⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣋⠩⣄⠂⡔⣈⡑⡈⢄⡐⠠⠌⣡⢉⡓⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⣡⠐⢋⠄⡃⢂⡑⢂⠒⡑⡈⢄⣈⡑⠌⡄⠌⡙⢶⣠⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⠅⣊⠰⡘⠈⢆⡑⢂⡰⢠⠁⢆⡰⢂⡄⠛⣦⡐⠌⠔⢢⡙⣷⣌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣟⠿⢋⠛⠓⡓⠠⠉⣼⢁⠂⠤⢁⣉⣤⣤⢧⠿⠶⠿⢷⠺⠴⠦⠬⣭⡽⢷⠽⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⠠⠀⣄⠔⢻⠋⠙⠛⠛⡋⣉⣉⣉⣉⣙⣿⣯⠽⢯⣭⡭⠥⠤⠤⠤⢤⣤⠤⣷⠖⠒⠒⠚⣷⠒⠦⢤⣀⠳⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠠⠠⠐⠒⠛⠓⢈⢃⣠⡴⢶⡿⣉⣡⢌⣼⣇⣾⡷⣿⡀⠤⣷⣿⣔⢦⣂⠱⣦⡙⢿⣮⣷⠨⠡⣁⠚⣇⢻⣦⡜⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣾⣿⡟⢼⣿⢷⣣⡝⣾⣿⣿⣿⢻⣿⣷⡘⢻⣿⣟⠻⣿⣷⣯⣱⣮⢿⣿⡳⠡⢘⢣⣿⠌⣘⣧⡹⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣿⣿⣯⣟⣿⣿⣹⣿⣿⣏⣿⡏⠸⣿⢻⣷⡨⣿⢻⣧⡼⠿⢿⣿⢿⣯⣿⣧⠑⢢⢸⡿⣇⠤⢻⣧⢿⠿⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⢻⡧⣿⣺⣿⣧⣿⣿⡇⢻⡏⠉⢻⡆⠻⣿⣿⣇⠙⢧⡀⢐⣬⣷⡿⢿⣿⣯⡔⣪⣿⣿⣘⢦⣿⣞⣇⠀⠉⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⡏⢸⣟⣿⣿⣿⣿⣿⣿⢤⣼⣷⣶⣾⣷⡤⠩⠿⣿⣆⠈⠹⢻⣙⣿⣿⡮⢽⣿⡇⣿⣿⡏⢫⡟⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣾⠏⢯⣽⣿⣿⡛⠀⠀⠈⠛⠂⠀⠘⡟⢿⣿⣿⠐⣿⡇⣿⣿⡇⠘⣏⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠯⠀⠘⣿⣿⣿⣿⣿⣿⣿⡍⠇⠘⡿⢻⣿⣧⠀⠀⠀⠀⠁⠀⠀⠠⠾⠟⠛⢃⣿⣷⣿⣿⣇⣼⣿⣿⡎⠻⣿⠀⠀⠀⣖⣲⣄⡄⠀⠀⠀⢠⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⢿⣿⣿⣿⣿⡄⠀⠵⠾⠛⠉⠁⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠹⠀⠀⠀⠋⡃⠘⡧⠩⡓⢦⡜⡧
⠀⠀⠀⠀⡠⡤⡄⣐⣾⡟⠀⣾⣿⣿⣿⣿⣯⠢⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡏⢻⣿⡇⠀⠀⠀⠀⢰⣄⡇⠀⢻⡢⣡⡸⣇⣻
⡖⢹⡷⠻⢡⣊⣡⠚⡗⠇⢀⡏⢻⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣽⣿⣿⣿⡟⢻⡇⠀⠙⣧⠀⠀⠀⠀⠀⢻⡇⠀⠀⠈⠻⣿⡈⡙
⢧⣫⡟⢣⣾⠷⠃⣸⠀⣌⠎⠀⠘⣿⣿⠺⣿⣿⣿⡿⣶⣦⣤⣄⣀⡀⠀⠀⢀⠴⠊⠀⣰⣿⣿⠻⡿⠁⠀⡇⡀⣀⡈⠁⠀⠀⠀⠀⠓⠆⠀⠀⠀⠀⡇⡟⡄
⠈⢉⠿⡿⠁⠀⠀⣓⡼⠀⠀⠀⠀⢹⣿⡀⢿⢹⡿⠀⠙⣿⣿⠈⠻⣿⣿⡀⠀⠀⡠⠔⣿⠋⣿⣀⠃⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠈⣤⠀⠀⠈⠃⠀⡷
⢠⠊⢸⡁⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⢿⡆⠈⣿⠁⠈⠁⠉⠁⣰⡤⠽⠁⠁⠂⢉⡀⠴⠆⠊⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⢠⡏⠀⠀⠀⢀⡼⠃
⢻⠀⠘⠁⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠈⠃⡞⠈⠀⠀⠀⣠⡾⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⡂⠀⠀⠀⡰⠃⠀⠀⠀⢹
⠀⢂⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⣠⣤⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣋⡁⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⢸⠆⠕⠀⠀⡸⠁⠀⠀⠀⠀⡎
⠀⠸⡆⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢠⠂⢠⡾⠛⠓⣥⠀⠀⠀⠀⠀⠀⡠⠂⢀⣴⠟⠉⠈⢧⡀⠀⢠⡿⡂⠀⠀⠀⠀⠀⠝⠇⠀⡟⠀⠀⠀⠀⠀⡯⠀⠀
⠀⠀⡷⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢠⠃⡐⠉⠀⠀⠀⠈⢳⡄⠀⠀⢀⠊⣀⡴⠋⠀⣠⡂⠀⠀⢻⡄⢈⣧⠳⠀⠀⠀⠀⠀⠁⠘⣴⠁⠀⠀⠀⠀⠀⡇⠀⠀
⠀⠀⢸⡆⠀⠀⠀⠀⠈⢂⠀⠀⢠⠁⡘⠀⡿⠀⠀⠀⠀⠀⢪⡄⠀⢀⣴⠋⠀⠀⠀⠈⠀⠀⠀⠈⢷⡸⠘⡆⡇⠀⠀⠀⠀⠁⢠⠃⠀⠀⠀⠀⠀⢸⡇⠀⠀
⠀⠀⠈⡇⠀⠀⠀⠀⠀⠈⠄⠀⠎⠀⡇⠀⠁⠀⠀⠀⠀⠀⣈⣷⣤⣾⠉⠀⡠⠀⠀⠀⠐⠠⢄⣀⣸⡇⠀⣳⠁⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀
⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠹⡀⠂⣉⣼⣷⡶⠾⢿⣿⣿⢿⠳⡞⠳⠶⣤⣤⢀⡀⠀⠀⠀⢿⡼⠁⢃⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀
⠀⠀⠀⠈⡃⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⢹⡟⢋⡭⠚⠱⣆⡾⡦⢱⣿⣦⠟⠀⠀⠀⠊⠙⠋⠕⣦⣀⢸⠇⠀⠈⡄⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⣧⡇⠀⠀⠀
⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠀⠀⠀⠀⠀⠙⢷⡇⠀⠻⣿⠢⣀⠀⠀⠀⠀⠀⠀⠀⢱⠏⠀⠀⠀⠘⡄⢺⡁⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀
⠀⠀⠀⠀⠀⠱⣀⠀⠀⠀⠀⠀⢀⢇⠄⣹⡀⠀⠀⠐⠀⠀⡾⢇⠀⠠⣿⡀⠈⢧⡀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠱⣼⡀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠱⠐⢄⠀⠀⠀⡼⢀⡴⠹⠁⠀⠀⠀⠀⠀⠛⠋⠀⠀⠉⠀⠀⠈⢷⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⢀⣨⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠡⡀⠑⠤⠊⢁⠞⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⢆⡀⠀⠀⢰⠏⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⠔⠁⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡉⣒⣴⠏⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⢣⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠈⡆⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢱⠀⠀⠀⠀⠀⠈⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⢲⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣀⢀⣀⣼⣤⣴⡶⠶⣿⢿⢿⡿⣕⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡮⠃⢿⡄⠀⠈⠈⠉⠙⠛⢄⠽⠳⠤⠶⠦⠶⠶⠤⠶⠀⠸⠏⢘⣛⣿⠶⠋⠁⠀⠀⢀⡐⡸⢻⣿⠘⡼⣟⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣰⣣⢌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⠀⠀⠀⠀⠘⢆⢂⣸⣭⠨⣹⡘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⣌⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠵⣀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⡀⠀⠀⠀⠈⠢⣽⣇⠀⠁⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⣿⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠢⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⡮⠙⠀⠀⢻⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢙⡂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⣄⢑⡆⠒⣖⣲⠁⠀⠀⠀⠀⠀⢘⠌⠀⠀⠀⠀⠀⠳⢺⠀⡀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀