Juniper Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣤⡙⣤⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠳⠀⣧⠹⢸⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⢻⠀⢸⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⣦⡈⣼⡆⢸⠀⠀⠀⣦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣉⣻⣿⣿⡀⢤⠀⠋⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠛⠻⢿⡟⠁⣄⠀⣆⢀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢐⠛⠿⣿⣷⡀⡿⡰⢻⡜⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠉⠁⣠⠄⠰⠞⠃⣴⠾⣵⣿⡇⣈⣷⢠⠇⠀⢀⡄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠻⡀⠀⠀⠀⠀⢴⠃⠛⣭⠀⠀⣤⠀⠀⠀⠈⠳⡼⢀⡴⠋⠠⠀⠀⠀⠀⡠⠁⡀⠀⠀⠀⠠⠚⢀
⠀⠀⠀⠠⠄⠀⠀⠀⢀⠘⣄⠀⠁⠀⠀⠀⠀⠀⢠⠀⠀⠑⢀⠰⠀⣿⢠⢔⠑⠀⠀⢀⡀⠀⠙⡈⠐⢁⠀⠀⢀⡔⢈⠀⢀⢀⠀⢀⣀⠀⠀⢀⣀
⠀⠀⠀⠀⠈⠢⢑⠢⢄⣅⠈⣦⠀⠀⠀⢀⠀⢀⠀⠳⣶⡄⢀⣿⣶⣷⣣⠀⢢⣄⠀⠈⠁⢀⣰⠁⡄⡈⠀⡴⠛⢁⣠⣠⣠⣮⣶⠟⢛⣉⡀⠉
⠀⠀⠀⠀⠀⠈⠟⣋⡷⠭⠿⣾⣦⡀⠑⣈⢂⠀⠳⣄⡹⠟⠛⠈⠉⠑⢧⣀⣼⣧⠕⠒⠛⠻⣿⢀⣴⠁⡸⣱⠇⣨⣾⣟⠉⣉⠥⠀⠀⠤
⠀⢀⡄⠀⠰⢁⣈⡁⢒⣀⠶⠾⠛⠿⡿⢾⣷⣷⢤⣬⠁⠀⠀⡀⠀⠀⠀⢹⡏⠀⠀⣀⡀⠀⠘⣾⣷⣾⢁⣡⣎⡁
⠀⠀⠀⢀⡆⠀⠉⠀⠀⠀⠀⠀⠀⠔⢀⡼⠁⠀⢈⠛⠀⠀⠀⠛⠀⠀⠀⢠⣇⠀⠀⠈⠁⠀⠀⣿⡿⠿⠛⠯⣖⡒⠛⠉⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠂⠀⡀⢀⣶⠾⢧⣀⠀⠀⠀⠀⣠⡾⠿⣠⢄⠀⠀⠀⢠⠏⠀⠀⠀⠀⠹⠛⠛⠮⠭⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠴⠁⠀⠀⠐⠀⠈⠁⠀⠀⠈⠱⣶⡶⠛⠋⠀⠀⠈⠳⣶⡶⡻⠿⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣶⠀⢤⠀⠀⠀⠀⣽⣇⠀⠀⠀⠀⢀⠀⠸⠃⠈⠳⠄⠀⠴⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢂⢈⣁⠀⠀⣰⣿⣿⣷⣄⣀⡐⠃⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠓⠒⠲⠤⣉⡶⢶⡺⢽⣿⣟⠏⠑⢍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⡀⠀⢀⡀⢶⡏⠁⠀⠀⠸⢻⡄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣠⡎⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⣹⢱⡀⠺⠇⠀⠀⠀⠂⠁⠠⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠰⠃⢸⠟⠆⠀⠴⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠬⢀⣼⡀⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢢⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠃⢀⠀⠰⡆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⢆⡐⠀⠤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠄⠊⠀⠳⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣤⠁⠈⠓⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠀⠀⠀⠏⠋⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⣉⡀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣮⠼⠶⣁⣈⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠴⠁⢦⣿⡅⠚⠀⠈⠉⠐⠦⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⡘⢹⡿⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠌⠻⡾⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡰⢏⡜⠀⠃⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⣔⠓⠂⠀⠀⠴⢼⡃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠄⣤⢀⠁⠀⠀⠀⠠⠤⠂⢀⠀⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡹⢇⡑⠀⡼⠂⣤⡎⠉⠁⠀⢀⠠⡤⠈⡌⠈⠛⠘⠃⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⢉⡍⣱⡮⠁⢰⡁⣤⠟⢣⡊⠆⠀⠀⠀⠀⠀⡆⠀⠀⠛⠋⠁⠉⠒⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠈⠁⣉⠉⠁⢺⡍⢱⡟⣤⢀⡀⢠⠀⡂⠀⠉⠁⠁⠀⠀⠀⣴⣀⣁⣰⡤⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⣧⣄⣀⡀⣀⠐⠂⢠⣄⠀⠄⠀⠀⠘⢪⣙⡄⠀⠀⠹⠀⠀⢸⢷⡟⠀⠆⠈⠻⣏⠯⠁⣀⢀⠀⢠⡄⠶⣆⠘⢉⠽⠋⠁⠈⡛⠖
⠀⠀⠀⠀⠀⠀⠉⢙⡋⠛⠻⣤⡉⠀⣀⠙⠇⢀⠀⠀⠈⠱⢽⢿⣠⣾⠃⠘⠲⠏⢸⢤⠶⠐⠒⢰⣮⡶⣀⠿⠛⠐⣃⠉⢀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⡄⠀⠄⠀⠀⠐⢻⡅⠦⠈⢚⣽⡄⠂⠀⠸⠄⠻⠀⣦⡀⢨⣀⠙⠂⢽⡀⠘⠱⡆⠰⠂⡹⠀⠀⣻⠾⠃⠠⠄⣤⢉⡒⠀⠠
⠜⢦⠠⣶⣄⢲⡄⠀⢠⡄⡤⡈⣽⣦⠁⠰⣟⣛⡿⣟⢺⡸⢷⣩⡒⢿⣻⣹⢃⣴⣄⠄⠸⣇⠾⠦⣭⠶⠄⣁⠀⠺⣏⠀⠃⡀⡀⢐⡀
⠀⠀⠀⠀⢿⠄⠁⢠⣼⣟⢣⣽⢿⣿⠾⣶⠬⠻⠶⢏⣙⡦⣀⠉⣈⠁⣈⣟⠉⠅⣤⡦⣠⣿⣀⣴⠁⠶⠀⠀⠈⠘⠂⠄
⠀⠀⠀⠀⠀⠐⢦⣀⠘⢿⣾⠃⠀⡈⡁⠉⠲⢦⡀⠀⠹⠀⣿⠀⠙⠻⣬⣍⣴⠂⠙⣃⡿⠿⠭⠾⠠⠶
⠀⠀⠀⠀⠀⠀⠀⢡⣤⡴⠁⠳⠄⠰⠀⠀⠀⠀⠡⠦⣄⣰⠟⣦⡀⣠⡿⢠⠟⠁⣠⡿⣀⠀⢠⠀⢀⣄⣄⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⢤⣾⣠⠄⠀⠀⠀⡀⠀⠈⢳⣄⢘⡙⢻⡅⠸⣦⣼⠿⡆⠀⢲⢆⠀⣠⢼⠿⣿⣶⣖⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⣤⣄⣤⠛⠅⢺⠆⣀⣉⢿⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⡘⣶⣿⡛⣻⡟⠳⠟⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠙⢻⣉⠿⣿⣤⣤⣿⠃⠟⢹⡗⢹⣿⣘⠁⠀⠰⢤⢄⡶⣋⠉⠙⠛⠿⠟⠓⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠙⢻⡄⠀⢿⢿⣿⣤⣤⣾⢟⣿⢣⣟⠂⢣⣄⠼⠋⠡⠌⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⡄⠾⠀⠋⣽⡟⠷⢌⠙⢻⣾⠛⠉⣁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣹⣀⠀⢀⠀⠈⠂⡐⢀⣿⣤⣧⣀⣭⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠻⡀⢀⠀⠀⢰⡇⢸⣿⣼⣿⣿⠋⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⢄⡟⠀⠟⠸⠛⠟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣴⠆⢸⠀⠰⡆⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⣷⡇⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢢⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⡄⠀⠀⠀⠀⠀⢀⠀⠀⡀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣾⠇⠀⠀⠀⠀⠀⣸⣿⡰⢱⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣤⠂⠀⠀⡀⠀⣿⡼⣱⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⣼⡿⢁⠀⠀⠀⠱⣄⠣⡿⠖⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⡧⠊⠑⢦⡀⠀⠙⣧⣠⡤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⣿⣄⣠⡄⠁⠙⠿⣈⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠐⣾⠉⡀⠀⢠⡀⡔⠙⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠠⠀⠀⠀⢠⠀⡼⣇⣸⠄⠀⠀⠀⠀⠀⠀⠐⢦⣨⣿⠴⠊⠻⡷⠿⢤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⢠⡆⠀⠀⠀⠀⢃⠃⡏⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢅⢤⢀⣎⡤⠆
⠀⠀⠀⢀⠀⣄⠀⠀⠀⠈⠺⣷⠅⢀⡀⠀⢘⢰⡧⠀⠀⠀⠀⠀⠀⠀⠈⢄⠸⣟⠀⢈⠿⠟
⠀⠀⠀⠀⠀⠺⣆⠄⢀⠀⠀⢿⠀⡆⠈⢠⠘⡟⣡⠆⠀⠀⠀⠀⠀⠢⣄⠀⠲⢬⣷⣡
⠀⠀⠀⠀⠀⠀⠈⢻⣆⢢⠀⠱⢷⠃⠢⠰⣛⡟⢁⠄⠀⠀⠀⠀⠀⠈⠺⣦⡾⣀⢻⡟⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢓⣤⡀⢷⡆⢸⢮⠗⠃⠀⠀⠀⠀⠐⠒⢀⣿⣿⣷⣏⠤⠽⡤⠀⠀⠀⠀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⣢⡇⣼⣸⡄⣄⣰⠃⠀⠐⢖⣻⡿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⡀⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠀⠀⢻⣿⣰⣿⣧⡤⡤⣈⣛⣿⣇⣾⣿⣿⣿⣿⡿⠀⠀⠀⠰⣷⣰⠡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⡿⣿⣿⣿⣇⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡗⠂⠀⢀⠦⡼⢡⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣹⣿⣧⣼⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣤⣼⣿⡟⠈⠐⣀⢱⡴⢶⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢾⡟⢿⣿⣿⡿⠷⢻⣿⡿⣿⣿⡿⣟⡉⠉⠁⠈⠀⣿⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠳⣿⠿⣿⠙⣷⣼⣧⣤⣽⣿⠇⠀⠈⢉⣠⠤⢦⣇⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠟⣿⣿⡥⠖⣶⣿⣿⣿⣷⣾⣤⣍⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡠⠔⢻⡆⠀⠻⣿⣿⣿⣿⡉⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠙⠦⣬⣧⡔⣿⡋⢻⣿⡟⠻⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⡄⠈⢻⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣴⣯⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠾⢿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡙⠦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢄⠀⣠⠄⠀⠀⠊⠀⢀⠄⠀⠀⠀⠀⠀⢀⣀⣨⠗⠀⠀⠀⢀⠉⢹⡆⠀⠀⠀⢀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠚⠛⢠⡄
⠀⠀⠀⠀⠖⠉⣶⡅⠀⡠⠃⣀⡜⣡⠈⠐⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠳⠞⠃⠀⠀⠀⠈⣻⣶⠀⠀⠘⢿⣷⣄⠀⠀⢈⡀⢻⡇⢠⡄⠀⠀⢀⡤
⠀⠀⠀⠁⠀⠀⠇⠉⠻⡦⢞⣁⡇⠙⢿⢿⡧⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠉⢛⡆⠀⢀⣀⠈⣹⠆⣶⡦⢬⡽⢿⣿⠳⠂⠈⠉
⠀⠀⠀⠀⠀⠀⠄⠈⠃⢠⠄⢪⣽⠀⣹⡿⡖⠛⢋⠉⠉⠁⠀⠀⣷⣤⡾⠀⠀⢀⡈⣹⣷⠴⣶⡷⠂⣷⣤⠟⠛⣾⣟⣉⣇⠃⠈⠛⠛⣱⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⢾⣀⣼⡿⢰⠉⡇⠻⠴⠞⠀⠀⠀⠀⠀⠀⠁⠀⠀⡠⠏⠍⠀⠁⠐⠁⠘⠛⢛⣄⣤⢵⢿⠈⠹⢹⣂⠊⠀⠀⣼⠁
⠀⠀⠀⢀⡤⢤⠀⠀⠀⠀⠉⠁⠀⠊⠂⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠿⠷⡟⠿⠀⠀⠉⠋⠁⠀⠜⠛
⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡀⣠⢤⣀⣠⡐⢢⣠⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠆⠀⢀⣤⠷⣼⡋⢻⠚⣿⣉⣽⡿⢧⡷⣤⠄⠀⠠⡈⢶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⠀⢀⣠⣾⣠
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⣬⠗⣁⠀⢺⡉⠲⣿⣽⣟⣙⣿⣧⣼⣏⠾⠗⢁⡄⠀⠀⠀⢦⢔⡡⠀⢀⠀⠀⠀⠀⣤⢠⡀⠁⣽⡿⢀⠀⡙⠻⢫⠀⠀⠄
⠀⠀⠀⠐⢄⣛⣠⠊⠀⡀⣿⡷⠊⠀⠀⠀⠉⠉⠛⠒⠦⠤⠽⠄⠄⠐⠓⢻⣿⠀⠀⠈⠒⢿⡁⡔⠁⣺⠖⠀⣈⠲⣷⢈⠅⡐⢦⣧⠁⠀⢄⠽⠣⣤⠊
⠀⠀⠀⠀⠂⣼⡗⠠⠆⢘⣿⣤⠄⢀⠄⠀⢀⣆⠀⠀⠀⠀⠀⠀⠀⠠⠀⣸⡆⠀⠀⢠⣄⣸⣿⠷⣿⠭⡑⠄⠀⢤⢤⣮⡀⠶⢾⡿⢷⣤⣿⡸⡶⢛
⠀⠀⠀⠀⠀⡙⣶⣧⠀⠉⢫⢀⢣⡦⡂⠀⠀⠻⣇⢀⡀⡀⠀⡀⢁⡎⣰⠟⠀⠀⠀⠈⠚⠻⢾⠶⠿⠀⠀⠀⢤⡡⡈⢸⣧⣠⣾⠿⠈⠙⡀⣴⡓⢄
⠀⠀⠀⠀⢀⡌⠉⣻⣦⣾⣿⡼⠷⠓⠂⠀⠀⠀⠘⢿⠛⣷⣺⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡾⠗⡟⠉⣼⠀⠙⣹⣦⣄⠀⢰⣿⠘⠈
⠀⠀⠀⡰⠙⣿⣿⢿⣁⢀⠿⠧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠘⠿⠀⢊⣠⣾⣏⠃⠈⡹⡦⡉⠀⠉⠸⠇
⠀⠀⠀⠻⠾⢿⣯⡿⠬⠧⣄⠀⠀⠀⠀⠀⠀⢀⠀⢠⡀⢠⣤⠀⠄⠐⣂⣀⠤⠔⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠜⠃⠃⠀⠀⠀⠅⠈
⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠈⠃⠀⠀⠀⢀⣴⠛⢓⠈⣻⣇⣠⣶⣏⡜⠋⠁⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⣄⡀
⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣶⣟⢉⣧⡼⢿⣭⠊⠀⠀⠀⠀⠀⠀⣿⣀⣀⠔⠀⠀⠀⠀⠀⠀⠀⢸⡃⢀⡴⠀⠀⠀⠉⢈⡿
⠀⠀⠀⠀⣀⣀⡽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢽⣧⣤⣤⣎⢀⡶⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁
⠀⠀⠀⠀⠀⠈⢀⣠⣤⣶⣶⣦⣤⣀⠀⠀⠀⠀⢀⠀⢀⡀⠉⠉⠁⠀⠀⠀⠀⠀⠌⠒⡀⠀⠀⠀⠀⣀⠄⣻⡀⠀⠀⠀⢠⡄⡀⠀⠀⠀⣀⠀⢀⣤
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠸⠶⠾⠁⠀⠀⢀⠀⠠⢶⣧⠄⠀⠐⠰⢆⠀⠀⠐⠿⢟⣯⡐⣀⠀⢄⡐⢻⡤⢂⢀⠤⣿⣤⣿⣍
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠰⠦⢷⠶⢘⡓⣄⢀⠀⠈⢗⠖⠀⠀⠀⢙⣎⠻⡕⠀⣾⠟⣿⢟⡾⣿⢃⣈⣭⡗⠂
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢤⣆⣫⡔⢂⠘⠡⠨⠞⣢⡼⠣⠖⠌⡶⠈⡿⣷⣾⠶⠿⢖⣹⡟⠋⠉⣛⣭⡟⠒⠒⠢
⠀⠀⠀⣿⣿⠛⣿⢻⡟⢻⠛⣿⡟⢻⡟⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⡚⠁⠀⠀⠀⣶⡿⠃⠈⠀⠀⠀⣈⠛⠀⠉⠀⠀⢲⣾⡇⠁⠄⢹⠏⠀⠁
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⣀⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣦⠀⠀⠀⣨⡔⢟⣼⠂⠁
⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⢀⡈⠁⢉⡷⠀⠀⠀⠀⠀⠀⠰⠞⢶⡄⠀⠀⠘⠛⠃⠁⠀⠀⠈⠁⠸⠿⠈⠀⠀⠀⠀⠀⠀⣴⠀⣰
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢂⠀⠈⢂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢤⡈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⠚⠊⠉⠀⣀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡂
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣹⡗⠀⢃⠀⢠⡄⠀⠀⠀⢈⠆⠘⠛⠣⡀
⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠳⣄⠄⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠂⠀⠲⠿⠟⢻⣟⠀⢰⠀⢀⠀⠀⠀⠰⡄⠠⠀⠀⠀⡟⠀⡀
⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⡀⠀⠀⠀⠀⡀⣠⠿⠗⠒⠠⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⡦⠊⠁⠉⠣⡀⠀⠀⠹⣦⣤⢤⣴
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⣤⠗⠈⠁⠀⠉⠛⢷⠀⠀⠀⠰⠀⠠⠀⠀⠀⢀⠴⠀⠀⠀⠀⠀⠀⢠⣠⡀⠀⠀⠀⠀⡁⠀⠀⢈⠁⠌⣏⠥
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠃⠀⡀⠀⠀⠀⠀⡌⡆⠀⠀⠀⠀⠀⠆⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠷⢶⣶⠚⠁⠀⢠⠘⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠀⠀⠀⠀⢀⠅⣗⡀⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠸⡄⠌⣰⠋⠁⠈⠳⠂
⠀⠀⠀⠀⠀⠀⠦⠄⣀⣀⣹⣶⣤⢄⣀⣠⣺⠞⠋⠹⠡⠕⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⢀⣤⣤⡽⡇⢠⡇⡀⣿⡀⠀⡠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⢀⠠⠄⠛⣫⠞⠠⣈⠻⠿⠃⠚⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠿⠀⡆⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠡⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢐⠁⢰⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠓⢄⣠⡀⡀⢀⠀⠀⠀⠐⢤⡀⠀⢀⠈⠀⠀⠀⠹⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢹⣇⡠⠇⠀⠀⣀⣠⠀⠀⠀⠁⠀⠀⢀⠀⢱⡀⡆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠞⠁⠀⠀⠀⠈⠏⠁⠘⢄⠀⠀⠀⠰⠀⠾⢛⠎⡄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⣀⣀⢀⢀⢀⡀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⢀⢂⠃⠀⠀⢸⡄⠀⠀⠀⠀⠀⢸⠀⠃⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠣⠟⠽⠿⠿⠽⠋⠿⠆⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⠚⠋⠛⠷⠶⢶⣾⠿⠦⠀⠀⠀⢀⡬⢣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠐⠦⡤⠥⠒⠂⠉⠈⠛⠐⠒⠁⢉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠐⠀⠀⠂⠂⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⠃⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡠
⠀⠀⠀⢠⡀⠀⣈⢢⠄⡫⠊⠀⢀⡀
⠀⠀⢀⡠⠄⣤⠞⣿⣉⣀⣄⠤⠊
⠀⠀⠈⠉⠁⢠⡾⠚⠋⠐⠲⢶⠤⠤⠰⠦
⠀⠀⠀⣠⠎⡟⠀⠀⡵⠶⠀⠈⠷⠒⠲⢤⠀⠀⠀⢀⣀⠀⢰⡄⢀⣔⠃
⠀⠀⠀⠉⠰⢧⢀⠈⠀⠠⠠⠌⠐⡀⠀⡀⢱⠀⠒⢀⢍⠛⠒⢻⣮⣞⠚⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠓⠤⣈⣬⢤⠚⡄⡀⢄⠀⡼⠀⠀⠤⠬⠻⢶⢾⡁⠑⠭⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣙⣯⢦⠔⠊⠠⠒⣪⠽⠿⢽⣿⠷⢤⣀⠲⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⡋⢻⣂⠀⠠⡞⠀⠀⠀⠀⠙⡄⠀⠀⢱⡀⠀⠀⠀⠀⡀⢠⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⠞⠋⠈⠃⢣⡀⣤⡴⠀⠀⠀⠀⡇⠠⡀⢲⠋⠀⠀⠀⠀⢑⢌⣕⡠⡿⠁
⠀⠀⠀⠀⢳⢄⣠⠈⡃⡸⡁⠀⠀⠀⠀⠀⢻⠾⠿⣌⡀⢡⠚⠀⠀⠖⠁⠀⠀⢤⡀⡀⢧⠉⡿⠛⠴⠀⠒
⠀⠀⠀⠀⢠⣡⡃⠃⢠⠹⠍⡀⢐⠠⡀⠀⣝⢣⢸⡈⢧⣠⠼⠐⡡⠃⠀⠀⠀⠀⣅⠁⠻⣿⠥⣤⠠⠤⠆
⠀⠀⠀⠀⠀⠴⣟⣧⠾⢴⣷⠤⠬⠤⠤⠾⣿⢋⣀⠱⡀⠀⠉⠉⠀⣀⠤⠌⣓⠲⢘⣷⣴⣁⣀⡀⠁⢦
⠀⠀⠀⠀⠀⡴⠚⠎⢃⣼⠿⠀⠠⠀⠀⠀⢸⠼⠏⠑⢌⢦⠀⢀⠊⠀⢰⣄⡄⠱⡀⢨⠏⠛⠭⣄⡀
⠀⠀⠀⠀⠈⠀⠀⠚⠀⠘⠏⢄⠠⡆⠰⣈⠞⠘⡾⠀⣈⡧⣕⢾⠀⠀⠉⠀⠀⢐⡿⣿⡟⡑⠓⠦⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠳⠀⠉⠀⠈⠁⠀⠀⠁⣚⠝⠒⠛⠃⢵⠅⣀⣀⠤⠊⣠⡏⠈⠪⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢷⠮⣕⠢⣉⣽⡏⠓⠦⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢘⣇⡀⡀⠀⠌⡜⠀⠈⠑⠢⣄⡓⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢤⣉⣀⡥⠞⠀⠀⠀⠤⠄⢀⣍⣳⡤⢭⣐⠢⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡠⢛⣷⠄⠀⠉⠒⠦⢤⣉⡉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠈⠆⠙⠆⠀⠀⠀⠀⠀⠉⠑⠋

⠀⠀⠀⠀⢼⣶⡀⠀⢸⢠⠆⡔⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄
⠀⠀⠀⠀⠪⠻⣧⠤⣼⡏⢰⠗⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣧⡿⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠇
⠀⠀⠀⠀⣒⣤⣦⠴⢿⠉⣥⡤⢤⣠⠞⣡⠀⠀⠀⠀⠀⠢⣄⠀⠐⢻⡾⢧⣼⡷⢂⡆⠀⠀⠀⠀⠀⠀⠀⠑⣼⣡⡇⠀⢀⠀⢿⠤⠊
⠀⠀⠌⢦⣟⣿⡯⠓⢼⢻⣟⢷⡺⣭⠀⠀⠀⠀⠀⠀⠀⠀⠚⠢⠆⢸⠿⣿⣿⠀⡌⡇⡀⠀⠀⠘⣸⠰⠀⢀⡐⢚⡩⢄⣬⡷⡴
⠀⠀⠒⠪⢿⣯⣒⢻⣾⡞⣧⣾⡧⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡉⢀⠿⣷⠀⢏⣿⠃⠀⠀⠲⣽⡧⢂⡠⠸⢷⣀⣼⣟⡓⠊
⠀⠀⠀⠒⠩⠚⡟⢻⣯⣻⣧⣼⣷⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠘⡷⢿⡶⢿⣡⠀⠀⠀⢒⡡⢞⣟⡁⣲⢼⡷⠓⠂⠄
⠀⠀⠀⠀⢒⣶⢛⣻⠋⣿⡏⢸⠉⠂⠀⢙⠾⣆⣶⣆⢀⠀⠀⠀⠀⠀⠹⠓⠾⠃⠞⠁⠀⠀⠀⠀⡆⢸⡖⣫⣽⢈⣛⠁⠀⢠⠀⣴⢇⠂
⠀⠀⠀⠀⠙⢁⡏⢿⢙⢿⠀⠈⠀⠀⠀⠀⠈⠛⠛⢻⢻⠀⣀⢀⡀⠀⠀⠙⠛⢓⠇⠀⠀⢀⣀⠤⡷⣨⡟⢤⣼⠿⠕⠒⢁⢸⣅⡩
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠈⠛⠏⠀⢹⢸⡧⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣈⡴⢛⢫⣔⡀⠆⣨⣔⣷⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⠉⢻⣡⠞⣟⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⣄⠺⢥⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣠⡄⠀⠀⠈⠙⢫⣍⣉⡥⠂⠘⢁⡄⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠐⠒⠦⣇⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⠃⡛⠀⠀⠀⠀⠉⠉⠉⠀⠈⡄⠛⠉⠀⠀⠀⠀⠀⢢⣼⠇⠀⢀⣄⡀⠀⠀⠐⠠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢥⠆⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⠀⠀⢀⡸⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠿⢿⣧⠰⣶⣿⢛⣿⣿⣿⠋⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠁⠀⠀⠈⠻⣷⣼⣧⣾⣿⣿⣯⡀⠀⠀⠉⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⣠⣿⠟⠛⣿⣿⠆⠻⣿⣄⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⠀⠀⠀⠀⣴⣿⠃⠀⠀⠀⣿⠀⠀⠈⢿⡳⡄⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⢀⡀⢰⣿⣀⣀⠀⠀⢻⡹⣄⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣾⣿⠿⣃⢸⣿⡟⠻⠛⢿⣿⣿⣿⣦⣀⡀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⡾⣟⣽⠛⠉⠀⣠⣿⣿⣿⣿⣿⡄⠀⠀⠉⣿⣼⡍⠛⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡽⠛⣿⡏⠀⢀⣶⣿⣿⠀⣿⣿⣿⣿⣦⡀⠀⠘⣿⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⠁⠀⠈⣿⠀⠀⣘⣿⣿⣿⡄⢿⣿⣿⣿⠛⢿⣦⡀⠉⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⣿⢀⡤⣽⡿⠻⣿⡄⣿⣿⣿⣟⠛⠿⢿⣿⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⢀⣼⡿⣱⣿⠁⠀⠈⣡⣿⡁⠘⢿⣷⣄⠀⠀⢫⣜⠻⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠓⣿⡇⠀⠀⣠⡟⣿⣿⠦⢀⣵⡻⣦⡀⠀⢻⣇⠀⠉⠛⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⢸⣿⠀⠀⢠⣿⠁⠙⠁⠀⠈⣿⣷⡈⢷⣄⠀⢻⡆⠀⠀⠀⠉⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⢰⣼⡏⠀⠀⣾⡇⠀⠀⠀⢀⡾⠋⢉⣡⡄⠹⣆⠀⠻⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⢸⣿⠇⠀⣤⣿⠇⠀⠀⠀⣼⣿⣿⣿⣿⣿⣦⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⢸⡿⠀⠀⢻⣿⠀⠀⢰⣮⠻⣿⣿⣿⣿⠋⠀⢣⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⢸⡿⠀⢀⠰⢿⣦⣄⡀⠐⢿⡀⠀⠀⢣⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣴⠟⠁⢹⣧⡉⠻⢷⣾⣿⠀⠀⠀⠻⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⢀⣠⡄⣿⣿⣦⠀⠙⣿⣧⣄⠀⠀⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⣿⡇⠀⠈⢿⣇⣿⣧⢻⣷⠀⢹⣿⠋⢿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⢇⣿⡇⠀⠀⢸⣏⢸⣿⣏⢿⡇⠀⢿⠀⠀⠙⠻⠖⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠌⠁⠘⣿⠀⢲⣶⣿⣿⣦⢿⣿⡈⢿⣄⡈⠀⠀⠀⠀⠈⠑⢉⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠉⠀⠀⠀⠀⠀⠀⡹⣏⣿⡏⠸⡏⠀⠘⣿⡿⠶⣦⣄⡀⠀⠀⠀⠙⠲⢵⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⡠⠞⠀⢹⣿⡇⠀⣧⠀⠀⢻⣧⠀⠀⠉⠛⢷⣦⡀⠀⠀⠀⠈⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⢀⡼⠁⠀⠀⣸⡿⣧⠀⢸⣆⣀⡈⣿⡄⠀⠀⠀⠀⠉⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢀⠎⠀⠀⠀⢀⣿⠀⠹⠇⠀⢸⣿⡙⢿⣿⣶⣤⣀⡀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⢸⡿⠀⠀⢐⠀⠈⣟⠷⡢⠛⠀⠈⠉⠻⠷⣦⣄⡀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⣿⡇⠀⠀⡈⠀⠀⠙⡆⠙⠣⡀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⢰⣿⠃⠀⢀⡇⠀⠀⠀⠰⢆⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⠃⠀⠀⠀⠀⠘⣄⠀⠀⠘⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠙⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣠⠄⠈⢀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢰⡄⣠⣿⣿⠃⠀⢸⣁⢴⠀⣸⡀⠒⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣎⣷⡠⣤⣶⢀⡸⠸⠃⠠⠈⢿⣿⣿⡿⠀⠀⣿⠿⣫⡴⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢰⠛⠺⣟⣿⣿⣷⣷⣦⣿⣷⠿⠀⠀⠈⢣⢷⣾⣿⠃⣸⣿⣿⠻⠛⠋⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣡⠧⠘⢗⢸⣬⣻⣿⣿⣁⠀⠈⠀⡀⠀⢹⣽⡏⠑⣿⣿⡟⡖⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⡾⡍⣻⣿⣯⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣿⣿⢟⣫⣾⣾⣟⣶⣺⣫
⠀⠀⠀⢀⣄⣀⣤⣤⣤⣻⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣯⣄⣉⣌⡙⣏⣱⠏⢞⣿⣶⣖⣰⣁⡀
⠀⠀⠀⠠⢸⣿⣄⡔⠋⣠⣁⣨⣿⣟⣿⡿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠘⠉⠁⠀⠀⠀⠐⠉⠈⠉⠌⠙⠋⠉⣻⣿⣛⣿⣿⣿⣿⠁⠘⣹⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢛⡙⠻⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣾⣿⠿⣿⣿⣷⣶⣯⣸⣯⣼⣿⠛⠟⢛⣿⣿⠿⠿⠿⠟⠠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⣸⡿⠀⠈⣹⣿⣿⣿⣯⡥⡄⠀⠺⠋⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠺⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡆
⣤⡀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡄
⠉⢙⢟⡽⢣⣠⡄⠀⠀⠀⣀⣴⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣤⣴⣞⣿⣾⣿⡇
⠘⠂⢠⣔⣼⣿⠛⣛⡀⣠⠿⣵⣻⣿⣷⣶⣄⡀⠀⣀⣀⣀⣠⢶⣿⣿⣿⣿⣻⣿⣿⣷⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠶⡁⠨⠀⠈⠓⠐⡆⠀⢫⡙⣿⡆⠽⠿⠀⢭⣽⣻⣿⠿⣿⣿⣿⣿⣿⡿⠟⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿
⢂⠀⠾⠉⣼⢇⠉⣶⣱⣷⠔⠫⣅⠙⣻⡡⠄⢰⢂⠈⢉⡻⣶⣴⣿⣧⣿⣟⣠⣿⣿⣿⢛⣟⢿⣦⣦⣤⣴⢀⡴⠶⣾⣛⣛⡋⣹⡛⠁
⠀⢀⠀⠠⠇⣀⣸⣿⣿⠟⢀⣥⣈⡦⢻⣴⠶⣠⠿⣧⣰⣷⡾⠑⠁⣿⣿⡛⠋⠩⣼⣿⣽⣿⣿⣷⠿⢿⠅⠠⣤⢷⠾⠾⠃⠰⠋⢐⡗
⠀⠘⣠⠰⣿⠯⢻⣭⠉⣉⣤⣮⠉⠑⢬⣷⣆⠀⠈⢹⣿⡿⡇⣸⠋⢙⣃⡏⢀⡿⢋⡽⠿⠟⢻⡇⡶⠓⠿⠿⠘⢽⠖⠚⠛⠃⠻⡖⠓
⡀⠀⠁⣰⣿⡏⠻⠛⢋⣹⡟⠹⢍⣤⣘⡟⠻⢗⣻⣿⢷⠛⣿⡒⠢⣹⡉⡃⠀⠀⠀⣁⣈⡀⢸⣟⠟⠃⢀⢀⡀⠛⠒⣀⡘⠼⠃⠛⠂
⠀⠀⣘⢻⡟⠓⡿⣋⢿⡿⠟⠒⣴⡏⢉⣌⣀⡼⠿⠤⣾⣻⣿⡖⠊⣻⡀⠐⠿⣗⣈⡥⠄⠀⠈⢯⢿⣹⣦⡈⠉⠀⣠⠄⢩⣌⣥⡀⣼
⠆⢠⡛⢋⠖⣷⣾⠿⠿⠷⠂⠀⠉⠁⠈⠙⡙⢡⣤⠛⢛⢁⡈⢈⢛⠀⠋⣀⡓⠸⠏⣿⠰⣆⠀⠀⠀⠙⠿⢁⠀⣀⣀⣼⣯⣿⣶⣿⡟
⡤⢍⣨⡗⣉⣙⣠⢿⣦⡤⣴⠏⠀⢠⠤⠌⠓⢀⠈⡁⢛⡯⣠⣿⣿⠀⣿⣿⣡⣷⣎⡨⢨⣿⣦⠄⣀⠄⠀⠀⠙⢿⣿⣿⣷⣾⣯⣼⣿
⣤⠀⠉⠉⠁⠀⠀⠠⠈⣠⣦⢿⣾⣿⣿⢾⠖⢀⡀⢘⣃⣴⣿⡟⣯⡾⡋⠰⠘⣫⣀⡈⠈⣹⣿⣾⣿⣧⠀⠀⠀⢀⣭⢉⣭⠙⠛⢳⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣶⡋⢁⣠⡤⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠠⡄⢹⣿⣿⣿⣿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢷⣸⡛⢛⣉⢁⣤⠐⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⣇⠈⡠⡖⢿⣷⣄⣘⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠙⣾⣿⣄⣀⣿⠿⠋⠀⠀⠀⠀⠀⠀⡆⣧⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣄⣠⣄⠈⡀⢿⠀⠿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⢀⣄⠹⢋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⣿⣿⣿⡇⡹⡜⠳⢛⡛⢉⣃⡄⠐⠶⠒⠀⠀⠀⢹⠸⡾⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠘⡣⠱⣸⠋⠉⡉⢉⣥⣉⣀⠀⠀⠀⠀⠀⠈⢦⣴⣏⡀⢐⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢐⠀⠀⢀⣨⣝⣿⡿⣠⠿⢋⢁⣚⣉⣉⣀⡀⠀⠀⠀⣀⠴⠾⡄⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⣿⣿⣿⣿⢹⣿⣶⣶⣏⠉⠉⠁⠀⠀⠀⡲⢿⣷⣄⣀⣴⣤⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠐⡀⠈⠛⢛⡟⠁⠘⠿⠿⠛⠁⠀⠀⠀⠶⢩⣥⡀⠀⠙⢳⣿⣿⣇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣴⣷⣶⣿⡆⣠⠴⠘⠻⣆⣴⠞⠁⠀⠀⠀⠀⠀⡠⢐⢨⣁⣾⣶⣼⣥⣀⠀⠈⠛⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢹⣿⣿⣿⡿⣿⣧⣼⣷⣾⠛⠀⢘⡦⠒⠂⠺⢧⣀⠀⢸⣿⡿⢿⣿⣭⡛⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠈⠙⣋⣉⣹⣿⣿⣿⣿⠠⣴⣾⣧⣍⣉⡀⠀⠉⠑⠠⣉⡛⠋⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡿⣻⣷⣿⣿⣿⣿⡿⠿⢟⡈⠩⣭⣁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡩⠾⠋⠀⠀⢸⣿⣹⣤⣠⣴⣍⠰⣭⡓⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠶⠿⣶⣾⣿⣿⡿⢿⣿⣿⡿⠋⠀⠀⠙⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⣧⣷⣷⣤⣿⣿⣟⠉⠀⠘⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣾⡇⡇⠘⣿⣿⣿⣿⣿⢧⣝⠷⣄⠀⠀⢌⠛⠦⡄
⠀⠀⠀⠀⠀⠀⣤⠴⠛⠉⢰⣿⢻⣷⡇⢿⢸⡍⠄⠙⣧⡙⢷⣌⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣿⢸⠏⡇⢸⡆⠳⠀⠀⠈⠳⡀⠈⠓⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣁⣁⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢠⡆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠢⢐⠻⢠⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⢠⠀⠀⠀⠀⠀⠈⠀⢿⢠⡆⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢠⡀⠀⠀⢀⣀⡟⠀⠀⠀⠀⠁⠀⠘⠀⡀⠀⠀⢀⠄⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠀⠀⠀⠀⡄⢡⠀⠀⠈⠟⠁⠀⠀⢀⠀⠀⠀⠇⡄⠃⠀⣶⡤⠦⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⣿⠘⠂⠀⠀⠀⠸⠀⠀⠔⢉⡶⠂⠀⠀⠈⢠⡤⠀⠀⠀⡀⠚⠋⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠧⢸⡃⠀⠀⣌⠀⠀⢀⠀⠀⠀⣸⠔⡶⡦⠀⠀⠀⠀⠀⢀⡄⠡⠄⠀⠤⢄⠀⠐⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⢇⠀⠀⠈⠀⠀⠘⡇⠀⢐⠏⠀⠀⠀⡆⠀⠀⠀⢠⣾⣷⠁⠀⠉⠀⠉⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⡄⣐⠀⢀⠀⠁⠀⢀⡆⢠⠂⡀⠐⣚⠒⠢⣀⣿⣿⠁⠀⠁
⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠤⢄⣀⢳⣿⡀⣸⣀⢠⠿⣾⣧⣎⡴⠁⠈⢠⡀⠀⠈⣿⣛⠁⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠄⠀⠀⠠⣤⠄⠀⠀⠀⠉⢛⣛⣷⡿⠀⠈⠀⠁⠘⠟⢃⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠁⢃⠂⠖⢂⣤⡤⠾⣻⠃⠀⠀⠀⠀⠀⡀⣌⡢⣀⣀⣀⣀⠀⠁⠀⠀⡀⠀⠀⠀⠉⠀⢾⣤⠀⢄⣀⢀⡀⢀⢀
⠀⠀⠀⠀⠀⠔⠀⠀⠈⠁⠐⠊⠀⣀⠘⠉⠁⠀⠀⢀⢰⡠⡄⢠⣿⠱⣽⣿⡿⢫⡶⠶⠄⠀⠀⠀⠀⢀⣤⣴⣥⠞⢊⣁⣩⡭
⠀⠀⠀⠀⠀⠠⠂⠀⠂⠀⠂⣩⠝⠙⠀⠀⠀⠀⠀⢈⢏⣻⢄⠂⠈⠀⣿⡞⠑⠄⠀⠀⠀⢸⣄⠀⠈⠉⠀⡠⠴⠾
⠀⠀⠀⠀⠀⡠⠂⠀⠀⢠⠄⠁⠀⠸⡦⢄⡀⠀⠀⠀⢈⣷⡄⠀⠀⣀⡘⢷⠀⠀⠀⠀⢠⣼⣧⣀⣀⣀⡀⠠⠄⠀⠀⠀⠢⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⢄⣤⡤⠤⠾⠀⠀⠐⠤⠈⢻⡌⠷⠠⣸⣯⣟⣭⣁⣀⣀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠀⠀⢠⣤⣀⣀⠤⣰⣖⣄⣀⠀⠀⣬⣧⠘⣦⡟⠉⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠠⠂⠁⢤⠤⠉⠉⠉⠉⣩⣽⠿⢧⡀⣰⠒⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠥⠒⠀⠀⠂⢀⡴⡣⠊⠀⠀⠀⠈⣁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣊⠌⠁⠀⠀⠀⠀⠀⠀⠙⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⢠⡆⢠⠀⠈⠢⣀⢠⡄⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠘⢶⣦⡑⡄⠀⠀⢸⡇⠀⣃⠀⠀⢸⡻⠇⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢘⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠊⢽⡳⠙⢦⣅⣀⣃⣸⣏⣤⠀⢸⡗⢠⡇⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⡬⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⠀⣰⠏⣹⠄⠜⢉⠞⠛⠘⠿⣴⢣⠂⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡾⠿⢃⡠⠐⠁⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠁⠈⢁⠼⠋⠀⠀⢣⣴⠦⣦⠆⢻⣗⢄⠀⢇⠔⣿⠀⠀⠀⠀⠠⠄⠀⠀⠀⣾⠙⠆⠀⠈⠻⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠁⠀⠀⠀⠀⠏⠑⠁⢸⡗⠀⠏⢛⠷⢤⡀⣿⠀⠀⠀⠰⣄⠁⠄⠀⢨⡔⣄⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣹⣷⠜⠁⢠⡲⠟⣿⠀⠀⠀⠀⢈⡆⠀⢱⡞⠀⣟⠢⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠋⠀⠀⠀⢠⣀⣻⡇⠀⠀⠀⣼⠃⢠⣾⣄⠈⣉⠐⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣷⡸⠀⠀⢻⡇⡾⢸⡚⢯⡄⠉⠉⠓⡄⠀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠹⣆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢂⢸⠀⠀⢄⣹⣧⠤⠈⠁⣠⣟⠘⡅⠄⠀⢀⣀⡘⢀⣤⣨⣧⠀⠀⠀⠀⠀⣱⠆
⠀⠀⠀⠀⠀⠠⣌⠙⢶⣦⡀⠀⢸⡄⣀⢻⢄⡠⠀⡀⠀⠀⢠⠆⠀⠀⠘⣦⡔⠐⠈⣻⣿⢀⣈⡔⣹⠃⠀⠃⠀⢰⡎⠉⢹⣿⠿⠿⠁⠀⠰⣆⢠⠆⢃
⠀⠀⠀⠀⠀⠀⠈⠳⡈⣧⣻⡄⢸⣷⣮⢏⣼⢃⠈⠡⠀⢠⠟⠀⠀⠀⠀⠈⣷⣄⠀⣿⣿⣈⡿⢠⣿⠻⣿⡦⣴⣾⢗⣶⣬⠿⡛⡉⠓⢦⣄⠏⠈⠰⠋⠐
⠀⠀⠀⠀⠠⠤⠄⠂⠸⢿⣿⡕⢛⢻⣿⣹⡏⠀⠀⣆⣴⠏⠀⠀⠀⠀⠀⠀⠙⠿⠆⢽⠸⣿⢋⣛⣿⣶⣾⣿⠃⢸⡆⠀⢠⡔⠀⣏⡀⠄⢸⢣⣤⡤⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠛⠀⣾⠋⠱⢈⡬⠻⢿⣇⣀⣴⡟⠁⠀⢀⡤⠀⠀⠀⢀⣀⠀⠳⢎⣱⣿⣉⣿⡿⠛⠉⠁⠀⢸⡇⣠⠏⠀⠈⢩⠀⢀⠞⡠⠋⡔⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⡤⠚⠉⠀⢰⠞⠀⢺⣷⣽⣿⡈⡇⢀⣤⣼⣇⠀⠀⠀⠈⠉⠛⠛⠋⠙⢼⣿⠋⠀⠀⠀⠀⠀⢸⡗⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⠟⢀⠞⠉⢻⠃⢾⣇⣠⣿⡇⠖⠐⠀⠀⠀⠀⠀⢀⣾⣣⡔⠀⠀⠀⢀⠀⠈⠁⠀⠀⡀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⢸⠁⣠⣿⠋⠦⢉⣙⠋⠘⠰⣤⠶⢏⣀⣀⣀⡚⠻⠩⠤⢀⡄⢀⢸⠀⠀⢠⡶⠈⣇⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⣿⠞⠉⠏⠘⠓⣼⣭⣶⠿⠷⢀⡙⠋⣩⠽⠉⣽⣧⠀⠀⢸⡇⠘⡴⠃⠠⣾⣶⠿⠛⠒
⠀⠀⠀⠀⠀⣄⠘⢷⡴⠋⡤⠀⠀⠀⠠⣍⠀⠀⠀⢚⣄⠻⣿⣷⣧⣀⣿⣧⠐⢶⣦⡀⢺⡇⠠⢀⡾⢀⣎⢀⣀⣤⣶⢘⣓⣶⠖⠧⠶⠄
⠀⠀⠀⠀⠀⠀⠙⠶⣧⠘⠁⠀⠀⠀⢠⠟⠉⠀⠠⣀⣻⡄⠀⠘⠻⣿⡿⠋⠀⠈⢷⣽⣿⠁⣠⣮⡿⠋⣹⢿⣿⣿⠛⠻⣦
⠀⠀⠀⠀⠀⠀⣠⠟⣩⣟⣤⣀⡴⠛⠁⠀⠀⠀⡄⠈⠛⠻⢧⣊⠀⢠⠀⣠⠎⠀⠀⣿⣷⡿⠋⡼⠃⠀⡏⠻⡄⠈⢧⠀⢹⡆
⠀⠀⠀⠀⠀⡴⠃⢰⠇⠹⣌⠙⠦⡀⢠⡀⠀⢠⡇⠀⠀⠀⠀⢩⠶⢯⡼⠋⢠⡞⢠⣿⡟⠀⢸⠇⠀⠄⠐⠄⠳⡰⠀⠐⠀⠁
⠀⠀⠀⠀⢰⠃⠀⠘⠀⢠⡜⠳⢤⡀⠀⢻⡄⢸⡇⠀⠀⣠⠔⠁⠀⣠⠟⢶⡟⠀⣿⣿⠀⠀⠈⠀⠐⠢⣄⢈⣦⣆⡒⢠
⠀⠀⠀⠀⠈⠀⠀⠀⣠⢏⣀⣤⣀⣸⠖⢺⣿⣼⡷⢾⡟⠛⠛⠳⣶⣏⠀⠌⢿⣤⣏⣿⣠⠞⢷⣶⢤⣤⣴⡛⠁⣸⠙⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠐⢁⡾⠿⣿⣿⣷⠀⠉⠀⢻⣧⡸⠃⠀⠄⠊⠁⠉⣳⣛⣊⣻⣿⡿⠋⠀⠈⡟⢸⡇⠉⠛⠳⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣾⣿⣿⡞⠀⠀⠀⠀⠹⣅⠀⠀⠀⢀⡴⠾⠋⢇⢿⡿⣾⠁⢀⡔⠈⠁⠀⠱⡀⠀⠀⠀⠉⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠉⠀⠀⠀⢸⢼⡇⣿⣰⠟⢅⠀⢀⣠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢿⣷⡿⠁⠀⠈⠻⡋⢡⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣾⣽⡇⠀⠀⠀⠀⢹⠳⡎⠉⠛⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⣥⠘⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⠈⢉⣈⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠉⣤⣀⠀⡠⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠈⠉⢠⡐⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠢⡤⠂⠀⠐⢺⠁⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢾⡇⢀⠀⠂⠀⢀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠠⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠈⠀⠀⠘⡦⣤⡌
⠀⠀⠀⠐⠀⠠⠽⠦⢤⡴⠗⠙⠓⠚⠁⠀⠀⠀⠀⠙⠦⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣤⠂⠀⠰⢀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⠠⠂⠀⠀⢀⢀⣀⣽⠀⡆⣜⠉⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠉⠜⡾⠃⠹⠶⠶
⠀⠀⠀⠀⠀⠀⠀⢀⢀⡐⠚⠒⠀⠀⠈⣦⡀⠀⠐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠠
⠀⠀⠀⠀⠀⠉⠓⠂⠸⠀⠀⠀⢀⠏⠀⠉⠴⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⡀⠆⠀⢀⠂⠀⠀⠀⠈⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡶⠶⠻⣄⢀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠅⠀⠀⠉⠀⠂⠀⠀⠀⠼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠦⢤⣾⡄⠀⠀⠰⢄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣀⣀⣄⠀⠀⠀⠀⠀⠀⠑⠂⠀⠁⠉⠐⠋⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈⠉⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣄⠀⠀⠀⠀⠀⠸⠀⠀⠡⣀⠀⢀⣠⡺⠀⠀⠀⢠⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠂⠀⠀⠀⢈⣷⣠⡄⣀⡰⠁⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠈⠳⣤⣀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣸⠤⠒⠀⠒⠷⣏⠀⠀⠀⡌⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠐⠒⠒⠒⠒⠛⠁⠀⠀⠀⠀⠀⡙⡄⠀⠀⠀⠀⠁⠠⠒⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⢀⢉⣇⠀⠀⠀⣠⠛⠉⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠨⠷⣦⡤⣄⣄⡿⠋⠋⠻⠶⠚⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠻⠆⠲⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠶⠤⢀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡤⠼⠤⢦⠀⠀⠈⠈⠳⡄⠀⡉⠈⠦⣄⠆
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⣀⠀⠀⠉⠒⢤⡶⠈⠀⠀⠀⠀⠀⣦⣀⠀⢼⣟⡟⠀⠐⠉⠀⠀⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⡂⠸⢇⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠾⣄⡉⠀⢀⣠⣼⠇⠀⠀⠀⠐⠈⡄⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠐⠊⠀⠀⢠⠂⠀⠀⣩⠟⠙⠟⠙⢷⣀⡀⣀⣤⠞⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢁⠀⠀⠀⠁⠀⠀⡀⠀⠀⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣆⣶⣶⣶⣶⡶⢴⡆⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠈⠉⠈⠄⠈⢁⠁⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠄⣀⠠⠀⠤⠀⠄


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⡛⡟⣛⢛⡛⢛⡿⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡴⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⣅⣼⢃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠁⣿⡽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢸⡆⢠⠀⠀⠀⠀⠀⠄⠀⣠⠆⠀⣀⡄⠀⢸⡅⣶⣿⣞⣛⣻⣟⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⡆⠀⠈⢼⣿⣼⣀⣠⡇⠌⣼⠀⢰⡇⢠⢪⣏⠀⣦⢸⣧⣿⣿⡿⢯⣄⡄
⠀⠀⠀⠀⠀⡀⢠⡀⠓⢘⡇⠀⢃⠀⣀⣈⢻⣿⣿⣿⠧⣿⡏⡠⣾⣇⣵⡿⠋⢐⠻⣸⣿⣿⣛⢅⡞⠁⠎⠀⠀⠈
⠀⠀⠀⣀⠀⠁⣄⢧⣘⣿⠿⠁⠘⠏⣩⣾⠯⠙⣿⣃⣀⣿⣿⣁⣻⡿⣫⢴⡁⠠⣠⣿⡁⠈⣿⠟⠁⢼⣴⣆
⠀⠀⠀⠀⠁⢐⠌⠀⠙⢁⣦⣰⣗⣩⣿⡿⠶⣿⣟⣛⣻⡟⠉⣹⡿⣟⡁⢂⣹⡿⣿⣿⣷⣶⣿⠉⢠⡟⠠⢸⣿⡥⢈⣷⠆⡀
⠀⠀⠀⠀⠀⠀⠙⢻⣿⡬⡿⠉⠉⣽⣻⡇⢠⣾⠿⠿⣇⣷⢜⡁⠘⢙⣿⡾⠟⠁⠘⠟⠃⠉⠁⣴⣾⣧⣼⠿⣿⣷⣾⠟⣻⡿⠗⠋
⠀⠀⠀⠀⠀⣠⠴⠛⢿⡀⢀⡆⢠⠿⣋⢀⣿⠷⠀⡀⢸⣿⣟⣑⠊⢀⠿⣆⢀⣀⠀⠀⠀⡀⠀⣼⠁⢻⣴⣤⣿⡿⣥⣿⣽⣿⣷⠤⢀⠈⠒
⠀⠀⠀⡶⠟⠁⠀⠀⠘⢇⣎⠀⢴⡆⠛⠋⠁⢠⡾⢿⣿⡿⠛⢻⡗⠚⢲⣿⣯⠀⣲⠤⠀⠀⠀⠁⠀⠀⠘⡿⠹⡇⠨⡟⠛⢷⠙⢖⠀⠍⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠹⠀⠈⠀⠈⠀⡰⠋⠀⠈⠁⠳⠂⢀⠀⣇⢸⣿⠖⡉⠀⠀⠐⠂⠀⠀⠄⠀⢠⠁⠀⠇⠀⠀⠀⠈⢳⡧⡑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢀⠀⠀⠀⠋⠀⠉⣰⠿⠙⠿⡿⠗⠲⠷⠛⠋⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢨⡗⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⡄⠀⠀⠀⠈⠀⠀⠀⠑⢄⡀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠂⠘⠆⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠉⠒⠂⠀⢀⡼⠤⠀⠀⠀⢀⡨⠅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⣠⠴⠟⠒⠒⡂⠋⠈⠁⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣿⣀⣴⠞⠋⠀⠀⠀⠀⠀⠑⠒⠃⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠅⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⣠⣄⣰⠏⠀⠀⠀⠀⠘⣧⣰⠻⡶⣄⠀⠀⠀⠀⠀⠀⠀⣠⠟⢦⡤⢶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣧⢀⣴⣿⠋⠳⡀⠀⠀⠀⢠⡞⢉⣯⠻⡄⠀⠀⣠⠶⠦⠞⠃⠸⣟⠎⠳⢄⠀⠀⠀⠀⣴⠋⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠙⣦⣤⡴⠋⢰⡿⣌⣢⠈⠳⢮⣁⣀⣀⠀⠀⠀⠹⡗⢀⣀⣵⢶⣷⡾⠁⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⣠⠴⣿⠁⠀⠀⠀⠘⢧⣀⡈⠑⠀⠀⠀⠀⠈⠑⠄⠀⠀⠉⠉⠁⠀⠸⣅⠙⠲⠦⢤⡴⢺⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡀⢀⣿⡄⠀⠀⠀⠀⣤⣻⡍⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣹⠇⢸⡄⠀⠀⡞⣰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇⢸⣿⣇⠀⠀⠀⠀⠸⣟⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠀⢠⠘⣆⣿⣇⠀⡼⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⠀⢹⠀⠈⠙⠢⠀⣤⠀⠀⠀⢀⣿⣷⠀⠀⠀⣿⡆⢸⣿⣿⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⢄⠀⠀⣾⡇⠸⡀⠀⠀⠀⢠⣿⡆⠀⠀⢸⣿⣿⡂⠀⢠⣿⡇⢸⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣦⣿⣦⣿⣷⠀⠙⢂⠀⠀⢸⣿⣇⠀⠀⣼⣿⣿⣇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡷⢦⡴⢄⣀⣿⣿⣿⣀⣀⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣷⣾⣧⣴⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠟⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⡀⠘⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠁⠀⠀⠀⠀⢠⠀⠀⠘⠦⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠊⠸⠈⠃⠀⠀⠀⠀⢀⠰⠀⠀⠀⠠⡀⠀⠀⠠⠀⠀⡀⠀⡀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⠀⠀⠀⠀⢀⠐⠒⠀⠀⠀⠀⠀⠔⡀⠀⡠⠀⠁⠀⠀⠲⠀⠀⠀⠀⣀⠀⠉⠀⠀⠀⠀⠐⡀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠠⠈⠀⠈⠢⠔⠊⠁⠀⠀⡀⠄⠂⠀⠀⢎⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠄⠀⠀⠀⢀⠇⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠒⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠤⠀⠀⣀⡠⠄⠀⠈⠁⠀⠀⠀⠀⠀⠠⠀⠉⠐⠢⢀⠀⢧
⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠐⠂⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠸⡆
⠀⠀⠀⠀⠀⣎⠠⠂⠁⠀⡀⠀⠀⠀⠀⠈⠁⠀⠡⢄⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⣀⣤⣄⣠⣬⣭⣷
⠀⠀⠀⠀⠀⡏⠡⠀⣀⣀⣀⣤⣄⣀⠀⠀⣠⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣇⠐⠋⠉⠉⠉⠀⠀⠀⣀⣀⠀⠀⠀⣀⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⡿⠿⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡟⠛⠳⢦⣠⣄⡀⠴⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⠀⠀⠀⠀⣤⣾⣦⡐⠛⠉⢁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢷⣦⡀⠀⠈⠁⠀⣀⣤⣾⣷⣦⣤⣄⣀⣀⣤⣦⣴⣶⣾⣷⣷⣶⣶⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠻⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢿⠀⠄⠺⠦⠴⠶⠴⠶⣤⣤⡤⣤⣤⣤⣤⣠⣤⣄⣀⣀⣀⣈⣉⣉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⠛⢻⠖⠊⠈⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣭⣬⣭⣭⢽⠶⠶⠶⠦⢤⠄⢀⣀⠀⠀⠀⠈⠉⠛⠛⢛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⡤⣿⠶⠶⣦⣄⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠠⠅⠀⠀⠀⠀⠀⠀⠉⠙⢩⣙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣏⡉⠀⠈⠛⠛⠻⠾⣮⣋⢝⡻⢛⡷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠇⡰⢠⠙⣧⠀⠀⠀⠀⠀⠀⠈⠻⢧⣶⡭⠶⠛⠋⠙⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⢰⠀⡘⢠⢁⠇⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡾⡏⢠⠃⠆⡰⢡⢃⠎⡐⢰⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⡽⣷⡄
⠀⠀⠀⠀⠀⠀⣼⡿⠡⢁⠇⡌⡰⢁⢂⠞⣄⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣄⣸⣯⣴⡾⢿⣆
⠀⠀⠀⠀⠀⣼⣿⢡⠃⠎⡌⡰⠁⢂⠎⢘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢟⠝⡩⣻⣯⢒⢝⣿⡆
⠀⠀⠀⠀⢰⡿⢸⠆⡜⡜⢰⠃⢀⠎⡰⢡⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣕⡡⣊⡼⠋⠙⠓⠛⠛⣿
⠀⠀⠀⠀⢸⡷⠿⣤⣵⢠⠃⢄⠎⡰⢡⣿⣵⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⣀⣴⢿⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠹⣇⣸⡎⡰⢱⣿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⢏⠕⡡⢻⡇
⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠉⠙⠓⠦⣭⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢕⣕⠡⠊⡠⣺⡇
⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡽⢷⠶⢲⠖⡶⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣣⡿⠃⡠⣮⠎⣽⠇
⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠎⢡⠃⠠⢁⠇⡈⢸⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠟⡠⢊⠔⢡⢾⡿
⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠹⣶⣰⠆⡰⠃⡌⢠⢃⠇⢌⡼⣿⠀⠀⠀⠀⠀⠀⠀⠈⣷⢊⠔⡡⠚⣡⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠈⢻⡔⠁⡜⢀⢆⠎⢜⣞⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢊⢔⣩⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠹⡎⢀⢂⢏⢎⢎⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣳⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⢹⣣⠏⢎⢎⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⣟⡘⠎⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⡀⣿⡌⡜⡿⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠾⠷⠶⠶⠶⠾⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⣤⣤⣶⠶⠶⢶⡖⢶⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠹⠋⠩⢡⣢⡤⠄⠒⠛⢶⣤⣶⣾⣿⣿⣯⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠆⠀⠀⠀⠀⠀⠀⣠⠹⣿⠀⠀⠀⢠⣿⣯⣼⣿⡿⣷⡝⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⢠⣴⡾⢻⡁⣰⡿⠀⢀⣠⠾⣣⠞⠟⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⣀⠀⠈⠿⠁⠘⢻⣿⣿⠞⢿⣿⡟⠿⢴⣶⡁⡀⢠⣆⠀⠀⠀⠀⠀⠀⠸⣤
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠼⠁⠀⠀⠀⠀⠻⡆⠀⠀⠀⣤⠴⢿⣏⣤⡄⢀⣀⣀⡎⡙⣿⠇⠛⠃⠀⡄⠀⡀⠀⠻⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠎⣠⠀⠀⠀⠀⠀⠀⠈⠙⣁⣴⣤⣥⠶⠟⠗⠀⠁⠈⢻⣿⣶⣗⠂⠀⠀⠀⢈⢀⠸⡿⠗⠀⠘⠂
⠀⠀⠀⠀⠀⡄⠀⢰⣿⠀⠈⠀⠀⠀⠀⢀⣼⣿⣿⠉⠀⠀⠀⠀⠀⢤⠀⠀⢹⣿⠿⠇⣠⣶⣾⡆⣄⢠⣄⣤⡀⠀⠠⣤⣾⡄
⠀⠀⠀⠀⠀⠀⢠⡿⣿⡇⢠⣿⣟⢳⡀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣤⡤⠝⠛⠋⢻⣿⡿⠟⠛⢣⡀⠀⠈⢸⣧
⠀⠀⠀⠀⠀⠀⢸⣗⣿⣆⠸⠋⣽⣭⡻⠿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠛⠁⣾⡿⠏⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠙⢆⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠘⠛⢿⢿⣿⡄⢹⡟⣿⣤⣌⣻⣶⣧⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⢻⡇
⠀⠀⠀⠀⠀⠀⡄⠀⠈⠈⢿⣷⣌⠛⠛⠛⢿⣿⣿⣿⡗⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⣸⠃⠰
⠀⠀⠀⠀⠀⠀⡅⠀⠀⠛⢠⣹⣿⠃⠀⡤⣄⢨⠙⠻⣧⡀⠀⠀⠀⢀⡿⢿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣤⡿⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠸⣦⠦⡀⠀⢻⡁⠞⠉⠀⠀⠀⠀⠀⠈⠧⣀⠀⠀⠀⠀⠈⠿⢇⣀⣀⣠⣤⣶⣤⡆⣿⠿⠏⠀⠖⢀⣿⡷
⠀⠀⠀⠀⠀⠀⠀⢄⠢⠈⠢⢀⣵⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠑⠢⣿⡷⠀⠀⠈⠉⠉⠿⢿⣿⣿⣇⢀⡖⠀⠀⡀⣠⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠙⢷⣄⠀⣆⣄⡀⠀⠁⠀⠀⠀⠀⣠⣿⡾⠆⠀⣀⠀⠀⠔⠙⣿⣿⠘⢶⡀⢀⣼⠟⣧
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠱⣬⡀⠀⠀⠹⡇⡿⠉⠿⣠⡄⠀⠀⠀⢾⣿⡟⠀⠀⠈⠁⠀⠀⠀⣧⣾⡇⠀⠀⢠⡞⠝⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡈⠳⣧⢃⠀⠀⠉⠀⣰⣿⣿⠋⠋⠀⠀⠀⠀⣀⡀⠴⠾⠿⣿⡇⢀⠔⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⣞⠻⣯⡀⠀⣠⣾⣿⣿⣋⣠⠤⠂⠀⢚⠁⠄⠀⢀⣀⣼⠿⢿⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠙⠻⠿⣦⣿⣿⣿⣿⣿⠿⠟⠛⠟⠛⠛⠛⠶⢿⠿⠟⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠀⠀⠀⠀⠀⠀⠐⠚⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢴⣶⡒⠶⣿⡙⠻⣿⣍⠉⠻⣿⡙⢿⡖⢶⡶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠺⢿⣭⣉⡛⠷⢮⣝⡲⢬⣙⠳⢤⡉⠳⢤⡈⠳⣄⠹⡌⢧⢹⢸⢇⡟⣹⠖⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⢯⣛⡿⠷⠶⢤⠤⠭⡙⠳⠦⣬⣙⠳⢮⣝⡲⠬⡓⠤⣙⠢⠈⠃⠈⠀⠈⠈⠈⠐⠡⢞⡡⠞⣋⣷⢤⡀
⠀⠀⠀⠀⢀⣴⣿⠿⠀⠀⠈⢲⡀⠀⣰⠃⠀⣨⠉⠓⣤⣌⣙⣒⠦⢍⡓⠤⢁⠀⠁⠀⠀⢠⠞⠉⠉⠉⠳⡄⠀⠐⠈⠥⠴⠶⢛⣳⣄
⠀⠀⢀⣴⣫⠔⠢⣅⣠⡞⠉⠳⢄⡀⠉⠀⠠⠉⠁⢀⣼⠷⢻⣿⣷⢶⡌⠉⠂⠀⢀⡴⢲⡇⠀⠀⠀⠀⠀⢸⣀⣠⠞⠛⣯⡩⠭⠭⠭⣷⣄
⠀⢠⣾⣿⣶⣾⣿⣿⣿⣷⣄⠰⠞⣉⡦⠴⠞⣳⠖⠋⠁⠀⠘⣿⡏⠀⠈⠓⣲⡯⣍⡀⣨⢷⠴⠚⠓⢦⣤⣾⣾⡃⠀⣼⡿⠻⢏⣹⢷⠚⠙⢦
⢠⣿⣿⣿⣿⣿⣇⣤⡤⣉⡛⠳⢿⡀⣀⣴⣞⠁⢀⠀⠀⠀⣩⣿⡿⠀⢀⡟⠙⠲⡀⠉⠳⢤⣴⡶⠾⡿⠟⡻⠉⠀⠀⣘⣷⠀⠀⢙⣷⣯⣴⣾⣇
⣼⣿⣿⣿⣿⣷⠟⣽⣻⣯⣿⣷⣶⡉⠛⢿⣿⣿⣶⡄⠀⣴⣿⠓⠀⣠⣿⣤⣤⣠⠞⢀⣀⢤⣬⣻⣮⣄⠈⠀⠀⢀⣠⣿⣿⣤⣾⣿⡿⣿⣟⣿⣿
⣿⣿⡟⣷⡿⣿⢻⡼⣧⣷⣿⣶⣿⣿⣶⣄⣌⡙⠻⢿⣼⣧⣤⣤⣾⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⢬⣿⣿⢿⡟⣿⣾⣿⣿⣻⣯⣿⣿
⣿⣿⢿⣿⣷⣞⣚⣿⣿⣭⣿⣿⣿⢻⣿⣶⡬⠝⠳⣤⣈⣙⠻⢿⣿⣿⣿⣟⡷⠾⠿⠛⢛⣋⣭⣭⣷⣶⣾⣷⣿⣿⠿⣾⣝⣗⣯⣷⣿⣿⢾⣿⣿
⢹⣿⣾⣿⣷⣮⣿⣧⣿⣭⣻⣏⡽⣻⣿⣿⣿⣷⣄⡙⠻⣿⡷⢯⣤⡿⠉⣿⣀⡀⠝⠻⠿⠷⠶⠯⢿⣧⢾⣛⣿⣶⡿⣿⣿⣿⣷⣿⣿⣹⣿⣿⣿
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡟⣿⣆⣤⣍⣳⣯⡳⣄⣈⠙⠛⠻⠶⣬⣵⣾⣿⠿⠷⠿⠯⠿⢶⣿⣟⣿⣞⢻⡿⣽⣿⣿⣿⠃
⠀⠀⠹⣿⡽⢿⣿⣿⣾⣿⣿⣿⣿⣿⣯⣗⣺⠭⣿⡷⣷⣿⣾⣳⣾⣿⣦⣽⡷⠀⠀⠀⠈⠹⣿⣿⣿⣯⠭⠽⢿⣟⣒⣶⣭⣝⣛⣿⣿⣿⡿⠁
⠀⠀⠀⠈⠻⣷⣾⣷⣺⣭⣿⣿⣿⣿⣭⢟⡻⢿⡲⢭⣓⢪⣽⣿⡿⠟⠋⠁⠀⠀⠀⡀⠀⠀⢸⣿⣛⣛⣯⣭⡿⢾⣛⣛⣿⣭⣭⣭⡽⠋
⠀⠀⠀⠀⠀⠈⠛⢿⣞⣛⡻⣍⣯⣭⣝⣿⣮⡓⣭⣷⡾⠟⠋⠁⠀⠀⠤⠀⠀⠀⡀⠀⠀⢀⣼⡿⣟⣻⠽⢗⣛⣭⠭⠴⣶⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣿⣽⣹⣯⣷⡿⣿⣭⠁⠄⠀⠀⠀⠒⡤⠀⠁⠀⠴⠀⣠⡶⣛⠽⣛⣭⠶⣚⡭⢵⣲⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣯⣶⣶⣖⢐⣠⣶⠖⠔⠐⠀⢢⣾⣷⣻⣵⣟⣯⣶⠿⠗⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠲⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣧⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⣠⣷⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣦⣆⣾⡄⠀⠀⠀⠀⠀⠀⠀⢠⡀⢶⣿⣷⣠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣟⠛⠛⠃⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣓⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠉⣻⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠒⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠀⠠⠄⠀⠀⠀⢤⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⣠⠾⠛⠃⠄⣀
⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡴⠟⠉⠀⢀⣾⣿⣿⣿⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⣿⣴⣷⣧⢀⢠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠰⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠉⠉⠃
⣿⣿⣿⣿⣿⣿⣷⣿⣠⡀⣦⣷⢠⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠰⠶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣼⣧⣧⣼⣰⣇⢠⡀⣿⡆⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡆⣠⢸⣷⣿⣦
⣿⣿⣿⣿⢿⢿⡿⠿⠛⠛⠟⠿⠛⠋⠛⠉⠉⠋⠉⠉⠋⠈⠉⠛⠁⠀⠋⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⡀⣸⣧⡆⢠⢀⣿⣾⣷⣿⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⣾⡧⢀⡀⣸⡇⣰⣠⣧⣴⣰⣿⣰⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⢸⣿⣿⣾⣧⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣽⣗⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣸⣷⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡿⠻⣿⣿⢿⡿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠃⠀⠀⠈⠁⠈⠀⠁⠈⠈⠉⠈⠉⠙⠛⠋⠉⠉⠛⠻⠟⠿⢻⡿⣿⠿
⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⢠
⣿⠃⢹⣿⣿⣿⣿⣿⡃⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣸⡇
⡟⠀⢐⣻⣿⣿⣿⣿⣷⣼⣿⣧⡀⢀⣦⠀⠀⠀⢰⣿⠄⠀⠀⠀⠀⠀⢸⣿⣯
⠉⠀⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣸⣿⡄⠀⢰⣾⣿⡇⠀⢰⡆⠀⠀⣿⣿⣿⡁⠀⠀⣾⠄⠀⠀⠀⢰⡧
⠂⠀⠴⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢺⣿⣿⣶⣤⣾⣿⠀⣘⣿⣿⣿⣶⠆⢸⣿⡇⠀⠀⠀⣿⣟
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣷⣘⣿⣿⣿⡟⢻⣿⣿⣦⣽⣿⣿⣿⣿⡥⣿⡿⠿⠀⠀⣽⣿⣿⠇
⠀⠀⠀⠀⣰⣿⣿⣿⣿⠃⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⣶⣤⣿⣿⣿⣷⡤⠀⠀⢰⡇
⠀⠀⠀⣤⣽⣿⣿⣿⡧⠀⠀⢽⣿⣿⣿⡿⢻⣿⣿⠦⠀⠠⢽⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠠⢽⣿⣿⣿⣿⣿⣧⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠤⠼⣿⡟⠓⠀⠀⠙⢿⣿⣿⡃⠀⣿⡟⠃⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⠀⠙⢿⣿⣿⣿⣿⣷⡖⣸⣿⣿⡅⠀⢀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠒⠚⣉⠁⠀⠀⢤⣿⣿⡟⠀⠀⠹⡏⠁⠀⠀⣤⣿⣿⡿⠁⢿⣿⣿⣏⠁⠀⠀⢤⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢁⣾⣇⠀⣸⣿⣟
⠀⠀⠀⠀⠠⠴⠒⠀⠀⠀⠀⠐⠚⠛⠂⠀⠀⠀⠀⠀⠀⠀⠐⠚⠻⠃⠀⠘⣿⣟⠀⠀⠀⠀⠐⢺⠻⣿⣿⠁⢻⣿⣿⡟⢻⣿⣽⣿⣧⣠⣿⣿⣿⣃⣾⠀⠀⣴
⠀⠀⠀⠀⠤⠦⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⢒⣿⠁⠀⢈⣿⣿⠀⠈⣿⠙⣿⣿⣿⠇⢸⣿⣿⣿⡆⢰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠉⣉⠅⠀⠀⠀⠙⠀⠀⠀⠋⠀⣿⣿⡿⠀⠈⣿⣿⣿⣷⣾⣿⣿

⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣾⣿⣶⡀
⠀⢰⠆⠀⠀⠀⠀⠀⠹⣿⠀⠀⠄⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⠀⠠⠀⢰⠆⠀⠀⠀⠀⠁⠠⣶⡿⠁
⠀⠘⣦⡀⠀⠀⠀⠀⣠⠟⠀⠀⠈⢃⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⢀⠉⡼⠁⠀⠘⣦⡀⠀⠀⠀⠀⣠⠟
⠀⠀⠈⠙⠺⡤⡴⠚⠁⠀⠀⠀⠀⠈⢳⣴⡿⠁⠀⠀⠀⠠⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠘⢾⢿⡞⠀⠀⠀⠀⠈⠙⠺⠤⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⠀⠀⠀⠀⡉⠁⣀⡀⢊⠀⠰⠉⠀⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸⢿⡟⠚⠖⠁⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠁⠘⠀⠀⡃
⠀⠀⠀⠀⠀⠤⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣤⣾⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠆
⠀⠀⠈⠐⠒⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠈⠐⠒⠒⠒⠒⠋

⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⡿⣿⣿⣷⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠹
⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣠⣿⣿⡟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⡀
⣼⠟⢤⠀⠀⠀⠀⠀⣀⣀⣠⣴⣽⣿⣿⣿⠟⠻⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⡟⢠⣯⡄⠀⠀⠘⠛⠻⠿⠿⣧⣀⣿⣿⣦⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡿⢿⢿⣄
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⣿⣿⣟⡉⢛⣉⡏⡀⣀⣸⣿⣿⡶⠿⠒
⠀⣾⡻⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⠃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣻⣿⣄⡄⠀⠀⠀⠀⠀⢸⡏⣿⣿⠟⢿⣿⠿⢿⣿⣿⣿⡀⢀⣠⣶⡀⣠⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣀⣄
⡟⠀⠙⠃⠀⠀⠀⠀⠀⢸⡿⢯⣿⣤⣀⣻⣆⣠⠿⣿⠟⣿⣯⣨⣿⣹⡟⣹⣿⣿⢿⣋⡀⣄⣽⢳⣻⡿⠁⠀⠀⠀⠀⣼⣇
⠅⠀⢰⡀⠀⠀⠀⠀⠀⠈⢄⣈⣭⣿⣿⣿⣣⡿⠷⣯⣟⣯⣅⠉⣻⣿⣾⣿⡿⠿⣫⡻⣿⠿⢻⣿⣼⣧⠀⠀⠀⠀⠀⠈⢿⣧⡀
⡟⠀⢺⡇⠀⠀⠀⠀⠀⠀⢘⣿⡿⡿⡼⡟⢶⣅⣄⣀⣿⣾⣿⣷⣿⣿⣿⣍⣴⣶⡿⠏⣥⣤⢿⣷⠖⢯⠀⠀⠀⠀⠀⠀⠈⢻⣷
⠁⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢗⡋⢱⣐⣯⣶⣽⣿⠿⠿⣻⣿⣿⣭⣔⡻⠿⢿⣾⣷⣚⣺⣛⣸⣗⡋⣻⡖⠀⠀⠀⠀⠀⠀⠘⢿
⠀⣿⣿⡇⢠⡄⠀⠀⠀⠾⣷⡿⠷⠛⢫⣽⠋⠡⡶⡖⠀⠛⠿⣿⣿⣿⣥⣤⡀⠈⠉⢻⣿⣿⣿⣿⣵⣾⣷⠄⠀⠀⠀⠀⠀⠀⠈
⢸⣿⡿⣰⣿⡇⠀⠀⠀⠈⠛⠿⣿⠿⣿⣯⣤⣤⡍⢵⣶⣿⣿⣿⣿⣿⣿⡿⠟⠙⣛⠋⠛⠋⢻⣾⡿⠟⠛
⣿⣿⣧⣿⣿⠃⠀⠀⠀⠀⢀⠀⠈⠙⢿⣿⣿⣻⣶⣿⣿⣿⣿⡿⠋⠁⠀⣠⣤⣤⣤⡼⣿⣿⡫⢋⣤⣀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠐⢿⠇⡀⠀⢸⣿⢿⣽⣿⣿⣿⣿⢿⣿⣶⡶⣾⠟⡓⣿⣫⣿⣿⠋⢀⣈⠛⠃⠀⠀⠀⠀⠀⠀⢰⣾⡇
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣀⡀⠉⠀⠈⣿⠐⠙⣿⣥⣿⣇⣦⢻⠻⠐⡿⣾⣿⣧⣼⢿⣿⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠁⠀⣈⠀⢿⣰⣿⣿⣿⣿⣿⣷⣼⠀⣰⣿⣿⣿⣿⠀⢘⣿⣤⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣴⣦⡉⠙⣿⣿⣿⢿⣿⣿⠀⣿⣿⡿⠛⠃⠀⢘⣿⠙⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⡏⢀⣶⣦⠀⠀⠀⠀⠈⠀⠀⠻⠟⠁⠀⣿⣿⣿⠀⢻⣿⣶⣿⠏⠀⠀⡀⠀⢸⣿⠀⢀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣝
⣿⣿⣿⣷⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣿⣿⣿⠀⢸⡟⠟⠡⣿⣿⡟⠀⠀⠘⠁⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠛⠿⠿⠀⠘⠇⠀⠀⠉⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣾⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⣴⣧⣴⡄⠀⢀⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣾⡿⢿⣿⣧⠀⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣶⣍⠛⠛⠻⢿⣿⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠹⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⣿⣿⣿⣿⡏⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⣴⣿⣷⣤⣄⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠟⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠻⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠉⠉⠁⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⢿⣍⣉⣉⣉⣉⣉⣉⣹⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⠋⢹⠉⢹⡟⢾⣝⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣸⣇⣞⣠⠟⢉⣷⣻
⠀⠀⠀⠀⠀⠀⠀⠘⢯⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⣛⣉⣨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣠⣾⡿⣭⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⠜⠁⠈⢿⢦⣈⡽⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⡤⠉⠀⠀⠀⠈⠳⣟⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣠⠎⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡈⢻⣿⣿⣿⣿⣿⣿⣿⡿⢁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠙⢿⣿⣿⣿⡿⠋⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠛⠛⠛⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠂⢲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣴⠀⢸⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠚⠋⠉⠉⠁⠀⢹⣀⣸⠀⠈⠉⠉⠉⠓⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⠿⠃
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡿⢟⣉⠄⣀⣬⣍⡙⠻⢿⣤⣀⠀⠀⠀⠀⠀⣴⠟⠋⠉⠉⠉⠙⠷
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⢸⣷⠟⠉⠁⠈⢻⣦⡄⠻⠟⠻⠟⠿⠟⠟⠃⢀⠠⠶⠶⠶⣦⡀⠈⢳⠄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⣾⡇⠠⠀⣠⡷⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⣿⡆⠀⣤
⠀⠀⠀⠀⠀⠀⠀⢿⣧⡀⠺⣧⡠⠾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢀⣿⠃⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣤⠀⠀⢠⣴⣿⣿⣿⣦⠀⠀⠀⠀⠀⢀⣴⡿⣿⣷⡀⠀⠈
⠀⠀⠀⠀⠀⠀⣀⣤⣼⣿⣷⣤⣀⠀⣿⣿⣽⣿⣿⣿⠀⠀⠀⠀⠀⢺⣟⢰⣬⣿⠇⠀⠀⠠
⠀⠀⠀⠀⠀⠺⠿⠓⢻⣿⣿⣛⣛⠀⠻⣿⣿⣿⣿⠛⠀⣀⣀⣀⡀⠈⠿⠿⠟⠉⠀⣴⣶⣶⣦⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⡿⠿⢇⠀⠀⠈⠉⠀⠀⠈⠻⣿⣿⠟⠀⠐⠛⠀⠐⠲⠾⢾⣿⡟⠻⠿⣶⠄
⠀⠀⠀⠀⠀⠐⠿⠿⠛⠛⣿⢿⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠏⠑⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠀⠀⠃⠀⠀⠀⠀⠠⠤⠤⠤⠴⠤⠤⠤⠤⠤⠒⠒⠒⠛⢻⡆
⠀⠀⢀⠀⡀⢀⠀⡀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠘⢿⣷⢷⡄⠀⣦⣹⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠘⣷⠀⠻⣼⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⡟
⠀⠀⠀⠀⢹⡇⠀⢻⣿⣿⡟⣿⡏⠋⠛⠙⠙⡷⠆⣶⠾⡇⠟⠺⠟⠛⠉⠉⠉⠉⠀⠙⡄⠻⣧⡄
⠀⠀⠀⠀⢸⡇⠀⢸⣿⠏⢠⣿⠁⠀⠀⠀⠀⣧⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠘⢿⡆
⠀⠀⠀⠀⢸⡇⠆⣸⣿⠀⢰⡿⠀⠀⠀⠀⢰⡏⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠁⢈⣧
⠀⠀⠀⠀⠸⣷⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⣾⠇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠾⠁
⠀⠀⠀⠀⠀⢿⣄⠻⢿⣤⣯⠀⠀⠀⠀⢀⣿⡄⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⠶⠂
⠀⠀⠀⠀⠀⠈⠛⢿⣦⣼⣏⠀⠀⠀⠀⢸⣿⠋⠋⠀⠀⠀⠀⢀⣀⣠⣄⣤⣴⣶⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣶⣶⣾⣿⡿⠿⠿⠟⠻⠿⡿⠛⠛⠛⢿⡿⠋⠉⠉⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⣇⣀⠀⠀⢾⡗⠀⠀⠀⠀⠲⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠙⠻⣧⠀⢸⡁⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣷⣦⣤⣤⣤⣤⡾⠶⠋⠀⠈⠓⠶⠶⠶⠶⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⠐⣿⠟⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣿⣿⣿⣶⣶⣶⡄⠘⠛⠀⣠⣶⣶⣿⣷⣾⣷⢦⣶⠆⠈⠙⠛⠷⠶⠄
⠀⠀⠀⠀⢸⡿⠊⠉⠉⠙⢻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢰⣦⠾⠿⠛⠃⠀⠀⠀⠀⠐⠓
⠀⠀⠀⠀⣶⡿⠀⠀⠀⢀⣾⠛⠋⠻⣿⣿⣤⣄⣀⣀⡰⠖⠀⠀⣀⣀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢀⣤⣶⣤⣴⣿⠋⠀⠀⠀⠈⢻⣿⣿⣿⣿⡇⠀⠀⠘⢿⣿⣿⣿⠀⠀⠈⠻⣿⣶⣷⡆
⠀⠀⠀⠀⣻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠙⢿⣿⠏⠀⠀⠀⠀⠀⠻⡿⠁⠀⠀⠀⠀⠀⠙⢿⣄⡀⠀⣠⣤
⠀⠀⠀⠀⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿
⠀⠀⠀⠀⠸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠤⢀⡀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠀⠀⠲⣦⠀⠀⠀⠀⠸⠆⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠋⠀⠀⠀⠋

⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠧⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠘⠛⠛⠀⠂⠙⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠂⠀⠾⠐⠀⠆⠀⠀⠀⡠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡿⠿⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠛⠉⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠁⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠀⠀⠀⢾⣭⣤⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠉⠁⠀⠀⢀⣤⣤⣘⣿⣧
⠀⠀⠀⠀⠀⢀⣿⡿⠶⠶⠶⠶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⠀⠀⣤⣄⢸⡇⠀⠉⢻⣿⡆
⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠻⡾⠇⠙⡆⠀⠀⣿⣷
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠚⢶⡄⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣤⣤⣤⠾⠃⠀⠀⢹⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⣠⠟⠉⠛⠶⣤⡀⠀⠀⠀⠀⠀⣀⣠⡼⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⣾⠉⠀⠀⠀⠀⠀⠀⠈⠷⠴⠶⣦⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⡇⣠⠞⠛⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠘⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⣄⠀⢸⣿
⠀⠀⠀⠀⠀⢹⣿⠏⠀⠀⢻⡄⣀⣤⣄⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠈⠛⠶⣤⡀⢠⡴⠖⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣼⡿
⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠈⠛⠋⠀⣼⠃⠀⠀⠀⠀⠀⣯⡈⠙⠛⠻⣆⠀⠀⠀⢻⣾⡄⠀⢀⡴⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢡⣿⠇
⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⠀⠀⠀⠙⢶⡄⠀⠀⠀⠀⠈⠷⣤⣤⣤⠟⠀⠀⠀⣾⠀⠙⠓⠋⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⢳⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⣀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⢰⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⣠⣿⠀⣠⡴⠶⣄⠀⠀⠀⠀⠀⣰⠟⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣄⠀⠹⢦⡀⣹⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣤⡙⠋⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣷⣶⣤⣄⣀⣀⣼⣇⣀⣀⣈⣀⣀⣀⣤⣤⣶⡾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠠⣄⠹⣿⣿⣷⣦⣤⣴⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡷⠎⠀⠹⣿⣿⣿⣿⡿⢋⣩⣥⣉⠉⠛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣢⠀⡡⠀⢠⣤⠸⣿⣿⡏⠀⢾⣿⣿⣿⠀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠁⢈⢻⣏⣯⢻⣿⣧⣀⡈⠛⢛⣫⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢄⢲⠀⠉⡁⠋⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡤⢭⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠠⠆⠁⡀⡦⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢄⠀⠘⠁⢿⠆⣠⡎⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡠⠐⠾⣷⢖⡐⠤⠇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⡘⠀⠃⠸⢃⣈⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠰⠳⠇⣠⣶⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⡴⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡄⠀⠀⠲⢈⢻⠏⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠇⡀⣀⡙⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⠀⠀⠊⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠠⢀⢀⠋⠀⠀⠸⣿⣆⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢹⣿⣿⠇⠀⢀⡄⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⡦⠀⢰⠀⠀⡀⠁⢹⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⠀⢢⠂⠘⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⠀⠑⠦⢄⡱⠄⠀⢀⣸⣇⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⡌⠀⠂⡁⠐⢐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠤⠄⠀⠈⠉⣄⠀⠀⢻⠟⠂⠀⣸⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⠃⣶⡆⠀⠀⠀⠀⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⠀⠀⠀⠀⢀⠈⠀⢁⠀⠀⠀⠈⠁⠉⠉⠛⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⠛⠀⠀⠀⢠⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠀⠀⠀⠀⠓⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠈⠢⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢁⣀⠄
⠀⠀⠀⠀⠈⠢⡀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡤⠞⠪⢊⠛⠒⣂⠄⡆
⠀⠀⠀⠀⠀⠀⠈⠲⣴⡧⠤⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⠙⠌⣀⢸⣷⡬⠼⠃⠀⠀⠀⠂⢾⣙⢠⡏⢸⢸⠇⠘
⠀⠀⠀⠀⠄⠀⢀⡀⢻⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⠉⠻⠥⣢⡈⠛⡩⢁⠀⢀⠀⠀⠀⠀⠢⠬⠿⢹⠼⠬⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠀⠀⠀⠀⠀⣀⢠⣤⡀⠉⢱⠛⠋⠅⠭⠅⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣙⣔⡀⡙⠛⢙⠢⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⡙⢷⡀⠉⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠠⠺⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠋⠛⠂⠀⠀⠀⠀⠠⠀⠀⠄⠑⠠⢌⡞⢁⡄⠁⠀⡀
⠀⠀⠀⠀⠀⠠⠢⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣄⡀⢀⡀⠀⠀⠀⠀⠀⠒⢤⣀⢀⢰⠉⡠⢁⣤⠘⠉
⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⣠⣰⠲⠆⠬⢶⡾⣿⠀⠀⠀⠀⠀⠀⢀⡐⠉⠑⣯⣦⡷⠃⠄⠁⠁
⠀⠀⠀⠨⢄⠶⢐⠀⠉⠁⠆⠀⠀⠀⠀⠀⠀⣠⠤⠲⢌⣳⣶⢞⣻⣿⣝⠄⠀⠀⠀⠠⠐⢄⠀⣶⡼⣿⢿⠓
⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣈⡺⢿⠋⠤⡟⡀⠿⠀⠀⠀⠀⠀⠀⠐⠡⠈⡡⠼⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠐⠄⠋⡬⠹⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣭⣴⣭⣯⣽⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⢀⠀⢘⢸⡇⠀⠀⠀⠀⠀⠀⢸⣾⣷⢠⠀⠀⠀⣾⠄⡀⠀⠀⠀⠀⠀⡀⠀⡀⠠⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡇⢘⣨⡀⣟⠚⠛⠒⠀⠀⠀⠀⠀⢸⣿⠟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡇⢀⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⣿⣿⣷⣾⣿⣷⣿⣷⣄⣀⡀⣀⠀⠀⠀⠀⠀⠀⠠⡄⠐⣾⣿⣦⡤⢀⣒⠀⠀⢸⣿⣾⣿⡿⢿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣾⣿⣿⣿⣟⠛⠛⠛⠻⠿⢿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣄⣠⣼⣿⣿⣿⣴⣦⡀⠈⢈⣇⣤⣤⣶⣶⣶⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠁⠉⠉⠉⠉⠉⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⣶⣦⣄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠙⢩⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠘⠛⠙⠟⠙⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢹⡇⠸⠀⠀⠺⢘⡏⠀⠀⠀⠀⠀⠀⢸⢿⣷⢰⠀⠀⠀⣤⠀⠄⠀⠀⠀⠀⠀⠄⠀⠇⢀⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠘⠸⠃⠀⠀⠀⠀⠀⠀⠘⢸⡏⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣺⣦⡤⠖⠚⠛⢳⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠘⠋⠁⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠏⠀⠀⣀⣀⣤⣤⣀⡀⠀⠀⠀⢀⣈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⠁⠀⠀⡴⢾⣿⣿⣭⠀⠀⣠⢞⣿⣷⣤⡀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠟⠉⠀⠀⣴⡷⠁⠈⠛⢻⠇⢀⡇⠀⠉⣸⠙⣇⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⣀⠀⠀⠀⡼⠁⣇⠀⢀⡴⠋⠀⠈⠙⠓⢾⡁⠀⢹⡀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⠋⠀⠀⣾⠁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣀⣽⠂⢸⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⠿⠟⣁⠀⢸⣷⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⠀⠘⠛⣦⠀⣏⠀⢠⡴⢯⣀⠀⠀⠀⠀⠀⠀⢠⡟⠙⠘⢋⣄⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢶⢾⡇⠀⢠⡞⠿⡟⠀⠀⠀⠀⠈⠛⠶⢄⠀⠀⡰⠋⠀⠀⠀⢏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⢉⣷⠀⠸⡗⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣰⠖⠋⠉⠙⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⣩⣭⣿⠀⠀⠻⣄⣙⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢛⡿⠷⠦⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⡇⣿⢸⠀⢀⣠⣬⠭⠼⠿⠷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣯⡴⠶⠆⢸⡗⠒⠒⠒⠒⠶⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⢿⢸⠀⣾⠀⠀⠀⠀⠀⠀⡀⠹⣦⡄⠀⠀⢀⣴⠻⣏⣠⡴⠆⢸⠃⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢸⢸⠀⡇⠀⠀⠀⠀⡴⠋⠀⠀⣿⡇⠀⠀⢸⣿⠻⡞⠁⣀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣸⠀⣇⣤⠖⠒⠺⠇⠀⠀⠀⠀⠙⠒⠚⠛⠋⠀⠻⡶⠛⢀⣿⣄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣿⣧⢸⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠚⠋⠁⠙⣆⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠹⣿⢸⣿⢦⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠘⣿⡏⠀⣿⠀⡞⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣟⠘⢧⡀⢀⣿⣧⣴⠏⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢸⡇
⢀⣀⣀⣀⣀⣀⣠⡞⣻⠁⢿⠀⠀⢻⣿⣝⣿⣇⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⣸⡇⠀⠀⢸⠃
⠀⠀⠀⠀⠉⠹⣿⣀⣽⣄⠘⣧⣀⣼⣿⣭⣽⣏⣉⡙⠛⠒⠒⠶⠶⠶⠶⠶⠤⠤⣤⣄⣀⣀⣀⣀⠀⠉⣹⣿⣿⠛⠛⠁⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⢛⣩⠿⠋⠀⢻⡟⠀⠈⠉⠉⠛⠓⠒⠲⠶⠶⠶⠦⠤⠤⠤⠤⠤⠭⠭⣭⡍⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠃
⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠋⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠶⠿⠯⠤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏

⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣙⠻⣿⣿⡿⠛⢻⡛⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠋⠉⠀⠈⠻⠃⠾⠃⠇
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⠯⠉⣀⣤⡔⠓
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣝⠛⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠈⠋⠛⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣶⣤⣼⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⡶⠂⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠛⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣄⣀⣀⣈⣭⣿⣯⣭⣤⣆⣠⣄⣀⢀⣀⡀⢰⣟⣡⣴⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡇⣙⣻⣿⣿⣆⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⡇⠀⠈⠙⠻⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⢿⣿⣏⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⣿⣦⣄⠀⠈⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⢿⣿⢸⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠃⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠟⠁⢴⣿⣿⢺⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⡟⠁⢀⣴⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣻⣿⣷⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷⣿⣾⢿⠿⠿⠿⢿⣏⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⢿⣿⡿⣯⣾⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⠏⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠹⠛⣉⣴⡿⣫⣾⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣠⡾⡻⣫⣾⣿⣿⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣰⣿⡏⢔⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠉⠈⠁⠉⠉⠉⠉⠉⠀⠉⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⡀
⠀⡀⡇⠀⠀⡀⣼⡀⠀⢠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐
⢰⡇⣿⣿⠀⣧⣿⡇⢰⢾⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠈⠛⠿⣿⣀⣿⣿⡇⣼⣾⡆⣷⢀⡇⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢿⡧⠇⢄⡀
⠀⠀⠀⠀⠓⠉⠛⠓⠿⠿⢯⣿⣤⣧⣾⣠⡆⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⠇⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠇⠀⠀⢳⠀⢸⠄⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠉⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠑⠄⠀⡬⡆⢀⠂⢠⠈
⠀⠀⠆⠠⠒⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⢀⠀⠀⠒⠣⣀⡤⡀⣸⣭⠆⠠
⠐⡆⢰⣛⣒⡀⣤⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢘⠌⢨⠅⣀⠹⣷⣥⣕⠂⢓

⠀⠀⠀⢀⡀⠀⠄⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠻⠀⠀⠀⠀⠀⣀⡩⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠉⠁⠟⠀⠀⠀⠘⠠⢤⣤⢤⠜⠃⠀⠀⠙⠃⠀⢠⠴⣯⠈
⠀⠀⠀⠀⠀⠈⢆⠀⡬⠁⠀⠀⠀⢰⡦⠤⠀⠀⠄⣴⠀⠀⢀⠀⠀⠀⠘⡆⠀⠀⠀⢻⡾⠿⣀⠀⠀⠀⠀⠀⠈⠂⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠈⠀⠀⠠⠄⠀⠀⠀⠀⠘⠇⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠼⠀⠀⠀⠀⠐⠺⠉⠀⠀⠀⠀⠧⡀⣈
⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⠖⠀⠀⠀⠀⠀⣓⡀⠀⠀⠀⠀⠷⠆⢀⠀⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⢩⠙⢢
⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠊⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⢏⡇
⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠶⠿⠶⠦⣴⡀⠀⠀⠀⠀⠀⠠⠀⠀⢠⠀⠀⠈⢄⠀⠀⠀⢠⣴⠃
⠀⠀⠀⠀⠀⢀⣞⢑⡀⠀⠀⣈⣱⠀⠀⠀⠀⠀⠐⠧⠀⠀⠀⢻⠃⠀⠀⠈⠀⠄⢀⢠⠀⢸⠀⠀⠀⠀⠁⠲⡄⠊⠁
⠀⠀⠀⠀⠀⣨⡉⠲⠅⠠⠀⢨⣟⠀⠀⠆⠀⠀⠀⠐⣴⠀⡜⠀⠀⠀⠀⠈⠁⢰⡀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⢹⣇⣀⣀⡀⠀⢸⣿⠀⠄⠘⠀⠀⠀⠀⠛⠀⠃⠀⠀⠀⠀⠀⠰⠐⠤⠚⠀⠊
⠀⠀⠀⠀⠀⠀⠙⠦⠧⠤⠤⠼⠋⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠚⠀⠀⠀⠀⠀⠲⠤⠴

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⡙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠁⣻⣷⠻⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢼⣿⣧⠘⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣇⢸⣿⣿⡆⠸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣸⣿⣿⣿⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣾⣿⣿⣤⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣁⠈⣛⡿⠀⠀⠀⠀⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠁⠁⣄⠀⠀⡀⠀⠃⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⠊⠁⠀⠀⠀⠀⠀⠐⣴⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠠⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⢀⠀⡾⠁⠀⠀⣤⣶⣾⣶⣶⣿⠛⢙⣃⣀
⠐⠦⢤⣴⣶⣶⣦⣤⣀⣘⣦⣀⣀⠀⠙⠚⢷⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⢠⡿⠁⠀⠈⢻⣥⣶⡟⣿⢿⣿⣷⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢷⣄⠀⣬⣷⣆⡀⠀⠀⠀⣿⡃⠀⠀⢠⣾⠟⠁⠠⣤⣾⣾⠿⠋⣼⣷⣄⣻⣿⣦⠙⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣼⣿⠏⠀⠀⠀⢼⢹⠁⠀⠀⠸⡇⠀⠀⣸⣿⠋⠁⠀⠀⠀⠹⣿⢧⢸⣿⡇⣾⡂
⠀⠀⠀⠀⠨⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣧⠀⠀⠀⢤⣷⠀⢠⡄⣠⣷⠀⠀⢿⡇⠀⠀⠀⠀⢀⡴⣃⡄⠈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⣄⣊⠽⠛⠀⠀⠀⢹⠃⢀⣼⣿⠏⠉⣷⡀⠈⢷⣄⠀⠀⠀⠈⢀⣿⣴⣴⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⡀⠆⠠⠾⠿⣿⣷⣶⡟⠀⠀⠨⡧⣼⠟⠁⠀⠀⣿⣇⠀⢸⡿⠀⣀⣠⠖⠋⠉⠉⠀⠀⠹⣿⡶
⠀⠀⠀⠀⠀⠀⠀⢈⠍⠁⠀⠀⠀⠀⠀⣀⣈⡽⠃⠐⠂⢰⣿⣶⡏⠀⠀⠀⢀⣼⣿⣧⢸⣷⠾⠋⠁⠀⠀⠀⠀⠀⢀⣤⣌⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣭⠝⠋⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⢀⣾⡋⠙⠛⠉⠀⠀⣀⣤⠼⢿⠦⠶⠟⣿⡟⣩⡄⠛⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠘⠃⠀⠀⠀⠀⠠⠀⢸⢙⣿⠀⠀⠀⠸⠿⢛⢷⣤⣤⠴⠞⠋⠀⠀⢀⢤⡀⢨⣿⣶⠿⠏⡡⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠊⠀⠀⠀⠀⢀⠤⢴⣦⣄⣸⣾⣿⡇⢠⡎⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠿⢃⣀⡜
⠀⠀⠀⠀⠀⢠⠒⠂⠺⠃⠀⠀⠀⠀⢀⣾⣤⡦⠀⠙⠉⠛⠛⠿⣿⣿⡅⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠁⠀⠈⠋
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠘⣿⣫⣶⡟⠓⠒⠐⠒⢶⡿⠉⢿⣿⣶⣠⡤⠀⣀⣀⣀⣤⣤⡶⠟⠃
⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠉⠉⠿⠻⡿⠻⠄⠀⣠⡿⠃⠀⠈⢿⣿⣿⣿⣿⠿⣿⡿⠿⢶⡀
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠇⠀⠀⠀⠈⣿⣿⠀⠙⠒⠀⠂⣀⣼⠇
⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠟⠀⣀⣀⡴⢁⣸⣿⣿⠀⠀⠀⢀⡾⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⢀⡞⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣀⣄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⢯⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⡸⡃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣇⣾⣷⣿⣿⣧⣁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⣫⣶⣾⡿⢿⣽⣿⣯⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣥⣽⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣴⣼⣜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣽⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⣯⣿⣿⡿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠻⠿⠿⠿⠿⠟⠛⠛⠿⠛⠿⠟⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⢤⣰⠤⠒⡲⢩⣉⠙⣯⡸⢇⣵⣶⣦⠆⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠋⠓⠶⠾⠞⠛⠷⠖⠋⠉⠉⠀⠀⠀⠀⠀⠀⠈⠁⠐⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣀⡤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡞⠁⠠⡀⠀⠭⠛⠦⡤⠒⠶⠦⢚⣩⠉⣩⠅⡠⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⣇⠀⠀⡀⠄⠀⠀⠀⠁⠀⠘⣟⠰⣦⣙⠋⣄⣀⡡⣬⡶⡀
⠀⠀⠀⠀⠀⢀⣴⠿⣬⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠁⠀⠀⠀⡀⠀⣄⢀⡉⠀⠉⠉⡤⣄⣬⠨⠏⢀⢠⣦
⠀⠀⠀⠀⣀⠘⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⠇⠀⢄⡄⠀⠀⠑⢀⠛⠆⠀⠢⠸⣀⠀⠉⡀⠁⠘⠀⠠⠄⠈
⠀⠀⠀⢠⣤⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡧⠪⠚⣶⣁⠀⠁⠀⠀⠂⠈⠀⠉⠀⠀⠀⢀⠀⡄⠁⠀⠀⠀⠀⠠⠀⢨⢶⣸⠇
⠀⠀⢠⢄⣠⠀⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⢀⣴⢏⠟⠇⢄⠀⠩⠀⠖⠁⠈⡁⠰⠆⢀⡀⠀⠀⠀⠀⠄⠀⠈⠐⠀⠀⢁⠀⠰⣬⠿⡴⠆
⠀⠀⣀⣴⢶⠀⠁⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣿⣟⣸⠏⢂⠀⠲⣬⣀⠈⡀⢀⠑⠂⠀⠊⠀⠀⠅⡀⠀⠀⡀⠢⠠⠌⠀⠈⠀⠁⠄⠁⠉⠺⠳⠃
⠀⣰⠿⠶⠦⠊⠠⣢⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⢣⡤⠀⠒⢊⠴⡖⣟⠐⠉⠢⠶⠀⠐⢒⢶⠌⢠⠀⣤⠀⣠⠀⠀⠈⠐⢠⠐⠀⢂⠀⠇⠀⠘
⠰⠃⣠⣤⢲⡖⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢼⡯⢉⡭⠴⢴⡊⢉⣂⡐⢀⠓⠢⠴⠊⠀⠐⠎⠀⢧⠈⠁⡈⢀⡦⢀⠀⠈⠀⠀⠠⠀⡀
⠰⢃⠶⠢⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣃⠀⢰⡋⢀⣀⣤⡹⣞⠉⠳⢤⣀⠘⡲⠘⠆⡀⠰⡼⣠⠄⢘⡃⠀⡰⠁⠀⠛⠈⠉⢅⡐⡄⠀⢀
⠈⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠿⠿⠟⠋⠝⢫⡋⠙⢿⣶⣶⣬⡓⢠⢄⠘⠀⡀⢿⡙⠶⠼⣧⠀⠏⠀⠀⠀⠰⡖⠄⣫⢍
⠂⢹⡆⣀⠀⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠛⠅⠀⠀⠁⠀⠀⠌⠀⣀⡄⠀⡘⠀⢻⡙⣆⠀⠱⡶⠠⠹⣀⠀⠙⢾⡆⢀⣠⡇⢀⠴⢓⡯⢱⣤⢂
⠳⣏⣟⣩⣧⣠⣴⡄⠀⣀⡀⠀⠀⠀⠀⠀⠀⢠⡟⣇⡀⠀⢀⠗⣀⢀⠐⠐⠀⠠⠆⠈⢠⡀⠱⠜⣏⠳⣴⣷⡤⠖⠉⠀⠀⠀⠉⠻⣼⣽⠏⠀⢸⡔⣿⠁
⠀⠀⣼⢩⡧⡴⠎⡷⢶⠾⣇⠀⡀⠀⠀⠐⠀⢸⡇⠋⠀⠉⠙⠋⠉⠉⠀⠀⠀⠀⠀⢀⠀⠋⠁⠢⠉⠓⠶⠲⢟⡳⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠹⣿⡇
⠀⠀⠳⣽⢄⡘⠀⠏⡼⢼⣭⡟⡛⣦⠀⠀⠀⠀⠱⣦⣐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠉⠛
⠀⠀⠀⢸⡂⡨⠘⢃⡇⢖⣢⣞⣙⣼⠃⠀⠀⠀⠐⠀⠉⠛⠛⠛⠲⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⢀⣳⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⡓⠂⠐⡋
⠀⠀⠀⣾⢃⡀⣴⠢⡔⡆⠫⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠐⠐⠦⣂⡼⠶⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣨⠿⠀⠂
⠀⠀⠀⠹⣆⣤⣿⢠⢀⣏⢲⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠐⠲⢤⡂⠜⢻⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢥⡀⠂
⠀⠀⠀⠀⠹⡄⠙⠿⠾⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣿⡀⠑⠬⠯⠺⠟⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉
⠀⠀⠀⠀⠀⢻⠀⠐⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸⣏⠡⢉⡂⡂⢌⡿⠁⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠐⢾⣤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡄⢤⣰⠟⠁⢾⠅⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣶⠶⠲⢶⣿⠯⣿⣖⣲⣤⣠⣤⣤⡁⣀⠴⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣰⣤⣀⣀⡀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⣶⡏⣿⣿⣿⣿⠛⠛⣡⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣨⣤⣤⣝⣾⣧⣈⠀⠿⠾⠀⢠⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣧⠙⢻⣿⣿⣷⣦⣤⣴⣿⣿⣿⣿⣿⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣦⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⢧⡀⣸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⣿⣿⡟⠿⣿⣷⣄⠀⠉⠛⠻⠿⠿⣿⢻⣷⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⡇⠀⣀⠸⠿⣿⣿⣿⣿⣦⣶⣿⣿⣸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣼⣻⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠃⢰⣿⠿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⢄⣨⡄⠴⠛⠁⠈⠃⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣻⣿⡖⣶⣿⣶⠠⢄⣀⠀⠀⠀⠈⢹⣿
⠀⠀⠀⠀⠀⠀⠰⠁⠒⠀⠀⠀⠀⠀⠀⠀⠈⠸⣿⣿⣿⣦⣴⣿⣿⣿⣿⡿⣦⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡆⠀⣿⣿⣿⣿⣿⣿⣿⢿⠿⠿⡿⠃⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡿⠿⣿⡿⠿⠻⠉⠁⠀⠀⡁⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢀⣶⣦⠛⠁⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣼⣙⠃⠀⠰⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢠⣿⣿⣇⣤⡴⠂⣤⣤⠀⡀⠀⠤⣤⣘⣻⣄⠀⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⣾⣿⣿⣦⣤⣀⡀⠀⠸⠏⢤⠀⢰⣯⣿⣿⣿⣄⠃⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⣀⠉⠉⠉⠛⠛⠛⠛⠓⠀⠀⠘⠛⠚⠛⠛⠿⠿⢿⣦⠳⡄⠀⠀⠀⢀⣥⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠒⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠦⠄⠠⠿⠿⠛⠁

⡀⠛⠀⠀⠀⠀⠀⠀⡀⠀⠀⣜⣿⡛⡁⠀⢐⣇⣤⠾⠃⢤⠀⣀⢀⠠⡌⠀⠀⠀⠉⠀⠀⠈⠛⠛⠃⠀⠇⠐⠘⡤⠄⢀⡄⣀⠁
⠁⠠⣄⠐⠂⠀⠀⠀⢿⣦⢸⣾⣿⡟⠃⠀⠸⠅⠀⠀⠀⠈⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⡂⠀⠄⠀⠀⠀⠀⠻⠀⢁⠀⡈
⠀⠀⠀⠀⠀⠀⠂⠀⢿⣿⢸⡿⠏⠁⣀⣠⠄⠀⢸⣷⡄⠀⣀⣀⠀⠀⠰⠌⣰⠀⠀⠀⠙⠀⠆⢰⠀⠠⣀⠀⠁⢘⠀⠁⠀⠰⠆
⠀⢤⠀⡤⠀⠀⠀⠀⠘⣿⡝⡆⢠⠞⠋⠁⠀⠀⣶⣿⠿⣿⣿⠏⠀⠀⢈⠓⣙⠃⠀⢠⠄⠹⡆⠺⠆⠀⠀⠀⠀⠈⠀⠰⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢠⣷⣤⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠉⠀⠶⢀⠾⠀⠙⠀⡀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉
⠀⠀⠀⠛⠃⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄⠂⠀⠀⠀⡀⠀⠈⢆⣠⠀⠀⠄⢁⠐⡄⡶⠘⢀⠘⠳⠄⠀⠀⠀⠠⠀⠀⡀⠀⡀
⠀⠀⢄⡀⠀⠀⠈⠰⠀⠀⠀⠰⠆⠀⠀⠀⡀⠀⣙⢀⣤⠀⠀⠈⠀⣄⠈⠃⡀⣸⠀⠙⠃⠀⠀⠦⠈⠙⠀⠀⠀⠀⠀⠀⡀⠀⠅⠠⠄
⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣘⡁⠌⠀⠄⠀⠀⠉⠉⠸⠟⣧⠐⢻⣅⠁⠀⡀⠄⠀⠀⢀⠀⠀⠀⠀⠰⠀⣤⢠
⠀⠈⠃⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⠒⠀⠀⠀⠉⠀⠀⠀⠀⠀⢠⡴⠶⣶⠀⠀⠛⠃⠉⠀⠀⠉⠈⠀⣀⠀⠀⡀⠀⠀⠀⠲⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⠻⣿⠧⣦⠀⠘⠀⠀⠀⠀⠀⠀⣈⠀⠈⠝⠂
⠀⠀⡀⠈⠇⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠐⢀⡄⠀⠀⣀⠀⠀⠀⠀⠀⢹⡼⠃⠀⠀⠐⠀⠁⠀⢁⠀⠀⠀⠀⠤
⠀⠀⠙⠀⠀⠀⡀⠀⠴⠛⣧⠀⠐⠢⢀⠀⠀⠀⠀⢀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡆⡘⠀⠈⠠⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⡀
⣀⣀⣀⠀⠀⠀⠀⠀⢀⡘⣼⠆⢀⡀⠀⠀⠠⠄⠀⠈⠁⠀⠀⠈⠀⠀⢸⠀⢸⠟⣠⣷⠇⠀⠄⠀⠀⠀⢐⡀⠀⠀⠀⠀⠀⠁⠀⠘⠃
⣻⣽⣿⡇⠀⣴⠀⠐⠀⠀⠀⡐⠉⣅⠀⠀⠀⠁⠰⠄⠀⣤⠄⠀⠀⠀⠘⡇⡇⢀⢸⡏⢰⠁⢀⠀⠀⠀⠩⠱⠀⠀⠰⣿⣽⠂⣸⡆
⠉⠉⠉⠀⡼⠃⠀⠀⠀⠀⠀⠀⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀⢰⡿⣄⡹⣡⠞⠊⠁⠀⠀⠀⠀⠀⠀⠀⢠⡈⠻⠀⢩⣷⡀
⣀⠀⠀⡾⠃⠀⠀⠀⠁⠰⠀⠀⠁⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⢹⡀⣧⢹⣤⡥⢀⡄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠹⣄⠡⠈⣿⠄
⠟⠀⡾⠃⠀⠀⠀⠀⠀⡀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⣀⠘⡀⠀⠁⠀⢸⡀⠘⢺⠃⢠⠞⠀⠀⠀⠀⠰⠀⠄⠀⠀⠀⠐⠻⣦⡑⢄⢘
⢀⠞⠀⠀⠀⠀⠘⠁⡈⠇⠘⠂⠀⠄⠀⠀⠠⠀⠀⠀⠀⣁⠁⠂⠀⠀⠀⠱⣄⣟⠴⠋⠀⠀⢀⠱⡆⠀⠀⠀⠀⠀⠀⣠⣄⡈⠛⢶⡙
⠀⠀⠀⠀⠀⠈⣐⠀⠈⠄⠁⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠈⠰⡠⠀⠀⠐⢲⣿⡥⠤⠄⠀⠀⠈⢇⠁⠀⠀⠀⠀⠀⠀⠙⠋⢀⡤⣤⣌
⠀⢠⣶⠀⠀⠀⢀⣘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠧⠀⠀⣠⣄⡀⠀⠈⠳⣦⠀⠀⠀⠀⠀⠀⠠⠛⠀⠀⠘
⣀⣾⣿⣄⠀⠀⠐⠉⠀⠀⠀⠀⠂⠀⠈⠀⠐⠀⠀⠂⠀⠀⠐⠀⠀⢀⣤⣴⠄⠀⠀⠘⢯⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⣿⠛⠙⠋⠀⠀⢀⡀⠀⠀⠀⠠⠄⠀⠀⡄⡀⠠⠄⠀⠀⠀⠀⡀⠀⠈⠁⠃⠀⠀⢀⣈⣉⠠⡀⠾⡮⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠛
⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠉⠁⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠈⠴⠉⡱⠛⠁⠀⠀⠀⠀⡄⠀⠁⣀⠃⠀⠠⡀
⡐⠀⠰⠀⠒⠀⠀⠐⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠠⠀⠐⠋⠀⢀⣡⡇⠀⢠⡄⠀⠂⠀⠰⡎⠱⠂⠀⣸⡄⣠⡄
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢀⣴⡀⠀⠀⠀⠀⠒⠄⠀⠠⠀⠀⠂⠀⠀⠀⠜⠙⡿⠧⣄⣻⣿⣆⠀⠀⠐⡀⠐⠦⢀⡤⢀⡤⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣷⣿⣿⣿⣿⣶⣄⣀
⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⣠⠔⠁⠀⠀⠙⢲⡄⠀⠀⠀⠀⠀⡀⡀⠀⢀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⢰⢃⣤⡤⠠⠤⣀⢀⣷⠀⠀⠀⠀⠀⠀⣠⠀⠘⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠄
⠀⠀⢸⠚⠛⠚⠀⠲⠦⢱⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⢀⠀⠀⠀⠀⠀⣨⠃⠀⠀⠀⠛⠁⠈⠛⠂⠀⠰⢿⣿⣿⣿⣿⠟⣥⡴⠖⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠗
⠀⠀⠀⠀⡆⠀⠀⠀⣰⡷⣄⠀⠀⠀⢀⣀⡀⠀⢀⠴⣦⠈⣿⣿⠿⠃⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⢀⡷⠀⠀⠀⢹⠀⠸⡄⢀⠔⠋⢸⠁⠀⠘⠀⠁⠀⣻⡟⢰⠆⠀⠄⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⢀⡤⠚⠁⠀⠀⠀⠀⣿⠀⠀⢰⠃⠀⢀⡆⠀⠀⠀⠀⠀⢸⣏⣯⢠⣤⢴⣦⣄⣾⢿⣿⣿⣿⣿⣿⣿⢟⣗⣿
⡘⠉⠐⠚⠉⠈⠁⠀⠸⠀⠀⣬⠀⢀⡼⠁⠀⠀⢀⣠⣤⣼⣇⢻⡀⠙⠓⠁⠈⠉⠀⠀⠁⠹⠿⠋⠀⢸⣹⠃
⠀⢠⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⢿⡆⠀⠀⣾⣿⣿⣿⣿⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣾⣷⣦⡀
⠀⠈⣇⠀⠀⢸⠀⡀⡀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠐⠤⠄⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣯
⠇⠀⠸⣧⠀⡰⡆⠤⠤⠀⠀⢤⠦⢀⠧⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⢿⣆⣶⡆⢀⣦⠀⠀⠀⢠⠎⢸⣿⣿⣿⣿⣿⣿
⠐⠀⠀⠉⠀⠈⡅⠀⠀⢡⠜⠡⡀⣈⠘⠀⠀⠀⠻⣿⣿⣿⡿⣿⠿⡄⠉⠀⠹⣴⠀⠁⢀⡴⢃⣤⣾⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⢘⠇⠀⢨⠀⠀⠑⡇⠀⠀⠀⠀⠀⠙⢿⣿⣷⠈⠀⠀⠀⠀⠀⠘⠁⢘⣩⣴⣿⣿⣿⣿⣿⢟⠉
⠀⢀⡀⠠⠄⠀⠸⡀⠀⢸⣇⠀⠀⡇⢒⠒⠶⢶⣄⡠⣌⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠟⠁
⢠⠞⠁⠀⢈⡄⠀⠈⠀⡟⢹⣷⠞⠣⢤⠁⠀⠀⢨⠙⣿⠀⠙⣿⣷⡀⠀⠀⡠⠐⣫⣿⣿⣿⣿⡿⠋
⢸⠀⠀⢀⠎⠀⠀⢀⣸⠀⠈⣇⠀⠀⠘⣳⣦⣀⡀⠀⡏⠀⠀⠈⠛⢿⣶⣂⠄⠊⠁⠹⣿⠟⠉⠀⠀⠀⠀⠂⡀
⠈⠃⢠⠎⠀⠀⠁⠈⠉⠀⠀⠉⠀⠀⠀⢀⣀⣜⠂⠨⠃⠀⠀⠀⠀⠀⠙⠧⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⣶⡾⠿⠛⠛⣿⠛⠛⠿⣶⡶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠋⢁⡴⠟⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⠳⣄⠉⠛⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⢀⡾⠁⠀⠀⣀⣀⣀⣠⣤⣤⣤⣿⣤⣤⣤⣀⣀⣀⡀⠀⠀⠙⢂⣀⣀⣀⣀⣀⠹⢦
⠀⠀⠀⣰⠏⠀⠀⠀⠀⢀⣀⣼⠷⠖⠛⠛⠉⠉⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠉⢉⣥⣶⠿⠛⠛⠛⠛⠛⠛⠿⢷⣦⣄
⠀⠀⣼⠃⠀⠀⣠⡴⠞⠋⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⣰⠏⢀⣴⠞⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀
⢠⡟⣴⠟⠁⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⣸⡿⠁⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⠤⠤⠤⠤⠤⠤⠤⠤⣿⠤⠤⠤⠤⢤⠤⠤⠤⠤⠤⠤⠤⢴⣿⠤⡄⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⢻⣧⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋
⠸⣟⢷⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⡆
⠀⢻⡄⠙⢦⣄⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⣶⣶⡾⠿⠟⡛⢿⡿⠟⣡⣶⣷⣄
⠀⠀⢷⡀⠀⠉⠳⢦⣄⡀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⣀⣤⠶⠋⠁⠀⠰⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⢳⡀⠀⠀⠀⠈⠙⠻⢷⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⡿⠛⠉⠀⠀⠀⠀⣰⠃⠘⢿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠘⣧⠀⠉⠙⠛⠛⠓⠶⠶⠶⠶⣿⠶⠶⠶⠖⠚⠛⠛⠉⠉⢠⡟⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠈⠻⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⠀⠙⢦⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣠⠞⠁⢀⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⣽⣷⣦⣄⣀⣀⣿⣀⣀⣤⣶⣿⡵⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠿
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠑⠀⠸⠀⠀⠆⣀⠀⠀⠀⠀⠊⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢠⡆⠀⠀⠀⢀⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢲⡀⠀⠀⠠⠀⠀⠀⠞⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣠⡴⠋⠓⠀⢰⡄⠀⠀⡀⠀⠀⠻⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⢄⠀⠂⢲⠿⣧⣀⠀⠀⠀⡇⠀⠀⡄⠀⡇⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣄⡀⠀⡈⠁⡐⠁⠀⣉⠂⢈⡹⠆⠀⠀⠀⠀⠀⠣⠀⢉⡠⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡈⠁⠀⠉⠈⢀⠉⠣⡄⠹⠂⠀⠀⣄⠀⠂⠀⣠⠊⠇⠀⠠⠐⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠂⢀⡀⢁⠈⣃⡀⠙⢦⡴⠒⠂⠀⠠⠔⠏⠀⡄⠇⠀⠀⠐⠀⠀⣠⠁⠀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⣬⣄⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠾⣀⣉⣉⣭⠊⢠⡀⠀⠀⠀⠀⡰⠂⠀⠀⠐⠁⠀⣠⡬⠠⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠠⣂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⣹⠃⠀⠈⠀⠀⠀⣠⣄⢁⣀⠄⢈⡀⢀⠌⠀⠀⠀⠀⠀⠀⠴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣴⣤⣄⡀⠀⣀⣠⡄⢁⠠⠀⠀⠀⠰⠊⠀⠇⠈⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⠛⠋⠉⠉⠀⠀⠏⠈⠙⢶⣦⡠⠔⠀⢀⠀⠀⠀⠀⢰⢀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⠟⠁⡠⠀⠀⠀⠀⠀⡸⠚⠉⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠤⠀⠀⠐⣢⣤⣀⣄⣤⡴⠛⠁⠠⠀⠀⠀⠀⠀⠤⠈⠀⠀⠀⠁⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⠀⠀⠀⠈⠀⠀⠠⠶⠁⠀⢀⢀⣀⣀⡤⠄⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠴⡾⠃⠀⠀⠀⠀⣠⡤⠔⠴⠚⠉⠔⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠒⠃⠂⠀⢈⠵⡀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠠⣠⡂
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡴⠛⢲⠒⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⡚⠀⠀⠀⢆⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠰⠃⠀⠀⠀⢸⠀⠁⠀⠀⠀⠠⠐⠀⠀⠀⢀⠠⠖⢹⡟⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⣿⠄⠀⠐⠆⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢾⠇⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠍⡄⠀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡴⠂⡏⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠐⣆⠀⠙⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⢀⡼⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠤⠤⠶⠦⠤⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⡀
⠀⠀⠀⠀⠀⠀⢀⡤⢾⣁⣀⡠⠤⠖⠒⠲⠦⠤⠴⠒⠛⠛⠓⠲⠤⠤⠤⠴⠒⠲⠼⠷⣄
⠀⠀⠀⠀⠀⣠⠿⠤⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣄
⠀⠀⠀⢀⡼⠁⠀⠀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠓⠒⠦⠤⠤⠔⠚⠓⠒⠒⠋⠁⠈⢧⡀
⠀⠀⢀⡾⠁⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⡾⠁⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⣸⠇⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠘⣇
⠀⣿⠀⠀⠀⢀⣠⠎⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠈⡇⠀⠀⡞⠹⡄⢻
⢰⡏⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣀⠤⣄⠀⠀⠀⠀⠀⢠⠞⠀⠀⠳⣀⡏⠀⠀⡇⢸
⢸⡇⠀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠖⠁⠀⠈⣇⠀⠀⠀⢀⡟⠀⠀⠀⠀⠈⠁⠀⢠⠇⢸
⠀⣷⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣼
⠀⢹⡄⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠁⢠⡏
⠀⠀⢷⠀⠀⠙⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⡼⠀⡾
⠀⠀⠈⢧⡀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⢀⡰⠃⡼⠁
⠀⠀⠀⠈⢳⡄⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⡼⠁
⠀⠀⠀⠀⠀⠙⢦⡳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⠤⠖⠋⠓⠶⠖⠒⢲⡄⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣴⣒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠒⠒⠒⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⡄⠻⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⣀⣰⠆⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡦
⠀⠀⠢⡀⠁⠀⠰⠀⠀⠀⠀⠀⣀⠸⠆⡀⠀⠀⠀⢨⢋⣽⣶⡛⠲⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠾⠄⡆⠘⠿⠞⠀⠀⠀⠀⠀⢀⣶⣾⣿⡄⠀⠀⠀⢀⣤
⠀⠀⠀⢹⠀⠀⠀⢠⠄⠀⠀⠀⣠⣌⢨⡂⠀⠀⠀⢨⡁⠈⢻⡄⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠘⠛⡀⢠⣤⣤⠀⠀⠀⠀⢸⣿⣾⣿⣧⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠂⠀⠀⠏⠀⠀⠀⠀⠉⠀⠨⢍⠆⠀⠀⠀⢳⡀⠸⡇⠀⢀⠀⠀⠀⠀⢾⡧⠀⠀⠀⠀⠀⠠⣿⡞⠋⠉⠀⠀⠀⢀⣿⡏⢻⢁⣿⣷⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠒⡶⠀⠀⠀⠀⠈⢷⠀⠀⢠⠎⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠘⠢⣈⠙⠁⢸⡀⠙⠃⢠⠏
⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⡂⣿⠄⡇⣐⣔⠄⠀⠀⠀⠸⣷⣤⠏⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣠⡼⠑⠄⠀⠀⠀⠀⠙⣢⠀⢸⡏⠀⢀⡟
⠀⠀⢀⡀⠀⢳⢼⠃⠀⠀⠀⠁⣈⢭⣀⠀⠈⠇⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠐⡇⡘
⠀⠀⠀⠃⡐⢘⠛⠀⠀⠀⠀⠀⠀⠠⠈⢒⣀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⣴⣾⡿⣯⡉⠛⠂⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠐⠯⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠋⠈⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠘⠃⠀⠀⠀⠀⠀⠀⢹⡇⣦⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣢⢄⠀⠀⢸⣇⣿⠀⠀⠀⠀⠀⠀⠤⠳⠂
⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢶⣾⢢⠤⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⣤⠀⠀⠀⠀⠀⢠⣶⣶⣤⡀⠀⠀⠀⠀⠸⡟⢿⡿⠂⠀⠈⣿⡏⠀⠀⠀⠀⠀⠂
⠀⠀⢰⣿⣿⣿⣿⡀⠀⠀⠠⠈⠃⢫⡀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠼⠟⢋⣤⠀⠀⠀⠀⠀⠉⠁⠁⠀⠀⠀⢻⡇⠀⠀⠀⠀⠠⠀⠘
⠀⠄⠀⠀⠘⣿⣟⡀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠘⠛⣀⠀⠀⠀⠀⠐⠆⠀⠆⠀⠀⠘⠁⠀⠈⠈⠋⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⠀⠀⠀⠠⡄⠀⠀⠋
⠀⡀⠀⣠⣀⣁⡀⠉⠀⠀⠀⠾⠄⡀⠀⠀⠀⠀⠀⢤⠉⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠠⣿⣽⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠉⠉⠀⠀⠀⠈⠏⠿⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐
⠀⠄⠀⠻⠟⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠂

⠀⠂

⠀⠀⠀⠀⠡⠀⠀⢀⠀⠀⠀⠀⠀⠀⠔⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀
⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣿⣷⣤⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣮⣩⡄⠀⣥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⢳⠀⠀⠠⠀⡀⠀⠀⣤⣀⠈⢸⠘⢻⠛⠟⠛
⠀⠠⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠐⠀⠀⠱⡄⢸⣿⡿⠟⢻⡆⠖
⢲⣾⣻⠯⢀⣀⡀⢀⣄⠀⠀⢀⠄⠂⠀⢠⠀⠀⠀⠐⡄⢠⡈⠑⠄⣀⠀⢹⢨⣤⣤⡈⢻⠐⣿⣷⣦
⠘⡇⢨⠀⠫⠀⠀⢹⣿⠀⠀⣠⠀⠀⠴⠋⠀⣾⠀⠀⡅⠀⠁⠀⠀⠻⣤⡀⢢⠀⠙⠑⢾⣠⠐⠈⠁
⠠⠶⠾⠶⠴⣤⡄⠉⡉⠀⠘⣿⠗⠀⠈⠀⣤⣤⣤⣦⡥⣤⣤⣤⣥⣴⣴⣶⣶⣷⣶⣶⣼⡷⠶⠛⢻⣯
⢰⣦⢞⣀⣠⡀⡀⣀⣀⣒⣸⣿⣧⣤⣤⡤⠬⣿⠭⢭⠽⠟⢻⣿⣿⣿⣻⣿⣿⣿⡟⡙⣹⡄⢠⡀⣾⣿
⣀⡶⢚⣻⢲⣿⠌⣱⣿⠟⣿⡿⠿⠿⠃⠀⠀⠠⠴⠫⣷⣖⠶⢿⢿⣿⣽⠟⠉⠈⢰⡟⣿⡥⣿⣄⣿⣿⠄
⠡⠺⠯⠖⠁⠀⠀⠿⠷⠏⠁⠀⠐⠂⢚⡁⢀⣉⡉⠉⡉⠙⠛⠿⠓⡬⣁⠉⠃⠀⣷⡀⣿⣗⠥⣸⣿⣿⣦
⠀⠀⢐⣒⣀⠀⠀⠀⠀⠀⠒⠀⠠⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠸⠦⠉⠀⠀⢠⣿⠀⣼⠁⠀⣿⣿⣿⣧
⢴⣾⠏⣽⡿⢿⡶⣴⣏⣿⠦⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⢟⡃⢹⠅⠈⣻⡿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠐⠆⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡂
⠀⠀⠀⠀⠀⠀⠘⢀⡠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠉⠀⢁⠈
⠀⠀⠀⠀⠀⠀⠀⢺⡇⣰⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠘⠃⠛⣃⢀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣸⣧⡆⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠘⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⣸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣿⣿⣠⡟⢠⣆⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣾⡏⠉⠜⢁⠂⠤⢂⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠐⢷⣾⡿⠿⠋⠀⣠⠉⠾⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⢀⠀⣸⣧⣩⣤⡌⠉⠃⠀⢽⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⣴⢫⣟⠰⢆⣿⡟⠋⣩⡀⠗⠂⠀⠃⠀⠐⣰⣿⠀⠀⠀⠀⠀⠀⠀⣤⠀⡿⠃⠂⠀⠁⢻⣈⢉⠉⠀⢠⣽⣀⠾⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣃⢲⣀⡀⠀⠸⡿⠇⠰⠀⠀⣰⣥⡆⠁⣶⣄⣈⣹⠀⠀⢠⠉⠉⠀⠀⠀⠀⠀⠀⣼⠏⢘⠁⠀⠀⠀⢶⣟⠛⡉⠁⠡⢀⡿⠟⠛⠃
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣠⡴⠻⣇⡙⠀⠀⠄⠀⡀⢠⡌⢻⠁⠶⠄⢀⡉⢻⠆⠀⡜⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠀⢠⡄⢀⡾⡇⣾⣧⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢠⣴⠾⠧⠀⣤⣄⣾⣧⡈⠀⢸⠀⠀⠀⠨⠑⠼⠀⠀⠁⠀⠀⠀⠰⡋⢙⡄⢀⡤⠀⠀⠀⢊⣱⡁⠀⠑⡶⠚⢁⡾⠉⢳⣿⣧⣶⣄
⠀⠄⠠⣄⢀⣀⡀⠀⣶⠃⠀⢴⣟⠀⠀⠀⠹⣿⣿⣷⡄⠀⠀⢠⠃⡀⠀⠀⠆⠀⠀⠀⠀⢀⠏⠀⢰⣴⡆⣾⣾⣧⣜⣿⡇⠿⣙⡆⡈⠀⠀⡾⠁⣰⣶⣶⡯⣿⣥
⠀⠀⠀⢨⠛⠻⣇⡀⠁⠀⣡⣶⠒⣻⣶⢀⣺⣿⡿⠀⠂⠀⢰⡇⠀⠹⣦⠀⡄⠀⠃⠀⠀⠸⠆⠀⢘⣿⣧⣾⡿⣿⣿⣷⣄⠀⠼⠏⠿⢀⡞⠀⣠⣬⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠐⠻⠄⣠⣶⠟⢸⠏⠈⢻⣾⣿⡟⠧⠀⢲⡀⢼⠃⠀⠀⠨⣧⠀⣀⣀⠀⠀⠒⠀⠀⠋⢹⣿⣿⣿⡟⢹⣿⣿⣷⡄⠀⢀⡼⢶⣰⠿⠇⠀⠁⣿⡀⠔
⡀⠶⠀⣀⠀⠀⠀⣹⣿⠁⢀⡿⠀⠀⠋⢦⠽⠀⠀⠆⢀⣁⣾⠀⠀⢀⣤⣌⡠⣿⣇⣴⡄⠀⠙⢀⣰⣿⣿⣿⣿⡇⠛⢹⣿⡟⢻⣀⣾⣻⠿⠁⠀⠶⠠⠀⣩⣴⡇
⠀⠃⣘⣡⡄⠀⠠⠙⢻⣆⡠⣄⠀⠀⣠⣴⣶⣶⣤⣾⣝⣿⣿⠓⠀⠘⠛⢻⣅⢘⡘⣿⡇⠐⠶⢾⣻⠻⠿⣿⡿⢯⣤⣿⢘⠁⢘⣿⢿⣿⠶⣀⣴⣧⣴⣾⡿⠛⣷
⠀⠀⢸⡿⡉⠶⠦⣤⣾⣧⠀⣼⣿⠁⠀⣾⡇⠈⠙⠻⡿⢿⣿⠀⠀⠀⠀⠺⡿⢚⢠⣿⣷⣄⠀⠀⠟⠓⠘⠋⠀⠘⣿⠋⠀⣰⢟⣵⣾⣞⣳⣿⣿⣿⣿⣿⡿⢿⡿
⠀⠀⠀⠘⠀⡇⠈⢢⣿⡟⠸⣿⣿⠀⣾⣿⠁⠀⠀⠀⢱⠀⣿⠀⠀⠀⠀⠀⡶⠋⣡⠆⠹⠋⠀⢸⠀⠀⠀⠀⡀⠈⠁⠀⠀⢡⡾⠛⢻⣿⠿⢿⣿⣿⣿⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣌⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣌⠻⣿⣧⡀⠀⠀⠀⠀⢀⣾⡿⢱⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧⡙⣿⣷⡀⠀⠀⠀⣾⣿⢃⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣌⢿⣇⠀⠀⢰⣿⡟⣸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣶⣶⣶⣦⣤⣀⡀⠀⠈⠻⣿⣦⢿⡄⠀⢸⣿⠇⣿⣿⠏⠀⠀⠀⠀⠀⠀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣶⣮⣭⣛⠻⠿⣿⣷⣦⡀⠀⠙⠿⣧⠀⠘⣿⢸⣿⡏⠀⠀⠀⠀⣠⣴⡿⣻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣷⣦⣍⡛⢿⣦⡀⠀⠘⣷⡀⡿⣾⠟⠀⠀⠀⣠⣾⡿⢋⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⢿⣷⣮⣿⣦⡀⠘⣧⣼⠋⠀⠀⠀⣴⣿⠟⣰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⣀⡀⠀⠀⠀⠈⠉⠉⠉⠓⢾⡏⠀⠀⠀⣼⣿⠏⣼⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣝⡻⢿⣷⣦⣄⠀⠀⠀⠀⠀⠀⢿⡀⠀⢠⣿⢏⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣌⡻⢿⣿⣦⡀⠀⠀⠀⢸⡇⠀⢸⣏⣾⡿⠋⠀⠀⠀⢀⣀⣤⣴⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣦⡙⢿⣷⡄⠀⠀⠀⣿⠀⣬⡿⠋⠀⠀⢀⣠⣶⡿⢛⣵⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣦⣙⢿⡄⠀⠀⢻⡾⠋⠀⠀⠀⣴⣿⡿⢋⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣀⡀⠀⠀⠀⠉⠛⠿⣦⣿⡄⠀⢸⡇⠀⠀⢠⣾⣿⢋⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣯⣙⠿⣿⣦⣄⠀⠀⠀⠀⠉⠛⢦⣸⡇⠀⢀⣿⡟⣡⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣮⡛⢿⣷⣦⠀⠀⠀⠀⠀⢹⡇⠀⣸⢟⣼⣿⠟⠋⠀⢀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡙⢿⣷⡄⠀⠀⠀⢸⡇⢠⣿⡾⠟⠁⣀⣴⣾⢟⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣦⡻⣿⡀⠀⠀⢸⣷⠟⠁⠀⣠⣾⡿⢋⣵⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⢦⣄⡀⠀⠈⠛⠿⣷⣜⢷⡀⠀⢸⡇⠀⢀⣾⣿⠟⣵⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡝⢿⣶⣄⠀⠀⠈⠙⠻⢧⡀⢸⡇⢀⣾⡿⣡⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⡹⣿⣷⡄⠀⠀⠀⠀⠙⣿⠁⣸⡟⣴⣿⡿⠋⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡌⢿⣿⡄⠀⠀⠀⠀⣿⢀⣿⣾⠟⠋⢀⣤⣶⣾⠿⣻⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣎⢻⣿⡄⠀⠀⢠⣿⠞⠋⠁⣠⣶⣿⠿⣋⣵⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆⢻⡇⠀⠀⢸⡏⠀⢀⣼⣿⠟⣡⣾⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣎⣿⠀⠀⣿⠀⢀⣾⢟⣥⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄⢠⡏⢀⣾⣣⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣡⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⣀⣀⣀⣀⡀⣀⡀⢀⣀⣀⣀⣀⣀⣀⡀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⢀⡀⡀⣀⣀⡀⢀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠠⣤⣼⣢⠞⠀⠀⠘⢦⣸⠃⠀⠀⠀⠹⡄⠀⡄⠀⣠⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣇⣀⣀⣻⡀⠈⢿⡀⠀⢀⣠⡖⢷⡄⢻⠞⠃⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢦⣀⠈⡏⠉⣩⠍⠻⣄⠀⣷⠞⠋⠳⢤⠀⢳⣈⡆⠀⠀⢸⣀⣠⠀⠀⠠⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠓⣷⠚⠁⠀⠀⠙⢾⡇⠀⡀⠀⣄⠀⠈⣿⣀⣤⠶⠛⠙⠀⠁⠀⠀⢀⡤⠚⠻⣝⢩⠻⡏⣏⡙⠟⢦⣀⠀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢼⠄⠀⠀⠀⠀⣿⡤⠞⠁⠀⠘⣇⣸⠋⠁⠖⡄⠀⡃⠀⠀⠀⡴⢋⣀⣀⠀⠈⢻⣴⣬⣤⣧⣷⣧⣽⣦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣸⣇⣤⣦⣶⣠⣏⣀⣀⣠⣶⠀⣸⠏⠀⡄⠡⣧⣤⢶⡤⣄⣼⡇⣯⢉⢹⡇⠀⠀⡇⣶⢶⠲⣆⠀⠀⣷⣤⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣡⣬⣤⣤⣤⣤⣤⣥⣄⣉⣉⠉⠉⠛⠛⠛⠛⠛⢳⠲⠶⠷⠦⠼⣯⣍⣇⣹⡈⠀⡇⠀⠀⡇⠿⠾⠴⠟⠀⠀⣿⣉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠛⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠙⢓⠲⠷⠦⣤⣤⣀⡀⠀⣻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡶⠶⠶⠶⠴⠦⠶⠤⠤⣤⣤⣄⣤⣀⣀⡀⠠⢀⠀⠀⠈⠉⠙⠻⢦⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠶⠤⣤⣄⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡤⠤⠤⠤⢤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣶⡿⢳⡄⠀⠀⠀⠀⠀⠀⠀⠳⣌⠠⠀⠀⠀⢀⠀⠈⠉⠉⠉⠙⠛⡓⠲⠦⠤⢤⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⢀⡞⠳⣬⡛⣛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⢶⣶⣶⠓⠚⡛⠓⠓⠺⠶⡶⢴⣤⣄⡀⠀⢂⠌⡃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⡀⠃⠀⢸⡳⣄⡞⣉⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡌⠁⣀⡀⠁⡀⢀⠈⡀⣵⠟⢳⡄⢍⡓⠦⣄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣞⣯⣹⢦⡀⠈⢇⣸⢉⡽⠋⠀⠀⢀⣀⣠⣤⣤⣤⡄⠀⠠⠀⠙⠲⢤⣄⠠⠀⣀⠀⣀⠙⢳⡟⠂⠠⡸⠁⢨⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⡈⢹⡃⡴⠛⣿⣾⡟⣭⡄⢀⡤⢾⡏⠸⣇⡼⠿⡿⠁⠀⠀⠁⠀⢰⡟⠻⡟⢦⣠⣄⢀⡀⣹⡏⣈⡴⢦⣴⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠉⠛⣟⠁⡴⢻⠍⣿⡿⣠⢾⡇⢸⣧⣞⡉⣩⠟⠁⠀⠀⠀⠀⠀⠈⡿⠛⠁⠈⣧⡶⣄⣤⣼⣏⣉⠷⣴⠛⠖⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⡙⠲⢿⠞⠁⡼⠀⢸⣠⢿⣠⠿⢯⣤⡤⠛⠁⠀⠄⠀⠀⡀⠀⠀⢰⣇⣤⡀⠀⢸⣿⠉⡄⢼⡇⢉⣨⣏⢉⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠙⠲⢬⣇⠼⡁⠘⢿⡿⠿⠛⠛⠉⠁⠀⠀⢐⠀⠀⣻⡟⠛⢦⠀⣸⢿⣬⡿⠀⢸⣿⠤⠴⢾⡇⣤⠼⣯⢠⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢴⣤⣿⣾⠷⣦⣄⣻⣠⣤⢀⣄⡀⠂⠈⡆⠈⠳⢹⣷⣊⡼⠀⢻⠀⣿⠀⠀⢸⢿⠄⠰⢸⡧⠄⠄⡧⠤⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠈⠹⡇⠀⠀⠉⠻⣏⣿⠟⠋⢠⠀⢀⡗⠀⢠⠏⠠⢻⠀⠀⢸⠀⢸⠀⠀⢸⢸⠖⠒⢺⡗⠶⠒⡷⠲⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠈⠀⠀⠀⣹⡀⠀⠀⠀⣷⠀⠀⣴⣀⣿⡿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣄⣠⠶⠛⠉⠁⠀⡀⠀⣿⣷⢠⡿⣿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠁⠸⣇⠀⣿⣿⡾⢡⡏⡿⠈⠙⢄⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠞⡄⠀⢀⠔⠀⠐⠂⠐
⡄⠀⠀⠀⠀⠀⠀⠹⣷⣿⣿⢁⣿⣿⠃⠀⠁⠈⢈⡀⢠⣤⣤⣤⣧⡔⠁⢸⠋⡴⠁⡀⢸⡇
⠂⠀⠀⠀⠀⠄⠀⠀⠈⣿⢾⣼⣻⡏⣠⣴⡆⣪⣶⣗⠉⣉⠛⣯⣝⠳⢄⣾⢉⣁⣠⢀⠏⠈⠐⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⣹⣿⣿⣿⣿⣿⡟⠟⠛⠛⠒⠛⠛⢯⡛⠓⣾⢿⣍⣀⣠⠟⠀⡀⠀⠀⠀⠉⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⡟⠁⢸⡀⠀⠀⠀⠀⠀⠹⢾⣃⠘⠓⡾⠉⠳⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠋⠀⠁⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⢈⡄⠀⠀⠀⠀⠘⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣄⡀⠀⠀⠀⠀⡀⣼⣿⣿⣿⠀⠀⠀⢸⠁
⠛⠒⠀⠘⠀⠼⢹⣿⣿⣿⡇⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢤⠤⠤⠴⠤⠤⠤⠤⠤⠤⠄
⡀⠀⢠⠀⢀⡀⣾⣿⣿⣿⠀⡀⢠⠈⠀⠀⠀⠋⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂
⠙⣶⠞⢣⠸⡇⡸⠋⠉⠙⣨⣡⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣆⠀⠀⢠⠶⠆⡒⣂⠀⢀⠀⣔⢉⠰
⡷⠛⣿⡾⢻⣶⣧⡄⣠⠃⡟⢁⡶⠂⠀⢠⡐⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢉⠚⠀⠌⠘⠛⠉⠀⠀⠀⠀⠈⠉⠀⠉⠀⡐⠋⠀⠀⠉⠘⠻⡟⠛⠂
⣁⣤⣯⣧⣾⣿⣾⡖⣹⣰⢴⠋⠀⠀⠀⠀⢙⣆⠙⣄⣀⣀⣀⣀⢀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⣈⣡⣄⣈⢀⣀⣀⣄⣀⡀⢀⣄⣀⠀⣀
⠉⡁⣿⢹⠗⣿⣷⣧⣽⣿⡏⣉⡔⠁⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠂⠀⠤⢭⣝⣯⡙⠛⣛⠛⠛⠛⠧⠷
⠀⢰⢻⢸⡄⣿⠂⣿⢟⣸⠔⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠘⡇⠸⣿⠃⠈⣯⣌⠃⠖⡾
⠲⣶⣼⣆⢻⢠⣾⡟⣿⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠
⣼⣾⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⡠⠆
⣽⠿⠿⢿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠔⠁⠈
⠀⠀⠀⠘⣋⢩⣿⡛⣯⠈⠙⢢⣴⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⢀⡂⢠⠮⠂⣴⠤⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⡀⠀⠐⢒⠉⢤⣼⣿⣾⣷⣽⣿⡯⠭⢭⠹⣿⣿⡂⢀⣴⣦⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⡜⡩⠋⠀⠀⠿⠤⠄⠠⠀⠘⠀⠀⠀⢸⣻⡅⢀⣀⠺⠄
⡄⠔⠀⠀⠀⡀⢭⣥⣥⣨⣿⣷⣤⡶⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢀⣄⣠⢆⣀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢡
⣄⣸⢀⣶⣭⣀⡨⠙⣻⣿⣯⣿⣇⣠⣼⣶⣾⠋⢻⠿⠿⢟⣿⣻⣿⡿⢿⣿⣿⣿⠋⣩⡿⠉⠀⠀⡀⢀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠁
⣯⢿⣿⣿⣿⣿⣿⣿⣾⠿⠻⠛⠙⣿⣾⣿⡿⣿⣿⠤⣴⢿⣭⣭⣿⣿⣿⣿⣿⣿⠀⠟⢿⣧⣤⠛⠉⠙⠺⠇⠀⠀⢈⣤⣄⡀⠀⠀⠀⠀⠀⢀⡀
⣟⣿⣿⣿⢿⣿⠋⡭⣄⠀⣀⣀⣼⣿⣿⣍⣟⠉⣃⡾⢣⣴⣶⠶⣶⡿⣿⣿⡏⠉⠉⠀⠀⠻⡇⠀⠠⣤⣤⢆⠀⠀⣸⠄⠈⠁⠀⠀⠀⠀⠀⠀⠈⠲⣤⣄⣀⣀
⣿⣿⣿⣷⠿⠁⢀⣀⣙⣿⡟⠋⠉⠙⣿⣿⣿⣿⣿⣾⡿⠛⠛⠛⠻⣿⣿⣿⣠⣤⣤⣤⣀⣀⣼⣶⣾⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠙⠛⠿⠿
⣻⣿⣟⣱⣟⠻⣧⡄⠀⠀⢈⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡶⠒⠒⢛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡀⣠⣤⣤⣤⣤⣴⣶⣟⠋⠉⠀⢄⣤⣄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⣦⡌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⢄⡀⠀⠀⠀⢸⢻⣴⢈⠈⢻⡄⠀⠐⡇⠀⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠤⠤⠤⠤⠄⣴⠇⢠⡠⠀⠹⠀⠀⠀⢸⣿⠆⠗⢃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢑⣄⠀⠀⠀⠀⠀⣀⢀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⢘⡛⠀⠀⠀⠀⠀⠀⠀⠺⣶⣺⡛⠤⡼⠀⠀⠀⠠⣿⡇⣤⣄⢀⠐⢺⠈⡇⠀⠀⠀⠀⠀⠀⠈⠻⣿⣯⠤⢀⣈⠈⠁⢹⠋
⠀⠀⠀⠀⠀⠀⢸⡟⡛⠋⢻⠆⠀⠈⠀⠛⣹⢿⠛⠉⠀⠀⠀⠀⠀⣿⡏⣦⠈⣤⣄⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣆⠘⢩⣄⡴⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⣠⣖⣧⣤⣀⡄⠺⡏⠀⠀⠀⠀⠀⠀⠀⣽⡉⣄⡿⠀⢀⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠁⢂⠎
⠀⠀⠀⠀⠀⠀⠀⢻⣯⡭⠤⠈⢠⡈⠉⢰⢢⠃⠀⠀⠀⠀⠀⠀⠀⣿⢷⡴⠟⠩⢾⡃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⠀⠀⠈⣧⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⣭⣿⡿⣤⣀⠀⠶⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⠀⠀⠀⠿⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⠯⠀⠀⠀⢖⣠⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⠀⢀⣀⣀⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡷⢀⡌⠁⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠢⣠⡀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣌⣄⣴⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠒⠂⣀⠶⠶⠔⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⡆⠀⠀⠀⠀⠘⢛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣤⠄⠉⠉⢀⠀⠉⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⢿⡄⠐⠐⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡀⢀⡀⠀⣄⠈⠇⣐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠟⢿⡆⠀⠀⠰⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⣀⣶⠒⣳⣦⣿⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⢾⣷⣄⣤⣼⢘⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣯⣩⠋⠁⠀⢨⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢟⡉⠁⠀⠀⠈⠃⠀⠀⠀⠀⠀⢰⡧⠳⠢⠔⡀⢀⡸⠰⠆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⡛⢁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⢰⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⠜⠁⠀⠀⠙⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣰⣀⣄⠄⠈⢀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡋⠁⠘⠈⠺⠋⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⠀⠀⢀⣴⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠚⠇⠀⠐⢢⡀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠒⠒⠒⠛⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⡁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠲⠒⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠉⠚⠂⠀⠀⠐⠀⠈

⡀⠀⠀⡠⠔⠒⠒⠒⠢⢤⡀⠀⠀⠀⢹⠉⢃⠒⠦⣀⠀⠀⠀⠀⢠⡧⣀⡀
⠙⠦⡍⠐⠂⠶⠄⣌⣂⡐⠌⢢⠀⠀⡞⢀⠀⡆⠀⠈⠳⡀⠀⡰⠁⣄⠄⡈⢂
⠀⠀⠈⢦⡀⠀⠀⠄⠐⠙⡛⢮⣇⠀⢣⠀⠀⠹⣆⡠⠀⢧⠀⠃⠐⢧⠀⠀⠀⠃
⠀⠀⠀⠀⠈⠓⠤⠤⠬⠤⠒⠉⠈⠙⠲⢕⣦⣘⠛⢦⡀⠇⠀⠡⣀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠀⢒⠀⢄⣤⠞⠓⢬⣍⣻⡄⠀⠀⠈⠂⠤⠻⣄⠁
⠀⠀⠀⠀⠀⠀⢀⠜⢀⢀⣈⠖⠒⠋⠉⡸⠀⠀⠀⠀⠈⠉⠓⠒⠦⢄⣀⡀⠈⣧⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢸⠀⠠⢞⠀⠀⠂⢀⡴⠁⠀⠀⡀⠔⠊⠉⠁⣒⢺⠉⠈⠉⠉⠉⠉⠀⠀⠉⠉⠉⠓⠦⣤
⠀⠀⠀⠀⠀⠀⠈⢄⠐⢀⡠⠤⠔⠉⠀⠀⢀⡎⠀⠀⠀⢒⠴⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⡸⠁⠤⠤⠛⠋⢘⡴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠲⢶⣚⢽⠛⠀⠀⠒⠒⢦⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠙⣛⡾⣿⣸⡣⢝⡙⡆⠀⠀⠀⣸⠀⣠⢝⡓⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⣾⡅⡪⠒⢫⣔⡭⢚⣿⢀⣴⡫⢔⡫⢖⡩⢚⡥⠂⣱⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠗⠁⢀⡤⠷⣺⢕⡫⢟⡭⢚⡡⢔⡡⠞⢉⠔⠛⣴⢊⠵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢀⡼⢓⡩⢒⡥⢚⡥⢒⡭⠒⣉⣤⡚⣏⠀⠀⠓⠊⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡀⠀⠀⠀⠀⠀⠀⢀⡾⢔⡥⣚⠥⢚⡥⠚⣡⠔⣪⣿⣷⣹⠈⢖⣹⠃⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡩⢇⠀⠀⠀⡀⢠⣤⡘⢶⡥⢊⡥⠚⣡⠔⣫⠔⣫⡴⠟⣫⢔⡏⠁⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠐⣹⣄⡀⠸⣿⣝⡩⢘⡩⠞⣡⠔⣉⢔⡫⣰⠟⣋⢔⣪⠏⠉⡔⣰⢦⢫⢹⡄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣊⠔⣋⠝⣆⠀⠫⡖⣉⠔⢋⢔⡪⣶⠕⣊⠴⣪⠗⣡⢜⠶⣼⢟⡵⢋⡵⠝⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡥⢊⡥⠚⣩⣷⠀⠀⠀⠉⠙⢍⣋⡴⢊⠵⣋⠴⡪⠖⡡⢺⠉⠁⠉⠉⠀⠀⣀⠀⡰⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣩⠔⢋⡤⢿⡀⠀⠀⠀⠀⠀⠈⢘⡕⢊⡴⢊⠔⣩⢔⡇⠀⠀⠀⠀⠀⠀⠘⣷⢟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢊⡡⠒⡩⠟⣦⢤⡀⠀⠀⠀⣼⠔⡡⢚⡥⣪⢕⡽⢛⠶⠲⣄⠀⠀⠀⠀⣇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⡴⢋⠴⢊⠤⢊⡽⡄⢀⣴⠗⣋⢔⡫⢚⠕⣡⠖⢁⠀⢋⠙⠃⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣖⣡⠖⣡⠔⣇⢸⡡⢚⡵⣋⠔⡡⠚⢁⠔⢫⡜⣉⢈⣹⣻⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠺⠤⣚⣹⢺⡶⣫⢞⡵⠋⣠⠞⢁⠔⢁⠜⣡⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⠓⠫⠤⠾⠥⠴⠇⠒⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⡠⢀⡠⢀⡀⣀⢀⡀⢀⠀⣀⠀⡀⠀⡀⠀⡠⠀⣠⢀⡀⢀⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠉⠀⠉⠈⠁⠉⠈⠁⠈⠁⠉⠀⠈⠀⠈⠀⠈⠁⠈⠀⠀⠀⠉

⠀⢀⣀⣤⣀
⠀⢹⣟⣿⠿⠀⠀⠀⠀⠀⠀⠀⣤⡀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠴⠒⠂
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⠉⣠⣭⣿⣿⠿⣖⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠋⠀⠔⠈⣩⣈⢿⣿⣀⣿⡥⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠳⣀⣄⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡾⠻⢩⠟⠀⠩⠛⠉⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠘⣷⡁⠉⠀⠀⣠⡄⠐⠀⠀⣀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⡎⠒⠋⠀⠀⢠⣄⣤⠀⠀⣀⠀⠀⠀⠀⠛⠀⠀⠀⣀⠅⣤⠄⠀⢐
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠂⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠋⢀⡀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⡀⠆⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠦⢻⠋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣅⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡐⠓⠉⣁⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⣴⣦⣴⡆⠀⠀⠀⠀⠀⠈⠁⠉⠙⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⠀⡍⠢⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⣍⣄⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠄
⠀⠀⠀⠀⣁⠀⠀⠀⠀⠈⠁⢤⠽⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣿⠏⠀⠀⠉⠘⠂
⠀⠀⠀⠀⠨⢁⠀⠀⠀⠨⢳⡌⢂⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠧⠀⠀⠀⠚⠁⢀⠀⢀⡤⠀⠀⠀⠋⠀⠘⠃⠀⡀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠐⠂⠐⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⣘⠃⠰⣂⠀⠀⠀⠈⠶⡀⠈⠉⠀⠀⠀⠛⠁⠀⠀⠂⠠⣀⠀⠀⠀⠀⢉⡤
⠀⠀⠀⠀⠛⡓⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠄⠀⠀⠀⠈⠛⠁⠰⠄⠀⠀⠀⠀⠃⠉⠀⠀⠀⠐⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⢤⡄
⠀⠀⠉⠐⠂

⢴⠤⡦⢾⠤⡧⢼⠒⠧⠘⠤⣇⠜⢤⠛⡤⢧⠼⡄⢻⣆⠀⠀⠀⠀⠀⣠⡀⠀⠙⢿⠆⠀⠀⠀⠀⠀⠢⠆⠀⣠⠞⠁
⡾⠛⠉⠛⢶⡇⣪⡏⠀⠀⠀⢸⣋⡧⠟⠹⢎⡱⣩⠦⣻⣦⣁⠀⢂⣾⠃⢠⡄⠀⠀⠀⠀⠀⠈⠀⣀⣤⡶⠛⢁⣴⡁
⡁⠀⠰⠀⠀⣿⠸⣽⢦⡀⡤⢢⣬⠁⠀⠄⠀⣇⣾⠒⡥⢿⣿⠀⠘⣧⠀⠛⠃⠀⠀⢀⣠⡴⠞⠛⠉⢁⣤⡆⣉⡥⢰
⣷⣄⣀⣠⡼⢿⠒⡽⣪⡵⠭⣟⢤⢓⠄⣤⣔⢳⢔⣭⠾⠋⠉⠀⠱⠛⠀⠀⠀⠀⠐⠉⠀⡀⣤⣶⣯⣈⡵⢰⡿⠧⠈
⠦⢮⢩⣍⡗⢾⢹⣽⠃⠀⠀⠈⣷⣫⡖⢣⢜⣵⠟⠁⠄⠁⠀⠀⠀⠀⠀⠀⠀⡀⡤⢸⣿⣇⣉⡝⢾⣿⠇⣤⣴⠁⢾
⣓⠷⠘⠒⠻⣉⣟⡙⣦⣀⣀⣴⠏⢦⠜⣵⡿⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣲⠹⣿⣧⣤⣾⠸⠿⠟⢠⣤
⣴⣶⡶⠀⢀⡷⡔⢲⡓⣿⠏⣆⡹⢊⣾⠟⠀⠂⠀⠁⠀⠀⠀⠀⠈⣠⠁⠀⠀⠀⣪⣏⡶⣽⢼⠸⠿⣟⣴⣾⡇⠻⢛
⠛⠛⠁⣀⣉⣀⠉⢫⣨⣤⣏⡹⢡⣿⠏⡀⠀⠀⠀⠀⠀⠀⢀⠢⢠⡏⠀⠀⠀⢐⢿⣻⡻⠾⠽⠱⣪⢅⠙⣫⢾⣿⡿
⡿⢀⣼⣿⣿⠟⡠⠶⠔⠷⡜⣱⣿⠃⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⣾⠁⢀⠀⠀⢸⡿⣰⠚⠉⠉⠛⣮⢺⣽⣺⢈⣉⡔
⣀⣉⣀⠀⠀⢠⣶⣿⣷⠀⣰⡿⢁⠈⠀⡀⠁⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⢸⡷⡇⠀⠀⠄⠀⢸⣗⣲⣺⣟⣾⡵
⣿⡿⠟⠃⠠⢚⡉⠙⢃⣴⠟⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⠀⠀⠀⡷⡻⣤⣀⣀⣠⣞⣟⢞⣞⣮⣷⡽
⣡⣤⡄⠀⣸⠿⠟⣠⡿⠋⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⣿⠀⠀⠁⠀⠀⢙⣴⢷⣻⣻⣽⢽⢹⡟⠉⠀⠈⠙
⠿⠛⠁⢘⠁⣠⣾⠏⡀⠀⠄⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⢿⡆⠀⢀⠀⠀⠐⣺⡽⣺⠚⠚⠫⢿⡇⠀⠐
⣾⣿⠆⣠⡾⠟⠡⠀⠀⠀⠀⠀⠀⢀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠂⠈⠀⠘⣽⣃⠀⠀⢠⢯⣻⣦⣤⣤⡾
⠀⠄⠚⡋⠀⢀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⡐⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠄⠑⣗⣯⢯⢖⣾⡽⣗⡷⠓⠛
⠀⡁⡨⠀⠔⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⠀⠔⠀⣀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠂⠈⢪⢯⡟⠁⠈⢱⡄
⡠⠀⠂⠐⠀⠀⠀⠀⢀⣴⡿⠟⠁⠀⠀⠀⠀⠐⢻⡆⠀⠀⠀⠀⠀⠀⢹⣿⣆⠀⠀⠀⠄⠐⠀⠙⢧⣀⣀⣜⡾⡽⡽
⠀⠀⠀⠀⠀⠀⠀⠠⠞⠉⠀⠀⢀⡔⠱⠐⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⠀⠀⠀⠀⢀⠀⠀⠙⢯⢷⣻⢽⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢸⡇⠈⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⡀⠀⠀⠀⠐⠀⠁⠀⠉⠪⣷⣿
⠀⠀⠀⠀⠀⣀⡀⠠⠈⠀⡀⢠⡿⢁⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣆⠀⠀⠀⠀⠐⠄⠁⢀⠈⢪
⠄⠂⠀⠀⢁⡼⠃⠚⢦⡀⠈⠋⠄⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠁⠰⠞⠋⠄⠉⠀⠈⣷⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡤⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣾⣿⡿⢻⣷⠈⢿⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⢺⡀⠀⠀⣿⠟⠁⠘⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⢸⡇⠀⣀⠀⠀⡀⠸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣀⠀⠀⠠⣿⣿⣿⡿⠀⠸⢇⣀⣿⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠆⠀⠀⢿⣿⣿⣧⢰⡿⠞⠛⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣤⣶⠀⢸⣿⣿⣇⠘⠁⣠⠖⠈⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣿⣿⣿⡤⠀⠻⣿⣿⣶⣶⣧⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣴⣿⣿⣿⣧⣿⣸⣤⣾⠿⠋⠉⠁⣾⠿⠿⠃⠀⣠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣧⣼⡟⠃⣿⠇⠐⠟⠃⠀⠀⢀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⠈⢻⣿⣷⣾⣄⢁⡀⠲⠀⠿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣇⠀⣱⡟⠃⠉⠉⢙⣤⠜⢺⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣶⡿⠷⠾⣷⣆⣹⣧⣼⣿⣿⣿⡿⠿⠟⠛⠃
⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡿⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠘⠟⠙⢣⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀⠀⠀⢸⡇⢠⣾⡅
⠀⠳⣄⡀⢸⣿⡀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⡀⢸⣿⣾⠟⠁⢀⡄⠀⣠⡆
⠀⠀⣿⡇⠘⣿⡁⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣀⣽⣧⡴⠞⠋⢀⣶⡟
⠀⠀⠉⢁⡴⠈⠉⠀⠀⠁⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠻⣿⣿⣿⣿⡟⠉⠁⣠⠶⣠⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠑⣄⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⢿⣆⠈⠻⣿⣿⣷⣮⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠻⣶⠃⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠀⣿⡇⠀⣹⠿⠟⠋⠉⠀⠀⡀⠀⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠘⠃⠀⠀⣶⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡜⢷⢀⣿⠀⠀⢠⡤⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣼⣿⢁⣴⡗⠉⠀⢀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⡀⠉⠀⠀⠀⠠⣤⡀⠀⢸⣿⣿⠉⢿⣿⣯⡘⠻⣧⣴⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣼⡿⣨⢻⣶⣌⣿⠃⠀⠀⠹⠁⣠⡾⢛⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣶⣿⣿⣠⣆⠀⠀⠿⠄⢹⣿⡏⠀⠆⠀⠀⣴⣟⣵⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣆⡀⠀⣤⡀⢻⣿⠀⢺⡆⠀⣼⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠻⣿⠉⠈⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⡟⠉⠉⠉⠉⠀⠹⣿⡾⣿⣇⠈⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣇⢰⣦⣤⣄⠀⠈⠁⢸⣿⣤⣿⠇⠀⠀⢀⣤⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⣰⠁⣀⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡌⢻⣿⣿⣷⣤⡀⠈⠻⣿⠇⢀⡀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡇⢰⡿⠻⣿⣿⡿⢡⠀⣿⢠⡜⢣⣾⠏⠀⠀⠀⠀⠀⢠⣄⡀⠀⠐⠀⠀⠺⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⠜⠁⠀⠘⢿⠇⢸⡄⠸⣿⣧⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⠈⠁⠀⣴⠁⠈⢠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣄⠀⠀⢠⠆⠀⢀⣤⡀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡌⠙⢧⡀⢰⣈⣷⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀⣾⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣶⣼⣿⠊⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠟⠀⠀⠀⠺⣿⡄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣏⠈⣿⣿⣇⣀⣤⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⡀⠈⣤⠀⢀⣴⡀⣿⣧⠁⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢸⣿⡀⢃⣼⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠛⢷⣄⡹⡀⢸⣿⠇⢻⣿⠀⠀⠀⠀⠀⠀⠀⠠⠿⠁⠀⠘⡷⠗⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣶⣤⠀⠀⠀⣉⠛⣿⠈⠻⠀⢈⡟⠀⠀⠀⠀⠿⠃⢠⠀⣀⠀⠀⠾⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣶⣤⣘⠿⣤⡄⠀⠀⠸⠟⠀⠀⠀⠀⠀⡂⠁⢠⣿⡆⠘⣀⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠿⣿⣷⣽⣿⣴⡀⢰⡆⠀⠀⠀⢠⣾⣿⣿⠀⢻⣿⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠈⠻⣿⣿⣿⠀⢸⡇⢠⡆⢠⡿⠋⠈⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠈⠻⣿⣤⣤⡆⢀⣶⠏⠀⣶⣾⣿⡀⠀⠀⠀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡄⠀⠀⠈⠙⣿⠁⠈⣡⡄⠘⣿⡇⢀⠁⣴⣾⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄⠀⠀⠀⠸⡀⢀⣿⡇⢸⣿⣇⡈⣡⡭⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⠀⠀⢰⣧⠘⣿⡇⠈⢻⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⣿⣿⣿⣄⠀⢸⣿⠀⠸⠇⠀⠏⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⢿⣿⣿⣿⣿⢈⣿⠀⠀⠀⠀⢸⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⢻⣿⣇⠈⠸⠿⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣌⢹⣿⣆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢣⣼⣿⡀⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣟⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠿⠛⠛⠉⠉⠉⠛⠛⠿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠉⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⠉⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠈⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣷⣤⣤⣀⡀⠀⠀⠀⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣁⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣦⣍⡙⠛⠻⠿⠟⠛⣋⣩⣴⣾⣿⠿⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠙⠟⣡⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋




⣀⣀⣀⠀⣀⣀⣀⢀⢀⣀⣀⣀⡀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡀⠄⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⡀
⠀⠀⢀⠠⠄⠀⠀⠀⠀⣀⢵⠀⠀⢀⠄⠀⠀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠴⣾⣿⣿⣿⣿⣿⡆⠀⢀⣠⣴⣤⣀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⡁⠀⠀⠀⠠⠄⠊⠀⠈⠠⠊⠀⠀⠀⠔⠙⠀⠔⠢⠀⠀⠀⠁⠄⠈⠀⠀⠈⠛⢿⣿⣿⠟⠃⠴⣿⣿⣿⣿⣿⠃⡠⠒⢴⣶⠿⠛⠠
⠀⠀⠀⠂⠄⠀⠠⠖⠀⠀⠀⠀⠀⠀⡎⠔⠁⠀⠐⠀⠀⠀⡂⠀⠀⠀⠀⠐⠢⠀⠀⠼⠋⠀⠀⠀⠀⠀⢹⡿⠋⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠈⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠁⣀⣀
⠀⠀⠐⠂⠀⠀⠀⠀⠀⡀⠀⠀⢀⠐⢀⠀⠀⠀⠀⡌⠀⠈⢻⢦⠀⠶⠀⠒⠀⠀⠀⠀⠀⢀⠀⢀⣀⣠⣤⡀⠀⠀⠀⢨⠁⠀⠙⣷⡄⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⢨⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠟⠛⠉⠉⠀⡀⠀⠀⠀⠀⢀⣠⣭⣅
⠀⠀⠀⠀⠢⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠙⣷⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠘⠻⣿⣿⣿⡦
⠀⠀⠀⠀⠐⠊⠈⢂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠯⠐⠉⠀⠀⠀⠀⠺⣿⣷⡄⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠟⠋⠁
⢀⠀⠀⠀⠀⠀⢐⠜⠀⠄⠀⠈⠀⡀⢰⠄⠀⡄⠐⢖⠀⠀⠀⠄⠀⡀⠂⠀⠀⠀⠈⡿⠃⠠⢶⣾⣿⣿⣷⡦⠀⢠⠀⠲⣶⣶⣶⣤⡀
⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⡢⠴⠚⠉⠀⠀⢃⡀⠀⡒⠒⢒⡚⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⢹⠿⠟⠋⠁⠀⠸⣀⠀⢸⣿⣿⣿⠇
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⣀⠀⠀⠀⠀⠀⣠⠄⢀⣀⣀⠀⣄⣀⠀⠀⠀⠀⣠⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢈⡭⠭⠀⡷⣏⠀⢀⣞⣇⠀⠙⢲⡀⠧⠄⢀⡇⢾⠀⠀⠀⠀⢸⣠⠟⢼⠉⣹⢸⠉⣳⢸⠉⣷⢰⠀⠀⢹⠭
⠀⠀⠀⠀⠀⠀⠓⠚⠈⠃⠈⠓⠈⠀⠈⠙⠛⠋⠈⠓⠋⠈⠁⠈⠓⠂⠀⠐⠋⢁⠀⠈⠉⠁⠈⠉⠁⠺⠛⠁⠘⠓⠓⠘⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠲⠀⠀⠀⠃⠙⠁⠒⠀⠀⠀⠀⠐⠙⠀⠐⠘⠀⠀⠂⠉⠀⠀⠀⠔⠀⠂

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣤⠲⠶⢶⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣯⠿⣿⣿⣿⣿⣶⣴⣶⣤⣶⣍⠻⣿⣿⣻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⢿⣾⣿⣶⣿⣿⣟⠉⠁⠈⠁⢠⣾⣿⡇⠻⣿⣤⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠿⠶⢻⣬⣿⣿⣿⣿⣿⣷⡄⣰⣿⣷⣍⣉⠀⠀⠀⠙⠿⠋⠀⠀⠀⠈⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⡿⠉⠈⠁⢹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠻⣿⢿⣿⣯⡀⣠⡀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢟⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠻⣿⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣏⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠻⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⡟⠁⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠈⠁⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⢦⣤⣀⣀⢀⡀⠀⠀⠈⣡⣤⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠾⠷⠿⠶⠶⠶⠚⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠠⡀⣶⣿⣿⣷⡴
⠀⠀⠀⠀⢰⠀⠀⠀⠀⢿⡆⠈⢁⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠹⣀⠀⠀⠀⠻⠀⠀⠐⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢀⠴⠶⠷⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠘⡇⠀⠀⠀⠀⠀⠲⣤⣤⣄⠀⠀⠀⠀⠀⠁⢀⠀⠀⢀⡀⡀
⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠉⢳⠀⠀⠀⠀⠀⠀⠙⣋⠈⠛⠙⢶⣄⡀
⠀⠀⠀⠈⠐⢆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢦⡄⠀⠀⠀⠀⠀⠐⢂⠺⠄⠈⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠈⢳⡀⠀⠄⠀⠀⢀⠀⡠⡾⠎⠙⢷⣧⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠂⠻⠀⠀⠀⠀⣸⣼⣗⠀⠀⠀⠀⠈⠃⠀⢀⠀⠀⢄⣻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⠀⠄⠉⠉⡆⢠⠇⠀⠀⠀⠀⠀⠀⣉
⠀⠀⠀⠀⠀⠀⠻⢷⡾⣿⣿⣿⣷⣶⢀⠀⡀⠀⠿⡟⠀⠀⠀⠀⠀⣼⣿⣴⡶⠴⠞⠀⠂
⠀⠀⠀⠀⠐⢤⡀⠛⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⡌⠀⠀⠉⠙⠉⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⣿⣿⣿⠟⣻⣟⠀⠀⠀⠀⠁⠀⠋⠻⣷⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠘⠙⡧⠀⠀⠐⠀⢰⣿⢛⣿⣿⡦⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡐⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡀⠀⠀⠀⢺⣿⣿⣿⣿⣷⣾⣄
⠀⠀⠀⠀⠀⠀⠈⠵⢷⣄⣀⣀⠀⠀⣀⣀⡑⡀⠀⠀⠀⠀⠀⢀⡀⠑⠀⠐⠀⠄⠈⠁⠨⣽⠿⡿⢳⣯⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠤⣿⣿⠉⢯⡄⠈⢻⣷⣦⡄⠀⠀⢀⠤⠛⠓⣸⠓⠀⣀⠴⠂⠀⢄⣁⢈⡗⠈⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⠁⠈⠁⠀⠈⠁⠙⣿⡌⠂⠀⠈⠀⣀⣉⠄⠀⠉⠓⠀⠀⠀⠀⠀⠉⠓⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⢹⣟⠂⠀⠐⠋⠉⠛⢻⡆⢀⣀⣤⡄⠸⢤⣴⣤⣤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠨⠛⣦⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠻⠆
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣛⣙⣻⣉⡟⢻⡟⣹⡿⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢈⠃⠀⠰
⠀⠀⠀⠀⠀⢠⠀⣠⣴⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢰⣿⠛⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠄⢺⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⡴⠊⠁
⠀⠀⠀⠀⣰⠏⠀⠆⠀⠀⠀⠀⠀⠀⠀⣠⠾⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⣤⡿⠏
⠀⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠨⣗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡦⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋
⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⢀⡈⠁⠀⠀⠀⠀⠀⣼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠈⠉⠀⠀⠀⠀⢡⣴⡟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠰⣀⣿⠿⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⣦⣿⣯⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠹⢷⣦⣄⡁
⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧
⠀⠀⠀⠀⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠻⣧
⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠘⠓⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡤⣶⣤⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠁
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⡀⠀⠒
⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠙⠛⡷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀⡤
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⣴⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠙⢿⣧⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⣠⣿⠏⠀⠀⠀⠀⣤⠛⠙⣿⣄⠀⠐⣄⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢰⣶⣤⡤⠀⡀⣠⣿⣿⣿⠋⠀⡀⠀⣠⣴⣞⠁⡀⠀⠀⠙⣳⣶⣿⣿⣿⠷⠦⠀⠸⠇⣷⣤⡄
⠀⠀⠀⠤⠈⣋⣾⣿⣯⣿⣿⣿⣿⡟⣿⣿⣶⠿⢛⡿⠟⠁⢤⣾⣿⣿⣿⣿⣯⠱⠶⢶⣄⠀⢀⠈⠁⠀⠄
⠀⠀⠀⠰⣾⣯⣿⣿⣾⣾⣿⣿⣿⣷⣿⣿⣿⣷⣾⣷⣾⣾⡆⠉⢻⣽⣿⣿⠿⠁⠀⢀⣨⣿⣿⣿⣷⣶⣦⣤⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣾⣟⠙⠿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠉⠙⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⠋
⠀⠀⠀⠀⠀⠀⠻⠿⡿⣿⣿⣿⣿⣿⣏⠙⠻⣿⣿⣿⣿⡛⠀⠂⢀⣀⣠⣬⣯⣭⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣶⣶⡀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⣿⠻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠿⠿⠿⠟⠙⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⡼⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣇⣀⠈⠙⠻⠧⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡤⠖⠲⠾⣅⠀⠀⠀⠉⠛⠶⣄⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠉⢻⣧⡀⠀⠀⠈⠙⢦⣀⠀⠀⠀⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⣠⠖⠉⢈⣷⣄⠀⠀⠀⠀⠈⠙⠓⢾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡴⠋⠁⢀⡴⠋⠀⠙⣷⡄⠀⠀⠀⠀⠀⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⢦⡠⠞⠁⠀⢀⡤⠚⠁⠙⣶⡤⣤⣤⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣹⠀⣀⣿⣦⣠⠖⠉⠀⢀⡤⠚⢉⣿⣦⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠉⢹⠉⠁⡇⢈⣳⣄⡠⠞⠁⢀⡴⠋⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣿⠒⠒⢻⠒⠛⣟⠉⠀⡟⢷⣀⠔⠁⣀⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠛⠦⣿⣄⣀⣸⣀⣀⣽⣤⣴⡿⠛⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣹⠀⠀⣿⠀⠀⢸⣁⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⢹⠙⠓⣻⣶⠿⠛⠉
⠀⠀⠀⠀⠀⢀⡾⢻⡉⣓⣾⠶⠛⠉
⠀⠀⠀⠀⢠⣿⣵⠾⠛⠉
⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠘⡆⠀⠀⣠⠤⠖⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⢀⠞⠁⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠓⣄⠀⠀⠀⠀⠀⠰⣶⠦⡘⢆⣠⠃⣼⣀⣠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⡄⢀⡤⠒⠛⡇⠈⠒⠼⠾⣯⠞⣁⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⢀⣠⠃⡟⠀⠀⢀⡇⠀⠀⣠⠞⠳⢏⡁⠀⠀⠀⠀⠈⣱⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠤⠤⠤⣀⠙⣄⣇⠀⣠⠞⠀⡠⠚⠁⢀⣀⣀⣉⡒⠤⠤⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⣦⠸⡟⠉⠀⣠⣾⣖⠒⠺⣏⠀⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣈⠉⠉⠳⣇⣠⠞⠁⡏⠀⠈⢳⠈⠑⠒⠒⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣄⠀⠀⠀⠀⠀⢰⠋⢣⠀⢀⡟⠁⠀⠀⢣⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⡄⠀⠀⠀⠘⢆⣸⢠⣯⠖⠋⠉⣹⠂⠈⠑⠒⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⣼⠀⣠⠞⡇⠀⢻⡿⢇⣀⠠⠔⢁⡤⠤⠀⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠦⣌⠑⢾⢸⡇⠀⢰⢀⡏⠀⠀⡞⢦⢠⠏⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⣄⠀⢈⡹⠮⣿⣇⠀⡘⢸⠀⢀⣤⠳⣼⡾⠤⠤⣔⣊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣟⠒⢾⣿⡶⠃⣿⠀⡏⠈⢧⠈⢇⣠⠋⠁⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠚⠻⠾⣿⣧⡀⢿⠀⡇⠀⢸⣸⠏⠙⠢⠤⠤⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢨⠏⠉⢻⠀⠸⡄⣼⡟⢀⠔⠊⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠤⠤⠔⢚⣁⣀⡀⠈⣇⠀⠘⣿⡵⠃⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠉⣳⠼⣦⠞⠉⠓⠢⠤⠄⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⠶⠶⠒⠊⣡⣦⠈⢦⡀⠀⠀⣰⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⠈⢆⠀⠑⢄⠀⣇⠀⢘⢀⡤⢴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠒⠲⢤⡳⣀⣸⢀⣖⣚⡷⢮⡲⣼⣏⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢄⣀⣀⣀⡠⠟⠪⣿⠼⠗⠛⠒⠒⠛⠛⠦⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣤⣤⠴⢚⣧⡀⠀⠀⡠⠚⠛⠓⠒⡶⠋⠉⠙⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣠⠃⠀⡞⠀⠙⣆⠀⡇⠀⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣠⠞⠁⠀⠀⠧⡀⠀⡼⠀⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠙⢦⡇

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

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


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

⠀⠀⠙⣩⣿⢿⣹⣿⣿⣿⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣾⣭⣕⣋⣉⢿⠛⡿⢿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣟⡿⣿⠋⠉
⠀⠀⠀⣿⠇⠐⠛⠀⠀⠀⢾⣿⡿⢫⣥⡂⢨⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣷⣾⣤⣄⣀⣠⠝⠋⠻⠈⠁⠀⠀⠁⠀⠉⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣰⣿⣿⣿⣿⣿⣾⣿⣿⣿⣧⣿⣿⣿⣿⢿⣿⣿⣿⣿⣶⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣰⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⡀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢢⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⣀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⡀
⠀⠀⠀⠀⠀⠀⣼⣻⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠚⢉⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣦⣤⣄⠀⢹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣟⣱⣶⣶⣶⣦⣤⣤⡀⠀⢀⣾⣿⡿⠟⠋⢉⡉⠛⠃⣸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣭⣭⣙⡛⢿⣇⠀⠀⠻⣿⣿⢿⣿⡿⣿⣶⡄⣻⡟⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⢿⣿⠀⠀⠀⠙⠛⠿⠟⠛⠛⠉⠁⠉⠐⣹⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡩⠿⠿⠋⠋⠉⠁⢈⣿⡇⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⢿⠗⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡄⠀⠀⠀⠀⣠⣴⣿⣿⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠈⢻⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣷⣦⠀⠸⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⡀⠛⠿⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠟⠉⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣧⣀⠀⣀⣀⣀⣀⣀⣀⣴⣿⠟⠁⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠉⠀⠀⠀⠀⠀⠀⢸⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣾⣿⣧⡉⠛⠛⣿⣿⣷⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣯⡖⠀⠙⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⢀⣿⣿⣿⣿⣿⣄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢠⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣿⣿⣆⡀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⡼⣷⡄⢤⡀
⠀⠀⠀⠀⢀⣤⣼⡷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣿⣾⣿⣄⢿⣷⡀
⡀⢴⣦⠀⣸⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⣿⣿⣿⣿⣿⣙⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣖⣢⡤⠄⠀⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢌⣻⣉⣠⠎⢹⡯⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣴⣾⣆⠘⣏⣿⡏⠀⣎⠀⠀⣡⠙⠳⣤⡁⠐⠂⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠟⠉⠙⠣⣿⡿⠁⠀⠙⠷⠾⠃⠀⢸⣿⡇⣠⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⢀⣾⡧⠀⠀⠀⠀⢠⣀⡼⢿⢏⣾⠋⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠀⠀⠀⠀⢀⣿⡏⠉⠁⠀⠀⠀⠀⠀⢀⣯⣿⣇⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⠀⠀⠀⠀⠐⣿⣿⣷⡄⠀⠀⠀⠂⢠⢡⣾⠏⣿⣿⡎⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠐⠲⣿⣿⣾⣷⠀⣴⣿⣿⣿⡿⠶⠘⠓⠛⢲⣾⡟⠐⣿⡯⠷⠒⠚⠚⡋⡽
⠀⠀⠀⠀⠀⠈⠀⢀⢀⣊⠉⠛⠛⠛⠛⠿⣿⣿⣿⣶⣶⣖⣰⣶⣿⡷⠞⠋⠀⠀⠠⡀⢕⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡄⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⣠⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠙⣿⣿⣤⣀⣠⡀⣨⢤⣿⣿⠿⠋⠀⣸⣿⣤⣶⣶⣼⣿⣏
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈⢧⣙⢿⣿⣿⣿⠃⠀⢹⢃⢀⠀⣰⣿⣿⣿⡿⡛⠉⠉⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢸⣆⢲⡿⠿⢿⣷⣦⣇⣎⣎⣾⣿⣿⣿⣿⡿⠁⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠈⢡⣽⠶⠦⠔⠚⢿⣏⣛⢳⢽⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⡃⢰⡖⠁
⠀⠀⠀⠀⠀⠀⠈⠹⠶⣶⣶⡶⣾⡟⣡⣿⢬⠙⣻⣽⠛⡫⢛⡻⣻⣿⣿⠇⠀⠀⢀⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠹⠻⣷⣾⣿⡿⠓⣿⢛⠎⠻⠟⢓⣟⠳⢛⢻⣿⠀⠀⠀⢸⡿⠊⢓⡒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣓⠀⢉⣉⠄⠟⣳⣿⣿⣼⡦⠿⠿⠿⠤⠦⢾⡇⠀⠀⢀⡿⠽⠳⠊⠑
⠀⠀⠀⠀⠀⢀⠀⠘⠂⠠⠀⣊⣵⠭⠟⠉⠀⠀⠁⠚⠿⣤⣄⡀⠀⣸⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠐⠦⠀⠄⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⢯⡀⢠⣯⠤⠠⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣷⣿⣿⡏⠁⠨

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⠘⠛⠀⠀⠀⠀⢀⣺⣿⠄⠀⠀⠀⠀⢀⡦⣶⣦⠀⠀⠀⠀⢨⣅⣀⠀⠀⠀⠀⠀⠘⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡬⠄⠂⠀⠀⠀⠀⠉⠉⠀⠀⢰⢂⡀⠀⠳⠞⠻⠄⠀⠀⠠⣼⣿⣷⠆⠀⠀⠈⠀⠀⠀⡄
⠀⠀⠀⠐⠀⡀⠀⠐⠦⡜⣻⡄⠀⠀⠀⠀⠀⠀⠀⠲⣷⣮⠦⠀⠁⠀⠀⠀⢀⡀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⡂
⠀⠀⠀⡀⠠⠁⠀⠀⠈⠁⠈⣤⣦⠀⢀⡀⠀⠀⣄⠀⠉⠙⠀⠀⠀⠀⢀⠀⠘⠷⣴⠁⠀⠖⠀⠀⠀⡀⠀⠀⢸⡳
⠀⠀⠀⠨⠀⠈⠀⠀⠒⠴⠂⡈⠁⠀⠈⠁⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠈⠃⠀⠀⡄⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡤⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⠤⠈⠀⢀⠤⠀⡀
⠀⠀⠙⠣⠀⠀⠀⠀⠀⠀⠀⠀⠴⠷⠀⠀⠀⠀⠀⠀⣐⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠰⠀⠸⠄⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⢓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠮⡀⠛⠀⠀⠀⡀⠀⠊⠁⠂
⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⢬⠔⠀⠀⠀⠀⠀⢤⠀⠀⠀⠒⠀⠱⠛
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣴⠾⢲⣞⠂⠀⠀⣴⠾⢲⡿⡶⢶⡟⠂⣼⣆⢰⣷⣰⣶⠀⠀⢰⣗⣰⠾⠃⣴⠾⢶⣴⣶⣰⡶⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠻⠶⠺⠷⠂⠀⠀⠻⠶⠚⠛⠓⠺⠶⠺⠛⠿⠺⠟⠋⠿⠀⠀⠸⠟⠻⠶⠎⠻⠶⠛⠹⠋⠿⠗⠾⠟⠃


⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠙⡿⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠉⣿⣾⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣻⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡟⠛⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣄⢀⡟⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⡸⠿⢿⡿⠛⠢⡄⠀⠀⠀⠀⠀⠀⠀⣼⣧⣿⣟⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⣉⣉⣉⣛⣟⠁⠀⠀⠀⠀⠀⠀⢞⠀⠀⠀⠀⡳
⠀⠀⠀⠀⠀⣠⠼⠤⣤⡬⠤⠤⠖⠚⠓⡄⠀⠀⠀⠀⢤⣽⣛⣛⣛⣻⣭⡆⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠿⡹⠭⢭⣉⣩⣉⠭⠽⠟⢋⡹⠃⠀⠀⠀⠀⠈⠉⣩⡍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣺⣿⡟
⠀⠀⠀⠀⠀⠸⠿⠤⠤⠤⠤⠤⠤⠼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠾⠷⠇


⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⣀⡔⠁⢠⠄⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⡴⠇⠀⣶⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣶⣿⣷⣦⡜⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣍⠉⠉⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢀⣰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣿⣿⣿⣿⣿⡷⠄⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠛⠯⣉⢀⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠱⣝⢿⣿⣿⣿⡱⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⢻⣉⠈⠛⣨⠗⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⡎⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠧⠤⠷⠀⠀⠀⠀⠀⠀⠀⠘⠦⢤⣍⣙⣋⣡⡤⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠴⣧⡵⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⠀⠀⣳⢜⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⡀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠙⠒⣄⠀⠀⠀⠀⢠⡬⣷⠂
⠀⢘⠀⠀⠀⠤⢹⡈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⡘⠀⠀⠀⠂⠀⠀⢀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⡃⣠⠀⠀⠀⠀⠀⠐
⠀⠸⠀⠀⠀⠀⠀⠁⡄⡀⣘⠀⠀⡃⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢬⡁⢰⠁⡄
⠀⠀⠄⠀⠈⠂⠀⡌⠀⣷⠉⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⢠⡀⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡈⠁⢰⠇⢀
⠀⠀⠀⠀⠀⢀⢀⡈⠙⢁⠀⠰⢧⠄⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠄⠀⢀⡀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣀⠈⠑⠉⢀⡜
⠀⡀⠀⠀⠀⠐⠝⠈⠀⠘⢁⠘⣀⢀⢎⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢁⠀⠁⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠠⠋⢀⣀⠄
⠀⠀⠠⠀⠀⢁⣠⢀⠑⢦⠊⠀⠟⠣⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠈⠠⡀⠂⠀⠐⠀⠀⠀⠀⡀⠀⠈⠉⠑⡐⡠⢀⡐⠉
⠀⠀⠀⠈⠀⠢⡅⠁⠀⢸⠀⠀⢰⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡄⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠂⣠⠀⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⢧⠂⠀⠀⡐⠂⠁⠀⠣⠐⠙⠀⠀⠀⠀⠀⢜⠀⠀⠀⠀⠐⢺⣤⡄⠀⠀⡄⠀⠀⢙⢁⠖⢢⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠘⣷⡤⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣄⣡⠤⠀⠀⠀⡀⠚⠀⢠⡆⣈⠀⠛⠋⠠⠈⠀⠀⠠⠁⡄⠀⠒⠛⠢⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠤⣤⢡⠤⠖⠀⠀⠀⠀⠀⠀⢣⡄⢠⢁⠀⡴⠐⣆⠀⣰⠤⠘⠀⢀⠀⠀⠀⠀⠀⠀⣄⠱⠵⠄⠀⠀⠀⠀⠂
⠀⠀⠀⢸⠇⠀⠀⠀⠀⢢⡭⠥⠀⠀⠀⠀⡀⡀⠁⠀⠠⡤⠀⠀⠀⠠⠶⠦⠄⣁⢤⡴⠃⠐⠃⠌⡏⠀⠁⠀⡀⠀⠀⠀⠀⢀⡄⠈⠳⠒⠂
⠀⠀⠀⣸⠂⠀⠀⠠⣄⠻⢚⣁⣀⠀⠀⠈⠉⠉⠃⠀⠁⠀⡀⠀⠀⠀⠀⠀⢵⠆⢚⠀⠰⢶⡈⢤⣤⣅⡀⠁⠂⠀⠀⠀⢀⠈⠀⠁⠠⠐⠐
⠀⠀⠀⡿⠀⠀⠠⣀⠈⠛⣉⣅⡤⠂⠐⠀⠀⠀⠀⠀⢰⠐⠁⠀⠀⠀⠀⠂⠀⡘⠙⠛⣤⠌⠤⠊⡇⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⢆⡰⠿⠓⠤
⠀⠀⢰⠃⠀⠀⣀⡘⠷⠿⠗⣀⣀⣀⡀⠈⠙⡢⡠⡀⠂⠒⠀⠰⠀⠀⠀⠀⠀⠁⠀⠀⡉⢂⠀⠃⠀⠷⡈⡈⠁⠈⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠙⢲⣷⡾⢙⠛⠁⠀⠀⡀⠀⠀⠀⠀⠀⢢⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁
⠀⠀⠀⠠⠄⠀⠠⠀⠀⠀⠑⠀⠀⠠⠈⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡟⡩⡿⠟⠿⠿⠿⠿⠟⠿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢲⣶
⠀⠀⠀⢠⣴⣿⣿⠋⣀⣀⣾⠛⣷⠂⠀⠀⠀⠀⢰⣶⣶⣿⣷⠀⠀⠀⠀⠀⠀⠀⢠⡼⢼⣿⣤
⠀⠀⠀⠀⠀⠀⢠⣦⣤⣤⣤⠘⠏⠀⠀⠀⠀⠀⠘⡇⠉⢉⡇⠀⠀⠀⠀⠀⠀⠀⣿⠶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⣿⡆⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠹⡳⡄⠀⡜⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣯⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢽⠟⠀⠀⠀⠀⢠⠀⠀⢀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⡶⣶⣶⣶⠛⠁⠀⠀⠀⠀⢰⣶⣲⣶⣾⣿⣆
⠀⠀⠀⠀⠀⡠⠤⠤⠄⠴⠠⠄⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠉⠀⠀⠀⠀⠀⠀⢿⣍⠁⠉⠉⠉⡹
⠀⠀⠀⠀⠀⢸⣧⠴⢿⣦⢴⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠶⠒⠉
⠀⠀⠀⠀⠀⢈⣧⣤⣼⣄⣨⣏⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠠⠶⠿⠷⠦⠀⠀⠀⠀⠀⠀⠀⠰⠶⠿⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣤⣤⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣠⡾⠿⠻⠛⠻⠉⠉⠉⠁⠁⠉⠙⠋⠋⢻⡿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠐⠛⣻⣠⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⡴⠁⡿⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⢀⣀⢀⣀⣒⣶⣶⡦⢀⢀⠀⠀⠀⠀⠘⠛⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⣠⣴⣦⣦⣿⣿⣿⣿⣿⣿⣶⣾⣿⣷⡦⠀⠀⠀⠘⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⢈⣿⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣄⣀⠀⠐⠟⠻⠀⠀⠀⣼⣶⣦⡄⠀⠀⠀⠀⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⢿⢿⣿⣷⣾⢿⣿⣻⡻⠈⠁⠘⠛⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣟⢿⣽⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⣧⣸⠁⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣳⣿⡡⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠏⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣿⢤⡄⠀⠀⠀⠀⢀⣼⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡘⠁⠀⢀⣠⣴⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢿⠷⠶⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⢿⣿⡿⠛⣿⡟⠛⢿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⢠⣿⠏⠀⠀⣿⡇⠀⠀⠹⣿⡌⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡁⠀⢠⣿⡏⠀⠀⠀⣿⡇⠀⠀⠀⠹⣿⡄⢀⣽⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠛⠿⣷⣾⣿⣤⣤⣀⣀⣿⣧⣀⣤⣤⣴⣿⣿⠿⠛⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⢀⣿⡏⠉⠛⠛⠛⣿⡟⠛⠛⠉⠉⢹⣿⡀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣤⣤⣤⣼⣿⣥⣤⣤⣤⣤⣿⣧⣤⣤⣤⣤⣼⣿⣧⣤⣤⣬⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⠉⠉⢹⣿⠋⠉⠉⠉⠉⣿⡟⠉⠉⠉⠉⠉⠛⠋⠉⠉⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠘⣿⣇⣀⣤⣤⣤⡟⠁⣠⣾⣿⢿⣿⣶⡄⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣠⣶⣾⣿⣿⠛⠛⠋⠉⠀⢸⣿⠏⠀⠀⠈⢻⣿⡄⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠉⠀⠸⣿⡄⠀⠀⠀⠀⠸⣿⣆⠀⠀⢀⣼⣿⠁⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⢻⣷⡄⠀⠀⣦⡀⠙⠿⣿⣿⣿⣿⣇⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣽⣿⣦⡀⣿⡇⢀⣤⣄⣀⡀⠙⣿⣧⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠿⠛⠋⠀⠀⢼⣽⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣂⣤⡀⣤⣠⢄⡤⢰⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠈⠉⠉⠀⠙⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣤⡀⠀⡀
⠀⠀⠀⠀⠀⡀⡴⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⢹⠤⠃
⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠦
⠀⠀⠀⠀⠀⠿⠀⠀⢰⠂⠀⠀⠀⠀⠀⢴⣆⣀⣀⡀⠛⠂⢄⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣀⣤⠆
⠀⠀⠀⠀⠀⠀⡀⣀⡜⠀⠀⠀⠀⠀⠀⣀⠟⠋⠈⠛⣦⡀⠘⠇⢸⠙⠁⠀⠀⠀⠀⠀⠈⠉⢁⡄
⠀⠀⠀⠀⠀⠀⠻⡏⠀⠀⠀⠀⠀⠀⠈⠟⠃⢀⡴⠆⠈⠃⠀⠰⣿⠀⠀⠀⠀⠀⠀⠈⠉⠋⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠞⠃⢉⣄⣀⣀⣤⣤⣄⠙⠁⠀⠀⠀⠀⠀⠀⣄⠒⡟
⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠛⠉⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠑⣴⠦⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠋⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠙⠳⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡋⣤⢤⣤⣠⠖⠚⠛⠛⠓⢦⡀⠀⠀⠀⣀⣤⡶⠛⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣿⣥⣬⣛⣃⣛⠲⡄⠀⠀⠀⠘⢧⠀⠀⠐⢿⣤⣤⣤⡀⠀⠈⠙⢶⣄
⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠉⠉⢹⡇⠉⡇⠀⠀⣀⡾⠀⠀⣀⣀⣀⣠⡼⠃⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢧⣀⡾⠁⠀⠀⢺⡏⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠈⠀⠀⠀⣀⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⢀⡟⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⢸⡷⣄⠀⢠⡞⠋⠉⠛⠓⠶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠛⠳⢦⣿⠷⢦⡄⠀⠀⠀⣠⣀⠀⠀⠀⠸⡇
⠀⣾⠀⢸⡇⢸⡇⠸⠿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠛⠀⠀⠀⢀⣼⣇⣈⢻⣆⠀⢸⡇⠘⠳⣄⠀⠀⣿
⠀⣿⠀⠀⣻⠞⠒⠒⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠳⣿⣿⣿⣥⠀⠐⣇⢠⡴⡿
⠀⣿⠀⢸⡏⠀⠀⠀⠀⠉⠙⢷⡀⠀⠀⠀⠀⠀⠈⠿⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠙⠋⢀⡇
⠀⢹⣄⡿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⣀⣀⣸⠇
⠀⠘⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢸⡏⠀⠀⣠⡞⠉⠁⣽
⠀⠀⠸⣆⠙⠳⢦⣀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠘⣷⣶⡄⢿⠀⠀⣼⠏
⠀⠀⠀⠹⣆⠀⠀⠙⢷⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣰⣿⣿⠀⣿⣄⣼⠏
⠀⠀⠀⠀⠘⢧⡀⠀⢸⡇⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⢀⡾⠁⠿⠃⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢦⣈⠳⣄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡴⠟⠁⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣬⣳⣟⣲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⢀⠄⠀⢀⡀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢻⣷⢸⣿⡇⠀⠀⠁⠀⠀⠈⠀⠀⠀⠁⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⡀⠘⣿⣾⣿⡇⠀⠀⠠⠀⠈⠀⡀⠁⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡶⣿⣿⣿⡧⠴⡶⠶⠶⠶⠚⠋⠀⠀⣀⢀⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠊⠙⠛⠓⠀⠀⠀⠀⢀⣀⠀⠀⡆⣿⣾⣠⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢀⠀⣠⡀⢁⠀⠰⢎⠁⣦⢸⣿⣿⣿⣿⣥⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡶⣤⣤⡀⠇⠃⠐⠀⠁⠀⠀⠜⠀⠸⡇⢿⣿⣿⡿⣿⣿⣖⠋
⠀⠀⠀⠀⠀⠀⣶⣶⣿⣿⣏⢻⣿⣇⠹⣿⣿⠄⠀⠀⠐⠀⠐⠀⠀⠄⢠⡇⠈⠇⢃⣨⣿⣿⠅
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣿⣿⣿⣿⣿⣿⣧⣀⠂⠀⣄⠀⠀⠀⢀⣠⠏⠐⡄⢈⣴⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠈⠻⠿⢛⣻⣿⣿⣟⣿⣻⣤⡆⠀⢈⣭⠿⣿⣿⣯⣄⠰⠀⢈⣽⢿⣯⣷⣿⣿⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠟⢋⣿⣿⢻⣿⣷⡇⣀⠋⠀⠐⠀⠈⠛⣿⡆⠀⠸⢿⣟⡿⢿⣯⣥⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⣿⣿⢏⣼⣿⢩⣏⠉⠢⡀⠀⡳⠀⠂⢸⡇⢀⣿⠾⣿⡗⢶⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⢿⣿⣿⣿⣿⡗⠤⢬⢅⣤⣽⡤⠤⠖⢻⠿⠷⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠴⠰⠄⠠⠦⠐⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡆⠀⠀⠀⠀⠀⣀⠀⢠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣰⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢠⣿⠃⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣷⣤⡐⠀⠀⠀⠀⠀⠀⠀⡿⠑⠃⠽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠀⣾⣿⣿⡿⡿⢿⣿⣿⣿⣿⣻⣿⣦⡴⠀⠀⠀⠀⣼⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣼⣿⣿⣿⣿⢟⣣⣟⡶⠿⠛⠻⣏⣿⣿⡧⠀⠀⠀⠀⠸⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣾⣿⣾⣿⠋⡉⢹⠟⣶⠤⠐⢿⠿⠛⠁⠀⠀⠀⠀⠀⢾⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⣯⣿⡾⢆⡋⠛⠁⢀⣴⡋⠝⢳⡀⠀⠀⠀⠀⠠⣾⣸⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⡄⠈⠩⢻⠙⠯⣙⣉⠛⣿⠛⠋⠀⠀⠀⣿⡆⠀⠀⠀⠀⢿⡟⠘⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⣿⠀⠲⣼⡟⠳⠶⡎⠀⣼⣀⣠⠞⠙⠋⣿⡄⠀⠀⠀⢻⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢈⡏⠀⠀⣄⣿⡴⠾⠀⠀⡾⠀⢱⢶⣤⣶⡿⠋⠀⠀⠰⣟⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⠀⠀⠈⣠⢳⡇⠀⠀⡿⠿⣿⠀⠀⢸⣇⠀⠀⠀⠀⢻⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠀⠀⣴⡟⢉⠹⠆⠈⠈⠁⠀⢀⣀⣞⣸⣆⠀⠀⠀⢸⣯⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠈⣷⣴⡛⣰⡁⢠⡤⠶⣶⡶⠟⠋⠉⠉⠻⠿⡆⠀⠀⡎⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠁⣷⠟⠿⠛⠛⠁⢠⡽⠃⠀⠀⠀⠀⠀⠀⠹⣦⡈⠳⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⢓⠖⣶⢃⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⡻⡦⠼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡽⣿⣶⢳⣄⣀⠀⢠⠁⠀⠀⠀⠀⢳⠀⠀⢠⡴⣆⠧⣻⠗⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡇⠈⠸⠿⣧⣀⡀⢠⣤⣄⠀⠈⡇⢠⡜⠇⠋⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠰⠻⠋⠙⠉⠈⠉⠓⠦⠰⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣤⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣷⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁



⠀⠀⢶⢶⠄

⠀⠀⠀⠀⠀⠈⣒⢳⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠘⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠉⠓⡒⣯⣽⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠋⠋⠀⠈⠉⠀⢀⣀⡉⠁
⠄⠠⢄⣠⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣆⠀⠀⡀⠀⠀⠀⢀⣴⣶⣦⡀
⠀⢠⣀⣄⣀⠀⢈⡉⠉⠑⠚⠒⠒⠤⠀⠀⣼⣿⣿⣿⣿⣿⡄⢀⣀⡉⠀⣲⣿⣿⣿⣿⣿⢠⣶⣶⣶⣀⡀
⠄⠆⠈⠈⠋⠑⠀⠀⠀⢀⣀⠀⠤⠀⠒⠈⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⠃⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠐
⠀⠠⠄⢀⠄⡀⠀⠄⠀⢀⡄⣀⢀⣂⣀⣀⣟⣛⣿⣿⢙⠉⣛⠈⠃⠀⣠⣴⣾⣿⣿⣿⣿⡟⣿⣿⣿⣷⣦
⣿⣿⣿⣿⣾⣿⣶⣶⣦⣶⣦⣤⣬⣬⣤⣄⣆⣈⣩⣣⣚⣁⣝⢋⢁⠩⢙⡉⡛⣛⢛⢟⡿⠿⠿⠿⠿⠿⢿
⠈⠘⠙⠛⠟⠛⠿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣷⣶⣾⣿⣴⣤⣼⣴⣵⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠉⠉⠋⠉⠉⠉⠛⠛⠛⠟⠻⠿⠻⢿⠿⠿⠿⠿


⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⢀⠀⠀⡀⠀⡀⠀⠀⠀⠄⡠⠄⠀⠄⠀⡠⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠄
⠀⠀⠀⠠⣤⠒⣼⣷⢿⣾⣷⣷⣦⣴⣶⣿⢻⣿⣰⡾⣷⣏⣷⣰⣿⣧⣶⣷⣶⣾⣷⠇⣷⣦⣶⣴⣤⣦⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡆⠀⠀⢀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠁⢀⣴⣿⣿⡛⠁⠀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢐⡄⠀⠀⢀
⠀⠀⠀⠀⠀⣶⡄⠀⠀⢸⣧⠴⠚⠉⢿⣿⣦⣍⠉⠛⠿⣿⡿⠿⠛⠉⠀⢀⠀⠀⠀⠀⢀⠀⠠⠍⣿⠀⠀⠀⠠
⠀⠀⠀⠀⠀⣿⡇⠀⢠⠎⠻⣿⣷⣦⠀⠙⠻⠿⢷⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠚⢆⠀⠀⠀⠁⢀
⠀⠀⠀⣀⠀⠈⠻⢦⠇⠀⠀⠈⠛⠻⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠹⠂⠀⠀⠀⠀⠄
⠀⠀⠀⠹⣿⣦⠀⢸⠀⣠⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠥
⠀⠀⠀⠀⠙⠛⠷⠾⣾⠉⠉⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⢒⡂⠀⠀⠀⠔⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⣼⡿⠏⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣮⣷⣾⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠐⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠓⠂⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠠⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠐⣀⢂⢻⡀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠖⠉⢈⡇⠀⠀⠀⠀⠀⠐⠀⢍⢻⡋⠕⠩⠀⠀⠀⠀⠀⠘⣷⣄⢸⡀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢳⡀⠀⡟⠀⣠⠞⠀⠀⠀⠀⠀⠀⡀⠈⡀⢻⠃⠁⠒⠀⠀⠀⠀⠀⠀⠈⢻⡜⡇⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⡇⢠⣥⠞⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⡈⢺⠓⡢⠑⡆⠀⠀⠀⠀⢠⣄⠀⠙⣷⠎
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢧⢠⣇⠞⣡⠤⠤⠶⣶⠀⠀⠀⠀⠠⠒⠂⠈⢺⠃⢀⠨⠠⠀⠀⠀⠀⠀⠻⣇⠀⢹⠀⢠⣾⠃
⠀⠀⠀⠀⠀⢸⠙⢦⡀⠈⣿⣷⣎⣁⣀⡤⠞⠁⠀⠀⠀⠀⠠⠠⠀⠘⢺⠓⠂⠀⠤⡄⠀⠀⠀⠀⠀⠈⢦⣸⠀⡾⠃
⠀⠀⠀⠀⠀⢸⡀⠀⢳⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠦⠀⠂⢻⠋⠐⠀⠀⡁⠀⠀⠀⠀⣦⡀⠀⢻⡞⠁
⠀⠀⠀⠀⠀⠀⠳⣄⣾⡥⠖⠉⠉⢉⡿⠂⠀⠀⠀⠀⠀⠀⠅⠦⠀⠈⢻⠃⠁⠀⠀⠁⠀⠀⠀⠀⠙⢷⡀⢸⠀⠀⣴⡶
⠀⠀⠀⠀⠀⠀⠀⢸⠟⠛⠓⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠁⢾⠦⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢳⣜⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⢺⠖⠀⠀⠒⡄⠀⠀⠀⠀⠀⠀⢀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣂⣠⣄⠀⠀⠀⠀⠄⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⢀⣴⣀⡔⣄⣄⠀⠀⠀⠀⠀⠇⠀⠤⠄⢿⠀⠀⠀⠀⠀⠀⠉⠀⠻⠀⠀⠀⠀⢀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠘⠯⠝⠉⢹⣿⠀⠀⠀⠀⠀⡄⠐⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⠻⡇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡟⠀⠀⠀⠀⠀⡄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⡆⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⡁⢠⣀⣸⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣁⠀⠄⣸⠃⠀⠀⠀⠀⠀⣄⠀⠀⢀⣸⠀⠀⠀⠀⣀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⡀⠀⣠⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠀⠀⠀⠀⠀⠈⠀⠐⠋⠁⠀⠀⠀⠀⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠐⠛⠋⠀⠀⠀⠀⠀⠀⠉⠀⠒⠒⠂⠉⠁

⠀⠀⠀⠀⢠⡄⠀⠀⣲⡕⣠⠀⠀⠀⠀⠔⠂⠀⠐⠂⠐⡆⠀⠀⢎⠁⠈⠉⠁⢹⡖⠀⠀⢰⠖⠂⣀⢐⡶⠶⡦
⠀⠀⠀⠀⠈⠳⠀⠀⠀⡿⠃⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠃⠀⠀⠀⢣⠀⠀⢠⠋⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢣⡀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠰⠆⠀⠀⠀⠀⠈⠀⠀⠀⠈⠙⣷⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣋⣀⠃
⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠟⠁
⠀⠀⠀⠀⠀⠢⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠌⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠘⠀⠀⠀⠀⠼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⢸⠚⠀⠀⠤⠤⢽⡆⠀⠀⠀⠠⠧⠴⠿⠷⣶⡄⠀⠀⠀⠄⠀⠂⠀⣤⠀⠀⠀⠀⢸⠋⠀⠀⢾⡇⠀⠀⠀⠀⣖⠀⣀⠨⡲⢶⡄
⠀⠀⠀⠀⠈⡄⠘⠀⠚⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⠀⠀⢸⡇⠀⠀⠀⠀⡍⠴⢻⡖⣵⠸⡇
⠀⠀⠀⠀⠀⢱⠄⠐⠂⠁⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡟⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠁⠀⠀⠀⠀⠯⠂⣑⣁⠞⠐⠇
⠀⠀⠀⠀⠀⠈⣄⠀⠀⣠⠁⠀⠀⠀⠀⠀⢦⠀⢀⣠⡇⠀⠀⠀⠀⠦⠠⠀⣠⡆⠀⠀⠀⠀⠀⠃⠀⠀⣸⠀⠀⠀⠀⠀⣤⠀⠀⠀⣠⣬⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣠⡾⢛⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⠋⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⢲⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⠀⢹⣾⣿⣿⣷⡄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⡾⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⣼⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⢰⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣽⣿⣿⡀
⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⡀
⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⠿⢿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣿⣿⣧⣀⠉⣨⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣦⣀⣠⡄⠀⠀⡀⢀⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢹⣇⣠⡄⠊⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣎⣉⣉⢛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠟⠿⣿⡿⠿⠿⠟⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢻⡿⠛⠋⢛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠄⠀⠀⠀⠐⢻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡻⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢻⡇⠀⠀⠀⡀⣀⣤⣴⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣥⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠘⡇⣸⣤⣤⣶⣾⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣤⣶⣤⣴⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣯⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠛⠋⠉⠉⠉⠉⠉⠛⠿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣶⣤⣀⣀⠀⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠈⠻⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠟





⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⣀⣀⣄⡀⠀⠈⢤⣠⣌⠹⢷⠖⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠁⠉⠉⠛⣉⣱⡦⢀⡀⠉⠉⠛⣶⣶⣤⣦⡈⢯⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠶⣤⢧⠀⠀⠀⠹⣿⣷⣤⠀⠀⠀⠐⣶⢿⣟⣉⣷⣿⣿⣟⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⡠⠀⠀⢂⣴⣹⣆⠀⢀⣴⣿⣿⣿⣇⠀⠉⣹⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⢀⣀⠐⠁⠀⢀⣀⣥⣾⣿⣿⣶⣶⣿⣿⣿⣽⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠉⠉⣿⣷⡄
⠀⠀⠀⣾⣿⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⠏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⡀
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠟⠿⠛⣟⣡⣼⣿⠁⢠⣴⣿⣿⣿⣿⣿⣿⡟⣛⣫⡿⠛⠟⠛⣋⠛⢿⣿⣿⣷
⠀⢠⣿⣿⣿⣿⣿⣼⣿⣿⣀⠀⠞⠉⠛⢻⣿⠀⢩⣱⢿⠟⢛⡛⢻⣿⣀⣳⣖⣩⣄⡈⢁⣿⣷⠀⠸⣿⣿⡆
⠀⠈⣾⣽⣾⢿⢿⣬⣿⡫⠒⠠⠮⢂⣅⣺⡟⢡⣤⣏⣹⣿⣿⣿⢷⡄⣬⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡇
⠀⢀⣯⡿⣿⣷⣿⣿⣿⣿⣶⠠⣤⣾⡿⡟⢃⣹⣿⣿⣿⣿⣻⣿⣾⣿⣿⣿⣿⣾⣿⡭⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⡿⡿⣩⣏⢸⣿⣷⣾⠛⠋⠀⠩⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣟⣿⣜⢋⣿⣿⣿⣿⣿⡇
⠀⠀⢹⣿⣿⣿⣿⣋⣻⣿⢉⡙⢣⢸⡯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣻⣿⠻⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠘⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣝⠉⠳⠴⠚⣿⠯⠟⠾⠿⠟⣿⣿⣿⣿⣿⣿⣙⣿⠇⠹⢿⣿⣿⣿⡏
⠀⠀⠀⠈⣏⠦⠁⠸⣿⣿⣋⠀⠉⠈⠡⢿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡯⢴⣿⣿⣧⠀⠸⢹⣿⡟
⠀⠀⠀⠀⠈⠁⢤⠀⠸⣿⣿⣿⣭⣬⡓⢿⡿⣿⠟⣿⣶⡆⠀⠀⠀⠀⠀⠈⣰⡿⣃⠋⠋⠀⠀⣾⠟
⠀⠀⠀⠀⠀⠀⠛⠷⠀⠙⢯⡝⡿⣿⣷⠄⠁⣿⡐⠛⠛⠁⣤⠀⠀⣄⠀⠀⠸⠟⠉⠀⠤⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢵⣵⠶⠌⠓⠂⠘⠁⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠛
⠀⠀⠀⠀⠀⠀⢀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠴⣦⢴⡴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠪⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠂⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠋⠄⡿⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀
⠀⠀⠀⠂⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢄⡀⠀⠀⠀⠀⠀⠂⠀⠰⡀⠀⠀⠀⠀⢀⡠⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠈⠐⠄⡀⠀⡜⠀⠀⠀⢣⠀⢀⡠⠒⠁⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⣈⠵⠖⠒⠒⠒⠠⠦⣁⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣆⡴⢊⡤⢲⠋⠁⠀⠀⠀⠀⠀⠙⢦⣰⣁⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⢠⠎⡰⠃⠀⡆⠀⢸⣉⠆⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⡌⢰⠁⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠛⡇⢸⠀⠀⠀⠀⠀⠉⠒⠢⢄⡀⠀⠀⠀⠀⢸⠛⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠀⠀⢱⡈⢆⠀⠀⠀⠀⡴⠒⢄⠀⢱⠀⠀⠀⠀⡎⠀⠀⠈⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠥⠤⠤⠤⠤⠤⠤⡷⣌⠢⣄⠀⠀⠳⠤⠊⠀⡸⠀⠀⣠⢮⠤⠤⠤⠤⠤⠤⠬⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠈⠓⠤⣉⡒⠒⠒⠒⢉⣀⠤⠚⠁⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⣠⠔⠈⠅⠈⠉⠉⠡⠁⠢⢄⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡠⠔⠊⠀⠀⠀⠘⡀⠀⢀⠇⠀⠀⠀⠉⠢⢄⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢡⠀⡌⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒



⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡀⠀⠀⠀⠀⢀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⢿⣆⠀⢸⣆⠘⠋⣀⠀⠀⠀⣷⡆⢀⠀⢠⣄⠀⠰⠀⠀⠀⢀⠘⠇⠀⠀⠀⢠⣄⠁⣀⠀⠀⠠⣤⣁⣼⠂⠀⠀⢀⠛⢀⠂
⠀⠀⠈⠻⠆⠈⡛⠀⠾⠋⠀⠀⠀⠈⠁⡿⠁⠀⠁⠲⠀⠀⠀⢀⡘⢧⡶⠂⠀⠀⠀⣉⠾⠋⠀⠀⠀⣬⡙⠁⣀⠀⠀⠈⣈⠺⠈
⠀⠀⢶⣦⡀⣰⠆⢀⣴⡶⠀⠀⠀⠀⠃⠀⠀⠀⠰⡀⢀⡀⠀⠈⣿⣆⣴⡇⠀⠀⠈⢻⣧⡶⠀⠀⠀⠉⠛⣿⡟⠀⠀⠀⠉⠲
⠀⠀⠀⠉⠀⡿⠀⠈⠁⠀⠀⠀⠀⠶⠶⠂⠀⠀⠀⡹⠈⠁⠀⠀⠈⠉⠋⠀⠀⠀⠸⣇⣡⡄⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⢈⡙⡔⠗
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠈⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⢀⡀⠀⠀⠉⠀⠀⠀⠰⣶⡄⢠⡄⠀⠰⢾⣶⢄⢿⠄⠀⠀⠸⣶⣾⣯⡔⠀⠀⢀⡀⠀⡄⣧⡀⠀⢰⣿⠀⠀⡀⠀⠀⢠⡄⠀⠀⡀
⠀⠀⢈⡄⠈⠀⠃⣀⠀⠀⢠⣌⠃⢸⠇⢀⡀⠀⠺⠃⠙⢇⣴⡆⠀⠀⠁⠀⠀⠀⠰⣌⢰⣧⡿⠆⠀⢠⣜⣡⠀⠀⠙⠳⠀⠘⠃⠀⢚⡁
⠀⠀⢀⠀⡄⠀⠀⣨⠀⢰⣬⡛⠀⡆⢰⣿⠃⠀⠀⠀⠶⢌⣾⠀⠀⠀⣠⠀⠀⠀⢳⣶⡧⠍⠀⠀⠀⢦⡟⢋⠀⠀⢠⣤⣦⣶⣤⠛⠉⠁
⠀⠀⠉⠀⠀⠐⠀⠀⠀⠘⠿⣷⠀⠄⠈⣁⣀⠀⠀⠀⠀⠀⠁⠀⠐⢶⣿⣶⠆⠀⢸⠃⠀⠀⠀⠀⠠⣤⣷⣿⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠐⠛⠛⠛⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣶⣆⠀⣀
⠀⠀⠀⠀⠀⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⣠⠞⢀⠀⠀⠀⠄⢁⠀⠀⠀⠀⠀⠀⣬⣿⣼⡿⣃⡀
⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⣴⣦⣸⣋⣀⠀⠀⠀⢢⠀⢸⠃⣠⠏⠀⠀⠀⠀⠉⢣⡶⠀⠀⠀⠀⠙⣿⣭⣾⡿⢃
⠀⠀⢶⣦⣤⡈⢻⢠⣶⣿⠇⠀⠀⢀⣬⡉⡟⣻⡟⠀⠀⠀⢸⡆⡴⠞⢃⡄⠀⠀⠀⠠⠀⣄⣇⠀⠀⠀⠀⠀⢉⣭⣭⣶⡿⠃
⠀⠀⠀⠙⠻⠿⠾⠟⠛⣁⡄⠀⠀⢈⣛⢻⡟⠻⠛⠀⠀⢸⡄⠙⠂⣠⡞⠀⠀⠀⠀⠀⠀⠀⢹⣠⠄⠀⠀⠀⠈⣭⣍⣧⣶⡷
⠀⠀⠻⣿⣶⣄⢀⣴⣿⡿⠁⠀⠀⠻⠿⢿⠿⢿⡿⠀⠐⣄⢿⣄⠋⠁⣴⠃⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣹⣛⣋⣡⣤⡀
⠀⠀⣀⣈⣙⠛⢻⠉⣁⣠⣤⡄⠀⣾⣷⣾⣥⣶⣶⡄⠀⠸⣆⠈⡤⠞⣡⠎⠀⠀⠀⠀⠀⠉⢹⠀⠀⠀⠀⠀⠀⣙⣛⣛⣩⣭⡀
⠀⠀⠙⠿⠿⠷⠾⠾⠿⠟⠋⠀⠀⠉⠉⠁⡏⠙⠋⠀⠀⠀⠈⠓⣀⡴⠋⠀⠀⠀⠀⠀⠀⣀⠸⠛⠀⠀⠀⠀⠈⣛⡛⣟⣻⣭
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⠀⠀⠀⠀⠀⠈⠻⠟⡟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⡆⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⢠⡀⠀⡀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⢀⠀⣇⣿⣰⠏⠀⠀⠀⠀⠀⡀⠀⠠⢎⢞⡇⠀⠀⠀⠀⢸⢀⠞⠉⠀⠀⠀⠈⣿⣷⡀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⡠
⠀⠀⠀⠈⣧⢿⣿⣫⡶⠟⠁⠀⠀⠀⢝⢯⣢⡿⠚⣤⡄⠀⢸⠃⢠⢃⡰⠇⠀⠀⠀⢠⡙⢿⢃⣴⡗⠀⠀⢀⡀⠀⣀⠀⠀⢀
⠀⠀⢀⢳⣼⣿⣯⣴⣾⣛⡁⠀⠀⣤⡤⢅⣸⡫⠍⣊⠀⠀⠘⢄⡞⠉⢀⡀⠀⠀⠀⠸⣿⣦⠿⠟⣁⠀⠀⠀⠁⠂⣬⡄⠀⠉
⠀⠀⣈⢷⣝⢿⣛⣿⣯⠵⠂⠀⢀⣈⡊⠽⡿⢒⠭⠋⠀⣀⠀⢨⠃⡠⠊⠁⠀⠀⠸⣦⣌⢹⣶⣿⠏⠀⠀⠐⠆⠀⣈⡀⠀⠞
⠀⠀⣌⡳⣝⠢⠥⢶⣞⣛⡁⠀⠙⣕⠮⢵⡖⡭⢽⡿⠀⠙⣄⢸⠉⠀⠀⠀⠀⠀⢠⣙⠻⢟⣩⣶⡏⠀⠀⢠⡀⠀⠙⠃⢀⡄
⠀⠐⢮⣝⡾⣝⣛⣻⣭⡥⠄⠀⢶⢖⡋⢭⡭⢛⣃⣤⠠⣦⠀⢹⣀⡼⠋⠀⠀⠀⢈⠿⣷⡞⣛⣭⣴⠀⠀⠀⠀⠀⠲⠗⠁
⠀⠀⠀⠉⠛⠺⠵⠶⠟⠛⠁⠀⠈⠑⠎⠹⠋⠍⠚⠁⠀⠈⠢⣼⠁⠀⠀⠀⠀⠀⠘⢿⣶⣼⠛⠛⠁⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠣⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣧
⠀⠀⠀⣠⣾⣿⠟⠛⠛⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠛⠛⢿⣿⣿⡆
⠀⠀⣰⣿⣿⣿⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣏⠀⠀⠀⢸⣿⣿⡇
⠀⢠⣿⣿⣿⣿⣷⣤⣴⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣶⣤⣴⣿⣿⡿⠁
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢀⣠⣤⡀
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡇⠀⢾⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡇⠀⠀⣾⣿⣿⣿
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡇⠀⠈⠛⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡇⠀⠀⠈⠻⠟⠋
⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⣠⣿⣿⡟⠉⠀⠉⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⠉⠈⠙⢿⣿⣿⣿⡄
⠀⠀⣸⣿⣿⣿⣇⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣇⠀⠀⠀⣼⣿⣿⣿⠃
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡿⠃
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠈⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⣀⣀⣀
⠀⠀⠹⣿⣿⣿⣿⣿⡏⠀⠀⢰⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡏⠀⠀⠀⢰⣿⣿⣿⡇
⠀⠀⠀⠘⢿⣿⣿⣿⣧⠀⠀⠀⠛⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧⠀⠀⠀⠈⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠉⠛⠿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣄⡀

⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠉⠒⡄⠀⠀⠀⠀⠀⣴⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣬⣅
⠀⠀⠀⣴⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣠⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢻⠿⠿⠿⠇
⠀⠀⠀⢿⣿⣿⣿⣥⣤⣤⣤⣤⣠⣄⣄⠀⠀⠀⠀⠀⣿⣾⣷⣿⣿⣿⡏⠛⠷⠀⠀⠀⠀⠀⢩⢿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠉⠉⠻⢿⣿⣿⣿⣿⡿⠿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣍⣉⣭⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣿⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠂⠤
⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣶⣶⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⠤⠤⠤⠤⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠉⠟⠁⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣛⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠿⠿⠶⠆⠀⠀⠀⠀⠀⠀⠀⠰⠶⠿⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠿⠷⠦
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿