Jug Emojis & Symbols

⠀⠀⠀⠀⠀⣠⠖⠚⠉⠉⠉⠙⠒⠲⠦⣤⣀⡀
⠀⠀⠀⠀⠸⣅⣠⡖⠛⠉⠉⠉⠙⠓⠒⠦⢤⣉⣙⠒⠲⠤⢤⣀⣀⣀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠲⠶⠶⠶⣶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⣀⣤⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠁⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢀⣀⣀⣤⣤⣀⣀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡵⠚⠉⠁⠀⠀⠀⠀⠉⠳⣄⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⡇⢀
⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⢸⠃⣼
⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⢀⡟⢰⠇
⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⡼⢡⠏
⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠈⢧⡼⣡⠏
⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠘⣷⠃
⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣸
⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡟
⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⣸⠃
⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⣰⠃
⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣰⠋
⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⢀⡼⠁
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠈⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣧
⣤⣤⣤⣤⣤⣤⣤⣤⣼⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⡖⠚⠛⠉⠉⠉⢉⣈⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡗⠒⠲⠾⠛⠋⢩⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠏⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠔⣒⣚⣿⣉⣡⣼⣇⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢈⠘⠉⠀⠀⠉⠉⡍⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⡼⠅⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣼⣟⠃⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠛⠚⠛⠛⠉⠋⠘⠛⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠸⣌⠳⡀⠀⣿⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡛⣿⠒⢒⣒⠒⠒⠒⠻⣯⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠰⢿⣧⡄⠀⠀⠀⠀⠀⠀⡤⠶⣛⡷⠿⠶⢾⣷⣶⣶⠶⠾⠿⠞⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣼⣇⢀⠀⠀⠀⠀⢰⠿⢲⣟⠻⢷⣶⣶⢏⡌⡣⣤⣦⡽⣻⠗⠦⣾⠛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣏⠘⡀⠀⠀⠀⠀⠤⢿⣯⡙⠀⠈⢀⠔⠃⡉⠚⢿⡋⠁⠀⠱⣿
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⣳⣤⣮⣁⠀⢠⡤⣽⣿⣿⣿⣿⠃⣠⣾⣿⣷⡄⢹⣶⣶⣿⣿
⠀⠀⠀⠀⠀⣎⡈⠋⠦⠀⠀⠀⠀⠀⠹⠿⢿⣿⣿⣿⣿⣿⣿⣟⠛⠿⣆⠹⣿⣿⡿⠃⣸⠿⠿⣿⠇
⠀⠀⠀⠀⠀⠈⠙⠛⠓⠲⣬⣒⡤⣀⠀⠐⠒⠂⠀⢀⣐⣒⣂⢻⣮⠀⠘⠳⢤⣭⠤⠞⠁⢉⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⣤⡀⢀⣀⡀⠂⠀⠀⠀⢹⣷⣤⣤⣤⣤⣄⣤⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠉⠙⠲⢤⡀⠈⠳⠶⠦⠬⠬⠤⠤⠿⠋⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣎⣂⡀⠀⣛⡆⠀⠈⠉⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⣁⠠⠄⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⢀⣀⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢦⠤⠤⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡠⠒⠁⡠⠤⢄⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠉⠉⠉⠀⠀⠈⠉⠁⡏⢀⠴⠋⠀⠀⠀⠱⡀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⢣⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⡸⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢠⠃⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢠⠏⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠖⠁⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠤⠤⢀⣀⣀⣀⠠⠤⠒⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣽⣿⣭⣽⣭⣿

⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⢠⣶⣿⡿⣿⣿⣿⣿⣿⣿⢿⣿⣂⣤⣤⣄⡀⠀⠀⠀⠀⠀⣔⣺⡫⠍⠉⣀⣀⣀⡈⠩⢽⣆⡠⠤⠤⢄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⣿⠟⠋⠉⠙⢿⣆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡏⡡⠒⠈⠁⢢⠱⡀
⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢫⠀⠀⠀⠀⠀⠂⠇
⠀⠀⢠⢃⣤⣤⣴⣶⣶⣦⣤⣤⣼⡄⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⡜⡠⠤⠤⠤⠤⠤⠤⠤⢄⢃⠀⠀⠀⡜⡰
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢀⣰⣿⠋⠀⠀⠀⠀⠀⠰⡘⠉⠁⠀⠒⠒⠒⠀⠈⠉⠃⡄⡠⢊⡔⠁
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣋⠔⠊
⠀⠀⢱⣻⣿⣿⣿⣿⣿⣿⣿⣿⣟⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⢄⠀⠀⠀⠀⠀⠀⠀⡠⡣⠁
⠀⠀⠀⠑⢯⣻⡿⠿⠿⢿⣟⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣕⡠⠄⠀⠠⢄⣪⠐⠁
⠀⠀⠀⠀⠈⠛⠻⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠩⠭⠅⠐⠊
⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡁⠉⠛⢶⣤⣤⣤⣤⣤⣤⠤⠤⠤⠤⠤⠤⠄⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠗⠀⠀⢿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⣀⣀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠟⠛⣿⣿⠛⠛⠛⠃⠘⠛⠋⠉⠉⠙⢿⣏⠀⠀⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⡃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢸⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⣸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣧⠀⢠⣿⣿⣇⣶⣦⣤⣤⣤⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⡄⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣘⠻⢿⣿⣶⣦⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠷⠿⠿⠟⠋



⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠉⠉⠉⠉⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⡠⠖⠋⠉⠉⠙⠲⢄⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⡞⠀⡴⠊⠉⠉⠑⢄⠀⢣⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢰⠀⠀⠀⠀⠀⠈⡆⠈⡆⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠘⡆⠀⢆⡔⠊⠐⢲⠉⠁⠲⡁⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠈⠦⣀⠀⠀⠀⠈⡆⠀⠀⢣⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡈⠀⠀⠀⠀⣏⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡏⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⢀⣸⣆⠀⡏⠀⠀⠰⠋⠉⠉⠉⠉⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⢹⠉⠀⠀⢰⡀⠀⠀⠀⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠐⠂⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡤⠓⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡜⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠇⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣠⠛⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠎⠩⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠜⠀⢀⡀⠀⠧⠤⢄
⠀⠀⠀⠀⡠⠚⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠰⣾⣷⠄⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⢄
⠀⠀⢀⠞⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⢤⠀⢸⡇⠀⡤⠤⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠣⡀
⠀⢀⠎⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢪⠀⢸⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠱⡀
⠀⡜⠀⣰⡆⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢸⠀⣼⣧⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢰⣆⠀⢣
⠸⠀⢀⣿⣿⡄⠀⠣⡠⠊⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢰⢠⣿⣿⡄⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠜⠀⢠⣿⣿⡄
⡇⠀⣸⣿⣿⣿⡆⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡌⣾⣿⣿⣧⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⢰⣿⣿⣿⡇
⡇⠀⠈⠉⠻⣿⠀⠀⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⢰⣹⣿⣿⣿⣿⣆⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢲⠀⢀⣿⠟⠉⠁
⠰⣀⠔⠢⡀⠈⠁⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⣸⣿⣿⣿⣿⣿⣿⣆⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠎⠀⠀⠁⢀⠔⠢⣀
⠀⠀⠀⠀⠈⠦⡀⠀⠀⠈⠑⠢⠤⠄⢀⣀⠀⠤⠤⠒⠙⢄⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠑⠒⠤⠤⠀⣀⡀⠠⠤⠔⠊⠁⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣶⣿⣄⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠄⣀⠀⠉⠉⠛⠻⠿⣿⣿⣿⣿⣄⡀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⣀⠠⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠠⠤⣀⡀⠈⠙⠻⢿⣧⣄⠀⠀⠀⠈⠹⢿⣿⠟⠋⠁⢀⡀⠤⠔⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⠀⠉⠻⢿⣆⡀⠀⠀⠀⠀⠋⡱⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠙⠋⠐⡢⠄⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡠⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡰⠋⢠⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢈⣉⠘⠀⡰⠃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠉⢡⠀⠀⠀⠀⠀⠀⡎⠰⠁⠐⠐⠋⡠⠊
⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡠⠀⢀⡜⣆⣰⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⡐⠁⢀⠊⢇⠳⡈⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠐⠀⠀⠀⠀⢀⡜⠀⢠⠃⡰⢈⠜⡸⠆⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⡰⠁⠀⡇⢸⠘⣀⠇⠀⢀⠔⠁⠀⡇
⠀⠀⠀⠀⠀⢸⠀⢠⠁⠀⠀⠀⠀⠀⠀⡠⠃⠀⡔⠁⠀⢰⡉⡞⢀⠇⢀⠴⠃⠀⠀⢰
⠀⠀⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⡐⠁⢀⠜⠀⠀⠀⡎⢸⣠⡎⠠⠁⠀⠀⠈⠀⡌
⠀⠀⠀⠀⠀⢠⠀⢸⠀⠀⠀⠀⢀⠜⠀⢀⠎⠀⠀⠀⠀⢇⠸⠉⢃⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠸⡀⠈⡄⠀⠀⢀⠊⠀⠠⠃⠀⠀⠀⠀⠀⢨⢜⠇⢸⢀
⠀⠀⠀⠀⠀⠀⠣⠀⠈⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⣀⡳⠟⣁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⠒⠒⢈⣁⠴⢺⣍⢦⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠤⠤⠤⠤⠄⠒⠊⠁⠀⠀⠀⠛⠛⠲⣶⢝⢲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠸⢃⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢿⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢸⠀⠀⢀⣀⡀⠄⠀⠀⠀⠀⠀⠀⠒⠈⠁⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣠⣤⠤⠞⠓⠈⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠉⠁⠀⠀⠀⠀⠀⠀⠘⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆⠀⠀⠀⠀⠠⢀⡀⣀⣠⣵⠶⠗⠛⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠞⠒⢻⡏⠁⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⡏⠁⠀⠀⠀⠀⠀⣺⠀⠀⠀⠀⠀⠀⠀⠃⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡇⠀⠀⠀⠀⠀⣸⡇⣤⣀⣠⠤⠒⣺⠈⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⣀⣤⠤⠴⠒⣿⠉⠁⠀⠀⠀⠀⡈⠠⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⡟⠉⠁⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠁⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠙⣧⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠃⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡄⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠈⠄⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠆⠐⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢡⡶⠏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣠⡴⢀⣴⡟⠁⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣬⣴⢖⣋⢁⠀⣱⠋⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡄⣸⡂⣤⣤⠤⠒⣿⡩⠐⠉⢀⣀⣸⢠⡋⢣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⢠⡴⠓⠋⣿⠀⠀⠠⠒⢁⣤⣶⣾⣿⣿⣿⣾⣿⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠀⢀⣁⠼⠒⢈⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣽⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣏⣩⣤⣴⣾⣿⣿⣿⣿⠿⠋⠻⡿⣿⣿⣿⣟⡁⣀⣀⣀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡤⠠⠝⠻⠿⠿⠛⠛⣛⣋⡉⠷⢛⣿⣶⡓⣿⣻⣯⡍⠻⣿⣿⣿⣿⠿⠷⣶⣤⣿⣿⣶⣛⠋
⠀⠀⠀⠀⠀⠺⠛⠀⠀⠀⠐⠶⠚⠃⣲⣖⢚⣳⣶⣿⣢⣾⣿⣟⣃⣠⣾⣿⠿⢿⣿⡿⠿⢋⣤⠘⠃⡙⠳⠾⠿⠿⣿⣿⣾⣿⣦⣭⣿⣟⣻⠿⠿⠋⣀⡀
⠀⠀⠀⠀⠀⠀⣤⣤⣄⠸⠧⠤⠾⠟⠛⠉⢻⣿⣿⠿⠷⠖⠒⢻⣿⣿⠿⠿⠿⢃⡄⠐⣺⣿⡿⠂⠀⣝⡛⠷⣲⠂⠀⠈⣛⣋⣩⣍⣉⠛⠛⠛⠁⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠓⢀⠲⠿⠦⠤⠄⠈⠛⠋⣀⣤⣤⣤⣉⠁⠀⠰⢿⣫⠄⠀⠀⠘⠿⣷⣯⣭⠖⠛⠙⠻⠿⠿⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣿⡿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣷⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣻⣿⡿⠟⠻⢿⣿⣮⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡏⠀⠀⠀⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢻⡷⠔⠒⢖⢲⣞⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠋⣤⡗⠛⣯⢹⣿⣵⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢳⠛⡇⢰⣿⣾⢻⡅⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠰⡗⠺⣴⡇⠀⣿⣿⣦⡇⢸⠁⢀⡤⠤⢄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣉⣹⠶⠶⠶⢶⣒⣋⣹⣄⡹⢧⣀⣿⣿⣦⣏⢻⣷⣿⣿⣷⠐⣷⣏⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⢳⣮⣉⠛⣿⣟⡧⢽⡶⢳⣼⣿⣼⠃⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢙⣿⣶⡏⠻⡧⠶⠞⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⣿⡇⣿⡆⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠨⣿⡇⣿⣱⠞
⠀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠀⢘⣿⡇⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀
⠀⠀⠀⠀⣠⢋⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⣴⡟⠀⢸⣿⣿⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠻⣄
⠀⠀⢀⣼⠇⣼⡟⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣐⣳⣇⡀⠸⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢳⣜⢧⡀
⠀⢀⣾⠏⢰⣿⣿⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣄⡋⢙⠿⢿⣼⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣎⢷⡀
⠀⣼⡏⠀⣼⣿⣿⡄⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠉⠻⣷⣿⢦⣞⡉⣣⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣼⣿⣿⣿⣧
⢰⡟⠀⢀⣿⣿⣿⣷⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢨⣿⣿⠀⣏⣿⣹⠆⠀⠀⠀⠀⠀⠀⢠⡟⠀⣠⣿⣿⣿⣿⣿
⡼⢀⣴⠿⡿⣿⣿⡛⠷⢽⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠠⣿⣿⡆⣿⣡⠏⠀⠀⠀⠀⠀⠀⢠⡾⠷⢛⡿⣹⢻⠿⣟⠻
⡷⠋⠁⠀⢳⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡏⠸⠦⠐⣿⣿⣿⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⣰⡿⡟⠀⠈⠙
⠀⠀⠀⠀⠘⣆⢈⡻⢿⣿⣶⣤⣄⣀⣀⣠⠞⣧⡞⠀⠀⠀⡼⢿⣿⣿⣧⡑⠦⢤⣀⣀⣀⣤⠤⠖⠉⣠⣾⡿⣱⠃
⠀⠀⠀⠀⠀⠘⣞⣿⣷⣍⡛⠿⠿⣿⣿⡟⢤⣿⣖⣀⠤⠚⠀⠀⠙⠿⣿⣿⣷⣶⣤⣤⣤⣴⣶⣷⣿⣿⢟⣴⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣝⡛⠛⠁⠀⠀⠀⣷⢬⡏⠉⠙⠲⡄⠀⠀⢠⠴⠛⠛⣻⣿⣿⣿⣿⣿⡿⢟⣫⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣄⣀⣹⡦⣽⡀⢀⣠⣾⣆⢰⠃⠀⠀⠐⠛⣛⣫⣭⣿⠾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⠼⡗⠿⢿⣿⣿⡇⣀⡤⠶⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠢⡇⠀⠀⠈⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢛⣛⣛⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣡⣾⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⣿⣿⡏⠁⠀⠉⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⣿⣄⠀⢀⣤⢾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡟⣡⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡇⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⣿⡇
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⣠⣾⣿⣷⣶⣖⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡃⢙⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣲⣿⣿⣷⣄
⠀⠰⣿⣯⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⣿⣿⠆
⠀⠀⠈⠻⣿⡿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⡇⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⡇⣿⣿⡄
⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣧⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿
⣿⡇⠙⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿
⣿⣧⣿⣮⡻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣾⢿⣏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⢻⣿⣿⣿⣿
⢹⡟⠉⢻⣿⣿⣦⡙⢷⣤⡀⠀⠀⠀⠀⠀⣿⠇⣼⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⣤⠾⢋⣴⣿⣿⡟⠉⠻
⠀⠀⠀⠀⠻⣿⣿⣿⣶⣌⡙⠳⠶⣤⣶⡾⠟⠀⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣟⣋⣥⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣦⣤⣀⣀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣛⣭⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣷⣬⣙⠻⠟⣡⣴⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣷⣾⣿⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢦⡀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠙⠀⠀⠀⣀⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⢸⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠑⠄⠀⠀⠀⠀⢰⠃⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣷⡲⢦⡬⠯⣈⡃⠄⠀⠀⠀⠀⠀⠠⡇⠀⠀⠠⠃⠀⠀⠀⠼⡀⢐⣠⠉⠁
⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠘⣿⣧⣼⣷⣟⣒⣒⣶⠒⣶⠀⠰⠇⠠⣄⣀⣐⣤⣬⣵⣶⠿⡿⠁
⠀⠀⠀⠀⢤⣤⣤⣽⣿⠿⣾⣿⣿⣿⣶⣿⣿⠿⣿⣿⣭⢭⣙⣛⣙⢛⣋⣩⡭⣉⣿⡿⠷⢶⣤⣭⠎⢡⠄⠀⠀⠂
⠀⠉⠁⠀⠚⠋⠁⠹⠿⠟⠛⠁⠺⠿⠶⠬⣉⡐⠛⠿⠿⠶⠽⠟⠛⢠⣾⣿⡥⠀⠩⣍⠀⠉⠉⠑⠈⠛⠀⠀⢐⣒⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠒⠂⠘⠛⠛⠧⠄⠀⠐⠒⠀⢲⡤⠉⠉⠡⠤⠀⠀⠉⠅⠀⠀⠁
⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⢲⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠰⠞⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠗⠠⠞⠿⣀
⠀⠀⠀⠀⠀⠀⠀⠒⠷⠧⠤⣤⡶⠶⠷⠗⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠷⢧⣤⣤⣶⠶⠷⠗⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠈⡏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣦⣇
⠀⠀⠀⠀⠀⡏⣄⠀⠀⠀⠀⠈⡇⠀⠀⠀⢀⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣄⠀⠀⢀⠀⠉⡏⠙⢶⡀⢀⡔⡇
⠀⠀⠀⠀⠀⠳⡉⠀⠀⠀⠀⠀⠳⠀⠀⠀⢈⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡉⠀⠰⡏⠀⠀⠻⠀⢸⠇⢈⡽⠃
⠀⠀⠀⠀⠀⠀⠙⢦⣥⣀⣀⠀⢀⣁⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣬⢋⣀⠀⢀⣉⣯⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢤⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⢴⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣥⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟⣷⣟⡿
⠀⠀⠀⠀⠀⠀⠰⠾⠶⠶⢌⠉⣹⠦⠶⢦⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶⣌⠉⣹⠷⠶⢦⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⣀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣿⡇
⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⣿⣟⠳⣤⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠛⣏⠀⠀⠀⠐⠈⢿⠀⠀⠀⠈⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣏⠀⢰⠞⠀⠘⠿⠀⢸⡇⠈⡿⠃
⠀⠀⠀⠀⠀⠀⠙⢧⣤⣀⣤⠤⡦⣐⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣜⣃⣠⠤⣦⣐⣛⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢰⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢰⡿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠒⠉⠉⠉⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡤⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠞⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⢀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⡿⠋⠀⠀⠀⠀⣀⠤⠤⠒⠲⠦⢤⡀⠀⠀⢄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⢤⠀⠀⠀⠀⠀⢠⡀⠙⡇⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠕⠁⠀⠀⠀⠠⢐⠁⠀⠀⠀⣀⡴⢞⣿⠟⢦⣀⣈⣧⣤⣴⡃⠀⠠⠀⠀⠀⠀⢀⣀⣠⠇⠀⠀⠀⣿⠀⡀⠀⠀⠑⣬⡇⠀⢸
⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⢀⡞⢋⡶⠋⢁⣴⣾⣿⣿⢿⣟⡞⣹⢶⣴⢧⡀⠀⠀⣼⣿⡟⢢⡀⠀⠀⡿⠀⠘⡄⠀⠀⠘⡇⠒⠃
⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⣠⠋⣰⠋⠀⣠⣾⡿⣳⣿⠏⣾⡎⣰⢯⡞⠁⠀⡑⡶⣾⢿⠏⣰⠟⠉⠓⠊⠁⡎⢃⠘⢆⣀⣴⠃
⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠰⠁⡼⠁⢀⡼⣹⠏⢰⣿⡿⢸⣿⢠⡟⣾⠁⠀⠌⢸⢹⡏⡼⠀⡿⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⢠⡀
⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⣸⠃⢀⡞⢰⡏⢀⣿⣿⠇⢸⡏⢸⣷⡇⠀⠀⠇⠆⡇⢰⢠⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠇
⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡏⠀⡼⠀⣾⠀⢸⡇⣿⢰⢸⣿⢸⡿⡇⠀⠀⠂⠀⢰⠀⢣⠡⠘⣆
⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢸⠃⡆⠀⠀⠀⠀⠀⠇⠀⡇⠀⡏⠀⢸⠇⠘⠈⡆⠟⠈⣇⣷⣤⣀⣀⡤⠴⠶⠦⢤⣤⠼⠦⢤⣀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢡⠀⠰⠀⠀⠀⢀⣔⡁⡀⠀⠀⠀⠙⠦⢤⣀⠀⠀⠈⠙⠢⠄⠈⠙⠲⢤⣄⣄⠀⠉⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠉⠉⠉⠉⠉⠂⠄⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⡄⠀⠀⢈⡄⣀⣀⡇⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⣹⡿⣻⢿⣆⠀⠀⠀⢀⡆⣁⠀⣧⡴⢅⠀⠀⢸⡷⠋⠀⠀⡈⠐⠶⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⢣⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢏⣾⠞⠉⠈⣻⠶⣶⣿⣷⢣⡞⠁⠀⠀⠑⠶⠞⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢢⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢠⡟⢸⡇⣿⡇⢸⡇⠀⠸⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⢣⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠸⡇⢸⡇⢻⣳⡀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙⢄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⢳⡀⢳⡈⠳⡑⢄⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠲⢭⣢⠌⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠓⠒⣺⣟⡐⠒⠤⢸⠀⠀⠀⠀⠀⣀⣠⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣳⣄⣀⣼⣤⣴⣶⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠲⠤⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢟⣛⠿⣷⡄⠀⠀⠀⠀⠀⠠⢄⡀⠠⠤
⠀⠀⠀⠁⣀⣈⣉⣽⣶⣶⣶⣿⣭⣽⣯⣭⣿⡿⠿⣿⢿⠿⣿⣛⣛⣫⣥⣤⣴⣶⣿⣿⣿⣿⣖⣦⠀⠀⠠⠤⠤⠤⠤⠄
⠀⢀⣀⣭⣍⣉⣉⡻⣯⣥⣶⣾⣿⢟⣿⣿⡿⠧⠾⡿⣏⣠⣭⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣭⣥⣤⠤⠐⠒⠒
⠈⠉⣉⡉⠛⠛⠛⠛⠻⢌⣉⣉⣐⣋⠉⠻⣿⣿⡛⠉⣴⣿⣿⡯⠛⠓⠊⠀⠉⠩⠭⠭⠭⠽⣿⣿
⠀⠀⠈⠉⠉⠉⠁⠀⠲⠦⢬⣍⣉⣁⡀⠀⠉⠉⠉⠥⠬⢽⣿⠧⠀⠀⠛⠓⠒⠚⢊⣉⣩⡭⠉⡉⠀⠐⠚⠛⠓⠒⠲⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠒⠒⠂⠀⠘⠿⠃⠀⠀⠀⠑⠒⠂⠨⠁⠀⠀⠉⠙⠒⠲⠶⠟⠋

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀
⠿⠿⠿⠿⠿⠿⠿⠿⠛⠿⠿⠿⠿⢾⣿⣿⠿⠟⠻⠿⡿⠍⠛⠛⠛⠓⠚⠒⠛⠛⠛⠛⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣤⣀
⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠑⣿⣿⣿⣓⣂⣋⣁⣀⣠⡤⠤⠤
⠈⠉⠉⠉⠉⠉⠉⢉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢨⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠠⠤⠤⠤⠤⠤⠀⠄⠀⠖⠒⠿⠧⠤⠤⠀⠀⠀⠀⠀⠀⠀⠈⠒⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠲⠮⠵⠶⠦⠤⠄⠀⠀⠈⠀⠤⠤⠶⠶⠶
⣤⣤⣤⣤⣄⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢀⣀⣀⡀⢀⠀⠀⢤⣄⣤⡄⠀⣤⣤⣤⣤⣤
⣾⣿⣶⣶⣷⣶⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣿⣿⣿⣶⡀⢀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⣿⡛⢛⣻⢿⠿⢿⣿⣿⣿⡿⣛⣛⣛⣛⣿⣿⣿⣿⣿⣯⣿⡖⠀⠈⠙⠻⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⠛⠛⠻⠛⠛⠛⠉⠁⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠂⠀⣈⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠧⢤⣀⣌⢱⢛⢹⣻⢻⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠰⠚⣻⠿⠿⠶⡶⣶⣮⣿⣿⣿⣿⣿⣿⣿⣛⢛⣿⣾⣿⣶⣤⣙⢲⡶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡟⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠤⠄⣈⣐⣛⡻⠿⢿⣿⡿⣟⣛⠛⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢠⠤⣤⣤⢤⣤⣤⣤⣀⣦⣤⣍⡛⠛⣿⣿⣿⣿⣿⣿⡿⠾⠏⠙⠿⠿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⡴⠿⠟⣩⡿⣻⣿⣿⣿⣯⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠹⠋⠉⠉⣿⣦⣘⠛⣋⣉⣉⠁⠘⢶⣶⣮⣷⣶⣭⣍⣙⡛⣿⣿⣿⣟⣙⣋⡻⠿⣿⣿⣭⣿⣿⣋⣉⣭⣭⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠾⠿⠿⠛⠛⠉⠛⠛⠛⢿⣿⣿⣿⣯⣽⣭⣁⡀⠀⢀⣹⢿⣿⣿⣿⣿⣷⣦⣬⣭⣟⡿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⢤⣤⣶⡶⣿⣷⣤⣭⣭⣤⣄⣀⠀⠀⠀⣠⣤⣶⣤⣭⣭⣭⣭⣭⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘






⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠛⠻⢿⣿⣶⣤⣬⣉⣍⣉⣭⣭⣤⣤⣤⣤⣶⣶⣤⣤⣴⣶⠶⢶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿
⣶⣤⣶⣾⣿⣿⣿⣿⡙⠛⠛⠛⠛⠏⠉⠉⠉⠉⠉⢉⣉⠁⠀⠄⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠀⠀⠘⢿⣿⣿⣿⣿
⣿⣿⣟⣛⠛⢻⣯⣙⠛⠃⠀⠀⠐⠒⠒⠒⠂⠀⠙⠈⠋⠙⠉⠉⠈⠁⠀⠀⠀⠀⠀⣈⣉⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢉
⣿⣿⣿⣿⣾⣿⣿⡭⠀⠀⠀⠐⠒⠒⠛⢒⣠⡶⠖⢂⣠⣤⣶⣶⣦⣤⣀⣀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⡿⣟⡛⠛⠛⠁⣀⣀⣤⣶⡶⠖⣛⣃⣬⣭⣵⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⡤⠤⠤⠤⠶⠶⠶⠶⢶⣶
⣿⣽⣿⣥⣼⣿⣿⣟⣋⣉⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣄⣀⣠⣤⣤⣤⣤⣤⣤⣬⣭
⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠉⠉⠉⡉⠙⠿⠟⠛⠉⣩⣾⣿⣿⣿⣿⡿⠟⠛⠉⠉⠉⠉⠉⣉
⣶⣤⣤⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⣀⠀⣄⣲⣶⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣯⣍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣴⣶⣏⣉⣠⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⣟⣉⣁⣀⣀⣠⣤⣤⣦⣤⣤⣶⣶⣶⣦⣄⣀⣀⣀⣀⣤⣤⣤⣶⣶⣶⣾⣿⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢸⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡆⠈⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣀⣤⡤⠼⠗⠘⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢴⠊⠁⠀⠀⠀⠀⠀⠀⠀⣆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡈⠀⠀⠀⢀⠀⠀⣀⢀⣀⣼⠦⠻⠚⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠴⠖⠒⠛⡟⠁⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣼⠉⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡿⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⡍⣀⡀⠀⠀⠶⢺⡏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⣀⣠⣤⠶⠖⣶⠏⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠉⠉⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢹⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡍⢷⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⢸⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡇⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠘⠀⠂⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢳⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠁⠀⢀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⡠⢀⣴⠿⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠤⣂⣠⡾⠋⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡄⠀⠀⠀⠀⢀⣰⣤⡶⢞⡁⡀⢈⡟⠁⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣃⣤⣤⠴⠆⢻⡏⠄⠋⢀⣀⣠⣄⣼⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⢢⡤⠟⠞⢹⡃⠀⡠⠔⠈⣠⣴⣶⣿⣿⣿⣷⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⣀⠸⠗⢈⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣨⣥⣴⣾⣿⣿⣿⣿⡿⠟⠙⣿⢿⣿⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡭⠛⠟⠛⠛⠛⢛⣛⣉⠻⠞⣻⣗⡜⢼⣟⣿⣍⠛⢿⣶⣾⣿⡛⠿⢿⣤⣴⣬⣶⣶⣋⠁
⠀⠀⠀⠀⠀⠺⠛⠀⠀⠀⠀⠶⠖⠋⢲⣶⠚⣳⣶⢿⣦⡾⢿⣿⣋⣀⣴⣿⡿⠿⣿⡿⠿⢛⣡⡘⠋⢘⠛⠶⠿⠿⢿⣿⣶⣿⣷⣬⣽⣿⣿⣿⡿⠟⠁⣀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⠠⠥⠤⠶⠾⠛⠋⠹⣿⣿⣿⠿⠂⠒⠻⣭⣿⠿⠿⠿⢓⡠⠀⢾⣿⣿⠇⠀⢨⡛⠻⢶⣶⠀⠀⠛⢛⣉⣩⣍⡙⠛⠛⠃⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠛⢀⣙⠫⠥⠤⠄⠈⠉⠋⢡⣤⣤⣤⣉⡁⠀⠐⢛⣫⡵⠀⠀⠈⠻⣷⣾⣯⡭⠉⠉⠛⠿⣿⡿⠧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠲⠖⠂

⣿⣿⣿⡿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠚⠛⠛⠳⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⣠⣴⡶⠷⠶⠦⣄⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⣿⠏⠀⠀⠀⠀⠈⣧⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣀⣠⠤⠤⢤⣀⠟⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠛⢰⡇⠀⠀⢸⣿⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢽⠿⡇⠀⠀⢸⣿⠟⠋⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠄⢲⣳⡟⠊⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠶⢰⣯⣀⡀⢀⣀⣀⣀⡀⢀⠀⠀⠀⡀⣄⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣆⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣴⣾⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⢻⡜⣿⠛⢻⠻⢿⣿⣿⡛⠋⠉⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣹⣦⣸⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣿⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢻⣿⡟⠳⣟⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⣿⡇⠀⠈⢧⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠸⢯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣸⣲⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣷⡀
⠀⢸⠟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⣠⢳⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡅⢹⣧
⠀⣾⣇⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠟⢻⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣽⢁⣎⣿
⠀⣿⣿⡈⢻⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢠⣄⣷⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⣾⠃⣸⣯⣿
⠀⣿⣿⣷⡌⢿⣧⡵⠶⠀⠀⠀⠀⠀⠀⠀⢀⡠⣣⠴⠊⣵⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠃⣰⣿⣹⡿
⠀⠸⡿⣿⣿⣆⠳⣄⠀⠀⠀⠀⠀⠀⢀⣔⣣⠿⣇⡤⠆⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠙⢿⠃
⠀⠀⠹⠏⠘⢿⣧⡀⠑⠦⣀⠀⢀⡔⣣⠟⢁⡜⠉⠀⠀⢸⣧⣿⣿⣦⡀⠀⠀⢀⡀⠄⠂⠀⣠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠉⠉⠙⠓⠊⠁⠀⠀⠀⠀⢸⣿⡟⠝⠛⠉⠀⠉⠁⠀⢀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⠀⠀⠀⠀⢀⣀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣷⣶⣤⣀⠀⠀⠀⠀⢸⣿⡫⣴⣦⣾⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣿⠀⠀⠉⠛⠿⣿⣶⣤⡀⣸⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠷⠀⠀⠀⠀⠀⠉⠻⣿⣷⣿⡿⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠛⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠨⡗⠀⠀⠀⠀⠀⠀⣀⡀⠼⠗⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⢀⡀⠋⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣹⠶⠛⠊⠉⠀⠀⠀⠀⢀⣧⣄⣲⡬⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠀⠀⠀⠀⠀⢀⣶⣬⠄⠒⠘⠉⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⠀⠀⠀⠀⠀⠀⣠⡲⠟⠛⠛⠀⠀⢠⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠠⠔⠂⠈⠁⠀⠀⠀⠠⣷⠀⢨⢀⣀⣀⠤⠤⠤⢦⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣰⠦⠖⠒⠂⠀⠀⠀⠀⢀⠀⠀⣸⡿⠀⠚⠋⠉⠁⠀⠀⠀⠈⢆⡀⠸⢷⡀⠀⠀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⠀⠀⠆⢠⣇⠤⢶⡞⠛⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣧⡀⠀⠉⠛⠛⠃⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠍⢂⠂⢉⡴⠷⠖⠋⠄⢀⠀⠀⠀⣀⣠⣄⡤⠷⠿⠿⣇⣀⣀⡀⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠒⠶⠖⢂⠍⠁⠀⠀⠀⢻⠀⠠⢰⢂⡠⠐⠒⠋⢹⡄⢛⣷⣴⣶⣿⣿⣿⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢰⠀⠈⠀⠀⠀⠀⠀⠸⡆⠀⢷⡛⠀⠀⠀⢀⣨⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣆⠀⡀⢀⡀⠀⣀⣀⣷⢀⣨⣵⣾⣿⣿⣿⣿⣿⣛⣹⠛⠿⣿⠿⣷⣷⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⣳⠾⠿⠶⠶⠿⠿⢏⣉⣡⣤⣽⣿⣿⣿⣿⣿⣿⣿⠛⣿⣾⣶⣷⣦⡀⢹⣿⣿⣿⣷⠆
⠀⠀⠘⠿⣿⣶⣦⣄⣀⠈⠻⠿⢏⣠⣞⡻⣿⣿⡿⠿⠿⠯⠮⣭⡭⠛⠛⠉⣩⣷⣾⣿⣿⣿⣿⣿⢸⡿⠿⢿⡷
⠀⠀⠀⠐⠉⠉⠉⠉⠻⠿⠿⠿⢟⠍⠉⠁⠹⣄⠀⠤⠤⠆⠀⠤⠶⡷⢶⠿⠿⣿⣿⣻⣿⢿⣿⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠙⠁

⠀⡀⣈

⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡞⠀⠐⣀⣀⠀⢄⠀⠰
⠀⠀⠀⠀⠀⣄⣠⣷⠎⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⣀
⠄⠀⠀⠠⡀⢀⠁⠀⢀⠄⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⡀⢤⡀⠀⠀⠀⠍⢻⠁⢀⡴⢃⣀⠈⠃⠐⠀⣠⣤⡘
⠉⠉⠘⠀⠀⠀⢘⢘⠀⠊⠀⢴⣆⣤⠠⢠⣖⣿⣿⡓
⠀⠠⠇⠀⠀⠀⣰⢣⠀⠀⠴⡶⢟⣠⡶⠟⠋⠉⢉⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠓
⠃⢘⡜⠁⢠⢠⣿⡴⡒⠂⡾⣿⠟⠁⠀⣠⠖⠓⢦⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢸
⢀⡏⠀⠠⢃⡾⠑⠃⢁⡞⣰⠏⠀⠀⢠⠁⠀⣤⡼⠀⠀⠈⢃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠆⠀⡆
⣸⠃⠀⢀⡞⠁⡀⢀⡞⣄⡇⠀⠀⠀⠈⡄⠈⠓⠒⠂⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⣴⣆⡀⠀⠃
⡇⠀⠀⠔⢊⠐⠀⠾⠣⡈⣧⠀⠀⠀⠀⠈⠂⢀⢤⡴⡶⣦⠀⢀⡀⠀⠺⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⠂⠿⠀⠀⠀⠀⠀⠙⡣
⠇⠀⠀⢀⡏⠀⠀⠃⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⣠⡤⠤⢀⠈⠉⠃⢾⡧⠾⠤⠀⡀⠀⠀⠊⣿⠇⡀⠘
⣦⠰⡁⠸⠁⠀⠀⡅⡄⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⠀⠀⣩⠅⠀⠑⠀⠱⣆⡔⠀⠀⠀⠀⠀⠉⠢⡄⠀⠈⠒⠁⠤⡀
⡟⠁⢇⠎⠀⠀⠀⢧⠀⠀⠀⢀⠙⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠴⠶⠋⠉⠈⠡⣞⠁⠀⣺⠀⠀⠀⠀⠀⠠⡶⢀⡼⠈⠂⠀⠀⡀⠈
⡇⠀⠀⠀⠀⡄⠀⢰⣌⠢⠀⠀⠉⠐⠈⠑⢤⣄⣀⠀⠀⠀⣀⣤⠤⠶⠊⠀⠀⠀⠀⠀⠀⠀⠈⣤⠐⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣷⠀⡀⠀⠀⠁⡄⠘⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠑⠂⠀⠀⠀⠀⠀⠀⡠⠒⠂⠂⠔⢘⠃⠀⠀⡀
⠹⠗⠀⠀⠀⠀⠀⠠⠀⠙⢦⣈⡁⠀⠀⡀⠀⠀⠘⠢⢤⣀⡀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠒⢢⡈⣄⠀⠀⠀⢈⢀⡀⠀⠀⠀⠀⠀⣀⡤⠄⢀
⠀⠒⠀⠀⠀⠐⡀⠀⠅⠀⠃⠙⠶⣀⣈⠢⠐⠒⠐⢀⡰⠊⠀⠀⠀⡠⢚⠕⠀⠀⠀⠠⠀⠀⠻⢼⡇⠀⠀⠀⠹⢄⠀⠀⠀⠂⠨⠥⡀⠀⠀⠐⠤
⡀⠀⢦⠈⠀⠀⢁⠀⡄⠀⠑⢄⠐⠈⠙⠷⠶⡶⢦⡤⡤⢄⡶⠊⠁⠄⠀⠀⠀⠀⠀⣀⠤⠀⠈⢛⢷⠀⠠⠀⠀⠀⠉⠀⠀⠠⠤⣤⠤⠄
⠀⠈⠐⣷⡀⠀⠠⠀⠑⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⠞⢀⠊⠀⠀⠉⠀⠀⠀⠀⠀⢀⠠⠐⠚⠺⢦⠀⠀⠀⠀⢀⣀⡳⠠⠐⠂⠀⠘⠢
⠀⠀⠀⠘⢷⣜⠀⠀⠀⠀⠄⠂⠐⠀⠀⠁⠒⠈⠁⠀⠈⠀⠀⠀⠀⠀⠐⠈⠀⡠⠀⠈⠀⠀⠀⠀⠀⠀⠀⢄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠹⣄⠀⠀⠘⡄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⣈⡉⣤⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⠀⢫⠒⠒⠰⠶⠖⠂⠀⠀⠒⣄
⠒⠀⠀⢄⠀⠀⠈⠛⢤⡀⠘⢆⠂⠑⢀⠀⠠⠤⣀⠀⠀⠀⢀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⡀⠀⠀⠀⠒⠀⠀⠀⢸
⠀⠀⠀⠀⠑⡐⠆⣀⠀⢉⡲⠮⣷⣦⣤⣬⣕⡢⠤⠍⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠲⣿⠀⠀⠋
⠀⠈⠀⠀⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐
⡿⢿⠿⠿⡿⡻⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣿⣿⣿⢿⣿⢿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢩⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣀⣄⡀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⣼⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⠤⠤⠴⠿⡿⠷⠤⠤⢤⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠷⠶⠤⣶⣶⣶⣶⡶⠴⠶⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣶⣶⣶⣶⣶⣴⣦⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⢦⠀⠀⠀⠀⠀⠀⠀⡇⣸⡇⠀⠀⠀⠀⠀⠀⢠⢦⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿
⠀⣸⠀⢧⡀⠀⠀⠀⠀⠀⡇⢹⣷⠀⠀⠀⠀⠀⢠⠏⢸⡆⠀⠀⣿⣿⣿⠀⢻⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡟⠈⣿
⠀⣿⡄⠉⣳⡄⠀⠀⠀⢀⡇⢸⣿⠀⠀⠀⠀⣰⡋⠀⣼⡇⠀⠀⣿⣿⣿⡆⠀⣹⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡋⠀⣸⣿
⠀⠻⠻⣄⠈⠦⣀⠀⢀⣸⠃⣼⣿⣄⠀⢀⣠⠞⢁⣾⠻⠃⠀⠀⣿⣿⣿⣿⣄⠈⠻⣿⣿⣿⣿⠃⢸⣿⣿⣿⣿⣿⠟⢁⣴⣿⣿
⠀⠀⠀⠙⢷⣄⡈⠛⠛⠁⠀⣿⣿⣿⣭⣭⣤⣶⠟⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣄⡈⠛⠛⠁⠀⣿⣿⣿⣭⣩⣤⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠛⠷⣶⣶⣴⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⢿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⣀⡄⠀⠀⠀⡆⠀⠀⠠⠃⠀⢀⠀⠀⠀⠟⢀⠁⠀⠈⢀⣤⡀⠀⠀⢘⣀⡀⠀⠐⠃⠀⠀⠀⠀⠀⠊⡀⠀⠠
⣀⢠⠠⢄⡀⠀⢸⠃⢠⠩⣄⡀⡄⠥⡀⠀⠡⡏⢀⡌⢄⣀⣤⠤⢅⠀⠀⣼⠀⡄⠡⣀⡠⡠⠤⡀⠈⢡⠃⢠⠉
⢙⡉⠀⣼⠤⠐⢂⠀⠬⠀⠈⡋⠀⢠⠥⠄⠒⡀⠤⠁⠀⢉⠁⠀⡬⠄⠐⠀⠠⠄⠀⠉⡉⠀⣴⠥⠀⠂⠀⠬
⠀⢰⠤⠷⠀⠂⡼⠀⠀⡤⠀⠀⣦⠼⠄⠀⠠⠂⠀⢦⠄⠀⣸⡴⠇⠐⠀⠖⠀⠰⡤⠀⢰⣤⠾⠀⠂⡸⠂⠀⡦
⠀⠠⠀⠀⠉⠐⢳⢃⡀⢲⠀⠀⠅⠀⠈⠀⠛⠏⣀⠐⡆⠀⠨⠀⠀⠁⠘⠾⢀⡑⢷⢀⠠⠅⠀⠉⠐⠳⠇⡀⢺

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡖⠶⠔⠂⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠉⠉⠉⠑⣀⣘⣉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣛⣓⣼⣔⣶⣒⣤⡇⡃⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢣⠀⠀⠀⠀⣼⡧⠤⢶⣶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠷⠶⣐⣾⠤⢤⡤⠤⣄⡇⠀⠀⠈⠯⡒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡲⠤⣼⠀⠀⠈⠫⠉⠉⠉⢭⣽⠷⠶⠶⣶⣤⣙⣦⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠞⠒⠋⠑⠒⡇⠀⠀⠀⠈⢆⠀⠀⠈⠻⡲⢄⣣⠀⠀⢈⣑⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⢱⠠⢼⣇⣾⣻⣟⣥⡲⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡇⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⡼⠀⣀⣀⣀⠀⡇⣠⢏⣘⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢸⢃⣠⡴⢒⠾⡟⢻⣿⠿⣽⣽⠗⣋⢭⣿⣌⡝⣻⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⡾⣛⡛⡓⡚⣶⣷⣾⣇⣀⣾⣿⣿⣿⣿⣾⣷⣷⣿⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣵⣓⣐⣐⣸⡿⢿⠿⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⠄⠠⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⣂⣐⡿⢿⣿⣿⣿⣷⣾⣿⣿⡿⠿⠿⠟⠋⠉⣭⣩⢉⣁⢻⡑⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠭⠭⠥⠷⠒⣦⡀⠙⠛⠓⠠⠤⠍⠀⠐⠂⠀⠴⠚⠃⠼⠃⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠈⠀⠀⠉⠁⠀⠒⠒⠒⠂⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣿⣷⣦⣤⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⢀⠈⠹⠛⢩⣿⣿⣿⣿⣿⣿⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢈⠋⠁⢸⣿⠻⠋⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⢨⠀⢠⢫⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⠿⠤⠖⠒⠇⠀⢀⣀⣸⡇⠘⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠄
⠀⠀⠀⡠⢀⣤⣤⣤⣄⣀⣀⣀⣀⣀⣸⡇⢀⠀⠚⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠹⣷
⠀⠀⠀⣴⢿⣿⠙⠋⠿⠸⡟⡛⢿⠟⠉⠁⠀⡿⠋⠁⢀⡤⣄⡠⠀⠀⢀⣤⣶⠟⠉⣀⣠⣄⡀⠀⠉⠁
⠀⠀⠀⠀⠀⠃⢀⣀⣀⡀⠡⣧⣄⡀⠠⣴⡾⠃⣰⡶⠺⣇⠯⣀⣀⡾⠛⠋⠀⠀⢀⣈⣿⣿⣿⠖⠒⠀⣀⡀
⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠈⠉⠉⠛⠀⠈⢿⣧⣀⠈⠑⠈⠙⠋⠉⠀⠀⣀⡀⠈⠉⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠉⠉⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⠔⠈⠁⠢⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⡐⠂⠒⠓⠀⠬⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠈⠒⠵⠤⠄⠀⣀⡐⠁⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠒⢁⠶⢀⢀⡠⠁⠈⢀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠄⠂⠀⠐⠐⠀⢦⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠈⢅⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡐⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠄⠊⠀⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠋⠀⠀⡠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠀⠁⠀⠀⡄⠂⠁
⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠁⠀⡠⠐⠉⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⡔⠀⢷⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠋⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⡘⠀⠀⠈⠕⠄⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡠⢢⠂⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠌
⠰⠁⠀⠀⠀⠈⢆⠂⡀⠀⡇⠀⠰⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡰⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠜⠀⢀⠌
⡀⠀⡀⠀⠀⢔⢦⠓⠁⠀⢁⠀⠀⠂⢀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠊⠤⡤⠀⠀⠀⠀⡃⠀⠀⠀⠀⡠⠂⠀⠠⠂
⠗⠈⠱⠀⠀⠀⠢⡑⢄⠀⠀⠄⠀⢈⡎⠀⠀⠀⠀⠀⠘⣆⡀⠀⠀⠀⠀⡠⢀⠔⠁⠀⠀⠉⠐⠂⠀⠀⡠⠊⠀⢀⠔⠁
⠀⠀⠀⠱⡀⠀⠀⠈⠐⠬⠀⠈⠗⠋⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⣀⠀⠥⠒⠁⠀⠀⠀⠌⠀⠀⢀⠠⠊⠀⢀⠀⠁
⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠠⠄⠊⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⡀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠊⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠀⠀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⢀⠊

⠀⠀⠀⠀⣿⡿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠿⠿⠿⠿⠿⠿⠿⠿⠷⣦⡀
⠀⠀⠀⠀⢿⣧⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣿⡇
⠀⠀⠀⠀⠸⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⢻⡟
⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡛⠛⠛⠓⠒⠶⣤⣀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠉⢳⡄⠀⠀⢹⡆
⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣿
⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⣸⡏
⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣤⡤⠾⠋⠁⠀⢀⣼⠏
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢀⣠⡾⠟⠁
⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠾⠛⠉
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠁
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⡿⠿⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⢾⣿
⠀⠀⠀⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡟
⠀⠀⠀⢻⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⣩⡟⠁
⠀⠀⠀⠀⠉⠛⠶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡶⠶⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠂⠰⠢⠠⠁⠐⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣦⠰⠀⡃⡄⠀⠐⠀⠠⠄⠀⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⡔⢀⠋⡇⡈⠀⠀⠐⠈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣶⢰⠼⠌⣍⠖⠃⠀⢀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣄⠠⠀⠠⠀⡐⠠⠀⠀⠀⣠⢌⣷⢎⠁⡉⢈⡮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⢀⣈⢤
⠀⠀⠀⠀⡀⠀⢀⠀⢀⡲⣞⣁⠀⡡⠀⢀⢘⠀⠀⠀⠈⣡⠧⠑⣿⢮⣀⣙⠳⢷⣝⢄⠀⠀⢀⡠⠄⡀⠀⢀⢈⡀⡘⣾⡇⠄⠈⡡⢛⣀⠂⠀⠤⡀
⠀⠀⠀⢰⢲⠞⣁⡓⠏⡔⢿⣿⡿⠟⡊⢁⣠⠜⢦⠠⣬⣟⡣⠒⠉⠁⠉⠉⠳⣦⡙⠃⠙⠲⣷⣼⣷⡶⠛⣡⣤⠫⣉⣜⣗⣶⣶⣛⣾⠿⣖⣒⣋⣳⡶
⠀⠀⠀⣜⣛⡥⠥⠒⠒⠚⡳⠮⠝⣍⡉⠛⢫⢀⣕⣰⡼⠎⠀⠘⣧⣟⣦⣃⣣⢘⣐⣤⡴⠮⣿⠛⠿⠛⠻⠿⣟⡟⠿⡿⠟⢛⣃⣾⣷⣟⣿⣿⣷⣶⡍
⠀⠀⠀⢘⠂⠋⢉⣙⠑⢦⡙⠭⡒⢬⡛⠛⠲⣛⠋⣍⡴⡋⠠⣞⡲⠃⠻⣶⢿⣿⡞⠥⠤⣙⣾⢦⠶⠶⠦⡶⠶⠽⠷⠴⠿⡿⠿⠿⢿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⢨⣤⠤⠞⠉⠻⡶⣭⣝⢾⣆⢹⡓⠂⠺⠯⠴⠦⠖⣊⡥⠍⢳⣖⣛⣋⡉⠓⠦⣰⣬⣿⡷⠷⢶⡶⠷⣾⣾⣶⣶⣶⡶⠶⢶⠾⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⣀⠔⠙⢦⣑⢲⢖⢲⡈⠛⠾⠧⣶⡖⠒⠚⡋⠁⠂⠀⠳⠭⠷⠤⠘⠲⠦⠼⣿⣧⠴⣚⠥⠒⠱⢤⡉⠉⢉⡿⠊⠉⢠⣙⠑⢲⣈⢀⡠
⠀⠀⠀⠙⠉⠀⠀⠀⠀⠈⠁⠀⠙⠙⠐⠒⠚⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠩⠙⠉⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⢀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢈⣿⣤⣤⡀⡐⠀⡀⠈⠀⠈⠈⠏⠂⢄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⡷⠶⣶⠅⠀⠀⠀⠀⠆⠀⠀⠀⡂⢾⠃⠀⠀⠀⠣⢀⡒⢦⣤⡠⠀⠀⠀⠐⠢⠀⠠⠀⢠⢄⡀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⣼⡷⠶⡦⠀⠁⠀⠀⠀⠀⠀⠀⣐⣱⡗⠀⠀⢶⡀⠐⠈⠂⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤⣠⡷⠛⠀⠀⠀⠀⠀⠀⠶⢷
⠀⠀⠀⠀⠀⣿⠛⠷⠿⢣⣄⠀⠀⠀⠀⠀⢰⣿⠟⠁⠀⠀⠀⠙⠒⠒⠋⠁⠀⠀⠀⠀⠀⢀⡠⠀⢤⣶⣄⣜⠃⠀⠀⠀⠀⠀⠀⠀⠂⣸
⠀⠀⠀⠀⠠⣿⠛⠿⠂⠀⠹⢿⣄⡀⣀⡴⠛⠁⠀⠀⠀⠀⢀⠠⣀⣄⣄⣀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⢰⡟⠛⢛⣠⣤⣤⡀⠈⠛⠋⠀⠀⠀⠀⠀⢠⠴⣚⡏⣇⣇⣏⡇⢩⡒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⢠⡄⣀⣀
⠀⠀⠀⠀⢸⡟⣻⣿⣿⠛⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢨⢖⡡⣴⣩⣍⢉⣭⠥⢤⣡⠇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣄⣬⡠⠳⠋⠈⣀⣦⡀
⠀⠀⠀⠀⣾⡛⣿⣿⣝⣀⠁⡼⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣚⣣⢤⡭⣭⢶⣓⣰⠎⠀⠀⠀⠀⠀⠀⠀⠀⡿⠔⠌⢃⣉⡔⠟⣠⣾⣛⣿⣟⣂
⠀⠀⠀⠀⣿⠛⠛⢿⣿⡿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢞⡵⣿⣿⣞⣭⣿⣿⣿⣿⣷⣿
⠀⠀⠀⠰⣿⠛⠛⠃⠀⠀⣀⣴⣾⣷⣖⣦⣀⡀⠀⠀⠀⠈⠁⠉⠙⠛⠛⠉⠉⠋⠀⠀⠀⣀⣀⣀⣀⠀⠀⠘⠻⣴⣿⣿⣿⣿⣼⣿⣿⣿⡷⠃
⠀⠀⠀⢰⡟⠛⠛⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣍⣻⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠉⠙⠛⠻⠻⠛⠛⠉
⠀⠀⠀⢼⡟⠛⠛⢠⣿⣿⣿⣿⣿⣿⣿⡿⢊⠕⣉⡽⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣟⡏⢭⣾⠿⣿⡀⠀⠀⠀⠀⠀⣀⣤⣴⡿⣶⢦⢄⡀
⠀⠀⠀⣾⠛⠻⠗⢸⣿⣿⣿⣿⣿⣿⣛⢺⣽⢾⣻⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠻⠻⠛⢤⠊⠉⠀⢸⡇⠀⠀⠀⠀⢰⣿⣿⡟⠁⠐⠀⠠⣽⣂
⠀⠀⠀⣿⠛⠿⠇⠈⢻⣿⣿⣿⣟⣛⣚⣭⣷⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⢷⣶⣠⠇⡖⣹⠁⠀⠀⠀⠀⢸⡿⣿⠃⠂⠀⠀⣀⠙⡟
⠀⠀⢠⡿⠶⠶⠆⠀⠀⠉⠛⠛⠻⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣮⣿⢌⡾⠃⠀⠀⠀⠀⠀⠀⠻⣤⢠⠈
⠀⠀⢸⡿⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⢠⡟⡀⠈⠉⠂⠐⠂
⠀⠀⣸⡷⣶⣦⠀⠀⠀⣀⣠⡤⢤⣖⢺⣟⠇⠀⠀⠀⢰⣤⡀⠀⠀⠀⠀⢀⣀⣴⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡏⣷⣄
⠀⠀⣿⣦⣤⣤⢀⣴⡿⣏⢽⡧⢊⡱⢶⡞⠀⠀⠀⠀⠀⢻⣿⣷⣄⢀⣴⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢠⣤⡶⠾⣿⣿⣿⣥⣴⣿⡿⡿⣷⡶
⠀⠀⣿⣤⣤⡀⣾⡽⢘⡤⢋⢔⣹⢿⡏⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣶⣿⣳⠯⣦⡶⢿⡿
⠀⠘⣯⣈⣉⣡⠛⢀⢀⣹⣏⣺⣿⠟⠁⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⢋⣿⣷⣘⠛⣿⡃
⠀⢸⣏⣉⡛⢣⣰⡾⢿⣿⡿⣿⡟⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠛⢹⣿⣿⡈⠉⠉⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⢡⣾⠿⣿⣿⣷⣾⣷
⠀⣾⡛⠛⠛⠋⠛⠀⠚⠛⠋⠉⠀⠀⠀⠀⠀⠀⠰⠟⠋⠁⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠁⠀⠀⠀⠉⠙⠛⠻
⠀⣾⠛⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠻⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠠⠐⠒⠀⠀⠀⠀⠀⠚⠀⠀⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⣀⠀⠀⠛⠀⠀⠀⣀⣠⣀⣀⣀⠀⠂⠀⢀⡰⡆⠀⠨⢀⠠⠤⠌⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣶⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠒⠒⠒⠒⠒⠒⠂⠀⠀⠉⠁⠀⠁⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢹⠐⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠲⠊⠁
⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠂⣸⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠩⡧⢑⡏⠀⠀⠀⠀⠈⠀⠐⣂⡀⠠⠀⠹⣀
⠀⠀⠀⢸⣛⣻⠶⡶⠦⠶⠆⣆⣰⣴⣶⣤⣤⣼⣄⣜⣁⣀⣠⣦⣤⣤⣤⣷⣿⣴⣶⣷⠾⢷⣦⣴⣶⣾⣿⢷⣿⡿⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠉⠛⠛⠲⣶⣦⣤⣤⣤⣎⣃⣇⣁⣀⣉⣉⣉⣉⣉⣉⣁⣖⣐⣜⣐⣐⣒⣉⣭⣭⣭⣭⠤⠖⣂⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⡛⠿⠟⠛⠿⠿⠿⠿⠿⠿⠿⠿⢿⡿⢿⡿⠿⠿⠿⠿⠿⠿⠷⠴⠀⠴⠡⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠃⠀⠈⠉⠐⠈⠉⠓⠾⠟⠋⠘⠷⠖⠀⠉⠉⠙⠓⠒⠛⠂⠂⠉⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⣠⠄⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣸⡏⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡿⠛⠛⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⠀⠀⡀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠸⡟⠀⢀⠀⠠⠤⠆⠀⠀⠀⠀⣀⠀⡠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠞⠚⠈⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣤⢶⣟⡻⢿⡆⠀⠀⠙⠚⣆⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⡴⠿⡏⢻⣁⣿⣤⠀⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡷⠶⠚⠛⠉⠉⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠀⡁⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠊⠁⢀⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⢶⣶⠆⠀⠀⠀⣶⠤⢶⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢸⠀⠀⠀⣀⣴⣞⣩⡤⣿⣄⣚⣁⣰⣄⣀⣀⣈⣦⣴⣯⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⣼⢠⣤⡾⠿⠟⣏⣹⣅⣼⣿⡭⠿⠿⠋⠁⠀⠀⢸⣿⠃⠈⠁⠀⠀⠀⠈⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢀⣿⣫⣥⠔⠖⠋⠉⠉⠁⠀⠈⠆⠀⠀⠀⠀⢠⡄⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⢠⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⢤⡄⠻⠶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣶⣦⣶⣿⣿⣻⣿⡗⠶⣷⣄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣀⣤⢴⣿⢸⣧⣤⡾⢛⣽⣿⣿⣿⣿⡟⠿⣶⣽⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢠⠾⠋⠀⢸⣿⣰⠟⣉⣶⣹⣭⣾⣿⣿⣿⣟⣿⣾⡉⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣔⣂⣠⣾⢟⣹⢞⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣽⡋⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⣭⣷⣿⡞⠉⠽⠿⠟⠛⠿⣿⣿⣷⠟⠉⠉⠃⡀
⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⠎⠀⠀⠀⠠⠈⠀⠘⣿⠃⠀⠈⠀⡀
⠀⢸⠀⠀⠀⠀⠀⠀⠶⠖⣴⡶⠶⠒⠛⠙⠛⠶⢶⣶⡶⠿⠿⠟⠃⢴⣶⣦⡀⣀⣠⣙⣷⣿⡀⠀⣆⣘⣃⠘⠶⠞⠃⠀⠀⠀⠀⠙⠛⠛⠑⠒
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⠟⣱⣿⣶⣶⣶⣦⡤⠀⠐⠒⣮⣵⣾⠆⠨⣵⣾⣿⣿⢁⡀⠙⢿⣿⣿⣋⣠⡄
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠁⣤⣭⣍⡙⠳⠂⢀⡤⠿⣿⡿⠀⠀⠞⣿⢟⣫⡘⢿⣿⣦⡈⠉⠉⠁⠀⠀⠈⠁⠐⠶⠆
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠘⠛⠛⠃⠀⠁⠀⠀⠀⠐⠒⠋⠙⠃⠀⠙⠿⠿⠴⠆
⠀⢀
⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡙⣀⡖⠉⠁⠒⠀⠐⠀⠀⢤⣤⣤⣶⣶⠶⠟⣛⠻⢛⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⣩⣌⡑⢦⠀⠈⡇⠤⣍⠒⠲⠶⠶⠿⠿⠛⠛⠛⠉⣁⠠⠴⠎⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⠞⠀⠀⠹⣆⢳⣰⠧⣶⣬⣉⠉⠀⠒⠒⠒⠒⠒⠈⠉⣀⣠⣶⠏⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⡼⠀⠀⠀⣠⢿⢼⡏⠀⠉⠛⠛⠿⠍⠉⠀⠀⠀⠀⠀⠀⠈⠙⠛⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⢧⠀⠀⠀⠻⢄⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠜⠟⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡇⠸⡀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⡀⠐⠠⠠⡄⠒⠀⠀⠘⠄⠄⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥⠀⢧⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⡈⠿⠆⠀⠀⠀⠀⠘⠐⠠⠀⠀⠀⡀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠘⣆⠀⠀⡇⠀⠀⠀⠀⠀⠀⢘⡻⠆⠀⢠⠀⠀⢀⠀⢰⣆⠀⠀⣗⣶⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠹⣆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡛⠃⠀⢨⠀⠀⠉⠡⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠸⣧⡇⠀⠀⠀⣬⠄⠀⡀⠀⢀⡀⢀⠀⡀⢴⣤⣀⠜⢁⡟⠰⠆⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡌⣷⠀⠀⠀⠤⠀⠠⠡⠄⠀⠀⠀⠀⠀⠜⠁⠠⠦⠄⢁⢠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠻⡆⠀⠀⠀⠀⠀⢰⠖⠛⠀⢀⠀⣠⠀⠀⡐⠋⠀⠀⠌⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠼⡄⠀⠀⠀⠀⠈⢀⠰⠆⠀⢰⠦⡄⠀⠁⠠⡤⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢶⠄⠀⠈⠛⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣜⡉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣁⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠚⠽⠥⠆⠴⠰⠆⠦⠯⠖⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠜⢁⠀⡼⠁⣆⠀⠀⠀⠀⠉⠠⠠⢠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠃⠁⠀⠈⠀⠀⠀⠐⠁⠘⠉⠈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣿⣿⡿⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣤⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣼⢄⣀⠀⢿⠿⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠋⠙⠉⠈⠈⠀⠀⠆
⠀⠀⠈⠀⠀⠀⠀⢠⣴⡆⠀⠀⠀⠀⠀⠀⠆⠸
⠀⠀⠀⠀⠀⠀⢰⠇⠉⠀⠓
⠀⠀⠠⣾⣤⣀⢸⠀⠰⠀⠰⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠉⢻⠛⠛⣿⢀⠀⠀⢀⡀⠀⠀⠐⡄⠸⡀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⢀⡤
⠀⣶⡆⠈⡀⠀⠈⠸⡀⠀⢄⠀⢄⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣰⡏⠙⠀⣇⠀⠀⠀⠙⡆⠈⠣⠈⠂⠠⡀⠑⠀⠀⠀⠀⠑⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⣀⣤⡈⠹⢧⠁⠄
⠋⠐⠀⠀⠸⠑⠀⢆⠀⠉⠀⠀⠀⠀⣦⠁⠀⠀⠀⠀⢂⠀⢀⠈⢁⡤⣄⣀⣀⠔⠊⠀⢀⡼⢹⢦⣸⡟⠏⠁⠧⠀⠣⠋⠀⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔
⠷⣦⣦⠀⠰⡀⠀⠀⣦⡀⡄⠀⠀⣀⠻⠁⠀⠀⠈⠒⢀⣹⣤⣄⠳⠋⠉⠁⠀⠀⢀⣤⡼⠗⠈⠀⢋⠁⢼⡄⠀⠙⠢⣀⣀⣹⡿⠶⠖⠂⠀⣀⣀⠀⠈⠁⡜
⠀⠀⠙⠳⣆⣴⠅⡿⠟⠿⠚⠉⢻⠯⠻⠦⠴⠶⠦⢾⠿⠋⠀⠈⠀⠀⣀⣶⠻⠃⠀⠇⡈⠈⡙⠆⠉⣠⡐⠗⠋⢈⣩⣄⠴⠠⡄⠐⠶⢛⢺⣉⠃⠁⠀⣼⣠
⠀⣀⡀⠀⠀⣀⠀⠀⠀⢠⣤⡀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠓⠘⠒⠛⠚⠛⠚⠛⣉⡀⢺⢡⡋⣝⣀⠠⠖⠉⠐⠓⠒⠛⠉⠙⠢⠤⠛⢉
⣴⡿⢁⠀⠀⢻⣿⣡⣆⠀⠛⣿⡤⢀⢿⣆⠀⠀⡄⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⠀⠀⠘⠊⢻⡿⠚⢤⠀⣤⣿⠿⠉⣀⠀⢀⣠⡀⠀⣤⣶⣄⠀⠀⠀⢈⡈
⠈⠀⠀⠌⠀⠈⠁⠉⡟⢢⣈⠉⢁⠀⠞⢡⠄⠎⠥⢆⡀⠀⢀⣀⠠⡘⠋⡰⠀⠚⢱⠊⣰⠃⡾⠀⠀⡀⠺⠏⠁⠀⣸⡋⠳⠞⢿⡃⠀⡳⡙⢿⡀⠀⠶⠙⢉
⠀⢀⠀⠀⠀⣠⠆⠜⠀⠠⠃⠀⢀⠄⢠⠁⠀⡀⡤⠊⠁⡠⠈⢀⡄⠀⢘⡡⢀⡴⠏⠚⠋⢐⡠⠤⠃⠁⢀⣤⢀⠀⠏⢃⠀⡜⡄⠳⡄⡱⡜⠈⠳⠙⢄⡀
⠀⠀⢀⡀⠘⢁⡀⣀⡖⢀⣀⣀⣀⣰⣎⣰⢋⡔⢁⣠⢖⡃⠀⢋⡰⡚⠋⠀⠈⠀⢀⣀⡔⢉⣴⠶⣟⠰⡙⢻⣈⠉⠂⠀⢣⡀⠈⠀⠀⠀⠈⣄⣀⢀⠀⣉
⣿⡟⣛⡿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣭⡛⢶⣄⣠⠾⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⢿⣿⣿⠶⠋⠁⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠁⠀⠀⡾⠃⣠⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⡀⠀⠀⢷⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣸⣿⡿⠛⠋⠀⣠⣏⣀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⣱⣾⣯⣭⣭⣤⣴⠟⠉⠀⣠⣤⣶⣿⣯⣭⣭⣭⣭⣛⣶⣿⣿⣌⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠿⢿⣿⣿⡿⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⠁⢠⡞⢹⣿⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠰⡇⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠙⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣿⣿⣿⣿⣿⡷⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣀⠘⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠙⢲⡄⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⡙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠙⣇⠀⢷⡀⠀⠀⠀⠀⢀⣤⠞⣻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡄⠈⢓⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠸⡆⠀⣷⠀⣀⣤⣾⡛⢁⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⢿⣿⣍⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⡧⠀⢸⠀⠈⠉⢉⣽⣿⡿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠻⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⡷⠀⢸⠂⢀⣴⣾⣿⠟⠀⠘⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡙⢦⣀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣿⡇⠀⠀⠀⣰⠃⠀⣼⠖⣻⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣌⠳⢤⣠⠞⠋⠉⡉⠉⠙⠛⠿⠷⢤⠴⠞⠁⢀⣼⣿⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣉⠓⠾⣏⣻⣿⣷⣶⣤⣤⣠⣤⣤⣶⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢿⣿⣿⣷⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠲⠛⠁⣠⡼⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⠦⠶⠛⠉⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡀⠀⠀⢀⣀⣠⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣷⣤⣤⣠⣿⢿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣦⠤⠄⢤⣴⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠉⠁⠀⠀⠀⠀⣻⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢠⢴⡤⠶⠖⠒⠙⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⠁⠀⠀⠀⠀⢠⣾⣿⣿⡇⠀⠀⠀⠀⠞⠚⠂⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢦⣔⣦⣀⣼⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⢿⣿⣷⢶⣖⣶⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⠈⠁⠀⠀⠁⠩⠏⠻⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⡕⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣼⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢠⣿⣿⣟⣿⣛⢗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢙⣾⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣐⠂⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣿⣿⡇⠀⠀⠠
⠀⠀⠀⠀⢸⣿⣿⣿⣺⠷⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣽⣿⣿⣿⣿⠇⠀⠀⠀⠀⢀⠀⠄⠐
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣭⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣾⣿⣿⣿⡿⠀⠀⠀⠀⠐⠁⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣎⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⡿⠁⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡶⣦⡠⡀⠀⢀⢄⢀⣄⢌⣴⣿⣿⣿⡿⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣾⣷⣦⣄⣩⡄⣢⣿⣽⣶⣷⣿⣿⠟⠀⠀⠈⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⣤⣤⡀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡟⠀⠀⢹⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠈⠲⠖⠋⢠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡶⢶⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⣀⡀⣀⣸⡇⢸⣇⣀⢀⣀⣴⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⣀⣀⣠⣭⣭⣍⣹⡇⢸⡏⣉⣭⣭⣅⣀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⢹⡇⢸⡏⠁⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⣿
⠀⠀⠀⠀⠀⠀⢀⠴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡄
⠀⠀⠀⠀⠀⢀⡎⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠸⡄
⠀⠀⠀⠀⠀⣸⠁⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢷
⠀⠀⠀⠀⢀⡟⠀⠀⢄⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⣠⠀⠀⢸⡆
⠀⠀⠀⠀⢸⡇⢀⡄⠘⣧⡴⠾⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠶⣴⠏⠀⠀⠘⡇
⠀⠀⠀⠀⢸⣧⠞⣧⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⣸⠓⢦⡇
⠀⠀⠀⠀⠈⠀⠀⠘⣧⡀⠀⠈⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⣠⡴⠟⠉⠀⠀⣰⠏⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠈⠙⠛⠷⠶⢶⣤⣤⠁⠀⠀⠀⣧⣤⡴⠶⠾⠛⠋⠁⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⡀⠀⠀⠀⡞⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣭⣭⣭⣭⣥⣬⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣶⣷⣶⣾⣿⣿⣶⣶⣶⣶⣾⣶⣾

⠀⠀⠀⠀⠀⠀⠀⢺⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⡿
⠀⠀⠀⠀⠀⣀⣀⣠⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣥⣀⣀⡀
⠀⠀⣠⡶⠛⠉⠉⠀⠀⠙⢷⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠁⠀⠉⠉⠙⠳⣦⡀
⢀⡾⠋⠀⠀⣠⣤⡤⠤⣤⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⠤⣤⣤⡀⠀⠈⢻⣆
⣼⠃⠀⢀⣾⠋⠀⠀⠀⠀⢨⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⠀⠀⠈⢻⡆⠀⠀⢻
⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⣿⠀⠀⢸
⣿⠀⠀⠸⣇⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⢠⡿⠀⠀⢸
⠸⣧⠀⠀⠙⠷⠦⣤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⠶⠟⠁⠀⢠⡟
⠀⠘⢷⣄⡀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⣀⣴⠟
⠀⠀⠀⠉⣽⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⡁
⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣼⠟⠛⠛⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠚⠛⠛⠻⣧
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⣀⣀⣀⣀⣀⣀⣀⣙⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣋⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⠀⠉⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠳⠒⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⢸⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠂⠤⠤⢀⡠⠀⠜⢀⡈⠀⠀⣕⠈⣄⠀⠀⠘⠘⡌⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣴⣥⣶⡄⣼⣧⠀⠀⢸⠀⢸⡄⠀⠀⠀⢻⡀⠑⠝⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢀⣧⠈⠉⠁⠉⠉⠀⠀⠳⣄⢸⡇⠀⠀⠀⠈⢷⠀⠸⣦⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠉⠛⠛⠛⡄⠀⠀⠀⠀⠀⠀⠈⣿⣿⡣⠠⣦⡤⠀⢈⣇⠀⠈⢷⡀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠈⠉⠉⠂⡄⠀⠈⣷⡀⠀⠀⠹⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⣼⠇⠀⠀⠀⠀⢠⠀⢀⣴⣷⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠈⡓⣲⣶⣖⠀⠭⠵⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠂⢀⣼⣿⠀⠀⢀⣂⣀⠈⣒⣿⣯⡇⢠⠃⡄⠀⠀⠀⠀⢘⡏⢉⠞⠑⠒⠒⠛⢷⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠐⠈⠉⢹⡷⠚⠋⣿⠀⡇⠀⠀⠀⠹⠁⢒⣡⣴⡖⣶⠊⠁⣹⣣⢃⣀⣠⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣟⠻⠿⠶⡾⣷⢶⡶⣅⣀⣧⣀⣠⣤⣧⣬⣤⠿⠿⠷⢾⣷⣶⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠤⠀⠂⠐⠐⠂⠂⠐⠀⠂⠂⠀⠀⠒⠊⠉⢁⣀⣬⣽⣶⣴⣶⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠁⠘⠛⠉⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠲⠦⠤⠤⢤⣤⣤⣄⣀⣀⣀⣀⣀⢀⣀⡀⢀
⠛⠛⠛⠛⠋⠉⢉⣉⣉⣉⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⡀⣀⢀⣀⣀⡀⠀⣀⣀⣀⣀⣀⣀⣀⢀⡀⡀⢀⣀⣀⣀⣀⣈⡉⠉⠉⠉⠉
⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣈⣷⠶⠿⠟⢳⣤⣀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⠶⣦⠀⠀⣼⣿⣉⡆⠀⠀⣿⣥⣿⡆⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣄⠻⣿⣿⣿⣄⣤⣿⣿⣿⣡⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠛⠉⠉⠛⠻⢿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣴⣿⣟⠀⠀⠀⠀⠀⢠⣼⣿⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⢉⡽⠛⠛⠛⣿⣷⣶⣤⣄⣦⢾⢿⠿⠻⠻⣟⠛⠃⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠋⢀⡴⠏⠀⣠⣾⣿⠟⠁⠀⠀⠈⣀⣘⣿⣿⣦⡀⠙⢧⠀⠈⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⣼⡇⣿⣿⣿⣶⣷⣿⣿⣿⣿⢿⠛⣷⠀⠀⠳⣄⠀⠈⠷
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠀⠀⠀⢀⣿⡗⣧⣀⣉⣉⣉⣉⣉⣩⣤⣾⡄⢿⣆⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢠⣾⠏⠀⣏⠻⠿⠿⠿⠿⠿⠿⠿⣿⠃⠈⢿⣦⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⢹⡷⣶⣶⣶⣶⣾⣿⠿⡿⠀⢰⡀⠉⠳⣄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⢻⣷⣶⣴⣶⣶⣾⣿⠇⠀⠈⠂⠀⠀⠈⠿⠂⠀⠀⣂
⠀⠀⠀⠀⠀⠻⠤⠤⠖⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⣥⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⣄⡀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠾⠛⠛⠿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡏⠀⡀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⠛⠛⠁⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⡼⠁
⠀⠀⢀⠴⠒⠛⠛⠒⠲⠤⣄⡀⠀⠀⠀⠀⢀⡼⠟⠉⠉⠻⢧⡀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠛⠛⠒⠦⡀
⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠦⣄⣠⠏⠀⠀⠀⠀⠀⠀⠹⣄⣠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣤⣤⣤⣤⣤⣤⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠙⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣽⠟⠛⠛⠻⣯⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋
⠀⠀⠀⠀⠙⠒⠦⠤⠤⠤⣴⡶⠶⠛⠋⠉⢠⣿⣶⣶⣶⣶⣿⡄⠉⠙⠛⠶⢶⣤⠤⠤⠤⠴⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⣠⠋⠉⠉⠉⠉⠉⠉⠛⣄⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢀⣴⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣦⡀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠖⠋⠈⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⠃⠙⠲⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⣤⡄⠀⣨⢁⠀⢀⣴⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠁⠀⠀⢦⣤⣺⡿⢧⣾⣇⠈⢿⣿⣿⡆⣿⣤⡄⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡀⡀⢀⣼⣿⠟⠻⢎⢻⡏⠀⢸⣿⣿⣿⣹⣿⣋⣿⣦⠔⡅⠀⠁⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠘⠋⠀⠘⠻⠿⠉⠀⠀⠌⢇⠀⠀⣯⠉⡿⠛⡿⢿⣿⣿⣿⣤⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠘⠛⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠉⠀⠁⠀⠁⠸⣿⣿⣿⡿⠁⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⡈⣸⢀⡿⣸⣿⣷⣤⡤⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⣠⣤⣆⣴⣿⡿⠛⠋⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⣸⣥⣿⣿⣿⣿⣇⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢟⠉⠉⠉⢛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⢀⠏⠀⣸⣿⣿⣿⣿⣿⣿⣿⡾⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⣄⣀⣄⣀⣠⡟⠀⢀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠏⠀⠀⠀⠀⠀⠀⠀⠻⠋⢷⣿⣿⣷⣤⡿⠁
⠀⠀⠀⠀⠀⢀⣠⡤⠶⠓⠀⠛⠉⠉⠉⣭⠉⠈⡙⠿⠛⠛⠳⠶⠆⠀⠤⢶⠶⠀⢀⣠⣭⡿⠃⠀⡀⠀⠀⣠⣦⣀⣴⠋⣀⡼⠋⠹⣿⡟
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠩⠤⠤⠥⠤⠀⠀⠙⠻⠤⣀⣀⠀⠸⢿⠇⠀⠀⢀⡻⠋⢁⣤⠊⠀⠘⠋⠘⣫⡯⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⢀⣶⠶⠒⠚⠛⠛⠛⠒⠀⠤⣀⣀⡀⠀⠀⠀⠀⠶⠚⠁⠤⠀⠀⠉⠰⠶⠛⢋⣡⢾⣫⣴⣮⣤⣀⣬⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠑⠂⠀⠀⠀⠀⠤⠈⠉⠉⠁⠀⠀⠀⠴⠚⠛⠉⣿⣿⣿⡿⠋⠙⠛⠿⠿⡿⠿⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠞⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⣀⣼⠃⡇⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡋⠙⠷⠠⠀⠀⠐⠤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⣀⡀⢀⢸⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠤⢬⣿⡀⠀⡧⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣸⠂⠈⠉⠛⢶⠿⢤⣑⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⡄⠀⠀⢀⡤⠀⠀⢀⢿⡏⠀⠀⠀⠀⠈⣇⠀⠈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣁⣘⡂⠘⠀⠦⣽⠇⢀⣴⣾⣿⣁⣀⣀⢀⡀⠀⢹⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⠉⠙⠛⠻⠿⣿⣿⣿⣾⣿⣿⡏⠉⠙⠛⠿⠤⣀⣿⠚⡇⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠂⠀⠀⠀⠀⠀⠀⠈⠈⠉⠛⢻⠃⠀⠀⠀⠀⠀⠀⠈⠙⡟⠲⠦⢄⣱⡀⠀⠀⠀⠀⠀⠀⣠⡶
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣼⠏⡐⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠇⠀⠀⠄⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣣⠠⢂⢀⣄⣠⣀⣤⢄⣴⣿⠋⠀⣥⢀⡄⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⢀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣉⢐⣱⠷⣿⢿⣿⣿⣿⣿⣿⣿⣡⣤⣤⣤⣤⣀⡀⠀⠀⠀⣠⣿⠇⣀⡽⠛⠦⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⡛⠛⠿⣿⣿⣿⣧⣽⣿⠟⠽⢻⡿⣿⣿⣿⣿⣯⣅⣀⣈⣷⣿⣿⣴⣯⣤⣀⣴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⡿⠤⣌⡙⠛⠛⠀⠀⢹⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣀⣴⣞⣛⢿⣿⣷⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⠏⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⣀⡜⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⢸⡛⠹⢿⣾⣿⣿⣿⣿⣿⣿⢋⣥⠶⠦⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠰⣻⣭⣴⣶⣷⣾⣿⣿⣿⣿⣿⣿⣩⣾⣿⠄⠀⠀⠛⠿⡻⡛⠛⢛⣇⠘⠛⠀⠀⢀⣀⣤⡾⠁
⠀⠀⠀⠀⠀⠈⠳⢦⣬⣿⡿⠛⠟⢛⣻⡿⠿⠿⢿⡿⢭⣍⣛⠧⣀⠀⠀⢀⣐⣛⢦⣘⠿⣿⣦⣴⣦⣤⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠁⠀⠈⠉⠴⣶⣿⣿⣿⣷⣦⡌⠛⢷⣬⡀⠐⠷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢶⡶⠶⢟⣉⣙⣿⣿⣿⣿⣿⡧⢀⣌⠻⣄⡀⠢⢭⣯⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠋⠙⠿⣿⣿⣟⠹⠇⢀⣿⣦⣀⣙⠛⠲⠶⠾⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠊⠹⠿⠷⣤⡐⠈⠙⠻⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⣨⣈⡑⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢀⣴⣶⣦⣀⣶⣤⣄⠀⠀⠠⣶⣶⣠⣤⣤⠀⠀⠀⣥⡾⢿⣿⣿⣿⠏⠂⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣌⣿⣿⢿⣿⣿⠟⡛⠀⣿⣄⠨⣙⠛⣋⠅⣠⣿⠐⢛⠻⣿⣿⣿⣿⣶⣶⣦⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⣏⣿⣿⣿⣿⣆⡛⣄⠙⠁⣶⣭⣿⣿⣷⠈⠋⣠⠛⣠⣿⣿⣿⣿⣿⣿⠟⠓⢸
⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⠛⠋⠁⠉⠉⠉⠛⠛⠉⠀⣋⣠⠼⡶⢙⣟⣿⣆⣸⠀⠉⠊⠉⠉⠉⠉⠉⠀⠀⠀⠀⠑⠠⢀
⠀⠀⠀⡠⠐⠨⣴⡆⡂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣽⡄⣤⣾⣿⣿⣯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⠙⡗⠂⠈⠐⡄
⠀⠀⢀⠁⠀⢀⣘⡷⣬⡐⢄⠀⣀⠀⡠⠁⠐⠒⠂⠀⡠⢿⣿⣿⣿⣿⣿⡿⢀⠀⠐⠒⠂⠈⢀⠀⣀⠀⡀⢒⣥⣺⣤⣤⣆⣷⢰
⠀⠀⢸⠀⠀⣾⣿⣯⣿⣿⣶⣶⣤⠀⠀⢠⡴⢛⠠⣦⣠⠴⠿⠙⠿⣿⣿⠧⣄⣴⠄⡛⢦⡄⣀⠀⣄⣶⣶⣿⣿⣿⣾⣿⡿⠻⠘
⠀⠀⢀⠀⠀⣽⣿⠻⣿⣿⣿⡿⢟⣀⠠⢞⡥⣿⡗⢨⠧⣴⣤⣤⣶⣾⣿⣠⠼⡅⢺⣻⣌⠳⠌⣀⣙⠉⢻⣛⣿⣿⣿⣅⠀⠀⡠
⠀⠀⠀⠑⠦⠈⠙⠿⠟⣋⣥⣶⡿⠟⠛⠢⢌⡛⠀⡆⢀⡟⠉⣠⣤⣬⣿⡛⠀⢸⠀⢋⠡⠀⠙⠻⢿⣷⣦⣌⠙⠛⠋⠁⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⠛⠉⠀⠀⠀⠀⠀⠀⢳⢇⡜⣿⣋⣉⣉⣻⣿⣻⢃⡸⡞⠀⠀⠀⠀⠀⠀⠈⠙⠛⠃⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣦⠋⢸⡇⢻⠛⠛⠛⢿⡟⡏⣼⡇⠙⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠚⠁⡇⢸⣗⡌⢷⣶⣿⣿⡟⣡⣿⡇⠂⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⢇⠘⠽⣶⣄⡙⢛⣫⣴⡟⠁⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⡈⠙⠻⠛⠛⠉⡀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⠀⡴⠊⠁⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠶⠶⠶⠶⠶⣤⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠐⢢⣞⣋⣉⠀⣀⣤⣶⣆⣀⣀⣀⣀⣭⣭⣽⣷⣶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⠝⠛⠛⠛⢿⣿⡟⠛⠛⢉⣭⣤⡬⢭⡿⠿⠿⠿⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣯⣿⣿⣿⣿⣶⣾⣶⣶⣦⣭⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⢻⢿⣿⡿⣿⣿⣿⣏⣉⣻⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡋⡋⡁⠁⠀⠀⠉⠁⠘⠉⠀⢻⣿⠛⣷⣝⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣷⣼⣢⣶⣄⡇⠀⠀⢰⡆⠀⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣦⣾⡇⣀⢀⣿⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣸⡏⢸⣿⣿⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⢯⣿⣿⣿⣾⠏⠀⣼⣿⡿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⢿⣿⣷⠟⢁⣼⣿⣿⣿⣽⡿⣿⣷⣿⠟⠁⣀⣴⣿⡿⢃⣿⠏
⠀⠀⠀⠀⢀⣠⣤⣤⣶⣾⣟⣻⣯⣇⣀⣩⣵⣎⣁⣛⣛⣭⣟⠛⠩⣾⠟⢛⣁⣤⣾⣿⠟⣋⣠⡼⠋
⠀⠀⠈⠙⠛⠛⠛⠿⠿⢿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠻⣿⣯⣿⣿⣿⣿⣿⡿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠴⠶⠖⠒⠒⠛⠛⠛⠛⠛⠒⠒⠲⠶⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡙⠛⠒⠒⠶⠶⠤⠤⠤⠤⠤⠤⠴⠶⠶⠶⠒⠒⠛⠋⠉⠉⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡛⣠⢤⡀⡴⢋⣷⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣀⡀⠠⠾⢷⡇⠀⣿⡥⠖⢶⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠪⠅⣸⣤⣀⢱⣴⣏⣠⣤⣞⠂⢘⡢⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡀⠀⠀⠀⠀⠉⠉⢀⡩⢷⠟⢤⣄⣈⡡⠈⠁⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢰⠃⠀⠀⢠⡦⣈⣽⠋⠀⣸⡄⠈⢃⠙⣦⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠠⠏⠀⠀⠀⠀⠉⠁⠘⠳⢞⣹⡟⠲⠋⠘⠛⠁⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠒⠒⠶⠶⠶⠶⠤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠒⠒⠒⠒⠛⠛⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠲⠶⠶⠦⠤⠤⠤⠤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠒⠒⠒⠛⠛⠋⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢚⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠶⠦⠤⠤⠤⣤⠤⠤⠤⠶⠶⠖⠚⠉


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣇⣠⢀⡀⣄⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢄⡶⢁⡼⢣⠟⡹⢏⢻⡜⢧⡈⢶⠠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⣇⣎⣎⢴⢏⠔⢋⡜⡅⢨⢣⡙⠢⡹⡦⣱⣱⣸⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣸⠛⣿⢻⡧⣾⠃⠀⣼⢰⡇⢸⡆⣧⠀⠘⣷⢼⡟⣿⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢦⢰⠃⣰⡷⣋⢴⣯⠀⠊⠉⣻⣇⣸⣏⠁⠑⠀⣽⡦⣙⢿⣆⠘⡆⡴⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⣾⢰⠏⡴⡟⢿⡇⢣⣛⠛⡱⢋⡙⢎⠛⣋⡜⢸⡿⢻⢦⠹⡆⣿⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢹⣮⣻⡹⡟⢸⢳⣷⡀⠙⢶⣿⢿⠁⠀⠀⠈⡿⣿⡶⠋⢀⣾⡞⡇⢻⠏⣟⣵⡏⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⠀⠈⢷⡇⣯⣹⠇⠻⣦⣤⣙⣿⡌⠀⠀⢡⣷⣋⣤⣴⠟⠸⣏⣽⢸⡾⠁⠀⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢓⣿⡇⣀⠸⣇⢳⢼⣦⠽⣷⣳⣄⠉⠻⣆⣰⠟⠉⣠⣞⣮⠯⣴⡷⡞⣸⠃⣀⢸⣿⡊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣮⠉⢻⣻⣭⢿⣜⣏⣷⡶⠋⠉⠉⠉⠙⠻⠟⠋⠋⠉⠉⠙⢶⣾⣹⣣⡿⣭⣝⡟⠉⣵⠂
⠀⠀⠀⠀⠀⠀⠀⢠⣀⡿⢢⡀⠹⡘⢳⡙⢯⢲⣧⣘⣒⠃⠠⣤⠆⠰⣤⠄⠘⣒⣃⣼⡖⡽⢋⡞⢃⠏⢀⡔⢿⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⠄⢀⣀⠇⡀⢱⡈⢿⡏⠑⠶⢭⣭⣿⡀⢀⣿⣭⣭⠶⠊⢹⡾⢁⡎⢀⠸⣈⡀⠠⡀⢿
⠀⠀⠀⠀⠀⠀⠈⢻⠃⠈⠀⠀⠉⢻⣧⠀⣷⣄⠙⠶⠞⠩⢀⠉⠉⠉⠉⡀⠍⠳⡶⠋⣠⣾⠀⣼⡟⠉⠀⠀⠁⠘⡟
⠀⠀⠀⠀⠀⠀⠐⣾⠀⠀⠀⠀⠀⠈⢯⢦⣹⡀⡹⠡⢠⠊⠁⠀⠀⠀⠀⠈⠑⡄⠌⢏⢀⣏⡔⡽⠀⠀⠀⠀⠀⠀⣷⠂
⠀⠀⠀⠀⠀⠀⠠⢼⡆⢃⠀⠀⠀⠀⠈⢷⡬⣙⠃⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠘⣋⢥⡾⠁⠀⠀⠀⠀⡘⢰⡧⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢂⠀⠀⠀⠀⠀⢹⢺⡀⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢃⢀⡗⡏⠀⠀⠀⠀⠀⡐⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠈⣳⡀⠢⡀⠀⠀⠀⣘⣽⣾⢋⣳⡀⠀⠀⠀⠀⠀⠀⢀⣎⠙⣷⣯⣃⠀⠀⠀⢀⠔⢀⣞⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣲⣈⢲⣔⡿⣾⠡⣮⡷⣟⠬⣶⣤⣤⣤⣤⣶⠥⣻⢾⣕⠌⣶⢷⣢⡖⣁⣖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠻⣰⣟⡿⢻⣄⢪⡞⠣⢟⣇⣸⡻⠜⢳⡕⣤⡟⢿⣻⣆⠟⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⣠⠋⣴⡿⣝⡁⢹⠋⠙⡏⢈⣫⢿⣦⠙⣌⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣼⢰⣍⠈⠻⣄⣠⠟⠁⣩⡇⣧⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠈⠙⠚⠈⠁⠛⠉⠁⠱

⠀⠀⠀⠀⠀⠐⠀⡴⣤⣴⡄⢘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣸⣿⣗⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⠳⠆⠶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢾⠋⢻⡆⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠈⢻⠛⠓⠒⠶⠾⣧⠤⠤⠤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠁⠉⠛⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠐⡐⠒⣶⣾⠃
⠀⠀⠀⠀⠀⠀⠐⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡃⣠⣤⣶⣶⣶⣾⣿⣖⣆⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠺⠟⠿⠿⠯⠭⠼⢿⣟⣿⣿⣻⣇⣀⣀⣀⠀⠀⠄⠀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠰⠞⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⢓⡿
⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠰⠦⠒⠒⠒⢾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡠⠴⠐⣿⡏⠉⣉⣁⣂⣿⣧⣤⡤⡀⠤⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠶⠷⠒⠛⠋⢻⣿⠉⠉⢉⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣦⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡻⣿⣿⣿⣿⣯⣾⣿⢻⣷⡄⠙⢤⣶⢖⣖⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠇⣛⣭⣭⣿⣶⣾⣿⠷⣄⣰⣿⣮⣷⠿⠋⠉⠁⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⢈⣿⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣿⣿⠛⠻⣿⣯⡉⣻⡿⣿⣿⡟⠀⣼⣿⡇⣿⢀
⠀⠀⠀⠀⠀⠀⠀⣀⡿⣿⣿⣿⣿⣧⣜⡀⠚⠻⣃⡁⠀⠈⠻⢧⡶⣿⠿⢶⣿⣆⢷⣄
⠀⠀⠀⠀⠀⢀⣤⣿⡄⢈⡁⠉⠿⠛⠛⠛⠦⡀⠉⠑⠙⣀⡀⠀⠀⣉⠀⠀⠻⣿⣷⣭⡛⠒⠀⣀⣠⣤⣀⣀
⠀⠀⠀⠠⣴⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣀⡐⠦⠉⠉⠻⣶⣄⡉⠛⠛⠛⠋⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣷⣶⣤⣤⣬⣭⢤⣀⠐⠳⢿⣿⣷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈
⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣧⣒⣯⣭⣭⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢻⡄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠩⠁⠀⢀⠀⠀⠰⠦⠔⠂
⠀⠀⠀⠀⠘⠆⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠉⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠈⠀⠈⡄
⠀⠀⠀⠀⠀⠀⣧⢸⠂⡆⢸⠇⡇⣞⠱⣿⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡄⠀⠀⠀⠀⣤⡤⣤⠤⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⢠⢐⣒⡂⣀⠀⠀⢀⠀⣀⡞
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠁⠠⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⣰⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠃⢄
⠀⠀⠀⠀⠀⠀⠴⠓⠒⠔⢲⢄⠀⠀⠀⠀⠀⠈⠉⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣳
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⣠⠞⣿⢃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠆⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢸⡧⣾⢁⡤⢰⡆⣶⢠⡞⢂⣴⠂⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠊⠀⠀⠀⠀⠀⣼⠀⢠⢾⡇⠛⠈⠂⠈⠉⠃⠈⠃⠀⠉⠀⢺⠗⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠉⠘⠁⠀⠀⠄⠀⠀⠀⠀⠔⠁⠀⠀⣿⠀⠠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢈⢩⠗⠀⠀⢸⠀⠀⢀⣼⠃
⠀⠀⠀⠀⡞⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠋⠁
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢢⠀⠈⠂⠀⠀⠀⠀⠀⠰⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠀⠈⠑⠄⠈⠂⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡩⠃
⠀⠀⠀⠀⠀⠀⡤⠤⠄⠠⠤⡔⠀⠀⠀⠀⠀⠀⠐⢤⠀⠂⢤⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⢲
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⡀⠠⢀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡇⢀⣜⠀⡠⠄⣴
⠀⠀⠀⠀⠧⠄⢀⡀⠀⡀⠀⠀⠤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⢷⠀⠴⠃⠛⠀⠑
⠀⠀⠀⠀⠆⠐⠀⢀⠀⠀⠀⠰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢦⡀⠀⠀⠀⠀⠀⠀⠻⣀⡇⠀⠀⠀⣀⢠⣠⡆⣾⡈⣇
⠀⠀⠀⠀⠀⡀⠀⠈⠐⠀⠀⠀⠀⠃⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣸⡆⠇⠘⠘⠏⠃⠀⠁⢹⠈⢦
⠀⠀⠀⠀⡄⠀⣀⡀⠀⡀⡀⠂⣁⡄⠀⠀⠀⠰⡛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⠶⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠾⠛⠛⠛⠷⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⣀⣀⣀⡀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢡⠂⢠⡞⠉⠉⠉⠛⣦⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⢸⠀⣿⡀⢀⣀⣀⡀⣸⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡘⣤⡾⢿⣋⣿⣉⣿⠟⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠋⣷⡾⢟⠉⠉⠀⠀⣀⣼⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣁⣿⣿⠀⠧⠀⠀⠀⣹⣿⣯⠽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢇⣼⣃⣿⣀⣋⣠⣀⣀⣹⣇⣹⣿⢿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣾⡟⢤⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣧⠀⢹⡶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡟⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⣷⣿⣇⢺⣇⠀⢀⣀⣀⣀⣤⣄⣀⠀⠀⣿⠀⢸⡇⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣍⠽⣧⣼⣿⣿⣿⣿⣿⣭⣤⣤⡟⠀⣼⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡞⣻⣿⡟⠋⠙⠟⢹⠅⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣴⠛⣿⣦⣄⣺⡅
⠀⠀⠀⢀⡾⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⡿⣷⣴⣋⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠳⣆
⠀⠀⢀⣾⠃⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡅⢈⠉⠻⣶⡊⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢹⣆
⠀⠀⣼⠃⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠂⠀⠀⠀⢸⣿⡎⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⡀⡇⠀⢻⡄
⠀⣸⣿⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⠙⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣿⠀⠈⣷
⢠⣿⠋⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠇⢀⠀⠀⢺⣧⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠀⠀⢿⠀⠀⠸⣇
⣼⠃⠀⠀⠀⠀⠀⡀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⡄⠀⠀⠀⢸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡀⠀⠀⠀⠀⠀⢻
⣿⡀⠀⠀⠀⠀⢠⠇⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣿⠀⠄⠰⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢃⠀⣨⠀⠀⠀⣸
⠈⢷⣆⠀⠀⡄⢸⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢉⣷⣿⢀⣄⠘⠀⠀⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣾⠀⡇⠀⢠⣶⠋
⠀⠀⣿⡀⠀⢳⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠈⣿⠃⣿⢀⡄⢀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢰⠁⠀⣸⠇
⠀⠀⠸⣧⠀⠈⣆⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⢻⡇⢠⡟⠀⠀⠠⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⠏⠠⢠⡟
⠀⠀⠀⠹⣧⠀⠘⢆⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣹⣷⣼⣃⣘⣆⣀⣀⣀⣠⣿⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⢀⣰⠋⢀⠎⠀⢠⡟⠁
⠀⠀⠀⠀⠘⢷⡀⠈⠣⡀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠛⢷⣦⣋⣉⣟⣉⡟⣉⣟⣉⣼⣭⣠⣿⣤⣟⣉⣹⠿⣦⠀⢀⣴⠟⠁⡰⠋⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠛⣦⡀⠈⠢⣄⠉⠛⠶⣤⣄⣀⣀⣀⣤⠾⢋⣽⡿⠋⢛⡿⣋⠉⠉⠙⠉⠻⢦⣄⣉⣉⣻⣦⡾⠟⠋⢀⡤⠊⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⠉⠒⣤⢀⡉⠉⠉⠉⠀⠐⠋⠻⠛⢈⡼⠁⠸⣆⡑⠀⠀⠀⠀⠈⠉⠉⢉⣀⣠⠔⠊⢁⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣌⣉⠒⠒⠒⠒⠶⡶⠶⢂⣁⠀⠀⠀⢈⣉⠲⠶⠶⠖⠂⠚⠒⢈⣁⣤⣴⣾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢦⣄⡀⠀⠀⠲⣬⠳⠄⡴⢋⡴⠂⠀⠀⣠⣤⠶⠟⠋⢩⣿⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⠈⠁⣾⠁⠈⢀⣤⠶⠛⠉⠀⠀⠀⢀⡾⣇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣆⣘⣠⡾⠋⠁⠀⠀⠀⠀⠀⢠⡿⣇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠀⠀⠀⢠⡿⣀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣅⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣿
⣿⣿⣿⡿⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣄⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣯⡿⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢰⠟⠻⠃⠈⢁⡴⠆⣀⣆⠀⠀⠀⠴⠄⠀⢠⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢻⠄⠀⠀⠤⣀⡐⠂⠙⠋⠶⠄⠀⠀⠀⠀⠈⠀⠋⠀⠀⠠⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠴⣒⣢⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⠀⠀⢸⠁⠀⠐⠒⠒⠒⠀⠠⠄⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠒⠒⠉⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠈⣉⠉⠙⠃⠀⠀⠉⠀⠀⠀⠂⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠉⠉⠑⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⠀⠠⠤⠀⣠⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠈⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢾⠆⠀⠀⠀⠀⠀⠀⠙⢦⠀⠈⠀⡄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣁⡀⠉⠛⠋⠀⠐⠀⠀⢀⠀⢀⣀⣀⣀⡀⠀⣁⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠋⠀⢉⠉⠁⠀⠀⣠⠚⠋⠤⠉⣿⣿⣿⣿⣿⣿⣿⣇⣶⣄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠘⠲⠈⠀⠀⠀⠌⠀⠀⠀⢠⣾⣿⡿⠟⠉⠀⠀⢨⣿⣷⣏⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⢀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⢀⠐⠆⠀⠘⣿⡿⢿⣶⡀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠀⠁⠀⠀⢈⠀⠤⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⣄⠀⠈⠂⠰⠀⠀⠹⣇⠀⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠛⠠⣀⠀⠘⢃⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠰⠀⠀⠀⠀⠈⠃⠀⠀⠄⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄⠁⠀⢀⠀⠿⠀⢤⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⡄⠠⠤⠤⠠⡴⠤⢠⡔⠤⠄⢠⠆⠀⠀⠀⠒⠒⠒⠒⠚⠁⠉⠐⠋⠉⠉⠈⠉⠑⠈⠐⠛⠛⠛⠓⠋⠙⠓⠛⠚⠛⠛⠚⠛⠛⠛⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠉⠈⠁⢀⢀⠈⡃⠘⠀⡀⠀⠈⠁⠋⠀⠀⠈⠁⠉⠀⠀⠉⠉⠁⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⢸⠀⡇⢈⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⡀⣀⣀⣀⣄⣀⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⡄⠘⠘⠀⡄⢸⠀⠓⠘⠀⠀⢠⠀⠦⠂⠀⠀⡄⠘⠃⠀⠀⠀⠠⠄⠀⠃
⠀⣁⣀⣀⣀⣠⣄⣀⣀⣀⣀⣤⡀⠀⢀⣀⠀⠈⣿⠆⠉⠉⠀⠐⠋⢘⠛⠺⠶⠚⠛⠚⠳⠶⢶⣾⠷⢿⡶⢶⠶⢿⠷⠿⠶⠷⢾⣿⡷⠶⠿⠾⠷⡶
⠀⠋⠉⠉⠀⠀⠀⠀⠉⠀⢤⣤⠀⠀⠀⠀⠀⠀⢾⡤⠤⠶⠶⣤⣤⣼⣶⣶⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⢶⣶⣶⣶⣦⣤⣦⣦⣴⣶⡶
⠀⠆⠀⡀⠀⠀⠀⠀⠀⠀⡄⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⠟⠻⣾⡧⣥⣭⠴⠧⣤⠀⠀⠀⠀⠀⠀⠀⣤⠀⠰⢦⡁⠉⣤⡼⠶⡄⠀⠀⠠⠄
⠀⢀⠈⠁⠀⠀⠀⠐⠒⢀⣉⡀⠴⠁⠄⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡈⠁⢠⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢸⡇
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⠀⡀
⠀⠀⠐⠀⠀⠀⠀⡐⢀⣾⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣰⣿⣿⣆⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣧⠀⠄
⠀⠀⠀⠀⠀⠀⣤⢠⣿⣿⣿⣿⣿⣿⣿⣦⡈⠓⢀⠀⠀⠀⠀⠌⢀⣾⣿⣿⣿⣿⣷⡄⠉⢂⡀⠀⠀⣠⠔⢁⣠⣼⣿⣿⣿⣿⣿⣇⠘
⠀⠀⠀⠀⢠⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⢀⣴⣠⣦⡁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣦⢀⣴⣤⣬⣡⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣄⣼
⠀⠀⠀⠀⠀⠀⣄⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⠀⣄⡀⣀⣀⣀⣀⣄⣀⣄⣀⡀⢠⡄⠀⡀⣀⢀⣀⣀⣀⡀⢠⣀⡀⢀⢀⡀⢀⡄
⠀⠀⠀⠀⠀⠀⠘⢿⣇⠈⠁⠀⠀⠈⠉⠉⠉⠀⠉⢁⢀⣄⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣖⡀⠀⠀⠀⠀⠀⢀⢸⣿⣿⡆⠠⠀⠀⠀⠀⠀⠀⢀⠀⣸⣿⡇⠀⠀⠀⠀⠀⢀⣤⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣮⣄⠀⠀⠀⠀⠈⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⡌⣰⣿⣿⡇⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣶⣤⡀⠂⢹⣿⣿⣿⣧⡀⠀⠀⢀⢂⣼⣿⣿⡿⣀⢀⣼⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⣶⣿⣿⣿⣿⣿⣦⡀⣠⣾⣿⣿⣿⣷⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⠿⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠘⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡀⠀⠀⠀⠀⡇⠀⠀⣀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡜⣷⢄⡀⠀⣿⠀⠀⢸⣷⡦⢀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢷⠸⡆⠙⢢⣿⠀⠀⢘⡟⡈⠂⠱⣤⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⠀⢚⣿⠃⠀⢸⠅⠀⠀⠀⠉⣻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠚⣿⡄⠀⢨⠀⠀⠀⠀⠐⢉⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣿⢹⠀⡆⠀⠀⠀⠀⠀⠀⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡯⡶⣴⠁⠀⠀⠀⠀⠀⠀⣿⢎⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⢸⣿⣇⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⢁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠀⠀⠀⠀⠃⣸⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠸⡀⠀⠀⠀⠀⠀⠂⠰⠒⠎⠢⠒⠘
⠀⠀⠀⠀⠀⠀⠀⣈⡥⠀⠀⠀⠀⢀⣈⣺⣿⣇⡤⠄⠂⡀⠀⠀⠀⠀⢺⣿⡀⡇⠀⠀⠀⠀⠀⣫⡤
⠀⠀⠀⠀⠀⠀⣰⡿⠁⢀⠄⠀⠀⠉⠉⢻⣯⣥⣞⣟⣛⡛⣀⡀⠀⠀⢘⣿⣧⡓⠀⠈⣁⣤⣾⡟⠁
⠀⠀⠀⠀⠀⢸⠝⢻⣟⡐⠒⠒⠒⠒⠛⠾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⣩⠽⣉⣥⣶⣿⣿⣿⡿⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡯⠟⠉⢻⣶⠂⢀⣀⣤⣿⣿⣿⣿⡿⠿⠙⠙⠿⠃⡠⠇
⣀⣀⣀⡠⠀⣀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣬⣭⣶⠗⠂⣰⡆⢘⣿⡆⢀⣄
⣿⣿⣿⠿⢷⠂⠑⠳⠄⢰⣦⣁⡘⠻⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣍⠉⣈⣱⣶⣿⣿⣿⡇⢸⡿⠛⢸⣿⣦⣤⣀⣀⣠⠔⡂
⠉⠉⠉⠑⠀⠀⠀⠘⠷⣦⣤⡉⠙⠻⣿⣾⣶⣿⡿⣿⣷⣾⣿⠿⠿⠿⠿⣿⡛⠿⠿⠿⠿⠃⠘⠇⡀⢸⣿⣿⣏⠛⠿⣿⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠢⠄⠉⠉⠈⠛⠳⠶⠆⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡇⠀⠘⢷⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣢⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢡⠞⠉⠉⠲⢄⡀⠀⠀⠀⡎⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡬⠖⠒⠦⠬⢦⣙⢦⣀⡾⣿⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣇⠀⢀⡭⢌⣐⣌⣉⠆⣿⣦⠆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢀⣞⠻⣦⡗⡾⠉⠁⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣝⡿⢻⠸⠀⠀⠀⠀⠀⣴⣯⡙⠒⠂⠀⠁⠀⠁⢀⠟⠟⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠁⠈⠀⠀⠀⠀⠀⠘⣟⠉⠉⠙⠛⠛⠓⠚⠚⠛⣆⠀⠠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⢶⡶⠶⠶⣶⣦⣶⠶⠶⢿⣄⠨⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠯⢆⣀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢹⠘⣯⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠦⠄⠀⠀⠀⠀⣀⠀⣤⣀⡀⠀⠀⠸⣤⣿⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⣄⣀⣀⣰⠋⠀⣠⣄⡀⠙⢧⡀⠀⣰⢾⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠃⢰⣾⣿⣿⣿⣦⠀⣿⣿⣿⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡟⠻⢿⣇⠘⣿⣿⣿⣿⠏⣀⡿⠿⠿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣮⣏⠀⠙⠿⣆⣉⣉⣀⡼⠋⠐⠊⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠉⠻⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣆⡀⠀⠀⢉⡉⠁⠀⠀⣤⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣦⣤⣍⣀⠢⣄⡀⠀⠀⠀⠀⠀⢸⣟⠛⠻⠻⠿⠿⠗⠛⢿⣿⡇⠠⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠭⠝⠲⢭⣕⡢⢄⡀⠀⠀⠈⠉⠛⢛⣛⠛⢛⣛⣛⣉⠀⠐⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢙⠻⢾⣄⡢⠤⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠢⡀⠙⠛⢳⢦⡀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡜⠶⠶⠄⡡⠟⡆⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⡶⠶⠛⠑⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠤⠤⠭⠉⠄

⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠠⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⡐⠋⠀⠒
⠀⠀⠈⠙⠛⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⡀⠐⠀⠒⠀⠀⠀⢸⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⣤⠀⠀⠁⢚⡁
⠀⠀⠀⠀⠐⠒⠆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠐⠂⠀⡀⢻⡀⢳⠀⡆⡀⠀⠀⣤⠟⠀⣾⡆⡄⠀⠀⣼⠀⠀⠀⠀⠒⠀⠀⢠⡄⠀⠈
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣼⠀⡀⠀⡄⠀⡀⢇⣆⡋⣤⢣⠉⣎⢡⢸⡇⠛⠀⠀⠿⠀⠐⣿⡀⢀⣾⡁⠔⢝⢤⢈⣠⡼⠀⠀⠀⠀⢀⠸⣇
⠀⠀⣀⡋⢁⠘⠃⣴⠀⣰⡞⠁⣾⠀⠀⢸⡀⡿⢸⣿⡍⠏⡌⠀⠀⠀⠊⠉⠀⠀⠀⠙⠂⠀⣿⣇⠈⣿⡌⣇⣸⢸⣾⣿⠁⡇⠀⠄⢀⢿⡀⠙⣷⠀⢰⡄
⠀⠤⠀⠀⠀⢀⡷⣷⠀⣽⠖⢴⣏⣂⠀⣸⡝⣧⠸⠏⠛⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣿⣥⣿⣿⡾⢿⣸⠂⣠⣸⡂⣀⢸⡷⢐⣿⡀⣾⠇
⠀⡀⠀⡀⢀⡜⠁⣿⣆⢻⡇⣼⢂⣽⣿⣽⣿⣦⠇⡀⠀⠀⢀⣠⠄⠀⠀⡇⠀⠀⡀⣀⣤⣴⣷⣾⣾⣷⣿⣿⣿⣦⠿⣧⣤⣽⣿⡟⢁⣸⣧⣸⣋⣼⠇⠐
⠀⢀⠸⣇⢈⣧⣰⣷⣾⣶⣶⣿⣿⠿⠿⠋⠈⠛⠿⣿⣶⣶⣿⣿⣶⣦⣶⣿⣦⣤⣼⣿⡟⢻⠻⣿⣿⣿⣿⣿⠿⠏⠀⠘⠛⠋⢙⣿⣿⣿⣿⣿⣿⣿⣦⣧⣄
⣶⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡟⠛⢁⡀⢀⠀⣤⣾⣿⣿⣿⣿⣾⣿⣿⣿⡃⠀⢿⣿⣿⡇⠘⠱⣿⢣⠎⣽⣧⣀⡀⠀⠀⠀⠀⡀⠁⠈⣻⣿⢻⣾⣿⣿⣿⣿
⣿⡿⣛⠻⣏⠳⣤⡆⣬⠿⢿⣿⣷⣿⣿⣿⣿⢣⣿⢫⡇⡿⢿⢹⣿⣟⠟⠁⠀⠘⣿⢿⠃⠀⡇⠉⣴⠿⣻⣿⠟⡟⢠⣴⡗⣴⣿⣷⣶⣿⠧⣾⢿⣿⣿⣾⣿
⠛⣧⣌⡋⢿⣿⠄⣸⣿⢆⣏⣿⣿⡿⠓⠘⡋⠞⣵⠟⣨⣴⠏⠈⠀⠈⠁⠀⠀⠀⠛⠂⠀⠀⠀⠘⠁⠀⢧⡳⣴⢸⡆⠟⣹⡟⢡⣿⣿⣿⣠⠶⠿⠿⣿⣿⣿
⡇⣿⣿⣷⠀⠙⠀⢋⡤⡐⢶⢆⡔⠗⠴⠆⠠⡌⠰⠁⠝⠒⢀⡀⠀⢃⠀⠀⠀⠀⠀⠰⡄⠀⠀⢄⢤⠙⠪⠃⠍⠆⢃⡌⣻⡿⢸⡟⣧⢁⣿⣇⣴⢂⡨⢿⣿
⣄⠹⡿⢉⠀⠀⠐⡏⠁⠀⠐⣶⡆⠀⣴⢣⠶⠆⠀⢀⡀⡀⢱⠀⣜⠉⠀⠀⠀⠀⢀⡐⠀⠀⢀⣩⠅⡄⠀⠀⢀⡐⣸⢿⡈⠏⣬⡉⠧⠘⣿⡇⠀⣬⣵⣞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⣄⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣦⡀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠐⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⣠⣶⣶⣶⣼⣭⣥⣀⡠⠀⠀⠁⠀⠒⠛⠋⠉⠈⠈⠉⠉⠀⠀⠀⠤⠤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⣶⣦⠈⠀⠉⠉⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⠀⣠⣿⣿⡁⠀⠀⢀⣀⣰⠾⠟⠉⠀⠀⠀⠀⢀⣼⡤⡄⠰⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠿⠦⣴⣷⣿⣿⣿⣿⣿⣦⣿⣿⡟⣠⣤⡤⠾⠛⠩⠰⠤⠴⠤⠤⠶⣤⣤⣴⣉⣉⣀⣐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⣴⣿⣠⡟⣿⠏⠛⠉⠉⠉⠉⠉⠉⢭⡥⠿⠉⠀⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⠻⠋⠉⠉⢛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡟⣳⣿⠞⠁⠀⠀⠀⠀⠀⣀⣰⣶⣿⠀⠀⠀⠀⢲⡴⢶⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠁⠀⠀⠀⠀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡷⠿⠟⣋⣸⣿⢿⣿⣶⣤⡤⣶⡾⣾⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠐⠦⠴⠷⠏⠀⠀⠄⠀⢰⡄⢠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠚⠟⠭⢄⠚⠉⣀⠀⢀⣏⣰⣚⣡⣶⣴⣦⣦⣤⣴⣤⣀⢀⡀⣼⣀⣀⣀⣀⡀⢀⣤⣤⣤⣬⣼⣏⣩⣩⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣿⣻⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣉⣉⣛⣛⣫⣭⣿⠿⠿⢿⠿⣟⣻⣯⣭⠭⢭⠿⠛⠉⠉⠋⣉⣉⠉⠡⠋⠉⠙⢡⢶⡮⣿⣿⣿⣿⣳⣶⣶⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⣶⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣏⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣦⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⢹⠟⢿⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣴⣟⣃⣾⣿⢃⠀⢀⣄⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠯⠛⠛⣛⣈⣧⣤⣷⡶⠀⠀⠀⠀⠀⠁⠈⠀⠈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣋⣤⣿⣦⣶⣶⣾⣿⡿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠠⠤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⣭⡛⠉⢿⣿⠜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡥⠀⢸⣿⣇⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠊⣿⣿⠄⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡄
⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⢀⡀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠀⠀⠶⠀⠐⠃⠀⠀⠀⠀⠀⠀⠿⣻⣷⡾⣝⠂⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣾⡀
⠀⠀⠀⠀⠀⠠⠄⠀⠡⠒⠀⠀⣿⣇⠀⠠⠀⠀⠄⠀⠀⠀⠀⢸⡟⢣⡔⢀⣤⠀⠀⠀⠀⠀⠀⠀⠺⠓⠻⣿⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣝⢣⡀⠀⠀⠰⠆⠀⠀⠀⠐⠠⠡⣄⣴⣷⡠⠆⠒⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⢹⢀⠀⠀⠀⠀⠀⠀⣀⣷⣴⣿⡁⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣿⢻⡾⠷⠀⠀⠀⢀⣰⡟⠉⣿⣿⣧⠀⢹⡄⠀⠀⠀⠀⠀⠀⢠⢯⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠿⣆⠀⠀⠀⠀⠀⢬⡗⠀⠀⠹⣿⣿⡆⠀⢣⡀⠀⠀⠀⠀⠠⣠⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠁⠀⠹⣧⡀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⣷⡀⣀⢡⣀⢀⡠⢈⣠⣼⠏
⠀⠀⠀⠀⠀⢤⣤⢀⠞⣏⢹⡅⠀⠀⠀⠀⠙⢿⣤⡀⠀⢹⣿⠀⠀⠀⣿⣿⣿⡇⠛⠀⣹⢍⣂⣴⣿⠏
⠀⠀⠀⠀⠀⠘⠛⠈⢃⣹⠘⢳⡄⠀⠀⠀⠀⠈⠹⢿⣶⣄⣉⠒⠀⠒⢿⣴⡿⣿⡄⣴⣴⣿⣿⡿⠁⠀⠀⠴⠶⠖⠀⠀⣾⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠹⣿⣿⢷⣶⣶⡿⣿⣿⣷⣾⣿⣼⠋⠀⠀⠀⠀⡠⠄⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⣿⡿⠷⠿⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⢶⡶⣠⣤⣭⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠉⠐⢛⣵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉


⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠘⠛⠫⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⢀⣀⣀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢤⣎⠉⢹⣿⡟⠁
⠀⠀⠐⠁⠀⠈⠃⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠄⢬⢻⡏⠛⠛⡋⡥⢤⡀⠀⠸⠁⠀⠀⠠⠆⢹⣿
⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠀⢀⡟⠀⠀⠀⠙⡀⠀⡇⠀⠰⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢈⣧⠊⠀⠀⠀⠀⠀⠙⢾⠇⠀⠀⠱⡄⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣼⡯⠑⠀⠀⠀⠨⠭⠐⢼⣧⠀⠀⠀⠙⡆⠀⠀⢊⣻⣿⣦
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⡮⠀⣿⣯⡀⠀⠀⠀⣈⢉⡢⢽⣿⠀⠀⠀⠹⠀⠀⠀⢀⣻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣮⣿⢿⠁⠀⠘⣆⠀⠀⠀⠁⠀⠀⠉⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠠⡀⢀⣠⣤⣾⣿⠟⠁⠀⠀⠀⠈⢻⣶⣦⣤⣤⣴⡞⠁⠀⠀⠀⠀⠀⠀⣤⣠⣔⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠉⠛⠷⠿⠿⠟⠃⠀⠀⠀⠀⠈⠛⠛⠛⠋⠙⠛⠛⠃⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠁

⠀⠀⠀⠀⢀⣴⠾⣿⡿⢷⣦⠀⠀⠀⠀⠀⠈⢫⠉⠉⠉⣏⣀⠀⠀⠀⠀⠀⠙⡏⠉⠉⢩⣇⣀
⠀⠀⠀⠀⣾⣿⠇⠛⠿⠹⣻⣧⠀⠀⠀⠀⢠⣿⣶⣦⣤⣷⣀⠇⠀⠀⠀⠀⣠⠓⠒⠲⠴⢧⣨⠃
⠀⠀⠀⠀⢿⣿⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠛⠿⢶⡶⠿⠋⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠱⢤⣀⣀⣠⠔⠁

⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠈⢱⠀⠀⠀⣏⠤⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡁⠀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡤⠠⡀
⠀⠀⠀⠀⢠⠚⠒⠒⠒⠛⣄⠇⠀⠀⠀⠀⠠⠃⠀⠀⠀⠒⡀⠇⠀⠀⠀⠀⡀⠀⠀⠒⠐⠳⣀⠃
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠘⠦⠤⠤⠤⠴⠃⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠓⢦⣄⠀⠀⠀⣠⠖⠉⠉⠓⢦⣀⠀⢀⡤⠚⠉⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠀⠀⠀⠀⠈⢹⢀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡏⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⠀⠀⠀⣠⠜⣸⣄⡀⠀⠀⠀⠀⠀⢀⡿⡇⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠈⠳⢽⣳⠤⠴⢪⠵⠊⠉⠛⢿⣶⣄⣀⣠⢔⡯⠐⠿⣷⣦⣤⣴⣾⠟⠀⠘⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠈⢳⡆⠀⠀⠀⠀⠀⠀⠈⢳⢩⠖⠁⠀⠀⠀⠀⠙⡆⡎⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⢀⡼⣇⠀⠀⠀⠀⠀⠀⢀⡜⣇⠀⠀⠀⠀⠀⠀⢀⡇⣇⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⣠⠔⠚⠺⣽⡦⣄⣀⡤⣞⠍⠀⠨⣝⡲⣄⣀⡴⣾⠟⠒⠺⣽⡲⣄⣀⡤⣞⡵⠚⠺⣷⡦⣄⣠⠔⠋⠁
⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠉⣧⡰⠋⠁⠀⠀⠀⠈⠙⢦⣵⠋⠀⠀⠀⠀⠀⠙⢲⡉⡞⠁⠀⠀⠀⠀⠙⢦
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢇⠀⠀⠀⠀⠀⠀⢀⡼⠀⢀⠀⠀⠀⠀⠀⠀⣸⠃⣇⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠓⢦⣀⣀⡴⣊⠅⠈⣝⣲⢤⣀⡠⢖⡩⠖⠓⢮⡓⢦⣀⣀⡴⣪⠴⠒⠮⣗⠦⠀⢠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡵⠊⠁⠀⠀⠀⠉⠛⡌⠰⠋⠀⠀⠀⠀⠙⠂⡈⣱⠊⠁⠀⠀⠀⠈⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢀⠇⢧⡀⠀⠀⠀⠀⠀⢀⠇⢳⣀⠀⠀⠀⠀⣀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⢀⡠⠖⠉⠀⠈⠙⠲⢤⣤⠴⠚⠁⠀⠀⠉⠓⠦⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⣿⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠿⣿⣿⣿⣶⡆⠀⢀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣦⣤⡤⠤⢤⣤⣤⣤⣤⣤⣤⡄
⠀⣿⡀⣀⠀⠀⢀⣼⣟⢹⠈⢻⣿⣿⣿⣿⣿⣧⣴⣾⣿⡿⠋⠁⠀⠀⣽⣿⣿⣿⣿⡯⠉⠉⠀⠉⠉⠉⠉⠉⠛⠿⠇
⠀⣿⣿⣿⣤⣤⣘⢉⣿⣿⣤⣿⣿⣿⣷⣾⣿⣿⡿⠋⠉⠀⠀⠀⠀⠚⠛⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣭⣩⣿⣿⣿⣛⠋⠉⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⡿⠿⠿⠿⠟⠋⠙⠛⠛⠛⠛⠃
⠀⣿⣿⣿⣿⣿⡛⠻⠟⠁⠼⡉⠉⠉⠀⠴⣿⡇⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⢀⣿⣷⣶⡶⠀⢠⣤⣶⣦⣀⣤⣄
⠀⣿⡉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣶⣀⣀⣐⣒⣳⣒⣈⣳⣷⣿⣿⣿⣿⣾⣶⣤⣤⣼⣿⣿⣿⣿⣷⡄
⠀⣿⠁⠀⠀⠀⠠⡀⣤⠤⠒⣠⡆⠀⠀⠄⣿⣿⣿⣿⣿⢿⣿⠿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⡆⢸⡿⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⢟⣛⣋⠉⣽⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⡂⠀⠀⠀⣀⠀⠀⢃⣾⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣯⠝⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⡅⠀⠀⢠⣿⡆⢠⣿⣿⡷⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⡟⢋⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠛⠛⠛⠒⠚⠛⠓⠛⠛⠛⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢲⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠎⠀⠀⠈⣿⡄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡁⠀⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡖⡚⣿⡙⢧⣢⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢙⣧⣿⣿⣿⣜⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣄⣇⣀⣠⣀⣠⣽⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣄⣤⣤⣿⣿⣧⣤⣤⣤⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⣿⣿⣏⢋⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠛⠛⠛⣿⣽⡏⠉⠉⠙⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡿⢣⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢪⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠖⡀⠀⠀⠀⠀⠀⡰⡱⢻⠃⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⡇⠙⠦⠀⠀⠀⡐⡱⠁⢸⠀⡟⣿⠀⠀⠀⠀⠀⢀⣤⡞⢸⡇⠀⠀⢰⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶
⠀⠀⠀⢷⣰⣄⠚⡁⠰⣰⠁⠀⢸⠀⡇⢸⠀⠀⠀⠀⠀⣹⣾⣗⣾⠁⠀⠀⠈⣿⣮⡲⠄⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢴⣾⣿⡏
⠀⠀⠀⠈⠋⠈⠻⣦⣙⠛⠤⣤⡾⢠⡇⠘⢤⠤⠀⣲⣿⠿⠋⠘⠃⠀⠀⠀⠀⠹⠟⢿⣦⣄⡀⠀⠀⠀⣸⣿⣿⣿⡀⠀⠀⠀⣀⣤⣾⠟⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣦⣄⡤⣾⣿⣤⣦⣶⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣟⣿⣶⣿⣿⣿⣿⣿⣽⣶⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡙⠻⣮⣵⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣀⡈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣍⣭⣫⣤⣍⣽⣉⣼⣏⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢠⡇⠀⡿⠛⢷⣏⠀⠾⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢂⠀⠀⠀⠀⣼⠀⠀⠙⣯⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠄⠀⠙⢦⠀⢠⡇⠀⠀⠀⢨⣿⣄⠀⠀⠀⡟⠀⣀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣰⠿⠞⢀⠀⠀⠀⠁⢼⠃⠀⠀⠀⡼⠁⠀⠳⠄⢸⡏⠀⠀⡟⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠤⡀⠰⣯⠉⠐⠶⢌⡦⡀⠀⠀⠀⠑⣄⣠⠞⠑⠂⠀⠀⠀⠻⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣿⠀⠠⡀⠀⠀⠉⠄⣤⣴⣞⣋⠁⠾⠖⠄⠂⠀⠀⠀⠈⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣿⠀⠀⠈⠳⢄⠀⠀⣼⡍⠻⣿⣿⣷⡆⠠⢍⡂⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⢠⣿⢠⠀⠀⠀⠀⠑⢾⡇⢸⠖⠉⢻⣿⡷⠀⠀⠈⣷⢄⡀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⠄⢎⣰⢂⡧⣀⡀⠀⠀⠀⠀⢠⣹⡇⢲⠆⠀⠈⠻⣄⠀⢰⡿⠀⠈⠂⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠊⠠⢡⣿⢇⣙⠃⠀⠀⠀⠀⣼⣿⢇⠙⠀⠀⠀⠀⠀⠳⢾⡟⠀⠀⠀⠈⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠓⢏⡉⠁⠀⣸⡟⠀⣏⠑⠂⡦⠀⣰⣿⢏⠂⠀⠁⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣴⠛⠂⠀⠀⠐⠾⢹⣮⡿⡓⠦⠟⡤⣂⣜⠃⡀⠀⠀⠀⠀⠀⠓⢄⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠃⠀⡻⢾⣏⠀⣽⣯⣭⣤⣈⣀⣊⡴⠿⣀⡗⠼⣋⢀⠄⠀⠀⠀⠀⠑⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠿⢦⣤⣻⢷⣿⠙⠻⣦⣉⠻⠛⢻⡷⣶⠠⣀⡃⠌⢿⢠⣟⡄⠀⣆⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⢠⡆⠀⠀⠉⠛⠻⣿⣷⠀⠀⠈⠛⠻⢦⣝⡃⠀⠀⢸⡗⠀⡀⠸⣧⢀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠰⠀⠀⣸⠧⠀⠀⠀⠀⠂⢈⣿⠀⡀⢀⠀⠀⠀⠈⠛⠳⠦⣼⡁⠀⠀⠳⣍⠃⠘⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢰⣿⢦⡆⠀⠀⠀⢈⡏⣽⡇⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⣦⣀⠙⣇⠲⢾⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠺⡸⡆⣱⡿⣿⡏⠿⣠⠂⢂⡞⢰⣿⣷⠇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠉⣿⣿⡁⠀⠈⠓⠠⣀⠀⠀⢀⡠⠛⠁
⠀⠀⠀⠀⠀⠀⢀⠊⠁⢙⢮⡿⣽⣿⡹⢻⣼⣹⠿⣡⣿⣿⣿⠀⠀⢺⠀⠈⠀⠀⠀⢀⠀⠀⣤⡆⢈⠿⣿⢼⡆⠄⢸⡄⠀⠙⠒⠿⡁
⠀⠀⠀⠀⠀⠰⠆⠤⠤⠤⢾⣿⣻⣿⣟⢾⣧⡿⠂⣱⣿⣿⣿⢰⡆⢸⡆⣸⣶⢀⡄⠘⠁⠀⢼⡇⣾⠀⣿⣾⡿⠇⢸⢁⡀⢠⡄⠀⠈⠓⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣱⠛⠣⠄⢟⡛⣂⢲⣟⠿⣿⣿⡼⠿⠺⡇⢺⡇⠸⡀⠷⠀⠠⢻⣷⣃⣼⠟⠛⠃⣀⣊⠈⣥⡟⡅⠀⣄⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⢀⢶⣴⣶⣾⣧⡔⣶⡟⠋⠉⠉⠿⣽⣳⢾⠇⢛⣀⣤⣶⣤⣶⠶⠶⠚⠛⢻⣿⣿⣿⣿⣿⣿⣾⣿⠃⠀⠘⠁⢂⠐⣃⢀⣴⠇
⠀⠀⠀⠀⠀⠀⣸⠏⢚⣹⠻⠻⢿⡿⣿⢶⣶⣾⡞⠀⢂⣷⢚⣩⠿⢟⣋⡥⣴⡞⠛⠛⠛⠉⠉⠛⢫⣹⣟⣻⡟⣹⣷⡀⠀⠀⠀⠈⠌⣿⠋
⠀⠀⠀⠀⠀⢠⡏⢀⣘⠭⠉⣀⠀⠀⠿⠿⣿⣛⣿⣚⣏⣻⣵⣲⡌⠛⠈⠁⠀⠀⠐⣈⣁⠐⠈⠒⣲⣿⣿⡿⢡⢿⡿⣷⠀⠀⠀⡀⠞⠁
⠀⠀⠀⠀⠀⠘⣧⠀⢥⣀⠀⠲⠤⠌⠁⣚⠋⢡⣺⣷⢿⣿⠛⠟⠋⡀⠀⠀⠀⠀⡉⠤⠤⠤⢤⣬⣭⣽⡟⣡⡿⣄⡿⠋⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢻⣄⠒⠠⢤⠉⡑⠒⠠⠔⠀⠬⢡⣄⣻⠈⠷⣤⡀⠀⠐⠀⠀⠀⠀⠒⠶⠦⠤⣽⣿⢯⣾⠏⡳⠈
⠀⠀⠀⠀⠀⢀⣟⣿⡉⣑⡀⠒⢬⣭⡤⠉⢁⣋⣀⠀⣭⣾⢿⡶⠄⠐⢋⡀⠀⠄⠐⢐⣲⣖⣶⡿⣳⡿⢃⠜
⠀⠀⠀⠀⠀⠸⠶⠟⠓⠹⠿⣿⣒⠲⠖⠒⠈⠬⣅⣀⣴⠶⠟⠛⠚⠷⢟⣢⣴⣮⣭⡥⢶⣾⣿⣱⡏⢱⣮⡄⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠉⠉⠙⠛⠋⠁⠉⠉⠁⠀⠈⠉⠛⠉⠙⠛⠻⠿⠿⠤⠀⠘⠿⢯⣿⣯⣽⠕⠛⠛⠛⠛
⠀⠈
⠀⢸
⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢠⣦⠀⣰⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⢰⣿⣿⣾⣿⣿⣿⣿⣿⣧⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠏⡾⠛⠂⡀⠀⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣂⣦⣤⣴⣾⢸⣿⣷⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠟⠛⠈⠜⢻⣿⠶⠦⡔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⡙⣿⣿⢿⣧⣝⡻⣮⣦⣤⣀⣶⣶⣶⣅⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠹⡏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⣾⣿⣷⣤⣻⣿⣿⣿⠿⣿⠛⠛⠿⠛⠿⠛⢷⣦⡶⠾⡀
⠀⠀⠀⠀⠀⠀⠀⠉⢉⡛⢿⣆⣿⠘⣿⣿⣿⣿⡿⠮⠻⡿⠻⢿⣿⣿⡻⠟⠛⠻⠯⣉⣀⣀⠀⠦⠄⠀⠀⢄⣤⣀⣼⣧⣲⣦⣀⡀
⠀⠀⠀⠀⠀⠠⡠⢒⣀⣿⣼⣿⣟⡰⢩⣿⠟⠃⠃⠀⠀⠈⠀⠀⠀⠈⠉⠛⠒⠀⠀⣠⣤⠤⠴⠶⣶⣶⣶⣾⡿⡿⠟⠛⠛⠣⠀⠈⠓⠲
⠀⠀⠀⠀⠀⠀⡉⠶⣽⣿⣿⡟⠈⠀⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⣀⣴⣶⡿⣿⠻⣿⢿⣿⣛⣿⣿⣇⡠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⡛⠋⢉⣈⠷⢿⣿⣿⣿⣮⣿⣿⣷⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⣀⣤⢾⡟⣧⡀⠈⠀⠀⣠⢋⠔⠉⣩⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⣿⣵⡶⡀⠘⣿⡇⠀⡼⠁⠂⣰⡀⢩⢹⣧⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⢠⣤⣤⣤⣤⠴⠒⣶⣶⠉⠀⠀⠸⣿⡟⣧⡀⠙⣧⣳⡁⣸⣺⣿⣶⣾⣿⡿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⢿⠀⠈⠁⠀⠀⠀⠀⠈⢿⠟⣃⠀⠀⠀⠈⢻⣩⡏⣹⣿⣿⣏⣛⣻⡽⣿⣯⣿⣷⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡻⠀⠔⡆⠐⡀⠀⢁⡀⠄⠁⠉⣿⣫⡤⡀⠀⠀⡨⠄⡉⠀⠀⢹⣿⣿⡿⠟⠋⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⢻⠀⠸⢷⠀⠂⠚⠉⢀⠄⠈⠀⠈⠭⡾⠟⠁⠀⠀⠈⠀⠀⢠⣯⡉⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⢀⡜⣿⡏⣆⠈⡆⠀⠐⢳⡀⠡⠈⠁⠀⠀⠘⣦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⣞⢞⠿⣻⡜⣄⠈⢆⠀⢀⡑⠤⠈⠠⢤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢸⣿
⠀⠀⠀⠀⠀⢸⡿⡌⡌⢹⣽⣎⠣⡀⠢⠈⠀⠀⠙⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⣿⡁⠀⠈⢋⠁⠾⣷⣔⣂⡈⠢⣦⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⣿⣿⣄⠀⡁⣍⢮⡙⢿⣿⣿⣶⣮⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄



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

⠀⠀⠀⠀⠰⢟⣲⢦⣄
⠀⠀⠀⠀⠀⠀⠈⠻⢿⡷⣦⣀
⠀⠀⢠⣾⣷⣶⣤⣄⣀⣙⣿⣽⣿⣶⣄⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠙⢿⣿⠟⢯⣿⣿⣿⣿⣝⢻⣿⣿⣶⣄⢀⡾⠋⠀⠀⠀⣤⠞⣋⣭⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠙⢦⣤⡈⠁⠋⠛⠿⣿⣿⣎⣍⣿⣿⣷⡤⠀⢀⠞⢠⡾⠃⠀⠀⠈⠙⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠉⠙⠿⣿⣧⣤⣤⣤⣿⣀⡿⠀⠀⠀⠀⠀⠀⠀⢹⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠀⠀⠀⠀⠀⠐⣱⣿⣿⢿⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⠠⢤⡤⣼⣿⣿⣿⣶⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠂⠀⠀⡄⠀⢨⣸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⠀⠀⠠⡁⠀⠈⣿⣿⣿⣿⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⢀⠇⠀⡆⢿⣿⣿⣿⡿⣶⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡅⠀⠀⠈⢳⠸⣿⣼⣿⣿⣿⡏⣹⠀⠀⠀⠀⠀⠀⠀⢀⡿⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠙⠀⠀⠀⠈⠀⠀⡿⠁⠿⢿⣿⡟⠀⡆⠀⠀⠀⠀⠀⠀⣼⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠠⡝⠀⠀⠀⠀⠀⠀⢈⡇⣶⣾⣾⣿⡆⠀⢡⠀⠀⠀⠀⠀⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣾⣃⠆⠀⠀⠀⠀⠀⢸⣇⣿⣿⣿⣿⣧⠀⠸⠀⠀⠀⠀⢠⣟⡟
⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠸⢿⢹⣿⣿⣿⣧⠄⠀⡆⠀⠀⢀⣾⡾⠁
⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⡀⠀⢰⣸⣿⣿⣿⣿⡟⠀⣸⠀⠀⣼⢻⠁
⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⠄⠀⠁⡆⠘⣿⣿⣿⣿⣿⣧⠆⢉⠀⣸⢧⠃⠀⠀⠀⠒⠒⠶⠶⣤⡀
⠀⠀⠀⠀⠀⡸⢹⡧⠀⠀⠀⠀⠀⠀⢰⡇⣷⣿⣿⣿⣿⣿⣿⠀⡈⣴⣿⡇⠀⣀⡀⠀⠀⠀⠀⠈⠈⡇
⠀⠀⠀⠀⢀⣷⣼⣷⠀⠀⠀⠀⠀⠀⢸⠀⣧⣿⣿⣿⣿⣿⣿⣷⢰⣿⣿⠢⠤⠤⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣸⢛⣿⠁⠀⠀⠀⠀⠀⠀⡌⢸⡇⣿⣿⣿⣿⣿⣿⣼⣿⣿⣯⣽⣽⣯⡍⠩⢅⡀⠀⠀⠀⡇
⠀⠀⠀⠀⣿⣼⢱⠁⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⣿⣿⣿⣿⠑⠍⡗⣿⣿⣽⣿⣶⣿⣷⣶⡴⢲⡀
⠀⠀⠀⣸⣇⣯⢤⠂⠀⠀⠀⠸⠀⠀⠀⡿⢸⣿⣿⣿⣿⣿⣿⣿⣆⠀⡆⣻⣿⣿⣿⣿⣿⣿⣿⣿⢀⡄
⠀⠀⠀⣧⣸⡷⠎⠤⠀⢀⡀⠀⠀⠀⢀⡃⣼⣿⡿⣿⣿⣿⣟⣻⣭⠲⠟⢩⣿⣿⣿⣿⣿⣿⡿⠞⠁
⠀⠀⠀⠙⠻⢇⣀⠀⠀⠀⠀⠉⠉⠉⢹⠉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣂⣴⣾⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⠤⠤⠤⣤⣬⣤⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠉


⣿⣿⡿⠻⠿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠕⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠁⠀⠀⠀⠀⠀⣠⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡴⠖⣁⣠⢤⠤⣤⣐⣶⡆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠒⠈⣛⣻⣿⡟⠒⣚⠋⠶⠛⣀⣈⣁⡤⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣚⡁⠀⠤⠙⠋⣁⣑⣿⣥⣤⣤⣀⡤⠭⠽⠧⠉⠒⠛⠛⠛⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⣠⡾⠁⡈⠈⣁⣀⠄⢸⣿⣿⣟⡓⠄⠉⠻⢿⣿⡄⠨⠉⠉⠈⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⠀⠀⠀⣠⢆⠃⣾⢀⡼⠁⠀⠀⠅⢀⣼⡃⠻⠟⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⡹⡇⠀⣠⠞⢉⠊⣴⡏⣼⠃⢀⣤⣥⣤⡮⠟⠀⠀⠀⢾⢻⡻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠡⣽⣾⣥⡴⢋⢾⡟⠱⢼⣷⣿⡿⣿⣯⡀⠀⠀⠀⠀⠀⢸⡇⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⢀⠯⡼⠁⠀⢨⠸⡟⠀⢻⣳⣽⢀⠀⣀⠀⡀⠸⡿⠃
⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⣠⠃⢀⠀⣾⢴⣄⠀⠀⠙⠿⣍⠻⢶⣖⢊⡤
⠀⠀⠀⠀⠸⢷⣦⣄⣀⠀⠀⣘⢁⠏⠀⢸⡃⡇⢸⣿⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠉⠝⠛⠿⠋⠀⠀⠸⠘⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣀⡀⣀⣀⠒⠇⠀⠰⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠁⠀⣠⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠚⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣄⡄⠀⠀⠀⠀⠀⠀⠶⠯⠕⢲⠀⠀⠀⠀⠀⠀⠀⠀⡈⢹⠀⠀⠀⠀⡀⠜⠓⢄⡀⠀⠀⢀⠦⠤⠄⠀⠤⠀⢇
⠀⠀⠀⠀⡍⢦⡔⡁⡑⣤⠌⡑⠀⠀⣸⣓⣒⣲⣚⣇⠀⠀⠀⠓⠒⠒⠂⠓⡙⠀⠀⠀⠯⠁⠀⠀⠀⢹⠃⠀⠆⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢸⡀⣀⣀⣀⣀⢘⡄⠀⠀⠹⠒⡖⢶⢀⠃⠀⠀⠀⠘⡉⠊⠑⢹⠃⠀⠀⢠⠧⠤⠤⠤⠤⠤⠆⠀⠘⡦⠒⠖⠢⠒⠶⠎
⠀⠀⠀⠀⠘⠭⢈⣁⣉⡀⠝⠀⠀⠀⠀⢦⢭⡭⡼⠀⠀⠀⠀⠀⢘⣀⣒⡇⠀⠀⠀⠰⢄⣀⣀⣀⣀⡀⠄⠀⠀⢱⣀⣀⣀⣀⡸
⠀⠀⠀⠀⠀⠠⡤⠤⢤⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⠤⠤⣤⡄⡀⠀⠀⠀⢀⡤⠤⡀⠀⠀⠀⠀⠀⠀⢠⢤⡀⡀
⠀⠀⠀⠀⠀⢀⠇⠀⠸⡀⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠤⠤⠄⢃⣸⠅⠀⠐⠞⠛⠄⠒⠊⠱⡆⠀⠀⣠⠄⠛⠈⠛⢮⡀
⠀⠀⠀⠀⢀⠃⠀⠀⠀⠘⡃⠀⠀⡠⣄⠤⠎⠱⠤⣠⢄⠀⠠⠊⠉⠉⠀⠙⡆⠀⠀⠸⡦⠲⠤⠦⠔⠤⠆⠀⠀⠟⠉⠊⠑⠉⠛⠃
⠀⠀⠀⠀⠘⡋⠙⠉⠋⢹⠃⠀⠀⣿⠥⠤⠤⢤⠤⢼⣷⠀⠸⡊⠒⠊⠒⠑⠇⠀⠀⠀⠐⢄⠀⠀⢠⠊⠀⠀⠀⠱⡄⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠈⠒⠚⠙⠁⠀⠀⠀⠈⠒⠒⠒⠒⠒⠊⠀⠀⠀⠈⠒⠒⠒⠊⠀⠀⠀⠀⠀⠘⠙⠉⠋⠀⠀⠀⠀⠀⠘⠋⠉⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠲⠆⣰⡆⠀⢀⣶⣦⠐⣦⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣴⠆⢽⣧⡤⢼⠗⠿⡆⠸⡏

⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⣄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⡄⠀⠀⠀⠀⣄⠀⣠⣠⡀
⠀⠀⠀⠀⠀⣔⣤⢣⠡⣔⣤⠀⠀⢠⠔⠐⠓⠈⠂⠢⡀⠀⡀⣤⢤⡤⣤⡤⢤⢀⠀⠀⡠⠐⠃⠈⠒⢄⡀⠀⠀⣽⢗⠿⠘⢗⠲⡄
⠀⠀⠀⠀⠀⡏⡉⢸⠆⣉⢳⠀⠀⢈⠒⠑⠊⠓⠉⠊⡅⠀⢱⣉⣉⣉⣉⣁⣈⡆⠀⣌⣀⣀⣀⣀⣀⡀⣱⠀⢰⢧⡋⢚⠙⡎⢱⠇
⠀⠀⠀⠀⠀⠫⡈⠁⠉⢈⠏⠀⠀⠘⢄⠀⠀⠀⠀⡠⠁⠀⠐⡄⠀⠀⠀⠀⢠⠃⠀⠡⡀⠀⠀⠀⠀⠀⡜⠀⠀⠑⠳⢄⣉⠤⠃
⠀⠀⠀⠀⠀⠀⠈⠒⠊⠁⠀⠀⠀⠀⠀⠑⠒⠒⠈⠀⠀⠀⠀⠈⠒⠒⠒⠒⠁⠀⠀⠀⠁⠒⠒⠒⠒⠊⠀⠀⠀⠀⠀⠙⠛⠃
⠀⠀⠀⠀⠀⣠⣀⣤⣶⡶⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠽⠀⠀⠀⠀⣀⡀⢄⡠⢄⡠⢄⣠
⠀⠀⠀⠀⠀⡿⢿⣧⣤⣄⡸⠀⠀⣠⡤⠤⠤⠤⠤⢤⣀⠀⠀⠀⡔⠓⠚⢢⠀⠀⠀⢙⡀⣀⠁⡀⢀⡀⣽⠀⠀⡛⢻⣀⡠⢤⣰⣤
⠀⠀⠀⠀⠸⣆⣀⣀⣈⣹⡇⠀⠀⠈⡗⠶⠲⢶⠶⣺⠁⠀⠀⠀⡍⠊⠑⢹⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡅⠀⠰⡤⢤⢷⠊⠁
⠀⠀⠀⠀⠈⢅⣈⣀⣁⡨⠁⠀⠀⠀⠻⣼⣸⣄⡧⠋⠀⠀⠀⠀⠰⣀⣀⠆⠀⠀⠀⠀⣇⠖⢢⡶⠒⣈⠄⠀⠀⠉⠉

⠄⢣⡶⢦⠀⠀⢠⡾⣷⠀⠀⠀⠀⠀⠘⠃⠈⣾⠻⡆⠀⠀⣾⣿⡦⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⡜
⠀⠈⣓⡋⠀⠀⠈⣻⠛⠀⠀⠀⠀⠀⢰⣿⣿⣝⣿⠁⠀⠀⢨⡟⡁⠀⠀⠀⠀⠀⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁
⠀⠘⠉⠀⣠⣤⡘⣇⡰⠂⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⠶⣦⠳⢤⠟⣤⣤⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠳⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠟⠀⠀⠈⠻⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡘⠈⢳⣄⣀⣀⣀⣀⣀⣀⡀⣀⡀⣠⣀⡀
⠀⠀⠀⢆⡀⠀⠀⠀⠀⠈⠉⠁⠀⠀⣤⣄⣲⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠖⠛⠓⡩⠤⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣀⣀⣀⡀⣀⣤
⠀⠀⠀⠀⠰⣯⣤⠀⠘⡇⠀⠀⠀⠈⠉⠑⠒⢤⣀⠀⠈⠁⠀⠆⣀⠀⠀⠀⠀⠠⣿⡟⠁⠀⠀⠀⠀⠈⣿
⠀⠀⠀⢀⠜⠁⠈⡀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠈⠉⠁⠀⠀⠈⠁⠀⠀⠀⠁⠙⠽⠂⠀⣠⠀⠀⠀⠊⠁
⠀⠀⠀⢸⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⢠⠄⠀⠐⠻⡥⡀⣀⠀⠀⠀⣿⡏⢹⡄⠀⠀⠀⠀⠀⢠⣮⣼⠗⠀⠂⠀⠀⢀⠀⠀⠀⠀⣰⣿⣿⣴⣀⡀
⠀⠀⠀⠰⡄⠀⠀⠀⠉⠁⢹⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢐⣞⡶⠀⠀⠀⠠⢫⢿⢝⣯⠁
⠀⠀⠀⠀⠁⠠⠀⠀⠀⣠⠆⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠀⠀⠀⠀⠀⠘⠀⠙⠑

⣟⣛⡛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉
⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠓⠲⢲⣿⠛⠛⠛⠛⠒⢛⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢻⣿⣿⣾⣶⣶⣶⣶⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠛⠉⣭⡉⠉⠉⠙⠛⠻⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢤⣾⣧⡀⠀⠐⣚⣂⠀⢀⣠⣀
⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⠛⢹⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁
⠀⠀⠀⠀⢀⡿⣷⣄⠀⠀⠀⠀⣠⠈⠀⠊⠁⢨⣿
⠀⠀⠀⠀⠀⠁⠹⡎⠓⡀⠀⠀⠉⠀⠀⠀⢀⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⣴⣧⣶⠚⠒⠂
⠀⠀⠀⠀⠀⡀⠀⢻⡄⠀⣴⣄⠀⠀⠀⢀⣾⣿⣿⣿⣷⣶⣶⣤⣤⣠⢤⣬⣿⣿⡆
⠀⠀⠀⡌⠀⡡⢠⣤⣷⣼⡫⣮⠑⣄⠀⠜⡋⣹⣿⣿⢸⣿⠙⠻⢿⣿⣿⣿⣿⣿⣿⡄
⠀⢀⣼⠃⠀⢰⣿⣿⣿⣿⣿⣿⣿⣬⡳⣌⣀⣿⣿⣿⣾⣿⣤⣀⣀⣈⡘⢝⣿⣿⣿⣿⡀⠀⡀
⠀⣾⠃⠀⠠⡿⢋⡿⠋⠙⣿⣿⣿⡟⢹⣿⣿⣿⡿⣿⠛⠉⠁⢨⣟⠈⠉⠙⠲⠮⣿⣿⣷⣀⣿⡄
⠀⠏⠰⠀⣤⣷⣬⣤⣤⣤⣼⣦⣭⣵⣿⣿⣿⣷⣶⣿⣶⣤⣤⣾⣿⣦⡤⣤⣤⣤⣤⣤⣽⡿⢿⣿⣆
⠀⠀⠀⠀⠀⣾⠟⠛⠉⠉⠁⣿⣿⠀⠀⠀⠌⠁⡜⠁⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢻⣯⠀⠀⠌⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⠖⠒⠒⡒
⡧⠶⠤⠤⠤⣿⡤⠾⠿⠿⠿⠿⢿⣿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠾⡶⠀⠀⠀⠀⠀⠀⠀⠀⣮⣿⡆⠀⠀⠁
⠂⠀⠀⠀⠀⢧⣘⣀⡀⠀⠀⡀⢘⣿⣦⡜⣳⢆⣀⣀⣀⣀⣀⣸⣃⡀⠀⠀⢀⣀⠀⢰⣟⣼⣿⠇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣾⣿⠿⠟⢿⣿⣿⠿⢻⣿⢿⣿⡿⠻⣿⣿⣿⠿⣿⣿⣿⣷⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠘⣿⠿⠛⠉⠀⠀⠀⠈⢿⣆⠀⢸⣿⠀⠀⠀⠀⡿⢹⣄⠀⠀⠁⠀⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣠⣤⣾⣿⣿⣼⣿⣦⡀⣤⢠⣇⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡠⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠚⠷⠶⣤⣶⡦⢤⣀⡀
⠀⠀⠀⠀⢀⣀⣀⣀⣩⣿⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⣀⣀⣀⣀⡉⠉
⠀⠀⠀⠈⠙⡻⠛⢛⣉⣉⣀⡙⠙⠛⠛⠛⢛⣻⣿⣛⣛⣛⣿⣿⣿⣿⠿⠟⠛⠛⠛⢿⡛⠋⠉
⠀⠀⠀⠀⠀⢰⠿⠛⠉⠈⠉⠙⠛⠓⠚⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡓⠬⣒⠬⡢⡑⠜⣧⡀⠀⠀⠀⠀⠀⠀⢤⡀⢀⠀⠀⠄⠄⠀⠠⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡒⠤⣱⡈⢳⡘⡆⡏⠶⡀⠀⠀⠀⠀⠏⠀⠷⠦⠰⠀⠶⠠⠂⠀⠿⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣕⠬⣑⢄⡙⠄⠁⠃⠁⠀⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⢿⡲⠦⠍⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣀⡴⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡀⠉⠀⠀⠀⠀⠀⠀⠣⠄⣀⢀⠟⠁⠀⠒⣯
⠀⠀⠀⠀⠀⠐⠶⣒⠠⠤⣖⣒⠨⣇⣤⠤⠖⠒⠓⢄⠀⠀⠀⠄⠈⠀⠊⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢟⠦⣤⣈⠙⠢⡙⢟⠓⢄⠀⢠⡇⠀⠀⠀⠀⠀⠀⠑⢄⠀⣀⠤⠔⣰⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢶⡀⠉⠁⠀⠀⠳⠀⢢⡈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠙⠁⠀⠈⠔⠚⠛⢓⡂⡀
⠀⠀⠀⠠⠤⠶⢒⣒⡀⠐⠂⠖⠢⠦⡤⣄⣈⡁⠘⠁⠀⢀⢀⡰⠊⠀⠉⠁⡦⠀⠀⠀⠄⠀⣐⠒⠒⠮⣕⡄
⠀⠻⢶⣦⣤⣤⣘⣊⣩⡭⠀⠀⢀⣠⡽⠖⠋⠁⢀⠠⠐⡛⠅⡀⠀⠀⠀⠀⠖⠄⠀⠀⢐⢌⠲⣭⣙⠲⣌⡳⡀
⠀⠀⠀⠀⠉⠙⠒⠲⠬⠤⣥⡀⠀⠤⠤⣶⡞⠋⢙⠑⠚⠃⢤⡣⡀⠀⠀⠀⠙⢴⡀⡄⡈⢦⠑⢬⡒⠌⡢⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢕⠄⠀⢠⠃⠀⠸⠀⢠⠐⡄⡙⠞⠀⠀⠀⠀⠀⠑⠳⢵⣌⢗⠦⣙⠓⣮⡬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⢃⢀⠴⠁⠀⠀⠐⢆⠸⡆⠹⡘⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣀⡠⢐⡡⠔⠉⠁⠀⠀⠀⠀⠀⠈⡄⢇⠰⠱⠡
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠼⡄⠆⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠊⠉⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠐⠂⢤⠀⠀⠀⠀⣼⣤⠀⠀⠀⠀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⠢⣄⠼⣿⠓⠀⠀⠹⡇⠀⠀⠀⠀⠀⢠⣶⣀⠤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡀⠀⠀⠀⢿⡀⠀⠴⠖⠒⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⡏⠳⢄⠀⠈⣇⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠠⠤⠔⠃
⠀⠀⠀⠀⠀⠀⠐⠤⣿⣆⣠⣤⡀⠀⠀⢳⠀⠀⠑⢆⢻⡀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⡠⠆
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣽⣿⣿⣷⡤⣤⣼⣣⣤⠀⣸⠈⣧⠀⠀⠀⠀⠀⠈⣇⠀⣀⣈⡠⣮⠞
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡍⢹⢚⣻⡍⣭⠉⢤⡈⡦⣀⡀⠀⣠⠴⣺⣟⣋⣡⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⣶⣦⠄⠦⠠⠼⠿⢷⣾⣿⣶⣾⣷⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄⢀⡀⢀⣈⣐⣿⣿⣛⣿⡛⣿⣿⣏⣹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⡿⠩⢉⡩⠉⣶⠾⣒⣿⣩⣯⡟⣭⣭⠯⡭⢤⡄⠁⠢⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⣉⣹⡛⠿⠋⢉⠉⡷⠶⢗⣋⠭⠋⠭⡭⠭⠍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠂

⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠃
⠀⠀⠀⠙⠿⠟⠛⠻⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠻⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠀⠙⣿⣿⠟⠛⠿⣿⣿⣿⠿⣿⡟⠉⠉⠛⠟⠛⢻⣿⠋⠀⠙⣿⡟⠛⠻⠿⠃⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠀⠀⠀⠀⣿⣯⠀⠀⠀⠀⠉⠀⠀⣽⣧⠀⠀⠀⠀⠀⢼⣿⡄⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠈⠉⠀⠀⠘⣿⠇⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠟⢿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⣠⣶⣦⣀⣴⣶⣄⣠⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡻⣿⡿⣛⢿⣿⢟⡻⣿⡿⣃⠀⠀⠀⠀⣴⣶⣦⣠⣶⣶⡄⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⣼⣿⣿⡇⣿⣿⣿⠀⠀⠀⠻⣿⢟⣘⢿⡿⣃⡻⡿
⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠙⣫⣷⣿⣛⣽⣮⣟⣫⣷⣿⡛⠉⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⡇
⠀⠀⠈⣿⣿⣿⣿⡿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⢀⡻⣿⡿⣛⢿⣿⢟⡻⣿⡿⣃⠀⠀⠀⠀⠀⠀⠉⢫⣷⣾⡝⠉
⠀⠀⢀⣿⡇⠀⣿⠁⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⡈⠁⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⣾⣿⣿⡇⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⢿⠿⠃
⠀⠀⢸⣿⡇⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠈⠙⣫⣷⣯⣛⣽⣾⣝⣫⣷⣿⡛⠉
⠀⠀⠀⠉⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠛⢿⣿⠟⠻⣿⡿⠃
⠀⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⢠⡿⠻⠿⠛⠻⠿⠻⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣠⣤⣶⣾⣷
⠀⠀⠀⠀⠀⠀⢸⣤⣤⣤⣤⣤⣤⣤⣼⡄⠀⠀⠀⠀⠀⠀⢰⠃⢿⡟⣷⣯⣏⣯⠛⠉⠉
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⠌⢡⠚⢡⣾⠇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠆⠈⠁⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡆⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⣠⣴⢾⣿⣿⣿⣟⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣰⡿⣫⡾⠛⠉⠀⠀⠙⣷⡘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⡾⠋⠀⠀⠀⠀⠀⢀⣿⢡⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⠀⠀⠀⢠⣾⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡾⣷⡄⠀⠀⠀⢠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢸⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⣤⣾⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄
⠀⠉⠛⠀⠀⠀⠠⠄⢒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡆⠀⠀⠀⠀⠀⠀⣀⠈⠛⠂⠀⠀⠀⠤⠄
⠀⠀⠐⠶⠀⡄⠈⠀⠀⠀⠀⢰⠀⠀⠀⠀⢠⡄⢠⡆⣄⣿⡜⢰⠀⡀⢀⠀⠀⠀⠀⠀⠛⠂⠰⠀⠀⠀⠀⢀⣀⠥
⠀⠀⡀⢀⡴⠃⣆⠀⠀⢠⢳⣆⡗⡆⡅⠂⢾⠀⣾⠇⠛⠛⠧⢹⣀⣰⣠⠇⠾⣦⣄⠀⠀⠈
⠀⠀⣧⢨⣇⢰⡇⢇⣧⣌⠸⣽⢷⣿⡇⠁⠚⣄⡇⠀⠀⢀⠂⠀⢻⡿⠷⠋⠰⣏⡷⣆⠀⠀⠉⠀⡆⢣⢠
⠀⢀⣼⣷⣽⣾⣷⡄⢳⢹⡷⡹⣳⣿⠁⠐⣿⣿⣷⣀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠛⣼⣧⣴⣄⡀⢀⣇⣟⣞⣠⣄⣄
⣴⣿⣿⡿⠙⣿⣿⣷⠀⡎⠃⣽⢟⣿⡀⠀⠉⠸⠟⠉⣀⡀⠀⠠⠀⠀⠲⠶⠀⢀⡟⢉⡛⢿⣿⣿⣿⣿⣿⠙⣿⣿⣶
⢿⠟⢿⠇⢀⡟⠉⡓⠀⠀⠀⠟⣪⣿⣿⠀⠀⠈⡀⠘⠛⠁⠀⠀⠀⠐⠀⠀⢠⣄⠟⠉⡷⠞⠻⣿⣿⠿⡿⠀⣺⡿⠻
⡞⡃⠀⠀⠸⠁⢀⠆⡄⠀⠀⣺⢟⣽⣿⣄⠀⢸⣿⣿⡀⢀⠀⠦⣴⣄⡘⠃⠈⢿⣿⡀⠠⢂⣒⢸⣦⠓⡃⠀⡽⠁⡀
⢋⣅⠀⠀⠀⢰⡄⠘⡀⠀⢨⡀⠛⠟⠉⠀⣴⣿⡿⣿⣷⠀⡇⣠⣈⣛⠛⠀⠀⠘⢻⡇⠀⣼⣛⣏⣛⠆⠷⠀⡞⢰⡿
⠀⠻⠆⠀⠆⠸⠿⠀⠝⠀⢘⣿⠀⠀⠐⠀⠀⢽⣿⣶⣶⣾⣅⣿⣿⢿⡁⠀⠀⠀⡴⠇⠀⠟⠋⠙⠛⠓⠀⠠⡤⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⢀⠘⠋⠉⠛⠋⠑⠈⠉⠉
⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠢⠀⠀⠀⠀⠀⢀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠬⠉⣉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠓⠂⠀⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡎⠉⠈⠓⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠰⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠂⠀⠀⢃⠀⠀⠀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢀⡜⢀⠃⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⡠⠊⠀⠀⠀⠐⢒⠤⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡀⣀⠠⠤⢄⠀⠀⣴⡶⠝⣆⣇
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠈⠉⠀⠀⠀⠀⠀⠪⡓⢆⠤⠏⢀⠿⡈⠡⡔⠀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠑⠀⠒⠉⠢⠀⠀⢰⠀⠀⠐⠀⢱⠀⡀
⡔⠂⠠⢄⡀⠀⠀⠀⠈⠂⠠⠤⣠⣄⣘⣯⠺⠿⢅⣀⡀⠀⠀⠀⠀⠀⣶⠂⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠿⣧⣭⡿⠁
⠈⠒⠤⢀⣀⠁⠢⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢧⡀⠀⠀⠀⠀⣟⣠⠤⣀⠀⢠⠃⠀⠠⠄⡠⠴⠛⠛⠀⠀⠐⠀⡜⠁
⠀⠀⠀⠀⠈⠉⠓⠒⠤⠀⠀⠀⣠⠀⠀⠀⢀⠀⠀⠠⡌⠙⠳⠤⣤⠼⠛⠁⠀⢀⡿⠋⠀⠀⠀⠀⣠⡄⠤⠒⠒⠃⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡕⠀⠀⠀⠈⠃⢢⠀⠈⠐⠀⠄⣀⠀⠀⡀⡠⢊⠠⠄⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⣄⠀⠀⣴⠃⠀⠀⠀⠀⠀⠁⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣅⣭⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣟⣛⣛⣛⣛⣛⣻⠿⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡍⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⣸⣿⢉⣉⣉⣉⣉⣉⣉⣛⠛⠶⣯⣝⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠿⢷⣦⣍⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣌⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠁⠀⠉⠛⠿⢿⣿⣶⣶⣶⣤⣤⣤⣤⣤⣤⣴⣶⣶⣿⡿⠿⠛⠉⠀⠈⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀⠀⠀⠀⠀⠀⢸⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⢠⣿⣇⡟
⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⣾⡿⣸⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⣼⣿⢣⡟
⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⢷⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠿⣿⣟⣴⣿⢏⡾
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠉⠛⠻⠷⣶⣶⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣶⣶⠾⠟⠛⠉⠁⠀⠀⣿⡿⣿⢏⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢣⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣶⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠋⠉⠉⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⣉⣿⠟⠿⠿⠿⠟⠻⠟⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠲⠦⠴⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡤⠰⡆⠈⠳⠖⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠚⠀⠀⠐⠐⠒⠲⣷⠶⠦⠤⠀⠀⠀⠀⠀⠤⠀⢸⣄⣀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠲⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡖⠒⠒⠛⠛⢲⡄⢲⡄⠀⠀⠀⠀⡀⠀⠸⣇⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣄⣠⡤⠀⠀⠂⠀⠀⠀⠀⠀⠈⣷⡀⠙⠀⠀⠀⠀⡇⠀⠀⠘⣷⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠂⠀⠀⠀⠹⡌⠂⠰
⠀⠀⠀⠀⠐⠀⠠⠀⠀⠐⠀⠀⠉⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⡠⡦⣿⡆⠀⠀⠀⣠⠀⠀⠀⠀⠀⣻⡀⠑⡀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⡀⠀⣤⣾⣶⣇⣀⣰⣿⣷⣾⣿⡀⠀⢰⡿⠀⠀⠀⠀⠀⢻⣇⠀⠈⢀
⠀⠀⠀⠀⠀⠈⠦⡀⠀⠀⠀⣠⢠⡾⣮⣇⣼⣿⣟⣿⣿⣏⣹⣯⣉⣉⡈⠃⠀⣰⣧⣤⣶⣶⣷⡛⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠈⢀⣤⡶⢳⣾⡟⠛⢿⠀⠀⠀⠈⣿⡟⣼⣏⠁⠀⠀⠈⣴⣿⣿⣿⣿⣿⣿⡟⢛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡯⣸⣿⡇⠀⠀⠀⠀⠀⢸⣿⣷⣾⡇⠀⠀⠀⠀⣿⠿⠟⠋⠉⠋⢻⡇⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠿⡇⠀⠀⣦⠶⠞⣿⣿⠓⠻⣷⠀⠀⢀⠎⠁⠐⠀⠀⠀⠀⢸⡇⠀⠀⠀⡿⣦⠄⣀⣀⠀⠳⡄⢀⣠⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣗⠀⠈⠁⠀⠀⣿⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠁⠀⠀⠀⠀⣉⣵⣿⠟
⠀⠀⠀⠀⠲⠦⠄⣠⠀⠤⢤⣿⣄⣿⠀⠀⠀⠀⢰⣿⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⣴⠾⡿⠋⠁⠀⠸
⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣼⣿⣄⣿⣀⣀⣀⣀⣬⣏⣥⣤⣤⣿⡀⠀⣀⣀⣐⣀⣀⣀⣀⣀⣸⣧⣤⣤⡶⠶⠿⠉⠄⡿
⠀⠀⠀⠀⠀⠀⠀⠙⢯⣛⣛⣿⡟⠛⠋⠛⠛⢻⡿⠛⠛⣿⡟⠛⢿⣿⠉⣙⣿⣏⣉⣉⣿⣯⣤⣤⣭⣯⣤⣶⣶⡾
⠀⠀⠀⠀⣀⡀⠀⠀⠀⡈⠿⣿⣿⣿⠿⣿⠿⢿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⢿⠿⣿⣿⣿⣿⣟⣅⣌⡻⢿⡿⣯
⠀⠸⣅⡀⠄⠒⠖⠒⠂⢐⣒⣿⣿⣶⣤⠁⠴⠒⠒⣲⣶⡾⠛⣛⠿⢉⣟⠛⣀⣬⣿⡿⣿⣿⣿⣿⣿⣿⠾⣿⣿⣌⠙⠓⠠⣔⣀
⠀⠀⠀⠀⠀⠂⠀⠤⠶⠚⠛⠛⠛⠛⠿⠟⠃⠠⠿⠇⠀⢖⣛⣿⣛⠀⢀⣤⣬⣭⡍⠐⠦⠤⠌⠉⠉⠁⠉⠚⠿⠿⠛⠻⠿⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡠⠠⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢠⣅⠀⠀⠑⠀⠈⢀⣰⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠊⠁⠂⠀⠀⣺⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢠⠄⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⡤⠄⠀⠀⠀⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠥⠒⢇⣀⠔⡠⠆⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⢰⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰⣀⡠⠘⢻⣿⡿⠟⠻⠿⢿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢀⠀⠀⠀⠀⠀⠉⠀⣀⣤⣄⣈⢻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡁⠀⣷⣃⣠⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠚⠋⠁⠂⠂⠤⠀⠀⢀⠔⠀⠀⠀⠀⠢⢤⡤⠐⠒⠐⢁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠉⠀⠀⠂⠀⠀⠀⠀⠀⠀⡈⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠄⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣀⠴⠒⠚⢗⡶⠦⠀⠀⠀⣠⠴⣶⣖⡶⠤⠀⠀⠀⣠⠴⣶⣶⣄⡀⠀⠀⠀⡀⠤⣶⠶⠄⠀⠀⠀⠀⠀⣠⣴⣶⣤⡀
⠀⠀⠀⢠⠎⠁⠰⢓⠀⣞⠀⠀⠀⣀⡾⡡⠀⢸⠋⠀⠀⠀⣠⣾⣿⢠⠏⠀⠀⠀⠀⣠⠞⡅⣾⠁⠀⠀⠀⠀⢀⣴⢻⣿⡿⠁
⠀⠀⠀⢻⠀⠀⢸⠘⠀⠸⡶⢶⢚⣿⣏⣷⣿⢸⣦⢤⢴⢚⣿⣿⣏⠘⣦⣤⣤⡖⣿⢸⡇⠇⠘⣦⢤⡤⣤⡖⠙⣎⢸⡟⣿⠦⣤⣤⡴⠆
⠀⠀⠀⣸⠀⠖⠊⢩⠻⢃⡦⠊⢩⢝⣿⠟⢋⡿⣿⡼⢋⡜⣿⣿⠟⢁⢻⣷⣿⠟⢉⣾⠇⣇⠔⠋⣹⣷⠁⡟⠈⣡⣿⣆⡟⠋⣩⣿⢷⡆
⠀⠀⠀⡽⠀⠀⠀⠇⠠⠊⠀⠀⠠⡬⠋⠀⠰⠟⠟⠀⠘⠦⠟⠁⠀⢧⡼⠏⠁⠀⢾⣿⠆⠃⠀⠸⣸⡿⠋⠀⠀⢗⡾⠋⠀⠰⢿⣿⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡦⢰⠔⡀⠠⠰⢀⡤⢂⢀⠄⣠⡀⠲⢀⠄⡂⠀⠀⠀⠀⡀⠠⠀⡄⠐⠀⡄⢀⣄⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠄⢠⠰⠡⠆⠚⠉⠅⠺⠀⠏⠀⠄⠪⢴⠥⠁⠀⠀⠠⠑⠇⠈⠇⠦⠀⠃⠎⢭⠀⢈⡸⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠛⣛⠳⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⢁⣴⣾⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢰⣿⡟⠉⠀⠀⠙⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣇⠀⠀⠀⠀⣿⠃⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣹⣿⣿⡷⣤⣴⠾⢋⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣶⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⣿⠛⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⣠⣶⣾⣿⡶⠶⠶⠶⠶⠶⣿⠀⣿⣿⣿⡷⠶⠶⠶⠶⢶⣾⠋⣤⣶⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡿⠟⠉⠉⠉⠉⠉⢙⣿⠀⣿⣿⣿⡏⠉⠉⠉⠉⠉⠙⠿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⣿⣿
⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦
⠀⠀⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡄
⠀⢰⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧
⠀⣼⠃⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢹⣿⣿⣿⡄
⢠⡿⠀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣾⣿⣿⣿⣧
⣼⣧⣤⣼⣿⣿⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠷⣶⣿⣿⣿⣿⣿
⠉⠀⠘⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⠀⠈
⠀⠀⠀⢻⣿⣿⣿⡻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣻⣿⣿⣿⠃
⠀⠀⠀⠈⢿⣿⣿⣿⣮⣙⠻⠶⢦⣤⣤⣄⣀⣀⣀⣸⠇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣤⣤⣤⠶⠾⢛⣩⣾⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣶⣦⣤⣀⣈⠉⠉⠉⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠛⠋⠉⢉⣁⣠⣤⣶⣾⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⠀⠠⣿⣿⡿⢛⣡⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡅⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠸⠩⠱⠄⢠⠖⠀⢰⡒⢤⣄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠈⠉⠀⠘⠂⠀⠀⠀⠀⠀⢀⡠⢠⡄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡀⠀⠴⠀⢃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⠀⠠⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠠⠀⠀⢸⡄⠀⠄⣀⢀⢁⣀⠀⢰⣶⣶⡀⠠⠂⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠈⠠⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢢⡂⠀⣶⣶⠢⠾⣿⡶⢄⠈⠻⠿⠿⣄⠀⠀⣆⠀⠀⡰⠇⠀⣿⣷⣆⠁⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠈⢳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠃⠐⢦⣽⡿⢧⠀⠽⠀⠘⢦⡀⠀⠀⠈⠓⢄⡀⠀⠈⠁⠀⠀⠈⠋⠙⠳⠦⡀⠀⠀⠀⠈⠓⠓⢄⠀⠀⠀⠓⢛⡦
⠀⠀⠀⠰⣤⣠⡴⠂⠀⠀⢈⡿⣧⠀⢀⠀⠀⠀⣾⣿⣶⢦⢄⠠⢀⠑⠠⠀⠀⠀⣰⣶⣾⠠⠀⠀⠀⠀⠒⠤⠀⠀⠀⠈⠂⠀⣠⠖⠁
⠀⠀⠀⠀⢿⠉⠙⠂⠀⠄⣰⡿⣿⠀⢆⡀⢲⣾⣿⣿⣿⡌⠣⢶⠞⠀⠀⠈⠃⢠⣿⣿⣿⠀⠀⠀⠀⠀⡀⠀⠀⠐⠖⠀⠰⠟⠁
⠀⠀⠀⢤⢤⡀⣀⡴⢆⣠⣿⡃⣿⠃⠁⠀⢤⣿⣿⢿⣿⡇⠀⠀⠀⡀⠀⠀⢀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⢆⠀⠀⠀⠀⡠
⠀⠀⠀⠀⢸⣿⣿⣿⣾⣾⣿⣇⣿⣶⣀⣴⣿⣿⣯⣭⣉⣃⠀⠤⠼⠇⡠⢠⣿⣿⣿⣿⣿⣆⡀⣀⣁⠴⣺⣷⣫⠀⠈
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿
⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣫⣽⢻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣾⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣾⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢻⠿⠿⠿⣿⣿⣿⡿⣿⣿⣿⢿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⢛⣛⣻⣉⠉⠉⠁⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠐⢂⣀⣀⣀⣤⣬⣭⣤⣤⣤⣤⣤⣶⣶⣶⣶⣿⣶⣞⣻⣛⣛⠿⠿⠿⠿⠿⠷⠖
⠀⠀⠀⠀⠀⠐⠒⠛⠛⠛⠛⠋⠉⠉⢈⣉⣉⣉⣉⣉⣉⣉⣥⣬⣭⣭⣭⣴⠆⠀⠀⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠛⠓⠒⠒⠒⠒⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠍⠁⠈⠀⠈⠀⠄⠀⠀⠒