Western Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣶⠀⠀⣤⣤⣀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣶⣾⣿⣿⠿⠟⠛⢻⡏⠋⢘⣼⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄⠀⣠⣶⣾⣿⣿⠿⠛⠛⠛⠉⠁⠀⠀⠀⢰⣿⣶⣾⣿⡿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⣀⠀⠀⠹⠿⢿⠟⠃⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⠟⠉⠽⠿⠭⠁
⠀⠀⠀⠀⠀⠀⣀⣄⡠⠖⢋⣁⣤⣶⣶⣶⣤⣤⣀⣀⢀⡀⠀⠀⠀⣠⣾⢻⡏⣷⡄⠀⢐⣳⣦⡀
⠀⠀⢀⣠⣶⣾⡿⢋⣰⣾⣿⠿⣿⣿⡲⣿⣿⠉⠀⠀⠀⠀⣴⣾⣿⣿⡇⠈⣸⡿⣿⣄⠈⠙⠋⠁
⢀⣾⣿⣿⣿⡏⣴⣯⣿⣿⣿⡞⡋⠁⠀⠀⠸⠅⠀⠀⠀⠀⠀⠻⣿⢿⣇⣰⣿⠃⠈⢿⣦⡀
⠘⢱⣿⣿⣿⠇⣾⡿⠟⣹⡏⠉⢻⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡞⢿⣿⡟⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⣀⣀⡀
⢠⣽⡿⠟⠁⠀⠀⠀⣨⣿⣧⡀⠈⣿⢛⠓⠒⠁⠀⠀⠀⠀⣠⠌⠣⠁⠀⠻⣿⡠⣄⠀⠀⠀⠙⢿⣶⣄⣠⡾⠟⠉⠁
⡿⠋⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣦⣈⣸⡆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠓⠂⣶⡆⡖⠀⢈⣛⠈⠁⢀
⠃⢀⣴⣿⣷⣤⣴⣾⠁⠀⠈⠉⠛⠛⢿⣿⣶⣦⡀⠀⠤⡄⠀⠀⠀⢠⣴⣢⡄⠀⠀⠀⠈⠳⣄⠠⣿⢟⡀⠀⣿⡆
⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣯⣻⣿⡄⡼⠀⠀⠀⠀⢺⣯⠔⠌⢀⠀⠀⠀⠀⠈⠛⣙⠘⢷⡀⠙⠿
⠀⢀⣾⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠠⡀⠘⢿⣿⢹⣿⠁⠀⠀⠀⠔⠈⣿⣟⠁⣾⠀⠀⠀⠀⠀⠀⠖⠁⢄⠳⣤⣄
⠀⠸⠱⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣧⠀⠀⢻⣷⠙⠀⠀⠀⠀⠀⢀⣿⡟⠘⣿⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠛⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢹⠄⠀⠀⠹⠈⣇⠀⠀⠀⠸⣷⣿⣿⠀⢻⡇⠀⠀⠀⢠⡞⠉
⠀⠀⠀⠀⢹⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡿⢀⣠⠄⠀⢸⣏⣿⠀⠀⠻⣦⣤⣤⠟
⠀⠀⠀⠀⠈⠛⢿⣿⣇⣷⠀⠀⠀⠀⠀⢷⠁⠀⠀⠀⢿⠉⠀⠀⠀⢸⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⡌⣇⠀⠀⠀⠀⠀⣇⠀⠀⠀⠸⠀⠀⠀⠀⢸⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⠳⡄⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⢸⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠙⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢼⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣧⠄⣰⠷⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣷⡦⣤⡄⠀⢀⡠⣶⢾⠻⣇⡑⠶⠒⠲⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⣿⣿⡷⣿⢇⢴⣆⠀⠀⠛⢧⣹⣷⣷⣾⣶⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣍⡍⠉⢿⣻⣧⡈⡻⡇⣩⣳⡈⢀⠰⢩⡿⢿⣿⡿⡿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣧⣀⣀⣯⣍⣷⡘⢿⣟⣿⡗⢸⣷⠎⠀⠈⣿⠃⡁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣾⣿⠀⠀⠀⢰⡏⠀⡜⠻⠟⠙⠛⣿⡻⣿⠃⠀⢹⣦⣾⣷⡏⠀⠀⠀⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣎⠳⣤⣿⣆⠢⠀⢠⣿⣶⣾⣿⣷⣿⠃⠀⢸⣿⣭⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣿⣿⣿⣿⣿⡻⠆⢠⣴⠟⠉⠉⠛⠛⠿⢟⡂⠀⣸⣻⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⣿⣿⢟⣿⠃⣴⠟⠁⠀⠀⠀⠀⠀⢀⣆⣡⣴⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢅⣿⣯⡾⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠸⣿⠛⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣼⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠰⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄⠈⢻⡀⠀⠀⠀⠀⠀⠀⣄⢇⡀⢀⠂⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠛⠟⠋⠁⠀⢈⣧⡜⠃⡦⠑⠷⠦⣿⠶⠽⠟⠳⠋⠻⠷⠶⠦⠤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠘⢷⣴⣷⣿⣿⠇⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉

⠰⠀⣐⣰⣆⢀⠂⢲⠀⠀⠀⠖⠀⢀⠀⠀⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠰⠀⢬⣿⢮⡙⠀⣸⠁⡆⡄⣹⠀⣯⠀⠀⠀⠀⠁⠉⠉⠉⠀⠈⠉⠉⠋⠉⠋⠋⠉⠘⠉⠛⠊⠹⠏⠿⣻⣿⣿⣿⣿
⠸⡆⢸⠛⡾⣴⡀⢹⡆⣣⡄⣟⠀⢸⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⣾⣿
⢶⡀⢸⣷⡇⢻⡟⠿⡇⣹⡂⢡⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠉⢱⣿
⣸⡁⢸⣿⡏⢸⡓⠀⣇⡿⠇⠉⠀⣼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⢀⡀⣦⢴⢂⠀⠀⠼⠷
⢸⡇⢸⣿⡧⣘⡇⠠⢾⡇⡀⣵⠰⠀⡄⢚⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢤⣼⣦⡿⣥⣿⣿⣾⡆⠀⠀⢲
⢸⠀⢠⡿⠓⢿⣿⠀⢺⡇⡇⢿⣀⠰⡇⣬⢀⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠈⠛⠛⠿⠏⠙⠟⠀⠀⣉
⢸⡀⠲⣿⠀⠈⠈⠂⢼⡏⠓⠾⣿⢴⣿⣿⠀⣟⢦⣶⣿⠟⠁⠈⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡘⢏
⢸⡄⠀⡟⠀⢀⠀⠀⢸⡇⠀⠘⠀⠾⡇⠀⠈⠙⠾⠗⠐⠆⢀⣠⣶⡄⠀⠀⢀⡠⣄⣢⣄⡠⠾⠸⠂⠀⠁⢸⡿⠿⠿
⠸⠇⠾⣇⠀⠀⠀⠀⢨⡇⢀⠀⠀⠲⣷⠀⠠⠀⠀⠀⠀⠈⠻⣟⠁⠀⠀⠀⠈⠹⠿⣋⣉⢀⡀⢤⡀⢶⣤⡞⣷⡢⡷
⠀⠀⠠⢿⠀⠀⠀⠦⡀⡇⠀⠀⠀⠀⠷⠀⠀⠀⢤⡀⢠⠀⠀⠸⣿⣮⠀⠀⢀⠀⠀⠀⠉⠙⠛⠛⠓⠛⠿⠿⠶⢤⠯
⠀⠀⠀⢺⡄⠀⠀⠀⢰⣷⠀⠀⠀⠀⠻⠀⠀⡠⠀⢄⠈⠁⠀⠀⠈⠙⠓⠧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⣀⣀⠀⡀⠀⠀⠄⠀⠀⡀⠐⢾⣦⡁⢁⠀⡀⢠⡶⠂⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠉
⢀⢠⠀⠀⠀⠀⠀⠛⠀⢸⠀⠀⠀⠈⠁⠀⠀⠀⠻⠰⡄⠀⠀⠈⡁⢸⣦⡴⡈
⢈⠉⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⣠⣿⠄⠀⠀⠠
⢰⡄⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠀⢠⣾⣻⠀⢷⡇⠀⠀⣿⣿⣿⡄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾
⢸⡇⠀⠀⠀⠀⠀⠄⠀⢸⣧⠀⠀⠀⠀⢹⣛⣿⠀⠈⡧⣆⢀⣾⣿⠃⠁⠀⠘⣿⣿⡖⠀⠀⠀⠀⠀⠀⠀⠈⢋⣿⣿
⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢰⣾⡟⠁⠀⠀⣷⣿⣿⣿⣿⣤⣆⣤⡄⠹⠟
⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢩⠁
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⢸⡿⢿⣿⣿⣿⣿⣿⣿⣧⣤⣼⣿⣀
⠀⠙⢿⠀⠀⠀⡄⠀⣠⣄⠐⠂⠀⣂⢰⠀⠀⠀⠂⠈⠠⠄⢹⣿⣿⣿⣿⣿⣿⠋⠿⣿⣿⡆
⠀⠀⢿⡅⠞⠃⠀⠀⠀⠀⠁⠀⠴⡈⣿⠁⠀⢦⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⡗⢿⣿⣿⣆
⠀⠀⣼⡇⠀⠀⠀⠚⠁⠀⠀⠐⠤⣬⡽⠀⠀⣸⠶⢷⣤⠴⠛⢉⣿⣿⠟⠻⣿⣿⣴⣶⣿⣿⡽
⠀⠀⣸⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⡇⠈⠀⡾⠉⠀⠒⠀⣿⣿⣿⢾⣿⣿⡇
⠀⠐⠉⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣼⡇⠀⠀⠀⢀⡼⣿⣿⠘⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⠀⠀⢰⣿⡷⠀⠀⠉⠀⠘⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠀⠀⡰⣹⠇⠀⠀⠀⠀⠀⠹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠙⠠⠄⠀⠀⠀⠀⠀⠀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠢⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢠⠼⠁⠀⠀⢷⣤⠀⠀⠔⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠋⠀⠀⢘⠒⠒⠐⠦
⠀⠀⠀⢀⡞⠋⠀⠀⠀⠈⠉⣇⢀⣀⠀⠀⠀⠀⢰⣟⡆⢸⣻⠀⠀⠀⢱⠀⠀⠀⠀⠀⢀⠤⣤⢼⣿⣤⣥⣤⣤⢤⣤⣯⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠧⠀⠐⠸⠀⠂⠒⠠⠄⠘⠰⠸
⠀⠈⡞⠀⠗⠃⠘⢺⠖⠀⠘⢿⠋⠙⠍⠀⠐⡖⠖⠀⣖⣒⣒⠒⠂⠀⢰⠀⠀⠀⡰⠀⠐⠀⢹⠈⠈⢸⡏⠉⠉⠸⡇⠁⠈⠹
⠀⠀⡇⣺⢓⣷⢰⡟⠛⡇⢰⣟⣰⠀⠀⠀⠀⡅⠀⢸⡏⠉⢽⠤⠀⠀⠘⠀⢀⠜⠄⠀⠀⠀⣿⣍⣉⣿⢺⠁⠈⠀⡇⣍⣭⣘
⠀⠀⡇⠈⢹⠉⠸⡇⠀⡇⠀⢹⠀⠀⠀⠀⠀⡇⠀⠸⡇⠀⢸⠀⠂⠀⠀⠀⠀⠆⠀⠀⠀⠀⢺⠀⠀⢸⡐⠀⠀⠀⡇⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠐⠓⠛⠛⠛⠛⠓⠒⢒⠒⠒⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠐⠐⡖⠀⠀⠎⠁⠀⠐⠋⠛⠒⢲⠒⠒⠒
⠀⠀⠘⠛⠋⠙⢻⠛⠛⠛⢻⠛⠛⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠛⠋⠉⠀⠐⠃⠁⠀⠘⠛⠚⠙⠃
⠀⠀⠰⡶⠶⠶⢶⡶⠶⠶⡾⢦⡴⢧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⢁⠀⡀⠀⠀⠀⠀⠀⠀⠉⠁⢀
⠀⠀⠀⢠⡤⣤⠸⠁⣀⣀⡈⢈⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣤⣤⣤⣤⠀⠀⠀⠐⠀⠀⣤⢤⣤⡄⠓⡏
⠀⠀⠀⠨⣇⣸⠨⠨⡇⠈⡇⢰⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣤⣤⣿⠀⠀⠀⠀⠀⠀⠿⠤⠤⠇⠀⡇
⠀⠀⠀⠀⠀⠉⠨⠠⡇⠀⡇⠀⠉⠉⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠂⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⣠⣤⣤⣤⣤
⠀⠀⠀⠋⠃⠇⠟⠛⠂⠀⠀⣀⣀⢀⢀⡄⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠐⠒⠒⠠⠀⠀⠀⠀⢠⠀⠀⠆⠐⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠙⠘⠛⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⡰⢠⠀⠴⡀⠀⠀⠀⠀⣔⠀⠀⠀⠀⠀⠄
⠀⠀⠀⣀⣀⣀⣀⣀⡄⠀⠀⣤⣤⡂⡆⢴⠀⠀⠀⠈⠀⠀⠀⠀⠀⢰⠀⠀⠁⡠⠊⠀⠀⠴⠀⠀⠀⠀⠀⡇⠀⠀⠤
⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠉⠈⠁⠉⠈⠀⠀⠀⠐⠂⠀⠀⠐⠀⠁⠀⠀⠈⠳⠬⠀⠐⠀⠀⠀⠈⠀⠀⠃⠀⠀⠒⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣶⣖⣒⣒⢲⣶⢶⣶⡶⠶⠶⠶⠶⠶⢶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣾⡷⠀⠀⣶⣶
⠀⠀⣀⡀⠀⠤⠤⠀⠐⠒⠒⠂⠄⠠⠐⠀⡀⣿⣿⣿⣿⣿⡿⠿⠷⡀⠀⠀⠠⠀⠤⠤⠤⠤⠄⠐⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⣿⣿
⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠊⡠⠘⠦⠴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⠔⢈⣆⠀⠀⠀⠀⠀⠠⠿
⠀⠀⠘⣿⠀⠀⠀⡀⠠⠐⠊⠉⠂⠜⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⣤⡤⡀⠀⠀⠀⠂⢀⣠⣾⣿⡄⠀⠠⣤⡄⠀⠀⠀⠀⢸
⠀⠀⠐⠟⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⡇⢹⠀⠔⠀⠀⠀⣿⡄⠀⠀⠀⠀⢿⡀⢀⣘⠿⠷⠾⠿⠿⠿⠿⢓⣀⠀⣼⠃⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⣀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣾⣿⢱⠀⠀⠀⢿⡸⣄⠀⠀⠀⠈⠳⢜⡿⣿⣶⣤⣤⣤⣴⣾⠿⣫⠞⠁⠀⠀⠀⢀⡞⣾
⠀⠀⠀⢰⡟⢸⡟⢸⡏⠀⠀⠀⠀⠀⠀⡼⢿⣿⣿⡿⢠⠀⠀⠀⠘⣷⣌⠳⣄⣀⠀⠀⠀⠈⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⠴⢋⣼⠏
⠀⠀⠀⠠⡧⠸⡧⢸⡆⠀⠀⠀⣴⠀⠀⠐⠊⠉⠉⢷⣾⣀⣀⡄⠀⠘⢿⣷⣦⣭⣍⣉⢻⢿⠖⠢⠴⣿⠢⠄⠒⠞⡟⣉⣉⣩⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⡀⠀⠀⢰⠛⢷⠀⠀⡀⣹⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠠⣄⣠⠀⣂⠤⠤⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣴⡄⠀⠀⠀⠀⠈⣾⡷⠀⠀⠈⡠⡈⠀⠀⢞⣷⠅⠀⠀⠀⠠⣾⢟
⠀⠀⠀⠀⠀⠀⠀⢸⠿⡃⠠⠤⢼⣿⡷⣺⣷⣷⣶⣿⣷⠶⠶⠶⠞⠉⠁⠀⠀⠀⠀⠀⢿⡇⠀⠀⠈⠀⠁⠀⠀⢈⣿⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⣀⠴⠚⢻⣿⡿⠿⡿⠟⠋⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⠘⣿⡇⠀⠈⢹⡖⠀⠀⠀⡰⣿⠁⠀⠀⠀⠀⣤⡄⠀⢰⣶⢷
⠀⠀⠀⣰⠁⢠⠞⠉⠣⡑⡠⠃⠀⠀⠀⠐⢎⠀⠐⢎⠀⠒⢎⠀⠀⠸⣝⠀⠀⠀⢸⢹⠀⠀⠀⠱⣧⣀⣸⡇⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠉⠁
⠀⠀⣴⠃⡀⢸⠀⠀⠀⢀⠤⢤⡀⠀⠀⢀⠤⢄⠀⠀⠀⠀⠀⠀⠀⢡⡸⢄⣀⣠⠞⡸⠀⠀⠀⢰⣹⣿⣿⡰⠀⢀⡀⠾⠀⢀⠀⠀⣀⣀
⠀⠈⠛⠉⠉⠉⠁⠀⠀⡏⠀⠀⠉⠩⠍⠁⠀⣼⣧⠀⠀⠀⠀⠀⠀⠀⠙⠺⠛⠻⠊⠁⠀⠀⠀⠀⠛⠛⠛⠃⠀⠈⢛⣈⣈⠋⠀⣹⣿⣿
⠀⠀⣠⣤⣀⠀⠀⠀⡸⠀⠀⠤⠂⠀⠀⢲⣾⣿⣿⡄⠀⠀⠀⣠⣤⡀⠀⣠⣴⣤⣀⣀⣀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠉⠀⠈⠀⠀⣽⠿⣿
⠀⠀⣿⡏⠉⠑⢤⣀⣧⣤⣀⣀⣤⣴⣾⣿⣿⣿⣿⣷⣀⠴⠋⠉⢻⣿⢰⡿⣿⣿⣿⣿⢧⣴⡄⠀⠀⠀⡸⣿⣿⣇⡀⠀⠀⠀⠀⠛⠛⠛⠃
⠀⠀⢻⣧⠀⢀⣤⣈⠛⠟⠛⠛⠛⠛⠛⠛⠛⠛⠟⢋⣡⣤⠀⢀⣾⠇⠀⠿⢻⣻⠟⣁⡈⠓⠁⠀⠀⠈⢉⣿⣿⡿⠟⠀⠀⠀⠐⠃⠀⠀⠃
⠀⠀⠀⠙⢷⡘⢿⣿⠉⠲⠤⣤⣄⣀⣀⣤⣠⠤⠒⢹⣿⠟⣠⡿⠃⠀⠀⠀⠘⠋⠀⠙⠃⢰⠀⠀⠀⢰⣌⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⡅
⠀⠀⠀⠀⠀⠉⠳⠬⣦⣅⡤⠂⢄⠀⠀⠤⠀⣄⣼⣾⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰⣿⣿⡏⠁⠀⠀⢹⣿⣿⣛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡞⢱⣟⣶⣤⠀⢀⣴⣺⣷⡎⡇⠀⠀⠀⠀⠀⠀⠀⠋⠰⣀⠀⠀⠀⠀⠀⠀⠀⢐⠓⣿⣿⣿⡷⠀⠀⢸⠿⠿⠿⠯⠿⣿
⠀⠀⠀⠀⠀⡀⠀⠀⠙⣤⣉⠛⠛⡅⡈⠛⠋⣉⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⣠⠰⠀⠀⠠⣘⣿⣿⣧⡄⠀⠀⢹⣶⣦⣤⣤⣶⣿
⠀⠀⣀⠴⣯⣬⡷⢄⠀⢸⡉⠳⠐⠀⠐⠀⠛⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠐⠀⠂⠐⠀⡀⠀⠀⢡⣿⣿⡟⠋⠀⠀⢸⣿⣯⣿⣿⣿⣿
⠀⠀⠘⡾⣭⣭⠷⡚⠀⠈⠳⣤⠁⢤⠄⢡⡤⠞⠀⠀⠀⢰⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣾⠃⠀⠠⠮⣿⣿⣿⡆⠀⢠⠟⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠐⠚⠓⠒⠀⠀⠀⠀⠈⢧⣀⣀⠞⠀⠀⠀⠀⠀⠀⠈⠿⣦⣤⣄⣀⣀⣠⣴⠞⠛⠋⠁⠀⠀⠈⠰⠿⠿⠏⠀⠀⠀⢀⣤⣤⣴⣴⣤⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣆⡄⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠐⣿⣧⠴⠀⠀⠀⠀⣴⣿⡶⠆⠀⠀⠀⠀⠀⢈⣽⠿⠋⠀⠀⠀⠀⠀⠺⣿⣟⠷⠃
⠀⠀⠀⠀⠀⠀⡠⠖⠋⠁⡀⠀⠀⠀⣠⡒⠉⡀⠀⠀⠀⠀⠀⣰⣇⠀⠀⠀⠀⡀⠀⠀⣰⣯⡉⢛⠀⢀
⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⢠⡀⢠⣿⣧⢨⠀⠀⣀⠀⠀⠀⣿⣿⡄⠍⠃⠀⣹⡀⢰⣿⣧⠀⡾⠀⠌⣶⡀
⠀⠀⠀⠐⠒⠒⠛⣿⣷⣞⠓⠚⠳⡞⠛⢳⣶⣶⡄⠛⠋⠒⠒⠚⠻⣿⣾⣷⠖⠻⠗⠚⠛⢻⣾⣇⣠⡐⠛⠁
⠀⠀⠀⠀⢀⣀⣀⣿⣿⣏⣤⣀⣀⣇⣀⣿⣿⣻⣹⡄⢤⣀⠀⣀⣰⣿⣿⣧⣇⢀⡁⠀⠀⣾⣷⣿⢻⠄
⠀⠀⠀⠀⠀⠀⣼⣿⡟⢹⣿⡏⠉⡏⢹⣿⣿⠉⠉⠛⣺⣿⡅⠀⢹⣿⡟⢻⡇⠈⠁⠀⣸⣿⣿⠿⣿⣷
⠀⠀⠀⠀⠤⢤⣿⣿⠧⠬⣿⡇⠴⡧⢼⣿⣿⠤⠤⣴⣿⣿⣷⠤⢼⣿⡧⢼⣏⣴⡦⣴⣿⣿⠇⠠⢹⣇
⠀⠀⠀⠀⠀⣸⣿⣿⡄⢀⣿⣗⠀⠁⠸⣿⣿⡄⣸⣿⡇⠻⠋⠀⢸⠿⠀⠈⣿⣿⡀⣿⣿⣿⠀⠀⢸⡏
⠀⠀⠀⠉⢉⣿⣿⣿⠉⢹⣿⣿⠈⡏⢹⣿⣿⡋⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⣿⣿⡋⣿⣿⡏⠉⠉⠉⡇
⠀⠀⠀⠠⠼⠿⡿⠛⠤⠼⠿⡿⠶⡧⠼⠯⢿⠻⠤⠤⠤⠠⠤⠤⢿⡇⠤⠤⠿⠿⠷⢿⡿⠅⠀⠤⠤⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠁⣀⠀⠀⣀⣀⣀⣀⣀⡀⢀⠀⣀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀
⠀⠀⠀⠉⠁⠉⠀⠈⠁⠈⠁⠀⠈⠀⠉⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⢀
⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢄⡀⠀⠀⠼⠁⠀⡳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠀⠀⠀⢐
⠀⠀⠀⠁⠀⠀⡀⣤⣀⡀⠀⣡⣄⣀⠐⠀⠀⠀⠀⠈⠀⠠⠘⠛⠈⠋⠀⠀⠀⠀⠀⠂⠐⢀⠂⠐⠶⠢⠶⠀⠄⡠⠀⠤⡀⠀⠀⠘
⠀⠀⠀⡆⠀⠁⢈⠀⢈⠀⠀⡀⠀⣸⠐⠂⠐⡆⠀⠲⠖⠶⠤⢠⢀⡤⠴⣀⡀⠀⠀⠀⢀⡀⢋⣉⠛⠋⠛⣛⡛⠀⠀⠀⠈⠂⠀⢰
⠀⠀⠀⠀⠀⠀⢸⠦⢼⠀⠀⡇⠴⢸⣿⡷⢸⠉⡇⠸⠑⢀⠀⣼⠈⢹⠈⠯⠝⠀⠀⠀⠛⠛⠀⠾⠀⠉⠁⠺⠄⠀⠀⠐⠛⠃
⠀⠀⠀⠀⠀⠀⠘⠀⠘⠀⠀⠃⠀⠘⠀⠀⠈⠀⠁⠀⠀⠘⠀⠙⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⣿⢟⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣤⡏⣿⡉⠠⠀⠀⡄⠀⠀⢀⣴⢆⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢌⠑⡿⠀⠀⠀⠀⢿⣦⣶⣾⣿⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠏⠀⠀⠀⠁⠀⠀⠀⠤⣾⣿⣿⡗⠂⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠒⠶⠆⠖⠀⠀⠠⢤⣤⣴⣿⣿⣿⣿⡀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢢⣄⠀⠀⠀⠀⠀⠁⣿⣿⣿⣿⣿⣷⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠙⠢⣄⡀⠠⠀⣿⣿⣿⣿⣿⣿⡀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⢰⣆⣉⠳⢤⣿⣿⣿⣿⣿⡏⢹⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣠⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣁⣉⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡟⠀⠀⠈⣋⣝⠓⡿⣽⣿⣿⣿⣿⣿⣿⣿⣯⢻⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⢹⣿⣀⣿⡿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠞⠁⢳⡀⠀⣸⣿⣿⣿⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡠⡀⠀⠀⠞⢀⠀⠸⡇⣀⣿⣿⣿⣾⣎⡿⣯⣽⣟⣿⣻⣿⣿⣿⡿⣿⣿⣿⣿⢀
⠀⠀⠀⠀⠀⢀⣴⣾⣥⡟⠀⠁⠀⣾⡁⠀⠐⠚⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢹⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⠀⠐⣯⠱⠄⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄
⠀⠀⠀⠀⢐⣿⣿⣿⣿⠛⣿⣾⠀⢸⡄⠀⢱⢌⣹⠋⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣷
⠀⠀⠀⠐⠀⠚⠹⢿⠁⠀⠹⣿⡷⠺⢻⣕⠁⢸⣿⣎⣾⣿⣉⣻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣟⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠙⣿⡀⠀⠿⢿⣿⣿⣿⣧⣿⣉⢩⣿⣿⣿⣿⣿⣿⢿⣿⣿⠟⠿⠿⠿⠿⠿⠟⠻⣿⣷⠊⠙⠿⠆
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⢿⣀⣴⣸⣿⣿⣿⣿⠁⢹⡏⡉⠻⢿⠫⣧⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣧⣿⣿⣿⣿⣷⣟⠁⠀⠀⠈⠓⠟⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⢿⣿⣧⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⡆⠀⠀⡀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠤⠀⠀⠀⣀⡀⣀⡄⠀⣀⣠⠴⡿⢁⠐⢯⡥⠆⠐⣧⡶⣋⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠒⠐⠒⠆⠐⠐⠠⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡄⢸⣧⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢆⣰⣅⣴⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⢹⢦⠃⢸⢇⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⠠⠤⠤⢀⡈⠛⢿⡃⠁⣗⠄⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠃⠈⠉⠀⠀⠀⠏⠀⠃⠚⠁⠑⠲⢄⡰⠉⠁⠠⠆⠀⢀⠘⢀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠁⢺⡁⠀⡋⠠⠂⠀⠀⠱⠂⠉⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢤⡁⠀⣨⣁⠌⠉⠉⠀⠀⠀⠈⠈⠁⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢸⡗⡀⠁⡀⠂⠀⠤⠀⠀⠀⢠⠀⠀⠀⡀⠀⠀⠀⠀⠰⠀⠀⡄⡁⠀⠀⠀⠀⢿⣇⠈⡟⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠃⢸⡏⠁⠀⡇⠇⢀⣡⠇⠀⠀⠀⠀⠀⠀⠋⢀⡀⠀⢀⠀⠀⠀⢸⡇⠀⠀⣦⠀⠀⠉⠤⡇⠀⠀⠀⠀⠀⠀⠻⣶
⠀⠀⠀⢀⡀⠀⣤⢴⡤⠀⢰⡈⢺⡿⠴⡤⢀⡆⠀⢹⡇⡀⠄⠀⠀⠀⠀⠀⠀⣲⣿⣾⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣷⠀⢀⢠⠈⠁⠉⠉⠛⠒⠒⠂
⠀⠀⠀⢰⢻⢠⣸⠞⠁⠀⠀⠙⢲⡇⠀⡿⠋⠀⠔⢉⡁⠀⣀⠀⠀⠀⠀⠈⡭⣻⡿⡻⣿⣷⠀⠀⠀⠀⠚⠃⠀⠀⠘⣶⣿⠘⢾⣧⠏
⠀⠀⠀⠈⢻⣀⡩⠤⣀⠰⣶⠀⢸⣇⠀⡇⠀⠀⢔⠉⠑⢤⣹⠆⡴⣦⣴⡅⢈⣿⢿⡇⠹⡗⠀⠈⠀⠀⠀⠀⠀⠀⠸⠛⠁⠀⠀⠀⠀⢀⣐⣸⣷⠄
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠱⡇⠺⡸⣿⣾⠷⠀⠀⠈⠳⣦⡜⠚⢤⠉⠀⠁⣰⣾⣿⡻⠀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡴⠉⠀⠀⠱⡄
⠀⠀⠀⠀⢀⣀⣿⣶⣤⣼⠷⠞⠁⠉⠁⠀⠀⠐⠀⠀⠀⢇⠀⣈⣷⣶⠾⢻⣿⣿⣷⣧⣀⣠⡧⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢰⡟⣉⣇⠀⠀⢀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠄⠚⠻⠿⠟⠂⠁⠀⠀⠘⠓⠈⠛⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠻⠟⠉⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣟⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣼⣿⣿⢿⣷⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣌⣛⣿⣿⣿⣶⣿⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣴⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣤⢱
⠀⠀⠀⣀⣠⡤⢤⢴⣾⣯⣀⠀⠈⠀⢀⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠚⠋⠁⠀⣾⠾⠿⣿⣿⣿⡀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⡴
⠀⠀⠀⠀⠀⢀⣿⡇⠀⠃⠹⣿⣧⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠋⠈⠁⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣾⣿⣶⣶
⠀⠀⢀⣏⡠⠈⢿⣿⣶⡄⠀⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣾⣿⣿⣭⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡶⠋⠵⣿⣿⣿⣿⣿⠿⡆
⠀⠀⢸⣽⣿⠀⠸⣿⣿⡗⢦⠀⢸⣿⡿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣀⠿⡆⠀⢀⣀⣤⣤⣶⣿⣿⣿⣿⣀⣤⠄⣿⣿⣿⣿⣧⣀⣢
⠀⠀⢻⣿⣾⣄⠀⣿⣿⣷⡾⠷⠚⣿⡄⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡵⠃⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⣻⣿⡿⣿⣿⣿⣿⣇
⠀⠀⣿⡿⠉⠛⢿⣿⣿⡏⠀⠀⠁⠹⣷⣶⣾⢛⣡⣾⣿⣿⣿⣿⣿⣿⢋⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⢿⠿⣿⣿⣷⣬⣿⣧
⠀⠀⠏⠀⠀⣤⣾⣿⣿⣿⣄⣠⣤⠀⢹⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⡿⢿⣿⣏⣿⣿⡇⢸⢿⣿⣿⣿⣿⣿⠟⠛⠿⣾⣷⣿⣾⣿⣿⣿⣿⣦
⠀⠀⣠⣀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣼⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣄⣿⣿⡿⣯⣿⠿⠃⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⣿⣿⡆
⠀⠀⣼⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠃
⠀⠀⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡛⠿⠟⠈⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢀⣀⣠⠄⠂⠁⡀⠉⠁
⠀⠀⢹⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣾⣿⣿⣿⡿
⠀⠀⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⢻⣿⣿⠁
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠺⣿⣿⣿⣰⣿⣿⡏
⠀⠀⣿⣿⣿⣿⠏⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣰⣿⣿⣿⣿⣿⡏
⠀⠀⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⠃⢀⡵⠋⢸⣿⣿⣿⠏⠉⣽⣿⣿⣿⠃
⠀⠀⣿⣿⣿⡏⣸⣿⣿⣿⠟⠀⠀⠀⠈⠉⠉⠀⠀⠁⠀⠀⢸⣿⣿⣿⠀⠐⣿⣿⣿⡇
⠀⠀⣿⣿⡿⣽⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⡿
⠀⠀⠛⠛⠃⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡾⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⣀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠴⠆⠀⢀⣤⣿⣷⢿⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠲⣧⣿⣾⣿⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⠯⣸⣿⡁⠀⣱⣄⣀⣀⡀⠴⣾⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⣾⣿⣿⣿⡏⠀⠈⣿⣿⣿⡄⣈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣴⣿⣿⣿⡟⠀⠀⠀⠘⣮⡿⢿⣟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠁⠀⠀⠀⠀⠈⢇⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣿⣁⡟⢀⠀⠀⠀⣥⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠰⠷⣿⠿⣷⢻⣄⠁⠀⣡⣠⣦⡄⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡎⠹⣷⠟⠻⢁⡄⠿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠿⣷⣀⠀⠀⠀⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢻⡀⠀⢸⣳⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣗⣿⣥⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠹⢿⡇⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠾


⣿⣿⠛⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⡆⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣯⣝⣟⣻⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣬⠀⣽⢸⠉⢩⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣟⠀⢛⠛⠄⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣹⣿⣌⠉⠉⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣾⣿⣿⣾⣿⣿⣿⣿⡿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢀⠠⣤⣿⣿⣿⣿⣧⣻⣿⠛⢛⣯⣴⣿⣿⣿⣧⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣸⣸⡧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣻⣿⣿⣿⣿⣿⡆⣦⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠛⠛⢹⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢠⡀⢠⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⣿⢉⣿⣿⣿⣴⣼⣿⣿⣿⣿⣿⣿⡇⣷
⠀⠀⠀⠀⠀⡑⠁⠆⠀⡼⣿⡾⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⡿⠟⠘⠷
⠀⠀⠀⠀⣼⢃⠀⢁⠀⠀⣿⠇⠀⢹⣿⣿⣿⣿⣿⣿⣿⠻⣿⣾⣿⣿⣿⣿⣿⡿⠀⠀⢿⣿⡿⠿⣒⢲
⠀⠀⠀⠀⠹⣯⣿⣤⣄⠦⠘⠃⠠⣼⣟⣭⣭⣭⣿⣿⣿⣅⣧⣿⣥⣿⣿⣿⣿⡇⠀⠀⣿⡿⢏⡙⣡⡿
⠀⠀⠀⠀⠀⠹⣿⣿⣶⣿⣾⣳⡖⢟⠻⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⠟⢻⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠉⢉⣿⣯⣛⣭⣿⣶⣿⠉⠀⠰⢾⣿⣿⠩⣿⣿⣿⣿⣿⣿⣿⠇⠀⢰⣥⠴⣟⣽⠇
⠀⠀⠀⠀⠀⠀⣴⡿⣽⣿⣿⣿⣿⣿⣿⣧⠠⢅⣐⣾⣿⣻⣟⠻⣿⣿⣿⣿⣿⡀⢠⠏⢘⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣾⣯⣼⣿⣿⣿⣿⢿⣥⣸⡀⠈⠿⠁
⠀⠀⠀⠀⠀⠠⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣿⡷⠘⠈⡀
⠀⠀⠀⠀⠀⢀⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠄
⠀⠀⠀⠀⠀⠻⣿⡆⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣧⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡟⣋⣠⣴⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠚⠛⠛⠛⠛⠛⠛⠛⠛⠈⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⠶⠶⠶⢶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⠇⣷⢰⡗⢰⡇⢿⣤⣤⣤⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀⣠⡶⠿⠶⢾⣟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠟⠈⠃⠘⠃⠀⠀⠀⢸⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠀⠀⠀⠀⢻⡟⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠻⣶⡘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠶⠶⠶⠶⠶⠀⠀⠘⢷⣤⡾⠁⠙⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠃⠀⠀⠀⠀⢸⡇⠀⢀⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⣤⠶⠶⠶⠶⠶⠶⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣭⡿⠷⠶⠶⠿⣿⠛⠛⠛⠛⠛⠛⣿⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⣀⡀⠀⠀⠀⢀⣤⡶⣿⠋⢿⡀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠸⠿⠟⢀⣤⠾⣯⠁⠀⠸⣧⣘⣧⠀⢀⣴⠿⠷⠶⠶⠶⠶⠶⠾⠋
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢠⡾⠁⠀⠙⠷⣦⣄⣈⣿⣵⡾⠛⠁
⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⢠⡟⠀⠀⢠⣶⣦⠀⠀⠀⠀⢹⡄
⠀⠀⣾⠁⠀⠀⠈⠛⠃⠀⠀⠀⠀⠘⣷
⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡼⠃
⠀⠀⠈⠛⠻⠶⠶⠟⠛⠛⠛⠋⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣻⡛⡛⢻⡟⣻⢿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⢁⠴⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡉⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⡠⠞⢁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠙⠢⣈⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⡠⠞⢁⡴⠋⡠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠳⢄⠉⠢⣀⠀⠀⢀⡴⠋⣠⠖⢁⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠙⠦⣈⣳⠞⢁⡤⠊⡡⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡶⠋⣀⠔⢋⡤⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢁⡠⠚⣡⢔⠙⢦⡈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⢥⡴⢋⡠⠊⠀⠀⠙⠦⡉⠲⣄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡩⢊⠁⠀⡱⠃⠀⠀⠀⠀⠀⠀⠈⠲⣌⠗⠁⠬⡳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠊⠁⠘⠒⣁⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠐⢌⡢⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡞⡑⢄⢀⡠⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡁⠀⠀⢈⢶⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠯⠽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⡶⣴⠋⠂⠣
⠀⠀⠀⠀⠀⠀⣀⡤⢼⢀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠇⠀⢀
⠀⠀⠀⠀⢀⡜⡁⡰⣡⣤⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠠⠘⠿⡒⢦
⠀⠀⠀⣠⢊⠎⠰⠁⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠊⠊⠢⠳⡀
⠀⢀⡴⠁⠂⠀⠁⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠈⠄⠑⡙⢄
⡠⠊⡜⠀⠄⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠠⢨⠳⡀
⠑⢄⡁⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠁⠀⠃⢘
⠀⠀⠈⠓⠢⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⣀⡠⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠘⢛⢃⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠁⠀⠒⢒⠓⠓⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠈⢉⢁⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠂⠀⠐⠚⠘⠘⣤⡀⠀⠀⠄⢀⣀⠀⠀⠀⠀⢰⠶⢀⣠⡴⠄⠁⠀⠀⣍⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⡤⠆⠲⡄⠀⣌⡓⠀⠤⠔⠒⠒⠉⠉⠉⠀⠈⠉⠉⢉⠙⢒⠤⢤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⡀⠀⢀⠀⣀⣀⣷⠀⠁⠀⠀⠀⠀⠀⢀⡀⠄⠤⠶⢶⡟⠛⠁⢀⣠⣧⡬⢄⣀⠈⠁⠒⠆⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⡀⠀⠀⢰⠀⠀⠀⠀⠨⠁⢀⡾⠀⠀⠀⠀⢀⠀⣈⣒⡤⢓⣂⠄⢐⡀⠄⢈⡀⠠⠥⠀⠠⢉⡉⠥⠄⠌⠐⣀⡠⠽⠧
⠀⠀⠀⠀⠀⡍⠀⠀⠐⠊⣳⢸⠀⠈⡀⠈⠀⠀⢈⢉⠉⢰⠒⣶⣌⣘⠳⡖⡭⠐⠚⠭⠤⣍⡉⠚⢯⠥⢞⡩⠤⣀⠈⠠⣄⣲⣤⠶⠍⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡀⠀⠀⢀⣈⣁⣸⣄⣌⠂⠤⠤⠤⠤⠤⠆⠹⠒⠀⠀⢆⠻⡇⢈⠁⠀⠈⠁⠀⠈⠑⠒⠠⠔⠠⠀⠐⠀⠈⠁⠤⣐⠤⣐⡢⢀⣀⣵⠠⠔
⠀⠀⠀⠀⠈⠁⠒⠒⢺⠙⠉⠐⠒⢜⠂⠀⠀⠀⠀⢀⠁⢾⣣⠄⠀⠰⣒⡟⢛⠾⠛⠴⠖⠢⠾⢷⣶⣦⣴⣄⡀⠇⠴⠖⢒⣉⡁⠥⠔⠒⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⠥⠬⠼⠦⠤⣀⣀⣀⣠⡴⠋⠉⠉⠋⠉⣕⠖⠅⠿⠁⠀⠀⢀⠀⠀⠈⠈⠉⠓⠑⠠⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠉⠀⠀⠈⡍⠁⡟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠐⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⣠⣶⣯⣽⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣖⣶⣶⣶⣶⣶⣶⣲⣶⡶⠶⠶⡶⡾⠿⣿⣶⣆⣀⣀⡀
⠀⠀⠀⠁⣇⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⡇⡇⠀⠨⡶⠋⠁
⠀⠀⠀⣇⡏⠉⠉⠉⠉⠉⠑⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠓⠒⠒⠀⠰⣇
⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣶⣶⣶⣶⣶⢶⣶⣄⠀⣶⣿⡿⣿⣷⡀⠀⠀⢀⡹⣒⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢠⡏⠀⠀⢹⡆⠘⣿⠀⢹⣿⣷⣿⣿⣧⠀⡰⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠣⣄⣀⡼⠃⢸⣇⡂⠀⢿⣿⣿⣿⣿⣇⢇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⢽⣷⡘⣿⣿⣿⣿⣿⡞⣟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⢻⣿⣿⣿⣿⣿⡽⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡌⣿⣿⣿⣿⣿⣷⢱⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠸⣿⣿⣿⣿⣿⣧⢣⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⢻⣿⣷⣼⣿⣿⡎⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣬⠯⠭⠭⠗⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣠⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⣤⣺⣟⣟⣻⣿⣿⣿⣭⣭⣽⣿⣟⣋⣀⣀⣀⣀⣐⡷⠶⠿⣗⡤⠤⢴⣻
⠀⠀⠀⢀⣀⣀⣤⣴⣶⡶⠶⠛⢻⣿⡿⣿⣿⣿⣿⣿⣿⡉⠉⠉⠉⠉⠉⠁
⠀⠀⢿⣿⣿⠿⠟⠋⠁⠀⠀⠀⣿⣿⠇⠀⠀⠀⠘⣿⣿⣧⡀
⠀⠀⠘⠈⠁⠀⠀⠀⠀⠀⠀⠀⠻⠿⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣇⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⡿⠟⠛⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠋⠛⠁⠀⠀⠐⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⣶⣦⡀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠓⠶⣤⣄⣀⠀⠀⢰⣄⠀⠀⠀⠀⠘⣿⣿⣿⠢⣀⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠀⢠⣼⣿⡄⠀⠀⠀⣀⢻⣿⣿⠀⠈⠻⠒⣿⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⣀⣹⡇⠀⠀⠀⠈⠀⢹⣿⡇⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⡇⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⢃⣀⡤⠀⠠⣦⣌⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠘⡇⠀⣄⡰⢇⡀
⠀⠀⠀⠀⠀⢀⠐⣀⢀⠀⠀⠀⠀⠀⣤⡤⠈⠀⠁⠀⠀⠐⢿⣿⠀⠀⠈⠀⠀⠀⠀⠀⠱⠄⠰⠃⠀⠈⣇⣼⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⠀⡼⠋⠀⢰⣤⣤⣔⣤⡄⠈⢿⡀⠀⠀⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠝
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⢀⣼⡿⠛⠙⠃⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣔⡛⠋⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡝⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⢀⣀⣀⠀⠀⢲⣶
⠀⠀⠀⠀⠀⠀⠀⠺⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡈⣿⡧⣶⣶⣾⠿⠛⠋⠚⠒⠀⠀⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⡀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣶⣶⠿⠿⠿⠟⠛⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⢀⡀⢠⣴⡶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣥⣴⡶⠿⠿⠛⠋⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠻⡃⣴⡄⠀⢀⣭⣾⡶⠿⠈⠀⡈
⠀⠀⠀⠀⠳⠤⠤⠔⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣶⣴⣶⠟⠛⠉⡀⠉
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢾⠏⠉⠂⠄⠀⠁
⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠀⠀⠀⠀⠈⣶⣶⡐⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⢹⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣥⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣼⠇⢻⣧⣷⠀⠀⠀⠀⢀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠾⣿⣿⣿⣄⠀⠀⠀⣾⣿⡿⠿⠦⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣿⣧⡄⠙⠷⢿⡶⠤⣤⣀⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣹⣏⣻⡟⡗⠀⡅⠀⠀⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢿⡟⢡⣮⣇⣀⣉⡰⠀⠉⣉⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⡟⢈⡿⣿⣿⣍⣋⠒⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢡⡼⠟⠿⣳⣾⣿⣿⠿⢿⣾⣿⠀⠠⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠘⣿⣇⠀⠀⣤⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡿⠿⣿⣿⣿⣷⣶⡿⠛⠳⠀⠀⠙⢿⣧⣘⠛⠿⠶⠴⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⢦⡀⢀⡀⢤⣤⣦⣴⡄⢀⠰⢀⣦⡈⠻⡿⠋⠻⣶⡾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠘⠛⢿⣿⡿⢋⣼⠃⠈⠀⠈⣿⣽⠿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡄⠀⠲⠶⢾⣯⣼⡿⠉⠀⠀⠀⢴⡟⢠⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣣⣿⡇⠀⠀⠼⣿⣿⠟⠁⠀⠀⠀⠀⢹⡅⢸⡿
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠃⢠⣿⣧⣤⣿⠇⠀⠀⢨⡿⠁⠀⠀⠀⠀⠀⠀⢻⡇⢀⣤⡞
⠀⠀⠀⠀⠀⢦⣈⣻⣶⣶⠇⠀⠀⢸⣿⣿⣏⠤⣠⣶⣠⣾⡇⠀⠘⠀⠀⠀⠀⠀⢸⣧⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⠀⢀⠀⠸⣿⣿⠏⣼⣿⣿⣿⣿⡇⠀⠐⠀⢂⠀⠀⠀⠈⣹⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡿⠀⠀⠃⠀⠀⠁⠈⡀⣿⣿⣿⣿⣿⡇⠈⣿⣻⣤⣤⠄⢀⡆⢸⣾⣿⢴⡂⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠁⠀⣤⣾⣯⡇⢻⣿⣿⣿⣿⣿⠀⠸⣿⡯⠉⠀⣼⡟⠀⢻⡟⠉⢻⡏⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⣾⣿⣿⠛⢹⣿⠿⢿⢿⣿⢼⣿⣧⠀⠈⠱⠂⢀⣉⣀⠀⣾⡄⢀⣾⠇⢰⣹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠁⠀⢼⣻⡄⢙⣻⣿⣦⣼⣿⣧⠀⠀⠀⢺⣿⣤⣿⣿⣤⣿⠃⠀⠀⠙⣿⣦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⢠⣼⢃⣵⠀⠉⢻⣿⣿⠿⠿⢀⡀⠀⠀⠙⠻⠛⠻⢿⣿⣦⣀⠀⠀⠈⠿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡄⣌⠻⣿⡧⢠⣤⣾⣿⣟⣣⣄⠈⢿⣷⣶⣄⠠⣶⣂⠀⠙⠛⢿⣗⣀⠲⣄⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣽⣮⠞⠁⠀⠀⣿⣿⣿⣿⣓⣼⠀⠀⠉⠉⠉⢿⣿⣿⠀⠀⠀⠀⠿⠋⢹⠿⣦⡴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡵⠛⠁⠀⠀⠀⠀⠈⠙⢿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠉⣿⣆⡀⠀⠀⠀⠀⠈⠀⠈⠁⠜⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡋⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠙⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡤⣡⣄⡀⠀⢀⣼⡦⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠘⢶⣶⣿⣿⣿⣿⣽⣿⣿⡟⢁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠾⠉⠉⠉⠉⠹⣧⣤⡉⠉⢿⣿⣿⣿⣍⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠆⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣀⠒⠲⢤⣤⣤⣠⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢻⣿⣿⣿⣶⣤⡉⠙⠛⠁⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣆⠀⢀⣚⣿⣿⡏⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⢠⣄⠀⠀⠀⠀⠀⣟⣿⣿⣿⣿⣿⣿⣿⣷⠀⣿⣿⣿⣿⣷⡤⠒⠉⠁⠀⠀⠀⠉⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠊⣿⣿⣦⣤⣤⣼⠉⠉⠉⠉⠉⠉⠉⠉⣉⣶⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⢂⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠉⠻⢿⣯⣭⣽⣿⣿⣭⣽⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⢠⣿⣿⣿⢻⣿⣧⣤⣀⡀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠏⠘⢹⣿⣻⠿⠟⠃⠁⣹⣿⣿⢫⣿⣿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢻⣿⡇⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⡀⢻⡇⠀⠀⠀⠐⠛⠛⢁⣉⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠉⢸⡿⡇⠀⠀⠀⠀⠀⠀⢹⣷⡶⠶⠏⠀⠙⠿⠞⠁⠀⠀⠀⣸⠟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⡶⡽⠆⠀⠀⠀⠀⠀⠀⠀⢸⣟⡓⠂⢶⡄⠀⠀⠀⠀⠀⣀⣰⠟⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠲⣿⢸⡇⣶⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠦⠀⠀⠀⠀⠀⢠⡶⠟⠁⠀⣠⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣻⣾⣧⣠⣆⡀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣄⠀⠀⣀⣤⣴⠇⠀⠀⢀⠔⡌⠀⠀⢠⠄
⠀⠀⠀⠀⠀⡀⠀⡄⠀⢨⣤⣶⣿⣿⣿⣿⡿⢷⣄⡀⠀⠀⠀⠀⠀⠈⠉⠙⣿⣿⣿⣿⣿⣷⣏⠴⠃⡸⠀⠀⠀⠘⡀
⠀⠀⠀⡆⠀⡇⠠⡗⢧⣈⡉⠈⢹⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠟⠁⠀⡰⠁⠀⠀⠀⠀⠈⠢
⠀⠀⣸⣿⣦⣷⣴⣿⣿⣿⠇⠀⠸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⢋⣠⡶⠀⠞⠀⠀⠀⠀⣠⠁
⠀⠈⠋⣿⣿⣿⣿⠟⠁⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⠶⢟⣁⡀⠀⠀⠀⠀⠀⡴⠃
⠀⠈⡄⠈⠿⡟⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⡁⣰⣶⠂⢸⠉⢆⠀⠀⣠⠊
⠀⠀⢱⡀⠀⠀⠀⠀⠀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⠿⢋⣿⠟⠹⠁⠀⠘⠀⠀⢣⡼⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣡⠬⠿⡿⠁⠀⣴⢿⡇⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢛⣿⠃⠀⡼⢣⣿⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⢈⡴⠉⣿⡆⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⣁⠤⠀⠒⠉⠀⠀⠸⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⢔⡲⠈⠐⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣂⣥⣾⡷⠛⠁⠀⠀⠀⠀⣠⣭⡒⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠊⠙⠲⢿⡛⠻⢥⡀⠀⠀⠀⢀⣤⣮⡥⢋⡴⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢐⣠⣭⣿⣶⣤⣀⠀⠀⠀⠈⠙⠲⢽⣦⣴⣾⠿⣛⣥⣺⢵⣚⣭⣶⡆
⠀⠀⠀⠀⠀⠀⣀⣠⢤⢲⣻⠿⠟⠋⠉⠘⢦⡉⠛⢦⣄⡀⠀⠀⠀⠈⣉⢲⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠈⠐⠠⢉⡛⠋⠁⠀⠀⠀⠀⢀⣠⣤⣿⣷⡶⣞⣻⣷⡆⣾⣿⣿⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢄⡀⠀⠀⣊⠕⡢⢄⡠⢶⣯⡥⢞⣛⣭⣶⣿⣿⣿⣽⡇⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠛⢷⣶⣍⡂⠀⠉⠓⢶⣾⣿⣿⣿⣿⣿⡿⢿⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠑⢤⡀⠨⡛⢿⣷⣤⣀⢸⣿⡿⢿⣻⣭⣷⣿⡿⣟⣯⣷⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠈⢹⣾⣭⣶⣿⣿⠿⣛⣭⣷⣿⡿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠢⣀⠈⡳⢦⣸⡿⢟⣛⣭⢶⣻⣿⣿⣿⡿⣟⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢮⡂⠙⠿⣟⣭⣾⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⢸⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⢀⣏⣑⣲⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡠⠤⠤⠤⠤⢤⠤⡀⠀⠀⡗⠒⢦⡀⢀⡖⠓⢦
⢻⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⠀⠿⠀⠀⠀⠀⢸⠀⠈⠀⠀⢳⠀⠀⠉⣉⠔⠉⠙⠃
⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⣷⡒⠋⠁
⠀⠀⠀⠘⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢿⡀⠀⡿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠻⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢀⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢱⠁⠀⠀⣀⣿⠓⡄⢠⡴⠖⠒⠋⠁⠀⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣅⡀⠀⠙⣉⡰⢃⣾⣿⡀⠀⠀⢀⠠⠒⠊⠒⠄⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠟⠁⣸⡇⠀⠀⡀⠀⠀⠀⠀⢈⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠇⠀⠸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠁⠀⠀⠈⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠐⠤⠤⠤⠤⠤⠄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⠿⠿⠿⠿⠿⠿⣿⡿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠿⠛⠋⠉⢀⣴⣶⡝⣷⣶⣶⣶⣶⣾⣿⠙⠛⠿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠯⠋⠀⠀⠀⢀⣴⠿⢛⡿⣳⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠈⠙⠿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⢠⣣⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠈⠻⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠁⠀⠀⠀⡀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⢀⠴⣊⣋⣩⣭⣍⣉⡛⠿⣿⣿⢻⣿⣿⣟⣻⣿⣿⣿⣿⣿⣛⣿⣆⠀⠀⠀⠀⠀⠀⣀⡻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⢰⣡⣿⣽⣿⣿⣿⣿⣿⣿⣿⣶⣬⣝⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣴⣶⣿⣿⡏⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠯⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣤⣤⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⠏⠀⠀⢽⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣷⣿⡏⠸⠻⠛⣛⣷⣭⣴⡿⣿⣿⣶⡾⠯⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⠝⣿⣿⡀⠀⠀⠈⠉⢱⠿⠛⠇⢸⢿⠿⠉⠀⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣆⠙⠿⣷⣄⠀⠀⠀⠀⢀⡤⠀⠘⣄⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢺⣿⡋⠀⠀⠀⣀⣨⣿⣿⣾⣯⣄⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠰⣾⡿⠟⠛⠛⠛⠛⢿⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠙⢦⣀⣸⣇⠀⠘⠻⠛⢃⣴⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠀⠀⠙⠿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣿⣿⡳⢤⡀⠀⠀⠀⠀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⡸⣿⣿⣿⣦⡀⠀⠀⠈⠻⣿⣿⣿⣿⡿⣻⣿⣿⣿⣿⣄⢙⣿⣶⣲⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣄⣠⡤⠴⣶⣿⢣⣿⣿⣿⠙⢷⣴⣄⣠⣶⣠⣾⡿⠋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣦⣵⡏⠀⠉⠙⣻⣯⣤⣾⣿⣿⣿⡿⠋⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣭⡛⢿⣟⣿⣿⣜⢾⣿⣿⣿⣿⣿⣿⣿⡿⣱⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣦⠹⣿⡏⡿⣿⣿⠟⢿⣿⣿⣿⡿⣼⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣧⣿⣿⠃⠁⢹⠀⢀⣿⣿⣯⡾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⢴⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠲⠚⣛⣭⣿⣶⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⣴⣾⣿⣿⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠿⢻⡏⠀⡸⠟⠁⠉⠛⠛⠓⠠⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⠘⠻⠟⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣙⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣿⣽⣯⡷⣦⡀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣽⠟⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣯⣽⣿⣿⣿⣶⣤⠶⠂⠀⠀⠀⢼⣵⣶⣶⣾⣿⣿⣿⡩⢛⣿⣿⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢿⣟⣷⣾⣿⣟⣻⣿⣿⣀⠉⠯⢷⣄⡀⠀⠠⣾⣿⣿⡿⢿⣿⣿⣷⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢠⣿⣿⣿⡟⣿⣿⣿⣿⣶⣿⣷⣤⣴⣿⣿⡄⡠⠿⠿⠛⠀⠀⠤⠮⣿⡿⢿⣿⣿⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣇⣩⣿⣿⡟⠋⠁⠘⢿⣾⣿⣿⡙⠻⢿⣿⣿⣧⠀⠀⠀⠀⠀⠒⠒⢾⠟⢛⣫⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣹⣿⣿⣿⣿⣿⡟⠀⢀⢀⡀⡀⣻⣿⣻⣷⣦⣄⡉⠉⠫⡑⠀⠀⡀⣀⣛⣭⣹⠁⣼⣾⠁
⠀⠀⢀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⠁⣴⡹⣿⣿⣿⣿⢿⣷⡿⣿⣿⣿⡶⠀⢻⠟⠟⠛⠻⣿⣿⠏⠀⢻⣿
⠀⠀⠻⣍⠁⠒⢶⢿⣿⣷⣿⣿⣿⣿⡿⠋⠀⠟⣿⣬⣿⣿⣿⣼⣿⣧⠈⠉⢺⡄⢿⣿⡇⠀⠀⠀⢀⠋⠀⠀⣿⡟
⠀⠀⠀⠀⠉⠒⠠⡿⠿⢿⣿⣿⣿⠟⠶⠀⢀⠀⣹⣿⣿⣿⡿⠛⠿⣿⣷⣀⣻⠧⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⣽⠁
⠀⠀⠀⠓⠂⠂⠀⠒⠒⢚⡻⣿⠿⣿⣷⣶⣿⠓⠚⠟⢻⣿⣿⡿⠛⠻⠛⠟⠁⠠⡾⣿⣿⣿⣿⣿⣿⡿⠿⣿⡿
⠀⠀⠀⢀⣀⣠⣤⣤⣄⣼⣿⣿⠀⠀⠙⠂⠉⠻⢿⡶⠾⢿⡿⠀⠀⠀⠀⠀⢲⣆⠀⠹⣿⣿⣿⣿⣿⣶⣠⣽⡇
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠛⠋⠀⠀⠀⠀⣴⣶⣶⣾⣿⠟⠛⠀⠀⠀⠀⠀⢸⠿⡆⠀⠀⠛⠛⠛⠛⠛⠛⢿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠓⠒⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⣿⣿⡿⠿⠾⣤⣶⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠈⠉⠛⠋⠉⠻⠶⢶⣦⣭⣿⣯⡍⠉⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾
⠀⠀⠀⠀⠀⠀⠀⡀⣤⣤⣼⣭⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⣽⠆⠀⡀⣀⡀⠀⠷⠶⠷⠇⣀⣠⣴⣦⡀⣀⣀⠄⣤⠀⠴⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠛⠁⢀⣀⣩⣭⣤⣶⣶⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⡄⢀⣼⣿⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡟⣋⣉⣩⣭⡀⠀⠈⠉⠁
⠀⠀⠀⠀⣠⣶⠿⠿⣟⣚⣧⣬⣵⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢺⡇
⠀⠀⠀⢰⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣻⣟⣛⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣾⣿⣿⣿⡿⠿⠿⠟⠚⠁
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠜⠉⠁
⠀⠀⠀⠙⠛⠋⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠁⠀⢹⣿⠀⠀⠀⢠⣖⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣯⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠉⠻⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣤⣤⣾⣏⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠈⠙⠛⠛⠛⠋⠉⠉⢀⡀⢀⣠⣤⣄⣰⣶⣶⣾⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣤⣶⣶⣆⣼⣿⠀⣿⡇⣿⡟⢹⣿⠋⢹⣿⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⣿⠁⠿⠇⣿⣿⣸⣿⢷⣿⡇⢸⣿⠀⣾⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠺⠿⢿⣶⡆⣿⡟⢻⣿⢸⣿⠁⣾⡿⠀⣿⡏⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣶⣆⣸⣿⢹⣿⠇⣼⣿⠸⢿⠿⠿⠃⠀⠛⠃⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠛⠛⠛⠋⠈⢉⡀⢠⣤⠀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣾⠂⠸⠟⠀⠛⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣈⣀⣀⣤⣀⣤⣴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣶⣦⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣾⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠿⡛⢻⠛⠛⡟⠻⡿⠇⠀⠀⠀⠀⢀⣠⣤
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣷⣶⣤⣄⣠⣶⣿⣿⣿⡇
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡏⢉⣿⡇⠀⠀⠘⢻⣿⣷⣶⡞⢻⣿⣿⣿⣿⣿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣟⠘⢼⣿⡀⠀⠀⠀⠘⠟⠛⠁⠸⣿⠋⠙⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣈⣿⣿⣆⠀⠀⠀⣠⠄⠀⠀⢿⡀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣇⠀⠙⣆⠀⣨⣤⣭⣿⣶⣿⡇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⣄⢀⣿⣼⡟⠛⣙⣛⣻⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢼⣿⠀⠹⣿⣿⣿⣅⡀⠛⠛⣿⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⡇⠀⠙⠿⣿⣿⣿⣷⣶⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠏⠀⢿⣧⠀⠀⠀⠀⢹⣿⢿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⡇⠀⢠⣼⣿⡄⠀⠀⣶⣿⠃⣰⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀
⠀⠀⢀⣴⣾⡿⣟⣛⣫⣿⣿⣿⡇⠀⣿⣿⣿⣿⣄⠀⠘⠃⣰⣿⣿⣧⡸⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⣰⡿⠛⠁⠀⠹⣿⣿⣿⣿⣿⡇⢠⣿⣿⣿⣿⣿⣧⡀⠀⣹⣿⠃⡼⣳⣿⣿⣿⣿⣿⣿⠏⣠⣿⡿⢦⡀
⢰⣿⣷⡀⠀⣿⣄⠹⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣾⣷⣾⡿⠁⣪⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⣷
⣿⢿⣿⢳⢠⣿⣿⣷⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⡿⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣸⣿⣇
⣿⣠⣿⢸⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢾⣿⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⡆
⡇⠀⣼⡟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣸⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠈⡇⠀⣿⠛⢛⣿⣿⣿⣿⣿⣿⣿⠛⣿⡇⢻⣿⡇⣾⣿⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠟⠉⠒⠢⠬⣝⡉⣩⣤⣶⣾⣿⡇⢸⣿⡇⣿⣿⣿⣷⣦⣭⣛⠿⠿⠿⠿⢟⣓⣿⣿⣿⡿⢿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⣝⠿⡏⣿⡇⢾⣿⡇⠿⣿⣿⣿⣿⣿⢿⣿⠗⠒⠙⠿⢟⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠹⠿⠁⣦⠟⠋⠁⠈⠉⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠉⠑⠒⠒⠤⠤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡚⠛⠉⠉⠉⠉⠒⠤⣔⣲⠶⣤⡀⠀⠀⠀⠀⠀⢀⣤⠞⣉⣉⣽⡶⠆
⠀⠀⠀⠀⠙⠛⠿⣯⣉⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣽⣗⡢⢄⣀⠀⠠⢀⡀⠈⠉⠓⡓⠤⢤⣀⣴⣾⣯⣷⣿⣿⣿⣷⣶⣿
⠀⠀⠀⠀⠀⣿⠒⠦⣌⡙⠓⠦⣄⣀⠀⠀⠀⠀⠀⠈⢙⣛⣛⣈⣉⣲⣤⣤⠝⠒⠋⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆
⠀⠀⠀⠀⠀⢸⡷⠦⣄⡉⠙⠲⢤⣉⡛⢒⣛⣛⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢰⡀⣻⡿⠿⠿⠿⠿⠯⠅⠀⠐⠒
⠀⠀⠀⠀⠀⠈⣗⠀⠀⠀⠀⠀⠀⠀⠉⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣷⣤⠶⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠠⣟⡒⠤⣀⠀⠀⠀⠀⣹⣟⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⢍⡓⠦⢍⣀⠒⠠⠄⠙⣩⠵⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠈⠁⠀⠉⠓⠢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣍⣙⠒⠒⠖⠋⣱⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠐⠒⠛⢣⣄⡀⠀⠈⠐⠛⠷⠆⠀⠀⠀⣀⣠⣤⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⢟⠂⠤⣀⠀⠀⠀⠀⠀⠈⠙⢋⣉⣐⢤⡶⠒⠋⠀⣻⣿⣿⣿⡿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢤⣤⣤⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠀⠀⠉⠒⠲⠶⣾⣿⣿⣿⣿⣿⣽⡇⢀⣀⣤⣿⣿⣯⣴⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠈⣉⡒⠾⣿⠿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶⠤⣀⠀⠈⢿⣿⣿⣿⠿⠟⢛⣳⣯⠛⠉⢁⣼⣿⣯⡽
⠀⠀⠀⠀⠀⢈⡀⠠⢤⣴⣿⠢⠬⣭⣿⣿⣿⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡠⠿⢤⣀⡈⠀⠒⠛⠉⢀⣴⣾⣿⣿⡿⠒⠚⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠬⢭⣛⣿⣊⣁⡒⠺⣿⡿⣿⡀⠤⢤⣴⣿⡆⠀⠀⠀⠀⢸⡏⠑⠒⠤⢍⣀⡐⠀⠀⢼⣻⣿⣿⣿⡷
⠀⠀⠀⠀⠀⢀⣑⠒⠶⣿⡟⠢⢬⣽⣿⣿⣿⠌⣙⣻⣿⣛⣧⠄⠀⠀⠀⠈⠣⠄⣀⠀⠀⠀⠉⠉⣒⣭⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠬⢉⣛⣿⣛⡁⠒⠲⢿⣯⣿⡓⠲⢾⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠒⠶⠚⠁
⠀⠀⠀⠀⠀⠐⠓⠶⢾⣿⣧⠬⢐⣒⣶⣿⣿⡅⠙⠛⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠲⢾⣿⣿⠿






⡿⢿⡿⠿⣿⡿⢿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣿⣿⣿⣿⣟⣛⣿⣿

⠀⣠⣶⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡿⣤⣀
⠾⣆⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⢀⡀⡀⢀⠀⡀⠳⢿⣬⣗⠲⠒⢢
⠻⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⡇⠁⠁⠈⣹⣿⣿⣿⣧⠀⠀⠛⠄⣟⠀⠀⠀⠀⠀⠀⠸⠸⠅⠧⠸⠀⠏⠗⠀⠀⠉⣴
⠀⢸⡛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠹⡟⠹⡿⣿⣿⡟⠛⢛⣿⡟⢋⣛⣛⣛⣛⠉⠉⠉⣿⣀
⠀⠈⠙⠙⠛⠛⠛⠛⠛⠛⠛⠛⡋⠉⠉⠉⠉⠉⢛⣿⢿⣿⣷⡀⠀⠡⠀⢰⣶⣶⣿⣿⣿⣿⣿⣧⠭⠭⠭⠵⠀⢠⣾⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣛⠿⠛⠛⠻⢿⣿⣿⣿⣿⣷⡻⡉⠋⠐⠤⠄⠤⠤⠂⠊⢏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠈⣆⡟⠀⠙⣿⣷⡘⡀⠄⢑⢀⠠⡠⠂⠰⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⣴⠙⠁⠀⢀⣿⣿⡇⢱⠀⡀⠀⠀⠈⠂⠀⠉⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠶⠖⠚⠁⠈⠙⢾⡎⢊⠂⠐⠐⠄⠀⡠⡀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡌⣆⠀⡀⠠⠀⠈⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠘⡈⠪⠀⠀⠔⠅⠌⠠⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⢡⠀⠀⡠⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡈⡆⠀⠂⠢⠀⠀⠕⠄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢰⠀⠀⠀⠀⡀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⣨⣀⣁⣀⣀⣊⣂⣀⣠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣄⡀⣤⣴⣶⣖⣠⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⠀⠙⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⣤⣦⠀⠻⣿⣿⣿⠤⠀⠨⢃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠉⠀⠀⠀⠉⠛⠉⠒⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠤⠄⠀⠀⢠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⣀⣀⣴⠟⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣤⣶⣾⡟⠛⠉⠁⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠁⠀⠹⣿⣻⣿⣧⠀⣀⣠⡞⠁⢀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠉⠉⢿⣿⣾⡿⠿⠛⠴⢁⡤⠈⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⢀⣠⣾⡿⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢀⣿⠀⢀⡀⢸⣿⣿⣿⣿⠂⠀⠀⠀⣿⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠄⠀⣼⠏⠀⠀⢠⣿⣿⢁⣿⠏⠀⠂⠂⢈⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠙⣶⠋⢹⡆⠀⠀⠿⠋⠁⣼⠋⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢰⡏⠀⡾⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢾⡿⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⣿⡃⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢹⡦⢤⡀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⠂⠁⣀⡉⠀⣀⡀⠂⠀⠀⠀⠀⠀⣀⣠⡾⠆⠀⠈⠁⠙⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⠀⠀⠉⠿⡆⠀⠄⠀⢲⣄⠿⠓⠛⠉⢀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣀⣀⣀⣡⡄⠊⠁⠀⠀⠀⢀⣠⣤⣤⣶⣒⣛⣿⣿⣤⣶⡄⢢⣤⡄
⠀⡀⠀⣾⣷⣶⣶⣶⣿⣿⡄⠐⢮⣿⣿⣿⡿⠁⠀⠀⠀⣴⣿⣿⢿⣿⣿⡿⢷⣯⡧⠿⠿⠟⠀⡟⠛⠃
⠀⠀⠀⠉⠛⠛⠛⠉⠉⠉⡿⠁⠈⡉⢉⠉⠁⠀⠀⠀⠈⠣⣄⣀⢀⡇⠀⠀⡸⢠⡇⠀⠀⢃
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⡇⣾⡇⠀⠀⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠉⡆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠃⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡅⢿⠀⠀⠀⢰⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠈⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀

⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢸⡼⣻⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠒⠃⠀⠀⠀⠀⠀⠀⣈⠀⠈⠘⣿⣿⡇⠀⠀⠀⠀⢀⠚⠁⠀⠀⠀⠀⢉⢑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣢⡌⢈⡉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇⠀⢻⣿⣿⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠓⠚⠋⢸⣿
⠀⠀⡠⠖⢒⣛⣛⣛⠒⠲⡄⠀⠀⠀⣾⣿⡿⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢷⡦⢐⣷⡆
⠀⠀⡓⠶⠤⠭⠭⢭⢤⠔⣻⠀⠀⠈⠻⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡴⡂⠀⠐⠹⠆⠀⠀⠀⠈⠢⡑⢄⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠇
⠀⠀⢏⠒⠦⠤⠤⠤⠀⢚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠾⠗⠷⠚⠛⠒⣄⠀⠀⠀⠀⠀⠀⠈⠲⡕⠄⠀⠀⠀⢸⣿⣶⣿⡿
⠀⠀⠀⠁⠒⠀⠂⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠸⡇⢠⠤⠔⠒⢲⡄⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠸⡜⢋⢛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⢹⠀⠀⠀⠀⠀⠀⢻⡈⠉⠁⠀⠈⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⠄⠄
⠀⠀⠀⠀⠀⠰⡶⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢀⣀⣰⢶⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠖⠚⠁⠀⠀⠀⠀⠀⠈⠻⣏⣀⡇
⢀⠤⠚⠆⠀⠀⠀⠀⠀⣴⣾⣿⣦⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⢹⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⢠⠑⠁⠀⠈⣳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠰⡒
⠀⠀⠄⠀⢃⠀⠀⠀⠀⠀⠈⠉⠛⢦⣘⣷⠀⡀⣠⣛⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠰⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣰⢾⣿⣿⣿⣦⠀⠀⠀⠘
⢂⠉⣀⡀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⠛⠛⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠂⠁⠀⠀⠀⠀⠀⠀⠀⢷⣺⣿⣿⣿⡿
⠀⢣⠄⢀⠈⢸⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠀⠀⠀⠀⠀⠀⠀⠀⢙⠿⠛⠛
⠀⠀⠳⡤⢤⠊⡧⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠙⠒⠋⠁⠀⠀⠀⠀⢀⠀⠰⠷⣶⣶⡶⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠁⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠘⡝⢶⡆⠀⠀⠀⠀⠀⠀⠁⠔⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠊⠀⠀⠀⢀⢄⠀⠀⠀⠀⠀⠀⠙⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠤⠒⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡉⠢⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠑⠒⠀⠉
⠀⢰⠀⠀⠀⠀⠀⠤⢂⠔⠁⠀⠀⠀⠀⠀⠀⠀⣠⠊⡀⢀⠀⠼⠂⣈⠢⡀⠀⠀⠀⠀⠀⠀⠀⡾⠑⠒⠓⠒⠬⣉⠹
⠀⠀⠓⠀⠀⠀⠀⡀⠙⢄⡠⠀⠀⠀⠀⠀⠀⡔⠁⢠⠀⠈⡀⢀⢠⠀⢑⡬⠂⠀⠀⠀⠀⠀⠀⠁⣀⣠⣶⣒⣓⣓⣶⣀
⠀⠀⠀⠀⢀⡴⠛⠷⡤⠊⠀⠀⠀⠀⠀⢠⠎⠠⠀⠀⠠⠄⠀⢠⡤⠒⠁⠀⠀⠀⣀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠢⡀⠀⠒⢀⡤⠚⠁⠀⠀⠀⠀⢀⣸⡍⢆⠀⠀⠀⠘⣿⣿⣉⣉⡉⠛⠛⣿⣿⡇⠀⠀⠀⢠⣤⡴⢦
⠀⠀⠘⢄⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡒⢈⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠊⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠈⢧⣠⠏
⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣁⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠲⢦⣤⣤⠶⠚⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣈⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣾⣿⣋⣉⠉⠉⠉⠉⠒⢒⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣍⡲⠤⣄⠀⠀⢾⣿⠟⠛⠛⠉⠉⠀⠀⢀⣠⠴⣚⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⠤⠤⣒⠋⠉⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠻⢿⣿⣯⣯⠁⢡⣶⣾⣿⣍⠀⢸⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠄⠉⠈⠁⠽⠃⠀⠙⠀⠀⠀⠀⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢶⣿⠠⠄⠀⠀⠀⠀⠠⡀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⡆⢰⠿⠆⠻⠻⠶⠶⠋⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣠⣶⣧⠸⣶⣄⠐⠂⣀⠈⠉⠈⠀⢀⢠⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠙⠦⣄⡤⠐⣧⡀⠀⠀⠀⠀⠀⢀⣷⣿⡱⠉⠀⣦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⣴⠇⠘⠓⢶⣤⡀⠀⢠⠞⣿⣄⣀⣠⣄⡠⠜⠁⣿⣇⠠⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⣾⡄⢀⢀⣘⣇⠀⢀⣠⣴⣿⣿⣿⡋⠙⠀⠀⢰⣿⢻⣶⣎⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢄⠀⣿⡇⠀⠀⠉⣿⠁⠀⢀⣸⣿⣿⣿⢿⡶⠤⣠⣾⠇⣸⣿⣿⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠀⠙⢻⡄⠈⠙⡟⠓⠉⣩⢿⣿⣿⢿⣦⡀⠀⠀⠀⣴⣿⠋⣿⠁⠀⠀⠀⠐⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⠀⢸⣷⣤⡤⠾⠒⠊⠁⠈⢻⣿⣮⣻⣿⣆⣼⣼⣿⣧⠈⠁⠀⠀⠈⡇⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣼⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡙⠇⣿⢿⣿⡟⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⣤⣿⣿⠃⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⣽⡄⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣮⡿⠃⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠰⣴⡄⠀⠸⡿⣿⢿⡀⣰⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⢀⡴⣫⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⣤⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣴⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣒⡢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣵⣿⣿⣿⣿⣷⡵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⣾⣿⣿⣿⣿⣿⣿⣿⣮⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣵⣄⠀⠀⠀⠀⢀⣠⢴⣖⣶⣶⣶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢧⡠⣔⣪⣷⣾⣿⣿⣿⣿⢿⣿⣝⣆
⠀⠀⠀⠀⢀⣴⣯⣿⣿⣽⣲⠤⣄⣀⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣶⡿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼
⠀⠀⠀⠀⣞⣾⣿⣿⣿⣿⣿⣿⣷⣶⣷⣿⣿⣿⣋⣩⣤⣤⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣽⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⣿⣻⣿⣯⣿⣿⣿⣿⠋⢻⣿⢿⣟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻
⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⡿⣿⣿⠟⣿⣟⣩⣾⣿⣿⠏⠀⠈⣟⠲⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⠏
⠀⠀⠀⠀⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡟⣼⣿⣧⣾⡿⢛⠟⣿⠟⠃⣴⡿⠿⠿⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⠋
⠀⠀⠀⠀⠘⢞⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⡀⢁⡚⠁⠀⣸⠏⡠⣂⠁⠀⠁⢹⣿⠻⢿⣿⣿⣿⡿⣫⠞⠁
⠀⠀⠀⠀⠀⠈⠳⣝⢿⣿⣿⣿⣿⠟⣙⠻⡿⠛⠁⠀⣿⡿⠛⠛⠁⠀⠀⠁⠈⢈⠛⠁⠀⠀⠈⢃⠈⠀⡿⣻⠽⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⣟⣿⢿⡘⠀⠩⡀⠀⠀⠀⢤⠤⠂⠀⣀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠃⣀⡾⡵⠁⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣝⢦⣀⠁⠀⠀⠀⠀⠀⣠⣴⣾⣦⣴⣿⣤⡀⠀⠀⠀⠀⢰⡞⣹⠊⠀⠀⠀⠀⠀⠀⠀⡸⣵⣯⡳⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢯⢻⡆⠀⠀⢀⢼⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠲⠂⣿⣰⠁⠀⠀⠀⠀⠀⢀⡤⣺⡵⠛⢇⡽⡹
⠀⠀⠀⡸⣽⣝⢆⠀⠀⠀⠀⠀⠀⠀⢀⣞⣼⣿⣤⣴⣿⣿⣿⠿⠋⠂⠽⢛⣥⣴⣾⣧⣀⣰⣷⣽⢦⡀⠀⠀⣠⣞⡵⠟⣅⣠⢶⡫⠞⠁
⠀⠀⢰⢱⣛⠟⠷⣽⣲⠤⣀⠀⠀⢠⣯⣾⣿⣿⣿⣿⣿⢍⡀⠀⠀⠀⠀⠀⠈⠉⣛⠿⣿⣿⣿⣿⣷⡳⡄⢸⣽⠋⢣⣄⣜⡿⣼
⠀⠀⠈⠻⢽⣲⢦⣄⣸⠛⠳⢭⡲⡄⢯⣿⣿⣿⣿⣿⣿⡄⠈⠳⡒⠦⠦⠖⣲⢋⣁⢴⣿⣿⣿⣿⣿⣿⢣⡼⣹⠖⡁⣄⣿⡱⡅
⠀⠀⠀⠀⠀⠈⢣⢿⣏⡴⢎⣀⣽⢸⠘⡟⣿⣿⣿⣿⣿⣧⠐⠠⢉⣦⣤⠾⠕⠊⠀⢸⣿⣿⣿⣿⣿⣦⣠⣞⢻⣀⠹⠷⢶⡏⡇
⠀⠀⠀⠀⠀⠀⠈⣳⣽⡇⣆⡤⠹⡾⣽⢋⣿⣿⣿⡽⣿⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⣟⡅⠴⠶⣶⢧⠇
⠀⠀⠀⠀⠀⠀⠀⢻⣻⡞⠚⠓⠲⡧⣻⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠉⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠳⣯⡭⣽⢯⠎
⠀⠀⠀⠀⠀⠀⠀⠘⣯⢷⣚⡫⠤⢢⡉⣿⣿⣿⣿⣿⣿⣿⡏⢄⠀⠀⠀⠀⠀⡐⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣠⣟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣟⡶⣾⡋⢀⣿⣿⣿⣿⣿⣿⣿⡇⠈⢢⠀⠀⢀⠎⠀⠀⢿⣿⣿⣿⣿⣿⣿⢹⣟⣛⣿⠽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠑⠴⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢻⢺⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡞⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣝⢿⣿⣿⣿⣷⣄⣠⣤⣄⣀⣀⣠⣴⣞⢿⣿⣿⣿⡿⣟⡯⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⠭⢟⣻⠴⠒⠚⠛⠿⠿⠓⠋⠉⠚⠛⠓⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠰⠍⠉⠉⠋⠙⠭⡆⠀⠀⡠⠤⠤⣤⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⡆⠀⠀⣰⠂⠀⠰⠂⠀⠀⠀⠀⠋⠁
⠀⠀⠀⣠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢲⡐⣀⣿⠀⠀⠈⢹⠁⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠄⠁⠀⣠⢀⡀⠀⠀⠈⠀⠁⠁⠀⠀⠀⢸
⠀⢀⡴⠀⢀⠠⡀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠂⠈⠉⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠠⠆⠀⠀⠀⢸⠀⠀⠀⠁⠀⠁⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠰⠃⠀⠀⠐⠀⠀⠀⠠⣂⠄⠀⠀⢰⠀⠀⠃⠀⠀⠰⠘⡄⠀⠀⣗⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⡠⠄⠈⠁⠀⠀⠀⠘⠀⠀⡀⠀⠀⠀⠹⠁⠀⠀⠀⠀⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⠀⠠⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⡀⠀⠀⠀⠀⢀⡠⠒⠉⣠⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄
⠀⠀⠀⠀⠀⡀⠀⠁⠀⠌⠂⠀⠀⠀⣀⠔⠉⢀⡴⠊⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠇⠄⠂
⠀⠀⠀⠄⠈⠀⠠⠀⠀⠀⠀⢀⠤⠊⢀⡠⠞⠁⠀⡠⠔⠋⠀⠀⠀⠀⠀⠀⣀⡠⡠⠁
⠤⠈⠀⡀⠄⠀⠀⠀⠀⡠⠖⠁⣀⠔⠋⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠐⠈⠕⠂⠀⠀⠀⠒⠀⠀⠀⣴⣾⣧
⠘⠁⠀⠀⠀⠀⣀⠔⠊⢀⡤⠚⠁⠀⠀⠂⠀⠀⢀⠔⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠃⠀⠸⣿⣿⡡
⠀⠀⠀⠀⡠⠞⣡⢀⠖⠉⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠰⠋⠁⠀⣠⠺⠁⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⢈⠀⣀⠔⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⣀⡀⡄⠀⣀⠈
⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣤⣤⠤⠔⠒⠈⠉⠁⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣆
⠀⠀⠀⠀⠀⠀⠀⠐⠈⣏⣁⡀⠀⠀⠀⠀⠀⠀⡉⠯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣤⣄⠀⠰⠇
⠀⠀⠀⠉⣻⡏⣲⢿⣿⠿⣷⡄⠀⠀⠀⠀⠀⠀⣄⢈⣭⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⢉⡖⠁
⠀⠀⠀⠀⠘⣛⡿⠘⡟⢆⡷⢺⡆⠀⠀⠀⠀⡿⡤⢼⣷⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣆⢒⣟⠄
⠀⠀⠀⠀⠀⠀⢁⣰⣧⡘⣻⡂⡃⠀⠀⠠⣠⠾⠟⣛⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠗⢹⣉⣟⣶⣆⣀⡀
⠀⠀⠀⠀⠀⢀⣯⣿⣿⣿⣷⣶⠁⠀⢰⣰⢳⠆⣄⣁⢸⣜⠄⠀⠀⠀⠀⠀⢀⡤⢶⠕⣂⠘⡟⠋⠉⠉⠀⠉⠛⢷⣦
⠀⠀⠀⠀⠀⠀⠃⠀⣷⠀⢀⠞⠀⠀⣼⣽⣃⡀⠉⢟⠏⣿⠇⠀⠀⠀⠀⡰⢋⡀⠁⢠⠐⡀⡰⠀⠀⠀⠰⡄⠀⠊⣿⡄
⠀⠀⠀⠀⠀⢰⠂⢸⢇⣄⠈⠀⠀⠀⠀⠀⠀⢟⠀⢸⠀⣿⠀⠀⠀⠀⢘⠇⠀⣱⠊⠈⠂⠨⠇⠀⠀⠐⠛⢀⠄⣞⣿⠃
⠀⠀⠀⠀⠀⠈⠀⠏⠚⠹⠼⣥⠂⠀⠀⠀⠀⠈⠀⠁⠀⣧⡀⠀⠀⠀⠼⡀⢼⠇⠀⠀⠀⡧⠇⠀⠀⠀⠀⠀⢣⢹⠏
⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠘⢼⡀⠀⠀⠀⢠⠚
⠀⠀⠀⠀⠀⠚⠋⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠓⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⢴⡶⢫⢻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣭⡍⢩⡍⡉⠉⡿⠉⡍⢩⣷⣿⣿⣿⠿⡟⢸⣿⠟⣥⣯⣭⣭⣿⣿⣿⣿⣿⣿⣶⠸⣧⠻⠴⠛⠚⢻⢀⣿⡿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣣⠾⠛⠓⠈⠁⢀⠈⠉⠁⠈⠉⣉⠝⣡⡖⢁⡇⡟⣵⠋⠀⢋⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠛⠓⠀⢸⢸⢿⡵⢿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠜⠁⠀⠀⢀⣤⠠⢄⠀⠀⠀⠀⡜⣠⣾⣿⢃⡼⢱⡿⣿⠀⠀⢸⠀⠀⠀⠀⣄⣠⣤⣠⢤⠀⠈⠀⠀⠀⠀⡜⡜⣼⣏⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢀⢎⡴⠁⠠⠄⠀⠀⠸⣌⣉⠴⠀⠀⠀⢸⣵⣿⣟⣡⢃⡆⣼⣼⣿⡄⠀⡼⠀⠀⠀⣸⢻⣿⠿⢯⠏⠀⠀⠀⠀⠀⡰⢱⠡⣽⣾⣏⡝
⠀⠀⠀⠀⠀⠀⡠⢣⠎⠀⠀⠀⢀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠘⡻⣿⡿⢁⢾⡇⣿⣾⡿⠁⡴⠁⠀⠀⣆⠀⢀⠄⣀⡀⣤⣤⣤⣴⡖⣰⢣⢣⡜⣽⣿⡿⠁
⠀⠀⠀⠀⠀⡔⣱⣿⣶⣶⣶⡖⠰⡆⣴⣠⣞⣪⠉⡥⣤⣤⣤⣿⡟⢠⢋⣾⡇⣿⣧⣶⡧⢤⣤⢤⣤⢀⣤⣤⠄⠤⢤⡥⣭⠵⠒⠺⢃⣯⢻⣰⣿⡿⠁
⠀⠀⠀⠀⢾⢤⠴⣶⣶⣶⣶⣶⣶⣶⣶⣶⡦⣤⣤⣭⣉⣉⣉⣹⣡⣏⣤⣼⣄⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⣭⣭⣭⡴⠶⠒⣒⣛⡿⣱⣷⣿⡿⠁
⠀⠀⠀⠀⢘⡿⣿⡿⢗⣒⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡷⠤⢤⣬⣿⣿⣋⠈⠀⡐⠒⠲⠶⠶⠶⢴⣶⣿⣽⣿⠃
⠀⠀⠀⠀⠸⣯⣭⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢷⣽⢿⣯⠿⠿⠿⠿⠿⠿⠦⠶⠶⠿⢿⣾⣿⣿⣿⣿⣿⢿⣻⡿⣿⣿⠇
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠻⠿⠋

⠀⠀⠀⡭⠭⠭⠭⠭⠭⠭⠭⠭⣽⢃⣀⣀⣀⣀⣀⣀⣀⣄⣏⢧⣿⣤⣤⣤⡤⠿⠛⢛⣛⡃⣀⣀⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣇⣸⣿⡆
⠀⢀⠈⠉⠁⠉⠉⠉⠁⠀⠉⠉⠈⠉⠉⠉⠉⠉⢿⠙⠉⠉⠁⠈⣹⢩⡽⢻⡟⣡⣤⣬⣭⣭⣭⣍⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⣾⣿⣿⣿⠃
⠀⠈⣳⣄⣀⣀⣴⣷⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⣾⠗⠋⣀⡾⠁⣋⣁⣠⣉⠉⢉⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⣿⣿⣿⡟
⠀⠘⣷⣈⡛⢻⡿⣯⡆⣿⣦⠀⠀⠀⡆⠀⠀⠀⠀⣯⠲⠶⠶⠶⠶⣾⠟⠛⠻⣿⡿⣿⣿⣿⡶⠾⠛⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⣿⡷⣿⡍⢱⣧
⠀⠀⣧⠻⢿⣿⡿⣝⣧⢸⣿⣆⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠠⣽⣧⠀⠠⠶⠖⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⢸⡿⣧⣿⣿⣧⣿⣧⣼⣹
⠀⠀⢿⣏⡞⣾⣷⣿⡞⢆⠻⣿⡄⢸⣿⠀⠀⢀⣀⡄⡤⣞⣭⣿⡇⠀⠀⠀⠀⢘⣿⣆⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⢀⠤⠒⢏⡁⣿⣿⣿⢸⣿⣻⣿⣿
⠀⠀⠪⣼⢸⣿⠗⣶⢭⡛⡇⢹⣿⣿⣿⣿⣿⣿⣷⡷⣿⡿⠻⣿⡇⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⣻⡆⠀⠀⢧⡀⠀⠈⣷⣿⣿⣿⢸⣿⢼⣿⣿
⠀⠀⢸⢣⣾⣿⡜⣿⣥⣽⢀⠀⣿⣿⢿⡟⠛⠁⢹⡧⢿⠃⠀⢹⡇⠀⠀⠀⠀⠀⠹⠿⣿⡶⣿⣿⣿⣿⢿⠟⠃⠀⠀⡀⠳⠦⣼⡏⣿⣿⣿⢉⢹⣺⣿⢾
⠀⠀⣿⢸⣿⣿⣿⢡⣥⣸⢸⣇⢸⣇⣘⣻⣆⢵⠌⠴⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠳⢀⠠⠟⠁⣿⣹⣿⣎⣘⣿⣿⢿⡀
⠀⠀⣿⡈⣿⣿⣿⡠⣿⣿⡆⣿⣿⣿⢹⠙⢻⣖⡶⠲⠒⠒⠂⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣋⣽⠶⢤⣀⠀⠀⠀⠀⠀⠈⢢⢊⢏⢩⣿⣿⡟⢩⣍⡾⠇
⠀⠀⣿⢰⣿⣿⣿⣷⣜⢶⣄⣿⣿⣿⡉⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣮⡿⡿⣎⡽⢳⡦⣤⣤⡴⣫⣤⣼⣸⣿⣽⣵⣰⡿⠃
⠀⠀⢻⡎⣿⣿⢟⣿⣿⣷⣽⣿⣿⡏⠁⠀⠀⠀⣤⠴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣵⡟⣈⠇⣼⣇⣬⠒⢛⣮⠷⣼⡿⣿⠛⣹⣿⡀
⠀⠀⠋⣿⣹⣷⣏⣴⣿⣿⡿⣿⣿⣶⠀⢠⡎⡘⠀⢛⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⢋⢹⡕⢿⠓⣷⣼⢨⡇⣶⣶⡉⢯⣽⣷⠟⠀⢻
⠀⠀⠀⢸⣿⣯⣻⣿⣿⣿⣽⣿⠏⠀⠀⣿⡇⠃⢠⣾⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣿⡿⣭⠾⠶⠾⠏⠚⠛⠉⠉⠉⠛⠳⠿⠀⠀⠈
⠀⠀⠀⠀⠻⣿⣯⠙⣛⣭⣿⠏⠀⠀⠀⣿⠇⢦⡀⠉⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢴⠶⠶⠶⣤⡀
⠀⠀⠀⠀⢠⣿⣿⡇⢻⣷⣿⡆⠀⠀⠀⢹⡀⠀⣸⡲⡀⠀⢜⢿⡄⠀⠀⠀⠀⠀⢀⢮⠋⠀⢾⠀⠀⠀⢨⡗
⠀⠀⠀⠀⠈⢻⣿⡄⢠⣿⣿⠃⠀⠀⠀⠈⢧⡀⠸⣷⡈⠀⠀⠈⠻⣦⣀⠀⠀⠀⡨⣾⡀⣠⣎⣻⣦⣴⢟⣥⣴⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠐⢿⣿⣇⢀⣿⣯⠇⠀⠀⠀⠀⠈⠳⣦⣬⣉⠳⠀⢄⠀⠩⠝⠟⡿⢡⣤⣤⢥⡄⢤⣤⣴⡆⢸⣿⣿⣟⣘⣛⣋⣙⣟⣛⣓⣆⣐⣂⣐⣲⣰⡃
⠀⠀⠀⠀⠀⠀⣿⠃⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⠰⣦⣔⡠⡄⢠⡇⠷⣿⣽⣯⡟⣾⣷⣿⡇⢿⠿⡟⣟⢹⠋⠉⠉⢉⢉⠙⠋⢉⡙⠋⠛⠛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⠿⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡿⠟⠋⠀⣀⣤⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⠿⠛⠁⢀⣠⣴⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⡿⠟⠋⠁⣀⣤⣶⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⡿⠛⠉⢀⣠⣴⣾⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣧⣀⣤⣶⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠁⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⣶⣿⣿⠃⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠀⣿⣿⣿⣿⣷⣄⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠉⠛⢿⣄⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠙⠛⠷⠶⣂⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⣀⣀⣠⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠻⣿⠇⠀⠀⠀⠸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⢀⣠⣶⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⣀⣴⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢤⣄⣀⣀⢙⠆⣄⡀⢀⡄⠀⠀⠀⢀⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣛⡦⢍⣉⠁⢈⣀⠀⡜⢋⠀⠀⠠⣤⠯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠠⢺⣿⣧⣀⢜⡀⢡⣻⡀⠀⠀⠘⠯⢬⣃⠤⠎⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⢠⡟⠀⢸⡭⡆⠀⠀⣤⡾⣾⡇⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⣼⡇⠀⠀⣷⣞⠷⡘⣏⢠⣿⠁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⣼⣿⠃⠀⠀⠿⢠⣄⣝⡿⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡏⠀⡀⠀⠀⣿⡅⡀⠙⠳⢦⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢹⣿⣿⣿⠁⠐⡃⠀⠀⠻⡆⠃⡀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡿⠿⣿⣿⣧⠄⠀⠁⠀⠀⠀⠀⠀⠹⢶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣰⣿⠀⣟⣀⠀⠀⠀⠀⠀⠀⠘⠳⠇⠰⢦⡄⠂⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠹⣆⠈⠉⠙⠲⣶⣞⣛⡥⠂⠀⠀⠀⠀⠀⠀⠴⠁⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣽⣆⠀⠀⠀⠙⢫⣥⣶⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠀⠈⠻⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣾⡆⠀⠀⠀⠀⠀⠈⠙⣿⣿⣶⣆⠀⣀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣏⢻⣿⣿⡆⣿⡄⠀⠀⠀⢀⣿⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠻⢿⡇⣿⡇⠀⠀⡀⢸⣿⡇⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠁⢻⡇⠀⢸⠁⢸⣿⡇⠀⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠈⣷⠀⠸⠀⠈⢿⡗⠀⠀⢠⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠸⣆⠀⠀⠀⠀⠙⠷⣄⣀⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡷⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠟⠁⢀⣴⠟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠀⠀⢠⣿⠷


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣹⣟⣻⣙⣏⣬⣿⣿⣋⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣿⣽⣿⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣒⣦⣤⣤⣶⣶⡓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠿⠟⠛⠛⠛⠛⣽⣿⣿⣿⣿⣧⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣀⣀⣀⣀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⡄⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣇⣀⡀
⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣿⠿⢿⣿⣶⣦⣄
⠀⡴⠞⣉⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣶⣤⣄⡉⠙⠿⣷⣄
⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠉⠉⠓⠄⠈⢿
⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⢼⣿⣿⣿⣿⠃⣹⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣴⣾⠿⠗
⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣼⡸⣯⣿⣿⠀⢱⣾⣿⣿⣿⣿⣿⣿⠸⣿⡟⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣕⣻⡌⢿⢰⡇⠙⢿⣿⣿⣿⣿⣿⡇⠹⠇⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣻⣿⠉⢱⠘⠇⣿⠀⠀⢹⣿⣿⣿⣿⣯⣄⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⣿⣿⣿⠀⢸⣆⠀⠉⠀⠀⢞⣽⣿⣿⣿⠻⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡍⠉⢼⣿⣿⢷⣆⢸⣿⣷⣤⡀⠈⠻⣿⣿⣿⣿⣷⠀⡟⠁
⠀⠀⠀⠀⠀⠀⣀⡼⠁⠀⠀⣙⠻⣷⣿⡻⣿⣿⣿⣿⣷⣦⣘⠛⠛⣛⣛⡸⠁
⠀⠀⠀⠀⢀⣠⠿⠀⣰⣿⣶⣌⠳⣌⠻⣿⣾⣝⢿⣿⣿⣿⣿⣿⢏⠉⠉
⠀⠀⣠⡴⢛⣡⣴⣿⣿⣿⣿⣿⣷⣽⣷⣮⡻⢿⣷⣝⢿⣿⣿⡟⣼⣆⣀⢤⡀
⠀⠚⢡⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣙⠻⣷⣝⢿⣧⣿⣿⣿⣿⣻⣷⡄
⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣷⡌⠻⣧⣿⣿⣿⣿⣿⣿⣿⡷⠂
⠘⠉⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⣿⣿⣿⣿⣆⠈⢿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠙⢿⣿⡏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⣷⡑⠀⠀⠹⣿⢹⠁

⢠⣴⣶⣿⣷⣶⣾⣷⣶⣤⡄⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣷⣦⣶⣶⣆⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣄
⠀⠻⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⣴⣾⣿⣿⣟⣛⡉⠙⣋⣻⡇⠀⠀⠀⠀⠀⣿⣿⣿⣭⣭⣽⣿⣿
⠀⠀⠈⠙⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⢀⣠⣀⣀⣀⡀
⢀⣴⡿⠛⠛⠛⠛⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣦⡀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣤⣴⣴⣤⣤
⢾⣿⠀⠀⠙⠿⠿⢿⣿⣷⠀⠀⠀⠀⠀⣰⣿⣿⣯⣁⣉⣀⣽⣿⣿⣦⠀⠀⠀⠀⠻⣿⣶⣤⣭⣥⣶⣾⡿
⠘⢿⣷⣤⣤⣤⡤⣾⣿⠃⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣷⣿⡿⠋⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠟
⠀⠀⠙⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣠⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣠⣄
⢠⣶⠿⠛⠻⠿⠛⠻⢷⣦⡀⠀⠀⠀⠀⠰⣿⣿⣛⣛⣛⣛⣛⣿⣿⡆⠀⠀⠀⠀⣴⣿⣟⡙⠛⠛⣛⣿⣷
⠀⠉⠻⣾⣶⣶⣶⣿⣿⡟⠁⠀⠀⠀⠀⠀⢻⣿⣟⠛⠛⠙⢛⣿⡟⠀⠀⠀⠀⠘⣿⣿⣿⣭⣉⣩⣽⣿⣿
⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣴⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡀
⠠⣾⣿⣯⣿⣿⣽⣿⣿⡦⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠠⣶⣿⣿⣛⣿⣟⣿⣿⣷
⠀⢹⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⣲⠶⠤⢤⡀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠛⠉⠉⠀⠀⠰⡅⢃⠴⣣⡼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠲⠭⠙⠛⠫⠷⢦⣄⠀⠀⣠⣾⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡶⠦⣤⣀⣀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⡿⠥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡐⠌⠊⠁⠀⠣⡠⠀⠐⢉⠝⣓⡶⢤⢠⠐⣸⣿⣿⠿⠏⣆⣴⣾⡆
⠀⠀⠀⠀⠀⣠⠐⢊⣥⠞⠁⠛⠀⠀⠀⠀⠀⠈⠉⠉⢉⢥⡪⣾⣿⡀⣳⣯⣿⢍⣡⣾⣿⣿⢏⡇
⠀⠀⠀⠀⠀⣾⢛⠒⠤⣼⣒⠤⣁⠀⡀⠤⣤⠄⢤⣲⣵⣿⣿⣿⡿⢐⣿⡛⠈⠀⡉⣿⡾⣻⢏⡀
⠀⠀⠀⠀⠀⡷⠈⠹⠷⣒⠭⠹⣶⣯⣠⣚⣭⣶⣽⣾⣿⣿⣿⡿⢟⣋⣀⡀⢠⢴⡃⣿⣿⠿⣤⠉⠹⡖⠤⢄⡀
⠀⠀⠀⠀⠀⠑⠢⠤⣐⡢⠭⣝⡂⠝⡓⠮⣿⣯⣷⣿⠿⠋⢉⣁⡠⢴⢻⣻⢮⠷⢻⡿⠖⠚⠹⣧⢤⡐⢋⣵⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣗⡲⢬⣙⡒⢧⣿⣿⡋⠭⠃⠀⠀⠀⡔⣵⣿⣿⠲⣱⡋⠀⢀⠔⠀⠀⠁⢚⠻⠿⡿⢿⡀
⠀⠀⠀⠀⠀⠀⣤⠒⠩⠍⠁⣂⡐⠀⢍⡛⠑⠁⠀⣀⢀⣀⡠⣪⣾⣿⣿⣿⣼⠁⠀⠒⠁⠀⠀⠀⠤⠀⠀⠀⠃⠰⣨⠉⣲⡄
⠀⠀⠀⠀⠀⠠⣗⠂⠈⠩⠥⠐⣂⡈⠓⠲⠤⢔⣉⣐⣊⢔⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⣀⢕⡤⢖⡫⢔⣷
⠀⠀⠀⠀⠀⠀⢿⣊⡠⠬⠍⠁⠐⠒⢒⣀⣤⣬⢐⣶⣂⣭⣾⣿⣿⣧⣤⡠⠤⣤⠄⣀⠀⢀⡀⠠⣐⡮⢝⣪⣵⣾⣿⣾⢿⡃
⠀⠀⠀⠀⠀⠀⠉⣶⢖⠶⡮⢭⠥⡤⡥⣤⢶⢖⡶⡶⣲⢻⣿⡫⡃⠀⠙⠻⣖⠬⣙⣠⣭⣶⣦⣟⣾⣿⣿⡿⣟⣊⣥⣒⣿
⠀⠀⠀⠀⠀⠀⠈⡯⠁⠑⠊⠂⢄⣈⡘⠐⠑⠠⠊⠄⠔⠔⡠⣻⣮⡀⠀⠀⠉⠯⢃⣬⣭⣍⣬⣿⣻⢿⣷⣾⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠈⢻⢦⡉⠭⠽⢓⣾⡿⣿⣿⣿⢿⠶⠄⡀⠀⠰⣷⡶⣶⡷⠶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠀⠀⢯⢍⠑⢾⣻⣷⣿⣿⠟⠋⠁⠀⠀⠀⠑⠒⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⢀⠃⠐⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠢⠰⢀⠄⡠⡀⢀⠀⠀⢐⠅⡨⢀⠀⡠⠀⠔⠁⠀⢀⠔⠡⠊⠀⢀⢀⠑⡡⠂⢀⠔⡁⡠⢀⢔⢕⡔⡰⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠐⠡⠪⠀⠅⠨⠀⠕⡨⢂⠔⠀⡨⠀⠀⡠⠪⢀⢀⠔⡡⠂⠂⠐⠁⠊⠌⠐⠁⠃⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⣀⣀
⠀⠀⠀⣸⣿⠈⠉⠓⠲⢤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣶⡖⠒⠒⠒⠶⠶⠶⠦⠤⢤⠀⣀⣀⣀
⠀⠀⢀⣿⠿⣀⣛⣛⣋⣓⣒⣐⣀⣀⣂⣀⣀⣀⣂⣀⣀⣀⣀⣠⣦⡼⢧⡤⠘⢿⣛⣛⣻⣟⡷⠈⢛⣡⣶⠿⠃
⠀⢠⣿⢿⡝⡖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣈⣷⣀⣸⣦⣀⣀⣀⣁⡀⡌⢿⣁
⠀⠘⣿⣌⣋⣿⣶⣶⣶⣶⣶⣶⣶⡾⠿⠿⠿⠿⠿⠟⠛⠛⠛⣛⣿⣭⢻⡯⠉⢻⣿⠿⠿⠟⢻⢿⠖⠂⠈⠉⠳⡀
⠀⠀⢨⣿⣿⣜⠛⠒⠉⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣶⣶⣶⣶⣟⣼⠷⠶⣾⣿⣀⣠⣤⣤⡞⠀⠀⠀⣴⣷⡳⡄
⠀⠀⠘⣿⣈⢿⣴⣶⣶⣶⣶⡿⠿⠿⠿⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠈⣿⣧⠀⠛⠛⢛⠋⠉⠉⣠⣤⡄⣸⣿⡿⣿⣿⣆
⠀⠀⠀⠈⠙⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣽⣧⣶⣼⣿⣿⣧⢰⣿⠋⣷⣿⣿⣿⢿⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⣿⡌⢿⣟⣼⡇⢠⢿⣿⣿⣿⣗⡾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠤⠭⠽⠒⠉⢸⣿⡿⣿⣷⢽⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠃⣿⣿⣯⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⣿⣿⡿⢿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣟⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⡾⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠖⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠳⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡗⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠸⢥⣿⣿⢻⣿⣷⡄⠀⠊⡙⢢⡀⠀⠀⠀⠀⠀⢀⣀⣠⣾⠇⠄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣾⡏⢈⣀⣽⣿⠿⡆⠀⠈⠂⣙⣦⡀⠀⠁⠀⠰⠛⠻⠻⠁⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣸⣿⢃⣸⡤⠤⠒⠀⠀⠁⠀⠁⠀⠀⢀⡀⠀⣠⣴⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡏⠉⠈⡄⠀⢰⡀⠀⠠⠾⠿⠿⢇⣹⡿⠀⠀⠈⢹⣯⣼⡆⣷⣶⡿⠃⠀⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠃⠀⠀⢡⠀⣸⠇⠀⠀⠀⠀⠀⣾⠏⠀⠀⢀⣴⣿⣿⡿⣹⠟⠉⠀⠀⠀⠀⠀⠀⠈⣷⡝⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⡟⠀⠐⠀⠀⠀⠀⠉⠀⢀⣴⣿⠿⢟⣫⠶⠁⠀⠀⠘⣆⠀⠀⠀⠀⠀⣸⣷⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠛⠀⠀⠀⠀⠰⣀⠰⠿⣟⠛⠛⠉⠁⣀⣀⣤⡦⠀⢻⠀⠀⠀⠀⣼⣿⠟⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠄⠀⠀⠀⣧⣾⣀⢀⣻⡆⠀⢰⣾⣿⣿⠟⠷⣷⣦⠀⠀⠀⢰⣿⣿⠁⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢀⣿⣿⣿⠀⢙⣷⠀⠈⠉⠉⠀⠀⠀⣿⣿⡆⠀⠀⣼⣿⠇⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠄⠀⠿⣿⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⡂⠀⣿⡿⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠙⠒⠒⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠸⣿⠀⠀⢻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠄⢿⣇⠀⠀⠙⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠄⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡏⢁⠀⠀⠀⣠⣼⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣴⣾⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡿⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢼⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⡄⠀⠀⣀⣾⣦⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢥⣄⣄⣼⣻⣿⡿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⢹⣷⠲⠀⠠⡀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣹⣿⣿⣿⣿⠉⡻⢿⣷⣤⡼⠛⢗⠒⠋⠉⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢁⣿⣿⣿⣿⠃⠀⠀⠻⣟⢿⣟⣴⣮⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⡿⠉⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⣿⢳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠘⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢿⣿⠀⠀⠀⠀⣀⠀⠀⢀⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣿⣿⣼⡉⠤⡀⠀⡀⠶⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⢻⣧⣻⢤⣀⡄⣰⣶⣷⠖⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢘⣿⣿⡇⠘⡿⠛⠙⢀⡇⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⠿⠛⣷⣄⠀⠀⢀⡼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣻⡀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣵⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠹⣿⡟⠀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠐⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠘⠂⠇⠀⠀⠈⠈⡃⠀⠀⠀⠃⠀⠀⠀⢀⠄⠼⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠤⠀⠄⠂⠀⠀⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣤
⠀⠀⠴⠒⠒⠒⠒⠲⠒⠒⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠉⢱⡆⠀⠀⠀⠀⠀⠄
⠀⠀⡇⢀⠖⠉⠁⠉⠢⡀⢰⠉⠉⡆⡊⠀⠙⠄⠀⠀⠀⠀⠀⠀⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠞⠃⠀⢀⠀⠀⠀⢀
⠀⠀⡇⡌⠀⠀⠀⠀⠀⠱⠈⡲⣒⠁⢙⡶⡊⠀⠀⠀⠀⠀⠀⠀⣀⠀⣃⠀⠀⠠⢀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠛⠄
⠀⠀⡇⠡⡀⠀⠀⠀⠀⡔⢸⠀⠀⡇⣇⠀⢈⠀⠀⠀⠀⠀⠀⠀⡅⠀⠬⠀⠀⢀⣀⣀⡀⠀⠠⠀⠰⠀⠀⠀⠀⠁⠤⠀⠄
⠀⠀⣇⠀⠑⠢⡤⠄⠊⠀⠀⠉⠉⠀⠈⠉⠁⢸⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⢀⠀⠤⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠻⠇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢈⠌⡆⠀⠀⠀⡇⠘⠀⠀⠀⠀⠅⡀⠀⠀⠀⠀⡈⠂⠀⠰⠰⠂⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠁⠀⠀⠁⠀⠀⢮⡰⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⡾⢩⠏⢿⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢤⢀⠠⢄⡀
⠀⠀⠀⠆⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠄⠀⠀⠀⡅⡆⠀⠹⢍⡘⢹⠀⢀⠔⠛⠋⠲⡄⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡁⠀⠀⢻⡀⢀⢈⡬⢷⡾⠀⢸⠀⠀⠀⠀⡸⠀⠀⠀⠀⡇⠀⢠⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠔⠢⡉⠓⠛⠛⠉⠀⠀⠀⠓⠤⠤⠖⠁⠀⠀⠀⠀⡇⠀⢸⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⢰⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠈⢢⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠠⢄⠀⠀⡇⠀⢸⢸⠀⠀⡇
⠀⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⠒⠂⠒⠀⠘⠀⠀⠀⠀⠸⠤⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠔⠀⠀⡇⠀⢸⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄⠀⠄⠀⠀⠀⠀⡀⠀⠀⠡⡀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠸⠘⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣾⣿⡿⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣶⡿⢿⡿⠀⠀⠈⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠿⠿⠟⠋⠁⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣾⣿⣤⣈⣁⣪⡍⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⠿⠿⠿⠿⠿⠿⠿⠷⢦⣤⣭⣉⣙⡛⠶⣦⣄⡀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⢶⣭⡻⢶⣄⠀⠀⠘⣿
⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡙⢷⣦⡀⣿⠆
⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡉⢿⣿⡀
⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠂⠀⠀⠈⠻⣦⡙⢷⣄
⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠈⢿⣆⠙⢷⡀
⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠞⠛⠛⠀⠹⣦⡀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⠓⠶⠶⢦⣄⣙⠛⠛⠛⠛⠛⠛⠛⢦⣙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣋
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠉⠁⠀⠀⠀⠀⠈⠙⠻⢷⣶⡀⠒⠀⠒⠚⠛⠳⣮⣙⠳⠶⣦⣤⣤⡶⠾⠋⣠⡾⠋
⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠛⢶⣦⣤⣄⣀⣀⣀⣉⣻⣶⣶⣶⠶⠶⠚⢛⣻⡏
⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⢤⡄⠀⠈⠉⢻⣧⣀⣀⣀⣀⣀⠀⠀⠀⠈⣉⣉⣉⣩⣉⠁⢠⣿
⠀⠀⠘⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⣠⡀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠚⠷⣼⣧⣤⣭⣭⣭⣭⣭⣭⣍⣉⣉⣉⡒⠶⠖⢀⣿⠃
⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠟⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠀⣠⢠⣟⠛⠲⠶⠖⠶⠶⠶⠶⠖⠒⠒⠉⠉⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠈⠛⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣬⣤⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⢸⡟⣼⡋⠉⠒⠒⠒⠲⠶⠶⠶⠶⠒⠒⠉⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣄⣀⣀⠀⠀⣀⣠⣞⠉⠀⣤⣀⣀⣠⣤⡴⠋⠁⠀⠀⣀⠐⠻⣦⠀⢹⣏⠙⠛⠓⠒⠒⠒⠒⠒⠞⠒⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⢤⣀⡉⠉⠙⣛⣛⣉⣉⣻⣄⠀⠀⢹⠉⢁⡸⢧⣄⠀⠀⠰⣿⣷⣴⠟⠀⢸⡟⢛⠛⠒⢒⣠⣶⣶⣶⠾⠛⠉
⠀⠀⢀⣠⠴⠛⠉⠛⠃⢠⣤⠼⠟⣻⡆⢹⣿⣩⣿⠉⠙⠻⣆⠘⣿⣉⠀⣀⠽⠛⢦⣤⣤⣤⣴⠆⠀⣼⣷⣿⣿⠿⠿⠛⠋⠉
⠰⣿⣿⡇⠰⠟⠀⠴⠶⢀⣁⣤⣾⡟⠛⢾⣿⣿⠿⣷⠙⠃⢿⡄⠈⠛⠻⠧⠴⠴⠛⠋⣁⣈⣥⠴⠛⠉
⠀⠈⠉⠻⣿⣾⣿⡿⠿⠛⢋⣥⣼⠀⣠⣿⠿⣿⡆⢸⡧⣶⡌⢻⣦⣤⣀⣀⣀⣀⡴⠟⠋⠉
⠀⠀⠀⠀⡿⢛⣥⡀⠾⠆⠀⢉⣽⣶⠿⣿⠶⣿⢇⠈⣷⠀⠀⣠⡍⠻⣿⡟⠙⢿⡆
⠀⠀⠀⠰⣧⣠⡠⠶⠒⠈⢙⠟⢻⡷⣾⣿⠾⣿⡆⠀⢹⣶⣿⡀⠁⠀⠠⠿⢦⡾⠃
⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣾⣟⢛⣿⣿⣿⡴⠟⠀⠀⣸⢿⣿⡇⠘⠿⠃⢠⣌⢻⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡏⢙⣟⢻⠟⠁⠀⠀⠀⠀⠀⣿⢶⣿⣿⡿⠇⣤⣦⠀⣠⡿
⠀⠀⠀⠀⠀⢘⡿⢉⣹⡿⠿⠟⣿⡆⠀⡀⠀⠀⠀⣸⣿⣼⣧⣼⣷⠀⣀⠀⢸⣿
⠀⠀⠀⠀⠀⢈⡿⠋⢀⡀⢠⠞⠻⣷⣌⠛⠀⠀⣰⣿⣷⣾⣷⣾⣽⣿⣋⣀⣿⠟
⠀⠀⠀⠀⠀⠀⠻⣆⠉⠀⢀⣠⣤⣿⠙⢷⣤⡾⠿⣿⣛⠛⠛⠛⣿⣿⣿⠋⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣤⡴⠟⠋⠉⠙⢆⠀⠈⠀⠀⠀⠙⢿⡛⠛⣹⣏⢩⣥⣤⡈⢙⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠂⠀⠀⠙⣶⣏⠁⠀⠀⢠⡞⠋⠁⢈⣻⡄
⠀⠀⠀⠀⠀⠀⠘⢃⣀⣀⣀⣀⡤⢤⡾⠀⠀⢀⣀⣤⣤⣤⣀⡈⢻⣅⠙⠛⢁⠙⠲⠶⠿⠛⠻⣦
⠀⠀⠀⠀⣀⣄⣐⣀⡙⠁⠀⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠛⣦⠹⣇⡉⣉⡁⠀⣤⠀⠀⠀⠘⡇
⠀⠀⠀⢰⡏⢉⣿⣧⣹⠂⠀⠀⠀⠀⠀⠻⣦⣀⣀⣀⡰⠀⠠⠤⢿⡆⢻⡅⠤⠄⠀⠈⠳⣄⣀⣀⡜
⠀⠀⠀⠘⠿⠿⢄⣈⡁⠀⠀⠀⠀⠀⠀⠀⣿⠁⠈⠻⢿⣼⡄⠀⠸⠇⠈⣷⠆⠀⠀⠀⠀⠀⠉⠉⢠⡀
⠀⠀⠀⠀⠀⠀⠀⢈⡟⠀⠀⠀⠀⠀⠀⠀⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡗⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⢀⡛⠿⣿⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠁⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⢻⣤⣴⣿⣿⣡⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠦⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⠉⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠆⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⢀⣤⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⣞⠉⣹⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣼⣧⣄⡀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⢹⣖⣉⣉⡤⠲⠆⠀⣠⣤⣤⣤⣤⣤⣤⣴⣦⡀⠀⣀⣴⠟⢻⣿⣿⣿⣷⣶⣤⣤⣴⠊⠁⠀⠀⠀⣀
⠀⠀⠈⠉⠉⠁⠀⠀⠄⠀⠹⣿⣿⣿⡿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠿⠿⠏⠹⣿⣿⣿⣿⣷⣄⠀⢠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠘⢿⣿⣿⣿⣿⠄⠀⠀⢀⣀⣐⡖⠀⠘⠿⠛⠛⠿⠏⠀⠀⠀⠠⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡾⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠿⠛⠋⠉⠀⠀⠈⣿⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣠⣴⠶⠿⠿⠿⢿⣤⡀⠀⠀⡀⠀⠀⠀⠀⣉⣉⣠⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠾⢿⡿⢋⠀⠰⠄⠀⠀⠒⣈⣉⣤⣤⣴⣶⠶⠿⠟⠛⠋⢿⣇
⠀⠀⠀⠀⠀⠀⣈⣤⣴⣶⠿⠛⠋⠁⠀⢀⣉⣉⣥⣤⣶⡶⠾⠛⠛⠿⢿⣭⡁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠘⠉⣡⣤⣤⣶⡶⠿⠿⠛⠛⠉⣹⡟⡁⠀⢐⣦⣀⠀⠀⠈⢻⣆⠀⠀⡀⢀⣀⡀⡀⣿⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⣩⣥⣤⣤⣤⣤⣤⣬⣭⣬⣤⣤⣭⣭⣬⣥⣤⣤⣥⣭⣤⣤⣥⣭⣭⣥⣥⣭⣭⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⣿⣟⠉⠉⠉⠉⠉⠉⠉⠉⣩⡿⠛⠉⠉⡉⠉⠙⠻⣯⡉⠉⠉⠉⠉⠉⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⡿⠀⠀⢀⡀⠀⠀⠀⣼⠋⠀⠀⣠⠾⡗⣦⠀⠀⠈⢿⡄⠀⠀⠀⣀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⣿⡟⠀⢠⣶⣦⠀⠀⢸⡏⠀⠀⠀⠻⣶⣇⡉⠀⠀⠀⠘⣷⠀⠀⢠⣶⣄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⣿⣷⠀⠈⠛⠋⠀⠀⢸⣧⠀⠀⠀⣤⡌⡟⢻⡆⠀⠀⢠⡿⠀⠀⠈⠛⠋⠄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠙⠲⡷⠛⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣠⣀⣀⣀⣀⣀⣀⣀⣹⣷⣦⣄⣀⣁⣀⣤⣾⣟⣡⣄⣀⣀⣀⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣢⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡞⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡠⠄⠣⠄⠀⠀⠀⠀⢀⣠⡤⠖⠚⠉⠉⣡⣽⣏⠬⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣠⣿⠓⠶⣤⠀⡀⢦⠐⠚⠉⠀⠀⠀⠀⣴⣾⡿⣿⣈⣢⡄⣄
⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⠀⠀⠀⠀⡀⣹⡟⠻⣿⣦⣀⠀⠀⠀⠀⠑⢤⡀⠀⠀⣠⣾⠟⠁⠀⠀⠉⢉⠹⣿⠆
⠀⠀⠀⠀⢠⣿⡟⢰⣿⠁⠀⠀⠀⠘⢹⣿⡆⣸⠃⡈⠙⣿⢰⠀⠀⠀⠀⠙⣷⣾⡿⠣⠄⠠⠄⡠⠞⠁
⠀⠀⠀⠀⣾⣿⡇⢸⣿⡄⠀⠀⠀⠀⣸⣿⣷⣏⣀⠈⡾⠓⠉⠀⠀⠀⠀⠀⠘⡟⠀⠁⠀⠈⠁
⠀⠀⠀⠀⢿⣿⣷⠀⢿⣷⡀⠀⠀⣼⡿⠛⠿⣿⣿⣿⣇⡸⣷⣶⣆⡀⠀⢀⣰⠇
⠀⠀⠀⠀⢸⣿⣿⠀⣸⡿⢀⣤⣾⠟⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⣿⡇⢀⣿⡇
⠀⠀⠀⠀⠘⣿⣿⡾⡿⢴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢺⣿⠇
⠀⠀⠀⠀⠀⠙⣿⠞⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣿
⠀⠀⠀⠀⠀⢀⡟⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠃⠀⣿
⠀⠀⠀⠀⠀⢼⣇⠀⠀⣈⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡄
⠀⠀⠀⠀⣀⣤⣿⡀⣴⡿⠟⠛⠻⠟⠶⠦⠶⠄⠀⠀⠀⠀⣀⣬⣿⡄⣴⣾⣗⣀⣶⣶⡶⡞⠄
⠀⠀⠀⠀⠈⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠟⠂⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⣀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⣿⣿⡄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣕⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣤⡀⠀⠀⠀⢰⣿⡿⢋⣬⢻⣿⣿⣿⣿⣿⣿⣿⣿⣥⣂⣠⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣷⣦⡄⢙⡿⣷⠟⠻⣷⣼⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣿⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠙⢻⣿⢟⡭⢻⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⠛⣻⡟⠁⠉⠟⠛⠛⠁⠀⠘⠁⠺⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠆⣙⣒⠊⠁⠢⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⢿⠙⠋⠙⠛⡦⠀⠀⠀⣠⠏⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣻⣿⣿⡁⠀⠀⠀⠈⠁⢀⣤⡾⢃⣼⢏⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⡀⢦⢠⣤⣴⣼⣋⣴⣿⣷⢟⣠⣴⣾⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢿⣿⣿⣿⣬⣽⣿⣿⣮⣶⣿⣿⣿⣿⣿⣿⣿⡏⢉⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣧⣿⣿⣿⣿⣿⠀⢀⣹⣏⠤⣸⡟⠛⠋⣹⣿⣿⣾⣿⡿⣻⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣾⣿⣿⣿⣿⣏⣴⣿⣿⡟⢋⠀⠙⢦⣰⣿⣿⣿⣿⣿⣷⣿⣿⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣿⣿⣿⣿⡟⠉⣵⣿⡟⠁⢸⡄⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠀⠊⡼⠋⠁⠀⠈⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡟⠁⠀⠚⡁⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⢀⣬⣼⠏⠀⠀⠀⠀⠀⣿⣅⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣥⣾⣿⣿⠀⠀⠀⠀⠀⣼⣿⣿⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠛⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣿⢸⣿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⠿⠛⠀⠀⠈⠀⡞⠙⠛⠟⠙⠳⠛⠻⢿⣿⣿⡿⠿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⡉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⡀⠀⡀⠄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠉⠀⠐⠉⠀⠀⠀⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠐⠁⠀⠀⣀⡤⠤⢀⡀⠄⢠⠄⣀⡀⠀⠈⢤⡀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠈⠻⣔⠒⠒⠚⠋⠤⠄⡀⠀⠀⠀⢀⣀⠄⠃⠚⠓⠂⠹⠋⠀⠀⠀⠻⠟⠒⠒⠒⠐⠂⠈⠛⠃⠀⠀⠬⠓⠉⠉⠀⠒⠀⢒⠓⠃
⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠈⠉⠁⠉⠁⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⡠⠊⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠁⠀⠀⠈⠂⠀⢀⡠⠄⠠⡀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠉⠀⠈⠁⠊⠁⠀⠈⠲⣄
⠀⠀⠀⠀⠀⡀⠐⠁⣀⡄⠐⡐⠂⠒⡴⠀⢩⠔⠈⢰⠑⢄⠀⠄⠀⠀⠀⠀⠀⢀⠄⠊⢀⣀⡄⠰⠒⠒⢰⠆⠒⠰⡆⠄⡀⠑⢤⡀
⠀⠀⠀⠐⠚⠁⠲⢭⣜⢆⣀⡶⡄⠀⠇⠀⣸⠷⠂⠈⠀⠈⠉⡩⠒⠀⠀⠀⠻⣋⠚⠈⠁⠀⣁⣉⣀⠒⠊⡁⣀⣈⣁⣀⠉⠐⠒⠚⡻
⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠢⣖⠀⠁⠀⠀⠉⠐⠛⠒⠈⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⣀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠠⠤⣀⡠⠔⠒⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⣀⡀⢀⡀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠘⠈⠀⡄⠀⠒⠢⠆⡀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⡰⠃⠀⣀⣀⢠⠂⠁⠀⠀⠀⠀⣓⣦⣀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⣠⠤⠈⡇⠀⢸⠀⠀⠀⡇⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠶⡄⠀⠈⠁⠀⠀⠀⠑⠂⠂⠐⠁⠀⠀⠀⠈⠁⠀⢒⡳⠆⠀⠀⠈⠀⠀⠀⣀⠈⠀⠉⠀⠂⠐⢣⠀⢀⠁⠒⡈⠀⡜⠂
⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢀⠀⠀⠀⠀⠠⠼⠀⢀⡥⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠤⠀⠀⠠⠤⠤⢀⡀⠠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠒⢉⡁⢰⣶⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠒⠒⠋⣉⡠⠤⠖⠒⠋⠉⠈⢇⠀⠿⠇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠔⠒⠚⣉⣁⠤⠤⠶⢊⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠘⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠒⠂⣉⣉⡤⠤⠔⢂⠟⠋⠁⢀⣤⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣧
⠀⠀⠀⠀⠀⡔⠒⢈⠉⢠⡤⠤⠒⠊⠉⠁⠀⠀⠀⡔⠁⠀⠀⣴⣾⣿⡿⣷⡆⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⡀
⠀⠀⠀⠀⠀⢱⠀⠘⣢⠀⡇⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢸⣿⡁⢹⣷⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇
⠀⠀⠀⠀⠀⠘⡆⢀⠥⠔⠁⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠈⠻⣿⣾⣿⣦⣤⣀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢹
⠀⠀⠀⠀⠀⠀⢡⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢹⣿⠙⢻⣷⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠀⠏⡇
⠀⠀⠀⠀⠀⠀⠈⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⢠⣿⡇⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠰⡇⠀⢧⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠐⠿⣷⣶⣿⣿⡿⠋⠀⠀⣠⠋⠀⠀⠀⠀⣀⣠⠤⠄⠓⠀⣉⣠⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠈⠛⠁⢀⣠⣞⠥⠤⠒⠒⢉⣁⠤⠤⠖⠒⠈⠉⠀⣳⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⣆⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣒⡲⠶⠶⠒⠊⣉⣁⠤⠔⠒⢺⠋⠁⠀⠀⠀⠀⠀⢀⠞⢁⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢴⠀⠙⡆⠀⣀⣀⠤⠄⠒⠚⢉⣁⠠⠤⠶⣶⣿⣿⠟⠁⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⢀⢸⡄⠈⡳⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠠⠇⠀⢉⣉⡁⠤⠔⠒⠋⠉⠓⢤⣀⠀⠀⠀⠈⠉⢀⣀⣴⣋⡠⠤⠤⠒⠒⠂⢋⣉⣉⡡⠤⠤⠰⡖⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠧⢲⠊⠙⡍⠀⠀⠀⠀⠀⣀⣀⣀⠠⠬⠽⠛⠒⠊⢉⣉⣁⡠⠤⠤⢲⠖⠂⠉⠉⠉⠀⠀⠀⠀⣀⣀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⡷⠀⠻⠒⠒⠊⣉⣁⣤⠤⠤⢤⠒⠒⠊⠉⠻⠟⠉⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⢠⠊⢠⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠦⠤⠖⠲⠶⠬⠭⠥⠤⠤⠤⠤⠤⠤⠭⠶⠦⠤⠤⠤⠤⠶⠮⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠀⠨⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣤⠿⣿⠻⡿⠿⢿⣯⡭⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠻⠶⣿⣛⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣮⡳⡀⠀⢸⣭⣭⣭⣭⠷⠶⠋⠁
⠀⠀⠀⠀⠀⠀⣰⣟⣿⣿⣿⠿⣦⡔⠻⡟⠋⠻⡟⠉⠁
⠀⠀⠀⠀⠀⣼⢏⣾⣿⢿⠃⠀⠈⢧⡀⠳⣄⣠⠃
⠀⠀⠀⠀⢰⡟⣾⣿⣿⣿⠀⠀⠀⠀⠉⠀⠉
⠀⠀⠀⠀⣿⣇⠿⠿⠿⣻⡇
⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠃




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢫⡄⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣧⠄⣰⠷⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣶⡦⣦⡄⠀⠀⡠⢖⢾⠻⣇⡑⠶⠒⠲⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⣿⣿⡿⡿⢇⢶⣄⠀⠀⠚⢧⣻⣿⣶⣾⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣯⡍⠀⢻⣿⣧⡀⠻⡇⣧⢳⡄⢀⠘⣹⠟⢿⣿⡟⡿⣧
⠀⠀⠀⠀⢀⠄⠀⣀⠀⠀⠀⠀⢀⣠⣦⣿⣧⣀⣀⣯⣉⣷⡈⢿⣏⢿⠇⢸⣷⠃⠀⠘⣿⢀⠀⠈
⠀⠀⠀⠀⣼⣶⣿⣿⠀⠀⠀⢸⡏⠀⠘⠛⢟⠉⢋⣻⡛⣿⠃⠀⢹⣦⣾⣽⠇⠀⠀⠀⠟⠋
⠀⠀⠀⠀⢹⣿⣿⣿⣧⡑⣤⣾⡷⠰⠀⢠⣿⣶⣿⣿⣷⣿⠁⠀⢸⣿⣤⡟
⠀⠀⠀⠀⣾⡏⢿⣿⣿⣿⣿⡟⠆⢐⣴⠟⠀⠉⠉⠛⠛⢛⡀⠀⣨⣩⣟
⠀⠀⠀⠀⣿⠀⡀⣿⣿⢟⣿⣋⣴⠏⠀⠀⠀⠀⠀⠀⢀⣮⣤⢾⣿⣿⡇
⠀⠀⠀⠀⣿⠆⢰⣿⣯⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁⠸⡿⠉⠻⣤
⠀⠀⠀⠀⣿⣆⣮⣿⣿⡿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠘⠁
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣦⠈⠹⡄⠀⡀⠀⡀⠀⠀⣦⢢⣄⣠⡋⣀⣀⡀
⠀⠀⠀⠀⠀⢹⠙⢛⠉⠁⠁⢀⣷⡜⢀⡦⠜⠛⠒⠿⠷⠿⠋⠛⠋⠛⠛⠒⠒⠆
⠀⠀⠀⠀⠀⢸⣦⡈⠻⣶⣿⡿⣿⠇⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣄⣀⣤⣤⢠⣤⠀⣠⣄⣠⣀⣤⢤⣤⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⣇⠄⠀⠀⠀⠀⠟⠟⠛⠳⠸⠚⠀⠳⠐⠛⠛⠆⠘⠘⠟⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠁⠶
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⡀⠀⠀⠸⡆⢸⢨⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠨⠀⠀⠀⠛⠿⠘⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠴⢠⡄⢿
⠀⠀⠀⠀⠇⢀⠰⠀⠀⢀⣸⠋⠀⠀⠀⠀⠀⢰⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⠀⠀⠆⠀⢐⠀⣾⠀⢾⡇⠀⡇⢲
⠀⠀⠀⠀⠛⠈⠓⢀⠚⠻⠿⠀⠀⠀⠄⠀⠂⠉⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠘⠓⣾⡿⢹⢲⣼⡇⡆⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠒⠏⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⠀⠁⠸⣼⢸⣿⢹⣡⡇⡄
⠀⠀⠀⠀⠀⠀⣀⡀⣀⡲⡄⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⣀⠠⠴⠀⠄⠠⠐⠚⠂⠀⠀⠀⢀⡀⠀⢀⡹⠎⠋⠈⠋⠾⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠋⠁⠈⠉⠉⠠⠀⠖⢊⠁⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣳⣤⡤⠖⠂⠀⣈⣦⣴⣏⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⢶⡶⡶⣾⠟⢿⣿⣿⣿⠿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡿⡄⠀⠀⢘⠀⠀⠀⠀⠀⢠⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣻⣂⡀⠀⠻⢤⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣻⡿⠉⣅⣷⠰⠒⠒⠠⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣿⡷⠄⣿⡁⠀⠉⠁⠀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡦⢤⣁⠁⠀⣿⣶⣤⣴⠞⣹⣿⣧⡀
⠀⠀⠀⠀⠀⠀⣠⠒⠠⠤⠤⠤⠂⠀⣿⣷⠰⠷⠄⠀⡿⠛⠛⠁⢀⣿⣿⡿⣿⠒⢤⣀
⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⢿⢿⣧⣘⣡⣴⣿⣦⡀⠀⣾⣿⢿⠃⣿⡆⠀⠈⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⢉⣽⣿⣿⣯⣛⢾⣿⣋⣈⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⠶⠋⠈⢿⣿⣽⣿⣷⣦⣍⣿⣷⣿⣿
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣴⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃

⠀⠀⠀⠀⣠⢤⣀⡀
⠀⠀⠀⣼⣿⣀⣀⣈⣉⣓⠒⢲⣿⡻⠭⠩⣍⣙⣓⣒⣒⣒⠲⠦⠤⠤⣀⣄⣀
⠀⠀⢸⡿⢿⣧⠀⠈⠉⠉⠉⠉⠙⣿⠀⠀⢹⡯⠭⠭⠭⢿⣿⡛⢇⠀⢰⣿⡿⠃
⠀⠀⠈⠻⣶⣿⠛⠛⠻⠿⢿⣿⣿⢿⣶⡶⣾⢿⣷⣤⣤⣤⣤⣥⣤⡄⠀⢧⡀
⠀⠀⠀⠀⣿⣻⣤⣤⣤⣤⣬⣭⣿⣀⣀⣀⣿⣶⠿⠶⠶⠶⠶⢦⣀⠀⡠⠄⠋⠷⠤⢄⡀
⠀⠀⠀⠀⠈⠛⠛⠛⠋⠉⠈⠹⣿⣏⠉⢹⡿⢿⣆⣀⣀⣀⣀⣈⣉⠟⠈⠁⠄⠀⢀⣤⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠛⢻⣿⣿⣿⣿⡿⠿⠿⠃⠀⠀⠀⠀⠀⣸⣿⣧⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣤⣤⣤⣤⣤⣤⡴⠦⠤⣶⣶⣶⣤⣴⢆⣿⣿⣿⣎⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣿⣿⣿⠋⠉⢿⣷⠀⣴⣿⡿⣿⣿⢟⣾⣿⡿⢿⣿⣮⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠘⡆⠀⢸⣿⢀⣿⠿⠇⠀⣬⣾⣿⣿⣶⣶⣿⣿⣧⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠱⣄⢸⣿⣜⡝⠀⣠⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠬⠭⠭⠖⠒⠚⠁⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠏⣿⣿⣿⣿⠿⢿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⣿⣿⣾⣾⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⡀⠹⢿⣿⣿⣿⣿⠿⣟⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠂⠈⠉⠉⠀⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⣂⣀⡀⠲⠂⣢⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠉⠉⠀⠂⠥⠀⢠⣁⡒⠤⢡
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⡟⢅⡒⠤⠀⣀⠀⠀⠀⠈⢉⣹⣾⡷⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢰⠁⡰⢀⠔⠉⠓⠒⠦⠶⠮⠟⠛⠻⠯⢤⡋⠀⢀
⠀⠀⠀⠀⠀⠀⠴⣥⡮⠐⠁⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⣿⠉
⠀⠀⠀⠀⠀⠀⠀⢎⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠗⠢⠀⠉⠩⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠁⠈⢢⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣇⣀⣀⣠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠀⠀⠀⠂⠤⢀⣄⣬⣷⣒⣒⡲⠶⢶⣊⣭⡤⣖⢺⡝⣏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣁⠢⠤⠄⠀⠀⢠⣔⣿⠭⠔⣒⡹⡟⣿⡼⠧⢟⣚⣩⢥⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠶⢶⠠⠀⢤⣀⠀⠀⠀⢱⣊⠭⠔⡲⢳⡸⣶⠻⡏⣷⠸⣏⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢸⠔⢒⣉⢭⠅⠀⠀⢀⠀⡠⢊⡴⠋⢦⢿⠧⣗⠺⠭⠝⠊⠁
⠀⠀⠀⣀⠤⠔⠒⠒⠒⠤⣄⠀⡠⢊⡥⢸⠋⠉⠠⠺⠀⠀⠀⠸⠊⡠⠊⠀⠀⣨⡎⡧⠒⠒⠤⢤⡀
⠀⣠⠊⢁⡤⡖⠒⡖⠒⡤⡤⢋⣴⠋⢰⢸⠀⡐⠁⡃⢀⣀⠀⠴⡎⠀⠀⡴⠋⣀⡵⡸⣲⠒⢢⢄⡈⠳⣄
⣰⠁⣰⣃⠀⠸⡀⣇⣼⢞⡥⢺⡀⢳⣸⢸⣈⣻⣽⡿⣶⣤⣀⣠⠇⠀⡞⠀⡜⠁⠙⣧⢻⢠⠏⠀⢙⣆⠈
⡇⠀⣇⣈⣑⣦⡷⢻⡥⠾⠞⠛⠓⠉⠉⠉⠀⢏⢸⡝⡻⠞⠀⠀⠀⢰⠁⢸⠈⣉⣒⣼⠞⢧⣒⣋⣁⣸
⣇⠀⡇⢀⡠⠚⡿⡾⡛⢇⡒⢒⠖⢲⠖⠒⠒⠒⠛⠉⠀⠀⠀⠀⠀⠸⡀⢸⣉⡠⠖⡹⠳⢏⠑⠢⢄⣸
⠘⣄⠘⢯⡀⡼⠀⡇⠱⡀⣨⠏⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠻⣄⡴⠁⡟⠘⡆⢀⡰⠃⣠
⠀⠈⠳⣄⠉⠓⠒⠓⠒⠉⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡈⠑⠒⠓⠐⠚⠉⣠⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡶⠦⠄⠀⠀⠀⠀⠀⠀⢀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⢋⣠⣤⣤⣀⠀⠀⠰⠶⠂⠀⠀⠀⠀⢰⠟⠁
⠀⠀⠀⠀⠀⠀⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠶⣾⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠠⡀⣤
⠀⠀⣀⠀⠤⠀⠴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠛⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⢀⣤⠀⠤⠆⠀⡀
⠒⠁⠀⠀⠑⠈⠩⠖⣨⡥⠔⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠁
⠀⠐⠊⠁⠈⠔⣊⠁⢀⡴⠖⢉⡤⠄
⠆⡠⠖⣂⡴⠏⣡⡶⠋⣴⠞⢉⣴⠞⢋⣡⣤⠀⢀
⣉⡤⠞⠡⠤⠖⠋⢀⡼⠁⠔⢉⡴⠚⣉⠥⠴⠚⢋⣠⠴⠆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠟⠀⣁⠖⢚⣩⠶⠋⣠⠶⠒⣩⡤⠶⠒⣠⡤⠞⠛⢁⣴⡶⠗⠋⣉⣤⠔⢂⠄⢀⣀
⣤⡿⢃⡴⢋⣤⠶⠛⣉⡤⠞⣋⠴⣠⡿⢋⣤⠾⠁⣸⠟⠀⣠⡾⠋⣠⡾⠋⣠⡾⠋⢠⣼⠃⣠⡤⢀⣠
⣩⣔⣯⣖⣋⣴⣞⣉⣡⣶⣛⣡⣞⣩⣴⣛⣈⣴⣚⣁⣤⣞⣉⣤⣾⣋⣠⣾⣛⣡⣴⣛⣀⣜⣡⣾⣋⣡⣤⣟⣃⣤⣶⣂⣠⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠠⠴⠤⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⢻⠿⠋⠀⠀⠀⠀⠀⠔⠀⠀⠀⠠⠂⠀⡠⠶⠖⠀⠖⠀⠆⠀⠆⠀⠛⠒⠠⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸
⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⣀⣈⠁⠀⢠⠂⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⣀⠀⢐⠀⠀⠁⠀⠀⠄⢀⣠⡆⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠐⠄⣀⠀⠀⠀⠀⠀⠀⠀⡄⢀⢀⡯⠖⠀⠀⢹⠁
⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⢸⠉⠃⠀⠀⠀⠀⠀⠀⠀⠓⠀⢒⠒⠐⠂⠀⠀⠀⠈⠉⠓⠒⠒⠒⠒⠓⠋⠉
⠀⠀⠀⠀⠠⡀⢰⠀⠀⡆⡆⠘⠀⠀⡇⠀⠀⡏⠀⠀⡔⢸⣽⢲⡄⢁⣈⠀⠀⢀⡀⠐⠃⠀⢀⠉⠀⠀⠘⡁⠀⠒⠒⢀⡀
⠀⠀⠀⠀⠀⠁⠈⢀⠟⠛⢻⡁⠀⢠⡇⠄⢠⠃⠀⠸⣚⣚⣗⣰⡇⢀⠀⠀⠀⠈⠀⠀⠀⣰⠛⢿⡄⣴⠀⠀⠀⠀⣄⢸⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⢸⡇⠀⠈⠁⠀⠀⠙⢋⠏⠃⠄⢀⠁⠀⡀⢠⠀⠀⠀⠘⠳⠾⠃⢸⠀⠀⠀⢰⡟⠊⠃⠀⢠
⠀⠀⠀⠀⠄⠀⠀⢸⠀⠀⠀⣿⠀⠘⠓⠀⠀⢸⢠⡖⠀⠸⠀⢰⠀⢸⠀⢀⣇⢀⠀⠀⢰⠛⠉⠀⠀⠁⠀⠀⠀⢸⡇⠀⠀⠈
⠀⠀⠊⠀⠀⠀⠀⠈⠆⠀⢰⠏⣀⠯⠀⠀⠀⠈⢸⡇⠀⠀⠀⢸⠀⠈⠀⠐⠶⢺⡀⢀⡀⠧⠔⠰⠶⠆⠀⠀⠐⠀⡇⠀⣷
⠀⠀⠀⠀⠀⠐⠀⠀⠀⢒⡋⣀⢹⠀⠀⡄⠀⠀⠠⠁⠆⡏⠀⢸⠀⠀⢠⡆⠀⠀⠀⠈⢱⢸⠀⠀⢆⠉⠀⠀⠀⢰⡇⠀⠁
⠀⠀⠛⠒⠋⠁⠀⠀⠀⠸⡝⠹⢸⡀⠀⠳⠀⠀⡄⠀⠀⠁⠴⠞⠀⠤⢹⠀⠘⠌⠀⠀⠈⢘⠀⠀⢸
⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⠑⠃⠀⢡⣤⣤⡄⢰⠁⠀⠀⠀⢠⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠇⠀⠀⠀⣀
⠐⠒⠒⠒⠒⠒⠒⠀⠒⠒⠓⠒⢢⣤⣤⣤⡄⢸⠀⠀⠀⠀⠘⡇⠀⠁⠀⠀⠀⠃⠀⠀⠀⠀⡄⠀⢀⡃⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠘⠀⠀⠂⠀⠀⠏⠀⠀⠀⠀⠀⠋⠀⠀⠄⠀⠃⠀⠀⠳⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠘⠛⠀⠤⢀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠄⠛⠛⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣿⣿⣿⡿⣿⡿⠷⢶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢾⣿⣿⣿⢿⣿⣿⣿⡁⠀⣀⣈⠙⣿⣷⣾⣾⡂⡀⠀⠀⢤⣀⣄⡂⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⢻⣿⣽⣿⣶⣾⣧⡤⠀⠉⠉⠉⠵⢞⣻⣿⣟⡛⠓⠾⠶⣿⣿⣿⣦⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣿⣿⣆⠀⠀⠀⠉⠻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣿⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠛⢿⣿⣿⣄⠀⠈⠙⠓⢻⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⢻⣿⣿⢛⢻⣿⣿⠟⠃⠀⣠⣤⣬⡙⢿⣿⣶⣄⡀⠀⠈⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗⠇⠀⠙⠋⠀⠀⠘⠛⠟⢛⣿⠗⢋⣸⣿⣻⡟⠶⠶⣤⣤⣤⣤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⢙⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠛⠀⠀⠀⠀⠀⠀⠀⣠⣒⡀⠀⣟⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⠓⢃⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⣿⡇⠀⠀⠘⣦⠀⠀⠀⡀⢷⣤⣤⣤⣄⠐⠛⠠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠿⠀⣸⣿⣧⡀⠀⠀⠙⣦⣄⠀⣷⣤⣀⣀⣈⠀⠉⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣋⣤⣿⣷⣤⣹⣿⣿⣿⣦⣀⠀⠘⠻⢿⣤⣀⠉⠉⠉⠛⠀⠀⠀⡀⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠉⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⢿⣿⣷⣶⣦⣿⣉⠛⠛⠒⠀⠀⠄⠈
⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠉⢹⣿⣿⡶⣤⣤⠄
⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡁⣯⠉
⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢸⣿⣿⡜⣧⣀⠀⠀⠀⠀⠀⠀⡂
⠀⣰⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣣⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⡎
⣴⠋⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡹⣿⣿⠓⢦⡴⠖⠋⣿⡙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣧⢻⣿⠀⠀⠀⠑⠤⠾⣿⡈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣎⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃
⣿⣿⣿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⢿⡿⣿⣿⣶⣶⣦⣤⣤⡄⢠⣄⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣘⣿⣯⣲⣦⠀⠀⠀⠀⢀⠄⠁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣇⠈⣧⠀⠀⢠⣿⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡈⢿⣿⣿⣆⣿⡇⢠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣉⣽⣿⣿⣿⣿⣿⠷⠚⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⣩⣿⣿⣿⣾⣶⡶⢲⣾⡑⢝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣧⣤⣾⣿⣿⣿⣿⠟⣸⠃⠈⠻⡿⣄⠁⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⡀⠋⢠⡄⣤⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠋⠀⢀⣴⣾⣿⣿⣶⣄⡀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⣽⣯⠀⠀⣿⣿⡟⡇⣿⠿⠛⠃⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡗⡀⠀⢿⣿⣿⣧⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⡇⠀⠀⠛⢹⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠒⠀⢠⣤⣿⢸⡟⣿⣿⡄⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⢈⣹⣻⣿⣿⣷⡶⠶⢦⣤⣄⣀⣀⡀⠀⠀⠀⢀⠠⠀⠀⠀⢀⣿⣿⣼⣧⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠉⠙⠻⢿⣿⣷⣤⣮⣿⣿⣿⣿⣿⣿⣷⣂⣀⡤⠖⣾⣿⣏⢻⣿⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠁⢷
⠀⠀⠀⠀⢶⣾⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣽⣿⠿⢛⣁⣀⣀⣿⣿⣷⡬⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠛⠛⣿⣯⣍⣉⣉⣉⡙⠛⠛⠻⠿⡯⠔⠚⠋⠉⠉⣙⡻⣿⣿⣧⢀⡀⢰⣁⠀⠀⠀⡆⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠐⠿⠿⠿⣿⣿⣿⣻⣻⣶⣶⣦⣤⣄⣀⡀⠐⠉⠁⣀⣿⣿⣿⣧⡬⠙⠿⠿⡿⠀⣂⣠⡤⠞⠋⠁
⠀⠀⠀⠀⢸⣶⣦⣤⣀⣀⡀⠊⠉⠉⠛⠛⠻⠿⣿⣯⡤⠴⣚⣉⣽⡿⠿⣿⣿⣿⣿⣿⣻⣟⠁⡿⠟
⠀⠀⠀⠀⠙⠿⠷⠾⣯⣿⣿⣿⣶⣶⣦⣤⣀⣤⠼⠗⠊⢉⣉⡠⠼⠁⠀⠀⣌⠁⠀⠀⢠⡿⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⣿⣧⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⡼⢽⡧⠀⠼⣏⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣻⡗⢀⡚⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠷⢦⣯⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣦⡤⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣒⣒⣢⠉⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠄⠀⠀⢠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣆⠛⢛⣿⣾⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠉⢍⡴⠋⠀⢰⡉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠹⠛⠿⢿⣿⠀⠀⠀⠀⠀⣀⣠⡾⠉⠀⢠⡆⣼⡇⠲⢠⣤⣁⢼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⢼⡁⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⢈⠿⠉⠹⡄⠀⣀⠈⠃⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠷⣦⣴⡿⠿⠗⠆⠀⣀⣤⣿⡇⠀⠀⠀⣎⣀⢠⡀⠹⣧⣄⠉⣴⠿⢓
⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠙⢆⠀⠀⠀⠘⣓⠈⢿⡄⠀⠀⠀⠏⣛⠂⠹⣶⣿⣉⡾⠃⢀⣉⣷
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⣿⣷⣮⠙⢦⡀⠀⠀⠁⠆⠈⠁⣄⠙⣷⠰⣎⠙⢋
⠀⠀⠀⠀⠀⢀⠆⠀⠐⣄⠀⠀⠀⠀⠀⠀⠄⠀⢠⣿⣿⣿⠀⠀⠙⠶⣄⡀⠈⠄⣀⣸⣿⢁⡸⠌⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣀⠀⠀⠀⠀⠀⠠⠀⣸⣟⡿⣿⡄⠀⠀⠀⠈⡙⠓⠺⠛⠛⠀⠙⣅
⠀⠀⠀⠀⠀⠀⣀⡀⠉⢻⡏⠛⠀⠀⠀⠀⢠⢦⣿⠿⠠⣿⠀⡓⠄⠀⠶⠴⠞⠃⢶⡄⠀⠰⡋⠛⠠⣄
⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠙⣷⣤⡄⠀⠀⣠⠞⠁⠀⠀⢸⣧⠀⠉⠳⣄⠀⠐⠦⠄⠼⠀⠀⢹⠁⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣹⠟⠅⢠⠚⠀⣶⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠒⠀⢲⡄⢅⢁⡀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⣿⠃⣴⠇⠀⢀⣩⣤⣤⡈⣿⡟⠀⠀⣄⠀⠠⡄⠀⠀⢀⠀⠀⣀⢠⠀⠙⠠⠀⠶⠀⠀⠀⢠⡄⠐⢘
⠀⠀⠀⠀⠀⣶⠈⠷⠀⠀⢏⠁⠀⠉⠀⠀⢸⡇⣿⣿⡿⡏⢠⠀⠀⠀⠁⠀⡀⠀⠞⣩⠃⣨⡀⠉⠀⠀⣠⠀⠀⠒⠀⣰⠞⠃⠀⡎
⠀⠀⠀⠀⠀⠈⠀⠀⢠⣀⣼⡀⢰⡄⠀⠀⣽⡧⣿⣿⡇⢸⡄⠃⢠⡅⠀⠠⠂⠀⠀⠈⠀⠈⠉⢀⠀⣠⠃⠚⠀⠀⡠⠂⠀⢠⢀
⠀⠀⠀⠀⠀⠀⠐⠀⣀⠸⣤⣿⣿⣿⣷⣿⣿⣧⣿⣿⣇⠀⠻⡄⠈⣧⡀⡀⢀⠀⠛⢦⡘⠠⣠⡄⣠⣠⣄⣈⣴⠋⠀⠀⣠⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣝⣡⡿⣿⣿⡟⣩⡟⠁⠘⢦⣘⠇⠀⠈⠦⣄⠀⠱⣜⣧⣩⡎⠁⠀⠀⠀⣠⠶⠔⠃⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡉⢹⡍⠈⣿⣿⠗⠟⠀⠰⢧⣄⡉⠛⠶⣇⠢⠄⠁⠀⠙⢯⡝⠀⠀⠀⠀⠘⠠⣀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⢸⡃⠈⠀⠀⣿⣿⠃⠙⠛⠙⠁⠀⠙⢢⣦⣌⣙⣠⣤⣄⡠⠀⠓⠀⡀⠀⠀⠘⠦⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠘⢡⠆⠀⠀⠹⠅⠈⠁⠀⠀⠀⢀⣴⠟⣿⣿⣿⣿⣿⠏⠁⠀⠐⣄⠙⠄
⠀⠀⠀⠀⣠⣤⣄⣀⣰⣿⠀⠀⠀⡀⢘⣄⠀⠀⠀⡀⠀⠀⢤⠘⠀⠁⢀⠿⢿⡿⠟⠁⠀⠀⠀⠙⢤⣀⠸
⠀⠀⢠⡾⠉⣩⢽⣿⣿⣿⠀⠀⠀⠀⠨⡉⢿⢛⣶⠶⢾⣗⣠⣤⣴⣺⣿⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⢡
⠀⠀⢻⣷⣾⣿⣦⣈⡛⣿⡀⠀⠀⠀⠠⢀⣌⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⢣
⠀⠀⣴⣿⣿⣿⣏⣋⣿⣾⡷⠀⠀⠀⠀⠈⣸⡿⠿⢿⣿⣿⣿⣿⡿⠋⠙⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢳
⠀⠈⠉⠻⠿⠛⠿⣦⣍⣟⣷⢀⠀⠀⠀⠀⣼⠃⠀⠈⠻⣿⣿⣿⠇⠀⠀⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢸
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣆⠁⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⠟⠀⣠⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢀⠦
⠀⠀⠀⠀⠀⢠⣴⡿⢿⡟⣹⣿⠀⠁⢀⠞⠁⠀⠀⠀⢀⠿⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⠻⢾⡀
⠀⠀⠀⠀⠀⣿⣿⣤⣤⣿⣿⡇⠀⠀⡸⠀⠀⠀⠀⠀⠀⡄⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠰⠛
⠀⠀⠀⠀⢠⣿⡟⠛⠋⠀⢉⠀⠀⢾⠁⠀⠀⠀⠀⠀⠢⣁⠀⠘⡄
⠀⠀⠀⠀⠀⠹⡅⠀⠀⠀⠈⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢁⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠒⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⣀⡰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣄⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣤⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣥⠼⢺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⡿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠻⢛⠛⣿⠋⢹⠛⢛⡟⠟⣛⣿⢟⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣿⣷⣿⣷⣿⣶⣿⣿⣷⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠞⠛⠛⠲⢦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠒⠲⣄
⠀⠀⠀⠀⠀⠀⣀⣤⣶⠶⠾⠿⠿⠿⠶⠶⢶⣦⣾⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠽⠃⠈⢳⡄
⠀⠀⠀⠀⠀⠀⢿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠒⢦⡀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⠟⠳⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢀⡀⠘⣿⡟⠓⢦⣄⠀⠀⠀⠀⠹⣦⡀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣪⢋⣁⠀⣿⠀⠀⠹⡮⡙⣶⣄⡀⠀⠹⣿⢿⣾⣿⡟⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢟⣯⡅⠀⣿⠀⠳⣦⣱⠱⡏⢎⢻⠲⣄⠈⢷⣿⣿⡆⠀⠀⠈⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢐⠿⠃⠀⡇⠰⡆⠓⠘⠂⠁⠈⣿⢲⡟⣽⢦⣝⠻⣿⣦⡀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⣽⡄⠀⢸⠇⠀⠈⢢⣀⠀⠀⢀⡏⢦⡁⠉⢸⠀⠉⠚⠿⣿⣦⣄⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣲⡷⠀⣾⠀⢀⠿⠄⠀⠀⡄⢸⣛⠯⠉⠀⣼⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠹⡳⣸⠃⠀⢫⠟⠃⢠⢀⠻⢿⠞⢁⡀⠾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢷⡇⡿⠀⡰⠁⣠⢆⡸⠏⣠⣯⠤⠊⣫⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢹⡇⠀⡧⠞⠁⢻⡴⠞⠃⣇⠀⣿⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⢸⣇⠀⠀⠀⠀⢸⡇⠀⠀⢹⣾⠇⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠈⠉⠙⣿⣤⣤⣀⣴⡿⠃⠀⠀⠀⢳⡀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠈⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⡀⠀⠀⠀⠀⠈⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡯⢝⣺⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠋⠳⣄⡀⠀⠀⠀⠀⠀⠉⠉⠛⠳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠈⠉⠒⢯⡙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠙⠳⣦⣤⡤⠾⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⣤⣸⣿⣦⣝⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⠿⡖⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠈⠓⢤⣉⠛⠒⠶⠶⠶⠴⠶⠶⠶⠒⠚⠛⣉⣡⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠒⠒⠒⠒⠒⠚⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠣⠀⠀⠀⠃⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠈⣀⠀⠀⠀⠀⡆⠀⠀⠀⡤⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡇⠀⠃⠀⠁⠀⠀⠈⠀⠀⠀⢀⠀⠈⠁
⠀⠀⠀⢀⠀⡏⢱⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⡜⣙⣎⣸⢇⡇⡀⠀⡀⠆⠀⡔⠀⠀⠀⠈⠁⠀⠀⠀⣀⡀⠀⢠⠖⢦⠀⠀⣀
⠀⠀⠀⡋⣇⡇⢸⣠⢏⡇⠀⠀⠀⢠⠶⡄⠀⡇⠉⢩⠦⢊⡀⠘⢆⣡⡠⠜⣀⡔⠉⢀⠀⠠⠖⠀⠀⠁⢀⠀⢸⠁⠸⠀⡰⠁⠱⡀⠀⠠⡄⢀⡀
⠀⠀⠀⠳⠤⡄⢰⠓⠊⠉⠉⠉⠓⢣⠀⡧⢬⡇⠦⣿⢰⢳⠈⢲⠊⠀⠀⠀⠀⠙⣏⠥⠒⠐⠒⡀⠀⠉⠁⠀⠸⡄⠆⢧⣇⠀⠀⡇⠀⡦⡇⢨⠀⢶
⠀⠀⠀⠀⠀⠁⢸⡄⠀⠀⠀⠀⠀⠈⣆⠉⠒⠁⠀⠻⠞⣸⢅⣣⠀⢀⣀⠤⠔⠒⠛⠂⠉⠉⠀⠀⣫⡄⡿⠋⠉⠙⠤⢠⡀⠀⠀⠳⢖⣡⢇⣈⡄⡇
⠀⠀⠀⠀⠀⠘⠈⠀⠀⠰⠷⠆⠀⠀⠈⠑⠒⡄⠈⡴⠉⠁⠀⠉⠙⠢⠤⢀⣀⣀⣀⣀⢀⠀⠀⠀⠀⠈⠁⠠⠄⠀⠀⠀⢣⠀⠀⡏⠉⠀⠀⠈⠇⠀⢤⣀
⠀⠀⢀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃⣶⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠁⠒⠀⠒⠀⢠⠀⢠⢸⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⠈⠁
⠘⠚⠚⠓⠚⠒⠒⠛⠛⠛⠚⠒⠒⠚⠒⠚⠒⠓⠂⠛⠓⠚⠚⠚⠓⠓⠚⠓⠒⠚⠒⠒⠓⠒⠒⠒⠒⠒⠚⠒⠛⠚⠒⠓⠛⠒⠒⠓⠛⠒⠒⠒⠒⠚⠛⠚⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⣾⡥⣄⠀⠀⢄⠀⢀⣠⠀⠀⠉⠁⠈⠉⠉⠉⠉⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⠁⠀⠀⢦⠀⠀⠱⠂⠀⢀⣀⣀⣨⠀⠀⠀⠀⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠱⠶⠖⠁⠀⠙⠛⠁⠀⢀⣀⠁⠊⡠⠂⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢿⣸⡇⠈⠀⠀⠋⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠖⠉⠛⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢀⣼⡇⠀⠀⠐⠒⠉⡄⠀⠸⣠⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠸⠿⠇⠀⠀⠀⠀⠀⠸⠆⠀⠛⠀⠀⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠸⠯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠚⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠠⠀⠁⠀⠀⠀⠡⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⡀⠀⠀⠀⠀⠀⡀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠚⠛⠃⠀⠀⠀⠈⠀⠀⣀⡜⠁⣀⠆⠀⠀⠀⠀⠀⠘⣄⡀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣠⠆⠘⠀⠀⠀⠀⠀⢄⡀⠀⠌⠀⣶⡄⠀⠀⠀⠀⠀⢀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⡵⠋⠀⠀⠀⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢶⡺⠛⠛⠓⠶⣤⠀⠀⢸⡟⠲⠦⠤⠤⠴⢶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣋⠀⠀⠀⠀⣀⣼⡇⠀⣿⣝⠷⠶⠤⠤⠶⣾⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣧⣉⡙⠛⠛⢋⣹⡿⠃⠀⢻⣍⡙⠛⠛⠛⠋⣿⡟⠀⠀⠀⣀⣤⠴⠚⣋⣥⣙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⣥⣍⣉⣛⣫⣽⣿⠀⠀⠀⠀⠈⠙⠛⢛⣛⣫⣥⣤⠶⠚⠋⠉⠀⠀⠉⠁⠀⠉⠓⠌⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⣀⣀⣤⠾⠛⢀⣀⣀⣤⡤⠶⠒⠛⠻⣿⣿⣿⣷⣄⠐⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣟⠋⠉⠉⠀⠀⠀⠀⠀⠀⣷⠈⠻⣿⣿⣿⣷⣄⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⣀⡴⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡶⠟⠳⣄⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠈⠻⣿⣿⣿⣿⣦⡛⠀⠀⠀⢀⣠⡴⢚⣩⡴⣾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⡛⢦⡀⠈⠳⣄⡀⢄⡀⠀⠀⠀⠉⠻⠷⣦⣬⠙⢿⣿⣿⣿⣦⡴⢞⣫⡵⣞⣫⠵⢶⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⡛⢦⡙⢦⣄⠈⠛⢦⡙⠶⣤⣤⠤⠤⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⠿⢛⣋⡭⠶⣚⣻⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡙⢦⡙⢦⡙⠷⣄⠀⠙⠶⠶⠶⣒⣛⣯⠽⣷⣿⡿⢿⣿⣿⣿⠶⣛⣭⠵⠖⠋⢩⡴⣞⠛⠛⠛⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣝⢦⣙⢦⣈⠳⠶⠖⢛⣋⣭⠵⠖⣛⣭⠴⢖⣻⣿⣿⠟⠋⠁⠀⠀⠀⠀⣿⣜⣃⠀⠀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⢦⣝⠷⠶⢚⣋⣥⠴⢞⣫⡭⠶⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣴⢻⣤⣉⠙⠛⢋⣉⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⠷⣚⣫⡥⠖⠛⠉⠁⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀⠀⠀⣿⠦⣤⣍⣉⣩⣭⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢹⠒⠁⠀⠀⠈⠙⣆⠀⠙⠶⣤⣤⣤⣤⡤⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠳⠦⢤⣤⠤⠶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⢤⣤⡤⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣖⠺⣄⡙⢷⣦
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠉⠓⠿⣷⣾⣄⣀
⠀⠀⠀⠀⠀⠀⢸⣁⣀⠀⠀⠀⠀⠉⠙⠻⠿⣖⣢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⣰⠁⠉⠛⠶⢤⡀⠀⠀⠀⠀⠀⠉⠛⠶⢭⣑⠲⢤⡀
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢰⠷⠴⣦⣤⣀⠀⠀⠀⠀⠀⠈⠙⠚⠧⣴⣂⢄⣀
⠀⠀⠀⠀⠀⠀⠙⠓⠦⣤⣀⡾⠀⠀⠀⠉⠙⠳⠤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢽⣍⡓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⡲⠤⣀⠀⠀⠀⠀⠀⠉⠑⠢⣄⣀⡤⣀⡀⠀⠀⠀⠈⠙⠓⡦⣭⣓⡢⢤⣀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣭⡓⠦⣤⡀⠀⠀⢼⡁⠀⣽⠶⡭⣕⠲⡄⠀⠀⢠⠁⡆⢈⠉⡷⢬⣝⣲⣴⣊⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡀⢯⣀⠀⠀⠉⠛⠻⠾⣤⣠⠟⢓⡖⠦⣼⠰⠇⢸⠆⡇⡇⢰⠉⡏⠳⠦⢭⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢩⡟⠲⠦⢤⣄⣀⠀⣆⠀⠺⣄⠀⢈⡿⢲⡞⠛⡷⡷⣸⢸⡇⣇⠀⠀⣿⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣼⠁⠀⠀⠀⠀⢈⡟⠳⢦⡀⠀⢙⡶⠟⠀⠱⣄⣉⠉⣹⣮⡇⣿⠀⢠⡷⠒⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢧⡇⠀⠀⠀⢀⣠⣞⣀⣠⠼⡇⠀⣸⠁⠀⠀⠀⠀⠀⠉⠁⠀⠙⠳⡏⢹⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡀⢿⣄⠀⠀⠉⠉⠉⠉⠀⠀⣷⠀⣸⠇⢀⣤⣤⣄⠀⠀⠀⢀⣠⢶⡿⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠿⣿⣶⣤⣄⡀⠀⠀⠀⣿⣠⢧⡀⠘⠿⠤⠟⠀⠀⣠⢋⡶⠲⠶⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢽⣟⣲⠾⠃⢻⠞⠁⠀⠀⠀⠀⠀⢠⡇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⠀⠸⣤⠀⠀⠀⠀⠀⠀⢸⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡾⠀⠀⠀⠀⠀⠀⣼⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢠⡇⠀⠀⠀⠀⠀⠀⣿⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣸⠁⠀⠀⠀⠀⠀⠀⢿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡟⠀⠀⡞⣹⢷⡄⠀⢸⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣖⡛⠺⢧⣄⡀⠙⠷⠞⠁⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⠶⢭⣽⣓⡲⣆⢀⡾⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⠛

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣷⣠⣷⣶⣤⣤⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣁⣠⣄
⠀⠀⠀⠀⢀⣀⠀⠤⣉⣩⣿⣿⣿⣿⣿⡿⠛⠛⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠈⠙⠃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⣥⣶⣴⡛⣛⡿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡍⠀⠀⠀⠐⠋⠛⡀
⠀⠀⠀⠈⠛⠋⠙⠛⠟⠋⠁⠉⠙⠉⠟⢛⡓⠋⠛⠛⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣆⠀⢀⠀⠗⠀⠑⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠏⢭⣙⠃⠚⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⣷⣦⣽⣷⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⡀⢀⠈⠃⠀⠀⣸⣿⣿⣿⡄⠀⢀⡀⢰
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠂⢻⠳⠖⠛⠉⠉⠈⠁⠀⠀⢠⠽⣄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇
⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⣑⠢⣄
⠀⠄⠀⠐⠛⠛⠛⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠓⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠿⠿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀⠀⠀⠀⢀⡀⠀⠀⠀⣠⣤⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠇⠻⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⢙⣿⣿⣿⣿⣧⡔⠂⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢷⣄⣠⣼⣴⣀⡀⠀⣾⣿⣿⣿⣿⣿⡉⠃⠤⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣷⣾⣿⣿⣿⣻⣏⣙⣿⣶⣽⡿⣿⣟⣿⣿⣶⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⢿⣟⡻⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣯⡻⢿⣽⣿⠿⣿⣻⣿⠆⠀⠀⠀⣀⣀⣴⣦⣽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡻⣶⣷⣜⣿⣥⣤⣤⣶⣾⡟⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡙⣹⣿⣆⢾⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣹⣿⡻⣷⠌⠛⢻⣼⣷⣝⣿⣿⣿⡿⠟⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⣾⡀⢞⣀⠀⠈⠈⠙⠛⠁⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣻⣿⣿⣿⣯⡛⣹⣿⣿⡇⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣥⣶⣿⣾⣿⣿⣿⣿⣿⣿⣯⣾⢿⢟⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠂⡲⣀⡹⢿⣿⣿⣿⡟⠁⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣯⣏⣀⡘⠋⣼⣿⣿⣿⣷⡆⠁⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣟⠻⣿⣿⡇⠐⣿⣿⣿⠿⣿⡄⠀⣸⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⡧⢹⣿⣷⣿⡯⢀⠐⢿⡇⢆⠣⠙⢿⡛⣰⣆⠉⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣷⣾⠿⠿⠟⠁⠘⠀⠀⣶⠀⠀⠰⣿⢻⣿⠇⠀⢸⣇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡛⢻⠿⠀⠀⢀⡐⠀⠀⠀⡩⠀⠀⠀⢙⣼⣿⠀⠀⠾⣿⣿⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠀⠀⠀⠀⠀⠈⠁⠀⠀⠚⢡⠴⠶⠖⣾⣿⡟⣠⣾⣧⠉⠉⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣤⠀⠀⠀⠀⠀⠀⠀⣄⡖⣠⣬⢀⡘⠀⣿⣿⣿⡟⢠⡟⠀⠀⢀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶
⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠿⠟⢁⡀⠀⠀⠀⠀⢀⠱⡿⠁⣿⣿⠎⢀⣼⣿⣼⠟⠶⢸⡇⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⣤⣤⣿⣿⣿
⠀⠀⠀⠀⠀⠻⠿⢛⣫⣵⣶⠴⣾⣿⡇⠀⠀⠀⠋⣠⣼⢠⣿⣿⠃⢼⣿⣿⠋⠀⠀⢸⡇⣤⠀⠠⣿⣿⣠⣀⢀⣶⣾⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡴⡖⠋⠀⣨⣟⣀⠈⠋⠀⢀⡶⠀⠀⢀⣀⣨⡿⢧⣾⣿⣿⠃⠀⠀⠀⢸⣿⣿⣿⢃⣿⣿⣯⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣠⣤⣠⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⢻⡿⠿⠿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⣿⣿⡇⠀⢰⡇
⠀⠀⠀⢸⣷⣶⣶⣶⣾⠃⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠅⠀⠀⠀⣿⣿⣷⠀⠸⠇⠀⠀⠀⠀⢀⣀⡀⡀
⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⣤⣾⠀⠀⣶⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣿⣿⠀⠀⡇⠀⠀⠀⠀⢸⣿⡇⡏
⠀⠀⠀⠀⣿⣿⣿⡏⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣾⡞⠂⠀⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡷⣿⠀⠀⡇⠀⠀⠀⠀⢸⣿⡇⡇
⠀⠀⠀⠀⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⣴⣧⠀⣿⡇⠀⠀⠀⠀⣿⣇⡀⠀⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣷⣿⢰⣾⣷⡆⠀⠀⢀⣘⣿⣧⣇⡀
⠀⠀⠀⠀⢿⣿⣿⠃⡇⠀⠀⠀⠀⠀⣿⣿⠀⣿⡇⠀⠀⠀⢸⣿⡇⡇⠀⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢸⣿⣿⢠⡇⠀⠀⠀⠀⠀⣿⣿⠀⣿⡇⠀⠀⠀⢸⣿⡇⡇⢸⣿⢈⡇⠀⠀⠘⣿⠂⠀⠀⠀⣿⣿⡇⠀⣿⣿⡇⠀⠀⠘⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠈⣿⣿⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⣿⣿⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣠
⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⣿⣿⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⣀⣤⣄⣀
⠀⠀⠀⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣿⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⢹⣶⣾⣿⣿⣿⣿⣷⣶⣶
⠀⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⢀⡾⠋⠀⠀⠙⢷⡄⠀⠀⠀⠀⣀⣙⣛⣛⣛⣛⣛⣋⣠⠀⠀⠀⠈⠻⠟⠁⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⣾⠁⠀⠀⠀⠀⠘⣷
⠀⠀⠀⣿⣿⢶⣶⣶⠀⠀⣿⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠉⠉⠉⠉⠋
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⢹⡄⠀⠀⠀⢀⢾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠻⠲⠒⠓⠓⠚
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠙⢶⣤⣾⡷⠏⠀⠀⢀⣤⣤⣤⡄
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠾⠿⢿⣿⣿⣿⠿⠷⠆⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⣿⣿⢚⣛⣾⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⣿⡆⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢀⣈⡇⠀⠀⠀⠀⠀⣾⠉⠙⠻⣿⣿⠟⠋⠉⣷
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠘⡆⠀⠀⠀⢠⣾⣿⣿⣧⡀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⣿⣿⠇⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⠦⣄⣰⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠘⢷⣤⣀⣀⣀⣀⣴⡾⠃
⠀⠀⠀⠿⠿⠛⠛⠛⠁⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠙⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣹⣿⣷⣤⣀⠀⠾⣿⣼⣿⡿
⠀⠀⠸⠤⠤⠥⡄⠀⠀⠁⠆⢨⣾⣿⣟⣿⣿⣿⣿⣿⣤⣾⣿⠿⠇⢰⣦⣤⡀⡀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡀⣀⣀
⠀⠀⠸⠿⠿⠭⠅⠭⠭⠥⠅⠈⠙⠛⠻⠘⣿⣿⣿⣿⣼⣿⣷⣶⣧⣼⣿⣿⣷⠈⠙⠉⠁⠉⠙⠉⠉⠉⠙⠉⠛⠃
⠀⠀⠀⠀⠌⠁⠁⠺⠉⠉⠁⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠿⠻⢿⣿⣿⡇
⠀⠀⠘⠁⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠉⠁⠀⠀⠀⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠉⠁⠂
⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⢀⡀⠀⡀
⠀⠀⠰⣾⣤⣦⣴⣴⣿⣽⣿⣿⣿⣦⣤⣿⡇⣶⣤⣀⣠⡄⣀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠀⣀⣀⠤⠁
⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠿⠋⠻⠿⢿⣿⣷⣿⣿⣿⣿⣿⠋⠀⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣄⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣌⡛⠿⣿⣿⠟⢃⣤⣴⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣧⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡏⣿⣏⣿⡟⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣇⣿⣷⣿⣽⣿⣰⣿⣿⢛⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⢀⣜⣣⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⢿⣿⣿⠿⢿⣿⠟⣣⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⣵⣶⣶⣶⠞⣢⣴⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣡⣾⣿⠿⣋⣵⣾⣿⠟⣋⡛⠿⠿⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣋⣼⡿⢋⣵⣾⣿⠟⣋⣴⣿⣿⣿⣿⣿⣷⡌⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⢛⣥⣾⣿⣿⣯⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⢟⣡⣾⣿⣿⣿⠻⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣱⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢟⣩⣶⣿⡿⢛⣥⣿⣿⠃⣠⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢠⣾⠟⣫⣴⣿⡿⢟⣡⣶⣿⣿⣿⣿⣄⠙⠛⢉⣉⠙⢿⣿⣿⣿⣿⣿⠿⣋⣴⣿⡿⠟⠉
⠀⠀⠀⢀⣴⡿⢱⣿⣿⣿⣿⡐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢨⣿⣿⠿⣋⣵⣾⣿⠟⠉
⠀⠀⠀⣾⣿⣿⡘⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣉⡀⢺⣿⣿⣾⣿⠟⠋
⠀⠀⠀⠻⣿⣿⣿⣌⢿⣿⣧⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣾⣿⠿⠋
⠀⠀⠀⠀⠈⠙⢿⣿⣦⡹⣿⣷⡜⢿⣿⣿⣿⣿⣿⡿⢟⣩⣶⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡜⢿⣿⣮⣥⣤⣭⡛⣩⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣎⠻⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠿⠿⠟⠉

⠀⢀⣀⣀⣀
⠀⡇⠤⠤⠤⠬⠭⠿⠿⠷⠞⡏⠉⠉⢉⣉⣉⣉⣉⣁⣀⣀⣈⣉⠉⠉⢦⣶⠖⠒
⠀⡷⣶⡶⠶⣶⣶⣶⣶⡶⢆⠀⠀⠀⢸⠀⠀⠒⠶⠤⡄⠀⠀⢩⠉⢆⠀⠣⣤⣄
⠀⢸⣀⣀⡗⣀⣀⣀⣀⣀⣘⡿⠿⠿⢿⠀⠀⠀⠀⠈⠁⠀⠀⢸⠀⢸⠀⠀⠀⠉⠒⠶⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢸⠀⠈⠙⠛⢲⣞⣿⣿⢞⣠⡟⠠⢂⡔⢄⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⠘⠒⠒⠒⠒⠒⠚⠛⠛⠚⠋⠀⠈⠉⠒⠢⢃⣠⣤⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠓⠢⡀⡠⠒⠒⠀⠸⣶⡶⠢⡄⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡅⠀⠀⠀⠀⣸⠃⠀⢸⢀⣠⣾⣿⣿⣿⣿⣿⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⣒⠁⣀⣀⣼⠛⠛⠛⢻⣿⣿⣿⣿⣶⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠐⠈⢣⢀⡨⠀⡀⠐⠈⣀⠅⠠⢇⠀⢀⡢⢸⠀⡀⠁⢀⠃⠄⢁⡊⠀⠀⢹⣿⣯⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⠙⠛⠛⠛⠛⠟⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢉⣴⣶⣶⠎⣩⡴⠚⣋⣭⣭⣍⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⣿⣿⡿⢁⡾⠋⣠⣾⣿⣿⣿⣿⣿⣎⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢸⣿⣟⣰⠟⠁⣼⣿⣿⠿⠟⠛⢙⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠸⣿⠿⠋⠀⢸⣿⡿⠉⠀⠀⠀⠀⠘⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⢤⣤⣤⣤⣤⠇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣠⣖⣯⣷⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣀⡤⢴⣲⣶⣦
⠀⠀⠀⠀⠀⣠⢮⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣟⠛⠛⠛⠛⠚⣳⢞⣭⣾⣿⣿⣿⠃
⠀⠀⠀⠀⢰⢧⣿⣿⣿⣿⡿⠿⠋⠉⠉⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣫⡀⠀⢀⡠⢞⣵⣿⣿⡻⣻⡿⠁
⠀⠀⠀⠀⡞⣾⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣤⣿⣿⡿⣿⢯⣽⣷⣿⣿⢟⣵⣿⡿⣣⢃⣴⠏
⠀⠀⠀⠀⡇⣿⣿⣟⡄⠀⠀⠀⠀⠀⠀⠀⢸⠪⣟⠦⠤⠄⠀⠀⢤⣀⣀⣀⣤⡼⠾⣛⣫⡵⣾⣥⣿⣿⣿⠟⣡⣾⡿⠋⠀⣰⠟⠁
⠀⠀⠀⠀⢻⢿⣿⣯⡃⠀⠀⠀⠀⠀⠀⠀⠸⣝⢾⣿⡃⢸⡉⢁⢀⣿⣶⣒⣺⣯⣽⣿⠷⣟⣿⣿⢟⡿⣣⣾⡻⢟⠀⣠⠞⠁
⠀⠀⠀⠀⠘⣞⣿⣟⢅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢥⣘⠛⠛⣈⣿⣛⣻⣯⣭⡷⠾⠟⠛⢋⡴⢋⣼⣿⡏⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠘⣞⢿⡪⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣠⠔⣋⣴⣿⡿⣠⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢚⣥⢾⣟⣹⡷⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢬⣓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⣋⣥⣾⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠬⣭⣉⣛⣓⣒⣒⣛⣋⣉⣥⣴⠾⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣠
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠐⠈⠉⠀⠰⠀⢀⠂⠀⢶⡔⠡⠐⡄⠀⠀⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠁⣀⠐⣄⠈⠲⣦⠀⠀⠆⠀⠀⠀⢀⠈⠀⡐⠀⠈⠂⠀⠀⠄⢮⣱
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠊⡀⠀⠀⠀⠘⠂⠀⠈⠁⢀⣀⠀⠀⠄⠡⠀⠀⠀⠁⠀⠐⠀⠀⢀⠈⠱⡃
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⣀⠑⢐⣀⣀⠠⠀⢠⠁⠁⠀⡀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠱⡄⠀⠂⠀⠀⠀⠀⠀⠈⠁⠀⣀⣁⣤⣼⡶⠂⠀⠀⠀⠈⢐⡄⠀⠀⠐⠀⢀⠄
⠀⠀⠀⠀⠀⡀⡀⠀⠈⠀⠀⠀⠀⠄⠀⠈⠉⠘⠿⠋⠊⣀⠀⠘⠀⣄⠀⠁⠀⠀⡀⠀⠀⠀⢠⠐⠀⠤⠀⠀⠀⠀⢐⠀⠀⢀
⠀⠀⠀⠀⢐⠀⠀⠀⡄⠀⠀⠀⠀⠒⠃⠉⠀⠀⠀⠁⠐⠡⠀⠀⠀⠀⠂⠀⠈⠘⡤⢔⡊⠁⠘⠠⠀⠀⠀⠂⠀⠀⠈⠠⡀⠘
⠠⠀⠌⠴⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠛⠆⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠁⢈⣦⠞⠡⠀⠉⢦⡀⠀⠈⠁⠀⠀⠠⠑⠂
⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⢈⣠⠄⠀⠀⠀⢠⠈⣂⠀⠀⠀⠀⠀⠀⠀⣴⡟⠣⠀⠀⠀⠀⢀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠒⠂⣤⣀⣀⣀⠀⠀⠀⠠⠀⠀⠐⠂⠀⠂⠀⠀⠁⠀⢀⣴⣾⡏⠁⠀⠀⠀⠀⠀⠉⠉⠙⢦⠀⠀⢀⠀⡄⠀⠤⢀⠠
⠀⠀⠀⠀⠈⠙⠛⣿⣿⡆⠀⠀⠀⠀⠐⠠⠄⠂⠀⠐⠒⠠⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣰⣶⠶⣶⣶⣦⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠉⠁⠉⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣰⣛⠸
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣉⢿⠛
⠀⠀⠀⠀⠀⠀⠀⢻⣦⠰⣶⡦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⠣
⠀⠀⠀⠀⠀⠀⠀⡼⠋⡽⠳⣽⣮⡙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡋⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣴⠟⣽⢁⣴⣿⣿⣿⠀⠈⠙⢦⣀⠀⠀⠀⡤⢤⠚⡛⠿⠟⠁⢀⡜⠁
⠀⠀⠀⠀⠀⣰⡛⣾⣿⠿⠋⠉⢿⢿⣆⠀⠀⠀⠙⢳⣤⡴⠛⠐⠡⠄⠐⢰⢾⣯⡄⣀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠻⠿⠟⠉⠛⠶⠤⣼⣿⣿⣤⡤⠔⠊⠉⠈⠉⠑⣢⡄⣿⣶⠟⠀⠃⣽⣷⣦⡤⠤⠶⠟⢉⡄⠀⠈⠉⠓⠲⣖⠲⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⣿⣇⢀⣠⡖⠀⠀⢴⢿⣷⠛⠁⣀⣴⣾⣿⣿⡿⠀⠀⠀⢠⣼⣿⣄⠘⣦⠀⠀⢸⡆⢸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡏⠀⠀⠀⠸⣾⠧⠀⣾⣿⣿⣿⡿⠏⠁⠀⠀⢀⣿⣿⣿⠛⠃⠙⣧⠀⠀⣿⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⢀⣷⡏⠀⣸⢉⢹⣉⡁⡀⡀⡀⣶⣶⣿⣿⣿⡝⠀⠀⠀⠈⠀⠀⡟⠀⢹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⢿⣿⠀⢠⣿⣼⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⣸⠃⠀⠘⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣦⣯⣙⣋⠲⠻⠿⠿⠿⠿⡿⠿⠟⠛⠋⠉⣻⣿⣿⣿⠗⢦⣤⡀⠀⣧⠀⠀⠀⠹⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠋⢻⣯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣋⡁⠀⠀⠹⣧⣤⠘⣧⠀⠀⠀⠈⠻⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠟⠉⠀⠀⠀⠹⣆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣍⢧⠀⠀⠀⠈⠻⣴⣮⡁⡄⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠊⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡧⠀⠀⠀⠀⠀⠀⠈⠙⣇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⢀⢀⣀⣠⣿⣿⣁⣀⣀⣀⣀⣀⡀⡀⠀⠀⠀⠀⠀⠙⠿⣶⣦⣀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡴⠤⠦⠦⠤⠤⠀⠀⠀⠀⠾⠿⠶⠶⠶⠶⠖⠁⠏⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠖⠛⠛⠋⠉⠉⠉⠉⠉⠙⠛⠓⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⣧⣄⣠⣤⠤⠖⠚⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⣚⠭⠥⠶⢺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⡀
⠀⠀⠀⠀⠀⣠⠖⡫⠞⠉⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⠉⠛⠲⢤⣀
⠀⠀⠀⢀⡞⠁⠊⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⡶⠶⠖⠒⠒⠛⠛⠶⠦⣄⣀
⠀⠀⢠⠏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢖⣻⠽⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠁⠒⠍⡛⢦⡀
⠀⢠⡏⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢞⠋⠑⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⣾⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⡥⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⢀⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠿⢤⣉⣉⠛⠛⠒⠒⠖⠒⠖⠒⢲⡞⢋⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡤⠤⠤⠶⠶⠶⠶⠶⠶⠖⠋⠁
⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠒⠒⣲⡾⢋⠔⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠞⠛⠉⠁
⠀⣇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣫⠔⠁⠀⠀⠀⠀⠀⣀⡤⠶⠛⠉
⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⠵⠋⠀⠀⠀⢀⣠⡴⠖⠋⠁
⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣫⠖⠁⠀⣀⣤⠴⠚⠋⠁
⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⣑⣈⡤⠴⠚⠋⠉
⠀⠀⠀⠀⠈⠛⠶⣤⣥⣒⣒⣶⣭⠷⠶⠚⠋⠉⠁

⠀⠆⠀⠃
⠀⠑⠒⠃