⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⡽⣷⡄⠀⠀⣀⣀⣤⠤⠤⠤⠤⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣟⣳⢿⣏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠈⢿⣯⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢼⠃⠀⠀⠀⠘⢄⠀⠀⠀⠘⡄⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣀⣀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⡔⠀⠀⠀⠀⡄⢸⢳⡀⠀⠀⠀⠈⢦⡀⠀⠀⢣⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠺⡗⣷⣤⣀⣀⠎⠀⡄⠀⠀⠀⡇⠀⠀⠀⠀⡇⢸⠀⠑⢄⠀⠀⠀⠀⠱⡄⠀⠸⡆⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀
⠀⢳⣻⣽⣻⡏⠀⢀⡇⠀⠀⠀⡇⠀⠀⠀⠀⡇⢸⠄⠀⠈⠳⣄⠀⠀⠀⠈⢆⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀
⠀⠀⠹⢽⣿⠁⠀⣸⠁⠀⠀⢠⡇⠀⠀⠀⢰⢧⠈⡇⣀⣠⡤⠬⠦⡀⠀⠀⠈⢦⢻⠀⠀⠀⠀⡇⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣿⠀⠀⣿⡆⠀⠀⢸⣿⡄⠀⠀⢸⣼⠀⡟⠉⠀⠀⠀⠀⠈⢢⠀⠀⠀⢻⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀
⠀⠀⠀⣾⡏⢐⠀⡇⡇⠀⠀⢸⣿⢱⡄⠀⢸⠋⡇⡇⠀⣀⢤⣖⣦⣤⡤⠵⣄⠀⠘⡆⠀⠀⠀⢸⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀
⠀⠀⠀⣿⣇⠀⠀⡇⢳⠀⠀⢸⡟⡿⣷⡀⠘⣇⢹⡘⣫⣿⣿⡿⣻⢿⣷⣴⡟⠑⢄⡇⠀⠀⠀⢸⠀⠀⠀⠀⢸⠖⠊⠉⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀
⠀⠀⠀⢹⢻⠀⠀⡇⠸⡀⠀⠸⡇⢱⣇⢳⡀⠇⢠⣾⡟⠁⠘⢛⣷⣾⠿⠋⠀⠀⠀⢳⠀⠀⠀⠈⡀⠀⠀⠀⢸⡤⠖⠊⠉⠢⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀
⠀⠀⠀⣸⠘⡆⠀⣷⠆⢧⠀⠀⣇⠀⠻⣆⠈⠀⢺⡏⠀⠀⢀⣿⠟⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⢸⠻⣆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀
⠀⠀⠀⣿⠀⣷⡀⠸⡙⠹⣆⡀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⠀⠀⠀⡗⠀⠀⠀⠀⠀⡀⠀⠀⠀⣇⠀⠀⠀
⠀⠀⠀⡿⡆⣇⠁⣠⣷⣶⣶⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡴⠋⠀⢀⠾⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⢹⠀⠀⠀
⠀⠀⠀⡇⢹⣹⣶⡿⢻⡿⣿⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡿⠀⢀⡴⡃⠀⡇⠀⠀⠀⠀⠀⢿⠀⠀⠀⢸⠀⠀⠀
⠀⠀⠀⡇⢀⠈⡟⣷⡀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢲⠋⠀⡇⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀
⠀⠀⠀⣣⠀⠐⡀⣟⠙⠟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⣇⠈⡇⠀⡇⠀⠀⠀⠀⠀⠈⡇⠀⠂⢠⠀⠀⠀
⠀⠀⠀⢰⠀⠐⠠⢹⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⡆⠁⠀⡇⠀⠀⠀⠀⠀⠀⣇⠀⢀⠨⠀⠀⠀
⠀⠀⠀⠸⠀⢈⠂⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢿⡀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠸⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⠘⠸⡇⠀⠀⠀⠀⠀⠀⢀⡠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⣝⠀⣧⠀⠀⠀⠀⠀⠀⠸⡄⠀⠆⠀⠀⠀
⠀⠀⠀⠀⡄⠂⡁⠎⢻⡀⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⡇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢷⡀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀
⠀⠀⠀⠀⠀⠐⡡⢌⠠⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⡇⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠸⡆⠘⣆⠀⠀⠀⠀⠀⠀⢰⠀⠘⠀⠀⠀
⠀⠀⠀⠀⢠⠡⡐⠌⠤⢁⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠨⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢷⠀⠙⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆⠀⠀
⠀⠀⠀⠀⠀⠰⢈⠜⠠⢃⡐⢈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢸⣳⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⠡⠌⡑⢂⠤⢱⡎⣏⠳⡀⠀⠀⠀⠀⠀⣠⠔⠋⠀⣀⠀⠀⠀⠀⠀⢀⣀⣿⠀⠀⠀⠀⠀⡞⢺⡀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀
⠀⠀⠀⠀⠀⠠⠑⡌⠰⡁⢢⢸⡷⣇⢂⠙⢦⣀⣠⣶⠯⠷⠖⣺⡏⠁⠀⠀⣠⠴⠚⠉⠀⢸⠀⠀⠀⠀⠀⡇⠈⣇⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀
⠀⠀⠀⠀⠀⢀⠣⡈⢅⠂⡅⢺⣿⡇⠢⣑⡂⠰⡀⢄⠒⡠⠃⠤⢻⡄⢠⠞⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠐⠸⣄⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⡄
⠀⠀⠀⠀⠀⠀⢆⡑⣈⠒⡨⣼⣿⣏⠡⡘⡧⠑⡰⢈⠤⠑⣈⢲⡟⣷⠏⠀⠀⠀⠀⢀⡠⢾⠀⠀⠀⠀⢸⠀⠀⠀⠈⣑⠦⠤⠤⢤⠽⠤⠤⡤⠤⣄⡤⢄⣤⢤⣷⣻
⠀⠀⠀⠀⠀⠈⠤⡐⠄⢣⠰⣿⡏⢻⢄⠡⣧⠃⡔⡁⢢⠑⡄⢺⣇⡏⠀⠀⠀⣠⠖⠉⠀⢸⠀⠀⠀⠀⢸⠀⠀⡰⠊⠀⠐⠓⠈⣁⠤⣴⠭⠒⠉⣠⢴⣛⢮⣛⢶⣣
⠀⠀⠀⠀⠀⠈⡐⢌⡘⠄⢳⣿⡏⢸⡎⡐⢿⣁⠒⢨⠐⠢⠌⢒⣿⣿⡀⡴⠊⠀⠀⠀⠀⢨⠄⠀⠀⠀⡇⠀⠀⠑⠤⢄⣀⣀⡎⠀⡼⠀⠀⢀⣴⢯⡳⣝⡮⣝⢮⣳
⠀⠀⠀⠀⠀⠀⢌⠰⡈⢌⢻⡿⡇⠀⣷⠈⠼⣇⠸⣀⠣⠑⡌⣺⡿⠟⣳⡀⠀⠀⠀⠀⠀⢸⠄⠀⠀⢠⠇⠀⠀⠀⠀⣼⣿⣿⣡⣾⣿⠆⢠⡾⣝⢮⡳⣝⢾⡹⣎⢷
⠀⠀⠀⠀⠀⠀⡘⠄⢣⠘⣼⣧⠀⠀⢸⡣⠄⣿⠀⡄⠸⢠⣸⡿⢃⠤⠀⣧⠀⠀⠀⠀⠀⣸⠇⠀⠀⣸⠀⠀⠀⠀⣼⣿⣿⠃⣿⣿⡟⢀⣿⢣⡟⣼⣣⢟⡼⣣⢟⡼
⠀⠀⠀⠀⠀⠀⠤⢉⠤⠹⣏⡇⠀⠀⠈⢧⡘⢹⣌⣤⠗⣭⣿⡇⠀⡦⠛⠁⣀⠤⠒⠊⠉⢹⠀⠀⠀⡇⠀⠀⢠⣾⣿⣟⡗⣤⣿⡟⢠⣟⡼⣣⢟⡶⣹⢮⣳⡝⣮⢽
⠀⠀⠀⠀⠀⠀⢢⠁⢎⣹⢿⠁⠀⠀⠀⠈⣧⡼⠋⢠⠞⠁⡼⢃⢹⡁⠀⠁⠀⠀⠀⠀⠀⡏⠀⠀⢰⠁⠀⣰⣿⣿⣟⣿⢰⡿⡟⢀⡿⣎⢷⣹⢮⡳⣏⢾⣱⢻⡜⣯
⠀⠀⠀⠀⠀⢀⠢⢉⢲⡿⡏⠀⢀⣠⠔⠊⠉⢀⠴⠁⢀⡞⢡⠊⢽⡇⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⡏⢠⠞⠈⠛⠛⠉⣯⣿⣠⢁⣾⢻⡜⣯⢞⣧⢻⡜⣯⢞⣧⢻⡼
⠀⠀⠀⠀⢰⢠⠘⢌⢻⢿⣤⣖⠗⠁⠀⣀⠴⠋⠀⢠⢎⠰⡁⠎⡌⢧⠀⠀⠀⠀⠀⠀⡜⠀⠀⢠⡗⠉⠣⣀⠤⠔⢺⢱⢇⡆⣼⣏⢷⣹⢎⡿⣜⢧⡟⣮⢻⡜⣧⢻
⠀⠀⣠⣾⣿⠐⢡⠊⡟⠁⠀⠀⣠⣴⠊⠁⠀⠀⣠⢋⢂⠱⣈⠒⡌⠼⣆⣀⣤⠴⣶⢺⠃⠀⠀⣾⣧⠀⠀⠀⠀⣠⡼⣺⡟⢰⡿⣜⣧⣛⢾⣿⢞⡽⣺⠵⣏⡾⣭⢻
⢀⣼⣿⢣⡿⢈⠆⣸⠃⣠⣴⣿⣿⡇⠀⠀⠀⡴⢃⠌⠠⡑⢄⣣⣼⡾⣟⣳⢭⣛⢶⡟⠀⠀⣸⢋⣼⣧⡴⡶⣟⣻⡇⣿⣇⣿⣿⣞⢶⣫⢯⣟⡮⣗⢯⣻⣜⣳⣭⢻
⣾⣞⢮⣻⡇⡆⣘⣿⣾⣿⣿⢟⣻⠀⣀⣀⣼⣁⠞⢀⣦⣵⣾⢻⢧⠿⣜⣧⢟⣼⡿⣠⠀⢠⣧⣾⡟⣧⡻⣵⢫⣾⣙⣯⣿⣿⣞⡿⣷⣯⣿⡽⣺⡝⣮⢗⡮⣗⢾⣹
⣿⣭⢳⡿⠀⠇⣹⣿⣿⢟⠕⣩⣞⢯⣽⢺⡝⡿⣞⡿⣳⣝⢮⡟⣾⣽⣿⣿⣫⡾⢷⠃⠀⣿⣿⢻⡼⣳⡝⣞⢧⣿⢸⡷⣿⣿⣯⢿⣽⣻⢷⣫⢷⡹⢧⢯⣳⠽