⠀⠀⠀⠀⠔⠁⣠⠞⠕⠁⢠⣾⢿⣻⣿⡿⠀⠀⣰⢀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣧⡀⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀
⠀⠀⠀⠀⠀⠀⠀⠌⢠⡞⢁⠊⠀⣰⠛⠁⢪⢺⣿⡇⠀⠀⣿⠘⣤⡀⠀⠢⣅⠂⢀⠀⠈⢻⣿⡕⡀⠀⠈⡟⢿⡻⣟⠛⣿⣿⣿⣿⣿⡀
⠐⠒⠤⠤⠤⢄⠂⡶⢋⠀⠀⢀⠀⡵⠀⢠⡟⠸⣿⢱⡇⢀⣿⣶⣯⡇⠀⢀⡉⣿⣦⣕⣤⡀⢙⣷⡌⣆⠀⠘⣦⠑⡜⣆⢱⡌⣿⣿⠙⣷
⠀⠀⠀⠀⠀⠎⡼⢡⣾⠁⠠⢡⡾⠁⢀⣿⣇⠀⣿⣿⣷⢸⣿⣿⡿⢿⣤⣮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣾⣆⡄⠈⢷⡈⢞⡄⠹⡄⢣⠀⣿
⠀⠀⠀⠀⣜⣼⣵⣿⡇⡠⣡⡿⠡⠀⣾⣿⣿⠀⢿⣿⣿⡐⢹⣿⣵⠀⠀⠈⠙⠪⡻⣿⣿⣿⣷⣝⠬⠃⠈⠂⠀⠈⢷⣻⣷⡀⢻⡌⡄⣼
⠀⠀⠀⢰⠹⣿⣿⣿⣳⣿⣿⠃⠇⢰⣿⢿⡿⣧⢸⣿⣿⣧⡌⣿⣿⡇⠀⠀⠀⠀⠹⣿⣿⣿⣷⡹⡟⢮⡢⡀⠀⠀⠈⣷⣿⣧⠘⣿⣡⣿
⠀⠀⠀⣆⣳⡿⢻⣿⡳⣿⡛⢸⠀⣾⡏⠸⣷⢻⣇⣿⢻⣿⡇⢿⣯⣿⡄⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣧⡉⡻⠤⣀⠀⠘⣿⣿⣇⢿⣿⣿
⠀⠀⢰⣿⡿⠁⢸⣿⣿⣿⣇⣤⣠⣿⠃⠀⠹⣧⢻⣿⣏⣿⣷⠸⡿⣾⣿⡄⠀⠀⠀⠀⠈⢚⣿⣿⣿⣿⣷⡸⣄⠠⠙⠆⠘⣿⣿⣼⣿⣿
⠀⠀⣸⣿⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠱⡻⣿⣿⣾⣿⣆⢳⠙⢿⣿⣆⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣽⣦⠡⡘⣦⢚⣿⣇⣿⣿
⠄⡀⢻⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⡉⠓⠺⢌⣻⣿⣿⣿⣾⣆⠀⠙⢝⠳⣄⠀⠀⠀⠀⢿⣿⡟⡟⣿⣿⡿⣷⡐⠼⣷⣻⣿⢿⣿
⠀⠈⠻⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣋⣡⣄⣀⡀⠀⠢⡙⢿⣿⣿⣿⣆⠀⠀⠁⢊⠱⠦⠤⠖⠚⣿⣿⠘⡘⢿⣿⣞⣿⣮⣻⣿⣿⣿⣿
⠀⠀⠀⠈⠄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⡏⠹⡍⢳⢆⠀⠙⢿⣿⣿⣷⡀⠀⠀⠀⢠⢞⣓⣠⣿⣿⣆⣡⣊⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⣸⣿⣿⢹⣿⣿⣿⣿⣿⣧⠉⠓⠺⠶⠶⠋⠀⠁⠀⠈⢻⣿⡌⠙⠲⣄⠀⠁⡕⠋⢃⠘⠃⡿⠈⣿⠟⢻⣿⣿⣿⣿⣯⢻⣿
⠀⠀⠀⠀⠀⢻⣿⣯⢀⢻⣿⣟⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⣷⡀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠓⠛⢣⠀⠘⣿⣿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠸⣿⢿⣄⠹⣿⣿⠘⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣽⣿⣸⣿⣾⢪⡟
⠀⠀⠀⠀⠀⠀⣿⠸⣿⣦⡈⢻⡇⠹⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣿⣫⡼⠁⣽⡏
⠀⠀⠀⠀⠀⠀⣉⡇⢻⣿⣷⣄⣎⠀⠹⡌⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⠜⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⡎⠁⠈⣿⣹⣿⣿⡄⠀⢳⠀⠙⠧⠀⠀⠀⠀⠀⠀⠂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢹⣼⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⣧⣿⣿⣿⡄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣤⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⠃⠀⠀⣰⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠠⢤⣤⡤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠸⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡠⠚⣹⣿⣿⣿⣿⣿⣿⣿⣿⣧⠙⢯
⠀⠀⠀⠀⢀⠃⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣏⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⡫⠀⢀⣿⣿⣿⣿⣿⣿⣧⠙⢷⡹⣇⠀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣓⠤⠌⠲⢤⡀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣯⠞⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣆⠀⠃⠘⠀
⠀⠀⡀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠙⠷⣦⣤⣤⣶⣿⡿⠟⠋⠁⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀
⣠⣀⣤⣤⣶⣾⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⠞⢻⣿⣿⣿⣿⣿⢿⠿⢿⣿⡀





⢀⠀⡤⢤⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⢶⠞⢩⣧⡨⠿⠿⢿⡝⠯⠛⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀ ⠀⠀⠀⢀⣶⠟⠍⠁⢒⠿⡠⠖⠉⠉⢙⣷⠀⠀⠀⠈⠩⣲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢤⡿⣥⡖⣲⣿⣿⣞⣁⣀⠴⢚⣿⠛⣷⡈⣆⠀⠱⡌⠉⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢰⡿⢛⣶⣿⣿⣿⠋⣹⣟⣁⣴⣾⠃⢀⡏⠇⠸⡀⠀⢱⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣿⡇⡘⣾⣿⣿⡇⣸⡯⠽⠟⢋⣉⠑⡞⠀⡼⢠⢧⠀⠀⡇⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠐⡿⢰⢁⡟⠀⠉⣰⠙⡿⣷⣶⢦⡄⢰⠁⢰⠃⣸⡌⠀⢸⠃⢀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀ ⠀⠀⣷⢸⢸⢧⡰⢼⣿⡀⠉⠀⠈⠀⠀⠀⢧⢇⣸⣳⠁⡰⢃⠀⣸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢿⣿⡸⣼⡝⢦⠣⠁⠀⠀⠀⠀⠀⠀⠘⠙⠻⢥⠞⢁⠜⣰⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⢿⢿⣼⣇⠘⣧⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣼⣧⣾⡷⠛⢿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠸⠺⣿⣿⣇⣿⠙⢦⡀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⡤⠶⠶⠿⢿⣿⡇⠀⠀⠈⠓⠤⣤⡤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡴⠋⠀⠀⠀⠀⠀⠙⠓⠤⠄⣀⡀⠀⢸⣷⣦⡤⠤⠖⠒⠒⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⠃⠀⠀⠀⠀⠀⢀⢆⡀⠀⠂⠒⠒⠒⠻⠦⣄⡀⠀⢀⠢⠤⠤⢄⡹⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡚⠀⠀⠀⠀⠀⠀⡸⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠈⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀
⢹⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⢬⣱⣄⠀⠀⠀⠀⠀⠀⠀
⠀⣇⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠻⡄⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣷⠀⠀⠀⠀⠈⡦⣀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠖⠋⠈⠳⣄⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠹⡄⠀⠀⠀⠀⢸⠈⠉⠒⠒⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠈⠳⣆⡀⠀⢀⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢥⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⡄⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⣷⠀⠀⠀⠀⡹⣿⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠛⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⠄⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠭⠄⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠸⡆⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣄⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡱⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁⠙⡄⠀⠀
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠹⡄⠀
⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀�