⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠉⠉⡉⠒⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⢀⣴⠦⠄⠀⠀⠀⠈⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣴⣶⣾⠛⠛⢛⡶⢤⢤⣀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠁⠈⣿⣶⣶⣃⡎⠉⠙⣝⣿⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣶⣾⣿⠃⠉⣷⣷⣤⣾⡿⠋⢹⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢀⡴⠳⠦⣴⡾⢢⡀⡀⠀⠀⡮⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠄⣼⠒⣀⣤⣟⡑⢒⢷⠸⠀⢸⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣇⠈⣇⠘⠧⠤⢤⡖⠀⢹⢆⠴⠃⡘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠚⠁⠁⠘⣆⠈⢳⠈⠀⠀⠈⠚⠉⠀⢀⡔⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠛⠿⣟⠓⢤⣀⠀⠀⠙⢦⣄⣀⣀⣀⠀⢀⣠⡔⡋⠀⠀⣷⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⣈⠷⣄⠙⣕⢦⡀⠀⠈⢓⣒⣚⣹⣭⣤⡖⠁⠀⢰⠉⢿⡌⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⣀⣀⣀⡠⠼⠓⠒⢚⢧⡈⢫⡉⢙⠖⣒⢤⣄⣀⣢⢸⡇⢠⢀⣧⠤⡼⣿⣶⡼⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡠⠖⠊⠉⠁⢸⠂⡠⠀⠀⠀⠀⠀⠀⠹⣆⠑⣄⡞⠁⠾⡂⡐⣯⡙⠳⣾⠿⣧⣽⣇⡌⣻⡳⡈⠮⠻⢕⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⢷⡈⢧⠀⠊⣧⣟⡒⢽⡃⡈⢻⣿⣇⠻⡏⣝⢷⡙⡜⢷⣤⠙⢎⠓⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠔⡀⠀⢻⣌⠱⣞⢿⢻⣻⠡⣧⣷⠀⣿⣻⡋⠹⣿⠽⢿⡝⡏⡏⣭⣶⣤⣜⡄⠀⠀⠀⠀⠀
⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣴⢁⣴⣮⣿⣦⠈⢯⡿⢻⡕⣦⢯⣶⡿⣇⠯⢶⣂⠔⡩⠻⡜⣗⠓⠁⠉⠑⠯⡲⡄⠀⠀⠀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⣿⣿⣧⣻⣟⣷⡀⢳⡟⣁⣿⣣⣾⢷⣛⣶⡋⣩⢛⡵⠚⢻⡘⣆⠀⠀⠀⠀⠙⠌⢢⠀⠀
⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⠿⠆⠟⠗⠀⠓⠛⡏⢻⣀⣳⣵⡟⣙⡛⠼⣒⡉⢻⠵⣹⣯⠄⠀⢷⠘⡆⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⢀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣿⡿⣿⠋⠀⠈⠂⠐⡊⠍⣷⣿⣔⠀⠙⢿⠿⣿⠆⠀⠀⠀⠀⠀⠀⡇
⠀⢰⠋⠀⠀⠀⠀⠐⠲⢤⡄⠀⠤⠒⠋⠙⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⢧⠸⡄⠀⠀⠀⠀⠸⣯⣿⣿⣟⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⢠⠇⠀⠀⠀⠀⠀⠀⠀⠉⠈⠢⣄⡀⠀⠀⠄⠙⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣸⣀⣷⣤⠀⠀⢀⡤⣔⡿⡇⠻⣀⣤⣾⣰⣅⡀⠀⠀⠀⠀⠀⠀⠸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡆⠘⣧⠀⠀⣨⠾⣾⢣⣧⠀⢃⣿⠃⢸⠟⠁⠀⠀⠀⠀⠀⠀⡇
⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢦⡀⠀⠃⠀⠀⠀⢹⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⣷⡀⢸⡆⢠⢄⣷⡇⡏⢲⡀⣸⡇⣠⡟⠀⠈⠁⠀⠁⠀⢀⡜⠀
⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠈⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡾⡟⣽⣳⠿⠿⠿⠷⠶⢿⣿⣿⣿⢉⣔⢂⣠⣖⣦⣤⡴⢫⠀⠀
⠀⠀⠑⢤⣀⠀⠀⠀⠀⠀⢠⣶⡏⠙⣆⠀⠀⠀⠀⠀⢸⣧⣤⣄⣀⣀⣴⣾⣟⣫⢭⡽⣿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⡟⠚⢛⡟⠛⡿⠛⠁⠀⠀⠇⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠉⠉⡉⠒⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⢀⣴⠦⠄⠀⠀⠀⠈⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣴⣶⣾⠛⠛⢛⡶⢤⢤⣀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠁⠈⣿⣶⣶⣃⡎⠉⠙⣝⣿⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣶⣾⣿⠃⠉⣷⣷⣤⣾⡿⠋⢹⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢀⡴⠳⠦⣴⡾⢢⡀⡀⠀⠀⡮⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠄⣼⠒⣀⣤⣟⡑⢒⢷⠸⠀⢸⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣇⠈⣇⠘⠧⠤⢤⡖⠀⢹⢆⠴⠃⡘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠚⠁⠁⠘⣆⠈⢳⠈⠀⠀⠈⠚⠉⠀⢀⡔⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠛⠿⣟⠓⢤⣀⠀⠀⠙⢦⣄⣀⣀⣀⠀⢀⣠⡔⡋⠀⠀⣷⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⣈⠷⣄⠙⣕⢦⡀⠀⠈⢓⣒⣚⣹⣭⣤⡖⠁⠀⢰⠉⢿⡌⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⣀⣀⣀⡠⠼⠓⠒⢚⢧⡈⢫⡉⢙⠖⣒⢤⣄⣀⣢⢸⡇⢠⢀⣧⠤⡼⣿⣶⡼⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⡠⠖⠊⠉⠁⢸⠂⡠⠀⠀⠀⠀⠀⠀⠹⣆⠑⣄⡞⠁⠾⡂⡐⣯⡙⠳⣾⠿⣧⣽⣇⡌⣻⡳⡈⠮⠻⢕⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⢷⡈⢧⠀⠊⣧⣟⡒⢽⡃⡈⢻⣿⣇⠻⡏⣝⢷⡙⡜⢷⣤⠙⢎⠓⡄⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠔⡀⠀⢻⣌⠱⣞⢿⢻⣻⠡⣧⣷⠀⣿⣻⡋⠹⣿⠽⢿⡝⡏⡏⣭⣶⣤⣜⡄⠀⠀⠀⠀⠀ ⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣴⢁⣴⣮⣿⣦⠈⢯⡿⢻⡕⣦⢯⣶⡿⣇⠯⢶⣂⠔⡩⠻⡜⣗⠓⠁⠉⠑⠯⡲⡄⠀⠀⠀ ⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⣿⣿⣧⣻⣟⣷⡀⢳⡟⣁⣿⣣⣾⢷⣛⣶⡋⣩⢛⡵⠚⢻⡘⣆⠀⠀⠀⠀⠙⠌⢢⠀⠀ ⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⠿⠆⠟⠗⠀⠓⠛⡏⢻⣀⣳⣵⡟⣙⡛⠼⣒⡉⢻⠵⣹⣯⠄⠀⢷⠘⡆⠀⠀⠀⠀⠀⠀⢱⡀ ⠀⠀⢀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣿⡿⣿⠋⠀⠈⠂⠐⡊⠍⣷⣿⣔⠀⠙⢿⠿⣿⠆⠀⠀⠀⠀⠀⠀⡇ ⠀⢰⠋⠀⠀⠀⠀⠐⠲⢤⡄⠀⠤⠒⠋⠙⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⢧⠸⡄⠀⠀⠀⠀⠸⣯⣿⣿⣟⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸ ⢠⠇⠀⠀⠀⠀⠀⠀⠀⠉⠈⠢⣄⡀⠀⠀⠄⠙⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣸⣀⣷⣤⠀⠀⢀⡤⣔⡿⡇⠻⣀⣤⣾⣰⣅⡀⠀⠀⠀⠀⠀⠀⠸ ⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡆⠘⣧⠀⠀⣨⠾⣾⢣⣧⠀⢃⣿⠃⢸⠟⠁⠀⠀⠀⠀⠀⠀⡇ ⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢦⡀⠀⠃⠀⠀⠀⢹⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⣷⡀⢸⡆⢠⢄⣷⡇⡏⢲⡀⣸⡇⣠⡟⠀⠈⠁⠀⠁⠀⢀⡜⠀ ⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠈⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡾⡟⣽⣳⠿⠿⠿⠷⠶⢿⣿⣿⣿⢉⣔⢂⣠⣖⣦⣤⡴⢫⠀⠀ ⠀⠀⠑⢤⣀⠀⠀⠀⠀⠀⢠⣶⡏⠙⣆⠀⠀⠀⠀⠀⢸⣧⣤⣄⣀⣀⣴⣾⣟⣫⢭⡽⣿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⡟⠚⢛⡟⠛⡿⠛⠁⠀⠀⠇⠀