Pp go boom Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⡀⠀⠀⠀⠀⢀⡏⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢻⠀⠀⠀⠀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⢰⠷⣄⠀⠀⠀⠀⠀⡟⠀⠹⣄⠀⠀⠀⡼⠁⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠸⡇⠀⠀⠀⢀⡴⠃⣷
⠀⠀⠀⠀⠀⠀⢸⠀⠈⢷⣄⠀⠀⢰⠇⠀⠀⠘⢶⠟⢶⠃⠀⠀⠀⠈⢷⡀⢀⣶⡄⠀⣰⠋⠀⠀⠀⢿⠀⠀⣠⠞⠀⠀⢹
⠀⢿⢦⡄⠀⠀⣾⠀⠀⠀⠙⢷⣀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡞⠀⠹⠞⠁⠀⠀⠀⠀⠘⣧⡾⠁⠀⠀⠀⠘⡇⠀⠀⠀⠀⣀⡀
⠀⠘⡆⠙⢷⣤⡇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢿⠀⢀⡴⠚⣹⠃
⠀⠀⢹⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠉⠀⢠⡏
⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⣶⣾⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢀⡀
⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡆⠀⠀⠀⠘⠛⣿⠁
⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠳⡀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡠⠊⢰⠁⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠈⣧⠀⠀⠀⡄⠀⠀⠙⣄⠀⣠⣀⠀⡼⠀⠀⠀⠀⠘⣆⠀⢀⡀⠀⢠⠛⣆⠀⢠⠞⠀⠀⡸⠀⠈⠀⠀⢠⡇
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢨⠳⣄⠀⠃⠀⠀⠀⠈⠳⠁⠈⠛⠀⠀⠀⠀⠀⠀⠘⢦⠎⠹⠴⠃⠀⠸⡴⠃⠀⠀⠀⡏⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠃⠘⢶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣸⠁
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣹⣄⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢠⡏⢰⡛⢛⡓⠀⠈⢉⣑⠒
⠀⣠⠴⢒⣉⠭⠤⠤⠤⠤⠤⣉⡓⢦⡰⠀⢀⡠⠔⠒⡒⠒⠲⢤⡀⠀⣠⠴⠚⣉⣉⣉⠑⢦⣠⠤⠤⠤⠤⢤⡀⣇⡤⢤⣾⡀⠸⡇⢸⠀⠀⠀⠀⠸
⡞⠡⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠹⠶⣫⠔⠉⠉⠉⠉⠑⠦⡈⠚⢁⠔⠉⠁⠀⠀⠉⠢⡁⠰⠉⠉⠉⢃⠻⢣⠖⠒⠢⢹⠀⡇⠘⠀⠀⠀⠀⠂⢰
⢹⠠⠀⠀⠀⢠⠆⠐⢆⠀⠀⠀⠀⠘⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠐⣀⠂⠀⠀⠀⠀⠀⠀⠀⢱⡆⠀⠀⠀⠈⠖⠁⠀⠀⠀⠀⢧⣧⠀⠀⠀⠀⠀⠀⣸
⠘⡇⡄⠀⠀⠀⠢⠤⠞⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⣿⠀⠀⠀⠀⡀⠀⡏
⠀⢻⢡⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⢁⡀⠀⠀⠀⠀⠠⢢⠀⠀⠀⠀⠈⠀⠀⠀⠰⡉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⣀⠀⢀⠁⢸⠁
⠀⠸⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠘⠊⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢀⢬⡉⢰⠋
⠀⠀⣷⠆⠀⠀⠀⠠⠖⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⠀⠀⠶⠀⠀⠀⠀⠀⠁⠀⠢⡤⠃⡼
⠀⠀⢹⠀⠀⠀⠀⠀⠆⠒⠁⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⣄⡐⠤⠀⠀⠀⠠⡆⠀⠀⠀⠀⢉⣉⣉⠀⢃⠀⠀⠀⠀⠀⠈⡗⠒⠋
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢀⡀⠁⠒⠀⠐⢒⣉⠴⠊⠁⠀⠉⠓⠒⠒⢲⠀⡁⢀⣀⣀⡌⡾⠀⠈⣇⠘⡀⠀⠀⠀⠀⠀⢹
⠀⠀⠈⡇⢄⣀⣀⠀⢀⣀⡠⠤⠂⣁⠔⠋⠉⠙⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⣤⣤⣭⣭⡼⠃⠀⠀⠸⣆⣉⣈⣉⣉⣉⣁⡼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣚⡭⠟⠛⠻⠿⢿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣦⠤⣄⡀⠀⠀⠀⠀⠀⠊⠜⠁⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄
⠀⠀⠀⠀⠀⡀⢀⣴⣿⡿⠟⠛⠉⠉⠉⠉⠓⠬⡑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢝⢿⣷⡀
⠀⠀⠀⠀⡐⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢪⣿⣿⡄
⠀⠀⠀⡐⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢰⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀
⠀⠀⠀⣷⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⢰⡿⣆⠀⣷⠀⠀⣸⣇⠀⢸⡀⠀⠀⠀⠀⠀⠀⣿⣿⣣
⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⣼⡇⠀⢰⡏⢹⡄⢸⠀⢹⡄⣿⣷⠀⣿⣿⡄⠘⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⡇⣆⡀⢸⡇⠀⣷⢸⡀⠈⣇⢿⡘⣇⣸⡟⣷⠀⣧⠀⠀⠀⠀⠀⠀⣾⣿⠧
⠀⠀⠀⣿⣿⢀⠀⠀⠀⠀⠀⠀⠸⣷⡿⢿⡘⡇⠀⢻⠸⡇⠀⣿⢸⡇⠸⣿⡇⠹⡇⠀⠀⠀⠀⠀⢀⣼⣿⡟
⠀⠀⠀⢻⣿⣏⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢻⡀⢸⡄⢻⣄⣿⠀⣧⠀⠙⠇⠀⠛⠸⠄⠀⠀⣰⡿⣫⠏
⠀⠀⠀⠘⣿⣿⣄⠀⠀⠀⠀⠀⠀⢸⡀⣸⡇⠈⢷⣾⠀⠈⠿⠿⠀⠈⠀⠀⠀⠀⠀⠀⣀⣴⠿⣫⠞⠁
⠀⠀⠀⠀⠘⢿⣯⡢⣀⠀⠀⠀⠀⠈⠿⠟⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⡾⠋⠁⠨⠀⠁
⠀⠀⠀⠀⠀⠀⠙⠿⣮⣓⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠺⠥⣐⡀⠀⡀⠀⠀⠀⢠⠏⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡐⢄⠀⡏⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠙⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀
⠀⠀⢀⣀⡠⠂⠀⠱⠀⠈⠀⠉⠌⠉⢂⠀⠀⠀⠀⠀⠀⠀⠾⠇⠀⡠⡔⣎⠉⢙⡧⡀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⠐⠒⢆⠀⢠⠀⠤⠄⠢
⠀⠀⠋⠀⠀⠀⠈⢀⠁⡀⠀⣄⢀⡠⡤⠃⢀⡄⠀⠀⠀⠀⠀⠠⠂⠘⢦⣼⣧⣾⣤⡤⠡⢤⡄⡄⠄⠀⠀⠀⠀⠘⠁⠐⢦⣨⡤⢮⡴⠇⠤⠴⠆⡖⠒
⠀⠀⡄⠀⠀⠄⡞⠠⢠⠇⠀⡇⠈⠁⢠⡝⢺⡀⠀⠀⠀⢀⡖⢡⠈⣭⠋⠉⠸⠁⢨⡀⠶⢈⢸⡀⠀⠀⠀⠠⢰⠁⠒⠒⠲⣹⠁⡂⢰⠆⠀⠀⠠⢥⢧
⠀⠀⠀⠀⠀⢠⡇⠘⡀⣠⣀⣡⣆⢴⣈⠱⢮⠅⠀⢀⡐⠘⣇⢘⢃⣈⣀⣤⣠⣤⣬⣥⡶⠞⣶⠆⠀⠀⠀⢣⠹⣄⠠⠖⠛⣻⢬⠱⣤⠺⢖⠓⢳⠊⠜
⠀⠀⠀⠀⡀⠈⠋⠉⠉⠉⠉⠀⠀⢀⠨⢄⢸⡄⠀⠀⠑⠐⠋⠚⠁⠘⠉⠀⡿⠉⠛⠀⠁⠌⢡⡄⠀⠀⠀⠀⠁⢸⡃⠀⡼⠁⢸⠃⠘⠄⠀⠰⠂
⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⢂⠀⠀⣄⡠⠚⠁⠀⠀⠀⠀⠀⢀⡄⠒⢤⠀⠁⠀⠀⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠶⢦⣀⣠⠔⠳⠖⠁
⠀⠀⠀⠀⠈⠓⠄⠤⠶⠐⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⣀⠀⣀⣴⡀⠰⠤⠿⢀⡀⢠⡋⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢐⣤⠀⠀⢀⠈⠀⠀⠀⢀⡀⣀⡠⠤⢤⠄⠠⣀⠀⡈⠛⡀
⠀⢠⠁⠀⠈⢦⡌⠀⠀⠀⠀⢹⡋⠑⠄⢴⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡝⠀⠀⢹⣟⣿⡋⢹⠁⠀⠀⠀⢖⠃⠐⠻⠀⠀⠀⠀⠀⠀⢐⠀⢊⠀⢀⡀⣀
⠀⠀⢱⠔⠐⠓⢷⣤⠀⣀⣠⣾⣀⣀⣀⡾⢤⢴⠀⠀⠀⠀⠀⠸⠀⠀⠰⣏⠀⠀⠀⠀⠈⠀⡘⠀⠀⠀⠀⢸⠐⠠⠄⡠⠀⢠⠔⠐⡖⡉⠻⠁⠋⢸⠇⡎
⠀⠀⢈⣳⢠⡤⡎⢀⠛⠳⢋⠙⢡⠌⠀⠁⢿⡄⠀⠀⠀⠀⠰⡅⠀⠀⠀⠸⠿⠿⡿⠶⡶⠏⠁⠀⠀⠀⠀⠀⡦⠀⢰⠃⡀⠂⡾⢀⠰⢏⠀⡀⣀⡘⣺
⠀⠀⡄⠀⡞⠀⣉⡉⠐⣄⠋⠤⠩⠆⣲⡖⠒⠁⠀⠀⠀⠀⡖⣁⢰⠀⢈⠀⠂⠀⡐⢲⠃⠂⠀⠀⠀⠀⠀⠰⡀⠀⣼⣤⠴⠦⠔⠛⠒⠛⠛⢻⠏⡉⠁
⠀⠀⠤⠶⢶⡃⢁⣘⣁⣤⠒⠁⠐⡏⠉⠀⠀⠀⠀⠀⠀⢀⠇⠆⠃⡀⠈⣀⢠⡀⢃⣜⡃⠀⠀⠀⠀⠀⠀⠀⢙⡐⡄⠀⠀⠀⠀⠀⣀⢠⡔⠫⠐⢤⡘⠓
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⢆⣀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠘⠒⠘⠋⠉⠉⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠑⠒⠙⠢⢿⢈⠓

⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠄⣀
⠀⠀⠀⠀⠀⣀⠔⠓⠂⠀⠀⠀⠀⢱⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠉⠐⠂⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠲⣄⠀⣀⣀⣦⡐⠀⠀⠠⠃⣀⡔⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢤⠀⡄⣀⠐⠠⡄
⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡵⠀⠀⠀⠀⠀⠀⢌⣹⠉⠀⣠⠟⠉⠒⢲⠎⠀⠁⠑⡄⠀⠀⠀⠀⠰⠀⠀⠐⠆⠀⠀⠉⠀⠁⠀⢃⠀⡀
⠀⠀⠀⠀⠀⠉⠓⠉⠓⣶⣶⡶⠶⠞⠋⠉⠀⠀⠀⠀⠀⢀⣀⠀⢈⠆⢚⠁⠀⠀⣀⠈⡦⠄⠀⡜⡯⠠⠀⠀⠀⠀⠀⠀⠁⢀⠄⠀⠄⠠⢄⡰⠊⠃⡽⠈
⠀⠀⠀⠀⠀⠀⡔⠈⠁⠁⠘⠃⠀⠐⣆⣀⣠⢤⡔⡆⠀⠀⠀⠀⠸⡆⢠⡏⠘⠀⠇⠀⣄⠸⢂⠀⣇⠀⠀⠀⠀⠀⠓⡀⠀⠸⠠⢀⠼⣀⢘⣁⣠⣤⣍⠁
⠀⠀⠀⢀⣀⣠⢼⠓⡴⠂⠲⢊⠙⢣⠌⠁⠀⢸⡰⠀⠀⠀⣁⠤⠚⣛⣠⣠⠤⠤⠴⠖⢦⠶⠺⡓⠂⠁⠀⠀⠀⠀⢀⡄⠀⠚⠛⠉⠉⠈⠉⠁⠘
⠀⠀⣸⠉⣼⠁⢘⣃⡁⠀⡀⢋⣄⣉⡦⠤⠶⣖⠂⠀⠀⠀⠈⡆⠀⠀⠹⢧⡀⣀⡴⢶⠋⠀⠀⡵⠀⠀⠀⠀⠀⠈⠁⠀⠦⠀⢀⠀⠀⠀⡀⠶⠁⢀
⠀⠰⣇⣤⡬⠶⠊⠉⠃⠉⠉⠁⢀⠀⠀⠀⠀⠀⢑⡄⠀⠀⠀⠈⠼⠋⢀⠆⠉⠉⡆⠸⠁⠚⠂⠙⠦⠀⠀⠀⠀⠀⠀⠀⣠⠁⠈⠀⠒⠀⠀⠀⡠⠁
⠀⠀⠈⠓⠄⠒⠒⠐⠒⠒⠈⠢⠐⠚⠂⠒⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⣸⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⢀⡴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⡴⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣤⡀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⢸⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⢰⢄⠀⡰⠋⠱⣄⢠⠞⠁⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣤⡄
⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⡀⠀⠙⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠴⡀⠀⠀⠀⠀⠀⠀⠛⠁⣧
⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠜⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣼⢁⣀⣠⡤⢤⣄⣀⠀⢀⡏⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠞⠁⠀⠀⠘⡄⠀⠀⠀⠀⣀⣀⣀⣿
⢀⡴⠋⠉⠀⠀⠀⠀⠀⠈⠱⡜⠀⢀⠔⠋⠁⠀⠀⠉⠙⢦⠀⣠⠖⠋⠉⠉⠓⢆⡴⠛⠓⠲⢤⢀⣇⣀⣀⠀⣰⢛⠀⠈⠉⠙⠒
⢸⡇⠀⠀⠀⣆⠆⠀⠀⠀⢀⡇⢠⠏⠀⠀⠀⠀⠀⠀⠀⠈⢷⠃⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠸⠞⠁⠀⠈⢿⠁⡈⠀⠀⠀⠀⣰
⡏⢰⠀⠀⠀⠀⠀⠀⢀⣠⣾⣀⢸⠀⠀⠀⢠⣒⡆⠀⠀⠀⣼⠀⠀⣤⢄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⠀⠀⢠⡇
⣇⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⠀⠀⠁⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⡇⠀⠀⠀⡼
⠸⡄⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⢀⡠⣺⠙⠤⣀⣀⣀⣀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⠀⠀⢠⠃
⠀⢧⠘⡀⠀⠀⠀⢏⡱⠀⠀⠀⠀⠀⣿⣍⠒⠂⢉⣁⡴⠻⢤⣀⣀⣀⣴⠿⠀⠀⠀⠀⣄⠀⠀⢀⠀⠀⠀⠀⢨⠷⠤⣤⠟
⠀⠸⡄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢰⢃⠃⠀⠀⠀⡜⠈⢦⠔⡹⠇⠀⠀⠀⠰⠖⠈⠻⡀
⠀⠀⣇⠰⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢼⣀⠀⢀⡜⢆⢀⢂⡜⠁⡀⠀⠀⠀⢰⡄⠀⣰⠃
⠀⠀⢸⡀⢇⣀⣀⣀⣀⠤⠔⠊⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣄⡈⠉⡿⠀⠈⠉⠁⢹⠀⠣⠤⠤⠤⣬⣡⠟⠁
⠀⠀⠈⢧⣀⠀⠀⢀⣀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠘⠲⠶⠤⠤⠴⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡙⣟⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⡽⢃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠈⢷⠀⠉⠒⠲⠤⠴⠶⠞⠳⠶⠶⠤⠒⠊⠁⠀⠾⠁⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⢤⠤⠤⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠈⠒⠒⠲⠶⢤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡙⠲⢤⣄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⢩⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⠀⠀⠈⠢⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠒⢤⡀⠈⢢⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡿⠀⠀⠀⠀⠀⠈⠲⠀⠱⠄⠈⠃⠀⠀⠀⠀⠀⠀⠀⠃⠀⠚⠀⠀⠀⣀⡀⠀⢠⡄⠸⡟⢦⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣄⡀⠀⠲⣶⣄⠀⠀⣸⣿⠀⠀⠘⡇⠀⢁⣀⣈⣉⣲⣤⡀
⠀⠀⠀⠀⠀⠀⠈⠻⣉⠉⠙⠛⠋⣀⣀⣠⣤⣄⡀⠀⣠⡖⠉⠙⢶⣄⢰⣿⠀⠀⠈⣿⣆⠀⡟⢿⣧⣠⠛⣿⡀⠀⠀⠷⠀⠀⠉⠉⠀⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠈⣿⡆⢀⣼⡟⠰⣿⡀⠀⠀⢸⣿⠺⣿⡀⠀⠀⣿⠏⠀⢳⠀⠻⡟⢀⡿⠧⠀⠀⠚⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢿⣏⠉⠉⣿⡇⠻⢧⣀⣀⡼⠋⠀⠉⠓⠒⠚⠉⠀⠈⠉⠉⠀⠀⠀⠀⠀⠐⠶⢬⣛⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠠⠼⠿⠖⠚⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⢀⡀⠀⠀⠀⣠⠖⠀⡰⠃⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀⠙⢦⡀
⠀⠀⠀⠀⠀⢀⡴⣟⡥⠶⠛⠋⠁⠀⠀⠀⠁⠀⠘⠁⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⢝⣶⣄
⠀⠀⠀⠀⢰⣿⡯⠴⠖⠒⠒⠲⣄⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⢀⡖⠋⠉⠉⠉⠛⠛⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣴⠋⠠⠐⠒⠲⢦⡄⠀⠀⠀⠀⣀⡤⠤⢤⣀⠀⠀⠀⠀⢠⠶⠚⣋⣐⡀⠙⢦⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡜⣡⠴⠛⠉⠙⠳⣄⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⠳⣄⠀⠀⠀⣰⠛⠉⠉⠙⠳⣤⡱⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠞⠁⠀⠀⠀⠀⠀⠙⣆⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡸⠃⠀⠀⠀⠀⠀⠈⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⢸⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣸⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⢄⠀⠀⣹⣶⡖⠒⠀⢀⣤⣴⣾⡿⠛⠉⠉⠉⠛⠿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⠦⣠⣾⣿⣿⣿⡿⠿⣿⣯⡤⠾⣁⠀⠀⠀⠀⠙⠉⠛⢿⣷⣀⡀⠒⡤⣂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣈⣳⣌⢻⡿⠋⢀⡖⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⣤⣿⣿⡿⢋⣞⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⡟⠀⠀⠘⠘⠀⠀⠀⠀⣀⣠⣔⣀⠀⠀⠀⠐⠲⢿⡟⠿⢷⣿⣿⡆⠀⠀⠀⠀⣀⣠⣤⠴⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣿⣇⡆⠀⠀⠀⠀⢀⡴⠋⠉⠀⠀⠈⠉⠒⠀⠀⠀⠀⠈⠀⠀⠉⢻⣷⣴⣶⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣰⠟⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⢄⢸⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢀⣿⣿⡿⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⣿⡟⠀⠊⠙⢾⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠿⣇⠀⠀⠀⠈⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⣦⠀⠀⠀⠀⠀⠚⠳⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⡆⢀⡾⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣿⠢⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣾⣿⣧⡒⠱⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣋⣴⣿⣧⡀⠀⠀⠀⢄⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠤⠞⣿⡿⠃⠀⠉⠛⠷⣤⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠵⢺⠙⢿⣿⣿⣿⣶⣼⣦⣀⡙⠲⠶⠷⢤⣀⣀⣀⣤⣞⣁⡀⠀⠀⢀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠈⠳⣤⣉⡛⠛⢿⣯⣉⣳⣦⣄⣀⣈⣩⣴⣿⣷⣦⣤⣤⣤⣤⣶⣟⢻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠙⠛⠋⠉⠉⡙⠀⡈⠙⠻⠿⠿⠿⠟⠛⠉⠉⠣⡳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠈⠓⠶⢤⣤⠤⠖⠋⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃



⣶⣶⣶⣶⣶⡾⢶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⡿⠿⠿⠿⠿⠷⣶⡷⠶⢶⠾⠶⢶

⠀⠀⠀⠀⠀⠀⡠⠐⠒⠒⠶⣏⠢⠾⠇⠀⠀⠀⢀⣠⣤⡀⠀⢠⡶⠶⣄⠐⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⣶⣒⡄
⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⣸⣇⠀⠠⣴⣶⣿⣿⠙⡟⠁⣠⣿⣿⣤⡈⠛⠶⠃⠀⢀⠀⠀⠀⠀⢀⠺⣿⠉⠁
⠀⠀⠀⠀⠀⣿⣿⣷⣶⣶⣾⣿⡿⢰⣶⣿⣿⣿⣿⡇⠀⣼⣿⣿⣿⣿⣷⠀⠐⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⠛⠿⢿⠿⠿⠛⠁⠈⠿⣿⣿⣿⡿⠁⠀⠘⢿⣿⣿⣿⠏⣀⠀⠀⠀⠀⠀⠀⠹⣿⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠉⣁⠠⠠⠀⠀⠀⠈⠉⠀⠘⡟⠃⠀⢠⣤⠀⠀⠀⠠⣍⡡⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢰⠃⠀⠀⣤⣾⣿⣆⢀⡀⠀⠀⢀⣶⣚⡖⠂⠀⠀⠀⠁⠀⠀⠀⣶⣿⣖⣠⣹⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠠⠀⠘⡁⢸⣼⣼⣿⣿⣏⢸⠀⢀⣾⣿⡿⠀⠀⣠⣶⣶⣀⠀⠀⢸⣿⣿⣻⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠴⠀⡇⠀⣤⣿⢿⣿⡋⠈⠉⠀⣼⣿⡿⠁⠀⣾⠿⠛⠻⠿⣧⠀⠀⠻⠿⠩⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⢀⣤⠈⠉⠁⠀⠀⠸⣿⡿⠁⠀⠀⢿⣻⣭⣥⣟⠏⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡠⠚⢉⣤⣤⣤⠀⣤⣤⡠⡀⠀⠀⠀⠀⠉⠛⡛⠉⠀⠀⠀⠀⢠⠡⡝⢃⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⣿⣷⣏⠇⠀⢰⣿⡶⠛⠛⢿⣿⡇⢻⣦⡄⠀⣀⢼⠶⠟⠺⣦⠄⠀⠐⠻⣿⡿⠋
⠀⠀⠀⠀⠀⣠⣶⣿⣷⣄⠻⣿⠿⠀⠀⢺⣿⣧⡀⢀⣼⣿⡇⠺⢣⡿⠀⣥⣤⣀⣀⣀⣤⠌⠁
⠀⠀⠀⠀⠀⢱⢟⣿⣿⡿⢗⣁⢀⣀⠀⣀⠉⠀⠉⠻⢯⣭⣴⣆⠟⠀⠀⠀⠹⠛⠛⠟⠙⠂⢀⢀⢀⢠⢄⡀
⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⠿⠿⢿⣧⡉⠉⣀⣠⣄⡀⠀⠀⠀⣶⡦⠰⣾⢶⣶⣔⣤⠀⢀⢤⣺⣺⣿⣭⠪⡢⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠭⠽⠿⠯⣽⡇⣼⣿⣿⣿⣿⣾⣷⠂⠀⠀⠀⢫⠛⣭⠛⣯⡆⠠⠊⠹⣿⣿⣿⠛⣣
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣶⣶⣶⣶⡾⠃⠻⣿⣿⣿⣿⠇⠀⠀⠀⠀⣀⣤⣀⣉⣴⣿⣁⠐⠪⠖⠆⡍⣄⠮⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⢀⣶⡀⢻⠁⢰⣿⢀⣿⠇⢾⣇⠀⢾⣇⢦⡴⣾⢓⠀⣾⢳⠐⣖⢰⡟⠀⢰⡿⢶⠛⡦⡿⠀⠰⣿⠇⣧⣴⡞⣃
⠀⠀⠀⠈⠉⠁⠋⠀⠈⠙⠀⠁⠀⠘⠉⠀⠘⠉⠈⠁⠈⠋⠀⠙⠉⠈⠉⠈⠀⠀⠈⠀⠈⠛⠁⠁⠀⠀⠁⠀⠁⠉⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣀
⠀⠀⠀⡠⠔⠶⠒⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠉⠀⠀⠀⢙
⠀⠀⣔⠃⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⣀⢄⠌
⠀⠀⠈⠳⠤⠦⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠳⠀⠀⠀⠀⢰⣄⠀⠀⠀⣸⠃⠙⢦⡀⠀⢀⣀⡤⠚⠁⠀⡾⠀⠀⠀⠀⡤⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⠈⠓⠶⠞⠁⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⢸⡇⠀⢀⠀⠀⠃⠤⠜
⠀⠀⠀⠒⢶⣤⣤⣀⣒⣢⡤⢖⣊⡵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠁
⠀⠀⠀⠀⠀⠉⠢⡀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⣶⡴⠦
⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂
⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⡀
⠀⠀⠀⠀⠀⠀⠉⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⣶⠶⡦⠿⠦
⠀⠀⠀⢄⢀⡤⠦⠀⣞⣠⠴⠒⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣻⠗⠛⠉⠉⠁
⠀⠀⢇⡠⠇⠀⠀⠸⠋⠀⣠⠎⠁⠸⡇⠀⣀⡴⠞⠋⠙⠓⢦⡀⠀⣼⠙⠃⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣧⠞⠉⠀⠀⠀⠀⠀⠀⠙⢶⡇⠀⠀⠈⢢⣀
⠀⠀⠀⢠⠴⠉⠒⠲⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠑⠇⠀⣀⣀⣀
⠀⠀⠠⡋⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠋⠁⠀⠸⢄
⠀⠀⠀⠈⠒⠤⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠀⡀⠀⢠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣄⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣦⣄⡀⠀⠀⢠⣿⣿⣿⣧⡀⠀⣠⣶⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣶⣶⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⡇
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠟⠛⠋⠉⠙⣿⣿⡿⠟⠛⢿⡿⠋⠉⠉⡿⠋⠉⠀⠈⢻⡆⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠈⢿⠀⠀⢸⠀⠀⢠⣶⣦⣾⣧⠀⠀⢸
⠀⠀⢀⡄⠀⠀⠀⠀⠈⠹⡇⠀⠀⢠⡀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡏⠁⠀⣿⣄⣀⣼⠂
⠀⠀⠀⣇⠀⠀⠸⠆⠀⢀⠁⠀⠀⠀⡇⠀⠀⠀⢹⠀⠀⠀⢀⠀⠀⠀⢆⠀⠀⢀⠀⠀⢸⡏⠁⠈⣦⡀
⠀⠀⠚⢿⠀⠀⠀⣀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⣆⠀⠀⠈⣶⣶⣾⣷⣿⣿⣿⣶⠾⠛⠉
⠀⠀⠀⢸⡄⠀⠀⠛⠀⠘⠀⠀⠀⠀⣠⣄⠀⠀⠀⠃⢤⣴⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠈⣧⣤⣤⣶⣶⣧⣀⣀⣀⣤⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠛⠁⠀⠙⢿⣿⣿⡟⠀⠀⠀⠉⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠙⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠈⠀⢄⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⣄⡀⠀⠀⠀⢰⣄⣠⡾⠁⢹⡄⠀⠀⠀⡠⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣿⣤⠾⠟⠛⠛⠛⠷⢬⡁⠀⠀⠀⢙⣦⡴⠋⣇⣀⣀⡤⢶⡟
⠀⠀⠀⠀⢤⣀⡀⠀⠀⠀⣠⣤⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⠦⠖⠛⠋⠉⠙⠛⠻⣍⣀⠀⣾
⠀⠀⠀⠀⠀⠙⣯⠉⣹⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠿⣶⣶⣋⡀
⠀⠀⠀⠀⠀⠀⢸⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣷⠒⠛⣿⣼⣄⣀⣀
⠀⠀⠀⠀⣠⠖⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠶⠿⢯⣹⠟⠛⢻⡎⣿⡄⠀⣿⣿⣧⠞⠉
⠀⠀⣀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⢴⣖⢤⣠⡶⠿⠛⠿⣕⣺⣿⠀⠀⠈⠋⠀⠀⢸⣯⣷⢻⣀⣿⡿⣇⡀
⠙⠻⢽⡇⠀⢀⣠⣤⣤⣤⣤⢤⣼⠏⠀⠀⠀⠈⢻⡿⠀⠀⠀⠀⠈⣷⣹⡇⠀⠀⠀⠀⠀⠸⣿⣿⣿⣁⣈⣷⣹⠅⠠
⠀⠀⠀⢳⣠⣿⠉⠁⠀⠀⠈⢻⣹⡀⠀⢰⣦⠀⢸⣷⠀⠀⣿⡆⠀⢸⣿⣇⠀⣷⣄⣼⣇⣠⣿⣿⠑⠭⠯⢵⠻⢦⣀
⠀⠀⠠⠤⡿⢞⡇⠀⠀⠶⠀⣸⣏⢧⡀⠈⠛⠀⣸⣟⢦⡀⠈⠁⣠⡿⣹⢿⣻⣿⠭⠿⠽⠿⠗⠋⠀⠀⢀⣾⣠⡤⠼
⠀⠀⠀⣠⣷⠸⣿⡀⠀⠰⠆⢸⣿⣷⢝⡲⠶⢿⣫⠟⠺⢽⣛⣛⣯⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠋⠁
⣀⣤⡾⠥⢽⣧⡝⣧⣤⣶⡶⣟⣷⠋⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡰⡀⠀⠀⠀⠀⠀⣈⡿⠛⠉⣿⠉⠉
⠁⠀⠀⠀⠀⠀⢙⡶⢶⡿⠛⠋⠈⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠠⠴⢚⡵⠃⠀⠀⠒⡲⠛⠁⠀⠀⠀⠹⣆
⠀⠠⡴⠊⠀⠐⠚⠉⢹⡇⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠳⣄⡀⠑⠛⠉⠀⠀⠀⣠⡼⢡⣶⠶⠟⠛⠒⠶⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣃⣠⡤⠶⣶⠀⣀⣀⠀⠀⠀⠀⠈⠙⢷⣶⡶⠶⠶⠛⠁⣾⠛⢧⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠰⠛⠉⠀⠀⠀⡿⠋⠁⠙⢷⡀⠀⣠⠾⣦⠏⠉⢛⠛⢷⣄⢸⡇⠀⠀⠀⠀⢠⣼⡢⢤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠞⠁⠀⠘⠀⠀⠘⡀⠀⠈⠻⡇⠀⠀⠀⠀⠀⠇⡩⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊
⠀⡁⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡆⠀⠠⠄⠀⠀⠠⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢰⣿⣿⣿⣿⣿⣿⡿⠟⠰⠆⠀⠀⠰⠆⠀⠀⠰⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⡟⠉⠈⠐⠆⠀⠀⠰⠆⠀⠀⠰⠂⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠷⠀⠱⠐⠿⠀⠀⠸⢷⣴⣾⣿⣿⣷⣦⠺⠂⠀⠀⠲⠀⠀⠀⠒⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠿⠀⠀⠸⠿⠁⣀⠸⠿⣴⣿⡿⢿⣿⣿⣿⣿⣆⠀⠿⠃⠀⠀⠻⠀⠀⠀⣾⣿⣿⣿⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⠂⠀⠀⠿⠂⠀⠘⣿⣷⣿⣿⣿⣷⣾⣿⣿⠁⢸⣿⣿⣿⣿⡿⠗⣀⡀⠿⠃⠀⠀⠟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠛⠂⣀⠀⠟⣶⡿⠋⠀⠀⢻⣿⣿⣿⡏⠀⣼⣿⣿⣿⣿⣇⡀⣿⣇⣀⣀⠛⠀⣀⠀⡿⠉⠛⢻⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢀⣀⣼⣅⡀⠀⠙⠃⣀⠀⢻⣃⣀⣿⣷⣄⣀⣀⣾⣿⣿⣿⠃⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠛⠀⣁⠈⠛⠀⠉⠛⠛⠋
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠃⢀⠀⠛⣷⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⡿⠛⠛⢿⣿⣿⣿⣿⣄⠈⠛⠀⡀⠀⠃
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡀⠙⠃⣀⣿⣿⣿⡀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⡌⠋⠀⠀⠈⠁
⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⡛⠁⣀⡀⠛⢛⣿⣿⣷⡀⠀⠀⠀⠻⠋⠉⠙⢿⣿⣿⣿⣿⣿⣿⣷⣤⣀⡀⠀⠉⠉⠻⣿⣿⣿⣿⣿⣀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢗⡒⠛⠃⣠⣼⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⢀⣽⣿⣿⣿⡟⠉⠀⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠈⠉⢉⡁⠈⠁⣠⡀⢻⣿⣿⡟⠉⠀⠀⠀⠀⠀⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣶⣿⣿⣿⣿⣿⣿⣿⣦⠀⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⢀⡀⠉⠁⣼⣿⣿⣷⣄⡀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⣠⡀⠙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷⣦⣅⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⣤⡄⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠈⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢠⡀⠉⠁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⣀⣿⣿⣿⣿⣿⣿⠁⠀⠀⢀⣠⣤⣶⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠿⠿⠿⠿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⠿⠿⠿⢿⣿⣿⡿⠿⠿⣿⣿⠿⠿⠿⣿⡿⠿⣿⡿⠿⢿⣿⠿⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣴⡆⠀⣸⡟⠁⢀⣤⡄⠀⢹⡟⠁⢀⣤⡄⠀⢹⣿⠀⠀⠀⣿⡏⠀⠀⢀⣿⠃⢀⣿⡇⠀⣾⡏⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣤⡀⠘⢿⠅⠀⣾⣿⠇⠀⣸⡃⠀⢾⣿⠇⠀⢸⡏⠀⣤⠀⠛⢀⡎⠀⣸⣿⣀⣸⣿⣀⣰⣿⣇⣀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠈⠉⣀⣠⡿⣧⣀⠈⠁⣀⣴⡿⢧⣀⠈⠉⣀⣴⣿⠃⢀⣿⠀⢀⣾⠁⢀⣿⡋⠉⣻⣏⠉⣹⣟⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠉⠉⠀⠀⠉⠛⠟⠋⠁⠀⠄⠉⠛⠻⠏⠉⠛⠻⠟⠛⠛⠻⠛⠛⠛⠛⠙⠛⠋⠉⠛⠋⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠤⠀⠀⠀⠆⠀⠀⠠⠆⠀⠀⠰⠆⠀⠀⠰⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⢠⠖⠒⠆⠀⠀⠂⠀⠀⠀⠐⠀⠂⠐⠀⠂⠀⠂⣰⡄⠂⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡶⠛⠋⠳⣄⣨⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠜⠉⠁⠀⠰⠒⠀⢤⣀⠀⡁⢈⠀⡁⢈⠀⡁⢈⠀⣉⢈⣡⣿⣿⣇⣈⠀⡁⢈⠀⡁⢈⠀⡇⢈⠀⡃⠀⠀⠀⠛⠁⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠀⠀⠀⠀⡠⠀⠄⠠⠀⠄⠠⠀⠄⠠⠄⠤⣠⣿⠏⠘⢿⣷⣄⠄⠠⠀⠄⠠⠀⣤⡴⠏⠀⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⣠⠀⢀⣠⣤⣶⣤⣆⣐⠀⠂⠐⠂⡒⣐⣶⣾⡿⠋⠀⠀⠀⠙⢿⣷⣶⠀⠂⠐⠐⡏⠀⠀⠀⠀⠀⠀⡄⠀⠀⣱⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡉⢉⢹⣿⡉⠙⠛⠿⠿⠿⠿⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣾⣄⡽⠃⢀⡄⠀⠀⠀⠁⢀⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⠀⠀⠀⠀⠀⢉⠟⢀⡴⠋⣠⢤⣀⣠⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⡟⣆⠀⠀⠀⠀⣠⠊⠀⣉⣀⣠⡀⠸⠁⣤⣤⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡃⢸⡊⠡⠀⠂⢣⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣈⣧⣀⣄⠐⠗⡁⢸⣿⣿⣿⣷⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠡⠿⠇⠀⠆⢻⡔⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣄⠈⠀⢸⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣤⡀⠀⠘⣦⣴⣿⣿⣿⣿⡒⠦⣄⠀⠀⡰⢻⣿⣿⣿⣿⣿⣿⣿⣷⡄⢸⣿⣿⡿⣿⣿⣿⣿⢿⣿⣿⣷
⠀⠀⠀⠀⠲⣶⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⣾⣿⣿⣿⣿⣿⣿⣿⣷⡈⢷⢠⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⡇⠹⣿⣿⡿⠈⣿⣿⣿⡄
⠀⠀⠀⠀⢀⠀⠀⢿⣿⣿⡇⠀⣸⣿⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⡇⣧⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⡇⠀⠹⣿⠃⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⠇⠘⢧⣘⢿⣿⣿⣿⣿⣿⠟⠀⢸⣿⣿⡇⠀⠀⠉⠀⡇⠸⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡍⠀⢈⣿⣿⣷⠙⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠉⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠿⠂⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣶⣿⣿⡿⠃⡴⠀⠙⠛⠻⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢟⠉⡁⢈⠁⡁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠟⠛⠉⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⡅⢨⠀⠄⠠⠀⠄
⠀⠀⠀⠀⠀⠀⡠⠞⠀⡀⠀⠀⠘⠀⢠⠎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⣀⠀⠀⠀⢠⣿⡿⠀⠆⠐⠀⠂
⠀⠀⠀⠀⠘⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⡿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⣠⣾⠿⡛⢛⠛⡟⢻⣿⣷⣤⣘⣿⣷⠀⢆⢀⠀⠂
⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⢀⣨⡏⢉⠀⡁⢨⠈⡙⢿⣷⣤⡀⠀⠀⣼⣿⣯⢈⡅⣨⠈⡅⣩⠀⡍⢹⠏⠉⠀⠀⠘⢦⠀⡁
⠀⠀⠀⠀⠀⠳⣄⣀⣰⡆⠀⠀⢀⡄⠄⠸⠀⠄⠠⠀⠄⠰⠈⠿⣿⣆⣸⣿⠇⠤⠠⠄⠤⠀⠆⠤⠀⠄⢦⠀⠀⠀⠰⡀⣼⠀⠄
⠀⠀⠀⠀⠀⠀⠠⠭⠀⠙⠶⠶⠚⠁⠂⠐⠀⠂⠐⠀⠂⠐⠀⠒⠘⢿⣿⡟⠂⠒⠐⠂⠒⠀⠂⠒⠀⠂⠘⠄⡀⠀⡄⠀⠛⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⢈⠀⡁⢈⠀⡁⢈⠀⡁⢈⠀⡹⢁⡁⣉⢈⡁⢈⠀⡁⢈⠀⠁⠊⠀⡁⢈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠄⠠⠀⠄⠠⠄⠤⠀⠄⠠⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠐

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡅⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠢⣄⣀⣀⡀⣸⢆⣤⠞⠀⢳⡀⠀⠀⣠⠄⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣼⣷⠞⠛⠉⠉⠙⠛⠶⣄⠀⠀⣀⣽⣶⣎⣹⣤⡤⠴⠚⡟
⠀⠀⠀⠀⠢⡤⣀⣀⣀⣠⡶⠞⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠉⠀⠀⠉⣉⣳⣤⣄⣸⡇⠀⢀
⠀⠀⠀⠀⠀⠘⣦⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢿⣿⣃⢤⡀
⠀⠀⠀⠀⠀⣀⣸⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣤⣤⡰⣿⡍⠉⢸⣿⡗⢲⡶
⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⣀⣤⡤⢀⡸⣟⠉⠉⢷⡟⠁⠀⣿⢾⢷⡀⢸⣿⡷⠋
⢤⣤⢴⡏⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠛⠳⣽⣶⠟⠉⠉⠙⢳⡝⣿⠀⠀⠀⠀⠀⠀⣿⢸⣏⡷⠾⣯⢳⠦
⠀⠈⠙⡇⠠⣴⠶⠟⠛⠓⢯⣆⡇⠀⢀⣀⠀⠈⣿⠀⠀⣤⡀⠀⣿⣸⡀⠀⣄⠀⣤⠀⢻⣿⣯⡷⣶⢿⣽⡀⠁
⠀⠀⠀⢙⣾⢿⠀⠀⢀⡀⠀⣧⣧⠀⠘⢿⠆⢀⣻⡄⠀⠻⠃⢀⣿⣿⣷⣶⡿⠾⢟⣿⣿⣿⠃⠉⠉⢹⡇⠙⣲
⠀⠀⠀⢹⡇⠻⡇⠀⠀⣅⠘⣿⣮⡳⣤⣤⣤⡾⣳⣝⠶⣤⣴⢾⣻⠏⠙⠋⠉⠉⠉⠉⠁⠀⠀⠀⣠⡿⠟⠋⠉
⠀⢀⡴⡚⣿⣜⢿⣀⣀⣠⣴⣿⡿⠉⠒⠛⠓⠛⠁⠀⠉⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⠶⢿⡟⠢⠄
⠚⠉⠉⠉⠉⠛⢯⣽⣭⣭⠷⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⣳⠀⠀⢀⡤⡤⠖⠋⠀⠀⠸⡇
⠀⢀⣠⠔⠀⠠⠞⠒⢾⠀⠀⠀⠀⠉⠛⠶⠶⠤⠴⢦⡀⠀⠠⠶⠚⠁⠀⠀⢀⡼⣁⣤⣴⠶⢦⢤⣽⣦
⠀⠈⠁⠀⠀⠀⠀⠀⡾⢀⣀⣤⣤⠀⠀⡀⠀⠀⠀⠀⠙⠳⢦⣤⣤⣤⡴⠞⢋⡶⢿⡁⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⡼⠛⠉⠁⠀⢸⠖⠋⠙⢧⡀⠀⢀⣴⣤⠞⠛⠿⠶⣤⡀⢸⠃⠀⠀⠀⠀⣠⡦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡴⠋⠀⠙⠀⠀⠀⠀⠈⠙⢿⠀⠀⠀⠀⠀⠹⢙⠄⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠉⠉⠉⢉⡝⠋⠛⠒⣆⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⠛⠛⠛⠛⢻⡿⢿⣿⣿⣷⣶⣶⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣟⣿⣿⣛⣻⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠉⠉⠀⠀⠀⠀⠹⣧⡼⠁⠈⠙⠿⠞⢋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠤⠀⠒⠲⠒⢂⠠⠤⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⠿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⣀⠀⣀⢼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣛⣛⣦⡀
⠀⠀⠀⣀⡤⠶⣶⠈⢻⡄⠠⠀⠀⠈⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠋⠀⠀⠀⡀⠀⢶⢈⠉⢩⣿⠁
⠀⠀⠘⣿⠖⠛⢉⣆⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⣀⠼⠆⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣤⠀⢠⠀⠀⠀⠘⠃⠾⠃
⠀⠀⠘⢿⣆⠰⠊⠀⠀⠀⠀⣌⣁⠀⠀⣄⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠇⣀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠊⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠀⠃⠀⠐⢴⠐⣶⣆⡉⣻⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠳⣆⠉⢁⣶⣦⡀⠺⠉⠉⠉⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠂⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠿⢋⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠏⠠⡤⣰⣥⣥⡀⠀⠀⠀⠄⠀⠀⠃⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠁⢀⠀⣿⡆⠀⣷⣄⠀⠀⠀⠘⣿⠛⢻⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⣴⠋⠉⠉⠻⢷⣶⣤⠀⠠⠔⠈⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠻⡇⠀⠈⠛⠀⠀⠀⠀⣹⣶⣶⠀⠀⠀⠀⠠⠤⠀⠀⠄⠀⠀⠀⣴⣤⣾⠟⠂⠀⠀⠀⢠⣿⠃⠀⠀⠂⠀⣶⡀
⠀⠀⠀⠀⠈⠀⠀⠀⣴⠂⠉⠀⢰⣴⣶⣾⣿⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⣀⣠⣤⣦⡄⢈⣡⠀⠀⠁⢀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠤⢿⡇⠠⢸⣿⣿⡟⣹⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠛⢻⣄⣸⡟⠀⠀⠀⠈⢰⣿⠀⠀⠘
⠀⠀⠀⠀⠀⠚⠁⠀⣸⡇⠀⣼⡟⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠠⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⡛⠋⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠈⠏⠶⠄⢠⣶⣆⡀⠰⠶⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠆⠀⣤⣶⣶⡄⠀⢸⡆⢠⣤⡀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣴⠂⠀⠛⠀⠉⠙⠋⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠈⠛⠃⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢠⣦⣤⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⣀⣀
⠀⠀⠀⠀⢸⡛⢿⡟⠓⡤⠀⠀⣀⢀⡄⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⣧⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠈⣃⣄⡤⣀⠲⠈⠉⠂⣰⡿⠅⠀⠀⠀⠀⠀⠀⠨⠖⣅⠃⠀⠒⠃⢠⢸⣖⠂
⠀⠀⠀⠀⠰⠚⠋⠉⠉⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠥⠭⠝⠛⠛⠋⠘

⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠈
⠀⠀⠀⠀⠀⢀⠀⠄⣄⠀⠀⣠⠀⣀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠘⣀⠁⡀⠘⠃⢠⠏⣰⣾⣿⡂⠀⠀⠀⠀⠀⠀⠀⠶⡌⠂⠀⢀⣤⡴⠆⠀⠀⠀⠈⠃
⠀⠀⠀⢠⡬⠧⣘⣣⡶⠷⠿⠖⠋⠈⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠉⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⡆⠀⠀⠀⠀⠀⠀⢰⣬⠿⠰⡆⠎⣷⠰⡆⢠⡴⢆⡤⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠘⢀⢰⡇⠀⠀⠀⠀⠠⣽⣶⣦⠸⠐⠃⢠⠈⢠⠀⢘⡆⢾⣵⣤⡴⠂
⠀⠀⠀⠀⠀⢀⡈⠆⡁⢠⣘⣰⠇⠛⠚⠁⠀⠀⠀⠠⠶⣿⢷⣤⡤⢶⣶⣒⣾⡦⠾⠾⠥⠬⣻⡿⠿
⠀⠀⠀⠀⠀⠈⢻⣀⣟⣘⣠⣤⣤⠀⠀⠀⠀⣀⠀⠀⠀⢀⠞⠋⣴⠰⣍⠩⣠⡀⣀⠰⣄⢈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠌⢇⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⢠⡦⠀⠀⣠⡼⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣀⠀⢻⡀⠀⣀⠀⠀⠀⠀⠀⠀⠹⡇⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠡⠀⢹⣿⣦⡀⠀⠀⠀⠀⠁⠰⠃⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⡀⠈⠀⠀⠈⣿⡎⠛⢦⡀⠀⣠⡀⢁⣾⠏⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠲⢦⣤⣿⡇⠀⠀⠀⠐⣿⢳⣼⠏⠀⠀⠹⣷⡦⠤⣀⠀⠔⣾⠀⣤⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⠬⣿⡄⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠹⠿⠛⠉⠀⢠⡟⠘⠃⠀⢀⡴⠛⠃
⠀⠀⠀⠠⢤⣤⣤⣄⣀⣀⠀⠈⡻⣿⣿⡛⠛⠛⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⣤⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠛⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠴⠥⠤⢶⣶⠖⠀⢠⣤
⠀⠀⠀⠤⢤⣄⣂⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠈⠀⠀⠀⠈⠋⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⡻⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠒⠋⠉⠉⠩⠛⠉
⠀⠀⠀⠀⠘⠃⠀⢀⡠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠆
⠀⠀⠀⣀⣤⡤⠚⣁⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣶⣤⣤⣀⡀⠀⠀⠄
⠀⠀⠘⠋⠉⠉⠉⠉⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⠑⠚⢛⡿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣠⡤⠶⠤⠂⣁⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣠⡤⠄⠂⠀⠠⠞⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣤⣽⣿⣶⣶⣖⡺⠛⠉⠉⠙⠃
⠀⠀⠀⠠⠐⠙⠉⠁⠀⠀⠀⠀⠀⠠⣿⠀⠀⢀⣠⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⡶⠾⠟⠻⠻⠟⣫⣍⡓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⡤⠒⠉⠭⠈⢿⡀⠀⢀⣶⡿⢤⣶⣦⣀⠀⠀⢀⡟⠿⣶⣦⡀⠙⢿⡆⠐⠒⠂⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⣴⠟⠁⡀⠈⢧⠀⣾⡿⠀⠈⠃⠀⠙⠳⣄⢸⣧⠀⠀⠉⠙⠒⠚⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢋⡞⢰⡗⠀⠘⢷⡿⠃⠀⠀⠀⠀⣠⠶⢌⠻⢿⡇⢰⡌⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢸⡇⠀⣤⡀⠀⠀⠀⠀⠀⠀⠙⠶⠋⠀⠀⠀⠀⢿⡌⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣿⠇⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛

⠠⠠⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠠⠠⠁⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠌⠂⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣀⡾⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⢀⣴⠶⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠐⢦⣄⣾⠏⠀⠀⠀⠴⠶⠶⠶⠷⣦⣴⠿⠛⠛⠛⠋⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣟⠁
⠐⠄⠀⠹⣧⠀⢀⣼⠃⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⢴⡄⠀⠀⠀⠀⠀⠀⠉⣿
⠐⠀⠀⠀⠈⢷⣿⡁⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⣁⣴⠟⠀⠀⠀⠀⣀
⡀⠀⠀⠀⠀⠀⠈⠛⠳⢶⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣠⣤⣀⠀⠀⠀⠉⠛⠳⢶⣄⢀⣴⠞⠛⠻⣧⠈⠂⠀⠀⠻⣦⢑⣬⡶⠛⠁⠀⠀⠀⠀⢘⣿
⠻⣦⠀⠲⣆⠀⠀⠀⠀⠀⢍⢷⡈⡀⠀⠀⣿⠀⠀⣼⠋⣵⣤⡙⢷⡀⠀⠀⠀⠠⢀⣽⠟⣠⣾⣷⠄⣿⠅⡀⠀⠀⠀⠹⣿⠁⠀⠀⠀⠀⠀⠀⠀⢼⡇
⠀⠘⣧⣀⣹⣧⠀⢀⠐⢌⠂⢸⣷⢰⠰⠀⣿⠀⠀⣿⢠⣿⣿⣷⠈⣷⠀⠀⠀⣴⠟⢡⡾⣻⣿⣿⡆⣿⠂⠌⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⡈⢺⡇⠀⠀⠀⠂
⠀⠀⠈⠉⠉⠙⢷⣄⡑⢄⣢⣴⢿⡜⠀⢸⡇⢀⣰⠏⣼⣿⣿⣿⡆⣿⢀⠀⢀⣿⢀⣿⡅⠙⣿⣿⡇⣿⡃⠀⠈⠘⠀⠀⠀⣿⠀⠀⠀⠀⢀⢀⣠⡿⠷⠶⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠛⣾⣿⣿⣮⣿⡄⢸⣇⢸⣿⣼⣿⣿⣿⣿⣷⣌⠛⠛⠛⣡⣾⠃⡀⢸⣿⣿⣇⠹⣧⣤⣀⣁⣠⣴⠾⠻⣦⣀⢀⣤⠞⠋⠉⢀⠀⠀⠀⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⣿⣼⡟⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⠟⢠⠁⣾⣿⠿⣿⣶⣤⣭⣭⣯⣭⣴⣾⣿⣦⣝⢿⡁⠀⠀⠀⣿⠆⠀⠀⣿
⠙⠛⠳⡶⣤⣄⠀⡐⢀⣿⣿⣿⣿⣿⣿⣛⣻⣯⣾⠿⠛⠛⠛⠒⠿⣿⣿⣿⣿⠃⠠⠀⡼⢋⣁⣠⣬⣤⡀⣴⠶⠿⢷⣶⡌⢹⣿⣿⡞⣷⠀⣀⣴⡟⠁⠀⠀⠹⣦
⠈⠀⠌⢀⠔⠙⢷⣄⣾⠃⣿⣿⣿⣿⣿⡿⢿⠟⣡⡶⠟⠛⠛⢷⣦⡈⣿⣿⠇⣘⠀⣴⡆⢻⡉⠁⠀⠈⠿⠏⠀⠀⠀⣿⣷⢸⣿⣿⣷⢸⡿⠋
⠀⠀⠀⠁⠐⡡⠀⢻⣿⠀⣿⣿⡿⢋⣡⣤⡄⣴⠏⢀⣤⣤⡀⠀⣿⣧⢸⠟⡴⠃⣼⠟⣡⣬⣥⣤⡀⠀⠀⠀⠀⠀⠀⣿⣿⢘⣿⣿⡧⣸⠇⡀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⢀⣀⠀⣽⡇⣻⣿⡀⣿⡉⠀⣿⡏⠀⣾⣟⣽⠃⢀⣿⣿⣠⣼⣿⣄⡀⠸⣿⣿⣿⣿⣿⣦⠀⠀⣶⡄⠀⢹⣿⠀⣿⠟⣼⠏⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⡀⢠⡾⠋⢰⣿⣿⣇⠸⣇⠀⠈⠀⠘⠿⠟⠁⢀⣼⣿⢯⣿⣿⣿⣿⣽⡄⠸⣿⣿⣿⣿⣿⡇⢸⣿⡇⠀⢸⣿⡇⠋⣼⠃⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠈⠛⠾⣦⡶⠟⠛⠶⢶⡄⢿⣿⡆⢿⡄⠀⠀⠀⠀⠠⣶⣿⣿⣯⡻⣿⣿⣿⣷⣿⣿⠀⣿⣿⣿⣿⣽⡇⣿⣿⣷⠀⠸⣿⡇⢸⡇⣀⣀⣀⣀⣀⣠⣴⢟⠃
⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⣠⣿⣿⣇⠘⣧⠀⠀⠀⠀⠀⣀⣀⠉⠻⣿⡝⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⢱⡟⠛⣿⡀⠀⣿⣇⢸⡟⠉⠭⠉⠉⠉⠉⠹⣦
⠀⠀⠀⠀⠀⢀⣾⠃⣰⣾⣿⣿⣿⣿⡆⢻⡄⠀⠀⠀⠘⢿⡛⣿⡆⠘⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣿⠏⣿⣿⠀⢸⡇⠀⢻⣿⠈⢷⣄⡀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠐⠠⠬⣿⢸⣿⣿⣿⣿⣿⣿⣧⠈⣷⠀⠀⠀⢀⡈⠙⠛⠁⢠⡟⣿⣿⣿⠟⠁⠾⣿⡿⢛⠁⠀⣿⣿⡀⡀⢿⣤⣼⣿⡇⢰⣌⠻⣶⡂⠀⣁⠀⠸⣇
⠀⠀⠄⠀⢀⣀⣿⣄⣀⣋⣉⣙⠻⣿⣿⡀⢹⡇⠀⠀⠘⣿⠶⣤⠶⠟⠀⠉⢉⣠⡀⠀⣀⡿⠃⢿⣤⣴⣿⣿⢀⣷⠌⠙⣛⡉⠁⠿⣿⣷⣌⢿⡄⢀⡀⠀⠻⣦⣀
⠀⠀⠀⠀⠀⣰⡏⠉⠉⠉⣉⣽⡇⢸⣿⣀⡀⠻⠶⠛⠛⠋⠀⣀⣠⣶⣞⣶⣿⣯⣖⢰⣯⣞⡆⠈⠙⠛⢋⣁⠜⡁⠐⢀⡤⢤⣔⡂⠈⣿⣿⡎⣿⠀⠀⠀⢠⡟⠙
⠻⣦⡀⠀⠀⣿⠀⠀⠀⣴⢏⣿⠁⣿⣿⣿⣿⣄⣠⣶⣐⣾⣿⣿⣟⣿⣿⣿⣿⣿⣯⣄⠉⠛⠖⠚⠉⠦⠀⡅⠈⢁⡼⣫⣶⡶⣦⣉⠛⠿⠿⢃⣿⠀⠀⠀⢼⡇
⠀⠈⣿⡄⠀⣿⠀⠀⣸⠏⢨⣏⢠⣿⣿⣿⣿⣿⣿⡿⠛⠿⠋⠛⠿⣿⣿⣿⣿⣿⣿⡿⠛⣛⣉⣛⠛⢶⣶⣴⣾⣿⣿⡏⠀⢻⣌⢉⠻⠶⢶⠛⢃⠈⠀⠀⠈⠻⣦
⠀⢊⠜⠻⣤⣿⠀⠀⢹⡆⠀⢻⡄⠺⣿⣿⣿⡿⢫⡶⠟⢻⠛⣷⣦⡈⠻⣿⣿⣿⠟⢁⣾⠏⠉⢙⡿⢶⡘⢿⣿⣿⣿⠃⠀⠀⠙⣦⡀⠀⠀⠑⠀⠁⠀⠀⠀⠠⠉
⠀⠀⢢⣀⣼⠇⠀⢀⡾⣷⠀⡌⠻⣦⣙⣙⣫⣴⠟⠁⠀⠈⣸⠇⠈⢿⡄⠙⠛⢁⣠⣾⡏⠀⠀⣾⠃⠈⢿⣄⠙⠋⣿⠀⠀⠀⠀⠈⠻⣦⣤⣤⣀⡀
⠀⠰⠟⠉⢀⣀⣤⡾⠁⢸⡇⠀⠄⠈⢙⠉⠉⠀⠀⠀⠰⠀⣿⠀⠀⠈⠛⠛⠛⣿⢋⠚⠀⠀⣼⠇⠀⠀⠀⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠛⢶⣄⡀
⠀⠀⣠⡾⠋⠉⣁⣠⣴⠟⠷⣤⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⣽⡇⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡿⣦⣄⡀⠀⠉⠻⣦
⠀⣰⠏⠀⠀⠞⠋⢹⣇⠀⠀⠈⠙⠛⠶⡄⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⣸⠏⠻⣦⠀⠈⢻⣶⣤⣤⣶⢶⣤⠀⠀⠀⠀⣸⠏⠀⠀⣾⠁⠀⠉⠻⣦
⠾⠃⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣵⠟⠁⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⣼⠃⠁⠀⠀⠀⠈⠀⠀⠀⢰⠟⠀⠀⠀⠈⠀⠀⠀⠀⠘⢷⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣄⣀⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⠶⣶⡿⠁⠙⠛⢩⡻⠂⣤⡀⣼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢰⡀⠀⣀⢳⡀⢀⠉⡀⢻⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠘⣡⠬⡍⠁⢨⠄⣁⣸⠘⠣⣤⠥⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⡤⣤⣶⣿⣶⣭⣠⣿⠿⠿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⠉⠈⠉⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡼⠷⠼⠇
⠀⠀⠀⠀⠀⠀⠀⢦⣼⡁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣽⣿⣿⣿⡄⠀⠦⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠒⠛⡶⠓⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⣦⣾⣿⣿⣿⣿⣿⣧⣀⣴⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣦⠀⠀⠀⢀⠀⠀⠠⣤⣸⡿⢿⣿⡏⢙⣿⡿⢿⣿⣿⢿⠿⣯⣻⡇
⠀⠀⠀⠀⠠⢀⠀⠀⠀⣸⢛⣿⣿⣿⣿⡆⢠⣴⠃⠀⠀⠀⠀⠹⡷⠿⠛⠿⠿⠟⢿⣿⠿⠿⢀⣠⣿⣯⠅
⠀⠀⠀⠀⠀⠀⠙⢶⣤⣽⣾⣿⡟⡋⠈⠇⢄⢧⣀⠄⠀⠀⠀⢠⠷⠛⠉⢣⡴⠶⣄⡴⠛⠳⢾⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢩⢹⣰⡚⡟⡗⡈⣦⣸⣼⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠐⠒⢆⠶⢱⠅⠟⠐⣫⣭⢿⣟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣀⣴⠿⠶⠿⣿⣷⡾⠛⢧⠀⠈
⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⢸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⢀⣾⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠔⠋⢹⠈⡏⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠈⣿⣤⣶⣿⣿⠀⣿⠷⠶⠦⣄⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⠢⣄⣠⣤⣤⡤⣿⣿⡿⠛⠛⠛⠁⠀⠀⠀⠈⢧⣀⢐⣶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣌⣻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢰⣀⣀⠀⠀⠀⠀⠀⠉⠩⠟⠲⠶⢤⡀⠈⢿⣿⣿⣦⣄⠀⠀⠀⠀⠀⢀⣀⣠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⠈⠉⠙⢿⡿⠷⠖⣒⡯⠝⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠟⣿⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣠⡟⠀⠀⠘⠆⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠃
⠀⠀⠀⠀⠀⠀⠉⠉⠉⢿⠙⣇⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢾⡀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⢰⢀⠀⢸⠇⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍⠂⠀⠀⠙⠌⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⣼⠏⣠⠟⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣦⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣷⡾⠁⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⡿⢋⣴⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢒⠟⣩⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡫⣖⠙⠻⠿⠟⠛⢶⣤⣤⣶⣧⣀⠀⠀⠀⢀⣡⣴⡛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠁⠀⠀⠀⠀⠀⠉⣿⡟⠛⠛⠋⠙⠛⠉⠁⠀⠱⡜⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠸⠁⠙⠢⡄⠃⠄
⠀⠀⠀⠀⢀⠔⠠⡀⡀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡰⡇⠀⠐⡁⠀⠀⠠⡀⣀⠀⠀⠀⠁
⠀⠀⢠⠒⠚⠀⠀⠀⠈⠀⠀⠀⠀⠠⠤⡀⠀⠀⡤⠖⠒⠢⡀⠀⠀⢰⠤⠜⠂⠊⠀⠱⠥⠘⠃⡀⡀⢀⠂⢈⠅
⠀⠀⠸⢄⠀⠀⡠⠒⠀⠀⡠⠖⠒⠒⠢⣄⢀⡸⣀⠀⠀⠀⠈⠊⠉⠙⢄⣠⠄⠀⢄⣠⢤⠮⡀⠉⢀⢂⠔⠁
⠀⠀⠀⠈⠓⠋⠀⠑⠀⡞⠀⠀⠀⢀⢀⠈⠁⠀⠀⠑⠒⠚⠁⠀⠀⣀⣈⣁⠀⠀⠀⠑⡫⢔⣒⣚⡳⣅⢖⡀⠀⠐⢒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠼⠀⡀⡸⠢⠴⣄⡼⡃⠀⢀⡤⢒⢒⠋⠅⠒⠢⡑⢄⡠⢊⠔⠁⠀⠀⣿⠌⡘⠀⠀⢀⡾⢡⠃
⠀⠀⠀⠀⠀⠀⢠⠆⠀⠁⠘⠂⠉⠁⠀⡠⠔⠐⠷⡶⢣⠊⠁⠀⠘⠂⠀⠀⠙⢮⠔⠁⠀⠀⠀⢰⣿⢀⠃⠀⢀⣾⢣⣏⠀⢀
⠀⠀⠀⠀⠦⠀⢾⠀⠀⡠⠖⣒⠢⢤⠎⠔⠁⠀⠑⢦⠁⠀⣴⣦⠀⠄⠀⢠⡀⠀⠀⣠⡄⠀⠀⢸⡏⠘⠀⠀⣼⢃⠇⠀⡄⠑⡄
⠀⠀⠀⠀⠂⠀⢈⠗⡩⠂⣁⠀⠈⣧⠈⠀⣴⡶⠀⠈⠀⠸⣏⠜⠀⡇⠀⢸⡟⠶⣾⠿⠀⠀⠀⣿⠇⠃⠀⣼⢃⠎⠀⢀⠱⢴⠃
⠀⠀⠉⠢⠀⠀⠸⡸⡀⠘⠿⠃⠼⠯⣆⠀⠻⠔⠀⢰⠀⠀⠀⠀⠀⡇⠀⠈⡧⢴⣶⢸⠀⠀⢀⣿⢸⣤⣼⢏⣾⠴⠖⠧⠀⡠
⠀⠈⠙⠄⠀⠀⠀⢇⢡⠀⢀⣤⢆⠀⠸⠀⠀⠀⠀⣸⣧⠀⠀⢀⣴⣇⠀⠀⣷⠈⠹⡎⠀⠀⢸⡿⠁⠈⡄⣎⠀⠀⠀⡏⠉
⠀⠀⠔⡥⢄⠀⡀⠘⡌⡄⠀⠛⠀⠀⢀⣷⠤⣤⣴⠟⣡⣝⣛⣛⣩⢬⢛⣛⣋⠌⠰⣙⠻⠷⠿⠟⢦⡾⢃⠗⠀⠈⠋
⠀⠀⠀⡇⠐⠁⢸⡎⢳⢰⠀⠀⢀⣤⠾⣡⠖⠲⠖⠊⠀⠀⠉⣅⡎⣀⠀⠀⠀⠀⠀⠀⠉⣍⡉⠉⠑⠚⠉⠀⠀⠀⡀
⠀⠀⠀⠈⠲⡤⠀⠁⠨⢆⡲⢞⣋⡥⠞⠁⠀⠀⠀⠄⠀⠀⠀⠁⠠⠞⢲⢀⢤⠀⡖⠲⠀⠀⢫⡀⠀⠀⣠⠤⠴⠊
⠀⠀⠀⠀⢰⠀⠀⢀⡤⠀⠉⠉⠁⠀⠀⠀⠌⠀⠀⣀⠆⠀⠀⠀⠀⠀⢸⠒⠀⢻⠀⠀⢀⡠⠈⠀⡂⠀⢘⡀
⠀⠀⠀⠀⠀⠣⣅⢀⣀⢀⠀⠀⠀⠀⢠⡆⠀⠀⢀⢄⡀⢀⠤⠤⠂⠀⠀⠀⠠⠀⠑⠒⠋⠢⣄⣀⣀⡴⠋⠱⡄
⠀⠀⠀⠀⠔⠒⠁⠈⠀⠀⠙⠲⠤⠆⠈⠈⠙⠒⢏⠀⠈⠁⠀⠀⠠⠤⡠⠴⠀⠀⣠⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠙⠒⠒⠒⠁⠀⠀⠀⠀⠎⠈⠀⠐⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢄⠀⢀⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⢀⢀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢷⣂⣀⣀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⡀⢀⣿⠿⣚⣷⣟⡷⠈⠉⠀⠀⠒⡿⠒⢠⣤⣤⣶⣤⣄⣀⠈⠁⣰⣿⠟⠻⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠠⠼⡟⠃⢀⡆⠀⠀⢀⣀⣤⣤⣯⣟⣈⣿⣿⣿⢿⣿⣿⣶⣴⣿⣿⣷⣷⣤⣿⣿⠉⠙⠿⣿⣿⣷⣦⣿⡏⠀⠀⠀⢿⣿⣿
⠀⠀⠀⠀⠀⢠⣿⣶⣶⣶⣶⣷⣶⣴⣉⢀⣶⣿⠟⠉⠙⠻⣿⣿⡿⠋⠀⠀⠈⠙⢿⣿⡿⠉⠻⣿⣿⣿⡏⠀⠀⠀⠙⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⢸⣿⣿⠉⠉⠉⠉⠉⠛⠻⣿⣿⡏⠀⢀⣄⠀⢹⣿⠇⠀⢸⣿⡄⠀⠘⣿⡇⠀⠀⠈⢿⣿⡇⠀⡀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⣾⣿⣿⠁
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣾⣿⣿⣦⠀⠙⣿⡇⠀⣿⣿⡆⠀⢿⡄⠀⣿⣿⣿⡀⠀⣿⡇⠀⢠⡀⠀⠹⠃⠀⣷⠀⠀⠀⠹⣿⣿⠀⠀⠀⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⢹⣿⣿⠏⠀⢀⣿⡇⠀⢹⣿⣷⠀⢸⡇⠀⢸⣿⣿⡇⠀⢹⡇⠀⢸⣿⠀⠀⠀⢰⣿⣧⠀⠀⠀⢹⣿⡀⠀⢀⣾⣿⣏⠃
⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠻⢿⣧⠀⠘⣿⣿⠀⠈⣧⠀⠈⣿⣿⡇⠀⢸⡇⠀⠸⣿⡇⠀⠀⢸⣿⣿⣇⠀⠀⢈⣿⠿⠿⠿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠰⣶⣶⣦⣄⠀⠀⢻⡄⠀⢿⣿⠆⠀⣿⡆⠀⠸⠿⠁⠀⢸⡇⠀⠀⣿⣿⡄⠀⢸⣿⣿⣿⣧⣀⣸⣷⠀⠀⠀⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⣿⣿⣿⣿⠇⠀⠀⣧⡀⠈⠋⠀⢠⣿⣿⡀⠀⠀⠀⣰⣿⣿⠀⠀⢸⣿⣿⣤⣼⣿⣿⠋⢿⣿⣿⣿⣷⣶⣾⣿⣿⣯⠃
⠀⠀⠀⠀⠀⠀⣻⣿⣇⠀⠀⠙⠛⠋⠁⠀⢀⣾⣿⣷⠀⠀⣀⣾⣿⣿⣿⣶⣤⣾⣿⣿⣿⣷⣶⣿⠋⠛⢿⣿⣿⠃⠀⠀⠉⠙⠁⠙⢿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣦⣤⣤⣤⣤⣤⣾⣿⡿⢿⣿⣿⣿⣿⣿⠇⠘⠿⣿⣿⣿⣿⠿⠳⢭⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⡿⠿⠿⠿⠟⠉⠈⠳⠿⠿⠿⠟⠉⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁

⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆
⠀⠀⠀⠀⢀⡤⠉⠈⢙⠀⠀⠀⢠⣀⡀⠀⠀⠀⠄⠀⡀⠀⣸⠟⠙⣆⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠒⢦⡀
⠀⠀⠀⠀⠸⠀⠢⠀⠘⢦⠀⠀⠀⢿⡍⠉⠓⠒⠤⠤⠤⠞⠁⠀⠀⠈⢆⠀⡀⠀⠀⠀⠀⣀⠴⠚⠡⠂⠀⠁⡄
⠀⠀⠀⠀⠀⠀⠑⢤⠀⠀⠳⡀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⡥⠀⠈⠀⢀⣀⡤⠦⠀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⢤⣵⣄⠐⢌⢨⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣰⡾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⠤⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡑⠄⠢⣀
⠀⠀⠀⢀⡔⠂⠴⠒⠋⠁⠀⡈⢠⠆⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣼⣷⠘⢆⠀⠈⠳⣄
⠀⠀⠀⠘⢀⠀⢠⢄⣠⠔⠊⠀⣾⣠⡶⠟⠋⠁⠊⠻⣦⠀⠀⠀⣴⠟⠑⠀⠀⠀⠀⠉⠁⠈⢳⡀⠐⢪⠀⡆
⠀⠀⠀⠀⠈⠢⠤⠈⠁⠀⠀⠠⠟⠉⠀⠀⠀⠐⠀⠀⠈⠳⣄⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⢀⠀⢰⢦⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡀
⠀⠀⠀⠀⣀⡀⢯⡇⠘⠚⠃⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠄⠄⠥⢨⣎⠔⠀⠀⠀⠀⠀⠀⠐⠃⠈⠡⠞⣻⢃⣤⣄
⠀⠀⢀⣼⠓⠁⠀⠀⠀⢀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣾⣦
⠀⠀⠈⠁⠀⢀⠀⠀⠀⢈⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⣀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠀⠀⠀⠀⠑⠉⠁
⠀⠀⠀⠀⠀⡀⠀⢀⣀⡉⠀⠠⠀⣠⠀⠀⠀⠀⠀⠀⠀⠐⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠈⢠
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⣦⠤⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢀⣀⣀⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢾⣿⠁⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠙⠚⠃⠀⠀⠀⠀⠀⠀⠀⣴⡎⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⡆
⠀⡀⠀⠀⠀⠐⢠⠀⠄⢀⠂⠄⠀⠀⠀⠀⠀⠀⠀⢠⣦⡴⠁⠀⠀⠀⠀⣀⡼⠙⢞⠙⠣⠄⠀⠀⠀⢀⠀⠀⠀⠄
⠀⢷⠄⣀⣹⣬⣠⣤⣄⡞⠀⠀⠀⠀⠀⠀⠀⠀⡏⢂⣼⠖⠂⠀⠀⠀⠀⠉⠀⢀⠈⢈⣼⠰⠂⠀⠀⠈⣧⣀⣜⣒⢓⡒⠢⠂⣀⣴⡾
⠀⠀⠳⣬⣥⣤⣤⡀⢈⣁⣤⠼⠁⠀⠀⠀⠀⠀⠖⠛⠀⠀⠀⠀⠀⠀⠀⠚⢯⣤⡤⢴⡏⠁⠀⠀⠀⠀⠈⠉⠙⠉⠉⠛⠒⠛⠋
⠀⠀⠀⠀⠀⠛⡟⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠀⢀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⡄⡔⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠒⠒⠒⠀⠒⠀⠀⠐⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀⡤⣤⣠⣤⡄⢀⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠉⠧⠘⠧⠋⠟⠛⡱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⢰⠀⠘⠈⠁⢠⡠⢂⡤⠐⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⣀⣶⣿⣶⣀⣰⣿⣷⣤⣶⣦⣦⣦⣤⣾⣿⣿⣷⣀⣷⣾⣷⣀⣀⣀⣀⣀⣀⣀⣶⣷⣆⣠⣴⣿⣶⣄⣠⣤⣿⣟⣠⣤⣤⣄⣀⣀⣀⣀
⣿⣿⠉⡉⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠓⠚⠉⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠴⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⠃⠀⡞⠉⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠀⠸⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⢠⡞⠁⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⡰⢆⠀⠀⠀⢀⡀⠀⠀⢀⡾⢹⣇⣠⠴⢾⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠇⢀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⠁⠀⠀⡄⠢⡀⠀⠀⣿⢷⡀⡰⠁⠈⢣⣀⢠⠎⠱⡄⣰⠿⡤⠯⣿⠀⠀⢸⠀⠀⠀⠀⣠⠔⣉⠔⠀⠀⠁⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⢆⣱⣄⣴⣏⡍⢻⡀⠀⢀⣼⢛⣧⡀⢀⡼⠋⣛⡫⠏⣿⠀⠀⣸⣤⣶⣄⣊⣴⣊⠀⠀⠀⠈⠢⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡀⡷⡈⣿⣷⣒⣒⣳⣄⣾⣳⡓⣒⣳⣟⣶⠞⠁⠙⢳⣺⣖⣛⣽⣾⡿⢋⡡⢾⠀⢀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣄⠀⢸⠀⠑⠢⣟⢧⣜⣿⣻⣖⣒⣒⣛⣒⣾⣦⣒⠒⣓⣛⣶⣤⣤⢟⢒⣞⣛⡞⣻⣧⣤⠶⣟⣛⣛⣹⣏⣀⣀⠠⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣎⣙⣾⣤⡀⢠⣷⣶⣽⡿⣳⡿⢶⣶⠞⠉⠀⠀⠉⠳⣶⠋⠀⠀⠙⠷⠒⣦⡶⠛⠛⠻⣶⣶⣿⣿⣷⠏⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠫⠬⠮⠽⢻⡿⠟⠛⠻⠋⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢹⢿⢏⡀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠉⢻⡭⠭⠭⠭⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀⠀⣠⣴⣶⣤⣄⢀⣶⣶⡄⣾⣿⣆⠀⠀⢀⣿⠤⠤⠤⠭⠔⠳⢶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠼⠷⢾⠯⠭⣭⠟⠉⣠⣶⣿⣿⣷⡆⢠⣾⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠈⠻⣯⠭⠭⠭⢭⡽⠫⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣮⣼⣿⣿⣾⠃⠀⠀⢹⣿⣧⣿⣿⣇⣸⣿⣿⠟⢻⢿⠿⡿⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢸⣇⣯⣥⢛⣶⣮⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢶⡋⠁⣘⣷⣛⣛⣻⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣾⣶⣧⣽⣶⣿⣿⣿⣿⡟⠙⠏⢻⣿⣿⣇⢀⢀⣾⣳⣳⡟⠉⠙⣼⢒⣈⣭⠿⠓
⠀⠀⠀⠀⢀⣀⣤⣴⣟⣛⣛⡶⠲⠶⣖⣻⣶⠂⠀⣿⣿⣤⣼⣿⣿⡿⣿⣿⣿⣿⠟⠛⠿⣿⡿⠟⠻⠿⠃⠀⠀⠀⠛⠛⠉⠀⣿⣐⣒⣒⣇⣰⡾⠟⢿⡉
⠀⠀⠀⠈⠛⠷⣷⣒⡚⢛⢧⣀⠀⣰⠓⣺⠇⠀⠀⠻⠿⠟⠛⠛⠉⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⠀⠀⠀⠀⠀⣿⢚⣚⡚⠒⣚⢳⡄⠀⣙⡦
⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣷⣞⣉⡕⡦⣿⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⣾⠀⠀⠀⢠⣷⣄⣀⣤⢿⡶⠾⢶⢞⢦⣤⠾⣷⣶⠷⡶⡤⣷⠶⠿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠉⢻⡾⣩⡽⠷⣤⡤⠾⣆⠀⠀⡾⢧⣄⣀⣀⡴⠯⠳⠶⠶⠯⠭⠯⠭⣍⣯⡭⠭⣿⠛⠿⠛⡯⠿⣽⢷⣽⡿⠿⣭⣏⡹⢷⡀
⠀⠀⠀⠀⠀⠠⠾⠅⠒⠒⢒⣶⣿⢟⡿⠿⢭⣡⠬⠭⠬⠷⠟⠭⠭⠭⠽⠭⢭⡩⢭⠭⠭⠭⠭⠭⠥⣿⣿⡟⠽⠏⠀⡀⢠⣟⠭⢬⣿⣽⢿⡀⠀⢏⠉⠛⠻⠆
⠀⣠⠤⣤⠊⢱⠀⠀⠀⢂⡽⣫⣿⡯⡿⠛⢉⡿⣭⣭⠭⠭⠭⠭⠭⠭⡭⠽⠋⠉⠉⣿⠩⠭⠭⠭⢭⡭⠭⠭⠥⢤⣦⣤⢴⡏⠻⣯⡙⠿⠿⣿⢶⠚
⡞⠁⠀⠀⠀⠀⠀⡠⢒⠿⠋⠉⠀⢰⡡⠖⠹⠿⠉⣿⣺⣆⣗⣤⣶⣊⣧⣀⠀⢀⣞⣇⡿⢷⣜⣰⡟⣷⣑⣊⢢⡟⠀⠹⠿⠉⠚⠀⠙⠳⣤⣸⡷⡑⢄⠀⠀⢀
⠙⡲⠀⢰⡀⠀⠀⠔⠁⠀⠀⠀⠀⠉⠀⠀⠀⠿⠚⣿⢒⣶⠟⠁⠘⣷⣒⣛⣶⣛⣓⣾⠁⡼⠙⠋⢆⢽⣖⣒⡟⢣⠀⡇⠀⠀⠀⠀⠀⠀⠈⠙⠿⠈⠃⠀⠀⣸
⠠⡇⠀⢠⡇⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⢱⠀⢠⠎⢹⣖⣒⣒⣲⡟⠀⠱⠃⠀⠀⠀⠘⣷⡾⠁⠈⢧⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⡄⠀⣷
⠀⠙⠦⢤⠑⠒⠒⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠃⠀⠀⢻⡖⣲⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⢀⣠⠴⠁
⠀⠀⠀⠈⠣⣄⣠⡤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⡤⠏
⣶⣶⡶⠶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⡐⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢠⡞⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠓⢦⣀⠈⠀⠃⠀⠀⣼⣄⣀⡴⠏⠀⢹⣄⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣀⣤⡿⠿⠿⠿⠷⠾⣧⣙⠋⠀⠀⠀⠀⠻⣦⣀⡴⠞⣷⠀⠀⢀⣀⡤⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣤⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⢀⣤⡶⠶⠿⠿⠷⢶⣿⡛⠛⠉⠁⢠⡟
⠀⠀⠀⠀⠀⠈⢳⣦⡤⣤⣀⣀⣴⡾⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣀⠬⠽⠶⣤⣤⣼⡇⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⢿⣥⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣷⠖⠛⠛⣿⡎⣧⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢀⣤⠶⠚⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣮⣕⢶⡽⠿⠶⣶⡍⣿⣇⠀⠀⣿⡇⣯⣠⡼⠋⠁
⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡤⢤⣀⠀⠀⢀⣤⣶⣒⣒⠦⣄⡏⣿⠀⠀⠀⠹⣾⠁⠀⠀⢹⡏⣿⢻⡆⠀⣿⡇⣿⠋
⣀⣤⡤⢴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡽⠚⠛⠛⠳⣮⣹⢆⡟⠋⠉⠀⠉⠻⣦⡹⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡧⣻⠶⠿⣧⣻⣤⠤
⠀⠉⠓⢿⡇⠀⢀⢤⣔⣶⣶⣬⣭⣗⡲⣮⡟⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⠀⠀⠀⠀⠘⣷⢹⡇⠀⠀⠀⠀⠀⡀⠀⢸⣿⢸⣏⢿⣄⣠⣿⢇⣷⠐⠒⠈
⠀⠀⠀⠈⢷⠀⣷⣏⠉⠀⠀⠀⠀⠈⢻⣿⡇⠀⠀⢰⣶⡄⠀⢶⣿⠀⠀⢸⣿⡇⠀⠀⣿⣟⣧⠀⠘⣷⣄⣰⣿⣀⣸⣿⢸⡏⠪⣽⣯⣵⡟⢷⣤⡀
⠀⠀⠀⢀⣀⣿⢇⣿⡀⠀⠀⢠⡄⠀⢸⣇⢿⡄⠀⠘⠿⠇⠀⣸⡼⣧⠀⠀⠛⠃⠀⣰⣿⢻⣿⠿⠿⢟⣛⣿⣛⣛⣿⣯⠾⠁⠀⠀⠀⣸⡇⢀⣈⣹⣷
⠀⠀⠀⠀⠈⣿⠘⣼⣇⠀⠀⠈⣁⠀⠻⣿⢦⡻⣦⣄⣀⣠⣴⡿⣳⣌⠻⣦⣤⣴⣾⢿⣷⠟⠉⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠛⠉⠉
⠀⠀⠀⣀⡼⢿⣆⢫⢿⡀⠀⠀⠋⢀⣰⣿⢸⡟⠢⢭⣟⣯⡭⠾⠋⠀⠙⠲⠾⠷⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣴⣾⣿⣄⡀
⣀⣤⣾⠭⠶⠾⣿⣾⡼⣷⣶⣶⠿⢿⣻⣵⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠉⠉⢹⣿⠀⠀⠈
⠁⠄⠀⠂⠀⠀⠀⠈⢛⣶⠶⣾⠿⠟⠋⠈⠻⣷⣄⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠤⠤⠴⠛⣰⠇⠀⠀⠠⠴⢲⠟⠋⠁⠀⠀⠀⠀⢻⡆
⠀⠀⢀⣠⠖⠀⠀⡠⠾⠓⠺⣿⠀⠀⠀⠀⠀⠀⠙⠻⠷⠶⠶⠴⠶⢷⣄⠀⠀⠢⠶⠶⠛⠁⠀⠀⠀⠀⣠⠏⣀⣠⣴⣶⣶⣴⣤⣤⣻⣦⡀
⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣀⡀⠀⣀⣀⣠⣴⠟⣡⣼⣿⠉⠀⠀⠀⠀⠀⠀⠉⠙⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡴⠞⠛⠛⢻⣇⣠⡴⣦⡤⡀⠀⠀⠀⠀⢀⠀⣹⣿⣿⣿⣟⠋⠉⠀⢠⡿⠁⠙⠆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠋⠁⠀⠀⠀⠀⣸⠛⠁⠀⠈⠻⣷⠀⠀⣠⡼⠿⣖⠏⠉⠉⡏⠛⠿⣶⡀⢸⡇⠀⠀⠀⠀⠀⠐⣴⣍⡲⢦⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⡼⠋⠀⠀⠙⠀⠀⠀⢰⠀⠀⠈⠹⣾⡇⠀⠀⠀⠀⠀⠀⢹⠎⡡⠄⠞⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠈⠷⠀⠀⠀⠀⠀⠀⠠⠘⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢀⡀⢀⡬
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣀
⣿⡟⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣶⣾⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⠀⢀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣠⠀⠀⢀⠆⠀⢰⡄⠤⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠧⠀⠀⠐⢢⠴⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⢀⠀⠈⣧⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠀⢀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠂⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⠀⠀⢲⠴⠋⠉⠀⠀⠀⠀⠀⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣠⢾⠁⠀⠀⠀⠀⠊⠁⢀⡠⠤⠒⠀⠀⠀⣀⡴⠁
⠀⠀⠀⢀⠁⠀⠀⠀⣀⠀⠀⠀⠀⢄⠀⠀⢸⠛⢦⣀⠀⣀⡴⠑⢄⡀⠀⠀⣀⠤⠊⠁⠸⡀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠆⠀⢀⡠⠃
⠀⠀⠀⠘⠀⠀⠠⢄⡀⠑⠄⠀⠀⠈⠆⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⠄⣀⠀⠀⠀⣀⡀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡁⠀⠀⠀⢄⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢳⡀⠀⡵⠋⠁⠀⠉⢻⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣰⣷⠀⠐⠬⣀⣀⣀⣠⡤⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢖⠤⠤⠜⠁⢰⠇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⢰⡟⣿⠀⠀⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢸⠀⠀⠀⠀⠀⣰⡇⠀⣤⡄⠀⢸⣷⡀⢸⡌⣧⡟⠀⢸⡄⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠎⠀⠀⠀⠸⡄⠀⠀⢀⣴⠟⠀⡜⠈⣿⠀⣸⠘⡇⠘⠇⠈⠀⠀⠘⠇⠀⠀⠈⠢⣄
⠀⠀⠀⠀⠰⠿⠿⠭⠭⣅⡀⠀⠀⠀⠀⠀⣇⣤⣾⡟⠁⠀⠀⣇⠀⣿⠀⠸⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⢻⣁⣼⠇⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠮⣅⡀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢱⠂⠉⠉⠉⠉⠉⠓⢷⣄
⠀⠀⠀⢀⡠⠠⠀⠀⢀⣴⡮⠕⠂⠀⢄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠈⡢⢠⠔⠤⠒⡄
⠀⠀⡠⡇⠀⠀⢁⣀⠫⠅⠀⠀⠀⠀⠚⠉⠉⢳⠀⣠⠔⠚⠛⡆⠀⡠⠚⠉⠉⠳⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠈⠺⣄⠀⠀⠚⠒⢆
⠀⠨⡀⠀⠀⠀⠉⠀⣀⡔⠀⠀⠀⠀⠀⠀⠀⣸⠞⠁⠀⠀⠀⣷⠊⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄⠀⠘⠀⠀⠀⠀⡸
⠀⠀⠠⠀⠀⠀⠈⠙⠀⢈⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠁⠌⠀⠈⠀⠀⠀⠰⡁
⠀⠀⠐⠂⠢⡄⠀⠐⠆⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⢀⠀⠀⠀⠀⣀⡠⠃
⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠦⠀⠌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠸⠀⠉⠢⠌⠐
⠀⠀⠀⠀⠠⠒⠢⠤⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡃⠀⢚⡀⠀⠀⠢⠤⢄⡀⠀⠈
⠀⠀⡐⠉⠁⠀⠀⠀⠘⠀⠀⠀⠀⠠⠤⠀⠀⡠⠒⠉⠐⢠⠀⠀⢸⠂⠘⠁⠊⠀⠑⠒⢈⢡⣀⠀⠠⠀⡸⠤⠄⠐
⠀⠀⠑⢄⠀⢀⡖⠃⠀⣠⠖⠋⠉⠙⠢⣀⡰⠧⡀⠀⠀⡀⠉⠀⠈⢢⡠⠒⠒⠢⡠⣶⠶⡄⢄⠠⠡⠊
⠀⠀⠀⠈⠉⠁⠀⠀⢠⠇⢀⠀⠀⣄⠄⠈⠀⣀⢈⠉⠉⠀⠀⢀⣤⠠⠤⡀⠀⠀⢈⠕⡂⠉⠭⡑⢴⢩⠉⠉⠉⣍⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⡃⠠⣠⠏⠒⠻⠴⠋⠀⢀⠔⡋⠭⢉⡒⠀⠑⢎⢢⡔⡡⠊⠀⠀⢠⣿⠀⠇⠀⠀⣰⠏⡜
⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠸⠁⠀⠀⡠⢒⠩⢙⠻⢣⠊⠀⠀⠀⠃⠀⠀⠈⠳⠊⠀⠀⠀⠀⢸⡟⢸⠀⠀⣰⠏⡼⠅⢀⠄
⠀⠀⠀⠀⠓⠀⢧⢀⡠⢚⠩⠭⣐⠚⡐⠁⠀⠀⠙⠇⠀⣴⠟⠆⠀⠀⢰⣄⠀⢀⣴⠂⠀⠀⣼⠇⠆⠀⢠⡟⡜⠀⠰⡁⢱
⠀⠠⢤⠀⠁⠀⠰⠡⠊⢡⣤⠀⣸⣧⠀⢰⡟⡃⢰⠀⠀⠻⠊⠀⠀⠀⠈⣏⢙⡛⢻⠀⠀⠀⣿⢠⠀⢠⡟⡘⠀⣀⠄⠙⠏
⠀⠀⣀⠁⠀⠀⠰⡰⠀⠘⠁⠀⠛⠲⠀⠈⠛⠀⠀⡄⠀⠀⠀⠀⡄⠀⠀⣿⢨⠸⠈⠀⠀⢸⣏⠸⠶⠟⣰⠟⠙⢈⣃⡤⠂
⠀⠀⢀⠁⠀⠀⠀⢇⢃⠀⢰⣟⠆⠀⢡⡀⠀⠀⣰⡿⠠⣀⣠⣾⢧⣤⣤⣿⢸⠇⢥⣄⣀⣸⣏⠀⣸⡎⡇⠀⣀⡼
⠀⠀⢉⡏⢰⠄⢄⡸⡌⡄⠀⠁⠀⢀⣼⢋⣒⠚⣋⡔⠓⠮⠥⢶⠒⠒⠒⠒⠊⠈⠢⠭⡍⠙⠬⣛⣋⠜⠁
⠀⠀⠀⢧⡀⠀⠸⠀⢣⢡⣀⣤⡶⢛⡵⠋⠈⠉⠁⠀⠀⠀⠴⠛⣰⣀⠀⠀⠀⢀⣀⠀⠲⡤⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⢨⠋⠀⠀⠈⠳⠭⠴⠚⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⢸⣴⠋⣦⠇⠈⠀⠀⠹⠀⡀⠀⢺⠉⠁
⠀⠀⠀⠀⠸⡄⠄⠒⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠘⠁⠀⠘⢄⣀⠶⡊⠀⠀⠀⢀⠴⣄
⠀⠀⠀⠀⣀⣨⠇⠔⠂⠱⢄⣀⣀⠄⠟⠦⢤⣤⠎⠒⠖⠊⠈⠀⠀⠀⢀⠈⠐⠈⠀⠀⠈⠓⠒⠚⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡀⡄⠀⠈⠲⢄⣀⣀⠤⠀⠈⠉⠉⡠⠰⠚⢀⣈⠇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⣤⣴⡄⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⠙⢿⡿⠓⣿⣿⣿⠟⣿⠀⢛⡧⠀⠀⠀⢀⡆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⣿⣿⠇⠀⠈⠀⠀⠀⣿⣿⣿⣷⡼⠃⠀⠈⠃⠀⠰⣶⣿⣿⣿⣿⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠋⠁⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣇⣠
⠀⠀⠀⠀⠀⠀⠘⢁⡠⡔⠚⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣰⣾⡁⠰⠀⠀⠀⠀⠀⠀⠁⢄⡀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⢿⣷⣾⣿⣿⣿⠿⠿⣦⠄
⠀⠀⠀⠀⠀⢼⠋⠈⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣡⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⣾⣿⣟⣽⣿⣿⣀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⣴⣧⣤⣴⠀⠀⠀⠀⠐⠲⢤⣌⣹⣷⣻⡥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⡿⣀⠀⣶⡄⢁⡂
⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣦⡈⠙⠛⠿⣿⣿⣯⣿⡿⣟⣀⠦⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⠁⠈⣼⠻⠎⠙⠓
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠛⠁⣉⣙⣿⣿⣿⣿⣿⡟⣣⣀⣠⠀⡀⠀⠂⠀⠀⣀⣀⣀⠀⠘⠿⠋⠀⠘⣿⠀⠃
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣶⡟⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⠯⠔⢁⠐⢿⣿⠇
⠀⠀⠀⠀⠀⠻⠿⠛⠻⣿⣿⣿⣿⣧⠉⢁⣴⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⡧⠀⢠⡀⠄⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠉⠟⠋⠁⢀⡾⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠳⠀⠀⣀⡠⢤⣶⠀⠀⠀⣿⣿⡧
⠀⠀⠀⠀⠀⢀⠀⠀⣠⣿⣿⡏⠂⠀⠀⠀⠀⠈⠀⠞⠡⠟⣿⡿⢿⣿⡿⡟⢏⠚⣆⠀⠀⠛⠁⡀⢫⡤⠰⠀⠈⠉
⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡿⢿⡿⠀⠀⠀⠀⠀⣴⣦⣤⠀⣿⠁⠸⡏⠃⠀⣨⣄⡈⢶⣤⣠⣴⣏⡀⠐⣲⢶⡀
⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⠗⠀⢠⣤⡄⠀⠀⠀⠘⣿⠛⠀⠁⠀⠀⠁⠀⣾⣿⣿⣶⣾⣿⣿⣿⣧⣥⣾⣥⣿⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢉⣩⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠲⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣹⢿⣿⣿⣿⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡄⠀⠀⠙⠿⠟⣋⡛⠻⣿⡿⠟⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⠃⠀⠀⠀⠙⠁⠿⠿⠟

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣋⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠙⣧⠀⠀⠀⠀⠀⠰⠷⠀⠀⣴⡞⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠘⣇⠀⢠⡀⠀⠀⠀⠀⠀⠘⡇⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡈⠀⠘⣿⣷⣄⠀⠀⠀⠀⠀⡘⢁⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⣶⠦⢀⠀⠀⠀⢿⡇⠉⠓⠄⠀⣴⡄⢰⣿⠁⠹⣷⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⠀⠉⠛⠶⢾⡇⠀⠀⠀⠈⠿⠹⣿⠃⠀⠀⠹⣿⣐⣦⠄⠘⢹⡇⣴⠂⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣬⣿⣗⠻⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⢀⡿⠁⠁⠀⣠⠾⠋
⠀⠀⠀⠘⠓⠶⠶⠶⠖⠀⠀⠈⠽⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⣶⠂⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠟⠋⠀⠰⣶⠄
⠀⠀⠀⠚⠛⠳⠶⢶⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣀⣀⣠⠤⢤⠀⣤⡤⠄
⠀⠀⠀⠀⢠⠄⠀⠀⠀⣀⠤⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠊
⠀⠀⠀⠀⠀⢀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣖⡛⠛⠁
⠀⠀⠠⠾⠿⠿⠶⠖⠒⠒⠒⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠙⠛⠷⠶⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣮⣥⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁⠀⠀⣠⣾⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠙⠿⣭⣉⠉⠀⠒⠚⠓⠛⠆
⠀⠀⠀⠀⢀⠶⠞⠋⠁⠀⠀⠀⠉⢰⡏⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡌⣿⣭⠍⠵⠻⠿⠛⠂
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⡀⠴⢛⡋⢿⡀⠀⠀⣠⣶⡀⣤⣄⠀⠀⠀⢀⡤⣤⣄⡀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⢀⣴⠆⠀⠈⢧⠀⢰⣿⠃⠘⠏⠉⠳⢦⡀⢸⣇⠀⠉⠛⠶⠠⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡿⣡⠆⣾⠀⠈⢷⣾⠏⠀⠀⠀⠀⣠⢤⡙⢾⣿⠄⢦⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⠋⠀⡿⠀⢀⡀⠁⠀⠀⠀⠀⠀⠳⠤⠏⠀⠈⠀⠘⣧⠙⠀⠀⢴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⢸⡇⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⢀⡀⠀⠀⠀⡤⡀⠀⡾⠈⢦⠀⣠⡄⣠⠞⢹⠀⠀⠀⠀⠀⠪⠚
⠀⠀⠀⠀⠀⠈⠑⠦⣉⠀⠀⢿⡑⠤⡀⡇⠈⠲⠃⠀⠈⠋⠀⠛⠁⠀⠾⠐⠈⡇⠀⣀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠳⡄⠈⠁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣒⡉⠉⠉⠀⢸⡇⣻⠀⢠⣿⠀⣶⡀⢠⡄⢠⣤⡀⢀⡀⠉⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠈⠙⡷⠀⠀⢸⠿⣇⢠⣏⣽⠀⡟⣧⣼⢀⡏⠈⠁⣸⠁⢠⡶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⣮⣄⡀⠀⠘⠒⠋⠞⠀⠹⠀⠇⠸⡏⢸⡇⣶⠀⡋⠀⣀⣱⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⢠⠃⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⣮⠖⠋⢹⢀⡴⣆⠀⣠⣆⠀⢠⣄⠀⡠⣤⡀⢻⠛⠒⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠸⠋⠀⠘⠞⠁⢹⣠⠃⠈⠛⠀⠈⠻⣮⡆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠈⠁

⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣆⠀⠀⠀⠀⠀⠀⠀⣀⣶⣶⣶⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⢸⣿⣿⣿⡏⠻⡿⠟⢿⡿⣿⣿⠀⡀⠀⠀⠀⠚⠿⠿⠿⢿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⣿
⢸⣿⡧⠀⠀⠁⠀⠙⠂⠀⢿⣿⣇⠀⠀⢴⡄⣤⢀⣀⣀⠀⡀⠀⠉⠁⠀⠀⠢⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠸
⢸⣿⣆⡐⠀⠀⠀⠀⠀⠈⢸⣿⣿⠀⠀⢘⡿⠃⣼⣯⡅⢾⡙⠇⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠄⠀⢀⣤⢽⣷
⠘⠋⠁⠁⠐⠷⠀⡤⠀⢿⠐⠆⠹⠀⡀⠘⠃⠀⢿⣤⠰⣆⣹⡆⠀⠀⢀⠄⡴⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⣄⣴⡆⣿⢸⣿⠰
⠀⠀⠸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⣬⣑⣒⠒⠬⠍⠠⠤⠔⢁⡞⠁⠀⠀⠀⢠⣶⣾⣦⢀⣤⣄⣴⢷⣾⡏⢿⠈⣷⠿⡨⠛⠀⠁
⢠⣤⣄⣠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡿⢿⠟⠿⠟⢿⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣷⣾⣧⡿⠻⠞⠋⢁⠀⣠⣴⠖⠁
⢸⣿⣿⣿⣿⣶⣀⠀⠉⠁⠀⠸⣶⣶⠒⠶⠀⠀⠀⠀⠀⠀⢈⣂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠿⠿⠻⡀⣁⠀⣀⣀⣴⣾⠿⠟⠁
⠸⠟⠋⠁⠝⠛⠻⡇⢤⣀⠀⠀⠀⠀⠀⢀⠀⠆⢠⣴⣶⣶⠏⠉⠓⠞⠉⠈⠓⠂⠂⡀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠉
⢰⣼⣦⡄⠀⠀⠐⠁⠀⠀⠀⠈⠉⢸⠒⠚⠷⢿⣿⣿⣫⢟⡟⠉⡡⠂⠀⠀⠄⡴⠌⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠬⠄⠄
⢸⣿⣧⡄⢠⠐⣄⣄⠠⠀⠀⠀⠀⠀⡥⠂⠀⣸⣿⣿⡉⠸⢆⣤⣤⣬⣥⡀⣀⡀⣀⡀⢤⣀⠂⢀⡀⠀⠀⠲⠀⢀⠀⠈
⢸⣿⣿⠇⣤⣇⠹⠋⠓⠢⢤⠠⠂⠀⠵⠄⠀⡿⠛⠛⡿⣶⠒⠶⢾⣉⣁⣉⡉⠜⠛⡱⠋⠀⠀⠀⠀⠠⠀⣤⠀⢠⠀⢆⢀⢀⡀⠠⠄
⢸⣿⣿⣌⣛⣿⣀⠀⠀⠀⠀⠐⠈⣰⣖⣬⢠⠀⠀⠀⢹⡽⣏⣀⣤⠶⢤⡀⡤⠶⠀⢹⠂⠀⠀⢀⣀⣀⣀⠐⣚⠓⠒⠨⠤⠴⣡⣇⣾⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣷⡶⣤⡶⠦⠾⠛⠛⢋⢘⣒⣂⣒⣉⣩⣭⣭⣤⣤⣤⣽⣶⣶⣷⣾⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣴⣶⣶⣾⣿⣿⣿⣿
⢈⣉⣉⣩⣭⣥⣤⣤⣶⣶⣶⣶⣿⣧⣼⠇⠿⢿⣿⠿⠿⢿⣿⣿⠟⢛⠛⢿⠿⠛⠛⠋⠉⠉⠉⣉⡀⠀⠑⠙⠛⠻⡿⠛⢿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⡿⠿⠿⠿⠿⠛⠻⠟⠛⠛⠻⠃⠀⠀⠀⠀⠀⠀⠀⢻⠃⣼⡏⢻⣦⠀⠀⠀⠀⠀⠀⠀⣿⡇⣀⠀⠄⣶⡆⢱⡷⡘⠉⠛⢿⣿⣿⣿
⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⡇⢸⡿⢸⡇⣿⡇⣼⡏⠀⣿⡇⣿⡇⠀⠿⠇⠸⠇⡆⠀⠀⠀⢻⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣹⡦⠀⠀⠀⠀⠀⠀⣠⣼⣧⣈⠛⢛⣡⡈⠓⠛⢁⠘⠓⠀⠛⠃⠛⠃⢀⠚⢂⠙⢀⠇⠀⠀⣠⣾⣿⣿
⢸⣶⣴⠀⣴⣶⣶⣿⣿⣿⣿⠿⠟⠿⠇⣤⣴⣦⣴⣦⣼⣯⣿⣿⣿⠿⢷⣤⣌⣙⡋⠁⠙⠒⠚⠉⠲⠶⠖⠋⠉⠉⣙⣉⣤⣴⣾⣿⣿⣿⣿
⠘⠛⠃⠘⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠶⠊⠀⠹⠿⠿⢿⡿⢷⡀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣾⠶⣶⣆⣤⢠⣤⡶⠶⣦⠄⣤⢀⡘⣿⣿⣤⠀⠀⠘⣀⠀⠀⠀⠀⠈⠨⠒⠀⢳⠋⠂⢺⣷⠀⠂⢀⠲
⠀⠀⠀⠀⣸⣿⣤⣾⡟⠷⠾⠙⠷⠶⠟⣻⢷⣟⡴⢻⣿⡿⠂⠀⠘⠚⠷⢤⡄⣤⣠⣤⠀⠀⠀⢦⡀⠀⢻⣦⣴⣿⣀⣠⡾⠃⢀⡀
⠀⠀⠀⠸⣧⠈⠉⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠈⢁⣼⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠈⠻⠶⡾⣿⡟⢻⡟⢻⠖⠖⠈
⠀⠀⠀⠰⢿⣿⣦⡀⢀⣠⣤⣆⠁⠠⣦⣤⣀⣤⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣀⣳⣴⣷⣼⣷⣶⣷⣾⣶⠶⠾⠆
⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⠿⠟⢳⣶⣮⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢠⣿⠉⠿⠉⢹⡏⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠿⠿⠇⠀⠀⠀⠀⠨⠿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠃⠀⠈⠋

⠀⠀⠀⠲⣄⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔
⠀⠀⠀⠀⠈⢳⡈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⡠⠚⠁
⠀⠀⠀⠀⢤⡀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠜⢠⠎⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠈⠀⠰⣄⠀⢳⣄⠀⠀⠹⡄⢠⡀⠀⠀⢸⡇⠀⡄⠀⠀⠀⠀⠀⢀⣴⠃⠀⢀⠔⣁⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠉⠈⠳⢤⣀⡙⡀⣇⣀⡤⠞⠓⠚⠉⠢⢤⣤⡤⠖⠁⡾
⠀⠀⠀⠀⠀⠀⠀⠦⡤⠚⠀⠀⠀⠀⠀⠉⠙⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⢴⠖
⠀⠀⠀⠀⡀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⣇⢸⡇⢂⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠉⠓⢶⣖⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⠇⠀⠉⠉⢀⡾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠨⣳⣤⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⣸⡇⠈⡇⡿⢸⠀⠀⠀⠀⢏
⠀⠀⠀⠀⠀⠀⠈⠉⠉⡿⠀⠀⠀⡏⣸⡀⢀⡾⢷⡄⢠⣾⣆⣧⢻⡟⡇⠀⡇⡆⢸⠀⠀⠀⠀⢈⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠶⠞⣁⠀⠀⠀⣿⠟⣷⢸⡁⠀⢿⣾⠀⣿⢻⠈⠀⣿⠀⠇⠃⠘⠀⠀⣀⣴⣁
⠀⠀⠀⠀⠀⠀⠀⣀⣠⡜⠀⠀⠀⣧⣤⡟⠘⣧⣀⡿⠘⠿⠋⠈⠀⠀⠀⠀⢀⡄⢸⠀⠀⠒⣧⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⠀⠀⢿⠋⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠁⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡘⢦
⠀⠀⠀⠀⠀⣠⡴⠟⠓⠊⠒⡄⠀⢀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⣀⠀⠀⠀⣀⣀⣀⣸⠁⠀⠉⠙⢷⣄
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣰⠗⢚⢏⡾⢄⠀⢠⠞⠉⠀⠀⠈⢿⠁⠈⠳⡄⡞⠀⠀⠀⠙⠂⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠈⢧⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⢄⠢⣈⠲⢤⡀
⠀⠀⢀⡴⢦⠀⠀⡠⠂⡰⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⢦⠈⠳⡄⠁
⠀⠀⣯⠀⠈⢽⡋⠀⣜⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠘⢦
⠀⢀⡗⠀⢀⠀⠑⠀⠉⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠳
⠀⠀⠉⠻⣅⣠⣄⣀⣄⡠⠃

⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣠⠴⠒⠚⠉⠉⠉⠙⠢⡄⠀⠀⢀⡴⠒⠚⠓⠲⢤⡀⠀⢀⡠⠴⠖⠦⣄⢀⡤⠤⣄⣀⠀⠀⠀⠀⠀⢠⡞⠓⠒⠦⠤⣄
⢸⡅⠀⠀⠠⠤⡄⠀⠀⠀⢹⠀⡴⠁⠀⠀⠀⠀⠀⠀⠙⣴⠋⠀⠀⠀⠀⠈⢻⠀⠀⠀⠘⣧⠋⠙⠒⣶⠃⡇⠀⠀⠀⠀⡼
⡎⠁⠀⠀⠀⠉⠀⠀⠀⣠⠏⢰⠃⠀⠀⢀⠤⡀⠀⠀⠀⡟⠀⠀⣀⡀⠀⠀⠀⠃⠀⠀⠀⠁⠀⠀⠀⢨⠀⡇⠀⠀⠀⢰⠃
⡇⠸⡀⠀⠀⠀⠀⠀⠉⠉⠉⠛⢇⠀⠀⠀⠉⠁⠀⠀⠀⡇⠀⠀⠑⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⡏
⢹⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⢀⡠⣺⠳⢄⡀⠀⠀⢀⡠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⣸
⠀⣇⢸⠀⠀⠀⠰⣉⠆⠀⠀⠀⠀⢸⣗⠒⠒⠈⣁⡴⠳⣄⣀⣀⣠⡼⠏⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠧⠤⣠⠇
⠀⢸⡈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠉⠉⠉⠁⠀⠀⠀⠀⠀⡼⡸⠀⠀⠀⡰⠁⢣⡔⡺⠇⠀⠀⠀⠸⠒⠚⢣
⠀⠀⣇⢠⠀⠀⠀⠀⠀⠀⠀⡀⠎⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢇⣀⡀⡰⠳⣀⣎⣴⠁⡄⠀⠀⠀⢰⣀⣀⡜
⠀⠀⢸⠈⢆⣀⣀⡠⠤⠔⠊⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⣀⣸⠃⠀⠈⠁⢸⡀⠱⠤⠤⠤⣼⠴⠋
⠀⠀⠈⠣⣄⣀⣀⣠⠤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠲⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠲⠀⡔⠐⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢉⠀⠼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡔⠁⠀⠀⠀⠀⠀⣤⡀⠀⠠⢢⠀⠀⠀⡀⠀⠀⣠⣟⣷⡠⠖⡇⠀⠀⠀⠀⠀⡠⠀⠀⠃⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⠄⠀⠀⢄⢢⠀⢀⣟⣷⡠⠁⠀⢳⣤⠞⠈⢢⣼⣣⣗⣿⠀⠀⡇⠀⠀⢀⠔⡨⠐⠁⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠳⣾⡟⣿⣷⡀⣠⢿⣿⠣⣠⢟⡿⠿⣽⣻⣇⡤⣿⣾⡿⠖⣻⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⠓⠤⣰⣮⡑⣾⣿⣾⣦⣿⢯⣶⣷⢶⣷⣿⣤⣀⣰⣷⣵⣾⡿⣿⣀⣡⣼⡴⣶⣿⠁⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢓⠾⣄⡀⢸⡟⢟⠟⣻⡿⣿⡿⠋⠉⠉⠻⣿⠟⠁⠈⢿⠿⢿⡿⠟⠛⢿⣿⣿⣿⣿⠋⠉⠁⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣾⣿⣿⣿⣿⠟⠛⠛⠃⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢹⢿⣁⣀⡀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⣿⣛⣛⣻⡧⠀⣀⣀⣀⠀⢀⣤⣤⣤⣄⣠⣶⣶⣶⣄⣼⣿⣦⣾⣿⡄⠀⢠⣿⣚⣛⣋⣛⢛⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡟⠁⠐⣿⣟⣿⣿⡆⣾⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠈⢿⣿⣿⣿⣋⡱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠚⢻⣿⣿⣿⡁⠀⠀⣿⣿⣿⣿⣷⣿⣿⣇⣸⣗⣻⣻⡄⣿⣿⣿⣿⠻⡟⣿⣿⣧⠀⠀⣼⣿⣿⠖⢻⣿⣿⣶⣦⠤
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣷⣶⣻⡽⠯⣽⣷⣤⠂⢸⣿⣉⣽⣿⣿⣿⣿⣿⣿⠛⠿⣿⣿⠿⠿⠏⠀⠀⠸⠟⠛⠀⢾⢻⣏⣿⣠⡶⠿⣟⠉
⠀⠀⠀⠀⠀⠈⠛⠿⣶⣶⢾⣧⣄⣠⣿⣾⠃⠀⠘⠟⠛⠛⠋⠁⠀⠉⠁⡀⠀⠀⠀⡀⠀⠀⢠⣤⣴⠀⠀⠀⢀⣾⣶⣾⣶⣾⣷⣄⣈⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡻⠿⣯⣟⣟⢿⡀⠀⠀⡀⠀⠀⣄⠀⠀⠀⢀⣧⡀⠀⣰⣷⣦⣴⣞⣀⣛⣞⣛⣶⣛⣛⣿⣛⣛⡻⣛⣻⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡄⣊⣀⣀⣠⣿⣿⣿⣿⣿⣿⣷⣄⣰⣿⣷⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠁⢿⣿⣿⣿⣿⡙⠻⡿⢮⣷⡀
⠀⠀⠀⠀⣀⣀⠤⠌⠀⠀⠠⣲⣟⣿⣯⠵⢿⣟⣟⣛⣟⣺⣿⣟⣛⣯⠟⠛⢳⣟⣟⣿⣝⣿⣻⣿⣯⣻⣟⠺⡿⢮⡛⢿⣿⣷⣤⠧⠀⠀⠁
⠀⠀⢠⠎⠀⠈⠀⠀⢀⠄⡾⠛⠋⢁⣇⠔⠺⡟⢻⣿⣿⣿⣿⣿⣿⡀⠀⢠⣾⣿⢿⡿⣿⢿⣿⣿⢿⠏⠙⠷⠇⠰⠉⠲⣼⣻⡦⡢⠀⠀⢀
⠀⠀⠈⢳⠆⠠⡀⠀⠐⠈⠀⠀⠀⠈⠁⠀⠀⠓⢻⣗⣴⠟⠁⢻⣶⡿⡶⣿⣶⠇⡸⠛⠛⠼⣷⢦⡟⢆⢸⠀⠀⠀⠀⠀⠀⠙⠷⠈⠀⠀⢸⡀
⠀⠀⠀⢸⠀⠰⡅⠀⡀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠸⠋⠹⡀⠔⠉⢿⣻⣟⡿⠁⠈⠃⠀⠀⠀⢻⡟⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⡗
⠀⠀⠀⠀⠉⠳⡈⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⣷⡾⠁⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢀⠖⠊
⠀⠀⠀⠀⠀⠀⠈⠑⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⢀⠀⠀⠀⠀⢀⡰
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢠⡉⢄⠀⠀⠀⣼⠁⠀⠀⡤⡄⡄⠀⠀⢀⡼⠆
⠋⢀⡀⠀⠀⣀⡐⡄⠀⠀⢐⠀⠀⠀⠀⢹⢠⠀⠀⠸⡇⠀⠀⠀⠀⡗⡄⠀⠀⣸⠍⠀⠀⠀⠀⠀⠀⠀⣤⠊
⠘⠂⡀⠀⠀⠀⠝⠁⠠⠃⠰⠀⠀⠀⠀⠨⡶⠀⠀⠀⠁⠀⠀⠀⢀⡇⠀⠀⠀⠥⠁⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠐⠞⠁⠀⠀⣀⠁⠀⠀⢀⠀⠀⠀⠁
⠀⠀⠀⠄⠀⠀⠐⣀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠛⠀⠀⠀⠀⡀⠀⠀⠀⡌
⠀⠀⠀⠠⠃⠀⠀⠁⠹⠀⠀⠀⠘⢄⠀⠀⠀⣰⠁⠘⠄⠀⢀⡀⠀⠀⠒⠛⠀⠀⠀⠚⠒⠀⠀⢀⣾⠃⠀⠀⡘
⠀⠀⠀⠀⠎⠀⠀⠀⠀⢀⡠⡔⠀⠀⠈⠉⠈⠁⠀⠀⠈⠒⠚⠛⠛⢻⡋⠙⠓⠲⢻⢛⠁⠀⢠⠛⠋⠀⠀⠈⠁
⠀⠀⠀⠐⡐⠄⠀⢂⠀⠀⠪⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠲⣄⠁⢂⠖⢄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⠛⠳⢦⣀⣠⣤⠤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠛⠳⠾⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡽⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⢸⠉⢹⡆⢀⡾⠛⣧⠀⢠⡾⠻⣆⠀⣰⡆⠀⣰⡄⠀⣤⠀⠀⠺⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣾⣀⡼⠃⢸⡇⠀⢻⡀⣿⠁⠀⣿⠀⣿⣿⢠⣿⡇⠀⣿⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⣿⠋⠻⡆⣿⡁⠀⢸⠀⣿⠀⠀⣿⢰⡇⢹⣿⢹⡇⠀⣿⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⣿⠀⢀⡇⢸⡆⢀⡿⠀⣿⡀⢀⡟⣸⡇⠀⠁⢸⡇⠀⠛⠀⠀⠀⢀⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠙⠛⠋⠁⠈⠛⠛⠀⠀⠈⠳⠞⠁⠛⠀⠀⠀⠘⠇⠀⠖⠀⠀⢀⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⣠⠤⠚⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⡴⣄⠀⠀⠀⠀⠀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠳⣤⣀⣀⣀⡤⠞⠁⠈⠛⠶⠶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⣍⠁⢠⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢿⠀⢸⡇⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢸⡀⢸⡇⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠸⡇⢸⡇⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⠴⠃⠀⠀⣿⠀⠀⠀⠀⠘⠦⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⣠⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡉⠉⠉⡓⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡆⠀⠘⣧⣠⡤⢤⣀⠀⢀⡠⠤⣄⡀⠀⡀⠀⠊⠛⠒⠚⠁⠀⠀⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣁⣤⡶⢄⣀⣀⣠⣌⡉⠉⠀⢀⡈⠉⠉⠀⢀⡶⣄⣀⣀⣠⠞⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠉⠀⠀⠛⠶⠖⠉⠻⠶⠤⠶⠋⠀⠈⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠋⠀⡰⠣⡀⡐⢾⣿⡿⢾⡿⠿⠀⠀⠀⠀⠴⡃⠀⠀⠀⠀⢰⣇⠀⠀⠸⣿⠿⠃⠀⠀⠀⠠⢎⠀⠀⠀⢀
⣦⡼⠀⢸⢆⠈⠀⠉⢙⡀⠀⠀⠀⠀⢀⠀⠐⠛⠃⠀⢸⣿⡛⣻⣶⣦⡃⠀⠀⠀⠀⠀⡀⠀⠚⠚⠀⠀⠛⠂⠀⣀⣤
⣯⣽⠬⡀⠐⠼⢠⠈⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠱⢰⣿⣿⣿⣿⣿⣿⣁⢀⠄⢀⡄⠴⠋⠀⣀⣀⠀⠀⢀⠀⠀⠙⠙
⡿⠛⠊⣨⣬⠇⢂⢀⣠⣴⣤⣈⡀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣷⣿⣿⠋⣰⣶⣦⣤⣥⣀⣀⠀⠀⠈⠂⡠⠀⠀⠠⠀⠀⠀⠦
⠀⠀⠀⠈⠁⣿⣷⣿⣿⣿⣿⣿⣿⣆⠀⢈⣀⡄⣀⠀⢻⣿⣿⣿⣿⣀⠋⠀⠉⠙⠛⠿⣿⢿⣦⡀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⡀
⡷⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⠯⢐⠇⠆⠸⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀⠀⠀⠀⠈⠉⠃⢰⠃⠀⣶⣶⡎⠀⠀⣰⣿⣿⣿
⠁⠦⠐⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣠⣼⠖⠀⣠⣿⢻⡿⢿⣤⣀⣠⠙⠛⠛⢿⣷⢶⣤⣀⣀⣜⠀⢂⠙⣭⣥⡌⢀⡻⢿⣿⡷
⠐⠀⠁⠀⣙⣻⣿⣿⣿⠟⢿⠛⠻⠿⠿⠿⠻⢟⣸⣾⣿⣿⡿⠀⠠⠀⠋⠹⣄⠀⠀⠀⡀⠀⢹⢿⠟⠙⠃⠀⢢⡉⢿⣡⣸⣿⡯⠄
⠀⠲⠀⣾⣿⠟⠋⠿⡿⠆⠘⠃⢈⣥⢶⣤⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠉⠛⠛⠀⠀⠉⠓⠺⠋⢠⣿⠍⠂⣬⣿⣶⣿⣯⡥⠄⠀⢀
⡟⠠⡆⠀⠁⠀⠀⠀⢀⣤⡤⠤⣾⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠈⠀⢠⣤⣿⢙⣴⣿⣿⢟⢵⡟⣿⣿⡔⠂
⠀⢀⠠⠄⠰⠄⢲⣦⣿⡇⠈⢀⢁⣸⠋⠙⢈⡙⠻⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣶⣤⣠⣾⣿⣧⣛⡛⠉⠐⠋⣼⣍⡈⢿⣿⡄⣻
⣷⠀⠖⢀⢀⡘⢾⣿⣿⡿⠶⠋⠂⠰⠦⠀⠀⠁⠀⢈⣿⣿⣄⠈⠀⠀⠀⠙⠟⠃⣿⡏⢹⣿⣿⣿⣿⡏⠃⠈⢃⣼⣿⣿⡿⡀⣙⡓
⠀⠀⠀⣾⡆⠀⢨⢙⣏⣴⣿⣿⠉⠁⣀⡀⠀⣀⠀⣸⣿⣿⣿⣿⣿⣾⣶⣤⣀⣰⣿⣿⣿⣿⣿⣿⣿⣷⠂⢀⡾⡁⠉⢽⣶⣿⣤⣺⣿
⠀⠀⠀⢸⣷⠀⠀⠁⠀⣿⡏⠀⣀⣄⠙⠀⠤⠋⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⠋⢀⣅⠈⠢⠙⠓⠄⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠈⢳⡄⢀⣀⣤⠴⠖⠛⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⠉⠉⠉⠛⠓⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢀⣠⣴⣾⣟⣽⡿⣾⣿⣿⣷⡀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⢀⣴⢾⣻⣭⣾⣿⠿⣿⣿⣾⣻⣿⣿⣷⣄⠀⠀⠀⠀⠻⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣾⣵⣿⣿⣿⣽⣿⢟⣻⣿⣿⣽⣮⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣧⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣾⣿⣿⣿⣿⣿⢹⣯⡻⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢯⣿⣿⣿⣿⣿⣻⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⣿⣿⣿⣿⡋⣲⣿⣽⣿⣿⣼⣿⣿⣿⣿⣿⡿⠁⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⣟⢿⣽⠿⣯⣼⡽⣿⣿⣿⣿⣿⣿⣿⡿⣟⣽⠟⠁⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⢽⣿⣿⣿⡓⡉⢞⣾⣿⣿⣿⣿⣿⣿⣿⣿⢗⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡱⣟⢿⢿⠿⣿⣫⣼⣿⣿⢿⣯⣛⣛⣛⣿⣯⡿⠛⠁⠉⠛⠶⢤⣤⣄⣀⣀⣀⣀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⡃⢊⣵⢟⣿⣿⣿⣿⢟⣽⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡳⣿⣿⣿⣿⡷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣷⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠛⠒⠦⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠀⠢⣌⢦⣠⣈⣦⡶⠚⠛⣿⣿⣤⣶⡞⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠷⠖⣀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠰⣿⠇⠀⠀⠀⢄⣠⡿⠛⢛⣋⣩⣄⡀⠀⠈⠉⠘⠿⢷⣾⣯⣶⣋⠀⠀⠀⠀⠐⢾⣭⣀⠀⢠⠛⣦⣤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠤⣤⣦⣤⡴⠛⠋⠉⠉⠀⠀⠀⠙⠢⣄⣠⠴⠚⠉⠉⠁⠉⠙⠒⠦⣀⠀⢰⡆⠀⠙⣦⡏⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣄⣄⠀⠀⠈⠀⠀⠛
⠀⠀⠀⢀⡀⠀⠀⣀⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⣀
⠀⠀⠀⠀⠁⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠁
⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣀⡀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⢀⡀
⠀⠀⠈⠛⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠃⠘⠋
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠻⣷⣦⣀⡀
⠀⠀⠀⢰⣶⡆⠈⠳⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⣤⣿⣯⣍⠁
⠀⠀⠀⠀⠉⠀⠀⠀⠈⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠰⣿⠦⣄⠀⠐
⠀⠀⠀⠀⣀⡀⢠⠟⢶⡀⠘⠳⢶⣤⣀⣀⣤⣤⣶⡾⢿⣿⣧⣀⣀⣠⣀⣀⠀⣀⣴⡿⠟⢋⣁⣤⣿⣗⠂
⠀⠀⢸⠉⠁⠙⢋⣠⠤⠟⠂⠀⠀⠀⠉⡟⠋⠉⠀⠐⠩⠋⠟⡿⣻⣽⣽⣿⢿⠿⣿⡿⠦⣿⡏⠙⠳⡌⠑⠂
⠀⠀⢸⣀⡀⠀⢈⣿⡆⡀⠘⡏⢉⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠟⠉⠃⠀⠀⠈⠁⠀⠙⢷⠀⢀
⠀⠀⠀⠀⠙⠓⠛⠁⠀⢠⡞⠛⠚⢳⡆⠀⠀⠀⢠⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢴⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣤⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢹⡇⠀⠀⠀⠀⠀⠀⠀⠼⣿⡷⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⣴⠈⠀⢀⡾⠁⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠳⣄⠀⠀⢸⣦⡀⠀⠀⢀⡾⠁⠀⠀⣿⠀⠀⠀⣰⣆⠀⡸⠃⠀⡰⠋⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠙⢷⡄⢸⡇⠙⢦⣀⡾⠁⠀⠀⠀⢸⡇⣠⠞⠁⣿⡰⠁⠀⠈⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠙⢾⡇⠀⠀⠙⠃⠀⠀⠀⠀⠸⡟⠁⠀⠀⢻⡃⠀⣀⣚⣁⣀⣀⣀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣤⣀⣀⢹⣆⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠉⠉⠉⢉⣡⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠲⠶⠶⠤⣽⣦⡉⠉⠙⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠁
⠀⠀⠀⢠⣴⣆⠀⠀⠶⠶⠶⠖⠒⠒⠒⠊⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣟⢛⢐
⠀⠀⠀⠈⠙⠁⠀⠀⢠⣀⡄⢀⣠⡤⠴⢖⣶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⠷⠈⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⢶⠾⠻⠋⠍⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣡⣤⣤⠶⠶⠶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⢿⣿⢽⢨⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠁⠀⠀⠀⠀⠀⣿⠀⠀⢀⣴⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣤⣀⠀⢻⣞⢐⢐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣴⣿⣿⣿⡆⠀⣄⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣶⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠏⣽⠭⢽⣿⣼⣿⣆⠀⠀⢠⣿⣿⣿⣿⡯⡯⠿⠽⠬⠹⠣⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⣡⡇⠉⢸⣿⣿⣿⣿⣆⠀⣸⣿⣻⣿⣿⡏⡅⡁⠉⠈⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡟⠀⢀⣘⣘⢚⡂⡓⣿⣆⣿⣛⢚⣘⡒⡂⡂⡂⣒⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠸⠃⠀⢼⣿⣿⠀⠀⠂⠆⢿⡿⠰⠰⠐⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠉⠀⠀⠀⠀⠅⠍⠍⠨⠨⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⠂



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

⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉⠉⠉⠈⠉⠁⠉⠉⠁⠀⠈⠀⠉⠉⢻⠟⣴⣿⠟⠛⠿⠿⠿⡏⠉⠍⢹⠃⣽⣿⣿⠻⠿⣼⣷⢀⣼⠿⢻⡀
⠀⠀⠀⣄⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣰⠾⢿⣿⣷⣦⡠⢀⢀⠺⣿⣿⠀⠀⠀⠀⠀⣇⣘⣂⣼⣧⣍⣻⣿⣷⣆⢻⠟⣿⡟⠋⠛
⠀⠀⠀⡅⠀⡀⢸⠀⠀⠀⠀⠀⢸⣤⣯⠁⠀⠀⠀⠀⢹⡗⢺⢸⣷⡜⣿⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢹⣿⡿⠳⣿⡿⠛⠿⣇⠀⣰⡁
⠀⠀⠀⣻⣇⡤⡼⠄⠀⠀⠀⠀⠸⠤⠏⠀⢤⠀⠀⣀⠘⠃⠛⣸⣯⡸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠈⠹⠀⠀⠀⡀⠈
⠀⠀⠀⡃⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢠⠄⠀⢹⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢡⣆
⠀⠀⠀⡇⠀⠀⠀⠐⠶⠬⠤⠀⠀⠀⠀⢁⠀⠀⠀⣠⡅⠀⠀⢈⣿⣿⣿⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⣴⡟⠁
⠀⠀⠀⡗⠀⠀⠉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠋⣠⠀⠀⠛⠻⣟⣿⣠⣆⣤⣤⣤⣾⣿⣧⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠃⠀⠑⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⡃⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⢈⡻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣄⠀⠀⠁
⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⣾⠿⣶⣺⣶⢰⣾⡖⠐⣖⠆⣤⣤⠀⠀⠀⠀⣒⡖⠀⠀⠀⠀⠂⠐⠀⠐⠻⣿⣿
⠀⠀⠀⡶⠷⠒⣒⣠⣤⡤⢀⣀⣀⡀⢀⣿⠿⠞⠁⢹⡇⣿⣿⢸⣿⡗⢰⡊⠄⠈⢛⣀⠀⠀⠀⡯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃
⠀⠀⠀⠑⣄⡠⠀⠀⠐⠉⠀⠌⠉⠕⠸⠋⠀⠀⢠⢸⡇⣿⣿⢸⣿⡇⠙⢥⠀⠀⠀⡤⠇⠀⠀⡾⠧
⠀⠀⠀⠀⠨⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣤⡄⠀⠀⠀⠀⠀⠀⢤⠀⠈⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡟⠛⢢⣀⠀⠀⠀⠀⠀⠀⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠠⠤⠀⠀⠀⢀⠤⠄⠀⠀⠀⠀⠀⠸⠓⠢⣀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⣀⣀⣀⣀⢀⡀⣀⣀⣉⣉⣈⣉⣉⡉⣉⣉⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠈⢃
⠀⠀⠀⣿⢻⣿⡇⢸⣀⣽⣹⣏⣿⣍⣿⣹⣯⣿⢹⡏⠀⠀⠀⠀⠀⠀⠠⠿⠋⠁⠀⠠⠏⠀⠀⠀⠀⢠⣤⣾⣶⣾⡿⠃⣨⣤⣌⡉
⠀⠀⠀⣿⢸⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⣯⣿⣿⠈⡇⣤⡀⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡭⠀⢸⡋⣻⣡⣶⡟⣛⠾⠗
⠀⠀⠀⣿⢸⣿⡇⢸⠀⠀⢠⣤⠀⠀⠀⣥⣿⣿⢠⡇⢀⠙⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⡇⠀⠈⣹⣿⡏⠭⠋⠹⣷⡆
⠀⠀⠀⡿⢴⢿⡧⢼⠃⠀⣾⢿⠇⠀⠰⣿⣿⣿⢠⡇⢈⡇⠀⠀⠂⠀⠀⠀⠀⠀⢤⣾⣇⡀⠀⠀⣿⣿⠀⢾⣿⣿⡀⠀⠀⣴⣤⣀
⠀⠀⠀⣷⢾⣿⡷⢾⠀⠈⠁⠛⠃⠀⠰⣿⣿⣿⠄⠁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠇⠀⠀⠉⠉⠀⠀⠘⠛⠁⠀⠀⠀⣀⣠
⠀⠀⠀⣿⢸⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠈⡇⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⢿⣿
⠀⠀⠀⣿⢸⣿⡇⢸⠀⠀⢀⣤⡄⠀⠀⣿⣿⣿⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⣿⢸⣿⡇⢸⠀⠀⣸⣿⣇⠀⠀⣿⣿⣿⠀⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠉⠈⠉⠁⠈⠁⠀⠉⠉⠉⠀⠀⠉⠉⠉⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⣤⡀⠀⠀⠀⠀⠀⣠⠎⡄⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠙⠷⣄⡀⢀⡼⠁⠀⠸⡄⢀⣀⣤⣴⠾⠛⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠈⠉⠉⠈⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣄⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠙⠶⣤⣀⠀⣼⠇⠀⠀⠀⠠⠶⠿⠛⠛⠉⠉⢹⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠈⠙⠳⠉⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢸⡆⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠈⡅⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⠀⠀⡟⠁⠀⠀⠀⢀⠀⠀⠀⡠⠴⠶⠦⣄⠀⠀⠘⣧⠀⠀⠀⣷⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠉⠁⠀⠁⠀⠀⠀⣾⠁⠀⣀⠊⠀⠀⠀⠀⠈⢳⡄⠀⠸⡇⠀⠀⣿⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠐⠊⠳⣄⠀⢹⡀⠀⡏⠀⢠⠞⠉⠙⠲⠾⠓⠀⠀⢷⡀⠀⣿⠀⢸⡀
⠀⠀⠀⠀⠀⠲⣖⠒⠒⠚⠛⠛⢛⣉⡁⠀⠀⡜⠲⣆⢰⠀⠀⠉⠳⣼⡇⠀⣿⢐⡇⠀⠀⣠⡤⠤⠤⠴⢦⠸⡇⠀⣿⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⡏⠉⢉⠳⡄⡇⢠⠈⢿⡇⠈⡷⡄⠈⠃⠀⣿⠘⣇⠀⠀⢸⣄⡀⠀⡄⢸⡆⢹⡤⠟⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⢷⠀⠁⠼⠷⢧⡈⢁⡈⠛⢆⢡⠈⠳⡀⠀⢿⡄⠙⢦⡀⠀⣀⡇⠀⣿⠟⠁⣠⠔⠢⣄⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⣸⣇⠀⠠⠀⣨⡇⡏⠻⡦⠾⣿⠀⠀⠉⢦⠜⠿⣦⣄⣉⠉⠉⣀⡴⠁⠀⠀⢿⣶⣢⠞⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢿⣀⣠⡾⠿⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠀⠀⣀⣠⡴⠞⠊⠁⢲⣄⠀⢀⡴⠟⠀⠀⢦⡀⠀⠀⠀⢀⣴⠏⠀⠈⠈⠑⠒⠦⢤⣄⣀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢀⣠⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠙⠗⠉⠀⠀⠀⠀⠈⢳⣄⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⠏




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⣀⠀⠀⠀⠀⢠⣶⣿⣶⡄⢀⠀⢰⡄⢀⠀⠀⠀⠀⠈⠁⠠
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠃⢀⠁⠀⠀⢀⡀⢁⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡜⠉⠀⢠⠀⠀⠀⠀⡘⠀⠀⠀⠀⠰⠙⢇⡐⠀⠀⠶⣾⣿⣾⡂⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢊⣷⠁⠀⡄⠘⠀⠀⠀⠀⠀⠀⠀⣀⠀⢡⣀⠀⠀⠠⠈⠂⢹⢻⣿⠁⠀⠃
⠀⠀⠀⠀⠀⠀⡤⠀⠀⣴⣿⣤⡄⠀⠘⠂⠞⠁⠀⠀⢀⠀⠀⠀⠀⢺⣿⠶⣾⡿⡇⢤⣶⠒⣠⠄⢀⣼⣷⣶⡄
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠉⠉⠁⠰⡥⠀⠀⠀⠀⠀⠀⢰⣴⠆⠠⠀⠃⢀⡸⠇⠀⠘⠃⠀⢛⠿⣿⣿⣿⡿⠟⠦
⠀⠀⠀⠀⠀⠈⠛⢂⠐⠚⢶⣆⣠⢁⡐⠐⣲⣤⠄⣠⢾⡿⠃⠀⠀⠄⠐⠿⠆⢀⡀⠀⠀⠈⢸⣗⣚⣿⣿
⠀⠀⠀⠀⠀⣶⣶⣧⠀⠀⢀⠟⠚⢂⠛⠐⠁⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢌⣀⡄⠀⠀⠀⠈⠙⠉⠛
⠀⠀⠯⠢⢸⣿⣿⣎⣤⠶⠄⠚⠢⠌⠟⢶⠠⠶⣤⠤⠤⢤⡤⠤⠤⠴⠶⢶⡄⠁
⠀⠀⠀⠠⢄⠉⠀⠙⠁⠀⢀⡄⠀⠀⠒⠺⢾⣿⢧⣴⡿⢻⡇⠚⠛⢷⠴⠖⠛⠛⠀⠀⡔⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢼⣷⣦⣾⣿⡦⠄⠀⠀⠀⠀⠀⠀⠑⠀⠉⠈⠉⠀⠀⠰⠖⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠠⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣀⠀⡀⠀⢀⣀⡀⠀⠀⠰⠠⠀⠀⠀⠀⠀⣀⡤⠔⠒⠒⠒⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠋⠀⠀⠀⠉⠀⠉⠈⠉⠉⠲⢏⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡲⠆⠀⠀⡄⣤⠀⣀⣠⠰⠀⠆⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠆⠀⠈⡯⣧⢰⠷⠰⠿⠏⠇⠟⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠘⣄⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠊⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠦⡄⢀⡠⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠞⠳⣤⣀⣠⣤⣤⣤⣀⣀⡠⠔⠉⠧⠀⠀⠀⠀⠀⠀⠐⢁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠀⠀⠀⣀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠤⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣀⠄⢠⡄⠈⡷⣠⢼⡔⠄
⠀⠀⠀⠈⣉⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠢⢤⣀⣴⣿⣿⣿⣿⣷⣦⠔⠁⢸⡇⠀⠀⠀⢀⠀⠘⣹⢸⡗⢉⢂⢠⢃⠧⡘⣩⢛⠊
⠀⠀⠀⠸⠀⠒⠀⢀⡀⠃⠀⠀⢀⡀⠀⠀⠀⢀⣀⠀⠀⠓⠀⠀⡉⠙⠋⣉⡉⢿⠋⢹⠃⠀⠾⠇⠠⢉⡴⠁⠀⢸⢹⣘⠇⣌⡆⢬⠇⢀⣻⠷⡟
⠀⠀⠀⠀⠈⠀⠁⠄⠀⠀⠀⠀⠀⠈⠒⠦⣄⡀⠀⠀⠈⠀⠀⠀⣧⠀⠸⠿⠃⣼⢀⠀⢀⠀⠀⢀⠘⠻⠤⢄⡀⠀⠁⠀⠀⠈⠑⠒⠒⠋⠹⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠟⠀⠀⠀⠀⠀⠀⠲⣵⣶⣾⣿⡾⠇⠈⠀⠊⢀⠈
⠀⠀⠀⠀⠀⠀⣄⠀⡀⠀⢘⠀⡀⠀⠀⠀⠴⠁⠀⠐⠒⠒⣶⠀⠀⢀⣠⠉⠿⠁⠀⣰⠟⠲⠦⠼⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠈⠆⠁⢰⡌⠀⠁⣴⠀⠐⠁⠀⠀⠀⠀⠀⢹⣀⠴⠋⠈⢧⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢨⠤⠞⠓⠋⠙⠒⢧⡀
⠀⠀⠀⠒⠄⠈⢷⠄⣀⣼⠉⠒⠊⠵⢀⣀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠣⠊⠀⠀⠀⠀⠠⠀⡄⠀⠀⡀⠀⠀⣀⡴⠃⠀⠀⠀⠈⢠⠐⢘⡽⠂
⠀⢀⡀⠂⠐⠻⡏⠀⠀⠀⠀⠀⠀⠀⢾⣁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⠀⠀⠦⠀⠀⠀⢑⠏⢣⡁⣄⠂⡀⠀⠀⢑⠏
⠀⠀⠀⠀⢤⠺⢧⠁⠀⠀⠀⠀⠀⣀⣾⡁⠀⠀⠠⠒⠀⠒⠐⠆⠀⠀⠀⠀⠀⠀⠀⠤⠀⣠⡴⠲⠀⠀⢐⠄⠀⠀⠙⠦⢤⡀⠤⡀⠀⠄⡇⠰
⠀⠀⠐⡠⠀⠠⠾⠗⣦⣠⣄⡤⠨⣅⠀⠤⣀⡔⠀⠀⠀⣀⠀⠈⢦⢀⠠⠄⠀⠀⣀⠴⠊⠙⠒⠁⠀⡠⠎
⠀⠀⠀⠀⠀⡀⢠⠀⠛⠁⠀⠓⠀⣀⠀⠀⠀⡇⠀⣠⠠⠘⣤⠀⠀⠀⢰⠒⠀⠁⠀⠀⠓⠲⠤⠂⠊⠀⡠⠐⠀⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠸⠀⠀⠀⠈⣓⠤⠐⠫⠅⣀⡀⢐⡘⠂⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠊⠀⠀⠀⠀⠀⣀⠀⠀⢠⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠂⠄⣒⢡⠉⠙⠋⠀⠀⣠⠼⠃⠀⠀⠀⠀⠀⠀⠠⠀⢁⠠⡎⣷⢰⠖⢹⡟⢨⣳⡇⢸⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠰⣄⡤⠢⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢀⡟⠁⠮⠵⠋⠈⠉⠈⠀⣩⠓⠢⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣈⠉⠂⠀⠀⢀⣀⣤⣁⣀⡀⣡⡟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠊⠁⠀⠀⠀⠂⠈⠉⠉⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠄⠀⡀⠀⠀⡀⠀⠀⠀⠰⢶⣦⣄⡀⠀⠀⢀⣾⡇⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠘⠁⠀⠀⠀⣤⠀⠈⠉⠀⠀⢀⣾⣿⠃⢀⠃⠀⣿⠀⠀⠀⠀⠀⠀⠰⢇⣄⠛⠀⠐⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠂⣤⡈⡀⠀⠀⠀⣿⡆⠘⠁⠀⠀⠘⣿⣿⠀⣸⠀⠀⢸⡀⠀⠀⠀⠀⠀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠛⠉⡑⢯⡀⠀⠙⠃⠀⠈⢷⡦⠀⢿⣿⠀⡇⠀⠀⢀⣡⣴⡞⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡦⠀⠀⠀⠛⣀⢸⠛⢄⠀⠐⠻⠊⠷⠀⢸⡇⢸⢁⡄⢰⣿⣿⡟⢠⣶⡿⡡⠊⠀⠀⠾⠃⠆⠈⠐
⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠐⠀⣀⣵⣢⡀⠀⠀⠁⣀⣀⣀⣀⠀⢸⣿⡿⠁⠀⡹⠊⢀⠜
⠀⠀⠀⠀⠀⠀⠙⠶⠀⠀⠀⠈⠛⢿⣿⣷⣄⠀⠀⠈⠿⠟⠁⣠⣶⣿⣿⣿⣿⣿⣿⣦⡌⠁⡴⠊⠀⠀⢊⣠⡴⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢒⡤⢄⣀⡙⠛⠿⠷⣦⡀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢀⣤⣶⠞⢋⣡⡄⠀⠀⠀⡇
⠀⠀⠀⠀⡀⠶⠀⠀⠀⠀⠘⠃⠙⠀⠉⢙⠓⠲⠆⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠋⠀⠀⠛⠉⠤⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠉⠈⠙⠻⣿⣷⣦⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡙⠻⠿⢿⠀⠂⠐⠦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⡀⠀⠀⠀⢀⣀⡭⠅⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢠⣿⣷⣤⠁⠠⣄⠀⠀⠀⠀⢰⡦⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠿⠆⠸⡏⣈⡤⠖⠚⠉⣀⣤⣴⣶⠀⡝⠻⢿⣿⣿⣿⣿⠿⠿⠁⢀⠈⠁⣠⣄⠂⠀⠈⠻⠶⣄⡀⠄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠤⢒⣾⡭⠀⠀⢘⠁⣿⣿⠿⠋⡡⢊⣤⢠⣄⣀⡀⠀⢠⣤⣤⣄⠀⣱⡞⠿⠁⠘⡆⠀⢀⠀⠀⠉⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠐⢁⠀⠉⢀⣀⠜⢁⡈⠉⢸⣿⣿⠁⠀⣄⢻⣿⡿⠀⣉⠩⠦⡀⣤⠉⠢⠀⠳⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⢀⠝⠁⠀⠸⣀⠀⠀⣿⡏⠀⠀⢹⡀⢿⣧⠀⠀⠀⠾⠻⢆⠂⠀⠘⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡠⠊⠀⠟⠀⠀⠀⠀⠀⢿⠀⠀⠀⠈⡇⠘⣿⣧⠀⢀⣶⠆⣤⠕⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣫⠊⠀⢠⠏⠀⢀⣠⡤⠖⠂⠀⠀⠀⢠⢢⣷⠁⠘⣿⠀⠈⠁⠀⢡⠀⠀⡡⡀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⢀⠔⢁⠀⠀⠀⠀⠀⠉⠁⠀⠸⠇⠀⠉⢀⡇⢸⡟⡆⠀⠁⠀⠀⠀⠰⠃⢀⠀⠑⡎⠢⡈⡙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⡰⠁⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠉⠁⢇⠀⠀⠀⠀⠀⠀⠀⠈⠺⠡⣆⠀⠈⣃⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⢈⡾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡯⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠘⠻⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡶⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⢎⡵⠀⢀⠎⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⡇⠀⠀⣠⡜⠙⠀⠀⠀⠀⠀⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⡤⣄⠀⠀⠀⠀⢠⠎⠀⠀⠀⠘⡄⠀⠀⣠⠀⠀⢀⠞⠁⢰⠁⠀⢠⠃⠀⠀⠀⠀⠀⠀⢸⡵⠊
⠀⠀⠀⠀⠀⡆⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠳⣄⠀⢠⠃⠀⠀⠀⠀⠀⠱⠴⠋⠸⡀⡴⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠉⠫⣾⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠲⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠒⠒⠒⢲⠆⠀⢠⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⢩⠑⠲⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠧⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣖⠒⠒⠒⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⣴⣶⣦⡀⠀⣴⡄⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠈⠉⣉⠴⠖
⠀⠀⠀⠀⠀⣠⣶⡆⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⣶⡿⠿⣶⡄⢠⣿⠋⢿⣇⢸⡿⠀⢻⣇⢸⣿⣿⡄⣼⣿⡇⠀⠀⠀⢀⡠⠴⠒⠉
⠀⠀⠀⠀⠀⠻⠷⠁⠀⠀⠀⢀⡠⠜⠃⠀⠀⠀⠀⢻⣧⣴⣿⠃⢸⣿⠀⢸⣿⣿⡇⠀⢸⣿⣼⡇⠹⣿⡟⣿⡇⠀⠐⢎⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠿⣶⡸⣿⡀⢸⣿⠸⣿⣀⣾⠇⣿⠇⠀⠈⠀⣿⡇⠀⠀⣈⠵⠀⠀⠀⠒⠒⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠢⠤⢄⣀⠀⠀⠀⠀⠀⠘⣿⣀⣠⣿⠇⠻⢷⠿⠃⠀⠉⠛⠉⠈⠛⠀⠀⠀⠀⠙⠁⠀⣀⡹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡴⡄⠀⠀⢀⡠⠟⠀⠀⠀⠀⠀⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠰⠯⠄⠐⠒⢒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡴⢄⠀⠀⠀⠀⠘⡆
⠀⠀⣀⢀⡤⣰⠆⠖⠒⠂⠀⠀⠀⠀⠀⠀⠞⠒⠒⠉⡹⠀⠀⠀⠀⢀⠀⠀⠀⡠⡄⠀⠀⡸⠑⢤⠁⠀⠙⠦⡀⠀⠀⠸⡄⠀⠈⡧⡄⡀
⠀⠀⠈⠙⠀⠀⠀⢀⡠⢔⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⠔⠉⡇⢀⡜⠁⢸⠀⢀⡇⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⢱⠀⠀⠀⠘⢱⣤
⠀⠀⠀⠀⠀⠀⢞⡧⠖⠉⠀⠀⠀⢀⠄⠀⠀⠀⠀⡏⠀⢀⠔⠁⠀⠀⠰⠊⠀⠀⠀⠇⢸⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⠢⣧⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠸⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣴⠄⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢸⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠼⠃⠀⠀⠀⠀⡱⠶⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠛⠀⠀⢸⡇⠀⢀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠀⠀⠸⣇⠀⠀⢸⡇⠀⢸⡇⠀⣼⠇⠀⠘⠃⢀⡿⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⢶⡄⠀⢿⡀⠀⢸⡇⠀⣿⠁⠀⣿⠀⣸⡇⠀⣼⢃⡀⢀⣼⠋⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⢻⡄⢸⣇⠀⢸⡇⢰⡏⠀⠠⡿⢠⡿⠀⠸⠃⠛⢁⡾⠁⢀⣠⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠹⣦⠀⠻⠀⠿⠀⠘⠁⠈⠀⠀⠀⠀⠀⣀⡀⠀⠀⣿⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡘⠷⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⣶⠀⣼⠋⠁⠀⢰⡿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡈⠉⠉⠙⠀⣾⠛⢻⡆⣰⠏⢻⡄⢸⣿⡇⠀⣿⠀⣿⠾⢻⡆⠘⠇⠀⠀⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⡛⠃⠀⣿⡶⣿⡀⣿⠞⢻⡇⢸⡏⣷⣀⡿⠀⠹⠶⠟⠁⠐⠆⠀⠶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠶⠾⠛⢋⣤⠀⣿⢀⣼⠇⣿⠀⠘⠇⠸⠇⠈⠉⠁⠀⠀⠀⣤⣀⠀⠰⣦⣄⠀⠀⠉⠁⠀⠶
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⣠⡾⠛⠁⠀⠙⠛⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢻⣆⠙⢷⣄⠀⠙⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋⠀⢀⣴⢂⡀⢠⡄⠀⣾⠀⣀⠀⣿⠀⠀⣿⡀⢻⡄⠀⠉⢀⡀⠙⢷⣄⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠈⢁⣾⠁⣸⡇⠀⣿⠀⣿⠀⣿⡀⠀⠘⠇⠈⢿⡀⠀⠈⢷⣄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠇⠀⣼⠇⠀⣿⠀⠀⣉⠀⣿⠀⢸⡇⠀⠀⠀⡤⠘⣷⠀⠀⠀⠻⣆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⢰⡏⠀⠀⠛⠀⠀⠁⠀⣿⠀⠘⣧⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⢹⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠁⠀⠀⠀⠘⠃⠀⠹⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠄⢠⣤⣤⣠⡐⠒⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠉⡙⠁⠉⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⠛⠣⠰⠶⠴⠶⡀⠸⠀⠀⠀⢰⢀⠜⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⢸⣐⠒⣒⡒⠒⠀⠈⠀⠈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠻⣄⣹⣻⡤⢄⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠴⠶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣀⡀⠤⠤⠄⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠐⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡗⠚⠷⠶⠶⠶⠿⡉⠑⠂⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠋⠈⠒⢢⠁⢀⡄⠠⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⡴⠒⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠈⡇⠀⣤⣤⠤⠤⠤⠤⠤⠤⠴⠂⠤⠔⠚⠉⠉⠅⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢀⡀⠤⠜⠀⠁⠑⠐⠚⠒⠚⢡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⣤⡖⠂⢲⡔⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⢀⠋⠀⠀⡤⠊⠁⠈⠿⣧⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⡌⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⣇⣀⣀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠤⠐⠒⠀⠒⠀⠀⠀⠀⠀⠘⠂⠐⠒⠒⠾

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢛⣛⣋⣉⣉⣉⡛⠓
⠀⣀⠴⠒⣋⣭⠭⠤⠤⠤⠤⢬⣉⠳⣄⠀⠀⠀⣠⠴⠒⠒⠒⠒⠦⣄⠀⠀⣠⠖⠛⢉⣉⡉⠓⢦⣀⣤⡤⠤⠤⠤⣄⠀⣀⣤⣤⣀⡀⢸⡇⢸⠁⠀⠀⠀⠀⡇⢸
⡞⢡⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡌⢳⡴⢋⠔⠚⠉⠉⠉⠓⠢⣌⠳⠞⢁⠔⠋⠉⠀⠈⠙⠢⡉⢠⠒⠒⠒⠐⠞⠾⣡⠒⠲⢦⢳⠈⡇⠀⠀⠀⠀⠀⠘⠀⣾
⢹⠈⠀⠀⠀⠀⡤⠀⠤⡀⠀⠀⠀⠀⢹⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠈⢂⢠⠁⠀⠀⠀⠀⠀⠀⠀⠱⡼⠀⠀⠀⠀⢣⡔⠁⠀⠀⠘⡌⣇⣧⢰⠀⠀⠀⠀⠁⠀⡇
⠘⡇⡁⠀⠀⠀⢣⣀⣠⠇⠀⠀⠀⠀⢸⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠘⣿⢸⠀⠀⠀⢠⠀⢸⠃
⠀⢳⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⢰⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠁⠀⠀⠀⢰⠛⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⢸⠀⠀⠀⡄⠀⡟
⠀⢸⡜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠲⡀⠀⠀⠀⠘⠤⠇⠀⠀⠀⠀⠘⠀⠀⠀⠈⠉⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠈⠑⠒⠒⢁⡴⠃
⠀⠀⣇⠇⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⡆⠀⣾⠉⣷⢘⡇
⠀⠀⢹⠀⠀⠀⠀⠀⢃⢀⣨⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⠢⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⢰⠶⠦⠾⠻⠀⠈⠁⠀⠀⢿⠠⣬⣿⣥⠞⠁
⠀⠀⢸⣇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠋⠢⢀⠀⠀⠀⣀⣀⠤⢀⡤⠛⠲⣤⣁⣒⣒⣒⠈⡟⠀⠀⠀⢀⢋⡞⠛⢶⠈⠇⠀⠀⠀⠀⠘⠀⢻
⠀⠀⠀⡏⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡠⢤⣄⣀⣀⣉⡭⠴⠚⠉⠀⠀⠀⠀⠀⠉⠉⢹⠀⠧⠤⠤⠤⠜⣸⠁⠀⠘⣇⠸⣀⣀⣀⡀⢠⠅⢸
⠀⠀⠀⢧⡓⠦⠤⠤⠤⠤⠴⠒⢊⣡⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠶⠶⠖⠒⠃⠀⠀⠀⠸⠦⢤⣤⡤⠤⠤⠶⠚

⣿⣵⢯⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣤⣼⣷⣜⡧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣽⣿
⢻⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢻⣿⣿⣿⣷⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠿⣿⡘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡀⠀⢀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠋⠛⠁⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣼⣿⣿⣷⢶⣿⣆⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠨⣷⣿⣿⣿⣿⣿⣿⣙⣶⣛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⣷⣤⣀⣄⣠
⣀⡀⠀⠀⠀⠀⠀⠀⠀⠛⢿⡿⠷⢿⣿⢾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡮⣾⣻⣟⣿⣿⣙
⣻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠛⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠠⣤⣤⣳⣀⣰⠀⠀⠀⠀⠀⢀⠈⠉⣸⣿⣟⣿⣿⣷⣿
⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣾⣏⣿⣻⣆⡀⠀⠀⠀⠀⠀⠀⠉⠈⢹⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡟⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣽⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢄⣴⣧⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⢿⣽⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠞⣿⣻⣹⣟⣿⣿⠶⠆⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠉
⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⣿⣿⣿⣿⣿⣶⣤⡄
⣤⣯⣷⣥⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣾⣿⣿⠄
⣿⣿⣿⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⡈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢠⣄⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣙⣛⡻⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢼⣿⣿⣿⣄⣤⣶⠀⠀⠒⠀⠀⠀⠀⠀⣸
⠙⣿⣿⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣻⣻⣳⣿⣿⣽⢿⡿⢲⡟⠀⠀⠀⠀⠀⠀⣀
⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⣿⠿⣾⣿⣿⣿⣶⡟⣭⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⢿⠟⠿⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣶⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣯⣿⣷⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⢀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⣿⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣶⣿
⣿⣿⣿⣋⣀⣄⣀⣀⣀⣀⣀⣀⣰⣆⣀⣿⣻⣿⣏⣚⣗⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣿⣿⣿
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠟⠛⠉⠉⠉⠛⠛⠿⣿⣿⣿⣷⣦⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⢁⣴⣶⣿⣿⣿⣿⣿⣷⣦⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠟⢁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣈⠉⠉⠉⠛⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠟⠛⠛⠁⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⢿⠇⠀⠀⠀⣴⣷⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠸⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠀⠀⠀⢰⣿⠇⠀⠀⠀⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠈⢹⣿⣿⣴⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢠⣶⣦⢤⡀⣀⣦⡀⣠⣤⡀⢸⣿⣿⠉⠀⠀⠉⢻⣿⣿⡇
⠀⠀⠀⠀⣰⣿⣿⣿⠟⠙⣉⡛⢿⣿⠟⠋⠉⠉⢳⠉⠟⠙⠋⠙⠀⠉⠀⠈⠒⠀⣼⡟⠱⡆⠀⠅⠀⠈⣿⣿⡇
⠀⠀⠀⢀⣿⡟⢿⠟⠁⠐⠿⠛⣼⣿⡀⠀⢀⣀⢰⡦⣀⡀⠀⣄⠀⣈⠁⠀⢀⣼⣿⡧⢀⣀⠀⠀⠀⠀⣿⣈⡇
⠀⠀⠀⢸⣿⠟⣏⠀⠀⠀⠀⣄⣿⣿⡀⠙⠛⠁⢀⣰⠋⠀⠀⠀⠀⠉⣰⣾⣿⣿⣹⣧⣾⣿⡆⠀⠀⣠⠇
⠀⠀⠀⠈⠁⠀⣿⣦⣄⠈⢽⣿⣿⣿⣷⣤⣀⣀⠪⣿⣆⠀⠁⠀⣠⢦⣿⢿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣯⡀
⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣷⠈⢻⣿⣿⣿⠟⢏⢷⣶⣾⣛⡵⣻⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣷⡄
⠀⠀⠀⠰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⣦⣈⣼⠟⠉⠁⣠⣀⠜⠛⠯⣿⣿⣿⣿⣿⣿⣿⡟⠂⠙⠂
⠀⠀⠀⠀⠔⡱⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠆⢩⡇⠀⢀⠀⡝⠁⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢈⣼⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠁⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⣠⣄⡀
⠀⠀⠀⢸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⠆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠁
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⢈⡁⣼⢿⣿⣈⠀⠀⣴⣶⣿⣏⣴⣭⢿⣯⣤⣌⣹⣧⡄⠀⢰⣾⣿⣿⠟⢦⠁⠚⠀⢠⡦⣤⣍⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠞⢻⣿⣦⣤⣾⣿⣿⣟⣡⡶⡹⢟⠲⠾⠛⢿⢿⡾⡿⠛⠋⢹⣧⡀⠀⠠⢾⣟⣿⠏⠑⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢺⡇⠠⡇⠀⡃⠀⢿⠁⣾⣟⢿⣟⢿⣿⣇⣿⡋⠠⢴⣶⡦⠹⣿⣿⣠⣴⣥⣿⣿⡇⠀⠀⠀⡀⠀⠈⠀⠴⠆⠀⠄
⠀⠀⠀⠀⠀⠀⠈⢳⣄⢳⡄⢙⡀⢶⣄⠉⠁⡴⠋⣿⣿⣦⣬⠟⠰⢾⣿⣷⣄⣿⣿⣿⣷⠟⠻⠋⢿⣷⣼⣷⣾⡓⣶⢿⣿⣌⣤⣤⠄⡂⠆
⠀⠀⡄⠀⠀⠄⠐⠀⠡⠁⠛⢤⣅⣾⣿⠈⣡⠀⣶⣿⣿⣿⡿⣿⣾⣿⣟⣿⣿⣿⣿⣿⣤⣤⠶⣶⠿⠛⠟⢿⣿⣃⣸⣾⣿⣿⣿⡏⢀⡀⠀⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡘⣶⢃⣶⡠⠉⠚⠲⣾⠿⣿⣿⡿⢛⣿⣧⣙⠟⣋⣸⣿⣿⣿⣿⣿⣽⣵⣾⣏⠁⣸⣿⣿⣿⣿⣞⣿⣿⣿⣿⣿⡾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢻⠿⠀⠐⠓⣤⢥⢂⠔⣨⠿⣿⣷⢻⣿⣯⣻⢿⣿⣿⣿⣿⣛⣽⣿⣿⣿⣿⣿⣿⣿⠏⢸⣿⣿⣿⣏⣉⣿⣿⣷⣁⢺⣿⣇⣴
⠀⠀⠀⠉⠰⠁⠀⠀⠀⣀⠐⠛⠇⣈⠙⣇⣁⣿⠿⣾⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣛⠉⠥⠂⣋⣡⣤⣿⣿⣿⣛⣿⣿⢛⣷⣿⣿⣴⠘⢿⣿
⠀⠀⠀⢨⣏⡄⠰⢀⡴⠟⠴⢤⠂⣿⠒⠉⠀⢀⡤⠟⢿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣶⣿⣿⣻⣽⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⠃⠀⠘⠋⠀⡀
⠀⠀⠀⣀⣿⠀⠀⠈⣱⣷⢠⣷⡮⠟⠀⠄⠀⠈⠀⢀⡤⠐⢉⠛⣿⣿⣿⣿⣿⣿⡿⠿⡏⠹⡿⠿⠿⠛⠱⣻⣾⣿⣽⠟⠀⠀⠉⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠙⣾⠆⠀⡌⠃⢈⣟⣿⡇⣀⣀⡠⠀⣨⣿⣿⣿⣿⣷⡾⣿⠊⠓⠛⠛⢛⡀⢁⠈⠉⣹⣶⡖⠛⠡⣻⣯⠊
⠀⠀⢄⣤⡙⣧⡌⠀⣀⣾⡿⣼⣷⡟⠉⠁⢴⢿⣿⣿⣧⣿⣧⣻⣄⡈⣠⣤⣀⡌⠉⠾⠹⠜⡏⠙⣓⠀⡠⠛⠀⠀⠀⠀⠀⠀⠀⠐⠘⠒⠄⠀⠀⠀⠆
⠀⠀⡀⠀⢦⣾⣿⣯⣻⣿⡅⢹⣿⡀⠐⠰⠃⣸⣿⣶⣿⣿⡟⣹⣿⣶⣿⣿⣿⣷⠄⠀⢀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢀⠀⠀⠆
⠀⠀⠁⠼⠃⠁⠙⢻⣿⣿⣬⣀⠃⠀⣾⣳⣖⣿⣿⣻⣿⣧⣴⣿⣯⣿⣿⣿⣿⣿⢇⢈⣈⣁⡀⠀⠄⠀⠀⠀⢠⡤⠀⠀⠀⠀⠾⠌
⠀⠀⠀⢀⣠⣤⣤⣸⣿⣍⠀⣷⣂⣴⣰⣿⣿⣿⠋⣻⢋⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣒⡎⣏⢹⣿⣦⡂⣶⣶⣮⠳⢟⣦⡐⠐⣆⠀⢠⠀⢠⣄⠠⠃
⠀⠣⠿⠫⢿⡿⣿⢶⡿⢻⣾⣿⣿⣿⣿⣿⣟⣢⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣽⣷⠖⢋⡩⠰⡼⣿⡥⢻⣮⡗⠁⠀⠀⠈⣱⣼⣩⡄⠐⠆
⠀⠀⠀⣴⣛⣹⡇⢀⡴⠒⣹⣻⠟⣿⣫⣿⣯⡨⠛⠋⠸⣬⢩⠿⣿⣿⡽⢏⡁⠛⠿⠼⣣⡟⠘⢾⡓⠻⢋⣩⠁⠀⠀⠀⢻⡘⠋
⠀⠀⠀⠉⠙⢡⠰⣶⣿⣶⣎⣥⣴⢸⡿⣞⢿⣩⣿⣅⣀⣤⣸⣷⣹⠟⠿⠻⢿⣾⣄⡀⠐⣁⠛⠃⠀⠀⠀⠁⠀⠀⠀⢀⢈⣵⠦⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣞⣀⣹⡯⠙⠳⣿⠣⢦⡛⠛⠛⠫⠿⠷⠉⠙⠛⢃⠀⣧⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠋⣉⣠⡔
⠀⢀⠀⠀⠀⠀⠀⢺⣌⣻⢿⣿⡅⠈⠃⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠈⠀⠀⠀⠀⠀⠀⠉⠉⠛⠈⡋⠙⠢⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣖⣖⣄⣰⡦⠤⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣛⣛⣛⠛⢿⣿⣿⣿⣶⣦⡀⣉⠒⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣿⣿⣿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⣋⣩⣽⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠛⠁⠋⠈⠁
⠀⠀⠀⠀⠀⠈⠻⠟⠛⠻⠿⠟⠛⠋⠁⠀⠀⢀⣀⣀⣄⣀⣐⣂⠘⠛⢻⣿⣿⣿⣿⣿⣿⣿⡿⣇⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⣿⣿⣿⣿⣿⣿⣧⣀⣤⠤⠼⠿⠛⠚⠉⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⢶⣻⣿⡿⠿⠛⠋⣉⣠⣠⣤⣤⡤⠀⠆⠒⠐⢺⡷⠚⠋⠋⠛⠉⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠖⠚⣉⣡⣤⣴⣶⡾⠿⠿⠟⠛⠹⠷⢞⣟⠋⣛⣿⣭⣍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣥⡴⠾⠛⢻⣿⣿⣿⣷⣦⣀⡠⠄⠀⠐⣻⣿⣿⣟⣫⣿⢭⢉⣀⣓⣲⣀
⠀⠀⠀⠀⠀⠀⠀⠲⠾⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠉⢻⣿⣿⣿⣷⣦⣤⣤⣄⣈⡉⠭⠽⠋⠓⠀⣴⠋⢁⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⡿⠿⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⠴⠒⠫⠍⠙⠸⠷⠤⠽⢯⠭⠿⠿⢿⠾⣟⣫⣍⣥⡀⠒⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠸⢿⣿⣿⣿⣶⣶⣶⣶⡶⢶⣶⣶⣾⣴⣴⣦⣼⠇⠀⠈⣉⣩⡿⣷⠔⠶⠐⠒⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠⠄⢈⣅⢀⠀⠀⠀⠀⠤⠸⣯⣿⣿⣿⣿⠤⠀⠀⠿⣽⣤⣑⣤⠿⠆⠀⢠⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡘⠃⠀⠘⠉⠈⠀⠈⠁⠀⢠⣤⣼⣿⣿⣿⠷⢤⡀⢀⣠⣿⣿⣭⣭⣴⣶⣀⡈
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢼⡇⠀⠀⠀⠀⢁⣀⡀⢤⠠⣴⡷⢻⣿⣿⣿⣷⣿⣿⣿⣿⣿⣛⣻⣿⡍⠶⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠘⣿⡈⠀⡀⠘⠋⠛⠹⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠯⣽⣿⣿⣿⡿⢿
⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⢸⣿⣦⣉⡀⢀⣶⡄⡀⠉⠹⠏⣩⣿⢿⣿⣿⣿⣿⣿⣿⣟⠿⢧⡄⣀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠎⡛⠛⠛⠻⣷⣿⣿⣿⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⡄⢸⠀⠀⠂⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⣒⠰⣄⡺⣽⣿⣿⣿⣿⡿⣿⠿⠋⠉⣿⣏⠁⠧⠈⠀⠀⢀⠀⠁⢄⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠈⢻⡿⣙⣹⣿⠛⢫⢿⣿⣀⣄⠘⠿⢶⡄⢀⠀⠐⠂⠰⣰⡄⠀⠰⠄⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠉⢻⣷⣿⣿⣽⣿⣿⣶⡆⢸⣧⡤⠀⠀⠀⠀⠁⠈⠀⡈⠀⠀⢙⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⠿⣿⣿⣿⣿⡿⠶⠇⠀⣀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠈⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⣿⡿⠏⠀⠀⠈⠀⠱⣶⡆⠠⠀⠀⢀⡤⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣧⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠘⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⣶⢠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⢹⠈⡇⠀⠀⠉⠙⠛⣦
⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠰⣄⣤⠾⠀⠿⠶⣶⣧⣀⢠⡟⠀⠀⣠⣄⣀⣀
⠀⠀⠀⣠⡾⠋⠁⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣿⠙⢷⡄⠀⠀⡟⠉⠉⠛⣷⠀⢠⡾⠁⠀⠀⠀⠀⠀⠈⠙⠟⠁⠀⢀⡟⠀⠈⠉⠛⠳⢦⡄
⠀⠀⢰⣟⠀⢀⣀⣀⠀⠀⢠⡇⠀⠀⣤⠶⠛⠉⠻⣆⠀⠀⢹⡆⠀⢻⣄⠀⣇⠀⠀⠀⢸⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⢻⡆⠈⣧⡽⠀⠀⢹⡇⠀⠀⡟⢀⣄⡀⠀⠹⣧⠀⠘⣇⠀⠀⠹⣆⣿⣀⡀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢀⣀⣾⠁
⠀⠀⠀⠀⣿⠀⠀⠀⣀⣴⠟⠀⠀⢰⡇⢸⠋⠙⠶⣄⠈⢷⡀⣿⠀⠀⠀⠙⣿⡋⠀⠀⢸⡆⣧⠀⢀⡟⠛⠛⠶⢦⣤⣀⠀⠀⢀⡏⠀⠀⢠⣀⣹⠃
⠀⠀⠀⠀⢸⡷⠾⠛⠛⠛⠓⢶⡴⢾⣇⣠⠀⢀⣠⡼⠇⠀⢻⣿⠀⠀⠀⠀⠘⠃⠀⠒⢺⡇⠹⣆⠸⣧⣤⡀⠀⠀⠈⠉⣿⠀⢸⣧⣀⠀⠀⣸⠏
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⢻⡉⣁⠀⠀⠐⠋⠹⡆⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠓⢶⠾⠃⠀⣀⣀⢠⡟⠀⣿⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠘⣟⡃⠀⢰⠛⠛⠓⢦⡄⠀⠀⠀⣷⢻⠻⢦⣄⠀⠀⠘⣦⠀⣿⡀⠀⠀⠀⢸⣷⣤⣄⣀⣨⠀⠀⢰⠏⠉⠛⠁⠀⠛⠷⢶⣤⡟
⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⠸⡆⠀⠀⢸⢸⠀⠀⠙⢷⣄⣴⠃⠀⣿⣷⠋⠀⠀⢸⡇⠀⠈⠉⣿⠀⠀⣿⠀⠀⠀⠀⣠⡴⠶⣤⠀⠁
⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⢀⣼⠃⠀⠀⡿⢸⡇⠀⠀⣿⣈⣁⣀⣤⡿⠸⡷⠶⠞⠛⠁⠀⠀⠀⠛⠛⠛⠋⠀⠀⠀⢸⡟⠂⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡄⠀⠘⠒⠒⠚⠛⠁⠀⠀⣸⣥⣼⡇⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠾⠛
⠀⠀⠀⠀⠀⢸⡇⠀⣠⠀⠀⠀⢰⣄⣹⡾⠋
⠀⠀⠀⠀⠀⠘⠿⠶⠟⠛⠛⠛⠛⠋⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠿⠿⠛⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠙⠉⠁⠀⠀⠀⠀⠀⠈⠈⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠙⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣶⣶⡄⠀⠀⠀⠀⠀⣀⣠⣶⣚⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⡻⣦⣾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡄⠉⠀⠀⠀⠀⠀⢘⣇⠧⡄⠀⠀⣴⣶⡄⠀⣠⣾⣶
⠀⠀⠀⠀⠀⠀⣠⠾⣿⣿⡙⢉⣽⣿⣿⣿⣿⣿⣷⣤⡀⣠⣶⠿⣿⣿⣷⣦⠀⢀⣤⡶⣾⣿⣶⣄⠀⣰⣿⣿⣧⢰⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢀⣤⠄⢻⣿⣷⣿⣿⠟⠉⠀⠀⠈⠙⣿⣿⡿⠁⠀⢻⣌⠹⣿⣷⣿⠋⠀⠨⣿⡛⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣼⣿⠀⣺⣿⣿⢻⡏⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢈⠀⠀⣿⣿⡇⠀⠀⢠⠄⠀⢸⣿⣿⣿⢿⣿⣿⡟⢻⣿⣷
⠀⠀⠀⠀⠀⣿⣿⣧⣿⣿⡏⠘⠀⠀⠀⠀⠀⠀⣰⣿⣿⣇⠀⠀⠀⠀⣰⣿⣿⣇⠀⠀⠈⠀⠀⣾⣿⣿⡏⠈⢻⡟⠀⠘⣿⣿⡆
⠀⠀⠀⠀⠀⢻⠿⣿⣿⣿⣅⣀⣀⣀⣀⣀⣤⣾⣿⡿⢻⣿⣦⣄⣠⣴⣿⠟⠹⣿⣦⣀⣀⣠⣾⣿⣿⡿⠀⠀⠘⠃⠀⠀⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠘⠀⢹⡟⢿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠙⠛⢿⣿⠟⠉⠀⠀⠈⢻⣿⡿⠿⢻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⢹⣏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠈⠏⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢸⢀⠆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡄⢀⣧⣯⣾⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠑⠋⢀⣈⣁⣀⠈⠻⣿⣉⣩⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡰⠓⠋⠋⠛⠀⠀⠀⠀⢹⣿⣿⣭⡤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢀⠃⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠁⠜⠃⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⣿⣷⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⣇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠟⠓⠀⠀⢫⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⢡⠉⡆⢠⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠀⠀⠸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢠⠁⠀⡀⣠⡴⡾⠶⠁⠀⠀⠀⠀⠀⠀⠀⠀⡘⢻⠉⣧⠀⢀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠀⠀⡇⢠⠀⠠⡀⢀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠃⢻⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⡇⢸⠀⠀⠈⠢⡙⢦⣀⠀⠀⢘⠄⠀⠀⠀⠁⠈⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢀⡆⠀⢰⡇⢸⡀⠀⠀⠀⠙⠲⣌⡉⡁⠋⠀⠀⠀⠁⠆⢸⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠋⠀⠀⠇⠀⢸⠂⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠀⠙⢄⠀⠘⡀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡿⢀⠊⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠙⢦⢡⠀⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⣸⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⢻⠸⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⡇⢿⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠀⡇⢰⡀⠙⠲⣄⠀⠀⣀⡀⠀⠀⠈⣉⢁⠁⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⣀⢀⣄⣄⢘⡆⠀⠙⠇⠈⠉⠀⠀⠀⠈⠹⠁⡟⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡇⠈⠀⠙⠋⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣀⠀⠈⠇⠀⠀⠀⡇⠀⢀⠄⠑⡀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⢸⠁⠀⠃⠀⠀⠀⡷⠊⠁⠀⠀⢹⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠇⢢⡀⠈⡆⠀⠀⢀⢴⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⢸⣀⣓⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠙⡄⠀⠀⢰⠀⠙⠦⢌⠦⣦⠈⢸⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⡎⡟⢹⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠑⠢⠀⠱⢤⣤⡤⠚⢩⠀⠸⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢸⠁⡇⢸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠒⠤⠤⢄⡰⠋⠐⢳⠀⠀⠀⡄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢸⠀⢠⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⠄⠂⢈⠤⠀⠈⠉⢸⠁⠀⠀⡇⠀⠀⠀⠀⠀⣅⣀⣀⣀⣀⠀⣀⠆⠀⠈⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠐⠁⠀⠀⠀⠀⡀⠀⠀⡀⡇⠀⠀⠀⠀⠀⠀⡞⠓⠚⠓⠒⠤⢄⠀⠀⡥⠾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⡀⠀⠘⠁⠀⠈⡍⠉⠁⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠉⢢⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠥⠠⠞⠀⠀⠀⠈⠂⠠⠤⠄⠀⠀⠀⣀⣀⡠⠓⠖⠃

⠃⠀⠐⠀⠢⠀⠄⠈⠀⠀⠚⠲⠐⠁⠚
⠈⠐⠂⠀⠀⠠⠐⠄⠠⠀⠀⠐⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠐⡀⠂⠒⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠐⢀⣀⣤⣤⣶⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣧⣾⣿⣿⡟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠂⠀⠀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠒⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣠⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⠹⣷⠀⠀⠠⠤⠤
⠀⠀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣴⣿⠀⠀⠈⠉⠀⠈
⠀⢸⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿
⠀⢹⣿⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿
⠀⠀⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠸⠀⠀⠀⢆⠀⠀⠀⠄⠀⠆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠁⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢄⡠⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠑⠊⠈⠑⢢⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⡰⠀⠀⢀
⠀⠀⠀⠀⠀⢠⣀⡀⠀⠁⡀⠀⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢈⢞⣫⢳⠠⠦⠝⠀⠀⠀⢀⣀⡤⠀⠀⠉⠈⠁⠀⠄⠀⢄⠀⠀⠀⠀⣀⢪⣔⠀⣤
⠀⠀⠀⠀⠀⠀⢀⠰⠀⠈⠒⠒⠁⢀⡀⠀⠀⠀⢰⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠁⠙⠬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠐⢆⠀⠓⠂⠀⠀⠀⠈⡉⠉⠉⠉⢉⠉⠉⠉⡉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠊⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠀⠰⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠈⠉⠉⠉⠁⠒⠢⣄⠀⠀⠀⠀⠀⠀⠀⠂⢀⠤⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠠⠀⠀⣀⢀⠀⠀⠈⠔⡂⠩⠤⠤⢈⣉⠒⢤⡀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠂
⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠈⢲⠑⠁⠀⠀⠀⠀⠀⠀⠉⠒⢬⠂⠁⠉⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠃⠀⠢⡀
⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠀⠀⠀⠀⠀⠀⠀⠸⢅⣀⠀⠀⠀⣀⣀⣀⣀⠬⠀⠀⠀⠀⠂⠈⠉⠙⡄
⠀⠀⠀⠀⠉⠐⠒⠒⠒⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⡰⢉⡱⠀⠀⠀⠀⣀⠠⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢀⡄⢀⡠⠄⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠈⠲⠤⠤⠒⠊⠉⠀⢠⠡⢤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠂⠉⠈⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⢀⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠄⢀⡠⠄⠀⡀⢀⡊⠀⠀⠠⠀⠂⢀⠀⠀⢀⣀⠐⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⡎⠀⠀⡄⢈⢀⡀⠂⠈⢀⣠⠄⠅⡀⠀⡩⠐⠁⣑⠄⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠄⠒⠁⠈⠨⢤⡔⠋⠀⠠⠀⠈⢎⡂⠀⠀⠀⣿⠀⠀⢀⣟⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢦⡀⠐⠂⢀⠄⠀⠀⠐⠆⢠⡶⠀⠀⢠⡀⢀⡤⠀⢀⡏⠀⢀⡞⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠆⢸⣇⠀⡿⠀⠂⠈⠃⠀⠀⢸⡏⠍⠁⠀⢸⢇⣀⡾⣤⡤⡄⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢤⠆⠀⡀⠀⢀⡶⢀⣀⣴⣄⣀⡇⠈⣀⣀⣸⠃⢩⢰⠁⢀⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⠆⢠⢠⠀⠈⠀⣠⢏⢴⣻⠕⠓⠛⡖⠀⠀⠀⠀⠊⠍⠉⠙⠫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠈⠀⠆⣒⡫⠕⠁⠀⠀⠀⠀⠈⠡⠇⡠⢄⡐⠂⠈⢃⠀⠀⡤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠒⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠁⠀⢡⠠⢄⠀⠀⡠⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠉⠈⠒⠤⠈⠁⠒⢊⡀⠀⠀⠀⠤⠄⢀⡀⠀⡀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⡀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠


⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠰⠂⡀⠀⠀⠀⡰⠁⠀⠀⣀⡀⢃⠀⠀⠀⢰⡫
⠜⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠌⠀⠀⠀⠘⢆⢀⠀⠀⢰⠃⠀⠀⠀⠁⢷⠀⠀⠀⠀⣏⠂⠀⠀⠀⠀⠀⠀⠀⣀⠔
⠠⠈⠁⠀⠀⠐⢲⡌⠀⡠⠀⡅⠀⠀⠀⠀⢸⠘⠀⠀⢀⠂⠀⠀⠀⠀⡏⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⢼⠇⠀⠀⠐⠀⠀⠀⢀⣰⠋⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠸⠋
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⢀⡁
⠀⠀⠀⠡⠀⠀⠀⠒⠢⡀⠀⠀⠀⣅⠁⠀⠀⠀⣠⠊⢃⠀⠀⠀⠀⠀⠀⢠⣬⡄⠀⠀⢀⠭⠇⠀⠀⠀⣴⠂⠀⠀⢰⠁
⠀⠀⠀⠀⣌⠀⠀⠀⠈⠁⠀⠀⡀⠀⠁⢤⠈⠠⠊⠀⠀⠢⣀⢀⠀⣀⡦⡛⠰⢀⠀⠀⠖⠒⠀⠀⢀⠚⠋⠀⠀⠀⠃
⠀⠀⠀⠀⠁⠀⠀⠀⡀⠠⢔⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠐⠢⢀⠤⠀⢡⠃⠢⣀⡠⡎
⠀⠀⠀⠀⢃⠁⠈⠁⠐⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠆⠨⢘⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣻⢸⡓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠘⢦⣀⣠⠀⠀⠀⢀⣠⣶⡿⠿⠾⠿⢷⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢕⠦⡔⢋⣿⣶⣶⣶⣾⣿⣿⢋⣠⠀⠀⠀⠀⠙⢿⠿⢿⣧⡀⠀⠀⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⡀⠑⣌⠛⢿⡿⠛⢩⢉⡽⠋⠉⠉⠉⠂⠀⠀⠀⠐⢄⣀⣹⣿⣷⠶⣩⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣿⣷⡏⠀⠀⠇⠊⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠠⢽⣿⢿⣷⣾⣿⡆⠁⠀⠀⠀⠀⣀⣠⡤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⣿⣠⡆⠀⠀⠀⠀⣠⠖⠋⠉⠉⠉⠓⠦⠀⠀⠀⠈⠂⠀⠙⢻⣿⣤⣴⣶⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣸⠟⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠻⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢂⣴⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⢄⢸⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⠿⢷⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⡿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⡏⠐⠁⠉⡏⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠁⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠒⠛⠛⣇⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣬⣇⠀⠀⠀⠀⠈⠉⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⣟⣒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢛⣯⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⡿⠻⢷⣬⡈⠓⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡫⣴⣿⣿⣷⣄⡀⣀⠈⠦⣒⡶⣄⠀⠀⠀⠀⠀⡠⠀⠀⠀⠒⢉⣾⡟⠁⠀⠀⠈⠙⠓⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠹⣌⠻⠿⣿⣿⣿⠿⢷⡂⠈⠉⠉⠝⢲⣶⣖⡛⠂⠀⠀⣀⣤⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠈⠙⠒⠂⠀⠙⠻⠶⠿⠟⢲⠲⠋⠛⢿⣿⣿⣿⣿⣿⠿⠛⢮⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⠀⠐⢤⣀⣉⣉⣁⡤⠂⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⠀⠀⠀⠀⠀⢀⡤⣤⠊⠉⠉⠱⠏⠙⢢⡀
⠀⠀⠀⠀⠐⠁⢀⣉⣁⡀⠈⠃⠀⠀⠀⢠⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣦⠀⠀⠀⠀⢀⠀⠄⠀⠂⢄
⠀⠀⠀⠀⢀⢰⠃⠀⠀⠈⡆⠀⢠⠀⠀⢮⡁⠀⡀⠀⠀⠀⠀⠀⠉⣀⠀⠈⣱⠀⠀⠀⠐⢪⠊⠉⠉⠉⢦⢃
⠀⠀⠀⠀⠜⠌⢦⣀⣀⡠⢇⠁⠀⠀⠀⠀⠈⠉⠙⠤⠤⠔⠋⠒⠒⠋⠉⠉⠁⠀⠀⠀⠒⠚⠒⠒⠒⠒⠚⠛⠃
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠤⠊⠀⠀⠀⠀⠀⠀⠰⠀⠀⡆⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢖⠉⡀⠂⠀⠀⣀⣝⣿⡷
⠀⠀⠀⡐⠉⠉⠿⠀⠀⠈⠉⠈⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠁⠀⠀⠀⠀⢀⡀⠁
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣳⠀⠀⠀⠀⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡔⠂⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠙⠒⠿⣤⠤⠶⣤⡤⠦⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠓⡦⠀⠀⠀⠀⣉⣽⠂
⠀⠀⠀⠀⠀⠀⠀⢠⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠥⢤⣀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⢹⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⠒⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢧⡐⠁
⠀⠀⠀⠀⠀⠀⠠⠋⠀⣀⠤⢀⡀⠀⠀⠀⠠⢀⠖⠉⡇⠀⠀⠈⠖⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠈⢣⡀⠀⠀⠚⠒⠒⠁⠀⠀⠀⠀⣈⣉⣉⣉⣉⣉⣉⠁
⠀⠀⠀⠀⡠⠀⠠⡄⠀⠀⠀⠀⠀⠠⡗⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢓⡆⠀⠀⠀⠀⠀⠀⢀⢀⡠⠐⠒⠒⠤⡀⠀⠀⠀⠀⣤⣤⠀⠀⡀
⠀⠀⠀⠘⠦⠤⠤⠤⠔⢀⠀⠐⢰⠒⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠁⠀⠏⠀⠀⠀⠀⠀⠘⣄⡀⠀⠀⠈⠉⠀⠼⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠉⠑⠃⠀⠀⠀⠀⠀⠀⠋⠉⠑⣄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢀⠀⠀⢀⠀⡀⠀⣀⣀⡩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠃⠀⠀⠀⠀⠀⠀⠑⠚⠑⠂⠉⢰⠁⠀⠸⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈
⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢰⠀⠀⠜⠒⠘⠁⠁⠒⢄
⠀⠀⠀⡄⠉⠁⠀⠀⠀⠀⠱⠆⠊⠀⠀⠉⠀⢀⡀⠈⢵⣿⣀⣁⣉⡃⠀⠠⠽⠂⠰⣟⡑⠂⠀⢀⣤⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠈⠀⠀⠀⠒⠶⠀⠀⠀⠰⠂⠀⠂⠀⠀⠐⢇⠀⠸⠁⠉⠉⠁⠁⢀⠄⠀⢀⠀⠀⠀⠀⠀⡀⠈⡛⠀⢩⡉⠀⠀⢀
⠀⠀⠀⠀⠐⡅⠀⣚⣶⡤⢀⣤⠘⠀⠀⠀⠀⠀⣼⠀⠀⣀⡀⠀⣠⠄⠀⠀⠹⡥⠤⠐⠀⠀⡀⠀⠀⠅⠀⠀⠀⠀⠀⠈⠈⠁
⠀⠀⠘⠳⡀⠀⠀⠀⠈⠁⠹⠋⠀⠀⠒⠻⠛⠁⠁⢀⡔⠛⠁⠀⠘⠋⠀⢁⠀⣀⣀⡤⠞⠋⠀⠀⠀⢷⠌⠀⠀⠀⠀⠠⠉⠁
⠀⠄⠀⠀⠀⠀⠀⠀⠀⢐⣤⠜⠃⡀⠀⠀⠀⠀⢰⡾⠆⠀⠀⡀⠀⠀⠃⡄⠐⡏⢿⡄⠀⠀⠀⠀⠀⠘⣖⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠀⢀⠀⠂⠠⠀⢀⡄⢴⠐⢄⠈⠉⠁⣠⠀⠘⡀⢁⡀⠉⠂⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢱⡆⠀⠉⡀⠐⠠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⡦
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡤⣆⠀⠀⠀⡈⡅⠀⡀⠀⠀⠀⠄⠛⢯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠆⠀⢠⠆⠀⠀⠈⠀⢀⠔⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⢢⠀⠀⠃⠀⢀⡤⠆⡉⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⡁⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠇⠀⠀⢠⠘⠆
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣾⡀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⠞⠛⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠈⠉⠀⠐⡄⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠤⠴⣄⠉⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣈⠓⠒⠚⠁
⠀⠀⠀⠀⠀⠀⢀⣤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⡆⠀⠀⠀⠀⠀⠀⠀⢰⣧⡟
⠀⠀⠀⠀⣠⡴⠏⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣅⠀⠀⠀⠀⠀⢸⡇⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⡧
⠀⠀⠀⢸⡇⠀⠀⠀⠑⠀⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠀⣸⠀⠀⠀⠀⠀⢸⠇⣿⡿⠀⠀⠀⠀⠀⢰⡟⠉⢳⡌
⠀⠀⠀⠀⠈⣿⠀⠀⢀⣤⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢁⣀⣀⣀⣀⠀⠈⠘⠙⠁⠀⠀⠀⠀⠀⠈⠳⠤⠞⠁⠀⠀⠀⠀⢀⣀⣠⠤⣄
⠀⠀⠀⠀⠀⠈⠑⠖⠋⠀⣤⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠈⠑⢦⡴⠚⠉⠉⠉⠓⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠈⠁⠀⢸⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠙⠳⣤⡘⠷⣤⡀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣀⠹⣄⣀⠀⡆⡀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⡙⠷⣤⡀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣠⡴⢞⣫⣥⠶⠀⢿⡀⠈⣡⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠈⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⡀⠀⠀⠀⢀⡇⠈⠉⠞⣋⣥⡴⠶⠀⠈⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢸⡄⠀⠀⢠⠎⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠉⠙⠲⣜⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣇⣠⠇⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠽⡞⠉⢲⠀⢠⡏⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢷⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠊⠀⣿⠀⠀⠀⠀⠀⠀⢸⠑⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢸⠇⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠐⠀⠀⠀⠀⠠⣤⣀⡠⠀⠀⠀⠀⠀⠀⠀⡾⠀⠟⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡘⢧⡀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⠄⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣠⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠛⠀⠀⠀⢀⡙⠒⠲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠁⠀⠳⣤⡴⣣⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⣋⡵⠖⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠳⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⣁⣤⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣀⠠⣄⣤
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠳⡄⠀⠐⠋⣁⣴⠞⠋⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠁⠘⢧⠙⣾⢷⡀
⠀⠀⠀⠀⠀⠀⢘⡦⢀⡀⠛⠛⢦⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⣒⠲⠶⠶⠖⠒⠋⣡⡄⠀⠀⠘⣧⠘⢯⡁
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠉⢠⣤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠉⠻⡄⠀⠠⠶⠾⣫⠄⠀⠀⠀⠘⢷⠈⢷⡄⣠⠴⠦⡄
⠀⠀⠀⠀⠀⠀⠹⢦⣀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡳⠤⠞⠁⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⡠⠇⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣩⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⡖⠀⠀⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠁⣀⣀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠊⠁


⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠶⢦⣄⠀⠈⡇⢀⡀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠋⠈⠀⠀⠀⢀⡿⠀⠀⢿⣸⠀⠀⠀⠀⠀⣀⢸⠀⠀⣰⠞⠛⠋⠉⠻⣦⣤⡀
⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⠀⠀⣀⣴⠇⠀⠀⠀⠏⠀⠀⠀⠀⠀⢸⠘⠀⠀⢻⡄⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⠶⡞⠹⡄⠀⠀⠀⢀⣠⡴⠶⠶⢦⣄⡀⠀⠀⠀⠈⢷⣄⣀⠀⠀⠀⠀⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠟⠛⠛⠳⠟⠁⠀⠀⠀⠀⠈⠻⠶⠖⠲⢶⠞⠉⡽⠳⠦⠶⠛⠋⠁
⠀⠀⠀⢀⡀⠀⠀⠀⢠⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠼⠷⣄⠀⠀⠀⢀⣴⠆
⠀⠀⠀⠈⠉⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢀⣴⠿⠆⣠⣤⡀
⢰⡟⠳⢶⣄⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⣀⣾⠋⠀⢻
⣿⡀⠀⠀⢿⡥⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠒⢿⡄⠀⣸
⠸⣇⣀⢀⡾⠁⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠙⠛⠁
⠀⠀⠙⠋⠁⠀⠀⠀⠈⢛⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠶⠋⠀⠠⢤⣀
⠀⠀⠀⠀⠀⠀⠀⣶⠟⠁⠀⠈⠙⠳⠶⠶⢶⣄⠀⠀⠀⠀⠀⣀⡶⠶⢤⢴⠖⢋
⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⢀⣀⣀⠔⢀⠀⠉⠛⠶⠶⠶⠛⠉⠀⠀⠀⠀⢳⣈⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠋⠉⠛⠳⣇⠀⠀⠀⠀⡄⢀⠀⠀⠀⠀⠀⢰⠞⠋⠀⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⣻⠀⠀⠀⠸⢹⠸⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠙⠛⠳⠖⠋⠉
⣿⡿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⢠⢢⠀⠀⡴⠁⢱⠀⠀⠊⠀⠀⠀⠀⠀⣠⢾⠀⠀⠀⢠⠶⠉⠁⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⣠⠀⠀⠀⠉⠀⡜⠀⠀⠀⢣⠀⠀⢀⠀⠀⢀⠔⠁⠸⠀⠀⡞⠁⠀⠀⠀⠀⠀⣾⠼⠁
⠀⠀⠀⠀⢰⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⢦⠀⢀⡜⠀⠀⠀⠀⠈⢧⠔⠉⡆⡠⠋⠀⠀⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠻⢵⣃⡀⠀⠀⠀⡀⠀⢀⠀⠀⡇⠀⠀⠑⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠓⠒⢒⠖⠀⠠⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⠉⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠤⠤⢄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢍⠉⠉⠉⠉⠀⠀⢀⣀⣀⠀⠀⢠⣴⣤⡀⢠⣾⢿⣆⠀⣶⣆⠀⢠⣶⠀⠀⠀⠀⠀⠀⢀⣠⠬⠛⠁
⠀⠀⠀⠀⢰⣿⡷⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⣿⡟⢻⣷⠀⣿⠃⢹⣧⣾⡇⠀⣿⢠⣿⢿⣦⣿⣿⠀⠀⣀⠤⠒⠊⠁
⠀⠀⠀⠀⠀⠉⠀⠀⢀⡠⠖⠊⠁⠀⠀⠀⠀⢻⣷⣿⣧⡘⣿⠀⢸⣿⢿⡇⢀⣿⢻⡟⠈⠻⢫⣿⠀⠀⠙⢆⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠐⠮⢅⣀⡀⠀⠀⠀⠀⠀⠀⢸⣷⠀⣹⣿⢻⣧⣼⠏⠈⠿⠿⠋⢿⠇⠀⠀⠘⠿⠀⠀⠐⣍
⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠉⠉⡱⠆⠀⠀⠀⠘⠿⠿⠛⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠴⠯⠤⠤⠤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⢧
⠀⠀⡀⢠⢤⡆⠴⠒⠂⠀⠀⠀⠀⠀⠸⠓⠒⠉⡇⠀⠀⠀⢀⡀⠀⠀⡠⡀⠀⢠⠛⢦⠃⠈⠑⢤⡀⠀⠈⡆⠀⠈⡦⣀
⠀⠀⠈⠁⠀⠀⣀⠤⣒⠆⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⠔⠉⢃⢀⠞⠀⢣⠀⡜⠀⠀⠀⠀⠀⠀⠉⠢⣄⠘⡄⠀⠀⠉⠷⡄
⠀⠀⠀⠀⠀⠸⠝⠊⠁⠀⠀⢀⠄⠀⠀⠀⢀⠇⢀⠔⠁⠀⠀⠘⠃⠀⠀⠘⣆⠇⠀⠀⠀⠀⢠⠀⠀⠀⠀⠑⠷⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⣜⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣢⠄⠀⠀⠀⢸⣛⠁⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠔⠂⠀⠀⠀⠀⠀⠀⠈⠯⠟⠁