⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⢸⣿⡀⡀⡀⡀⡀⡀⣿⢨⡒⣧⣼⣿⡇⣿⡇⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣿⡇⡀⡀⡀⡀⡀⡀⣿⡎⡏⣿⣿⣿⡇⢻⣧⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣿⣇⣀⡀⡀⡀⡀⡀⢿⡇⣇⢿⣿⣿⣇⠘⣿⣦⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⣠⣤⣤⣀⣀⡀⡀⡀⣤⣿⠋⠉⠉⡀⡀⡀⡀⡀⠘⢿⣮⣻⣼⡿⠃⡀⡀⠻⣷⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⠹⣧⠈⣍⣉⠙⠛⠛⠻⠿⠛⠻⠟⠛⣿⣿⠿⠿⣿⣿⠿⠿⠿⠿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⠇⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⠻⣧⠢⠈⢹⠻⠿⣶⠟⠛⠻⣄⣿⢋⡾⣭⣝⢷⡙⣿⡀⡀⡀⡀⠈⣿⣏⣶⣭⢶⠻⣿⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⠻⣦⡀⢀⡷⣿⡏⡀⡀⡀⣿⣿⢸⢸⣿⣿⡧⣷⣿⡿⠿⣷⡀⡀⠘⣿⢹⣿⡇⡇⣿⡇⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⠹⣷⡀⣷⡿⡀⡀⡀⡀⢹⣿⠈⢦⣛⣛⣴⠃⣿⠃⡀⠘⡀⡀⡀⣿⣘⣋⡾⢡⣿⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⠙⣷⣿⡀⡀⣀⣤⣤⣤⣿⣿⣤⣌⣥⣴⣿⣿⣾⣶⣶⣶⣶⣶⣿⣤⣤⣶⣟⠛⠿⣶⣄⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⣿⠇⢀⣾⠟⠉⡀⡀⡀⡀⠈⠉⠉⠁⡀⡀⡀⡀⡀⡀⡀⡀⠈⠉⠉⡀⠙⣿⡄⡀⠙⢿⣦⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⣀⣿⡀⣿⠏⡀⡀⡀⡀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣀⡀⠘⣿⡀⡀⡀⠈⢿⣦⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⣴⡿⣿⡇⢰⣿⡀⡀⢀⣴⣿⠿⠋⠉⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠉⠉⠻⣿⡀⣿⡀⡀⡀⡀⡀⠙⣿⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⣿⠏⠄⣿⠆⣾⡏⢀⣾⠿⠉⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⢻⣧⣿⡀⡀⡀⡀⡀⡀⠈⣿⣄⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⣼⡿⡰⠆⣿⡀⢿⣷⡿⠁⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠈⢿⠟⡀⡀⡀⡀⡀⡀⡀⠈⣿⡄⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⣿⠃⠕⠂⣿⡇⡀⠁⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣰⣾⣿⣿⡿⠿⠿⣿⡀⡀⡀
⡀⡀⣿⣠⣾⢿⡻⣿⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣴⣿⣤⣤⣤⣶⣶⣤⣿⣦⣄⡀
⡀⡀⣿⡫⣾⣿⣧⣿⣦⡀⡀⡀⡀⣠⣤⣤⣶⣶⣶⣶⣶⣶⣿⣿⣿⠿⣿⣿⠿⠿⠿⠛⠛⠛⡛⢛⡋⢉⢉⡉⢉⡉⠩⢁⡈⠠⢄⡠⢄⡠⢄⡀⣿⡀
⡀⡀⣿⡇⡿⣝⣷⡿⣿⣆⡀⡀⡀⣿⣏⣀⣀⣀⣀⣀⣤⣤⣤⣤⣿⡆⡄⠹⣿⠈⡑⠪⢕⡪⢕⡪⢕⡪⢕⡪⢜⡱⢪⢕⡪⢜⡱⢢⢌⡑⠪⠅⣿⡆
⡀⡀⢹⣷⣾⣿⢁⡪⠈⣿⣦⡀⡀⣿⡏⠩⠍⠩⠍⠩⠍⠩⠌⠅⣿⡇⡇⡀⣿⡀⡣⢎⡕⡢⢌⡑⠪⢕⡪⢕⡪⢕⡪⢕⡪⢜⡱⢪⢕⡪⢜⡑⣿⡇
⡀⡀⠈⣿⣿⢀⡱⢎⡱⠈⣿⣿⡄⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡇⣡⣾⠟⠔⠂⠨⠑⠊⠑⠃⠊⠑⡂⣈⣁⣈⣁⣈⣁⣈⠑⡪⢕⡪⢜⡑⣿⡇
⡀⡀⡀⣿⠃⡣⢜⡣⢜⡣⠌⢿⣿⣿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣛⣛⣛⡛⠛⠛⠛⣿⣁⣥⣤⣤⣶⣶⣾⡇
⡀⡀⡀⣿⡀⢕⡪⢕⡪⢡⣿⠟⠛⠛⠛⠻⠟⠿⠿⣿⡄⡈⣿⡀⢪⢕⡪⢕⡪⢕⡣⢎⡕⡪⢕⠊⣾⠿⠛⢛⡛⣿⡇⠃⢰⣿⠛⠋⠉⠉⡀⡀⡀⡀
⡀⡀⡀⠉⢿⣦⡁⢜⡃⣿⡇⡎⢱⡎⢱⢰⣿⣿⣿⡿⠃⡣⢸⣿⣌⣁⣈⣁⣈⣥⣤⣤⣤⣴⣶⣶⣿⠈⢱⡎⢱⢸⣿⠿⠿⠃⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⠈⢿⣷⡀⣿⡄⡕⢪⡕⢪⡀⣿⠠⢔⡪⠕⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣤⣁⣤⣤⣾⡟⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⠈⠻⣿⣧⣁⣈⣁⣨⣴⣿⣤⣤⣥⣤⣿⠁⢐⡰⢆⡠⢄⡠⢄⡠⡀⣿⢸⣽⣽⡇⠈⠉⠉⠉⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣿⢹⣛⣛⣹⣭⣭⣭⣿⣯⢿⣀⠱⢎⡱⢎⡱⢎⡱⢎⡱⡀⣿⡿⡁⣿⡇⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣿⣾⣿⣿⣶⣭⣿⣿⣹⣿⣤⡉⠻⣶⣌⠃⢸⡇⢸⡇⠘⣠⣿⢐⠂⣿⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣿⡀⢸⡆⢀⢻⣿⢿⣿⡇⠛⢿⣿⣦⣍⠻⣷⣤⣈⣴⣿⣿⢁⡣⢰⣿⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣿⡀⢸⡇⢸⡂⣿⢘⣿⠁⡀⡀⡀⠉⠿⣿⣷⣿⣿⣿⡾⠿⡸⢇⣼⡿⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⢀⣿⠐⢪⡕⠪⣰⣿⣸⣿⡀⡀⡀⡀⡀⡀⡀⠈⢻⣷⢢⡕⢪⡕⠂⣿⠇⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⢸⣿⢐⡪⢕⡪⢍⡛⣻⡟⡀⡀⡀⡀⡀⡀⡀⡀⠘⣿⠐⠪⠕⡪⢄⠻⣿⣶⣀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⣠⣿⠃⢎⡱⠎⢑⣈⣤⣿⠃⡀⡀⡀⡀⡀⡀⡀⡀⡀⠿⢿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣦⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⣴⣿⣥⣴⣶⣿⠿⠛⠛⠉⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⠙⠛⠋⠉⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀

⠄⣿⣿⣿⣿⠘⡿⢛⣿⣿⣿⣿⣿⣧⢻⣿⣿⠃⠸⣿⣿⣿⠄⠄⠄⠄⠄
⠄⠄⣿⣿⣿⣿⢀⠼⣛⣛⣭⢭⣟⣛⣛⣛⠿⠿⢆⡠⢿⣿⣿⠄⠄⠄⠄⠄
⠄⠄⠸⣿⣿⢣⢶⣟⣿⣖⣿⣷⣻⣮⡿⣽⣿⣻⣖⣶⣤⣭⡉⠄⠄⠄⠄⠄
⠄⠄⠄⢹⠣⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣽⡧⡄⠄⠄⠄
⠄⠄⠄⠄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⢘⣿⣷⣿⣿⣿⣿⣀⠄⠄
⠄⠄⠄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣿⣿⣿⣿⣿⣿⣿⡆⠄
⠄⠄⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠄⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠣⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠑⣿⣮⣝⣛⠿⠿⣿⣿⣿⣿⠄
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠄⠄⠄⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠄


⠄⠄⢸⣿⣿⡿⠟⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⡇⠄⣶⠒⣦⠄
⠄⠄⠘⣿⣿⠁⠄⣠⠒⡄⢹⣿⣿⣿⣿⣿⣿⣷⡀⠉⠉⣁⣴
⠄⠄⠄⣿⣿⣦⡀⠉⠊⠁⠸⣿⣿⣿⣿⣿⣿⣿⣟⣭⣿⣿⣿
⠄⠄⠄⣿⣿⣿⣿⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠄⠄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠄⠄⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠄⠄⠄⠘⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠄⠄⠄⠄⠈⢿⣿⣿⣄⠄⠉⠻⠿⠿⣛⣛⣫⣭⣿⣯⠴⠄⠄
⠄⠄⠄⠄⠄⠄⠙⠿⣿⣆⠄⠄⠐⣯⣿⣿⣿⣿⣿⠉⡀⠄⠄
⠄⠄⠄⠄⠄⠄⠄⢄⠉⢻⣷⣄⠄⠐⢄⣨⣥⣤⡷⠖⠁⠄⠄
⠄⠄⠄⠄⢠⠄⠄⢸⣷⣤⡀⠈⠂⠄⠄⠄⠄⠄⠄⠄⠄⠄