Respiration Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠦⣄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡤⠶⠶⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢸⡇⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣼⣆⠀⠀⠀⡏⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣀⡴⠛⠁⠈⠓⠶⣤⣷⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣇
⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡟⠉⠀⠀⠀⠀⠀⠀⠀⣿⡟⢦⡸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⡄
⠀⢠⡟⠓⠶⠦⣤⣤⣤⣀⣀⣀⣤⣤⡤⠶⣾⠟⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠹⡄⠁⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠈⣧
⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⡄⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢻⡄
⠀⡏⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢻⡀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠘⡇
⢸⡇⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠘⡇⣸⠁⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣿
⣸⠁⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢹
⣿⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢀⣠⠶⠛⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠸⡇⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⢸
⢿⡀⠀⠀⠀⠀⠀⣿⠀⣀⡤⠞⠋⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⢸
⠈⢧⡀⣀⣀⣤⠴⠟⠋⠉⠀⠀⢀⣀⣠⣤⠴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣤⣀⣀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⣠⠏
⣀⣀⣙⣻⣷⣶⣴⣖⣒⣚⣛⣋⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣛⣒⣒⣶⣤⣶⣶⣛⣁⣀
⣿⣿⣉⣋⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⣠⣶⣦⡀
⠀⠀⠀⢀⠡⢸⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⡇
⠀⠀⠀⠘⠂⠾⢷⣶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢚⣶⠁
⠀⠀⠀⠀⠒⠒⣺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⣻⣿⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⢰⠂⠂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠙⡁⢠⠀⠀⠀⢀⡌⣇⠀⠀⠀⠀⠜⢆⡘⠀⢀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠘⠃⣠⠴⠋⠀⠘⠢⣄⡀⠀⠀⢈⡟⠑⠣⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠂⢨⡏⠀⠀⠀⠀⠀⠀⠀⠉⢱⠀⠉⠑⡄⠄⠑⣄⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡇⠀⠀⠔⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠈⠀⠅⠘⠘⠀⠧⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠃⠀⠙⠀⠀⢀⣇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠐⠄⠀⢀⠂⠑⠁⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⠦⣖⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠐⠀⠀⣀⠴⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠈⢅⠙⠒⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⢀⡀⢀⡀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⢶⠀⠀⠠⢋⢦⡁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠁⠀⠀⠀⠉⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠀⠀⠈⠉⠀⣀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⣀⡀⣀⣀⡀⡀⡄⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠛⠛⠓⠲⣄⠀⠀⣸⣿⣿⡀⠀⢀⡴⠚⠛⠛⠒⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⡀⠀⠀⢸⣷⣾⣿⡿⣿⣿⣶⣿⠀⠀⠀⡀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⣀⣠⠈⠙⢛⣶⣶⡿⡿⠋⠁⠀⠀⠉⠻⡿⣷⣶⡞⠛⠉⢠⣀⡀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⡀⠀⣤⣬⣿⡶⠚⣹⣿⡟⠀⡇⠀⠀⠀⠀⠀⠀⡇⠘⣿⣿⡛⠲⣾⣯⣤⡄⠀⡀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⢀⡽⠛⠉⣴⡏⠀⣰⠟⣰⣇⠀⡇⠀⠀⠀⠀⠀⠀⡇⢀⣿⡘⢷⡄⠈⣷⡌⠙⠻⣅⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣀⠀⠘⠁⠃⣰⢿⡄⠉⡟⠂⡇⠀⠀⠀⠀⠀⠀⡇⠚⡏⠁⣼⢿⡄⠃⠙⠀⠀⡀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢶⣄⣀⣭⡷⠶⠶⣺⠏⢰⣷⣄⡙⠀⡇⠀⠀⠀⠀⠀⠀⡇⠘⣁⣴⣷⠈⢿⡒⠶⠶⣿⣅⣀⣶⠆⠈⢧
⠀⠀⠀⠀⠀⠀⢀⡟⠀⢠⣴⠋⢡⠿⠄⠀⣰⡇⠀⠞⠈⣧⡉⠀⡇⠀⠀⠀⠀⠀⠀⡇⠈⣡⡏⠘⠆⠀⣷⡀⠀⠼⢧⠈⢳⣤⠀⠘⣇
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⣀⠈⠀⢀⣴⡇⢹⣄⠀⠀⠋⠁⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠉⠁⠀⢀⣼⠁⣷⡄⠀⠈⢀⡀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⢰⠇⠀⠀⠲⠦⣽⡦⢶⡟⢁⡇⠀⢻⡛⠂⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠚⣻⠃⠀⣇⠙⣶⠤⣾⡵⠶⠂⠀⠀⢷
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠶⠞⠁⣠⡏⠀⣼⣷⡀⢰⡟⢦⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢻⡀⠀⢠⠞⣿⠀⣰⣿⡄⠈⣧⡀⠙⠶⠆⠀⠀⢸⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠠⠴⣾⠋⣇⠀⢁⡏⠁⠈⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣈⠀⠈⠀⠉⣇⠁⢀⡏⢻⡶⠤⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠾⠃⠀⠈⣀⣨⡥⠴⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠤⣭⣀⡈⠀⠀⠻⠆⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠙⠲⠦⠤⠶⠖⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⠤⠤⠶⠚⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡆⠀⠀⠀⠀⠀⠀⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⠿⠿⠿⢿⣶⣤⠀⠀⢰⣿⠃⠀⠀⠀⠀⢻⣿⠀⠀⢠⣴⣾⠿⠿⠿⢿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠉⠀⠀⠀⠀⠀⠈⢻⣷⡀⣼⣿⠀⠀⠀⠀⠀⠸⣿⡄⣰⣿⠋⠀⠀⠀⠀⠀⠈⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⠇⠀⠀⠀⠀⠀⠀⢿⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⢀⣠⣀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⠀⠀⣠⣾⠿⠛⠻⢿⣦⡀⠀⠀⠹⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⠀⠀⣠⣾⡿⠁⠀⠀⠀⠀⠹⣿⣦⡀⠀⠀⠛⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠁⠀⢀⣤⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⣄⠀⠀⠉⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⣤⣾⡿⠛⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠻⣿⣦⣄⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⠋⠁⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠉⠛⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠙⠿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⡿⠟⠁
⣶⣶⣶⢶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⡀⠀⡀⢀⡀⢠⣀⠀⡀⢀⠀⠀⢀⠀⣀⠀⣀⠀⢀⡀⠀⠀⡀⢀⠀⣀⠠⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⠷⠺⠹⠸⠁⠯⠽⠸⠬⠇⠗⠯⠅⠀⠏⠸⠿⠋⠿⠆⡧⠽⠺⠸⠁⠟⠇⠯⠸⠺⠭⠇⠇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠒


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡤⢠⣤⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠄⠄⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⠀⣀⢀⠁⠈⠁⠈
⠀⠀⠀⠀⠀⠈⠈⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠁⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣀⢀⡠⠂



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉


⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣥⠒⠉⠀⡀⠐⠊⠉⠠⠐⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢠⡾⠛⠁⠠⠀⠈⠈⠀⠀⠀⢀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⠰⠃⠀⡀⠋⠁⠀⠀⠀⠄⠀⠀⠀⠀⠈⢻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠃⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⣽⡟⠀⣀⠂⠀⢀⡀⠈⠀⢀⠀⠁
⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠁⢀⡏⣹⠋⠉⢸⠉⠀⠈⠉⠙⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠀⠀⠈⠃⠀⠀⠀⠀⠀⠺⠀⠀⠀⢀⡶⢦⣁⠂⢰⠞⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⢀⣙⡀⠀⠀⠀⠀⠄⠀⠱⣄⠀⣀⣠⣟⣪⣻⣷⣾⣧⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢹⡿⠁⣠⣟⣧⣀⠀⡄⡆⠀⢀⠟⣠⠟⡩⠄⢁⣼⡿⠛⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡾⠀⢀⣙⣻⢷⣿⢿⣾⣔⠦⣇⠘⠣⡀⠄⠐⠛⠾⢿⠶⣾⠟⠿⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣧⠀⢼⣿⣿⣷⣿⣿⣿⣿⣷⣄⣤⣶⣒⠀⠀⠀⣀⣠⠶⢏⣤⣾⡧
⠀⠀⠠⠀⠀⠀⠀⠀⠀⣸⣿⡟⢰⣿⣿⣛⢻⣿⣿⣟⣻⣿⣿⣾⣿⣿⣶⡿⠟⣛⡈⣶⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢱⡟⠁⢠⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⡿⣯⣽⣶⡛⠻⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢸⣧⢸⠃⠀⣻⣿⡿⣿⢿⣻⢿⠟⣢⣿⣿⣿⣾⣟⣽⣿⣯⢹⢺⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣷⢘⡀⢰⣿⣿⣿⣷⣾⠟⢊⣾⣿⣯⣿⡇⣻⣿⢸⣿⣿⣿⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠆⠀⡼⣿⢿⡇⢸⣿⣿⠟⣩⣷⠾⣋⡭⠺⢿⣿⡇⣿⣿⣶⣽⡏⢟⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⠀⢸⣇⣿⠿⠀⣾⣟⣥⣛⢫⣄⣾⠏⣴⠃⣸⣿⣧⣿⣿⣿⣿⢁⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⠀⢸⡇⣿⠀⢀⣿⣿⣧⡿⣧⠿⣯⣼⡿⣄⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣸⣿⣼⡇⢸⣿⣿⣿⣈⣠⣾⣿⣿⠁⠿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡆⠀⣿⣿⣿⡇⢸⣿⣿⣿⣿⡟⢿⣿⠿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠇⠀⣿⢻⡟⠀⣼⣿⣿⣿⣿⣿⣿⡿⢆⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠠⠄
⠀⠀⠀⠀⠀⠀⣿⣽⡇⠀⣿⣿⣿⣿⣿⣿⣿⣆⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠐
⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣾⣹⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⢤⠀⢸⡿⣿⡿⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠐⠀⢸⡷⣿⡁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠈⣿⢹⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⢻⣾⡗⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⡄⠀⠘⣿⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠇⠀⠀⠉⠁⠆⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⢰⠀⠀⠀⠀⢀⠈⠙⠛⠋⠁
⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠸⡀

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⠤⣟⣋⡛⣛⣛⣛⠛⣋⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣲⣄⣀⣄⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠤⢠⣀⣮⣿⢿⡋⢙⡵⠛⠁⠀⣀⡀⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠙⠳⢟⣺⡛⢻⡾⣦⣸⠀⢀⠷⢷⡿⢇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠁⠀⠀⠀⠈⢫⢳⠋⠘⠈⠓⠛⠄⠀⠀⠀⢠⣄⠔⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠘⣢⣠⣤⣼⣫⠀⠀⣠⣤⢰⡦⣆⣠⠄⢎⣁⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠄⣴⣓⡶⢺⡋⡏⡷⠳⣮⣹⡀⢀⡷⡗⣾⢯⢽⡚⡶⢽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁⠀⠈⠓⡴⠃⠀⠈⢣⢻⠋⠛⠷⡏⠻⣠⡃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠦⡗⠄⣠⡀⠀⠀⡸⡹⠀⡠⡄⣀⢀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢉⡶⢍⣶⠷⣴⣄⢀⣷⣥⠴⠿⡗⢯⢹⡝⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⠒⠲⠖⣼⠧⣋⠶⠀⠀⣿⠏⡜⠁⠀⠀⠉⠤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠒⡟⠊⠀⠀⠀⠀⢸⢰⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠒⠴⠒⠒⠒⠐⠒⡟⢸⠖⠲⠒⠦⠄⠲⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣤⣄⡘⢦⡤⢒⠭⡆⣮⡣⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡤⣼⠯⠖⢻⠏⠀⣟⣆⠈⢢⣾⡿⠑⠶⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣼⡆⡸⡄⠀⢼⡟⡷⣜⠏⢿⡶⡄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡼⠋⠃⢀⡾⠇⠀⠀⠁⢈⡿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⠀⠠⠎⠃⠀⠀⠀⠀⠈⠳


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⣠⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣭⣀⣀⣤⣽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠛⠛⠁⠀⠉⠉⠿⠿⣧⡄⢀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⣠⡴⠛⢁⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣄⠈⠙⠛⠛⠿⣿⣶⣤⣴⠾⠋⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⠀⠀⠀⠀⠀⡄⠀⠘⣿⠛⠿⣶⣦⣤⣀⣀⣨⣿⡁⠀⠀⠀⢀⣽⣿⠀⠀⠀⠀⠀⣶⠟⠶⢶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣰⣿⠀⠀⠀⠉⠙⠛⠛⣿⠛⠛⠛⢿⣶⣾⣿⣿⣠⣶⣦⣀⣀⣿⣤⣶⣿⣿⠹⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠀⠀⣀⣿⠏⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⢀⣴⣿⡿⠛⠛⠛⠻⣿⣿⡿⣿⣿⣿⣿⣿⣄⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠀⠀⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⢀⡜⠀⣠⣶⣿⡿⠋⠁⠀⠀⠀⠀⠀⠹⣷⣾⣥⣏⣙⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢁⣼⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠻⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡸⠃⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⣤⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡅⠀⠀⠀⠀⠀⢀⠋⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠀⠀⠀⢸⣇⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⣸⡇⠀⠀⠀⠀⠀⠀⠛⢻⣿⣯⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⡀⣼⠀⠀⠀⠀⣀⣀⠛⠻⡇⠀⠠⠀⢀⣤⠀⠶⢿⣿⣏⣨⣤⣭⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢀⣤⣶⡾⠿⠿⢃⣠⣾⡀⠀⢸⣿⠿⠂⠘⠲⠾⠿⠾⠿⠿⠿⠿⠟⠻⠟⠛⠉⠙⠛⠉⠁⣠⣽⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⣾⡟⠃⠀⠀⠀⠈⢿⣷⣧⡀⠛⠛⠿⠛⠒⠐⠛⠓⠐⠲⠶⠶⠶⠶⣶⣾⣿⣿⣿⣷⣾⣿⣯⠀⠀⠙⢿⣿⣧⡀
⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⢲⣶⣶⠦⠄⠠⠤⠀⠤⠤⣤⣤⣾⡧⠘⠛⠛⠛⠛⠉⠁⢠⠀⠀⢈⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⣦⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢿⣿⡰⢆⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣾⣿⣿⣿⠧⣤⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠻⢿⡏⠹⠿⣤⠈⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢀⠀⠁⠀⠰⣿⣷⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢄⠀⣹⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⣠⠖⠁⠀⠂⠀⠸⠋⣻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢨⣦⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⣀⣶⠀⠀⠀⠀⠀⠀⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⣠⣾⣿⠏⠀⠀⠀⣹⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⣦⣤⣴⣧⣰⣿⣷⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⣿⣯⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⡅
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⢀⣠⣤⣤⣼⠁⠐⠖⠀⠰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠂⠐⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⣀⢠⣶⣶⣦⣤⣤⣼⣿⣿⡇
⣶⠛⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⠀⠀⠀⠒⠲⠀⠀⠀⢀⡀⠀⣤⣤⣤⣆⠀⠂⠀⠀⠀⠘⣿⣠⣤⣼⣿⣿⣿⣿⣿⣿⣧⣼⡀⠀⣤
⠉⠿⡆⠀⠀⠀⠀⠀⢀⡀⠀⠈⠉⠁⠠⠄⠀⠀⣶⠄⠀⣀⠀⠉⠛⠚⣾⠙⠿⠿⠶⠀⠀⠀⠈⢠⣿⡏⠉⠉⡉⠉⣬⣿⣿⣿⣤⠈⢁⣰⡿
⠀⠀⠻⣷⡀⠀⠀⠀⠀⣿⡀⠀⠀⠀⣠⣶⣆⠀⠉⠀⢰⣿⣄⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠁⣤⣾⣧⣤⣼⣿⣦⣤⣿⣶⣿⠋⠀⠀⣀⣀
⠀⣼⠿⠃⠀⠀⠐⠂⠘⣿⡔⠂⠀⠀⠘⠿⠻⠷⠀⠀⠿⢿⠿⢿⣷⡾⣿⣷⣦⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣷
⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠀⠀⠤⠀⠀⠙⠉⠀⠀⠸⠿⠇⠀⢹⣿⣻⣿⣿⣿⣿⣿⣿⣿
⣤⣜⣿⡶⠶⣠⣤⣤⣤⣤⣀⣀⣀⣀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠁⠀⠉⠉⠉⠛⠉⠉⠙⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠋⣩⡿⠛⠛⠋⠉⠉⠉⠉⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠴⠶⠆⠀⠀⠀⠀⠀⠀⠉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣿⣿⣶⣶⣶⣶⣶⣾⣿⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⡿⠛⠛⠿⠿⠻⠟⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⢀⣀⣀⣀⣀⣀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠎⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⡿⠛⠶⢶⢶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⣹⣶⠐⠺⠿⢶⣦⣤⣀⢀⡴⠋⠁⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠈⠀⠀⠀⠀⠄⠀⣽⠿⣦⣤⣀⣀⣩⣿⣁⡀⠀⢀⣼⡟⠀⠀⠀⢠⡶⢶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⠀⠀⢡⡿⠀⠀⠈⠉⠉⢹⠋⠙⠛⢿⣿⣿⣷⣶⣤⣤⣤⣧⣼⣿⡏⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⡀⠀⠀⢀⣤⠟⠁⠀⠀⠀⠀⠀⠟⠀⣠⣴⣿⠟⠁⠈⠉⠛⢿⣍⡛⠿⠿⣷⣆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⣸⡏⠀⠀⠀⠀⠀⠀⢀⡞⢀⣾⡿⠛⠁⠀⠀⠀⠀⠀⠈⠛⠻⣶⡦⠼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⡎⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡃⠀⠀⠀⠀⠀⣴⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠃⠀⠀⠀⠀⢰⡇⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⡜⠁⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣷⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⢿⡀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠒⡀⠀⠀⠀⠀⣤⣾⠀⠀⠀⢀⡀⢸⣤⣿⡗⡚⠻⡷⣶⣶⣤⣤
⠀⠀⠀⠀⠀⢀⣠⣤⠿⠿⣀⣀⠀⠀⢰⡿⠆⠙⠂⠼⠶⠼⠿⠾⠧⠻⠷⠛⠛⠟⠛⢉⣻⣿⣄⡀
⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠸⢿⣎⣀⠙⠛⠛⠀⠀⠃⠀⠐⠖⠲⠓⢶⡾⠿⠿⣶⣿⣿⠁⠁⠹⣿⣷⡀
⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⠁⠀⠘⠛⠛⠛⠀⠀⠀⠀⠰⠛⠾⠋⠙⠁⠉⠉⠀⢀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢸⣿⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢸⣿⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣡⢾⣧⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢻⣾⠻⣿⡁⠲⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣴⣿⠇⠈⠙⠃⠀⣷⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⣽⣿⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⡋⡐⠂⠀⠀⠠⠋⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣿⣿⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠹⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠏⢀⣴⡶⠀⠀⠀⠀⠘⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠒⠛⣟⣀⣤⣀⣼⣧⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣯⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⣀⣀⠠⡤⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠄⠀⠀⠤⠀⠀⠠⢤⣤⣤⣤⢰⣿⣿⡀
⢶⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⣀⡀⣶⣦⢦⠐⠀⠀⠀⠀⢓⣦⣄⣭⡥⢾⣿⣿⠦⣧⠀⣀
⠀⠻⣄⠀⠀⠀⠀⢠⠀⠀⠀⠀⢒⡀⠀⡇⠀⣴⠀⠀⠀⠃⠈⠛⠋⠀⠀⠀⣠⡿⠁⢀⣤⠀⢴⣟⠛⢰⢀⣴⠟
⠀⣠⡟⠑⠢⠤⠀⢸⡃⠀⠀⠀⣿⣷⡄⠀⢸⣿⣷⣦⣠⣾⣤⣀⣀⣀⣴⣴⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣴⣶⣶⡀
⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠙⠉⠁⠘⠛⠉⠹⠿⣿⣿⣯⣿⣯⣿⡅
⢤⡜⣋⡰⠀⠶⢦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣾⣿⣷⣾⣿⣿⣿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠉⠉⠙⣋⡉⠁⠀⠀⠉⠠⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣐⠇⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡠⠖⠉⠉⠢⡀⠀⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⠋⠀⠀⠀⠀⢠⡇⢸⡃⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⡀⠀⠀⣼⡧⢼⣷⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠒⠸⠫⠕⠒⣿⠁⠂⣟⠀⠀⢠⣄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠠⢊⠼⠀⠘⠛⢦⡤⣼⣏⠀⠀⠀⠘
⠀⢀⠀⠀⢀⡀⠀⠀⠀⡇⠀⢀⠀⠀⠀⢈⠁⣂⠰⠇⠀⠀⠀⠁⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠍⢂⠀⠀⠈⠉⠉⠁⠀⠀⠇⠈⡀⠀⠀⠸⣷⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠃⠀⠀⠄⠀⠀⠀⢈⠁⢀⠉⢆
⠀⢐⠀⠀⢀⡀⠀⠀⢺⠐⠛⠂⠐⠄⠀⠀⠀⢀⡜⠀⠈⡀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣀⡤⠶⠚⠛⠛⠛⠣⠤⠄⣀⣁⣠⣈⣢⣀⣀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡴⠋⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠃⣀⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣤⠄⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃
⠀⠀⠁⠀⠀⠀⠀⠀⢄⣴⡟⢆⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⣶⣾⠃⣟⣏⣻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡼⢿⣿⣻⣯⣿⡟⢿⣽⡿⣏⣠⠀⠀⠀⠀⠀⠛⡄⠀⠀⠀⠀⠀⢀⠀⠘⢛⡷
⠀⠀⠀⠀⠀⢰⣷⢺⣽⣿⡼⣩⠁⢻⡿⣿⣾⣿⠃⠀⠀⠀⢀⠆⠃⣤⣶⣿⣿⣿⣦⠀⠀⠈⢀⠦
⠀⠀⠀⠀⠀⠀⢻⣱⣿⣿⣿⠏⠀⠀⠉⠛⠃⠁⠀⠀⠀⢠⣽⠃⣾⣿⣿⣿⣿⣿⣿⠃⠀⡨⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢾⣿⠀⣿⣿⣿⣿⣿⣿⠏⡰⣫⢿⢇⠁
⠀⠀⠠⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣳⠗⣌⢻⠿⠿⡏⣡⢞⣤⠙⠛⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⣿⣿⠾⣻⣿⣇⣼⡹⢠⡿⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠝⠲⠶⠿⢉⣷⣶⠆⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠚⠋⠉⠉⠉⠛⠛⠒⠒⠶⠦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠉⠛⠲⠦⠤⠤⠤⠴⠺⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⢀⡏⢰⢷⢀⡤⠄⠀⠀⠀⠀⠀⠀⠤⢤⡀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢸⢁⡘⠛⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⣠⠖⠛⢲⡘⣿⡷⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⠁⠀⠀⠀⣷⡇⠀⣤⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠘⢦⣀⣀⠀⠉⠀⠐⢯⠇⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠈⣧⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⢀⡟⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⠦⣤⣀⣀⣀⣀⣀⣀⣤⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠦⣄⠀⠈⠹⡯⢭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠻⢤⣀⠀⠈⠙⠲⣄⣻⣼⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠈⢻⡓⠶⢦⣠⡽⢿⡀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢠⢳⡀⠀⠀⠀⢸⡇⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢠⡏⠀⢹⠀⠀⠀⠀⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠤⠤⠤⠤⠼⠤⠤⠾⠤⠤⠤⠴⠧⠤⠼⠿⠆

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀
⠀⠀⠀⠀⢠⠏⠓⡾⠁⠀⢹⡀⠀⠀⠀⡀⠀⠀⢀⡤⠶⠒⠲⣤⡀
⣠⣤⢠⣄⡾⠀⣰⡁⢸⣡⠀⣧⠀⢀⡴⠁⢀⡔⠁⠀⠀⠀⠀⠈⢷
⠷⢫⢩⣄⠀⠀⠀⠈⠳⡄⣔⣿⡆⢉⡤⠲⢾⢀⡀⠀⠀⠀⠀⠀⣸
⠀⠈⠁⠘⣆⢀⡴⠲⢤⡝⢮⠁⢹⡏⠀⠀⢨⡿⠟⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⡀
⠀⠀⠀⠀⠈⠉⠀⠀⣀⣙⣭⡇⠈⠳⢤⡤⣾⠦⣄⣀⣀⣀⣴⣏⣀⣠⣤⡤⠤⠤⠤⠤⣤⣀⠀⠀⠀⠀⠀⡖⠲⢤⡀⡀⣄
⠀⠀⠀⠀⠀⠀⢰⠋⣭⣾⢭⠳⠀⡀⠀⠔⠁⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⢳⡀⣸⠳⠷⡏
⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣤⡄⠰⠵⠀⡠⢆⠀⡀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣏⣹⣿⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢀⠀⠙⠋⠀⣍⡆⠀⠀⠙⠲⢤⣀⠐⠠⠄⣄⢀⡀⣀⣠⠟⠀⠀⣠⢴⡁⠈⣯⡞⡫⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢨⣱⢀⠴⡀⢀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡖⠚⠒⢻⣹⠁⠀⠀⠀⠙⠀⣉⣽⣩⠧⡄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⠶⡇⠀⠀⠈⠛⠁⠧⠇⠀⠀⠀⣀⣀⣀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣄⡷⠴⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡛⠤⣿⡀⠀⠀⠀⠀⢀⡤⠶⡎⠙⠍⠳⡄⠀⠀⢸⣄⠀⠀⠀⠀⢀⣀⡤⠔⠊⠁⣹⠲⢲⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣌⣐⠈⠉⠒⠒⠾⠭⠄⣀⣙⣦⣶⣚⣁⣀⠀⠈⣧⠿⠍⠉⠉⢧⠀⠀⣠⠞⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⠙⠓⠶⢤⣀⣀⠀⠀⠀⠀⠀⠀⢠⠀⠙⣶⣿⠀⢤⡀⠀⣸⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠉⠉⠓⠲⠤⢤⣀⡾⠀⠀⠀⠘⠀⠀⠳⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢹⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣧⡠⠤⢤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢴⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠸⡄⠀⢸⡀⠀⠀⠀⢀⣀⣀⣀⣤⡤⢤⡖⣲⠋⠉⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠻⡀⠈⡏⠉⠉⠉⠉⠉⠀⠀⠀⠀⡜⢠⠇⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣎⠓⠀⢱⡀⠀⠀⠀⣀⣀⣀⣠⣎⣠⣋⣀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⢥⡄⣀⡜⠁⠀⠀⠀⠀⠊⠉⣿⣉⣉⡥⠤⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣶⣶⡒⠒⠒⠚⠹⣏⠉⠉⠉⠉⢉⡩⠕⠛⠧⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡽⠛⠛⠻⣝⢦⣀⣀⡼⡆⠀⠀⠀⡯⣀⣀⣠⠴⣒⡿⠿⢍⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢼⣒⣛⣛⣩⠬⠭⠖⠊⠁⠀⠀⠀⠑⠲⠤⣉⣉⡓⠒⠶⠤⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠲⢶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⣀⡴⠛⠉⡠⠒⢁⡞⣠⡴⠶⠛⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⢧⠞⠁⠀⣠⠊⠀⣀⠾⠋⠁⢀⣀⣀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢀⡼⢁⣀⣴⣯⣤⡴⠛⠉⠉⠉⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⡄⠀⣀⡤⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⢠⡎⠈⠑⠦⡀⠀⠀⢰⣦⣟⠀⢸⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⡠⠚⠁⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠛⠀⢀⡾⠁⢈⡷⠋⠀⠀⠀⢠⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠛⠓⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⣤⠒⠒⣲⠄⠀⠀⣼⡀⢀⡜⠁⠀⠀⠀⠀⠻⢋⣉⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣄⠀⢻⠀⠀⠀⠠⣔⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠹⠴⠛⠁⠀⠀⠀⡿⠌⢸⡇⠀⠀⠀⠀⣠⠞⠋⠉⠉⠳⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡁⠀⠘⣇⠀⠀⠀⠈⠛⠦⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠞⠹⡇⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⡆⠀⠀⣴⠉⠀⠙⣆⠀⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡤⢿⠀⠀⠀⢿⣤⡤⠶⠶⠶⠟⢋⡇⠀⠀⠀⡄⠀⠀⢹⡙⠛⢻⠏⠀⢸⡍⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠘⢇⠀⠀⠘⢧⡀⠀⠀⠀⣠⠞⠀⠀⠀⠀⢳⡄⠀⠀⠉⠈⠁⠀⠀⢸⣇⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠷⣤⣘⡄⠀⠀⠀⠙⠛⠚⠛⠁⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⠶⠺⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣤⣴⠾⠋⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⣰⣿⣿⣷⣭⣿⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⢠⣇⡘⠻⣿⣿⡏⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⢰⠋⢙⡷⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠁⠉⠙⠻⠿⠀⠀⠀⢠⡾⠋⠀⠀⠀⢠⡞⠙⠓⢾⠁⢀⠟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⣛⡭⠿⠿⢷⣶⡦⠤⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢸⡆⠂⢴⡿⠤⡎⣀⡾⠁
⠀⠀⠀⣠⠶⠚⠛⠛⠉⢁⡴⠚⠁⠀⠀⠀⠀⠸⣤⣤⣬⠿⣤⣀⠀⠀⠀⠀⢀⣠⡴⣏⠀⠀⠀⠀⠀⠀⠀⢀⡽⠀⢀⡀⠠⠘⠁
⠀⠀⣾⠁⡀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠈⡼⠉⠛⠛⠛⠛⠋⣹⣦⢘⣷⠀⠀⠀⠀⢀⣤⠞⣡⡴⠋
⠀⠀⠉⢶⠁⢀⡀⢀⣀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⡇⠀⢀⠀⠀⠀⠀⠌⢿⠟⠙⠻⢶⣶⣾⡯⠶⠚⠁
⠀⠀⠀⠈⠛⣿⡀⣸⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢠⠀⠀⣹⠗⢐⣶⠀⡄⢸⡇
⠀⠀⠀⠀⠀⠈⠉⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣘⠀⠙⠃⠀⠾⠏⢀⠂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡾⢻⣦⠀⠀⠀⠀⠀⢀⣿⣤⣀⣀⠈⠛⠛⠶⠶⠶⠶⠖⠀⢸⡇
⠀⠀⠀⠀⠀⢀⣼⠏⡰⠃⢸⣇⠀⠀⠀⠀⢸⡗⠀⠉⠉⠙⠛⠓⠶⠤⣤⣄⣀⣀⣼⠁
⠀⠀⠀⠀⣠⡿⢡⠞⣀⣴⣏⣹⡄⠀⢀⣴⠞⢻⣇⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⣹⠏
⠀⠀⢀⣾⠋⡴⠃⠰⣿⡟⢿⡀⢻⣄⣼⣿⣄⠀⢙⡷⣴⣆⡀⠀⠀⠈⠙⠛⣺⡧
⠀⣰⠟⣡⠞⢦⢀⣴⠋⠀⠈⢷⣀⣩⡼⠋⠙⠛⠉⠀⠀⠈⠙⠳⠶⣤⡶⠿⣿⣟⡳⠶⣤⡀
⣼⣋⣴⣁⣠⡾⠟⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⠆⢀⡿
⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣥⣼⣿⣶⣶⢦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠧⣄⣀⣀⣙⠉⠁⠀⠀⡔⠋⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠶⣶⣦⣭⣤⣭⣉⣉⣛⡓⠶⠶⡤⢿⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣙⣛⣛⣿⣷⣿⣇⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⠤⠤⡤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⣶⠦⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⣶⡶⣦⣤⣀
⠀⠀⠀⢀⡔⠋⠀⠀⠀⡇⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢠⡶⠋⠁⠀⠀⣿⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⢀⣶⠟⠉⠀⠀⣿⡆⠀⠈⠙⢷⣄
⠀⠀⢠⠏⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠱⡀⠀⠀⢰⠏⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢻⡄⠀⠀⢠⡿⠁⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢻⣇
⠀⠀⡞⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣇⠀⠀⡿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣧⠀⠀⣾⠃⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡄
⠀⠀⢇⠀⠀⠀⠀⢀⡔⡿⢦⡀⠀⠀⠀⠀⡟⠀⠀⣷⠀⠀⠀⠀⢀⡴⣿⢦⡀⠀⠀⠀⠀⡿⠀⠀⣿⡄⠀⠀⠀⢀⣴⣿⣷⣄⠀⠀⠀⠀⣿⠃
⠀⠀⠘⡄⠀⢀⡔⠋⠀⡇⠀⠙⢦⡀⠀⣰⠃⠀⠀⠹⣆⠀⢀⡴⠋⠀⣿⠀⠙⢦⡀⠀⣸⠃⠀⠀⠸⣧⠀⢀⣴⠟⠁⣿⡇⠙⢷⣄⠀⣰⡟
⠀⠀⠀⠘⢶⡋⠀⠀⠀⡇⠀⠀⠀⢙⠖⠁⠀⠀⠀⠀⠙⢶⣋⠀⠀⠀⣿⠀⠀⠀⣹⡾⠃⠀⠀⠀⠀⠙⢿⣟⡁⠀⠀⣿⡇⠀⠀⣹⣿⠟
⠀⠀⠀⠀⠀⠈⠓⠢⠤⠷⠤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⠿⠴⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⣿⡷⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠶⠶⠦⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⠶⠦⣤⣀
⠀⠀⠀⢀⣴⡿⠟⠋⢹⣿⠉⠙⠻⢷⣦⡀⠀⠀⠀⠀⡠⠚⣡⠴⠒⢳⠀⡗⠒⢤⡉⠢⡀⠀⠀⠀⠀⣠⠞⣋⡴⠖⢻⡆⣿⠓⠦⣍⠳⣆
⠀⠀⣰⣿⠋⠀⠀⠀⢸⣿⠀⠀⠀⠀⠹⣿⡄⠀⠀⡜⢡⠞⠁⠀⠀⢸⠀⡇⠀⠀⠈⢢⠘⣆⠀⠀⣰⢋⡼⠋⠀⠀⢸⡇⣿⠀⠀⠈⠳⡌⢷⡀
⠀⢀⣿⠇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢹⣿⠀⢸⠁⡏⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⢧⠸⡄⢰⡇⣼⠁⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⢻⡈⣇
⠀⢸⣿⠀⠀⠀⠀⢀⣼⣿⣦⡀⠀⠀⠀⢀⣿⠀⢸⠸⡇⠀⠀⠀⣠⠞⠀⠣⡀⠀⠀⠀⢸⠀⡇⢸⡇⣏⠀⠀⠀⣠⡾⠁⠻⣄⠀⠀⠀⢸⡇⣿
⠀⠀⣿⣆⠀⢀⣴⡿⢻⣿⠛⢿⣦⡀⠀⣸⡿⠀⠸⡀⢧⠀⣠⠞⣡⢶⠀⡦⡈⠢⡀⢀⠏⢸⠁⠘⣇⢻⡄⣠⡾⣫⣶⡆⣶⣌⠳⣄⠀⡼⢠⡏
⠀⠀⠘⢿⣶⡿⠋⠀⢸⣿⠀⠀⠙⣿⣾⡿⠁⠀⠀⠱⡌⠛⣡⠞⠁⢸⠀⡇⠈⢢⡈⠋⡰⠃⠀⠀⠙⣦⠛⢫⡶⠋⢸⡇⣿⠈⠳⣌⠛⣡⠟
⠀⠀⠀⠀⠙⠿⣶⣦⣼⣿⣤⣴⣾⠿⠋⠀⠀⠀⠀⠀⠈⠢⣌⡑⠒⠞⠀⠷⠒⣉⡤⠚⠁⠀⠀⠀⠀⠈⠳⢮⣙⠳⠾⠃⠿⠖⣛⣡⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠈⠓⠒⠒⠒⠤⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠀⠀⣀⣤⢤⣤⣀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⢀⣴⣿⣿⣿⠷⠿⠟⣿⣦⣦⣄⠀⡀⠄⠀⠀⠀⠀⠉⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢰⣿⣒⣶⣿⡿⠛⠃⠀⠀⠀⠀⠀⢤⣭⣻⡧⠀⠐⠂⡀
⠀⠀⠀⠀⢀⣀⡀⠀⣠⠤⠒⠠⢌⢀⣄⣿⣿⣿⣿⣯⡇⠀⠀⡄⠑⡄⠀⠀⠈⢹⣯⡧⠔⠚⠉
⠀⠀⠐⠋⠉⠉⠿⣿⣶⣤⡀⠀⠀⠙⣿⣿⣿⣿⡿⠁⠀⠀⠀⠉⠀⣀⠠⢄⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠛⣶⣥⣀⡈⠹⢿⣿⣦⡀⢀⣇⠈⠁⠀⢸⡀⠀⢠⣄⠀⡄⠐⠆⡬⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠈⠙⢿⣷⣤⠄⠙⠯⣿⣧⡏⠀⡅⠀⠀⠱⡀⠀⠙⣃⢤⡲⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠈⠻⣶⣤⢂⠈⠛⢷⣤⠀⡘⠛⠿⣦⡁⢠⣴⡾⠟⠉⠁⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠈⠙⣷⣆⠀⠀⠉⠛⠢⢤⣄⣀⠉⠒⠥⣀⣠⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠡⢆⠀⠀⠀⠈⠻⣷⣤⡄⡀⠉⣦⣄⠀⠲⢤⣀⠀⠀⠀⠀⣠⣾⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢶⣀⠀⠙⠻⣷⣄⠠⠌⠻⣷⡄⠀⠈⠑⠀⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠉⠻⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣴⣿⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠿⢿⣿⣿⠏⠀⠀⠈⠃⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⠀⠀⠛⠁⠀⠀⠀⠀⣾⠇⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠠⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⢻⣿⣿⣿⣿⣷⣦⣀
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⣙⠿⣿⣿⣿⣿⣿⣷⣄
⠀⢨⣿⠃⢠⣤⣀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣮⣝⠻⢿⣿⣿⣿⣦
⠀⢸⡏⠀⠈⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠛⢿⣦⡙⣿⣿⣿⡄
⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣿⣷⠸⣿⣿⣇
⠀⢸⠁⠐⠐⠛⠋⠙⠛⠻⢿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢹⣿⡇⢻⣿⣿⡀
⢀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⢿⣿⠸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠉⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⡤⠶⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⢄⠀⢸⠀⠀⡇⢀⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠈⡇⢸⠀⠀⡇⢸⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢸⠀⠀⠐⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠄⠀⠀⠰⣄⠔⠁⠀⡇⢸⠀⠀⡇⠘⡇⠀⠀⠀⠀⠀⠀⢀⠘⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠄⠀⠠⠈⠀⡏⢀⠀⢰⡇⡾⠀⠀⡇⢀⡇⠄⠐⠀⠀⠑⠈⢁⠀⡀⠙⢆
⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠁⡀⠀⠀⠀⠀⡇⠀⠀⠀⠟⢁⣀⣀⠙⠺⠇⠀⠀⠀⠀⠀⠠⠈⠀⠀⡀⠀⢧
⠀⠀⠀⠀⠀⢠⠎⠄⠀⠀⡀⢀⠉⠒⢄⡀⢘⣤⡤⡶⡖⠉⠀⠈⠓⡄⠀⢌⡀⢁⢀⣐⣃⢀⠠⠐⠀⠠⠀⢣
⠀⠀⠀⠀⢀⠏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⡩⠁⢰⠃⣇⠀⠀⠀⠀⡇⢳⠀⡀⠉⡁⠀⢀⠈⠉⠐⠄⠑⠄⠀⢣
⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡟⠀⣿⠀⠀⠀⠀⢷⠘⡆⢀⠁⠂⠀⠂⠀⢀⠀⡀⠀⠀⠀⠈⢇
⠀⠀⠀⡼⠀⠀⠀⠀⡀⠀⠌⠀⠀⢀⠀⠀⠂⣺⠁⢀⡏⠀⠀⠀⠀⢸⡄⢧⠀⠀⠂⠀⠍⠀⠀⠀⢀⠀⠀⠀⢀⠘⣆
⠀⠀⢰⠃⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⡰⠃⠀⣸⠁⠀⠀⠀⠀⠈⡇⠈⠳⡀⠐⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠸⡄
⠀⢀⠏⠀⠀⠂⠀⠀⠀⣀⠀⢐⣀⡠⣤⠞⠁⠀⢀⢻⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⢧⣄⡀⠐⠀⠀⢀⡈⠀⠀⢀⠀⠀⢳
⠀⡼⠀⠀⠀⡠⠖⠋⠉⠁⠀⠀⠀⠠⠃⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠄⠀⡈⢇⠀⠈⠉⠉⠀⠁⠆⠀⠀⠀⠀⠈⡇
⢠⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⠂⢀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣇⠀⡀⠀⠀⠈⠀⠀⠀⠃⠂⠀⠀⠢⡀⠈⠀⠀⢸
⣼⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⢻⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠢⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⣈⡴⠃⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠠
⡇⠀⠀⠀⠀⠂⠀⢂⣀⣠⡥⠤⠔⠒⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠶⠤⠤⣤⣈⣀⠀⠀⠀⠀⠂⠀⠠
⠙⣤⣤⣤⠶⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⢤⣤⣤⡴
⣿⣿⠛⠛⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⢸⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠄⠀⡉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣠⣿⡟⠀⠀⢸⡧⠉⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠁⡞⠁⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠠⠀⠐⠒⡄⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡿⠃⠀⢄⡀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⣂⡔⠁⠐⠤⠸⠁⠀⠀⠀⠈⠈⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⡤⠀⠀⠀⢠⠖⡉⠂⠠⠀⠀⢀⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠛⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠄⡽⠀⡀⠀⢀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠉⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠍⠀⠀⠀⠙⠨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⢀⠒⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢄⡀⠁⠀⠀⠀⠀⠀⢀⠄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢀⠚⠣⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢀
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣄⠐⠓⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠃
⠀⠀⠀⠀⡄⠁⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠄⢀⡀
⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠈⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⣠⣤⣴⠶⠶⠿⠛⠛⠛⠛⠛⠿⠶⢶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢻⣇⣤⡶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠈⠻⢅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⣠⠟⠳⠶⢤⣤⣄⣀⣀⡀⠀⠴⠶⠶⠶⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣴⠃⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢋⡿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠇⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠰⢶⡦⣤⠀⢼⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢿⠀⡆⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⠀⠀⠀⠤⠬⣿⣷⣶⣤⣉⠋⠀⠘⠀⠙⠚⢋⣉⣀⣠⣤⣤⣤⠄⠀⠀⢸⠃⠘⡇⠁⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⣸⡏⢀⣀⡇⠀⠀⠀⠐⠒⠒⠚⠛⠛⠉⠀⠀⠀⠀⠀⠛⠛⠿⠿⣿⣟⣛⠛⠓⠀⠀⡿⠀⠀⣿⠀⠀⢀⣰⣿⣄
⠀⠀⠀⠀⠀⠀⠸⢧⣤⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣧⠀⠀⣿⣀⣔⣭⠄⠀⢻
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⠀⢀⡁⢀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣄⡀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣬⡿⠟⢿⣄⣠⣄⡀⣠⠿⠒⢒⡿⠛⠿⣍⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠶⠖⠛⠋⠁⠀⠀⠀⠀⠉⠛⠿⠿⠷⠶⠶⠛⠁⠀⠀⠀⠉⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠞⣦
⢀⣠⣤⣾⠙⣦⡴⠞⠉⠈⢿⡈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⠷⣄⡀⠀⠀⣀
⢺⣥⣌⠉⠀⠀⠀⢀⣤⡴⠟⣷⡀⢀⣠⣤⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣴⣇⠀⠈⠙⠶⠞⢹⣧⣤
⠿⢥⣴⠆⣠⠶⠛⠉⠀⠀⠀⠙⠛⠛⠉⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠳⢦⣀⡼⠃⠈⠻⢦⣄⠀⠀⠈⢁⣤
⠀⠀⢿⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠀⠀⠀⠀⠀⠈⠑⢦⠰⣦⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠋⠛⠛⠛⠲⠶⠶⠶⠶⠶⣤⣤⣤⣤⡴⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⡶⠶⠶⠦⢤⣤⣤⣤⣤⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣀⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣏⡀⠀⠈⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡛⠉⢉⣀⣹⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡀⢀⣀⣀⠀⠀⠁⠀⢀⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⢀⣈⣉⠀⠀⢠⠈⡈⠉⠙⢄⡈⠀⠰⠐⠷⠠⠐⡔⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠠⠼⠿⠿⠷⢿⣿⣁⡤⠤⣴⡷⣀⠡⠄⠀⠈⠳⢼⡆⣱⠠⠄⠸⡑⠀⢄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⣀⡐⢠⣶⣴⣾⣿⡿⠦⠞⠛⡀⣄⡀⠤⣤⢞⣶⣌⢻⣦⣶⢞⠃⠀⢳⣄⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣉⣑⣦⣿⣷⣼⠧⣤⣷⠶⢾⣧⣬⣿⣿⣵⣞⣛⠻⣿⣿⡇⠘⣤⠠⠀⠀⢀⡷⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⠍⢭⡡⣴⣍⣽⢉⢶⣋⣥⠵⠾⠛⠋⠛⣙⣛⣿⣿⣶⣿⣿⣿⣶⣶⠶⠟⠛⠋⡴⠃⠁
⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⢛⣿⣿⠛⢯⠥⠀⠺⠶⠿⠒⠛⠻⠿⣍⣙⠻⣿⡏⢿⣿⠗⠊
⠀⠀⠀⠀⠀⠀⠰⠍⠉⠣⠀⠀⠘⠿⠿⠸⠫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣎⡇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠈⠁⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⡇
⡇
⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠉⠂⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠲⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⠄⠀⢀⠀⠀⠐⠁⠤⠤⠤⢀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⣀⠂⠈⠁⠀⠈⠁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠙⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠂⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠢⠄⢨⣿⣤⠀⠓⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⢻⠀⠀⠀⡀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢰⠀⠀⠀⢸⠇⠀⠀⠃⡀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣺⢣⠀⠀⡄⢡⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⠘⠀⠀⠰⠆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠠⠀⠀⠀⠀⠀⢶⠘⡇⠀⠀⠸⡈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠐⠀⠀⠀⢀⠂⠈⢃⠀⠀⠀⢃⠡⡀⠀⠉⠦⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠈⢥⣈⡄⠆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⡁⠀⢹⠀⠀⠀⠸⠀⠈⠢⣀⠀⠈⢓⣴⡀⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠉⢉⠐⠂⢤⠈⠀⠀⠀⢰⠀⠀⠀⠀⠈⢳⡀⠆⠀⠀⠀⢠⠀⠀⠀⠀⠱⠄⠀⢠⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⠀⠀⠠⢕⡠⢀⠀⠀⣀⡀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣈⠀⠠⠄⠀⠔⠂⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠈⠁⠞⡴⠋⠛⠿⠿⠿⠿⢃⠿⠿⠛⠋⡙⠂⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠈⠁⠐⠃⢠⡄⠠⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠈⠲⢄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠙⣿⣷⣶⣿⣍⠹⣦⡀⠀⠀⡀⠠⠐⠈⢀⣐⡳⣶⣦⡉⠀⠆
⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣵⣴⣧⣬⣉⣤⣤⣤⣤⣄⣀⣠⣌⣉⣤⣤⣤⣤⣬⣿⣿⣷⣌⣠⣤⣤⣤⣤⣤
⣿⣿⣿⠛⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣀⣠⣤⡴⠶⠶⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠞⠋⠉⠀⢀⣠⡄⠀⠀⠀⠠⣌⡙⠷⣄
⠀⠀⠀⠀⠀⠀⠠⢾⣿⠁⠀⠀⠀⠀⠿⢋⠀⡠⠀⢠⡀⠘⠷⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⢰⢃⣞⢣⠀⢰⢻⣦⣂⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⣿⡿⠆⠀⢰⠄⠀⣸⣿⡏⢸⠀⢸⠘⣿⣿⣄⠀⠀⠸⣦⠔⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⢠⢿⠀⠈⢳⡀⠀⡞⠀⢾⣆⠀⢰⠋⠀⢰⠀⣿⣞⣽
⠀⠀⠀⠀⠀⠀⡞⠡⡙⠦⠘⠋⠀⠈⠣⣭⠉⢀⡀⠈⢧⡉⠵⡋⠀⢸⠃⢹⣍⡇⠀⠀⡤⠖⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢻⡿⢹⣆⠀⠀⣿⠀⠉⠁⡀⢨⣤⣄⣀⠙⠲⣄⠀⠘⠃⠘⣿⠁⠀⠀⢀⡩⠴⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⣀⢰⣿⠀⠀⣰⠃⠀⠀⠀⠈⠙⠒⢚⠀⠀⠀⠀⣿⠀⢀⡴⠋⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⠘⠛⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⣸⡏⣰⠋⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⣧⡴⠖⢒⣒⣀⠒⠲⢤⣸⠀⣴⡟⢰⠃⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠹⣧⡀⠘⢁⠀⠀⠀⠀⠀⠀⠀⢀⣬⠾⠋⠀⢸⡀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⣠⣝⡓⠲⣿⣦⣤⣤⣤⣤⡶⠋⠀⠀⠀⢀⡀⢉⢾⣤⣀⣀⠴⠋⣡⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡜⡴⠓⣋⡤⣾⣧⢭⣽⣍⡗⢦⠀⠀⡰⣇⢙⢿⣁⡏⠀⠠⠔⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣿⠆⠀⠀⠸⢞⡟⣿⠓⠄⠋⠀⣯⡳⣄⣷⢈⠏⠀⠈⢳⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠋⠉⠁⢀⣀⣀⣠⡞⢁⣿⠀⠀⠀⠐⢿⡵⢉⡹⣌⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠠⣇⡀⣀⡴⠋⠀⠀⢸⣤⣾⣿⣧⣥⣤⣄⣸⡇⢾⠀⠈⢡⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢠⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡿⣤⡶⠶⠶⣶⡌⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢿⣙⢿⡃⢻⡇⠀⠀⣿⠁⣼⣋⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⠭⠤⠖⠒⠚⠃⠀⠀⠀⠀⠈⠈⠷⠤⠤⣈⡓⠂⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⣦⣤⣠⣿⣷⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⢿⡇⠀⢠⡀⠀⠀⠀⢀⣠⠀⠀⣄⡀⠀⠀⠀⢀⡄⠀⢸⣿⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠘⣧⠀⠀⠙⠻⠶⠾⠛⠁⠀⠀⠈⠛⠷⠶⠟⠋⠀⠀⣼⠏⠀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣽⡀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⠀⠀⠀⠀⢀⣯⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠈⢉⡁⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠿⠟⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣬⣿⠦⢄⣀⡀⠀⣀⡠⠴⣿⣥⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⢉⠂⠀⠀⠀⠀⠀⠀⠀⡉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⡄⠀⠀⠀⠀⠀⠀⢀⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠶⠶⠶⠶⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠶⠶⠶⠶⠶⠶⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⢀⠇⡇⠀⡠⠴⡄⢀⢠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⠀⡇⠘⠾⠀⠘⠊⠀⠀⠙⠛⢤⠀⠙⢆⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢇⠓⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣙⠚⠁⠀⠀⠀⢀⡠⠔⠒⠉⠑⠒⡶⠀⠀⠀⠀⢨⡆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡗⡒⠉⠁⢀⣠⣢⠀⠀⡸⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠜⢢⠀⠀⣞⠀⠈⡇⢀⢧⡆⠀⣀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆⡜⠀⠀⢿⢀⡴⠃⣸⠁⠹⠏⣅⢸⡄⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠙⠊⠐⠉⠀⠀⠉⢀⣴⣿⡄⠀⢀⠡⢞⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠐⠢⣤⣀⣠⣤⣾⡟⠋⢹⠇⠀⢀⣤⠞⢛⣣⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠈⢿⡿⠛⡟⣀⡴⠋⠀⡴⠋⣟⠉⢹⠈⢩⠃⠁⠀⢩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢯⠟⠒⡄⠈⠓⠤⣍⣛⣉⣁⣠⠤⠞⠁⢀⣸⠤⠞⠀⢇⣠⠆⢠⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠒⢇⡔⠰⡇⠀⠀⠀⢀⣀⡴⠋⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠑⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⠍⠊⠳⣌⡑⠒⣒⣳⠎⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢡⣏⣀⣀⣀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢫⡉⠉⠉⠁⠀⠀⠀⠀⠈⣹⠃⠀⠀⠀⣠⠦⠤⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢨⢗⢳⠀⣀⡤⡀⠀⢀⠀⠀⠉⠒⣦⠀⡠⢷⡒⢦⢀⠀⡝⢝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣾⠀⠙⠊⠁⠀⠈⠉⠻⣄⠀⢀⡴⣃⡰⢋⡵⠟⠳⢮⣎⡠⠞⠁
⠀⠀⠀⠀⠀⢀⡴⠾⢛⣛⡿⢝⡢⢼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠲⣬⠖⠁
⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠈⠑⠪⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⠤⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⠀⠀⠀⠀⡇⠀⠀⣠⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣿⣿⣿⣿⣿⣦⣀⠀⢐⣇⠀⠀⠀⠉⠉⢙⠻⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡃⣸⣿⣿⣿⣿⣿⡿⢻⣿⡿⢸⢻⠀⢄⠀⠀⡈⢊⠀⢹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡩⠉⣿⣿⣿⣿⣿⡿⡿⠈⣧⣤⣼⡿⠿⠿⠇⢀⠀⠠⢤⠞⠈⠻⣧
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠀⠄⠀⠈⣿⣯⣿⣿⠏⠀⠿⠀⣼⣿⠊⠀⣦⣀⣠⡴⠋⡄⠂⠀⠀⠠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣿⣿⣿⡟⠀⠸⠀⢀⣽⠃⠀⠀⣿⣿⣏⣤⣚⠁⠀⠉⠉⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠐⠀⢐⠀⣿⣿⣿⡇⠀⠳⠀⢸⣿⡀⠀⠀⣿⣿⣿⣿⠈⣁⠰⠠⠇⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠤⡀⢸⠀⣿⣿⣿⠇⠀⠀⠀⢠⡝⠃⠀⠀⠀⠳⠹⣝⠺⡏⠂⠉⠀⢤⡗⡄⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠀⠉⠨⡮⢈⠭⣿⣿⣿⠀⢀⣠⣴⠏⠁⠀⠀⠀⠀⠠⠀⠂⠀⠘⡄⡌⠁⠰⠠⢉⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡠⡈⢀⡁⠂⢒⣿⣿⣿⣷⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠁⠃⠀⢳⠀⠦⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⢰⠐⠡⢱⠀⣈⠂⣒⣮⣿⣿⣿⣿⣿⣞⢧⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⡎⢁⠕⠊⠓⠒⠊⡉⠀⠸⣿⣿⣿⣿⣿⣿⡷⡂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠒⠀⠀⢰⠆⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠈⡟⠃⠐⡃⠢⠐⠏⠆⢨⡆⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⣀⡈⠀⠀⠘⢄⠀⠀⠀⠀⣴⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢶⣠⣁⢈⠑⠘⣴⣥⣴⣵⣿⢿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⠀⠀⠂⠀⣀⣤⣶⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠛⠛⠒⠛⠻⠟⠛⠛⠻⠿⠿⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣦⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡏⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⢴⠁⠀⠈⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠘⢿⠏⠀⢠⣶⣶⣄⠀⠀⠀⠀⠀⢀⠁⠀⠢⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢠⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⠑⠒⠀⠘⠋⠀⠀⣀
⠀⠀⠀⠀⠀⣰⣿⣿⣷⣤⣸⣿⣿⣿⠏⠀⠀⠀⠀⠁⠀⠐⠄⠈⠀⢀⠀⠀⠀⠀⠃⠀⡀⠀⠀⠀⠀⠀⣴⣶⣶
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠈⠁⠀⠀⠼⢀⠀⠀⠀⠀⡐⠁⠀⢀⠀⠀⣀⣤⡀⢰⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣄⠐⠀⠐⠁⢀⡀⠑⢄⡀⠄⠙⠄⠀⠀⠀⢰⣿⣿⣿⢀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⣸⣿⣿⣿⠟⠀⠀⠀⠀⠘⣿⣦⠀⢁⣴⣶⠃⠀⠀⣀⡈⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠛⠁⣰⣿⣿⣿⣷⣄⠈⠉⠈⠈⠛⠁⠀⠀⢸⣿⣿⡆⢈⣡⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠛⠛⠉⠀⠢⠀⠀⣿⣿⣿⣿⣿⡿⠀⢀⠀⠀⠀⢠⣿⣿⡆⠻⣿⠇⠛⠿⢿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⣤⣄⡑⠀⣿⣿⣿⣿⠟⠁⢀⠂⠀⣠⣶⣄⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⢀⡀⢻⣿⡿⢀⠝⢷⣾⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⣿⡿⢀⠹⣷⣤⣤⣄⠉⠛⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣥⣀⡀⠙⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢉⠀⠤⠘⣿⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⢦⡀⠀⠀⠀⠰⡏⠃⠀⠀⠀⣿⠀⢀⡶⡫⠭⠉⡛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢹⡆⠀⠀⠀⡷⠀⠀⠠⠀⡇⠀⣿⠈⠀⠀⠀⠈⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣷⠀⠀⠀⠰⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⣄⠀⠀⢨⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⢀⠀⠀⠀⢻⠀⠀⠀⢸⡇⠀⠀⠀⡇⢸⠃⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢰⡈⡏⠀⠀⠀⠘⣧⠀⠀⣼⡃⠀⠀⠀⣷⡍⠀⠀⠀⠀⠀⠘⠇⢀⠶⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠈⣻⠶⠿⡀⠐⠠⠖⠸⡇⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢠⠀⠀⠀⢿⣆⠀⠀⢀⡼⠋⠀⠐⠆⣀⠟⠀⠀⠘⢦⡀⠀⢀⡴⣱⠇⠀⣠⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠘⡆⠀⠀⠀⠘⣏⠓⢾⠏⠀⠀⠀⣠⠞⠁⠱⣤⡀⠀⠈⠙⠲⠞⢡⡿⠂⠁⠀⢀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠐⣲⢶⣷⣶⠶⠒⠲⠓⠚⠁⠑⢀⣴⣿⠁⠀⠀⠀⢈⡗⢦⣀⠚⠀⠀⠸⣇⠀⠀⠀⣏⡤⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠉⠓⠦⢤⣀⣀⣆⠂⣼⠇⢸⡆⠀⠀⠀⢸⠃⠀⠻⠀⠀⢠⣤⣬⣙⡛⢾⡋⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣠⠟⢻⡆⣿⠀⠀⣷⠀⠀⠀⣾⠀⠀⠀⣆⢠⢿⢦⡀⠈⠉⠉⠻⡤⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡀⠉⣱⣾⣡⠴⣻⢅⡏⠀⠀⢸⡆⠀⠀⡇⠀⠀⠀⡷⡜⠀⠙⠻⣶⣀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠂⠈⠁⡹⠟⠁⠀⢠⡟⣸⠇⠀⠀⠈⡇⠀⠀⡇⠀⠀⢀⣦⠥⠀⠀⠀⠠⠈⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⣾⠶⠋⠀⠀⠀⠀⡏⠀⠀⣇⠀⠀⠁⡉⠒⠓⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢀⣤⠤⠖⠂⣨⠟⠁⠀⠀⠀⠀⠂⣼⠃⠀⠀⠹⣆⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⢀⣀⡴⠞⠁⠀⠀⠀⠀⠈⠳⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⣀⣠⠴⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⢤⣄⣀⡀⠀⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉




⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠆⠶⠤⠀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠍⠋⢁⠀⠀⢠⡄⠀⠈⠉⠉⠱⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠈⠀⢀⠈⠀⡀⠀⠚⠐⠗⠀⠋⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡛⠃⠀⠀⠁⠛⠲⠀⡈⠠⠄⠁⡀⠠⡄⠀⢀⡀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡇⠀⠀⠠⠄⢀⠰⢀⣆⡦⡈⠅⠂⢠⠀⠀⠀⠁⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠈⠁⢀⡀⠈⠰⠠⢙⠿⣗⡥⢀⠂⠀⡐⠀⠀⠔⠀⠼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡄⠀⠀⢀⠀⠀⠈⠘⠀⠀⠀⠹⠗⠔⠂⠁⡀⠐⠂⣀⡀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⠁⠀⠉⠀⠀⠀⢃⠩⠈⡀⠡⠉⡈⠠⡄⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣦⠀⠀⠠⠀⠐⠃⠀⠈⣀⠐⠀⠘⠀⢣⡀⣀⢀⡀⠀⠹⡡⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⣄⠀⠀⠀⢀⠰⠆⠉⠀⠀⠈⠀⡀⠀⠀⠀⠀⣠⠞⠡
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠋⠙⠷⣤⣠⡌⢀⠀⠀⠘⠂⠀⠀⠉⠀⢀⡨⠚⡉⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠃⠁⠀⠀⡧⠀⠀⠈⠉⠳⠭⠦⣴⣶⣤⣤⣶⠶⠚⠋⢀⠀⠅⠀⠀⠀⠀⡼⠇
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣀⣀⠡⠀⠆⠀⠐⠀⠀⠀⣠⠀⠂⠀⠑⠀⠀⢸⠷⠄⢀⠀⠠
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣈⣤⠐⠶⠄⠀⠀⠀⠀⠀⡚⠀⠀⠀⢬⠀⠀⠀⠀⠀⠛⢭⡄
⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⡟⡁⠀⠀⠀⠀⠀⠐⢃⣠⢁⠐⠆⠀⠀⠄⠀⠁⠀⠀⠀⢀⠁
⠀⠀⠀⠀⡠⠞⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⢸⡟⠃⠿⣶⡄⠀⡰⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠁⠀⠀⠁⠘⠁⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠂⠀⠤⠀⠀⠸⡀⢰⠑⡀⠀⠀⠀⠹⠀⠀⠁⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠖⠀⠄⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣻⣿⢿⠿⠟⠛⠉⠉⢡⣴⣿⣯⡀⠀⠈⠉⠉⠛⠿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷
⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⡿⠋⠁⣿⣿⣿⣿⠀⠙⠻⠿⣿⣿⣿⣷⣤⡀⠀⠀⠀⣸⣄
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⣀⣤⣼⣿⣿⠏⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠹⢿⣿⣿⣷⣦⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⣴⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣄⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣤⣀⣀⠀⠀⢻⣿⠻⡿⠀⠀⣀⣠⣴⣾⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⠀⠀⠀⢈⡷⣄⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣷⡀⠀⠀⠀⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢀⡤⠒⠁⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⣠⣶⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⡠⠐⠛⠢⡺⣯⡆⠀⠀⠀⢰⣀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣸⠀⢠⠉⠈⣀⠙⣷⡀⠀⠀⢸⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡄
⠀⠀⠀⠀⠀⠐⢹⠀⣿⠀⢸⣿⡑⢾⡇⠀⠀⠀⣿⠃⠀⠐⠁⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣍⣿⣿⣿⣿⣿⣿⡟⠉⠛⠃
⠀⠀⠀⠀⠀⠀⢸⠀⢿⣀⣠⠄⠐⢸⣧⣄⣀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⡿⠛⢛⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠨⠀⠀⢸⡟⠉⠉⠉⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⡿⠛⠛⠒⠚⠀⠀⠘⡇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣧⠀⠈⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠰⠁⠀⠀⢠⣙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠻⣧⡶⣛⠶⠥⠾⠶⢦⡴⡟⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣥⣼⣿⣿⡿⣿⣿⣷
⠀⢸⢹⢳⡒⣿⣥⣄⡴⠛⣁⠀⠀⠄⠤⠊⠀⡇⠀⡇⠀⠀⠀⠀⠀⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠁⣸⣿⡇
⠀⢸⣏⠀⠀⢸⡟⣡⠂⢩⠃⠀⠀⠀⠀⠀⠀⢇⠀⡇⠀⠀⠀⠀⠀⣿⡟⠁⢸⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠘⠋⠀⠀⢸⢸⡇⡐⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦⠰⠀⠀⠀⠀⠀⠛⠀⠀⠸⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠿⢯⣑⣠⡄⠀⠀⠀⠀⠀⠀⠀⢸⡧⠓⢒⡲⠀⠀⠀⠀⠀⠀⣽⣿⣿⣦⡤⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠈⣦⠈⣃⡀⢀⣀⠁⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣉⣀⣀⣀⡁⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⡤⠃⡞⠁⠉⠋⠀⢱⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣛⣋⡐⢿⣿⣿⣿⣿⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠊⠀⠙⠒⢦⡀⣀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣧⠙⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢰⠀⣤⡄⣰⠀⢳⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣠⡄⠈⠒⢻⠉⠀⠉⢻⠒⠋⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣋⣼⣿⣿⣿⣧⣙⣿⠆
⠀⠀⠀⠀⠀⡜⠀⣠⠴⢒⡋⠀⠀⠀⠘⣒⠶⢤⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⢀⣿⠸⣿⣿⣿⣟⠿⣿⣿⣿⣿⠇⣷⡀
⠀⠀⠀⠀⡼⠁⠀⠳⠤⠥⠭⠷⠶⠶⠭⠬⠤⠜⠁⠀⢱⡀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣶⣤⣤⣶⣶⣶⣦⣤⣶⣾⣿⣷
⠀⠀⠀⠀⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠁⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠛⠃⣻⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⣉⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠈⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡇⠠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡆⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢰⠀⡀⠀⢘⠇⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠑⠢⣇⠇⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠦⠤⠱⢟⣼⣶⠒⢐⠂⠀⣾⣇⠀⠀⠀⠀⢠⢰⠃⣼⠂⢀⡀
⠀⠀⠀⠀⠐⠐⢑⣤⣶⡀⠀⢀⣀⠀⢈⣽⣁⡴⠏⣠⣄⡈⠹⣦⣄⠀⠀⠸⣾⠆⣟⣠⡟⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢉⠉⠙⠛⠴⠋⠉⣟⠋⢠⣾⠿⠛⠳⢦⣀⡈⠝⠳⣦⣄⣈⣿⣿⠒⠊⢁⣠
⠀⠀⠀⠀⠀⠀⠀⠐⠊⢿⣁⣄⣀⠠⠔⢫⣼⡟⠁⠀⠀⠀⠀⠈⠉⠻⢤⣜⠿⣿⢿⠧⡀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢩⠴⠖⣻⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢻⣎⡀⠈⠓⠲⠤⡀
⠀⠀⠀⠀⠀⠉⣤⠖⠀⠰⢏⡰⢫⡝⣿⣦⠤⠦⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⣷⣎⢿⣷⢄⡀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⡟⡜⣹⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⠉⣸⣶⣶⣙⣿⣇⣀
⠀⠀⠀⠀⠀⠀⢃⠔⠀⢁⢰⣿⠿⣽⣧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢧⣸⣿⡇⣼⡇⢫⠉⢦⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠼⡸⡧⢷⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠸⠇⢷⠸⢳⡘⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠘⠃⢸⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢈⠁⠀⠠⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠼⣆⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⡀

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢦⣄⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠛⠿⠿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠉⠀⠀⠀⠀⠙⢿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢠⡤⠀⠀⠀⠀⠀⠀⠛⠋⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⢻⣿⣿⣿⣿⣿⣿⣤⣶⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢠⣿⠇⠀⠀⠀⢀⣶⣄⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⣾⣿⣿⣿⡿⠙⢿⣿⠟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⡄⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⣿⣿⣿⣧⣄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⣤⣀⡀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣙⠛⠿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣿⡏⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠀⣸⣿⣿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⢀⣼⣿⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠃⢀⣿⣿⡏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢀⣴⣿⣿⣿⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠿⠋⠀⠀⠀⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣷⣶⠦⠤⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⠿⠛⠛⠛⠿⠿⠿⠟⠋⠁⠀⠀⠀⡀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠙⢄⢢⠠⠀⠈⢢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠡⡆⢺⠆⢈⣿⣷⡦⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⣷⣿⠀⢸⣿⣿⣿⢿⣿⣕⣦
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣦⠀⠀⠀⠠⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠤⢆⣹⡏⡀⠀⠙⢿⠟⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠂⠀⠀⠀⠀⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠉⠺⣁⢤⢶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠏⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⣀⡤⠚⢍⠉⠉⠁⢶⠢⠤⣄⡀⠀⠈⠢⠍⠾⢤
⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠈⠻⣿⣿⠿⡿⠏⢹⠀⠀⠀⠱⡀⠀⢸⠀⠀⠀⠈⠑⠒⠤⣀⡠⠔⠃
⠀⠀⠀⠀⠀⠀⡆⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠣⡀⠀⠀⢸⡄⠨⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⡇⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣠⠾⠗⣸⠋⠁⢀⢫⡟
⠀⠀⠀⠀⠀⠀⢃⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⢒⡪⠥⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠈⠮⡑⠒⠤⠤⠤⠄⠒⠛⢛⣢⣴⠞⢉⠽⡓⡂⣁⣀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠈⠑⠒⠒⠒⠒⠲⢾⠿⠟⣁⢴⠁⠀⠀⠀⢀⡀⠀⢀⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⢠⣀⣀⠀⣀⣀⡠⠤⠒⠉⠀⠀⠢⠤⠒⠉⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠
⠀⠀⠀⠀⠀⠀⠈⠙⠛⡀⢀⣆⠀⠀⢠⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⢿
⠀⠀⠀⣴⡿⠃⠀⠀⢸⣷⣾⣿⣆⠀⠀⠙⢿⣟⣦⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⣿⣿⠏⠃⠀⠀⢰⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠻⠟⠦⠀⠀⠀⠉⠻⣿⠀⠀⠀⠀⠀⠀⣠⡀⢀⡠⢽⣿⣿⣿⣶⠃⠶⠀⠀⠊⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣼⣿⡿⠀⠀⠀⠈⠻⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠘⣿⡿⠆⢈⠛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⢀⡠⣴⣾⣿⣿⡆⠀⠀⢀⣴⣶⣾⣀
⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣶⠀⠻⣿⠋⠀⠀⠀⠀⠈⢹⣿⣿⣿⡿⠀⠀⢴⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢀⠙⣻⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣼⣿⣏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠟⢻⠿⣇⠀⠀⠀⢹⣿⣿⠿⠏
⠀⠀⠀⠀⠁⠛⠳⢿⡗⣍⠙⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢠⣾⠇⠉⢀⣀⣐⡤⠀⠀⠀⣶⣶⣧⡀⠀⠀⠀⠀⠈⠑⢦⣠⣴⡟⠤⠀⠤⠤
⠀⠀⠀⢤⡞⣤⣠⣿⡟⣿⠴⠆⠀⠀⠀⠀⠀⠀⣺⡇⠀⠀⠀⢙⣄⠀⣻⣷⣶⡃⠀⠀⠉⣹⠿⣿⡟⠳⠂⠀⠒⠒⠉⠁⠀⠈⢧⡀
⠀⠀⠀⡄⠁⣉⣁⣽⡗⣿⢠⡆⠀⢀⠀⠀⢀⠀⣜⡁⠀⠀⠀⣿⣿⠓⠄⣸⡁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⣀⣤⣤⠀⠉⠒⠦⠤
⠀⠀⠀⠀⠜⠉⢁⣼⡇⠙⠛⠛⠂⠀⠀⠀⢀⡉⣿⣇⠈⠀⠀⠉⠀⠀⠀⠉⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡘⠻⣿⣏
⠀⠀⠀⠀⣶⣀⠈⣿⣿⣷⠶⡇⠀⠀⠀⠀⠀⠀⣼⡏⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡶⢤⠈⠉⠀⠤⢀⠚⠈
⠀⠀⠈⠀⣹⡿⠦⢀⣿⣷⣾⡷⠄⠀⠀⠀⠰⠶⣿⡟⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛
⠀⠀⠀⣤⢀⡰⣤⣽⣿⣿⡦⣤⡤⠄⠀⠀⢠⣦⣹⣿⣁
⠀⠀⠀⠈⠛⠃⠁⣨⣿⡷⢶⡦⠀⠀⠀⣀⠸⣿⣽⣿⣄⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⣀⡀⡄⢷⣠⣄⠈⣀⡀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠊⠿⢡⠄⠙⠀⣻⡾⠏⠉⠀⠀⠀⠀⢺⠉⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⢀⢀⠀⠀⠀⠀⣠⠀⠀⠀⠁⢠⠀⠠⠞⡁⠀⠀⠀⠀⢈⡅⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⣶⣄⠀⠙⠁⠀⢀⡄⣸⡏⠐⣆⡀⢀⡀⢀⢠⡀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢰⣃⣹⡧⠀⠀⡆⠀⣰⡇⠀⠀⢏⠳⠂⠀⠧⠀⠂
⢽⣿⣲⣶⣤⣠⣟⣍⣿⣇⣠⣼⣇⣾⣟⢠⢅⣿⠀⣠⡄⠀⢀⣐⣟⣀⠀⠒⠄⣈⣟⣷⣴⠆⣱⣾⣤⢽⢧⢖⣀⣾⣃⠀⢀⣰⡄⠀⣄⡀
⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡎⢃⣷⡾⣿⣧⡿⢾⣿⣾⣿⣿⣾⣯⣾⣿⣿⣿⣷⣿⣾⣿⣿⡏⣴⣷⣿⣿⣤⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣬⣯⣭⣤⣿⣶⣬⣿⣥⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣷⣾⣿⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠠⡐⠒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢨⢠⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⢀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣧⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⢨⠁⠩⡤⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠣⣀⡀⠀⡼⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢚⡰⠾
⠀⠀⠀⠀⠀⠀⠀⠰⢻⠋⠹⠂⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣿⢡⠋⣉⣉⢉⡉⠁⡀
⣦⡖⠊⠂⠀⠀⢠⠂⠐⠀⠐⠀⠀⢤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣤⠀⠀⠀⠈⡏⠁⡏⠀⡇
⢿⢷⡇⡞⠁⠒⠁⠀⣇⠀⠀⠀⠀⢠⣇⣀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣯⠔⡖⠒⠊⠈⣮⡄⠀⠀⠘⣯⠉⠙⠃
⠈⠉⠉⠓⢤⣀⣀⠾⡏⡄⠀⠀⡼⣁⠀⡹⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠒⠓⠤⣀⠔⠁⣿⠀⠀⠀⠻⢷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⣆⠀⠈⠛⠙⠁⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡄⠄⠐⠊⠁⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠘⠓⠒⠀⠒⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡞⠁⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⣀⠔⠈⠀⠑⠦⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⣠⠞⠫⡀⠀⠀⠀⠉⠉⠉⠑⣲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⡴⠋⠈⠢⣀⠀⠀⠀⠀⠀⠈⠑⣤⣠⣀⡀
⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⡾⠁⠀⠀⠈⠲⢄⣀⡠⠤⠶⣄⣏⠠⢛⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⡜⠀⠀⠀⠀⠈⠉⠒⠒⠲⠦⠤⠴⣧⡇⢠⢿
⠀⠀⠀⠀⢀⡼⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⠇⠀⠀⢀⠀⠀⠀⠀⡼⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣮⠏
⠀⣀⣀⣾⣉⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠝⠀⠀⠀⠸⣏⠳⠆⢰⣙⡶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⠏
⠑⠪⢿⣍⣉⠙⡷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡢⢄⣨⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢀⡀⢀⣀⠀⠐⠠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠀⠀⣂⠀⠈⡉⠀⡽⠈⠉⠉⠐⠒⠚⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⡇⢀⡀⠙⡄⠀⠥⢰⢁⡄⠀⠀⠀⠙⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⠁⠈⠉⠀⠀⣯⠉⡀⠀⠐⠊⠀⠉⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡃⠀⢠⠃⠀⠈⢣⠀⢠⡀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠤⠀⢀⠯⠇⠀⠸⡀⠀⠀⣀⣇⢉⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠔⠶⠄⠶⠄⠈⠀⢀⡀⠳⡀⠀⢁⡏⠉⠛⠒⠒⠒⠒⠒⠒⠒⠲⠶⠦⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠶⠶⠶⢶⠀⠀⣎⡁⠀⠈⠉⠕⠃⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠙⢢⣄⠀⠄⠀⡀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠠⡁⠀⠐⠄⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠉⠑⠒⠀⠤⠤⠤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡑⣤⠄⠀⠀⢀⡠⡴⠆⠀⣲⡆⠀⢀⣀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠘⠗⠁⠀⠀⠀⠊⠰⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢺⠀⠀⡄⠀⠀⣀⣠⠂⠀⢠⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡀⠘⣂⢰⣷⣮⡏⢠⡃⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢡⠞⠀⠀⣿⣿⠟⢸⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠙⠠⠎⡀⠀⠀⣇⡏⡀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⡀⠰⠋⠁⠀⠀⣿⢠⡄⢷⠚⠲⢦⠤⣤⠴⣄⠄⣄
⠀⠀⠀⠀⠀⠠⡋⠇⠀⡸⡔⠀⠀⠀⠀⠀⠀⠀⡘⢁⠅⠀⠀⠎⠀⣀⢠⣄⠐⣅
⠀⠀⠀⠀⠀⠦⣋⠁⡈⠹⠀⠀⢀⡴⠀⠀⢠⠀⠁⡜⠀⠀⠀⢀⠃⠋⠉⠤⠀⠉
⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠄⠀⠀⠀⢠⡜⠀⢰⠀⠀⠀⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⣀⡏⠀⠠⢷⣧⠁⠀⠘⡆⢠⠃⢀⠃⡜
⠀⠀⠀⠀⠀⢀⠀⡀⣸⡆⠸⡇⠀⠀⠆⠈⢹⠶⠞⠂⡞⠀⢸⠒⠃
⠀⠀⠀⠀⠀⠀⠘⠀⠧⠥⠀⠀⠠⠊⠀⠀⠘⠀⠀⠀⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠤⠄⠀⠠⠤⠤⡄

⠀⠀⠰⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⠒⠒⠒⡖⡖⠒⠒⠒⢲⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠶⠒⠒⠆
⠀⠀⠀⠀⡟⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⢸⠀⠀⠀⡇⡇⠀⠀⠀⢸⠘⢲⠚⠒⠚⠛⠛⠛⠛⠓⣧⢰⠀⠀⠀⠐⠂⠐⠒⠐⢰⠄⡘
⠀⠀⠀⠀⢁⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢸⠀⠀⢸⠀⠀⠀⡇⡇⠀⠀⠀⢸⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⠀⠀⠀⠆⠀⠀⠀⠀⠸⠀⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢔⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⢐⡇⢸⠀⠀⢠⠄⠀⠀⠀⠀⡀⠀⠇
⠀⠀⠀⠀⠀⡀⠀⠀⠀⣄⡈⠀⠀⠀⠡⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸
⠀⠀⠀⠀⠀⠥⠀⠀⠀⣃⡄⠀⠑⠢⠄⠀⠆⠀⠀⠼⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠠⢰⠀⣦
⠀⠀⠀⠀⠀⢠⡀⠀⡠⠃⠀⠀⠀⠉⠉⠀⠈⠂⠀⡘⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⢀⣼⠤⢦⡀⠀⠀⢀⡀⠀⠀⠈⠀⡋
⠀⠀⠀⠀⠀⠀⣤⠀⡀⠤⠤⠀⠀⠐⠒⠂⠂⠀⠉⠁⠀⠀⠀⡆⠀⠀⡇⢸⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠐⢥⣴⠖⡇⠀⠀⠀⠀⠀⠀⠀⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⢸⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠐⠢⠀⠀⢀⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⢸⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⡀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠃⢸⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠁⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣧⣤⣄⣤⢸⠀⠀⠀⢸⠀⢼⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⣠⠖⢊⣡⡼⠋⠻⠥⣀⠀⠀⠉⠚⢦⣤⠤⢼⣀⣾⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠀⢰⠀⢰
⠀⠀⠀⠀⠀⠀⠀⡐⢂⠜⠒⠂⢤⣠⠞⠀⢀⠰⠋⠀⠀⠀⠀⠀⠈⠁⠐⠲⠤⢌⢢⣀⠄⠘⠓⢦⣄⣀⣀⣀⣤⡇⢸⠀⠀⠀⠀⠀⠀⠸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣇⠃⠀⡠⠴⢾⢃⡤⠊⠁⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠈⣿⠒⠤⣀⠀⢩⡉⠉⠁⠀⠁⠈⠀⠒⠒⠒⠒⠀⠃⠀⡌
⠀⠀⠀⠀⠀⠀⠀⢿⡤⠏⠀⠀⠈⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠢⣼⠉⠉⢉⡩⠭⠭⠌⣉⠉⠈⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⢀⡤⢾⠇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⣄⠀⠀⠁⠀⠀⠏⠐⠠⠀⠂⢹
⠀⠀⠀⠀⠀⠀⠀⠀⡀⡅⠸⣀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡸⠀⢸⠀⠀⡘⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠀⠈⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⢡⠀⢸⠀⡤⡟⠒⠠⠤⠤⠤⠴⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⢸⠀⢸⢐⡇⠀⢱⣦⠶⢶⡖⠈⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡼⠀⠀⠀⠀⠀⢠⠝⠶⠐⠒⠚⠃⠐⠒⠶⠤⢶⣉⣀⠀⠀⠀⠀⠀⢣⡈⢸⠱⠦⠴⠶⠿⠿⠦⠤⠦⠴⠶⠦
⠀⠀⢀⣀⡀⡀⢀⣀⢈⣷⠒⠀⣀⠞⠉⠀⢀⣄⣀⡀⠠⠄⣀⣀⠀⠀⢀⡀⠉⠫⣄⡈⠉⠈⣯⣵⢼⣄⣀⣀⣀⡀⣀⠁⡀⢀⠀⣀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣀⣰⠁⠀⠀⠰⡿⠏⣀⡀⠤⠄⢀⣀⠉⠑⡾⢬⡇⠀⠘⡇⠀⠀⠈⢁⠛⠋⠘⠛⠙⠛⠛⠟⠋⠙⢻⢾⡃⠉⠀⠉⠈⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡘⢁⣀⣈⣦⣠⠶⠅⡀⣠⠊⢠⠀⢀⠀⠀⢈⠙⠀⢀⠨⠶⣶⣦⢤⣤⣤⠈⣶⡦⠀⣀⣀⣀⠄⠀⣀⣀⢸⡟⠋⠛⠛⢷⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢨⠛⠀⠈⠁⠀⠃⢀⡴⠉⠀⠀⢀⡂⣈⡁⠨⣄⠐⠀⠁⠀⠀⡴⠦⣀⡤⣄⠀⠉⣴⣄⣀⣄⣀⣀⣀⣏⣻⣿⣃⣤⣤⣤⣼⣧⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⡃⡆⢀⡔⠀⠄⠀⣭⣠⢀⡀⠀⠀⠀⠀⠀⠁⡌⠀⣤⢤⣰⡄⠈⠆⢸⡡⠖⠀⠀⠈⠅⠀⠀⠀⠀⣀⣀⠠⠀⣀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⡾⠁⠰⢀⣐⣀⠠⡎⠀⠀⡾⠶⠸⢤⣀⡅⣤⣧⡄⠠⠘⡀⠀⠈⠦⠠⢀⠀⠀⠃⠐⠂⠚⡄⠀⠠⢎⠀⠀⡀⣀⠤⠃⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⡸⠓⠛⠂⠉⠉⢀⢀⣣⣤⣌⣀⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠸⠦⠴⠆⣀⣀⣀⣈⣉⣉⠈⠀⠙⣄⠀⢀⡄⠛⠛⠁⠒⠓⢠⠀⠀⠀⠈⢆
⠀⠀⠀⢠⠒⠉⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠐⠊⠉⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠀⠑⠒⠐⠒⠒⠂⠁⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠈⡄⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠦⡠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠶
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠢⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠠⠄⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠉⠉⠉⠉⠙⠉⠉⠀⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄
⠀⠀⠀⠀⢀⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⡀
⠀⠀⠀⢠⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡄
⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄
⠀⠀⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷
⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇
⠀⠀⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿
⠀⠀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃
⠀⠀⠀⠘⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠻⡅⠀⠀⠀⠀⠀⠀⢨⠟⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠃
⠀⠀⠀⠀⠈⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣾⣥⣤⣽⡄⠀⠀⠀⠀⢠⣯⣤⣬⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⢸⠀⠀⠀⢨⠃⠀⠀⠀⠀⠘⡅⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⣄⠀⠀⠀⠣⣀⡤⠋⠀⠀⠀⠀⠀⠀⠙⢤⣀⠜⠀⠀⠀⣠⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠈⣿⣿⣿⣿⠁⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣷⡀
⠀⠀⠀⢀⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡀
⠀⠀⠀⣾⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷
⠀⠀⢸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇
⠀⠀⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷
⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⢿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿
⠀⠀⢸⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⣿⣿⣿⣿⠻⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇
⠀⠀⠀⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠁⠀⣿⣿⣿⣿⠀⠈⠻⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿
⠀⠀⠀⠈⣿⣿⣿⣿⣆⠀⠀⢀⣴⣿⣿⣿⣿⠟⠁⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⡀⠀⠀⣰⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣧⣴⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⣼⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⢀⣿⣿⣿⣿⡀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠳⠀⠠⠄⠠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠒⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡠⠄⠤⠄⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠀⠀⣀⡴⠚⠉⠉⠉⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠡⠀⠀⠠⠤⠁⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡇⢠⢤⣀⠀⠓⠈⠙⠛⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⣇⠀⡀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠈⠀⠀⡄⠀⠀⠀⠀⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠄⠀⠀⠀⡠⣊⡂⠰⠀⠀⡀⠄⠀⢀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢄
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠠⠄⠀⠉⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠈⠀⢄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠑⠤⣀⡀⠤⠀⠀⠀⢀⣀⡀⠀⠀⠀⠐⠒⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠠⠊⣰
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠦⠤⠤⠤⠒⠒⠒⠀⠀⠀⠀⠀⠈⠒⠢⠤⠤⠤⠀⠠⠤⠤⠤⠴⠖⠿⠦⠤⠤⢤⣤⣤⣬⣅⣤⣤⣤⠤⠤⠤⠤⠴⠖⠒⠊⠉⠁

⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢸⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⣤⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣾⠊⠁⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣤⣆⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⡟⠉⠉⠉⠹⣿⣬⣿⠀⣏⠂⠄⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣯⡴⠛⠢⣴⡆⣤⡀⠼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣅⠀⠀⠀⠀⣈⣿⣿⣿⡶⠀⢻⡗⠛⠀⣤⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⠀⠀⣿⣿⣿⣿⣿⣧⣴⣏⠀⣦⣤⡼⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠀⠀⠹⠿⠛⠿⣿⣿⣿⣿⣶⣿⣯⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣃⡀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡁⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⡏⢻⡿⠗⠂⠀⠀⢀⣾⠛⢿⣿⣿⣾⣿⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢘⣉⣀⢨⣿⣷⣶⠄⠀⠈⠁⠀⣸⣿⣿⡿⣿⣿⡇⢠⢰⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⢀⡙⣿⡟⢻⡍⠁⠀⢀⣠⣴⣿⠛⣿⣿⣿⣿⣿⣿⣿⣾⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣼⣶⡆⢸⣿⠻⡯⠈⢻⣿⣿⣿⣿⣿⠟⠀⣿⣽⣿⣏⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡗⣀⡹⠶⠀⠋⢹⣿⣿⡟⠛⠁⠀⠀⠹⣿⣿⣟⢿⣭⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣿⣿⣿⡄⢀⣿⣿⠏⠁⠀⠀⠀⠀⡀⠀⢿⣿⣶⠟⠰⢦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠃⣾⣿⠟⠃⠀⠀⠀⡀⠀⣉⣴⣿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⣝⣿⣿⡟⠉⠆⠀⠀⠀⠈⢀⣺⣿⠙⢿⣏⢰⠄⠀⠈⢷⠄⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⡟⢿⣿⣿⡏⢷⣼⡌⡇⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⣀⢐⣄⣀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠁⠈⣿⡟⠀⠞⠻⡇⢁⡀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠤⠀⠸⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡇
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⢄⠀⠀⢸⠀⢸⠀⠀⢀⠤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠑⡄⢸⠀⢸⠀⡔⠁⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⡾⠀⠸⡼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢇⡀⠀⠀⠀⠀⢀⣀⡀⠤⠔⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⢤⣀⣀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠉⠑⠒⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠒⠉⠁


⣏⣈⣯⣯⣟⣯⣿⣽⣽⣽⣉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⣦⣄⡀⢰⡟⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣭⡶⢷⣦⠉⠻⣾⡷⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⣦⣄⡀⢠⡿⠛⢻⡄⠀⠀⣴⠟⣁⣀⣀⡈⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣦⣉⠻⣾⡷⣦⠾⢃⡘⠷⠶⠾⠋⣉⣉⠻⠶⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⡈⢻⡆⠀⠈⠻⠶⠶⠾⠛⠋⠙⠷⢶⣶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⣠⡾⢿⣄⠛⣛⣿⡇⠀⠀⠀⠀⣴⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⡇⠀⣿⠀⠀⠙⠛⢋⣉⣤⣤⣶⠶⣶⣿⣄⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣿⠆⠀⠀⢰⡟⠉⠁⣀⣀⣤⡙⣷⣭⣥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢋⣾⠋⠀⠀⠀⠀⠈⠻⣦⣀⠙⢿⡝⢷⣮⣵⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⢠⡶⠟⠛⠛⠷⣮⠙⣷⠈⣿⢀⣴⠾⠛⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠹⡇⢿⠀⠿⢼⠇⠀⠀⠀⠀⠈⢿

⠀⠀⠀⠀⠀⠀⠀⣠⣄⡤⣤⣠⣤⣤⣠⣤⣤⣄⣄⣠⣤⣄⡤⣤⣠⣤⣤⣤⢤⣄⣤⣀⣤⣄⣄⣀⣀⣀⣀⣀⣠⣀⣀⣤⣤⡄⠀⢀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⣹⢿⡷⣷⢯⣿⣿⠷⢿⣿⡽⣿⣿⣯⣿⢶⣿⠭⢧⡿⣿⣿⢿⣿⠾⣻⣮⡿⣻⡛⢽⡿⣿⣿⣾⣷⣿⣞⣼⣿⣿⣿⣷⣄⡠
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡷⡴⣿⣿⣿⡿⡿⢳⢛⢛⣟⡛⣿⣿⣭⡿⢿⣹⣿⣹⣾⢳⣾⡖⣧⠻⢿⣷⣿⡿⢶⣿⡻⢝⠿⠋⢽⣦⡤⣿⣿⣟⣻⡟⠃
⠀⠀⠀⠀⠀⠀⠀⣤⣌⣷⣶⣟⣯⣿⣿⣣⡏⣼⣷⣼⣵⡿⣾⣿⣿⣿⡽⢻⠗⣿⢼⣿⡿⣷⣿⣨⠟⣁⡘⣿⣷⣤⣾⣩⣴⡂⣼⠻⣻⣽⡿⢟
⠀⠀⠀⠀⠀⠀⠀⣃⡽⡮⢁⣹⣻⣟⣿⣿⣀⣿⣿⣿⡿⢷⣚⠊⣽⡿⣿⣿⠷⢸⣿⣿⣿⢟⣭⡟⣡⣜⣝⠻⣽⣗⡸⣻⢾⡟⣻⣗⣿⡿⢷⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⣿⡛⢿⣿⠌⢫⣿⠗⢙⣟⣿⣯⠹⠓⢈⣥⠞⣛⡟⣿⣿⣁⢘⣻⣭⣴⣛⡯⣽⡻⣶⠭⣠⠶⢡⣤⠞⣻⣧⣿⡞⣷⣞⡷⢞⣉⡃
⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⡿⢡⣾⡛⣾⣿⡻⠙⢀⣤⣶⣿⣶⣶⣿⣧⣿⣿⣿⣿⣻⣏⣿⣿⣿⣐⣷⣿⣿⣾⣦⡾⣧⡴⣟⣿⢻⣅⣿⡟⣃⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠻⡎⢿⠈⣿⠻⣿⣿⠏⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡿⣶⡿⡿⣿⡬⣵⣲⣿⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢩⡄⠀⢹⡧⠙⣯⣼⣿⣿⣿⢥⣄⣷⣸⣿⣿⣿⣿⣿⣿⣯⢸⣿⣸⣿⣿⣿⣿⣿⡿⢿⢿⣿⣷⣉⣿⡾⢡⢶⣷⣿⠻⠄
⠀⠀⠀⠀⠀⠀⠀⢂⠀⡄⠀⢸⡇⠸⠇⣿⣿⣿⣿⠀⠀⠈⠉⣿⣿⣿⣿⣿⡏⠁⠉⠀⣽⡆⣿⣿⣿⣿⣿⣧⣜⣿⣿⣿⣿⣿⣾⠡⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣯⡐⢣⠀⠀⢁⠀⠰⣿⣿⣿⡿⢦⡐⠀⠴⣿⣿⣿⣿⣿⣷⣄⠀⠀⠺⣷⡿⢿⣿⡿⠿⣿⣿⣿⣽⣿⡿⢟⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠘⠐⠚⠀⠀⠀⠁⢸⠿⠛⠁⠀⠀⠀⠣⠄⠀⠈⢻⣿⡿⠛⠍⠛⣶⠤⠀⠀⠈⠿⠀⢀⣿⡿⢁⣭⣶⣷⠉⠁⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⣤⠤⢀⡒⠀⠄⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡆⠀⠀⠀⠀⠀⣨⠄⠀⢠⣀⢻⡇⣼⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠉⠂⠹⣾⠿⡡⢤⣄⡀⠀⣀⡀⢀⡄⠘⣿⣿⡏⠀⠀⠀⠀⠀⢻⠀⠀⢘⣿⣿⣧⢹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡷⢼⣿⠋⠛⠉⠉⠉⠉⠁⢀⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⡆⢁⣀⣄⠀⢄⡈⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠄⠀⢀⣧⠀⠀⢰⣈⣰⡶⢀⣀⣄⡀⢀⠀⡀⣃⡀⠶⠛⠛⣿⡇⢸⢿⠭⠀⠦⡱⠂⠿⠂⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⠀⠀⣿⣿⣿⣶⡶⠀⠸⣷⠀⠀⢿⣿⣷⣄⠀⠈⠀⠠⡏⠀⠀⠀⠁⠙⡧⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠀⠀⣀⠀⠁⠀⠈⠉⠙⢷⣳⠦⠀⠘⢻⣄⣠⣀⣰⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠱⠄⠘⠻⣯⣿⣿⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⢠⠀⠀⢸⣯⡟⢆⠸⡏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣵⣿⠀⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢸⡟⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⠀⠀⢸⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⢼⣿⣾⣳⠀⣠⠧⣘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠢⡀⠼⢺⣿⢿⣿⣏⣀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⡿⢳⣾⡟⣶⢻⡟⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠉⠿⣷⣾⡿⣿⢿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢀⣼⡿⠃⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠊

⣿⣿⡿⠻⠟⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⢀⠀⠀⠈⡃⠐⡇⠀⠀⢰⣾⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠈⠀⠀⠈⢇⠀⠀⠇⠚⡇⠀⢐⣿⡿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡙⠀⠀⠀⠀⠀⢘⠆⠀⡇⢸⡇⢰⣷⣿⣏⢸⣝⢛⣛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠁⠀⡔⣇⡀⠉⠈⠡⢾⣿⣷⣦⣬⠛⠸⢯⣿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡔⢸⡆⠀⠀⠀⠀⠀⠘⠀⣿⣠⠞⠙⠂⣤⣀⡈⠉⢉⣤⣻⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢀⢯⠀⠀⠀⢀⣿⡿⣿⡦⠈⠛⢛⢉⣽⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢻⣄⠀⡀⠀⠀⠀⠃⢸⡀⠀⠀⢸⣿⢷⡂⣴⡗⠐⣭⡛⢿⣻⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠙⢷⣀⡀⠀⠀⡀⢸⡇⠀⠀⠸⣿⣿⣷⣧⣼⡇⣇⠸⣯⠿⡿⣧⠆
⠀⠀⠀⠀⠀⠀⠰⠠⢪⠓⠀⠀⠀⠈⢙⠳⣼⣿⡿⠁⠀⠀⠀⢿⣿⣷⣿⠟⣁⣿⢀⢜⠻⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠂⣬⣷⡀⠀⣀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢈⣿⣿⣷⢿⣿⢣⣿⠘⣏⣷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢰⣼⠈⢳⣄⠈⠻⡓⠰⢴⣼⠀⠀⠀⠀⠀⠈⢾⡤⣿⣿⡏⣾⣿⣰⣜⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠂⡀⠹⢶⠘⠪⣭⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣽⣽⣧⣿⡿⢻⣿⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠈⣧⣆⠙⠠⠈⣷⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣠⢁⣤⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣦⣴⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⠀⠀⠀⠄⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣴⣷⣶⣾⣿⣶⢠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠠⠀⠀⠀⠆⠀⣿⣄⠀⠘⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⢸⡏⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⢿⡏⠟⠀⢿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣙⣬⡋⣿⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⡄⢀⠀⠀⢘⢠⠀⠀⠀⣻⢁⠁⠀⠘⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⢹⣷⢿⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⣷⢀⢳⢸⠀⠀⠈⢳⣇⠀⠀⣿⢸⣷⢀⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣾⣿⣟⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠄⣿⠘⡀⠛⠄⠀⣬⠀⣿⠀⠀⣹⣧⣟⡌⠿⣂⠙⡛⠿⢿⣿⢿⡟⠟⣿⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠐⠀⠀⢈⡖⠀⣿⡐⣧⠐⢦⡠⠿⠇⠸⢣⡸⣻⡞⢻⣧⣧⠈⠃⠁⡜⡛⠻⠛⠀⠶⠤⢠⣭⣉⣛⡛⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡆⣿⣿⣽⣦⠋⠀⠀⠀⠈⠛⣷⣿⣡⠐⠸⡃⠀⡇⠸⢷⣿⣶⣦⡄⠀⠀⠀⠀⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⠟⠛⣻⣀⣠⣠⡀⠀⠀⠀⠬⢻⠉⠀⠀⠁⠀⠇⠀⠘⣻⡿⡿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣸⠻⡛⠛⣂⣴⣿⣿⡟⣿⣿⣿⣦⡀⠀⠀⠈⠀⠃⠀⡆⠀⠁⠀⠰⠃⠘⡇⡀⠀⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠰⡤⢼⣿⡿⠉⠉⠛⠿⠛⠈⣿⣷⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡞⢀⠀⠃⠁
⠀⠀⠀⠀⠀⠠⠁⠀⠀⠉⠻⠿⠁⠀⡀⠀⠀⠀⠘⠙⢿⣇⠀⠀⠀⠀⠀⡀⠘⠀⢢⠁⠸⠆
⠀⠀⠀⠂⡔⠁⠀⠀⠀⠀⠀⠀⠑⡀⢀⠀⠀⠀⠀⠀⠉⠀⠀⠄⠀⠚⠀⠈⠀⠀⠀⠰⠐
⠀⠀⠀⠀⠋⠓⠒⣬⠁⠀⠀⠀⠈⣖⢏⡀⠠⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡜⠀⣄⢀⠀⠀⠀⠀⢀⣉⠂⠘⣷⡶⠀⠀⠨⣀⢀⠀⠀⠀⠀⠀⠀⡀⠚⠲⠂
⠀⠀⠊⡅⠀⡆⠀⡤⠀⠀⠀⣤⠙⠇⠀⠀⣄⠀⠀⠀⠉⠉⠀⠀⠀⢀⡠⠄⠐⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣠⢻⠐⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠨⠀⠁⠀⠀⡁⠀⠀⠇⠀⠠⠱⠀⢀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠇⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠈⠉⠉⠉⠁⠒⢤⡤⠒⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣠⣤⣶⠶⠶⠶⠶⣶⣤⡈⢿⡆⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠋⡁⠤⠤⢀⡀⠀⠀⠈⢻⣌⠻⠦⡀⠀⠀⠑⢄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡞⣁⡀⠀⠀⠤⣄⠀⠈⢆⠂⠀⡆⠆⠈⠉⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠰⣦⠀⠀⠶⠀⠀⠀⠘⢤⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡠⡤⠄⠊⠅⠀⠀⠀⢀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⡀⠉⠀⠀⣀⠔⠋⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢈⣩⠀⠀⠀⣒⠖⠂⠁⠀⠸⠉⠉⠀⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠈⠀⠀⠊⢠⠀⠀⠀⠀⡧⠀⠀⠠⢄⠀⠈⠢⡀
⡔⠀⠚⢱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢸⠀⠀⠀⠀⠀⠀⠁⠂⠀⡞⠀⠀⠀⠀⠀⠑⢄⠀⠑⡀
⠁⠀⢠⣿⠉⠐⠀⠠⢀⡀⠀⠀⢀⠄⠈⠑⢪⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢆⠀⠐⡀
⠀⠀⠀⠀⠁⠢⢄⡀⠀⠀⠈⠉⠀⠀⡠⠔⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠐⠂⠁⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⡰⠉⠆⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠒⢄⠀⠀⣀⠤⠊⠀⠀⠈⢆⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⢏⠁⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⡰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠎⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⣱⣄⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⣿⠃⠀⣱⠆⠀⠀⠀⠀⠀⠀⢂⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣶⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⣠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⠁⠈⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢏⡴⡚⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⢖⠻⠊
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⠒⠀⠀⠀⠒⠂⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣊⠁⠀⣀⠤⠄⠄⠀⠠⠄⠤⢀⡀⠀⠁⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⢸⡿⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢀⠔⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠲⡀⠀⠣⡀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⡜⠀⢀⡊⠀⠴⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠈⢄⠀⠱⡀
⠀⠀⠀⠀⠀⢺⣿⡆⢰⢀⢠⠏⢴⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢅⠀⢠⣄⠀⠈⡄⠀⢣
⠀⠀⠀⠀⠀⠈⠻⠇⡖⠁⠸⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠦⠢⣸⠟⠀⠀⢰⡄⠘
⠀⠀⠀⠐⢿⣿⠦⡘⡇⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠃⠀⠈⠢⡀⠀⠀⣠⣤⣤⠐⢄⠀⣿⡟⠀⡂
⠀⠀⠀⠀⢠⣴⡦⠠⡇⠀⢠⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⡀⠀⠀⠀⠀⠑⠄⡉⠉⠁⠀⠀⠀⢺⠀⣾⠟
⠀⠀⠀⠀⠈⠁⠀⠀⢳⠀⠈⡄⠀⠀⢀⠔⠉⠀⢀⠔⠊⠁⠀⠀⠉⠢⣄⠀⠈⠢⢄⠀⠀⢀⠆⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠐⡤⠊⠁⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠑⠤⠜⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⢀⡔⠉⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠑⠬⡂⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⢤⣿⣶⡤⠤⠄⠀⠠⠤⠤⠒⠊⠁⠀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⠄⣤⣀⣀⣀⣀⡀⠤⢔⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⠟⠁⢂⣤⡶⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢻⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣄⠀⢰⠆⢃⠀⢀⡔⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠘⡚⣏⡄⠈⣇⡇⠀⣜⡟⡚⠁⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣰⣼⣦⣤⣽⡀⠀⢹⠀⠀⣸⣥⣤⢾⣴⡸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⢿⣿⣿⣷⣿⣿⣶⠞⢶⣿⣿⣷⣿⣿⡿⠧⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣩⣽⣿⣿⠁⠀⠀⢹⣿⣿⣭⣻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣟⠿⢛⣯⣿⣿⣿⡿⢂⠀⢀⡺⣿⣿⣿⣯⣟⠻⠿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠉⠰⢀⣰⣯⣿⣿⣼⣧⠛⠀⠘⢡⣿⣼⣿⣏⣷⣄⠰⠈⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣶⣾⣿⣿⣿⡻⣿⠟⠀⠀⠀⠘⢿⡿⣻⣿⣿⣿⣶⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡍⣍⣽⣇⠂⠹⣿⡇⠀⠀⠀⠀⠀⠀⣿⡿⠀⠊⣿⣍⡌⣅⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⣛⠃⠈⠂⠰⠈⠀⠀⠀⠀⠀⠀⠀⠈⠡⠀⠊⠀⢛⣧⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏⠈⠻⠿⠏⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⡇⣿⣿⣿⡏⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⢀⣠⣴⣶⣶⣶⣦⡘⢿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣛⠿⢿⣿⣿⣿⠿⢿⡇⣿⡿⣋⣭⣬⣭⡻⠟⣩⣥⣭⣝⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⡇⣿⢱⣿⣿⣿⣿⣷⡞⢿⣿⣿⣿⣇⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠛⠛⠉⣴⣿⢸⣿⣿⣿⣿⣿⢣⣎⠿⠿⢿⡏⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⣤⡀⣿⣿⣆⢳⣶⣶⡙⡟⣸⣿⣿⣿⡿⣱⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⣭⡝⣿⣷⣌⠻⣿⣤⣿⣿⠟⣫⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⢸⣿⢸⣿⢠⣤⣤⣤⣤⣤⡄⣿⡇⣿⡇⣛⢻⣿⣿⣮⡙⢟⣵⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⢸⣿⠈⠛⠛⠛⠛⠛⠁⣿⡇⣿⡇⣭⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⣛⣡⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠛⠁⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⡀⠀⠀⠀⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⢀⣀⠈⠑⡤⢌⠁⠀⠀⠀⠀⣀⠤⠔⠒⠶⣶⠦⠤⠠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠄⠀⠀⠐⠣⡀⠀⠀⡜⠀⠀⠀⠀⠀⠀⡘⠛⠕⣢⠀⠀⠉⠀⠀⠀⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⠐⢲⠀⠈⠁⠀⠀⠀⠀⡈⠀⠀⠀⠘⡤⢸⠤⢀⠀⠀⠀⠀⢠⠁⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠈⠑⠢
⠀⠀⠀⠀⠀⠀⡠⠔⢊⠅⠒⠀⠀⠒⠢⠉⠁⠒⠤⢌⡀⡄⠈⠀⠀⠃⠀⣳⠀⠀⠀⣉⠢⣀⣀⢸⡀⠀⠀⠀⠀⠀⠈⠒⢄
⠀⠀⠀⠀⡠⠊⠀⡰⢋⣰⠆⣠⡄⢴⣈⡙⠄⠀⠀⠀⠀⠁⠂⠤⢀⣠⣶⠟⠀⣐⠁⠀⠀⠀⠀⠈⠉⠓⠤⡀⠀⠀⠀⠀⠀⠁⠢
⠀⠀⠀⢰⠁⠀⠀⠳⣻⠟⠏⠙⢉⠻⢿⡟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢩⡤⠄⠉⠒⠠⠀⠠⠤⠀⠀⠘⠓⠈⠉⠁⠀⠀⠀⢡
⠀⠀⠰⠁⠠⠤⠤⣴⠋⠀⠐⢄⠆⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠰⢯⢊⠅⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔
⠀⠀⠀⠀⠒⠒⠒⢺⣅⠀⠀⠈⠀⠀⠀⣸⠗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠒⠛⠉⠈⠛⠢⡀⠀⠀⠈⠉⠀⠀⠉⠁⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡄⢀⠀⣠⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠉⠉⠉⠉⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⢀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣠⡀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣀⣠⣾⣿⣿⣷⣤⣾⣿⣿⣷⣄⠀⣠⣾⣿⣿⣤⣴⣆
⠀⠀⢀⣤⠶⢾⣿⣿⡟⠛⠒⢦⣄⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⣰⠟⣡⡴⠾⠿⠿⠿⠶⣤⡀⠙⣧⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⢀⡏⣰⠃⠀⠀⠀⠀⠀⠀⠀⠻⡄⢸⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠋⠉⢹⣿⣿⣿⡄
⠈⣧⠙⢦⣴⢦⡴⢦⣴⢶⡞⢾⠇⣸⠇⠀⠀⢠⣿⣿⠀⡄⣄⠀⠀⠉⢉⡉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣷⠗⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠰⢿⡉⠛⢶⡄
⠀⠘⢷⣼⡷⠞⠷⠞⢳⡞⠛⢻⣶⠋⠀⠀⠀⢸⣿⡇⠀⠿⠋⠀⠀⠀⠸⠷⠇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⠀⠀⠙⠶⡴⠇
⠀⠀⠀⢈⡙⣷⠶⠴⣏⠉⠻⠏⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠰⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠀⠀⠀⡀⢹⣿⣿⡀
⠀⠀⢠⣿⡇⠹⢦⡀⠈⡷⠀⠀⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⠛⠶⡄⢰⠶⠀⠰⠟⠋⠀⠀⠀⠀⠀⠀⠘⣷⠇⠈⣿⣿⣇⡀
⠀⠀⠀⠉⠀⠀⢀⣹⣆⣀⣠⣤⣧⡀⠀⢀⡤⢼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢿⠋⠀⠹⡆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⣧⢰⡏⠀⠀⡗⠀⠀⠀⠀⠙⠲⣶⣤⡀⠀⠀⠀⠀⠰⠾⣿⣭⣥⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠘⣧⣤⣤⠤⠴⠖⢿⡀⠳⣄⣀⡇⠀⡄⢠⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠈⣧⠴⠟⠁⠀⠠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠈⣧⠀⠈⠙⡇⠀⠛⠋⠀⠀⠀⠀⠠⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠘⣧⣙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠹⣆⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⢹⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠙⢦⣀⣿⠀⢠⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣸
⠀⠀⠀⠀⢀⣠⣴⠖⠀⠀⠻⣆⠀⠀⠀⠀⠀⠉⠹⣆⠈⠛⠀⠀⠰⠞⢦⣠⠴⢦⣤⠶⠶⠴⠟⠓⠛⠛⠂⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠰⣏⣴⠏⠀⠀⠀⠀⠘⢦⡀⠀⢺⣴⣦⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠁⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠠⣄⡤⠀⠀⠀⠀⠀⠀⠀⠘⠛⢀⣠⠾⠋⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⣤⣄⣀⠀⠀⠀⠙⠓⠶⢤⣤⣉⣀⣀⣀⣀⣀⣀⣤⡴⠶⠛⠉⠀⠀⠀⠀⠘⢧⠀⠀⠀⢠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠉⠙⠳⣦⡀⠀⠀⠀⣿⣤⢼⣿⣿⣭⣉⢈⡇⠀⠀⠀⠀⠀⠐⣦⣤⠀⠘⣧⠀⠀⠀⠻⣌⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠘⣿⣿⡏⠈⠉⠁⠀⠀⠀⠀⢰⡀⠈⠁⠀⠀⢸⡆⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢰⣿⣿⣷⠀⠀⣄⢀⡀⠀⠀⠈⢷⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⣸⣿⣿⣿⡄⠀⠙⠞⠻⣤⡖⠀⢸⣇⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢈⣿⡀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⣶⡦⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣾⡇⢻⠖⠛⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣇⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⠆⢸⡷⠶⠶⠒⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣟⣠⡄⠀⠀⠀⢸⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⡷⡴⠶⠶⠋⢹⣿⣷⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣿⣿⣧⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⠃⢈⣿⣿⣿⣿⣿⣿⡟⠉⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣴⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠟⠋⠁⠀⠸⣿⣿⣿⣿⣿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠈⠓⢀⠔⠊⠉⠉⠉⠉⠓⠦⣔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢰⠁⣀⡀⠀⠀⠀⠀⠀⠀⣠⡂⠸⡇⢀⡀
⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⢀⡇
⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠉⠁
⠀⠀⠀⢀⠀⠀⠰⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠑⢄⡀⠀⠀⠀⠀⣀⡴⠟⠁⠦⡄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠈⠉⣉⠉⠉⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄
⠀⠀⠀⠀⠀⠂⠀⠀⢀⡴⠋⠀⠀⠀⠀⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠛⡩⠢⠈⠀⣰⠀⣠⠀⡀⠈⣆⠀⠂⠀⠀⠒⢄⠀⠀⠀⠀⠀⠢⢄⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣶⠀⠈⠀⢁⡀⢰⠃⠀⠀⠀⠀⠀⢨⢃⠀⠀⠀⣀⢀⠈⠀⠀⠀⠐⠠⠚⠀⠒⠤⠀⠠⢇
⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠃⠀⠀⠀⠀⠀⠠⢞⠁⠄⠁⠉⠉⠠⢈⠁⢀⠁⠈⠉⠀⢈⡂⢰⡄⠁⠀⠀⠀⠉⢠⠀⠀⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠃⢸⡀⠀⠀⠆⠰⠀⡀⢸⠀⠀⠀⠀⡀⠀⠜⢢⡀⠀⠀⡀⠠⠘⠁⠀⠀⠀⢤⣀⠀⠀⠀⠀⢺⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠠⠀⠀⠀⠀⠃⢠⡆⠐⠂⠀⠁⠐⠀⠴⠓⣶⠀⠙⠖⠀⣥⡀⠀⠀⠀⠀⠠⠀⠀⣀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠀⠀⠀⠀⠁⠸⡇⢰⠀⠀⠀⠀⣗⠉⠒⠟⠀⠐⢨⡊⠁⠀⠘⠀⠀⠀⠰⡀⠀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠈⠀⣇⢤⠀⠀⠀⠀⠼⣄⡄⠀⠀⠀⠀⠰⡤⠤⠤⠀⠀⠀⢀⠇⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢹⠁⠀⠀⠠⠒⠀⠠⠃⠀⠀⠴⠒⠀⠄⠁⠀⠀⠴⠂⢀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠱⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠊⠉⠉⠉⠉⠉⠓⠒⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢀⠠⠄⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⣾⠀⠸⠀⢀⢆⣠⠀⢀⣤⣠⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠰⡀⠀⠀⡷⢀⠛⣄⡌⣶⠁⠾⠀⣄⢆⢠⡆⠈⠟⣧⠎⠘⡃⠱⢟
⠀⠀⠀⠀⠀⢠⠐⢀⡠⠶⠂⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠘⠀⠘⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢰
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠀⠀⡘⠸⠀⠀⣠⠤⠤⣤⣔⠁⠀⠛⠀⠀⠙⣆⣀⣀⡐⣼⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⠛⠳⢤⡀⠀⢠⢁⠐⣠⡮⠤⠖⠊⠀⠈⢳⠀⠓⠁⢠⠞⠁⠙⠦⣽⣆⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠠⠒⢢⠹⣤⡸⢀⠀⣏⠀⠀⠀⠀⠀⣾⠆⡇⠀⠀⡇⠀⠀⠀⢸⡇⢸⣸⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡧⠒⡆⠀⠉⠛⡘⠀⠹⡀⠀⠀⠀⠀⠙⡱⠀⠀⠀⢣⣀⠀⠀⠈⢁⠜⠳⣟⡮⠀⠀⠤⠤⢄⢒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⢀⣺⡄⠀⠀⢀⠷⠀⠀⠙⠲⠤⠤⠴⠊⠀⠀⢠⠤⢶⠈⠛⠒⠚⠁⢠⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⠄⠀⠀⡎⠐⠀⠀⠀⠃⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⡀⢄⠖⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠁⠀⠀⢰⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣎⠔⠂⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠺⠀⢀⡠⠒⠉⠲⡀⠀⠀⠀⠀⠶⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠘⠖⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠋⠉⠉⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠋⠀⠀⠀⠀⠀⠘⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⢀⠀⠀⠀⠀⠸⡀⠉⠉⠒⠒⠶⠦⣤⣤⣀⣤⣄⠠⠤⡔⠲⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢠⠆⠀⢀⠀⠀⠀⠙⢄⠀⢀⠔⠚⠳⣧⡰⠗⢦⡀⠀⣠⠃⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠋⠀⢀⠃⠀⠀⠀⠀⠈⠓⠁⠀⠀⠀⠸⠁⠀⠈⡧⠚⡅⠀⠀⠀⠈⣇⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠙⠲⠦⢄⣀⡘⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⢱⠀⠀⠀⠀⣸⡀⠀⠀⠀⡵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢈⠃⠀⠀⢠⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡁⠀⠀⡗⠒⠈⠉⠁⢇⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠰⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢱⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⢠⣄⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⣠⣶⣿⣧⣀⣼⣿⣿⣿⣿⠋⠐⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⣴⣷⣶⣿⡿⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡁⢠⣾⣿⠉⢹⠏⠀⠀⠀⠀⠀⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⡏⢈⣿⣿⣿⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⡴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠙⠿⣿⠿⢷⠀⣶⣤⣤⣤⣤⣀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠾⣷⣶⣶⣶⣆⣠⣄⣈⣁⡘⢳⣝⠿⣿⣿⡿⠟⠀⠀⣠⣴⣿⢿⣿⣿⣿⣷⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣶⣿⣿⣿⣿⠋⠉⣹⣿⣿⣿⣬⣇⠀⠶⠶⣤⠴⣶⣾⣿⣿⣶⡿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣾⣿⠛⣿⣿⣿⣿⣿⣉⣿⣿⣇⣄⣤⣿⣿⡿⢿⣿⣿⣿⣶⣦⠀⠀⠻⠿⢿⣿⣷⣤⡾⠛⠉
⠀⠀⠀⠀⢀⣤⣶⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣹⣿⣿⣿⣷⡘⠀⢩⣿⣿⣿⣆⣀⡀⠀⠀⠀⠀⠈⠃
⠀⢠⣄⣴⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣤⣿⣿⣿⣿⣿⡿⢿⣿⣿⢿⣷⣠⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠈⢻⣧⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣴⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣷⣄⣀
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⠋⠉
⠀⠀⠀⠀⠀⠙⢿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣥⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⢿⣿⣿⣿⣿⣀⡽⣻⣿⣿⣿⣿⢻⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣼⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣀⣀⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡄⠀⠀⠀⠀⢨⣯⣿⣿⣿⠀⠀⠀⠀⠀⢠⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣝⢿⣿⣷⡀⣠⡖⢲⣾⣿⣿⣿⣿⣷⡖⢲⣄⣰⡟⣼⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠹⣎⢿⣿⡷⢻⠛⠛⠉⠱⠎⠀⠱⡉⠛⠛⠟⢉⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠛⢿⡟⢺⡷⢠⡥⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠉⡛⣽⡷⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠻⡿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⡌⣁⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⢄⣀⣀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣤⣆⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⣽⠀⠈⠀⠀⠀⠀⠀⠀⠀⠆⢰⣰⣤⡖⢦⣤⣢⠀⠃⠀⠀⠀⠀⠀⠀⠀⠉⠰⣏⣩⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠶⢶⣶⣶⣾⣻⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠈⠰⣄⣰⠈⢃⠐⠁⠀⠀⠀⠀⠀⠀⠀⢀⢤⡾⠟⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡁⠁⠀⠀⠀⠀⠀⠀⠀⢾⡁⢀⣺⡇⣿⣿⢘⢘⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⠈⢘⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢫⣿⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣄⠡⠉⠩⠈⡭⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣪⣿⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣍⣽⠀⣀⠀⠀⠀⠀⠀⠀⠀⡄⠘⠩⠛⠷⠞⠛⠉⠀⡄⠀⠀⠀⠀⠀⠀⢀⣀⢐⣏⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠸⠛⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠊⠉⠉⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢠⡌⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠲⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⢿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⠶⠿⡟⢘⡓⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣀⣥⡻⠷⢴⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢏⣾⣿⡷⣼⡴⢦⣤⡰⢂⣀⡐⣥⡴⢴⣦⣈⢿⣿⣿⡄⠈⠛⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣫⣾⣿⡿⠁⠙⠷⠾⢻⣿⣿⣿⣿⠟⠷⠾⠃⠙⣧⢻⣿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⠂⠀⠀⠀⠀⢘⣛⣿⣿⣿⠀⠀⠀⠀⠀⠘⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠉⠉⠡

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣴⣾⠛⠃⡐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢪⣾⣿⣿⡁⠀⡄⢉⠸⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣿⣿⣿⣿⣧⠀⠁⡠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⢿⣿⣿⡟⢀⣴⡅⠰⢐⠖⠁⣹⣖⠈⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠑⢾⣻⣧⠀⡚⠚⠉⡽⢌⣁⣁⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠛⠟⠀⠈⠲⠊⠀⠀⠀⠉⠀⠀⠀⠀⠐⠒⠒⠒⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠋⠁⠀⠀⡄⠀⠀⠀⠀⠀⠘⠀⠀⢀⠀⠀⠀⠀⠆⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢩⠀⠀⠀⠀⠄⠀⢢⣄⠀⠀⠀⠈⠀⢰⠀⠀⠀⠀⡀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡄⠀⠀⠀⠀⠠⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⣁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠁⠠⠄⠀⠃⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⠀⠀⡀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡰⠁⠔⠚⠈⠈⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠧⣀⠀⢆⣀⡠⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠉⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠐⠲⠖⠒⠚⢱⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠹⠉⡗⠒⠂⠀⠀⠀⠀⠀⠀⠀⠤⣴⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⠃⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠄⡀⡌⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠚⠁⠘⠧⢔⡒⠠⠤⠀⠀⣀⣀⣀⠀⠀⠀⣀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⣀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⡀⠀⢠⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⣁⠉⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠰⡚⣃⡭⠳⡤⠋⡆⢹⠀⠀⠒⡖⠆⠀⢀⡠⠊⠓⢄⡠⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑⢐⡇⠑⢄⠀⠀⡇⠀⢣⠀⢀⠀⠀⢠⠁⠀⢥⠂⡄⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢸⠁⠹⣯⡧⢠⠇⠀⢸⡆⠉⣀⠀⣸⡀⣀⠈⠘⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢒⠞⠀⠀⢀⣙⣞⣡⣴⠲⣻⠙⣯⣽⣼⢛⠋⣫⢀⣉⢠⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠹⠀⠀⠙⠛⠿⠿⠿⣯⠴⣷⣾⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⠿⠿⠿⠿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠍⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣟⣴⣿⣿⣿⣿⡿⣿⣿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣀⣀⣀⣀⣤⣤⣤⣴⣦⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⠿⡿⠿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⣟⠋⠙⠉⠉⠁⠀⠀⠀⠀⠀⠛⠟⠻⢷⣶⣿⢶⡤⠴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠿⠿⣿⣿⣿⣿⣿⠿⠿⠋⠁
⡯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠀⠀⠀⠙⠟⠋⠉⠉⠉⠊⠁⠀⠁⠘⠃⠀⠀⢀⣀⠊⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣴⡿⡄⠀⠀⠿⢶⡶
⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀
⣶⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠢⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⢰⠹⡀⠀⣀⢤⡄⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢸⠀⠣⡟⠀⠳⠞⠀⠀⠳⠞⣄⡇⠙⢦⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠣⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⡟⠓⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠤⠼⠆⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⢄⣀⡀⠀⠀⠀⠀⢀⡐⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠴⢤⣀⠤⠔⠚⠁⠀⡀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣄⠀⠀⠀⡴⠛⢲⡀⠀⠈⢀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠈⡇⠀⢰⣧⠀⢀⡇⢀⡸⢏⣀⠤⠶⡄⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⠟⡰⢁⡄⠘⢧⡤⠞⠀⣼⠀⠀⠋⢴⠄⡿⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡀⠉⠀⠁⠀⠀⠀⢀⣠⣾⣿⡆⠀⠀⠐⠚⣅⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⠑⠲⣤⣤⣴⣶⣿⠟⠁⢨⠇⠀⢀⣤⠖⣉⣓⣒⡢⢄⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠹⣿⠟⠫⡇⣠⡴⠋⠀⣴⠋⢸⠉⠉⡇⠀⡸⠃⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣯⠝⠓⡆⠀⠙⠲⢬⣉⣉⣉⣁⣀⡤⠞⠁⠀⣀⡧⠴⠃⠘⣅⣰⠆⢠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠒⢎⡠⠀⠇⠀⠀⠀⠀⠀⢀⡽⠋⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠣⣤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣐⠪⠔⠲⣄⠑⠦⠤⠼⣯⠏⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠋⢹⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢖⠒⠒⠶⠖⠋⠉⠉⠉⠉⠑⢶⡇⠀⠀⠀⠀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠚⣒⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⢸⡩⣉⠉⠉⠓⣯⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠙⢬⣧⠴⠊⠙⠤⠤⣖⠀⠀⠀⢀⡽⠀⣰⢭⣳⣤⣇⠄⡠⠃⣱⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣴⡓⢺⡀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⠴⣋⠳⢞⡡⠚⠉⠀⠉⠛⠒⠊⠁
⠀⠀⠀⠀⠀⠀⢾⠛⠒⠒⠭⢵⣒⠭⣚⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⠀⠀⠈⠙⠪⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣓⣒⣒⣚⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠐⢍⡉⠓⠤⠬⠿⣄⣀
⠀⠀⠀⢖⠒⠢⠴⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠒⣄⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠉⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠉⠑⠒⢤⡎⠀⠀⠈⠉⠒⢦⡀
⠀⠀⠀⠀⠠⣔⡒⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠐⣟⠒⠤⠤⠖⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⠉⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡨⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢰⡓⠤⡤⠤⠖⠉⠉⠉⠑⠦⠤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠘⣷⣝⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠢⠦⣄⠀⠉⠀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢤⣀⢀⣀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⠲⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢀⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⢈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⣿⠤⠤⣿⣿⠀⠀⠀⠀⠀⠀⣀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠄⠁⠀⣏⠀⠀⠀⠀⠀⣿⠒⠲⣿⣿⠀⠀⠀⠀⠀⠀⠷⠀⠀⠴⠇⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡰⡞⠂⠀⢈⡏⢠⠀⠀⠀⣿⡉⢙⣿⣿⠀⠀⠀⠀⡄⣫⡅⠀⠙⢣⠰⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⠀⣹⠀⠚⣇⣈⡟⢁⡀⠀⠀⢿⠀⣹⣿⣿⠀⠀⢀⣠⡈⢯⢀⡞⠃⢸⡇⠀⣐⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣀⡙⢦⢿⠀⠀⠈⣾⡠⣽⠋⢀⢀⣿⠀⣼⣿⣿⣆⠀⡌⠙⣧⢬⣇⠈⠀⢸⡇⡞⢁⣀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣀⣄⣸⣧⣜⠘⣇⢀⢀⣭⡹⢅⠑⣫⠞⠃⣀⣼⣿⡿⢛⠷⣄⣉⠔⢢⢥⣀⠀⡾⠁⣤⡿⣠⢤⣀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⡁⠸⠟⠛⠻⣄⠛⣴⡏⢀⣷⠴⠛⢁⣦⣷⣿⣿⣿⣷⣧⣆⠘⠙⠶⣯⡀⢹⡖⠃⣸⠟⠛⠻⣢⣄⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠇⠠⡄⣀⣼⣄⠛⡟⢋⠁⣄⣼⣿⡿⠟⠉⠀⠀⠙⠻⢿⣿⣦⣄⠈⡙⠿⡃⣤⣧⣄⠠⠀⠹⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠞⠉⠉⢻⡏⣽⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣾⣿⣆⣹⡏⠀⠈⠧⠁
⠀⠀⠀⠀⠀⠀⠀⠐⣁⠀⠀⢦⡀⠐⠿⢆⣀⠤⣊⣫⣍⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⣥⣆⣑⠤⣀⡼⠛⠀⢠⠆⠀⢀⡎
⠀⠀⠀⠀⠀⠤⠀⢢⣼⣗⣠⣤⣻⣤⣾⠛⣤⠞⠛⣉⣾⡏⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⢻⣷⡙⠛⠳⣄⢻⣦⣼⣍⣤⣀⣾⣧⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡽⠋⠁⣠⣾⡇⢀⣆⣽⡟⠀⢀⡖⠂⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⢳⠀⠈⣿⣏⣵⡀⢹⣶⣤⡈⠙⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠤⠞⣭⠟⣡⣶⠟⣹⠋⢡⣮⣼⡿⠟⠶⢎⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⡀⡽⠖⠻⣿⣦⣷⠈⠹⡏⠻⡶⡌⢻⣬⠱⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠞⠀⢀⡀⠀⣀⢐⣵⠏⣼⠁⠀⣴⠋⡍⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⠛⡦⠀⠘⣇⠛⣦⡀⡀⠀⢀⠠⠀⢃⠐⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⣀⣬⣵⡗⢦⡙⠁⢷⣿⠀⠁⠛⠄⠙⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠃⠘⠣⠈⠀⣿⢰⠈⢯⡽⣶⣯⣤⣄⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠁⠘⠛⡞⠉⢀⣿⣹⢆⠀⣼⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠷⣧⣀⠸⣝⣗⠀⠉⠻⠙⠠⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣴⠋⠛⠀⠙⢋⢝⣿⠙⢄⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⢀⡼⢹⣿⣩⠙⠃⠀⠉⠹⣦⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠏⢹⠀⡀⠀⡠⢪⡿⢹⡆⠨⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠀⣸⠟⢻⡡⠀⠀⣀⠐⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣈⣻⡿⣻⡏⠀⣱⣎⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡆⠀⣿⡏⣷⣋⣀
⠀⠀⠀⠀⠀⠀⠀⠲⡦⠔⠺⠋⠠⣱⠾⢷⡀⠀⠩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⣠⢏⢿⣔⠈⠙⠃⠲⢄⠐
⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⡠⣠⡾⣿⠀⠈⣟⠑⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡴⢋⡟⠀⠀⣿⢷⣠⡄⣀⣀⣉
⠀⠀⠀⠀⠀⠀⠐⠊⠉⠀⢰⡏⠀⠃⠀⠀⢮⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠿⠀⠀⠉⠀⢻⠄⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠾⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠠⠊⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣶⣶⣶⣶⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠟⠛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠉⢻⡇⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠂⠸⠃⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠧⡀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠆⡀⠈⠁⠒⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⠀⠂⠐⠢⡑⠄⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠄⠠⢗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣀⠠⠊⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡠⠂⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⠴⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠈⠀⡀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⠀⠂⠄⢀⠀⠀⠀⢀⠀⢀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠤⠀⢀⠀⠈⠁⠈⠁⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢈⠦⢀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣀⣀⡀⠀⣀⣠⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢿⣿⣿⣿⣿⣿⠛⠻⠋⠐⢄⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠃⠀⠀⠙⣿⣿⠏⠀⠀⠀⠀⠀⠉⠢⣀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊⠁⠀⠀⠀⠀⠌⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠈
⠀⠀⠀⠀⢀⡠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⢀⣀⠄⢆⠁⠀⣀⠠⠄⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠘⡀
⢻⡀⢲⡞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⢡
⠀⢳⡈⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠆
⠀⠀⠻⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⢼⣄⣀⡤⠒
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠴⠛⣋⣤⠖⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣯⣭⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡤⠀⡀⠐⡈⡉⠤⠄⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠭⢑⡑⠧⠘⢀⠞⠁⠀⢀⣤⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣁⣀⡀⠉⠀⠀⠀⠀⠀⠀⢀⣀⠀⢡⠔⠂⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⠔⠊⢉⡀⠀⢀⠠⠐⢈⡁⠄⠤⠤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⢔⢞⡉⢠⠖⢁⠤⠊⠁⠠⠐⢀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⠁⠀⢀⠷⢅⡐⠀⢸⡇⠴⠊⠙⠒⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠨⠑⠀⠀⠀⠁⠤⡴⠙⠂⠉⠉⠀⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⠘⠀⡇⠀⠀⠀⡀⣈⠀⢀⣀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠠⡀⡃⠀⢀⡾⠉⠉⢀⠎⠀⠈⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⡤⣾⡿⡀⠨⠃⠁⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠰⠉⣴⠋⠀⠙⠢⠄⠊⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠈⠉⠈⠉⢀⡄⢀⠄⠀⠐⠒⠃⠀⠘⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠹⣆⢀⠄⠂⠉⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡘⢷⣤⣀⠌⠀⠀⠒⠂⢤⠆
⠀⠀⠀⠀⠀⠀⢠⠤⣀⡱⡆⠹⣉⠒⠠⠤⠄⣤⠜
⠀⠀⠀⠀⠀⠀⠎⢠⠏⢩⢧⡀⠘⠝⠁⣎⠉⢽⠋
⠀⠀⠀⠀⠀⣸⢦⣴⣀⠔⠋⠳⠄⢈⠀⠈⠁⠁
⠀⠀⠀⠀⣴⠉⢈⡾⠍⡲⡶⢶⣤⢷⣄
⠀⠀⢀⣤⣴⣿⣷⣷⣶⣿⣶⣚⣋⣅⡀⢙⣿⣿⣷⣶⣶⣤⣤
⠀⠀⠈⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠂⡲⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢬⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠦⠀⠀⠀⠀⠀⠀⢸⠀⠠⢿⠀⠀⠀⠀⣴⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠰⠀⡀⠀⢸⡀⠀⠀⢸⡀⡴⣿⠀⠀⢀⣾⢿⣿⣿⣿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡅⠆⠀⠀⠀⠀⠈⢳⡀⠀⢨⡄⢅⡇⠀⢀⣴⣾⢿⠙⣿⣥⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠰⠀⠀⡀⠀⢈⠄⠀⠠⡁⠀⠸⡆⢨⡇⠀⣼⣿⡟⠿⡆⢻⣦⢉⣥⣬⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣄⠂⡀⠀⠀⠀⠈⠀⢀⡖⢻⣠⠀⣀⡀⠐⠶⠿⣿⣿⣟⣤⣌⠉⠸⠟⣼⠿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢸⡆⢀⢁⠀⠈⡀⠈⢀⡝⠀⣹⣄⣨⠿⠙⠳⢤⣀⣀⡀⠈⠉⢡⣤⠸⣿⣮⣻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢸⣇⠈⠀⠂⠁⠀⡀⠀⢀⠀⣻⠋⠁⠀⠀⠀⢸⣿⣿⣿⣦⡀⠸⡿⠿⠛⠛⢻⣽⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠈⣿⠄⡈⠄⠀⠀⠀⠄⢀⡞⠹⡆⠀⠀⠀⠀⣿⣿⣇⡙⢉⣥⡄⢁⣐⢶⣿⣿⣷⣿⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⠀⠀⠈⢿⣤⡀⠆⠀⠀⠀⡘⠀⠈⣇⠀⠀⠀⠀⣿⣏⣽⡇⣾⠟⣡⡄⡿⠳⣌⢿⣾⣿⣿⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⢦⠀⠀⠠⠈⠷⣦⣌⡀⠐⢈⣄⣘⣿⠀⠀⠀⠀⢿⣿⣿⡿⢿⣷⣿⠇⣿⡀⡻⣷⣙⣝⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡈⣇⠑⡀⠂⠢⠀⠀⢩⠛⢶⣿⣿⣿⠃⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠋⣀⣿⠇⡐⢞⢛⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢀⣘⣧⠀⠀⡀⠀⡃⠁⠈⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡾⣹⣿⡋⣼⡇⠸⣎⣷⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣯⠙⣷⡀⠌⠻⣶⠶⠆⣄⣰⣺⡇⠀⠀⠀⠀⠀⠀⢸⡽⣋⢻⣷⣿⡟⣸⣿⡇⢦⡹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⣂⠈⢻⣦⣠⢤⡃⢞⢣⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⡿⣿⡇⣿⣿⣿⣯⣷⣹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢐⢀⣶⠄⠙⢷⣌⣤⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣿⣿⣿⣾⣫⣽⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⢧⡈⡑⠂⠐⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣤⡍⣠⣶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⢿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣻⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠞⠁

⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⡐⠈⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⡀⠂⠁⠀⠁⠙⠄⠀⠄⢀⠀⠀⠌⠀⠈⠈⠉⡀
⠀⠀⠡⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠆⠀⢀⠀⠐⠀⠀⠠⠀⠀⠀⠀⠀⠐⠄
⠀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠄⠀⠀⠣⠀⠀⠀⠈⠀⠀⡄⠀⠐⡀⠀⠂⠐⠀⠀⠈⠈⠀⠀⢀⡠
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠎⠐⠠⠄⡀
⠀⠀⠔⠀⠙⠄⠀⠐⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠂⡤⠘⠁⠲⠈⠠⢄
⠀⠀⠠⠄⠠⠈⠈⢡⠀⠤⠄⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⡅⠰⠀⡄⢀⠀⢈⠁⡃
⠀⠀⢄⠀⠀⠜⠀⢃⠀⠀⡠⠀⠀⠀⡠⠈⠀⠀⠀⠀⠄⡄⠀⠀⢠⠠⠀⠀⠀⠀⠑⢀⠀⠀⠀⠐⣀⠃⢤⠈⠤⠈⠢⠑
⠀⠀⠈⠀⠸⠀⡀⢈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇⠀⠀⢸⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠐⠐⡠⠄⠔⠌⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠁⠀⡀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡄⠀⠀⠀⠐
⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠡
⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠐⠀⠀⠀⢀⠀⣀⠀⠘⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡆⠀⠀⢰⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠐⢠⡂
⠀⠀⡔⠂⣁⠈⠄⠁⠃⡀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⣡⢀⠐⠁
⠀⠠⠅⠔⠁⠀⠀⠈⣄⠠⢄⠀⠀⠀⠀⡀⠀⠀⠁⠀⠠⠇⠀⠀⠸⠄⠀⠈⠀⠀⢀⠔⠀⠀⠀⠀⠉⠁⠘⡀⢀⠆⠤⢤⠘
⠀⠰⠀⠲⡀⠒⠘⢀⡄⣈⠅⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⠊⠀⠩⡀⠀⡀⠁
⠀⠀⠡⠤⢈⠀⡚⢀⡰⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠀⠄⠀⠁⠒
⠀⠠⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠒⠄⠀⠠⠀⠀⠠⠀⠀⠀⠄⠤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠆⠀⠠⠀⠀⡀⠀⡀⠈⡀⠠⠀⠀⢀⠂⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠰⠀⠁⠀⠀⠈⠀⠈⠀⢀⠀⠀⢀⠈⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⢄⠀⡀⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢁⢀⠄⠀⠀⠂⠀⣀⠀⠁⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠠⡀⠠
⠀⠐⠀⠀⠈⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⠟⢿⠛⠿⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⠠⢤⣴⣶⣿⣿⣿⣷⣦⡤⠤⠄⠒⠒⡄⠀⠀⠀⠀⠀⡀⠠⠤⠄⠠⠔⠁⡥⠀⠀⡀
⠀⣀⠔⠊⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠱⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠸⡥⡤⠔
⢀⠁⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠈⠷⡤⠀⠓⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢀⣉⠁⠀⢀⡀⠀⠐
⢸⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠻⠀⠀⠁⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⡠
⠈⠀⠀⠉⣿⣿⣿⣿⣿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁⣴⣾
⠀⠣⠀⠀⠙⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢢⡐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢔⣡⣾⠟⠁
⠀⠀⠑⢄⠀⠀⠙⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠄⠈⠈⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣴⡯⠛⠁
⠀⠀⠀⠀⡶⠠⠄⠀⠉⢉⡋⠗⡄⡠⣤⠒⢒⠚⠐⠀⢤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠉
⠀⠀⠀⠀⠀⠀⠂⠀⠄⢸⠀⠀⠈⠘⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠐⠦⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⡄⢠⣦⡾⡄⠀⠀⢳⣶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠊⣀⢴⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠈⠙⠄⠀⠊⣛⡿⣿⣿⣿⣿⣯⣿⡇⠀⣿⣿⠟⢣⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠁⠀⠀⠉⠉⠛⠛⠇⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣿⣿⣿⣻⣿⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠄⠀⠀⠀⠠⢤⡤⣖⠂⠀⠀⠠⠄⠴⠶⠦⠤⠀⠀⣠⠀⠀⠀⠠⠀⠀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⣀⣤⠄⠀⠑⠀⠃⢀⠁⢬⠀⠀⠉⠀⠀⣿⣿⣿⣿⡆⠀⠀⢀⣴⣶⣾⣃⡌⠀⠀⣤⣄⣾⣾⣷⡖⠐⢀⣀⢀⠀⠀⡀
⠀⠀⠀⠄⠀⠀⠈⠀⢠⡽⣷⣿⣷⡄⣄⠀⠀⠀⡄⡀⠀⠀⠐⢣⣐⣒⢛⡁⠈⠛⠫⡄⢀⣩⢿⡭⣿⣷⣄⡘⢷⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⠿⠛⠁
⠀⠀⡀⣀⡀⣂⡀⠀⠈⠉⠙⢻⣿⣷⣿⡑⠄⠀⠈⠐⣀⢒⡊⠳⢦⣬⣽⣲⣦⠀⠀⠀⠀⠙⠷⣾⠛⣿⣿⢥⣿⣿⣿⣿⣽⣿⣿⡿⡟⠛⠋⠁⠀⡀
⠀⠀⠁⠈⠙⢿⡗⢀⣀⠠⠺⣿⣿⡿⣿⣆⣐⡀⠚⣤⣞⢎⣻⠟⠋⠉⠉⠁⠀⠀⠀⣀⣀⠀⢈⠊⢻⣿⣿⣟⣛⣿⡿⢿⣏⢿⣿⢾⣿⣿⣵⣾⣶⡀⠀⠂
⠀⠀⠀⡀⣀⣠⢷⣈⣵⣾⣷⣿⣻⣿⣿⣿⣿⣽⡿⠿⣆⣠⣼⣶⣤⣤⣤⣤⣶⣶⣿⣿⣿⡆⠀⠀⡀⣿⣿⣾⣿⣿⣿⣿⣷⣿⣧⣿⣿⣿⣯⣿⣿⣷⣶⣶⣦⣄
⠀⠀⢁⠈⠵⡄⢘⣛⢘⣿⣿⡿⠯⢿⣿⡍⢉⢽⣆⣐⣋⣽⣮⢻⣮⣭⣶⡈⠉⠉⠉⠉⢿⣿⣶⡷⠛⠙⢿⣿⣿⣿⣿⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⢀⣀⣁⠉⠾⢿⣯⣭⣭⣯⣭⣍⢿⣆⣔⣀⣉⣁⣚⣳⣾⣷⡙⢿⡟⠳⠎⠉⠉⠀⠈⠛⢿⡷⣄⠀⢸⣿⣾⣿⣿⣿⣿⣿⣿⣿⡻⣿⣛⣾⠿⢿⠟⠉
⠀⠐⠛⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠛⠙⠛⠛⠛⠋⠉⢡⡿⢆⠀⠀⠀⠀⠀⠀⠀⠐⠀⠛⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓
⠀⢀⣀⣀⣀⣘⣻⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⣀⢑⠒⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠉⠓⠟⠛⠓⠐⠛⠾⠶⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡘⢿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣭⠹⠟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠒⠂⠂⠐⠂⠐⠀⠂⠀⠀⠀⠀⠐⠒⠀⠀⠂⠀⠒⠀⠀⠀⡂⡀⠓⢂⡐⠂⣀⣀⣂⡀⠂⠒⠐⠀⠂⠐⠒⠐⠒⠂⠐⠒⠀⠀⠒⠀⠒⠒⠒⠒⠒⠐
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣉⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⢀⡴⠚⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⡰⢀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⣠⠢⠀⢠⢷⣬⣄⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠁⠈⠀⣸⢿⠛⠀⠤⠐⢤⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠟⠃⠀⠀⠀⠀⢀⠀⠠⠔⠒⠀⠐⠲⠦⡈⠀⠑⠂⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠁⠀⠀⢀⡤⠆⣩⠥⢬⢸⢷⣠⠠⡄⢀⠭⠐⠂⠛⠀⠠⠔⡒⠦⢍⣱⡤⠔⠂⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠎⠀⠀⠀⠞⢸⠃⡄⠘⠶⠁⠚⢿⠼⢵⣱⢀⠀⠀⠀⠀⠀⠐⠀⠙⣇⠀⠘⠃
⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠀⠀⢀⠂⢰⠀⠟⠀⠀⡄⠀⠄⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠉⢻⡀⠀⢦⣲⡀
⠀⠀⠁⠀⠀⠘⠀⠀⠀⢸⡀⠄⢹⣠⣲⣦⡾⠂⠀⠀⠈⠀⠀⡀⠀⠀⡇⠀⣈⠣⡀⠴⠆⠀⠀⠈⢣⡧⢨⢻⠇⡄
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢣⡀⢳⢘⡿⣿⠁⢀⠀⡁⠀⠚⠰⠗⠀⢀⣇⠂⣀⡰⠟⠉⠂⠀⠀⠀⢘⣇⡿⣿⣷⠂⢀
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠁⠾⠏⠷⠞⠒⠮⣐⠃⠀⠀⢀⠤⠀⠚⠛⠖⠁⠀⠀⠀⠀⠀⢲⣄⡼⡟⠱⢇⠏⢀⠌
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠁⠀⠢⠁⣀⣀⡀⠀⠉⠉⠙⠛⠚⠦⡀⠀⠀⠀⠀⢀⣀⣤⡴⠻⠿⠛⣱⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄⠀⠀⠀⠀⠀⠀⠀⡖⠛⠻⠭⣽⠟⣽⡄⠄⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⠞⠋⠑⠲⠄⢀⣠⠤⠞⠁⠀⠀⡼⠃⠀⠟⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⡜⠀⢠⣞⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠠⠶⠶⠶⡶⠄⠀⠀⠸⠃⠀⣌⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣡⣭⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠶⠿⠛⠛⠛⠛⠛⠻⠷⠶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⡀
⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠉⣿⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠛⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀
⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⣿⠁⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⢹⡇
⢸⡏⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠈⣿
⣿⠃⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⢿
⣿⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⢸
⢿⡆⠀⠀⠀⠀⠀⠀⢸⣧⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣿⠀⠀⠀⠀⠀⠀⠀⣾
⢸⣧⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⢠⣿
⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⣿⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠶⠟⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣛⣿⣶⣶⣶⣶⣶⣶⣶⣾⣛⣛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠰⢖⡶⠶⣶⠿⠒⠒⠲⠖⣠⣠⡖⢳⡞⠲⣲⠶⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠂⠀⠀⠘⣼⣶⣶⡶⡄⠿⠣⡴⡂⠠⠀⢠⡄⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⣤⢀⣀⣀⡿⣴⡏⢠⡄⠀⠔⣳⣿⣤⣼⢶⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠈⣱⣭⣽⣿⣦⣌⣵⡶⠀⠉⢀⣿⣿⣼⡅⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠘⠀⠀⢻⣤⣺⣟⣹⡷⣾⣶⣄⣟⠴⣿⣿⡇⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⢀⣀⠤⢿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢛⡿⢿⣟⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⣠⠀⢂⣼⣿⣿⣿⡟⣿⣿⣿⡿⣷⡿⠶⣾⠃⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠞⠀⠰⢎⡹⣿⣿⣿⣿⣿⣿⣿⠏⠀⣀⢸⡇⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⢡⣧⣽⣿⣿⣟⣿⣿⣿⡻⠛⠻⢻⠄⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⢀⠀⠀⣼⡾⢻⡽⣯⠿⡧⣿⣿⣯⠌⣄⢰⠂⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠸⠓⠚⣿⢿⣷⢱⣿⡆⣅⢹⣿⣿⡝⠘⠻⠁⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⡄⠀⢀⣿⣿⣿⡄⠙⠁⠋⣸⣿⣿⣄⣼⣤⡀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠓⢀⠈⠟⣻⠏⢁⣼⣿⣄⠹⡿⢧⡩⣿⣿⠇⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠠⠖⠀⣿⠂⢼⡀⠈⠁⠉⠉⠀⡥⢸⠃⢉⣿⡄⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠈⠁⠀⠀⢏⢰⡷⠂⠀⠀⠀⠲⡆⠀⠀⠈⢹⠃⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠁⠔⡑⢦⠃⠀⠀⠀⠀⠀⠀⢹⠠⡄⠠⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⡅⠂⡛⢻⠀⠀⠀⠀⠀⠀⠀⢸⣄⠃⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⡀⠀⢰⡶⠄⠀⠀⠀⠀⠀⠀⠀⠻⣤⡤⣔⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⣯⢹⣧⠀⠀⠰⠇⠀⠋⠀⠀⠉⠀⠉⡿⢘⡇⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⡃⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠀⣤⣸⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠾⠆⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⣿⣻⣿⣽⣷⡦⣤⣄⣀⡠⢤⡖⠚⠁⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⣿⠛⠚⠿⣟⣿⡏⠛⠛⠿⠉⠟⠋⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠀⠈⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣄⡀⠀⢸⡇⠀⡇⠀⢀⣠⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣠⢀⣀⠹⡄⢸⡇⠀⡇⢠⠏⢀⡀⣄⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⣀⠹⣼⡇⠀⣧⡼⠀⠀⢳⣼⠀⠸⣧⠏⣀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠈⢻⣛⣻⣄⠀⣸⣀⡤⠤⣄⣇⠀⢠⣟⣛⡟⠁⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣤⠀⠈⠁⢀⡼⠛⠹⡏⠀⠀⢸⠏⠙⢯⡀⠈⠁⠀⣤⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢰⡦⣭⣹⣦⡴⠋⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠙⢦⣴⣏⣭⢴⡆⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣦⠴⢋⡴⢻⠁⠀⠀⠀⠀⣿⠀⠀⡿⠀⠀⠀⠀⠈⣟⢦⡙⠦⣴⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢀⡞⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⡟⠀⢳⡀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⡼⣇⠀⠘⡟⠢⣄⢀⠀⠸⡇⢸⠇⠀⡀⣠⠴⢻⠇⠀⣸⢷⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠁⢀⡤⣾⠁⠸⣆⠀⢹⡀⠈⠯⠀⢀⡇⢸⠀⠀⠹⠁⠀⡟⠀⢰⡇⠈⣷⢤⡀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢠⢷⡀⢸⠿⡆⠈⣇⠀⠀⠀⣸⠁⠈⣇⠀⠀⠀⣸⠁⢠⠟⡇⢀⡾⡆⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡾⠀⣷⠀⢀⣿⡀⣿⠦⢀⡴⠃⠀⠀⠘⢦⡀⠰⢿⢀⣿⡀⠀⣼⠀⢻⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡇⢰⣇⠀⠟⠦⠈⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠁⠰⠻⠀⣸⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠘⠉⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠉⠃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠎⠀⠀⠉⠁⠈⠓⠒⠓⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠖⢲⡀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣵⡾⣿⣶⣄⢷⠀⣰⠇⣠⠞⠙⢦⡲⣄⠀⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠫⢎⡥⠖⢺⡽⡏⠁⢧⠀⠀⠀⠀⠀⢩⡄⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠈⢠⣞⣉⡤⣇⠀⠀⣶⠀⢤⣀⡀⠈⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⢀⡟⠉⠈⢉⡁⠀⠀⠀⠀⣀⠈⠀⠀⠛⠻⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⡴⢿⠉⠀⠀⠀⠘⠋⠀⢀⡀⠈⠙⠀⠀⠀⣠⡴⠃
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⢀⡞⠁⠈⠙⢲⡀⣦⠀⢀⡤⣤⡤⢤⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢸⠁⠀⠀⠀⠈⢳⡀⠀⠈⠓⠚⠓⠚⠁⠸⢆⡞⠀⠀⠀⡐⡆
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠙⢲⡤⣤⡄⠀⠀⠤⢶⠋⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠙⡾⠲⢤⣀⡤⢼⡁⠀⠀⠀⠀⠀⢈⡧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⡼⠁⠀⠀⢻⣀⣠⡽⣆⠀⠀⢀⡴⣫⣀⣀⡟⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⡞⠁⠀⠀⠀⠈⠁⠀⠓⠮⣷⣤⡮⠴⠏⠀⠉⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠦⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡄⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠈⠉⠉⠁⠀⣇⣀⣀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡏⠉⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡅⠀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣔⡀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣎⠳⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣗⣒⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⠶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⢥⡿⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⢤⡀⡽⠆⠀⠀⣿⣿⢥⣿⠀⠀⠀⣸⡇⠀⣦⣦⠂⢀⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢑⠚⠓⠾⣿⣤⣀⠻⣇⡸⠃⠀⣿⣿⣾⣷⠀⣸⡇⠘⣧⣀⡯⣎⠀⣪⡛⢁⡠⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡌⢸⡏⢀⠀⣀⣀⡚⣯⠠⠛⣡⡴⢀⣿⡿⢒⣷⠀⠨⠵⣝⡄⢯⠁⣣⢢⠅⠰⢊⡠⢸⣫⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡐⣷⣤⣷⣼⣄⡉⠋⢩⣭⠻⠀⢯⣴⡾⠋⣉⣄⣹⠷⢄⡀⠸⠷⣼⡄⠇⣨⣶⣿⢁⡀⠚⡧⠾⢷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣋⡩⠟⠛⡹⢿⣾⣿⣬⢶⣷⠛⢻⡧⠆⣠⡾⠋⠙⠻⣶⣤⣹⠻⠦⠾⠃⠘⠧⠤⣀⣨⠑⠋⢀⣤⡢⠂
⠀⠀⠀⠀⠀⠀⢀⡈⡀⠠⣀⣠⣀⡀⢉⢀⣴⣿⣋⣥⡶⠛⣰⠞⠉⠀⡄⠀⢀⣠⢍⠛⢿⡷⢸⡟⠛⣷⣤⣓⡠⣤⣼⡽⠷⠀⣶⣀⡀
⠀⠀⠀⠀⠀⣀⣉⣱⣳⣤⣐⣮⣽⣶⡌⣹⣭⠕⠛⢦⣶⠦⣧⣤⣶⡧⡄⠀⠈⣿⣮⣄⣨⣷⠘⠷⣄⠘⠻⣏⠙⢷⡴⢶⡒⠻⣏⣁⡀
⠀⠀⠀⠀⠀⠈⠉⠉⣡⣠⠟⠡⠛⠩⢟⣵⣏⣠⣾⣿⣿⠈⠻⡏⠓⠀⠁⠀⠀⠈⣨⣩⡿⢻⣿⣦⣤⣝⠒⠈⣁⡄⠑⠀⠀⠀⢀⠘⠋⠂
⠀⠀⠀⠀⠀⠰⠶⡄⠈⢡⠀⠠⣀⡐⣉⣽⣼⣹⣿⣿⠿⣀⢀⣤⡔⠀⠀⠀⠀⠀⠁⠟⠃⣸⣾⢻⠸⣿⣯⠭⡻⢴⣾⣇⣀⣶⡯⠷⢾⠃
⠀⠀⠀⠀⠑⢤⣈⣙⢀⣻⣿⣶⣿⣿⠟⢯⣿⢋⣿⣷⢀⣇⣟⠛⠣⠀⠀⠀⠀⠀⠀⠀⣴⡃⣿⣸⣇⣿⠘⡆⠱⠠⠌⠉⠂⠀⢸⢷⡄
⠀⠀⠀⠀⠀⣤⡿⠛⠛⠉⡉⢽⣣⡏⢠⣾⠟⣻⡟⣿⢺⡾⣏⡀⠀⠀⠀⠀⠀⠀⠠⡞⢻⠃⣿⠻⡅⠹⢳⣧⡶⠀⣌⠙⠛⠃⠀⢀⡄
⠀⠀⠀⠀⢀⣠⣤⡴⢦⡷⢿⣿⣿⠈⠹⢃⣼⣿⡇⣿⡘⢰⣯⠀⠀⠀⠀⠀⠀⠀⠘⢃⣿⢸⡿⣾⠈⢀⡟⠃⡗⠐⠻⣷⣧⠴⣾⡻⣅
⠀⠀⠀⠀⠀⡤⣼⡖⠡⠶⣾⠧⠙⢂⣸⣿⡉⠛⡇⠿⣇⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⢸⡷⠈⣧⡈⠋⠀⡵⡯⠁⠘⢿⠓⡄⡉⠁⠠
⠀⠀⠀⠀⠀⢈⣤⣴⢶⣾⣟⠂⢠⠞⣸⡇⡅⠘⢣⣰⠏⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣴⡏⠍⢱⠀⢷⠀⣝⡓⠈⣧⠈⠩⠆
⠀⠀⠀⠀⠀⠀⣠⠴⣮⣇⣬⠀⠋⠀⠉⠀⠉⡒⠀⠋⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⢠⠣⠈⠀⡊⠑⠘⠘⠂⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠼⢇⡙⠌⣡⢄⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⢾⡅⡄⠐⣒⠲⣄
⠀⠀⠀⠀⠐⠀⠀⠞⠳⠟⠆⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠊⠈⠟⠄⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣭⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⣟⣶⣶⣶⠀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡿⣿⣿⣄⠀⣠⢤⣠⣿⣿⣿⣿⣠⡤⣄⠀⣼⢫⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡈⢻⡝⣿⣿⡞⡿⠶⠛⢫⡝⠉⢩⠛⠶⢿⠟⣣⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡷⢾⣧⢨⡅⠈⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠲⠉⠻⢻⡶⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡦⢥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢤⣶⣤⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠥⠶⠏⠳⢦⡄⢀⠁⠀⠀⠀⠀⠀⠀⠀⠘⣧⣞⠙⠉
⠀⠀⠀⠀⠀⠉⣽⣯⣭⣥⡍⠞⠀⠀⠀⠀⠀⠀⢀⡠⠀⢄⡄⣷⣾⠀⡥⠀⢠⠀⠀⠀⡄⠀⠀⢠⠶⢩⣭⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠿⠿⠿⣿⣇⣄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠃⠿⠿⠀⠆⠀⠘⠀⠀⠀⠃⠀⠀⠘⣠⣸⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⡾⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⡚⢦⣤⣼⠴⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢧
⠀⠀⠀⠀⠀⠀⠸⠿⠟⠛⠿⠗⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⠘⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠿⡟⢘⡃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣀⣬⡿⠷⢴⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⣿⣿⢧⣷⠶⣤⣜⣃⣀⣘⣤⠶⣶⣦⡹⣿⣿⣆⠈⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣵⣿⣿⠏⠀⠙⠚⠋⣿⣿⣿⣿⠙⠓⠋⠀⢻⡜⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠀⠀⠀⠀⠀⣯⢿⠿⠿⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠻⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⠦⠀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠈⠉⠀⠈⠉⠉⠉⠉⣠⠁⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡏⠀⠠⣤⣜⡓⠘⠰⣶⣂⣀⣀⠀⢠⠋⡄⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣰⠃⠀⠈⠉⠉⠀⠀⠀⠉⠉⠛⠋⠁⢸⠀⣇⣠⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣀⣀⠀⠀⢀⠀⢀⣀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠄⠒⠋⠉⠒⠶⠮⠜⠉⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠢⠤⠚⣇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡦⡠⠖⠋⠙⠶⠊⡇⠀⠀⠀⠀⠀⠀⠀⡶⢄⠜⠲⢄⡉⠙⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⠼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠒⠂⠤⠤⠤⠤⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡼⠉⠉⠉⢹⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⢤⡇⠀⠀⠀⠀⣇⡸⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠀⠀⠀⠀⠀⠓⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠾⠿⠛⠛⠛⠻⠿⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀
⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⣿⣆⠀⣈⠛⢷⣄⡀⠀⠀⠀⠀⠀⠘⣷
⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠟⠋⠁⠀⠀⠀⠻⠋⢸⣿⡆⠀⠉⠻⢶⣤⣀⠀⠀⠀⢹⣇
⢸⡏⠀⠀⣀⣀⣀⣠⣤⣴⡶⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠈⠙⠻⠶⢶⣶⣿
⣿⠟⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣿⠀⠀⠀⣀⠀⠀⠀⠠⡶⠾⠟⠛⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠻⠷⠂⠀⠀⠀⠀⣴⡄⠀⢸
⣿⠀⠀⢰⣿⡇⠀⠀⠀⢠⡶⠾⠛⠛⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠛⠿⠶⣶⠀⠀⠀⠀⢸⣟⣿⠀⢸
⢻⡇⠀⠹⣿⠟⠀⠀⠀⠸⣧⣀⣀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣀⣀⣠⡿⠀⠀⠀⠀⠈⠙⠋⠀⣿
⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠀⠀⠀⢰⡆⠀⠀⣶⠀⠀⠀⠈⠙⠋⠉⠀⠀⠀⠀⣤⡄⠀⠀⢠⡿
⠀⢻⡆⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⣾⠃
⠀⠈⢿⡄⠀⠀⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣼⠏
⠀⠀⠈⢿⣄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠻⠾⠟⠻⣶⠟⠻⡶⠟⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠈⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣀⣀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡶⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠦⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣠⠤⠐⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠨⠤⠒⢂⠀⣉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⣠⣤⣴⣯⣿⠿⠟⢡⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣥⣼⣿⣏⣈⠈⣍⣁⡠⠂⠑⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢰⠀⠐⡖⠘⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣤⣄⡙⢿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠙⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢠⠿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠀⠙⣿⡄⢿⡏⠙⠉⠁⠀⠀⠀⡀⠀⠀⠉⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⣿⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⢈⣿⣾⣷⡔⠀⠀⠀⠀⠀⢹⣷⣦⣄⣠⣎⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⡏⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡄⠀⠀⠉⠻⢿⣤⣀⣀⣰⣶⣶⠊⠀⠀⠈⠉⠙⠻⣿⣄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣥⣀⡀⠀⠀⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡇⣼⣷⣶⡶⢂⠀⠀⠀⠀⠀⠀⢀⡄⠀⢸⣴⡧⠼⢷⣆⠀⠆⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⠀⠀⣸⣿⠆⠀⠀⠙⣆⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢉⡽⣷⣿⣿⣦⣤⣄⣤⣴⣿⣿⣆⣄⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⣿⣿⣿⣿⣿⢿⠋⣿⣿⣿⣷⣦⡜⣿⣻⣿⣿⣿⣟⣻⣥⡟⣾⣿⣿⣿⣿⣿⣿⣇⣀⠂⠸
⠀⠀⠀⠀⠀⠀⠀⠂⠨⠥⣀⣄⣈⣿⣿⣿⣿⣁⢀⣼⣿⣿⣿⣿⣿⣾⣿⡿⣿⣿⣿⣿⣿⣿⣧⣿⣿⡿⢶⣿⣿⣿⠛⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⢿⣿⣛⢙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⡿⣿⡿⠠⣤⡙⢍⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠉⠒⠁⣼⣿⣿⣿⣿⣁⣰⣾⠆⠀⠛⢧⡄⠈⠻⢏⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣦⣌⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣴⣤⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⣄⠀⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣿⣏⣿⣿⣿⣿⣿⡙⡡⠤⠸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡛⠉⠀⠀⠒⠛⠛⠉⠉⠙⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠛⠉⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠉⠛⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⣾⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷
⠀⠀⠀⠀⣼⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧
⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀
⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇
⠀⠀⠀⠈⣿⣿⣿⡇⠀⠀⣀⣴⣾⣿⣿⣿⣿⠿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣷⣦⣀⠀⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⢹⣿⣿⣿⣶⣿⣿⣿⣿⡿⠟⠋⠁⠀⣿⣿⣿⣿⠀⠈⠙⠻⢿⣿⣿⣿⣿⣶⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣤⣀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣀⣤⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠓⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠄⠀⠀⠈⠁⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢀⣠⣶⡾⢀⠀⣀⣀⣼⣦⢹⣷⣤⡀⠀⢠⣶⣶⣶⣿⠁⠀⠀⠀⠀⠰⠾⠃⠠⠄
⠀⠀⠀⠀⠀⠀⣀⣠⡴⠛⠃⠀⢀⣀⣼⠿⢿⣿⣷⣾⣿⣿⣽⡿⢉⣿⣿⣿⣼⠇⢀⠙⢿⣿⣧⡄⢠⣴⣤⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠐⠈⠙⠋⣀⠀⣤⣴⣿⠟⠁⢠⣾⣿⣿⣿⣿⣿⣿⣆⣿⣿⣿⣿⣿⣒⣿⣷⣼⠛⢷⣥⣀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠉⠁⠀⠁⠀⢼⣿⠿⣿⣿⣧⣿⣿⣿⣿⣿⠿⠿⣿⠿⠿⣿⣟⡧⠀⠈⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠉⠋⢰⣿⡿⠟⠉⠛⠀⠀⠀⠀⠀⠀⠀⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠗⠀⠑⢂⡀⢶⣄⠀⠀⠀⠀⢀⣤⡴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦⡆⠀⠶⠴⠄⠀⠀⠄
⣸⡟⣛⠛⠻⣾⡋⠀⠐⣠⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠸⡿⣴⡄⣴⣷⡐⠀⢀⣠⡀
⠋⠈⠁⠀⢸⣿⡄⠰⠾⠿⠿⠶⠒⠦⠤⠤⠀⠀⠀⠀⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣘⣷⡄⠐⠠⢈⣠⣈⠃⣱⣤⠘⣿⣿
⠀⠀⠀⠐⠚⢻⣶⠶⠶⠶⠶⠶⠤⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⢠⡾⠿⠿⠿⠿⠷⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣧⣹⡷⣶⢟⣰⡿⣉⣏⠙⢀⣻⣿⠵
⠀⠀⠀⢀⡐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠂⠙⠙⣿⣇⠤⣼⢟⡇⠈⠄⠀⠀⣿⣿⣜
⠀⠰⠋⡛⠛⠿⠷⠶⠶⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⡠⠉⠙⠛⠛⠿⢶⣄⠀⠀⠀⠀⠀⠀⣀⡠⠀⠀⠀⠀⠈⠀⢹⣿⣿⡾⠟⠛⠒⠀⠠⠄⠿⠏⠁
⠀⠀⠞⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠀⠀⠀⠀⠀⠀⠀⠉⠆⡀⠀⠀⠐⠋⣉⠀⠀⠀⢀⣤⣶⣶⣧⣤⣤⡀
⢠⡾⠛⠾⠶⠶⠄⢀⣠⠾⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⣠⡞⠁⢀⣤⣶⣦⣤⣄⣀⣀⡀⠀⠈
⣼⣥⣀⠀⠀⢀⡴⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣤⣤⡴⠖⠂⠀⠀⠐⠒⠒⠀⢸⠟⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠉⠉⠙⣷⣦⣨⣤⣤⡴⠶⠀⢠⣴⠋⢠⣧⠂⠀⣠⣶⣿⡿⠟⠋⠁⠀⣀⣤⠤⠤⠀⠀⠀⠀⠀⠀⣀⣀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⠋⢁⣀⡀
⠀⠀⣼⡗⠈⠉⠁⠀⢀⡄⠰⠋⠀⠀⠈⠁⠀⠀⡙⠉⣿⠃⢀⡜⠀⠈⠁⠀⠀⠀⠀⠀⢠⣴⡶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠘⠛⠉⠉⠉
⠀⠀⠟⢀⣠⡴⠊⠀⣾⠀⠀⠀⢀⣁⠖⠁⠀⣾⣿⣆⠻⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠺⠁
⠠⢶⠾⠟⠉⠀⠀⠘⠃⣠⠄⠀⢹⠃⠀⢠⣿⡿⣿⣿⡀⠀⢀⡐⢃⠀⠀⠀⠀⠰⠿⢿⡿⠿⠿⠟⠛⠉⠁⢀⠀⠀⠀⠀⠀⢸⣥⣴⣴⠶⠛⠉⠀⢠⡀
⠀⠀⠀⠀⠀⠀⢠⠶⠟⠁⠀⠀⢀⣴⠎⢹⣿⠃⠸⣿⡇⠀⠀⠐⠉⠁⠀⠀⢶⣦⡀⣰⣇⡠⣤⣴⠶⣶⣶⠶⣶⣶⣶⡷⢶⣿⣅⣀⣀⣐⣄⣀⠀⠘⠁⠠⠀⢀
⠀⠀⣀⠀⣸⡄⣀⣴⣾⢤⠄⠀⠉⢹⡄⢸⣿⠀⠀⠙⣷⣼⠷⢀⣤⠀⠀⡀⠀⣈⡻⢿⣿⡇⠘⢿⣧⡈⠻⣧⡈⠛⠿⠷⢮⣉⠉⠉⣏⠹⣿⡏⠃⠀⠀⠀⠛⣾
⠀⠀⠀⡄⣹⡿⠋⠉⢩⣤⠀⠀⠀⢸⣿⣿⣿⣘⣀⠀⣟⠻⢶⣀⣿⡁⠀⠀⣸⣿⠇⣸⣿⠋⢦⠀⠻⢿⡄⣈⡿⢧⡴⢀⣤⣆⣿⣾⣧⣤⣬⣷⣤⣹⣆⠀⠀⢁
⣤⡆⣰⣏⣇⣄⠀⠀⣨⣽⣤⣤⡷⢺⢿⣷⣿⣿⣧⡀⢾⣶⣿⣹⣀⡂⢰⣾⣿⣿⣀⣽⣿⣠⠄⡀⣶⣄⡙⢠⣭⡴⣶⣼⣿⣿⣿⣿⣧⣬⣽⣿⣡⣥⣽⣠⣴⣿
⠛⣷⣿⣿⣷⣤⣶⡎⣹⣿⣿⣿⣷⣿⣤⣿⣿⣿⣽⣾⣬⣿⣷⣷⣿⣿⣾⣿⣿⣿⣯⣹⣿⡟⢴⣧⠛⢿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿
⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⡿⢦⣿⣿⠏⠁⢿⣻⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠘⣿⣿⣧⣿⠹⣿
⣿⣿⣿⣿⣷⣿⠟⠛⣿⠛⣿⣿⣿⣿⣿⠿⡟⢻⡿⠇⠘⠻⠛⢻⣿⣿⠃⠘⠃⣸⣿⣿⣤⣶⢸⣿⣿⣿⣿⠻⠛⣿⡇⠙⢿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠿⠁⣼⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⢀⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⢀⣼⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⣿⣿⣷⡀⠀⠀⣀⣀⣀⣀⣀⣀⠀⣀⣀
⠀⢿⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡿⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡆
⠀⠀⠀⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠙⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣏⠙⠉⠉⠀⠀⠀⢀⣠⣤⣤⣤⡀
⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⠿⠛⠛⢻⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⠿⣿⣿⡿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣦⣤⣼⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠀⠀⠙⢿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢧⡀⠀⢠⡆⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠈⠛⠿⠋⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣦⣀⠀⢀⡀⣠⣾⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠉⢉⡡⣿⣯⣸⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⣿⣿⡻⣿⠀⠈⣷⣿⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠀⠡⣟⣿⡇⣷⡿⠀⠈⢻⡟⢸⠸⣿⠿⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⠶⠶⣦⡉⠀⣿⣷⡄⣨⡿⠋⢉⣤⡶⠾⠿⢒⢀⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠈⠀⠳⠀⣹⡄⠙⠟⠛⣉⣴⠾⠛⣽⡆⠀⠀⣿⠈⣛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠂⠀⠀⠀⠀⠰⢰⣿⣷⠀⣠⣞⠟⠁⠀⠀⠘⣷⠀⠐⣯⠄⠁⠀⠁⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣴⢹⡇⢠⣿⠃⢀⠰⠞⡻⠇⣿⣿⠀⣽⡆⠀⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠠⠞⠃⣾⠁⣿⡏⠀⣮⡾⠉⣻⠶⣿⡛⢷⢻⡟⢃⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣅⡋⠈⠀⠀⠀⠀⠂⢠⠤⠄⢰⣿⣸⡿⠁⢰⠏⠀⣼⣿⣶⣿⣿⣿⣿⣷⡄⠁⠀⠰⠀⢴⠆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢠⡟⣥⡛⠃⠀⠀⠀⠀⠀⠐⢠⣦⣴⣿⣿⢻⣇⠀⢾⠀⠀⢩⣿⣿⣬⣟⡙⠛⣿⡙⢶⠀⠀⠀⠤⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠠⠘⠿⠤⠁⠀⠀⠀⠈⠀⣴⢶⣿⢻⣿⠿⠁⢠⣾⡄⢸⠀⡄⢈⣿⣿⣿⣿⣿⣶⣌⣷⠄⢠⡀⠀⠴⢮⠁⡤⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⣷⣆⣠⣾⠁⠀⠀⠀⣤⡇⠘⠀⠈⠏⠀⠿⢿⡿⣿⣿⣿⣇⠘⢠⡄⠀⠀⠠⠖⠏⢰⣹⡇
⠀⠀⠀⠀⣰⠎⠀⠐⠀⠘⠇⠀⠀⠀⢀⣠⡿⣿⣿⣿⣦⣀⣛⣠⡾⠃⡀⠀⠀⣠⠀⠀⠈⠛⠊⢱⣙⣿⡿⢲⣇⣀⢁⣀⡆⣀⣐⣀⣷
⠀⠀⠀⠀⣼⣂⠈⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠟⣿⠏⣩⡟⠛⣡⠄⠀⠀⢈⣠⣤⡆⠀⠂⠆⢸⡛⢻⣋⣻⠛⣉⣇⠈⢡⡈⠍⠹⠋⢡
⠀⠀⠀⢸⣯⡅⠒⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣧⣲⡽⣧⡂⠶⠖⣀⠈⠰⣯⠚⣟⣿⠇⠀⠀⠀⠀⠉⢻⣿⣿⣿⣼⣏⣟⣠⠛⠂⢄⠒⠫⣦
⠀⠀⢀⣾⡿⡟⠀⠀⠀⠀⠀⠀⠀⣿⣤⣽⣿⣿⣟⣿⣿⣧⡀⢀⡘⠂⠀⠃⠁⢘⣿⠂⠀⠀⠀⠀⠚⣿⣿⣿⣽⣟⣿⣍⡿⢳⣶⡟⢳⣾⠿
⠀⠀⣿⣿⠇⠻⣆⠀⠀⠀⠀⠀⠀⢉⣹⣿⣿⣟⣭⣿⣧⣿⣿⣤⠉⠁⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⣹⣿⠟⢿⣿⣿⣿⡃⢸⠿⠞⠉⢀⣾⡄
⠀⢠⣿⣿⡰⠈⢿⡄⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⡓⡾⠿⣟⣿⣿⣆⡆⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣹⡟⠀⠀⠀⢬⣿⡇⠀⠀⠀⠀⣾⠻⡇
⠀⢸⣿⡿⠁⠠⠘⣷⡀⠀⠀⢀⠀⠘⢿⣿⣿⣿⡹⠅⣤⣹⡟⠙⠻⣿⣿⣷⣦⣾⣷⣶⣶⣶⡄⢠⣤⣿⠁⠀⠀⠀⢺⠿⠁⠀⠤⣂⣾⡏⣿⣿⡆
⢠⣾⣿⣿⡤⠁⠀⢿⡧⠀⠀⠀⠤⢤⣿⣿⣿⣿⣓⣾⣿⡿⢡⣀⣄⠈⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⢩⠀⠄⢹⢵⣿⠆⠀⠀⢀⣿⡟⢂⣼⢻⡇
⠈⣿⣿⡞⠀⡄⠀⠘⢣⡛⠀⣀⠀⢀⣿⣿⣿⣯⣼⣿⠉⢀⣴⣿⣿⡔⠂⠀⠠⣾⡩⣛⠿⣿⣿⣿⢿⣦⠴⣖⣌⣿⡏⠀⠀⣿⡿⠋⠉⠁⠉⣳⠄
⢠⣿⡿⣷⠟⠀⠀⣀⠘⣇⠀⣤⣴⣿⣿⣿⢟⡋⠛⠀⣴⡟⠁⣠⠄⠀⠀⠀⢠⢼⣿⣿⣤⡀⢀⠘⢣⣽⣷⣿⣼⣿⣷⡾⠟⠛⢂⣀⠀⠙⣳⢿⣇
⣺⣿⣿⣦⠋⠀⠒⠀⠀⠹⣦⠆⣿⡿⠿⠉⠙⠁⠀⠀⠉⠁⠞⠁⠀⠀⠀⠀⠀⠈⢿⣆⠙⠿⣿⣷⣼⣿⣿⣿⣿⣿⣿⠔⠀⠀⠀⠀⢀⣈⠍⠳⣿
⣿⣿⡿⢿⡏⠃⠸⡋⠀⡄⠹⣾⡿⠁⢀⣴⠃⣴⣤⣶⡔⠁⠀⠀⠀⠀⠀⠀⠉⠀⠈⢿⣷⣶⣿⡟⢿⡏⢹⣿⣿⣿⡯⠛⠲⡀⠀⠀⠄⠁⠰⣦⡿
⢻⣿⣷⡟⣗⣠⡾⠡⡔⢄⣰⡟⢁⠴⠟⠁⢠⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⡆⠀⠹⣿⠿⣿⣿⣄⣀⣀⠷⣲⣆⣠⣟⣊⣹⡷
⢺⣿⣿⣯⣿⣷⣾⣿⣿⣛⡿⢱⠊⠀⠀⠀⣰⠄⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⣹⣄⢀⠀⠀⣸⢿⣿⣿⣿⡆⠸⠤⠞⢿⢿⣿⣿
⢸⣿⣿⣿⣿⣿⡿⠟⠉⠉⠀⠀⠠⡄⠀⠀⠁⠀⢂⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣼⡉⠛⠿⣧⣿⡿⣧⣭⣿⠿⣿
⢸⣿⣿⣿⠙⠋⠁⠀⣀⡀⣰⣿⡁⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠻⠿⡿⣿⣿⣶⣀⣹⣿⣿⣿⣷⣯⣿⣿
⠀⣿⣿⣉⣥⣶⣤⣤⣿⣿⣿⡿⢃⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣙⣿⣿⣿⣿⣿⣿⣿⣿⡟⣹⡏
⠀⢹⡟⠻⢿⣿⣿⣿⠿⠛⠩⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠻⣿⣿⣤⣿⣿⠟⠋⢰⡿
⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠢⠉⡉⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣖⢐⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠟⠁⠘⠿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⠀⠙⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⢂⣴⣯⣵⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⡟⠀⣶⣿⣿⣽⣦⣀⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣾⡟⢃⣄⣯⣾⣿⡿⣿⣿⣟⣿⣿⡇⠀⠀⠀⠀⠀⢸⢁⣾⡟⣿⣿⣽⡿⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣼⡿⣢⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣾⣿⢷⡽⣱⣿⣾⠟⣿⣷⣽⣿⣿⣆
⠀⠀⠀⠀⠀⣼⡿⠁⢺⣈⢿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠘⠻⠷⠋⢀⣯⣤⣶⣿⣿⠿⠛⡙⢻⣿⣆
⠀⠀⠀⠀⣾⡿⠀⣠⣾⣿⣆⣸⣿⣿⡽⠩⢁⣤⠟⠋⠀⣀⡄⢐⡢⠄⣄⡀⠀⠺⣿⣿⣿⣿⠟⣁⣠⣾⣿⣆⢹⣿⣧⡀
⠀⠀⠀⣼⣿⠀⣼⣿⡿⢿⠛⠿⢿⣟⣀⡼⠃⠀⣀⣴⣾⠟⠉⠁⠀⠀⣀⣈⣳⣄⠈⠻⣿⣿⣶⢿⣟⣿⣿⣿⡜⢿⣿⣷⡀
⠀⠀⣸⣿⠇⢰⣿⣿⣿⣿⣿⣿⡷⠌⢁⠀⣠⣾⡏⠀⠈⠁⠀⠀⠀⠉⠉⠉⠁⠌⣷⣄⡼⣯⣽⣿⣟⣟⡺⢿⣷⣼⣿⣿⣧
⠀⢠⣿⠇⠀⠀⣿⣿⣿⣿⠟⢃⣤⣶⣟⣶⠍⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣩⣴⡶⢶⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡆
⠀⣸⣷⠂⠀⠀⣿⣟⣯⣴⢣⣾⣿⡟⢣⣾⠟⣩⣶⠞⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣶⣬⣻⢷⡄⠙⠻⢷⣿⣿⣯⡍⢸⣿⣿⣧
⠀⣿⡯⠀⠀⠀⣿⣿⣿⣧⣿⢟⣵⡇⢸⡟⣴⡿⠷⢤⣀⡀⠀⠀⠀⠀⠀⢠⢃⣹⣿⣿⡸⣷⣽⡆⣦⡙⣷⣦⣾⣿⣸⣿⣿⣿⡄
⢰⣿⡷⠀⠀⠀⠞⣿⣿⣿⣿⣿⣿⣲⣿⣿⣿⠁⠀⠐⠂⠀⠀⠀⠐⠐⡷⢆⣸⣿⣿⣿⣯⣿⡿⡇⣽⣷⡈⣝⠿⣄⢙⣿⣿⣿⣇
⢸⣿⣿⠆⠀⠀⢸⢻⣿⡿⢻⣷⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠙⠛⣿⡏⣿⣿⣿⢣⣿⣿⣧⠹⣿⡗⢹⣿⣿⣿⣿
⢸⣿⣿⡅⠀⠀⠈⣼⣿⣿⡿⣾⣿⣿⣿⣿⡿⠀⠀⡀⠀⠀⠀⠀⢀⠎⠀⠀⠀⣼⣿⣇⠈⢿⣏⡦⣿⣿⣿⢹⡯⣻⣾⣿⣿⣿⣿
⠘⣿⣿⣷⡄⠀⠀⠈⢪⣿⣽⡯⣿⠟⠉⣸⣧⠀⠀⠀⠀⠀⡀⢀⠎⠀⢀⣠⣾⣿⣿⡈⠀⠀⠈⠛⠻⠛⠿⣈⣱⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⡇⠀⠀⠀⠀⣳⣽⣿⣏⣤⣾⣿⣿⣷⣶⣴⣶⡦⠊⠥⠀⠐⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢠⣾⣾⣿⣿⣿⣿⣿
⠀⢻⣿⣿⣿⠄⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿
⠀⠘⢿⣿⣿⣦⢀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡧⠀⠀⠀⠀⠀⠀⡀⣸⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠈⠛⢿⣯⣿⡟⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣟⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠉⠻⢷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠦⢀⡭⠾⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⢀⣼⣿⡟⢀⡤⠂⠀⠀⠀⣠⠄⠀⠀⠀⠀⣠⡤⠖⠀⠀⠀⠀⢐⣿⢂⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⣀⣴⡿⠟⠋⠀⠈⠛⢉⠗⠀⢰⣿⠀⠀⠀⢀⣾⠋⠀⠀⣶⡀⠀⠀⡼⠋⣿⠃⢀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠈⠛⠂⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⠘⠁⠀⠈
⠀⠀⠀⠈⠉⠛⠋⠉




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿