⡪⡪⡪⡪⡪⡪⡪⡪⡪⡪⡪⡪⡘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⣿⣿⣿⣿⣿⣿⢿⠻⢿⣿⡿⠃⠀⠀⢀⢱⢱⢱⢱⢱⢱⢱⢱⢱⢱⢱⢱⢱⢱⢱⢱⢱⢱⢱⢱
⡪⡪⡪⡪⡪⡪⡪⡪⡪⡪⡪⡪⣊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣟⣿⣿⣿⣿⡍⢶⣖⠮⣻⡿⠁⠀⠀⠀⠀⠅⡇⡇⡇⡇⡇⡇⡇⡇⡇⡇⡇⡇⡇⡇⡇⡇⡇⡇⡇
⡪⡪⡪⡪⡪⡪⡪⡪⡪⡪⡪⡪⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠞⡿⣿⣿⣯⣶⣿⡷⠁⠀⠀⠀⠀⠀⠪⡪⡪⡪⡪⡪⡪⡪⡪⡪⡪⡪⡪⡪⡪⡪⡪⡪⡪⡪
⡪⡪⡪⡪⡪⡪⡪⡪⡪⡪⡪⡂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⠸⠸⠸⢸⢸⢸⢸⢸⢸⢸⢸⢸⢸⢸⢸⢸⢸⢸⢘
⡪⡪⡪⡪⡪⡪⡪⡪⢪⢊⢪⢤⡐⠰⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢀⢀⣠⡂⡄⠀⠀⠀⠀⠀⣊⣯⣿⣿⣿⣿⣦⡣⢣⢣⢱⢱⢱⢑⢕⢕⢕⢕⢕⢕⢕
⡪⡪⡪⡪⡪⡪⡪⡪⡪⣲⢯⣻⣺⣄⠰⠀⠀⠀⠀⠀⠰⠀⠀⠀⠨⡸⡘⡔⡄⡄⣔⣵⣸⣿⣷⡆⠀⠀⠀⠀⢀⣿⣽⣿⣿⣿⣿⣿⣷⡡⡣⡣⡣⡣⡣⡣⡣⡣⡱⡱⡑⡕
⡪⡪⡪⡪⡪⡪⡪⡪⢼⢽⣝⣾⣾⣾⣆⠁⠀⠀⠀⠀⣒⠀⠀⠀⠐⡸⣘⢾⣭⢟⣾⢱⢿⣟⢿⢠⠀⠀⠀⠀⢸⣻⣿⣿⣿⣿⣿⣿⣿⡦⢑⢕⢕⢕⢕⢕⢕⢕⢕⢕⢕⢕
⡪⡪⡪⡪⡪⡪⡪⡊⣯⢗⣷⣿⣿⣿⣷⡆⠀⠀⠀⠀⢮⠀⠀⠀⢘⢶⣄⣾⣿⣷⣿⣜⣥⣦⡷⣜⠀⠀⠀⠀⣸⣺⢽⣿⣿⣿⣿⣿⣿⣿⢨⢪⢪⢪⢪⢪⢪⢪⢪⢪⢪⢪
⡪⡪⡪⡪⡪⡪⡪⡊⣞⡽⣾⣿⣿⣿⣿⡅⠀⠀⠀⠀⢙⠀⠀⠀⢰⣹⢽⢿⣿⣿⣿⣟⢽⡵⡵⠆⠀⠀⠀⢀⡷⡽⣽⣿⣿⠿⠿⠛⠛⠻⢘⢜⢜⢜⢜⢜⠬⣒⢕⢕⢕⢕
⡪⡪⡪⡪⡪⡪⡪⡪⡒⣽⢞⣿⣿⣿⣧⠂⠀⠀⠀⠀⠀⠀⠀⠀⡼⣾⣿⣿⣿⣿⣿⣾⣿⢏⠷⠁⠀⠀⠀⢸⣺⢽⡿⠋⣠⣴⣶⣶⣶⢄⠀⠱⡱⡱⡱⡱⡱⡱⡱⡱⡱⣑
⡪⡪⡪⡪⡪⡪⡪⡸⡸⡘⣯⣻⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣼⡏⠀⠀⠀⠀⠞⠚⢉⣠⣾⣿⢿⢿⢿⡟⡸⡰⠀⢕⢕⢕⢕⢕⢕⢕⢕⢜⢔
⡪⡪⡪⡪⡪⡪⡪⡪⡪⡪⡸⣺⣻⣿⡿⡄⠀⠐⠄⠀⠀⠀⠀⢠⢻⣿⣿⣿⣿⣿⣿⣿⣿⣇⠃⠀⠀⠀⠀⡤⣮⣻⣽⣷⣿⣿⣿⣯⠠⠣⠃⢀⢜⢜⢜⢜⠬⡒⡕⡕⡕⡕
⡪⡪⡪⡪⡪⡪⡪⡪⡪⡊⡆⣟⢾⣿⣿⠇⠀⠀⡇⠀⠀⠀⠀⠘⠝⡽⣟⣿⢽⣟⣟⡽⡝⠇⠀⠀⠀⠀⢀⡯⣗⣿⣿⣿⣿⣿⣿⡇⠀⡠⡪⡪⡪⡪⡪⡪⡪⡪⡪⡪⡪⡪
⡪⡪⡪⡪⡪⡪⡪⡸⡸⡠⡁⢯⣟⣿⣿⣯⠀⠀⠐⡂⠀⠀⠀⠀⠀⠀⠙⠈⠋⠚⠊⠃⠀⠀⠀⠀⠀⠀⢸⢽⢽⣿⣿⣿⣿⣿⣿⠣⠀⡕⡜⡜⡜⡜⡜⡜⡜⡜⡜⡜⡜⡜
⡪⡪⡪⡪⡪⡪⡪⡪⡪⡢⠅⢱⣗⣿⣿⣿⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢽⢽⣿⣿⣿⣿⣿⣿⠀⢐⢕⢱⢱⢱⢱⢱⢱⢱⢱⢱⢱⢱⠱
⡪⡪⡪⡪⡪⡪⡪⡪⡪⡢⡁⠘⣞⣞⢿⣿⣷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣝⣗⣿⣿⣿⣿⣿⡇⠀⢄⢇⡃⡇⡇⡇⡇⡣⡕⡕⡕⢕⢕⢍
⡪⡪⡪⡪⡪⡪⡪⡸⡸⡐⠀⠀⢵⣫⢿⣿⣿⣯⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡺⡮⣿⣿⣿⣿⣿⠃⠀⠀⢆⢇⢕⢕⢕⢕⢕⢕⢕⢪⢱⢱⢱
⡪⡪⡪⡪⡪⡪⡪⡪⡪⡪⠀⠀⠐⣽⡳⣿⣿⣿⣷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⣯⣻⣿⣿⣿⣿⡟⠀⠀⠀⠁⠕⡕⡕⡕⡕⡕⡕⡕⡕⡕⡕⡕
⡪⡪⡪⡪⡪⡪⡪⡪⡪⡪⠀⠀⠀⠵⡯⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣳⣽⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠁⠁⠁⠃⠣⠣⡣⡣⡣
⡪⡪⡪⡪⡪⡪⡪⡸⡨⡪⠀⠀⠀⠘⣽⣺⣿⣿⣿⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡺⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⢇
⡪⡪⡪⡪⡪⡪⡪⡪⡪⡪⠂⠀⠀⠀⣗⣗⣿⣿⡫⣯⠇⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡳⡯⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀