⣠⠴⠒⢛⣋⣉⣉⣉⣉⣛⠓⠦⣤⣀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⠴⢚⡭⠖⠚⠉⠉⠁⠀⠀⠀⠀⠉⠉⠓⠲⢭⡓⢦⡀⠀⠀⠀
⠀⠀⠀⠀⣠⢞⡥⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡝⢦⠀⠀
⠀⠀⠀⣰⣫⠞⡤⡀⠀⢀⠀⠀⠀⠀⡀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⣎⢧⠀
⠀⠀⢰⣷⠇⠈⠛⣵⠄⠀⢁⡆⠀⢠⠇⠸⢼⡀⠾⠻⠢⡄⠀⠀⠀⠀⠀⠘⡾⡆
⠀⠀⡏⡼⠀⠀⣼⠏⠀⢠⣾⠀⢀⡞⣳⡀⠈⣧⠀⠀⠀⢹⡄⢀⡶⣛⣓⣦⢹⣷
⠀⢸⠁⡇⠀⣸⠁⠀⣠⢯⡇⢀⣾⣀⠀⢳⡄⢻⡄⠀⠀⠈⣿⣸⣸⠁⠀⢸⠈⢹
⠀⢸⠀⡇⢠⠇⢀⡜⢁⣟⡴⠋⠀⠉⠉⠓⠻⡜⡇⠀⢀⣖⡿⠿⠋⠀⠀⠋⠉⢻
⠀⢿⠀⣇⢸⡞⠁⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠸⣟⠁⠀⠀⡟⢦⠀⠀⣸
⠀⢸⠀⠹⡾⡯⠿⠦⡄⠀⠀⠀⢀⣤⣦⣄⡀⠀⣧⠀⠀⠈⡷⠆⠀⣇⠸⡆⢾⡷
⠀⢸⠀⠀⡇⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⣿⠀⠀⠸⣇⠀⢀⣿⠀⠹⣾⠃
⠀⢸⠀⠀⡇⣿⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣿⠉⢸⠀⠀⢙⠀
⠀⠈⡇⠀⢧⢻⡀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⣿⠀⠀⢸⠉⢸⠀⠈⡇⠀⠀⢧
⠀⠀⢹⠀⢸⠈⢷⣄⠀⠈⠃⠘⠉⠁⠀⠀⠀⣠⣿⠀⠀⡏⠀⠘⡀⠀⢿⠀⠀⠈
⠀⠀⠀⢇⠀⢇⠀⠙⠳⣤⡀⠀⢀⣠⡤⢖⣋⡥⡇⠀⣸⠀⠀⠀⣇⠀⢹⡇⠀⠀
⠀⠀⠀⠈⢇⠘⣄⠀⠀⠀⠉⢩⢿⣷⣿⡉⠀⢸⠇⢸⠟⢳⠤⠤⠼⣄⡘⣿⡀⠀
⠀⠀⠀⠀⠈⣧⡙⣴⠋⠉⠉⣻⣾⣇⣿⣧⠀⣼⣰⠃⣰⠋⢀⡤⠖⠂⠈⢳⣧⠀
⠀⠀⠀⢠⡞⣫⠽⢿⣇⡴⡏⢻⣿⣿⣟⣟⣿⡷⠃⣰⠃⡴⠋⠀⠀⠀⠀⠀⣿⢧
⠀⠀⠀⢸⡇⠐⠚⠡⣿⢧⠁⠀⠛⢻⠄⡏⠞⠀⢰⢇⡞⠁⠀⠀⢀⠤⡀⠀⣿⠘
⠀⠀⠀⣸⡇⠀⠀⢸⡏⠈⠀⠀⠀⢸⣶⠛⢦⠴⣻⡞⣀⣀⣰⠮⠶⠚⠁⠀⣿⡄
⠀⢀⡏⡿⢧⡀⠀⠀⣧⠀⠀⢀⣠⠞⣿⣤⠞⢰⠋⠉⠁⠀⠀⠀⢀⣤⠀⢸⡏⢧
⠀⢸⠁⠙⢦⣙⠦⠼⣿⡖⠚⣉⣠⠖⠁⠸⡄⡿⠀⠀⠀⠀⠠⠚⠉⠀⠀⡞⡇⢸
⠀⡟⠀⠀⠀⠎⡍⠉⢹⢿⡏⠁⠀⠀⠀⠀⢹⡅⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⣷⠀
⡼⠃⠀⠀⠀⠆⡇⢀⠟⡆⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⣸⠁⢸⢻⠀
⠁⠀⠀⠀⠀⠀⠁⡾⠀⢹⡘⠃⠀⠀⠀⠀⠀⠈⢣⢤⡀⠀⠀⠀⢠⠇⠀⢸⢸⡆
⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⢧⠙⠆⠀⢠⠏⠀⠀⡟⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⠏⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡤⠀⠀⠀⢀⣠⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣯⢰⣶⣶⣴⢬⣋⣽⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣾⣿⣿⣿⣿⣿⣿⣿⣧⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⣿⠙⠛⢿⣿⣿⠿⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⢿⡟⠀⠠⠀⠀⣼⠻⣧⠀⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣇⡈⢿⠀⣤⡤⣾⣿⠀⠹⢷⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⢣⢹⣿⣟⠗⠊⠺⣿⣿⠀⠀⠀⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠁⠀⠀⠟⠀⠠⠦⠀⠀⢻⡅⠀⠀⠘⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⡀⡀⡀⠀⣀⣠⣄⠀⢘⣃⣀⣀⣼⠇⠀⠀⠀⠈⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⠉⠉⠉⠋⠁⠀⠈⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠻⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠁⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠛⠿⠏⢦⣀⣀⣀⣀⣀⣀⡈⠻⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠈⣿⣿⣿⣿⣿⣿⣿⣄⠈⠙⠿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⣠⣾⣇⠘⠄⠀⢀⡀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠈⠛⢿⣿⣿⣿⣦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡅⠀⠀⠀⠀⣰⣿⣿⣿⣿⣶⣶⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⡿⠿⠃⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⠀⠀⠀⠀⠘⠛⣿⡟⠛⠙⠛⠙⠛⢻⣿⣿⣿⣿⣿⡿⠛⠁⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠉⠀⠀⠀⠀⠈⢻⣆⠀⠀⣴⣶⠶⡄⠀⠀⢸⣿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡄⠀⠘⠛⣿⣿⡆⢀⣿⣿⣿⣧⣦⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠁⠀⠁⠀⣠⣾⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣤⣾⣿⠟⠀⢸⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠲⣦⣤⡾⠃⠀⠀⠻⣿⣿⣿⣿⡿⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠏⠉⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀