Anna Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠂⠉⠉⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠎⠡⠄⠀⠀⠀⠀⠀⠀⠄⡈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢀⢀⡤⠔⠒⠉⠉⠉⠉⠓⢄
⠀⠀⠀⠀⠀⢸⠃⠒⡒⠀⢄⡶⢊⡵⠋⠀⠀⣀⡤⠒⠋⠉⠉⣀⠓⡄
⠀⠀⠀⠀⠀⣼⣒⠡⠔⡊⠡⠐⠁⢀⡠⠔⠊⠁⢀⡀⣤⠔⢉⠀⢀⠰⣄
⠀⠀⠀⢀⡔⠉⠀⠀⠀⠈⢉⠙⠋⣐⡒⠒⠚⠛⢦⣾⡁⠁⠀⠀⠀⠀⣸
⠀⠀⠀⡞⠀⠀⣀⠴⡖⠉⠀⠀⠀⠈⣿⠉⠀⠀⠀⠀⠀⠀⠐⠒⡆⠜⠁
⠀⠀⠀⠇⣠⠊⢡⠊⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠦⠌
⠀⠀⠀⢸⠃⠀⠋⠀⠀⢀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂
⠀⠀⠀⢸⡀⠀⢐⠀⢠⠋⢮⢄⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠳⣄⡀⠁⠃⠀⠈⡆⠣⡀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⢠⠀⠉⠑⠦⡀⠀⡿⢷⣌⢢⣄⠀⢀⠠⠶⠁⠈⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⢄
⠀⠀⠀⠀⠘⣆⢀⢀⠀⠈⠺⠉⠲⣽⣾⣯⠀⠀⠀⠀⠀⠠⡅⠀⠀⠂⣀⠀⠀⠀⡀⠀⠐⡆⠂⠠⢀⡀
⠀⠀⠀⠀⠀⣈⠓⠳⠦⠤⣀⡂⠀⠘⣟⣿⡉⠒⠒⢢⡀⠀⠀⠀⠈⣡⠛⣌⣀⢤⡉⠀⠀⢣⡀⠀⠀⢨⡀
⠀⠀⢀⣤⣮⡿⠓⣄⠐⡀⠀⠀⠀⢀⡻⣞⢳⡀⠀⠀⠙⣶⢤⡤⠚⠁⠈⠀⠀⠀⠈⠒⠤⣤⠉⠽⠓⠄⠇
⠀⠀⠀⠈⠻⢧⣄⢾⢳⡈⠒⠶⠤⢬⡉⢻⡞⢣⠀⠀⠀⠹⢯⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠞⡄⠀⠁
⠀⣸⠀⠀⠀⠈⢛⣿⣿⢿⣄⠸⣄⠀⠘⣀⣿⡏⢇⠀⠀⢀⣷⣇⠀⠑⢦⡀⠀⠀⠀⠀⣸⠳⠶⠾⣷
⠀⡾⠀⠀⠀⠀⠈⢻⢿⣼⣿⣦⣬⣷⠦⢧⠱⡙⠉⠀⣾⣟⡗⠛⣇⠀⠀⠙⠦⣀⠀⢠⠇⠁⠀⠈⢸⡆
⠀⡇⠀⠀⠀⠀⠀⠈⣿⣿⡟⣿⠿⠻⡄⠀⠃⣧⡀⢸⣏⡎⣧⣠⣿⠀⠀⠀⠀⠈⣓⡞⠀⠀⠀⠀⠀⡇
⢠⠇⠀⠀⠀⠀⠀⠀⢹⣿⣇⠹⣦⣠⣾⡆⢰⡿⣿⣼⣏⠣⣯⢟⡟⠀⠀⠀⠠⠚⢱⠃⠀⠀⠀⠀⠀⣿
⢸⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣶⣬⣙⣻⠋⠀⢹⡫⣿⡇⢠⡔⣿⡁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡟
⣼⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣳⡜⢰⢫⣽⡏⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⡏⠀⠀⠀⠀⠀⠀⢸⠈⢻⣿⣿⣿⣿⣿⡄⢀⣾⣻⣧⡸⣿⣼⠁⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⢀⡼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠄⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣤⣤⣤⣤⣄⡀⠀⡂⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⠋⢉⣡⣤⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣁⡼⠃⠀⠈⠉⣡⢶⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⢿⣿⠓⠂⠀⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡯⢀⡴⠀⠀⠀⣠⠞⠉⠟⠁⠀⠀⠀⠀⠀⠉⠙⠀⠀⣀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⢿⠃⡞⠁⠀⣠⠞⠋⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣦⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡞⢠⡔⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠂⠀⠀⠀⠀⠀⠀⠘⣿⠉⠂⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⢃⡀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⢹⣧⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⡀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡉⠉⠉⠛⢷⠀⠀⠈⣿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣧⢰⣿⣿⣿⡿⠳⡄⠀⠀⠀⠀⠀⠀⠸⠛⠃⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣾⣿⠈⠏⠘⠛⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⢾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠐⠶⠶⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⣿⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠙⠻⣿⣿⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠁⠉⠉⣭⡿⠳⣤⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣿⠇⠀⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢿⡄⠹⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢙⣿⠂⠄⠀⠀⠀⠀⠀⠀⠀⠛⣷⠉⠲⡄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡦⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣼⣿⠿⠋⣾⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣼⠟⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣟⣻⣿⣿⣷⣄⣀⠀⣞⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⢉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣿⣿⣿⣿⣿⣿⣿⠛⣶⡛⠢⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠸⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣤⣀⠀⠙⠿⣿⣿⣿⣽⣷⣤⡁⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⠀⢹⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⣿⣄⠀⠙⢿⣿⣿⣿⣿⣿⣶⢿⣇⠀⠀⢠⣀⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠻⣿⣿⣿⣿⣿⡿⠿⢶⡄⢀⣠⣤⣬⣅⣐⡀⢀⠀⠀⠂⠀⣯⠈⠙⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⢀⠙⣿⣿⣿⣿⣿⣶⣤⢻⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣿⣷⣆⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⢸⣿⣿⣿⣿⣿⣿⡄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⣿⣿⣿⣦⡀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠈⣿⣿⣿⣿⣿⣿⢡⣿⣿⣟⠛⡘⣻⡟⣽⣿⣿⣿⡄⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢀⣿⣿⣿⣿⣿⠇⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣾⣿⣿⣿⣿⣿⠀⢰⡧⣾⣦⣭⣿⣿⣿⣝⣛⢿⣿⣿⣿⣿⣿⣿⣧⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠿⠿⠃⠀⠀⠐⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠊⠀⠀⠀⠀⠉⠐⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠁⢀⠔⠀⢀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⢀⠔⠁⠀⣀⠀⠃⣰⠏⠀⠀⠈⠄⠀⠀⠀⠌⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⢀⡔⣁⣤⠴⣞⣁⠀⢠⠁⠀⡀⠀⠀⠈⡄⠓⡀⠘⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⠋⠈⣉⣀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠈⠲⡄⠀⠱⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢠⠛⣿⢛⠿⠟⢦⡀⠀⠀⠀⢀⡴⠾⡛⠛⡿⠂⢁⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣠⠃⠀⠘⠢⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠔⠁⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠛⡆⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣹⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⠀⠀⢱⡀⠀⠀⠀⠀⠀⠇⠀⠄⠀⠀⠀⠀⠀⠀⡸⠈⢀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡰⠁⠀⣸⡇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⢢⡀⠀⠀⠐⠀⠀⠀⠀⢀⡤⠊⠀⢠⠀⢻⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠆⠉⠧⣄⡀⠀⢀⡠⡖⢈⣆⠀⠀⠸⢀⠟⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠘⠀⠀⣷⣌⠍⠌⣙⣠⣶⣿⡎⠑⢀⠋⢀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⡵⠂⢀⠿⣿⣿⣾⣿⠿⠟⢻⠻⡄⣸⠗⢋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠆⠸⣋⠔⠋⠀⠈⠻⡟⠁⠀⢁⠸⡄⡰⠁⣠⠼⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⡈⢧⠀⡴⢁⡐⠁⠀⠢⠨⠐⠂⢡⠀⡀⣽⠀⢈⡥⠀⠀⠙⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠳⡈⢡⠖⢁⠄⡀⠂⠀⠀⠀⢀⢸⡀⡟⠀⠂⠉⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢒⠉⣠⠖⠁⠐⢁⠔⡡⠂⢂⢸⣀⠟⠀⢈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢸⣾⠃⠔⠡⢊⡅⢊⠀⢀⠘⡄⠃⣒⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢠⠼⢆⢈⡤⠊⡅⠀⡈⠀⠃⠐⡍⠙⢇⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣾⣇⠀⣦⣱⡂⠁⣠⢣⠀⡄⢠⠀⠠⡸⣌⡦⢀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⢿⡀⠘⠻⡟⠋⠁⠘⡔⠀⠘⡀⠀⢣⠀⢳⡈⡁⠲⠄⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⢧⣱⠉⠀⠀⠀⠀⠀⠀⠀⠀⡗⡀⢸⠀⣸⣅⣤⠀⠀⠈⠁⠂⠠⡀⠀⠑⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⡘⢅⡀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠖⠊⠁⠀⠘⠤⢶⡀⡀⠀⠀⢨⠉⠳⣄⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣅⢠⡆⠀⠀⠀⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠖⣇⠈⣀⠀⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠳⡀⠄⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠒⠛⠛⠛⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣀⡤⠖⠛⠉⣉⠅⠉⡭⠷⠦⠀⡸⠁⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡴⠞⠁⠀⢀⡴⠊⠀⢠⠊⠀⠀⢠⠏⠉⠳⣦⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣰⠟⠀⠀⠀⡠⠋⠀⠀⣠⠏⠀⠀⣠⠟⢳⡀⠀⠈⣧⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⡏⠀⠀⢀⠜⠁⢀⡴⢺⠏⠀⣠⠾⠃⠀⠀⢷⠀⠀⢹⡀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠁⠀⠠⢊⣠⠾⠿⣴⣫⠴⠛⠁⠄⠦⠴⠒⠚⢧⡀⢸⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⠇⣀⡴⣖⣩⣤⣤⡈⠁⠀⠀⠀⡐⠀⠀⣠⣤⣤⣀⣙⣤⡇⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⢀⡿⣇⠙⡋⠁⣾⡿⡄⠀⠀⢰⠁⠀⡼⢿⡧⠉⡙⢹⡷⢹⡄⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣇⢹⠀⠘⠂⠂⠀⠀⠀⠀⠘⠀⠀⠀⠀⠒⠒⠁⢸⢴⠇⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⠟⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣰⠏⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⡀⠀⠙⣦⡀⠀⠈⠛⠥⢮⡽⠋⠀⠀⣠⡴⠋⣄⡴⠋⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠙⢦⡴⠋⠙⣦⣄⠀⠀⠀⠀⣠⡴⣿⢿⡀⠀⠸⡆⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⢸⡇⠀⢀⣟⠻⣟⠗⠒⢛⡿⡫⢸⠈⡷⣄⠀⣇⡴⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠈⠻⡅⡰⢿⡅⠙⢌⢿⣦⣟⠎⠀⠈⣿⡇⠈⠳⡇⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠚⠀⣸⠇⡀⠈⠞⣿⠈⠀⢠⢎⡼⠳⡄⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡏⠲⣤⡾⡏⠳⣌⡂⣞⣿⣃⢒⣡⠞⢱⠀⠘⣆⠴⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠈⢧⡀⠈⠀⢹⠀⠜⠉⠈⣿⠀⠉⢙⡂⠸⣆⠀⢸⠀⢸⠏⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢠⡛⠶⣄⣼⠀⠈⢓⠖⣿⠲⢺⠉⠀⣼⠙⢆⡘⣠⠟⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠘⢧⠀⠈⠉⡇⠠⠈⠀⣿⢰⠈⠆⠄⢻⡀⠈⡏⢸⡆⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢨⡗⢄⣸⡃⢀⠆⠀⣿⡀⠈⡆⢰⠋⢷⠀⣡⠊⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠘⢧⡀⠃⢱⠀⠟⢻⡿⡇⠃⢇⡘⢧⠈⢀⡟⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⡇⠀⣄⢸⠇⢳⠰⡴⠈⣼⠳⠋⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢀⡜⠛⣇⣠⢌⡾⠀⠘⣆⢥⣠⠋⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⢀⣞⠀⠀⣿⢏⡼⠁⠀⠀⠸⣼⠏⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢠⣿⠞⠀⠀⠀⠀⢰⣿⠀⢀⣶⡓⢀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⣀⣤⡈⡿⢀⠞⠁⠀⠀⠀⠀⠀⠀⢻⡀⠾⣝⢭⡐⠤⠐⠠⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢀⣟⠐⣤⡠⢐⡴⠖⣊⡥⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⠳⢬⣒⠠⢑⡒⠢⢄⡀⢄⣸⡄
⠀⠀⠀⠀⠀⠀⠀⡼⠁⣀⣐⣭⠵⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠋⠩⠕⠉⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠿⢿⣿⣯⣉⢻⡷⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣇⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣛⠛⠻⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡄⡀⠀⠠⠟⠛⠛⢿⣿⣿⣟⠉⠛⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠉⠻⢷⣄⣠⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⢀⣠⣤⣤⣤⡀⠙⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠿⠛⠛⠓⠀⠀⠀⠀⠀⠉⠉⣀⣤⣤⣄⠀⠀⢹⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣠⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⣾⣿⣏⠉⠛⠗⠀⠈⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠛⣿⣿⠉⠀⠀⠀⠀⠀⠀⠿⠿⠇⡠⠔⠀⠀⠀⢹⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠈⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣴⣶⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣦⠀⠀⠀⠐⠶⠶⠶⠚⠋⠁⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶⣤⡀⢀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣮⣭⣙⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣏⣽⣿⡂⠤⣀⡀⠀⢸⣿⣿⣿⢛⢻⡿
⠀⠀⠀⠀⠀⠀⠘⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⢿⡿⠹⡇⠀⠀⠙⢦⡈⣿⣿⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡗⢠⣇⠀⠀⠀⠀⠙⠿⢿⣿⡻⣿⣿⣿⡧
⠀⠀⠀⢀⣴⣶⣿⣷⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣏⡉⠻⣿⢷⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⣴⣦⡀
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⣿⣾⣇⠀⠀⠀⠀⠀⠈⠛⠿⣿⡟⣿⠿⠇
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣾⠟⠲⣄⡀⠀⠀⠀⠀⠀⠀⢉⡻⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦⣠⣤
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣦⠀⠋⠀⠀⠈⠳⣆⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠣⡈⠛⢿⣼⡿⠇
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠈⠿⣿⡆⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣿⢿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢻⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⡸⢸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠛⠋⠉⠛⠒⢶⡶⣒⣶⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣛⣩⠏⠀⠀⠀⠀⠀⠉⢳⡘⣓⠦⣄⠈⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠬⢲⡏⠄⠀⠀⣠⠔⠚⠙⢧⡀⠀⠁⠢⡉⠂⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⣴⢃⠊⠁⠀⠈⣇⢀⣄⡞⠁⠀⠀⠀⠀⠳⣄⠐⣄⠈⢢⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⣸⠃⠁⠀⠀⣠⠔⠛⣾⣿⠀⠀⠀⢀⣀⣀⣤⣈⣳⣼⢧⡀⢣⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣴⠛⠛⠓⠚⠓⠁⠀⠀⠈⠉⠀⠀⡀⠀⠹⡇⢳⡀⠆⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢸⡇⠄⠀⢰⣿⣿⣿⡳⣆⠀⠀⠀⠀⠀⢀⣾⣿⡛⠟⠻⣦⡀⣇⠀⣀⠴⠒⣄⢸⡄
⠀⠀⠀⠀⠀⠈⣧⠀⠀⣿⡧⡙⢋⡠⠘⡆⠀⠀⠀⠀⢼⡛⠛⡡⠂⣠⠞⠁⢸⡼⢁⠔⠀⢸⣸⡇
⠀⠀⠀⠀⠀⠀⢹⡀⢠⣿⠉⠘⠓⠂⠈⡰⠀⠀⠀⠀⠀⠉⠉⠀⠉⠀⠀⠀⠘⠀⢸⠲⠀⣸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢳⡸⣿⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⢠⠇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢠⠷⣤⡤⠖⡛⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠙⢦⡀⠀⠲⡀⠀⢀⡸⠋⠀⠀⠀⠀⠀⢀⣴⠏⡀⡜⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡖⠋⢀⠊⡜⠀⠀⠀⢱⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣄⠀⠀⠈⢳⣄⠀⠀⢀⣠⠴⠚⠉⢸⡇⠠⢁⠎⠀⠀⠀⠀⢸⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠳⢦⣠⠏⢸⠙⠋⠉⠀⠀⠀⠀⢸⡇⠁⠎⠀⠀⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠉⠀⢸⣆⠀⠀⠀⠀⠀⠀⣸⣷⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⣀⠀⠀⣸⢧⣉⠒⠦⣄⠀⡴⢁⣿⣄⡀⠀⠀⠀⠀⢯⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢿⠀⠉⠙⠂⠋⠈⢿⠙⢷⡈⢷⠁⣾⠁⠈⣿⡳⠶⠶⠤⢼⡄⠀⢳⡀
⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠈⢷⡀⠀⠀⠀⠀⢈⢷⡌⣧⠸⡀⢏⠔⠋⠀⠙⢦⣀⠠⡀⠀⠀⢸⡿⣦⡀
⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⢹⡟⠦⣄⡀⠘⠀⣷⠈⠃⣇⠈⠀⠀⠀⠀⢸⡏⠙⠺⢤⡀⣼⡁⠈⢷⡀
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠁⠀⢸⠀⠀⢸⠐⠀⠀⠀⠀⠸⡇⠀⠀⠀⠙⠁⢷⠀⠘⡇
⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⡾⠀⠀⠟⡇⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⣼⠀⠀⣿
⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡶⢦⣀⢸⢳⡀⠀⡀⡇⠀⠀⠀⠀⠀⠀⣸⢣⡀⠀⣰⠏⠀⠀⢻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⣿⣿⣿⣿⣿⣦⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢿⣿⣿⣿⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠈⠙⠛⠛⠛⠁⠀⠀⠈⠛⠿⠇⢾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣾⣿⣿⣿⣷⡶⢻⣆⠀⠀⠀⠀⢠⣴⣶⣾⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠃⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⣠⣈⣿⣿⣿⣿⣿⣿⣿⣿⠆⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠈⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣾⣷⣾⣧⣤⣶⠀⠀⠀⠀⠀⠀⠸⣿⠄
⠀⠀⠀⠀⠀⠀⣤⣀⠘⢿⣿⣿⣿⣿⣿⡿⠃⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡄⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠸⠛⠋⣰⣾⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣰⣿⣿⣷
⠀⠀⠀⠀⠀⠀⡀⣸⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⡀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢰⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣾⣦⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢿⣯⡝⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣅⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠈⠙⣇⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣽⡿⢿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠸⠛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⠛⢹⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⠿⣿⣿⠟⠀⠀⢰⣶⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠯⠛⠻⣿⣿⣿⡿⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠈⠉⠀⠀⢰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠀⠀⠸⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⡋



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣛⣛⣛⣛⣙⣛⣋⣈⣛⣻⣛⣛⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣯⣭⣭⣭⣭⣭⣿⣿⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⣿⡟⢀⣿⣶⣾⠧⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⡤⣤⣾⡆⠀⣤⠀⣬⠀⣼⡿⣿⣿⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⣜⣿⣤⡤⠿⠷⣼⠁⠀⣸⠿⢏⡠⢈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣧⡀⢀⣸⣶⠚⠓⢶⣿⠿⠿⠄⢀⣀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⣤⣾⠋⠻⣿⡏⠁⠀⣦⣀⣸⣿⣿⣿⣃⠘⠹⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⢿⣿⣿⣿⣿⣏⠈⠙⢀⣸⣿⣿⣶⣾⣿⠟⢉⣿⣷⣿⣬⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢠⣿⣿⣿⣿⣻⣿⠟⠀⣨⠁⣹⣏⠉⣆⣯⠴⠿⣿⡟⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣾⣿⣿⣿⣿⣿⣿⣷⣾⣿⣷⣿⣿⣿⠋⠁⢥⣀⣼⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢋⠈⢿⡏⠉⢻⠇⠈⢿⣿⣿⣿⣿⣿⣿⣦⣤⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣧⣼⣷⣾⣧⣄⣾⣤⣤⣾⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣬⠹⢿⣿⣶⣶⣄⠀⢘⣿⣿⣷⡀⣸⣣⣼⡏⣩⡿⢿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠏⠰⢾⣿⣿⣿⠏⢸⣿⣿⡉⠁⠉⠋⠉⠁⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢸⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢠⠏⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣀⡴⠋⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡟⠁⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢀⡿⠟⠛⢿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⡴⠏⢴⣶⣾⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄

⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⠒⠒⢄⠀⠀⠀⢠⣴⣿⣿⣿
⣿⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠚⠉⠉⠍⠁⠢⡀⠀⠀⢠⠞⠉⠀⠀⡀⠀⠀⠓⢄⠀⠀⠘⣿⣿⣿
⣿⣿⣿⡿⠒⠀⠀⠀⠀⠀⠀⢀⡿⠀⠠⠀⠂⠈⠀⠀⠰⠀⠀⢸⠀⡊⠀⠀⠀⠀⠑⠄⢼⠀⠐⢿⣿⣿⣿
⠍⠉⠛⣤⡀⠀⠀⠀⠀⠀⠀⠀⡇⠴⢶⠀⠀⠾⠛⢠⢈⠃⠀⠸⣼⠛⠷⠀⠀⠛⠙⢡⢸⡀⢀⣠⠉⠉⠉
⣷⣶⣶⡋⠁⠀⠀⠀⠀⠀⠀⠀⢣⣄⠀⢀⣀⠀⠀⠈⡰⠀⠀⠀⠙⡆⠀⠀⠠⠀⠀⢔⡏⠀⠈⠉⣴⣶⣶
⣿⣿⣿⣿⠇⠀⠀⠀⠀⡠⠊⠉⠉⠀⠰⠤⠀⠀⣦⠘⡀⠀⠀⠀⠀⠑⣄⠀⠀⢰⠖⠨⡃⠀⠰⢾⣿⣿⣿
⠉⠉⠉⣅⡀⣰⢾⡍⠉⠀⠀⠀⡀⠀⠄⠀⠐⣀⡪⣤⣁⠀⠀⠀⣀⡠⠾⠀⢀⠃⠀⠰⣁⡀⢀⣠⠉⠉⠉
⡶⠶⣶⠛⣉⣵⡯⣤⡬⠒⠀⠀⠀⠀⠀⠀⠀⢹⡟⠀⠈⠑⣴⠋⠁⠀⡀⢀⢈⠀⡀⠀⠀⠉⣏⠙⢦⠦⠶
⣿⣿⣿⣷⠤⠀⢰⠿⠒⠈⡿⠄⣀⠀⠀⠀⠀⠄⠙⠄⠀⠀⢹⠀⠀⠀⠈⠉⣀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿
⣽⣭⣽⣅⠀⠀⠀⠀⠀⣸⢻⠄⠉⢠⣤⡄⠀⢀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢫⣽⢍⢯
⣿⢿⣿⠟⠋⠀⠀⠀⣰⠃⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠑⠆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⡈⣿⡾⢾

⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⡇
⠀⡇
⠀⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠶⠶⣤⣄⠀⠀⣀⣀⣀⣀
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡾⠋⠁⢀⣤⠤⠤⣤⣈⠛⠛⠛⠋⠙⠛⠛⠿⣶⣤⡀
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠁⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⡙⠃⠀⠀⠀⠀⠀⠐⠠⠀⠉⠻⣦⡀
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣠⢾⡇⠀⠀⠀⠀⠀⢀⣤⡶⠾⣧⡀⠀⢀⠈⠑⢦⡀⠀⠀⠐⢄⠈⢿⡄
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⣿⡀⠀⠀⡄⢀⡴⠋⠀⠀⠀⠘⣷⡄⠈⠻⣄⠀⠈⢳⣄⠀⠈⢳⡀⢿⡄
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣼⣇⠀⢰⡇⣾⠁⠀⠀⠀⠀⠀⠈⠻⢦⡀⠹⣧⠀⠀⠙⣧⠀⠀⢻⡌⢿⡀
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⡀⠀⠀⣠⣞⣁⣹⣦⡘⣿⣿⠀⠀⠠⢤⡴⠶⠶⠶⠾⢿⣆⣻⣦⠀⠀⠘⣧⠀⠀⢷⡈⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⣼⠁⠀⣰⡟⣉⠉⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠸⣷⠙⣧⠀⠀⠸⡇⠀⠈⣧⢸⡄
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢸⠇⠀⢰⣿⣿⡟⢛⠷⣦⠀⠀⠀⠀⠀⠀⠀⢰⣾⡟⠛⢻⡛⢿⣿⣄⢹⡆⠀⢀⣷⠶⠶⣿⡌⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⠀⣿⣿⡻⠿⠟⢠⠟⠇⠀⠀⠀⠀⠀⠀⣿⠿⡿⠋⣠⠃⢀⡼⠟⢛⡇⣰⠟⢡⣶⡆⢸⣷⣿
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⠀⠀⣿⠿⠿⠶⠒⠁⠠⠄⣘⠀⠀⠀⠀⠀⠛⠶⠀⠒⠥⠒⠋⠀⠀⢰⡿⠃⢀⣿⣰⡇⣸⢻⣿
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⣿⠀⠀⠀⠀⠀⣠⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢸⢹⠟⢠⡟⢸⣿
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⣿⠀⠀⠀⠀⠘⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢠⡾⠁⢸⡟
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢿⣇⠀⠀⠀⠀⠙⠓⠀⠘⠃⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣾⢷⣤⣤⣶⡟⠁⠀⠀⣷
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣹⣦⠀⠀⠐⢶⡮⠄⡩⠭⠶⠲⠋⠁⠀⠀⠀⠀⠀⠀⢠⡾⢣⡾⠉⠉⣸⡟⠀⠀⠆⢻
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠉⠙⠷⣤⡀⠀⠓⠶⠤⠦⠚⠁⠀⠀⠀⠀⠀⠀⣀⣴⠟⣱⡟⠡⠀⢰⠏⢸⡆⠀⢸⣿⡆
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠈⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⡏⠁⣴⠋⡰⠁⣰⠏⠀⢸⡇⠀⢸⣿
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⠀⢘⣿⣦⣤⣤⣤⣤⡶⠾⠛⠉⠁⣸⡇⢰⠇⢠⠇⢀⠋⠀⠀⢰⣿⣠⣿⠃
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠳⢤⡀⢠⡞⢹⡇⠈⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠀⣾⠀⠈⠀⠀⠀⣸⡟⠉
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠉⠻⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠈⠀⢿⠀⠀⠀⠀⣰⡟⠁
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⢠⣯⡙⠷⣤⡀⡀⠀⠀⢠⡾⠋⣿⡄⠀⠘⣧⠀⠀⠀⢻⡿⢷⣄
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⣤⣄⣀⠀⢸⠛⢿⣤⣀⡈⠙⢦⡀⡾⢡⡾⠋⠻⣦⣄⣈⣳⣤⡀⠈⢷⡀⠹⣧
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⢿⣇⠀⠈⠉⠁⠊⠀⠈⢿⡍⠛⢶⡀⣿⡇⣾⠀⠀⠀⢹⣏⠉⠉⠉⠉⠓⢾⠇⠀⢹⣇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠈⣷⣄⠈⣧⠸⡇⣿⠀⣠⠞⠋⠙⠷⣤⣀⠀⠀⠀⠀⠀⠀⣿⢶⣤⡀
⠀⡇⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣶⣤⣀⡀⠀⢀⠃⢻⡷⣾⡆⠀⢸⣾⠁⠀⠀⠀⠀⣼⡏⠛⠶⣤⡀⠀⢰⡇⠀⠙⣿⡄
⠀⡇⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠙⠳⣾⠀⠘⣷⠘⣷⠀⠈⣇⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠉⢢⠞⣷⡀⠀⠸⣿
⠀⡇⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡟⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⣿⡇
⠀⡇⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣿⠀⣾⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⢀⣿⠀⠀⢹⡇
⠀⡇⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡷⣦⣄⠀⠀⠟⣧⠀⠀⢻⡀⣿⠀⠀⠀⠀⠀⠀⠀⣠⠟⢦⡀⠀⠀⣼⠃⠀⠀⢸⡇
⠀⡇⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠁⠂⠀⠘⠃⠀⠸⠇⠻⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠻⠀⠊⠓⠀⠀⠀⠈⠃
⠀⡇
⠀⡇
⠀⡇
⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠚⠓⠒⠒⠂⠙⠃⠀⠒⠛⠉⠆⠁⠙⠒⠂⠲⠔⠂⠂⠐⠒⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⣠⠶⠛⠉⠀⢀⠴⠒⠈⡡⠊⠉⠉⣠⠴⣅⡀⠈⢷⡀⠀⠀⠀⠀⠄⠀⣸⣿⣿⣯⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⣴⠛⠁⠀⠀⢠⠞⠁⠀⢀⡞⠀⠀⠀⢠⡇⠀⠀⠙⣦⠈⣷⠀⠀⠀⢀⣄⣠⣹⣿⣿⣹⣷⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢠⡟⠁⠀⠀⠀⡴⠁⠀⠀⣠⡾⠀⠀⢀⣴⠏⠘⣆⠀⠀⠘⣇⠸⡆⠀⠀⠾⠿⠿⠟⠛⠛⠻⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⡾⠀⠀⠀⢀⠎⠀⢀⣤⠞⡿⠁⢀⣤⠞⠁⠀⠀⠹⡄⠀⠀⢻⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢸⠃⠀⠀⡠⢁⣠⠶⠿⣤⢞⣡⠶⠋⠁⣄⡠⠤⠶⠦⠹⣄⠀⢸⡄⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⡏⠀⣀⡬⠖⢋⣡⣄⡀⠋⠉⠀⠀⠀⡀⠀⠀⡀⣤⣀⡀⠈⢢⡘⡇⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠀⣺⢻⠘⠿⠛⠋⣹⣷⡄⠀⠀⠀⣰⠁⠀⢠⣾⣏⠙⠛⠿⢻⣙⢣⡀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢰⣏⢸⡆⠘⢤⡀⠙⠁⠉⠀⠀⠀⡇⠀⠀⠁⠈⠋⢀⡴⠁⣿⢡⡌⢣⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⡿⣌⣇⠀⠀⠀⠉⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠙⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠘⢷⡀⠀⠀⠀⠀⡀⠀⠀⠀⣈⣀⠀⠀⠀⠀⢀⡾⠁⠀⢠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣦⡀⠀⠈⠻⣦⡀⠀⠀⠉⠭⢍⡥⠞⠁⠀⠀⣀⡴⠛⠣⣠⠴⠋⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠉⠳⣄⡞⠁⠙⣦⣄⠀⠀⠀⠀⠀⣀⡴⢾⣿⡄⠀⠀⢹⡀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⣸⠁⠀⢀⣿⠭⡟⠶⠶⠖⠾⡿⢛⠽⡇⢻⣄⠀⠈⡇⣠⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠳⣿⠀⣠⣾⡏⠑⢌⢧⣤⣆⡞⡕⠁⠈⣧⣟⠈⠳⣤⠋⠁⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠈⡶⠋⢸⡇⠀⠈⢆⢿⡟⡜⠀⠀⢠⢚⣷⡀⠀⠈⡆⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢄⡀⠀⣠⠾⣍⠢⣀⠘⢸⡇⠀⣀⠔⣡⠟⡄⠙⣆⠀⢃⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣿⠀⠙⢾⠋⣆⠈⠳⣦⠭⢼⡧⠫⢤⣾⠋⢰⡇⠀⠘⡖⠉⢰⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠹⣦⡀⠀⠀⢸⠀⢀⡀⠀⠨⡅⠀⢀⡨⠄⠀⣷⡀⠀⡇⠀⣸⠇⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⣎⠙⠳⣄⣾⡀⠀⠈⠛⡚⡟⠊⢙⠀⠀⢸⠃⠹⣄⢃⣴⠋⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⢻⡄⠀⠘⠀⡇⠀⠀⠁⡇⣇⠀⠈⠀⠀⢸⡆⠀⢹⠁⢸⠄⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⣽⠢⡀⣰⠃⢀⢀⠀⡃⣿⢠⠈⡀⢀⡏⢹⡄⠈⣠⠏⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢻⣄⠘⠇⣧⠀⢈⢆⢻⢿⡈⡀⡄⠈⢧⠀⠃⢈⡇⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⣀⡁⠀⠂⢸⣸⠘⡇⢃⠀⠀⠈⣧⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢬⡇⠠⢢⢣⡏⠀⢻⠘⡗⡄⡼⢤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⣷⣲⢇⡾⠀⠀⠈⣧⢻⣾⠁⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⠀⠀⣿⢋⡾⠁⠀⠀⠀⠘⣾⡏⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢀⠀⢹⣧⠀⢠⡿⠋⠀⠀⠀⠀⠀⢸⢹⠀⠀⣼⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇

⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠉⢦⣐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠀⠠⡰⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⣄⣤⣈⠀⡀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡟⢫⣵⠄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⣿⣿⣿⣷⣦⣄⠀⢀⣾⣿⣿⣿⣾⣿⣿⣦⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠾⣿⣿⣿⡿⣻⣿⣿⣿⠿⣿⡟⢫⣼⣿⣛⢿⣿⡿⠋⠥⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⠿⠿⠟⠃⠐⣭⣼⠉⠉⠛⠉⣨⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⠉
⠀⠀⠀⣤⢄⡀
⠀⠀⠀⣸⣦⡹⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣷⣦⡀
⠀⠀⠀⢽⣿⣿⣿⣿⣿⣷⣤⣤⣶⣶⣶⣾⣿⢿⠏⠀⠘⠛⠝⠋⠐⠠⠤⠄⣀⡀
⠀⠀⠀⠘⢻⣿⣿⠿⣿⣿⣿⡗⡯⠟⣤⡏⠿⣦⠀⠀⢀
⠀⠀⠀⠀⠈⡁⠀⠀⢀⣠⣿⣿⣾⣷⣿⣷⠴⠾⢀⣴⡯⡆
⠀⠀⠀⠀⠀⠙⠙⠋⢛⣻⣿⣹⣿⣷⡤⣶⣶⡾⠹⠟⣳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⣌⡑⠺⠿⠋⠀⣸⡿⠛
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣥⡆⠀⢀⡾⠍⠀⠰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢈⢵⣿⣿⣷⣾⣽⣿⢿⡞
⠀⠀⠀⠀⠀⠀⠀⠸⠺⢿⣠⣴⡿⣿⣷⠏⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⡿⠋⠁⠅⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⠃⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠉⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣰⣿⡿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠈⠻⣿⣿⣆⠀⠀⠀⢀⣼⣿⠟⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠈⢻⣿⣷⡄⢠⣿⣿⠃⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⣴⣄⡀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡿⠃⠀⠀⠀⠀⠈⣿⣷⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣄⠀⢰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⢀⣠⣴⣾⡿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⣿⣏⢻⣿⣷⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢿⣷⣾⣿⣿⠟⠉⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠙⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠃⠸⣿⣄⡀⠀⠀⠀⠀⠀⣸⣿⡿⠋⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⣿⣿⢿⣷⣦⣤⡀⠀⠀⠀⠸⣿⣷⣄⣀⣀⣼⣿⠇⠀⠀⠀⣀⣴⣿⣿⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⢀⣿⡟⠀⠉⣿⣿⠁⠀⠀⠀⠀⠘⠻⣿⣿⣿⠿⠋⠀⠀⢠⣿⡟⠋⢻⣿⡀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠺⣿⣷⣶⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣼⣿⣷⡄⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠭⠉⠉⠉⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣦⣤⣬⣭⣽⣿⣿⣿⣶⣾⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣽⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠞⠛⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠈⠉⠓⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢀⠀⣠⣄⢡⣄⡀⢀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠖⠚⢳⠀⠀⠀⢹⣿⢸⡟⠀⠀⠀⢰⠟⠐⠶⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠉⠀⠀⠀⢀⣀⣼⣧⣶⡾⠎⠁⠈⠉⠷⣶⣦⣿⡴⣀⠀⠀⠀⠈⢉⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢰⣿⠛⠉⠉⡀⢀⡀⠃⠈⢃⡀⢀⡉⠉⠘⢻⣧⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⣷⠀⠀⠀⠙⠓⠖⠢⡔⠲⠶⠟⠀⠀⠀⢸⣿⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⣿⠀⠀⠴⡄⡀⠀⠸⠇⠀⠀⠀⡄⡀⠀⢸⣿⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⣿⠀⠀⠀⠻⣄⡀⠀⠀⠀⣡⠞⠀⠀⠄⣾⡇⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠑⠀⠄⠀⠈⡈⠳⡁⡱⠞⠉⠢⠀⠀⠀⠙⠇⠀⢸⠁⠀⠀⠀⠀⠀⠀⡠⣴⣷⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡴⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⢀⣀⣠⣄⣤⡄⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠛⠀⠀⢀⡀⠀⠀⣿⡇⠀⠀⠘⠛⠿⣿⣿⣷⠀⣿⠀⠀⠀⠀⠀⠀⠀⠠⠤⢄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣇⡀⠈⠀⢠⡾⠃⠀⠀⠿⠃⠀⠀⠀⠠⠀⣸⣿⣿⣀⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⣿⣶⡶⢉⡤⠦⣄⠀⠀⠀⢠⠔⠒⠀⠸⣿⣿⣏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⣠⣤⣤⣤⣤⣤⣤⡀⣬⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡆⠙⠛⠛⠛⠛⠛⠋⠀⠙⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠁⠀⠀⣠⡆⠀⠀⠰⢦⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠈⠙⠓⢿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣍⠶⢆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⡿⡟⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠠⠀⣼⣿⡇⠠⠤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⢻⠃
⠀⠀⠀⠀⠀⠀⠀⡄⠛⠛⠁⠀⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡣
⠀⠀⠀⠀⠀⠔⡀⠀⠀⠀⠀⠀⡀⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠀⣦⣤⡔⢀⢴⠦⢀⡀
⠀⠀⠀⡠⠆⢀⠀⡶⠀⡄⠀⠄⠀⠄⠄⠲⠀⠀⠀⠀⠀⠀⠀⠀⠹⣼⡇⠈⠁⢠⣳⣏⠞⠉⠙⠂
⠀⠀⠀⣥⣘⡀⠠⠀⢁⠀⠂⡀⠀⠠⠀⢐⡠⠀⣀⡰⠀⠀⠠⢄⠀⠹⠇⠀⢠⢳⢧⣟⠀⠀⠀⠀⠂
⠀⠀⢰⣿⣿⣿⣿⣿⣶⣶⠖⡂⠀⠁⠹⣿⣿⣤⡤⠫⠁⠰⡦⡀⠑⢌⡄⢀⡏⣿⣼⡫⠄⠀⠀⠀⠰⡀
⠀⠀⣼⣿⣿⣿⣿⣿⡿⣿⠀⠁⠀⠀⢣⠘⣿⣿⣶⠶⠆⠀⠇⠈⠦⣾⣿⣼⣟⣽⣿⣯⠅⠐⠶⣤⣬⡅
⠀⠀⢿⣿⣿⠁⢰⠁⠀⡾⠘⠿⠿⠿⠿⣧⠈⣰⣤⠘⠀⠀⠠⠀⣾⣿⣏⣹⣾⣿⡿⠋⠀⣀⠔⠉⠁
⠀⠀⠘⣿⣇⠀⢾⢈⣿⡇⠀⠀⠀⠀⠀⢻⡇⠹⡇⠂⠀⠀⢠⠀⢻⣿⣿⣿⡿⠟⠁⣠⣾⣿⣆
⠀⠀⢸⣿⣿⣄⣀⣼⡿⠁⠀⠀⠀⠐⠀⠘⣿⠈⠉⠀⠀⢀⣶⡆⢠⣿⣿⡟⠁⢀⣺⣿⣿⣿⣿⡆
⠀⠀⠿⠿⠿⠿⠿⠿⠁⠦⠤⠤⠠⠤⠤⠴⠿⠇⠀⠀⠀⠼⠿⠿⠿⠿⠿⠷⠶⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢀⣇⢠⡿⣦⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢠⣧⠀⢰⠆
⠀⠀⠀⠀⠀⢸⠚⣿⠉⣛⡏⢻⡟⠞⠀⠀⠀⠀⠀⠀⠀⣆⢸⣇⢰⣇⢰⡇⣰⠀⠀⠀⠀⠀⠀⠘⢧⣼⣷⠟⡙⢷⣾⣤⠟
⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣷⣾⣷⡇⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⡟⢻⡿⠿⠟⠀⠀⠀⠀⠀⠀⠀⢸⣶⣿⣶⣷⣶⣷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠇⠠⠿⠾⠸⠣⠷⠈⠳⠀⠀⠀⠀⠀⢤⣼⣿⣿⣉⣿⣽⣽⣿⣹⣧⡤⠀⠀⠀⠀⠐⠌⠸⠦⠆⠟⠞⠆⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠂⠀⠀⠀⠆⠄⠀⠐⠚⠓⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠸⠀⢀⡀
⠀⠀⠀⠀⠀⢶⡄⢀⣹⢤⢾⡁⠀⣴⠂⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣷⡠⠀⠀⠀⠀⠀⠀⠀⠰⡀⣸⣄⢼⢄⣼⡀⡰
⠀⠀⠀⠀⠀⢘⣾⣁⣹⣿⣿⣁⣹⡟⠀⠀⠀⠀⠀⠀⠀⣴⣩⣍⣯⣼⣭⣭⣥⠀⠀⠀⠀⠀⠀⠀⠀⣿⣭⣤⠶⣢⣬⣽⡇
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣉⣉⣉⣉⣉⣉⣁
⠀⠀⠀⠀⠠⡆⢸⠿⠾⠸⠩⠿⠈⢳⡄⠀⠀⠀⠀⠀⠐⠏⠹⠷⠆⠞⠾⠌⠳⠄⠀⠀⠀⠀⠀⢶⡿⠽⠯⠽⠿⠯⠽⠯⠿⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠀⠀⠀⠘⠀⠀⠀⠘⠉⠃

⠀⠀⠀⠀⠀⠀⢀⣀⡀⢼⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢠⣿⣿⢿⣿⡿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣠⣿⣧⣼⡄⡀
⠀⠀⠀⠀⠀⠀⢻⣇⣀⣿⣀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⢹⣿⠀⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣽⣅⣿⣇⣽⣭⠁
⠀⠀⠀⠀⠀⠀⠘⠛⠛⢛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣾⣯⣷⡿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢛⠛⠛⠛
⠀⠀⠀⠀⢀⡄⢠⣤⣤⢠⢄⣄⠤⣄⠄⠀⠀⠀⠀⢀⣰⣶⣶⣒⣶⣲⣲⣶⢲⣆⣀⠀⠀⠀⠀⠀⡠⢠⣠⣠⢠⣤⣤⠠⣄
⠀⠀⠀⠀⠀⠁⠈⠉⣉⠈⢈⠈⠀⠈⠁⠀⠀⠀⠀⠼⠿⠍⠉⢉⠉⡍⡉⠉⠩⠿⠧⠀⠀⠀⠀⠀⠓⠈⠉⠉⢈⠈⡉⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁

⣿⠿⢿⢷⢿⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿⣿⡿⣿⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⢴⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⡶⣠⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠬⢇⡠⠁⢦⣀⠘⣦⡀⠀⣰⣧⠀⠀⣠⠖⠀⣀⡌⢠⡸⡯⠘
⠀⠀⠀⠀⠀⠀⠀⢰⡤⠶⣟⢃⡄⠀⠙⣷⣿⣿⣶⣿⣿⣷⣿⣿⣴⠟⠁⢠⡘⣛⠷⠦⣖
⠀⠀⠀⠀⠀⢀⡀⠋⠰⣦⣉⠌⣿⡶⣦⠈⣿⣿⣿⣿⣿⣿⣿⣿⠃⣠⣶⣿⠃⣌⣱⠆⠸⢀⡀
⠀⠀⠀⢀⡀⠬⣻⠎⠲⠀⣼⣏⡺⣿⣿⣇⠘⠛⠉⠉⠉⠉⠛⠃⢨⣿⣻⡷⣙⣯⡀⠔⠱⢟⠥⠀⡀
⠀⠀⢴⡻⣿⣾⣇⡚⢷⣌⣀⡝⠁⣿⣏⣙⠦⣄⡀⠀⠀⢀⣀⡴⢋⣙⣿⠀⢻⣄⣈⡴⢛⣸⣷⣿⠟⡷
⠀⠀⠀⠈⠋⢿⡨⠅⠈⢿⢌⣁⣽⣠⡽⡉⠿⢁⡉⠀⠀⢈⣁⢿⠉⢹⣥⣻⣈⡡⣾⠃⠀⣍⡾⠏⠃
⠀⠀⠀⠀⠀⠉⠓⠀⠐⠚⠘⠂⢀⢮⡄⢹⡦⣾⣣⡄⢠⣌⣳⢴⡞⠀⣭⠄⠀⠂⠙⠂⠀⠺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠿⠏⠙⠃⠀⠉⠁⠈⠉⠁⠈⠋⠹⠿⠞⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠥⠤⠠⠤⠴⡆
⠀⠀⠀⠀⠀⠀⣠⠤⠤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⡏
⠀⠀⠀⢶⣒⠊⠀⢀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⠤⠤⠤⢶⠗
⠀⠀⠀⠀⠈⠉⠉⠉⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠎⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢳⡀
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢳⡀
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⢠⣎⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣷⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⡇⠘⠳⠞⠀⠻⠶⠃⠀⠳⠞⠀⡇⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⢠⣏⣀⣀⣀⣀⣀⣀⣀⣀⣷⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⣏⣀⣀⣀⣀⣀⣀⣀⣀⣷
⠀⠀⠀⠀⢸⡇⠙⠒⠃⠘⠒⠃⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠁⠛⠚⠃⠙⠒⠃⢸
⠀⠀⠀⠀⢸⠇⠀⠀⠀⠠⠀⠀⢸⣤⣤⡀⠀⣠⣤⡀⡇⣠⣄⡀⠀⣠⣤⡄⠀⣠⣠⣄⡇⢠⣤⣄⠀⢠⣠⣼⠀⠀⢀⠀⠠⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠒⠒⠀⠀⢸⠀⠀⢧⣤⡇⠀⣧⣧⡇⠀⢧⣤⡇⠀⢧⣤⡗⠀⢯⣧⡼⠀⢿⣤⡼⠀⢸⠀⠀⠈⠒⠚⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⢸⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠈⢢⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠔⠢⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠐⠊⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢻⠉⠀⠀⠀⠉⡟⠦⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠂⠐⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⢠⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⡴⠃⠀⢸⠀⠀⠀⠀⠀⡇⠀⢹⣆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⡄⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣸⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⢸⠸⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠃⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⢸⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⢀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣀⣇⣀⣀⣀⣸⣀⣀⣀⣀⣀⣇⣀⣸⣀⣷⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠉⠉⠀⠈⠈⠁⠈⠀⠁⠈⠁⠀⠁⠈⠀⠀⠀⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠟⠛⠶⣄⢳⣸⢃⡴⠛⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⡏⠀⢀⣠⠼⠎⠧⠯⢤⣀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠛⣿⡟⠛⣉⣀⠀⠾⠿⠆⢀⣉⠛⠻⣿⠛⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⢀⡟⢀⣶⣿⣿⡿⢣⣆⢿⣿⣿⣵⡄⢻⡄⠀⠈⡿
⠀⠀⠀⢀⣠⣶⡾⠶⠶⣤⣈⠳⢄⣾⢀⡀⢻⡿⢀⣀⣿⣿⣆⠀⢿⣿⢁⡀⢻⣠⠞⣁⣤⡶⠶⢶⣶⣤⡀
⠀⠀⠀⢿⣿⣯⡀⠀⠀⣀⣨⠽⢗⣂⢸⡿⢀⡺⢿⣿⣿⣿⣿⣿⠷⣃⢺⡗⢈⡲⠯⣍⣀⠀⠀⢀⣸⣿⡿⠂
⠀⠀⠀⠀⠉⠛⠻⠛⠛⠉⢠⡞⠁⢿⡀⢀⣾⣿⣀⡀⣿⡿⢀⣀⣾⣿⡄⠀⡼⠉⠳⡄⠉⠛⠻⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣄⠀⠈⣷⡀⠛⣿⣿⠿⠞⠳⠿⣿⣿⠛⠃⣼⠃⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣿⡷⠶⣤⡀⠀⠻⠟⠂⣀⣤⠶⢶⣿⣶⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠉⣹⢳⡞⢏⠉⠀⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠶⠛⢡⡏⢹⡌⠛⠶⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠁⠀⠀⠀⠀⢀⠔⠀⠀⡤⠊⠁⠀⠠⠾⣖⣋⠉⠀⠀⠠⠂⠄⠸⡄
⠀⠀⠀⠀⠀⢄⡀⢠⣄⠲⢶⣶⡀⠁⠰⣶⠁⠔⢖⡋⠀⢶⣀⡀⠒⠒⠂⠒⠻⠫⣙⡃⠋⠠⡀⠒⠲⣄⠰⠒⠒⠒⠒⢲⡆⡀⡒⡆
⠀⠀⠀⠀⠀⠀⠑⠤⠀⠢⢀⣙⢣⠀⠀⠉⢧⠀⠈⠱⡄⠀⠀⠈⠁⠀⠀⠀⠀⠁⠏⡆⠀⠀⠀⠀⠉⠀⠉⠁⢠⠀⠀⠀⠐⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠂⠀⣀⡀⠈⠈⠂⠀⠀⠀⠈⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠐⢶⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠀⢀⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠤⠀⢀⠱⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⠉⠀⠑⠄⠘⠡⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡶⠀⡂⠀⠙⠒⣠⡀⠈⠢⢈⡆⠀⠀⢀⣀⠀⠀⠀⠄⠀⢀⣀⡀⡀⠀⢠⡀⠀⠀⠀⢿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠙⠒⠒⠒⠒⠚⠛⠚⠛⠛⠀⠀⣿⣿⣧⠀⠀⠐⠚⠃⠤⠉⠛⣠⠝⡳⠆⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⡀⠀⠠⠁⠀⠀⠀⠐⠊⠀⠔⠁⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⢒⣎⡀⠡⡀⠀⠄⠀⠁⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡐⠁⢶⡀⣼⡿⢣⣦⢀⣦⣾⡸⣃⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣣⡄⠀⠀⠌⠀⣀⠈⠻⢌⣰⣿⠀⠂⢻⠿⠻⠿⠿⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢋⠰⢦⣀⢉⣛⢶⣼⢏⡴⠊⠀⠄⠀⠀⠀⠀⠀⠀⠉⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣹⠟⡿⣷⠏⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠲⢶⣶⡀⢂⣤⡀⠀⡀⠀⠜⠿⣷⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠉⠛⠀⢠⡙⠿⡆⠩⢻⣆⠈⠳⣶⡝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠦⠶⠦⠀⣋⣙⣄⡈⣿⣷⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠱⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠟⠛⠛⢿⣾⣿⣿⣛⣀⣴⡄⠀⣄⠀⡀⠀⠀⠀⠀⠀⠀⠈⠐⢿⣦⡀⠀⢰⡄⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠉⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠿⢾⡲⢀⡀⠀⠀⠀⠐⠦⡀⠀⠁⠀⠰⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠀⠀⢀⢠⡄⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⢉⡿⠁⠢⠁⠠⣤⠅⠀⢀⠀⠀⣀⠀⠀⠀⠹⣤⡀⠀⠀⠲⠀⠀⠀⠀⠰⢻⣎⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠖⠉⠉⣥⠁⣔⠰⡀⣸⡌⠀⣾⠀⠤⠿⣦⠀⠀⠀⣿⣷⣭⣤⣠⣮⣵⡄⣀⢶⣻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣸⠇⠘⠋⠘⣿⣤⡥⠟⢧⠶⣯⣤⣤⠛⠁⠀⠀⢰⣿⠟⠛⣁⣠⡤⠀⠀⢉⣩⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠀⣀⡠⠴⢿⣿⣳⣤⣄⢰⠘⠁⢀⣆⠀⠀⢀⡟⣽⠿⢛⡫⣅⣀⣤⣾⡿⠧⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢿⡽⠁⠂⠈⠙⢿⢬⡁⠈⢹⣷⣀⣸⠋⠀⢠⠊⢘⣿⣶⠱⣾⣿⠿⠿⠿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡿⠖⢀⣉⣽⠤⠮⢽⡗⠠⣵⠟⠁⢀⡴⠁⢀⣼⣛⡋⠚⣛⣤⠴⠶⠶⢾⠾⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢀⠙⠁⢁⣩⣅⠠⠴⡉⠟⠀⠁⢀⠔⠁⠀⣤⣿⠿⠃⣾⢑⣐⣀⡉⢠⣄⠤⠤⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠁⠀⠋⠀⠐⠚⠋⠁⠀⣠⠾⠋⠀⠀⣸⢿⡏⣶⣬⡣⡈⢿⠁⢰⡆⣀⠀⢠⣉⣉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⣀⠀⠀⠀⠀⣀⣀⣤⡶⠟⠃⣀⣠⣴⣾⡋⠈⠀⠙⠿⣷⡜⣞⠳⢌⢿⣄⠀⠑⠋⡁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣯⣿⣭⣴⠶⠶⠛⣿⣿⡈⠡⢁⡌⠀⠀⠀⠙⣻⣾⣓⠀⠀⢈⣀⠀⠻⣤⢁⡀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢃⠐⠊⠀⠀⠀⠀⠀⠀⠉⠙⠷⠒⠤⠍⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣦⣤⣖⢻⡟⣲⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⣀⡀⠀⠀⠀⠀⣷⣿⣿⣯⣻⣿⣿⣿⣟⣟⣼⣿⣗⡿⡀⠀⠀⠀⢀⡀⢐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣈⠀⢠⡀⠀⢿⣿⣿⣦⣶⣶⣶⣾⣷⣶⣶⣶⣼⣿⣿⡿⠀⢀⡄⠀⣀⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⡉⣛⠓⢢⢹⡆⠀⢻⣟⣿⣭⣽⣧⣼⣧⣼⣯⣽⣿⣻⡟⠀⢰⡏⠰⠚⣛⡉⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣟⠁⠠⡿⢿⣃⡼⢁⣤⣻⠿⠿⣿⠋⢻⠙⠁⡋⠙⣿⡿⠿⣟⣤⡘⢇⣛⡿⢿⠄⠈⣛⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡳⣶⣞⣼⣿⣷⣿⣛⠅⣐⣰⣤⣤⣰⣴⣷⣦⣤⣤⣆⣂⠘⡋⣿⣮⣿⣧⣳⣶⢞⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣶⢂⣾⣿⡍⡿⣯⣿⣿⣿⣿⣳⣾⣿⣿⠋⠉⠁⣙⣋⠀⠉⠙⢿⣿⣿⣗⣿⣿⣿⣿⣽⢿⠩⣿⣷⡐⡦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣽⣸⡁⠋⡶⢱⣿⣿⣿⠟⠋⣍⣉⣾⣧⣀⣠⣀⠈⠁⢀⣄⣀⣼⣷⣉⣭⠹⠻⣿⣿⣿⠞⢿⠙⢘⣇⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⣯⣷⠀⠀⢸⣿⣿⣟⣰⡿⢉⣿⣷⡄⠘⡛⡇⠀⠀⢸⢛⠋⢠⣾⣿⡙⢿⣦⣻⣿⢻⡇⠀⠀⣾⣽⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣧⣿⡀⣤⣼⣧⣿⣟⣏⣁⣩⣴⣿⣽⣾⠃⣀⠀⢀⣀⠘⣵⣯⣿⣮⣉⣉⣹⣻⣿⣿⣧⣦⢀⣿⣼⣦⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⣬⡁⠸⣇⣿⣿⡝⢻⣿⠀⢸⣿⣿⣿⠀⠙⠳⠞⠋⠀⣿⣿⣿⡏⠀⣿⠿⢩⣿⣿⣸⠇⣨⣥⡟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠈⢲⣤⣹⠿⡄⠙⣧⣿⣿⡴⢲⣦⡼⡟⢻⣿⣶⠤⢾⣷⠦⣴⣿⡟⠻⠧⣴⡶⢦⣾⣿⣼⠋⢠⠟⣏⣤⡞⠁⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⣙⣷⣦⣿⣀⣿⣿⣻⣧⢺⠋⠀⢁⡾⡿⣿⣦⣤⣠⣴⣿⠿⢳⡀⠀⠹⡇⣸⣟⣿⣿⢀⣿⣵⣿⣋⣡⠜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣤⡼⠟⢻⣿⣿⣿⣿⣽⡌⣿⣏⠘⣧⡀⢻⡿⢿⡿⢿⡟⢀⣾⠃⣹⣿⢡⣯⣿⣿⣿⣿⠟⠻⢧⣤⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠉⠻⠿⡿⣿⣿⣿⣿⣿⣿⣬⣄⣢⡹⠟⢳⣤⣴⠞⠹⢏⡜⣰⣥⣿⣿⣿⣿⣿⣿⢿⠿⠟⠉⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣴⣧⣿⣿⣿⣿⣿⣿⣿⣿⣷⣧⣾⣅⡈⢁⣨⣷⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣼⣦⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠘⣿⣿⣿⣟⣿⣿⢹⡿⣿⣿⣿⣿⣾⣿⣿⣿⢿⡏⣻⣿⣻⣿⣿⡿⠁⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⢴⣤⣸⣿⣿⣿⣯⣻⣿⣷⣿⣿⣿⡇⢸⣿⣿⣿⣾⣿⣏⣿⣿⣿⣿⣃⣤⡦⠀⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣃⣶⢿⣟⣿⣿⣿⣶⣿⣾⣿⣟⠻⠃⡘⠿⣻⣿⣷⣿⣾⣿⣿⣿⣻⠿⣶⣺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣤⣿⣾⣿⣿⣿⣿⣯⡄⢉⡏⢀⣿⣿⣿⣿⣿⡿⣿⣤⣄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠋⠁⠈⠻⢿⣿⣷⣼⣦⣴⣧⣾⣿⡿⠟⠁⠈⠙⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡶⠶⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⡠⠔⠀⠀⠀⠀⠤⠀⠐⠒⠀⠀⠄⢀⡀⠀⠀⡇⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠁⠈⠁⠀⠹⣧
⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠲⣄⠀⠀⠀⠘⣧
⠀⠀⠀⠀⢠⣿⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁⠀⠀⠀⠙⡄⠀⠀⢸⡆
⠀⠀⠀⢀⣿⠃⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⢾⠟⠁⠀⠀⠠⠀⠀⠀⢸⡀⠀⠈⡇
⠀⠀⠀⣾⠇⠀⠠⠃⠀⠀⠀⠀⡄⠀⣠⡴⠚⡶⠺⠛⠚⠛⠉⠉⠁⠀⠀⠀⠀⠀⠦⠌⠂⠀⠀⠀⢸⡇⠀⠀⣇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣐⣠⠞⠁⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠛⠿⡿⠇⠀⢰⡇
⠀⠀⠀⢻⣄⠃⠀⠀⠀⠀⠜⠉⠉⠀⠀⠀⣤⡴⠛⠛⠛⠓⢤⠀⠀⠀⠀⠀⠀⠀⠀⢡⣼⣷⠀⠀⡀⣿⠀⢸⡇
⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠈⡀⠀⠰⣾⠇⠈⠡⠀⠀⠀⠀⠀⠂⠀⠀⠉⠁⠀⠈⠀⢹⡄⣼⠁
⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠙⠂⠤⠄⠒⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⡷⠏
⠀⠀⠀⠀⠘⣷⠀⠀⣆⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠌⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠈⠂⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠉⡀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⢀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠂⠙⠁⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⠎⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣴⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⠀⠉⠓⠶⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠈⠢⡀⠀⡀⠀⠀⠈⠉⠑⠒⠒⢤⡴⠚⣟⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⢀⡀⠀⠠⡐⠀⠀⠲⠂⠀⠤⢄⡀⠀⠀⠀⣏⠀⠂⢄⠉⠛⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⠃⠀⠀⠀⠀⠀⠀⢀⠃⠀⠐⠀⠀⠑⠦⡀⢹⠂⠀⠘⡀⠀⠘⡷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠐⠁⠀⡄⠀⠀⠀⠀⠀⠀⢺⡜⠀⠀⢣⠀⠀⠄⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⡟⠐⠀⠀⡄⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠁⠈⠂⢀⠑⣄⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⣄⠀⠀⠀⠀⢠⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⢂⠘⡁⠀⠙⢆⢀⠉⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⠏⠀⠀⠀⠀⠀⠉⠐⠀⠀⠈⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠡⠐⢤⡀⠀⠙⠄⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠄⠁⢠⡀⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠉⠂⠤⠤⠄⠀⡿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡿⢻⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠈⠃⡀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠏⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠃⠈⠠⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⢳⠤⣀⠀⠠⣷
⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⡀⠀⠠⠈⠉⠀⠀⠀⠀⠀⠀⠸⠀⠈⠁⠀⠙⣦
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠈⡑⠊⠀⠀⠀⠀⠀⠀⠀⠀⡐⢴⡀⠀⢙⠢⢄⠀⢠⣽
⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠠⠀⠀⠺⠁⢈⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⢰⠀⠀⠀⠀⠀⠢⠠⡜⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢐⠓⠀⠙⢷⡄
⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⡄⠀⠀⢸⠈⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠄⠁⠀⠀⠀⢠⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢸⣦⡀⢠⡟
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣷⡞
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣼⡇⠈
⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⣆⠀⠀⠀⠀⠀⠐⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⣉⣤⠒⠒⢲⠒⠠⢌⡑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡰⠋⠐⣎⢦⠀⠘⡀⠀⣠⠜⣦⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢠⠃⣼⣁⠀⡀⠈⣆⠻⠀⡦⢊⠴⠊⠀⠱⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠒⠂⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡖⠂⠠⢤⣄⡉⠀⡀⠀⡋⠒⠒⠀⢠⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠲⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣁⠀⠀⣠⣞⡵⠞⣳⡌⢫⠉⠁⠀⢐⠂⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠻⠙⢫⠋⠁⠈⠀⡇⢿⠉⢦⡱⡀⢀⠎⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠉⠀⡀⠀⢀⡸⠀⢀⡱⠞⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠈⠈⣿⡉⢤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣀⠀⠀⠀⢀⣀⡀⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⠁⠀⠉⠉⠉⠁⠀⡀⠄⠂⠙⡗⠲⣤⣀⣀⣀⣤⠔⠚⠉⠀⠀⠀⠀⠀⠙⢆⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡄⠀⠀⠀⢀⠔⠈⠁⠀⡄⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡇⠀⠀⠀⡈⠀⠠⠂⠁⠑⢄⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢣⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⠀⠀⣇⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢀⡔⢃⠀⠀⣀⣀⣀⣀⠀⢦⠀⠀⠀⠀⢸⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠇⠀⠀⠀⠈⢠⡡⣀⠤⠒⠉⠀⡀⠀⠀⡏⠉⠉⠉⠀⠀⢱⠀⠀⠀⠿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠂⠀⠀⠇⠀⠀⠀⠀⠀⢈⠀⠀⢀⡇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⠀⠀⠀⠀⠀⠀⡠⠉⠀⠉⠓⠴⢦⡂⠀⠀⠀⠀⠀⡕⠸⠀⠀⢸⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢸⡄⠀⠀⠀⠀⡰⠁⠀⢠⠧⢄⡀⠀⢩⠁⠒⠒⠂⠐⠈⠁⠀⠀⡶⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⣁⠀⠰⠃⠀⠀⠁⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⠀⠀⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢸⢂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠸⡀⠀⠀⠀⠈⠢⢀⡀⠀⢀⠄⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⠀⠀⠀⠀⠀⠀⠈⠑⠂⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⣄⠙⠶⢤⣀⣀⣀⣀⡀⢀⣀⣀⡠⠴⠚⠋⣠⢶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡈⠙⠲⠤⣤⣀⣀⣈⣉⣀⣀⣤⠤⠖⠒⠉⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠒⠒⠊⠉


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣟⣿⣻⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣰⡿⠋⠛⣧⠀⠀⠀⠀⣿⠋⠀⠀⠙⣷⠀⠀⣾⡏⠉⠙⣷⠀⠀⢰⡿⠋⠉⢿⡄⠀⠀⣴⡿⠛⠻⣦⠀⠀⠀⠀⠀⣠⣴⠶⠶⣶⡄
⠀⠀⠀⠀⠀⢿⣇⠀⢀⣿⠀⠀⠀⠀⠻⣧⣤⣤⣼⠟⠀⠀⢿⣧⣠⣴⡿⠀⠀⢸⣧⡀⠀⣸⡇⠀⠀⢿⣄⡀⢀⣿⠀⠀⠀⠀⠀⣿⡁⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠈⠻⢿⣿⡁⠀⠀⠀⠀⠀⢸⡿⢿⣄⠀⠀⠀⢠⡿⠛⣿⡀⠀⠀⠀⢹⣿⣿⣏⠀⠀⠀⢈⣿⠿⣿⠁⠀⠀⠀⠀⢀⣽⣿⣶⣶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⣦⡀⠀⠀⢀⣿⠁⠈⢻⣦⠀⢠⡿⠁⠀⠘⣷⡄⠀⣠⡿⠁⠈⠻⣧⠀⢠⡿⠃⠀⠹⣷⡀⠀⠀⣴⡟⠋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⢿⣦⢀⣾⠃⠀⠀⠀⠹⣷⡿⠁⠀⠀⠀⠘⣿⣴⡟⠀⠀⠀⠀⢹⣷⣿⠃⠀⠀⠀⠹⣷⣤⡾⠃⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⢿⠃⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠉⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣤⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⣷⠀⠀⠀⠀⠀⠀⣰⠞⢋⡿⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⢦⡀⠀⠀⠀⠀⠀⠀⢀⡴⠞⣷⠶⣿⣇
⠀⠀⠀⠀⠀⣸⡏⠀⠰⣄⣿⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠟⣿⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢹⡆⠀⠀⠀⠀⢀⡟⠀⠈⠁⠐⠃⢹⡇
⠀⠀⠀⠀⠀⢻⣧⣀⣳⣼⠏⠀⠀⠀⠀⠀⠘⣷⡀⠀⢀⣰⠏⠀⠀⠀⠀⠀⠀⠀⣿⡴⢀⡄⠀⣼⠃⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠙⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⠷⠛⠃⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⢿⣇
⠀⠀⠀⠀⠀⢠⣾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣦
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠘⢿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠷⠶⠶⠶⠶⣶⣶⣶⣤⣤⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣴⠶⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⢚⡓⡒⠲⠢⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⠀⠒⣾⠑⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⢿⠿⣿⣿⣶⣾⣧⡁⠀⠀⠐⠄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢛⢇⢀⠈⢤⣾⣽⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠈⢿⡇⢺⢿⣾⢣⣄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡄⠑⢄⠀⠀⠀⡀⡆⢰⡀⠚⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠙⠂⡀⢵⣣⣦⡓⡿⣼⡧⣧⣴⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢛⡚⢹⣿⡷⠿⠟⠻⠻⠟⠫⣆⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣿⡿⡉⢿⡿⠓⣾⡾⣿⣿⣦⠛⣅⡦⠄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢸⣿⣋⠱⣎⣸⣯⣽⠻⣯⣤⣶⣧⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⠳⣨⠉⠻⣿⣿⡧⡹⡿⠛⣦⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠟⠚⡿⣿⣾⢉⢙⠃⠌⠂⠀⡪⣁⡀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⣼⣯⣶⣽⣤⣤⣧⣾⣿⠑⢌⠀⠀⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠟⢀⠁⠄⠀⠀⢀⠀⠀⢈⠆⠀⢁⠈⠉⠁⣍⠀⠁⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡶⠀⠀⠀⠘⡉⠹⠒⠤⠄⢀⠐⣀⢴⠀⠁⠀⠀⢰⡏⠀⢈⢀⡈⣀⣈⠀⠁⢿⣾⢣⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡠⠤⠀⠲⠋⠁⣀⠤⠐⠒⠒⠛⠻⢦⡀⡏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠆
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⢀⠤⠀⠁⠀⠀⠀⠀⠀⣀⠀⠀⠁⢺⠐⣴⠁
⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠸⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠜⠴⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠥⠄⠀⠂⠀⠀⠁⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠐⢦⠀⢀⠐⡼⡅⠀⠤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠂⠰⢦⡀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠓⠢⣤⠤⡄⠴⠂⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠑⡌⠀⠀⠀⠀⠀⠈⠄⠤⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠂⠀⠀⠀⣤⡀⠀⠈⠁⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠈⠂⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠀⠀⠀⠑⢌⣻⣶⣦⠀⠀⠀⠀⡀⠣⢄⡞⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⠂⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣭⣤⣴⣈⡁⠰⠞⠉⠀⠀⠀⠠⠦⡐⠃⠀⠀⠀⢆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠐⡄⠄⠊⠁⠂⠈⠀⠀⣀⠠⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠭⠁⠀⠀⣀⠠⠄⠀⠒⠀⠀⠈⠆⠀⠀⠀⠈⡄⠰
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢀⡐⠉⠀⠀⠀⠱⡀⠀⠀⠰⣄⠐⠀⡀⠀⠀⢀⠈⠄⠀⠀⠀⠀⠀⡀⠀⣼⠁⠀⠀⢐⠀⠀⠀⠀⡇⠰
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡀⢀⡤⠁⠀⡴⠀⠀⠀⠙⢄⠀⠀⠈⠱⠖⠀⠄⠂⠉⠀⠀⠀⠀⠀⠀⠀⠹⢼⠇⠀⠀⠀⠈⠀⠀⠐⢠⠃⢘
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⡀⠄⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⢐
⠀⠀⠀⠀⠀⢀⠀⡠⠊⠁⠀⠀⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠂⠰⡀⠀⠀⠀⠀⠀⠈⠳⠶⢴⠄⠀⢀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠈⢄⠀⢀⠌⠀⢀⡆⠀⠀⠀⠈⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠒⠶⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠄⣀⢠⠁⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠘⠂⡾⠀⢀⣼⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠘⠠⣀⡀⢀⣀⣀⣀⣤⣤⡀⢀⣀⡀⣀⡀⠔⠀⠈⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠹⠹⠀⠀⠀⠀⠀⠀⠀⠀⠙⡛⠛⠛⠳⠒⢆⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠻⢤⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠢⡔⢧⣤⠚⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⡀⠠⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠈⠀⠀⠀⠀⠂⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠺⣍⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠⠈⠀⣄⠀⠀⠉⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡒⠀⠀⠁⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠰⠂⠘⢦⠈⡡⠛⠒⢠⠤⠄⢠⣄⣠⡄⠈⢢⡀⠀⡀⠳⠄⠠⣄⠀⠀⠀⠀⠀⠀⠀⠸⠁⢄⡀⠀⠀⠀⠉⠁⠀⢸
⠀⠀⠀⠀⠀⢀⠄⠌⡀⠀⠀⠀⠀⠳⡄⢄⠀⡄⠉⠀⠀⠰⡄⠈⠀⠉⠹⣄⠐⠄⠀⠰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠝⠂⠴⠀⠀⠀⠈⣸
⠀⠀⠀⠠⢀⠃⠀⠈⠀⠀⠀⠀⠀⠀⠈⠊⠁⠘⠁⠀⡄⠀⠈⢢⠀⠀⠀⠈⡷⡀⠀⠀⠙⠐⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠓⠀⠀⢠⣸
⢠⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠄⠁⢀⠀⡀⠑⣄⠀⠒⠇⠹⢦⡀⠀⣀⠀⢰⣀⣀⡤⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⢄⢸
⢰⠄⠀⠀⠀⡄⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠑⣆⡀⠀⠀⠤⠤⠈⠁⠈⠁⠈⠉⠉⠁⠉⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠠⠁⠈⠀⠀⠀⣻
⠨⢄⣄⡀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⢟⢤⡀⠀⠀⠀⠀⠀⠀⠢⢀⠀⠐⠀⠀⠈⠀⠀⠰⠄⠴⠂⠀⠀⠀⠀⠃⠀⠀⠀⢻
⠀⠀⠀⠀⠲⡄⠁⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢢⠑⢄⠀⠀⠀⠀⠀⠀⠀⣐⣠⡤⠔⠒⠚⠋⢷⢀⠀⠀⢀⡤⠀⠼⠦⠀⠀⠀⢻
⠠⡀⠀⢀⣈⣾⠃⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⣇⢨⡙⣦⣤⣦⡖⠒⠉⣉⢫⣄⢀⣄⡈⠀⣒⢁⡉⣛⣴⡀⠁⠀⠀⠈⢢⣀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠊⠀⢀⡀⠀⠀⠀⠀⠀⣰⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⡀⠀⠀⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣄⣹⠇⠀⠀⠀⠀⠀⠀⠀⢦⣤⣾⢃⣠⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠸⣿⠟⠋⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠛⠛⢋⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⢱⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣤⣤⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⢀⡸⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠚⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⢠⣿⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⣤⣴⣾⣿⡄
⠀⢀⣼⠏⡀⠁⡈⠁⡀⢀⡀⢀⡈⢓⡦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷
⠀⣸⠋⡄⠁⡄⠁⡄⠁⡄⠁⡄⠁⡄⢁⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇
⢀⠃⠄⠀⠄⠀⠄⠠⠄⠠⠄⠠⠄⠠⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿
⢸⠂⠀⠂⠀⠂⠀⠂⠐⠂⠐⠂⠐⠂⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇
⠚⣷⡃⡀⠃⣀⣧⣄⠃⡐⠃⡀⠃⡐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣷⡀
⠀⠈⠙⠷⣶⠟⠁⠉⠿⣅⡀⢁⡀⢁⣈⣥⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠹⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠛⠛⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠟⠁⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠇⠈⠿⠟⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡾⢷⠀⠀⠀⠀⠀⠀⠘⠛⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠄

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⡴⢶⡽⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠄⢀⡀⠀⠀⠀⢀⣠⣠⣀⡀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⢀⣼⢇⡄
⠀⠀⠀⠀⠨⠵⢞⣳⡟⠻⠂⠀⠀⠀⣠⣀⣤⡀⠀⠀⠀⠈⢢⣄⣠⡄⠃⠀⠀⠀⣿⣿⡟⠁⠀⣀⣾⣿⢋⠈⠀⠀⠀⠀⠀⠀⠐⣯⠁⡄
⠀⠀⠀⠀⠀⠀⠁⢻⠀⠀⠀⠀⠀⠰⣿⠿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡇⠠⠔
⠀⠀⠀⠀⠀⠀⠀⡎⠰⠃⠀⠀⠀⠈⢾⣿⣽⡞⡗⠀⠀⠀⠀⢲⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⢀⠀⠀⠀⠀⠚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠉⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⣤⠀⢘⣦⠾⠁⠀⠠⠐⣦⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⢠⣤⢞⠀⠀⢀⠈⡄⢠⠀⣀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠔⠋⠀⠊⠀⠀⠀⠀⠀⠀⠅⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡟⠋⠉⠁⠀⠀⠈⠀⢑⡺⡍⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⡇⠀⠀⠀⠀⠀⠈⠰⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⢀⡄⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣨⠁⣉⠀⠀⠀⣶⡶⠀⠀⠀⡇⠀⡿⠋⠁⠀⠀⠀⠀⠀⠀⡀⢉⠀⠀⠀⠀⠆
⠀⠄⠀⠀⠀⠀⣀⠀⡦⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⡃⡿⣃⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⣿⠀⠀⠀⠾⡷⠄
⠀⠂⠀⠀⠀⠀⠈⢰⡖⠚⠀⠀⠸⠿⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠉⠳⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣾⠆⠄
⠀⠂⠀⠀⠀⠀⠀⠘⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉
⠀⡂
⠀⠁
⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠻⠛⢛⡷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣬⣉⠙⢀⡄⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡧⠥⡬⠀⠥⠬⠼⢿⣄⡀
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⠊⠀⠰⠀⠀⠀⠈⠀⠀⠀⠙⢷⣄
⠀⠀⠀⢰⠞⠉⠉⠉⢳⡀⠀⠀⢀⡞⣁⣀⣀⡠⠤⠤⠖⠒⠀⠒⠢⠐⠄⠳⢤⣙⣧
⠀⡴⠖⠛⠂⠀⠀⠀⣸⡃⠀⣴⠛⠏⠠⠀⠀⡆⠀⠈⠀⠀⢘⠀⢰⠀⢠⠀⠘⡀⠉⣧
⢸⡇⠀⠀⠀⠈⢢⠀⠈⢿⡀⡏⢠⠀⢈⠀⠀⠁⠀⠀⠂⠀⠈⠀⠀⡄⠀⠀⠀⡇⠘⣸
⠀⣿⠂⠀⠀⡀⠈⠀⠀⣸⡿⠛⠀⢤⣚⣩⣭⣩⣒⡉⠉⠀⠴⠖⠒⢆⣀⡐⣖⣳⠂⠉⢳⡄
⢀⣸⡇⠀⠀⠁⠀⣤⠞⢹⠃⠀⠀⢰⠀⢀⠊⣴⡿⢷⠀⠀⠀⡴⢷⣦⠑⡄⠈⡇⠀⠀⠈⣷
⡟⠀⠀⠀⣆⠀⠀⡹⠶⠛⠠⡀⢀⡸⢠⣬⣄⣛⣿⠞⣠⣤⣄⢻⣿⠟⡠⣁⢀⡃⠀⠀⠀⣿
⠻⢤⣠⣤⠁⣀⢀⣠⡶⠓⠲⣦⣠⣾⣻⠟⠟⠩⢋⠕⠛⣏⠙⢌⠫⠑⠟⢿⢙⣷⣤⡤⠞⠁
⠀⠀⠀⠈⠉⠉⠉⠁⢀⣴⠞⠋⠉⠙⡇⠀⠀⠀⠀⠀⠚⡍⠂⠀⠀⠀⠀⠀⡋⠉⠉⠙⠦⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⠤⠔⠊⠘⠀⠀⠀⠀⠀⠀⠜⠙⢆⠀⠀⠀⠀⠀⡈⠉⠑⠤⢤⡙⣆
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⣀⣀⡀⢳⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠇⠀⠀⠀⠀⣈⣢⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⡸⠁⠀⠀⠙⢦⡀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣦⠄⠊⠁⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢸⠒⠂⢲⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠤⢽⡇⠀⠐⠒⠀⠤⢼⠀⠀⠀⠀⠀⠀⡸⠄⠒⠊⣿⣤⡴⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠁⠀⠐⠒⠂⢎⣰⣣⣆⣴⣰⣠⠗⠒⠈⠉⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣅⠀⡽⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣛⣛⣦⣤⣀⣀⣀⣤⣐⣿⣾⣓⣠⣄⣀⣀⣠⣴⣞⣯⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠉⠉⠀⠀⠀⠈⠀⣼⣿⡀⠈⠀⠀⠀⠈⠉⠁⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⣀⠴⣿⢷⡳⢄⡀⠀⠀⠀⠀⠀⠀⢀⡼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠻⢼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠕⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠰⠆⣤⢠⡄⣠⣶⣯⣤⢀⡄⢠⢠⡄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣒⣲⣇⡀
⠀⠀⣀⠠⠤⠠⡀⣀⡤⣾⠃⡀⠀⠠⣤⠤⣄⠀⠀⠤⠁⠊⠉⣐⣀⣀⡂⡇⣻⣹⠓⣒⣒⣒⠐⠢⢳⡦⠀⠀⠀⣀⣴⠾⠛⠉⡏⣿⡋⠛⠷⣦⡀
⢨⠃⠀⠀⠀⠀⠀⠒⢹⣿⡜⠑⠂⣁⡀⠉⠢⢂⠀⢒⢀⠀⠀⠛⣿⠁⠀⠁⢐⠀⢰⣿⠟⠻⣶⠀⢸⠓⠀⢠⡾⠛⠁⠀⠀⠀⡇⡗⡇⠀⠀⠈⠻⣆⡀
⠺⢠⠚⣏⢹⣤⣻⣿⣟⣫⠾⡅⡜⠿⢻⣦⣔⡼⠂⠘⢈⢇⠀⢸⣟⣦⣂⢧⢨⣶⣴⢠⠈⣿⡿⠇⠚⠁⠀⡼⠡⠀⢰⠄⣄⠀⡀⣰⠋⠄⣀⢰⡀⠝⢿
⠀⢣⣽⣾⢼⣿⣿⣷⣶⣿⣶⣿⣿⣧⣼⣿⣵⠁⠀⠀⢿⣴⣿⣦⣿⣿⣿⣶⣿⣦⣷⣿⣺⣯⣭⣽⡧⠀⠀⢷⡀⠋⣽⡏⠃⠞⣺⣷⠀⠖⢩⡼⠁⠈⡿
⠀⠸⢟⠛⢻⣿⣉⠛⡉⣿⡛⣿⣿⣿⣿⢿⠇⠀⠀⠀⢼⣫⣭⣭⣭⣥⠽⠿⡿⠿⢶⣿⣿⣯⣭⣿⡇⠀⠀⢸⠽⢼⣽⣯⣤⣼⢫⣨⣯⡐⢹⡽⣅⢌⡥
⠀⠨⣿⣭⣭⣽⣬⣦⣾⠿⡷⠿⠿⢿⣿⣿⠁⠀⠀⠀⢤⠶⡥⠤⠶⠸⠄⢤⠴⠶⠸⢿⣤⣿⣿⣿⡇⠀⠀⡼⠁⠉⠛⢈⡀⠃⢘⠋⠘⣃⠹⢿⡿⢿⡧
⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠣⣤⣥⣤⣬⣤⣤⣥⣤⣬⣶⡥⠼⠶⠼⢭⡷⠀⠀⠉⠛⠐⠒⠒⠐⠀⠀⠀⠐⠒⠐⠒⠑⠊
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠦⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⠀⠀⢤⡀
⠀⠀⠀⠠⢠⡆⢠⡆⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠈⠀⠀⢠⣄⠁
⠀⠀⠀⡄⢸⡇⢸⡇⢠⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⡿⣷⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⣀⠀⣿⣿⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣶⣄⢹⣯
⠀⠀⠀⣷⣸⣟⣸⣗⣼⡄⠀⠀⠀⠀⠀⣸⣿⠀⣶⣿⣿⣦⢠⣿⡇⠀⠀⠀⢀⣀⠀⣿⠀⣿⣿⠀⣿⠀⣀⡀⠀⠀⣰⣆⢸⣷⢿⡿⡿⡮⡿⢠⣦
⠀⠀⠀⢿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⣠⣿⣿⢴⣶⣿⣷⣶⣾⣿⣷⡀⠀⠀⢹⣯⣼⣿⣿⣿⣿⣿⣿⣧⣿⡇⠀⠀⢻⣿⣶⣾⣾⣷⣷⡗⣶⣿⣿
⠀⠀⠀⢸⣿⡏⢹⣿⣧⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡇⠀⠀⣼⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⡇⠀⠀⣼⣿⣿⣿⣿⠟⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢦⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠇⢀⡀
⠀⠀⢰⣶⣶⡆⠰⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⢀⡈⣾⣦⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣾⡆⠀⠓
⠀⠀⠠⣯⣼⣤⣤⣿⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸⣇⣏⣧⣼⠁⠀⠀⠀⠀⠀⠀⢻⣟⣀⠀⣿⣿⢀⣀⣻⡗⠀⠀⠀⠀⠀⠀⣦⡼⡿⢿⣴⠀⡀
⠀⣠⣠⣿⣿⠛⡟⣿⣿⣅⣄⠀⠀⠀⠀⣀⣺⣀⡟⠛⠛⠿⣰⣇⡀⠀⠀⠀⢀⣾⣿⣉⣉⣿⣿⢉⣉⡿⣇⡀⠀⠀⠀⢸⣧⣬⣻⣿⣟⣡⣤⣷
⠀⢸⣿⣿⣿⣿⢿⣿⣿⣿⡏⠀⠀⠀⠀⠹⣿⣿⣷⡶⣷⣾⣿⣿⠃⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠸⠿⠿⠿⠇⠈⠿⠿⠿⠿⠀⠀⠀⠀⠐⠿⠿⠿⠃⠸⠿⠿⠿⠀⠀⠀⠀⠾⠿⠿⠿⠿⠇⠸⠿⠿⠿⠿⠷⠀⠀⠀⠀⠿⠿⠿⠁⠿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠃⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⢴⠄⠀⠐⠆⠀⣶
⠀⠀⠀⠀⢰⡇⢹⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⢀⠠⡆⠐⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⢶⣶⠀⣿⡇⠀⠀⠀⠀⠀⠠⠾⢧⠀⣼⠄⠰⠿⠆
⠀⠀⠀⣶⢸⣷⣾⣇⢰⠀⠀⠀⠀⠀⠀⠀⠀⣧⢼⡿⣿⢧⡇⠀⠀⠀⠀⠀⠀⠀⣠⣤⠁⣾⣟⢠⣿⡇⠀⠀⠀⠀⠀⠘⣟⡟⣤⣿⣥⢹⣻⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣀⣸⣿⣾⣷⣿⣞⣿⣀⣀⡀⠀⠀⢠⡄⢸⣿⣶⣿⣿⣼⣿⡇⢠⠄⠀⠀⠀⣘⣛⣟⣛⣟⣛⣻⣟⣓⡀
⠀⠠⣤⣿⣿⣿⣿⣿⣿⣤⡄⠀⠀⠀⢘⣟⣿⣛⣟⣛⣛⣟⣻⣛⣻⡁⠀⠀⢰⣷⣾⣿⣿⣿⣿⣿⣿⣷⣾⡄⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⢀⣿⡟⣿⡿⢻⣿⢻⣿⡇⠀⠀⠀⣸⣿⣿⣿⣿⣟⡿⠛⠟⠛⢙⠁⠀⠀⠈⣿⡟⠻⣟⠿⢿⢿⡟⢿⣿⠀⠀⠀⠀⢰⣿⣿⣿⡏⣿⣿⣿⣇
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉⠉⠉⠀⠈⠉⠉⠁
⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄
⣤⣤⠀⠀⠈⣿⣿⡇⠀⠀⣠⣤⠀⠀⠀⢴⣶⡦⢀⣿⣿⠀⣴⣶⡆⠀⠀⠀⠀⣤⠄⣈⣿⣿⣿⣿⡀⠠⣤⠀⠀⠀⠀⠀⠀⢰⣦⢀⣿⡦⢼⣿⠁
⢸⣿⣀⣀⣀⣿⣿⣇⣀⣀⣸⣿⠀⠀⠀⢸⣿⠷⠶⠿⠿⠶⣿⡾⡇⠀⠀⠀⠀⣿⡀⣟⣛⣛⣛⣛⣛⢀⣿⠀⠀⠀⠀⣿⡇⢸⣯⢸⣿⡇⢸⣿⠂
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠠⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣯⣤⠀⠀⠀⣿⣷⣶⣷⣶⣷⣶⣾⣿⠇
⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⣿⡟⣻⣿⣿⣿⡇⠀⠀⠀⣸⣿⣿⢻⣿⣿⣿⣿⠟⣿⣿⣇⠀⠀⠀⣾⣿⣿⣿⣿⣿⡏⢹⣿
⠉⠉⠀⠀⠈⠁⠀⠁⠀⠀⠈⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠁⠀⠉⠉⠉⠉⠀⠈⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠙⠶⢶⣄⣀⠀⠀⠀⠀⣀⡶⠛⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠀⠀⠀⠀⠀⠉⠙⠷⠲⢦⡼⢧⡀⠦⢄⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣧⡀⠀⠀⠀⠔⠲⡤⢄⢀⠀⠀⢀⣈⢥⠀⢈⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠁⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠈⠀⠀⢀⠉⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢠⠒⠃⠀⠀⢠⠀⠀⠀⠀⢀⣄⣼⠃⠰⠚⠁⠁⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⣀⡴⢿⣰⠀⢀⣘⣿⣄⣠⣄⠀⣀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣴⠗⠚⡅⠐⠂⢹⡆⣸⠋⠁⠀⠀⣬⡟⠉⣟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠉⠀⠀⠀⠓⠋⠀⠀⢠⠄⠀⠀⠀⢻⣧⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢀⠀⠀⠀⠀⠀⠫⠇⠀⠀⠀⠀⠀⠀⡀⢺⡧⠤⢤⡈⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⣸⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⡆⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠄⣿⡗⠒⠚⢁⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣟⠒⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠋⡦⣿⡏⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣑⣑⣚⣘⣃⣄⣤⣤⣤⠤⠤⠬⠶⠮⢕⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠒⠛⠛⠛⠛⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⢀⣀⣀⣀⢀⣀⡀⠀⠀⡣⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⣠⠴⠚⠉⢀⠤⠂⡠⠒⠙⢣⡴⣇⡀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣠⠞⠁⠀⢀⠔⠁⠀⡰⠁⠀⢀⡾⡀⠀⠹⣆⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣸⠃⠀⠀⡠⠃⠀⣠⣾⠃⠀⣠⠟⠀⢱⡀⠀⢹⡈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⡇⠀⠀⠜⣀⣴⢏⡵⣃⠴⠛⡁⣀⡤⠤⢷⡀⠘⡇⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠸⠀⣠⠖⣊⣥⣄⠘⠉⠀⠀⡀⠀⠀⣠⣤⣀⠙⣄⡇⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠁⡽⣷⠙⡋⠡⣿⢧⠀⠀⢠⠁⠀⡼⣿⠌⢉⠋⣷⢹⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣷⣸⡀⠈⠒⠀⠀⠀⠀⠘⠀⠀⠀⠀⠒⠉⢸⣷⠇⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠙⠙⣇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣸⠁⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠙⣆⠀⠀⠀⣄⣀⣀⢈⡤⠀⠀⠀⣠⠏⠀⣠⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢦⡀⢈⠷⢄⡀⠀⠉⠛⠋⠀⣀⣴⡞⠁⠱⡞⠁⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⢸⡏⠀⢸⣟⡶⡤⠤⢴⣾⡿⣏⢳⣄⠀⢣⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⢾⠃⣠⣟⠃⢌⢶⣆⡼⡗⠁⢸⣾⠉⠳⡞⠁⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠚⠁⣸⠀⠀⠇⣿⠙⠀⢀⢎⡿⣆⠀⢡⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠒⢤⠾⡉⠦⣕⣸⣿⣂⢚⣵⠋⡄⠈⢣⠰⢺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠻⣄⠈⠀⡇⠠⡉⠀⢻⠀⢈⣹⠀⣳⡀⢸⠀⣸⠏⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⡍⠓⢤⢧⠀⠈⡇⢻⠉⢹⠀⢠⡇⠙⣌⠴⡋⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠳⣄⠀⢸⠀⠀⠀⣿⡔⠀⠉⢀⢷⠀⡙⢰⠃⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠸⡏⠳⠛⡀⢪⢠⣸⡇⣄⡄⢳⠈⠃⢰⠁⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠉⢢⣠⠁⠈⡌⡏⢧⢠⡁⠂⣥⡴⠋⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⡰⢾⡀⡨⣸⠁⠘⡌⣍⣸⠓⡄⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢀⣞⠀⢘⡿⣱⠃⠀⠀⠹⣼⠃⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠘⣿⠀⣸⡷⠁⠀⠀⠀⢸⠟⠀⣰⣟⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢤⣤⠅⣟⡰⠋⠀⠀⠀⠀⠀⠀⠻⡐⠫⣑⢄⠒⠄⡀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢀⡿⠂⢲⠆⠵⣛⡥⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠓⠮⢔⣈⡍⡡⠤⢀⢴⡇
⠀⠀⠀⠀⠀⠀⠾⠉⠤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⣾⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠙⠳⣦⠀⠀⠻⣤⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⢸⡇⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⡴⠟⠀⠀⠀⠈⠙⢶⣄⣀⣀⣻⠄⠀⠀⠀⠀⠀⠈⠻⣄⠀⣀⡴⠏⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣋⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⣦⡀⢀⣹⣤⣦
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⢤⠸⠋⢠⡴⢦⡈⠉⢻⡁⠀⢹⡀
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⠶⣤⠞⠁⣠⢦⡄⠀⠀⠘⠷⠾⠃⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣀⡼⠋⠀⠀⠀⣠⣤⣄⠀⠳⠴⠏⢠⣤⣄⠀⢀⣤⣄⢸⡇⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠉⠉⠙⠉⢠⡶⢶⡄⠀⠻⣤⠟⠀⣀⣀⠀⠹⢦⠾⠀⠻⣤⡿⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡿⠀⢀⡶⠲⡆⠈⠳⠞⠃⣠⣤⡄⠀⢸⣏⣹⠇⠀⠀⣤⣤⡀⠀⢀⡿⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢿⡉⠁⠻⣆⠀⠛⠚⠃⠀⠀⠀⠀⠻⣤⠟⠀⠀⠈⠁⠀⠀⠘⢧⡼⠇⢀⡾⠁⠠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢠⣄⡀⠀⠀⠀⠈⠻⣄⠀⠈⠳⣄⠀⠀⢰⡟⢻⡆⠀⢀⣤⣄⠀⠀⣼⠛⣷⠀⠀⢀⣴⠟⠁⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣄⣀⠀⠈⠳⣄⠀⠈⠻⢦⣄⠙⠋⠀⠀⠸⣧⡾⠁⠀⠈⠛⠉⣀⡴⠟⠁⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠾⠿⢦⣄⠀⠁⠀⠲⠶⣤⣤⣤⣤⣤⣤⡴⠶⠛⠉⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⣄⣶⡐⠨⠁⠈⠑⠄⠁⡤⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⡒⠃⠀⠀⠀⠀⠀⢂⠀⡀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢠⢈⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⢀⡀⠀⠈⠂⠀⠐⠀⠀⡠⠄⠀⠈⡆⠀⠀⢀⠈⠈⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⣀⠠⠪⠀⠀⠀⣠⡁⠀⠀⠈⢑⠀⡀⠠⡇⢀⠔⠁⠱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣐⡄⢰⣦⠀⠀⠘⣿⠀⠀⠀⢀⠢⠀⠆⠘⠁⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠈⠿⠃⠀⠀⠀⠀⠀⠀⡈⠀⡀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠃⠐⠀⢠⠊⢀⠰⠁⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⢠⠀⢉⠠⣥⠒⢁⠌⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠃⢠⠀⠀⢀⣀⠀⠤⠄⠒⠂⠉⠄⣦⣤⡆⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⡕⠂⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠉⠀⠀⠀⠈⠉⠉⠉⡙⠀⠀⠘⠋⠉⢹⠤⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠄⠀⢸⠷⠿⠿⢿⠀⡀⢀⠄⠂⠀⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⢠⣷⢶⣤⣤⣼⡆⡀⠌⠳⠤⢔⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢲⠙⠿⢿⠠⢍⣯⣽⣌⢁⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⠈⠋⠉⠉⠈⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠉⠁⠚⠊⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠒⠀⠁⢢⣀⡀⠀⠀⢡⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡔⠁⠑⣼⠄⠀⠈⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⡶⠃⢀⡇⢀⠥⠀⠊⣡⠤⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⢀⣠⠶⠋⠀⠰⠀⡀⠊⠁⠀⢀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⠉⠀⠀⠀⠀⠀⠁⠈⠈⠀⠉⠄⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢸⠏⠀⣼⡇⠈⢻⠈⢹⡆
⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠲⠤⢤⣀⡤⠂⠀⣀⣀⣤⡀⠀⠘⢣⣞⣦⣥⣿⣧⣧⣞⣦⠚⠀⠀⣠⣄⣀⡀⠀⠢⣄⣠⠤⠴⠒⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠒⠲⣄⠀⡎⠉⢳⠮⠤⠠⠏⠉⠛⠋⠉⣿⣿⣻⣻⣛⣻⣛⡏⠉⠛⠋⠉⠧⠤⠬⢶⠊⠉⡇⢀⡴⠒⠒⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢧⠀⠈⢄⠀⠀⣎⡗⠀⠀⣀⣤⡤⠴⠦⠶⠤⣤⣄⣀⠀⠀⢏⡆⠀⢀⠈⠀⢠⠇⢸⡀
⠀⠀⠀⠀⠀⠀⠲⠤⣀⡀⠠⠔⠃⠀⠈⠓⠦⣤⣤⣬⣧⡶⠚⠋⠉⠀⠀⠀⢐⠀⠀⠀⠈⠉⠛⠲⣦⣯⣤⣥⡤⠖⠋⠀⠀⠑⠤⣀⠀⡀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠒⠈⠉⠉⠑⠒⠒⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⣴⡿⣷⡄⢀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣅⣀⣹⣿⣿⣁⣀⣭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠉⠉⠓⠲⠤⣄⠀⠀⠀⠀⠀⢩⢭⣭⣭⣯⣽⣽⣭⢯⠁⠀⠀⠀⠀⢀⡤⠴⠒⠋⠉⠉⠁⠐
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠒⠂⠂⠀⠐⠲⡄⠀⠀⠈⠳⣤⠖⠋⠉⠒⠒⠒⢒⡖⣖⠓⠒⠒⠊⠉⠓⢦⡴⠋⠀⠀⠀⡴⠒⠀⠀⠂⠒⠲⣄⡀
⠀⠀⠀⠀⠀⢠⠎⠀⠘⢦⣐⢠⡀⢀⡴⠃⠀⠀⠀⣼⠉⠳⣀⠀⠀⠀⠐⠈⠁⠉⠐⠀⠀⠀⢀⡰⠋⢹⡀⠀⠀⠀⠳⣄⡀⣠⢐⣠⠞⠀⠈⢦
⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠈⢉⡏⠁⠀⠀⠀⠀⠀⠸⣄⠀⠈⠓⠤⣆⡀⣦⠀⢠⡆⣀⡤⠔⠋⠀⢀⡼⠁⠀⠀⠀⠀⠈⠉⣏⠉⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠉⠒⠦⠴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠄⣀⠠⠶⠃⠀⠀⠳⠦⢀⡀⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠖⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢂⠩⠤⠰⢒⣲⣲⣟⣉⣡⣆⣖⣶⡶⡤⢄⣉⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠐⣿⡷⠴⠛⠻⠿⠿⢿⢟⢻⢛⡛⠛⠃⢠⣤⣾⠃⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠃⠂⠈⠑⢾⣿⢿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣶⡤⠊⠀⡠⣿⡟⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠁⠕⠶⣤⣀⡀⠙⠻⣿⣿⣿⣹⣿⣿⣿⣿⠿⠋⠀⣀⣬⣤⢟⢛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣛⡿⣿⣶⣤⣀⡉⠛⢷⣆⡈⠛⢿⣿⣽⡿⠛⢡⠴⠶⠛⢉⣩⣴⠞⠋⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢡⣴⣬⡿⠀⠀⠉⠛⠛⠿⢷⣬⣥⡶⢤⣽⢫⣴⢶⣀⣤⡴⠶⠛⠉⠀⠀⠈⠁⠈⠉⠛⢦
⠀⠀⠀⠀⠀⠀⠀⢀⣞⣱⠅⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⣻⡷⠊⠋⠉⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠠⠟⠻⢏⢻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠲⢎⢽⡄
⠀⠀⠀⠀⠀⠀⣗⠰⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠻⡀
⠀⠀⠀⠀⠀⢸⣶⠁⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡰⠀⠀⠈⠥⠀⣧
⠀⠀⠀⠀⠀⣿⠘⠄⠄⠈⠳⢘⠲⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⠋⢻⡤⠂⠐⠠⢰⣽
⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠐⢮⢀⣦⢠⡀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠺⠁⠀⠀⠀⠀⠀⠀⣀⢰⣿⣿⠞⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⡿⠃⠁⠀⠀⠀⠀⣴⣆⣿⡜⠇⢠⡐⡠⣠⠀⠘⢁⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢑⢻⣾⡿⠄⡖⡶⠆⠀⠀⠐⣿
⠀⠀⠀⠀⠀⣯⡆⠀⠀⠀⠀⠚⠟⢻⡿⣟⣿⣒⣛⣛⣈⣀⢀⠏⢐⢄⡟⠁⡽⠋⠉⠱⣖⠶⠺⡗⡻⢿⡇⡇⠇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⢇⡗⠀⠐⠠⠶⢻⣿⡝⠫⠍⠀⢹⠟⠛⠇⡮⠀⠴⠮⠒⠀⠀⠀⠀⠈⠁⠀⠀⠫⠞⢹⡗⠦⠄⠀⠀⠀⠀⢰⡾
⠀⠀⠀⠀⠀⢸⠜⠉⠀⠀⠀⠀⠸⠋⠙⠀⠁⠀⠀⠀⠀⠀⢳⣦⣾⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠐⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⣥⣽⢿⣦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢺
⠀⠀⠀⠀⠀⠀⢻⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡑⢀⡻⣿⠛⣛⠩⣶⢤⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠆
⠀⠀⠀⠀⠀⠀⠀⠣⣄⣠⡄⡀⠀⠀⡀⠄⠀⢨⡴⣟⣫⣻⡿⡖⠱⠘⠈⠁⠹⠃⢽⡠⣢⣠⣔⠀⢀⡀⠀⠀⠀⠀⣧⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣀⡲⡶⣿⡳⠛⠟⡿⠃⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢧⡙⣲⠤⢆⣠⣤⣶⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢩⠏⠁⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠉⠉⢡⢁⣞⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠋⠄⠀⠀⠀⢿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢸⠀⠀⠀⠈⣹⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠀⠈⠀⠚⠛⠛⠒⠒⠒⠠⠒⠚⠟⠉⠉⠛⠊⠉⠁⠀⠀⠀⠀⢾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠾⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠶⠄⠄⠖⢦⠐⢶⡶⠶⣶⣠⢶⣶⣶⡶⠲⠹⠟⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠀⠀⠀⣿⡀⠀⠀⠰⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢶⠀⠀⠖⢒⣻⣷⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄⠀⠀⢀⣸⡆⡀⠋⠉⢻⡏⢀⣀⣸⡆⠀⠀⠀⠀⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠘⢻⡇⣧⠀⠀⣸⣷⢸⠘⢻⡇⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⢠⣼⣧⢰⡄⣸⢿⣿⣼⣄⣼⣧⠀⠀⠀⠋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣤⣿⣷⡄⠀⢲⣿⣿⡾⣰⡟⣏⣿⣿⣿⢿⣿⠀⢠⣮⣤⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣾⣿⢠⡀⠃⣹⣿⢉⢹⣿⣿⣿⣿⡏⢸⣿⢀⡄⣤⣤⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢀⣿⡏⣹⣿⠠⣞⡟⠾⠿⡟⡿⢙⣿⢃⣿⣆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠉⠙⡇⣻⡿⠀⣀⣤⣤⣀⠀⠀⢻⣿⡟⣹⣿⣧⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢹⡃⠀⢘⡇⢸⡇⢸⣿⣿⣿⣿⠂⠘⠀⣿⡧⣻⣿⡏⣽⣇⡀
⠀⠀⠀⠀⠀⠀⠀⣴⡿⡇⣶⣿⣿⣄⣼⡇⣹⡇⠠⣿⣿⢻⣿⠀⠒⢈⣿⣧⣾⣿⣿⣿⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠶⢶⣶⣿⣧⣶⣻⣿⣛⣻⡷⠞⠋⠉⠉⠉⠳⢾⣛⣛⣿⣓⣦⣼⣿⣶⡶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣽⣿⠁⠀⠀⠀⠀⠀⠀⠀⠙⢯⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢀⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣿⣿⡿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡂⠁⢉⠉⢙⣥⡤⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⡆⣷⣿⠋⠀⠀⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠀⠀⣿⡉⡷⣇⣀⠀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⢙⠷⡀⡷⣬⣼⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡶⢚⣾⠃⢳⠶⠾⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣷⡼⡠⣶⣿⣰⡿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⣤⣶⡶⢶⣾⠟⣻⠯⠈⢡⠷⣿⡅⠉⠲⠿⠻⣿⣶⣶⣶⣶⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⠛⣋⡵⠤⠒⢞⣻⠶⢚⣩⠐⠊⠀⠀⠀⠃⢀⡏⢻⠀⠀⠀⠀⠈⠙⠻⢭⢻⣯⠻⠹⢿⣿⢿⣿⣰⣤⣀
⠀⠀⠀⠀⠀⠀⠦⠞⠃⠀⠚⠁⠀⠀⠀⣠⠞⢁⡴⠋⠀⠀⠀⠀⠀⡀⠀⣊⣧⠘⡇⠀⠀⠀⠀⠀⠀⠀⠙⢪⡽⠀⠀⠀⠁⠈⠛⢿⣿⣷⣄⡀
⠀⠀⠀⠀⡶⠎⠁⠐⠀⠀⠀⠀⠀⢀⣰⠏⠰⣰⣶⣶⣶⣿⣿⠟⢁⠃⠀⣀⣽⡆⢻⠽⠻⡟⠓⣶⣶⣄⠀⠀⠛⣉⡇⡀⠀⠀⠀⠀⠙⢿⣿⢧⡄
⠀⠀⠀⠶⠃⠀⠀⠀⠀⠀⢀⣤⠀⢤⡁⢠⢁⣿⣿⣿⠟⠉⠀⠀⠈⠀⠀⡀⣈⡁⠸⡆⢀⣠⣶⣯⣏⣹⡇⢀⠀⡁⢾⡤⣄⠀⠀⠀⠀⠀⢻⣟⣷⡄
⠀⢀⠐⠂⢠⠀⠀⠀⣠⣶⠟⠀⠀⢼⡆⠈⢸⣿⣿⠃⠀⠀⠀⠀⠀⢀⡜⠁⠘⢷⠀⣼⣾⣿⣿⣿⣿⣿⣿⢠⠄⠀⢼⡇⣿⣷⣦⡀⠀⠀⠈⣿⣽⣦
⠀⣘⢺⢀⠈⠀⢀⣜⣿⠂⠀⠐⠢⢶⡇⠀⢸⣿⡧⠀⠆⢀⡀⠀⢨⠃⠀⠀⠀⠀⠰⣿⣿⣿⡿⠏⠉⣄⢭⢸⡃⡠⠶⠗⢻⣿⣿⣷⣄⠀⠠⣽⣿⣷
⠀⢈⡻⢈⠈⠀⣾⣉⠋⠀⠀⠀⠀⠀⠀⢰⠸⣿⣦⣀⠆⠈⢡⠔⠘⠄⣀⠀⠀⢄⡴⠻⢿⣿⣃⢹⣯⡿⡖⣸⠃⠀⠀⢠⡸⣿⣿⣿⣿⡆⠀⣼⣾⣿⠄
⠀⣈⣽⣽⡄⣾⣷⡟⠂⠀⠀⠀⠀⠀⠀⡦⠃⡠⠀⠑⣧⣀⡎⢠⡴⠀⠈⢀⠉⠂⣴⠒⠀⢸⣭⡥⢋⡤⠄⠙⣴⣶⣠⡟⣳⣿⣿⣿⣿⡇⠤⣿⣿⡷
⠀⠉⠉⠛⠻⡹⠷⠿⠦⢤⠀⡤⠀⠠⠘⠃⢸⢘⣦⠀⣾⠗⠁⢻⡀⢻⠀⠠⠀⣸⣯⡷⢀⡬⠻⡇⢺⣶⣦⠀⡿⣳⢏⣵⣦⢸⣿⡿⠿⠫⣼⠟⠛⢃⡼
⠀⠀⠀⢤⡀⡁⣰⢲⠀⡺⢌⠧⣴⡆⡄⠠⠈⠇⢻⣶⡟⠒⠀⣾⣿⣉⣁⢀⣀⣈⢹⣿⣿⠃⠐⢿⣾⠇⢻⣀⠁⢁⣾⡏⣸⣿⡿⢡⣶⣇⣿⣾⡿⠟
⠀⠀⠀⠀⠉⢁⡘⡩⠣⡧⢰⠛⣾⣧⡤⠖⠖⠻⠞⣿⣿⠏⠻⠿⡿⠇⣡⡈⣧⡈⠻⠿⠿⠋⠻⣿⣏⠰⡿⢻⠷⣾⣿⣟⣿⠠⣧⣸⡾⢳⠟⠁
⠀⠀⠀⠀⠀⠀⠩⣘⣤⣵⡾⢉⡛⢙⡁⠖⠒⠋⠒⠠⠰⠊⠙⠲⠆⠈⠐⠸⠇⠈⠄⠐⠆⠿⠶⠤⡄⣨⣷⣼⣷⣿⡟⣛⢹⣿⠯⣤⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡵⠃⣀⢀⠤⠤⠀⠄⠒⠂⠂⠀⠀⠀⠀⠀⠒⠒⠖⢲⡲⢚⢒⣢⣤⣤⣴⢖⠰⣟⣿⣿⣿⣼⣿⣯⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠞⠉⣀⣀⠀⠀⢪⣭⢽⡆⠘⢣⣶⣜⠀⠠⢠⡴⠾⠒⣿⣰⢮⡵⣷⣿⠛⢠⡾⠉⣿⣟⠿⣿⢿⡟⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⠈⢿⡁⡂⠀⠾⠷⠶⠃⠀⠚⠁⠘⠀⠀⢽⢭⠛⠫⠓⠸⠾⠶⣦⢿⣀⣾⣤⣠⣿⣟⣾⡽⣿⠃⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⢁⣀⢀⣀⠤⠄⠄⢀⡄⠠⠄⠀⠀⠐⠀⠂⠀⠐⠀⠄⠀⠀⠂⠐⠧⠤⠠⠭⣿⣯⣿⣿⣿⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣅⣿⣁⠀⢨⡦⠄⢠⣨⢇⡰⠀⠐⠢⠄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⡽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠈⠉⠊⠛⠛⠛⠛⠛⠙⠂⠐⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⡽⠿⠿⣅⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡴⣦⣴⢞⡿⢋⡙⠛⢧⣼⡀⠻⠃⣸⣤⠟⠛⣉⠻⣟⢶⣤⡶⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⠁⢀⣯⣬⡿⠛⠋⠉⠉⠀⣿⡃⣿⡇⣿⡇⠈⠉⠙⠛⠻⣯⣬⣇⠀⢹⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⢸⣤⣥⡇⠀⠀⠀⠀⠀⡟⢳⣶⣶⠞⡇⠀⠀⠀⠀⠀⣧⣥⣼⠀⢠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⡀⢸⣬⣅⡇⠀⠀⠀⠀⠀⢧⠈⣩⡉⢰⠃⠀⠀⠀⠀⠀⣇⣭⣼⠀⢸⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠃⠈⢉⣅⠡⠀⣰⣷⣄⠀⣠⡜⢿⠿⣤⡀⢀⣴⡷⡀⠀⢀⣍⠈⠀⢛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣴⣾⡿⡷⢾⡷⡿⢿⠟⢿⣟⢯⢟⣿⠟⢿⢻⡷⣿⠶⡿⣯⣧⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣬⣤⣵⣯⣤⣤⡷⢿⠶⢶⣾⢿⣾⣶⢿⣶⠶⢾⠷⣽⣦⣬⣧⣧⣬⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠟⡹⢂⣿⣐⣷⣾⣠⣷⣜⣡⣙⣴⣧⣸⣶⣗⣸⣇⠺⡙⠇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠩⠟⠛⠣





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⠻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⣶⣶⢶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⢍⡱⠋⠉⠀⠀⠉⠒⢰⠆⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢻⠻⠿⠓⢟⣀⠬⠀⠈⡇⠀⠀⠀⠀⠀⠀⠈⣳⡜
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⢿⣠⣤⡀⢼⣇⠀⢀⣞⠀⢲⣾⣿⣶⠛⠃⠀⣤⡙⡂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⣿⣿⣷⣾⣿⣿⣿⣧⣶⡿⠊⣡⣤⣤⡐⢀⣀⡤⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣏⢹⠉⠁⣞⣵⣷⣤⡎⣿⣿⣿⣿⣿⠯⠃⠉
⠀⠀⠀⠠⠤⠤⡄⢸⠀⠀⣆⡖⠀⠁⠃⣸⠘⣿⢻⡋⣰⡏⠻⠛⡌⣽⣿⣿⣿⡅⠎
⠀⠀⠀⠀⠈⠂⠀⢸⣇⠀⣡⠁⠀⠀⠂⠘⠆⢻⣆⣿⣿⣧⢀⣰⢧⣻⣿⠯⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡖⠛⠗⠀⠀⠀⠀⠐⠂⠹⣦⣽⣿⣮⡿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢰⣶⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠓⠀⡤⠊⠁⠀⠀⠀⠀⣄⣾⣿⣿⣿⠒⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⢿⡧⠄⠀⠤⠀⠀⠀⠀⠀⠘⢛⣿⠛⣿⣽⣿⡿⠘⢹⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣤⣠⣼⠷⠶⠓⠒⠀⠀⠀⠀⠀⢀⡟⠙⢯⣜⣥⣿⢥⣞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠂⠈⠉⠙⠙⠆
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠐⢠⠀⠀⠀⠀⠀⠐⢀⠸⡇⠈⠑⢄⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠈⠳⡀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢂
⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⡈⠃⠀⠀⠤⠀⠀⠈⠀⢀⠀⠱⡀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡄
⠀⠀⠀⢀⠐⠂⠐⠂⡃⢸⠀⠀⠈⠀⠀⢀⢀⡀⡀⠀⠀⠙⠢⣽⣀⡌⠀⣀⠶⡤⢤⣀⠀⠀⠀⣶⢰⠟
⠀⠀⠀⠀⡠⠐⠀⠀⠁⢨⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣣⠏⠀⡄⠀⠀⠀⠀⠤⣴⣯⣀⢀⡀⠀⢀⠶⡃
⠀⠀⠀⠀⠀⣠⠖⣃⠀⠀⢀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣯⣭⣭⠛⠀⠁
⠀⠀⠀⠀⠀⠨⠁⠙⡂⠀⠈⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠛⢿⠁
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠂⠀⠈⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠢⣀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⡀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⡿⠟⣃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣅⣿⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠷⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣾⣿⣿⣿⣿⣷⣾⣷⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⡠⣤⠌⣋⣉⣻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣿⡿⣿⣿⣿⣟⣿⣿⣯⡇⣼⠀⠀⠀⠀⠀⠀⠘⠋⡖⠀⠀⠉⠈⠻⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡽⡇⣿⢹⣿⣟⣿⣿⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣽⣿⡇⣿⣿⢿⣿⣇⣿⠆⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡤⢆⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣸⣿⣿⣿⣿⣿⣿⣿⠯⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⠈⠉⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣯⡤⠤⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢠⠀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠿⠿⠏⠀⠀⠀⡀⡄⣴⣦⣹⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⡈⠐⠚⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢠⠀⠉⠻⢿⣿⣿
⣿⣿⣿⣿⣿⠂⢀⢠⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⠿⠂⠀⠀⢈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣧⣠⣿⣾⣧⣰⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣤⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⣠⣤⣿⣸⣿⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⣀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠖⠂⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⣦⡤⣀⣄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠋⠀⢀⠀⠀⢤⡤⣀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠃⠉⠳⣤⣀⠈⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡃⢽⡶⣄⠈⠑⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⡀⠀⠀⠀⠀⠰⠂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠀⠀⠀⠀⠀⢯⡛⠶⣿⣯⣌⣲⣤⣄⠲⣴⡆⣀⡀⢤⣤⠤⠀⣀⡀⠀⡠⠀⠀⠈⣊⣤⠎⠳⠠⠄⠀⠙⠡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠓⠊⢤⣴⡒⠦⣛⣴⣤⣤⣄⡀⠀⠈⠓⣤⣤⣀⠞⠀⠀⡀⠚⢩⡼⠂⠀⠁⠀⣀⠴⢚⠤⣴⠶⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠁⠀⠀⡐⠦⣤⣀⡀⠀⠈⠛⠷⠾⠇⣻⢿⣤⣿⣿⣟⡷⣮⣞⠉⠉⠒⠊⠀⠀⣀⣀⠠⠀⠂⠩⠐⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠆⠠⠄⣸⠍⠙⠻⠻⢏⣹⠷⢋⣙⠛⠛⠁⠀⠀⠠⠖⠋⠉⠀⠀⠀⠒⠀⠁⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⠤⡀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠈⠀⠈⠉⠉⠴⠶⠾⠀⠀⠉⠀⠀⡀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠙⠛⠓⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠊⠳⡈⠳⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠤⠤⠤⠀⠀⠀⠒⠂⠈⠉⠉⠉⠉⠐⣀⠤⠷⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢿⡀⠸⡄⠀⡀⠐⠠⣄⣀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⢫⣖⠴⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠁⠀⣧⣄⣈⣓⠒⠀⠉⠉⠒⠠⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠐⠂⠀⠀⠀⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠃⠀⠤⠄⠉⠉⠉⠈⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠼⠴⢦
⠀⠀⠀⠀⠀⠀⠀⡴⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⣴⢉⡁⠁⣦⣄⠶⡄
⠀⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡂⠀⠀⠠⣀⠋
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠁⠀⠀⠀⠈


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⣝⡛⣋⡙⢻⡟⣫⣿⢻⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡤⢀⠴⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⡾⠛⠉⠠⠤⠤⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠳⣤⣤⣤⡤⠴⠚⢁
⠀⠀⠀⢠⠁⠀⠈⠁⠀⠀⢀⣀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠟⠛⠀⠑⠒⠒⠒⠛⠋⠉⡠⣄
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠠⠴⠶⠶⠒⠚⠉⠁⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡤⠶⠶⠖⠒⠒⠒⠒⠶⠤⢬⣽⠓⠦⢤⣤⣀⣀⣀⣀⣀⣤⠴⠶⠛⡏
⠀⠀⠀⢰⠀⠀⠀⠀⠈⠁⠀⣤⠀⠹⠓⠒⠶⠦⠤⠤⠤⠤⠶⠒⠚⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⢠⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠈⠄⠀⠀⠀⠶⠀⠀⠀⠀⠿⠀⠀⠀⠘⠃⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠇⠀⠀⠀⠀⠰⠆⠀⠠⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠛⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⣿⣟⠀⠀⢳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠰⡀⠀⠀⠀⠀⠀⣡⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠡⠃⡂⠀⢀⣼⢻⠟⣀⣀⠀⠀⣼⣿⣿⣿⠿⠛⠻⠿⣷⣄⡀⠻⠀⠀⠀⠀⠀⠄⠀⠀⠀⣴⠀⠀⣼⠏⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⡆⣴⠀⢀⣾⠃⢸⠇⠘⠇⠀⢺⣿⣿⣿⣯⣤⣤⣄⣄⡀⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⣸⡿⠀⢹⣷⡀
⠀⠀⠀⠀⠀⡀⢀⡟⣀⠀⠸⣵⡏⠀⠀⠩⡀⠚⠀⠈⡷⠰⣾⣿⠿⠟⠛⠉⠉⠉⠙⠙⠟⠀⣠⣦⡀⠀⠀⣤⠀⠀⠻⡆⠉⣶⣿⡇⠀⠀⠙⣧
⠀⠀⠀⠀⣼⠁⣾⠡⠁⠀⢸⣿⠀⠀⠀⠀⠙⣿⣿⡿⠉⠭⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠇⠀⠀⠀⢹⣷⣿⡇⠃⠀⠀⠀⠈⢀
⠀⠀⠀⠀⢻⣻⣿⣿⣿⣿⣿⡿⡟⢓⠒⠖⠶⢿⡿⠁⠀⢀⣹⣿⠀⠀⠓⠀⠀⠀⢰⠀⣷⣶⣦⣀⡀⣀⣀⣀⣀⣀⠈⢀⢙⣿⠀⢀⣀⣀⣀⣀⣀⣆
⠀⠀⠀⠀⢸⣿⣧⠬⢴⣯⣾⣿⠀⠀⣠⡄⠀⠚⠀⠀⠀⠀⢉⣷⠀⠀⠀⠀⠀⠀⠀⠀⣫⠏⠛⢻⣟⠛⠛⠛⣟⠉⠉⢹⡋⠉⠂⠸⣿⡍⠉⠉⠉⠛
⠀⠀⠀⠀⠈⣛⣽⣆⠈⠛⣿⣿⠀⠀⠹⠃⣴⡶⠖⢿⣿⣿⣿⣿⠀⣀⠀⠀⠀⠀⣄⠀⡟⠀⠀⢘⠛⢀⣀⣈⣛⣀⣀⣈⡉⢀⡆⢸⡿⠷⠆⠙⠛⠁
⠀⠀⠀⠀⠀⣬⣿⡟⢸⣿⢽⣷⠀⠀⠀⢀⡙⠆⠐⠖⠂⠙⢛⣿⡀⠙⠀⠀⠀⠀⠉⢀⠁⠀⠈⠻⠿⠿⠃⠙⠈⠉⠉⠙⠙⡻⠇⢸⡿
⠀⠀⠀⠀⠀⡹⢿⣿⠸⠿⣿⣿⠀⣿⠀⠸⣄⣀⣀⣠⣤⣤⣬⣿⡇⠀⠀⠀⠀⠀⠀⠹⠿⠁⢰⠬⠃⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⣈⢟
⠀⠀⠀⠀⠀⠿⣽⣿⢐⡻⢯⣿⠀⠀⠀⠀⢽⡟⠉⠉⣿⣿⣿⣽⡇⢸⡆⠀⠀⠀⡗⠈⠀⠀⢸⠶⣬⣤⠀⠀⠀⣤⣤⠀⠀⠯⣻⣛⠿⢀
⠀⠀⠀⠀⠀⣰⣾⣿⠀⣋⣹⣿⠀⠀⠀⠀⢘⡇⢠⠀⢸⣿⣿⣞⣇⠸⡇⠀⠀⠀⠉⠀⠀⠀⠀⠀⣥⣭⠀⠀⠀⣷⣽⠀⠀⣶⡿⣿⠅
⠀⠀⠀⠀⠀⣈⣚⣿⠀⣿⣿⣿⠀⣿⠀⠀⢨⡇⠀⠀⠀⠀⠫⣼⣿⠀⡀⠀⠀⠀⠀⠀⠹⣋⢟⡭⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⣳⣖⣂
⠀⠀⠀⠀⠀⠈⠛⣿⡏⠼⣿⣿⡄⠀⠀⣿⣿⣛⠀⠀⠐⡿⢿⣷⣿⢠⡄⠀⠀⠀⣶⠀⠀⡟⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡷⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠈⢠⡿⡿⣿⡃⠀⠀⠈⠉⣿⠀⠀⠰⣿⣾⣟⣿⠐⡇⠀⢀⠀⠛⠀⠀⣷⠀⠀⢰⡄⠀⠀⠀⢴⡆⠀⠀⠸⢿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⠾⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠒⠘⠃⠀⠀⠀⠘⠁⠀⠀⢸⣇⠿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⣥⣈⡅⠀⠀⠀⢀⣿⠀⠀⠀⢐⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣝⣧⢄⣀⣤⣤⣼⣤⣤⣡⣬⡷⡿⠿⠧⡤⢤⠤⠀⠀⡀⠾⠛⠛⠿⠿⠿⠷⠶⠶⠶⠶⠦⠴⠾⠛⠛⠛⠂
⠀⠀⠀⠀⣤⡶⠶⠻⠟⠛⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢺⠟⢟⠀⠤⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢀⡀⠒⠒⠄
⠀⠀⠀⠀⠀⠀⢀⠈⠂⠈⠆⠀⠀⠀⢀⠅⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⡟⡉⠉⠂⢰⠤⠐⠓
⠀⠀⠀⠀⠀⠀⠈⠁⠸⢄⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠈⠃⠀⠠⠈⢄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠁⢀⠀⠀⠀⠠⡀
⠀⠉⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀
⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡄⠀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⡠⠀⠤⠶⠈⠊⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀⠀⡆⢀⡀⠑⠠⠐⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⢦⠧⢄⠉⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⣿⡇⣠⣴⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠛⣿⣿⣿⡟⢛⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⠀⠁⠀⣸⣿⣿⡆⠀⢀⠤⣒⢦⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⢹⣿⣷⣶⣤⣿⣿⣿⣷⣾⣵⣿⣿⡏⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠹⣶⣄⠀⠀⠀⠀⠀⠀⢸⣿⣦⠀⠀⠀⢟⣿⣿⣿⣿⡇⢻⣿⣿⣷⡟⡟⠀⠀⢀⣴⣿⠃⠀⠀⠀⠀⠀⢀⣴⣾⠋
⠀⠀⠀⠀⠀⠙⣿⡀⠀⠀⠀⠀⠀⠈⣿⣿⣷⡀⢀⣎⢮⣹⣿⣿⡀⣠⣿⡿⣩⡔⣠⣀⣠⣿⣿⡿⠀⣀⡀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⢠⠊⡷⠀⣿⡟⢻⣟⢿⣿⣯⢟⢿⣿⣿⣿⢟⡞⣝⣼⣿⣯⡿⠋⣿⡇⠘⠃⢸⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣀⠓⣤⣔⠻⣧⡀⣿⣼⣿⣿⣷⡣⣯⢿⣏⣎⢮⣾⣿⣿⣿⣀⣼⠏⣽⣶⣮⢁⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣾⣿⣯⣾⣟⢿⣿⣹⣿⣿⣿⣿⣟⠎⡸⣳⣿⣿⣿⣦⡽⣟⣻⣷⢽⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣭⡽⣿⣞⣻⣿⣿⣭⣭⣭⠎⠰⡹⣿⣿⣿⣿⣿⣿⣶⣷⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⢿⣛⣫⣭⣵⠶⣶⣿⣿⣛⡧⣴⣹⣟⣛⣶⠾⣯⣽⣟⡷⢿⣽⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣟⣿⣽⠿⠞⠛⠛⠉⠉⠉⠉⢞⡟⠉⠉⠉⠉⠛⠛⠶⢿⣽⣻⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣤⣀⣀⣠⣄⣀⣀⢀⣄⣀⣀⣀⣠⣀⡀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠐⠶⠦⠤⠴⠀⠤⠆⢤⠤⠤⠴⢤⢄⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠘⠒⠒⠚⡒⠆⢀⠖⢒⠒⠛⠚⠒⠂⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠘⠉⠉⠉⠉⠉⠈⠀⠉⠉⠉⠉⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿


⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠁⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠃⠀⠀⢡⣠⡆⠀⠀⢀⣀⠀⠀⠀⠀⠐⠀⠀⠀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⣄⢔⢸⣿⣦⡀⡧⠤⢤⠀⠄⠀⠀⢙⠀⠀⠘⣿⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣼⠿⢿⣿⡇⠀⠀⡀⠀⠀⠀⠃⠀⠀⠀⢿
⠀⠀⢂⡀⠀⠀⠀⠠⠀⠉⠳⢯⡛⢿⠆⠛⠎⠁⠀⠀⠈⠁⠀⡁⠀⠀⠀⠀⠠⠚
⠀⠠⢀⠁⠀⠐⠀⠒⠒⡄⢷⣶⣶⡥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠈⠀⠀⠀⠀⠀⠰⢰⡘⠿⠿⢄⠀⠈⠀⠀⠀⠀⠀⠤⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⢶⣾⡄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⠇⠂⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢉⠾⠻⠿⡿⣶⣖⣂⣤⢴⣒⣾⣿⠿⠟⠋⠉⠀⠈⠁
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠀⠀⢰⣿⣿⣿⠙⠋⠁
⠀⠀⠀⠀⠰⠀⠀⠠⠐⢄⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠃⠉
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠑⠢⠀⠀⠀⠨⠐⠂⠤⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠖⣚⣉⣭⠭⠭⠤⠶⠶⠶⠶⠶⠶⠶⠤⠬⠭⣭⣍⣛⣛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⡶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢮⡻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢦⣙⠲⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠇⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠛⠲⢦⣭⣍⣛⣓⣲⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠶⠖⣒⣚⣩⡵⠟⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠐⠒⣠⠖⠋⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⢀⡴⠶⢦⣄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⢰⣯⣀⠀⠠⠤⣀⣀⣀⣀⣠⡤⢞⣿⠔⠋⢀⣀⣀⣈⢳⣄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⡾⠈⠉⠛⠓⠲⢶⠶⠾⠧⣤⣤⡾⠧⡤⠤⠤⠴⠶⢒⡶⢿⡏⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠳⣄⠀⠀⠀⠀⣼⢀⣠⠞⢻⡟⢧⡀⠀⠀⣀⡴⠞⠉⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠑⠦⣄⣀⣿⠟⠁⠀⣿⠃⠀⠙⢷⡞⠁⠀⠀⠀⣠⠞⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣠⠞⠉⠢⣀⠀⠸⡇⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣟⡠⠊⠁⠀⠀⠀⠈⠙⢶⡧⠔⠋⠀⠀⠤⠀⠀⠀⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠻⢄⡀⠀⠀⠀⢀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠳⢤⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣠⠤⠴⠶⠖⠚⠛⠛⠛⠓⠂⠀⠀⠀⠀⠐⠲⠤⢤⣀⡀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠠⢄⣀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠈⠉⠙⠛⠲⠶⠶⠤⣤⣤⣤⣤⣤⣤⣤⠤⠴⠖⠒⠊⢁⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠶⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⠻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⠀⠀⠀⠀⢀⡠⣿⣏⣹⣿⣄⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⣠⣿⣯⢿⣿⣶⡤⢤⣄⠐⣿⡿⢽⣧⣿⡭⢿⣿⠒⣤⡤⢠⣶⣿⡿⣴⣿⣄⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠙⣷⣶⢤⡄⣤⡎⢿⡏⢹⣿⣏⠙⠿⣾⣿⣶⡝⠆⢨⣿⣿⡁⠸⢯⣶⣿⢧⢿⠋⣽⣿⡟⢹⡿⣰⣆⢤⣤⣶⣿⠏
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣞⣿⣿⣍⣴⣿⣿⣷⣶⣿⣿⣿⣿⣾⣿⣿⣾⣿⣿⣾⣿⣿⣿⣶⣾⣿⣿⣦⣬⣿⣿⣳⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣟⣳⡿⣟⢻⢹⣟⣛⣻⡽⠯⣿⡿⢭⢩⣿⣛⣻⡟⡝⣻⢿⣾⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣴⣾⠷⠖⠛⠓⠚⣚⣋⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣙⣓⣒⠚⠛⠾⠿⢷⣶⣿⣯⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠟⣋⣭⣥⣶⣶⠟⠛⠋⠉⠉⠉⠉⠉⠁⠀⠀⠈⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠛⠳⣶⣶⣬⣭⣛⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠲⠿⣿⣿⣿⣿⠶⠶⢶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡶⠶⠶⣿⣿⣿⣿⠿⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣸⣧⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⡄⠀⠀⠀⣠⣿⣶⣤⣤⣤⣾⣿⢿⣷⣤⣤⣤⣴⣿⣄⠀⠀⠀⢀
⠀⠈⢀⣤⡀⠀⠈⠃⠈⠀⠠⠀⠈⠁⠀⠄⠀⠀⠘⠁⠀⠀⣠⣄⠁⠀⠀⠘⠷⠲⣾⣿⣿⣽⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣯⣿⣿⣷⠖⢾⠃
⠀⠀⢨⡿⢉⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⡉⠻⡇⠀⠀⠀⠀⠘⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣄⠃
⠀⠀⠐⠋⣭⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣭⡝⠃⠀⠀⠀⠀⢠⣿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⢠⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⣿⡄⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠔⠁⢠⠀⡀⢠⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠤⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠈⠠⠃⠈⠑⠘⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡶⠀⠀⣀⣤⠆⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⡻⠻⠆⠀⠀⠀⠀⠀⠀⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⢀⠀⠀⢀⣠⣆⣠⠀⠀⠃⠀⢀⣲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣽⡿⠁⠀⢠⡋⠉⠹⢃⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠶⠆⠂⠀⠺⢿⢇⡀⡀⠀⡸⠿⠛⢀⡿⢶⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⢻⡃⢠⠇⠀⢠⠂⠀⠠⠀⠀⠀⣴⣿⠟⠲⣄⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢠⠏⠀⠀⡇⠀⢠⣿⠟⠻⠀⠀⢹⠆⠀⠀⠶⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⢠⣄⠀⠘⢀⡄⠀⠃⠀⠉⠉⠀⡴⠞⠀⠀⠀⠀⢠⡆⠈⢢⡀
⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠈⠀⠀⠀⠞⢷⡛⠱⠄⠀⠀⡴⠀⠀⠀⠀⠀⠀⠐⠿⠃⣀⣸⡟⠦
⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⡄⢲⡆⠀⢠⣷⢀⡀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠀⠀⢸⡋⠁
⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⡎⠑⠇⠀⢘⠉⠘⡇⢢⣿⡶⠊⠀⠀⠀⠀⠐⠂⠀⠀⠁
⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠃⠀⠠⡘⢾⡄⠀⣏⡼⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠇⠀⣷⡄⡟⢀⣶⣇⠀⠀⠀⠲⢀⣄⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⡀⢠⡄⢱⠙⡄⢸⣿⢋⡀⣘⣀⠀⠘⠛⠛⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠱⣾⡇⢸⡄⠁⠀⡇⢈⠷⠛⡟⢃⣠⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢸⡇⢸⡇⠀⢀⣤⣾⣶⡷⠴⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⡀⠀⠈⡇⣸⣷⣷⡿⠟⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠀⠀⢳⡏⣿⠥⣶⣶⣽⣿⡴⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠠⣄⡀⠀⠀⠀⣸⠇⣿⣠⣨⣿⣿⣿⣿⣋
⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠈⠙⢦⡀⠀⣿⡔⡿⠃⠛⠛⢿⣼⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⢿⡆⢿⡿⡇⠀⢀⣀⡀⠋⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣤⣿⠾⠃⠀⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠊⠉⠀⠀⠀⠀⠀⠈⢆⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢨⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠑⠀⠐⠐⠘⠀⠈⠐⠒⠒⠂⠠⢄⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠱⠀⠀⠑⠢⢄⣀⣀⡀⡤⠀⠉⢁⡀⠀⠀⠀⠀⠀⠈⢃⡐⠀⠀⠀⠀⠀⡂⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠄⠀⠀⢀⡀⠀⠉⠒⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠑⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠤⠀⠀⠀⠈⢈⣲⣶⣤⠀⠀⠀⠀⡀⠢⣀⠆⠀⠀⠀⢀⡀⠁⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠴⣭⣤⣔⣉⠀⠶⠛⠀⠈⠀⠀⠄⡐⠃⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠢⠀⠊⠁⠀⠈⠀⠀⡀⠄⠗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠩⠭⠀⠀⢀⠠⠤⠀⠒⡀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡄⢁⠀⠈⠀⠣⡀⠀⠀⢦⡀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⢠⠀⣸⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠴⠀⠀⡎⠀⠀⠀⠑⢄⠀⠀⠀⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡏⠀⠀⠀⠈⠀⠀⠁⠆
⠀⠀⠀⠀⠀⠔⠀⠀⡀⠄⠀⢠⠇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠄⠐⠁⠀⠀⠀⠸⠀⠀⠀⢠⠀⠀⠀⠀⠀⠁⠀⠀⠁⠘⠆⣀⠀⠀⠀⠀⠀⠙⠛⠓⠀⠀⢄⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠘⡄⢀⠋⠀⢀⡇⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠂⠈⢉⠖⠸⠄
⠀⠀⠀⠀⠀⢠⠀⢸⠁⢠⢾⠀⠀⠀⠀⠀⠂⠀⠠⣤⢤⣤⣤⣀⣀⠨⠶⠆⠐⠒⠒⠒⠙⠓⢴⡆⡖⠐⠂⠁⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠠⡠⣴⢈⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⡀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠎⠹⡖⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠓⠀⠀⠀⠒⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡤⠀⠀⠂⠀⠀⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢰⢀⡄⠙⣄⠐⢂⠶⠀⢠⣄⠀⣀⡀⣀⠈⢢⡀⠀⠀⠧⣀⣀⡀⠀⠀⠀⠀⠀⠀⢠⠗⣄⠀⠀⠀⠀⠒⢀⠈
⠀⠀⠀⠀⠀⣀⠀⠉⣀⠄⠀⠈⠲⡀⣀⠀⠈⠂⠀⠉⢌⠉⠠⠈⠓⣄⠐⠄⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠬⠑⠆⠤⠄
⠀⠀⠀⡄⣚⠄⠄⠂⠀⠀⡀⠀⠀⠘⢞⠂⠙⠀⢀⠀⠀⠣⡀⠠⠀⠈⣦⡀⠢⠈⠓⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠂⠤⠀⣤
⢠⠀⠀⠀⢰⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠑⢄⡠⠈⠁⠀⣀⠈⢆⠀⠐⠇⢹⣦⡀⠀⣀⠀⣂⣀⣠⠄⠀⠀⠀⠀⠀⡇⠀⠀⠠⡀
⢰⠀⠀⠀⢠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠠⠤⠀⠉⠈⠉⠉⠉⠁⠁⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⢈
⠸⠠⣤⠀⠀⢂⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⠀⠀⠀⠀⠀⠀⠒⠠⠀⠈⠀⠀⠁⢀⣀⠃⠴⠂⠀⠀⠀⠈⠁⠀⠀⠈
⠀⠀⠀⠀⡷⢈⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠈⢎⠱⣄⠀⠀⠀⠀⠀⣀⣤⡴⠒⠈⠉⡉⠸⡇⣀⣀⡠⠖⡀⠚⠢⡀
⠐⠄⠠⠠⠬⠏⠀⠠⠿⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠸⠤⠆⠱⠶⠻⠏⠡⠀⠌⠴⠄⠴⠤⠀⠸⠅⠄⠹⠿⠄⠀⠀⠀⠀⠦⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠁⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠄⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⠦⡀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠉⢠⣄⣀⠀⠀⠀⠀⢱⣼⣷⣾⠄⠀⠀⠀⠀⢀⡀⠀⠂⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢶⣤⡀⢤⣚⣿⢿⡟⠀⠀⠀⠈⠀⠀⠀⠑⠄⠀⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠑⠤⣄⡀⠀⠀⠀⠀⠀⢈⡄⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠔⠊⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣠⣿⣿⣻⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣤⡠⡀⠀⠀⠂⠄⠐⠰⢄⣀⢀⡠⠀⠠⠉⠀⠐⠀⠀⢹⢿⣿⡿⠄
⠀⠀⠀⠀⠀⠸⠅⠀⠀⢀⠄⠉⠀⠀⢰⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠏⠃⠀⠀⠐⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠼⠁⠀⠀⠀⠸⡺⣄⠀⡀⠀⠀⢃⠀⡀⠀⠀⣄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠤⠄⣀⠈⠻⢿⣾⣷⡤⠂⠁⠀⠀⠸⠁⠀⣤⣀⠀⢀⡀
⠀⠀⠀⠈⠀⠀⠀⡠⢮⡿⠁⡀⠀⠠⠬⠿⢻⣿⣿⢷⡦⠤⠖⠚⠁⠙⢷⡏⠿⠛⠀⠀⡁⠀⠐
⠀⠀⠀⠾⠁⠉⠀⠀⠈⠁⠀⠀⡄⢀⣨⠶⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⡀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠜⠃⢨⠀⣠⠀⠀⠂⠀⠀⠀⠀⠀⠀⣰⠌⠀⢀⡏⢓⠢⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠀⠀⠀⠀⠠⣿⡃⠀⠀⠑⠒⠀⠒⠂⠁⠀⠀⠀⠩⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣏
⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠢
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⣿⣧⣀⣠⣼⣇⣀⣴⣄⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⡀⠀⣿⣷⣶⣿⣿⣿⣿⣿⣯⡿⡿⠼⢿⠿⣿⣿⣿⣿⣇⠀⢰⣶⣶⡄
⠀⠀⣀⣠⣤⣀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣷⠿⠿⠿⠿⠿⣿⡟⢻⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠶⢖⢲⣶⣞⣾⣶⣲⣾⣿⣿⣷⣿⣿⣲⢒⣶⣶⣶⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠿⢿⣬⢭⣭⣍⣭⣽⣿⣛⣿⣿⣿⣿⣾⣿⣭⣿⣿⣿⣿⡿⡝⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠙⣿⡟⠛⠻⣿⢿⣿⡿⢟⣿⣻⣿⣿⣿⣷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢴⣾⣆⣩⣴⡺⣿⣾⡶⠶⠟⠋⠉⠉⠅⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢹⠶⠂⠦⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⠀⠒⠷⣦⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⣿⠖⣤⡜⠛⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠈⣟⢦⣽⣎⣒⢠⣈⡉⢄⣀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠫⠆⡠⢄⣀⠀⠈⠶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠓⠪⠋⣙⡯⣭⡹⣠⣉⣀⡉⠉⠉⠣⣀⣁⠀⠔⠁⠄⠠⠞⣡⠅⠀⠀⠀⠀⡮⢄⣠⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠑⠀⠀⠐⢤⣄⡀⠀⠈⠛⠷⠼⡏⣻⣯⣽⣿⣷⣶⣼⡭⠙⠂⠒⠀⠀⠀⢁⡀⠀⠐⠃⠅⠂⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠳⣄⠀⠀⠀⠀⠀⡀⠈⣙⠳⢂⣦⣜⣎⡙⠛⠻⣛⣿⢾⣍⢛⡛⠁⠀⠀⠠⠚⠉⠁⠀⠀⠂⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣷⢤⣀⠀⠀⠀⠀⠀⠙⠒⠀⠈⠀⠀⢈⠐⡓⠺⠛⠃⠙⠁⠠⠤⠀⣀⣤⣅⡀⠀⠈⠀⡘⠋⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⢧⠈⢷⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠓⠒⠒⠂⣀⠈⠉⠉⠀⠀⠈⠀⠀⣤⡐⠛⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠙⢧⠀⡇⠀⢠⣈⠒⢦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠸⡗⠃⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠟⠓⠲⠤⠭⠀⠀⠀⠈⠶⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⢄⡀⣀⡈⠁⠉
⠀⠀⠀⠀⠀⠀⡉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⠍⠉⠀⡈⠝⠀⠁
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠈⠃⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡾⠋⡉⠤⠤⢬⣙⠛⠛⠛⠛⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⣰⡏⠀⠀⠀⠀⣀⣠⡾⠀⠀⠄⡀⠀⠂⢌⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠈⣿⠀⢀⡄⣠⠞⠉⠁⢻⣄⠐⢦⠈⠳⣄⠀⠳⡘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢀⡾⣆⢸⣷⡇⠀⠀⠀⣀⣙⣷⡌⢧⡀⠈⢧⡀⠹⡼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⡔⠀⣰⠿⠶⠿⠶⠿⠃⠀⠉⠛⠉⠉⠉⠛⣾⢷⡀⠈⣧⠀⢳⣹
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣸⠁⢰⣿⣿⣻⢶⣄⠀⠀⠀⠀⢰⣿⣛⢻⠻⣿⣼⣇⠀⣼⡶⠾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⣿⣿⣝⡭⠊⡛⡆⠀⠀⠀⣿⣟⣣⣘⡤⠟⠉⣿⣾⠏⠉⣦⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⣿⠉⠉⠁⢀⡴⠃⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠟⠁⠀⠀⣸⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣿⡀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢀⣴⠏⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⢤⣤⣳⣦⣤⡴⠒⠀⠀⠀⠀⢀⣾⣽⠟⢻⣿⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠻⢦⣀⠙⠒⠒⠚⠁⠀⠀⠀⣀⣴⢟⡿⡋⢀⡾⢹⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠙⣷⣄⣀⣀⣀⣤⡴⠞⣿⠃⡞⢰⠁⡞⠀⢸⣇⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠓⢦⡀⣰⢻⡏⠉⠉⠉⠁⠀⠀⣿⠸⠁⡟⠀⠀⠀⣸⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠈⠃⢸⣷⣄⡀⠀⠀⠀⢠⣿⡆⠀⣷⠀⠀⢰⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣀⡀⢸⠿⣎⣉⠒⢄⢀⢏⡾⠿⣤⣘⣧⣄⠈⣯⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⢿⡄⠉⠉⠒⠀⠹⡏⠳⡌⣿⣿⠀⢀⣹⣏⠉⠉⠙⠿⠂⠸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠈⠻⣦⣀⡀⠀⢀⡿⣦⣽⡙⢹⣴⠋⠁⠈⢻⡶⣤⡀⠀⢠⡟⠻⣦
⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⣼⡏⠉⠓⢶⡀⢿⠙⣧⠈⡇⠀⠀⠀⢸⡇⠀⠈⠢⠞⣧⠀⢹⣇
⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⣿⠀⢻⡀⣿⠀⠀⠀⠘⢷⡀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⣄⡀⢠⢷⡀⠸⡇⣿⠀⠀⠀⠀⠀⡿⢄⠀⠀⡼⠃⠀⢿
⠀⠀⠀⠀⠀⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠒⠛⠛⠚⠓⠚⠓⠛⠒⠒⠒⠒⠚⠛⠚⠓⠚⠛⠒⠒⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣴⣶⡆⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⣿⣿⣧⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢠⣿⣿⣿⣶⠀⠀⠀⠈⢻⣿⠿⣿⣿⡟⣱⣷⣶⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣾⡿⣿⣿⣿⡄⠀⠀⠀⠀⢁⣤⣻⠁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⢠⣶⣿⡇⠋⠀⢸⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠀⣨⣍⡛⢧⡹⣿⣿⣿⣿⣇⡇⠀⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠐⠻⢿⣿⢿⡏⢸⣿⣿⣿⣷⣄⣸⣿⣿⣿⢹⡇⠀⣸⣦⣽⣿⣿⣿⣿⡟⠿⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⢘⣯⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⡿⢻⣿⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣷⣿⣿⡿⠟⠁⠀⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣯⣥⣷⣾⣧⣀⠀⣠⢽⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣶⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢻⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠘⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣙⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣫⡽⠖⠀⠀⠘⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣰⣿⣍⠳⣜⠿⣿⠿⣟⣛⣛⡿⣿⣿⡿⣫⡵⢋⣥⣶⣿⡧⠀⠀⠀⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣷⣍⡛⢛⣋⣭⣭⣭⣙⠲⠶⠚⣫⣴⣿⡿⠋⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡀
⠀⠀⣠⣤⣴⣿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⡄
⠀⢺⣿⣿⠟⠁⠀⠀⠀⣰⣿⣿⣿⣿⣿⠏⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⢠⣾⡿⠋⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⡿⢰⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⣿⣿⣿
⠘⣿⠇⠀⠀⢀⣴⣶⣾⣿⣿⣿⣿⣿⡇⠈⠟⠛⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⡿⠟⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠛⠓⠒⠛⠛⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠛⠛⠙⠛⠉⠙⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⠛⢻⡦⣾⠛⢻⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡘⠛⠋⠀⠈⠛⠛⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠚⠶⠞⢂⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠻⣷⡤⣴⡿⢻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⢻⡀⢈⣷⣏⠀⣸⠋⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠀⠀⠀⠀⠀⠙⠛⠠⠘⠛⠁⠀⠀⠀⠀⠈⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⢸⡆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⡀⣀⣀⣸⣇⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁



⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠘⠃⠀⠤⠂⠀⢀⠀⢀⡀⠚⡀⠀⠁⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠁⠀⠠⡀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠐⠀⠂⠀⠀⠀⠒⠀⠸⡀⠄⠁⠀⠀⠀⠈⠀⠀⡀⢀⠀⠀⠀⠠⡆⠀⠀⡀⠊
⠀⠀⠀⠠⠄⠂⠀⠀⣂⠂⠀⠁⠁⢀⠤⠄⠀⠂⣀⠄⠀⠢⢄⠀⡴⡄⠀⠀⠀⠉⡉⠁
⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⠀⠨⠃⠀⠀⠂⠀⠒⠁⠀⠀⣂⠀⠀⠀⠀⠔⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣤⣴⠃⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡆⣦⣿⣟⣠⡟⢀⣴⣰⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⠟⢻⣿⣿⠿⣿⠿⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢸⠁⠇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠤⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠀⣁⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠠⢀⡀⡀
⠀⠀⠐⠘⣆⠀⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡃⠀⠀⢈⠇⠐⠀⠀⡧⢴⡷⢱⣂⠃⠀⡄⢄⡀
⠢⡄⠀⠀⠈⠀⠀⠀⠀⠜⢡⠁⠀⠐⠻⠿⠟⠀⠀⠀⠀⠀⠙⠁⠀⢠⡏⠀⠀⠀⠀⠳⡮⢻⠞⠿⡾⡟⣱⣈⠨⡦⡀
⠀⠀⠠⠀⢤⣬⣶⣦⣴⡀⠀⠐⠛⠀⠀⠀⢀⡤⠄⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⠊⠹⣸⡟⡔⡛⠉⢈⠰⣛⣧⡀
⠀⠀⠀⢀⢿⣟⣷⣟⣿⡿⠀⠀⠀⠀⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠐⠛⠀⣉⠀⢈⠉⢁⠋⠉⠘⢋⡐⠞⢥⣷⡄
⠐⠂⠉⠉⠛⣟⢿⣿⡿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣄⠀⠀⢘⢻⡞⡊⡥⣈⠉⠀⠺⠟⠓⠚⢴⣻⡇
⠀⠀⠀⠀⠀⠐⠟⠓⣧⠄⠒⠂⠦⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⡄⠀⠊⠅⡶⣼⣇⡀⠈⡐⠀⢲⢄⣼⠟
⠀⠀⠀⠀⠐⠀⠔⠠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣄⠀⠐⠃⢾⣷⣀⡿⠞⠹⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠆⠀⠀⠐⠀⠀⢠⠀⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣶⣤⣈⡀⠀⠀⠐⠊⢁⣠⠆
⠀⠀⠀⠀⠀⠐⠀⠀⣾⡖⢒⠀⠀⠀⠀⠀⠠⠀⠀⠉⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣯⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣍⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⣴⣶⣾⣿⣿⣿⣿⣿⣿⠏⢿⠀⠉⠙⢿⡆
⠀⠀⠀⠸⣿⡇⣿⣿⡟⠘⠋⠋⠅⠀⠀⠀⠉⠀⠈⡇⠀⠀⢀
⠀⡀⠀⢀⣿⣧⣸⣿⡇⠀⠀⠀⡆⠀⠀⠀⠂⠁⠀⡇⠀⠀⡄⠂⡄
⢆⠀⣠⣾⣿⣿⠏⢻⡇⠀⠀⠀⢀⠀⠀⠀⠀⡠⠀⢧
⠈⠛⠛⠛⠛⠁⠀⠀⢹⣄⢀⠀⠀⢸⣶⣶⡇⠀⠀⡸⠀⢀⠁⠇
⠀⠀⠀⠨⠐⠄⠀⠀⠘⠮⣅⡀⠀⠈⠁⠁⠀⣀⡜⠓⠆⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠄
⠀⠀⠀⠐⠂⠨⡢⠀⠀⡀⡔⠉⠑⠒⠒⠒⠋⠁⠘⡆⠁⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⠀⠀⠛⠀⠠⠖
⠀⠀⠀⠀⠀⠀⠈⠂⠄⠂⠉⢲⠁⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠶⠎⠀⠀⠁⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣀⣤⠤⠤⠶⢲⠾⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠒⠒⠲⣶⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠀⠀⢀⣀⣴⣏⣀⣾⠄⡼⠀⠀⠀⠀⢸⣿⣭⣭⣍⣉⣹⣿⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢑⢦⠞⠉⠉⠁⠀⠀⣠⢣⠎⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠱⡀⠀⠀⠀⠀⠀⠘⠛⠻⠷⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⣵
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡇⠀⠀⠀⠀⠈⠡⠀⢷⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣤⣦⣤⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠢⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢏⡜⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠂⠈⠙⡆⠈⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢠⡼⣄⣀⡂⠀⠀⠀⣇⠀⠀⠠⡔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣡⠤⠶⠚⣟⠁⠀⠀⢠⠝⠀⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠾⠵⠒⠶⣅⣀⢸⡆⡀⠀⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠁⠈⢣⡀⠀⢰⠀⠈⠉⠑⢢⡤⠂⠀⠀⠀⡤⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⢦⡤⠀⠀⢀⠀⡀⠈⠳⡀⠀⠀⢸⡆⠀⠀⢁⢀⠀⠀⠇⠀⠀⠀⡔⡠⠀⠀⠉⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣠⠀⠀⠀⠀⣀⣀⡀⠐⠄⠀⠀⠙⠀⠐⢌⡴⢂⠀⠀⠀⠀⣐⣎⠋⠀⠉⠂⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠈⠢⠀⠀⠸⡄⠀⠐⠀⠀⠈⢢⠹⢦⣾⠠⠂⠀⠀⠠⠀⠐⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠁⢀⡊⠀⠀⡄⠤⠆⠀⣤⡀⠁⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⢡⢀⠀⢠⡀⠀⠀⠀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠘⠓⠁⠠⠶⠂⠒⡻⢆⠀⠤⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠂⣷⡋⠂
⠀⠀⠀⠀⠀⠀⠰⠀⠀⡇⠐⠚⠞⠁⠀⣈⣀⠤⠤⠬⢶⡀⠀⠀⠀⠀⣤⢤⡀⠀⠀⠀⠀⠆⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⢀⣀⡤⠂⠉⠁⠀⡀⠠⠀⠐⠳⠆⣀⠀⠀⡇⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠈⢀⠄⡜⠱⠶⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢄⡄⠤⣠⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠖⠉⢶⠞⠁⠀⠀⠀⠀⠀⠀⠡⡀⠀⠙⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢨⠦⣴⡟⠠⠄
⠀⠀⠀⠀⢀⠀⠂⢚⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢀⠆⠀⠀⠁⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣁⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⡈⣤⠀⣤⣄⡂⠀⠀⠀⠀⠀⢀⢤⠔⠳⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣒⣲⣶⣿⣿⣿⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⢷⣀⣽⣶⣿⡿⣿⣿⣵⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣴⣿⣿⣿⣿⡛⠋⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣿⣿⣿⣿⣟⡭⢭⡒⠄⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣯⢿⣦⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣯⠭⠙⠛⠶⠝⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠈⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣤⡄⠀⢤⣤⣶⣦⢤⡀⠀⠁⢿⣿⣿⣿⡿⠿⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠐⠓⠓⠲⠀⠀⠀⠈⢻⣿⠋⣴⠙⢆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠰⠘⢃⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⠛⠛⠒⠆⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠛⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣤⣤⣤⣤⣤⡴⠾⠋⠀⠀⠀⠀⠀⠀⠈⢂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⡠⠊⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⣇⠔⠋⡠⠊⠀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣟⡠⢁⡴⢊⡠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠛⠋⠀⠉⠐⠋

⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⣤⣀⡀
⠀⠀⠀⢶⣄⠀⢠⠞⠁⡡⠦⠼⢄⠀⢀⠤⠤⣄⠀⠰⣿⣿⣋⣽⠀⠀⠀⠀⠀⠀⣰⣿⣴⡿⡿⢿⣍⠁⠀⠀⢀⣾⠁⢿⡄⠈⢻⢯⣷⣄
⠀⢤⡿⣷⢫⣇⠺⣄⣸⠁⠀⠀⠐⡄⡇⠀⠀⢸⡇⣿⣟⣻⣟⡿⠇⢀⠄⠀⠀⡄⢿⣿⠻⡇⠡⠅⢻⣤⠀⠀⢸⣿⡦⡈⠳⠦⢿⣐⣬⣿
⠀⢺⣾⣎⣿⣿⠁⢾⣇⢁⡤⢰⡶⠁⢀⠀⢈⣨⠇⢹⣟⣿⣻⡟⡴⢿⠀⠀⠀⡿⠸⣿⢿⡿⢷⡞⣻⡟⠀⠀⠸⣿⣿⣷⣦⡤⣾⣿⣿⡏
⠀⠀⣿⣻⣺⡇⠀⠀⢿⣿⣿⡟⠀⠀⠘⣝⡿⡏⠀⠀⣿⣿⣿⠁⠱⣦⡤⠴⣶⠆⠀⠙⠿⣿⣾⡷⠟⠁⠀⠀⠀⣿⡷⠾⢽⡭⠿⣾⣿⠃
⠀⠀⢹⣭⡿⠀⠀⠀⠸⣿⡳⠆⠀⠀⠀⣷⠃⠀⠀⠀⢸⣯⡿⠀⠀⢻⡶⣶⡏⠀⠀⠀⣴⣿⠭⣦⡄⠀⠀⠀⠀⢹⣄⠀⠠⠄⠀⢸⡇
⠀⠀⠘⣷⠇⠀⠀⠀⠀⢯⣽⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⡗⡇⠀⠀⢸⣟⣻⡇⠀⠀⠀⠉⠉⠀⠉⣠⣴⠖⠒⠀⠘⢧⣀⡀⢐⣀⣼⠏
⠀⠀⠀⠉⠀⠀⠀⠀⣀⠈⠃⠀⣿⣯⣻⠆⢰⠟⢿⡄⠀⢛⣃⣀⠀⠀⠃⠛⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡀⠀⢀⡂⠀⠈⠉⠉⠉
⠀⠀⠀⠃⢰⠀⠀⣾⣿⣟⢦⠘⣿⣮⣿⠆⢸⠀⢸⡇⢰⣿⠛⣿⡷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣭⣁⠂⣺⣿⣿⣿⣷⣶⣶⣤⣄
⠀⠀⠀⠀⠀⡆⢸⣿⣿⣿⠹⢀⣿⠿⣿⡄⢸⠀⢸⡇⣿⡇⠀⢻⣷⣷⠀⠀⠀⠀⠀⠀⠀⠰⣷⡀⢈⣓⠿⠛⠁⠀⠀⠀⠀⠀⠉⠉⠛
⠀⣤⠀⠀⠀⣹⢸⣿⣿⣿⠰⠈⣿⣿⣿⡅⢸⠀⢸⡇⣿⣇⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⡿⣦⣾⠀⠀⡀
⠀⣯⣁⣀⣿⣿⢿⣿⣿⣿⠀⠀⣿⣟⣿⠇⢸⡀⣼⡇⠘⠿⣷⣾⠿⢃⣠⣤⣤⣖⣶⢤⠀⠀⠎⣿⠟⡿⢷⣿⡛⠓⠈⣧⠀⠀⠔⠴⢳
⠀⠀⠈⠉⠉⠀⠈⠛⠛⠛⠁⠀⠉⠛⠋⠀⠘⠓⠟⠃⠀⠀⠀⡁⠀⣼⣿⣤⣼⣿⣿⣿⠃⣇⣤⣇⣀⡀⣬⣿⠇⢸⡆⢹⣶⡿⣾⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢼⢿⣿⣿⣿⡿⠁⠀⠈⢙⣿⠷⣿⣿⠃⠀⠀⣿⣮⢾⡸⣿⠃⢏⠼⡀
⠀⡴⠊⠉⠑⣆⠀⢀⡤⠤⢤⣀⠀⢀⠤⠂⠤⡀⠀⣠⠴⠶⢤⡀⣷⣬⣼⣿⣏⣛⣧⡆⠀⠀⠘⠋⠀⣠⠏⠀⠀⠀⢿⣿⣷⣿⣿⣿⣿⣾⠇
⠀⣧⡀⠀⠀⣺⡀⣼⠀⠀⠀⢸⡆⣮⠀⠀⠀⢸⢸⡁⠀⠀⣠⡇⢸⡿⠉⠉⠛⣿⢻⠁⠀⠀⢠⢾⢇⣟⡀⠀⠀⠀⠈⢹⣿⣶⣿⣴⣿⠋
⠀⠛⠭⠴⠾⠟⠁⠙⠤⠠⠤⠟⠀⠻⠤⠤⠶⠛⠀⠛⠆⠴⠟⠃⠘⠳⠤⠀⠤⠿⠛⠀⠀⠀⠈⠒⠀⠛⠁⠀⠀⠀⠀⠘⠻⠽⢬⠷⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠊⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⠐
⠀⠀⠀⠀⠂⣠⠈⠛⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡝⠀⣄
⠀⠀⠀⠀⠙⢟⢁⠀⢳⠑⢄⠘⢉⡆⠀⠀⠀⢸⣿⣿⣷⡄⠀⠀⠀⡐⠢⠀⠀⠀⣠⣾⣿⣿⡀⠀⠀⠀⢐⣈⠃⡠⢾⣿⠀⠀⡛⠋
⠀⠀⠀⠀⠀⠀⢣⣣⠸⡇⠈⣦⣻⣧⡀⠀⢠⣿⣿⣿⣿⣿⢦⡀⠀⢛⡆⠀⠀⠈⠛⠻⢿⣿⢷⡀⠀⢀⣼⣯⠜⠀⣼⣇⣀⠌
⠀⠀⠀⠀⠀⠀⠀⢻⣿⢧⠀⣿⣿⣿⣿⣴⢟⣼⣿⣿⣿⣿⣷⣝⡷⡉⠉⠁⠀⠀⠀⠀⢸⣿⣯⠻⣴⣘⣭⡆⠀⣴⠘⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢁⡘⠀⢹⣿⣿⠿⠺⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣾⣽⡞⠀⢸⣿⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢼⣟⣛⡤⢤⣌⣩⣭⣿⣭⣭⣭⣭⣍⣁⣀⣀⣀⣀⠉⠉⢉⣙⡛⠿⣿⠿⠃⢠⣾⠇⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡙⢩⣽⣿⠉⠉⠀⢾⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣾⣿⣽⠛⠒⣚⡛⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⣸⣿⣷⣄⡀⣠⣾⣿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠻⢿⠿⠿⢿⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠙⠲⠾⣿⣿⣿⣿⣿⣿⣛⣛⣛⣋⣓⠂⠀⠀⣴⣤⣤⣤⣤⣤⣬⣟⣩⣅⣉⣉⣉⣥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠻⣿⣟⣛⠛⠛⠿⠏⠉⠉⠉⠉⠉⠀⠠⠿⠿⠿⠿⢿⣿⣿⣟⣿⣿⣯⣭⡷⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠓⠒⠚⠚⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉








⣿⢿⢿⠿⢿⡻⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿

⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠊⠀⢴⣾⣿⣿⠀⠘⠈⠙⠓⠀⠀⠒⠀⠀⠀⠐⠁⠀⠀⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠋⣼⣿⡿⠀⡀
⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠘⠀⠀⠀⠄⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⠄⠀⠀⠀⠀⠀⠀⠀⠋⡟⠁⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣾
⢀⠔⠈⠁⠈⠁⣀⠀⠀⠀⠀⢠⣖⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠲⢀⣾⣏
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠤⠀⡴⠊⠉⠉⠐⢄⠀⠀⠀⠁⢄⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⣀⡀⢀⣤⣿⣿⣟⡡
⠁⠉⠁⠉⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠄⠦⣄⡠⠲⠒⠖⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣋⣀
⡀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠈⠑⣤⢄⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⡻⣿⣿⣷
⠙⠂⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠀⠴⠊⠁⠀⠀⢀⣿⡟⠃⠀⠃⠈
⠀⠀⠀⠀⠀⠁⠀⠀⠠⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠙⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣽⣷⡀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⡀
⠀⠀⠀⠈⢱⣄⡀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣄⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣹⠁
⠀⠀⠀⠀⠈⢆⠀⠉⠁⠀⢄⠵⠶⣦⠀⠀⠀⠀⠀⢀⠀⠀⢽⣿⣿⡇⠀⣾⠇⠀⢀⠄⠀⠄⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⢀⣀⢀⡀⠉⠉⢻⣿⣿⣧⢠⣿⣷⣾⣿⡶⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠄⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠙⠻⣿⣿⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠀⠀⠀⠀⠀⠳⡀⠀⣰⠁⠀⠠⡄
⠀⡄⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⡀⠐⠃⠀⠁⠈
⠀⠐⢄⢿⡄⡀⠀⠀⠀⠀⠕⠀⠄⠀⠀⠀⠀⠀⠀⠒⢂⣩⡴⠋⡉⠀⠙⢷⡀⠈⠀⠀⠐⠒⠒⠄⠠⠄⠀⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠆
⠀⠀⠀⠀⠙⠁⠀⡎⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠋⠅⡔⠁⠀⠋⠀⠁⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠈⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣤⣴⣶⣤⣬⡄⠀⠀⠀⠀⠀⠀⠀⡀
⡠⠖⠀⠀⠀⠀⠀⠀⠁⠰⣀⠀⢀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⡀⠀⠀⠀⠈⠙⠙⠓⠚⠠⢄⢀
⠀⠀⠀⠀⠀⠆⠀⠀⠂⠀⠀⠉⠉⠒⠆⠒⠘⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⣦⢠⠀⢀⣴⣿
⠀⠀⠀⠀⠇⠑⠂⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢣⡀⣄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⢺⣧⣠⣿⣿⣿⣿⠏
⠒⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢿⣿⣄⡀⠀⠸⠁⠀⠀⢀⣤⣞⡀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣟⠋⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠁⠀⢀⣀⠀⠀⠀⠈⢻⣿⣷⣀⢠⠀⠀⣤⣿⣿⠋⢀⠀⠠⠠⠀⠠⠄⠀⡸⠉⠐⠙⠧⣐⠀⠤⠤
⠲⢀⠈⠁⠀⠁⠀⠀⠠⠀⠀⠀⢀⠂⠁⠀⠀⠀⠀⠀⠀⠙⠢⠀⠀⠈⠻⣿⣿⠛⠃⢸⣿⣿⡡⠊⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠙⠒⠒
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠄⠀⢈⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣾⣷⣶⣦⣉⡀⠃⢌⣿⣟⡡⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠎⠀⠀⠀⠐⠚⠉⠉⠉⠻⠛⠛⠛⢻⡿⢿⠁⣠⡆⢫⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠒⠒⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣀⠤⠀⠒⠒⠠⠄⣠⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡡⠊⠁⡠⠂⢀⠀⠀⡴⠀⠱⡜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢳⠁⠀⠀⣠⢴⢏⡠⠊⠀⢡⠀⢡⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣠⣔⣊⡉⠃⠉⠀⠈⣁⣀⣁⣈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⢣⠉⠐⠛⠀⠀⠀⠐⠛⠀⠉⣎⠆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠑⣄⠀⠀⠠⠤⠀⢀⡠⢋⡠⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⠰⠁⣱⢤⡀⢠⡴⠛⢆⠀⡄⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢃⡴⡏⠈⢶⡞⠁⠰⡏⠑⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⢄⡴⠣⡀⢸⡇⢀⠴⠉⢂⠠⣊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠹⢄⠀⠇⠈⣈⡄⠉⠀⡧⠀⢠⠏⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢆⠉⡧⠀⠀⡇⠀⠘⡇⠙⠹⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠸⡐⠃⠀⢸⡇⠄⠰⠐⠰⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠐⠆⠐⡞⠸⠐⠠⠟⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⢰⠀⢸⡳⠁⠀⢠⠇⢀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⣀⢸⢂⠟⠀⠀⠀⠨⡠⡞⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⣐⠢⠕⠈⠀⠀⠀⠀⠀⠀⠁⠈⠑⠠⠄⠀⠤⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣴⠂⢠⣤⣤⣄⣀⠀⠀⣰⣧⣬⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣽⣿⣶⣿⣿⣿⣿⣿⣆⡀⣿⣯⣄⢠⣀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠹⠇⠹⢿⣿⣿⣿⣷⠸⡅⣩⠀⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡉⠀⣘⠃⠸⣿⣿⣿⣿⠤⣧⡙⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣦⣁⡴⠀⠻⣿⣿⣿⣶⠟⣿⠟⢀⣾⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⠻⢻⠯⣤⡟⠀⠀⣀⣈⣋⠀⠀⠀⠘⣇⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⠄⠀⠀⠀⠀⠀⠀⠀⠄⣌⢠⢠⠀⢈⠄⠀⠐⠙⠿⠈⠳⡆⠀⠀⢻⣿⣿⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⡾⠃⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⢇⠀⠀⢸⣿⣿⡆⠀⢌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣰⣤⣞⠰⣧⠚⣷⣄⣄⠀⠀⠃⠀⢀⣲⡏⢳⣿⠀⢸⢸⣿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠈⢿⡟⠁⢠⣿⣿⡿⠀⢰⠄⠀⣨⢷⡅⢈⣻⠀⢸⣾⣿⣿⡇⢠⣟⠃⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⣠⠟⠈⢻⡀⣰⠃⠀⣴⠋⣾⣯⣨⣿⠀⢘⣿⣿⣿⡇⠀⠈⢁⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠻⠶⢀⠈⠃⠋⠐⠈⣧⠚⠉⠛⢿⡿⢀⣾⠛⠉⢹⡇⠐⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠓⠧⣀⣠⡀⡇⠀⢀⠄⠈⡠⠊⠃⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡛⣡⣤⣴⡾⠁⠀⠀⢀⣴⡟⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⢠⣤⣴⠋⠀⠀⠀⠀⠠⠽⠋⠙⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⢀⠀⣿⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⣰⡏⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠟⠀⠀⠀⢠⠀⠀⠀⠀⠱⡀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⣀⢀⡀⠉⠀⣀⠴⠦⠀⠀⠀⠀⠀⠀⠕⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠠⣻⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠻⠇⠀⡟⠃⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣭⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀⠀⣗⡆⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣤⣿⣧⡀⢫⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣷⡛⢃⣺⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⡟⠉⠀⢠⡤⢠⡄⠀⠀⠀⠀⣾⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠛⠻⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠔⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⣀⡀⠤⠤⢀⡀⠈⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣠⠔⠉⢀⠄⠀⠀⠀⢁⠚⠢⡌⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡰⠁⠀⠀⠁⣠⡞⠀⡠⠋⢠⠀⠹⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠃⠀⢀⠴⠮⡝⠴⠊⠠⠤⠈⢧⠀⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣰⠪⠴⢲⣄⠀⠀⠀⠀⣴⡲⠦⠔⢆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣇⡇⠀⠈⠁⠀⠀⠀⠈⠈⠁⠀⣯⠊⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠳⣄⠀⠀⠒⠒⠂⢀⡤⠋⡠⠚⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⡎⠀⡷⣦⠄⢤⣶⢫⢣⡀⢰⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢂⠾⡇⠈⢶⡞⠁⠀⣾⠈⠊⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠤⣀⡴⢇⠀⠸⣃⠀⡰⡋⠣⠈⢔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠙⢇⡈⢸⠀⠉⢘⡁⣉⠀⢡⠀⠀⣸⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢢⠉⠚⡄⠀⢘⡇⠀⠀⡇⠰⠐⡁⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⡇⠤⠁⠀⢸⡇⠀⢰⠙⠀⠔⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⢄⡀⢀⣸⢡⢀⠀⡠⠄⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠈⣧⢠⠇⠀⡆⡦⠉⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⣇⠀⡯⠊⠀⠀⠸⠃⢠⡇⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⠠⢄⡏⠘⠀⠀⠀⠀⠈⢆⠫⣀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣀⠠⠕⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠃⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠋⠉⠉⠉⠉⠉⠛⠛⠷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⡤⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⠎⠀⠀⠈⠛⢷⣤⣤⣤⣤⣠⣤⠀⢤⢴⡒⠲⠤⣄⢀⠀⠀⠀⠀⠠⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⠀⠀⠀⡤⠠⡀⠈⠀⠀⠀⠀⠀⠀⠀⣀⡀⠁⠀⠈⠇⠑⠠⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢸⣀⣠⣤⢤⡀⠀⠀⠀⠀⠀⠀⢀⠐⣉⠉⠉⠛⠀⠀⠐⡆⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠟⠋⠀⣶⣄⠀⠀⠀⠀⠀⠀⠈⠀⡙⣃⠠⠂⠀⠀⠀⠁⠀⠀⠰⠆⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⣎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⢀⠉⠀⠀⢠⡄⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡏⠳⣤⡀⠀⠀⢀⡠⠖⠋⠁⠠⢄⠀⠀⠀⢀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣿⠙⠛⠁⠀⠀⠀⡈⠀⠀⠃⠀⠀⢘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠲⠀⢸⡄⢀⠄⠁⠀⠀⠘⠢⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⢸⠀⠀⡧⠁⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠛⠶⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠘⠄⠀⠀⠠⠂⠀⠀⠀⠉⠻⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢰⠎⠐⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠄⠀⠀⠂⠄⠀⠀⠀⠀⠠⠐⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠄⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠠⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⠻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣀⠠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠱⠀⠀⡐⠀⠀⠀⠀⠀⡀⠄⠒⠰⠀⠐⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡠⠊⠁⠀⠀⠀⠀⠁⠐⠒⠒⠠⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠊⠀⡀⠀⣦⠀⠐⠁⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢤⡠⠊⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠁⡄⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⣰⣇⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠔⠒⠒⠲⠤⢄⡤⠴⠒⠒⠒⠒⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠊⠉⠉⠉⠉⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⡆⠀⠀⠀⠀⠀⣠⢋⡜⠀⠀⢠⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠈⠀⡴⠒⢄⢀⣠⠗⠉⢀⣀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡄⠑⠒⣺⢡⠃⠀⢀⡎⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠘⢄⣀⡤⠊⠑⠤⠖⠁⠘⠦⠴⠋⠳⠔⠃⠘⠦⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢢⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠆⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠊⠀⠀⠀⠀⠀⠀⣠⠎
⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⣀⡀⢰⡟⠀⡠⡄⢠⣤⠀⠀⣠⡉⠀⠀⣠⡦⢦⠀⠈⣿⡔⠒⠆⠀⠀⠀⠀⠈⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠁⠀⣸⠁⠈⠀⠈⡏⠀⠀⢀⡷⠁⠀⠀⡸⠀⡼⠀⢀⠇⠘⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠱⡀⠀⠀⠀⠀⠀⠘⠤⠚⠀⠣⠔⠂⠀⠣⠤⠴⠊⠳⠴⠚⠐⠃⠀⠘⠒⠊⠀⠀⠉⠂⠀⠀⠀⠀⡸
⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠊⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠦⠤⠤⠤⠤⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠤⠤⠤⠔⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢲⡒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢁⡀⢀⡀⢀⡀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡲⠆⠰⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⢨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠉⠉⠛⠶⣤⣠⣀⢻⡖⠐⠂⢲⣖⣁⣠⣤⠖⠛⠉⠉⠉⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⢀⣠⣤⣶⣦⣤⣤⣄⣀⣉⣹⡇⣿⣿⢸⣏⣉⣀⣤⣤⣤⣴⣦⣄⡀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⣿⣿⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣤⣤⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡘⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⢿⢇⠛⠛⠸⠿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⢇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢹⣿⡿⢶⣾⣿⣿⣿⡏⢿⣦⡀⠶⠶⢀⣴⠟⢹⣿⣿⣿⢱⣦⣿⣿⡟⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢾⠟⠔⠜⣥⣄⢹⡇⢠⡌⠁⣴⣦⠉⢡⡄⢸⠏⣶⡌⠣⠦⢻⡿⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣀⡀⢨⡮⢸⡇⢈⣡⡦⠙⠋⣴⣌⠀⢸⡆⣶⡄⣀⣀⡿⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣐⣋⣀⡀⠘⢧⣿⣷⠿⠋⠀⠛⠓⠈⠛⠷⣼⣷⡼⠁⢀⢀⣛⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠩⠿⢟⠛⠲⠶⠤⠤⠤⠤⢤⠤⠤⠤⠤⠤⠖⠖⠛⠻⠯⠍⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠐⠟⣢⠀⡌⢵⡌⡀⠀⢰⣿⣷⠀⠀⡀⣴⡍⠄⠀⠐⡻⠀⢸
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠓⠚⠿⠤⠤⣤⣜⣈⣀⣀⣘⣻⣛⣀⣀⣈⣐⣣⣤⡤⠴⠿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⡉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠄⠀⣀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⣀⠀⡄⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠋⠉⠉⠉⠉⠉⠛⠛⠉⠉⠉⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⢈⡄⠐⠰⠄⠐⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠐⠛⠀⣀⣀⠤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠉⠁⠀⣤⠀⡤⣀⠀⡔⢀⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢈⡚⠀⢠⣤⠶⠚⠉⠁⠀⠀⠀⠑⠈⠀⠀⠀⠀⢐
⠀⠀⠀⢀⣄⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⠀⠀⠀⠁⡀
⠀⠀⠀⡈⠀⡊⠐⢠⡇⠐⠐⢦⡀⠀⠀⠀⢀⡀⠀⠀⠐⠁⠀⠀⠀⠀⠀⡄⠁⠁
⠀⠀⠈⠵⠀⣀⠀⣼⠁⠸⣻⣿⡀⠀⠀⢩⣭⣤⡀⠀⠀⠀⠄⠀⢀⠀⠄⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⡴⠛⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠄⠁
⠀⠀⠀⠀⠀⠐⠃⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⠂⠀⠀⠀⠋⢀⡀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡣⠐⠈⠀⠀⠀⠀⠀⡄⠀⠀⠠⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠳⠂⠂⠂⠀⢄⡈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠐⢶⡠⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡀⠲⢾⡙⠠⡀⠠⠐⠀⠈⡀⠐⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠷⠂⠿⡃⠜⠀⠀⠀⠠⠄⠀⠀⠟⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⢀⡂⢸⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠋⠉⠉⠉⠉⠛⠒⠶⢤⣀⡀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⠋⠉⠉⠉⠛⠓⢲⢤⣄⡀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⣀⠀⠀⠀⠀⢠⣦⠰⣿⠇⠀⣭⡓⠦⣄⡀⠀⠀⠀⠉⢳⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠥⠼⠃⠀⢸⣻⣮⠟⠛⠙⠲⢼⡟⠁⠀⡀⠉⠓⢦⡀⢀⡟⣦⣼⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡤⡄⢀⣤⢀⣸⠃⠀⠀⠀⠀⠀⠓⢤⡿⠋⠀⠀⠀⣙⡾⠀⣇⣀⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠖⠃⠀⢹⡏⠀⠽⠂⠀⠀⠠⢶⡆⠈⢣⡴⠶⠂⠈⠉⣷⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⣯⡿⠁⣠⣄⣠⣄⣀⣠⡀⢀⡀⢸⠀⠀⠀⣄⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠳⡀⠀⠀⠁⠀⠉⠀⠈⠉⠀⢈⡷⠤⠼⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡶⠤⢤⣤⣄⣠⡤⣶⡿⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⠋⠙⠲⢶⣶⠶⢾⣋⣉⠀⢸⣧⠳⣄⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⣨⣧⣄⣠⣼⣧⣀⣀⣈⣉⣠⣼⡇⢷⡀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⠀⣿⢻⠀⢠⡄⠀⣄⠀⠀⠸⠀⠀⠀⣴⢃⡜⠉⠛⠒⣶⠶⢭⡉⠁⠀⢸⡇⠀⢳⡀⢘⡆⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣧⡇⠘⡆⣸⢧⢠⢿⡀⠀⠀⠀⣠⠂⣿⢿⡁⠀⠀⢸⠁⠀⠀⠙⠒⠦⣼⡇⠀⠀⠙⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⣳⠈⢠⡄⠙⠃⠘⠋⣸⠁⠀⠀⠐⠇⠀⢁⣸⠧⠤⠤⣼⡀⠀⠀⢰⣧⡖⠃⣩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠙⠀⡼⠙⣆⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠙⠒⣾⣲⣶⣏⣙⣛⣻⣛⣲⠭⣟⡷⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡾⠁⠀⠘⡆⠀⢻⠀⠀⠀⠀⠀⠀⠀⠰⣯⣥⣤⠤⣾⠿⠭⢞⡿⠁⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣧⣀⣀⣠⣧⣤⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⢡⢾⠉⡏⢇⠙⣍⠪⣝⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠳⣴⠃⣰⠖⢾⡘⡶⠋⣳⠸⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠃⠀⠉⠒⠋⠀⠉⠛⠁