Bicycling Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⣆⣀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⠀⠀⢠⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⢤⡧
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠀⠠⢾⣿⣯⡀⠀⢸⣟⡁⠀⡀⠀⠀⠀⠀⠀⢀⣿⠙⠣⡀
⠀⠀⣄⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡧⠀⠐⠀⠀⠙⠙⠛⠀⢠⡌⠓⠀⠀⠀⡀⠀⠀⢠⣼⣿⠄⠐⡋
⠞⠛⠋⠙⢷⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⢸⡀⠒⠂⠀⠀⠀⠀⢀⣴⣿⣿⣤⣾⠇⠀⠀⠀⠁⠀⠀⠿⠿⣷⣄⠀⢼⡧⠀⠀⣤⣿⣿⣦⡄⠡
⠀⠀⠀⠀⠀⠻⣾⡁⠀⠈⠀⣰⠧⠀⠐⠀⠀⠀⢀⡜⠀⠈⢦⠀⠀⠀⠀⠉⠿⣿⣿⣿⣿⣷⣦⠄⠀⠀⠐⠀⠀⠀⠈⠉⢀⣿⠇⠢⠐⣾⣿⣃⡘⠟⠟
⣅⣰⢄⡀⠀⠀⢻⣶⣤⣠⣚⣀⠿⢿⣦⠀⠄⠀⣾⣄⢀⣄⣈⡆⠀⠀⣀⠀⠀⠘⠻⠛⣿⣿⡤⠀⠀⠀⠀⠡⡀⠀⠀⠀⣬⡟⡓⢡⣴⣟⣿⢿⣧⢺⣦
⢿⢿⡿⠧⠘⠂⠀⠉⠙⠗⠉⠈⣢⠀⢰⡄⢀⠈⡃⠉⢻⠟⢻⠟⠒⠀⠉⠡⠀⠀⠀⢀⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣾⡿⠤⠤⢹⡏⣴⠆⢁⣆⢡
⣼⣮⡀⠀⠀⠀⠘⠲⠂⡀⠀⢺⣿⠀⠈⠰⠀⠀⠰⡀⠀⢀⡞⠀⠁⠀⠀⠀⠀⠤⠄⠈⠽⠿⣶⣀⣀⠀⠀⠒⠀⣄⣤⣿⢟⡉⠘⠻⣀⣾⣷⣴⣮⣽⡷
⢿⣿⡇⠆⠆⢘⣁⠠⠀⢆⠀⠀⠀⢀⡞⠁⠀⢀⡤⠴⠒⠁⠀⠀⠀⠀⠀⠠⣆⠀⠀⠀⢤⣀⡀⠀⣀⣄⢀⠀⠀⠀⠴⣿⣷⣶⣄⠀⠘⠛⠙⠛⠛⠛⠉
⠀⠈⠉⠁⠀⠘⠁⠀⠀⣈⣀⣀⡠⠊⠀⠀⠀⠈⠐⠆⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⡀⠀⠀⠉⠀⠠⢿⣿⣿⡇⠀⠀⠀⠈⠉⢹⣤⠀⠀⠀⠀⣀⡴⠆⡀
⠸⠗⠛⠉⠀⠀⠙⠠⠀⠠⠜⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠈⢉⡀⠀⠀⠈⠉⠀⠐⣤⠤⠄⠀⠈⠉⠙⠀⠠⢤⣦⣄⠙⠧⠔⠀⠀⣠⣴⣷⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣓⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣰⠂⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠛⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠶⠊⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢦⣴⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠠⠤⠀⢀⣼⠃⠀⠐⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢷⣦⣄⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣄⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠁⠀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⡠⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠀⠀⠀⠰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠐⠒⠎⠀⠀⡀⠐⠀⣁⠠⠤⠤⢀⡉⠑⢄⡀⠀⢀⠌⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⢄⡤⠄⢀⡠⠒⠁⠀⠀⠀⠀⡊⠀⠐⢄⠑⡄⡌⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠠⠆⠁⠀⠀⠀⠀⡄⠂⠁⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠈⠆⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠨⣀⣠⣄⢤⢠⠠⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠘⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠉⠀⠀⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠓⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⡛⠀⢀⡀⠀⢈⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⢀⠠⠀⡏⠀⠈⠁⢀⠒⠀⡏⠀⢀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢻⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠅⠀⠀⠀⠠⠂⠀⣁⠀⠀⢢⡤⠖⠚⠒⠒⠒⠒⠒⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠄⡄⠀⠀⠄⢀⠀⠀⢠⢹⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢠⠉⡇⠀⠀⠀⠀⠀⠀⡇⢠⡇⠰⠋⠀⠀⠀⢀⣮⠚⡀⠀⠀⣠⠾⡟⠉⠛⠶⣄⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢡⠀⠀⡀⠤⢄⠀⡇⠈⠁⠀⠐⡀⠀⠀⡜⠁⠀⡇⠀⠟⠁⢰⠇⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠀⠰⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡞⠀⠀⢀⣷⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠘⡀⠀⣧⠁⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⡾⠀⠀⠀⡞⠙⡅⠰⢠⡆⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠙⢄⣸⠀⠀⠀⠀⠀⢀⣇⣀⡴⠀⢰⡇⠀⠀⢰⡁⠀⢻⠐⠃⢇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⢹⡀⠤⠔⠂⠈⠁⠀⠐⡄⠁⢸⠀⠀⠀⣸⠅⢀⠸⡆⠐⠀⡆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⣿⠀⠀⠀⣿⠀⠀⠀⠩⡄⢤⡄⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⡀⠠⣤⠒⠁⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢏⣀⣀⠜⠀⠀⠀⢸⡀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⡇⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠈⠳⣄⡀⠀⠀⠀⣀⡴⠞⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⠉⠙⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⠀⢀⣀⣤⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣏⢁⣴⣦⣄⡙⢿⡁⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣛⣛⣛⣛⡛⠿⠿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⢛⣯⣝⢿⣿⡿⠋⠀⠀⣰⣇⠙⢷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣾⣿⣿⣷⣧⠀⠀⠀⠺⢿⣟⠀⠈⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣌⡁⠀⠀⠀⠀⢻⣄⡀⠀⠀⠈⠳⣄⣠⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠙⠷⣄⠀⠀⠀⢘⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡿⠄⠀⠀⠀⠀⠈⠳⣶⣴⣯⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡛⠿⣿⣿⣧⡀⠀⠀⠀⠀⠀⢀⡌⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢀⣴⣻⡿⠇⠀⠐⠀⠈⠀⠀⡀⢺⣦⣶
⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣦⣤⣀⠀⠀⠀⣹⣿⣿⠿⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢹⣸⡠⣤⣤⣤⣄⡀
⠀⠀⠀⣠⡶⠟⠉⠀⠀⠀⠀⠀⠈⢙⠷⣦⡾⠟⣯⠀⠀⣶⠀⠀⠀⢀⠜⠁⠀⢀⣴⠿⣯⣿⡟⠀⠈⠉⠻⠷⣄
⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣷⡄⠸⡆⠀⣿⠄⠀⡠⠁⠀⠀⢀⣾⠋⠀⠘⣿⢷⡀⠀⠀⠀⠈⠙⢷⡀
⠀⢰⡟⠀⠀⠀⠀⠀⢀⣠⢀⡀⠀⠈⢿⣿⣿⣿⡀⣣⣀⠉⠀⠊⠀⠀⠀⠀⣸⡇⠀⠀⠀⢸⣞⣧⠀⠀⠀⠀⠀⠙⣷
⠀⢸⣇⠀⠀⠀⠀⠀⢾⡿⣾⡇⠀⠀⠀⠛⠟⢻⣦⣾⣿⣄⣠⣶⣦⣤⣀⣀⢻⡇⠀⠀⠀⠀⠻⠿⠷⠀⠀⠀⠀⠀⢻⡆
⠀⢸⣧⠀⠀⠀⠀⠀⠈⠀⠛⡷⠀⠀⠀⠀⢠⣿⠉⢿⣿⣿⠿⠿⠿⠿⠟⠉⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠰⣷⠀⠀⠀⢴⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣀⡀⠀⢀⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠉⠛⠻⠶⠶⠶⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢚⠬⠤⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢘⠄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣸⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡄⠀⡤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠀⠘⢄
⠀⠀⠀⠀⢀⡠⠤⠒⠒⠒⠒⠤⣴⠁⡈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠓⣌⠣⣀⠤⠔⠒⠒⠒⠢⠤⣀
⠀⠀⢀⠔⢉⡤⠒⠊⠉⠉⠑⠲⠄⠈⠪⡢⡈⢢⡀⠀⠀⠀⠀⠀⢰⠁⡎⠀⡨⠃⡀⢔⠂⠉⠉⠉⠒⠢⣌⠑⢄
⠀⢠⠃⡰⠃⠀⠀⠀⠀⠀⢀⡜⣠⠛⢆⠑⡜⢆⠙⢄⠀⠀⠀⢀⠇⡸⠀⡜⢠⠊⠙⢌⠣⡀⠀⠀⠀⠀⠀⠳⡀⢣
⠀⡇⢰⠁⠀⠀⠀⠀⢀⡠⢊⠔⠁⠀⠈⡆⢱⠀⠑⢄⠑⣤⣴⡞⢠⠃⢸⠀⡎⠀⠀⠈⠣⡙⢄⠀⠀⠀⠀⠀⢱⠀⡇
⠀⡅⢸⠀⠀⠀⠀⠀⠧⠴⠁⠀⠀⠀⠀⡇⢸⠀⠀⠀⠳⡈⠁⠁⣞⣀⡸⠀⠧⠐⠒⢒⣊⣁⠀⠇⠀⠀⠀⠀⢸⠀⡇
⠀⢇⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡸⠀⠀⠀⠀⠈⣄⠀⡠⠤⠰⡀⢲⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⠇
⠀⠈⢆⠙⢄⠀⠀⠀⠀⠀⠀⠀⢀⡰⠃⡰⠁⠀⠀⠀⠀⠀⠈⣟⡄⠀⠀⠱⡀⠣⣀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⠎
⠀⠀⠀⠑⢄⡉⠒⠤⠤⠤⠤⠒⢉⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠓⠛⠀⠀⠀⠈⠢⣈⠑⠢⠤⠤⠤⠔⠊⣁⠔⠃
⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣉⣉⣿⣉⣙⣛⣛⡿⠷⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢶⡛⣉⣩⡤⠖⢺⡭⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠈⠉⠉⠛⢒⣲⡄⠀⠀⠀⠀⠓⠒⢛⣉⣥⠴⠖⢚⣉⣧⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢳⡞⢻⠋⠉⠁⠀⢀⣀⡤⠴⠒⠋⣉⣡⠴⠖⠚⣩⠟⢙⡦⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡤⠴⠒⠋⣉⣠⠤⠖⠚⠉⠁⠀⠀⣠⠞⢁⡴⠋⢳⣤⣷
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣠⠞⡇⢸⡤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⣠⠞⠁⡴⠋⠀⠀⠀⢧⠸⣆⣀⣀⣀⣀
⠀⠀⠀⢀⡤⠖⠋⠉⠀⠀⠀⠀⠉⠙⠲⢤⣠⠞⢁⡴⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠊⠀⠀⠀⣀⠴⠚⡆⢻⠀⠀⠀⠉⠉⠓⠦⣀
⠀⢀⡴⠋⠀⣠⠴⣞⠋⢹⡏⠙⢳⠦⣤⠞⢁⣴⡋⠀⡇⢸⡀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠊⠀⠀⠀⢠⠞⠁⢀⣠⢿⡈⣏⣿⠙⠓⡶⢄⡀⠈⠳⡄
⢀⡞⠀⢠⠞⠁⠀⠹⡄⢸⡇⣠⣯⠞⢁⡴⡏⠀⢳⡀⡇⢸⡇⠀⠀⠀⡀⣠⠞⠁⡠⠊⠀⠀⠀⠀⣰⠃⠀⡰⠋⠈⢣⣧⠸⣿⠀⡼⠁⠀⠙⣆⠀⠘⣆
⣼⠀⢠⡏⠓⠦⣄⠀⢳⣸⣷⠟⢁⣴⣫⡤⣿⣀⣠⠧⠷⠾⠗⠒⡖⢉⣍⠑⣤⠊⠀⠀⠀⠀⠀⢠⡇⠀⣸⠓⠦⣄⣀⠹⣆⣿⡼⢁⣀⡤⠴⠚⣇⠀⢸
⡇⠀⢸⠤⠤⠤⠬⠽⣾⠉⢹⡶⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⢧⡘⣎⢣⡏⠀⠀⠀⠀⠀⠀⢸⠀⠀⣏⣀⣀⣤⡬⢽⣟⠉⣻⣭⣥⣤⡤⠤⣽
⣧⠀⠸⡆⠀⢀⣠⠖⢫⢿⢿⡛⢿⣛⠓⠒⢲⠖⠶⡶⠶⠶⠶⠶⠒⠛⣻⢶⣻⠷⠀⠀⠀⠀⠀⢸⡄⠀⢿⠀⢀⡤⠖⢫⠟⡟⣏⠉⠲⢤⣀⠀⡾⠀⢠
⠸⡄⠀⠹⡔⠋⠀⢠⠏⢸⡄⢳⡀⠈⠙⢦⠏⠀⢰⠇⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠘⢟⠉⠀⢠⠏⢀⡇⠸⡄⠀⠀⠈⡽⠃⠀⡼
⠀⠙⣆⠀⠙⠢⣄⡞⠀⠈⡇⠀⢹⣤⠔⠁⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠈⠳⢴⡃⠀⢸⠇⠀⢻⣀⡤⠞⠁⢀⡜⠁
⠀⠀⠈⠓⢤⣀⠀⠉⠙⠛⠛⠋⠉⠀⣀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠉⠙⠛⠓⠚⠋⠉⠀⣠⠴⠋

⠀⠀⠀⠀⠀⣤⣤⣀⣀⣠⣀
⠀⠀⠀⠀⠀⠻⠶⠮⣽⣿⣟⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡆⣿⠀⠀⠀⠀⠀⠀⠀⣠⠞⣻⣏⣿⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢷⣤⣤⣤⣀⡀⠀⣰⠇⡾⠙⠾⠋⠛⠓⠶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣟⣀⠀⠈⣿⣛⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣹⡟⠋⠉⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣾⠦⣤⣄⣀⠀⠀⠀⠀⢿⣿⣦⠤⣄⣀⣉⣉⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢠⣿⡓⣶⠶⢭⣽⣛⡲⠦⣤⣄⣉⣿⢳⡶⠦⣤⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠓⣺⠃⢳⡌⢷⡀⠀⠉⠉⠛⠲⠦⣽⢃⣾⣅⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⠴⠖⠛⢻⡇⠀⣿⣄⡀⠻⡄⢳⡄⠀⠀⠀⠀⢰⡏⣼⠳⢮⡳⢦⣠⡴⠛⣉⣤⡴⠶⠶⡦⣤⣈⠙⠷⣄
⠀⠀⠀⠀⣠⠞⠋⣀⣤⠴⡶⣾⠀⢠⣇⡈⠙⢶⣽⣆⠻⣄⠀⠀⢀⡟⣰⠇⠀⠀⢹⣦⡙⢶⡟⣍⠀⢇⠀⢸⠁⢠⠏⠛⢦⡈⠻⣄
⠀⠀⠀⡾⠁⢠⡞⠉⢣⠀⢳⡏⠀⣸⣫⠛⣦⡀⠹⣿⣆⠙⣦⠀⣾⢡⡏⠀⠀⢠⡟⢠⣿⣦⣝⠿⣦⢸⡀⢸⠀⡞⠀⢀⡴⠻⡄⠘⣧
⠀⠀⣼⠁⢰⡏⠙⠢⣄⠳⣼⠃⠀⣿⡡⢊⡬⢷⡄⠸⣿⣦⣘⡿⠧⣾⡦⠤⠤⠾⣤⣿⣤⣬⣿⣷⣬⣿⣧⣟⣼⠀⡰⠋⠀⣀⢿⡄⠘⡇
⠀⠀⡏⠀⣾⠉⠙⠒⠢⣽⣿⢠⢸⡿⠿⠗⠒⠊⢿⠀⣿⢸⢯⢖⣒⢦⢻⡦⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣀⣈⠉⠻⣦⠒⠉⠀⠘⡇⠀⡇
⠀⠸⡇⠀⢿⠊⠉⢉⡩⠖⡿⡶⡟⢿⠫⢍⡉⠉⢹⠆⣿⢸⡌⣾⠿⠼⣰⠷⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⠤⠽⠃⢀⣿⡓⠒⠲⢴⡇⢀⡇
⠀⠀⢻⡄⠘⣷⠚⠉⢠⠞⢀⡇⢧⠀⠱⣄⠈⢑⡾⢠⡟⠀⣹⠟⠲⠾⠿⠶⠶⠶⣶⠶⣶⠶⡶⠶⢶⠶⢶⠶⢶⠛⣅⠙⠢⣤⡞⠀⣼⠃
⠀⠀⠀⠹⣆⠘⠷⣴⠃⠀⢸⠀⢸⡀⠀⢈⣳⠟⢁⡾⠉⠛⠷⠄⠀⠀⠀⠀⠀⠀⠹⣆⠙⢿⣄⡠⠋⠀⡼⠀⠸⡄⠈⣦⡴⠋⢀⡼⠃
⠀⠀⠀⠀⠘⠷⣄⡈⠛⠶⠷⣤⣤⠷⠖⠋⣁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡈⠙⠲⠦⠧⠤⠴⠷⠛⠉⣀⡴⠛
⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠤⠤⠤⠶⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠦⣤⣤⡤⠶⠞⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⢒⣊⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠭⢿⣭⣭⠭⠟⣛⣯⡥⣴⠲⣏⢻⣎⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠦⠤⠤⣀⣀⠀⠀⠀⢰⣊⡽⠤⢒⣊⡭⢿⠻⡷⠽⠷⣛⣫⣭⣶⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⢤⠴⡖⠒⠉⠀⢀⣀⠤⠔⢒⣋⡭⠴⢒⠟⣦⢷⣏⢿⡌⣷⡸⣧⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡧⠖⢒⣉⡥⠴⠒⠋⠉⠀⣠⠖⢁⠔⠹⡤⣿⠬⣗⣚⠯⠝⠒⠉
⠀⠀⠀⠀⣀⡤⠤⠴⠤⠤⢄⡀⠀⢀⠔⢹⠀⡗⠋⠉⠀⠀⠀⠀⠀⣠⠞⢁⠔⠁⠀⠀⣱⠸⡭⠤⠤⠤⣀⡀
⠀⢀⡴⠋⢀⣠⠤⢶⠦⠤⣄⢈⠗⢁⠔⢹⠀⡇⠀⠀⠀⠀⠀⣠⠞⢁⠔⠁⠀⢀⡴⠋⢈⣇⢳⢤⠤⢤⣀⠉⠲⣄
⢠⠎⢀⡔⠁⠘⡆⢸⠀⣼⡵⢁⢔⠉⢣⢸⠀⡇⠀⠀⢰⣶⢾⣁⠔⠁⠀⠀⢠⠋⢀⠔⠙⢾⡜⣿⢀⡜⠈⠳⣄⠈⢆
⡎⠀⡞⠙⠢⢄⣸⣼⡾⢃⣴⣥⣾⠧⠤⠿⠒⠓⢲⠏⡹⡹⡟⠁⠀⠀⠀⠀⡇⠀⡞⠒⠤⣌⣳⣹⣞⣠⠤⠖⠛⡄⠘
⡇⠀⡟⠒⠒⢒⡿⣄⣽⣭⢥⣀⣀⣀⣀⣀⣀⣀⣈⣳⡽⠵⠃⠀⠀⠀⠀⠀⡇⠐⡷⠒⠒⣲⢿⣤⡽⢶⡒⠒⠒⡇
⢳⠀⠱⡤⠖⠁⡼⢹⠱⡌⠑⠦⣴⠃⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢣⠖⠉⢠⠋⡇⢳⠀⠈⠑⡶⠁⢰
⠀⠳⡀⠙⠦⣴⡁⢸⡄⢘⣤⠖⠁⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠑⢴⣃⠀⡇⠀⣧⡠⠞⠁⡠⠃
⠀⠀⠈⠲⢤⣀⡉⠉⠉⢁⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡈⠉⠉⠉⢁⣀⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠛⠒⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⢤⡈⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠹⡄
⠀⠀⠀⣠⠴⠖⠒⠓⠒⠲⣴⣇⠀⠀⠉⠲⣄⠈⠙⠲⠤⣄⣀⣀⣤⠔⡿⠀⠀⠙⢦⣠⠤⠴⠶⠦⢤⣀
⠀⣠⠊⠁⠀⠀⠀⠀⠀⣰⠃⠈⠳⣄⠀⠀⠀⠙⠦⣀⠀⢀⡄⠀⠀⣸⠁⠀⢀⡴⠋⠑⣄⠀⠀⠀⠀⠈⠙⢦⡀
⣸⠁⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠘⡆⠀⠀⠀⠀⠈⠓⢬⣳⣤⣤⡇⠀⢠⠏⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⢳
⡇⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢹⠋⣻⠓⠒⠦⢤⣀⡀⢳⠀⠀⠀⠀⠀⠈
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠙⠲⠖⢯⡀⢸⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠀⢠
⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⠀⠀⠀⢀⣠⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⢀⣴⠁⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⣠⡄⠁⢀⡼⠁⣠⠟⠁⠀⠄⠀⢀⠔⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣤⠀⠀⠀⠀⠀⠀⠁⢀⣴⣿⢄⣼⣿⠃⢠⡾⢀⡾⠁⠀⠀⠀⠀⠐⢡⢆⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣴⡏⢀⠀⠀⠀⠀⠀⢠⣾⣿⢿⣾⣿⠟⢡⣋⠶⠋⠁⠀⠀⠀⢀⡄⢀⣷⢟⣡⣾⠆
⠀⠀⠀⠀⠀⢠⡶⢶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠅⢹⡏⢠⡾⠀⣀⣴⣎⣤⣿⡿⣡⣿⣿⠃⢀⡌⠀⠀⠀⠀⠀⠀⡴⠋⣴⢞⣡⣿⠟⠁
⠀⠀⠀⠀⠀⣿⣿⠿⠟⢀⣷⣶⣴⡄⠀⢀⠀⢠⢄⡀⠀⠀⡟⢰⠟⢠⡿⣱⢆⣽⣛⣿⣿⣿⣿⣿⣿⣯⣶⠜⠃⠀⠀⠠⠀⣠⢞⣡⣞⣵⡿⠟⢁⡠⣲⣶⠇
⠀⠀⠀⠀⠀⢛⣿⣿⣷⠆⠈⣿⣿⣧⠄⠀⠀⠀⠀⠘⡉⠟⠁⠿⢰⡿⢱⣧⡿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡅⠀⠀⠀⢀⣴⠞⣠⣿⣿⣿⣟⣱⣆⣯⣾⠷
⠀⠀⠀⠀⠀⠀⠈⢹⣷⣦⣴⣿⣿⣿⣿⠀⠀⠀⠀⠐⠁⠀⡀⠀⠙⣷⣧⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⢀⣄⡴⠋⣵⣾⣿⣿⠟⠹⠍⣹⣏⢁⣤⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⣧⣀⣀⣠⣯⡀⠀⠀⢸⣿⣿⣿⣿⣟⣾⣿⣯⣿⣿⡅⣣⣞⠛⣉⢀⣼⣿⣿⣿⡷⢿⣶⠞⣹⡷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⠉⡟⠀⠀⣿⣿⣿⣾⡿⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⣯⢈⣼⣿⣿⠿⢋⣭⣿⣵⢟⣱⠞⣋⣤⠶⠂
⠀⠀⠀⠀⠀⠀⠀⡿⢀⠀⠀⠀⠀⢰⠁⠀⢀⣿⣿⣷⣿⣿⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣩⡿⣿⣩⣥⣿⣿⣿⣡⣾⣿⡿⣫⡶⣛⣴⠟⠛⠁⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢀⡠⠁⠀⠀⢸⣿⣿⣿⡿⡿⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⣻⣏⣴⣾⣿⣿⡿⢡⣾⣿⣿⢽⣾⡯⠙⠋⠀⠀⣠⣴⣿⠿⠋⢁⣀⠄
⠀⠀⠀⠀⠀⠀⢸⠀⠄⠀⢰⣯⠘⠃⠀⠀⢈⣿⣿⣿⢾⡇⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣽⣿⣿⣿⡇⣠⠞⠋⠀⠀⣠⣔⣟⢉⣋⠴⢖⡚⠁
⠀⠀⠀⠀⠀⠀⢸⠇⠀⣠⡞⠙⠀⠀⠀⢀⣸⣿⣿⡿⣿⣿⡀⠀⠙⣿⣿⣿⣿⢿⣾⣿⣿⣿⣻⣿⣿⣿⠿⣭⠉⠛⠀⢀⣠⣦⣽⣿⣿⣿⣿⣿⣟⠋⣁⡀
⠀⠀⠀⠀⠀⠀⢘⣴⡞⢻⡿⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣷⣤⠀⠈⣿⡟⣱⣿⣿⣿⣻⢛⣿⡿⣟⡭⠈⠁⣀⣠⣾⣿⣿⣿⣿⣿⣿⡿⠟⠉⢀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⠑⠷⣤⣄⣴⢈⣿⣿⡿⡏⠻⡿⣿⣿⡀⠀⢹⣥⣿⣿⣿⠝⣿⣿⡿⠐⢋⣡⣴⣾⣿⣿⣷⣿⣿⣿⠿⠋⣁⣤⡴⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠓⢾⣿⠻⢿⣿⣿⣄⣀⠘⣷⠏⣼⣧⠀⠈⢉⡉⠛⢷⣶⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣽⣟⣡⡖⠛⠉⠁⠄⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⡼⠛⢷⣴⢿⡏⠈⢹⠿⣿⣾⣿⣹⡄⠀⢍⣶⣿⣦⠙⢿⣿⣿⣟⣩⣿⣷⣷⣿⣿⣿⣿⣿⠿⡿⢤⣤⣤⣤⣬⡿⠛
⠀⠀⢀⣠⣶⠿⠟⠛⢛⢻⡷⣶⣄⡚⠃⣹⣿⣇⠀⢸⣿⣿⣿⣿⣿⣷⡦⠀⢻⣏⣻⣧⠘⣿⠿⣏⣭⣾⣿⣿⣿⣿⣿⣧⣤⣼⣿⠿⠍⠙⠛⠈⢀⣀⡤⠄
⠀⣰⣿⠋⠁⠀⠀⠀⢸⣿⡇⠀⠙⠻⣶⣿⣿⠛⢷⣼⢉⣻⣿⡾⠿⣿⠃⢠⣿⡿⣇⢤⡄⢸⣿⢿⣿⣿⣿⣿⡿⢟⣿⡟⣿⢿⣿⣀⣠⢴⣖⣊⠉⠁
⢰⣿⠁⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⡙⣿⣥⣤⣤⣼⣿⣽⣿⣷⡾⠉⣠⡾⡟⢾⣧⣾⡇⢸⣿⣿⣿⣟⣯⡕⠚⠻⠟⠒⠛⠛⠋⠉⠉⠁
⣿⡇⠀⠀⠀⠀⠀⠀⣸⢿⡀⠀⠀⠀⠁⢸⣿⣿⣿⣿⣿⣿⠿⣿⣎⠛⣋⠄⣴⣪⣽⣿⠃⣾⣭⣴⣿⠿⠟⠋
⣿⡇⠀⠀⠀⠀⠀⠸⠟⠈⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣥⣽⣿⣷⣥⣛⠿⠿⠟⣡⣾⠿⠯⠙⠃
⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣨⣿⣿⡟⠟⠛⠿⠿⠟⠋⠁
⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠿⢿⣿⣿⠟⠋
⠀⠀⠈⠻⣷⣧⣤⣀⡀⠀⢀⣀⣤⣾⠟⠁⠀⠐⠛⠉
⣤⣤⣤⣤⣤⣍⣽⣿⣿⣿⣿⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⠤⣟⣋⣛⣛⣋⣛⣙⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⢰⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⢿⣦⡀⠀⢀⣀⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠟⠛⠛⠛⠻⠿⣷⣿⡏⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⢙⣷⣾⡿⠛⠛⠉⠉⠉⠛⠻⢿⣶⣄
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⣼⡿⢿⣧⡀⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⣠⣿⠟⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀
⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠻⣿⡄⠈⠻⣷⣄⠀⠀⠀⠀⢠⣿⠃⠀⢰⣿⠃⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠟⠀⠀⠀⠀⢿⣷⠀⠀⠈⠻⣷⣄⢶⡦⣿⡟⠀⠀⣿⡏⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠸⢿⠟⠁⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠙⢿⣿⣿⣿⠁⠀⣀⣿⣧⣤⣤⣶⣶⠾⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⢿⣿⣿⠿⠛⠛⢻⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠈⠙⡏⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠤⠀⠀⠀⠀⠻⢿⣦⣄⡀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣶⣤⣴⣶⣾⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣶⣶⣾⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢀⠴⠂⠒⠒⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣶⣶⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡠⢀⣸⠏⠳⣄⠈⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢢⡀⠀⠀⣀⣠⣤⣦⣤⣴⡤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠛⠉⠉⠛⠛⢿⣿⣕⣤⡈⢷⣄⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣹⣶⣿⡿⠟⠛⠋⠉⠛⠛⠻⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⢀⡞⠀⠙⢿⣿⣄⠙⢷⣄⠀⠀⠙⠳⣄⡀⠀⠀⡾⠀⢀⣼⣿⠟⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣦
⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⢻⣿⣆⠀⠙⢧⡀⠀⠀⠈⠙⢦⣼⠃⠀⣾⣿⠃⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧
⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⢀⣀⡼⠁⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠙⢦⡀⠀⣀⣰⡇⠀⣸⣿⡇⠀⠀⠀⠀⠀⠱⣄⣀⠀⠀⠀⠀⠀⠀⠀⢻⡏
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠐⠺⠿⠂⠀⠀⠀⠀⠀⠀⢸⡿⠇⠀⠀⠀⠀⣽⢿⣉⡟⠳⡄⢛⣿⣁⣀⣈⡤⠥⠶⢾⠿⠿⠂⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⠀⠀⠀⠀⠀⣧⣠⠿⢷⣉⡏⠉⢻⣇⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⣀⣠⠞⠛⠶⠞⠛⢦⡀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠈⠻⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠉⠉⠁⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣦⣤⣤⣤⣶⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣶⣤⣤⣤⣶⠾⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⢉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡠⠤⢊⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣉⡠⠔⠒⠒⢻⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣕⠢⠤⠤⠤⢽⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⡱⠋⠉⠉⠀⠀⠀⠀⢀⣀⠤⣒⠭⠓⢋⡵⡻⣸
⠀⠀⠀⠀⠀⠀⣀⠤⢐⣒⣒⣒⣒⠰⢵⡁⠀⠀⠀⣀⡠⣔⣪⢽⠊⠉⠀⢀⢔⡯⠊⣀⠧⠧⠤⠤⠤⢀⡀
⠀⠀⠀⠀⡴⣪⠔⠉⠅⠀⢠⠀⢠⣿⠓⢬⡲⣖⠭⠒⠉⠁⢸⢸⠀⢀⢔⡵⣫⢔⡩⠒⢺⢹⠉⠉⠉⠒⠪⣑⢤⡀
⠀⠀⢀⢎⠜⠁⠠⡀⠈⠀⠠⢠⣿⠃⡠⠊⠙⢌⢆⠀⣀⠤⠼⠾⣔⠕⠋⡔⡱⠛⢄⠀⠘⡏⣆⠀⠠⠁⢀⠌⠳⡑⡄
⠀⠀⡎⡜⠐⠠⠀⠈⠢⠐⢰⣿⡧⠊⠀⠀⠄⠚⡎⡾⠁⢀⣀⠀⠈⡆⡜⡼⠠⢀⠀⠑⢄⢹⢹⠀⢀⠔⠁⠀⠀⠸⡘⡄
⠀⠀⡇⡇⠄⠀⠀⠀⠁⢰⡛⣻⣶⣿⡶⠭⠭⠭⡧⡟⠲⢞⠖⠁⠀⣿⡇⡇⠀⠀⠀⠀⠂⢱⠿⢧⠁⠐⠀⠀⠀⠀⡇⡇
⠀⠀⡇⡇⠀⠀⠄⠂⡨⢊⠙⢉⠀⡐⠀⠄⡀⠀⡇⡏⠓⢄⡱⠔⠊⠀⡇⡇⠀⠀⠀⠄⢀⠘⠓⠛⢄⠀⠀⡀⠀⠀⡇⡇
⠀⠀⠸⡘⡄⠀⡠⠊⠀⠂⢀⠀⠄⠀⠠⡀⠀⡜⡸⠀⠀⠀⠀⠀⠀⠀⢱⠱⡈⠀⢀⠔⠁⠌⠀⠈⠀⠑⢄⠀⠀⡰⢱⠁
⠀⠀⠀⠘⢌⠪⣀⠀⠄⠀⢈⠀⠘⡀⢀⡠⢊⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⡑⢄⠁⠀⡐⠀⠀⠀⠐⠀⠀⢁⠔⡱⠁
⠀⠀⠀⠀⠀⠙⠪⢝⣒⠢⠴⠤⢒⣚⠭⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢝⡒⠤⢄⣠⣀⠤⠗⣚⠵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠉⢐⠆⣉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠉⠁⠀⠈⠉⢐⡄⠀⠀⠀⠀⠀⢠⢣⠎⠀⠑⡌⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⢤⣤⡤⠤⠒⠉⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⣄⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠆⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⣀⡎⡴⡘⢍⠉⠉⠉⠉⠉⠉⠉⠉⠉⢙⣨⠤⠤⠤⠤⢄⡀
⠀⠀⠀⠀⠀⢀⠴⢊⠤⠒⠊⢹⠉⢐⠦⣌⡓⢌⢎⢢⡀⠀⠀⠀⠀⠀⡠⢒⡡⢴⢲⠉⡏⠑⠒⠤⡑⠦⡀
⠀⠀⠀⠀⢠⢋⠔⠣⡀⠀⠀⢸⠀⡜⡜⡠⠛⢆⠳⡣⡑⡄⠀⠀⢀⠜⡰⠛⢄⠀⠆⢃⡇⠀⠀⢀⠜⠢⡙⡄
⠀⠀⠀⢀⠃⡎⠀⠀⠈⠢⡀⢸⡜⡰⠋⠀⠀⠈⠆⢣⠑⡌⢆⠀⡜⠰⠀⠀⠀⠑⢼⡘⡇⢀⠔⠁⠀⠀⢱⠘⡀
⠀⠀⠀⢸⠀⡧⠤⠤⠤⠤⠬⣾⣿⣷⣶⣶⣶⣶⣾⣺⣒⣚⠖⠉⢨⡧⠤⠤⠤⠤⠤⢵⣷⠥⠤⠤⠤⠤⢼⠀⡇
⠀⠀⠀⠘⡀⡇⠀⠀⠀⡠⠊⢸⠈⠢⡀⠀⠀⠀⡌⡸⠀⠘⠆⠚⢇⢡⠀⠀⠀⢀⠔⠁⡇⠑⢄⠀⠀⠀⢰⢀⠃
⠀⠀⠀⠀⠱⡘⢆⡠⠊⠀⠀⢸⠀⠀⠈⠢⣀⠜⡰⠁⠀⠀⠀⠀⠈⢆⠣⣀⠔⠁⠀⠀⡇⠀⠀⠑⢄⡰⢃⠎
⠀⠀⠀⠀⠀⠈⠢⡑⠢⣄⣀⣸⣀⣀⡠⠔⣡⠞⠁⠀⠀⠀⠀⠀⠀⠈⠳⢌⠢⢄⣀⣀⣇⣀⡠⠔⢊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠠⠤⠤⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⠤⠄⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⠷⢶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠈⢿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣶⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⡇⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⠄⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⠦⣤⣄⡀⠀⢠⣿⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⢀⡤⠒⢀⣸⣷⡶⠾⠛⠛⠛⠓⠲⠦⢄
⠀⠀⠀⠀⣠⣶⣿⣿⣿⡿⠷⠒⠒⠒⠒⠤⢄⣀⣼⣿⠟⢷⡀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⣀⣴⡿⣋⣽⡖⠛⠉⠉⠉⠉⠒⠢⢄
⠀⠀⢠⣾⣿⣿⡿⠋⣁⠄⠀⠀⠀⠀⠀⢀⣴⡿⠛⠀⠀⠈⠳⡄⠀⣀⣀⣠⣾⠟⠁⠀⣴⣿⢫⡞⠉⠀⢳⡀
⠀⠰⢿⣿⣿⠟⣠⠞⠁⠀⠀⠀⠀⠀⢠⣾⠏⠀⣀⣀⣤⠄⢀⣼⡿⡿⣿⡿⠁⠀⠀⣼⣿⣳⠏⡀⠀⠀⠀⢣
⠀⠀⠀⢠⡟⣼⡏⠀⠀⠀⠀⠀⠀⣰⠟⣷⠟⠛⠉⠁⠀⠀⢸⣟⠸⣿⡿⠁⠀⠀⢰⣿⣿⡟⣰⠇⠀⠀⠀⠀⣵⣿⣿⠃
⠀⠀⠀⢸⣧⣿⡇⠀⠀⠀⣰⠁⣸⣿⣾⣿⠤⠶⠒⠒⠀⠀⠀⠻⢦⣬⣤⠞⠀⠀⠈⠉⢹⡇⣿⡄⠀⠀⠀⠸⣿⣿⣃⡀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠸⣷⣶⣿⣿⣯⣤⣤⠤⠴⢾⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⢿⣿⣿⣧⣀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡏
⠀⠀⠀⠀⠈⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⣄⣀⣀⣀⣀⣤⣶⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣦⣤⣤⣤⣤⣶⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢸⣦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢿⡀
⠀⠀⠀⠀⠀⠀⠈⠙⣉⣭⣭⣭⣭⣭⣽⡉⣰⠋⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢷⣀⣠⣤⣤⣤
⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⠟⠛⠛⢿⢿⣿⣷⣤⡀⢻⣆⠀⠀⠀⠀⠀⠀⠀⣠⣾⢟⣵⣾⣿⠿⠿⠛⠛⠻⠷⣦⣄⡀
⠀⠀⠀⢀⣼⣿⡿⠋⠀⢡⠀⠀⠀⡌⡼⠉⠻⣿⣿⣄⢻⣆⠀⠀⠀⠀⣠⣾⠟⣱⣿⡿⠋⠹⣆⠀⠀⠀⠀⠀⠈⠙⢿⣆
⠀⠀⠀⣼⣿⠟⠠⢀⠀⠀⢣⠀⢰⡞⠁⡀⠄⠛⣿⣿⡄⠹⣆⢲⢂⣴⠟⠁⢰⣿⡿⠁⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⢩⣿⠀⠀⠀⠈⠒⠤⣦⣿⣐⣉⣀⣀⣀⣸⣿⣁⣀⣿⣿⣿⠋⠀⠀⣿⣿⠇⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢸⡟⠀⠈⢿⣿⡿⠀⠀⠀⠉⣿⡆⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⢀⣧⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠹⣷⣄⡀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⣶⣶⣶⡶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣶⣶⣶⣶⠾⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⣀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠁⡀⠉⣸⣿⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⠳⣤⣾⠟⣷⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣆⠸⠀⠀⠀⠀⢹⡟⠁⢸⡀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⣙⣶⠟⠁⠘⠶⠀⠀⠀⣰⡞⠀⠀⢸⣧⡀⠀⠀⢀⡾⠣⠊⠀⠀⠠⡲⠁⠀⠀⠰⠁⡀⠀⢀⡤⠖⠁
⠀⠀⠀⠀⠀⢀⣞⡽⠁⠀⠀⠀⠀⡀⠀⠀⣹⣇⠀⠀⣸⣿⠵⡀⡺⠋⠀⠀⠀⠀⢀⠈⠁⠀⠀⠀⠀⣀⣠⠾⠋⠀⠀⣀
⠀⠀⠀⠀⣡⡿⠋⠀⠀⠀⠀⠀⠀⠐⠀⠀⠿⣿⠀⢸⣿⣷⡖⡳⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠁⠀⣀⠴⠚⠁
⢀⠀⢀⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠀⢸⣿⣷⣿⡅⢀⣔⠀⠀⢀⠀⠀⠀⢀⡤⠾⠁⠀⠀⡶⠚⠁
⠈⠐⢼⠗⢀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⠀⠀⢀⣿⣿⠿⠋⠠⠜⠁⠀⠀⠔⠀⣠⠶⠃⠀⠀⠀⢀⣠⡤⢂
⠀⠀⠀⠀⢸⡄⡀⠀⠀⠀⠀⠀⢀⣁⣠⣴⠟⠀⡾⠟⠁⠀⠀⠀⠀⠀⠀⣠⠂⠈⠀⠀⠀⠀⡰⠖⠋⠁⠂⠁⠀⠀⣀⢀⠄⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠘⡇⠀⠀⠁⠐⠀⣠⢾⣹⣿⣅⢲⣤⡞⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⣀⠀⠊⠀⡀⠀⠀⠀⠀⣀⠰⠋⠀⠀⠀⠠⠐⠂⠈
⠀⠀⠀⠀⠀⠣⣀⢀⠀⢀⠔⠁⣾⣿⡝⣿⣿⣷⡀⠈⠀⢀⡤⠞⠋⢀⡀⠀⠐⠀⠉⠁⠀⣀⠔⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⣟⠀⠁⠀⢰⣿⣿⣏⣹⣿⣿⢲⣦⣶⡉⠀⣀⠐⠋⡠⠀⠁⠀⠀⠀⠀⠉⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣸⣏⡛⠓⠲⢦⣿⣿⣿⣼⠋⣰⡿⢑⠙⣷⠀⠀⠺⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢠⣞⣿⡌⠳⢦⡀⣿⣿⣿⣽⣿⣦⠟⣱⠋⠁⣿⣶⡻⠂
⠀⠀⠀⢠⣿⡟⢻⠋⠠⣾⠿⣿⣿⣷⣯⣬⣥⣶⣿⣄⢀⡿⠉
⠀⠀⢠⣿⡟⢀⣿⣦⣏⣡⣰⣅⣽⣿⣿⣿⡻⠉⠀⠋⣽⠃
⠤⢤⣿⣟⢀⣿⣿⣶⣤⡴⢏⢍⣥⣼⣿⡁⠀⢀⣠⡾⠁
⠀⣼⣿⢿⡟⣿⣭⣟⣃⣭⡿⠟⠛⠉⠙⠛⠛⠛⠁
⢠⣿⣧⠞⣰⣿⣿⠿⡛⠉
⠘⣿⣇⡴⠛⠉⠉
⣶⡾⠿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣽⣷⣶⣶⣶⣦⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⡄⠀⢀⠀⢆⡀⡼⡀⠀⢰⡀⠀⠰⣀⢠⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⢐⢮⠄⠎⡙⣷⢣⢌⣁⣞⡁⢆⡅⢈⠋⢹⡅⠩⡍⠺⡤⢠⡃⡄⠶⣀⠶
⠀⠀⠀⠀⠀⠀⠥⡤⠛⣬⣱⡊⡭⢁⡀⠚⢠⢞⣔⣐⡢⠔⢬⠙⠂⣼⣃⠈⢥⠒⠳⡀⢌⡉⠬⣚⢀⣀⡄
⠀⠸⣀⠀⢨⡧⡎⣅⡌⠑⣂⢑⠂⠀⢷⡶⡀⢠⢈⠔⠆⠉⣤⢰⡃⠲⠍⠃⢳⡅⡙⠘⡉⣥⠛⢭⠎⠱⣗⢒⢀⡸
⠀⢛⡵⡼⡥⠈⠁⠃⡀⡧⣆⠄⠈⠧⢸⡍⣃⡶⣉⠀⢀⢳⡐⡂⣺⢀⡓⠉⠤⣀⠐⠂⠁⣬⠲⢈⠡⠄⠠⡽⡴⣍⠃
⠦⣬⠤⠇⢚⡈⢩⡉⠁⠤⢥⢉⠚⢀⡬⠱⠦⠭⠅⠐⠎⠵⠁⣍⢖⠾⠷⢉⡥⢈⡐⣂⠉⢀⡘⢐⢒⠀⣘⠂⠧⢬⡤
⠀⠒⣛⡿⢎⡴⠀⣼⢱⢀⠛⢘⣰⢄⠐⠂⡉⠀⢹⠠⢘⠁⠀⠲⠨⠏⠁⣌⠴⠀⣲⠀⠀⢎⢹⢨⡍⠰⣜⢪⣍⡃⠓
⠀⠀⠀⠀⠎⣡⠅⠆⢺⠈⡵⠀⡄⠼⡤⢀⠀⠀⠢⣈⠹⣆⠀⡴⠀⢀⡤⠂⣜⠓⠇⠀⠈⡁⢰⠄⣆⣧⠙⠆
⠀⠀⠀⠀⠀⠠⠎⠱⢆⢣⡈⠦⣐⢆⡇⠌⠓⠶⣶⣿⣷⣿⣿⣷⣾⠿⠖⠊⠅⣇⢖⡠⠎⡬⢣⠶⠉⠦
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠖⠘⠁⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠁⠛⠐⠇⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣛⣛⣿⣿⣿⣟⣻⣿⣿⣗⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣻⣶⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡘⠁⠀⢹⡄⠀⠀⠀⢀⡟⠉⢑⡄
⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⣧⢠⠄⠀⢧⠀⠀⠀⠸⡇⡾⠻⣸⡄
⠀⠀⠀⠀⠸⡁⠘⠏⢢⠀⠀⠀⠀⠸⣿⠛⣠⣼⡀⠀⠀⠀⡇⠀⠾⣻⡇
⠀⠀⠀⠀⠀⢡⠀⠀⢀⣣⠀⠀⠀⠀⢧⠘⠁⠈⢡⠀⠀⠀⣷⣆⢰⡿⢿
⠀⠀⠀⠀⠀⠀⢣⡻⠀⠁⢱⡀⠀⠀⠘⠀⠀⠀⠀⢇⠀⠀⠋⢀⣼⡋⠘⡀
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠱⡄⠀⠀⡄⠀⠔⠈⢹⡀⠀⡆⢘⠁⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢀⡠⡙⢆⠀⢹⡐⠀⠀⠀⡇⠀⡇⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤
⡴⠓⠲⢄⠀⠀⠀⠀⠀⢇⠏⢁⠈⠀⣆⠀⠇⠑⠤⠤⠾⡤⠃⠔⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⡠⢀
⢳⣠⣠⣤⣷⣄⠀⠀⠀⠈⣦⠀⠀⠀⠘⠖⠀⠀⢀⠀⠀⠀⠀⠀⠹⣆⠀⣷⠀⠀⠀⠀⠀⠀⠀⡄⢠⠀⣠⢄⡜
⠀⠙⢆⡉⠉⠉⠑⢦⣄⠀⠈⣧⠀⢾⣤⠀⠀⠈⠟⢀⢀⠀⢤⡀⣔⣈⡶⠿⡀⠀⠀⠀⠀⠀⠼⠹⢻⡾⠿⡞
⠀⠀⠀⠉⠢⡄⣸⡯⠈⠵⠒⠋⠐⡄⠀⢀⠀⢀⣴⣿⠿⠷⠛⠊⣿⠛⡀⠐⡿⢆⠀⠀⠀⣼⠒⡛⢐⠄⡼⠁
⠀⠀⠀⠀⠀⠈⢧⡄⠨⠐⠢⠚⠀⠀⠐⠇⠤⠛⠉⠶⢶⣶⣟⡄⣳⣇⡛⣰⡏⢙⡳⠶⠾⡅⣐⢴⠌⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡄⣄⠠⠀⠀⠀⠀⠀⠄⠀⣄⠊⠁⣫⡷⢗⡰⢈⡇⡿⣌⢁⢀⣀⠀⢄⡿⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡜⠷⠆⠀⠤⠢⠈⠀⠈⢻⣷⣦⡉⠀⢀⣴⡄⠀⠥⢟⣼⠽⠋⠃⢼⣁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⣤⣤⣷⣿⠶⠀⠀⠐⠃⢹⡷⠀⠀⠈⠁⠨⢉⡙⠈⡿⡷⠀⣩⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⡿⣿⠻⠆⢰⠀⠀⠀⠀⠀⠀⠒⢘⡟⢦⡿⠂⠀⣤⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢈⠠⠀⠈⢆⠀⠀⠐⢦⣤⡠⠀⠀⠊⡗⢰⣿⣧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣉⠀⢀⣀⣀⠀⠤⠦⠴⠄⠀⠐⠀⠐⠻⠶⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡎⣹⣞⡉⠉⠉⣤⣤⣿⠀⠀⠛⠒⠒⢲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣠⣤⡀⠡⢤⠄⠀⢹⡿⠓⠤⠤⠄⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠌⡡⠐⣰⠶⠶⠶⠀⠀⠘⠀⠛⣹⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡛⢳⡏⡄⣰⠆⢜⡇⡚⡄⣠⡏⢸⡆
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣤⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣊⠩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠐⣒⣒⣥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠈⠁⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⢌⣉⣀⣒⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣥⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⢶⠒⠒⠊⠀⠀⠀⠀⢀⡠⣔⡲⠝⣊⢽⣼
⠀⠀⠀⠀⠀⢀⠤⢔⣒⣒⣒⡠⠦⡃⠀⠀⠀⣀⢤⣲⣯⠓⠉⠀⣠⣾⠟⣁⠯⠦⠤⠤⠄⣀
⠀⠀⠀⡠⢪⠕⠉⠁⠀⠀⢀⣾⠟⠪⡑⣮⠛⠋⠁⠀⣿⠀⡠⡾⢋⢕⡩⠒⢻⢯⠉⠉⢉⠒⢍⡢⡀
⠀⠀⡐⡕⠁⠐⢄⠀⠄⢀⣼⠏⠠⠂⠈⢎⢆⢠⠔⠚⠛⢮⠊⢠⠣⠊⠀⠀⠈⣇⡇⠀⠂⠀⠀⠙⢜⠄
⠀⢠⢱⠀⠀⠐⠀⠁⢀⣾⣟⣈⣀⣀⠀⠨⡜⣇⢀⣖⡄⠀⣧⢇⠋⠐⠀⠀⠀⢸⣧⡀⠀⡀⠀⠐⠘⡜⡀
⠀⢸⢸⠀⠀⠀⠀⠄⠻⢴⠟⠛⢉⠉⠉⠉⣟⠦⡉⢫⣀⡴⢹⢸⠂⠀⠀⠀⠀⢼⣭⡗⢀⠀⠀⠀⠐⡇⡇
⠀⠀⢇⢆⠈⠀⠀⠂⠄⠀⠀⠀⢀⠀⠐⡼⡸⠀⠈⠉⠀⠀⠀⡏⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢱
⠀⠀⠈⢪⡲⡀⠀⠀⠀⠀⠀⠀⠀⣁⢞⠕⠁⠀⠀⠀⠀⠀⠀⠘⢌⢦⠀⠀⠐⠀⠀⠀⠀⠀⠀⡴⡣⠃
⠀⠀⠀⠀⠉⠪⠝⣒⡠⠤⢄⣚⠪⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠭⣒⠤⠄⠤⠀⠴⣒⠭⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣄⡀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠓
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⢼⡿⢿⡣⠀⠀⣠⣤⣤⠀⠀⢇⠄⠀⠶⡄⠂
⠀⠀⠀⠀⠀⠀⠒⡀⠐⠀⠀⠀⠂⠐⠀⠀⠘⠁⠀⠉⠉⠀⠀⠀⢠⠰⠆⠉⠀⠀⡐⠂⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡔⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⠗⠈⠓⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠜⣏⠁⢄⡀
⠀⠀⠀⠀⠢⣤⠠⡄⠾⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣠⠀⠀⠀⢀⠖⡾⠂⠀⣀⠇⠀⠀⠳⣆⠀⠀⠘⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠓⠀⠀⠀⠀⠀⠀⠙⠰⠃⠀⠀⠀⢣⠀⠀⢁⠀⠀⠀⠀⠈⠻⠄⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡸⠖⠋⠀⡠⠄⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⢃⢀⡤⠈⠀⠀⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⡀⠀⠈⠀⠑⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⡀⢀⡄⠀⠀⢀⠀⠀⢀⡀⢀⡄⠀⠀⠀⠀⠀⠀⠠⠶⠤⠉⠓⠚⢤⣀⣀⣀⣀⠤⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠈⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⡤⢶⡶⠦⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠃⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠿⢦⠄⢀⢀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⠄⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠤⢴⡒⠛⠉⢙⣉⣡⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⡁⣐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣏⠀⢦⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠓





⠀⠀⠀⠀⠀⠀⠀⠠⢠⣤⣤⡀⠀⣦⣄⣀⣠⣤⠀⢠⣄⣀⣠⣠⡄⢠⣀⣄⣀⠀⣤⣤⣠⣀⡄⢠⣄⣀⣄⣀⣤⣤⣀⣠⡀
⠀⢀⣀⣀⣀⠀⣀⣀⣀⣀⣁⣀⡀⢁⣀⣁⣀⡀⣉⣀⣀⣡⣀⡀⢀⣈⡀⢀⣈⡀⣤⣀⣀⣀⡁⣀⢀⣀⡁⢀⣁⣀⣀⣩⣁⣀⣀⣤⠤⠄
⠀⠈⠉⠉⠉⠀⠉⠉⠉⠈⠉⠈⠁⠈⠀⠁⠈⠀⠉⠉⠉⠉⠉⠁⠈⠉⠁⠈⠉⠀⠉⠉⠉⠉⠀⠈⠈⠉⠀⠀⠁⠈⠀⠉⠉⠉⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⢈⣉⣉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠶⠶⠦⣄⠀⠀⠀⢀⡴⠞⠋⠛⢦⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠈⣷⠶⠞⠋⠀⠀⠀⠀⠀⠈⠉⠀⠀⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠐⠀⠀⠒⠐⣄⠀⠀⠀⠐⠃⣠⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠢⠀⠀⠀⠀⠀⠻⠷⣶
⠀⠀⠀⠀⣠⡶⠶⠚⠛⠳⠐⠒⠒⠦⠀⠀⠀⠀⠀⠀⠠⠁⠂⠀⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠲⠤⠠⠄⠀⣤⣦⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⣦⣦⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⠿⠁⣠⣆⣀⢀⣠⠼⠶⠄⠀⠀⠀⠀⠈⣿
⠀⠀⣹⡶⠀⠀⠀⠀⠀⢀⡀⠀⠀⢉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⢀⡿⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣤⣀⣤⡄⠀⠀⠀⠀⠀⠈⠀⠀⠱⣀⠀⠀⠀⠀⣠⣤⡿
⠀⠈⢷⣤⣤⣤⣄⡀⠐⠦⠀⠀⠀⠐⠀⠀⢠⠀⠀⠖⠻⣿⠋⠈⠀⠀⠀⠀⠀⢂⠂⠀⠀⠀⠉⣹⡂⠀⠉⠙⠛⣷
⠀⠀⠀⠀⠀⠀⠙⣿⠄⣀⠀⠀⠒⠦⠶⣴⡶⣶⡤⣶⣶⣿⣤⣀⡀⣀⣀⣀⠀⢀⡐⠂⠺⣶⡊⠁⠀⠀⠠⠀⢠⡿
⠀⠀⠀⠀⠀⢀⣴⠏⠈⠁⠀⠀⠀⢀⢀⡈⠑⠈⡻⡟⠚⠋⠈⠿⠉⠛⠱⠿⣿⡙⠒⠀⠀⠈⠓⠲⢿⣶⣶⠶⠛⠁
⠀⠀⠀⠀⠀⠿⣤⣬⡴⠶⠶⢶⡥⠈⠀⠀⠑⠀⠀⠈⣠⣤⣀⣀⠀⠀⢺⡷⢦⣀⠀⠀⢀⣀⣀⣀⣠⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠞⠛⠛⠛⢿⣿⣝⠛⢊⣹⡷⠾⠋⠁⠀⠉⠛⠳⠾⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠐⡠⠀⠘⠉⠀⠀⠀⠈⠀⠀⡀⠀⢀⠄⠔⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢤⠄⠀⠠⡀⠀⠁⠀⠀⣀⣠⠤⠤⠤⠤⠤⢄⡀
⢰⢦⣂⣙⣙⣉⣉⣉⣉⣉⣉⣉⣉⣢⣄⠀⢀⡤⠚⠉⢀⡠⠤⠰⠄⠀⠈⢢⢀⡴⠋⠁⣠⠤⠖⠒⠛⠛⠛⠛⠷⣤⡀
⢘⠀⣹⠉⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠊⢄⡴⠊⠁⠀⠀⠀⠀⠀⠀⢸⡋⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⡛⠀⣿⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣸⠇⢀⡞⠀⠀⠀⠀⢀⡠⠤⠤⢠⠼⠂⣸⠁⠀⠀⠀⣠⠔⠐⠒⢦⣄⠀⠀⠀⠀⠹
⡇⠀⡏⠀⠀⠀⠀⠙⠛⠛⠛⡄⠀⠀⡎⠀⣸⠀⠀⠀⠀⡔⠁⠀⠀⠀⣼⠀⢀⡇⠀⠀⠀⢸⠁⠀⠀⠀⠀⡏⢳⡀
⡇⢀⡇⠀⠀⠀⢀⡀⣀⣀⡠⠇⠀⠀⣯⠀⣟⠀⠀⠀⠰⣇⡀⠀⠀⠀⢻⠀⢸⡅⠀⠀⠀⢸⠀⠀⠀⢀⡼⠁⢸⡇
⡇⠨⡇⠀⠀⠀⢸⠖⣒⡒⠚⠛⢒⡓⢿⡀⢳⠀⠀⠀⠀⠫⣌⣉⣉⡉⠙⣆⠀⣇⠀⠀⠀⠈⠻⣒⢊⠉⣀⣠⠟⠀⠀⠀⢀
⣗⢠⡇⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⢹⣜⣆⠀⠀⠀⠀⠀⠍⠉⠉⠉⠙⣦⡘⢦⡀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⢠⠞
⠙⢮⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠙⢶⣄⣀⡀⠀⠀⠀⠀⠀⠀⣇⠙⠶⣝⣦⣄⡀⠀⠀⠀⠀⠀⠀⣠⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡿⣿⡶⣶⣀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡱⠟⢽⠯⠋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⢀⣘⣧⣰⣤⣶⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⢾⣥⣿⣿⣷⣾⣯⡈⠉⠛⠻⠞⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⣿⣷⣶⣦⣄⡀⠀⠀⠀⠀⠀⠜⣱⣾⣿⣿⣿⣿⠟⠛⣿⣿⣦⣤⡀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣶⣌⠲⡦⣸⣾⡏⠀⠀⠀⠀⠀⠀⢀⡿⣽⣿⣿⣷⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⠄⢀⡀⠀⠀⠀⠀⠉⠻⡿⢃⡔⢿⣿⣷⡾⠧⠀⠀⠀⠀⣸⣷⣿⣿⣿⠟⡁⠀⡄
⠀⠀⠀⠀⠀⠀⢀⡐⢊⡤⠖⣛⣛⣛⠻⢷⣦⣀⠠⡤⣎⣴⢻⣿⣸⣿⡇⠐⠒⣶⠶⠂⠈⠉⠉⠉⡉⠋⠉⠠⡤
⠀⠀⠀⠀⠀⠀⠎⢠⠏⡐⠀⠀⢀⠀⢙⠒⣌⡻⠃⢠⣿⣿⣇⢿⣧⢣⡿⡀⣦⡈⠀⠀⠀⢐⡀⠀⠈⠀⣠⠄
⠀⠀⠀⠀⠀⢐⠀⡟⠀⠀⠐⢄⡴⠂⠒⣂⠈⠀⡴⢿⣽⣿⣿⣼⣿⡆⢰⣣⣿⣿⠶⣄⠢⣀⠿⣀⡀⠔⠁
⠀⠀⠀⠀⠀⢸⣾⡇⠁⠀⠄⣾⠀⣾⣟⡧⣀⡈⡄⠘⣿⠫⠿⢧⣤⢻⣮⢾⣿⣿⣞⢧⠀⠀⠀⡿⢶⣤⣀
⠀⠀⠀⠀⠀⠸⣿⣿⡀⡀⢀⠼⢧⡈⠛⠋⡉⠀⠀⠀⣿⡆⠴⣿⣿⠈⣧⢰⢾⣭⣬⣆⠂⠀⠈⠀⠀⠈⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠹⣿⣷⡐⠀⠀⠌⠉⠹⠁⠀⠁⡂⢀⣿⢟⣸⣽⠾⠴⠧⢖⡛⣿⣾⣿⡆⠀⠒⠀⠀⠀⠀⢀⣤⣴⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠤⠀⠀⠀⡀⠔⣡⣾⠏⠀⢸⢀⣤⣄⠶⣘⣇⣾⣟⣻⣥⠀⠀⠀⢰⣒⣾⡀⠈⠉⠛⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣶⣦⣴⣶⣾⡿⠋⠀⠀⠀⠈⡅⠠⠈⠻⢿⡮⣿⣯⢶⣆⣴⣾⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣝⣾⣿⣟⡿⣭⡟⠏⠙⠄⠉⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⡷⣿⡀⠀⢴⠤⠦⠈⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⣿⣿⣶⣄⠘⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠙⢻⣿⡎⡑⠖⠈⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⠀⠁⠘⠳⡄⠡⠆⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⡀⠄⠀⠀⠀⢀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣦⣥⣶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢒⣈⣹⠖⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡴⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠠⠤⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠎⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠠⢴⡴⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⢒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⣂⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⠤⢀⡀⠀⠀⠚⢋⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⡩⢂⠡⠐⣁⢅⠠⠤⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⣡⡤⠶⠖⠒⠶⢶⣤⣁⠢⣀⢞⢤⠱⡀⠀⠀⠀⠀⠀⠀⡠⢊⠔⡡⢊⢕⠨⢊⣹⡜⡶⠲⡶⠶⠶⡣⢄
⠀⠀⠀⠀⠀⠀⣠⢊⡴⠋⠉⢂⠀⠀⠀⠀⡎⠀⢉⢟⣵⠻⡀⠡⡱⡀⠀⠀⣀⠔⡪⠚⢁⢎⠔⡱⢡⣶⠋⠁⢁⢡⠀⡇⠀⠀⠀⠑⣽⢦
⠀⠀⠀⠀⠀⠀⡘⠝⠁⠀⠀⠀⢢⠀⠀⡸⢀⢔⠕⠁⣈⣷⡘⡄⠱⡈⡨⠥⠒⠁⠀⡠⡡⠃⡜⣰⠏⠀⠙⢦⡀⢆⢢⡇⠀⠀⡠⠊⠀⠑⣕⡀
⠀⠀⠀⠀⠀⠠⡘⠃⠂⠤⢀⡀⠀⢣⣠⢗⣵⠡⠒⠉⠀⠘⣇⢰⠀⢱⣴⠤⣤⣠⢊⠔⠀⢰⢀⡟⠀⠀⠀⠀⠉⢺⡜⡇⡠⠊⠀⠀⠀⠀⠘⡤
⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⢉⢾⣁⡛⣁⣄⡀⠀⠀⠀⣉⢈⠁⢗⡅⠠⡈⡟⠁⠀⠀⢀⣸⡗⠒⠒⠒⠒⠒⠒⣳⣽⡐⠒⠒⠒⠒⠒⠒⡇
⠀⠀⠀⠀⠀⠐⣱⠀⣀⠤⠒⠈⠀⡘⠀⢢⠀⠀⠁⠂⠒⢰⡙⠛⠀⠈⠛⠒⠻⡀⠀⡀⠀⠈⠙⡰⠀⠀⠀⠀⡠⠊⠀⡇⠙⢦⡀⠀⠀⠀⢠⡁
⠀⠀⠀⠀⠀⠀⠱⡹⡀⠀⠀⠀⢀⠃⠀⠀⢣⠀⠀⠀⢠⠱⠁⠀⠀⠀⠀⠀⠀⠈⠚⠚⠃⠀⠀⠱⣧⡀⡠⠊⠀⠀⠀⡇⠀⠀⠙⢦⡀⡠⡱⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠮⡢⢀⠀⡌⠀⠀⠀⠀⢃⡠⣂⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢝⠦⣀⠀⠀⠀⡇⠀⠀⢀⠠⡫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠫⠵⠒⠒⠒⠬⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠫⠔⠒⠓⠒⠬⠒⠉






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠒⠊⠉⠉⠉⠉⠉⠉⠓⠒⠲⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠉⣀⣠⣶⢀⡴⣛⣥⠾⠋⠉⠉⢻⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠻⠟⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⢻⣾⣿⣏⣡⢼⡟⠉⠳⡄⠀⠀⠀⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡃⠀⠀⠘⠛⠻⢿⣿⡧⣟⠀⢰⠇⠀⠀⢀⡾⠀⠀⠀⠀⠀⢠⡶⠶⠶⢤⡴⠋⠙⠓⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢃⡟⠀⠀⠀⢀⡴⠛⢉⢿⣰⠋⠀⡏⠀⡴⠞⠋⣤⣤⠀⠀⠀⠀⣸⠃⣴⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡾⠀⠀⠀⠀⢾⡀⣴⠋⠦⡧⠀⠈⠧⠞⠀⠀⠀⠉⠉⠀⠀⠀⢿⣁⠐⡏⢳⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣼⠁⠀⠀⠀⠀⠀⣿⠃⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡀⢉⣙⣧⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⢀⡴⠒⢦⡀⠀⠀⢠⠃⢠⡏⠀⠀⠀⠀⠀⠀⠀⠠⢶⡞⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢯⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢀⡴⠶⢦
⡞⠀⠀⠀⢹⡄⠀⣼⠀⢸⠃⠀⠀⠀⠀⠀⠀⣠⠴⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣪⠀⠀⣀⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⢠⡏
⡇⠀⠀⠀⠀⣇⠀⡇⠀⢸⣆⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣰⠎⣁⡈⢿⠷⣮⠓⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸
⡇⠀⠀⠀⠀⢸⠀⡇⠀⠘⣿⠀⡴⠒⢦⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠛⠉⠁⠙⢦⣠⠶⠲⠖⣁⠀⢰⡛⣦⡴⢦⠀⣾⣿⠀⡇
⡇⠀⠀⠀⠀⢸⡀⡇⠀⠀⠸⣆⣧⣤⢶⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣝⢇⠀⠉⢩⠇⠸⣿⣿⣿⠀⡇
⡇⠀⠀⠀⠀⠈⡇⢳⠀⠀⠀⠈⠳⣼⡆⠙⠷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡀⣠⠞⠀⠀⣿⢹⡏⢸⠁
⡇⠀⠀⠀⠀⠀⡇⠘⡀⢠⠔⠢⣄⠈⠻⣄⣴⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣇⣠⠖⢦⣄⣼⠁⢸
⡇⠀⠀⠀⠀⠀⢳⠀⢻⡏⠀⠀⠈⢣⠀⣸⠇⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⢹⡟⠀⡾
⡇⠀⠀⠀⠀⠀⢸⠀⠸⡇⠀⠀⠀⠈⡆⢧⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⠀⠀⠀⠀⠀⢀⡞⣻⠀⠀⠀⠀⢸⠇⠀⡇
⢳⠀⠀⠀⠀⠀⢸⠀⠀⢷⠀⠀⠀⠀⠸⡌⢧⡀⠀⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣼⣡⠇⠀⠀⠀⠀⡿⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠘⡆⠀⠘⡇⠀⠀⠀⠀⠹⣆⠙⢦⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢀⡜⣿⠏⠀⠀⠀⠀⣸⠃⠀⢰⠃
⢸⠀⠀⠀⠀⠀⠀⢧⡀⠀⢻⡀⠀⠀⠀⠀⠙⢦⡈⢇⠀⠀⠀⣴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡴⢋⡼⠃⠀⠀⠀⠀⠀⡟⠀⢀⡼⠀⠀⠀⠀⠀⢀
⠈⡇⠀⠀⠀⠀⠀⠀⠉⠲⣜⣇⠀⠀⠀⠀⠀⠀⠙⢾⣆⢀⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢋⡴⠋⠀⠀⠀⠀⠀⠀⣸⣣⠖⠉⠀⠀⠀⠀⠀⠀⢸
⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⠤⣤⣤⣄⣀⣀⣠⣤⣤⠤⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⠈⣉⣁⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠆⠠⠀⠀⠃⠀⠑⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⢰⠐⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⢀⠀⠐⠂⡰⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠤⠐⠚⠁⠀⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡐⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠳⠤⢄⠀⢀⠠⠊⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡋⠀⠀⠀⢐⠂⠀⠀⠀⠀⠸⠶⢖⠒⠲⠦⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠠⠀⠐⠒⠦⢤⠁⠀⠀⠀⢀⡀⠀⠀⠀⢀⡃⡀⠀⠐⠄⢀⡀⢈⡒⣄⠀⠀⠀⠀⠀⠀⠀⠈⠒⣕⢄
⠀⠠⢂⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⡀⠙⠢⡀⠀⣠⡁⠀⠀⠀⠀⠐⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠔⠃⣄⡄⠀⠉⠒⠤⢁⣴⠋⠀⠀⠀⠀⠐⠄⢀⠀⠀⠀⠀⠀⡄
⢰⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠹⠂⠀⠀⠘⠭⢶⠏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠈⡈⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠊⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠱⡐⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠈⠃⡀⠀⠀⠀⠀⠀⢀⠀⡢⠂
⠀⠀⠈⠪⠐⠄⢀⠀⠀⠀⡀⠡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠒⠉
⣀⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣭⣿⣯⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡦⣄⠀⠀⠀⠀⠀⠀⠀⡠⠄⡖⣿⢐⠺⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠛⠛⠀⢀⠀⠀⣰⡊⠀⢸⣇⠐⠂⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⡈⢿⠿⡇⠀⠀⠀⠀⠁⢀⣴⠉⠀⠀⢠⡎⠀⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⠘⠂⠈⡀⠀⡀⢸⡋⠀⠀⠀⢸⠻⡄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⡙⠒⠲⠾⡀⣟⡄⠀⠀⠀⠈⢧⣿⣤⡴⠛⠙
⠀⠀⠀⠀⢀⠄⠠⠳⠀⠸⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡸⠀⠀⢸⡇⠀⠀⠀⠠⣄⣘⣿⠤⢀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠒⢈⠀⡆⠄⠘⣧⠀⠀⠀⠀⠀⠀⠁⠁⠌⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠰⠀⠎⠀⠀⠀⡀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⡨⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠂
⠀⠀⠀⠀⠀⡄⢀⠀⠀⠹⠀⠀⠠⣀⡠⠆⠐⣶⠀⠀⠤⠀⠈⠀⠀⠄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⡤⠈⠀⠀⠀⠍⠀⢰⣶⡄⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣼⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⢰⣿⣿⡀⠀⠀⠀⠀⣘⠀⠀⠀⠀⠌⠁⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⢹⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠅⠀⠁⠀⠀⠀⠁⠀⠀⠉⠉⠁⠂⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠠⡀⠀⠐⠊⠀⠀⠀⠴⠉⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠰⠀⠀⠉⠰⠂⠀⠀⠀⠀⠀⠠⠄⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠚⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⢿⣷⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣒⠠⠖⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢷⡔⠋⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸⣷⣄⣀⣀⣤⣤⣶⣶⠿⠿⠛⠛⠛⢿⣿⡆
⠀⠀⣾⡟⠒⣀⣀⣐⣚⣿⠿⠟⠒⠒⢦⣴⣿⣿⠛⠋⠉⠉⠀⠀⠀⠀⠀⡠⣾⠞⢿⠿⣄⠀⠀⠀⠄
⠀⠀⡨⣿⣖⡩⠄⠐⠒⠀⠀⠩⣒⣴⣾⠟⠹⡿⡀⠀⠀⠀⠀⠀⠀⢠⢮⡾⢁⣴⠞⢣⠹⡄⠐⠂
⢠⢎⢞⠔⠙⠄⠀⠀⠀⠀⠀⣀⡾⠛⠉⢂⠀⢻⣇⠀⠀⠀⠀⠀⣔⡵⠋⣰⡿⠑⠁⠀⢠⢳⠀⠀⠀⠀⠀⠀⠊⡂
⣤⣮⣎⣀⡀⠀⠢⡰⠀⣠⢾⠟⠁⠀⠀⠀⠃⠘⣿⡄⠀⠀⡠⣢⠟⠁⢐⡿⣄⣀⠀⠀⠀⢣⣣⡀⠀⠀⠀⠀⠀⠈
⠅⣼⠁⠈⠉⠙⠾⣿⣫⡕⣯⣤⣤⣤⣤⠤⠭⣤⡽⢧⣤⢎⡞⠁⠀⠀⣿⠀⠀⠀⠉⠉⠒⢢⣿⣷⡂⠀⠀⠀⠀⠀⠐
⠀⢿⠀⠀⠀⠀⠰⠙⣿⠿⠛⠓⠛⠛⠩⠿⠿⣟⡀⣹⣏⠔⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠈⠋⠢
⠀⠘⠇⠀⠀⠀⠀⠀⠈⢹⡀⠀⡀⠀⢀⣐⣀⠘⣍⣶⣅⡈⠀⠛⠛⠃⠀⠀⠱⡀
⠀⠀⠀⠰⣀⠀⠀⠀⠀⠈⠉⠋⠉⠭⠉⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠍⡂⠀⠄⠠⠀⠂⠠⠜⠋
⣶⣶⠶⠶⠶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣾⣿⣿⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⣀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣾⣿⠛⠛⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⡍⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢿⡟⠠⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠁⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⢀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠹⢬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠈⠀⠐⠀⠀⠀⢀⠀⠀⠀⠄⠀⡤⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤⠄⠀⠀⠀⠀⣸⡆⠀⠀⠐⡀⠁⠀⠈⠀⣠⠊⠁⢘⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣷⠶⠶⣦⣤⣀⠀⢀⠴⠃⠹⠈⠀⠄⢵⡂⠀⠀⣰⣡⠔⠊⢉⣳⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⣠⡾⠋⠁⢹⠀⠀⠀⠈⢹⣷⣏⠀⠀⠀⠀⠣⣀⠄⠉⠀⡰⠋⠀⣠⡾⠛⠙⡏⠉⠉⠉⠛⢷⣄
⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⡇⠀⢀⠔⠁⠀⣻⣆⣀⣀⣈⣠⡇⠀⠀⡼⠁⠀⣼⠏⠀⠀⠀⢹⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⣿⣾⣍⣉⣉⣍⣽⣿⣤⣴⣯⠘⡬⡶⠊⠀⠀⢸⡟⠀⠀⠀⠀⣨⣷⣄⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⣿⡄⠀⠀⠀⠈⠛⠛⠒⠒⠒⠲⢶⡿⠤⠼⠷⡯⠴⠁⠀⠀⠀⢸⣇⠀⠀⠀⠀⠹⠿⠟⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⡐⠩⠁⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠈⠛⢶⣤⣄⣀⣀⣠⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣄⡀⠀⠀⠀⢀⣠⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠐⠐⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠂⠀⠀⠀⠉⠂⢄
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡰⠀⠈⠈⠀⠀⠀⢀⣀⠀⠢⠈⠀⠀⠀⠠⠄⠉⠑⠀⠁⠈⠄⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠠⠊⡐⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⠂⠐⠳⡄⠀⡀⠘⡀
⠀⠀⠀⠀⠠⠁⠌⠠⠂⠀⠀⠀⠊⠀⠐⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠱⠀⠀⢙⣈⠁⠀⠀⢡
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢹⣃⠀⠸
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠄⠁⠀⠀⡤⠂⠀⠀⠀⠀⠈⠀⠀⢀⣤⠦⣬⠀⠀⠀⢸⡆⠐⡈⣀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠁⠀⠀⠀⢠⠣⠀⠀⠈⠂⠂⠀⠁⠀⠀⠈⠑⠈⡑⢫⠋⠩⠂⠈⠙⣗⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠀⠀⠀⠀⠀⠁⠈⠈⠙⠉⡊⠈⠀⠀⠀⠀⠀⡆⠀⠀⠀⡀⠀⠀⠀⠉⢪
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⡠⡀⣀⠂⠀⡁⠀⠀⠁⡀⠀⠁⠉⠡⡞
⠀⠀⠀⠀⡂⠀⠀⠀⠈⠀⠀⠂⡀⠂⠀⠀⠐⠀⠀⠀⠀⠎⠁⠉⠈⠉⠉⠠⠀⠀⠈⠑⡦⠂⠀⢠⡁
⠀⠀⠀⠀⠅⠀⠀⠀⠀⠐⢠⠐⠈⠳⠤⠜⠱⢃⢑⣦⡌⢀⠐⠀⠱⠰⠄⠖⠄⠤⡤⠄⠀⠀⠘⣠⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⢀⠀⠀⠉⢏⠀⠈⡛⠋⠋⢻⣀⠀⠠⠂⠐⢂⠀⠐⠀⠐⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⡜⠃⠠⠔⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⡀⢀⠀⢀⢀⠄⠀⠀⠠⠄⡀⢡⡄⠀⡀⠀⠀⠀⠀⠀⠀⠂⠄⡀
⠀⠀⠀⠀⡄⠀⠂⠀⠂⠀⡄⠆⡬⢡⢀⣶⠄⠀⠀⠀⠀⠈⠀⠀⣀⠙⠦⡀⠸⠘⠀⠠⡀⠂⡦⡤⢀⠀⢠⡀
⠀⠀⠀⠀⡇⠀⠀⠳⠖⢂⡤⠤⠄⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣁⣀⣁⣀⡀⠠⠤⣤⡰⠶⠾⢊⡇
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠀⡄⠈⡧
⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡄⠀⡧⠀⡠⢀⠄
⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡠⢁⠌⡀⠀⠂⠀⡀⠁⠀⠀⠄⠀⠀⠀⡠⠊⠐⡁⠐⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⣤⢄⣀⡀⠀⠀⠀⠀⠄⠡⠊⠔⠁⠈⠀⠡⠂⠈⠀⢀⣀⡄⡤⡶⢚⠁⢂⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠒⠒⠒⠲⠒⠐⠒⠒⠒⠃⠙⠩⠀⠉⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣍⡉⠉⠁⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣾⣿⣷⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⢿⣿⣿⣖⣛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⢀⣨⣿⣿⣿⣿⠀⠈⠉⢛⡿⠿⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢏⣿⣿⠟⣸⣿⣧⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⢻⣿⣿⣿⣿⡟⠁⣰⡿⣻⡿⣯⠉⠉⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠠⢻⣿⣿⣿⢻⣿⣴⡿⢱⡟⣧⠸⡆⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⣅⢨⣿⣿⣹⣾⣻⡿⠁⣿⠀⢹⡄⢻⡀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⠿⠭⣿⣿⣿⣿⡿⠁⠠⣿⠀⠈⢧⣤⠧⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠈⠃⢠⣿⣿⣿⡿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⢈⣿⣿⣿⡅⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁⠉⠉⠀⠀⠀⠀⠀⠹⣦⣀⠀⠀⣀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣤⡴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢶⡛⢣⢬⢤⠛⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢛⠩⢙⠒⡄⠀⣴⣋⡰⠤⠴⣈⣻⡄⠐⠲⣖⡶⣒⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠁⠀⠈⢱⢹⠆⠀⠉⣧⢾⢦⡏⠁⠀⠠⠾⢽⣗⢅⠜⣼⣆
⠀⠀⠀⠀⠀⢠⣶⣦⠘⢧⡕⢠⡔⢣⡼⢀⣶⣦⡀⢸⠀⢠⣶⣦⠀⠀⠸⡏⢹⡏⢩⠛⣆
⠀⠀⠀⠀⠀⣿⢽⡽⡇⠀⠈⠹⡏⠉⠀⣿⢽⡿⡇⢸⠀⡿⣽⢽⡇⠀⠀⣇⠀⢧⠀⠆⠘⣆
⠀⠀⠀⠀⠀⠘⢿⣿⠃⠀⠀⠀⡇⠀⠀⠙⣿⡿⠃⢸⠀⠻⣿⡿⠁⠀⠀⢈⡽⢺⠗⠚⠒⠚⠂
⠀⠀⠀⠀⠀⠀⡬⡧⣧⣷⢼⠤⠧⠼⠴⠦⠼⠧⠧⠼⠤⢧⣾⠤⠾⠼⠤⠾⢴⢧⡤⡧⡾⣤
⠀⠀⠀⠀⠀⠀⣇⢷⢹⢸⡘⡆⢸⡃⢰⡶⡲⢄⠀⣀⣴⢿⣿⠷⣄⡀⢎⠆⡼⣸⢱⢣⡇⡏
⠀⠀⠀⠀⠀⠀⠸⡜⣎⢧⢳⡹⡄⢜⡆⣹⢾⣾⣿⡱⢸⢾⢈⣙⢮⣿⠆⡴⣣⢃⢏⡞⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣌⢮⡳⡙⢌⠢⣐⠷⠵⠊⡠⡉⠻⣾⣿⡞⢋⡡⢞⡴⡡⢋⢞⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣕⢬⡲⣝⡪⢝⣲⠤⠵⣥⡤⠤⢵⣚⡭⣚⡥⣊⠕⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢭⣒⠭⣗⣺⠭⠭⠭⠭⠭⢽⣒⡯⢕⣪⠕⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠚⠭⠭⠭⠭⠭⠽⠒⠊⠉

⣏⣈⣿⣽⣿⣯⣯⣽⣽⣿⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⢀⣾⣿⡟⠻⣿⣿⡄⠀⠀⠀⠀⢀⣀⣀⣸⣉⣉⣹⣆⣀⣀⡀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⣠⣶⣿⡟⠛⢻⣿⣶⡀
⠀⠀⠀⢸⡿⠟⠛⠚⠛⢿⡇⠀⠀⠀⢸⣿⣿⣿⡿⠛⡛⠻⣿⣿⣿⣿⠀⠀⠀⠀⣀⣀⣸⣿⣇⣀⣀⠀⠀⠀⢰⣿⠿⠿⠇⠀⠸⠿⠿⣿⡄
⠀⠀⠀⠈⠀⠀⣀⣀⡀⠀⠁⠀⠀⠀⢸⣿⣿⣿⢠⣴⣿⣤⢸⣿⣿⣿⠀⠀⠀⠀⠿⠿⢿⣿⡿⠿⠿⠀⠀⠀⢸⣿⣀⣀⡀⠀⢀⣀⣀⣿⡇
⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣦⣀⢛⣠⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠘⢿⣿⣿⡇⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠈⠻⣿⣿⠿⠃⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⡿⠿⠟⠋


⠀⢰⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⣤⣾⣿⣷⣆
⠀⢸⡇⢰⣦⠀⢀⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⡿
⠀⢸⡇⠸⡿⠀⠸⡯⠀⠸⡧⠀⣿⠀⠀⠀⠀⠀⠀⣹⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣏⣿⣿⣷⡀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡿⠁
⠀⠸⠗⠚⠛⠒⠚⠛⠒⠚⠓⠚⠿⠀⠀⠀⠀⠀⠀⢾⣻⠇⠀⠀⠀⠀⠀⠀⠀⠠⣟⣿⣿⣤⣿⣿⣷⠀⠀⠀⠀⠀⠀⠉⠛⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⠀⠀⠀⢠⣾⣿⣿⣷⣶⢰⣶⣿⣿⣿⣧⠀⠀⠀⠀⣴⣶⡟⢛⠒⣦⠀⠀⠀⠀⠀⠀⠎⠉⣠⡴⣭⡑
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠃⠀⠀⠀⠻⠿⣿⣿⡏⠀⣿⣿⡿⠿⠋⠀⠀⠀⠀⣿⣿⣌⠻⠋⣼⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡼⠃
⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢅⡀⠀⠀⢀⡠
⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠠⠤⣄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⣠⣶⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣴⠟⢷⡜⠀⠀⠀⠀⠀⠀⣿⣿⠋⣴⡌⢻
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢠⣤⣴⣶⣶⣶⣶⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠙⠶⠞⢱⠀⠀⠀⠀⠀⠀⣿⣿⣧⣍⣥⣿
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠘⠛⠿⠿⠿⠿⠿⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠙⠻⠿⢿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣤⣀⣀⡤⠚⠀⠀⠀⠀⠀⠀⠙⢻⡿⢿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⣶⡶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠥⣽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⣰⣧⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⡠⠤⠤⠄⣄⣤⣄⡀⠠⠤⡄
⠀⠀⠀⠀⠀⠀⠠⢖⣫⣭⣶⣶⣶⣮⣭⣵⡿⣌⢷⡴⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢯⣜⣡⣖⣮⣽⣿⣶⣾⣿⣭⣷⡧⣄
⠀⠀⠀⠀⠀⣠⣾⣿⠿⠛⠋⠉⠉⠉⢻⡿⣿⣦⡙⣿⣆⠘⢷⡄⠀⠀⠀⠀⠀⠀⣼⠃⣴⣿⣿⡿⠟⠛⠉⠉⠉⠉⢻⠿⣿⣮⡳⣄
⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⢠⡟⠁⠈⠻⣿⣮⢟⢷⣤⠟⣦⡀⠀⠀⢀⣼⢃⣾⣿⣿⠛⠳⣄⠀⠀⠀⠀⢠⠃⠀⠈⠻⣿⣞⢦
⠀⠀⠀⣾⣿⠅⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢹⣿⡇⠀⠻⣆⠠⢽⣭⠉⣹⠃⠘⣿⣿⠁⠀⠀⠈⢳⡄⠀⢠⠇⠀⠀⠀⠀⢻⣿⡎
⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⣴⣿⣿⡆⠀⠀⠀⠀⠀⣿⣿⠀⠀⠘⢷⣨⡤⣴⠧⠤⢼⣿⡗⠒⠖⠂⣒⣒⣿⣿⣿⡆⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣿⣯⠀⠀⠀⠀⠀⠘⠻⠟⠁⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⣎⣿⣿⣿⣷⡤⠾⣿⣿⠟⢛⣋⡩⠽⠿⠛⠛⠁⠀⠀⠀⠀⢐⣿⡇
⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠃⠀⠀⠀⠣⣱⢿⣛⡾⠠⠄⢻⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡿⠁
⠀⠀⠀⠀⠙⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⣿⡿⠃⠀⠀⠀⠀⠀⠀⠉⣅⡀⠀⠀⠀⠹⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⣶⣤⣶⣶⣾⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⣶⣴⣦⣶⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠋⠁⠀⠈⠉⠻⡿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠹⡇⠀⠀⠀⠀⠀⠀⢱⠹⣿⣿⣿⢴⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⠀⠀⠺⠇⠀⠀⠀⣿⣿⣿⣿⣿⡛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠘⠙⡿⣿⣿⣯⣇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠖⣈⣴⣿⣿⣾⠏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡉⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠿⠦⣀⣬⠋⢀⣄⠀⣀⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⢃⣴⣶⠾⠿⢷⣿⡏⠀⠀⠀⠀⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣾⡏⣠⣴⡶⣮⠙⠃⠀⠒⠰⠤⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣦⣀⡉⠙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠶⠦⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠂⠒⠀⠆⠀⠀⠒⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢏⢀⡀⣈⢀⡁⢈⡉⣙⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⡁⣨⢀⡇⢨⡀⣩⢸
⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠠⠠⠦⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣀⡀
⠀⠀⠀⢀⣤⣤⣶⠿⠛⠛⠘⠛⠻⢷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠘⡛⡛⠛⠛⠻⢿⣿⣦⣄
⠀⠀⣠⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⣷⡀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧
⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⢠⠀⠀⠴⠀⠀⢹⣧⠀⠀⠀⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠁⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠹⣿⣧
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠛⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⢀⣶⣶⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣿⣿
⠀⠀⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⣿⠏⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠇
⠀⠀⠈⠻⣿⣷⣄⡀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⡀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠋
⠀⠀⠀⠀⠈⠛⠿⣿⣿⣶⣶⣾⣿⡿⠟⠋⠀⠀⠀⠂⠄⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⣶⣶⣶⣿⡿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⠨⡉⠀⠀⠐⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⡆⠈
⠀⠀⠀⣀⣤⣤⣀⣀⠜⢱⠤⠂⡀⠁⢀⢔⣱⡤⣤⣂⢀⣠⣤⣄⡀⢀⠜⠀⠀⠀⠀⣰⠡⢸⣀⣀⡀
⠀⢀⡞⠁⠀⠀⡠⠛⣆⠀⣇⡆⢀⠔⣱⠋⠀⠄⠀⣸⢻⡀⠀⢀⠝⢧⠀⢀⡰⠀⡐⢁⡞⠉⠃⠀⠉⢲⡀
⠀⢸⠀⠀⠐⠿⠀⠀⣿⢒⣯⠘⠁⠀⣇⠀⠀⠻⠂⡇⢨⡇⠾⠗⠒⢺⡗⣷⡙⠊⠀⢸⠀⠀⠸⠆⠀⠀⡇
⠀⠀⠳⣄⡀⠀⣀⡴⠃⠀⠈⠁⠀⠀⠘⢦⣀⠀⢀⣹⢞⡀⠀⢀⣠⠞⠀⠐⠁⠀⠀⠘⢦⡀⠀⠀⢀⡴⠃
⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⣿⠀⢶⣴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠠⣴⣿⣾⡂⠀⠀⠁⠀⠀⠀⠀⢸⡇⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⠏⢀⣬⡄⠹⢿⡟⠁⠀⠰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢶⣄⣹⡏⠀⠈⠛⠛⢀⣤⡄⢇⣀⠀⠀⠀⠀⢴⣿⣿⣿⣦⠀⠀⠀⠀⡄⠈⠀⠀⠀⠈⠀⡿⢃⣤⡀⣀⠀⠀⠀⡴⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠚⣿⣭⣻⡷⠖⣰⢃⣼⣶⣷⡄⠀⢀⠀⢒⠂⢻⣿⣿⣿⡿⠃⠀⠀⠈⠀⢀⣄⠈⠶⠄⠈⠁⣘⠛⠙⠿⠁⠞⣿⠁⠀⢠⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠸⠈⣿⠿⠏⠁⢀⡉⢸⣿⣿⣿⡿⢀⡃⠀⣀⠀⠀⠙⠙⢿⡠⡆⠀⠐⠛⠐⣤⡈⠑⠀⢀⡢⢠⡟⢰⣇⣠⠄⠀⢤⣀⡙⢾⠀⠀⢙⡃
⠀⠀⠀⠀⣠⠟⢠⣶⣷⠀⠐⣶⡿⣧⡤⠉⠛⠉⣠⠘⢧⣀⠁⠰⠐⠆⠀⠸⠗⢀⣠⡄⠀⠀⠀⠉⠀⣀⠀⠀⠛⠠⢾⣯⣿⣾⣿⣷⡄⠀⣠⣦⡀⠀⣛⠄
⠀⠀⠀⠀⠿⠀⣿⣿⢏⡀⢿⣿⣿⣿⡄⠂⠀⢾⠃⠀⢀⠙⠀⣠⣤⡀⢀⣀⠙⠉⠉⠀⠀⠀⠀⠀⠀⠙⠿⠲⠄⣰⣿⣿⣿⣿⣿⣿⣿⡀⢿⣿⠏⠀⠀⠃
⠀⠀⠀⠀⠀⢸⣿⠏⠸⡇⢸⣿⣿⠋⠀⢄⠀⠀⠀⣀⠸⣇⠀⢨⣽⣷⣿⣿⣷⡀⠀⡙⠁⠀⠀⠀⠄⢠⢀⠀⢀⡘⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⡆⠄
⠀⠀⠀⢀⡀⠘⣃⠀⣤⣆⠀⠈⡀⠀⣀⢚⡄⠧⠻⠌⠀⠉⠀⠸⣿⣿⣿⣿⣿⠇⠀⠁⠀⠀⣄⠀⠀⠀⠀⠀⠘⠁⢈⠻⣿⣿⣿⣿⠿⠃⢰⠀⠈⠀⠀⠹
⠀⠀⠀⠈⣇⠈⣿⣷⣈⢻⣿⣠⣤⡀⠈⣉⣁⡀⢀⠘⠓⠚⠛⠀⠙⠻⠛⠛⠁⢰⣤⣴⡆⠀⣬⡃⠀⠀⠀⠀⣰⣿⣿⠄⠈⠛⠋⢉⡴⠖⠘⢄⠀⠄⠀⠈⣤
⠀⠀⠀⠀⠙⠁⠛⠛⠛⠷⣿⣏⡉⢀⡾⢹⡉⢻⡝⠷⠀⢀⢈⠁⢀⣠⣬⠑⠒⢈⡀⠈⣠⣄⠈⠛⠶⠀⢀⡀⠈⠙⠏⠀⣰⣷⠀⠙⢁⣤⣤⣠⣤⡄⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⣷⠀⣾⣿⣿⡿⠘⢧⣈⢁⣼⣯⠑⣀⣤⣾⣄⠘⣿⣿⠃⠀⠘⠿⢾⣿⠏⠀⠞⠀⢠⠞⣿⣄⠀⣠⣶⣿⣿⠀⠀⢻⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠙⢻⠃⠀⢀⣉⣉⠀⠉⢸⡏⢸⡄⢻⡄⣈⠁⣠⣾⣿⣿⣿⣷⡆⢸⠀⠣⡈⠛⠛⠁⠀⠻⣿⣿⠟⠀⠀⠄⠙⢿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣖⣄⠀⠘⣿⣿⡟⠀⠀⠈⠳⢦⡤⠟⢠⡟⣼⣿⣿⣿⣿⣿⣿⠃⠰⡄⠀⠉⢠⣼⣿⡆⠀⠀⣠⠀⠞⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣬⠓⣄⡀⠀⠁⠘⠀⠀⠐⠀⢠⠀⠋⠀⠈⠻⣿⣿⡿⠿⠋⠀⣠⣴⣾⣆⠙⠿⠟⠁⠚⠋⢩⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢯⣽⡶⠖⠀⠀⠀⡀⣼⡼⠆⠀⠀⠀⠀⣠⣶⣾⡂⠈⢳⣿⣿⠏⠰⣾⣿⣿⠀⣾⣿⡄⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢶⡌⢳⠀⠀⢰⣶⣶⡆⠀⢉⠛⠋⠃⠀⠀⢈⡀⠠⣄⠈⠋⠋⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠈⡄⠀⠀⠀⠀⠻⠏⠀⠀⠁⠈⢰⢄⠴⠀⢸⠀⠀⠀⢰⠇⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⢠⣶⣥⣤⡀⠀⠀⢌⠡⠇⠀⠀⠀⠀⠈⠳⠄⠀⢠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢿⣿⣿⣶⣤⡀⠀⠀⣴⣿⣿⣿⣷⡆⠀⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠃⠀⠀⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⣤⣤⡈⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⠷



⣿⣿⡿⠿⣿⡿⢿⢿⡿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣷⣿⣵⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠈⠁⢒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⡜⢩⣦⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠰⢧⣤⣿⣇⠀⡎⢀⠜⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠃⠀⠀⠀⠀⠀⢠⠃⠈⢿⣿⣿⠀⠳⠧⠤⠊⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⡎⠀⡠⠤⣉⠀⠀⠀⠀⢀⣴⠋⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡸⠀⢘⢀⠔⠉⡇⠀⠀⣼⣿⣿⣷⠯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣰⠃⠀⠈⠓⠒⠊⠀⠀⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⠀⠈⠑⠒⠁⠀⣸⠀⠈⠑⠒⠉⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⡀⠀⢀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠈⠉⠁⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠒⠒⠒⠒⠒⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣶⣤⣤⣤⣶⣾⣿⣿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡿⡟⠿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⢷⡎⠀⠀⠀⠀⣀⣠⡶⢲⣶⠶⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⠇⠀⣀⡤⠚⢧⣤⡴⠾⢧⣁⡤⠽⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡾⢿⣿⣿⣠⠎⣉⡸⠖⠉⠀⠀⠀⣼⣿⢀⣀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⡈⠣⣀⢯⣿⡵⠞⠉⠀⠀⠀⠀⠀⢠⡇⢸⣣⡀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⣿⣿⣿⣿⣿⠗⠾⣯⣿⣄⣀⣀⣀⣀⣀⣀⣠⡞⢷⡀⣳⣷⣖⢋⡹⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠄⠠⣶⡿⠿⢶⣼⣿⣿⣿⣿⣧⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⠉⠉⠙⠳⢼⡏⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠙⠛⠛⠻⢻⣿⣿⣸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠙⠷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣭⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⡰⢡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⡷⠋⢻⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⡰⠁⣿⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⢰⣿⣯⣽⡅⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⢰⣤⣀⡀⠶⠚⠁⢀⣿⡄⠀⠀⠀⠀⠀⢸⣄⡀⠀⠈⠉⠙⠋⠁⠀⠀⠘⢷⠀⠀⡆⠀⠀⠀⠈⠲⣦⡹⣄
⢺⡇⠀⠀⠀⢀⣠⠺⠛⢧⠀⠀⠀⠀⠀⣾⣿⣿⠷⣶⣦⣤⣀⣀⠀⢀⣀⣈⡷⠶⠷⢤⣄⣤⡀⠀⠈⠙⠻⠷⣄
⠀⠉⠿⢧⠔⠛⠁⠀⠀⠘⡆⠀⠀⢀⡾⠁⠀⠙⢧⠀⠙⣇⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⢤⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣠⠏⠀⠀⠀⠀⠈⡇⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⡿⠁⠲⠋⡼⢦⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⢰⠏⠀⠀⠀⠀⠀⠀⣿⡀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⠎⠀⠘⡇⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣈⣢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣍⣛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣆⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣥⠬⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⣿⡄
⣟⠁⢹⣋⣛⣛⣛⣛⣏⣛⣛⣛⣟⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣻⣟⣻⣿⣟⣛⣻

⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⣿⠶⣭⡻⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢌⠉⠁⠐⠉⠟⠋⠉⠁⠉⠉⡉⠙⠉⠉⠉⠉⠩⠇⠀⢉⠭⡉⠬⠋⠉⡁⠻⠛⢁⠈⣴⠥⣁⡁⠉⠉⣿
⠀⣿⠀⠙⠧⠀⠀⠂⢄⣁⠐⠂⠀⠀⣂⠀⢈⠀⠈⠀⠀⢀⡇⠀⠀⢀⠀⡔⠰⠀⢀⠀⠀⠀⠐⣠⡘⠀⠀⠘⣀⡀⠀⣿⣴⣍⣦⠧⠀⣠⠤⠃⠀⠆⣿
⠀⣿⠀⠀⢣⠀⠀⠀⠀⢁⠄⠈⠀⠉⠤⢜⠠⢈⠁⢉⠇⢀⠈⠄⠀⠀⠂⠑⠀⠁⠀⠘⠄⠠⠎⠁⠀⠀⠠⠀⠔⢚⣿⡟⠉⢩⡖⠊⠀⠀⠕⡄⠠⠈⣿
⠀⣿⢄⢠⠈⠦⠀⡈⠢⣀⠛⠚⠃⠈⠂⠀⠀⠘⠀⠘⠀⣦⠄⠀⢀⠁⡠⡀⠐⠁⠀⡀⠀⠐⠄⣀⣡⠐⠈⠠⠀⢒⢽⣿⠤⡺⣿⣇⢄⡀⢈⠒⡠⠀⣿
⠀⣿⡄⠀⠐⠀⠀⢈⢅⡀⠏⠀⠀⠐⠀⠀⡀⢀⠈⡠⠀⢁⠤⠆⣈⠑⡌⠀⠀⡺⣀⣄⢀⠔⠈⠠⣏⣀⠀⢨⠀⠁⢿⡅⠾⡷⠉⢌⢠⡠⣈⠀⠀⠤⣿
⠀⣿⠀⠀⠀⠒⠤⣇⡀⠂⠀⠀⠂⠀⢀⣀⣿⠀⠀⠄⡈⠀⠀⠈⠀⢠⠀⠀⠈⠎⢈⠙⢻⠀⡀⠄⠈⠉⠁⠠⣀⢊⠆⠑⠀⠋⠀⠈⡀⢀⠒⠆⠀⢀⣿
⠀⣿⠀⠀⠀⠉⠲⠉⡉⠀⢀⣀⣈⠰⢶⠋⠉⢲⣦⡵⢆⣩⣤⣴⣾⣷⣶⣤⣆⠀⠀⠀⠖⠳⡴⣀⠔⠁⠈⠂⣈⠂⠀⠢⣋⠀⠆⠶⣈⣉⠂⠀⠃⠉⣿
⠀⣿⠀⠀⠀⠄⠐⠋⡱⠂⡸⠛⠍⡐⠏⡳⣴⠏⣩⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠐⠀⠛⠒⠁⣈⣀⠍⠳⠤⡴⠁⠐⠓⠆⠱⠄⠉⠉⠀⢀⣿
⠀⣿⠀⠀⠀⠀⡭⠀⠒⠋⠀⠁⠠⠉⠔⠐⡁⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢠⣀⠠⠀⢈⡀⢄⠤⠖⢤⠋⠂⢀⠠⢘⠏⠀⠈⠀⠀⣿
⠀⣿⠀⠄⢂⠠⠑⠀⡀⠻⢄⣀⠠⠋⠀⢀⠃⣿⣿⣿⣿⣿⢛⣩⠀⠀⠀⡝⢿⣿⣿⣿⠁⠚⢄⣛⠋⢷⠖⠁⠀⠀⠀⠀⣈⣹⡴⢂⠊⠀⠀⢀⠠⡀⣿
⠀⣿⠀⠈⠀⠂⢁⣰⠤⣦⡉⠀⡀⢄⠠⢂⠀⢿⣿⣿⣿⠃⣴⣶⠀⠀⠠⣶⣆⣿⣿⡟⠀⡁⠀⡈⢲⠤⠀⠂⢀⠀⠛⠀⠀⠱⠄⠀⡁⠀⠈⠢⠐⠈⣿
⠀⣿⠁⠀⠠⠊⠉⣤⠙⠲⠛⠋⡉⠉⠠⠀⠀⠊⠻⣿⣿⡄⠙⢫⡔⠠⢀⠉⢁⠿⠋⠁⠀⡈⠀⠈⠢⠐⠁⢦⠀⠁⠀⢀⡤⠀⠃⠀⠈⠀⠀⠀⢀⠀⣿
⠀⣿⠀⣠⠁⠐⢌⠀⠈⠚⠁⠀⡀⠀⠀⠑⠀⠀⠀⠘⠛⠿⢦⡄⢉⣀⣁⠀⠁⠀⠀⠈⠀⡈⠢⠀⢂⠐⡈⠀⢲⡄⠐⠈⠃⠀⡎⠀⠀⠀⡀⢀⠀⠈⣿
⠀⣿⠂⠝⠀⠴⠀⠀⠀⠀⠠⠀⢀⠀⣀⡠⠥⠤⠦⠤⠠⣴⣿⣧⣸⣿⣤⠂⡀⠀⠀⠀⠄⠈⠄⠐⡀⠀⠠⠐⠀⡤⢀⣀⠡⠘⠀⠀⠁⡐⠀⢌⠀⠀⣿
⠀⣿⠀⠘⠀⠀⠀⠀⠀⠀⢈⠄⡩⢋⠀⠐⠀⢀⢀⠀⠀⢻⣿⢿⣿⣿⣿⡇⠈⢆⠀⢂⠀⠂⠈⡀⢡⠀⠀⡀⠀⠠⠀⠈⢦⣐⠦⠆⠤⢄⡀⠀⠀⠀⣿
⠀⣿⠀⠀⠠⡐⠀⠌⠀⠠⠁⠀⠠⠀⠄⢈⠀⠄⠤⠀⢀⣼⣷⡀⠘⢿⡧⠈⠠⡀⠑⠀⡄⢠⡀⣀⣀⣀⣄⡄⠀⡈⠀⡀⠀⠈⡄⠀⠂⢨⣀⠑⠶⠼⣿
⠀⣿⠀⠀⠀⠈⠆⠄⣀⣇⠠⠄⠀⢒⠐⠠⢤⠀⠀⢠⣼⣿⣿⣿⣷⣄⠙⢦⠀⠈⠂⠀⠑⢄⠐⡀⠀⠂⠀⠀⠉⢑⠂⢄⡀⠀⢰⠀⢀⠀⠁⠀⠀⠄⣿
⠀⣿⠀⠀⠀⢀⡠⠚⠁⢀⠨⠀⠀⠀⠀⠀⠀⠈⣶⣏⠻⢿⣿⣿⡿⠏⠁⠦⡙⢧⣀⡀⠁⣠⣽⣶⣦⠀⠀⠈⠁⠂⠌⡐⠁⠢⣤⡈⠀⠀⠀⠀⣼⠀⣿
⠀⣿⢀⠀⠲⠙⠁⠀⠐⠠⠂⠉⠀⠀⠀⠈⢀⣺⣿⣿⣿⣶⣮⣽⣌⡀⠀⠀⠈⠦⠯⡝⢒⣿⠋⠙⠚⢅⠁⠒⠠⢈⠂⢈⠀⠀⠈⠢⡄⠀⠀⠁⠈⠀⣿
⠀⣿⠉⠄⠠⠁⣀⣴⣶⠓⡚⠲⢄⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⡄⠀⡏⠉⡘⢡⠄⠀⠈⡀⣀⠰⠤⠤⠤⢁⡈⠀⠀⠳⡄⠢⠉⠀⠌⣿
⠀⣿⠐⠀⡠⣴⠋⡤⠶⠤⠭⠍⣢⣽⣤⠈⠐⠄⠈⠻⢿⣿⣿⡿⢿⣿⣿⣿⠀⠠⢾⣷⠛⢓⠗⡆⡠⠉⠉⠁⠀⠀⣀⠀⠂⢌⠲⣀⠀⡏⠑⠂⠑⠀⣿
⠀⣿⠠⠟⠀⠁⠘⡁⠀⣀⠐⠢⠈⠋⢸⠂⢈⠀⠀⡔⢸⣿⡿⠀⣾⣿⣿⡟⠀⠀⠈⣿⣆⠠⢘⡩⠚⡁⠀⡁⠀⠀⠀⠀⢄⠀⠀⠈⣾⣁⣀⡈⠀⠠⣿
⠀⣿⠑⠠⠀⠁⠈⠀⠈⠄⣀⣤⣶⣷⣿⣶⣦⣦⣀⠀⣿⣿⡇⢸⣿⣿⣿⠵⢖⣾⡿⢛⣿⣬⣥⣴⣦⣤⣀⡐⠈⠀⠀⡁⠀⠁⠀⠀⢸⠄⠀⠈⠋⠀⣿
⠀⣿⠀⠀⠀⡀⠄⠒⣰⣾⣿⠟⢿⣉⣿⠉⠻⣷⣿⡷⣞⡽⡏⠩⣿⣿⠇⣤⣼⣯⣪⣶⣿⣷⠞⠻⠿⢶⣯⣛⢧⡆⡴⠈⠲⢶⢴⣤⡘⠀⠀⠄⠀⠀⣿
⠀⣿⠐⠒⠟⠀⠈⣽⣿⡿⡙⢧⡄⠆⡗⢒⣾⠟⣙⣷⣽⣇⣩⣀⣿⣿⠟⠉⢠⣿⣯⡿⠛⣿⡀⢀⠃⣀⡮⠻⣧⡹⣆⠀⣨⠋⠀⠈⢻⡄⢠⠀⠀⠀⣿
⠀⡿⡀⠀⢀⠀⢰⣿⣿⠷⢤⣄⣿⣾⣷⣿⣡⣞⣥⣾⣿⣽⢖⣦⠛⣧⢀⠄⣼⢧⡟⠳⣦⡘⣿⣜⣰⡿⢁⠀⠹⣿⣿⡆⡉⠒⢥⠐⣭⠄⠠⢀⡀⣠⣿
⠀⣿⠀⡀⢀⠀⢸⣷⣟⠓⠒⢀⣈⣿⣿⣿⣿⣿⣗⣀⣿⣹⡿⢏⡇⢸⡇⢸⣿⣸⣅⣘⣛⠿⣿⣿⣟⡀⠦⣀⠐⣻⣿⡧⢦⠁⠀⠀⢩⠳⡶⠋⠁⠀⣿
⠀⣿⢀⡸⠤⠴⠘⣿⣿⣉⣠⠞⣹⡿⢹⣿⣏⠛⢿⢿⣿⣿⣛⣋⣡⡾⠣⢸⣿⢻⡄⢀⣠⣶⣾⢿⣿⡛⠷⢦⣄⣸⣿⡇⠀⡁⠀⠀⠆⠀⠀⢀⠀⠐⣿
⠀⣿⡁⠀⣀⢀⡀⠻⣿⣿⣳⡔⢡⠃⢀⠘⡏⢷⣤⣿⣿⡇⠉⢘⡀⠦⢲⡟⣿⡞⣧⣉⡾⠛⡏⠀⠉⢿⣮⠟⣿⣿⣿⠃⠄⡃⠄⠊⠠⠛⠀⠊⢀⠀⣿
⠀⣿⠀⠁⠀⠀⠀⠀⠻⣿⣿⣧⣼⣄⣈⣤⣽⢾⣿⠟⠁⣸⠆⠀⢤⢨⡀⠈⠘⢿⣜⢷⣄⠀⠅⠀⠠⠀⢻⣿⣿⣿⠃⠀⢠⠀⠀⢂⡐⠐⠰⠀⠁⢀⣿
⠀⣿⠀⠂⡈⠀⠀⢬⠀⠈⠙⢿⠿⠿⠿⠿⠟⠛⠁⠎⠀⢸⡏⠀⠈⡄⠄⠐⠀⠀⠙⠳⣿⣻⡶⣦⣴⣶⣿⡿⠛⢃⡄⠠⠃⠀⠀⠀⠐⠠⠠⠀⠀⠁⣿
⠀⣿⠄⠄⡀⠀⣀⠂⠀⠀⣀⠎⠀⠀⠠⠀⠀⠠⣔⣴⠆⡄⣿⡀⢔⣠⡆⠅⠀⠀⠀⠐⠀⠉⠛⠛⠛⠛⠉⠀⣠⣋⢴⠄⡠⠀⠀⠀⠀⠅⠀⠀⠀⠀⣿
⠀⣿⡀⠀⡀⢠⣀⠀⡀⣀⡌⠀⠀⠀⣀⠀⡐⣀⣻⣿⣇⣗⣻⣿⣿⣿⣇⣈⣀⣄⣀⣀⠚⣁⣀⣁⣀⡀⣐⡚⠀⠁⠛⣛⣦⣦⡀⠀⠁⣀⡀⢀⡈⣘⣿
⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢃⣤⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢾⣟⣁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣸⣿⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣯⣭⣅⠀⠉⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⢀⣀⣤⡿⠀⠀⠀⠀⣾⡛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢷⣄⠸⢯⣤⠀⠀⠀⠀⠀⣸⠇⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣇⠀⠈⣷⠀⠀⠀⣀⣽⠇⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠙⣷⣺⡏⠀⢠⡾⠋⢀⣠⣤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⣀⣬⡷⠶⠶⢦⣼⣿⣁⠀⠈⠷⡴⠾⠁⣼⣏⣤⠶⠳⣦
⠀⠀⠀⠀⠀⠀⢸⡏⣩⣭⡉⣿⠉⠁⠀⠠⣤⣀⠀⠉⠉⠛⢷⠀⢀⣠⣾⠟⠋⠁⠀⣀⣼⠃
⠀⠀⠀⠀⠀⠀⢸⡇⠻⠾⠃⣿⠀⠀⠀⠀⠀⠉⠛⠛⠶⠶⠿⠛⠋⠉⠀⣀⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠘⢷⣤⣤⣴⠿⠛⠉⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢲⣊⣯⠭⠭⠭⠭⣹⣃⡖⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢒⡭⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⢭⡒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢊⡐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢂⡑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢬⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠉⠉⠉⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡇⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢀⣤⣤⡀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡇⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣿⡉⢉⣿⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢸⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡅⠀⠀⠀⠀⠀⠀⠀⠀⢇⢀⣾⡟⠛⠛⢻⣷⡀⡸⠀⠀⠀⠀⠀⠀⠀⠀⢨⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢼⡀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⡁⠀⠀⢈⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⢀⡯⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢊⢣⠀⠀⠀⠀⠀⠀⣰⣿⣱⠁⠈⠉⠉⠁⠈⣏⢿⣆⠀⠀⠀⠀⠀⠀⡜⡒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⠳⡀⠀⠀⠀⣼⣿⣳⠃⠀⠀⠀⠀⠀⠀⠘⣞⣿⣧⠀⠀⠀⢀⠞⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠽⠦⣀⣾⣿⢣⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⣿⣷⣀⠴⠩⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢫⡟⠊⠭⣑⣒⡲⠲⠖⣖⢒⣊⠏⠑⢻⡝⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢇⡞⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⢳⡸⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣘⣛⣻⣯⣾⣀⣀⣀⣀⣀⣤⣶⣾⣷⣶⣤⣀⣀⣀⣀⣀⣷⣽⣟⣛⣓⣀⣀
⠀⠀⠀⠀⠀⣀⣤⣀⣀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣀⣀⣤⣀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠈⠻⠿⠿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠙⠻⠿⠿⠟⠉⠉⠉⠙⠻⢿⣿⣿⡿⠟⠉⠉⠉⠉⠻⠿⠿⠟⠋⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠤⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⠐⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⠁⠒⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠁⠀⠠⣈⡠⠂⠀⠀⠀⠀⠀⠀⢀⠤⠤⠂⠀⠀⣀⣀⠀⡠⢀⡀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠔⢀⠠⠃⠀⠀⠀⠀⠀⣀⠤⠔⠈⠐⠁⠀⠀⠀⠀⠊⠀⠈⡀⠀⠅⣂⠁⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⠔⢁⠖⠁⠀⢀⠄⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠁⠀⣈⣗⡀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆⠀⠃⠀⠀⢠⠆⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⠀⠈⣥⣂⠡⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⡈⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⡀⠀⠠⠂⠀⠀⠀⠄⠔⢀⡆⠀⠀⠈⢏⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠁⠀⠀⢲⠂⠀⠀⠀⠀⠀⢈⠀⢀⣠⣴⢢⢤⠁⢊⡂⣈⠇⠈⡭⣅⠮⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡐⠃⠀⠀⠀⠀⠀⠜⠢⡈⡀⢀⡐⢁⣀⠀⠀⠀⠀⢉⠐⠉⠩⠉⠁⠒⠈⠉⣳⠭⡂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠐⠄⠀⠀⠀⠀⠀⠈⠀⠁⠩⠍⡉⣁⡂⡠⠀⠀⢀⡀⠀⠀⠂⠁⠀⠀⠀⣈⠻⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣌⡤⣨⣌⠀⣀⡀⠀⠀⠁⡀⠀⠒⠉⠁⢾⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠐⢄⠐⢂⠁⠠⠊⠀⠐⠀⠀⠌⠊⠁⠈⠉⠉⠄⢄⠀⡄⠑⢦⠐⠐⠠⡎⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡕⠊⢛⠦⡴⠋⠛⣘⣹⣭⡁⢨⠂⠀⠖⡲⢀⠖⠔⠤⣤⠄⠠⠊⠚⢀⢗⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠍⠊⠀⠈⠳⣅⠀⢉⠚⠉⠩⣒⣀⠁⠨⠊⠐⢂⡈⠀⠊⠈⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠈⢐⠛⠂⠈⠓⠐⠀⠀⠁⠀⠀⠐⠂⡀⠠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⠐⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠒⣻⣿⣶⠃⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠌⣣⠀⠀⠀⠀⡋⠀⠀⠀⠐⠛⠛⠛⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠆⠀⠀⠄⣄⣲⢠⢟⡀⢣⣄⠐⠁⠀⠀⠈⢒⠀⠁⠈⠻⣍⠙⢻⠻⠛⢁⠃⡆⣶⢡⡉⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠤⣀⡤⠀⠀⣈⣈⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠈⠉⠀⠃⠃⣂⣁⡀⠉⢠⣄⣡⡤⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠐⠚⠛⠋⠉⠉⠁⠀⠨⠭⠭⠍⠉⠉⠉⠐⠒⠒⠂⠈⠉⠉⡁⠀⢠⢁⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄⠀⠀⠀⠀⠀⠀⢆⢀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⡐⢁⠈⡐⡠⢂⠈⡐⠀⠠⠁⢀⠀⠊⡡⢋⡠⢊⠄⡀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⣀⡀⠀⠀⠀⠠⠐⠀⠈⠀⡠⢊⠔⡠⠊⠐⠡⢂⠔⡠⠁⠌⠐⠐⢀⣁⡤⠖⡈⠐⡁⢀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠛⠒⠦⠦⠤⣤⣬⣄⠁⣈⡀⢈⣤⢤⣤⠄⡠⠶⠂⠜⠛⠉⠊⠔⠈⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢠⣾⣿⠁⠀⠀⠀⢰⣷⡄⠀⠀⠐⠦⢄⠴⠖⠀⠀⣀⣶⣷⡆⡀⠀⢀⡀⠘⠿⠀⠀⠠⠚⠁⠈⠁⡆
⠀⠀⠘⠸⠟⠀⠀⠀⡀⣈⢛⡁⡀⠀⠀⠀⢸⠀⠀⠀⠀⠨⠻⡿⠣⠀⠀⠸⠋⠠⣴⡤⠀⠀⠈⠉⢩⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⢀⣴⠙⢁⣦⠀⠀⣿⣿⡇⠀⠀⠀⠀⣠⡞⢶⡆⠀⠀⣀⣀⢀⢀⡀⠀⠀⠀⣤⡅⠂⠀⢠⣤⣤
⠀⠘⡩⡄⠘⣏⠃⠀⣻⣿⣻⣿⡇⠀⠈⣿⠀⢀⡟⠀⠰⣏⣿⣸⣮⡇⠀⠈⠈⣿⡃⠌⠀⠀⣾⣿⠟⠻⠆
⠀⠀⠈⠑⠀⠁⠀⠀⠛⠀⠛⠛⠃⠀⠀⠛⠉⠛⠃⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠉⠉⠀⠀⠃
⠀⠀⠀⡐⢂⡀⠀⠀⠀⢀⡗⢠⠀⠀⠀⠀⣠⠀⠀⠀⠀⢀⡴⢰⣀⠀⠀⠀⡤⠀⢄⡀⠀⠀⢀
⠀⠀⢀⡁⣉⠀⠀⠀⠀⣾⡉⠛⠀⠀⢀⣴⣧⣀⠡⠀⠀⢹⢐⡍⠔⠂⠀⠘⠌⣮⢚⠃⠀⢠⣾⡂⢐⣿⠂
⠀⠀⠈⠻⠟⠀⠀⠀⠰⠃⠃⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠻⠛⠛⠉
⠀⠀⡀⠀⠀⣀⠀⠀⠀⢠⡤⣄⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⡀
⠀⠘⠿⣀⣘⠿⠃⠀⢰⣿⠟⠈⠀⠀⠀⣤⣿⣿⡇⠀⠀⣔⡀⠰⣄⠀⠀⣶⠒⢲⡄⠀⠀⠀⣿⢰⡆⣤
⠀⠀⠀⠧⠟⠀⠀⠀⠸⠏⠀⠁⠃⠀⠀⠸⠦⠤⠃⠀⠀⠻⠛⠛⠿⠃⠀⠻⠗⠚⠛⠿⠂⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⣄⣤⣠
⠀⠀⢠⢖⡢⡄⠀⠀⠀⢸⡆⢹⠀⠀⠈⠃⠸⠀⠛⠀⠀⠒⣿⠷⠄⠀⠀⠀⢠⡾⠅⠀⠀⠀⠈⣿⢁⣀
⠀⠀⠣⣘⣃⠎⠀⠀⠀⢼⡧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡄⣶⠂⠀⠀⠀⠙⠀⠀⠀⠀⢰⢻⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⡴⢣⡐⢦⠀⠀⣴⣧⢀⣿⡆⠀⠀⢰⣶⡇⠀⠀⠀⡔⢉⣥⠛⡄⠀⠀⣠⣼⣶⡄⠀⢀⣾⡏⢹⣷⡄
⠀⠀⢣⡉⢃⠞⠀⠀⢣⣨⣶⣀⠇⠀⠀⠲⠌⠶⠃⠀⠀⢧⡘⢛⣠⠃⠀⠀⠟⠁⡼⠃⠀⠈⢿⣿⣺⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣶⣷⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣶⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣟⠋⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⡧⠙⠛⠛⠛⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⢀⣠⣤⣶⣶⣤⣤⣀⠀⢀⣴⡟⠙⣷⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣠⣾⡿⠟⠋⠉⠉⠉⠙⢻⣿⣿⡟⠀⠀⢹⣧⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣼⣿⠋⠀⠀⠀⠀⠀⠀⣠⣾⣽⢿⣷⡀⠀⠀⢿⡆⠀⠈⠻⣆⠀⠈⠻⣦⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣸⣿⠃⠀⠀⠀⠀⠀⠀⣴⣯⡾⠁⠈⣿⣧⠀⢰⣾⣿⡀⠀⠀⠙⢷⣤⣤⣿⣿⣿⣿⣿⣿⠿⠻⠟⠛⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⣿⣿⠀⠀⠀⠀⠀⢾⣿⡿⠋⠀⠀⠀⣿⣿⠀⠘⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣄⣠⣴⣾⣿⣿⣿⣿⣿⣿⠿⠛⠉
⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠈⢻⣿⣿⣿⣿⡿⢿⣿⣿⡿⠟⠛⠛⠛⠻⣿⣿⣿⣿⣿⡿⠟⠋⠉
⠘⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠀⠀⠀⠀⠀⢻⣿⣿⣿⣴⣿⡟⠛⣿⣄⠀⠀⢠⣶⣿⣿⣿⣿⣏
⠀⠈⠻⣿⣦⣄⣀⠀⢀⣀⣤⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⣿⢿⣾⣿⣟⣀⠀⠘⣿⣦⣴⣿⣿⣿⣿⠟⢿⣿⡆
⠀⠀⠀⠈⠙⠻⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⣿⣇⣉⢙⣳⣾⣿⣿⣿⡿⠟⠁⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣽⣋⠀⠀⠀⠀⣀⣤⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣧⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠋⣘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⡀⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢛⣀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠒⠈⠈⠛⠿⠲⡆⠀⠀⢀⣀⡀⠀⠀⢀⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⣀⠼⠃⡀⣶⣿⠃⠀⠀⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⣼⠇⢀⡞⠁⠀⣾⣇⡅⢻⣧⡀⠀⠀⣠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠛⡟⠛⠓⢻⡄⠈⣿⡿⠦⢈⡼⠁⢀⣿⣿⣠⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⠟⠉⢀⠾⣾⣿⡄⢸⡇⠸⠿⠿⠋⠉⢦⣾⣿⢿⣿⣿⣿⢟
⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⣠⣾⣾⣿⡟⠁⠸⠇⠀⣿⣷⣴⡆⢸⠟⠁⠸⡿⠙⢡⣿⣀
⠀⠀⠀⠀⢀⣶⢟⣁⣤⣶⣿⣿⣿⣿⣿⣧⠈⠐⠚⢉⣿⣿⡿⠇⠀⠀⠀⠀⠈⠲⡌⠙⡿⠃
⠀⠀⠀⠰⢮⣻⣾⠬⢿⣿⠋⠁⠀⠀⠀⠹⡇⠀⢀⣾⠛⠉⠀⠀⠀⠀⠀⠀⠀⣠⣵⠞⠁
⠀⠀⠀⠀⠀⠻⣽⣶⣾⣿⡄⠀⠀⠀⠀⣸⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⣴⠟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣳⠄⠈⠀⢀⠏⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠀⠀⠀⣸⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠁⠚⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣄⠀⣠⡀
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣀⣠⣜⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣟⡀⠀⠉⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⡀⣀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣀⡄
⠀⠀⡼⢫⣿⠟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠶⠾⠀⡀⠈⠁⠀⠀⠀⠀⠀⠀⢽⣿⣻⠛⣄⣀⣸⣇⠀⠀⠀⠀⠀⠀⣠⣤⠀⣾⣿⣤⣼⡶⣖⣦⡦
⠀⠀⠀⣿⣧⠀⠈⠳⢤⣼⡋⠀⠀⠀⠀⠀⠀⠾⢹⣿⡇⠀⠀⠀⣯⣀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣾⣿⣿⣿⠿⠆⠀⠀⠀⠀⠀⠥⢿⡿⣿⣼⣷⣾⠷⠟
⠀⠀⣸⠏⠹⣇⠀⣀⡾⣏⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣡⣤⣦⣭⣁⡉⠉⠀⠀⠀⠀⠀⠐⠛⠋⢀⣀⣹⣯⣀⣀
⠀⠐⠋⠀⠀⠻⠄⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⠇⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠇

⠀⠀⢀⡔⠺⠁⢙⢈⠓⢢⡀⠀⠀⠀⠀⠀⠀⣠⠾⠟⠛⠿⠿⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⢠⠏⢀⠂⠁⠀⠈⠐⢈⠩⡆⠀⠀⠀⠀⠀⢄⠀⠀⠀⣀⡀⠀⠈⠙⡆⠀⠀⠀⠀⠀⠀⢀⢀⣴⣿⣿⣿⣦⡀⡀⠀⠀⠀⠀⠟⠉⠉⠹⡄⢀⣀⡀
⠀⣬⠀⠄⠀⢸⣿⡧⠀⠠⠄⣽⠀⠀⠀⠀⣤⠀⠷⠐⢾⣿⣿⣤⣀⡄⠁⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⡿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⢠⠜⠏⠀⠙⣆⣠⠏⠉⢳
⠀⢸⡄⠀⢄⡞⠉⢳⡠⠈⢠⡆⠀⠀⠀⠀⠀⠈⠁⠚⢺⣿⣿⠤⠌⠱⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⣀⠀⠀⠀⠀⠀⠀⠁⠒⠊⠀⠀⠸⢆⠀⠀⠀⠀⢀⡼
⠀⠀⠸⢦⣞⠁⢀⠈⣳⡴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠃⢻⣿⡇⠙⠶⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⠤⠒⠉
⠀⠀⢠⡎⠉⠈⠛⠁⠉⢹⡄⠀⠀⠀⠀⠀⠰⠿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠀⠀⠀⠀⠀⠰⠾⣿⣿⣿⣿⣿⣿⣿⠷⠆⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢠⢠⡄⠀⠀⠀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠆⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡒⣤⣶⡞⠁
⠀⠀⢸⣾⣆⣀⠀⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢿⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⣍⢾⡿⣉⣯⣥⣤⣄
⠀⠀⢸⡟⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⠀⠀⠀⠀⢸⢹⠀⠀⠀⠀⠀⠀⠸⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢟⣴⣾⠕⡩⠭⢗⡁
⠀⠀⢸⣀⣀⣘⣀⣀⣀⣀⡁⠀⠀⠀⠀⠀⠀⢀⣾⠷⣤⡀⠀⠀⠀⢸⣺⠀⠀⠀⠀⠀⠀⢸⣇⣿⣷⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⣫⠖⣩⠔⠊⠁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⡾⢯⡀⠀⠉⠐⠀⣴⢸⢼⠀⠀⠀⠀⠀⠀⠿⡿⣾⠛⠃⠀⠛⡟⠂⠀⠀⠀⠀⠀⠘⡿⢋⡤⠋
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢴⠛⠁⢸⡇⠀⠀⢠⣤⣿⣾⣿⣤⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠈⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠶⠶⡶⠶⠶⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢴⡀⢺⣇⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡻⠗⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢠⣶⣄⣤⣤⢴⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠙⣹⣿⣉⣉⣀⣀⣀⣀⣈⣁⠀⠀⠀⠉⢠⣿⢿⡄⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠘⣿⣿⢡⢐⠑⢻⣿⡇⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⣠⣄⣠⣤
⠀⠀⠀⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢷⡀⠙⠋⠀⣀⣀⣠⣠⣾⣿⣿⡇⠀⠀⠀⠀⠈⠛⠷⢶⡴⠿⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢴⠏⠈⣧⡀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⢸⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⡠⢄⡠⢼⡧⢄⡠⡄⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⢤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⣩⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢃⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠠⣤⡏⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢻⡀⠀⠀⣀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠄⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⡈⣿⠈⠁⠀⠀⠀⠀⠀⠈⠁⠂⢦
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⡄⠀⢻⡄⠀⠀⠀⠀⠀⠀⢠⡴⠋⠀⠀⠂⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣦
⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢻⡀⠈⢷⡀⢣⠀⠀⣠⡶⠋⠀⠀⢀⠁⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡀⢡⣶⣿⣤⣤⣤⠤⠤⠀⠠⢬⡧⠀⠼⣧⣼⣠⡶⠋⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⢷⣄⡀⠁⠀⠀⠀⠀⠀⠀⠀⢩
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠻⠏⠭⡉⡙⠛⠛⠟⠻⠶⡿⠶⣾⣼⣿⠻⡇⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠹⡷⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⣺⠅⠀⠻⣥⣏⡳⠃⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣶⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⣦⣤⣤⣤⣤⣶⣿⣿⣾⣦⡶⠒⢂⠀⡀⠀⡂⠀⢠⡤⢦⣐⠮⠤⠦⣤⡀⢿⣳⣮⣍⣒⣒⣒⣲⣬⣴⣿⣏⡤
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠋⠉⠉⠉⠉⠉⠉⠉⠃⠈⠐⠈⠉⠀⠀⠀⠄⠂⠀⠛⠃⠜⠁⠀⣀⣈⠡⠀⠙⠋⠉⠉⠉⠙⠉⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠿⠿⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡙⣉⣦⣶⣶⣶⣶⣾⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢟⠋⠄⢹⡖⢈⠙⠻⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⡏⠀⠘⠃⡠⠈⠁⡀⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣽⠀⠲⠄⠀⠀⠀⠐⠁⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡏⡦⣀⠀⠀⠀⠉⢀⡔⠁⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⠀⠀⠀⠀⣀⣴⡢⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡙⢰⣿⣿⣷⣄⣸⣿⣿⡃⠀⠀⠠⣄⣤⢦⣧⡆⣸⣯⣬⣦⠰⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⢅⣾⣿⣿⡏⠉⢻⡁⣙⣷⣤⣶⣿⣾⣿⣿⡟⢣⣿⠻⢽⣿⡿⢽⡹⣻⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢹⡇⠖⠃⠕⠦⢫⢻⡗⠦⠄⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠷⡎⠙⠛⢹⣿⣿⠿⠛⣿⣿⣿⣿⣿⡟⠁⠀⣸⣧⡄⢰⡖⡰⢻⣶⡌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠒⠀⠀⢀⣾⡿⠁⢀⣴⣶⣿⣿⣿⡏⠀⠀⠀⡟⢱⡀⠀⣰⠁⠧⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣽⡇⠀⣠⣾⡏⠀⠀⣠⣽⣿⡿⠋⠀⠙⢦⡀⠀⢿⠈⣱⡜⣋⣀⣀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡓⠗⠾⢾⠿⣿⠀⢀⣼⣿⠿⠋⠀⠀⠀⠀⠀⠉⠳⢌⣳⣴⡛⣋⣉⡁⢙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⠎⠀⠀⠰⠁⣼⡏⢀⣾⣿⠋⠀⠀⢄⠀⠀⠀⠀⡰⠁⢀⢛⣻⠟⠸⡿⠀⠠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠉⡆⠀⠀⢠⠁⢸⣿⣷⣿⣿⢡⠀⠈⠢⠀⠆⠀⠀⠠⠁⠀⡎⠀⠛⢾⠦⠀⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⠀⠀⠀⠘⠀⣾⣿⣿⣷⡇⢀⠀⠀⠀⠁⢸⠀⠀⠀⠀⡌⠈⡄⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠃⡰⡄⠀⠀⠰⣀⢹⣿⣿⢹⡿⠈⠀⠀⠀⠀⠌⠀⠀⠀⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠀⢡⢰⡀⠀⠡⡀⠑⣿⡿⣿⣴⠀⠀⢀⣉⠐⠀⣤⣤⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢋⣉⠉⢂⠀⠈⠦⢉⠁⠻⡁⢆⠀⠀⠀⠀⣠⣿⣿⡿⢦⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡎⢨⠀⣤⠛⣄⣴⡎⠈⠁⠉⠉⠑⠐⠛⠀⠠⡄⠉⠢⣀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡱⠋⠱⠈⣿⣆⡹⣿⣉⣉⠉⠑⠄⠀⠀⠀⠀⢰⠀⠀⡐⠉⢢⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢛⠅⢺⠀⠀⢸⠛⡇⠀⠀⠀⠑⠀⠈⣢⠀⠀⠀⣤⣤⡔⠁⠀⠸⠤⠤⠘⡄
⠀⠀⣰⣼⣤⢠⠀⠀⠀⠀⡜⡧⢤⠂⠀⠀⢰⠀⢡⠀⠀⠀⠀⢣⡈⢁⡵⠀⢠⣿⣿⠇⠀⠀⠐⠄⣀⡀⠔
⠀⣻⣿⣿⣿⣿⣄⠀⣀⣴⣿⠔⠂⠀⠀⠀⠈⡀⠨⢤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡋
⠀⠙⠻⠿⠿⠿⠟⠃⠘⠋⠁⠀⠀⠀⠀⠀⠀⠃⠀⠄⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣭⠉⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⡙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠇⠀⢸⠓⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠓⠖⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⣀⣧⡀⠀⠀⠀⠃⠠⣰⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠀⠀⠀⠀⠻⡦⠄⡄⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⣰⢟⣀⣀⢄⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡠⢢⣿⠟⣫⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠘⠓⠦⣄⣀⣠⠤⠒⢲⣶⣿⠟⠠⠋⢤⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⠀⠀⠀⠀⠉⢁⠀⠀⢀⣀⣽⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠶⠶⠤⣤⣄⣨⡀⠘⠻⡋⢹⠀⠀⢀⣴⣒⣛⣛⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠳⣄⠀⠀⠀⠀⠁⢁⡄⠀⠘⢿⢷⣤⣿⠟⠉⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠛⢛⠟⣦⡈⠳⡄⠀⠀⠇⠀⢀⠀⠀⢀⣷⣝⢇⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠸⢠⡏⠙⣧⠙⢦⡀⠀⠀⠈⠀⠀⠈⣾⡏⠪⠠⣠⣄⠄⠀⠀⣷
⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⡇⢸⠁⠀⠹⣇⠀⠙⢦⡀⠐⠀⠀⣿⣿⠃⠀⣘⣾⣿⢮⠀⠀⣿
⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⣼⠁⡟⠀⠀⠀⣿⠀⠀⢀⣿⣾⢶⣠⣿⣿⠄⠀⣻⣼⠤⠃⠀⢰⡇
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⣴⣷⡾⠁⠀⠀⠀⣿⠀⠀⢸⣿⣿⡿⠈⠀⢹⡄⠀⠀⠀⠀⠀⢀⡾⠀⠀⢀⣀⠤⠄⠀⠂
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⠭⠄⠀⠀⠀⢸⠇⢀⣠⣿⡟⠋⠀⠀⠀⠀⢷⣤⡀⠀⢀⣠⠿⠒⠚⠉⠁
⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣀⣀⣤⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⡹⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡠⠘⠚⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⢄⣠⡞⠠⢊⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣧⢀⣠⣴⣶⣊⡩⠔⢲⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠋⢁⠔⠛⠛⠃⠀⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣿⠃
⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠀⡔⠁⢠⡞⠐⣴⣯⣉⣤⠠⠖⠁⠀⠀⠀⠀⠀⠀⣿⣿⠟⠛⠋⠁
⠀⠀⠀⠀⢰⣿⣿⠏⠀⢀⣠⣾⢁⣤⡾⠠⣼⠃⠀⠀⠀⠀⢀⣀⣴⠀⠀⢠⡄⢀⣿⡟
⠀⠀⠀⠀⣸⡿⠁⢠⣴⣿⠿⢿⢺⣏⣃⠘⣿⣶⣶⣾⣿⣿⣿⣿⡍⠀⠀⣰⡟⠛⠋
⠀⠀⠀⠀⣿⠃⠰⠿⡏⠀⠀⣸⣿⠀⠨⠁⠀⠉⠉⠉⠉⠉⣿⣿⡇⢀⣤⡟
⠀⠀⠀⠿⢿⡒⢀⣻⣷⠀⣠⣿⠋⣰⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡧⠻⣿⡀
⠀⠀⠀⠀⠈⠻⣯⣾⣟⠉⢹⡏⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡄⢩⠆
⠀⠀⠀⠀⠀⠀⠈⢿⣟⠃⠀⣷⡘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⣠⡏⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡗⠙⡗⡄⠀⠀⠀⠀⠀⠀⢠⣾⠃⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠻⠇⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠟⠛⠻⠓⠚⠐⠲⠶⠾⠛⠛⠛⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠟⣃⣤⣶⣶⡿⠿⠿⠿⠿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡏⠀⣴⣾⠟⣋⣥⣄⠀⠀⠀⠀⠀⠀⠈⠙⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⡀⠉⠁⣠⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣤⣤⣉⡙⠻⢿⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠸⣷⣌⠻⣿⣿⡷⠂⣡⣴⢶⣿⡟⢛⡏
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢿⣿⣷⣿⠟⣠⣾⣿⣿⠘⢁⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⢿⣿⡟⢠⣿⣿⠟⢁⣴⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣹⣇⢸⡟⢁⣤⣬⣭⣽⠟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⡀⠛⠛⠛⠉⠀⠀⠀⠀⣠⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⣠⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢡⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⢠⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠋⣴⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠋⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⢀⣀⢀⣀
⠀⠀⠀⠀⠰⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀⠦⠀⠀⢀⠀⠀⠀⠀⠀⠀⢾⠄⠀⠀⢐⡀
⠀⠀⠀⠀⠀⠈⠻⣿⠟⠁⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠈⠀⠰⠉⠀⠆⠁⣸⠆⠀⠀⠀⠰⢶⣶⣶⣦⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢉⣻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠉⠁⢀⠀⣴
⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⡄⠀⠈⢫⢿⣿
⠀⠀⠀⠀⠀⣶⣾⣿⣶⡆⠀⢀⠀⠀⠀⠀⢙⣿⣿⣿⠋⠀⢀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⢸⡀⠀⠀⠀⠀⢸⣾⠃⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⢀⡞⠀⠀⠀⠀⠻⠿⠛⠿⠋⠀⡼⠀⠀⠀⠀⠸⠛⠁⠀⠀⢀⣾⠃⠀⠀⠀⠀⡾⠙⢷⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠾⠁⠀⠀⠀⠀⠀⣀⣀⣀⣤⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠤⠒⠈⠉⠉⠛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣄⣦⠀⠀⠀⠀⠀⠀⢀⣀⣤⣠⣄⣀⠀⠀⠀⠀⠀⠀⢠⠴⠶⠦⡀⠀⠠⠀⣠⠏⠀⠀⣀⣤⣤⣤⡀⢹⣿⣿
⠀⠀⠀⠀⠀⢠⡞⢻⣏⠀⠀⢰⠀⠀⠀⢸⣿⣇⠆⣽⣿⠀⢰⠀⠀⠀⠀⠠⠀⠀⠀⡄⠀⢠⣧⢴⠀⢀⣶⣾⣿⣬⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠈⠛⠟⠃⠀⢀⡞⠀⠀⠀⠀⠉⠉⠉⠉⠉⣠⡿⠀⠀⠀⠀⠐⠤⠤⠤⠂⢀⣾⡟⣿⣆⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠤⠴⠞⠋⠀⠀⠀⠀⠐⠀⢀⣠⣤⡶⢿⡉⢸⣿⣿⣷⣦⣤⣤⣤⣶⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠚⣴⣾⣿⣿⠟⠉⢀⣀⣀⠉⠻⣿⣿⣿
⠀⠀⠀⠀⠀⣰⣾⢿⣷⡆⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣄⠀⠀⠀⠠⡇⠀⠀⣶⣶⣶⡄⠀⠈⣿⣿⡏⠀⠀⢸⣿⡿⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠺⣷⣴⣿⠋⠀⢰⠀⠀⠀⠀⠛⠛⠛⠛⠛⠀⢰⡇⠀⡄⠀⠀⢹⡟⣿⠀⠀⢀⣿⣿⡇⠀⠀⣶⣿⣷⡄⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢆⡘⡔⠀⠀⠈⠀⠉⠀⣠⣾⣿⣿⣷⢄⠀⠿⠿⠿⠃⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠐⠒⠚⠛⠋⠀⠏⠁⣤⠾⣷⢶⠶⡶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣄⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠿⣳⡿⢋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡦⢠⣻⣤⣤⣶⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡐⣷⣿⣿⣿⣷⣿⣦⡀⠉⠛⠋⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⢿⣿⣶⣄⡠⣀⡀⣌⡜⠛⠛⠛⠛⠋⢸⣿⣿⣷⣆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠈⠛⠿⣿⢌⣽⣿⣧⡤⡄⠀⠀⠀⣸⣽⣿⣿⡿⠀⢀
⠀⠀⠀⠀⠀⢀⠔⣩⠶⣚⣛⡻⠷⣤⣀⢤⢔⣱⢿⣇⣿⡏⠐⠲⡶⠂⠀⠋⠉⠙⠛⠁⢠⠄
⠀⠀⠀⠀⠠⠂⡞⠁⠀⠀⠀⠈⠗⢬⠛⢀⣾⣿⡞⣿⡟⠟⡄⣦⡁⠀⠀⢠⡀⠀⠀⡴
⠀⠀⠀⠀⢨⣸⠃⠀⠈⣰⢋⣬⡕⠀⠰⠛⣿⣿⣿⠟⢿⡸⣽⣿⣏⣷⠠⠀⢡⣄⠈
⠀⠀⠀⠀⢸⣿⡇⠀⠁⣻⡘⠿⠾⠒⠀⠀⢻⡅⣡⣻⡞⢿⣹⣿⣛⣌⡀⠀⠈⠉⠛⠶⣄
⠀⠀⠀⠀⠀⢿⣿⡄⠈⠀⠝⠓⡂⠁⢀⠀⣼⡗⣸⣿⠧⣎⢜⡿⣿⣿⡄⠀⠀⠀⠀⠀⢀⣡⣤⣄
⠀⠀⠀⠀⠀⠀⠹⣿⣦⣀⠀⠀⠀⠠⢊⣴⠟⠀⢹⣠⣤⢒⣼⣷⣿⣻⡧⠀⠀⢰⣶⣦⠀⠉⠙⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣶⣿⠿⠋⠀⠀⠀⠰⠀⠀⠙⠿⣿⣷⣿⣶⢿⣿⠿⠿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣿⣿⣿⣷⠅⠈⣀⣀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢹⣷⣤⡉⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠈⠻⣿⡉⠲⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⠘⠳⠄⠐⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣄⡀⢀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢖⠀⠀⠈⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡄⠒⠒⠂⠈⠉⠹⡖⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢀⣠⠇⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢱⠀⠀⠀⠀⠀⢠⠧⡀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡏⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣼⠤⠤⠒⠒⠊⠁⠀⢷⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⣀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢷⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠸⡀⠀⢀⣤⢠⡀⠀⣸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⢀⡀⠀⠀⢀⣀⡤⠴⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠙⠒⠛⢻⡇⠘⡏⢁⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣻⢿⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⣾⢿⡧⣼⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⣀⠀⡔⣱⠃⠸⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠔⠚⠁⡗⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡠⠔⠈⠁⠀⠀⠀⠀⠀⠀⠀⡼⢱⢧⡀⠀⢇⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠋⢉⡠⠔⡾⠃⣠⡟⢿⡡⢽⠦⠤⠤⠤⠤⠄⢀⡀
⠀⠀⠀⢀⠔⠋⠀⢀⡤⠖⠊⠉⠉⠉⠉⠑⡾⢡⠃⠀⠙⢦⠈⠀⢃⠀⠀⠀⠠⠆⠐⠈⢉⣀⠤⠒⠋⠁⣠⠎⢀⡞⠉⣇⠸⡇⣀⣤⣤⣤⣀⣀⠀⠀⠈⠑⢄
⢀⠔⣤⠊⠀⢀⠖⠁⠀⠀⢀⣤⠤⠤⢤⡜⢱⠋⠳⡀⠀⠈⠃⠀⠙⠂⠁⠀⢠⠤⠐⠊⠁⠀⠀⠀⢀⡔⠁⣰⠋⢠⠂⠍⠀⣟⠉⠀⠀⠀⠀⠉⠙⠲⣄⠀⠀⠱⡄
⡞⣸⠇⠀⢠⠋⠀⠀⢀⠔⠁⠀⠀⠀⡼⢠⠛⡄⠀⠙⣄⠀⠀⢳⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⠜⠁⣰⠃⣠⢗⠀⢻⢀⡤⠖⠒⠒⠤⡀⠀⠈⠳⣄⣀⡽
⠳⣿⠀⢠⠇⠀⠀⣰⠃⠀⠀⠀⠀⡸⢡⠃⠀⠘⣆⠀⠘⣆⠀⠀⣇⠸⡀⠀⠀⠀⠀⠀⠀⠀⡴⠁⣠⠋⠀⢠⠃⢰⠃⢸⠀⢸⠏⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠈⣇
⠀⣇⠀⡟⠀⠀⢰⠃⠀⠀⠀⠀⡰⢡⠏⠀⠀⠀⠘⡄⠀⠘⡄⠀⢹⡀⡇⠀⠀⠀⠀⠀⢀⠊⠀⡀⠁⠀⠀⡎⢠⡇⠀⠈⡇⠘⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⢸
⠀⠸⢴⡇⠀⠀⡞⠀⠀⠀⠀⣰⢣⠏⠀⠀⢀⣀⡤⠷⠐⠒⠛⠲⢌⣧⣸⣄⣀⡀⠀⡰⠋⢀⠞⠁⠀⠀⢀⠇⣼⠀⠀⢠⠿⡀⢇⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⡇⠀⠀⡇⠀⠀⣠⠴⣧⣯⣤⣴⣾⡁⢨⠆⠀⠀⠀⠀⠀⣼⢻⠁⠀⠈⠻⣾⠁⣠⠏⠀⠀⠀⠀⢸⠀⡏⠀⠀⣾⠀⢳⠘⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⡇⠀⢰⡷⣉⢌⡇⠀⡏⠧⣼⠞⠀⠀⠀⣀⣤⣾⣿⠈⢇⠀⠀⠀⢸⡶⠁⠀⠀⠀⠀⠀⠈⣤⠃⠀⠀⡿⠀⠈⢧⠹⣄⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠇⠀⠀⣧⠀⠈⠣⠥⢼⣇⡀⠓⠾⠿⠶⠒⡿⠋⠉⠀⣿⠉⡇⢸⡦⡀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⣿⠀⠀⠀⠻⣽⣸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠸⡄⠀⠀⠀⠀⠈⠉⠛⠶⣤⣀⢀⡇⠀⠀⢠⢻⠀⠘⢾⣡⠇⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⢆⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡻⠳⢤⣄⡾⠀⣇⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠘⢆⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⢸⠋⠓⠿⢦⣀⣀⣀⣴⠏⢱⣀⡀⠠⠤⠤⢤⡀⠀⠘⡄⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠈⢣⡀⠀⠈⠲⣄⣀⣀⣀⡤⠞⠁⠀⠀⡰⠃⠀⠀⠀⠀⠀⠉⠉⡴⢆⠈⡇⢳⠀⣀⣀⡤⠇⠀⠀⠹⡄⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠈⠉⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠹⣤⣀⣀⣀⣠⠞⠁⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠤⠤⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀⠀⠀⠉⠉⠉⠁⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢤⣀⣀⠀⣀⣠⠔⠋

⠀⠀⠀⠀⠀⡀⠔⠒⠈⠁⠈⠉⠉⢉⡒⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⡀
⠀⠀⢀⣔⣉⠀⠀⠀⣀⣤⣴⠟⣛⣋⡰⣾⣿⣿⣦⡀⠀⠀⠀⠀⡠⠔⠀⠉⠁⠀⠀⠀⠀⠀⠉⠁⠢⣄
⠀⣴⣿⣿⣿⣷⣖⣦⣬⣭⣤⣼⠟⠙⠓⠀⠻⣿⣿⠟⡄⠀⡠⠊⠀⢀⠰⠂⡸⠀⢲⠀⡆⠠⠀⠒⠀⠀⢢
⣸⣿⣿⣿⣟⣽⣉⣌⣿⣯⣀⠀⣀⣽⣿⣷⢦⠹⠅⠀⠘⡈⠢⡀⠀⣈⠃⡀⡇⢤⠈⣇⠀⠀⠀⠂
⡿⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢣⠀⢰⠀⠈⠉⠀⠉⢨⠁⠈⠁⠀⠉⠉
⡇⠀⠀⠈⣿⣿⣿⣿⣿⠟⢉⣉⣁⣤⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⠀⠀⡇⠈⡀⢲⠀⠀⠀⠀⠀⠀⠐
⢳⠀⠀⠀⢿⣿⣿⣿⣦⣤⣥⠀⢡⣤⣠⣤⣤⣀⠀⠀⠀⡜⠀⠀⠐⢄⠀⠀⠧⠛⠧⠸⠤⠆⠀⡀⠔
⠈⣆⠀⠀⠀⠉⠋⠉⠉⠉⠉⢠⣾⣿⣿⣿⣿⣿⣿⠇⡰⠃⠀⠀⠀⠀⠑⠂⠀⠀⠀⠂⠂
⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⣿⡿⠏⠁⡴⠁
⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢀⠤⠊
⣀⣀⣀⣀⣀⣈⣑⣒⣤⣤⣤⣤⣤⣤⣒⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠖⢚⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣒⣓⣖⠒⠒⣒⣲⡶⠶⣞⣉⣩⡤⣶⠒⣿⡹⣧⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠭⢿⢛⣉⡥⠤⢲⡾⣿⣧⢻⣧⣻⣧⠽⠗⣋⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⠀⠉⠉⠙⢓⣲⡆⠀⠀⠀⠈⠓⠒⣋⣩⠤⠖⢚⣉⣷⠸⣟⣩⣭⡴⣶⡞⢿⡏⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⠉⡏⠉⠀⢀⣀⡤⠤⠖⢛⣉⡥⠴⠒⠋⣩⠞⢉⡷⠻⣿⡸⣿⡘⣿⣘⣿⠼⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡷⠒⠋⣉⣠⠤⠖⠚⠉⠀⠀⠀⣠⠞⢁⡴⠋⠹⡴⢿⠷⣛⣛⠭⠵⠒⠋⠉
⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⠤⢤⣀⡀⠀⠀⣠⠞⢹⠀⡗⠚⠉⠁⠀⠀⠀⠀⠀⠀⣠⠞⢁⡴⠋⠀⠀⠀⢳⠘⣯⠤⠤⠤⣄⣀
⠀⠀⢀⡴⠊⠁⢀⣀⣤⣤⣄⣀⠀⠉⣲⠞⢁⠔⢻⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡴⠋⠀⠀⢀⡴⠚⠉⣇⢳⣠⣤⣀⣀⠈⠙⠲⣄
⠀⣰⠋⠀⡠⠚⠹⡄⠀⡇⠀⡸⣻⠞⢁⠔⠻⡀⢸⠀⡇⠀⠀⠀⢠⣀⣠⠞⢁⡴⠋⠀⠀⠀⣰⠋⠀⡠⠾⡹⡌⣧⡇⠀⡸⠛⢦⡀⠈⢣⡀
⣰⠃⢀⡾⣄⠀⠀⢳⡀⡇⣼⠟⢁⠔⢙⣆⠀⢹⣸⣀⣷⣀⣀⡤⢽⢲⣭⡷⠋⠀⠀⠀⠀⣰⠃⢀⣞⡁⠀⠙⢷⠸⡇⡼⠁⠀⢀⣹⡆⠀⢳
⡏⠀⢸⣀⣀⠙⡲⣤⡷⠿⣁⣴⠿⠾⠿⠛⠚⠋⠉⠉⠀⠀⣏⢠⣏⡞⡏⠀⠀⠀⠀⠀⠀⡏⠀⣸⠀⠉⠙⠲⣬⡷⠿⡥⠖⠚⠉⠀⢹⠀⠈
⡇⠀⢸⠉⠉⠉⣉⠽⣧⣴⠷⣦⠤⠤⠤⣤⣄⣀⣀⣀⣀⣀⣬⡷⠿⠚⠁⠀⠀⠀⠀⠀⠀⡇⠀⢹⠛⠉⢉⡭⢟⣷⣴⠛⠯⣍⡉⠉⢹
⢹⡀⠈⢧⡴⠚⠁⣰⠁⡏⢣⡈⠙⠢⣤⠎⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠘⣦⠚⠉⢀⠎⢸⠉⢧⠀⠀⠉⢲⠏⠀⣸
⠀⠳⡀⠈⠳⣄⣰⠃⠀⣇⠀⠳⣀⡴⠋⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠈⠣⣤⠏⠀⢸⠀⠘⣆⢀⡴⠋⠀⡰⠃
⠀⠀⠘⠢⣄⠀⠉⠓⠒⠛⠒⠋⠁⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠈⠙⠒⠚⠒⠒⠋⠉⢀⡠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣁⣀⣀⣀⣀⣀⣀⣀⣈⣉⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡿⢷⣄⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠛⠛⠻⠷⣾⣟⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢙⣷⡾⠟⠛⠋⠛⠛⠿⣶⣄
⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⢠⣿⠛⣷⡌⠻⣧⡀⠀⠀⠀⠀⣸⡏⠀⣰⡿⠋⠻⣆⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠘⣿⡀⠈⠻⣦⡀⣀⣠⣿⠁⢠⣿⠁⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠰⡿⠋⠀⠀⠀⠀⣿⡇⠀⠀⠈⢻⣾⣿⡇⢀⣸⣯⣤⣤⣴⠶⠾⠿⠇⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⢿⣿⠟⠛⠙⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣤⣤⣤⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣶⣶⣶⡶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠛⠉⠉⠙⠛⠿⠿⠿⠛⢿⣿⣇⠀⢀⡀⠀⠀⠀⠀⠀⠐⠁⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⡈⠂⢤⣤⣶⣾⣿⣷⣦⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⡄⠀⢀⠀⠙⠟⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠸⠛⠠⠀⠀⠀⠠⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡓⠃⠀⢒⠂⠃⠀⡀⠀⠀⠀⠉⠀⠀⠀⠘⠃⠀⠰⠃⣴⣷⣄⡀⠀⠐⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠀⠃⡀⠀⠀⠱⡀⠀⠀⢳⣀⡀⣁⠀⠀⢀⠂⠰⣿⣿⣿⣿⣦⠀⠀⠀⠉⠉⠉⠋⠙⠻⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠈⠢⢀⠀⠈⠠⡀⠈⠻⠟⠃⡀⠔⠁⠀⢀⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠉⠀⠀⠚⠀⠀⠐⠦⡀⠀⠀⠀⣼⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠻⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠂⠘⡄⢠⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠈⣁⣐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⠀⢰⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⠿⠿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈⢰⣾⣿⢿⡿⠃⠀⠀⠀⠀⠡⠀⠀⠀⣠⣾⣿⠟⠋⠁⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠒⠚⠀⠀⠀⠀⠀⠀⢀⣈⣄⡽⢿⣿⣦⣀⡀⠀⠀⠀⠀⠐⢀⣼⣿⠏⠀⠀⠀⠀⠂⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⠖⠁⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠈⢻⣿⡇⠀⠉⠛⠿⢿⣷⣶⣶⣶⣿⣿⠿⡇⠀⠢⡀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦⣄⡀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠳⠄⣁⠀⠀⠘⠁⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣶⣾⣿⣿⣧⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⣿⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠒⠒⠀⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠻⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠉⠀⠀⠀⠈⠂⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡇⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⢀⠀⠈⢿⣿⡄⠀⢀⠤⠘⢄⠀⠀⠀⠀⠀⠸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠃⠀⠀⠀⠀⠀⠀⠠⠀⢀⠊⠀⠁⠉⠉⠉⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⡇
⢄⡀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⡐⠀⠀⠀⠀⠠⠈⢀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧
⠈⢿⣶⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⠃⢠⠋⠀⠀⠀⠀⠐⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿
⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠑⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠘⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣠⣬⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣶⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢰⡆⠀⢀⡀
⠀⠀⠀⠀⠀⠰⠶⠶⣶⡾⠿⠿⣿⡶⣶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠴⠂⠐⠤⡉
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⡀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢇⣄⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡸⣿⣷⣶⣿⢡⠂
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡁⠸⡟⠃⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⠉⠉⢱⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⠠⠜
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠾⠀⠄⠾⠸⠲

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣳⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠙⠟⡇⣠⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠐⡂⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡷⣄⣿⠻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⣤⣀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣴⣌⡃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣀⠀⠀⠀⢀⣀⣾⠃
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⣶⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⡿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⠀⢸⡿⠛⠁
⠀⠀⠀⠀⠀⠀⡀⣠⢠⠄⡀⢀⠄⣠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⡉⢀⠈⠁⣀
⠀⠀⠀⠀⠀⠀⠇⠧⠸⠜⠧⠸⠄⠿⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⠄⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠓⠀⠛⠛⠀⠒⠛⠚⠂⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠶⡄⣴⠄⣠⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠒⠖⣡⣾⢟⠕⢮⡻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⠶⠦⠴⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠾⢿⠗⢡⢀⢠⠘⢮⠻⠷⠀⠀⠀⠀⠀⢀⡴⠚⢻⣿⡑⠤⡀⢀⢃⣽⡞⠛⢶⡄⠀⠀⠀⠀⠀⣿⣿⣿⣏⠈⠙⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⡠⠄⠚⠀⠀⡀⠀⠀⠀⠀⢸⡅⠀⠋⢀⡗⠀⠈⠛⢺⡗⠛⠂⠀⣿⠀⠀⠀⠀⠀⣿⣿⠏⢹⣿⡶⠛⢿⣿⡇
⠀⠀⠀⠀⠀⠁⠶⢀⠀⣀⢈⠀⢀⣤⢄⠲⠈⠀⠀⠀⠀⠈⠻⠶⠶⠟⠁⠀⠀⠀⠀⠛⠶⠶⠞⠁⠀⠀⠀⠀⠀⣿⣿⣦⣤⣿⣤⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⡿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣰⣦⢀⡀⠀⣀⢀⣶⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣀⣰⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣻⠏⠀⢿⣿⣽⣟⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡽⠛⠉⢉⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣿⣶⣁⣀⣀⣀⣹⣶⣎⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢻⣽⣿⣿⣿⣿⣟⣯⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡆⢸⣿⣟⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣛⣿⣟⣾⣿⣻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠋⠛⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⣀⣤⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣼⢿⡦⣶⠀⢶⣦⡿⣷⢴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⣸⣤⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡟⠁⠀⠈⢻⣿⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⢰⣿⣿⡿⣿⡿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣾⣿⣿⣿⣿⣿⣳⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠸⡿⠿⠻⡿⣶⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣾⣿⣿⣿⣷⣿⣷⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⣿⣿⡿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠺⠿⠿⠾⠛⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⡿⣿⣾⡿⣿⡿⠿⠿⠿⠿⠯





⡿⢿⠿⠿⠿⢻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣦⠙⢿⣿⣿⣿⣿⣿⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣧⡈⠻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣄⠀⠙⠿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣆⠀⠈⠻⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⠤⡀⠉⠉⣉⠀⠀⠁⠀⠀⡉⠉⠩⢿⣿⣿⣿⣿⣿⣿⡟⠻⠟⠻⣿⣿⣿⣿⣿⣿
⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⠿⣟⣛⡛⠛⠿⣷⣶⣾⣿⣿⣿⣟⣷⣶⣦⣽⣿⣿⣿⣿⣿⣦⣤⣴⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣾⣷⣶⣴⣶⣶⣿⣿⣿⣿⣿⣷⣤⣙⡛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣃⡀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣌⣿⣿⣭⣥⣶⣦⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⣶⡆⠀⣄⣠⠄⠀⠀⠀⠀⠀⠀⢠⣤⣤⣾⣷⠀⠸⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣸⠿⠿⠿⠟⠋⠀⠰⠬⡵⠂⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⡿⠀⢠⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠈⠙⢆
⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠴⠋⠀⠀⠀⠀⠀⠀⡠⢠⠀⠈⠃⢀⠘⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠒⠂⠽⠆
⠀⠀⠐⢰⠃⡠⠃⠀⠀⠀⠀⠀⠑⠸⡾⠶⠀⠀⠀⠀⠀⠀⢀⣴⣊⡡⠀⠠⠚⠹⠿⣿⣿⣿⣿⠤⠄⠀⠀⡔⠂⠀⠀⣤⣄⣴⡀
⠀⠀⠀⢿⠎⠀⣿⠟⠀⠀⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⢀⡄⠀⠈⠉⠛⠇⠁⠀⠈⢀⣀⠀⢀⣈⡛⢿⠃
⠀⢀⣶⡄⠀⣤⠄⠀⠀⠀⢌⡁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠀⠈⠙⠇⠈⣀
⠀⠘⠏⠀⠀⠑⠀⠀⡀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠙⠀⠀⠀⠀⠼⣴⡶⠀⠀⠀⠀⢈⣿⣦⣄⠀⠀⠀⢰⠁⠐⠹⠄
⠀⣬⠄⢀⠀⠀⠀⣼⠁⠐⠀⡄⢀⡖⡄⢶⠦⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠉⠉⢰⠀⠀⠈⠉⠑⠈⠟⠀⠀⠀⠘⢀⢰⠢
⠀⠦⠆⠀⠐⠊⠩⠃⠀⠀⠈⠄⢂⡧⠃⠀⡀⠀⠀⠀⠀⠚⣢⣤⠢⠤⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡄⠀⠀⠴⠶⢈⠘⢷⡟
⠀⠓⠟⠀⠀⢰⡇⣀⠀⠴⡲⠖⠼⠃⠀⠀⠀⠀⠀⠀⡼⣮⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠘⠃
⠀⢈⡛⠃⠀⠀⠑⢧⣁⣀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠁⠀⢹⡄⠘⠿⠂⡀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠂⠀⠠⠄⠀⠺⠶⠟⠃
⠀⠨⡭⠀⠀⠀⢈⠈⠉⡁⣠⢘⡀⠀⠀⣀⠀⠀⠐⠀⠀⠀⢆⡿⠂⠐⡀⠐⠊⠀⢀⠔⠡⠀⠅⠑⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⢸⣄
⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢐⢻⠆⣰⣄⡀⠀⠀⠈⠾⡀⠀⠁⠀⠙⠉⢁⣠⣖⡭⠴⠋⢱⠄⠀⠀⠀⠀⠀⠀⠀⠀⣜⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⠐⠿⠿⠁⠃⠀⠀⠈⠚⠉⠀⠀⠀⡴⢋⠭⠖⠒⠒⢲⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠠⠤⠤⠼⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⣀⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠉⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠔⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡤⠤⠃⠀⠀⠀⠀⠀⠀⠀⡰⡖⠒⠲⠀⠀⠒⠒⢲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢀⠔⢃⠀⢀⡴⠖⠛⠓⠶⣤⠊⠀⢱⡀⢀⠢⠀⡠⠚⣡⠼⡖⠒⠲⢦⡀
⠀⠀⠀⠀⣀⣬⡤⢤⣌⡀⢀⠞⠃⠀⠈⡀⠀⢠⡢⠔⣚⣴⣏⣀⡀⠀⢀⠞⢉⣳⡀⢀⣧⣇⠴⠊⠀⡾⠁⠀⠰⠀⠀⠀⠹⣆
⠀⠀⢠⡞⠁⠀⡀⠀⣠⠟⣧⠀⠀⠀⡄⠀⢠⠋⣠⠞⠉⣿⠀⠈⠙⢿⡕⢒⣒⣺⡗⣺⣪⠉⠀⠀⢸⡇⠀⠀⠸⠷⠀⠀⠀⣿
⠀⠀⣾⠀⠀⠀⣴⡞⠁⠀⢹⡇⢨⡙⢥⠔⠁⢰⡏⠀⠀⢸⣦⡀⠀⠀⣷⠀⢀⡞⠀⠀⠛⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⢻⡄⠀⠀⠀⠉⠁⠀⣾⠃⠚⡿⠈⠀⠀⠘⣇⠀⠀⠈⠉⠙⠲⢶⡿⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⠤⠶⠚⠁
⠀⠀⠀⠙⠦⣄⣀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⣀⣀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣭⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⡶⣶⣤⣄
⠀⠀⠀⠀⠀⢶⡦⢀⣀⣀⡀⠀⠀⠀⠠⠴⠋⢈⡉⣻⣷⠀⠀⠀⠀⠈⠿⢇⣤⣤⣤⠀⠀⠀⠄⠐⠋⢀⡤⣈⣿⠇
⠀⠀⠀⠀⠀⠀⣰⠿⠛⠙⢻⣦⠀⣀⠀⠐⢒⡿⣾⡋⢀⣀⡀⣀⣀⣀⣠⣟⣋⡉⢉⡟⠀⡒⠒⢢⣉⣹⠻⣯⣄⠀⠠⡄
⠀⠀⠀⣠⠠⠾⠿⠞⢉⡽⠛⠀⢴⣶⡆⢀⣿⣿⡇⣷⣬⣿⣿⢹⣹⣿⣛⠛⠉⠴⠛⢁⣴⣄⢀⠺⠛⠛⠒⠿⡈⠁⠀⢸
⠀⠀⣠⠴⠶⢿⣃⣴⣶⠀⣶⡷⠶⢭⣿⡿⠿⠛⠛⢿⡻⢿⣧⣿⣿⣿⣿⣴⣿⡇⠐⢿⣿⠿⠏⢀⡤⠶⠶⢶⡌⣷⠶⡅
⠀⢰⠃⠀⠀⠀⠙⡆⠀⢹⢇⠁⣠⡾⠋⣇⢀⣀⠀⣡⣿⠀⡞⠀⠀⠀⠀⠘⡆⠀⢄⠀⣀⠶⠛⢀⣇⡤⢖⣀⣾⠇
⠀⠸⡀⠀⠀⠀⢀⠇⠀⠘⢧⠚⠡⠶⣶⡿⣧⣴⡿⠟⠃⠀⠹⣄⠀⠀⠀⠜⠁⠀⠈⣂⣁⡚⢹⡿⠛⣶⣿⡟⢉⣀⣀⣀⡀
⠀⠀⠉⠲⠶⠂⠀⠀⢠⡴⠚⠛⠓⢶⡏⠀⠸⣿⣓⠶⡚⠋⠛⠻⢯⡉⠀⠀⠀⠴⠋⠉⠈⠉⠻⡅⠀⠸⣿⡽⠻⣅⠀⠈⠙⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠹⡆⠀⠙⣏⠀⠈⠂⠀⠀⠀⣧⠀⠀⢸⠃⠀⠀⠀⠀⠀⢸⡄⠀⢹⡇⠀⠀⠁⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠃⠀⠀⠀⠀⠀⢀⠏⠀⠀⠸⣆⠀⠀⠀⠀⠀⡸⠁⠀⠀⠣⡀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣀⠄⠊⠀⠀⠀⠀⠑⠤⣤⡤⠴⠋⠀⠀⠀⠀⠈⠳⠦⠤⠖⠈⠀⠀⠀⠀⠀⠈⠒⠖⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⣿⡆⢸⠯⡀⠀⣙⡤⠀⠀⠀⠀⠠⣄⣀⣤⣀⣀⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣇⣿⣿⣤⢸⣞⠆⠆⣹⡇⠀⠀⠀⠀⠀⠹⣟⡻⢾⣿⣷⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⢀⣴⣌⣿⣟⣛⣩⠟⣾⡤⠖⠒⠓⠯⣌⠀⢀⠀⣼⣣⣿⣽⣿⣿⠄⠀⢐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⣧⣿⣼⡿⢿⣿⡻⣶⣶⡗⠚⠂⡆⢀⠲⢶⣿⣿⣿⢿⣽⣿⣇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡠⠀⣠⠤⣶⣿⣿⣿⣆⣀⣽⠛⠉⠉⢯⠹⠁⠓⡲⠖⠏⢴⠞⢃⡼⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⡶⠟⠲⢭⡿⠋⣹⣿⣿⠂⠀⢠⣶⣾⣶⣦⣤⣄⣨⡤⠤⠴⣿⡀⣱⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡞⣿⣔⢙⣀⣨⣼⠛⠋⣹⣷⠖⠉⠁⠈⠹⣏⠻⣿⡇⠀⠀⠀⣀⡼⣿⣿⠟⠂⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢺⣃⢹⣍⣿⣭⣤⠟⠢⢀⣿⡅⠀⢸⡇⠀⠀⢹⣆⣯⣤⡴⠶⢾⡁⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⣯⢘⣿⡇⠤⠄⠀⣀⠞⢿⡧⠹⣿⡇⠀⠀⢸⠟⠉⠀⠀⠀⠀⠙⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠃⠀⠀⠀⢀⣾⢇⠘⣏⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣴⡿⣸⠄⠙⣤⣤⠀⠽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢁⡾⠀⢷⠀⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠿⠛⠿⢿⣿⣿⠿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠈⠀⢀⣤⣶⣶⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠆⠀⠀⠈⠋⠀⠀⢤⠀⠀⢀⡀⠀⠉⠑⡀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡉⠈⠀⠄⠀⠀⢀⠀⠀⠀⠁⠀⠁⣼⣿⣦⣄⠀⠀⠸⠿⠿⠿⢿⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠂⡀⠘⢷⠶⠁⡀⠂⠀⣹⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠘⠀⠀⠲⢀⠀⠀⢠⣿⣿⣿⡟⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⢂⠀⠈⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢢⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠉⣀⣠⣤⣤⣤⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢠⣽⣿⣿⠏⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠛⠉⠉⠉⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠀⠀⠀⠀⠀⣀⣨⡿⣿⣦⣀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⣰⣄⠀⠀⠀⠀⠙⣿⡇⠈⠉⠛⠿⠿⠿⠿⠿⠟⠃⡀⠢⠀⠀⢀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣦⣤⣀⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠄⠊
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⢻⣿⣿⠛⠃
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣿⣧⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠊⠁⠀⠀⠀⠀⠀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠘⠛⠃⠀⠁⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠃
⢄⡀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⠀⢀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣆
⠈⢻⣶⣶⣴⣶⣿⣿⣿⣿⣿⣿⡟⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠈⠙⠻⠿
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠛⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣻⣟⣓⠀⢤⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⡀⢀⣿⡩⠽⠷⠮⣕⠢⡀⡞
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠹⢉⣻⣟⠈⠀⠀⢉⡗⣜⢇
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢀⡬⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⠉⠙⣿⣇⣀⣠⡞⠈⠉⠳⠄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣎⣤⣤⡒⢠⣀⠀⠉⠀⡀⠙⠓⠀⠀⢤⢤⣴⣋⢙⣿⠟⠃⠙⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢳⣀⡸⣿⣿⣿⡿⠷⠈⠁⠀⠀⣃⣀⠀⠀⠀⠀⠀⠈⠉⣿⣿⡇⠀⠐⠂
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⣡⠀⢀⢀⣀⠀⢉⣿⣟⠛⡃⠀⠀⠔⡁⠀⠐
⠀⠀⠀⠀⠀⠂⠀⠰⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠈⠘⠋⠛⣿⠿⠻⠟⠀⠀⣿⠀⠘⠉⠀⠠⠤⣄
⠀⠀⠶⠀⠀⠀⠀⣄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠀⠀⠀⠐⠈⣹⣿⣿⣶⡶⠁⠀⠈⠁
⠀⠀⠀⠑⢤⣼⡞⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣠⡠⣄⣀⠴⣞⡯⠞⠋⠁
⠀⠐⢆⠀⠀⠈⠁⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠐⠛⣿⠏⠞⠋⠰⠊⠁⣀⠠⣤⣶⣾⡦⠔⠋⠄
⠀⠀⠈⣆⠑⣤⣀⠈⠻⣧⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠹⠷⠄⠀⠀⠰⣮⣴⣶⣿⣿⣿⠟⠂
⠀⠀⠀⠘⢉⡙⠉⠢⣾⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡟⠁⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠈⠀⠀⢰⢃⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⡀⠀⣀⡀⢀⡰⠁
⠀⠀⢀⣤⡄⠀⢀⡏⢨⡤⠤⠤⠀⠂⣴⠀⠀⣐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠉⣰⡿⠀⠃
⠀⠀⠀⠀⠐⣁⠨⢯⣁⣈⣁⠄⠀⠈⠙⠛⠷⠿⠷⡀⠀⠀⠀⠀⠀⢀⣀⡴⠀⠀⠀⠀⠀⢀⣠⣞⠉⠈⢁⡧⠦⠄
⠀⠀⠀⠀⠤⢌⠒⠀⠀⠀⠀⡤⠴⠤⠀⠀⠀⠀⠀⠀⠘⠒⠾⠿⠿⠟⠛⠉⠀⢀⣤⠤⠖⠋⠉⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣷⣾⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠿⠤⠀⣀⣤⣤⣤⣤⣤⠤⠤⠼⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠂⠈⠉⠉⠉⠁⠀⠂⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⣷⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣀⠀⠀⢀⠤⠂⠐⠠⠀⢀⠀⠀⠀⣠⠀⠀⠀⠁⠀⠡⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠁
⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣷⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢀⢯⣣⠙⠁⢀⡇⠀⠀⣀⠀⠀⠀⠐⠣⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣠⡈⠀⠀⠀⢉⠉⡇⠀⠀⠉⠀⠀⠀⢠⠄⠀⠀⠀⡀⠀⠀⠀⢀⣿⠇⠠⠁
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠿⠿⠿⠿⠃⠀⠀⢀⣤⣾⣿⣿⡄⢡⠀⠘⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⠏⡠⠁
⠀⠀⠀⠻⢿⠿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠁⠀⢂⠀⠈⠪⣄⠀⠰⣤⡤⠒⠀⠀⠀⠀⠀⣠⣴⣿⡿⠃⡔⠁
⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣇⠀⠀⠑⢀⠀⣰⣷⣤⡀⠀⠀⠀⠀⣀⢤⣾⣿⣿⠟⠀⠈
⠀⠀⠀⢀⢊⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠀⠀⠀⠀⠱⣿⣿⣿⣤⣤⣾⣿⣿⣷⠀⢻⣿⡇
⠀⠀⠠⠃⠘⠁⠀⠀⠀⠾⠂⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⡀⠀⠀⠀⠀⢀⣿⣿⣿⠟⢿⣿⣿⠏⠀⠀⠉⠓⠀⠀⠘
⠀⠀⣾⣿⣿⣷⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠸⣿⣿⣿⣷⣀⠀⠀⠄⠼⠟⠋⠁⠀⣠⣿⣯
⠀⢀⡀⠀⠀⠀⠉⠙⠻⢿⣿⣦⠀⠀⠀⠀⠁⠀⠀⠀⠙⢿⣿⣿⡿⠛⠄⢀⣀⣠⣤⣶⣿⣿⣿⣿
⠀⠈⠋⠀⠀⠈⠒⠀⠀⠀⠙⢿⣷⡄⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣅⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠿⠿⢷⣶⣶⣶⡾⠿⠛⠉⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠲⣶⣿⣶⣶⣄
⠀⠀⠘⠠⡀⠉⠀⠀⠀⠀⠰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠠⠒⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡏
⠀⠀⠊⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠁
⠀⢀⠀⠀⠎⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⡇
⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⡐⠉⠂⠄⠀⣴⣿⡟⢀⠠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⢸⣿⣿⣿⣿⣿⣿
⠀⢘⣀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⠀⢀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄
⢠⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠈⠂⡀⠘⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⣀⣴
⠾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠆⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠃⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣐⠁⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠂
⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣏⣀
⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠉⠉⠁⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⣀⣐⡐⣖⠒⢄⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠛⠋⠑⠀⠌⠔⡠⢌⡐⠠⢄⢂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢹⢧⡂⠄⢀⡀⠀⠀⠀⠈⠉⠛⣲⣾⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠌⢀⠌⢠⠉⠓⠶⠤⣭⢤⣴⠶⠾⢿⣋⣡⠗⠈⢀⠁
⠀⠀⠀⠀⠀⠀⢠⣌⣄⡊⠔⠁⠀⠀⠀⡂⢠⠀⠀⠀⠀⠀⠀⠘⣗⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢤⡀⠀⠒⠒⠤⡒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡡⠀⠤⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢰⠀⠀⠀⢁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⡖⠒⠂⢪⠀⠆⠀⠀⢠⣄⣀⣀⣀⣀⣀⡀⠤⠄⢒⣈⡥⡼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠠⣀⠀⠀⠁⠁⠐⠒⠻⣿⠶⠖⣚⣩⣽⣶⢺⣏⢿⣈⣧⠿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣠⡤⠐⠂⠉⠉⠉⠀⠀⠒⠿⣟⣋⠭⠔⣚⣇⢻⣓⣋⡭⣵⠲⡞⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡏⡏⠉⣀⡲⠦⠀⠀⠀⠸⠖⠚⣩⠞⣹⡞⢿⡌⣷⣹⣇⠿⢜⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⣟⣉⠥⠔⡾⠀⠀⠀⢠⣠⠞⣡⠞⠁⢹⡛⡯⠭⠑⠒⠉⠁
⠀⠀⢀⡤⠒⠋⠉⠉⠉⠙⠲⢤⡠⢊⡴⡇⡇⠀⢀⠌⡁⠀⠀⠀⠘⣡⠞⠁⠀⣀⠴⢧⢻⠉⠉⠉⠒⠦⣀
⠀⡴⠁⣠⠖⢫⠉⣏⠉⡵⡲⢊⡴⢏⠀⡇⡇⢠⡂⢠⣀⠠⠤⠀⠚⡅⠀⢀⠞⠁⡠⣞⡌⣯⠏⢉⠖⢄⠈⠳⡀
⡼⠀⡰⢥⡀⠈⡆⢇⡼⢋⡴⢋⣆⠈⣧⣇⣧⠨⡷⢾⢛⣷⢦⠤⠴⠃⢀⡏⢀⣎⣀⠈⢿⡸⣧⠏⠀⣀⡵⡀⠹
⡇⠀⡧⠤⠬⠵⣞⠛⡴⠟⠛⠋⠉⠉⠁⠀⠀⠀⢇⠮⢊⠗⠃⠀⠀⠀⢸⠀⢸⣀⣈⡭⢶⡟⢳⣎⠭⠤⠤⡇
⢇⠀⢇⢀⡤⠚⡽⡟⣝⠫⢍⡉⡶⠓⡶⠒⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠸⡄⠸⣀⡤⠚⣱⢻⢻⠈⠓⠦⣀⠇⢀
⠘⢆⠈⠣⡀⡰⠁⢳⠘⢆⣀⠝⠁⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠙⢄⡰⠁⢸⠀⢧⠀⣠⠎⢀⠞
⠀⠈⠳⢄⡈⠙⠒⠚⠒⠋⢁⡠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠉⠒⠚⠒⠚⠉⣀⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡛⠉⠉⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠿⣿⣿⣿⣿⣿⡙⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠉⢿⡿⠾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣀⣀⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠽⠛⠉⣿⠁⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣹⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⢿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⣶⣶⢠⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡻⢎⣻⣿⡈⢿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣾⣿⢟⣵⣾⣿⣾⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠃⢿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠃⠀⠀⠀⠀⠀⠀⠀⣀⠀⠸⠿⣿⣿⣿⣿⡿⠿⢃⣀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣶⣦⣤⣀⡀⣠⣶⣿⠟⠁⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣿⣯⣭⣶⣭⣭⣵⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠉⠙⠛⠻⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦
⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⢋⣟⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣾⣿⣿⣿⣶⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⡀⢻⣿⣿⣿⠟⠃⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡟⢛⣷⠛⣻⡚⢻⡟⢠⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢏⡛⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠐⣿⠀⣿⡇⢸⡇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⡄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡿⠀⢿⠇⢸⡇⢻⣿⣿⣿⠇
⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠿⠟⠁⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠟⠋⢀⣀⡄⢿⣿⠟⣄⣀
⠀⢀⣠⣴⣿⣿⣿⣿⣿⣶⣦⣤⣄⢠⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠀⣿⢠⣿⣿⣿
⠺⣿⣿⣿⣿⢿⣿⣿⣶⣶⣶⣶⣶⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣿⣼⣿⣿⣿
⠀⠀⠉⠁⠀⠀⠀⠈⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⢶⣶⣤⣤⣤⣤⣤⣤⣶⡶⠾⠟⠛⠉⠁
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠂⠀⠀⠀⠈⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠⠤⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡠⠤⠤⢤⡼⣅⡀⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠉⠉⠉⠻⣦⡀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠹⡱⡀⠀⠀⢠⣤⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡀⠤⠄⠀⠀⠀⡴⠁⠱⡀⠀⡾⠈⠃⠀⠀⠀⢱⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⡠⠀⣀⠀⢞⠀⠀⠀⠱⣀⠇⠀⠈⠀⠀⠀⠀⢉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠢⢶⣷⣠⡄⠁⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠠⠤⠬⢿⠀⣀⠈⡀⠉⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⢉⡖⠀⠀⠀⠀⠄⠈⡆⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠣⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢠⠦⣵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠐⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠠⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠈
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠐⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠉⣰⠔⠀⠀⠀⠀⠑⡄⠀⠀⠀⠲⠏⠀⡀
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⢎⣠⡤⠤⢠⣤⡄⢀⠐⠀⢀⠀⠀⢀⡜⠹⣇⣉⣀⣁⣀⣀⣈⣂⣤⠴⠦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢣⠀⢀⢮⠊⠀⠀⠀⠀⠀⠀⢸⠀⠀⠆⠠⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣜⠈⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⡀⠀⠀⡀⠀⠄⠀⢠⠀⠀⠀⠈⢳⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣧⡤⠀⠀⠀⠀⠀⠀⣢⠄⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠑⠀⠉⠲⠀⠐⠚⠀⠀⠈⠀⣄⠀⠀⠀⠙⢄⠀⣿⣀⣀⡀⣤⠤⠖⣋⣵⢊⢿⣶⠀⠀⢀⠄⡑⢉⣄⣰⣈⠱
⣘⠀⠐⣈⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⡤⠀⠈⠢⢀⠀⠀⠀⠳⡇⠀⠀⠀⢨⠄⢚⣿⣿⡷⣔⣊⢉⠒⠀⠚⡋⢟⣿⣿⡥⠄
⠙⠐⠂⠪⠛⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠏⠁⠀⠀⠈⠉⠂⠂⠈⠀⠀⠀⠀⠀⠓⠥⣨⡀⠤⠋⠀⠈⠄⠀⠀⠙⢬⣌⣁⣩⠞
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡒⠀⢈⠁⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠄⣠⣰⣴⠤⠓⠂⠉⠁⠀⠈⠉⠒⡈⢠⠞⢡⠼⢪⠀⠀⠀⠀⠀⠀⢰⡎
⠀⠀⠀⠀⠀⠀⠐⠀⣄⣀⡼⡲⡽⠋⠁⠀⠀⠀⠐⠀⠀⠀⠀⠸⠀⡡⠀⡎⢸⡈⣄⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠨⠙⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠂⠎⢽⣄⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⢀⡠⢒⠑⠁⠀⠀⠀⠀⠀⠀⣈⣁⣠⣀⠐⢃⠋⢾⡀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢘⢵⡈⣽⢱⡼⠀⠀⠀⠀⠀⠈⣧⣺⣭⠫⡑⠀⢊⣠⣀⠀⠀⠀⢠⡧
⠀⠀⠀⠀⠀⢀⠀⡆⡆⠀⠀⠀⠇⢘⣤⠝⠻⠁⠀⠀⠀⠀⠀⠀⠉⢿⢭⣤⠔⠀⠈⣣⠌⠃
⠀⠀⠀⠀⣧⣿⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⢀⠼⠄
⠀⠀⠀⠀⠣⣤⢠⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠐⡆
⠀⠀⠀⠀⠀⠀⠀⡇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡘
⠀⠀⠐⠀⠁⠀⠓⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠊⠎
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢁⠠⡺⠞⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂⢀⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠈⠀⠑⠀⡀⢠⡖⠀⠀⠀⠀⠀⡤⠤⠌⠛⠃⡎⢀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠒⢀⡴⣺⠀⠀⠠⠄⣀⡀⢉⣑⡒⣲⣄⡶⠈⡼⡔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠠⣼⣦⣄⠞⠀⢠⠖⠛⡅⡐⣹⢧⣄⠘⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈