Racing Emojis & Symbols

⠀⠀⠀⠀⠀⢰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠀⢄
⠀⠀⠀⠀⠀⠀⣃⢣⠠⣀⠄⠠⡀⠀⡠⢔⣲⣦⣤⣭⣭⣭⢭⣴⣷⣌⠀⠀⠀⠨⠃⠰⠀⢄⣀
⠀⠀⠀⠀⠀⠘⠹⢸⢰⠋⢀⡐⡡⣫⣾⣿⣿⣿⣿⣿⣿⣯⣞⣾⣿⣿⣷⣄⣀⡀⠈⠀⠀⠀⠜⠁⢢
⠀⠀⠀⠀⠀⢠⠀⠸⣶⠐⠃⣔⠱⠷⠿⠻⣟⣛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⢖⡒⠒⠚⠲⠤⣄⣀
⠀⠀⠀⠀⠀⣺⣆⠀⢧⠳⡎⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠭⣐⠀⠠⢀⠀⠀⠀⠀⠀⠠⠄⠉⠒⢄⠠⣀⠀⠉⠙⢦⣄
⠀⠀⠀⠀⠀⣸⡌⠙⠆⠀⠙⣦⢄⣀⠀⠀⣠⡄⠀⠀⠀⠀⠀⣀⣙⣦⡀⠊⠐⡄⠀⠀⢀⣀⡘⣄⢀⣱⣌⣳⢰⢻⣿⢽⠆
⠀⠀⠀⠀⠨⡀⠘⢦⡀⠀⠀⢸⣆⡀⠉⢺⣿⣄⠢⡀⠀⠀⢠⡟⣽⣯⠁⠀⣨⠀⠙⣻⣭⣭⣥⣽⣤⣴⢾⡧⠸⠼⢶⡟
⠀⠀⠀⠀⢀⡩⢄⢸⣯⠲⣽⣾⠷⣿⡦⢸⣿⣿⣷⡐⣆⣀⠈⠷⠷⠶⠂⢠⠎⠀⢰⣿⣿⣿⣿⣿⣿⡏⣸⣷⢣⡿⠛⠃
⠀⠀⠀⠀⠁⢠⠚⢸⣸⡀⡟⠿⠸⣿⣿⡦⡿⢁⣼⣻⣆⠠⡙⠀⣒⣢⠜⢉⠈⢠⡘⠟⠛⠿⣻⠟⣋⣼⣯⣿⣯⣀
⠀⠀⠀⠀⡄⠨⠄⠀⡜⣷⣷⣌⣤⠘⠓⢲⣿⡿⢿⣿⡿⢷⣬⣽⣽⠶⠶⠶⠿⠤⢭⢭⣭⢠⣴⣶⣿⣿⣿⡇⢹⣿⣿⡄
⠀⠀⠀⠀⠁⠑⠀⠄⢳⣿⣿⠈⢪⣢⣹⣿⠋⠞⠿⢿⡇⠈⠿⠛⣿⣤⢻⣿⣿⣿⣋⠐⡏⠣⠧⠿⡟⣹⣿⡇⡘⠀⢉⡇
⠀⠀⠀⠀⠀⢸⠀⡄⠘⣿⣿⣠⣾⣽⣿⡏⠀⢀⣇⣼⠇⠀⠀⠀⣯⣶⣤⣭⣤⠖⠀⠐⠈⢲⣾⣿⣶⡿⠋⣠⣣⣤⡾
⠀⠀⠀⠀⠀⠈⠂⠓⠤⠻⠿⠿⠛⠛⢻⣷⣾⣷⡿⠋⢠⢢⣠⡞⠉⠉⠉⠉⠁⠀⠀⠀⠀⠘⠿⣿⣿⣶⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣤⣶⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡿⠦⣀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⣯⣅⠀⣾⣴⡄⢀⡀⠀⠀⠠⠴⢶⠆⠀⢀⠀⠙⠋⠀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠙⠛⠾⣧⠃⠀⠜⠀⣠⣀⡦⠀⣤⢡⡈⠻⢀⣤⠤⢹⡶⢦⠀⠁
⠀⠀⠀⠀⠀⢀⣀⣠⣤⢘⠈⠀⢀⠀⠍⢀⡙⢋⠀⡠⠀⣿⠏⠀⢠⠏⣤⣇⡀⣠⣯⣏⣩⡿⡟⠓⠓
⠀⠀⠀⠀⠀⣨⣧⠀⠈⢻⡇⠀⠊⠀⢀⡠⠂⠀⠀⠀⢀⠟⠀⠠⣏⣸⣿⡃⣱⣤⣸⣿⡟⠳⢬⢀⣢⣾⢷⠶
⠀⠀⠀⠀⠀⡿⠋⠀⠀⢠⣷⢀⣠⡴⠛⢥⣴⡆⠀⠀⣮⠴⢀⡚⣿⣿⣾⣧⡛⠹⠉⠻⢿⣤⣌⣻⣿⣿⣧⠳⡀⠀⣀
⠀⠀⠀⠀⠈⠁⠀⣠⣮⡞⢷⢀⡙⢿⣿⣶⣿⣷⣼⡟⠿⠆⠊⡁⢸⣿⣧⣿⣷⣤⡄⠀⠀⠙⢯⣉⢹⣿⢿⣷⠾⠆⠿
⠀⠀⠀⠀⣤⣡⡄⡋⢬⣿⣎⢧⢳⣀⡛⠙⣿⣿⣿⣼⡶⠀⠹⡜⢸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠹⣟⣯⡟⠁⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠘⠃⠀⠀⠸⣿⣿⡌⠣⠻⡇⠀⠘⢿⣯⣹⡯⠀⣦⣿⣿⣿⣿⡄⣿⣿⠀⠀⠀⠀⠀⢤⠼⣟⡅⠀⢠⡾⠆⠀⣰⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠁⠀⣠⠃⠀⠀⠈⢻⣿⣷⡖⣿⣿⣿⣿⣻⣇⢿⡿⠀⠀⠀⠀⠀⠸⢠⠏⠀⠰⢿⣧⡄⠀⠙⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⡆⠁⠀⠀⠀⠀⠈⣿⣿⣣⠿⣿⣿⣿⠙⢿⡌⣿⣆⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⢸⡇⠀⢰⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠋⡇⠸⣶⣄⡀⠀⠀⣿⠟⠉⠀⣨⠁⠻⣶⣤⣥⣌⣽⣦⣀⠀⢹⠏⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⣸⣄⠈⢿⣷⠀⢠⣁⠈⣷⣾⡿⠇⠒⣾⠻⠉⠛⠛⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣥⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡏⠛⠻⠁⠀⠀⠉⠑⢿⣿⡇⠀⠀⢹⣦⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢀⣤⢾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢿⣇⠀⠀⠈⣿⠀⠀⠀⢠⣄⣼⢳⡶⠀⠀⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡄⠀⠀⠀⠘⣿⡄⠀⠀⠸⣧⡄⠀⢀⣿⠀⠀⢀⣿⣼⣷⠏⠀⠀⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢿⣿⣀⣀⡀⠻⣿⣦⡘⣿⣁⡠⢿⠟⠁⠀⠀⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠉⠁⠉⠛⠁⠈⠉⠁⠹⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠴⠄⠀⠀⠀⠀⣀⡤⠄⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⠒⠉⠉⠉⡩⢟⣿⣷⣶⣶⣤⣀⡠⣴⣖⡒⠋⠉⣉⣉⣉⡀⣀⠀⠀⠚⠋⠙⠢⠤⢤⣄⣀⡀
⠀⠀⠀⠀⢠⡴⠚⠉⢉⣠⣤⣤⣭⣌⠁⢾⢶⠻⠛⠛⠻⣿⣿⡭⢠⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢤⡀⠛⣰⢤⠀⠀⠀⠀⠈⣃⣶⡀
⠀⠀⠀⠀⣿⠁⠀⢶⣾⣿⠟⠛⡻⣿⣷⡄⢼⠀⠀⠀⠈⣁⣸⡄⠀⢀⣤⣤⠀⣀⣴⣾⣿⠿⢛⡿⣿⣄⠈⣷⠢⠼⠃⠔⣠⣴⣶⣾⣿⠁
⠀⠀⠀⠀⠘⢧⣀⣸⣿⡇⠸⢿⠇⢹⣿⣷⠈⡑⠤⢀⣀⡠⠠⠀⠀⠀⠀⠀⠀⢈⣿⣿⡇⢰⣿⠆⢸⣿⣧⠀⠀⠃⢀⠈⢿⣿⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣤⣤⣶⣿⣏⡼⠣⣬⣤⣀⣀⣀⣀⣀⣤⠤⠤⠤⠤⠤⠽⠿⣿⣤⣍⣤⣾⣟⡿⠦⠤⣶⣶⣶⣶⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠉⠀⠀⠀⠈⠙⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣻⣮⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠎⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠿⣠⣤⣤⣶⣶⠈⠉⠉⠁⠀⠀⢠⡤⣤⣤⣤⣀⡀⢀⣾⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠋⠁⠀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⣿⣹⣿⣿⣿⡿⠋⠀⠉⠙⠻⢿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣷⡄⢀⣀⣀⣘⠿⠟⠛⠛⣤⣤⣀⣀⠀⠿⠿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⢛⡟⠉
⠀⠀⠀⠀⠀⠀⠀⣀⡙⠉⠉⠀⢻⣿⣿⣿⡀⠀⠀⠠⣿⣿⠯⠇⠀⠀⠀⠀⢨⣷⣦⣤⣀⡀⠀⠀⢀⣠⣶⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠺⢿⡿⢆⣀⣀⣤⣽⣿⡿⠶⠒⠚⠛⠋⠁⠀⠀⢀⣀⣀⡐⠛⣿⢿⣿⣿⡿⠛⠘⠻⣿⣿⣿⣿⣿⢿⣄⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣀⣉⣩⡗⠲⣄⠀⣼⡿⠁⠀⠀⠀⠐⢠⣬⡉⠒⢬⡙⢯⣅⠀⠀⠀⠀⠀⠙⠻⠟⠉⠁⠀⠀⢻⣧⣿
⠀⠀⡠⠤⠀⠀⠀⠀⠀⠠⠄⢐⠒⠒⠦⠤⢻⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⢾⣷⡄⢙⡦⠈⠛⢓⠒⠄⠒⠿⣿⣦⡀⠀⠀⠀⠠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣘⡄⠀⠀⢀⡀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠠⠴⠀⠈⠉⠉⠀⠀⠀⠀⠀⠑⡄⠀⠀⡈⠛⠛⠀⣤⣴⣶⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢿⠁⠀⠀⠀⠀⠐⢤⣄⡀⠉⠓⠲⢤⣉⠒⢄⡀⠀⠀⠀⠀⠀⠀⠘⢆⠀⣂⣤⠀⠀⢻⣿⡿
⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⣦⣌⡻⢷⡀⣶⣯⣿⡄⢬⠻⣿⣉⣿⣶⢦⣌⡙⠲⣭⡳⢤⣀⣀⣀⣿⠀⠸⡆⢹⠘⡷⣿⡇⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠿⣶⣵⣿⡿⣿⣤⠸⣆⠈⠉⠛⠛⠺⠿⠿⣶⣄⡙⢦⡉⠙⠛⠉⠀⠀⠑⠈⢀⠟⢿⡀⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠉⠛⠿⣿⠿⢯⠄⠈⠛⠦⠄⠀⠀⠀⠀⠀⠈⠉⠂⠀⠀⠀⠀⠀⢀⣴⣖⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⢾⣷⣶⡀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠒⠂⣸⡷⢂⡾⢛⣀⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⠀⠠⠀⠀⠀⠀⠀⠐⠒⣛⠋⠁⠀⠐⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠛⠛⠛⢻⠟⠭⣫⡙⣿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠌⠋⠛⠛⠛⠛⡛⠁⠀⠀⠈⢻⢸⡟⠛⣷⠶⠶⠾⠿⠿⠷⠶⠶⠶⠀⠀⠂
⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠈⠉⠉⠁⠀⠀⢠⡆⠀⠀⠀⠀⠀⢷⣶⣶⣶⣿⣿⠀⠰⣤⣿⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠂⡤⠤⠤⠤⠤⠤⢀⡀
⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠆⠀⠀⣼⡷⠤⢐⣶⣾⣾⣏⠛⠛⣉⣁⣼⣿⣆⣿⢛⣦⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⢆⣿⠀⠀⠀⠀⠀⢸⣧
⠀⢀⡀⠀⠀⠼⠄⡀⠀⠀⠀⠀⡌⠀⠀⣼⣿⣿⣿⣿⣿⣿⠿⠭⠾⠼⠧⢽⣿⣿⣯⢻⣿⠙⣿⣿⣿⣿⣿⣷⣇⠀⢀⢸⣾⣿⡆⠀⠀⠀⠀⢸⣿
⢀⡏⠀⠀⠀⠀⢰⣿⣄⣀⡀⠌⣀⣀⣀⣈⣿⣿⣿⠟⣿⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⢸⣿⠀⢰⣿⣿⣿⣿⣿⡏⢀⠔⠛⠉⠙⠃⢢⡀⠀⠀⢸⣿
⢸⠀⠀⠀⠀⠀⢸⣿⣿⠿⠁⣰⣿⣿⣿⣿⣿⣿⡇⠸⠿⠤⠶⠶⠶⢶⣶⣤⣤⣾⡿⠈⠀⣠⣿⣿⣿⣿⣿⣿⡇⡎⠀⠀⠀⠀⠀⣾⡗⠀⠀⢸⣿
⢸⠀⠀⠀⠀⠀⢸⣿⣿⡿⠷⠾⠯⢽⣿⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣻⣿⣿⣿⣿⢿⠶⡇⠀⠀⠀⠀⢠⣿⣿⢀⡀⢸⢿
⣼⠀⠀⠀⠀⠀⣿⣿⣿⡷⢸⣶⣶⣶⣾⣦⣭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣤⣶⣏⣉⣉⣉⣉⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄
⢿⣀⣀⣀⣀⣤⣿⣿⠿⢧⣾⣿⡿⠿⠿⠿⢿⣿⡆⢂⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⡿⠿⠿⠟⠁
⢸⣿⣿⣿⣿⣿⣿⡿⠀⠈⠁⠀⠈⠉⠉⠉⠉⠀⠂⠀⠄⣀⣀⣀⣀⣀⢴⣿⣿⣿⠿⠒⠒⠒⠒⠒⠒⠒⠒⢪⢙⣇⡀⣀⣀⣀⣼⣿⡇
⠀⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠤⠤⠤⠤⠤⠤⠈⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢤⠶⣄
⠀⢠⣶⡖⠒⠒⠚⠛⠉⢉⣁⣠⠼⣀⠈⠛⠶⢤⡀
⠀⢸⠉⠻⢶⣶⣉⡉⢩⢍⠑⠀⡀⠈⠑⠦⣄⣸⡇
⠀⠸⣄⡀⠀⠈⠉⣿⢄⡹⠝⢀⣨⣵⣶⡶⠟⣿⠁
⠀⠀⠘⡏⠓⠲⣤⠿⠶⠾⡟⠛⣿⠁⠈⢧⣀⣽⡦⡤⢤⣤⣄
⠀⠀⠀⠸⡄⠀⢻⡄⠀⠀⠳⣄⣹⠶⠞⠋⠉⠀⡐⢀⣾⣿⣿
⠀⠀⠀⠀⠙⢦⡀⠻⠦⠴⠞⠉⠀⠨⢀⠀⠀⢀⠀⢸⠛⢿⠛⢶⣶⣄
⠀⠀⢀⣤⣶⣾⣿⣿⣷⣤⣄⠀⠀⠀⠀⠀⠀⠈⠀⢸⣴⣧⣸⣿⢿⣿
⠀⣰⣿⣿⣿⣿⣻⣿⣿⣿⣿⡽⠁⠀⠀⠀⠀⠀⠀⠈⠻⣁⠙⢟⢿⠟⠉⡉⡳⣦⣀⣤⣶⣶⣦⣄
⠀⣿⣿⡟⣿⣿⣧⣿⣿⣿⣿⣷⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢸⠀⠓⢠⠘⠻⢿⣿⣿⣿⡆
⠀⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠟⠦⣄⡉⠓⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⣀⣉⡡⠊⠻⣿⣿⣿⣀⡀
⠀⠈⠿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠉⠛⠲⢤⣤⣿⡿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠉⠉⢉⣳
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣧⢄⣀⡠⠆⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⣿⣿⣿⣿⣿⡿⢿⡧⠤⣀⣀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠉⠙⠒⠛⠛⠿⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠔⠒⠈⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠵⢶⣒⣠⣤⣄⣀⡀⠀⢀⡠⠀⠀⡥⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢶⣉⠠⠄⠞⢼⣿⣯⢿⣷⡶⢰⢡⣶⣿⡢⠷⠋⠡⠀⣀⠉⢐⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣽⣥⠈⠁⡁⠂⠲⢤⣥⣟⠀⠰⠜⠛⠋⢀⡄⠀⢀⣾⢼⠃⠈
⠀⠀⠀⠀⠀⠀⢰⠻⠀⠀⠀⣿⣽⢗⣾⢿⠅⡀⠀⠀⠉⠉⠁⠶⣴⣾⢆⣾⡇⠀⡟⠁⣬⠆
⠀⠀⠀⠀⠠⠤⠈⠁⠀⠀⠀⠀⠀⠈⢡⠔⠈⠀⠀⠀⠀⣀⢐⠄⣽⣻⡎⢿⡀⠾⠤⠚⠁
⠀⠀⠀⠀⠀⠀⣠⣀⡀⠀⠀⠀⠀⠌⠂⠈⠀⠀⠀⣰⣻⡭⢂⠈⢋⠉⠚⠁
⠀⠀⠀⢻⣷⠄⠀⠉⠙⠛⠒⠠⠌⠀⠀⠀⠀⠒⢰⢱⡾⢀⣴⠋⠁
⠀⠀⠀⠀⠈⠑⠠⢀⡀⠀⢀⣤⣀⡀⠨⠀⣀⢰⣃⠸⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⠀⣀⡀⠀⡀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣿⡀⠈⢁⠀⢛⡝⠾⠁⠀⠀⣀⣤⣤⣤⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⠿⠀⢸⡴⠀⠀⢀⣀⣤⣿⠿⠛⠉⠙⢆⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⠈⢁⠀⠿⣄⣠⣶⣿⡿⠋⠁⠀⠀⠀⠀⠈⣯⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣞⣉⠉⠉⠑⡆⠀⠐⠒⢤⣤⡄⢁⣴⡏⠁⠠⢤⡉⢙⣥⣀⣀⣴⠋⢠⣾⢦⠀⣸⣟⣦⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⡿⠛⠋⠉⠉⠁⠀⣀⡇⠀⠀⠀⠈⠟⣤⣼⡿⠃⠘⠛⠚⠛⠋⢀⣤⠎⠀⢠⡿⠉⠉⠉⠑⠿⣿⡿⠰
⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⠟⠉⢹⣿⠃⠀⠀⠀⠀⠀⠀⡸⣍⣁⣀⠀⠀⠢⢤⢙⡄⠘⠂⠀⠀⠀⠀⠙⠁⠀⣰⡿⠁⠀⠀⠀⠀⠀⠈⠋
⠀⠲⣿⣿⣿⣿⣿⣿⣿⢿⡿⠃⠀⠀⢸⣿⣦⣀⡂⠀⠀⠀⠀⢸⠀⠀⠈⠉⠓⠂⠙⠨⡅⠀⠀⠀⠀⠈⢻⣦⣤⣿⠋
⠀⠈⠛⠿⠿⠟⠉⠽⠗⠊⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣏⠀⠀⢨⢸⣶⣄⡀⠀⠀⢸⡇⢠⡖⠈⠀⠀⢀⡈⢿⡿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠤⡈⠁⠀⠘⢰⡮⠟⣿⣶⣦⣼⡇⠀⣱⢀⠔⠀⣉⢀⣼⢗⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⣿⠀⠀⠉⠛⠿⠿⣷⣿⣻⠄⠈⠻⣿⣋⠈⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠙⠛⠛⠋⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠶⢀⣈⣁⣀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣤⡷⠀⠈⠛⢿⣦⣤⣤⣤⡚⠆⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⣀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢠⣿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣴⣶⡞⠂⠀⣀⣤⠴⠤⡄⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣤⣤⣶⠁⢀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠘⢬⡉⠉⢷⡶⠞⠁⣀⠠⣀⣄⣲⣯⣿⠿⠷⢦⡄⠀⢀⣠⣶⣶⠀⢀⣀⠀⣀⠄⠈⠩⠷⠶⢫⣿⠿⠤⣆
⠀⠀⠀⠀⠐⠶⠿⠟⠋⠀⠁⠀⠀⠀⢸⡾⣥⣶⡄⠀⠀⢈⣶⣰⣿⣿⡝⠋⠁⠀⠀⠀⠀⠛⠻⠛⠁⣽⠀⢘⣧⣼⣿⣊⣷⣆⠀⢠⡨⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⣚⢯⣾⠿⠃⠀⢰⢿⣿⣿⣧⣿⡚⠀⠀⠀⠀⠀⠀⠀⢀⢶⡖⠥⠞⠋⠀⠈⠙⠛⢯⡹⠓⠞⠁
⠀⠀⠀⠀⠀⠀⢠⠖⣋⠄⠀⣀⠤⣚⡿⠝⠉⠉⠙⠷⡆⠀⠀⠀⡀⢻⡌⠛⠢⢀⡀⠀⢀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠘⢀
⠀⠀⠀⢀⡠⠔⡁⠈⠀⠀⠈⢡⠌⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠲⠳⡀⠀⠀⠉⠀⠘⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠤⣯⡅⢀⡠⢀⣀⣀⣀⣀⣀⢀⣀⣀⣢⣀⠀⣀⣈⠂⠀⠀⠠⠤⠀⠀⢠⣤⣄⢠⣤⢤⣤⡤⣅⣤⣤⠤⠤⠤⠤⣤⠀⠄
⠄⠄⠀⠉⠓⠓⠛⠛⠛⠒⠒⠐⠙⠛⠓⠛⠛⠋⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠉⠋⠉⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣟⣿⣷⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⣠⠏⠉⢻⣿⣿⣿⠉⠉⠉⢁⡏⠉⠉⠉⠻⣷⡉⠛⣿⣶⣄⡀
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠨⢿⣿⠿⠛⠛⢻⠿⠿⠿⠿⠟⠛⠳⠾⠷⣺⠶⠆⣀⣹⣿⣄⠹⠾⠿⠿⠶⠄
⢠⢤⣮⣅⣄⣦⣀⣛⣛⣁⣈⣛⣛⣛⣙⣋⣽⣶⣤⣤⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠠⠤
⢸⣧⡤⢣⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣇⣈⠿⣁⣈⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘
⠀⡍⡬⠏⠈⠋⠭⠉⠉⢩⣍⠉⠋⠛⠛⠛⠛⠬⣥⢤⠭⠻⠶⠀⠀⠰⢿⡿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢿⣷⠀⠀⠰
⠀⠁⠀⢠⣤⣀⣀⣀⣀⣀⣛⣃⣀⣀⣀⣀⣀⣀⣉⣈⣀⣀⣀⠀⢠⡆⠟⣩⣮⢿⡇⠀⠀⠶⠶⠶⠖⠒⠒⠒⠒⠒⢒⣾⡧⠾⢿⡧⠄
⠀⠀⠀⠀⣿⣿⣷⡿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡀⢀⡙⠛⣸⣧⣤⣤⣤⣤⣤⣤⡤⠴⠶⠶⢶⣾⣿⣟⠶⣸⠃
⠀⠀⠀⠀⠘⠿⣿⣿⣾⣷⡿⠟⠀⠀⠀⠉⠉⠉⠉⠀⠀⠘⢿⣿⣿⣿⣘⣛⣵⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠋

⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠠⡄⡀⠀⣤⢀⣠⣤⣤⣤⢀⣤⡄⠀⠀⠀⡀⠀⠀⠀⣀
⠀⠀⢀⣴⣾⣿⣟⣻⣿⣟⣿⣿⣷⣄⠀⠀⠀⠀⢀⢾⡀⣿⡤⣀⡆⣨⢂⠿⢸⣿⣿⣿⣿⢸⣿⠇⠀⣔⣹⣿⣦⣀⣾⣿⣑⣄⠀⢁⠀⡠⠀⠄
⠀⢠⣿⡟⣿⣿⠟⢻⣻⡟⢻⣷⣽⣿⣷⠀⠀⠀⣾⠠⢄⡠⠆⠨⣠⠤⢤⡆⠈⠙⠛⠛⠛⠈⡁⠀⠘⢿⣿⣷⡯⣨⢿⣮⣿⡿⠃⠈⢄⠰⡆
⠀⣾⣿⣿⡿⢭⣿⠝⠛⢹⣯⠜⣿⣿⣿⡇⠀⠀⣿⢻⠉⠉⠉⠉⠉⢹⢻⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣭⣌⣻⣏⣛⣬⣏⠀⠀⡠⠞⠛⠛⢄
⠀⣿⣿⣿⡇⢨⣯⡘⠴⢂⣷⡀⣿⣿⣿⡏⠀⠀⢸⡙⢢⡤⣄⡤⣤⠚⣹⠀⠀⠀⠀⢈⣿⣿⠹⠁⢠⣿⡋⢨⣿⢽⣿⡉⢻⣷⢰⣧⣤⣟⣁⣤⡇
⠀⢻⣿⣿⣿⣯⠛⢳⢿⠟⢛⣼⣿⣿⣿⠇⠀⠀⠀⢫⡛⣟⠀⢘⡿⣫⠃⠀⠀⠀⠀⢈⣤⣤⣴⡧⢤⣙⠻⠿⠋⠀⠙⢿⠟⠃⠀⠻⣿⣿⣿⠟
⠀⠀⠻⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠙⠢⠧⠧⠚⠁⠀⠀⠀⠀⡸⠉⠀⠃⠀⠀⠀⠏⠣⡀⠀⠀⠀⣸⣷⡤⠠⣤⣤⢤⡀
⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⢀⠨⠁⣶⣶⣶⣶⣶⣶⣦⠈⠅⡀⠀⠱⡆⠀⠂⠀⠀⠀⡄⢄⡈⢆⠀⢰⡏⢉⣷⣖⡉⣷⡔⢣⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠼⢉⡿⡿⢛⠛⡛⡿⣏⠈⠇⢡⠀⠰⡇⠀⠀⠀⠀⢸⢁⠋⠠⡼⢀⡿⠿⣃⡔⢻⣅⡸⣗⢉⣧⡈
⠀⠀⢯⣶⣄⣒⣒⣿⣿⣤⡄⠀⠀⠀⠀⠸⡇⡖⠚⣃⣶⣴⣶⣲⣆⡘⠓⡆⢸⢐⠪⡇⠀⠀⠀⡠⠏⠉⡙⠞⠁⡜⠡⠘⠛⢲⠾⣿⡄⠿⣃⡝
⣠⣴⣾⣿⣭⣍⣛⣿⣯⣝⣿⣿⣷⣄⣀⠀⢧⢽⠾⢿⣿⡽⠿⢯⣿⠿⢶⢧⡆⣮⣴⠃⠀⠀⠀⠈⢻⡆⠀⠀⡸⠁⠀⠀⠀⢈⣈⣛⣓⠺⢿⣀
⢿⣿⣿⣿⣾⠿⣿⣉⣭⡻⣿⣿⣿⣿⣿⡅⠈⠐⠚⠉⣀⠀⣀⣀⠀⠉⠓⠀⠀⠌⣷⠀⠀⢠⠀⠀⠀⢳⠀⢰⠃⢼⠈⠛⣿⠀⢸⣿⡇⢸⠟⢙⣧
⠘⠿⢿⢿⡟⣻⣿⣿⣟⢿⣿⣿⣦⡀⠈⠻⣦⣄⣀⣾⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠛⠡⠀⠀⡳⠀⠀⠘⡄⠏⠀⠘⢷⡍⣿⡀⢸⣿⡇⣸⣻⡿⠃
⠀⠀⠈⢾⣿⣿⣿⣿⣿⣿⣿⣿⠿⢷⣄⠀⠈⢿⣿⣿⣹⣿⣿⣿⣽⣧⣄⡀⠀⠀⠀⠀⠀⠀⡇⢀⡀⠀⣷⠀⠀⠀⠀⣻⣿⡇⢸⣿⠇⣿⣟⠁
⠀⠀⠀⢸⣿⣿⣿⣭⣭⣽⠿⠟⠙⣠⠻⣧⢠⡘⢿⣿⣿⣿⣏⡭⠿⣛⣿⣿⣦⡢⠀⠠⠀⡈⡇⠘⡄⠀⠻⠀⠀⠀⠀⣿⣿⣧⡠⣿⣼⣿⣿
⠀⠀⠀⠸⣿⣿⡏⡹⣿⣿⣿⣭⣳⣆⡀⠙⣯⢳⡆⠀⣹⣿⣽⠞⢋⣭⣴⢿⣾⣷⡅⢰⡇⣛⡧⠀⢠⣀⢺⣆⠀⠀⠀⠈⠉⠘⡷⣿⠁⠉⠁
⠀⠀⠀⠀⠙⢿⣿⣶⠹⣿⣶⡶⣤⠤⠽⣷⣾⣧⢿⣾⣿⣿⢧⡞⣋⣥⠴⠛⠋⠁⠀⢀⣿⡿⣇⠀⠈⢲⠿⣏⠀⠀⠀⠀⢀⣠⢿⣯⣀⡀
⠀⠀⠀⠀⠀⠀⠙⣿⡆⣿⣷⡶⠶⠒⢛⣛⣛⣛⣉⣩⣿⣿⣿⠟⠉⠀⠀⠀⠀⠠⣶⢧⣬⠧⠟⠀⠀⣸⣀⣿⠀⠀⠀⠀⣸⣤⣯⢵⣤⣇
⠀⠀⠀⠀⠀⠀⠀⠈⠷⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠀⠀⠀⠀⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠀⢲⣄⡀
⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠐⠒⠈⠉⠉⠀⠀⠀⠀⠀⠠⠘⣿⣿
⠀⠀⠀⠀⢨⠁⢠⣀⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⢀⣬⣽⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⡆⢀⣀⣀⡀⠤⠄⠒⠒⠀⠈⠉⠀⠀⠘⣿⡿⠓⠓⡺⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠆⠀⡏⠁⠀⠀⠀⠀⣄⣦⣀⣀⣀⣤⣤⡴⠖⠉⠀⠀⢸⣷⡹⣿⡿⠿⠿⠿⢆⣤
⠀⢠⣴⣶⣾⣿⣿⣿⣦⣤⣶⠖⢻⣿⣟⠋⢩⡝⠋⠁⠀⠀⠀⡀⠀⠈⡩⠐⠛⠠⡀⢰⠊⢻⠈⠁
⢠⡿⣿⣿⣿⣿⣿⣿⣿⣷⠻⢭⠿⠿⢿⡻⣭⡀⠀⠀⠀⢤⣀⡘⠉⣰⠁⠀⢀⣀⣨⡌⠃⣸⣄⢀⢀⣤⣄⡀
⢸⢺⣿⣿⣿⣿⣿⣿⠑⡄⠀⠳⣶⢾⡧⠐⡈⠁⠀⠀⠀⠀⠀⠀⠠⡹⣄⠙⠛⠟⠋⡅⠀⠡⠐⠛⣎⣠⡤⠞⢳⡄
⠸⡾⠏⢻⣿⣿⣿⣿⣄⠹⡄⢀⡹⠷⢠⣠⡁⠀⠀⠀⠀⠀⠀⠷⠶⠚⠯⣑⠲⠦⠾⢟⣶⣤⠦⠄⠁⠴⢛⣉⡍⢻
⠀⢧⠀⢸⣿⣿⣿⣿⣿⣿⡟⣄⡉⠳⣎⠁⠂⢀⠀⣤⠤⠴⠴⠶⠧⠤⠶⠠⠌⠘⠋⠉⠀⠀⠀⠀⠀⠈⠛⢿⡇⢸⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣶⣄
⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣇⣛⣿⣷⠺⠀⠀⠀⠁⠀⢀⣤⣤⣶⠆⠀⠀⠁⠀⠀⠀⢠⡄⠤⣀⡀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⣿⣿⣯⣻⣿⣿⣿⣷⡀
⠀⠀⠀⠙⠛⠛⠛⠛⠋⠛⠿⣿⡻⢿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⢀⣀⣀⣀⣀⠀⠀⠅⠀⠀⣨⠖⠠⡀⠀⠀⠀⠈⢢⣄⣀⣀⡠⣿⣿⣿⢷⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣄⠀⢀⠄⠀⠀⠘⠛⢛⠟⣉⢿⣿⣿⣿⣷⡦⢄⣀⡞⠁⠀⠀⠀⠀⢄⠀⠀⢀⡙⢯⠂⠲⣾⣻⣿⡟⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣤⣀⣤⠴⠚⣹⣻⠻⣿⣿⣿⣿⣿⣷⡈⣉⣳⣶⣿⣟⠀⠸⣳⠠⡀⠻⠭⠇⠱⣤⣽⣿⣿⣷⡟⢻⢿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠒⢫⣿⣿⣿⣿⣿⣿⣿⣿⣷⡾⢿⣿⣿⣿⣿⣶⡯⣄⠈⢂⠀⠀⠀⠘⡆⠻⠟⠋⠉⠁⠀⠑⣌⠑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⣿⣿⣿⣿⣿⣿⡟⢲⡛⢛⣛⣛⣛⠁⠀⠀⠙⣦⠑⣄⡀⢀⡼⠁⠀⠀⡀⠠⠐⠒⠘⢢⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣊⡿⣿⣿⣿⣿⡏⠀⠀⠑⠄⠀⠀⠀⠉⠒⠊⠁⢸⠀⣯⡤⢼⡗⠂⠁⣀⡤⠤⠖⠒⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣽⣿⣿⣿⣿⡧⣀⠈⠺⠆⡇⠀⠀⣀⣀⣉⡭⠤⠵⠶⠔⠒⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠙⠲⠤⠏⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡴⠟⠁⢈⡷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⢃⣴⠟⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⠃⣀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠠⣀⡀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠤⠤⠬⣀⠄⣷⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢦⠀⣀⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠑⠦⡀⠀⠀⠀⠀⠀⠤⣀⢀⣠⣤⣤⣤⣤⣀⣀⣀⠀⠀⠠⠤⠄⣌⣷⣃⡀⠁
⠀⠀⠀⠀⠊⠰⢆⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠘⡆⣄⠀⠀⣀⠀⠈⠉⠉⠉⠁⠈⠉⠻⠿⠿⠧⠀⠒⠢⠄⠀⠀⠙⠢
⠀⠀⠀⠀⠀⢇⣼⡇⠀⠀⡄⠀⠀⣠⡶⢲⡀⠈⠀⢐⠂⠘⢋⣒⡤⢝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠢⠐⣾⣿⠃⠃⠂⠃⣠⢺⢃⡼⠃⡆⠄⠀⠈⠑⠦⠈⠋⢀⠎⣏⢿⣿⣿⣷⠖⣰⠖⣶⣲⢿⡆⢀⣀⣀⣀⡀⣤⢖⡠⠇⢳
⠀⠀⠁⠀⣸⡼⡿⠀⠀⠀⠺⠃⡮⡜⢰⠈⠇⡄⣀⣀⣉⣒⠤⢀⠀⠀⠀⠀⠀⠈⠉⠐⠿⠦⢤⡠⣼⣇⠸⠿⠟⠛⠸⠚⠲⠉⠄⢬⡀
⠀⠀⠀⠘⠹⣇⣆⠨⠁⠀⠀⠀⡇⡄⠞⢠⠀⡡⢜⣛⠋⢻⣿⡄⢳⡄⣾⢻⣿⣟⡛⠶⣖⣶⠶⢶⣦⡤⢤⣤⣤⣀⣀⠀⠀⠀⠀⠈⡁⡄
⠀⠀⠀⠐⠦⠽⢿⡟⠂⠀⠀⠀⡁⢷⡄⡌⠀⠈⠀⠀⠈⠻⠿⣿⣮⡇⣿⣸⣷⣯⠟⠷⣮⣝⢻⣾⣿⡻⢶⣿⣿⠀⣿⢁⣼⣿⣿⠉⡅⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢳⣌⣽⣿⠶⠶⣶⣤⠄⣠⣀⣀⡀⠀⠈⠉⠉⠉⠉⠛⠓⠒⠛⠲⠾⠿⠿⠿⢿⣼⣿⠾⣿⠿⠯⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠙⠛⠿⠧⠎⠈⠉⠉⠙⠒⠈⠙⠓⠷⠶⣶⢶⣶⣤⣤⣤⣄⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠾⡿⠿⠿⢿⣭⣿⣟⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡀⠉⠉⠉⠉⠉⠉⠉⣉⣉⣍⣉⣉⣉⣉⣉⠉⠛⠛⠛⠓⠒⢒⡶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡹⠀⠀⡰⠋⠉⠈⠉⠉⠍⠓⢒⡦⢤⣸⣆⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⢥⡀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⠳⠦⢤⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢙⠅⠀⠀⠀⣀⣹⣧⣄⡀⠀⡠⠖⠒⠂⠒⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⢉⢩⣉⠥⠤⠴⣅⡐⠜⠀⠒⠈⠉⠁⠀⠀⠀⢩⠏⠀⠀⠀⣠⠖⠋⠛⢢⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡖⠋⢩⣭⡧⠒⠲⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢰⠃⠀⠄⠄⠀⡇⡇
⠀⠀⣀⡠⠤⡴⠠⣄⠀⠀⠀⣀⠤⠚⠉⠉⠙⠛⠯⣀⣶⣶⣸⠣⠀⠀⠀⠀⠀⣀⣀⠤⢰⡇⠀⠀⢠⢼⠀⠀⠤⠀⠀⡃⡇
⣰⠊⠁⠀⡎⠀⠑⡌⣦⠴⠊⠁⠀⠀⠴⣶⠢⠄⠀⠀⠙⢿⡏⠀⡀⠤⠐⠊⠉⠀⠀⠀⢸⢳⡀⠀⢨⢸⡄⠀⠀⠀⢰⢣⠃
⡇⠀⠀⠀⣆⣬⠖⠋⠀⠀⠀⠀⠀⠀⡰⢻⡥⠖⠉⠁⠉⠙⠲⡍⠀⠒⡠⣀⠀⠀⠀⠀⠀⢑⣿⠦⣄⣢⡙⠦⠤⠴⣣⠋
⣇⠀⢠⡊⠁⠀⠛⠉⢒⢤⡀⠀⣀⡔⣡⠟⠀⠀⢀⠀⣴⡋⠑⡌⡆⠀⠈⠚⢵⣤⠴⠒⠉⠁⠀⠀⠀⠈⠉⠙⠋⠉
⠹⡄⠀⠙⢦⡀⠀⠀⢀⣇⡗⠈⠙⢲⡏⠀⠀⠀⡀⣸⢻⡇⢠⡇⡇
⠀⠈⠓⠒⠒⠛⠛⠛⠛⠉⠀⠀⠀⢸⡀⠀⠀⠀⡇⡏⠁⠀⡜⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢠⡙⠒⠊⢀⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣶⢿⣯⡓⠒⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⢙⡓⠚⠁⠛⠿⠋⠈⢻⣌⣀⠀⠀⠄⢉⣁⠂⣢⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠼⣄⣠⣞⠉⠀⠉⠒⡀⠀⠀⠀⠈⡏⠻⣽⡟⠒⢦⠴⠋⡡⠔⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⡀⢀⠴⣊⣠⠤⣤⣤⣤⣿⣿⣿⣿⡷⠀⣰⠆⠀⠀⠀⡇⠀⢸⡇⢀⣼⠖⡻⠶⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡚⢍⡿⡒⠦⣄⣀⠀⠀⠀⠀⠀⠀⢈⣐⠼⠛⠉⠁⠀⠀⠉⠻⠿⠬⠙⣛⣛⡡⠖⠁⠀⠀⠀⠀⠁⣄⠀⣸⠟⣡⢊⠊⢀⢖⡵⣵
⠀⠀⠀⠀⠀⠀⠀⣰⢋⢎⢞⠕⡡⢊⣤⠶⠯⢝⡦⠀⣀⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⢈⣢⡯⠰⢡⢁⢡⣿⡧⠀⠸
⠀⠀⠀⠀⠀⠀⢠⡇⡎⡎⡎⡜⢰⠏⣠⣶⣶⣤⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣶⣍⠛⠃⠀⠀⠀⠀⠀⡜⢸⡇⠀⠘⠀⡜⡿⢃⢂⡇
⠀⠀⠀⢀⡠⠔⢼⣃⠇⢁⢠⠁⡇⠀⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠠⣦⠀⠀⠀⣠⢴⢖⠶⣦⢽⣯⡇⠀⠀⠀⣠⣴⣾⠿⠻⣇⡀⠀⠀⡇⡶⢃⡞
⠀⢀⠔⢉⡠⠖⡉⠉⠙⠺⢬⣄⣇⡜⠁⠀⠀⠀⢀⡠⠀⢀⡠⠔⢛⣿⣦⣄⢻⣇⢠⢞⠔⡵⡩⢊⠔⣈⠔⢛⣓⣶⡿⠟⠋⠀⠀⠀⠈⠉⠛⠲⠶⠜⠋
⢰⣣⠔⠛⠦⣄⡀⠉⠒⠠⢄⣀⡏⠀⠀⠀⡠⠔⡁⠔⣊⡥⠖⠻⢿⣤⣈⠛⢿⣿⠏⡎⡞⡰⢡⢊⡼⢁⡖⡍⡆⣿
⠈⠙⠲⢤⣀⠀⠉⠓⠦⢄⡀⠈⣷⣄⣀⣊⣀⡥⠖⠻⢥⣀⡀⠀⠀⠈⠙⠻⢶⣾⠸⢸⢠⠁⠏⡼⢡⣿⠇⡇⡇⣿
⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠈⠑⡧⠽⢍⡇⠀⡟⠒⠤⣄⡀⠉⠓⠲⠤⣀⡀⠀⡇⠀⡆⠀⢸⢠⡇⢸⠏⡜⡰⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⠀⠷⠊⠑⠲⢤⣀⠉⠓⠲⢤⣀⠀⠉⠑⣓⡶⢧⡄⢸⢸⡇⢰⢊⠔⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⡀⠀⠀⠉⠒⠦⣄⡈⠉⡷⠚⠁⢀⠼⠳⢼⣌⣧⡈⣥⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢤⣀⠀⠀⢩⠟⣠⠴⠊⠁⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢍⡤⣾⣧⣤⡀⢤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣄⠐⡧⠸⠛⠋⠁⠺⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⣀⣴⢀⠄⠛⠀⠉⠀⠀⠀⣤⣦⢀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡾⣿⣿⡿⣷⣶⣄⡀⠀⠀⠲⣾⡿⣾⣿⡆⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣰⣷⠘⣿⠉⠀⠀⢀⠝⠉⠈⠉⠉⠓⢤⣄⠗⢈⡷⠬⢻⡇⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⡀
⠀⠀⠀⠀⠰⣿⠫⢽⡲⡋⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡤⣤⣦⠠⠄⠠⣼⡿⠛⠛⠿⢿⠿⢿⣿⠿⣿⣿⠟⠋⠁
⠀⠠⠠⠄⠀⠐⣬⢽⡗⣰⢔⠔⣳⣤⣄⢀⣀⡁⠀⠀⠀⢸⠻⣿⣷⠰⢡⠴⡀⠀⠈⠑⠄
⠀⠘⠓⠒⠩⠯⠧⠴⠿⠛⠁⠀⠻⢿⠀⢸⣿⡃⠀⠀⠀⡘⣻⠟⠒⠒⠋⠀⠰⡀⠀⠀⠈⠅⢂
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢖⢠⡍⠁⠤⣀⣴⡴⠋⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠈⠀⡀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⠀⠀⣹⣷⠤⠤⠤⠤⡖⠊⢉⠲⣄⠀⠀⠀⡰⡁⠐⡈⠙⠏⠛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠉⠐⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣒⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠌⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣿⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⣠⠰⠹⢿⣿⣿⣿⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠀⢀⡴⠲⣶⣶⢶⣄⡀⢀⣿⡶⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠁⣠⢤⡀⠂⠈⠹⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⢏⡈⡔⢡⣁⠊⠑⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⠄⣸⣥⣤⣤⣤⣭⣿⡿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠐⢶⣿⣷⣾⣿⣿⡿⢃⠀⠁⢹⠛⡁⢁⣤⡄⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⣠⡾⠁⢈⡼⢣⡈⠀⠁⠈⣌⠁⠀⠀⢿⣿⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢢⢄⡄⠀⠀⢸⠁⠉⡀⠉⠒⠤⣄⣀⡀⠀⠀⠀⠀⠀⡠⠈⠹⠧⠖⠋⠀⢀⠑⠀⠀⡔⠙⠀⠀⣤⠈⣿
⠀⠀⠀⠀⠀⠀⠐⠀⠸⣿⣿⣿⡏⢡⢇⡃⠃⠀⠀⠈⠀⠘⣧⠀⣀⣠⠤⠚⠁⠀⠀⠀⠠⣎⡀⢀⣀⠠⠤⠔⡚⢿⢃⡤⠊⠀⢀⠃⠀⠰⠀⡉
⠀⠀⠀⢄⡶⡾⣆⠀⠀⠹⣿⡟⡅⣵⡎⠙⢓⢠⠄⠀⠀⡀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡸⠀⠀⠀⠀⡇
⠀⠀⠀⠾⣿⢾⣧⠀⠀⠀⠙⢷⣴⡹⡇⠂⠈⠁⠀⠀⠀⣿⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⢰⠁
⠀⠀⠀⠰⣿⣷⠁⠀⠀⠀⠀⠀⣿⣇⠈⣺⠀⠀⠀⠀⠀⠈⢳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠁⠀⢸⡆⠇⠀⠀⠀⣀⠔⠁⠀⠀⠀⢀⡌
⠀⠀⢀⣄⠘⣻⡅⠀⠀⠀⢀⣉⠁⠹⢆⣿⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡑⢌⣣⠸⢸⠀⠀⢰⠀⠐⠃⠀⠀⢀⡠⠖⣹⣷⡄⠀⠀⠐⣉
⠀⠀⠀⠏⠳⠹⣗⠀⠀⠈⠙⠛⠋⠀⡜⠛⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠑⡄⣏⠦⡘⠀⠀⠸⣄⢸⣤⡠⠖⠁⢀⣼⣿⣿⣷⠀⠀⠀⠈⠐
⠀⢠⣶⡆⠀⠀⢋⣄⠀⠀⠀⠀⢰⣾⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⣠⠀⠠⢸⠄⠃⠀⠀⢠⣿⣿⣯⠀⠀⢠⣾⣿⣿⣿⣿⡄⠀⠀⢠⠂⠠
⠀⠀⠻⠇⠀⠶⡀⠉⠀⢀⡠⠴⢏⡈⠙⠳⢷⢶⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⢀⡧⠔⢻⡄⠀⠘⢃⣿⠻⠀⣰⣿⣿⣿⣿⣿⣿⣿⡀⠀⠉⠀⠉
⠀⠀⠀⠀⠀⢸⠁⠀⣿⠙⠓⣿⠌⠉⣲⣤⣀⣀⣀⣹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠾⢿⡧⠶⠀⠸⣶⣾⣿⣷⡀⠀⣸⣿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⡰⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⣸⡀⢼⡷⠟⠀⣈⢻⢫⣮⢿⡄⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠀⠀⢻⣿⣿⣿⣷⣠⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⢠⠟⣡⣤⠀⢰⣯⢓⠾⡀⠡⠊⢀⣀⣀⠀⠀⠀⢀⡴⠴⠦⠼⢄⣀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢠⡄⢀⡎⠀⠀⠀⡎⢸⠯⠜⡲⠆⠞⠁⠀⣀⠤⣊⠖⠲⣭⠻⢆⠀⠈⣞⣙⣋⠱⠂⠀⡝⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⣠⢿⠀⡜⠀⠀⠀⣸⢁⣤⣦⠞⠀⣴⣶⣶⣿⣶⣶⣶⣾⣷⣿⣶⣾⣶⣶⣿⣷⣶⣶⣶⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⣰⣷⡾⣷⠁⠀⠀⠀⠧⣛⣃⣍⠀⠀⠈⠉⠹⠋⠉⠉⣉⣉⣭⣭⠭⢭⣿⡉⠉⠉⠉⠉⢩⣿⣿⠉⠉⠀⠀⢀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠉⢉⣿⣸⣄⣀⣀⣀⣀⣀⢠⡭⣍⡁⠚⠻⢞⣁⢾⡏⢩⣟⠷⠀⢠⣞⠛⣿⡀⠀⠄⠀⢸⡇⡀⠀⠀⠀⣠⠊⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠙⠋⠛⠛⠛⠛
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢⡠⢀⢽⢏⡀⣤⠀⢀⣐⡄⠀⠀⢹⣯⠆⢤⣴⣾⣿⡇⠀⠀⣀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠓⢾⡎⣧⡻⠃⠀⠺⠙⠁⠀⣾⠎⢹⣯⣱⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣦⣸⡆⢹⢁⣶⡿⠂⢠⣶⡇⠀⣴⡿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢻⣇⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣷⣾⣿⣷⣇⡈⢉⣤⠤⠽⡿⠒⠋⠙⣇⠁⠀⠉⢻⣿⣿⣿⣿⣿⣏⠙⣷⢀⠤⠀⠰⠄⠂⢀⣩⣽⡮⠁⠈⠈⠁⠚
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢿⠉⠀⠀⡰⠉⢦⠀⠀⠹⣆⠀⠀⣾⠉⢿⣿⣿⣿⣿⣦⢹⡄⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢻⠛⠉⠀⡆⠀⢠⡅⠀⠀⠳⡀⠀⠹⣦⣀⣿⠀⠀⠙⣿⣿⣿⣿⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⣇⣀⣀⡈⢻⡇⠀⠀⠀⠈⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢠⣁⠀⠘⣆⠀⠀⠀⠙⣿⣿⢿⣿⣿⣶⣤⣤⣀⢀⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡞⠀⠀⠀⢸⣿⠻⢿⣿⣶⣄⣀⠀⠈⣧⠀⣼⠀⠉⠛⣻⣿⣿⡿⠋⠈⠋⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠓⠚⠁⠀⠀⠀⠀⠃⠀⠀⠈⠙⠛⠿⠿⠗⠛⠚⠋⠠⠶⠟⠛⠋⠁⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⣿⣿⣿⣿⣿⢿⣿⣿⣿⣷⠚⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣤⣶⣿⣶⣯⣭⣿⣧⣿⣿⡿⣿⣿⡧⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢿⣿⣿⠟⠹⠿⠟⢻⡟⢻⣿⡶⠖⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣼⡷⡌⣿⣿⡆⠀⠀⠀⢈⠉⠉⠈⢳⡦⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣿⣿⣿⠀⠀⢺⣿⣿⣿⠀⣸⣇⡇⠀⠀⠀⠀⠀⣶⣾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢨⣿⣿⣿⣿⡟⠀⠀⠀⠉⠉⠛⢀⣿⠿⠁⠀⠀⠀⠀⠀⠹⣿⣿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠗⠀⢤⣾⣿⣻⢋⣿⣿⣶⣤⣀⡀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⢰⣿⡽⣱⣿⣼⣿⡇⠙⠋⠉⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠙⢶⣤⣿⣿⣷⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿⣿⣿⣿⢁⠀⢀⣠⣤⣴⣶⠿⣿⡿⣇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡃⠀⠀⠘⡟⢜⢿⣿⣿⣷⡻⣧⠄⠀⠀⣾⣧⣭⣿⣿⣿⣿⡟⣼⣿⢟⣿⣿⣿⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⣰⡿⢸⠘⠿⣿⣿⢿⣟⠛⢦⣠⠏⠀⢺⣿⣿⣿⠟⣱⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⢀⣼⣍⠀⠀⠀⢠⣿⢣⣯⣴⡇⠨⣿⣿⡿⠃⠀⠘⠂⢀⠀⢹⡿⢻⣾⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⢀⣠⣾⣿⣿⣿⣿⣿⠀⠚⠁⠀⠀⠀⠀⠁⠘⡀⠀⠃⠈⠉
⠀⠀⠀⠀⠀⠀⠘⢿⣇⣀⣩⣿⠟⠉⢻⣿⣿⡏⢇⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⢿⣿⣇⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢣⣿⣿⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣭⡤⠀⠀⠀⠰⠀⢀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠛⠶⠿⡥⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠉⠉⠉⠉⠉⠉⠒⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡰⠳⣄⠀⠀⠀⠀⠀⣠⠞⠑⣄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⠖⠉⠙⠦⣀⠀⠀⠀⢸⠁⠀⠈⠓⠦⣄⡀⢰⠃⠀⠀⠈⢧⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⠙⠲⠤⣬⡦⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡰⠓⢶⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣠⠶⠚⡇
⠀⠀⠀⠀⠀⠇⠀⠀⠘⠦⣀⡇⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠉⠁⠀⠀⠇
⠀⠀⠀⠀⠀⢢⠀⣀⠀⠀⠈⠁⠀⡜⠉⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠈⢳⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⢹⡹⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡿
⠀⣴⣤⣶⢀⠀⠈⢇⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢰⠃⠀⡀⣶⣤⣦
⣻⣽⠞⠻⢯⠇⠀⠘⢎⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⠃⠀⠸⡽⠟⠳⣿
⠙⢯⡀⠀⠸⢤⡀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠶⠶⠶⠶⠶⢦⠀⠀⠀⠀⠀⠀⠀⢀⡜⡱⠃⠀⢀⡤⠃⠀⢀⡽
⠀⠀⠈⠳⡀⠀⠙⠦⣀⠀⠙⢿⣄⡀⠀⠀⠀⠀⠀⠹⡍⠉⠉⠉⠉⢉⠏⠀⠀⠀⠀⠀⢀⣠⣯⠞⠀⣀⠴⠋⠀⢀⠞⠁
⠀⠀⠀⠀⠙⢦⠀⠀⠈⠳⢄⡀⠙⠻⣦⣄⠀⠀⠀⠀⠈⠓⠒⠒⠚⠁⠀⠀⠀⠀⣠⣴⠿⠋⢀⡠⠞⠁⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠉⠳⢤⡀⠉⠛⠷⢦⣤⣤⣀⣀⣀⣀⣤⣤⡴⠾⠛⠋⢁⡤⠞⠉⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠉⠳⡶⣄⠀⠀⢘⡇⠀⠀⢸⡃⠀⠀⣠⢶⠞⠉⠀⠀⠀⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⡰⠃⠀⠙⠒⠋⠱⢤⡤⠞⠙⠒⠋⠀⠈⢆⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣾⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⡀⠀⠦⠶⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣀⣤⣿⡉⢻⣷⣦⣄⣠⣶⣦⣾⠁⣠⡿⠀⣼⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣦⣴⢶⣮⣉⣥⣟⣿⣾⣿⠏⠀⡙⣿⣿⣿⣿⡟⢀⣼⣯⡞⠀⠀⢀⣾⣿⡄
⠀⠀⠀⠀⢠⣼⢶⡀⠀⠀⠀⣀⣤⡶⠉⠉⠉⠙⣿⣷⣿⣿⣿⠛⠁⢀⣤⣴⣾⠿⢛⣫⣴⣿⣿⣿⡇⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⣾⣿⣷⡧⢤⠂⠙⠛⠛⠙⠛⠛⠓⠚⠛⠋⢩⡵⠀⠀⠀⣴⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣸⡙⠿⠁
⠀⣄⠀⣠⣿⣿⣿⠧⣼⡗⠀⠀⠀⠀⣀⣀⣴⣾⣿⣿⣿⡇⠀⠀⢠⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠙⠿⠿⠿⠿⠷⠂
⠀⠘⠻⠿⠿⠿⠋⠀⠀⠀⠤⢄⣴⣿⣿⣿⡿⠿⠿⠿⠿⢇⣀⣀⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠛⠛⠿⠟⠋
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⠋⠉⣇⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡹⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣍⠀⠀⠀⣿⡈⠀⠀⡀⠀⠀⠀⠀⠀⡀⠆⠀⠔⠈⠀⡇⠀⠀⠀⠀⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⡏⠐⢤⣴⣿⣿⡆⠀⠀⠀⣀⠀⠀⠀⠀⠘⣦⠀⠀⢀⠃⠠⠀⢀⠔⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠋⠉⠁⠀⠈⠙⢻⣿⣿⠆⠀⠰⠂⠈⠀⠄⡀⠀⠁⠐⠠⠊⠀⠔⠀⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⡀⠀⠀⠀⢹⠀⠘⣿⡷⠀⠀⠀⠀⠄⠀⠀⠁⠄⠀⠀⠀⠙⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠠⠁⠀⠀⠀⡆⠀⠀⠡⠀⠀⠀⠁⠂⠀⡔⠀⠀⠀⠀⠀⠀⠀⢡⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢜⠀⠀⠀⣰⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠏⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣤⣤⣤⣼⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠖⠀⠈⠒⠇⠀⠀⠀⢠⣤⣤⣀⣀⣀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠿⠿⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⣿⣿⣿⣿⣿⠿⠛⠉⠙⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⢉⠙⠉⠀⠈⠁⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⢀⠆⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠰⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣊
⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢋⡀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⠁⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠴⢚⡿⠀⠀⠀⢀⣶⣰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠉⢳⡄⠀⠀⢰⡏⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⢿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠒⠺⡏⠙⠹⠓⠗⠛⠉⡷⠦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠇⠀⠀⢀⣠⡴
⠀⣀⡀⠀⠀⠀⠀⠈⢳⡝⢦⡀⠀⠀⠀⠀⠀⠀⢀⡟⢠⡄⠀⣧⠀⠀⠀⠀⠀⣸⠃⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⣾⠤⠶⣾⢕⡠⠶⠚
⢀⣬⡿⣷⣶⡖⠶⠒⠉⠀⠀⠙⡆⠀⠀⠀⠀⢀⡾⠀⡾⠀⠀⢸⡀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠠⡿⠟⡒⠒
⠀⢬⣭⣻⣯⠶⣠⡤⠦⢤⡀⠀⣷⠀⠀⠀⠀⣼⠃⢸⠇⠀⠀⠈⡷⠶⠶⢶⡞⠀⠀⠀⢹⠀⠀⢻⡄⠀⠀⠀⠀⠀⣸⠃⠀⡴⠛⠉⠉⠉⠉⠙⠛
⠀⠀⠀⠾⠖⠋⠁⠀⠀⠀⣧⠀⢸⡄⠀⠀⢰⡏⠀⢾⠀⠀⠀⠀⢷⠀⠀⣸⠁⠀⠀⠀⣿⠀⠀⠈⣷⠀⠀⠀⠀⢠⡏⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣇⠀⠀⣿⢄⡀⠘⣆⠀⠀⠀⢸⡄⣰⠃⠀⠀⠀⢀⣿⠀⠀⠀⠸⡆⠀⠀⠀⡞⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢹⠀⢰⠇⠀⠉⣱⠘⢧⣤⠤⢬⣧⣏⣀⣀⡀⢀⡾⠙⣧⣤⠴⠞⣧⠀⠀⣸⠁⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣇⡏⠀⠀⣰⠏⠀⣎⣵⣤⡴⠷⠾⢤⣀⡿⡾⠁⠀⠘⣇⠀⠀⠸⡆⢠⠏⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⡿⠀⠀⢠⡏⠀⠀⠀⢈⣆⣀⣀⣀⠀⠈⣹⠇⠀⠀⠀⠘⣆⠀⠀⠙⠞⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡾⠀⠀⣠⡴⠋⠁⠀⠉⠉⠉⠉⠛⢦⡀⠀⠀⠀⠘⣆⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢀⡞⠁⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠈⢦⡀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⡾⠁⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠾⠃⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡠⡻⠲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⡦⡴⡞⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡼⡜
⠀⠀⠀⠀⠀⠉⠀⠑⠠⣀⠀⠀⣀⠤⠒⠩⣒⠤⣀⠀⠀⣀⠔⠊⠑⠢⢄⠀⠀⢀⠤⠀⡄⡄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡎⠁⠀⠀⠀⠀⠉⠊⢱⠉⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠘⠙⠁⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢀⡠⣸⠀⠀⠀⠀⠀⣠⠀⠀⡀⢀⠀⠀⠀⣀⢠⢀⣤
⠀⠀⠀⡀⣀⡤⡀⣀⠴⠊⠁⠈⠑⠢⣄⠀⣢⠕⠊⠁⠀⠀⠀⠀⡞⠈⣀⢰⠓⡎⡀⡀⢰⢫⡞⡏⡸⡃
⠀⠀⠸⠩⠹⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣄⠀⠀⠀⣉⠁⠁⠈⠀⠁⠃⠃⠀⠀⠀⠉⠀⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⣿⠀⢀⣰⠇⡀⢀⠀⣀⡀⢀⠀⠀⠀⢀⣀⠀⢰⠇⠖⢀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⡏⣼⣫⡿⢸⠟⠉⣾⣓⢃⣿⠞⡇⣰⣯⢿⣀⡟⣼⢃⣿⠞⡇⢸⣓⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠋⠀⠀⠁⠈⠁⠉⠈⠀⠀⠈⠉⠀⠀⠀⠁⠈⠁⠀⠉⠁⠉⠀⠀⠀⠁⠈⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⣰⠛⠉⠁⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡀⠀⢀⡟⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣸⠛⠋⠁⢰⢟⡶⣰⢠⣧⢶⠀⡾⠽⢰⢟⡶⣼⡵⣶⢠⡿⠚⣰⢆⣾⢴⡆⣼⠽⠃⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠘⠓⠚⢁⡟⠉⠀⠉⠈⠁⠈⠁⠙⠋⡟⠉⠀⠉⠀⠉⠈⠁⠀⠉⠈⠁⠈⠁⠙⠋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠛⣿⣿⣿⣿⡟⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠻⡿⠰⢀⠐⠸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⣤⣤⣀⣾⠋⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠙⠻⢷⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣆⣀⣀⣀⣀⣀⣠⣾⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣦⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣰⣾⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣎⠻⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⢡⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣴⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡸⢳⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣾⣽⣼⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⣞⠛⠉⣈⡛⢿⣦
⠀⠀⠀⠀⠀⠀⠈⠉⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⡠⠉⠊⡳⣟⣽⠷⠿⠟
⠀⠀⠀⠀⠀⠀⠀⣄⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠈⠀⠈⣿⡂⠐⠢⡄
⠀⠀⠀⠀⠀⢰⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠛⣱⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡤⢤⠐⠁⠀⢈⠀⠙⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣿⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⣀⠀⠀⠀⠀⠁⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⢰⣿⠾⣄⣠⠷⣾⡧⠀⠀⡀⢀⡠⠊
⠀⠀⠀⠀⠀⢀⡿⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠉⠛⠃⢲⠀⠛⠁⠉⠀⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⣤⠀⠀⠀⡀⢠⡄⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⢹⣿⡟⠷⡶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢀⣤⢀⡞
⠀⠀⠀⠀⠀⣀⢀⠔⠦⡁⠀⣾⣿⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣡⡞
⠀⠀⠀⠀⠜⠁⠉⢀⣀⠉⢁⣽⠿⣷⣌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣾⡟⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⣯
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣶⣶⣶⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣼⣇
⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⣰⣿⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⢿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⢿⣿⣶⣦⣄
⠀⠀⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⣀⣤⣌⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣿⠇⠀⠀⠀⠀⠀⣀⠤⣴⢍⠳⣾⣏⣼⡟⣉⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢸⡏⠀⠀⠀⠀⠀⡼⠁⠀⡏⠀⠈⠃⠈⢹⣿⢄⡀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢇⠀⠀⡀⢀⢠⡤⠤⢼⡦⠤⠬⠤⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠴⠊⠀⠀⠀⠁⠀⢀⡀⠔⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣛⣟⣿⣫⣛⣟⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣭⣭⣭⣽⣿⣯⣿⣽⣭⣯⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣶⣶⣦⣼⣿⡛⣛⣻⠿⠷⢶⢶⠶⠶⠶⠶⠶⣶
⠀⠀⠀⠀⠀⠀⠴⠶⠖⠂⠒⢷⣂⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⡿⠷⢿⠸⠀⠀⠀⢀⡀⠉
⠀⠀⠀⠠⡺⣆⠀⠀⠀⠴⠆⣾⣿⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⢛⣛⣩⣭⠉⢀⣼⣿⠗⠠⣤⣤⣀⠀⠀⣴⣇
⠀⠀⢠⢰⣷⡹⡄⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠐⠀⣉⠤⠶⢚⣋⡭⠚⠁⢀⣿⣾⣿⡆⠀⠈⢿⣿⣷⢰⣿⣿
⠀⠀⣼⣧⣝⡻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣭⡶⠶⢾⡿⠋⢀⡠⠴⢾⣿⣿⢿⡃⠀⢰⣿⣿⠏⢸⣿⣇
⠀⢀⣿⣿⣿⣿⣾⣶⣶⣶⣶⣶⣶⣤⡀⠛⠳⣶⣶⣾⣿⣿⣦⡈⠀⠀⠀⣿⣿⣯⣿⠥⢴⣿⠿⠋⠀⣸⣿⡇
⠀⣘⣭⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⠻⢿⣿⣿⣿⣷⡀⠀⠀⣿⣿⣿⣿⣇⣤⣴⣶⡾⠿⠛⠋
⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⠿⠿⠿⠛⠛⠉⠁
⣿⡿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⠀⠀⢀⢀⠀⡀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⣀⣀⣒⣂⣒⡂⣒⢒⣲⠇⡀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⠀⠀⢀⡐⠦⢤⡄⠀⢤⠀⠀⠀⠈⠀⠀⡴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⠤⣭⡉⠀⠈⠽⢞⠭⠔⣁⡦⠊⠀⣀⠀⠀⠀⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢠⣶⣬⣷⠄⠐⠋⣁⠤⠒⠈⠀⠀⠀⠈⣇⠀⢨⡃⠂⡀
⠀⠀⠀⠀⡠⠀⠀⠀⢰⣦⡀⣁⡗⠛⠻⢶⠷⠞⡡⠴⠒⣉⣀⡀⠀⠀⠀⢠⣆⠀⡿⡀⢨⡿
⠀⠀⠀⢠⢮⣿⣿⡧⢸⣟⠋⠀⣀⣀⣠⠐⠈⠉⢉⡀⢀⢻⣿⣿⠀⠀⠀⠀⠀⢀⣗⣤⡂⠀⣀⣤⠄
⠀⢸⠖⠸⣿⣿⣿⡿⠛⠀⡀⠲⣶⣾⡇⠀⠀⠀⢾⣿⠂⣼⣿⣿⣤⣴⣴⣶⠚⠛⠉⠛⠉⠋⠉
⠀⢅⣀⣂⣄⡄⠉⢀⣀⠠⠭⠙⠛⡛⠋⠈⢖⠀⢟⠉⢠⣿⠀⠀⠈
⠀⠁⠀⠉⠁⠘⠛⠛⠉⠈⠒⠈⠙⠂⠐⠿⠾⢿⣦⣶⡿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⡄⣿⡀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣬⡻⠟⠋⠁⡱⠒⡑⠊⣁⣠⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⡠⠚⣡⣾⣿⣿⡿⢃⠀⠀⠀⠀⠀⠀⠀⢠⡤⣄⠰⢯⣟⡣⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠗⣠⡅⣰⠟⠋⠀⢀⣤⠀⠈⣹⠶⡀⠀⠀⠀⠀⢈⡎⠱⡄⠉⡿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢡⡴⠀⠀⠋⢀⠀⠀⢄⢰⠜⢆⠀⠀⠀⢀⣷⢠⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡘⠇⠀⠀⠀⠙⠁⠀⣤⣉⠆⠈⣇⠀⢀⣾⣷⣙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡀⠀⡄⡰⢊⡁⠀⣀⡠⠤⠀⣿⣠⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡄⠁⠁⠸⣶⣶⣶⣾⠏⢀⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣦⡐⢄⠈⠛⠛⢉⢄⣾⣿⣿⣿⣿⠃⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣦⣥⡒⢂⣵⣿⣿⣿⣿⣿⢵⣾⢫⣯⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣿⣿⣿⡇⠈⠑⠼⡏⠙⠉⠙⢟⠛⠛⠻⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠁⢃⡀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠛⠛⠿⠿⣿⣿⣿⣶⠶⢤⣤⣀⠀⠀⠀⠒⠒⠰⠀⠓⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢁⠇⠀⠈⠸⣩⢻⢦⡀⠀⣠⡼⢤⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣼⡀⢀⢃⢇⠀⠄⢋⢞⠃⢀⠀⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢻⢩⠇⡸⠌⠈⡄⣴⠟⡌⣄⣴⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⡏⠈⠊⡤⢇⣄⣠⠟⢁⣤⡾⡿⡱⡘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⢰⢰⠀⢡⣠⣟⡥⢶⣏⠁⠁⠁⠃⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣤⡆⠆⢾⣠⡾⠛⠁⠀⠀⠙⣶⡰⡆⠀⡀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣶⡞⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⣴⣥⣼⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠛⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠙⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣆⠀⠀⠶⢿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠿⠛⠉⠁⠀⠀⠀⠀⠈⠙⠃⠀⠀⠈⠉⠙⠻⠿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠙⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⡀
⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀
⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷
⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠇
⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⢠⣤⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷
⠀⠀⠀⠹⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⢛⣿⣿⣿⣿⣶⣶⣶⣶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⡀
⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⢻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣦⣄⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣶⣤⣤⣀⣀⣰⣦⣄⡀⠀⠀⠀⣠⣴⣶⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⢿⣿⣿⠿⠛⠀⠀⠘⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉

⠀⠀⠀⠀⠀⠀⣀⣀⣠⣄⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠠⢤⡀⠀⠀⣤⠀⢠⣤⣤⣤⣄⢠⣤⡀
⠀⠀⠀⢀⣴⣿⡿⠿⣛⣿⣻⢿⣿⣷⣄⡀⠀⠀⠀⠀⢀⣴⡁⣼⣆⣀⢠⡆⣱⠄⢿⠀⢸⣿⣿⣿⣿⢸⣿⠃⠀⣠⠒⣿⣦⡀⢠⣾⣟⠲⣄⠀⢀⠀⢄⠄⠈
⠀⠀⣴⣿⢟⣥⣿⠿⢿⢿⡿⢿⣷⣝⢿⣷⡄⠀⠀⠀⣼⣀⣁⠈⣩⠀⡏⠀⣉⣈⡄⠀⠸⠿⠿⠿⠿⠘⠋⠀⠰⣿⣿⣫⣼⡋⢹⣧⣝⣿⣿⠆⠀⠂⠈⣤⠄
⠀⣸⣿⣿⣞⣿⢣⣤⠻⣿⢷⡴⣛⣿⣾⣿⣿⡄⠀⠀⣿⠶⡶⠾⠷⠶⠾⠶⢶⢶⡇⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠈⢻⡋⢳⣚⠳⢟⣙⡟⠁⠀⠀⣸⡶⢾⣆
⠀⣿⣿⣿⣿⠁⢛⡟⢰⣠⡌⢿⠃⢸⣿⣿⣿⡇⠀⠀⢹⠴⣇⠀⠀⠀⠀⢀⡼⢸⠃⠀⠀⠀⠐⢠⣤⣤⣶⢂⠀⣴⡿⠛⢶⡿⣷⡖⠛⣿⣦⡀⡞⡀⢤⠤⠀⢱
⠀⢿⣿⣿⣿⣤⢿⣿⢮⣭⣴⣿⡳⣼⣿⣿⣿⠇⠀⠀⠘⡷⢼⡯⡍⠉⡭⣿⠤⡞⠀⠀⠀⠀⠀⠘⠛⠋⢀⡄⠈⢻⣧⣤⣿⠏⠹⣷⣤⣼⡟⠁⢿⣿⣿⣷⣾⣾⠃
⠀⠘⣿⣿⣿⣿⣯⣄⣙⣻⣇⣠⣿⣽⣿⣿⡟⠀⠀⠀⠀⠘⢦⡙⠢⢀⠞⣡⠞⠀⠀⠀⠀⠀⡤⠜⠿⠟⠛⠋⠑⣲⣌⠋⠁⠀⠀⠈⢻⡁⠀⠀⠀⠙⠻⠟⠛⠁
⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⣀⡀⠀⠀⠉⣛⡛⠋⠁⠀⢀⡀⠀⠀⠸⠁⠀⢀⠀⠀⠀⠀⡃⠈⠢⡀⠀⠀⢀⣿⣿⡟⢀⠸⢿⠛⢦
⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⡔⢄⠀⣽⣾⣿⣿⣿⣿⣷⣇⢀⡄⢄⠀⠀⢻⠀⠈⠀⠀⠀⠀⡆⢂⣤⠘⡆⠀⣼⣇⡀⠿⣯⣤⢾⣷⠉⣵⡶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠺⢠⡽⢻⠻⠉⠩⠛⡛⣧⠀⠇⢸⠀⠀⢻⠀⠀⠆⠀⠀⢈⡃⡘⡀⠲⠇⣸⠹⢿⣥⡔⢛⣧⡰⢿⣃⢨⣧
⠀⠀⠳⣼⣿⣒⣒⣲⣿⣧⣦⡄⠀⠀⠀⠀⠀⢸⠀⡖⠛⣃⣼⣦⣶⣶⣶⣇⡟⠛⣶⢸⠀⡒⢽⠀⠀⠀⠀⠀⠏⠁⢈⠟⠃⢠⠟⠀⠘⠛⢳⡿⣿⣧⠚⢟⣀⠁
⢠⣴⣶⣿⣏⣩⢍⡻⣷⣭⣝⣿⣿⣷⣄⣀⠀⠈⣧⢹⢾⠯⣿⣿⣿⣿⣹⣿⢷⢶⢧⡞⠸⣵⠞⠀⠀⠂⠀⠐⢺⡆⠀⠀⢀⡟⠀⠀⠀⠀⠀⠉⠙⠓⠒⢿⡟
⢻⣿⣿⣿⣦⣼⣿⣭⣉⣝⠻⣯⣷⣟⣿⣿⣋⡆⠈⠠⠗⠋⠁⠀⠀⠀⠀⠉⠑⠮⠀⠀⢈⢹⠀⠀⠀⢀⠀⠀⠀⢹⠀⠀⡼⠀⡌⠙⠒⣿⠁⢸⣿⣿⠉⡷⠟⢻⡂
⠸⣿⣿⣿⣾⠷⢾⣿⣿⡿⢷⣿⣿⣦⡉⠈⠙⢧⣄⡀⠀⣰⣿⣷⣿⣿⣷⣆⠀⠀⠀⠀⠀⠺⠃⡀⠈⠈⡆⠀⠀⠈⡇⣸⠁⠀⢻⢖⠓⣿⡀⢸⣿⣿⢀⡗⣳⣾⠃
⠀⠀⠀⢦⣿⣾⣿⣿⣿⣿⣾⣻⣿⣿⣿⣄⠀⠀⠹⣿⡿⣿⠿⣿⣿⣻⣛⣻⣀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⣷⠁⠀⠀⠀⠘⢳⣽⡇⠀⣿⡏⢸⡟⠗⠁
⠀⠀⠀⢸⣿⣻⡿⡿⠟⠛⢿⣿⣯⡭⡔⢻⣷⡀⠀⢹⣿⣿⣿⣾⣿⠿⢛⣿⣯⣿⣦⣄⢄⠀⠀⠀⠀⠼⡇⢰⠂⠀⠹⡆⠀⠀⠀⠰⣾⣻⣧⠀⣿⡇⣾⣻⡶
⠀⠀⠀⠀⣿⣹⣿⠿⣿⣿⣿⣥⠤⢤⣐⠃⠙⢷⡹⣆⡘⠛⠻⣿⠿⠄⣩⡵⠖⡛⢟⣻⣦⠀⠀⢠⢐⣵⡇⠀⠇⠀⢀⣇⠀⠀⠀⠀⠷⠾⠻⣏⣿⡿⠻⠷⠃
⠀⠀⠀⠀⢻⣿⣿⣄⠜⢿⣿⠛⠓⠒⠻⠳⢤⡈⣷⡽⣧⣀⣴⣻⣿⢋⣁⡤⠶⢛⣩⣿⠿⠚⠀⠻⣴⣼⠃⠀⢰⣒⣨⡿⠀⠀⠀⠀⠀⠀⠀⢿⣿⠄
⠀⠀⠀⠀⠀⠉⠻⣿⣯⠘⣟⣛⡿⠛⠛⢛⣛⣿⠿⣷⣹⣿⣿⣷⣿⣤⣡⡴⠞⠋⠉⠀⠀⠀⣠⠾⢯⠝⣇⠀⠀⢹⡋⢷⠄⠀⠀⠀⠀⢰⠖⠛⢟⠒⢶
⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇⠿⠿⣟⣛⣋⣉⣉⣭⣭⣭⣤⣤⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠻⠟⠶⠟⠛⠋⠀⠀⢸⣥⣼⡇⠀⠀⠀⠀⣼⠶⣷⠾⣶⣾⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣈⣓⣛⣋⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣛⣁⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⣀
⠀⠹⠏⠉⠉⠙⠛⣿⣛⣿⣤⣺⣿⣿⣻⣤⣄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣖⠻⠏⠿⠿⠉⠉⠉⠉⠛⠟⢷⣄⣲⠶⣶⣶⣤⣀⡀
⢠⣆⡀⢲⣄⣿⡙⢿⣿⣿⡿⡐⠒⠶⠤⢬⣛⣓⠨⠵⣿⠦⣌⠉⠛⠿⣷⣶⣄⡄⢀⣀⠥⣠⣀⡀
⠰⣟⣿⣿⡇⢋⣉⣍⠘⠿⢧⡀⣄⠠⠴⠠⢮⡏⣽⣯⠶⣦⣬⣅⠀⠀⠀⠙⠻⠿⢾⣝⢿⠉⠻⣿⣦
⠀⢍⣿⣿⣇⠹⣿⣿⣷⣀⣀⢴⣖⡪⡷⣤⣀⠙⢿⣿⣿⣿⣿⣌⡇⠀⣶⣦⣤⣀⡀⠀⠀⠀⢂⣿⣿⢿⣷⣦⣤
⠀⠀⠀⠀⠈⠛⠭⠥⠴⠿⡏⡙⠻⠃⣏⢊⢿⣿⣿⣧⣀⣈⣉⣁⣀⡀⣿⡿⠟⠻⢿⣿⠀⠐⠉⡗⠠⠒⡈⠁⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠿⢿⠃⠨⠌⠈⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠻⠷⠶⠿⡀⢰⣶⠿⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡜⢷⣸⠲⢆⣿⣿⣿⡟⣉⣿⣿⣿⣿⣦⢄⣀⣤⣴⣶⣿⣯⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣬⡧⢞⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠋⠉⠀⠀⠀⠉⠉⠉⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠛⠛⠛⠋⠁
⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⠶⣶⡶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⢀⣤⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⠀⣾⣿⣿⣿⣿⡟⣠⣾⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡇⣼⣿⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⢠⣿⡿⠿⠛⠛⠿⢃⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⠈⣿⠿⠟⢋⣉⣤⣤⣶⣾⣿⣷⡀⢺⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⣿⣿⣿⣿⠇⠹⣿⣿⣿⣿⣿⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣈⣉⣙⢋⣠⣄⡙⠛⠻⠿⠟⠀⠹⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣦⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣧⠀⣉⣉⣉⣙⣛⠁⣼⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⠛⠛⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⠿⠿⠿⠻⠿⢿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡛⣛⣛⡛⢛⣻⣿⣿⠿⠿⢿⡿⠿⣿⣿⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣒⣭⣷⣶⣿⣛⣟⡻⣿⣷⣾⣭⣖⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⣾⣿⣿⣿⣿⢿⣿⣿⣧⡘⢿⣿⡏⠹⣷⣾⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠻⠿⠿⣿⣯⣴⣿⣿⣿⣿⣿⣶⣮⣍⣀⢛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⡿⠋⠉⠙⠻⣿⣿⣿⣿⣿⣿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⠿⠿⠧⠀⠀⠀⠀⠤⠬⠭⡙⢿⣿⣇⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⡿⢁⣤⣤⡀⢶⡀⠀⠀⢀⣤⣀⡀⠀⢿⣿⣿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢁⣹⣿⣿⡇⠘⠋⠀⠀⠀⠁⠀⠀⠉⠀⠈⠉⠀⢸⣿⣿⣄⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣿⣿⣿⢿⠀⠀⠀⠀⠀⢾⣇⠰⠄⢀⠀⠀⠀⠀⣾⣿⣿⣿⢾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣾⡇⠀⠀⢀⣤⠤⠀⠶⠦⣌⢳⠀⠀⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠈⠻⢿⠿⣿⣿⣷⠀⠀⠘⣿⣿⣿⣿⣿⡾⠋⠀⠀⣸⣿⣿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⣴⣞⠀⠀⣠⣿⣿⣿⣆⠀⠀⠈⣉⣉⡋⠍⠀⠀⠀⢠⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠉⠁⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣦⣤⣤⣴⠾⠟⠈⣿⣿⣿⣿⣉⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣷⣤⣴⣤⣤⣤⣿⣿⣿⡟⠻⣿⠋⠉⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⣿⡿⣿⣿⣿⣿⣿⣿⣿⠃⠃⠀⠛⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⢈⢷⣹⣿⣿⡏⠘⠻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⠙⢿⣿⣿⣿⠟⠛⠁⠑⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⢀⣦⡸⡞⣇⢻⣿⣦⡀⠑⠺⢻⣆⠀⠀⠀⠀⠀⠀⣴⣿⠟⠀⠀⢸⣿⢿⠟⠀⠀⣀⡄⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣾⣿⣿⣿⣹⡜⣿⣿⣿⣷⣦⣄⡙⠳⡄⠀⠀⢀⣼⠟⠁⢀⣠⣶⣿⡿⠀⠀⠀⣼⣿⣵⠏⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣻⣿⣧⢻⡏⢻⣿⣿⢿⣿⣿⣷⣌⠢⣠⣞⣡⣶⣾⣿⠛⡉⡁⠀⠀⣇⣼⣿⣿⡏⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⠃⠈⠋⠛⠁⠂⢻⠛⣿⣷⣽⣏⠋⠉⠁⠀⠈⠀⡐⠀⣆⣿⣿⣿⡾⠀⣤⣤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠈⠀⠀⠐⠀⠈⠃⠈⠃⠘⠛⠀⠀⠀⠀⠀⠘⠁⠀⠛⠛⠛⠛⠁⠘⠙⠋⠐⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣤⠀⠀⠀⠀⠀⢸⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡏⠀⠀⠀⠀⠀⡿⠀⠀⠈⣷
⠀⠀⠀⠀⠀⢀⣀⠀⢰⠛⢦⠀⠀⣀⡤⠖⠚⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠓⠶⣤⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠇⢀⡴⠛⠁
⠀⠀⠀⠀⠀⡏⠉⠳⣾⠀⢸⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⡴⠶⠦⣄⡀⠀⠀⠸⠶⠋
⠀⠀⠀⠀⠀⢻⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⣀⣤⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⠞⠈⣇
⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⡴⣄⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⡴⠋⠀⠀⢹
⢠⡏⠀⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⢰⣄⣀⡸⠁⠈⠒⠀⠚⠁⠈⠑⠦⠴⠈⠳⢄⣀⣀⣀⣠⡶⠋⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⢘
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⢀⡴⠒⠒⠦⡀⠀⠀⠀⠀⠘⡆⢀⡤⠚⠛⢤⡘⡇⠀⠀⠀⠀⠸⠧⠶⠶⠒⠛⠋⠉
⡏⠀⠀⠦⠼⠂⠀⠀⠀⠢⡄⠀⠐⣦⣤⠜⠀⠀⡎⠀⣠⣤⡀⠹⠀⠀⠀⠀⠀⢹⡏⠀⣶⣶⠀⢷⡇
⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢁⠀⠀⠀⠀⢧⠀⠙⠿⠃⢠⠇⠀⠀⠀⠀⠈⣧⡀⠙⠋⢀⡿⡇
⢻⡀⠀⢀⠀⠀⢠⠞⠉⠀⠈⠦⠤⠎⠀⠀⠀⠀⠈⠳⠤⠤⠴⠋⠀⠀⠀⠀⠀⠀⣧⠉⠒⠒⠉⠀⢹⠀⠀⠀⠀⠠⣶⠒⠒⠲⠶⠤⢤
⠈⣧⠀⠀⠁⠀⡇⠀⢀⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⣸
⠀⠈⢧⡀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠳⡄⠀⢰⠇
⠀⠀⠀⠙⢦⣀⠀⠙⠓⢲⡆⠀⠀⠀⠀⠀⢸⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠁⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠹⣦⡏
⠀⠀⠀⠀⠀⠉⠛⢦⡄⢸⠇⠀⠀⠀⠀⠀⠸⡄⠀⠀⠈⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠶⣯⡞⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠈⠙⠒⠤⢤⣄⣀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠁⠀⠀⢀⡾⠃
⠀⠀⠀⠀⢀⣤⠤⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⢑⡞⠁⠀⠀⠀⢀⡴⠋
⠀⠀⠀⣴⠋⠀⢰⠋⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⡾⢄⠀⠀⠀⢠⠞⠁
⠀⠀⢰⡇⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⣉⣀⠜⠀⠀⢠⠏
⠀⠀⠀⢷⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠘⡇⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⢠⡞⠳⣄⡀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠑⢄⡀⠀⢀⡴⣏⠀⠀⠈⠙⠳⢤⣄⣀⣠⡴⠟
⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠙⣢⠞⠀⠘⡆
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⣠⣾⣁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣧⢀⣠⠞⠁⠀⠈⠙⠶⣄⣸⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠿⠿⠿⣿⡟⣛⢷⣶⣶⣶⣶⣶⣶⣤⣤⣤⣬
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢘⠂⠀⠈⠛⠛⠛⠛⠃⠀⠀⢈⡾⡟⠳⡶⠶⠶⠶⠶⠶⠶⠇
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠈⠀⢀⣇⠀⣀⣀⣠⢷⣿⡿⠿⣇⡘⣾⣇⠀⠀⠀⠀⠀⠈⠈⠁⢂⡀⠀⠀⠒⠠⡀
⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢀⠂⢀⣼⣿⣥⣿⣿⣿⣣⣦⣖⣿⣿⣷⣿⡞⣿⣿⣿⣿⣧⡄⠀⢠⣼⣇⠀⠀⠀⢸⣷
⠀⠀⢠⠊⠀⠀⠀⣿⣄⣀⡠⢂⣀⣀⣿⣿⡿⢻⠏⠀⠀⠀⠀⠈⢻⣿⢹⠇⢠⣿⣿⣿⣿⠇⡠⠚⠛⠛⢠⡀⠀⢸⣿
⠀⠀⢸⠀⠀⠀⠀⣿⣿⣟⣠⣿⣿⣿⠿⠿⠇⠛⠒⠒⠛⠛⠶⠶⠾⠟⠀⣠⣿⣿⣿⣿⣿⣣⠀⠀⠀⠀⣸⡇⠀⢸⡟
⠀⠀⢸⠀⠀⠀⠀⣿⣿⡏⣯⣭⣍⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠟⠛⠛⣿⣽⠀⠀⠀⠀⣿⣷⣶⣼⡃
⠀⠀⢸⣀⣀⣀⣠⣿⡿⣧⣿⣿⠿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣠⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⡿⠿⠿⠃
⠀⠀⠸⣿⣿⣿⣿⣿⠁⠈⠀⠀⠀⠁⠀⠈⠂⠀⠠⠤⠤⠤⢴⣿⣿⠟⠂⠀⠀⠀⠀⠂⠪⢹⣄⣠⣤⣴⣿⡇
⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠀⠒⠒⠒⠒⠒⠂⠤⠤⡄⢀⠠⠤⣄⠀⢀⣀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡅⠀⠀⠀⠉⠻⠉⠉⠉⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠥⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠃⠤⢤⡴⠶⠂⠀⠂⠀⠀⠠⡀
⠀⠀⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⢠⣶⡀
⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣖⠈⣿⣶⣾⢆⠜⠀⠀⠀⣴⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠇⣿⣿⡇
⠀⠀⠀⠀⣶⣶⢶⣦⣤⣤⣤⣤⣀⠀⠀⣀⣀⣀⣤⣤⣤⣤⠀⠀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠈⢸⣿⣿⡀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠈⠿⣿⡏⠀⠸⣻⣿⡿⣿⣿⠀⠀⠀⠀⢀⣠⣤⣶⡶⠶⠿⠟
⠀⢐⣉⣈⣉⡛⠛⠛⠛⠿⠿⠿⠿⠿⠇⠀⠀⣀⣈⣈⣉⣀⡀⢺⣿⣿⡿⣿⢯⣤⣶⠾⠿⠛⠋⠉
⠀⠙⠛⠛⠿⠿⠿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣬⣭⣭⣭⣭⣽⣶⣾⠿⠿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣽⣯⣿⢷⡄⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣄⢀⣀⣴⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⣿⣿⡏⢘⣿⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣿⣿⣧⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣻⣿⣋⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⢿⡿⢿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⣾⣿⣿⣿⡆⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠾⡿⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠙⠿⡿⠟⠁⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠒⡗⠂⣠⣾⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣦⣼⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠀⠀⠰⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣽⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠁

⠀⡀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣷⡄⠻⣿⣿⣿⣖⣒⠒⠀⠀⠀⠀⠀⣤⣄
⢸⣿⡄⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣿⠍
⢸⣻⣿⠁⢂⡀⠀⠀⠀⠀⠈⠉⠛⠊⠉⢀⣀⣤⣀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂
⠘⢿⣿⣀⡞⠛⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢟⣛⣿⣷⣄⠀⠀⠀⠀⢠⣤⣄⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠈⠛⠓⠿⣿⣷⣦⣄⡀⡀⠀⠀⠀⣾⣿⣽⣿⣿⣿⣿⣧⠀⠐⠲⠒⢻⣹⣿⣿⠗⢦⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⡁⠀⠀⠀⣿⣇⣿⡮⢽⣿⣿⣿⡆⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⠓⠄⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀⣿⣿⢈⣶⣮⣽⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣯⣙⣾⣿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣧⣽⣿⡿⢿⣤⣤⣤⣄⣀⣀⣀⣈⣉⣉⣭⣭⣽⣿⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠙⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠈⠉⠉⠉⠉⠉⡍⢠⠒⠭⡖
⠀⠀⠀⠀⣀⡤⠖⠒⠒⠒⠒⠀⠀⠁⠀⠀⠀⠐⠒⠒⠒⠒⣒⣒⠒⠒⠓⠚⠒⠚⠒⠒⠀⡤⠌⠉⠉⣉⡳⡀
⠀⠀⠠⢊⡽⠀⠀⠀⠀⠀⠀⢀⠀⢒⢸⡉⠉⠃⠀⠀⣴⡿⠓⢦⡙⢆⠈⠀⠀⠁⠀⠀⢀⠁⠀⠐⢸⢿⣽⡇
⠀⢀⠐⠋⠀⠀⠁⠀⠀⠠⠐⠐⠒⠉⠉⠀⠀⣀⠒⡾⡟⢢⢂⣎⣧⢸⠈⠀⠀⠠⠐⡨⠁⠀⠄⠀⢸⡽⣾⡇
⠀⠘⡃⣤⠀⠄⠀⠀⠀⡇⠀⢴⠒⢀⠀⠀⠀⠀⠀⢧⡏⢒⣿⢏⢹⠘⠀⠀⠀⠀⢀⣂⣀⣁⣀⣀⣘⣦⡽⠁
⠀⠀⠀⢈⣑⣒⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠂⠒⠀⠀⢷⠉⡏⢃⡾⣠⡔⢶⣾⠿⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠋⠛⠛⠉⠀⠀⠀⠓⠒⠋

⠀⠀⢀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢀⡀
⠀⠀⢸⡀⠈⠉⠉⠛⠒⠒⣶⠀⠀⣠⠴⠖⠒⠚⠛⠙⣿⣷⡞⢉⠉⢳⠀⠀⠀⢀⠠⣄⡀
⠀⠀⠀⠙⠒⢒⡶⠚⠉⠉⠉⠙⢿⣍⠀⠢⡀⠀⠀⠀⠻⠿⠇⠉⠴⠺⠷⠶⠶⠥⠤⠜⠉⠓⠦⢤⣀⡀
⠀⠀⠀⠀⠀⡞⠀⠐⡍⠀⠪⡢⠀⢹⡆⠀⠡⠐⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢋⢉⠓⢦⣤⣄⣀
⠀⠀⠀⠀⠀⣇⠀⢃⢄⡀⡠⠣⠀⢸⡇⠀⠀⢩⡄⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠘⠐⠐⠑⠀⡇⠀⠈⠙⠓⠲⢤⣄⣀
⠀⠀⠀⠀⠀⠘⢦⡀⠁⠀⠈⢁⣠⡟⠁⠀⠀⠀⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠤⠤⢧⡈⠪⠠⠊⣠⣯⡁⠀⠀⠐⢠⡤⠤⠾
⠐⠒⠂⠒⠒⠒⠂⠙⠛⠛⠛⠛⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁⠂⠂⠀⠂⠙⠛⠚⠛⠁⠊⠉⠉⠉⠉⠉⠒⠒⠒⠒⠒⠒
⣶⣶⡶⠷⠶⠾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⡄⠀⡀⠀⣀⣤⣀⣤⣀⣤⣤⠄⣀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⡀⠀⢀⠀⠈⣁⣤⠷⣛⣿⣿⣿⢿⣻⡭⠖⢛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣀⣠⣯⠶⣿⣯⣾⣿⣿⡿⠷⠊⣉⣥⣴⣾⣿⠿⠟⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣠⣦⡎⠀⠈⢀⠤⠖⣋⣭⣷⣿⠿⠛⢋⣩⣤⣶⣾⠿⠟⠛⠉⢉⣀⣤⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⠆⠆⠀⠀⠃⠠⠟⠈⠁⠀⠀⢴⠾⠿⠛⣋⣡⣴⣶⠿⢟⡛⢩⣅⣀⣤⣽⣶⠿⠟⣻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⢂⠙⢶⣶⣆⠶⠶⠶⠖⠲⠶⠭⠛⢉⣻⣧⣼⣶⡿⠿⡿⣿⣭⣤⣶⡾⣿⣟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⢉⡙⡉⠉⢋⣉⣹⣯⣤⣾⣿⣷⡶⣤⣼⣿⣁⣿⣿⣿⣯⢬⣍⡛⠍⠓⠯⣕⢺⣿⡿⠿⠟⢛⣿⣿
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣦⣶⣬⣿⣶⣶⣾⠿⢿⣿⣿
⠀⠀⠈⢲⡄⠀⢻⣿⣿⣿⣿⣿⣿⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣙⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿
⠀⠀⠀⠐⠻⠶⣾⣿⣿⣿⣉⣩⣭⣭⣽⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠍⠹⠿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⡿⣿⣭⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠟⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⢩⣿⣓⡒⣲⣶⣿⣿⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣭⣽⣬⣭⣯⣾⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠬⠀⠀⠀⣀⡀
⠀⠀⢠⣄⣾⣦⡀⣠⠤⡹⠻⣆
⠀⢀⣠⣼⣿⢿⠄⠂⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁
⠖⠓⠘⠛⣿⢰⡄⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢈⣧⣿⣞⣋⠉⠈⠉⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠐⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣤⠾⢻⡙⣿⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣽⣿⡃⠉⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸
⠁⠀⠸⢳⣿⢳⡦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠚⢙⢹⣧⡄⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠑⠒⢰⣾⣖
⠀⣀⣴⣶⣿⠻⠇⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢾⠯⠅⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢈⣽⡟⠙⠢
⠛⠁⢘⠹⣿⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠰⢸⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠘⠀⠉⠨⣿⣿⣤⡀
⠀⠀⡝⢂⣿⠄⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⢴⣾⣟⠇⠀⠙⠆⠀⠀⡠⠖⡀⣠⣤⠀⠀⠀⠀⢀⣤⢤⣾⡶⠀⠈
⠀⠀⡄⠘⣿⠈⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢄⢹⡏⠃⠀⠀⠀⠀⠀⠀⣀⡀⣹⡿⠲⢄⡀⠀⠁⠀⢘⣽⡿⠖⡤
⠀⠰⠈⢤⣿⠃⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⡇⠃⠀⠀⠀⠀⠀⠘⠉⠉⢿⣿⣤⡀⠀⠀⠀⠀⢀⢸⡇⠂
⠀⠐⠀⢸⣿⡂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⡇⠤⠀⠀⠀⠀⠀⠀⡠⠀⢺⡏⠀⠈⠀⠀⠀⠀⡀⢼⡇
⠀⠁⠈⣺⢿⠉⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣸⡇⢈⡀⠀⠀⠀⠀⠀⠀⠀⢾⡟⡉⠂⠀⠀⠀⠀⠁⣸⡇⠰
⢸⣳⣄⢼⣾⡧⣀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡀⣭⣧⡀⠂⠀⠀⠀⠀⠀⠀⠀⢊⡇⠀⠀⠀⠀⠀⠠⠁⣼⡇⢀⡄
⣻⣀⣈⣻⣻⣃⣉⣹⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣏⣈⣿⣗⣉⣵⣀⣀⣀⣀⣀⣀⣀⣠⣇⣀⣀⣀⣀⣀⣸⣉⣻⣟⣉⣇
⣿⣿⠋⠛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⢀⣀⢀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⢫⣾⣿⣧⡏⣏⡯⡿⡽⣍⣷⣤⠉⠛⠳⢶⣤⣄⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠿⠟⠛⠛⠛⠉⠁⠉⠉⠉⠉⠉⠉⠉⠀⠀⣀⣠⣤⣀
⠀⠀⠀⣠⠔⠠⢠⢖⣭⡭⣕⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣩⡲⡖⣮⠳⡄
⠀⠀⣴⠄⠀⠀⣇⢾⣿⡿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣯⠿⡿⣭⢃⣧
⠀⠐⠀⠀⠀⠐⠉⠈⠙⠛⠁⠊⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠈⠉⠙⠉⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣶⣶⣾⣶⠀⡀⠀⠀⠠⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠠⢀⡤⣤⡢⠐⠀⠤⠀⠀⠀⠀⠤⠤⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣷⠂⡔⠂⡑⠢⡀⣦
⠀⠀⠀⠀⢠⡄⠀⡀⡏⢴⠆⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡅⠸⠟⢀⠇⣼⣿
⠀⠐⠀⠀⠀⠀⠀⠁⠙⠒⠒⠁⠂⠀⠀⠀⠀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠈⠓⠒⠁⠴⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⣴⣿⢻⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⢀⡴⣢⠤⠤⠤⣄⠐⠀⠀⠰⠿⠛⠃⠘⠛⠛⠛⠛⠛⠀⠘⠛⠛⠛⠟⠉⠀⠀⠀⡠⠒⣒⣒⠢⣄
⠀⠀⠀⠀⠎⡜⣱⡞⣿⢗⣎⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣰⣿⢙⡛⣽⡌⢇
⠀⠀⡀⠀⠀⡇⢯⣝⠄⣉⡻⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢫⣥⡕⣢⣝⠇⡼
⠀⠸⠷⠀⠀⠛⠈⠙⠚⠋⠁⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠂⠉⠓⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⣤⣲⠿⠀⢀⠈⠻⢷⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡇⣿⣿⣧⣵⣾⡿⠶⠂⡿⣇⡤⣾⣿⣷⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡟⠛⠛⢻⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡜⠀⢀⣄⣾⡿⠋⡉⣫⠡⠀⢨⣾⡏⣦⠀⣠⣶⣧⠀⠸⣿⣿⡛⢿⣿⢿⣯
⠀⠀⠀⠀⠀⠀⣰⣿⣥⣼⣧⡾⣻⣋⡁⠀⠈⠁⠹⣦⣤⠈⣯⠀⠉⢉⣿⡏⠉⢂⠀⢸⣿⣿⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⢁⣼⣿⣿⣇⠀⣰⣶⣴⣿⣿⢿⣿⣶⣜⣿⣉⣷⣀⣤⣤⣄⡀⠈⠻⣟⣛⣿⡇
⠀⠀⠀⠀⠀⠀⣾⣿⡿⠷⠾⠛⠁⢿⣿⣾⣿⣿⢟⣹⣶⣛⣿⣿⣿⣯⡓⠀⡹⣿⠉⢙⠟⠁⠀⢹⣿⡿⠁
⠀⠀⠀⠀⢀⣤⣿⠏⡶⣶⣤⣶⡴⠛⢿⠟⢫⣴⣿⣟⡿⣣⣭⣿⣻⣿⣷⡘⠶⣉⣀⠙⠴⣶⣢⣾⣿⡇
⠀⠀⢀⡴⢋⣽⣧⡄⠄⢉⠀⣿⠃⣴⢂⣶⢾⣧⡾⣻⣾⣿⣿⣿⣿⣿⣿⢇⣾⣷⡤⣄⣠⣬⣿⣿⡟
⠀⣰⣯⣴⣿⣿⠋⠈⠀⣠⣤⣯⣶⣿⣿⣳⣿⣿⣿⣿⣿⣿⣿⣾⣿⣟⣩⡈⣙⣿⣷⡁⢹⣿⠛⠋
⠀⢿⣿⣿⣿⣿⣷⣴⣛⣿⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⣀⣬⣿⣿⣿⣿⣿⣷⠀⠈
⠀⠈⠻⣿⣿⣿⠿⢻⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠉⠁⠀⠹⣿⣿⣿⣿⣿⣿⢿⡭⠉⠉⠩⠀⠀⠀⠀⠀⡙⠛⠻⢿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠃⠀⠀⠉⠁⠀⠀⠚⠂⠐⠾⠁⠀⠀⠘⠉⠙⠋⠈⠫⠤⠤⠉

⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⢠⡈⠳⣄⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠱⠂⣠⡞⠁
⠀⠀⠀⠀⠱⣄⠈⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⡞⠉
⠀⠀⠀⠀⠀⠈⢡⠊⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⠶⣿⠷⢦⡄⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠈⢿⣦⠀⠀⠀⠀⠀⢸⠃⡀⠀⢀⡀⢿⡄⠀⠀⠀⠀⣠⡾⠋⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠹⣷⡀⠀⠀⠠⢾⠀⠁⠂⠈⠁⣸⢧⠀⠀⣴⡾⠋⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠈⠻⢦⡀⠀⢻⡄⠺⣯⡗⢀⡿⠃⢀⣾⠟⠀⠀⣠⠞⣠⣴⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠙⢦⡀⢻⠦⣤⡴⣿⣠⣴⡿⠋⠀⣠⣾⢫⡞⡀⠀⣀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣦⠀⠀⠈⢷⡈⢰⣦⣴⠛⠛⠁⠀⢀⣾⠛⠁⢸⠀⣉⣁⡀⠀⢙⠆⠀⢤⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠁⠈⠀⠉⠀⠀⠀⢀⡾⠁⠀⠀⠸⣆⠻⠾⢁⢠⡋⠀⠀⡂⢿⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡀⠀⠀⠈⢓⡖⠉⠸⣀⠀⠀⢹⡄⠀⡼⠁
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣆⣤⣶⣋⢳⣤⡾⠉⠳⣶⠟⠀⡸⠁
⠀⠘⡕⢦⣄⣏⢳⡀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣻⣿⢋⠀⠀⠀⠙⢿⣿⠀⠤⠤⠄⠒⠒⣲⡾⠁⠀⣼⠃
⠀⠀⠙⣦⡈⠁⠘⣇⡀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⣼⣿⡇⢠⡄⡄⢠⡦⠸⣿⣿⡆⢒⣛⣛⣻⠏⠀⢀⡾⠃
⠀⠀⠀⠈⠙⠶⣤⡈⠛⠷⣤⣄⡀⠀⠀⢿⣇⠀⠀⠀⣿⣿⡇⠈⠁⠀⠀⠀⠀⢻⣿⡏⣉⣁⣼⠏⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠉⠛⠷⣦⣸⣿⠲⢤⣀⣙⢿⣿⡀⢿⣥⣽⠃⣰⣿⠟⢀⣠⡿⠃⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠙⠿⣧⣤⣀⣀⠀⠉⠳⣄⣉⣡⢴⡏⠀⣴⣿⡋⠙⢦⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⡿⠙⢻⣿⣷⣦⣼⠉⠁⢘⣧⣾⣿⣿⣿⣦⡾⠋⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢧⣠⣿⣿⣿⣿⡏⠛⣶⠛⠀⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠉⠻⣿⣿⣿⣷⣾⣿⣷⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢶⡟⠉⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡴⠛⠳⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣈⠙⠛⠲⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠉⠛⢦⣄⠀⠀⠉⠙⠛⠶⢦⣤⣠⡶⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⢿⣄⢀⣸⣧⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣀⠰⡆⠀⠀⠀⣀⣴⣛⣉⣿⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠉⠙⠛⠶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠀⠀⠀⢀⡾⠋⢉⠙⢿⡁⠀⠀⠀⠀⠀⠀⢰⡏⢰⡆⠀⠀⠀⠈⠛⠳⠶⣤⣍⣙⠳⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⣿⠁⠀⠈⠀⢸⡇⠀⠀⠀⠀⣀⣤⣾⠁⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⣿⠀⠀⠀⠀⣸⠇⠀⠀⠀⣰⠏⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⠾⣷⣤⣤⡴⠞⠁⠀⠀⠀⣴⠏⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢻⣄⣼⢷⣄⠀⢀⣠⡾⠁⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣠⡤⢤⡤⠿⠷⠛⢷⡀⣠⡾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣀⠀⠀⠀⣀⣠⡼⠟⠉⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⠉⠛⠛⠋⠉⢻⡄⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⣠⡼⠷⣤⠾⠃⢀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣀⣤⠶⠛⠉⢁⣠⣴⠿⠞⠛⠉⠉⠀⠀⠈⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⣀⣤⡴⠞⠋⠁⠀⠀⠀⠀⠀⠘⢷⣀⣤⠀⠀⠙⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⡟⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠛⠻⣧⡀⠀⠀⠀⢈⣿⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⢀⣠⡴⠞⠋⠁⠈⠳⣦⡀⠀⢙⣷⣄⣴⠟⠁⠀⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⢶⣟⠁⠉⠳⣤⡀⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⣤⠞⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠲⠤
⣴⡠⣄⣠⣤⣀⢀⣤⣦⠀⠉⠉⢈⣴⣿⠟⣰⣦⡀⠀⠑⠦⣄⣀
⢠⠄⣉⡛⠿⣄⣀⡈⠁⠑⠠⣐⣫⡉⢛⡀⢿⣿⣿⣆⠀⣦⡈⠻⣿⣶⣴⣤⣤⡤⠄⠠⠤⠄
⢸⠍⡏⠻⢿⣶⣿⣿⣶⣤⣤⡥⠭⢻⣿⡇⠀⠙⠿⣿⠧⢍⣿⣦⣈⣹⣿⣿⣷⠤⡄⠀⠀⠀⠀⠀⠀⢀⡀
⠈⢹⣼⠀⠀⣿⣾⣿⣿⢿⡾⣽⡛⠻⠿⢧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡭⠔⠓⠀⠀⠀⠀⠀⠀⠈⠛⠷
⠀⠛⠻⢷⣤⣙⠳⠶⠿⣿⣧⠀⠉⢻⣿⣿⠿⠿⣶⣤⡀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⣠
⠀⠀⠀⠀⠈⠙⠻⣦⣄⡈⠛⣧⠀⠀⣿⡏⠔⢡⡘⢿⣷⣷⣄⠀⢸⣿⣷⣄⠐⠤⣀⠀⠀⠀⠀⠀⠀⠠⠟⠋⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣄⡀⣿⡇⠸⠇⡁⡎⣿⣿⣿⣧⡀⠀⠋⠿⠀⠀⠈⠁⠀⠀⣀⣤⠶⠓⡆⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣄⣿⢡⡄⣹⣿⣿⣿⣷⣄⣀⣀⣠⡤⢶⣾⣿⣿⠋⣀⣴⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣮⣤⣿⣿⣿⣿⣿⣿⣯⣥⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⣤⣴⣶⡟⠉⠹⣿⠀⢨⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⢊⣉⣩⡍⠙⠛⠻⣆⠸⣿⣧⣼⣿⣗⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⢁⡒⠛⠛⠿⢿⣧⢠⣆⣿⡛⣯⡭⠏⠛⠉⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⡈⣽⣵⣁⡃⢀⣸⣿⣦⡿⣿⣿⣷⣶⣶⣶⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣿⣿⣿⣿⢿⢺⡇⣿⣯⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⣸⣘⣿⣿⣿⣿⣿⣿⣿⣿⠻⠀⠉⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣟⠈⢁⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡟⡕⡻⢻⣿⣿⣿⣿⠙⢿⣍⠢⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣹⢏⠌⠀⠉⣟⢅⠙⣆⠻⣢⣽⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡎⣀⡀⢀⣬⣺⣶⣌⣷⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣷⣶⣾⣿⣿⣿⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⡿⣿⢸⣿⡟⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣷⣿⣽⡿⣧⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠉⠹⡇⣿⣿⢹⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢿⣿⠀⢘⡇⠻⠋⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⣾⡇⠄⢸⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⠇⠀⢸⡄⡄⠀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡷⢸⠀⠀⠈⡇⡇⠀⣿⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⣷⡇⠀⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⡇⠀⢸⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡇⠀⠀⠀⠀⠘⣿⠀⠘⣷⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⢹⠀⠀⢸⡼⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣃⣀⣀⣀⣀⣀⣈⣀⣀⣀⣓⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠛⠛⠛⠓⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠄⠂⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⣠⣤⡶⠶⣤⣄⡈⠙⠲⣤⡀⠀⠀⠀⡠⠄⠚⠉⠁⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⡇⣸⠋⠀⠀⠀⠀⠀⠉⠓⢦⣀⠉⠓⢦⣄⡄⠀⠀⣀⠠⣄⣲⠌⠋⠀⢀⣀⡀
⠀⠀⠀⠀⠀⢀⣀⣤⣼⣇⠘⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠈⠛⠷⣭⣅⣉⣧⣼⠶⠞⠛⠋⠉⠹⡆
⠀⠀⠀⠀⣰⠿⣧⠀⠈⠻⣄⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⣤⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⣠⣴⡓⠛⠛⣷⡀⠀⠈⠳⣄⠙⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⣴⡿⠛⠶⣦⣄⡀
⠀⠀⡏⣀⠻⡄⠀⠹⣷⡀⠀⠀⠈⠑⠦⣉⠛⠿⣷⣶⣶⣶⡶⠟⠋⠀⠀⣀⣠⣤⣤⣤⣤⣴⣯⣀⠀⠈⠻⣅⠀⠀⠀⠈⠹⡝⠳⣦⡀
⠀⠀⣇⠁⠀⢻⡀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠉⠓⠢⠄⣀⠀⠀⠀⠀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠈⢣⡀⠀⠀⠀⠘⣆⢼⢻⣷⡀
⠀⠀⢻⡄⠀⠈⡇⠀⠀⢸⠛⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⠶⠶⠶⣤⣀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠱⡀⠀⠀⠀⠘⡇⠀⢹⣿⡄
⠀⠀⠈⢷⠓⠀⡷⠀⠀⣸⠀⠀⠈⠙⠻⢶⣄⠀⠀⠀⠀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⢿⣾⣿⣽⣿
⠀⠀⠀⠈⠳⠴⠧⠶⠚⠁⠀⠀⠀⠀⠀⠀⠈⠳⡄⣠⣶⡾⢿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠉⢡⠏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣉⣴⣞⣋⡉⠉⠉⠛⢷⣄⠀⠀⢠⣶⠦⠤⢑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⣁⣠⣌⣉⠳⢤⡀⠀⠙⢷⣄⠸⡇⠀⢀⣠⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡜⡽⠋⠀⠉⠳⣄⠹⣄⠀⠀⢹⣷⣌⠢⢿⣿⡿⠃⠀⠀⠀⢀⣀⣠⡤⠖⠋⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡇⡇⠀⠀⠀⠀⠸⡆⠸⡆⠀⠀⣧⠙⢷⣄⣠⡴⠖⠒⠛⠛⠉⠉⠀⠀⢀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢳⡱⣄⠀⠀⠀⠀⡿⠀⣿⠀⢀⡏⠀⠀⠛⢿⣿⠶⣤⣤⣤⣤⠤⢶⣻⡽⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡛⢿⣿⣶⣶⡿⠃⢀⡟⣠⠞⠀⠀⠀⠀⠀⠉⠛⠒⠶⠶⠒⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣬⣉⣁⣀⣠⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣴⡶⠶⠶⠦⠤⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⢋⡩⠗⠋⠉⠉⠉⢉⡙⠲⢦⣀⠀⠀⠈⠉⠙⠲⠤⣀
⠀⠀⠀⠀⠀⠀⢀⣀⠰⠞⠋⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠘⠦⢀⣀⣈⣳⣦⣤⣤⣤⣄⠀⠀⠁
⠀⠀⠀⠀⠀⡈⢳⡄⠈⢉⢉⡛⠓⠒⠒⠶⠶⠶⠶⠶⠶⠶⠟⠛⠛⠉⠉⠁⣠⡴⠖⢛⣛⡯⠂
⠀⠀⠀⠀⣘⠱⡂⢿⠀⠙⢾⣄⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠹⢴⠇⣸⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢀⠰⡸⡴⣰⢀⢻⣦⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⣤⣴⠏⠀⠀⠀⠀⠀⠤⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢀⣙⠏⠙⡟⣃⡀⣿⡆⠀⢀⠠⣍⡙⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠂⠤⠄⣀⣉⠉⠙⠛⠳⠶⢶⣤⣤⣸⡇⠀⠠⠿⡃⠀⡟⠧⠄⣹⡇⠀⠈⡆⠙⠿⣷⣶⣼⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠦⠤⣀⡀⠉⠙⠃⠀⠀⠞⡟⣾⢻⠳⠀⣿⡇⠀⠀⡇⠀⠠⣤⣄⣉⣁⣠⣤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣤⣀⠀⠀⠃⠆⢁⣼⡿⠁⠀⣀⡇⠀⠀⠈⠻⢾⣯⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣶⣶⣿⠛⠁⣠⣤⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣄⡀⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⠂

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

⠀⠀⢀⡀⠀⡀⠄
⠀⠀⣀⠀⠀⠂⢠⡔⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠒⠀⡀⢤
⠀⠈⢹⠀⠸⡇⠘⠁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢲⡘⠀⠞
⠀⠐⠀⠀⠀⠤⠄⣠⡁⠄⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢴⡄⠈⠋⠠⠖⠐⢀⠖
⠀⠀⢣⠀⠀⠀⠀⠀⠁⡰⠶⠾⠀⠀⠀⠀⠀⠰⣶⠀⣀⠀⠐⣀⠀⠶⠄⠠⢆⠀⠀⠀⠀⠈⠓⢦⠀⢥⠀⠀⠀⢀⠻⢦⡀
⠀⠀⠀⠓⣤⡀⠀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠘⠀⠀⠁⠀⠀⠞⠁⠈⣃⠀⠀⠀⠀⠀⢸⡀⠀⠙⠀⠀⡜⠁⠒⠛
⠀⠀⠀⠀⠀⠹⠂⠸⡄⠀⠀⠀⠀⠀⠀⢀⠰⠎⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣐⠆⠀⠀⠀⢱⠄⢀⡠⠴⠁
⠀⠀⠀⠀⠀⠀⣧⠀⢿⡀⠀⠀⠀⠀⠀⣘⡷⠄⠚⠓⠂⠘⠛⠒⠤⣀⠀⠀⠀⠀⢞⠁⠀⠀⠀⡰⠃⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠙⡄⠘⣧⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠳⢄⠀⠀⠈⢓⠀⢀⡞⠁⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢃⡀⢘⣄⣠⠏⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠀⠀⡀⠀⠈⠆⠀⠀⣸⣄⠎⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⢰⡞⠉⠉⢹⠇⠀⠀⠀⠀⠀⠊⠁⠙⠀⠀⣀⡤⠚⡁⠀⠀⠸⠀⠀⣿⠏⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⡎⠀⠀⠀⣤⣤⠤⠤⠤⠔⠒⠋⠁⣠⣼⡇⠀⠀⠐⠓⢶⣟⡀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⡇⠀⠀⠀⢻⣿⣄⣀⣤⣤⠤⠚⠛⢻⡟⠀⠀⠀⢦⡤⠎⠀⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢰⠀⠀⠀⠈⢿⣿⡿⠿⠇⠀⠀⣸⠋⠀⠀⠀⢠⠟⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢳⡀⠀⠀⠀⠙⠓⠀⠴⠖⠊⠀⠀⠀⢀⣴⠋⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠙⠢⣄⣀⡀⠀⠀⠀⠀⢀⣀⣤⠶⠋⠀⠀⠀⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠉⠉⠉⠙⠛⠒⠒⠢⠤⠤⣄⣀⣀⣀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢻⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠰⠒⠒⠂⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠉⠉⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢣⡀⠀⢀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠉⠁⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣤⣤⠤⠒⣇⣀⣀⠀⠀⠀⠀⠀⠀⢀⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⡄⠀⠀⠉⠉⢹⠉⠉⠚⠟⠓⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠇⠀⠀⠀⠀⠸⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣰⠀⠀⠀⠀⠀⠀⣧⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⣳⠀⠀⠀⠀⣠⠜⠁⠀⢠⡄
⠀⠀⠀⣀⠀⠀⢀⣤⠔⠊⠁⠀⠀⠀⢻⣀⠠⠔⠊⠁⠀⠀⠀⢸
⠀⠀⢘⠁⠄⢀⣈⣃⣀⣀⠀⢀⡠⠛⠊⠉⠣⠀⠀⠀⠀⡀⠁⢰
⠀⠀⠈⠁⠀⠤⠤⠤⠤⠤⠀⠿⣈⠀⠄⠀⠀⠰⠀⠤⠄⣛⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠄⠠⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⢀⣀⢀⣀⡀⠀⠀⢀⠀⡀⠀⡠⠔⢰⣦⣀⣀
⠀⠀⠀⠀⣿⢹⠉⠉⠉⠉⠉⠉⣿⣷⣋⠀⠀⢠⣼⣧⣤⣤⣤⡄⠀⠀⠀⣀⣀⣀⣀
⠀⠀⣠⣤⣿⡟⣄⣀⣀⣀⣤⠾⣛⣉⣀⣀⣀⣀⣛⣛⣂⠛⠻⠋⡉⠭⢭⣉⠉⢉⡉⠉⢉⠛⠓⢶⣦⣄⣀⣀
⠀⠀⣷⣼⣿⠃⠸⣒⣲⣿⠛⣛⢩⡉⣁⠀⠀⣩⣭⣽⣿⣿⣶⡾⠿⢿⣷⣄⠉⠋⠁⠀⣀⣉⠒⢤⣀⠈⠙⠻⢿⣦⡀
⠀⠀⡿⢻⣿⣿⣶⣿⣿⣿⣇⣿⡸⣇⢿⣸⣦⡻⠿⣿⣿⣿⠃⣾⣟⣷⡜⣿⡆⠀⢿⣭⣍⣉⡀⠠⢈⡑⠦⣄⡀⠈⠛⢷⣤⣀
⠀⠀⠙⠛⠛⠛⠃⠀⠀⠉⠙⠛⠛⠻⠿⠿⠿⠿⠛⠛⠙⣿⠛⢿⣿⣟⡗⣿⣧⣴⠚⣿⣿⣝⡻⠿⣶⣾⣭⣶⣌⣁⡀⣀⢉⣿⣿⣷⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣝⣛⣛⣵⣿⣿⠏⠀⣿⣿⣿⣿⣷⣶⣾⣿⢿⣛⡿⠿⠝⠛⠛⠉⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⣤⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⢾⣿⠿⣻⣿⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣭⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⡿⣫⣾⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣥⣤⡌⠿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⣮⣤⣬⣟⣛⣛⣛⡛⢛⣛⣛⣛⣛⣭⣭⣭⣭⣭⣥⣴⣶⣶⣶⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣝⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⣍⠙⣿⣿⣷⣽⠟⠻⢿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢶⣌⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣼⣷⣿⣿⣿⣿⢠⡟⢿⣿⣿⣿⣟⠁⠀⠀⠀⠀⠀⠈⠙⠲⣌⣙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢻⣷⣤⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣼⠀⠻⢯⣝⡿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠈⢻⣷⣬⣛⠛⣛⣛⣛⣛⣉⣭⣤⣶⣿⣿⡿⢟⣥⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⣻⠀⣧⢸⣿⣿⣷⣯⣟⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢟⣛⣯⣵⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡂⠁⡘⡿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⢻⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⢋⣉⣤⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⢀⣉⡛⠿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⣟⣩⣤⣤⣴⣶⣶⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⣮⣭⣭⣭⣭⣭⣭⣭⣭⣿⣶⣿⡿⠿⠿⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⡿⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡼⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⣀⣤⠤⠶⠒⠒⠛⠛⠛⠛⠛⠓⠲⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⣾⠁⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⣴⠶⣄⠀⣿⠀⢹⡆⢀⣤⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣠⡶⠋⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠈⠙⣿⠀⠸⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡴⠛⠉⠛⠲⣤⡀⠀⠀⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⢀⣠⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⡀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⢠⡴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠚⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢹⡄
⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⣷
⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⢦⣀⠀⠀⣠⠏⠳⣄⠀⠀⢀⠿⣄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⢸
⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡧⣤⣤⠎⠀⠀⠈⠈⠉⠁⠀⠀⠈⠙⠓⢋⠀⠈⠳⠤⣤⣤⣤⠴⣞⠁⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⣀⣀⣸
⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢠⠴⠛⠛⠓⢦⡀⠀⠀⠀⠀⠀⠘⡆⠀⡤⠒⠋⠉⢶⡀⣿⠀⠀⠀⠀⠀⠀⠿⠶⠶⠶⠛⠛⠉⠉⠁
⣿⠀⠀⠰⣄⡸⠂⠀⠀⠀⠐⢆⣠⠀⠀⢰⣄⣀⡴⠁⠀⢀⡇⠀⢀⣤⣀⠀⠱⠀⠀⠀⠀⠀⠀⢹⡼⠀⣠⣶⣦⠀⢻⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⠀⠀⠸⡇⠀⠺⣿⡿⠀⢀⡇⠀⠀⠀⠀⠀⠈⣧⠀⠙⠿⠛⠀⣸⣿
⢿⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠲⣆⠀⢀⡇⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⣀⡞⠀⠀⠀⠀⠀⠀⠀⢹⠳⢤⣀⡀⠴⠋⢹⡄⠀⠀⠀⠀⠀⢀⣤⣤⣀⣀⣀⡀⣀
⠸⣇⠀⠀⠢⠀⠀⢰⠋⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠹⣧⠀⠀⠈⠉⠉⠉⢹
⠀⢹⡄⠀⠀⠀⠀⢸⠀⠀⠠⠽⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⡾
⠀⠀⠹⣦⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⣼⠁
⠀⠀⠀⠈⠳⣤⡀⠀⠈⠛⠲⠤⡤⠀⠀⠀⠀⠀⠀⠀⡇⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠚⠁⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣰⠇
⠀⠀⠀⠀⠀⠈⠙⠳⣤⣀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡇⣼⠁⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⣲⠶⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⣀⠀⢀⣠⠴⠋⠁⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⣀⣤⠶⢦⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⢉⡟⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⢀⡾⠋⠀⢀⡾⠁⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⢀⡾⢄⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⢸⠃⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⣸⠇⠀⠀⢠⡞⠁
⠀⠀⠀⢸⡆⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⢠⡟
⠀⠀⠀⠀⢷⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠘⣇⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⣠⡟⠉⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠙⢦⡀⠀⠀⢀⡼⣯⠀⠀⠀⠀⠉⠛⠶⣤⣤⣤⣤⡴⠞⠁
⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠉⠳⣴⠏⠀⠘⣧
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣴⠟⠉⠛⠶⣤⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣆⣴⠟⠁⠀⠀⠀⠀⠀⠉⠛⠶⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⣀⠀⠀⠀⠠⠆⠰⠄⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⡻⠿⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⠿⣏⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⢿⡿⢤⣤⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣤⠰⣿⡟⠻⡃
⠀⠀⠀⠀⠀⠀⢼⠷⡿⢴⣶⣀⣨⡟⠛⠻⣆⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠛⣯⣀⢠⣷⡼⣷⢿⡆
⠀⠀⠀⢀⣤⣦⣶⣠⡆⣸⡉⠘⡿⠿⢶⣶⣾⣦⠀⠀⠀⠀⠀⢀⣾⣷⣶⠜⠿⡟⠉⢹⣀⣷⣼⣧⣶⣄
⠀⠀⢀⣤⣏⣠⡇⢘⠋⠿⠓⣾⣷⣤⣤⣟⠉⠉⢣⠀⠀⠀⢠⠟⠉⢛⣧⣄⢠⣿⡗⠻⡟⢻⡋⣯⡈⣥⣄
⠀⠀⠈⠛⠉⠿⠛⣿⢶⣷⣄⣼⣅⠈⠻⠛⠃⠀⠀⢷⡀⣠⠏⠀⠀⠛⠻⠋⠀⣸⣄⣰⣷⢾⡿⠻⠏⠉⠋
⠀⠀⠀⠀⣾⣷⣠⣆⢀⣟⠈⠿⠋⠀⠀⠀⠀⠀⠀⠈⣷⡏⠀⠀⠀⠀⠀⠀⠈⠻⠏⢈⣏⢀⣧⣰⣿⡆
⠀⠀⠀⠀⠈⠀⠹⠟⢻⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⠛⠻⠁⠈⠁
⠀⠀⠀⠀⠀⣶⣿⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⣷⡆
⠀⠀⠀⠀⠀⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⣤⣄⣀⣀⣀⣀⣀⣀⡀⣀⠴⠂⡶⣆⣤
⠀⠀⠀⠀⣇⡏⠁⠀⠀⠈⢉⣿⣿⠥⠀⠀⢤⣿⠶⣶⣴⣦⣀⣀⣀⣤⣄⣀⣀⣀⣀⡀
⠀⠀⡴⣶⣿⠑⡶⠤⢤⣶⣻⣭⣤⠶⠶⠶⢤⣭⣥⣌⡉⢃⣆⣒⡒⠤⣀⡠⠀⠀⠬⣉⠙⠳⠶⣤⣤⣄
⠀⠀⣗⣿⣏⣀⣿⣽⢿⡍⣷⢰⣦⣦⣤⢰⣶⣶⣿⣿⡿⢛⣯⣟⢿⣦⠀⠀⡤⠤⠤⠌⠑⠦⣄⡀⠉⠛⢷⣄⡀
⠀⠀⢧⣿⣿⣿⠛⠛⠿⢷⣽⣮⣧⣽⣾⣷⣭⣽⡿⣿⠃⣼⣿⣿⡇⣿⡇⣀⣿⣷⡶⣦⣤⣔⣢⠭⣒⠤⠀⠈⠻⢷⣦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠸⣇⠹⣿⡿⣣⣿⣿⠏⢸⣿⣿⣶⣭⣙⣻⣿⣿⣷⡶⢶⣦⡾⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠶⠾⠿⠟⠋⠀⠈⠛⠛⠛⠿⠿⠿⠿⠘⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⢀⡤⠔⠒⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⢠⡄⢀⡀⠀⠀⢀⣿⠃
⠀⠁⡠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⢈⠇⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⡄⠀⠀⣀⡀⠀⠀⢄⠈⠢⡀
⠀⠎⠀⠀⠀⢰⠓⢦⡀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⣧⡀⠀⢀⣠⣴⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⣠⠞⢉⡿⠀⠀⠀⢳⠀⠹
⠀⠀⢠⡴⠤⣈⢳⡀⠳⡄⣾⠁⣠⠟⠀⠀⠀⠀⠀⠀⠀⣴⠟⣉⡀⠀⠀⠀⠀⠈⠉⠀⠉⠙⢷⣄⠀⠀⠀⠀⠀⢀⡴⠒⠒⢦⣟⠈⠙⠃⣴⢛⣀⣀⠀⠀⠘⡇
⠀⠀⠀⠱⢦⡈⠳⠟⠁⣹⠿⢰⡃⠀⠀⠀⠀⠀⠀⠀⣸⠇⠚⣧⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣦⡙⣧⠀⠀⠀⠀⠘⠷⠒⠛⡦⠈⠻⣦⠀⠘⢋⣠⡼⠃⠀⠀⡇
⠀⠀⢀⡤⠴⠨⠗⠀⡾⠁⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡏⢧⣀⠀⠀⠀⠀⠀⠀⢀⣴⢿⠁⠸⣇⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠘⠆⠀⠛⢤⡀⠀⠀⢸⠇
⠀⠀⠈⠑⠒⢲⠖⠀⠁⠀⠀⠰⢏⠀⠀⠀⠀⠀⠀⢸⡟⠀⢸⡇⠀⠙⠓⠶⠤⠤⠴⠖⠛⠁⢸⡀⠀⣿⡀⠀⠀⠀⠀⠀⢀⡽⠂⣤⣠⡴⠟⠓⠲⠃⠀⠀⠉
⠀⠀⠀⠀⠰⡥⠞⠛⠶⠶⠞⢧⡀⠳⣄⠀⠀⠀⢠⡞⡛⠂⠸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠛⠍⢳⠀⠀⠀⣰⠏⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠳⣄⠀⠈⢳⡶⠀⠀⣿⣿⣷⣤⣄⣀⣀⣀⣀⣤⣴⣿⡇⠀⢲⣴⠟⠀⢠⡞⠁⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠈⠳⣄⠈⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⣗⢀⡴⠋⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠳⣿⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣷⠞⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣿⠀⠀⠀⢹⡟⠿⢿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢸⣇⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⣡⡿⠃⠀⠀⠀⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣄⠀⠙⠳⠶⠶⠞⠋⠀⡀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⡀⠀⠈⠳⣦⣄⣀⣀⣠⣴⠞⠁⠀⠀⠀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⠳⠦⣄⣀⣉⠉⠉⠉⢀⣀⣀⣤⠴⠚⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⣠⣤⡄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠋⢠⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣠⢦⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠢⡀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⡇⠘⣇⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣧⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢿⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⣼⠃⠀⡟⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠙⣆⠀⠀⠀⠸⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡇⠀⠀⢀⣠⠶⠛⠙⠃⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠈⠳⣄⠀⠀⢠⠟⣻⠻⠶⠶⢶⣶⣶⣶⣶⡶⠶⠾⠟⠻⣏⠻⣄⢀⡴⠛⠁⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠈⠳⠶⠋⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣈⠋⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢀⣤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⡟⠀⠀⠀⠀⢸⠃⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠶⣄⠀⠀⣀⡤⠶⠒⠚⠉⠉⠉⠉⠉⠙⠒⠶⢤⣀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡾⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠑⢾⠀⢸⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡴⠶⠦⣄⠀⠀⠀⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡠⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⡠⣦
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⢀⡞⠁⠸⡄
⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⡀⠀⢠⠙⠦⠀⠠⠞⠑⢄⣀⡰⠳⡄⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⡰⠋⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠒⠁⣠⠤⠤⡀⠀⠀⠀⠀⢠⡀⢈⡑⠒⢖⠹⡇⠀⠀⠀⠠⣞⣠⣤⠤⠶⠖⠋
⠀⠀⠀⠀⠀⠀⣾⠀⠠⠄⠴⠀⠀⠀⢄⡀⠀⢠⣄⡠⠃⠀⡜⠀⣠⣄⠈⡆⠀⠀⠀⠀⢷⠃⣠⣦⡈⢷⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⡄⠀⠀⠀⢣⠀⠻⠟⠀⡆⠀⠀⠀⠀⢸⣄⠈⠋⢀⡾⡇
⠀⠀⠀⠀⠀⠀⢸⡄⠀⢄⡀⢀⠞⠁⠀⠈⠲⠔⠁⠀⠀⠀⠀⠑⠤⠴⠊⠀⠀⠀⠀⠀⠀⡆⠉⠀⠉⠀⢳⠀⠀⠀⠀⢾⡛⠓⠒⠒⠶⡦
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⢸⠀⠀⠰⠄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠱⣄⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠢⣄⣀⡀⠀⠀⠀⠀⢸⡝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠥⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠘⣆⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀⢨⡇⠀⠀⠀⠀⠈⡄⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡼⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠙⠒⠤⢄⣀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⢀⡤⠋⠁⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠁⡞⠁⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⡎⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠈⠲⢤⡾⢣⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢣⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⢱⣄⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠈⠢⡀⠀⠀⠀⣴⠋⠈⠓⠦⣄⡀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠈⠲⣤⠎⠙⣇⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⣠⣞⡁⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣠⠞⠁⠀⠉⠓⠦⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⣀⡀⠀⢀⣀⣀⣀⠀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣀⠀⠀⢀⣀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣍⣉⠉⢉⣉⣻⡇⣾⠋⠉⢻⡇⢹⡏⣉⣹⡇⠉⢹⣿⠉⠁⠀⢰⡟⠉⠉⠁⢠⡿⢻⣆⠀⣿⡏⣉⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣭⣿⢲⣿⠉⠉⠀⢿⣦⣤⣼⠇⢼⡇⠻⣧⡀⠀⢸⣿⠀⠀⠀⠸⣷⣤⣤⣀⣿⠟⠛⢿⡆⣿⡏⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢀⡀⡤⣀⣀⣀⣠⡀⢠⣄⣤⢀⡀⣀⣠
⠀⠳⠦⣄⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠁⠉⠉⠋⠙⠁⠈⠀⠋⠈⠉⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⡤⠖⠂
⠀⠀⠀⠀⠉⠓⠦⢝⡳⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⣋⠥⠒⠋⠁
⠀⠀⠀⠀⠀⣷⣶⡄⠈⠁⠈⠑⠢⣄⠀⠀⠀⠀⡸⠿⠟⣩⡍⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⣭⡙⡿⠿⢀⠀⠀⠀⢀⡤⠚⠁⠂⠁⠀⣶⣾⡇
⠀⠀⠀⠀⠀⠿⣿⡇⠀⢸⣿⣷⠀⢠⣄⠀⠀⠂⢀⣀⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣅⣀⠀⣀⠀⢀⣥⠀⢰⣿⡇⠀⠀⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⢱⣶⡜⠛⢀⢀⣡⠤⠈⠟⠛⠉⠈⠁⠀⠀⠀⠀⠀⠈⠁⠈⠉⠛⠯⠤⢠⣁⢀⠘⢻⣴⡎⠉⢹⣿⣿
⠀⠀⠀⠀⠀⣤⣤⡼⠿⢧⣤⣼⠿⠇⠀⠀⣾⠃⡀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⡀⢹⡇⠀⠘⠿⢧⣤⡼⠿⠿⣤⣤⡄
⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣧⣀⡀⡜⠉⠹⠎⢇⣴⣆⣰⣤⠂⣤⣤⣤⣤⣤⣤⣤⣤⢂⣴⣄⣰⣄⠝⠶⠋⠙⠆⣀⣸⣿⡇⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠛⠉⢱⣾⡇⠀⢸⣿⡇⠀⠄⢀⡀⠀⠀⢀⣈⣉⢐⣛⣛⣛⣛⣛⣛⣛⣛⠈⣉⣁⠀⠀⠀⣀⡀⠄⠸⣿⡇⠀⢸⣷⣶⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠸⠟⣃⡤⠀⠀⠀⢐⡀⠙⠛⠷⠀⠿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏⣿⣿⠇⠰⠟⠛⠉⢐⠃⠀⠀⠤⣘⡻⠿
⠀⠀⠀⠀⠀⠀⠰⠚⡋⠁⠀⠀⠀⠀⠀⠉⠙⠛⠓⠒⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠒⠚⠛⠉⠉⠀⠀⠀⠀⠀⠍⡓⠶
⠀⠀⠀⠀⣀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠰⢶⡒⠶⠶⠤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠯⠭⣉⣉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠛⠷⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣟⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣊⣤⠤⢄⠀⠀⢠⡬⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣻⠋⠀⠀⠀⠱⣄⠀⠉⠓⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⡏⠀⠀⠀⠀⠀⢹⡀⠀⣀⡬⢍⠂⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⡇⠀⠀⠀⢀⠀⢘⡷⠋⠁⠀⠈⡇⠀⠀⠈⠁⠒⣲⠂⠀⠀⠀⠀⠀⠀⠀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣄⠈⣇⠀⠀⢰⣏⣡⠎⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⣌⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠁⠹⣄⠀⠈⠋⣟⣤⠀⠀⢀⡀⠞⠀⠀⠀⡍⠁⢸⣇⣠⢤⣄⣀⠀⠀⠀⠀⢹⡌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠓⠠⢌⡓⠆⠚⢇⠉⣠⠖⠉⠀⠀⠀⠀⠼⠀⠀⠠⣤⡴⠦⣤⡈⠱⡏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠱⠀⠀⠈⠿⠁⠀⠀⠀⠀⢀⠎⣰⠀⠀⢀⠀⠉⠙⠋⠃⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⡏⠀⠀⠌⠙⠓⠶⠤⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⠀⠀⠀⠀⠀⢠⠀⠑⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⠀⠀⠀⠐⣤⠞⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠊⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡄⠲⢤⣀⠀⠀⠀⠀⠀⣧⣀⡀⠀⠀⠘⠧⠀⠀⠀⠈⠀⠀⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⡀⠐⠃⠄⠀⠤⣼⠑⠢⢄⡀⠀⡟⠉⠀⠀⠀⠸⣷⠁⠀⠀⠘⡄⢿⠲⠼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣡⠞⠐⠉⠉⠉⠹⣄⡤⢴⠈⠒⠁⠀⠀⠐⡆⠚⠛⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠑⠚⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣄⣀⣀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⡍⠙⠉⠉⠉⠉⢉⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⠗⠓⠂⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢚⡤⠤⣼⣀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⣾⣤⢶⣹⣽⣭⠉⠑⠒⠒⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠓⠛⠓⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠐⠂⠀⠀⠀⠂⠀⠂⠂⠁⠂⠂⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠈⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠀⢶⣤⡀
⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠐⠒⠈⠁⠈⠀⠀⠀⠀⠀⠄⠘⣿⣿
⠀⠀⠀⠀⠨⠁⢠⣀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣀⢠⣬⣽⡀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡆⢀⣀⣀⠠⠤⠄⠒⠂⠈⠉⠁⠀⠀⠘⣿⡿⠓⠓⡛⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠄⠀⡏⠀⠀⠀⠀⠀⣖⣦⣀⣀⣠⣤⣤⠴⠖⠁⠀⠀⢸⣿⡹⣿⡿⠿⠿⠿⠆⣤⡀
⠀⣠⣴⣶⣾⣿⣿⣿⣦⣴⠶⢖⢻⣿⣟⠋⢩⠝⠋⠀⠀⠀⠀⣀⠀⠈⡡⠐⠙⠢⡀⢰⠉⢻⠉⠁
⢰⡿⣿⣿⣿⣿⣿⣿⣿⡷⠏⡭⠿⠻⢿⠛⣧⡀⠐⠀⠠⢤⣀⡘⠁⣸⠁⠀⣀⣀⣸⡌⠣⣸⣄⣀⢠⣤⣄⣀
⢸⢸⣷⣿⣿⣿⣿⣿⠑⡄⠀⠳⣾⠿⠧⠐⡈⠁⠀⠀⠀⠀⠀⠀⠠⡙⣆⠙⠛⠛⠋⡍⠀⠠⠘⠛⣇⣤⠤⠎⣳⣄
⠸⡜⠋⢻⣿⣿⣿⣿⣆⡘⡄⠀⡹⠷⢤⣤⠅⠀⠠⠀⠂⠀⠀⠟⠾⠚⠧⢝⡒⠲⢾⢛⣶⠤⠶⠄⠁⣔⣋⣭⡭⢻
⠀⢷⠀⣸⣿⣿⣿⣿⣿⣿⡟⣦⡈⠻⣦⠁⠒⢄⠀⡤⠴⠶⣶⣾⠷⡶⠶⠢⠬⢙⠋⠁⠀⠀⠀⠀⠀⠈⠙⢿⡇⢸⠀⠀⠀⠀⠀⠀⢀⣴⢶⣶⣶⣶⣄
⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣇⣛⣿⣷⠺⠀⠈⠀⠙⠀⢩⣴⣶⣶⠆⠀⠈⠁⠀⠀⠈⢐⡖⠠⢄⣀⠀⠀⠀⠀⠉⠲⡀⠀⠀⠀⠀⠀⣿⣿⣿⣻⣿⣿⣿⣷⡀
⠀⠀⠀⠉⠛⠛⠛⠋⠉⠙⠻⢯⣻⢿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢀⣀⣠⣤⣀⠀⠀⠄⠀⠀⣠⠇⠢⢀⠀⠀⠀⠈⢲⣤⣀⣀⠠⢿⣿⣿⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣄⠀⠀⠆⠀⠀⠈⠛⢋⠏⣍⢻⣿⣿⣿⣷⡖⢄⣠⡞⠁⠀⠀⠀⠑⣀⠀⠀⢀⣘⢳⡂⠚⣿⣽⣿⡟⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣤⡤⠔⠋⢹⣯⠻⣿⣿⣿⣿⣿⣿⡍⣬⣽⣿⣿⣟⡀⠸⣻⠑⡄⠻⠼⠃⠙⣤⣾⣿⣿⣷⣟⡹⢻⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠑⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⠾⢿⣿⣿⣿⣿⣷⠯⢄⡈⠢⠀⠀⠀⠈⠆⠹⠛⠉⠉⠀⠀⠑⠄⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⢿⣿⣿⣿⣿⣿⣿⡟⢻⣉⣛⡛⠛⢋⡀⠀⠀⣈⣷⠑⣄⡀⢀⡜⠁⠀⠀⠀⠀⠂⠀⠋⣢⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣢⡿⣿⣿⣿⣿⡇⠀⠀⠐⠠⡀⠀⠀⠈⠉⠉⣁⣸⡀⢧⠤⠾⠇⠉⢀⣠⠤⠔⠒⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣼⣿⣿⣿⣿⠧⢄⡈⠻⠂⡇⠉⣀⣀⡠⠤⠤⠤⠕⠓⠒⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠉⠓⠶⠏⠉
⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢉⣀⢼⣇⣤⣀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⣤⠐⣯⠈⠿⠟⠘⠢⢅⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣤⢀⡀⠿⠀⠛⠀⠀⠀⢀⣡⡀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣤⣾⣿⣿⣿⣦⣤⡀⠀⠀⠀⠠⣼⣿⣇⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣴⡄⢾⡟⠋⠉⠀⠙⡟⠓⠙⠛⠻⢧⣀⡀⠲⠜⢷⣌⣿⣿⠀⠀⠀⠀⣰⣴⣾⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⢀⣴⡾⡛⣧⣼⠃⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⢈⣿⣧⣔⣉⣉⣀⠉⠀⠀⠀⣼⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠖
⠀⠀⠀⠀⠈⠙⡁⢵⣷⡑⡄⡤⡤⢏⡀⠀⠀⠀⠎⠀⠀⠀⢹⣿⣿⣿⡉⡿⠁⡀⠀⠉⠛⢅⠀⠀⠁⠉⠈⠉⠙⠈⠉⠉
⠀⠨⡤⣁⡴⣠⢿⠋⣹⣴⡯⠊⠀⣿⣿⠃⣴⣷⠂⠐⠢⠀⢀⣧⣿⣿⣁⣡⠏⠉⡀⠀⠀⠀⠣⠄⡀⠀⡀
⠀⠀⠀⠌⠀⠉⠉⠉⠉⠁⠀⠀⠀⠉⢫⢄⠘⠿⠦⠀⠀⠀⢆⣼⠃⠀⠀⠀⠀⢀⠁⠄⠀⠀⠀⠥⠒⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠱⣆⠀⠉⠒⢾⣞⠁⠀⠀⠀⠀⣀⠥⢄⡀⠀⠀⠀⠀⢀⢆⢀⠰⣶⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠀⠂⠼⠟⠃⠒⠒⠊⠙⠄⠠⠔⠙⠦⣀⠀⡠⠊⠀⠀⠲⡀⠉⠀⠹⠟