Entree Emojis & Symbols

⠀⢀⣀⠴⢳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠸⢯⡟⣙⡾⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿
⠘⢲⣼⢏⣋⡟⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⠀⠀⠀⠀⠀⢀⣼⣿⠏
⠀⠨⠏⡴⣭⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⠧⡀⠀⠀⠀⠀⠀⢹⣿⣿⣦⠀⠀⣀⣴⣿⣿⣿⠃
⠀⠀⠐⠋⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣿⣿⣿⣿⣿⣿⣿⣾
⠀⠀⠀⣼⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣾⣷⣿⣿⡟
⠀⠀⠀⢹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠋⠙⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡆⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⠀⠙⢿⠿⡟⠁
⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠻⣿⡄⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢧⡠⠧⠤
⠀⠀⠀⠀⠙

⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠉
⣾⣿⡋⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣀⣢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⢁⣠⡄⠒⡄
⠙⠛⠻⣿⣿⣿⡏⢧⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣀⡈⠂⠠⢤⢀⡨⠀⠀⠀⠀⠈⡟⠉⣴⠆⢷⣄⣀⣤
⠀⠀⢸⣿⣿⣟⣷⣼⠀⠀⠀⠀⠀⠀⠀⡀⠙⠛⠛⠋⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠛⠀⡀⠗⠛⠉
⠀⠀⣿⣿⣿⣿⣿⣿⠤⣿⡦⠀⠀⠀⠀⢹⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢰⠂⣠
⠀⠀⢿⣿⣿⣟⣴⠟⠛⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠸
⠀⠀⣼⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠈⡁⠸⠄⢀⣀⡄⠀⠀⠀⠀⢀⠀⢋⣠⡴⢃⡞
⠀⠀⣽⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠀⠤⠀⠠⠀⠠⠤⠴⠂⠀⠀⠀⠈⣤⠈⠉⠀⠈⠲⡴⠖
⠀⠀⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠠⠶⠞⠀⠘⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⣿⣿⣿⡿⣿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠟⠛⠉⠉⣁⣀⣀⣀⣀⠀⠀⠉⠉⠙⠛⠻⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⣀⡤⠶⠒⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠁⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹


⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢉⡙⠻⢿⣍⡉⢿⠻⡍⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⣩⡽⠋⢉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠙⠿⣾⣆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡻⣦⡀⠈⠛⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣌⠛⢦⡀⠀⠈⠙⠛⠿⠷⠶⠶⠾⠟⠋⠁⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡉⠂⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⡀⠀⠀⠀⠀⠀⠀⢀⡏⢹⡟⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⡀⠀⠀⣠⠎⠀⡼⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡶⠛⠁⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢩⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠰⠚⠲⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠩⠀⠀⠤⣈⣉⣁⣀⣀⣐⠩⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢀⣻⣽⣿⣿⣿⣿⣿⣿⣿⣷⣿⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠤⢬⡭⣭⢭⣭⣭⣿⣿⣟⣛⣛⣛⣛⣛⣛⣛⣛⣻⣿⣯⣭⣭⣭⡭⠥⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⡛⠻⣿⡿⢿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣈⠛⢿⣦⡈⢢⣀⠀⠀⠀⠀⠀⠀⢀⡟⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣄⠙⢳⡀⠹⣷⣦⣤⣀⣤⣴⡞⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⢧⡀⠛⠿⣿⣿⣿⡏⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠉⠙⠳⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⡀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣤⠴⠂⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠘⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⡥⠈⠀⠀⣠⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⢦⣀⣤⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡴⠿⠻⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


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

⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⡀
⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣄
⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠻⣤⣀⣸⣇⣀⣴⠏⠀⠀⠀⠀⢀⣴⣿⣿⠟⣋⣭⣴⣶⣶⣶⣶⣶⣬⣙⠻⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⢹⡏⠉⠀⠀⠀⠀⠀⣰⣿⣿⠟⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣦⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣼⣿⡿⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⢻⣿⣷⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⢰⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢸⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢸⣿⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠸⣿⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣾⣿⡏⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⢻⣿⣷⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣼⣿⡿⠁⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠻⣿⣷⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⣾⣿⡟⠁⠀⠀⠀⠀⠻⢿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣦⣙⠛⠿⠿⣿⣿⠿⠿⠟⣋⣥⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠹⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⡤⢤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣔⣒⠭⠔⠊⠩⢁⠮⢑⢬⢳⠀⠀⣀⣴⡾⠛⠉⢛⣤⠤⠈⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡷⣴⣢⢽⣿⣿⣽⣿⣷⠟⣼⣠⠞⠉⡁⠑⢆⠀⠀⢀⡕⢴⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢶⣍⣩⡭⠍⠩⢶⣶⣿⠟⠁⠉⠧⡄⠐⠀⠱⠀⢀⣀⣨⢀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠷⢶⡾⠻⠇⠚⢣⡀⠀⠀⠘⠟⠀⠉⠁⣄⠀⠡⠊⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⣬⡴⠚⠋⠀⢱⣔⠶⠆⠚⠎⢀⠄⠈⢂⠀⣼⣻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠃⠠⠚⠁⢂⣹⣝⡒⠊⠉⠐⠄⠀⠀⢀⡀⠴⠰⣋⠆⣸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠀⢈⠳⡖⠋⠀⠀⠈⢇⠤⣀⠂⠤⠐⡩⠍⢠⢂⣾⣍⡾⢁⣠⣤⣶⣶⣆⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡥⠐⠻⣅⢀⠘⢆⡂⠴⠒⠈⢣⡀⠀⠀⠧⠀⠀⣹⠃⣽⢋⣴⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⣰⡏⢈⠀⠀⢀⡈⠾⠋⠉⠳⡀⠀⠈⠀⢢⡀⣀⣀⡶⠞⠁⣡⠿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⢸⣿⢷⢴⣮⣅⣀⡀⠐⠃⠀⠀⠰⠤⣠⣰⡞⢫⢁⣰⠼⢫⣤⠿⢻⢽⣿⣿⣿⣿⣽⣿⠟⠃
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠄⠀⠀⠉⠓⠒⣺⡟⠓⠚⠋⢿⣠⣾⣯⣴⣾⠋⣧⣶⣿⣿⣿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⡀⠃⠀⠀⣀⣋⣿⣁⣆⣠⢴⣮⠽⣟⣭⣿⡷⢿⣯⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⣠⡤⣿⣿⣶⣻⣅⣼⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⣷⡾⣯⣷⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣿⠿⣿⣿⣟⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣫⣷⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⡷⠁⠈⢿⣿⣿⣷⣦⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⡮⠀⠀⢀⢲⣶⡔⢲⠀⠀⠀⠀⠀⠀⣰⠀⢈⣤⣽⣦⣤⡄⠸⠏⣡⣶⣿⣿⡄⠘⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠨⠀⠂⠀⠸⠾⠙⢹⠄⠀⠄⠀⠀⠀⠀⠻⣄⣨⠛⢛⠿⠻⠋⠀⠚⠛⠟⠉⠯⢀⣱⠜⠁
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠻⠦⠧⠬⢤⠰⠤⠷⠖⠂⠉
⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠐⣧⣀
⠀⠀⠀⠀⠀⠀⡠⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠉⠁⠉⠉⡛⠋⡉⠔⣂⣺⣶
⠀⠀⠀⠀⠀⠈⠙⠂⠄⠀⣀⡀⢀⢀⠄⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣎⢰⣰⠏⠀⠼⡂⢹⣷⠸⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⠉⢀⠉⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣷⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣄
⠀⢠⣾⠟⢉⠭⠒⢂⣄⡀⠀⠀⠀⠀⠀⠀⠀⢰⠋⡴⠚⠉⢻⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣄
⠀⠈⢿⠀⡏⢀⠞⠁⠀⢈⣑⡄⠀⠀⠀⠀⠀⠘⢾⠀⠀⠀⡟⠀⠉⣳⠦⣄⠀⠀⠀⠀⠀⠀⢠⡂⠙⠿⣿⡿⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠⡂⠿⠟⠋⠻⠋⢰⡄
⠀⠀⠀⠑⢧⣸⡀⠀⣴⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠑⢄⡀⢇⠀⠘⠁⠀⢘⠆⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣾⣿⠂
⠀⠀⠀⠀⠀⠉⠓⠤⠽⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠒⠦⣄⣠⠞⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⡿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠏⠀⣤⣤⡄⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠈⣲⣿⣿⠆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⡀⠃⠀⠀⢹⠟⠀⠀⠀⠀⠀⠉⠲⢤⡀⠀⢀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠈⣳⠶⣉⠉⠉⢉⣓⣦
⠀⠀⠘⠻⠏⠂⠤⢀⣽⣶⣦⡀⠀⠀⠀⠀⠀⡀⢠⠈⠓⢨⣔⠀⠀⡠⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠳⠌⠁⠆⢷⡀⠀⠀⠀⠀⠀⠀⢇⠀⠈⠀⣾⣿⡦⠤⡀
⠀⠀⠀⠀⠐⠀⡀⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠬⠠⠔⠇⠈⠁⠈⣤⡁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⣐⠲⠣⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⣟⠿⣉⠆⠱
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠈⠑⠂⠀⠀⠀⠀⠀⠈⠠⠤⠤⠐⠉⠀⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠹⠧⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠁

⠀⠀⠰⣄
⠀⠰⢤⡈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠢⣄
⠀⠀⠀⠉⠢⡙⠄⠀⡕⠰⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⡶⢶⠶⣖⢦⣄⠀⠀⠀⠀⠀⢀⡰⠷⠾⠓⠒⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠙⣦⡈⢦
⠀⠀⡠⢺⣵⠀⢃⣊⠠⡀⠄⠄⣤⠀⠀⠀⠀⠀⡿⣧⡙⠦⢽⣲⡯⠴⣻⢷⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠘⠟⠷⡆⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢿⣿⡼⡇
⠀⠀⣇⠙⠿⣿⣦⣶⣤⣬⡤⠇⢀⠇⠀⠀⠀⠀⠳⣄⠉⠛⠒⠒⠚⠋⣡⠟⠀⠀⠀⠀⠀⠀⠀⠙⠣⢤⠐⠇⢾⣿⠇⡆⠀⠀⠀⠀⠘⠶⣤⡀⠀⠀⢸⣿⣷⣷
⠀⠀⠈⠒⠤⣤⣀⣀⣀⣠⠤⠔⠋⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠖⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠴⠶⠂⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣸⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⣠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⡆
⠀⡠⠒⢲⣑⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⠀⡀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣴⣾⣿⣿⣶⡀⠀⠀⠀⠀⠀⡀⢀⣿⣿⣿⣿⣿⣆⡴⠲
⠀⠈⠺⣁⡬⠲⣎⢉⠝⢒⡤⣀⠀⠀⠀⠀⠀⠀⡌⠁⠘⠻⢿⣿⡿⠃⠈⠂⠀⠀⠀⠀⢰⡏⣠⡏⣭⣮⣻⣿⣿⣿⣿⡀⠀⠀⠀⢨⣟⡿⣽⣿⣿⣿⣿⡯⠤⠄
⠀⠀⠀⠈⠢⡖⢉⡻⣶⠏⣠⠊⣵⠀⠀⠀⠀⠀⣱⣄⣀⡀⠀⠀⢀⣀⣠⡆⠀⠀⠀⠈⢿⠿⣍⣻⣌⣁⣿⣿⣿⠿⠿⡏⠀⠀⠀⢻⠾⢟⣿⣾⣿⠋⣐⣈⡦⠤
⠀⠀⠀⠀⠀⠈⠣⣠⠜⣳⠀⠉⠁⠀⠀⠀⠀⠀⢇⠉⠛⠻⠿⠿⠛⠛⢉⠄⠀⠀⠀⠀⠘⢄⠈⠀⠀⠈⠀⠀⠈⢀⠜⠀⠀⠀⠀⠀⢂⠀⠀⠀⠈⠁⠀⠀⠁⡰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⣀⡠⠄⠀⠈⠀⠀⠀⠐⠒⡉⠁⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣀⠀⠀⠀⠈⠐⠀⠀⠀⠸⣷⡄⠀⣼⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢰⠃⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣯⡆⡄⠀⠀⠀⠂⠀⢀⣏⣷⣴⣟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⢐⠊⠀⢀⡄⢀⠈⠐⠛⡰⢤⡀⠀⢠⠀⠀⠀⠀⠀⢀⠙⡖⠀⠀⢀⣸⢠⢎⣿⣿⣻⠀⠀⢀
⠀⢀⠔⠀⠀⠄⠐⡀⢂⡶⠂⠀⢀⡄⠀⠀⠀⠀⠈⠀⢉⠀⠉⠁⠀⠀⠀⠠⠈⠀⠀⣀⢤⡺⠿⠒⠛⡥⠌⠁⠀⡀⣺
⠐⠁⠀⢀⡄⡀⠆⡀⠄⠌⠀⠀⠈⠉⠈⠉⣠⠄⠀⠀⠐⠀⠀⠀⠐⠀⠈⠀⠐⠚⠋⠉⢀⣀⠠⠒⠂⠀⠀⠀⣀⡼⠃
⡀⠀⠀⢌⠃⠀⠈⠀⠈⠀⣠⡀⠀⣠⠆⠀⠀⠀⠐⢋⡀⠤⠐⠀⠀⠀⠀⢀⡤⠔⢂⠉⠈⠁⢀⣁⣤⣴⠾⠛⠁
⠁⡀⠀⠈⠁⠀⠀⠀⠀⠰⠋⠉⠁⠤⠠⠓⠀⠤⠈⠀⠥⠄⠚⠛⠂⠀⣀⣀⣠⣤⣤⡶⠶⠛⠛⠉⠁
⠀⠈⠒⢤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣴⣶⣶⠿⠿⠛⠛⠉⠉⠁
⣤⣤⣤⣤⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠀⠀⠐⣿⠶⣶⠀⠴⡄⠂⡀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣸⣿⣛⣧⡀⠁⢰⡆⠀⠀⡙⠓⠒⡜⠛⣱⠀⠀⠀⣠⣿⢺⣿⡧⠀⡄⠀⠀⠀⠀⠀⠀⠀⡀⠲⢀⣤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠨⡴⠿⡇⣿⣷⣄⡁⢠⣀⡃⠘⢃⣬⠙⢟⡀⢶⠶⣿⢇⣿⠟⢡⡄⡳⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠠⢴⣯⣿⣿⣿⣧⠙⠗⠀⠀⣇⢀⡈⡈⠀⢀⣀⠀⢇⢘⠻⡿⠂⢈⣥⠀⠀⠀⠈⠀⠖⠦⣴⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⣿⣿⣿⠁⠰⠶⠀⣸⣿⡿⠃⢰⡆⠀⠉⠳⠀⠘⠿⢶⡾⠋⠁⠀⠀⠀⣀⣤⠶⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠐⠀⠐⣄⠙⠿⣿⣿⣶⣶⣶⣿⣿⣀⣄⡈⣷⡀⢸⣦⣿⣶⣦⣀⠙⠷⣀⣠⣴⡾⠋⠙⣤⣄⠀⢰⡀⠀⠓⢆⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣶⣾⣆⣏⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⠹⣿⣿⣷⣄⡈⠻⣿⠇⣤⣴⠋⢩⠘⠃⠀⢀⣰⣏⣠⣤⠆
⠀⠀⠀⣀⢀⣀⠤⠄⠀⠀⠀⣀⠀⠀⠀⠀⠙⠛⠿⠀⠀⠀⡀⠀⣠⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⡟⢻⣤⣄⣄⣸⣤⣀⣤⣆⣀⣤⣿⠟⣿⠏
⠀⠀⠀⠈⠻⠁⠀⠀⣴⠋⠨⣧⠀⠲⣶⣶⣶⣾⠃⡀⠀⠰⣷⣾⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⣹⠁⠀⡼⠁
⠀⠀⠀⠀⠀⢸⣧⣼⣇⢂⣠⡷⠀⠀⣴⣼⡿⠇⣰⣿⣿⣾⣿⣁⡀⢨⣷⣾⣿⣿⣿⣿⢻⣏⠁⠀⠀⣥⣤⣄⣀⡀⠀⠀⠀⠼⢿⡔⠾⠃⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⣿⣏⣠⣮⡏⢠⣿⣿⣧⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠛⠁⠀⢠⣿⣿⡿⠿⣿⠶⠤⣤⣤⣄⣤⣄⠀⠀⠲⡏
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣧⣤⣿⣟⣧⣷⣠⣽⣿⣿⣿⣿⣻⣷⣦⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⢸⣿⠀⠀⠀⠀⣠⣦⣤⣄⠉⣽⠋⠁⠀⠀⡨
⠀⠀⠀⠀⠀⠀⠀⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⡇⠀⠀⣸⣿⣷⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢀⡷⡄⢀⡉⡉⠻⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⢈⣿⣿⣿⢸⣿⣿⣿⣇⠀⠀⣿⣿⡿⠟⠛⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⡏⣿⣿⡆⣿⡏⢿⣿⣦⣤⢿⠏
⠀⠀⠀⠀⠐⠂⢀⣀⠈⠀⠀⣿⣿⣿⠿⠟⠛⢻⡏⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡇⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠉⠘⠷⠺⠋⠁⠀⠀⠀⠀⠘⠁⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣇⢰⣶⣦⣄⣀⣤⣴⣿⣷⣿⣷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⡿⠿⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣾⣿⣿⣿⣷⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡼⣿⣿⣿⣿⣿⣿⡆⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⢈⢿⣿⣿⣿⣯⣿⣿⣿⣿⣟⣻⣿⣩⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⡿⢋⣾⣿⣯⣿⣿⢎⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡿⠿⣟⣠⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃
⠀⠀⠀⠀⠀⠀⠀⢀⡿⢻⣿⣿⣿⣿⡟⠁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢀⠀⠀⣼⣷⣾⣿⣿⣿⡟⠀⣾⣿⣥⣼⣿⣿⣿⣟⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣏⣿⣇⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢻⣿⣾⣿⣟⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠀⠈⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣏⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⢿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣄⣈⣻⣿⣿⣿⡿⠿⠟⠛⠀⠉⠉⠉⠀⢿⣿⠛⠋⠀⠻⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⣿⠻⠟⠋⠁⠀⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣄⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠑⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠙⢀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⣴⡖⠻⠃⠂⠀⠐⠐⠢⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠙⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⡀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡀⠈⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠐⠁⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠈
⠀⠈⡁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⢀⠁⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⡀⠗⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠒⠀⠀⠔⠁⠀⠀⠀⡘⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠃⠀⠀⠈⠐⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠒⠈⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠾⡂⠀⠀⠀⠀⠀⠀⠀⠄
⠸⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⡁⠀⢢⣀⠂⠐⢆⢀⠀⢯⡀⢐⠴⠰⠂⠀⣤⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠐⠈⠀⠀⠠⠃⠀⠀⠀⠀⢤⠀⡀⠠⠒⠋⣉⣈⠉⠍⠀⠀⠀⠙⢁⠀⠉⠀⣀⠎⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠷⠐⠦⠘⠧⠀⠉⢱⠔⠶⠀⡀⠀⠈⣷⠐⠒⠈⠁⠀⠀⠀⠄⡰⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⢤⣄⢀⡴⠀⠛⠠⠀⠈⠉⡄⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠓⠠⠄⣀⠤⠄⢀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠑⠢⠤⠤⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⡀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⣀⡤⠤⠒⠂⢸⣋⡋⠉⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⢀⠀⢰⠖
⠀⠀⠃⠠⠀⠄⢓⡄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⡀⠀⢰⢂⠈⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠒⠊⠁
⠀⠘⠒⠀⠀⠀⠚⠀⠀⠀⠀⢒⠀⠠⠄⠀⠀⠀⠀⠀⠸⠆⢀⣠⠀⢀⡀⣄⠈⠠⠈⠶⠀⠀⠀⢀⣀⡀⢀⠀⠀⠂⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠇⠀⠈⢿⢷⠀⢁⡀⠀⡀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠃
⠀⠄⠀⠀⠀⠀⠀⠸⠀⠀⠰⡜⠂⠀⠀⡆⠀⠀⠀⠀⠘⠀⠀⠈⠃⠀⢦⡀⠀⠒⠀⠀⠀⠀⠐⠀⠂⢀⣀⠀⠄⠊
⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠂⠠⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠐⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣄⡀⠀⢀⣤⣶⣾⣟⣛⣻⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣿⣿⡿⠋⠉⠻⣿⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣇⠀⢀⣿⡟⠀⠀⠀⠀⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⢸⢼⢽⢸⣿⡇⠀⢨⣿⡇⠀⠀⠀⠀⣸⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⡄⠀⠀⢸⣿⡇⠀⢸⣿⣧⡀⠀⠀⣰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⢠⣾⣿⡇⠀⢸⣿⣿⣿⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠈⣿⣿⣷⠀⢨⣿⣿⡇⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⠀⢸⣿⣿⠁⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⢸⣿⣿⠀⢸⣿⡏⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠈⣿⡟⠀⢸⣿⠁⠀⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⣿⣿⠀⠀⢹⡇⠀⢸⡇⠀⠀⣿⡿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠀⠀⠀⢀⠄⠀⠀⠀⢠⠀⡄⠀⠀⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⠟⠀⠀⠀⢸⠆⠀⠀⠀⢘⢃⡇⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠁⠀⠀⠘⠒⠀⠀⠀⠈⠀⠃⠀⠀⠀⠑⠊





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

⡧⠀⠀⢠⡖⠦⡁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿
⠀⠀⠀⠀⠙⡆⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢴⣿⣴⣶⣄⠀⠀⠀⢀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠈
⠀⠀⠂⠀⣀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⡄⠀⣶⣝⡃⢸⣂⢘⢓⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣈⡩⣥⢄⡀⠿⣿⣿⣦⣭⣾⣿⡇⠀⠀⠘⠟⠀⠀⠀⠀⠾⠃
⠀⠈⠋⠀⠀⠂⠀⠀⠀⠀⠀⠈⣷⣷⣶⣷⣿⠀⠀⠙⠿⠿⠋⢁⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠉⠴⠀⢶⠀⠀⡤⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄
⠀⢠⣶⣷⣦⠄⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠀⠐⠀⠀⠀⠠⠀⠀⢸⣿⡿⠗
⠀⠸⣿⠟⠁⢸⣷⣬⣥⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠙⠿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠘⠁⠀⠀⠀⠀⠐⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠲⠀⢄⠀⠀⠀⠀⠙⠿⠃⢀⠆⠀⠠⠗⠀⡀⠤⠤
⣁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⣷⣦⣤⣤
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢠⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⠀⠀⠀⠀⠉⠄⣤⠛⠿⠿⣿
⠀⠠⠠⠀⠀⠀⠀⠀⢀⡤⠀⠀⢠⣿⣿⣿⣿⣿⣿⣦⡀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣠⣶⣶⣾⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠰⠛⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⡀⠈⠀⠬
⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠁⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠃⠀⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠸⠆⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶

⠀⠀⠀⡀⠀⠠⣴⠥⠠⢄⣠⡤⡄⡀⡀
⠤⠾⠿⠠⠗⠳⠥⠾⠦⠇⠅⠵⠩⠀⠴⠔⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢤⣤⣤⣤⣤⣤⠤⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠆⠀⣿⠰⠰⢸⡇⠆⠆⣿⠇⢀⠀⢴⠖⠶⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠈⠉⠉⡁⠀⠉⠈⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠭⠭⠭⠤⠤⠤⠄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠤⠒⠂⣉⣀⠀⠤⠄⠐⠂
⠀⠀⠀⠀⠀⠶⠶⠵⠂⠒⠋⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣔⣶⣽⣯⣿⣿⣿⣝⢿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣇⡟⣿⢫⣿⣿⠿⣫⣻⣿⢫⡿⣿⢿⣩⣦
⠀⠀⠀⢰⣶⣶⠂⢠⣤⣤⣤⣤⡀⠳⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⢀⠀⣿⣿⡇⠀⣿⣿⣿⣿⣿⣇⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠛⠛⠓⠀⠿⣿⣿⣿⣿⠿⠐⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⠿⠛⠁⢀⡀⠤⠐⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⢊⢉⣀⠠⠤⠄⠒⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢰⣾⣿⠟⢒⣚⣉⣉⣁⡀⣀⣀⣀⣀⡀⣀
⣿⣿⠛⢛⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣴⡖⣶⣶⡶⢶⣄⠀⠀⢀⣤⣟⢳⢀⡟⢴⠟⣙⠇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⢀⡀
⠀⠀⠀⠀⢀⣾⣿⣹⣿⠾⢿⣶⣵⡅⠀⣿⡻⣯⣑⠤⢀⣿⣤⣥⣤⠀⠀⠀⠀⠀⢀⠀⢉⠀⠀⠀⠐⠈⠑⢄⠀⠀⢠⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠚⠈⠁⠀⠀⢀⠈⠻⡾⡇⠀⠹⢷⣤⣉⣛⣻⣿⣿⠏⠀⠀⠀⠀⣀⠀⢸⣷⣦⣅⡐⠀⠀⣁⣴⣿⠀⠀⢸⡿⢹⠹⢩⡿⢸⠻
⠀⠀⠀⠀⡁⠀⠀⡠⢐⠀⡇⠀⣿⡇⠀⢰⠉⠉⠉⠛⢻⣿⡟⠒⠒⠂⠀⠀⠀⠄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢼⡯⢮⡄⠺⣿⣿⣸
⠀⠀⠀⠀⢠⠀⠠⠀⠇⠸⠂⠀⣿⠏⠀⠠⠀⠀⠀⠀⠸⣿⣿⡀⠀⠂⠀⠀⠠⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠒⠀⠠⠤⠀⠀⠚⠁⠀⠀⠀⠀⠂⠐⠀⠀⠛⠛⠛⠒⠒⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡄⠀⠀⠀⢀⠀⠀⠀⠀⢀⡤⡶⣶⣶⣦⣤⡀⠀⠀⠀⠀⢀⡤⠀⢠⠤⢤⣤⣄⡀⠀⣀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⡼⠟⠛⠛⠋⠛⢿⣿⣷⠀⠀⠀⡄⠀⠐⠀⠟⢹⣿⣿⠓⠐⡠⠮⠉⠑⢄
⠀⠀⠀⠀⠀⣀⣀⣀⣰⣿⣿⣇⣀⡀⣀⢀⣀⠴⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⠀⠀⠀⣇⣨⠤⢀⠁⣿⣿⣿⠉⠀⠀⠈⠐⢢⠸⣇
⠀⠀⠀⠀⠀⡊⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⡆⣸⡟⠀⠀⠀⠈⡇⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⢸⠃⢆⣧⡀
⠀⠀⠀⠀⠀⠸⢀⠀⠀⢿⣿⣧⡀⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠑⠂⠂⠃⠈⠴⠋⠀⠀⠀⠀⠀⠘⠤⡀⣠⠀⢻⣿⣿⡀⠀⠠⠔⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⢀⣀⣞⣿⣿⡷⣶⢿⠋⢀⡀⠈⠉⢠⡀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⣿⣫⣭⣋⣩⣽⠋⠀⠠⠾⡼⠾⣁⣀⣠⣴⠛⢟⣆⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⣾⠋⠁⠠⠜⠏⠀⡉⠐⠒⣈⡉⢲⠆⢈⠀⣻⡎⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠻⣦⠠⢧⡀⠱⠆⠈⠶⠀⣀⡁⢐⠰⢘⣾⡿⣣⡎⣾⠿⡆⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⣈⠙⠓⠲⠭⢽⡷⠷⠦⠠⠒⠚⠹⠛⠋⠉⠟⡐⠇⡎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠏⠿⣿⣿⣻⢶⣧⣤⣤⣤⣤⣤⡖⠒⣛⣁⠄⢀⢀⣄⣼⠕⠀⢀⡄⠰
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠢⢙⠻⢽⠭⣿⣿⣿⣿⣦⣄⣀⠈⠑⠒⠈⢀⢥⡟⠉⠀⠈⠈⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠀⠘⠛⠛⠛⠛⠛⠛⠛⠷⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⢠⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠏⠑⠛⠒⡿⠚⠙⠋⠊⠄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡨

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠤⢤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠖⠒⠒⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢋⣡⣴⣶⣾⣿⣿⣿⣿⣷⣶⣤⣉⠓⢦⡀
⠀⠀⠀⢀⡴⠋⠁⠀⣀⣤⣶⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⣶⣿⡿⠟⠋⠉⠀⠀⠀⠈⠉⠙⠻⢿⣿⣦⡙⢦⡀
⠀⠀⣠⠟⠁⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠘⢷⡀⠀⠀⠀⠀⡴⢃⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⠹⣄
⠀⢰⡏⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⠟⠁⠈⠻⣿⣿⣆⠀⠈⢷⡀⠀⠀⣼⢁⣾⣿⠋⠀⢀⠀⠀⡀⠀⠀⣾⠀⠀⣴⣿⣷⣄⠀⠀⠻⣿⣧⠸⡆
⠀⡿⠀⠀⣼⣿⣿⡟⡏⣿⢻⣿⡟⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⡆⠀⠘⡇⠀⢰⠇⣼⣿⠇⠀⠀⢸⣦⣤⡇⠀⢸⣿⠀⢠⣿⣿⣿⣿⠀⠀⠀⢻⣿⡇⢻
⢰⡇⠀⢀⣿⣿⣿⣇⠀⠀⢸⣿⡇⠀⣿⣿⣄⠀⠀⣠⣿⣿⣿⣷⠀⠀⣿⠀⢸⠀⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⣼⣿⠀⠀⢿⣿⣿⠟⠀⠀⠀⢸⣿⣷⢸
⠸⡇⠀⠈⣿⣿⣿⣿⡆⢰⣿⣿⡇⠀⣿⣿⣿⡆⢰⣿⣿⣿⣿⡿⠀⠀⣿⠀⢸⠀⣿⣿⠀⠀⠀⠀⢻⡏⠀⠀⣿⣿⠀⠀⠀⢿⡏⠀⠀⠀⠀⢸⣿⡿⢸
⠀⢿⠀⠀⢻⣿⣿⣿⡇⢸⣿⣿⣇⠀⣿⣿⣿⡇⢸⣿⣿⣿⣿⠇⠀⢠⡇⠀⠸⡇⢻⣿⡆⠀⠀⠀⢸⡇⠀⠀⢿⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⣿⡇⣼
⠀⠘⣇⠀⠈⢿⣿⣿⡇⢸⣿⣿⣿⡇⣿⣿⣿⡇⢸⣿⣿⣿⠏⠀⢀⡾⠀⠀⠀⢻⡈⢿⣿⣄⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣴⣿⡟⢰⠇
⠀⠀⠘⣦⡀⠀⠙⢿⡇⢸⣿⣿⣿⠇⢿⣿⣿⡇⢸⣿⠿⠃⠀⢠⡞⠁⠀⠀⠀⠀⠳⡌⢻⣿⣦⡀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣸⡇⠀⣠⣾⣿⠏⣰⠋
⠀⠀⠀⠈⠳⣤⡀⠀⠀⠈⠿⢿⣿⠀⢸⣿⠿⠇⠈⠁⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⠻⣿⣿⣷⣄⣀⣀⣿⡇⣀⣀⣿⣷⣿⣿⠟⣡⠞⠁
⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢬⣙⠛⠿⠿⣿⣿⣿⣿⠿⠿⠛⣉⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠶⠶⠶⠖⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⢐⣤⣤⠀⠀⠀⠀⠀⠠⠀⣠⠶⠆⣤⠠⢶⠦⣶⠦⣴⠶⣠⡦⣄⢠⢠⡴⠆⢰⠒⠒⠒⠒⠒⠀⠀⢤⠤⡦⠰
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣾⣷⡴⠆⠀⠀⢀⠀⠀⠀⣿⠀⢰⢏⡆⢸⠀⣸⠁⣾⢀⡷⡆⡿⣶⣿⠁⡦⢸⠀⠀⠀⠀⡀⠀⠤⣨⡦⣦⣼
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢙⣿⣏⡀⠀⠀⠈⠀⠀⠀⠹⠤⠾⠛⠷⠾⠀⠷⠤⠾⠙⠶⠧⠇⠻⠹⠦⠇⢈⠀⠐⠾⣿⣧⢀⣶⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣷⣶⣾⣿⣽⣽⣿⣷⢶⡄⠀⠀⠀⢲⡴⡆⢲⠆⢰⠂⣴⡆⡆⡆⣶⣴⢴⢢⡰⣦⠀⠀⠀⠈⠀⠈⣥⣾⣿⢿⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠁⠛⠛⣿⣿⡀⡀⠀⠀⠀⢠⠀⠀⣬⠠⣌⣉⡤⠩⢠⡅⢤⣤⠄⣭⢠⡉⠀⠀⠀⠀⣀⣀⣾⣿⣃⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣟⡁⠄⠀⠀⣛⣻⣇⣤⣤⣤⠀⢈⠀⠁⢈⠁⡈⢋⢙⡘⠛⢓⠋⠁⡀⠉⠀⠉⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣰⣾⣿⣿⣿⣿⠉⠛⠿⠀⠸⠀⠂⠸⠈⠛⠻⠸⠙⠎⠸⠀⠀⠇⠀⠈⠓⠀⠀⠀⣼⣿⣿⣿⣿⣻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠚⠓⠒⠒⠂⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠐⠒⠒⠛⠒⠛⠛⠛⠛⠛⠋⠉⠋⠙⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⠟⣙⡏⢹⠟⡟⢫⠙⡏⢩⡙⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣰⢯⢁⣾⡇⡼⣷⢏⡞⣩⢇⡾⣰⢃⡾⠍⢸⠏⠁⢏
⠀⠀⠀⠀⠀⠀⠀⢸⣦⣭⣧⣿⣆⣷⡤⣸⣧⣼⣶⣿⢿⣿⡿⢻⡿⣿⡇⠀⠀⠀⣿⣿⣿⣿⠛⠓⠿⢼⣿⣿⣼⣿⣧⣾⣷⣦⣿⣤⣿⣾
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠷⢙⣿⣸⡇⣿⠰⠿⡿⡎⡷⢿⡇⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠍⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠉⠉⠉⠒⠡⠀⠉⠩⠈⠙⠉⢁⠁⠃⢘⡇⠀⠀⠀⣿⣿⡿⠀⠖⠀⠀⠀⣴⣦⣀⡀⠀⠀⠀⠈⠉⠛⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⣀⠃⠀⢠⠁⠀⠀⠀⡇⠀⠀⠀⣿⣿⠁⠀⠀⠠⠄⢉⢻⣿⣼⢟⠀⠀⠀⠴⠢⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⣀⡀⡀⢄⠠⢄⡀⠱⠀⠀⢀⠀⡇⠀⠀⠀⣿⣇⣀⠀⠀⠀⠀⢀⡈⠉⠑⠚⠀⢀⡀⣚⠁⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠄⠈⠉⠉⣍⡀⠐⢉⠁⠀⣤⣴⣿⣧⡄⠀⠀⠀⣿⣿⣿⣷⡶⣤⣤⣾⡁⠀⠀⠀⣤⣾⡿⠻⠇⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠐⠀⠂⠐⠀⠀⠀⠠⣾⣷⣴⣿⣿⣿⣿⡇⠀⠀⠀⣿⡀⡀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠛⠻⢿
⠀⠀⠀⠀⠀⠀⠀⢻⣇⡓⣀⡀⠈⠀⠀⠉⠀⣀⣀⣈⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣯⣯⣧⣤⣭⣌⣀⣀⣀⣈⣀⣀⣀⣀⣄⣀⣁⣈⣨⣌
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣏⣉⠉⣍⠉⢉⢩⡥⣉⣭⠉⠉⠉⠉⠉⠉⣉⢹⡇⠀⠀⠀⣿⣿⡿⠛⠛⠛⠛⠛⠛⢏⠻⠛⠐⣤⣤⣦⣦⣔⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠛⠓⠀⠊⠁⠀⠀⠀⠀⠠⠂⠶⠊⠉⠛⢰⡇⠀⠀⠀⡿⠋⠰⠲⣶⣆⣶⠆⢢⣾⠺⠂⠀⣛⣛⠋⠃⠋⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢺⣿⣷⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣷⣦⡈⠀⠁⠈⠠⠀⢀⣘⣃⡀⢉⠉⢈⣡⡴⠶⣾
⠀⠀⠀⠀⠀⠀⠀⢸⣷⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⡇⠀⠀⠀⣿⣿⣿⣷⣶⣿⡧⠄⠘⣿⡟⠰⠀⢤⠀⠀⡀⠂⢹
⠀⠀⠀⠀⠀⠀⠀⢸⡿⡿⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⣿⡿⠋⠉⠀⠀⠀⠀⢻⣇⣙⣉⡉⣙⣤⣭⠉⣾
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⣿⣇⠀⠀⠁⡅⠀⢀⣀⣿⣦⣀⣤⣬⣤⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣄⠀⠀⡠⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⣿⣿⡷⠀⢀⠀⠀⢾⣿⣿⣿⣿⣛⡙⠻⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠰⠦⢀⡐⠀⠀⢸⡇⠀⠀⠀⣿⣿⣿⣤⣤⣿⣷⡄⣰⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠐⠂⠀⠀⠀⠂⠀⠂⠂⠁⠂⠃⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠃⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⠤⠀⡀⠀⠀⠀⠀⠤⢄⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⡀⠒⠐⢂⡉⠉⢄⠉⠐⡆⠤⣀
⠀⠀⠀⠀⠄⣄⣠⣀⣀⣀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠈⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢅⡀⠀⣤⠀⠘⣖⠀⣸⣆⡀⣸⠄⢻⠲⡀
⠀⠀⠀⢁⠚⢸⡃⡩⡆⠀⡗⠁⢸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⢰⠎⠉⡖⠒⠀⠀⠀⠀⠀⠀⠀⠀⡰⣄⡀⠈⠳⡂⠸⠒⠈⠉⠐⠀⠈⠁⠈⠑⢇⣰⠅
⠀⠀⠀⣖⣂⡯⠀⢴⡁⣨⣧⠀⢸⠀⠀⠀⠀⠀⠀⠀⢲⡀⢂⡸⠀⢘⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠅⠀⠠⠄⠀⠀⠀⠀⠐⠂⢈⠈⠋
⠀⠀⠀⢠⡠⠕⢞⡷⣿⡿⢾⡀⢨⠀⠀⠀⠀⠀⡆⢃⠡⠴⠀⠘⠾⠇⠀⠉⠛⠢⣄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠐⠂⠀⠀⠐⠀⠀⠄⠆
⠀⠀⢠⠈⠲⠆⠘⠉⠻⠛⠉⠠⢀⠠⠀⠀⠀⣜⠩⠀⠾⣀⣀⡀⠀⠀⠀⠎⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡁⢀⠀⠀⠀⠉⣿⠀⠀⡌⠱⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠇⠈⢰⡯⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠂⠘⠉⢁⢈⠁⢈⢐⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⢄⣹⢤⠛⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⠜⠁⡌⡄⠘⠈⠢⠈⢸⣄⢀⡸⠀⠀⣸⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠙⡅⠀⠀⡴⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢦⠀⢇⠀⡨⠤⠄⠉⠀⠀⡘⠉⠙⢀⠙⠷⠺⢍⡔⠠⢤⡀
⠀⠀⠀⠠⡎⠀⠰⠃⠀⡠⠁⠀⡴⠁⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠘⠉⠉⢃⠐⠀⠀⠡⠂⠀⠴⠂⠈⠀⠐⠀⠀⠀⠙⢦⣀⡴⠆
⠀⠀⣿⠀⠉⡒⠚⠁⠒⠓⠄⣠⠃⠀⡞⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⣾⡀⢠⠄⡘⢤⣶⢀⡄⢹⢀⠜⢁⡼⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠙⡀⠘⠀⡹⠂⢠⠚⠃⠘⡧⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠁⠐⢤⢱⠈⠰⡤⠠⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡄⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⣘⠁⠀⠀⢸⡇⠀⠐⠉⠂⠖⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣌⠁⠀⠀⠀⠀⠀⠀⠐⢄⣀⡀⠀⡇⠀⠀⠀⢸⠤⠬⠿⠻⠃⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠗⠛⠛⠛⠂⠹⢻⠀⠀⠀⢸⠙⠈⠳⠚⢃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⢺⠤⢮⠹⢟⣂⣀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡠⠀⣄⠀⠨⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠁⠀⠀⠀⣴⠩⠗⣈⡑⢦⠈⢂⡬⡀⡎⢹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⠂⠈⠱⡀⠀⠀⠀⠀⠀⠅⠈⠚⠤⠝⠘⠏⠹⠀⠎⠐⠑
⠀⠀⢀⢀⠀⣁⣉⠉⠙⣏⣩⡟⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⣄⡄⠀⡀⢠⡄⠀⣴⢀⢠⡄⡀⢀⡖⠲⢰⡆⠢
⠀⠀⢸⠀⠀⠉⢠⣷⣋⠉⠁⣀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⡹⣿⠞⣋⡉⢶⠄⢉⣴⠒⣁⣤⠉⠀⠄⣊⢨⠟
⠀⠀⠀⠀⠀⠰⢸⡥⠐⠙⠿⠍⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡍⠄⠘⠆⠀⠀⡜⠀⠀⢰⠈⠘⡆⢵⢀⢰⠃⠛
⠀⢀⠀⠀⠠⠀⠉⠀⠈⠀⠀⠀⠀⠰⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠄⠱⠀⡴⠀⠀⠀⠃⠀⣠⠃⠀⠐⡀⠀⢟⠃⢸⠂
⠀⠀⠂⠄⠀⠀⠀⠄⠀⠀⠀⠔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠃⠰⠁⡴⠁⠀⡔⠁⣠⠀⠁⣀⠊⣼⡅⡠⠴⠇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⡇⢸⡇⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣷⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⡇⢸⡇⠀⠀⠀⣠⣶⣿⣿⠿⠛⣉⣉⣉⣉⣉⡛⠻⢿⣿⣷⣄⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⡇⢸⡇⠀⢀⣼⣿⡿⢋⣤⣶⠿⠛⠋⠉⠙⠛⠻⢷⣦⡉⠻⣿⣷⡄⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣰⣷⣾⠇⢀⣾⣿⠏⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠙⣿⣿⡄⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⠟⠋⠀⣼⣿⡟⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠸⣿⣿⠀⠀⠿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⣿⣿⠇⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⡄⠀⠀⠸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⣿⣿⡇⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⠃⠀⠀⢰⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢹⣿⣷⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⣸⣿⡟⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢻⣿⣧⡈⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣴⣿⡿⠁⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠹⣿⣿⣦⣉⠻⢷⣶⣦⣤⣤⣶⡾⠟⢋⣠⣾⣿⠟⠁⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣶⣤⣤⣤⣤⣤⣴⣾⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀
⠀⠀⠀⠀⢠⣖⣨⣭⣷⢶⣶⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠉⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡆⠀⠀⢀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⡯⠛⠿⠨⠀⠒⠂⢻⡆⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⢬⡓⢄⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠈⠠⣇⠀⣰⠁⠘⡇⠀⢸⢷⡻⡜⠀⠀⠀⠘⡋⠙⠋
⠀⠀⠀⢫⡃⢠⠤⠤⠐⠒⡘⣻⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⡆⠀⠀⡴⠁⠈⢢⡱⡀⠀⠀⢸⠀⣿⠀⠹⣄⢀⠗⠀⠸⡙⢙⡧⠀⠀⠀⢀⠇⠀⡷
⠀⠀⠀⠸⡧⠐⠒⠂⣉⣉⠤⢋⡇⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⣀⠜⠀⢀⠎⢀⢆⢆⠀⠑⣵⡀⠀⠈⡀⠻⡄⠀⠸⢹⠀⠀⠀⡇⡏⠀⠀⠀⠀⡸⠀⠀⢱⡀
⠀⠀⠀⠀⢧⢀⣉⠡⠤⠐⣒⠨⢷⠀⠀⠀⢹⠉⡏⢹⠉⢏⢹⠀⠀⢠⠃⠀⠀⠈⠀⠀⠀⠈⢷⠀⠀⠓⡞⠃⠀⡄⢸⠀⠀⢰⢀⠀⠀⠀⠀⡞⠁⠀⠀⠘⡟⡆
⠀⠀⠀⠀⠸⣑⣒⣊⡩⠥⠄⠒⠉⠀⠀⠀⢸⡦⠤⠤⠶⠦⢸⡀⠀⠘⠚⠛⠋⠛⠛⠛⠛⠓⠊⠀⠀⠀⠡⠾⠀⠰⠞⠀⠀⠘⠚⠀⠀⠀⢀⠃⠒⠒⠒⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡢⠀⠀⢸⠀⠐⡷⠣
⠀⠀⠀⠀⠀⣀⠤⠒⢒⡒⠂⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣦⣀⠀⠀⠀⠀⠀⠀⠈⡩⠹⢭⠯⠭⠍⠩⠭⠭⣽⠭⢍⠉⠉⠁⠀⠀⢸⠆⠤⠤⠤⠤⠠⠆
⠀⠀⠀⢀⠞⣡⠖⠉⠁⠀⠉⠲⣌⢢⠀⠀⠀⠀⠀⡠⢖⠉⡠⠀⠀⠀⠉⠲⡄⠀⠀⠀⠀⡇⡈⠉⢡⡶⠛⠛⠷⣌⠉⢹⢸⠀⠀⠀⠀⠀⠈⢄⣀⣀⡀⣀⣠⠇
⠀⠀⠀⡎⡜⡡⠒⣦⠀⣀⢀⠀⠈⡆⢇⠀⠀⠀⡘⠕⠁⢰⠁⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⡇⠀⠀⣟⣷⣶⢠⣠⣿⠀⢸⢸⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⡇⢷⠁⠀⢼⠀⠏⣾⢸⠇⡄⠸⠀⢀⣀⣃⣀⣀⣈⣀⣠⣀⣀⣠⣠⣤⣸⣀⡀⠀⡇⠀⠀⠘⢿⣥⣬⣿⠋⠀⢸⢸⠀⢀⢃⡦⡄⣔⣄⣴⠄⣴⡢⢦⡿⡀
⠀⠀⠀⠡⡘⡳⡴⠊⠀⢯⡉⣫⡾⢡⠃⠀⠹⠖⠶⠶⠖⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠋⠀⢇⠰⠤⠄⢘⢁⠀⠙⠀⠤⠼⢸⠀⢸⣈⣑⣓⣛⣂⣁⣈⣘⡐⣫⠧⠇
⠀⠀⠀⠀⢙⠃⡗⠢⠤⠤⡅⡁⠔⠁⡠⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢄⡀⠀⠀⠀⣀⡀⠠⠤⠤⠀⡀
⠀⠀⠀⠀⢼⣰⠈⠉⠁⠀⠀⡇⠀⢸⣍⣐⣒⢾⠀⠀⢀⠔⢚⠋⡉⡈⡉⠡⣲⠄⣵⠀⠀⠘⠃⠒⠒⠒⠒⠒⢛⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠓⠀⠀⡇⠇⠇⡍⡏⠀⠀⢨⠀⡎⢸⠠⡇⠣⠀⢀⡴⣫⠀⠀⠀⣷⡛⠋⢩⣿⣟⡏⠀⠀⠘⢛⠗⠿⠛⠿⠿⠟⠟⠟⠻⠟⠯⡛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣄⣀⣰⠁⠀⠀⠸⡗⡤⢤⠤⡥⢴⣎⡿⠔⠁⠀⠀⠀⢼⢀⣤⡽⠟⢻⡃⠀⠀⢸⡎⠀⡞⢹⣶⣿⠖⡯⢰⡷⣼⠀⡇⡇
⠀⠀⠀⢠⠖⠒⠒⠂⠒⢦⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠈⠉⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣏⡙⣋⣀⣼⠇⠀⠀⢸⠘⠦⠝⠋⠬⠡⠄⠙⠈⠡⠉⠴⠃⡇
⠀⠀⠀⡈⢙⡒⢒⢒⣂⠹⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⡀⡀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠉⠉⠉⠉⠉⡀⠀⢀⡈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⡆⡿⢒⠒⢒⠉⡇⡀⠀⠀⠈⢹⢠⠤⠤⠤⠤⠴⠥⠬⠭⠭⠤⠧⠤⠤⠤⠤⢤⢸⠀⠀⠀⠀⡔⠒⢊⣈⣊⣁⣙⡒⠲⠔⢆⣠⡀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⢣⠣⡁⠂⢀⡴⡡⠁⠀⠀⠀⢸⠈⠋⠙⠙⠛⠙⠙⢛⠋⢻⠉⠋⠋⠛⠛⠉⠋⢸⠀⠀⠀⠀⡱⣊⠉⡇⢠⠀⡄⡌⢙⠲⢄⣰⠗⡂⠀⡴⠥⠤⠨⢧
⠀⠀⠀⠀⠑⠢⣭⡅⠊⠀⠀⠀⠀⠀⢨⡭⣭⣍⢹⠀⠀⠀⢸⠀⢸⠀⠀⠀⠘⢩⠉⠍⢩⠀⠀⠀⠀⣇⠀⠀⡇⢸⠀⡇⡇⠸⢀⡤⢧⡀⡅⠀⡓⡖⡖⠒⢺
⠀⠀⠀⠀⠀⠀⣞⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⢸⠄⠀⠀⢸⠀⢸⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠓⣕⣲⠧⠤⠤⠤⡔⢊⠽⠀⠀⠙⠂⠀⡆⠃⠃⠘⢸
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠰⠜⠀⠀⠸⠸⠂⠀⠀⠸⠤⠞⠀⠀⣀⠒⠚⠀⠀⠘⠒⠀⠀⠀⠀⠈⠀⠉⠉⠙⠉⠀⠁⠀⠀⠀⠀⠀⠀⠙⠶⠶⠦⠞
⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠆⠀⠀⠀⠀⣀⣀⠤⠄⣀⡤⢴⡡⠗⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣀⡤⠄⣒⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠆⠇⠱⠻⠉⣉⠹⠏⠁⠀⠀⠀⠀⠀⢀⠔⢩⢴⠀⠀⡔⠉⠠⠀⡵⡀⠀⠀⠀⠀⠀⠀⠀⠈⢽⣿⣅⡼⠉⠉⠀⠀⠀⠀⠀⠀⠀⡤⢶⡛⣳⠤⣄
⠀⠀⡖⠒⠚⠓⠛⠓⠒⠚⠃⢺⠀⠀⠀⠀⢠⠃⢰⠃⣎⡀⠀⠣⣀⣠⠞⡀⣷⠀⠀⠀⠀⠀⠀⣀⣀⣘⣿⣉⣀⣀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⣉⠥⠴⠚⣃
⠀⠀⠱⡜⢄⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠸⣄⠈⢆⡈⣻⠀⠀⢀⠤⢊⡴⠃⠀⠀⠀⠀⢰⢫⣾⣭⡷⣽⣶⣭⣟⢗⡄⠀⠀⠀⠀⠀⡟⠊⣩⢤⡴⠛⠊⡸
⠀⠀⠰⣎⡒⠥⠤⠤⠤⢖⣛⠆⠀⠀⠀⠫⢄⣉⣀⣈⣉⣉⣀⣉⣀⣀⣉⣀⠼⠃⠐⢆⠒⠛⠛⠛⠙⠛⠛⠛⠙⠛⠛⠛⠒⣲⠆⠀⠘⣇⣉⡥⠤⠖⠂⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠉⠉⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠁⠠⣶⣶⠀⡶⠿⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠁⠈⠉⠻⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠿⠇⠀⠈⠀⠀⢹⡄⢸⡆⠀⠀⠀⠀⠀⠀⢀⠞⠁⠄⠀⠀⠀⠀⠀⠀⠙⢿⠿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⢠⡤⠀⡄⢸⠃⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠎⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⡀⠀⠈⠉⢀⡿⠀⠀⠀⢠⡴⣶⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡗⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢿⣀⠉⠳⣶⣶⣾⣅⠀⠀⠀⠈⠛⠲⠾⢭⣽⣿⣿⣿⣿⣷⣖⣒⣋⣉⣩⣭⣭⣥⠴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⡷⡀⠈⣙⡂⣀⣙⣵⡟⢿⣷⣶⣄⡀⠀⠀⠀⠀⠉⣿⣿⣉⣩⣭⢟⣽⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢿⣿⣿⣿⣿⣿⣿⡌⠀⣿⣯⠿⠿⠿⠿⢸⣿⣿⣿⠛⢷⡄⠀⠀⠠⡇⠨⠛⠋⣡⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⡋⢻⣿⣿⣿⣿⣿⡄⠈⠁⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠹⣦⡀⢠⡇⠀⣴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⡄⠀⠈⢻⣼⡍⠛⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠠⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⣿⣿⣿⣿⣿⡆⠀⠀⠀⢹⣷⠀⠀⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡆⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣸⣿⣿⣿⣿⣿⣇⠄⠀⠀⢸⡇⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣮⢻⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⢿⣮⢠⠀⣿⣇⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣟⣦⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠙⢿⣸⠈⢿⡌⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠀⠀⢠⣤⣿⡿⣿⣿⡿⣿⠿⠿⠿⣿⣿⣿⣿⡿⠿⠻⠿⣿⣻⡇⠀⠈⠻⢶⣌⠑⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⢹⣇⢠⡀⠀⠀⠀⠀⠀⠈⡟⠉⠀⠀⢐⡄⠠⠤⠬⣍⣷⠀⠀⠀⠙⠳⣤⣄⣀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⣿⢸⢁⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⡼⠀⣈⠒⣒⡶⠋⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⣞⣰⣡⢂⠀⣿⠘⡟⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⣷⣿⠀⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀⣀⣀⣤⣤⣤⣤⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠿⢟⣛⣿⣿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠁

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

⠼⢀⡀⣀⡄⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢘⢀⠰⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠐⠡⠔⢈⠠⠀⠂⠁
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠔⠑⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀
⣉⣷⠀⠀⠀⠀⠀⠀⠀⡀⠄⠰⢄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⡠⢀⡉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡖⠂⢁⠀⠀⠀⢀⡠⠲⠀⠅⡁
⠁⢸⠀⠀⠀⢀⠄⠂⠉⠁⠢⢄⠀⠀⢂⡐⡿⠇⠀⠀⠀⠀⢡⠊⠀⢅⡀⠈⢂⠐⠆⠀⠁⠀⠀⠀⠈⠀⠂⠀⠀⠈⡄⠀⠀⡫⡅⢑⣘⣽⠽
⢀⠌⠀⠀⠀⠁⠀⠀⠀⠀⢀⣄⡄⠀⢈⠻⠆⠀⠀⠀⠀⠀⠘⣀⠼⠤⢼⠀⠈⠀⡃⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⢴⠤⠤⢔⠛⠒⢁⠠⠆⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠌⠒⠤⣀⣠⠞⣠⠑⠔⠁⠀⠀⠀⠠⢀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠋⠁⠀⠀⠀⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⢈⠐⠂⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠴⠈⡀⠀⠀⠀⡉⠓⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡀⠐⠈⠀⠠⢉⠁⠀⠀⠀⠀⠀⠀⠀⡀⠐⠈⠀⠈
⠀⡡⣠⠔⠨⠀⠀⠀⠀⠀⠀⠔⠀⠄⠀⠀⢀⠠⠀⠰⢶⠤⢴⠀⠀⠀⠀⢀⣔⠄⠂⠈⠀⠀⣀⠔⠃⠈⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠈⢢
⠠⠂⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢀⠀⠂⢀⠀⠄⠒⠁⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠒⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡐⠀⢀⡀⠀⠄⠒⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠⠄⠊⠃⠒⠈⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠠⠐⠁⠀⠀⠀⠀⠀⠤⠄⠐⠉⠁⠀⠀⠂⠁
⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⠢⣀⠨⠳⠍⠻⠶⣤⠀⠈⠀⠀⠀⢀⣠⣤⣄⠤⠤⠄⠀⠄⣤⣄⠀⠀⠀⠀⠀⠀⢀
⠠⠀⠀⠀⠀⢒⣀⠴⠀⠀⠀⠀⠀⠀⠠⠆⡁⠀⠀⠀⠈⢸⠄⠐⠃⠀⠒⠚⠀⠀⠀⠀⠀⠀⡀⠀⠈⠁⠀⠉⠁⠀⠀⢠⠀⠀⠀⠀⠀⠐⠁
⠐⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠈⠠⢀⣇⡠⠀⠀⠀⠀⢀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣀⠊⠐
⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠡⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⣀⠉⠛⠶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠦⠀⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⢀⡀
⠄⠀⠀⠀⢀⡠⠒⠀⠀⠀⠀⢉⠖⡂⠈⣍⠉⠙⠒⠀⠀⠀⠀⢀⠔⠈⠀⠀⠠⡀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⢀⡠⠐⠁
⠦⠂⠀⠀⠣⠀⠑⠂⠀⡀⠂⠀⠰⠋⢀⠄⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠅⠂⠀⣒⠖⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠂⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⢤⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠠⣀⠀⣀⠀⠀⡁
⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠁
⠢⡈⠙⡲⡦⣄⣀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⢀⡀⠄⠊⠀⠀⠀⣀⡠⠬⣒⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢊
⡄⢘⡄⠥⠑⠖⠓⠓⠀⠀⠀⠀⠛⠛⠛⠛⠋⠁⠀⠀⢀⠀⠀⠀⠀⠀⡂⠁⠀⠀⠀⣀⠄⢈⠠⠀⠀⢀⠟⠛⠛⠀⠀⠀⠀⠀⠀⡐⠃⢄
⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠈⠳⠦⡄⣈⠀⠂⠀⠄⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠃⠀⡀⠱
⠁⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠈⠀⠂⠈⠁⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠈⠂⠐⠉⠐⠐
⡠⠐⢀⡤⠤⠒⠒⠒⢛⢶⣦⡶⠚⠳⠀⠀⠀⢀⠠⠐⠢⡀⠀⠀⢄⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠚⠀⠀⠀⠀⢁⡂⠤⢄⡀
⠀⡄⠉⠀⠀⢀⠔⠁⠀⢀⠇⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠑⡀⠀⠇⠠⡤⠵⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⡠⠂⢉⠀⠄⢁⡤⠤⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣿⢹⣰⢲⡔⡄⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⠤⠤⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡎⡼⠇⡆⠇⣾⠀⠇⠀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⣄⠀⠀⢠⠃⠀⢀⣀⣀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⡸⢠⢹⢸⠱⢸⢻⢸⡤⠊⠁⠀⠀⠀⢀⣀⡠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣸⠀⢠⠋⠀⠀⠙⣄⠈⡆
⠀⠀⠀⠀⠀⠀⢠⠃⣇⡆⣜⡇⡜⡆⡈⠀⠀⠀⣠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢸⠀⠀⠀⠀⠘⡀⡇
⠀⠀⠀⠀⠀⠀⠸⡀⠉⠒⠊⠤⠝⠃⡇⠀⡰⠊⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⣆⠀⠀⠀⢀⢣⠃
⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⢀⡠⠎⢀⠞⠀⠀⠀⠀⢀⣤⣾⣭⣶⣶⣶⣶⣶⣮⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣦⣴⠲⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⣿⠉⠀⢀⣏⡠⠤⠒⢖⠓⠉⠉⠉⠉⠉⠙⠛⠛⠿⢿⣀⠀⠈⢆⠀⣀⠤⠒⠒⠓⢚⠖⢣⠸⣶⣇
⠀⠀⠀⠀⠀⠀⠀⢀⠏⢻⡇⢀⡴⠊⠁⡠⢤⠀⠀⢧⠀⠰⡀⢡⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠟⠁⠀⠀⠀⠀⢸⡀⠘⠆⡟⡘⡀
⠀⠀⠀⠀⠀⠀⠀⡸⡘⣼⠃⠎⠀⠀⠀⠻⠟⠁⠀⠘⡀⠀⠇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢲⠀⠀⠀⠀⡇⠀⡇⢡⢃⢣
⠀⠀⠀⠀⠀⠀⢀⢃⠃⡏⡆⢿⠓⠀⠀⠀⠀⠀⠀⢠⠁⠀⢀⣴⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⡇⠀⡇⢸⠘⡜⡄
⠀⠀⠀⠀⠀⠀⡌⣸⢰⠀⢇⠈⠳⢄⡀⠀⠀⠀⠀⡼⠀⠀⠸⣿⣿⣿⣿⣿⣧⠀⠀⣠⠔⢳⣄⠈⠁⠀⠀⠀⢰⠃⢸⠃⠀⡆⣇⢱
⠀⠀⠀⠀⠀⢰⠁⡇⡌⠀⠸⡄⠀⠀⠉⠒⠲⠤⠼⣁⣀⣀⣀⣈⣛⣿⠿⣿⣿⠖⠉⢀⡴⠃⠈⠓⠤⣀⣀⠀⠘⣄⡎⠀⠀⢃⢸⠈⡇
⠀⠀⠀⠀⠀⡎⢸⢀⠇⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢄⣀⣀⣀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁⡜⠀⠀⠀⠸⠀⡇⢱
⠀⠀⠀⠀⠸⠀⡇⢸⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⡇⢸⠀⢇
⠀⠀⠀⠀⠇⠸⠁⡞⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⢣⠈⠇⡸
⠀⠀⠀⠀⠈⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠦⢄⣀⡀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠋⠁⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⢀⣀⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣙⢦⣀
⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⣤⣶⣶⣾⠯⠟⠓⠒⠒⠒⣂⡈⢳
⠀⢸⢇⠀⠀⠀⠀⢀⡠⠔⣒⠭⠕⢒⣉⠽⣿⣿⣿⣿⣍⣉⠩⠭⠭⠤⠤⠌⡷⣄
⠀⠈⢎⢣⡀⠀⠀⣿⡈⠩⠴⠒⣉⣩⠤⠒⢊⣻⣿⣿⣿⡶⠒⠒⢒⡒⣒⠊⢹⠈⠣⡀
⠀⠀⠈⢣⡱⡄⠀⢰⠈⠓⠢⠬⠥⠒⠒⢩⡍⢸⣿⣿⠿⣇⠀⢈⠥⠭⣍⢁⣼⣄⡀⠈⠢⡀
⠀⠀⠀⠀⠱⡜⢆⠸⡀⠐⠒⠒⠀⠀⠀⢀⣠⡚⢻⠀⠀⠀⣟⣇⠀⠀⢈⡇⠀⠀⢸⠉⠙⢿⢦
⠀⠀⠀⠀⠀⠘⢎⢣⣧⡀⠀⠀⠤⠆⢰⠁⢠⠝⠚⠢⣤⡞⠉⠈⣷⠊⠉⠳⣶⠖⠋⠳⣄⡸⠉⡗⢄
⠀⠀⠀⠀⠀⠀⠈⢢⠱⡉⠓⠒⠒⠚⣿⣦⣧⣀⠀⠀⣸⢳⡤⠤⠻⣀⠀⣠⢧⣀⣀⣴⣿⣤⣶⡇⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡙⣄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣌⢦⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⢛⡿⣟⠓⠒⠒⠤⢤⣈⢦⣤⣤⡤⢒⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠳⡀⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⢡⠖⢉⡥⣀⠉⠲⠼⠃⢸⠀⠀⠀⠀⠀⠈⣗⣉⠶⣍⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡑⣄⠉⠛⠻⠿⣻⠟⢋⢄⠀⠀⢸⢀⠋⠀⠀⠉⠑⢒⣒⣁⠀⠀⠀⣀⣀⣀⢟⠛⠛⠋⢟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡜⢆⠀⡴⠋⠀⠀⠸⣌⠲⠤⠞⣸⠀⢀⡠⠔⠊⠉⢀⠀⠀⠀⠀⠀⢀⠀⠘⡄⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⢣⣧⠀⠀⠀⠀⠈⠉⣛⡩⠕⠚⠉⠀⠰⠒⠒⠉⠁⠀⠀⠀⢠⠋⠀⢰⠁⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠹⡗⠤⠤⠔⢒⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣠⠔⠁⠀⠀⠀⢀⣠⠤⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡘⢾⡄⠀⠈⠙⠓⠀⠀⠀⠀⣀⣀⣤⠤⠤⠖⠒⠉⠉⠀⠀⠀⣀⡤⠖⠊⠁⣀⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣌⢫⡳⠤⠴⠖⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⢀⡠⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠁⣀⠤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡘⢆⠀⠀⠀⣀⡠⠔⠚⠉⢀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⢳⠒⠉⠁⣀⡠⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠤⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢠⣿⣹⣆⣀⢀⣰⣦⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⡆
⠀⠀⢸⣿⢻⣏⣿⣾⣽⣷⡟⣿⠀⠀⠀⢀⡤⠊⠀⠀⢀⠀⠀⠀⠴⡖⢀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠈⠛⠈⠋⠉⠙⠋⠋⠋⠉⠀⠀⠀⡎⠀⠀⠐⡀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠘⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢊⣁⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠂⠤⠀⠀⠀⠀⠏⠀⠀⠀⡆
⠀⠀⠀⠀⢀⣀⠐⠐⠪⠿⢮⡄⠀⡶⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠤⠬⠌⠧⠄⠀⠀⠆⠘⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠤⡁⠀⠐⠢⠀⠈⠉⠁⠀⠀⡀⠆⠋⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠛⠘⠣⡠⠒⠊⠀⠀⠀⡰⠀⠀⢀⡤⠒⠚⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠄⡀⠀⠀⠀⠀⠀⠙⠀⠀⣀⣠⠊⠀⠀⢰⡇⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⡲⠀⠑⠆⡺⠷⣋⣀⣐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⡇⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠺⠿⠂⢰⡀⠠⠊⢄⡈⠈⠉⠤⠜⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⣤⠈⠋⠀⢂⠄⠠⡄⠀⢐⡋⠀⢠⠇⠁⠀⠀⠀⠀⠀⠀⠀⡼⡇
⠀⠀⠀⠀⠀⠀⠈⠥⣡⡘⡒⠤⢐⠖⠈⡕⡱⠶⠄⠉⣥⡄⠂⠀⢀⠌⠀⠀⠀⠀⠀⡇⣧⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠂⠚⠁⠒⠾⠴⠦⣶⣶⠍⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡇⣿⠀⢸⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠋

⣿⣟⣛⣛⣛⣻⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣀⣄⣤⠤⢠⢤⡤⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠠⠤⢤⣀⡀⠀⠀⠀⢀⡀⡄⠤⢠
⠀⠋⢉⣵⣶⣶⣆⠢⢄⢤⠀⠈⠀⠀⡄⢲⢦⠶⣶⣶⠶⠰⠀⠀⠚⢉⣶⣿⢿⣿⡷⣦⢉⠀⠀⠀⡠⡔⢄⢶⣦⡀
⠀⠠⡘⠿⠿⠿⠿⠗⠘⡠⠃⠀⠀⢀⢼⣦⣶⣾⣖⡷⣤⠠⠀⠀⡀⠸⠿⠿⠿⠾⠿⡿⢀⡀⠀⠀⢈⡟⠠⠊⣋⡸
⠀⠀⠈⠩⣉⡭⠥⠤⠅⠀⠀⠀⢀⠀⠳⡸⠍⠹⢉⣉⠈⠀⠀⠀⠈⠁⠖⢂⣀⣐⡀⠀⠁⠀⠅⠄⠀⠂⠭⠭⠄
⠀⠀⠀⠀⡤⠤⠤⠤⠄⠀⠀⠀⠀⠐⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡄⠤⢀⡀
⠀⢀⣀⣤⢮⣤⣦⣤⣤⡐⡀⢀⠀⣤⣤⠀⠀⢀⠤⠴⡄⢀⣸⣇⡄⠀⣀⢀⡀⢀⡴⣿⡳⣄⡀⠈⡵⣴⠀⡶⡄
⠀⡀⠻⠅⢀⠀⠈⢠⡤⡀⢠⠄⠀⠘⣋⠀⠀⢸⠐⠚⡇⢠⢿⡟⢦⡘⠵⠣⠃⠀⢼⠿⣻⠌⠁⠀⠘⠛⠙⠊
⠀⠓⠒⠒⠒⡀⠀⠒⠒⠒⠒⠒⠂⠀⠀⠁⠀⠈⠙⠉⡁⠁⠀⠁⠀⠁⠀⠀⠀⠀⠘⠁⠉⠛⠁⢀⣀⠴⡆⢋⣉⣅
⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠀⠀⠀⠀⡀⣀⣀⢀⠀⠠⠠⠃⣀⢠⢠⠰⠰⠰⠦⠦⠄⠂⠀⠀⠀⠀⠁⠐⠈⠉⠁
⠀⢠⡤⢆⢊⡩⣉⢓⠒⠄⠀⠀⠒⠋⠉⠠⠈⠈⠉⠀⠀⢀⢁⠔⠀⠰⠋⠀⠶⠆⢠⢄⠘⣀⠀⠀⠀⣀⣩⠀⠄
⠀⢀⠄⠆⠈⢉⠉⠀⡁⠀⠀⠠⢂⠒⠠⠀⠠⠈⠁⠐⠀⡎⠮⢂⠌⠀⠀⠀⠀⠀⠁⢨⢈⣿⡇⠀⢲⠙⠉⠤⠐
⠀⠘⠤⠀⣀⣀⠀⠀⠄⠊⠀⠀⠒⠠⠄⠈⠀⠐⠂⠂⠀⠘⠢⠈⢀⣂⣢⣀⠂⠈⠑⠀⠈⠉⠁⠀⠈⠠⠀⠐⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠴⣒⣋⣩⣤⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⠂⢀⡬⠅⠒⠚⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠭⣼⡪⣆⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣇⣿⣷⣞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢟⣼⡧⣿⣖⠋⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠈⠉⠧⣩⡟⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣆⣀⣰⡿⢫⡱⣦⣤⠊⠈⠓⠚⣤⣤⡰⠇⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡒⢒⡺⣿⠟⠋⠉⢟⡘⣼⠁⠁⠙⠁⠀⠀⠀⠀⣚⠓⠿⢴⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢴⣾⢷⠤⠾⣿⠂⠀⠀⠀⠀⠿⡏⠀⠀⢈⣰⠲⠄⠀⠚⣏⠀⠀⢰⡟⠚⠦⠤⣤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠋⠀⠀⠀⠀⡸⠦⣀⣀⣠⣤⠴⢛⠕⢦⣙⣙⠁⠀⠂⠀⢈⠷⣦⠊⠀⡒⠀⠀⣛⡶⠀⠝⡆
⠀⠀⠀⠀⠀⢀⡴⠊⠉⢏⠀⠀⠠⠀⣈⢰⣿⢿⢿⢜⣧⣀⡄⠚⠒⣻⣿⡶⠤⢄⡤⠃⠨⡈⠢⠤⡔⠃⠤⠙⠃⠀⡠⠃
⠀⠀⠀⠀⠀⠎⠀⡀⠀⢸⡃⠢⡐⠲⣗⣦⣴⣾⣿⣿⣯⣻⣧⣌⡚⠁⠋⠓⠄⣨⣭⣆⣠⣸⢦⡸⠉⠚⠔⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠈⠒⢭⣆⣀⠙⠒⠒⠀⠲⢶⣏⣓⣠⠭⠿⢿⣥⣭⣷⣀⣐⡦⢿⠟⠉⠒⠋⠀⠀⠀⠀⣀⣠⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢷⡒⠀⠒⠂⠉⠭⠁⠉⠺⠶⠭⠝⠋⠁⠉⠀⢁⣀⣀⡠⠤⠤⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⢤⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣤⠄
⠀⠀⠀⠀⢀⠀⠛⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⡏⢸⡆⢻⡄⠀⠀⢔⠁⠀⠀⠀⠀⠀⢀⣾⣵⣛⠶⠀⠀⠀⠀⠀⠀⣀
⠀⠐⠚⠓⠈⢇⡗⡧⠀⠠⣭⡭⣥⠆⠀⠀⠀⣷⠘⢃⠀⡿⠀⠀⣀⠦⠠⡀⠀⠀⠴⣻⠿⣿⠟⠁⠀⠀⠀⠀⢠⡿⣹⠇
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠚⠁⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⢀⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀⣄⣠⣀⠀⠀⠀⡠⠄⡐⠉⠄⠀⠀⠀⣀⣀⣠⡀
⠀⠀⠘⠄⠀⠀⠀⠀⠀⢀⠄⢠⡲⠄⠀⠀⠠⣌⠻⠈⣠⠇⠀⠀⠈⠙⠛⠁⠀⠀⠀⣡⠆⠀⠐⢦⠀⠀⠘⠛⢿⡟⠛
⠀⠀⠀⠀⠶⠀⠀⠀⠠⡄⢂⣉⡀⠀⠀⠀⢀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⢀⠶⠀⠸⢋⣀⣉⣀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠰⠖⠢⡀⠀⠀⠈⠛⠠⣤⠀⠀⠀⠀⠰⠦⠆⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠍⠙⠳⣦
⠀⠀⠀⠀⣠⣀⡀⠀⢠⡆⠀⠀⠀⠀⣶⠀⠨⠃⠀⡠⠈⠁⠀⢀⠀⠀⠀⢀⠀⠀⠐⣲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠚⠁
⠀⠀⠀⠀⠾⣟⠏⠀⠀⠁⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠀⠀⠀⠀⠀⠛⠁
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⡀⢠
⠀⠀⠴⠆⠀⠶⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠂⠀⠀⠘⣡⠀⠀⢀⠀⠆⣘⡴⠃⠀⠀⠀⠐⠀⠐⠀⠀⢰⠶⠶⠶⠶⢤
⠀⢠⠀⠀⠀⠆⠀⠀⠀⠀⠄⠀⢤⠀⠀⠀⠀⠤⠠⠤⠦⠜⠀⠐⠒⠒⠋⠀⠀⠀⠀⠂⠂⠀⠤⠀⠀⠀⠈⠂⠄⠀⣊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⠇⠀⠰⠒⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠂⠀⠀⠐⠂⢸⠀⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡄⠁⠀⠀⠀⢤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠂⠀⠄⠂⠀⠐⢯⠾⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠻⣼⣧⡽
⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠐⠀⠀⠀⣴⡶⣞⢶⣆⠀⠀⠀⢀⣀⠀⠀⠀⠀⠶⠆⠀⠠⠶⠀⠀⢀⡤⢈⢛⠀⡀⠀⠀⢠⣴⣦⡄⠰⠆
⠀⠙⢶⣶⣶⡶⠏⠀⠀⠙⠳⠿⠷⠛⠀⠀⣄⡈⠀⣁⡄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠠⢺⠏⢺⣅⣱⠀⠛⠳⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⠚⠉⠉
⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠰⡆
⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⢴⠆⠀⠀⠀⠘⢻⣟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠅⠆⠀⠀⠀⠀⢡⡂⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⣠⣾⡀⠀⠀⠀⠀⠀⣄⣀⣀⣸⠟
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠆⠨⠀⠘⡄⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⣬⡓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤
⠀⠀⠐⠃⡤⠾⡅⠀⠐⠓⠂⠒⠑⠀⠀⠀⣶⣽⣷⠶⡀⠀⠀⠀⢀⡆⢀⡷⠀⠀⢀⣤⠀⠀⠁⠀⠀⠀⠰⠖⠛⠑⠛
⠀⠀⠀⠀⠈⠀⠁⠀⠀⠠⣤⡄⠀⠀⠀⠒⠊⠐⠃⠙⠃⠀⠀⠠⠁⠁⠽⠀⠀⠀⠀⠉⢴⠇⠙⠂⠀⠀⢃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠛⠛
⠀⠀⠦⠀⢠⠀⠀⠀⠀⠀⠉⢡⡀⠀⠀⠀⣠⣤⡶⠶⠀⡀⠀⡀⠀⢀⠰⠆⠀⠀⠀⠐⠲⠖⠠⠄⠀⠀⢘⢃⠀⠠⣞⢳⡀
⠀⠀⠸⠿⠛⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠾⠉⠁⠀⠀⠀⠀⠀⠀⠐⢺⠗⠀⠀⠀⣤⣄⣠⣧⠖⠈⠀⠀⠀⡠⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣟⠏⠛⣿⠉⡟⠛⣟⠟⣻⡿⣛⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⡠⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣁⡤⠄⠀⢀⣤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⡠⠊⢉⢠⡄⣀⠔⠁⠀⠙⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣵⡶⠿⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡡⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⢠⣤⢴⡾⣿⠶⠚⠛⠶⣶⣤⣄⣀⣤⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢁⡴⢋⠔⠁⠀⣰⠤⠿⠛⠛⠛⠛⠛⠓⠒⠛⠛⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⡴⠟⠡⠃⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⢀⠄⠊
⠀⠀⠠⣊⠔⠁
⠀⠀⠈

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣈⠙⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣿⣿⣿⣿⣿⣿⡿⠟⠿⣿⣿⣿⣿⣄⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣹⡏⣿⢻⡸⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⡿⠋⠀⠀⠀⠈⢿⣿⣿⣿⣦⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡇⣧⢻⢸⡇⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣇⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇⢿⠸⡆⣧⢸⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⣿⣿⣿⣧⢸⠀⠇⠸⠈⣿⣿⣿⠁⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⢀⣿⣿⣿⣿⣿⡇⠀⢳
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⢠⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⢠⠃⣼⣿⣿⣿⣿⣿⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⣿⣿⣿⣿⣿⣿⣿⡌⢆⠀⠀⠀⣼⣿⣿⢸⠀⠀⠀⣿⣿⣿⣿⣦⡀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠘⣿⣿⣿⣿⣿⣿⣿⣿⣬⡀⠀⣴⣿⣿⣿⢸⠀⠀⢰⣿⣿⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⠸⠀⠀⢸⣿⣿⣿⣿⣿⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠻⣿⣿⣿⣿⣿⣿⣿⣿⡄⢻⣿⣿⣿⣶⣤⠀⢸⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠙⢿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠛⢿⣿⣿⣿⣿⣧⠀⣿⣿⣿⣿⣿⠀⠘⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠉⠻⢿⣿⣿⠀⢹⣿⣿⣿⣏⠀⠀⣿⣿⣿⡏⠀⣼⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠈⠉⠀⠈⠿⠿⢿⡇⠀⠀⠿⠿⠿⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂
⠀⣀⡠⠤⢤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠊⠁⣼⣿⣶⠈⢻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠀⢀⣀⠀⠀⠀⢀⣀⡀⠀⠀⣀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⣸⣿⣿⡟⠀⢸⣿⣿⡆⠀⣴⣿⣿⠟⠂⢠⣿⣿⠇⠈⠀⠸⡿⠃⠀⣴⣿⡟⠉⡂⠀⠿⠟⠀⣴⣿⡿⢋⡀⢀⣾⣿⡆⠀⢠⣤⡀⢸⣿⡀
⢠⣿⣿⣿⠁⠀⢸⣿⣿⠃⣼⣿⣿⠇⢠⠂⣾⣿⡟⠀⠀⣸⣷⡆⠀⣼⣿⣿⠁⠀⢁⣾⣷⡄⢰⣿⣿⠃⠈⠋⢸⣿⣿⠃⠀⣿⣿⠇⠘⣿⣿⣄
⣸⣿⣿⠇⠀⠀⣿⣿⡟⢰⣿⣿⣟⠐⠁⢀⣿⣿⡁⠀⢀⣿⣿⠇⢠⣿⣿⡇⠀⠀⢸⣿⡿⠀⣿⣿⡟⠀⢀⠀⣼⣿⡿⠀⢸⣿⡿⠀⠀⠙⣿⣿
⣿⣿⡟⠀⠀⣸⣿⡿⠁⢸⣿⣿⠋⠀⠀⢸⣿⡟⠀⠀⣸⣿⡿⠀⢸⣿⣿⠁⠀⠀⢾⣿⠇⠀⣿⣿⡇⠀⠌⠀⢿⣿⠇⠀⣿⣿⡇⠠⠀⠌⠀⠻
⢿⣿⡇⠀⣰⣿⠟⠀⠀⠻⣿⣿⡀⡀⠔⠙⢿⣧⠤⠊⠀⢿⠧⠀⠀⠻⠿⠤⠂⠀⠘⠿⠄⠀⠈⠻⠧⠈⠀⠀⠘⠛⠂⠀⠘⠛⠋⠀⠀⠢⠄⠤
⠈⠛⠓⠚⠉⠁⠀⠀⠀⠀⠉⠉⠁⠀⢤⣄⡀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⠂⠘⣿⣷⡠⢰⣤⡀⠀⠀⣴⣿⠏⠁⠀⣿⣷⠀⢹⣷⡀⢠⣿⣿⠇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⠀⢸⣿⣷⠀⣼⣿⡟⠀⠀⢸⣿⣿⠀⢸⣿⠃⢸⣿⣿⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⣼⣿⡟⠀⣼⣿⡇⠀⣿⣿⡇⠀⠀⣼⣿⡏⠀⢸⣿⠀⢸⣿⡏⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣿⣿⠇⡐⢸⣿⠇⠀⣿⣿⠃⠀⣿⣿⠀⢀⠂⢸⣿⠁⠀⢸⣿⡄⠹⣿⡇⠀⠹⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣸⣿⡟⢠⢀⣿⠟⠀⠀⢿⣿⠠⠊⠹⠿⠟⠁⠀⠀⠉⠀⠀⠘⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣶⣿⣋⣀⣀⣙⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⡴⣒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠒⠒⣦⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡄⠀⠀⠀⣴
⠀⠀⠀⡴⣾⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠘⢿⠃⠺⡶⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⢀⣴⡆
⠀⠀⣼⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠘⠳⡖⣶⣶⣶⣶⣶⣶⣶⣶⣶⠚⠃⠀⠀⠀⠀⢸⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣯⡀
⠀⠤⠭⠯⡭⠭⠭⠭⠭⠭⠥⠤⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⠀⢠⡇⠀⢸⡄⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣠⣴⣾⣿⣿⣷⣶⣶⡾⠃⠀⠀⠀⠀⠀⠀⣇⣻⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣿⠀⢸⣧⠀⣾⡇⠀⠀⠀⢠⣤⣌⠛⠛⣿⣿⣿⣿⣿⣿⣵⣶
⣿⣿⡿⠏⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣟⠂⢸⣻⠀⣟⡇⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠈⠁⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢸⡇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣂⣀⣰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⣀⠤⠤⠀⢻⢺⡆⠤⢤⡀⠀⠀⠀⠀⢰⣟⣿⣿⣿⣿⣿⣿⣂⣠⣤⠤⠶⠒⠂⠀⠘⢿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⠿⠟⠉⠉⠉⠉⠉⠉⠉⣏⠉
⠀⣿⣦⣤⣤⣀⣈⣤⣤⣴⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢤⣤⣽⣿⣿⣥⡤⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠶
⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢸⣻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠶⣶⣾⣶⣶⣶⣶⣶⠶⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡇⠙
⠀⠀⠀⠙⢿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⣭⣭⣭⣭⣭⣭⣭⡀⠀⠀⠀⠀⠘⠻⢿⠿⠿⠿⠿⠿⣿⣿⡿⠟⠃⠀⢸⣤⣤⣤⣤⣴⣶⣿⣿⣿⣯⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀
⠀⠀⠀⡴⠋⠁⡀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⡀⣴⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢹⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣿⣿⣷⣶⣤
⢠⣀⠸⠁⣠⣴⣷⣤⡄⠹⠀⠀⠀⠀⠀⠀⠀⠹⡜⣎⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠈⣿⣿⣿⣾⣶⣶⣷⣿⣶⡀⠀⠀⢀⣀⣀⣀⣀⣁⣁⣈⣀⣀⣀⣀⣤⠶⠾⠿⠛⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠉⢻⢻⣿⣿⣿⣿⡟⠁
⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠈⢏⢿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⡇
⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⣁⣐⣒⣒⣒⣒⣒⣲⡂⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⢀⠀⠀⠀⢤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠲⣶⣶⡤⢀
⠀⠀⠀⣠⠠⠀⣐⣤⠀⠈⠀⠉⠀⠀⠀⠐⠁⠔⡴⣿⡛⠦⣤⣀⠀⠀⠀⠀⠀⣼⡧⠬⠁⠠⠃⢀⣩⣷⡀⠀⠀⠀⠀⣀⠀⠌
⠀⠀⢻⡂⠙⣏⣄⣀⣦⣽⣖⣳⠂⠀⠀⠀⠐⣺⠀⣬⣽⡉⣍⠯⣠⠀⠀⠀⠀⣝⠳⢶⣿⣦⣴⡯⠾⢃⠁⠀⠀⠀⠀⠀⠐⠀⡨⠃
⠀⠀⠈⠁⠐⠃⠟⠿⠿⠿⠋⠙⠁⠀⠀⠀⠈⠉⠉⠁⠈⠁⢈⠁⠁⠀⠀⠀⠀⠙⠛⢛⠰⠆⠀⢁⠀⠉⠀⠀⠀⠰⢯⡈⠁⢀⢃⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠚⠚⠂⠊⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⠀⢀⣈⠐⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀
⠀⠀⠀⠀⣠⣤⣔⠖⡤⣤⣈⡀⢀⠀⠈⠈⡀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠰⣿⣿⣿⣾⣍⡋⢷⡀⠀⠀⠀⠀⡄
⠀⠀⣠⡖⠾⠀⡈⢠⡏⢲⢟⠉⡂⡀⠀⠀⠠⠢⣄⠀⠀⠀⣸⠀⠠⠀⠀⠀⠀⠀⠈⠏⠫⢿⣿⣥⣝⣿⣆
⠀⠀⠘⠢⠤⠀⣠⡲⡏⠱⣺⠲⠟⠀⠀⠀⠠⢠⠀⠈⠀⠈⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠀⠀⠈⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠐⠒⠊⠃
⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⡀⣄⠀⠰⡢⠐⠀⠉⢁⣀⠀⠀⠀⠠⢚⠓⠂⠀⠉⠾⠟⠂⠀⠀⠀⠀⢀⠀⠔⣈⠀⠐⠀⣀⠀⠀⢰⣏⣩⣿⠿⣿⣼⣷⡦⣄⡀
⠀⠀⠐⠘⠈⡥⠰⣀⣤⣤⠔⠀⠊⠀⠀⠀⠻⣖⠒⡐⡃⠀⢀⡀⠡⠂⠀⠀⠔⢀⢀⠀⠀⠒⢀⡀⠀⠐⡀⠀⠈⠙⠱⠤⡼⡿⡊⢢⡵⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠉⠅⠐⠉⠀⠀⠀⠈⠙⠶⢤⡀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠣⠀⠀⠀⢀⡴⠂⠉⠀⠀⠁⠀⢄⠀⢈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠣⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠁⠀⠀⠰⠀⠀⠀⠀⠸⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠻⢦⣀⣀⣀⣤⠞⠀⠀⡀⠀⢸
⠐⠄⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠈⠂⢄⠀⠀⠈⠒⠠⠄⠀⠀⠐⡩⠂⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣠⠔⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡃
⠘⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢄⠈⠐⠂⠦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠄⠀⡔⠉
⠀⠀⠀⢀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢠⠀⠇
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠠⠼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣉⢰⠟⠉⠻⡆⣴⠋⠉⢳⡀⡟⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠸⢦⣠⡼⠃⠻⣤⣠⡾⠁⣧⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⢀⡄⠀⣠⢤⡀⣀⣠⠀⣠⣤⠀⣠⣄⢀⣠⣄⠀⡄⢀⢀⡀⢀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⠆⡾⠻⡀⢧⣀⠄⡿⢧⠘⢇⣲⠆⠿⢎⠸⣄⣸⠃⢇⡼⢸⠝⣾⢸⣇⡿


⢀⠰⣉⡝⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⢁⢡⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁
⠀⠀⢨⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠠⢄⡀⠁⠂⢄⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡀⠀⠀⠀⠈
⠤⠒⠁⠀⠀⠈⠉⠑⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠉⠀⠀⠀⠨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠔⠂⠀⡀⡀⢜⠠⠀⠀⠐⠢⠤⢀⠀⠀⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠆⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢀⡴⠖⠛⠓⠶⣄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⡁⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢳⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠑⠄⢀⠀⠀⢀⡀⠔⠁⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣦⡆⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣧⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡇⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣷⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⣠⣾⣷⣄⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠋
⠀⠀⠀⠀⢠⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣮⢙⠿⢿⣿⣿⡿⠿⣫⣵⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⣛⣵⣿⣿⣿⣶⣶⣜⡻⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠋⣅⠀⢻⡆⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣯⠝⡛⢭⣿⣿⡿⠀⢀⣼⢂⡼⠃⣸⠇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣿⣿⣷⡿⠟⠁⠀⢸⡇⠸⣇⠸⣇⠀⣸⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣭⣷⡀⠀⠀⢀⣤⣤⠄⣀⣀⣀⢾⠟⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣦⡻⣿⣿⣿⣿⢫⣤⣤⣿⡟⣡⣾⣿⣿⣿⡇⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣮⣝⡟⣥⣭⣭⣭⣭⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣤
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣝⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠶⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣠⠀⡀
⠀⠀⠀⠀⠀⢀⡠⣴⣿⣷⣶⣄⢀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⢷⣶⣧⣼⡀⢠⠆
⠀⠀⠀⠀⠔⠃⠀⠀⢸⠋⢹⡿⣿⣷⣄⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠐⣿⠀⣿⢸⡏⢹⡟⢻⣦⡾⠂
⠀⠀⠀⠀⠤⠒⠒⠤⢤⣀⠸⣇⣿⣿⣿⣷⡀⠀⢸⡀⠀⠀⠀⠀⠀⠐⠁⠰⠃⡼⢃⣠⠇⣺⡷⡿⠚⠯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣿⣿⣿⣿⣿⣷⣾⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠿⣄⣁⣦⡀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⢻⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⠟⠃
⠀⠀⠀⠀⠀⢀⣤⠤⠴⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣀⣴⣾⣷⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⢤⣀⠀⠀⠀⢀⣴⣿⣿⠟⠀⠀⠈⠁⢠⠀⠀⠀⠀⠀⠀⠀⢰⣿⢯⣾⢻⣷⣤⣄⣀
⠀⠀⠀⠀⠈⢓⡦⣴⣟⣿⠟⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢸⣷⣿⣖⣿⣿⣿⣿⣿⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡗⣤⣀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠴⠒⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠶⠛⢛⠛⣛⡿⠖⠛⢋⣉⡤⠄⢖⣊⣳⡂⠑⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠞⠛⠉⠁⠀⠀⠀⠀⢸⠘⠷⠶⠖⠛⠉⠁⠀⣀⣀⣤⣬⣽⣦⠘⣄⣠⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⠛⠛⠛⠻⠷⢖⡤⡒⠋⠉⢉⣒⣂⣠⣬⣧⠘⣦⣤⣄⣬⡍⢻⡆
⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠉⠁⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⣈⣛⣛⣛⠻⠿⠿⠿⠿⣛⣛⣛⣫⣭⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠁⠘⠲⠤⣤⣤⣀⣤⡔⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠚⣧⠈⣉⡉⢁⠀⢀⡀⣀⣀⣀⣀⠤⠴⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠲⠶⢦⣤⣤⣤⣤⣤⣤⠴⠞⠋⠁⠀⠀⠀⠀⢿⡀⠀⠀⠈⠉⠉⠁⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⢀⣠⣼⣇⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⠶⣦⣤⣤⣄⣀⠀⢀⣠⣤⠤⠀⠀⠤⠤⠤⣤⣤⣤⣤⣴⣾⣍⣀⣀⣻⡀⠉⠉⠉⠉⠉⡼⠋⠉⠈⠁⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣭⡿⠛⠁⠀⢠⡰⣄⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⢿⡍⠉⠛⠛⠛⠲⡾⠷⠶⢤⣤⣤⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠟⠋⠁⠀⠀⠀⠀⠀⠱⡌⢳⡀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠻⣇⣤⡶⠶⠟⣾⠓⠚⠓⠲⢦⡀⠀⠈⠉⡛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⣅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠳⣄⠀⠀⢀⣤⠶⠚⣩⣶⡾⠛⠁⠀⠀⠁⠶⠞⠃⠀⠀⠀⢹⣦⣤⠞⠁⢸
⠀⠀⠀⠀⠀⢠⣶⡛⠋⠙⣇⠀⠉⠉⠛⠛⠛⠷⠶⢶⣤⣤⣤⣤⠼⡧⡙⡂⠜⣋⡤⠖⢿⣿⡟⢀⠔⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡇⠙⠛⠛⠾⠿⠿⣿⣷⣶⣶⣤⣤⣤⣄⣀⣀⣤⣤⣷⣽⣿⠞⢉⣤⣴⠋⣹⡇⠼⠀⠀⠀⠀⣀⣤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢤⣈⠉⠉⠉⠛⠛⠛⠛⠻⠿⠶⠾⣷⣶⣾⣍⣷⡀⣀⣀⠐⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡼⠂⣰⢂⡿⠂⠀⠀⠀⠀⠀⡔⠒⠲⠦⣄⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⢀⡴⠁⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠈⡷⠟⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⣤⡜⢀⠀⠈⠣⣄⣀⣀⣀⣀⡀⢀⣀⣴⠏⠀⣴⠃⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣙⣒⣒⣂⠀⠀⠀⠀⠈⠉⠉⢀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⠲⠶⠶⠞⠛⠉⠀⠉⠉⠙⠓⠲⠶⠦⠤⠴⠶⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠖⠂⠠⢄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢒⡄⠰⢐⡱⠀⠀⢀⠁⠁⠈⠊⠉⠒⠀⡀⠔⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⢁⠀⣘⣌⣛⣓⠁⠥⠠⢄⡂⠀⠐⠴⢈⡁⠂⢤⢠⠤⢈⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡔⠮⡤⠾⢐⣁⣃⡐⠄⣀⡁⢠⡀⡁⢂⠁⠀⢉⢁⠅⣀⣀⠀⠂⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡐⠬⠀⠀⠬⠁⢆⡀⢄⠠⢀⠠⣧⡐⠁⠈⠐⠚⠅⠀⡧⠀⣤⠌⠁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠑⠀⠠⠄⡚⠰⣃⠌⠓⠢⠭⠥⣈⡩⠴⢎⠭⠒⠖⠆⠔⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢘⡊⢠⡀⢁⠋⠂⠉⢀⡂⡸⠶⠭⢀⡩⡤⠌⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠓⠓⠈⠀⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣤⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠧⢴⡷⣤⢄⣴⡆⣶⢏⡴⣦⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠿⠞⠿⠺⠽⠟⠿⠺⠗⠿⠚

⠀⠀⠀⠀⣀⣤⣴⡶⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠄⣲⣶⣄⠀⠀⠀⢀⢤⠄⠶⠀⠖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀
⠀⠀⠀⢀⣉⡉⢉⡉⠉⢢⠀⠀⡁⠀⠀⠀⠀⠀⢠⡴⣏⣤⣾⣿⢟⠋⠹⠃⠀⠀⣧⣿⣶⡾⠛⠛⣋⡙⢻⣆⠀⠀⠀⠀⠀⣰⡷⠺⣿⣯⡭⣿⣆
⠀⠀⠀⢸⣿⢶⢻⣽⠀⢸⣷⣾⡇⠀⠀⠀⢀⣴⣛⢠⣶⣿⡟⠁⠰⠈⠀⠀⠀⠀⠈⠉⠙⢀⢰⣿⣻⣿⠀⠃⠀⠀⠀⢀⣾⠏⠀⢂⢬⣿⣷⡄⠾⣧
⠀⠀⢠⠐⣯⣿⣷⣿⡆⢼⣿⣿⠇⠀⠀⢠⣾⣓⣮⣿⣽⢟⡤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠗⠉⠀⠀⡀⠀⠀⢠⣿⠟⠰⠤⠜⡷⠬⣿⣷⢶⡿⠃
⠀⠀⠈⠢⠀⠀⠈⠉⠠⠞⠋⠁⠀⠀⠀⠉⠲⠾⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠀⠁⠠⠒⠃⠀⠀⠈⠛⠒⠒⠂⠀⠀⠀⠼⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⢠⠄
⠀⠀⠀⠀⠀⢀⣠⡶⠞⠓⠀⡀⠀⠀⠀⠀⠀⣴⡚⢫⡭⣭⡬⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣴⣧⣅⣼⠦⡀⠀⠀⠀⠀⣀⢠⠒⢠⠀⠀⢶⣶⢂⡄
⠀⠀⠀⢰⠋⠀⠀⠑⢈⡧⢤⣴⣪⠄⠀⠀⠀⣟⢷⡈⠙⠛⠁⠠⠄⢳⠀⠀⠀⠀⠀⠜⠁⣀⠄⠀⠈⠹⣠⣧⠀⠀⠀⠀⡏⠨⠁⠀⠀⠀⠲⣿⡿⡇
⠀⠀⠀⣅⠡⠶⠒⠀⠘⢃⠘⢁⢼⠂⠀⠀⠀⢃⠈⠿⢶⣆⣴⡴⡶⢤⡆⠀⠀⠀⢀⠀⣸⡭⡳⣆⠀⢸⡏⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢾⡗
⠀⠀⠀⢶⡒⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⣤⣈⡉⠈⠃⠀⣸⠇⠀⠀⠀⠘⠀⢿⣿⣎⢯⡇⠈⣷⠏⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣷⡿⠃
⠀⠀⠀⠘⠓⠒⠒⠚⠃⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠋⠛⠓⠛⠃⠁⠀⠀⠀⠀⠀⠦⡀⠉⠉⣩⢅⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣄⠂⢀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⠄⠀⠀⠀⠀⠀⠀⠀⣠⠀⠑⢂⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠰⣾⠿⠁⡆
⠀⠀⣀⢈⡉⠀⠈⠱⢿⣄⣤⡷⠗⠋⠍⠀⠀⢀⠀⢠⡄⠴⣖⡫⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣶⡞⠁⠀⠀⠀⠀⠀⣾⣁⠎⣁⣹⣥⣿⣾⡇
⠀⠀⠑⠫⠤⠆⠐⠟⣽⠿⠟⠋⠀⠀⠀⠀⠀⠈⠀⠀⠉⠇⠸⠟⠁⠀⡐⠀⠀⠈⠁⠀⠤⠠⠄⣰⣇⢸⡗⠈⠀⠐⠀⠀⠀⡿⢘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠛⠃⠐⠈⠀⠀⠀⠀⣇⢸⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⣠⠔⠄⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠰⠀⠒⠰⠆⠀⠀⠀⠀⠀⠀⡀⠐⡠⠒
⠀⠀⠀⠀⢠⠀⠄⠄⠀⠄⢰⡆⠆⠀⠀⠀⠀⢰⠤⠀⠒⠐⠒⣀⣈⣀⣀⠀⠀⠀⠀⠀⣤⠈⠀⠀⡀⣀⣤⠽⠞⠛⠀⠀⢰⠁⠀⠑⠀⣄⡀⠀⢀
⠀⠀⠀⠀⢸⣄⡀⣀⣀⣰⠀⣿⣤⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢀⡜⢁⡠⠴⠞⠋⠁⠈⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢉⡐⢀⠀⢰
⠀⠀⠀⠀⢀⠁⠀⣿⣿⣿⠀⣿⡿⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠘⠳⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⣀⠐⠁⠐⣈⣴⡿
⠀⠀⠀⠀⠀⠁⠈⠉⠉⠉⠰⠋⠀⠀⠀⠀⠀⠈⠙⠒⠤⠤⠼⠿⠿⠟⠃⠀⠀⠸⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠄⠀⠀⠠⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣀⣀⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢸⣀⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁
⠀⠀⠠⣾⠿⠿⣿⠿⠿⣿⠦⠀⠀⠀⠀⠀⠀⢳⣸⣬⠛⣿⡏⠀⠀⠀⠀⠀⠀⠀⠠⣤⣴⣧⣤⡤
⠀⠀⢨⣭⣵⣯⣭⣵⣭⣭⣭⠀⠀⠀⠀⠀⠀⣷⣯⣬⣥⣵⡟⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿
⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿

⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⡀
⠀⠀⠀⠀⠀⡇⢳⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣻⣿⣷⡄⠀⠀⠀⠀⠀⠀⣠⣶⣿⢿⣿⣿⣦⡀
⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⠒⡆⠀⠀⠀⠀⠀⠀⢛⣻⣯⣷⡄⠀⠀⠀⠀⠀⢰⣿⣿⣿⠟⠻⣽⣿⣿
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠊⠀⠀⠀⠀⠀⠀⠀⢾⣶⣷⣶⠆⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⣴⣿⣿⣿
⠀⠀⠀⠤⠽⠿⠿⠿⠯⠤⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠳⣝⠛⢿⠟⢛⡯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⡀
⠀⠀⠀⠀⣾⣿⣯⣟⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⣻⢟⠀⠀⠀⠀⠀⠀⠀⢈⣿⣾⣿⣿⣆
⠀⠀⠀⠀⣽⡟⣿⢿⣿⣮⡳⠄⠀⠀⠀⠀⣠⣾⣿⢿⣧⢟⣵⡿⠀⠀⠀⠀⢠⣦⣴⣿⣿⣽⣿⢿⣿⡄
⠀⠀⠀⠀⣶⢾⡿⡿⠟⠋⠁⠀⠀⠀⠀⠀⢻⣟⣻⢟⣵⡿⠋⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣼⢿⣿⡿
⠀⠀⠀⠀⠿⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠱⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠾⠿⠾⠋

⣿⣟⣛⣟⣛⣻⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⠤⠀⣀⠀⣰⡧⠴⡗⠚⣻⣁⣸⣦⡤⠖
⠀⠀⠀⠀⠀⠀⠤⠴⣿⣚⡽⠛⠁⢸⠀⢠⡿⡆⢸⠀⢰⡇⢰⡇⢈⡽⠁⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣠⣴⣟⣁⡤⢤⡤⢼⠁⢸⠃⢻⣼⠀⢸⡇⠀⠛⠋⢠⠀⠀⢀⠀⠀⢀⡼⣣⠖⠁
⠀⠀⠀⠀⠀⠈⠉⠉⣇⡤⠖⠋⠀⠻⠴⠺⠀⠀⠛⣤⠀⢠⡄⣷⠖⣾⡴⣦⠞⠀⢀⣞⢞⡵⠊
⠀⠀⠀⠀⠀⢀⡤⠾⡏⠀⠀⠀⠀⠀⢠⣀⣿⣠⠄⡟⠿⠿⢹⡏⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣤⣦⢶⠾⠆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⢀⣀⣤⣾⡜⠳⣻⠿⠂⠧⠖⠂⢀⣀⢀⣀⣤⣴⠶⣖⣉⣉⣉⠉⠒⢬⡁⠁
⠀⠀⠀⠀⠀⠀⠀⣷⣲⣾⡧⠛⠉⡏⠟⠈⠋⠀⠀⣀⡠⠊⠁⠀⠫⣴⢣⣀⠷⠈⡧⢄⠈⢳⠀⠀⠈⠳⢄⠀⠀⠐⠂
⠀⠀⠀⠀⠀⡰⠂⡏⠀⠸⠀⠀⠀⠁⢀⣠⣴⡖⠉⡱⠁⠀⠀⠀⠁⠙⢗⣭⠷⣴⡳⠌⡇⢸⠯⡉⢳⡀⠀⠙⠦⣀
⠀⠀⠀⠀⠀⠑⠞⠁⠀⠀⣀⡤⠖⠋⠁⡟⠋⡻⢤⡃⠀⡔⠒⠂⠉⢂⡜⠁⠂⠀⠙⣞⣠⠷⣄⡹⢸⡣⢚⣶⠤⡈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⡀⢀⡀⠄⠠⡈⠒⠌⠒⢦⣶⣆⡀⡌⠀⠀⠀⣀⣀⠈⢏⡲⠗⠧⣞⠔⣱⠋⡰⣁⠠⢠⠈⠲⡀
⠀⠀⠀⠀⠀⠀⣼⠡⣴⢤⠐⠢⣀⠢⡕⠂⢁⠠⠐⡧⠠⠠⡈⠛⢶⡃⠀⡎⠀⠀⠈⣤⠋⠀⠀⠀⠈⢾⠃⢜⠔⢁⠔⡦⢀⡦⢿
⠀⠀⠀⠀⠀⠀⡟⣆⠀⠸⡇⠦⠀⠂⠒⣈⠠⡄⠰⣄⠢⠀⣴⣦⣶⡌⠑⠧⣶⡄⢐⠁⢠⠄⠐⠒⢢⠈⢟⠳⣪⡭⠞⠋⠁⠀⡼
⠀⠀⠀⠀⠀⠀⡇⠀⠙⢦⡃⠘⠙⠣⠉⢮⢹⣈⠃⢈⣀⠴⢯⡁⡠⠛⠉⠙⢪⡑⠧⡀⢢⢠⢰⣄⣤⠤⠚⠋⡁⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠉⠳⣍⡀⡳⡜⠛⠛⡿⠉⠀⠀⠀⡸⢁⡴⠒⢢⠀⣇⣀⣨⠽⠚⠋⠁⠀⠀⠀⠀⠀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠉⠳⢤⣀⢸⠁⢠⠊⠑⡄⣇⣸⣠⡤⠜⠒⠉⠁⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠈⠉⠓⠛⠒⣚⠋⠉⠁⠀⠀⠒⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠁⠁⢀⣀⡤⠴⠒⠋⠁⠀⢀⠀⣠⡀⣴⣠⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠈⠙⠒⠒⠚⠋⠉⠀⠀⠀⠀⠀⠀⢠⠟⣷⠁⠿⠘⠃⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⡀⠀⠀⠀⡀
⣿⣯⢭⣟⡛⣛⣛⠛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⣴⣦⣀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⡇⢸⣧⡄⠀⠀⠙⣷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⠀⡇⢸⣯⡄⠀⠀⠀⣨⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣄⣬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⠀⣧⢸⣯⣥⣤⣤⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⣇⢻⡀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢏⣼⢋⣿⡘⣧⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣡⠟⣡⠟⠉⢷⡹⣦⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢧⠾⠋⠀⠀⠈⠳⣌⠷⣌⠳⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣳⣬⣙⡲⠮⣭⣛⡻⠶⣶⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠶⢛⣻⠿⣿⡿⢟⣻⡿⠿⣛⣶⡮⣭⣛⠶⣭⡛⢿⡿⣿⣿⣿⣿⡛⠶⠶⢦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠉⠁⠀⠀⢀⡴⢛⣵⠟⣩⡶⢛⣥⠶⣛⣭⣶⣿⡿⠿⠿⣦⡙⢦⡙⢶⡙⢿⡻⣿⢷⣄⠀⠀⠈⠉⠛⠳⢶⣄⡀
⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⢀⠀⣰⠟⣴⠟⣠⠞⣁⣴⡿⠗⣛⣋⣭⡴⠶⠖⣛⣛⣩⣿⣆⠹⣦⠹⣦⠻⣌⢷⡹⣆⠀⡀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⢷⣯⣼⣣⡼⠟⣛⣭⠴⠖⣛⣩⣥⠴⠶⢛⣛⣩⣭⡤⠶⠿⣇⠘⣧⠸⣇⢻⡌⣷⣽⡾⠃⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠹⢦⣄⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣶⣿⣿⣽⣷⣶⣿⣿⣯⣭⣷⣶⣶⣟⣻⣿⣤⣿⡶⠿⠞⠛⠋⠁⠀⠀⠀⠀⠀⠀⣠⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣷⣶⡶⠶⢤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⡤⠴⠶⣶⣾⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠶⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠚⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⢿⡛⣿⣋⣿⣿⣴⣿⠇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢶⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡐⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠘⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠋⠁⠀⠀⡀⠀⡟⡉⠐⢘⡁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣴⠆⠀⠀⣦⠦⠿⢀⠁⢨⡉⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⠿⠇⠀⠀⠁⠀⡉⠘⠀⡱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⣼⡿⠛⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠂⠀⣴⣶⣶⣶⣶⣶⣶⣾⣷⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⢟⢱⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠠⠀⠀⠀⠂⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠙
⠀⠀⠀⠀⠄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡄⡇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠰⢆⠀⠄⠀⠀⣀
⠀⠂⡀⡀⠀⠀⠃⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠿⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠑⠲⠒⠂⠉
⠀⠀⠀⠉⠀⠀⠀⠀⠘⠿⠿⠿⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡜⠠⠑
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣦⣤⣀⡀⠐⡀⣰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠤⢚⠭⠉⠑⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠒⠀⢀⡀⠀⠀⠀⡀⠤⠀⠀⠀⢀⠐⠁⠀⡀⠀⠀⠀⠈⠈⠐⠖⠁⠐⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠱⠆⡉⠀⠉⠐⠀⠀⠀⠀⠀⠸⠀⡠⠈⠂⠀⠐⠀⢦⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⣤⠐⠒⠢⠤⣀⠀⠀⠀⠐⢀⡀⠀⠀⠀⠀⠀⠀⠈⠲⣀⠀⠳⣄⠀⠒⢀⡠⢞⠐⠈⠀⠀⠉⠈⠲⡕⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠶⣦⣄⣀
⠀⠀⠀⠀⠈⠛⠒⠀⠈⠁⠀⠀⠀⠀⠁⠂⡀⠀⠀⠀⠀⠀⢀⡘⠇⠤⣬⢉⠋⣩⢀⠀⠀⡠⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠳⣄⠀⠀⡔⣿⣹⢳⣙⣷⢺⡤⡍⢸⡀⠀⠀⠀⠀⠤⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠧⣀⣋⠲⢿⢯⠽⣊⠷⣟⣺⢟⡆⠀⡗⠀⠀⠀⠀⠀⠀⠈⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠡⡆⠙⡝⠲⣶⡉⣷⣿⣉⣟⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⢼⡇⢈⡽⠠⠼⠧⣩⠈⠼⠀⡙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠙
⢀⣠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⢀⡠⠂⡑⠀⠀⠀⠀⠈⠨⠛⠃⠳⠄⠤⠀⠤⠰⠇⠀⠀⠀⠀⠀⣠⡤⠄
⣅⣀⠀⠀⠻⠿⣿⣶⡤⠀⠙⠲⣄⠀⠀⠀⠠⡀⠀⠀⢄⠀⡈⠁⠀⣅⠀⠀⠀⠀⠀⢰⠈⠄⠀⠰⠠⠀⢀⠎⠀⡠⠀⠀⠀⢨⣀⠀⠀⢴⣦⡄⠳⠆⠀⠉⠳
⣿⣿⣿⣶⣦⣤⣀⣀⣀⣠⣤⣴⡿⠀⠀⠀⠀⠈⠢⣀⠀⠉⠘⣦⠀⠀⠓⠦⠦⠴⠴⠃⠴⠆⠀⠁⠠⠔⠁⡠⠎⠀⠀⠀⠀⢸⣿⣿⣷⣦⣤⣉⣁⣀⣀⣤⣴
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠁⠀⠉⠁⠒⣂⠁⠂⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⠤⣤⣤⡤⠤⠶⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⢿⣿⣿⣿⣿⣿⠿
⣶⣶⣶⡶⣶⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣶⣶
⣿⣿⣿⣀⣀⣀⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢃⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣠⡟⢡⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡋⠉⠛⠛⠋⠉⠉⠳⣄⠀⠀⣴⠏⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⣉⣤⠶
⠀⠀⠀⠀⠀⠀⠀⢰⠟⠛⠛⠛⠋⠛⠶⠄⠀⠀⠀⠀⠀⠀⠀⠉⢻⡿⢁⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⢋⣡⡴⠟⠉
⠀⠀⠀⢀⣠⣤⢶⣿⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⠀⠀⣰⠟⣰⡟⠉⠙⠛⢶⡄⠀⠀⠀⠀⠀⢀⣠⡶⠛⣉⣤⠾⠋⠁
⠀⢀⡴⠛⣁⣤⠾⣗⠶⠀⠀⣤⠶⠄⠀⠀⠀⠀⠿⠀⠀⢀⡾⢋⣼⠋⠀⠀⠀⠀⠀⡿⠳⣦⣀⣴⠞⢋⣡⡴⠛⠉
⣰⡏⣠⡾⢯⡜⠓⠉⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣡⡿⠁⠀⠀⠀⣀⠀⢘⣻⡶⠟⣉⣴⠞⠋⠁
⣿⠀⣿⣹⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣴⠏⠀⠀⠀⠀⣀⣼⠷⢋⣥⡶⢻⡏⢸⡇
⢻⣦⡟⠷⠀⠀⠀⣠⡤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⣠⡴⠞⣩⡴⠞⠋⠁⠀⢸⡇⢸⡇
⠈⣿⣿⣗⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣥⠶⠛⠉⠀⠀⠀⠀⠀⢈⣹⣿⠁
⠀⠘⣿⡙⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠛⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⣿⠃
⠀⠀⠘⢷⡝⠶⠍⠛⠻⠶⣦⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣶⠶⠟⠛⠉⠀⣼⠏
⠀⠀⠀⠈⠻⣾⠛⠦⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠙⠷⣝⡛⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢯⣭⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡿⠶⣦⣤⣤⣤⣤⣤⣤⣤⣤⡶⠶⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⢦⣤⣤⣤⣤⣤⣤⠶⠶⠟⠋

⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿
⠀⠀⡏⢙⡻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣷⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢒⣶⣾⣷⣦⡀
⠀⢸⠻⢿⡟⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⢠⢎⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢸⢱⣿⣿⣿⣿⣿⣷
⠀⡇⣤⡀⠀⣻⣿⡎⢻⡄⠀⠀⠀⠀⠀⠀⣺⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠸⣼⣿⣿⣿⣿⣿⡟
⣸⠐⠟⠟⣀⣉⠥⠥⠚⠟⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠟⠁
⠓⠒⠋⠉⠁


⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢀⡤⠧⡀⡸⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⢠⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡭⣀⣄
⠀⠘⣀⢀⣛⣀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⠀⠋⠿⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠄⠰⠿⣿⡆⣀⡀
⠀⠀⠺⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠘⠒⣢⣶⠄⠩⣾⢟⣧
⠀⠀⠀⢿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛⠹⢟⣤⣦⠈⠿⢟⡀
⠀⠀⠀⠘⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⣿⣦⡄
⠀⠀⠀⠀⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠣


⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿
⠀⠀⠀⣠⣴⡿⣫⢚⡛⣂⠀⠀⠀⠀⠀⢀⣀⢀⡀⠀⠀⠀⠀⣀⣠⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⣶⣶⣦
⠀⢠⣾⡿⣫⡎⣭⣨⣿⡟⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣿⣿⣿⣿⡿⠟⠋⣀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣻⣥⣄
⢰⡿⡫⠾⠰⣦⣾⣿⠟⠀⠀⠀⠀⠀⠀⢀⣀⢀⠈⠉⠉⠉⠉⢁⣠⣰⣿⣿⠁⠀⠀⠀⠀⠀⠀⣼⣭⣭⣻⡟⣤⣄⡀
⠉⣚⡃⣻⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣷⣿⣿⣾⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠘⠿⡿⢨⣾⣿⣿⡧⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢀⠀⠀⠴⣾⠋⣽⣹⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⠖⢋⣩⠖⠁⠀⠀⣀⠀⣾⣰⡇⡜⣿⢸⣇⠞⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢀⣠⠖⡋⠁⣧⠴⠟⠒⠛⣻⠆⢰⡷⡟⡿⠑⠟⠀⠏⠀⡁⠀⢰⠇⣰⠇⠀⢀⣠
⠀⠀⠀⠀⣴⡯⠖⠋⠀⠀⣧⠀⠀⣠⠞⠁⠈⠙⠚⢠⠀⢀⢠⠀⣤⢸⣷⢛⣾⠞⢹⠀⠀⢋⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⠚⠁⠀⠀⠀⠀⠀⣸⠀⠿⠻⠞⣿⣼⠙⠋⣿⠀⠘⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣿⠀⠀⠀⣠⢤⡖⠀⠀⣿⣠⡴⠀⢸⡇⠛
⠀⠀⠀⠀⠀⠠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠻⠂⠀⠀⠋⠁⠀⠀⠀⢀⣀⡠⢤⠔⠶⣒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠛⠉⠁⢢⡗⠀⠒⢠⠙⢆⠈⢲⠤⣀
⠀⠀⠀⠀⠀⠀⢀⣤⠖⠊⠁⠀⠀⣠⡄⡰⠒⠾⢭⡢⢤⡊⢀⣠⠤⠃⡠⠗⠒⠐⢸⠀⡰⢀⡬⠔⢊⣉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⡠⣄⠎⠲⠃⠈⠁⠀⠀⣀⠓⠂⠉⠒⢤⣠⠊⢀⡀⠴⣒⠡⠶⠵⢍⠀⢎⢹⣂⠇⠀⣈⡆
⠀⠀⠀⠀⠀⠀⠀⡰⠒⠃⡄⠀⠀⠀⠀⠀⠀⠤⠏⠁⣨⡤⠒⠂⣏⢙⠤⣠⠊⢠⠂⡏⠌⡲⠑⣤⣈⡤⠖⠊⠁⡇
⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠐⠀⢀⣀⣶⣲⠃⠀⠀⡇⢀⣀⣰⣾⠄⠑⢩⠛⠲⣝⣓⣨⠴⠚⠉⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠉⠳⢤⣀⡔⡉⠀⠀⠀⣱⠼⠾⣔⠋⠀⠀⠈⢣⣀⣠⠷⠒⠋⠁⠀⠀⠀⠀⠀⠀⣀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠈⠙⠧⣄⣀⡞⠠⠀⠀⠈⣧⣠⠴⠒⠉⠁⠀⠀⠀⠈⠁⠀⢀⣠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠢⣄⠀⠀⠀⠀⠀⠈⠉⢛⣒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣈⣁⣀⡤⠴⠚⠉⠁⠀⠀⠀⣀⡴⡴⠖⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠙⠆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢼⣿⡧⠄⠀⠐⠊⠃
⠀⠀⠀⠠⣒⣁⠰⠃⢀⡠⠄⠀⠀⠀⠀⠀⠀⠄⠀⠸⣿⣇⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⠆⠀⢤⡀
⠀⠀⠀⠀⠋⠈⠉⠁⠲⢄⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⡎⣿⣿⡀⠈⠀⠀⢪⠉⣲⡄
⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠻⣿⣧⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⡀⡀⠀⠈⠲⡏⠀⠀⠙⢿⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣄⠀⠀⠠⠈⡀⠀⠀⠀⠐⠂⠈
⠠⢴⠀⢰⠒⠰⠢⣀⠀⠀⠀⠀⣀⣤⣤⣤⣄⠀⠀⠀⣾⣿⣿⣿⣷⣶⣶⣿⠃
⡲⠚⠛⠉⠐⠒⣄⠉⠒⠀⠀⠀⠈⠉⠉⠉⠁⠁⠀⠀⠙⢿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠛⠛⠉⠁⠀⠀⠀⠠⠄⠀⠖⠠⢠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⢠⠈⠀⡰⠁⠈⠙⠀⡀
⠀⠀⠀⠀⠀⠀⡾⢭⠹⢸⡁⣊⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠋⠉⠁⠐⠓⠠⢆⠀⠠⢐⡀
⠀⠀⠀⠀⠀⣾⢿⣕⣐⣨⣿⣿⠁⠀⠀⠐⠂⠀⠀⠀⠀⠀⣀⢤⣀⠤⠤⡀⠀⠀⠠⠀⠀⠀⠀⠀⠈⢳⣀⠠
⡿⠊⠀⠀⠀⠙⠷⣦⠴⠶⣿⣿⠈⠁⠀⠀⠀⠀⠀⠀⠀⡴⡎⣀⣄⣄⣵⣬⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠈⢿⣷⣤⠤⠀⠀⠀⠀⠀⢀⡽⠛⠉⠉⠉⠙⠛⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛
⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢾⡀⢠⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⡀⠂⣀⣀⣤⡾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⡀⠀⡀
⠀⠀⠀⠀⠒⠂⠉⠁⠙⠒⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⢀⡀⢀⠄
⠀⠀⠄⠤⠤⠤⠤⢤⣀⣀⠀⠀⠰⠀⠀⠀⠀⠀⠺⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣦⣠⠊⡁⠁⠀⠀⠹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⣠⣴⢤⣴⣤⢦⣄⡀⠀⠀⢀⣠⡿⢧⠈⡷⣠⠟⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣰⣿⡟⣾⣇⣿⢦⣍⣵⠀⢠⣿⢻⣏⠦⣈⠀⣾⣀⣉⣀⡀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠌⠑⠢⡀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⢰⠙⠋⠉⠉⠀⠀⠻⣿⣿⡀⠘⣿⣦⡉⠛⠶⢾⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⣷⣄⣑⠂⠄⠀⠀⠃⣠⡇⠀⠀⢸⡿⢿⢻⢛⣿⢿⢿⠅
⠀⠀⠀⠀⠒⠀⠀⠀⣀⠄⢱⠀⢸⣼⠃⠀⡰⠛⠛⠻⠿⣿⣿⡿⠤⠤⠄⠀⠀⠊⠂⠀⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⠀⠀⣾⡗⣚⡂⢼⣿⣼⣇
⠀⠀⠀⠀⢘⠀⠀⡈⢰⠁⡮⠀⢸⣿⠇⠀⡁⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣧⣼⣥⣤⣽⣿⣾
⠀⠀⠀⠀⠘⢀⣀⠀⡈⠈⢀⠠⠾⠋⠀⠀⠁⠠⠄⣀⠀⡈⣿⣿⣧⣀⣤⠀⠄⠊⠀⠀⠙⠿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣽⡿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠚⣡⠠⡿⢥⣿⣿⣟⡀⠀⡲⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠋⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠋⠉⠀⠂⠀⠀⠉⢿⣿⡇⠀⠀⢸⠀⠈⠀⢀⠠⣾⣿⣿⠄⠀⠎⠩⢀⡀⠙⡤
⠀⠀⠀⠀⠀⢰⠤⠴⠖⣿⣿⣿⠒⠒⠒⠒⠒⠒⠋⠛⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠘⣿⡇⠀⠀⠘⢶⠞⠁⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠘⡇⠑⢄⠂
⠀⠀⠀⠀⠀⢸⡀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠈⢄⠀⠀⢔⢀⠌⣼⡟⠀⠀⠀⠀⠸⡀⠀⠀⠀⠹⣿⣿⣇⠀⠀⠀⠀⡰⠁⠘⢼⡖
⠀⠀⠀⠀⠀⠈⠓⠠⠀⠈⣿⣿⣧⣀⠀⡀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠓⠀⠦⠀⠹⠿⠿⠆⠀⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⢀⠤⠔⠒⠒⠢⢄⡀⠀⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⣀⠄⠒⠋⠁⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀⠀⠈⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⡀⠀⠀⠑⠢⢄
⠀⠀⠀⠀⣀⠤⠒⠁⠀⠀⢀⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠔⠒⠢⠤⠋⠙⣦⣀
⠀⡠⠔⠋⠀⠀⠀⣀⠤⠊⠀⠀⠀⠀⠀⡄⠀⠀⠀⠊⠁⠑⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣀⠜⡠⠔⠒⠒⠠⠄⣶⠤⣑⠂⣄
⠸⣀⠀⠀⢠⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠃⠀⠨⠀⠀⠀⡠⡔⠋⠀⠀⠂⠈⡹⠯⠶⡄⡀
⠀⠈⠑⢢⢄⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⢀⠤⠂⢡⠁⠀⡄⠀⠂⢇⡔⠉⠀⠀⠀⢋⠤⠒⠉⠠⠄⠀⢹⣌⡑⠢⣄
⠀⠀⠀⢸⠀⠉⠒⠤⡈⠒⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡄⠤⠀⠀⠘⠢⡀⠀⠄⢀⣠⣦⡲⠀⡀⠂⠁⠀⠈⢀⡀⠄⠊⠁⣠⢹⠀⠀⠉⠂⢄
⠀⠀⠀⠈⠢⢄⡀⠀⠀⠑⠢⢌⠑⠢⢀⠀⠀⠀⠀⠀⢀⠤⠒⡋⠀⣰⠥⠴⢦⣤⣌⡂⢁⠈⠢⢄⢩⣴⡶⢶⠵⣄⠑⢀⣞⡉⠀⠂⠀⠀⠁⠀⡇⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠈⠒⢤⡀⠀⠀⠉⠒⠄⡉⠒⠀⠒⠊⠀⡰⠧⢭⣚⡥⢤⣥⣇⣀⣼⠚⠉⠉⠂⠆⠀⠀⢰⠁⠀⠀⠙⠉⠁⠙⢢⣀⣀⠀⠀⣀⢇⡠⢶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⠀⠀⠈⠑⠠⢀⠀⠀⠑⢤⢾⡁⠀⠀⠀⠀⠉⠛⢦⠀⢀⠀⢠⣄⠠⡀⢀⠠⠴⡁⠀⠀⠀⠈⠂⠈⡉⠄⠊⠁⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡀⠀⠀⠉⠒⢄⡀⠉⠃⠀⠀⠀⠀⢀⡀⠜⣀⣀⢠⣌⡨⠏⣘⠁⡀⢠⠈⠁⠀⢀⡠⠒⠉⠀⠀⡀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠈⠑⠠⣀⠀⠀⠀⠁⠀⡎⠀⢀⠀⠀⢙⡄⠃⠀⠓⠁⣀⠤⠊⠁⠀⢀⡠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀⠀⠀⠁⠢⢄⡀⠀⠃⠐⠀⠀⠀⠈⠃⢀⡠⠔⠈⠀⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀⠀⠀⠈⠐⠤⣀⠀⠀⣀⠤⠂⠁⠀⢀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠈⠁⠀⠀⣀⠤⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⡀⠠⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⣶⢟⣻⣟⠻⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⢉⣡⠴⠦⣄⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢜⣫⣶⣿⣿⣿⡇⣾⣦⣴⣿⣿⡆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣣⠖⠋⠀⠀⠀⣸⠉⠛⠛⠀⠀⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣴⣿⣿⣿⡿⣛⠁⠈⠉⠉⠉⠉⣽⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠋⠀⠀⠀⣠⡤⣾⣿⣿⣿⣿⣿⠇⢸⣿⣿⣿⣿⣿
⠀⠀⠀⣠⠽⠿⠯⠵⠾⠿⠷⠤⠤⠤⠤⠾⠿⠿⠤⣄⠀⠀⠀⢸⣿⣿⣿⢟⣃⣀⣐⣚⣀⣀⣐⣚⣛⣛⣛⣁⣀⣘⣛⢻⣿⣿⣿
⠀⠀⠀⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠞⠀⠀⠀⢸⣿⣿⣿⣜⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠶⠶⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⠀⠀⠀⠀⠀⠐⠦⢤⣈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠺⣥⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠒⡞⢣⠒⢀⠔⠆⠀⠀⠀⡀⠘⠻⠿⠿⠛⢷⣄⣀⣀⣴⠃⠐⣶⡶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠖⠶⠖⠒⠚⢸⣄⡇⢸⡀⠛⠛⠛⠋⠉⠉⡿⠀⠀⢰⠀⠀⠀⠉⠛⠋⠁⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣯⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡸⠹⣍⡙⢛⣛⣩⣭⣥⣄⡀⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⢠⠃⡔⣪⡉⢻⡯⢄⠀⠀⢧⠙⣆⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⢀⠏⠀⠁⡴⠃⠈⠁⠉⠀⠀⣸⠀⣿⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⢸⠀⢀⠀⢥⣄⠀⠀⠀⠀⢠⡟⢀⡏⠀⠀⣦⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠸⡀⠀⠘⢆⠐⠒⠉⠀⢀⣴⠟⠀⣼⠁⠀⠀⡏⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢳⣀⣋⡏⠳⠤⠴⠚⣹⠋⡴⠀⡇⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠺⢷⡄⠀⠀⠀⠀⢠⡏⠸⣧⡀⢷⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⠴⠦⢄⣠⡠⢠⡄⢄⠀⠀⠀⡇⠀⠀⠙⠀⠀⠀⠀⠀⠻⠦⠼⠟⠁⠑⠾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠢⠟⠀⠙⠉⠒⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠄
⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⢾⣷⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡦⠀⠀⠀⠀⠀⢰⣶⡆
⠀⠀⠀⢘⠿⡃⠀⠀⠹⣇⣀⠿⣃⠀⠀⠀⠀⣸⣟⡀⠀⠀⠀⢀⣿⣃⠀⠀⠀⠀⢀⡿⠇⠀⠀⠀⠀⣸⠿⠀⠀⠀⠀⠀⠀⢀⣹⠟⠀⢠⠄
⠀⠀⢠⣿⣿⢿⡆⠀⠀⠉⢹⣿⣿⡆⠀⠀⣼⣿⣿⣷⠀⠀⢀⡟⠈⢹⡆⠀⠀⠀⣿⣷⣄⡶⠆⠀⢸⣿⣷⡲⣄⡀⠀⠀⠀⢸⣿⣷⡚⠑⠒
⠀⠀⠈⢻⣿⡎⠃⠀⠀⠀⢸⣿⡟⠃⠀⢠⠟⣿⣿⠛⠀⠀⠘⢃⠀⡘⠃⠀⠀⠀⣿⣧⡀⠀⠀⠀⠈⣿⠀⣿⣿⣿⣿⡆⠀⠈⣷⠘⣿⣿⣿⣿⡆
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢠⡿⠹⡇⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⡇⠀⠀⣿⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⢰⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⢰⣿⣴⡆⠀⠀⠀⢠⡐⣿⠳⡄⠀⠀⠙⠶⣿⣼⣄⠀⠀⠀⠀⠀⠀⢿⡷⡄⠀⠀⠀⠀⠀⠀⢀⡀⢿⡷⣄⠀⠀⢸⣯⣿⡤⠄⢦⡀⢺⡷⡄
⠀⠀⠘⣿⠀⣀⣀⣀⣀⣤⡭⢿⣇⡇⠀⠀⠀⠐⣿⣿⡈⢙⣸⣇⣀⣠⣤⢼⣇⡆⠀⠀⢀⣶⣀⣠⣦⣽⠿⣇⡶⠀⠀⢾⣿⣇⣀⣀⣀⡭⢽⣿⠇
⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⡿⡿⠿⡆⠀⠀⠀⠀⣼⠋⣿⣿⣿⣿⣿⣿⣿⣿⠿⡇⠀⠀⢸⣿⣿⣿⣿⣿⢿⠿⣇⠀⠀⢸⣿⣿⣿⣿⣿⡿⡿⠿⡇
⠀⠀⠀⠋⠀⠉⠉⠉⠉⠉⠉⠉⠈⠁⠀⠀⠀⠈⠃⠀⠉⠈⠉⠉⠉⠉⠙⠋⠀⠁⠀⠀⠈⠉⠉⠉⠉⠙⠉⠀⠉⠀⠀⠘⠉⠋⠉⠉⠉⠙⠁⠈⠃
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠘⢣⣋⣀⡻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⣤⣟⢉⣀
⠀⠀⠀⠈⣽⣿⠟⢦⡀⣀⠀⣀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣿⡿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣅⡀⠀⠀⠀⠀⠰⣿⠿⣯⣥⣀⡀
⠀⠀⠀⠀⢹⣿⢧⡀⣿⣿⣷⣿⣶⡆⢸⠀⠀⠀⢀⡀⠀⢸⡿⣦⠁⠀⠀⠀⠀⢀⣀⠀⠀⠀⣠⣠⣄⣿⣛⣿⣽⣿⣤⠄⠀⠀⠀⢹⡆⣿⣿⣿⡇
⠀⠀⠀⠀⣼⠃⢸⠇⠿⠿⠿⠿⠿⠋⠹⠀⠀⠀⠿⠿⠀⠾⠁⠻⠀⠀⠴⠦⠀⠼⠿⠆⠀⠀⠙⠿⠿⠻⣿⣿⡟⠉⠉⠀⠀⠀⠀⢸⠇⣿⣿⣿⡇

⠀⠀⠀⠀⠀⢀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣶⡄⠀⠀⠀⠙⠻⠿⣿⣿⣿⣶⡀⠀⠀⠙⠻⠿⢿⣿⣶⡀⠀⠀⠀⠀⢠⣾⣆⠀⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠇⠀⠀⠀⠀⠀⠀⠀⠉⢿⣦⡀⠀⠀⠾⣿⡿⠄⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⣀⠀⠀⣠⣤⠄⠀⠀⠀⠀⢀⠀⠀⢀⣤⡤⠀⠀⠀⠀⠀⣤⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⣤⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠈⠙⣿⡿⢥⣄⣀⡄⠀⠀⠀⠈⢻⣿⠯⢤⡀⠀⠀⠀⠀⠈⢹⣿⠿⠿⣿⣿⡆⠀⠀⠀⠈⠙⣿⡿⠿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠁⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣀
⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⢀⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢠⣐⣖⡸⢶⡶⡇⠀⠀⠀⠀⠀⠀⣀⡈⠹⣿⣧
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠘⠿⡿⣷⣯⣵⡆⠀⠀⠀⠀⠀⠀⠉⠁⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉


⣿⣿⢟⠿⠛⠿⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠄⠀⠀⠀⠀⠀⢀⢠⣴⣶⣶⣶⢶⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣒⣒⣒⢒⠒⣒⣒⣒⣒⣒⣲⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠠⠤⠴⠒⠒⣚⢒⢒⣐⣐⡇⠀⠠⠤⠤⠤⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠉⠉⠉⠉⠉⠀⠐⠒⠒⠒⠒⠒⠂⠠⠬⠭⠭⠭⠭⠍⠉⣹⣷⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠐⠒⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣉⣛⣟⣻⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠰⣏⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠶⢄⣀⡀
⠛⠛⠟⠛⠛⠛⠟⢛⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠁⠀⠈⠀⢙⡆⠀⠀⠀⠀⠀⢠⣠⡀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⣅⣤⣤⣄⡅⠈⠓⠁⠀⠀⠀⠀⠀⠀⠀⣠⠤⣤⣩⣥⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⠀⠸⣿⡇⣴⠾⣛⣛⢷⣄⢸
⠀⣾⣿⠿⠚⠿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⣴⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⣿⠉⡍⠉⣭⢙⡷⠀⠀⠀⠀⠀⠀⠀⣿⢸⡏⣾⣿⣿⡇⣿
⠀⠙⠻⠦⠤⠾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠴⠶⢤⡼⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠻⣮⣛⣻⣵⠟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠉⠀⠀⠈


⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣤⣀
⠀⡰⣪⣽⢿⡍⣰⣮⢦⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⠀⢀⣴⡫⣪⠀⠀⠀⠀⠀⠀⠀⢠⠎⣴⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⠀⢀⣼⣧⡀
⢸⣼⠷⣿⣟⣐⣛⠓⠛⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣧⣸⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣾⠶⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏⠀⢈⢻⣿⡁
⠸⡼⣿⣿⢿⣯⡽⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⡉⣿⣿⣿⣿⣧⠾⠋⠀⠀⠀⠀⠀⠀⠀⢠⢸⣿⣿⠀⡂⣾⣿⣧
⠀⠈⠪⠽⠿⠭⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠈⠻⠆⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⠿⠀⠼⠿⠿⠧


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠢⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀
⠀⠀⠀⡔⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⠭⠈⠄⣀⣀⢡⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣁⢠⣤⣤⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⣉⡉⣯⣀
⠀⠀⢠⡙⢛⣛⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠻⡈⣿⡯⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡺⣸⣿⣿⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⡇⢸⠇
⠀⠀⠀⣙⣛⣛⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣭⠗⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣿⣯⠏⠁
⠀⠀⠀⠈⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡃
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁


⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣦
⠀⢧⣾⣿⣿⣿⣿⣦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢋⠦⢤⡀
⠀⢸⣏⣉⣉⣉⣏⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡖⢲⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠘⣧⡤⣿⣿⣿⣿⣦⡖⠲⡄⠀⠀⠀⠀⠀⢠⠋⠀⠠⠄⠐⠀⡍⡆
⠀⣸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢣⠸⣿⣿⣿⣿⠜⠁⠀⠀⠀⠀⠀⠀⢿⡀⣿⣿⣿⣿⣿⣇⡼⠃⠀⠀⠀⠀⠀⠀⢳⣾⣶⣾⣶⣷⡶⠁

⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⡎⠀⠀⠀⡀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⣄⢠⠀⣄⠀⣀⠀⠀⠀⡇⠀⢀⡀⠀⠀⢀⡀⠀⢸⠀⠀⡇⠀⠸⣿⣷⢶⢆⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠀⠁⡨⠟⠋⠀⠀⠀⠀⡇⠀⠉⠉⠈⠈⠈⠋⠀⢸⠀⠀⡇⠀⠀⢶⣤⣤⣴⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠉⠉⠁⠀⠀⢸
⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⡔⠈⠁⠁⠀⠀⠀⠈⠁⢂⠀⠀⡐⠀⠉⠀⠀⠀⠀⠈⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⡀⢀⠀⠀⠀⢸
⠀⠀⠀⠀⣔⠒⠀⠀⡀⢀⠀⠐⠀⠀⠀⡇⠀⢀⣄⣀⣀⣴⠆⠀⢸⠀⠀⡇⠀⠀⠠⣦⣱⡆⠀⠀⢸
⠀⠀⠀⠀⠉⠛⠓⠒⠛⠉⠀⠀⠀⠀⠀⡇⠀⠀⠙⠛⠙⠁⠀⠀⢸⠀⠀⡇⠀⠀⠀⠿⠿⠃⠀⠀⢸
⠀⠀⠀⢆⡀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⢃⡀⠀⠀⠀⠀⠀⠀⣀⠜⠀⠀⠣⣀⢀⠀⠀⠀⠀⠀⣀⠘
⠀⠀⠀⢀⠤⠤⠤⠤⠤⠤⠤⠤⡀⠀⠀⢀⠤⠤⠤⠤⠤⠤⠤⠤⡀⠀⠀⢀⠤⠤⠤⠤⠤⠤⠤⠤⣀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡇⠀⠀⣀⣀⣤⠀⠀⠀⢸⠀⠀⡇⠀⠀⣀⣀⣀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠰⢶⡤⠄
⠀⠀⠀⡇⠀⠀⠈⠉⠐⠋⠀⠀⢸⠀⠀⡇⠀⠘⠛⠛⠛⠛⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠐⠚⠁
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠒⠒⠒⠒⠒⠒⠒⠂⠁⠀⠀⠈⠐⠒⠒⠒⠒⠒⠒⠒⠁⠀⠀⠈⠐⠒⠒⠒⠒⠒⠒⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣮⣦⣤⣀⣀⣀⡀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣺⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣠⣤⣤⣤⣤⣤⣶⣶⣶⣶⣿⣿⣿⣿⣯⡢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣲⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⣾⣿⣿⣿⡿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠙⣿⣿⣿⣿⠟⢉⣉⣈⣹⡿⢃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢰⣿⣿⣿⣤⣿⡇⢺⡇⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣠⡄⠘⣿⣿⠁⢠⣾⣿⠿⢋⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢱⡘⣿⣿⣿⠀⣿⡇⢻⡇⢸⡇⢸⢿⡧⢾⡏⠴⣦⣽⣧⠈⣶⣾⣿⡿⠁⢰⣿⣿⡄⠘⣿⠀⠸⣟⢁⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠻⣿⠇⣿⡃⢸⡇⠘⡇⠀⠊⠁⢸⡟⠳⠖⢸⣷⠀⠿⢿⡿⢧⣰⠿⠟⠛⠻⠀⠘⡧⣀⠈⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡘⢿⣿⠿⠋⠁⠂⠀⠐⠒⠁⠈⠻⢷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣠⠶⠋⣠⢞⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⡴⣋⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⡿⠋⣠⡶⡫⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠿⠉⣤⢞⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡘⠷⣄⡀⡀⠀⠀⠀⣀⡴⠞⢁⡴⢚⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢮⣀⣉⣛⣛⣛⣋⣉⡤⠶⣋⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣉⠉⠉⠉⠉⢁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠠⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⡆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⣠⡀⠀⠀⠘⠇
⠀⠀⠀⠀⡠⠄⠄⠤⢼⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣻⠀⠀⠀⠀⠠⣪⣞⣯⣭⣭⡍⠍⢙⣾⣢
⠀⠀⠀⢸⣄⣀⣀⣀⣼⡇⣩⣷⣄⣀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠾⣧⡑⢤⡀⠀⠀⠀⠀⢸⠿⣿⣻⣷⢻⣯⣸⣯⡯⡟
⠀⠀⡆⢰⠿⠛⢿⢿⠽⠿⠿⠟⠛⡏⢓⠀⠀⢠⠺⢿⡃⣼⠗⠰⢊⣛⣦⡷⡄⠀⠀⠀⠈⢦⠀⠀⠀⠉⠉⠁⠀⡠⠃
⠀⠀⠙⠤⣱⣶⣔⣦⣥⣀⠀⠀⠉⠐⠁⠀⠀⠀⢨⡏⠉⠛⠛⠂⠛⠉⠉⠈⠀⠀⠀⠀⠀⠀⠉⠐⠀⠤⠤⠔⠋

⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⣄⠘⠨⠇
⠀⠀⢀⠀⠴⢟⣋⠡⢰⡛⢳⣶⣖⢌⠁⠀⠀⠀⠀⠀⠀⢠⠔⠶⣴⣶⣦⣭⠀⠀⠀⠀⢀⣤⠖⠊⠀⠀⢠⣤⣶⣽⣿⣕⠢⡀
⠀⠐⢡⣾⣿⣟⣿⠿⣷⠤⢹⣻⡿⡆⣷⠀⠀⢀⣔⣤⣴⠶⠾⠛⣣⠚⠙⡀⠁⠀⠀⠀⣏⢟⡷⠟⢿⣿⡿⠿⠿⠿⢛⡻⢀⠇
⠀⠘⢄⡁⠀⠉⠀⠈⠉⠁⠁⣐⣊⡴⠁⠀⠀⠰⣮⡏⢀⠀⠤⠚⠒⠊⠉⠁⠀⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⠀⠍⠵⠒⠁
⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣄⣄⡀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⠀⣠⡈⠀⠀⠀⠀⠀⢀⣠⠄⠐⣂⣒⠲⠒⠀⠰⡶⡄
⠀⢀⣴⢂⡴⣟⣧⣗⡿⣟⡿⣿⡇⣳⡀⠀⠀⠀⡠⠒⠉⢙⠻⡻⢏⠼⣣⣀⣸⡄⠀⠀⠀⢰⢻⣿⣟⣉⠙⢚⡤⣞⡓⢦⠽⡽⡆
⠀⡊⢲⡖⠷⣦⠶⣅⣷⣛⣭⡟⠛⢻⢹⠄⠀⢰⡀⠑⠲⣤⠅⣡⣼⣴⣿⢛⠟⢻⠀⠀⠀⠸⡘⠿⢛⡛⡍⠉⠁⠺⡝⠁⢠⢣⠇
⠀⠈⠢⣍⡓⠾⠶⠎⠉⢉⠉⠀⢐⣡⠞⠀⠀⠈⠿⡐⢶⣟⠀⠈⠀⠀⠊⠁⡠⠏⠀⠀⠀⠀⠙⠢⢄⡀⠑⠈⠀⠀⣐⡊⠥⠊
⠀⠀⠀⠀⠈⠁⠂⠀⠒⠒⠂⠋⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠰⢰⡂
⠀⠀⢠⠒⠛⣭⢻⣉⣷⢤⣤⣤⣀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣠⠤⣤⣤⣀⡈⠁⠀⠀⠀⠀⠀⣀⡤⣤⡾⢽⣿⠿⣶⣤⡀
⢠⢾⠟⠳⡴⢋⠁⢱⡉⣛⣯⣟⠻⡷⡄⠀⠀⢠⣶⢾⣁⡤⣍⠽⠻⣶⡏⡱⣟⢮⣳⠀⠀⠀⢠⠎⣵⠛⢿⣇⠀⠀⠀⡾⠽⡏⣦
⠈⢧⡀⠀⠸⣦⣀⣐⣃⡑⣒⣉⣦⣿⠟⠀⠀⢻⠳⡀⠘⠮⠄⠠⠰⠿⠏⢐⣸⢖⣼⠃⠀⠀⢈⠀⢘⢏⠁⢀⡝⡺⣻⠗⢀⢇⡏
⠀⠀⠉⠛⠲⠮⠭⢍⣉⣥⣤⣴⠚⠁⠀⠀⠀⠀⠑⠢⠬⠀⣀⣀⣀⡈⠩⠄⠐⠋⠀⠀⠀⠀⠀⠑⠌⢑⠀⠉⠀⠉⠀⠐⠁⠊

⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣠⠤
⠀⣌⠁⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⠁⠀⠀⠀⠀⠀⠈⢧⠈⠈⠠⠄⠀⣶⡘⠉⠀⠀⡀⡄⢀⣀⡤⠤⠾⣿⡶⢶⣦⠤⠄⡀⢆⠀⠀⠀⠀⢀⢀⠀⡄
⠀⠈⠀⠠⠲⠆⠀⢠⠴⣶⣷⠆⠀⠀⠟⠉⠀⣒⣤⣰⣤⣤⣶⣏⣢⡚⠀⠤⠄⠁⠁⠁⠈⠀⠀⠀⠀⠘⠃⠠⢤⣇⣤⣤⣤⣶⣟⡟⣹⠀⡀⠀⠀⠒⠚⠲⠢⠆⣀
⠀⢠⣶⠖⠞⢀⣰⡟⣀⢌⠴⠉⠀⠀⠀⠀⠆⠛⠿⠋⢿⡟⠁⠀⠙⠥⠀⠀⢠⠀⠂⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠉⠉⠓⢧⢈⣄⢀⠀⠀⠔⠒⠀⢰⠄⠈
⠀⠀⡠⠙⠊⠀⠉⠑⢲⡘⠲⠏⠁⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣠⢸⣅⣀⣒⠒⠀⠐⣂⣓⣻⡄⠂⠀⠀⠀⢀⠀⠀⠀⢀⣹⡇⢬⠠⡄⠀⠀⠀⠰⠈⢁
⠀⠀⢵⠆⠀⠀⠀⠿⢡⣄⡄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢻⡇⡠⠤⣤⢤⠚⡻⠜⣿⢻⠟⠛⣓⢻⡛⢉⡿⠩⡍⠑⣦⠊⠁⠀⠀⠀⠀⢉⡠⠾⠀⢩⣤⠖⠛⠓⢆⠈
⠀⠦⠸⠷⢀⠀⠀⠙⠌⠁⣀⣀⣀⣤⣀⣀⠃⠀⠉⠉⠈⠉⣿⡆⠇⡄⠀⣥⢸⡀⢠⠐⡇⠛⠓⣀⣘⣷⠀⢈⠆⠁⠀⠀⠀⠀⠀⡤⣀⡀⢡⠈⠉⠋⠀⡟⠀⠈
⠀⠀⠀⣰⠔⠃⠀⠀⠀⠀⢈⡭⢽⣿⣿⣿⡇⠀⠀⠀⠀⠀⣷⣶⠀⣠⡄⢁⣀⠙⢁⣄⠀⠀⣦⡈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡉⠀⠐⣠⠞⠳⣄⢀⡀
⠀⠀⠈⠉⢙⠃⠀⠀⠀⠀⣽⠦⠰⢿⣉⣡⡇⠀⠈⠁⠀⠀⡏⢸⠀⢰⡄⠸⠿⠀⠘⠿⠀⠀⠻⠃⣸⠃⢀⡀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⡙⠃⠀⢐⡀⣉⡄
⠀⠀⠀⢔⡄⠉⠀⠀⠀⠀⢤⡀⠀⠄⠈⠁⠈⠠⣤⢄⠤⢚⡗⢈⣤⣤⠗⠤⡤⠞⠢⠶⠺⠶⠖⠚⠁⠤⠉⠀⠀⠀⠋⢠⠄⠀⠁⡀⠀⠀⠈⠻⠇⠀⠐⠤⣀⠀⠁
⠀⠀⢀⣾⣿⡅⠀⠀⠀⠀⠰⠁⠀⢀⡀⠀⠀⠀⠘⢬⠢⠁⠀⠠⠀⠀⡰⠈⠣⡀⠠⣀⣁⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣩⡤⠀⠉⠁⠠⣀⡀⠀⣈⢄⠀⠀⠈⢀⠉
⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈⡀⠊⠈⠉⠒⠦⢇⠀⠀⠀⠁⠀⠉⠀⠐⢏⡿⠧⠛⠿⠷⢶⣶⣿⠇⠀⠀⠀⠀⣿⡇⢂⠠⠶⣇⠙⠌⠹⣿⠈⠃⡀⡀⠸⡄
⠀⠀⠀⠀⠀⠀⢀⡄⠠⠀⠀⠀⠉⢠⡀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⡀⡀⠀⠀⠙⠁⠀⠀⠀⠀⢰⣿⣁⣴⣤⣤⣤⣴⣿⣷⣯⠋⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⣦⡾⠛⠀⢿⣧⠆⢫⡾⣷⣦⣤⣴⣶⣿⠿⢸⡗⠀⠀⠀⠀⠀⠒⠄⠀⠀⢙⡃⠀⢀⠰⠇⠘⠀⠐⠠⠀⠉⢿⣿⣿⣿⣿⣿⣿⠿⡿⢀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠐⠉⠀⠀⠀⠐⠛⠁⠀⠀⠙⠛⠛⠛⠉⠉⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠁⠀⠈⠂⠒⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠙⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠀⠉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠀⠀⠀⠠⠀⠂⢄⠀⠀⠈⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⠀⢀⠠⠂⠁⠀⢀⡠⠖⠃⠂⢄⠀⠀⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⡀⡀⣤⣦⣀⠀⡤⠒⠁⠀⠀⢀⠠⠚⠁⠂⠄⠀⠀⠀⠈⠓⠦⣀
⠀⠀⠀⠀⠀⠀⣀⠤⠚⠉⠀⠀⠀⡄⣤⣤⢤⣀⠈⢁⡬⡝⣦⣤⣤⡀⠂⠁⠀⠀⢀⠠⠊⠉⠑⠀⡀⠀⠀⠀⠙⠢⢄⡀
⠀⠀⠀⢠⠔⠋⠁⠀⠀⢀⣤⣤⣤⣀⠈⠁⡬⡜⢶⣶⢧⣅⠈⠁⡀⡥⡤⢄⡀⠊⠀⠀⠀⢀⣠⡞⠉⣹⡤⠠⡀⠀⠀⠉⠓⢤
⠀⠀⠀⢸⠀⠀⠀⠀⠰⠋⠉⠛⢉⣽⣧⣄⣡⡄⠀⠁⠀⢈⢀⣺⠉⠀⠀⠀⠈⡆⠀⣠⡞⠅⣹⠭⢾⠿⢀⣴⠷⠉⠁⠀⠀⠸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠐⠁⠉⠋⠁⡌⠗⠚⠲⢤⠋⠀⠈⢱⠒⠀⠨⠤⣕⢤⣌⣩⣞⣧⢀⣼⠗⠋⠁⠀⠀⠀⠀⠀⢠
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠂⠀⠀⢀⡀⠇⣀⡄⠘⠠⢀⠀⠀⢢⠤⠙⢩⡄⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠆⠀⠈⠉⠁⡠⢒⣂⠳⢜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠁⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⠠⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠙⡀⡎⠀⠀⢀⣶⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⣷⣓⢷⡰⣾⡃⠐⢲⣿⣽⠻⣯⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠋⠉⢹⡿⣻⣌⢻⡛⣺⣷⣾⣦⣴⣷⣿⣿⣷⣆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢃⠻⢗⣶⡞⣷⠙⣷⣤⣧⣿⣿⣿⣿⢿⣿⢣⡿⡽⢷⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠛⣤⡠⠤⣼⡛⢓⣾⣧⡇⢾⣧⠝⣫⡟⣻⣼⣾⣟⢼⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⠢⠶⠖⢋⣑⣒⡊⢷⠹⣾⣿⠎⡿⢃⡍⠀⢛⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠞⠳⠀⠛⡃⠸⠇⢀⣀⡀⢒⡋⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⢲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠤⢋⡤⠒⠂⢀⠀⠀⢙⡆⠀⠀⠀⠀⢀⣠⠔⠚⢉⠀⠀⠀⡉⠉⢑⠀⠀⠀⠀⢀⣤⣴⣶⣮⣍⣉⠉⠒⢤
⠀⠀⠀⠀⠀⢀⠎⣀⠈⠁⠀⢀⢀⣨⠇⠔⡊⢠⠀⠀⣠⡖⢩⠀⠀⠀⠀⡨⠖⠈⠀⠀⠁⠀⠀⡠⠂⠉⠉⠉⢛⠤⣀⣀⣱⡆⠈⢱
⠀⠀⠀⠀⣠⣧⡊⢀⡰⠆⠒⡉⡠⠀⣀⠬⠐⠁⠀⢰⡡⠄⠒⠒⡠⠔⠉⠀⠀⢀⣁⠔⠀⠀⢸⡁⠀⠀⢀⠔⣡⠤⠺⣿⣿⣿⠔⠉
⠀⠀⠀⡜⠘⣡⠞⠲⠤⣌⠭⠉⠒⠉⠀⠀⠀⠀⠀⠸⣉⠒⠒⠩⠄⠀⢐⡮⠔⠁⠀⠀⠀⠀⠘⢆⠉⠀⡉⠉⠀⢀⡘⠿⠛⠁
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠛⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⠤⣞⣩⠊⣙⣯⠵⡤⢄⠀⠀⠀⠀⠀⢀⡴⢉⡯⢥⠘⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⠀⠠⠤⢤⣴⣀⡀
⠀⠀⠀⠀⠀⡴⢳⡴⣏⡠⢏⣀⡽⢤⠚⠚⡄⠀⠀⠀⠠⣾⠀⠊⣀⣠⢥⡴⡗⠲⠦⡀⠀⠀⠀⢀⣤⡄⡀⢠⣙⣃⠨⡀⠽⣃⢠
⠀⠀⠀⠀⠀⠑⣾⣀⣸⣀⣀⣽⣄⣠⠶⡞⠀⠀⠀⠀⢠⠛⡚⠿⢸⢉⣳⡾⠢⢤⡈⠀⠀⠀⠀⢀⠝⣶⣾⣵⣮⣷⣬⣥⢲⣮⠞⠳
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠄⠀⠇⠀⠀⠀⠈⣈⠂⡘⠦⠊⣎⣠⠓⢀⡾⠁⢠⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⡄⠸
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠇⣀⠀⠀⠀⠀⠀⠈⠁⢧⡐⠀⠓⠛⠉⣁⡤⠒⠁⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠐⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⡀⠀⠋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⡠⠊⡠⢊⠐⢀⠀⠈⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠈⠀⢊⠔⠈⠔⢁⠔⡡⠂⡀⠂⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠴⠐⠁⠀⡡⠀⠀⠊⡠⠊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠉⠢⡀⠑⠀⡀⠀⠀⡠⠊⠀⡠⠈⠉⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠈⠢⡈⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢄⠀⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠐⢄⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⠄⠁⠀⠀⠁⢄⠀⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠁⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠊



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⣀⡀⣠⠶⠛⠻⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠭⠭⠭⠉⠁⠀⠀⣼⠋⠀⠈⠁⠀⠀⠀⠀⠛⠛⠲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⡀⠀⠀⡀⠀⠀⠈⡇⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠞⠀⠉⠛⠉⣧⡀⢀⠀⠁⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⣠⠦⡄⠀⠀⠀⠀⠀⠀⣰⠃⣀⣀⣀⠀⠀⠈⠉⠉⢳⣔⣚⣼⠇
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⢇⠀⡇⠀⠀⠀⠀⠀⠀⡟⠉⣁⠀⠉⢙⠓⢦⣶⢼⡄⠉⠉⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⣠⠀⠀⠈⠁⠀⠈⠳⠟
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡀⠀⢠⠏⢤⢷⡀⠀⣀⠀⠀⠑⢒⢒⣇⠀⠀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠃⠀⢸⠓⠒⣻⡟⠒⠉⠀⢳⡗⠒⣺⠂⠀⠀⠉⠩⣗⡚⠓⠂⠉⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢠⠏⡏⠉⢹⠀⣸⠡⡀⡞⠀⠐⢦⡴⠢⡀⢀⣀⡤⠖⣷⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡇⠀⠜⠀⣿⠀⣿⠀⠀⠐⠲⠶⠶⠎⠀⡀⠒⠶⢼⡇⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⣤⣀⡔⠓⠤⠙⠒⠀⠐⠃⠤⠛⠦⠀⢠⣤⠀⢠⠆⡄⢹⡄⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣛⣛⣛⠓⠒⠒⠚⠛⣿⢛⡛⠉⠡⠴⠯⠚⣇⡼⠁⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡖⠠⣍⡁⠒⠊⣉⠤⠐⠃⠀⠀⠀⠀⠼⠋⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢭⠛⠒⠀⠀⠀⠀⣀⣀⣠⡤⠤⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⡄⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠙⠋⠉⠉⠉⠁⠘⠿⠟⠃⠀⠀⠀⢠⡤⢀⣾⣿⣿⣿⣿⣿⣷⡄⢰⡇⠀⠀⠀⢸⣿⡇⣠⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣾⣿⣿⣿⣿⣿⣿⣿⣷⠘⡇⠀⠀⠀⠀⢙⠀⢘⠁⠸⡇
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⣿⣿⣿⣿⣿⣿⣿⠏⠀⡇⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠙⠃⠀⠉⠉⠉⠙⠃⠀⠀⠀⠀⡇⠀⠉⠻⠿⠿⠿⠟⠋⠀⠀⡇⠀⠀⠀⠀⣸⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠘⠀⠀⠃

⠀⠀⠀⢀⡃⠀⠀⠀⠀⣸⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠶⠆⢀⡀⢤⡄⣿⠸⣿⢃⣄⠀⠀⠀⠀⠀⠀⠘⠿⠁⢶⠂⣿⠀⠀⠀⠀⠀⠀⠠⠀⢿⡄⠀⢠⡄⠀⠄
⠀⠀⠡⣥⡅⠈⠃⢸⠀⢻⠀⡄⠸⡿⠀⠀⠀⠀⠀⠀⢀⣦⣀⣼⣀⣨⣄⠀⠀⠀⠀⠀⢁⠀⠈⠻⣤⠊⠀⠀⠈
⠀⠀⠀⢂⡂⠀⡆⢸⠀⢸⠀⡇⠀⡆⠀⠀⠀⠀⠀⣿⡿⡿⠿⢿⠿⢿⠿⣿⠀⠀⠀⠀⠈⣆⠀⡴⠙⣄⠀⣰⠁
⠀⠀⠀⠈⠅⢸⡇⢸⠀⢸⠀⣧⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠁⠈⠀⠈⠀⠁⠀⠁

⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⠋⠏⠙⡇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣨⣥⣶⣤⣄⡀⠀⠀⡇⣼⣧⠀⠀⢀⡀⡈⠀⡄⢀⡄⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⢀
⠀⠀⡴⠘⠏⢠⣿⣿⣿⣿⣿⣿⣿⡄⢸⡇⠙⠋⠀⠀⢸⡇⣷⣄⣿⣾⡅⠀⠀⠀⠀⠀⠀⠀⠻⣆⢀⠟⠃
⠀⠀⠀⠀⡇⢾⣿⣿⣿⣿⣿⣿⣿⣿⠈⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⣶⣾⣿⣿⡧⣨⢇⢾⣿⣿⣷⣶
⠀⠀⣆⠀⡇⠘⢿⣿⣿⣿⣿⣿⣿⠃⠀⡇⢸⡇⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠛⠁⠁⢀⡴⠁⠈⢧⠀⠀⠈⠛
⠀⠀⠟⠀⠇⠀⠀⠙⠛⠛⠛⠋⠁⠀⠀⠇⠸⠇⠀⠀⠸⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠹⣿⣷⠀⠀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣿⡟⢀⣴⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡇⣼⠏⣠⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢳⡄⠈⢱⡟⢰⠿⠿⠿⠿⠿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⢱⡀⠈⣆⠀⢿⣆⠈⠀⡇⠀⠀⠀⠀⠀⠀⠈⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⢳⡄⠘⣧⠈⢿⣧⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠈⣿⡄⠸⣧⡈⢿⣇⠹⣄⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠘⣿⣄⠹⣷⣌⣿⡄⠈⠳⣤⣤⣤⠶⠛⠁⠀⠀⠀⡀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠘⢿⣧⣹⡿⠶⠏⠈⠀⠁⠀⠐⢩⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡸⠀⠀⠀⠈⠻⠍⠀⡀⡀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⢀⣀⣀⣀⣠⣷⣤⣤⣿⣦⣦⣤⣴⣦⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡤⠤⣴⣶⣓⣒⣺⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢸⣇⣀⠤⠤⠒⣒⣋⣭⣭⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣠⣀⣀⡀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠟⣓⣋⣭⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠟⠿⠿⠋⣿⣿⣿⡟⡤⠊
⠀⠀⠀⠀⠀⠀⠀⣿⢡⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣛⣛⣛⣩⣭⣭⣥⠶⠶⣶⣶⠖⢒⣿⣿⠛⣭⣭⡭⢉⣿⣏⡻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣭⡽⠷⣶⣾⣿⣿⣯⠉⠉⠀⠀⠀⠀⠀⣴⣿⠏⢀⣾⣿⠃⢠⠟⣫⣾⣿⣿⣟⣛⣒⣩⠦
⠀⠀⠀⠀⠀⢀⣀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣏⠉⠉⠀⠀⠀⢸⠁⢿⠁⢻⣿⣧⠀⠀⠀⠀⣠⣾⡿⠋⢠⣾⣿⠏⠀⠀⠈⠁
⠀⢤⡤⢦⣼⣿⣿⢿⠿⣟⣛⣯⡅⠙⣿⣷⣤⡀⠙⢿⣷⣦⣄⡀⢠⡆⢀⡟⠀⠀⢻⡉⢣⣠⣴⣾⣿⠟⢀⣴⣿⣿⠋
⠀⠀⠈⠓⡌⢻⣿⣶⣾⣝⣛⡿⢿⡀⠈⠻⣿⣿⣦⡀⠙⠻⠿⣿⡿⠀⢰⣷⣶⣶⣾⣧⠀⢿⡟⠋⣡⣴⣿⣿⡿⠃
⠀⠀⠀⣸⠱⣿⡫⠝⠛⠉⠉⠉⠀⠀⠀⠀⠈⠻⢿⣿⣷⣦⣤⣼⠇⠀⣺⠛⠛⠋⣉⣉⣆⠈⣻⣿⣿⣿⠟⠉
⠀⠀⠀⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⡟⠀⣈⣿⣿⣿⣿⣿⣿⣿⣆⠀⠿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠠⣼⡟⠛⠛⠛⠛⠉⠉⠹⡆⠈⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣤⣾⠁⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠒⠐⠂⠀⠀⠀⠂⠂⠂⠂⠃⠂⠂⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠁⠈⠀⠃⠈⠈⠁⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣉⢉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⠶⠒⠒⠒⠒⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢲⠀⠻⡇⠁⠀⠀⠀⠀⠀⣠⠴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠙⠀⠐⢠⠀⠀⠇⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠁⠀⠐⡧⠀⣀⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠂⠠⠣⡃⠀⠉⠈⠀⠀⠀⠐⠀⠠⠀⣀
⠀⠀⠀⠀⠀⠀⢀⠠⠐⠈⠀⠀⠀⠀⣆⠀⢀⠀⢢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀
⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⣠⠐⡀⠀⠃⡁⠨⠀⡾⡛⠒⢧⡠⠥⠖⡂⠀⠀⠀⠀⠀⠉⠢
⠀⠀⠀⠀⡌⠀⠀⠀⠀⢨⠀⠄⠽⠄⠠⠆⠨⣺⢀⡧⡐⠠⣙⡴⣤⣀⡀⠀⠂⠄⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠁⠀⠀⡰⡀⠀⡠⠚⠛⠂⡠⠏⠕⡡⡔⠀⡰⡀⢀⣀⠀⠙⠟⠓⠩⢤⡀⠀⠀⠀⡇
⠀⠀⠀⠀⡆⠀⠀⠀⡄⢈⠄⣀⠀⠐⡸⠤⢀⣤⠴⠊⠁⠈⢲⠀⠀⡄⠔⡠⠾⠀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠸⡀⠀⠸⠈⠚⠘⢁⠔⠤⠠⠔⠃⠈⡛⠶⠠⡴⠿⡈⠀⠀⠀⠂⠀⠘⠁⠀⠀⡼
⠀⠀⠀⠀⠀⠙⣄⠀⠀⠠⢴⠃⡀⢀⠀⠄⠀⠈⠁⠀⠰⠔⠓⠂⠖⡶⢂⣄⠂⠀⠀⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠑⠰⣅⠀⢀⠐⢀⠐⣳⠄⠀⠀⠀⠊⠀⠈⠀⢀⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠰⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠈⠁⠂

⠀⠀⠀⠀⠤⡤⢀⡀⠀⠀⠴⠶⢤⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂
⠀⠀⠰⢤⡤⣀⣀⢀⠃⠒⠒⢦⢈⠗⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠘⢩⠏⡧⢿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠁⠀⠀⠠⣵⣦⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠃⠀⠀⣠⠞⠀⠑⢬⡂⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⠀⠀⠆⠀⡹⠊⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⡀⠉⠦⠒⠉⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠀⣴⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⢀⣿⠗⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠟⠋⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⡄⣀⠀⠀⣠⠴⢥⣌⢳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸
⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣆⣀⡀⣸⠵⠓⠲⣄⣹⢾⠈⠉⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡿⠃⠀⠀⠀⡽⣽⢻⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠨⢤⡍⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣀⣀⠀⠠⠀⠂⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⡠⠂⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣼⣷⣿⣧⠀⣀⠀⢷⠃⠀⠀⠀⠀⡀⠄⠀⠉⡁⠀⢃⣬⡁⣀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠉⠁⠀⠉⠀⠁⠀⠀⠀⢀⠈⠀⡄⠀⠀⢰⣦⡼⡿⠛⠉⠀⢄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢰⣶⣴⠿⠛⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠟⠒⠠⠃⠒⠦⠿⠿⠿⠿⠛⢷⣝⣾

⠀⠀⠀⡠⣀⣦⠦⡠⡀⠀⠀⠀⠀⠀⡰⡄⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⠀⠀⠀⠀⠀⠂⠄⠀⣰⣦⡀⠀⠀⠀⠀⡄⡄⡄
⠀⠀⠀⠈⠻⢔⠔⣩⡒⠀⠀⠀⠀⡐⠤⠙⠄⠀⠀⠀⢀⡮⣤⡤⢤⣄⢵⡀⠀⠀⠁⠤⢾⣜⠬⠋⠀⠀⠀⠀⠐⠂⠒⠒⠄
⠀⠀⠀⡰⡖⢣⠑⡤⠰⠀⠀⠀⢠⢫⡔⠭⡨⡄⠀⠀⠈⢟⠉⠙⠛⠉⡺⠁⠀⠀⢀⢔⠝⠁⠐⠄⠀⠀⠀⠀⠀⠀⠀⣀⠆
⠀⠀⠀⠈⠐⠀⠐⠈⠀⠀⠀⠀⠈⠙⠛⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠂⠃⠀⠀⠂⠈⠁⠈⠉⠀⠂
⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⢔⠩⣰⠈⠁⢸⠀⠀⠀⢀⣹⣿⣿⡏⠀⠀⠀⠀⠰⠛⠆⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠃⠀⠀⠀⠀⣠⢴⡶⠦⠤⣄⡀
⠀⠀⢰⠀⠈⢸⠀⡀⢸⠀⠀⠀⢀⣸⡻⣿⣁⠁⠀⠀⠀⢠⢀⡼⠉⠑⡄⠀⠀⠀⠀⠐⠤⢇⠴⠀⠀⠀⠀⣇⣈⣀⡤⠖⢊⠆
⠀⠀⠐⠄⡃⢀⠈⡃⠼⠀⠀⠀⠈⢶⣦⣴⡾⠀⠀⠀⠠⢞⣚⣓⣒⣛⡲⠄⠀⠀⠀⠉⢐⡚⠈⠀⠀⠀⠀⠑⠀⠐⠈⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈
⠀⠀⠀⢀⠄⠠⢀⡀⠀⠀⠀⠀⠀⢀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡠⠂⢄⡀
⠀⠀⠀⠔⢀⡀⠀⠀⠑⠀⠀⠀⠠⣼⠿⢩⣦⠀⠀⠀⠀⢀⡤⣾⣇⣁⠀⠀⠀⠀⠘⣤⣢⣔⡆⡀⠀⠀⠀⡠⠤⠐⠀⠀⠁⡄
⠀⠀⢘⠠⡃⡈⢀⠠⠈⠀⠀⠀⠀⠂⢳⠁⠀⠀⠀⠀⠠⡼⠿⠮⠿⠿⢧⠄⠀⠀⠀⣿⢿⢿⣇⠄⠀⠀⠀⣯⣭⣭⡿⣼⢷⡇
⠀⠀⠀⠐⠠⠐⠂⠀⠀⠀⠀⠀⠀⠰⠼⠤⠃⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠲⠤⠴⠁

⠀⠀⠀⡠⠀⠀⠀⠠⡀⠀⠀⠀⠀⠴⠒⠟⠆⠀⠀⠀⠀⢀⠀⠠⣶⠀⡀⠀⠀⠀⢀⡴⠒⡉⠀⠂⠀⠀⠀⠀⢰⠘⢀⠱
⠀⠀⢰⣷⢳⣖⡖⢲⣷⠀⠀⠀⠀⡒⠊⠐⠂⠀⠀⠀⠀⣼⠤⠦⣿⡴⣧⠀⠀⠀⠸⠀⠀⢇⠀⡀⠀⠀⠀⠀⠠⠔⠀⠠
⠀⠀⠈⠫⠉⠉⠉⠩⠋⠀⠀⠀⠀⣅⣈⣁⡀⠀⠀⠀⠠⣟⣋⣉⣉⣙⣻⠄⠀⠀⡘⢉⠀⠀⡉⠑⡄⠀⠀⠀⠀⡨⠑⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⣀⣀⡄⠀⠀⠀⢠⡤⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⡄⠠⢄⡀⠀⠀⠀⠀⠀⡀⡀⡀
⠀⠀⠀⢀⡁⠀⠅⣁⠀⠀⠀⠀⠀⠐⣶⣶⠂⠁⠀⠀⢸⢀⠔⢢⠀⣤⣄⠀⠀⠀⡞⢙⣦⢶⡘⢻⠄⠀⠀⢠⡤⠥⠥⠤⣤
⠀⠀⠀⠨⠄⠢⠤⠤⠀⠀⠀⠀⠀⠀⠳⠆⠀⠀⠀⠀⢸⣈⣍⠁⢁⣍⣉⡆⠀⠀⣷⠾⢦⡬⠧⣼⠆⠀⠀⠀⠶⠄⠠⠴⠂
⠀⠀⠀⠀⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠙⠋⠀⠀⠙⠋⠀⠀⠀⠈⠓⠶⠴⠒⠋⠀⠀⠀⠘⠓⠒⠒⠒⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠖⠛⠛⠛⠛⠛⠛⠛⠛⠲⠶⠦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⣶⢸⢰⡆⡆⠀⠀⠀⠀⣴⣷⠀⠀⠀⠀⣠⣴⣶⣶⣄⠀⠀⠀⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⡏⢸⢸⡇⡇⣠⡤⠤⣸⣿⣿⠠⠤⠄⣴⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⡇⡇⠀⠀⠀⣿⣿⣿⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⢀⡠⠞⠀⡇⢸⢸⡇⣿⠀⠀⢸⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⡀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⣇⣸⢸⡇⣿⠀⠀⢸⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠙⢦⡀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⢻⡄
⠀⢀⡟⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⢻⡄
⠀⣾⠁⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡿⠁⠀⠀⣾⣿⣿⣿⠀⠀⠀⠈⠙⢻⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠈⣷
⢰⡇⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⢸
⣼⠁⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠈
⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢈⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⡀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢠
⠸⣇⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢸
⠀⢻⡀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢀⡟
⠀⠈⢧⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⣿⣇⣀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⣀⣸⣿⣿⠈⠁⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠈⠉⠙⠓⣿⣿⣿⠐⠛⠋⠉⠁⢸⣿⣿⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠀⠿⠿⠇⠀⠀⠀⠀⠻⠿⠿⠀⠀⠀⠀⠀⠸⠿⠿⠀⠀⠀⢀⣠⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⠲⢶⣶⣤⣤⣤⣤⣤⣤⣶⣶⠒⣛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠛⣉⣦⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⣀⣤⠶⠛⠉⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⣁⣠⡴⠞⠋⠉⠀⠀⢀⣀⣤⠶⡞⠋⢉⣹⠆
⠀⠀⠀⠀⠀⠀⢰⡶⢤⣤⣤⣄⠀⠀⣀⣤⠶⠛⣁⡤⠶⠛⠉⠀⠀⢀⣀⣤⠶⠚⠋⣁⣸⡤⠷⠛⠋⠁
⠀⠀⠀⢠⡾⢋⢻⢥⣂⣰⡄⢸⣷⣛⣩⡴⠖⠋⠉⠀⣀⣠⡤⠶⠛⠋⣁⣤⡴⠞⠋⠉
⠀⠀⣰⠏⣤⡍⢊⠀⣉⡙⠙⡄⢸⡏⠁⢀⣠⡤⠶⠛⢉⣡⣤⠶⠚⠋⠉
⠀⠈⢻⡆⢔⣩⠖⢢⡄⠰⣌⣡⡼⠷⠚⢋⣡⣤⠶⠚⠋⠁
⠀⠀⠈⣷⠰⢍⣡⡼⠶⠛⣉⣡⡴⠶⠛⠉⠁
⠀⠀⠀⠈⢻⣏⣥⣶⠾⠟⠋

⠀⠀⠀⠀⠀⣀⣀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣒⣒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡄⣠⣆
⠀⠀⠀⠀⢼⣿⣿⡿⠛⠛⣷⡄⠀⠀⠀⠀⠀⠀⠀⣧⢿⣿⣼⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿
⠀⠀⠀⠀⠈⠻⣿⣷⣄⣨⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⣡⡶⠂⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣟⣡⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠟
⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⣐⡀⢤⠠⡀⢦⡑⢄⠀⠀⠀⠀⠀⠀⣄⠀⢠⡄⣤⠦⠘⣦⠀⠀⠀⠀⢀⣾⣿⠛⣛⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠉⠛⠻⠷⠾⠷⠛⠋⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢿⣾⣿⠋⠀⠀⠀⠀⠸⣿⣿⡾⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣾⣆
⠀⠀⠀⠀⠀⠀⢨⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣀⡤⣤⣤⣀⡠⢤⣄⡀⠀⠀⠀⠀⠸⢿⣶⣶⣶⣶⡿⠇⠀⠀⠀⠀⠀⠀⢹⣿⣿⣻⡶
⠀⠀⠀⠀⠀⠀⠸⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣶⣾⣷⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⢨⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣟⡛⠻⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡖⢋⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠿⢛⣷⡄
⠀⠀⠀⠀⠀⠀⠋⣵⡆⠂⠙⠁⠀⠀⠀⠀⠀⠰⣿⡿⠟⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣀⣰⣛⣿⠇⠀⠀⠀⠀⣠⡀⠀⠀⡀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⡀⣶⠆⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠙⠻⠿⠟⠋⠀⠀⠀⠀⠀⠈⠋⢨⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋


⡿⠿⡿⠿⡿⠿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣻⣿⣛⣿⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢧⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠈⢣⠀⠙⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⡻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣿⣃⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠻⠿⠿⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠿⠿⠿⠿⠟⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣀⣀⣤⣶⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢲⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣧⢻⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠿⠋⠁