⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠈⣾⣿⣿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡞⠐⢷⡞⣰⠟⠁⢀⣾⣿⣿⣿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠩⣩⠎⠀⣹⢪⢤⣠⣯⠞⢷⣄⣼⡿⠿⠿⠿⢾⣶⣶⣤⠀⢠⣴⣾⠷⠿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣏⢁⡀⡿⠁⢀⣿⡣⠄⠈⢣⢠⣴⣻⡏⠀⠰⣶⡿⣿⣿⣿⠉⠀⢽⣿⣿⣿⣿⡆⠈⡿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⣿⣿⡀⣸⣇⣨⡟⢦⣴⣀⡀⣰⢻⢷⣷⠀⠀⠀⠀⠉⢉⣠⠏⠀⠀⢯⠳⠀⠀⠀⠀⠀⣧⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⡠⠖⣾⡷⠒⠉⣰⣿⣿⣿⣷⣿⡧⠟⠉⠀⠀⠰⡼⠁⠀⣏⣳⠀⠀⠀⠀⠀⠀⠀⡤⢂⡀⢈⡣⡀⠀⠀⠀⠀⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠤⠎⢈⠠⡘⣞⣳⣯⣽⣿⣿⣿⣿⢿⣿⣄⠀⠀⠀⠀⠳⠄⠠⡼⠵⢇⡀⠀⠀⠀⠀⠀⠳⠿⢶⢿⠟⠃⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠋⠃⠳⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣄⢀⣴⣅⣀⣀⣀⣆⣀⣀⣹⠀⠀⠀⢀⣀⣀⢦⣽⣬⣥⡒⡠⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⠀⡀⠃⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡿⠉⠉⠀⠀⠀⠀⠀⠘⣧⣄⣀⠰⠎⠿⢿⢿⣙⣿⡿⠛⠃⢀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠠⣴⢀⢁⣂⢠⣿⣽⣿⣿⡿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢿⣿⣿⣷⣶⣶⣿⣯⣷⡶⠶⠿⠿⠾⠤⠤⠄⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠘⡿⠿⣾⣿⣿⣽⠿⠻⣶⠶⣖⠒⠶⣪⠟⢿⡲⡤⢄⡠⠒⢿⠏⠓⣷⣿⣿⣿⣿⣿⣿⣿⡿⣃⠀⢰⠇⠀⣆⣶⣟⠄⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢄⣴⣡⡔⣮⣿⣾⡽⣾⣽⣍⣇⢎⣅⣶⡾⣷⡄⢱⣬⣛⡯⣗⠫⠠⢄⠙⡻⠿⢿⣿⢿⣿⣿⣄⣏⡄⢸⣄⣐⢿⣿⡃⠀⠀⠘⣶⣶⡦⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡤⣴⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⣻⣿⣯⡭⠲⢊⡒⠹⣷⠆⣈⠑⠀⠈⠁⠠⣾⣶⣶⣿⡇⣿⣧⡫⢷⢶⣌⣻⣿⣯⠇⠀⠀⣴⣷⡽⣌⠉⠉⠉⠈⢿⣶⣦⡀⠀⠀⠀⠀⠀⠀
⠈⣿⢿⣿⣿⣟⣞⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠋⠐⠁⠀⠈⠹⣦⡱⠑⠄⠀⢠⢀⣍⣿⡿⣿⣟⣿⣻⣿⣶⣄⠈⠙⠋⠀⡀⢀⣼⣿⣿⣿⣮⠛⢆⣀⣰⣟⣌⣝⠛⡲⣦⠀⠀⠀⠀
⣻⠟⢸⡿⣿⣟⠌⠿⣿⣿⣇⢿⡿⡓⠀⠈⠂⢀⡀⠀⠀⠀⠀⣸⣧⡈⠈⠀⠐⠛⠚⠁⠁⢍⡯⣿⣻⠛⢯⠻⢷⣄⣀⠀⣠⣾⣿⣿⣿⣿⣿⡭⣀⢩⣉⢛⢀⢊⠷⡛⢙⢷⡀⠀⠀
⠃⠀⢸⣯⣿⡟⢀⡨⣲⣿⡿⢾⠆⢐⢀⡀⣈⣀⣌⣅⠀⠰⠤⠤⡹⣿⡄⣀⠀⠀⠁⠀⠀⠠⣯⣏⣿⠂⣬⢜⡹⠓⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠻⠑⠀⠤⠈⠻⡌⠙⠦
⠀⠀⠀⣿⠼⣻⡄⠈⣿⡿⣟⡌⡍⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠿⣷⡀⠀⠀⠀⠀⢀⣸⣸⣗⣯⣿⣿⣮⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡎⠻⡀⠀⠀⠀⠀⠀
⠔⡀⠐⣿⠺⡽⡣⠸⡾⡇⠄⡷⡇⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⡏⠀⠻⣷⡈⠢⠁⠀⢀⣨⢸⢾⣷⣿⣿⢋⡄⠒⠀⢈⣿⠿⣷⣾⣱⣽⣿⣿⣿⠋⠽⠂⡿⢹⠳⠈⠘⢢⠄⠀⠀
⡻⡮⢨⣿⢒⣷⣿⢸⠷⠂⠀⠈⠹⢿⡻⠍⠉⠛⠉⠨⠊⠀⠀⠀⡀⠀⠀⠹⣇⡀⠀⠀⠩⣿⣼⣿⡿⣿⣟⠈⠁⠀⠀⣸⡏⠀⣠⣿⣍⢵⣿⠟⣥⣁⠤⠀⢨⢃⠀⠀⠰⠁⠀⠀⠀
⣿⣿⣾⣿⣿⢯⢯⡿⠆⠀⠀⠐⠀⠀⡁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢯⡀⠠⢤⡧⡿⣾⣽⣿⢦⡀⠀⠀⠨⠝⢀⣼⣿⣿⡿⠟⠁⢤⢷⠷⠋⠇⠐⠄⠀⡄⠀⠀⠀⠀⠀
⠛⠿⢿⣿⣿⣻⡟⠁⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠂⠀⢀⡈⠀⢈⡥⣏⡿⢸⠟⠃⠀⠀⠀⠀⣀⣺⣿⣿⠿⠑⠂⢙⣍⠦⢓⡤⠂⠈⠀⠀⠀⠀⠀⠀⣀⠀