Horse Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⢀⡀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣅⠀⠀⠸⠙⢆⠈⠙⢦⣈⠓⢦⣀
⠀⠀⠀⠀⠀⢀⡴⠋⠙⠦⠁⢀⠀⠀⠀⠀⡀⢀⡉⠙⠦⣍⠇
⠀⠀⠀⠀⠀⢾⠋⠀⠀⢀⣠⠎⠀⠀⣀⠀⠙⣿⣿⣦⠀⠈⢇⠢⡀
⠀⠀⠀⠀⠀⢸⠃⠰⣳⠋⠁⠀⠀⠀⠛⠀⠀⣿⢿⣿⡆⠀⠈⢀⡇
⠀⠀⠀⠀⢀⡟⠀⠸⡏⠀⠈⠄⠀⠀⢠⠀⣠⣿⠸⣿⡇⠀⠀⠀⠙⡄
⠀⠀⠀⠀⢸⠀⣾⠀⠱⡀⠀⡄⠀⠀⠀⣸⠏⣿⠀⡿⠁⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠘⢦⣧⢢⠀⠅⠀⡅⢰⡄⠀⣿⠀⢹⠀⠀⠀⠀⣀⡠⠞⠉⠢⢄⢀
⠀⠀⠀⠀⠀⠀⠈⠓⡇⠁⠀⠘⠿⣀⣴⣿⡆⣸⠀⢀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠐⠒⠒⠦⣄⡀
⠀⠀⠀⠀⣠⠴⠶⠾⠁⠀⠀⠀⠀⣿⣿⣿⠁⢁⠴⠁⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣄
⠀⠀⠀⢸⡡⢒⡶⠒⠀⠀⠀⠀⢀⣿⡿⠁⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠘⣿⣦
⠀⠀⢠⣾⡷⣋⢀⢔⡤⠀⠄⠐⢁⢿⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⣦⠀⠀⠀⠀⠀⠉⢿⡇
⠀⠀⢠⣿⢞⡏⡾⢁⠤⠤⠐⠒⠁⣠⣿⣸⣆⣀⡘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠂⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠈⠿⣾⠀⠀⢇⡀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⠟⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⢸⡿⠦⣤⣀⡀
⠀⠀⠀⠀⠹⡆⢰⠟⠀⠀⠄⠒⠁⠘⠛⠛⠛⠁⡸⠋⠀⠀⠀⢀⣴⣦⣄⣀⣤⣤⣤⣤⣄⣀⣤⠴⣿⣷⣾⣦⡀⠀⠀⠀⠀⠾⡇⠀⠀⠈⠙⢶⡄
⠀⠀⠀⠀⠀⣹⢸⠀⠐⠀⠀⠀⠀⠀⠀⠀⣠⠎⢀⡀⠄⠂⠘⠛⠛⠛⠛⠛⠛⠛⠉⠻⣿⣷⣶⣿⣿⣿⡿⢀⡍⢦⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠘⠶⠾⠁⠀⡆⠠⠀⠀⠀⠀⠀⠀⣸⡅⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠏⠀⢸⡇⠀⠳⣤⡴⠀⡄⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⡁⠀⡀⠀⠀⠀⠀⠀⡛⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⠻⣿⡄⠀⠀⢧⠀⠀⠹⡇⠀⠀⠀⠀⠀⠀⢠⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠙⢿⠃⣶⡆⠀⠀⢠⠁⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠙⣷⠀⠀⠈⢧⡀⠀⣿⡀⠀⠂⠀⠀⠀⠸⡧⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡇⠀⠀⠃⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠘⣷⠀⠀⠈⢳⡀⢻⡇⠀⠀⠀⠀⠀⠀⢻⡌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠿⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢾⡟⠀⠀⠀⠀⢻⡄⢿⠃⡀⠀⠀⣼⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡞⠀⠀⠀⠀⠀⠀⣷⢸⠀⠇⠀⠀⣽⢷⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⡾⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⣿⢸⣧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⢷⣴⡇⠀⠀⠀⠀⠀⢀⡴⠃⠘⣦⣼⠃⣰⢃⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⣁⣀⡇⠀⠀⣀⡤⠤⠴⠚⠉⠀⠀⢰⣯⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⢁⡴⠻⠒⠒⠒⠒⠲⠶⠖⠚⠻⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿
⠀⠀⠀⠀⠀⢰⡄⢸⣆⣀⡀
⠀⠀⠀⠀⠀⢀⣿⠛⢷⣿⡿⣾⣿⣷⣶⣖⣂⡀
⠀⠀⠀⠀⢰⠟⠛⠂⠀⠀⠀⠈⠙⢿⣿⣿⣵⡤⠄
⠀⠀⢀⡔⠁⠀⢠⠀⠀⡀⠀⠀⠀⠀⠙⢿⣿⡿⠃
⠀⠀⣯⡖⢀⡤⠐⠓⠐⣿⡀⠀⠀⠀⠀⠀⢻⣿⣷⡄
⠀⠀⠈⠛⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠈⠉⠙⠃⠀⠀⢀⣀⣀⣀⣀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣷⣥⡀⠀⠀⠀⠀⠀⠀⠇⠀⠻⠿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣿⣷⣦⣴⣇⡀⢀⣀⣀⣤⣤⣶⡿⠿⠿⠛⠛⠛⢿⣿⣦⡀⠀⠀⠀⢠⡆⠀⠀⠀⠈⠛⠿⢿⠷⡤
⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠈⠉⠉⠙⠋⠁⠉⠻⣿⣿⡅⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⢦⡀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠀⠤⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢧⣀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠁⠀⠀⠀⠈⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠹⣿⡄
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀
⡛⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁
⣿⣦⣈⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃
⣿⢿⢿⢿⠿⠷⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣷⢶⣶⣶⣶⡶⢶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢰⣄⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢖⣺⠟⠉⠈⠀⠉⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡠⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣔⣻⡿⠋⠀⡀⠈⠀⢰⡇⠀⠀⠀⠠⢤⡽⠛⠋⠀⠀⠀⠉⠙⠀⠀⢀⣀⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⢀⡤⠖⠛⠛⠲⠶⠤⠬⠽⠟⠋⠀⠀⠀⢻⣄⠀⠘⣇⠤⠶⠖⠂⠀⠀⠀⠀⠀⣄⠀⠠⠀⠁⠈⠉⠉⠉⠀⠀⠀⠘⠷⠄⠃
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸⠈⢧⠀⡹⠀⠀⠀⠀⡀⠀⠀⠀⣼⠉⢣⡀⠀⡆⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠙
⠀⠀⠀⣀⣾⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡆⢸⠆⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⢰⠃⠀⠀⠹⡄⡙⠀⠀⣠⢆⡶⠋⠉⠧⣄⠙⠀⡆
⢀⡤⢞⣭⢞⣿⡄⠈⢷⣄⠀⠀⠀⠳⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⡀⠀⠀⠀⠀⡞⠁⡉⠀⠀⠀⠀⠈⢃⢀⢹
⡞⡟⠉⠐⠉⠀⠉⢀⠀⢹⡟⠲⠤⠠⡄⠀⠀⢠⡆⠀⠐⢦⣠⣤⣀⠀⡄⠀⠀⣦⣬⠞⣙⠒⠺⡇⠀⠀⢀⡄⠠⠾⢧⣀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⣷⡼⠀⡄⠻⣄⠀⣿⠈⠛⠲⠦⡉⠃⠀⠀⠳⣄⠀⣿⠙⢦⡘⢦⠀⢷⠀⢰⠋⠀⠈⠓⠲⡌⢙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠘⡇⢸⠁⠀⠙⢦⠘⣆⠀⠀⢰⠁⡇⠀⠀⠀⠈⠃⢸⡆⠀⠈⢈⠁⠘⡆⢸⠀⠀⠠⢤⠜⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢷⣈⢣⠀⠀⠈⡇⣼⠀⣀⣨⢀⡇⠀⠀⣀⡔⣠⠏⢀⣀⡄⣸⠀⢰⢃⡞⠀⠀⠈⠑⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⣿⠸⡇⠀⢀⠁⡿⠘⡧⠖⠋⠀⠀⠘⠵⠖⠁⠀⠨⠖⠋⢁⡄⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⠣⠁⠀⣇⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣳⣤⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⢀⣤⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣤⣤⣄⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢠⣿⣿⣿⣿⣿⢿⣿⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠟⠛⠉⠁⠀⠀⢹⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⣰⡀⠀⠀⠀⠀⠀⢠⣦⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠸⣿⣇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠢
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⢿⡄⡄⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠛⢿⣦⣴⣶⣴⡇⣿⣿⣿⣿⣿⣦⡙⠢⠄
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣴⣿⣿⣿⠿⠿⢧⡙⠓⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠻⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣀⣀⣀⣠⡤⠐⠀⣀⣠⣎⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣽⣤⣶⣶⣿⣿⡿⠋⡀
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⣊⣁⣀⣀⡀
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠘⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⣾⣿⣿⣿⡈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣀⣀⠀⣾⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣀⠀⢀⣼⣿⣿⣿⣿⡇⠀⠙⢌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠙⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡉
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⡘⠉⠀⠈⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠮⠙⠿⡟⠛⢿⣯⡛⠛⠿⡿⢿⣯⡝
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣷⡄⠀⠀⠀⢰⡇⠀⠀⠀⠀⠻⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠋⠀⢹⣿⣿⣿⣿⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠀⠀⢿⣿⣿⣿⢠⣆⠀⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠻⣿⣿⣼⣿⡀⡿⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⠀⠀⠈⢿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣆⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠀⠀⠘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣶⣤⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣧⣸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⠋⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡀
⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⡄
⣿⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⣿⣦⡀⠱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠋
⣿⣾⣷⣄⠀⠑⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡿⠿⢛⣋⣉⠉⠉⠉⠙⠻⢷⣶⣿⠀⠀⣠⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⣋⣥⣶⣿⠟⠉⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⢟⣥⣾⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⠟⣵⣿⣿⣿⣿⣿⣿⡷⠆⠀⠀⠀⠀⣀⠀⠀⠁⢀⣤⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⢛⣩⣿⣿⠟⢡⣾⣿⡿⠿⠛⠛⠛⠛⠂⠀⠀⢀⣠⡞⢁⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⠿⠋⢠⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⣰⡏⠀⠀⠀⠺⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠃⣰⣿⡇⠀⠀⠀⠸⣿⠈⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠃⣰⣿⣿⣷⡀⠀⣀⠀⢿⠀⠈⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡟⢰⣿⣿⣿⣿⣷⠀⢿⣆⠈⠆⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⡟⠀⠀⠙⢧⡀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠉⠉⠛⠿⠟⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⢳⡀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢻⣿⣿⣿⣿⣿⣿⠟⠁⣠⣄⠀⠈⠃⠸⣧⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠀⣰⣿⣿⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣼⣧⣀⣀⣤⣤⣶⣶⣦⢻⣿⣿⡿⢸⣿⣿⣿⣧⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠁⣿⣿⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢻⣿⡟⠛⣋⣵⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⣿⡏⠀⣿⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢈⣿⣷⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠋⣿⣿⠃⠀⣿⠇⠀⣿⣇⢠⣿⣿⣷⣾⣷⣦⣤⣤⣤⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⢸⣿⣿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⢠⡿⠀⠀⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢀⣴⣦⠀⢠⣤⣿⡿⠁
⠀⠀⠀⠀⠀⢠⣿⣿⣿⡆⢸⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣼⣿⣿⣆⣠⠿⠛⠁
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣇⢸⠃⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠛⠋⠉
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠀⢰⣿⣿⣿⠏⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⣴⣿⣿⣿⣿⡿⠃⠀⠀⢠⣿⣿⠟⠁⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡿⠁
⠀⢀⡼⢿⣿⣿⡿⠋⡠⠀⠀⢠⣿⠟⠁⠀⠀⠀⠀⠀⡀⢸⣿⣿⣿⣿⠏
⠀⢉⣼⣿⡿⠋⠀⢸⠇⠀⣠⡿⠃⠀⠀⠀⠀⠀⢀⣼⠇⠀⢿⣿⡿⠋
⢠⣿⠟⠁⠀⠀⠀⠘⠀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠨⠋⢀⣠⣾⠟⠁
⡿⠋⠀⠀⠀⠀⠀⢸⠃⣿⡏⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠁
⠃⠀⠀⠀⠀⠀⠀⠈⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⢼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣠⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⣿⣆⣰⣿⠃⠀⠀⠀⠀⢀⣴⠀⢠⣾⡿⠁
⠀⠀⠀⠀⢻⣿⠟⠁⠀⠀⠀⠀⠀⢸⣿⣧⣼⠟

⠀⠀⠀⠀⠀⠀⢠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂
⠀⠀⠀⠀⠀⠀⢠⣶⣆⠀⠀⠀⠀⠀⣠⣴⣶⣷⣿⡟⠛⠛⠛⠓⠲⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢈⣿⣿⣷⣄⣤⣴⡿⠋⠁⣿⢻⣿⡇⠀⠀⢀⣢⣶⣦⣰⣬⣙⢷⣦⡀
⠀⠀⠀⠀⣠⢞⡿⠛⣯⣻⣿⣿⡟⠀⢼⠀⠙⣿⣿⡇⠀⠀⠀⠉⠻⣿⣿⢿⣿⢷⣬⣿⣦⡀
⠀⠀⠀⠈⠁⠃⣠⠞⠻⠛⣫⣿⣇⠀⠈⢻⣀⢀⠙⢷⣄⡀⠀⠀⠀⠙⣿⣆⠻⣇⠙⢿⣿⣿⣄
⠀⠀⠀⠀⠀⢰⣿⣠⡴⠛⢡⣿⡘⣄⣄⠈⠻⣾⣶⡄⢈⠛⢦⣣⡰⡄⠘⣿⡄⠹⡄⠀⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⢸⣧⠳⣌⠻⢦⣀⠀⠙⠷⣌⢷⣄⠻⣧⢻⡄⢸⣷⠀⠀⠀⡀⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠁⠙⠚⠷⣤⣿⣽⣒⠤⡈⠙⠿⣷⣻⡞⣧⠀⣿⡀⣧⠀⢷⠸⣿⣿⢿⡀
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣯⠛⠮⣑⣄⠘⢿⣷⣿⠀⣿⡇⣿⡇⢸⡇⣿⣿⣇⢃
⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠻⣿⢿⠈⢧⠀⠈⢿⡆⠸⣿⣿⠀⣽⣇⣿⡇⢸⣷⣿⣿⣿⡎
⠀⠀⠀⠀⠀⠀⠈⡇⠐⡀⠀⠀⠀⠀⠀⠀⢴⠆⠀⢸⡆⠀⠘⣿⠀⣿⣿⠀⣿⢿⣿⣧⡾⢻⡏⣿⣿⢻⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣿⢸⣿⠇⢰⡏⡼⠁⣿⠃⠸⢹⣿⣿⠸⠃
⠀⠀⠀⠀⠀⠀⠀⢹⡄⢸⠀⢀⡀⠀⠀⠀⠀⠀⢀⣿⠀⡀⢸⣿⣿⡟⢀⡿⠀⠀⠀⠋⡆⢠⣿⣿⡿⢸
⠀⠀⠀⠀⠀⠀⠀⠈⣷⣸⠀⢸⡇⠀⢠⠀⢠⡶⢫⡇⣰⠀⢞⣿⡿⢁⣾⠃⡀⢠⠀⣸⣣⢟⣿⣿⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⢸⡇⠀⣾⠀⡞⠀⣾⠀⡇⠀⣼⣿⢁⣾⠃⡼⢀⣾⣴⡿⢫⣾⣿⣿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⠀⠈⣷⠀⢸⡜⠀⢸⡇⠀⡇⢸⣿⢣⣾⢏⡾⣡⣿⡿⢋⣴⠟⣽⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⣇⠀⣾⡇⠀⣇⣿⣇⣿⣫⣾⣿⠟⢋⡴⠟⢁⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡷⣄⠀⠀⢀⡿⠷⠏⠀⢿⡇⢁⣿⣿⣿⣿⠿⠋⠁⠀⠉⠀⣠⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡘⠀⠀⠘⠀⣠⠆⠀⢸⡇⡘⣿⣿⠋⠀⠀⠀⠀⡠⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠦⠴⠟⠉⠀⠀⢠⣷⣰⣻⣇⣴⢎⢀⡴⢋⣴⢋⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠘⣧⣿⣿⣷⡷⢋⣴⠟⣱⠋⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⠜⣿⣿⣯⡴⠛⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⣆⣌⡿⣿⣤⡞⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣰⣿⡏⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⣿⣿⣿⣿⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠉⣿⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠈⣿⣦⡀
⠀⠀⠀⠀⠀⣿⣟⡆⠀⠀⠀⣿⣿⢿⡄⠤⠖⠂⠀⠀⠐⠲⠤⠀⡀⠀⠀⢀⣀⡠⠄
⠀⠀⠀⠀⠀⠹⣿⡏⠃⠀⠀⢻⣿⡞⣧⠀⢀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⢹⡟⠄⠀⠀⠀⠙⠷⠏⠀⠀⠉⠳⠦⠤⠤⠛⠓⠒⢤⡀⠀⠀⠀⠀⠠⠀⠡⣀⡤⠖
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣀⡀⠀⠈⠓⣆⡀⠀⠐⢂⣹⠷⠊⠁
⠀⠀⠀⠀⠀⠈⠀⠀⣀⠀⠀⠀⠀⠀⠀⡉⠉⠉⠀⠉⠻⣄⠀⠛⠙⠿⠷⠶⠟⠿⣿⣿⣿⣷⣼⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⡳⣄⡀⠀⠐⠤⡀⠀⢄⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⡆⢹⠳⣄⡀
⠀⠀⠀⠀⣿⣦⠀⠛⠋⠛⠁⡛⠛⠣⢄⠀⠀⠉⠢⣿⣾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡇⠸⠃⢸⠓⡄⠀⠈⢄⠀⠀⠐⠂⢤⡀
⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⣶⣶⣾⡗⠀⢲⣼⣾⣿⣿⠻⢷⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⢷⡜⠀⠀⠀⠄⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢸⡇⠀⠘⠃⠀⠀⠀⠻⢿⣿⠇⣦⢠⣼⣿⣿⣿⣷⡀⠈⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠁⠀⠀⠀⠈⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠹⡄
⠀⠀⠀⠀⢸⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣾⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠑⡄⠀⢰
⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⣀⡀⢁⢀⠀⢀⠈⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢠⠏⢠⡀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⢸⣿⣿⣿⣿⣿⣿⠻⢿⣷⣡⣄⣴⡈⣰⡏⠀⠀⡠⠀⠀⠀⡠⠂⡴⠁⢠⠀⠃⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠂⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠙⣿⡛⢳⣴⣿⠀⠀⠊⠀⠀⢀⠔⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⣆⣴⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠄⠀⠀⠀⠠⠊⠀⠀⠐⠋⠄⢀⡤
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⣾⠟⣩⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⡔⠁⠀⠐⠁⠀⢀⣴⡿⠋⠀⠀⠀⠀⢀
⠀⠀⠁⠀⠀⢤⡄⠀⠀⠀⠞⢋⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃⠀⠀⠀⠈⠀⠀⠀⠀⣠⣾⡿⠋⣀⠄⠀⠀⢀⠔⠁
⡰⠃⠀⠀⠀⠙⠰⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠎⠀⠀⠀⠀⠀⠀⠀⡤⠚⠋⣁⠴⠊⠁⠀⢀⡴⠃
⠑⡀⠀⡼⣷⣶⡀⡆⣄⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⢂⡤⠊⠀⠀⠀⠀⠀⠋
⠀⢷⣿⣆⣿⠿⠿⣧⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠙⢿⣿⣷⣶⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣃⣉⣋⣀⣏⣉⣯⣋⣭⣟⣽⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢐⣙⣿⡚⢉⣽⣷⣾⠿⠶⣿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣷⣮⣽⣿⠟⠋⠁⢀⣀⡤⢚⠉⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⣿⣿⣿⣟⣻⠟⠁⠈⣠⠾⣫⣿⠀⠀⠟⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣽⣿⠿⠛⠁⠀⢠⡾⢁⣾⣿⠋⠳⢴⣗⠇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡠⢔⢒⡒⠀⠒⠒⠒⠚⠁⠀⠀⢤⣴⠟⢠⡿⣼⠇⠀⠀⠀⠙⢄⢠⣤
⠀⠀⠀⢀⣲⣌⣳⣤⣴⣊⣿⣿⠋⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢼⣧⣿⣆⠀⠀⠀⠀⠈⠛⠋
⠀⠀⠀⢀⡈⠻⢿⡿⢁⣿⣿⠇⠀⡎⡇⣇⠀⠀⠀⠀⠀⢀⢀⠁⠰⢦⡀⣇⢸⣿⣿⣿⡆
⠀⠀⠀⠈⢽⣿⣑⣺⣿⣿⣿⡆⠀⠣⢻⣜⣷⣤⣀⡀⣀⣸⣿⣷⡄⠀⣠⣿⣼⣯⣿⣿⣤⣀
⠀⠀⠀⠀⠠⣬⣭⣛⣙⣾⡿⠛⢷⣤⣄⣙⢿⣿⣿⣿⣿⣿⣿⣿⣷⡴⡃⠀⢌⠛⢿⣿⣿⣶⣷⣶⢤
⠀⠀⠀⠀⠀⠐⠛⠟⠁⠀⠀⠀⠀⠈⢙⣤⣀⢹⡿⣃⠀⠀⠀⠀⠀⠈⠋⠑⠒⠛⠓⠯⣟⣦⠀⠉⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⢿⣌⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⣹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢍⣳⣄⠀⠀⠀⠀⠀⠀⠀⢰⣶⠇⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠬⣷⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠤⡄⠀⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣭⡙⠿⢿⣦⣿⣯⣻⣿⣶⣤⣤⣄⣤⣤⣶⣦⣤⣤⣀⣴⣿⣿⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⣀⠘⠛⠿⣧⣿⣶⣤⣬⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿
⠀⠀⠀⠀⠀⣠⣤⣤⠶⠄⠀⠈⠉⠿⣿⣷⣦⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⣠⠖⠈⠉⠀⠘⠒⠲⢶⣿⣧⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠠⢀⣴⣶⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠖⠛⣛⣩⣭⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣪⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷
⠀⠀⣠⣾⣿⣿⢟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢿⣿⣿⠇
⠀⠀⠹⠛⢉⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡏⠀⢾⣿⣿⡆
⠀⠀⠀⣠⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣷⣄⡄⠀⢿⣿⠀⢇
⠀⠀⠀⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠙⢿⣿⠀⠈⢻⣆⠈⡀
⢀⣤⡾⠛⢙⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⢹⣧⣄⠈⡋⠀⢱
⠀⠁⠴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣷⣇⡀⢀⢷⡄
⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠙⠻⢿⣿⣿⣿⣿⣿⣆⠹⣿⣿⣏⢻⣷⣄⠾⣿⡀
⠀⢀⣋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡷⠈⠙⣿⣦⣿⣿⡂⠹⣷
⠀⠀⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣼⣿⠂
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⣼⣿⣿⣻⣿⡿⠃
⠀⠀⠀⠀⡟⠻⠿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠁⠀⠛⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠐⠀⠀⠀⠀⠀⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⠀⠘⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣱⡿
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠈⠛⠙⣿⣿⣿⣿⣿⢯⡿⢿⣿⣿⣿⣿⣿⡏⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠱⠜⢀⡑⠝⠫⠝⠻⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⠀⠀⠀⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣇⠀⢠⣶⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡄⠀⠀⠀⠀⠀⠀⠀⢀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⢆⡞⢿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣽⣿⣿⣧⢼⣿⠗⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⣴⣶⣶⣤⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣴⢊⣵⣟⣵⣾⡿⣋⢠⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣿⣿⣿⣿⣿⣿⣏⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡿⣿⡿⢉⢹⡿⣠⣆⡀⡀
⠀⠀⠀⠀⠀⠀⢀⣼⠟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢰⠯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡇⣿⣿⣁⡛⡍⢀⣃⡤
⠀⠀⠀⠀⣴⣶⣿⠃⢸⡇⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⡄⠀⠀⠀⠀⢰⣃⣸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣀⣋⣱⠦
⠀⠀⣤⣾⣿⡟⠁⠀⣼⣷⣿⣿⣿⣿⣿⣿⣏⡀⢀⣷⣿⡂⠀⠀⠀⢸⣿⣿⣿⡄⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡽⡵⣊
⣤⣿⣿⡿⠁⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣼⣿⣿⣿⣤⣿⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠋⠉⣵⣴⣯⠞⢹⣿⣿⣯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣰⣿⠟⢹⠏⠉⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣷
⣩⣶⣿⣿⣿⠟⠀⢻⣿⣿⣿⣽⡿⣿⣿⡿⢿⣿⡿⢻⣿⠀⠀⣴⣿⠛⣠⣿⡇⢰⣿⣿⣿⣟⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⣿⣿⣿⡿⠀⠀⢸⣻⣿⣿⣿⢷⣼⠇⡹⡢⠘⣇⠀⡃⠀⡐⣈⣿⣴⣟⣿⣀⢺⣿⣿⣿⣿⣿⣿⣿⠿⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢈⣵⡃⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⠀⣱⣆⠛⠰⣷⣴⣿⣿⡿⣩⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠙⢷⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋
⠀⢸⣿⡇⠀⠀⠀⣾⣿⣿⣿⣿⣿⡟⢿⢿⠀⠀⡀⠀⠸⣻⣿⣿⣿⣿⣾⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⢠⣿⣿⣷⠀⠀⢠⣿⣿⣿⣿⣿⠏⠀⠈⠇⠐⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⣿⣿⣿⣿⣿⣿⠏⠀⡿
⠀⢻⣿⣿⡆⠀⢻⣿⣿⣿⣿⠏⠀⠀⠀⠀⠰⣽⣿⣿⣿⡿⠁⠙⠛⠁⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠁⠌⠈⠀⢀
⠀⠠⠈⢿⣧⣴⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⢻⣿⣿⡟⣸⡟⠂⠛⡛
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⣿⣿⡿⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢠⣼⣦⠶⠶⠶⠶⢤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⣦⡀⠀⠀⠀⢀⣠⣶⡿⢿⡿⣿⣯⣿⠀⠀⠀⣀⠀⠀⣀⠉⢉⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣼⣿⣿⣦⣶⣶⣿⠟⠁⠌⠘⣿⢻⣿⣿⠁⠀⠀⠐⢬⣿⣾⣿⣶⣿⣷⣦⣙⠿⣷⣄
⠀⠀⠀⠀⠀⢠⡶⣟⠿⠟⢿⣍⣿⣿⣿⣿⡏⠀⢸⠀⡄⠹⣾⣿⣿⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⣿⣿⡻⣷⣌⢻⣷⣄
⠀⠀⠀⠀⠐⠏⡴⠁⢀⣴⠿⡿⠟⠿⣻⣿⡇⠀⠘⢆⡇⡀⠈⠻⢿⣦⡀⠀⠀⠀⠀⠀⠘⢿⣿⡌⢻⣷⡈⠻⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡟⡁⠀⣠⣴⠟⣿⡏⢧⢠⡀⠈⠿⣦⣠⡀⡄⠉⠻⢦⣄⠠⡀⢄⠀⠈⢿⣿⡀⠹⣧⠀⠈⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⠞⠉⠀⢰⡟⣿⡘⢧⣷⡀⠀⠈⠻⣿⣷⡀⢶⣄⠙⢷⣽⡌⢷⡀⠈⣿⣧⠀⠙⠆⠀⠈⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠘⣷⣌⠷⣄⠙⠿⣦⣀⠀⠀⠙⠿⣦⡻⣷⡄⢻⣿⡘⣷⠀⢸⣿⡀⠀⠀⠀⢀⠈⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠙⠈⠓⠼⣳⣦⣠⣯⣛⡶⢄⣀⠈⠙⠿⣿⣧⢻⣇⢿⡇⠀⣿⡇⢸⣇⠀⢸⡆⢹⡿⣿⡿⣧
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⠷⢮⣕⠢⡀⠈⢻⣾⣿⣼⣿⠀⢻⣿⠀⣿⡄⠘⣿⢸⣷⢿⣷⠸⡆
⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡟⣧⡀⠉⠳⣌⢦⠀⢹⣿⣿⣿⠀⢸⣿⢠⣿⣧⠀⣿⣇⣿⢺⣿⣇⠷
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡷⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠟⠃⠘⣧⠀⠀⠹⣯⡇⠀⣿⣿⣿⠀⢸⣿⣼⣿⣿⢠⣿⣿⣿⣸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗⠀⢣⠀⠀⠀⠀⠃⠀⠀⠀⠀⣶⠏⠀⠀⢻⡀⠀⠀⣿⣿⠀⣿⣿⡏⠀⣾⢿⣿⣿⣿⣼⠋⣿⡏⣿⣿⣿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠁⠀⠀⠀⣸⠇⠀⠀⢸⣿⢠⣿⣿⠃⢠⣿⢠⡿⠈⣿⡏⠀⣿⢱⣿⣿⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣷⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣾⣏⣾⣿⡏⠀⣾⠇⠘⠁⠀⡿⢀⠀⢁⣾⣿⣿⡇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⣿⠀⠀⣾⠀⠀⠀⡄⠀⢀⠀⣠⣾⡏⢀⠀⢀⣿⣿⣿⡟⠀⣼⡟⠀⠀⠀⠀⠁⣾⢀⣾⣿⣿⣿⠃⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣿⠀⠀⣿⠀⠀⢠⡇⠀⣸⡿⠋⣾⠁⣼⠀⠸⢣⣿⡿⠁⣼⡿⠀⡄⠀⣶⠀⣼⣇⡾⢻⣿⣿⡟⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⠀⠀⢸⠀⢠⠟⠀⣼⡇⢠⡇⠀⢠⣿⣿⠃⣼⣿⢁⣼⠁⣼⣏⣼⣿⠟⣱⣿⣿⣿⡱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⡇⠀⢸⣇⡞⠀⢠⣿⠁⢸⡇⢀⣿⣿⠃⣼⡿⢃⣾⢃⣼⣿⣿⠟⢁⣾⠟⣽⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠛⠀⠀⠙⠋⠀⠀⣿⠀⠀⣸⡏⠀⢸⡇⢸⣿⢇⣼⡿⢡⣾⣣⣾⣿⠟⢁⣴⠟⢁⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠀⡀⢹⡇⠀⣿⡇⠀⢸⣧⣿⣿⣼⣿⣵⣿⣿⠿⠋⣠⡴⠛⠁⢠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡀⠀⠀⠀⣸⠟⠿⢸⠂⠀⣿⡇⠀⣮⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠁⠀⢀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠛⠀⠀⠀⠛⠀⠀⡿⠀⠀⢹⣇⠰⢹⣿⣿⡟⠋⠁⠀⠀⠀⠀⢀⠔⣠⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣄⣀⣀⣠⣶⠾⠃⠀⠀⠈⣿⡀⡞⣿⡟⠃⣀⡴⢀⠀⣠⡶⢁⣼⢟⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⣸⠿⣧⢸⣹⣿⣾⣯⡖⣡⠞⢁⣴⠟⣡⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⣧⣿⣿⣿⣿⠞⢁⣴⠟⢁⡾⠋⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣰⠻⣿⣿⣿⣥⣴⠟⢁⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡇⡄⡘⢿⣿⣿⠀⣴⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⣿⣿⣷⠋⣹⣿⡿⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣿⣿⣿⣿⣰⣿⣿⠁⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠃⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡈⣿⣦⡀⠀⣀⣤⣤⣤⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣅⣀⣠⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣶⣞⡫⠽⢛⣛⣶⣀
⠀⠀⠀⠀⠀⠀⠀⣼⢵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⣛⣉⠁⠀⠈⠓⠂
⠀⠀⠀⠀⠀⠀⠘⢉⣿⣿⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⡓⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠸⠋⢸⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠮⢭⡭⠭⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡤⠭⠲⣬⣓⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡷⢶⣶⣤⡉⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡿⠄⢈⠋⠻⣷⣶⠤⠔
⠀⠀⠀⠀⠀⠀⠀⣨⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡂⠒⠉⠁
⠀⠀⠀⠀⠀⠀⣰⣿⠿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠹⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠬⠍⠉⠉
⠀⠀⠀⠀⠀⢰⣿⣏⣄⣼⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣏⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡿⢛⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⢋⡴⣻⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠛⠡⠊⠰⠃⣸⣿⣿⣿⣿⣿⢟⣿⣿⣿⡿⣿⢿⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⣿⣿⣟⣁⣀⣀⣠⣤⣾⣿⣿⣿⡿⣫⣯⣾⣿⡿⢋⡼⢃⠞⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⣿⠿⠟⣛⣩⣭⣭⣶⣿⣿⣿⣿⣿⣿⣯⣥⣾⣿⠿⠛⢉⡴⠟⢁⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣋⣥⣴⣿⣿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣏⣉⣁⣀⣤⠴⠚⠉⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣿⣿⠿⠛⣉⣡⠤⠤⠶⠒⠒⠒⠲⠖⠶⠖⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢞⠽⠿⠛⠁⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⡄⣀⣿⣧⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣻⣿⣿⣿⣿⠶⣶⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣍⡙⠻⢦⣧⠨⣝⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⡟⠉⠻⠛⣾⣿⠃⠀⠀⢸⣿⣿⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⡿⢡⣾⣦⠜⠃⢹⡇⠀⠀⠀⢹⣿⢿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠋⠰⠿⠟⢃⠀⠀⠀⢸⣆⠀⠀⠘⠋⠀⠹⣿⣏⠁
⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠴⠃⠐⣨⠞⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⣿⣿⣄⡀
⠀⠀⠀⠀⠀⢠⣾⡿⠁⠴⣠⠆⢘⣅⠀⠀⠀⣠⡟⠀⠻⣦⡀⠀⠀⠀⣿⣿⣿⣿⣇
⠀⠀⠀⠀⣤⡿⠛⢠⠀⠘⠃⢀⣨⣿⢶⠶⢾⣿⣦⡀⠀⠈⠓⠀⠀⢸⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⡇⠿⠚⣁⣸⡷⠟⠋⠁⠀⠀⢠⣿⣿⠁⠀⢠⠀⠀⠀⠀⠀⢿⣿⣿⣿⣁⣀⡀
⠀⠀⠀⠀⠻⣧⣡⣔⣸⠏⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⢸⠀⠀⠀⠀⠀⢈⡍⠻⠋⠉⠉⠙⠛⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⡏⠀⠀⢸⠀⠀⠀⠀⠀⣼⡇⠘⢦⡀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⢸⠀⠀⠀⠀⠜⣻⠇⠀⠈⠣⠀⠀⠀⠀⡀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡏⢸⣧⠀⠀⣼⠀⠀⠀⠀⢠⡟⠀⠀⠀⡀⠀⠀⠀⠀⢻⣄⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⠈⠁⠀⣿⠀⠀⣿⠀⠀⠀⢠⠟⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢿⣧⠀⠈⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⡀⢀⢻⡆⢠⡟⠀⠀⠀⣇⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⣼⢿⠀⠀⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠁⢸⠈⠁⠈⠀⣦⠀⠀⠙⠛⠃⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⢰⠏⠠⠛⠑⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢠⣿⠀⠀⠀⣼⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⠀⠀⠀⣿⠃⠀⢀⣾⣿⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠓⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠷⠤⠀⠀⣸⡟⢀⣠⣾⣿⠇⠀⠀⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⢘⣷⣤⣼⣷⣄⣀⣀⣼⠏⠀⠀⡄⠀⢐⣿⣿⣿⣿⣿⠏⠀⠀⣼⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⡴⢿⣿⠛⣿⣿⣿⣿⣿⣿⠀⠰⠋⠁⣠⣿⠋⠉⣿⣿⣿⠀⠀⣰⣿⣿⣿⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⣴⠟⠁⢀⣿⣿⣿⣿⣿⠇⠀⣠⣾⠟⠋⠁⠀⠀⣹⣿⠃⢀⣼⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⣾⠞⠁⠀⠀⢸⣿⣿⣿⡟⠁⠀⢺⡿⠁⠀⠀⠀⠀⠀⣿⡏⣰⡟⢿⣿⣿⣿⣿⣿⣿⡛⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠘⣿⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⣾⡇⠀⠀⠀⠀⢀⣼⣿⣿⠟⠁⠈⠋⠙⢿⣿⣿⣿⣿⡄⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⣠⡟⠀⠀⠀⠀⠘⣿⣿⣿⣿⣄⢰⣿⠁⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠘⢿⡟⠣⠹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃⣿⠁⠀⠀⠀⠀⠀⢻⣿⣿⣟⢻⡀⣿⡀⠀⠀⠀⠀⢻⣿⡿⠃⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣼⡇⢿⣇⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡴⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣧⡀⣹⣇⣀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣆⠀⠀⠀⠀⠀⣸⣿⣿⡿⠀⠙⣿⣿⣿⡟⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⣿⠀⠀⠀⠀⢀⣿⣿⠏⠀⠀⠀⠸⣿⣿⡅⢀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡌⢻⣷⡄⠀⣼⣿⡿⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣬⣿⣿⣰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⢻⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠟⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣛⣟⣛⣛⣛⣛⣁⣛⣛⣟⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣽⣭⣭⣭⣭⣿⣿⣿⣭⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣦⣶⣼⣶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣦⣐⣲⣾⣿⣿⣿⣽⣿⣿⣷⣶⣄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣝⣿⣿⣿⡿⠛⢉⠭⠦⡀⠁⠏⣾⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣖⢿⣿⣿⡿⣿⠟⠉⠀⠀⠀⣠⡔⠁⢦⣤⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⣹⣿⣿⣿⣿⣫⡿⠋⠀⠀⠀⠀⢰⣿⡇⠀⠀⢌⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⢋⣾⠷⣀⣠⡄⠠⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⠤⠒⠒⠒⠒⠲⠖⠒⢺⡿⠁⠀⠀⢳⠀⠀⠀⢀⠀⠀⣾⡇⠀⠀⠙⢾⢂⡨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿⠁⢀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢸⡘⢧⠀⠰⠋⠀⢸⣿⡁⠀⠀⠀⠈⢏⣟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⠀⢰⡏⡀⠀⠀⠀⠀⠀⠀⠈⢀⢠⠀⠈⠳⣶⣤⢀⣠⠀⣼⢿⣇
⠀⠀⠀⣠⡴⣶⣿⣿⣿⣿⣿⣿⢟⠿⠃⣿⠀⠘⣷⡇⠀⠀⠀⠀⠀⠀⠀⣬⠸⠇⠀⠀⠈⠀⠸⣿⢠⡟⠀⣿
⠀⠀⣾⣥⠾⡻⠿⣓⣛⣿⠿⠋⠀⠀⠀⢹⣧⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠞⠃⠀⠛⠘⠀⢰⣿
⠀⡼⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠂⠀⢙⣦⣄⣀⣀⠀⢸⣿⠀⠀⢀⡤⠲⡏⠆⠀⡄⣸⣾⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣦⡈⣿⣿⣿⣿⣿⣿⣿⣄⣀⠉⡚⠢⣔⢤⣤⣤⣾⣟⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡏⠻⣿⣿⣿⠀⠀⠉⠒⠢⠤⣼⡀⡁⡿⢦⡈⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⢧⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣧⣧⡏⠀⠑⢤⡙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣎⠱⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣻⠏⠀⠀⠀⢸⣧⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡈⠻⣿⣆⠀⢀⣠⣄⡠⢢⡟⠀⠀⠀⠀⢌⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠹⣷⣾⣿⠿⠷⠟⣡⣶⣦⡠⢞⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠈⠻⣿⣷⡀⠀⠀⠛⠿⠿⠷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠽⡏⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠠⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣄⣀⣤⣤⣤⣤⡀⠀⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣤⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⡿⠟⢿⣿⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⠛⠁⠀⢸⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠻⣿⡟⠁⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⣰⣿⣿⣧⣀⣀⠀⠀⣠⣤⣿⣿⣿⣿⡻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣁⣐⣿⣿⣿⣿⣿⣿⣄⣴⣾⣿⣿⣿⣿⣿⣷⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢀⡀
⣴⣿⣿⣿⣶⣶⣶⣶⣦⣄⣄⣤⡈⣿⣿⣿⣿⣿⣿⠰⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣦⣷⣷⣶⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⡏⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢳⣿⣿⠟⠉⠡⠀⠀⠸⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⢺⣿⣷⣆⣤⣤⣤⣀⡀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠟⢁⣠⣤⡶⠀⠀⣭⣿⣿⡿⠿⢻⠟⠿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⢋⣶⣿⣿⠏⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣼⣿⠟⠁⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⣿⣿⣿⡉⠛⠛⠛⠛⠿⠿⠿⠿⢿⠿⣿⣿
⣿⣿⣿⣿⣿⠃⠀⠀⢀⣀⠀⠀⠀⣿⡇⠀⡀⠀⠀⠀⣼⡇⠀⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠉⠀⠀⠉⠁⠀⣠⣶⣿⡟⢀⠀⢠⡿⣁⣸⡇⣀⢀⣸⣿⣁⣀⡘⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣴⣿⣦⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣆
⣀⡀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠈⣿⣶⣶⣤⣤⣶
⣉⡇⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠉⠿⡿⠿⢿⣿
⢿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣠⣶⣦⣤⣤⣿
⣀⡀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠙⣻⣿⣿⣿⣿
⡉⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢻⣿⣿⣿⣿
⣇⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⣷⣾⠷⠛⣻⣫⢷⡶⣶⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠟⠁⠀⠐⢋⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⠃⠀⠀⣠⠀⠋⢀⡀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠃⠀⠀⠀⠃⠀⠈⠉⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⣆⠀⢀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⠷⠶⠤⠽⠿⠟⠉⠀⠀⠀⠀⠀⠀⢠⣿⣦⣾⡄
⠀⠀⠀⠀⠀⣀⣤⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠙⢷⡘⣠⡀⠈⡄
⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢤⣸⣿⣿⡇⠀⠀⢸⣏⣿⠟⠁
⠀⠀⢀⣴⣿⣿⡏⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⣿⣿⣿⣿⣷⡀⠀⠀⠈
⠀⠠⣾⣿⣿⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⣰⣿⣿⣿⣿⣿⣷
⠀⠀⣿⣿⣿⣿⣷⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠈⢿⣿⣿⣿⣧⡀⠀⠙⢶⣄⠀⠈⢦⣄⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢻⣿⣿⡿⠃⠀⠀⠘⣿⣿⣦⠈⠻⣿⣶⣤⣸⣿⣿⣇⠀⠀⢀⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⢠⣤⣶⣿⠿⠟⠃⠀⠀⠀⠀⠈⢿⣿⡇⠀⠀⢙⣿⣿⢿⣿⣿⡆⣠⠀⢻⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿
⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡀⢀⠀⣿⣿⠀⠀⠀⠀⠙⠀⠠⣽⣿⣧⡀⠀⠉⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣼⢰⣿⠇⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⡳⣄⠀⠀⠙⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣜⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢹⣾⣿⣆⠀⢀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡟⠀⠀⢣⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠙⣦⡀⠀⠀⠀⠀⠀⠀⢰⣿⠀⣿⡧⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣴⠛⢳⣄⠀⠀⠀⠀⠀⠳⠿⠟⠋⠀⠀⠀⠀⠀⠀⠹⠛⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣄⢶⣿⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠻⣷⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⡀⠀⡄⠀⠀⠐⠛⠛⡻⠛⠿⡤
⠀⠀⠀⠀⠀⠀⠤⠾⠋⢾⣿⡻⢿⣦⣤⣄⣠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⣀⡛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⠀⠀⠉⣭⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣄⣒⣶⣶
⠀⠀⠀⠀⠀⠈⠛⢷⣔⣿⣷⣴⣾⠟⠟⠏⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⣾⣿⣿⣿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠉⠁⠘⠃⠀⠀⢤⣤⣼⣦⣤⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⢿⠿⣯⣿⣿⣿⣂⣘⣷⣚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡷⠿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⡅⡟⠀⠀⣀⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣾⠁⡎⠀⡿⠛⠛⠓⣒⣒⠯⡱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⢰⡇⢸⠀⢰⡇⡀⠀⠀⠀⠉⢙⡲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠈⡃⠁⠀⠋⠈⠙⢢⡀⠀⢄⢴⣽⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⡠⠋⠀⠀⠀⠀⣀⣀⡸⢃⡼⢸⢸⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠁⢀⠀⣞⠃⠀⠀⠀⠀⣊⡭⡭⣉⠉⠀⢱⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢈⠀⡏⡸⠀⠀⠀⠈⣷⣿⣍⣿⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠘⡄⡇⣇⠀⠀⠀⠀⠘⠻⠟⠁⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣷⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣠⡇⠀⠀⠀⠉⠂⠀⣄⡀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⢀⡔⠉⠀⠀⢀⣀⠹⣄⡤⠖⠒⠲⠤⣤⣀⣠⠽⣡⣢⠟⠀⠀⠀⠀⠀⠀⢸⡇⠙⡖⡀⠀⠀⡴⡆⠘⡆
⠀⠀⠀⠀⡞⠀⠀⠀⡰⠁⢈⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢳⡅⠀⠀⠙⢀⣠⠇
⠀⠀⠀⢸⠁⠀⠀⠀⡇⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠙⠲⠶⠖⠉
⠀⠀⠀⠘⡆⠀⠀⠀⢧⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠓⠦⣀
⠀⠀⠀⠀⢣⠀⠀⠀⢸⡀⢹⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⢀⠀⣸⢁⠞⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⠂⣠⣀⠤⡄⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⢠⡼⡰⢳⠏⠀⠀⠀⠀⢀⣠⠖⠛⠦⢤⣀⡀⠀⠀⠀⠘⡄⠀⠀⠀⠘⡟⢁⠤⠐⠁⠀⠀⢰⠋
⠀⠀⠀⠀⠀⠿⠋⠀⢸⠀⠀⠀⠀⡞⠁⠹⡀⠀⠀⠘⡇⠉⠓⠒⠲⠖⢻⡀⠀⠀⠀⢣⠘⣇⠀⠈⠢⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⡇⠀⠀⠙⡄⠀⠀⠘⡄⠀⠀⠀⠀⠀⣧⠀⠀⠀⠈⢧⡘⣆⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠀⠤⢼⠀⠀⠀⡗⠐⠒⠀⢻⠀⠀⠀⠀⠀⢹⡄⠠⠤⠄⠈⢧⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣀⣀⣨⡇⠀⠀⠳⠤⠤⠤⠴⠇⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣸⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⢀⡶⠟⠉⠁⣠⣶⣚⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡿⣿⠻⠏⠀⠀⠀⠘⢛⣿⣟⣛⣃⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡟⠲⠃⠈⣠⠆⠀⠀⠀⠀⠋⠉⠉⠉⠉⢉⣽⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠯⠶⠶⣤⡴⢻⡟⠛⠛⠓⠶⢶⣤⣀⡀⠀⠀⠛⠛⠛⠿⠶⣶⣭⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡁⠀⠙⠁⠀⠀⠀⠀⠀⠀⠉⠛⢷⣤⡀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⣉⡉⠉⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠐⢤⡉⠻⣷⣄⠀⠀⣰⡾⠿⠶⣽⠄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠖⠙⠋⠀⠀⠀⠀⠸⣿⣷⣄⠀⠀⠀⠀⠀⠀⠹⣦⡈⢿⣧⡀⠀⠉⠙⢶⣬⡀
⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠛⣧⠀⠀⠀⠀⠀⠀⠙⣷⡀⠹⣷⡄⠀⠀⠀⠙⢿⣆
⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠘⣧⠀⠀⠀⠀⠀⠀⠘⣧⠀⢹⣿⡀⠐⠲⣦⡀⢻⡆
⠀⣴⠟⡁⠀⠀⠀⡤⠖⠛⢻⣧⣀⣀⣀⣀⣠⡾⠿⣿⣷⡀⢻⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣿⡇⠀⠀⢹⣿⣆⣿
⠀⣿⣿⠟⠀⢠⣤⣤⣶⣶⠿⠟⠛⠛⠛⠋⠁⠀⠀⢸⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣿⣿⠀⠀⠀⣿⣿⡿
⠀⢿⣷⣴⡶⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⣿⡇⠀⠀⠀⣿⡿⠃
⠀⠈⠉⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠁⢀⡄⠀⠀⠀⠀⣀⣀⣠⣴⠟⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⣾⡀⢧⣀⡀⠀⢙⣮⣭⠀⠀⡀⢀⣿⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠙⢷⣄⠙⠛⠛⠛⠋⠁⣠⣴⣿⣼⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣴⡾⠋⠁⠀⠀⣀⠀⢠⣾⠋⠘⠉⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸⡏⠀⢠⣶⠟⠋⣿⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢠⡟⠁⠀⠀⠻⣄⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡁⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣿⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣂⡛⠊⠻⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠚⠉⠉⠀⠀⠻⢸⡶⠶⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠈⠓⠀⠀⠀⠀⠉⠻⣆⠈⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⡅⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠈⠧⠈⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⢀⡀⠀⠀⢀⣀⣀⣀⠶⠁⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⢾⣯⣡⠏⠉⠉⢹⠍⡫⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠲⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⢠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣌⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠶⠦⢄
⠀⠀⠀⠀⠀⠀⣠⠤⠤⢄⡀⠀⠀⢀⣴⠋⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢳⢀⡵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠠⠴⠀⠀⢉⠙⠲⣄
⠀⠀⠀⠀⢀⡴⠁⢒⡚⠂⠉⠒⠦⣼⠀⢠⠊⠔⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠙⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠒⠀⠀⢸⡄
⠀⠀⢀⡤⠋⠀⡼⠉⠉⠳⡀⠀⠀⣨⠀⡆⠐⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠓⠀⠀⠈⠉⠐⠒⠒⠒⠒⠒⠲⠦⣄⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⡴⠋⢠⣄⡼⢁⠔⠠⢀⡘⣦⣻⡇⢠⠃⠀⠀⠀⠀⠀⠀⢀⢜⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⣀⠼⠁⢠⠶⡐⠀⠀⠀⠀⠀⢱
⠈⡇⠀⠳⣄⢰⠃⠰⢀⡀⠀⠉⠉⠁⠺⠆⢀⠀⠀⠀⠀⠀⠑⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⠤⠔⠃⢀⠆⠀⠀⠀⠀⡄⢀⠁
⠀⡇⠀⠀⣸⠘⠀⢰⠋⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⡆⠀⢀⡎⠀⠀⠀⠀⢰⢁⡎
⠀⠙⠤⣠⡏⠠⠀⠘⡆⠀⠀⠈⠳⢤⡀⠀⠀⠀⢀⡀⣠⡖⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⠀⣿⠀⠈⣇⠀⠀⡀⠀⠸⣎⣣⣠⠄
⠀⠀⠀⠀⠀⢰⠀⠀⣹⠀⠀⠀⠀⠀⠉⠙⠓⠦⣭⠞⠻⢭⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢻⠀⠀⠈⠢⡱⣄⢌⣆⠹⡍⠉
⠀⠀⠀⠀⠀⢸⡄⠸⣅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠦⠤⠤⠤⠴⠶⠶⢶⣶⣀⣀⡀⢀⡀⠀⢀⠀⠂⠐⠄⢸⠂⠀⠀⣀⣷⢟⡼⢿⢂⡇
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠯⠭⢷⣄⣈⠉⠃⠂⠀⠈⠀⡙⢄⠀⠀⠀⠀⣉⣁⣼⠞
⠀⠀⠀⠀⠀⠀⠈⢄⣁⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢦⡀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡗⢶⣤⣄⡀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢿⠙⢆⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠘⣆⠈⢧⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⠀⣀⣸⡄⠘⡅⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢠⠏⠀⠀⢀⡇⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⣷⠀⠀⠀⣸⠓⠂⣠⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣀⣀⣤⠴⠋⠀⢠⠞⠁⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢘⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⣶⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⢸⣿⡆⠀⠀⣿⡇⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡇⠀⠹⣿⣶⣦⣤⣶⣶⣿⣿⣿⣿⠿⠿⠿⠛⣿⣿⠀⢸⣿⣥⠾⠋⠀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣦⣀⡈⠛⣿⣿⣿⣿⡿⠛⣁⣤⠶⠋⠁⠘⠋⠁⠀⣠⣔⠶⣤⣶⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢉⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣄⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⠋⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⢁⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠙⣿⣿⣿⣿⣿⡟⠀⣼⣿⠁⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣶⣿⣿⣿⣿⠏⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢰⣿⣿⣿⡇
⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⢀⡴⠀⠀⠈⠛⠿⠿⠿⠟⠁⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⣼⣿⢿⣿⡇
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⠎⠀⢸⣿⡀⠀⠀⠀⠀⠀⣴⠆⠀⠀⠙⢿⣿⣿⣿⡟⠈⣿⡇
⠀⢹⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠋⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠈⣿⣇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⣿⡇
⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⢹⣷⣄⣠⣿⡇
⠀⠀⢹⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠇
⠀⠀⠈⣿⠈⢿⣿⣿⣧⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠈⠙⠋
⠀⠀⠰⠃⠀⠈⢿⣿⣿⣇⠀⠀⠀⠸⣷⢸⣇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠘⣿⠙⣿⠀⠀⠀⠀⣿⡀⣿⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠋⠀⠀⠀⠀⢻⣇⠘⡇⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠸⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠋⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣶⡘⣆⠀⠀⠀⠀⠀⢀⣴⠒⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢹⡄⠀⠀⠀⢰⣿⠁⠀⠀⠀⠠⡀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢻⡄⠀⠀⣿⡟⠀⠀⠀⠀⠀⣷⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⢿⡄⠀⣿⣿⠀⠀⠀⠀⠀⠘⣷⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠈⢿⣄⣿⣿⡆⠀⠀⠀⠀⠀⢹⣧⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠘⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⢹⣿⣿⣿⣿⡄⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠁⠀⠀⠀⠀⠀⣨⣿⣿⡟⠀⠀⠀⠸⣿⣿⣿⣿⡇⠀⠀⠀⣼⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡇⠀⠀⠀⠀⢀⣴⣿⣿⣿⡇⠀⠀⠀⠀⢻⣿⣿⣿⡇⣠⣤⣴⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢻⣿⣿⡿⣿⡿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠀⠀⠀⠀⠀⢻⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠙⠳⣜⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠼⣄⢰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣷⣮⡁⣒⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⣿⣿⣿⣿⣿⣿⡇⣿⣷⣌⢿⡝⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠟⠛⠛⠻⠿⣿⣶⣸⣿⣿⡝⣿⣏⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢯⣿⣿⣿⣿⢷⣜⣿⣿⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢟⣵⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣢
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣷⣶⣶⣶⣶⢦⣄⡀⠀⠀⠀⣠⣾⣷⣿⣿⣿⣿⣿⡿⢻⢾⣮⣭⠥⣟⣹⠇
⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠈⠉⠙⠻⣿⣿⣶⣄⣼⡿⣿⡿⠿⢿⣿⣿⣿⣷⣿⣿⡿⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⡙⠻⣿⠿⢻⣾⣿⣿⣿⣶⣦⣸⣿⣿⠟⢁⣤⣯⣭⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢿⣿⣿⣿⡷⣶⣤⣬⣭⣿⣧⣼⣿⣿⣿⣿⣯⣤⡎⣡⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣽⣛⣿⣥⣻⣽⣾⣿⣿⢟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⣛⣿⣿⣿⣿⣿⣿⣿⢸⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡟⣿⢏⣴⣿⣿⣿⣿⣿⣿⣐⡺⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡁⢠⣾⣿⣿⣿⣿⣿⣿⡇⠈⣻⣝⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⢸⣿⣿⣿⣿⣿⣿⠟⠀⠀⠹⣽⣦⠻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠈⠻⢿⣿⣿⡿⠋⠀⠀⠀⠐⠹⣿⣰⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠘⣿⣿⡿⠁⠀⠀⠀⠀⢰⣧⣿⣿⣿⡿⢿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣮⡿⢹⡇⠀⠀⠀⠀⠀⣸⣿⣮⢿⣮⣟⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⣿⣿⡆⠀⠀⠀⣠⣶⣿⣿⣷⣍⣿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠇⠀⠀⠀⢰⡿⠋⠁⠐⠒⠚⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣅⠀⠀⠀⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠾⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣧⡈⣿⣆⠀⢀⣤⣤⣤⣴⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡶⢶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣼⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣌⠠⠔⠛⠢
⠀⠀⠀⠀⠀⠸⢻⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠦⢤⣀
⠀⠀⠀⠀⠀⠀⡿⠙⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢶⣤⡈
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡬⠁⠢⣙⡒⠀⠄
⠀⠀⠀⠀⠀⠀⠀⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⣶⣤⡁
⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣟⠀⠈⢉⡻⠿⠖⠂
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠄⠠
⠀⠀⠀⠀⢀⣾⠏⢁⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠁
⠀⠀⠀⢀⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⣼⣿⣿⣿⣿⡟⣽⣿⣿⡿⠋⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠟⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠈⠿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⢡⢾⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡿⠋⠀⠁⠈⣠⣿⣿⣿⣿⣿⣿⣿⣿⢟⡿⣹⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠿⠿⢶⣶⣶⣶⣿⣿⣿⠿⣋⣾⣾⡿⠟⣡⠟⢀⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡿⢛⣉⣤⣴⣾⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⢉⣠⡴⠚⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⠿⠛⣋⣉⣥⡤⠤⠤⣬⣭⣭⠿⠿⠟⠛⠉⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢾⡿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡄⠀⠀⠀⠀⠀⣰⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⢸⡇⢠⡇⠀⠀⠀⠀⠘⣇⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡗⠶⢶⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⢸⠃⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⢄⣠⠱⠀⠀⠀⠀⢻⡿⣿⣦⣀⠀⠀⢸⠀⠘⠀⢀⡄⠀⠀⠀⠀⠀⠀⢀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣿⡇⠈⠻⢿⣷⣦⣬⣄⣥⣴⣿⣷⣄⣀⣀⣠⣴⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠇⠀⠀⠀⠉⠙⠛⠛⠛⠉⠈⠙⠛⠟⠛⠋⣿⣿⠁⠀⣤⠤⠴⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⣶⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣿⣧⣀⠀⢻⡀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣆⣠⠞⠉⠀⠀⠀⠀⠉⢻⡿⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠋⠀⢀⣿⣇⣀⣀⠀⢠⡶⢶⡀⠀⠀⠀⡏⠀⠘⣿⡅⣴⡾⢛⣻⣷⠶⠿⢶⣿⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢶⣶⣶⡾⠛⠉⠁⠀⠉⠙⢾⣿⡧⠙⢦⡀⢰⡇⠀⠀⠹⣷⣏⠀⢿⠀⠀⠀⠀⢸⣿⣆⣀⣸⡓⠂
⠀⠀⠲⡶⠺⡄⠀⠀⠀⠀⠀⢹⣿⡠⢀⣤⣴⣦⣄⡀⠀⠙⣯⢂⠈⠳⣼⠀⠀⡀⠀⢻⣟⠦⣼⡷⠟⠁⠀⣼⠹⡏⠀⠈⣷⣤⣤⣀⡀
⠀⠀⠸⠓⢴⠆⠀⠀⠀⠀⠀⣾⣿⣏⣾⠋⠀⠀⠉⢷⠀⠀⢸⡷⣄⠀⠈⠀⢠⣧⠀⠀⢿⣄⠀⠀⣀⣤⡾⣣⠾⠷⠶⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣷⣷⡀⠀⠀⠀⣸⠃⠀⢸⣯⣿⣷⡄⠀⢸⠙⣧⣠⣬⡿⠻⠛⠛⠉⠰⡏⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⣀⡤⠶⠾⠿⠿⠿⢿⣿⣿⣯⠻⠦⠤⠞⠁⠀⢀⣾⣿⣿⡈⠻⡶⠟⠀⠉⠉⠁⠀⢄⣑⣄⣀⣀⣹⣄⠀⠀⠀⠠⢎⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⡹⢿⣿⣧⣀⡀⢀⣀⣴⣿⣟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠳⡌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⣠⠞⠁⠀⢈⠿⠛⠛⠛⠋⠁⠈⠻⣧⠀⠀⠀⢀⡴⠞⠛⠓⠲⣄⡂⠀⠙⢦⠈⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣰⠋⠀⣠⠴⠶⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⠔⢃⡀⠀⠀⠀⠀⠀⣿⣰⠃⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣃⡀⢰⠊⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠛⠀⠀⠀⠀⠀⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⠩⣽⣿⣿⣿⡟⠙⢻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⣀⠀⠀⠀⣀⣀⣀⣀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠿⠧⠤⠴⣿⣿⣿⣿⣟⣿⣿
⠀⠀⠀⠀⠀⠀⣠⣬⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⢏⣾⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠋⠛⠉⠉⠉⠉⢿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⣷⣄⡿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⣭⣿⣿⣿⣿⣧⡍⣋⠠⣶⠆⣹⣿⣿⣾⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⡴⠋⢐⣯⣽⣿⣿⣿⣿⣿⣷⡘⣿⣶⣤⣾⣷⣿⡿⢯⣍⣻
⠀⠀⠀⠀⠀⠀⠀⣰⢫⢿⣛⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣵⡿⠷⠋⣿⣏⠁⠉⣉⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠴⠋⠘⠀⠀⣴⣿⠛⢡⣾⣻⠿⢿⣯⣻⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠱⠋⠀⠋⠀⠀⠀⠉⣵⣿⣷⣶⣶⣶⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢺⣿⣿⣗⣲⣤⡀⠀⢀⣦⣴⡄⢿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⡆⢸⣿⣿⡧⢾⣿⣿⠋⠈⢸⣿⣿⣿⣿⠟⣻⣿⣿⣿⣿⣿⢻⣿⣿⣿⣤⣀
⠀⠀⣠⠀⠀⠀⠉⠉⢦⣸⣌⣿⣿⣿⣿⣿⣿⣿⣦⡙⠷⠰⣿⣿⡇⠀⠀⣿⣿⣿⡿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⣰⠁⣠⣤⠤⠲⠶⢛⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣶⣽⣿⣷⣶⣴⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣽⣿⣿⣿⣿⣦⣾
⠀⣧⠀⣿⢀⣴⣷⣾⣿⣿⣿⡿⢿⢻⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢏⣿⣿⣿⣿⣿⣿⣿
⠐⢻⡄⢸⣽⣯⣀⢀⡴⠳⣞⣷⣮⣿⣷⣿⣿⣿⣿⣿⣰⢀⢀⣀⣠⠔⠒⠀⠀⠈⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⣰⣿⣿⣏⣿⣿⠉⢻⣦⣶⣽⣿⠻⣿⣿⣷⣿⣿⣿⣿⠻⣤⣿⡉⣠⠞⠁⢀⣤⠤⣄⡀⠻⣿⣿⣿⣿⣿⣿⡿⠛⠿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⢿⡭⠉⠀⢿⣿⣿⣿⣿⣿⠁⠀⠀⠓⠻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣧⠀⠀⠈⠻⣿⣿⣿⣿⣧⡀⠀⠀⢰⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡟⣿⠍⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⢀⡰⠿⠿⢿⡯⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠦⠤⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⡦⢄⡀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⡀⠀⠀⠀⢀⠔⠈⠀⠠⠈⠀⢀⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢀⠸⠀⢆⠀⠄⠀⡀⠐⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠘⠻⠒⡽⠃⠈⠂
⠀⠀⠀⡠⠂⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠤⠤⠤⠤⢄⣀⠈⠁⠀⠀⠀⠄⠦⠤⠤⠤⠀⠀⡀
⠀⠀⢀⡁⠀⠀⠀⠂⢩⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⠐⠠⡀⠀⠈⠑⠢⣄
⠀⠀⢼⠀⠀⠀⠂⠐⠋⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠈⢦
⠀⠀⠘⡀⠀⠀⠀⠈⢼⠁⠀⠀⠀⠀⠀⠀⠁⠈⠀⡈⠍⠛⠀⠀⠈⠀⠁⠈⠉⡉⠉⢉⠉⠉⠉⠁⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠂
⠀⠀⠀⠳⠀⠀⠀⣀⠙⠀⠀⠀⠀⠀⠀⠄⠈⠢⠠⠤⠂⠡⠔⠠⠀⠠⠀⠀⢄⡀⠐⠢⠤⠀⡀⡀⢤⣶⠀⠀⠀⣀⠤⠄⠒⠈⢀
⠀⠀⠀⠀⠉⠂⠀⢀⠀⠀⠀⠀⠀⡃⠀⢀⢬⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠒⠀⠀⠈⠁⠈
⠄⠀⠤⠁⠀⠐⠂⣁⣀⡀⠀⠀⠀⠁⠊⠁⠀⠉⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠁⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠠⠄⠈⠍⠉⢀⡀⢀⣐⣒⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⡠⠘⠣⣀⢀⣈⡍⠁⠈⠀⠀⠀⠀⠀⢰⠂⠐⠒⠒⠊⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢷⠀⡀⠀⡔⠋⡀⢀⠀⡬⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⠔⠂⠈⠒⠑⠊⠀⠀⠣⠘⠒⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡨⠄
⠀⠀⢀⠂⠤⠀⣀⡀⠀⠘⠀⡠⠔⣊⠤⠊⠀⠀⠀⠀⡀⠀⡀⠀⠐⠉⠉⠉⢨⡂⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡠⠊⡠⠊
⠀⠀⠀⠈⠑⠒⠦⠬⣉⣹⣁⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠠⠐⣐⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡠⠊⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠄⠀⠜⠀⠰⠤⢀⣉⠐⠒⠠⠤⠤⣸⡄⠊⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠢⠤⠆⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣏⣀⡀⣀⣤⣴⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠏⢨⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡻⢿⣿⣿⣿⣇⠀⣼⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣤⣤⣄⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣆⣿⣿⣿⣿⡶⠛⣿⣿⣿⣿⣿⣿⣿⣿⣆⣸⣿⣿⣿⣶⣶⣶⣿⣶⣾⣿⣧⣤⣾⣿⠿⠿⠛⠛⠛
⣴⣶⣾⣷⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⢹⣿⣿⡹⣿⣿⣿⣿⣧⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠋⠉⠉
⣿⣿⣿⣿⡿⠿⠛⠓⠀⠀⠀⠀⠀⠀⠀⠐⠀⢰⣀⠹⣿⣿⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡉
⣿⣿⣷⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣝⣷⡌⠛⢷⣬⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣄⣀
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣤⠀⠀⢧⣀⣿⡟⠃⠀⢸⡂⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠉⠙⠛⠓⠛⠛⠛
⡿⠿⠿⠋⠀⠉⠀⠀⠀⠀⠀⠸⠿⠀⠀⠈⣿⣿⠀⢰⣿⡿⡇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡏⠙⢿⣿⣿⣿⣿⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿
⣿⣿⣿⣿⡇⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡻⣧⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣭⣀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣽⣿⣿
⣿⣿⣿⣿⣷⣴⣿⡿⠀⠀⠀⣀⠀⠀⠀⢻⡎⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠯⣶⠗⠄⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠘⠿⠇⠀⠀⠈⣿⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣍⡀⠀⠀⢀⣀⣰⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠉⠀⢀⣶⣶⣄⠀⠀⠀⠀⣿⡇⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⢿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⢿⣧⠀⠀⣸⠿⠋⠀⠄⢠⡀⠀⠀⠀⠀⢿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠇⠈⠉⠀⠈⠃⠀⠀⠀⠀⠈⣿⡆⣀⡁⠀⠸⣿⣿⣿⣄⢈⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡿⠶⠄⠀⠀⠀⢤⡀⢀⡀⠀⠀⠹⣿⡿⠁⠀⢸⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡈⢣⢸⢇⠎⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣢⣿⣿⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣵⣿⣯⣽⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢧⣟⡿⡾⢾⠶⣟⣹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠙⡾⢽⣭⣟⣹⡏⢹⣅⡼⡆⠀⠀⣠⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣏⣹⠓⡿⢤⣯⡼⣯⡽⢧⡤⢷⢀⡴⢋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⢧⠀⠀⣸⠦⣯⣹⣿⣾⡟⠚⣟⠶⢻⠒⠛⡏⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠳⣄⣟⢺⠧⢼⡇⠙⣏⢛⣏⠓⣻⡓⢚⣧⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢹⣍⣽⠓⢺⡇⠀⢹⣏⣿⣙⣹⣏⣉⣼⢰⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⢦⣻⡀⢼⠢⣿⣝⣽⠇⠀⠈⡯⢾⣍⣉⣿⣭⢼⣾⡏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠻⡇⣾⣉⡿⢤⡟⠀⠀⠀⡷⢺⡯⡭⢿⠦⠾⡿⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠹⡷⢭⣽⣾⠁⠀⠀⢰⣟⣹⡗⠶⣿⡓⣾⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⡖⢿⡏⠀⠀⠀⢸⣧⣤⣏⣋⣹⣿⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠈⢿⣹⠇⠀⠀⠀⢸⠶⠾⣧⣉⡽⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⢿⠀⠀⠀⠀⢸⣓⢚⣷⠏⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠸⡇⠀⠀⠀⠸⣯⡟⠁⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⢻⡀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⢳⣄⠀⠀⠘⡇⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣍⣷⣤⣼⡧⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡀⢼⣄⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡾⣧⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡇⣿⡞⣿⠈⠁⠀⠀⠈⠉⠉⠙⠛⠿⢶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⢿⣿⣦⡀⠀⢀⣠⣴⣿⠟⠋⠁⣿⡇⣿⣷⢹⡇⠀⠀⠀⠂⠤⣤⣒⣤⣀⢢⣤⣉⠻⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢀⡄⠀⢻⣇⢿⣿⣼⡇⠀⠀⠀⠉⠲⣾⣽⣿⣿⣿⣿⣿⣷⣦⣝⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣴⠿⣫⡿⠛⠙⢿⣤⣿⣿⣿⣿⣿⠁⠀⢸⠄⡆⠀⠻⣾⣿⣿⡀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡻⣿⣿⡝⢿⣷⣌⢿⣿⣦
⠀⠀⠀⠀⠀⠘⠋⡴⠁⠀⣠⡾⢛⡿⠟⠿⣋⣿⣿⠀⠀⠈⠠⣧⠀⠀⠈⠻⢿⣧⡀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡈⠻⣿⡄⠙⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢰⠀⢀⣾⠋⠀⠈⢀⣤⡾⢻⣿⠿⡆⢠⠀⠀⠻⣶⣃⡷⠀⡀⠙⠻⢷⣄⠀⢢⠀⢄⠀⠀⠹⣿⣷⠀⠹⣷⠀⠀⠹⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣠⡾⠛⠁⠀⣿⢿⡆⢻⣌⢧⠀⠀⠙⠻⣾⣧⣷⡀⢠⣄⠙⠳⣆⣷⡈⣷⡀⠀⢹⣿⣇⠀⠙⡇⠀⠀⠘⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏⠀⠀⠀⠀⣿⡈⢿⣄⠙⢾⣷⣄⠀⠀⠈⠛⢿⣷⡈⢻⣧⡀⠙⣿⣷⡘⣷⡀⠀⢿⣿⡀⠀⠁⠀⠀⠀⠹⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⠀⠀⠀⠀⠀⢿⠻⢦⡙⢷⣄⠈⠛⠷⢦⣀⠀⠀⠈⠻⢶⣽⣿⣤⡈⣿⣧⢹⣧⠀⢸⣿⡇⠀⣄⠀⠀⣶⠀⢿⡾⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠃⠀⠀⠀⠀⠀⠀⠁⠀⠈⠋⠽⠿⣦⣟⣷⣦⣝⡳⠦⣄⠀⠈⠻⢿⣿⣸⣿⡜⣿⡄⠀⣿⣷⠀⣿⡆⠀⢹⣇⢸⣇⣿⣿⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⠻⠷⣦⡙⢦⡀⠀⠻⣿⣿⣧⣿⡇⠀⢻⣿⠀⣿⣿⠀⠸⣿⠀⣿⢸⣿⣧⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠺⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⠟⣷⡀⠀⠙⢦⡙⣆⠀⠸⣿⣿⣿⡇⠀⢸⣿⠀⣿⣿⡆⢰⣿⡇⣿⠸⣿⣿⡆⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⢿⠀⠸⣷⠀⠀⠀⢿⣹⡆⠀⣿⣿⣿⡇⠀⢸⣿⣆⣿⣿⡇⢸⣿⣷⣿⢠⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡅⠀⣇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢈⣤⠃⠀⠀⢿⡇⠀⠀⠈⣿⣿⠀⢸⣿⣿⡇⠀⣿⣿⣿⣿⣿⡇⣼⠟⣿⡿⢸⣿⣿⡟⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢹⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⢸⡇⠀⠀⠀⣿⣿⠀⣿⣿⣿⠁⢀⣿⠃⣿⡏⢿⣷⡟⠀⣿⡇⣾⣿⣿⡇⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠸⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⣿⣿⢠⣿⣿⠏⠀⣼⡟⢠⡿⠀⢸⣿⠃⠀⠿⢰⣿⣿⣿⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⣿⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⣿⣧⣾⣿⡟⠀⢠⣿⠁⠀⠀⠀⢸⠏⣰⠀⢀⣾⣿⣿⣿⠇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢰⠀⣰⣿⡟⠀⡸⠀⢰⡿⣻⣿⣿⠁⢠⣿⠏⠀⠀⠀⠀⠀⢠⡿⢀⣾⢿⣿⣿⣿⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣿⠀⠀⢸⡇⠀⠀⢠⡄⠀⢀⣾⡿⠋⣾⠃⢸⠇⠀⢾⣱⣿⣿⠃⢠⣿⡏⢀⡄⠀⣰⠄⢀⣾⣃⡾⢋⣾⣿⣿⠇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣇⠀⠀⢸⡇⠀⣼⠏⠀⣸⡟⠀⣿⠀⠀⣠⣿⣿⠇⢠⣿⡟⠀⣾⠃⢠⣿⢀⣾⣿⠟⣡⣿⣿⣿⡟⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⠀⠀⢸⡇⢠⠋⠀⢠⣿⠁⢠⣿⠀⢠⣿⣿⡏⢠⣿⡿⢀⣾⠋⣠⣿⣿⣿⡿⢋⣴⡿⢫⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⣿⠀⠀⢸⣿⠆⠀⠘⣷⡎⠀⠀⣼⡟⠀⢸⣿⠀⣼⣿⡟⣠⣿⡿⢁⣾⠏⣴⣿⣿⡿⠋⣠⣾⠟⣡⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠈⠀⠀⠈⠁⠀⠀⠀⢻⡇⠀⠀⣿⡇⠀⠸⣿⢠⣿⣿⢰⣿⡟⣠⣿⣿⣾⣿⠟⠁⣠⣾⠟⠁⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⠀⠀⠀⠀⠀⢠⣀⣀⢸⡇⠀⢸⣿⡇⢰⠀⣿⣸⣿⣧⣿⣿⣼⣿⣿⠿⠋⠀⣤⠾⠋⠀⢀⣼⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⡄⠀⠀⠀⠀⣼⠟⠻⠀⡇⠀⠘⣿⡇⠀⢤⢻⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠈⠀⠀⠀⣠⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠿⠀⠀⠀⠘⠧⠀⠀⣠⠀⠀⠀⣿⡇⠀⢻⣸⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⢀⠄⢀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⢠⣤⣶⠟⠀⠀⠀⢸⣿⠰⡇⣿⣿⠿⠀⠀⣠⠀⠀⠀⣠⡴⠃⣰⡿⣻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⠶⠶⠛⠋⠀⠀⠀⠀⠀⣸⣿⣆⢸⢸⣿⣄⣴⡟⣡⠞⣠⡾⠋⣠⡾⢋⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣿⣄⣿⣿⣿⣯⣾⣧⡾⠋⣠⣾⠏⣠⡾⠋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⣿⣼⣿⣿⣿⠟⠁⣠⡾⠏⢁⣴⠏⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣴⠛⢿⣿⣿⣿⣴⡾⠋⠀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⢧⡐⣈⠻⣿⣿⣯⠀⣠⢾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣸⣷⣯⡾⠛⣿⣿⣿⠋⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣿⣿⣿⣿⠇⣼⣿⣿⠇⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴⣿⣿⣿⣿⣦⣿⣿⡟⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⢻⣿⡇⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠈⣿⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣇⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣡⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁

⠀⠀⠀⣄⠀⠀⡇⣤⣤⣤⣄⣀⡀
⠀⠀⠀⢸⣿⣿⣦⠾⠿⣿⣿⣿⣿⣷⠄
⠀⠀⠀⠋⠉⠛⠋⠳⠀⠀⠙⠻⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠰⠒⠀⠀⠈⢧⠀⠀⠀⠙⢿⣿⣿⣦⡄
⠀⠀⠂⠀⠀⠄⠀⠀⠀⢸⠇⠀⠀⠀⠀⠙⠿⣿⣿⡦
⠀⠆⠀⠀⠀⠀⢤⡤⠴⡇⠸⠀⠀⠀⠀⠀⠀⠀⠉⠛⢓⡀
⠄⢤⠀⡀⠀⠔⠉⠀⠀⢹⡄⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣠⣴⣦⣄⡀
⠀⠀⠠⠵⠀⠀⠀⠀⠀⠀⡇⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠶⣾⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠘⡏⠙⣿⣿⢿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢹⡀⠘⠟⢿⡙⢿⣿⣿⣿⣯⣽⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣄⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⠈⢻⢿⡿⠿⢿⣿⠋
⠀⠀⠀⠀⠀⢀⠠⠒⠂⠈⠙⢦⣀⠙⢄⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠰⠁⢀⠄⠛⠶⠶⠿⠿⢦⣤⣿⣶⠒⠀⠀⠀⣼⣧⣀⡀⢀⣀⣀⣀⣤⣤⣶⣾⣿⣿⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⡴⠚⠻⠿⠿⠟⠛⠛⠛⠉⠉⢿⠉⠛⢿⣿⢧⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⡹⡱⡣⣄⠀⠈⢧
⠀⠀⠀⠀⠸⠀⠻⢦⡄⠀⠀⠀⠀⢀⠂⢐⡡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠘⣎⢣⡀⠘⣷⡄
⠀⠀⠀⠀⠀⠈⢠⣿⡇⠀⠀⠀⠀⢸⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⡰⠃⠀⢣⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠈⠳⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⢀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡘⠋⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣰⠋⠀⠀⠀⠀⠘⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⡱⡶⠁⠀⠀⠀⠀⠠⠆⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠀⠀⠀⠀⣾⣶⣶⠉

⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⣤⡀⠀⣿⢣⠀⣀⡀
⠀⠀⠀⡸⠿⢷⡟⠚⠿⣿⣿⣷⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡄⠀⠀⠠⣤⣀⢀⣶⡄
⠀⠀⠘⠀⠀⠈⠳⠦⠄⠀⠙⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡆⢠⣶⣤⠈⢹⣿⣇⠀⠀⢀
⠀⠀⣀⠀⠀⢰⣷⠆⠀⢠⡀⠀⠈⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣵⠿⡏⢿⣄⡚⠁⢸⣷⣞⡁⠀⠀⣀
⠀⠀⠘⠀⠀⠀⠈⠀⠀⠈⣿⣦⡀⠀⠙⣿⣿⣅⡒⠒⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠦⣬⣥⣀⠠⣿⣿⣿⡟⠛⠁
⠀⠠⠀⠀⠀⠀⢀⣄⣀⣠⣿⣿⣿⣦⡀⠈⢿⠿⡛⠉⠁⠀⠀⠀⠀⠠⣦⠀⠀⠀⠀⢰⣶⠌⣽⣦⠻⣿⣿⣶⣤⡀
⡄⢀⣤⡀⣀⣠⡾⠟⢻⣿⣿⣿⣿⣿⡇⠀⠈⢷⣌⠳⡒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣿⣿⡀⢹⣿⣿⣿⣯⡁
⢿⣿⣿⣿⠏⠀⠀⠀⢸⣿⣿⣿⣿⠟⠀⠀⠀⠀⠻⣷⣙⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⢿⣿⣿⣿⣥⣀⠀⠀⠀⠀⠀⢀⣀
⠀⠉⠉⠉⠀⠀⠀⠀⢸⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠈⢿⡿⡷⠶⠂⠀⢸⠀⠀⠀⢠⣀⣠⣾⣿⣿⡟⠀⠈⢻⣿⣿⣯⣭⣍⠁⠀⢠⡾⢛⣛⡛⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⣸⣄⣤⣤⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⢱⠓⠒⠛⠢⢄⣰⣯⣾⣿⣿⣿⣷⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⡝⣿⣿⣿⣿⡇⠀⠀⠀⠀⠃⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣼⣿⣿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣽⣿⠇⡇⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡉⠉⠻⣿⡏⣿⡆⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡇⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢹⡇⢸⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⠀⠹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⡇⠀⠀⢸⡇⣼⢡⡟
⠀⠀⠀⠀⠀⠀⡀⠄⣀⣾⣿⡤⠀⠀⠀⠀⠀⢠⣿⣄⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⣿⡄⠀⠈⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⡟⣸⣿⡇⠀⠀⢸⡇⣿⢸⠃
⠀⠀⠀⠀⠂⣡⣴⣿⣿⣿⠏⠀⠀⠀⠀⣀⣴⣿⣽⣿⣿⣶⣤⣤⣄⠼⠀⠀⣠⣶⣿⣿⣧⠀⠀⠀⠀⠀⢀⡎⢠⣿⣦⣤⣶⡟⣰⣿⣿⠀⠀⠀⢸⡇⠸⠈
⠀⠀⠀⠀⣴⣿⠿⣿⠟⠁⢀⣠⣤⣶⡿⠟⠛⠃⠉⠙⠛⠛⢩⠛⣡⣴⣶⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣼⢇⣾⣿⣿⡿⣋⣼⣿⣿⡏⠀⠀⠀⢸⠁
⠀⠀⠀⠀⢹⡇⢰⠁⢀⣾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠿⠿⠿⠟⠛⣻⣿⡇⠀⠀⢀⣾⣿⣾⣿⣿⣿⣾⣿⣿⣿⡿⠁⠀⠀⠀⠀⠇
⠀⠀⠀⠀⢸⡇⣨⣶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⡟⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣰⣿⡿⣿⡏⠁⠸⣿⣿⣿⣿⣿⠃
⠀⠀⠀⢀⠛⡻⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⣼⡿⢿⠀⣿⡇⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠈⠳⠁⣿⠉⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣿⡇⢸⣿⣿⠀⠀⠀⣿⣿⣿⠏
⠀⠀⠀⠀⠀⣴⠟⠠⣙⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⡄⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠀⢸⣿⡇⠀⠀⢠⣿⣿⡟
⠀⠀⠀⠀⣸⣿⣆⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⠁⣠⠇⠀⠀⠀⠀⠀⠈⢠⢿⣿⠃⠀⣾⣿⣇⠀⢠⡾⢿⡿
⠀⠀⠀⢀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠀⠀⠀⠀⡆⠈⠀⣿⠀⣸⢿⣿⣿⠀⢸⢁⣾⡇
⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⢁⣸⣿⠀⢿⣸⡿⠏⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢸⠀⣹⡏⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⣿⣧⠀⠀⠀⠀⠀⢀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶⣶⣧⣿⣿⣿⠀⠀⠀⠀⢀⠦⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⢿⣿⣿⡿⠀⠀⠀⠀⢌⡀⣸⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠁⠒⠠⠄⠸⡄⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠊⠉⠉⠉⠁⠒⠲⢄⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣄⠀⠻⣄⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠈⡳⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠉⠓⠤⡴⠁⡴⠁⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢂⠀⢀⣀⠀⠀⠀⠀⠘⠣⡀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⡠⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢿⡎⠔⣵⢶⡇⡀⠀⠀⠀⠀⠈⠢⣀⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠘⠠⣄⠀⠁⠂⠤⣀⠀⢀⡤⠊⠀⠒⠊⠰⡀⠛⡻⢁⡅⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⣸
⠀⠂⠒⠒⠒⠠⢀⠀⣀⡾⠋⠀⠀⠀⠀⠀⠀⠈⠉⢀⠎⣵⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣿⠀⠀⠀⠀⠀⠀⡏
⠀⠀⡔⠀⠀⠀⠀⠚⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⣰⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠞⣁⠔⠁⠀⠉⠐⠒⠣⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⢻⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠙⠦⣤⣀⣀⣠⠎⠀⠀⠀⠀⣀⡤⠚⣩⣌⡉⣹⠶⠦⢤⣀⡀⠀⠀⠀⠈⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢸⠀⠀⠘⣆⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠉⠂⠠⠄⠔⠂⡹⠁⢠⠂⠀⠣⢄⣀⣠⠔⠊⠁⠉⠓⠲⢤⡴⠋⠀⠀⠀⠀⠀⠀⢀⣀⡤⡾⠉⠀⠀⣸⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠋⠉⢀⠔⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠉⠳⣆⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⣀⡼⠀⠀⠀⠀⠀⠀⡇⠀⠀⠳⣄⣀⣢⠀⠀⠀⠀⢸⠀⠀⠔⠉⠉⠁⠀⢲⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠠⠤⠴⠃⠀⠀⡜⠁⠈⠙⡇⠀⠀⠀⢸⠀⠀⠙⠧⠤⠼⠆⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠇
⠀⠀⠀⠀⢠⠋⠁⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⣀⣀⣀⡤⠤⠤⠷⠶⠦⠤⠿⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠃⠀⠀⢸
⠀⠀⠀⠀⠈⠒⠧⠴⣅⡴⣒⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣠⠀⢀⣄⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣀⣠⠂⠀⢰⠒⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠆⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠀⠃⠀⠀⠀⠊⢀⢀⠄⠰⡆⢀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠐⠛⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠢⠂⠀⡀⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⣠⠔⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⣀⡠⠀⠀⠀⠀⠂⠀⠀⠀⠀⡌⠀⠀⠁⠀⠀⠀⠀⢀⡴⠇
⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⠃⠐⠧⠒⠒⠈⠀⠈⠒⠂⠚⠀⠄⠄⠐⠈⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠁⠀⡈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣀⠤⠀⠤⠄⠠⠤⠤⠄⠀⣀⡀⠀⠀⠀⠀⣀⡠⠔⠀⠀⠀⠀⠀⠉⠓⢄⡀⡀⠀⠀⠀⠀⠴⢠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣧⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢻⡀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⣀⣤⣤⣤⣀⣀⣠⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⢻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠙⠲⢖⠲⣄
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠉⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡃⢹
⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠿⠿⠧⠤⠄⠀⠚⠉⠉⠀⠀⠉⠙⠲⣆⠘⠶⠖⢀⡼
⠀⣾⣿⡿⠟⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠠⣌⠇
⢠⡟⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⡑⠶⡿
⠈⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠸⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⡀⠀⠀⠀⠀⠀⠀⡇⢰⠄⠀⠀⢨⡇⠉
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠿⠃⠀⠈⠉⢉⠝⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣠⠴⠧⠞⠀⠀⢰⣸⠃
⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⠛⠁⠀⠀⠀⠓⣕⠊⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣡⣶⢆⠀⠀⠀⢊⡿
⠀⠀⠀⢸⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⡏⠀⠔⠀⠀⣀⣤⠄⠴⠂⠀⠀⠀⠀⠀⠀⢀⣷⣤⣤⡶⢶⡿⢿⡇⠀⠣⣀⢀⠈⡇
⠀⠀⠀⢸⣿⢹⣿⡟⠀⠀⠀⠀⠀⢹⡲⠜⠈⡆⠀⣠⠞⠁⠘⢢⠀⠀⣀⣦⣤⡄⠀⠀⢸⠀⠀⠀⠀⠈⢆⠀⢧⠀⠀⠈⢇⠀⢸
⠀⠀⠀⣾⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⣵⠀⠦⢀⠠⡏⠀⠀⠀⠈⡄⠐⢸⣿⡟⢰⠀⢸⣸⠀⠀⠀⠀⠀⢨⡆⢼⠇⠀⠀⠈⢳⣦⢳
⠀⠀⢰⣿⣇⣿⡇⠀⠀⠀⠀⠀⠀⢰⠁⠧⢀⣬⠀⢸⠀⠀⠀⠀⢱⠀⢸⣿⡇⠀⢃⠀⡏⠀⠀⠀⠀⠀⠈⡆⡞⠀⠀⠀⠀⠀⡄⣸⠄
⠀⠀⠈⣯⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⡃⠀⡈⠀⠀⣰⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⢸⢀⢇⠀⠀⠀⠀⠀⢠⢣⠃⠀⠀⠀⠀⠀⢸⢹
⠀⠀⠀⠓⠲⣾⢿⣆⠀⠀⠀⠀⠀⠀⢰⠀⠁⠀⣸⠁⠀⠀⠀⠀⠘⡄⣜⢩⠿⠳⡘⣄⣸⠀⠀⠀⠀⠀⡜⢸⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢸⠀⡁⢠⡇⠀⠀⠀⠀⠀⠀⡇⡇⠀⠉⠉⠁⢸⢙⠀⠀⠀⠀⢠⢃⠞⠀⠀⠀⠀⠀⠀⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡡⠒⠁⠀⠀⠀⠀⠀⢠⠁⡇⠀⠀⠀⠀⠈⡌⡄⠀⠀⢰⠃⣨⠆⠀⠀⠀⠀⠀⠀⡸⠹⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢺⠁⠀⠀⠀⠀⠀⡇⣸⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢸⣇⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠒⠋⠀⠀⠀⠀⠀⠐⠁⢸⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⢟⣽⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⢸⣧⣤⣤⣤⣤⣤⡖⠁⠀⣼⢏⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⣿⣿⣦⣰⡟⠸⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⢻⣿⣿⣿⠞⠁⢿⡿⣿⣿⣿⣿⣿⣿⣿⣷⣖⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏⠀⠈⢋⣿⣁⠀⠀⠈⠻⣬⣛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠋⣠⡾⢡⡾⠋⠁⠀⠀⠀⠀⠈⠙⠻⠷⣶⣦⣭⣭⣭⣭⣻⣿⣿⣿⣿⡟⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣣⣾⡿⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢻⣿⠛⠛⠻⠿⢿⣶⣤⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠟⠁⣠⠂⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣇⠀⠀⠀⠀⠀⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠏⠀⠀⢸⡏⠈⢳⡄⠀⠀⠀⢀⣀⡘⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⣿⣄⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠁⠀⠀⠀⢸⠁⠀⠀⠹⣦⣈⣩⣭⡙⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠁⠀⠀⢰⡆⠈⠀⠀⠀⠀⢸⡀⣾⣿⣷⡀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠁⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠘⣿⣈⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⣸⣿⣿⣿⡟⠀⠀⠀⠀⢷⡀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠈⣿
⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⢷⣼⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠸⡇
⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⢹⡀
⠀⠀⠀⢠⣿⡟⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢷
⠀⠀⠀⣸⣿⠁⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠸⡇
⠀⠀⠀⣿⡏⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷⣶⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠐⣄⣻⡀
⠀⠀⢰⣿⠁⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢹⣿⣿⣄
⠀⠀⢸⡏⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢻⣿⠃⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⣿⣿⣿⡇
⠀⠀⣾⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⢸⣿⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⢀⣴⣶⣷⠲⢬⢻⣿⣿⣿⣿
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⣿⣿⠀⠀⠀⠀⠀⠀⠳⡈⢶⣄⠘⢿⣿⣿⣷⡄⠏⣿⣿⣿⡉
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⢡⡀⢻⣷⡀⠙⠛⠋⣱⣶⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠹⣿⣷⣶⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠃⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠛⠻⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⢸⣿⣿⣿⡄
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⢸⣿⣿⣿⣧
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⢸⣿⡏⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⢸⣿⠁⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢸⡏⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⣿⡇
⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⢡⣦⣴⠻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠻⠻⣇⠡⠞⠛⠛⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢷⣦⣄⡀⠛⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣿⣿⣿⡿⠁⠀⡄⠻⠂⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣥⣿⣿⣿⣿⠃⠀⠀⡇⠈⠃⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⣿⣿⣿⣿⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢛⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⣿⣄⣴⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⣿⣿⠿⠏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠶⠶⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⣭⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⢀⣄⣠⣀⢸⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠙⣷⣤⣀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠙⣿⡿⣿⣿⡿⢦⡀
⠀⠀⠀⠀⠀⠘⠛⠛⠛⣋⣽⠿⠟⣻⠿⠟⠁⠀⠀⠀⠀⠀⢶⣆⠀⠀⠘⣿⡟⠻⣄⡠⣀⠀⠀⣿⣆⡀⠀⠀⠀⠀⠀⠊⠀⠀⠙⠻⣦⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⢹⠃⠀⠀⠀⠉⠙⣶⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢸⡀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠇⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⢳⣄⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⢸⡇⠀⠀⠀⣠⣿⣿⣿⡿⠋⠀⠀⠀⠈⠳⣄⠘⢧⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⣇⠀⠀⠀⠹⣿⣿⣿⠅⠀⠀⠀⠀⠀⠀⠈⣷⣬⣿⡄⠀⠀⠀⠀⠀⠉⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠛⠇⠀⠀⠀⠻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡇⠀⠀⠀⡀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢠⡄⠐⣷⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠘⣿⣧⣿⣷⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠉⠻⢦⡀⠀⠀⠀⠀⠘⢿⣿⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢐⡀⠀⠀⠀⠀⠈⢳⡀⠛⣦⠀⠀⠀⠷⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠘⠿⣿⣿⠆

⠀⠀⠀⠀⠀⠀⣴⠃⢀⡴⢖⣶
⠀⠀⠀⠀⠀⠀⣿⡀⢸⡅⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤
⠀⠀⠀⠀⢀⣠⣬⣿⣦⣝⣶⣦⣴⣶⣾⣿⣿⣶⣾⣿⣿⣿⣯⣤⣤⣤⣀⣀
⢀⠀⢀⡴⢋⣴⣶⣿⣿⣿⠿⠛⠉⠁⢠⡿⠋⠀⠀⠀⠈⠻⣧⠀⠈⠉⠉⠛⠛⠷⢶⣦⣤⣀⡀
⠙⠶⠋⣰⡿⣻⣿⣿⠟⠁⠀⠀⠀⠀⣿⡇⣾⡆⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣦⣄⡀
⣀⣀⣴⠟⣱⢿⣿⠋⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣤⡀
⠀⠀⣠⠼⠁⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢶⣶⡶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⡄
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆
⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢀⣤⣿⣧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⢰⣿⠏⠁⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠰⣶⣦⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡿⠃
⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠉⠉⠉⢻⡿⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉
⠀⠀⠻⣿⣶⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠠⠤⠶⠾⠛⠋⠻⣦⣀⠀⠀⠀⣠⡆⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠏⠉⠉⠉⠛⢿⣦⡀
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠛⠋⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣀⡄⠀⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠀⠀⠀⢈⣿⣴⣶⣶⣦⣤⣀
⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠉⠻⣷
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⣿⣷⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⡶⠾⠿⠛⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⡠⢫⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⡐⡱⠁⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋
⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⡀⠀⠀⠀⠀⠀⢴⣦⣈⠏⡴⣤⣤⣤⣤⣤⡶⠶⠞⠛⠉⠁
⠀⠀⠀⢰⣿⠁⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠹⢿⣦⡀⠀⠀⠀⣠⣿⠋⠊
⠀⠀⠀⣸⡟⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠙⠿⣷⣶⣿⡿⣷⣷⣤⡄
⠀⠀⠀⣿⠇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⢿⣟⣾⣷⣯⣭⡶
⠀⠀⢠⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⢠⢊⠟⠿⣾⠏⠉
⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠁
⠀⠀⢸⣿⣤⣤⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣾⠿
⠀⣀⣨⡍⠉⠉⣿⠋⠉⠉⠉⣿⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⠁
⠀⠻⣿⣯⣤⣤⡿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠉⢉⣡⣤⣤⡤⠤⠤⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣧⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⢰⡏⠁⠀⡄⠀⢀⣀⡄⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⣤⣀⠀⠀⢰⠀⠈⣻⡆
⣀⣀⣀⣈⣻⣶⣶⣷⣚⣋⣉⣷⣶⣛⣀⣀⣀⣀⣀⣀⣀⣀⣘⣷⣶⣏⣙⣛⣷⣾⣶⣾⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠻⣿⣿⣿⣿⡇⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢿⣿⡇⠀⠀⠀⢘⡔⠚⠛⢃⠸⠓⣟⠟⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡇⠀⠀⠀⠘⠙⠛⠛⠛⠛⠂⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠠⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠶⠀⠈⠣⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣦⣌⣁⣴⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣏⠉⠚⢱⣾⣿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣄⣀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⣴⣀⣄⣄⠀⠀⠀⢿⢿⠁⠀⠀⠀⠻⠿⠃
⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣧⠘⠛⠀⠀⠀⢰⣶⣾⣶⣶⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣧⡀⠘⠂⠀⠐⠂
⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠉⠉⠋⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⣿⣿⣿⡿⠿⡫⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣶⣀⣀⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⠹⠍⠋⡃
⠀⠀⠀⠀⠻⣿⣇⢻⡀⣷⣼⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⡿⣟⣿⢿⣏⣻⠛⠛⢿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣼⣀⡔⠁
⠀⠀⠀⠀⠀⣽⠿⠿⠿⠛⠻⣿⣿⣿⣿⣿⡃⠘⣿⢯⣯⠿⣅⠿⣿⣮⣯⣯⣳⠒⠈⠘⠿⠟⠛⣹⣿⡋⠉⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠸⣷⣶⣶⣷⣶⣿⣿⣿⣿⣽⡉⠲⠈⠁⠙⠃⡀⡉⢘⠃⠒⠎⢳⣾⣿⣶⣾⣾⣿⣿⣿⣿⣿⣷⣿⡿⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⢶⠆⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⡀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⢀⡀⠀⠐⠀⡇⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠆⠀⢻⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠘⠀⠀⠒⠊⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡄⣠⡀⠀⠀⠀⠀⡀⠑⠲⠴⠒⠁⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠁⠀⠁⠈⠐⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⢀⠄⠀⢄⠀⠀⠀⠀⠀⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠉⠢⢄⣀⣀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠘⡀⠀⠀⠄⠁⢠⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠀⡐⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⡄⠀⠀⠀⠀⠡⣬⠀⠸⡉⠀⡜⠀⠀⠀⠀⠀⠀⡠⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢀⡜
⠳⠄⠀⠜⠁⠐⠀⡅⠉⠉⠈⠀⠀⠀⠀⠀⡠⠊⡜⠓⠂⠤⢤⠊⠉⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⢠⡄⠠⠤⠔⠈⣠⣾⣤⣀⡀⠀⠀⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠈⢦⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠤⠤⠐⠋⠁⠙⠁⡐⠀⢹⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⡄⠀⠀⠀⠑⢤⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠀⠀⠘⢦⣀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⣀⠵⠀⠀⢀⡤⠊⣀⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠂⠤⠤⠤⠄⣀⡨⠀⢨⠁⠒⠚⡄⠱⡤⠤⠄⠒⠊⡡⠐⠂⠉⠀⠀⢀⠔⠋⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⠄⢀⠀⢠⠤⠤⠀⣠⠖⠒⠚⠒⠒⠢⢼⡄⠱⡤⣄⣄⣀⡟⠉⠉⠒⠒⠞⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢾⣀⣼⣀⡸⠀⠀⠀⡇⠀⡄⠀⠀⡀⠀⣄⠈⠂⠘⢄⠀⠸⣀⡆⠀⡄⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⣆⣀⣀⡧⠤⠇⠈⠒⠢⠚⠀⠀⠀⠈⠉⠙⠒⠚⠁
⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢫⢶⣇⣀⣠⣄⡀⣠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⡿⢠⠃⢸⠋⠉⠉⠉⠭⣕⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⡇⠘⠀⡞⠲⢤⠀⠀⠀⣐⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⡤⠃⠀⠀⠀⠀⠀⡇⢠⢱⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡺⠂⠀⢠⡞⠀⠀⠀⠀⣚⣉⣙⠒⠁⢺⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⢡⠃⠀⠀⠈⣷⣯⣸⠁⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⡸⢸⡀⠀⠀⠀⠙⠛⠋⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⡇⠀⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⢀⡤⠒⠚⠓⠦⡀⠀⣀⣀⣀⠀⠀⠀⢰⡇⣀⡾⠀⠀⠀⠀⠀⢰⠦⣄⠀⠀⠀⢀⠀⢧
⠀⠀⠀⢠⠎⠀⠀⢠⠒⢢⡽⠋⠁⠀⠈⠉⠛⠒⢫⠾⠟⠁⠀⠀⠀⠀⠀⠸⡄⠸⡔⠀⠀⠫⠁⢸
⠀⠀⠀⡞⠀⠀⠀⡇⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠑⠤⣤⠤⠒⠁
⠀⠀⠀⢷⠀⠀⠀⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠒⠤⡀
⠀⠀⠀⠘⡄⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⢀⠀⡼⡴⠃⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⢸⣀⠴⠤⠶⠀⠀⠀⡼⠁
⠀⠀⠀⠀⢰⣾⠜⣹⠀⠀⠀⢀⡤⣶⠋⠉⠓⡦⢤⣀⣀⠀⣹⡄⠀⠀⠘⡇⢴⠂⠁⡀⠀⡸⠁
⠀⠀⠀⠀⠈⠁⠀⢸⠀⠀⠀⠉⠀⠘⢆⠀⠀⠹⡀⠀⠈⠉⠉⢳⠀⠀⠀⢱⡘⣆⠀⢈⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢸⠀⠀⠈⡅⠀⠀⢵⠀⠀⠀⠀⠸⡆⠀⡀⠀⢹⡈⠓⠋
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣈⡇⠀⠐⣧⣀⣀⣤⠇⠀⠀⠀⠀⢏⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣇⠀⢀⡀⢀⡀⢀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⣿⣿⣿⠏⠀⠀⠙⠛⢁⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡿⠛⠀⠀⠀⠀⢿⠋⠁⠀⠀⠀⠀⠀⠀⠩⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠴⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠘⠃⠀⠀⠀⠀⠀⢠⠊⠀⠀⢤⠶⠛⠁⢸⠀⣴⡘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡰⠁⠀⠀⠀⢸⠀⠃⠹⡄⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸⡎⣆⠀⠀⡞⠀⠀⡴⠀⣀⡀⠀⠀⠀⡟⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢷⠈⠓⠐⠀⠀⠘⠁⢾⢯⡝⡷⠀⠀⠁⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣜⡆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠇⠀⠀⢸⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠁⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣤⣀⠀⢀⣀⣤⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠀⠙⠻⠿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣄⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡄⠀⠀⠀⣾⠿⢄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠰⣿⣄⣼⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣧⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣹⣦⣭⣿⡿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢻⣮⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣀⣀⣰⣾⣐⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠒⠶⣾⣿⣿⣿⡿⠟⠉⠀⠁⠈⣿⢏⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠄⠀⣀⠀⠀⠀⠀⠀⠀⠲⣶⣿⣿⡿⠿⠛⠉⠀⠀⠀⣴⠀⠀⢠⡈⠢
⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⣿⠷⠤⢼⣤⣀⠘⡦
⠀⠀⠤⠭⣿⣿⣿⣿⣿⣼⡏⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠃⠀⠀⠀⠈⠉⠱⠔⠊
⠀⢠⣤⣾⣿⣿⣿⣿⠏⢿⣧⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⢠⣾⣿⣿⣿⡏⠀⠘⢏⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⡴⡿
⠀⠀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⡀⠀⠀⣠⢿⣷⣄⡀⠀⠀⠀⣼⡆⠀⠀⣷⠤⠀⣠⣤⠇
⠀⠀⢿⢿⣿⣿⣿⡻⠀⠀⢿⡐⣧⠶⢛⣡⣿⣿⠟⠈⠙⠒⢲⠛⠃⣠⣠⣀⣀⡪⡔⠁
⠀⠀⠀⠸⠛⣿⡿⠃⠀⠀⠈⠙⠓⠀⠸⣅⠙⠁⠀⠀⠀⠀⡌⢀⣀⡀⠀⠻⡍
⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⣀⡈⡱⠛⠄⠁⠀⠀⢱⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠀⠀⠀⣽⣰⡿⠛⠁⠀⠀⠀⠀⠚⠧⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠿⠿⠿⠟⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⢑⣰⠇
⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⠂⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠂⡀⢀⠀⠐
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣶⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠙⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠠⡀⠈⢧⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⡞⠑⠠⣀
⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⣰⡁⠀⡉⡆⠀⢘⠀⠁⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢴⡀⣀⠀⣢⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⡴⣂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠀⠀⠀⠀⠀⠈⠈⠸⡷⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⣀⣤⣤⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⢰⡿⠀⢀⣾⡿⠋⠁⠀⠀⠀⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠑⠧⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⢠⣿⠃⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠘⠐⠲⠞⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠆⠁⠀⠀⠀⠀⠀⢠⣾⠏⠀⢀⣾⠇⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⢀⣾⠏⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⢀⣾⠏⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠃⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⣠⣿⠃⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣅⠀⣴⡿⠁⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣁⣨⣿⠟⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠀⠀⣿⡇⠀⠀⠀⠘⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⡿⠛⠁⠀⠀⣠⣶⡿⠋⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⣨⡷⠀⢀⣴⣾⠟⠉⠀⠀⢀⣿⠟⠉⠉⠙⣦⠀⠀⠀⠀⠀⠀⣴⣿⣷⣦⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣡⣴⠟⠁⢀⣾⡟⠁⠀⠀⠀⠀⠈⣿⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠻⣿⣿⣿⠇⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠁⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⡾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⡿⢷⡄⠀⠀⠀⣸⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠘⣿⣿⣿⣿⠀⠀⣴⣾⣶⣤⠀⠀⠘⢿⣿⣿⠏⠀⢀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⠹⣆⠀⠀⠈⠙⠋⠁⠀⠀⢻⣿⣿⣿⠇⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⣀⣤⣾⠿⠋⠀⠀⠀⠀⠀⠀⠨⢄⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⣦⣤⣤⣤⣤⣴⣶⡾⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠿⠋
⣦⣤⠀⠀⠀⠀⠀⣤⠾⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣤⣄⣀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠈⠙⠛⠿⠿⠛⢫⣅⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⣴⣶⢞
⣤⣄⣤⣤⣤⣀⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⠊⠀⠀⠀⠀⠶⠿⠿⠿⣿
⠟⠛⣿⣿⣟⣋⣉⣉⣉⣁⣀⣀
⠀⠾⣿⡿⠿⠿⠛⠛⠛⠋⠉⠉⠀⠀⢀⣠⢀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣤⣤⡄⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⡀
⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⠟⠉⠛⣦⣄⣠⣶⠂⡉⣿⡁⡼⠻⣿⣿⣿⣶⣿⡟⠻⣀⣠⣤⣤⣴⣿⣿⣿⣿⡿⠛⠻⠄
⠀⠀⠀⠞⡴⡿⠋⠉⠛⠿⡏⠀⠠⠿⠿⠻⠿⣿⢿⠋⠙⠓⢿⡿⠼⣿⠻⠛⢿⡧⡞⣹⢿⡿⢿⠿⠿⣿⡿⢿⣇
⠀⠀⠀⠠⠂⢰⠀⠀⡜⠒⠁⠠⠀⠀⠀⠀⢲⡇⠀⠀⠀⠀⠀⠇⠀⠈⠂⠀⠀⠁⡜⠁⠾⠀⢹⡄⠀⠱⣼⠈⡆⢷
⠀⠀⠀⠀⠘⠋⠙⠀⠁⠛⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠉⠁
⠂⠀⠀⠀⠚⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠾⠿⠿⣿⣿⣿⣶⣶⣤⣄⡀⠀⠀⣀⡴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣠⣴⣶⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣶⡞⣩⣾⢃⡠⢶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠡⢞⣿⣿⣿⠟⠉⣀⣤⠴⣶⣶⣾⣿⣿⢏⣾⢿⣿⣿⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⡇⣰⣿⣿⣿⢋⣴⠟⡩⢖⠋⠉⠀⠀⠀⠉⠈⠁⠛⠛⠶⣌⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢻⢻⣿⣿⣿⡿⠁⣨⡾⠃⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠙⣿⠢
⠀⠀⠀⠀⠀⢸⣧⠀⡇⠀⣆⣿⣿⣿⣿⠇⢰⣿⠁⠀⠀⢀⡴⠃⠀⠀⠀⣀⣿⣧⠀⠀⠄⢈
⠀⠀⠀⠀⠀⠀⡏⠻⣿⢸⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢠⣿⣷⠃⠀⢀⣀⠽⠿⠟⢻⡄⠈⢹⡀
⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⣿⣿⣿⠋⠹⡇⠈⠃⣼⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⢷
⠀⠀⠀⠀⢀⠀⢿⡀⣷⣼⣿⣿⣿⡇⠀⠈⠀⢰⣿⠀⣿⣿⣿⣷⣶⣶⣦⣤⡤⠀⠀⠀⠑⠀⠈⢇
⠀⠀⠀⠀⠘⣷⣼⣧⢸⣿⣿⣿⣿⣿⡀⠀⠀⠘⣿⠀⢻⣿⣿⣿⣿⣏⠀⠈⢿⣦⠀⠀⠠⢄⠈⠘⢦⡀
⠀⠀⠀⠀⠀⠘⣆⠙⠈⣿⣿⡿⢿⣿⣷⡄⠈⢄⠈⠣⠘⣿⣿⣿⣿⣿⣆⠀⠈⣿⣀⠀⠘⣿⠀⠀⠀⡇
⠀⠀⠀⠀⢀⡀⠘⣆⢠⣼⣿⡇⠀⢈⠙⠿⣦⡀⠳⣦⡀⠘⠋⠀⠈⢻⣿⣦⠀⣏⣈⠓⠄⣂⣀⣠⣴⡟
⠀⠀⠀⠀⠀⠹⣦⣘⣦⠻⣿⣿⡄⠈⢿⣶⣤⣍⣑⡈⠻⢷⣄⠀⠀⠀⠹⣿⣷⡈⠙⠷⠶⠤⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠘⢿⡻⢷⡿⣿⣿⣄⠈⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣠⣬⣿⣿⣦⠈⠻⣿⣷⣬⡉⠉⠛⠛⠛⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⣿⣿⣿⠷⣄⠙⢻⣿⣿⣶⣶⣤⣤⡄⠀⠀⣿⡟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣦⣄⡀⣈⣻⣿⣿⠿⠟⠀⠀⠀⠛⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣶⣤⣤⣀⣠⣤⠆⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⣿⣿⣯⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⣠⠄⠀⠀⠁
⠀⠀⠀⡀⣀⠀⠀⠰⠁⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠍⣶⠐⠆⠀⠀⠘⡄
⡔⠈⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡄⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⡈⠁⠀⠀⠀⠀⡅
⠑⠄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡶⢿⠆⠂⠀⢴⠀⠃
⠀⠀⢆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⢈⠣⠀⢀⣀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠀⠀⠈⠤⡀⠀⠀⠀⡌⠀⠄
⠀⠀⠀⠄⠀⡀⠘⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⠰⠀⢀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⡄⠀⠀⠀⠢⠤⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⡀⠀⠀⠀⠀⠀⢀⡠⠞
⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠤⠀⠉⠑⠒⠒⠀⠋⠭⠀⡁⢀⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣔⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢦⣀⣄⣀⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢦⠀⢀⣀⣘⣶⣿⣿⣿⢿⣿⣿⣿⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⣭⣽⣿⣿⣿⣿⣷⡿⠿⠿⡻⢻⡷⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣤⣽⣿⣿⣿⣿⠿⠋⢁⠴⠶⢖⡀⠁⠸⢸⣷⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⠶⣿⣮⣿⣿⡿⣿⣿⠛⠁⠀⠀⠀⠀⣠⡜⠈⢤⣐⡨⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠪⠿⣿⣿⣾⣿⠟⠫⣾⠟⠁⠀⠀⠀⠀⠀⡼⣿⠀⠀⠀⡉⠃⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣾⣿⣿⣿⣿⣿⡷⠟⠁⠀⠀⠀⠀⠀⠀⣸⢟⣿⡀⠀⠀⠘⠂⠘⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⣛⣿⡿⠋⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⢣⣾⠏⠙⢦⣴⠿⡂⢓⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣤⣀⡴⠒⠋⠉⠉⠉⠉⠓⠒⠚⠛⣿⡏⠀⠀⢀⠘⣇⠀⠀⠀⠀⡀⠀⠀⣾⣿⠀⠀⠀⠈⠳⡟⢃⠼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣇⠻⣆⠀⢠⡾⠁⠀⢰⣿⡇⠀⠀⠀⠀⠀⢹⣏⡟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡿⠁⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠀⠈⢻⣄⢌⡀⠀⠀⢀⠀⣸⣿⣷⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⣾⣿⣿⣿⣿⢿⣿⢿⡇⠀⠘⣿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⡄⠀⠀⠙⠪⠛⢰⣶⡏⠀⣿⠋⢿⡇
⠀⠀⠀⣠⡴⢛⣻⣿⣿⠿⣿⣿⣿⣿⡿⠣⠛⠈⢸⣷⡀⠀⠹⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠸⠁⠀⠀⠀⠀⣠⠀⢿⡇⣸⠇⠀⢸⡷
⠀⠀⣰⣯⣶⠟⣫⠿⠛⣛⣛⣯⠿⠋⠀⠀⠀⠀⠀⢿⣧⡀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠰⠋⠁⠀⠘⠃⠋⢀⠀⣿⡇
⠀⢠⠟⠝⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡦⠀⠈⠻⣦⣄⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⡠⢴⡟⡴⠀⢠⠀⣸⣼⣿⡃
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⡀⠘⣿⣿⣿⣿⣿⣶⣾⣿⣷⠀⠀⠔⢋⠦⣀⠡⡀⠀⣘⣀⡣⣿⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣿⣿⣿⣾⣿⣿⣿⡟⠛⠳⠾⠀⠲⠦⠌⢳⡝⢿⣛⣿⡻⣿⢮⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⠀⢹⣿⣿⣿⠇⠀⠀⠀⠀⠀⠐⠶⠶⠟⠷⢨⡟⠻⣦⢴⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⠸⣄⠈⢿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⠃⠀⠈⠳⣏⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠉⢦⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠃⠀⠀⠀⠀⣸⣇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⡄⠻⢾⣿⣧⠀⠀⠀⠀⡀⠀⢀⣀⣾⠇⠀⠀⠀⠀⢀⢟⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⠿⣷⡀⢠⣾⣿⣷⣍⣼⡟⠀⣀⠀⠀⢀⢂⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢀⣹⣿⣿⣿⠟⠛⠛⠋⣴⣿⣿⣷⡖⢋⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠀⠹⣿⣿⣇⠀⠀⠀⠉⠛⠛⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣹⡏⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢷⠁⢘⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⡄⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠐⠀⢀⠀⢄⠀⠀⠀⠀⠀⢀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠨⢀⠌⠃⠀⠀⠀⠁⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⡄⢀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢠⣤⣤⣤⠀⠀⠀⠀⠀⠀⢣⠀⠀⠈⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢡⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠂⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠂⡀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢄⠁⠀⠠⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀
⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠐⠀⢀⠔⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠤⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢄⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠈⠀⠀⠀⠀⠉⠙⠀⠀⠀⠁⠀⠀⠂⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠢⠋⠀⡠⠊⠈⢐⡄⠀⢄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡀⠰⡄⠀⠀⠀⠀⠀⠀⡀⠀⢸⠄⠈⠜⠀⠀⠀⠀⠀⠀⠉⠐⠠⠄⠀⠀⠠⠄⠀⠀⠀⠄⠿⢀⣆⢠⠎⠀⠀⠀⡜⠀⠀⠀⡅⡀
⠀⠐⠀⠀⢃⠀⠈⢆⠀⠀⢀⠀⠀⡟⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢣⡀⠀⢀⡌⠀⠀⢀⡜⠀⣸⠂⠐⠐⠒⠂
⠀⠀⠀⠀⠀⠐⠤⠸⠦⣀⠜⠁⠈⠀⠀⠀⠀⠀⠀⢀⠀⠄⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠉⠉⠁⠑⠤⠤⠋⠔⠚⢁⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣵⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣷⣶⣶⣾⣿⣼⣿⣶⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣄⠀⠀⠀⠀⠀⠀⢀⡾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⣿⣶⣶⣶⣶⣅⠀⡞⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⣿⣿⠿⠻⡿⣿⣿⣿⣿⣿⣧⣉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⡈⣱⡯⠀⠀⠙⢮⣟⣻⠿⢿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⣡⡞⠕⠋⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠚⢾⡿⠿⢿⣯⣍⡑
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠟⢡⠄⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠁⠀⣿⠀⠳⣄⠠⠤⢤⣬⡄⠀⠀⠀⠀⠀⠠⣼⣷
⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⢀⡄⠃⠀⠀⠈⠈⣿⣷⡈⢷⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⢰⣿⣿⡏⠀⠀⢸⡇⠀⠀⠀⠀⣿⣍⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢀⣿⣿⡿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠘⣦⡀⠀⠘⣇
⠀⠀⠀⢰⣿⣿⡿⠀⠀⠀⢳⣸⣷⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠘⣷⠀⠀⢹⡀
⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢧
⠀⠀⢰⣿⢹⣿⠁⠀⠀⠀⠀⠀⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠘⡆
⠀⠀⣾⠇⢸⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣦⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠘⡄⢀⢹⡀
⠀⢀⡿⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡟⢷⡄⠀⠀⠀⠀⠀⠀⠀⢱⡈⣿⣷⡄
⠀⢸⡇⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⡇⠀⠈⢢⡀⠀⠀⠀⢀⣀⣈⠳⣿⣿⣿⡀
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⡇⠀⠀⠀⠱⡀⣄⠰⢿⣿⣧⡑⢿⣿⡟⠃
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢰⣿⡇⠀⠀⠀⠀⠠⡈⢷⣌⢉⣋⣵⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⣸⣿⡇⠀⠀⠀⠀⠀⠙⠦⠿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢸⡿⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⢸⠇⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⣴⣦⡤⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠀⣠⡤⠤⠤⠂
⠀⠀⠀⠀⢸⣿⣿⣽⣧⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠞⣡⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠘⢛⣭⣷⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣾⠛⡩⠭⠭⣙⠻⣷⡄
⠀⠀⠀⠀⠀⠈⠙⢻⡏⠀⠀⠀⠀⠀⠀⠐⢷⣦⣀⠀⢿⣿⣧⣎⡤⠒⠢⣌⠳⣌⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⣀⣤⣤⣤⣶⣶⣶⣾⣿⣿⣾⣿⣿⣷⣬⣁⣉⢎⠣⠈⢦⡙⢿⣦⡀
⠀⠀⠀⠀⠀⣠⣴⡾⠿⠛⠛⠉⠉⠀⠀⠀⢠⡾⠋⠁⠀⣈⡙⢿⡉⠙⠛⠿⣷⣵⣄⠙⢦⠙⣿⣄
⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠿⠍⣼⠇⠀⠀⠀⠈⠙⣷⣗⡄⢳⢸⣿
⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⣤⣤⡾⠋⠀⠀⠀⠀⠀⠀⠈⢿⣾⣸⢸⣿⡄⢀
⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡸⣧⠙⠛
⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠴⣦⣤⣀⠀⠀⠀⠀⢻⣿⣷⣽⣷⡄
⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣷⠉⠀⠀⠀⡾⠟⠻⣿⣿⠿⠊
⠀⠀⠀⠈⠙⠛⠿⠶⠶⣶⣶⣶⣶⣶⣶⡶⠶⠿⠆⠀⠀⠀⢀⣼⠋⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢀⣀⣀⣠⣤⠶⠛⠁⠀⢀⣼⡇⠀⠀⠀⢿⣦⣶⡿⠋⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠹⣯⠀⠀⠀⣠⣴⣿⣿⠃⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⣀⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠹⣿⣿⡿⠛⠁⣰⠟⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⢠⣶⣦⣤⣾⣧⣾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠈⠛⠶⠶⠞⠋⢀⣾⠀⠀⠀⢀⣿⠁⠀⠀⠀⠈⠉⢉⣿⣿⣿⡿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⠁⠀⠀⠀⣸⡟⠀⠀⠀⠀⣀⣴⡿⠋⠉⠉⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣍⣉⣛⡓⠒⠒⠶⣆⠀⣠⠤⠦⠴⠶⠶⠶⠲⣿⡃⣀⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠒⠒⠒⠶⠶⠶⢾⣿⠛⠋⠉
⠀⠀⣠⣶⣄⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣸⡇⠀⣹⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⣿⠀⢀⡟⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣤⠿⠆⣼⠁⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠿⠴⠞⠛⠛⠿⢶⣦⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠿⠷⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⠷⠶⠶⢶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⣄⣀⣀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣤⠈⢿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⢸⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣷⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣶⣦⣄⡀⠀⠀⠀⠀⠀⢀⢀⣰⠇
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣉⣛⣻⣿⣿⣴⣶⣶⠂⠉⣼⠃
⢀⠒⠦⠶⣾⣿⣿⣿⣿⣿⣿⣭⣽⣿⣾⣯⣭⣥⣤⡾⢏⡀⠀⠀⢄
⠀⢉⣛⣛⠛⣛⣿⣻⣿⣿⣿⣿⣿⡿⣡⠏⠉⢉⡍⠀⠀⠙⠛⢿⣿⡁
⣴⡿⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⠏⣴⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⢈⡁
⠉⠙⣻⣿⣟⣛⣛⣿⣿⣿⣿⠏⣰⠃⠀⠀⣀⠸⣷⣤⣀⠀⣀⣠⣿⡗⠀⡀⠀⣴⡄
⠶⠿⣟⣻⣿⣿⣿⣿⣿⣿⡟⠀⡃⠀⠀⠀⣿⡆⠹⡇⠉⠉⠉⠉⠉⠻⢮⡱⣄⢭⡇
⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢹⣿⠀⢇⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉
⣴⣾⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢻⣇⠸
⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢻⡀⢃
⢉⣻⣿⣿⣿⣿⣿⣿⣿⣿⠃⣠⣶⣶⣶⡶⠆⠀⠈⠻⣶⣦⣄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠹⣿⣷⡄⠀⠀⢀⣴⣶⣤⣴⣤⡀
⠛⢻⣿⣿⡿⢿⣿⠋⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠃⠀⠀⠀⠀⠀⢈⠉⠉⠛⠄
⠀⠀⠀⠙⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⢀⡴⠚⠛⠛⠻⣶⡄⠀⠸⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⠀⠀⠀⠀⠛⠉⢀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠙⢧⣀⡈⠻⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⠀⠀⣠⠾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣮⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡿⠃⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢡⠖⠒⠢⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⢹⡀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠊⠀⠀⠈⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠠⠂⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠊⠉⠉⠉⠉⠒⢄⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠁⠀⠀⠀⠉⠳⢤⡁⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⣀⣀⠀⠀⠈⠀⠀⠀⢀⠋⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢠⠀⠀⠀⡌⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠚⠤⡀⢠⠁⢀⠟⠒⠲⠤⠤⣀⣀⡣⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢞⠓⠀⠀⠰⡄⠀⠉⠰⢏⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⣰⠁⠀⢙⡆⠀⠀⠀⠀⠠⢲⡆⢨⡴⠊⠘⡄⠃⠀⠀⠀⠀⢣⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⡏
⠀⠀⠀⢀⡸⠄⠀⠘⠄⣀⠀⢀⠀⠓⠈⠐⡙⢾⠅⢰⠀⡆⠀⠀⠀⠀⠀⠧⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡃⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⢏⠀⢀⣄⡀⠀⢀⠜⠁⠀⠀⠀⠀⠘⠦⠔⠃⡸⠑⡀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡇⠀⠀⠀⠀⠀⡞
⠀⢠⠀⠈⠉⠁⠒⠰⢟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠈⡇⠀⠀⠀⠀⠀⡇
⠀⡆⠸⠂⠀⠀⢀⡀⠀⡗⠀⠀⠀⠀⠀⠀⠀⢀⠔⡀⠈⠐⠢⠄⣀⠀⠀⣠⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⣧⠀⠀⠀⠀⠀⡇
⠀⠣⡎⠀⠀⢀⠋⠂⠀⡇⠀⠀⠀⠀⠀⠀⡠⠊⠴⢦⣤⣀⡀⠀⠀⠉⠘⠃⠀⠀⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢻⠂⠀⠀⠸⡄⠀⠀⠀⠀⢻⡀
⠀⠀⠈⠑⠲⠼⣦⣤⡴⠁⠀⠀⢀⣠⡴⠊⠀⢠⠐⠂⠁⠀⠈⠁⢲⠀⠀⠀⡶⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⢸⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠛⢩⠁⠀⠀⠀⠁⠉⠉⢸⠀⠀⠀⢸⡀⠀⠀⢱⠀⠀⠀⠀⠀⢀⣠⠴⠖⠋⠉⣣⠀⠀⢸⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢣⠀⠀⡘⠀⣄⠀⠀⠙⠒⠚⠁⠀⠀⢸⣠⠞⠉⠀⠀⢀⡤⠊⠀⣀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⡠⠎⠀⢠⠇⠀⠈⣱⠒⠦⢴⡀⠀⠀⠀⢸⠁⠀⢀⠊⡉⠉⡀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⡏
⠀⠀⠀⠀⠀⢀⡠⠤⠠⠤⠤⠤⠄⠀⠀⠤⢤⠤⠖⠋⠀⠀⢰⠃⠀⠀⠀⠇⠀⠀⠀⢸⠀⠀⠘⠒⢦⠤⢇⡠⠔⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠶⠦⠾⠀⠀⣸⠁
⠀⠀⠀⠀⢠⠎⢠⠀⡀⠀⠀⠀⠀⣀⠤⠤⠤⠤⠤⠤⠤⠤⠿⠀⠀⠒⠒⠚⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢰⠀⠀⡄⠀⠀⡇
⠀⠀⠀⠀⢺⣀⡃⢠⡇⠀⡷⡶⢋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠾⣀⣠⣇⠀⢀⡝
⠀⠀⠀⠀⠀⠀⠉⠙⠓⠚⠹⣴⠋⠀⡞⠀⠀⠸⡏⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠦⠤⠤⠾⠃
⣿⣷⣶⣶⣶⣾⢷⣾⣷⣷⣿⣶⣶⣿⣶⣶⣶⣿⣷⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣷⣾⢷⠾⠷⠿⠷⢾⣿⠿⠿⠿⠿⠿⣿⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣷⣿⣷⣶⣆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣟⣿⣙⢿⣿⣿⣿⣤⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠹⣿⣷⣿⡟⣩⣾⣿⣟⣉⡀⡤
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣠⣿⡿⠛⠋⠉⢩⡝⠉⠻⢿⣥⣼⡖
⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⠋⠉⠀⣿⣿⣿⡿⠀⠠⣤⠖⠸⢷⠀⠀⠀⠙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⠠⣄⡁⢀⣴⡍⠀⠁⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⢿⣥⣦⢤⣶⣿⣿⣿⠃⠀⠈⠀⠀⠀⠀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣶⣿⣿⣿⣿⣷⣄⢻⣿⠋⠁⠀⢀⡀⠀⠀⠀⠀⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢸⣿⣿⣿⣿⠽⢿⡇⠀⢁⣨⣽⣷⠘⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣾⡷⢾⠿⣷⡸⣷⠇⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢿⣿⣿⣿⡷⣿⣶⣽⣷⡀⠁⠀⠐
⠀⠀⠀⠀⠀⠀⢰⣾⣿⣯⣿⠿⠿⠿⠿⢿⣶⣿⣿⣿⣿⣷⣐⣠⣶⠃⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⢳⡟⣸⠟⠁⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣯⠻⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠸⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠷⢿⢯⣖⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣯⣗⣤⣀⣀⡠⠄⠀⠀⠀⠀⠘⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⣿⣿⣿⣇⠀⠀⠀⠀⠀⠈⠋⢹⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠟⢷⡀⠀⠀⠀⠀⠀⠉⢿⣧⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡠⠊⠀⣠⠘⣿⢁⡄⠀⠙⠶⣄⠰⣄⠀⠀⣸⡗⠟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣆⠠⠂⠀⢠⢄⣄⠀⢀⠎⡀⡀⣰⣿⢀⣼⣧⢶⡄⠀⡀⠈⢳⡟⣦⡔⢫⠥⡄⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⠀⢀⣀⣈⢀⠱⠆⣤⠀⣸⣿⣠⣴⡊⣡⣿⣷⡟⣿⣾⢿⣯⣾⡷⠊⢤⡄⠀⡇⣭⠛⢁⠀⠊
⠀⠀⠀⠀⠀⡀⢰⣾⣷⣶⣾⣿⣿⣿⣿⣧⢾⠷⣦⣿⣷⣠⡟⣿⣿⣿⣿⣿⣟⣡⣾⡗⢻⣄⣵⣯⡟⢻⢁⡷⠋⠀⡀
⠀⠀⠀⠀⠀⠳⠿⠛⠘⠋⠽⠉⠉⢉⣽⣽⣷⣿⣼⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣧⣾⣿⣿⣡⣯⣾⣡⣤⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠛⠛⠡⢴⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡿⣿⣿⡿⣿⣯⡝⠾⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿⠿⠿⠿⠿⠛⠛⠛⠃⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⣄⠀⣠⡄⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣘⣦⣾⣷⣾⣿⣿⣿⣿⣿⣿⣧⣾⣃⣼⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢆⣸⣖⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠛⠻⠿⣿⣿⣿⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢬⣲⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣽⣶⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣧⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢻⣿⣿⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣾⣿⣿⡿⠁⠀⣀⣤⣤⣤⣄⣀⠀⠀⣀⣤⠖⠛⠉⠉⠉⠙⠻⣶⣄⠀⠀⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠁⣠⣾⣿⠟⠋⠁⠀⠀⠉⠉⠉⠀⠀⣀⣀⣀⣀⠀⠀⢹⣿⣧⠀⠀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢴⣿⣿⣿⠁⣼⣿⡿⠃⠀⣠⣴⠾⠛⠳⢶⣄⣠⠞⠉⠁⠀⠉⠻⣆⠀⠙⠛⠀⠀⠸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⣿⡇⠀⠛⠋⠀⢠⡾⠋⠀⢀⣀⠀⠀⢻⡏⠀⠀⢠⣤⡀⠀⢻⡆⠀⠀⠀⠀⠀⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⠿⠃⠀⠈⣿⡀⠀⠈⠉⠀⠀⢸⡷⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⢀⣼⠛⢿⣦⣄⡀⢀⣠⣾⠃⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠈⠻⣦⣤⣤⣴⣾⣿⠁⠀⠀⠹⣿⣿⣿⣏⣁⡀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠨⢾⡿⠋⣏⠀⠀⠀⠀⠹⣿⡙⠳⠦⠀⠀⠀⠀⠀⠀⠆⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⢀⣠⣿⠀⠀⠀⠀⠀⢻⣏⠳⣄⠀⠀⠀⠀⢀⠈⠁⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣿⡀⠀⠀⠀⠀⣼⡿⠀⣿⠀⠀⢀⠂⠀⠐⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠾⢷⣤⣀⣤⣾⡿⠻⠿⠛⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢀⠀⠰⠀⠠⠀⠀⠨⢙⣟⢻⠁⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⢀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⣨⣷⣾⣧⠀⠀⠀⠐⠀⠂⠀⢀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠠⣿⣿⣿⣿⣧⠀⠀⠠⠀⠀⠀⠀⠀⢀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣿⠏⠂⠀⠀⠀⠀⠀⢀⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀⠀⡄⠀⢀⠀⠂⠀⠀⠉⠀⠈⠙⠛⠳⣦⠀⠀⠀⠤⠄⢀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣥⣀⠁⠄⡀⠀⠀⠀⠀⠄⠀⠘⠄⠸⣷⢀⣠⣤⣶⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠻⠿⠷⠶⠶⠾⠶⠿⠿⠟⠛⠋⠁


⠀⠀⡀
⠀⠁⠀⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣶⡗⣧⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣼⣯⣾⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣰⣿⣿⣿⣿⣿⠿⣵⣆⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣌⣤⣿⣿⣿⣿⡿⣿⡆⠀⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⣿⣻⣯⠿⢧⡹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢭⡿⣿⣿⣦⣄⣉⠓⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⣺⣿⣿⣿⣿⣿⣿⣿⣷⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣭⣴⣾⣿⣿⣿⣿⣿⡍⣿⣿⣿⣟⣡⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⡴⠉⣴⣿⣭⣀⣑⠦⣄⣠⣴⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣾⡽⠟⠉⠈⣿
⠀⠀⠀⠀⠀⢀⡜⠀⢀⣿⠟⠩⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⠟⣿⣿⢿⣿⣿⣿⣿⣿⠟⠛⠿⣧⣦⠀⠀⣀⣿
⠀⠀⢄⣰⡶⢋⣠⣴⣿⠏⠀⠀⠀⠀⠀⠘⣿⡻⣿⣿⣿⡟⢸⣿⣼⡷⠾⠿⠛⠉⣀⣄⢀⣀⣠⡿⠀⠸⠟⠁
⠀⠐⠾⣿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⢹⣿⣾⣿⣿⡇⣸⠃⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋⠁
⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣣⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠻⢗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢀⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠄
⣶⣶⣶⣶⢶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡶⢶⣶⣶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⣴⣆⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠠⣄⢸⣿⣿⣿⣷⣶⣶⡞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢫⣾⣿⡿⠿⠛⠛⢿⣷⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠋⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠋⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⣿⢇⣴⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿⡆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡋⠁
⠀⠀⠀⠀⠀⠀⡾⠀⠀⠙⢿⣐⡀⠀⠀⠈⣿⣿⣿⣿⣿⡿⣦⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⠻⣾⣿⣿⢃⣿⣿⣿⣿⣿⢦⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠉⠙⠂⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠻⣿⣿⣿⡀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠀⠀⠀⠀⠀⢻⣤⣀⣀⣀⣠⠤⠀⢀⣠⣌⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣶⣿⣿⡿⢋⡀
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣤⣤⡄
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀
⠀⠀⠀⠀⢻⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠈⢿⣿⣷⡄⠀⠀⢠⣿⣄⠀⠀⠀⠀⠀⠀⠀⣸⣆⠀⠀⠀⣼⣿⣿⣿⡇⠘⢿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣾⣷⣼⣿⣿⣶⡄⠀⠀⠀⠀⠀⢹⣿⣷⣶⣾⣿⣿⣿⣿⡇⠀⠀⠑⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡓
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⠛⠉⠛⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠉⠻⢝⠻⢿⡿⢿⣿⡿⠿⢿⣿⢿⣟⠒
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣧⡀⠀⠀⢀⣿⠀⠀⠀⠘⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠀⢸⣿⣿⣿⡇⠀⠀⣼⡇⠀⠀⠀⠀⠈⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠇⠀⠀⢻⣿⣿⣇⢰⡄⣿⠇⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⠙⢿⣿⣿⣧⡿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠈⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡇⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠈⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣶⣤⡀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣄⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⢹⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣦⣍⣻⡿⣿⣿⣦⡌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠒⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⠉⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⣈⣉⣩⣽⣿⣿⣿⠟⠡⣻⣿⣿⣿⡇⠀⢼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⢟⡂⠀⠈⠙⠛⠛⠟⠋⠁⠀⢰⣿⣿⣿⣿⡁⠀⠸⡇⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⢋⣴⡟⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠈⣿⣿⣿⣿⣷⠸⣆⠁⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⡟⢠⣿⣿⠇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⣿⣿⣿⡟⠀⠈⠱⡀⠀⣿
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⢃⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠸⣿⣿⠁⠀⠀⠀⠁⢸⣾
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡟⣿⣿⣿⠘⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣰⠿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠸⣿⣦⣤⡄⢠⡟⣿⡷
⠀⢀⣴⠟⠋⣠⣾⢟⣿⣿⣿⡇⠸⣿⣷⠋⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⣼⡿⠟⠋⢠⠟⣰⠿⠁
⠀⠀⠀⢀⡾⠛⠁⣼⣿⣿⠏⣿⠠⠟⠁⠀⠀⠀⣰⠈⠙⠻⠿⠦⠄⠀⢸⠋⠀⠙⢶⣿⡿⡃
⠀⠀⠀⠈⠀⠀⢠⣿⣿⡟⠀⣻⠀⡀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢡⡾⠁
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⠀⣿⢰⠃⠀⢠⣾⣿⣿⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣤⣠⣅
⠀⠀⠀⠀⠀⠀⠀⢿⡻⣟⠾⣿⣿⡄⠀⠈⢻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣿⡿⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠈⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⣬⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠘⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠻⣿⣼⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠚⠛⠛⠲⣄⠀⡼⣧⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠛⠛⠳⠱⣌⣁⣀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠶⣤⣤⣀⣀⣀⡤⠀⠀⠀⠈⠉⠀⢀⣬⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣤⣤⣀⣤⠞⢉⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣀⣠⠄⠁⣀⡴⠋⠀⠀⠀⠀⠀⠀⡤⠀⡀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠿⠛⠛⠛⠻⠿⠿⠶⢶⣶⣦⣤⣤⣤⣭⣴⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠈⠙⠲⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢶⣤⢤⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠈⠛⠛⠁
⠀⠀⠀⢤⣀⡀⢀⣠⡾⠋⢀⣴⢟⣵⢿⡏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠢⣄⡀⠀⠈⠉⠉⠁⠀⣠⠟⢁⡾⠁⢸⡇⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠉⠓⠦⣤⣀⣠⡾⠋⢀⡾⠁⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⡃
⠀⠀⠀⠑⠢⣤⣉⣁⣀⣤⠟⠁⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⣿⠉⢻⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠈⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠈⠉⢉⣡⡞⠁⠀⠀⠀⠀⠀⠀⠉⢻⡀⠀⠀⣰⣯⡴⠏⠀⠀⠉⠉⠙⠛⠛⠛⠛⠻⣟⠀⠀⠩⣍⠙⠛⢻⡆⣽
⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡇⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣄⠀⠘⡆⣠⠞⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢤⣇⠀⣾⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⣞⣀⣤⣥⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡜⣯⠙⠳⣦⠀⠀⠀⠀⠀⠀⢰⣁⣀⣙⠋⣉⣡⠤⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣜⢧⡀⡏⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠎⠻⢦
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣚⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣫⣽⣿⣿⣿⡿⠷⣒⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⣠⡞⢡⣿⠀⠀⠀⠀⠀⠀⣀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿⣱⣿⡟⣽⡿⢋⣼⣿⣦⣾⠏⢀⣾⣿⣷⡞⠛⠉⠋⠉⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣾⡿⣿⣿⣿⢻⣿⡽⣿⢋⡼⠋⣠⡾⠉⠻⣿⠃⢰⣿⣿⣿⣿⣿⣆⠀⣨⠄
⠀⠀⠀⠀⠀⠀⠀⣼⣼⣿⣿⣿⡟⡿⣸⣿⣿⣿⣿⢻⢰⣿⢳⡿⣽⠇⣾⣿⢳⠃⡼⠁⣴⣿⢿⣦⣀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡉⠁
⠀⠀⠀⠀⠀⠀⢰⢳⣿⣿⣿⣿⢹⡇⣿⣿⣿⣿⡇⠇⢸⡏⢸⠃⡿⠀⡟⠁⢋⡄⢀⣼⢻⠇⠀⠈⠛⠻⢷⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣾⣾⣿⣿⣿⡿⢸⠃⣿⢸⢹⠿⡇⠀⢸⡇⠾⠀⠃⠀⢀⣴⡏⢀⡾⠁⣿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡟⣿⣿⣿⣿⡇⢸⠀⣿⠘⠘⠀⡄⠀⠘⠃⠀⠀⢠⡄⣿⣿⠀⣾⠁⢠⣿⠀⠀⢠⡀⠀⠀⠐⢻⣿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⡇⢸⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣸⠇⠀⠈⣿⠀⠀⢿⣟⣦⠠⠄⠀⢻⡇⣿⣿⠋⡇
⠀⠀⠀⠀⠀⠘⡇⠏⠹⡿⠻⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋⢿⡟⠀⠀⠀⣿⠀⠀⠈⠉⠋⠀⠀⠀⠀⠇⣿⡟⡘
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠈⢷⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠅⣱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠙⢦⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠑⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣦⣤⣤⣄⡀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⡉⠙⠛⠻⢷⣶⣶⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡆⡀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢆⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⡟⠁⣢⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡇⢣⠀⣯⠀⣠⣾⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠳⣿⠀⢿⡿⣆⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⡀⠈⠉⠀⠀⠁

⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣦⣷⣴⣾⣤⣿⣶⣿⣽⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠈⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⣦⢀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣸⣿⡿⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠇⣰⣿⣿⣧⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢡⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀⣏⣶⣿⣿⣿⣿⡷⠃
⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⣠⠴⢒⣋⣭⣥⣴⣦⣬⣕⠾⣾⣿⣿⣿⠟⠁
⠀⣾⣿⣷⣮⣭⣭⣁⡒⠤⢀⠀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢠⠞⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣀⣀⡠⠠⠀⣄⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡰⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠶⣾⣿⣿⡿⠿⠟⠿⣿⣿⣿⣿⣿⡿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⡜⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⠈⣿⣿⡿⣿⣿⣿⣿⣿⣿⡿⢿⣪⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣾⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣯⣶⣿⠟⠀⢰⣿⣿⣿⣾⣿⣿⣇⢿⡿⡗⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⣿⣿⣧⠀⠆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣭⡆⠀⠘⣿⢿⣏⡽⣿⣿⣿⣯⣁⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢱⠇⢿⣿⣿⣿⢿⣿⠘⡄
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠖⠋⠀⠀⠀⠙⣦⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⠇⡼⠀⡞⣿⣿⣿⠈⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠋⠹⠿⠛⠀⢀⣀⣀⡀⠀⠘⠛⠉⠙⣿⣿⣯⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⢠⠃⠀⠸⡿⠙⣿⡀⠀⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡀⠀⢀⠆⢰⣿⣛⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣰⠎⠀⠀⠀⠹⡄⠈⠃⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠙⢛⡛⠛⢀⡠⠃⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡴⠋⠀⠀⠀⠀⠀⠈⠲⢤⡀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡘⠻⡉⠻⠝⡇⠠⣾⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⢟⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢳⡀⣀⠜⣶⡘⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⡿⢋⡸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿⡼⣿⣿⣿⣿⣿⣿⣸⣿⣿⢟⠕⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡤⢴⣶⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣇⡩⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠐⠀⠀⢣
⠀⠀⠀⡄⠀⠀⠀⠂⠀⠀⠀⠐⢟⣻⣿⠿⣟⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠴⣄⣀⣰⣤⡼⠃
⠀⠀⢨⡁⠀⠀⠀⢀⡠⠄⢒⠨⠀⠈⠀⠀⠀⠈⠀⠛⠋⣉⣁⣋⣙⠋⠵⠃
⠀⠀⠀⠈⠀⠒⠐⠋⠀⣆⠀⠀⠀⠀⠀⡠⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠧⣤⣦⡰⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣿⣥⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣿⣶⣾⣿⣯⣿⣷⣾⣿⣾

⣽⣿⣷⣛⣀⣀⣀⣠⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠙⠀⠀⠐⣀⣠⣴⣶⣿⣶⣷⣦⣤⣴⣶⣶⣦⣤⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠓⠀⠐⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠶⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠯⡉⠉⠉⠐⠂⠀⠀⠀⠘⢶⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠫⢿⣿⣿⣿⣿⠿⠛⠛⠉⠀⠀⠀⠉⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣇⠀⠂⠉⠛⠋
⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡀
⠀⠀⠈⠈⠉⠀⠀⠈⠙⠛⠙⠻⠿⠟⠛⠿⠟⠁⣠⣶⣾⡟⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣶⣿⠟⢻⣿⠟⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢸⡟⠀⠀⠸⣿⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⠀⢀⣠⡴⠟⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠈⠉⠀⠀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⠀⢀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢠⠤⣴⣶⣿⣿⣿⣦⣤⣤⡀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠟⣤⣤⡼⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠾⠿⢿⡈⣠⠿⠻⠇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣲⣦⣄⠀⠰⠟⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣶⣶⣿⣿⡏⠀⢀⠰⡦⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣋⣽⣿⣿⣿⡟⠀⠀⢸⠀⠲⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢘⣿⣿⣿⣿⡟⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⢶⣿⣿⣿⣿⠟⠀⠀⠀⠀⢸⣇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣿⣿⡿⠛⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣿⣧⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠻⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣤⣄⣄⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠤⣔⣒⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⢶⣴⣶⣦⣾⣿⣇
⠀⠀⠀⠀⠀⠀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠟⠃⠀⠀⠀⢀⠀⠀⠈⢿⣷⣶⣄⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⢙⡿⠛⠿⣿⣟⠳⡄
⠀⠀⠀⠀⠀⠈⠀⠀⠩⠤⠶⠛⠉⠡⠛⠉⠁⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠈⢿⡏⠉⠲⠌⠤⣄⡀⢸⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⢸⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡄⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟⠀⠀⠀⣄⠀⠀⣄⠀⠀⠀⠀⠀⠀⠈⠹⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⢸⡇⠀⠀⠀⢰⣿⣿⣿⡿⠋⠀⠀⠀⠀⠈⠳⣄⠘⢧⡀⠀⠀⠀⠀⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣧⡀⠀⠀⠈⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣷⣾⣿⡀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠉⠃⠀⠀⠀⠘⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠁⠀⠀⢀⡀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⣴⠀⢈⣷⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⣹⣿⣷⣿⣷⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠙⠳⣆⠀⠀⠀⠀⠀⠙⣿⣧⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⢘⡄⠀⠀⠀⠀⠀⠹⣆⠈⢳⡄⠀⠀⠘⢶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠀⠀⠀⠀⠀⠀⠻⢿⠿⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠉⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡁⠀⣀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠊⠀⠀⠈⠀⠀⠀⠉⠀⠀⠈⣧⠠⠤⠔⠒⢒⠶⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠠⣉⣡⣀⣁⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠠⠋⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠠⠤⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠁⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠃⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⡇⢠⠃⠀⠀⠀⠀⠀⡰⠃
⠀⠀⢀⠀⠐⠈⠑⢀⣴⡁⠀⠀⠠⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠂⠀⠀⠀⠀⠀⡜⠁
⠀⢀⠔⠒⠶⢄⠀⡜⠠⢸⠀⠀⠄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⡼
⠀⡌⠀⠀⠀⢠⠁⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⡤⠆⠀⠠⠋⠀⠐⠊⠈⠉⠉⠑⢤⡀⠀⠀⢰⠁
⢠⠁⠀⠀⠀⡆⠀⢘⠂⡀⠀⡀⠀⠃⠀⠀⠀⢠⠃⠈⠁⠀⠀⡇⠀⠉⠉⠉⠉⠉⠀⢹⠀⢀⠇
⡌⠀⠀⠀⢠⠃⠀⢨⣀⡡⠂⠄⠀⣂⡀⠀⠀⣸⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠊⠁⠈⠉⠉⢹⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡇
⠣⢄⣀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠇
⠀⠀⠀⠉⠀⠉⠒⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢰⣃⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠤⠤⠤⣇⠀⠀⠀⠀⡇⠀⠀⢀⣀⣀⡤⠤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠄

⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠈⠻⣆⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠀⠠⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠁⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡠⠄⠒⠒⠒⠢⣄⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢋⠑⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠊⡀⠑⠄⠀⠀⠀⠘⡀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁⠀⠀⠀⠀⠀⠁⠢⡞
⠀⠀⠀⠀⠀⠀⠠⠊⠱⡀⠀⠢⠀⠀⢀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡠⢔⠋⠉⠉⠉⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡠⢄⠁⠀⠀⠀⠀⠀⠘⠓⠠⡄⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠐⢦⡀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⢀⠄
⠀⠎⠀⠀⠐⠀⠀⠀⠀⠀⠛⠁⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢠⠄
⠀⢆⣣⣀⠴⠀⠀⠀⠀⠀⠀⠀⢀⡄⣰⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢀⠁
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣠⠎⢠⡿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠈⠢⠤⣀⡠⠤⠴⡊⢁⠔⠻⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠔⠀⡠⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢸⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⡸⡇⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠂⠀⠀⠀⠀⠀⠀⠄⠀⠀⢆⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⡰⢡⠃⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⣀⡴⢋⠔⠁⠀⠰⢁⡜⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠠⠤⠒⠉⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠀⠀⠱⡖⠊⠁⢮⣀⡄⢀⣠⠗⠈⠀⠀⢀⠎
⠀⢀⠆⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠤⠒⠓⠀⠀⠀⠀⢱⠀⠀⠀⠀⠉⠡⡅⡀⠀⠀⢀⡜
⠀⠈⠮⣀⣀⣀⣀⡀⠀⠤⠤⠐⠒⠒⠒⠒⡲⠛⠉⠀⠀⠀⠀⠀⠀⣀⣠⡤⠞⠀⠀⠀⠀⠀⠀⠀⠑⠂⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣀⣀⠀⣠⣀⡤⠒⠊⠉