Spirituality Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣷⣾⣷⣾⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣼⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣡⣌⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣙⢛⣠⣿⣿⣿⣿⣿⣿⠿⣻⣿⣟⡿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣯⣶⢝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⡏⣶⢽⣿⣮⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢰⣾⣷⡆⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠻⠟⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣬⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⣿⣿⣿⣿⡛⠉⡿⠋⠉⠙⣿⣿⡿⢿⣿⡿⠟⢿⣿⣿⣿⣿⣿⡟⢡⡌⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡅⠶⣸⣿⣿⣿⣿⣾⡇⠀⠀⢠⣿⡏⠀⠀⠉⡇⠀⢘⣿⠾⣿⣿⣿⣿⣦⣴⣿⣿⣿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⠉⠙⢿⣿⣦⣤⣿⣯⣴⣶⣶⣾⣷⣶⡿⠋⠀⣠⠜⣩⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠃⠀⣠⠴⠿⣿⣿⣿⣿⣿⣿⣿⣭⣉⣀⣤⠶⠾⠷⠚⣿⣿⣿⣏⠤⢘⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠛⠟⠋⠛⠋⠁⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢇⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣾⡟⣼⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠉⠙⠋⠙⠛⠉⠋⠙⠋⠉⠛⠛⠛⠋⠉⠉⠛⠛

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⠀⠀⠻⢦⡄⠀⠀⠀⠀⠀⠠⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠘⡏⢸⡁⠀⢀⣣
⠀⠀⠀⠀⠀⠀⢠⣾⡆⢀⡖⠠⠄⠀⡀⠀⠀⠁⠀⠀⣀⠘⢳⠀⠀⣄⠀⠠⠶⡴⠀⠀⢲⠂⠀⠐⡀⠨⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠟⢸⠃⠀⠀⠴⠀⠀⠀⠀⠀⢦⠹⠁⠈⠚⠣⠈⣃⣄⠀⠀⠀⠀⠘⠀⠀⠀⠱⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡠⠁⠀⠀⠀⠀⡀⣏⠈⠁⠀⢀⠀⠂⠐⠍⣉⢀⠀⠀⠀⠀⠀⠁⢄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢠⠀⠉⠀⠂⠀⠒⠀⠀⡂⡄⠀⠙⣼⠀⠀⠃⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⡄⠠⡆⠀⠀⠼⠀⢀⠀⡀⢁⡄⠀⠘⣡⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠎⠀⠀⠒⡀⠀⡄⠀⠐⠀⠀⠀⢰⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠈⠐⠠⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢦⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢲⠂⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠀⠶⠀⠀⠀⠀⠀⠀⡠⠀⠀⢨⠀⠠⠈⢀⠀⠀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢡⠀⠂⢀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣸⠀⠀⠀⠀⠀⡈⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⣀⡀⠀⠀⡤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⠁⢠⡤⠦⠆⠨⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢤⠀⠀⠐⠏⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠐⠐⠃⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠁⠀⠀⠀⠀⠀⠀⠐⠀⠄⠈⠀⠀⡁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢤⠀⠀⡀⠄⠀⠀⠀⠂⠠⠈⠁⠀⠀⠠⠀⠀⠀⠀⠀⠈⠛⢄
⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢄⡁⠀⠐⠀⠀⢄⡀⠔⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠁
⠀⠀⠀⢀⠀⠰⠄⠀⠤⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⢀⠈⠘⠳⣄⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠁⣤⢁⡄⠀⠀⠀⣠⡶⠂⢠⡄⠁⠀⠩⠄⢠⡀⠀⠀⠀⠀⠲⣀⠨⣷⠀⠀⠀⢀⢄⠀⠤⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠐⡅⢴⠄⠀⠀⢸⡃⠄⠹⢤⡀⠢⠀⠀⠀⠀⠀⠀⠀⠂⡜⠉⠁⢸⠆⠀⠀⣒⠋⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡄⠀⠨⣇⠀⠀⢤⠟⣡⠀⠀⠀⠀⠀⠀⠀⠜⠛⠂⠀⠜⢀⣴⠾⠁⠀⠀⠀⢠⠀⡀
⠀⡃⠀⠀⠀⢀⠈⠀⠀⠀⠀⡔⠹⢃⠀⠈⠠⠤⠔⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢉⠴⠈⢠⡄⠁
⠀⠄⠀⠀⠀⠉⠀⠀⠀⠀⠀⢁⠀⢀⠠⠔⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠭⠐⠀⠋⠒⠀⡀
⠀⠠⠀⠀⠀⠀⠀⠀⠠⡀⠉⠈⢉⡀⠀⢀⠁⠀⠀⠄⠠⠀⡄⠀⠁⠀⠀⠀⠀⠀⠔⠀⠀⠀⠄⢀⠀⠀⠐
⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠠⠢⠀⠊⡠⠂⠀⠀⢁⠈⠀⠀⠴⠀⠀⠁
⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠰⠉⠀⠀⠀⠄⠀⠐⠀⠀⠈
⠀⣾

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠃⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉⠀⠀⠀⠀⠀⣴⣾⣶⡄⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣧⠀⠀⠈⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿
⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠏⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣾⣦⣄⣠⣴⣿⣿⣿⠟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠹⣿⣿⣿⣶⣤⣀⣴⣷⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣥⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣀⣀⣠⣤⣤⣤⣤⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⣶⣤⣄⣀⣀⢀⡀⠀⢀⣸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠗⠿⣿⡯⠽⠊⠽⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⡛⠛⠳⢦⡴⠒⠒⠒⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡹⣿⣟⣿⣛⡿⠿⠙⣿⡟⠻⡉⠙⠛⠒⠶⠷⠶⢶⣶⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣧⡈⢹⡟⠛⠳⣦⠀⠀⣻⣾⡛⠛⠶⢶⣤⡤⠶⢛⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⡿⣄⠙⠷⣤⣿⡀⠀⢸⣧⣴⠟⠈⠙⡷⠶⠦⣄⠳⣼⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢹⣷⣶⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⢠⡟⠳⢦⡀⠉⡟⠓⢾⡁⠘⣄⡀⠀⡷⠤⣀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡿⢿⡻⠛⢃⣼⠃⢼⠀⠀⠀⠘⣦⠇⠀⡟⢳⡀⠀⠈⠐⣇⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⠂⠀⢣⣠⠏⠹⡆⢸⢦⡀⠀⠀⠘⡳⠮⢵⣄⡳⢤⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡴⠞⠋⠀⠀⢻⡼⠀⠁⠀⠀⠘⠁⠀⣀⣱⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣏⣧⣄⡀⢤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣦⣠⣿⣀⣼⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠱⣼⣷⣿⣟⣋⣙⣻⣿⣵⢎⣀⡀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⣧⠦⠀⢬⣿⣿⠛⠿⣏⣿⣿⠟⢙⣿⣿⣅⡀⠐⢶⣷⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣀⠀⠚⠋⠙⠂⠀⠠⠴⢟⣿⣿⣶⣖⣲⣶⣿⣿⡍⠁⠀⠀⠛⠈⠋⠀⢤⣄⡖
⠀⠀⠀⠀⠀⠀⠀⠘⣻⡿⠥⠀⠀⠀⠀⡀⠀⢄⠀⠾⡋⣧⡿⣿⣿⡟⢿⠀⠙⠀⡠⠀⠀⠀⠀⠀⠴⠟⣿⠉
⠀⠀⠀⠀⣄⢀⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⡵⢁⣉⣈⠃⠀⠔⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⣦⣠⠆
⠀⠀⠀⠠⠾⣿⠥⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⢢⠐⠀⢴⣿⠿⣿⡆⣠⠐⡡⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢻⡟⠓
⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⡘⣠⣾⣿⣶⣿⣶⣿⠮⠂⠐⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⣀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣷⣠⠄
⠀⠀⠀⠀⢨⡽⣷⠒⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠁⣠⣿⣯⣿⣿⣿⣿⣿⣆⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⠳⠆
⠀⠀⠀⠀⠀⠀⠛⠀⢀⡀⠀⡀⢠⠖⠒⠦⠴⣾⢿⣧⣙⣛⣛⣿⣩⣿⣿⣿⠿⠻⠟⠂⡰⣲⣦⠀⣤⣀⡴⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠤⣽⣿⣁⠀⠛⠶⠆⢧⣤⣤⡿⣋⣩⢹⣿⡍⣼⣡⠰⡄⠤⢲⡭⠗⠛⠋⠐⠾⢿⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠛⢻⣭⣿⠛⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⠶⠂⠉⠈⢉⠍⢳⡈⠁⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠈⠀⡔⠉⠳⡀⠠⠆⣄⠀⢐⠹⡉⠉⠉⠉⠈⠁⣢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠚⠄⢀⠐⠁⢀⡄⠸⠂⠈⢠⣅⣀⠀⣀⣀⣀⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⢸⠄⠀⢨⠃⠄⢠⠔⠈⠐⠺⡅⢇⠀⠀⠀⠀⢀⣉⣀⣀⣈⣣⡀
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⢿⠀⠀⠃⠀⢆⠙⠀⠀⠀⠀⡟⠊⠀⠈⠈⡉⠀⠉⠉⠋⢉⠜⠑⡀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠸⠶⠄⠠⣒⣬⠕⠓⠤⠤⠊⠀⠈⠀⠈⠁⠉⠀⠑⢀⢀⠂⠀⠀⢰⡀
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠣⡠⠖⠉⢦⠀⠀⠀⠑⠀⢀⣳⣀⣠⡀⠈⠠⠄⠸⠏⡄⠀⠀⠀⢇
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⣤⡀⠓⠢⡀⠠⣄⠀⠀⠈⡍⠁⠀⠒⠒⣼⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠚⠢⠤⢴⠄⠄⠙⢆⠀⠀⢠⠁⠀⠀⠀⠘⠀⢸
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⡠⠍⠄⡈⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠝⢠⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠤⣀⠀⠀⠀⠀⠀⠘⡞⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠥⠌⠓⢜⡆⠀⠀⠀⠀⠀⠇⠀⠀⠆⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣤⡈⠞⠁⠀⠀⠀⠀⠀⠇⠀⡀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⢌⡯⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠤⢀⡀⠀⠀⠀⣀⣠⠴⠬⠛⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⡀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣢⣲⣭⣵⣽⣦⣶⣯⣭⣖⣢⣤⣐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣵⣾⠿⠛⠋⠁⣼⣿⣿⣿⣟⣿⣷⡈⠉⠛⠶⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⢯⣀⠀⠀⠀⠈⣯⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢙⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢮⡑⠲⢤⣀⡻⣿⣿⣿⣿⣯⡞⠀⣀⣠⠴⢚⣭⡒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠐⠉⠒⢤⢌⣉⣙⠛⠛⣉⣉⢩⠡⠖⠜⠢⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠀⠈⠸⠀⠈⠈⠂⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠚⠛⠛⠛⠒⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠉⠉⠉⠉⠉⠛⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⢀⠠⠄⢐⣀⠀⠤⢀⠀⠉⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⣐⣐⣐⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠋⠉⣿⣿⣿⣿⡟⠉⠛⠦⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠴⡛⠉⢿⣿⣿⣯⣿⠉⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠛⠛⠚⠐⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠁⠀⠀⠀⠀⠈⠘⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⡀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⣀⠠⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠉⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⡀⠀⠈⠁⠀⠀⡀⠤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠉⠈⠒⢄


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡆⠀⡔⢀⡆⠀⡆⠀⠠⠂⠠⠆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠞⠀⠀⠱⡄⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠄⠘⠂⠀⢀⠈⢀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠘⠀⠀⠀⠐⢀⠀⠠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⡰⠃⢀⣘⣠⠀⠀⣀⡆⠀⠀⢆⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⢱⠀⠀⡐⠁⠁⠈⠀⠁⠀⠀⠈⠐⠢⣄⡌⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⢄⡜⠀⢰⠂⠀⢀⢤⠤⠤⣤⣀⠀⠀⠀⡴⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠜⠢⣤⡇⠀⢠⠶⡟⢩⣝⠳⢯⡁⠀⠀⠃⠀⠱⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠁⠠⣧⣙⣫⣬⡤⠝⠃⠀⠆⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠠⡄⠀⠀⠀⠀⠀⠀⠘⠓⠓⠓⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠈⢆
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠈⢤⡀⢀⠀⠀⠀⡀⠀⠀⠁⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⠤⠄⠊⠰⣀⠀⠠⢤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠃⠀⠀⠀⠀⠀⠈⠀⠝⠋⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⣆⠀⠀⠀⠀⡀⠀⠀⠀⢀⠋⠀⠔⠁
⠀⠀⠀⠀⠀⠀⡀⠠⠤⠀⠀⠤⠤⣄⣀⠀⠀⠀⠀⠈⠢⢌⠙⠃⠂⠸⡏⠐⠘⠛⣉⡴⠊⠀⠀⠀⠀⣀⣠⡤⠄⠄⠀⠤⠄⢀
⠀⠀⠀⢰⣦⡅⠀⠀⠀⠀⠀⠄⠒⠂⠉⠉⡙⠂⣀⡀⠀⠈⠣⠔⢷⣴⣷⡴⠣⠔⠁⠀⠀⣀⠀⠘⡌⠉⠀⠒⠂⠀⠈⠀⠀⠀⠀⣵⣶
⠀⠀⠀⠁⠰⣊⠄⣠⠄⠡⠀⠀⠀⠀⢀⡀⠄⠀⠉⣉⢉⠖⠶⣾⡿⠞⠙⢾⣿⡶⠖⣋⢉⡉⠠⠀⠄⣀⠀⠀⠀⠀⠀⠁⣄⠀⣌⠲
⠀⠀⠀⠀⠀⢱⡾⠖⠀⠀⠴⠖⠂⣈⡵⠚⠁⠐⠊⠀⣉⣉⣻⡿⠂⠔⠓⠀⠻⣿⣋⠉⠁⠐⠂⠈⠙⠲⣭⡀⠲⠦⠄⠀⠲⠿⣖⠈
⠀⠀⠀⠀⠀⠈⠐⢺⣷⡂⢀⠀⠄⠋⣴⠞⠛⠂⠀⡐⡯⡛⠿⢢⣞⠀⠀⢸⢔⠾⢃⡙⣖⠀⠀⠚⠳⢶⡈⠃⠀⢀⠀⣶⡟⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠫⠊⡆⠀⠀⣀⠀⠉⣁⡄⠾⠁⠤⠠⣬⡦⢠⣯⣤⠤⠆⠹⠆⣄⡉⠀⢀⠀⢱⡳⠌⠚⠃⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠂⠐⡾⠋⠀⠐⠦⢰⣾⠇⠥⠠⠈⢳⣦⠠⠖⠀⠉⠳⡆⠠⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠐⠊⡠⠉⠜⠀⠆⠀⡄⠸⡍⠁⠘⡒⠂⠀⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⢀⢀⠀⡶⢐⠃⣐⠳⣀⢆⡼⢃⠀⣗⠠⠻⠦⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠀⠸⠀⠀⠋⢼⠻⠿⢻⠿⠀⠀⠀⠀⠙⠀⠀⢈⠀⠀⢠
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠦⠀⠙⠘⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄
⠀⠀⠀⠀⠀⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡐⠀⠐⠀⠀⠀⢀⣼⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠀⢂⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠙⠁⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⣿⡿⠛⣻⡿⣿⣿⣿⠿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣥⣽⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡄⠀⠀⠀⠀⠠⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠐⠀⠀⣀⡀⠤⠄⠀⡀⠀⠀⠂⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠁⠀⣠⣶⣿⣿⣿⣷⣄⠀⠀⠘⠢⣄⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠂⢠⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠈⠑⠀⠐⢈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠘⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡇⠀⢀⡁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⢀⡇⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⢀⡾⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠌⠳⡀⠈⢻⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⣴⣴⠟⠁⢠⠜⠠⠖⠀⠁
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠈⠻⠭⣄⣀⣀⣀⣤⣴⠿⠟⠁⠀⠄⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠠⡄⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠘⠤⢀⠀⡀⠉⠈⠉⠁⠀⠀⡀⠤⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠
⠀⠀⠩⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠌
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢠⠀⠀⠒⠀⠀⠀⠀⢐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡨⠀⠀⠀⠀⠒⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠙⢂⠀⠀⠀⠈⠁⠀⠀⠀⡔⠉⡠
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠢⡈⠢⠤⠤⢰⡆⠠⠤⠔⣀⠌
⠀⠀⣤⡆⠀⠀⠒⠐⠀⠄⠩⠑⠂⠄⠀⠀⠈⠑⡠⢤⣸⣄⡤⣀⠎⠁⠀⠀⠠⢐⠚⠭⠤⠐⠀⠀⠀⠀⣂⣄
⠀⠀⠨⢂⡀⣠⠀⠀⠀⠀⠀⡀⠀⠀⠘⡃⠒⠴⣶⣾⠟⠻⣷⡶⠆⣒⡚⠃⠀⠀⡀⠀⠀⠀⠀⣀⡀⢀⡸⠉
⠀⠀⠀⠠⠾⢆⠀⠨⠓⠂⣡⠞⠉⠀⠊⠈⣉⣙⡿⠂⠘⠂⠸⢿⣋⣉⠉⠂⠀⠙⠲⣅⠐⠢⠀⠀⠸⢾⠉
⠀⠀⠀⠀⠀⠹⡯⢠⠀⠀⢁⠞⠉⣁⡀⠼⡎⠙⠻⢮⠀⠀⡷⠛⠋⡗⠄⢀⡙⠓⠷⠈⠀⢀⡠⣷⠏⠉
⠀⠀⠀⠀⠀⠀⠰⠐⠁⠂⠀⠀⠂⢀⡶⠊⡁⠐⣒⡾⠇⠺⣷⡒⠃⡘⠰⣦⣀⠐⠀⠀⠲⠐⠡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⢀⠜⡱⢿⠀⡁⠈⢰⠿⡈⢣⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡀⡄⠁⣶⢠⠈⡦⠁⡬⢀⠢⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠁⠈⢾⢿⡿⡏⠀⠁⠀⠀⠆⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⣴⣧⠀⠀⠐⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠐⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠋

⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠤⠀⠀⠀⠆⠀⠆⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢁⣀⡀⠀⠀⠀⠀⠀⠆⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠤⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⢰⠆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡠⠊⠐⠃⠀⢠⣀⠤⣄⠀⠀⠀⠀⠀⠄⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠃⢀⠌⣥⡔⣈⡭⣤⡛⢂⠀⢀⣀⡂⠀⢀⠞⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⣂⠠⠄⢋⣠⠤⣌⠡⣆⠘⠁⢀⡀⠂⠀⠔⢠⠀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠐⠀⡄⡹⡆⠉⢠⡟⢸⠁⠘⠇⠈⠂⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠇⠁⠀⠈⠛⠇⠀⡤⠁⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀⠀⠀⠀⠀⠀⠀⠁⠀⡐⠛⡀⠛⡠⡠⠞⠀⢀⡈⠂⠀⡠⠐⠁⠀⠀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠁⢖⠀⣤⡛⠐⠒⠉⠉⡸⠟⢛⣥⠀⠀⠈⣁⡱⢶⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢰⠀⠀⠈⠛⠙⠶⡄⢻⠄⢠⡄⢹⠀⡴⠋⠀⠁⠈⠀⠀⠸⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢣⠐⠀⠀⠆⠀⠀⠀⠠⠀⢴⠀⠀⠀⠀⠈⢀⡅⠀⡀⠁⠀⠀⠀⠀⠆⠀⠀⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠡⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⣈⡇⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡆⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠋⠁⣴⣶⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⣠⠄⠀⠀⠀⢄⠐⠗⠌⢁⡀⢡⡀⠀⠀⠀⠰⣦⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⢠⡜⠧⠀⠀⠀⢀⠃⠀⠀⠁⠀⠢⠈⣧⠀⠀⠀⠀⠈⠣⠂⠀⢧⠀⠐⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠉⠂⠀⠀⠀⣈⠀⠄⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠁⢀⣤⠀⣀⢠⣄⠀⢿⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠏⠀⡿⠁⠙⠛⣿⠙⠟⢸⠂⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⣤⢰⣷⠀⠀⠀⣻⡄⠃⢸⡆⠀⠄⠀⠀⠀⠈⠛⠓⠐⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠃⠀⠀⢹⢴⠉⢳⡀⠀⠘⣷⡿⢸⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠀⠀⢀⡼⡀⢀⠳⠀⠀⡻⢷⣸⡄⠀⠀⠀⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⢸⣰⠟⢰⠟⠀⣰⠇⢨⣼⠃⠀⢠⣤⣇⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⡀⠀⠼⠀⢀⣠⢄⡴⢊⡙⢠⡟⠀⡰⠛⢀⣾⡿⡂⡀⠫⡀⠀⣠⡀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠊⢅⠀⠰⠦⡄⠉⠠⠈⠰⠋⢀⣼⠀⠸⡇⠀⢸⡇⡷⣇⠀⠀⠈⠀⢻⡀⠠⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⡀⢤⣀⣱⠤⠘⠐⠢⡅⠀⣤⠴⠋⠛⡀⢠⣿⡀⠈⠳⠽⣏⠤⠄⢀⡖⠤⠹⢆⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡄⠀⠀⠃⠀⠐⠋⣠⠔⢣⠀⠀⠛⡀⠠⡐⠀⠀⡇⢸⠷⣴⡂⢠⣀⣤⣤⣶⠚⠀⠀⠰⣄⠲⠟⠀⠡⠤⠴⠋⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠐⠂⠀⠀⠀⠙⠂⡞⠀⠤⠂⠠⣤⣈⠉⣀⣀⠰⠄⢹⠃⠌⡉⠁⠀⠀⠙⠂⠀⠀⠈⣀⠀⠋⠔⠀⠀⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠈⠥⠀⠁⠠⢀⡀⢰⠋⠡⠸⡀⣶⠖⠉⠉⠋⣙⠃⠊⠁⠘⢦⣟⡡⠀⠀⠀⠀⢸⣀⡇⠦⠠⣬⡁⠈⠙⠋⠶⠂
⠀⠀⠀⠀⠀⠀⢆⣰⣆⡀⠀⠈⡠⠂⠘⡒⠲⠞⢡⡞⠁⠁⠀⠆⠑⠒⠒⠀⠀⠈⢄⡘⠧⡀⢈⡀⠁⠤⣦⡔⠉⠤⠄⠀⢂⡰⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠻⠃⠀⠢⠺⠉⢀⠀⠀⡌⠀⠀⢠⡀⡙⠀⠀⠛⡀⢸⡎⠁⡶⠍⠋⠀⡀⠆⠈⡁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤
⣿⣿⠿⢿⣿⡿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣥⣬⣽⣿⣯⣼⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠠⣞⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠈⠁⠀⠛⣀⣠⡤
⠀⠀⠀⠀⠀⠉⠍⠀⠄⡀⠀⢀⡀⠀⠀⠀⠠⠌⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣉⠆⠀⠀⢉⣤⠖⠉⠛⠛
⠀⠀⠀⠀⠀⠒⣂⠛⣦⣮⣀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⢀⠀⠢⠐⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⠻⢛⣾⣿⣯⣠⣫⡕⠂⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢤⠃⠀⢣⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠠⠞⢡⡄⠀⢄⣀⡀⣀⠑
⠀⠀⠀⠀⠀⠀⠀⢫⣿⣿⡾⠛⢻⣏⠠⡀⠀⠔⠀⠀⠀⠀⠀⠐⢀⢀⡀⠀⠀⠀⠀⠁⡴⣦⠀⠀⠀⠀⠀⠀⠀⣀⡶⠉⡟⠧⠎⠙⠉
⠀⠀⠀⠀⠀⠀⠈⠁⣿⣠⣘⣟⡳⠏⠰⠄⠀⠔⣣⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰⠷⡄⣀⠀⢤
⠀⠀⠀⠀⠀⠠⠀⠀⢈⣹⣦⣤⣿⡤⠀⠀⠄⠸⠁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠘⠀⠀⠀⠀⠀⠀⢂⣠⣴⡆⠀⠴⠷⠖⠒⠁
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠹⢟⣳⣘⠻⠎⠀⠲⣄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠉⢀
⠀⠀⠀⠀⠀⠂⠀⠀⠰⠀⠈⠛⣛⣙⣃⠀⠰⢿⡎⠦⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⢣⠀⠀⠀⠀⠀⠐⣉⠛⡉⠹⠂⠀⣸⣈⣀⡄⢠⠀⠀⡀⠀⠤
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠙⣯⣽⠀⠀⡿⣷⡀⢂⠀⠀⠀⡀⠀⠀⠀⠿⠐⠧⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⡼⠃⠐
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠂⠀⠀⠀⢹⡆⠀⠐⠋⠁⠈⠀⠀⠘⡅⠀⠀⠀⡀⠀⡀⠀⡄⠀⢀⡀⢀⣀⠠⠤⠄⠒⣀⠀⡌⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠙⢦⣤⣤⠤⠤⠤⠒⠀⠀⠀⠠⠀⠓⣈⡁⠴⡿⠀⠈⠉⠫⡉⠁⠂⠠⣤⡴⠶⠄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡠⠀⠀⠀⠀⠀⠘⠲⠆⠀⠀⢀⣀⠀⠊⢀⡜⠋⢠⠙⠋⠀⠦⡄⢀⡀⠉⠀⢀⠀⡀⡀⠤⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠁⢈⠀⠢⢉⡙⠏⡯⢉⣐⢀⠘⠄⠙⠒⢤⡀⠈⠁⢀⡀⠄⠂⠀⠀⢀⠀⠀⠁⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠄⠤⠊⠀⠀⠀⠀⠀⢀⠀⠀⢰⡏⣳⣆⣸⣿⠄⠀⠐⠄⠀⠀⠉⠀⠛⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⣉⣩⣠⣿⢀⠀⢀⠄⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠈⠀⠈⢀⡤⠀⠀⠀⠀⠐⢻⡿⠥⢄⣈⠛⠭⠉⠥⠀⢰⠎⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠾⣷⢶⠶⠤⠛⡦⠰⠖⠀⠘⠂⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠦⠀⢀⡀⠐⢁⣠⠶⡄⠀⠐⠒⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠠⠞⠉⠉⠉⠁⠀⡔⣴⡄⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⣿⠉⣷⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⢠⡄⠀⠀⠀⠀⠀⠛⢠⣼⠋⢲⣄⡰⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡼⠀⠀⠀⢀⡀⠀⠀⢘⡃⠀⠀⢿⣧⢈⡀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢈⠁⠀⠀⢸⣏⠀⠀⢀⠀⠈⢣⠀⠀⠀⠂⠀⠀⠀⠀⠘⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⣾⠀⠀⠀⢨⡻⠄⠀⠐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⠀⠈⠀⠀⡿⠀⠀⠀⠘⣵⠆⠀⠀⣿⠀⠘⠠⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠠⠤⠆⠀⠦⠆⠴⠒⠠⠿⠀⡜⠀⠀⠀⠀⣇⠀⠀⠀⠀⢹⠆⠀⠀⢹⠄⠀⠴⠆⠛⠷⠀⠤⠀⠐⠠⠂⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠇⠀⠂⠀⢀⣬⠀⠀⠀⡀⣿⠇⠀⠀⠘⠃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠋⠀⠀⠉⢁⠙⠋⠹⠀⠀⠀⢁⠀⢠⡀⠉⠁⠀⠀⠀⠀⢠⠀⠂⠀⠀⢆
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠘⠀⠾⠀⡀⢀⡆⠘⠆⢨⠧⠀⢀⠀⠀⠰⢜⡧
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⡠⠀⠀⠠⠂⠁⠠⢀⠁⠜⠁⠀⢀⡁⠀⠈⠃
⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⣄⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠇⠀⠄⠀⠀⠆⠀⠋⠀⠈⠡⠀⠘⠉⠀⠀⠀⠀⠠⠀⠄
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠠⠀⠈⠠⠊⠀⠀⠀⠀⠀⠀⠀⣁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢈⡷⠀⠀⠰⣿⡆⠀⠀⠀⠀⠀⠒⠛⠊⢠⡀⠲⠶⢄⠀⠁⠲⠊⠑⢀⡀⠀⣀⠴⠾⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⣿⠆⠀⠀⠸⡅
⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⡺⠃⠀⣼⡟⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠈⠔⠂⠘⠈⡄⠉⠡⢅⠀⠄⠈⢁⣀⡀⠀⠀⢸⣧⡀⠀⠛⠄
⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣠⣤⠘⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠃⢤⣄⡀⠘
⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⢸⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠤⠀⠀⢀⡀⠀⠠⣄⡀⢤⠰⠂⠀⠀⠀⠀⠠⢀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐⠃⠀⠀⣀⢘⡲⠖⢃⣠⠶⠞⢀⠀⣀⣸⠥⢔⠒⡀⡈⠀⣸⠆⠀⠀⢀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣺⡥⠀⠀⠀⠙⠀⢄⡀⠋⠡⠂⡏⠁⠐⠊⠌⠡⢈⡀⠂⠀⡝⠻⡄⠀⠄⠀⠒⡀⠁⠀⠀⠀⠈⢿⣇
⠀⠀⠀⠀⠀⠀⢀⡸⢏⠀⡋⢁⣀⣀⠀⠀⠀⡨⢁⣀⡃⠀⣀⡀⠀⠙⠋⢁⠈⠀⠀⠀⠁⠘⣪⢠⣄⠀⠀⠀⠀⠀⠀⠀⢿⣤⡀
⠀⠀⠀⠀⢀⡴⠟⠛⠉⠀⠁⢸⠀⢩⡶⠛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⢀⠀⠀⠁⠌⡾⠁⠀⠀⠀⠀⠀⠬⠛⠛⠄
⠀⠀⠀⠀⠛⠁⠰⠖⢲⣤⠀⢠⣰⢛⡠⠖⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⢈⣄⡄⠀⠆⣀⡤⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⢀⣼⢇⠀⢀⣊⠁⠀⠐⠀⠄⠀⠀⠀⠈⠀⠀⠀⠂⠀⠄⠀⠈⠀⠀⠀⠈⢢⠠⣜⣷⡥⢻⠅⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠸⠏⢀⡼⠋⠽⢏⣠⣐⠂⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘⢨⠅⢸⠀⠀⣠⡁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⢋⠀⣀⢠⠔⠉⠉⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⢸⣧⢃⡿⣰⣄⠰⢏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢏⣴⢁⣕⣮⣯⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⢈⡎⢳⢿⡌⢿⡤
⠀⠀⠀⠀⠂⡃⠀⢀⡾⢃⡼⣿⣻⣯⣿⠏⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⣼⠁⠁⠈⡇
⠀⠀⠀⠀⢠⡂⢠⣾⣓⢿⣵⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠁⠀⠀⠎⡬⢿⣄⣠⡴⠁
⠀⠀⠀⠀⠀⢡⣿⣷⣻⡷⣾⣿⣽⠋⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠁⣀⡀⠀⠰⠀⠀⠀⠀⣰⠻⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣟⢻⡏⠁⠀⠲⢀⠀⠀⠀⡐⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠈⣹⠀⡘⢣⣄⠀⣶⠈
⠀⠀⠀⣴⢩⣽⠟⠋⣠⣏⡸⠙⠂⠀⢣⡀⠐⢀⠀⠀⣀⠀⠄⢠⠀⠀⠀⢀⠀⢂⠀⡁⠰⣄⠀⡁⠞⣩⠀⠠⠀⠀⢀⡀
⠀⠀⠀⣛⠟⠁⠚⠿⣴⠶⢇⠀⠀⠀⢠⣇⣶⡈⣐⡄⠈⠁⣀⡀⣀⡀⣀⣀⠁⠀⠘⢳⡀⠀⠹⠿⠅⠀⠱⡆⠴⡿⠉
⠀⠀⠀⢁⡀⡋⠛⣠⣿⢠⣸⠀⠀⠀⠀⠸⠒⠸⠟⢢⡔⠉⢉⡀⠛⠁⠀⠈⠁⠑⠆⠉⣠⠠⢀⠀⠹⣆⡀⠸⠆⠄⢀
⠀⠀⠀⠀⠉⢀⣸⠟⠛⠙⡉⠀⠀⠀⠀⠀⠀⠀⣦⠞⣿⡛⠎⠠⠦⠀⠀⠐⠀⠄⠀⠀⣀⠀⡀⠄⠀⠁⠀⠀⠀⡀⠀⠒
⠀⠀⠇⠋⡐⢾⠻⡀⢹⣋⣗⠀⠀⠀⠀⢠⣤⣾⣵⡟⡠⠁⠀⠀⠀⠀⢀⢀⠀⠀⠀⠁⡤⠈⠀⠀⠀⠀⠀⠀⣀⣡
⠀⠀⠀⢰⠃⠸⣄⣇⣈⣭⣿⡄⢠⠆⠀⣸⣧⡎⠘⠏⡀⠁⠀⠀⠀⠀⠁⠚⠃⠀⢁⠚⡶⠂⢀⠀⠀⡀⢀⢂⠜⠁⠐
⠀⠀⢀⡃⠀⠀⠛⠛⡛⢺⣯⣿⡾⠰⣦⣟⡿⠁⠀⡔⠁⢄⠀⠀⠀⠠⠆⠀⣠⣠⠀⣀⠀⠀⠀⢀⡀⠀⠀⠊⠄⠀⠀⢀⣴⡧
⠀⠀⠈⢛⠀⠉⠉⠉⣩⠟⣦⣟⡄⠀⠋⡉⠁⢠⡄⠀⠀⢀⠀⠈⢠⠀⠤⠆⣿⠁⠀⠄⠀⣅⡀⡔⢦⣼⣷⢀⡔⠁⠶⡁⠠⠔
⠀⠀⢀⡈⢁⠀⠀⠠⣿⣄⢻⠭⣾⠤⡄⣬⡖⢀⣀⣦⡐⠾⠖⠐⠶⣶⣴⠀⣿⣦⡀⠐⣉⢹⡆⡤⣌⣳⣞⠏⡾⢳⣼⡿⣤
⠀⠀⢉⠀⠛⢡⣤⢤⡄⠙⠈⣿⠹⠛⠀⢪⢥⣆⣽⣭⡘⢶⣟⣞⣬⡵⢇⠼⠃⢠⣟⠖⣧⣶⣼⡗⠻⠉⣯⢻⡼⣿⣿⣎⠙⠆
⠀⠀⠙⡀⡦⣬⠙⠀⠀⠀⠀⣮⢰⡌⠶⠿⠾⠟⠿⠋⠟⠛⠋⢉⠉⠁⠙⠉⠙⠛⠻⢦⣽⣂⣿⠁⠀⠀⠀⠀⡏⣼⠛⠛⡀
⠀⡀⠀⢹⡵⢺⣇⠀⠀⠀⠀⠙⣧⡀⠀⠀⠐⠀⠀⢀⣠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⢈⡽⠁⠀⠀⠀⢀⡐⠀⢨⠶⢼⡇
⠀⠁⠀⠀⠀⢤⠹⠀⠀⠀⣤⡿⠿⢓⡄⠀⠀⠈⠐⠁⠀⡀⠈⠀⠀⠀⠀⠂⠀⠀⠀⢁⠶⣿⡠⡄⢀⢰⠿⣷⡎⢾⠖⢶⠃
⠀⠀⠀⠈⠁⠀⢧⠰⡌⡭⠛⠿⠁⠀⢀⠀⠀⠀⢀⣤⡈⢥⣠⣄⣸⣤⣄⠀⠄⠀⡀⠈⢤⡉⢹⣿⣯⡈⢬⡖⢛⢮⠄
⠀⠐⠀⠈⢉⣄⢁⡀⢻⡀⠀⢛⠀⠀⠇⢀⠤⡄⠼⡯⡃⢾⣧⣤⣧⣴⢋⣁⡀⢤⣴⣀⣠⣧⡄⢹⠟⡰⢃⣠⣴⠦⠈
⠀⠾⠀⠀⠀⠛⠃⢉⣩⠡⣶⡏⡗⠐⢒⣍⠀⣲⣭⢙⣛⢫⡉⠉⠉⠑⣎⢨⢁⡞⢫⣽⣯⣾⡲⠿⠖⠋⠚⢶⠧
⠀⣶⠀⠀⠀⠀⠰⠆⠀⡱⢬⠤⠄⢀⡉⠙⠀⠀⠊⢀⡛⠃⠀⠀⠀⠐⠛⠛⠫⠑⠛⠻⠶⠇⠀⢹⠟
⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⢄⣠⠀⠀⠀⠀⠐⠀⠀⠀⡄⠀⠀⠀⠒⡀⠀⠀⢦⡴⠀⠀⢆⡐⠀⠀⢶⡆⠀⠀⠀⢀⠄⡀⠀⠀⢺⣟
⠀⠀⠀⠰⠆⠀⠀⠐⠂⠀⠀⠀⠈⠀⠀⠀⠟⠀⠀⠀⠀⠃⠀⠀⠊⠐⠀⠀⠉⠑⠚⠀⠀⠁⠀⠀⠀⠀⠘⠿
⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠠⢤⠆⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴
⠀⠀⠀⠈⠀⠀⠈⠉⠉⠁⠀⠈⠀⠀⠀⠀⠓⠂⠀⠀⠨⠁⠀⠀⠀⠆⠀⠀⠂⠀⠁⠀⠀⠁⠐⠀⠀⠀⣾⣻
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⡀⠀⠀⢠⡄⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠰⣀⠜⠀⠀⠚⠓⠀⠀⠀⡆⠀⠀⢈⡈⠀⠀⠀⠀⠹⠀⠀⢐⡊⠀⠀⠈⡍⠀⠀⠠⢀⡀⠂⠀⠨⠉⠛
⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣉⣹⣸⠀⠀⢈⠁⠀⠀⠀⡁⠀⠀⠀⣓⠀⠀⠀⢦⠴⠀⠀⠤⠤⠀⠀⢀⠀⠁⢀⠒⡆⠉⠒⡆⠀⣖⣱
⠀⠀⠀⠈⠀⠀⠀⠉⠉⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠁
⠀⠀⣤⣠⣀⠀⠀⢠⡦⠀⠀⠀⢠⡄⠀⠐⠀⠐⠀⠀⠀⣤⡀⠀⢠⡄⠀⠀⠰⠤⠀⠀⠀⢀⠆⠀⠀⠀⣶⢰
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠈⠁⠁⠀⠀⠛⠀⠀⠀⠊⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛
⠀⠀⠠⠄⠀⠀⠀⡤⢤⠀⠀⢀⡄⠀⠀⠀⣤⠀⠀⠀⡤⠤⠀⠀⠀⢀⠀⠀⣤⣦⡄⠀⠀⢠⣦⡀⠀⠀⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠂⠀⠸⠹⠆⠀⠀⠀⠐⠀⠠⠔⠌⠁⠀⠻⠿⠃⠀⠀⠀⠹⠃⠀⠀⢸⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⠀⠀⠲⢄⠀⠀⢀⢀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠴⠂⠀⠀⠀⢀⠀⢀⠀⠀⠀⠠⢴⣿⡦⠄⠀⠀⢀⡤⣀⠀⠄⠀⠀⠀⢤⡈⡣
⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠁⠼⠧⠐⠀⠀⠀⡠⠊⠦⡀⠀⠀⠀⢬⣉⡤⡄⠀⠀⠀⠨⠀⡍⠀⠀⠀⠀⠸⠭⠁⠀⠀⠀⠀⠘⠙⠀⠀⠀⠀⠀⠚⢽⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠈⡤⠆⠀⠀⠀⢠⡀⣀⣐⡠⠀⠀⠘⠀⡼⠛⠋⠀⠀⠀⡤⡻⠣⠀⠀⠀⠐⢇⢐⢀⡀⠀⠀⠈⠀⠀⠦⡀⠀⠀⠠⠀⣀⣹⡄⠀⠀⠀⡁⠠⠗⢄
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁⠃⠀⠀⠀⠀⠈⠉⠚⠃⠀⠀⠀⠁⠀⠐⠇⠀⠀⠀⠘⠀⠐⠀⠀⠀⠘⠊⠁⠈⠉

⠀⠀⠀⢀⣤⠤⠐⠀⠀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⣀⠂⠀⠀⠀⠓⣶⠖⠀⠀⠀⠀⠐⢦⣀⠀⠀⠀⠀⠀⡠⠾⠀⠀⠀⠀⠖⠀⠒⠀⠀⠀⢀⡀⣰⡄
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠄⠐⠠⡄⠀⠀⠀⢀⠐⠉⡁⠀⠀⠀⣀⣯⣄⡀⠀⠀⠀⠀⡐⠈⠂⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⢀⣁⠂⠀⠀⠀⠈⣀⣡⣬⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠈⡁⠀⢘⠀⠀⠀⠀⢦⠀⡵⠂⠀⠀⠀⢀⢠⡄⠀⠀⠀⠀⡴⠷⢶⠄⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠛⠒⡔⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠋⠚⠊⠁⠀⠀⠀⠉⠀⠉⠁⠀⠀⠀⠀⠙⠒⠂⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐

⠀⠀⠀⢤⡤⣄⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠒⣤⠂⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⣤⠒⡀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⢠⢤⠄⠀⠀⠀⠀⠠⠷⡄
⠀⠀⠀⢀⡀⠋⠄⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⢈⠋⠁⠀⠀⠀⠀⠸⠀⠆⠀⠀⠀⠀⢀⣵⡤⠀⠀⠀⠀⠈⠁⢄⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠚⣒⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠐⠀⠒⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⡦⠀⠀⠀⠀⣆⡀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⣤⡰⠤⡁⠤
⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⡀⠂⠀⠀⠀⠐⢀⡂⡀⠀⠀⠀⠀⣀⢀⣀
⠀⠀⠀⠠⢺⠠⠀⠀⠀⠐⠹⠯⠌⠃⠀⠀⠀⠸⠄⠔⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡀⠐⠂⠀⠀⠀⠀⠠⣀⠔⠁⠀⠀⠈⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠐⠒⠖⠀⠀⠀⠀⠀⠊⠀⠆⠀⠀⠐⢠⡉⡅⠋⠀⠀⠀⠀⠈⣹⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠰⠈⠈⢁⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⠊⠀⠀⠀⠀⠂⠉⠲⠀⠀⠀⠀⠈⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀
⢀⣠⣇⡀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡄
⢨⣿⣿⡅⣄⣀⣀⣀⣀⣀⣄⢹⣿⣿⠀⠀⠀⠀⠀⠀⠉⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⡶⢠⣿⡿⢩⣭⣥⣤⣤⣤⣤
⢠⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣇⢻⡿⠁⠛⢛⣛⣛⢻⣿⡟
⢸⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣦⣄⡀⠀⢀⣠⣾⣿⣦⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⣿⣧⠁⠀⠀⠀⠻⣿⣧⡛
⢈⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠛⠛⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣧⣉⣉⣁⣠⡀⣼⣷⡹⣿⣷⡀
⠐⣿⣿⠇⠿⣿⣿⣿⣿⣿⠿⢺⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣶⣶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠿⠟⣼⣿⡟⣴⣿⣿⣷
⠈⠙⡟⠁⠀⠀⠀⠀⠀⠀⠀⠉⢻⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⣤⣀
⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⣷⣿⣿⣷⣮⣝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⣿⣿⠿⠟⣁⠻⡿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣏⣾⣿⣿⣿⣾⣿⣿⣿⣧⣹⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⣷⠀⢾⣷⠀⢸⠆⢸⠆⢸
⠀⠀⢿⣿⡇⠸⣿⣿⡹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡾⣿⣿⠟⣩⣭⣍⠻⣿⣿⢶⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿⠀⢸⣿⠀⢸⡇⢺⡇⣾
⠀⠀⠈⢿⣶⣦⣀⣴⣷⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣿⡏⣼⡏⠀⢹⡇⢻⣿⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡀⢻⣦⣼⣿⣤⡾⠃⣼⠇⣼
⠀⠀⠀⠀⠈⠙⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⠛⠶⠻⡇⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠘⢷⣤⣀⣹⣿⣁⣤⣾⠟⢠⡿
⠀⠀⠀⠀⠀⢠⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡋⣻⡇⣿⠈⢸⠀⡇⢸⣟⡙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣌⣉⣻⣿⣉⣉⣤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣹⣧⣿⣦⣾⣴⣧⣼⣇⣐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢻⣿⠛⠋⠉
⠀⠀⠀⠶⠶⠾⠿⠷⠶⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢦⢸⣁⣋⡰⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⢀⣀⣀⣀
⠀⠀⣀⡤⣤⣤⣤⣤⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣴⢹⣿⡿⣧⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⢭⡉⠙⢷⣴⡟⢫⣭⡍⢻⣆
⠀⢠⣿⡿⣮⣝⢿⣽⣳⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⢸⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠃⠀⠈⣿⠀⢘⣋⡃⠀⣿
⠀⣾⣿⣧⡈⢻⣷⢨⣯⣿⣯⣭⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢬⡿⣿⢸⣿⡇⣿⢟⡥⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⠇⠀⠀⣿⠀⠐⣾⠃⠀⣿
⢸⡟⠻⣯⡟⣼⢇⣿⣿⣄⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣼⣿⣿⣿⣿⣿⣾⣿⣷⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⢻⠆⠀⣿⠀⠀⠿⠆⠀⣿
⠨⣝⢦⡀⢰⡟⣾⣿⠿⠃⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢲⣶⠄⠀⣿⠀⠈⠻⠃⠀⣿
⠀⠹⣷⣝⠿⢱⣿⡶⠾⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠗⠻⠟⠗⠾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠠⡅⠀⠀⣿⠀⢸⠿⡆⠀⣿
⠀⠀⠙⣿⠃⢶⣾⠟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣶⣦⣤⣴⣶⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠾⠿⠶⠶⠶⠶⠶⠟

⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣄⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⣤⠐⠂⠀⠀⣄⠤⣄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠈⠟⠀⠀⠀⠈⠉⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠒⠴⠀⠀⠀⣀⠀⣒⣄⠀⠠⣤⣄⣀⠀⠀⠀⣭⣬⡀⠀⠀⣴⠀⢠⡀⠀⠀⣔⣶⠀⠀⠀⠀⣀⡄⠀⠀⠀⢀⣠⠄⠀⠀⠀⣠⣤⠀⠀⠀⢠⣤⣤
⠀⠀⠐⠀⠸⠂⠀⠈⠛⠛⠛⠛⠀⠀⠀⠉⠀⠀⠀⠀⣯⣿⠃⠀⠀⠈⠀⠋⠀⠀⠀⠛⠛⠁⠀⠀⠀⠈⠁⠀⠀⠀⠘⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠘⠋⠛

⠀⠀⠀⣠⡶⠀⠀⠀⣄⣠⣤⡄⠀⠀⠚⢻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⣶⣶⠀⠀⠀⣤⣦⣤⠀⠀⠀⠀⠄
⠀⠀⠀⠉⠀⠀⠀⠀⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠈⠁⠉⠀⠀⠀⠛⠛⠛

⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⡶⡆⠀⠀⠀⠀⡔⠀⠀⠀⠀⠠⠞⠀⠀⠀⢤⠤⠂⠀⠀⣠⣤⣤⣤⠀⠀⢠⣶⡄⠀⠀⠀⡀⢀⡄⠀⠀⢀⣦
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠈⠁⠀⠀⠈⠛⠃
⠀⠀⠀⠀⡀
⠀⠀⠀⢸⢽⠃⠀⠀⢠⣤⣶⡄⠀⠀⢜⠉⣳⠀⠀⠀⡐⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡟⡇⠀⠀⢠⣿⣷⡀⠀⣀⣃⣐⣠⡀⠀⠀⠀⠆
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠛⠈⠁⠀⠀⠀⣿⣇⠀⠀⠀⠀⣿⠄⠀⠀⠠⣤⣤⠄⠀⠀⠀⢴⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⣤⡴⠀⠀⠀⢰⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠔⠛⠂⠀⠀⠀⣀⡀⠀⠀⠀⣠⣰⣶⠀⠀⠀⢸⡿⠀⠀⠀⠘⠶⠣⠀⠀⠀⢨⠹⠃⠀⠀⠀⠺⠇⠀⠀⠀⠺⢷⣶⣤⠀⠛⠃⠰⠅⠀⠀⠸⠏⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⠀⡄⠀⠀⠀⣤⡤⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣤
⠀⠀⠀⠿⠿⠗⠀⠀⢰⣿⣿⠀⠀⠀⢹⡴⠇⠀⠀⠀⡿⣟⠀⠀⠀⠐⢿⣆⠀⠀⠀⠾⠿⠆⠀⠐⠟⠟⠛⠻⠀⠠⠖⠴⠷⠀⠀⠼⠿⠷⠶⠂⠀⢸⣦⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⡸⣆⠀⠀⢸⠶⠿⠿⠆⠀⡷⠾⠿⠷⠀⢸⠶⠿⠿⠧⠀⣟⣻⣃⣻⡇⠀⠀⣾⣷⠀⢴⣷⡆⣴⣶⠀⣿⠀⠀⠀⠀⢀⣾⣫⣯⣫⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠙⠋⠀⠙⠙⠁⠉⠉⠀⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⣀⠀⠀⢀⡠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⡄⡀⠀⢠⣤⠀⢤⣤
⠀⠀⠸⠿⠇⠿⠇⠠⠿⠀⠸⠷⠄⠸⠆⠶⠰⠇⠀⠀⠀⠀⠀⠀⠀⠺⠂⠾⠂⠀⠀⠐⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠂⠺⠆⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⢳⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠉⠂⠑⡀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⡰⣄⠀⠘⢄⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⡜⡀⣸⢆⠀⠈⢆⠈⢆
⠀⠀⠀⠀⠰⢒⠒⠀⠒⠒⠒⠒⠒⠀⠈⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠉⠈⠁⠉⠉⠉⠉⠉⠉⠉⠉⠿⢉⠗⢤⡀
⠀⠀⠀⠀⠀⠡⠁⠀⢠⣤⡤⣤⡤⠀⠀⢠⣤⣤⢤⠤⡤⡤⠤⢤⡀⠀⢠⣤⡤⠤⠤⡤⣤⠄⠀⠀⡠⠋⢀⡞⠁
⠀⠀⠀⠀⠀⠀⢣⠀⠀⢫⠻⡝⠁⠀⣰⠣⢂⡫⠛⠓⠛⠋⠛⠚⠳⡀⠀⠻⠙⢿⠙⢻⠃⠀⢀⡞⠁⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠑⠀⠀⡴⠃⠄⡼⠁⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠱⡈⡳⠃⠀⣠⠗⠀⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⣼⠃⠠⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠘⠁⠀⡴⠋⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠹⡛⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠄⠀⠀⢼⣄⣰⠋
⠀⠀⠀⠀⠀⠀⢀⠎⠀⢀⣦⠀⠀⠱⡈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠈⢋⠻⡄
⠀⠀⠀⠀⠀⢠⠂⠀⢀⠞⠘⣡⠀⠀⠰⡀⢣⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⡰⠻⣄⠀⠀⢣⠈⢆
⠀⠀⠀⠀⠠⠃⠀⠀⠛⠚⠓⠃⠃⠀⠀⠘⠒⠓⠒⠒⠒⠒⠒⠒⠊⠀⠀⠘⠓⠒⠟⠂⠀⠀⡱⡈⢆
⠀⠀⠀⢠⣀⣉⣀⣠⣤⣤⢤⣤⢤⢤⡀⠀⠀⣤⣤⣤⣤⣤⡄⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠐⠘⡌⢢
⠀⠀⠀⠀⠈⠛⠾⠾⠶⠶⠞⠚⠒⠓⠳⠀⠀⠸⠙⣗⢛⠞⠀⠀⡴⠉⣹⠾⠽⠽⠽⠽⠭⠽⢩⣩⣝⣗⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠣⣨⠎⠀⢀⡞⠐⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠁⠀⢠⠋⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⡰⠃⢠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡼⠁⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠃
⠀⠀⠀⠐⠔⠠⠂⠔⠥⠪⠀⠀⠅⠀⢊⠔⠀⠢⢂⠔⡠⢀⠔⡠⠀⢀⠔⡠⠂⢀⢔⠔⡠⠂⢊⠔⡡⢂⢀⠔⠔⠴⠔⠔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠁⠈

⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⡠⠚⠉⡽⢏⠉⠳⣄⠀⠀⢀⡴⠊⠉⠁⠉⠙⠢⡀⠀⠀⣠⠖⠋⠉⠉⠉⠓⢤⠀⠀⠀⡤⠚⠉⠉⠉⢉⠲⣄
⠀⣼⠱⣒⡞⠓⠚⢷⣲⠞⡆⠀⡎⠀⠀⢀⣾⡄⠀⠀⠹⡄⢰⠃⣀⣠⠤⢤⣀⠀⡀⢣⠀⡞⠀⢸⣧⣤⣤⣿⡆⠘⡆
⠀⣧⢀⡿⡄⠀⠀⣰⢧⡀⡟⠸⡇⠀⢀⣾⠋⢿⣄⠀⢀⡇⢸⠘⢯⣀⣀⣀⡤⠟⠇⢸⠀⡇⠀⢸⣿⣿⣿⣿⡇⠀⡇
⠀⠘⢎⡉⠙⣏⡹⠋⢉⡽⠁⠀⠳⣔⠛⠓⠒⠚⠛⢒⠜⠀⠈⢧⡀⠀⠀⠀⠀⠀⣠⠏⠀⠙⣄⠘⠇⠀⠀⠹⢁⡼⠁
⠀⠀⠀⠙⠒⠾⠗⠒⠉⠀⠀⠀⠀⠈⠓⠲⠤⠶⠚⠁⠀⠀⠀⠀⠙⠒⠦⠤⠖⠊⠁⠀⠀⠀⠈⠑⠲⠤⠴⠒⠋
⠀⠀⠀⢀⣠⠤⠤⣄⡀⠀⠀⠀⠀⠀⣠⠤⠤⠤⣄⡀⠀⠀⠀⠀⢀⡤⠤⠦⠤⣄⠀⠀⠀⠀⠀⣀⡤⠤⠤⣄⡀
⠀⢀⠞⠁⠀⠀⠀⠀⠙⢦⠀⠀⣰⠋⠀⠀⠀⠀⠀⠙⢆⠀⠀⡴⠉⠀⢠⢦⡀⠀⠙⣄⠀⢀⠞⠁⠀⠀⢀⣀⠉⢢⡀
⠀⡞⠈⠳⣄⠀⠀⠀⠀⠀⣇⢠⡇⡀⣀⣀⢠⢀⣀⣀⠘⡆⢸⠁⠘⣯⠋⠉⠹⡝⠀⢸⠀⡏⠀⠰⠎⢹⢡⣋⣠⠀⣇
⠀⢧⠀⠀⠹⣶⣴⣾⣦⢀⡏⠈⣇⠘⢝⢝⣿⡻⣾⠎⢰⠃⠸⡄⠰⠯⢦⢠⠾⠵⠀⡸⠀⢧⠀⠷⠾⢺⠀⠉⠁⢀⡏
⠀⠈⠣⣀⠀⠈⠉⠉⣠⠞⠀⠀⠘⠦⣀⠉⠻⠉⣀⡴⠋⠀⠀⠙⢦⣀⠀⠋⢀⣠⠜⠁⠀⠈⠳⣄⠀⠉⠁⢀⣠⠞
⠀⠀⠀⠈⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⣠⠴⠒⠚⠒⠲⢄⠀⠀⠀⢀⡠⠖⠒⠒⠒⠦⣀⠀⠀⠀⠀⣠⠔⠒⠒⠒⠦⡀⠀⠀⠀⢀⡤⠖⠒⠒⠲⢤⡀
⠀⡼⠁⠀⠀⣰⠀⠀⠀⢳⡀⢠⠎⢀⢀⣄⣀⣀⡀⠈⢆⠀⢀⡞⠀⠀⠰⡦⠀⠀⠈⢦⠀⢰⠋⠀⣰⣶⣷⣄⠀⠙⡄
⠀⡇⠀⠀⠀⣾⡆⠀⠀⠀⡇⢸⠀⠘⠃⣿⠉⢹⡇⠀⢸⠀⢸⠀⠀⠀⣰⣧⡀⠀⠀⢸⠀⡏⠀⠈⠻⢿⡿⣿⠀⠀⣿
⠀⢣⡀⠀⠀⣿⡇⠀⠀⣰⠃⠘⣆⠀⠀⠿⠀⠸⡇⢀⡞⠀⠘⣆⠀⠀⣿⣿⡃⠀⢀⡜⠀⠹⡄⠀⠀⢰⠀⠀⠀⢠⠏
⠀⠀⠙⠦⣄⣀⣠⠤⠞⠁⠀⠀⠈⠳⢤⣄⣀⡠⠴⠋⠀⠀⠀⠈⠳⢤⣈⣉⣠⠴⠊⠀⠀⠀⠙⠢⢤⣀⣀⡤⠔⠁
⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⢠⠞⠉⠀⢀⡄⠈⠳⣄⠀⠀⣠⠚⠁⠀⡄⠀⠉⠢⡀⠀⠀⡰⠊⠁⠀⠀⠈⠙⢦⡀⠀⢀⡴⠋⠁⠀⠀⠈⠑⢄
⠀⡏⠀⠀⣴⣾⣷⡄⠀⠘⡆⢰⠃⠀⠀⢸⣿⠀⠀⠀⢹⠀⢸⠁⢀⡠⣤⣤⢤⡀⠀⢧⠀⡞⠀⠀⢸⣿⣿⡇⠀⠈⡇
⠈⡇⠀⠘⣿⣷⣿⠇⠀⢠⠇⢸⡀⠀⢸⣿⢿⣿⠀⠀⢸⠀⢹⠀⠘⠤⣿⣾⡧⠋⠀⡼⠀⣇⠀⠀⠀⠻⠏⠀⠀⠀⡇
⠀⠙⢄⡀⠘⠟⠁⠀⣠⠎⠀⠀⠳⣄⠘⠟⠐⠟⢀⡰⠃⠀⠈⢣⡀⠀⠀⠀⠀⢀⡴⠃⠀⠘⢦⡀⠀⠴⠄⠀⢀⠜⠁
⠀⠀⠀⠉⠓⠒⠚⠉⠀⠀⠀⠀⠀⠈⠙⠒⠒⠚⠉⠀⠀⠀⠀⠀⠈⠓⠒⠒⠚⠉⠀⠀⠀⠀⠀⠉⠓⠒⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡶⠶⠶⠶⣤⣀⣴⡇⢸⣶⣀⣤⠶⠶⠶⢶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠶⠶⠶⢭⣷⣆⠰⢦⣍⣸⡃⢸⣇⣩⡴⠆⣰⣾⡭⠶⠶⠶⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠳⢦⣄⠒⡍⠉⠒⠒⢦⡉⠃⢸⡇⠙⣇⣸⠋⢸⡇⠘⢉⡤⠒⠒⠉⢡⠒⣠⡴⠞⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠀⠹⡆⠘⠀⠀⢀⡏⠀⢀⣘⣓⣦⡌⢡⣴⣚⣃⡀⠀⢹⡀⠀⠀⠃⢰⠏⠀⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠘⢧⣸⠇⠠⠀⠀⠸⠗⢋⣩⡭⠭⢭⣭⣭⡭⠭⢭⣍⡙⠺⠇⠀⠀⠄⠸⣇⡼⠃⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣈⠀⢀⡀⠁⠀⠀⠀⢀⣀⣰⣯⠀⠀⢀⡾⠁⠈⢷⡀⠀⠀⣽⣆⣀⡀⠀⠀⠀⠈⢀⡀⠀⣁⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣡⠞⣻⠁⢀⣠⠶⣚⣩⣭⠄⠀⠀⠀⠺⡅⠀⠀⢨⠗⠀⠀⠀⠠⣭⣍⣓⠶⣄⡀⠈⣟⠳⣌⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡞⣡⣆⢸⡴⢋⡴⠚⢩⠟⠁⠀⠀⠀⠀⠀⢹⡀⢀⡏⠀⠀⠀⠀⠀⠈⠻⡍⠛⢦⡙⢦⡏⣰⣎⠳⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣼⠃⢹⡌⢡⡟⠁⠀⡏⠀⢠⣤⣤⣄⡀⠀⢈⡇⢸⡁⠀⢀⣠⣤⣤⡄⠀⢹⠀⠈⢻⡌⢡⡏⠘⣧⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠻⡟⠀⠀⠀⢿⠀⠀⠷⣌⠻⣷⣄⠈⠃⠘⠁⣠⣾⠟⣡⠾⠀⠀⣿⠀⠀⠀⢻⠟⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣀⠤⠤⣤⠀⠀⠀⡀⠘⢧⠀⠉⠒⠲⡌⣿⠇⠀⠀⠰⣿⢡⠖⠒⠉⠀⡼⠃⢀⠀⠀⠀⣤⠤⠄⣀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣥⢶⠀⡟⢰⡤⠤⠟⠀⢸⡆⠀⠀⠀⢱⠏⠀⠀⠀⠀⠹⡆⠀⠀⠀⢰⡇⠀⠻⠤⢤⡆⢹⠀⡶⣬⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢸⡆⣇⠈⣇⠀⠀⠀⠘⢧⡀⢧⠀⣹⣠⣠⣄⣠⣤⣀⣇⠀⡜⢀⡼⠃⠀⠀⠀⣸⠁⣸⢰⡇⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⡌⣧⠸⡀⠈⠢⣄⣀⡀⠈⠳⣤⠇⢣⡁⠹⣄⣰⠏⢈⡜⠸⣤⠞⠁⢀⣀⣠⠔⠁⢀⠇⣼⢡⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠹⣾⣆⠳⣀⠤⢤⠀⠀⠀⠀⣿⠪⣐⠀⠀⢸⡇⠀⠀⣂⠑⣿⠀⠀⠀⠀⡤⢤⣀⠞⣰⣷⠏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠉⣠⣥⠀⠘⣆⢠⠀⠀⠘⢶⣶⣶⠶⠋⠙⠶⣶⣶⡶⠋⠀⠀⡀⣰⠃⠀⡬⣄⡉⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⡰⢋⣤⢸⡇⠀⠘⢶⡀⠀⠀⢸⣷⠂⠀⠀⠀⠀⠐⣾⡇⠀⠀⠀⡶⠃⠀⢰⡇⣤⡙⢆⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⡏⢸⢸⡇⠀⠀⠀⠀⠠⡄⠈⢿⣶⣄⣠⣀⣠⣶⡿⠁⢠⠄⠀⠀⠀⠀⢸⡇⡇⢹⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⡞⣼⠁⣠⡴⢶⡆⢸⣁⣀⠀⠟⠻⣿⣿⠟⠻⠀⣀⣈⡇⢰⡶⢦⣄⠘⣧⢳⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠱⣏⢸⡏⠀⢸⡇⣨⡄⢸⠀⠀⠀⠘⠃⠀⠀⠀⡇⢨⣅⢸⡇⠀⢹⡇⣹⠎⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣝⢿⡇⠀⠘⣿⠁⣿⠘⡆⠀⠀⠀⠀⠀⠀⢰⠃⣿⠈⣿⠃⠀⢸⡿⣫⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠀⠀⠹⠀⠸⣆⠙⠇⠀⠀⠀⠀⠸⠋⣰⠇⠀⠟⠀⠀⠠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠻⣆⠀⡴⠋⠙⢦⠀⣰⠟⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣧⠀⠀⣼⠞⠁

⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄
⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⣰⣿⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣇⣀⠀⠀⠀⠀⠀⠈⣿⡟
⠀⠀⠀⠀⢿⡿⠁⠀⠀⠀⠀⠀⠀⢩⡿⠁⠀⠀⠀⠀⠀⠀⠀⠉⣸⡿⠁⠀⠀⠀⠀⠀⢰⠟
⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠞


⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠰⣿⣿⠆⠀⠀⠀⠀⢠⣿⣿⣥⠂⠀⠀⠀⠀⠀⠀⢚⣿⡟⠁⠀⠀⠀⠀⠀⣰⣿⣀
⠀⠀⠀⠀⢠⣿⣏⠀⠀⠀⠀⠀⠈⣽⡿⠁⠀⠀⠀⠀⠀⠀⠀⠺⣿⡖⠀⠀⠀⠀⠀⠀⠋⣿⠃
⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠀⠀⢸⠏
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠄⢀⣘⠆⠀⠀⠤⢔⡄⡀⠀⠀⠀⣀⠀⠀⠀⠘⣆⢐⠀⠀⠀⠠⠄⠤⠀⠀⠀⢺⢿⠒⠀⠀⠐⢢⡆⠂⠀⠀⢐⡮⡽
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠐⠉⠈⠑⠀⠀⠈⠒⠊⠁⠀⠀⠐⠀⠃⠀⠀⠀⠨⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠤⠆⠀⠀⠀⣀⣀⣂⠄⠀⠘⢰⠟⠋⠀⠀⠠⢼⠧⠀⠀⠀⢒⣐⢱⠀⠀⠈⠀⡀⠄⠀⠀⠠⢄⢘⡆⠀⠀⢠⠉⠗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉
⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠰⠟⠁⠀⠀⠀⣁⠀⢈⠀⠀⠀⠀⠐⠃⠀⠀⠀⢸⣃⠀⠀⠀⠀⠘⠲⠀⠀⠀⠀⠩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⣓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠉⠉
⠀⠀⢀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠
⠀⠀⣠⣠⡨⠀⠀⠀⡸⠠⠯⠀⠀⠀⠴⣶⠀⠀⠀⠐⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠁⠁⠀⠀⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⢀⣠⡀⠀⠀⠀⢤⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠤⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡴⡄
⠀⠀⠀⡅⠃⠀⠀⠀⠀⠠⠀⠀⠀⠀⢨⠩⠀⠀⠀⠀⠂⠘⠀⠀⠀⠀⢭⠄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢂⠓

⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡖⡆⡀
⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠛⠀⠀⠀⠀⠀⠻⠟⠀⠀⠈⠉⠑⠏⠉

⠀⠀⠀⢀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢄⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⣄⣠
⠀⠀⠀⠊⠐⠀⠀⠀⠙⠒⠀⠀⠀⠀⠙⠐⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠤⠀⠀⠀⠀⠑⠁⠀⠀⠐⢐⢃⠐⠀⠀⠀⠘⢑⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠄⠀⠀⠠⠊⠁⡀⠀⠀⠀⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⠀⠀⠀⠊⠑⠂⠀⠀⠀⠙⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⠘⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣾⣿⣁⣀⣀⣀⣀⣈⣿⣷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠹⣿⣿⠛⠛⠛⠛⠛⠛⢻⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⡟⠛⠛⠛⠛⠛⠛⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠹⣿⣦⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⣰⣿⠟⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠻⣿⣆
⠀⠀⠀⠀⠀⣰⣿⣿⣤⣤⣤⣤⣤⣤⣼⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣧⣤⣤⣤⣤⣤⣤⣿⣿⣆
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢿⣿⡉⠉⠉⠉⠉⢉⣿⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄⢠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⢀⣴⣿⣿⠿⠛⠉
⠀⢀⣴⡿⠛⠉
⠐⠋⠁

⠀⠀⠀⠀⠀⠀⠀⣶⣶
⠀⠀⠀⠀⠀⠀⢀⣤⣥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⡄⢀⡄⠀⠰⠄⠀⣄⠀⣄⠀⣄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣄⣀⣀⣀⣀⣀⣀⣀⣠⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⠰⣧⢰⣧⠀⢺⡇⠀⣾⠀⣾⠀⣷
⠀⠀⠀⠀⣀⡤⠟⣋⣝⡻⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢿⡄⠻⣦⣼⣧⡾⠋⣰⡏⣸⡏
⠀⠀⠐⢫⣵⣾⣿⣿⣿⣿⣷⣮⣍⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠛⢶⣤⣤⣤⣤⠾⢋⣴⠟
⠀⠀⠀⢸⣿⣿⣿⠛⠛⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠶⣤⣤⣤⣤⠶⠛⠁
⠀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⣿⣿⡿⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣿⣷⣶⣦⡄



⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠾⠛⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣆⠰⣷⣦⣤⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣄⣀⣀
⠀⠀⠀⢻⣟⢛⡟⠛⠛⠘⣷⡛⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠉⣻⡟⠛⠛⠛⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⣽⣿⣿⣿⣿⣍⠻⢦⣄
⠀⠀⠀⠀⢹⡿⠁⠀⠀⠀⠈⣽⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠁⠀⢸⣿⣿⠉⢙⣿⣿⠀⠀⢙⣷
⠀⠀⠀⣰⣟⡸⣧⢀⣀⣀⣾⣋⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣈⢿⣿⣷⣿⣿⠟⣀⣴⠟⠁
⠀⠀⠀⠉⠉⠉⠙⣷⣉⣽⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠈⠻⠿⠿⢶⣤⡀
⡠⠒⠂⢒⡀⠀⠀⢰⣿⡿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⢿⣿⣿⣟⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣷⣮⣻⣦
⢧⣤⣴⣾⣷⣄⡀⢘⣿⣷⣿⣿⠇⠀⠀⣠⡦⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣝⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣤⣴⣿⠷
⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣤⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣻⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠉⠛⠛⠿⠟⠛⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⢿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⣿⣿⣿⡿⠟⠻⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠿⢿⣿⣿⣿⣿⣿⣿⠛⣤⠉⡀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠉⢇⠘⣇⠰⢷⣐⠃⠀⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⠀⠀⠀⢀⠀⠈⠓⢬⡀⠀⠉⠤⠎⠰⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣯⠛⠁⠚⠛⠉⠉⠉⠑⠺⠧⠤⣀⡀⣬⣿⡧⡖⣾⡛⠛⠪⣀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠤⡀⠜⣀⠖⣰⠄⡄⠀⠀⠀⠀⠀⠀⠀⠬⠛⠒⠈⠁⠀⠀⠀⣿⡄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠈⠀⠚⠁⠀⠀⠂⠒⠂⠠⢀⠀⠀⢀⡀⠀⠘⠀⠀⠀⠀⢀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢠⡄⠀⢠⣶⡄⠀⠀⠀⠈⠡⠉⠁⠀⠀⠀⠈⠃⠀⠀⠰⠤⠀⠀⣼⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠟⠈⣼⣿⣿⣶⣶⣤⣤⣄⣀⡀⠀⠀⢀⣤⣤⠂⠀⠀⠀⠛⣸⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠻⠃⢿⡴⢻⠩⣿⡀⠀⠀⠀⢰⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠈⣿⣶⣶⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⣀⠀⠱⠀⠀⠈⠃⠀⠀⢀⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠺⠄⣴⣚⣿⡟⣿⢻⠋⠛⠻⠋⠟⣷⣶⣶⢬⣵⣦⢤⣀⡀⢁⠈⠀⠀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⡬⢀⣀⣚⠻⣿⣾⣳⢃⠎⠀⠀⠀⡠⢸⣿⣟⡷⠀⢿⠧⠀⢿⣿⣿⡶⢂⠀⢸⠁
⠀⠀⠀⠀⠀⠀⢷⣆⠀⠈⢉⣉⣉⣉⠈⠉⠁⠀⠘⠳⠿⠯⠿⣁⡌⢨⣧⢀⣿⡿⡿⠃⣼⡄⠈
⠀⠀⠀⠀⠀⠀⠈⢎⣇⢠⡾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠭⣅⣁⡀⠀⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢳⠏⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⠒⣀⡠⠀⠀⠀⠀⠀⠀⠸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠲⠫⠦⠍

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠁⠀⠀⠄⠒⠀⠀⠈⠃⠐
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠈⠀⢀⢀⣀⠈⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠄⠀⢸⠞⠉⢆⠀⠀⠀⢀⠉⠀⢀⠘⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⡀⢀⠈⠙⠖⡋⠘⠂⠂⠀⠚⢂⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠄⠉⠐⡀⠚⡅⠛⠐⠧⡤⠀⢶⠀⠆⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠸⠖⠌⠀⠸⠧⡇⠀⡀⠠⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢀⠷⡀⡁⠞⠀⠀⠀⠀⢀⠀⠆⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠀⠐⡄⠀⡐⠂⠄⠈⠙⢱⠾⠶⣁⠀⠀⠒⣈⠦⡲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⠘⠛⠲⣄⢸⡆⢠⡄⣵⢀⡔⠋⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⠀⠀⠈⠀⠀⠀⠀⠄⢸⠀⠀⠀⠀⠈⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⠀⠠⠀⠀⠀⠀⢀⠊⠘⡁⡀⠀⠀⢀⣸⠀⠂⠀⠀⠀⠀⠀⣀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⠸⠷⠈⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣠⠞⠀⠀⠀⢱⠈⠑⠔⠀⡆⢥⡀⠀⠀⠀⠻⣄⠀⠀⡾⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠙⢄⠁⠀⠀⠀⠄⠀⠀⠀⠀⠁⢀⢧⠀⠀⠀⠀⠈⠀⠀⠙⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠟⠴⡆⣿⢳⡄⠧⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢸⢀⣬⠀⠀⠀⢿⠠⠃⢰⠀⢀⠀⠀⠀⠐⠶⠄⠀⠠⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⣁⠛⢢⡀⠀⠸⣧⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣍⡄⠀⠳⠀⠀⡿⣇⢰⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣧⡟⢰⡗⠀⣰⠇⣇⣿⠀⢀⣀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⠁⠀⠀⠈⠀⢀⣤⡠⠞⡩⠀⡞⠀⡐⠛⣠⣿⠟⠀⠸⣁⠉⣠⡀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠂⠂⠀⠴⠦⡈⠀⠄⠀⠊⢀⣼⠁⠐⣇⠀⢼⡸⣟⡄⠁⠀⠀⢨⣄⠠⠀⠀⢤⡀
⠀⡀⠀⠀⠀⠀⠀⠘⣃⡀⠀⢄⡑⠆⠀⠑⠲⡆⠰⠶⠋⠉⠃⢰⡿⠄⠀⠓⠿⠄⠒⠀⡴⠕⠘⠧⠀⣀⠀⠀⠀⢀⡀
⠀⠅⠀⠀⠀⡠⠄⠀⠈⠀⠘⠉⣔⠊⡇⠀⠀⠱⠀⢢⡀⠁⠸⠸⠫⢶⡀⢢⣴⠲⠶⢏⡁⠀⠈⠦⠘⠋⡈⢰⠐⠊
⠀⠂⠀⠀⠀⠀⠤⠀⠀⠠⠀⠀⠈⡠⠇⣘⠂⢀⡲⡦⢄⢠⡴⢐⠁⢺⠀⣠⠀⠀⠐⢀⠱⠀⠁⠀⠆⠀⠐⢣⣄⠀⡁
⠀⠀⠀⠀⠀⠀⠘⡂⠀⠘⠑⠄⠘⣠⠠⣀⡼⠏⠀⠀⠀⡈⢤⢀⣀⠈⠻⠑⡀⢠⠀⢀⠠⠃⠂⠉⠤⡷⠄⠀⠀⠈⠉
⠀⠃⠀⠀⠀⠈⠒⠓⠖⣠⡄⠐⠀⡠⠏⠈⠀⠚⠀⠀⠀⠈⠠⠠⠈⡼⠀⢈⠲⢈⡕⠠⠆⠠⡄⠹⠏⠀⠉⠁⠠⠐⠐⠒
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠃⠀⠘⠂⠀⠀⠀⠀⠀⠚⠒⠈⠁⠀⠁⠀⠉
⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⡿⠿⠶⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⣥⣾⣿⣿⣿⣿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠚⠉⠁⠈⠉⠻⣿⣇⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠸⣿⡄⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏⢀⣶⣿⣿⣿⣷⣄⠀⣿⣷⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠏⢀⣾⣿⣿⠟⠛⠛⠻⠀⢸⣿⣇⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⣀⣀⣀⣀⣀⢠⡿⢿⠟⢀⣾⣿⡿⠁⢀⣄⣅⣀⢸⡶⣿⣿⣜⣧
⠀⠀⠀⠀⣀⣀⡀⡀⠿⠷⠶⠶⠤⠤⠬⠭⠉⠛⠒⠊⠁⢠⣾⣿⡿⠁⢀⣼⠿⠿⣿⣿⡇⡟⢿⣿⣿⣧⣀⣠⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀
⠀⣠⣾⣿⣿⠏⢰⠇⠀⣠⣶⣿⣿⣿⠶⢶⣶⠆⣤⣤⣶⡿⢿⡿⠁⣠⠟⠁⣀⣤⣈⣿⠳⣇⠀⠈⠉⠉⠙⠛⠛⠛⠛⠿⢿⣿⣿⣿⣷⣌⠉⠳⣄
⣸⣿⣿⣿⣿⠀⠈⣆⢰⣿⣿⣿⠋⠀⢀⣾⣿⡟⠃⠐⣛⠻⠷⣿⠿⠋⢀⣼⣿⣿⣿⣿⠀⠋⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡄⠀⠈
⣿⣿⣿⣿⣿⡀⠀⠈⠺⢿⣿⣿⠀⠀⠘⣿⣿⡀⢠⣿⣿⣿⣷⣦⣤⣴⣿⣿⣿⣿⣿⣿⣦⣄⣙⠛⠿⢿⣿⣿⣿⣿⣦⡀⠀⠀⣹⣿⣿⣿⠃
⠹⣿⣿⣿⣿⣿⣄⡀⠀⠀⠉⢻⣦⡀⠀⠈⢻⣿⡄⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⠛⠛⠿⠿⣿⣶⡄⠀⠘⣿⡻⣷⠀⣠⣿⣿⡿⠋⠀⠀⢠
⠀⠙⠿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠙⢿⣦⡀⠀⠙⢿⡆⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣄⠘⣿⡟⠀⢀⣿⠿⣿⣾⠿⠋⠁⠀⢀⣠⠶⠋
⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣷⣤⡀⠀⢸⣿⣦⠀⠀⠃⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠓⣋⡤⢔⣩⠴⠒⠋⠀⠀⢀⣠⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣦⣼⣿⣿⡴⣵⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡻⠛⢛⣿⣿⢛⣚⣭⡖⠀⠀⠀⣠⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⢏⣶⣿⠿⡞⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⠟⠉⣿⣿⣿⡇⠀⢀⡾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢏⡾⣸⠏⣰⡕⠀⠙⠋⠉⣀⣤⣴⣯⢿⢿⣿⣿⣿⣿⡇⠀⣿⡆⠀⢻⣿⣿⣇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⢁⠃⠀⢻⣿⣷⣶⣾⣿⣿⠿⠛⠛⠛⠻⠻⣿⣿⡿⠃⠀⢸⡇⠀⠸⣿⣿⣿⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠃⣾⡀⠀⠀⠙⠻⠟⠛⣉⣤⣶⣶⣄⢢⡀⠙⢿⣍⣀⣀⣤⣿⣧⠀⠀⣿⣿⣿⣇⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣾⢿⣷⣦⣄⢀⣤⣾⣿⣿⡿⠟⠉⠀⠈⠳⣝⠶⣝⢿⣿⣿⡿⢋⠀⠀⣸⣿⣿⣿⡀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠙⢷⣶⣿⣾⣿⣿⡿⠛⠁⠀⠀⠀⠀⣀⣠⣬⣳⣌⠳⣤⡀⣀⠞⠀⣰⣿⣿⣿⣿⠁⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠉⠙⠛⠉⠁⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣍⠁⠑⠶⣿⣿⣿⣿⠿⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠛⠉⠉⠉⠉⠛⢿⣿⣿⣿⣦⡀⠀⠉⠉⠀⠀⠀⠀⠀⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣤⣤⣤⣤⣶⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣷⣄⡀⠀⠀⠀⣀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣿⣿⣿⣟⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣻⣿⣿⣿⣛⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠔⠚⠉⠀⠀⠀⢀⢀⣀⠀⠀⠀⠀⠀⠰⢀⡠⡆
⠀⠀⠀⠀⠀⠘⠆⠀⠀⢧⡀⠐⠁⠀⢠⠀⠀⠀⠀⠔⢆⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠠⠬⠯⠤⠁⠀⠀⠰⢨⢯⣗⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠒⠒⠚⠁⡀⠀⠒⠂⢀⡜⠀⠀⠀⣄⣀⣀⣬⣷⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠂⠀⠀⠒⠈⠸
⠀⠀⠀⠀⠀⠀⠄⣀⣀⠀⠀⠀⠀⠀⠀⢀⠸⠴⠤⠑⠁⠀⠀⠠⡁⠀⠀⠀⠉⠄⠀⠀⠀⠀⠒⠀⠀⠀⠀⡀⠀⠀⠀⢰⠄
⠀⠀⠀⠀⠀⢀⠉⠀⠀⠑⠀⢀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⡗⠐⠂⡒⡐⠒⢲⠀⠀⠀⠈⠒⠠⠀⠄⠊⠀⡠⠂⠁⠘⠠
⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢇⠁⠀⠡⢁⠀⡸⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⡄⡄⠀⠘
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡴⠭⢦⢤⡄⠀⠡⣀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠁⢀⠁⠀⠀⡀⠐⠀⠀⡀⡴⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡁⠀⢈⣾⠀⠀⠀⠀⠉⠋⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⡍⢳⠈⡇⢳⣃⡇⠀⠀⠀⠘⠓⠻⣒⠞⠚⠃⠀⠀⠀⠀⠀⢠⣴⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠒⣬⠒⠂⠀⠀⠀⠱⠃⠀⠀⢀⠀⣁⡈⢈⣁⠈⣁⢀⢀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠜⢀⠁
⠀⠀⠀⠀⠀⡐⠈⠀⠀⡀⠢⠀⡀⠀⠀⠀⠡⠀⠀⠸⠀⡇⢹⢠⣤⣧⠼⠸⣤⠇⠀⣀⠗⢆⠄⠀⠸⡷⠀⠀⢀⠈⢀⠃
⠀⠀⠀⠀⢐⠀⠀⡨⠀⡀⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠚⠁⠀⠀⡄⡃⢀⠄⡠⠁⠀⠀⢀
⠀⠀⠀⠀⠘⡀⠄⠀⠀⠈⠀⢀⠄⠀⠀⠠⠀⠀⠀⠠⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⢈⠐⠁⠀⢰⣾⣚⠂
⠀⠀⠀⠀⠀⠈⠒⠠⠤⠤⠔⠁⠀⠠⠀⠀⠀⠀⠀⠂⢰⠉⢈⠆⢄⠀⠀⠀⣠⢄⠌⠀⠀⠀⠀⡙⠀⢠⠀⠀⠀⠀⢼⡾⠵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣑⠦⣑⣃⠭⢜⠕⠁⠀⠀⠀⢤⠀⠈⠂⡀⠀⠀⠀⠀⡙⠋⠁
⠀⠀⠀⠀⠀⠰⢀⠀⠀⠀⢀⣀⡀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠑⠃⠈⠉⠉⠉⠁⠁⠀⠀⢘⣄⠜⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣦⡅⣀⡐⠀⠀⠀⠀⠀⠀⠀⢄⠇⣊⡴⠓⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠒⠀⠉⠀⠀⠠⣈⡁⡴⠀⠂⡈⠀⠀⠀⠑⠊⠉⠻⢟⡓⠀⠀⠀⠀⠀⠀⣨⣊⠔⠒⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡐⠉⠀⠉⠃
⠀⠀⠀⠀⢀⣜⡀⣀⣁⣀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⡠⠒⠠⡄⠈⠃⢠⠐⠀⢀⠀⠀⠀⠄⠀⠐⠀⡠
⠀⠀⠀⠀⠐⠆⠶⠶⠶⠶⠶⠒⠩⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣀⠀⣠⣀⡀⠀⢀⡇⠀⠀⠂⠀⠀⢈⡔⠒⠈⠀⠀⠀⠚⠁

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⢠⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⠛⣇⣀⣀⣀⡀⠀⠀⠀⠀⠀⡎⢡⠀⠀⣀⡀⠀⠀⠀⠀⢠⠏⣆⣀⣀
⠀⠤⡄⠈⠀⠃⢠⠆⠀⠀⠀⠀⣀⠀⠀⠀⠀⠙⠳⡭⠀⣀⣶⠉⠀⠀⠀⠀⠘⠻⣭⠆⠈⢩⠟⠋⠀⠀⠀⠀⠉⠣⡠⢠⠊⠁
⠀⣠⠭⠐⠒⠢⠷⣄⠀⠀⠀⣰⠛⣆⣀⣀⡀⠀⣼⠵⠋⠙⠺⣧⠀⠀⠀⠀⠀⣰⠥⠒⠙⠦⣳⡄⠀⠀⠀⠀⠀⠏⠀⠈⠃
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢙⠞⠷⢥⠖⠉⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡰⢆⠀⠀⠈⠙⠆
⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⣈⠔⠳⣜⠀⠀⠀⠠⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⣀⡼⠘⠆⢀⡄⠀⠀⠀⠉⠀⠀⠈⠀⡀⢠⠇⠀⠑⠤⠀⠀⢠⠄⠰⢄⠀⠚⠀⠀⠂⠀⠉⠁⢀⡵⠋⠀⠀⢠⠏⢧
⠀⢠⡷⠲⣾⠁⠀⠀⠀⠀⣀⠀⠀⠙⡄⢁⠀⠀⠕⡀⠀⢀⠔⠁⠀⠀⠀⠑⢢⠁⠀⠉⠀⡀⠤⡋⠀⠐⠶⣤⡎⠀⠀⠡⣠⠜⠃
⠀⠀⠀⠀⠀⠀⠀⡐⡆⠐⠉⠀⠀⠀⠈⡆⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⡌⠀⠀⠨⠂⠄⡀⢙⢄⠀⠀⢰⢧⡶⠲⣄⣇
⠀⠀⠀⠀⠀⠀⡐⠁⠛⠀⠀⠀⠀⠀⠀⢹⠐⡠⠒⢦⠸⠀⠀⠀⠀⠀⠀⠰⠍⠀⠀⠀⠀⠀⠀⠁⠀⠦⠠⠟⠁⠀⠀⠈⠻
⠀⠐⠶⢖⡒⠐⠁⠀⠀⠓⠂⣀⠴⠂⠀⠀⠞⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⢀⢑⠠⠶⠦⠰⣯⡁⠀⠀⠀⠀⠀⠀⠀⠠⠄⢄⡤⡠⢄⡠⢄⠤⡠⣤⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠰⠂
⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⢢⠁⡆⠀⠸⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠻⠄⠏⠼⠐⠷⠨⡇⠀⠔⠂⠀⠀⠀⠀⠀⠀⣠
⠀⠀⢰⡀⠀⠀⠀⠀⠀⠳⢀⠀⡀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⢀⡄⡠⠀⢤⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠋⢇
⠀⠀⠻⠏⠀⠀⠀⠀⠀⢵⠀⠚⠃⢁⡂⠀⠀⠀⠀⢨⠂⠀⠀⠀⠃⠙⠀⠈⠀⠀⠀⠆⠁⠀⠀⠀⠠⣤⠼⢀⡘⠦⠄⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠴⣋⡂⠀⠀⠊⡡⠂⠀⠀⠀⠐⡧⡀⠀⠀⠀⠶⠆⠀⠀⠀⠠⠆⡁⠀⠀⠀⠀⠀⡟⡨⢍⡲⡁
⠀⠀⠀⠀⠀⠀⠠⠄⠠⠊⡤⢤⠌⠉⠉⠿⣄⠀⠀⠀⠈⠸⢦⣀⠀⠀⠀⠀⠤⡸⠐⠁⠀⠀⠀⠀⠀⠚⠉⠀⠀⠉⠛
⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠑⢳⡦⡊⠒⠂
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⡀⠀⠀⢤⡟⡀⠂⠀⠀⠀⠀⢀⣮⡄
⠀⠀⠀⠀⢰⢫⣀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠌⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣧⠀⠀⠀⠉⠁⠀⠀⠀⠀⠰⢵⠉⣛⣤⠶⠂
⠀⠘⠿⣉⡟⠀⢫⡩⠝⠉⠀⢤⡰⠐⠂⢢⠀⡀⠙⠀⠉⢉⡲⠛⠀⠀⠀⢰⠇⠸⣆⣠⣀⣤⡄⠀⠀⠀⠀⣤⡯⠬⠟⢧⡀
⠀⠀⢀⠼⠿⠛⠫⣧⠀⠀⠀⠀⠀⠀⢠⠠⠛⠓⠑⠒⠰⡋⠀⠀⠐⠾⢟⣫⡀⢀⡓⣢⠞⠉⠀⠀⠀⠀⠈
⠀⠀⠚⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⢠⠃⢀⠄⠒⠠⢤⣀⡐⠄⠀⠀⠀⢀⣾⡥⠚⠢⢷⣄⠀⠀⢘⣾⡗⠀⢠⡀
⠀⠀⠀⠀⢀⡚⣤⠄⠀⠀⠀⠀⢠⡿⠊⠁⠀⠀⠀⠀⡀⠉⠙⠂⠀⠀⠞⠉⠀⠀⠀⠀⠉⠳⠀⠀⠀⠀⢠⡀⢷⡀
⠀⠀⣀⣄⡛⠀⠘⣄⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠜⠁⠀⠑⠶⠄⢠⠴
⠀⠀⠀⡒⡦⠀⣀⠀⢚⡉⠀⠀⡀⠀⠀⠀⠀⢤⣤⡞⠀⠒⠤⡀⠄⠀⠀⠀⠀⢠⣦⠄⠀⠀⠀⠀⠀⠀⡦⠀⣀⠀⠴⠉
⠀⠀⠀⡿⠓⠋⠓⠉⠛⠱⠀⠠⣖⡆⠀⠀⠀⠀⢈⢆⠀⠰⣎⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣰⠣⠊⠁⠑⢄⡂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣊⣀⣀⣀⣀⣑⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣁⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣀⢀⡀⠀⠀⡐⢉⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠹⠀⠀⠀⠀⠙⢌⡣⠊⠁⠀⠀⠀⡎⠢⠐⠀⠀⢀⡀
⣶⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠐⠀⢃⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠑
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁
⠀⢀⡀⠀⠀⢀⡴⣚⡂⣄⠀⠀⠀⢤⡀⠀⠀⢰⣄⠀⠀⣠⠦⢤⡀⠀⠀⠀⠀⠠⠄⠀⣠⠷⠀⠀⣀⡀
⠀⠰⠋⠀⠀⠀⠙⠮⠚⠁⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠈⢢⠀⠁⠈⠘⠀⠀⠠⢤⠀⡦⠀⠀⠀⠴⠋
⠐⠪⠅⠀⠀⠆⠀⠠⠆⠐⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠣⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠺⡒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣤⠀⢠⠀⠀⠀⠀⠀⠀⢀⣀⡴⢦⣀
⠀⠀⢠⣤⣤⣤⣄⣄⣤⠀⠀⠀⢠⢳⢀⠚⣀⠞⡄⠉⢁⠉⠀⠀⣮⣠⣤⣤⠥⣵⡀⠀⢀⣀⡀⣀⣀⣀⢀
⠀⠀⠈⠀⠀⠀⢠⣥⣤⠀⠀⠀⠛⠒⠛⠐⠛⠒⠃⠀⠀⠀⠀⠀⠄⠀⠄⠴⠠⠠⠀⠀⠀⠉⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣄⣀⣀⡀⠀⠀⢀⢀⡀⡀
⢄⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠸⣾⣷⡇⠀⠀⠀⠀⢆⣦⡆⠀⠀⠀⠀⠀⠀⣀⡠
⠒⠋⠀⠀⠐⠩⠀⠬⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠱⠃
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠊⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠂
⢀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⢚⡱⡇⡓⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠊⡇⠀⠀⠀⢀⠠⠀⢀⠀⠀⠀⠈⠑⠊⠃⠀⠀⠌⢱⠆⢡⠀⠀⠉⠙⠙⠋⠉⠀⠀⢠⣏⣧⠀⠀⠀⠛⢣
⠵⠇⠀⠐⠀⠁⠂⠒⠉⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⠒⠐⡁⠀⠀⠀⠀⠀⠆⠀⠀⠀⠓⠒⢒⠃⠀⠀⠰⠢
⠀⠀⠀⠀⠀⣈⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⠄
⣉⡉⠉⠀⠀⠀⠀⠀⣠⠀⠀⠘⣀⡛⣤⠃⠀⠀⣶⣛⣶⠀⠀⢀⣀⢘⡓⣄⡀⠀⢀⡼⣻⠛⠀⠀⠀⠉⣉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⣀⣤⣴⣶⣶⣿⣿⣿⣷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠿⠟⠛⠛⠋⠉⠉⠁⠀⠀⠂⠀⠈⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠈⠅⠐⠀⠀⠀⠠⣤⠄⠀⢠⠤⠄⢀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠈⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⢀⣤⣶⣿⠀⠀⠀⠈⠀⠀⠀⠶⠀⠈⠋⠙⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠐⠀⠐⠀⣀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠄⠀⠀⠀⠉⠻⠋⠀⠀⠀⠀⠀⣤⠄⠠⠤⠶⠄⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠈⠻⠆⠀⠀⢀⣠⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠈⡉⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠹⣷⣶⣿⣿⠶⢤⣤⣤⣤⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠈⠀⠀⠀⠉⠁⠀⠀⠀⠙⠛⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠰⢄⠀⠀⠀⠀⣰⢋⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠀⠈⡇⠀⢑⠖⠀⠺⡗⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⡀⠄⠂⠀⠀⠀⠀⠀⠂⠸⡀⢩⢀⢀⠸⢄⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠻⣧⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢇⠈⠀⠀⠀⠀⢁⠀⠈⠷⠇⢀⠀⠀⠀⠀⢠⡀⡀⠀⢠⡀⠀⢸⣀⣠⠂⢠
⠀⠀⠀⡀⠐⠂⠑⢄⡀⠀⢀⢾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡇⢁⠀⢸⣿⣿⢅⡀⡀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠂⢍⠐⠃⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐⠀⠁⢀⡀⠀⠀⠀⠀⠀⠀⣄⡘⠄⣸⠛⠻⠄⠠⠀⠀⠁
⠣⢀⡤⠾⠟⠀⢠⠀⢴⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⡀⢈⠉⠀⠀⠀⠀⠀⠐⠉
⠀⠈⠐⠀⠀⠀⠣⣀⡠⠀⠀⠇⠈⠲⡄⠄⠀⣀⣀⣀⢀⣀⣀⡀⠀⠗⠀⠀⠀⠀⠀⣏⣀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣹⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢀⠎⡴⠀⠀⢀⠀⠀⠀⠀⠀⠰⣶⠞⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⢀⠎⢰⠃⠀⢀⠃⠀⠀⠀⠀⠀⠀⠰⠀⠀⢠
⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⡠⠃⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠈⡆⠀⠀⠀⠀⠀⠀⡰⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠸⡀⠀⠀⠀⠀⡔⠁⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠴⠐⠈⠁⠀⠀⠁⠀⠀⡠⠊⠀⠉⠉⠁⠒⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣁⣀⣠⣤⣤⠤⠤⠤⠶⠶⠚⠀⠉⠐⠓⠠⠦⠤⣤⣀⣀⣈⡐⢄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣸⠉⢹⠇⠀⠀⠀⠀⠀⣠⠛⢲⠆⠀⠀⠀⠀⠀⢀⡼⠉⣩⠟⠀⠀⠀⡤⠦⠤⣤⠀⠀⠀⠀⠀⡰⢡⠏⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⢠⡇
⠀⢀⡇⠀⣯⠀⠀⠀⠀⠀⡴⢁⡴⠃⠀⠀⠀⠀⠀⣰⠋⠀⣾⠆⠀⠀⠀⣰⠃⠀⣰⠃⠀⠀⠀⠀⡰⠁⢹⠏⠀⠀⠀⠀⠀⠀⡴⢻⣁⠀⠀⠀⠀⢸⢇⣼⠃
⠀⠘⡦⢠⡎⠀⠀⠀⠀⡾⠤⠬⢿⢯⠟⠁⠀⠀⣠⠋⢐⡿⠂⠀⠀⠀⢠⠇⠀⢘⡷⠀⠀⠀⠀⠐⡿⠱⡷⠀⠀⠀⠀⠀⠀⢾⢀⡴⠁⠀⠀⠀⠀⢠⣌⢹
⠀⠠⡇⡸⠃⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⣠⠟⢰⡿⠀⠀⠀⠀⠀⢈⠇⡴⠋⠀⠀⠀⠀⠀⠘⡟⣾⠁⠀⠀⠀⠀⠀⢠⣯⠞⠀⠀⠀⠀⠀⠀⣸⠙⡇
⠀⢠⡟⠁⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⣿⠶⠋⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠘⣷⠃⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣠⠞⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⣸⢰⠃⣀⠀⠀⠀⠀⢰⠃⠀⡴⠃⠀⠀⠀⠀⠀⣰⠃⠸⢥⡄⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⢰⢒⠆
⠀⢀⡞⠹⢦⠀⠀⠀⠀⠀⡏⠘⠋⡟⠀⠀⠀⢀⣏⣀⠘⢛⡗⠀⠀⠀⠀⠘⣳⢖⣴⡋⠀⠀⠀⠀⠀⣠⣿⠁⠀⠀⠀⠀⠀⣼⠁⡷⠀⠀⠀⠀⠀⢀⠇⠸⡄
⠀⠘⡗⢠⡏⠀⠀⠀⠀⢰⣃⢴⠀⢹⠃⠀⠀⠀⢁⡏⣀⠘⣲⠂⠀⠀⠀⠾⣥⠄⡽⠃⠀⠀⠀⠀⣴⡃⢩⠇⠀⠀⠀⠀⠀⡟⠐⣷⠀⠀⠀⠀⠀⠘⢳⢰⠁
⠀⠸⣥⣠⠋⠀⠀⠀⠀⠈⠁⠾⢶⡜⠀⠀⠀⠀⠈⠉⡝⡰⠁⠀⠀⠀⢀⣔⠁⢾⠄⠀⠀⠀⠀⠀⢹⡷⠃⠀⠀⠀⠀⠀⠀⡽⠰⡇⠀⠀⠀⠀⠀⠀⣧⡽⠂
⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⢉⣟⡴⠃⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠞⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⠶⢤⡀⠀⣠⡴⠖⠶⠶⣤⡀
⠀⠀⢸⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣸⣿⣿⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣿⣥⣽⣧⣤⣤⣤⡄⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⢻⣼⠃⠀⠈⠁⠀⠈⣷
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠙⣧⣠⡟⠀⠀⠀⠘⣧⣠⡟⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠄⠀⠀⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⣨⣿⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣼⣏⣈⣷⣄⣀⣀⣴⣏⣈⣿⡄⠀⠀⠀⠀⢸⡇⠀⠀⠈⠀⠀⢸⣿⠀⠀⠀⠁⠀⠀⣿
⠀⠀⠘⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢻⣌⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠁⠀⠀⣿
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⢀⠀⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠙⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⠀⡇⢸⠀⡇⢸⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⡇⢸⠀⡇⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⢸⠀⡇⢸⠀⡇⢸⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣤⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⠀⠀⢀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠅⠸⣀⠱⠤⠔⢁⡸⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⢃⠹⣿⣿
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⡻⣿⡽⣷⡀⠀⠀⠀⠀⠀⠀⠈⠢⢌⡑⣚⡃⣉⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣶⠿⢶⣿⡿
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣽⣿⠿⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢰⣆⢀⣴⣾⣿⢿⣿⣶⣤⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡀⠀⠀⢀⣿⡀
⠀⠀⠀⣿⢿⣶⣿⣿⣿⣴⣿⣿⣿⣴⠿⡆⠀⠀⠀⠀⠀⠀⢶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⣠⣾⣿⡇
⠀⠀⠀⣿⣾⣿⣿⣟⢛⣌⢛⣿⣿⣽⣶⡇⠀⠀⠀⠀⠀⠀⠀⢹⡟⠛⠿⢛⠿⠟⠛⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⢹⣿⣿⣯⣬⣛⣬⣿⣿⣿⠙⡇⠀⠀⠀⠀⠀⠀⠀⢸⢣⠈⠒⠈⠐⠊⢠⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⢹⣿⣿⣿⣿⠏⣠⣄
⠀⠀⠀⣿⣿⣿⠟⢻⡟⠛⣿⠛⢻⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⢸⣷⣾⣛⣻⣶⣿⠘⠀⠀⠀⠀⠀⠀⠀⠀⡰⢾⣳⡌⢿⣾⣿⣿⣿⣧⣾⣿⣿
⠀⠀⠀⢿⡿⠿⠀⠸⡇⠀⡿⠀⢸⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠇⠃⠁⠈⠛⠛⠛⠉⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⣶⣶⣶⣶⢶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⣤⣄⣀
⠀⠀⠀⠀⠀⣿⣿⡏⢩⣬⡛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣾⣁⣹⣆⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⠀⠀⠻⣿⣿⡧
⠀⠀⠀⠀⠀⠸⣿⣧⣝⢛⣤⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⢩⡟⠁⠉⠉⠙⣯⢩⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⡀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣯⠀⠀⠀⠀⠀⣈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡞⠙⢿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣏⣘⣷⣀⣀⣀⣰⣟⣈⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠙⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢿⡍⣽⠏⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⠀⠀⠀⠻⣿⣧
⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠋⠀⠀⠀⠀⠘⠃
⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⠿⠿⠟⠻⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⡿⢿⢿⣿⣿⠿⠿⠿⢿⡿⠿⠿⠿⠿⣿

⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⢄⡀⠀⠀⠀⠀⢀⡀⣠⣤⣠⣀⡀⠀⠀⠀⢀⣤⠤⢄⠀⠀⠀⠀⠒⠢
⠀⠀⠀⠀⢁⣘⡀⠰⠀⠠⠀⡰⢋⠂⠀⠀⠀⢆⡀⠀⠀⠸⠑⠀⠀⢠⡆⠁⠀⠀⠀⢠⢠⢉⣳⠀⠀⠂⠄⠀⢂⠁⠀⠊⠄⡄⢨⠰⠆⠀⠀⠰
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠑⠒⠊⠁⠀⠀⠀⠈⠓⠦⠤⣤⣤⠤⠤⠤⠊⠀⠀⠈⠑⠀⠐⠀⠀⠀⠐⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠠⠀⣶⡛⠄⠒⠂⠀⠐⠢⡐⠀⣴⡟⠉⢀⡀⠀⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⣰⡆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⢠⣧⡄⠀⠀⠀⣠⠏
⠀⠀⠀⠼⠑⠤⠈⠳⠀⠀⠀⠀⢀⡞⠃⠌⠀⠀⠻⡄⠀⠀⠀⢐⢌⣂⣐⡑⡁⠀⢠⡄⠀⠠⢀⠊⠦⢰⠀⠀⠀⠈⢋⠟⠀⠀⠚⣹⢁⠀⠀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⠭⠉⠍⠃⠀⠘⠀⠀⠐⡶⣷⠷⠟⠀⠀⠀⠠⠮⡤⠀⠀⢠⢿⡏⠀⢀⣾⡷
⠀⠀⠀⠀⠀⠀⣄⠀⡀⠀⠀⠀⠠⡄⠤⣤⢕⡴⢂⠴⠀⠀⠀⠀⡄⢠⢀⠀⠀⠀⠂⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⡼⠀⠀⠀⡾⠁
⠀⠀⠀⠃⠵⠂⠄⠱⠌⠀⠀⠀⠀⡘⠗⢮⣩⠔⠟⢆⠀⠀⠀⡗⠈⠩⢯⠀⠘⣤⠀⢠⠊⠅⠦⠒⠚⠒⠴⠀⠀⠘⠀⠀⠀⠀⠀⠁⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠈⠑⠀⠀⠓⠀⠀⠈⠀⢰⠠⠤⢐⣴⠀⠉⠀⠀⠀⠛⠀⢠⠀⡀⡤⠀⠀⢀⣠⠄⠀⢀⠄⠀⠀⣠⠂
⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⣉⠀⠀⠀⠀⠀⠀⠀⠈⢫⠞⠁⠀⢀⣾⣇⠀⢠⣮⡤⠀⠐⣻⠃
⠀⠀⠘⠿⠁⠀⠘⡶⠁⠀⢩⡏⠀⠀⢈⡥⠒⢢⡇⠀⠀⣄⣴⣠⠀⠀⣤⠀⠈⠈⠀⠘⠀⠀⠰⣤⣦⡄⠀⠀⠀⢰⠇⠀⠀⢘⣿⡇⠀⢀⣾⡅⠀⠰⣻⠟
⠀⠀⠶⠶⠆⠀⠰⠷⠀⠀⠚⠓⠀⠀⢌⡓⢚⣫⠏⠀⠀⠰⠿⠇⠀⢠⡀⡴⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠸⠇⠀⠀⢉⡟⠁⠀⠈⡟⠁⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⢀⡂⠀⡀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠃⢩⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⢀⠀⡤⠄⠀⠂⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠋⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠁⠠⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠨⡢
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⢀⠃⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣦⠐⢂⡀⠰⠆⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠠⠀⠈⠀⠀⠤⠀⠀⠠⠀⠀⠀⠀⠂⠁⡉⠙⠂⣈⡐⠁⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠅⠀⠄⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠉⢉⠂⢠⠠⠄⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁
⠀⠀⠨⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠢⠄⠬⡀⢦⢀⡀⠀⠀⢠⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡂
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢀⡄⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠰⢄⡨⠄⠀⠀⠀⠀⠐⠀⢈⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠃⠐⠠⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠁⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠂⡅⡀⠃⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⡰⠂⠀⠀⠀⡁⠀⠆⢁⡈⠀⠐⠁⠸⡄⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀⢤⠁⠀⠀⠀⠀⠀⠁⠚⠀⠅⢀⡝⠒⠀⢠⠀⠘⠛⠀⠒⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠂⡈⠁⠀⠈⠘⢁⠀⢀⠀⠀⡊⠀⡀⠀⠀⠁⢠⠂⠀⠀⠀⠀⡀⠀⣀⠀⠀⢐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⣀⡳⣄⣷⣰⣇⣤⣂⡄⠁⠀⠀⠀⠠⠀⠀⠀⠄⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⡀⢩⢓⠐⠀⠉⠡⠤⠀⢤⡀⠂⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠈⠀⠠⠁⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣷⠀⠰⠀⠘⠁⠀⠀⠀⠛⢋⣿⣷⢆⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣷⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠙⢷⣄⡐⡀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢈⠲⡛⠉⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⢧⠄⠛⠀⠀⠁⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡠⠀⠀⠀⣀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠠⠼
⠄⡄⠀⠰⠂⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠊⠠⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⡅⠀⠀⠀⠔⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢧⡰⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢰⣷⣼⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣿⡆⠀⠀⠀⠀⠓⠀⠁⠀⠀⠀⡀⠅⠀⡀⠀⠄⠀⠀⠀⠻⠏⠈⡉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠘⢀⡄⠄
⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⡽⠁⠀⠀⠀⠀⠀⠀⠠⠀⠉⠲⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⣯⣿⡆⠀⠀⠀⣀⡠⠴⠀⠀⢀⡶⠋⠀⠄⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠄⠂⠉⠀⠀⠀⠈⠁⠀⢀⡔⣀⣤⣦⣄⣀⣠⠃⠀⣼⡄
⠀⡉⢩⡍⡉⠀⠘⠉⠀⠀⠁⠀⠈⠁⠀⠠⠄⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠤⠌⠀⠀⠀⠀⢀⡀⠀⣄⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⣠
⠀⢀⠈⠁⠀⡠⠴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡈⣿⡷⣴⣶⡄⣽⣷⠆⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⢠⣀⡀⠀⣀⡈⢈⣴⢖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠘⠿⠇⠀⠀⠙⠻⠃⠟⠁⣀⣀⣀⣉⣁⡉⢉⡀⢉⠈⡉⠁⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⡄
⠀⠀⣶⠤⡤⠴⠖⣛⡿⠟⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⠿⡅⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⡶⠶⠶⠶⠶⣶⣤⡄
⠀⠀⢿⡀⣤⢀⡔⢉⣴⣀⣾⠇⣴⣷⣤⣤⡀⠀⠀⠀⠀⠀⢀⡼⣹⢫⠟⢿⣦⠀⠀⠀⠀⠀⠀⠸⣿⠀⣤⣤⣤⣤⣤⣀⠛
⠀⠀⠸⣧⣿⣾⡅⠚⢛⣋⣥⣼⣿⣿⠟⣿⡇⠀⠀⠀⠀⠀⣸⢹⢳⡏⣾⠎⢿⢳⠀⠀⠀⠀⠀⠀⠙⣷⡈⠃⠀⣠⡿⠃
⠀⠀⠀⠘⢿⡏⢀⣾⡿⠙⣿⡧⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⣿⢸⠀⠿⣯⡶⢸⡎⡇⠀⠀⠀⠀⠀⠀⠈⢿⣦⡘⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠞⠁⠀⢠⠟⢁⡿⠁⢸⠇⠀⠀⠀⠀⠀⠀⠸⣜⣇⢾⡿⡹⣸⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠦⣼⣱⣫⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡆
⠀⠀⠀⠀⠀⢰⡄⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠇⡄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⢀⡀
⠀⠀⠀⠀⠀⡀⢰⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⠛⠉⠉⠛⠿⣿⣆⣶⠖⠀⠀⠀⠀⢶⣤⡀⠀⠀⠀⠀⠀⣠⣶⡷
⠀⠀⠀⢠⣀⡷⣸⣷⠰⣇⣠⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣄⣸⣷⠆⣠⡼⣿⣧⠀⠀⠀⠀⢀⡾⣿⢁⣾⡀⠀⣸⡆⢹⡛⣧⡀
⠀⠀⠀⠀⣿⢡⣴⣶⣄⢹⡇⠀⠀⠀⠀⠀⠀⣶⣶⠀⠘⢛⣩⠴⢛⣥⠀⠻⠿⠀⠀⠀⠀⢾⡅⢻⡄⠻⣿⣿⡿⠃⣸⠃⣸⡇
⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⣼⣿⣶⠟⠋⢵⣾⠛⠟⣀⡀⠀⠀⠀⠀⠀⠀⠙⣦⠙⢶⣤⢠⣴⠞⣡⡞⠋
⠀⠀⠀⠀⠸⢿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠉⠁⠘⠿⣿⢶⣤⣤⣶⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⣽⠆⣼⠋⠈⢳⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢛⡉⣙⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⢀⡀⠙⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣄⠁⠈⠀⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀
⠀⠀⢀⣀⡀⢀⣸⣷⣶⣿⣧⣤⣤⡀⠀⣀⠀⠀⠀⠘⢏⠀⠀⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⠻⠷⣄
⠀⠐⣿⡿⠿⣿⣿⣋⠀⣛⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⢀⣀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡄⣿⡄⢀⠈⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⡇⠀⠀⠉⠀⠀⠀⠀⠀⠀⢨⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣟⣡⣿⡇⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⡿⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢟⣿⣷⣦⣄⣀
⠀⠀⠀⠀⣀⣀⣴⣿⣷⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⢿⣿⣿⣉⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⣿⠿⠁⠻⠋⠸⡿⣿⠀⠀⠀⠀⠀⡀⠀⠈⠉⢀⡌⢻⣿⣿⣿⡿⠋⠀⠀⠀⠀⠙⣶⣄⣀⣼⢂⣄⢻⣄⣀⣴⡞⠁
⠀⠀⣴⣿⣷⣤⠐⣶⣶⠀⣤⣿⣷⡄⠀⠀⠀⢻⣄⠀⠀⠀⠀⢿⣿⣏⠁⠀⠀⠀⠀⠈⠉⢻⡏⢸⡿⢡⣾⣿⣦⠹⣏⠈⣿⠋⠉
⠀⠀⠈⠻⣯⣤⠈⢉⠉⢀⣬⣿⠋⠀⠀⠀⠀⠈⢿⣿⣶⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠤⢼⣧⣸⠇⣿⣿⣿⣿⡇⣿⣠⡿⠤⠄
⠀⠀⠀⠀⠿⠿⣦⣿⣷⡾⠿⠟⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡅⠉⠉⠉⠉⠀⠀⠀⠀⠀⣠⠟⠉⠀⠹⣏⢉⡿⠁⠈⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠈⠁

⠀⠀⠀⠀⣠⡤⢄⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠄⠘⢄⢠⢁⠼⠛⠦⣷⡜⠁⠼⠧⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠘⣧⣓⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠙⠣⠤⠤⠒⠉⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠈⠉⡁⠀⠀⠀⠀⠠⠾⠆⠀⢀⢢⡓⠉⠀⡄⠀⠀⠀⠐⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠀⠀⢠⣴⣶⣄
⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⡍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠸⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠺⠆⠀⠀⠀⠀⠉⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⠀⠐⠂⠀⡀
⠀⠐⠀⠀⠀⠀⠀⠛⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠠⠉⠀⠀⠀⣀⣤⡤
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⣾⣿⠋
⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⠀⠀⠂⢸⣿⣿
⠀⠀⢀⠃⡖⠢⠀⢀⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⣤⠀⠀⠈⢋⡀⠀⠀⠈⢿⣿⣧
⠀⢀⠀⠀⠀⠀⠠⣌⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣤⡦⠀⠀⠀⠿⠀⠀⠀⡿⣝⣆
⠀⠀⠀⠀⠀⠀⠠⣌⠀⠀⠀⠈⠄⠀⠀⠁⠀⠀⠀⠀⢈⣡⡄⠀⠀⠀⠀⠠⣤⠄⠙⢿⠇⠀⠀⠒
⠀⡀⠀⠀⠐⠿⠂⠈⠀⠀⡄⠀⠀⠀⡀⠀⢴⣦⠀⠀⠻⠛⠁⠀⠀⠀⠀⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣤
⠈⠠⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠹⠃⠀⠀⠁⠀⣀⣀⡀⢠⣴⣦⠀⢸⣽⢣⡷⠀⠚⡁⠀⠀⠀⠀⠈⠋
⠀⠐⢃⠀⢿⣿⡿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠂⠀⠉⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠈⠄⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⡦⡌⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠆⠀⠀⠂⠀⠀⠀⠀⠀⡀
⠀⠀⠠⠀⠀⠀⠂⠀⠀⠠⠀⠀⡆⠀⠄⠀⠀⠀⠀⠀⡠⠔⡶⡒⠒⢤⡀⠀⠀⠀⠰⡀⠶⠀⢀
⡰⠂⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣎⣀⣠⣇⣱⣄⠤⣽⡄⠀⠀⠄⠁⠀⠀⠀⠆⠀⢀⠠
⠁⠇⠐⠀⠀⢶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠙⠲⣾⡀⠀⣱⡖⠁⡇⠀⠈⠂⠀⠀⠉⢀⣐⡂⠘

⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠃⠁
⠀⣿⣿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⡀⠀⠄⠀⠀⠀⠀⠐⠿⠿⠀⠀⢸⣯⡀
⠀⠀⠀⠀⠀⢿⣿⣿⡟⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈
⠀⣄⢀⣄⢀⣤⢀⣀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈
⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠄⠀⠀⠀⠀⠀⢮⠇⠀⠀⠐⠁⣰
⠀⠀⠀⠀⠀⠉⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠠⠀⠔⢢⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠛⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠂⠀⠈⠁⠀⠸⠤⠼⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣴⣄⣤⣤⠀⠀⠀⠀⠀⠀⠀⠄⡄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠉⠉⠙⢛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣚⣒⣐⣒⣚⡨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢦⣶⣶⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢣⠀⠂⠀⠀⠀⣯⢣⠀⢠⠄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡔⢒⡇⣧⠀⠀⠀⠀⠀⠋⡆⢇⡸⣃⣀⠬⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠿⢿⢿⣧⠀⠀⠀⠀⠄⠄⢼⠊⣹⠀⠀⢴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣦⣿⣾⣿⢧⠀⠀⠀⠈⢢⠔⠊⠐⠉⠉⠀⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⠖⣾⣿⣿⣿⣿⣿⣿⡿⣡⡀⠀⠤⠤⠤⠀⠠⠤⣈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢾⣾⣿⣿⣿⣿⣿⣿⣧⠛⠒⠤⠤⠤⠤⠤⢴⣀⡠⠜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⡞⡄⣿⣿⣿⣿⣿⣿⡞⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣧⣰⣿⡹⣿⣿⣿⣿⣧⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠟⣽⡿⡇⣿⣿⣿⣿⣿⢳⣝⠀⠀⠈⠁⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠐⠙⠾⣛⣛⣛⡡⠜⠈⠓⠂⠶⠲⠆⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


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

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



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

⠀⠀⠀⠀⠀⣴⣿⣿⠇⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⣰⣿⣿⡿⠀⠀⠀⠀⣾⣿⣿⡗
⠀⠀⠀⠀⢰⣿⣿⠏⠀⣀⡀⠀⢀⣾⣿⣁⣠⣤⣤⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⢠⣿⣿⡿⠁⠀⠀⠀⢠⣿⣿⡿⠁
⠀⠀⠀⠀⣾⣿⣿⣾⣿⠏⠀⠀⡼⠿⠟⠛⣻⡿⠁⠀⠀⠀⠀⣠⣿⡟⠀⠀⠀⠀⢀⣿⣿⣿⣷⡞⠀⠀⠀⣸⣿⣿⠃⠀⣀⡀
⠀⠀⠀⠸⠛⠋⣹⣿⠏⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⢀⣼⣿⣿⣷⣶⠖⠀⠀⠘⢉⣿⣿⡟⠀⠀⠀⠀⣿⣿⣿⣾⣿⠏
⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⢀⣾⣃⣤⡤⣤⠀⠀⠛⠋⢹⣿⣿⠋⠀⠀⠀⠀⣾⣿⣿⡤⠀⠀⠀⢸⡿⠟⢩⡿⠃
⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⡴⠁⠀⠀⠀⠀⣾⡟⠁⠀⠀⠀⠀⠘⠉⣿⠟⠁⠀⠀⠀⠈⠀⢠⡟⠁
⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⢀⣠⣄⡀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⢀⣾⠋⠙⣷⡀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⢠⡿⠁⠀⠀⠘⢿⡄⠀⠈⢿⣄
⣴⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣄⠀⠈⢻⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻
⠹⣧⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣆⠀⠀⢻⣦⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣼
⠀⠙⣷⡀⠀⠀⣾⡏⠉⠉⢉⣿⠏⠉⠉⣿⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣧⠀⠀⠹⣿⠉⠉⠉⢹⡧⠀⠀⢀⣾⠋
⠀⠀⠘⢷⡀⠀⠈⢿⡄⢀⣾⠋⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠹⣷⡀⢠⡿⠁⠀⢠⡾⠃
⠀⠀⠀⠈⢷⡄⠀⠈⢿⣾⠃⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠘⣷⡟⠁⠀⣠⡾⠁
⠀⠀⠀⠀⠈⢿⣆⢠⡿⠁⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⣰⡟⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⣻⡟⠁⠀⣠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠏⠀⠀⣴⣟
⠀⠀⠀⠀⠀⣰⡟⠀⠀⣰⡟⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⢀⣼⠏⢻⣆
⠀⠀⠀⠀⣼⠏⠀⠀⣴⣿⡀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⢀⣾⣏⠀⠀⠹⣧
⠀⠀⢀⣾⠏⠀⠀⣼⠏⠘⣷⡀⠀⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⢀⣾⠃⠹⣧⠀⠀⠹⣧⡀
⠀⢀⣾⠃⠀⠀⣾⡏⠀⠀⠘⣿⡄⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⣠⣿⠁⠀⠀⢹⡧⠀⠀⠘⣷⡀
⢠⡿⠃⠀⠀⠀⠈⠛⠛⠛⠛⠛⢿⣄⠀⠈⢻⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠘⢷
⢿⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣆⠀⠀⢻⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨
⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣧⠀⠀⠹⣯⠉⠉⠉⢉⣽⠏⠉⠉⣽⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠙⣷⡀⢀⣾⠋⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠘⠻⠟⠃⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⣰⡟⠁
⣶⣶⡶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⢀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠴⠄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠈⠐⠓⠀⣀⡁⠀⠀⠀⠀⠀⣴⡋⠰⢺⣇⡻⠃⠀⣠⡾⠟⠁⢠⣶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⡀⠁⠛⠀⠀⠀⢀⣠⡿⢿⣋⣻⠀⠄⠀⠀⢤⣧⡩⢤⣤⡉⠀⣠⡾⢋⣠⡴⠞⠋⣀⣤⣶⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⠀⢀⣄⡉⠁⠀⡀⠐⠂⠁⠀⢠⡖⡼⢛⣏⣙⠁⠀⠀⣤⢀⣝⢻⡟⢋⢻⣿⣿⣿⣟⡷⠤⠖⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⣧⣄⠀⢘⣶⣷⣈⡳⢼⢛⡿⡿⢀⠈⣴⠇⢸⡇⠈⡡⠈⠁⣷⣫⣾⣟⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⣀⣠⣌⡉⠀⢻⣿⣿⡈⢻⡀⠏⠿⢷⠿⣇⢸⣇⠸⣇⣸⡇⢠⡿⢻⡟⣱⣿⣯⣿⠁⠐⠂⠠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢰⡿⠷⢠⣼⣻⡿⣧⣠⢻⣤⠘⣆⡸⣩⣟⢹⣦⣿⣿⣴⡿⣵⠏⣀⣿⣿⣿⠃⢠⡆⠀⠘⠃⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣀⣀⠀⠀⢀⠜⣿⠉⠠⠈⠙⣦⣿⣏⠚⠉⠁⠈⠃⠉⢉⡟⢙⡿⣏⣰⣿⢫⣿⠟⠀⠀⠀⠀⠈⠂⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠘⣙⡶⠁⡀⠐⢀⠋⠉⠲⣼⣿⡁⠄⠀⠀⢀⠆⠀⠀⡧⠘⠟⢠⣿⠃⠀⠀⠀⡘⠀⠘⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢹⠀⠀⠠⡔⠻⣷⣄⡀⠋⢤⣿⣦⠀⠐⠏⠀⠀⢀⠶⠀⠘⡟⠁⠀⠀⠀⠸⠁⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠞⢣⠀⠀⠚⢿⣦⣀⡙⠻⣶⣄⡙⣾⢧⡖⠀⠀⠀⡎⠁⠀⡾⠀⠀⠀⠀⢀⠁⢀⣠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠈⠗⠀⠀⠀⡩⠛⠿⣶⣦⣭⣙⣿⣿⣿⣶⣤⣾⣤⣤⣼⣷⣷⣶⣿⣿⣿⠿⣟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣁⠂⠀⠀⢀⠔⠁⠀⠀⠀⠉⢛⣿⣿⣭⠿⢿⡿⠿⣿⡿⠿⢿⠿⡟⠛⠉⠀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⠚⠁⠀⠀⡠⠀⠀⠀⠀⠀⣀⣴⡿⠛⠁⠀⢀⠞⠀⢠⠟⠁⠈⣇⠸⡷⠀⠰⣀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠸⠃⠀⠀⢀⡼⠁⠀⣀⣤⡶⠟⠋⡙⠁⠀⠀⢠⠏⠀⣠⣯⠏⠀⣰⣾⠃⢀⣾⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠀⠀⠀⠠⠋⢀⣴⡿⠛⠉⠀⢀⠎⠀⠀⠀⣠⠃⠀⣠⠿⠁⣠⢠⣿⣦⣼⡿⢁⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠜⢁⡴⡿⠋⠀⠀⠀⢀⠎⠐⠀⠀⢰⠃⠀⣰⡏⠒⣠⠟⣻⡟⢽⣿⣄⠺⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣠⣥⣾⡿⠛⠀⠀⠀⠀⢠⠃⠀⠀⠀⣰⠃⠀⣰⠟⣠⣌⣳⡾⠋⠀⢎⣼⣨⣾⠇
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⣴⣿⡿⠛⠁⠀⠀⠀⠀⢀⣴⠃⠐⠀⠀⡼⠁⠀⣼⠏⠀⣷⠈⡿⠁⠀⢠⡾⢠⣿⡏
⠀⠀⠀⠀⠀⠀⢸⡆⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⢠⡞⠁⠌⠀⢀⡾⠁⠀⣼⠏⠄⢀⣯⠱⠀⠀⢰⣮⣑⠣⡟
⠀⠀⠀⠀⠀⠀⠈⣷⣤⡿⠃⠀⠂⠀⠀⠀⠀⢀⢰⠟⠀⠊⠀⢀⡾⠁⠀⣼⢟⡀⠀⡸⢣⠆⠀⢠⡟⡹⢉⡵⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠁⠀⠄⠀⠀⠀⣠⠋⠀⠀⠀⢀⡞⠁⢀⣾⢇⡶⠁⠀⣧⢈⠘⠀⣾⢃⡣⣾⠃
⠀⠀⠀⠀⠀⠀⣾⡟⠳⣟⡄⠀⠄⠀⠀⠀⡴⠃⠀⠀⠀⢠⡞⠀⢀⣾⠧⣶⡃⣀⠂⡄⠀⠀⣰⡟⡎⣽⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠹⣶⡄⠀⠀⢠⡾⠁⡐⠀⠀⣠⠏⠀⢀⣿⣇⣀⣠⣬⣻⡶⣷⣄⣀⠛⠠⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⡀⢠⡟⠀⠌⠀⠀⣰⠏⠀⢰⡿⢛⡛⣿⣭⣯⣿⣿⡷⢾⠉⠛⠷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣝⡟⠀⠀⠀⠀⡴⠃⠀⣰⡟⢀⣛⠉⡅⣄⠰⣦⣇⣰⡴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⢀⡼⠁⠂⣴⠏⠀⠘⠛⠑⠋⠋⠀⠁⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⡞⠁⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠷⠾⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢋⣄⣠⣷⠤⠞⠒⠛⠋⠥⠤⠬⠤⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠛⠁⠀⠀⠰⠆⠀⢀⠛⡀⠄⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣩⠔⠀⠠⣴⠀⡀⢠⡶⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⣠⣾⠝⠀⠀⢀⠀⠐⡴⠋⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⢡⡖⠀⠀⠀⠀⣄⠉⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡁⢠⡌⠀⠺⠧⠀⠀⢸⣀⡓⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣀⠀⢧⠀⡀⠠⣾⠦⣬⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⢺⢁⡏⠃⠈⡄⠁⠄⢠⠀⢻⠇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠲⢸⡇⠈⠁⡀⠀⠹⠀⠀⣀⠀⢸⢠⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠁⢸⠇⠶⠁⠁⣀⠀⢀⠀⠋⠀⢀⢠⡟⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢾⢘⣦⠠⡆⠙⠀⠀⠀⣤⠀⣨⡘⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⣾⡈⣁⠶⠆⠛⠀⢀⠀⠀⠸⠝⠇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⡉⠣⣴⠰⠶⢌⠡⢀⠀⢪⡸⡄⠈⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠡⠘⠔⢦⡓⠲⢤⢤⣛⣃⣤⣽⢿⣄⢲⣬⡅⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠀⢀⠀⢸⡀⢫⡐⣦⠠⡤⡄⣠⡍⣹⣿⣆⠻⣧⠀⠑⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣄⢠⣴⡇⠚⣠⣤⣤⡍⢺⠧⠠⣭⣷⣌⠳⠾⠷⠂⠀⢀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠃⠙⠦⠁⡄⣸⢋⣥⢬⡛⣏⠡⠐⠦⠄⠀⠰⢦⠉⠛⠷⠤⡹⣆⠠⠤⣀⠑⠦⢄⢰⡿⣡⠴⢚⣫⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢾⢰⡄⣾⣯⡟⠀⢈⣅⢰⣿⠀⠀⡘⢣⡖⣲⢽⠮⣶⣖⠀⣀⣀⣴⢋⣾⣿⠰⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠄⠀⠀⠀⠠⣀⠲⣯⡀⢀⡀⠿⠋⠀⣠⠀⣠⡀⠈⢰⠡⡗⠀⠸⠆⢀⠑⢺⢃⠚⣯⡅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠀⢰⠎⣱⡮⢝⠳⣤⢂⣀⣈⠀⠀⠉⡁⢀⠈⠁⠠⡆⢴⠆⠀⣴⠏⠆⢙⡿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠳⠽⢦⡦⠽⠓⠭⣌⣣⡼⠃⢢⡜⣦⣤⣲⣐⠙⠻⢛⣫⢉⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠺⠀⠀⣠⠜⣋⡉⠃⠲⣶⢚⠓⠚⠯⠙⠉⢀⣴⡄⠛⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠙⠨⢁⠃⠀⠙⠈⠀⠀⠀⠀⠀⠴⠛⡿





⣶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣷⣏⣻⣧⣤⣤⣴⣤⣬⣤⣭⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢠⠤⠄⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⠶⢧⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣤⡀⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢵⠶⣴⣿⣷⡴⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠋⠛⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠠⠀⠠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡔⠀⠀⢸⣶⣞⢬⣷⣶⣾⠠⠀⡐⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⢿⣡⠄⡀⢠⣼⣏⣷⣝⣿⣤⠀⡀⢤⣹⢿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⠁⠀⠉⢲⣾⠟⣿⡟⢻⣶⠊⠁⠀⣽⣾⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⢻⡇⠀⠀⠘⡇⠄⠈⠀⢄⠏⠀⠀⠀⡿⠋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣉⠂⠃⠀⠀⠀⠘⡂⢈⠀⡊⠀⠀⠀⠀⠃⢊⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠰⢁⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠋⠁⠀⠀⡀⢀⠀⢈⢋⡃⠀⠀⢀⡀⠀⠀⠉⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡄⠀⠀⠀⠀⠿⡀⡄⢬⡁⣩⢄⡄⣸⠃⠀⠀⠀⠠⡤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢜⡴⠛⠁⠀⠠⣤⡐⠂⠄⠮⢮⣟⣧⠾⢆⡠⠚⣠⣤⠀⠀⠙⠳⣜⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⠁⢀⡘⠃⠚⠉⡄⢶⡀⡀⢴⡶⣿⡷⣶⢄⡀⣱⠖⡌⠙⠀⠋⣀⠁⠉⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠒⠒⢲⠏⠀⠐⢶⣆⠀⠘⣧⠀⢠⠴⠻⠳⢄⠀⢸⡟⠄⢀⣶⠦⠀⠈⢶⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣯⣀⠀⠈⠋⠁⣀⣩⣱⢂⡗⢰⡀⣰⢐⣇⢳⡩⣁⡀⠙⠁⠀⢀⣨⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⠋⡩⠛⠻⢂⣶⣀⣷⡏⠘⠋⠚⠷⠟⠎⠛⠈⣷⣆⣘⣆⠚⠛⠫⡈⠣⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠐⠂⠂⠓⠁⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠙⠂⠂⠒⠈⠒


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠆⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢰⠀⠀⣤⠀⢀⠀⠀⢰⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠔⠀⠁⠒⠣⠀⠓⠀⠉⠘⠁⠘⠚⠀⠀⠈⠉⢨⠀⠃⠀⠈⠁⠓⡃

⠀⠀⣀⣠⣾⣄⣀⡀⠀⠀⠀⠀⢠⢢⣄⠀⠀⠰⡄⠀⠀⠀⠀⢶⡐⣶⡶⢶⣶⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤
⠀⠀⢸⣏⠀⢸⣏⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⢧⠀⠀⠀⠀⠀⠁⣿⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠮⠆
⠀⠀⠉⠉⢿⠋⠉⠁⠀⠀⠀⠰⠷⠶⠶⠴⠶⠶⠿⠀⠀⠀⠀⠀⠀⠿⠁⠀⠿⠀⠀⠀⠀⠘⠉⠉⠉⠉⠉⠙


⠀⠀⣰⡀⠀⠀⣠⡄⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠀⠀⠀⣠⣤⡄⢠⣤⣄
⠀⠀⣿⣿⢫⡛⣿⡇⠀⠀⠀⠀⣀⣸⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⡴⣾⣿⡷⡆⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿
⠀⠀⣿⡿⠶⠶⣿⡇⠀⠀⠀⠀⠀⠇⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⡿⣯⣉⡷⡇⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠁⠁⠀⠀⠀⠀⠀⠉⠉⠉⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⡆⡤⣸⡠⡄⡆⠀⠀⠀⠀⣀⢤⣶⣶⣶⠤⡀⠀⠀⠀⠀⠀⢸⣟⣿⡏⠀⠀⠀⠀⠀⠀⣤⣶⠿⢶⣦⡀
⠀⠀⠙⢽⣺⡺⠝⠁⠀⠀⠀⠈⠣⢼⣿⣴⣿⡠⠚⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣾⣿⣿
⠀⠀⠀⠠⠾⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠧⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠉⠁


⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⢀⡀⠀⠀⠀⠀⠃⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⣠⣦⣠⠖
⠀⠀⠀⠘⣦⣀⣀⣤⠀⠀⠀⠀⢾⡉⠀⠀⢉⣷⣏⠀⠀⠀⠀⠀⠀⠑⠒⠊⠀⠀⠀⠀⠀⠀⢠⣾⠡⠏⣿⠆
⠀⠀⠀⠀⠘⠻⠿⠋⠁⠀⠀⠀⠀⠙⠛⠛⠋⠀⠈⠁⠀⠀⠀⠀⠀⠠⣶⠄⠀⠀⠀⠀⠀⠀⢸⣿⣿⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠋⠁⠀⠀⠀⠀⠈⠉⠛⢷⣶⠾⠛⠛⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠉⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠘⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠻⣷⡀
⠀⠀⠀⠀⢠⣾⠟⠋⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠋⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣤⡶⠟⠉⠀⠀⠀⣼⡇
⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠈⢿⣦⣀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⢠⡿⠉⠙⠻⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢲⣦⠀⠀⣀⣠⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠛⢿⣿⣷⣶⣶⣾⠛⠁⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⣷⣦⣄⣀⣀⣸⡟⠀⠀⠀⠀⣰⣿⣇⡀⠀⠀⠀⠀⣀⣠⣴⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⢿⣿⠁⠀⠀⠀⣰⣿⠛⠿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⢷⣆⠀⠀⣼⠇⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⢻⣆⢰⡿⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠿⣦⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠻⣿⡇⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠶⠟⠋⠁⠀⠀⣿⠀⠀⠀⠾⠃⠀⣠⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⢠⣶⣶⣤⡀⠉⠀⠀⣰⣿⠃⠀⠀⢀⣶⠟⠛⠛⠛⢿⡟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣀⡀⠀⠀⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣰⣿⠟⠿⣿⣷⡀⠀⣰⣿⠃⠀⠀⠀⢸⡏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣵⣿⠃⠀⠀⠹⣿⣿⣶⣿⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⠀⠀⠀⠈⣿⣄⣠⣤⣤⣼⣧⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠉⠙⣿⠈⠉⠉⠛⠻⠶⣦⣄⡀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢹⡇⠀⠀⠀⠀⠀⠀⠉⠛⠷⣶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃

⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣻
⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿
⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠉⠁

⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⣿⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣿
⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿
⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣻⣿⣿⣿⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠁

⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⣟⣿⣻⣻⣛⣛⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣻⣟⣿⣟⣟⣿⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢿⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⢀⣴⣝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠡⠃⠀⣰⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⣯⡵⣀⣤⠞⣰⢺⠟⣐⣦⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢪⣤⣌⡄⠀⠞⢁⢘⡟⠀⡘⠛⢛⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣿⠿⡟⣀⣀⡀⢸⡜⠠⣾⠃⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣃⡤⠂⠋⠜⣿⣿⢿⡆⣸⣧⣴⠇⣾⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢊⣰⡿⣠⠀⠀⢠⠿⣿⣦⣶⣿⡟⠻⣀⣿⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠒⠛⠛⢿⠙⣃⣤⣥⣤⡈⢙⡳⢤⡁⢤⣹⡼⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⣴⢯⡾⣫⣤⣤⡈⢻⡎⣿⣦⠑⢸⣿⠷⢸⣯⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡿⠋⢠⠏⢸⣿⡏⠙⠏⡼⢃⣿⣿⡆⠈⣽⡷⢈⡙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⠁⠀⠀⠀⠸⣿⣧⡀⠲⠶⠿⣿⣿⠇⠚⢋⡃⣶⡹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⣀⣘⠻⣿⣶⣶⣿⣿⣿⠿⣷⣄⠙⣿⢁⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿⠁⠀⠈⣿⡇⠘⠼⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠶⣶⠿⢁⣷⣄⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠐⢿⡋⢻⡌⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏⠀⠀⠀⠙⣿⣿⣿⣿⣿⠟⠋⠙⢷⣄⢻⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⡀⠀⠀⠘⣿⣿⣿⡇⠀⢠⡀⠀⢻⣇⠇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣇⠀⠀⠀⣿⣿⣿⣷⣄⣸⣿⣄⠘⣿⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⢿⣦⣤⣼⣿⣿⣿⣿⡿⢿⣿⣿⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠸⣿⡙⠛⠁⢸⣿⡏⠀⠀⠘⣿⡧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣇⠀⠹⣿⣄⠀⢸⣿⡇⠀⠀⠀⣿⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠘⢿⣷⣄⢿⣧⠀⠀⠀⡟⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠀⠀⠙⢿⣿⣿⣧⠀⠈⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠙⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⢀⢰⡦⢀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠈⢓⣾⡖⠃⠀⠀⠀⠘⣳⢿⡛⠃⠀⠀⠀⠘⣻⢿⡛⠁⠀⠀⠀⠙⣿⣿⡛⠁⠀⠀⠀⠙⡲⢗⠋⠀⠀⠀⠀⢛⣿⣿⠋⠀⠀⠀⠈⢛⡾⣟⠋
⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠉⠀⠋⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠈⠉⠈⠁⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠈⠀⠉


⠀⠀⠀⢠⣄⣻⣠⡄⠀⠀⠀⢠⣄⣟⣠⡄⠀⠀⠀⢠⣜⣯⣴⠀⠀⠀⠀⣤⣸⣿⣠⡀⠀⠀⠀⠶⣤⣏⣀⠀⠀⠀⠀⣤⡸⢇⣤⠀⠀⠀⠀⣄⣻⣠⡄
⠀⠀⠀⠀⢴⠛⡧⠀⠀⠀⠀⠀⣴⠙⣦⠀⠀⠀⠀⠀⢾⠹⡶⠀⠀⠀⠀⠹⣿⢻⠯⠀⠀⠀⠀⠚⠉⡧⠉⠀⠀⠀⠀⢠⠆⢲⡄⠀⠀⠀⠀⢴⠏⠷


⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⢀⡄⢠⡀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⣀⢠⢀⡀
⠀⠀⠀⠀⣡⢾⢎⡁⠀⠀⠀⢈⣳⣿⣎⠀⠀⠀⠀⠠⢱⣷⠡⠀⠀⠀⠀⢀⡽⡦⠚⠀⠀⠀⠀⠶⣿⡿⠶⠀⠀⠀⠀⠾⣿⣿⠦⠀⠀⠀⠀⢉⣿⣏⠁
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠁⠃⠁⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠀⠀⠘⠃⠙⠁⠀⠀⠀⠀⠉⠘⠈⠁


⠀⠀⠀⠀⣤⣶⣤⡄⠀⠀⠀⠀⢦⣰⣀⠄⠀⠀⠀⠀⡀⣆⣀⠀⠀⠀⠀⠀⢦⣠⠄⠀⠀⠀⠀⠀⢦⣴⠀⠀⠀⠀⠀⢠⣄⡆⠀⠀⠀⠀⠀⣀⢰⢀⡀
⠀⠀⠀⠠⠾⣿⠿⠄⠀⠀⠀⠐⠚⣟⠣⠀⠀⠀⠀⠠⠞⡟⠦⠀⠀⠀⠀⠚⣿⢿⡛⠀⠀⠀⠀⠘⣿⣿⠃⠀⠀⠀⠀⠖⢻⢿⡄⠀⠀⠀⠀⠴⢙⠳⠄
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⣓⣾⣚⠀⠀⠀⠀⠀⢐⢾⠒⠂⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠤⢿⣿⠤⠀⠀⠀⠀⢤⣿⣿⠤⠀⠀⠀⠀⠤⣽⡾⡥⠀⠀⠀⠀⣾⣟⣿⡄
⠀⠀⠀⠀⠁⠚⠈⠁⠀⠀⠀⠐⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠃⠀⠀⠀⠀⠀⠋⠛⠀⠀⠀⠀⠀⠘⠁⠓⠀⠀⠀⠀⠀⠚⠉⠛


⠀⠀⠀⠀⢢⣴⡤⠀⠀⠀⠀⠠⣄⣶⣠⠄⠀⠀⠀⠀⣀⣆⣀⠀⠀⠀⠀⢀⠰⢀⠄⠀⠀⠀⠀⠀⢦⡴⠀⠀⠀⠀⠀⢠⣰⣦⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠈⠽⢿⠫⠁⠀⠀⠀⠠⠞⣿⠳⠄⠀⠀⠀⠀⠚⡟⠂⠀⠀⠀⠀⠈⠕⢣⠐⠀⠀⠀⠀⠙⡟⢿⠋⠀⠀⠀⠀⠈⢿⢯⠁⠀⠀⠀⠀⠾⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣀⣠⣤⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠿⠛⡿⢻⡿⢛⡿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⣼⠀⠀⠀⢬⠂⠀⢰⡜⠀⢠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢛⠀⠀⢸⢸⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀⢀⣾⡀⠀⠀⣻⣀⠀⢸⣾⠀⣰⣇⠀⠀⠀⠀⠀⠀⠀⠺⣿⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⣿⠁⠀⠈⣿⠁⠀⢸⡏⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡀⠀⠀⣿⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠛⠛⠛⣿⠛⠛⠛⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠛⢹⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣤⣤⣤⣤⣄⣿⣆⣀⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⡆⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢉⣩⣿⣏⡉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣠⠞⠁⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⢸⡏⠙⠛⠲⠦⣤⣀⡀⠀⠀⣠⠞⠁⠀⠀⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠉⠛⣶⣧⣤⣀⣀⠀⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠐⠿⠛⠛⠛⠒⠒⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡟⠻⠿⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠀⠀⣤⣤⣤⣤⣤⣮⣽⣦⣀⣀⢀⠀⠀⠀⢘⡿⠿⢿⣿⣿⣿⠀⠀⠀⠀⢠⠏⠙⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⡀⠀⠀⠈⠳⣄⠀⡞⠀⠀⠀⠉⠳⣼⠁⠀⠀⢻⣷⣤⣠⠚⣹⣿⠀⢀⣠⠔⠁⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⠙⣿⣿⣷⡄⠀⠀⠀⠈⣿⡁⠀⠀⠀⠀⢠⠏⠳⢄⠀⠀⡂⢸⣿⡋⠉⠛⠋⠉⠀⣀⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡺⣄⠘⢷⡈⢿⣿⣿⣦⠀⠀⣸⣁⣹⣦⠤⠤⠖⡞⠒⠉⠉⠀⢸⠁⣼⣿⡇⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠵⣌⢦⡀⠻⣄⠹⣿⣿⣷⡄⠈⠀⠀⠀⠱⡄⠰⠀⠀⠀⠀⠀⠀⢹⣟⡯⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⡺⢦⡱⣄⠙⢦⡈⢿⣿⣿⣆⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⡀⠔⢚⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣌⢧⡈⢳⡄⠹⣿⣿⣷⡀⠀⠀⠀⠀⣀⠤⠒⠁⠀⠀⠮⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢔⠹⢦⡹⣄⠙⣆⠈⢿⣿⣿⣆⠠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠅⠳⡌⢷⡈⢳⡤⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣨⣽⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡋⢙⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣤⣀⠀⠀⠀⠀⠀⣾⣿⣿⣏⣹⣿⣿⣿⣿⣯⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣧⣽⣿⣷⣄⠀⠀⠀⠘⠛⠛⠛⠛⢿⠟⠛⢻⡟⠉⠉⠉⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢿⣿⣿⣿⣿⢿⣿⣦⠀⠀⠀⠀⣀⣀⠀⣼⠀⠀⢸⡇⢀⣴⣿⡿⠟⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣾⣿⣿⣿⣷⣾⣿⣿⠀⢀⣴⣿⠟⣿⣿⣿⡀⠀⢸⣧⣾⣿⣿⣿⣦⣿⣟⣻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢙⡍⠙⡏⠉⠉⠀⣴⣿⣿⣷⣦⣾⣿⣿⣿⡄⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⢦⣄⠀⠈⡇⠀⡇⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡸⣿⣷⣼⣿⣿⣿⣶⣿⣿⣿⣾⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⣾⣿⣦⠀⢳⠀⢿⠀⣼⣿⣿⣿⣿⣯⡸⣿⣿⣿⣿⣿⡇⡟⠛⠛⣻⠛⠛⠛⡟⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣙⣿⣟⣡⣿⣿⣷⠘⡇⠘⠀⠘⠛⠛⠛⠻⡿⠿⠿⣿⠿⠿⠟⢡⡇⠀⠀⡏⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⣃⣄⣀⣾⠂⠀⠀⠀⠀⡇⠀⠀⡟⠀⠀⠠⢸⠇⠀⢸⠱⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡆⠀⠀⠁⢫⠀⠙⡆⠀⠀⠀⢰⡇⠀⢰⡇⠀⢀⡀⣾⠀⢀⡏⠃⢀⣴⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢱⣴⣶⣦⡘⡆⠀⢸⡀⠀⠀⢸⠀⠀⠘⣡⡾⣿⣿⡄⠀⡼⠀⢠⣾⣟⣿⣅⣽⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⣿⣿⣿⣷⠀⠈⡇⠀⠀⡞⠀⢠⣾⣿⣿⡿⣿⣿⡄⠁⠀⠘⣻⠛⠿⠿⡿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠿⣿⠿⡟⠋⡇⠀⢸⣶⣶⡇⠀⢸⡟⣻⠛⢻⠿⢿⠇⠀⠀⢠⠇⡌⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢹⡀⣹⡄⢹⢀⣿⣧⣿⣿⠆⢸⢁⠇⠀⣾⡴⠃⠀⠀⢠⠏⡸⠁⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣾⣄⢧⠘⢷⡘⡆⠀⡏⢸⠀⠀⣿⡜⠀⣴⠋⠀⠀⡠⣠⢻⣌⠁⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣾⣿⠈⢇⠀⢳⣇⠀⡇⢸⠀⢠⡿⠁⢰⡏⠀⣠⢞⡴⠇⡁⠈⢳⢤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣆⠈⢦⠘⣆⠀⠙⣄⡧⢸⠀⡼⠁⢀⡞⠀⡪⣐⢭⠔⠊⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣄⠀⠀⠈⠣⡈⠧⠈⢧⠀⠈⠋⠘⠋⠀⠀⠞⠋⠉⠈⠀⠀⣠⠔⠋⠀⠀⢠⡤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡀⠀⠀⠀⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠆⠀⢸⣿⣷⡄⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠰⠀⠈⠛⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡇⠀⡀⠀⣼⣿⣿⣿⣿⣷⠀⢀⠀⠸⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠋⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠙⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠁

⠀⣠⣀⣄⣀⣀⣄⣀⣤⣤⣤⣀⣠⣤⣀⣀⣤⣤⣤⣤⣄⣤⣀⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣤⣤⣤⣤⣤⣤
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⡿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡿⣫⠽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣟⣿⣿⣿⣿⣿⣿⡿⠋⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⡿⠁⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠀⠀⠀⣀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣏⣾⣻⠃⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⢻⣿⠀⠀⣰⣿⣧⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⢈⣿⣀⣴⡿⢿⣿⣇⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠂
⠀⣿⣿⣿⣿⣿⠀⡅⠰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⠧⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣾⣛⣿⣿⠀⡁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⠸⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⠀⡇⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣟⢿⡄⣿⠀⢠⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⠃⠀⠀⠀⣿⢿⣿⣿⣿⣿⣿⣿
⠀⠙⠻⠡⠈⠨⢣⠘⣆⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⢿⡐⠃⠀⠀⠀⣸⡟⢘⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠈⠀⠈⢷⡘⣆⠀⠀⠑⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡍⠛⠉⠭⠏⠋⠁⠀⠀⠀⣰⣿⠃⠂⠈⠉⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡙⢷⡀⠈⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡑⢦⣀⠑⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣾⡷⣫⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢝⠳⣄⠁⡈⠟⠳⠒⠒⠤⣤⣤⣤⣤⢤⣴⣶⣒⠛⣭⠷⢛⣭⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠪⡑⠈⠀⠀⠀⠀⠀⠘⠃⠀⠉⠋⢈⠻⣧⠀⠛⠏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢠⠤⠚⠉⢻⣿⣿⣿⢦⣛⠿⠿⣟⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢰⠏⠀⠀⠀⢸⡟⢿⡇⠀⠉⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢸⠀⠀⠀⠀⢨⡇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⡄⠀⠀⠀⣾⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠇⠀⠀⠀⣿⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢸⡀⠀⠀⢀⡿⠸⣇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠉⡈⢁⣀⠀⢹⣀⣸⡏⣠⣤⣄⠀⣍⠺⣧⣀
⠀⠀⠀⠀⠀⠐⣶⣶⣶⣾⣿⣿⡟⠫⠵⡠⠀⣿⣿⣇⠘⠿⣬⣤⣿⣿⣿⡀⠿⢾⣏⣿⣿⣿⣿⣿⣿⣿⡷⠶⠂⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢶⣥⡴⠾⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠓⠚⠋⠉⠁⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣩⠐⠆⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠀⠀⠀⠀⠙⢿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠙⣿⣿⣧⡀⠀⠀⠀⠀⠉⠛⠿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢻⡇⠀⠀⠘⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢠⡞⠁⠀⠀⠀⠈⣿⣿⣿⣄⠀⠀⠀⠀⠀⠙⠛⠛⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠈⢳⡄⠀⠀⠀⠀⣿⠙⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣴⠟⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⢀⡰⠋⠀⠀⠀⣠⣾⠟⠀⠈⢿⣿⣷⡀⠀⠀⠀⠀⠀⢸⡄⠀⠙⣇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⢿⣿⠀⡠⠋⠀⣀⣠⣴⡿⠋⠀⠀⠀⠀⠀⢻⣿⣷⡀⠀⠀⢠⠶⠋⠀⠀⠀⠉⠉⠉⣷⣄
⠀⠀⠀⣠⣤⣤⣤⣤⣶⠞⠋⠁⠀⢸⡇⠀⠀⠔⠊⠀⠀⢸⣷⠀⠀⠀⠀⠀⣠⣿⣿⣿⣧⠀⠀⠽⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⣇⣀⣀
⠀⠀⣼⠋⠀⠀⣠⡾⠃⠀⠀⠀⠰⣏⠀⠀⠀⠀⢀⣠⣴⡿⠋⠀⠀⠀⠀⣰⣿⡿⠁⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠈⠉⠙⣇
⠀⢀⡏⠀⠀⢸⡏⠀⠀⠀⠀⣠⠶⠾⡷⠶⠶⠟⠛⠋⠁⠀⠀⠀⠀⢀⣼⣿⡟⠁⠀⠀⢹⣿⠶⠶⠶⠿⠛⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠔⠋⠀⠀⠀⢈⡇⠀⠀⢠⠊⠀⢠⠞⠁⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣾⣿⡏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡌⠀⠀⠀⠈⠀⡠⠃⠀⢀⡤⠶⢶⡟⠛⠋⣿⠏⠁⠀⠘⣿⣿⡄⠀⠀⠀⠈⠙⠛⢶⡄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣰⠏⠀⠀⢠⣿⠀⠀⠀⠀⠘⢿⣿⡄⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠙⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠘⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⠸⣿⣿⠛⠿⢷⣶⣶⡒⠚⠛⢫⡉⠉⠂⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⠉⠛⣷⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠛⠁⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡇⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡆⠀⠀⠀⠀⠀⢹⣧⣤⣤⣀⣀⣀⣀⣀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⡚⠛⠋⢩⡞⠁⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣆⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⢿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⡀⢻⣆⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠾⢋⣽⣿⠟⠁⠀⣿⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⢀⣴⣿⢿⡅⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⣴⣿⠟⠁⠘⣿⡀⠀⠀⠀⠹⣦⡴⠖⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠞⠁⠀⢠⣾⡟⠋⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠲⠁⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠈⢧⡀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⠁⠀⠀⠀⠙⡆⠀⠀⠀⠀⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠀⢸⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡌⣧⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠈⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁

⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢀⠀⠀⠀⢠⠖⠡⠀⠀⠀⢦⠠⠆⠀⠀⠀⢀⠄⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠻⠰⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢡⠂⠀⠀⠀⠀⠰⠀⠿⡂⠃⠀⠘⣌⠄⡔⠅
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⠘⠀⠀⠀⢀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠠⠉⠀⠀⠀⡈⠁⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐
⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢀⠠⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠧⠀⠀⠠⠀⠀⡠⠀⠀⠀⠀⠀⠀⢀
⠀⠈⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠀⠀⠀⠀⠀⠔⠁⠀⠀⠠⠈⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠆⠀⠀⠘⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠂
⠀⠀⠀⠀⠀⢠⡀⠀⠀⢀⠀⠈⢆⢀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢩⡌⠀⠢⠀⠀⠀⠐⢠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠀⠀⠀⠀⠀⠀⠠⠤⠤⠐⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠤⢄
⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⡀⠑⢄⠀⠦⢠⠄⡀⠀⠀⠘⠁⢠⡶⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠁⠂⣘⠀⠉⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⣁⡈⣦⣠⡤⠁⠀⠀⠀⠀⠐⠃⠀⢸⡀⠚⠠
⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⡀
⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠉⠙⠶⣄⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠀⠀⠀⠈⠙⠳⣄
⠀⢀⡾⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⢠⠟⠀⠀⠀⢀⡀⡀⣀⠀⠀⠀⠻⡄⠀⠀⠀⡼⠃⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⢸⠃⠀⠀⠀⢋⡔⢰⣿⣷⠀⠀⠀⠸⡇⠀⠀⡟⠀⠀⢰⣌⠲⠥⠿⠈⠣⠀⠀⠀⢻⠀⠀⢸⠇⠀⠀⠀⠀⠀⠈⣒⡠⣀⠀⠀⠘⡇
⠀⢸⠀⠀⠀⠀⢸⡄⠈⢛⠅⠀⠀⠀⢀⡇⠀⠀⣇⠀⠀⠀⢻⣷⣶⣶⣿⣶⠀⠀⠀⣸⠀⠀⢸⡄⠀⠀⠀⢳⣤⡳⣦⣴⣿⡄⠀⠀⡏
⠀⠘⣧⠀⠀⠀⠓⠙⠛⠑⠿⠀⠀⠀⣼⠁⠀⠀⢻⡄⠀⠀⠀⠙⠻⢿⡿⠟⠀⠀⢠⡟⠀⠀⠈⣧⠀⠀⠀⠀⠀⠴⠿⠛⠛⠁⠀⣸⠃
⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠹⢦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠈⠛⠲⠶⠶⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠶⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠶⠶⠖⠛⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⢀⡴⠚⠉⠉⠉⠉⠙⠳⢦⡀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠉⠉⠉⠉⠙⠲⣄⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠉⠉⠉⠓⢦⣄
⠀⠀⣴⠋⠀⠀⠀⠀⣀⠀⠀⠀⠀⠙⣆⠀⠀⠀⢠⡞⠁⠀⠀⠀⢀⡀⠀⠀⠀⠈⢳⡄⠀⠀⠀⣰⠋⠀⠀⣀⣀⣀⣀⣀⠀⠀⠙⣦
⠀⢸⠃⠀⠀⠀⡆⠾⢃⢰⡏⠁⠀⠀⢸⡆⠀⠀⡿⠀⠀⠀⣠⡀⠟⡡⠤⣄⡀⠀⠀⢿⠀⠀⢰⡇⠀⠀⠰⡌⠻⣿⡟⣩⡄⠀⠀⠘⡇
⠀⢸⠀⠀⠀⠰⣝⣛⣡⣿⢣⠀⠀⠀⢀⡇⠀⠀⡇⠀⠀⢰⡇⠿⢆⡴⠒⠀⠀⠀⠀⢸⠀⠀⢸⡁⠀⠀⠀⠁⡇⠿⠸⢋⠁⠀⠀⠀⡟
⠀⠸⣇⠀⠀⠀⠈⠉⢭⡴⠋⠀⠀⠀⣸⠃⠀⠀⢻⡀⠀⠈⠣⠰⠎⠀⠚⠂⠀⠀⢀⡿⠀⠀⠘⣇⠀⠀⠀⠼⠇⠀⠀⠹⠆⠀⠀⣰⠇
⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠙⠳⠦⣤⣤⣤⠴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣤⣤⡤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⣤⣤⠴⠞⠋

⠀⠀⠀⠀⣠⡴⠶⠚⠛⠒⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠚⠓⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠛⠒⠶⢤⣀
⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠉⢳⡄⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⣀⡀⠀⠈⠙⣦⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⢰⡏⠀⠀⠀⠐⢤⠀⢀⡤⠀⠀⠀⢻⡄⠀⠀⣼⠁⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠈⣧⠀⠀⢠⡟⠀⠀⠀⠀⢀⣀⠀⣬⠀⠀⠀⢹⡆
⠀⢸⠀⠀⠀⠀⣠⢼⠤⣾⣠⠄⠀⠀⠈⡇⠀⠀⡇⠀⠀⠀⠤⠄⠈⢥⣶⠀⠀⠀⠀⢸⠀⠀⢸⠁⠀⠀⠀⢠⣤⢹⡇⣿⠀⠀⠀⠀⣧
⠀⢸⡄⠀⠀⠀⠀⠈⠠⠌⠂⠀⠀⠀⢸⡇⠀⠀⣷⠀⠀⠀⠀⠀⢺⣷⡄⠀⠀⠀⠀⣼⠀⠀⠸⡇⠀⠀⠀⠀⣔⣫⣼⣿⠀⠀⠀⢠⡇
⠀⠀⢳⣄⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⠟⠀⠀⠀⠘⢧⡀⠀⠀⠀⠈⠛⠃⠀⠀⢀⡼⠃⠀⠀⠀⠻⣄⠀⠀⠀⠉⠉⠉⠁⠀⠀⢠⡞
⠀⠀⠀⠙⠳⣤⣀⣀⠀⣀⣀⣤⠞⠋⠀⠀⠀⠀⠀⠈⠛⢦⣄⣀⡀⢀⣀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠘⠳⣤⣀⣀⠀⣀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡁
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⡀⠀⡈⢄⠀⠀⠀⠀⡀
⠀⠀⠈⢿⡟⠛⢛⡿⠛⠛⠛⠛⢿⡛⠛⣻⡟⠀⠀⠀⠀⠘⣿⡛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⣿⠏⠀⠀⠀⠀⢀⠠⠀⠀⠀⠄⠀⠀⠀⠄⠄⠀⠠⠀⠁
⠀⠀⠀⠀⢻⣤⡟⠁⠀⠀⠀⠀⠈⢿⣴⠟⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠂⠒⠠
⠀⠀⠀⠀⣠⡿⣆⠀⠀⠀⠀⠀⠀⣼⢿⣄⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠
⠀⠀⠀⣰⡟⠀⢹⣦⡀⠀⠀⢀⣼⠋⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠐⢠⠀⠀⡐⡀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠈⠛⠛⠛⠛⠛⣿⡛⢛⣿⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠀⠐⡀⠂⠂⠆⠔⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢿⣆
⠀⠀⠠⠀⠀⠀⠀⠰⠠⠀⠀⢄⢠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠀⠰⠀⠄⠀⠀⠀⢠⠀⠀⠀⠠⠄⠀⠀⠤⢤⣿⣧⣤⣿⣧⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠁⠀⠌⠉⠁⠀⠀⠈⢂⠁⠀⠁⠀⠀⠀⠀⠀⠈⠄⢁⢁⢁⣼⣿⣿⣿⣿⡌⠁⠠⠁⠀⠀⠀⠀⠀⠀⠈⡈⡩⡹⠉⠉⠉⠉⠹⣯⠉⣽⡟
⠀⠀⠀⠀⠠⠜⡀⠀⠀⠀⠀⠀⠀⢢⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣾⣿⣿⣿⣿⣿⣿⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⡜⡑⠀⠀⠀⠀⠀⠀⢹⣿⡏
⠀⠀⠀⠀⠀⠘⡐⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠹⣿⣿⣿⣿⣿⡟⠁⠂⠀⠀⠀⠀⠀⠀⠀⠌⠔⢕⡄⠀⠀⠀⠀⢀⣾⠟⣷⡄
⠀⠀⢀⠀⠂⠀⠰⡀⠄⠀⠀⠀⠁⠀⠀⠐⠐⡀⠀⠀⠀⡀⠁⠂⠀⠂⠙⠿⠟⠟⠟⠐⠀⠀⠂⢡⡀⠀⠀⢀⠀⠰⠤⡬⣞⣤⣤⣤⣤⣾⣧⣤⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠌⢄⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡈⢀⠈⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣇⢠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠉⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣟⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠠⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⡀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠏⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⡶⣶⣷⣾⡏⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⡀
⠀⠀⠀⠀⠚⠻⣿⡍⠻⣶⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣇
⠀⠀⣠⣄⢀⣠⣿⠷⣄⡙⣿⣷⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢀⡀⠀⠀⠀⣠⣿⣿⣦⡀⠀⠀⠀⡀
⠀⠀⢹⠟⠛⢻⡏⡀⢈⠻⣿⡿⣆⠀⠀⠀⠀⠀⠀⢦⡌⢳⣄⠀⠀⢀⣴⡇⠀⣠⡞⢁⣠⠀⠀⠀⠀⣿⣾⣷⣾⣿⠋⠉⢿⣿⣶⣷⣾⡇
⠀⠀⠀⠀⣰⣾⣾⣿⣾⢷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢿⡀⠀⢀⣾⠀⢀⡿⠀⠉⠁⠀⠀⠀⠀⢻⣿⣿⣿⠄⠀⠀⠀⣽⣿⣿⣿⠁
⠀⠀⣀⣠⢿⠟⢹⣿⠟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠿⢀⣸⣿⠀⠿⠀⠶⠋⠀⠀⠀⠀⠀⢸⣿⣿⣷⣶⣾⣷⣶⣶⣿⣿⡿
⠀⠀⣿⣯⠀⠀⠛⢿⠀⢼⣿⣷⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢻⡏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠙⠋⠙⢹⣿⠙⠉⠛⠙⠇
⠀⠀⠈⠻⢷⣤⣀⣀⣀⣀⣤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠟⠃⡞⠰⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⣿⣤
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⣴⣶⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠹⠟⠛⠁
⠀⠀⠀⠀⢰⡿⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠀⠀⣿⣿⠀⠀⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠈⢀⣴⣤⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⢸⡿⢿⡇⠀⢸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠈⠁⠈⠁⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⣰⠁
⠀⠀⠀⣦⡀⣀⣿⣿⣦⣀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⡆⢰⣶⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⢁⣾⠏
⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢛⠋⢴⣿⠏⢠⠆
⠀⠀⠀⠀⠀⠈⠻⠀⠉⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠋⣿⠈⢋⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣛⠛⠋⠼⠛⣁⣴
⠀⠀⠀⠀⢰⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣆⠻⠏
⠀⠀⢠⡀⢸⣿⣦⣸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣿⣷⡀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏
⠀⠀⢸⣿⣤⣿⠈⢻⡇⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣀⣿⣿⠃⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⢸⣷⡈⠻⣧⣄⣿⣼⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⡾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡟⡏⠹⣿⢸⡏⢩⡇
⠀⠀⠈⠛⢷⣦⣿⣿⠛⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣬⠀⣧⣼⠛⠘⢳⣂⡇
⠀⠀⠀⠀⣀⣽⡿⠟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⠟⠙⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣷⢶⣿⠛⣿⠛⠛⡇
⠀⠀⠀⠀⠉⠛⠷⠶⠶⢶⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠿⠘⢿⠄⣉⢹⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⡟⠛⠛⢀⣿⣸⣇⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢀⡄⢸⡇⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠐⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣠
⠀⠀⠀⠀⣠⣿⣧⣾⣿⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣴⣟⠀⠀⣀
⠀⠀⠀⣰⡟⠉⢹⣿⠀⠻⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣴⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢹⡿⠿⠿⠛⠁
⠀⠀⢠⠏⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢃⣶⣖⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⢀⣴
⠀⠀⠞⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⠙⠉⢠⣿⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⡿⢻⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣿⠀⣿⢸⡿⠀⠀⠀⠀⠀⠀⢀⣀⣈⣁⣴⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠙⢀⡿⠈⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣇⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⠷⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠁⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡊⠁⠈⠀⠀⠀⠀⠀⠀⢀⠼⡗⠂⠓⠢⣤⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⢰⣿⣷⡴⢄⠀⠀⠀⠀⠀⣰⠁⠀⢃⠀⡴⢿⣟⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠸⣿⣿⠃⠀⠀⠀⠀⡄⠀⡇⠀⢀⣼⣏⠀⣼⠈⢹⣦⣄⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⣄⠀⢰⣄⠁⠀⢳⠞⠉⠀⡇⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⢈⣿⣿⣷⡀⠀⠀⠀⠀⢸⢸⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠌⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡏⣸⣦⠀⠀⠀⠸⡷⠤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣰⣿⠏⣷⡄⠀⠀⠇⠀⠀⠀⠀⡀⠄⠀⠈⢀⡠⣴⡾⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⣼⣿⣿⣦⡀⢀⣠⠤⠒⠋⠀⠀⠀⡲⠵⠺⠗⠀⠄⣫⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⣞⣏⡤⠐⢀⣠⣤⣶⣇⠀⠀⣐⡥⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⢑⡆⠀⠀⠀⣹⣿⣿⡿⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⡤⠞⠙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣼⣿⣆⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢖⠊⣼⣿⡿⣿⣧⠑⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢪⠁⣼⣿⠟⣀⠹⣿⣷⡘⡜⡔⠢⠄⡀
⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣢⣶⣁⣨⢃⣾⣿⠏⣰⣋⣧⠙⣿⣷⡈⠬⠁⠀⠈⠐⠂⠀⠠⠤⠠⠄⠤⡤
⠀⠀⢫⠉⢤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⣿⣿⣤⣤⣶⣶⣴⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣠⠃
⠀⠀⠀⠳⠚⣿⣿⡟⢻⠛⣛⣛⡛⠛⠙⣻⢛⣋⡏⢛⣛⣉⣙⠋⣛⢿⣿⣿⠉⢻⣉⡝⠁⣿⣿⡟⣡⠈⡇
⠀⠀⠆⠘⠀⠊⢻⡟⠈⠀⠛⢺⡇⠁⠀⠉⣺⠉⠁⠉⠉⣏⠉⠈⠉⠹⠋⠉⠁⠉⣏⠁⠀⠉⣹⡃⠀⠀⠀⠜
⠀⠀⠀⡤⠀⢰⣏⢻⣦⠀⣰⣰⣿⣦⠀⣰⢣⢵⡂⢰⡮⠈⡇⢀⡴⣿⣼⠆⣠⣮⠛⣇⢀⡼⡜⢹⠀⡠⠂
⠀⠀⠀⠈⠂⠸⡩⢆⠻⣧⡘⢿⡿⢡⣷⣀⠿⡄⡵⢆⢩⢩⡖⣉⡬⢞⣀⣎⠹⣿⡿⣡⢏⠜⣁⠎⠔
⠀⠀⠀⠀⠀⠷⢍⢦⡂⣽⣿⣦⠃⢻⡕⢻⡙⠒⠨⣽⡀⣏⡉⠥⠒⠝⢣⡽⠃⢩⣾⣿⣘⡖⡡⠎
⠀⠀⠀⠀⠀⠀⣸⠓⣍⡻⣿⣿⣶⡲⣬⣙⡒⠧⢯⠈⠉⠉⡹⠦⢀⣋⣩⢖⣷⡿⢿⠟⣡⡞⢿⠄
⠀⠀⠀⠀⠀⡿⢁⣾⣿⠟⣆⠀⣙⠷⢌⣳⣽⡛⣒⣿⠀⢻⠁⠈⢺⣴⣁⠾⢟⣄⣠⡛⢿⣿⡌⢢
⠀⠀⠀⠀⡔⢁⣾⣿⠏⠼⠌⠽⠌⢿⣶⡦⠭⠭⢙⡁⠀⠀⣙⡉⠭⢴⣶⣾⠃⠺⠵⠿⠈⣿⣿⣆⠱
⠀⠀⢀⠞⢠⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣶⣷⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠱⡀
⠀⠀⠼⢤⠤⡤⠤⣤⡤⢤⣤⣤⡤⣤⣄⠹⣿⣯⠠⡇⠀⢸⡤⣸⣿⡟⢀⣤⣠⣠⡠⢄⠠⠄⠄⠤⠤⡴⠗
⠀⠀⠠⠞⠚⠓⠢⡴⠓⡉⠉⠀⠥⠱⡘⢦⠹⣿⣧⡅⠀⢘⣵⣿⠟⢀⡊⠋⠨⠃⠁⠀⠐⠂⠂⡠⠒⠀⠦⣄
⠀⠠⠀⠐⠒⠄⠀⠇⠀⠀⠘⠀⠀⠜⠑⠈⢧⠙⣿⠃⠀⠀⣿⠏⢠⢊⢙⢂⠀⠰⠂⠀⡀⢨⠈⡠⠒⠀⠀⡄
⠀⠐⠀⠀⠠⠘⠀⠀⠦⠠⠄⠐⠢⠄⠀⠀⠈⡷⠁⠀⠀⠀⠈⢲⣗⠀⠀⠁⠀⠁⡀⢌⢀⠀⠈⠀⠀⠀⠤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠺⢀⣀⡀⠀⠀⣴⡷⠈⢁⠁⠀⠉⠀⡀⢈⠌⢣⡀⡀⢀⠀⢁⡰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠫⠉⠑⠀⠀⠀⠚⠑⠐⠀⠀⠒⠒⠛⣿⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⢀⣤⣤⡤⢤⡤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⢸⣿⣀⣠⠄
⠀⠀⣠⠆⠉⠉⠀⠀⠀⠀⠀⠀⠀⠛⠳⢦⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠆⠈⢉⣴⠿⠿⠿⠀⠀⢀⣀⣀⡠⠶⠶⠶⠦⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣼⡇⠀⠀⠀⠀⠀⢀⣾⣟⣿⣀⢰⣿⣿⣶⣄⣤⡀⠟⠈⠉⠉⠀⠀⠀⠀⠀⠈⠁⠁⣄
⠀⡤⠀⠀⠀⠀⢸⡿⠛⠉⠉⠙⢷⣴⢋⡬⠃⠀⠀⠀⠀⣰⡾⣿⡟⢻⣿⡏⢡⠟⠛⠛⠛⣄⠀⢰⡄⠀⣀⣀⣀⣀⣘⠀⠀⠀⠘⡗
⠀⢠⡄⠀⠀⠀⠋⠀⠀⠁⠀⠀⣨⣽⠞⠁⠀⠀⣀⡴⠞⠉⢀⣿⠇⣾⣿⠗⠀⠀⠀⠀⠀⢿⠀⠈⣷⠞⠋⠉⠉⠙⠛⢶⡄⠀⠀⢸⡇
⠀⠀⠑⠆⢀⠀⠒⠂⠙⠒⠒⠂⠘⢿⣤⣠⠶⠟⠋⣷⠆⠀⢸⣿⠀⢿⣷⠀⠺⠆⢠⡀⠀⠈⢧⡜⠋⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠋⠀⠀⣠⡟⠀⢠⣿⠇⠀⢸⣿⡆⠘⡆⠈⢷⣶⠂⠛⠓⠄⠤⠀⠠⣄⠀⠀⠀⠀⣀⠠⠞
⠀⠀⠀⠀⢀⠴⠤⠀⠦⠰⠶⠀⣠⡿⠁⠀⠀⢠⡟⠀⣰⣿⠏⠀⠀⢺⣿⣷⠀⢹⡆⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⣄
⠀⠀⡐⠂⠀⠀⠀⠀⠀⠀⢠⡼⣏⠀⠀⠀⣰⡏⢀⣾⣿⠋⠀⠀⠀⢾⣿⣿⡄⠸⣇⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣠⣤
⠀⠀⠃⠀⠀⠀⠀⠀⣀⣴⠏⠁⠀⠀⠀⣰⡟⢠⣾⡟⠁⠀⠀⠀⠀⠀⠻⣿⣿⡀⢻⡆⠀⠀⠀⢰⡄⢀⣤⣤⠀⢠⣤⡀⠀⠀⣠⠛⠻⣿⠿
⠀⠦⠀⠀⠀⠀⣶⠋⠉⢻⡆⠀⣾⠏⣻⡟⣱⣿⣿⣤⠀⠀⠀⠀⠀⠀⣰⣿⣿⣧⠀⢷⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠘⣠⡴
⣶⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⣼⠃⣸⣿⣼⣿⣿⣽⠁⠀⠀⠀⠀⠀⠀⣿⠉⢿⣿⡆⣾⠀⠀⠀⢀⠘⣿⣄⠀⠀⢀⣄⠀⢿
⣿⠀⠀⠀⠀⠂⠀⠀⠀⢠⣬⠉⣤⠛⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣿⣷⠆⠀⠀⠀⠸⡿⢷⣴⠂⠀⠀⣾
⠁⡀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣰⣏⣼⣿⣿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣇⠀⣿⡀⠀⣀⡿⠀⠁⠀⠀⠀⠈
⠀⡇⠀⠀⠀⠀⠀⠀⠛⡴⠞⢣⣸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠟⠀⢸⡇⠈⠉
⠀⠀⠀⠐⠀⠐⠀⠀⢰⣷⠀⢀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠂⣄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠄⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠆⠉⠛⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⣿⢉⣿⣿⣿⡀⠀⣠⠟⠀⠄
⠀⣾⡆⠀⠀⠀⠲⠂⠀⠀⠸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣿⢻⣿⣿⣿⡿⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠛⡄⠀⠀⠀⠀⠀⠈⠁⢀⠰⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣄⣿⣦⣄⠀⠀⠀⠀⣰⡉⢿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣴⣆⣀⣾⣏⣠⣀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠈⠻⣿⡻⢿⣿⣿⣿⣿⣿⡿⠿⣿⣿⠟⢻⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⢹⣿⡏⠀⠀⢠⣿⠏⢀⣾⡟
⠀⠀⣰⣶⣤⣀⠀⢀⠀⠀⠀⣿⣿⠀⠀⠘⣿⠆⠀⣸⣿⡇⠀⠀⣿⠃⠀⠘⡟⠁
⠀⠈⠿⠊⠛⠛⠀⠀⠀⠀⠀⢿⣇⡀⠀⠀⢀⣴⡞⢛⠛⣷⡄⢠⡟⠀⠀⢸⣷⠀⠀⣴⣄⠰⠆
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸⣿⠃⠀⠀⢸⣿⡇⠘⠀⠙⣧⣼⠁⠀⠀⣼⠃⠀⠀⠀⠁
⣀⣿⢿⣿⠀⠉⠉⠀⠀⠀⠀⢿⣿⣷⠆⠀⠀⣿⣧⠀⠀⣶⣿⠏⠀⠀⢠⡏
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⡀⠀⠻⣿⣿⣿⡟⠀⠀⠀⣴⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣀⠀⠈⠉⠉⠁⠀⢀⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⣦⡀⢀⣶⣶⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⢻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠏⠀⠀⢸⣿⡅⣤⣤⣤⡄⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠛⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⢹⣷⣶⣿⣿⠿⠿⠿⠿⣷⣤⣄⠐⢺⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠘⢿⣏⣀⣀⣀⣀⣀⠀⠟⠛⣿⣦⠀⠁⠀⠀⠀⠀⠀⠠⣤⠐⠂⠚⠷⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡄⠀⠈⢻⣿⠛⠛⠛⢿⣷⡄⠀⢻⣿⡆⠀⠀⢀⡤⠞⠉⠀⠀⠀⡀⠀⠀⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠉⠀⢉⣿⣿⣤⡀⠈⣿⣷⣤⣤⣼⣿⠇⠀⢻⣿⡇⠀⣰⠏⠁⣀⠐⠆⠀⡀⠀⢠⣤⣠⣥⣀⣀⣀
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡿⠋⣠⣤⡾⠟⠋⠉⠻⣿⣿⣦⣈⠉⠉⠉⠈⣀⣴⣿⠟⢀⡼⠃⣠⣦⠀⠀⣀⠀⣠⣀⠀⠀⠈⠙⢻⡿⠟
⠀⠀⠸⠇⠸⠀⠀⠤⢀⣠⣶⣟⣤⣿⣏⣀⣀⡀⠀⢴⣶⣿⡟⠛⠿⢷⣶⣿⡿⠿⠛⠁⠠⡿⣴⣿⣦⡤⢤⡼⠻⠟⠉⠙⣻⣉⡙⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢿⣏⠁⠀⠉⠙⠛⠛⠿⢿⣿⡿⠲⢀⣤⣴⣶⣿⣶⣶⣾⣿⣾⣿⣿⣋⣉⣵⣀⣀⣀⣠⣤⣤⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡿⢷⣤⡀⠀⠀⠀⠀⠀⠏⠀⠛⢿⣤⣽⣿⣿⠟⣿⣿⣛⣿⣿⣿⣿⣿⠿⠽⣿⡁⠈⠉⠠⣄
⠀⠀⢠⡞⢁⡤⣄⣄⠀⡀⠙⣷⣄⡈⠻⢦⣤⢾⣷⠀⣴⣻⡇⠈⣿⡉⢻⣿⣿⣦⣴⣟⣻⣿⠛⠋⢡⣶⣤⠉⠉
⠀⠀⠘⠷⠚⠷⠿⠟⠀⠉⠀⠉⠛⢿⣶⡌⢻⡆⠀⠀⠈⠉⠁⠀⠙⣿⣶⣿⣷⣼⣷⣿⣿⣿⣶⣶⣦⣿⡿⣿⣶⠒⠐⠂⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣷⡄⠀⠀⠀⠀⠀⠀⢾⠉⠉⣿⣿⣿⣿⡟⣿⣼⣻⣭⣽⣷⡞⠻⣷⣶⠟⠛⠛⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⡄⠀⠀⠀⠀⢠⠈⠁⢠⡜⢻⣾⣿⡿⢿⡟⠻⣿⣿⣤⣿⣦⣈⠙⣻⣦⠠⠄
⠀⠀⠀⠀⠀⡶⢶⣿⠀⠀⠀⠀⠀⠀⣰⣿⡿⣿⣿⡇⠀⠀⠀⠀⠀⢴⠀⠈⠿⠺⣿⣿⣦⠈⠙⠢⠘⢶⣿⠀⠈⠛⠋⠉⠛⠇
⠀⠀⠀⠀⠀⠀⠺⠃⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠻⣦⣀⡀⠈⠙⢿⣆⠀⠀⠀⠀⠈⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠘⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠒⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣄⣀⡀⠀⠀⣸⢿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⡀⠙⡆
⣀⣀⣀⣀⣀⣀⣻⣟⣻⣯⣙⣃⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣦⣼⡇⡇⢿⣿⣿⡈⣷⠸⣰⣃⡴⢚⠹⠶⣆⡏⠀⣄⡸⠀⠀⢸⡇⢡⡯⣄⣾⣷⡏⢯⠉⢸⣿⣿⣿⢸⣿⣿⡀⣿⣷
⡾⣿⣧⢹⢿⣿⣿⣷⠋⣧⠟⢹⠁⣀⡾⢃⡟⠴⠐⣉⣄⣰⠂⠀⢿⡌⢧⡯⣿⢹⡶⡈⢿⣼⣿⣿⣿⣼⣿⡇⣍⣿⣾
⠹⣯⢹⣯⣸⣿⣿⣏⣼⢹⣄⣟⡴⢨⡵⢋⠀⠀⠄⢲⡄⣙⢿⣤⠈⢷⣄⡐⢻⣧⢳⡀⡈⢻⣿⣿⣿⣿⣿⡏⢻⣿⣿
⡄⠾⣿⠉⢿⣿⣿⣿⢧⣶⣿⣭⠞⠉⣄⠢⠃⡀⢁⣂⡈⢍⠒⢮⣿⣮⣷⡘⠛⢼⣧⡹⢷⡀⣿⣿⣿⣿⣿⡇⣼⣿⣯
⣯⣲⣿⢀⣼⣿⣿⣿⣿⢹⣿⠁⡀⠶⠍⢡⢞⣭⠄⠉⠉⠄⡭⠉⠉⠙⠻⢿⣽⡲⢮⣿⣤⡿⣿⣿⣿⣿⣿⣿⣿⡿⡿
⢻⡿⣾⣿⣾⣿⣿⡟⢄⣿⣏⡴⠛⣚⠟⠚⠂⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣍⢻⣧⢻⣿⣿⣿⣿⣿⣟⠛⣷
⠈⣷⣿⠟⣿⣿⣿⣇⣾⡯⢿⣾⡷⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿
⠀⢸⣷⣴⣿⣿⣿⣿⣿⣵⣟⣛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟⣸
⢀⡿⢁⢻⣿⣿⣿⣿⣷⣛⠮⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠁⢿
⣼⠃⢹⣿⣿⣿⣿⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡆⢸
⣿⠀⡾⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢞⣹⣿⣿⣿⣿⣿⡇⣸
⣿⡿⠁⣿⣿⣿⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢮⢺⣿⣿⣿⣿⣿⣷⣿
⣿⠁⠀⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢾⣿⣿⣿⣿⣿⣿⡏
⣿⣆⠀⣿⣿⣿⣿⣿⡁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡆⠛⣿⣿⣿⣿⣿⣿⣿⡇
⠹⣿⣆⣿⣿⣿⣿⣿⣯⢶⡹⠍⢹⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⡗⣀⣿⣿⣿⣿⣿⣿⣿⡇
⣴⣽⣿⣿⣿⣿⣿⣿⣿⣗⡟⢤⣿⡴⣴⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⣿⣧⣿⡿⣿⣿⣿⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣻⡯⠳⣯⢄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿
⡯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣈⣿⡱⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢲⣀⣷⣿⣿⣿⣿⣿⡏⣿⣿⣿⠸⣿⣿
⣠⡿⢻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣳⡁⠀⢰⢀⣦⣤⣶⣶⣶⠶⢶⣺⣿⣿⣿⣿⠋⢸⣿⣿⣿⢾⣿⣿⢧⣻⣿
⣿⠃⣸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣶⣿⠛⡉⢁⡶⢻⣥⢶⢿⣿⣿⣿⣿⡟⣴⣿⣿⡟⣿⢸⢹⣿⣾⣷⣿
⡟⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⠟⣱⣾⣵⣿⣾⣿⣿⣿⣿⢳⣿⣿⣿⣿⣿⣿⣾⣿⣿⡟⣿
⣷⠀⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⢿⣿⣷⣿⣿⣿⣿⣿⢩⣿⣿⣿⢸⣿⣿⣿⣏⣿⠏⣿⣿⣿⣷⢹

⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢂⠖⠐⠢⡀⠀⠀⠀⠀⠀⠀⢠⢔⠉⡏⠉⡳⣄
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠑⡀⠀⠀⠀⠀⠀⠀⠀⡰⠂⢀⠊⠀⠐⠀⠘⡆⠀⠀⠀⠀⠀⡧⠤⢵⣷⡮⠤⠼
⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⢇⠀⠃⠀⠀⠀⠜⠀⡅⠀⠀⠀⠀⠀⠃⡴⠋⣬⠑⢤⡜
⠀⠀⠀⠀⢀⡞⢐⠙⠻⠛⢑⠘⡄⠀⠀⠀⠀⠀⠈⢧⡈⠃⠀⠊⢀⠝⠁⠀⠀⠀⠀⠀⠀⠈⠑⠙⠚⠁
⠀⠀⠀⢀⠎⠀⠀⠉⠀⠈⠁⠀⠈⢄⡀⠀⠀⠀⠀⠀⠈⠉⠁⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⣿⣷⣦⣄
⠀⠀⠠⠊⠔⠢⠢⠄⠔⠄⠢⠂⠔⠜⠃⠀⠀⠀⠀⠀⡀⢠⠀⡴⢀⠂⠀⠀⠀⠀⠀⠀⡈⠘⣿⣯⣽⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣆⣹⠞⠒⠓⠺⣤⡄⠀⠀⠀⠀⠀⡀⠀⠈⠻⠟⠛⠉⠉⠹⣿
⠀⠀⠀⠀⠀⢀⣤⡶⠋⠉⠉⠀⠀⠀⠀⠀⠀⣈⠓⡿⠑⠖⠁⠘⠷⠊⣿⣁⢀⡀⠀⠀⠠⡀⠀⠀⠀⠀⠰⠇⢀⡟
⠀⠀⠀⠀⢠⢃⣎⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠹⣷⠀⠀⢀⠀⠀⢀⣿⡍⠀⡀⠀⠀⠀⠀⠂⠤⠀⠀⠀⠐
⠀⠀⠀⠀⡏⢸⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠜⢧⣄⣙⣁⡤⠾⠁⠀⠄
⠀⠀⠀⠀⠹⡄⠻⣄⠁⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠐⠁⠼⠈⠁⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣆⣤⡄⣀⡀
⠀⠀⠀⠀⠀⠙⠦⣈⡓⠒⢤⣤⠶⠋⠀⠀⢀⢤⡀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠠⣔⡪⠋⢯⠿⠄⠸⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡆⣅⠀⠸⣄⠃⢊⣷⣂
⠀⠀⠀⠀⠀⣀⡤⠴⠶⠶⣤⡀⠀⠀⣀⢀⣸⠺⣀⢠⡄⠀⢀⢀⣴⠋⠀⠈⠰⡄⠀⠀⠀⠐⣘⢦⣧⢉⣁⠽⣵⠂
⠀⠀⢠⣾⣯⣥⡖⠋⣟⡏⢲⡅⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⣿⣄⡀⢀⣼⡟⠀⠀⠀⠀⠀⠃⠾⠙⠘⠆⠀⠑
⠀⠀⠀⢠⠤⠀⢉⣱⣿⣿⠉⠀⠀⠀⠀⠀⢘⢱⠀⠀⠀⠀⠀⠐⠈⢙⠛⠋⠁
⠀⠀⠀⠀⠉⠉⠉⠉⠀⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢠⠻⡷⠒⠇⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⡠⢤⠤⢄⠀⠀⠀⠀⠀⠀⣀⢀⠀⡀⠀⠀⠀⠀⠘⠀⠘⠀⢸⠀⠀⠀⠀⠀⠠⠐⠉⠉⠀⠂⣀
⠀⠀⠀⠀⢰⠁⠀⢸⠀⠀⠱⠀⠀⠀⠀⠀⠸⣾⣤⠁⠀⠀⠀⢀⡇⠀⠀⠀⢀⣦⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠠⠀⣠⢂⢂⠀⠘⠀⠀⠀⠀⠀⠀⢹⠃⠀⠀⠀⠀⠙⠙⠀⢸⠀⠈⠈⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠘⢃⣼⣀⡟⠊⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

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

⠀⠀⠀⣀⣀⣤⣤⡤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡆⠀⠀⠀⠀⠀⠀⠀⣰⠙⠛⠉⢰⠂⠀⠀⠀⠀⠀⠀⢀⡤⡶
⠀⠀⠀⡏⢠⠴⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢠⠞⣼⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡴⠛⣱⠁
⠀⠀⢀⡇⠁⠁⠐⢻⠇⠀⠀⠀⠀⠀⠀⡰⠃⢠⡏⠀⣀⣀⡀⠀⠀⠀⢸⠁⠀⠀⢸⠁⠀⠀⠀⠀⣠⠊⠀⠀⠓⠲⠤⣤
⠀⠀⢸⡇⠀⠀⠀⢸⣀⣀⡀⠀⠀⠀⡼⠁⠀⠈⠉⠉⠉⣹⠃⠀⠀⠀⡾⠀⠀⠀⠈⢡⡟⠀⠀⢰⣁⣤⣤⡀⠀⢀⡞⠁
⠀⠀⢸⠀⠀⠀⠀⠈⢀⡜⠀⠀⠀⣸⢃⣀⡤⢤⠀⠀⣴⠃⠀⠀⠀⠀⠓⢲⠆⠀⠀⡿⠀⠀⠀⠀⠀⣠⠏⠀⣠⠏
⠀⠀⠘⠛⢙⡇⠀⢀⡞⠀⠀⠀⠀⠿⠛⠉⠀⣿⠀⣼⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⣼⠁⠀⠀⠀⠀⣴⠁⠀⠰⠷⣶
⠀⠀⠀⠀⢸⡁⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⣴⠃⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠉⡸⠁⠀⠀⠘⠟⠒⡆⠀⡼⠁
⠀⠀⠀⠀⢸⡇⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠓⢲⠀⣰⠃⠀⠀⠀⠀⠀⢸⢣⡞⠁
⠀⠀⠀⠀⢸⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣰⠃⠀⠀⠀⠀⠀⠀⣾⠟
⠀⠀⠀⠀⢸⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⣶⠺⠛⢛⣳⠀⠀⠀⠀⠀⢸⢆⠀⠀⠀⠀⠀⠀⢀⠤⠤⠶⠖⡦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⢀⣤⠇⠀⠀⢘⡏⠀⠀⠀⠀⠀⢸⠘⣆⠀⠀⠀⠀⠀⡼⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⢀⡾⠋⠋⠉⠉⣴⠟
⠀⠀⠀⠀⢸⡇⠀⠀⢠⡴⠃⠀⠀⠀⢀⡴⠊⠀⠈⢆⠀⠀⠀⠀⠇⠀⠀⢠⠇⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⢠⡾⠁
⠀⠀⠀⣶⠚⠀⠀⠀⣸⠃⠀⠀⢀⡴⠋⠀⠀⢐⡴⠋⠀⠀⠀⢸⠀⠀⢀⡟⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⣰⣏⣀⣠⡀
⠀⠀⠀⢻⡄⠀⠀⡶⠚⠀⠀⠀⠙⡄⠀⠀⠐⡏⠀⠀⠀⠀⠀⣘⠀⠀⢸⠇⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⣨⠇⠀⠘⣇⠀⠀⠀⠀⠀⠘⣄⠀⠀⠱⣄⠀⠀⠀⠀⡏⠀⠀⠈⢛⠟⠀⠀⢰⣏⣤⣤⡄⠀⠀⣠⠞
⠀⠀⠀⠹⡄⠀⢾⠉⠀⠀⠀⠀⠀⠀⠈⠦⢤⢂⢸⠀⠀⠀⠀⠓⠒⡿⣠⠋⠀⠀⠀⠈⠀⠀⢰⠃⠀⡴⠃
⠀⠀⠀⠀⠻⡄⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢸⡄⠀⠀⠀⠀⣸⣱⠁⠀⠀⠀⠀⠀⠀⠀⣿⢀⡞⠁
⠀⠀⠀⠀⠀⠹⣌⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟
⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋

⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢺⣿⠟⠉⠉⠙⢻⣷⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠎⣿⡏⠀⠀⠀⠀⢀⣿⣇
⠀⠀⠀⠀⡠⠔⠂⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠨⢻⣧⠀⠀⠀⢀⣸⣿⡎
⠀⠀⡴⠊⢀⡴⠀⣊⣉⡒⠢⢄⠀⠀⠀⠀⠀⠀⠈⠹⡆⠀⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀
⠀⡼⠁⡰⢁⣴⠛⣉⣉⡙⠳⣄⠑⢎⠀⠀⠁⣴⡀⠀⢿⡀⢱⣿⣿⡇⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠈⠙⣧
⡆⡇⢐⠃⢸⡇⣿⠉⣌⠹⢦⣈⢷⣅⠁⠀⠀⡃⢱⡤⣸⡇⢺⣿⣿⠀⠀⠀⢀⣴⣬⣿⠟⠀⠀⠀⠀⠀⠀⢸⠇
⠀⣇⠀⢢⡘⣧⡙⠓⠶⣶⣬⣷⣦⣙⢷⣖⢶⣇⣸⣏⠘⣷⣾⣿⡇⢀⠠⢂⣺⣾⣿⠇⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠜⢦⡀⠙⠌⠙⠓⠶⠶⢦⣽⣮⣻⣿⡿⣿⣿⣽⡟⢳⣿⣿⣿⣿⣿⣴⣿⠿⠋⠁⢀⣀⣠⣤⠤⠶⠚⠋
⠀⠈⠢⡁⠢⢤⣀⣉⠛⠒⠶⠴⢬⣙⣛⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣵⡴⠶⠛⠋⠉⠁
⠀⠀⠀⠀⠁⠂⠬⢉⣭⣛⣛⠿⠿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⠁
⠀⠀⢀⠄⠀⠀⠀⠠⠤⠀⠈⠉⠔⠷⣾⣛⣿⣷⣿⣿⣿⣿⣿⡏⠛⠻⠿⠶⣤⣄⣠⣶
⠀⡀⠁⠁⠀⠀⠀⠀⣀⣀⣠⣼⡾⢾⠿⢛⣹⣿⣿⠟⡿⣿⣿⡇⡀⠀⠀⠀⠀⢨⣿⡟⠓⠲⠦⠤⣤⣤⣦
⠀⡀⠀⠀⠀⣰⠾⣟⣋⡩⣥⣴⣶⣿⣽⠟⣋⠻⢹⢀⣷⣿⣿⡗⡇⡀⠀⠈⠢⣈⠛⠃⠀⠀⠀⠀⢠⣿⠛⢆
⠀⠀⠀⠀⢸⡏⢠⢩⢒⣐⣿⡯⣥⠟⠑⠊⡀⠆⡼⡫⣿⡿⢻⣧⠀⡃⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠘⣿⡉⠷⠿⠯⣫⡾⠋⠀⠀⢀⣴⡄⠉⠰⣻⡇⣿⢹⢰⢑⠀⠀⠀⠀⠀⠿⣷⣄⡀⠀⠀⢠⣄⣼⠁
⠀⠀⠀⠀⠀⠈⠛⠳⠶⠛⠉⠀⠀⠀⠀⣾⠋⠀⠂⢤⡿⢸⢃⡞⣏⢇⠀⠀⠀⠀⠀⠀⠨⣿⣿⡯⡶⠿⠿⠇
⠀⠀⠨⡢⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠘⠀⠠⠀⣐⣾⢃⡟⡸⣳⣹⡎⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⢰⢵⣧⠆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡹⠠⢥⠼⣠⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠛⠶⠶⠾⠫⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣀⣀⣀⡀
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠐⠆⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠄⡀⢀⠈⡁⠀⠀⠠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⡀⢰⢰⠂⢫⣄⣁⣼⠠⢒⢠⡀⣀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣨⣀⡆⢘⣷⣾⠿⠛⠛⠿⣷⣷⡯⢰⣆⣉⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⣀⣈⠉⠙⠻⣿⣥⣾⠿⠿⢷⣤⣻⠿⠛⠉⢉⣀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⢋⡁⠉⢀⣀⠙⠲⢄⠙⢤⣾⣶⣦⠟⣡⠖⠋⣁⡀⠀⢡⡙⢃⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⣉⣀⣉⠓⠦⡑⣌⠃⠀⣁⠌⡠⠚⣉⣀⣉⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⢻⡿⣷⣿⠳⣔⣘⣃⣙⣁⢎⣴⣣⣼⢿⡿⠛⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢰⡖⠀⢸⡶⣿⠿⠛⠡⠤⢦⠌⠙⠻⣿⢶⡞⠀⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠲⠀⠄⢿⣏⢤⢄⠏⠀⠀⠙⣗⠅⡸⡿⠧⠄⠶⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠄⡡⠤⠠⠤⡐⣧⠁⠀⠀⠀⠀⠈⣼⢐⡤⠄⠠⢄⠠⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⡜⠡⣾⣭⣦⠌⢿⠀⠀⠀⠀⠀⠀⣿⢃⢔⣦⣴⡆⠱⠐⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢇⠀⢿⣾⣻⣶⢾⡆⠀⠀⠀⠀⢠⣧⢶⣿⣻⡿⠃⢰⡈⡸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⠬⠘⠨⢦⣈⣿⣼⣿⣹⡷⠀⠀⠀⠀⠷⣏⡿⣿⣿⣇⡴⠦⠃⠥⣄⠬⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠐⣥⣬⣢⣙⣿⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⣿⣏⣐⣪⣤⡁⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣡⣧⣌⡷⣿⢹⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠿⡿⣿⣣⣼⣥⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣫⡽⠶⠿⠿⢿⣿⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣿⡷⠿⠷⠷⢮⣍⡓⢀
⠀⠀⠀⠀⠀⠀⠀⡜⢱⡞⠩⣶⣶⣋⠀⢒⠌⣿⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢿⡡⣔⠀⠙⣷⣶⠌⢻⡖⣳
⠀⠀⠀⠀⠀⠀⢰⣏⣟⣛⡛⢦⡈⠀⢄⢬⣿⣿⠀⠀⣿⠂⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⣿⡯⢂⠂⢊⡴⠞⢋⣛⣋⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠙⢊⠷⣄⢾⢺⣿⣿⣄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⣿⣿⡏⡦⢂⡾⠳⠛⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠲⣙⢷⣌⣹⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⣿⣃⣴⢋⠖⠁⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠉⣿⣶⣤⣄⢤⣴⣶⠞⢻⣿⣽⣿⡳⢶⣶⣤⢄⣤⣶⣾⠉⣇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣘⡴⢋⣃⣍⠶⠞⣡⢳⠂⠘⣿⣿⠋⠌⢺⣪⡛⠶⣡⣍⡍⢣⣘⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡌⠙⠉⠡⠤⠀⠐⠈⠁⡈⢇⠄⢹⡏⠀⣨⠃⡌⠁⠒⠀⠄⠄⠉⠙⢀⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀⠀⢀⠀⠀⠀⠀⠀⢄⠐⡈⢳⣸⠇⡶⠁⠌⠠⠀⠀⠀⠀⠀⡀⠀⢀⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠘⠃⠈⢢⠹⠟⡠⠂⠐⠛⠀⠀⠀⠀⠀⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂

⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣀⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿