⡀⠀⢀⠠⠒⠈⠀⠀⠀⠀⠀⠀⠁⠒⠄⡀⡄⠀⠀⠀⠀⠒⢍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠤⢀⡀⠀⡠⠛⡁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡲⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⠀⠓⡖⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⢡⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⠀⠀⠂⠈⠒⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⢾⣷⠀⠀⠀⢀⡇⠉⠉⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠓⢴⡄⠀⠀⠀⠉⠀⠀⠠⠄⠀⠈⠁⠀⠀⠀⠈⠀⠚⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠉⠋⠂⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⡠⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠀⠤⠂⠀⡀⠀⠀⠀⠀⠐⠛⠁⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠤⠤⡤⠤⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⢰⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠁⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⣄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠛⠀⠀⠀⠀⠀⠀⠉⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡰⠀⠘⠀⠀⠀⠀⠀⠀⠀⢄⡨⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠓⠠⡆⠀⠀⠀⠀⠀⠀⠀⢨⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⣀⠀⠀⠀⠀⢀⣀⢀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⣠⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠃⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠀⣤⠀⠀⣇⠈⠐⡦⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠣⠤⢄⡪⠃⠀⠀⠑⠬⣉⣀⣀⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⡀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠂⠉⠀⠀⠀⠀⠈⢾⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢁⠤⠚⠙⠛⠂⠀⠀⣀⠈⠀⢬⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠰⠋⠀⠈⠳⡀⠠⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢆⠀⠘⣆⠀⠀⠀⠀⠀⠊⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢂⠈⠝⠀⠀⠀⠈⠀⠈⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⢀⠀⡄⠀⠀⠀⠐⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠉⠒⢄⢀⢆⠀⠀⠀⣠⡀⠀⢀⠆⠀⠠⠄⠀⠈⠀⠀⠀⠀⠑⢶⠒⠀⠀⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⠭⣑⡐⠨⠾⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⡹⠋⢠⠒⠉⡆⠀⠆
⠀⠀⠀⠀⠀⠀⠁⠀⠐⠆⠠⢤⣀⡀⢈⠈⢑⠒⠂⢤⣀⠃⠀⠀⠀⠀⠀⢰⣿⣦⠀⠀⠀⠀⠀⠀⠀⠈⢣⠤⠃⠀⠀⠘⡀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠉⠉⠉⠉⠉⣹⣿⣆⠀⠀⢀⣀⣠⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⡆⠈⡠⠔⠈⠑⣤⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡋⠣⡀⠀⠀⠀⠀⠀⠸⣿⣿⠿⠟⠛⠟⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⢠⠃⠋⠀⠒⠛⢀⠃⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡉⠉⠳⡕⢄⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⢀⣀⣠⠎⡰⠢⣀⣀⠴⢃⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡠⠻⡌⠢⡀⠀⠀⠀⠐⣄⣀⢀⠐⠀⠀⠀⠐⠋⠬⠁⠁⣩⠔⠁⡰⢁⠔⠉⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠒⠚⠂⠈⠲⠆⠒⠈⢉⠄⠀⡀⠀⠀⠀⠀⣀⠑⠂⠀⠀⠀⠺⠎⠁⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⢆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⠚⣆⡠⠗⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡔⠂⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⢸⣘⡇⡗⠢⣼⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝⠾⠀⠀⠀⠀⢾⢹⠉⠁⠀⠆⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠁⠀⠀⠀⠀⣠⠈⠀⠀⡜⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠂⠀⠀⠀⠀⠀⢄⡀⣿⣿⡇⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣀⣀⠀⠤⠄⠀⠈⢣⠈⢆⠀⠀⣀⠞⠀⠀⢧⡟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢃⣈⡄⠈⠁⠀⠀⠀⠈⠸⡃⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢄⠀⠀⠀⠀⠀⠀⠈⠁⠱⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⣄⡀⠀⠀⠀⣀⡤⢖⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢶⡛⢯⣿⣛⣛⢻⡿⠿⠭⢶⢽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢀⣀⣏⣐⣀⠀⠉⠉⠩⠭⠿⠟⠛⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠄⠤⠀⠀⠐⠒⠒⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠠⠤⠀⠄⠀⢠⠠⠠⠀⠀⠠⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠆⠆⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠒⠒⠒⠢⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡻⠄⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠠⡖⠈⠉⠉⠉⠉⠉⠉⠉⠉⠒⢖⠺⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢀⠜⠀⠰⠊⠈⠂⠀⠀⠀⠈⠙⠆⢨⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡀⡀⠘⠀⠀⠀⢠⡶⣦⡄⠀⠀⠀⢴⣶⣦⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠒⠌⠒⠀⠀⠀⣮⠿⠿⡇⠀⠀⢀⣸⠿⢿⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠋⣀⠀⠀⠀⠀⠀⣤⣀⠀⠀⣀⣧⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠢⣄⡀⠀⠀⠈⠢⠬⠭⠤⠃⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢕⡖⠲⠤⢤⣤⡐⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠻⡀⠀⠏⠐⠒⠊⠀⠈⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢃⠀⢸⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡸⠘⠀⠀⣄⣀⣀⣀⠀⢠⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠈⢲⠴⠁⢠⢇⣰⠁⠀⠀⠸⣆⣯⡗⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⢠⠗⠁⠀⠀⠀⡀⢀⠀⠀⠈⢣⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⢁⠀⠀⠀⠣⡀⠀⠀⢀⠜⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡄⢸⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⢸⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣼⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠷⡏⠀⠀⠀⠀⠀⢰⠊⠉⠀⠀⠈⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡇⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠐⠂⠠⠄⠒⢹⠁⠒⠒⠂⢁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⡖⠒⡴⠒⠋⠉⡆⠀⣎⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣯⠍⡧⠳⡀⠀⡔⠓⢮⠀⠸⣢⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠉⠉⠐⡝⢁⠴⡇⠀⢟⡢⢀⠑⠎⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⣀⠠⢚⡡⠊⠀⠀⠀⠈⠂⠌⣑⡂⠀⣂⠇⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣠⣴⣿⣿⡛⢶⣿⣶⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣦⡄⠈⢯⣕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠈⠙⠻⣿⡦⠀⢿⣀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⡀⢀⡀⡀⠀⠘⣿⣿⣶⣮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠃⠀⠙⠛⠁⠀⠙⣷⡿⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⠀⠀⠀⠀⠀⢰⣿⢡⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣸⡀⠣⠀⣼⠇⠀⢀⠀⠀⠸⣿⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⡀⠀⠀⠶⠘⠏⠀⠀⠀⢻⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣶⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠉⠀⢰⠆⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⢹⡆⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠁⠀⠀⢸⡧⣰⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⠇⢻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠃⠀⠀⠈⠉⠙⠻⢿⡿⢿⡿⠿⠿⠿⠛⠋⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢸⠇⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⣾⠀⠀⠀⠀⠀⡰⠀⠀⢠⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⣾⠀⠀⠀⠀⠀⠃⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⢀⡟⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠶⠶⠶⠶⠶⠤⢤⣤⡤⠶⠒⠊⠉⠉⠉⠉⠙⠓⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⢠⠇⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⢸⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢻
⠀⠀⠀⠀⡏⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⡇⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⢣⠀⠀⠀⠀⢀⣠⣤⡤⠀⠀⣀⣀⣀⣀⣀⣈⣀⠀⠀⠀⠐⠶⢶⣤⠀⠀⢀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠣⣄⠤⠶⠛⠛⢋⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣛⣓⠒⠶⢬⣅⣀⠻⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠉⠙⠛⢦⡀⠀⠀⠀⢀⡴⠞⠉⠉⠉⠉⠙⠲⣄⠁⠘⠒⠀⠐⠊⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣤⠦⢤⡀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢠⡟⠀⠀⠀⠀⣀⣤⣤⣀⠘⣇⠀⠀⠀⠀⢠⡗⠘⢷⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢧⡀⠈⢡⠀⠀⠀⣿⠀⠀⠀⣴⣾⣿⡿⠷⣄⣷⠀⣾⠀⠀⠀⣠⣿⣿⣿⡟⠛⢷⣿⠀⠀⠀⠀⣾⢀⣠⠟⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠛⣦⠴⣆⠀⠀⢻⡄⠀⠀⣿⣿⣯⣥⣴⣿⠇⠀⢻⡄⠀⠀⣿⣿⣿⣯⣷⣤⣾⡿⠀⠀⠀⣼⠋⠉⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠹⣆⠀⠀⠻⢦⣀⡙⣿⣿⣿⡿⠋⠀⠀⠈⢳⣄⠀⠘⢿⣿⣿⣿⣿⡿⠁⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠙⣦⠀⠀⠀⠉⠉⠉⠉⠁⣴⠒⠂⠀⠀⠀⠙⠛⠶⠶⠿⠿⠛⠉⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠳⣤⢠⡤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⠀⠀⠀⢦⣄⣤⠆⠀⠀⠀⠀⢀⣠⣤⠞⠛⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⡶⠶⠦⢤⣤⣤⠶⠶⠖⠛⠉⠁⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩
⠀⠀⠀⠈⠙⠳⢦⣄⣀⣀⠀⠀⠀⢀⣀⣠⣤⠖⠋⠀⠀⠀⢸⠆⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢤⣤⣀⣀⣀⣀⣤⣤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⡀⠎⠀⠀⡸⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠺⠁⠀⠁⠉⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣴⠃⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⢂⠔⠋⠘⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠉⠒⢠⠀⠀⠀⡤⠴
⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⣀⠀⠀⠀⠴⠈⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⢁⠀⠀⠀⢀⡤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⡠⠁⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⣀⣰⡅⠀⠀⠀⠀⠀⠀⠑⠀⠄⠉⠐⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢄⣠⠀⡄⠠⠊⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠂⠙⡉⠍⠉⢉⠋⠟⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠘⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢰⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠘⡇⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⠁⠁⠀⠀⢀⣧⡀⠀⠀⢀⠝⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⣉⣁⠢⠜⠋⠁⠙⠶⣆⠁⠀⠀⣀⠼⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢀⣀⣀⣀⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄
⠀⠀⢀⣼⣿⣿⡷⠆⠀⠀⠀⠀⠀⠠⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠂⠀⠀⠀⠀⠀⠀⣰⣿⣿⣦
⠀⠀⠀⠈⣿⠟⠀⠀⠀⠀⠀⠀⠀⠤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟
⠀⢠⣶⣶⣿⣦⣤⠀⠀⠀⠀⠀⢀⠀⠿⠄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣿⣀⡀
⠀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⡿⠀⠀⠂⢠⠀⠀⣀⣤⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⢿⡆
⢀⡿⠘⣿⣿⣿⣿⠀⠀⠀⢰⡏⠀⠀⢀⠐⠻⣇⡼⠃⠀⠀⠀⣿⢻⣿⣿⡿⠃⠀⠀⠀⠀⢰⡟⢹⣿⣿⡟⠘⣧
⠀⢿⡀⠘⣭⣿⣿⠀⠀⠀⠈⢷⡀⠀⠀⢱⡏⠉⠀⠀⠀⠀⠀⣿⡀⢿⣿⣷⣤⠀⠀⠀⠀⠸⣧⡀⢿⣏⠡⢄⡿
⠀⠈⣇⠘⢿⣿⣿⡇⠀⠀⠀⠀⠳⣤⠀⠀⠇⠑⠀⠀⠀⠀⠀⠹⣇⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⠗⠠⠀⠐⠈⠃
⠀⠈⠻⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠐⠀⠀⠄⢀⠀⡀
⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡇⠀⠀⠀⠀⠈⠹⡆⠀⠈
⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠂
⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⢠⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⣼⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢠⠏⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣄⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⣰⡟⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠰⠟⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠀⠁⠀⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡈⢲⠀⠀⣀⠁⠀⢀⠄⠈⡀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠻⠟⠂⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣀⣀⣀⠤⠚⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⢠⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠊⠐⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠠
⠣⠤⢰⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠢⠀⠀⠀⠋⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠠⠂⡂⠐⠁⠠⠤⠐⠀⠨⠄⠀⠀⠀⠀⠈⠒⠠⠄⣀⠦⠀⠑⠐⣺⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢰⠁⠐⠦⣀⣀⣤⠫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣀⣀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡤⠀⠀⠀⠀⠂⠀⠁⡒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⠰⡀⠀⠀⠀⠰⠀⡐⠀⠀⠀⠁⠠⠐⠒⢂⡀⠀⠁⠄⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠁⠀⠀⠀⠃⠀⠈⠁⠸⠀⠀⠀⠀⠀⠑⠀⠄⣠⡇⠈⠈⡰⠀⠀⠇
⠀⠀⠀⠀⠀⠀⣀⠠⠘⠓⠒⠔⠒⠃⠀⢘⠒⠒⠒⠈⠉⣄⠀⠀⠀⠀⠀⠀⠀⠱⠊⠈⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠃⠤⠤⠀⠤⠄⠠⠘⠀⠘⠀⠂⢀⡀⠠⣨⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠁⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡊⡤⢀⠀⢠⡌⠳⠦⢤⣀⣀⣀⣠⡤⠖⠉⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⠅⠸⠃
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠸⣿⣿⣿⡗⠀⠀⣀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣠⣶⣦⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣇⠀⠀⠀⣠⣤⣤⠙⠳⣶⣼⣿⣿⣴⠶⠞⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣿⣿⣿⠃⠀⣠⠶⠂
⠀⠀⠰⢦⣈⣻⣿⣿⠟⠀⠀⠀⢿⣿⡿⠃⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⣠⣾⣶⣆⠀⠀⠙⠒⠶⣦⣵⣿⣿⣶⠟⠁
⠀⠀⠀⠀⠈⠛⣻⣿⣿⣶⠀⢠⣾⣿⣷⣄⠀⢰⣿⣿⣿⡄⠀⠀⠀⠀⠈⣻⣿⣏⣤⣤⠀⠀⠀⠀⢸⣿⣿⣅
⠀⠀⠀⠀⠀⢀⣿⣿⣿⡅⢰⣿⣿⣿⡟⢿⡆⢸⣿⣿⣿⡇⠀⠀⠀⢀⣾⣿⣿⣏⠁⠀⠀⠀⠀⢠⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢰⣿⠻⣿⡄⣼⡿⢿⣿⡇⠈⠁⢸⣿⠈⣿⡇⠀⠀⠀⠸⢋⣾⣿⣿⣦⡀⠀⠀⠀⠈⢹⣿⢻⡟⠋
⠀⠀⠀⠀⢠⣾⡟⠀⣿⡇⠙⠃⠰⣿⠃⠀⠀⢸⣿⠢⣿⣇⠀⠀⠀⠀⠻⡏⠉⠉⣹⡇⠀⠀⠀⠀⠸⣇⠸⡇
⠀⠀⠀⠀⠘⢿⠇⠰⣿⠆⠀⠀⠀⢿⠄⠀⠀⠈⠉⠙⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠸⡇
⠀⠀⠀⠀⠰⣿⡇⠀⣿⠁⢀⡀⠀⣯⠀⠀⠀⢰⣶⠀⣿⠇⠀⠀⠀⠀⢀⠄⠀⠀⢠⡀⠀⠀⠀⠀⠀⡄⢀⡇
⠀⠀⠀⠀⠀⠘⣿⣤⣾⠁⢻⣶⣴⣿⡁⢀⡀⠸⣯⢀⣿⡇⠀⠀⠀⠰⣌⢲⡿⣿⠛⠀⠀⠀⠀⢠⣰⣷⣼⣧⡄
⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄⠀⠻⢿⣿⣷⠟⠀⠀⣝⣿⣿⠂⠀⠀⠀⠀⠘⣻⣿⠆⠠⠤⠀⠀⠀⠀⠙⣾⣿⠟⠁
⠀⠀⠀⠀⠈⣡⣾⣿⣦⠀⠀⠀⣶⣿⣷⠀⠀⠀⣻⣿⡇⡀⠀⠀⠀⠀⠘⠻⠿⠃⠀⠀⠠⠀⠀⠒⣚⣿⣿⠓⠤⡀
⠀⠀⠀⠀⠘⠛⠛⠟⠃⠀⠀⠀⠀⠀⠡⠔⠋⢉⣬⣿⡿⡉⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠿⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠂⠀⠀⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢴⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢳⡄⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⢉⢷⣼⠁⠀⠀⠀⠀⠀⠀⢰⢷⠀⠀⠀⡿⢢⠀⢷⡼⠯⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢠⣿⡛⠂⠀⠀⠀⠀⢀⡴⠃⢸⡀⢠⣠⠇⢸⡇⣿⡈⠆⠀⠀⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢠⠀⠀⡟⠘⣇⣠⠄⠠⠔⠚⠁⠀⠀⠟⠒⠋⢁⣀⠾⢻⣇⠙⣄⠀⠀⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣿⣆⠸⡇⠀⠸⣇⢆⠀⠀⠐⠻⠏⠀⠀⠀⠘⠟⠈⠁⣸⣸⠇⣹⣀⠜⠁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠓⠛⠀⠀⠈⠑⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠙⢦⡀⠀⠀⠀⠐⠂⠀⠀⡠⠺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⠤⡀⠉⠒⠠⠤⠤⠄⠒⠉⠀⢀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠳⡀⠀⠀⠀⠀⢀⡼⠁⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢄⠀⠙⠢⢀⡀⡠⠎⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠃⠀⠉⠒⠤⡀⠈⠁⡤⠚⠁⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠻⠦⠄⠃⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⣶⠦⠠⢤⣀⣠⣄⡠⠤⡴⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣿⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣀⡏⠀⢀⡏⠁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢀⣹⠦⣼⡤⠤⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠭⠗⠈⣹⠁⠈⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠚⠉⠉⠉⠉

⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣲⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠲⢤⡀⠀⠀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠈⠉⠙⠲⢄⡠⠋⠁⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⡼⠁⢀⣀⣀⠀⠙⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠖⠚⠓⠲⢄⠘⣆
⠀⠀⠀⠀⠀⡼⢡⠞⠉⠀⠀⠙⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢳⡸⡄
⠀⠀⠀⠀⢰⢣⠏⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢧⠀⠀⠀⠀⢷⢳
⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⡄⠀⠀⠀⠘⣾
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⣠⠀⠀⠀⡇⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⢻⡀⠀⠀⠈⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⢸⠃⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣠⠏⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⣀⡴⠋⠓⠤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣧⡤⠔⠚⠧⣄⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢠⠇⠀⠀⠈⠓⢄⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢌⣇⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣸⠁⠀⡀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⣀⠴⠧⠤⢤⣘⠲⢤⣦⡤⠞⠙⠦⣤⣠⣼⣭⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⡀⠀⠀⢠⠇⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠁⠀⠀⢸⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⣸⢤⣤⣴⠴⠶⢤⣤⣾⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣤⠄⠈⠉⠛⠋⠉⠉⣀⠠⠄⠀⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁⠀⠀⠀⠀⣤⡀⠀⠀⠒⠒⠒⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⢠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠤⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠠⣄⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⠴⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⠀⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠗⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⢐⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠠⡿⠀⠀⡴⠋⠉⠉⠙⠳⡄⠀⠀⠀⠀⠀⢀⡰⠞⠁⠀⠀⠙⢳⡄⠀⠀⠀⠀⢸⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⣸⣤⣠⣶⣶⡶⢶⣿⠆⠀⠀⠀⢀⡾⠷⠶⣶⣤⣤⣤⣤⣤⠀⠀⠀⠀⠘⢃⣀⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠹⠋⠀⠻⣿⡷⢷⠌⠀⠀⠀⠀⠀⢁⠀⠀⣿⣿⣿⣤⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠛⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠛⠠⠄⠘⠋⠀⢾⠀⠀⠀⠀⠘⢷⣄⣈⣛⣛⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣉⠉⠀⠀⠀⠀⠀⠀⢀⣠⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣦⣤⣄⠀⡀⢀⠀⣀⣀⣀⣤⣤⡴⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⣻⡋⠀⠀⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠊⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⠐⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄⠚⠀⠀⠂⠀⠀⠀⠚⠓⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⢀⡀⠀⠚⢠⣤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⢦⣀⠀⠀⠀⠀⠀⠸⢇⣀⠀⠙⠤⠚⠋⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠉⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⣠⠇⠀⢠⣤⠤⠄⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠇⠀⠀⢼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠛⣄⠀⠀⠀⢰⠿⣀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⣾⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠹⡆⢀⡴⠃⠀⠙⢿
⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⣼⠃⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢳⡄⠀⠀⠀⠘⠿⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⣦⠀⠀⠀⠀⣠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠻⣤⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠉⠁⠈⠉⠀⠀⠀⠀⠀⠐⠒⠒⠒⠶⠒⠚⠓⠂⠈⠉⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⣈⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣏⣀⣀⣀⣀⣀⣀⡀⢀⣶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⠶⠶⠶⠖⠈⠙⠛⠁
⠀⠀⠀⠀⠀⢀⣤⠚⢋⡈⠙⢉⠉⠉⠉⠁⠈⢱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢂⡀⠀⠀⠀⠀⢀⡘⠋⢠⣄
⠀⠀⢀⣤⠎⠀⠈⠢⡄⠙⠆⠘⠀⠀⠄⢡⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⢀⠀⠀⠀⠆⠀⠔⠀⠀⠀⢂⡀
⠀⡔⠁⠀⠀⠉⠤⠀⠀⠉⠀⠀⠤⣀⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠈⠁⠀⠠⠈⠉⠈⠉⠠⠄
⡆⠀⠀⠀⠀⠀⠀⠁⡀⠀⠐⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠂⣀⠈⠀⠀⠀⠀⠀⠀⠘⢀
⣿⠀⣀⡀⠀⣀⣀⠀⣧⡼⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢢⣄⠑⠀⠀⠀⠀⠀⠀⢀⣼
⣈⣉⣠⣤⣄⣈⣙⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣃⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡯⠉⠀⠍⠁⠀⠀⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣴⣶⣶⣶⣶⣶⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠛⠛⠛⠛⠛⠛⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣴⠂⠀⠀⣀⣒⡂⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⡄⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣦⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢠⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠎⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠘⠟⠋⠀⠀⠧⠭⣿⣿⣿⣿⣿⢃⡄⡠⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠆⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠇⠀⠀⣀⣤⣦⣀⣀⣬⣿⣿⣿⣿⣿⣾⣵⣾⣿⣿⣥⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠀⣀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⣻⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡎⢀⣻⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣔⣅⣶⣤⣰⣿⣿⣿⣿⡟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠓⣛⠛⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣱⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⡋⢹⣿⣿⣿⣿⣿⣿⣿⢏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠾⢋⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⡟⣹⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠄⣸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢟⣴⣿⢿⣿⠏⣰⠟⢻⣿⣿⡿⠟⠁⠀⣽⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠁⠀⣀⡾⠁⠀⠀⢻⣿⡀⠀⠀⠀⠴⠒⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠟⠀⢀⡴⠟⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟

⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⢀⠔⣫⢶⡙⢣⡤⠄⣀⡀⠀⠀⠀⠀⠀⢀⠈⠿⠿⠤⢄⡀⠀⢺⢣⡤⡀⢀⢀⢄⢠⣤⣀⣀⢤⠎⠟⢀⣤⡶⠖⠛⠛⠓⠶⣄⠀⠀⢄⠠⠒⣠⡤⠖⠶⠤⢀⣲⡟
⠀⠂⢡⠾⡠⡠⠀⠢⡂⠭⠳⠀⠀⠀⠔⠁⢀⠀⠀⢀⠀⠈⢆⠈⠀⠈⠎⠀⠈⠉⠉⠁⠀⠙⡄⠀⠀⡼⠁⠀⢀⠀⠀⡀⠀⠘⠆⠒⢉⢮⠎⠈⠀⠀⠀⠀⠀⠹⣹
⠀⢠⡇⠛⠈⠐⠀⠀⠈⢁⠁⣦⠀⢾⠀⡄⠀⠀⠀⠀⠀⠄⢸⠠⠀⢾⠀⡀⠀⠀⠀⠀⠀⡀⡒⠂⠲⡇⠠⠀⠀⠀⠀⠀⢀⠄⡜⠂⠱⠟⠀⠀⠁⠀⠀⠀⣰⠀⠋
⠀⠀⠱⣌⠢⡀⠀⠀⣠⢎⠜⠀⠀⠀⠢⡀⠢⠤⠤⠤⢊⡠⠃⠀⠀⠀⠣⡑⠤⠀⠀⡠⢊⡔⠁⠀⠀⠘⢦⡐⠀⠤⠤⠔⡡⠎⠀⠀⠀⠈⢦⡁⠤⠀⠠⠔⣁⠜
⠀⠀⠀⠈⠑⡲⠬⠵⡂⠁⠀⠀⠀⠀⠀⣀⠙⠖⠲⠚⡇⠄⣀⢀⢀⠀⡀⢀⡁⠲⠖⢪⠁⡀⠀⠀⠀⢀⡠⠌⠝⠒⠞⠩⡀⢄⠀⡀⠀⢀⡠⠌⠶⠴⠖⢏⡀
⠀⠀⠀⡄⠊⠁⠀⠀⠱⡈⠁⠚⠲⠈⠁⠀⡃⠀⢃⣀⡇⠀⠀⠉⠈⠈⠉⢀⠀⠀⠀⠈⡆⠀⠁⠋⠘⠉⠀⡘⠀⠀⠀⠀⢡⠀⠀⠁⠉⠀⠀⢠⠀⠀⠀⠈⡎⢂⡀
⠀⠀⠀⠀⢰⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⢇⣨⠅⣨⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠑⡆⢐⡖⢒⡆⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠈⠀⡇⠐⠋⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⡁⠀⠀⠀⠀⠀⠀⠘⠂⢰⠒⠒⠂⠉
⠀⠀⠀⠀⠀⠚⠃⠀⠛⠃⠀⠀⠀⠀⠀⠒⠛⠀⠀⠘⠒⠀⠀⠀⠀⠀⠀⠐⠛⠃⠘⠛⠀⠀⠀⠀⠀⠀⠀⠚⠛⠀⠀⠓⠋⠀⠀⠀⠀⠀⠀⠘⠋⠈⠛