Stringed Instrument Emojis & Symbols

⠀⠀⠀⣠⠞⠙⢲⡶⣆⣀⡀
⢀⡴⠛⠁⠀⠀⠈⠁⢠⡍⠳⣶⡄
⠘⣧⠀⣠⡀⠀⠀⠀⠀⠈⠳⣈⠳
⢀⣸⠷⣤⡀⠓⠂⣄⡀⠀⠀⠈⠢⣇
⢧⡨⠀⡼⠟⢿⣶⣦⣍⣁⠢⡀⠀⠘⣦
⠀⠀⠈⠃⠀⠸⠁⠀⠉⠣⣝⢶⡄⠀⣈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡽⡿⠉⢳⡻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠱⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡸⣎⠀⠱⡈⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡸⣆⠀⠁⠂⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢹⡄⠀⠀⠂⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢻⡄⠀⠀⠂⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢳⡀⠀⠈⠄⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⣷⡀⠀⠀⢀⡀⠘⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸⠷⡀⠂⠈⠣⠀⠹⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠡⡆⠱⡀⠀⠀⢣⠀⠹⣿⠛⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⡿⢰⠁⠄⠠⠀⠀⠀⢀⠀⢻⠇⠙⠌⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠇⣘⠀⠀⠀⠀⠀⠀⠀⣤⣠⡗⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⡏⢠⠐⣷⡇⠀⠱⠀⢀⡤⠖⢻⠄⡀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠘⢸⡇⡳⠀⠀⠑⠊⠁⠀⣀⣷⣤⣤⣤⢄⡀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠘⡌⠀⡄⢸⡿⠃⠀⠀⠀⢀⣴⣿⠙⣏⣛⡀⠀⣸⣽⣧⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⡇⠀⡇⢸⠁⠀⠀⠀⠀⣼⣿⣿⣧⣿⣯⣿⣷⣿⣿⣿⡇⡀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⡀⠀⠀⠁⠈⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣇⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢋⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⠿⠟⠀⢠⡀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢀⡀⠀⠀⠀⠀⠀⠀⠄⠐⡁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠁⠀⠀⢰⣤⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⡀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣴⠀⡇⠀⠀⠀⠆⠀⠐⣣⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠐⠫⠑⠀⠜⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣅⠀⠀⡀⠀⠀⢂⠶⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠠⠀⢁⣄⣠⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣆⠐⣇⡀⣷⢈⣌⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢃⠨⡜⢳⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣽⣷⣹⣷⣿⣿⣿⣧⠹⣿⣆⡐⢆⠀⠀⠀⠀⠀⠠⠀⠐⡈⡀⢰⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣤⣽⣷⣙⣿⣿⣷⣾⣿⣿⣷⣄⡀⠀⠀⠀⠀⢓⣶⣀⣰⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣴⣥⡶⠾⠛⠁
⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣻⠋⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⣠⡏⠀⠀⠀⠀⠀⠀⣸⠛⠁⠈⠀⠀⠀⠀⢐⣼⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⣿⢳⠄⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⢀⡠⣘⣯⠂⠀⠀⠀⠀⢀⣿⢧⡀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡏
⠀⠀⠀⠀⣦⣰⡿⣹⠀⠄⠀⠀⠀⣸⣿⡞⠁⠁⠀⠀⠀⠐⣾⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠈⠛⠛⠋⠸⠀⠀⠲⢹⣾⣏⠀⠀⠀⠀⠀⠀⢠⣿⠀⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢦⣿⢧⣿⠀⠀⠀⠀⠀⠀⣿⡇⣴⠃⠠⠀⠀⠀⠀⠀⠠⠤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠴⠀⠀⢀⠞⣀⡙⡄⠁⠀⡀⠀⠀⠀⠈⠀⠀⢀⣿⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡙⠑⢷⡄⠰⠀⠀⠀⣠⠀⠀⢀⣿⣿⠀⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠗⣠⣿⠀⡄⠀⠀⠀⢠⢀⡀⠺⠿⢿⣻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣾⠿⣶⠿⠏⢀⠀⠀⠄⣄⣰⠃⡀⠀⠀⣿⡟⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈⠁⠀⠳⠞⢡⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⣶⣶⠀⠀⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢁⠄⠀⡀⡺⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠗⠁⠀⣤⡿⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠋⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣧⣴⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣿⣧⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣤⣾⣿⣏⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣯⣤⣿⢦⣬⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠹⡟⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⢸⢻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢰⡆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⡀⡌⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠘⠛⢁⠇⢰⣿⢟⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⢸⠀⢸⡟⠀⠀⠈⢾⡛⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⠀⠀⠀⡞⠀⢸⡇⠀⠀⠀⠀⣿⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢇⠀⠀⠀⢠⠀⠀⣸⠇⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⢻⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢸⡇⠀⠀⠐⠒⠀⠛⠀⠀⠀⠀⣾⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠼⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠈⣇⣀⡀⠀⢀⣠⡀⢀⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⢠⣿⠈⠁⠀⠈⠉⣷⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢠⠀⠀⢀⠜⣿⠀⠀⠀⠀⢰⡿⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠛⠦⠄⠋⠀⣿⠀⠀⠀⠀⢾⡅⠀⠚⢿⢷⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⢀⠎⠀⢀⣄⠀⣰⠏⠀⠀⠀⠀⣿⡇⢀⡀⠀⢻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⡎⠀⠀⠀⠙⠚⠁⢺⠟⣿⠗⠀⠈⠓⠞⠁⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢸⢸⡏⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢀⡴⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⠏⢹⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣴⠁⠶⡎⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⢠⠈⠁⠀⠉⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠐⢠⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣀⠀⠀⢰⢸⠄⠀⠀⠀⣀⡀⣀⣼⣿⠟⠁
⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣷⣾⣾⣿⣶⣤⣬⣵⡾⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠛⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠙⠛⠛⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣾⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢀⣤⡴⠾⠫⡢⡀
⢘⠀⠀⠀⢠⡌⠪⡢⡀
⢾⢠⣤⠀⠀⠀⣶⠜⢌⢄
⠀⠑⢦⡀⣤⣄⡑⢤⣤⡡⣢⡀
⠀⠀⠀⠈⠢⡀⣬⣦⡙⢝⢺⣽⠦⡀
⠀⠀⠀⠀⠀⠈⠑⢌⡻⣦⡟⢍⠳⣬⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡺⣦⣑⢌⡪⡪⣢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡛⢷⣝⠪⡢⡙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡙⢷⣌⠪⡑⢝⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡑⢕⢌⠢⡑⢌⠻⣾⣉⠩⢉⣑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡑⢅⢄⠠⡐⢌⠻⡆⠈⠉⠻⡶⣵⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡑⢕⢌⣢⣿⢶⠒⠠⠤⡀⠈⠳⣕⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠒⠛⣟⣤⡖⢝⠩⣢⠘⡆⠀⠈⠑⠒⠤⢥⣂⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠆⢹⢝⠢⡑⣼⠧⡇⠀⠀⠀⠀⠀⠄⠀⠉⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⠦⣙⠓⢞⣣⡕⢅⠀⡀⠀⠀⠀⠀⠀⡄⠃⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠉⠁⠙⠮⡢⡑⢌⠢⣀⠖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠉⠮⣢⠝⣡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡢⠀⠀⠀⠀⠰⡥⠊⠉⠪⡢⡀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠪⣢⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠮⡂⢀⡀⠀⠀⣀⠀⣀⠔⠛⠊
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣵⣾⣶⣿⣦⣭⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠒⠉⢰⠉⠑⠚⠋⠀⠀⢸⠀⠀⠀⠀⠀⢸⠇⠀⠈⣿⢿⣦⠀⠀⠀⣾⡇⠘⠻⠛⠛⠻⠛⠛⠛⠛⠛⠻⠛
⠀⠀⠶⠀⠀⠀⢸⡀⠀⢼⠂⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⢻⣆⣴⡶⢸⡇⠐⠒⠲⠶⠶⣶⠶⠶⠿⠶⠶⠶
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⢰⣿⣿⡄⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢈⣼⠃⢸⡇⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠠⢿⣧⣹⡇⢸⣿⣿⠀⣸⣿⣿⡀⠀⠀⢀⣼⣄⡀⠀⡇⠀⢠⣾⠃⣠⣿⣷⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⣿⣿⡇⣨⣿⣿⡀⣿⣿⣿⡷⠀⢰⣿⣿⣿⣿⡄⡇⣰⣿⠃⢰⣿⣶⣾⣿⡆⠸⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⣀⠀⠀⢿⣿⡇⢿⣿⣿⠇⠈⠙⠉⠀⢤⠘⢿⣿⣿⠟⢠⣿⣿⡃⠀⠘⢿⣿⣿⡿⠃⠀⢿⣿⣿⣿⠿⢿⣿⠏
⣴⣴⣦⣤⣤⣤⣴⣦⣤⠐⠶⠶⠶⠶⠶⠶⠖⣀⢀⣀⣤⡄⠀⠀⢀⠀⣀⣀⣤⣄⣀⣀⣠⣤⢰⣶⣶⣦⡀⠀⣴⣶⣦
⠀⡇⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠒⣒⠀⠀⠀⠀⠀⠐⠒⠿⠗⠛⣂⠀⠈⢸⣿⣿⣿⣿⠂⣿⣿⣿
⠀⡇⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠆⠀⣞⣥⣴⡟⠀⣸⣷⣄⠀⣠⣤⣤⣤⣤⣶⣶⣿⠤⢾⣿⣿⣿⣿⣿⠆⣿⣿⣿
⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠣⣿⣿⣿⡿⠋⠉⠉⠸⣇⣀⣉⣹⣿⣇⣹⠙⠀⢀⡇⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠁⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⣀⠤⠔⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⣀⠀⠀⠀⠀⣀⠤⠔
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⠀⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⣾⣿⣷⡀⢰⣶⣷⡂⢀
⠀⠀⠈⠉⠉⠀⠀⠀⣤⣤⣤⠙⢛⣷⣿⣮⡉⢩⣶⣶⡆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠩⣽⣿⣿⣾⣾⣿⣿⣷⠈
⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⠀⣿⣿⣿⣿⣿⠀⠀⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⡏⠠⢀

⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⢠⢺⠤⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠃
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠂
⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⣰⣿⣿⣿⣦⠀⡆⠀⠀⠀⣀⣤⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠸⣿⣿⣿⠟⠀⠇⠀⠀⠀⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣠⣿⣿⣿⣄⠀⠀⠀⠀⠀⠻⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢻⣿
⢰⣿⣿⣿⣿⣿⡄⠀⠀⠀⢀⣼⣿⣿⣿⣧⡀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⢸⣿
⠈⠻⢿⣿⡿⠟⠠⠄⠀⠀⣼⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠘⠃⠀⠀⣠⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣞⣿⣿⣿⣿⣿⣇⣟⣶⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⡿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⣿⣟⡟⣿⣿⣿⣿⣟⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢸⣿⣿⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⣿⢸⣿⣿⣶⣶⣾⣿⡇⢻⡿⣦⣤⡀
⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣇
⠀⣠⣾⣿⣿⣿⣿⣿⣦⣤⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡯⠭⢽⡷⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠁⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠹⣿⣿⣿⣿⣿⣿⠟⠋⠛⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡏⠁

⢠⠶⡈⠁⠢⡀
⡘⢃⠹⡀⠀⢹⠀⣠⣠⣔
⠉⠺⡤⠧⠔⠋⢉⣩⣿⣎⠘
⠐⠒⣳⡠⢗⣄⠻⣿⣿⣿⣷⣤⡀
⠐⠲⡉⢀⡃⠁⠢⠽⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠈⠉⠁⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢝⢿⣿⣟⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣝⢿⣷⣿⢿⣿⣷⣤⡴⠚⠉⠉⠉⠉⠉⠉⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⣿⣷⣝⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠙⢿⣳⣝⢿⣻⣟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠈⠻⣿⡷⣝⢿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣝⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢀⡟⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣽⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⢧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣽⣿⠛⡁⢄⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠀⠢⡈⠢⡑⢴⣯⣉⣑⢦⡈⠳⢦⣄⡀⠀⠀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⢢⡴⠞⡙⢅⢸⠈⢿⣆⠀⠈⠙⠛⠛⠉⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠈⢷⣠⡤⣤⣀⠀⠀⠀⠀⢀⠴⣤⡀⠈⠯⡈⢢⡤⠶⡛⢇⠀⠻⣧⣀⠘⢛⡷⠀⠀⠀⠉⠛⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣸⠀⠀⠀⠉⠳⣄⠀⠀⠸⡄⠉⠀⠀⣦⠞⠫⡀⢀⠈⠢⡑⢄⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⠛⠓⠲⣤⡈⢳⡀⠀⠹⣦⡀⠀⠉⠀⠀⠀⢂⠑⢄⡐⢞⡽⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠹⡄⠀⠈⢷⡄⠀⠀⠀⠀⠀⢹⡦⠞⠋⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣿⠀⣀⡈⣷⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡷⠃⠀⠻⠴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⠻⣄⠀⠀⠀⠀⠀⠀⠀⡼⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⢤⣄⣀⠀⠈⢧⡙⢦⡀⠀⠀⠀⣠⠞⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠘⣧⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠈⠳⣦⠀⢧⠀⠙⣆⣠⠞⠁⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠘⢷⡀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣈⣷⠛⠙⡇⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠈⠻⣦⣀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋⠘⣇⠀⣹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠙⠻⠾⣷⣦⣤⣤⣤⣶⠶⠟⠛⠉⠀⠀⣠⡶⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⠥⠄⠀⠀⠀⠸⠷⣶
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⡧⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣻⣿⢬⠀⠀⠀⠀⠀⠀⠀⠀⠠⢼⣿⠒⠂⠀⠀⠀⠐⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢈⣩⣿⡷⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣶⣦⣄⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠛⣿⣿⣿⣿⣿⣿⡟⠃⠀⢰⠀⡇
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡇
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⣴⣿⣼⣿⣿⣿⣿⣿⣦⠄⢰⠀⣿
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣾⣿⣿⣿⣿⡟⢹⣿⣿⣿⣧⢸⠀⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣿⣿⣿⣿⣿⢧⣿⣿⣿⣿⡟⠀⠀⣿
⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⣀⣴⢿⢻⣿⣿⣿⣿⣇⢻⢷⣄⡀⠘⠱⠛⠛⠛⠷⢹⣿⣿⠟⠁⢰⠀⣿
⠀⠀⠀⠀⢀⣿⣿⡿⣿⣿⣿⡿⣿⣿⡇⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢈⠀⣿
⠀⠀⠀⢀⣼⣿⡇⣿⣿⣿⣿⣿⡷⣿⣧⡀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠁⣸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣿
⠀⢠⣾⣿⣿⣿⡇⣿⣿⣿⣿⣿⣇⢻⣿⣿⣿⡆⢻⣿⣿⣿⣿⣿⣿⣠⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣾⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈
⠀⣿⣿⣿⣿⣿⣿⣿⢿⡟⠁⣸⣿⣿⣿⣿⣿⣿⡇⠀⠈⠛⠛⠻⠿⠿⠟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣴⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇⢠⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⣿⠁
⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣲⣿
⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⢿
⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠉⠉
⣿⣿⣿⣿⣿⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣿⣛⣛⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠶⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀
⠀⠀⠀⠉⠋⠀⠁⠀⠀⠀⢀⣐⡀⠀⠀⠀⠀⠀⠀⠇⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⡤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡵⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⠀⠀⢠⡀⠖⠀⠰⠈⢂⣀⡀⠀⠀⣤⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢾⡧⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠐⣁⠀⡅⠀⢠⣇⣀⣀⣀⢀⠀⢀⡽⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⣾⣧⠀⠀⠀⠀⠀⢠⠊⠉⠋⠁⠀⠀⠀⠀⠓⢀⠀⠒⢊⠙⠺⠅⠟⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠀⠀⠄⠀⠀⢂⠀⠚⠃⠀⡘⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠂⠀⠀⠀⠐⠒⠂⠈⠀⠀⠀⠈⠀⠀⠛⠓⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠲⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠂⠀⠀⠀⠀⢣⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠂⠀⠀⠀⠀⡆⠈⠙⢦⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠰⠉⠇⠀⠀⠀⠀⠀⠀⠀⠲⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⡄⠀⠀⠀⡇⢸⠀⠀⠀⣠⢃⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠃⢀⠆⢰⠁⠀⠀⠀⠃⢀⠀⠀⣰⠃⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢠⠃⠀⡾⠀⠀⠀⠀⠐⡘⠀⡰⠃⠀⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⠀⢠⠇⠀⠀⠀⠀⠀⠐⠐⡁⠀⢀
⠀⠀⠀⠀⠀⠀⠂⡘⡖⠠⠆⠀⠈⢧⠀⠀⠀⢂⠸⡇⠄⠀⠀⠀⠀⠈⠂⠀⢿⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⢨⠆⠀⠀⠀⠀⠀⠰⠖⠅⢀⠦
⠀⠀⠀⠀⠀⠀⠠⢘⡏⠀⠀⠀⠀⠘⢀⠀⠐⠂⠀⣇⠐⠉⠀⠀⠀⠀⠀⠂⡾⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠀⠀⡀⠀⠀⡄⠀⠀⠓⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠠⢀⣸⣀⠠⠜⠀⠀⠀⠀⠀⠀⠐⠒⠸⠀⠀⠀⠀⠀⠐⠄⣀⣀⠀⠴⠂⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢱⣿⣯⠐⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠈⠁⠁⠁⠁⠁⠀⠁⠈⠒⣤⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠐⠢⠤⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⣄⠴⢀⠄⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⢴⣯⣿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⢀⠰⣌⣷⢿⣷⡀
⠀⠀⣀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡄⠆⠈⣺⣿⣦⡄⢿⣷⡀
⠀⣼⣷⣏⠆⠀⠀⠀⠀⠈⠛⠻⠿⠷⠗⠀⠀⠀⠀⠀⠀⢰⣾⣥⡇⣴⢀⡄⣤⢠⠄⣄⢠⠀⡄⢲⣸⡆⠁⡈⠸⣿⡆⠸⣿⣿⣿⠰⠎⢿⣧
⢠⡿⠱⣾⣯⡐⣴⡠⠠⢀⢀⠀⠀⠀⠀⠀⡀⡀⣄⣦⠀⢸⣿⠋⣩⣾⡏⣿⡄⠀⣿⡏⣷⣦⣕⡜⣿⣷⢸⡀⠃⢿⣿⡐⢷⣼⣿⣟⡷⠘⣿
⣼⢏⣹⣿⢻⣡⣿⠰⢆⣼⡖⣙⣡⣶⡖⢡⣷⣟⠛⣿⡄⢸⣿⠁⠻⣿⡇⣻⡆⠀⣿⣇⣿⣿⡿⠍⢻⣿⡎⣖⠘⠸⣿⣧⠘⡿⢿⣿⡐⠒⢹
⣿⣬⣿⣿⡇⣼⡿⢤⣵⣿⡗⣯⣽⣿⡇⢨⡟⣿⠇⢻⡇⢸⡏⠖⠀⣿⡇⣿⡆⠀⣿⣯⠹⣿⠯⠁⠸⣿⣿⠀⢈⠀⣿⣿⡌⣵⡞⣿⡇⡷⠊
⡇⣤⣿⣧⡗⣿⡇⠊⢹⣿⠁⠽⣟⣻⡇⠰⣏⣿⠐⢺⡇⠸⣇⠀⣶⣿⣧⢉⡖⠀⣿⣿⢨⣦⣬⣢⠀⣿⣿⡦⡯⠰⢹⣿⡇⣙⡃⣿⡇⠀⠤
⢠⠄⣿⢛⣁⣿⡇⢱⣼⣿⢠⣄⣾⣿⡇⢸⣤⣿⠗⢸⠃⠀⠇⠀⠈⢹⣿⣸⡿⠀⢹⣿⣎⡬⣿⡿⠆⢸⣿⡇⣓⡴⠸⣿⣷⢩⣷⢿⡇⣤⡐
⠀⣤⣿⣺⣽⣿⢓⠯⣿⡟⠸⣿⣿⣿⡇⣸⢻⡿⠀⠌⠀⠀⠀⡉⠀⣸⣿⣧⢸⡃⠸⣿⣯⠻⣿⡷⠁⢸⣿⣿⠀⡈⡀⣿⣿⣘⠟⣿⡇⡯⠁
⠀⢹⣻⡟⠿⣿⢠⣄⣿⡇⢈⣻⢡⣿⠣⣾⣿⡿⢀⠀⠀⠀⠀⠀⢆⠙⠿⣿⣿⣿⠀⢻⣿⣮⣷⣬⣴⡀⣿⣿⢸⠅⡇⣿⣿⣼⣆⣿⡷⠁
⠀⠀⠹⣿⡿⣿⡼⣜⣿⣷⣿⣿⣿⣿⣰⣿⡿⠤⠀⠀⠀⠀⠀⠀⠀⠐⢞⠹⣿⣿⣧⣌⢿⣿⣿⣿⣿⢭⣿⣿⠼⠅⢃⣿⡯⣝⣿⡿
⠀⠠⠄⠹⠯⣿⣧⠈⣿⡟⢻⣻⣿⣽⣿⠟⢻⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠻⠛⠻⢿⠿⠎⢿⣿⣽⡟⣉⣿⣿⡆⢈⣼⣿⠧⠞⡋⠀⢐⠓⠂
⠀⠀⡃⠀⡃⠈⡉⠉⡍⠉⡍⠀⠭⠀⠩⠀⠨⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⣭⠀⠀⢩⠀⠀⢨⡅⠀⢨⡍⠉⢉⡍⠀⠀⡍⠀⢈⡅⠀⢨⡁⢀
⠀⠀⠆⠀⠆⠀⡆⠀⡂⠀⢒⠀⢒⠀⢘⠀⢘⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠶⠀⠀⠰⠀⠀⠰⠀⠀⠰⠆⠀⠰⠆⠀⠀⠆⠀⠰⠆⠀⠰⠠
⠀⠀⠈⠀⠁⠀⠥⠄⠥⠀⠼⠀⠴⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⡛⣀⠀⣘⠀⠀⢘⠀⠀⢘⡃⠀⢈⡃⠀⠘⠇⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⣀⡀⠀⠀⣼⠁
⠀⠀⢸⣿⣈⡇⠀⠀⢀⠀⢀⠀⠀⠀⣀⢀⣀⢀⣹⣦⣄⣿⣿⣤⣴⣟⠀⢀⣀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⢀⣤⣤
⠀⣀⢺⣿⣿⣿⣮⣲⣿⣃⣿⣧⣠⣤⣿⣸⣿⣾⣧⣾⣿⣿⣿⣿⣥⣿⣤⣸⣿⣶⠀⣀⣿⣤⣤⣼⣿⣿⡄⢀⣿⣿⣰⣿
⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠻⢻⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿
⠀⠹⠿⣿⣿⣿⣿⡿⢓⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⢿⣿⣿⠋⢈⣺⣿⡟⢿⡿⠃⡛⢾⣿⡿⢿⣿⠿⢻⠻⣿⡿⡟⠿⠏
⠀⠀⠚⠛⣻⣿⠛⠛⠒⠀⠈⣾⠟⣿⠃⢘⡙⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⠑⢀⡉⣙⣿⡇⠀⠉⠀⣀⣀⢙⣿⣦
⠀⠀⠀⠀⠙⠛⠃⠐⠋⠁⠀⠉⠀⠉⠁⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠠⠾⠓⠢⣄
⡄⠀⠀⠀⡸⠀⠠⣶⣶⣿⡆
⠀⠑⠄⠈⠙⠻⣿⣮⣯⣍⠁
⣴⣶⣾⣿⣴⡆⠙⢿⣿⣿⣷⣄
⠙⠛⠿⠟⠀⠈⠂⠌⠻⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠤⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠈⢓⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⢿⣝⢷⣿⣷⣄⠀⠀⠀⠀⢰⡏⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⢷⣽⣿⠟⢗⠢⡀⠀⠀⠳⣟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡷⠫⡀⠐⢄⠑⠌⣠⡤⢤⡈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠑⢄⣩⣶⣷⡏⠳⣄⠈⠑⠦⠤⠤⠖⠘⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠙⢢⡀⢀⣀⠀⠀⠀⠀⠀⢀⢤⡀⠀⢈⣶⣿⣿⣿⠿⠏⢂⠈⠳⢤⣀⠰⣶⠀⠀⠀⠉⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⣀⠉⠉⠉⠙⠲⣄⠀⠀⢿⡀⠁⠀⠸⣿⠟⢉⠈⢀⠐⢄⠱⡀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡴⠒⠶⣤⡈⠳⡀⠘⢷⣄⠀⠀⠁⠀⠀⠀⡀⠑⡀⢂⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⣆⢹⡀⠀⠙⣧⡀⠀⠀⠀⠀⠈⢢⣼⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠈⣷⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠐⠿⠤⠋⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣄⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣦⡀⠀⠀⠀⢀⡼⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⢀⡴⢋⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣫⣵⢟⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡌⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠾⠛⢉⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠉⠛⠷⠶⢶⣶⣶⣶⠶⠾⠟⠛⠉⠁⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣤⣀⣀⠀⠀⢀⣀⣀⣤⠴⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣶⠔⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠸⣟⠹⢿⡠⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣧⡄⢹⣧⠀⠛⣦⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢟⠋⡄⠀⢀⣡⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡞⢫⣽⣸⠃⠵⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣖⢤⢸⡇⢿⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣴⢺⡍⣦⢹⢸⢿⠇⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢲⡝⣧⢹⣮⡇⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⣤⡖⠛⠙⠓⠢⢄⡀⠀⠀⣀⠴⠚⣧⢹⡄⣷⣿⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢡⣾⡟⠀⠀⠀⠀⠀⠈⠓⠖⠋⠁⠈⡇⢸⣸⡷⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⣾⣿⠁⠀⠀⢠⡄⡆⢿⡀⠀⠀⠀⢀⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⣿⡇⠀⠀⢸⡿⣏⢿⡈⠃⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⢹⣇⠀⠀⠘⠷⠉⠁⠀⠀⠀⢀⡴⠀⠀⠀⠀⢀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⢠⡾⢃⠠⢔⡠⠤⠐⠀⠀⠠⡙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠈⠀⢀⡀⠀⠀⢀⣴⠟⣿⠧⠀⠀⠒⠀⣶⠀⠀⢀⣹⣤⣙⣄⣀⣀
⠀⠀⠀⠀⠀⢀⡠⠴⠖⠛⠉⠛⣓⣶⣶⣤⣽⣟⣛⣋⡁⠀⠿⠀⠀⠀⠀⠀⢸⡄⡴⢿⣤⣀⣉⣓⠒⠈⠉⠁⠀⡄
⠀⢠⣄⠀⠀⠀⠀⠀⠑⠒⣒⣤⣭⣍⣉⣙⠉⠋⠷⠻⠿⣫⣿⣶⣶⣶⠤⠄⠀⠛⠀⢀⠏⠉⢹⠏⠉⠹⡇⠀⠀⢻⡀
⠀⠀⢻⣟⠳⠶⠤⣤⣀⣀⠀⠈⠁⠀⠉⠉⠉⠛⠛⠓⠒⠀⠠⠤⣭⣄⣀⣀⣀⠂⠀⣀⠀⠀⠀⠄⠀⠀⣇⠀⠀⠘⡇
⠀⠀⠀⢻⣶⣶⡦⠤⣤⣍⣉⣛⡒⠒⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠛⠋⠉⣼⡗⠲⢶⠤⠤⠤⣿⠀⠀⢀⡥
⠀⠀⠀⠈⣧⠈⠁⠀⠀⢰⡆⣴⡀⣬⠁⠀⠠⣤⠤⠤⠀⢶⡛⠋⣉⡤⠤⠀⠀⠀⠀⣿⡇⠀⡾⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢶⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢰⣄⣀⣤⠤⠀⠀⢹⣇⢸⠃⠀⠀⠀⠀⢀⡞
⠀⠀⠀⢀⡈⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠉⠁⠀⠀⠀⠀⠘⣿⣟⣒⢒⡒⠐⠀⣸⠄⠤⣄⣀⣀⣀⣀
⠀⠒⠋⠉⠓⠉⠛⠛⠿⠿⢶⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢀⣠⣿⠀⠀⠈⠉⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠲⠦⠤⠀⠀⣷⠀⣀⣠⡤⠶⠿⠿⠛⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣽⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣥⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣀⡴⠚⠉⠉⠉⠉⠁⠒⠒⠂⠤⠤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⣴⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⠀⠀⠀⡎⠐⠒⠊⠁⠀⠀⠀⠈⠑⢦⡀
⠀⣼⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠠⣟⡃⠀⠈⠉⠀⠒⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⢰
⣸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⣠⣄
⢿⣿⣿⣿⣿⣿⠃⣀⣤⣴⣶⣶⡀⠀⠀⠀⠀⢲⡄⠈⠉⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⡿⠟⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⣿⠉⣿⠈⠙⠋
⠈⠉⠁⠀⠀⠀⠀⠀⠉⠉⠛⠛⠃⠀⠀⠀⠀⠙⠁⠀⢲⡄⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠸⠿⠇
⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠛⠛⠛⠛⢉⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠀⡠⠒⠉⠉⠀⠀⠀⢳⢀⡠⠀⣀⣀⠀⠀⣀⡠⠔⠁
⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⡅⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣀⡐⠀⠔⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⢊⡀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠀⠀⠉⢛⠂⠀⠘⠀⢠⡀⠙⢦
⠀⠀⠀⠀⡀⠋⠀⠀⠀⠀⠀⣠⠜⠁⠠⠀⢀⡀⠊⠀⠀⠀⠀⠀⠀⠀⢠⠂⣀⡀⠀⠀⠀⠀⣠⠴⠂⢁⣴⠄⠀⠀⠉⠵⠂⡀⠘⠏⠀⠨⠄
⠀⠀⠀⠲⢧⡄⠡⢣⢀⣤⠚⠁⠀⠀⠄⠀⠁⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠰⠛⣷⠈⠁⠀⠆⠀⠠⠔⠋⠁⠀⠀⣠⠆
⠀⠀⢈⡶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡔⠒⠋⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⡄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠄⠚⠁⠀⡀⠀⠀⠠⠄⠀⠀⠆⠀⠈⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠴⠒⠂
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⢒⠀⠜⠁
⠀⠀⠀⠀⠀⠀⣾⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣅⣠⠌⠈
⠀⠀⢻⡄⠀⠀⢙⡤⠀⠀⠀⢰⡀⠀⠀⣄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠢⡀⠀⢉⡀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⢻⡆⠀⡟⢱⡶⠶⢦⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⢀⡴⣏⢣
⠀⠀⠀⠀⢻⡆⠳⢈⡀⢡⣠⠟⠀⣤⠀⢠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⡌⠹⣤⣄⠂
⠀⠀⠀⠀⠀⠻⡆⠀⠀⣴⢞⡄⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠹⠂⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠖⠒
⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⡷⠀⠀⠀⠀⠘⠀⠀⢀
⠀⠀⠀⠀⠀⢀⢃⠀⠠⠀⡀⣤⠀⠉⡀⠀⠀⠀⠀⡀⢀⡅⠀⠀⠀⠀⠆⠀⠖⠒⠊⠁⠀⠀⠀⠤⣀⣀⣀⡀⢀⣤⡤⠗⠀⠈
⠀⠀⠀⠀⠀⢨⠄⠀⣈⠀⠂⠀⠂⠁⡄⠐⠀⠀⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⢠⣫⡶⣦⠐⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⣠⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠶⠆⠀⠃⠀⠠⠌⡀⠀⠁⠈⠱⠆
⠀⠀⠀⠀⡏⠞⡇⡽⠈⠀⠀⠀⠀⠀⠀⠁⠞⠀⡀⢠⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠁⣴⠀⠀⣀⠀⠀⠀⠄
⠀⠀⠀⠀⠣⠤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠁⠸⠇⠀⣠⣤
⠀⠀⠀⠀⠀⡾⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠱⠖⢦⡀⠀⠉⠁
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢷⣅⣀⣯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣸⢷⣾⣿⣿⣷⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢳⣄⣸⣿⣿⣧⡼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⢹⣿⣿⡟⠘⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢳⣿⠇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣿⡟⠀⠀⠀⢗⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⣼⣿⣮⣿⡁⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⠫⣂⠀⠀⢠⣿⡿⣏⡏⠘⢷⣄⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢣⠊⠀⠀⠀⠀⣾⣿⡽⡼⠀⠀⠀⠙⢧⠸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⡸⣿⢧⢳⡃⠀⠀⠀⠀⠘⡇⡏⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⢰⣻⣏⣧⡿⠀⠀⠀⠀⠀⢠⡇⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⢀⣧⡿⡜⣼⠃⠀⠀⠀⠀⠀⡼⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⣮⣾⢳⡝⡟⠀⠀⠠⣤⢀⡼⠁⠀⢃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⣤⡀⣸⢿⢻⠛⠁⠀⢈⠕⠚⠚⠁⠀⠀⢸⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⡴⠋⢀⡾⠉⢀⡟⣎⠇⢴⣦⢀⡏⠀⠀⠀⠀⠀⠀⢸⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠈⠉⢀⢀⡾⠁⠀⣼⡙⡜⠀⢠⡟⠸⡅⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢋⢄⠀⠀⠛⠛⠁⠀⢠⢷⢱⠀⠀⢸⠇⠀⢣⡀⠀⠀⠀⠀⠀⠈⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡱⠁⠈⠀⠀⠀⠀⠀⣶⣯⣇⡇⠀⠀⠸⣶⠆⠀⠉⡏⠀⠀⠀⠀⠀⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⠀⠀⠀⠀⢐⣿⣏⣍⡝⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡀⢨⠃⢈⡟⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢳⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢠⠃⣸⠀⡾⠀⠀⠀⠀⠀⠀⠀⣤⡎⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⢀⠇⣰⠁⠀⠀⠀⠀⢀⢤⣼⡞⠀⠀⠀⠀⠀⠀⠀⢸⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠾⣠⣏⣀⠀⠀⣀⣀⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢀⠌
⠈⠑⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⢶⣾⡆
⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠄⠂⠀⢀⢀⠀⠀⠀⡀⠄⠐⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢿⣩⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠺⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠃⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠚⠁⠀⠀⠀⢀⠀⢒⡄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⡀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠉⠀⠀⠀⢀⣠⠤⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠋⠀⠀⠀⢇⡄⠀⠀⠀⠀⣀⡤⠖⠚⠉
⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣼⠟⠉⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠉⠁
⠀⠀⠀⠀⠈⠳⣴⢖⣴⡿⠁⠀⠀⠀⣀⡤⠴⠚⠉⠁⠈⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠻⠟⢀⡤⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠂
⠀⠀⠀⠀⣀⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⡀⡀⠀⣀⠀⢀
⠠⠖⠺⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠤⠀⠀⠀⠈⠠

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠
⠀⠀⡀⢰⡆⢀⡄⠀⣼⣿⣿⣿⡇⠀⠀⠀⠙⢦⣤⡴⠶⠶⠖⠚⠋⠁⢀⣿⣿⣶⣶⣤⣀⣀⡀⠀⢰⣿⡟
⠀⠀⢁⣾⠃⠸⠃⠀⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢅⣀⣿⣿⢻⠻⢻⡆⠸⣿⣧⡀⠀⠀⡶
⠀⠀⠉⠁⠀⡇⠀⢠⣿⣿⣿⣿⣿⣿⠀⠘⠓⡖⠀⠀⠀⠀⠀⠘⢳⠄⠀⠀⠀⢠⣌⡉⠙⠛⠋⠀⠀⠈⠉⠁⠀⠰
⠀⠀⠀⢀⣴⣷⣄⠀⠁⠉⠉⠉⠙⠋⠀⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⢀⠄⠀⠀⣼⠿⣿⣦⠀⠀⠀⠀⠀⠠⠀⠠
⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⠀⠘⣦⡀⠀⠀⠀⡖⢺⢠⡀⣀⣤⡴⡴⡾⠀⡀⠀⡿⠀⠘⣿⠀⠀⣀
⠀⠀⠀⣸⣿⣿⣿⡁⠀⠀⢀⣴⠿⠻⠆⠀⠡⠁⣉⣉⣉⣁⡰⣀⣅⠦⠈⠁⢀⡇⠀⠀⠹⠷⣿⡟
⠀⠀⣼⣿⠻⣿⣿⣿⡀⢀⡞⣻⣼⢷⡀⠀⠀⠀⠟⠟⠛⠛⢒⠋⠉⠀⠀⠀⢸⠃⠀⠀⣠⣾⠟
⠀⠀⠻⣿⣾⣿⣿⡿⠀⠈⢧⣿⠿⣴⠃⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⣼⠀⢀⣴⣿⠏⠀⠀⠀⠀⠀⡰⠆
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡆⠀⠀⠀⣿⣴⣿⣿⠃⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⣦⣦⣤⠤⠤⠤⠠⠦⠀⠲⠒⠶⠶⠆⠀⣿⣿⣏⣿⣿⡿⠃⠀⠀⣼⣿⣿⡟⠁⠐⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠛⠋⠀⠀⠀⠀⠉⠛⠉⠁⠀⠀⠀⠈⠁
⣶⣶⢶⣶⢶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡧⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣸⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠠⢼⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⣶⣾⣷⣶⣄⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠃⠀⠀⠀⠀⢠⣶⣿⣷⣆⠀⠀⠀⠀⠀⣀⣧⣀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢠⣿⣿⣿⣿⣄⠀⠀⠀⠀⠸⣿⣿⣿⡟⠀⠀⠀⠀⢸⣿⣿⣿⠆
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢀⣼⣿⣿⡄⠀⠀⠀⠀⠀⣿⣿⣏
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⣿⣿⡄⠀⠀⠀⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠈⠛⠿⠿⠿⡿⠿⠿⠛⠁⠀⠀⠀⠈⠻⠿⢿⡿⠿⠟⠁⠀⠀⠀⠙⠿⠿⠿⠟⠀⠀⠀⠀⠙⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠟⠛⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⣠⠴⢠⡤⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⣴⠟⡁⠀⠸⣤⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⢉⡝⠉⠛⣋⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⡠⠋⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡡⢂⢞⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢫⠞⡰⢣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⡴⢃⠎⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⣠⠞⢡⠎⡴⢁⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠁⠀⢀⡼⠋⡰⢁⠊⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⣲⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⣰⠟⠀⠀⡐⠁⠀⠈⠻⣦⡀⠀⠀⠀⠀⢀⣠⡤⠖⠋⣁⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁⠀⠀⢠⡾⠃⠀⢀⠊⠀⠀⠀⠀⠀⠘⣷⡄⣤⡶⠟⠋⢁⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢀⣴⠟⠁⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⡏⣷⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡾⠁⠀⣠⡾⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠒⠛⠉⠀⢀⣼⡟⢁⠊⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⡴⠲⡄⠀⣨⠏⡝⠛⠶⠦⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⠴⠆⠀⡇⠈⠁⡰⢁⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡟⠋⠉⣀⣤⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠘⠦⠤⠞⠁⠀⠈⡠⠃⠀⠀⠀⠰⠏⢱⡄⠀⠀⢸⡏⠀⠀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠘⣷⠀⢸⣏
⠀⠀⠀⠀⠀⣀⣼⡇⠀⠀⠀⠀⢰⡷⠶⣦⣄⡀⠀⠀⠀⠀⠀⣏⠀⠴⡄⠀⠀⣠⣿⠷⠀⢻⡀
⠀⠀⣀⣴⠞⠋⢹⣇⠀⠀⠀⠀⡟⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠘⠲⠶⠃⠀⣼⡟⢁⣤⡾⠋
⠴⠞⠻⠔⠂⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣍⠂⠀⠀⠀⠀⠀⠀⠀⠤⠴⠚⠉⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣤⣄⣀⣀⣀⣀⣠⣤⠴⠶⠚⠉
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢹⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣴⢠⢃⣴⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⢉⣿⣿⣯⣽⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⣿⣿⢯⣀⡙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡟⠁⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠈⠀⠀⠒⠠⠄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠉⠀⣰⣿⣿⣿⠟⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⣉⣹⣷⠤⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⢀⣾⣿⣿⣿⠋⠰⢄⠀⠀⠂⠀⠀⢀⣀⡠⠤⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣰⣿⣿⣿⡿⠁⠀⠀⠀⢣⡀⠠⠖⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⢀⣾⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⣬⡁⠀⠀⠀⣰⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣒⠬⠐⠀⠙⢇⣹⣀⠀⠾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢔⡬⠕⠊⠁⣀⣤⠔⠚⠈⢀⠏⣸⠗⠶⣤⡙⠛⠁⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⢊⡡⠖⠋⢁⣠⡴⠖⠋⢉⠉⠒⠖⠒⢡⡊⠸⠏⢻⡾⠃⢰⠚⠃⢀⠤⡤⢄⠀⣿⠁
⠀⣀⣀⡤⠖⠚⠉⠀⠀⣰⣋⣤⠴⠛⠉⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⡀⣸⠀⢰⡡⠤⠦⢄⣿⡇
⠻⠛⠋⠓⠄⠈⣉⣠⠴⠛⠋⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢠⣴⣤⣄⡀⠀⠀⠀⠙⠁⠀⡟⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠂⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⣠⡴⠋⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠷⣾⣏⡀⠀⠀⠀⠀⠀⣀⣤⣾⠏⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢼⡙⠛⠿⠿⠿⠿⠟⠛⠋⣀⡤⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣠⡀⡈⠢⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣡⣤⠀⠁⢄⠉⢠⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⡾⠁⠀⠀⠀⢈⣠⠺⢷⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢳⡶⠆⠙⢋⣳⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣋⣀⣀⠀⠉⣷⠿⢓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠿⢿⣭⠙⠇⠻⡄⠀⢺⢽⣻⣄⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡆⠀⠀⠀⠀⣄⠐⠩⢗⡻⣦⢿⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣉⣤⣶⣆⠀⠀⠀⡙⣆⠀⠹⣾⠽⣦⣰⣯⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⢡⣶⠘⣿⡎⠀⠀⢒⡫⢞⣯⡙⢠⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣵⣿⣿⣤⡟⠁⠀⠀⠀⢐⡽⣞⣽⡻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠨⠵⣪⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢟⣭⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⠤⠶⠶⢦⣤⣄⠀⠀⠀⠀⠀⠀⠘⣫⢷⣾⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡇⠀⠀⠀⠀⠀⠀⠘⠭⢺⣭⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⢀⣤⣶⡒⣐⣦⠀⠀⠀⠀⠀⣀⣴⠿⣿⣆⠀⠀⠀⠀⠀⠀⠈⢵⣺⡿⣳⡄
⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠘⠻⠿⠿⠛⠋⠀⠀⣀⣠⣴⡞⠛⣿⡄⠈⢻⣆⠀⠀⠀⠀⠀⠀⠈⢶⡿⣟⡿⡄
⠀⠀⠀⠀⠀⠀⠘⣿⣷⣤⣀⣀⣀⣀⣀⣀⣤⣤⠶⣾⣿⠉⠉⢿⣧⠀⠘⣿⡄⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⢪⢽⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣿⠁⢹⣿⠉⢹⣿⠉⠀⣿⣇⠀⢻⣿⡇⠀⠘⣿⡆⠀⠘⣿⡄⠀⠀⠀⢀⣀⡀⣀⡀⠀⠀⠱⣚⣯⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⣿⡀⢸⣿⡇⠀⢸⣿⠀⠈⣿⣿⠀⠀⢹⣿⣀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠈⣹⡇⠀⠀⠁⠀⣀⣠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⣿⡇⠀⠸⣿⡄⠀⠸⣿⡇⠀⣴⠏⠁⠠⣶⣿⠶⠇⠀⠀⣀⡤⢾⡋⠻⣶⠀⠀⠙⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⢿⣇⠀⠀⢿⣷⠈⣿⢦⣤⣄⣠⠤⣤⠲⣶⡏⠹⣷⠈⢿⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⢸⣧⡀⠀⠸⣿⡄⠀⠘⣿⡆⠃⠀⣇⢸⣷⠀⣿⠀⢹⣧⠀⢹⣇⣼⣿⠁⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⢷⣄⠀⠘⣿⡄⠀⢻⡇⠀⠀⣿⠀⣿⠀⢹⡇⠀⣿⡄⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣦⡀⠙⢷⣄⠘⢿⡄⠸⡇⠀⠀⣿⠀⣿⡀⠈⣷⠀⠸⣧⣾⢳⣻⡇⠀⠀⠀⠀⠀⠀⠀⣴⣶⠿⣟⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⣤⠿⠦⣤⣈⠳⣤⡙⣆⢃⣠⣀⠈⠳⡌⠳⣄⠘⣧⠀⢿⣿⡜⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⡏⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢼⡇⠀⢰⠇⠀⢨⠁⠉⣿⠀⠀⠙⡇⡤⢛⡶⣌⠓⣌⢳⣸⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢰⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠈⢄⠀⢘⡀⠀⣈⢀⡀⠐⠃⠀⢾⡀⣼⠁⠈⠉⠃⠙⢿⣿⣿⣟⡿⢷⣤⣄⣀⣀⣀⣀⣀⣼⡷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣾⣻⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⡿⢿⡿⠻⡿⣿⢿⠿⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣷⣾⣶⣾⣷⣾⣿⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⣿⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣴⡤⠼⠿⠿⣿⠟
⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣧⡴⠾⣷⡶⠖⠏
⠀⠉⠛⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠟⠁⠉⠁
⠀⠀⠀⠀⠀⠉⠛⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⢟⣹⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣴⣾⠟⢋⣵⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⣠⡴⠞⠉⢁⣴⣿⡟⠫⣲⣼⡿⠋⠁⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣄⣠⡞⠁⢀⣤⣾⣿⣻⣵⣿⡾⠟⠉⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠛⠶⣄⣙⠍⠛⠻⠟⠋⠉⠀⠀⠀⠀⠀⢀⣰⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⠤⠄⠀⠀⠀⠉⠓⠶⣤⣀⠀⠀⠀⠀⠀⣠⣴⡿⠟⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣶⣦⡀⠇⠀⠀⠠⠤⡀⠉⠂⠬⣑⡲⢿⣏⠋⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡴⣾⡿⠋⠉⠉⠉⠛⠚⠀⠀⢀⣴⢇⣤⣾⡿⠂⠩⣒⡤⣉⡛⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁⠀⢈⣽⣷⣦⣤⡀⠀⠀⠀⠀⢀⣾⠟⢫⣿⣿⣧⣤⠶⠛⠉⠉⠁⠈⠑⠢⢬⣀⠀⠄
⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁⠀⠀⠀⢠⣿⢛⣿⣿⡿⠁⠀⠀⠀⠀⠈⢧⣴⢞⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢤⣐⠲⠤⣀
⠀⠀⠀⢀⣴⣿⣟⠀⠀⠀⠀⠀⣴⡿⣵⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣫⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⣉⠒⠦⣄⡀
⠀⠀⣠⣾⣿⣿⣧⣌⣵⣾⣷⣿⣟⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⠁⣼⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣧⣽⡃
⠀⠀⣿⡏⠹⢿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⢄⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁
⠀⠀⢿⣿⣀⢨⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠐⢂⣠⣴⣾⣿⣿⡿⢋⣠⡾⠃
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣿⠟⢉⣠⣿⠟
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢋⣡⣤⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣍⢹⠛⢽⡏⡏⡏⣫⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣿⠿⢻⣻⣿⣿⢿⣿⡻⣿⣿⢻⡟⠟⡿⢻⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢰⣾⣷⡆⠀⠀⠛⠒⠒⠒⠚⠒⠚⠛⠛⠛⠛⠛⠛⠚⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⢠⣿⣿⡆⠀⠀⠀⠀⠀⢀⣴⣾⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢀⣯⣿⡀⠀⠀⠀⠘⠛⢺⡿⠀⠀⠀⠀⠀⣯⣾⣿⣿⣷⣹⠄⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠨⠽⣗⣺⡏⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢿⣿⢿⡟⠀⠀⠀⠀⠀⠀⠐⢽⣇⡐⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢰⡶⡖⠀⠀⠀⠀⣠⠀⢺⠄⠀⠀⠀⠀⢀⣸⢿⢿⣿⡿⣇⡀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⣧⡇⠀⠀⠀⠀⠉⠀⢸⠀⠀⠀⠀⠀⣼⣷⣾⣾⣿⣷⣾⣧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⢰⣿⣿⣿⣧⢸⣿⣿⣿⡆⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣸⣾⣷⡆⠀⠀⠀⠀⢰⣿⣿⡀⠀⠀⠈⠿⣿⣿⣿⢸⣿⣿⠿⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢿⣿⣿⡟⠀⠀⠀⠀⣿⠇⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠇⠀⠀⠀⠀⠀⡇
⠀⠀⠆⢸⣿⣿⡇⠀⠀⠀⠀⠻⣖⣼⠇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠼⠿⠿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢻⣇⠀⠀⠀⢠⡄⠀⡇
⠀⠀⢠⣶⠤⠴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡾⠿⠀⠀⠀⠀⠡⡎⡋⡶
⠀⠀⠈⠉⠂⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁⠀⠀⠀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣍⢈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⡠⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⢸⣃⡃⢀⣇⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠁⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢀⣘⣄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣦⣤⡀⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣁⢹⣁⣏⣿⣶⣶⣤⣶⣶⣗⣠⣄⡿⣿⣿⣿⢿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣤⣾⣿⣿⣿⡟⣉⣿⣿⣿⣿⣿⣿⣷⣶⣬⣝⠻⠷⡈⠁⠙⠚⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣿⣿⣿⡟⣽⣿⡿⢿⣿⣿⣿⢿⣿⣿⠦⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣗⣿⣿⣿⣿⣿⣿⣷⣼⣎⣥⣿⣿⣿⡧⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⠛⠉⠉⢿⣛⣯⣷⣾⣿⣿⣿⣿⣿⣿⣿⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣯⠄⢀⡀⢸⣴⣿⡋⠉⠉⠙⣏⠏⢥⣿⣟⣿⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣮⣴⣶⣴⣾⣿⣿⠿⠀⠀⠀⠀⢸⡄⣾⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢿⠋⠛⠻⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⣻⣿⣿⣷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠋⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠉⠻⠙⣧⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠹
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣽⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠇⣰⡗⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡆⠀⢀⠀⢀⠀⠀⠀⣀⠀⠉⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣴⣶⣄⠀⢀⡀⠀⠀⢢⠀⠀⢣⠀⠀⣓⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⠀⠘⠛⠁⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣋⣼⡿⠛⠂⠀⠀⢧⠀⠘⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠀⣴⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠻⡟⠋⠑⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡂⠀⡄⠀⡌⠀⠀⠈⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣤⡀⠘⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⡚⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠹⠆⠀⠀⠐⠆⠀⠀⠀⠀⢀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠁⠀⢤⠀⠐⡆⠘⠂⠘⠀⠋⠀⡴⢀⡄⠀⠀⢀⡄⠀⠀⠀⢈⣁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠠⡄⠀⠠⡄⠀⠠⡄⠀⣀⠀⠀⠀⠠⡄⢀⡄⠀⠀⠈⠀⠀⠀⠉⢀⣀⠀⠀⠈⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠤⠄⠀⠶⠀⠠⡄⠀⣀⠈⠁⣄⠀⠀⠁⠈⠁⠀⠀⠀⠠⡴⠂⠀⠘⠁⠀⠀⠀⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢤⠄⢀⣀⠀⣀⡀⢀⠈⠁⠐⠆⠀⠀⠀⠀⠀⠠⡴⠆⠀⣄⡄⠀⠀⠛⠁⠀⠀⠾⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠤⠀⠠⠄⠀⠤⠀⠨⠅⠀⠭⠁⠀⠀⠀⠀⠀⠀⠶⠦⠀⠁⠀⠀⠀⠶⠖⠀⠀⢶⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠉⠉⠉⠰⠀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⣀⢠⡀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠔⠁⠀⠀⠘⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠸⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠗⠦⠤⠀⢀⠤⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢠⢀⠆⠐⠃⠀⠐⠠⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⠀⠀⠄⠀⢠⣇⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣠⠔⡆⠀⠀⠀⠀⢰⠀⠬⠃⢀⡰⢃⠎⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡄⠀⠀⠀⢸⠀⡠⠀⠁⠂⠀⡐⢁⠔⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠁⡇⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡠⠒⠁⢀⡀
⠀⠀⠀⠀⠀⠀⡀⠐⣀⣀⣀⣀⣀⠀⡇⠀⠀⠀⠀⠞⡇⠀⠀⠒⠒⢦⡀⠀⠀⡀⠐⠉⠀⠀⠰⠀⠀⠀⠆⠀⠀⠀⠀⠄⢀
⠀⡀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⣀⣉⣉⠙⠒⠒⠬⣉⡂⠒⠀⠁⠢⢐⡄⠈⠀⠀⢀⠠⠊⠐⠁⠀⠀⠓⠀⠀⠀⠐⢦⠠⢀⠀⡀⠠
⢘⣀⠀⠀⢀⣀⣀⣤⠤⠖⠒⠙⣉⠍⠀⠈⠑⠀⠀⠀⠀⠉⡀⠄⠀⠐⢄⢀⡀⣈⣉⡡⠔⠁⢠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠠
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⠠⠋⠀⠀⠀⠀⢀⠄⢀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠔⠁⠀⠀⠈⠀⠀⠀⠈⢅⠀⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⠤⢤⠤⠖⠚⠁⢀⠆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⠄⠊⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢢⠄⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⣭⠉⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⢼⢹⡍⡳⣄⠀⣈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣁⣠⡤⢒⡡⠻⡌⠓⢴⡘⣮⡌⠳⢬⣑⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⢋⣴⠞⠋⠀⠀⢳⡀⠀⢳⡈⢷⣄⣀⡤⠒⠈⢓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣰⣿⣿⡐⠄⠀⠠⢒⣳⣄⣀⠙⠾⢶⣢⣀⣀⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣿⣿⡟⠛⠁⠀⠀⠁⠛⠛⢿⣿⣿⣿⣯⡫⢻⣿⡿⣦⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡿⢫⣧⠀⠐⠀⠀⠀⠀⠀⠈⡿⢷⡿⢹⢟⣤⣥⣯⣻⡟⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣆⢰⡶⠶⠄⠀⠀⡠⣣⠞⢡⠎⣾⣿⣿⢞⡙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡼⢉⡟⢿⣿⣆⠈⠀⢀⣠⣾⠟⣡⠖⡡⠂⢻⠿⣿⣿⢡⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠸⡇⢯⣻⣿⣿⡿⣿⢟⣡⣞⠕⣩⣶⣩⡵⣛⣭⣶⣿⣣⣖⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡟⢄⠹⡀⠉⢻⡿⢡⢇⣾⡕⣡⠖⢡⣼⢟⣼⣿⣿⣿⣯⠮⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣳⢡⣿⡆⢹⡀⢋⣴⣿⣾⡿⣴⣣⡖⣺⣿⣾⣿⡟⠁⢳⠰⡝⢮⡳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢸⠘⡟⣀⡷⢛⠏⣾⣿⣧⣿⣡⡞⢰⣿⣿⡟⠀⠀⠀⢣⠙⢦⣙⢦⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⣨⠶⠛⠉⠀⡰⣰⣿⡏⣿⣿⡿⢁⣾⣿⠉⠀⠀⠀⠀⠈⣧⡈⠫⡓⠽⣎⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠵⣒⣊⣥⣾⣼⡟⣿⢳⣻⡿⠃⣼⠇⡟⠀⠀⠀⠀⠀⠀⢹⣿⢦⣈⠲⢬⡸
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣥⣶⣿⣻⣟⣛⡿⢏⣼⡏⢎⡟⠁⣰⣿⢠⡇⠀⠀⠀⠀⠀⡄⠀⣿⡷⣌⢻⠍⠁
⠀⠀⠀⠀⠀⠀⠀⣴⠻⢡⣿⡟⢥⡰⣿⢻⣿⣋⣤⡿⠁⢠⣿⡇⣼⠀⠀⠀⠀⢀⣼⣠⡶⠋⢛⡾⡫⠊
⠀⠀⠀⠀⠀⠀⠀⠙⢬⣂⢿⣧⣞⠧⣀⡿⠿⣿⡾⠀⣠⡾⣻⡏⢭⣒⢒⣒⣉⣀⣀⡩⠤⣠⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣮⠍⡛⠒⠛⠫⠥⢶⡿⣋⣴⢯⡟⠋⠉⠉⠁⢠⠿⡋⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣟⠓⣲⣾⠷⠛⢇⢜⣮⣁⠤⠔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉
⣶⣶⣶⣶⡶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡶⠶⢶⣶⣶⠶⢶⡶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢿⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⣾⢸⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣿⢸⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣿⣿⡇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⡇⣿⣿⡇⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠃⣿⣿⡇⢻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠀⣿⣿⣿⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⢸⣿⣿⣿⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⢸⣿⣿⣿⢸⣿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣾⣿⣿⣿⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢫⣿⣿⣿⣿⣭⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⡇⢸⣿⣿⣿⣿⣿⠇⣿⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⢸⣿⠿⠿⠿⣿⠀⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣻⣣⣿⣿⠒⠒⢲⣿⣧⡛⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⡇⠀⣿⠿⠿⠟⠋

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣽⣭⣯⣽⣧⣾⣭⣯⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣶⣷⣶⣾⣿⣷⣶⣶⣶⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣯⠸⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⠲⠾⠟⠉⠉⠉⢻⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠿⠂⢒⡛⢡⡀⠀⢀⡆⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠐⢐⣩⠊⠐⠟⠁⠘⠁⣄⣀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡇⠀⠀⡀⢀⣢⣵⠾⢛⣴⠄⠀⠐⠢⣈⠘⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡗⠀⠀⢀⢠⣂⣰⣾⣿⣋⣁⣀⣈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠾⡗⠀⠀⢀⢠⣰⣼⣾⣿⢿⣯⣽⢿⢿⣿⡯⠌⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠛⠟⠀⠀⠠⡐⣄⣾⣿⡿⠟⠋⣰⠟⣊⠻⠿⠿⣋⠰⠂⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣔⠀⠈⠋⠀⡠⠔⢊⣣⣿⣿⡿⠛⠉⠀⠀⠀⣿⣄⠉⠀⠛⠂⠉⢀⣤⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠂⠀⠀⢀⠤⢒⣡⣴⣾⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠘⣿⣷⣶⣶⣶⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⡤⡤⠤⠤⠀⠀⠀⡠⠐⣈⣤⣶⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠛⠋
⠀⠀⠀⢀⡤⠒⠉⠁⢀⣀⠀⣠⣶⣾⡶⠀⠁⣤⣿⣿⠿⠋⠁
⠀⠀⡴⠋⠀⠀⢀⠀⠩⠛⠀⠛⠛⠋⢁⣄⣿⣿⣿⠋
⠀⡼⠁⠀⠀⠀⠀⢳⡀⠀⣀⡀⠀⢠⣦⣿⣿⣿⡏
⢘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⢠⣦⣾⣿⣿⣿⡇
⣼⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣿⣿⣿⣿⠃
⠈⢾⡓⣤⡀⢀⠀⢀⡀⢰⣶⣾⣿⣿⣿⣿⣿⡿
⠀⠀⠻⣿⣶⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣍⣮⣭⣤⣬⣩⣏⣴⣿⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢱⣾⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⣡⣿⠋⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣿⡇⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠴⣾⢱⣿⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⣸⠇⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⡾⠰⠂⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡄⠀⢸⣃⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠙⠛⠛⠃⣰⡦⣄⣹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠆⠀⣤⠀⡀⣼⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣴⢹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⡊⠀⣿⠀⠀⠀⡼⠁⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠘⠋⠘⠁⠀⠀⠼⠁⢶⠀⠛⢳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⢀⣈⡉⠃⢂⡀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⡄⡄⠀⠀⠀⠈⠉⠙⣻⣿⣿⣿⣷⣾⣧⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢣⢣⣀⠀⠀⠀⠀⣴⠿⠿⢿⣿⣿⣿⡇⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣦⠀⠀⠀⠀⢉⡻⡀⠀⠀⠾⠁⠀⠀⠀⢀⡿⢋⡾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣿⡀⠀⠀⠀⠀⠙⠲⣦⣤⣤⣀⣀⣀⣠⣥⣶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇

⣀⣀⣀⣀⡀⣀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⢿⡿⡿⠿⢟⠻⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣛⣛⣿⣿⣛⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠔⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠺⠛⠳⡀⠀⠆⠠⠔⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢹⠃⣠⠎⠀⠀⣹⣆⡛⠁⣠⠘⡏⠙⠆
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢄⡀⠀⠀⢰⡿⠲⣄⠀⢴⡴⠋⠀⠀⠀⠉⠨⡁⠈⠀⡸
⠀⠀⢰⡞⣩⠟⡆⠀⣀⡀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠙⠶⣄⠿⣅⠀⠈⢳⣄⣀⣄⠀⠀⠀⠀⠃⠀⠀⡐⠀⡠
⠀⠀⠈⠳⣤⣾⡏⣶⢀⠿⠀⠀⠀⠀⠀⠀⡇⠀⢀⡴⠂⠀⠘⢷⡈⠳⣄⠘⠿⠁⡀⠀⢀⡷⣄⠀⢀⠀⡀⠂
⠀⠀⠀⢀⣴⢙⡁⣇⢰⡦⠶⠀⠀⠀⠀⠀⡇⢠⡟⠀⠀⠀⠀⠀⢹⣄⢸⣷⠀⠀⠀⠀⠈⠁⠈⢶⡀
⠀⠀⠦⠼⠃⡟⠛⠘⢇⠀⠀⠀⠀⠀⠀⢠⠇⠈⠀⠀⠀⠀⠀⠀⠀⠹⣎⠉⠀⠀⠀⠀⠀⠀⠀⢼⠗
⠀⠀⠀⣀⣤⠾⣤⠖⢇⠀⠀⠀⠀⠀⣠⠞⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠻⡆
⠀⠀⠀⠉⠁⠀⠱⡀⠈⢧⠀⠀⣠⠞⠁⢀⡤⠞⠋⠉⠁⠀⢀⣀⣀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠈⣷⠞⢁⠄⠊⠁⠀⠀⠀⠀⠀⠀⠈⠙⠿⠷⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣨⣧⠀⠘⣇⠁⠀⠀⠀⠀⣀⣀⣀⡀⠀⠠⣤⡀⠀⠀⠀⣧⠰⠄
⠀⠀⠀⠀⠀⣠⠞⠁⠘⣇⠀⠘⣦⠆⢠⡴⠛⠉⠈⠉⠛⠷⣶⣄⡀⢀⡀⠀⡷
⠀⠀⠀⢠⠞⣡⣄⠠⢌⣹⡆⠀⠈⣦⢟⣛⣛⡻⠷⣦⣄⡀⠀⠙⠷⣮⣤⡴⠃
⠀⠀⡰⢫⠞⠉⠀⡰⣡⡶⣻⡄⠀⠘⡄⠀⠀⠈⠑⢮⡻⣿⣄
⠀⡼⣥⡃⠈⠁⠀⢲⢯⠞⠀⢱⡀⠀⠸⡀⠀⠀⠀⠀⠙⡌⢿⣆
⢠⠁⣿⠀⠀⠀⢀⡀⠇⠀⠀⠀⢣⠀⠀⠹⡀⠀⠀⠀⠀⢹⠘⣿⡀
⡎⢠⠋⠀⠄⢐⠄⠁⡆⠀⠀⠀⠈⢇⠀⠀⢻⡄⠀⠀⠀⠀⠆⢿⣷⡀
⡇⠈⠇⠀⣀⠶⠂⢸⣣⠀⠀⠀⠀⠘⡆⠀⠀⢻⡀⠀⠀⢀⠌⣶⠏
⡇⠸⠴⣂⠠⢐⡖⣅⡷⣳⡄⠀⠀⠀⠸⡆⠀⠀⢳⡀⠀⢸⠈⣿⡀
⢹⡈⢰⡙⠁⠏⠀⠔⠥⠹⢿⣄⡀⠀⠀⢻⡄⣠⡴⠗⢠⡄⢣⠹⣷⡄
⠀⠳⡄⠹⣇⠠⡌⠸⠂⡄⢻⠶⠊⡳⡄⠀⣛⠁⠀⠀⠈⠀⠀⠳⣌⠻⣶⣤⣤⢖⣶
⠀⠀⠙⢦⡈⠣⠙⣀⠖⠀⠈⠂⠀⠸⡱⠀⠉⠀⢀⣤⣴⣶⡀⠰⡌⢹⡆⠉⠛⣎⢻⣷⡀
⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠀⠀⠀⡇⠘⠀⠐⠿⡏⠁⠉⠁⠀⠈⠒⠋⠀⠀⠈⠳⣝⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⠦⠄⠀⡰⣸⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣮⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡝⡌⠳⠃⠀⠀⠀⢠⣴⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⠀⠀⠀⠀⠀⠈⠹⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡷⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠴⢣⠕⠊⠉⠊⠷⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⠀⣀⣀⣤⣤⠄⠀⣼⣿⣷⣄⠀⡀⣴⡞⡁⠀⣽⠀⠀⠀⠀⠄⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⠿⢛⣩⣥⣶⣾⣷⠦⠉⣹⠛⠉⢹⠸⣧⣴⠀⢰⡇⠀⡀⠀⠀⠀⠀⠠⣙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢱⡌⣶⣦⠉⡁⠉⡥⠀⠀⠀⠀⠀⢐⢸⡇⢻⢿⡆⠀⣸⠃⠈⣱⠄⠀⢀⣤⣬⣼⣧⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢣⡿⡼⣿⣯⡿⡻⡀⣶⠘⠀⠀⠀⠀⠘⢸⣿⡌⡿⣿⣆⠀⢇⠀⢉⣀⠀⢛⡛⠛⠛⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⣾⠱⢱⣿⣿⣃⢧⢱⣿⢰⠃⠀⢀⠀⠀⠀⠘⣧⠙⢉⣬⣧⡘⢦⣌⣻⣿⣿⣶⣶⣤⣤⡿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣍⠸⡿⠃⡟⣼⢿⣿⣿⣸⡏⢸⡇⠀⠘⢠⠀⠀⣀⠙⢿⣿⣿⣿⣿⣦⣝⢛⢿⣿⣿⣿⣿⣿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣀⢠⣌⢋⡁⠋⠃⠘⢧⡔⣶⡠⡁⢦⡘⢿⣦⡙⠿⣿⣿⣿⣿⣷⣄⠉⠻⢿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣿⣿⡽⣄⠀⢦⣈⡻⣎⠻⢞⢿⣟⣦⡙⢿⡦⡈⠻⣿⣿⡿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⠻⣷⡹⣷⣌⢻⣿⣮⢧⡑⠁⠙⠷⣻⣆⣹⡎⠞⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠋⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢸⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠁⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣤⣄⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⢀⡇⣤⢀⣰⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⢻⡿⣿⢿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣩⢶⣮⠠⣿⢽⣿⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠊⠉⠛⠋⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢸⡿⢰⡧⠐⠿⣿⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢩⣶⣄⠀⠀⣤⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣣⡟⡻⢀⡄⠀⠶⣿⡇⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠇⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢰⣿⣿⣿⣿⡏⠁⢀⡄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⡄⠀⠀⠀⠀⠂⠰⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⣿⣿⡟⢀⠀⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣀⠀⠐⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠟⠠⠀⠀⢂⢠⠄⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⣴⡜⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣀⡀⠃⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠿⣿⣿⣿⠃⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣤⠶⠿⠶⢶⣄⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⣴⠂⠀⣠⣴⣶⣦⡀⠀⠱⣶⠀⠀⠀⠀⠀⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⡁⠀⢸⣿⣿⠀⣼⣿⣿⣿⣿⣿⡄⠀⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⠀⢿⣿⡄⠀⠸⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⡿⠀⠘⢿⣿⣄⡀⠉⠛⠛⢉⡀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣾⣷⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣦⣦⠀⠀⠀⠀⠀⠈⣹⣿⣿⠋⠀⠀⠀⠀⠙⠻⠿⠒⠛⠿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⣿⡟⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠋⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣷⡄⠈⠓⠶⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠰⣞⠃⠀⠘⢿⣿⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⡐⠀⠀⠀⠶⢄⡀⢠⣤⣤⣤⣤⡄⠘⣇⠀⠀⠀⠈⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⠉⠁⠉⠛⣿⣿⣿⣿⣿⣷⣀⡀⠀⠀⠀⠸⣿⣿⣿⣿⣷⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢿⣿⡿⠁⠀⠀⠀⠀⠈⠉⠛⠉⢫⣭⣭⣤⣤⠤⠤⠴⠿⣿⣿⣿⣉⣀⣀⣀⣀⣀⣀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⡀⠀⠣⡀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠠⠀⠀⣠⠀⡟⢡⠐⡄⠀⠀⢀⠀⠀⠀⠀⠀⡗⢨⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡾⠂
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠀⠀⡀⠈⠁⠀⡀⠀⡆⠈⠀⠟⢥⠀⢀⠀⡰⠃⠀⠀⠙⠠⢶⣀⠀⣆⠀⢱⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡾⠛⠁
⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠁⠀⠀⠀⠀⠚⡧⢼⡠⠀⠀⢀⠀⡀⠠⡄⠀⠶⣤⠀⠀⠂⠈⢓⡅⠀⠱⣬⠿⠀⢰⠙⢆⠀⠀⠀⠀⠀⠀⠀⢠⡼⠤⠄⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣀⣈⡠⠂⢀⠀⢠⠋⠉⠀⠀⠀⣇⢀⣇⡀⠀⠸⢿⡦⠀⠀⠘⠫⣞⡏⠈⢧⢰⡄⠱⣀⡕⠀⠀⠀⢀⠔⠲⢿⢠⠓⠐
⠀⠀⠀⠀⠀⠀⠠⠇⢀⠀⠀⢻⠀⠀⢰⠁⢛⠃⠀⢱⡖⠂⡖⢹⡞⠈⢳⡀⢰⠀⠤⠤⢤⡰⠏⢳⡈⠀⡼⢠⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠆⠈⡠⠀⡏⡧⠤⠃⠘⠁⣴⠀⠉⠘⠀⠀⠀⣈⡡⣠⢎⠉⠊⠀⡏⠀⠀⠀⠀⢀⡜⠣⠀⣔⡠⠊⠁⣄⢀⠰
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⠀⠀⣼⡙⠗⠀⠐⡇⢁⡄⠀⡀⢴⠁⢼⠀⠀⡀⠀⠀⠈⠃⠈⢸⡄⠀⢀⠟⡀⠐⠃⠀⠺⠿⢧⡴⢻⢡⡞⢧⡀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⣤⠀⡈⣴⠀⡠⠴⠁⠈⢐⡶⠊⠈⣀⣾⠃⠉⢿⠍⢀⠇⢀⠄⠈⢃⠛⠋⠚⡇⠀⠁⠐⠁⢀⡞⢀⡞⢸⠛⣤⣹⣿
⠀⠀⠀⠀⠀⠀⠀⢄⡤⠂⠀⠀⠈⠄⣇⢸⠀⡗⡂⠐⠀⠏⠁⠀⠀⣬⢼⣶⡂⠀⢠⡁⣴⡇⠀⠀⢸⠀⠀⢸⡣⢄⡁⠨⢷⠯⠠⠞⠳⠏⠈⢉⠀⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠸⠔⡇⠃⠀⢇⢰⢀⢠⡀⠀⠈⠈⠉⠁⠈⠁⠀⠀⠀⠀⠘⠀⠀⠘⠁⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠰⠀⠚⢤⠔⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡆⠀⢀⣤⢀⡄⡼⣿⣿⣇⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⢀⡤⢀⠀⠀⠀⣀⡀⠀⠈⢿⣿⠇⠀⠈⣿⣷⡄⢻⣿⣿⣿⡇⠀⠀⢿⡆⣳⣤⣾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⠃⢀⣠⣤⣿⣧⡄⠀⣾⣷⠀⢘⣿⣿⣿⡇⢘⣿⣿⣿⣷⣀⣠⣶⣿⢿⡟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠛⢿⣿⣿⠃⠀⣿⣿⣶⣿⣿⣿⣿⣷⣀⣿⣿⣿⣿⡿⠛⠉⠀⠀⠛⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⣤⣤⣴⣦⣸⣿⣧⣶⣿⣿⣿⣿⣿⣿⣿⡟⡀⣉⣿⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠿⠉⢍⣍⣴⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠆⢐⢠⡀
⠀⠀⠀⠀⠠⠄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠃⠐⠌⠢
⠀⡠⣞⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠆⠒⡀
⠐⠉⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠛⠀⠀⠀⠈⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠈⠑⠀⠀⠈⠳⠞⢁⠀⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠙⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢤⣤⣶⣧⠖⠀⠀⠀⠄⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⣤⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⢹⣿⣿⣿⡄⠀⠀⠐⠂⠀⠀⠳⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠁⡀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠁⠀⠒⠀⡄⢀⠀⠀⠀⠀⠀⠀⣡⠂
⠀⠀⠀⠀⠤⡀⠀⢠⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⠀⠂⠀⠀⠀⠘⠁⠀⠀⠀⠘⠀⠰⠒⠁⠀⠀⣀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠐⠂⠈
⠀⠀⠀⠀⠒⠐⠂⢀⣁⠀⣰⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢠⠴⠀⣤⠀⠀⠀⢁⡀⠄
⠀⠀⠀⠀⠀⠠⠤⣈⠉⢁⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⠀⠈⠀⠀⠀⠀⠀⠉⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠮⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⢀⢁⡀⢀⠀⠀⠀⠀⠚⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣂⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠊⠁⠈⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⣠⠀⠀⠀⠉⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠬⠤⠤⠈⠀⠛⠒⠁⠀⠀⠀⠤⢿⣀⠀⣤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠈⠑⠠⠀⠆⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣆⣴⠀⣀⣔⣀⠃⠰⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⡄⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡉⠉⠀⠾⠀⠀⠁⠉⠉⠉⠓⡶⠀⠀⠀⠐⠦⠇⣈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠊⠀⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣨⠀⠰⡀⠠⡤⠀⢀⣀⣀⣀⡀⠀⡀⠐⠠⣤⡆⠠⠀⠀⢀⠀⠀⢠⣄⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣍⡘⢛⡋⢤⢛⣇⠈⠉⣼⡖⣀⡂⠠⢠⡀⣀⣄⠀⢒⣒⠶⠒⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠉⠹⠿⠶⠾⢯⣄⣉⡁⠉⠀⠂⠀⠨⠁⠨⠀⠀⠀⢀⣄⠀⠀⠈⠁⠀⢠⡄⠀⠀⠚⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠰⠊⠀⢰⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠀⠀⠀⠘⠛⠲⠶⢾⣶⣤⣄⣅⣀⠀⣼⠀⣒⢲⣶⣂⡻⠤⠀⣾⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⢀⠷⢰⢀⣠⡤⠄⢸⠟⠀⢀⣄⣥⣥⣾⠀⠀⠀⠀⠐⠉⠍⣽⠛⠓⠒⠶⠤⣤⡛⠛⡁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⢐⠸⠿⢾⠉⠉⢀⣺⣁⠥⠮⠐⠒⣶⡆⠋⠽⠏⠉⠁⣀⠁⠀⠀⠰⠎⠠⠀⠀⠀⠀⠤⠀⠸⡃⠀⣰
⠀⠀⠀⠀⠀⢀⡀⠀⠈⡀⠐⢊⠀⠀⠀⠀⠀⠉⠁⠈⠀⠀⠹⠶⢶⣦⣄⣀⣅⠀⠀⢘⠋⠀⠀⠀⣤⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⡀⠙
⠀⠀⠀⠀⠐⠀⠁⠀⠊⠅⢐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠉⠉⠙⠛⠒⠶⠶⠶⢧⣤⣤⣀⠆⠀⠀⠀⠀⠶⠶⠶⠖⠂
⠀⡀⠀⢂⠄⠀⠈⠡⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠁⠀⠀⠈⣀⠂
⠰⠃⡄⠈⠔⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢠⠲⡈⠑⢠
⣘⢁⠱⠀⢀⡇⣠⣖⣖
⠀⠙⡞⠒⠉⢀⣴⣿⣦⣉
⢈⢉⠽⢨⠢⠎⠻⣿⣿⣿⣷⣄
⠀⠙⠤⠼⠀⠀⠉⠙⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⢿⣿⣿⣿⣿⣦⣄⣠⠤⠖⠒⠒⠒⠒⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣝⢿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠙⢿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠹⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠈⠻⣾⣷⣽⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢸⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣝⣿⣿⣿⣦⠀⠀⠀⠀⠘⣆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⠿⠋⠢⡀⠀⢀⣀⣀⡈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⣀⠢⡀⢢⣹⡷⠶⡝⢷⡈⠙⠶⢤⣤⣤⠾⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠈⠓⣦⠀⣀⡀⠀⠀⠀⠀⠀⢀⣀⠈⠂⣼⠫⡁⢮⣠⣧⠈⢿⣆⠀⢠⣶⡄⠀⠈⠛⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠈⡏⠉⠙⠳⣄⠀⠀⠀⣏⠙⠃⠀⡇⡴⠞⠋⠁⢈⠠⡀⠙⠳⠶⠞⠁⠀⠀⠀⠀⠈⠉⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⠴⠦⢤⡈⠳⣄⠀⠙⣦⠀⠀⠛⠀⠈⠂⡀⠀⠐⢤⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡘⢦⠀⠈⢷⡄⠀⠀⠀⠀⠀⣢⡥⠖⠉⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢸⡆⢀⡀⢿⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠟⠀⠘⠷⠋⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢦⣄⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠳⡀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠸⣆⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠓⠲⢤⣀⠈⢧⠙⢦⡀⠀⣀⡴⠁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠹⣆⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠈⢧⡈⣇⣠⠽⡞⠁⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠙⢧⡀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⠟⣇⠀⢷⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠙⠷⣤⣷⣄⣀⣀⣀⣠⣤⣴⠶⠛⠉⠀⣀⡼⢦⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠤⠤⠤⠤⠶⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠸⡆⠀⠀⠀⠀⠀⢀⠠⠐⠄⠀⠀⣹⠄⠀⠀⠀⠀⠆⢀⠤⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠒⠀⠸⢃⠐⠂⠀⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡞⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠦⣄⠑⠌⠂⠀⣀⠀⠀⡀⠀⠀⠀⠙⠈⠀⠱⠀⠂⠀⠀⢀⡐⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⣰⡀⠔⢠⡛⠗⠉⠧⢩⠇⠀⢠⣠⣀⠌⠀⠀⠀⠀⠀⠈⡀⠀⠼⠀⠀⣴⣔⡉
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢈⣴⠿⢀⢤⠋⢣⠙⢄⡀⠀⠂⢼⣿⣿⣿⣷⡤⠄⠀⣪⠀⠀⠈⠁⣠⠄⠈⢉⡀⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠁⢀⣠⠀⡈⠀⠘⠀⠀⠀⠁⠀⠑⠀⠀⠘⢛⢿⠟⡁⠀⠀⠀⠇⢘⠀⠃⠀⠀⠀⢱⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠿⠁⢘⠛⠓⠶⠄⠀⠀⠀⠀⠀⣴⠂⠀⠀⠀⠀⠀⣈⠷⠈⠀⢀⢢⣰⣄⡀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠣⠛⠒⠄⠀⠠⠄⣳⠾⠅⠻⠃⠂⠰⠤⠴⢍⡀⠤⣀⣬⣿⣿⣿⣿⡦⠀⠀⠐
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⣠⢀⣠⠛⠀⠀⠉⠂⠈⠀⠀⠀⠀⠀⠀⠘⠉⣄⠀⠐⢟⠛⠻⠏⠂⠀⡀⠂⠁
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠒⣴⣿⣿⡏⠀⡞⠄⠀⠀⠀⠀⠐⡀⣠⠀⠀⠀⠀⠀⡠⠀⡛⠁⣀⡁⠃⠀⠈⠉⣁⡄⣀⡀
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠑⣿⣿⣿⣿⣦⠠⡀⠀⠀⠀⠀⠀⣷⣿⣤⣄⠀⠀⠀⠀⠈⢉⣠⡡⠑⠘⠛⢈⡒⢁⠋⠁⠀⢀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠋⠙⠉⠈⢁⣁⡀⠀⠀⠠⠤⢼⣿⣿⣿⣿⠔⠒⠀⠂⠀⣨⡵⢢⣄⠀⠀⠀⠈⢈⠄⠀⣖⠈⢩⡐
⠀⠀⠀⠀⠀⢄⡒⡐⠈⠀⠀⡀⡀⣀⡠⣤⡀⠀⠀⢀⠤⠺⠛⣿⠛⠁⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⠀⢀⣐⡾⠀⠐⡘⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠀⢤⡈⡀⣴⠆⠆⠀⢈⠞⠠⠜⠁⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⢀⣠⡀⠂⠀⠀⠀⡰⠟⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠦⠁⢲⠀⡀⠀⠀⠘⠀⡀⠀⠀⣄⡝⠡⢥⡶⡉⠀⣼⠀⢤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣉⠝⠉⠀⣠⠀⠀⠘⠁⠃⠀⠰⠣⠀⠁⠐⠒⠉⣀⠖⢄⣙⡀⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠀⠀⠢⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣿⣟⢽⣟⠻⣿⢛⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣽⣿⡏⠟⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⡳⢔⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣹⡿⢏⣼⡇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡶⣠⣴⢞⣫⣵⣾⣿⣿⣡⣞⣵⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⣱⣿⣿⣿⣿⣟⡀⢩⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣯⣤⠀⠀⠀⠀⣴⢒⡴⢠⣿⡿⣼⣿⣿⡿⠛⠉⠉⢡⡬⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⠛⢻⠠⣦⢴⡆⠀⠀⠙⢩⣧⠞⣻⣿⣿⣿⡟⠀⠀⠀⣦⣬⡆⢸⠛⣢⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠺⣳⣧⢢⡾⠘⣇⠀⠀⢠⣿⠇⣰⣿⣿⣿⡿⠀⠀⠀⢠⡇⢸⡆⢨⣶⠻⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠛⠶⠘⠻⣿⣿⣶⣠⡿⠋⢰⣿⠟⠛⠛⠁⠀⠀⣴⣿⣻⡿⢋⠰⠞⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠙⠖⠺⣞⣚⢿⣿⡟⠓⢶⡿⠟⠀⠀⠠⣤⠖⠛⣿⡿⢛⣘⡾⠲⠛⠁⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⢶⠗⡁⠈⠉⣯⢀⡛⢛⣋⠹⠇⢶⣆⣶⠂⠟⢙⣟⢻⣃⢸⡉⠱⢈⡁⢱⣞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣤⣿⣈⡉⠻⣯⡉⠠⠿⢟⡃⠙⣿⠛⠛⠛⣻⡛⢁⣛⠾⠥⠉⣹⠿⠉⣍⣼⣧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠍⠷⠿⣌⡄⠀⠠⠗⠑⠀⣨⣻⣮⠉⠀⠀⠀⠉⠸⣞⣩⠀⠐⠐⠇⠀⠰⣨⡾⠻⠏⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯⣿⣿⢟⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡤⠿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠸⠿⣿⣿⡇⣇⢈⡇⠀⠀⠀⡾⢤⣾⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⣣⡔⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣳⣤⣤⣤⣤⣼⣿⠿⣡⡾⠁⡀⠀⠀⢑⣬⡓⠋
⠀⠀⠀⠀⠀⠀⠀⣼⠞⠛⣯⢡⣆⠀⣼⠀⠀⠀⣾⣍⡥⢄⣤⣾⡿⢛⣭⣶⣶⣷⣶⣾⣾⠟⠀⠀⣹⠀⢰⡎⢩⡟⡓⢾⡀
⠀⠀⠀⠀⠀⢠⣄⢁⠘⠧⢤⣂⢻⣿⣿⡄⠀⠀⣠⣼⠤⠲⣿⣫⣾⣿⣷⣿⣿⣿⣿⠛⠹⣾⣯⣵⣿⣿⣿⢁⣢⠴⠏⣀⢡⡄
⠀⠀⢀⣀⣄⣘⡿⣃⡉⠀⠺⣧⠝⣿⣿⣧⣴⡾⠛⢁⣴⣿⣿⣿⣿⡿⠋⠁⠀⠀⠙⠃⠀⠀⠀⣠⢿⣿⡟⢤⡟⢂⠈⣈⠹⣟⣀⣄⡀
⠀⠀⠘⢍⡟⣽⢧⡌⢹⡶⢴⡎⣀⠿⢷⣬⣻⠦⣴⣿⠿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⡤⣾⣡⣴⠿⢇⡈⡶⢴⡾⠉⣤⢿⡿⡏⠟
⠀⠀⠀⠀⠀⢻⣖⠂⠀⣻⢂⠰⠿⠶⠆⣄⠛⢠⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠘⢃⡔⠷⠾⠦⢐⢿⠁⠀⣲⣾⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠈⠁⣐⣿⣄⣼⡟⠿⢶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠾⠛⣿⢄⣾⣳⡀⠉⠀⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⡟⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣟⠻⣿⢺⣿⣾⣿⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⠟⠛⡟⠛⣿⣿⣿⣿⣿⡇⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠠⠤⠀⠀⡠⠌⠉⢻⣿⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠐⠒⠀⠺⠉⡄⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⣀⠜⢀⡾⠋⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⡴⠚⠑⠢⠤⡴⢾⡧⣹⡿⣩⣷⣾⣯⣅⣀⠀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⢦⠀⠀⠀⠆⠀⠀⢀⣴⠏⠀⠉⠙⠛⠿⠉⠉⠙⠛⣿⣧⣄⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡄⠀⠀⠐⠀⠀⠀⠈⣠⣤⣤⣾⣿⣷⠀⠀⢀⣀⣄⠀⠀⠀⠀⠙⠻⢿⣿⣿⣯⣴⣆
⠀⠀⠀⠀⠀⠀⠀⡠⡃⠀⠁⠀⠀⠈⡄⠀⠀⠀⡟⣻⡿⠻⠋⠉⠀⢿⣿⣿⠿⠀⠀⠀⠀⢠⠀⠰⠋⠙⣿⣿⣿
⠀⠀⠀⠀⠀⡠⠊⠀⠑⠀⢠⠀⠀⠀⠱⡀⠀⠀⣼⢏⠂⠀⠀⢀⡎⠤⠄⠂⢰⣧⠀⠀⠀⠈⠂⠀⠀⠈⠁⠈⠉
⠀⠀⠠⠖⢉⣒⠠⠀⢤⠤⣸⠀⠀⠀⠀⠣⣀⡾⢳⡎⠀⠀⣠⠃⠀⠒⠀⠁⠘⡈⠀⠀⠀⠀⢰
⢠⠀⠀⠉⠀⠈⠁⡄⠀⠀⣸⡀⠀⠀⠀⣠⡿⠀⠀⠃⢀⡜⠁⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢸⡀
⢸⠀⠀⠀⠀⠀⠀⠘⢦⠀⠁⡇⠀⠀⣼⠟⠀⠡⡀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠂⢁
⠀⢦⠀⠀⠀⠀⡔⠋⠉⠁⠀⠓⢤⣀⠎⠀⠀⠀⣸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⠀⠀⠀⠀⢀⡜
⠀⠀⠑⠦⡀⢸⣦⡀⠠⠀⠄⠰⢀⠁⠀⠀⠀⠀⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡖⠤⠤⠴⠛
⠀⠀⠀⠀⠈⠺⣿⣿⣦⢡⠈⢀⠋⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢻⣁⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⡈⠀⠀⠉⡁⠀⠀⢠⣿⢀⣀⠀⠀⠀⢀⠉⠉⠉⠉⠉⠉⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠤⠀⠊⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠘⠦⠤⠤⠤⠤⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣳⣿⣿⠿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⡏⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣆⣠⡄⡀⢀⡠⡀⠀⠀⠀⠀⠀⠈⠛⠻⢿⡽⣧⡙⠂⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣯⠉⠘⠃⠿⣶⠶⣃⣀⣤⣤⣴⠾⠿⠞⣢⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠙⠆⠘⢿⣿⣿⣟⠃⠀⠀⠀⣠⣴⣿⠿⣋⣥⣶⣶⣾⣿⣿⣿⠟⣀⣤⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠆⠀⣠⣿⠀⠀⠀⠈⢺⣿⣉⣡⠖⣠⢾⣻⠟⣠⣾⣯⣿⣿⣿⣿⡿⠟⠹⣾⣿⣵⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢱⡷⡁⠀⠀⠀⠀⠉⣩⣇⣐⠿⣿⣫⣾⣿⣿⣿⡿⠟⠏⠻⢷⡆⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⣀⣹⠃⠀⠀⠀⠀⣠⣞⢿⠉⢀⣾⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣐⣛⢛⣽⡇⠀⠀⢀⢜⠕⡁⢁⣰⣿⣿⣿⣿⣿⠏⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣯⠆⠉⢹⠇⢀⡴⡷⡫⠊⣠⣾⡿⠋⠙⠿⠛⠁⠀⠀⣀⣦⣼⣿⣿⣆⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⢿⣿⣿⣿⢶⢀⣟⡴⡫⠊⠊⠀⢰⣿⠟⠁⠀⠀⠀⠀⢣⡜⠊⣿⣿⣿⣿⣿⡇⠐⠃⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⠠⣄⡟⠛⣃⣴⡿⠋⠀⠀⠀⠀⠀⡼⠃⠀⠀⠰⠦⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⠆⠀⠀⢪⣝⣟⠂
⠀⠀⠀⠀⣠⣴⡶⠒⠲⢶⡾⣦⣿⡷⠶⣺⣵⠮⠶⣿⣷⣴⢢⡶⣠⣀⠚⡀⢀⣠⡀⣦⢲⣴⣾⡷⠆⠢⢤⡀⠀⣴⣦⡄⣠⠔⠓⠾⣿⣧⡀
⠀⠀⠀⢰⠛⠿⣗⠀⢀⡀⡨⠻⣿⣇⢸⡿⠿⡄⢠⣼⠈⢹⣿⢿⣿⣿⣿⣿⣿⣿⣿⢽⣻⠉⢸⣤⠀⠸⠿⣿⣄⣿⡿⠫⣀⢀⠀⢀⡿⠿⢳
⠀⠀⠀⠸⠀⢠⡼⣶⣾⣳⢷⡆⠈⠫⣿⡀⠀⠾⠿⢟⣤⠈⠳⣾⣿⣿⣿⣿⣿⣿⣟⡶⠋⠠⣌⠻⠿⠆⠀⣸⡯⠛⠀⢶⠿⣻⣶⡾⣥⠀⢸
⠀⠀⠀⠀⠑⠘⢷⢜⣫⡐⠳⠃⠀⠀⠈⠙⠲⢤⣀⣈⣻⡷⠖⢹⠿⣿⣯⢹⣿⡿⢿⠑⠶⣿⣋⣀⣠⠴⠚⠉⠀⠀⠀⠱⠗⣨⣛⢳⠟⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠜⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⣮⡙⠿⢿⡇⢠⣾⣿⣼⠃⠀⠀⠀⠀⠀⠀⠀⠈⢉⡿⣿⠻⠀⠉⢁⡀⠀⣞⣷⢋⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠁⣠⡄⠀⢨⣿⣿⣿⡀⠴⠛⢿⡄⠀⠀⠀⠐⣻⠿⠋⠀⣠⣾⠿⣃⠀⣻⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠎⡽⠇⢠⢻⣿⣿⡟⡇⠀⠀⢸⡟⠀⣀⣴⡟⠉⠀⠀⢞⣫⠵⠚⠃⠀⠉⠉⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢠⡽⣛⠀⠀⠊⢰⡿⣿⣿⡏⠠⣄⡞⢀⣴⡿⠋⠀⠀⠴⢛⣉⢄⡶⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣤⠀⠀⠀⠀⠀⠘⣧⣬⣿⠀⡆⠀⢸⡎⢻⡏⠀⠛⣰⣿⠋⡠⠁⠀⢠⣤⠖⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠺⣛⡿⢾⣷⡄⠀⠀⠀⠰⠆⠀⣿⣿⣿⠀⢘⠀⢸⡟⠆⢳⡄⣿⣿⣩⠞⠁⠀⠾⡳⠀⢀⣠⡞⠀⣰⠏⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣀⠀⣄⢀⣿⢿⣿⡄⠀⠀⠀⠀⠀⠹⣿⠏⣀⠸⢀⡿⠀⠀⠀⣼⣿⢡⠏⡘⠀⠈⠀⠀⣰⣿⣫⣴⠋⠁⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠚⢉⣰⡾⢋⡿⣿⣿⣿⠀⢰⠀⠘⠀⠀⠁⣰⣿⠇⢾⣷⣄⠔⣰⣿⡿⠁⠈⣀⣤⠀⠰⠞⠱⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⡀⠀⠀⣠⡶⠾⣿⡿⣿⣷⣿⣿⡟⣿⠘⠀⠀⠀⢀⣾⣽⠋⠀⢾⠇⠈⣴⣿⠟⠁⠀⠀⠀⡄⠐⠀⠀⣀
⡂⢳⡄⠀⢋⣀⡙⣿⣿⣾⣥⣿⣿⠀⠀⣀⢄⡆⠀⢠⣿⠇⢰⣧⡀⠀⠲⠛⠁⢀⡜⢁⡾⠃⠁⠄⠀⢸⡿
⠃⠈⣷⠀⠈⠋⣀⣿⠛⢿⣿⣿⡟⠀⠀⢥⡘⠀⠀⣿⡟⢠⣼⡟⠀⠀⠀⠀⣴⢛⡟⠋⠀⠀⠀⠀⣰⣟⡡
⣶⠀⢹⡇⠀⠀⣼⣏⠳⠈⣿⣿⣿⣾⢶⠞⡁⠀⠀⣿⠀⣿⠿⠀⠀⠀⠀⡃⠉⠻⣠⣾⠂⠀⢀⡴⠛⠟⠁
⣿⠀⠀⠀⠀⠘⣿⣋⣄⡀⣿⣿⣿⡌⠧⠀⠀⠀⠀⠀⣤⠀⠀⣀⠀⠀⣸⡇⢀⡀⠹⠇⠀⣰⠞⠁⠀⠀⢀⣆⠀⠀⢀⣄⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊
⠉⡀⠀⠀⠀⠀⠘⠉⠀⠀⠀⢘⣿⣿⣧⡐⣴⠆⠀⠀⠈⠀⣸⣿⡆⠀⠿⣏⣈⡿⠀⢀⡜⠻⣄⠀⠀⠀⠘⣿⠀⣴⢟⣬⣭⣌⣿⡄
⠀⠀⠀⠀⠄⢰⠆⣠⣰⡾⡿⢦⣿⣿⣿⣿⣿⠆⠀⠘⣤⡀⠈⠿⡇⠘⣷⣿⠉⢠⡄⢸⣇⣄⣫⡞⡞⠧⠀⣨⠃⠻⣿⣿⣏⣿⣿⣇⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠸⣦⠛⠛⠟⣿⠈⣘⣿⠟⣿⣿⡇⠙⢆⣸⣧⠀⠀⠀⢀⣿⣿⡴⠋⠃⢸⣿⣿⣿⠈⡿⠀⠀⣿⡀⣰⣿⣿⣿⠿⠛⠃
⠃⡀⠉⠠⠀⢷⣌⠑⠓⠛⠋⢀⣿⣿⣩⣷⣿⡿⠃⠀⢹⣿⡄⠀⠀⠀⣿⡟⠀⡰⠀⠂⢋⣬⣿⠠⠀⠀⢁⣽⡿⠿⠛⠋⠀⠀⠀⢠⣀⡛⠀⡄⠀⠀⠀⠀⠠⠊
⡿⡧⠀⡄⠀⠘⠛⠛⠃⠀⢠⣿⣿⡛⠷⣶⣏⢀⠀⠀⢈⣿⡇⠀⠀⠀⣿⣇⣿⠇⢠⡶⢿⣛⢿⣷⣀⣴⣿⢉⣹⣶⡟⠀⠀⠀⢀⣼⣟⣥⣶⣶⣆⠀⠠⠾⠃⠂
⠃⣇⠘⣥⡖⠀⠀⠀⠀⢠⣿⣿⣿⠱⠆⣿⠟⠃⠀⠚⠈⢿⣧⡅⢰⡆⢸⢿⠏⢠⣿⢖⡉⡀⢳⣿⡟⠃⣿⠘⢫⣾⠀⠀⠀⠀⣶⣿⡿⠛⢏⠉⠛⣠⠖⠚⠁
⢰⣌⠻⣄⢳⣄⠠⠔⢺⣻⡿⠟⠉⠀⠀⠀⣚⠀⠀⠀⢀⡀⣼⡇⠀⡇⢸⡎⢀⣰⠏⠸⠋⢰⣿⡿⢀⡐⠃⣰⡏⠉⠀⡤⠀⣾⣿⣿⣿⣦⡿⠒⠦⣿⠄
⠲⣿⠀⠀⣾⣿⠀⣰⠿⠛⠲⠞⠉⠀⠀⠀⠈⠑⢀⢰⠸⢡⣟⣧⠀⠃⠈⣡⢪⡟⡴⢶⣶⣿⣿⣽⣿⣿⣴⡋⣀⠀⣀⠀⠀⣿⡇⠛⠷⣯⠁⢀⡄⣿
⣰⣿⠀⡷⢹⣿⣶⣥⣤⣴⣦⠄⠀⠀⠀⢀⠀⠀⠀⠿⠆⢸⣿⣿⣇⠘⠀⢡⣿⣁⠛⣿⣿⣿⣿⣿⣿⣿⣁⣼⣿⣿⣿⠁⠘⣿⡷⠖⠀⠘⣀⠼⢣⡏⢀
⠁⢹⡄⠃⢀⣿⣿⣿⣾⣿⡞⠣⠀⢰⣾⠃⠀⠰⣌⡀⠀⣾⣿⣿⣿⠀⠇⢸⣧⠋⣰⣿⡋⠘⣿⣿⣿⢿⡿⢿⣿⣿⣋⣷⣾⠏⡡⢒⢈⠁⠀⠴⠋⠜⢃⡤⠈
⣧⡄⠀⠀⠘⢿⡏⣻⡯⢀⠐⡀⠀⣼⣿⡟⢄⣻⠀⠃⠀⢹⣿⣿⡿⠁⢀⠸⠃⣸⡿⠉⢁⣸⣿⣿⣿⣻⣷⣄⡾⢻⣿⣿⢏⠚⠴⡟⠉⠀⢀⣴⣶⣏⡾⠁⡀
⠛⢿⠀⠀⢦⡾⢷⣘⣿⣦⡀⠀⠑⢈⠛⣿⡈⣿⡆⠀⠀⣿⣿⡗⠀⠀⠈⠀⣼⣏⠀⢠⣾⣿⣿⣿⣿⡿⣿⣿⡀⣄⣿⠟⣬⡷⠂⠀⣴⣾⡿⢏⡽⠟⠁⠀⠀⢀⣠
⡀⢸⣷⢰⣾⣛⣀⣿⣿⣿⣿⣿⣶⣄⠁⠈⣷⣿⣇⠀⣸⢿⣿⠁⡠⣀⠖⠀⠘⣩⣁⣾⣏⢹⣿⢠⣻⣷⣿⣿⣷⢿⣾⡧⠟⠁⣠⣾⣿⣿⣿⣿⠀⢀⠀⠀⠘⢩
⣹⠈⢻⠾⣷⣿⣿⡿⣿⣿⣿⣾⣿⣿⣏⠞⢊⣿⣷⡾⢣⠀⢿⡾⠠⣿⡀⠰⠀⣿⠉⢿⣿⣿⢃⡾⢛⣻⣿⣿⡀⣻⠟⠁⣠⢾⣿⣿⣿⣿⠁⠀⠛⠛⣂⣾⣿⣶⣾
⡉⣃⢼⣿⣿⣿⣿⣿⣿⡿⢿⣿⣷⡟⠁⢀⠀⠻⣮⡃⢸⡿⣾⣷⣤⣿⡇⠀⢸⡿⠂⢸⣿⢿⣿⣷⣾⣿⡿⢻⠗⣁⣴⣿⣏⣸⣿⡟⢹⣿⣧⣤⣴⠿⠋⠉⣟⣿
⣿⣿⣶⣿⣿⣿⣿⣩⣿⣄⣀⠙⡿⠁⢠⡎⠀⠀⠘⠃⠀⣻⡿⣿⣿⣿⡟⠀⡏⠀⢀⣄⣥⣾⠏⣶⣿⣿⣿⣯⣶⡿⣿⣟⡉⢛⣻⣶⣿⣿⡏⠁⠘⢷⠀⣠⡴⢋⣿
⣿⣿⣿⣿⣿⣿⣿⣧⠙⣿⣿⣖⠁⣰⣈⣷⡀⣆⠀⠀⠀⢿⡇⢸⣿⣿⠁⢰⣷⢠⣿⡏⢻⣿⢠⣿⣿⣿⣏⣾⠟⢻⣾⣿⣿⣮⣽⣿⣿⣏⣴⣾⡾⠋⡀⢸⡿⠋⡁
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⡟⣠⡙⣿⢻⡇⣿⣦⠁⢸⢰⡇⢺⣿⠏⠀⣾⡗⢸⠟⠀⣿⡉⢾⣿⣿⣟⣿⣿⡇⣼⣧⣿⠊⣿⣿⣿⣿⠟⠋⠹⠞⢻⣧⡾⢇⣾⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⠺⣿⣿⣿⡈⢻⣿⣿⣇⣼⣿⡇⢻⣇⠀⣰⣿⠀⠋⢠⣠⠦⠀⣾⣿⣿⣿⣿⣿⣼⣿⣷⣣⡾⠻⢿⣯⡵⣶⠶⣶⡾⣿⣧⣀⡾⣁⡸
⠿⠿⠟⠿⢿⡟⢿⣿⣿⣾⣿⣦⣈⠛⣿⣿⡄⢻⣿⡿⡉⢿⠃⢸⡟⠀⢠⡏⢠⠀⠃⣴⠀⠀⢹⣿⡟⣿⣿⣩⣬⠟⠉⠁⣀⣤⣼⠃⠐⠀⣠⣝⠁⣼⣋⠛⠙⠿⢷
⣀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣷⣿⣿⡓⠛⢻⣷⠈⠹⣇⢠⡄⠀⢰⠇⢠⡄⠃⠈⠀⠰⠇⠀⠀⣀⠻⣿⠿⣿⣿⠏⠀⣴⣾⣿⣿⡏⢂⡇⠐⢺⣿⣾⡟⠉⠐⣢⣄⡾
⠹⣦⢛⣿⣟⠿⣿⣿⣿⡇⢘⣿⣿⡗⠀⢠⠛⢀⠀⠈⠻⡇⠀⠈⡄⠀⠃⠰⠀⠀⠠⠀⠄⣰⣶⡿⣱⣿⡻⢧⣴⣶⣿⣿⣿⡏⠀⣊⢈⣿⡿⡛⠛⣁⣴⣿⡿⢛⣻
⣦⣉⡝⣿⣯⣶⣤⡹⣿⡇⠈⡵⢌⠁⠀⠀⡤⠄⠂⠐⢆⠁⠀⠀⢿⠇⠀⠃⠀⠀⠀⠀⣼⣿⣿⣿⡿⢋⡱⣻⠿⢿⣿⣿⠋⡠⣞⣽⠟⠉⢠⣄⢴⡿⠟⢁⠸⣿⣿
⣿⣿⣄⠙⢻⣿⣿⣿⠀⣿⡆⠀⠀⠀⢀⡀⠀⡄⠀⢆⠈⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⢸⣿⣿⡿⢋⠴⢫⣶⣿⣿⣧⣿⠛⠁⣼⣟⣡⣶⡏⣽⣿⠟⠁⠀⣤⡶⠞⢻
⠉⣉⠙⠛⢮⣟⢿⣿⣶⡇⣥⡀⢻⡄⠀⡀⠀⠀⠚⢪⣡⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⣼⠿⣫⠟⠁⠀⣠⣾⣿⣥⣭⣷⣿⣷⣿⣽⣿⣟⣴⣿⢿⠛⣽⣿⣷⣷⣶⣦
⡿⠛⢻⡏⠻⠉⠳⡝⣟⠳⣹⣧⠈⣷⡀⠀⠀⢀⡏⠀⠐⢢⡘⡄⠀⠀⠀⠀⠀⡇⢀⣿⠟⠡⣶⡿⠐⠿⠉⢹⣿⠇⠀⣿⣿⣉⣹⠿⣿⣷⢛⣼⣶⣿⣿⢿⣿⠟⣷
⣶⣾⣷⢿⡄⠀⠀⣇⠈⠀⠸⣿⡆⠘⣧⠀⠀⠈⠁⢦⡆⠀⢉⡀⠀⠀⠀⠁⠀⠹⣾⢧⡆⢠⣾⠀⣴⠶⢦⣾⠃⠀⣸⣿⣿⣿⣽⣶⣍⣹⣯⣿⣿⣿⣿⣿⡙⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠈⠉⠉⠁⠦⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⡀⣄⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⢀⠎⠁⠀⠀⠱⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⠦⣤⡀⠀⢀⣠⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠴⠀⠀⠤⢤⡜⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣿⠀⠀⢀⡀⠀⢠⠃⠀⠀⠀⠄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⣀⢴⠀⠀⠀⠀⠀⡉⢀⣒⠄⠀⡠⢋⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡜⠀⠀⠀⠀⠀⡇⠀⠀⠄⠪⠀⠉⠔⠀⡠⠂⠀⠀⠁⠀⠀⠀⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⢺⠀⠀⠀⠀⠇⠈⠀⠀⠀⠀⠈⠄⠁⠀⠀⠀⠀⡀⢐⠠⠊⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢳⣀⢀⡠⠤⢄⡀⠀⠀⠀⠀⡠⠐⠁⠀⢠⠈⠁⠁⡄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠴⠒⠒⠒⠀⠈⠀⠐⠒⠚⠒⠤⠤⢤⣘⡜⠢⢠⠀⠤⠤⢉⠠⠀⠀⠀⠀⠀⢀⠠⢠⠂⠀⠀⠤⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀
⡠⠊⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠴⢲⠚⠓⠒⠄⠀⠀⠉⠑⢆⠀⠀⠀⢥⠈⠂⠠⠄⠒⠂⡡⠀⠰⠀⠀⠀⡀⠀⠀⠀⠀⠘⠀⠀⠂⠂⠢⡀
⠘⠒⠒⠒⠶⠒⠊⠋⠉⠁⠀⠀⠰⠂⠀⠀⠀⠀⢀⠀⠀⠀⡘⠀⠀⠀⠀⠁⠒⠒⠒⠊⠉⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠒⠂⠔⠀⠀⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣖⠊⠀⠀⠀⠀⣀⠤⢈⠁⠃⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠈⡈⠢⠍⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠋⠉⠀⢀⠤⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠊⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡡⠊⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁

⠀⠀⠀⠀⢀⠀⠆⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⢰⠊⠀⠀⠀⠀⠀⠀⠀⠠⠄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤
⠀⠀⠀⠀⣨⠀⠀⠀⠀⠀⠈⠀⣀⢀⠆⠀⠚⠥⠀⡀⠰⠁⠀⠙⠠⣦⣀⢄⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⠋
⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠈⠛⠸⡄⠀⠤⠠⠀⢢⠀⠳⣦⡀⠀⠈⣧⣀⡳⣟⠀⠪⡙⢄⠀⠀⠀⢀⣠⡞⣑⢀⠃
⠀⠀⠀⠀⠀⣠⠀⠀⢳⠒⠂⠀⢡⠌⠀⣤⠄⡄⣧⠒⢦⠀⡍⢋⣀⣀⡴⢟⠂⢸⢚⠃⠑⠃⠀⢀⡴⠁⠈⢃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠷⠈⠠⠰⢧⠠⠃⠋⢰⠈⠁⠃⠀⠀⣉⢀⢜⠑⠀⡏⠀⠀⠀⢀⠺⠅⣄⠀⠉⠀⣀⠆
⠀⠀⠀⠀⠀⠀⠃⠀⠸⠀⠀⠮⠛⠀⢸⠀⡆⠀⡔⠄⢺⡀⣠⡀⢀⠝⠈⠸⠄⠠⢗⡀⠀⠀⠘⠁⠔⠩⢰⡑⠀⣿
⠀⠀⠀⠀⠀⠀⢀⠀⠈⠛⠀⡚⡄⠆⡅⠀⠤⠋⠀⢐⣻⣀⠘⠃⠌⣀⠀⠀⡄⠀⢀⣀⠀⠀⣄⡀⣠⢆⡀⠤⠀⣟
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠇⠇⡆⠀⠀⢰⠀⢀⠈⠘⠙⠒⠘⠊⠁⠀⠀⠃⠀⠘⠃⠋⠀⠃⠀⠀⠀⠀⠀⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠤⠚

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣦⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⣰⣿⣃⣿⣿⣿⡏⠁⠈⠛⠉⠹⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀⠘⠛⠲⡆⢰⡋⠁⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⢠⣿⣿⠁⠉⠄⠀⠈⠒⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⣼⣿⣿⡆⠀⠀⠀⠀⠀⠀⠐⢂⠠⢰⣷
⠀⠀⠀⠀⣿⣿⣿⣿⡏⣦⡀⠀⠈⠀⠀⠀⢀⡿⠟
⠀⠀⠀⠀⢻⣿⣿⣿⣧⠀⠙⢿⣶⣤⣤⣤⣊⠀⣀⣀⡿⢧⡀⡀
⠀⠀⠀⠀⢀⣤⣿⣿⣿⡀⠀⠀⠚⠓⠀⠀⣀⣀⠩⠉⠁⠀⣁⣀⣀⣀
⠀⢀⣤⣾⣿⣿⣿⣿⣿⣧⢰⣲⣦⠀⠀⠈⠃⡈⠑⣒⣋⣈⠛⠻⠿⣿⣷⣤⣀⠐⠄
⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⡆⠋⢻⣿⣿⣄⣾⢻⡧⡿⣿⣯⣷⣆⠀⠀⠉⠹⠙⡆⡆⢠⢠⡀
⢀⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡄⠰⠿⢭⣻⠿⢺⣦⣿⣷⣹⣯⣿⠿⣹⣿⣿⠀⠆⠉⠈⠘⠿⢷⣦⣄⣀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢀⡀⠁⠀⠀⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠉⠿⣿⣷⣤⡦⣄
⠐⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⣀⠐⠆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⡿
⠀⠀⢀⣠⣄⣀⠀⠀⣐⣄⠂⢤⡁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣰⡄⠀⠀⡀
⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⣷⡌⣷⣸⡀⠀⠀⠀⢹⡿⣿⠿⠿⠛⠟⣱⣿⣷⡄⠐⢧
⠀⠀⠘⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠈⠁⠀⠀⠀⣰⣾⣿⣿⣿⣿⣆⡈
⠀⠀⠀⠀⠀⠀⠙⠋⠛⠿⠛⠉⠛⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠉⠁
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣤⣿⣷⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣌⣿⣿⣿⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡿⠟⢿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⡱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⡐⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⠀⣴⣿⣿⠏⠀⠀⢀⣀⣤⠤⠖⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⢀⣾⣿⣿⡧⠶⢶⡛⠉⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣀⣠⠤⢶⣿⣿⡿⠓⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡼⢷⠖⠋⠉⠀⠀⣴⣿⣿⠏⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠚⠋⠉⠀⠀⢸⠀⣀⡀⢀⣾⣿⣿⠋⠀⠀⠀⠀⠀⢰⢻⠁
⠀⠀⠀⠠⠴⢶⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⠎⠀⠀⠀⠘⠻⠿⢃⣄⠀⠀⠀⠀⠀⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠈⠀⠉⠀⢀⡀⠀⠀⠀⠀⠀⠀⡙⢀⠴⣒⣒⣒⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢠⣤⣠⠂⠈⠙⠻⠶⣦⣄⠀⠄⣰⡷⠋⠉⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢁⠂⠀⢿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⠀⠀⢱⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⠏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⢀⣠⡶⣋⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠉⠙⠒⠶⠿⠿⠵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⣦⠀⣀⣿⣚⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣤⣼⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣞⣿⣿⡿⣟⠉⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⣿⡿⠟⠋⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢪⡵⠢⢄⡀⠘⠹⣦⣄⢀⣠⣔⣽⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢿⣴⠏⠀⠀⣾⣦⢴⣤⣶⣟⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡸⢡⡿⠻⠀⠀⣘⣽⣿⣿⣿⣿⣿⠿⣯
⠀⠀⠀⠀⠀⣀⣀⣠⠴⡉⣷⣄⣀⣀⠤⠾⠖⠾⠃⢠⣶⣟⡯⣿⢿⣿⢿⠿⠂⠆⠸⡄
⠀⠀⡠⠒⠋⠉⡠⠖⢢⠁⣞⣋⣁⡴⠛⠉⢇⣤⡒⠽⣻⣵⣾⠟⡉⠊⠀⠀⠀⠂⣸⠇
⢀⣞⣼⣧⠀⠀⠀⠀⠈⠀⠀⢀⣤⢤⡄⣻⣿⣿⣿⠮⢛⠏⠁⠀⠀⠀⠀⡠⠂⣠⠏
⣼⣿⣿⣷⣄⠀⠀⠀⠀⣠⣦⣁⡾⣿⣫⣶⣻⣏⣛⡖⢻⢠⢐⠂⢀⣂⡴⠶⠛⠁
⣿⣿⣿⣿⣿⣦⠀⣠⣾⣿⣿⣷⡻⠛⠋⢐⣿⠟⢟⡷⠋⠉⠉⠃⠋⠁
⠸⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠉⠀⠀⠀⣎⡙⡵⠃
⠀⠹⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢠⢧
⠀⠀⠉⠻⣿⡿⢀⠆⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉
⠀⠀⠀⠀⠈⠻⣤⡈⠀⠀⠀⢀⢀⣀⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⣀⠀⠀⠀⠀⠀⣸⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⡗⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⣿⣇⣉⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⣿⣿⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠴⣿⡟⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣿⣿⠛⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣧⣀⣀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣼⣿⣿⣤⣄⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⢿⡇⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢀⣼⣿⣸⣿⣿⣿⣿⢸⣧⣀⡀⠀⠀⢈⣿
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠃⠀⢨⣿⣷⣷⣿⣿⣿⡿⣿⣿⣿⣿⣇⠀⠀⠈⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣿⡿⣿⣿⡇⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⣿⣿⡄⠀⠀⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢀⣿⠷⣿⣿⣿⣿⣿⣿⢻⣇⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⣾⣿⣿⣿⣿⣿⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣀⣀⣾⣿⢀⣿⣿⣿⣿⣿⣧⢸⣿⣦⣀⣀⠘⠣⠖⠛⠛⠛⢷⡎⣿⣿⣿⡿⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣹⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⡷⡿⠿⠏⠀⠀⠈⠐⠀⠀⠀⠚⠛⠋⠉⠀⠀⠀⣿⠀⣿⡆
⠀⠀⠀⠀⠀⢠⣿⣿⢻⣮⣿⣿⣿⣿⣵⡟⣿⣧⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡿⠉⢹⣿⣿⣷⡶⠷⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇
⠀⠀⢀⣀⣴⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⣿⣿⣦⣀⣀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⣿⡇
⠀⢀⣾⣿⣿⣿⢿⡟⣸⣿⣿⣿⣿⣿⣿⣇⢻⡿⣿⣿⣿⡆⢹⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⣸⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠙⠻⠿⠿⠿⣯⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⣀⣿
⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⣿
⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢿
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣿⣿⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣛⣿⣿⣿⣟⣛⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⣿⣿⣿⡟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠛⢻⡿⠛⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣷⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠀⣿⣿⣿⡇⢸⣿⣷⣦⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⠀⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⠀⠀⣿⠀⠀⢀⣠⣤⣤⣤⣄⡀⠀⠀⠀⣀⡀⢀⣸⣇⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠀⣿⣿⣿⡇⢸⣿⣿⠟⠀⠀⠀⠀⣿⣠⣾⠟⢋⣉⣉⣉⡙⠻⣷⣄⠀⣿⡏⢸⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⣿⣿⣿⡇⢸⣿⡏⠀⠀⠀⠀⠀⣿⡿⢁⣾⣿⣿⣿⣿⣿⣷⡌⢻⣧⠘⠇⠸⣿⣿⠇⠸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⣿⣿⣿⡇⢸⣿⣷⠀⠀⠀⠀⢠⣿⠁⣾⣿⣿⣿⣿⣿⣿⣿⣷⠈⣿⡆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣦⣤⣤⣤⣤⣾⣿⣿⣷⡄⠀⠀⠈⣿⡄⢻⣿⣿⣿⣿⣿⣿⣿⡿⢀⣿⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠿⠛⣛⣛⠻⢿⣿⣿⣿⣿⣆⠀⠀⠹⣷⣌⠻⣿⣿⣿⣿⣿⠟⣡⣾⠟⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠃⣴⣿⣿⣿⣷⣄⢹⣿⣿⣿⣿⡆⠀⠀⠈⠻⣷⣤⣬⣉⣥⣤⣾⠿⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⣿⠀⣠⣤⣤⣼⣧⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣄⠹⢿⣿⣿⠿⢃⣼⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⣶⣤⣤⣶⣿⣿⣿⣿⣿⣿⣁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠉⠙⠛⠛⠿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠟⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠶⠶⠶⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⣼⣿⣦⣄⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⣴
⣷⣠⣿⣷⡄⢀⣤⣠⣾⣿⣿⣶⣤⡄⠀⢠⣶⣤⠀⢰⣶⣶⣄⣿⣿⣿⣿⡧⡄⢀⣀⠀⠀⣶⡆⢀⣴⣿⣿⣶⣰⣶⣰⣶⠀⣤⣴⣾⣿⣿⣿
⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣏⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣾⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⢀⣼⣿⣿⣷⠟⠛⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⠿⣿⣿⡏⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⢀⣼⠿⣿⢿⡟⠋⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠛⢿⣿⡿⠛⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⢴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⣿⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⠿⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⢸⣿⣿⠀⣿⣿⡆⢠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠟⠛⢩⣿⣾⣿⣿⠀⣿⣿⠁⠀⠀⠉⠣⣴⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⣸⡇⣿⣿⡏⢰⣿⣿⠀⠀⠀⠀⠀⠙⣾⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⣿⢿⣿⡏⣇⢸⣿⡏⠀⠀⠀⠀⠀⠀⠸⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⢰⣷⣼⣿⢱⢳⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠏⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣿⣿⣿⡏⢻⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⣿⣿⣿⢳⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢸⣿⣿⣿⢸⣇⣿⣿⡿⠀⠀⠀⠀⠀⠀⠐⢧⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣦⣀⢀⣾⣿⣿⡟⣼⣧⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⢺⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⢹⣏⣿⣿⣿⣿⡇⣿⣽⣿⣿⡇⠀⠀⠀⠀⠀⠀⠠⠈⠁⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⢡⣿⣽⣿⣿⠇⠀⠀⠀⢠⣒⠘⢁⡾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⡏⠰⣄⠐⣼⣿⣟⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣄⠰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠈⠙⠿⣿⠰⠿⢽⣿⣿⣿⢹⣿⣿⣿⣿⠁⢿
⠀⠀⠀⠀⠀⣰⡤⢶⣾⠋⣌⣿⣿⣷⡧⢴⣤⣼⣿⣿⡿⢹⣿⡟⠸⣏⠀⠈
⠀⠀⠀⠀⠀⣿⠀⣾⣿⣇⣼⣿⣿⢿⣷⣸⡇⣿⣿⣿⠗⢿⣿⢀⣀⡘⢦⣄⡀
⠀⠀⠀⠀⢠⡿⠀⠘⠿⢿⣿⣿⣏⢸⣿⣿⣷⣿⣿⡏⠀⢸⣿⢸⣿⡿⠀⠀⢣⣼⣿⣿⠃
⠀⢀⡆⠀⣼⡇⠀⠀⠀⣾⣿⣿⣿⣤⣿⣽⣿⣿⣿⡇⠰⠀⠙⠛⠋⠀⠀⠀⢸⡏⠙⣿
⠀⠀⡄⢀⣿⠀⠀⠀⠀⣿⠿⣽⣿⣿⣿⣿⣯⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡆⢸⡆
⠀⠀⠀⢸⠇⠀⠀⠀⠀⣿⣶⡿⢿⣿⣿⡏⠈⣿⣯⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⣧
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠁⠀⠸⣿⣿⡇⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆
⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠇⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣿⣿⠃
⠀⠀⠆⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⣿⡏⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⡄⠀⠘⣷⣄⠀⠀⠀⠀⠀⠀⠸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢻⣿⠏
⠀⠀⡀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢟⣻⡿⠋
⠀⠀⠃⠀⠀⠀⠀⠀⠉⠻⣦⣄⠀⠀⢿⡶⣿⠃⠀⠀⠀⠀⠀⢀⣀⡠⣾⣵⡿⠋
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠾⣄⣩⣤⣤⣤⣤⣤⡐⠋⠥⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠤⠀⡇⡆⠀⠀⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣞⣀⣤⢤⡇⣇⣀⣤⣀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡤⢦⡸⠁⠙⠋⠀⡀⢻⡇⣷⠈⠙⢿⠀⢡⠶⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣀⠘⢁⠇⠀⠀⠀⡇⢸⡇⣿⠀⠀⠈⢰⡈⠀⡀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣂⠘⠷⢠⣍⠋⠀⠀⠀⠐⢃⢘⣧⣏⠀⠀⠀⠀⠋⣡⠴⠾⢁⡂
⠀⠀⠀⠀⠀⠠⠂⠀⣰⠆⠐⠲⡘⢃⡀⠀⠀⢀⡄⡏⠤⠌⡇⣄⠀⠀⡺⣣⠖⠆⢰⡄⠀⠐⡀
⠀⠀⠀⡀⠀⠉⢰⠁⠊⡖⠀⢀⡿⠘⣿⣤⡀⠀⠘⢡⣶⡖⠉⠨⠀⡄⡁⢏⢀⠐⠆⠑⢸⠀⠁⠀⡀
⠀⢀⡀⠀⠰⠆⠺⢤⡃⢀⣲⢊⣴⢿⣹⣿⣷⡄⠀⠈⣿⡇⠀⠀⣴⠟⡷⣌⢱⣂⡄⢧⠼⠃⠺⡄⣠⢀⡀
⠀⠀⠀⠈⠅⡤⠖⠉⠁⠘⠥⠥⠖⠾⢿⡛⠛⢋⡴⡺⣿⠇⢀⡼⠯⠤⣄⣄⣠⠭⠤⡄⢀⡠⠞⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠐⠑⠈⠚⠁⠂⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⡠⠾⠃
⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⢠⠊⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⡄⠉⡉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠎⠀⢐⠀⠀⠀⠀⠀⠀⢠⠄⢚⣉⡄⠀⠁
⠀⠀⠀⠰⠃⠀⣦⡴⠀⠂⡠⠀⠀⠀⠀⠀⠀⠀⠈⠒⠐⠊⠀⠀⠀⠀⠂⠀⠀⠑⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠟⣻⡟⠁
⠀⠀⠀⠀⡠⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⢒⢚⣱⠟⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠐⠊⠀⠀⠄⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⢀⣄⣄⠀⢠⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢆⣧⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⡇⡰⠊
⠀⠀⠀⠈⠀⠀⠐⠘⡇⠀⠱⡀⠀⠀⠀⠀⠀⠀⠸⣼⡆⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢰⠈⠁
⠀⠀⠀⠀⢀⠀⠀⢂⡰⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠆⠀⠈⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠨⠀⡁⠴⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣿⣤⣼⡄⠀⠀⠀⠀⠀⠀⠨⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⣰⣶⣾⣿⠿⠿⠟⠟⠿⠟⠋⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⡄⣠⣤⣿⣿⣻⠟⠛⠿⠯⢀⣀⡀⢀⣀⢰⣶⢰⣿⣷⠀⠀⠀⢀⣆⣀⡀⠶⠈⣴⣆⣈⣀⣀⣤⡆
⠀⠓⠯⣙⣛⠿⠿⣿⣿⣶⣶⣾⣿⣷⣾⣿⢸⣷⣸⡿⠏⠀⠀⠀⠸⡎⠉⠙⠛⢻⡟⠛⠉⠉⠉⣼
⠀⠀⠀⠀⢹⡟⠓⠶⣮⣭⣙⣻⣿⣿⣿⣿⠸⢻⠃⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠘⡇⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠸⡇⠀⠀⣿⠀⠉⣿⠋⠉⠉⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⣇⢀⠀⢰⠏
⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⡟⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⣰⣎⣀⣤⣿⣤⣤⣼⣆
⠀⠀⠀⠀⠀⠃⠀⠀⠛⠂⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠙⣟⠉⠉⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣯⣟⣿⣻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡀⠀⢦⡦⠀⠀⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢦⠀⠀⠀⢀⠂⠀⠀⠀⠀⠐⠦
⠀⠀⢀⣠⠀⢀⠀⠀⠀⠀⠀⠁⣦⡕⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠉⡱⠊⠤⣾⣷⠋⡀⠀⠀⠀⢼⣂⠐
⠀⠸⡿⠎⠀⠉⠀⡀⠠⣤⠀⠀⠁⠀⣑⠈⡅⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠒⠂⠊⠀⣺⠖⠀⠀⠙⠁⠀⡇⢀⢀⣴⣾⣿⢾
⠀⢰⣿⣇⠀⠀⠀⠁⣠⣽⢳⡂⡀⠄⠛⣹⡿⠛⠛⠛⠓⠒⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⣀⠀⠀⠰⠂⠀⠀⠁⠀⣰⣦⠀⠁⠂⢲⣿⣿⣷⣿
⠀⠘⠋⠘⢷⣤⠒⠂⠀⢿⡁⠙⣤⠤⢄⡨⠟⠋⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠙⠀⠁⠀⠈⠀⠀⡈⠁⣠⠄⠀⢀⣠⣿⣿⣿⣿⣿
⠀⠀⠀⠠⡄⠁⢀⡕⠀⠈⠉⠉⠠⠤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡄⠺⠀⠁⠀⠀⣼⣷⣼⣿⣿⣿⣿
⠀⠐⠶⣶⣤⣿⠶⠀⠒⠀⢨⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⡀⠀⣸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠙⠛⠛⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⠛⠛⠛⠛⠛⠛

⠀⠀⢠⢤⠀⠀⣠⠄⠀⠀⠀⠀⢤⣤⣤⣤⣀⠀⢠⠀⠠⢠⣤⣄⣀⣤⣄⠀⠀⢠⡤⢠⠀⣄⠀⠀⢀⡄⣀⢠⣄⠀⠀⣀⠀⢀⠀⣤⣄⣀⢤⣤⣄⠀⠀⠀⠀⢤⣀
⠀⠀⠘⠀⢀⠀⠑⢄⠀⠈⠱⠀⣦⢰⣟⠻⣿⣦⡀⡠⣄⠽⢣⠄⠀⠉⠀⠀⠀⠀⠀⠆⠘⢋⢴⡞⠩⣷⣜⣤⠐⢄⠈⠓⠂⣄⡀⢘⠙⠃⠛⢀⠽⣗⣂⠤⠀⢄⠠
⠀⠀⢠⡄⠈⠀⠀⠀⠑⠢⢠⡯⢿⣰⣌⣃⠀⡻⢿⣷⠌⠒⠀⠀⠀⠀⡀⠀⢂⣤⣀⢀⠀⠠⠁⠚⢻⡟⢛⢣⣔⡀⢱⠀⠀⠀⡘⠋⠀⠀⠀⢠⣷⡌⠻⠄⠀⣄⣱
⠀⢐⠀⠈⠳⠀⢀⣴⠆⠙⠻⣤⡄⣉⣍⠙⢱⡄⠊⠈⢳⣶⣆⠀⠀⢨⠀⠀⣘⡧⠈⠀⠀⣠⣄⢉⣸⣦⠁⠈⣖⢀⣴⣄⡀⠀⠀⠀⠀⠀⣠⢿⣥⠀⠀⣀⢀⢼⣬
⠀⢀⣨⠄⠀⠈⠛⠉⠲⣈⠀⠻⢾⡏⠈⢽⣿⣅⠈⠀⠀⠉⠙⢢⣀⡇⠀⠀⠳⠀⢀⡟⠘⢻⣌⢿⡟⠻⣿⡙⣯⣿⣏⣸⣿⣄⠠⣤⡴⠛⣷⠎⠁⠀⠸⣏⡼⣳⠿
⠀⠈⣅⡠⠂⢀⣀⣀⢀⣿⡀⡠⠜⠙⠇⠈⠐⠿⢓⡤⣤⣤⣖⠿⠛⠉⠉⠀⠀⠈⢿⡄⢼⡟⢶⡂⠉⠓⠤⡉⠻⠡⠉⢻⡁⠈⠁⠈⢉⡡⢉⣱⡄⠀⠀⣜⡒⠀⢀
⠀⠘⣿⣀⠀⠁⢺⣯⣼⣾⠎⠠⢤⣤⣶⣾⣀⣶⠊⡙⠛⠉⠉⠀⢠⠐⠐⠄⡀⠀⢸⣧⣀⣘⣇⠘⠀⠀⠐⢉⡴⠁⠀⠀⢳⡄⠀⢠⠠⣦⠞⢋⡀⠁⠀⠈⡁⠀⢠
⠀⠀⠾⠷⠷⠶⠿⠿⠾⠤⠤⠤⠴⠶⠾⠿⠿⠷⠶⠧⠴⠷⠤⠤⠴⠶⠶⠶⠴⠷⠾⠿⠿⠿⠿⠦⠴⠷⠶⠶⠶⠤⠤⠴⠤⠿⠦⠴⠤⠶⠶⠶⠿⠶⠶⠶⠿⠷⠾

⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⢀⡀
⠀⠈⠁⢻⠌⡀⠑⡴⢌⠂⠈⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠎⠀⢈⠼⡜⠁⡀⣼⠫⠩
⠀⢀⡅⠀⠀⠁⣠⠃⠈⣆⠀⠘⠀⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠈⠀⢀⡎⠀⢣⡀⠁⠀⠀⣅
⠀⠸⠋⠀⠀⣈⡯⣿⡛⠀⠀⢤⡁⣤⣀⡰⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⢀⣠⡄⣡⠤⡀⠘⣛⡭⣯⡀⠀⠘⢻
⠀⠈⠀⠁⠀⠋⠁⠉⠀⢢⣀⡀⠐⠛⢻⡾⣆⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⢀⣶⣷⠛⠃⠀⣀⡠⠂⠈⠉⠉⠃⠀⠁⢈
⠀⠠⣦⠀⠀⠂⡁⠀⠀⠀⣸⠇⠐⠁⠈⡇⠈⠣⡄⠀⠀⢀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⠁⠀⠀⡠⠉⠀⡏⠀⠑⠀⢿⡀⠀⠀⠀⡁⠀⠀⢠⣶
⠀⠀⠁⢀⠀⠉⢁⠀⡀⣴⣧⣤⠀⣠⣞⠁⠈⣶⠉⠒⠀⣈⡅⡀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⡽⡀⠀⠘⢱⡆⠈⢙⣦⡄⢠⣤⣧⡄⠀⢀⠉⠁⢀⠀⢁
⠀⠠⣀⣀⣄⣄⣱⣾⣿⣽⣿⣿⣷⣿⣉⣀⣀⣹⣦⣀⣸⣿⣾⣥⣤⣣⣴⣆⣠⣀⣀⣠⣀⣤⣤⣦⣤⣿⣾⣿⣀⣠⣾⣁⣀⣈⣹⣷⣿⣿⣯⣾⣿⣾⣁⣄⣄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠁⠀⠈⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠁⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠖⣢⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣍⢹⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⢉⣿⡿⠶⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣦⣼⣷⠷⠦⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠒⢛⣿⣿⡇⣨⠓⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⢸⣿⣿⡅⠈⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠘⣷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠐⢸⣿⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢻⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠿⢿⠇⠀⠀⠀⠠⣴⠟⠃⠀⠀⠀⠀⠀⠀⠀⠈⣿⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⡔⠂⠀⠀⠀⠖⡀⠀⢼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠀⢰⡇⣤⣀⣤⣀⢰⡇⠀⠘⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢻⠜⠀⠀⠀⠀⠀⠸⡇⢤⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢈⢸⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⣠⣄⣀⠀⠀⠈⠁⠀⠀⠘⢳⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢈⣹⣿⠀⠀⠀⠀⠀⠀⠀⠟⡆⠀⠀⠀⠀⠀⠀⠀⠈⠟⣇⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠈⣿⡏⢀⠀⠀⠀⠀⠀⢈⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⡆⣿⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⢈⣸⣿⣇⣀⣀⣀⣴⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣨⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣛⣛⣛⣋⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣻⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣿⠴⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣍⣀⣹⣿⠶⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢹⣿⣿⣿⣿⣿⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣶⣟⣿⣾⣿⣿⣿⣿⣿⣿⣟⣷⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⣿⠿⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⢛⢿⠛⡟⢿⡟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠈⠉⠀⠀⠀⠀⠀⠀⣰⢲⠀⠀⠀⠀⢠⡖⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⡼⠀⠀⠀⠀⢸⡀⠀⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣀⣴⠾⠋⠁⠀⠀⠀⠀⢠⠄⠀⢀⣤⠀⣴⠋⡟⠀⣿⡇⡼⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣰⠇⡿⢰⣿⢸⣷⡼⠙⠿⠃⠀⢀⣀⣀⡤⣄⠀⠀⠈⢦⢠⡏
⠀⠀⠀⢀⡤⠀⣠⢶⠀⠀⣠⣦⠀⠀⠀⠀⠀⠀⣶⠀⠀⣿⢀⣿⡄⣠⠟⠿⠃⠉⠀⠀⠀⠀⢀⡼⣻⣿⠸⠻⠆⠀⠀⢠⡟
⠀⠀⢀⣾⠁⡴⢣⠏⠀⣰⠇⡟⠀⠀⠀⢰⡀⢠⢿⡇⣸⢿⠞⠉⠋⠁⠀⣀⣀⣤⣤⣀⣠⡖⢋⣴⠋⠀⠀⠀⠀⠀⣠⠏⢱
⠀⠀⣼⣏⣼⡷⠋⠀⢰⡟⢰⢇⡴⣶⠀⣿⣧⠎⢸⡟⠃⠀⠀⠀⣠⡶⠛⠉⠁⣠⣴⢟⠁⢩⡿⢧⣀⠀⠀⠀⠀⣴⠏⠀⣸
⠀⠀⠘⣿⠟⢁⣄⠀⣾⢃⡞⣼⢃⣿⣸⢹⡏⠀⠉⠀⠀⠀⢀⡼⠋⠀⢀⣴⡾⢏⢕⠅⣴⠋⠀⠀⠙⣧⠀⠀⢰⡏⠀⢠⡇
⠀⠀⢸⡿⢠⡏⢈⢳⣿⠞⢰⣿⡾⠉⠁⠀⠀⠀⠀⠀⠀⢸⢧⠀⣠⠖⣹⢿⣴⣡⢪⡞⠁⠀⠀⠀⢀⣿⠀⠀⢸⣇⣠⡞
⠀⠀⣿⡇⢸⡇⡜⢸⣿⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⢞⡷⢾⣔⢻⣿⠿⠋⠀⠀⠀⠀⣠⠞⡿⠀⠀⠀⠉⠁
⠀⠀⣿⡇⢸⣿⠁⡎⠙⠋⠀⠀⠀⠀⠀⠀⣠⠤⠤⢖⡿⢾⣿⡞⣡⢏⡬⣻⢄⡀⠀⠀⣀⣠⣴⡯⠊⠁
⠀⠀⠈⠓⠋⠻⠖⠁⠀⠀⠀⠀⢀⣠⠴⠚⠁⠛⠓⠉⣠⣿⣏⣞⡵⣻⡖⠋⢈⡿⢠⡞⣋⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢶⠄
⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠛⠠⣄⡀⠀⠀⠀⣰⠋⠈⣻⠿⠂⠈⠁⢀⡞⠀⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⣄⢸⣿⣆⠄
⠀⠀⢧⠀⢠⠞⡆⠀⢀⡏⠀⠀⠀⠀⠀⢹⡆⢀⡼⢃⣴⣿⠟⠀⠀⠀⠀⠘⢧⠄⢑⡄⠀⠀⢀⢀⡆⢀⢸⣆⣿⡞⠛
⠀⠀⠈⠣⣿⡴⠇⠀⠈⣇⣄⠀⠀⠀⢀⡞⣠⢞⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⡇⠀⠀⣿⣸⠃⣼⣼⠿⠁
⠀⠀⠀⠀⢿⡀⠀⠀⠀⣿⣿⠦⣀⠀⣿⠕⣵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠊⠀⠀⢸⣿⣿⣀⡏⠉
⠀⠀⠀⠀⠸⣇⠀⠀⠀⡇⠙⢷⣄⡀⠈⠛⠅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡇⠀⠀⢸⢸⠀⠈⠛
⠀⠀⠀⠀⠀⢻⣆⠀⠀⠳⣄⠀⠙⠻⢶⣤⣀⢈⣧⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⡇⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠙⢷⣄⠀⠀⠈⠙⠻⠯⣤⣄⣀⣀⣀⣤⡶⠟⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠛⠶⣤⣀⣀⣀⠀⠈⠉⠉⠉⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠀⠀⠀⠀⠉⠉⠙⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠞⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄
⠀⠀⠀⣀⠀⠀⢰⣄⢄⣰⠾⠁⣤⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢖⣄⡀
⠀⠀⣾⠋⠀⣤⡄⢴⣼⣷⣄⠲⠿⠆⢸⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣿⢠⣽⣷
⠀⠀⣴⣤⢩⣩⣿⡞⠉⠙⣿⣷⡴⠶⢘⢧⡶⡄⡾⠂⠀⠀⠀⠂⢀⣯⣼⣿⠉⠀⣀⣀⣠⣴⣶⣿⣿⣶
⠀⠀⠀⠈⣣⣭⢾⣷⣖⣘⣞⣿⣷⣤⣜⣿⣵⣴⣿⣿⢡⡄⠀⠀⠈⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⢿⠏⢸⣿⣿⡿⠟⡿⠿⢿⣾⣿⣿⣿⣿⣿⣿⣤⣄⢀⡀⣿⡏⣩⣯⡍⠨⣿⠟⣿⣿⡿⠟⣋⣀⡀
⠀⠀⠀⠀⠺⣿⣶⣿⣧⣀⠘⠶⡆⠀⠀⣸⣿⣿⡯⠛⠛⠉⠉⠀⠙⣿⣿⣿⣿⣏⣛⣿⣿⣿⣿⠃⢘⠿⣿⡷
⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣶⣤⣼⣅⣼⣋⠁⠐⠤⠀⣀⢰⣼⣶⣿⣿⣿⣿⣿⣿⣿⣯⡰⣥⣿⣶⣦⣭⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣶⣤⣶⣿⣻⣿⣴⠀⠘⣿⣿⣿⣿⠀⠀⠀⢠⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡻⢿⣿⣿⡙⣛⣩⠀⠀⠐⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⢿⣦⣼⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠑⠈⠀⠤⠀⠦⠀⠆⠀⠈⠲⠏⠰⠀⠰⠐⢠⢂⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠉⠉⢿⣦⠀⠀⠀⠀⣤⠤⠤⠤⢴⡴⠎⠀⠀⠀⡄⢀⣼
⠀⠀⠀⠀⠴⣄⣀⣀⣀⠀⠤⠤⠤⠄⠀⠌⢈⠀⠀⠀⠂⠀⢛⡇⠀⠀⠀⠀⠀⣀⢀⠀⢆⠀⠀⠀⢰⢃⣽⡃
⠀⠀⠀⠒⠠⠁⠀⠀⠂⠀⠀⠀⠀⠀⠈⠘⡜⠀⠀⠀⢀⣴⣴⠇⠀⠀⠀⠀⠀⢿⢿⠆⠈⡄⠀⠀⠺⣼⠛
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠀⠀⡯⡇
⠀⠀⠀⠀⠀⣠⡝⣱⠀⠀⢠⣄⣠⣤⡀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠐⠲⢳⠚⠋⠀⠀⢠⡇⡁
⠀⠀⠀⠀⠀⠈⢹⡇⠀⠀⢀⡉⠋⠓⠮⣿⡶⠯⢄⣀⠀⢱⠠⠀⠀⣀⠀⠀⠈⠛⠉⠛⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡜⠁⠉⠲⠴⠉⣹⣿⡟⠒⠿⠀⠀⠀⠉⢂⠀⠀⠀⠀⢀⣿⢿⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⢧⣀⣀⣀⡀⢀⠛⠠⢫⠟⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⢸⣿⡊⢇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠁⠻⡥⠹⡆⠱⠀⠀⠀⠀⢤⣤⣴⠞⢁⠇⠀⠀⠀⣿⣿⣹⢿
⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢀⡴⣺⠿⢉⡶⠋⠀⢀⠀⠀⠉⠋⠉⠘⡃⢀⡄
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠾⠟⠛⠋⢀⡴⠋⠀⠀⠀⢸⠀⠀⢙⠓⠀⠀⠃⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣧⠀⢀⠀⠀⠀⠀⠀⣀⠤⠤⠄⣀⠀⠁⠐⠒⠉⠁⠀⠀⠀⠀⠀⠀⢃⠀⠼⠄⠀⠀⢇⠜
⠀⠀⠀⠀⠰⠆⢘⣇⠀⠛⢲⠀⠀⢠⠎⠀⠀⠀⠀⠈⠑⠒⠐⠊⠉⠒⣄⠀⠀⠀⠀⠀⠀⠑⠤⣀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠇⠀⢸⠀⠀⠀⢀⡀⠀⠀⣀⣤⣤⣄⣀⣸⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⠀⠀⠀⠸⡟⠀⠘⢿⣟⣻⠟⠿⠛⠛⠛⠛⠁⠀⠀⠐⠒⠚⠛⠟⠻⠃
⠀⠀⠀⠀⠀⠤⠀⢤⠤⠐⠊⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠸⣡⣊⢫⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣕⢛⣲⣶⡬⣝⠒⡿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⣾⣾⣿⡿⣿⡙⠫⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣖⢿⣿⡿⠃⠐⠁⠈⠡⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢋⣨⢞⠝⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⡤⠞⠛⠋⡋⠘⠁⢀⡴⣺⣍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣊⠚⠁⠐⡀⠔⠀⢀⣠⣞⡿⠋⠉⠑⢮⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⠅⠀⠀⠀⠀⣠⠴⣩⠞⠃⠀⠀⠀⠀⠀⢃⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣀⣀⣠⣞⢡⣦⣀⢀⡴⣞⡵⠛⠀⠀⠂⠠⠀⠀⢠⡶⢏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡋⠚⢛⣾⠯⢬⣍⡈⢿⡟⠉⠞⠋⠀⠀⠀⡠⣠⣄⣄⢀⢿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠻⡥⣴⠟⡉⢒⣅⣊⠷⣆⡁⠰⠿⣿⢦⠀⣼⣿⣩⣥⣽⣹⠍
⠀⠀⠀⠀⠀⠀⠀⠠⡲⠋⠀⠀⠀⠀⠚⡱⡦⣄⠀⠊⠉⠓⠛⠂⣀⣴⣻⠜⢸⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢀⠎⠀⢁⣩⠇⠀⠀⠀⣶⠟⣍⠉⢀⠀⠀⠻⠆⢠
⠀⠀⠀⠀⠀⠀⣼⠅⠀⠀⠀⠐⡔⠁⡠⠖⠋⠀⡄⠀⠀⠀⠙⠛⠛⠀⠈⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⣠⠃⣴⠊⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⡟
⠀⠀⠀⠀⠀⠀⠨⡁⠆⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡾⣻⠟
⠀⠀⠀⠀⠀⠀⠀⠳⠄⡐⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣤⡾⣫⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢨⡑⠂⠄⠔⢀⠨⢤⠤⠍⣀⡴⠾⣋⣵⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡒⠒⠐⠰⠤⠥⠔⠲⠞⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣦⣤⣀⠀⠀⠸⡼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠃⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣯⣝⠻⣿⣿⣿⡧⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠉⠁⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⢿⣿⡿⠀⠀⣠⠂⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠛⠁⠀⠈⢙⡆⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠃⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣄⠀⢀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⢸⡦⠖⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⣌⠳⣤⡸⠟⠒⠀⢀⣀⣤⣤⠀⠈⠛⠣⠀⣷⡤⠤⣠⣄⣀⣀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣦⡘⠟⢻⠇⠛⠛⠛⠛⠛⠀⠀⠀⠚⠓⠻⡿⠻⠷⠿⠿⠧⠄⠀⠀⢻⠀⣼⣿⠿⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠁⠀⠀⠀⠙⢿⣿⣿⣷⡀⢹⣦⡀⠀⠀⠀⠀⠀⠚⠉⣉⣀⡀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠋⠉⢿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠶⠒⠒⠂⠀⠙⢿⣿⣿⡄⠻⣿⣦⣀⣀⣀⣀⣠⠄⠉⠉⠉⠻⣷⣦⣤⡴⠂⠀⠀⠔⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠛⠉⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⠀⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⢹⣄⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⠀⠀⢱⣄⠀⠀⢀⡀⠈⣿⣿⣿⣿⣄⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣦⣤⣤⣤⣤⣤⠤⠄⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⢿⣿⣿⠏⠀⠀⢸⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠈⠋⠁⢀⠠⣄⠈⠉⠙⢛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣾⣷⠙⣇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠀⠀⣠⣿⣿⣿⡆⡏⠠⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠋⣿⣿⣿⣿⣷⣦⣤⣤⣤⣤⣤⣴⣾⣿⣿⣯⣴⣿⣿⣿⣿⣿⠇⠀⠀⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⣰⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠉⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁