Outer ear Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣴⣶⣦⣦⣆⣠⡀⢀⣀⡀⠛⠂⠀⠀⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣪⣿⣿⠟⠉⠉⠉⠉⠙⠛⠛⠿⣷⣮⣶⣦⣄⡠⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⢻⣿⡿⠾⣛⠢⠄⠄⠀⠀⠀⠀⠀⠀⠈⠻⣟⢿⣷⣂⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢸⢺⡏⠀⠄⢀⠀⠀⢐⠲⢄⠀⠀⠀⠀⠀⠀⠨⢷⣻⣿⣧⣐⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣼⣷⠏⠐⠈⠀⠀⠠⠁⠀⠷⡆⠀⠀⠀⠀⠀⠘⡆⣿⠛⢿⡄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢿⢻⣦⠎⠀⠤⠀⠀⢀⠀⣰⠇⠀⠀⠀⠀⠀⠀⣷⣯⠖⢺⡇⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠈⢿⣿⡄⠈⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⢻⣂⡬⢿⡇⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠈⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⣿⣇⠀⠀⡀⢹⡇
⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠈⢳⣻⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠼⣇⠀⠀⠁⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣿⣶⣶⡦⠤⣀⠀⠀⠀⠀⠀⢻⣾⣇⢇⢻⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠑⠻⢿⣿⣿⣿⠿⢶⣮⣶⣄⠀⠀⠘⣞⣿⠀⢽⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠰⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣦⣄⡈⠿⣷⡀⠀⠀⢱⣿⡶⠆⣿⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⠀⠉⠉⠛⣦⣝⣿⠀⠠⠌⢿⣇⢰⡟⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠭⠋⣝⣷⣄⡀⠀⠀⠱⣿⡇⠀⡀⢳⣿⣿⡇⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⡍⠀⢀⠠⠑⣿⣧⠀⠀⠀⣿⠃⠀⡀⢸⣿⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡗⠂⠀⡆⠠⢤⣽⣧⠀⢻⡿⠀⠀⠀⢌⣿⣿⠇⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣿⣿⠟⡥⠏⠀⠠⡀⣈⣿⣿⣶⡿⠃⢀⠀⠀⢹⣿⡿⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣠⡾⢛⡧⣶⣷⣶⣷⣶⣿⣿⣿⣿⣿⡥⠀⢀⣤⣶⣿⣿⡆⣠⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⡟⢉⢈⠫⢿⡋⠉⠉⠀⠨⠉⠀⠉⠁⠀⠀⢈⣉⣿⣿⢋⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠹⣿⡿⠤⣮⡀⢛⠘⠂⠛⠛⠛⠛⠉⠛⠚⠻⠿⣿⠟⣋⣴⣿⡿⠋
⠀⠀⠀⠀⠈⠈⠀⠁⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠙⠓⠛⠛⠚⣿⢿⡾⠟⠛⠃⠀⠐⣺⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠚⠂⠒⠂⠒⠐⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠃⠀⠒⠀⠛⠀⠂⠈⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⣀⣀⣀⣀⠤⠔⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⢀⢦⣤⠀⠘⡀⠀⠰⡆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⢸⡏⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠣⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣄⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡄⠐⢀⠈⠱⡢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣸⡟⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⡐⣰⡄⠙⠢⡟⡀⠀⢀⠑⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢽⣿⣷⣤⢶⠨⣄⡹⠃⠡⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠒⠲⣶⣶⣞⣿⣿⣿⣿⣿⣷⣾⣿⡿⠿⠳⢝⡂⠁⠐⢸⢀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⣀⠀⠀⠉⠻⠿⠟⠋⠁⠀⠀⠈⠉⠁⠀⠀⠀⠠⣿⣦⡀⠀⠸⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠣⡜⠠⡄⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⠀⠀⠀⠈⠻⠀⠁⡀⡔⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢉⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡔⢄⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⢢⣽⡷⢄⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠆⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡌⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣮⣳⡈⢆⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢤⣤⣛⡶⠠⢇⡀⢽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠁




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣰⣯⣍⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠄⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠁⠁⠘⠁⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⣀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠀⠁⠈⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡈⠀⡰⠐⠀⠀⠃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠢⠦⠀⠀⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠈⠁⣰⠅⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠢⢄⠀⠀⠀⠀⠉⠛⠒⠀⠀⠀⠀⠌⠃⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⢀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠤⠄⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⡀⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠁⠀⠀⠀⠉⠉⠉⠉⠛⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⣠⣤⡤⢶⣆⡠⡠⣤⣤⣀⡀⠀⠀⠈⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣟⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣽⣷⣦⣤⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⢰⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⢆
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⢻⡙⢿⣿⣿⣿⣿⣿⣆⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡑⠄⠀⠀⠁⠘⣿⣿⣿⣿⣿⣮⡄⠀⠀⢧
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠙⠟⠋⠑⠄⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⡆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠻⡿⣿⣿⣿⣾⠖⠆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣟⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣄⣀⣤⣤⣀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣻⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠈⢹⣿⣿⣿⣯⡀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠞⠻⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢰⣾⣿⣿⣿⢁⠀⠀⣘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠙⢿⡿⢿⣿⣿⣿⡆⠀⠈⠿⣿⣿⣿⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣾⣷⣿⣟⣷⣿⡻⣿⣿⡄⠀⢱⡸⣿⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⡇⠀⠀⢀⣿⡗⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢰⣿⠃⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⡟⠀⠀⠀⣸⡿⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⢀⣿⠇⠀⠀⠀⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⠟⠛⢋⠩⠍⠉⠉⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠿⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⢀⠄⡀⠌⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠛⠁⠀⠀⠀⠀⠀⠉⠑⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⢀⣤⣴⣶⣤⣤⣀⡀⠀⠀⠈⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⢞⡁⠆⢀⣴⡾⠛⠛⠉⠉⠉⠙⠻⣿⣷⢌⠁⠀⠀⢝⢦⡀
⠀⠀⠀⠀⠀⢠⠾⠁⢈⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⡀⠀⠈⠙⢷⡀
⠀⠀⠀⠀⢠⡏⠐⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⠀⢰⣳
⠀⠀⠀⠀⡾⣴⠃⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⣿⣇⠀⠀⠀⠀⣿⣇
⠀⠀⠀⢸⢷⡇⢸⠇⠤⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⠐⠀⠐⠀⢿⣿⡀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⡿⣾⠁⡾⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠁⢸⣿⣇⠀⠀⠀⢸⣿⡇
⠀⠀⢰⢷⡟⢸⠇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⣿⣿⡁⠀⠀⡟⣼⣷
⠀⠀⣾⣼⠃⡿⠀⠃⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣧⠀⠸⣗⢳⣿
⠀⢠⣿⣿⢰⡇⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡀⣻⡯⠙⣇
⠀⢸⢿⡇⣿⠄⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⠇⢿⢽⠀⠙
⠀⣼⣾⡇⡿⠀⢰⣿⠀⠀⠀⠀⠀⣠⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣼
⠀⣿⣿⡇⣏⠀⣻⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣰⡿⠃
⠀⣿⣿⣧⢿⣼⣿⣿⠀⠀⠀⢠⣿⣿⣿⠟⠛⠋⠉⠁⣺⣿⣿⣿⣿⡿⠀⣰⡟⣀
⠀⢸⣧⣿⡜⣿⣿⣿⡄⠠⠀⢸⣿⡿⠁⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣦⣴⢋⣸⠙
⠀⠘⣿⣽⣷⢹⣿⣿⣧⠀⠀⢸⣿⡇⠀⠀⠠⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠘⣿⣽⣧⠻⣿⣿⡄⠀⠀⢿⣧⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡛
⠀⠀⠀⠘⣿⣿⣧⠹⣿⣿⣆⠈⠈⢿⣷⣀⠀⠀⠀⣞⣽⣿⣿⣿⣿⣿⣿⡄⡀⠀⠀⣀
⠀⠀⠀⠀⠈⢻⣿⣷⡜⢿⣿⣷⣀⠀⡻⣿⣿⣷⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣾⣶⣤⣄⠈
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡙⢿⣿⣶⣄⠈⠛⢿⣿⣿⡋⠀⠄⠀⠠⢈⠙⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣮⡙⢿⣿⣷⣦⣀⣹⣿⣿⣦⠀⠐⠀⠀⠀⠙⢿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣿⣷⣌⠻⢿⣿⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣦⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠆⠄⠀⠀⠀⠀⣸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠁⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣶⣤⣤⣤⣤⣶⣾⠟⠁⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠁⠀⠀⠀⠀⠉⠑⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⢀⡤⠔⠒⢖⡒⠒⠤⢄⡀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⣠⠞⠁⠐⢄⡀⠀⠙⢢⡀⠀⠉⢦⡀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢰⠁⠀⢄⡀⠀⠙⢢⡀⠀⠙⣄⠀⠀⢳⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣄⠀⠀⠈⠙⠒⠒⠛⠀⠀⠘⡆⠀⠈⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⡌⠓⠦⠤⠤⠤⢤⣀⡀⠀⠀⢸⡀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⡇⢀⡇⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢈⠓⠦⠤⠤⠤⡀⠀⢸⡄⠀⢹⡼⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⢸⠃⠀⡸⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⡤⠚⢠⠎⠀⠀⠀⠀⢀⡴⠋⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡏⢀⡔⠁⠀⠀⠀⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⣄⣀⡠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠯⣰⣶⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠈

⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀
⠀⠀⠀⠀⢰⢣⣶⣤⠀⠀⠀⠀⠀⠀⡅⠀⡆⠀⣇
⠀⠀⠀⠀⠸⢿⡻⣿⡇⠀⠀⠀⠀⠀⡇⠀⡇⠀⢡⠳
⠀⠀⠀⠀⠀⢛⠇⢿⣿⡀⠀⠀⠘⠀⠳⠀⠀⠀⠘⢆⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠆⣾⡿⣿⣦⡀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠶⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠁⡸⠛⠻⣦⡇⠀⠀⠘⡀⠀⠀⠀⠀⠔⡆⠀⠀⠀⠀⡀⣼⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠁⠀⠀⠹⣧⠀⠀⠀⢃⠀⠀⠀⢀⣶⣻⡿⣦⣄⡄⠻⠿⠀⠛⠁⠁
⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⢀⣴⣿⡄⠀⠀⠀⠁⠀⢒⣀⣈⣉⣭⣭⣿⣿⣧⢰⣠⣦⣤⣀⠎⡭⠉⠂⢉⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⢰⢹⡄⠀⢀⣾⣿⣿⡃⠀⠀⠀⣠⣾⠟⠋⠉⠉⠉⠉⠈⢽⣆⠀⣷⣿⡇⠉⢳⣄⠐⢰⡿⠿⠋⠀⠨⠍⠓⠂
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣸⣿⣿⣽⣷⣶⣶⡿⠋⠠⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣌⣻⣿⣷⠿⠿⠷⣤⡙⠦⣄⢤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠁⢻⡀⢸⣿⣿⣿⣿⣿⣿⣧⠀⠀⢀⣀⣤⣤⡤⠤⠶⠛⠉⠉⠉⠉⠈⠛⠷⠄⠀⢀⡉⠢⣌⠙⢷⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⡀⠙⠿⣿⣿⣿⣿⣿⠿⠋⠙⢀⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⣷⠀⠀⠈⢿⣿⡿⠃⠀⡠⡔⠋⠉⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⣄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⣷⣤⣾⠋⠁⠀⠀⢰⠄⠀⠀⠀⠀⠀⠘⢆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠑⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠶⠖⠚⠁

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

⡀⠀⠀⠀⠄⠠⣤⡄⠀⠆⢠⡤⠀⠀⠀⡀⠀⠄⡴⡀⢤⠠⣤⡄⢰⠀⣤⠄⠀⠀⠀⠀⠤⠠⢠⣤⠀⢰⠀⣤⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⡀⣀⡀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠛⠁⠊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡆⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡴⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠢⠄⠤⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠈⠀⠀⢸⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡏⣴⣿⠆⢹⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢿⣿⣿⣿⣦⡀⠀⠀⢠⣀⣤⣾⣿⡟⢣⡌⢅⠀⠈⣿⣿⣿⣿⣶⣤⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣾⣾⣿⠟⢉⠙⡿⢃⣁⠀⠀⠀⠀⡠⠠⠀⠨⠻⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢽⣿⣿⣿⣿⣿⠇⣸⣆⣠⣄⢁⠀⠀⠀⠀⠀⠠⣤⣤⢸⣿⣿⣿⣦⡀⠄⠂⠂⠀⠐⠲⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⣀⣀⠀⠈⡉⠙⠭⠵⠂⠙⢟⣧⡀⠈⠀⠀⠀⠀⠀⠀⠀⡠⣢⣿⣿⣿⣿⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠃⣙⣃⣀⡀⠀⠀⠀⠉⠻⢷⣾⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡿⢿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠙⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠂⠀⣠⣶⣿⠿⠋⠉⠁⠀⠀⠀⠀⠉⠙⢿⣿⣷⣆⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠋⠙⠁⠀⠀⠈⠀⠈⠃⠓⠂⠘⠛⠙⠉⠫⠉⠑⠤⠀⠀⠀⡆⢀⣤⡐⢠⡄⠰⢠⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠴⠆⠀⠲⠀⠴⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⢀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠚⠐⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⡆⢠⢠⣀⡄⣤⠀⠄⠀⠰⠠⢤⡄⠀⢀⠀⡤⣤⡄⠀⣦⣤⡀⣄⢤⣠⡄⠀⣄⠤⣠⠀⢠⢄⠀⣤⡄⢠⠀⣠⡄

⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠸⠀⢀⠈⢣⡀⠈
⠀⠀⡇⠀⣿⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⠸⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⣠⠴⠶⢤⡀
⠀⠀⠇⠀⣿⠀⠀⠸⡄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄⠀⠸⡀⠘⣿⠀⠀⠀⠀⠀⣼⢣⠊⠉⣢⢹⢞⣫⣛⢶⡀
⠀⠀⠆⠀⡿⠀⠀⠐⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣴⡆⠲⡄⠘⡄⢸⡆⠀⠁⠀⣴⡟⢸⢔⠝⡖⢠⡜⡄⢈⡆⡇
⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⢠⡇⠀⣿⠿⣃⠀⣀⢾⣿⠇⠚⠲⠴⠁⣇⡔⣥⠞⡼⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠈⣟⠥⣐⠻⡿⣻⣿⠈⣧⠀⠀⠀⠠⠀⠈⠁⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⡀⠀⠀⢰⣿⣇⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣶⡟⠄⡟⠁⠈⠙⠓⠲⠒⠈⢿⣷⣦⣄⣀⣀⣀⢠⡞
⠀⠀⡇⠀⠀⣼⣿⡿⠻⠛⠉⠉⠋⠛⠛⠛⠛⠿⠶⠞⠟⠿⠛⠿⠉⠋⠉⠃⢠⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠩⠉⠛⠛⠃⠃
⠀⠀⢰⠀⠀⠘⠿⡃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣤⡶⠶⠶⠖⠲⠶⣤⣀
⠀⠀⠀⣇⠸⡄⠀⠸⠦⠴⠤⣼⠶⠚⠛⠋⠉⠙⠓⠂⠀⠐⠲⠒⠒⠒⠘⠛⠋⠙⣷⡀⠀⠀⠀⠀⠀⡀⠉⠛⠧⡀⢀⣠⡴⠶⠶⢶⣤⡉⢳⡄
⠀⠀⠀⠘⣆⠙⠆⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣆⠀⠀⠀⠀⠛⠿⢶⡴⠒⠉⢀⠴⣒⣒⣒⢌⢻⡆⢿
⠀⠀⠀⠀⠘⢆⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣄⢰⠀⠁⠀⠀⠀⢰⢣⣟⡥⡤⣬⣿⢸⡇⣼
⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣆⠀⠀⠀⠀⠘⣌⢿⣿⣶⣾⡵⢟⡴⠃
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢆⠀⠀⠀⠀⠈⠓⠒⠤⠴⠒⠋
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠕⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢐⠂⠀⠀⠀⠐⠐⢂⠀⠀⠀⠀⠀⠒⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠒⠒⠒⠒⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠁⠤⠁⠁⠀⠈⠀⠄⠈⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠭⠉⠉⠠⠀⠀⠀⢀⣀⠠⣀⠀⣀⣤⠔⡲⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⢠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠨⠁⠀⠄⠀⠀⠰⢁⣠⡶⢤⠱⡵⢈⣾⡅⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣠⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠍⠄⠄⠀⠀⠐⠄⠻⡇⣼⡀⢈⡛⠛⢀⣹⣶⠶⠒⠶⢟⣛⣛⣋⣭⣿⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠉⠉⠀⠀⠀⣳⣿⡁⠙⢾⢿⡦⠈⢿⣦⡊⢍⠱⡄⠿⠋⠉⠀⠈⠀⠀⢀⡀⡀⡐
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠢⢀⡀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡽⢷⠀⠀⠡⣼⣗⡈⠫⠿⡬⠜⣡⠇⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡳⢤⠀⠀⠰⠈⠉⠉⠓⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠁⠐⣒⢒⣐⠂⠈⠁⠐⠾⣄⢀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠐⠐⠒⠂⠂⡂⢀⣀⡀⢀⡀⠀⠰⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⡐⠀⢐⢒⡀⢐⣀⠀⠈⠂⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠀⢀
⠀⠀⠔⠁⠀⠀⠀⢀⣀⠀⠀⠀⢸⣿⣿⠀⠸⡆
⠀⠀⠀⣠⠶⢦⡀⠈⠉⠀⠀⠀⠀⣿⣿⡄⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞
⠀⡆⢰⠁⠀⠰⣿⡀⠀⠀⠀⠀⠀⠹⠏⠻⠀⠀⢈⡉⢙⡓⠒⠦⢄⡄⠀⠀⠀⠜⠁⠀⠄⠀⠀⢀
⠀⠇⢸⠀⠀⠘⠉⠛⢦⠀⠘⢂⠀⠀⠀⠀⠀⠀⣺⢨⣮⣷⣶⣦⣀⣍⢁⣀⡀⠀⣀⠠⡀⠀⢠
⠀⠀⠸⠀⠀⠀⠀⣠⣾⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠙⠙⠟⠉⢿⠋⢻⡿⠁⠂⠈⠀⠈⡐⠃⢀⠐⠆⠀⡀⠐⢂⡀
⠀⠀⠀⢃⠀⠀⢠⣿⠋⠓⢤⣀⣀⣘⠉⠉⠉⠁⣀⣤⣴⣶⣶⡶⣯⡸⣧⣷⠾⡶⣶⠀⠀⢈⣷⠀⠐⠛⢭⡵⠄⠀⠁
⠀⠀⠀⠈⢆⠀⠸⣷⠀⠀⠀⡍⠛⠛⠛⠛⠿⠿⠛⣛⠉⠉⠙⠻⣴⠙⢿⣿⣿⣿⣽⡄⠀⢸⡇⠀⠀⡀⡀
⠀⠀⠀⠀⠈⢆⠀⠻⣦⣀⠀⡕⠀⠀⠀⠀⢀⡀⠴⠻⠶⠶⢦⣤⣻⣷⣤⣿⣿⣿⡏⡻⣦⠄⠱⢌⣒⡡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠞⠛⠂⠂⠂⠉⠀⠀⠀⠀⠀⠰⠿⣀⡍⡙⠛⠛⠿⠿⠌⠙⠛⠷⣤⣉⠁
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣦⣶⣤⣥⣀⡁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠏⣻⢛⡿⢟⣻⠏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠂⢀⡄
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⠨⠝⣟⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡤
⠀⠀⠀⠀⠀⠀⠀⢢⠠⡄⠀⠀⠀⢾⣿⣶⣴⡃⢀⣎⡀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠤⢤⡬⣄⡹⠦⢀⡚⣿⣿⠟⠝⣲⣿⣝⢢⠀⠀⠀⠀⣠⠦⢀⡀⢴⣦⣹⣇⠼⠇
⠀⠀⠀⠀⠀⣰⡾⢉⣽⣿⣷⣶⡾⣽⣴⡶⣿⣿⣧⣿⠿⠋⠀⠀⠀⣿⣶⣾⣿⡀⡛⠛⠿⣀
⠀⠀⠀⠀⠇⠙⠃⡐⠻⣿⠛⢻⣿⣏⡉⣥⢁⡐⣴⣅⠸⣧⣤⡤⠀⠙⣻⠿⠿⠷⠿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣳⣿⣿⣷⣥⣿⣿⣿⣿⣿⣿⣿⣕⣌⠙⠓⡠⣤⡀⠀⢠⡀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢯⣭⣥⣿⢿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣷⡿⣻⣿⣷⣦⢇⠀⠘⠓
⠀⠀⠀⠀⠀⠀⠀⢠⡼⡯⣝⡿⠛⠉⣯⣿⣿⣿⣾⣽⡶⢳⣾⣿⢝⣿⣿⣿⣻⣻⠿⣷⡀⣀⣈⢠
⠀⠀⠀⠀⠀⠀⠀⡀⡤⢶⢠⣀⣤⣮⢰⣿⣿⣿⣿⣿⣿⣶⣿⣿⡂⣿⣿⣟⣟⣈⡁⠙⢿⣿⡽⠡⡀
⠀⠀⠀⠒⣫⡷⣾⢸⣋⣴⠾⢿⣿⠿⠛⢉⠛⣿⣿⣿⣿⣿⣿⡷⢦⣛⣯⠜⣿⠛⣽⣷⣀⣉⠁
⠀⠀⠀⠀⠴⠞⢳⠞⢛⡷⠗⠈⡟⠀⢁⣤⠉⠟⢿⣿⣿⣿⣟⠁⡈⠉⠀⠀⠋⠈⡿⠀⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⡤⠀⠀⠀⠘⠀⠄⠄⠀⣿⣿⡿⠃⠀⠀⠀⠀⠀⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣾⣿⣿⣿⣷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡲⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡷⡜⢀⠀⠀⠶⠃⠁⢠⠀⠀⠻⠆⠀⢀⠘⡴⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠸⠆⢀⠛⠄⠷⣠⠐⠀⢸⢀⠐⢠⡰⠇⡜⢃⠀⠾⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠤⡷⠀⠛⣢⡘⡂⠀⣌⠃⠀⡀⡀⣰⢼⡌⠔⡀⠀⠀⢋⡄⠀⡚⣠⡚⠃⠐⣦⠄
⠀⠀⠀⠀⠀⠀⠈⠛⣠⠄⢀⠡⣀⠀⡀⢙⣆⠀⠘⡃⠁⣸⠃⡀⡛⠀⢀⣞⠁⡀⢀⡠⢁⠀⢤⡘⠋
⠀⠀⠀⠀⠀⠀⣠⡄⢠⡤⠘⠛⠉⢇⠈⢲⡁⠹⢶⡇⠀⣿⠀⢠⣷⠾⠁⣱⠊⢀⠏⠙⠛⠠⣤⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠅⠀⠠⠀⢶⡖⠾⢧⣘⠋⠛⠢⢽⣆⣿⣠⣾⠥⠚⠋⢛⡠⠿⠖⣶⠆⠠⠀⠀⠍
⠀⠀⠀⠀⢀⡆⠀⠚⠢⡠⢔⡒⠳⠦⣤⢬⣽⣿⣶⣶⣿⣿⣿⣷⣶⣾⣿⣭⢤⡤⠶⠓⣒⠤⠠⠚⠂⠀⣄
⠀⠀⠀⠀⠈⠱⠶⡄⠀⠀⠈⢹⡄⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣼⠉⠀⠀⠀⡴⠶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⠴⠶⠾⠿⠿⠿⠿⠿⠷⠶⠦⣤⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠀⠈⣇⠀⠀⡅⠠⠄⠀⠆⠀⠀⠀⠀⢆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⣆⡴⠃⠀⠀⠎⠀⠀⠀⠠⠀⠀⡏⠀⡠⢇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⡸⠀⠁⠀⠀⠀⠈⠒⢇⠀⠀⠈⠢⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣈⡇⠀⠀⠀⠀⠀⠀⢄⢸⡇⢸⣰⠃⢃⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⡀⠀⠀⠀⢸⡇⠀⣼⠅⠀⠀⠀⢀⠀⠀⠀⢸⡇⣸⠏⣀⡆⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠛⢻⣧⠞⠁⠀⠀⠀⠀⠀⢣⡀⠀⢸⣧⠟⠉⠀⣈⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢾⣿⠀⠀⠈⠠⣀⠀⠀⣼⡇⠀⣼⡟⠛⠛⠫⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⡀⠀⠀⠀⢀⠀⠀⢸⣿⠀⠄⠀⠀⣸⠀⣠⣛⣇⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⠈⠙⠓⣶⠶⠶⣷⣄⣸⣿⠀⠈⢢⡴⠛⠛⠁⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠑⠦⣀⡨⠶⠋⠁⢀⠠⠈⠉⠻⣿⣆⢠⡿⠀⠀⠀⣠⣿⣿⣶⢶⣄⣀⠠⠄⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠈⣷⣿⠏⠀⠀⠀⠉⠛⠛⢶⣒⣒⢛⠓⠒⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡄⣼⣿⠏⠀⠀⢠⠀⠤⠀⠀⠀⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⠀⢀⡴⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡥⠖⠋⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇
⢀⣀⣸⣄⠀⠀⠀⢀⣤⣤⣴⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⡆⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⣄⢀⡄⠀⠀⠀⠀⠀⠀⢀⣠⣀⣠
⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣶⣶⣶⣶⣷⣾⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⣤⡀⢀⣤⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣀⠸⠿⠀⢿⣿⠺⢿⡶⢀⡄⣿⠇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⣿⣾⢛⠹⠷⠞⠀⢰⣦⡀⣴⡇⢿⠇⠀⢸⠟⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⠉⣤⡉⣿⠇⠀⣷⣦⢠⣝⠃⡙⠁⣠⣴⠆⣀⢼⡿⠇⢀⡄⢠⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣜⠃⣶⡠⢰⣬⣄⠈⣷⠄⡙⢀⡟⢣⣾⣶⠙⠋⣾⡿⢀⣴⣇⢻⠟⠛⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣤⡉⠉⠀⠉⠁⠳⣍⠙⣇⣠⠊⣴⡄⠶⣶⢧⣸⣴⣦⠉⠀⠘⢿⡣⠔⠀⠿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣃⣙⣿⠗⢾⡛⠋⠻⢾⣿⡄⠛⠀⠀⣤⣤⣹⣇⣹⣴⠗⠫⣿⣶⠆⠀⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣦⡜⠛⣋⡀⢈⣭⣝⠲⣾⣿⣋⢻⣇⠙⠛⠢⣉⣹⡿⢛⣭⣽⣓⠒⢰⡾⠀⣶⠶⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣭⢀⡘⢿⣆⡭⠴⣾⣿⣍⠽⠿⣿⣷⣤⣤⣾⣿⣷⣮⣭⡭⠁⠀⣶⡿⠃⠀⠘⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠏⠠⡶⠀⢀⡀⢺⣷⠀⠀⠈⢿⣿⣿⠏⠀⠀⠀⠙⠍⠛⣻⣯⣅⠀⠘⠿⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠈⠁⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢰⣌⠙⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠘⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⢻⣿⢿⣿⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠒⠖⣛⠿⢛⢿⡥⡴⠋⣹⡈⢯⡛⢤⣘⠿⣛⡛⠛⠛⠒⠲⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠸⡀⠀⠃⠈⠂⠉⠑⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣤⢸⣷⠀⠀⠀⣤⠀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣬⣿⣤⠉⣾⢰⣆⠻⣚⣛⠁⠀⢴⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⣀⣻⣿⡆⣶⠉⠓⣶⢣⡘⢿⢢⡭⢁⡀⣿⣌⢱⣿⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡘⣿⡇⢉⠩⣤⣤⡈⠀⣧⢉⡋⢻⡞⣰⣾⡿⢀⢿⣿⢈⡁⣰⡄⣶⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣧⣬⡀⠟⣶⠀⢭⣬⡶⣤⡸⢳⡎⣷⢛⡋⠀⢸⣷⠀⢛⡁⣿⣇⡉⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢤⣤⣄⡀⣦⡀⠠⣭⣤⡤⡉⣾⠀⢹⡇⠨⣤⣪⡁⣿⠛⢡⣦⠷⢈⣀⡍⣁⣼⣋⣛⠛⠛⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡗⠙⢹⡇⡀⠉⠁⢘⢦⡼⢻⣰⣷⠨⣥⡙⣿⢠⡞⢿⣴⣣⠴⢺⡅⠈⠛⠿⠓⠀⢸⣿⣀⣴⠂
⠀⠀⠀⠀⠀⣀⠀⠀⠈⠛⠻⠃⠠⡼⠇⠸⠆⣿⡆⡙⢾⣜⠁⢶⣌⠁⣿⢈⣷⠿⣷⣤⡛⢂⠀⣜⢣⡄⢰⣦⡘⠟⣩⣵⠆
⠀⠀⠀⠀⠀⠸⣷⡄⣦⠀⣌⠛⠃⢹⡍⣛⣇⡪⣹⠇⣶⡙⣷⡜⠿⢶⣿⣿⠃⢠⡌⢉⠉⢸⣷⣉⠴⢛⢈⠿⡷⠘⣛⡋
⠀⠀⠀⠀⠀⠰⢾⡄⢉⠰⣿⡆⣷⣄⢷⡈⠉⢤⡬⠓⠾⣤⣘⣿⣆⣼⣿⡏⣀⠳⠕⣫⡄⣦⠙⡿⢚⣋⡘⢇⡜⢙⣋⠁
⠀⠀⠀⠀⠀⠰⠾⠿⠗⠑⢮⣄⣙⠿⠈⢳⣄⣀⣚⣛⡛⠀⠹⣿⣿⣿⣿⢣⣼⠷⡞⣩⡤⣉⡾⢳⡿⢟⣥⠟⣴⣿⢏⣾⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⣃⣤⣭⡝⢛⣛⣛⢋⣹⠿⠿⠿⣶⣼⣿⣿⣿⠈⠿⣴⣧⣽⣿⣿⠷⠶⠶⠿⠧⠤⠽⢥⢠⣅⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠛⣠⣤⣉⠝⣭⠸⡆⠀⠀⠘⣿⣿⣿⣿⠀⣴⠟⠛⠉⢩⡍⣙⠲⣟⣧⡄⢾⣷⡌⠓⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣶⡆⢿⡇⠁⠀⠀⠀⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠛⢸⣿⢀⡓⠈⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠇⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⡿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠿⠿⠟⠛⠛⠛⠛⠿⠿⠿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠛⠉⠁⠀⠀⠀⠀⠉⠙⠛⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡆⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠛⠛⠛⠻⠿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⠁⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠃⠀⠀⢀⣠⣤⣤⣤⣤⣤⣶⣶⡿⠿⠛⠁⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠀⠀⢠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠁⠀⠀⢀⣾⡿⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣤⣀⣀⣀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟
⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣦⣤⣤⣄⣀⣀⣀⣤⣤⣴⣶⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣠⣶⣶⣶⣖⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⢤⣀⣴⣿⣿⣿⣿⣿⣯⣿⣿⣿⣻⣿⣆⣄⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣀⡄⣤⠼⣿⣾⣿⣟⣿⣾⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀
⠀⠀⠀⠀⢀⣠⣠⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣿⣶
⠀⠀⠀⢰⣾⠿⣿⢿⣿⣿⣞⣳⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣷⣛⠿⠛
⠀⠀⠀⠸⢿⣿⣿⣿⣿⣽⣿⢯⣯⣯⣿⣿⣩⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⣙⣛⣿⣿⣿⣶⣽⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣻⣿⣯⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡿⠟⠉
⠀⠀⠀⠀⢹⣾⣿⣿⢿⣿⣿⣿⡿⠿⣧⣿⡏⢻⣽⣿⣹⣿⡿⠋⢉⡽⣿⣟⣿⣿⣿⣾⣿⢿⡆
⠀⠀⠀⠀⠀⠀⠻⣿⣾⣷⣿⣿⠝⠒⠚⠛⢷⣾⣻⡿⣹⣿⣽⠟⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠀⠀⠀⠀⠀⠈⠏⠫⣷⣿⣿⡟⠀⠀⠀⠀⠀⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢰⣟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡀⣿⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠿⠿⠿⠧⠿⠿⠿⠿⠿⠶⠶⠦
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣥⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠉⣀⣠⣤⣤⣤⣄⡀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢠⣾⣿⡿⢿⣿⢿⡿⣻⢿⣷⣦⡀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⢰⣿⣿⣷⣿⣥⣂⢕⢌⠪⡫⠙⢻⣷⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⣿⣿⣿⣿⠟⠙⠻⢷⣕⠀⠀⠀⠀⣸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢸⣿⣿⡟⠁⠀⠀⠂⠀⠈⠂⠀⠀⠀⣻⡇⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠐⣿⣿⣿⣄⠀⡈⠀⠀⠀⠀⠀⠀⠀⣸⠃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠹⣿⣿⣿⡏⠳⡄⠀⠀⠀⠀⠀⠀⢻⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡅⠀⢹⡄⠀⠀⠰⡄⠀⣿⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠈⡇⠀⠀⠀⡇⠀⡿⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣟⠁⠀⠀⠠⡇⠀⠀⢠⡇⠀⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⣿⣿⡟⠣⢀⣠⣴⠾⠁⠀⢀⠞⠀⡀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠘⣿⠿⠷⠗⠛⠉⠁⠀⠀⠊⢀⡤⠂⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢿⣷⡀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⢧⣉⠀⣀⡤⠞⠉⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠈⠉⠁⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⢀⣠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠷⠿⠿⠛⠉

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠇⠀⠀⣀⠀⢰
⠀⠀⠀⢀⣀⠀⢩⢦⣀⣀⠀⠀⣿⡆⠘⣧⣄⡀⠀⠀⠀⡀⢀⡀
⠀⠀⢰⠏⠉⣳⡁⠀⣠⠁⠀⠀⢿⣃⠀⢙⠿⠿⣷⣆⣂⣀⠒⠀⠀⠀⢀⡀⢀⣀
⠀⠀⢸⠀⠀⠾⢿⡀⠅⠀⠀⠂⠸⣿⡆⠈⠻⣾⣞⡏⣭⡻⢿⠶⠶⠎⠁⢰
⠀⠀⢸⠀⠀⠉⢩⣥⠀⡈⠀⠈⠀⠙⠳⠄⠀⠈⠚⠱⢟⠹⣶⠦⣴⣦⣄⠘⢄⣀⠀⣀⣐⣲⣦⡀⠀⡀
⠀⠀⠘⠄⠀⣴⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠶⠶⣄⣐⣂⣠⣸⡷⣿⣿⣿⣿⡿⢹⡿⣋⠤⢌⠙⣧⠀⠐⠂
⠀⠀⠀⠀⠰⣿⣿⡿⠻⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⡿⠛⠛⠻⣿⡾⣿⣿⣿⣷⣤⡹⣷⣴⣦⣇⠃
⠀⠀⠀⠀⠀⠹⣿⡃⠀⠘⣿⠿⠛⠉⠋⠉⠀⠀⠸⠛⠒⠒⠒⠬⣷⠿⠿⠿⢿⣿⡍⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⡿⢄⠐⡎⠐⠛⠉⠀⠀⠀⠘⠀⠀⠂⠀⣠⡆⡂⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠂⠀⠐⠀⠀⠀⠰⠆⠀⠾⣿⡏⠀⠀⠀⠀⠈⢿⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠙⢆⢄⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠂⠘⠁⠙⣦⡀⠀⠀⠀⢀⣅⡀
⠀⠀⠀⠀⠀⠐⠄⢀⡀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡴⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣡⠾⠿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢸⣿⡿⡂⣦⣤
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠆⠀⠀⠲⠘⡇⠀⠈⢿⣷⡄⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠘⡇⠀⢀⠈⢿⣷⣿⣿⣿⡄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⣿⣿⠟⠉⠁⠠⢤⣤⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠻⠟⠋⠁⠀⠀⣼⣿⣦⠀⠢⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⡅⠀⡠⢾⣄⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⢞⣧⣠⣄⡀⠉⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⡿⠛⠋⠀⠀⠀⢀⣀⣠⠴⠶⠖⠢⠙⣿⣶⡾⠘⡁⢰⢹⡆⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇⠀⠀⠀⢲⣶⣏⠁⣀⣀⣀⣀⡀⢀⣑⣌⡻⠦⠈⣥⠾⢿⣆⠀⠲⣦⠀⠰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⢠⢸⣿⠟⠉⠹⠯⠙⠁⡙⠃⠀⢘⣺⣿⣿⣟⢛⡶⠈⢻⣖⡚⠓⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠞⠁⠀⠀⠀⠀⠀⠂⠀⠄⠀⠙⠋⠉⠁⠈⣉⡻⠀⠀⠙⠫⡮⣔⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣉⣁⡀⢀⣀⠀⢹⣮⠛⠮⠷⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠉⠀⠀⠀⠀⠀⠈⠙⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡿⠀⠀⠀⠀⢠⣶⣿⣿⣯⣭⣻⣷⣤⣄⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⣰⣿⠿⠋⠉⠛⠟⠗⢱⣾⣿⣷⣄⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠃⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⣿⡆⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠋⠀⠀⠀⠀⠀⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⢿⣷⡄⠀⠀⠀⠀⢀⣾⣿⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠈⢿⣿⡄⠀⠀⠀⣸⣿⡿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⢸⣿⣷⠀⠀⢰⣿⣿⠃⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣀⠄⠈⣿⣿⠀⠀⣸⣿⠃⠀⠀⣼⡇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⡰⡸⣻⡏⠀⢀⣿⠃⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠻⠿⢋⡖⠀⣰⡛⠁⠀⣼⠃⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⢀⣬⡽⠾⠟⠋⠀⠀⣼⠃⠀⣠⣿⡿⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⢠⡟⠋⠀⠀⠀⠀⠀⠐⠃⠀⣴⣿⣿⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣶⣦⣤⣤⣴⣶⣿⠧
⠀⠀⠀⠀⠀⠀⠀⢜⡟⡿⢫⠟⠁⠩⢛⡻⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠴⣣⠞⠁⠁

⡀⣀⢀⣀⢀⡀⣀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⡀⡀⢀⢀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣤⠀⢰⣦⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡙⠛⠃⠘⡿⠀⣾⣿⠃⠀⢀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⠛⠁⠀⣶⡄⠈⢢⡉⠀⠀⠀⣟⡰⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣄⡀⠀⠸⡇⠀⠀⠀⠁⢢⠀⣧⡴⠚⠋⠀⠀⠰⣷⣞⠿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⠐⢤⡃⠀⣴⡄⠀⠀⣷⠏⢠⡶⠀⠀⢀⣀⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⢧⡀⢿⠇⠀⢸⡿⠊⠀⠀⢀⠰⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠗⠐⠊⣩⡍⠉⠛⣶⣤⡀⢸⡇⠀⢀⣠⠾⠒⢒⣀⠀⠴⣿⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠂⣀⣉⣀⣀⠞⠁⠈⢻⣾⣿⣴⣯⠥⣤⠄⠈⠻⠷⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠉⠁⠀⠀⢠⣄⠀⠀⢻⣿⣿⢀⣀⠀⠀⠀⣤⠲⠦⠈⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠈⠀⠀⢸⣿⡇⠈⠉⣷⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣀⣤⣄⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣾⣿⣿⡿⣻⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⡽⠋⠐⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⡀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡿⢿⣿⣿⡇⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣥⡇⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⠤⢄⣀⣀⣠⣴⣾⣿⠟⠁⠀⢀⣾⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⣲⡷⠷⢿⣿⣿⣿⠟⠋⠉⠀⠀⠀⣠⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⢰⠟⠁⠀⣼⣿⣿⣿⠃⠀⠀⠀⣤⠾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡎⠀⠀⣼⣿⠿⠟⠁⠀⢀⡄⢞⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⡃⠀⠐⠛⠁⠀⠀⣀⣀⣤⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠸⡆⠀⠀⠀⣠⣾⠿⠋⠉⠉⠛⢿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⡄⠀⠀⢿⡀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢹⣀
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢧⡀⠀⠀⠻⢄⠀⡆⠀⠀⠀⠀⠀⠀⠀⠠⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠘⣷⣤⣀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠻⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡠⠐⠂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⢀⣠⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠛⠒⠉⠉

⡀⣀⢀⣀⢀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⠉⢆⠀⠀⠀⠀⡀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣆⠸⡄⠀⠀⠠⡇⠆⣿⣧⣏⣠⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⢷⣠⠐⣰⠠⢰⣿⣿⣿⣿⣿⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⢿⡇⡠⣙⠡⢸⠈⠈⠿⣿⣿⣿⣿⣿⣿⣇⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⣼⣷⡟⠈⡠⠨⠴⠘⢗⡈⡩⠉⣛⢻⣿⣿⣷⣄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠰⠛⣿⣄⢸⢾⢠⣊⣥⠀⢥⢠⠀⣠⠿⢿⣿⣿⣿⣄⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⢀⣶⣿⡿⠛⢚⠭⠉⣮⡭⣤⣬⣼⣿⣟⣓⣩⢿⡿⠿⣧⣀⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⢾⣿⡿⣯⣄⣄⣀⣴⣿⣿⣿⣿⣿⣿⣇⠹⣮⠹⣄⡹⢸⣿⡐⠜⣂⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡘⢿⡀⠈⣿⣿⣿⣿⣿⣿⣟⢛⣻⢿⠿⠓⣛⣳⣶⣶⡻⢿⣷⡔⢦⣤⣤⢀⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢻⣷⡄⣿⣘⣿⣿⣿⠟⢁⣺⣷⣿⡾⣟⣿⣿⣿⡟⢻⣿⣿⡴⣘⣷⣄⠻⡿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡙⠻⠿⠛⠁⠀⠨⠟⠛⠉⣽⣿⣼⡿⠉⠀⠘⣍⣿⡃⠙⢿⣿⣧⣑⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠘⠃⠁⡀⠀⠁⠀⠀⠀⠄⠀⢹⠀⣿⠁⠀⠀⠄⠈⠩⠀⠀⠄⠹⣿⣾⠳⣌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠰⠅⠸⠀⠀⠀⠀⠀⠸⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠠⣤⢀⡀⠉⠤⠈⡿⠴⢤⠀⠀⢄⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⢐⣛⡣⠀⢋⠛⠈⣡⡄⡔⠒⠊⢠⣐⢎⡉⢁⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣭⠀⣭⣉⢀⠣⠆⢀⣰⠀⠒⠁⠂⣦⡀⣚⣁⠜⠃⠾⠃⣦⣀
⠀⠀⠀⠀⠀⠀⢠⣤⣤⠆⡂⠩⢗⡉⠛⢠⡌⠻⢼⡩⠧⣽⠀⠘⠀⠿⠀⢤⡉⠈⡆⣙⠁⢷⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠹⢢⣅⢠⢨⠛⠖⡄⠰⠆⢺⠛⣶⣭⡓⣤⡃⣦⣻⠞⣁⠨⠶⢈⠷⠀⣄⣇⠤
⠀⠀⠀⠀⠀⠀⣤⡴⠘⢂⣑⠚⣄⠥⣤⢀⣤⢖⡻⣟⠰⡀⠀⣶⢶⡞⢱⣄⣿⣘⣦⣊⣒⠇⢠⡌⠔⠃
⠀⠀⠀⠀⠀⠠⠾⣤⠘⠿⢧⠇⢈⡱⢦⣌⡬⢻⡁⢻⣄⣴⠊⠷⣼⢠⡞⢭⢀⠖⣤⠐⢀⣵⡊⠁⡆⡒⠶
⠀⠀⠀⠀⠀⠀⢀⣀⠋⠴⠀⣂⡘⠁⣛⣉⢷⡖⠘⡙⣿⡑⠄⡽⣣⡟⡰⠦⣼⠳⠀⣶⢀⡘⣥⠰⠇
⠀⠀⠀⠀⠀⠀⠠⠝⢡⠤⣃⠤⣠⠴⠶⢤⣩⡟⢾⣅⢸⣧⣾⣵⠟⢸⣷⣎⡬⣕⠂⣽⣦⠉⡀⠠⠜⣤⡀
⠀⠀⠀⠀⠀⠀⠸⠇⣲⣀⠘⠂⣈⠋⣰⢿⣮⠳⣦⡙⣿⣿⣿⣷⡿⣿⡉⣱⡦⠝⢃⡄⢍⡚⢋⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠨⠌⢁⠱⠀⣧⠡⠀⣀⠈⢻⣾⣿⣿⡏⠀⠈⠃⢀⡰⠒⠀⠠⡌⠑⢇⡐⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠃⠀⣅⠷⠀⠰⡶⡉⠀⠀⢻⣿⣿⠀⠀⠀⠀⠨⠤⢀⣄⠃⠀⠠⠾⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠉⠀⠀⠘⠁⠙⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠿⠟⠛⠛⠛⠛⠛⠻⠿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⣠⣴⣶⣿⠿⠿⠿⢿⣶⣦⣀⠀⠀⠈⢻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⢠⣾⣿⠟⠁⣀⠀⠀⠀⠀⠈⠙⢿⣷⡄⠀⠀⠹⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⢠⣿⢫⡟⣠⣾⠃⠀⠀⠀⠀⠀⠀⢀⠙⣿⣦⠀⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣼⠟⣻⣧⢴⡇⠀⠀⠀⠀⠀⠀⢧⠈⣿⡆⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠛⠁⠾⠄⠒⠀⠀⠀⠀⠀⠸⣇⠘⣿⡀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠻⢿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠀⠀⠀⠻⣄⠀⠈⠉⠻⣿⣦⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⡀⠀⠈⠳⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⢸⢁⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠷⢤⣀⡀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⡎⢸⡇⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠉⠉⠀⠀⠀⠀⢻⡄⠀⠀⠀⠁⣿⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣸⠃⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⡠⠂⡼⠀⠀⠀⣰⠋⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠁⠀⢀⣀⣀⣀⣠⣤⠴⠚⠁⠈⠀⠀⠀⡴⠃⠀⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣶⡾⠿⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠋⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⠀⠀⢀⣀⣠⣤⠾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⡿⢿⣿⠿⢿⣿⣿⣿⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣤⣦⣽⣤⣶⣤⣼⣦⣴⣿⣵⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠶⠛⠛⠛⠛⠛⠛⠛⠶⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣬⣶⣶⣷⣶⣶⣤⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣭⣭⣭⣭⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⠉⠑⠒⠂⠀⠀⠀⠀⠀⠠⠀⠀⠉⣉⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡵⢊⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠚⠉⠽⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣏⣴⢇⠈⡱⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠔⠒⣬⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣏⡆⡾⢡⠞⢁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠩⠐⠚⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⣿⣿⢻⠁⡞⢠⡏⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄⠀⠒⠨⠅⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣫⡴⢛⡵⠋⣼⠇⣬⠀⠃⢰⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠂⠤⠉⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣟⠛⡡⠖⠉⠀⠊⠁⠜⠁⡰⢀⠄⡀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠂⠤⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠀⠀⠄⠐⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣷⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠠⠀⠢⣀⠺⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠁⢄⠁⢸⣇⠀⠀⠀⠀⠀⢄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄⡑⢄⠑⢆⣻⡆⠀⠀⠀⢀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣄⡀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠑⢄⠹⡇⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣯⡶⢟⡿⢟⡼⢫⡟⣻⣷⣦⡀⠘⢠⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠑⢄⠹⣿
⠀⠀⠀⠀⠃⠘⠘⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣯⡛⠉⠀⠈⠀⠋⡜⢃⡿⣹⣿⣆⠘⠁⡘⠀⠆⠀⠀⠀⠀⠀⠀⠀⠰⣀⠐⢄⠰⣿
⠀⠀⠀⠀⠀⠀⠃⠐⠘⠀⢀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣉⣁⠀⠀⠀⠀⠋⠴⢋⡾⢻⣿⣄⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢲⣿⠀⠀⠀⠀⠀⠂⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠀⠀⠀⠀⠀⠙⢿⣷⡶⢄⣀⠀⠀⠀⠀⠈⠠⠊⣼⣿⣧⡈⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡈⢢⣿⠀⠀⠀⠀⠠⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠙⠻⣶⣦⡀⠀⠀⠀⠀⠀⠈⢡⡟⣹⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠢⡌⢲⡟⠀⠀⠀⠀⠀⠀⠐⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠻⣦⡄⠀⠀⠀⠀⠀⠀⠀⠃⣸⣿⣦⠀⠀⠀⠀⠀⠀⢀⡈⢣⠉⣾⡇⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠈⠀⠻⣿⣧⠀⠀⠀⠀⠀⢀⠀⠀⢳⣼⠃⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⣿⣦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠲⡄⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢄⠁⡌⢸⡏⠀⠀⠀⠀⠀⠀⠀⢨⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡙⠈⠻⢮⡑⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⠐⢠⣾⠁⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠷⠀⠀⢳⣀⡀⡉⠢⠂⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠘⢃⣾⠇⠀⠀⠀⠀⠀⡆⢸⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⣠⣾⣿⣿⡿⢏⠂⣁⣤⣦⣿⣾⣾⣷⣷⣶⣶⣵⣦⣀⠀⢀⠀⣀⣴⠟⠀⠀⠀⠀⢀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠁⠘⣿⠁
⠀⠀⠀⠀⠀⠠⠀⢀⣤⣶⣿⣿⣿⣿⢷⣉⣮⡿⠋⠉⠀⠀⠀⠀⠀⠀⢄⠀⡉⠙⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠄⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢀⠀⣸⠀⡼⠀⡃⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⠿⠋⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠃⠐⠃⡜⢁⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡆⠀⠀⢰⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣄⣾⠃
⠀⠀⠈⣣⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢰⠀⠼⠀⡀⢀⣠⠾⠋⠀⠀⠀⠀⢸⠀⠀⠀⡀⣠⡿⠁
⠀⠀⠀⣿⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⣼⠶⠛⠁⠀⠀⠀⠀⠀⠀⢸⠀⡄⢀⣷⠟
⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠸⠀⡌⢀⣥⡾⠁
⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⢀⣣⡾⠋
⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠄⠁⠀⠀⠀⠀⢀⠀⢀⠀⢀⣥⡾⠋
⣷⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢰⠀⠀⢰⡄⡆⠘⢀⡈⢀⣌⣴⠟⠉
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⡇⢸⢀⡄⠈⢀⠃⢀⣬⣶⠾⠋⠁
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣿⢀⢃⠘⡘⢁⣧⡾⠞⠛⠉
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡆⢠⠃⣘⣤⡶⠟⠛⠁
⠈⣷⣀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⣾⠀⣫⣼⡾⠛⠉
⠀⠈⢿⣄⠈⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡈⡀⢡⣤⡾⠛⠁
⠀⠀⠀⠙⢷⣤⡁⠀⠀⢠⠀⠔⢀⠀⢀⠃⡘⣠⣥⡿⠛⠁
⠀⠀⠀⠀⠀⠈⠛⠻⠶⣶⣤⣤⣬⣤⣶⠶⠟⠛⠁

⠀⠀⠀⣀⡠⠄⡀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠤⢍⣩⠩⣽⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣣⡄⠀⠀⠀⠀⣾⣷⣾⣂⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣦⡀
⠀⠀⠀⠨⠿⠞⠛⣟⣻⣿⣿⣧⡤⣀⡀⠀⠀⠒⣻⣿⡿⠅⠀⠀⣰⣾⣿⣿⣿⣿⣶⣦⠀⠀⠼⣿⣿⣟⣻⡿⣿⣿⡉⠃
⠀⠀⠀⢀⣀⣤⣴⣿⣿⣟⢿⣿⣿⣟⡿⢷⣤⣀⠈⠇⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠈⠁⢸⡏⠁⠉
⠀⠀⠐⠋⠉⠉⡬⣭⣿⠿⣟⡯⢿⣿⣷⣾⠛⠁⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠘⢃⣀⡀⣐⣦⠤
⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠈⠻⠄⠀⠀⠀⠀⠀⢠⡄⢀⣀⠀⠉⠛⠉⠉⢹⠏⠉⠙⠋⠁⠀⠀⠀⠀⢠⣿⣿⣿⣧⣿⣯⣵⡆⡻⢶⡀
⠀⠰⣿⣿⣿⣿⣤⡄⠀⠀⠀⠀⠀⣠⣤⣶⣿⢐⣬⠶⢉⡇⠀⠀⠀⠀⠀⠸⠇⠀⠀⣀⣴⣶⣦⡄⠀⠀⠋⠛⠛⢿⡻⡿⠿⠿⠃
⠀⠘⡿⣽⣿⣿⡿⠄⠀⠀⠀⢠⣿⣾⣿⣿⣏⣿⣿⣶⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣜⢻⣿⣿⡟⠛⡃⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠘⠉⣉⣉⠁⠀⠀⠀⠀⠾⠘⠋⠿⢿⣿⣿⣿⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⢯⣈⣻⣿⣿⣭⣤⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠠⣽⠦⠀⠀⠀⣠⣤⣤⣤⣤⣠⣀⡀⠈⠉⠱⡿⠉⠀⠑⠀⠠⣤⣶⣦⣤
⠀⣠⣴⣿⣿⣿⣿⣿⣋⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣏⣻⣇⠤⠄⠀⠀⠀⠀⠀⣾⡿⢿⣿⣿⣿⣷⣦⡄
⠀⣿⣯⠽⠿⠿⠾⣿⣿⣿⣿⡟⠿⣿⣿⣿⣀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣄⣀⠀⣴⢶⡾⣿⣧⣾⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠈⠀⠀⠀⠀⠿⠿⠯⠟⣿⠧⢴⣿⣿⣿⣿⣦⠀⠀⠉⠈⠉⠻⣻⣧⡿⠋⠉⠈⠉⠉⠁⠀⠀⠀⠈⠁⠈⢻⡿⢩⠋⠻⠟⠻⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⢻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠡⠄⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠢⣖⡔⠂⠀⠀⠰⣠⢆⡀⣠⡀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠎⠀⢠⠊⠉⠉⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⣼⠀⠀⢸⡞⠀⢰⡏⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠚⠀⠀⠀⠀⠀⠀⢀⡟⠀⣿⠀⠐⠞⠀⠀⡾⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠇⠜⠊⠁⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣀⠤⠶⠛⠁⢰⡷⠀⠸⠎⠀⢸⠃⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⢀⡠⣴⣖⣋⢣⣍⢀⡀⠀⣐⠀⢸⡇⠀⠀⠀⠰⠇⠀⠀⠀⠀⠇⠀⠀⢠
⠀⠀⠀⠀⠀⡀⢀⢀⡀⠀⠀⠀⡀⠀⣈⣥⣤⣾⢏⣍⢻⣿⣿⣿⡇⡈⠀⣸⠉⠀⢨⠃⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠀⠀⢸
⠀⠀⠀⠀⠐⠁⠘⠘⠀⠀⠀⣸⣈⣛⣛⠃⠉⠏⠸⣟⡙⣶⣽⣯⠑⠀⣂⡤⠀⢠⠟⠀⠀⣀⢴⢱⡿⠉⠁⢦⠀⠈⠸⠀⢸
⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠐⠛⠉⢉⣙⡛⢇⡀⡀⢠⢾⣿⣿⢿⠀⣀⣀⣠⣶⡱⣞⣀⠀⠀⠈⠘⠇⠀⠀⢘⡆⠀⡈
⠀⠀⠇⠰⠶⠂⠀⡶⠞⠂⠉⠀⠀⠀⠳⢾⡿⢀⡩⠤⣠⣼⣵⣿⣽⣛⣛⣭⣾⣻⣿⠛⠒⢶⠆⢰⠀⠀⢀⡼⡁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠐⣡⠖⠉⠀⣤⣾⣿⠟⠟⠛⠛⠋⠙⠛⠋⠀⠀⠀⠸⠀⣾⡀⡴⠁
⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⠄⠀⣀⠔⠋⠀⠀⠀⢀⣠⣬⣭⣀⣈⣀⠒⠀⠀⠒⠒⠠⢄⡀⠸⡀⠛⠛⠁⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠸⠟⢁⡠⠞⠁⠀⠀⠒⠆⠐⠟⠉⠀⠀⠀⠀⠈⠉⠉⠉⠒⠒⠢⣀⠙⠢⣅⠀⠀⠀⠀⠀⠏⢐⠂⠼⠋⠘⠷⠃
⠀⠀⠀⠀⠀⠠⠊⣠⡖⠋⠀⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⡤⠚⠁⠀⠀⠀⠉⡻⠁⢰⠀⠀⠀⡸⠀⠆
⠀⠀⠀⠀⠀⠠⣾⠋⠀⠀⠀⠀⢠⡄⠀⠀⠐⢁⡲⠁⠀⠈⠀⣠⠋⠀⠀⠀⠀⠀⠀⢴⡰⠁⣰⠃⠀⠀⢀⠇⠰
⠀⠀⠀⠀⠀⠀⠈⢆⠀⢀⠄⠀⠊⠀⠈⠃⠀⠈⠀⠠⠊⢀⠔⠁⠀⣀⡄⠀⠀⣀⢀⠄⠁⠆⡎⠀⠀⢀⠎⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⡔⠉⠀⠀⠀⠎⠐⠆⠐⠘⠞⠀⠀⡁⠱⡀⠀⢈⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠀⠐⠁⡂⠉⡇⢨⣷⣄⡀
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠸⣆⠀⠀⢀⡆⠁⠀⠃⢸⡯⣽⡿⠖⠂⠄⣀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⠘⢧⣀⠆⡇⠀⡀⠀⢸⣿⡁⠀⠀⠀⣧⡛⢤⠀⠀⠀⢀⡆
⠀⠀⠀⣼⡯⠟⠁⠁⠀⠀⠀⠈⢙⢀⡇⠀⠀⠄⠘⣿⡀⡀⠀⠹⢻⢂⡜⠄⠀⠀⢸⠃
⠀⢀⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⡄⢸⡇⠀⠀⢐⠀⠘⣿⣀⡀⠀⠆⠈⠋⠀⠀⠀⣼⠀⠀⠀⠠⡀
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠸⣷⡀⠀⢓⡀⠀⠀⠀⠀⡆⠀⠀⠀⡄⠁
⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⡇⠀⠀⣿⣷⣶⡼⠒⠀⠀⠀⢠⡃⠀⣀⠈⠇⠀⠘⠇
⢀⡀⢀⢻⡇⠀⠀⠀⠀⠀⠀⠀⡄⢸⠿⡀⠀⠀⡃⠀⠀⢿⣿⣿⣅⣤⠔⠁⠀⢨⠉⣿⣭⣹⠁⣼⡁⠀⠀⠀⠀⡀
⠀⠀⠲⣜⡇⢀⠄⠀⠀⣠⡄⠀⣠⣿⠀⠱⠀⠀⠁⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⡙⢸⢿⣶⣽⣅⠃⠘⠦⢄⣀⠀⠃
⠀⠀⠀⢠⢿⠤⣀⣾⡿⣛⣧⣼⣿⡟⠀⠀⠁⠀⠢⠀⠀⠀⠀⠂⠀⠀⠀⠀⣠⡷⣛⣚⢿⣏⣿⡄⠀⡠⠞⢩⠙⢲⢄⡀
⠀⠀⠀⢸⢸⠀⣿⣿⣿⣿⡿⠟⡻⠁⠀⠀⠀⠀⠀⠠⡀⠐⠁⠀⠀⠀⠀⠀⠻⣿⢿⡿⢟⡻⣿⣿⣯⠶⠶⣤⡀⠀⠈⡳⡒⠄⣀⡀
⠀⠀⠀⠈⢸⡆⣿⣿⣿⠙⢧⡀⣤⣀⣀⡀⠀⢀⣠⣤⣤⠶⠿⠛⠛⠋⠀⠀⠠⢌⣺⣾⣏⡾⠓⢸⠣⢂⣰⡌⣿⠈⠃⠉⠨⠀⠀⠀⠁⠒⠢⡀
⠀⠀⡀⠀⣸⣟⣸⣿⣏⣀⣀⣨⠨⣾⣿⣿⣿⡿⠿⠑⠃⠀⠒⠐⠀⠀⠀⠀⠀⠈⠃⠈⠻⣷⡄⠈⠻⠾⠞⣳⣿⢧⠤⠄⠤⠄⠤⠤⠤⣄⠀⣤⠀⠠
⠀⠀⠀⠀⡍⠝⣇⠙⢿⣦⡀⠀⣀⣠⡉⣿⣿⣿⣶⠤⡴⢒⣴⣶⣾⡶⢦⣭⣓⡠⠦⢤⠶⠝⣿⢒⢒⣒⣞⠛⠷⢭⣆⢀⠀⣙⣶⠆⠀⠀⣶⣶⣄
⠀⠀⠀⠀⠸⡀⠘⠦⠄⠈⠛⣷⡼⢿⣣⣿⣿⠟⠁⡂⠐⢸⠳⠒⠀⡑⠁⠉⠉⠉⠉⢹⢿⠋⣾⣿⣿⡿⠿⠻⢦⣄⠙⣷⣦⠤⣬⣷⣤⣀⣘⣿⡿
⠀⠀⠀⠀⠀⠱⡄⠌⢷⡦⠀⢻⣻⣾⣿⠟⠃⢠⡤⠀⠠⡢⠤⠤⢄⣘⠁⠀⠀⠀⠀⠀⣼⢿⣙⠿⠋⢹⠶⠶⣾⠁⠀⡈⢿⣿⣦⡙⢍⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠁⠀⠀⡄⣀⠀⠐⠐⠶⠡⢴⠋⠀⠀⠀⠀⠀⠈⠃⠓⠀⠀⣠⣮⠘⠋⠀⠀⢸⣀⣀⣻⡆⠀⠘⠄⠙⢿⡙⠀⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠀⡥⠀⠀⠀⠀⠸⠘⠀⠋⠂⠂⠀⠇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⡇⡇⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠐⣌⠳⡄⠐⢧⣍⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢠⡇⡃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢿⣦⣤⠽⣷⢬⡵
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢸⡁⠃⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡸⣿⣿⡖
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⢀⣆⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠴⠶⠗⠉⠤⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡙⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣶⣄⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠙⠻⢿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢀⣤⣴⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠋⠁⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠟⠋⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠈⠻⢿⣿⣿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⠀⠀⠂⠚⠉⠘⠢⠔⠃⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠊⠀⠀⠀⠀⠀⠀⣀⣴⣄⣀⡀⠀⣀⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣠⡀⠀⠀⠀⠀⢀⣸⣯⢉⡙⠁⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠂
⠀⠀⠀⢀⡀⠤⠀⠒⠂⠚⠁⠹⠛⠆⢴⣏⣲⣿⣿⣿⣯⣧⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣲⣊⣄⢰⣿⡛⠁⠈⠉⠙⠐⢣⣤⡀⣄⢚⡃⠀⢠⠌⡧⠴⠛⠀⠀⠀⣤⠀⠀⡄
⠀⢀⣀⠀⣠⣀⣠⡀⢀⠀⠲⣼⣿⣿⣿⣿⡿⢆⣙⢶⣤⣆⡸⣷⣿⣯⣿⣦⠿⠷⣦⡄⣶⣬⣥⣀⣀⣾⣷⣊⡀⠂
⠀⠀⠁⠚⠒⠛⠿⣿⣿⣷⣿⣿⣿⣿⣿⣷⡶⠿⠿⣿⣿⣿⡿⢻⣿⣽⣥⣔⣒⡋⠁⠐⠋⠉⠋⠁⢛⣿⣿⠛⠁
⠀⠀⠀⠄⠠⠀⠂⠀⠒⠝⠛⢀⠛⠀⢜⢻⣅⢿⣯⣼⣿⣻⣷⣸⣿⢿⣷⣶⣷⣶⡆⢠⣄⣤⣄⣼⠛⠉⠁⠀⢀⠉⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⢓⣦⣀⡚⠓⠙⠛⢛⣿⡿⢷⣇⡿⠋⠈⢿⡿⣿⠿⣿⡷⡿⠀⠉⠀⠀⠀⠀⣀⣈⡀⠀⠀⢀
⠀⢐⣂⡄⠀⢀⣀⡀⠀⠠⠉⢩⡭⢉⢽⡶⢦⣼⣿⡟⢁⡿⢋⡀⣸⣠⡏⢀⣤⣬⣟⣂⣄⠂⠀⠀⢠⣬⠣⠘⣫⠄⣤⣇⣩⡁
⠀⠈⢻⣶⢆⣍⠿⠖⣛⡉⣀⣮⡟⢥⣿⣧⣦⣿⣿⣧⣼⡧⣿⠛⢻⣿⢻⡏⠀⠸⠆⢴⠀⠀⠴⢡⣶⣷⠮⣿⣿⣾⣿⡶⣸⠿
⠀⠀⠀⠀⠉⠐⡿⣾⣿⣷⣶⣿⣿⣷⣾⣿⣿⣎⣿⣿⣿⣇⣿⣴⣿⣧⣾⣯⣤⣾⣧⣶⡭⠾⢗⣋⣿⣿⣿⣿⡿⣛⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠿⠛⣫⠿⣿⣿⣿⣿⠟⢿⣿⣿⣾⣿⠻⣿⢟⣿⣿⣿⣿⣷⣎⢻⡁⢠⡾⠉⠘⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⣿⣿⡿⠙⡃⠐⠀⠉⠛⠻⠟⠛⠛⠛⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⠀⢀⣀⣤⣤⣴⠶⠿⠿⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠖⠉⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣠⣾⠯⠀⢀⣠⣤⣶⣶⣶⣾⣿⣿⣿⣛⢦⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠁⢀⣴⣿⡿⢿⣽⠉⠉⠛⠻⣿⣿⣿⣷⡙⢦⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⢀⡟⠉⠩⢷⡀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡞⡄⠀⠀⠘⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⣾⠃⠀⣶⣎⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⢸⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⢨⡏⠀⠰⣿⠀⣾⣿⡧⠀⠀⠀⠀⠀⠀⢐⣿⣻⣿⣿⣿⡿⡧⠀⠀⠀⣹⡄
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣸⡿⠂⣿⣿⠃⠀⠀⠀⠀⠀⢀⣴⣿⡿⣿⣿⣿⡟⣿⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⣿⡗⢐⣿⡟⠀⠀⠀⠀⠀⠀⠈⢈⠉⠲⣾⣿⣿⣿⣿⠀⠀⠀⢘⣿⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⡻⠀⣸⡟⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠹⣿⣿⣿⣿⠂⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⣄⠀⣿⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⠀⠀⢠⣿⣿⣇
⠀⠀⠀⠀⠀⢀⣿⠀⠀⢻⡅⢸⣿⠀⠀⠀⠀⢀⣤⣶⡄⢠⢠⣄⣴⣴⣿⣿⣿⡟⠀⠀⠀⢪⠟⠁
⠀⠀⠀⠀⠀⠘⣿⡀⠀⠩⠀⣼⡗⠀⠀⠀⣶⣯⣿⣟⣼⣾⣿⣿⣿⣿⣿⣧⡏⠀⣀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⣿⡇⠀⢲⠆⣿⣇⠀⠀⠀⣿⣿⣿⣿⠋⡹⢿⣿⣿⣿⣿⣿⣷⣶⡯
⠀⠀⠀⠀⠀⠀⠹⣿⡀⠸⢣⣸⣿⠀⠀⠀⣿⣿⠏⠉⠀⣡⣾⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠿⣧⠀⢱⣿⢹⣇⠀⠀⠹⣷⣷⣧⠄⠉⢛⡍⣻⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠛⠏⡽⣦⠀⠀⣴⣿⣷⣠⣔⣶⣾⣿⣿⣿⣿⣿⣿⣿⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠐⠕⢙⢷⣀⠈⠿⣿⣿⣿⣽⣿⣿⣿⠿⣿⣿⣿⣿⣿⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠳⣝⣵⣳⣤⡈⠉⠉⠉⠁⠔⠀⠀⠀⠉⠻⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣤⡀⠙⠛⣿⣿⡶⡾⡿⠶⠀⠀⠀⠀⠀⠀⠀⠛⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣦⣀⢀⠀⠀⠔⣀⢀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⠀⠀⠀⠀⠄⠀⠀⠀⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⠶⣤⣂⣀⡴⠞⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⡆⠀⠨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⡄⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⡠⠀⡇⠀⡆⠀⠸⠃⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⡐⠁⠀⠀⠰⠀⠀⡇⠀⠀⠀⠀⢰⡟⠀⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡅⠀⠀⣄⡀⠀⢘⠀⢀⡀⠀⠀⢿⣧⠄⠀⠀⠀⠀⠇⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⡇⠉⡁⠉⠹⣶⣠⡿⠇⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡾⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⢀⡀⠊⠐⠀⠀⠈⠧⠘⠁⠘⢀⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠊⠀⠁⠀⠀⠀⣀⡠⠀⠈⠁⠀⠀⠀⠰⠾⠳⠗⠀⠈⠐⠠⠄⣀⠀⠀⣠⠃⠀⠀⠀⣷
⠀⠀⠀⠀⠀⣠⠄⠀⠈⠀⠀⠀⠀⠀⢰⠀⠀⠄⠀⢀⣴⣻⠃⠀⢸⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠶⠿⣦⡀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⡀⠀⣹⣿⣿⠀⠀⠜⢀⠀⠀⠀⢀⠄⠤⠀⠀⠀⠀⠀⠀⠀⢹⠈⠀⠀⠂⠀⠀⣼
⠀⠀⠀⠀⠚⠛⠓⠺⠶⠶⠾⠛⠋⠁⠀⠀⠁⡠⢸⣿⢻⡏⠀⢠⠾⠟⠛⠁⠀⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢀⣠⡴⠊⡡⠪⠂⠈⠙⠻⠇⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠴⢤⣤⣀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⣠⠆⠀⠀⠀⢀⡀⠀⠄⠀⠁⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢢⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⡞⠁⢠⠆⠀⠀⠀⠀⠈⣤⠤⢴⡄⠀⠀⠀⢀⡀⠀⡛⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡇⠀⡆
⠀⠀⠀⠀⠀⠀⠀⡠⠈⢀⣼⠟⠁⠠⠃⠀⠀⠀⠀⠀⠀⢹⠀⠘⢼⡀⠐⠂⢸⡇⠀⠀⡇⠙⢦⡀⢀⣠⡶⠟⠙⢷⠀⠀⠀⠀⠀⠀⠁⠀⡇
⠀⠀⠀⠀⠀⢠⠈⠀⣴⣿⠋⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠘⢀⣀⣸⠇⠀⠀⢸⠇⠀⢠⠅⠀⠀⠈⠉⠉⠀⠀⠀⠸⡄⠀⠀⢰⠀⠀⠀⢠⡇
⠀⠀⠀⢀⡔⠀⣠⣾⡟⠁⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⣾⢀⣼⡿⠃⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⢀⡴⠃
⠀⠀⠀⠈⠉⠈⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠙⠋⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢐⣶⣾⣷⣿⣿⠶⠁⡺⣿⣿⢿⣿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣹⣿⠿⢉⣉⣿⣿⣤⣬⢻⣿⡭⢄⣻⣿⣿⣿⣁⣤⡖⠆
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣿⣿⣥⣼⣿⣯⣷⣦⣾⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣽⣿⣿⣏⣩⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⠠
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣼⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣭⣿⣯⣛⣛⣩⣴⣤⣀⡀
⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⢿⣿⣿⣧
⠀⠀⠀⠈⣹⠟⠚⠛⣿⣟⢻⣲⣷⣿⣿⣿⣷⣿⣿⣿⣿⠆⣙⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣻⣦
⠀⠀⠀⠀⠀⡀⣾⣿⣿⡟⠛⢉⠺⢛⡛⢻⣿⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⢷⣼⡿
⠀⠀⠀⠀⠀⠚⠫⠋⣀⣠⣄⠴⣾⣿⣿⢀⡘⢯⣻⡟⢿⣿⣿⣿⣿⡿⢿⣿⡟⡫⣿⣿⡽⣿⣯⡳⢿⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠁⢠⣽⣿⣿⣤⠀⠀⠙⢻⣦⣿⣿⡿⠉⠃⠀⠛⠀⠀⡓⠻⡆⠈⣿⠁⠹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠉⠈⢻⠇⠀⠀⠀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣴⣀⣀⣠⣠⣤⣤⣴⣶⣿⣿⣿⣿⣶⣿⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣤⣤⡶
⠀⠀⠀⠀⠂⡞⠿⡿⢿⠟⠛⢿⣿⡿⡿⡿⠿⠋⠉⠁⠠⠬⠭⠿⠿⡿⠿⠿⢿⡿⠻⠟⠻⠻⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠴⠤⠤⠤⠤⠄⠤⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣤⣄⢠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠛⠻⡟⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣀⡻⠿⣘⡿⢫⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⣄⣠⠧⣤⡜⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⠿⠗⣼⣿⣷⣜⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣀⣸⠃⠀⠈⢧⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢾⣿⣿⣿⣿⠇⠀⢠⣶⡄⣀⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⡁⠀⠀⠀⢀⣽⣿⡟⠛⢿⠿⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡜⠿⣃⢿⠿⣁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⢧⣠⢼⣀⣠⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣣⣾⣿⣇⢾⣿⠇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠞⠁⠀⢹⡁⢀⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⡋⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠸⠟⢁⣴⡦⠀⠀⠀⠀⠀⠀⠘⠻⠟⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠖⠒⢦⠘⢋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⣸⢶⣿⠗⠀⠀⠀⠀⣤⢠⣶⠀⠀⠀⠀⠀⠠⢾⣷⣬⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡀⠀⠀⠀⢀⡀⣠⡄⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⢀⣿⣤⡅⠻⠃⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣙⡑⢟⡀⠀⠀⢸⣃⣉⠰⠶⠀⠀⠀⠀⠀⣤⢴⢀⡀⠀⠀⠈⠻⣿⡿⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⡟⠋⠀⠀⠻⣿⣿⢶⡶⠀⠀⠀⠀⠠⣶⣶⣨⡄⠀⠀⠀⠀⠈⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠋⠀⠀⣀⢠⡄⠀⠀⠈⠋⠀⠀⠀⠀⠀⢀⡤⣼⡛⠱⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣯⣌⠱⠿⠂⠀⠀⡦⠴⢂⡀⠀⠈⣿⣿⡦⠂⢶⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣀⠀⠰⣿⠇⠙⠃⣾⡿⠀⠀⢀⡤⢴⠆⠀⠙⢿⣿⡷⠾⠂⠀⠰⣿⣷⣬⡄⠀⠀⠻⠭⠺⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⢩⠀⢠⣾⣿⣿⣷⣀⣴⡄⠀⢴⣷⣦⣚⡃⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠙⠻⣿⣿⠋⠋⠀⠀⠀⠻⠋⠉⠁⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠀⠀⠀⠈⠁⢀⣤⢀⣤⠀⠀⠀⠀⠀⠰⣷⠬
⠀⠀⠀⠀⠀⠀⣶⠀⣤⠀⣨⣧⣍⠞⠀⠀⠀⣠⡼⣥⡘⢩⣤
⠀⠀⠀⠀⠈⣛⢠⣶⣿⡆⠈⠻⠛⠋⠀⠀⠀⠘⣇⣠⣯⣬⡁
⠀⠀⠀⠀⠀⠙⠙⠿⠋⠀⣠⣀⣤⠀⠀⠀⠀⠀⠈⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⡉⠾⠆
⠀⠀⠀⢶⣃⣃⠀⠀⠀⠈⢻⣿⠷⠶
⠀⠀⠀⠘⠛⠋⠀⣀⠀⠀⠀⠁
⠀⠀⡀⠄⡀⠀⢸⣤⡥
⠀⠀⢶⣶⠄⠀⠀⠈⠁








⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⠐⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠉⠀⠀⣠⣤⣶⣶⣦⣴⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣠⣾⠛⠻⠛⠛⠛⠻⠟⢿⣿⡿⣦⢄⡀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠁⢉⢝⣿⣿⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⢸⣿⣶⣖⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⠃⢽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠆⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢂⣱⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⢠⣿⣯⣷⠼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⣿⢿⣿⣿⢿⠆⠀⠀⠀⠀⠀⠀⢸⣿⡷⡛⣤⣿⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⢠⣿⠋⠉⠁⠛⠀⠀⠀⠀⠀⠀⠀⠀⠻⠧⠙⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠐⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣍⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⡘⣿⣤⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢟⠈⡠⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⠀⡀⣈⣉⡙⢋⠛⣿⢿⣿⣿⡇⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠹⣿⣿⣄⣙⠡⣌⡩⠓⢽⡾⣿⣿⣿⡇⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣶⠈⢻⣿⣿⣶⣥⡴⣱⣀⢏⢿⣿⣿⡇⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠚⠀⣹⠀⠀⣿⣿⡿⠛⢷⣾⣿⡻⣾⣿⠏⠀⠀⠀⢨⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠡⢡⣠⠖⡉⠀⠀⣿⠁⠀⠛⢫⣷⣿⡿⠿⠃⠀⠀⠀⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠑⡡⠎⡱⠀⣰⠏⠀⢀⡴⠛⠉⠁⠀⠀⠀⠀⠀⣠⣿⣿⠟⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢄⠖⠁⣼⠋⠀⠀⣾⠁⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠟⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢹⡀⠀⢀⠿⠀⠀⠀⠀⠀⠈⠛⣹⣯⡟⠁⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣾⣦⡶⣶⣾⣶⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠁

⣴⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢉⣥⠸⣿⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠸⠇⣠⣶⣦⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⢈⡻⠿⠿⢃⣍⠁⠀⠀⠀⠀⠀⠀⠀⢰⣦
⡜⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⣴⡦⠰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠰⣿⡌⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⣴⣶⣜⡋⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⢣⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣸⣿⣷⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣬⢛⣣⣶⣦
⠀⠀⠀⠀⠀⣼⣿⣿⠇⣬⣭⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⠛⠛⢋⣬⡋⠀⠀⠀⠀⠀⠀⠀⠀⢉⣥⡘⠿⠟⣛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣩⣅⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠙⠋⣵⣦⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⣿⡇⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢱⡿⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣴⡌⠛⠃

⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣄
⣁⠛⣫⣤⡄⠀⠀⠀⠀⠀⠀⠀⢠⡘⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⡍⣡⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠺⠿⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠
⣿⡇⣙⡋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢋⣌⠛⣋⠸⡗⠀⠀⠀⠀⠀⠀⠀⠰⢿⡿⠿⢣⣬⣍⠀⠀⠀⠀⠀⠀⠀⠀⠺⢇⣾⣿⣷⡝⠛⠀⠀⠀⠀⠀⠀⠀⠈⢿
⠟⠘⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠙⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠌⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⣠⡄⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⣤
⠀⠀⠀⠀⠀⣀⣉⢡⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣶⣎⢛⣁⣙⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢿⣿⣿⠇⣶⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣥⣌⠱⣾⡇
⠀⠀⠀⠀⠐⠛⣋⣈⠛⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⢨⣅⢾⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠘⠿⢂⣭⡁⣶⡌⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⢇⣤⡄
⠀⠀⠀⠀⠀⠀⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠈⠉

⢠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⡠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣈⡛⢡⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⡌⣫⣥⡀⠀⠀⠀⠀⠀⠀⠀⠸⠛⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡧⢠⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⡌
⣿⣿⡌⠉⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡿⣃⡙⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢇⣭⢉⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣫⣍⢱⣶⡍⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱
⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⣴⣶⣦⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠿⢇⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢶⡟⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣾⣷⣤⣄
⠀⠀⠀⠀⠠⣶⠜⢿⣿⠟⣥⡀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⢰⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣩⣥⣶⣄⠛⣋⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠻⣿⡿⢟⠁
⠀⠀⠀⠀⠀⠁⣾⡦⣰⣦⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⣈⡻⢿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⣤⡄⠿⠃
⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⢸⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠆⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⡇⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣤⡖⢢⠤⢀⡠⣴⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⢡⡿⣇⠛⣛⣿⢿⣿⣿⠏⠻⢻⢷⣤⣀⣵⡀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣬⡫⠩⠝⠂⠍⣄⠘⣛⣦⣶⢶⣾⣿⣿⢻⠿⠿⢿⣾⣷⣄⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠐⠘⠀⠒⣶⠤⠤⣹⣾⣷⢾⡉⣾⡵⠻⢿⣤⢠⡄⢉⣽⣿⣿⣿⣯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡀⣾⣿⠩⠉⢀⠀⠁⠐⠂⠳⡄⠠⠙⡃⠹⣿⡴⣦⡟⠻⡆⢳⣌⢽⣿⣿⣿⠥⠠⡄
⠀⠀⠀⠀⠀⠀⠀⢴⣼⣤⠄⢿⣇⣆⢠⢈⠀⠃⠀⡀⣀⠀⡀⠤⠈⣧⠀⠀⢁⢰⣆⠞⣼⠼⣽⣿⣿⣷⣿⣗⡃
⠀⠀⠀⠀⠀⠀⠀⠰⢇⣻⣶⠾⠛⠻⡶⠂⠀⠤⠀⣠⣙⡷⢂⠀⡠⢄⡐⣄⣸⣛⡿⡻⠁⠐⠢⢻⣿⢿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠓⡂⠀⠀⣀⡆⠄⣒⣠⣾⣿⣿⣿⣴⣶⣿⣿⣿⣿⣿⣟⠳⢈⣷⣾⣿⡛⠿⣟⣻⣿⡇⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣾⣯⣠⣾⣷⣿⣯⢿⠿⠿⣿⣿⣿⣿⣿⣿⢿⣿⢻⣿⢿⢿⣿⣿⡿⠿⠷⣾⣽⣿⣿⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣿⣿⣿⣿⣿⣿⣿⣛⠛⠻⣿⣄⢨⣿⣿⣥⣴⠟⣻⣿⣾⣿⣿⣿⣦⣧⣼⣿⣿⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣾⢰⠾⡟⠛⠿⠟⠛⠛⠛⠻⣿⣾⠋⣿⡏⢁⣴⣿⠟⢩⣽⣿⣿⣿⣿⣿⣿⣿⣷⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢿⠟⣹⡿⠋⠀⢾⢿⣿⣿⣿⡿⣯⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⢀⠀⢰⣿⠁⠀⠀⠀⠀⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢸⢀⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡸⢺⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⣷⡜⠿⢿⢷⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠛⠛⠛⢋⠉⠛⠛⠛⠷⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⣿⣦⣤⣤⣤⡀⠘⠃⠀⠀⠀⠀⠀⠈⠙⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠟⠉⠃⠈⠚⣿⣴⣦⣄⣀⣀⣤⣤⣄⣤⣀⡀⠈⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣻⠏⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⠙⠛⠳⠄⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠋⠁⠀⣀⣤⣾⣿⣿⣿⣿⡿⠿⠻⠻⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠏⠉⠀⣀⣴⣾⣿⣿⣿⣿⣿⣭⠟⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣷⡀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢹⡏⠀⠀⣠⣾⣿⣿⣿⢻⣇⡞⠙⠙⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇⠀⣸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡟⠀⠀⣼⣿⣿⠟⢿⡇⣎⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⡟⢠⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⣼⣿⣿⢻⡄⢸⢣⡙⠎⠁⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⠃⢺⡏⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡀⠀⢰⣿⣿⠏⡇⠹⢈⠀⠟⡀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣷⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⣿⣿⡏⢸⠃⣴⢸⢀⡄⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⣿⣿⠃⢸⡄⢸⢸⠈⠁⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠟⠋⣹⣿⣿⣿⣿⣿⣏⠛⢻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡏⢰⡇⠀⣼⣿⡇⠀⢸⡇⣿⢸⠘⠀⢀⣼⣿⣿⣿⠿⠛⠋⠁⠀⠀⣠⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣿⣿⣧⠰⢸⡇⣿⠘⠀⢀⣾⣿⣿⡟⠁⠀⠀⣀⣰⣶⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⣿⢀⢸⡇⡿⠀⠀⣼⣿⣿⠇⠀⣠⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠂⢠⣿⣿⣿⣸⢸⠀⠃⠀⢰⣿⡿⢡⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣧⣀⠀⠀⢸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⡈⣿⣿⣿⣿⠈⠀⠀⠀⣼⣿⡷⣿⣿⢿⣿⣿⣟⣿⣿⣿⢿⣿⣨⣤⣼⣿⡉⣯⣱⣾⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠘⡃⢸⣿⣿⣿⡆⠀⠀⠀⣿⣿⣷⣿⡿⢼⣿⢿⣿⣿⣽⣯⣾⣿⣿⣿⣿⡿⠃⠉⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠁⠀⣿⣿⣿⡇⠀⠀⠀⢿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⡆⠘⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠈⣧⠀⠹⣿⣷⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣌⠀⠐⣶⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠹⣿⡆⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠻⡇⠀⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠻⣿⡄⠀⠀⠀⠀⠀⠉⢛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠸⠿⠇⠈⠻⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠰⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠶⣾⠇⠈⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⢸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠙⠐⠀⠀⢀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠃⢀⡈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠆⠀⠁⠈⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠘⡁⠀⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣔⣣⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⠘⢌⣦⡀⠀⠀⠀⠀⠀⠀⠖⠀⢰⠂⣀⢰⡀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⣻⣷⣙⣿⣷⣤⣀⣀⣀⣐⣈⠴⣂⠔⣡⣦⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣯⣭⣩⣵⣾⣵⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣦⡘⢃⣤⠄⠀⢀⡄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⣶⣄⠿⢃⡀⠀⠈⠛⠘⠉⢠⡄⠸⢟⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⣙⣃⠀⡈⠉⠐⠛⠋⠀⠀⢠⠀⠀⣿⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⠁⠀⠀⠙⠲⡀⠀⠀⢀⣿⠖⠉⠀⠀⠀⠀⠀⢠⣀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡻⠂⠀⠀⠀⢠⡼⣦⠀⣼⠁⠀⣿⣆⣴⠀⠀⠀⠈⠋⠒⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠸⠃⠀⠀⠸⠛⠀⠘⣷⡿⠘⠿⠮⠙⠋⠀⠀⠀⠀⠀⠀⢠⣶⠟
⠀⠀⠀⠀⠀⢀⣤⣀⠿⡆⠀⢸⡆⠀⠀⠀⢰⣿⢀⣀⠀⠀⣿⡇⠀⠀⢀⣀⠤⠒⠂⠈⠉⠁⠀⢴⡜⠿⠆
⠀⠀⠀⠀⡀⠈⠉⣽⠶⠂⠀⠀⠁⢀⡀⠻⢷⠅⠿⠃⠀⠀⣿⣷⠀⣠⠞⠁⠀⠀⣠⡄⠀⠀⠀⠈⢁⠀⣠⡄
⠀⢀⣤⣀⠿⡆⠀⣀⠀⠐⠈⠉⠉⠉⠛⠓⠶⢦⣤⡀⠀⠀⢻⣿⣼⠃⠀⠀⠀⢷⡝⡷⠆⠀⠀⠀⠻⢎⣯⡀⠀⠰⠿⣡⣤⡀
⠀⠀⠉⣩⡶⠀⣼⠟⠇⠀⠀⣀⣀⠀⠀⡀⠤⠶⠾⢿⣷⣄⠸⣿⡿⠀⠀⠀⠀⣠⣤⣤⣤⣤⡤⠤⠤⠬⠅⠀⠀⠀⢶⡌⠉⠁
⠀⠀⠀⠉⠀⠀⠈⠀⠀⠺⢶⡌⠉⠃⣼⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠀⢀⣴⡿⠋⠉⠀⠀⠀⠀⠀⣀⢾⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣆⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠉⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⣀⣴⣶⣿⣿⡿⢛⠋⠛⢿⢿⣿⣶⣤⣀⠀⠀⠈⢿⣧⡀
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣼⣿⣿⣿⣿⠏⠀⠀⠀⠈⠂⢸⣿⣿⣿⣿⣧⡀⠀⠀⠻⣿⡄
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸⣿⣿⣿⡿⠁⠀⠀⠀⠀⢀⡰⣫⣿⣿⣿⣿⣿⣧⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢸⣿⡏⠿⠁⠀⠀⠀⠀⠀⣠⢾⣹⣿⣿⣿⡿⠟⣿⡆⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⡸⣿⠙⠀⠀⠀⠀⢀⣴⠎⣣⣽⣿⣿⡿⠟⠁⢠⣿⡇⠀⠀⠀⠀⠙⢷
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠋⠀⠀⠀⠀⣰⠛⠮⣿⠿⠋⠉⠉⠀⢀⣴⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡧⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⡟⠀⠀⠠⠀⠈
⠀⠀⠀⠀⠀⢠⣿⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⢟⡿⣻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣴⣿⢿⡻⡿⠋⣉⣴⡿⠋⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡟⡀⠀⠀⠀⢀⣾⡿⡟⡁⠈⠀⠀⠛⠁
⠀⠀⠀⠀⠀⢸⣇⠀⠀⢸⣷⠃⠀⠀⠀⣼⡧⡘⠀⠀⠀⠰⢾⣷⣷⣶⣄
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠸⣿⡼⠀⠀⠀⢿⡇⠀⠀⠀⢠⠎⠀⢿⢯⣿⣿⣧
⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⢻⣧⡁⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⣿⣸⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠻⣧⡫⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠈⢿⣯⣷⣄⠀⠀⢰⣆⣀⠰⣤⢾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⠀⠀⠀⠈⠻⣿⣇⡀⠀⢻⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡄⠀⠀⠀⠈⠻⣿⡄⠀⠙⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆⠀⠀⠀⠀⠈⠻⠀⠀⠀⠙⠿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣕⠦⣀⡀⠀⠀⠀⠀⢀⢄⣤⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣾⣭⣭⣭⣿⣶⣿⣿⡟⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⠿⢻⣻⢹⣿⡌⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣳⢻⣿⡌⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⠹⣿⡌⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⢆⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣽⠿⠲
⠀⠀⠀⠀⠀⢀⠀⣠⣤⣤⡀⠀⠀⠠⡀⠀⡇⠀⠀⠀⢿⣿⡆⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠈⣸⣿⡟⠉⠁⠀⠀⠀⠀⠁⡏⠀⠀⠀⠈⣿⣿⣴⡸⣿⣿⡆
⠀⠀⠀⠀⠀⡆⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠘⣿⣿⡧⣽⣟⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠘⣿⣷⣾⣿⡅
⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⣿⣿⣿⠁⢀⠀⠀⠀⠀⢄⣂
⠀⠀⠀⠀⠀⠀⠈⢟⠀⠀⠀⠀⠀⠀⠀⢀⡏⠱⡀⠀⠀⠀⠀⠘⠿⠟⠋⠀⠀⠀⠀⢠⣿⣯⢷⡀⠈⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⣠⡴⡇⢠⣾⡧⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣟⢿⡎⢷⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡾⢵⣶⣿⣿⡿⠇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⢛⡉⠘⢷⡶⠖⠶⣄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⣗⣿⡟⠋⠀⠈⠁⠀⠀⠀⢀⣠⣤⠤⠶⠛⠛⠋⠀⠂⠈⡽⠾⣋⡾⠂⠘⠀⠀⡰⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢟⠿⠁⠀⠀⠈⠛⢿⣿⣾⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠈⢂⣠⣾⡃⠄⠀⠀⢠⣀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠳⢦⠀⠀⠀⢀⠀⣿⣿⣶⠆⠀⠂⡴⠖⢻⡟⠒⠠⠤⠐⠒⣚⡂⣩⣥⡔⣴⣦⣌⡉⠳⢤⡀⠀⠓⣄⠀⠀⠹⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠉⠳⣄⠺⢀⣿⡟⠁⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠉⢰⡿⣿⠿⢫⡀⣙⡟⠁⠈⢻⡆⠉⠙⠓⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠈⠓⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠷⠁⠀⠀⠀⠈⣿⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠲⠟⠛⠀⠀⠀⠀⠀⠙⣄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠶⠶⠦⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠟⠀⠀⠀⠀⠀⠀⢀⣤⣴⡶⣶⣶⣤⣄⣀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡿⠀⠀⠀⠀⠀⢀⣾⣿⣻⣻⣿⣿⣶⣮⢿⡿⣿⣦⣄⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣡⣾⡿⠁⠀⠀⠀⠀⢠⣿⣿⣏⣩⠭⢛⠹⢟⡵⠧⡻⣶⣿⣿⣷⣄⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣿⢻⣿⡆⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣽⠟⠁⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣿⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠚⢰⣿⣿⡏⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠌⢿⣿⣧⠀⠀⠀⠀⠰⢀⣾⣿⣿⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⡜⣿⣿⣇⠀⠀⠀⢠⣾⣿⣿⠇⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣦⣀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⣸⡿⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⡶⡀⢻⣿⣿⠀⠀⠀⣼⣿⡟⠀⠀⠀⢠⣿⠃⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⣿⣿⡿⢡⡇⡾⢻⣿⠀⠀⢀⣿⡟⠀⠀⠀⢠⣿⡿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⣿⣿⣿⡿⣡⠔⠘⣱⣿⠇⠀⠀⣼⡟⠀⠀⠀⣠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠚⠉⣋⣴⠋⠀⣴⣋⠏⠀⠀⣰⡟⠀⠀⠀⣴⣿⣿⡇⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⢁⣬⣭⣷⠿⠟⠋⠀⠀⠀⢰⡟⠀⠀⢠⣾⣿⡿⢏⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⢠⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣠⣿⣿⢏⡶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⠇⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⡭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⢀⣤⡾⡿⠛⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢿⣿⣿⣿⣿⣻⣿⣶⣶⣶⣶⣿⣿⣿⡿⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣻⠟⣷⠟⣿⠯⠀⡉⠽⢻⣿⣿⣿⣿⠛⠋⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣣⠞⠁⠞⠁⠀⠀⠀⠈⠁⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⣴⢲⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⣬⣦⣤⢤⣍⣻⣿⣿⡶⣿⠾⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣀⠰⠦⣿⣉⠀⠐⣶⣶⣿⠳⠞⣿⣿⠿⠋⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡦⣻⣿⣧⡳⣾⣿⡞⢲⠛⡻⣿⣤⣶⡾⣤⣤⠖⠺⣿⣯⣣⣤⢠⠒⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠿⣿⣿⣿⣿⣿⣿⣶⣬⠵⢿⣿⣧⣠⣃⣽⣶⠀⠀⢘⠛⠛⢲⠿⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⢿⠻⠟⠁⡟⠻⣿⣿⣽⣷⣿⣿⣿⣿⣿⣷⣴⣾⣦⡄⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠀⠠⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⡀⣤⣤⣥⣥⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⣷⣿⣿⠛⣿⣿⠿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⣤⢀⣀⣤⡟⢿⡿⠷⡌⢤⡛⠻⢿⢿⣿⣿⣿⣿⣻⣿⣿⡿⢝⠻⠿⢣⣖⢾⢾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠛⣻⣿⣟⣿⣿⣷⣦⣾⣿⣶⣦⣼⣿⣿⡿⣷⣿⣿⠐⣀⣉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠇⠈⢋⣻⣿⠇⢿⡻⣿⣿⣿⢫⣾⣿⣿⣿⡟⠻⠉⠀⢴⣴⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⣬⣿⣿⡿⣧⢹⣿⣿⣿⣿⣵⡆⢀⣷⣼⣿⣶⣤⡐⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣄⡀⢀⢀⡛⣿⣿⣿⠹⣿⢠⣿⢿⣿⣿⣿⣇⠀⢿⣯⡍⠍⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⢿⣿⣿⣮⣭⣯⣥⡉⠀⣿⣧⣶⣄⢋⣭⣿⣍⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠹⠷⠇⢡⣿⠉⠹⡟⠀⣿⣷⠛⠷⠻⣼⣿⣻⣿⣃⣤⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⡉⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⠒⠢⠀⡈⠙⠟⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣭⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠺⠛⠃⠃
⣿⣿⠿⠿⡿⢻⠿⣿⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡀
⠀⠀⠀⣠⣶⣶⣤⡀⠀⣸⣿⣿⣿⣷
⠀⠀⠀⣿⣿⣿⣿⣿⡄⢿⣿⣿⣿⣿⠀⣀⣤⣀
⠀⠀⠀⠘⣿⣿⣿⣿⣿⠘⢿⣿⣿⠏⢰⣿⣿⣿⡆
⠀⠀⣀⣀⠈⠛⠿⠿⠃⠀⠀⠉⠀⠀⢻⣿⣿⣿⡏
⠀⣾⣿⣿⣿⣶⡄⠀⢀⣴⣿⣿⣿⣶⣌⠿⣿⠟⠁
⠀⠘⢿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⡀
⠀⠀⠀⠉⠛⠛⠉⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣠⣤⣄⠀⠐⣿⣿⣿⣿⡟⣠⣾⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠿⠛⠛⠛⠛⠉⠀⠀⠀⠀⢰⣿⣿⣿⣷⠀⠹⣿⣿⠟⢡⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠿⠿⠿⠛⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢛⣤⣶⣿⣿⣿⣦⡀⠀⣰⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠉⠻⣶⡄⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠛⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⣼⣿⢸⣿⡄⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠉⠈⠉⠻⣦⡄⠘⣿⣧⡀⠀⠀⠀⠀⢠⣿⡇⠘⣿⣧⠀⠀⠀⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠈⢻⣆⠘⢿⣿⣦⣤⣤⣶⣿⠟⠀⠀⠹⣿⣷⣄⣀⣠⣴⣿⡟⠁⢀⣠⡴⠖⠛⠓⢶⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠋⠀⣰⡟⠁⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⢈⣿⡇⠀⠀⠀⢀⣠⠴⠶⠒⠲⠶⣤⣀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⠀⠀⠀⠀⣸⣿⠇⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⣶⣤⣴⣾⣿⠏⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣿⣿⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⣡⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣄⣀⣀⣠⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡛⠿⢿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣄⣀⣀⣀⣤⣤⣶⣶⣿⣿⣿⠿⢿⣿⣷⣦⣤⣀⣀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠟⠛⠛⠋⠉⠀⠀⠀⠀⠀⠉⠛⠻⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠤⠀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣀⠀⠀⠙⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡀⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠐⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠲⠌⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠄⢅⣀⣶⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠢⠮⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠃⠓⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄
⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣄⠀⠀⠀⠀⠀⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠀⠀⠀⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢶⣏⠉⠉⠉⠉⠛⠿⣶⣄⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠸⣿⡆⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄⠀⠀⠀⢻⣿⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢠⡶⠾⠿⠿⠷⣶⣦⣄⡀⠀⢀⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣟⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⠟⠁⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⣼
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣼⡿⠁⠀⠀⢠⣿⠟⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠟⠋⠀⠀⠀⢀⣼⡿⠃⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠉⠁⠀⠀⠀⠀⣠⣶⠿⠋⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⣠⣿⠟⠁
⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⣴⣿⠏
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢀⣼⡿⠃
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⢀⣴⣿⠟⠁
⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⣰⣿⠟⠁
⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠃
⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠶⠂⠀⠀⠀⠀⠀⢀⣠⣶⣿⠟⠉
⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠛⠋⠁⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠋
⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁
⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠏
⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣠⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣾⣿⣿⣿⣿⣟⠻⡿⣿⣿⣿⣿⡿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⣿⣿⣿⢾⣿⣿⢿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⢀⣀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢽⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢀⣀⣤⣀
⠀⠘⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠋⠁⣼⠁⠀⠙⣩⠿⠛⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣛⣁⣶⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⠽⠛⠚⠹⡇⠀⠰⣇⠀⠀⢰⠇⠀⠀⣤⣾⣿⣷⣝⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭
⠀⠀⢸⡿⢿⡿⠋⢻⢻⠿⢟⡟⣿⣿⣿⠿⠿⠿⣿⣉⡀⠀⢰⠏⠀⠀⠀⠀⡇⠀⠀⠙⢦⠀⡜⠀⠀⢸⡇⠀⠀⠀⠉⠻⠶⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣓
⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠙⠿⣷⣾⠀⠀⠀⠀⠀⣧⠀⠀⠀⣸⡗⠁⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠛⠛⠿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠈⠳⣤⠀⣿⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⣄⡀⠀⢀⣿⡿⠋⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⡀⠀⣰⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠻⠿⠷⠆

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


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠛⠛⠛⠷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣦⣤⡀⠀⠀⠘⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠋⠉⠙⠛⢿⣦⡀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠀⠰⢿⣷⣄⠀⠙⢿⡄⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠈⠻⣷⡀⠈⢷⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⣀⠀⠀⠀⠀⠹⣧⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡉⠉⠛⠷⣦⠀⠀⠀⣿⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠈⠳⡄⠀⠀⠈⢳⠀⠀⡇⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡀⠈⠀⠀⠀⠀⠃⠘⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠛⠒⠂⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⣀⣀⡀⠀⠀⠀⠀⠀⣠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⣀⡴⠟⠁⠀⠉⠉⠁⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⣤⣤⣴⠿⠋


⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣻⣻⣿⣟⣻⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠛⠋⠙⠛⠓⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⢤⣄⣀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠛⠉⠀⠀⠀⠈⠉⠳⢦⡀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠠⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣤⣀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⢀⡆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠂⠀⠀⠀⠀⠀⢹⡀⠀⠀⣸⠁⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡄⠀⠀⠀⠀⠀⠀⡇⠀⢠⡏⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠁⠀⠀⠀⠀⠀⢀⢰⡇⢀⡾⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⢠⣿⣿⢠⡞⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⢠⣿⣿⡷⠋⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡇⠀⢀⣠⣶⡾⠟⠋⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⠛⠉⠁⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠛⠁⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄⡀⠀⠀⠀⠀⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣷⣴⣶⠶⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠄⠀⠰⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠈⠐⠀⠐⠈⠀⣀⠀⠰⠀⠁⠈⠀⠀⠀⠀⠀⠠⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠀⠀⠉⠀⡄⢠⠀⠠⡄⠈⠈⠂⠀⠚⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠊⠩⠀⠘⠰⠀⠄⠈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠉⠈⠒⠢⣄⣀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⠁⠀⢀⠀⠈⡐⠀⠀⠀⠀⠀⠀⠀⠈⠠⢄⠀⠀⢐⣤⡜⠂⢎⡀⠀⠈⠁⠉⡒⠰⠖⢢
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⡀⡀⠂⠀⠈⡂⠀⠀⠈⠒⡇⡇⠲⠐⠛⠓⢒⡒⡎⠬⠁⠀⠀⠀⠣
⠀⠀⠀⠀⠀⢀⢀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠤⡄⠐⠒⠀⠀⠀⠀⠀⠀⠀⠐⢄⢠⠤⡇⠁⠈⠀⠀⠀⠈⢂⠀⡇
⠀⠀⠀⠀⠀⡖⠀⠀⢀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠠⠀⠀⠆⠤⠠⠁⠀⠒⢢
⠀⠀⠀⠀⠀⠈⠀⠃⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠀⠀⠂⠄⠂⠀⠀⠀⠀⢀⢁⠀⠀⠠⠤⠀⢠⠘⠁
⠀⠀⠀⠀⠀⠀⢀⠊⡀⠀⠀⢀⠀⢠⠀⠀⠀⠀⠀⠆⠈⠀⠂⠐⡀⠠⣌⣀⣀⣀⡀⠀⠀⠀⠐⠀⠈⠠⢄⠄⠀⠱
⠀⠀⠀⠀⠀⡈⠀⠀⠀⠁⠄⠐⠀⠁⠄⠀⠀⠀⣀⢀⠤⠤⣤⠈⠑⠤⢄⠠⡅⠠⠋⢏⠈⢃⣀⣀⡀⠀⠀⠀⡀⡠⠃⠂
⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠁⠀⠀⠈⠀⢄⣠⢿⠊⣈⠤⠀⠀⠀⠀⠀⠀⠠⠘⠄⣀⣈⣇⡀⠹⢂⠉⠉⠉⠀⠀⡀⠡⠨
⠀⠀⠀⠀⠀⠀⠀⢀⢈⠒⠈⠆⢀⠀⢀⢀⠠⠊⠀⠁⡄⢀⡀⡀⠀⡀⡄⡀⡀⡄⠠⢠⠓⠆⠐⠁⢀⠒⠤⠤⠰⠄
⠀⠀⠀⠀⠀⠂⠠⢀⠀⠀⢀⠈⠉⠀⡀⠀⠀⠀⠀⢠⢈⣀⡀⢈⠉⠈⢀⠁⠯⠙⣻⢀⢀⡀⢀⣠⠀⠢⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡌⠑⠒⢀⢂⠓⡲⠀⢄⠚⠈⠁⠃⡐⠂⢅⢀⣠⠀⢌⢠⡐⠃⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡌⠀⠂⠀⠀⠀⠐⠒⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠤⠤⠴⠀⡄⠄⢄⣄⡴⠄⣄⠤⠴⢰⣆⣠⣀⡀⢠⣴⣦⡄⣆⢤⣤⢠⣤⢀⣤⣄⣄⢠⣤⣤⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠉⠈⠀⠀⠀⠀⠁⠀⠁⠀⠀⠈⠀⠀⠁⠀⠈⠀⠉

⠀⠀⠀⠀⠀⠀⢀⣤⠶⠖⠛⠛⠲⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠛⠛⠓⠦⣄
⠀⠀⠀⠀⠀⣴⠏⠁⢀⣤⣤⣤⣄⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢀⣤⡤⣤⣄⠀⠈⢷⡀
⠀⠀⠀⠀⣸⠃⠀⣴⠏⠁⠀⠀⠉⢷⡀⠈⣧⠀⠀⠀⠀⠀⣸⠃⠀⣴⠋⠀⠀⠀⠈⢳⡄⠈⣷
⠀⠀⠀⠀⣿⠀⢰⡏⠀⢠⠞⠛⠛⢿⡇⠀⢹⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢠⠞⠋⠙⢲⣧⠀⢸
⠀⠀⠀⠀⣿⠀⢸⡇⠀⠏⠀⠀⢀⣼⠇⠀⠸⠀⠀⠀⠀⠀⣿⠀⢸⡆⠀⠋⠀⠀⢀⣼⠇⠀⠸
⠀⠀⠀⠀⢹⡄⠀⢷⠀⠀⣠⣾⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢷⠀⠀⢠⣶⣿⡁
⠀⠀⠀⠀⠀⢻⡄⠈⠳⠀⠹⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⠳⠀⠘⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠖⠲⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⣀⣤⣀⠀⠀⠈⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⣠⠊⠙⠋⢀⣽⠟⠳⣄⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⢰⠁⠀⠀⣠⠋⠀⢀⡜⠈⢳⡀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣏⢠⢀⠜⠉⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠃⢡⠋⠀⠀⠀⢀⣠⣤⠴⣿⣿⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⢢⡏⠀⢀⡴⠋⠉⠡⠀⠐⠁⡺⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⣿⠀⢠⡟⢾⠁⠀⠀⢀⡤⠊⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⣿⡀⢸⣇⡄⠀⡶⠛⠁⢤⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢹⣧⠘⢿⠟⡀⠀⠀⠀⣼⠛⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⢷⡳⡌⠳⢾⡦⠀⠀⢻⣶⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⢳⣌⠢⣀⠉⠀⣀⣀⡈⠻⠟⣦⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠙⣄⣌⡂⢀⡏⠈⠙⢶⡴⠃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁


⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⣀⣀⣤⣤⣶⣶⣶⣤⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⣿⣿⣿⣿⣿⣟⣟⣻⣽⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⢛⣩⣶⣿⡿⠟⠛⠉⠩⢤⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢉⣴⣿⡟⠋⠅⠀⣀⡤⣢⡤⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠋⣰⣿⣿⣿⣮⡵⠷⠋⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⣴⡟⢭⡿⠁⣾⣟⠿⣿⣿⣧⣶⣶⣾⣿⣿⣿⣶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⢱⣿⠁⠶⣈⣭⣿⣷⣍⠻⣿⠿⠛⠛⠛⠿⢷⣄⣀⠈⠑⠦⣀
⠀⠀⠀⢀⣤⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⣿⢿⠇⢠⣾⡿⣛⣻⡾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠈⠙⣷⣠⠰⠛⣧⡀
⠀⡴⠊⢻⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣼⣿⣭⢠⣿⣯⣾⣿⣿⡟⠈⠻⣿⣿⣧⡑⢄⠀⠀⠀⠀⠀⠻⣦⠀⠘⢷⡀
⢰⢿⡀⢨⣿⣿⣿⣻⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣥⣿⢘⣿⣏⢸⣿⡟⠀⠀⠀⠈⠻⣿⣿⡦⡑⠄⡀⠀⠀⠀⠘⣧⠀⠸⣷
⠀⠀⢣⠹⡿⡿⣿⣷⣝⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⣼⠟⡟⣠⡿⠛⠸⢿⡿⣴⣿⡇⠀⠀⠀⠀⠀⠈⠻⣷⣌⡲⡁⢄⠀⠀⠀⢸⠆⠀⣿
⠀⠀⠀⠱⡄⠙⣜⢿⣿⢮⡻⣿⣿⣶⣄⡀⠀⠄⠀⠀⠀⠀⠀⢀⣴⠞⠋⡠⠓⠁⠀⠀⠀⠈⠳⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠈⠻⣿⣌⠲⡑⡀⠀⢸⣔⣣⣿
⠀⠀⠀⠀⠹⣆⠈⠈⠻⣷⣼⣽⠿⣷⣌⠙⠓⠶⠦⣤⣤⣤⣬⣭⣤⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡜⣜⢤⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⢧⡀⠀⠈⢻⣿⣗⢌⢻⣷⣄⠀⠀⠃⠠⠴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⠻⣞⢷⡄⠀⠀⠀⠀⠀⠀⠀⢿⣷⡜⡞⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠦⡀⢻⣿⣶⣅⠛⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢒⣷⣽⣮⣿⣷⣦⣤⡀⢁⠀⢰⣾⣿⣧⢸⢹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠳⠢⣿⣿⣿⣦⣰⠝⣿⣿⡷⣦⣤⣤⣤⣀⣀⠀⠒⠈⠈⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣟⣛⣿⣷⣀⣀⣸⢿⣿⡁⢨⡈⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢩⡛⠙⣿⣿⣿⣮⣬⣛⠲⢮⣿⣛⠻⠿⢿⣿⣦⡀⢀⠀⠀⠀⠀⠀⢰⣾⣿⠿⠛⠉⢉⣉⡛⠿⡏⣵⣿⢳⢃⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣼⣯⠀⠸⣿⡜⣿⣿⣟⣛⣳⡮⠭⣿⣿⣷⣦⣬⣭⣤⣤⣤⣤⣤⠤⣄⣀⣠⣀⢠⣴⣿⠿⠿⣷⣌⣉⡴⢋⣾⠇
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⣽⣿⣿⡯⠐⠘⢝⣿⣿⣿⣵⣭⠉⠉⠠⡔⢦⣬⣉⣛⣛⣳⠶⣶⣶⣶⣶⣶⣬⣝⣻⠟⠁⠀⠀⠈⠛⢋⣴⡿⠃
⠀⠀⠀⠀⣤⣠⡾⠋⠐⡀⣤⣾⣿⢿⣿⢣⡀⢸⣿⣿⣿⣿⣿⣷⠀⠀⣿⡸⠿⢿⡿⠿⠟⠻⠿⠿⠿⠿⠿⠛⠛⠿⠿⠶⠦⠤⢶⣺⠛⠋
⠀⠀⠀⠀⣼⡟⢠⣥⡾⣿⣿⣾⣿⢢⠻⣇⣷⡈⠻⣟⢿⡿⠿⠋⠀⣠⡿⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⢠⣿⢀⣠⣴⣿⣿⣿⣿⣿⣞⠇⣿⠛⢁⠀⠙⠙⠀⣀⣤⣾⣿⠁
⠀⠀⠀⠸⣿⡌⠻⢿⣿⣿⢿⠛⢁⠀⠂⠀⠃⣀⣠⣤⣶⣿⣿⠟⠙⣿
⠀⠀⠀⠀⠹⣿⣷⣦⣤⣬⣤⣤⣤⣴⣶⢾⢿⡿⠟⠛⠋⢉⠀⣠⣼⠟
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠹⣿⣤⣌⣂⣀⣰⣤⣤⣶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠙⢿⢿⣿⢟⠻⠛⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣠⠔⡧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠔⢉⣉⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⣀⣈⠉⠀⣿⣿⣿⣦⠈⠦⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣾⣿⣿⣧⠀⢿⣿⣿⣿⡇⠀⣴⣦⡀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⣿⣿⣿⡇⠈⠻⣿⣿⠇⠘⣿⣿⣿⡄⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠜⢁⣀⡈⠻⣿⣿⠇⢠⣤⣄⠁⠀⠀⠹⣿⣿⡇⢀⠇
⠀⠀⠀⠀⠀⠀⢸⠀⢼⣿⣿⡄⠀⠀⢀⣿⣿⣿⣿⣦⣄⡀⠈⠉⣀⠜
⠀⠀⠀⠀⠀⠀⠈⡀⠸⣿⣿⣿⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⣄⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠑⣄⠙⠻⠛⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠶⠀⣿⣿⣿⣿⣿⠟⠛⠉⠉⢉⣠⠜⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠎⢀⣴⣿⣿⡆⠈⠋⣁⡈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⠛⠛⠋⢀⠔⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⣴⣄⠀⣼⣿⣿⣿⠇⢠⣾⣿⣿⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣰⣿⣿⡿⠀⣿⣿⣿⠟⠀⣾⣿⣿⣿⠇⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⣿⣿⣿⠃⠀⠈⢉⣡⣄⠀⢿⣿⣿⠏⢀⣀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠈⠋⠁⢀⣠⣴⣿⣿⣿⣧⠀⠉⠁⣴⣿⣿⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⣤⣾⣿⣿⣿⣿⣿⣿⣿⡀⠀⢰⣿⣿⣿⠁⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠘⠿⠟⠁⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡈⠙⠛⠛⠻⢿⣿⣿⣿⣿⡇⠀⠤⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠢⣄⠉⠛⠿⠛⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⢷⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣦⡀⠈⣷⡀⠀⣼⣿⣿⠰⣂
⠀⠀⠀⠀⠀⠀⠀⣼⡏⢡⠀⢹⣧⠀⢿⣿⣿⡆⣿⣆⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢡⡿⠀⢸⣇⠸⣿⡄⢸⣿⣿⡇⢸⣧⣅⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⡃⠀⠛⠻⡀⢸⣷⣼⣿⣿⣯⠠⠻⣿⣷⠈⣼⣿⣿⡿⢛⠲⡄
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢀⣷⠀⢻⣿⣿⣿⣿⠌⠢⠤⠘⠀⢹⡿⡡⡇⣿⡇⣧
⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⣸⣿⣧⠀⢿⣿⣿⣿⣆⠀⠁⠀⡀⣹⠗⡷⠛⣎⣥⢉⣠⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣿⣿⣿⣷⠘⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣘⣧⣴⣮⣝⢾⣏⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣤⣿⣿⣿⣿⢿⠟⠿⠋⠋⠙⢮⡻⣿⣿⣿⣿⡿⣿⡄⠐⠂⣤⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠹⣿⣿⣿⡿⠛⠉⠀⢀⠠⠄⠀⣐⣬⣽⣷⣿⣿⡏⠙⢿⣮⡙⢦⡙⠧⣍⣳⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡀⠀⢸⣿⣏⠀⠄⣤⠚⠉⢄⣴⣾⣿⣿⣿⡿⢙⡿⣇⣀⣀⡙⢿⣦⡈⠒⢿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠂⠀⠀⢸⣿⡄⢾⡿⢹⣿⡟⢆⠀⠀⠈⠉⠻⠿⣭⡁⠀⠈⠀⠀⡟⢹⡇⠈⠈⢿⣦⣀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⠈⢻⣧⠸⡆⠀⠀⠀⠀⠠⠹⣿⠀⠐⠀⠀⠧⠸⠇⠀⠀⠀⠻⣿⣷⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⣼⣿⡀⡇⠀⠀⠀⠀⠀⠡⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣴⣾⣿⠟⠃⠇⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⡿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠉⠛⠛⠿⠿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠁⠀⠀⠀⠀⠀⣀⣤⣤⣴⣦⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⠿⠿⠿⠿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠙⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠀⠀⠀⠀⢠⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣷⡄⠀⠀⠀⠀⠈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⢠⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆⠀⠀⠀⠀⠘⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡆⠀⠀⠀⠀⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣟⠻⢿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣤⣀⠀⠀⠙⢿⣿⣷⣤⡀⠉⠛⢿⣿⣷⡄⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣦⣀⠀⠙⣿⣿⣿⣶⣄⠀⢻⣿⣿⡀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⡀⠈⢿⣿⣿⣿⠀⠈⣿⣿⡇⠀⠀⠀⣸⣿⠏⠀⠀⠀⢰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠁⣿⣿⣷⠀⠈⠻⠟⠃⠀⢰⣿⣿⠇⠀⠀⢠⣿⡟⠀⠀⠀⢠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣟⠁⠀⠀⠙⠛⢁⣠⣤⣀⣀⣀⣤⣿⣿⡟⠀⠀⢀⣿⡟⠀⠀⠀⢠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡄⠀⠀⠀⣴⣿⣿⢿⣿⣿⣿⡿⠿⠋⠀⠀⢠⣾⠏⠀⠀⠀⣠⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣆⣀⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣯⠙⠿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⢠⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣦⣤⣶⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉




⣶⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣭⣭⣭⣯⣭⣭⣤⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣷⣶⣾⣿⣷⣧⣴⣶⣾⣷⣶⣼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠳⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⢀⣤⡶⠞⠛⠋⠉⠉⠁⠀⠀⠀⠈⠉⠉⠛⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⢻⡆
⠀⢀⣼⠋⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⢿⡄
⢀⡾⠃⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠈⣷
⠛⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⢀⣀⣀⣀⣤⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⣄⣀⣉⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠈⠉⠉⠙⠛⠛⠷⠶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠉⠉⠙⠛⠷⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾⠃⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣀⣠⡾⠋⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠶⠿⠟⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⢀⡼⠏⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠐⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢀⢄⣤⡴⡦⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⡶⢲⣄⢞⠘⢺⠔⢏⣢⣷⠗⢿⠂⠀⣏⣧⢤⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⡦⡄⣥⢘⣺⠼⡲⣠⣑⡶⢤⡙⡆⢩⣺⢅⢛⠛⣘⣣⢤⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣾⣹⣿⣷⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⢯⣗⣻⡿⠰⣀⢝⣹⣿⠜⡽⣴⢳⣩⠿⠽⢟⢊⣈⡭⣷⠂⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⢿⣏⠿⣫⣿⢿⣧⣿⣿⡌⢟⣛⣛⣻⣷⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡼⣅⢙⣦⣼⢼⡴⡇⢸⡋⢡⢟⣬⡇⣤⢺⣈⣱⢹⡽⢲⠰⡄⠀⠀⣿⣿⣿⣿⣿⣿⣷⣝⠃⣿⣿⢸⣇⢾⡟⣱⣿⣿⡿⣟⣽⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢖⠢⡝⣹⠛⡾⠊⢯⡁⠑⠦⣽⡿⢠⡾⠋⢡⡗⠆⣉⣄⢁⡤⡟⠁⠀⠀⣿⣿⣿⣿⣿⡿⣿⣿⣷⡘⢿⣮⣟⠂⢠⡿⢩⣵⣾⣿⣿⡿⣻⣿⣿⣿⣿⣿
⠀⠀⠸⣖⣶⢈⠟⢺⠹⡍⠛⠷⣦⣸⣷⡾⠓⢁⣏⡙⣴⣁⣿⣻⣾⡥⠀⠀⠀⣿⣿⣿⣿⣿⣷⣯⣿⣶⣶⣦⣌⡻⣷⠸⢃⣽⣿⣿⣶⠏⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠮⣝⣶⠦⠛⠛⠒⠻⣿⣿⣧⡴⠾⠷⠚⠁⠀⠀⠈⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣮⣜⣃⣤⣤⣽⣅⡀⠀⠘⣛⣫⣭⣭⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣭⣥⣿⣿⣿⣽⡁⠀⣴⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠿⠿⠷⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣀⣀⣈⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣤⣶⡦⢄
⠀⠀⠀⣼⣿⣿⣦
⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣆⠀⠀⢀⠀⠀⠀⢀⡃⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⣇⡀
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢡⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⢏⡴⡄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠘⡆⠀⠁⠀⠠⠀⡀⠀⠉⠀⢸⡎⣻⡘⣯⠋⠀⠀⠃⠀⠀⠘⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢹⡄⠘⠃⠠⠠⠉⠐⠀⢾⣷⠶⠿⢧⣈⠀⢀⠀⣀
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠳⡀⠓⠀⠈⠀⠁⠀⡲⣴⡤⢦⢰⣿⡿⠲⣿⣗⡒⠀⠱⣄⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣨⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⣿⣦⣛⢷⣦⣄⣶⡿⠿⠿⠿⠛⠛⠛⠉
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣠⣾⣿⣿⣿⠿⠿⠁⠀⠘⢻⣿⣿⣾⣿⣿⡉⠙⠓⠯⣟⠷⠖⠦⣀⠐⠦⣴⣤⣤⡀
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⡿⠟⠛⠀⠀⣀⠤⠔⠒⠚⠋⠉⣉⣉⣉⣇⢀⣀⡀⠈⠑⢦⡀⠈⠛⠷⣎⡻⣿⣿⠄
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⡟⠁⢀⡴⠂⠁⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⡌⣿⢿⠷⠶⢦⠙⢷⣄⠀⠈⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡀⠀⠙⢿⣿⡿⠓⢀⣠⠀⠀⠀⠀⢠⣾⣿⣿⠿⠻⠛⠛⢋⠘⠉⠀⠀⠀⠀⠀⠀⠙⢤⡙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣦⣀⠀⠀⡀⠴⠟⠁⠀⠀⢀⣼⠛⠛⠻⣷⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠻⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⠀⠐⠀⠀⢀⡔⠧⠀⡀⠀⠀⠹⣧⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡌⠻⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣷⣄⠀⠀⠀⣾⠃⠈⠀⠀⠀⠀⠀⢹⣧⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣆⠀⠀⢿⡁⠈⠀⠀⠀⠀⠀⠀⠙⢷⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣦⠀⠘⡇⢀⠟⠠⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⡿⠃⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄

⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣏⣛⡙⠿⠗⠀⠀⠀⠀⠀⠀⠀⣷⠺⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡻⠷⢠⣤⡀⠀⠀⠈⠛⠁⠻
⠀⠀⠀⠀⠀⡀⠀⠐⣤⣀⠀⠀⠀⣿⣷⢸⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠰⣾⣿⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⡋⢾⣿⣿⡇
⠀⠀⠀⠀⠀⠘⢿⣷⠹⠿⠀⣀⠀⠈⠁⠻⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢸⡾⠿⣴⠆⠀⠈⠛⣥⣦⡝⠛⠃
⠀⠀⠀⠀⠙⢿⣷⠀⢰⣶⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡆⣀⣀⡀⠀⠀⠀⠀⠷⣰⣿⣶⠄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢠⣶⡾⣆
⠀⠀⠀⠀⠀⠠⣶⣶⠌⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣮⣍⡉⢿⣿⡿⠃⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠠⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠉
⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠘⢿⣿⡟⣱⣶⣮⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣸⣿⢧⣿⡆
⣟⣙⠻⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣝⢛⣰⣶⡄⠀⠀⠀⠉⠀⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡿⢧⣭⣌⣛⡋
⠿⣿⠈⠉⠀⠀⠀⠀⢀⣄⠀⠀⠀⢬⣽⡅⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡈⣶⣤⠀⠀⠀⣿⡿⣸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣸⠟⣛⠾⠆⠀⠀⠉⠶⣷⠎⠉⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣈⡻⠿⠙⠛⣀⣤⡀⠀⠀⠙⠻⠛⠉⠀⠀⠀⠀⠀⣀⡀
⣤⣤⡄⠀⠀⠀⠀⠛⢾⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⣧⡀⠀⠀⠀⠀⠈⠻⠿⠀⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿
⠿⣟⡁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⠿⠀⠀⠀⠀⠀⠀⠐⠾⠿⠂⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣦⣄
⢿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿
⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⡼⣿⢳⣿⣇⠀⠀⠀⠀⠀⠀⠀⣀⣠⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠈⠻⣿⠏⣼
⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣼⣟⢳⣶⣦⣭⣅⠀⠀⠀⠀⠀⠀⢀⣯⣬⢹⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡮⣩⣾⣿⡆⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢦⣤⣄⢿⣶⠀⠀⠀⠻⠟⣼⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠻⠿⠀⠀⠀⠀⠀⢠⣠⣄⢀⠀⠀⠲⣾⠆⣙⢿⣿⡇
⠀⠀⠀⠀⢄⣤⣽⠛⢈⣉⣤⣶⣄⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣼⢫⣭⣻⡃⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠙⣛⣀⠸⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡻⠿⠿⣧⣶⣶⡆⠀⠀⠀⠈⠉⠩⠟⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿
⠀⠀⠀⠀⠀⠘⠛⠛⠁⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣦⠙⣻⣯⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⢏⣈⡻⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⣷⡄⣤⡀
⣦⢴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⢀⣀⠀⠀⠀⠙⠿⠃⢾⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡞⠿⠹⣿⡧
⣷⣶⠨⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠗⣴⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⣾⣯⢹⣿⣷⣶⣦
⠉⠛⠀⠀⠀⠀⠀⢠⣴⣷⣠⡀⠀⠙⠿⣃⣬⡻⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣶⣦⢻⣿⠀⠀⠀⠛⠋⢾⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⣮⣡⣶⣭⡀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⠀⠀⠀⠲⣶⣮⠉⢠⣤⣶⣿⡆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣤⣶⣦
⣿⣿⠇⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣳⣿⠃⠀⠀⠀⠀⠀⢈⣭⣤⡘⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣝⡛⠛
⣽⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠁⠉⠉⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷
⣿⡛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⢸⣿⣆⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⢫

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠛⠋⠉⠉⠉⠉⠉⠛⠛⠿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⣠⣾⣿⡿⠿⠿⠿⣿⣿⣦⡀⠀⠀⠀⠀⠈⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⣼⣿⡿⠁⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠛⠁⠀⠀⠀⠀⢀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣶⣤⣶⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠉

⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⢰⢠⣶⣄⡀
⠀⠀⠀⠀⠀⣠⣤⡀⠈⢦⡀⡆⠀⠸⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⡇⠀⠀⠻⣿⣿⣿⣿⣆⡄
⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⣧⠀⠰⠀⢹⣿⣿⡿⠛⣹⣦⣄
⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢀⡟⡄⠀⠀⠘⠿⠟⠋⠀⠀⣼⣿⡆⠀⢀
⠀⠀⠀⠀⠀⢨⠀⣰⣾⣧⣾⡇⣀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⣿⣆⣤⣈⠙⢢⠄
⠀⠀⠀⠀⠀⠈⠰⣿⣿⠋⢋⣁⣀⣀⣤⡤⠶⠚⠓⠒⠭⡿⣿⡟⢻⣀⡎⣧⡁
⠀⠀⠀⠀⠀⠀⠠⠹⣯⡀⠀⣟⣿⣿⣥⡤⣤⣴⣶⣴⣆⣼⣮⣛⣲⣶⣞⡛⢧⡄⠸⣶⡀⢲⣦
⠀⠀⠀⠀⠀⠀⠀⠁⢀⠙⢷⣿⣿⠏⠀⠈⠙⠿⣈⠂⠀⠀⣿⣿⠿⠻⠿⣿⠁⠙⢿⣍⢉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⡁⢰⡏⠉⠀⠈⠓⠛⠀⠀⠀⠙⣗⢯⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⢆⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣄⠀⠀⢠⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⢾⣿⣿⣿⠇⢀⡀
⠀⠀⠀⠀⠀⠀⢀⣄⡈⢿⣿⣿⠇⠀⠈⠻⠿⠋⣴⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠀⠀⣴⣿⣿⣦⡀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠹⣿⣿⡟⢀⣼⣿⣿⣿⣿⣿⣦⣈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡄⠀⣰⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠹⣿⣿⡿⠁⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠈⠉⣡⣤⣄⠘⠛⠛⣡⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠃⢀⣼⣿⣿⣿⣷⡀⠀⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣄⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠻⠿⠿⠿⠃

⠀⠀⠀⠀⠀⡠⠖⠛⠉⠉⠙⠲⣄
⠀⠀⠀⠀⠈⢀⡴⠋⠉⠉⠓⢦⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⣒⣒⣒⣂
⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠈⣇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠈⣿⣿⣿⣧⣼⣿⠛⠶⣄⡀
⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠢⣸⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠠⢀⣹⣷⣭⣭⣾⣿⣤⡶⠋⠉⠁
⠀⠀⠀⠀⠀⠈⠈⠉⠉⠙⢦⠀⢹⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⢰⣶⡄⠀⠀⠀⢸⠂⢀⠇⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣀⣀
⠀⠀⠀⠀⢀⣿⠁⢀⣠⠔⠁⠀⠘⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠸⢧⠔⠁⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠏⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⣠⠶⠶⡶⠀⠀⡠⠃⠀⢠⠋⠉⡆⠀⢸⠀⠀⢸⠀⠀⡰⠋⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠉⠀⠀⠀⠐⠊⠉⠀⠀⠀⢸⠀⠀⢸⠀⢸⠀⠀⢸⠀⢀⡇⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠤⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⡇⠀⢸⠀⢸⠀⠀⢸⠀⢸⠀⠀⢸
⠀⠀⠀⠀⢀⡠⠴⠦⠤⠤⠶⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⠀⡇⠀⠈⡇⢸⠀⠀⢸⠀⡏⠀⠀⡏
⠀⠀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⡇⣇⠀⠀⡇⢸⠀⠀⢸⢰⠁⠀⢀⡇
⠀⠀⠐⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢣⢸⠀⠀⢣⢸⠀⠀⢸⡸⠀⠀⢸⠁
⠀⠀⠀⠢⢄⣀⣀⣀⣀⣀⣀⣀⣀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣼⠀⠀⠘⠊⠀⠀⠘⠃⠀⠀⣸⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣴⠟⠉⠀⣀
⠀⠀⠀⢀⠔⠊⠉⠓⠲⠒⠊⠉⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣠⡞⠁⠀⣠⡾⠋
⠀⠀⡐⠁⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⡼⠋
⠀⠀⠀⣠⣦⡤⡆⣀⡨⢁⣀⠸⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⣤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⢠⡾⠛⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠈⠀⠛⠛⢦⡀⠀⠀⡠⠚⠛⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠶⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠶⠦⣤⣀
⠀⠀⠀⠀⣠⠞⠉⠀⣀⣀⣀⣀⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠁⢀⣀⣠⣀⣀⠀⠉⠻⣦
⠀⠀⠀⠈⠁⠀⣴⠟⠁⠀⠀⠀⠉⠳⣄⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⣠⡾⠉⠁⠀⠈⠉⠻⣦⠀⠈⢷⡀
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠄⠘⡆⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢀⠘⣧⠀⠈⣷
⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⢹⡀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⡆⢹⡆⠀⢹⡆
⠀⠀⠀⠀⠈⣇⣀⠀⠀⠀⠀⠀⠀⠈⢳⡀⣧⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣷⠀⢸⡇
⠀⠀⠀⠀⠀⠙⠉⠙⠓⠒⠒⠲⣦⠀⠀⢱⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢏⠙⠓⠒⠒⠒⢦⡀⠀⠹⣿⡀⣸⡇
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⢻⠃⣼⠁
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⡟⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⡆⠀⠀⠀⠀⠀⡼⠀⠀⣼⠀⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣀⡴⠋⠀⠀⢠⠁⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡌⠁⠀⠀⢀⡴⠚⠁⠀⢀⠇⢸⡏
⠀⠀⠀⠀⣼⡏⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢠⠖⠛⠉⠀⠀⠀⠀⠈⢀⡿⠁
⠀⠀⠀⠀⠘⠳⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠖⠋⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠒⠠⠤⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠴⠖⠋

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

⠀⠀⠀⠀⠀⢀⡴⠊⠛⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢀⡞⢠⣶⣦⠀⢹⡆⠀⠀⣿⡇⠀⠀⠾⡄⣿⣷
⠀⠀⠀⠀⢸⠁⣾⣿⣿⡄⠀⣷⠀⠀⢸⡇⠀⠀⢳⠰⣿⣿⡄
⠀⠀⠀⠀⢸⠀⣿⣿⣿⣇⠀⠘⣦⠀⣸⡇⠀⠀⠈⢸⣿⣿⣿⣄
⠀⠀⠀⠀⡿⠀⣿⣿⣿⣿⣧⡀⠀⠙⠋⠀⠀⠀⣶⡸⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⡇⠀⣿⣿⣿⠋⢻⣧⡀⠀⠀⠀⠀⠀⠭⠳⣤⣍⠻⠿⠿⠿⠷⢆⣦⣤⣤⢤⡀
⠀⠀⠀⠀⢳⠀⣿⣿⡇⠀⣾⣿⠃⠀⠀⠀⠀⠀⣫⣭⠀⠬⠁⣨⣙⢛⠁⢀⣿⢹⣮⢻⣿⢖⡤⢀
⠀⠀⠀⠀⠠⠀⣿⣿⠀⡼⠟⢻⡀⠀⠀⠀⠀⠀⠘⢟⠬⠄⡄⢫⢘⡧⢘⡭⣿⠘⣿⡎⢿⣿⠀⠎⠷⠂⡀
⠀⠀⠀⠀⠀⡀⢸⡏⠀⠀⣠⣾⣷⠀⠀⠀⠀⠀⠀⠈⠊⠿⣐⢒⡛⢲⣏⣾⡟⣿⣚⡱⢿⣿⣷⢶⣶⣄⡘⢻⣢⣄⡀
⠀⠀⠀⠀⠀⡇⢸⡇⢠⣾⣿⣿⣿⠀⠀⠀⠀⠂⠀⠀⣺⣽⣿⣶⣶⣤⣤⣽⠎⢿⣽⠶⠦⡏⣢⢴⡈⣿⢧⠀⣽⠯⢉⣈⠐⠒
⠀⠀⠀⠀⠀⡇⢸⡇⢸⣿⣿⠏⠻⣄⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆⢠⣿⣷⣄⡙⠻⢛⣽⣟⢘⢷⡄⢽⣋⠁⠑⠀⣀⠀⢶⣶⣦⡄
⠀⠀⠀⠀⠀⢠⠘⣷⠈⢿⣿⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠙⠛⢛⡙⡉⢉⣩⠉⠛⢾⡲⡄⢿⣿⣿⣯⣤⣤⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠈⡄⢻⣧⠀⠙⢷⣄⣠⣿⣿⣿⣿⣿⣿⠿⠛⠻⠋⢀⣀⡤⣴⣶⣀⣠⣄⣩⡉⠥⢟⠙⡷⠄⠀⢻⣦⡠⠙⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⢱⠈⣿⣷⡄⠘⣿⣿⠟⣱⣿⣿⠟⠁⠀⢠⣶⣯⠉⡠⠗⡏⠂⠘⠙⠛⠛⠀⠀⡋⠉⣿⡇⠙⢄⠙⢿⣧⡈⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠈⢆⠸⣿⣿⢦⠙⠿⠿⠿⠛⠁⠀⠀⠀⠉⠉⠉⠀⠃⣾⣧⣸⡀⠀⠀⠀⠀⠀⢷⣶⡿⠁⠀⠀⠑⣄⠹⣿⠓⠤⣍⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣖⡿⠃⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢿⣷⡀⠈⠓⣤⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⣿⣷⡀⡠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡹⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⡀⠀⣠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢚⠀⣌⡈⠀⣴⣀⠄⠐⠀⠆⠀⠀⠀⠀⠀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣦⣠⣴⣆⣜⢟⣤⢌⢓⣚⣡⠄⠀⡔⠤⠥⢘⢿⣅⡀⠀⢸⢮⠁
⠀⠀⠀⠀⠀⠀⠂⣨⡙⢳⣨⢹⣷⡻⣽⠏⣿⡎⣮⠁⣹⡃⠑⡓⣇⣧⣴⣏⢣⡉⣄⡠⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢒⣾⡿⣾⢿⣿⣛⣻⣿⠿⣷⣭⣜⣿⣿⣥⠈⠬⢉⡽⣿⡾⠁⣞⣩⣴⣦⢷⡮⠁
⠀⠀⠀⠀⠀⢀⡀⢽⠍⠠⠁⣾⣔⡿⣽⣿⣿⣿⣽⣿⣟⣻⡵⣹⠾⣿⣇⣿⣿⣾⢟⣟⡭⠼⢅⠈
⠀⠀⠀⠀⢠⠀⣙⣋⣶⡅⠈⡥⣻⢆⣴⡿⣿⣿⣗⣝⣻⣹⡿⣱⣼⣝⡏⢿⡿⢋⣼⣁⢩⡄⠘⢹⢲⣄⠀⡤
⠀⠀⠀⠀⠨⡔⣉⡫⠂⠌⠢⣽⣿⣤⢉⢃⣿⣿⡻⣳⣦⢾⢝⡥⡟⣿⠇⣠⣴⡯⢿⠃⣼⢟⢴⢹⠒⢉⢀⢡⠔⢠
⠀⠀⢠⠀⣀⠫⠚⢽⣧⣳⣂⡠⣿⣿⣋⢧⣧⣽⢽⣮⣿⢸⣨⣾⡟⢁⣾⣻⣾⣾⣧⣾⠶⠬⠙⣁⣯⡶⠾⢋⡰⠘
⠀⠈⢋⢰⣎⣮⣰⣱⡄⢗⡾⣿⣟⢶⢪⣧⠹⣙⣷⡙⣾⡆⠻⡿⢠⣺⣽⣿⡟⣽⣋⡰⣌⣶⣼⣿⡧⢢⡋⣤
⠀⠀⠀⠀⠙⡿⣸⡿⡟⢶⣶⣦⣈⣉⢁⣤⠄⡈⠈⠁⡉⠂⡐⡄⠛⠻⠒⢉⣀⣥⡶⢾⡭⣴⡏⠖⢹⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠍⠗⠛⡟⢟⠀⣹⢟⢛⣿⡠⠀⠙⢡⣞⡿⠻⣿⣿⣷⢿⢻⡠⠲⠤⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠁⠀⠀⠈⠀⣳⡧⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠐⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣠⣾⡖⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣿⣾⣿⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠉⠒⢛⣾⠷⣟⣴⣿⣷⣶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠈⠛⠀⠋⠁⠠⠔⠃⠀⠈⠀⠾⠷⠏⠾⠯⠈⠛⠶⠦⠙⠿⠿⠶⠛⠲⠦⠤⠠⠤⠤