Key Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠸⣄⡀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⠀⠉⢋⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⣠⡾⠻⡁⢳⣤⣶⣶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⣀⣴⠟⠉⢀⣠⠿⠛⠁⠀⠹⣦⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠈⢧⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⢀⣤⠾⠻⣄⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣠⠖⠒⠒⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠀⠀⠀⠀⠀⣠⡶⠛⠁⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⢀⡾⣟⠳⣦⣀⡴⢻⡟⠁
⠀⠀⠀⣠⠖⠋⠉⠉⠙⠛⠶⣦⣄⠈⠳⣦⡀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⣀⡴⠟⠁⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⣠⣾⢦⣀⡿⢮⡆⠀⠉⠛⠋
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠙⢦⡀⣠⠞⠁⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠘⣮⠻⡉⠓⠚⠁
⠀⣸⠃⠀⠀⠀⢠⡴⠶⢦⣄⠀⠀⠀⠈⠳⣤⡼⠛⠁⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢀⣠⠞⣻⠇
⠀⣿⠀⠀⠀⠀⣾⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠤⠴⠃
⠀⣿⠀⠀⠀⢸⡿⡄⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⢀⣤⣾⠋
⠀⢹⣆⠀⠀⠈⣇⠹⣄⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠙⣧⠘⣧
⠀⠀⢻⡄⠀⠀⠘⣦⠘⢷⣄⠀⠀⠀⠈⣷⠀⠀⠀⠀⢹⡄⣿
⠀⠀⠀⢻⡄⠀⠀⠈⠳⣄⠉⠳⣄⡀⢀⡽⠀⠀⠀⠀⢸⠇⣿
⠀⠀⠀⠀⠻⣆⠀⠀⠀⠈⠙⠓⠚⠛⠋⠁⠀⠀⠀⢀⣿⡾⠃
⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⣀⣀⣀⣤⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠉⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠠⠔⠒⠲⢦⡀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⢀⡞⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⣿
⠀⠀⠀⢀⡾⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⡇
⠀⠀⠀⣸⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣼⠁
⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⣸⡿⢦⣀
⠀⠀⠀⣿⠀⠀⠀⠈⠿⣦⣀⣀⣤⡾⠛⠀⠀⢀⣼⡏⠀⠀⠉⠛⠶⣄⡀
⠀⠀⠀⢹⡆⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⣠⡾⠉⠙⠷⣤⣀⠀⠀⠀⠙⠳⢤⣀
⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠀⠀⠀⠀⠀⠉⠻⢦⣄⡀⠀⠀⠈⠙⠦⣄⡀
⠀⠀⠀⠀⠀⠈⠛⠳⠶⠶⠶⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⠀⠀⠀⠉⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠈⠙⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⠀⠀⠀⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⡀⠀⠀⠈⠑⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢀⠎⡽⠀⠀⠀⠀⠀⠙⠳⢦⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⠎⡼⠁⠀⠀⠀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢠⢃⡞⠀⡰⠒⡄⢠⠋⢱⠆⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⢁⠞⠀⡜⢀⡞⡰⠃⣰⠋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠚⢀⠞⡰⠁⣰⢃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢰⣁⡞


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

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣀⣀⣉⣀⣀⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡤⠖⣋⠭⢥⢄⡀⠀⠀⠉⠉⠙⠶⣄⠀⠠⡀
⠀⠀⣠⢈⣀⡴⠉⢀⡯⠶⠒⠾⣶⢾⣆⠀⠀⠀⠠⠄⠈⠳⣄⠈⢄
⠀⣬⠇⠀⡟⠀⠀⠈⢄⠀⠀⠀⠀⡩⠋⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠄
⢨⣾⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰
⠀⣷⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢠⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠃⢀⠛
⠀⠸⣏⠦⡀⢁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠁⠀⠀⠀⢆⠂
⠀⠀⠜⢦⣙⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣄⡀⠀⢶⢄⠀⠑⠄
⠀⠀⠀⠈⠛⠛⢷⡦⣭⣩⣒⣒⣒⣒⣦⠀⠻⣷⡀⠑⢿⢢⡈⠳⡕⢄⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⢋⢿⣳⣤⣿⣷⣄⠀⠑⢌⢢⡈⠣⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠍⠩⠻⣾⣷⣄⠂⠙⢽⣦⣌⢮⣧⣄⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⣶⣶⠁⠤⠤⡄⠿⠷⠦⠵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣄⠀⠙⢟⡷⡘⣾⡢⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣓⣴⣶⣝⢜⣮⡙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡆⠑⢝⢦⡑⢟⢄⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧⡀⠀⠳⣽⣌⠳⣽⢦⡀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣶⣤⣼⣿⣷⡌⠷⢽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⣋⣭⠽⠶⠶⠿⢭⣟⡲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⡴⠛⠁⠀⠀⠀⠀⠀⠀⠈⠙⢶⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⢤⣄⣀⠀⠀⡏⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⣠⣤⣄⣈⠉⠙⡃⢸⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢿⣤⣌⣉⠙⠒⠇⠈⣧⣀⡉⠙⠲⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣦⣼⣌⣙⠛⣶⠀⢻⡀⠀⠀⢀⣠⠤⠖⢒⡟⢠⠟⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣀⡴⠶⠛⠛⠳⢶⣤⡀⠈⠉⠙⠛⠋⠀⣸⠇⢠⡴⠋⢠⣴⠿⢋⣴⣻⠄⠀⠀⠈⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠉⢿⡄⠀⠀⠀⠀⢰⣿⣴⠋⠀⠀⠘⠻⠶⠿⠛⠁⡀⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⡿⣻⠃⠀⠀⠀⠀⠀⣴⡶⠟⠛⠛⠳⣄⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠘⣆⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⢀⡾⢃⡟⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⣀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠈⠙⠲⠶⠶⠾⠟⠋⠀⠀⢀⣴⡿⠁⢸⡇⠀⠀⠀⠀⠀⠈⠻⠶⠞⠛⠉⣀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠻⠶⣦⣤⣄⣀⠀⠀⢀⣠⣴⠟⠉⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠉⠁⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⣦⠀⠀⠈⢉⣿⠟⠛⠛⠉⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⠄⣀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣀⣀⣤⣤⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢠⡟⡾⠉⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⢀⣠⡤⠶⠒⠋⠙⣦⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢀⡞⣼⠁⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⠀⢰⡟⠁⠀⠀⠀⠀⠀⢹⡄⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⡼⣱⢣⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⢀⣤⣴⠶⠛⠋⠁⠀⠘⡆⠘⢷⣤⣤⣄⣀⣀⠀⢨⡗⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡿⡸⢡⢧⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⣠⡾⠁⣀⣤⣤⣬⣉⡉⠛⠛⠋⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢰⡏⣰⢃⠏⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠹⣷⡀⠀⠀⣠⡾⠋⢠⡾⠋⠀⠀⠀⠈⠉⠑⢦⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⢠⡟⢠⢇⠎⢸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠘⠿⢶⠾⠋⠀⣠⡿⠁⠀⠀⠀⠀⣀⣤⡴⠞⠀⠀⢀⣾⠃
⠀⠀⠀⠀⢀⡾⢁⠏⡜⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⣿⣁⣠⣴⡶⠟⠛⠉⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⣼⠃⡌⡘⠁⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢀⣀⣤⣴⡶⠟⠋
⠀⠀⠀⣸⠇⡘⠠⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⣀⡀⣀⣀⣠⣤⣶⠶⠟⠛⠋⠉
⠀⠀⠀⠹⡄⢀⣡⠴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠘⠉


⣶⣶⣶⣶⣶⡾⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣾⣶⣿⠶⣶⠷⠾⠿⢷⣶⡶⠶⣾⠾⠶⢾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⠿⠿⠛⠛⠿⢿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⠛⠿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⢀⣤⣄⡀⠙⠻⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠀⠀⢿⣯⣿⣿⣷⡶⠾⠛⠿⣷⣶⡶⠾⠿⠷⢶⣦⣠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⢀⣽⡿⠋⠀⢀⣀⣠⣾⠟⠁⠀⠀⣀⠀⢀⣨⣿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣅⠀⠀⠀⢀⣾⡟⠀⠀⢀⣿⢿⣿⠏⠀⠀⣴⣿⣿⣿⣿⠟⠁⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣦⡀⢸⣿⠀⠀⠀⠀⠻⣿⣿⠀⠀⠀⢿⣿⣿⣿⠏⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣻⠏⢸⣿⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣵⠋⠀⣼⣿⣆⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⢀⡠⠔⣚⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡷⢀⣼⠏⠈⠻⣷⣦⡤⠤⠤⠬⠿⣷⣦⣄⣀⡤⠖⢊⣡⣴⡾⢿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠃⢀⣾⠃⠀⠀⠀⠀⢹⣧⣤⣤⣤⣤⣾⣿⠟⢻⣷⣿⡟⠉⠹⣧⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣾⣿⢣⡿⠁⠀⠀⠀⠀⠀⠈⣿⡇⠀⣹⠀⣿⠆⠀⠀⠉⠛⣿⡷⠀⠹⣇⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠉⣡⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⢻⠘⣿⠄⠀⠀⠀⠈⢿⣧⣄⠀⢻⡆⠈⢿⡄
⠀⠀⠀⠀⠀⠀⣰⣿⡿⠿⣱⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⠈⢻⡆⠀⠀⠀⠀⠈⠙⠿⣦⡀⢻⡄⠘⣷⡀
⠀⠀⠀⠀⠀⢠⣿⡟⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⣻⣷⠀⢿⡀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠙⠷⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⢸⣆⣿⠇⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠈⣧⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⡇⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣾⣷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⠋⢳⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣉⣉⠳⣿⡷⠛⣉⣙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⢠⠋⠁⢈⡇⢸⠀⢮⠁⠈⢳⠈⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⡴⢇⠘⣄⠀⠻⠴⠋⠳⠼⠃⠀⡼⢀⡷⣌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⡞⠀⠈⢦⡘⠦⣄⡀⠀⠀⢀⣤⠞⣡⠞⠀⠈⣇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣧⠀⠀⠀⠙⠢⢌⡉⠒⠚⣉⡤⠞⠁⠀⠀⢀⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⠣⣄⣀⢀⣀⡤⠛⡹⣏⠣⣄⣀⠀⣀⣠⠞⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣭⣭⣭⡴⢋⣠⣈⠓⣦⣭⣍⡩⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡉⢩⠤⢤⡉⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠻⣀⣠⠇⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⠤⠴⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠛⠛⠛⠛⠛⠛⢟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡭⠗⠀⢠⡖⠒⢟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢲⣆⠀⢸⡇⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠁⠀⠸⣧⣀⣽⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⣀⣀⣀⣀⣀⣯⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠶⠶⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⠶⠶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠤⠤⠞


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡟⣿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⣩⡴⠶⠒⠶⠮⣕⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢣⡞⠁⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠤⠒⠒⢤⣀⣤⣤⣤⣤⣯⣧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⣹⣿⣞⠛⠓⣦⠈⠳⣤⢤⡤⣤⣄⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠸⣅⣈⣻⣗⡶⣿⠃⠀⢸⣡⣞⡡⠞⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⡿⠟⠉⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢠⠋⢠⠏⠆⠀⢠⣴⠋⠀⠀⠀⢰⠀⠸⡄⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡞⠀⡞⠀⣠⠴⠋⠈⠢⣤⠴⠢⢄⢣⠀⠹⡄⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢱⠁⢰⠃⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢣⢣⠀⠹⡄⠷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡏⢀⡏⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢳⠀⠹⡄⠻⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣸⠁⣼⠰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣳⡀⠹⡜⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢧⠇⢠⠇⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣳⡀⠹⡼⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡞⡼⠀⡞⢶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣳⡀⠹⡄⢹⡆
⠀⠀⠀⠀⠀⠀⠀⣸⢳⠇⢸⠃⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣻⡄⢹⡞
⠀⠀⠀⠀⠀⠀⠀⠹⣟⣠⡿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠒⠛⠛⠛⠛⠛⠓⠒⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⢀⣠⣤⣤⡴⠶⠶⠶⠶⣶⣤⣤⡈⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⣠⡶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣈⡙⠳⣄⡀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣚⣭⣤⠬⢭⣷⣦⠹⢦⡈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠙⡇⠀⠳⡄⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢹⡀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢰⡇⠀⠀⡴⣛⣭⣭⣗⢦⡀⠀⠀⠀⠀⠀⠀⠙⠳⠦⠤⠤⠶⠋⠀⠀⠀⠈⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⢸⢡⡏⠀⠀⠈⠳⣝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡆⠀⠸⡞⡇⠀⠀⠀⠀⠈⢳⡱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠈⢧⠀⠀⠹⡝⢄⠀⠀⠀⠀⠀⠙⣎⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠘⣆⠀⠀⠈⢦⡳⡄⠀⠀⠀⠀⠈⣧⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣰⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢧⡀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠙⢬⡦⣄⣀⠀⢀⡼⣸⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⢇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣮⡿⠀⠀⠀⢀⡤⢶⡄⠈⠳⢤⡀⠀⠀⠉⠓⠿⠭⠭⠞⠁⠀⠀⠀⠀⠀⣠⡴⠋⠀⢀⡴⢃⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣴⠟⠀⠀⣠⠖⠋⣠⠏⠀⠀⠀⠀⠉⠳⢤⣄⣀⣀⣀⣀⣀⣀⣀⣤⠴⠖⠋⠁⠀⢀⡴⠋⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡴⠟⠀⠀⠀⢀⡴⠊⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣠⠴⠋⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⢀⣠⠞⠉⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⣀⡴⡶⠦⣤⣀⣀⣀⣀⣀⣀⣀⣠⠴⠖⠋⣡⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠒⠶⠛⠁⠀⣠⠴⠋⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣠⠶⢋⣤⠿⠶⣤⣤⣈⣉⣉⣉⣉⣁⣠⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⣠⠞⠁⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⣴⠟⠁⠀⠀⠀⠈⠉⠉⠉⠙⠛⠉⠉⠁
⠀⠀⣀⣤⣄⣠⡴⠏⠀⢀⡴⠋⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣀⡴⢋⣡⠾⠋
⣠⠾⠋⠀⠀⠀⠀⣠⠶⠋⢀⡰⠊⠁⠀⠀⠀⠀⠀⠀⢀⣠⠞⣡⡶⠟⠁
⣿⠀⠀⠀⠀⣠⠞⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣤⠞⠋
⣿⠀⠀⣠⠞⠁⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⢀⠴⣋⡴⠋⠁
⣿⡤⠊⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀⡠⠚⣡⡾⠋
⣿⣧⣤⣎⠁⠀⠀⠀⠀⠀⠀⣠⠴⢋⣴⠟⠋
⠈⠛⠶⣬⣝⣓⠲⠦⠤⡴⠛⣡⡾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣾⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⠋⠉⠀⠀⠀⠈⠙⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⠉⠻⢿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⢀⣤⣄⠀⠈⠛⠿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃⠀⠀⣾⣏⣿⣿⣶⣦⣴⣶⢿⣿⣷⣤⣤⣴⣶⣶⣶⣤⣤⣀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠁⠀⠀⠀⠈⢻⣿⡿⠟⠉⠀⠀⠀⣠⣿⡿⠋⠁⠀⠀⠀⠀⠉⣻⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⢀⣴⣾⣾⣿⠋⠀⠀⢠⣤⣶⣤⣤⣾⣿⠟⠉⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⣄⠀⠀⠀⢰⣿⠏⠀⠀⠀⢸⣟⣿⣿⠇⠀⠀⢰⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣦⡄⣿⣿⠀⠀⠀⠀⠈⠻⣿⣿⠀⠀⠀⠈⠿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢏⡟⠁⢿⣿⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢣⡞⠀⢀⣼⣿⣆⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠚⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣿⠀⢀⣾⠏⠙⢿⣷⣤⣀⣀⣀⣀⣈⣻⣿⣦⣀⠀⠀⣀⡤⠴⠚⣉⣤⣶⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⠟⠉⢠⣿⠃⠀⠀⠀⠉⢻⣿⣁⣀⣀⣀⣀⣉⣻⣿⣿⣿⣧⣤⣶⠿⠛⢿⡆⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⡄⣰⡿⠁⠀⠀⠀⠀⠀⠘⢿⣿⡟⠛⢻⠛⢻⣟⠋⠀⠸⠿⣿⣷⣤⡀⠈⣿⡄⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟⣱⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⠀⣼⡟⠀⠀⠀⠀⠈⣽⣿⠃⠀⠘⣷⡀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣏⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⠀⣿⡷⠀⠀⠀⠀⠀⠸⣿⣶⡄⠀⠸⣧⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⢏⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⡆⠙⣿⠀⠀⠀⠀⠀⠀⠀⠛⢻⣶⡀⠹⣧⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⢠⣿⡿⠃⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⢸⡇⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣷⠄⢹⡆⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⢸⡇⣻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣅⠀⠀⢻⡄⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⢸⡇⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀⠈⢿⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣧⣈⣇⣴⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠶⣦⣄⡀
⠀⠀⠀⠀⣴⠟⣁⣤⠶⠶⢦⣄⡉⠻⣦
⠀⠀⢠⡾⠁⣼⠋⠀⠀⠀⠀⠈⢻⡄⢹⡇
⠀⢠⡿⠁⠰⣇⠀⠀⠀⠀⠀⠀⠀⣷⢀⣿
⠀⣾⠁⢀⣤⠿⠆⠀⠀⠀⠀⠀⣰⡟⢸⡇
⢸⡇⢰⡟⠀⠀⠀⠀⠀⠐⣶⠛⠉⢀⣾⣦⣄⡀
⢸⡇⢸⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⡏⠀⠈⠛⠳⢶⣤⣀
⠘⣧⠘⣧⡀⠀⠀⠀⠀⣠⠏⢀⡾⠋⠛⠶⣤⣀⡀⠀⠈⠉⠛⠷⣦⣄⣀
⠀⠙⢦⣈⠙⠳⠶⠶⠛⢁⣴⠟⠁⠀⠀⠀⠀⠉⠛⠷⢦⣄⣀⠀⠀⠈⠙⠻⢶⣤⣀⡀
⠀⠀⠀⠙⠛⠶⠶⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⡀⠀⠀⠉⠛⠶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠦⣤⣀⠀⠀⠉⠙⠻⢶⣤⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣄⡀⠀⠀⠉⠛⣶⠋⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⣤⣴⠃⢠⠟⠛⠳⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡴⠻⢦⣄⣀⠀⠈⠙⠛⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣍⠻⠶⣤⣀⡀⠀⠉⠛⠳⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠦⠼⠃⠀⠀⢉⣙⠳⣦⣄⡀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⣀⡀⠀⠀⠺⣏⡙⠋⠈⠙⠳⠶⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡁⠀⣼⠋⢙⣷⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠃⠀⣾⠃⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠟

⠀⠀⠀⠀⢀⣤⡶⠚⠛⠛⠛⠻⠶⣤⣄⣀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⣴⠟⠁⠀⣀⠤⠤⠤⣄⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠶⣤⡀
⠀⠀⣸⡏⠀⠀⣾⡿⠛⠻⣦⡈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⣿⡁⡀⠸⣿⠀⠀⠀⢹⣷⡏⠀⠀⠀⣶⠶⠛⠛⠛⠓⠶⣄⡀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⣿⡇⢳⠀⠙⢷⣤⣤⠾⠋⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⢻⣆
⠀⠀⣿⣿⡞⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣿⡀
⠀⠀⢹⣿⠁⠀⠀⢀⣀⣤⠴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠆⣿
⠀⢀⣿⠃⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠌⣼⡟
⠀⣼⠃⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⣼⡿
⣸⡏⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡜⠁⠀⠀⠀⠀⠀⣿⣿⠁
⣿⡇⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⢩⣕⢦⠀⠀⠀⠀⠀⢿⣿⡀
⣿⣧⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⢀⡀⠄⠊⠁⠀⠈⡄⢿⡓⢅⢦⡀⠀⠀⠈⠻⢷⣄⣤⣄⡀
⠸⣿⢣⡀⠀⠀⠀⠀⠈⠉⠒⠒⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠂⠳⡌⠢⡑⢄⠀⠀⠀⠀⠀⠁⢹⣷⡀
⠀⠙⣷⣕⢤⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⡀⠀⠀⠀⢀⡴⠓⡜⢦⡑⢌⠣⡀⠀⠀⠀⡠⢸⣿⡇
⠀⠀⠈⠙⠻⠿⢾⣿⣍⣡⣠⣤⣤⣴⣶⣶⣤⣍⣳⣄⠀⠻⡀⠀⠈⢦⡱⣌⠢⡀⢄⠀⠀⢣⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣆⠀⠘⢆⠀⠀⠳⡌⢳⡔⢄⠢⡀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠈⠣⡀⠀⠈⢆⠙⣦⠱⡐⢄⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣤⣤⡙⢄⠀⠀⠱⡈⢳⡌⢂⠳⡀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⢻⣆⠓⢀⣀⣈⠢⡙⢦⡑⠜⢆⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣷⣶⣬⡇⠙⢮⠳⣄⠢⡳⡀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣷⠀⠀⠱⣌⢦⡀⢈⠂⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣭⡑⠢⡈⠣⡙⢦⠱⡕⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣧⠹⠀⠈⢪⠳⣌⢮⡢⡀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠑⢜⢧⡑⣄⢆⠀⠀⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣧⣴⣀⣀⣀⡀⠢⡙⣎⠢⡑⣄⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣤⣬⣥⠀⠘⢎⠳⣔⢌⠢⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⣿⣅⡉⡺⠳⡜⢧⡁⡀⢄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⠀⠈⢦⡙⣮⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣉⣉⣳⣼⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠂⣉⣉⣉⣉⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⡴⠊⠉⠁⠀⠀⠈⠉⠲⣄⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠦⠼⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡤⠤⠤⠤⢤⣒⠲⠷⢶⣠⣄⡒⢦⡀⠙⢢⡀
⠀⠀⠀⠀⠀⢀⢔⣥⠾⡄⢻⣭⠉⠙⠶⡜⣶⣄⠑⢌⢻⣽⡆⠻⠀⠀⠳⡀
⠀⠀⠀⠀⠀⡱⠋⠁⠀⠙⠈⠁⠀⠀⠄⡁⣿⡇⡀⠀⠀⢻⠀⠘⠀⠀⠬⡇
⠀⠀⠀⠀⡄⠇⡀⠀⠀⣀⡈⣀⠛⢁⠔⣴⡟⢁⠠⠀⠀⣸⠴⠁⠀⠀⠀⢈⢄⡀
⠀⠀⠀⠀⡱⢀⡀⠉⢶⠊⠴⠤⣐⣥⣾⡚⢌⠂⢀⡠⣣⠇⠊⡀⠰⠀⣠⣎⡲⢮⣑⠢⣀
⠀⠀⠀⢮⣀⣼⣳⡿⣰⡶⠾⢿⣟⡻⠷⠚⠒⢉⡀⢶⢻⣥⣤⣤⣦⡷⣅⡈⣙⠷⣭⣙⠶⣭⣂⢄⡀
⠀⠀⢀⢌⣾⡿⡑⣽⡟⠀⠀⠀⠉⠛⠓⢷⣹⡜⡹⡱⣧⠏⠁⠀⠀⠀⠈⠙⠛⠫⡶⢿⡿⢶⣭⡛⢮⣕⡢⢀
⠀⠀⢰⣞⡿⡑⣜⠏⠀⠀⠀⠀⠀⠀⠀⠈⣧⣧⡔⡱⣹⡞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠔⣾⠿⢷⣬⣙⣷⣯⣶⡀
⠀⡰⣿⡿⡱⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢷⣰⣧⢻⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠛⠛⠻⠿⠃
⣠⣿⣻⢟⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢻⣆⢧⢣
⢻⣳⣯⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣻⡜⣧⡆
⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠟⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋
⠀⠤⠀⢔⢄⠀⡂⢐⢄⠀⠀⡠⢐⠄⠀⠠⡀⢉⢐⠁⡪⣀⢑⠅⡪⡂⢐⢅⠂⡢⡢⢀⠅⠂⡂⢐⢄⡠⡀⡀⡀⢀⢄⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿
⠀⠀⠀⣤⣤⣤⣴⣶⣿⠛⣭⡉⠻⣿⣿⣿⣿⣧⠀⠀⠀⠀⢸⣿⣿⣿⠛⠛⠛⠛⠉⢁⣴⠲⣶⣤⠀⠀⠀⠀⠹⣿⣿⣿⣿
⠀⠀⠀⢹⣿⣿⣿⣿⡇⠘⠷⣶⡄⢸⣿⣿⣿⣿⣇⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⢸⣧⣈⡉⣿⡇⠀⠀⠀⠀⢹⣿⣿⣿
⠀⠀⠀⢷⢻⣿⣿⣿⣿⣦⣈⣙⣤⡿⠿⠛⠛⠛⠛⠀⠀⠀⢸⣿⣿⣿⡘⡄⠀⠀⠀⠈⠻⠷⠶⠛⢀⣀⣤⣤⣤⣤⣿⣿⣿
⠀⠀⠀⠈⣧⠻⠿⠿⠿⠿⠛⣋⣡⣴⣶⣿⠿⠿⠿⠀⠀⠀⢸⣿⣿⣿⣷⡹⣄⣀⡀⣀⣠⣤⡶⠿⠛⠉⣁⣀⣀⣈⣿⣿⣿
⠀⠀⠀⠀⠘⣷⣶⣶⣶⣶⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⠀⠐⠀⠉⠉⣁⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣴⡄⠀⠀⠀⠀⠀⡈⣤⣤⠜⠓⠀⠀⠀⠀⡀⠀⠀⠀⠀⣼⣿⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠉⠁⠀⠀⠀⠀⠀⠦⣾⠅⢻⡟⡆⢀⠀⠈⠡⠀⠀⠐⡿⣧⡝⠀⠩⠁⠀⠀⠀⠀⠀⠀⡐⣠⡘⣴⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠙⣳⢯⡷⠍⠀⠀⠀⠀⠀⠈⠻⡐⠉⠀⠀⠀⠀⠀⠀⠀⠀⢁⠠⡅⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⢀⡠⠔⠁⢀⡀⠀⠄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠐⠛⠩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠀⠤⢤⣤⣴⢃⣐⣏⠀⢀⡖⠁⠀⠤⣰⠀⢷⣈⠃⠀⠀⣀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠑⠒⢤⣄⠀⢠⠛⠦⠈⣉⣳⠜⡡⠃⠀⠈⠁⠄⠰⠿⣦⡠⡘⠃⠀⠀⠙
⠀⠀⠀⠠⠀⡚⢀⢺⡧⡶⣾⣷⣗⣂⡆⠻⣿⣏⠉⡆⠀⡏⠡⠀⠀⠀⠀⠄⠀⣧⡀⢶⡀⠰⠆⠀⠚
⠀⡀⠀⠀⠀⠙⠋⣈⢙⣁⡿⣠⠤⠀⢡⡿⠋⠀⠀⠁⡀⠀⠀⢰⡄⠀⣿⠀⠀⠀⠱⠄⠀⠀⠀⠀⠉
⠠⣩⣱⠲⢿⣼⣤⠆⡦⠻⢋⣻⣷⣞⡻⠛⠀⠀⠈⠇⠁⠐⣠⠏⠁⣈⣡⠒⠀⠀⠀⡐⠉⠻⡇
⠤⠸⣧⠀⣀⢹⠏⢠⠧⣴⣿⣿⣻⡏⠁⠀⠀⢀⢀⠀⡻⠀⣡⣜⠳⢶⡭⣤⠀⠀⠀⡿⡆⠀⠇⠀⠲⡄
⠀⠀⢿⣦⠰⢹⣖⣋⡤⠾⠷⣬⠋⠀⠀⠀⢠⣎⡄⢀⡈⢛⡿⠿⣶⠋⢙⣣⣤⠀⠀⠡⠀⠘⠀⡀⠀⠐⢦
⣤⢤⡜⢻⠷⠄⠉⠀⠀⠀⠐⠃⠀⠀⡀⢀⣘⡟⠧⠈⢀⡟⠃⠒⠀⢤⣤⣀⣄⡸⣇⠀⠀⠀⠃⠁⢰⠀⡀
⢈⡉⠁⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⢐⣾⠟⠩⠁⠀⠈⠉⠀⢹⣦⠂⠈⠙⠲⠀⢆⠀⡣⣀⣈⡨⡁⢀⠊⠁
⠛⠀⠀⠀⠀⠀⠀⢠⠖⠠⣾⠇⡰⠛⣋⣵⣻⣤⣤⣤⣤⣤⣼⣿⡀⠀⣀⢀⣀⡈⠉⠀⠠⣀⠀⠀⢀⠒⢤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣈⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣈⣠⣤⣶⣾⠿⣿⣧⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢀⣁⣤⣴⣶⠿⠟⠛⠉⠁⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠂⣁⣠⣤⣶⣿⣿⣿⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠛⠂⡉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⣀⣠⣤⣶⣾⡿⢛⠩⣀⠀⠀⠐⢨⠙⠿⠖⠐⠀⣀⣀⣠⣤⣴⣶⣶⠿⢿⣿
⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⢀⣀⣤⣶⣾⠿⠛⠛⢉⡼⠿⠴⠀⠀⠉⣀⣀⣤⣤⣴⣶⣶⠿⠿⠛⠛⠋⠉⠉⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠉⠀⢴⣶⣿⠿⠟⠉⠁⠀⢀⣈⣀⣤⣤⣶⣶⣾⣿⠿⠛⠉⠉⠉⠉⠙⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⣤⣶⣶⣾⡿⠟⠛⠛⠉⠉⢀⣰⡟⣁⠊⢀⣨⣧⣤⣄⠀⠈⢂⣻⣷⡀⠀⣀⢊⣠⣤⡈⣀⢸⣿⡆⢘⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠂⠈⡏⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⡿⠟⠋⠩⢉⠭⠙⠛⢿⣯⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢋⠀⠁⢀⣤⣿⣤⡀⠀⠐⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣿⣿⠀⠀⢀⣀⠀⠄⠀⠀⣿⠇⠀⠀⠀⣾⡁⣿⠸⠿⠀⠀⠈⠈⣿⡄⠀⢀⠐⣀⡀⠢⠀⢸⣿⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⣿⣿⢰⠀⣿⣿⡧⢈⠀⢸⣿⠀⠀⠀⠀⠈⠻⣿⣶⣄⠀⠀⠀⠀⣿⡇⠀⠆⢸⣿⣿⠆⡁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣿⣿⠀⠀⠈⠉⠠⠂⠀⠀⣿⡄⠀⠀⠀⣾⡆⣿⠈⣿⠀⠀⢀⢀⣿⠃⠀⠈⠀⠉⠁⠔⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣌⠀⡀⠈⠛⣿⠛⠁⠀⠠⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣹⣷⣦⣄⣒⣈⣐⣀⣤⣾⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⡄⠒⠒⠂⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣿⠋⠀⠀⣷⠀⠀⠀⠀⠀⠱⡀⢀⠀⠀⠀⠀⢀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⠀⠀⠀⠘⠀⠀⠀⢸⡄⠀⠂⠀⠀⠀⢻⠘⠳⣤⣶⠶⠆⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠘⠃⢸⠀⠀⠀⣀⡈⢢⢴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⣼⠇⠀⠲⠒⠠⣤⣀⢀⣀⡤⠴⠦⠴⠼⢏⡁⡀⠘⣿⣿⡄⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣭⡿⠄⢠⡤⠖⠋⠙⠉⠉⠁⠀⠀⠀⠐⠀⠀⠈⠱⢶⡙⣿⠃⠖⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⢿⡿⢾⣇⡰⠊⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠈⢻⣄⠠⡃⡀⠒⠐
⠀⠀⠀⠀⠀⠀⠀⣿⠤⠤⠴⠁⠀⠀⡾⢻⡏⠻⣦⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢻⣿⣯⡐⡀⠀⠄
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⡀⠸⣇⣸⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠿⢿⡁⠒⢀⠅
⠀⠀⢨⠀⠀⠀⠁⠉⠀⠀⣈⡩⠁⠀⠙⣿⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⣾⠟⠂⠉
⠀⠀⠀⠀⡄⠀⠀⣀⠀⠀⠂⠀⠄⢀⡀⢹⡀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠠⠘⠋⣠⣷⣦⣄⡀
⠀⠀⠀⠀⠐⢀⣐⠀⠀⠀⠀⠀⠀⠀⡳⣤⣇⣠⠟⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⢀⡆⢀⣴⡏⡁⠒⠍⣿⣷⠆
⠀⠀⠀⠀⠰⠀⣒⡎⠄⣀⡀⠀⠀⠀⢳⡀⠙⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣠⣾⣿⣿⣿⣕⡬⣭⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠯⢀⠒⠒⠀⣨⣄⣈⣿⣦⡀⠀⠀⠀⠀⠀⠀⠢⣿⡿⠋⣨⡿⠿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⠿⡇⠀⠀⠀⠀⠀⠞⢁⣾⡯⢇⡀⠀⠉⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠀⢰⣿⡏⢄⣛⣮⢶⣿⡟⠁⢸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣭⣵⣿⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⠿⠛⠛⠛⠉⠉⠉⠉⠉⠉⠛⠛⠶⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄
⠀⠀⠀⠀⠀⢀⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⢀⣴⡿⠋⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⢸⣄⣇⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⣀⠀⣠⡿⠋⠀⠀⠀⠰⣿⣷⣤⡀⠀⠀⠀⠀⠀⢺⣿⣇⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄
⡿⣷⡟⠁⠀⠀⠀⠀⠀⠈⢻⣿⣷⣄⠀⠀⠀⠀⠘⣿⣿⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⢿⡄
⣽⡟⠀⠀⠀⠀⠁⠀⠀⠀⠈⠋⠻⣿⣷⣄⢀⡤⠖⠋⠉⠁⠀⢠⡉⠉⠓⠢⣄⠀⠀⠀⠀⢨⠃⠀⠀⠀⠀⠀⢹⡄
⡟⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢛⠟⠁⠀⠀⢠⠀⠀⠀⣾⠇⠀⠀⠀⠀⠑⣄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣷
⠁⠀⠀⠀⠁⠀⢀⡀⠀⠀⠀⠀⠀⢠⠋⠀⠐⢀⠀⠉⠀⠀⠀⠿⠀⢠⡶⠃⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠐⠀⠸⣾⣷⠀⠀⠀⠀⢠⠃⠀⠀⠔⠀⠀⠠⠊⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠈⠀⠀⠀⠠⡄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠄⠀⠀⠏⠛⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⢰⠀⠀⡄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⡇⠀⠀⠀⠀⢸⠇
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⢠⠀⠀⣄⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⢀⣀⣀⣠⣤⡤⠴⠀⢿⠀⠀⠀⠀⠀⢹⣆⠀⠀⡀⠀⠀⠀⠀⠠⠀⠀⢀⠂⠀⠀⡀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⢀⡤
⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⡀⠙⢷⣄⡀⠀⠀⠀⠀⢀⣀⠔⠁⠀⠀⢸⠃⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠒⠀⠀⢿⣆⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠚⠋⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠄⠀⠈⣽⡄⠀⠀⠀⠀⠘⠈⢷⣄⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⣧⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢈⠳⢤⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠋
⠹⣆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣿⣿⣷⣶⣶⣤⣤⣤⣴⣶⣶⣿⣿⠿⠛⠉
⢆⠹⣦⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⡤⣄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠙⣷⡀⠀⠀⠀⠁⠀⠀⠀⠀⢹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⢦⡀⠀⠀⡀⠀⠀⠁
⠀⠀⠀⢌⢻⣦⡀⠀⠀⠀⠀⠀⡀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⣖⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠃⠀⠀⠙
⠀⠀⠀⠀⠳⣝⣷⣦⡀⠀⠀⠀⠁⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠂⠀⠀⠀⣠⣴
⣀⣀⣀⣀⣀⣈⣙⣛⣛⣦⣄⣀⣀⣀⣀⣀⣀⣂⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣁⣀⣀⣀⣀⣴⣾⣿⣿
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢀⡀
⠀⠀⠀⠀⠀⡄⠀⣄⢠⡴⠛⣟⠉⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠰⠞⢿⣷⣄⡀
⠀⠀⠀⠀⠀⣜⣟⣟⣢⡈⢉⡋⠀⠀⢀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡤⠀⢨⣥⣤⣘⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣉⣄⠉⠀⠀⣤⣤
⠀⠀⠀⠀⢀⣥⣮⣹⠏⣵⣶⠁⠠⠔⠀⠀⠀⠀⠉⢘⣿⣿⣿⣷⣶⣶⣾⣿⣿⢍⣁⣿⣦⡄⠀⠉⠑⠃⠀⢤⡼⣻⣭⡛⢀⣾⡿⡃
⠀⠀⠀⠀⠠⢿⠈⡛⠸⡯⡅⢀⣀⡄⣠⣶⣤⣴⣾⣽⣾⡿⠟⣛⣛⣋⣙⠛⠟⢿⣷⣿⡻⣅⡀⢠⡐⠄⠀⣾⣿⣯⣼⣿⠀⢵⣿
⠀⠀⠀⠀⢄⠈⠗⣧⠀⠀⢀⡿⠃⠀⢉⣟⣭⣿⢟⣫⣴⣾⣻⣯⣹⣿⣿⣿⡟⣷⣮⣍⣹⣾⣟⣏⢻⣮⠻⣿⣿⣠⣶⣹⡿⡀⠛⣀⢠
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠲⠀⠉⠀⢠⣯⣾⡛⣱⣿⠝⢛⣹⣉⢉⣉⣿⠿⢿⠙⠛⠉⠹⡿⣭⢿⣾⣳⡛⢧⠹⣿⣟⣫⡄⢀⣯⣀⡈
⠀⠀⠀⠀⠀⠀⣾⠏⠈⡁⢀⠀⢠⢏⣿⣟⣼⡟⣷⡽⣾⢇⡝⠻⠿⢿⣶⣿⣷⢠⣬⢻⡜⠛⢿⡩⣿⣹⡀⢳⣿⡍⢙⣥⠨⡟⠇
⠀⠀⠀⠀⠀⣨⣽⡴⠀⣯⣬⡶⣿⣿⣿⣿⡿⣿⣿⣧⣄⣀⡎⠀⠀⠀⡿⣿⣾⢷⣄⣀⠀⣶⣸⣧⢿⣛⣳⠀⡟⣅⢸⡿⠀⣳
⠀⠀⠀⠀⠀⢱⣶⡟⠛⣿⣯⢐⡇⣿⣿⣿⡭⣿⣿⣟⡟⠙⠗⠾⢶⣼⣁⣛⣿⢈⠻⣿⡉⣛⠉⣿⣸⣿⣿⣼⣿⣿⣿⡏⠀⣴⣶⡆
⠀⠀⠀⠀⠀⠈⣿⠃⠀⡟⢹⠸⡷⣿⡟⣷⡽⣿⣿⣧⣼⣿⣆⠀⢠⠏⣩⣿⣗⣦⠦⣭⣷⣤⣿⣿⣔⣯⣿⠉⣿⡏⢹⡟⠀⣿
⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠈⠀⣿⢿⣿⢻⣷⡿⣿⣼⣿⣿⣿⣷⡾⣾⣷⣟⣿⣿⡇⠀⠀⣹⢿⡿⣻⣿⣿⣿⡟⣷⢈⢠⣀⣨
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢄⠘⣷⣿⣯⢻⣷⣎⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢀⣴⣿⡿⢽⣿⣿⠻⣿⢱⡇⢸⣿⠿⣿
⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⢸⣌⢷⠘⣿⣿⢳⡿⣿⣶⣾⣿⣿⣯⣿⣿⣿⡿⠟⢀⣤⣾⣿⢟⣵⣿⣴⢯⣄⢀⣿⣇⢸⠥⡂⢸⡀
⠀⠀⠀⠀⠀⠀⡇⣠⢀⠀⢸⣿⣦⢠⡈⠻⣿⣿⡾⢻⣿⣿⣿⣾⡏⢿⣡⣾⣷⣼⠿⣻⣵⣾⣛⡿⠃⠟⡁⠁⣼⡇⢸⠿⠈⣽⠁
⠀⠀⠀⠀⠀⠀⠓⠛⠈⠁⠚⡅⠘⣄⠙⠿⣿⡟⢿⣿⣿⠇⣽⣿⣟⢻⣿⣯⣩⣾⣷⣿⣿⢟⣁⠄⣀⠄⠀⣴⡿⠶⢼⠳⠿⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠈⠓⠐⠮⢍⠀⠈⢉⣓⠶⡿⢿⣿⡿⠿⠷⠛⠛⠉⠀⢚⡡⠈⠁⢀⠈⣽⣟⢱⢸⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⡁⠹⠀⠎⠀⠻⣶⣥⣤⣀⠀⡀⡀⠀⢀⣉⣙⣛⣙⣛⣛⣛⣛⣡⣤⠴⣾⠋⢀⣀⣰⣬⣿⣿⣿⣾⢉⣦⡤⠶⡏
⠀⠀⠀⠀⠀⠐⠖⠶⠒⠲⠶⠖⠀⠀⠀⠀⠐⠂⠀⠀⠀⠶⠬⠭⠭⠌⠥⠅⠩⣥⡤⢀⡀⠀⠩⠄⠀⠀⠀⠁⣴⠶⠲⠖⠶⣶⣷⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⣤⣤⣤⣤⣤⣀⠀⠉⠑⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⡭⠋⠉⠉⠉⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠴⠦⢤⣤⣀⠀⠀⠀⣿⢮⣅⣀⣀⣀⣀⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡫⡞⠉⠀⠀⠀⠀⢙⡇⠀⢠⡿⣦⣄⣉⣉⣉⣁⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡛⠶⢤⣤⣤⡤⢶⣿⡇⠀⢸⡟⠶⣬⣭⣭⣭⡭⣾⣿⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡸⠟⠶⣤⣤⣤⣤⣾⡟⠁⠀⠀⠛⠦⣤⣤⣤⣤⡤⠿⠋⢀⣀⣤⠴⠞⠋⡡⠴⠞⠛⠦⣉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡙⠓⠶⠶⠶⠖⢻⣿⠇⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣿⣿⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠶⠶⠶⣛⣉⣤⣤⠶⠶⠚⠛⠋⠉⢀⡙⢿⣿⣿⣿⣷⣄⠉⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠙⢿⣿⣿⣿⣷⣄⠈⢻⡀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣟⠛⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠙⢿⣿⣿⣿⣷⣬⡃⠀⠀⠀⣀⡤⠞⣋⣥⠶⢿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⡛⠷⣄⠀⠙⢷⣄⠠⢄⡀⠀⠀⠀⠈⠛⠿⠶⠤⠌⠻⣿⣿⣿⣿⣦⢖⣛⣭⢶⣛⣭⠶⢚⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣛⢦⣌⠳⣄⠀⠉⠳⣄⠙⢶⣤⡤⠤⠤⠀⢀⣀⣤⡴⣾⣿⣿⣿⣿⡿⠿⣛⣩⡥⠶⣛⣻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡙⢷⣝⢷⣌⠳⣦⡀⠈⠻⠦⠶⠶⢖⣛⣻⡭⢿⣞⣿⠿⣿⣿⣿⡷⠖⣛⣭⡵⠖⠛⢉⡴⢖⡛⠛⠛⠛⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢷⣝⠷⣌⠻⣦⠴⠶⠚⣛⣩⡭⠶⠚⣋⣥⠴⢾⣿⣿⣿⠟⠋⠉⠀⠀⠀⠀⣾⣄⠋⠀⠀⠀⠀⢀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢷⣌⠳⣤⡴⠖⢛⣉⣥⠶⢚⣋⣭⠶⠚⠋⠋⠉⠀⠀⠀⠀⠀⠀⠀⣠⢿⣍⠙⠛⠒⠒⠛⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢷⣤⠶⢛⣫⡥⠶⠛⠉⠁⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⣿⣬⣉⡙⠛⠒⣚⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⠶⣻⠝⠉⠉⠉⠛⢶⡄⠀⠻⣄⣉⠉⠉⠉⢉⣹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣮⣁⣀⠀⠀⣀⣀⣴⡇⠀⠀⠈⠉⠙⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣉⡉⠉⠉⢉⣩⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠠⢄⣀⠀⠀⢀⣀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣓⠄⠰⡱⠙⠦⠠⢀⣩⠤⢀⠝⡦
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠈⠁⠀⠀⠉⠛⠒⠀⠒⠛⠋⠉⠁
⠀⠀⠀⣀⠤⣆⡒⠺⡺⢻⡅⡀⠲⣦⠴⠀⠀⠁⢐⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣤⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠆⠰⢤⣀⡀⠀⢀
⠀⠠⠎⠘⠒⠛⠒⠬⠠⣈⠽⠛⠚⠁⠀⠀⠀⠀⠀⠈⢡⠄⠀⠀⠀⠀⢀⠖⠀⠀⠠⠂⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠠⠔⢠⠞⠂⠉
⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡴⠿⣥⠭⡅⠂⠀⢀⢄⠞⠁⠀⠀⢀⠀⡂⠤⣀⠠⡀⢀⣼⠇⠒⠒⠀⣲⣤⣤⢀⣀⣠⡷⢶⠔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢘⠲⡀⠀⢀⠄⠠⠂⠁⠀⠀⠀⠐⠃⢈⡠⠤⢍⡿⢀⠾⢇⡀⠀⠀⠈⢚⣈⣡⠁⡠⢄⡀⠹⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠙⠒⠒⣩⠔⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠉⠁⠀⠀⠀⠢⠀⠀⠀⠉⠀⠀⡀⠞⠩⠝⡟⡳⠁⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠰⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢆⢀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⠀⠸⠀⠀⠀⠀⠀⠊⠀⠈⡷
⠀⠀⠀⠀⠀⠀⠢⢧⣀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⠲⠥⠠⠄⠀⠀⠀⠐⠒⠦⠒⠂⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠧⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠕⣒⠀⠀⢰⠽⠒⠰⡦⠤⣔⢃⢷⣮⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠖⠒⠒⠒⢀⣢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢝⠚⠉⠁⠊⠀⠑⠫⣢⡰⠂
⠀⠀⠀⠀⠀⠀⣀⠀⡀⣀⣀⡠⣄⠀⠺⠄⢲⣆⣀⠨⣟⠐⠯⠕⠶⠤⠤⢤⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢥⣀⠐⣀⠀⠀⠈⠑
⠀⠀⠀⠠⡊⠁⠀⠏⠃⠁⢠⣄⡜⢢⠀⠈⠀⠀⠀⠈⠁⠛⠬⠤⠤⣥⡤⠤⠍⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋⠀⠉⠀⠉⠐
⠀⠀⠀⢠⡁⠂⢅⠱⠒⠂⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⡖⠚⠒⠲⠄⡀⠄
⠀⠀⠀⠈⢏⡑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠐⠯⠷⡷⢄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠀⠉⠊⠸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠆⢁⣠⠬⡈⠦
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⠀⠀⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠒⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠐⠠⢀
⠀⠀⠀⠀⠀⠀⡠⠐⠁⢀⣠⣤⠶⠒⠛⠛⠛⠛⠛⠛⠛⠛⠲⠶⣤⣀⠈⠒⢄
⠀⠀⠀⠀⢀⠊⠀⣠⡴⠝⣩⠴⣚⣉⢙⡍⡑⣦⡀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠑⢄
⠀⠀⠀⡶⢡⣀⣼⠏⢀⠀⡵⠾⠛⠉⠙⠻⠽⣷⣵⠀⠀⠀⠈⠁⠒⠀⠀⠘⢷⡄⠀⢣
⠀⣘⣽⠀⠀⣼⠃⠀⠀⠂⢃⠀⠀⠀⠀⠀⠀⣡⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢃
⠀⢷⡏⠀⢰⠇⠀⠀⠀⠀⠀⠉⠒⠀⠐⠒⠈⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⡀
⠸⢿⡃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⡇
⠀⢹⣷⠀⠸⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⡘⠀⢐⡇
⠈⡼⣟⣄⠀⠑⡀⠀⠀⠀⠀⠀⠀⠐⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠀⢀⡎⠁
⠀⢥⢿⡘⣦⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠊⠁⢀⢀⠀⠀⡎⡸
⠀⠐⠈⢧⡅⠳⢄⠊⠈⠁⠀⠄⠀⠀⣀⣀⠀⢀⠀⠄⠒⠉⠀⠀⠀⠀⠀⠀⢠⣀⠀⠈⠣⡀
⠀⠀⠈⠁⣙⢶⣤⣉⡖⠦⠤⣄⣀⣀⠀⠀⠁⠀⠀⠀⢀⣠⡍⠂⢲⡶⣄⠀⠘⢯⡳⣄⠀⠈⢢
⠀⠀⠀⠀⠀⠈⠡⠝⠿⡗⢶⣦⣴⣭⣭⣉⣉⣏⣹⡄⠀⠻⢿⣆⠀⠙⢾⡙⣦⡀⠻⣬⠳⣔⠠⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠿⠽⠾⢛⢿⣞⢦⡤⡷⣿⣦⠀⠀⠙⢮⣛⢦⡈⠻⣜⢧⡀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠙⣛⡛⢉⣻⣮⡳⣤⡀⢠⡑⢷⣝⢦⡈⠳⣙⢦⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢝⣿⣮⣻⣦⣀⣀⡙⠾⠛⠂⠈⠺⡛⠆⠀⠙⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠍⠻⠿⢿⣏⣿⠀⠘⢫⣘⣆⡈⠲⡷⣿⠗⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡿⣦⠀⠀⠙⢷⣹⢦⡈⣪⣕⢄⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣙⣿⡷⢤⣤⣤⣝⢮⡳⣮⠙⢮⡵⣄⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠓⠚⠿⢯⣷⡙⢇⡝⣄⠙⢮⡳⣀⢂⠁⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢻⠀⠀⡑⢥⡳⣄⠹⣎⠲⣄⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣳⣄⠉⠀⠻⣮⡣⡈⠳⣝⢷⣄⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⢿⣳⡤⡤⢬⣷⡝⢦⠬⢷⣯⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠟⠽⡿⠻⡿⡟⢾⢯⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠏⡆⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⢦⡀⠀⠀⠀⠀⠀⠀⡜⡎⢰⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠙⢆⠳⡙⢆⠀⠀⢀⡤⡞⡜⠀⠚⠂⣿⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡜⠀⠀⢀⡱⠬⠖⠛⠉⠈⣻⣰⡥⠤⠖⠊⢫⢳⠉⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠴⠗⠒⠉⠁⣀⡠⠤⠖⠒⠉⠉⠈⢷⡀⠀⠀⠀⢇⢇⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⠒⠉⢁⣀⠤⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⢀⣀⡬⣧⢀⣀⠤⢾⣿⣠⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⡳⣤⡴⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⣀⣠⡤⣖⡺⠯⠗⠓⡏⠀⠀⠀⢸⣼⣿⡿⠟⢡⡾⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠜⣧⠼⢄⡀⠳⠗⠀⠀⠀⠀⠀⣀⡤⡴⣖⡹⠷⠛⠋⠁⠀⠀⠀⠀⣇⣀⣠⣴⣾⠟⠋⢀⡴⠋⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⣠⣾⣷⡜⠌⠓⢖⣉⡹⠀⢀⣤⢴⣲⣹⠓⠛⠉⠀⠀⠀⠀⣀⡠⠤⠔⠒⠻⡍⠳⡜⢯⠁⣠⠴⠋⠀⠀⠀⠀⠀⢀⣴⣷⡄
⠀⠀⠀⠀⣼⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠘⠚⠉⠁⠀⢀⣀⡤⠴⠒⢎⠉⠀⠀⠀⠀⠀⠀⠈⢦⡈⢢⡙⣅⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⣽⡄
⠀⠀⠀⠀⠻⢿⣿⣿⣿⠀⢀⠀⢀⣀⣀⡠⠤⠶⢶⣯⣭⣭⣭⣭⣭⡭⠭⠶⣒⡤⠄⣀⠀⠀⠀⠑⣄⠙⣌⣳⡀⠀⣀⣴⣿⣿⠟⣡⣾⡿⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⡿⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠈⠁⠒⠫⠵⠶⠶⠮⠍⠚⢉⣤⣾⣿⡿⠋⣁⣾⠟⠁⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⠟⠉⣠⡾⠋⠁⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⡿⠛⠉⣀⡴⠚⠁⠀⠀⠀⠀⢀⣴⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⠿⠟⠋⠁⣠⠴⠊⠁⠀⠀⠀⠀⠀⢀⣤⠿⠋⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣶⣶⣶⣶⣿⣿⣿⣿⠿⠿⠛⠋⠉⢀⡠⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⢀⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠛⠛⠛⠉⠉⢁⣀⣠⡤⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⣠⣶⡿⠋⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠶⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⣀⣴⡾⠛⠁⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠖⠋⠁⠀⢀⣠⣴⠾⠛⠁⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣤⣤⣶⡶⠶⠞⠛⠋⠁⠀⠀⣀⣠⣴⡾⠟⠋⠁⠀⠀⠀⢀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⢄⣀⣀⣀⣤⣴⣶⡾⠟⠛⠉⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠯⠥⠄⣀⣀⣀⣀⠠⠤⠴⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⠙⠻⣿⣦⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⢀⣀⠀⠈⢻⣿⣾⡿⠿⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⢀⣴⠟⠳⠶⠶⠛⢉⣷⣾⡿⠟⠉⠀⠀⠀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⣠⠞⠁⠀⠀⢀⣠⣶⣿⠟⠉⠀⡀⠀⠀⠀⠀⠀⠘⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⢠⡾⠋⠀⠀⣠⣶⣿⠟⠋⠀⣠⡴⠟⠙⠶⣤⣤⠶⣦⠀⠈⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⠀⠴⠏⠀⣠⣴⣿⠿⠋⠀⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⣀⣴⣾⠿⠋⠁⠀⠰⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠈⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⢰⣶⣾⡿⠛⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠇⠀⠀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⢀⣴⣾⡿⠟⠁⢀⣀⠀⠈⢿⣦⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢋⣠⣾⡿⠟⠉⢀⣠⡶⠋⠁⠀⠀⣾⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⠀⠀⠀⣀⣴⣿⠇
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣶⣿⠟⠉⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⢻⣷⣤⣴⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⢀⣴⣾⡿⠋⠁
⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠋⠀⠀⠰⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⣿⣧⠀⠀⠀⠀⣠⡴⠟⠁⢀⣤⣾⡿⠛⠁
⠀⠀⠀⢀⣾⡿⠋⣿⣇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣀⣤⣿⠇⠀⠀⠰⠞⠉⢀⣠⣾⡿⠟⠉
⠀⠀⠀⢸⣿⡁⠀⠙⣿⣦⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⢿⡆⠀⠀⢀⣠⣶⣿⠟⠉
⠀⠀⠀⠈⢿⣷⣦⡀⠈⢿⣷⡄⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⣠⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠻⣿⣆⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠁⣠⣴⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣜⣿⣷⡀⠈⢻⣤⣤⣤⣤⣀⢀⣠⡶⠋⠁⣀⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣄⠀⠁⠀⠀⠀⠙⠛⠁⢀⣴⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣦⡀⠀⠀⠀⢀⣤⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⣤⣾⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⠛⣻⣿⢿⡿⠷⠖⠒⠐⠶⠤⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣾⣿⣎⠻⡀⠀⢀⣴⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠻⣿⣿⡄⣷⠀⣼⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣍⣭⣿⣿⣿⠿⠿⠆⡉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⢡⣾⣿⣿⣿⢿⠋⢿⣧⠈⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⡿⠃⠞⣀⢈⡉⠃⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠁⢀⣤⣿⣾⣧⣄⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣽⡇⠀⣾⣿⣻⢸⡟⠛⠃⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠙⣿⣿⣾⣇⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⠿⠿⠄⠀⢀⢸⡟⣿⣿⣆⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠐⣷⣶⣶⣦⡤⢠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣇⣧⣿⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠈⠉⠛⠿⣿⣷⣴⣭⣿⣿⣿⣿⣷⣆⣠⠤⢲⣿⡏⢻⡿⡿⠛⠁⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⣸⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⡽⢛⣫⢤⣤⣼⣿⣷⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠆
⠀⠀⠀⠀⢈⣿⣷⣶⣶⡦⣤⣤⣉⡙⠃⠀⠀⣀⠤⠊⣹⣿⣦⠀⣸⣦⣤⡄⠀⢀⣀⣤⠖⠋
⠀⠀⠀⢠⣤⣀⣉⢉⠉⠛⠛⠻⠿⢿⣿⣟⣉⡤⠔⣲⣿⣿⣿⣿⣶⣺⣾⡆⣲⠿⠉
⠀⠀⠀⠘⠿⠿⢯⣿⣿⣶⣶⣤⣤⣀⡤⠟⠓⢊⡉⠭⠃⠀⠈⡉⠉⠉⢹⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠶⠒⠉⠁⠀⠀⠀⠀⠀⣓⡿⠀⣚⣅⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣂⣯⡀⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣀⣀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⡴⠦⠌⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣀⠐⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⢏⠉⠉⢹⠀⣀⣿⣇⣀⣨⡁⠉⢹⠓⠒⢤⣄⡀
⠀⠀⠀⠀⢀⣠⠖⠋⠉⢣⣀⡤⠼⠒⠋⠉⠉⠉⠀⠀⠈⠉⠉⠉⠙⠒⠦⢄⡀⠉⠗⠦⡀
⠀⠀⣠⡔⠋⠀⢳⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⣶⢰⡆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣤⡿⠓⢄
⣠⠞⠛⢿⣦⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⠀⠑⢤⡊⠑⣄
⠧⡀⣠⠞⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠁⠀⠀⠀⠀⠀⠀⢀⠞⡴⢋⠖⠀⠀⠙⢤⠊⢢
⢀⡼⠁⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⢏⢞⡴⠃⠀⠀⠀⠀⠈⢣⠀⢳⡀
⠞⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁⠀⠀⢠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢣⠄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀⠤⠖⣂⠀⢣⡶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣴⣊⣁⡀⠘⡀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣖⠒⠀⢰⣀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡎⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢳⠢⡄⠀⢀⠎⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠣⠊⠀⠀⠄⠠⢀⠀⠀⠀⣀⠀⡄⠀⠀⠀⠘⠆⠃⢀⡔⢡⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢁⠀⠀⠘⠠⢇⠚⡄⠀⠀⡇⡄⡏⠦⠀⠀⢀⡠⠂⡠⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⢰⡆⠤⠀⠀⠀⠀⠀⣤⠐⡂⡩⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠒⠒⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⢀⠔⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⣍⠀⠁⠀⢂⠲⢀⡀⠀⠀⠐⡡⢂⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⢊⠈⠁⠀⠀⢠⡀⣋⠀⠈⠉⠚⠠⠅⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠐⠐⠀⠄⢀
⠀⠀⠀⠀⠀⡀⢀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⣼⢤⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠀⢀⡀⠉⠀⠂⠀⣀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣀⣀⣸⡀⠀⠀⠀⠈⠈⠐⠘⠀⢰⣠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⡐⠈
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⢇⠀⠀⠀⣠⡿⢛⡿⠈⠐⡠⢄⣠⠀⠀⠀⠀⠀⠈⠉⠚⠃⢇⡖⡤⣀⠀⠀⠀⠀⠀⠑⠀⣀⠔⢢⠜⢠⡀
⠀⠀⠀⠀⡠⠸⣽⣿⠿⠗⢛⣴⣵⢄⢘⡶⣶⡇⢀⠄⡀⠀⠀⠉⠀⠀⠒⠠⠀⡀⠀⠀⠀⠀⠁⠙⠽⢃⢂⢤⠀⠀⠄⠠⠃⠀⠂⣴⣿⣦⡀
⠀⠀⠀⢠⠽⡺⢁⡰⣢⣿⡶⢣⡇⠘⡻⢀⣍⡗⠀⠊⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠈⠈⠀⠀⢄⣀⡀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⡆
⠀⠀⠀⡾⠄⢡⣳⠉⠹⡿⠂⢴⡟⢲⠁⠀⢖⠅⢀⡐⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠀⠠⠀⠀⠀⠀⠠⠿⠿⠿⠛⠁
⠀⠀⢰⣄⠀⣘⠉⢠⡔⠂⢹⣿⠰⠈⠓⠊⠀⠁⠀⠀⠐⠀⠀⠀⠉
⠀⠀⢸⠀⠩⡇⢐⡺⠋⠸⠃⢐⣤⡟⡄⠀⠀⠈
⠀⠀⠀⠀⠴⠇⢈⡟⢶⣤⠶⢦⣟⣴⠇
⠀⠀⠀⠈⠠⢺⡞⢛⠻⠮⡶⡾⢞⣁⠈
⠀⠀⠀⠀⠘⠿⠗⠀⠀⠒⢃⣍⣄⣠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠘⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡁⠄⠀⡦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣡⡀⢠⠴⠴⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢠⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠄⠞⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄⠒⢒⣉⠉⣶⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠔⠒⣂⡉⠤⠄⠒⠂⠉⠀⠈⢄⡚⢃⡈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄⠐⢂⣉⣡⠤⠒⠂⠉⠉⠒⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢱⡀
⠀⠀⠀⠀⡀⠤⠐⠒⣈⡁⠤⠐⠒⠈⠁⢀⠔⠁⠀⣠⣿⣶⣦⡀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠈⡈⡇
⠀⠀⠀⠀⢡⠀⢣⠀⡃⠀⠀⠀⠀⠀⢀⠂⠀⠀⢸⣏⠸⣷⠈⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⢣⢹
⠀⠀⠀⠀⠘⡄⡤⠒⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠻⠷⣿⣶⣤⣄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⡈⣆⡀
⠀⠀⠀⠀⠀⢡⢱⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⣿⠈⢻⡇⠀⠀⠰⠀⠀⠀⠀⠀⠀⡔⢩⠁⢠⡄
⠀⠀⠀⠀⠀⠈⡄⡇⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢰⣦⣤⣿⣷⡿⠃⠀⢠⠃⠀⠀⠀⠀⢀⣀⠇⠀⠇⢀⡇
⠀⠀⠀⠀⠀⠀⢣⠰⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠉⠉⠙⠛⠀⣀⣔⡁⠤⠄⠒⣊⡉⠥⠄⠒⠊⢹⠉⡇
⠀⠀⠀⠀⠀⠀⠸⡆⢇⠤⢀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠲⠶⠒⢒⣉⣥⠄⠒⢒⠏⠁⠀⠀⠀⠀⢠⠊⡤⠀⠇
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠲⡀⢡⠠⠤⠐⠂⣈⢩⡤⠀⠒⠚⠛⣿⠟⠁⠀⡠⠊⠀⠀⠀⣀⡀⠠⠼⠀⠘⠂⣸
⠀⠀⠀⠀⠀⠀⠀⣿⣄⣈⠡⡄⠒⠂⠉⠁⠀⠀⠈⠐⣀⣤⡤⠤⠤⠖⠛⢒⣈⣉⠡⠤⠀⠒⠒⠂⠉⠉⡇⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢘⡄⠸⠤⠤⠐⠒⢀⣈⢉⡤⠤⠀⠒⣶⠞⠋⠉⢀⠔⠁⠀⠀⠀⠀⠀⢀⠖⣀⠃⠄
⠀⠀⠀⠀⠀⠀⠀⡏⠦⠤⠄⠶⠦⠭⠭⠭⠤⠤⠤⠬⠴⠦⠤⠤⠤⠴⠮⠥⠤⠤⠤⠤⠤⠤⠤⠼⠀⠵⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠦⠔⠒⠒⠢⠴⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⢀⣠⢤⣔⣒⣒⣈⣉⣭⣭⣭⣀⣒⣒⡢⠤⣄⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠛⠉⠙⢻⣿⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣻⣿⣿⡀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣧⠘⢿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣀⣥⣭⣭⣅⢉⣾⣥⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⠟⣷⣄⠙⠻⢿⣿⣷⣶⣤⣤⣄⣀⣀⣠⣤⣤⣶⣿⣿⣷⡧⠄⣠⣴⠬⠶⣾⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣄⣀⣿⣿⣷⡄⣀⡉⠙⠿⢛⣻⣿⣿⣿⣿⣛⡛⠿⠛⢉⣀⣴⣾⣿⣿⣀⣠⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⠿⡿⣿⣿⣿⣿⣗⡏⠉⠑⠒⠲⠶⠤⠤⠤⠤⠶⠒⠒⠊⠉⢹⣿⣿⣿⣿⣿⢿⠿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⢻⡆⠇⢱⠀⠀⠙⠿⢿⣴⣒⡠⠀⠀⠀⠤⠄⠀⠀⠀⢄⣒⣾⡿⠿⠋⠀⠀⡎⠸⢰⡟⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⢸⡇⢰⠈⡄⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⢠⠀⡇⢸⡇⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠈⡇⢹⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠋⢸⠃⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠸⠀⠀⠀⡰⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⢦⠀⠀⠀⠇⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⢸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣓⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠛⠉⠙⠁⠀⠙⠛⠛⠛⠉⠉⠉⠀⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠛⠋⠁




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣡⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠛⠛⠉⠙⠛⠓⠶⢤⡀
⠀⠀⠀⠀⢀⡴⠋⠁⣀⣤⠴⠖⠒⠒⠶⢤⣀⠀⠙⢦⡀
⠀⠀⠀⣠⠏⠀⣠⠞⠉⠀⠀⣠⡤⢴⣇⠀⠉⢳⡀⠀⢹⡄
⠀⠀⢠⡏⠀⢰⠏⠀⠀⠀⠀⢯⡀⠀⠙⠂⠀⠀⣷⠀⠀⣷
⠀⠀⢸⠀⠀⢾⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⡿⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠛⣋⣉⣉⣉⣉⣙⠛⠒⠶⢤⣀
⠀⠀⠸⡆⠀⠸⣆⠀⠀⠳⣤⣤⡴⠃⠀⠀⢀⡼⠃⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⣤⠖⠛⣩⠉⠉⢉⡭⠿⠯⣍⣛⡲⣄⠈⢳
⠀⠀⠀⢻⣄⠀⠈⠳⣄⣐⠋⠀⠀⣀⣠⠴⠋⠀⢀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠳⢬⣉⣛⡲⠚⠋⠀⠀⢀⣋⣠⡤⠞⠀⣰
⠀⠀⠀⠀⠻⣷⣤⣀⠀⠉⠉⠉⠉⠉⠀⣀⣤⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣷⣦⣤⣀⣉⡉⠉⠉⠉⠉⢉⣁⣀⣤⣴⣾⣿
⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣶⣶⣶⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⣽⣛⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣛⣽⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣀⣀⡀⠀⢸⣷⣽⣻⠿⣿⣿⣶⣾⣿⣿⣷⣶⣿⣿⢿⣟⣯⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⣉⣩⣤⠤⠤⠤⠤⠤⢤⣬⣉⡙⠻⢷⣽⣻⠿⣿⣷⣾⣿⣿⣿⣿⣷⣾⣿⠿⣟⣯⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠠⡞⢩⣤⣖⣀⣀⡴⠛⠉⢛⡶⠾⠙⣦⠀⢹⡿⣿⣿⣶⣾⣿⣿⣿⣿⣷⣶⣿⣿⢿⣻⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⠉⠓⠶⠭⣭⣭⣤⣤⣤⡤⠤⠖⠛⠁⣠⣾⡿⣷⣾⣿⣭⣭⣭⣭⣭⣭⣿⣷⣾⠿⣻⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⣿⣿⣶⣶⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⣿⡿⣻⡿⣷⣾⣿⣭⣽⣿⣿⣯⣭⣿⣿⣾⠿⣛⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⣷⣯⣿⣛⣿⣿⡿⠿⠿⣿⣿⣟⣻⣯⣷⡿⣻⣿⣷⠾⠛⠉⠉⠉⠙⠛⠿⣽⣿⣾⡿⣟⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⣷⣯⣿⣻⣿⡿⠿⢿⠿⠿⣿⣿⣻⣯⣷⣾⣿⠏⠀⢀⣠⠴⠶⠶⠦⢤⣀⠀⠙⢷⣿⢟⣿
⠀⢀⣠⠴⠖⠛⣋⣉⣉⣉⣉⣉⡙⠛⠲⠦⣿⣿⣶⣭⣟⣻⣿⠿⠿⠿⠿⠿⠿⣿⣛⣯⣽⣿⠃⠀⣠⠏⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠹⣿⣿
⣴⠋⢠⠴⠛⣉⡍⠉⠀⣠⠶⠿⢭⣍⣟⢦⡀⠙⢯⣟⣻⡿⢿⣿⣿⣿⣿⣿⣿⠿⢿⣻⣿⡟⠀⢠⡇⠀⢠⡄⠀⠀⠀⣀⣀⠀⠘⣦⠀⠹⡿
⣷⡀⠘⠲⢭⣙⣓⡒⠛⠁⠀⣀⣘⣁⡤⠞⠃⢀⣾⣟⣻⠿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣧⠀⢸⡇⠓⢿⡀⠀⢀⡞⠁⠈⣧⣀⢸⡄⠀⣷
⣿⣿⣷⣦⣤⣀⣉⡉⠉⠉⠉⢉⣉⣀⣠⣴⣾⣿⢿⣟⣿⠿⣿⣿⣷⣶⣶⣿⣿⡿⢿⣟⣿⣿⡄⠀⢷⡀⠀⠙⠶⠋⠀⠀⠰⠏⠈⢸⠃⠀⣿
⣿⣿⣽⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⣷⣿⠿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⡿⠿⣿⣷⡄⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⢠⡇
⢿⣾⣽⣻⠿⣿⣿⣶⣶⣶⣶⣶⣿⣿⠿⣟⣫⣷⡿⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠙⢿⣿⣦⡀⠈⠓⠶⢤⣤⣤⠴⠚⠁⠀⣠⠞
⠀⠙⠿⣿⣿⣿⣶⣾⣾⣿⣷⣷⣶⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣶⣤⣀⣀⠀⠀⣀⣀⣤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣴⢂⣴⣾⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⣠⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⣠⣴⣾⡟⣼⡇⣾⣿⣿⣿⡇⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠚⢿⠟⠂⠀⠀⠀⠀⠀⣀⡤⣷⣿⣿⣿⣇⣿⣇⢿⣿⣿⣿⣿⣦⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⢖⣵⣿⣷⣾⣿⣿⣿⣿⣿⡸⣿⡌⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⢶⠄⠀⠀⠀⠀⠀⠀⣠⣶⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⣿⣦⡙⠿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢠⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣝⠿⠶⣤⠄
⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠓
⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⢿⣿⣿⣇⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⢸⣿⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣷⣿⣿⣿⣷⣌⠻⣿⣿⣿⡿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⢰
⠀⠀⠀⣾⣿⠀⠀⢀⣰⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⣿⠀⣶⣿⣿⣿⣿⣿⣿⡿⠟⢩⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⣿⡀⠙⠟⠉⠁⠈⠉⠁⠀⠀⣾⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⣼
⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⣼⣿⡟⠀⠈⠙⠉⠸⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣀⣰⣦⣀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⢹⣿⣧⠹⣿⣆⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⣰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠹⣿⡆⠀⠀⠀⠀⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣤⣀⡀⠀⠀⠀⠀⠙⣿⣇⠀⠀⠀⢻⣦⣤⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣷⣶⣶⣶⣾⣿⣶⡶⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠃⠀⠘⠀⠘⠃⠀⠛⠀⠀⠀⠸⠀⡂⠀⠀⢀⠀⠁⠘⢀⠀⠠⠀⠀⠐⠂⢐⠀⠀⠀⠐⠟⠀⠘⠃⠀⠃⠀⠐

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣬⣽⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣯⣿⣽⣿⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠿⣟⢿⣿⡿⡷⣶⡤⠤⠄⠤⣤⣤⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣿⣷⡽⢷⡄⠀⠀⠀⡠⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⡀⣷⡀⠀⣼⣿⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⢦⣼⣿⣿⣧⣿⣇⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⣾⣿⣿⣿⣛⣛⣙⣠⢄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡉⢀⣴⣿⣿⣿⣿⢻⡏⠘⣿⣦⠐⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⣿⣿⠃⠾⢁⡀⣈⠊⠃⠀⠀⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠁⠀⣠⣼⣷⣿⣤⡀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣟⣿⡁⠀⣼⣿⡿⡏⣿⠿⠿⠆⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⢀⠀⢻⣿⣿⣇⣿⡀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣾⠀⠀⠙⢻⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠀⠠⣤⣷⢸⡟⣿⣷⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⣩⣿⣿⣿⣿⣿⠓⢶⣦⣤⣤⣀⣀⡀⠀⠀⠄⠀⠀⣀⣠⣿⣿⣾⣷⣿⡿⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⣀⣙⣛⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣷⣤⠴⠚⢻⣿⣧⢙⣿⠻⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⢘⣿⣿⣿⣿⣿⣽⣽⣿⣿⣿⣿⣷⡾⢟⣫⣥⡤⣴⣾⣿⣿⡗⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇
⠀⠀⠀⠀⠀⣹⣿⣶⣶⣤⣦⣤⣄⣀⡉⠙⠓⠁⠀⠀⣀⠤⠈⣹⣿⣷⡄⢢⣽⣤⣤⡄⠐⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⣀⣈⠉⠉⠛⠛⠻⠾⠾⣿⣿⣿⣷⣶⠖⢀⣀⡤⢴⣾⣿⣿⣿⣯⣬⣭⣽⡭⢠⣾⡟⠉
⠀⠀⠀⠀⣿⣿⣟⣓⣾⣶⣤⣄⣀⣀⠀⠈⠉⣻⡛⠭⠖⠒⢛⡏⠉⠙⠛⠛⠛⠛⢻⡆⣘⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⢿⣿⣿⣿⣏⣁⡤⠔⠚⠉⠁⠀⠀⠀⣳⣢⡄⠀⣸⠄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣑⢼⠇⠠⡾⠯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣭⣏⡬⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣶⣶⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣀⡀⠀⢠⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣴⣿⠿⢿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠁⢀⣀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣾⣿⣇⠀⢠⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⠀⠐⣿⣿⠂⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠙⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣴⣶⣷⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣇⣀⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣇⣀⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⢼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠈⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⣴⠃⠀⠀⣀⠀⠈⢾⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠘⡟⠀⡽⠃⠀⠶⠛⠉⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠤⠄⠈⠿⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣇⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢨⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⣱⡄⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣤⠄⠀⠀⠀⢀⣭⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠀⠀⠀⠂⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣪⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⡔⠋⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠓⠚⠉⠀⢀⡎⠙⢀⡼⢀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡤⠺⠷⠤⣤⣤⣀⠀⠀⠀⠉⣨⣦⣴⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣁⠀⠈⠀⠀⠀⠀⠈⠉⠓⠲⠤⣾⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡢⠌⠋⢸⠈⠙⠛⡲⠦⢤⣄⣀⠀⠀⡀⠀⣿⣿⣿⣿⢿⠉⣩⣳
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⢒⠭⣦⠉⠀⠀⠀⠈⠳⢄⣠⣌⠦⠅⣺⣫⣽⠐⡕⡰⣿⡿⠿⢉⣼⣾⣿⡿⡄
⠀⠀⠀⠀⠀⢰⣎⡀⠰⠚⣡⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⣀⢔⣩⣾⣿⣿⠄⣼⣿⠿⠭⠴⣿⡷⣝⣟⡼⠃
⠀⠀⠀⠀⠀⢸⠚⢭⣙⡓⠮⣭⣒⠠⣠⠄⢒⣲⡖⣐⡬⣖⣿⣿⣿⣿⣟⡤⠿⠉⠀⠀⢠⢸⣯⣫⡞⠢⣄⣀
⠀⠀⠀⠀⠀⣸⠂⠀⠙⠪⢕⡒⣉⢿⣷⡲⢭⡴⣾⣶⢿⣿⣿⣿⣿⡛⠫⠥⣤⣀⣠⣼⢏⣸⣿⠟⠓⣤⠀⢘⠏⢒⡢⢄
⠀⠀⠀⠀⠀⠈⠑⠒⠠⢍⣒⠦⢭⣈⡈⠍⣛⣚⣯⣽⣿⣝⡋⠃⣀⠤⠤⢲⣫⣿⠗⡟⢊⡿⠟⠛⢉⠹⢖⠦⣐⣉⣵⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠲⠮⢍⡓⢃⣾⢿⡟⠩⠬⠀⠀⠀⢀⢔⣵⢿⣿⠙⢈⡞⠁⠀⡠⠂⠀⠀⠉⣋⠛⠿⣿⠿⣂
⠀⠀⠀⠀⠀⠀⢀⡔⠊⡡⠉⢠⡀⠐⠠⢍⣙⡉⠈⠀⢀⣀⣀⣀⢔⣩⣿⣷⣿⣿⣼⠯⠀⠐⠊⠀⠀⠀⠀⠤⠀⠀⠀⡃⠀⣆⡍⢁⣶
⠀⠀⠀⠀⠀⠀⣺⠦⠥⠌⠉⣀⢒⣂⠉⠱⠖⠦⠤⠂⣁⣰⣋⣴⡟⣫⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⢂⡱⣀⠤⢚⡭⢞⡇
⠀⠀⠀⠀⠀⠀⢸⣧⠍⠭⠄⠒⣒⣀⡀⠤⠍⠡⣤⠆⢤⣄⣐⣩⣮⣿⣿⣧⣀⣀⠠⣤⡀⣀⠀⠀⠀⢀⠀⠀⡐⡨⢔⣫⣴⣮⣵⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢋⣭⠩⣤⣔⣒⣒⣒⣒⣉⡩⣥⢭⣈⣭⡹⢫⣿⠟⠏⠉⠛⠷⡦⢌⡑⠎⣑⣲⣭⣐⠾⣽⣶⣿⣿⡿⠿⢛⡩⢔⡎
⠀⠀⠀⠀⠀⠀⠀⢸⡓⠑⠪⠢⢌⠐⠁⠩⠊⢔⡅⡩⢈⠑⡡⡩⠛⢶⣵⠀⠀⠀⠙⢳⠪⢹⣿⣧⣽⣷⡿⠿⢛⡋⠥⣒⣮⡾⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠼⡃⠀⠀⠀⠀⠉⠒⢷⡤⠄⠒⢒⣤⢄⣤⣶⣷⣿⣽⣷⣄⡀⠀⠈⠀⢽⢿⣟⠯⣭⣖⡾⡅⠯⠑⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀⠀⠀⠀⠈⢮⢷⠬⣩⣴⣾⣻⣿⡿⣿⡻⠕⠁⠈⠐⠤⣀⠽⠟⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⡀⠀⠈⢶⠀⣩⣷⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⢄⡼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⡀⢀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠐⠄⡠⢀⠄⡀⠐⡀⡠⠐⠠⢊⠄⡨⠀⠔⡡⢂⠄⠠⢂⠄⠀⡀⡀⢂⠔⠡⢂⠔⡡⢊⢄⠄⡠⢊⢔⠕⡵⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠊⠐⠁⠂⠁⠀⠂⠐⠨⢀⠄⠀⠔⠁⠪⢊⠀⠐⠡⠂⠔⠁⠊⠐⠁⠂⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⣀⣀⣤⣾⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟

⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⢰⣿⠋⠉⠉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿⣇⡀
⠀⠀⠀⠀⢸⣿⠀⢠⣾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣷⡀
⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⢠⡟⠛⠛⠛⠛⠛⠛⠛⠁⠀⣶⡄⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⢿⡀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠿⠿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣆⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠀⣿⡇
⠀⠀⠀⠀⠈⢿⣷⣼⣿⠀⠀⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠁⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠈⢹⣿⣀⣀⣀⣀⣛⣛⣛⣛⣛⣛⣛⣛⣂⣀⣙⣁⣀⣀⣘⣛⣛⣛⣛⣛⣛⣛⣛⣀⣀⣀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠒⠈⠉⠉⠉⠐⠒⠢⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟⢋⣷⢄⡀
⠀⠀⠀⠀⢀⣀⠠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⠛⢁⣴⠟⠁⠀⠉⠢⡀
⠀⠀⠀⠻⣥⠄⠒⠒⣉⣱⢤⡀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⠛⠉⢀⡴⠋⠀⠀⠀⠀⠀⣠⣿⣄
⠀⠀⠀⠀⠀⠑⢯⣁⡩⠶⣔⠛⢢⣀⠀⠀⠀⠀⡜⠀⠀⠀⣀⣠⣤⣶⣿⣿⡿⠟⠋⠁⠀⡠⠞⠁⠀⠀⠀⠀⠀⣠⣾⣿⡿⣩⣆
⠀⠀⠀⠀⠀⠀⠀⠉⠢⣖⠊⣛⣥⡨⠷⢄⠀⣼⣶⣶⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⢉⣼⡿⠋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢤⡤⠝⠶⣊⠙⣻⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠟⢁⣴⠟⠁⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⣠⠒⠉⠉⠙⠿⣖⣒⡭⢿⠤⠤⢐⣒⡺⡍⠑⡔⢢⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠟⠋⣠⠶⠋⠀⠀⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠙⡹⣆⠈⠆⢿⠿⣿⠛⢸⠀⢱⠴⠁⢀⣠⣤⣶⣿⣿⠿⠛⠉⣀⠴⠊⠁⠀⠀⠀⠀⢀⣼⠟⢉
⠀⠀⠀⠀⠀⠀⢈⠀⣀⣀⠤⢄⠀⠀⢡⢻⠀⢸⠘⢃⣉⣶⣼⣤⣾⣶⣿⣿⡿⠟⠛⠉⣀⡤⠔⠋⠀⠀⠀⠀⠀⠀⣠⡶⠋⢁⣴⡿
⠀⠀⡠⢤⣤⡲⢯⠭⠔⠒⠊⠉⠀⠀⢸⡿⠄⢺⣿⣿⠿⣉⡩⠷⢤⡚⢫⣄⡠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⢀⣴⡿⢛⠇
⠀⠀⠑⠉⠀⠀⠀⠣⡀⠀⠀⠀⠀⢠⠟⠀⠀⠀⢣⣤⣤⣼⠶⣖⠊⢙⣧⣤⠵⢤⡀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⣠⣶⠟⠋⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⠤⠖⠁⠀⠀⠀⠀⠀⠫⡀⠀⠀⠀⠙⢤⡤⠜⠳⣎⣉⡦⣤⠔⠚⠉⠀⢀⣠⣴⠟⠋⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⣤⣤⣤⡼⠶⣉⣉⠹⠤⠘⠓⢤⣴⠾⠛⠉⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢍⣀⣀⣀⣬⣷⣤⠴⠖⠚⠉⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠫⠥⠀⣀⣀⣀⠠⠤⠔⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠈⠛⠁⠀⠉⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣷⣦⣤⡀⠀⢀⣠⣴⣶⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⢿⣿⠀⣉⣿⣿⣷⣿⣿⣍⠀⢸⣿⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠘⣿⣶⣿⣯⣤⣄⣤⣽⣿⣷⣿⠏⠀⠀⢙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠆⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠐⢿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣏⠀⠀⢠⣿⠿⣿⣍⠻⣿⠟⣋⣾⡿⣿⣆⠀⠀⢨⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⣾⣿⠀⣉⣿⣿⣿⣿⣿⣋⡀⢸⣿⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⡿⠿⠛⠉⠀⠈⠙⠻⠿⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⠀⣠⡀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⣀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠿⠟⠋⠉⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⣦⡀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠉⠉⢹⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠋⠉⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠂⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠋⠀⠀⠀⠀⠀⠀⠉⠉⠑⠐⠂⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠤⣖⠊⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⠴⣏⣀⡀
⠀⠀⠀⠀⣄⣀⣀⠀⠀⠠⠤⠤⠄⠀⠀⠀⠚⠻⢥⣀⠀⠀⠀⠀⠉⠒⠩⠽⣳⡆⠀⠀⠀⢀⣠⣴⢯⣥⣶⣶⣛⣫⡀
⠀⠀⠀⠀⢠⣌⣙⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠿⣾⣯⠓⠦⢄⡀⠂⢤⣀⡠⠬⠉⠑⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⣧⣈⠙⠒⠦⣍⡙⠲⢤⣤⡤⠤⠤⣶⣶⣶⣶⣶⣾⣿⣿⡇⠀⠀⠀⢀⠀⣷⣿⣿⣿⣿⣿⣟⣛⣛⣉⠁
⠀⠀⠀⠀⠀⢹⣌⠉⠂⠀⠀⠉⠙⠲⣶⡶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢀⣀⣤⠼⠶⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠈⣱⠦⢄⡀⠀⠀⠀⠀⢲⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⣝⠲⢬⣑⠒⠤⢀⣉⠻⣛⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠐⠂⠉⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢍⣙⡒⠲⠤⠴⠚⣹⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠐⠚⣏⡉⠀⠉⠐⠲⠶⡄⠀⠀⠀⢀⣀⣤⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠾⠁⠀⠀⠀⠀⠀⠀⢘⠣⠤⣀⠀⠀⠀⠀⠀⠙⠛⢋⣀⡄⣦⠄⠒⠀⢸⣿⣿⣿⡿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⣤⣤⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⠂⠀⠉⠒⠴⢶⣾⣿⣿⣿⣿⡇⣯⣀⣀⣤⣾⣿⣯⣷⡶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢩⣑⣲⣿⡿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠤⢀⡀⠈⣿⣿⣿⡿⠟⢛⣹⣾⡞⠉⢀⣼⣿⣿⠗
⠀⠀⠀⠀⠀⢭⣐⣒⣶⣾⣓⠲⠬⣿⣿⣿⠀⠀⠀⣀⡀⠀⠀⠀⠀⢠⣔⠚⠣⢤⣀⠉⠈⠉⠀⣤⣾⣿⣿⣿⠓⠛⠛⠋⠉
⠀⠀⠀⠀⠀⢐⠂⠭⢽⣯⠤⢍⣛⣿⣿⣿⢄⣒⣶⣾⣿⠀⠀⠀⠀⠈⣇⠉⠉⠒⠢⠤⣀⡀⢐⣻⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠠⢌⣙⣻⣿⣑⡒⠶⣿⡿⣿⡀⠬⢿⣭⣽⡗⠀⠀⠀⠀⠉⠒⠠⠀⢀⡀⠀⠀⠽⣺⠟⠋
⠀⠀⠀⠀⠀⢘⠒⠲⣿⣿⠦⢈⣉⣛⣛⣿⠍⣙⣻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⣓⣒⣺⣿⣿⣿⠃⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣄⡀⠸⢀⣄⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⣀⣴⣾⡯⠤⠾⠛⠁⠀⠀⠀⢠⣿⣥⣿⣿⡷⠀⣷⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠛⠛⠋⠂⠈⠀⠀⠀⠀⠀⠀⠀⠉⢙⣿⣿⣿⣇⠄⠈⣾⠇⠀⢀⣀⡀⠀⠀⠀⠀⠀⣸⠀⠀⢠
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠏⠈⡄⠀⡠⠔⢏⠉⢹⣇⣇⣦⠖⠁⠀⣴⣿⣿⣿⡆⠀⠀⠀⠤⣿⡀⠃⣼⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢵⣢⣇⣴⠿⣦⣤⣼⣿⠾⣷⣆⠀⠹⣿⣿⡿⠃⠀⠀⠀⡖⣿⣏⣹⣿⢸
⠀⠀⠀⠀⣀⣀⡀⠀⠀⠄⠄⠀⠀⠈⠁⢻⣄⣽⣿⡿⠧⠤⠚⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⡇⢸⣿⣹
⠀⠀⡤⠀⠈⠙⠳⢦⣄⡀⠀⠀⠀⢰⠀⠀⢻⣥⣽⣷⠀⣠⣴⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⡷⡿⡇⢸⢿⢾
⠀⠀⣤⣀⣴⡇⠀⠀⠈⠉⠛⠶⢶⣾⣄⣤⡀⠀⠈⢿⡏⠀⣿⡓⣶⣵⡀⠀⠀⠀⠀⠀⠀⣿⠃⡇⢸⡎⢿⡀
⠀⢾⣿⠋⠙⠳⣄⠀⠙⠓⠶⠟⠋⠀⠈⠉⠉⠀⠀⠈⢿⣤⣴⡝⢿⣟⠁⠀⠀⠀⠀⢀⣾⣹⡿⢹⣾⢹⣿⣷⡄
⠀⢠⣿⣦⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡋⠁⠘⣉⠀⠀⠀⠀⠀⠀⠁⣟⣷⢿⣿⢾⣘⠆⠁
⠀⠉⠀⠀⠉⠳⢴⣄⣀⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠻⢥⡑⣄⠀⠀⠀⠀⠘⠁⠈⠁⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣈⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢂⠀⠀⠉⠁⠀⠀⠀⠀⣠⡖⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠈⠙⠲⢕⢄⠀⠀⠀⠀⠀⠀⠀⢹⣤⡆⠀⠀⣰⠃⠀⠘⠋⠉⠠⠂⡀⠐⠀⣆
⠀⠀⠀⠠⠐⠄⠀⠀⠀⠀⠀⠔⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠉⠀⢀⢈⠀⠀⠀⣄⠁⠰⠆⠀⡀⣐⣰⣿
⠀⠀⠀⠀⠘⢠⣤⡀⠀⣠⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣲⣤⠶⠷⠛⣨⠀⠀⠈⣶⣴⣰⣾⣥⣿⣿⠃
⠀⠀⠀⠀⠀⠈⠛⠃⠀⠛⠏⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⡤⠚⠋⠀⣀⣴⣾⣯⣄⣀⠀⠈⠙⠛⠛⠛⠛⠁⠀⡀
⠀⠀⠀⡀⠀⠀⠀⢀⢀⠀⠀⠀⠠⠀⠀⠀⠀⣤⠚⠋⠀⢀⣤⡼⢟⣋⠥⠭⠯⠥⠭⠵⠖⠀⠈
⠀⠀⠈⡄⠀⠀⠀⠈⠁⠀⠀⢀⠐⠁⠀⣠⢾⠀⣀⣴⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠤⣶⣾⠛⠁
⠀⠀⠀⠀⠀⠔⠂⠄⠀⠄⠒⠐⠁⠀⠰⠿⠷⠛⠋⠉⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⣸⣿⡟⠀⢠⣿⡟⠀⣰⣿⠟⠀⣴⣿⠟⠁⢀⣴⣿⠟⠉⠀⣀⣴⣾⣿⣿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿
⠀⢠⣿⡿⠀⢠⣿⡟⠀⣰⣿⠏⢀⣾⡿⠃⢀⣴⣿⠟⠁⢀⣤⣾⣿⠿⠛⠁⠀⠤⠄⠀⠶⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀⡀⠀⠈
⠀⣼⣿⠁⢀⡾⠿⠀⠸⣿⠃⢠⣾⡟⠁⢠⣾⡿⠃⠀⠀⠈⠉⠛⠁⠀⣠⣴⣇⠀⠀⠀⠀⠀⠈⠛⠉⠉⠉⠉⠉⠉⠛⠛⠻⠿⣿⣿⣿⣷⣶
⢠⣿⡏⠀⣠⠀⠀⠀⢀⠀⠀⠘⠻⠀⢠⣿⡟⠁⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠟⠀⣀⣀⠀⠀⠠⠀⠈⢿⣿⣿⣿⣷⣶⣶⣤⣄⣀⠈⠙⠻⢿
⢸⣿⠃⢀⣿⡆⠀⠀⠈⠣⠀⠀⠀⢰⣿⡟⠀⢀⠀⠀⠀⠀⠀⠻⣿⠟⠉⣠⣴⣿⣿⣿⠿⠆⠀⠀⠀⠈⠉⠉⠛⠋⠉⠉⠻⣿⣿⣿⣶⣤⡀
⣿⣿⠀⢸⣿⠁⢰⣶⡄⢀⣀⡄⠀⣾⣿⠁⢀⣿⣦⠀⠀⠀⢀⡠⠋⣠⣾⣿⡿⠛⠉⠀⢀⣳⣄⣠⣦⣤⣤⣄⡀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿
⣿⣿⠀⢸⣿⠀⢸⣿⠃⢸⣿⡇⠀⣿⠏⠀⠘⠻⠇⠀⢸⣿⡿⠁⣴⣿⡿⠋⠀⣀⣴⣾⣿⡿⠿⠟⠛⠛⠛⠿⠿⢶⣄⣀⠀⢀⣤⡀⠀⠈⠙
⣿⣿⠀⢸⣿⠀⢸⣿⡀⢸⣿⡅⠀⠁⠀⠀⠀⠀⠀⠀⣿⣿⠁⢸⣿⡟⠀⢠⣾⣿⠟⠉⢀⣀⣤⣤⣤⣤⣤⣤⣀⡀⠀⠉⠙⠻⣿⣿⣷⣄⡀
⣿⣿⠀⢸⣿⡄⠘⣿⡇⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⠃⣰⣿⠿⠁⠀⠘⠻⠟⠛⠛⠛⠛⠛⠛⠁⠀⣴⣶⣄⡀⠀⠙⠻⣿⣿
⢸⣿⡆⠀⣿⣇⠀⢻⣷⠀⢻⣧⠀⢲⣿⠀⠀⠀⠀⠀⢿⣿⡆⢻⣿⠀⣿⠇⣠⡷⠀⠀⠀⢰⡶⠿⠿⣷⡆⠀⠀⠀⠈⠙⠿⣿⣶⣄⠀⠈⠻
⠈⣿⣷⠀⠸⣿⡄⠘⣿⡆⠘⣿⣆⠈⢿⣧⠀⢻⣿⡄⠘⢿⣿⡌⢿⣇⣿⢰⠟⣠⠎⠀⠀⣤⣶⣶⣤⣤⣀⡀⠀⠀⣽⣦⡀⠈⠻⣿⣷⣄
⠀⠘⣿⣇⠀⢹⡿⠀⠀⠀⠀⠘⢿⣆⠈⢻⠗⠂⢹⣍⠀⠈⠙⠛⠮⡻⣿⢸⣼⡇⠀⠀⠴⠶⢦⣤⣌⣉⠙⠻⢷⣤⡀⠙⢿⣦⡀⠈⠻⣿⣦
⣆⠀⠹⣿⣆⠀⠀⠀⢀⠀⢶⣦⡈⠻⣷⣄⠁⠀⢸⡇⠀⠢⡀⠀⠀⠀⠙⠻⣿⣿⡯⢭⣙⡛⠶⣤⣉⠛⢷⡦⠀⠙⢿⣦⡀⠻⣿⣦⠀⠹⣿
⣿⣆⠀⠘⢿⣷⡀⠘⢿⣷⡀⠙⢿⣦⣈⠙⠻⢶⣾⡇⠀⢀⣙⡂⠀⠀⠀⠀⠘⣷⣝⠲⣌⠛⢶⣌⠙⢷⣄⠀⠀⠀⠈⢻⣷⡄⠘⣿⣧⠀⠘
⢻⣿⣧⡀⠈⠻⣿⣦⡀⠙⠟⠃⠀⠉⠛⠿⣶⣤⣼⣿⣼⣿⣿⣿⠀⠀⠀⠀⡠⣷⢻⣆⠙⡷⠀⠻⢷⡀⠻⣷⡀⠀⠁⠀⠙⢿⡄⠘⣿⣧
⠀⠙⢿⣿⣦⡀⠈⠻⣿⠀⠀⠀⠸⠿⣶⡦⠀⢠⣿⣿⣾⣿⠿⠟⣀⡴⢂⣿⠃⣿⠈⣿⠆⠀⠀⠀⠀⣻⡀⢹⣷⡀⢳⣤⠀⠈⣷⠀⢹⣿⡆
⣦⡀⠀⠛⢿⣿⣦⣀⠈⠀⠰⣾⣶⣤⣀⢀⣴⣿⣿⣿⣿⣿⣦⡾⠋⣠⣾⠇⢰⣿⠀⣿⡄⠀⣴⣦⠀⢿⣧⠀⣿⡇⠈⣿⡆⠈⣿⡇⠀⣿⣿
⣿⣿⣦⡀⠀⠙⠿⣿⣷⣦⣄⡀⠉⢙⣿⣿⣿⣿⣿⠟⠛⠙⢻⣿⣾⣟⠁⣠⣿⠇⢸⣿⡇⠀⣿⣿⠀⢸⣿⠀⠸⠿⠀⣻⣇⠀⢻⣇⠀⢹⣿
⠙⠿⣿⣿⣶⣄⡀⠠⠛⠋⠙⣻⣿⣿⣿⣿⣿⣿⣥⡶⠖⠛⠋⠉⠉⠙⣿⣷⡍⢠⣿⡿⠀⢠⣿⡏⠀⣼⣿⠀⠀⠀⠀⢹⣿⠀⢸⣿⠀⢸⣿
⣄⡀⠈⠛⢿⡟⠁⢀⣠⣴⣿⠟⠋⣡⣿⡿⠋⠟⠁⠀⠀⣠⠂⠀⠀⠀⠟⠁⣠⣿⡿⠁⢀⣾⣿⠃⠀⠟⠋⠀⠘⠁⠀⣼⡏⠀⣸⡿⠀⢸⣿
⣿⣿⣶⣄⣠⣶⠾⠟⠛⠉⠀⢘⣷⣿⣿⣶⣶⠀⠀⠀⠀⠁⠀⠠⠐⢁⣠⣾⡿⠟⠀⠀⠾⣿⠃⢠⣀⡀⠀⢠⣴⠆⢠⣿⡇⠀⣿⡇⠀⣿⣿
⠈⠙⠿⣿⣿⣿⣦⡀⠀⠀⣠⣿⠋⠙⠛⠛⠛⠃⠀⠀⠀⢀⣀⣤⣶⣿⡿⠛⠀⠀⠀⢠⡀⠀⠀⢾⣿⠃⢠⣿⡟⠀⣼⡿⠀⠸⠿⠃⢰⣿⡏
⣤⣀⠀⠀⠉⠛⠿⢿⣷⣼⣿⣥⣴⣷⣶⣶⣶⣶⣶⣾⣿⣿⡿⠿⠛⠁⠀⣀⣴⣦⡀⠀⠀⠀⠀⠀⠁⣠⣾⡟⠀⣸⣿⠃⠀⠀⠀⢀⣾⣿⠁
⣿⣿⣷⣶⣤⣀⠠⠴⠿⠟⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⢀⣠⣴⣾⣿⠿⠋⠀⣠⣾⣶⠖⠀⣰⣿⠟⠀⣴⣿⠃⠀⠀⠀⠀⣼⣿⠏
⠀⠉⠛⠿⢿⣿⣿⣿⣶⣦⣤⣤⣤⣀⣀⣀⣀⣤⣤⣤⣴⣶⣿⣿⡿⠟⠋⠁⢀⣤⣾⣿⠟⠁⣠⣾⡿⠋⢀⣼⡿⠃⢠⣾⡟⠀⣰⣿⡟⠀⢠
⣠⣤⣄⣀⠀⠀⠉⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠉⠀⢀⣠⣴⣾⣿⠿⠋⠁⣠⣾⡿⠏⠀⣴⣿⡟⠁⣰⣿⠟⠀⣴⣿⡟⠀⢀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣦⣤⣀⢀⣤⠶⠞⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠋⠉⢻⣿⣧⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡀⠀⠀⣼⢹⣿⣧⠀⠀⠀⠀⢠⣾⡷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⡼⣏⢿⣿⡆⠀⠀⠀⢛⡾⠁⣸⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⣅⣀⣤⡴⠟⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠈⠉⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣴⣿⣿⣿⣿⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⣾⣿⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡞⣱⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣅⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣰⢋⣼⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠙⣦⣴⠞⠉⠘⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⢀⡾⣡⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⣿⣿⣷⣄⣠⡶⠋⠻⣄⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⢠⣏⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠻⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⣹⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⢸⣿⡿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠠⠴⣶⠖⠢⠄⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠂⣠⠶⠛⠒⠆⡀⠐⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠒⠒⠁⡄⢠⡄⢠⠈⠒⠒⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢱⡦⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⢴⡎⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠌⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢡⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠐⠀⢀⣀⣅⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⡀⠀⠆⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⠁⡌⠈⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⢰⠀⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠄⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⡄⠠⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠔⠁⠀⠸⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠃⠀⠈⠢⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢁⠔⠁⠀⠀⡠⠂⠀⠀⠋⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠀⠀⠘⢤⡀⠀⠈⠢⡈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⢀⠔⠁⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⠢⡄⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠂⠁⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠈⠐⢄⠈⠳⣄
⠀⠀⠀⠀⢀⡴⠊⠁⡠⠊⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠧⠤⠄⠠⠤⠤⠠⠤⠤⢸⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠑⢄⠈⠑⢦⡀
⠀⠀⠀⣰⠋⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠦⠀⠠⠠⡔⢢⠤⠄⠀⠸⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠈⠢⡀⠙⣆
⠀⠀⠀⢹⠀⠃⠒⠀⠒⠒⠒⠀⠒⠂⠀⠒⠒⠒⠒⠒⠒⠂⠖⠁⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠈⠲⠐⠀⠒⠒⠒⠒⠀⠐⠂⠒⠂⠒⠒⠒⠂⠒⠘⠀⡏
⠀⠀⠀⠺⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠆⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠰⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠤⠤⠤⠤⠤⠤⠤⠤⡇
⠀⠀⠀⠀⠤⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣀⡀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠠⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠄⠚⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠃⢤⣀⣀⡀⣴⢿⡿⢠⣀⣀⣀⠠⡞⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣅⣠⣀⣭⣹⣯⣨⣅⣀⣈⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⡿⠋⣍⣀⣀⣀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⣿⣿⣿⠟⣉⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣤⣄
⠀⠀⠀⠀⠀⣿⣿⡏⣼⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⡟⢰⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡇⢸⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⡆⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⢋⣩⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣠⣄⡈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣍⡙⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢻⣿⣿⠇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⡇⣿⣿⣿⣶⣦⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⡉⢁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⠃⣿⣿⣿⣿⣿⣿⣘⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣇⣸⣿⣿⣟⣛⣛⣛⣛⣛⣛⣛⣛⣁⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏

⠀⠀⠀⠀⠀⠀⠀⣰⣿⡆
⠀⠀⠀⠀⠀⢠⣾⣿⣏⣀⣾⣷⣄
⠀⠀⠀⠀⣴⣿⣿⣿⡿⣿⣿⡿⠋⣠⣦⡀
⢀⣴⣤⡀⠙⢿⣿⣿⣦⡀⠙⠀⣾⣿⣿⡟⠀⣀
⠺⣿⣿⣿⣦⣾⣿⣿⣿⠃⣤⣄⠈⣻⣿⣦⣾⡿⠃
⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⠋⠻⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣦⣤⣤⣀⠀⣠⣤⣶⣶⣶⣶⣦⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠈⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣄⣀⠀⣀⣀⣴⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⣠⠤⠔⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠀⠀⢠⡤⣼⢾⢚⢚⡹⠀⠀⠀⠀⠀⢀⣀⠠⠆⠀⢠⠛⠠⣤⣄⣀⣀⠀⠀⠀⠀⠙⠃⠀⠚⠁
⠀⠀⠀⠀⣆⣏⣥⡦⠶⠒⠒⠀⠀⠀⠀⠀⠀⠘⠊⠉⠉⠀⢀⣠⠴⣺⠟⠋⠉⠀⠒⠀⠀⠀⠠⠞⢈⣹⠎⠍⣳⡄⠀⠀⠁
⠀⠀⠀⠀⠏⡧⠖⠗⠁⠀⠀⠳⣶⣤⣶⡏⠀⠀⠀⠀⠀⠀⢺⣧⣀⠈⠓⠒⠖⠒⠶⠒⠶⠒⠚⠋⢉⣀⣤⡾⠟⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠔⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠶⠶⠶⠶⠶⠶⠿⠿⠛⠛⠋⠉
⠀⠀⠀⠀⣀⢤⠶⠋⠁⠀⠀⠀⠀⢀⡀⠀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⢀⣀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⢀⠠⠀⠫⡽⢀⠘⠋⠀⣠⣀⠀⠀⠠⡀⠀⠀⣖⢻⢻⠚⡗⣦⡄⠓⠋⣼⡏⠇⢹⠸⠸⡇
⠀⠀⠀⠴⠶⠚⠃⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁⣵⠀⢸⣸⣸⡤⠷⠋⠀⢀⠘⠀⠙⠛⠚⠒⠗⠃
⠀⠀⠀⠀⠀⠀⣀⣬⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠋⠀⢀⠝⠀⠈⠈⠀⠀⠀⠀⠀⣸⣣⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠔⠐⠒⠀⠀⣀⡀⢀⣀⣀⡀⠀⣀⣘⠁⠀⠀⠀⠸⡀⠀⠀⠀⠺⡄
⠀⠀⠀⠀⠀⠀⠉⠓⢤⣸⣷⣦⠀⠀⠀⢀⣴⣶⣦⣰⡆⠀⣾⣷⠿⣿⡟⠃⣾⣯⣭⡍⣿⣏⣽⠇⠀⠀⠀⠀⡗⠒⠒⠀⡖⠀⠂
⠀⠀⠀⠀⣤⣀⠀⠀⠈⠀⠀⠙⠣⠀⠀⢼⡇⣸⣿⠹⣿⣤⣿⠇⠀⣿⡇⠀⣿⣥⣤⡀⣿⡟⠻⠦⠀⠄⣀⠀⢹⠚⠓⢺⡃
⠀⠀⠀⠀⠀⠈⠙⠡⠀⠀⣠⢤⡀⠀⠀⠘⠛⠛⢁⣀⡈⣩⣥⣄⠀⣬⣤⡀⠀⣠⣶⡄⣠⡴⠶⠆⠀⠀⠀⠀⣼⠦⠤⠼⣇⡈⠄⠤
⠀⠀⠀⡤⢤⣤⡄⡀⠀⢸⡁⢀⡽⠀⠀⠀⡀⠀⠿⣤⣅⢹⣧⣾⠗⣿⡾⣇⢸⣏⠀⡀⢿⡷⠚⠃⠀⠀⠀⠀⠀⠁⠒⠀⠉⠁
⠀⠀⠀⠀⠠⢶⣷⣤⡄⠀⠉⠉⠀⠀⠀⠁⡀⠀⠶⠶⠿⠸⠿⠀⠀⠛⠀⠉⠈⠛⠛⠁⠸⠷⠚⠛⠀⠀⠀⠀⠐⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠘⠈⢀⡄⠂⡂⠠
⠀⠀⠀⠀⠰⠄⣀⡉⠀⠀⠀⠀⠀⣀⠤⠆⠚⠀⠈⠒⠂⠀⠓⠲⣄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢰⡁⠀⡀⠧⠖⠀⠀⠤⠀⣀⢀⡼⠀⠀⠆⠂⠀⢀⠀⡐⢶⡀
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠑⠦⠦⣄⣀⡑⠠⠤⠠⠚⠉⢀⠀⠀⠈⠀⠀⠀⠀⠀⠸⠃⠀⠀⠰⠀⠑⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠈⣶⠆⠀⢤⠒⠈⠳⣤⡀⠀⠀⠀⠀⠐⠒⠘⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠑⠀⠀⠸⣦⣄⡒⠀⠠⠤⠀⠀⠂⣀⡜
⠀⠀⠀⠸⠇⠀⠀⠘⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠳⠶⠖⠈⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⢋⣙⡛⣛⣩⣌⠛⣫⣴⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⣡⣿⣿⣿⠟⣋⣴⣾⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⢣⣾⣿⠿⢋⣵⣾⣿⡿⢛⣥⣦⣍⣛⣋⣍⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣯⣴⡿⢋⣥⣾⣿⡿⢛⣡⣶⣿⣿⣿⣿⣿⣿⣿⣦⠹⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⢛⣡⣾⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⢟⣡⣶⣿⣿⣿⣟⠹⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢡⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⣩⣴⣿⣿⠟⣋⣴⣿⣿⠃⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⣠⣾⠟⣫⣴⣿⣿⠿⣋⣴⣾⣿⣿⣿⣿⡀⠙⠛⠋⠉⠙⢻⣿⣿⣿⣿⣿⣿⡿⢛⣥⣾⣿⡿⠛⠁
⠀⠀⠀⠀⢀⣾⡟⣴⣾⣿⣿⣿⡁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣿⣿⣿⡿⢛⣡⣶⣿⡿⠟⠁
⠀⠀⠀⣰⣿⣿⡇⢿⣿⣿⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⠛⠁⢠⣿⣿⣿⣶⣿⡿⠟⠉
⠀⠀⠀⢻⣿⣿⣿⣆⠻⣿⣿⣅⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠙⠿⣿⣿⣧⡙⣿⣿⣦⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣋⣴⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣌⢻⣿⣷⡘⠿⠟⠿⢿⣿⠿⣋⣵⣾⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠹⣿⣿⣾⣿⣿⣶⣤⣾⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡜⢿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠆⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⡀⠃⠐⠒⠀⠀⠀⠀⠀⠒⠘⢀⠃⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠎⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⠒⠒⢰⣏⣀⣀⣹⡤⠀⠒⠂⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠂⠀⢀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⢀⠀⢘⠠⢨⡿⠭⡓⠆⠃⠀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⡀⠈⠁⠁⠰⠆⠌⠈⠁⢀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⢹⡖⠂⠐⠂⠀⢲⠏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠀⠰⢀⣾⡇⠀⠀⠀⠀⢸⣷⡀⠂⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣾⢿⡃⠐⠀⠀⠀⢘⡻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣰⡏⢸⡇⠀⠀⠀⠀⢸⡅⢹⣆⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡿⠘⢙⡏⠉⠉⠉⠉⢹⡋⠃⢿⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣠⢖⠁⠁⡆⢘⡃⠀⠀⠀⠀⢘⡁⢸⠈⠨⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠕⠁⡠⠊⠀⢨⡇⠉⠉⠈⠉⢸⡇⠀⠑⢄⠈⠪⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢋⠔⢁⠄⠊⠀⠀⠀⢸⣇⢀⡀⢀⡀⢸⡇⠀⠀⠀⠁⠠⡈⠢⡙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⡞⡡⠂⠀⠀⠈⠀⠀⠀⠀⠀⠸⠃⠀⢴⡦⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⢳
⠀⠀⠀⠀⠀⠀⠀⣧⣉⣉⣉⣉⣭⣉⣉⣍⣉⣩⡇⠀⠀⢸⡇⠀⠀⢸⣍⣉⣉⣉⣉⣭⣉⣉⣩⣉⣼
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠤⠀⢀⡤⠀⣀⠀⠀⠀⠄⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⣀⠀⢠⡀⠀⠤⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠤⣤⠤⠬⡆⢲⠤⢤⣾⡷⣤⠤⠾⢰⠏⠤⣬⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢁⠈⣿⣿⡿⠿⣿⣿⡇⠘⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣤⣶⣤⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣶⣤⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⣶⡄⠀⠀⠀⠀⢀⣤⢤⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⣿⠅⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠟⠁⠙⣶⡄⣀⡈⠃⠀⠈⠓⢤⣟⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⣿⠀⠀⣿⡞⠁⠀⠀⣠⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⢸⡯⠜⠀⣿⠉⢠⣿⠑⢉⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠉⢿⡔⢻⡇⢸⣧⠼⡠⠀⠀⠀⠐⣿⣀⣀⣤⡾⠋⠈⠑⠊⠣⠿⠀⢸⣇⠔⠉⠉⠀⢀⠄⠐⠒⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠻⡆⠘⠿⠈⢧⡜⠛⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠶⣶⡶⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⣛⣛⠃⠀⠀⠀⠀⠀⠀⣀⡀⠀⡀⠠⣄⠀⠄⠀⠀⢄⠀⢀⣴⡶⠶⢶⣶⠀⢀⢀⠀⠀⠀⠀⠀⢩⡙⠳⢬⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⠞⢋⡭⠀⣀⠠⠄⠂⠈⠀⠀⣿⡆⠰⠀⢿⠀⡄⠀⠀⠀⠰⡿⠋⠀⠀⢰⡟⠀⠆⠀⠈⠐⠠⢀⠀⠀⢷⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠃⠀⢡⠀⢀⡀⠠⣶⠿⠛⠃⠀⠂⢸⡇⠁⠀⣄⡀⠀⠀⣠⡄⠀⣾⠃⠰⠀⠀⠀⠀⢀⣾⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⢻⠀⢿⡄⠀⠀⠈⡄⠈⣷⠸⠀⢹⡏⠲⠟⢉⠁⢰⡟⢀⠀⠀⠀⣼⠛⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠸⣧⠀⠀⠀⢁⣀⠹⠆⠂⠚⠃⠠⠄⢄⠀⣾⠁⠀⠀⠀⢈⣇⠰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⢹⣇⠀⠀⠈⢧⡀⠦⠽⠿⠏⢨⡀⣴⣰⡏⠠⠁⢀⣴⡾⠁⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡻⠆⡠⠊⠁⡠⠖⢢⠀⡀⠀⠈⠙⢝⠁⢷⡀⢈⣷⠀⠀⡐⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠄⠀⠀⠀⢀⠂⣠⠊⡜⠁⢀⡼⠀⡇⢀⠇⡜⠢⠱⡀⠙⠟⠁⠀⠐⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠒⣾⡟⠀⠀⣀⠄⠂⣰⡃⢸⠓⡞⠛⠢⡀⡇⣸⢰⠁⠀⡇⠰⠀⠀⠃⠊⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⢃⣼⢻⠁⠀⠀⡭⠀⢸⠀⡇⢘⢲⡆⣧⠇⡞⠂⢸⠂⠀⠀⠀⠀⣀⣈⣡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠚⠉⠁⠀⠀⡄⠹⣀⣠⠄⢣⠘⣼⠁⠛⢸⣅⠀⣘⠀⠰⠀⠀⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣬⣥⣥⣤⣀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡄⠀⢹⡍⢩⣭⠉⠁⠀⠀⣶⣶⣶⣶⣶⠌⠉⠉⠉⣩⣉⢛⠛⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣫⠗⣬⠀⠀⠈⢁⣀⣀⣀⣀⣀⣘⣛⣃⡀⠀⠀⠀⠀⠙⠋⠸⣂⣰⣇⡦⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣁⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣷⣶⣦⣦⠛⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡇⢤⣤⣴⡆⠀⠀⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⢀⣉⠁⠐⠆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠈⠉⠉⠁⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠸⠿⣿⡇⠲⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡰⣠⡀⢸⠀⠃⠀⠀⠀⠀⠀⠀⠀⠹⣿⣅⣠⣶⣤⣠⣼⡟⠁⠀⠀⠀⠀⠀⠀⠈⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⣄⠺⠖⠏⠈⡄⠰⠀⠀⠀⠀⠀⢠⡄⠀⠙⢛⡛⠙⠻⠿⠋⠀⠀⠀⢀⢀⠀⠀⠀⡀⠀⠙⡇⢰⣮⢁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠰⣁⣰⠔⡏⠈⠀⢖⣤⡶⠿⠓⠰⢂⡬⣿⢂⣷⣀⠰⢶⣻⣻⣦⣶⣄⠂⢄⣀⠲⣆⣤⣦⣲⣠⣾⢧⣄⠔⠂⢘⡷⣀⡠⢀
⠀⠀⠀⠀⠀⠀⠀⠠⢰⢟⠀⠁⠐⣴⡆⠨⠀⠙⢤⡶⡟⠁⣘⣶⢿⣿⣿⡖⣿⣠⡽⣻⣿⢿⣻⢿⡧⠍⠑⠃⢙⠳⣉⠀⠽⡶⣦⠄⠀⠏⠱⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠲⠀⠃⠀⠀⠐⠂⠈⠁⠠⡯⠍⠀⠀⠀⢟⠈⠏⡂⠣⠃⠈⠉⠡⢿⠿⡕⠀⠀⠀⢹⠶⠀⠐⠀⠦⡀⠀⠐⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠉⠂⠀⠀⠀⠀⠀⠈⠨⠂⠀⠀⠀⠈⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠭⣽⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⠿⠟⠋⡁⢄⣤⣶⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⡛⢉⡁⢀⣢⣤⢶⣿⣿⣿⣿⣿⠿⠟⠟⣩⣝⢿⣿⣿⣿⠟⠡⠀⠂⣀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣏⠰⠟⣡⣾⣿⣿⣿⣿⣿⣿⠿⠿⠀⠀⠀⡝⢏⢷⣻⣿⠰⠀⢀⠰⡆⠀⣂⣿⡇
⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠛⣿⣿⣷⣨⣿⣿⣿⣿⠛⣉⡀⠀⠀⢀⠀⠀⡄⣇⢸⡄⣏⣿⣷⣄⣈⣐⣆⣬⣿⣿⣷⡄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣧⣶⣿⣿⣿⡟⠋⠉⠁⣌⢰⣉⣽⢀⠀⢸⡄⠀⠸⡜⢾⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⣾⣿⣿⣴⣿⣿⠉⣟⣡⣴⣧⣤⣤⣶⣿⣼⣇⢹⣿⣿⣿⣿⣶⣶⣾⣢⣭⣿⣿⢿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⢏⠉⠛⢿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⡿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣽⣿⡿⢻⣿⠟⠿⣿⣿⡻⣿⣿⠛⢻⣿⢍⠚⠀⠈⠉⠑⠈⢻⣿⣿⣿⡄
⠀⠀⠀⢸⡿⣫⡟⠡⠤⠖⢀⠀⠀⠘⢻⣯⣻⣿⡿⣿⣧⣽⣶⡇⢰⣿⣿⣷⣜⠪⠐⢾⣿⣇⠠⠀⠀⠀⠀⠉⢈⣭⡛⢿⠁
⠀⠀⠀⠈⣱⣏⠀⠠⠀⠀⠀⠀⠀⣀⣤⠹⠷⢻⣷⣿⣿⠿⢿⣳⡊⠛⢿⣿⣿⣷⣧⣶⣦⣍⡛⠳⠶⢦⣤⣤⡄⠀⠐⡀⢄
⠀⠀⠀⢠⣿⠒⠖⠀⠀⢀⢀⣀⣠⡟⠈⢘⢸⣿⣿⣷⡝⣅⣀⣭⣭⣁⣟⢡⣿⣟⢻⣿⣿⣄⠙⠛⠳⢶⣦⣬⣤⣀⢀⣠⣾⣫
⠀⠀⠀⠸⣿⣳⣖⡻⢿⣿⡿⠟⠋⠀⠁⠀⣺⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⠘⣿⣿⢿⣿⣿⣿⣿⣶⣿⣿⣿⡟⢿⣿⡿
⠀⠀⠀⠀⢻⣅⡀⠉⠀⠉⠀⠀⡀⢀⣢⣞⣿⣿⣿⡇⡌⣿⣿⣿⣿⣿⣿⣇⢹⣿⣿⣿⣷⠛⠛⠛⢻⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠙⠿⣤⣀⠀⠠⠀⢶⣯⢷⣿⣿⣿⣿⢡⢿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⠿⣛⡀⠀⣸⣿⣿⣷⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢬⣭⣭⣭⣶⣿⣿⣿⣿⡏⣌⣾⣿⣿⡿⣿⣿⣿⣿⣇⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣾⣿⣿⣿⢶⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢦⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⡟⣼⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣧⢻⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠙⠛⠛⠛⢇⠛⠛⠛⠛⠉⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣶⣦⣔⡛⠋⠉⠉⠙⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠛⣩⠝⠻⣿⣦⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣇⠀⣿⠀⢀⣿⣿⣧⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣰⣾⣿⣿⣿⡆⠀⠀⠀⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⢷⣝⢿⣿⣿⠃⠀⠀⣠⣴⣾⢧⣀⣤⣴⣶⣶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⡳⢾⡭⣠⣴⡿⣟⣭⣶⣿⣿⣯⣿⣷⣿⣿⣟⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣮⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠈⢟⣫⣾⢟⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠫⠀⢨⣝⡻⠿⣿⣿⣿⠿⠟⠋⠁⢀⣴⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠑⠀⣴⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⣠⣾⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠂⣠⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⡄
⠀⠀⠀⠀⠀⣠⠞⠄⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣄
⠀⠀⠀⢀⡴⠁⠀⣠⣾⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣦
⠀⠀⣠⠎⠀⢀⣼⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣷⡀
⠀⠀⠀⠀⣠⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⣷⡄
⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣽⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣮⣿⣿⣿⣿⣷⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⠀⢂⣄⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠤⠔⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠠⠀⢐⢠⠠⠼⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡝⠀⠀⠐⡘⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⢀⣀⡀⢤⡉⢀⡢⠌⢀⡰⠤⠤⠒⠒⠃⠀⠚⠛⠒⠶⠤⣀⣁⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠉⠀⢠⡧⠈⠉⠀⠀⠀⠀⠀⠀⠀⣤⡤⣖⠶⠊⠀⢈⣆⣺⡉⡐⠂
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡆⠀⠀⠉⠉⠉⢁⡗⠀⠀⠀⠀⠀⢀⣒⡤⢒⣂⠀⣂⠀⢀⡀⠩⠭⠀⠉⣉⠉⡠⠬⠀⣀⡠⠹⠄
⠀⠀⠀⠀⢨⠁⠀⠀⠈⡱⠰⠀⣁⠀⠀⠀⢈⠉⠉⣰⠲⠦⠁⡛⣶⢩⠐⡚⠭⢬⡍⠑⠻⠽⢖⡭⠄⠠⠁⠤⣐⠦⠶⠍⠀⠀⠀⡀
⠀⠀⠀⠀⣠⠀⠁⠀⣩⣥⠼⠤⡍⠐⠒⠒⠒⠒⠂⢱⠀⠀⠒⠙⣟⣈⢀⠀⠉⠀⠀⢁⠀⠐⠒⠀⠀⠈⠀⠉⢐⣀⣐⠤⠖⡀⣠⡇⠰⠂
⠀⠀⠀⠀⠀⠈⠉⢹⠉⠀⡈⠑⢉⠀⠀⠀⠀⠠⠀⣿⣵⣀⣀⣫⠟⠉⣚⡙⠚⠋⠚⠛⠾⠿⣶⣦⠄⠀⠛⣉⠥⠥⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠋⠒⠒⠢⠒⠖⠛⠁⠀⠀⠀⠾⠩⠁⠂⠠⠤⠢⡄⠀⠀⠀⠀⠈⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⢀⣀⢀⠠⠼⠑⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⢋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠈⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠛⠀⠀⠀⢘⠃⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⡦⠀⠀⠀⢸⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠪⢄⠀⠀⠀⠀⣇⠀⠀⠀⢀⠀⠀⠀⠈⠈⠓⢄
⠀⠀⠀⢄⠀⠠⢀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⢀⠃⠀⣶⠀⠹⡀⠀⠀⠀⡁⠀⠀⡑⡀
⠀⠀⠀⠈⠐⢌⠲⣅⢄⡀⠀⠀⠁⠀⠉⠑⠒⠒⠒⠒⠒⡊⠉⠁⠈⠒⠀⡚⠀⠀⣿⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠐⢬⡓⣎⣢⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠴⣤⣤⣿⣤⢤⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠹⡶⡞⠀⠄⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠛⠛⠛⠿⠿
⠀⠀⢀⡤⠴⠀⠀⠃⠀⠿⠐⠀⢞⠻⣕⠢⠀⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠐⠀⠀⠊⠀⠀⠀⠀⠉⠂⡙⢭⡣⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠈⠐
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠶⠶⠶⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢁⡴⠞⠛⠳⢦⡉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢠⡟⠀⠀⠀⠀⠀⢹⡄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⡇⠀⠀⠀⠀⠀⣼⡁⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡄⠙⠦⣤⠄⠀⠀⠉⠙⢷⡀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠛⠉⠙⣷⡀⢰⡇⠀⠀⠀⠀⠀⢘⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠞⠋⠁⠀⢀⣠⠴⠞⠋⢳⡄⠳⣄⠀⠀⠀⢀⡼⠃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠛⠉⠀⢀⣠⡤⠶⠛⠉⠁⠀⠀⠀⠀⠙⢦⣌⡙⠛⠛⣋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠞⠋⠁⠀⣀⣠⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢶⣤⡤⠶⠛⠉⠀⢀⣠⡴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠿⡄⠹⡄⣀⡤⠶⠛⠉
⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠉⠁⢀⣠⠿⣤⡿⠁
⠀⣀⣤⡶⠞⠋⠁⢀⣀⣤⠶⢛⡽⠇
⢸⡋⠀⢀⣠⣴⣞⣫⣍⠀⠀⠘⠶⠾⣆
⠈⠛⠛⠉⠁⠀⢉⡴⠋⠀⣀⡤⣆⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢻⡄⠘⣧⣤⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⡴⠗

⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣾⣿⡶
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣛⣟⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣦⣄⡀⡀⠀⠀⣠⣶⣿⠿⣷⣦⣄⠀⢠⣴⣶⡿⢿⣶⣦
⠀⠀⠀⣼⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⡟⢸⣿⣿⣇⢹⡇⠀⠀⠀⣀⣤⣤⣸⣿⡿⢻⣿⣿⣿⣿⡿⠃⠀⠀⠀⠙⠿⠿⠿⠿⠿⢿⣤⣿⣛⠿⠿⠿⠿⠛
⠀⣴⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠁⣿⡏⢹⣿⠈⠁⠀⠀⠀⠹⣿⡿⠟⣿⣿⣸⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠏
⠀⠛⠛⠿⣿⡿⠛⠛⠃⠀⠀⠀⠀⠀⠘⡿⠁⠘⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠⡅⠀⠀⠀⣤⠀⠀⠀⠀⣥
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠛⠂

⠀⠀⠀⠰⣀⣀⢰⣂⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠙⣤⡄⠀⢀⣀⡀
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠈⠫⣿⡯⠋⣠⣤⡝⠆⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢘⣶⣿⣿⠛⣿⣿⡛⡄
⠀⠀⠀⠈⢻⣿⣅⣽⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⢠⣾⣿⣿⠇⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣴⡿⠽⣿⣿⣏⣀⠿⣿⠃
⠀⠀⠘⠀⣿⣿⣿⣿⣿⣿⠀⠃⠀⠀⠀⠀⠀⠰⣜⠿⠟⢁⣾⣿⣦⣄⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢈⣿⣧⣞⣙⠿⣿⣿⠖⠁
⠠⠮⠀⠀⠉⠉⠭⠍⠉⠉⠀⠀⠥⠄⠀⠀⠀⠀⠈⠀⠀⠀⠙⢿⠟⠁⠀⠀⠀⠀⠈⠛⠿⠿⡿⠿⠟⠁⠀⠀⠀⠀⠀⠈⠛⠴⠿⠟⠂⢾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗

⠀⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠤⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⡀
⠀⠀⠀⣍⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⡘⣿⣯⡀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠘⠳⣦⠄⢠⣴⣶⣶⣦⡀
⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⡷⠘⣿⣿⠷⣟⢀⡤⣾⡇⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠈⠑⣿⣿⢿⣿⣿⣿
⠀⣤⣾⣿⣻⣿⣦⡄⠀⠀⠀⠀⣰⣿⣿⣯⡻⠿⠤⠼⢟⣶⣿⣿⣆⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠯⣰⣤⣀
⠀⣿⠋⢸⣿⠈⢻⡇⠀⠀⠀⠀⠙⢿⣿⠟⠀⠀⠀⠀⠀⠻⣿⡿⠇⠀⠀⠀⠀⠀⠘⠛⠻⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉
⠀⠃⠀⠀⠇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀
⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣷⣦⣤⡀
⢀⣾⣿⣿⣿⣹⣿⣶⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣄⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⣿⣿⢿⣶⣤
⠸⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⢿⣟⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⢀⣀⣴⣹⣛⡟⢻⣛⣏⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢧⠀⠈⠉⠙
⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠉⢻⣽⣽⢿⣿⣿⠄⠀⠀⠀⠀⠿⣿⣿⣷⣿⣿⣿⣷⣾⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⡸⠠⡄⢆
⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣤⣼⣧⣬⡉⠁⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠉⢹⡏⠉⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⡆

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⣀⣀⣠⡾⢛⣛⣉⣛⡻⢶⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣲⣶⠋⢄⣀⠌⢳⣶⣒⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⢛⣿⣟⠿⢿⣶⣄
⠀⠀⣾⣿⡻⣅⡈⠉⠀⠛⠀⢉⢑⣹⢿⣿⣧⠀⠀⠀⠀⠀⠘⣟⣩⠝⢻⠷⢎⢉⡴⢾⢛⢭⣙⡛⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⢁⡴⣯⣿⣮⡷⡀⠹⣿⣆
⠀⠀⠈⠁⢁⠿⠉⠛⠳⠶⠞⠛⢩⢿⡈⠈⠁⠀⠀⠀⠀⠀⠀⠈⢠⣀⣼⠰⢛⣿⠳⢸⣄⢠⠈⠀⠀⠀⠀⠀⠀⢠⣾⡫⠟⠃⢸⣶⣼⣿⣯⣶⣿⠀⠻⠽⣳⣄
⠀⠀⢠⢂⣵⡽⠀⠀⠀⠀⠀⠀⠐⢿⣧⠐⡄⠀⠀⠀⠀⠀⢀⡴⢋⡠⠁⣂⡾⡛⣆⡂⠁⢌⠳⣄⠀⠀⠀⠀⠀⣿⡏⢀⣔⣻⡄⣠⣧⣿⣟⣇⠀⣾⣢⣀⢰⣿⡆
⠀⢀⢇⡞⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⢸⠀⠀⠀⠀⠀⡎⢹⠊⢠⣾⠀⢹⣿⠁⢹⣦⠈⢻⠉⡂⠀⠀⠀⠀⣿⡟⣽⣾⡯⠽⠿⢫⣿⣏⠿⠯⢽⣿⣮⣻⣿⠃
⠀⠸⢸⢿⠟⣠⣶⣦⡄⠀⣠⣴⣶⡄⢻⣿⢇⠀⠀⠀⠀⠀⣦⣸⣀⣸⣀⣴⡟⠟⢷⣄⣨⣂⣰⣠⡆⠀⠀⠀⠀⠙⢏⢿⣿⣵⣴⣢⡘⠿⢏⣴⣦⣾⣿⡿⡹⠋
⠀⠀⠣⠻⡾⣽⢾⣿⣿⣐⣿⣿⡿⣿⣷⢇⠌⠀⠀⠀⠀⠀⠈⠊⠝⠻⡿⠀⠀⡆⠀⠘⡿⠋⠍⠊⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⢟⢿⣦⣿⠯⠋⠉⠉⠁
⠀⠀⠀⠀⠈⠻⠻⣆⠀⠠⡿⣿⠻⠏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣸⡿⡿⣿⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⢹⣶⣶⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣾⣸
⠀⠀⠀⠀⠀⠀⠀⢼⣛⣛⣻⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠟⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠔⣿
⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⠷⣦⣤⣤⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⣀⡀⢸⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠠⢸⣿⡇⣿⢿⣿⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠁⡈⡋⣎⠀⢰⣶⣖⡂⠀⠀⠀⠀⠀⠀⠰⠶⢸⣿⣧⣼⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠸⢸⣿⠀⣿⣳⣒⣒⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠿⠿⠛⣿⠷⣯⣅⡨⠄⠀⠀⠀⠀⠀⠀⢘⣛⢹⣿⣿⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⡛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⣶⣿⣮⡃⠀⠈⠙⠓⠀⠀⠀⠀⠀⠀⠀⢈⣭⡬⠿⠏⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠥⠤⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠋⠉⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠛⠙⠛⠋⠋⠙⠉⠛⠋⠛⠋⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⢠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⠁⠀⠀⠀⣈⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡲⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣀⠄⠂⢉⡀⠀⡄⢰⠆⣤⢰⣆⣠⣤⠀⣠⢶⠄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⢻⡄⠀⡷⢿⡇⢸⣠⢿⡀⠿⣼⠏⣿⢉⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠈⢧⡰⠁⠸⣧⡎⠁⠀⠉⠀⠀⠀⠉⢸⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣄⣰⠀⢠⢦⠀⠄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢰⢺⡇⠌⠉⠒⠁⠀⠠⠀⠀⠘⠃⠄⡠⢤⠞⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠘⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠃⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⡶⠶⠒⠒⠂⠀⠀⣀⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⡴⢄⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠠⠃⠀⢀⠄⠀⠀⣇⠀⡆⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠤⠄⢠⡠⠃⠄⠀⢀⡀⠸⠜⠁⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠿⠛⠉⠁⠀⢀⣀⣀⣀⣀⠀⠈⠙⠛⠿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠛⠁⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⠈⠛⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠏⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠀⢹⣿⣿⣿⣿⣿⣿⣦⡀⠀⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠇⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⠋⠉⠉⠉⠉⠉⠁⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⡀⠈⠿⢛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⡄⢠⣿⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⡄⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣧⠀⠈⠉⠛⠛⠻⠿⠿⠛⠛⠋⠁⠀⠀⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⢿⣿⡇




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠆⢀⠀⠐⢊⡽⢿⣟⠒⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠿⢤⣤⡾⠃⠉⠛⠁⢀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠙⠻⠶⢶⣶⣶⣶⡶⣶⡾⠿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠤⣤⣤⣤⣤⣤⡄⠀⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣀⣀⡀⢉⣉⣁⣠⠀⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⣶⣷⡶⢤⣀⠀⠈⠉⠛⠛⠛⠛⠋⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠠⠾⢷⠾⠋⠈⢀⣠⣿⡆⠐⠶⠶⠶⠶⡶⠿⠀⢻⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠙⠛⠶⠶⠾⠿⠿⠟⠻⣿⣿⠠⣤⣤⣤⣤⣤⡤⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠒⠲⠶⠶⠟⠃⠀⣿⣏⢀⣀⣀⣈⣁⣩⣤⠀⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠘⠛⠛⠛⠛⠁⠀⣿⡟⢀⣀⣉⣉⣉⣉⣁⡀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠉⠛⠛⠛⠛⠁⢠⣿⡇⠀⠉⠙⠛⠛⠛⠉⠁⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠙⠛⠛⠛⠁⠀⣿⣇⠀⠛⠷⠶⠶⠶⠷⠂⣿⣧⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠈⠛⠛⠛⠛⠁⠀⣿⣿⠀⣀⣀⣠⣤⡴⠶⠖⠛⠛⠋⢉⠁⠀⠙⠳⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠑⣒⣶⣾⡿⠷⠶⠛⠛⠋⠉⠙⠛⠶⣤⣄⣀⡠⠄⠘⠲⢭⣥⣨⡖⠒⠠⡉⠛⠻⠷⣶⣤⣄⣀
⠀⠀⠀⠀⣀⣀⣀⣤⠤⠶⠒⠚⠋⣉⠽⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣦⣄⣀⠀⠀⠀⠈⠑⠲⣔⠲⠷⢦⠉⠙⠛⠻⠿⠶⠄
⢀⡀⠈⠉⠉⠁⢀⠀⣤⣙⡳⣤⠥⠖⠒⠛⠉⠉⠛⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢶⣤⣀⡠⠔⠚⢛⣛⣋⣤⠤⠶⢒⣀⣠⣧
⠘⡇⠈⠑⢦⡀⠀⠳⢤⣼⣿⡾⠲⢤⣀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣬⠿⡷⠚⢛⣉⡩⠤⠶⠚⣋⡩⠥⢸⣿
⠙⣷⣄⠀⠀⠉⠳⢤⡀⠀⠈⠙⠓⣶⢫⣍⠛⢷⣦⡴⠖⢚⣩⠤⠄⠉⠂⠀⣀⣠⣤⠶⠖⠛⠋⠉⠀⠀⠀⣷⠀⠀⠀⠐⠒⠋⣉⣠⡤⠴⣿⡿
⠀⣗⠈⠛⢦⣄⠀⠀⠉⠳⢦⣀⠀⠘⢦⣉⣿⣦⣿⡥⠖⠋⣀⣤⣤⡶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⡠⠴⠖⠛⠉⠁⣀⣠⣤⣿⡇
⠀⣿⠑⢦⣀⠈⠳⢦⡀⠀⠀⠈⠙⠦⣄⠈⠉⣉⣤⣴⠾⢟⡋⠉⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣠⡤⠶⠶⠟⠛⣉⣁⣬⣿⡇
⠀⢿⠲⣄⡈⠓⢤⡀⠙⠳⣄⡀⠀⠀⠀⢰⣟⣯⠵⠖⢚⣉⣤⡤⠖⢒⣁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣀⣠⡤⠶⠞⣛⠋⢍⣘⣿⣿⣶
⠀⠀⢤⣀⠙⠦⣄⠉⠒⢄⠀⠙⠲⣄⡀⢸⣷⠶⠚⢛⣉⣭⠴⠖⠛⠉⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡯⠍⣒⣺⣭⣵⣶⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⢺⣷⠶⠟⠛⠉⠀⠀⠀⠀⠀⡀⢸⡇⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣿⣿⣿⠿⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠉⠻⣦⣄⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⣀⣤⣴⡶⠿⠋⠀⢹⣇⣠⣤⣴⣶⣿⣿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣀⠀⠀⠀⢸⣿⠖⠚⠩⠛⣉⡥⢀⣠⣤⣶⣾⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄⡀⢸⣿⣤⡴⢖⣫⣵⣾⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢾⣏⡴⠞⠛⠉⠁
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⡄⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠱⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠆⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢰⠀⠘⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠰⠀⣇⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆⠠⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⢡⡃⠀⠉⠉⠀⠀⠀⢹⠁⠀⣧⠁⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⡏⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⢸⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⣰⠏⠈⢧⢸⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠂⠀⠀⠘⡇⠀⢰⠏⠀⠀⠀⢿⠀⠀⡟⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⢀⠃⠀⢀⣀⠀⠀⠆⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣀⣀⣀⣀⡇⢸⠰⢩⡿⠯⢏⠡⢸⠀⣇⢀⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⡀⠁⠱⠦⠦⠄⠁⢹⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡍⠙⡖⠒⠒⠒⠒⠒⣿⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠒⠒⠂⠒⡀⣾⡇⠀⠀⠀⠀⠀⣿⡆⡗⠐⠂⠒⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡀⣠⠀⠖⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⡿⠈⡇⠀⠀⠀⠀⠀⣷⠸⣧⡀⠀⠀⠀⣸
⠀⠀⠀⠀⠈⠃⢤⠀⠒⠀⠈⣀⡈⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠁⠈⢈⡟⢳⠚⡗⠒⠒⠒⠒⠚⣟⢲⠛⣅⠁⠁⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠑⠦⡉⠁⣙⢶⣤⠄⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠴⡫⠀⡀⢈⡇⠀⠀⠀⠀⠀⣿⠈⡈⠩⠳⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠌⠳⣶⡀⠉⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠞⡡⠊⢀⠔⠁⠸⡏⠉⠉⠉⠉⠉⣿⠀⠑⢄⠈⠢⡑⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⠴⠈⠛⢷⣀⠈⡡⢄⠀⠀⠀⠀⠀⢀⡴⢊⡵⠋⡠⠒⠁⠀⠀⠐⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠑⠤⡈⠢⣝⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠲⡂⠈⠉⠳⣄⣀⠱⠆⠀⢀⡴⣋⢔⣁⠀⡎⠀⠀⠀⠀⠀⢸⡇⠀⣀⣄⢀⠀⣿⠀⠀⠀⠀⠀⠈⡆⠈⠑⢌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠈⠉⠻⣧⣉⡡⠬⢷⠦⠄⠐⢚⣁⡙⠦⠀⠐⡗⠲⣖⠒⢒⢒⡎⠀⠀⢸⣿⠀⠀⠈⣖⠒⠒⠒⠒⠒⠓⠒⡒⠒⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠐⡉⢷⡀⠀⢂⢻⣿⡷⡾⠟⠃⢃⠀⢡⠤⠋⠉⠉⠉⡇⠀⠀⢸⣿⠀⠀⠀⡏⠉⠉⠉⠉⠉⡏⢉⣉⠉⠩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⣧⠀⠸⠀⠷⠀⠛⣤⡀⡘⢀⡜⠓⠀⠠⠤⡄⣇⣀⣀⣸⣿⣀⣀⣀⡧⡄⠤⠀⠐⠂⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠤⠀⠒⣈⡉⠆⠀⠀⠀⠀⡿⠀⢰⠤⠤⠀⢒⣊⠹⢧⣝⠛⠫⣯⠉⠉⢃⠸⣯⣭⣿⣿⣿⣭⣽⢀⠉⠈⠹⠉⠉⠹
⠀⠀⠀⢀⠀⡶⠖⠙⠒⢂⠈⠉⠀⠀⠀⠀⠀⠀⠀⡾⠋⠉⠀⠀⠀⠀⠙⢤⡄⠒⠙⢻⣄⡈⠵⢄⣘⠀⠉⠉⠉⠉⠉⠉⠉⢘⣀⣀⣀⡀⣀⣀⡆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣉⠁⢈⠻⣦⡄⠙⠢⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠙⠋⠛⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢀⡀⡀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣈⣻⣌⣀⠀⠙⠶⣄⠋⠓⣄⠀⠀⠀⠀⣃⣀⣀⣀⣀⣀⣀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡤⠀⠛⠻⠎⢀⣑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⢖⠀⠉⠀⣀⣀⠨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣙⣿⣶⠟⠀⠻⢶⣯⣃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⠉⢹⣿⡏⠁⠀⠀⠉⠿⡟⠉⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠛⠁⠀⢀⣀⡀⠀⠀⠑⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⢠⣶⠋⣉⠙⣷⡄⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣶⡿⠈⠈⠉⠡⠽⠿⣶⣶⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⡆⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣉⠉⢹⣧⣠⣿⣼⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣈⣿⡇⢀⣉⣹⣿⢾⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢈⣿⡏⣉⣉⣿⣾⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣉⣀⣿⡟⠛⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠔⠒⠒⠒⠖⠒⠲⠶⠶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀⠀⠀⠀⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠠⠄⡀⠀⠀⠀⠀⠀⠈⠻⣦⡀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⢤⣴⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⣠⣾⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣜⡀⠀⠀⠀⠀⣴⠟⠛⠁⠀⠀⡠⠔⠂⠈⠁⠀⠀⠀⠀⠈⠉⠙⠒⢤⣀⠀⠀⠀⠉⢢⣴⣾⠿⠋⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣶⣤⣾⠃⠀⠀⢀⠔⠊⠀⠀⢀⣠⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⣾⣿⠿⣇⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⠿⣷⣦⣤⣀⠀⠀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡏⠁⠀⠘⣧⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⣤⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠈⠙⠻⣿⣿⣾⣏⡀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄⣴⣿⣿⠟⠋⠀⢻⡄⠀⠀⠘⣇⠀⠀⠀⠀⢻⣇
⠀⠀⣿⣶⣿⣿⣿⣧⣤⡦⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⡽⠻⢿⣿⣷⣤⣀⣴⣶⣶⣶⣤⣴⣿⢿⡏⠁⠀⠀⠀⠀⣿⠀⠀⠀⢻⠀⠀⠀⠀⠸⣿
⠘⣿⣿⣏⡷⠾⣧⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠈⠙⢻⣿⡟⠉⠉⠙⣿⣿⠁⠀⢻⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⣿⡆
⠶⢿⣿⣽⣧⣠⣼⢷⣿⣏⡁⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢸⣿⣇⠀⠀⢀⣼⣿⠃⠀⢸⠃⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⣿⡇
⠀⣾⠿⣿⣾⣧⣿⣿⣟⠙⠁⣴⡾⠛⠻⣷⣄⢸⡀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠻⣿⣷⣶⣿⡿⠋⠀⢠⡿⠀⠀⠀⠀⠀⣼⠁⠀⠀⣸⠃⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠿⠋⠹⠏⠈⠋⠀⢰⣿⠀⠀⠀⢸⣿⠈⣇⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⢀⡼⠁⠈⠉⠀⠀⠀⢰⠟⠁⠀⠀⠀⠀⢠⠏⠀⠀⢠⡿⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣤⡾⠋⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠑⠄⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⣀⢀⣾⠃⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⣿⣿⣧⡀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠟⠋⠁⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡇⠀⠀⠀⠀⣀⣀⣤⣴⣾⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁