Ribbion Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠉⠓⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠛⠉⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢤⡈⠱⡏⠓⠛⣶⢁⠴⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢣⣤⡤⠤⠤⣀⣀⣀⢀⣉⡇⢠⠀⣿⣀⠀⠀⠀⠤⠤⠤⢶⣦⡸⡄
⠀⠀⢀⣀⣤⠤⠶⠒⠛⠋⠉⠉⠙⢻⡘⠯⠁⠀⠀⠀⣀⣠⡤⣾⣠⣮⡴⠳⡦⢤⣀⣀⠀⠀⠀⠀⠚⢁⡇
⠐⠛⠿⠦⠤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠭⠿⠟⠛⣋⣭⠟⠋⢀⠏⠀⠙⢮⠢⡙⡄⠙⢬⡉⠛⠿⣟⣛⣉⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⣀⣀⣀⣤⠤⠶⠒⠋⠉⠀⠀⠌⡼⠀⠀⠀⠀⢳⡈⢮⣢⡀⠙⠢⢄⠀⠀⠉⠉⠉⠙⠛⠋⣩⠏
⠀⠀⠀⠀⠸⡧⠖⠚⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡈⠄⡇⠀⠀⠀⠀⠀⢣⠀⠱⡝⠦⣀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣹⢠⡇⠀⠀⠀⠀⠀⠈⣇⠐⡝⣆⠈⠑⠦⢤⣀⣀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⢸⡇⠀⠀⠀⠀⠀⠀⠸⡄⢹⡘⣆⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⢳⠀⢷⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠘⡇⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢹⡀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣧⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠘⡄⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠱⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⣄⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠁⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⡰⠋⠹⡄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡧⠚⠁⠀⠀⢷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⣿⠲⣄⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⡏⠀⠈⠙⠦⣄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⡇⠀⠀⠀⠀⠈⠓⢾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠉⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠤⠔⠒⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠶⠒⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠴⠒⠚⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⢢⡀
⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠴⠒⠒⠒⠒⠒⠒⠒⠒⠤⠤⠤⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⣠⠴⠒⠒⠚⢻⠀⠀⠀⠀⣀⡿⠤⠶⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢳⠖⠦⢤⣀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢇⠀⠀⠀⠀⠸⣤⣖⣉⢹⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⡤⠤⣄⣀⠉⠑⠲⠤⣄⡇
⠀⠀⠀⢸⢰⣄⡀⠀⠀⠀⠀⢀⡰⠃⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣶⣖⣚⣋⡉⠉⠋⢯⣀⣠⡄⠀⠀⢹⡓⠒⠆⠐⠃
⠀⠀⠀⢸⡉⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠙⠒⠶⠶⣶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⣀⣀⡀⠀⠉⠉⠙⠒⠶⡇
⠀⠀⠀⠀⠉⠓⠒⠒⠒⠒⠉⠉⠉⠉⠐⠒⠢⣄⠙⢏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡯⠴⠒⠉⠉⠉⠉⠉⠉⠉⠑⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠶⠒⠒⠒⠒⠚⠛⠛⠛⠛⠒⠒⠒⠒⠲⠤⠤⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠤⢤⣀⣀
⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠲⡀
⠀⠀⠀⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠶⠒⠒⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠒⠲⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀
⠀⠀⠀⠀⠈⡇⠀⠀⣀⣠⠤⠖⠊⠉⢫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠤⣤⣀⣀⠀⢸⠀⠈⠉⠉⠲⡄
⠀⠀⠀⠀⢠⡷⠚⠉⢁⣠⠤⠖⠒⠒⠲⢷⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠒⠮⠷⠋⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠉⠉⠉⠉⡇⢀⣀⣠⣀⠤⠞⠁⠀⠠⢤⣄⠬⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠔⠚⠉⠙⠦⣄⣀⣀⣀⠤⠤⣄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣉⣀⣀⣀⣠⠤⠤⠤⠤⢄⣈⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠯⢍⣩⠅⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠛⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⠴⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠦⢤⣀⡀
⠀⠀⠀⠀⠀⢀⣠⠴⠖⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠤⣄⣀
⠀⠀⠀⢠⠞⠉⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠦⠤⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠈⠛⠓⠒⡦⠀⠈⢧⣀⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⡤⠖⠋⢡⡟
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⠙⠶⣄⡀⣻⠉⠉⠛⠲⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⡴⠋
⠀⠀⠀⠀⡴⠃⣀⡤⠶⠚⠛⠉⠉⠉⠉⠛⠋⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣞⡁
⠀⠀⠀⠘⠷⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠤⣤⣤⣀⣀⣀⣀⣸⠃⠀⠀⠈⠉⢉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢀⣀⡤⠞⠉⠀⠀⣀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⠉⠁




⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⡠⠀⠀⠀⢱⠀⠀⠀⠀⢀⣀⠠⠤⠔⠒⠒⠒⠒⠒⠒⠒⠢⠤⢤⣀⡀⠀⠀⠀⠀⡎⠀⠀⠀⢄
⠀⠀⠀⠀⢀⡄⠀⠀⠀⠙⡤⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⢤⠉⠁⠀⠀⢠⡀
⠀⠈⣑⣄⣈⡝⠈⠐⠩⣀⠌⠒⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠒⠡⣀⠝⠂⠁⢫⡁⣠⣊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠲⠀⠀⠤⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⢠⠀⠀⢰⠛⠛⠛⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣶⠶⠒⠆⠀⠤⠈⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢺⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⡒⠉⠁⠀⠀⠀⠀⠙⠒⠦⠀⠀⠠⠤⠤⠤⠄⠀⠀⠀⠀⠈⠁⠀⠀⠂⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣀⠀⠀⠀⠤⠀⠈⠁⠀⠀⠈⠐⠢⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠢⠤⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠤⠄⠐⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠒⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠖⠒⠚⠉⠉⠉⠉⠉⠉⠉⠙⠓⠒⠒⠒⠦⠤⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⡄⣡⡆⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢩⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠒⠢⠤⠴⠷⠾⠿⠛⢋⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠠⠃⢀⣀⣀⣤⣤⡤⠤⠤⠤⠤⠤⠄⢀⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡊⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⠉⠭⢭⡹⠀⠕⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠐⠒⠢⠤⠤⠀⢀⣀⣀⣀⣀⣀⡀⠬⠕⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠀⢀⠄⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠃⢀⠀⠀⠀⠀⢰⠒⠚⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠠⠤⠄⠤⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠦⠀⠀⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⢀⠔⠋
⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠃⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠐⢲⣶⡶⠖⢃⡀⠠⠤⠀⠀⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠿⠟⡻⠿⣿⠟⣿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠠

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⣀⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣄⣀⠀⠀⠀⠀⠀⢠⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤
⠈⣿⣿⢿⡿⠛⠛⠛⠛⠛⠛⠛⠛⢿⡿⣿⣿⠃⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠘⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠃⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠆⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠂
⠀⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁


⠀⠀⢀⣠⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣦⣶⣿⠃⠀⠀⠠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡤
⢀⣾⠿⠛⠋⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⡿⠿⠃⠀⠀⠀⢸⣿⠿⠟⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠁
⠠⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⡀⠀⠤⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠠⠄⠀⡀

⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣷⣦⣴⣾⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠸⣿⣿⣿⡿⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⡉⣠⣿⡖⠀⠀⢲⣿⣄⢉⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢟⣫⣥⠶⣫⣾⣿⠏⠀⠀⠀⠀⠹⣿⣷⣝⠶⣬⣝⡻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢰⢟⣫⣭⣭⣭⣭⣭⣭⣶⣶⣶⡿⠟⣋⣵⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣮⣙⠻⢿⣶⣶⣶⣭⣭⣭⣭⣭⣭⣝⡻⡆
⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⠿⠿⠛⣛⣭⣵⣾⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣶⣮⣭⣛⠛⠿⠿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠁⠀⡠⠄⠠⠍⣦⠀⠀⣀⣠⠔⠋⢻⠟
⠀⠀⠀⠀⢀⡠⠤⠒⠒⠚⠛⠛⠛⠛⠛⠒⠶⢤⣀⡀⠀⠀⠀⠀⠀⢿⡳⠀⠈⠳⠶⠴⠷⣓⣚⣭⣥⣄⣩⠔⠁
⠀⠀⠀⢸⠱⣾⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⠀⠀⢻⣩⣭⠉⠉⠉⠉⠀⠀⠐⣦⢹⡇
⠀⠀⠀⠘⡇⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠾⠧⣟⣀⣀⡀⠀⠀⠀⣀⡩⢾⠇
⠀⠀⠀⠀⡇⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣩⣴⠂⡜
⠀⠀⠀⢀⡇⢟⣣⡤⠤⠴⠤⠤⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⠁
⠀⠀⠀⣾⢖⠍⠀⠀⠀⠀⠀⠀⠈⣧⢸⠉⠛⠲⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢸⡆
⠀⠀⠐⣇⠻⣶⡀⠀⠀⢀⣀⣠⡤⠭⢥⣇⠀⠀⠀⠀⠉⠉⠛⠒⠦⠤⣤⣀⣀⣀⣀⣀⣀⣀⣉⡥⠞⠁
⠀⠀⠀⠙⠓⠒⠒⠒⣏⡹⠁⠀⠀⠀⠑⡎⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠜⣂⣀⣠⣤⠴⠊






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

⠀⠀⠀⠀⠀⠀⠀⡴⠛⠋⠉⠙⠛⠛⠲⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⢀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠦⣤⣀
⠀⢀⡤⠖⠛⠉⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢤⣄⣀
⣾⠉⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠶⠶⠤⠤⠤⠤⢤⡄⠀⠀⠀⠀⣀⣤⣤
⠈⠉⠙⢛⡆⠀⠀⠳⣤⣤⡤⢤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡤⠶⠚⠉⣠⠏
⠀⠀⣰⠏⠀⠀⠀⠀⠀⠉⠳⢦⣄⠀⣿⠈⠉⠙⠓⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡴⠃
⢀⡞⠁⣀⣤⠴⠖⠚⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⡏
⢿⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠦⣤⣀⣀⣀⠀⠀⠀⠀⢀⡇⠀⠀⠈⠉⠛⠛⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⣉⡽⠟⠋⠀⠀⠀⢀⣠⠴⠚⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣓⣶⣿⣥⣤⣤⣴⣶⣒⣛⣉⣀⣀⣀⣀
⣿⣿⡋⢙⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣠⠶⠛⠛⠛⠛⠛⠛⠲⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⠒⠒⠒⠶⠦⣄⡀
⠀⠀⠀⠀⢠⣾⠁⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⢀⡴⠟⠛⠋⠉⠉⠉⠉⠉⠉⠛⠓⠲⠦⣤⣀⡀⠀⠈⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⢀⣠⡤⠶⠖⠚⠛⠛⠉⠉⠛⠛⠛⠿⢤⡀
⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⣠⠞⠉⠀⣀⡤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⠙⣷⠴⠶⠶⢿⣁⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏
⠀⠀⣿⠻⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠁⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠞⢻⠁
⠀⠀⢸⠀⠀⠈⠙⠛⠳⠶⠦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⡤⠶⠶⠛⠛⠉⠀⠀⣿
⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣇⠀⠀⠀⠀⢀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣄⠀⠀⠀⣸⣷⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠖⠛⠉⣡⠞⡿⠀⠈⠉⠉⠉⠁⢸⡆⠀⠙⣟⠛⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠘⢧⣄⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡶⠞⠛⢉⡿⠁⠀⢀⡴⠋⢰⡇⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢹⡄⠀⠀⢻⡝⠛⠷⢶⣤⣄⣀⣀⡀⠀⠀⣀⣠⡴⠋
⠀⠀⠀⠀⠉⠛⠻⠿⠿⠟⠛⠛⠉⠉⠀⠀⠀⣠⠟⠀⠀⣠⡟⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣷⠀⠀⠀⢳⡀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⣴⢿⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠈⣿⡇⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡾⠃⡿⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⡀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⠀⠀⠀⣠⡟⠁⣸⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡿⣧⠀⠀⠀⠘⣧
⠀⠀⠀⠲⣶⣶⡶⠒⠛⠋⠉⠀⠀⠀⠀⢀⣼⠏⠀⣠⠏⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠈⣧⢻⡆⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⢻⡟⠀⢀⣴⠟⠁⢀⣴⠋⢀⣀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⡌⢿⡄⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢋⣠⡴⠟⢁⣠⡴⢛⣡⡾⢛⡟⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⢷⡈⢿⡄⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⣠⣴⣶⡿⠷⠞⠛⠉⠀⠈⠛⠛⠛⠋⠁⢀⡾⢁⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢳⡌⢻⣄⠀⠀⠀⠀⠀⠈⠛⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠙⢦⡙⢧⡀⠀⠘⢶⣤⣤⣤⣤⡽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡵⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⣀⣀⣀⣹⡿⠛⢷⣄⡈⠙⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⣿⠙⠉⠁⠀⠀⠀⠉⠛⠷⢶⣤⣭⣿⣶⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠘⠷⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⡬⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⢦⣄⠀⣠⡤⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠘⠟⠁⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠖⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣰⡶⠛⠉⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣤⣄⣹⣷⠞⠛⠻⣯⡉⣉⣉⣉⣉⣉⣹⣧⣴⠾⠟⠛⠛⣷⡀
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠈⠉⣿⠃⢀⡀⠀⠈⢿⡛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⢀⣼⣧⣤⣄⡴⠚⠛⠻⣏⢹⡆⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⡏⠀⠀⣠⣾⡁⠀⠀⠀⣽⠋⠀⠀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯
⠀⠀⠀⠀⠀⢻⡀⠀⠿⡼⢷⣤⣤⠶⠋⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⣿⣦⣀⣠⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣧⣤⣤⣄
⠀⠀⠘⠛⠛⢻⡟⠛⠂⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢀⣠⣼⣷⡤⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⢀⡤⠤⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠐⢲⣿⠷⠦⠄⢀⣴⠶⠛⠲⣦
⠀⠀⠀⠀⠀⠀⠹⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠶⣤⣾⡁⠀⠀⣴⠏⠀⠀⠀⣀⢸⡇
⠀⠀⠀⠀⢠⡴⠞⠛⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⠈⠙⠳⠤⣟⠀⢀⣀⣴⢏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣾⣿⣋⠁⠀⠀⠀⠀⣠⡶⣫⣤⣍⣭⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣶⣤⡾⢋⡟⠉⠉⣿⠉⠉⠉⠉⠉⠁⣿⠀⠀⢹⣆⢙⣷⠶⠦⣤⡾⣫⡾⠋
⠀⠀⠀⠀⠀⢀⡾⠋⠉⠀⠈⢛⡿⠁⠀⠀⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⢹⣿⠁⠀⠀⠘⡿⠋⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣠⡴⢦⣤⠶⢦⡄⠀⠀⠀⣾⠁⠀⠀⠀⣰⠇⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢰⡏⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣿⠀⠀⠀⠹⢦⣄⣠⡴⠋⠀⠀⢀⣸⡛⣻⣇⡀
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⣾⠁⠀⠀⠀⠀⠈⠻⣄⡀⠀⣀⣼⠃⠀⠀⠀⠀⠀⠈⣷⠀⠀⢀⡶⠛⠉⠙⠉⠉⣙⣳⣦
⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⡏⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣼⣧⡴⠞⠛⠳⠾⠋⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠶⠞⠛⣻⡟⠁⠀⠘⢻⡾⠻⠛⠛⠛⠛⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⡀⠀⠀⠀⠀⣀⣽⣇⣀⣀⣀⣀⣠⣤⡾⠏⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠋⠩⠭⠩⣉⣉⣉⣉⠀⢀⣀⣀⣀⡄⠄⠤⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠶⢶⣤⡀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠉⠙⠻⣶⣄⠀⢀⣀⣀⣀⣀⣀⣀⣠⡾⠋⠀⠀⠀⠈⠻⣷⣀⣴⡾⠛⠉⠁⠀⠀⠀⠙⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠛⠛⠛⠉⠉⠉⢉⣿⠁⠀⠀⠀⢀⣀⣀⣈⣿⣯⣤⣄⡀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢰⣟⠋⣽⡟⠋⠉⠈⠉⠻⣷⣶⠶⠶⠶⠿⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠈⠻⣷⣿⠀⠀⠀⠀⠀⠀⢸⣷⣦⡀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣄⣀⣀⣠⣬⣿⣆⠀⠀⠀⠀⢀⣼⡟⠙⣿⡄⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠙⠻⠶⠶⣶⡟⠛⠛⠛⠋⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⣠⣴⢿⣇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠸⣿
⠀⣀⣀⣀⣠⣤⣼⣷⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣴⣿⣤⣤⣀⣀⣀⡀
⠘⠛⠛⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⢰⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠙⠛⠛
⠀⠀⠀⠀⠀⠀⢸⣧⣤⣴⡶⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠇⠀⠀⠀⠀⠰⣶⣤⣤⣿⠁
⠀⠀⠀⠠⣶⠾⠟⢿⡏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠛⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⣽⡟⠿⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⣠⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⢶⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⣠⣼⠿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡿⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠿⠋⠁⠀⠈⠻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋⠀⠀⠉⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢶⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠶⠶⠶⠶⠿⠿⠿⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡷⢿⣿⣿⣷⣶⣶⣦⡿⠏⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠦⢾⣫⢻⡇⠸⡿⡿⠿⠻⡆⠈⢠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣡⣶⣶⠶⣬⣿⣿⣄⣄⠘⠛⠟⠁⢀⣼⣿⣿⣷⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⡭⠙⢻⣿⣿⣿⣷⣦⣤⣴⣾⣿⣿⣿⣿⣿⣍⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣄⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⡄⠀⣺⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣽⣿⣿⣷⡀⠀⠘⠻⣿⣿⣿⣿⣿⡟⣿⣮⣹⣶⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡟⢃⠘⠙⣿⣿⣿⣤⣤⣤⣤⡈⢉⣩⣽⣿⢸⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠝⠿⠗⠀⠠⠄⠛⢋⣿⣿⣿⣿⣿⣷⣿⣿⣿⣻⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣴⣯⡽⣯⣿⣿⣛⣵⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠃⢰⣿⣿⣿⣿⣿⡯⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⢸⠁⢀⣴⠟⠃⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⢠⠚⢿⣿⣄⣈⣙⣿⣏⠎⠀⠀⠀⠀⠀⠈⠐⠻⢷⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⡀⠀⠀⠂⠀⠐⠀⠀⠀⠀⡎⢠⣿⣿⣿⡘⠿⠿⣻⣤⣓⣶⣦⣄⡀⠀⠀⠀⠘⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢈⠃⠀⠉⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠘⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⠟⣷⡆⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢟⠷⣿⠟⡄⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⡿⠿⠟⠛⠀⠀⠀⠃⠂⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣅⡄⣹⠿⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣸⣯⠟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣰⣿⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣶⡌⠁⠀⠀⠧⢀⠀⠀⡀⠀⠀⠀⠀⢼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣁⣤⣴⠖⢢⣤⣤⠀⣀⠀⠀⢤⣅⡰⢟⣷⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠂⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣯⣅⣀⠀⠀⠀⠀⣩⡏⢁⡀⠀⢠⡟⣿⣿⣿⣷⡀⠅⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠰⣶⣷⣄⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⢷⣦⠄⠀⠈⢈⣷⣄⣽⣿⣿⣿⣿⣿⣷⡀⢦⡀⠀⠀⠀⠀⠀⢸⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣄⣀⡀⣰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢿⣎⣛⠀⠀⠀⠀⢠⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠒⠐⠛⠛⠉⠁

⢠⡶⠳⢦⡀⠀⠀⣠⠶⠳⣦⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⢠⠖⠒⣄⠀⠀⡠⠒⠲⡄
⠀⣧⠀⠀⢈⣶⣎⠁⠀⢰⡇⠀⠀⠀⠀⣸⠇⠈⠳⣀⡰⠋⠀⢿⡀⠀⠀⠀⠀⣟⠀⠀⠈⠂⣤⠐⠁⠀⠀⣻⠀⠀⠀⠀⢻⣀⣀⣈⣦⣴⡁⣀⣀⠇
⠀⠙⠛⠋⣱⠃⢻⡉⠛⠛⠁⠀⠀⠀⠀⠻⣤⡤⠖⡿⡗⠦⢤⡼⠃⠀⠀⠀⠀⠈⠁⣢⡤⠞⠉⠳⢤⣔⠈⠁⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠑⣄
⠀⠀⠀⠀⣿⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠁⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢹⡆
⠀⠀⠀⣸⡇⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⢰⡿⠟⠁⠀⠀⠙⠿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠋⠀⠀⠀⠈⠃
⠀⢠⡄⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⣰⣶⣤⡀⠀⠀⣀⣤⣶⡄⠀⠀⠀⠀⠀⢠⣶⠒⢄⠀⢀⠔⢳⣦⠀⠀⠀⠀⠀⣴⣔⡠⢄⡀⣀⡠⠄⣢⣦
⠀⢹⣿⣶⣤⣶⣶⣤⣶⣿⡇⠀⠀⠀⠀⢿⣿⣿⣽⣶⣾⣫⣿⣿⡟⠀⠀⠀⠀⠀⠸⢿⣷⣶⣶⣶⣶⣿⠿⠀⠀⠀⠀⠀⢿⣿⣿⠿⡿⢿⠻⣿⣿⣿
⠀⠸⠿⠿⢫⡿⢿⡝⠿⠿⠃⠀⠀⠀⠀⠸⠿⠛⢡⣿⣿⡉⠛⠿⠃⠀⠀⠀⠀⠀⠀⢀⣠⡾⠁⠱⣤⣀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣾⡇⠸⣷⡀⠈⠁
⠀⠀⠀⠰⠿⡇⢸⠟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠋⠀⠀⠀⠘⠻⢿⠇⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⣤⣀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⢀⣴⠲⣄⠀⢀⡴⢲⣄⠀⠀⠀⠀⠀⠀⢀⣴⠖⠻⡆⣰⠛⠲⣦
⠀⠀⣏⡀⠹⡆⢰⠋⢀⣻⠀⠀⠀⠀⢠⣿⣽⣷⣼⡆⣼⣴⣿⣽⣧⠀⠀⠀⠀⢀⣤⠿⣶⢄⢻⣇⡴⡶⠿⣤⡀⠀⠀⢰⡉⠉⠓⢄⠀⠀⡠⠒⠉⠉⡇
⠀⢰⠋⠛⠓⣷⡶⠚⠛⠙⡆⠀⠀⠀⠙⠦⣝⣻⣿⣷⣿⣿⣛⡥⠞⠀⠀⠀⠀⢿⣇⣀⣀⡁⣿⣿⣈⣀⣀⣸⡇⠀⠀⣠⠛⠛⠓⢒⠷⠶⡒⠒⠛⠛⢦
⠀⠘⠓⠒⢉⡟⢳⡑⠒⠚⠃⠀⠀⠀⠀⠀⣀⡴⠚⠉⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢀⡇⢳⠀⠉⠉⠉⠀⠀⠀⠈⠒⠒⡾⠋⠀⠀⠈⢷⡒⠒⠋
⠀⠀⠀⠠⠟⠀⠀⠷⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠙⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈
⠀⠀⠀⡠⠠⠀⡠⠢⡀⠀⠀⠀⠀⠀⠀⢀⣴⢄⠀⠀⠀⡀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⢠⣄
⠀⠀⠀⢣⠀⢀⠀⣰⠁⠀⠀⠀⠀⠀⢀⣼⣿⣆⢳⠀⡴⢡⣿⣿⡄⠀⠀⠀⠀⠀⢀⡞⠙⠓⢦⠀⠠⠚⠙⠳⡄⠀⠀⠀⠀⢸⣿⠀⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠃⣘⡐⠁⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣳⣶⣗⣻⣿⣿⣿⠀⠀⠀⠀⠀⠘⢧⡀⠀⣠⣷⣆⡀⢀⣠⠇⠀⠀⠀⢠⣾⣿⡷⡶⣶⢾⣿⣧
⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⢉⣼⡞⣷⡉⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠉⣩⠜⠙⡄⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⣿⣄
⠀⠀⠀⠀⠀⢦⠡⣄⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠀⣿⣿⠆⠀⠀⠀⣤⣄⣤⠖⠒⢛⠋⢁⣤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⢸⠿
⠀⠀⠀⠀⠀⠈⠳⠮⠉⠃⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠘⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀
⠀⠀⣾⣶⣄⣀⣤⣶⣶⠀⠀⠀⠀⣤⣤⣀⣀⣤⣤⠀⠀⠀⠀⢰⣦⣀⣀⣀⣴⡆⠀⠀⠀⠀⠐⠄⢀⣀⢀⡔⠂⠀⠀⠀⠀⢸⣿⣄⣀⣠⣾⡇
⠀⠀⠿⠿⠋⠉⠛⠿⠿⠀⠀⠀⠀⠛⠛⠋⠙⠛⠛⠀⠀⠀⠀⠸⠿⠛⠉⠛⠿⠇⠀⠀⠀⠀⠉⠔⠈⠉⠐⠦⠁⠀⠀⠀⠀⢾⠛⠉⠉⠉⠛⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠿⣷⣦⣄⢀⣴⣾⣿⡿⠟⠋⠙⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠙⢻⣿⡿⠛⠁⠀⠀⠀⠀⠀⢻⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣶⣾⣿⠃⠀⠀⠀⠀⣀⣀⠀⢹⣿⣄⣀⣀⠀⠀⠀⣀⣨⣿⣿⣤⣀
⠀⠀⠀⠀⠀⣰⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣶⣾⣿⣿⣿⠿⠿⠛⠛⠋⣿⡟⠀⠀⠀⢠⣾⠿⢿⣿⡿⠟⠛⠛⠿⣷⣾⡿⠛⠛⠉⠉⠙⢿⣧
⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢻⣿⡀⣾⡟⠀⠀⠀⠀⠀⠈⣿⣷⣤⣄⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠈⠻⠿⣿⡇⠀⠀⠀⠀⠀⠀⣼⡟⢙⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣀⣀⣀⣀⣤⣿⣿⣦⣀⣀⣀⣠⣾⡿⢿⡿⠟⠁⠀⢠⣿⠏
⠀⠀⠀⠀⠀⢿⣿⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠛⠛⠋⠁⠀⠉⠛⠛⠛⠛⢿⣦⣀⠀⠀⢀⣠⣾⣿⣦
⠀⠀⠀⠀⠀⠘⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⡿⠿⠟⠉⢻⣿⡆
⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣤⣶⣶⣶⣶
⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⠿⢿⣿⡏⠉⠉
⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠀⠀⠠⣶⣾⣿⣿⣿⠿⠿⠷
⠀⠀⠀⠀⠀⣿⣿⣄⣀⣀⡀⠀⠀⠀⠀⢀⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⢀⣤⣴⣾⣿⣿⣿⣿⠟⠛⠃⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣿⣿⣀⣀
⠈⠋⠉⠀⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣁⣀⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⡿⠛⠿⠿⣿⣶
⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣟
⠀⠀⠀⠀⠾⠿⠛⠉⠻⣿⣧⣀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠉⠻⠿⢿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⡏⠀⠀⠀⠀⠀⠈⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠛⠿⣿⣿⣷⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⠛⠉⠀⠘⣿⣿⣄⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠛⠛⠛⣿⡏⠁⠀⠻⣿⣦⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⢀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣿⣿⡟⠉⠁⣴⣿⠟⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠙⣿⣷⡀⠀⠀⠈⢻⣿⣧⣠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠛⠛⠙⣿⣿⣄⣾⣿⣯⣤⣤⣀⡀⠙⢿⣷⣤⣀⣀⣀⣀⣤⣾⡿⠋⠀⢀⣀⣤⣤⣼⣿⣷⣄⣀⣤⣴⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⣿⣿⣿⡿⠿⠿⠿⠿⣿⣷⣤⠉⠛⠛⠿⠿⠛⠛⠉⠀⣠⣾⣿⠿⠿⠿⠿⠿⣿⣿⣿⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣠⣾⡿⠋⠁⠀⠀⠀⠀⠀⠈⠻⣿⣷⡀⠀⠀⠀⠀⠀⢠⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣦⣤⣤⣤⣴⣾⣿⡿⠟⠛⠋⠉⠉⠉⠉⠉⠙⠛⠻⠿⣿⣷⣶⣦⣤⣴⣶⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠛⠻⣦⣤⣴⠶⠟⠛⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠻⠿⢶⣦⣤⣤⡴⠶⠶⣿⠃⠀⠀⣀⡈⣿⡀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢾⣉⡟⠋⠉⠛⣶⠛⠛⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⣀⣀⣽⡄⠀⠀⢀⡿⢳⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠙⠓⠶⣟⠉⠉⠀⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠛⠁⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢴⠶⣿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣤⣀⠀⠀⠀⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡷⠀⠀⠀⠀⣀⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⡅⠀⠀⠀⠀⠿⠟⠀⠀⠀⠀⣠⡤⢤⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢩⡿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠛⠻⠶⣦⣀⠀⠀⠀⠀⠀⠀⠛⠦⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⣡⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠳⢶⡇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣤⣀⣀⣤⣤⣤⣤⣤⣶⣶⣿⠿⠋⢁⣠⠴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⡌⠳⣤⣀⣤⣦⡀⠀⠀⠀⣸⡿⠉⣽⠋⠁⠀⠀⠉⠙⠁⢸⣷⡚⠉⠀⠀⢸⣿⡀⠀⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⣴⡟⠁⠉⠻⣶⠶⠟⠋⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠙⣿⠉⠉⠉⠉⣿⡏⠁⢾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢿⣦⣤⡴⢋⡀⠀⠀⠀⠀⠀⠈⠳⢦⣄⡀⠀⠀⠀⢀⣼⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⢤⣅⡀⢸⣿⣿⠆⠀⠀⠀⠀⠀⠀⠈⠉⠛⢻⡞⠋⠳⣤⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡏⠻⣦⡉⠀⠀⠀⢸⣿⣾⠆⠀⢀⣀⠀⠈⣧⠀⢸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣷⠀⠈⠙⠳⠶⢦⣄⠋⠁⠀⠀⠸⣿⡿⠃⢹⡀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡘⢷⡀⠀⠀⠀⠀⠉⠛⠲⠶⠦⣤⣅⣀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣽⣦⣀⠀⠀⠀⠀⠀⠀⢠⡿⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠶⠤⠤⠤⠶⠋

⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⣤⡀
⠀⠀⠀⠀⢠⡟⠉⠉⠙⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠈⢷
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣦⡴⠶⠶⠶⠶⢶⣞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣤⣀⣀⣀⣀⣤⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⢀⣠⡤⣶⠟⠉⠀⢰⠏⠀⠀⠈⢳⡄⠉⠙⠓⢶⣤⣄⣀⡀⠀⠀⠀⢠⡏
⠀⠀⠀⠈⠳⣤⡤⠶⠖⠛⠉⢁⡼⠃⠀⠀⢠⡟⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠻⣄⠉⠙⠛⠓⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⣾⠷⠶⢺⡇⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⢸⠟⠲⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡿⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣤⡐⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢿⣿⣿⡟⣰⣿⣷⣄⢶⣶⣶⣬⣭⣙⣛⠿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣷⣝⡿⢸⣿⡟⣩⣴⡌⣿⣿⣿⣿⣿⣿⣿⣶⠹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⡿⠛⠛⢻⡇⣼⣿⢸⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⡿⣇⢸⣿⡘⢿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⢇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣫⣾⣿⡜⣿⣿⣦⡭⢰⣿⣿⣿⣿⡿⠿⢛⡉⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⣛⣋⣀⣋⣩⣭⣥⣶⢰⣿⣿⡇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⠻⠟⣡⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣏⠶⢈⣿⣿⠰⠆⣹⣿⡇⠶⢸⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⣿⢛⣛⣫⣾⣿⣿⣿⣿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⠇⣼⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣩⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⢉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⣤⢠⢠⣄⢠⢀⡤⣄⢠⠀⡄⣤⡀⡄⣠⠤⢠⡤⢠⣄⢠⡄⣤⠄⣤⡀⡤⢤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠻⡄⡏⢿⠸⠏⢿⠸⣤⡾⠸⣤⠇⠿⠹⡇⢧⣄⢸⣯⠸⠻⠟⡇⣿⡅⠿⠹⡇⢸⠆

⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⡏⠒⠤⠄⡀⢠⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⠤⠄⠒⠉⡏
⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⣴⣶⣤⣈⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⣁⣤⣶⣾⡆⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠀⣿⠙⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⢹⡇⠠⣖⠁
⠀⠀⠀⠀⠀⢀⣀⡤⠚⠁⠀⣿⠀⠀⠈⠙⠿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠟⠋⠀⠀⢸⡇⠀⠈⠳⢤⡤
⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⠤⣿⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠤⠖⠚⠉⠁⠀⠀⠀⠀⠀⢸⡷⠶⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠲⠶⠴⠶⠶⠶⠖⠒⠋⠉⠁



⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠶⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢛⠢⠀⠀⠈⠙⠻⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠙⠋⠉⠁⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢴⠦⠀⠀⠀⢠⣄⣀⢈⠩⠿⣷⢿⣿⣿⣿⣶⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠙⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠩⠉⠒⠀⠀⡁⠈⢹⣿⣿⣶⣿⠈⠉⠹⠛⣿⣿⣿⣿⣿⠟⢙⠛⠁⠀⠀⢀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣉⠐⠀⠀⠀⠤⠲⠶⠿⠻⣿⣿⣠⣤⣶⣷⣿⣟⡟⠿⣷⣿⣶⣠⣀⠁⠈⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣖⡄⠀⠀⢀⡀⢠⣤⣴⣿⣿⠿⠿⠿⠿⠿⠿⢿⣓⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⡶⠾⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢦⣤⣤⣤⣀⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠃



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣩⣝⣭⣍⣽⣭⣿⣻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠊⠀⢴⣾⣿⣿⠀⠘⠈⠙⠓⠀⠀⠒⠀⠀⠀⠐⠁⠀⠀⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠋⣼⣿⡿⠀⡀
⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠘⠀⠀⠀⠄⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⠄⠀⠀⠀⠀⠀⠀⠀⠋⡟⠁⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣾
⢀⠔⠈⠁⠈⠁⣀⠀⠀⠀⠀⢠⣖⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠲⢀⣾⣏
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠤⠀⡴⠊⠉⠉⠐⢄⠀⠀⠀⠁⢄⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⣀⡀⢀⣤⣿⣿⣟⡡
⠁⠉⠁⠉⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠄⠦⣄⡠⠲⠒⠖⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣋⣀
⡀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠈⠑⣤⢄⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⡻⣿⣿⣷
⠙⠂⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠀⠴⠊⠁⠀⠀⢀⣿⡟⠃⠀⠃⠈
⠀⠀⠀⠀⠀⠁⠀⠀⠠⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠙⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣽⣷⡀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⡀
⠀⠀⠀⠈⢱⣄⡀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣄⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣹⠁
⠀⠀⠀⠀⠈⢆⠀⠉⠁⠀⢄⠵⠶⣦⠀⠀⠀⠀⠀⢀⠀⠀⢽⣿⣿⡇⠀⣾⠇⠀⢀⠄⠀⠄⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⢀⣀⢀⡀⠉⠉⢻⣿⣿⣧⢠⣿⣷⣾⣿⡶⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠄⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠙⠻⣿⣿⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠀⠀⠀⠀⠀⠳⡀⠀⣰⠁⠀⠠⡄
⠀⡄⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⡀⠐⠃⠀⠁⠈
⠀⠐⢄⢿⡄⡀⠀⠀⠀⠀⠕⠀⠄⠀⠀⠀⠀⠀⠀⠒⢂⣩⡴⠋⡉⠀⠙⢷⡀⠈⠀⠀⠐⠒⠒⠄⠠⠄⠀⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠆
⠀⠀⠀⠀⠙⠁⠀⡎⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠋⠅⡔⠁⠀⠋⠀⠁⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠈⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣤⣴⣶⣤⣬⡄⠀⠀⠀⠀⠀⠀⠀⡀
⡠⠖⠀⠀⠀⠀⠀⠀⠁⠰⣀⠀⢀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⡀⠀⠀⠀⠈⠙⠙⠓⠚⠠⢄⢀
⠀⠀⠀⠀⠀⠆⠀⠀⠂⠀⠀⠉⠉⠒⠆⠒⠘⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⣦⢠⠀⢀⣴⣿
⠀⠀⠀⠀⠇⠑⠂⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢣⡀⣄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⢺⣧⣠⣿⣿⣿⣿⠏
⠒⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢿⣿⣄⡀⠀⠸⠁⠀⠀⢀⣤⣞⡀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣟⠋⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠁⠀⢀⣀⠀⠀⠀⠈⢻⣿⣷⣀⢠⠀⠀⣤⣿⣿⠋⢀⠀⠠⠠⠀⠠⠄⠀⡸⠉⠐⠙⠧⣐⠀⠤⠤
⠲⢀⠈⠁⠀⠁⠀⠀⠠⠀⠀⠀⢀⠂⠁⠀⠀⠀⠀⠀⠀⠙⠢⠀⠀⠈⠻⣿⣿⠛⠃⢸⣿⣿⡡⠊⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠙⠒⠒
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠄⠀⢈⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣾⣷⣶⣦⣉⡀⠃⢌⣿⣟⡡⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠎⠀⠀⠀⠐⠚⠉⠉⠉⠻⠛⠛⠛⢻⡿⢿⠁⣠⡆⢫⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣰⠉⠁⠀⠀⠀⠀⠉⠉⠉⠒⠢⢤⣀⢠⡆⠐⠒⠒⠂⠠⠤⠤⠤⠄⠒⢒⠇
⠀⠀⠀⠺⢍⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠚⠷⢦⣤⣤⡤⠀⣐⣒⡨⣦⠺⣄
⠀⠀⠀⠀⠀⢹⠀⠀⠀⢸⠖⠒⠒⠒⠒⠲⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⣇⡠⠄⠛
⠀⠀⠀⠀⢀⢃⣀⣀⣀⣀⠑⠢⠤⠤⠤⠤⠤⠤⠤⠬⣹⠓⠒⠤⢤⣀⣀⣀⣀⣀⣀⣠⠤⠖⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠁⠒⠦⠤⣄⣀⣀⣀⡤⠞⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⣾⡟⠛⠛⠻⠷⣶⣤⣄⣠⣶⠟⠋⠉⠙⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠈⣹⠟⠁⠀⠀⠀⠀⠀⠹⣷⡶⢶⣶⣶⣶⣶⣤⣤⣾⠿⠛⠋⠉⠁⠀⠙⣿⡇
⠀⠀⠀⠀⠀⢀⣼⣧⣤⣀⡀⠀⣠⣴⠶⠿⣶⣶⠟⢷⡄⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⢠⡿⠋⠉⠉⠉⠙⣿⠟⠁⠀⠀⠀⠹⣧⣼⠇⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⡞⢿⡀⠀⠀⠀⠀⢀⣿⣅⣀⣀⣀⣀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⠇
⠀⠀⠀⠀⠸⣿⠀⠀⠀⠻⣦⣼⣷⣤⣀⣀⣤⡾⠋⠉⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⣻⣧⣀⠀⠀⢀⣠⣴⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⢸⣿⠉⠛⠻⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠿⠿⠿⠿⣿⡿⠶⣶⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣿⣀⣀
⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⡆⠀⠀⠀⠀⠛⠛⣻⡿⠛⠛⠛⠛⠂
⠀⣤⣶⣶⣾⣿⠿⠆⠀⠀⠀⠀⠀⠘⠿⠟⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⢀⣀⣀⣿⠇
⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠉⠉⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⡿⠷⠶⡦
⠀⠀⠀⢀⣀⣼⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣄⣼⡟
⠀⠀⠰⠛⠋⠁⠈⠻⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠟⠋⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠷⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣴⣶⡶⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠳⣤⡀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⡀⠬⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣶⣧⠀⠀⠀⢠⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣒⣿⠋⠀⠀⠀⠈⢻⣇⡈
⠀⠀⠀⠀⠀⠀⠂⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⠀⠀⠀⠀⠀⢸⣿⣅⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢂
⠀⠀⠀⠀⠀⣠⠋⠉⠉⠉⠙⠓⠒⠀⠀⠀⢘⣿⣿⠆⠠⠀⠀⠀⢿⣿⣟⠀⠀⠀⠀⠒⠛⠉⠉⠉⠉⢻⡆
⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠠⠸⣿⡿⠫⠃⠀⠀⠀⠀⠀⠀⠯⠻⣿⡿⠢⠀⠀⠀⠀⠈⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⢨⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⡀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠓⠒⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠒⠀⠱⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠶⠶⠶⠟⠛⠛⣉⣙⠳⢜⣿⣼⢸⢃⡟⢰⡏⢸⡇⢸⡇⣸⠇
⢶⣶⣶⡶⠿⠟⠛⠛⠛⢦⡇⡇⡟⣸⠃⣾⢁⣿⢁⣿⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⠇⡿⢸⡇⣸⠇⣼⠇⣼⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢸⡇⣿⢠⡟⢠⡟⢰⡟⠀⢀⡠⣐⢤⡒⢮⡙⢦⡙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢸⠠⡇⢸⡇⣾⢇⣘⡵⣿⣶⣿⢮⢧⢹⡌⢷⡈⣷⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡮⡮⣣⣿⡬⣗⡮⢗⡪⠭⣓⣚⣷⢏⡏⣇⢷⠘⣧⠸⣧⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡯⠯⣝⣚⡭⢗⣫⠵⢚⣫⠵⣒⣫⠭⢯⣽⡜⢧⡙⣧⡙⢧⡙⢧⣀⢤⣒⣒⡺⣝⢯⢢⡐⣄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⠭⠭⠶⣚⣩⠴⢚⣭⠶⠛⠉⠀⠀⢸⢷⢧⢮⡽⠢⢽⡲⣝⡬⢚⣩⡥⠶⢭⡞⡎⣇⢻⡘⣦⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣛⣩⡭⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠻⣼⣸⣖⣺⠵⣊⡵⠞⣋⣤⡶⠶⠶⡅⡇⢿⠘⣇⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢇⡯⠖⣛⣡⡴⠟⢋⣡⣴⡶⠶⣆⣿⢸⡀⣿⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣛⣭⣥⣴⠾⠛⣉⣤⣤⣤⣤⣿⢸⡇⣿⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣤⠶⠶⠿⠛⠋⠉⠉⣿⣿⡼⣷⢸⣇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡏⣇⢻⡌⢿⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⡯⣳⣙⠬⢗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣏⣭⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢉⣁⣤⣶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠋⢁⣠

⠀⠀⠀⣠⣤⣀
⠀⠀⣰⣿⣿⣿⣿⣶⣤⣄⡀
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡿⠿⠿⠿⠿⠿⠿⠿⠶⣷⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣶⠲⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠻⢿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣤⣀⡀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣷⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠈⠉⠉⠛⠻⠿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠂

⠀⠀⣻⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿
⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠄
⠀⠸⠛⠋⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣴⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁
⠀⠀⢶⣶⣿⣿⣷⣤⣤⣤⣄⣀⣀⡀⠀⠀⢀⣀⣀⣠⣤⠀⠀⠀⠀⢸⡿⠟⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠚⠛⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣄⣀⡀
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⣠⣿⣿⣿⣷⣦⣄⣀⣀⣀⣀⣀⣀⣀⣤⣴⣾⣿⣿⣿⣄⠀⠀⠀⠐⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠃
⠀⠀⠈⠹⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛
⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣛⣟⣿⣟⣛⣻

⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣶⣄⠀⣀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⣰⡿⠉⠉⠉⠙⠻⢷⣦⣴⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠀⠈⢻⡇
⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠐⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠿⠛⢻⣇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡀⢀
⠶⠻⠟⢻⣿⠛⠒⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠐⠛⢻⣿⠛⠛⠛
⠀⠀⠀⢘⣿⣄⣀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣠⣼⣿⣀⡀
⠀⠀⠛⠛⠻⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣉⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠉⠙⠁
⠀⠀⠀⠀⣀⣽⣿⣿⡋⠀⣀⣤⣤⣤⣤⣀⠀⠀⠀⠉⠁⠀⠀⠀⣀⣤⣤⣄⣀⡀⠀⠈⣿⣿⣷⣤⡀
⠀⠀⠀⠟⠋⠁⠀⠙⣿⡿⠋⠉⠀⠀⠉⠙⢷⡄⠀⠀⠀⠀⢠⡿⠋⠉⠉⠉⠉⠻⢷⣿⠋⠀⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣿⡁⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣀⣀⣠⣤⣀⣤⣾⠏⠀⠀⠀⠀⠹⢷⣤⣤⣤⣀⡀⢀⣴⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⠛⠛⠿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣄⣠⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⡆⠀⠀⢶⡶⠀⠀⢰⣶⣦⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣆⣠⣾⣷⣀⣰⣿⣿⣿⡇⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣀⣀⣹⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠶⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠈⠛⢶⣄⣀⣤⣤⣤⣤⣤⣤⣤⣶⠟⠛⠛⣷⣤⡶⠞⠛⠛⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣴⠟⠁⢀⣀⣄⣹⣇⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⣿⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⣴⡟⢻⡿⠿⠿⢦⣄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠠⢻⣿⣿⢠⡄⢀⡀⠹⣷⣶⠶⠿⠷⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣶⣶⣿⠟⣀⠁⠻⠟⣸⡟⢻⣦⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢶⣍⣱⣿⣿⡿⣷⡾⠏⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣾⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⢿⣇⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠈⠻⣷⣶⣾⣿⡤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⠛⣿⠉⠀⠀⠀⠀⢹⡟⠀⠀⠀⠀⠀⢀⡠⢄⡀⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⣘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⠔⠃⠀⠀⠀⠀⠀⠈⡀⠀⠀⢀⢨⡿⠉⠉⠓⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠙⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠟⠳⢤⣀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣻⣷⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣞⠋⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⡶⠾⠟⠛⢁⣿⠃⠀⠈⢿⡿⠏⠉⠉⠉⠉⠉⢿⡿⠁⠀⠘⣷⡝⠻⠶⣤⣄⣀⣀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠘⢿⣤⢀⠀⣿⠇⠀⠀⠀⠈⠳⣤⡄⠀⠀⣀⣤⠞⠁⠀⠀⠀⢹⣷⠁⢀⠐⣿⡛⠙⠳⣄
⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⢛⣷⣿⣿⠏⠀⢀⡴⠶⢤⠀⠈⠉⠛⠋⠉⠀⠀⢀⣠⣤⡀⠀⢿⣤⣿⣿⡏⠁⠀⠰⣿⡀
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢘⣋⣿⠏⠀⠀⠸⣄⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⣹⠀⠘⣿⣿⠃⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⡐⠁⣸⠏⠀⣠⡴⠾⠿⠿⢧⣄⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠁⠀⠀⠸⣯⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⢷⣤⣀⣎⠁⣼⠃⢠⡞⠁⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⢀⡀⠟⠻⢻⣿⣷⣦⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⢿⣤⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⡠⠊⣿⠃⠀⠀⠀⠄⠀⠀⠀⠀⠀⢻⠀⢠⣶⡀⠀⠀⣀⢀⣼⣿⢀⣤⣶⣼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⣿⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣄⡾⠿⢻⣷⣾⠿⠿⠛⠛⠛⠻⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⣿⣇⡄⠀⢀⠁⠀⠀⠀⠀⢀⣿⡟⠃⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⢐⠎⢻⣿⣒⡞⢀⠀⢠⢀⣠⡾⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏
⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⠃⠀⠙⠻⠷⣾⣶⠾⠿⠋⠀⠀⠈⠛⠳⠶⠶⠶⠶⠶⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡀⠀⠀⠀⠀⠀⡴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠰⢣⠀⢀⠀⢀⢀⡇⢹⣀⡠⢤⣄⡀⣃⢸⣂⣀⠤⣄⠠⡞⡀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⡼⢋⣉⣙⣷⣷⣈⣃⢸⣛⡷⣯⡋⢙⣥⠤⠻⠥⢤⡟⠉⣫⣤⣏⣀⣸⣃⣹⡶⠟⣫⠝⢷⡀
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠑⠲⢠⣮⣀⣱⡞⢟⡀⣏⣙⣯⣭⣶⢲⣤⣮⣭⣿⣉⣹⡛⣧⣼⣁⣹⣥⠚⠁⠀⠈⢷⡀
⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⣮⢤⡿⠾⣿⣀⣿⢛⣛⣿⡀⣹⠈⣅⣠⡌⢩⡝⢻⣆⣸⡷⠟⣻⣿⢳⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠐⠛⠓⠒⢢⢯⣀⣽⡞⠳⠤⠌⣽⠉⢹⡥⠞⠛⠛⣾⣅⡸⣧⠚⠀⠀⠙⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⡏⣰⠋⢤⣙⡆⠀⣼⠉⡟⣏⡑⣆⠀⢸⢀⠎⠳⣌⡱⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⠳⣴⠃⠀⠀⠀⠀⠀⠛⠁⠀⠀⠈⠁⠀⠿⠞⠀⠀⠉⠳⠀⠚⠋⠀⠀⠀⠉⠀⠀⠀⠀⠈⢷⡴⠋
⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠠⢤⣤⣤⣼⣇⣀⠀⠀⠀⠀⠀⢠⣤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣤⠀⠀⠀⠀⠀⢀⣀⣧⣤⣤⣤⠤
⠀⠀⠀⠀⠀⠸⡏⠉⠁⠀⠀⠀⠀⠀⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡇⠀⠀⠀⠀⠈⠉⢹⡏
⠀⠀⠀⣠⡤⠶⢿⡖⠒⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⡤⠤⢤⡀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠐⠒⣾⠷⢦⣄
⠀⠀⠈⠁⠀⠀⠈⢷⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⠏⠀⠀⠀⠁
⠀⠀⠀⢀⣠⡴⠶⠛⠻⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠳⠶⣤⡀
⠀⠀⠀⠋⠁⠀⠀⠀⠀⠈⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠁⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠈⠻⣿⡗⠶⠶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠶⢞⣿⡟⠋⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⠀⠸⣿⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⡟⠀⣠⣶⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠙⠿⣷⣄⠘⠾⠿⠟⠛⠻⢶⣦⣀⣤⣶⡿⠛⠛⠿⠿⠟⢀⣼⡿⠛⠉⢷
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠈⢻⣦⠤⠖⠚⠋⠉⠉⠉⠛⠉⠉⡉⠙⠙⠲⢦⢤⣿⠋⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣀⡀⠀⠀⣣⠀⠀⠀⠀⠁⢸⡇⠀⠀⠀⠀⠀⠀⠙⢷
⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⣼⣿⣨⢤⡀⠄⢡⣇⡽⣶⡾⣭⣧⠀⠀⣠⡭⣸⣿⡀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⣀⣀⣠⣾⡷⠟⠛⠻⣿⣮⣿⣿⣶⣤⣴⣾⣿⣿⣾⡟⠛⠿⢷⣿⣦⣀⣀⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣽⣿⠛⠁⠀⠀⠀⠈⠄⠁⠀⠀⠉⠉⠉⠁⠀⠀⠈⠀⠀⠀⠀⠨⠙⢻⣿⣥⣄⡀
⠀⠀⠀⠀⠀⢀⣴⠿⣶⣶⣄⠈⢦⣐⣀⣠⠤⠖⠒⠒⠒⢦⣀⠀⢀⡤⠔⠒⠒⠲⠦⢤⣄⣀⣰⠏⢠⣶⣶⣿⣶⣄
⠀⠀⠀⠀⠀⣾⣷⣿⣿⣿⣿⣦⣀⣉⣁⣤⣴⣾⣿⣭⡛⢦⡈⠙⠋⣠⢞⣩⣽⣿⣶⣤⣄⣉⣀⣠⣿⣿⣿⣿⣷⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢷⡾⠾⠛⠋⠉⠀⠀⠈⠙⢷⣽⣒⣻⣵⠟⠁⠀⠀⠈⠉⠛⠿⠿⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣾⣤⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠔⢄
⠀⠀⢠⠋⡱⢤⡴⢦⠀⠀⠀⠀⠀⡃⢾⡉⣷⡸⠀⠀⣀⣄⡀⠀⣀⣠⣄⠀⠀⠀⠀⠀⣀
⠀⠀⠘⠮⠧⢴⠃⡘⠀⠀⠀⣠⣤⣌⣒⣁⣀⣀⣤⡞⠁⢀⡙⣟⡉⠀⢈⣧⠀⠀⠀⡞⢉⣹⡤⣤⠔⠲⡄
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢰⡇⠀⠀⠈⠉⠉⠀⢸⡇⠀⢯⡟⠉⠙⣿⡉⠙⣆⠀⠀⣇⠈⣺⣀⣸⠷⢀⡇
⠀⢀⣀⡀⢀⡴⠲⡄⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠈⠓⠶⠚⠳⠤⣴⠓⠃⣠⡇⠀⠀⠈⠉⠁⠈⠉⠒⠋
⠀⢸⠀⣿⠉⢹⠆⣸⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⢹⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⠦⡬⠋⠉⠉⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠈⣹⣇⡀⠀⠀⠀⠀⠰⢒⡤⢮⡈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⣟⠃⠀⠰⡿⠀⠀⠀⣠⢤⡀⠀⠀⠙⠁⠀⢀⣸⣯⣷⣶⣶⣤⣄⠰⣈⡷⠚⠒⠁
⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣷⣿⣏⣀⠀⢀⣀⣀⡀⠉⠋⠁⢀⡴⠶⠶⢦⣼⣿⠿⠛⠛⠛⠻⢿⣷⣄
⠀⠀⠀⢠⣾⣿⠟⠋⠉⠉⠉⠙⠛⠿⡶⠋⠉⠉⠙⣦⣤⣤⣿⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧
⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣤⣤⡿⠿⠛⠛⠷⢾⣷⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣦⡀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⠋⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠈⣿⣧⠀⠀⠀⠀⣾⣿⠇
⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣴⡿⠃⠀⠀⠀⢀⣿⣿
⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠉⠀⠀⠀⠀⢀⣾⣿⠇
⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣷⣄⣀⣀⣀⣀⣤⣾⣿⣧⣄⣀⣀⣀⣠⣴⣿⡿⠋
⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠙⠛⠻⠿⠿⠛⠋⠉⠉⠛⠿⠿⢿⣿⠿⠟⠋⠀⢀⣀
⠀⢀⣀⠀⠀⠙⢿⣷⣦⣄⣀⣀⣤⣶⣿⡿⠋⠀⠀⠀⢀⠎⢉⣲⣄⣠⠴⢤⠀⠀⠀⠀⠀⡔⢒⡖⢪⡄⢣
⢠⡏⣠⡵⢤⡴⠢⡙⠻⠿⠿⠿⢿⡛⠁⡠⢤⠀⠀⠀⠸⡀⠘⣏⠀⣸⠆⢠⠇⠀⠀⠀⠀⠡⣈⡳⠐⠓⠃
⠀⠳⠼⠦⢴⠋⣠⠁⠀⠀⠀⠀⠉⢸⡋⢱⢀⠃⠀⠀⠀⠉⠋⠉⠉⠑⠖⠋⢀
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠲⠋⠁⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠘⠈⠈⠉⠀⠀⠘⠈⠠⠀⠃⠀⠀⠈

⠀⠀⠛⠶⢦⣤⣤⣤⣤⣤⣤⡶⠾⠃⠀⠿⠇⠀⢸⡇⠀⢸⠅⠀⠸⡷⠀⢀⣾⡄⠀⢼⠟⠁⠀⠸⠛⠛⠛⠛⠛⠛⠇⠀⠈⡏⠉⠉⠀⠈⠉⠉⠉⣩⣭⡽⠆
⠀⠀⣤⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⡄⠀⠀⠀⠀⠀⠀⣤⣀⣀⣀⣀⣀⣀⣤⠉⠀⢀⣀⣀⣀⣀⣀⣀⡄⠀⠀⠀⢀⡀⠀⠀⠃⠀⢸⡆⠐⠛⠛⠛⠃
⠀⠀⣀⠀⠈⠉⠉⠉⠉⠉⠉⠀⢀⡀⠀⠐⢷⡟⠀⠀⠘⠛⠛⠛⠛⠛⠛⠃⠀⠀⠈⢉⣻⣿⣿⣥⡄⠀⠀⠈⣛⣿⣅⡀⠀⠤⠤⠼⠃⢀⣠⣦⣀⠈⠉⢹⡆
⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠈⠀⠀⠀⠻⢷⣶⣶⣶⣶⣶⠟⠃⠀⠀⣠⣭⣽⣿⣶⣤⠀⠈⢙⣻⣯⣍⠁⠀⣤⠶⢦⡄⠈⠙⠟⠉⠀⠀⠘⠃
⠀⠀⠳⠶⠶⠶⣤⣤⣤⣤⠶⠶⠶⠞⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⢈⣙⣿⣦⡄⠀⢹⡆⣾⠁⠀⠀⠀⣤⣄⡀
⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠘⣛⣻⣿⣭⣉⠉⠉⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣻⣿⣥
⠀⠀⠀⠛⠛⠛⠿⠿⠿⠿⠛⠛⠋⠀⠀⢀⣀⡀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢼⣿⣭⠀⠀⢺⣿⡗⠀⣠⣬⣭⣽⣿⣶⣦⠄
⠀⠀⣠⣀⣤⣤⣦⣶⣶⣤⣤⣀⣄⠀⠀⢸⣿⡇⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠘⠻⠿⣿⣿⣛⠛⠃⠀⠀⠚⠛⠳⠀⠀⠈⠉⠋⠀⠀⠨⣽⣿⡶⠶
⠀⠀⠈⠉⠁⢀⣀⣤⣄⣀⠉⠉⠁⠀⠀⢸⣿⡇⠀⢀⣀⣀⣠⣤⣤⣀⣀⣀⠀⠀⠀⠶⠶⠾⠿⠿⠿⠗⠀⠀⢠⣤⣤⠀⠀⠀⣶⣷⣦⠀⠀⠰⠿⠿⣇⡀
⠀⠀⣀⣶⠚⠋⠉⠉⠉⠙⠓⢶⣄⠀⠀⢸⣿⡇⠀⠈⠛⠉⠉⠉⠉⠉⠙⠛⠀⠀⣠⣴⡶⠶⠶⠶⣦⣤⡄⠀⠶⠿⠿⠀⠀⠀⠛⠿⠋⠀⠀⠀⠀⢀⣀⡀
⠀⠐⠛⠁⠀⠀⣀⣀⣀⡀⠀⠈⠹⠇⠀⢸⣿⡇⠀⠾⠿⠿⠿⠿⠿⠿⠿⠿⠆⠀⠉⠁⠀⠀⠀⠀⠀⠉⠁⢀⣤⣤⣤⣤⡀⠀⠀⣤⣴⠶⠶⠀⠀⣿⣿⣿⠆
⠀⠀⣠⡾⠲⠟⠛⠋⠛⠛⠿⠺⣦⡄⠀⠈⠉⠁⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⢠⣴⣶⠶⠶⠦⠶⠆⠀⠠⠾⣿⣿⡶⠂⠀⢀⣀⣀⣀⣀⣀⣀⠈⠉⠁
⠀⠀⠈⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠙⠃
⠀⠀⠘⠻⠟⠛⠛⠛⠛⠛⠻⠿⠛⠀⠀⠀⠀⠀⠀⠻⠿⠶⠶⠶⠶⠶⠾⠿⠓⠀⠐⠟⠋⠉⠉⠉⠛⠁⠀⢠⠤⠶⠶⠾⠟⠓⠀⢰⣶⣶⣶⡆⠀⠠⣶⡶
⠀⠀⢴⣦⡀⠀⠀⠀⠀⠀⠀⣠⣶⠄⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⢠⡀⠀⢀⣀⣀⣠⣦⠀⠀⠀⠀⣀⣀⣤⣄⠀⠸⡿⠿⢿⠇⠀⠀⠹⠃
⠀⠀⠈⠙⠛⠿⠶⠶⠶⠾⠟⠛⠉⠀⠀⠀⠀⠀⠀⠛⠿⠟⠛⠛⠛⠉⠉⠉⠃⠀⠈⠛⠛⠛⠋⠉⠉⠁⠀⠀⠙⠋⠉⠉⠉⠉⠀⠀⢀⣀⣀
⠀⠀⣤⣤⣀⣤⣤⣤⣤⣤⣀⣤⡄⠀⢶⣶⣶⡖⠀⠀⣦⣤⣤⣤⣄⣀⣀⣀⠀⠀⣠⣤⣤⠶⠶⣦⣤⣤⡀⠀⢠⣤⣤⣶⣶⣤⣤⠀⣼⣿⣿⡆⠀⣾⣿⣇⡀
⠀⠀⠈⠉⠉⠉⠉⠀⠈⠉⠉⠉⠁⠀⠛⠛⠛⠃⠀⠀⠀⠀⠀⠉⠉⠉⠉⠋⠀⠀⠉⠉⠀⠀⠀⠀⠈⠈⠁⠀⠈⠉⠉⠛⠛⠉⠉⠀⠈⠙⠋⠁⠀⠙⠛⠛⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣮⣍⣤⣬⣭⣏⣵⣿⣽⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⣀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠿⠛⠿⣷⣶⣶⠿⠛⠉⠉⠉⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠛⠛⠛⠿⢿⣷⣶⣤⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠈⢻⣧⣀⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣼⡿⠁⠀⢀⣾⡿⢿⡿⠛⠛⠛⠻⢶⣶⡶⠾⠿⠿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⢿⣤⣾⠀⠀⠀⠀⠀⠈⣿⣄⡀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠙⣿⣄⠀⠀⠀⠀⢀⡟⢹⣷⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠶⠶⠟⠛⠻⢷⣶⣶⣶⠿⠿⠿⠋⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣠⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⢹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣶⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠐⠛⠛⣿⣿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣼⣿⣤⣤⡀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⢀⣿⣿⣄⣀⣀
⠀⠀⠀⠀⠀⠚⠛⠛⠋⠙⢿⡏⠁⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠈⢻⣿⡟⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣤⡄⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣏⠉⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠟⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡿⠿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣷⣿⠃⢀⣠⣶⣿⠿⠿⠿⢶⣤⡀⠀⠀⠀⠀⢀⣤⡾⠟⠛⠛⠻⠷⣦⡀⠀⢀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⠉⠛⢿⣿⣿⣯⣥⣀⠀⠀⠀⠉⢻⣦⠀⢀⣴⠟⠉⠀⠀⠀⠀⢀⣤⡾⠿⢾⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⢠⣾⡿⠋⠉⠉⠙⢿⡄⠀⠀⠀⠙⢿⡟⠁⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⣀⠀⣀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣤⣴⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠛⠻⢿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠛⠛⠉⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⣀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠿⣷⣄⠀⢀⣴⡿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⢀⣴⠿⠃⠀⠈⠻⣷⡿⠋⠀⠀⠙⢿⣦⡀⠀⠀⠀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠢⡀
⠀⠀⠀⠀⠀⠀⠀⡴⢿⠛⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⡌⠀⠈⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠻⣧
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠐⢭⡢⣄⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣣
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣵⣄⡀⢀⣀⣠⣾⣿⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢈⢿⡆
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣷⠋⠙⠳⣿⣿⣿⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⢀⢳⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⣿⣿⡇⠀⠀⢽⣿⣿⣿⣄⡤⠴⠂⠀⠀⠀⠀⠀⢀⣿⣿⡄
⠀⠀⠀⢸⢸⡀⠀⠀⠀⠀⠀⠒⠚⠛⣋⣹⣿⣿⡿⡧⠀⠀⣾⣿⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇
⠀⠀⠀⠘⣸⣷⠀⠀⠀⠀⠀⠀⠀⢉⣼⣿⣿⠛⡇⠀⠀⣠⣿⡿⣿⣿⣍⡉⠉⠀⠀⠀⠀⠀⣰⣿⣿⣿⠁
⠀⠀⠀⠀⡿⣿⣷⡄⠀⠀⠀⠀⣠⣾⡿⣿⣽⣿⣿⣶⣿⠿⣽⣾⣷⣿⣿⣿⣷⣤⣀⣀⣠⣴⣿⣿⣿⣿
⠀⠀⠀⠀⢱⣿⣿⣿⣦⣤⣶⣿⡿⣽⣿⣿⣿⠟⡿⣿⡇⠀⢿⠛⠿⡻⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢿⠿⣿⢿⡿⠛⢉⣴⣿⣿⣿⠃⠘⠁⣿⠀⠀⢸⠀⠀⠈⠈⢻⣿⣦⠈⠙⠻⢿⣿⠏⡼
⠀⠀⠀⠀⠀⠈⠦⠚⠁⠀⢀⣾⣿⣿⣿⠃⠀⠀⠀⢿⠀⠀⠘⡄⠀⠀⠀⠀⢹⣿⡳⡀⠀⠀⠉⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣿⣿⣿⡏⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⣼⣿⣇⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠻⣿⣿⣇⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⢀⢾⡻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣆⠀⠀⠀⡇⠀⠀⠀⠸⡀⣠⠕⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢝⣦⡀⠀⢇⠀⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⢦⣀⠀⠀⠀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢦⣌⡳⣄⠀⠀⠠⣴⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣮⡙⣎⠣⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡃⣿⣿⣿⣿⣿⣎⢳⡀⠀⠀⠀⠀⠀⢠⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣡⣧⢸⣿⣿⣿⣿⣿⣆⢳⠀⠀⠀⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣰⣿⣿⡆⢻⣿⣿⣿⣿⣿⡄⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣿⣿⣿⣿⣿⣄⠙⣿⣿⣿⣿⡇⡿⠀⠀⠀⠀⢠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣫⣍⠷⣤⣿⡛⣿⣿⣿⣿⠿⠿⠷⢈⣻⣿⣿⠿⠃⠀⠀⠀⠀⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⣧⡈⣟⠃⢠⡿⠾⢿⣟⣿⠞⠉⠉⠉⠀⠀⠀⣤⡤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢭⣿⣥⣾⣧⣤⣮⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡤⠞⠁⢈⡿⣝⡿⣿⢋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣠⠎⢃⢿⣿⠟⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠⡚⠁⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣤⣤⠠⣼⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣠⣧⣶⣮⢻⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡞⣫⣿⣿⢻⣿⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿⣿⠏⣾⣿⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣸⡿⣿⡇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⢣⠃⠘⡎⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠏⠀⠀⢹⣼⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⣇⣄⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⣀⡀⠿⣭⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠟⠩⠁⠀⠀⠀⠀⠐⠖⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄⠠⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡤⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠿⠙⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⣾⣿⠟⠙⠉⠀⠀⡀
⠀⠀⠀⠀⢀⣼⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣄⣼⣿⠏⠁⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⣼⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⠛⣀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⡀⡀
⠀⠀⠀⠀⣿⣿⣿⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠾⠛⠻⣿⣿⠇
⠀⠀⢠⣾⣿⣿⣿⣿⣦⡂⠀⠀⡀⢀⣀⠤⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠸⠋
⠀⠀⠘⡿⢿⣿⣿⣿⡿⠿⠾⠚⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠙⣆⢤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠺⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠌⡁
⠀⠀⠀⠀⠘⡎⠇⡀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡐
⠀⠀⠀⠀⠀⠑⢄⣀⣂⣀⣠⣤⣶⣾⠋⠉⠀⠀⠀⣀⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣠⣀⠠⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢝⠿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠛⠩⣿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣿⡇⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠋⠁⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⣠⣴⣿⣿⠇⠀⠀⠀⠀⢠⣴⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠙⠿⣿⣿⣷⣄⣤⢶⣿⠟⠙⡀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠁⠀⠒⠛

⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤
⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⢠⣾⣿⣿⣷⣤⡀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⣤⣾⣿⣿⣷⡄
⠀⠀⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠻⢿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣽⣿⣿⣏
⠀⠀⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣧
⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢠⣾⣿⣿⣷⣤⡀⢰
⠀⠀⠛⠛⠛⠛⠛⠛⠛⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣀⡀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢻⣿⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠙⠛⠿
⠀⠀⠀⠀⣤⣾⣿⣿⣿⣆⠉⢙⠛⠛⠿⠿⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠷⠾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡗⠺⢿⡓⠲⠦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⠀⠀⠀⠀⠀⠀⠉⠻⣝⠲⣄⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠶⠦⣤⣸⠃⠀⠈⠉⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⢄⡴⠋⠀⠀⠀⠀⠀⠈⣧⠠⣤⣤⣤⠤⠤⠴⠖⢲⠿⠒⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢰⡏⠀⠀⠀⠀⠀⠀⢲⣴⣠⡼⠧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠎⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢳⣄⡀⠀⢀⣀⣠⡴⠋⠁⠀⢀⣴⠟⠛⠳⠶⣤⣀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠈⣉⡽⠟⠋⠉⠷⣄⣀⣠⡾⣅⠀⠀⠀⠀⠀⠉⠳⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣠⠶⠛⠁⠀⠀⠀⠀⠀⠈⠉⣻⠀⠈⠑⠄⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⣠⠶⠛⠉⢹⡀
⠀⠀⠀⠀⠀⢴⠶⠿⠆⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠳⠶⠶⢚⣁⣀⣀⣀⣸⠇
⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣠⡴⠛⠉⠉⢯⠉⠉⠁
⠀⠀⣀⣠⣤⣼⢶⣿⣦⡤⠤⠀⠀⠀⠀⠀⠈⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠙⢾⣦
⠈⢉⣁⡤⠤⠴⠒⠛⢿⣉⣉⡡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠬⢷⠖
⠀⠀⠀⢀⣀⡤⠶⠒⠋⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋
⠀⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢤⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣄⣈⣹⠶⢯⡉⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡳⠶⠦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠞⠛⠳⣄⡀⠉⠳⢤⡀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⣀⠀⠀⠀⠈⠉⠉⠛⠛⠓⠲⠶⠶⠶⠶⠶⠶⠶⠖⢶⡛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠙⠳⢤⡀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⢻⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⣰⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⣰⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢀⡴⣰⠋⠏⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠈⠻⣆⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣰⠋⡿⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠲⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠈⠛⠳⢦⣄⠀⠀⠀⣀⣠⢾⣥⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⢀⣴⠿⣤⠤⠴⠖⢒⡛⠉⠉⡀⠉⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈

⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡏⠉⠀⠀⠀⠀⠉⢹⣿⣿⣿⡟⢻⠿⠟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠈⠁⠄
⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣼⡈⠀⠈⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⣿⣿⣧⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡟⠉⠛⠛⠛⠋⠛⠛⠛⠛⣿⣿⣿⣿⣷⠀⠀⠀⠘⠚⠀⠀⠀⠀⠀⠀⠀⠀⠸⠖⠁
⣿⣿⣿⣦⡀⠀⠀⠀⣰⣿⣿⣿⣿⠿⠓⠂⠶⠶⠖⠶⠶⠲⢶⣶⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐
⣿⣿⣿⣿⣷⣦⣤⣾⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣄⣠⣤⠜⣘⢻⣿⣿⣿⣯⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣿⣿⣿⣿⠛⡛⢙⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⣿⣿⣿⠡⠂⠀⠀⠈⠊⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠢
⣿⣿⣧⢀⠀⠀⠀⠀⠠⢠⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠀⠛⢢⣀⣸⣿⣿⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢀⠁
⣿⣿⣿⣶⣑⡀⠀⡒⣱⣾⣿⣿⣿⣿⠶⠶⠶⠶⠤⠶⠶⠦⠶⠶⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠒⠛⠋⠉⠉⠙⠚⠃⠀⠀⠀⠀⠀⠀⠀⠑⠢⠀⠔⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⢤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⡟⠛⠚⠑⠨⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠂⢄
⣿⣿⣟⠋⠀⠀⠀⠀⢘⢻⣿⣿⣿⣿⣿⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⡄⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠈
⣿⣿⣿⣗⣄⠀⠀⣀⣌⣿⣿⣿⣿⣿⣿⣀⡐⠋⠘⠛⠛⠛⠛⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⢄⡀⠀⢀⠔⠁
⣿⣿⣿⣿⣿⣴⣷⣾⣿⣿⣿⣿⣿⣿⣿⠿⠿⠶⠶⠶⠶⡶⠶⠿⠿⣿⣿⣿⣿⠀⠀⠀⠠⠤⢤⣶⠶⠶⠶⠶⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣿⣿⣿⠟⠉⠂⠀⠪⡝⣿⣿⣿⣿⣿⣿⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣄⠀⠠⠄⠀⠀⠀⠀⠀⡤⠂⠀⠂⢀
⣿⣿⣿⠈⠀⠀⠀⠀⢀⢸⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⡿⠿⣿⣿⣿⣿⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠄
⣿⣿⣿⣄⠀⡀⠀⡠⢂⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣡⣄⣿⣿⣿⣿⠀⠀⠀⠰⣈⣿⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⡀⠐
⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⣸⣿⡟⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⣿⠏⠀⠀⠀⠀⢀⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠛⠒⠲⠦⣄⡀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⣴⠞⠁⠐⠒⠠⣀⠉⠓⢦⣀⠀⠀⠀⠀⠀⢀⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠇⢸⣧⡀⠀⠀⠀⢀⡴⠉⠲⢀⠉⠳⣄⠀⠀⠀⠋⠀⠀⠀⠀⠀⣠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⣇⠈⢝⠻⣷⣦⣔⠋⡠⠊⠀⠀⠙⢦⡈⠳⣄⠀⠀⠀⠀⠀⣠⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠣⣿⡿⣋⣶⡄⡀⠊⠀⠀⠀⠉⢆⠘⢷⡄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⢼⣯⠞⣿⡅⡀⠀⠀⠀⠀⡤⠳⡀⠹⣦⡀⠀⠀⠀⢠⣾⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢾⢫⡪⢿⣦⣠⡊⠀⠔⠀⠘⢆⠘⣷⡀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢞⣿⢿⣿⡿⣦⣠⠔⡀⠈⣆⠸⣧⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠼⣿⡟⣷⣈⠀⢀⣼⠀⣿⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⡉⠛⠣⠴⠅⠃⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⣶⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠂⣀⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⣐⠂⣀⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠉⢁⣤⣶⠟⣯⠁
⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⠀⣢⠘⣧
⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⣤⡶⠟⠉⠀⠀⢀⡤⠚⢁⣠⠞
⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⣏⠁⠀⠀⣀⠤⠞⠉⣀⡴⠋⠁
⠀⠀⡜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡾⠛⠉⠙⢦⡘⠦⠖⠋⣁⣤⠶⠋⠁
⠀⡾⠁⢵⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠙⠶⠚⠛⠉
⠘⡇⠀⠀⠓⢤⣀⣀⣀⣀⡴⠟⠋
⠀⠈⠢⢀⣀⣠⣴⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣠⣾⣿⣿⣷⣄⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⡿⠛⣁⣤⣶⣾⣿⣿⣿⣿⣷⣶⣤⣈⠛⢿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠃⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠘⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢠⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⣾⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣷⣄⡉⠛⠿⢿⣿⣿⣿⣿⡿⠿⠛⢉⣠⣾⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣬⣥⣤⣤⣌⠛⢿⣿⣿⡿⠛⣡⣤⣤⣬⣥⣴⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣆⢠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠟⠛⢻⣿⣿⣿⣿⡿⠀⠀⠀⠸⣿⣿⣿⣿⣿⠋⠛⠿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⢻⣿⣿⣿⠁⠀⠀⠀⠀⢹⣿⣿⣿⠃⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠇⠀⠀⠀⠀⠀⠀⢿⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡝⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠈⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠀⠉⠙⠳⠦⢄⠠⣤⣤⣴⡚⠉⠉⠀⠀⠀⠀⠀⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢗⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠠⠈⠠⠤⠖⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⣀⣀⡠⠤⠤⠀⠒⠒⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⣀⠤⠖⠉⠁⠐⠢⠤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣰⠃
⠀⢀⡎⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠲⢯⣀⡀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⣠⠞⠁
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠒⠒⠀⠒⠲
⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠲⠦⠶⣎⣉⣀⠀⠀⠀⢀⣀⣀⠤
⢰⠁⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⡸
⡞⠈⠉⠉⠉⠉⠉⠉⠉⢁⣰⡦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠙⠦⠤⠄⢀⡄⠤⠐⠋⠁⠀⠀⠀⠀⠉⠐⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠀⠈⠉⠒⠦⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠁⠉⠉⠉⠉⠉⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠔⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁
⠀⠀⠀⠀⠀⠀⠙⠯⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢙⠛⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠔⠋
⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⢀⣠⡤⠖⠂

⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣷⣦⡀⣠⣤⣶⣶⣦⣄
⠀⠀⠀⠀⠀⢀⣾⠋⠁⠈⠉⠛⠻⣷⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡟⠉⠁⠀⠈⢻⣆
⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⢀⢘⣿
⠀⠀⠀⠀⠀⢿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢀⣀⣠⣤⣴⣿⣥⣤⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉⠛⣛⣿⣿⣿⣿⣿⣿⣿⠇
⠸⣿⣿⣿⣿⠏⠉⠉⠉⠉⠉⠉⠉⠛⠛⠿⣷⣶⣤⢦⣤⣤⣴⣶⡾⠿⠛⠛⠛⠛⠻⢿⣿⣿⢿⣿⣶⣶⣤
⠀⠙⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣼⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠚⠛⢻⣿⣿⠟⠓⠀⠀⠀⢀⣴⣦⠀⠀⠀⠀⣸⡟⠉⠉⠙⣿⡇⠀⠀⠀⣠⣤⠀⠀⠀⠀⠲⠶⣿⡿⣿⡿
⠀⠀⠈⢿⣿⣄⣀⠀⠀⠀⠘⢿⡿⠀⠀⠀⢠⡿⠁⠀⠀⠀⠹⣧⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⣀⣿⣧⡿
⠀⠀⠛⠛⠻⣿⣅⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⢰⡟⠛⣷⠀⠙⢷⣄⠀⠀⠁⠀⠀⠀⠀⠀⢹⣿⡿⠿⠶
⠀⠀⠀⠀⣀⣼⣿⣿⡛⠒⠖⠒⠚⠛⠉⠁⠀⠀⠈⠙⠛⠉⠀⠀⠀⠉⠛⠲⠶⠤⠤⠤⠶⣶⣿⣏
⠀⠀⠀⠾⠛⠉⠀⠙⠿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋⠙⠻⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣿⣿⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣶⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠛⢩⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⣹⣿⣿⣿⣿⡍⠻⢷⣤⡀
⠀⠀⠀⠀⠀⣠⣾⠟⠉⠻⣷⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣴⡿⠛⢿⣆
⠀⠀⠀⠀⢠⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠘⣿⡄⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠈⠻⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠛⠛⠛⠛⠛⠛⠛⢻⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣀⣀⣀⣀⣀⣀⣀⣼⣧⣀⣀⣀⣀⣀⣀⣠⣴⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠂⠀⠉⠉⠉⠉⠉⠁⠀⠐⠂⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣦⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣄⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢀⣄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⢰⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⣶⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡠⠄⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠤⡀
⠀⠀⠀⠘⣄⠀⠈⠢⢆⠀⠀⡔⠉⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⡐⠉⠁⡄⠀⢀⠦⠊⠀⢀⡘
⠠⠄⠤⣀⠀⠑⢄⡀⠀⠑⠤⡘⢄⡀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⢠⠃⠀⠀⢀⠠⠊⠀⡀⠜⡠⠔⠁⠀⣀⠔⠁⢀⡠⠄⠤
⠠⠀⠀⠀⠈⠑⠂⠌⣂⣄⠀⠈⠐⠬⠆⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⡰⠁⠀⢀⠔⠁⠀⠀⠮⠔⠊⠀⢀⣄⡪⠄⠒⠉⠀⠀⢀⡠
⡠⠀⠀⠀⢀⡤⣀⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠰⡤⠄⢀⣀⡀⠠⠄⠂⠁⣀⠔⠊⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⠠⣔⢂⠉⠤⠤
⠣⠀⣀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠁⠒⠀⠂⠐⠀⠂⠉⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠁⠀⠀⢀⣀⠤
⠀⠀⠀⡀⠌⠈⠁⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠩⠀⠀⡀
⠀⠀⠀⠂⠄⢀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠤⠀⣀⣀⠠⠔⠊⠉⠒⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠊⠉⠒⠤⢀⣀⡀⠠⠄⠐⠊
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣾⣿⣿⣿⣿⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠱⠈⠁⠒⠒⠀⠒⠂⠠⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠀⠒⠐⠒⠐⠈⠁⠆
⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⢠⡖⠒⠠⣀⢑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡊⣁⠄⠒⢲⡄⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⢱⠄⠀⠀⠀⢸⠑⠤⡀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢀⠤⠊⡇⠀⠀⠀⠠⡊
⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⡇⠀⠀⠁⠚⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠓⠈⠀⠀⢸⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠠⢆⠁⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⡱⠄
⠀⠀⠀⠈⠁⠀⠒⠒⠒⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠒⠐⠒⠒⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿
⣿⣿⣿⡗⣾⣯⣿⣿⣿⣛⣻⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣟⣛⣻⣿⣿⣽⣶⢺⣿⣿
⣿⣿⣿⣿⡽⣿⣿⣿⣿⣿⡿⢻⣟⣿⣶⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣶⣟⣻⡟⢿⣿⣿⣿⣿⣿⢯⣿⣿⣿
⣿⣿⣿⣿⣿⣞⣿⣿⣿⣿⡇⣾⡻⣿⣿⡆⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢲⣿⣿⢟⣷⢸⣿⣿⣿⣿⣳⣿⣿⣿⣿
⣿⣿⣿⣿⢟⣵⣿⣿⣿⣿⣿⢹⣿⣷⣿⣍⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣩⣽⣾⣿⡏⣿⣿⣿⣿⣿⣮⡻⣿⣿⣿
⣿⣟⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⣿
⣿⣿⣿⣾⣭⣽⣟⣛⣻⣿⣻⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣟⣟⣛⣛⣿⣿⣽⣷⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⡀

⠀⣾⣟⡛⠛⠳⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠞⠛⢛⣻⢷
⢸⡇⠈⠛⢦⣀⠀⠀⠈⠙⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⠁⠀⠀⣀⡴⠛⠁⢸
⣾⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠉⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⢀⣠⡴⠞⠉
⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⡀⠙⢷⣤⣄⠀⠀⣀⣠⡾⠋⢀⣀⣤⠴⠞⠋⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⣿⣬⣽⣻⣭⣼⡟⠛⠉⠉
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠛⠛⠛⢻⡇
⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢛⣛⣛⣛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⢉⣉⣉⣉⢻⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠋⠁⠙⠋⣽⢿⠉⠛⠉⠙⢷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⢷⣄⣠⣤⣤⠤⠶⠞⠛⠉⠁⣼⠃⠀⠀⠀⢠⡏⠸⡇⠀⠀⠀⠀⢳⡈⠉⠛⠳⠶⠤⣤⣤⣄⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣼⠁⠀⢿⡀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢰⡏⠀⠀⠘⣇⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢻⡀⠀⠀⡀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠶⠞⠛⢷⡀⢰⡇⠀⠀⠀⠀⠘⣇⢀⡾⠋⠛⠲⢮⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠻⡿⠀⠀⠀⠀⠀⠀⢹⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣹⣶⣿⠏⠙⠋⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡤⠈⠉⠀⠀⠀⠀⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠓⢤⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶
⠀⢀⣴⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠿⢇⡀⠀⠀⠀⠀⣤⠞⠉⣹⢶⡄
⠀⣌⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⣴⣷⣄⣠⣤⣾⣿⣆⣠⡴⠿⠃
⠀⠙⠷⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⠶⠾⢿⣿⣿⠿⢿⡍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠠⠁⠀⠀⠙⠂
⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⣰⡄⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⠉⠀⠀⠀⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣶⣦⣀⠀⠀⠀⠀⠀⣴⢛⣁⣠⡀⠀⠀⠉⣀⣴⡇
⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⣄⠀⠀⠀⠀⠀⠐⢦⣤⣀⡀⠀⠈⠛⠿⢿⣿⣶⣶⣿⣿⡟⠛⣉⣠⣤⡶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠴⡄
⠀⠀⠀⠀⠀⠀⠈⠻⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠻⠿⢿⠟⠛⠛⢿⡟⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠋⠉⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⣶⡄⠀⠀⠀⠀⠀⠀⣠⡶⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⡸⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡀⠀⠀⠀⠀⠀⠀⣠⠾⢿⡄
⠀⢠⡎⠉⠉⣠⠀⠀⠀⢠⣾⡉⠀⠘⣫⡦⠀⠀⠀⠀⠀⡤⠘⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠉⠁⠀⠀⠀⠀⣠⣟⠁⠀⠘⣻⡆
⠀⣿⣀⣀⣛⣿⣶⣶⣶⣿⡟⠿⠶⠛⠋⠀⠀⠀⠀⠀⠸⢁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣻⣿⣾⣶⣶⣿⠛⠷⠶⠛⠋
⠀⠈⠉⠉⠉⣉⠛⢻⠛⠙⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⣉⠛⣿⠋⠙⢇⡀
⠀⠀⠀⠀⢰⡭⠶⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠆⠀⠠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠶⢶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠛⠉⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠉⠠⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠈⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠘⢳⡶⠿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣷⣤⣄⣀⣤⣤⣴⣿⣥⣶⡶⠆⠀⢀⣠⣴⡿⠇⠀⠀⠀⠀⠀⠀⠀⣷⠤⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⣤⣄⣀⣈⣛⣛⣿⣿⣿⣿⣿⡥⠤⡶⠶⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠙⠋⠛⠁⠀⠀⠛⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠊⠉⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡗
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⣠⡴⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡀⠀⠀⠀⠀⠀⠀⣠⡾⢿⡄
⠀⢠⡞⠉⠉⠀⠀⠀⠀⢀⣾⣉⠀⠀⣩⡦⠀⠀⠀⠀⠀⠀⣀⠞⠻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠉⠋⠠⠀⠀⠀⢠⣞⠉⠀⢈⣹⡆
⠀⣿⡀⠀⠺⢿⣶⣷⣶⣿⣿⣉⠶⠾⠋⠁⠀⠀⠀⠀⢠⠛⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⡀⠀⠻⣷⣾⣶⣾⣿⣿⡿⠶⠞⠉⠁
⠀⠈⠁⠐⠀⢻⡿⠿⠋⠙⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢸⠟⠿⠋⠛⢤⡀
⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠉⢀⠀⠀⠀⠀⠀⠀⠀⣶⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠠⣄⣀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠆⠀⠰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠶⠖⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⠀⠀⠀⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠺⣿⣦⣤⣀⣀⣀⣠⣿⣤⣴⡶⠆⠀⢀⣤⣶⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠰⣤⣀⣤⠤⠗⠀⠀⠀⠀⠀⠈⠻⠷⠶⢤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣤⠤⠶⠶⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⣤⣠⣴⠾⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠻⠋⠈⠉⠛⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀
⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣚⠉⢀⣀⢀⣠⣴⣿⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣄⣀⣀⣀⣈⣓⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤
⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠋⠉⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠷⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠊
⠀⠀⠀⠀⠀⡀⠀⠀⠶⣺⣄⠀⠀⠓⢄⠠⢀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠄⠀⠀⢀⠔⠀⠠⠄⣀
⠀⠀⠀⠀⠸⣧⡀⠀⠠⠈⠉⠳⠄⠀⠀⠑⠄⡀⠀⠀⠀⠀⠐⢄⢲⡀⠀⠀⡼⢀⠔⠀⠀⠄⠐⢈⣉⠟⠁⠀⠀⠠⠎⠤⠀⡀⢀⣌⣷
⠀⠀⠀⠀⠈⡎⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⡸⣦⣄⠻⣷⡀⣼⡷⢁⠀⠀⢀⣠⠖⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠹⠋⡇
⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⣈⣹⣿⣿⠁⠉⠉⠉⠹⣿⣶⢞⣩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡠⠴⠒⠀⠀⠀⣀⣤⠿⠛⣿⠆⠀⠀⠀⢐⣿⡃⠀⠉⢻⡆⠀⠀⠶⢤⣂⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠊⠀⠀⣀⠄⠒⡙⠋⠀⢀⡴⠻⠷⠶⠐⢆⠊⠉⢌⢦⠀⠀⢉⠓⠦⣄⡀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠂⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢢⠀⠘⣉⠀⠀⠈⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠐⠛⠳⣄⠀⠀⠀⠀⠀⠀⠈⠛⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣠⣄⠀⠀⢣⣾⠗⠈⠉⡉⠉⠻⠷⡀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠼⣿⡷⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄⢈⣡⣶⠾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠙⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⣀⠄⢀⡠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠻⣦⠀⠈⠁⣀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠁⠀⠀⠘⣇⠀⠩⠤⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⢖⠋⠀⢸⣄⣀⣀⠀⢸⠀⠐⡒⠚⠈⣁⡀
⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⠿⠋⢹⠀⠉⠁⠁⠀⠸⣿⣿⠿⠀⢸⡇⠀⠤⠤⢄⣀
⠀⠀⠀⠀⠀⢸⡁⡟⠉⠁⠀⠀⠀⠸⡄⠀⠀⡀⠀⠀⢏⠁⠀⠀⢸⡇⠀⣈⠑⠂⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠈⠓⠸⢤⣶⣶⣦⣴⠖⠛⠀⠀⠒⠚⠲⠼⣦⠀⠀⢸⠇⢀⠀⠉⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡅⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠊⢠⡀⠑⠀⠈⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣄⠀⣀⣠⣤⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣄⣀⠀⠀⢀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⠋⠉⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠉⠙⠻⠛⠛⠋⠉⢩⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣴⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⡿⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢈⣿⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠘⠋⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢠⠴⠶⢿⡷⠒⠀⠀⠀⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠂⠀⠀⠀⢶⣿⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣤⠄⠀⠀⠈⠛⠁⠀⠀⠀⠀⣴⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠙⢷⣄⣠⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⣴⡾⢿⣿⣽⣟⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣿⡀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⣤⡾⠟⣻⣿⣿⣿⡿⠷⠶⠶⠿⠿⢿⣿⣿⣿⣅⠀⠈⢷⣄⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⣁⠀⣰⣿⣿⣿⣿⣿⣦⣄⣀⣤⣴⣿⣿⣿⣿⣿⣆⠀⠀⢻⣦⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⣿⠉⠉⠉⠉⠉⠉⠉⠁⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠶⠶⠶⠿⠛⠉⠛⠻⠷⠶⠾⠿⠛⠋

⠀⣠⠖⠋⠉⠉⠓⠦⡀⠀⠀⠀⠀⢀⡤⠚⠉⠉⠉⠓⢤⡀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠙⠒⢄⠀⠀⠀⠀⠀⢀⠴⠚⠉⠉⠙⠲⣄
⡼⠁⢀⣀⠀⢀⣀⡀⠙⡄⠀⠀⠀⡎⠀⢀⡀⠀⢀⡀⠀⢱⠀⠀⠀⢀⠏⠀⣀⡀⠀⢀⡀⠀⢳⠀⠀⠀⢠⠃⠀⣤⡀⠀⣠⡄⠈
⡇⠀⠁⠈⠁⠈⠀⠁⠀⡇⠀⠀⠸⡀⠀⠉⠁⠀⠈⠁⠀⢀⡇⠀⠀⢸⠀⠀⠉⢁⣀⠈⠁⠀⢸⠀⠀⠀⢸⠀⠀⣉⣀⣀⣈⡀
⠹⡄⠈⠑⠒⠒⠋⠀⡰⠃⠀⠀⠀⢣⡀⠈⠓⠒⠚⠁⢀⡜⠀⠀⠀⠈⢧⡀⠘⠉⠉⠉⠂⢀⠞⠀⠀⠀⠘⢆⠀⠳⣄⣀⠴⠁⢠
⠀⠈⠓⠤⠤⠤⠴⠊⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠙⠢⠤⠤⠤⠖⠉⠀⠀⠀⠀⠀⠀⠑⠦⠤⠤⠤⠚⠁

⠀⠀⣠⠤⠴⠤⢤⣀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠦⠤⣄⡀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠦⠤⣄
⢠⠋⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⡴⠁⠠⡀⠀⢠⠄⠙⣆⠀⠀⠀⢀⡜⢁⠀⠀⠀⠀⠀⠑
⡏⠀⠰⠗⠀⠐⠿⠀⠀⡇⠀⠀⢰⠃⠀⠦⠖⠀⠲⠴⠂⠈⡆⠀⠀⢸⠁⣀⣉⣁⣀⣉⣁⡀⢸⡀⠀⠀⢸⠀⢈⠽⠂⠀⠺⠅
⢧⠀⠐⠒⠒⠒⠒⠀⢀⠇⠀⠀⠈⣆⠀⢠⠖⠒⠢⡄⠀⣰⠁⠀⠀⠘⡄⠳⡀⠀⠀⠀⣠⠇⡸⠀⠀⠀⠸⡀⠀⡠⠖⠒⠦⡀
⠈⠳⢄⡀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠈⠢⣄⡀⠀⢀⣠⠔⠁⠀⠀⠀⠀⠘⠦⣈⠓⠒⢋⣡⠜⠁⠀⠀⠀⠀⠙⢦⣀⠀⠀⢀⡠⠞
⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⢀⡴⠊⣡⠀⢠⣉⠳⣄⠀⠀⠀⠀⢀⠔⠋⠀⠀⠈⠉⠢⡀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠈⠙⢢⡀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠈⠑⢦
⡞⠀⢠⣦⠀⠠⣦⠀⠈⡆⠀⠀⢀⠏⠀⢢⣄⠀⣠⡔⠀⠹⡀⠀⠀⢠⠃⠐⣦⡖⠀⢢⡔⠀⢱⠀⠀⠀⢰⠁⠀⢦⡄⠀⢠⡄
⣇⠀⠀⣀⣠⣀⡃⠀⠀⡇⠀⠀⠸⡄⠀⢉⠀⠀⠀⡉⠀⢀⠇⠀⠀⢸⡀⠈⠁⠉⠀⠁⡈⠀⢸⠁⠀⠀⢸⠀⠀⣉⠀⠀⢀⡁
⠘⢄⠈⠁⠀⠀⠉⢀⡜⠁⠀⠀⠀⠳⣄⠀⠉⠉⠉⠀⣠⠞⠀⠀⠀⠀⠳⡀⠉⠉⠉⠉⠁⣠⠏⠀⠀⠀⠈⢣⡀⠈⠉⠋⠉⠀⡠
⠀⠀⠙⠒⠲⠒⠚⠉⠀⠀⠀⠀⠀⠀⠈⠑⠒⠲⠒⠊⠁⠀⠀⠀⠀⠀⠀⠈⠓⠒⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠉⠓⠒⠖⠒⠋

⠀⢀⡤⠒⠒⠒⠢⢄⠀⠀⠀⠀⠀⠀⣠⠔⠒⠛⠒⠢⣄⠀⠀⠀⠀⠀⠀⣠⠔⠒⠒⠒⠦⣀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠒⠒⢤⡀
⣰⠃⠀⠖⠀⠐⠦⠀⠱⡄⠀⠀⠀⡞⠁⠐⢒⣀⡒⠂⠈⢳⠀⠀⠀⠀⡞⠁⠰⠤⠀⠴⠂⠈⢦⠀⠀⠀⢠⠎⠀⢠⡄⠀⣤⠀⠘
⡇⠀⠳⠔⠀⠘⠴⠃⠀⡇⠀⠀⢸⠀⠀⣰⠉⠀⠉⣆⠀⠀⡇⠀⠀⢸⠀⠀⠺⠆⠀⠸⠇⠀⢸⠀⠀⠀⢸⠀⠀⠛⠀⠀⠛⠃
⢳⡀⠈⠲⠤⠤⠊⠀⣠⠃⠀⠀⠀⢧⠀⠹⣞⣉⡳⠏⠀⡼⠀⠀⠀⠈⢆⠀⠰⠊⠉⠓⠄⠀⡼⠀⠀⠀⠘⣄⠀⠉⠉⠉⠉⠁⢀
⠀⠙⠦⣄⣀⣀⡤⠖⠁⠀⠀⠀⠀⠀⠑⠦⣄⣀⣤⠴⠊⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣠⠤⠊⠀⠀⠀⠀⠀⠈⠲⢤⣀⣀⣠⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⢀⣾⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⣼⡿⠋⠀⠀⠀⠀⠀⣴⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⡀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣁⣤⠤⠤⣤⣉⡑⠲⢤⡀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⣠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠈⠙⠳⢤⡙⠳⣄⠀⠀⠀⠈⠀⠀⠀⠀⢀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠙⢦⡀⠀⠀⠀⠀⣰⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠳⣄⠀⠀⠈⠀⠀⠀⠀⠀⢀⣠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⢳⡄⠀⠀⠀⠀⢀⡴⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠹⡄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⡀⠀⠁⠀⠙⢢⠀⠀⠀⠀⠀⠀⠙⣦⠈⢳⡀⠀⠀⠀⠀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠳⡤⠊⠀⠀⠀⢀⠀⢠⠑⡄⠀⠀⠀⠀⠀⠈⢧⡀⠱⡄⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠹⣄⠀⡀⠀⠀⠀⠠⣀⠄⠀⠀⠀⠀⠀⠀⠈⢷⡀⢹⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⢷⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢫⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⠦⣄⣀⠀⠀⠀⠀⠀⢀⣰⠃⣸⠃
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠑⠢⢤⣉⡙⠛⠒⠒⠛⠉⣀⡼⠃
⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋⠉
⠀⢀⡴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠠⡏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⣄⣀⣀⣴⣟⣀⣄⡀
⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠐⣲⡿⠻⢯⣭⣀⡅
⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⢷⣄
⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠒⠛⢦
⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠠⢤⡤⠞⠁⠈⠳⢄⣀⡤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⠙⠒⠒⠶⠶⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⡤⠞

⠀⠁⠀⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠉⠁⠀⠀⠀⣹⢦
⠀⠀⠀⠀⠀⠀⢀⠴⣏⠉⠉⠙⠒⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⢀⡠⠞⣥⠈⡇
⠀⠀⠀⠀⠀⢠⠃⠀⠈⠙⠶⢤⣀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣀⣀⣠⠤⠖⠊⠉⠀⠀⠀⠀⢱
⠀⠀⠀⠀⢀⡏⠀⠋⠅⠀⠀⣀⠀⠉⠓⠒⠤⢤⣽⣦⣤⠶⠒⠲⡤⢤⠦⣾⠓⠋⠉⠀⠀⠀⠀⠰⡲⠀⠀⠰⡶⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠛⠀⢀⡀⠀⠀⡠⡄⠀⠈⣆⠀⠀⠐⠀⠀⢹⠀⢀⣀⠐⠤⠂⠀⠀⠀⠄⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⢸⠀⠺⠃⠀⠠⠄⠀⠀⠈⠈⠀⠀⠈⠀⠀⠀⢸⠀⠀⠀⠀⠀⣸⠂⠈⠈⠀⢀⠀⠀⠁⠀⡤⠄⠀⠀⣀⠀⡇
⠀⠀⠀⠀⢸⡄⠀⠀⢀⠑⠂⠀⠂⠀⡤⠀⠀⠰⠕⠀⣠⡾⠲⠶⠚⢋⡟⢉⣙⢢⣄⡀⠑⠂⠀⠀⠀⠉⠁⠀⠀⠐⢠⠇
⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⡀⠀⠀⠉⠀⠀⢀⣤⠞⢡⠇⠀⠀⢀⡞⠀⠀⠁⢸⠀⠈⠉⠒⠲⠤⢤⣀⣀⣤⡤⠴⠋
⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠁⠁⠀⠀⢀⣤⠞⣻⠃⠈⢸⠀⠀⠀⣸⠀⠘⠍⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣤⠤⠖⠋⠀⣰⠃⠬⠀⢸⡄⠀⠀⣿⠠⡦⠀⠐⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠈⣧⠀⠀⢸⡀⠀⠀⡀⠂⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡤⠀⠀⠀⠈⢧⠀⠀⣧⠀⠸⠝⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂⠀⠀⡎⠂⢸⡇⠀⢸⡄⠄⠀⠀⠘⠃⢨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣠⠔⠋⠀⠀⠀⡇⠀⣔⠆⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠖⠋⠁⠀⠀⠀⠀⠀⠀⢷⠀⣠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠶⢶⣦⣄⠀⠀⠀⢀⣀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⣿⣶⣦⣤⣀⠀⠀⠀⢀⣀⣀⣀⣀⣴⡿⠋⠀⠀⠀⠈⠻⣷⣶⡿⠿⠛⠋⠉⠙⢿⡆
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠉⠙⠻⢿⡿⠿⠟⠛⠛⠛⢻⡿⠀⠀⠀⠀⣀⣀⡀⢹⣿⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡏⢩⣿⠿⠛⠻⢷⣦⣤⣴⣶⣶⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠘⠷⣿⠃⠀⠀⠀⠀⢻⣏⠉⠀⠈⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢿⣷⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣦⣤⣤⡾⣿⣆⠀⠀⠀⢀⣾⠟⣷⠀⠀⠀⣸⡗
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠻⢷⣶⣶⡿⠿⠶⠟⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄⣀⣀⣠⡾⢿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠀⠀⢻⣧
⠀⠀⠀⠀⠀⢰⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣤⣤⣤⣤
⠀⢠⣴⣶⡶⢿⣿⡿⠷⠶⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠉⠉⢹⣿⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣠⣤⣼⣿⣄⣀⣀
⠀⠀⠀⠀⣀⣤⣾⣿⡿⠛⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢠⣶⣿⣷⣦⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢹⣿⠋⠉⠛⠛⠁
⠀⠀⠀⠀⠉⠉⠀⠙⣿⣄⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣶⣿⣯⡀
⠀⠀⠀⠀⠀⠀⢀⣤⡾⠿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠙⠻⠷⡆
⠀⠀⠀⠀⠀⠺⠛⠉⠀⠀⠈⠙⠿⣷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿⠿⠿⠛⠋⠉

⠀⠀⢀⣤⣶⠶⠿⠛⠛⠛⠛⠿⠶⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠶⠿⠛⠛⠛⠛⠿⠶⣶⣤⡀
⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦
⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⣿⠀⠀⠀⣀⣠⣤⣤⣶⠶⠶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⠶⠶⣶⣤⣤⣄⣀
⣿⣠⣴⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⢻⡿⠿⠿⠿⠿⣿⡟⠀⠀⠀⠀⠀⣀⣴⡶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣦⣄
⣿⡋⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⡀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⢀⣴⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢙
⢹⣧⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣦⡀⢸⡇⠀⠀⠀⠀⢸⡇⣀⣴⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⣼
⠀⢻⣆⠀⠀⠀⠀⠈⠙⠻⠶⣶⣤⣤⣤⣤⣤⣶⠶⠿⠛⠉⠀⠈⠻⣿⡇⠀⠀⠀⠀⢸⣿⠟⠁⠀⠉⠛⠿⠶⣶⣤⣤⣤⣤⣤⣶⠶⠟⠋⠁⠀⠀⠀⠀⣰⡟
⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⣀⣀⣀⣀⣼⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠋⠉⠉⣻⣿⡉⠉⠙⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠙⠿⣦⣤⣀⡀⠀⠀⠀⣀⣀⣠⣤⣴⠾⠟⠋⣱⡿⠁⠀⢀⣼⠏⠘⢿⡄⠀⠈⢿⣎⠙⠻⠷⣦⣤⣄⣀⣀⠀⠀⠀⣀⣀⣤⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⢀⣴⠟⠀⠀⢠⣾⠃⠀⠀⠈⢿⡄⠀⠀⠻⣦⡀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠃⠀⠀⢠⣿⠃⠀⠀⠀⠀⠈⢿⡄⠀⠀⠘⠿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠾⠛⠁⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠈⠛⠷⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⣟⠋⠉⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠉⠙⣻⣿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣤⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣼⠏

⠀⢀⡠⠔⠒⠋⠉⠙⠓⠦⣀⠀⠀⠀⠀⠀⡠⠄⠒⠒⠒⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠒⠒⠦⣄⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠉⠙⠓⢦⣀
⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⣠⠎⠁⠀⠀⠀⠀⠀⠈⠱⣄⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⡏⠀⠠⣦⠀⠀⠀⠀⠷⠀⠀⢸⠀⢰⠃⠀⠀⠀⣄⠀⢠⡄⠀⠀⠀⠈⡆⠀⢠⠃⢀⣤⡄⠀⠀⠀⠶⠖⠀⠘⡆⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢷
⡇⠀⡄⠀⠀⠀⠀⠀⠀⡲⠀⢸⠀⢸⠀⠀⡀⠀⠉⠀⠈⠀⠀⢴⡁⠀⡇⠀⢸⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⠀⣿⠀⢸⠀⠘⡇⢠⣦⠀⠺⠗⢈⡇⠀⢸
⠹⡀⠈⠢⣄⣀⣀⣠⠔⠁⢀⡞⠀⠘⣆⠈⠣⠴⢄⡴⢄⠴⠒⠚⠁⢠⠃⠀⠘⡄⠈⢿⣭⣧⢼⡤⢿⡾⠋⢀⡏⠀⠈⢧⠀⠙⢦⣀⣀⣀⡤⠞⠀⢠⠟
⠀⠙⠦⣀⠀⠀⠀⠀⣀⡤⠊⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠘⠦⣀⠉⠛⠚⠛⠉⢀⣠⠞⠀⠀⠀⠀⠳⢤⣀⠈⠉⠁⣀⣠⠔⠋
⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⣀⡠⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⢤⣀
⠀⣠⠎⠁⠀⠀⠀⠀⠀⠉⠳⡀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⠈⠳⣄
⣰⠁⠀⠀⢀⡀⠀⣀⠀⠀⠀⠹⡄⠀⢰⠋⠀⠀⠀⠀⠀⢀⡀⠀⠀⢹⡀⠀⢀⠎⠀⠀⢀⡀⠀⠄⠀⠀⠀⠹⡄⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⡇⡀⠀⠀⠈⠁⠀⠁⠀⠀⢸⡄⡇⠀⡏⠀⠀⠟⠀⠀⠀⠈⠉⠀⠀⠀⡇⠀⣸⠀⠀⠀⠀⠀⠀⠀⣠⢤⠀⠀⢳⠀⢸⠁⠀⣠⣄⠀⠀⠀⢀⣤⠀⠀⣿
⣇⢹⡀⠀⠀⠀⠀⠀⠀⠀⣸⢡⠇⠀⣇⠀⢤⠤⠤⢒⣒⣒⣊⡟⠁⢰⠇⠀⢸⠀⠀⣠⡔⢞⠉⠉⠉⠙⠦⠀⡼⠀⠸⡄⠀⠈⠁⡠⠤⣄⠈⠉⠀⢀⡏
⠈⢦⡙⠦⣄⣀⣀⣀⣠⠔⣡⠎⠀⠀⠘⣆⠈⠻⢭⣉⣀⠤⠊⠀⣠⠏⠀⠀⠀⠣⡈⠀⠈⠉⠀⠀⠀⠀⢀⡴⠁⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠉⠒⠦⠤⠭⠤⠒⠋⠀⠀⠀⠀⠀⠈⠑⠦⢤⣤⣤⠤⠴⠊⠁⠀⠀⠀⠀⠀⠈⠒⠤⢤⣀⡠⠤⠖⠋⠀⠀⠀⠀⠀⠈⠙⠲⠤⠤⠴⠒⠋
⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⢀⡴⠚⠉⠉⠉⠉⠉⠓⢤⡀⠀⠀⠀⠀⢀⠔⠋⠉⠀⠀⠉⠳⢤⠀⠀⠀⠀⢀⡴⠊⠉⠁⠀⠉⠙⠲⣄⠀⠀⠀⠀⠀⣠⠔⠋⠉⠉⠀⠉⠙⠲⣄⡀
⢠⠏⠀⠀⠀⠀⠀⠀⡀⠀⠀⠱⡄⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢠⠎⠀⠀⠀⠀⠀⢀⠀⠀⠈⣧⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⡏⠀⠀⡹⠯⠀⠀⠿⠅⠀⠀⠀⢷⠀⢰⠇⠀⣠⠤⡄⠛⢀⠤⢄⠀⠈⡇⠀⢸⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⢸⡄⠀⢸⠃⠀⠀⠀⠀⠄⠰
⣇⠀⠀⢀⣀⣀⣀⣄⣤⡤⠄⠀⡿⠀⢸⡀⠀⠁⠀⠀⣠⡀⠀⠀⠀⢀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢸⡄⠀⠤⣤⡤⡴⠶⠒⠒⠋⠁
⠘⢆⠀⠀⠀⠀⠀⠀⠓⠁⢀⡴⠁⠀⠀⢧⠀⠀⠀⠀⠻⠃⠀⠀⢀⡞⠀⠀⠀⢣⡀⠀⠀⡴⠒⠒⠄⠀⢠⠞⠀⠀⠀⢳⡀⠀⠹⣠⠇⠀⠀⠀⠀⢀⡴
⠀⠈⠑⠦⢄⣀⣀⣀⡤⠴⠋⠀⠀⠀⠀⠀⠑⠦⣄⣀⣀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠙⠢⣄⣀⣀⣀⡤⠖⠁⠀⠀⠀⠀⠀⠉⠳⢤⣄⣀⣀⣤⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠐⢿⣷⣦⣄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠿⠷⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡷⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠉⢹⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠰⢿⠿⠋⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢒⠎⠀⠈⠀⠀⡀⠀⣀⠀⠐⠉⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⠀⠀⠀⠸⠀⠀⡀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠠⠄⠀⡠⠀⠀⠀⠀⠀⠐⠄⡀⠀⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

⣿⡿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠛⠛⠁⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠙⢷⣄⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣾⠶⢦⣤⣤⣤⡴⠾⠛⢻⡄⣿⠀⠀⠀⠀⠀⠀⢀⣾⠛⣷⠀⠀⠙⣧⡐⠛
⠀⠀⠀⠀⠀⠀⠀⣰⠿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⡆⠀⠀⠀⠀⣠⡾⠁⠀⢹⣆⠀⠀⠈⢷⡀⣠⡴⠖
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⠇⠀⠀⣠⣾⡏⠀⠀⠀⠀⢻⡟⠛⢷⡈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢀⣴⠟⠋⢸⡇⠀⠀⠀⠀⠀⢻⣄⣾⠃⠈⣷⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⢀⣾⠇⢰⡟⠁⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠹⣧⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣀⣀⣀⣴⡿⠋⢰⣿⡇⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⣀⣀⣘⣷⣄⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣉⣛⣉⡁⠀⠀⠘⢿⣿⣦⣀⣀⢀⣀⣀⣽⣶⠞⠛⠉⠉⠉⠉⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠛⠛⢿⣍⣩⡟⠙⠛⠶⣦⣀⠉⠉⠉⢻⣟⠉⠻⣷⣽⡆
⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⢀⡿⣿⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠻⣦⠀⠙⣿⡄
⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⣼⠃⠸⣧⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠙⣷⡀⢘⡿
⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⢀⣿⠂⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠈⠛⠛⠁
⠀⠀⠸⣧⣤⣤⣤⣀⣀⣀⣀⣀⣀⣈⣻⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠒⠲⢤⣀⣠⠚⠉⢳⡀⠀⠀⠀⠀⢀⣠⠖⠛⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣠⠼⣇⡤⡀⠀⢷⠒⠒⠒⠒⠛⠁⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⣹⡇⠀⠀⣷⡁⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢧⠀⠓⠛⡶⠚⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠑⠦⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⣼⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⣀⡀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠇⠀⠀⠒⢺⡧⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡟⠓⠒⢌⣳⡄⠀⠀⠀⠀⠀⠸⠉⠕⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⡀
⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀⠀⠀⠘⣇⠀⠀⠀⣹⣁⣀⣀⡤⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣐⣦⡞⠀⠈
⠀⠀⢰⡏⠹⠦⠤⠼⠋⢹⠀⠀⠀⠈⠳⣄⣀⣼⠈⠉⢙⡏⠉⠑⢮⡷⣶⠖⢶⡖⠚⠛⠉⠉⠀⠈
⠀⢠⠞⠁⣠⠤⠤⢤⡀⠈⢷⡀⠀⠀⠀⠀⢹⠉⠑⢤⣼⣀⣀⡴⢋⣰⠋⠙⡎⢧
⢠⠏⢀⡎⠸⠂⣚⠃⣼⢧⠈⢳⡀⠀⠀⠀⠘⢶⠖⠋⠙⣾⢟⠞⢏⡿⠤⠤⢾⠈⢧
⣿⠀⠘⣏⡃⠘⠤⠃⣤⣼⠀⠈⣷⠀⢰⡒⢦⣾⣀⡠⠞⣿⠈⠓⠺⢥⡀⠀⡼⠀⢸⣇⣀
⢷⡀⠈⠣⠷⠤⠤⠤⠔⠁⠀⣀⣿⣄⣀⣽⠟⢻⣅⠀⠀⣿⠀⠀⠀⠀⠉⠋⠁⣀⡾⠉⠉⢹⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⣇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣨⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢱⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠄⢀⡠⠤⠄⢀⡀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢀⣀⡠⠤⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡯⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠉⠈⢻⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⢀⣎⢒⣀⣶⠶⠒⠒⠒⣒⠶⠶⠒⠒⠒⠂⠐⠒⠒⠿⣿⣇⠀⠀⠀⠀⠀⠀⣸⣿⠛⠓⠒⠒⠒⠒⠒⠶⠶⠦⣖⡒⠒⠒⠶⢦⣤⣀⠱
⠀⠀⠼⠃⠀⠀⡠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠀⠘⡧
⠀⠀⠀⠈⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⠀⠀⠀⠀⠀⠀⣸⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠺⢫⠟⠋⠁⠀⠁⠀⠙⠋⠙⢷⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⣠⣴⠞⠋⠁⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢆⠀⠈⠻⠲⣦⣀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⢀⠤⢺⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠙⠓⣄⡀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠔⠀⡴⠃⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠘⣄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠂⠘⠁⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠢⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⣶⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣦⣄
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡋⠀⠙⠛⠛⠋⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⣰⣿⡿⢫⣾⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⡍⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠈⠙⠛⠛⠛⠋⠉⠁⠀⠀⠀⢀⣼⣿⠟⣵⣿⣿⡇⠀⠀⠀⠀⠀⠀⠸⣿⣿⡜⣿⣿⣿⡀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠋⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠃⠀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠈⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⠟⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠸⣿⣿⣿⣧
⠀⠀⠀⠀⠉⠙⠛⠋⣹⣿⡿⠃⠀⠀⣠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⢹⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣀⡼⠟⠉⠀⢀⣠⣾⠿⠋⣹⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⢻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠉⠀⠀⣰⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⠀⠹⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣷⣄⠈⠻⣿⣷⣉⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠉⠛⠋⠁⠀⠈⠙⠻⠷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠷⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⢴⣴⣶⣶⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣶⣶⣦⡆
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠉⠉⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠉⠹⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻
⠀⠀⠀⠀⣠⣴⣦⣶⣶⣾⣿⣿⣿⣿⣶⣶⣤⣶⣤⣀
⠀⠀⠴⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠸⠋⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣶⣶⣤⡀
⠀⠀⠐⣶⣤⣄⣀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠛⠙⠃
⠀⠀⠀⠈⠻⠿⠿⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣾⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣠⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠉⠉⠙⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠷⢶⣤⣀⣴⣾⠿⠛⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡾⠃⠀⠀⠀⠙⢿⡏⠁⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠿⠿⢿⣷⣶⣦⣤⣶⣾⣿⠿⠿⠛⣿⠃⠀⠀⣴⡶⣶⣾⠿⠿⢶⣤⡶⠾⠛⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⣿⠀⠀⠸⣿⣠⡿⠁⠀⠀⠀⢹⣷⣦⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠈⣙⣿⣄⠀⠀⢀⣼⣯⣼⡏⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠛⠉⠙⠻⠟⠻⣿⣉⠁⠀⣠⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠈⣿⣇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⠟⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠘⢻⣿⠟⠛⠛⠃
⠀⠀⠀⠀⢀⣠⣤⣶⣿⣷⠶⠆⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣀⣾⣟⡀
⠀⠀⠀⠀⠈⠉⠁⠀⠸⣿⣄⣀⡀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⣿⣅⣬⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡿⠛⠻⠿⠦
⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⣿⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⣻⣷⣿⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⠃⠀⠀⠉⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠛⠿⣿⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣾⠿⢿⣿⡉⠁⠈⢿⣧⡀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⣠⣴⡿⠟⢛⣿⠟⠉⢻⣟⠉⠉⠉⠉⠉⠀⣸⡿⠀⠀⠹⣷⡄⠀⠈⢻⣷⡄⣀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠛⠻⣷⣤⣿⣏⣤⣀⡈⠻⣦⣄⣀⣀⣠⣴⡿⠁⢀⣀⣤⣽⣿⣄⣀⣤⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣽⡿⠛⠛⠛⠛⠿⣷⣄⠉⠉⠉⠉⠁⣠⣾⠟⠛⠉⠙⠛⢿⣿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⢀⣤⣿⣿⣶⣶⣶⣶⣶⣿⣷⣤⡀⠀⠀⠀⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⠿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠿⠛⠋

⠀⠀⠀⠀⡤⠤⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠤⡄
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠉⠓⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠒⠋⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀⠤⠤⣤⠤⠤⣄⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠉⠉⠉⠉⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠘⣇⣀⣀⣠⡤⠤⠖⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠦⠤⣤⣀⣀⣀⡟

⣴⡟⣡⠎⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠈⢦⠘⣷
⠏⣰⠃⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢳⡈
⡰⠁⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢱
⠇⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⡇
⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⣿
⢸⠀⠀⠀⠀⠀⡀⠐⠀⢠⠄⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡆⢢⠤⠴⠇⠚⠓⠁⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠂⠊⠀⠈⠀⣄⠀⢸⡦⠴⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠸⡆⠀⠀⠀⠀⠀⠐⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⡆⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠘⡄⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⡼
⣧⠙⣆⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⡜⢠
⣙⣷⣈⣢⣈⣳⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣋⣀⣊⣰⣟

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⢦⣄⡀⠀⠀⠀⣀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⡾⠋⠉⠉⠛⠛⠶⣤⣀⠀⠀⠀⠀⣀⣀⣀⡾⠃⠀⠀⠀⠈⠻⣦⠶⠛⠉⠁⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠉⠉⣿⠁⠀⠀⠀⣀⣀⣀⣻⣤⣄⡀⠀⠀⠀⡀⣹
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣏⣹⠟⠉⠀⠈⠙⣷⠛⠋⠉⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⢙⣿⠀⠀⠀⠀⠀⣸⣦⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠹⣦⣀⠀⣀⣴⣟⣨⡇⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢧⣉⠉⠀⠀⣠⣾⡇
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣀⣀⣀⡀
⠀⠀⣤⣤⣤⣾⡷⠶⠶⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠈⠉⢩⡏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣤⣼⣇⣀⣀
⠀⠀⠀⠀⢀⣠⣿⡶⠖⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣴⠚⠛⠲⡄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⡿⠉⠉⠉⠁
⠀⠀⠀⠀⠉⠁⠈⢷⡀⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⣾⣁
⠀⠀⠀⠀⠀⢀⣠⣼⢿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⠙⠛⠶⠄
⠀⠀⠀⠀⠀⠛⠋⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⣤⣤⣤⣄⣀⣀⣀⣠⣤⣤⣤⠴⠶⠞⠛⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠂⠀⠈⠉⠉⠉⠉⠉⠁⠀⠂⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣶⣦⣄⡀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⣠⣴⣶⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠿⢦⡀⠀⠀⠀⠀⠀⠀⢀⡴⠿⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠁⠀⠉⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⡠⠒⠈⠉⠉⠐⠢⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⢉⠐⠄⡀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⢂⠀⠀⠁⠒⠠⢀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠐⠈
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠻⡒⠤⢄⣀⡀
⠻⢍⡁⠀⠀⠀⠈⠉⠉⠒⠒⠒⠒⠤⡀
⠀⠀⠈⠉⠑⠒⠒⡶⣶⣶⣦⣤⣤⣤⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠠⠤⠔⠒⠊⠉⠉⠉⠉⢆
⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠉⠛⠛⠛⠛⠃⠐⠒⠒⠒⠒⠒⠚⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⣶⣶⠾⠤⠒⠒⠈⠉⡩⠝⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⠤⣄⣀⣀⣀⣀⣀⠀⠤⠤⠤⠤⠤⠤⠒⠒⠒⠒⠋⠉⠉⢻⡿⠿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠑⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⣀⣀⣀⠤⠔⠒⠊⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠤⠤⠤⠤⠤⠤⠀⠀⠤⠤⠤⠤⠤⠤⠄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠐⠀⠀⠠⢨⡀⠀⣀⠠⠄⠒⠊⠉⠉⠉⠀⠀⠀⠀⠉⠉⠑⠒⠠⠄⣀⠀⢀⡅⠀⠀⠀⠂
⠀⠠⢀⠀⠼⡄⠀⡠⠀⠻⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠟⠀⢄⠀⢠⡧⠀⠀⠄
⠀⠀⠊⠐⠊⠀⠀⠈⠑⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠐⠊⠁⠀⠀⠑⠂⠑