Bagels Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠈⠉⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠘⢲⡄⠀⢡
⠀⠀⠀⠀⠀⠀⠀⢀⠄⢒⣀⠐⠒⢄⠀⠀⣸⡅⠀⠐⠀⠜⠃⠀⢸⡀⠊⠉⠉⠉⠢⡀
⠀⠀⠀⠀⠀⠀⡴⠡⣾⡟⢻⣎⣀⣰⠗⠉⡉⢉⡍⠲⣤⠔⡂⠙⣛⢇⠀⠀⠟⠛⡄⠈
⠀⠀⣀⠤⣒⠊⠉⠲⡤⠚⢉⢀⠀⠈⠑⢮⡟⢻⡿⠀⠘⠰⠿⠟⠈⠀⡂⠀⠀⠀⠁⠀⡅
⢠⢞⣵⣿⡟⠀⢀⢰⡆⠀⠛⠛⠀⠀⠀⢸⠈⠁⠀⢀⣼⡠⠤⠤⢀⣠⡟⠢⠀⠀⠠
⡏⠈⠉⠉⠀⢄⣠⠾⣄⣀⣀⢀⢀⣀⠴⠿⢿⣤⡶⠋⠀⠀⠀⠀⠀⠈⠣⡀
⠙⠢⠤⠤⠒⣋⣀⣀⣠⣤⠭⠍⠀⠀⠀⠂⠀⠤⢤⣄⡤⠤⠤⠤⢤⡀⠀⠙⠖⠀⠀⠀⠀⠐⢒⣄⡀
⠀⢀⣠⡊⠽⠯⢽⣧⣤⡄⠉⠙⠢⡀⠀⢈⢝⣿⣱⡦⡤⠶⠂⠀⢘⣼⠀⠀⠀⠲⣆⡀⠀⡀⠈⠁⣀⣦⡀
⢰⠋⠙⡝⠶⠤⡤⣄⣀⣠⣤⠤⢶⠇⠀⠸⡮⠥⠤⠤⠤⠤⠖⠚⢉⠼⣧⠀⠀⠀⠀⠑⠂⠀⠉⠢⡄⣀⡀
⠐⡀⠀⠈⠑⠠⠡⢤⣤⡤⠤⠤⠊⠀⠀⠀⠐⢢⣤⠤⠤⠴⠖⠊⠁⡠⡆⠀⡀⠤⠤⠄⢀⠀⠀⠀⠈⢂⠑⠠
⠀⠀⠑⠂⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠁⠀⠈⠺⠄⠀⠀⠠⠤⠾⠷⠄⠀⠀⡦⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⢉⣁⡀⡀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣀⠀⠈⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⣉⣁⣀⡀⢀⡀⠄⠀⣀⣀⠠⣄⣯⠅⠀⠀⠀⠀⠀⠤⠠⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠚⠛⠋⠉⠉⠛⠒⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢶⠛⠋⠛⠒⠲⢤⣀
⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⢹⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⣼⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢯⡳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⡿⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠳⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠈⠓⠤⠤⠆⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸
⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠠⣖
⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡴⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⢀
⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠰⠿
⣠⢤⠀⠀⠀⠀⠈⠳⢤⣀⡀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁⠀⠀⠀⢀⡤⣄⠀⠉⠓⠲⠤⢤⣤⣀⣀⣀⣤⠤⠴⠒⠉
⠉⠋⢀⣖⣦⠀⣤⣄⠀⠀⠉⠙⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠋⠀⠀⠀⢰⣶⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⢠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⡼⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⡞⠁⠉⡦⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⢦⡀
⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⢸⠀⠀⢀⡀⠀⠀⠀⠀⠀⢰⠋⠈⣹⠀⠀⠀⠈⠉⣠⠴⠦⡀
⠀⠀⠀⠀⠀⠈⠣⣀⣠⠟⠒⠉⠀⠀⣏⣹⠆⠀⠀⠀⠀⠈⢧⣀⡗⠀⠀⠀⡔⠉⠃⠀⠀⢸⡄
⡴⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠰⡖⠋
⠓⠞⠁⠀⠀⠀⠀⡴⠒⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣨⡇⠀⠀⠀⠗⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠔⠚⠛⠉⠉⠉⠉⠁⠉⠓⠓⠒⠲⢤⣄
⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠋⠁⠀⠀⠀⠀⣤⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢀⡀⠀⠀⡸⠀⠀⠀⠀⠀⠀⢀⣤⣤⠴⠛⠉⠁⠀⠀⠄⠀⢀⣀⠀⠀⠁⢀⢄⠀⠀⠀⠀⠀⠶⠀⠀⠀⠠⠄⠀⠀⠀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⣸⠛⠒⠚⠁⠀⠀⠀⣀⡴⠞⠋⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⣠⡄⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢀⡞⠁⠀⢠⡀⠀⠀⠀⠀⠀⠀⠙⠁⠠⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠋⠁⠀⠀⠘⠁⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠉⠀⢠⡄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢐⡳⠀⠀⠈⠁⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⢀⣀⣀⣀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠉⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⢀⣄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣤⠴⠖⠛⠉⠉⢀⡼⠁⠀⠀⠀⡶⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⡀⠀⠀⠘⠃⠀⠀⠠⠀⠀⠀⠀⠈⠋⠀⢀⡴⠋⠀⠀⢀⣀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠶⠶⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⣾⠀⠀⠀⠹⠃⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠪⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⡀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠉⠁⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁
⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠦⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠤⠤⠶⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⡀⠀⠀⠠⡶⠶⠆
⠀⠀⠀⠀⡆⠉⠲⢤⣀⣀⣀⣀⠀⠀⡂⠀⢡⢀
⠀⠀⠀⠀⢢⠐⠀⠀⠛⠛⢻⠁⢀⠄⠀⠀⠐⡆⠈⡠⠤⠒⠒⡈⠡⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠂⠠⠤⠔⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠀⢀⣀⡤⡊⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠄⠒⠂⠒⠠⢄⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠈⠭⠀⢀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⣀⠀⠀⠀⠱⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⣀⡠⠀⠷⡀⠀⠟
⠀⠀⠀⠀⠀⡞⠀⠀⢰⣭⣿⣷⡄⠀⠀⠀⠀⠀⠁⠀⠀⢀⡤⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⡜⠁⠒⢂⠔
⠀⠀⠀⠀⠀⠂⠀⠀⠈⠁⠀⠝⠃⠀⠀⡄⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠒⠁
⠀⠀⠀⠀⠀⢀⠠⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠄⠊⠉⠀⠀⠀⠀⠀⠈⠑⠀⠠⠀⠒⠋⠀⠀⠀⠀⠀⠀⢀⠔⠐⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢀⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠂⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠲⠄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠢⠤⠀⣀⡀⠀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠤⠤⠂⠉⠀⠀⠀⢁⡀⠘⢲⡤⢠⡀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠛⠀⡀⢄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠀⠀⠀⠀⠀⠀⡄⠀⠀⣴⠾⣦⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢿⠀⠀⠉⠢⠈⠀⠀⠀⠇⠀⠀⠀⠠⣄⢠⣄⢠⡤⠄⠰⠶⠶⠶
⠀⠀⠀⠀⠀⠈⢦⣀⡀⠀⠀⠀⣀⡤⠂⠀⠀⠈⠳⣄⣀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠤⠄⠀⠀⠀⠤⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠁⢀⡠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠈⠓⠤⠃⠀⠀⠀⠀⠀⡤⠤⢄⠀⠀⠀⠀⠀⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠊⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠁⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡆⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡰⠂⡆⠀⠀⠘⠤⠂⠀⠀⠀⠀⡆⠑⡄⠀⠀⠀⠀⠈⠓⠁⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⢁⣠⠄⠒⠒⠒⠠⣄⡀⠀⠈⠃⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠰⠦⡀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠉⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⢄⠀⠀⠀⠘⠆⠜⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⡠⠐⡂⠀⠀⡤⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣇⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠣⠔⠁⠀⠀⡀⣱⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⡉⠑⡄⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡔⠃⡿⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢧⠈⡆⠀⠀⠀⠈⠋⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠈⠋⠀⠀⠀⠀⠀⢀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⡹⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠰⢇⠼⠃⠀⠀⠀⢀⣀⣀⠈⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⢀⢠⡄⠀⠀⠉⠁⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠇⠀⠀⠀⢰⠂⡆⠀⠀⠀⠀⠀⢅⠼⠁⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠠⢖⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠁⠁⠀⠀⠀⠀⢠⠖⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠰⣅⢈⠳⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠀⠘⠖⠁⠀⠀⠀⠀⢪⣠⠜⠁⠀⠀⠀⠀⢀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠂⠤⠤⢀⣀⣀⣀⣀⣀⣀⣀⡀⠤⠄⠒⠈⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠄⢶⠤⠐⠂⠈⠛⣙⣙⠛⠛⠛⠋⢀⣠⡄⠶⠦⠠⢤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢊⣉⠀⠀⠀⠱⠤⠋⠀⠀⠀⠓⠼⠃⠀⠀⠀⠛⠴⠃⠀⠀⡴⢲⠄⠉⠉⣢⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠈⢆⡹⠀⠀⠀⡔⠂⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠒⡄⠉⠁⠀⠀⠪⠤⠜⢀⡉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢸⠉⡆⠀⣀⠀⠀⢀⠄⢄⠑⠠⠤⠄⠀⣀⣀⣀⣀⣀⠀⠤⠤⠤⠒⠃⣀⠔⢲⡄⠀⠀⠀⠠⠌⣣⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠛⠀⠸⣁⢳⡀⠀⠑⠒⠃⠀⠀⠀⡠⠒⡄⠀⠀⠀⢠⠤⠤⡀⠀⠀⠈⠉⠉⠀⠀⢴⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢰⢚⡇⠀⠐⠳⠞⠁⠀⠀⠀⠈⠓⠶⠃⠀⠀⢛⡑⡄⠀⠀⠀⠉⠋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠈⠉⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣟⣽⣯⣽⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠉⠙⠟⠒⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠀⠤⢄⣀⡴⠉⠀⠀⠇⣀⠐⠒⣀⠉⡅⢀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡖⠁⠲⠀⠰⠃⠘⠁⡜⠀⠘⠄⠲⠈⠁⢠⡄⠉⠀⠁⠘⢀⠸⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠠⠜⢠⠆⠀⠶⠈⠃⣴⠁⠀⠀⠴⠀⠘⠄⠈⠀⠀⠀⠀⠀⠸⡍⠐⠹⠌⢣⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠛⠋⣆⠀⠀⠀⠀⠀⠠⠆⢸⠀⠀⣠⠄⢲⠀⠐⠒⠀⡀⠀⠀⠀⠀⠀⠈⠁⠾⢋⠱⡀
⠀⠀⠀⠀⠀⠀⢰⣟⠐⢃⠀⠀⠀⠀⠀⡀⠀⠴⠘⡀⠀⠀⠴⠂⠀⡆⠀⡀⢙⢂⠀⠀⠀⠀⠸⠀⠓⠈⠁⢳
⠀⠀⠀⠀⠀⠀⠸⢉⡆⠘⠀⠀⠀⢀⢪⠁⠀⢠⡀⢃⠀⠀⠀⠠⠖⢀⡄⠉⠀⠠⡆⠀⠀⠀⠀⠀⡴⠆⢠⠈⡆
⠀⠀⠀⠀⠀⠀⡶⠀⢰⡄⠀⠀⠀⢸⠆⠀⢀⡀⣁⠈⠆⠀⠀⠀⡤⠈⣠⠄⠒⠂⠁⠀⠀⠀⢀⡀⠤⠀⠆⠀⡇
⠀⠀⠀⠀⠀⠀⢃⠛⠀⠀⠀⠀⠐⠁⠠⠗⠈⠃⠋⡀⠈⢂⠀⠀⠀⠀⠠⠆⠸⠀⡳⠀⠶⠦⢀⡉⣀⠸⠄⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡆⠙⠀⢀⡄⠀⢠⠄⠸⠃⠙⠀⠃⠀⡼⠳⣄⠀⠀⠀⠀⠀⠘⠃⢰⠀⢴⠈⠁⠘⠀⠀⡸
⠀⠀⠀⠚⠂⠀⡤⠱⡀⠇⢠⠀⢠⡀⠐⡆⠀⠴⠂⢀⠞⠀⠀⠈⠓⢤⡀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠁⢀⡔⠁⠀⠛⠀⠀⠀⠀⣀
⠀⠀⠀⠀⡠⣄⠀⠀⠙⢦⡈⠁⠀⠀⠂⠀⠀⡠⠖⠁⠀⢀⣀⠀⠀⢀⣉⠓⠦⢤⣀⣀⣀⣀⣀⠤⠖⠉⠀⣀⡀⠀⠀⠰⠆⠀⠉
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠑⠒⠒⠒⠊⠉⠀⠀⠀⠀⠀⠁⢠⢤⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠛⠉⠉⠀⠀⠉⠉⠛⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⡀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠙⢿⣦⡀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠹⣿⡄⠀⢻⣷⡀
⠀⠀⠀⠀⠀⣠⣶⡿⠟⠋⠉⠀⠀⠀⠀⢀⣤⣴⣶⣶⣾⣶⣶⣶⣦⣄⡀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⢹⣿⡀⠀⢻⣷⡀
⠀⠀⠀⠀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⡉⠀⠀⠀⠀⠀⠀⢉⣻⣿⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣀⣿⡇⠀⠀⢿⣧
⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣷⣶⣶⣾⠿⠿⠛⠁⠀⠀⠀⠀⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⢿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠘⣿⣏⠛⠿⢿⣶⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⣶⡿⠟⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠉⠉⠀⠀⢀⣴⡿⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⠀⠀⠛⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣦⣤⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣴⣶⡿⠟⠉⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣀⡀⠀⣀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠟⠋⠁

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


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⠶⠾⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⡿⠿⣶⣶⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⡟⣩⣿⣿⣿⡿⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣧⣈⣿⣿⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣤⣶⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠉⣿⣿⣿⣿⣿⣿⡿⠻⣿⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⣾⣿⠟⠛⣿⣿⣿⣿⣿⣦⣈⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡁⠙⣿⣿⣿⣶⣾⣿⣿⣿⣿⡿⠟⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣿⣿⣥⣴⣿⣿⣿⣿⣿⣷⣦⣴⣿⣿⣿⣟⠻⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠋⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣾⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠖⠒⠚⠉⠙⠛⠉⠙⠚⠒⠚⠓⠒⠒⠒⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠉⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣄
⠀⠀⠀⠀⠀⣀⠞⠁⠀⠀⠀⠶⠀⠀⠚⠅⠀⠀⠀⣄⠀⠀⠀⠀⠖⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠈⠓⢦
⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⠂⠀⠀⠰⠄⠀⠀⠈⢇
⠀⢀⡴⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄
⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠦⣀⣤⣤⠴⠶⠶⠦⢄⣀⠤⠄⠀⠀⠀⠀⡀⠀⠘⠓⠀⠀⠀⠀⢨⠆
⢰⠁⠀⠀⠀⠄⠀⢀⠀⠈⠁⠀⠀⢀⣄⠀⠀⠀⠀⠀⠉⠁⠢⠤⠤⠚⠁⠀⠀⠒⠂⢀⠀⠀⠀⠀⣀⠀⠐⠀⠀⠘⣆
⢸⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠀⠀⠀⠀⠈⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢸
⠈⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠐⠗⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⡜⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢴⡂⠀⠀⠀⠀⡤⠀⠀⠀⠈⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⡌⠙
⠓⢤⡈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠎⢀⡴
⠀⢸⠉⠀⠀⠁⠒⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠋⠁⠀⠈⣧
⠀⢸⠀⠓⠤⠤⠠⢄⠀⠀⠈⠉⠒⠒⠀⠤⠤⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠒⠋⠉⠀⠀⠀⠀⢀⡰⠋
⠀⠈⢦⡀⠀⠀⠀⠀⠉⠂⠀⠐⠂⠉⠉⠉⠑⠢⢄⡀⠀⠉⠉⠉⠉⠉⢉⣉⣉⠀⠀⠀⠀⠀⢀⡀⠄⠊⠁⠁⡏
⠀⠀⠀⠑⢤⡀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠈⠀⠀⠈⠑⠂⠀⠐⠒⠉⠁⠀⠀⠈⠉⠉⠈⠉⠁⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠈⠙⠒⠤⣄⣀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠶⠤⠤⣤⣄⣀⣈⣀⣀⣀⣀⣀⣠⡤⠤⠴⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⠤⠖⠒⠚⠉⠛⠛⠋⠙⠒⠓⠒⠒⠒⠒⠒⠲⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⢤⣀
⠀⠀⠀⠀⠀⠀⢀⠖⠋⠀⠀⠀⣤⠀⠀⠰⢒⠀⠀⠀⠀⡠⠀⠀⠀⠀⡔⠀⠀⠀⡀⠀⠈⠀⠈⠀⢠⡀⠀⠈⠳⢤⡀
⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠄⠀⠀⢀⡀⠀⠀⠀⢳⡀
⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠳⣄
⠀⣰⠋⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣖⠀⠀⠀⠀⠀⢳⡀
⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⠈⠀⠀⠀⠀⠀⠀⠑⠒⠻⠯⢉⡉⠉⠉⢉⡶⠊⠉⠀⣀⠀⠀⠀⠆⠀⠀⠀⠀⠀⡀⠀⠀⢾
⢸⠁⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠶⠀⠀⡠⡀⠀⣀⠀⠀⠈⠉⠉⠁⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠴⠆⠀⠀⠀⠀⠀⣧
⠸⡇⠀⠀⠀⠀⠀⠀⠀⢴⠄⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⠀⠀⠀⡀⠀⠀⠑⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠀⠀⠁⠀⠀⠈⠊⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇
⡞⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠎⠃⠀⠀⠀⠀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⠃⠘
⠙⠢⣄⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠁⢀⡴
⠀⢠⡇⡀⠀⠈⠑⠢⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠀⠀⠘⣇
⠀⢸⠀⠑⠤⠄⠀⢄⡀⠀⠀⠉⠑⠒⠂⠤⠤⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠚⠉⠁⠀⠀⠀⠀⢀⡰⠃
⠀⠈⢇⠀⠀⠀⠀⠀⠈⠒⠤⠠⠤⠀⠒⠈⠉⠒⠢⢄⡈⠉⠉⠉⠁⠐⠒⠒⠒⠚⠉⠉⠉⠁⠀⠀⠀⠀⢀⣀⠔⠒⠉⢉⠏
⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠀⠀⠂⠀⠈⠑⠦⠄⠀⠤⠤⠒⠒⠉⠉⠉⠐⠒⠒⠒⠒⠊⠁⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠙⠲⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⠤⣄⣀⣀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠖⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡴⠃
⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⡞⠁⠉⡧⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⢸⠂⠀⠀⡀⠀⠀⠀⠀⠀⢠⡏⠉⢹⠄⠀⠀⠀⠉⢀⠄⠠⣄
⠀⠀⠀⠀⠀⠈⠧⣀⣠⠟⠒⠋⠀⠀⢏⢙⡆⠀⠀⠀⠀⠀⢧⣀⡽⠀⠀⠀⢰⠊⠙⠀⠀⠈⡆
⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⠀⠀⡶⠞⠁
⠳⠞⠃⠀⠀⠀⠀⡴⠒⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⢀⣠⠇
⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠀⠷⠒⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠖⠚⠋⠉⠉⠉⠉⠉⠓⠒⠒⠢⠤⣄
⠀⠀⠀⠀⠀⠘⣅⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠉⠀⠀⠀⠀⣠⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢨⠃⠀⠀⠀⠀⠀⢀⣀⣠⡴⠞⠛⠉⠀⠀⠠⠀⠀⣀⡀⠀⠈⠀⢠⡀⠀⠀⠀⠀⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⣠⠉⠒⠒⠁⠀⠀⠀⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠀⠀⢀⣄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠈⢀⠀⠀⠀⠘⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠘⠃⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⣖⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠰⠖⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠃
⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⢀⣤⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⡤⠴⠒⠛⠉⠉⣠⠋⠀⠀⠀⢰⠆⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⡁⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣰⠚⠁⠀⠀⢀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⠧⠴⠶⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠆⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⣼⠀⠀⠀⠸⠆⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠃⠀⠀⠀⠀⠠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠤⠴⠚⠋⠁
⣶⣶⣤⣶⣤⣦⣴⣶⣶⣶⣶⣶⣶⣤⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣤⣴⣦⣦⣶⣶⣦
⣿⣿⣏⣀⣁⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠉⢀⠆⠀⣾⣶⡤⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠃⠀⠈⠁⠀⠀⢸⣧⣦⡀⠀⡴⠛⢢⡀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠻⣿⡟⠀⢿⣿⠞⠁⢀⣿⢀⠀⡀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠨⠤⠌⠰⢀⢻⠑⠐⠀⠋⣣⠀⣉⠁⠨⠉⢠⠀⠀⣼⡿⠷⣴⣿⣿⡆⢀⠞⡏
⣶⡟⣿⠁⠀⡀⣈⣨⡇⢀⠖⠉⢩⠛⢄⠀⠀⣠⢊⠜⢯⠀⠉⠢⡀⠀⠀⠈⠏⠀⠀⠹⠿⣶⡟⠁⠠⠁
⣿⡇⣿⠠⠀⠄⠠⠛⠀⢨⠀⠰⠿⠖⣺⠶⡾⠁⠸⡀⠈⠓⠀⢀⠇⡆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀
⢙⠃⠉⠀⠀⠀⠀⠀⠀⠀⢱⡤⠤⠴⠁⠀⠀⠀⠀⠵⠦⠤⠴⠎⠀⠀⠀⠁⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⣤⡀⠉⢉⢩⠉
⠉⠀⢠⠁⠀⠀⢀⠀⠀⠈⠀⠀⠈⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣟⣺⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠀⠀⠀⠀⡀⠐⠆⠀⠒⠂⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⡟⣽⡿⢹
⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠉⠐⠻
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠂⠐⠁⠘⠀⠀⠀⠁
⠀⠀⠁⢠⣄
⠀⠀⠀⣚⣻⣷⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢶⡶⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠯⡿⢫⡜⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⢹⠆⠀⠀⠀⠀⠀⠀⠈⠉⠈⠒⠚
⠀⠀⠨⣿⣷⠠⢮⣿⣟⣣⡄⢠⣄⠀⠀⠀⠀⠀⠈⢦⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⣀⠀⠀⣺⣿⣆⠘
⠀⠀⠀⠀⢮⠟⠀⠀⠈⠀⠀⠸⠁⠀⠀⢀⠀⠀⠚⠘⠓⠈⣀⣀⣀⣴⣶⣶⣦⡀⠀⠀⠈⣻⣿⣿⣿⡄⠀⠉⠻⣿⡆
⠀⠀⣀⣀⡈⠀⠀⠀⠀⠀⠀⠀⠀⢄⡄⠀⠐⠀⠀⠀⠀⠀⣿⡥⠽⢛⣫⣭⣥⣶⣦⠀⠘⠛⠛⠋⢻⡧⠀⠀⠀⠈⠙⠛⠋
⢀⣶⣿⣽⣭⡙⠰⣠⠄⠀⠐⠀⢚⠋⠀⡀⠀⠀⠀⢀⣐⠀⠘⣷⠁⠀⢈⣉⡁⠁⠀⠀⠀⠀⠀⠀⠾⠙
⠀⠉⠉⠛⠻⠷⠾⣿⣿⠆⠀⠀⠉⠁⠉⠁⠀⠀⠀⠈⠙⢀⠀⠿⡇⠀⣷⣛⠎⠆
⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⢰⡄⠶⠒⠀⠐⠀⠸⠘⠓⠉⠉⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢿⣾⡟⣷⣶
⡀⢤⡼⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠁
⢉⣷⢬⣿⣶⠶⠶⠿⠿⠿⠿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠘⠳⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣀⣀⡀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⣴⠀⣾⣉⣛⣆⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⣀⣠⣤⡀⠶⣶⣎⠙⢯⣧⣄⠸⣆⠈⢙⡓⠛⣫⡧⠀⠀⠀⠀⣀⠀⠀⠄
⠀⠀⣔⠀⠀⠀⠀⢠⣤⢤⡀⠚⢿⡈⠉⢮⡳⣶⢤⠀⠈⠻⢿⣢⣯⠓⠯⣿⠄⠈⠙⠉⣉⣀⡤⠶⠚⠋⠉⣀⡀⠰
⠀⢸⣉⠂⠀⠀⠀⠀⠘⢿⣿⠵⣤⣽⣶⣀⠑⣿⣦⣤⠿⠆⠀⠈⠁⢀⣀⣤⠤⠶⠚⠋⣉⣁⠀⠤⠐⠒⠉⡥⠄⡂
⢀⢮⣻⣦⠀⠀⠀⠀⠀⠀⠉⢳⣤⡭⠉⠉⠁⢉⣀⣀⣤⠴⠶⠚⠋⢉⣁⣠⣤⡒⠺⡩⠗⣌⡙⣲⠌⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠈⠳⣤⠙⢻⣦⡀⠆⠀⠀⣀⣨⡤⠶⠖⠚⠋⢉⡉⠀⠠⢔⡒⠊⠉⣽⠙⡶⣅⢼⣀⢙⠂⣚⢲⣱⣦⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐
⠀⠀⠈⠻⡠⡙⢿⣷⣔⠢⣌⠀⠀⠆⣒⡃⠥⠌⠀⠀⠀⢀⠙⠢⡀⢸⠆⣉⢛⢀⣉⣝⣾⢳⣫⠽⠟⣛⡁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⣒⠀⣰
⠀⠀⠀⠀⠈⢢⡀⡈⢹⢷⣄⢑⠀⠀⠁⠀⠀⠀⠀⠀⠉⠀⠀⠄⠙⣾⡎⣽⡯⠭⠓⣊⡉⡤⠔⠖⠊⢁⠀⠀⠀⡤⣄⠶⠸⢗⣂⣭⣽⣦⣴⢋⢠
⠀⠀⠀⠀⠀⠀⠈⠪⡀⡁⠙⢗⠦⡀⠄⡂⠄⠠⠀⠀⠀⠀⠀⠀⠀⠈⢤⠤⠖⢛⠁⠁⣀⣄⣤⡤⣧⣴⣦⠀⣬⣯⣶⣾⠿⡟⠻⠋⠋⠉⢁⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢮⠀⠠⠳⡈⠓⣬⣐⢄⠠⡀⢠⠤⠖⠓⡃⣀⢠⣤⣆⣶⢸⣿⣗⣿⣿⣿⣿⠿⠻⠒⠋⠉⠁⠀⠀⠀⠄⠐⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠐⠙⣄⠀⠈⢳⡔⠰⠀⣷⣶⣮⣧⠇⢹⣿⣷⣿⠿⠟⠛⠉⠉⠂⠀⠀⡀⠀⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⡊⡊⠓⠒⠚⠁⠀⣸⣿⡿⠷⠛⠒⠉⠉⠀⢀⠐⠠⠀⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠤⠂⣀⠤⠚⡫⠄⢀⡀⠠⠔⠊⠐⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡬⠥⠴⠶⠶⠶⠶⠶⠶⠶⠶⠤⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠉⢀⣴⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢫⡀⣀⣴⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣀⣀⡸⠣⠴⠒⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠉⠛⠻⠯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠞⣛⠩⠀⠀⣀⣀⣄⣀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠭⠭⠅⠐⠒⠂⠉⠁⠀⠀⠀⢸⠁⠀⠀⠀⠙⡆⢂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢀⡇⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⣠⣴⠟⢁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠾⠟⠛⢉⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠳⡀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⣀⣠⠤⣔⠊⠉
⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⠀⠙⢦⡙⢅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠂⢄⠀⠉⠓⢄
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡦⠤⣀⣀⣀⠀⠀⠀⢀⣀⣀⣤⢴⠖⠉⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⢳
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠢⠤⠬⠭⠭⠭⠭⠴⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒⠀⠀⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⠤⣀⣈⠁⠒⠂⠠⠤⠄⠀⠤⠠⠤⠀⠠⠤⠀⠐⠒⠈⢁⣀⣠⠤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠆⠤⠶⣄⠀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠦⠈⢄⠠⠀⣤⡶⠿⠀⣪⣍⠐⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣱⠄⠩⣠⡀⠀⠈⠉⠓⠾⠛⠂⠀⠒⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠁⠀⠀⠉⠛⠧⡀⠀⠀⠀⠀⠀⢤⣌⣡⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⣀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢐⡇⠀⠀⢰⢤⣴⣴⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠜⠉⠀⠀⠺⢷⣌⠻⢷⣦⣬⣀⠀⠀⠀⢁⢀⠸⠀⠀⠀⠙⠛⣿⣿⢿⢻⣳⠂⣾
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⠓⢹⣿⠛⠛⠛⠛⠒⠂⠉⠹⠀⠀⠀⠀⡀⡙⣿⣅⣾⢱⣼⠇
⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠐⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢐⢰⠰⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠶⠛⠁⠀⠀⠀⢶⣶⡄⡄⣄⢠⡠⡀⣈⢑⣂⣛⠥⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣆⣦⠤⢮⢹⡜⣿⣺⣶⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⡿⠯⠽⠞⠓⣿⠉

⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠒⠠⠄⣀⡔⠉⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠬⢙⡢⣄⡀
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⢠⢶⡀⢀⠀⠀⠈⠉⠛⠦⢄⣀
⠀⠀⠀⠀⠀⢰⠃⠐⠤⣄⡀⠀⠀⠀⠏⠀⢷⠏⡼⢅⠀⠀⡆⢀⡀⠀⠉⠐⠂⠤⢄⣀
⠀⠀⠀⠀⢠⢏⡾⣋⡥⠂⠉⡛⠲⠦⠤⢄⣀⣀⠃⠀⠀⠀⣧⠞⣷⣠⠏⣠⠠⠄⠀⠀⠀⠉⠐⠒⠤⣄
⠀⠀⠀⢠⢏⡾⠚⠁⣠⠴⠊⡡⠔⠀⢈⡆⢀⣀⡉⠑⠒⠤⢄⣀⠛⠁⣀⠈⡆⠀⠀⠀⠀⠀⠀⠀⢀⠞⣱⡆
⠀⠀⢀⢏⣎⡤⠖⠋⣡⠔⢋⠠⠔⢂⠞⠀⠀⣁⠈⠒⠢⢤⣄⣀⠉⠑⠒⠬⢄⣀⠀⠀⠀⠀⠀⢠⢎⣴⠥⣬⠂
⠀⢀⡞⡞⠁⣠⢔⠫⠒⠉⣀⡤⢴⠏⠀⠀⠒⠤⢈⡑⠂⠤⠀⠈⠉⠑⠒⠲⠤⠤⣉⡑⠢⠀⢠⣿⣫⠔⠊⠅
⠀⡞⡼⠁⠑⠊⣁⡤⠖⠋⠁⢀⡞⠀⠀⠈⡑⠲⠤⣀⡉⠐⠀⠀⠄⣀⠒⠠⠄⡀⠀⠉⠓⢠⣿⡿⠃⠀⠀⠀⢨
⣸⣱⠃⠀⠚⠉⢁⡤⠖⣚⣩⡞⠀⠀⠬⣑⠒⠬⢔⡂⠌⠁⠀⠒⠦⢄⣉⠐⠂⠤⠁⠀⣠⣿⣟⠑⠠⠤⠄⣴⡿
⡇⠈⠀⠀⠐⠠⢀⡖⣫⠥⣾⠀⢀⠑⠲⢤⣈⠐⠠⢌⡁⠀⡐⠢⠅⣒⠠⢉⠒⠂⠀⣰⣿⡃⠀⠉⠀⢠⣾⠿⠚
⠱⡀⠉⠑⠒⠂⠤⠬⣁⡀⠁⠐⠠⣉⡑⠢⠤⣉⡒⠢⠀⠢⢄⡉⠒⠤⢉⠑⠂⠀⣸⡿⠡⢉⠓⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠤⣌⣁⠒⠢⠄⠠⣀⠒⠤⢌⡙⠒⠤⣈⠁⠀⡿⣍⡑⠂⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⡀⠀⠀⠈⠉⠓⠒⠤⠤⣉⣒⠢⠬⣉⠐⠀⠀⣼⡉⠒⢁⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠀⠀⡼⢥⡈⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⡼⠉⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⠴⠋

⠀⠀⣀⣀⣤⣤⣤⣶⣶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣶⣤⣤⣤⣤⣀⡀
⠀⣾⣟⡯⠍⠓⠒⠒⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠭⣽⣿⣧
⠀⣿⠈⠉⠻⠓⠒⢶⣶⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢴⣶⣶⣶⡶⠟⠛⠉⢸⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣀⣠⣤⣤⣀⡀
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠟⠋⣉⣉⣉⣙⠻⣷⣄
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣴⠟⠋⠉⠉⠛⣷⡌⢿⣆
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⠃⠀⠀⠀⠀⠀⠘⣿⠸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠃⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿
⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⡟⢸⡿
⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⢀⣼⡟⢠⡿⠃
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⣠⣾⠋⣰⡿⠁
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣠⣴⡾⠋⣡⡾⠋
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⠛⠋⣁⣴⡿⠋
⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⣴⡾⠛⠁
⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠋⠁
⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠈⠻⣶⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⡾⠟⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣽⣭⣬⣽⣭⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠈⠀⠀⠀⡀⠀⠈⠁⠈⠁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠖⠒⠉⢉⣩⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠛⠙⠻⡏⠀⠀⠀⠀⢀⣀⣀⠀⠀⠓⢢⣄⠀⠈⠑⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣀⡀⠀⠀⠉⠲⢤⡀⠈⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡇⢀⡀⠁⠉⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠚⠉⠁⠀⠀⠀⠈⠙⠲⠶⠶⠶⠶⠶⠶⠛⠛⠉⠉⠀⠀⣀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠸⠇
⠀⠀⠀⠀⠘⠿⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡐⠲⠃⡠⠀⠀⠀⢹⠇
⠀⠀⣦⣄⠀⠀⠀⠉⠙⠛⠫⠧⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠐⠀⠀⠀⣀⠤⠚⠀⣀⢸⠀⠀⠸⠗⠀⠀⠀⣾
⠀⢄⡉⠛⠋⠶⠴⣦⠀⣿⠒⠒⠞⠋⠀⠀⠀⣀⣀⡀⠉⠂⠀⠀⠀⡀⠤⠒⠉⠀⠀⠀⡜⠉⠀⠳⣴⠁⠀⠀⠀⡔⠁
⠀⠈⠛⠷⠶⢠⣤⣄⠠⠏⠀⠀⠀⠀⠀⠒⠂⠀⠀⢁⣀⠠⠔⠂⠀⠀⠀⠀⠀⠀⠀⣠⣧⠀⢀⡤⠈⠀⠀⠐⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣁⣀⠀⠀⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢹⣆⠻⠀⠀⠀⣀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠋⠉⠀⠀⠀⢀⠀⠀⠀⠐⠒⠤⠄⡀⠀⠀⠀⠀⢡⣾⡏⠁⠀⠀⡹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⣀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠘⠓⠒⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⡀⠀⠀⢻⣿⣿⣿⠷⢤⣀⠀⠀⢀⠀⠀⢚⡄⠀⠀⠀⠀⠀⣀⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠉⠛⠿⣶⣬⡟⠃⣀⡠⠤⠟⠀⠀⠀⡠⠀⢌⡁⠀⠀⠀⠘⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠁⠀⠀⠀⠀⠀⢰⠀⠀⠠⠛⢛⡶⢶⡊⠀⢹⡄
⠀⠀⠘⢦⣀⣀⣀⡀⠀⢀⣀⣀⣠⡴⠃⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣠⠀⠀⠀⠘⡏⠀⠀⣸⠁
⠀⠀⠀⠀⠉⠙⠛⠓⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠤⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠹⢄⣄⣀⣤⡑⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠖⠒⠒⠒⠒⠒⠒⠒⠒⠲⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣚⡭⠵⠖⠒⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠓⠲⠤⣍⣓⠲⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠀⠀⢀⣀⣠⣤⠤⠤⠶⠒⠒⠒⠒⠒⠒⠶⠤⢤⣄⡀⠀⠉⠛⢦⡙⢦
⠀⠀⠀⠀⠀⢠⡏⡏⠀⢀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢤⡀⠀⠀⠀⠉⠙⠲⣄⠀⣿⢸⡆
⠀⠀⠀⠀⠀⢸⣇⢣⡞⠉⠀⠀⠀⠀⠀⢀⡤⠴⠒⠒⠒⠋⠀⡤⠤⡄⢹⡀⠀⠀⠀⠀⠀⢀⡿⣡⢾⡇⣀⣤⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⢸⡈⠳⣌⡳⢤⣀⡀⠀⠀⠸⡄⠀⠀⢰⠖⠒⠚⠁⠀⢹⢸⡇⠀⢀⣠⡴⢞⣡⠞⣡⠞⠋⠁⠀⣀⣀⣀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠉⠓⠮⣍⣛⠒⠶⠿⠦⢤⣬⣷⣤⣤⣤⠤⠿⠾⠒⢋⣩⡤⠖⠉⠀⠀⣧⣀⡤⠖⠋⠁⠀⠉⠙⣦⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠶⠶⠶⠶⠶⠶⠶⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢧⣄⣀⣀⣀⣀⡤⠞⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣨⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡀⠀⠈⠉⠉⠁⣀⡠⠔⠋
⠀⠀⠀⠀⠀⣠⠞⠉⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠻⣍⠛⠋⠉⠉⠁
⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⢸⡷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡏⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠙⢦⣌⡙⠲⠦⢤⣄⣀⣀⣀⣀⣀⣀⡤⠤⠖⠚⣋⡥⠞⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⢻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠤⠤⠤⠤⠤⠤⠴⠶⠒⠋⠉⠀⠀⠀⠀⢀⣠⣾⠟
⠀⠀⠀⠀⠀⠙⠪⣝⡲⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢴⣚⡭⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠺⠭⣍⣓⣒⠲⠦⠤⠤⠤⠤⠤⠤⠤⠴⣖⣒⣻⠭⠷⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠊⠘⠢⢤⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢆⠀⠳⣄⠀⠁⠀⡜⠈⠀⠀⠖⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠘⢦⠀⣹⠀⠀⠀⣧⠀⢠⡞⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⢧⣄⣿⣿⠀⠠⡀⠈⢠⢠⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠉⠻⣆⠀⢧⠀⠀⠸⠷⠎⠁⠀⠀⢀⡠⠄⠐
⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⢈⣄⠀⠀⣧⠀⠀⣀⡴⢫⠖⠀⠠⠂⠀⡄
⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢆⠟⠙⠒⠛⠒⠀⠀⠰⠃⢸⡠⠂⢀⡜
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠣⣄⣀⡀⠀⠋⢀⣠⡶⢚⣠⠴⠋⢀⠀⠈
⠀⠀⠀⢀⠇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⣀⠀⣯⠉⠉⠉⠑⠊⢉⣁⠔⢰
⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⠉⠙⠻⢯⣀⢀⠀⠈⠡⡴⠃
⠀⠀⢸⠁⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠯⠢
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠦⠤⠤⠤⠄⠤
⠀⢸⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣼⠃⠀⠀⠀⠀⠀⢀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡤⠞⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉
⣿⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠊⠉
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠒⠉⠉
⣀⣈⣳⣦⣤⣤⣤⣴⣔⣒⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡀⢀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀
⠀⠀⠀⠀⠠⣿⣠⡿⠛⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⢨⣷
⠀⠀⠰⠞⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⣠⡿⣤⣅⣀⠀⠀⠀⠀⠀⠀⠀⡟⢹⣿⠟⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠠⡀⠀⠀⣠
⠀⠀⠈⠀⠈⠀⠈⠁⠂⠀⢀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢼⠷⡀⢠⡋⠀⣀⠀⠀⠀⠸⣦⡈⢹⡟⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⢠⠊⠀⠀⠀⠒⠀⠈⣧⣤⡄
⠀⠐⠒⢀⠄⠀⠀⠈⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠚⠒⠚⠿⣿⠟⠀⠀⠀⠀⠄⠉⢹⣿⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⢋⠐⠄⠀⠄⠀⠀⠀⠻⣦⡶
⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⡀⠀⠀⠂⠀⠀⠀⡘⠁⠀⠀⠀⠁⠀⠀⠀⠰⠀⠘⠧
⠀⠀⠀⠀⠢⢍⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠒⠒⠀⠴⣶⠁⠀⠂⠀⠘
⠀⠀⠀⠀⠀⠀⠈⠒⠄⠀⠀⠀⠀⠀⠃⠒⠀⠁⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠁⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠉⢃⠼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠘⠋
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠓⢂⣠⣀⡀⠻⠆⠛⠻⠀⣵⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠲⢆⣀⣀⠚⠃⠀⠀⠀⠉⠉⠛⠛⢣⡞⡲⠞⣀
⠀⠀⠀⠀⣀⣻⣀⣤⠀⠀⠀⣀⣀⣀⡀⣠⡟⠉⠀⠀⠀⠀⠀⠀⠱⣀⠀⠍⠀⠀⠀⠀⠀⠀⣤⡼⠗⠛⠈⠉⠁⠈⠀⢢⠀⠂⠘⠀⠀⠀⠀⠷
⢀⣀⢀⡀⠿⠋⠀⠀⠀⠀⠀⠀⠉⠠⠶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⡀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠈⠂
⠀⣈⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠈⠀⢀⡄⠀⠀⠀⠘⠀⠀⠀⠄
⢞⣸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠠⠄⠀⠀⢠⢈⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⡤⠐⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠆⠀⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠈⠁⠉⠁⠀⠀⠀⠀⣠⡶⣄
⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠺⣧⣴⢾⡇⠀⣆⠀⠀⠀⠀⢀⠀⠀⠀⢀⣿⠀⢀⡀
⠀⠀⠀⠐⠁⠀⠀⠀⠒⠀⠈⠀⠈⠀⠀⠀⠀⣠⣄⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡀⠀⠀⢹⡎⣀⡀⠹⣷⣶⣄⢶⡿⢤⢀⣠⣾⠇⠔⠋
⠀⠀⠀⠁⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⠀⣸⠇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠻⠛⣾⣀⢉⡥⠟⣥⠴⡾⠋⠁⠀⠄⡀⠈⠉⠉⠙⠛
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠑⡆⢐⣦⠾⠛⠀⠀⠀⠈⠀⡀⠀⠀⠈⡀⠀⠀⠀⠀⡀⠀⠉⠋⠉⠈⡓⠒⣥⡀⠁⢁⡀⠂⠀⠄
⠀⠀⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠂⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠰⠀⠀⠀⠈⠷⠲⡀⠀⠁⠀⠂
⠀⠀⠀⠀⠦⠀⠉⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠒⠀⠐⠀⠀⠠⠀⠀⠀⠀⢈⠁⠀⠀⡁⠂⠄
⠀⠀⠀⠉⠲⠤⠀⠀⠀⠀⢀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠒⠁⢠⠀⠀⠂⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⢄⡄⠀⡀⠀⠀⢀⡠⠄⠒⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠄⠉⠀⠀⠀⠀⠀⠀⠈⠃⠄
⠀⠀⠀⠀⠀⠐⠄⠀⠣⠀⠀⠀⠀⠠⣤⡄⠀⠄⢠⣄⢿⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⣠⠤⠀⠂⠒⠟⠀⠈⠁⠉⠺⠋⠀⠀⣀⠀⢿⣮⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢀⠤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠈⡉⢉⠀⠄⡝⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣤⡀⢾⡳⢀⠀⠀⠀⠀⡀⡀
⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠀⠀⢀⡀⢈⠀⠀⡀⢚⢻⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠐⠈⠉⠀⠀⠉⠉⠓⠈⢣⡀⣸⡊⠁⢈
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠂⠐⠀⠀⠠⠀⠂⠄⠰⠋⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡼⠋⠠⠐⠀⢀⠁⠀⡀
⠀⠀⠘⡀⠀⠀⠄⠀⡀⠄⢰⠁⠂⠐⠀⠂⠀⠁⠂⠐⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠴⠀⠀⠀⠒⣈
⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⡄⠈⠁⠀⠉⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⡀⢈⠐⠀⠀⠀⠠⠀⡂⢀⡁⢀⠀⠠
⠀⠀⠀⠀⠀⠒⠀⠀⠐⠋⠁⢄⠉⠀⠀⠀⠁⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠑⢀⠖⠀⠆⢠⠐⠀⠄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠒⠐⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢠⡏⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣼⠁⠀⡾⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣰⠇⠀⣸⠃⠀⢠⠏⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢠⡟⠀⢰⠇⠀⢠⠏⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠖⣾⠀⢸⠃⠀⡿⠀⢠⡟⠀⢀⡏⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣛⣭⠴⠒⠀⠀⡏⠀⣿⠀⢸⡇⠀⢸⠁⠀⣼⠀⠀⠠⣌⣙⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢾⣿⣶⣞⣋⣥⣤⣿⣤⣿⢠⣼⡇⣤⣼⢠⣤⣿⣀⡀⠒⠦⣭⣛⠆⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⠈⢉⣠⠟⢑⣶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣍⠓⠂⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉⣉⣀⣤⡴⠾⣛⣵⡆⣚⣛⠧⣄⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣝⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⢋⣉⣉⣭⣴⡶⠿⣛⣭⠀⠉⠉⠙⣾⡇⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠃⢸⣟⡳⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣉⣉⣉⣥⣴⡶⠿⣛⡅⠈⠑⣂⡴⣻⡣⢠⡏
⠀⠀⠀⠀⠀⢀⣤⠞⣋⡥⢒⡠⠄⠀⠀⢻⣟⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣉⣉⣉⣩⣭⣴⠶⢟⡋⢡⣞⣛⣽⣾⢟⣵⠟⢶⣄
⠀⠀⠀⠀⣰⠟⣡⡞⢡⡞⢉⡴⠊⡀⠀⠈⢿⣟⠒⠄⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣉⣉⣭⣭⣴⡶⢿⡛⢡⡿⢿⣛⣭⡶⠛⠁⣀⢢⡈⢷⡄
⠀⠀⠀⠀⣿⠀⣿⠀⣿⠀⣿⡀⢾⡀⠀⠀⠀⠻⣯⡒⠀⠀⠀⠀⠀⠀⠀⠈⢉⣉⣭⣥⣤⠴⠶⣞⡛⣱⣿⣿⣿⣿⣿⢿⡌⢳⡈⣆⢳⠈⡇
⠀⠀⠀⠀⠻⣆⠹⣧⣌⠳⢮⣛⠶⣬⣁⠀⠀⠀⠈⠻⣥⣄⣀⠀⠀⠀⠀⠀⠤⠤⠶⢶⣶⣾⣿⣿⣿⣿⣿⢸⣿⡜⣿⢎⡇⢸⠇⡏⢀⡼⠃
⠀⠀⠀⠀⠀⠈⣳⣮⣙⠷⣦⣍⣙⠓⠦⠄⠀⠀⠀⠀⠈⠙⠻⢯⣿⣛⣛⣛⣛⣛⣯⣽⡿⠿⢛⣫⣵⡿⢋⣾⠟⣰⠟⠘⠁⢋⣠⣶⣿⠤
⠀⠀⠀⠀⠀⠀⠀⠨⠽⠿⠶⢮⣭⣙⣓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠽⠷⠶⠶⠶⣚⣛⣋⣭⡵⠾⠛⢁⣈⣡⣤⣶⣾⣿⣿⠿⠭⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣉⣉⣹⣿⣿⣷⣶⣶⣶⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣭⣭⣭⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣛⣉⣉⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠶⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⣛⣛⣛⣛⣛⣛⣛⣛⣻⣟⣛⣛⣛⣛⠛⠛⠛⠛⠛⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢉⣉⣩⣭⣭⣭⣭⣭⣭⣉⣉⡉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀⣲⣴⠶⣶⣴⡂⣰⡇
⠀⠀⠀⠀⠀⠀⣠⠀⣾⣶⣄⡠⣺⣤⣴⣒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠷⢽⣿⣿⢖⣾⣿⢿⣿⣇⢰⡄
⠀⠀⠀⠀⠀⠀⢹⡀⣿⣿⣿⣿⣿⣾⣿⠟⠛⢲⣄⢸⣞⢒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⡛⡉⠰⢣⡿⣵⣟⡿⣟⣦⣿⣿⣿⣀⣤
⠀⠀⠀⠀⠙⢿⣾⣷⣹⣿⣾⣿⠿⣯⣦⣛⣟⣾⣿⣮⣅⣷⢁⣘⠀⠀⠀⠀⠀⠀⠀⣴⣿⣏⡉⢈⡋⣀⣟⣡⣿⣿⡥⣿⣿⣿⣿⣿⣟⡁
⠀⠀⠀⣄⣀⣈⣙⢿⣿⣿⣿⣯⣸⣿⣿⣿⣿⣿⣧⣿⣿⠞⠃⢬⢉⡀⠀⠀⠀⠀⡘⡿⢶⢈⢁⣍⣠⣿⣷⣿⣿⣿⣦⣬⣹⣿⣿⣿⣿⢷⠶⠆
⠀⠀⠀⢸⣯⣿⣿⣿⣿⣿⣶⢠⣏⣿⣿⣿⣿⣿⣿⣼⣿⡌⠷⠠⡄⣌⡀⡀⠀⢘⠋⠆⡦⢈⢈⣇⠈⡻⣿⣿⣿⣿⣻⢛⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠈⢙⣿⣿⣿⣿⣟⢁⡰⣿⣿⣿⣿⣿⣿⣿⣧⣿⡖⣷⠴⡆⢤⣉⡀⠀⠈⣭⣖⢰⠆⢰⡇⢘⣣⣿⡗⣺⣿⣁⡦⣿⢽⠟⢿⠀⠁
⠀⠀⠀⣴⡿⣿⣿⣿⣷⡍⠈⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⣷⡲⠤⣤⠀⠀⠈⠙⢿⣷⣿⣷⣯⣽⣿⣿⣿⠿⣿⣿⣣⣿⡿⠃
⠀⠀⠀⠀⣀⣿⡟⢉⡀⢦⠙⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡷⢶⠆⡀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠁⠐⡎⠃⠸⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣶⣠
⠀⠀⠀⠀⠀⠀⠀⠳⠘⣿⡙⠿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠌⠿⣄⣙⣢⣙⣩⡭⢿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠤⠐⠒⠒⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠔⠒⠊⠉⠉⢉⣀⣀⠀⠠⠄⠀⠀⢂⠀⢇
⠀⠀⠀⠀⠀⡤⠄⠒⠈⠉⢁⣀⠠⠤⠄⣒⡒⢈⠭⣄⢰⢒⣤⠉⠀⠀⠀⠀⡀⠸⡄
⠀⠀⠀⠀⠀⢡⡀⠰⡊⠅⠀⢐⠌⡵⢅⢠⡝⠋⢋⠁⠙⣾⣹⠀⠀⠀⠀⡀⣅⠀⢧
⠀⠀⠀⠀⠀⠈⡧⠀⢀⠀⠀⠀⣰⠯⠌⠈⠙⠀⢋⣉⠠⠤⢐⣒⣤⢄⠀⠀⠫⠀⠘⡄
⠀⠀⠀⠀⠀⠀⢱⠀⠀⣀⠀⠀⠄⠀⣒⡂⠭⠅⡲⡒⠟⠋⠉⠁⠀⡀⠀⠀⠀⠁⠀⢳
⠀⠀⠀⠀⠀⠀⠘⡆⠀⢐⢶⣤⡘⡖⠁⠞⠃⠃⠀⢇⠀⠀⠀⠂⠊⠀⠐⠂⡀⡸⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⢳⠀⠈⠄⠃⠀⠀⠀⠀⡀⠀⠉⠀⠀⠄⣀⡤⠖⠒⠛⣉⡠⠀⢄⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢠⠀⠂⠈⠄⠀⢀⣂⠤⠤⡇⢠⠑⣒⠂⠍⠀⠠⠦⠐⠀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢦⠔⠒⡉⢋⣁⡒⡟⢿⣷⠈⡄⠀⠀⠀⠀⠀⠀⠈⠀⢡⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠸⠀⠂⠂⠈⠛⠓⠛⠉⠀⡆⢠⠈⠡⠀⠀⠀⠀⠀⠀⠘⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⡆⠒⠒⡈⠤⠀⢀⠢⠌⢤⠈⠀⠀⠀⠀⠀⠀⡐⣀⣠⡇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠩⠐⠀⠀⠡⠁⠀⠀⠀⠀⢃⠵⠶⣶⠚⡻⠋⠉⢁⠄⢡⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠄⠰⠠⠀⠀⢀⡀⠸⡀⠀⢹⠎⠠⠖⣀⠀⠀⠨⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠆⠀⠀⢀⡄⢧⢴⣓⠂⠬⠽⠿⡗⢶⡾⢟⣦⣄⡠⠠⠃⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠚⠛⠉⠉⠉⣰⣴⡀⠤⠤⠐⣑⣂⣉⣭⣥⣴⣴⠶⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠒⠂⣉⣉⣭⣤⣴⡶⠾⠟⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠶⠟⠛⠛⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠛⠋⠙⠛⠛⠛⠛⠻⠿⢿⣿⡶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡭⠶⠶⠶⠶⠶⠶⠦⣤⡀⠀⠀⠀⠀⠀⠉⠈⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠁⠉⠁⠀⠀⠀⠛⠛⠛⠲⠶⠶⣤⣤⣀⣀⠢⣄⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⢋⣁⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠁⠀⠉⠁⠀⠐⠒⠛⠲⠶⢦⣤⣈⣉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⢼⣶⣄⡀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠚⠉⣭⣉⣙⠛⠲⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣤⠾⠋
⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⠶⣬⣍⣛⡒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠉
⠀⠀⠀⠀⠰⠋⠠⠖⠚⠲⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⠶⣶⡄⠀⠀⠀⠀⣀⡴⠋⠁
⠀⠀⠀⢀⠀⠀⠀⠘⠓⠶⣤⣀⠈⠙⠳⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⣤⣄⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉
⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠷⠀⢀⣠⡶⠛⠁
⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣀⣀⣀⣀⣀⣤⡴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢀⡀⢀⠱⢈⣂⠀⠉⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⡄⡍⢤⣑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⠣⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢠⠙⠒⠀⣡⠬⢁⡃⠀⠀⠀⠀⠀⠀⠀⠀⡆⠁⠹⣁⣀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⡀⠍⠀⡔⢀⠒⠀⢀⠄⠀⠀⠀⠀⠀⠀⠘⢂⡖⠽⠞⠛⣶⡀⠀⠤⣤⠄⠠⢤⣤⣉
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠂⠀⠀⠐⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⡑⠶⢦⣲⠟⣂⠄⡂⠡⢆⠋⡰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠭⠌⠍⠀⠌⠑⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⢐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠄
⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠰⠂⡋⠰⠄⣀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⢮⡏⠉⠀⠀⠐⢦⡀⠈⠑⠠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⢼⠶⣄⡒⠤⠖⢾⠶⡀⠀⠀⠃⠀⢡
⠀⠀⠀⠀⠀⠀⠘⠃⠲⣤⡀⣠⡤⢪⠀⠀⠀⠀⡃⡼⡍⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢬⠻⣉⣉⡸⠃⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠗⠃⠒⠒⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠳⠀⠁⢉⢲⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠤⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢎⣀⣀⣀⣠⢠⣤⣴⢭⣧⣠⣤⡤⠶⢶⢞⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡠⠶
⠀⠀⠀⠀⠀⠀⠀⢂⢉⣽⠛⠑⠂⢠⣼⣧⣤⠂⠀⠀⢨⢓⠆⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠉⠃⠄⠀⠤⠄⠋⠀⠈⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢂⠄⠂⠐⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠉⢛⡛⠛⠻⠶⠶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⠁⣤⢴⣀⣀⣠⣤⠤⢤⣄⠀⠈⠉⠛⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⢉⣠⣀⠀⠀⠀⠳⣌⠉⠀⢠⡠⠤⣫⠴⢋⡠⠤⣄⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠋⠡⣤⣄⣀⡀⠉⠉⢿⡒⠲⠾⣦⠀⣀⡴⠋⡡⠖⠁⠀⠠⣦⣝⢦⡀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠉⢀⠤⠚⠉⢀⣄⠀⠈⠉⠙⠓⠲⠷⠀⣀⡴⠞⠁⠴⣋⢠⣤⡰⢶⣝⢷⣝⣷⡙⣄⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠿⠛⠉⢀⡀⠀⠙⠻⢶⣴⡊⠁⠀⠀⠀⢀⣤⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠉⠺⣟⢷⡝⣧⡻⢎⡵⠋⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠛⠉⠀⠀⠀⠀⠉⠙⠢⣄⠀⠀⠉⠻⢶⣤⡤⠔⢉⣠⠾⠋⡀⠔⠀⠂⠀⠀⠀⠀⠀⠀⠈⠳⡻⣮⠕⡫⢀⠄⡠⢸⡇
⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠁⠀⠀⠒⠿⣟⠻⠿⠶⠶⠶⠶⣿⣦⣄⡀⠈⠉⣠⡶⠛⢁⠔⡨⢀⠅⡂⢔⠠⢀⠀⡀⢀⠀⡀⠀⠘⡡⢊⠔⡡⠊⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⣄⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⠁⣠⡴⠟⢉⡠⡊⢔⠡⡂⠔⡨⢐⠥⡊⢔⠨⢂⠄⡠⢀⠄⠀⠀⠕⠀⢀⠴⢃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠐⣶⣾⠿⠿⣦⣄⡀⠀⠀⠀⠙⡷⠂⠀⣀⣴⠟⠋⡠⢐⠅⡪⢒⠅⡪⢔⡥⣊⢕⡪⢖⠥⣊⠕⣊⠔⡡⢊⠄⠀⢀⡴⠋⣰⠟⠁
⠀⠀⠀⠀⠀⢤⣤⣿⢿⣉⠓⢦⣀⡉⠻⢷⣦⣄⡀⠀⠀⠚⠋⠁⠔⡪⡂⢕⠥⡊⢔⡩⣒⠕⡪⢔⡩⢒⡥⢊⡥⢊⠴⢋⠔⣡⠂⡀⠈⣠⡾⠃
⠀⠀⠀⠴⣿⣟⠳⠮⣅⡈⠙⢒⠮⣝⡓⠶⢤⣭⣛⡳⠶⢤⣄⣀⡀⠀⠀⠑⠪⢖⠵⣪⢕⡩⢖⡩⢞⡡⠖⣡⠞⣡⠞⣡⢞⡥⠊⣠⡾⠋
⠀⢀⣴⣾⣟⣛⣛⣓⠢⢬⣑⠢⢄⡐⠨⣝⡒⠦⣭⣉⣛⠻⠶⢾⣭⣿⣲⣦⣤⣀⠈⠒⣩⠖⣩⠖⣫⠴⢋⡴⢊⡵⠚⡡⠊⣠⣾⠋
⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⣶⣯⣶⣬⣑⠦⣍⡒⢤⣉⠛⠻⢷⣶⣤⣉⠙⠻⢿⣿⡻⣦⣄⠋⡴⢚⣡⠞⣩⡴⢋⡴⠋⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⡾⣝⡷⣬⣛⢦⣄⠉⠛⠿⣿⣶⣄⠙⢿⣎⢻⣦⠀⢋⣥⠞⢉⡴⠋⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣮⡛⢾⣝⣷⣍⡻⣦⣄⠈⠙⢿⣷⣄⢹⡆⢻⡇⠈⣡⡶⠋⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠙⠿⣮⣛⢮⣙⠷⢦⣤⡿⠟⢠⠇⣼⡇⠀⠉⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠈⠻⣷⣝⠳⢦⣤⡤⠴⢋⣴⡿⢁⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣀⡙⠿⢶⣶⣶⣾⣿⣿⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⡿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠀⠐⠀⠀⠀⠤⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢠⡰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡈⢿⠧⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⢀⠠⠐⠂⠉⣀⣀⣀⣀⡀⢁⠶⠭⣓⠦⡀⠀⠙⠔⢅⠁⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡠⠖⠀⠠⠖⠛⠉⢉⡔⠣⢌⢙⠣⣍⡒⢌⠳⡄⣄⠀⠀⡀⡀⡢⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠌⠀⠴⠐⠁⠀⠀⠀⠀⠀⠑⠰⢯⡈⠑⠠⣁⠡⡄⢉⠀⢣⠙⣌⠳⣄⠀⠘⠄⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠑⠈⠢⠑⡇⠘⡄⠠⡀⠑⢬⣑⠲⣄⠀⠂⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⡀⠀⠘⡀⠡⡘⢤⡘⠂⡀⡀⠂⢤⡑⣌⢣⠁⡀⠈⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⢀⠂⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠒⠄⡁⠑⣄⠳⡜⢆⠱⢌⠢⢤⡂⠈⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠔⠀⠀⢀⠀⠀⠀⠀⠀⠀⠢⡙⢅⠈⢢⠙⠀⡑⠦⡽⠊⠀⢦⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠣⡀⢡⠈⠢⢌⡀⠀⠀⠀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⡐⠀⠌⠀⠀⠀⠈⠃⠠⡄⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⢆⡈⠑⢢⠍⠀⠀⠀⠀⠐⠀⠑⡀
⠀⠀⠀⠀⠀⠌⠀⠘⠄⠀⠀⠀⠀⠠⠀⢳⡀⢀⠀⠤⢤⠤⠂⠀⠲⠂⠰⠂⠐⠢⠄⢀⣀⠀⠀⠀⢠⠏⠢⠌⠛⠀⠀⠀⡀⠀⠀⠀⠀⠄⠈⠄
⠀⠀⠀⠀⠌⠀⡐⠀⠘⡀⠀⠀⠀⠀⡡⠤⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠰⣏⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠈⢂⠈⠂
⠀⠀⠀⠐⠀⡐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠠⠌⠀⠀⠀⠀⠀⠈⢄⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠉⠉⠉⣉⡉⠉⣉⠉⣉⣉⣉⣉⠉⢉⣁⠉⢉⡉⣉⣉⡉⠉⠉⠉⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣷⡀⣿⠀⣿⣉⡉⢹⡇⣸⣿⠀⣿⢸⣏⡉⠁⠀⠀⠀⢸⣿⣿⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⠈⢿⣿⠀⣿⠉⠁⠈⣷⡟⢸⣿⠇⢀⡈⢹⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠉⠀⠀⠉⠀⠉⠉⠉⠀⠉⠁⠀⠉⠀⠈⠙⠋⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣡⠶⠋⡹⠋⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⣠⠞⠂⠀⣼⠁⢠⠀⠀⠀⢈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠛⠀⠀⣴⢃⣠⡦⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠈⠂⠀⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⡤⠚⠀⣠⣦⠰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⢾⠆⠀⠀⠀⠀⠀⠀⠌⠃⠀⠁⠈⠉⢒⡆⠀⠙⣿⠀⡆⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠳⣤⢀⡉⠻⠀⢇⠀⠀⠁⠀⠀⠀⠀⠀⠀⣸⠇⠀⢸⡿⠀⠀⢠⠗⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠱⣤⠦⠈⠢⣀⣀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠏⣰⡿⠚⠁⠰⠟⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⢀⡤⠃⠀⠀⠉⠻⠆⠐⠶⠚⠾⠃⠀⠀⠶⠀⠀⠒⠛⠉⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠠⡟⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠘⣦⡀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⢠⠀⠀⠀⠈⠁⠀⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣅⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿⣧⠀⠀⠶⣦⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⠶⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⠉⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣧⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⠀⠀⠀⠀⠀⠀⠐⠒⠲⠚⠀⠀⠀⠀⠀⠀⢀⣤⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢠⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢤⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠀⠀⠘⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⠀⠀⠀⠘⠀⠀⢸⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠇⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣶⣶⣶⣶⣶⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘






⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣾⣶⣶⣶⣾⣶⣾⣶⣿⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣟⣻⡟⢿⡿⢿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢆⠀⠀⠀⣠⡀⠀⢀⡠⢫⠋⣤⡀⠀⠀⢀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢧⠀⠙⠒⠾⢄⡙⠲⣽⣡⡷⠚⣁⠼⠖⠚⠁⢠⠇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡙⠚⣦⣌⡶⠤⠤⠽⢶⣾⣿⣴⠾⠥⠤⢤⣖⣥⡚⠚⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⢋⡅⠀⠀⠃⠀⢀⡈⢉⡀⠀⠘⠀⠀⢨⡙⢶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⡴⠀⠀⠈⠁⠈⠁⠀⠀⢦⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡄⠀⢠⠀⠀⢀⠀⢰⠄⠠⡄⠀⡀⠀⠀⡄⠀⢠⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣀⠀⠀⠐⠃⠀⠀⠀⡄⠀⠀⢠⠀⠀⠀⠘⠂⠀⠀⣀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠈⠀⢤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡤⠀⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡱⡐⡄⠀⠒⠀⠀⢀⡀⢀⡀⠀⠀⠒⠀⠀⠂⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡽⣜⢎⠢⡀⢠⡀⠀⠀⢀⡄⠀⢀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⢧⡑⠬⡀⠀⠀⠀⠀⡀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢍⡳⢄⠀⠀⠀⠈⠁⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣤⣤⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⠀⠀⠠⠖⠿⢗⠀⠀⡐⠀⠂⠀⠀⠀⠀⠈⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⡴⠄⠀⠀⠀⠤⠀⢀⠀⢀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠠⠀⡀⠀⠀⠕⡆⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠁⠈⠐⠀⠠
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠰⠈⠀⠀⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⢀⠀⠀⠈⢀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡸⠄
⠀⠀⠀⠣⡀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡀⠤⠂⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⠟⠃
⠀⠀⠀⢆⠈⠂⠤⣀⠀⠀⠀⠉⠐⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠬⠌⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡠⠆⠊⠁
⠀⠀⠀⠀⠑⠢⣄⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⠀⠀⣀⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠈⠐⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⢤⠖⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢀⣀⠀⠈⠁⠐⠂⠀⠀⠀⠐⠂⠀⠀⠀⢀⣠⡤⠴⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠦⠤⣤⣤⠤⢤⣄⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀⢀⣀⣀⣀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⡠⠀⡀⠀⢀⠢⢤⡀⠀⠀⡏⠀⢻⢀⡤⣠⠊⠉⠀⣀⠤⠄⠀⢞⢡⠀⠑⢆⠀⠐⠤⡀⠀⠀⡞⢳⡄⠀⢠⠟⠓⠦⣔⠒⠓⠦⣄⣀
⠀⠀⢀⠰⢀⠑⣄⢻⠀⡠⠓⠀⠀⠈⠓⠁⠸⡀⣀⠇⠀⢸⣇⠀⠀⠀⠈⠉⠁⠉⠁⠤⠤⠒⠳⡄⠀⣧⣸⠃⢀⡟⠀⠀⠀⠈⠙⠒⠶⢤⡈⠉⠉⢱⡀
⠀⠀⠀⠂⢢⣽⣿⡻⠋⡠⠔⠒⠉⠉⠓⠦⣀⠀⠀⠀⠀⢸⣧⠤⢄⣀⡀⠀⠀⠀⠈⢉⣀⡈⢽⡃⠀⠼⠁⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⢀⡟⡄
⠀⠀⠀⢊⢀⠔⠁⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⢸⡅⢀⡀⠀⠈⣈⡉⢉⣉⣀⠀⠀⣰⠇⠀⠀⣠⠞⠀⠀⠀⠀⠁⠀⠀⣠⢎⡵⠁⠀⡼⡸
⠀⠀⠀⠀⠁⢀⡎⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠈⣇⠀⠙⠒⠦⠬⠤⠤⠄⠀⢀⣀⠠⠤⠘⠁⠀⠀⣧⣀⠀⠀⠐⠀⠀⠀⣰⢣⠏⠀⠀⢠⢇⡇
⠀⠀⠀⠀⠀⠸⣇⠠⡀⠀⠀⢀⡞⠁⠀⠀⠙⢦⠀⠀⡿⠀⡴⠾⠶⡆⠀⠀⠀⠀⠀⠠⢤⣄⣀⠀⠀⠀⠘⠫⢽⣲⡦⣄⠀⢠⠇⣾⣀⣀⣠⡼⢸⠁
⠀⠀⢀⣤⣄⠀⠙⢦⡀⠀⠀⠘⣇⠀⡀⢀⡄⣸⠀⢠⠇⢸⠁⠀⣰⠏⠀⢠⠒⠒⢦⡀⠀⠀⠈⣧⣀⣀⣀⣀⣀⡈⠙⠲⣝⣫⠞⠓⠶⠶⠶⠖⠋
⠀⠀⣾⠁⠈⠳⣄⠀⠉⢦⡀⠀⠈⠳⠬⠵⠞⠁⢠⠟⠀⠘⠒⠚⠁⠀⠀⠘⢮⣀⣠⡇⠀⢰⣯⣽⣏⣀⣠⣤⠤⠽⠇⠀⠀⠀⠀⠀⢠⠖⠒⠲⡄
⠀⠀⣿⡄⠀⠀⠈⠑⢤⣀⠑⢄⡀⠀⠀⢀⣠⡴⢋⣠⠴⠖⠒⢛⡓⠒⠲⢦⡀⠀⠀⠁⠀⢸⠀⠀⢹⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠏⠀⠀⠠⢹⡀
⠀⠀⠸⣷⡀⠀⢀⣤⣤⡈⠑⢦⡈⠉⠉⠉⠁⡴⢯⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠲⢄⡀⠀⠸⣄⠤⠤⢷⡤⠤⠤⣤⡇⠀⠀⠀⢠⣾⠀⠀⠀⠀⠃⢦⡀
⠀⠀⠀⡟⣧⠀⢻⠀⠀⡝⣦⠀⢹⡄⠀⠀⡞⠀⠈⣧⠈⠀⠀⠀⠀⠀⠀⡎⠀⠀⢠⢙⡆⠀⣟⠒⠒⠚⣷⠚⠛⢉⡇⠀⠀⠀⠸⡍⢙⠋⠙⠉⠉⣩⠁
⠀⠀⠀⢿⠘⣇⠀⠳⢤⣤⠟⠀⢰⠇⠀⢰⠳⣕⠀⠘⣆⢀⣒⣒⣀⣀⣸⣇⣀⠐⣡⠞⢳⠀⢻⠀⠀⠀⠸⣆⠀⢸⠃⠀⣀⡀⠀⠙⠾⣷⣤⡤⠞⠁
⠀⠀⠀⠈⢧⣈⠳⢤⣀⣀⣠⣴⠟⠀⠀⢿⣐⣨⠷⠞⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⢧⣀⣸⠇⠈⢧⣀⣀⣀⣉⣠⠜⠀⢯⠉⢧⠀⢀⣠⡴⠀⠢⣄⡀
⠀⠀⠀⠀⠀⠉⠛⠒⠒⠛⠋⣁⣀⣀⡀⠀⢉⣁⡀⠀⠀⠀⠸⡳⣶⣄⡇⠀⠀⠀⠀⠈⠁⢀⣴⣦⣍⡉⠉⠀⠀⠀⠀⠈⠓⠻⣄⣈⡓⠶⠴⠶⠞⠃
⠀⠀⠀⢰⡒⠦⣴⡒⠋⢉⣉⣀⠠⠀⠉⢶⡁⠀⡝⠀⠀⠀⣠⠽⠾⢿⣧⣠⠤⠤⣄⡀⠀⢸⣝⡻⠷⣯⣭⣍⣉⣉⣉⣛⠲⢤⣀⠈⠉⠓⠦⣄
⠀⠀⠀⠘⣧⠀⠱⢭⣒⣤⣀⠀⠠⠄⠀⠀⠱⡀⢨⠆⠀⣼⢡⠊⠈⠀⠈⠁⠀⠀⢠⠹⡄⠈⠻⢿⣷⣶⣶⣯⣭⣭⣽⣛⡻⠶⣌⢳⡄⠀⠀⠘⣆
⠀⠀⠀⢰⡿⡦⢤⡘⣿⣭⢻⢷⣦⣄⡀⠀⠀⠑⣌⠆⠀⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠶⠤⢤⣉⣉⣉⣉⡉⠛⠻⢷⣼⠋⠀⠀⠀⠀⣿
⠀⠀⢀⣀⠓⢬⣒⢵⡀⠛⠉⣟⠀⠛⢿⢗⠦⣄⡈⣧⠀⠸⡆⠀⠀⠀⠏⠘⠆⠀⠀⢰⠁⠀⢸⠭⠶⠶⠶⠶⠶⠦⣭⣍⡓⠤⡉⠀⠀⢰⠋⠉⠉⣹⠂
⠀⠀⠀⠈⣳⡀⠈⠑⠮⣓⢦⣈⠉⠳⡀⢀⡙⠲⠽⠇⠀⠀⠹⣔⠤⡀⠀⠀⠀⠄⣠⠏⠀⠀⠘⠿⢿⣒⣒⣒⣒⣒⠢⠌⡙⣾⠀⠀⢠⡟⡶⠒⢲⣿
⠀⠀⠀⠑⢿⣧⣤⡠⠴⡌⠑⢬⡓⢦⣉⣁⡸⠂⠁⢀⣀⠀⠀⠈⠓⠒⠛⠓⠒⠛⠁⠀⣀⣤⠴⠶⠲⠶⠦⣄⡀⠈⠙⢦⣄⣷⠀⠀⠸⠶⢷⣬⣿⣸
⠀⠀⣠⢾⡿⠋⠛⠒⠋⠀⠀⠀⠈⠓⠖⠋⢀⡠⠞⠉⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⢉⢳⡀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠈⠁
⠀⠀⠉⢁⡤⠤⠤⠄⢤⣄⣤⠤⠤⡀⠀⠀⡿⠤⠤⠤⠤⢤⡴⢦⣄⣈⣓⠲⢄⠀⢸⢿⠀⠀⠀⠀⠀⠀⠀⠀⡰⡠⠃⢀⡴⠞⣻⠋⠉⠉⠉⠓⢦
⠀⠀⡎⢹⠉⣿⠛⠛⠛⠛⠛⠛⠛⡏⠀⠀⣛⣷⠀⠀⣀⠀⠙⠋⠀⠀⠉⠉⢻⠀⠈⢧⡳⢤⡀⠀⠀⠀⠀⠀⢸⠁⢀⡞⠀⢎⠉⣠⠶⢶⣄⠀⠈⢧
⠀⠀⠹⣼⣧⠘⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⣯⡀⠀⠸⠽⠂⠀⣠⣤⣄⠀⠀⢳⠀⠀⠀⠙⣆⢷⠀⠀⠀⠀⠀⠘⡇⠸⡇⣠⠊⠘⣇⠀⣸⣽⠀⠀⣸
⠀⠀⠀⢠⡛⣦⣀⡀⠀⠀⠀⣠⠗⡄⠀⠀⣾⠁⠀⠀⠀⠀⠀⠛⠶⠟⠀⠀⢸⠀⠀⠀⠀⢿⢸⡆⠀⠀⠀⠀⠀⢹⠀⢷⡸⢄⣠⣌⢉⣩⣁⡖⣲⠏
⠀⠀⠀⠀⠉⠛⠿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠙⠒⠒⠒⠒⠒⠚⠙⠒⠒⠒⠒⠚⠀⠀⠀⠀⠈⠳⣧⣄⣀⣠⣤⡤⠞⠀⠀⠙⠶⣤⣈⣉⣠⡭⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⡾⠛⠾⠻⠞⠻⠞⠳⠟⠳⠟⠷⠛⠶⠛⠾⠻⣾⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢈⣤⣉⣤⣉⣤⣩⣤⣡⣌⣡⣌⣡⣌⣤⣍⣤⡉⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⡇⠀⠀⢠⣄⣤⢠⡤⢠⣤⣄⣤⣤⠦⠀⠀⠀⢸⣦⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⠛⡟⡟⢛⣟⠻⠛⠟⡛⢛⣿⣿⣿⡇⠘⠛⣿⣿⣿⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠸⠟⠿⠸⠯⠌⠿⠻⠇⠬⠿⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⣰⣄⣇⣐⣻⣄⣤⣠⣟⣂⣽⣿⣿⡇⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⢸⡇⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣽⣿⣿⡇⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⢸⡇⠀⠰⠶⠶⠶⠶⠖⠀⣴⣶⣶⡶⠶⣶⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⣿⣿⣉⣉⣉⣉⣉⣿⡏⠉⠉⢉⣉⡉⣿⣿⡇⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⢸⡇⠀⠰⠶⠶⠶⠶⠖⠀⠿⠋⠙⢁⣤⣿⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⣿⣿⣉⣉⣉⣉⣉⣿⣇⣴⣶⣴⠟⠃⣿⣿⡇⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⢸⡇⠀⠰⠶⠶⠶⠶⠖⠀⠰⠾⠿⠿⠿⠿⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⣿⣿⣉⣉⣉⣉⣉⣿⣿⣁⣀⣁⣀⣀⣿⣿⡇⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⢸⡇⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⣿⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿⣿⡇⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⢸⡇⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⣿⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿⣿⡇⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠸⢧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡿⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⢀⣰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⢀⣤⣀⠀⠈⠁⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠁⠀⠀⠀⠀⠀⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⣴⣿⣿⣿⡆⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠒⠤⣀⣀⣀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠛⠿⠿⠛⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⡼⠀⠀⠀⣀⣤⡤⠤⠶⠤⣄⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠀⠀⠀⠈⢺⡆⠀⠀⠀⠀⠀⢨⢣⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠇
⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠐⠒⠒⠒⠒⢠⡀⠀⠀⠀⠀⠀⢀⡤⠼⠧⠤⠤⠤⠤⠤⠼⠼⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠃
⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⢀⣀⡠⠤⠔⣊⡅⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂
⠀⠀⠀⠐⢬⠭⠉⠉⠭⠭⠁⠤⠐⠂⠉⠀⡂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠘⡄⢀⣀⣀⡀⠤⠤⠀⠐⠒⠈⢹⠀⠀⠀⠀⢀⣇⣀⣀⣀⣀⣀⢀⣀⣀⣀⠀⠻⣃⣟⠘⠃⠀⣠⠒⠢⠤⠭⠍⠉⠐⢲⠋⡇
⠀⠀⠀⠀⠘⡄⠀⠀⠀⣠⣤⣦⣄⠀⠀⠀⢸⠀⠀⠀⠀⠓⠐⠒⠀⠀⠀⠐⠒⠒⠒⠒⠒⠐⠀⠐⠃⠀⠀⡇⠀⠀⠢⠠⠐⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⢱⠀⠀⠰⣿⢿⣽⣿⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠁⢀⡀⠀⠠⠤⠤⠄⠚⠃⢰
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠙⠻⠟⠋⢀⣀⠠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⣔⡉⣑⢏⡠⢶⠀⠀⠀⠀⠘⠁⢠⢰⡴⠄⠶⠦⠀⡇⢠
⠀⠀⠀⠀⠀⠀⠐⢖⠒⠒⠂⠈⠉⠀⠀⠀⡄⠀⠀⠀⡠⠠⠤⢊⠭⠠⠕⠀⣈⠔⠨⠥⠄⠒⠤⠀⠀⠀⠀⢰⣈⡡⡀⠤⠤⠤⠤⠒⠃⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠄⣀⠀⠀⣀⡠⠄⠊⠀⠀⠀⠀⡗⢒⡒⢁⠔⠒⠊⠀⠀⠀⠀⣀⠀⠀⡀⡀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢰⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠓⠂⠑⢊⣐⢂⡮⠭⠱⢞⢄⡨⠮⡀⢴⠀⠀⠀⠀⠆⠀⠐⠒⠂⠀⠒⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠐⠠⣀⠀⠀⠀⠀⠀⢸⣁⢉⡈⡡⠤⠤⠊⠑⠒⠉⠑⠀⠒⠢⠼⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠀⠀⠀⠀⠀⠈⠐⠠⢄⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢀⠤⡐⠜⠋⠁
⠀⠀⠀⠀⢀⠴⠈⠠⠤⠤⠄⠤⠤⠤⠤⠄⠤⠤⠤⠬⠷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠄⠧⠤⠶⡈⢁⡀
⠀⠀⠀⠀⠘⣷⣮⡴⡤⠰⢶⣤⣤⣄⣄⣠⣡⣤⣐⣲⣶⠃⠀⠀⠀⠀⠀⠀⠀⣔⠈⡡⠄⠰⡀⠠⡤⢀⣐⠢⠤⣄⡉⢲⡄
⠀⠀⠀⠀⠀⢏⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⢋⡁⠃⠀⠀⠀⠀⠀⠀⠀⢙⠛⠠⠬⢵⣏⣘⣏⣁⣬⣡⠦⠶⠚⢛⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⠀⠀⠀⠀⠀⣀⣠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠂⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠐⠒⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢁⠈⠀⠀⠒⠂⠈⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠒⠂⠀⠈⠐⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠠⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠢⡀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠶⢺⡀⠀⢁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢂⣹⠀⠀⣏⠉⢵⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⢋⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⠒⠡⠐⢧⠀⠱⣀⡀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⣀⡀⠀⡈⡗⢤⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⡜⢀⢊⠐⠁⠀⠀⠀⠑⢤⡈⠑⠂⠷⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠴⠊⠉⠀⠀⠀⢀⠻⠲⡐⠄⠑⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠆⠀⠀⠀⠀⠀⠀⠀⡼⠉⠑⠂⠠⠀⢤⡀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠡⠁⠀⠀⠀⠀⠀⣠⠞⠀⠃⠀⢨⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢱⡈⠀⠀⠀⠀⠀⣠⠜⠁⠀⠀⠀⠀⠀⠀⠙⢒⠉⠀⠀⠒⠀⠀⠀⠀⠀⠀⢀⣁⠤⠖⣒⠼⠶⠶⠦⠤⠶⠚⠁⠀⠀⠀⠠⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠠⠆⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⣠⠴⠶⠒⢉⣡⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠂⠌⡁⠂⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠉⠀⠀⠔⠊⠁⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠴⠒⢈⠠⢐⡮⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⣀⠀⠀⠀⠈⠉⠁⠒⠒⠀⠤⠤⠤⠤⠤⠤⠠⠤⠤⠶⠶⠾⠛⠛⠓⠊⠉⠉⠠⠀⠂⢀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠠⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠃⠐⠒⠒⠒⠒⠒⠒⠒⠂⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠐⠒⠈⠒⠃⠀⠑⠰⠐⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣀⢀⠀⡀⡀⢀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣁⣉⣈⣉⣁⣈⣉⡝⠀⠀⠀⡆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠛⠸⠠⢆⢆⡆⢰⢀⢠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⠻⠈⣿⡿⠀⣌⢉⡀⠀⠀⠀⠀⠀⠀⠉⠙⠚⠀⠀⢹⡄⡄⠄⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢋⣸⠀⠀⣻⠇⠘⠽⠜⠞⡾⠄⣰⡀⣄⡀⡀⠀⠀⠀⠀⠀⢀⠁⠀⠃⠃⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⠀⠄⡟⠀⠀⠀⠀⠀⠈⠈⠉⠓⠙⠰⠹⢳⢷⢒⡂⢠⡄⣀⣀⠀⠀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠊⠹⠼⠂⣧⢦⠖⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢐⣶⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢁⡆⠛⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⣦⣤⣤⣀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠒⠶⠦⢤⣄⣀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠓⠛⠿⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓

⠀⠀⠀⠀⠀⢠⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⢤⢄⡀
⠀⠀⠀⢀⣢⣄⣿⣤⣼⡃⠀⠀⠀⠀⢀⢀⢀⠀⠀⠀⠀⡐⠈⢸⣸⠿⣇⠀⠀⣐⣿⡿⠇⠀⠀⠀⠦⣷⢿⣿⣿⣧
⠀⠀⣴⡃⡜⢈⢹⣻⣿⡳⠶⠆⠘⠢⣾⣾⣼⠠⣠⠀⣰⠇⣆⡿⠗⢚⡿⠀⠀⠳⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⢉⣵⣿⣶⣄
⠀⢠⠋⡁⠱⣸⣭⣿⣿⣿⠀⠀⠠⣌⡉⠽⢿⣧⡇⢀⣿⣼⣟⣣⠍⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠈⠁⠀⣪⣿⣿⢾⡇
⠀⢸⣁⠠⢸⣽⣿⣿⣿⠟⠀⠀⠀⠈⠓⠊⠉⠉⠳⠘⠛⠛⠁⠀⠀⠀⠀⢀⢴⣴⣷⣀⡴⣄⡈⠀⠀⠀⠀⠀⢈⡵⣿⣿⠃
⠀⠈⢤⣭⣾⣿⣯⠏⠉⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣒⣀⡀⠀⠐⠿⢿⣿⣁⡤⣿⣿⣭⣶⣟⣅⠀⠈⠀⣿⡉
⠀⠀⠈⠙⠋⠁⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡈⠹⣿⣷⢖⣫⠀⠀⠀⡟⠠⡸⣵⠿⣼⡿⣏⠛⣿⣷⡄⠀⠀⠁
⠀⠀⠀⠀⠀⠠⣦⡰⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠃⠛⡇⠈⠀⠀⠀⢗⣰⣿⣿⡠⠉⠷⠋⣻⣿⣿⠇⠀⠀⣄⠠⠶
⠀⠀⢀⣭⢿⡶⣿⣿⡷⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⢿⡿⠿⠗⠈⠄⣼⣿⣿⡟⠀⠀⣾⣏⡗⢀
⠀⢰⡿⠐⠆⠹⠆⣿⣿⣦⠀⣤⢢⣙⠀⠀⠀⠀⠀⠀⣄⣀⣠⣧⠴⠀⠀⠀⠀⠀⠀⣴⣡⣄⣀⠀⠚⠿⠋⠀⠀⠀⠉⠉⠁⠉
⠀⢺⣦⢉⡘⣧⣿⣿⣿⣿⠀⢲⡈⢿⣷⡀⣂⠀⠐⢲⣿⣿⣿⠟⡶⢦⡀⢀⣲⢶⣶⠟⡐⠀⠜⢳⡀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠘⣿⡤⠹⣿⣿⣿⣿⠏⠀⠀⠁⠽⣿⣷⠃⠀⢰⠋⣁⠈⣉⡀⠐⠊⢻⠀⡅⠀⠖⠈⠛⠂⠀⡀⢣⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠻⣿⣿⡿⠟⠁⠀⠀⠀⢀⣀⣤⠽⠀⠀⠸⡄⠀⠄⠙⠇⢘⠀⢸⠀⠻⢦⣄⣁⡈⡁⣁⣀⣨⠀⢀⣤⣄⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠛⠉⠃⠀⠀⠀⠀⠙⢦⡀⠂⠀⠀⢰⠃⠀⠀⠙⠤⠿⣟⠿⠟⠋⠀⠈⢿⣼⣯⡿⢋⣤⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠽⠛⠁⠙⠼⠿⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⢶⣿⣿⡿⠛⣿⣿⠛⠛⢷⠶⠶⠶⢦⣤⣀⡀
⠀⠀⣠⡶⠻⠉⠈⠀⠀⠠⠆⠁⠀⡀⣀⠉⠀⠀⡀⣿⠿⠓⠒⠶⠤⠉⠛⠷⣦⣀
⠀⣸⠟⠀⠪⠀⠀⡀⠀⠀⠀⠀⠈⠀⠉⠀⠠⠨⠀⠀⢀⣤⡴⠶⠛⢛⡛⠛⠛⠛⠿⢶⣄
⠀⣿⡌⠀⠀⡁⡜⡍⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⣠⠞⡩⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠠⠙⣧
⠀⢻⣷⣾⡆⣠⢰⢸⢰⢾⡜⡎⢀⢀⢬⠡⠠⣸⡏⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢻⠟⠿⢿⣿⣾⣿⣿⣿⣿⣧⣿⣾⣀⠸⣿⡇⠀⠀⠀⠀⠅⠀⠁⠀⣀⣀⣀⣀⡀⠀⣾⠃
⠀⠀⢸⡇⠀⠀⠀⠈⠉⠉⠛⠛⠿⠿⢿⣿⣿⣿⣧⠀⠀⠀⠀⢀⣠⣶⣿⠿⢛⡉⠉⠉⠙⠻⣄
⠀⠀⣸⠇⠂⠀⠀⠀⠀⠐⠀⠄⠐⠂⠦⠄⡈⠉⠙⢷⡰⡠⢠⣿⣿⠟⠁⠀⠀⠁⠀⠀⢀⣠⣽⡷⠟⢛⡉⠉⠩⠛⠳⣄
⠀⢀⣿⢀⡀⠰⠀⠃⠰⠀⠐⠀⠄⠀⠀⠀⠀⡄⠀⢨⣷⠀⣼⠿⡇⡄⠠⠀⠀⠀⢉⣴⣿⠿⠋⠀⠀⠀⠀⠀⠀⢂⠀⢸⡇
⠀⢼⣯⣼⡗⣆⢀⢀⠈⢡⠀⠀⠀⠀⠀⠀⢰⢀⡖⢾⢹⠀⢸⡆⢡⡱⠀⠀⠀⠀⣾⢻⡇⠀⢀⡀⠀⠂⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠙⠿⢿⣿⣿⣿⣷⣼⣰⡄⣴⡇⣀⠀⢸⢨⢻⣿⣿⠀⠀⠻⣿⣿⣶⣤⡐⠀⢿⡸⡄⠄⠠⠀⢀⠀⠀⠀⠀⠈⠐⠘⣇
⠀⠀⠀⠈⠉⠉⠛⢿⣟⣿⣿⣧⣿⣧⣿⣷⣧⣾⣿⣿⣯⠀⠀⠀⠈⠻⡏⠙⡇⠀⡈⠻⢿⣦⣄⡀⠀⠀⣀⠀⠁⠒⠂⢀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠩⠭⠭⠭⡿⢿⠿⣿⣿⣿⣿⣿⣿⣿⡹⠀⠀⠀⣀⣠⣿⣦⣵⢣⡀⠀⠀⢻⠛⢻⠀⠀⠒⠃⠀⠀⠀⠀⠉⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠓⠒⠒⠒⠒⠲⠛⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⣷⣝⣂⣀⣸⣷⣼⣇⢀⠀⠀⠀⠀⠀⠀⠀⠐⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⣛⣻⡟⠛⠛⠛⠛⠛⢻⣿⣿⣦⣓⡀⠀⢀⣀⣤⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠁⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠐⠒⠀⠀⠀⠶⠰⠒⠆⠐⠐⠂⠐⠐⠰⠀⠀⠂⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⡀
⠀⠀⠀⡖⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠒⢉⣁⡰⣦⠱⡄
⠀⠀⠀⣇⠀⠉⠒⠒⠀⠀⠒⠐⠒⠈⣉⣉⣀⠀⣀⣰⡖⢿⣬⣥⣤⡀⠘⣄
⠀⠀⠀⠸⡄⠐⡒⠂⠰⣶⣄⢷⣜⢯⣁⣉⠉⢦⣸⣿⢿⣆⣶⠤⠼⢏⠀⠈⢦⡀
⠀⠀⢰⡒⢳⡀⠁⡀⢤⡹⣿⡻⢿⣮⡻⣦⣤⠴⠍⡛⠀⠨⠵⠖⢑⣀⠥⢔⣦⠙⢦⡀
⠀⠀⢾⣷⡀⢳⡄⠈⠭⠅⣈⡿⢂⡩⠭⠭⠖⢚⣉⡤⣄⠰⣖⠯⢕⣒⣫⣭⠄⠀⡀⠈⠒⠤⢄⣀⣀⡀
⠀⠀⠀⢻⣿⡄⠹⣆⠀⢋⣉⣭⢥⣤⠀⢺⡉⠉⠅⠀⠈⠁⣀⠘⢻⣟⣭⡴⢞⣛⣽⠟⠋⠐⠒⠐⠢⢌⠑⡄
⠀⠀⠀⠀⠙⣿⣆⠈⢣⡀⠉⠙⠋⢍⣤⣄⡙⢦⡀⠀⠀⠀⠉⠳⢤⣌⠒⢟⠉⠉⠀⢀⠀⠀⠀⠀⠀⣨⣄⡇
⠀⠀⠀⠀⠀⠈⢻⣷⡀⠹⣆⠀⠀⠞⡛⢛⣩⣄⣙⢦⣀⡠⠔⠒⢉⣡⡴⠀⠑⡆⠀⢀⣀⣄⣶⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄⠈⠳⣄⠀⠙⢿⣿⢯⣝⠪⣤⠴⠶⣛⣯⣵⣦⡅⢠⣿⠶⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠊⠙⡶⣄⠐⠛⠂⣭⢖⣺⣽⣿⣻⠷⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⡀⠀⠑⠎⣧⠀⢼⣾⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣤⣤⠼⠖⠋


⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⡀⠀⢀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⡴⠀⠀⠰⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡎⠀⠀⠀⢀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠠⠀⡘⢡⡷⢤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢤⡄⢖⠀⣠⣴⣶⣶⣿⡿⢷⣼⣿⣿⣿⣿⣻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠷⢂⢹⣧⣤⣬⠍⠉⠉⠁⠉⠉⠉⠁⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠑⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⠠⠐⠞⠻⣦⡀⠀⠀⠀⠀⠀⢀⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⠷⣤⣍⣓⣶⣶⡤⠴⡋⠁⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣌⠻⣿⣿⣿⣿⣦⣌⣉⣉⣉⣁⡀⠔⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠈⠀⠀⠀⢠⡴⣿⣿⣿⣿⣷⣿⣿⣿⣿⡟⠋⠉⠉⠀⠀⠀⣀⣀⣤⡔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣟⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠈⠉⠁⠀⠈⠁


⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⠀⠀⠸⠛⠳⠛
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠀⠉
⠀⠀⠀⠀⠀⣠⡴⡿⠀⠀⠀⠀⠀⠀⠀⣠⠤⣇⠀⢀⡔⠀⠀⠀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠙⠻⠿⠀⠀⠀⠀⠀⡀⠤⠐⠒⠠⠴⡏⠀⠀⠀⠺⠟⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣧⡤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠻⡗⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠇⢀⣈⡃⢀⠉⠀⠀⠀⣀⠀⠀⠀⠀⠄⣀⠀⠀⠀⢀⡀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢙⣥⡉⣹⠂⠀⠠⢋⡟⠉⢂⢀⠀⠀⠒⠈⣧⠀⣰⣿⣧⣶⡟⠚⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣀⠁⠀⠉⣤⣄⣠⣾⠷⠷⠾⢎⠀⠀⠀⠀⠙⣦⠿⠟⠁⠀⠂⠀⠀⠈⣧⢀⣰⢀
⠀⠀⠀⠀⠀⠀⢠⣿⡋⠹⣷⣿⣿⣿⡲⣤⣠⣦⣤⡄⠀⠀⠆⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠘⠀⠶⢴⡟⠙⠂⣡⣉
⠀⠀⠀⠀⠀⢰⣿⠿⣍⣀⡘⢿⣿⣯⣷⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⣠⠆⠀⠀⠐⣻⣆⣣⡈⢻⠁
⠀⠀⠀⠀⠀⠈⣿⣗⠢⢍⣙⣋⡙⢿⣿⣿⣿⣷⠶⠶⠶⠂⠀⠀⢠⠇⠀⢠⡏⡀⠀⠁⠀⠀⠀⠀⠉⠛⢿⣷⣤⠤
⠀⠀⠀⠀⠀⠀⠛⣿⣷⢀⣀⣩⣿⣎⠻⡾⢿⣿⣿⡟⠒⠠⣤⣤⠹⡆⣠⡿⠻⠒⢦⠀⠀⠀⠀⠀⢀⣀⢀⡀
⠀⠀⠀⠀⢠⣤⣶⣿⣿⣷⣌⣛⣯⣽⣧⣅⣼⣿⠿⡇⠀⠀⠈⠉⢳⡅⠻⠓⠁⠀⠀⡆⠀⠀⠐⣿⣿⣿⣿⡷⣾⣶⣤
⠀⠀⠀⢠⣿⣿⠋⠙⢛⣭⠭⣩⢜⡿⣿⠿⠿⣿⣶⣦⡄⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⡇⠀⣠⣾⣿⣿⡿⡡⢞⡹⠛⠉⠶⢄⡀
⠀⠀⢠⣾⣩⠔⡊⠤⠛⢉⡾⣿⠈⠃⠀⢀⣼⣿⣿⣿⣿⣆⠀⣸⠃⢰⡄⢀⡄⠐⣿⣷⣾⣿⣿⣿⣿⣉⣀⣊⣀⣀⡀⠀⠈⠧
⠀⠀⠈⣿⣅⣠⣆⢠⣴⠿⡇⢻⣆⣴⣠⣿⣭⡿⠻⢍⡙⠛⠀⣟⠀⠘⣧⣻⢠⣖⠹⣿⣷⣿⣿⠛⠿⠟⠻⢿⣍⡉⠉⣿⣶⣾⠆
⠀⠀⠀⠸⣿⣿⠁⣾⣿⣧⣿⣾⣿⣿⣟⣉⣻⣗⠂⠀⣈⣁⣀⡘⠀⠄⠼⠿⠻⠿⠷⠿⠿⣿⣛⣷⡢⠀⠀⠐⠄⠉⠢⠀⡈⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠉⠙⢿⡿⠋⠈⢉⣉⡉⠉⣛⠿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣾⣯⣶⣀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣍⣿⣿⣯⣍⣿⣿⣶⡿⠋⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⣰⣏⠀⠀⠉⠙⢿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⡿⠋⠰⠆⢻⣿⣿⡿⡟⢿⠲⠀⠀⠀⠠⣾⣿⣿⣿⣦⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣽⡿⠟⠛⡙⠟⠁⠀⠀⠀⣤⣼⣿⣿⣿⣌⠻⣷⡄⠀⠀⢀⣏⢸⣿⣿⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢿⡷⠎⠁⠁⣀⣆⣴⣶⣿⡿⠛⡿⠋⢠⣿⣷⣿⣬⣤⣾⡟⡣⢺⣿⡿⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⢶⡿⠟⠛⠛⠋⠉⢀⣤⢻⡇⠀⣸⡟⠉⢿⣿⣿⣿⣾⣵⢆⣿⣧⣴⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⡇⢀⣿⠁⠀⠈⠛⠻⣿⣿⣧⣤⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⡇⡜⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠙⣯⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⣀⣤⣤⣄⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠁⠀⠀⠀⠀⠀⣀⣀⠤⠄⠒⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⡔⢲⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠌⠓⠲⢄⠀⡀⠤⠒⠒⡲⠦⣀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠋⠀⠑⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⠁⠀⠀⠀⠀⠀⠘⣦⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⢠⠀⠀⠀⢻⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡄⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠠⠀⠛⠺⡄⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⣽⠀⠀⠀⠀⠀⠀⠀⠀⣈⠃⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⣀⡀⣀⡀⠻⠗⠀⣣⠈⠀⠀⠀⠑⠄⣀⢀⠀⠀⠀⠀⠀⣴⠋⠊⢉⠉⠣⣀⠀⠀⡀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠐⠁⣠⢄⠹⢶⠤⠾⠃⠦⡀⠀⠀⠀⠀⠀⠈⠑⠂⠄⣰⣾⡀⠀⠀⠈⠀⠀⡤⠑⠛⠀⠀⠀⠀⠀⠀⠈⠓⠄⣀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠠⣀⡩⠞⣼⠁⠀⣠⠀⠘⢂⡤⠐⠒⠀⠐⠒⠠⢄⣹⡞⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠇⠀⡠⠄⠠⣄⠸⡀⣶⢌⢁⣠⡞⠉⣠⠀⠐⠀⡒⠀⠀⠀⠉⢧⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⢸⡆⢧⣅⣀⢀⡀⠀⠀⠈⠣⠈⠀⠁⠔⠁⠀⠀⠈⣳⣦⠤⢀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠥⠎⠀⡟⠁⠀⠈⡇⠀⣀⡈⠁⠀⠀⠀⠐⢂⠨⠓⠝⠋⠀⠀⠀⠀⢘⠀⡠⠂⠀⠀⠉⠒⠤⠤⣀⠀⠀⠀⠠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⣇⠸⠖⠀⣣⠀⠀⠤⠂⠀⡐⠠⡜⠁⠀⠀⠀⠀⠀⠀⠀⢀⡶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⣀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠾⠀⠓⠒⠒⠙⢄⠀⢂⠀⠀⠰⠀⡀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⣡⠀⠀⠀⠤⠐⢞⠈⠑⠤⣀⣀⠀⣀⣳⡀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢿⠁⠁⠀⠠⣄⠀⠀⠀⠀⠀⢨⠀⠀⠈⠢⠵⠀⣠⣄⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢀⡀⠀⠉⠉⠳⠂⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠈⠐⠂⢤⠀⠀⠀⠀⠀⠀⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⠀⢀⣀⠤⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡴⠉⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠐⢄⣀⣀⣀⠠⠜⢗⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣁⠠⠊⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⣠⠴⠖⠒⠉⠀⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠉⠉⠉⠁



⣿⣿⡿⠿⣿⡿⢿⢿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣶⣴⣿⣷⣿⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠂⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡈⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁⠀⠀⠀⡈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠄⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⠤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠐⢄⠊⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠦⣍⣘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠀⣸⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣠⠔⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⢀⢤⣶⠿⠛⠉⠀⠀⢀⢠⠃
⠀⠀⠀⠈⢇⠀⣈⣉⠐⠠⣀⢮⠢⡐⠄⠀⠀⠀⠀⠀⠙⠓⠲⢦⡴⢒⡵⠟⠁⠀⠀⠀⠀⢄⡴⠃
⠀⠀⠀⠀⠀⠑⠄⡀⠁⠲⣈⣧⠦⠬⠱⡾⣽⡴⡦⡄⣀⣰⣯⣥⣔⣉⡀⡀⠠⢀⡂⠬⠒⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣉⣗⣾⡿⣿⠬⡯⢑⢔⡉⣀⠛⣧⣶⡖⡖⠒⠂⠀⠀⠉
⠀⠀⠀⠀⠀⠤⠠⡿⠿⠹⣞⣛⢯⡉⣯⣽⣿⣘⢣⢻⣯⢏⣿⣾⣤⢀⡠⡀
⠀⠀⠀⠀⠀⠀⠜⠀⢘⢐⣝⡭⠨⣽⢛⠖⢚⣿⣓⠾⠏⠉⠉⠉⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⠤⠂⠈⠑⠋⠉⠒⠃⠘⠯⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠙⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⡄⠀⠀⠀⠀⠀⢀⠈⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠
⠀⠀⠀⠀⠀⠱⠂⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠁⠀⠀⠀⠀⠀⠚⠳⠖⠋⢹⠂⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠄⠠⣤⢦⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠉⡆⠄⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣰⠁⠈⠿⣽⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣀⠀⠀⠀⠀⣀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⢀⢰⠋⠈⠉⠑⠠⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡔⠐⡄
⠀⠠⢠⡌⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⡠⠀⠀⠀⠠⠈⠉⢉
⠀⠀⢈⠀⠀⠀⢀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⡤⠜⠚⠉⠓
⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣸⠯⠙
⠀⠀⠀⠀⠈⠂⠤⠠⢀⣀⠀⣠⠆⠀⠀⠐⠁⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢹⣀⠁⠀⠀⠄⠀⠀⠀⠀⠀⡠⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠀⠀⠀⠀⠈⠣⠄⡀⠀⠀⠀⠀⠀⡀⠋⠀⣀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠐⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠀⠀⠈⠢⢀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠋⠀⢀⠀⠀⠤⡄⠀⠀⠄
⠀⠀⠀⢁⠤⠠⡀⠁⠀⠀⠀⠈⠂⠀⠀⠀⠁⢀⠀⠁⠂⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢐⠁⠀⠐⠀⠀⠸⡀⠈⠳⡄
⠀⢀⠀⠸⠤⠞⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⡙⠳⡶⠶⡶⠶⠛⠀⠀⠀⠀⠀⠀⡋⠀⠐⠀⠀⠀⠀⠘⡆⠀⠈⢂
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠈⠓⠦⠄⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠋⠀⠀⠀⠀⣀⣠⣤⠶⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⢀⣠⠾⠋⣼⠃⠀⣀⣴⠶⠛⠋⠁⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣿⠃⠀⠀⠀⣰⠟⠁⠀⣼⣏⣴⠟⠋⠁⠀⠀⠀⠀⣠⣾⣋⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣿⠀⠀⠀⡼⠃⠀⠀⢠⡿⠋⠁⠀⠀⠀⢀⣠⣴⠿⠋⠉⠉⠉⠛⠛⠿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣀⠀⠀⢸⣇⢹⡄⠀⣸⠃⠀⠀⣴⠋⣠⠞⠋⢀⣴⠾⠛⢉⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣼⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⠛⠿⣾⣿⡘⣧⣰⠏⠀⢀⡾⠁⣴⠋⢀⣴⡿⠶⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣫⣿⠻⣶⡀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢹⣧⠸⣧⡀⠀⣾⠁⣼⠃⠀⣀⣤⠶⠒⠒⠀⠀⠀⠀⠀⢀⣀⣤⣴⡾⠛⠁⠀⠙⠋⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣽⣶⠾⠛⠻⣧⠘⢷⡜⠃⣼⣏⣤⠾⠋⠁⣀⣀⣤⣶⠶⠿⠟⠛⠛⠋⠉⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠁⣀⣤⣤⣤⣶⠿⣧⣈⣻⡆⢹⠟⠁⠀⣴⠟⠋⠉⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠸⣿⡄⠀⠀⠀⣿⣦⠀⠀⢿⡆
⠀⠀⠀⠀⠀⣴⠟⠁⢀⠴⠛⠉⠁⢀⣀⣤⠔⣨⡿⠛⠁⠋⠀⠀⠀⢸⡇⠀⢠⣄⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠈⠀⠀⠀⠀⠘⠿⠀⠀⢸⣿
⠀⠀⠀⠀⣼⠃⠀⠀⣀⣤⣴⠶⢻⣿⠟⢁⡾⠋⠀⣰⠂⠀⠀⠀⠀⣼⠇⠀⠈⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⣰⠃⣠⣶⠟⢻⡿⠁⣰⡿⠁⣠⠏⠀⠀⣰⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠈⣿⡇
⠀⠀⣰⣧⡿⠋⠁⢀⣿⠁⢰⡿⠁⠀⠏⠀⢀⡼⠁⢿⡄⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠆⠀⠀⠀⠀⣶⡄⠀⠀⠘⠿⠀⠀⣿⡇
⢀⣴⠿⠋⠀⠀⠀⣼⡇⠀⣿⡇⠀⠀⠀⣠⡟⠀⠀⠘⢿⣦⡀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⣿⡇
⠚⠁⠀⠀⠀⠀⢠⣿⠀⠀⣿⠀⠀⠀⣰⡟⠀⠀⢠⣀⠈⠙⢿⣦⡀⠀⣿⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⡇⠀⢠⡿⠀⠀⠀⠘⣿⡆⠀⠀⠙⢿⣦⢸⡇⠀⠀⠘⣿⡧⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⢷⠀⣾⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠹⣿⡇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⢸⣷⢰⣦⠀⠸⣧⣿⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⡄⢻⡇⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠁⠀⢸⡇⠀⠐⣶⡄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⠀⠘⣿⣦⠀⠀⠀⠀⢀⡀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠁⠀⠀⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡄⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⢸⣿⡀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣄⠀⠀⢻⣦⡀⠀⠀⠙⢿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣤⣌⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠘⢿⠇⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠘⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⢿⣿⣶⣦⣤⣀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠈⠻⣷⠄⠀⠀⠘⢿⡧⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⢶⣶⣶⣴⣶⠶⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢦⠹⡌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠘⡄⣿⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢰⠃⣿⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢀⠏⣸⠇⠀⢰⢿⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⡟⠀⠀⡞⢈⡇⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣾⠁⠀⡼⠁⣼⠃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣰⠏⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⡶⠶⠖⠚⢻⡀⠈⠁⣿⠀⠈⠀⠀⠀⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⣋⣥⣤⠴⠶⠶⠒⠈⠻⠦⡘⠙⢧⡘⠛⠳⠶⠶⠤⢤⣀⠉⢻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⣿⣥⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠾⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠻⠟⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⣀⣼⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢁⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⡇⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢹⡇⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠖⢋⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣡⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⣶⠶⠾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢰⡁
⠀⠀⠀⢀⣀⣤⣶⠾⠟⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀
⠀⢀⣴⡿⠋⠉⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⡶⠶⠖⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡄
⠀⢸⣿⠀⠀⠀⠀⣠⣶⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢤⣄⠀⠀⠀⠀⠀⣹
⠀⠈⢿⣦⡀⠀⠀⢿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠄⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠙⠿⣶⣄⣈⠙⠿⢷⣶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠞⠋⠁⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠿⠶⣶⣤⣬⣉⣉⡙⠛⠛⠛⠻⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠛⠛⠋⠉⠉⠁⣀⣀⣠⡤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠻⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠛⠛⠋⠉⠉⠉⠁

⣦⣶⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣽⣭⣭⣿⣵⣮⣽⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣶⣶⣾⣷⣶⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣶⠋⠉⠉⠉⢩⠏⣉⡉⠉⠉⡏⠉⠉⠉⠉⠙⢲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢧⣀⣰⡏⠉⠛⠋⠁⠉⠓⠶⠛⠓⠲⠒⠒⠶⣍⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⢠⡇⣠⠶⠞⠁⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠸⣇⣀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⠷⣽⠆⠀⠀⠀⣠⠴⠒⠛⠉⠉⠉⠉⠉⠉⠓⢦⡀⠀⠀⠀⢹⣽⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡗⡇⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠠⣎⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡟⣇⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢸⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣸⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⠷⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡾⠃⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣄⠀⠀⠀⠉⠉⠉⠀⠀⠈⠉⠉⠉⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⣄⣀⣠⠤⣤⡤⢤⣀⣠⠖⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢖⣛⣛⣭⣭⣛⣛⣒⠶⢤⣄⡀⠀⠀⢀⣠⡤⠶⣒⣛⣛⣭⣭⣛⣓⡲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠵⠋⠉⠀⠀⠀⠀⠀⠀⠈⠉⠓⠢⢭⣓⣚⡭⠔⠚⠉⠁⠀⠀⠀⠀⠀⠀⠉⠓⠮⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⡄⠀⠀⠀⠈⢦⢳⡀
⠀⠀⠀⠀⠀⠀⠀⡾⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣉⠉⠀⠀⠀⠺⠿⠆⠀⠀⠀⠀⠈⠈⠀⠘⠀⠀⠀⠘⡇⣷
⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠙⢆⠀⢀⠤⠤⢄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣟⢹
⠀⠀⠀⠀⠀⠀⠀⣷⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⣶⠁⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣎⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠛⠀⣇⠀⠃⠀⠀⢈⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⡗⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⣀⡔⠹⣄⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠉⠛⠋⠉⠀⠀⠈⠙⠛⠉⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢙⠇⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠸⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸
⠀⠀⠀⠀⠀⣀⡤⢶⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡶⢤⣀
⠀⠀⠀⣠⠞⠁⣠⢼⠇⡇⠀⠀⠠⠒⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡧⣄⠈⠳⣄
⠀⠀⣼⠁⢀⡞⠁⢸⢰⡇⠀⠀⠈⠒⠊⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⡇⠈⢳⡀⠈⣧
⠀⢠⡇⠀⢸⠀⠀⢸⢸⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⡇⠀⢸⡄
⠀⠘⡁⠀⠸⢦⡀⢸⡎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢱⡇⢀⡴⠃⠀⢈⠃
⢀⡞⠁⠀⠀⠀⢧⣀⣳⡹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⣞⣠⡼⠀⠀⠀⠈⢳
⣼⠀⢀⣴⡦⣄⣀⣠⠇⠙⢮⣑⡢⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠤⢔⣫⡵⠋⠸⣄⣀⣠⢴⣦⡀
⡇⠀⢸⠇⣧⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠶⠶⠶⠶⠶⠶⠶⣶⠶⡶⢶⠶⣶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⣾⠸⡇
⠙⠲⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⣿⠀⡇⢸⠀⣿⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠀⠀⠀⠈⠉⠙⠓⠿⠀⡇⢸⠀⠿⠚⠋⠉⠁⠀⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡗⢺⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡟
⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣺⣿⡻⢓⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⠿⠕⠒⠉⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⢀⣠⣾⣛⣹⣿⣶⣶⣿⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣛⣉⣭⣽⣿⣿⣿⣿⣿⣿⠛⠋⠉⠁⢀⣀⣀⠠⠠⠄⠒⠈⠁⠀⠀⠀⠀⠘⡆
⢀⣀⣠⣤⣤⣴⣶⣶⣾⣿⡿⠿⠟⠛⠛⠉⠉⠀⠀⠘⢿⣷⡀⠀⠉⠉⠁
⠈⣻⡛⢭⡉⠉⠉⠁⠀⠀⣀⣀⡀⠀⠀⠐⠊⠉⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⣾⣿⣿⣮⣿⣆⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⣀⠤⠚
⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⣀⣀⠤⠤⠔⠒⠊⠉⠉
⠹⣿⣏⢿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠷⠒⠊⠉⠉
⠀⠹⣿⣷⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠊⠉⠉
⠀⠀⠈⢿⣿⣿⣿⡟⠀⣀⡠⠴⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⠀⠀⠀⠙⢤⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⢆⠀⠀⢐⢒⣂⠀⠀⢿⡆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠈⡄⡧⠌⣴⣓⣀⠒⠂⣨⣷⣘⡇⠀⢀⡈⠀⠀⠀⠀⠀⣀⣀⡀⠠⣤⠤⠄⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠁⠐⡓⠷⢴⡏⠀⢙⡟⢻⡁⠀⢉⣷⣄⡤⢰⠶⠖⠚⢉⣉⠡⠄⠀⠒⠈⠙⠉⠌⡗
⠀⠀⠀⠀⠀⢋⢭⡉⠙⠛⠒⠲⠶⠤⢤⣀⣐⠠⠤⡀⢀⢀⡀⠀⠀⠀⠐⢏⢠⠀⠙⢒⣋⣠⣤⢿⢖⡋⠡⠄⠒⠀⠁⢈⢁⢀⡀⣤⠢⢺⠆⣯⢍⠀⢠
⠀⠀⠀⠀⠀⠈⡄⠅⠀⠈⠡⠀⠀⠀⠀⠀⠀⠉⣉⠒⠈⠑⠪⢽⡒⠴⡖⣤⣹⡠⠴⢚⠩⠑⠈⣀⠀⣤⣠⢤⠒⢥⡀⠤⠈⢋⡙⠙⢀⣨⣤⣀⠖⠂⢰
⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠄⠀⣡⠀⠀⠀⠈⠕⠊⠂⠀⠚⠉⠀⢶⣄⠁⢹⡂⠈⢃⠩⢘⡩⠏⠈⠙⣐⠘⠤⠄⠰⠒⡢⢖⠂⢠
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠆⠈⠌⡠⡂⣛⠼⢙⠂⢸⠀⠀⠀⠀⠄⠘⠀⠀⠀⢀⡈⠄⠒⠬⣈⠠⠄⠒⡀⠈⡉⠀⠰⢲⠰⣫⣑⠀⠃⢸⠀⠘
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠥⠄⠀⠀⠁⠒⠀⠀⡂⠀⢁⠁⢸⠀⠀⠀⢀⠀⠀⠀⠀⠊⣈⠄⠐⢊⡡⠤⢄⢠⠏⡁⡆⡴⠮⢀⢨⠢⠑⢠⠀⣛⡛⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢁⡠⢤⠈⠑⠀⠀⠬⢉⣃⠐⠦⠭⠄⠈⠁⠀⢐⠀⠀⠀⠀⠀⠀⡡⠔⢀⠄⠀⢘⡳⢁⠂⠀⡆⠑⢌⠐⣆⠐⠔⠤⢿⠀⣳⣉⠐⠳⡄
⠀⠀⠀⠀⠀⠀⠔⢈⠤⢾⡀⠀⠀⠀⠂⡤⢀⡙⠀⠠⠀⠈⠅⠀⠀⡀⠠⠀⢠⠀⠎⢠⠋⠑⠃⠀⠈⡁⢆⠀⡄⡃⣎⠓⢠⠻⠴⠈⢉⣡⡄⢸⣁⣁⠈⡇
⠀⠀⠀⠀⠀⠀⢣⠰⠶⡞⠀⣀⠀⠀⠒⠠⠡⠐⡀⠨⠥⠒⡀⠀⢀⠭⠄⠀⢸⢸⠀⢠⠀⠀⡀⢸⠈⠑⡈⡂⠣⠀⠈⣉⡤⠴⠂⠉⡁⠠⡔⢸⣉⡉⡸
⠀⠀⠀⠀⠀⠀⠨⡀⢶⠁⠀⢉⡙⠀⠒⠆⢰⣠⣀⠈⠁⠒⠀⠀⢐⡛⠰⠀⠸⠀⠣⣀⠉⡁⣈⠀⠚⡁⠈⠤⢖⡊⠄⠀⠒⠀⠇⡀⠄⠑⠀⠸⢧⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠟⠃⡠⠀⠄⡐⠂⠀⠐⠂⠠⠀⢁⠘⠖⠲⢠⣄⣈⠉⠀⢸⠀⠀⠈⣉⠉⠡⢞⡛⠇⠰⠈⠵⠲⠄⠀⣾⠱⠗⠿⣿⣻⡆⠰⡿
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠨⠁⠀⣀⠐⠀⠌⠀⢂⠀⠁⠀⠄⠉⡒⠀⢸⠀⠀⡉⠭⡐⠈⢝⣃⠀⣤⢀⣂⣤⣶⠦⣤⣠⣤⡼⠿⠛⠁⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀⠀⠀⠀⠈⠠⠄⣀⠠⠄⠁⠀⠀⣁⠀⠀⢸⠀⠄⢂⠨⠐⡨⢵⡆⠀⣤⣠⡬⠤⢦⣤⡤⠤⠴⣦⣤⠤⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠄⠀⠤⠍⠉⠁⠓⠲⠤⠄⠀⡀⡀⠀⠀⠠⢀⠈⠀⠘⠀⠩⢐⠀⠐⠀⠍⠀⠀⠭⠐⠀⠠⠀⠔⠠⠀⠈⠃⠀⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢃⠀⠁⠀⢀⣀⠀⠀⠤⠀⠭⡐⠂⠀⠉⠑⢲⠀⠀⠀⠀⡆⠀⠨⠐⠊⠤⠖⠂⠀⠀⢈⠀⠈⠀⠀⠒⠀⠀⠈⠑⣂⣀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠈⠀⠄⠀⠠⠈⠀⠀⢀⠁⠠⢒⡩⢀⠀⡁⠀⠀⢀⡠⠄⠒⠒⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⠤⠤⣀⣁⠀⠁⠀⠀⠁⠄⠀⠀⢸⠀⢐⡩⠐⠈⢁⡠⠔⠈⠁
⠀⠀⠀⠀⠀⠀⡠⢠⠠⣐⠠⢐⠀⢀⠀⠀⡀⠄⡀⠉⢐⠢⠤⣀⠀⠀⠀⢸⠀⢀⡠⠐⢈⠀⡀⢀⡠⢀⠀⢀⠠⣀⠄⡀⢐⠠⢐⠠⡀⠄⡠⢀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠁⠐⠁⠒⠨⠀⠄⢂⠄⡉⢕⡲⠰⠊⠁⠔⡫⠀⠅⠂⠅⠊⠔⠈⠔⠈⠐⠡⠂⠁⠂⠔⠈⠐⠁⠈⠀⠐⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠋⠉⠉⠉⠉⠉⠉⢛⣓⣲⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⣀⡤⠴⠒⠋⠉⠉⠀⣀⣀⡤⠤⠼⠓⠒⠒⠒⠶⠤⣄⡀
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣠⠖⠋⠁⣀⣠⠤⠖⠒⠋⠉⠀⠀⢀⣀⣠⠤⠴⠒⠒⠒⠒⠒⠛⢓⣢⣤⣀
⠀⠀⠀⠀⢸⡇⠀⠀⠠⠾⠕⠒⠚⠉⠁⠀⠀⠀⠀⣀⡴⠖⠋⠉⠁⠀⠀⣀⣠⠴⠖⠚⠉⠉⠉⠉⠉⠉⠉⠙⠒⢤⣀
⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⢀⣀⡠⠴⠒⠊⠉⠁⠀⠀⠀⣀⣤⠤⠖⠒⠒⠛⠛⠛⠒⠒⠮⠷⣄
⠀⠀⠀⠀⠀⢻⣥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠒⠒⠉⠉⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠤⠽⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠙⢷⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⣀⣀⣠⠤⠤⠖⠒⠒⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⡴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠴⠶⠶⠶⠦⠤⠤⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠯⣤⣀⠤⠀⣠⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠁⠀⠀⠀⣀⣀⣠⡤⠤⠤⠤⠤⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠾⢶⣇⣔⣁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠶⠶⠶⠒⠒⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣥⣔⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⠧⣄⣤⣀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠺⠷⠶⠶⠶⠾⠗⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⣊⠉⠛⠲⣄⡴⠀⠸⣄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣬⣧⣄⣳⡸⡇⢰⠀⠘⣦⠀⢀⡗⢠⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⡀⠉⠙⠳⢼⡆⠀⣿⡴⢛⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⢴⠆⠀⠀⠚⠋⠀⠀⠀⠀⠙⢦⡛⠉⠉⠛⠓⠒⠲⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢀⣄⠀⠀⢀⡤⢀⠀⠀⠀⠹⣄⣀⣀⣀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡡⠀⢠⡷⠀⠀⠈⠉⠀⠀⠀⠐⠁⠀⠞⡰⠀⢹⡉⢉⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⡄⠀⠀⠀⠀⡈⣧⣈⠢⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠐⠗⠀⠀⢞⠔⠀⠈⠕⠂⠀⠀⠈⠒⠁⢀⡴⡀⠇⣿⢹⡗⠒⠤⠌⠂⠀⠀⠀⠀⠀⠀⠀⢀⡏⣆⢀⡤
⠀⠀⠀⠀⠀⠀⢰⢧⠀⢀⣄⠀⠀⢀⠀⠀⠀⢠⡤⡀⠀⠀⣀⠀⠀⠊⠀⢠⡟⠂⠁⠀⠀⠀⠀⣠⠴⠖⠚⠛⠓⠶⣼⡇⢹⡏⢸
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠋⠀⠈⠫⠆⢀⠀⠈⠉⠀⠀⠈⠥⠊⣠⠔⣠⠟⠀⠀⠀⠀⢀⡴⠋⠁⠀⢀⡀⠀⠀⠀⠈⢳⡄⠃⣸
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠴⢂⠀⠀⢘⡋⠔⣀⣀⠤⠴⣖⣚⣭⠶⠛⠁⠀⠀⠀⠀⣠⠟⠀⠀⠀⠠⣤⡽⠗⠒⠀⠀⠀⠻⣶⣛⠲⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠳⢤⣀⣀⣀⣀⣀⣠⣬⠽⠶⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢀⣨⠶⠋⠁⠀⠀⢀⡴⠀⠀⠀⠈⢯⡙⠦⣄⡤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠠⣤⠞⢁⡤⠖⢊⣤⠞⠋⣀⢴⡂⠀⢘⢸⣧⣀⡂⠑⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⢈⡤⢚⣩⡴⢞⣉⣤⡖⠋⡇⠈⠃⠀⢊⣾⡇⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠄⠀⠛⠛⡋⠓⢚⠏⠀⠘⠀⠀⠀⣀⣠⡴⠛⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣐⣁⣈⣁⣤⣤⡤⠶⣶⠾⠿⠭⠉⠀⡠⢢⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠭⣭⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⠬⠞⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠂⠀⣠⠀⠀⠀⠀⠀⠀⢵⡆⠀⠀⠀⠀⠀⢀⠀⠤⠤⢀⣀⠤⢤⣤⡄⠐⠁
⠀⠀⠀⠀⠀⠈⢀⣤⡤⠔⠋⠁⠐⠋⢁⠀⠀⠀⣰⣤⣤⠟⣫⠩⠉⠀⠀⠀⢯⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⢀⡶
⠀⠀⠀⠀⠀⠐⣿⣥⣀⣠⠀⢀⣢⡤⢽⣖⣻⠋⣩⡿⠀⣼⠉⡠⠀⠀⠀⠀⠀⠙⠒⠶⠤⠤⠦⠤⠤⠶⠶⠚⠋
⠀⠀⠀⠀⠀⢀⣿⡿⣍⣛⠾⡭⠵⢶⠾⢉⣉⠉⠔⣀⣽⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⠤
⠀⠀⠀⠀⠀⠀⣿⡿⠿⣿⣾⣭⣤⣥⣤⠤⠤⠐⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠰⡛⠛⠛⠶⠀⠀⠀⠀⡽
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⡤⠤⣤⢤⣄⣤⣤⣤⣶⠟
⠀⠀⠀⠠⠶⠏⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠉⠉
⠀⠀⠀⠀⠒⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⣾⣶⠀⠀⠀⠙⠷⣭⠶
⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⣾⣻⣦⢄⣀⠀⠀⠀⠀⠀⢀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠻⠟⣿⣿⣿⣿⣿⣳⠲⣾⡿⣏⡉⠩⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢤
⠀⢀⠄⠀⠀⠀⣶⡒⣲⡷⠀⠀⠀⢠⣄⠀⠀⠀⠉⠙⠛⠛⠈⠻⠙⠋⠙⠉⠀⠀⠀⠀⠀⡀⢁⡔⠒⠓⢄⠀⠀⣄
⠀⠘⣆⠀⠀⠀⠉⠉⠉⣀⣀⣀⣤⡟⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⡿⠀⠀⠀⠀⠀⢲⡇
⠀⠀⢉⡷⠶⢶⢶⣦⣷⡴⢞⡹⠿⣛⣹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠐⢀⠀⠠⢸⡧
⠀⠀⠈⠁⠲⠿⣿⡚⠻⣿⠚⣽⠷⠟⠉⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢴⡄⡀⠀⠀⢀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠊⢀⣀⡐⣦⠀⠀⠂⠀⠀⠈⠁⠈⠙⠉⠉
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠘⠿⣦⣄⣀⡀⠀⠈⠁⠀⠀⠀⢠
⠀⠀⡔⠀⠀⠀⣠⠶⠀⠠⢴⣆⠀⠀⢢⠀⠀⠫⣥⠠⡄⠀⡀⠀⠊⢉⡹⣖⣶⠖⠛⠁⠀⢀⡤⠄⠒⠒⠂⠤⢤⣄
⠀⢰⡇⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⣼⡇⠀⠀⠘⠳⠤⠭⠥⠤⠾⠿⠿⠛⠁⠀⠀⠀⠀⠘⠃⣠⠴⠶⠦⠐⢲⡟
⠀⠈⢿⣦⣀⠀⠀⠀⠀⠀⠀⡀⠀⣨⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⠀⠐⠉⠉⠻⣤⣤⣤⡴⠟
⠀⠀⠀⠉⠛⠻⢶⣤⢭⣭⣤⠄⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡤⠖⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠀⠀⠀⠀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⠃⠀⠀⠀⠀⢠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠠⠆⠉⠈⡈⠉⣁⠉⠀⠀⠀⠀⣼⣿⣿⠿⣽⣶⣶⣶⣮⣉⠙⠁⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠴⠀⠀⠀⠀⢀⠀⢲⣂⣀⣤⣤⣶⠂⠀⡀⠀⠁⠘⠛⣩⠉⠿⣿⣿⡆⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠄⠀⣸⠃⠀⠀⠀⠀⠀⢈⡀⠁⣙⡂⠐⠛⢿⢷⣄⡀⠈⠃⣬⣿⡿⠁⢘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⢌⣙⠀⠤⣄⣀⠀⠈⠉⠀⠩⠁⣠⡀⠀⢐⠛⢛⣛⡻⠞⠋⣤⡞⢾⣏⣀⣄⣈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠻⣶⣶⠄⠀⠀⠀⡀⠐⠓⠛⠓⠂⣙⡋⠩⣷⣴⡀⢠⡿⣧⣿⡟⠁⠀⠙⣯⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠛⠁⠀⠀⠉⠄⠈⢸⠏⣿⠇⠀⠀⣠⡿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠻⠶⠶⠛⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠷⠟⠛⠿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣿⣿⣷⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢀⡴⣿⣗⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⢀⣶⣿⡿⢟⣳⡄⠈⠉⠛⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢘⣿⣉⣻⣷⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠘⠀⠀⣛⡾⣿⣟⣩⣿⣥⡀⠀⠱⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⢁⣯⣽⡟⠉⣻⠏⣿⣾⣧⣤⣀⣀⠀⣀⢀⠀⣀⣁⣠⣶⣿⣫⢊⡽⣒⣿⣿⣷⠞⠀⠀⠀⢘⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢛⣥⡀⣌⠉⢙⠛⠻⠿⠿⠿⠿⠿⡿⠿⠿⣿⣟⢍⣯⣱⣝⠋⠁⠀⠀⠀⢀⢰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠞⠁⠰⡾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⠉⠉⢁⠀⢀⡮⢶⡋⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠰⠤⠦⠤⠤⠤⣄⣢⣤⣄⣤⡐⠿⠯⠁⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠠⠠⣀⠀⠀⠀⠀⢀⡴⠖⠋⠉⠉⠉⠛⠛⢛⠒⠢⠤⡀⠀⠀⠀⠀⠀⣠⡤⠶⠲⢤⣄
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⣀⣠⣤⣨⡾⠳⠦⣄⠀⣏⢮⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣐⣈⣆⠀⢀⡤⠻⢿⡄⠀⠀⢸⡿⠓⠦⡀
⠀⠀⠀⠀⠀⠀⢿⡦⡊⠁⠀⠉⠟⠛⠀⡀⢀⠸⡆⠸⣷⠇⠀⢠⣴⣟⣗⡆⠀⠀⢨⣽⣿⠋⢀⡾⠆⠀⠀⠃⠀⠀⢸⠀⠀⣠⡿⡄
⠀⠀⠀⠀⠀⠀⠈⣷⡇⠀⠀⡸⢀⠀⠀⠈⠳⣿⠂⠀⢨⠀⠀⠀⠈⠏⠁⠀⠀⠀⠀⣹⠏⠀⣎⠀⣈⡶⠤⠾⠶⠶⠛⠒⠾⢧⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⣠⣜⣇⣑⣐⣀⣤⠟⠀⠀⠘⠳⢤⣤⣴⠀⢀⣀⣠⣤⡬⠏⠀⠀⠰⠜⠁⠀⠀⠀⢀⣀⣀⣀⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣴⣾⠛⢻⣿⣮⡵⣄
⠀⠀⠀⠀⠀⢀⡤⠤⡤⠀⡶⢤⣤⠖⠒⣶⢶⣶⠶⠋⢉⣩⣟⡲⠚⠉⣹⢒⣳⣶⠾⠉⠓⠒⢆⠀⢠⠓⠒⠉⠉⢻⣿⡟⠉⠙⠇⠀⢷
⠀⠀⠀⠀⠀⣏⡀⠠⠴⠋⠛⠁⠤⠒⠛⠛⠉⠀⠐⠚⠙⠉⡉⢀⣒⢉⡋⢉⡉⠀⢀⠀⣀⢄⡼⠀⠸⡀⠀⠀⠀⠟⠀⠙⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠙⠛⠛⠒⠓⠒⠓⠻⠿⠶⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠛⠛⠛⠉⠀⠀⠀⠀⠈⠒⠶⣤⣤⣀⣤⣤⣴⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⢒⡲⠶⠶⠖⢒⣢⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣠⠤⣤⣀⣀⣀⡤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣎⢀⣷⡦⣰⣯⣄⣀⡴⣮⢬⣉⣀⣀⣀⣱⡄⠀⠀⢀⠤⠂⠉⢡⣫⠟⠋⢁⣴⡫⠿⡋⢀⡴⡿⠟⠋⠁⠢⡀
⠀⠀⠀⠀⠀⠀⣾⢸⡟⠻⣆⠙⠽⢧⡌⠛⠷⣌⣀⠈⠳⢤⡈⠉⠙⢦⠀⢺⣠⠀⠀⠟⠁⠀⠀⣸⡿⠁⠀⠀⠚⠉⠂⠀⠀⡀⠩⡿
⠀⠀⠀⠀⠀⠀⠈⠛⠿⣄⣘⣤⣄⠈⠻⡀⠀⠈⠛⢅⡂⠀⣙⣦⠴⠋⠀⠀⠙⠷⣆⣀⠀⢀⣤⡅⠀⠀⢀⣀⣀⠀⣀⣤⣯⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠓⠒⠚⠑⠒⠛⠚⠉⠉⠉⠁⣀⣀⣀⣀⣠⣤⣀⡈⠉⠉⠙⠛⠒⠚⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⢤⡀⠀⠀⠀⠀⠀⠀⡠⠒⢗⠖⢛⠍⡭⠊⢡⠔⠂⠈⠛⡆⠀⠀⠀⠀⠀⣀⣤⡤⠀⡒⢢⣤⢄⡀
⠀⠀⠀⠀⠀⠀⣰⢇⡴⠊⠛⢄⠙⣆⠀⠀⠀⢀⠎⢰⠫⠊⡠⣣⣿⣴⠞⠁⠀⠀⠁⢠⠃⠀⠀⢀⢔⠟⢛⣽⠛⡄⡦⢠⢘⡦⠙⣆
⠀⠀⠀⠀⠀⠀⡏⣼⢇⣀⡤⠤⠶⢿⡀⠀⠀⣾⢾⡇⣾⠸⡃⡇⠘⣷⠀⠂⠀⠀⠂⠘⠀⠀⣴⠃⡆⢠⡋⣂⡬⢔⡥⠂⢼⡗⠀⢼
⠀⠀⠀⠀⠀⠀⣇⠸⡃⣌⣉⣓⣶⠶⡈⢦⠀⠹⣌⠣⡱⠄⠳⡘⢷⡝⡆⠀⠠⠀⠀⢰⠀⢸⠁⡀⠑⢀⣀⣀⣠⠵⠂⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠘⢦⢷⣅⠙⠈⠀⠀⢸⠋⢧⠀⠈⠱⡌⣎⢆⠘⡀⠃⡇⠀⠀⠀⢁⡎⠀⠘⣷⡈⠲⣦⣤⣤⣤⣤⡤⢴⣾⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢌⡁⠗⠰⠖⢁⡰⠃⠀⠀⠀⠸⣌⣘⡄⡇⡄⢿⡄⠀⡠⠊⠀⠀⠀⠈⠑⠲⠼⠿⠾⠷⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠓⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠖⠛⠛⠛⠛⠛⠛⠲⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⢁⣀⣤⠴⠶⠒⠒⠒⠒⠶⠦⣤⣀⡈⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢀⣤⠞⠋⠁⠀⠀⢀⣀⡴⠶⢤⡀⠀⣀⠈⠙⠳⣤⡀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡴⢋⣠⡴⠚⠓⠒⠛⠉⠉⢡⡤⠤⠙⠉⣉⡙⠳⢦⣀⠙⢦⡀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣠⠟⢁⡿⠀⠀⣟⣉⠀⠀⣀⡤⠦⢽⣥⡀⠞⢉⠉⠀⠀⠙⣦⠀⢻⣄⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⢰⠏⢠⡟⢡⡗⠀⠈⠉⣠⡾⠉⠀⠀⠀⠀⠀⣠⠟⢠⡿⣦⠀⢸⣶⠏⣽⡆⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⣿⣤⠟⢀⡾⠁⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⢰⡏⣠⠟⠀⠈⢳⡄⠀⠀⣿⢻⡀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣇⡀⢠⠟⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠋⠀⠀⠀⠀⢿⠀⡀⣿⠘⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡿⠃⢿⠀⠀⣸⠃⠀⠀⢀⡴⠶⣄⣠⠞⠁⠀⠀⠀⠀⠀⠀⣠⠟⠀⡇⣸⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣆⠀⠘⣇⠀⠹⡄⠠⠴⠟⠁⠀⠈⠡⣤⣀⡀⠀⠀⠀⠀⠸⣇⢀⡼⠃⡿⢰⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠈⣿⢷⣀⡀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣤⡄⠀⡿⠾⠁⣸⠇⣼⠁⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠘⣆⠉⠙⣦⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠃⣸⠃⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠘⢧⡀⠈⠛⠛⡷⠘⠷⣤⣄⣠⣤⣤⣤⣤⠤⠶⠃⣠⡖⢸⡇⢀⡼⠃⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠈⠻⣤⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⣋⡴⢟⣤⠟⠁⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠙⠾⣧⣀⠀⠀⠀⠀⠀⠀⠀⠘⢁⡴⣚⣩⠶⠋⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⡀⠉⠛⠓⠶⠶⠦⠴⠶⠶⠛⠛⠉⢀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⠶⢤⣤⣤⣤⣤⡤⠶⠶⠛⠉⠁






⣿⣟⣛⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠙⣙⣃⠈⠀⠅⠈⠀⠀⢀⠈⢠⠀⠸⢧⣀⡀⠠⣿⣭⠎⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠮⠵⠛⠀⠀⡀⠈⠒⠲⠖⠊⠁⠀⠀⣁⡧
⠀⣿⠁⠀⠀⠄⠀⠀⠀⠈⢀⠀⠀⠀⠀⠁⠉⠻⣿⣧⣀⠀⠀⠐⠀⠠⠰⡄⠀⠀⠉⠁⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠣⠀⠀⠀⠀⠀⠀⠀⠲⠖⠁
⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠤⠀⠄⠎⠁⠉⠉⠁⠀⠐⡾⠷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⠀⡄⣠⡤
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⣇⣑⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠉⣾⠁⠀⠀⠀⠀⠀⢠⡼⠏
⠀⠀⠃⠔⠀⠒⢄⠀⠀⢀⠀⠀⠀⠀⠀⠀⡞⠁⠉⠒⠀⠀⠀⠀⠀⠀⠑⠘⠃⠀⠀⠀⠀⠀⠈⠿⠟⠀⠀⠀⠀⢀⡺⠇⠀⠀⢀⡰⠖⠊⠉
⠀⠀⠆⠀⢐⣔⡼⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠠
⣡⣆⡀⠀⠀⢻⣡⠶⠞⠛⠁⠉⠉⠙⠃⢀⠀⠀⠀⢤⠀⠀⠀⠐⠀⠀⢠⣤⠀⠀⢀⡘⠳⢻⣷⡶⠑⠮⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠿⡷⠄⠀⠺⢣⣽⣦⠴⠤⢤⡀⣤⠊⠀⠘⠀⠀⠀⢀⣀⠀⣀⠐⠀⠀⠌⠛⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⢀
⠋⠠⣄⣤⣈⡿⡟⠃⣿⡷⠿⢿⣄⣁⡀⠈⠓⢠⠤⠿⠛⠓⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠂⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⣴⠂⠀⠈⠉⠐
⡄⣀⠝⣯⡯⠰⢇⠀⡿⠀⠀⠀⢢⢽⣧⠀⠀⠈⠀⠀⠀⡀⠀⠀⢉⡼⠛⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠆⠢⠀⠀⠀⣮⠾⠠⠄⠀⠀⢀
⠛⠋⢁⢿⣶⠶⡌⠑⢇⠀⠀⠚⡙⣸⡿⠀⠀⢀⣀⣀⡄⠠⢠⡶⠁⠀⠀⢰⢀⠀⠀⠀⠀⢀⡀⣁⡠⠀⠀⠀⠀⠀⠀⠀⠑⠦⠀⠈⠙⠂⠀⠀⠀⠀⠙
⠀⣠⠟⠣⠀⠊⠀⠀⠈⠆⢀⢉⣴⠟⠁⠀⠁⠁⣀⢼⡁⢤⡯⠀⢀⡠⢤⣏⣠⣠⣷⣦⣄⠀⠀⠉⠀⢀⠂⠌⠀⠁⠲⠄⠀⠀⢀⠀⣀⠀⠀⠀⣄⡀
⡰⠃⠀⠀⠀⠀⢠⣀⣀⠈⠓⠊⠁⠀⠀⠀⡀⠀⠈⢻⠗⠾⣶⣠⣯⣤⣿⠏⠁⠘⠲⡜⢻⣧⠀⠀⠀⡀⠀⠈⠡⣄⠀⠀⢲⠁⢀⡀⠈⠆⠀⠘⠻⠃
⠁⠀⠀⢀⣤⣶⠏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠸⠗⢪⣿⠃⠙⢻⡄⠀⠀⠼⢂⠘⣿⠀⠀⠀⠗⠀⠀⠀⠘⢦⡄⠁⠠⠎⠀⠀⠠⠀⠀⠀⠀⡠
⠀⠀⠀⠟⠈⠉⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣭⡠⠅⣈⣉⡟⢳⡦⣤⡻⠦⢀⠀⠈⠈⡿⠀⠀⠀⠸⢦⡀⠀⠀⠈⣿⣭⡇⠀⠀⠀⡈⠀⠀⠀⠀⠠
⣢⡆⠀⢀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣉⡙⠆⠰⢾⣽⣹⡆⣁⡥⡄⠀⠀⠉⠓⠚⠁⠀⢀⢀⠄⠐⠛⠒⠛⠒⠿⣿⣷⣤⣤⠄⠀⠀⠀⠲⢶⣖
⡉⠁⡀⠀⠀⠀⠀⠀⠀⠈⠠⠇⠀⠐⠦⢠⣤⠔⠈⠀⠀⠀⣢⠘⢟⠿⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠
⡀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢤⡺⠟⠀⠸⡟⠁⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁
⡡⠊⠀⠌⠀⠀⠀⠡⠀⠀⠀⠀⠿⠀⠑⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣇⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⢦⢤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣻⠀⠀⢀⠀⠀⠀⢄⡲⠿⣦⣄⠀⢸⠀⠀⠀⠻⡁⠀⠀⠀⠀⢀⠼⠇⠀⠀⠀⠁⠀⠈⠙⠃⠈⠁⠄⠀⠀⠀⢀⣤⠞
⠀⠀⣠⡀⠀⠀⠀⠀⠁⠀⠀⠀⢿⡀⠀⠠⠀⠀⠀⠀⠈⠑⠺⠟⡅⠈⠂⠀⠀⠀⠙⢄⡀⠀⠀⢼⡝⠁⠉⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠈⠉⠁
⡢⠞⠁⠈⡄⠀⠐⠠⢉⠀⠀⠀⢺⣧⠲⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠠⠄⠀⠀⠀⠀⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠁⠀⠀⢨⠇⠄⢠⡀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⣈⠄⠀⠀⠀⠁⠀⠀⠀⠀⠲⠀⠀⠀⡀⠀⢀⣶⢆⠀⠀⠀⡀⠁⣀⠂⠀⠤⠴⠖⣴⠆⠀⣴
⣠⠔⢢⣿⡶⠶⡛⡻⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⢠⠁⡀⠈⠀⠀⢠⣜⠁⠀⠆⠀⠀⠀⠉⠁⠀⠈⠀⠐⢢⣼⡄⣾⠃
⢷⢶⣿⡏⠀⠀⠀⡝⠨⣿⡆⠀⠀⢀⠠⠀⠄⢠⠀⠀⠀⠀⢺⣿⠐⠀⡆⠀⠀⠰⠃⠀⠀⠀⢀⡴⠋⠈⠀⠀⠄⠀⠀⣀⠀⠀⠀⠈⠄⠀⠀⠝⣉⣟⣻
⣧⢀⠈⢷⡀⠀⠈⠉⠀⣹⡇⠀⠀⡀⠀⠙⡄⠀⠈⠢⠀⠀⠠⠀⠀⠀⠡⡀⠠⠀⠀⠐⠀⢄⣼⠁⠀⠀⠀⠄⠀⠀⠐⠿⠇⠀⠀⠀⢠⣂⠤⠀⠙⠡⢾
⣀⠛⠸⠒⠙⠑⠢⣀⢀⡼⠁⠀⠰⠁⠀⠀⠈⡷⠀⠀⡗⠀⡑⡀⢦⠀⠀⠑⠀⠀⠀⠀⠀⣾⡇⣀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢿⣷⠂⠐⢠⣦
⠧⠠⠞⠳⠀⠀⠀⠀⠀⠀⠀⠠⠰⠆⠀⠀⠠⠱⠆⠠⠂⠴⠃⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠈⠆⠀⠰⠀⠀⠀⠀⠀⠀⠈⠔⠺⠁⠚⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠂⠐

⠀⠀⠀⠀⢀⣤⡀⣠⡀⣠⣄⢀⣤⡀⣤⡀⣠⣄⢀⣄⢀⣤⡀⣠⣄⢀⣄⢀⣤⡀⣠⡀
⠀⠀⠀⠀⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿
⠀⠀⠀⠀⣿⣿⠀⠀⢀⣀⡀⣀⡀⢀⣀⣀⡀⣀⡀⢀⣀⡀⣀⣀⢀⣀⣄⣀⠀⠀⢸⣿⢰⣶⣶⣶⣶⣶⣄
⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⣧⣿⡇⢸⣿⡟⠃⣿⣇⣾⣿⡇⣿⣿⢾⣿⡹⠿⠆⠀⢸⣿⠀⠀⠀⣿⡇⢸⣿⡆
⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⣿⣿⡇⢸⣿⡿⠇⣿⣿⣿⣿⣷⣿⡇⠘⠻⣿⣶⡀⠀⢸⣿⠀⠀⠀⣿⡇⢸⣿⡇
⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⠹⣿⡇⢸⣿⣧⡄⢸⣿⣿⠸⣿⣿⡇⢸⣿⣸⣿⠇⠀⢸⣿⠀⠀⠀⣿⡇⢸⣿⡇
⠀⠀⠀⠀⣿⣿⠀⠀⠈⠉⠀⠉⠁⠈⠉⠉⠁⠈⠉⠉⠀⠉⠉⠀⠀⠉⠉⠉⠀⠀⢸⣿⠀⠀⠀⣿⡇⢸⣿⡇
⠀⠀⠀⠀⣿⣿⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⠀⠀⠀⣿⡇⢸⣿⡇
⠀⠀⠀⠀⣿⣿⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⠀⠀⠀⣿⡇⢸⣿⡇
⠀⠀⠀⠀⢹⣿⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⠀⠀⢀⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠘⣿⣆⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠙⠛⠛⠛⠛⠛⠛⠃⠀⠘⣿⣇⠀⢸⣿⠃⢸⣿⡇
⠀⠀⠀⠀⠀⠙⢿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣿⠟⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠈⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠄⡀⡰⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠌⠁⣲⣾⣿⡏⠥⠉⠙⣦⣦⡀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠴⠖⠊⠁⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡞⣥⡶⠀⢙⠿⢯⠱⠄⠀⠀⠉⠶⣭⣔⠒⠂⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠂⠐⠒⡀⠀⠀⠼⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⠀⢰⡄⢫⠔⠟⠀⠀⠀⠀⢁⡼
⠀⠀⠀⠀⠀⠀⠀⢀⠐⢃⣀⠀⠀⠀⠴⠃⠄⠈⢷⡐⠀⡀⠀⠀⠀⠀⠀⠸⠑⡃⠘⣀⠶⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⣠⢀⡄⠀⠰⠀⠉⢠⢃⡀⠐⡀⠀⠀⠀⠀⠘⠓⠇⠜⠁⠀⠀⡀⠀⠀⢠⠀⡀⢀
⠀⠀⠀⠀⠀⠂⡄⠀⠐⠁⠀⠀⠀⠀⠚⠀⠀⠀⠇⢸⡃⠀⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣩⠶⠶⠗⢶⢤⣩⣄⠀⠤⠀⠥
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠒⢁⠀⠐⠋⠀⣀⠄⠒⡀⢳⠝⠂⠀⠀⠀⠓⠤⣆⡤⠄⣉⠔⠋⠁⠃⠀⠰⠀⢀⠀⠐⡈⢠⡀
⠀⠀⠀⠀⠀⢀⠋⠀⠀⡀⠉⠠⠔⠀⠀⡁⠀⠐⠁⠀⠀⠀⠀⠺⣠⠆⡔⢈⠙⡚⠁⠀⠇⠀⠨⠆⠀⢀⠘⠀⠀⠇⠐⡈⠂
⠀⠀⠀⠀⠀⠘⣀⠀⠊⠀⠀⠠⠄⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⡌⠠⢘⢃⢀⣰⡃⡆⠀⡀⠀⡂⠀⠀⠠⠀⠀⡐⠀⠀⠁⣴
⠀⠀⠀⠀⠈⠀⠁⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠈⡈⠤⣠⠇⠀⠀⠃⠀⠁⠀⢠⠀⢀⠄⠀⠀⠰⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⡄⠁⡄⠐⠀⠀⠸⠄⠀⠃⠀⡈⠀⠈⠀⠐⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠊⠀⠀⠀⠀⠀⠘⠀⠀⠃⠀⠃⠀⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⢀⠀⠇⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⢔⣹⣃⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠀⠀⠋
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀⢡⣍⣤⡀⢀⣀⠀⠠⠭⠀⠐⠀⠀⠙⠪⠢⠀⠀⠁⠐⠤⢤⣀⣀
⠀⠀⠀⠀⠁⠀⠀⠉⠉⠀⠉⠉⠀⠉⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⢒⠱⣧⣧⠭⣀⠀⠀⠀⠀⠉⠛⠿⣭⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣢⣀⡅⠀⠀⠀⢶⠂⠁⢠⠂⢡⠀⠀⠠⡍⠀⠀⠀⠀⠀⠀⡀⠙⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠛⡷⣿⡆⢅⣄⠀⢶⢂⠀⡤⠀⠀⢰⠐⠂⣤⠀⠀⠀⠀⠀⠀⡅⠀⠀⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠴⠁⠀⢰⠂⢹⣄⠝⠖⢲⡌⡁⢠⠉⠰⠆⠀⢧⡌⠀⢀⣠⣤⣦⣴⡂⠀⠀⢀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡀⣤⢊⢿⠀⠈⣉⢰⠔⠈⡄⠑⢄⠢⠐⠆⠜⠈⠀⠘⠏⠌⠁⠝⢿⡝⡶⠡⠞⡿⢿⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡉⣌⠙⠀⠰⢀⠈⠀⠘⠀⠁⠀⠀⠁⠠⠼⠂⠀⠐⠃⠃⠚⠀⠀⠺⣡⣷⠆⢠⡁⠸⠃⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢨⡀⠱⠀⠐⠂⠨⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⣩⠈⡄⢡⠟⣅⢿⢈⡁⡂⠙⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⡘⠆⠙⠀⢃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃⠄⣁⠶⠀⠁⠀⢥⡀⣌⠁⢣⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠁⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠠⠎⠌

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⢤⣤⡤⠐⠒⣒⣶⣤⠤⠤⠤⣀⣀⡀
⠀⠀⠀⠀⠀⡠⣺⠉⣀⠔⡪⢁⡠⢤⣤⣤⣄⣀⣀⡒⠋⢉⣉⠤⠬⢭⣶⠤⣀
⠀⠀⠀⠀⡀⣼⢃⠞⡱⠊⡔⠁⠼⡳⠄⢒⣒⣒⣚⣛⡳⠦⣄⣐⠀⠀⣀⣀⣀⠱⡀
⠀⠀⠀⠀⢇⠀⠁⠊⠀⢸⠀⠀⠤⠒⠉⠁⠀⠀⢀⡠⠤⠤⠤⠬⠍⣒⣤⣀⠀⠀⡇
⠀⠀⠀⠀⠈⢆⠀⠀⠀⠸⠀⠀⠀⠀⠤⠶⠤⠤⠒⠒⠉⢉⣉⣉⠀⠠⢀⣈⡑⠦⡁
⠀⠀⠀⠀⢀⠤⢀⠔⠀⠐⠢⣀⠀⠀⠀⠀⠀⢀⣠⣄⣀⠠⠤⠒⠒⠉⠉⠉⠛⠋⠺⠶⣄
⠀⠀⠀⠀⠇⢠⡁⠀⠀⠀⠀⠀⠑⠦⡀⠀⠀⠈⠁⠀⠀⠀⢀⡠⠤⠀⠀⢈⣉⣁⣒⠢⢬⡳⡄
⠀⠀⠀⠀⢗⡸⣈⠑⠢⣄⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠘⠋⠈⠉⠉⠉⠀⠀⠀⠀⠈⠉⠙⠚⢆
⠀⠀⠀⠀⠀⠑⢨⡛⠓⠺⣗⠤⠤⠤⠤⣔⡇⠠⢄⡀⠀⠀⠀⠀⠀⠀⠶⠶⢀⣀⣈⣉⡁⠒⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠈⣟⠿⠿⢭⣴⠞⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⠤⠔⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⠰⠶⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠖⣊⣉⠭⡤⠤⢤⠤⠬⢭⣙⡒⠦⣀⠀⠻⠷⣀⣀
⠀⠀⠀⠀⠀⡰⣡⣶⣋⣭⣥⣴⣶⣶⣶⣶⣧⣔⠀⠈⠑⠪⡻⣄⠀⠀⠈⠑⣄⣀⣠⠄⠀⠤⠄⠠⢄
⠀⠀⠀⠀⢀⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣺⠗⠀⠀⠀⡹⣏⣹⠦⠤⠚⡁⠀⣀⡴⠂⠈⠀⠠⠊
⡴⢊⡭⠭⣙⣿⣌⠻⢿⣿⣿⣿⣿⣿⣿⣿⢿⡇⢀⣠⠤⣚⣹⡟⠋⢀⠄⣸⣷⣚⡉⠉⠀⠀⣠⣆
⣇⢸⡀⢀⡼⣿⣿⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉⣍⣴⣿⣿⣿⢧⣤⡷⢞⣉⠁⠀⠀⠀⣴⣉⣠⣄⣙⡢⢄
⠘⢦⡙⠯⣄⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⣿⣿⣿⣿⠀⣌⢧⡀⠭⠀⢠⢶⡉⠻⠀⠉⠈⠉⠂⢘
⠀⣼⠙⡷⠦⣽⣟⠣⠀⠀⠀⠀⠀⠀⠀⠀⣌⠙⣿⣻⣿⣿⠇⢀⡿⣧⢿⣯⣭⣓⢦⣉⣀⣠⠤⠤⠶⠋⠉
⠀⢿⠠⠂⠀⠀⠙⢦⡀⣠⣄⣀⣄⣠⣀⣦⣿⣼⣿⣿⣿⣿⣶⣄⣠⡿⣻⣧⣩⣿⠟⠁
⠀⠘⢆⠠⠀⠀⠀⠀⠙⠺⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣶⠤⠆⣱⣿⣍⣧⣽⣀⡀
⠀⠀⠈⠓⢤⡀⠀⢀⡀⠀⠀⠈⠙⠛⠿⠟⠿⠿⠟⠛⠉⠉⢀⣠⣾⣿⠛⢽⣿⣭⣿⢿⡆
⠀⠀⠀⠀⠀⠈⠓⠲⠤⣍⣁⣀⣀⣀⢀⣀⣀⣀⣠⣤⣶⣞⣩⣿⣿⣿⢀⡘⠿⠤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠛⠉⠉⠉⠈⠿⣯⠭⠶⣿⢿⣝⢧⣀⣾⣯⣽⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣎⢿⣼⠏⠳⠬⡅⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠠⠎⡁⠊⠀⠀⠀⠤⢀⠀⡀⠀⠉⠀⠒⠀⠀⠠⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⠈⠀⠑⠁⠂⠠⢀⡢⠀⠀⡀⠀⠀⠀⠈⠐⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⢀⠀⣀⠀⠀⠀⠀⠀⠰⠁⡆⢚⠎⢠⢂⢔⡒⠀⠀⠄⠘⢀⠊⢁⠀⣖⠤⢀⡀⠀⠀⠀⠈⠁⠀⠐⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⣀⡈⠐⠂⠈⠠⢈⣁⠒⠨⠡⢀⡑⠂⠴⢁⠂⠀⠄⠀⢰⠀⢷⠘⠃⡀⡼⠁⠈⠠⣀⠹⠃⠀⠀⠠⠄⠀⠀⣠⡳⡄
⠀⠀⠀⠀⠀⠀⠔⠀⠐⠢⠬⢁⣉⡒⠠⠀⠀⠀⠀⠀⠂⠤⠃⠰⠀⡄⠄⡀⠀⠘⠤⠀⢰⣀⠟⢀⣂⠀⠠⠈⡇⠀⠀⠀⠀⠀⠀⣼⣾⣃⢥
⠀⠀⠀⠀⢀⠊⠀⠠⠀⠀⠁⠀⠀⠡⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠉⢀⠁⠒⠤⢉⠐⠂⠂⠠⠅⢈⡂⠁⠊⠀⠀⠀⠀⠀⢀⡾⡻⢻⡶⢫⡇
⠀⠀⠀⠀⡈⠀⠀⡠⠀⢂⠀⠁⠂⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠠⠄⠀⠀⠀⡔⠰⠀⡀⠁⠀⠒⠀⠄⠀⠛⢰⣄⠢⠀⣠⢎⠨⣡⣿⡾⢃⣷⡀
⠀⠀⠀⠀⡀⠀⠢⠄⣁⡀⠀⠁⠂⠀⢀⠀⠀⠠⠐⢊⡁⠀⠀⡀⠋⠁⠀⠀⠀⡔⢀⣄⣉⠋⡚⠓⠖⠠⠤⠀⠀⠀⢠⠞⡱⣡⣾⣟⢵⢃⣟⣞⠇
⠀⠀⠀⠀⠡⡀⠀⠁⠀⠀⠀⠀⠀⠀⢄⡀⠒⠨⠉⡀⠀⠀⢄⠤⠐⠂⢐⠂⠈⠀⠂⠀⠀⡈⠉⠑⠒⠂⠀⠀⢀⣪⢋⠈⣈⣾⡿⡡⣱⣿⠝⠁
⠀⠀⠀⠀⠀⠀⠐⠲⢶⣀⡄⠀⠀⠀⠂⠀⠉⠓⠠⠤⣌⠉⡐⠂⠍⠁⣂⠌⠀⠀⢀⡀⠄⠀⣂⠈⠐⠀⢀⢔⠕⢅⢤⣨⢞⠞⠝⣹⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠠⠄⠐⠦⠄⠀⠨⢐⡀⠀⠉⡁⠀⠀⠀⠀⠈⠀⢂⡐⠪⠤⢀⠀⠀⢀⡆⣡⣾⢟⠝⣡⠪⡪⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠤⡀⠀⠀⠐⠀⠅⠀⠀⠀⠄⠒⢀⠄⠀⠀⠀⠀⣞⡴⠩⣿⠋⠔⢁⢄⣴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠤⠀⠀⠀⢀⡐⠀⠢⠀⠁⠀⠀⢸⣾⠀⠀⠘⠫⠴⣿⠛
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠮⠤⣤⠀⠘⣎⡈⠂⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠠⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡨⠐⠨⠀⠀⠀⠄⠈⠀⡈⠀⠈⠀⢀⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠄⠀⠀⠀⠀⢀⠄⡀⠔⡀⠐⡀⠀⠀⠂⠀⡀⢀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠁⠊⠀⠀⠀⠀⠀⠐⠁⠊⠀⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⡛⠿⠛⠻⠻⡻⢛⣿⢿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⠶⠶⠶⣦⣤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣶⣟⡿⠋⡁⠄⠀⣠⡀⠈⠁⠒⠬⢝⡺⢷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠕⣿⡷⠉⢂⠕⠊⠀⠀⠈⣃⠐⠀⠀⠀⠐⢍⠲⣝⢷⣌⠷⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⡴⣫⠞⠁⣾⠏⡄⡔⠁⠠⠖⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⠈⠎⢿⣦⣤⠖⢘
⠀⠀⠀⠀⠀⢀⡞⡴⠁⠀⢸⣼⠘⠨⠀⠀⠀⠁⡀⠀⠀⣴⠄⡀⢁⠀⠀⠀⠀⠀⠀⠘⣼⡇⣠⢾⣧⡀
⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⣿⢿⢀⠀⠐⡀⡀⠀⠘⠦⠄⣀⣄⠀⠈⠀⠀⠸⠀⠀⠀⠀⡇⣿⣥⣀⡀⠀⠈⠀⠒⠄
⠀⠀⠀⠀⢰⠃⠁⠀⠀⠀⢻⠸⡈⠌⢆⢳⣕⢄⡀⠀⠀⢈⠃⡴⣋⢔⠵⠁⠀⠀⡸⢸⢹⣿⠟⠻⠉⢗⡶⠦⠤⠃
⠀⠀⠀⠀⢸⠠⠀⠀⠀⠀⠈⣧⠱⡀⠀⠀⠙⠻⢟⣓⠶⣤⡁⣴⡿⠃⠁⠀⢀⠜⡱⢣⣿⠛⠃⠀⠀⠈⣧
⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠘⣷⣙⢦⡈⠲⢄⡀⠀⠉⠢⢱⠇⠀⠀⠀⠔⣡⢞⣵⣿⠁⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⢍⡲⠤⣌⣁⣀⣀⣀⣀⣠⠤⢖⣫⠴⣻⠞⠁⠀⠀⠀⠂⠀⠀⡟
⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢂⠈⠻⡷⣍⣙⡒⠲⠾⠭⠶⠶⠒⢛⣉⡴⠞⠁⠀⡀⠠⠀⠌⠀⠀⢠⡇
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠠⡣⡈⠮⡛⠻⠷⠶⠶⠶⠖⠛⠋⠉⠀⠀⠠⠊⠀⢀⠌⠄⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠓⠀⢀⠀⠀⠀⠀⠀⠀⡀⠄⡪⢀⠕⡡⠊⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠈⠂⢅⡀⠀⢀⠀⠈⠀⠀⠀⠂⠉⢀⠁⠊⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣔⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢀⠄⣢⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣗⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⣖⣫⠵⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠯⠭⣿⣿⠿⠭⠭⠷⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⢦⡔⢀⠎⠀⣤⢀⠄⠀⡱⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠁⠀⠀⠀⠀⠁⠊⠀⢀⠁⠈⠀⢀⠅⡀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠀⠀⠀⠀⠀⢤⣡⠎⠀⢀⠀⠄⠀⢀⢎⠜⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⡄⠀⢀⡖⠀⠀⠀⡠⠂⠀⢀⠔⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⡃⢐⠎⠀⢀⠊⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣀⠀⠐⠀⠀⠀⠀⠁⠀⠠⠔⠁⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠎⡀⠀⠀⠀⡁⠀⢀⢂⠅⢄⠈⠒⠀⢠⠄⢡⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⡀⢖⠁⠄⠀⠀⠁⠀⢨⡌⠓⠃⠀⠀⠀⠈⠡⠂⠀⠀⠇⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠙⠸⠒⢁⡀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣀⡄⠤⢉⢀⡔⠘⠓⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠀⠈⡀⠀⠀⣰⠆⠀⠀⢠⠀⠀⠀⠀⠀⠀⠰⡀⢉⠀⠀⠄
⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠀⢀⠄⠉⠁⠻⣚⠠⠧⠄⣄⠀⠘⢆⠀⠀⠀⠀⣀⠀⠁⠈
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠈⠄⢀⠀⡠⠇⠀⠀⠀⡁⠀⠀⢄⡀⠀⠀⠀⠉⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠴⠃⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡇
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠍
⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠎
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⢄⡀⣀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣟⣿⣿⣻⣿⣿⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠴⢴⣄⡀⠀⠈⠀⢀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⢄⡠⠀⢘⣷⠶⠄⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠾⠀⠀⣻⠉⠉⢘⠀⠄⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⣿⠗⠵⠶⢲⡒⡖⢚⠩⠆
⠀⠀⠀⠄⡀⠀⠀⠀⣀⡴⠧⠷⠞⠛⠾⢦⠤⣘⣄⣿⣼⣤⣇⡀
⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⣤⠀⠀⠀⠀⣄⣀⠈⠋⠉⠉⠉⠉⠉⠙⠓⠒⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠓⠲⠤⠐⠞⠉⠀⠰⠈⢂⠀⠀⠀⠀⠀⠀⠀⣀⣀⡁⠂⠠
⠀⠀⠀⣠⠾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠷⠒⠈⠁⠀⢀⠀⠀⠀⠈⠉⠰⠀⠠⡀
⠀⠀⡾⠋⠀⢘⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡡⠀⠀⠀⣶⠆⠀⡀⠀⠀⢦⡼⠀⠀⠀⠀⠐
⠀⡼⠀⢠⡀⠉⠀⠉⡈⠓⠢⢤⣤⣄⣀⣀⣀⣐⣸⠀⠁⠀⢀⠀⠈⠓⠐⠗⠚⠛⠋⠀⠀⠐⠆⠀⠀⡈
⠀⡷⠀⠈⠁⠀⠀⠀⠉⢁⣀⡒⡄⠀⠀⠩⢩⠿⣇⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄
⢀⠁⠀⠀⠀⠀⠀⡤⠐⠛⢻⡧⠅⠀⠀⠀⢸⡄⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⡴⠂⢀
⠘⣇⠤⣤⣤⠠⠈⠀⢠⠈⠈⠀⠀⠀⢓⠲⠋⠄⠀⠘⠶⣕⡂⢤⣄⣀⣀⢄⣀⣄⠠⠠⠖⡊⠁⠀⢠⠞
⠀⠘⢷⡠⣤⣦⣄⣀⠀⣂⣤⣤⣴⡶⠿⣊⠕⠀⠀⠀⠀⠈⠙⠛⠶⠶⠧⣦⠤⠖⠒⠂⠀⠀⡠⠖⠁
⠀⠀⠀⠙⠻⣶⣬⣭⣉⣉⡩⠭⠴⠒⠛⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠉⠋⠝⢒⠤⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠠⠤⠤⠤⠤⠄⠒⠀⠀⡈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠒⠂⠉⢉⡀⡀⡀⠀⠠⠄⠠⠒⠒⡂⠀⣠⡇⣀
⠀⠰⠂⠀⠚⠉⠉⠉⠀⠠⠤⠀⢦⠀⠀⠀⢠⡠⠠⠐⠐⠒⣉⣉⡉⣄⠀⠠⡀⠀⠈⠁⢯⣀⡀⠀⠀⠀⠂
⠀⠀⠀⠐⢄⠈⠀⢳⡄⠤⠄⠖⠐⠁⠀⠀⠐⣦⡀⠈⡀⠀⢧⢀⠀⠘⢦⡀⢙⣄⡸⠀⢀⠉⢹⠆⠀⢀⣀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠑⢄⠀⢄⠀⠀⠀⠀⠀⠠⡀⠀⢹⡙⢦⣱⠀⠸⡏⠀⡀⠀⠳⠎⠈⠃⢀⡙⠂⠩⠄⠐⠂⠉⠀⠀⣄⠀⠀⠐
⠀⠀⠀⠀⠠⠀⠈⠣⠈⠣⠀⠀⠀⠀⠀⠐⠀⠀⠓⠀⠉⠀⡀⠁⠀⠁⠀⠒⠂⠈⠁⠀⠀⠀⠀⠀⠁⠐⠀⠀⢀⡀⢚
⠀⠀⠀⢠⠀⢁⠀⠀⠈⢄⠀⠀⠀⣀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠴⡈⠇⠸
⠀⠀⠰⠈⢣⠀⠡⠀⠀⠀⢠⠀⢄⣠⣤⡶⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠈⠀⠀⢄⡶⠄
⠀⠀⠀⠑⡄⠁⠀⠁⠀⠀⠀⠐⠀⠁⢀⠀⠀⢀⠐⠀⠀⠈⡀⠀⠀⠀⠈⠉⠁⠐⠂⠀⠀⠐⠂⠀⠀⠈⡄⠀⠀⠂⠑⡈⠀⡀⠀⠁
⠀⠀⠀⠀⠈⢆⠀⡀⠱⡀⠀⠀⠰⡀⠎⠐⠀⠈⢀⠀⠀⠀⠀⠀⠀⢰⣀⣀⠈⢀⣠⠳⠐⡂⠀⠀⠀⠀⠐⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠁⡀⢄⠘⣄⠀⠀⠐⡀⢀⠀⠀⠀⠂⠀⠀⠀⡀⠀⠈⠉⠹⠟⠛⠉⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠡⡈⢢⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠱⡀⠀⢠⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⢣⠈⢆⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⠐⣀⣀⡠⠄⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⣀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠤⠀⠐⠂⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⢈⣀⣀⡀⡀⠀⠤⠠⠀⠀⠀⠐⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⢀⠀⠈⠡⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄
⠀⠈⠁⠀⠀⢸⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⣴⠥⠀⠰
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠠⢄⣒⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠂

⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⣆⠄⢀⠀⠀⠀⢰⠀⣰⠀⠀⠀⢰
⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠏⠾⠈⠳⠁⠼⠸⠇⠘⠠⠿⠑⠼⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⡀⡀⠀⠀⠀⡀⠀⣀⡀⠀⢠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠱⡀⠈⠁⣀⣼⢀⡀⠜⠀⠀⠀⠀⠀⠀⠇⢀⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠠⠀⠀⠀⠀⠀⠀⠈⠉⡋⠉⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⣀⠄⠐⠀⢠⠀⠐⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠠
⢀⡀⠀⢰⢁⡈⢀⡶⢾⣹⡾⡿⠂⠀⠀⠄⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢐⣒⠀⠂
⠈⠻⠀⠈⠁⠠⢭⣙⣤⣄⣘⣭⡼⠶⠞⠀⠀⠀⢐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠐⠪⢙⠒⠀⠤⠤⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡠⠀⠀⠀⠀⠀⠀⠉⠂⠒⠒⠂⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠔⡒⠒⠋⠩⠉⠉⠉⠉⠉⠉⠙⠒⠒⠦⣄⡀
⠀⠀⠀⢀⡠⠔⠚⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⣄⡀
⠀⢀⠔⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣤⢤⣤⠄⡤⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦
⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣢⣥⠦⠤⠾⣮⣤⠞⠀⠄⠀⠀⠀⠁⠀⠀⠀⠈⢣
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢸
⢻⣄⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⠤⣤⣀⣀⣴⣿
⠈⢿⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠠⠀⠀⢀⡤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⢤⣀
⠀⠈⠻⡌⠑⢦⣄⣀⠀⠀⠀⠀⠁⠀⠀⠒⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠘⢦⣀⠀⠉⠏⠛⠲⠲⠲⠤⢤⣤⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠳⡄
⠀⠀⠀⠀⠀⠉⠳⢦⣐⣄⡀⠀⠠⠠⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠚⠒⢶⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠋⢉⣽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠰⡇⢀⠉⠀⠀⠙⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⡀⠀⠀⠀⠀⠈⢣⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣤⡤⠤⠚⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠠⠔⠀⠀⠀⠰⠖⠐⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⣠⢶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠷⣒⣒⡶⠦⡤⠤⣴⣖⣲⡫⠝⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢤⣤⡤⢤⣄⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⡤⡤⠤⣀
⠀⠀⠀⠀⠀⡀⢤⠒⠋⢹⡖⠂⠈⢳⡌⢂⠙⣿⠣⡀⠀⠀⠐⡟⠡⠔⣛⣫⣶⣦⠀⠀⢀⠎⠀⣀⠤⠜⠀⠧⣀⠀⠡⡀
⠀⠀⡠⠒⣭⢄⠸⡍⠳⡈⢳⡀⠀⠀⢻⡌⡄⠘⣄⡇⠀⢸⠀⠃⢠⠊⠉⠉⢉⣈⣁⠀⣸⡀⠚⠀⠀⡄⢠⠀⠀⠱⠀⡧
⠀⠰⠁⠀⢿⠀⠂⠹⣆⢱⡈⢷⡀⠀⠘⡇⣧⣶⠟⠀⠀⠘⢧⡀⠐⠔⠒⠉⠉⢉⡟⠀⢿⣧⡀⠀⠀⠸⠂⠀⠀⣀⣴⠇
⠀⢠⡀⠀⠈⠓⠄⣦⣘⢦⣧⣀⣷⣶⡶⠟⠋⠁⠀⠀⠀⠀⠀⠙⠲⠤⣤⣤⠴⠚⠁⠀⠈⠻⣿⣷⣶⣶⣶⣾⣿⡿⠋
⠀⠀⠉⠛⠓⠒⠛⠛⠛⠛⠛⠋⠉⠀⡠⠒⠀⠁⠀⠠⡀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢉⡩⠭⠟⠫⢤⣀⣀
⠀⠀⣠⠒⠫⣅⠉⠉⠱⣶⢤⠀⠀⡌⠀⠀⠀⠀⠀⠀⠳⠀⢀⠄⠀⠀⠙⡿⠉⠱⡀⠀⠀⢀⣴⡙⠀⠀⠀⠀⢨⠀⢨⣉⠵
⢀⣞⠣⠄⢈⢮⡀⠀⣠⡇⢠⠳⡄⡇⠁⠀⠀⠀⠀⠀⣸⠀⣸⠶⢦⣄⡰⠁⠀⠀⣷⠀⣰⡋⠀⠘⢎⢂⠀⠀⣸⠀⠊⠁
⠠⠁⠀⠉⣷⣿⣿⡾⠛⠶⠾⢤⠁⠘⢄⣀⣀⣀⣠⠴⠃⠀⡇⠀⠀⠘⡇⠀⠤⠞⡿⠠⠀⠬⢷⣤⠼⠿⠶⠛⠁
⠘⠦⠒⠛⣉⣁⠀⠄⡠⠠⣀⡀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠰⣤⡀⠀⠰⣄⠀⠚⠀⠀⠀⠀⠤⠶⠾⠥⣄⡀⠈⠙⠲⣄
⠀⠀⣴⠍⠁⠀⠀⠀⠀⠀⠛⣯⠀⠀⠀⠀⠀⠀⠀⠤⡤⢤⣤⣤⣭⣉⠉⠁⠀⠀⠀⢠⠚⠀⠀⠀⠀⠀⠀⠙⢲⡀⠀⠈⡆
⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣤⠈⠢⡀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢲⣧⡤⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⡴⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⡱⠀⠀⠠⠁⠀⢀⠀⠀⠀⠀⠀⢹⠁⠀⢸
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠑⠦⠄⣀⡀⠀⠀⠀⠀⠀⠀⠈⣿⡾⠃⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣸
⠀⠀⠀⠙⠢⠤⠴⠄⠂⠒⠚⠉⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠒⠒⠒⠒⠋⠀⠀⠀⠀⠘⠓⠢⠤⠤⣄⣀⡀⠀⡼⠖⠉
⡶⠒⢶⢶⡶⠶⠶⣶⢶⠶⠶⡶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⢶⣶⠶⠶⡶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠙⣀⠀⠀⠀⢀⡀⠀⣠⠖⠒⠲⠤⣤⣀⠀⠀⠀⠀⠀⠀⢠⡓⣀⠀⠀⠀⠀⠀⠀⠀⠠⢶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡌⠑⠀⠀⠀⠀⠀⠛⠛⠀⣙⢋⣛⣷⠶⡶⢄⢀⡀⠀⠀⠀⠀⠠⠀⠁⠀⣠⠀⠸⠿⠃⠀⠀⢠⣔⣒⡄⠀⠀⠀⠀⠀⠀⠘⢠⣿⡀
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠰⣦⠀⠀⠀⠀⢠⣼⣍⣭⣉⢈⠙⠛⠿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢴⣿⣷⡄⠀⢠⣾⠛⢛⡃⠀⠀⠀⠀⠀⠀⣠⣾⣿
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣄⠿⠯⠭⡝⠻⠷⠶⢶⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠈⠉⠀⠻⠟⠀⠀⠀⠀⠘⠓⢻⣿⣿
⠀⠀⠀⢛⣻⣶⣦⣤⣄⣀⠻⢿⡿⠇⠀⠉⣠⣶⣂⣀⣐⡢⠄⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏
⠀⠀⠀⠉⠋⠙⠛⠛⠛⠿⠿⠷⠶⠦⠀⠈⠳⣄⣉⣹⣿⣵⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠁⡆⡆⠛⠀⡇⢸⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠃⠇⠘⠀⡇⠀⢣⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠈⢦⡀⠠⠀⢠⠀⢇⢀⡸⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠶⠶⠶⢦⣬⡙⠻⢿⣿⣧
⠀⠀⠀⠀⠀⠀⢠⡴⠛⠁⠀⠔⠂⠂⠀⠀⠈⠉⠛⣷⣄⠙⢷⣾⣤
⠀⠀⠀⠀⠀⣠⡟⠁⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠉⢀⡉⠈⡉
⠀⠀⠀⠀⠀⣿⠃⡴⠄⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠜⢿⣇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠰⠖⠀⠀⢀⠀⠂⠑⠀⠀⣤⠄⠁⠀⠀⠀⠄
⠀⠀⣆⠀⢸⣿⠀⡇⢰⠃⡼⢂⠔⠀⡞⠁⢻⠀⠃⠀⠀⠦⠀⠀⠀⠁⠀⠀⠀⠀⠘⠗⠀⢠⣆⣀⡂⠐⠀⠘⠿⠀⠂⠀⠀⡀⠀⢀⡀⠀⠀⢰⡶⠶⠒⡆
⠀⠀⣿⣦⠀⣿⡀⢳⠀⠒⠀⠀⣂⡀⠀⢰⣿⡀⠄⠀⠀⠀⠛⠀⢋⡀⠠⠀⠀⠀⠀⠀⠀⡸⡯⣿⢷⠄⠀⣘⠣⣔⢀⡀⠔⠁⠀⠛⠁⠀⢰⣤⠟⠀⠈⡇
⠀⠀⠹⣿⣿⢿⣷⡈⠣⡀⠀⠀⠉⢁⣴⣿⡿⣷⣖⠴⠀⠀⠛⠀⠁⠀⠀⠊⢀⡀⠀⣷⡿⢣⠀⠁⣸⠆⠸⡇⣰⡿⠿⠆⠈⠀⠀⠀⠀⠀⠸⠣⠖⠀⠘⠁
⠀⠀⠀⠙⣿⣟⢿⣿⣦⡈⠑⢂⠀⠊⠙⠡⢮⣸⣿⣷⡀⠀⠀⠀⠀⣀⣤⣶⡿⠀⠀⠸⠿⠟⠀⠄⠴⣶⣾⣿⡧⠀⠀⠰⠄⠉⠁⡘⠀⡆⠁⠀⣀⠀⠈
⠀⠀⠀⠀⠈⠻⣿⣝⢿⣿⡦⠊⢴⡢⢀⡀⠈⡟⡽⢯⣿⣠⣤⣶⣾⣿⢿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠮⡅⠠⡀⠘⠣⢧⡀⠚⣲⠏⣾⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠶⠶⠵⠿⠛⠋


⣿⣿⣿⣿⣿⡷⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠶⢿⣶⠿⠿⠷⠿⠷⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⠤⠀⠀⠒⡂⠐⠒⠦⠤⣋⣉⠉⠩⠉⠙⠒⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣚⣋⡀⠀⠀⠀⠀⠒⠀⠀⠀⢀⠀⠀⠀⠀⠉⠓⢦⣀⣀⠀⢄⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠌⠠⠒⠔⠲⠈⠍⠩⠉⠙⠛⠛⠛⠛⠒⠒⠤⣌⡀⠠⡀⠉⠁⠀⠈⠙⠲⣄⢀⠀⡴⠐⠈⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡈⣀⢈⣥⣤⣀⡀⣀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠓⠼⠄⠀⠀⠀⠀⢠⡈⠛⠀⡠⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢠⣞⠽⠚⠁⠉⠀⠀⠐⠉⠍⠉⠉⠙⣛⠓⠚⠣⠦⣄⣀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠠⠀⠀⢀⠀⠀⠀⠨⢆
⠀⠀⠀⠀⠀⠀⡰⡷⠁⠀⠀⠂⠀⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠈⠑⠫⣢⡀⠀⠈⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠈⠀⡀⠀⠅⢪⡀
⠀⠀⠀⠀⠀⠀⠸⣧⡈⠀⠠⠀⠤⠉⠀⠀⠀⠀⠀⠤⠀⠈⠀⠀⠀⠅⠀⠀⣀⠊⠀⢈⡹⣷⣌⠂⠄⠀⢁⠀⠀⠆⠄⠆⠀⢁⡨⠀⠀⠀⠀⣿⣓⣒⡀⡀
⠀⠀⠀⠀⢀⣀⣴⠿⠿⣦⣄⣈⡀⠀⠀⠀⠄⠠⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⠻⣦⠀⡀⢂⡴⠀⠄⠀⠀⠃⠀⠀⠀⠀⠀⣴⡋⠐⠂⠠⠀⢀⡀
⠀⢀⣴⡾⠛⠉⠉⠉⠨⠉⣉⠉⠈⠛⠀⠛⠷⢶⢤⢬⣀⡀⠁⠀⠀⠠⠀⠐⠂⠀⠀⡀⢂⠂⠠⢳⡧⠐⠃⠀⠀⠐⠀⠂⠀⠀⢀⢀⡨⠞⢁⣀⠀⠐⠀⠀⠉
⢠⣨⠁⠀⡀⠀⠀⠈⠁⠀⠠⠀⠀⠀⠀⠅⢀⠀⠀⠨⠍⠐⠛⠿⢶⣤⡀⠀⠀⢀⠀⠀⠀⡩⠂⣾⠷⠔⠐⠶⠳⠶⠾⠶⣶⣷⣶⣾⣿⣶⣶⠶⣶⣶⣶⣷⣶⠖⠂
⢺⠉⠙⠳⢦⣤⣐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠡⠀⢛⢶⡀⠈⠡⠀⠀⢀⣴⡽⠲⠲⠒⢒⠳⠶⠖⠛⠛⠻⠿⠏⢻⣿⣿⣿⣿⣿⠟⠋⠉
⠈⠉⠁⠶⢄⡀⠉⠛⠶⢤⣀⠀⠁⠀⠐⡀⠀⠀⠀⠀⠤⠁⠀⠀⠀⠀⠄⢳⣿⡻⣷⠖⣞⣿⢫⠾⠟⠛⠛⠿⠏⣛⣛⠛⠛⠛⠒⠒⠚⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠉⠒⠲⣄⠈⠉⠓⠶⣠⡀⠀⠀⠀⠀⠀⠀⠆⠈⠤⠀⣀⣎⣼⠤⠩⠀⠀⠉⠑⠒⠒⠢⠥⠒⠒⠠⠤⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⠤⣤⣀⣙⠣⠗⠶⠒⢒⣲⣖⣼⡿⡛⣡⡮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠴⠶⠤⢴⣂⣲⣧⣴⠟⠛⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠀⠒⣶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⢠⠁⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢻⡷⠦⠤⢾⠧⢴⠚⠻⠛⠶⠤⠤⠤⢤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠠⠀⡸⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⠳⢤⡤⠃⠀⡎⠠⠀⡤⠄⠰⣄⣠⡀⠀⠙⣆⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠠⠀⡀⣤⠔⢀⣀⢴⠾⠀⠙⠀⠓⡘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡷⡇⠘⠛⠋⡀⢀⠀⢀⠀⠀⣅⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⡼⠂⠀⠘⢛⠘⢻⡀⢋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⢀⡇⠳⡄⠀⠈⠈⠁⠉⠉⠉⠁⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠇⠙⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⡀⠀⠀⡀⠀⠀⡇⠘⢄⡀⠀⠀⡠⠻⣀⣀⡠⠏⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢄⠀⠀⠑⠢⠄⣀⠀⠀⠀⠀⣾⠈⠭⠍⠩⠍⠁⠀⡇⠀⣀⢀⢉⣉⣀⡀⠀⣀⢀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠠⡀⠀⠀⠀⠉⠁⠐⠾⠿⢀⣟⡀⠘⠓⠀⣠⠀⠀⢩⡄⣬⠩⠉⠁⠁⠉⠀⠀⠸
⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⣀⣀⡑⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠛⠧⠰⠴⠶⠶⠶⣦
⠀⢀⡴⠛⠛⠙⠋⠁⠉⠙⢶⣄⠀⠀⣿⠶⠛⠋⠉⠉⠉⠙⠲⢬⡒⠦⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⡌⠀⠀⣀⣠⡀⢸
⢰⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⠷⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠉⠉⠉⠙⠂⠚⠉⠉⢀⠀⢻⠋⠀⠀⠈⢋⣉⡡⠼
⡏⠀⠀⠀⠀⣰⠋⠉⠙⢳⣶⡿⠘⠁⠀⣠⠖⠓⠉⠙⢦⡀⠀⢀⣀⡿⠷⠤⠴⠆⠀⠲⠒⠴⠶⠦⠤⢤⣴⡟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⡀
⠃⠀⠀⠀⠀⠇⠀⠰⠊⣸⡏⠀⠀⠀⠀⠻⣀⡀⣀⡤⠞⠋⠉⠀⠀⢀⣀⣀⠤⠤⠤⠤⠤⠄⣀⣀⡀⠀⠀⠉⠙⠒⠤⣀⠀⠀⠀⢀⡴⣋⡿⠷⢦⡀
⠰⡄⠀⠀⠀⠈⠓⠲⠟⠛⢧⠀⠀⠀⠀⠀⡼⠛⠁⠀⣀⡤⠖⠊⠉⢈⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠉⠒⠦⢄⡀⠀⠀⠑⠢⣴⣩⡟⣹⠤⠖⣶⠟⢧⣀
⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⢀⡷⢦⣀⡰⠋⠀⠀⡠⠊⡥⢤⣀⡔⠊⠉⠙⣿⠯⢤⣞⣿⡾⣻⣏⢉⡵⣶⢦⣀⠀⠉⠒⣄⠀⠀⠈⠻⣿⡧⣶⠋⢉⣽⠛⢯⣧
⠀⠀⠀⠀⠉⠑⠒⠒⠒⠚⠉⠀⠀⡟⠀⠀⢀⠞⠀⢀⡴⠾⠿⡆⠀⠀⣴⠃⠀⢠⠋⠉⠉⢉⠏⠀⠀⠘⡟⠋⠑⢦⠀⠀⢳⡀⠀⠀⠹⣿⣛⣻⣋⣉⣩⠟⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡎⠀⣰⢻⠀⠀⠀⣸⣏⣉⣽⣇⡴⡞⠀⢀⡰⠛⢳⡤⡤⣾⠥⠤⢄⡠⢳⡄⠀⢷⠀⠀⠀⢋⣸⣏⣋⡽⠉⠙⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡇⢠⣏⠙⠤⡤⠴⠛⠌⢻⡶⣏⠀⠈⠉⣁⣹⣀⣸⣄⠑⣷⠀⠀⠘⡷⠋⡿⣄⢸⠀⠀⠀⢸⣇⣀⣝⠹⢓⡶⠧⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢯⠹⡏⠹⢶⣿⣻⣦⠀⣀⣽⠛⢋⣩⣿⣏⣿⢯⢤⠀⣑⡛⡶⡖⣾⣁⣼⣾⣇⡎⠀⠀⠀⣾⣻⣏⡉⣹⢿⢦⣤⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⣀⣀⣈⣿⣷⡧⠴⢟⡛⣛⣉⠉⣉⡀⠘⠻⠾⠀⠈⠁⣼⣼⣭⡟⠓⢻⣁⠀⠀⣸⠟⠀⠀⠀⣼⠉⠑⣴⡙⠙⢳⣾⣁⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠉⢉⣩⢭⣭⣵⠶⠒⠛⢿⣍⠁⢽⠁⠀⠈⠛⡿⠟⠉⠒⠒⠚⡄⠀⠀⣷⠤⠿⣯⡷⠛⠁⠀⠀⣠⣞⠛⢳⢶⠋⢉⠟⢧⣀⡝⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⠒⢄⡀⠀⠈⠑⠂⠕⣶⣿⠞⢣⣜⠄⠒⠒⢲⣧⣄⡴⠥⠔⠊⠁⠀⢀⣠⣴⠿⡁⢈⠷⣅⢀⡽⠧⢤⣀⠟⠦⣤⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠄⠀⣀⠀⠀⠀⠀⠉⠈⠁⠉⠀⠀⠀⢀⣀⠠⠤⠒⢊⠉⠉⠑⣤⡜⢧⣀⣼⢋⠛⢣⡴⠛⣆⡠⠊⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠒⠒⠀⠉⠉⣁⡀⠀⠀⠀⢮⣋⠆⠀⠈⠣⣤⣏⡽⣻⡔⠛⢧⡠⠚⢷⡤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⣈⡼⢟⠻⢳⠀⣡⠦⡄⠀⠀⠀⠀⠟⠉⣀⡡⠋⠉⢳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣩⠟⠓⠋⠉⠁⠀⠛⠴⠀⠀⠀⠀⠀⠱⣴⠿⠧⣄⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⣳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⣠⠄⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⣼⡏⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⢻⣧⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠻⣷⡀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢹⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⡼⠃⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠀⠒⠒⠂⠒⠒⠒⠒⠐⠒⠒⠒⠒⠐⠒⠀⠠⠤⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⡖⢉⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⡍⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉⣇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠳⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⡰⠁⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣃⠤⠊⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣁⣀⣀⠤⠐⠁
⠀⠀⠀⠀⡠⠄⠒⠒⢈⣉⣭⣤⣬⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣧⣤⣤⣤⣉⡁⠒⠒⠠⢄
⠀⠀⠀⠀⠢⢄⣀⠀⠛⠿⠿⣿⣿⣿⣿⣿⣶⣤⣤⣄⣀⣀⣀⣀⣠⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⢀⣀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⢤⡀⠭⠭⠉⠉⠉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠭⠭⢀⡤⠒⠒⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠂⠀⠤⠤⠤⠤⠤⠤⠀⠐⠒⠒⠂⠉⠁