Purse Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢞⣿⠟⠋⢉⣉⣭⣭⣭⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⣴⠏⠀⣠⡞⠋⠀⠀⠀⠑⣌⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⣼⠃⠀⣰⠏⠀⠀⠀⠀⠀⠀⠸⡄⣆⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⢰⡏⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣼⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⠷⣆⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣾⣷⣿⡿⠦⣄⢀⣀⣠⣤⣤⣤⣤⣤⣿⣿⣧⠿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣾⣿⠿⠿⢿⡇⠣⠀⣿⠛⠋⠉⠉⠉⠉⠉⠉⠙⡇⠠⠴⣿⡇⠐⠛⢳⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠷⣾⡉⠀⢀⠀⣀⠈⠻⠶⠾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠁⠀⢯⡽⡇⣧⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⢻⣷⡀⢣⢔⡤⠃⠀⠀⠀⠀⠀⠀⠀⠰⡇⣵⡁⣰⠀⠀⠀⠀⣀⡀⠀⠀⠁⠃⢹⣧⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠸⣿⣷⠀⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠱⣄⠞⠁⠀⢰⢁⠺⡅⡷⠀⠀⠀⠀⠘⣿⡇⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⣿⣿⡄⠀⠀⣏⢉⡸⢅⢹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠔⡱⠁⠀⠀⠀⠀⠀⣿⣿⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣿⣟⠀⠀⠀⢸⣿⡇⠀⠀⠘⢦⡰⣣⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⢸⣿⡃⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⢸⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⣠⢄⡀⠀⡀⠀⠀⠀⠀⠀⠀⡖⠓⣧⢰⠀⠘⣿⡿⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢸⣿⠀⠀⠀⢈⣿⣿⢰⣏⠂⣀⠀⠀⠀⠀⠀⠀⢧⣴⠤⣠⠏⠀⠀⠀⠀⠀⠀⠙⠦⢺⠞⠀⠀⣿⣷⢟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⢸⡇⠀⠀⠀⠘⣿⣿⠸⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠓⠛⠃⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⡠⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⠄⢀⠀⠀⠀⠀⠀⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⢀⠄⠐⠗⣾⠀⠀⠀⡀⠀⡔⠀⡄⠀⠀⠈⢐⣩⠦⠋⠀⠀⠀⠀⠀⣿⣿⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡈⠢⡀⠀⠀⠀⣸⣿⣿⡇⠀⠈⠂⠁⠐⠚⠀⠀⠘⡁⠄⢉⣴⠃⠀⠀⠀⠉⠀⠀⠀⠀⣠⣤⠀⢰⣿⣿⢫⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠙⠢⣌⠢⣄⢠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠘⠧⠛⠻⠀⣼⣿⣷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⢦⣵⣦⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠜⡞⣟⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⠋⠉⠉⠉⠉⠉⠉⠙⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣪⠥⠖⠒⠒⠛⠉⠉⠉⠉⠉⢉⣿⠶⠶⠤⢌⣙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡔⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢀⡀⠈⠉⠛⣷⡄
⠀⠀⠀⠀⠀⠀⢀⡞⠋⢉⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⣀⠀⠳⠗⠀⣠⣠⠇⢻⡄
⠀⠀⠀⠀⠀⠀⡜⢀⢄⠈⠁⠀⠺⡳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠀⠀⠀⠀⡏⠁⠀⠀⢳⡀
⠀⠀⠀⠀⠀⣸⠃⣇⡸⡀⠀⠀⠀⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡢⠀⣠⡴⠃⠀⠀⠀⠈⣇
⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⣿⠀⠀⣠⠶⢤⠴⠖⠋⠀⠀⠀⠀⣠⠴⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣾
⠀⠀⠀⠀⡇⠙⠋⠁⠀⠀⠀⠀⠀⣀⣤⣀⢸⡅⠀⠀⢾⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠙⠃⣀⣂⠈⠉⠉⢳⠀⠀⠀⠀⠀⠀⠱⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠙⠦⣄⣀⣀⡀⠀⠸⣄⠀⠀⠒⠚⠧⡼⠀⠀⣠⠞⠀⠀⠀⠀⢀⣠⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠚⠿⢷⣄⣀⣀⣁⣀⣀⣿⣤⠤⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠖⠒⠛⠛⠛⠒⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠋⠀⠀⢀⣠⣤⠴⠶⠒⠒⠒⠒⠶⠶⢤⣄⡀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢠⡟⢀⣀⣀⣤⡤⠤⠤⠤⠄⠒⠒⠒⠒⠒⠒⠒⠲⠶⠦⢿⡄⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢴⠃⠀⠀⡼⠋⢉⣁⣀⠀⠀⠀⠐⠒⠋⠁⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⢳⠀⠀⠈⣿⠓⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⡥⠀⡞⠀⠀⢠⠃⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢸⡆⠀⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⢞⡴⠁⠀⢀⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣧⠀⠙⠆⠀⠑⡄
⠀⠀⠀⠀⠀⠀⢠⡏⠈⠀⠀⠀⡘⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣿⠀⠀⠀⢠⠀⠘⡄
⠀⠀⠀⠀⠀⠀⡾⢠⠀⠀⠀⢠⢁⡤⢤⡀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⡴⢦⠀⣿⠀⠀⠀⠈⢧⠀⢹⡄
⠀⠀⠀⠀⠀⢰⡇⡾⠀⠀⠀⢿⡸⠧⠼⢃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⠣⠚⣰⠇⠀⠀⠀⠀⠘⠃⠀⢷
⠀⠀⠀⠀⠀⣼⠁⠃⠀⠀⠀⠀⠳⣄⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠈⠳⣤⠔⠁⠀⠀⠀⠀⠀⠀⠀⡆⢸⡄
⠀⠀⠀⠀⠀⡟⢀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⣤⢤⣤⡤⣤⡤⢤⣤⡄⣶⡖⢶⣾⣩⠿⣽⣋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠼⠿⠆⠿⠧⠿⠏⠸⠿⠧⠼⠿⠞⠛⠚⣯⣀⣸⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣼⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣟⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢹
⠀⠀⠀⠀⣿⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢫⠏⡯⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸
⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⢠⡇⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣿⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⡎
⠀⠀⠀⠀⢸⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⣼⠁
⠀⠀⠀⠀⠀⢳⣄⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠙⠷⣤⣉⡓⠀⠀⠠⣤⣤⣤⡀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣤⠀⠀⠀⣀⣀⣠⡄⠀⠀⠀⠐⠒⠉⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠲⠶⠶⠤⠤⠤⠤⢤⣤⣤⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠶⠤⠤⠤⠴⠶⠶⠶⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢰⣯⡸⠏⠾⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣫⠟⠁⠀⠀⠀⠀⠈⠘⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢛⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠚⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣪⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢗⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣦⢭⡆⣀⠠⣤⣤⣤⢀⣀⡤⣤⣤⡤⡀⠤⣤⢶⡔⠾⠓⣶⣆⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣿⣎⢾⢳⠜⠃⡿⡇⠈⠋⣴⠿⣄⣴⣿⣧⣾⢉⣇⠀⠿⠸⠀⠀⠿⠋⠒⠼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⣿⣿⢏⠀⢀⣤⡄⠀⠀⠀⠤⠻⠷⠟⣫⣿⣿⡿⣚⣼⡄⢀⢰⣶⡆⡀⡀⢿⠳⣮⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡧⠞⣋⣥⡍⢸⣺⣿⣣⣷⡄⢀⠲⣀⠤⠎⠉⠽⠶⣮⣭⢻⠃⢹⣿⠿⠶⣻⠇⢸⢧⠈⢻⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡞⣼⠟⣋⣾⣄⠻⣾⣷⠾⠹⣯⠉⣡⣾⠟⣻⠽⠿⡘⢿⣦⠑⡄⠛⠿⠿⠋⣀⡎⠸⣇⠀⢐⢷
⠀⠀⠀⠀⠀⠀⠀⢠⡆⢺⠐⢁⣞⣥⡄⡿⠷⠶⠀⠄⠁⣤⣆⡇⣿⢰⣷⠉⣷⣼⢸⣟⢼⢻⢲⡆⠐⡻⢩⢣⣦⣿⡀⣿⣷⢀
⠀⠀⠀⠀⠀⠀⣤⡈⣧⠀⠁⣼⡾⣫⡾⠀⡤⠶⢶⣒⢐⡲⢦⣌⡳⠞⣿⣼⡏⣿⢘⣯⣁⡥⢛⢰⡾⠁⡎⠳⢋⣽⠃⢸⣿⢸⣧
⠀⠀⠀⠀⠀⢼⢇⡇⡘⣧⠀⠸⣦⡟⡕⢩⡴⢶⣦⣈⠳⣯⡼⠏⡻⢦⠙⢯⡗⢿⡘⢃⢠⡌⠻⠈⣱⠸⣟⣿⣻⠏⢀⣿⣏⣼⠁⡇
⠀⠀⠀⠀⠰⢃⢎⣰⢃⣿⠀⠰⠎⠁⠁⣿⣾⠿⠯⣝⢧⡈⠻⢿⣥⡾⠀⡄⠙⠿⢷⣜⠶⠖⠀⣴⠹⣷⣬⣭⡴⡦⠟⢙⠛⢿⣾⡇⠘
⠀⠀⠀⠀⣰⣿⣦⣵⣾⠏⢀⢰⣐⢃⢠⡘⣿⡰⠆⠈⢢⢻⣄⠀⢄⣤⡾⣣⡘⢶⣤⣤⣤⠴⢞⠅⠺⠢⠭⠁⠀⠀⠳⣬⠅⣼⠟⣵⢿⣶
⠀⠀⠀⠀⠘⠛⠛⠛⢁⣠⣬⣥⣉⠥⣴⣶⣌⡻⣦⡀⠀⢳⡻⣾⠶⣶⣾⣻⣝⡦⣌⣓⣒⣢⣄⡴⠋⢀⣠⣶⣾⣟⠛⣷⡆⠀⣤⠻⡶⠞
⠀⠀⠀⠀⠘⢶⣶⢰⣡⣴⣊⣭⣿⣷⡙⢆⣠⣙⠦⣍⡓⠤⣽⣿⣷⣿⢿⡮⢍⣉⣀⣨⣭⡟⠋⠀⣴⢋⡮⣭⣭⣉⣩⡭⣠⣤⣬⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠛⠛⠉⠉⠁⠐⠾⠙⠕⠊⠙⠷⠶⠭⠭⠟⠓⠞⠒⠒⠾⠷⠟⠋⠀⠘⠛⠈⠓⠛⠛⠉⠁⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢦⣠⠶⢄⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢀⣀⣀⣀⣠⣭⣶⠶⠷⠤⠭⠷⠶⠦⠤⠤⠤⠬⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢱⠞⠉⣉⠤⠤⠤⠄⠀⢤⡤⢤⠤⠶⠒⠒⠒⠒⠒⠲⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⠀⣼⠁⠀⠀⠀⠀⠀⠀⢷⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡏⡆⠀⠀⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠀⠀⠀⢻⣼⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢇⠣⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⣔⠊⠉⢩⣿⠏⣏⠰⠞⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣴⣶⡾⠗
⠀⠀⠀⣠⠤⠤⠟⠃⠠⡟⠀⠀⠈⠛⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠂⠀⠀⠀⠀⠤⡀
⠀⠀⢸⡆⠀⠀⢀⣀⣠⣷⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢀⣀⣀⣠⣤⣴⣧⡄
⠀⠀⠀⠉⠀⠀⠚⢚⣛⣛⣆⣑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣠⡿⠃⠙⠚⠛⠛⠛⠉⠁
⠀⠀⠀⠀⢠⡞⠉⠁⠀⠀⠀⠀⠈⠙⢳⣶⣴⣀⣀⣶⣶⡟⠉⠉⠀⠀⠈⠉⠙⢻⣦⣤⣤⣤⣬⡵⠞⠋
⠀⠀⠀⠀⠘⢗⡀⠀⢠⡬⣤⣤⣤⣖⣽⡿⠋⠉⠉⠉⠈⠳⠄⣐⣲⣲⣖⣶⡾⠿⠋⠀⠀⡔⠊⠉⠉⠀⠈⠁⠐⢢⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢀⠤⠖⠒⠓⠉⠘⠛⠓⠢⣄⣤⢤⣤⣤⣴⣾⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠁⠀⠀⠀⠀⠈⠲⡄⢾⣄⠀⠀⠂⠀⠀⠀⣀⣀⣼⣿⡟⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⠀⠀⠀⠀⣀⣀⣀⣤⣴⣿⡿⠓⠤⠤⠭⠭⠭⠭⠽⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠤⠬⠭⠵⠶⠞⠛⠉



⣶⣶⢶⢶⢶⠶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⢶⣶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣼⣿⣼⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠞⠛⠛⠛⠛⠤⡈⠻⠿⠧⣽⣆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣀⣴⡶⢶⢶⣶⣶⣶⣾⣶⣶⣾⣶⣶⣬⡿⣯⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣼⠣⣼⣷⣿⣿⣿⣿⣿⣿⠿⣿⣿⢿⣿⣿⣷⣿⡆⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢩⣾⣿⣿⣿⣵⣾⣿⣿⣿⣶⣿⣿⣯⣿⣿⣿⣿⣧⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⢡⣿⣿⣿⣿⣿⢿⡿⡿⣿⣿⣿⢦⣷⣿⡘⣿⣿⣿⡗⢻⣷⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⣾⣹⣟⣽⢷⣿⢘⡏⠃⠾⠙⠉⣿⢻⣿⣷⠙⣿⣿⣧⢎⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣴⠋⣰⣿⠀⣿⣿⣿⣿⣿⣽⣦⣾⣾⣿⣶⣷⣧⣾⣿⣿⡆⣿⣿⣿⠸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⡜⢡⣾⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⢿⢟
⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣛⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽⣛⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢻⣿⣿⣿⣯⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⠒⠒⠚⠋⠒⠒⠦⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠈⠁⠀⣀⡤⠴⠶⠒⠛⠛⠛⠓⠶⢤⣀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠈⠳⡀
⠀⠀⠀⠀⠀⢠⠃⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢦⡀
⠀⠀⠀⠀⠀⣸⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⢦
⠀⠀⠀⠀⠀⠘⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⡇
⠀⠀⠀⠀⠀⡇⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹
⠀⠀⠀⠀⠀⠇⠀⣿⠀⠀⠀⢀⣠⣤⣤⡤⠤⠴⠶⠶⠖⠒⠛⠛⠛⠛⠛⠛⠛⠛⠒⠲⠶⠶⠶⠶⠶⠤⠤⣄⠀⠀⣿⠀⣸
⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢯⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠘⡄⢸⡆⠀⡞⠉⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠈⢧⣰⠇⡘
⠀⠀⠀⠀⠀⠀⢱⠀⢧⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠐⠀⠀⠀⠀⠀⠛⢰⠃
⠀⠀⠀⠀⠀⠀⠘⡆⠘⠁⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠀⠀⠀⢀⡤⢤⢄⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋
⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠂⡀⠀⠀⠀⠁⡆⢸⠀⠆⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠢⠀⠀⠑⠬⠥⠔⣀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶
⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⠤⠤⣤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⣠⣾⡍⣷⡿⡾⣧⡀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠰⣟⣿⡿⠀⡀⠀⢘⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢿⡐⠀⠈⠁⠀⠛⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢼⠁⠀⠀⠠⡤⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢈⣩⣭⣛⣻⣟⣛⡣⠤⣄⣀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣀⠀⣀⣟⣛⠿⣿⣛⣻⠂
⠀⠀⠀⠀⠀⠀⠸⣟⣿⡾⠟⠛⠛⠻⣿⣿⡇⠀⠀⠀⠀⠀⠈⣿⣯⣵⣶⣻⡿⠿⣿⣿⣷⣿⣿⡁
⠀⠀⠀⠀⠀⠀⢰⣿⣉⡰⢾⣿⣟⣻⣿⣷⡇⠀⠀⠀⠀⠀⠀⢸⣯⣯⣿⣿⣿⣿⡿⢿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⢈⡛⢿⣿⣦⣌⠻⣿⣯⣿⠻⡄⠀⠀⠀⠀⠀⠈⣿⣽⣿⣿⣭⣤⣾⣿⣿⣯⣿
⠀⠀⠀⠀⠀⠀⢸⣇⣾⣿⣿⣟⣸⢷⣿⣿⣿⣥⡀⠀⠀⠀⠀⠀⠸⣿⣯⣿⣿⡛⣿⣿⡌⣯⢻⡇
⠀⠀⠀⠀⠀⠀⢸⡟⣿⣿⣿⠿⢦⣾⣿⣃⣘⠻⠷⢶⣤⣤⡤⠴⠶⠿⢧⣽⡿⢋⣿⣿⠖⣳⣿⠂
⠀⠀⠀⠀⠀⠀⢸⣯⣬⣿⣿⣿⣿⣗⠿⣽⡿⠿⠟⢷⣽⣶⣵⣾⣿⣿⣿⣿⣶⠟⣩⣭⣶⣿⡿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡷⣦⣤⣙⣻⣿⠿⠛⢻⣾⠿⢿⡻⢟⣛⣿⣿⣿⠓⢷⣞⣥⢗⣿⠟⢁
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣌⣙⣛⣿⣷⣞⣻⣿⣍⠉⠀⠈⠓⠾⠷⠚⠵⢶⣋⣤⡾⠟⠁⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣙⣿⣟⣛⠻⣮⣭⣭⣭⣽⣶⣦⣿⣶⡶⣶⣦⣤⣿⡿⠿⡿⣶⣿⣿⢏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠿⠛⠛⢋⣛⡛⠛⣋⣡⣴⣶⣾⣿⣷⣶⣿⣿⣿⣯⣵⣿⡁

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣶⣟⡛⢟⠢
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣸⡇⢀⡀⣸⣶⡦⣄
⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡀⠀⣰⣿⡏⠀⠀⢣
⠀⠀⢀⣜⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠃⢀⣾⣿⣷⡆⠀⠘⡄⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣴⠘⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠡⢤⡠⢻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⢻⣿⣿⣧⣠⣤⣀⡀⠀⠀⠀⣀⡀⠀⢴⣄⡀⠙⠒
⠀⠀⠀⠀⣼⣿⣿⡿⠋⠉⠉⠉⠁⠀⠀⣀⡀⠀⢸⠛⠋⠁⢀⣀⣿⣿⢶⣦⣄⠁⠀⠀⠀⠉⠀⠀⠀⡀
⠀⠀⠀⠀⡿⣿⣿⣦⣀⣠⣤⣤⠶⠾⠛⢱⠁⠀⡿⠀⢨⣿⣿⣿⣿⣿⣷⣹⣿⣯⣀⣶⣶⣶⣶⣶⣾⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠁⠙⠛⢻⣿⣿⣷⠄⠀⠀⠀⡟⠀⣼⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣽⡗⠀⠀⠀⢀⣴⡄
⠀⠀⠀⠀⠀⢾⡿⠈⣿⣿⠋⠀⣠⡷⢾⠇⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢷⣤⣤⣤⣤⡿⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠟⡀⠀⣴⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⢻⠿⢻⢿⣿⣿⣿⣿⠟⣷⣟⠈⢿⣻⡿⠛⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣀⠀⠃⠞⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣧⠾⣸⣿⢿⡿⣧⠸⣿⣿⡇⣿⠙⠷⢧⡉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠊⠀⠀⠓⣄⠈⠹⣟⣻⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣷⣿⣷⣿⣿⣷⣿⣶⢸⣿⣿⢿⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢾⣿⡧⠏⠀⠀⠸⡛⣿⣿⠟⣽⣿⣿⣿⣌⣿⣿⠽⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣎⣼⣟⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠰⢧⠘⢎⠉⠛⠻⢿⣿⣿⣷⣿⣿⣆⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡄⠀⠀⠸⣿⣿⣿⣽⣿⣿⠀⠀⢸⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣶⣤⣤⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣰⠀⠀⠀⠀⠀⢡⣾⠇⠀⠀⠀⢀⣿⣿⣿⣿⣷⡀⠀⢹⣿⣷⡈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⢠⣿⣿⡏⠻⣿⣿⣿⣦⣀⢹⣿⣿⣿⣿⣿⡟⠋⠉⠉⠉⠉⠉⠙⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠈⢣⣠⣿⠻⢿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⠀⠀⠈⠉⠛⠛⠛⠻⠿⣄⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢄⡠⢤⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠀⠀⠀⠋⣤⣾⡧⢬⣯⣡⠄⠀⠋⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢢⡤⢺⠁⠀⠀⠀⠙⠆⠚⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢹⠋⠁⢰⠦⠀⠀⠄⠲⢬⣌⢣⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠌⡀⣱⡾⠤⠀⠀⠤⠴⢾⡯⡜⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⡆⠀⢀⠘⢉⣿⠄⠀⠀⠠⠀⡀⢘⣇⠂⠉⠑⠒⠄⠀⠀⠀⠆⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠑⠀⠁⢀⡼⡝⡼⠀⠂⠐⠀⠀⣿⣖⡔⠀⣀⠀⠀⠇⠀⠠⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢁⠀⠙⠀⠀⠀⠈⠈⡁⢿⡝⠖⢮⡢⠊⠀⠀⠀⢀⠎⠀⠀⠀⠀⣠
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡛⠀⠸⡄⢀⠀⠀⠀⠀⠈⠔⣾⠃⠈⠀⠃⠄⠀⠀⠠⠃⠀⠀⠀⠀⠐⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢧⣀⠀⡉⢢⡄⠨⠀⠀⠇⡐⠏⢡⠀⠀⡌⠀⣄⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠓⣸⠆⢃⠀⠀⠀⠀⠀⠘⠞⠤⣞⠀⡜⢠⠂⠀⠀⠀⠀⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢣⠈⠀⠀⠗⠀⠀⠀⠀⠁⠀⢠⡤⠀⣤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡰⢂⠠⠚⣦⠀⠀⠀⠀⠀⢀⠀⠈⠻⠤⠤⠤⠠⠄⡄⠁⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠈⠀⢲⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠊⢠⡀⠐⠒⠒⣈⡉⠰⣌⢂⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠘⠐⠁⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠙⡁⠀⣄⠀⠀⠂⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⠀⢀⠁⠀⠀⢀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠘⢯⣀⡤⠊⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣤⣓⣀⠀⠁⠠⠄⢀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠚⠁⠀⡠⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣅⠀⠠⣀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠉⠈⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⡀⠀⡀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠁⠀⠀⠙⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋⠀⠒⠠⡀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡍⠁⠐⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠓⠀⠀⠒⠈⠉
⡿⢿⡿⠿⡿⡿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⢻⣿⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢿⣶⣾⣯⡼⢿⣿⡿⠓⢙⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣾⣿⠋⠃⠀⠀⣠⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣤⡀⢀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣭⣁⠉⢹⣶⠇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡔⠠⠉⠀⠀⢀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⡿⠃⠀⠀⠀⠠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠋⠉⠀⠀⢀⣴⣾⡄⠙⠆⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠈⠀⠀⠀⢀⣠⣾⣿⣿⣿⡄⠀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣉⠧⠤⣠⡄⢸⣿⣿⣿⣿⣿⣿⣿⣔⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣴⠋⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⣸⠋⠉⣶⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡆⢀⣠⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⢄⡼⠃⠀⢀⠈⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢩⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⢿⠃⣀⠄⠁⣀⢻⡤⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠨⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠾⣿⡿⠔⣿⡄⢠⣾⣶⠀⠙⠛⠿⣿⣿⡿⠟⠛⠰⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡾⠟⠁⢰⣿⡁⢀⣉⣨⠅⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⣇⢠⣾⡇⠘⣇⢻⣼⣿⣿⣄⡀⠖⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠼⣧⢷⡬⠛⣶⣿⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠠⡀⢀⡄⠀⡀⠀⠀⠈⠀⠁⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣠⣴⣤⣽⣇⠀⠀⡌⠀⢰⡿⢠⣤⣄⠤⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣽⡆⢀⣀⢠⣿⡀⠀⠃⠁⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿
⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡍⠁⠁⠀⠀⠉⠉⠉⠋⠋⠁⠒⠤⣄⣀
⠀⠀⠀⢀⣴⡶⢶⣄⠀⠀⠀⢀⣴⡾⠿⢷⣾⠿⠛⠟⠛⠛⠛⠉⠽⢟⡟⢰⣦⣤⣀⢸⣿⣿⣦⣤⣀
⠀⠀⠀⣾⡟⠀⠀⣿⡆⠀⢠⣿⠏⠀⢠⣾⠇⠀⠀⠀⠀⠀⠀⠀⣠⣎⡰⠁⠉⠉⠻⢿⣿⣿⣿⣿⣏⢳
⠀⠀⢰⡿⠁⠀⢠⢿⠇⢠⣿⠟⠀⢀⣼⡿⠇⠀⠀⠀⠀⠀⠀⢸⠟⣼⠅⠀⠀⠀⠀⢐⣿⣿⣿⣿⣇⣼
⠀⠀⣿⠃⠀⢀⡞⡜⢠⣾⠟⠀⠀⠹⣿⡏⠀⠀⠀⠀⠀⠀⠀⣾⣸⡏⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠟⠁
⠀⠀⠠⣷⣤⠞⣼⡶⠟⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⢿⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡏
⠀⠀⠀⠀⣸⢰⠁⠀⠀⠀⠀⠀⠀⢰⣯⣀⣤⣀⣀⣀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⣾⣿⣿⣿⣯
⠀⠀⠀⠀⣧⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⣀⠀⣾⣿⣿⡟⠋
⠀⠀⠀⠀⠸⣿⣦⣤⣀⠀⠀⠀⠀⠀⢈⣹⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠉⠙⠻⣷⣤⣀⣠⣴⠟⠋⠁⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⠿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢚⣩⡙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠦⣤⡃⡶⠋⠀⠙⣷⠜⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢠⡼⠛⠻⣦⡹⡆⠀⠀⠀⠘⣆⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣶⣋⠀⠀⢠⢐⣷⢹⠀⠀⠀⠀⢷⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡇⢨⣿⣿⣹⣿⣿⣾⣿⠈⣇⡀⠀⠀⢸⡅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣟⠿⣿⢀⣾⣿⣽⡿⢿⣿⣿⣿⠁⣿⣿⡿⣶⣾⣄⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠊⣩⠟⡇⠀⣽⣿⣟⠿⢿⣿⣿⣿⠀⢽⣿⣿⡿⢿⣽⣿⣧⣿⣷⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⠊⣧⠔⠇⣠⡇⡠⠎⠉⣻⠢⢭⣙⠀⣿⣿⣿⣿⣷⣻⣮⢿⡿⣿⣿⠽⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠐⠈⠀⢸⡧⣸⢋⡠⠘⠇⢀⠴⠂⠘⡀⢻⡶⠭⣝⣻⠛⠘⠒⠛⠉⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣀⡰⡆⠁⣹⡟⠉⢡⡠⠞⢧⠀⠠⣘⠧⣄⡠⢶⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⣠⠼⡏⢀⡱⡖⠁⢑⡤⠚⠣⢀⡸⠗⠛⣦⠔⠉⡇⠀⡔⠀⠀⠀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠀⢀⡤⠚⢇⢀⠸⢖⠁⢀⣤⠐⠙⣀⠜⢮⠀⡳⡞⡃⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⠚⢃⡈⢶⠈⢣⣀⠠⠲⡁⢀⠶⠊⠉⢀⡤⠛⣅⡤⠃⠀⢀⠀⠀⠀⠈⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠳⠈⢠⡠⠶⠉⠁⣤⠔⠛⣄⡀⠕⠆⢈⡡⡔⠉⡱⠆⠀⠞⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⡙⣄⣠⡔⠛⣄⢀⣤⠒⠉⢂⡠⠚⢣⣠⠜⢋⡀⢾⠀⠂⠀⠀⠀⢀⡄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠬⣦⠈⢢⡠⠷⡎⢁⡱⡖⠉⣈⢶⠂⡹⢦⡀⠀⡀⠀⠀⠘⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢉⢣⡠⠞⣋⡠⠒⣉⢤⡊⠃⠀⠀⠀⠀⠀⢀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠧⢼⣎⣁⠤⠛⠻⡄⣀⠀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠐⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠁⠉⠈⠉⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⣀⣀⣀⣀⡀⢙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠓⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠠⠀⠀⠀⠀⠀⣿⡧
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠠⠀⠐⠀⠃⠐⠂⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣷⣶⣿⣦⣤⣼⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠑⠒⠂⠒⠒⠀⠒⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⣼⣤⣽⣿⣿⣧⣤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣄⠀⣖⣰⢰⡂⢀⡄⢀⡦⠀⠀⠀⠀⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢢⣜⣹⠈⢗⣨⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡼⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠉⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⢶⠈⠙⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⣀⣀⣀⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢹⣤⣤⡇⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢠⣔⣫⣉⠉⠙⠿⠟⢨⣭⣳⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⠇⣿⣿⣿⡇⢻⣿⣷⣤⣤⣀
⠀⠀⠀⠀⠀⠀⢠⠎⠿⠷⡅⢸⡃⣤⣶⡯⢽⡟⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣴⡂⢀⠁⢃⡿⠻⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢪⡌⠁⠀⠉⠹⡅⢀⢸⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠑⠈⠀⠀⠂⠹⡣⣐
⠀⠀⠀⠀⠀⠀⣧⡇⣽⡀⠸⠃⠀⡴⠀⠈⡃⠸⡆⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠁⠀⠄⠂⠀⡠⠔⢅⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠠⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠥⢀⠈⢂⠑⡈⢄⠀⡀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⣀⠤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠬⢀⡐⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉

⣿⠿⠿⠿⡿⠟⠿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⣿⣟⣿⣿⣿⣟⣿⣟⣿⡿⣿⣻⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠛⠉⢳⡒⠢⡀
⠀⠀⠞⠿⣽⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⡔⣋⡉⣽⣿⡇⠀⠀⠀⠀⣄⠒⣩⠶⠦
⠀⠀⠀⠀⣸⣼⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡟⠀⣿⣟⣻⣿⠃⠀⠀⠀⢐⠀⣴⣯
⠀⠀⢀⣰⢿⣿⡟⣩⡍⠁⠲⡄⠀⠀⠀⠀⠀⠈⡛⢳⣜⣻⣿⣿⣆⠀⡰⠞⢠⣵⣿⣟⢷⡄
⠀⢰⣿⢿⣿⡟⣿⠛⠋⠛⠳⣾⠀⠀⢀⠴⢺⡋⠀⢐⡏⡏⠉⡹⠮⣄⣧⠶⠿⠿⣿⣾⣗⣿⣦⡀
⠀⣟⣻⣻⡿⢇⡈⢄⠀⠀⠀⠜⠀⢀⠌⠀⣤⠁⠀⢘⠇⠀⠘⠁⠀⡏⠧⡀⠀⠀⠀⠙⣿⣷⣾⢽⣄
⢸⣟⢭⣿⡟⠚⢷⡀⠈⠒⢤⡀⡰⠁⢀⣼⣿⠁⢀⣼⣀⠀⠀⣼⣦⣁⡀⠈⠢⡀⠀⠀⠘⣿⣿⣟⢿⣆
⢸⣖⢿⣿⡇⠀⠀⠙⢷⣄⣀⢹⣄⣰⣿⣿⢹⡾⠞⣿⡏⣤⣼⣿⡿⣟⠿⣶⣤⠘⠆⠀⠀⢹⣿⡏⠳⡻⡄
⢸⢞⣿⣿⣷⠀⠀⠀⠀⢙⣻⡿⠚⢋⣼⡼⡼⠃⠠⠾⡉⢈⣿⣿⠱⡿⢗⡈⢿⡄⠘⡀⠀⢸⣿⣿⢀⡏⡇
⢸⣓⡿⡿⣿⣆⠀⠀⡔⢏⣙⣮⣶⡗⣫⡟⠀⢀⠂⠐⠨⣿⣿⣿⡇⠣⡈⢙⠊⢻⡄⢠⡀⣼⣿⣿⡀⡁⣿
⠈⣦⣯⣸⡿⣿⣧⡠⠠⠏⢀⡼⠛⠭⣿⠀⠀⣸⠃⠀⣰⣿⡿⣿⣷⠈⡇⠻⠈⢄⠹⣴⡆⣿⣿⢿⢁⡃⡇
⠀⠻⣷⢯⣛⡻⣿⣿⣷⣤⣤⣯⣔⢂⠉⠳⣤⣅⣤⣴⣿⣿⣱⢿⢿⣶⣟⣾⢃⣾⣷⣿⣻⢿⠿⠾⢡⢧⡇⣀⡀
⣺⣷⣿⣿⣥⣭⣚⢚⢽⣻⣻⣿⡟⣻⣲⣎⡟⠹⡟⣿⣿⡟⢳⣄⢳⣦⣶⣿⣿⣿⡿⠿⢭⢴⠭⣰⢺⡾⣁⣀⣠⠤⠤
⢘⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣠⡇⠁⢸⣿⣏⢸⣿⢏⠟⠟⠿⢺⡛⡾⣹⡾⢅⠵⡾⠿⠟⢛⣛⡋⠠⠖
⢉⣀⣀⣠⣍⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠴⣿⠽⣟⣿⡿⣳⣲⣖⣄⣷⣿⡿⣻⣭⠧⣟⢓⣂⣨⣤⣤⠤⠤
⠈⠥⠀⠀⠨⠭⠟⠛⠉⠛⠛⠛⠛⠉⠉⠀⠸⠛⠋⠂⠀⠉⠙⠛⠻⠿⡿⠿⢞⣛⣿⣿⣭⣉⡩⠉⠀⠀⠀⠀⠂
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣶⣶⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⣶⣤⣴⡄⠀⠀⠸⠿⠿⠿⠏⠀⠀⠀⢰⣶⣦⠀⠀⠀⠀⣾⣶⣾⡇⠀⠀⢠⣾⣶⣿⣶⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣶⡆⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢠⣶⣦⠄⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢰⣶⣶⡄
⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠘⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢰⣿⣧⠀⠀⠀⠀⢰⣶⣦⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⣤⣤⠀⠀⠀⣤⣤⣤⣤⠀⠀⢠⣶⣶⣦⣦⡄
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠸⠿⠟⠀⠀⠀⠀⠸⠿⠿⠀⠀⠀⠀⠀⠾⠂⠀⠀⠀⠘⠷⠶⠶⠶⠀⠀⠀⠻⠿⠿⠏⠀⠀⠸⠿⠿⠿⠿⠇

⠀⠀⢠⣤⣤⣤⡄⠀⠀⢰⣶⣶⣶⡶⠀⠀⢠⣶⣤⣤⡄⠀⠀⢠⣶⣶⣶⣦⠀⠀⠠⣶⣶⣶⣶⠀⠀⢠⣶⣶⣶⣶⠀⠀⠀⣰⣶⣶⡆
⠀⠀⠸⣿⠿⠿⠗⠀⠀⠸⠿⠿⠿⠗⠀⠀⠘⢿⡿⠽⡷⠀⠀⠘⠿⠿⠿⠟⠀⠀⠀⠿⠿⠿⠿⠀⠀⠀⠛⠛⠛⠛⠀⠀⠀⢿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠾⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⠘⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠘⠻⢿⣶⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⣾⠏⠀⠀⠂⡀⣠⡴⠃⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢦⣠⣾⡿⠋⠀⠀⠀⠀⠘⠟⢀⣠⡾⠁⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠙⢿⣇⡴⠋⠠⣀⠄⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⠀⠀⠀⠺⠟⠀⠀⠾⠋⠀⠈⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠿⠛⢸⣿⠄⣰⣿⢿⣷⠀⠀⢀⣴⡿⠿⠇⢸⣿⣾⡇⠸⣷⢻⣿⣻⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠛⠿⢶⣌⣿⣿⣿⡁⢸⡿⠀⠀⢸⡏⠀⠀⢀⣼⣿⠹⣧⣴⡿⢿⡿⠿⠿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⣴⡿⣿⠀⠙⣷⢸⡇⠀⣀⡘⢷⣶⣶⡿⢻⣯⣤⣽⣿⣴⣾⣷⣶⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⣠⣾⡿⡿⡆⠑⠦⠀⠉⠉⠁⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⠉⠀⠀⣀⣶⠆⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡶⢰⣿⣿⣿⣿⣿⣿⣿⣗⠸⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⠿⠃⢸⣿⡿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠀⠀⠀⠸⣿⣇⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡧⠀⠀⠀⠀⠿⡿⠃⠀⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠈⠍⢀⣀⣤⣤⣤⣄⣀⡀⠀⠀⠀⢰⣦⣤⣽⣿⣿⣷⣭⣻⣿⣿⣿⣦⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠿⠷⠶⢶⣦⣭⣭⣍⣛⣛⣛⣿⡿⠿⣿⣿⠟⠛⠻⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⢠⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣯⣭⣭⣍⣉⣉⣁⣀⣀⣀⣛⣛⣛⣛⣛⠛⠛⠳⠶⠶⠶⠶⠶⣤⣤⣬⣭⣭⣭⣭⣍⣹⣆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢨⣻⡿⠿⠿⠿⢿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣾⣟⣍⡍⣽⣿⣿⣿⣿⣿⣯⣿⣿⣿⣟⣿⣿⣿⠺⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡿
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣟⠿⠇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣷⣶⣶⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⣷⣶⡟⠃⠀⠀⢠⣀⣀⣀⣀⣀⡀⢀⠀⠀⠛⠃⣾⡿⢿⢻⢷⣿⣿⣿⣿⣿⢸⣿⡌⡃
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣧⠂⠀⠀⡼⠀⠀⠀⠉⠉⠁⢸⢀⢸⣿⣿⡿⡇⠈⣸⢸⣿⣿⣿⣿⣿⢸⣿⣿⠹⣦
⠀⠀⢰⣿⣿⣿⣿⣿⡌⠿⠃⠈⢿⣿⠇⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⡏⠘⣸⣿⡏⡠⠁⠀⡟⣸⣿⣿⣿⣿⡏⣸⣿⣿⠀⢻⣆
⠀⠀⢸⣿⣿⣿⣿⣿⣷⣬⣒⠶⠤⠤⠄⣀⣀⠈⠉⠉⠈⠁⠒⠒⠚⠁⠀⠹⡿⠟⠁⢠⣱⠇⣿⣿⣿⣿⣿⡇⣿⣿⡇⠀⠀⢻⣧⡀
⠀⠀⠈⢯⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣬⣉⣀⣒⣒⠂⠀⠀⠀⠀⢤⡴⣿⠟⣰⣿⣿⣿⣿⣿⠁⣿⣿⡇⠀⠀⠀⠙⠿⣦⣄
⠀⠀⠀⠐⠀⠤⢈⡉⣁⠀⠚⠿⠿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⠁⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠂⠐⠂⠐⠀⠉⠉⠉⣛⠛⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⡿⣤⣄⠀⠀⠀⠀⠀⠀⠈⢻⣷
⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣈⣉⣉⣙⣛⣭⣥⣾⣿⣟⣁⣉⣙⣛⠳⣶⣤⣤⣀⣤⣾⢛
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠛⠛⣋⣉⣛⣛⡒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⢀⣠⠔⠚⠉⠉⠁⠀⠈⠉⠓⠮⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⡰⠃⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡘⠀⠀⠀⠀⢀⣠⡶⠟⠛⠛⠛⠛⠶⣤⡀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢠⠁⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡄⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣻⣀⡇⠀⠀⠀⣿⡷⠶⠶⠶⠶⠶⠿⠿⠿⠿⠿⠿⣿⠀⠀⠀⢸⠷⠶⢤⡀
⠀⣠⠶⠚⠛⠛⠋⠉⠉⠉⠉⠛⢛⠟⠣⡀⠀⠀⠀⠀⡇⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⢀⠞⠻⠛⠛⠛⠓⠒⠒⠒⠒⠶⠶⠶⢤⡀
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠢⣀⠀⠀⢡⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⣦⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠈⠓⡴⠛⠄⠀⠘⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠘⠀⠀⠈⠙⡆⠀⠀⠀⠇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⣠⡃⠀⠀⠀⠀⠀⠀⠀⣀⠔⠛⣄⡀⠀⣀⣠⠴⠁⠀⠀⠀⠸
⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠋⢻⠀⠈⠢⢄⡀⠀⣀⡤⠚⠁⠀⠀⠀⢙⠁⢸⠁⠀⠀⠀⠀⠀⠀⠇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠈⡣⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠐⣏⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠
⠈⡇⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⢰⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠘⡇⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⢷⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠘⡄⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⢻⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠸⡄⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⢱⡀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠹⣄⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣃⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠛⠒⠶⠶⠶⠶⠶⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠞⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⡋⠓⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢂⠞⠉⠙⢦⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣫⠏⠀⠀⠀⠈⢳⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⠟⠀⠀⠀⠀⠀⠀⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡎⠀⠀⣀⠀⠀⠀⠀⠈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢶⣾⢻⠹⣉⣷⣶⣶⣷⣾⣿⠟⢇⣤⢶⡤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡰⠾⢛⣁⡀⡁⡀⠉⠀⢌⠻⣿⣷⣷⡟⢀⠄⠀⢄⠀⡙⢿⣷⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣅⡟⣴⡴⢿⡧⣼⠈⡆⠠⠄⠁⠈⠙⠋⠀⠀⣾⢳⣈⠳⣯⡲⠏⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣇⠈⣿⣿⣝⢮⠄⡿⠀⡷⣳⡜⠠⡢⡀⢦⣴⣵⣿⠀⣿⢿⣮⡻⢂⣸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⢻⠀⡏⠀⠉⠓⠴⣇⠀⢹⡎⠈⠒⠕⢯⠫⣿⠟⣯⠛⢸⣵⠝⠒⠈⢸⠛⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡕⠀⡄⠀⠀⠀⠀⠀⢧⡉⠓⠁⡝⠃⠀⠀⠠⠀⠟⠻⣝⠛⠁⡀⠀⢰⠀⢸⡏⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⢃⡄⢠⠢⡀⠀⠀⢙⡛⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡋⠀⠀⢂⢇⢻⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣄⣿⠀⣦⡡⠀⠀⠀⣾⢻⠀⠀⠀⠀⢀⢌⠀⠀⠀⠀⣟⣇⠀⡠⢧⢆⡀⢡⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⠿⠾⢠⡙⢳⠀⠀⠀⣞⡛⠀⠀⠀⢀⢮⠫⠲⠀⠀⠀⡛⠁⠀⡁⠟⡾⢁⠀⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣙⠃⡜⢱⣬⡷⠀⠀⢤⠄⠀⠀⠀⠈⠣⠑⠆⠀⠀⠀⢋⣉⠀⠀⠠⠑⢧⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢤⣤⠀⠩⡅⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⡠⠤⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠉⠉




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

⠀⢀⡤⠖⠚⠛⠉⠉⠝⠳⢦⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⡤⠐⣄
⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡏⠪⡑⠢⣄⠀⡴⡅⠀⠀⢸
⣇⣾⠖⠒⠲⢄⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠈⢷⡏⠀⢠⠀⢸⡣⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠈⠃⠀⠀⠀⠈⢳⡐⢄⡀⠀⠀⣠⣄⣀⣀⣀⣀⠀⠈⣧⠀⠀⢀⡾⢁⠇⠱⡀⠀⠀⠀⠀⠀⠀⣠⣤⢀⡴⠊⢁⠠⠂⣸⡡⠤⠶⠦⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣍⣵⣾⣗⣤⣾⡻⣯⢣⢄⠀⣸⣆⡀⢸⠁⠘⠀⠀⡇⠀⠀⠀⠀⣠⠋⠁⢸⠎⠀⠀⠀⢀⡴⠋⠀⠠⠐⠈⣡⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣾⣉⣹⣿⣿⣏⣻⣿⠇⠀⢸⣿⠁⠐⡈⠀⠀⠀⣠⡧⠂⢫⠀⢰⠃⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣭⣸⣿⡙⣷⣿⣿⣤⣴⡟⠇⠀⠀⠁⠀⠀⠚⢯⣽⣷⣌⢣⡏⠀⠀⢀⡴⠊⠀⠀⢀⣀⣀⣀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⠿⠿⠍⠩⠶⠶⡤⢴⣙⣿⣶⣶⣶⣴⣍⠿⢹⣷⡄⠀⠊⡀⠀⠀⠀⠁⠤⠬⢵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⡋⠉⠀⠀⢀⣀⠀⠀⠰⠈⠀⠀⠀⠀⠉⠻⣝⢫⣙⢽⡆⣦⠹⢹⠀⡜⠀⠀⢀⣀⡤⠤⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣼⠋⠛⠻⢷⢷⠌⠉⠀⡈⢀⣄⣀⠁⢀⣠⣤⣀⠀⠘⣷⣿⢾⣇⣿⣷⡀⢸⠑⠀⣰⡯⣥⡤
⠀⠀⠀⠀⠀⢀⡴⢳⣦⡴⢶⣾⡋⢀⡰⠒⠃⡼⢿⣉⣿⣿⡉⠀⠀⠀⠈⣧⣟⣛⣿⣿⣿⠃⡞⠀⢰⠃⠜⠀⡆
⠀⠀⠀⠀⠀⡌⠀⠘⠯⢀⣀⣘⠻⡶⠤⢀⠈⢿⣾⣛⣡⣿⢆⠀⠀⠀⠀⣿⣯⣬⣽⣿⣿⠀⣷⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠐⠈⠫⡄⠀⠁⠀⠙⢏⠘⠀⠀⢡⠀⠀⠄⠛⠛⠋⠉⠉⡿⠀⠀⠙⠖⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠳⢤⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⡈⢍⣀⠈⠀⠀⠀⠀⣠⣤⣀⣴⠾⠋⠀⠀⠀⠀⠀⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⠒⠂⠀⠀⠀⠀⠀⠀⠀⠙⠀⣉⡈⠁⠀⠐⢺⣾⣿⠏⠁⠀⠀⣷⢀⠀⠀⠀⠀⠀⠘⠢⢄⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢤⡤⠤⠤⠀⠀⠀⠀⢈⠀⠀⠀⠀⣀⡀⠉⢯⣒⣄⠱⣄⣿⣿⣏⣤⠤⣤⣀⣀⣀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣠⠖⠁⠀⠀⡀⠀⠀⠀⠀⠀⣘⠂⠀⠉⠛⠛⠛⣿⠷⣥⣄⣐⣌⣻⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠇⠀⠤⠒⢹⡅⠠⣤⡤⠞⠁⠀⠔⠊⡀⠀⠀⡈⠁⠀⠀⠠⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⢠⢎⠀⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⡤⠀⠀⠀⠀⠀⢸⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠐⣜⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡴⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢠⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣡⡀⠀⠈⠶⠛⣏⡇⠠⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢣⣿⠷⠟⣼⠀⢶⠭⢁⠀⠸⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡀⠃⣀⣀⣀⣠⠧⣀⡀⠀⠀⠀⠐⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣘⣷⡰⣏⡼⠋⠁⠀⢉⣿⣇⠀⣶⣠⡇⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢉⡾⡇⣘⡇⠛⠀⠀⠋⠙⣿⡄⣼⡟⡰⢟⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⡀⠚⠁⢠⡀⠠⣤⠀⢠⣿⣾⣿⣿⣇⡙⠶⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢉⣀⡄⠀⣠⣿⣷⣤⠀⢰⣿⠉⢻⠏⠛⣷⢉⡀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣆⣰⣿⣷⣸⣿⠀⢈⡡⠠⡞⠃⠀⢀⡘⢿⣷⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⢀⣉⣡⡶⠟⠁⢉⡰⠋⠠⠘⠁⠀⠁⠀⠀⠀⢻⣿⣿⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠁⣀⠸⠋⠀⠐⠈⠁⠀⢀⣶⢷⣦⠀⠀⠀⠀⠀⠀⠻⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠂⢠⡾⠉⢠⣆⠀⠀⠠⠀⢺⣄⠛⠬⠁⠀⠀⢠⡀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢹⠁⠀⢀⡙⠃⠀⠰⡇⠀⠻⠿⣦⣀⣀⣤⡌⠀⡀⠀⠀⢸⣏⣳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⡀⠀⠀⠐⢖⠀⠘⠦⡀⠀⠀⢉⣛⠻⣄⡀⠀⠀⠀⠀⠉⠙⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣤⣄⡀⢱⡀⠀⠀⠀⠘⢻⡿⢿⡼⡇⠀⢀⠀⠀⠰⢂⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠘⡎⣿⡟⠛⣶⡷⠀⠀⠀⢀⠜⢡⠟⢰⣵⠄⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠁⢠⢿⠟⠁⢠⡿⠁⠀⠀⠀⠁⠀⠀⠀⠀⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢛⡛⠉⠀⠀⠀⠈⠀⠀⠀⠀⢠⠀⠀⠀⢡⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⠐⠀⠀⠀⠀⠀⠀⠀⠘⢷⣌⡑⢚⣿⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠂⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠌⠻⣶⡿⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⣿⠇⣈⠠⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡯⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠟⠛⠛⠛⢠⠾⣄⠓⠀⢀⡴⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⠀⠂⠀⠀⠀⠀⠀⠃⣠⣾⡟⠁⠀⣤⡀⠀⠀⠀⠀⠈⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⡁⠀⡀⠀⠀⠀⢰⡟⣢⣵⠞⠉⠀⠉⠳⡄⠀⠀⠀⠀⡀⢀⣄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠸⣿⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠈⠀⠀⠀⠀⠉⠿⣀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⣿⣦⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡠⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣶⣾⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡌⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢛⣾⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠲⠶⠤⠤⠀⠤⠤⠀⠃


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠉⠈⠉⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⠶⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⣫⡵⠶⠚⠛⠛⠛⠲⢭⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢃⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠵⠾⠷⠶⠦⠤⠤⠤⣤⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠓⣶⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢀⡶⠲⠶⠦⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣾⣅⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢼⣥⣤⣀⣰⠃⠀⠀⣀⣀⡀⠀⠀⢠⡟⡼⠁⣽⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⠀⠀⢸⡇⠄⡟⠁⠀⢀⡼⠉⠉⠉⡿⢀⡾⣼⣧⣼⢳⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣻⠿⠿⠿⠷⣴⣶⣾⠣⢰⣧⣀⣀⣘⣓⣶⣶⣾⣡⣾⠃⠙⠾⢣⡟
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⣞⣿⣦⣾⡏⠉⠛⠛⠛⠛⠛⠛⣹⠃⠀⠀⢀⡞
⠀⠄⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣠⠏⠝⡿⠋⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⡾⠁
⠀⠂⠀⠀⠀⠀⡾⠀⠀⠀⠀⢠⡾⠁⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠐⣯⣝⣒⢦⣄⣀⡙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣍⡉⠛⠓⠿⠿⠯⠭⠭⣿⣷⣶⣶⣶⠾⠁⣀⡼⠃⣰⠏
⠀⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠲⠶⢤⣄⣀⣀⣀⣀⣐⣓⣛⣛⣉⣤⠾⠃
⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠟⠛⢋⣉⣉⣐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠋⣠⠖⠋⠉⠉⠉⠻⣷⣄⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⢀⡞⠁⠀⠀⠀⠀⠀⠀⠹⣿⣧⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⢸⠀⠀⠀⠀⠀⠀⠀⣀⣀⣾⣃⡄⣸⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⡶⠶⠶⠿⠿⡘⡿⠛⠛⠛⠛⠛⠛⠛⣻⣏⡙⣱⡏⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠠⡴⣧⡿⢸⠀⠀⠀⠀⠀⠀⢻⣴⠔⢋⠴⠀⠀⠀⢹⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠻⡇⠀⠀⠀⠀⢳⠤⠐⠊⠀⠀⠀⠀⠀⠀⠀⡇⡥⠐⠆⠀⠀⠀⠀⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣶⣷⠀⠀⠀⠀⢸⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⢿⠰⠀⣄⠀⠀⠀⠀⠸⡏⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⣸⣿⣿⡆⠀⠀⠀⢸⡇⣀⡄⡀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠁⠀⠀⠀⠀⠀⢻⡀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⠀⢸⣿⠈⣿⡇⠀⠀⠀⠀⣟⠋⢛⠀⠀⠀⠀⠀⠀⠀⠈⣿⣀⠇⠀⠀⠀⠀⠀⠘⣇⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠓⢉⣿⠋⠀⠹⣿⠀⠀⠀⠀⢿⣤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠛⣣⢰⢰⡄⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⡀⠀⡐⠀⢀⣿⣿⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢰⠃⡐⠀⠀⢸⣿⣿⠋⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣾⢰⠀⠀⠀⠘⣿⣿⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⣿⡇⣠⣾⣿⣷⣿⣿⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⣼⣿⡿
⠀⠀⠀⠀⠀⣿⠁⢘⣿⣿⣿⣿⣧⣜⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣦⣦⣦⣤⣤⣤⣶⣶⠶⠃⠀⠀⠀⣰⣿⣿⠃
⠀⠀⠀⠀⠀⣿⠀⢺⣻⣿⣿⣿⣿⣿⣿⣿⣶⣤⣶⣶⣾⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉⠉⠛⠛⠓⠒⠐⠚⠁⠀⠀⠀⢀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⢸⡇⠈⠻⠋⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢋⣩⡤⣤⡤⠐⠤⣤⣤⣀⣀⡀⠀⠀⠀⠀⣀⣤⣶⣿⠿⠋
⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠑⠈⠙⠛⠛⠛⠛⠛⠋⠉⢉⡠⠒⡩⠴⠚⠋⠁⠀⠀⠉⠛⠛⠛⠛⠻⠿⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣤⣀⠀⠀⠀⢀⣀⡠⢔⣨⡿⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠿⠟⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⢊⣉⣉⣉⡭⡭⠽⣦⣄⣀⠤⠄⠐⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⣴⡤⠦⣤⣤⠀⣿⠓⢮⡪⡾⠋⠀⠀⠀⣀⣤⣴⢖⢮⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠉⣁⠘⠃⠼⠤⣾⠯⠮⠍⡽⢄⣺⣆⢘⠿⠒⢀⠤⠭⣏⠀⠈⠱⣕⢕⡩⣷
⠀⠀⠀⠀⠀⠀⡔⠁⠒⠒⡒⠒⠂⠀⠀⠀⠠⠯⠤⠼⠆⠀⣉⣘⡑⠚⢀⣠⣴⣮⢟⢿⡆⠀⠀⡇⡪⡪⣺⡇
⠀⠀⠀⠀⠀⠀⡅⠰⠂⢀⡉⢰⡝⢓⠘⡓⠚⢳⢢⠀⢃⢀⠀⡃⠀⢣⡞⡫⡢⡡⡱⡱⡇⠀⠀⡇⠨⡪⢸⡇
⠀⠀⠀⠀⠀⠀⢃⠀⠠⠤⠀⠸⣀⣊⠁⠀⠐⣏⡬⢄⡸⠀⠉⢀⢠⣸⡏⡪⠊⢊⢊⢎⡇⠀⠰⠀⠨⡪⣽⠁
⠀⠀⠀⠀⠀⠀⣯⡀⠀⠤⢤⠈⢩⢙⣛⣻⠛⠓⠁⠀⠀⡀⡤⠂⣸⣿⢇⢔⢔⢕⣵⠗⡇⢀⠃⠀⢨⣺⡟
⠀⠀⠀⠀⠀⠀⡏⢽⣟⢫⣿⣶⡖⠾⢤⡤⣤⣮⠒⣽⣧⡤⠴⠾⠛⠣⠔⠕⡡⣎⡿⡯⡇⡎⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⡗⠂⢈⣛⠛⠉⢠⡀⠀⠀⠛⢇⣚⣿⡿⠇⠀⡄⠀⢰⡨⣺⣽⢽⢻⣻⣏⡄⠀⢀⣾⡧
⠀⠀⠀⠀⠀⠀⣇⢀⣤⠁⠤⠤⠤⠄⠀⣠⣒⢂⣀⣀⣀⣊⣒⠀⠀⡌⣢⢷⣾⡛⣋⣿⢀⢌⠖⣾⡟⠁
⠀⠀⠀⠀⠀⠀⡇⠸⣤⣖⣀⣐⣃⡀⠐⢀⡤⠤⠀⡀⡀⠀⣺⣦⠀⡏⣾⢘⢾⠾⣵⣿⢠⣥⣾⠁
⠀⠀⠀⠀⠀⠀⡏⠁⠁⠆⢘⡁⠛⡩⡉⢹⠉⠉⠉⣉⡏⢻⣽⡯⢰⠭⡆⢰⢛⣂⣰⡿⠛⣷⢹⡒⢤⡀
⠀⠀⠀⠀⠀⢀⡏⠀⡄⠆⡬⠥⣀⠇⣈⠁⠀⠀⠀⢈⠅⢸⣻⡇⢸⡨⡧⣈⣩⣵⣟⣇⠀⠈⢛⠛⡉
⠀⠀⠀⠀⠀⠣⣵⣤⣜⣒⢶⢤⢤⢭⣬⣭⡭⡭⣥⣥⡤⣮⡼⡡⢒⢬⣪⣴⠵⠳⠵⢛⣐⣦⡴⠖⠒⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠓⠚⠋⠐⠚⠐⠒⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣾⣷⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉



⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣶⣶⣶⣶⣶⣾⣷⣶⣾⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣾⣿⣷⣿⣿⣿⣶⣶⣶⣶⣶⣾⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠠⡄⠀⠀⠀⠀⠀⡀⠀⡈⠃⠀⡠⠒⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣬⢐⣰⠇⠀⢀⠀⣴⡻⠟⠛⢶⣦⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡛⠁⢤⣭⠀⢠⠁⡟⣿⠀⢀⡺⠲⠭⣛⡄⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⢛⡂⠀⢸⡅⣻⠀⠀⣥⢠⣇⣨⣤⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢹⣧⠀⠈⡵⢻⣏⡿⣽⣃⣀⠸⣷⣿⠗⠻⣦⠖⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⡀⠀⠈⡼⠋⠈⠭⣴⣵⢟⣛⣢⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⢻⡄⠙⠘⢠⣤⠞⠁⠈⠐⠛⡄⢧⠙⡇⠙⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢙⡷⠀⠀⣿⠋⠀⠀⠀⠀⠘⠳⠿⢟⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⢰⠀⠀⠀⠀⡀⠀⠀⠾⠀⠱⠀⣄⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⠀⠀⠀⠀⠀⠀⢠⡀⠁⠀⠀⠀⠀⣰⣶⡖⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠲⠀⠉⠀⠀⠀⢻⡇⠀⢀⣾⣿⣾⣷⣦⣀⠀⠈⣿⠗⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⣿⢻⣀⣾⣿⣿⣿⣿⡯⠉⢷⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⢐⡃⠠⢨⡋⡼⣿⣿⣿⡃⠀⠀⠀⠟⠘⠛⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢈⣟⣿⣠⠠⠶⣯⠯⠭⡥⠀⠉⠀⠀⠀⠀⠀⠺⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣛⠑⠆⠛⢹⣿⣿⡟⠂⠄⢀⠀⡀⠰⠂⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⣿⠀⠀⣐⢻⣿⣟⠁⠀⠀⠈⠡⠈⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⢻⡄⠈⠀⣾⣿⡿⠑⡆⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠈⠀⠀⢰⣿⡟⠀⠀⠀⠀⠨⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠿⣬⣧⣀⠀⠀⣀⠀⠀⡺⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠈⢿⣷⣾⣿⣡⣿⣆⣾⣆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣢⣀⠀⠈⠹⢿⣿⣗⡀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠿⣿⡷⣤⣦⡀⠙⢛⡻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⣀⣾⢡⡾⠿⠀⠐⠈⢽⣷⣌⠁⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⡟⠴⢓⣌⠀⠀⣦⠹⢿⣿⣀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣣⣾⣿⣁⢐⣀⡷⣴⠏⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠉⠹⣦⣝⣛⣛⣴⣶⡖⠁⠀⡆⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣛⠟⠋⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢒⣩⠥⣤⣄⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⡴⠫⠀⣈⠔⠋⠳⡌⢻⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣜⣴⣷⠎⠀⠀⠀⠀⢳⠀⢘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢠⣿⣿⠏⠀⠀⠀⠀⠀⢸⡆⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿⡟⠀⠀⠀⠀⠀⠀⢠⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⠀⢿⠇⠀⠀⠀⠀⠀⠀⢸⣧⠘⡛⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⢸⣇⢀⣨⢃⡀⠀⠀⠀⠀⠀⣷⡤⠼⠀⣆⢈⡙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠈⢿⡛⣉⣎⡁⠀⠀⠀⠀⢤⣿⠦⠐⣶⡏⠈⠱⡈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠃⠀⠀⠀⠀⡨⠁⠀⠚⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠻⠀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⢀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠀⠀⠀⠀⠀⢠⡀⠀⠀⡼⠀⠀⠀⠀⠀⢲⣄⣀⣴⠏⠀⠀⠀⠀⠈⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⠷⠞⠁⠀⠀⠀⠀⠀⠀⠙⠻⠋⠀⠀⠀⠀⠀⠀⣿⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⣀⣀⠠⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⣦⣀⣀⣸⣀⣀⣀⣄⣤⠤⠀⢒⣒⣒⣐⣘⢐⣌⣙⣶⣶⣿⣷⡶⣾⣿⣷⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⠿⠫⢿⠯⠝⠒⠊⠙⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠋⠭⠾⠶⠶⠶⠦⠬⠭⠀⠀⠁⠀⠀⠀⠀⣀⠈




⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡞⡽⠛⠛⠛⢻⡝⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣹⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠶⠶⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢠⢿⡄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⡟⠓⠚⢻⡇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢀⡟⠀⢳⡈⢧
⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⢸⡇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡾⠀⠀⠈⢧⠘⣇
⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣇⠀⠀⠀⢠⠶⠶⠶⡇⢸⡷⠶⠶⢾⡇⢸⠶⠶⠶⡄⠀⠀⠀⢰⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⢧⣸⠃⠀⠀⠘⣇⡼⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⢀⡴⠋⠉⠲⠚⠛⠛⢦⡀⠀⠀⠀⠀⠀⢠⠖⠢⠤⢤
⢀⡶⠋⠀⡠⠀⠀⠀⠀⠀⠈⠋⠛⢶⡄⠀⢰⠋⠁⠀⠀⠸⣅
⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢈⠇⠀⠀⢳⣀⣀⠀⢰⠇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⢦⠀⠀⠀⠀⠈⠉⠉
⠘⠲⢤⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⢀⡤⢤⣤⠤⠤⡀
⠀⠀⠈⠳⠤⣀⣄⠀⠀⠀⢀⡤⠖⠋⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⠀⠹⢄
⠀⠀⠀⠀⠀⠀⠀⠉⢙⡋⠉⠀⠀⠀⠀⠀⠀⣣⠀⠀⠀⠀⠀⠀⠐⠀⣠⠧⠤⢤⢤⠤⢤⣀
⠀⠀⠀⠀⠀⣀⣤⡞⠉⠉⠙⠖⢦⠀⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⠎⠠⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠚⠛⠛⠓⣦⣄
⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠠⠧⢄⠀⠀⢈⣙⣆⣀⡤⢤⣤⠼⠁⠀⢀⡴⠋⢧⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣀⣤⠏⠀⢠⠖⢯⡁⠀⢸
⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣷⠋⠉⠉⠉⠉⠀⠒⠢⠤⠀⣀⠘⠳⠤⢼⠀⢀⡇⠀⠀⠀⢀⠴⠚⠋⠁⣀⣠⠶⠃⠀⢈⠃
⠀⠀⠀⡏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠦⡤⢉⡑⠒⠚⠁⠀⠀⢠⡞⠁⠀⠀⠀⠀⡼
⠀⠀⠀⠣⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡵⢦⠀⡿⠀⠀⠀⠀⠀⢰⠁⠀⠀⢠
⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣡⠖⠉⢸⡇⠀⠀⠀⠀⠀⢸⡀⠀⢀⠃
⠀⠀⠀⠀⠀⠙⠷⠦⠤⣤⣀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠡⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⡇⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠏⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢃⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠀⠀⠀⣴⢸⡇⠀⠀⠀⠀⠀⠀⣾⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠘⠁⠈⠇⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠈⠙⠲⡄⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠂⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⣸⠇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡁⢇⠶⢈⠆⡔⠒⠲⠤⣄⣀⡀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⢠⡗⠒⠒⠒⠲⠚⠃⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠑⠢⠭⠭⠔⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠛⠁⠀⠀⠀⠀⠀⢸⣇⡤⠴⠖⠶⠦⠤⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠲⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⢀⣠⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠦⢤⣄⣀⣀⣤⠤⠖⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣪⡿⠟⠉⣉⣉⣉⡉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣱⣿⠋⢀⠖⠊⠉⠙⠻⣿⣷⡄⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⡿⠁⢠⠃⠀⠀⠀⠀⠀⠹⣿⣏⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⢸⣇⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⢿⡄⠀⠀⠀⠀⠀⢰⣿⣇⣠⣠⣤⡴⠶⠊⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣿⡿⣷⣤⣤⠶⠶⢞⣛⣫⣿⡯⣷⣶⣶⣴⢴⣿⡧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡛⣉⣉⣩⣽⣿⣿⡄⠾⠿⠟⠛⠹⡿⣴⣿⠀⠁⠘⠻⣿⣷⠠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢿⢻⣿⡟⠻⠉⠓⣿⣉⣿⠀⠀⠀⠀⠀⠱⠹⣷⡇⠀⠀⠀⠀⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣶⣵⣿⣿⠀⠀⠀⠀⢛⢿⣧⠃⠀⠀⠀⠀⠀⠑⣾⡇⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣏⣙⢾⢸⣿⢿⣿⣆⠀⠀⠀⠀⢺⣿⡆⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⢻⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡟⣼⠋⣼⣿⣿⡄⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠹⢋⠁⠀⠀⠀⠀⠈⣿⢿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⣀⣀⠰⣼⡿⠋⢛⣿⣿⡟⠀⣥⣶⣿⣿⣿⣧⠀⠀⠀⡘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠈⠙⠻⢿⡍⢠⣄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⣴⣿⣿⣿⠀⠰⣿⢿⢸⠛⣿⣿⡆⠀⠀⠈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠉⠺⣿⣧
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣧⣀⣴⣼⡾⢸⢰⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣧⣸⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠛⢿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠚⠛⠁⠀⠀⠀⠀⠀⠀⢀⣴⡏
⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢋⠞
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠀⣀⣀⠀⠉⠛⢿⣿⣿⣿⣿⠃⢁⣀⣀⣤⠴⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⢿⣿⣷⣄⠀⠈⠉⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠉⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠐⠈⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣶⣤⣤⣤⣶⣶⡆⠀⠀⣀⠀⠤⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣇⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⠀⣰⠈⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⠀⠈⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢩⣿⠿⠿⠿⠋⠀⡄⣰⣾⡶⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢧⣀⣿⡇⠀⠀⠀⠀⠈⠰⣟⣿⡇⠈⢷⣦⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠈⠉⠑⢢⠀⠀⠀⢠⣴⣿⣿⣤⣀⢨⣟⣿⣯⡛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣀⢖⣿⡟⠛⢟⠋⠁⠀⠀⣸⠀⠏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠵⠉⠋⠛⠀⡿⠀⠀⠀⢀⣿⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⡇⠀⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠒⠲⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⣇⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠠⠚⠉⠉⠙⠳⣤⡀⠀⠀⠀⠀⢀⡀⠴⠊⣠⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢀⠀⣀⡴⠶⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣶⣤⣄⣤⣶⣶⣿⣿⠟⠁⠀⠀⠀⠀⢀⣀⠀⠄⡩⠆⠁⠀⠄⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⢀⣠⠴⠚⡁⢐⡲⠊⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢉⡉⠳⠀⠀⠀⢔⡀⠁⢠⣤⠴⠒⣉⠹⠇⠲⠦⠶⠋⠀⠀⠀⠠⠂⠀⠖⠀⢠⠆⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠸⢷⠏⠁⠉⣴⣢⠄⠀⠉⣀⠐⠈⣁⠀⢀⠀⠂⠀⢠⠔⠂⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠸⠀⠀⠊⠉⠋⣜⣨⠅⢀⢒⡎⣀⠬⠤⣀⠐⠊⠁⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣒⣐⣒⣀⣀⣀⣀⣈⣉⣉⣉⣀⣈⣁⣀⣈⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣐⣀⣀⣀
⣿⣿⣉⣩⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠈⠀⠀⠀⠤⠁⠒⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠿⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⢦⣤⣤⣟⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣫⣿⠛⠉⢠⣴⡤⢤⣀⣀⠀⠀⠚⠓⠢⠤⢠⣼⣻⣎⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠉⠀⠉⠀⠀⠀⠀⠀⠈⠙⠓⠫⢝⣉⣐⠛⠒⠒⣶⣾⠗⠋⠀⠂⢻⡆
⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⡈⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠏⠉⠉⠉⠀⠀⠀⠀⠈⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⣾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠃⠀⡆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡜⠀⠐⣴⠁⠀⠀⠀⣀⣀⣀⣀⡀⠀⡀⠀⠀⣀⣀⣀⠑⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠷⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠨⠋⣠⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠱⠃⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡂⠄⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⢀⣤⡶⠋⠀⢰⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠙⠛⠶⠶⣦⣴⣤⣿⠦⡠⢸⣦⣤⣴⠶⠟⠋⠁⠀⠀⠀⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣤⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠤⠀⠀⣀⡀⣀⣀⣀⠀⠀⠂⠀⠀⠀⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠉⠛⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠋⠉⠉⠙⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡎⠉⠉⠉⠉⠉⠉⠉⠐⠒⠒⠒⠒⡀⠀⡇
⠀⠀⠀⠀⠀⠀⣀⣀⡤⣤⠤⠤⣼⠀⠀⡧⠤⠤⠤⢤⡤⠐⠂⠦⡄⠤⠤⢤⡇⠀⣇⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢀⠞⠁⠉⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠓⠒⠛⠁⠀⠀⠈⠉⠉⠈⠉⡍⢢
⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠈⡇
⠀⠀⠀⠀⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⡄⠀⡇
⠀⠀⠀⠀⡇⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⡇⠀⣷
⠀⠀⠀⠀⡇⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⡇⠀⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⠀⠀⠀⠤⠒⠒⣀⠐⠓⠢⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠀⠀⠀⠃⠀⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠢⠤⠤⢴⢰⠂⠙⠤⠉⠙⠀⡗⠠⠤⠀⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠂⠤⠄⠐⠚⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠐⠒⠒⡒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠑⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠒⠒⠒⠒⠒⠒⠲⢤⣤⣀⡤⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠖⠚⠁

⠀⠀⢰⠖⠂⠀⣀⠶⣶⣶⣶⣶⠲⡶⢶⠶⠶⠶⡶⠶⠐⢂⠰⢷⣤⣤⡶⡖⡀⢦⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠢⠔⠠⠆
⠀⠀⢸⠀⠠⡀⡎⠀⠈⠉⠉⠉⠉⠀⠉⠈⠙⠻⢿⣶⢨⠟⣿⣿⢿⣿⣿⣿⡻⠀⠀⠀⢚⡻⠶⠶⢦⣤⣤⣴⣦⣄⡀⠀⡀⠀⠠⠆
⠀⠀⢰⠄⠀⠘⠖⢻⣔⠂⠀⠀⣀⡀⠀⣤⡀⣰⣿⢹⣿⣬⠟⣻⣾⣿⣿⣿⣣⠀⢰⢆⡜⠁⠐⠚⠛⠛⠿⠟⠛⠉⠉⠠⠉⠀⠠⠇
⠀⠀⢀⡀⠁⢀⡰⢉⠿⣤⣤⣤⣤⣤⣤⣬⣤⣤⣭⡝⠻⡝⢩⣽⠻⠟⣿⡇⢩⠻⢧⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⡥⠀⠠⢀⣀
⠀⠀⣍⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠋⠉⠉⠈⠉⠋⠉⠉⠉⠙⠋⠁⠀⠉⠉⠉⠉⠉⠉⠈⠁⠈⠛⠉⠉⠉⠋⣯
⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣀⠀⠀⠀⠹⠟⠿⠗⠂⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠳⠶⣶⣄⠀⠀⡄⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⣿⠋⠷⠀⠀⠀⠘⠇⠀⠀⠀⠀⣿
⠀⠀⡿⢠⡀⠀⠀⢀⣠⣀⡈⠁⣤⡀⠠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿
⠀⠀⣟⠙⠋⠁⠐⠚⢓⠊⠉⠛⠓⠋⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⣼⣿⡳⠇⣿
⠀⠀⣿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠃⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠽⠟⢳⣧⣿
⠀⠀⣿⠐⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⣀⣤⡶⢻⣾⣬⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠏⣿
⠀⠀⡏⢰⣶⡄⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣶⣯⣬⣵⣦⡀⠀⠀⠀⠀⠀⢀⣉⣀⣤⡄⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠸⡔⠃⠀⠛⠒⠆⠀⠀⠠⠀⣬⡍⠉⠽⠏⣉⢙⣫⣿⣿⣦⠀⠀⠀⠀⠀⠉⣟⠙⠇⣿
⠀⠀⣿⠀⠐⠂⠀⠀⠀⠈⠀⠀⠀⠀⠒⠁⣀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⣽⣏⠈⣹⣿⣷⡀⠀⠀⠀⠀⠀⣋⣼⣆⣿
⠀⠀⣿⢰⠶⣄⣰⡄⠀⢲⣤⣀⠀⢀⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠠⠔⢿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢹⣿⣧⣿
⠀⠀⣿⣨⣀⣸⣿⡇⠀⣈⡉⠉⣀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⡀⠂⡈⢋⡽⠿⣿⣿⣿⣿⣟⣛⣿⣏⠀⠀⠀⠀⠀⠈⠻⠇⣿
⠀⠀⣿⣴⣬⣿⣿⣶⡏⢉⣁⡀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⢄⡀⢺⣷⢿⣿⣿⣿⣿⣿⣿⠀⢀⡀⠐⣀⣈⣵⡆⣿
⠀⠀⣿⢻⣿⣿⣿⣿⣿⡛⠻⠻⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠠⣄⣀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇⠨⠉⢉⠁⠀⠀⠀⠸⢉⣴⣿⣇⣿
⠀⠀⣿⢿⠿⠿⠿⣮⠉⠁⠀⠀⠀⠘⠓⠂⠀⠀⠀⠀⠠⠤⣀⡜⢋⡁⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠋⣄⠋⣿
⠀⠀⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠋⠁⠛⠁⠲⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⣾⣿⣾⣿⣶⣿⡇⣿
⠀⠀⣷⣀⠐⣶⡄⠀⠈⠁⠀⠀⠀⠀⠙⠉⠉⠀⠀⠀⠀⠀⠀⢀⣴⡃⢰⢖⣦⣸⡀⠀⠀⠀⠂⠀⣘⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡋⣿⣷⣈⢻⣦⡷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣶⣿⣿⣯⣿⠟⣶⣆⠀⢾⣿⣏⣀⣰⣿⣿⠿⣿⣿⣿⣿⢍
⠀⠘⡇⢸⣿⣿⣿⡁⠐⡤⠀⠀⠀⠀⠀⠀⠺⠿⣷⣶⣶⣦⣻⣿⣿⣿⣿⣿⣿⣯⣩⣻⣿⡿⠃⠒⣉⣻⣿⣿⣿⣿⣿⣿⠛⣿⣟⢸⡇
⠀⢘⣇⣻⣌⣙⣙⣛⣧⣤⣠⣤⣀⣠⣄⣀⣀⣂⣚⣊⣙⣛⣛⣛⣛⣛⣛⣛⣛⣛⣟⣛⣐⣒⣒⣊⠫⢉⣉⣙⣻⣛⣛⣛⣻⣛⣋⣼⡇
⠀⢸⡵⣎⡁⠔⢫⠶⠗⣺⣿⣶⠶⢞⣶⣾⣶⡷⣿⢒⡉⠓⠈⣑⢿⡕⢦⡥⠀⢩⡶⠟⠚⠓⠛⠓⠛⠛⠓⠿⠻⠿⠷⣭⠳⠜⣽⢺⡇
⠀⢸⠃⡎⠻⢉⣼⣿⣛⠛⢻⣻⠇⣽⠿⠿⠿⣿⢻⣿⣷⣅⠛⠁⢸⡃⠄⠸⣨⣤⡿⢿⣿⣿⣿⣿⠟⠉⠉⠛⠻⠿⢿⣧⣅⣿⠀⢸⡇
⠀⢸⢣⡟⣶⣿⣿⣿⠿⠀⠼⠟⠁⠀⠀⡀⠀⠛⣾⣿⣿⣿⣿⢿⡶⢧⡿⣷⣏⣡⣴⣿⠿⢿⣿⠃⢀⣴⣶⣿⣶⣶⣶⣿⣿⢶⢿⣼⡇
⠀⢸⡞⣧⢿⣯⣭⣉⠐⣞⡲⠐⠀⠀⠀⠁⢰⣾⣟⣿⡟⣼⣿⣾⠷⠼⢧⡿⢿⣿⣿⣿⣷⣦⣄⣤⣾⡿⣿⣿⣿⢿⣿⢷⣿⣻⣾⣻⡇
⠀⢸⣿⠇⣤⡙⢿⡟⣓⣭⣛⣤⣶⣶⣻⢷⣶⣿⣿⣿⣿⡛⣌⠂⢸⡇⠆⣨⡜⢿⣏⠙⢲⣿⡿⢟⣿⣿⣉⣩⣷⣾⣧⣴⡟⣭⠙⢸⡇
⠀⢸⣳⣟⣃⣅⣘⣯⣭⣭⣭⣬⣥⣭⣥⣤⣤⣤⣤⣤⣃⣁⣁⣻⣸⣇⣟⢃⣀⣘⣯⣭⣭⣥⣤⣭⣭⣭⣭⣭⣭⣭⣭⣛⣡⣈⣻⣽⡇
⠀⠀⠉⠈⠉⠉⠉⠁⠀⠀⠒⠒⠒⠐⠒⠒⠒⠒⠒⠂⠉⠉⠉⠉⠈⠉⠀⠉⠉⠉⠁⠀⠀⠒⠐⠒⠒⠂⠒⠒⠒⠀⠀⠉⠁⠈⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⡟⠉⣻⣶⠟⠻⣦⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢾⣇⣀⣿⠷⠾⠿⠻⢦⡵⢿⣅⣦⣿⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣿⣦⣿⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠹⣧⣶⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⣽⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⣢⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢩⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⡛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣾⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣮⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡃⠄⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠶⠶⠶⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠶⠶⠶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣸⢇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠁⠹⣆⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣴⠏⠀⠈⣷
⠀⠀⠀⠀⠀⠀⢰⡟⠈⠀⠀⠙⣧⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢈⣾⠋⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⣸⠇⠁⠀⠀⠀⠈⠻⣦⡃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢂⣴⠟⠁⠀⠀⠀⠈⠸⣇
⠀⠀⠀⠀⠀⠀⣿⢀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⡂⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⢀⠀⢂⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠁⣿
⠀⠀⠀⠀⠀⠀⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣥⣤⣠⣿⣿⣿⣿⣆⣤⣥⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿
⠀⠀⠀⠀⠀⠀⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿
⠀⠀⠀⠀⠀⠀⢻⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡟
⠀⠀⠀⠀⠀⠀⠈⢷⡌⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣎⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣔⡀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⢀⣩⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠶⠶⢶⣮⣤⣤⣤⣬⣤⣥⣤⣉⣀⣈⣠⣤⣭⣤⣬⣥⣤⣤⠶⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠉⠉⠉⠉⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠀⠿⠿⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠻⠿⠀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠏⣴⣶⣶⡌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣴⣶⣦⡙⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡀⢿⣿⣿⠟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠸⣿⣿⣿⠃⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⣭⣥⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣭⣤⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⣈⢴⣀⣿
⠀⠀⠀⠀⣼⣷⠟⠤⠈⣁⢰
⠀⠀⠀⢰⣷⣸⣄⠀⠀⣀⠞
⠀⠀⢀⡎⣯⡛⢿⠀⠸⣿⣩⡀
⠀⢀⡾⣰⠀⠀⠈⠁⠀⠻⠀⣀⣀⠀⠀⠀⡄
⠀⠈⡁⣿⠀⠀⠀⡤⢀⠀⢺⣿⣟⠛⠐⠚⠃
⠀⠘⢻⣿⠀⠀⠀⠠⠀⠀⣼⣿⣿⣿⣖⢢⣤⠴⠤⠄⠀⠀⠀⢸⣧
⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⣿⠿⢦⣘⣇⠒⠊⠉⠑⠂⠀⠀⠀⠼⣿⠁
⠀⠀⠀⠈⠉⣧⠀⠄⠀⠀⠉⠓⠂⢈⠉⢇⠀⠀⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢉⠚⢦⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⠳⡀⠀⠀⡀⠀⢸⣏⠀⠀⠀⠀⡠⠂⠃⢄
⠀⠀⠀⠀⢰⣧⡀⠀⠀⠀⠀⠓⠄⡀⢐⣤⡘⣵⡙⣖⣲⣦⡄⢉⡛⠀⠀⠀⠀⢄⠀⠈⠁⠖⠐⠀⢲⣄
⠀⠀⠀⠀⢠⠛⢇⠀⠀⠀⠀⠀⠀⠀⠁⠡⠻⢿⣿⣜⣻⡏⠛⣌⣷⠦⡀⠀⠀⠀⠙⣆⣠⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠈⠂⠤⠵⢦⡦⣤⣄⣀⣰⣆⣂⠸⠀⠉⣙⠍⣈⣵⣿⡿⠀⣼⠀⠀⠀⣴⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠾⠿⠿⣿⠿⢿⣿⣷⣼⣨⣼⠿⣿⣇⣼⣿⣾⣿⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⠉⠉⠁⠈⠉⠉⠉⠉⠉

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

⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⡀⠀⣀⡀
⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⣿⡟⠻⡟⢸⣿⣿⣿⣿⣯⣆
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⣻⣿⡇⢀⢧⠀⣿⣿⣿⣿⢿⢿⣦
⠀⠀⠀⠀⠀⠉⠻⣿⣥⣭⠇⠉⠀⣰⣿⡾⢿⣿⣿⣼⣆⢧⣿⣿⣶⣽⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠻⣿⣿⣿⣯⣹⣷⣷⣽⣻⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣴⣿⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡙⢿⣏⢻⣿⣻⡇
⠀⠀⠀⢄⠀⠀⠀⠀⠲⣶⣶⣶⣿⣿⣾⣿⣿⣿⣿⣿⣿⣧⣀⢻⡟⣼⣿⣶⣾⣿⣷
⠀⠀⢀⣼⣿⣦⣀⠀⠀⠉⠻⢿⣿⣿⣿⣿⡿⣿⡿⢻⣿⣿⣿⣿⣶⣿⣿⣿⡟⣿⡏
⠀⠀⢸⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣇⣿⡷⣄⣿⡿⠻⣏⠛⠻⣿⣿⡿⠋
⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈⣿⣿⣦⣼⣷⣾⣶⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣱⣿⣿⣿⣿⣿⡟⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡿⣻⣿⣇⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⡿⠽⣿⡿⠿⠿⣿⡮⠿⣿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡶⣖⣮⣙⣒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣪⠟⠉⡸⢫⠞⠉⠻⣟⣮⠙⢦⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠃⡎⠀⠀⠀⠸⣘⡇⠀⢷⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢰⢰⠁⠀⠀⠀⠀⡇⣿⠀⠐⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⢸⢸⠀⠀⠀⠀⠀⢸⣺⠀⠀⣎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣹⡇⠀⢸⢸⡀⣀⣀⣀⣀⣸⣼⣄⣀⡔⣧⣀⣀⣀⣀⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⣤⣤⢺⠋⣉⣉⡉⠉⠉⠉⠉⠉⠩⢹⣼⠇⠄⠀⠀⠀⠀⠀⠀⠀⡀⡴⠂⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢙⠁⡟⢸⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣦⠀⠀⠀⠀⠀⠀⠀⢴⣷⣿⡆⠀⠀⠀⠀⠀⠰⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣵⠔⡇⢸⠀⠭⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⡠⢀⠄⡉⠁⠀⠀⠀⠀⠀⠀⣨⠆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠕⡨⡇⢸⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⠄⡰⠂⠀⠀⠀⠀⠀⠀⢐⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠡⢒⡇⢸⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡡⢊⡔⣡⢊⠄⠀⠀⠀⠀⣡⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢪⢕⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠡⠋⠜⡡⢀⠔⡀⠀⠀⠀⡕⡣⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠅⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠁⠈⠀⠀⠀⠀⢖⡕⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢔⡨⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠨⢀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⢂⢊⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⣊⢕⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠢⠃⡄⡠⢡⠃⢎⢊
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⢕⠭⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠔⠐⢁⢀⠞⡔⠡⢃⣮⢢⢢⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⠅⡨⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡔⢠⠊⡔⡑⠁⠊⠀⡰⢣⢃⠜⡰⢁⡲⢁⢣⢣⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠅⡪⢐⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⡠⢂⠌⡰⢁⠎⠀⠀⢠⢂⠜⡰⣡⠂⡜⡠⡕⠡⠀⠢⢁⠂⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡫⢐⠕⡇⠀⢀⣨⢖⠄⡄⢀⠀⠀⠀⠐⠁⠃⠐⠀⢀⢀⡔⡑⣡⢂⠞⡔⢁⠎⡔⡝⠠⠣⢁⢃⠂⠆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢒⠅⢊⡇⠀⢨⠐⡣⢊⠔⡡⢂⠔⡠⠀⡀⣀⢆⡔⡡⢋⢈⠔⡱⢃⠞⡠⢂⢎⠜⡰⠡⢡⠂⠎⠌⡔⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠥⣊⠕⡇⠀⢸⠐⡕⡡⢊⠔⢡⠞⡠⢊⠔⡡⢊⠜⡰⢡⠊⡔⢠⠊⡰⢡⢃⠎⡰⠡⢁⠂⠄⠌⡔⡐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣊⣔⣾⣯⡀⢸⢈⠎⡠⢃⡔⢁⠞⡱⢂⡔⡡⢋⠎⡰⢁⠎⡔⢁⠞⡴⢡⢊⡜⡠⢁⢆⠄⠌⡜⡐⡰⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠥⢌⣣⣼⠀⠈⠀⠋⠐⠃⠞⠡⠋⠜⠡⠃⠎⠰⠃⠞⢠⠃⠎⠰⠁⠎⠐⠡⠊⠊⣊⣈⣘⣀⣙⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⡀⠀⡠⠀⣀⠀⡀⠀⡀⠀⡀⢀⡠⢀⠄⣀⠄⣠⠄⢀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⡂⠕⡈⠔⡈⠔⡨⠐⡈⠔⡊⠔⡈⠔⡈⠐⡩⠔⡡⢒⠅⢊⠴⢊⠄⢊⠤⢊⠤⢊⠔⣉⠔⢉⠔⡡⠔⡡⠖⡡⠊⡡⠊⡡⢊⠔⡡⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠐⠋⠐⠋⠐⠋⠔⠊⠔⠊⠔⠉⠐⠋⠔⠋⠖⠡⠖⠁⠞⠡⠆⠁⠔⠡⠖⠁⠚⠁⠐⠁⠚⠀⠚⠀⠊⠀⠊⠀⠉⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠛⠲⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣠⣤⣤⣤⣤⣀⠀⠙⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⢀⡶⠋⠁⠀⠀⠀⠀⠉⠳⣆⠀⠙⣧
⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠶⢶⡆⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⢸⡏⠳⢦⣤⣤⣤⣤⣤⣿⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣼⣧⣤⣤⣤⣤⣤⠶⠋⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣥⣤⣬⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣤⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠒⠠⠀⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣀⠀⠒⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢈⡷⠀⠀⠸⠀⢀⡰⠏⠀⠺⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⢰⢧⣟⠀⠀⠃⠀⣲⠏⠈⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠂⣐⣄⠡⠜⠒⣄⠀⠀⢀⡟⢀⠈⠀⠄⡏⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠎⣀⠀⠀⠀⠀⢅⠀⠀⠀⠱⡆⠀⠐⡄⠤⢧⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⠸⠀⠀⠀⠀⠱⠀⠀⡰⠋⠀⠆⠠⣄⠀⠖⠀⠿⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠔⠻⠎⠀⠀⢀⡤⠶⠉⠀⡀⠐⢾⢸⠅⢀⣄⠈⠃⡤⠠⠌⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠭⠛⠀⡀⢀⡀⠄⠀⡄⠀⡀⢀⢠⡄⠀⠀⠀⠀⠀⠇⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢲⣀⢀⣠⣁⢘⠀⠣⠱⡆⠨⠚⠁⠹⠀⡀⢠⡁⠙⠲⣄⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⣻⢮⡄⠀⠈⠢⢀⡶⠁⣠⡄⠀⣠⣆⠘⠊⡅⢠⣤⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢷⣤⢀⡠⢀⡙⠢⠤⠶⠩⡔⢀⣿⠃⢠⣆⡈⣎⡸⠷⠂⣿⠐⢀⠒⠋⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡃⠀⢉⡽⢀⠀⣀⠀⠱⣼⡏⡠⡄⢺⢁⢫⣴⡉⣶⢡⠆⢈⠻⠉⡙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣎⠩⢤⣈⠀⢸⣇⠑⢿⣶⡟⢉⡴⢁⣨⡄⣿⠈⣶⢹⣆⣦⡄⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠃⠘⠒⠘⠛⠂⠐⠛⠙⠘⠁⠀⠃⠛⠈⠀⠁⠈⠀⠈

⠀⠀⢀⡴⠋⠉⠶⠚⠛⠓⢦⠀⠀⠀⠀⠀⠀⢠⠖⠦⠤⣄
⢀⡖⠋⢀⡠⠀⠀⠀⠀⠀⠈⠉⠛⢶⡀⠀⢰⠋⠀⠀⠀⠺⡄
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠈⢣⣀⡀⠀⣴⠁
⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠉⠉⠁
⠘⠲⢦⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⢀⡤⢤⡤⠦⢤⡀
⠀⠀⠈⠳⠤⣠⣄⠀⠀⠀⢀⡤⠖⠋⠀⠀⠀⡴⠶⠋⠀⠀⠀⠀⠀⠧⣀
⠀⠀⠀⠀⠀⠀⠀⠉⢙⡋⠉⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠐⠀⡨⠦⠤⡤⣤⠤⢤⡀
⠀⠀⠀⠀⠀⣀⣠⠞⠉⠉⠑⠖⢦⠀⠀⠀⠸⡁⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⣠⠃⠄⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠛⠛⠛⢲⣤⡀
⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠸⠦⣄⠀⠀⣈⢻⣀⣠⠤⢤⡤⠞⠁⠀⢀⡔⠙⡆⠀⠀⡟⠀⠀⠀⠀⠀⠀⢀⣀⡼⠁⠀⡰⠚⡏⠁⠀⣇
⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠋⠉⠉⠉⠉⠐⠒⠠⠤⠀⣀⠘⠶⠤⢇⠀⣸⠃⠀⠀⠀⣠⠖⠛⠉⢀⣀⡤⠞⠀⠀⡙⠀⠀⠸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠲⢤⠌⣉⡓⠒⠋⠀⠀⠀⣰⠋⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠣⣀⡀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⠷⡄⢸⡇⠀⠀⠀⠀⠀⡏⠀⠀⠀⡜
⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠈⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡥⠊⠁⣼⠀⠀⠀⠀⠀⠀⣇⠀⠀⡼
⠀⠀⠀⠀⠀⠙⠷⠤⠴⣄⣀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡌⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⡀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠄⢸⠀⠀⠀⠀⠀⠀⢀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡦⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠈⠙⠲⡄⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⢀⡾⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠀⣁⠶⢨⠀⠔⠒⠢⠤⣀⣀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⣸⠓⠓⠒⠒⠶⠛⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠑⠠⠭⠥⠜⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⡋⠀⠀⠀⠀⠀⠀⢿⣀⡤⠶⠲⠶⠤⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠦⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠤⣤⣄⣀⣠⡤⠴⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣟⡀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠻⠿⢂⡀
⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠲⠏⣀⡸⠀⠀⢸⠀⠀⠀⢝⠁⠐⠀⠀⠀⠀⠀⠀⡀⠩⠀⠉⠉⠁⡩⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠢⡀⠀⣠⠆⠀⠀⠈⢧⢀⡀⠁⠀⠀⠀⡰⠀⠀⠀⠈⠁⠀⠦⢀⠖⠈⠄⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠑⠂⠥⠤⠘⠂⠀⠀⠀⠀⠀⠀⠠⠀⡜⢀⠔⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠁⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⡔⡎⠀⡖⠁⠆⠀⠀⠤⠒⠒⠒⡆⠀⠀⠀⠀⠀⣠⣄⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠸⡄⠔⠈⣆⢈⡿⠃
⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠒⠀⠁
⠀⠀⠀⠀⠀⠀⢸⠁⠈⠳⡄⠀⠀⠀⠀⢀⢀⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣞
⠀⠀⠀⠀⠀⠀⢸⡖⠂⠁⡄⠀⠀⠀⠀⠀⠀⠁⠀⢢⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠌⣏
⠀⠀⠀⠀⠀⠀⢸⡁⠀⢰⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⢤⣄⣀⡀⠔⠀⠀⠀⠀⡆⠈⢀⣤⠀⠀⠸⣼
⠀⠀⠀⠀⠀⠀⢸⠷⠚⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡋⠁⡌⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢸⠖⠢⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠤⠀⠀⠀⠀⠀⡀⠀⠈⠀⠓⠚⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠘⠆⠀⠆⠠⠀⠀⠀⠀⠀⠆⠣⣄⡁⠀⠀⢀⡔⠋⠉⠁⠉⠑⢄⠀⠀⢀⠔⠋
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⡜⢠⠔⠠⡀⠀⠀⠀⠈⣠⠔⠚⠁⠀⠀⡌⠀⠈⢢⣆⡠⠠⠀⡇⠀⠆⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⠞⠁⣇⢀⠆⠀⣐⣤⣤⠀⠘⠶⠀⠀⠀⠀⢇⡱⠀⢨⠏⠂⠀⣰⠇⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠰⠚⠁⠀⠀⠈⠿⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡶⠎⠦⢤⡾⠊⠀⠘⠶⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣰⣿⣿⣶⣛⠉⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣛⣋⣙⡛⠛⠋⠙⠻⠿⣿⣿⡟⠛⣿⣧⡾⣿⣿⣿⠟⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⠛⠻⡿⠿⠀⠀⠀⠌⠀⠀⡿⣯⣤⣼⣿⣗⡿⢿⣿⠙⣛⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣙⡳⢿⣿⣿⣧⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⡏⠈⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⠃⢠⡀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠂⠀⡄⠀⠈⠑⠒⢀⣤⣴⣦⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠉⠁⠀⠐⠁⠀⠸⠗⢄⠀⠀⠘⠀⠀⠉⠙⢳⣧⠒⠁⣱⡀⣧⢀⠀⠱⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣶⣿⣷⡄⠀⠀⠀⠀⠠⠀⠀⠈⢧⠀⣆⠀⠀⠀⠀⣸⣏⠲⢾⣤⡗⠀⠀⠁⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠋⠀⣹⣿⡟⢀⣤⠖⠈⠀⠀⠀⠀⠘⠀⣁⠀⠀⠀⠜⠀⣹⣼⣿⠟⢀⡈⠀⠀⠄⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣿⠃⢀⣴⠟⡻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠀⠀⠰⠀⠾⢿⡿⢁⠀⠻⣿⣶⣶⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣴⣿⡿⠏⠀⠀⠀⠀⠀⠀⡅⠀⣴⠎⠀⣠⡇⠀⢀⣄⠀⠀⢀⠀⣿⡇⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣍⠁⠀⠀⢀⣀⡀⠀⢴⣶⣷⢸⡟⢀⣼⡿⠁⠀⠈⢿⣷⣤⡴⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠉⢉⡿⠉⢉⣽⣿⡿⠿⣿⣿⠾⠇⠼⠛⠁⡠⠀⠀⠘⣿⣿⣿⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢠⣾⠇⢰⣿⠟⠁⠀⠀⢈⣿⡦⣠⣤⣴⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⣿⠃⠀⢸⠏⣀⣤⣤⣴⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡇⢀⠂⡾⣼⠟⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡜⣼⠇⠻
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣾⣽⡟
⠀⠀⠀⠀⠤⠤⠴⢞⣿⣿⣿⡿⠏
⣤⣤⣭⣷⣾⣿⣿⣿⣿⣯⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣠⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠰⣿⠘⠇⠿⢿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡡⠍⠁⠀⠀⠀⠀⠀⠘⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣩⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣉⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣉⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢟⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣭⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣋⠛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠟⡫⠀⠇⠲⡄⠳⣶⢖⣦⠖⠒⣲⡶⣛⠤⡙⠟⡿⠁⠹⠘⣿⢞⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢖⣼⣿⠀⠇⠃⠀⠈⠛⠀⠁⣾⣍⣲⡿⣻⣾⣿⡰⣧⠀⠁⢀⡀⠈⠈⢀⣭⣛⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢒⣿⠿⠫⠆⣀⣿⡖⢠⠀⠀⢈⡈⠉⡁⠾⠿⡛⡫⠶⣿⠄⣠⣿⡿⣠⣴⠈⣯⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢛⡉⣡⣾⢟⡄⢿⣛⣳⣫⣿⡜⠮⠴⣉⣥⣤⠭⠝⢿⣷⡉⢀⠹⣮⣿⣭⠟⢄⡌⣧⠈⠇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡀⡾⠸⢋⡴⠚⢳⣦⣙⡛⠋⠨⠙⣆⡜⣿⢃⡞⠭⣍⠆⢻⡇⣨⣇⣈⠉⣡⣄⢾⢀⢹⡆⢰⣮⡃
⠀⠀⠀⠀⠀⠀⠀⢾⡄⠳⠀⣞⡾⣿⠞⠉⣁⣀⡌⢠⣙⣛⠳⢞⣸⣿⢠⡟⣿⢸⠳⠍⣏⡿⢁⡴⢂⢷⣻⢞⡧⢸⣿⡇⣆
⠀⠀⠀⠀⠀⣰⢇⠈⢷⡀⠀⣿⢱⢟⡤⢊⣉⣉⠓⢾⡲⢙⡟⢾⡅⢻⣿⣿⣿⢸⡟⠊⠡⣾⠙⡅⢨⣦⡤⣾⠃⢸⣿⢃⡟⣆
⠀⠀⠀⠀⢠⠏⡜⣠⢸⣷⠀⣨⡟⠚⢰⣏⣵⣟⠳⣤⠙⢿⣤⣂⣽⠀⠙⠧⣽⣆⢲⣘⡥⠀⡀⣷⣈⠛⢛⣡⣠⠿⢿⣾⣥⡟⢰⡀
⠀⠀⠀⠀⣠⣾⣙⣪⣾⠇⠀⡀⢤⠀⠘⣿⡁⣀⠙⢎⢷⡀⠉⠉⠉⣰⢇⠢⣌⣉⣑⣀⣠⠼⠃⣬⣛⡛⠋⠉⠱⣐⢆⢨⣿⢇⢤⣄
⠀⠀⠀⠀⠻⠿⠿⠟⠋⢀⣒⡙⢒⢃⣹⣈⠻⣅⠀⠈⢣⣻⣴⣍⣩⣭⢟⡣⣉⠛⠯⠍⠁⠀⣰⠆⠉⣀⣤⡴⠶⢯⣍⠊⢡⢺⣎⡹⠇
⠀⠀⠀⠀⠠⣤⡄⡴⢋⡭⢛⣻⢷⡜⢏⠉⡓⢬⡓⠦⣀⣻⣿⣦⣿⣿⣋⡛⠪⠉⢛⣛⠿⠋⢁⡴⢞⣗⣛⠻⠷⣞⡋⣀⣘⠱⠾
⠀⠀⠀⠀⠀⠈⠁⠉⠻⠳⠿⠟⠛⠋⠀⢭⣛⡶⠍⠓⣦⣭⣙⣛⠿⠮⠕⠫⠭⣽⣯⠿⠃⠀⠰⠷⠙⠼⠿⠟⠛⠉⠛⠛⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⣻⡟⣹⠛⣛⢛⡿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠋⣩⡿⠛⠛⠛⠿⣷⣄⠀⠀⠈⠙⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠋⠀⠀⣼⠏⠀⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠐⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⣤⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⣠⡀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⠃⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠛⠁⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠚⠗⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠐⠛⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠛⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠶⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢠⣤⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⣀⡀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠐⠛⠃⠀⠘⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠈⠋⠁⠀⢸⡇
⠀⠀⠀⠀⠀⢀⣠⣤⣾⣇⣀⣠⣤⣴⡶⠾⠟⠛⠛⠛⠿⠶⣶⣶⣾⣿⣿⣿⣿⣷⣶⣿⡇⠀⠀⠀⠀⠀⣸⣷⣄⡀
⠀⠀⠀⠀⢰⣿⡿⠋⠙⠛⢛⠉⠁⠀⣤⣤⣤⣀⡀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢸⣧⣤⣤⠶⠾⠛⠛⣿⠙⢿⣦
⠀⠀⠀⠀⢸⣿⡆⠀⠀⠐⠛⢀⠀⠀⠈⠉⠉⠙⠛⠿⠿⠿⠛⠛⠛⠛⠿⠿⠛⠛⠿⠿⠟⢻⣆⠀⢠⣶⡄⠘⣇⠀⢿⡇
⠀⠀⠀⢰⣿⡏⢻⡄⠀⠀⠀⠙⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠹⣆⠈⠻⠃⣀⣿⠀⠸⣿
⠀⠀⠀⠀⣻⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⢹⣶⠾⠛⠋⠉⠀⠀⢿⡇
⠀⠀⠀⣰⣿⠃⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⢰⣿⠃⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠸⣿⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⢠⣴⠶⣦⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠸⣧⣠⡿⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠈⠉⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠖⠀⠀⠀⠀⠈
⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⢀⡀
⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠘⠁⠀⠀⠀⣼
⠀⠀⠙⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⣾⠋
⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣴⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠿⠿⠿⠿⠿⣿⣷⣶⣶⣶⣶⡿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⢋⡾⣍⣛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⡟⠀⢸⡷⠆⣇⠀⠀⠀⠀⠀⣠⠞⠉⣳⣶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⣀⡀⢿⣀⠰⡇⠀⢸⣉⠁⡟⠀⠀⠀⣠⠆⠀⣠⠞⠁⢸⣟⡈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⡿⠋⠀⠈⠉⠙⠛⠓⠳⠶⠾⠶⠦⠷⠤⠤⠼⢵⣦⣾⣁⡀⠀⠀⢳⡶⠚⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⣽⣷⣾⣿⡆⠀⠀⣟⡁⠹⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⡴⠒⣦⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⢠⢯⠞⠋⠉⠉⠀⣹⠃⠀⠀⢩⣶⣾⡇
⠀⠀⠀⠀⠀⠀⣼⡟⠠⣏⠀⠀⣿⠀⠀⠀⠀⢠⠞⠁⠸⡗⠀⢠⢯⠯⠤⢤⠤⠖⢒⣧⠀⠀⢀⡟⢀⣿⡇
⠀⠀⠀⠀⠀⢸⢯⣀⡀⡟⠀⣞⣁⣠⣀⣀⠀⠘⡦⢀⡴⠃⢀⢏⡾⣷⡇⠀⡀⣀⡀⣿⣇⢀⡞⠀⣾⣿
⠀⠀⠀⠀⠀⠘⡷⠒⣾⡓⠚⣳⡟⠻⠯⢭⣝⣲⣃⣘⠣⢶⣫⠚⠀⢿⣏⠀⡟⠀⢳⡼⣻⣸⠁⣸⢧⠇
⠀⠀⠀⠀⠀⣸⠁⠀⡼⠇⢾⣽⠁⠀⠀⠀⠈⣧⠉⣽⣿⠉⠀⡇⠀⣜⢿⣿⡙⠓⣿⣶⠃⡇⢀⡏⣼
⠀⠀⠀⠀⠀⡟⠀⢀⡟⠒⣾⠋⠀⠀⠀⠀⠸⡧⠤⣽⠿⠀⢀⡇⠀⡾⠙⢹⣋⣈⣷⢸⠀⡇⢸⠃⢿
⠀⠀⠀⠀⡼⠁⠀⣸⠁⢸⡇⠀⠀⠀⠀⠀⠰⡇⠀⡟⠀⠀⢸⠀⠀⣷⠀⠀⡇⠉⡏⣼⠀⡇⢸⢀⡼
⠀⠀⠀⠀⡇⠀⣠⠇⢀⣾⠀⠀⠀⠀⠀⠀⠀⣇⠀⢧⠀⠀⠘⣀⠀⢻⠀⢰⣇⠀⢳⡏⢈⡇⢸
⠀⠀⠀⠀⡇⠀⢧⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⠀⠀⣆⠸⣷⡾⠿⠶⠟⠳⠾⢧⣰⡇⢸⡄
⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⣠⠞⠀⠀⡏⢀⣈⡀⠀⣀⡀⢀⡀⣨⣯⣦⣸⠇
⠀⠀⠀⠀⠀⠳⣆⠀⡀⢀⠀⠀⠀⠀⠀⢠⠀⡀⣀⢀⠀⠀⣇⠈⢉⣁⠉⢉⡉⠉⣉⢩⣍⠍⠉
⠀⠀⠀⠀⠀⠀⠉⠳⠷⣬⢧⣴⡦⡤⠶⠶⠶⠷⠼⣬⣧⣦⡻⣄⣠⣇⡴⣘⢸⠄⡟⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢣⠞⠁⠀⠀⠀⠀⠀⠀⠉⢉⣻⠟⡿⠛⠛⠛⠛⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠔⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣀⣀
⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣯⡙⠁⠀⢀⣀⡀
⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠉⣾⡟⠉⠡
⠀⠀⢴⠘⣿⣿⣿⣿⣿⣿⣿⣿⣇⣁⣸⣿⣷⠂⠀⠂⠀⠀⠀⠀⣠
⠀⠰⠈⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⢄
⠀⠀⠉⣡⣿⣿⠿⠛⠛⠛⠉⠉⠉⠀⡽⠿⠛⠈⣹⣶⣤⣀⠐⠀⠈⠛⠂
⠀⠀⠀⢿⣿⣧⣄⣤⠤⠴⠶⢋⠇⢀⡇⢀⣷⣿⣿⣿⣎⣿⣷⣠⣦⣤⣶⣶⣇
⠀⠀⠀⠀⣉⡙⣿⣿⠇⠀⡀⣸⠀⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⠀⠀⠀⢶⠆
⠀⠀⠀⠀⠉⠁⣿⣷⣤⡾⠋⢋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡟⣧⣿⣷⠏
⠀⠀⠀⠀⠀⠀⠈⠛⢀⡀⠣⠎⠹⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣏⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⢀⣈⢤⠈⠻⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⣠⡲⡽⠿⣵⣿⣿⣷⢿⣿⠋⣼⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⠀⠀⠸⣿⣿⣿⣿⡇⠀⢹⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳⣦⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠐⣾⠀⠀⠀⣨⣿⣿⣿⣧⠀⢸⣿⣆⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠈⠀⠀⠘⠛⢏⠈⣿⣿⣿⣦⣽⣿⣿⣿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⠇⠈⠙⠛⠟⠟⠿⢿⡀⠀⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠤⠴⠖⠒⠒⠒⠒⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠤⠤⢴⠏⠁⢀⣠⠤⠤⠤⠤⠤⠤⠤⠤⠤⡀⠀⢟⠙⠓⠒⠒⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣒⡊⠉⠁⠀⠀⠀⠀⠀⡼⣄⣀⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢾⣳⣦⣾⠗⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠉⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⣾⢠⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⢸⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⢿⡀⠀⡄⠀⢸⢸⢦⠀⠀⠀⠀⠀⢀⡤⢤⡤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⢠⢖⣒⣒⣒⡒⡄⠀⢀⡞⣽
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⢸⢿⠀⠙⠦⢤⣄⣀⣈⣳⡏⠉⠉⡷⠋⠀⠀⠀⠀⠀⠀⣀⣈⣙⡇⠀⠀⡿⠥⠒⠉⡇⣿
⠀⠀⠀⠀⠀⠀⢸⣇⠀⡇⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠉⢧⠀⠀⡏⠉⠉⠉⢹⠉⠉⠉⠉⠁⠈⢧⠀⢰⠇⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣇⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀⡇⡏
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢹⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⠀⢸⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⡀⠘⡇⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣽⣄⡙⠦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠞⠁⠈⠓⠦⠤⠤⠤⠶⠒⢒⣚⣩⠵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠾⠭⠭⠭⠭⠥⠤⠤⠤⠴⠖⠒⠒⠒⠛⠋⠉⠉⠉