⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠃⠀⠀⠀⢠⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⡼⡼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣣⡿⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣼⢿⣽⣳⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣰⣟⣿⣺⣽⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⢤⡾⣽⢮⣷⢻⡶⣯⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠙⢦⡄⠀⠀⠀⠀⣀⣠⣴⣿⣾⢯⡿⣽⣿⣾⣯⡿⣽⢯⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣤⣀⠀⠀⠀⣘⣷⣴⣶⣻⢿⡽⣞⡷⣯⢿⡽⣧⣿⠟⠋⠉⠉⠙⠺⣽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⢶⣾⣿⣹⢷⣯⢷⣻⡿⣽⢯⡽⣯⢿⡽⠁⠀⠀⠀⠀⠀⠀⢾⣭⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢟⣿⣿⣯⠿⣿⣿⢯⣿⣾⣟⣷⣻⡽⣯⢗⣿⣾⠁⠀⠀⠀⠀⠀⠀⣀⣾⢯⡷⠛⢤⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣶⣽⣿⣯⣽⣞⣷⣿⣞⣷⣻⣽⣻⢾⣿⢀⠀⠀⠀⠀⢀⣠⡿⣽⣫⠐⠁⠂⢃⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡛⠙⠉⠉⠿⢟⢿⣿⣮⣿⣿⣞⣷⣻⣞⠿⣞⠤⢙⣶⢿⡿⣿⣻⡽⣧⢷⣦⠶⠃⠠⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⡇⠀⠀⠀⠀⠀⠙⣻⣿⣿⣿⣿⣷⣻⣞⣿⣃⠀⠈⡸⣯⣟⣳⢯⡷⣯⠟⢮⡶⣙⠦⠑⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡆⠀⠀⠀⠀⣀⣦⣿⣻⣟⣿⣿⣿⣿⣻⢾⡽⢂⠞⢹⡷⣯⣟⡯⢷⣯⢟⣋⣀⠠⢀⠀⠡⣂⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣗⣲⣶⣶⣞⣽⣿⣳⣟⣾⣻⣿⢷⣫⠗⡋⠁⠀⠀⣤⣿⣷⡽⣞⡾⢉⣋⣼⠶⠞⠂⣈⠩⡵⣎⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣯⢿⣽⣻⢿⣿⣟⣷⣿⣾⠿⣿⠯⢷⡈⣤⠀⠀⣠⣿⢿⣳⣭⣴⠖⠋⠙⣀⣈⠀⡀⢀⠐⡰⣉⠆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢞⡿⣾⡽⣯⣿⣿⣿⣿⣿⣿⣷⣜⡯⣟⡆⢻⣰⢿⡽⢯⡷⣏⠶⠄⠂⠈⠀⠀⠀⠁⠉⠘⣱⢫⡛⠄⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢻⢻⡞⣯⣿⡜⢣⡌⠀⠁⠀⢢⠁⠀⠂⢱⡎⣵⠘⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣯⠀⠀⠉⠉⠁⠈⠉⠈⠀⠀⢠⠱⡈⠄⠠⢈⡒⢜⡰⢋⡲⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣤⢤⣲⣄⣀⢀⡀⣀⠀⢄⡘⢦⢣⠁⠆⣁⠰⡸⣤⢩⡟⣷⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠤⠒⠈⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡽⣯⣟⣷⣻⢮⣷⣻⢦⣋⠶⡙⣎⠲⣉⠒⡄⣹⠱⣌⣷⣻⣽⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣽⣻⢾⡽⣯⣳⢭⡱⣌⠳⣤⢋⠠⢓⠻⣽⣞⣷⣟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣟⣯⣟⡷⣯⢷⣻⢾⣙⡦⢧⣞⣷⣻⣷⣻⣯⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣻⡽⣯⣟⣯⢯⣟⡾⣽⢾⣟⣿⣟⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣷⣻⣞⢷⣻⣳⢯⡾⣽⢾⡽⣯⢿⣽⣻⢾⣟⣾⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣟⣿⣻⢿⡿⣿⢯⡷⣯⢷⣻⡽⣞⣳⢯⣧⢿⣽⣫⢷⣯⢿⡽⣯⣟⡾⣽⣻⢾⣯⢿