⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣶⣿⣬⣭⣧⠴⣠⠤⠤⣤⣤⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡪⣻⣶⣿⣿⣿⠯⠕⠤⢹⡶⣽⡱⡰⡺⣽⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡱⣻⢟⢿⣿⣿⢿⣷⢜⠴⣕⣤⡳⣮⣻⣮⢪⣿⠿⣮⡻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⢭⡾⣿⣾⣿⣿⠟⠛⠻⣿⣻⡈⣻⡻⣿⣿⣌⣷⣽⣷⣼⣽⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣏⡷⣻⢯⣿⡿⠃⠀⠀⠀⠘⢿⣷⣾⣛⢿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣿⣝⣿⡾⡑⢀⠀⠀⠀⠀⠀⠀⠛⣟⣗⡻⣿⣿⣿⣿⣿⣿⣿⣿⣽⣧⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢿⣻⡟⠀⠀⠀⣠⣴⠖⣛⡉⠁⠂⠘⡟⣷⣟⠿⡿⣿⡿⢻⣿⣿⣿⣿⣧⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢙⣿⡿⣲⣦⠀⠈⠋⠐⣿⣿⢓⡷⠦⠀⠸⠹⡟⣆⢹⡟⡝⠀⢿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣜⣿⣿⣿⠟⢷⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠟⣿⣿⡿⠵⢙⢸⢹⣿⣿⣟⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⢂⠀⠀⠀⢀⠌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣻⣿⠖⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢙⡈⢈⣡⣿⡿⣿⣿⣿⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠠⠀⠆⠀⢠⠃⢰⠀⠀⢀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣹⠀⠀⠄⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡕⡞⠖⣿⣿⣿⣿⣿⣿⡟⣿⣳⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡃⠘⡀⠆⠀⡆⢀⠔⠁⢀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣷⢧⠀⠁⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠰⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀
⠄⡃⢀⣷⡤⣤⡷⡥⣤⡔⢁⠠⠄⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣿⣧⠰⠆⡟⢓⠖⠋⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳⠀⠀⠀⠀⠀⠀
⠀⠸⣿⡟⣷⣿⢷⢀⣿⡿⢗⣶⡔⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⡕⣻⣷⡐⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡄⠀⠀⠀⠀⠀⠀
⠈⠺⣿⡣⢁⣿⠎⣸⡫⣪⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⣕⣿⣷⡄⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣧⠀⠀⠀⠀⠀⠀
⠀⠀⠘⣯⠰⣗⢻⣿⢱⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣿⣿⣮⠿⣿⣿⣿⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⡆⠀⠀⠀⠀
⠀⠀⠀⠘⡦⣟⣄⣽⡌⣘⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡛⡿⢻⣿⣯⣹⣿⣿⠿⢛⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀
⠀⠀⠀⠀⠈⢎⠚⠲⡿⢽⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⢡⣾⡝⡿⣿⡟⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣯⣇⠀⠀⠀⠀
⠀⠀⠀⢀⠔⠸⣦⣥⣠⣸⠃⠼⣿⣡⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⣻⣻⣇⣿⣿⣿⣿⠁⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⣾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡄⠀⠀
⠀⠀⠠⠎⣠⣼⡿⣿⣷⣾⡆⢀⡹⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣰⡿⣽⣿⢿⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣞⣯⢍⣥⣶⣿⣿⠯⠿⠿⢽⣿⣿⣯⣯⣷⠀⠀
⠀⠀⡴⣿⡌⢻⢿⣟⣿⣿⣿⡷⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡶⠋⣰⣿⣯⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⣱⣯⠻⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠉⠻⣿⣿⣼⣇⠀
⠀⠀⣿⡹⣿⣔⣘⣿⣿⣿⣿⣟⠿⡟⡀⠀⠀⠀⠀⠀⠀⡀⠜⠉⢀⣴⣿⣿⣿⣿⣿⣿⡄⠀⠀⣀⡠⠴⡒⢝⠫⠊⢰⢿⠷⣆⣿⡿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣿
⠀⢸⣉⣣⣿⠿⣏⠉⣹⣿⣿⣿⡅⣳⣢⠀⠀⠀⠀⣠⠊⠀⢀⣠⣞⢟⠿⠙⣿⣿⣿⣿⠟⣋⠩⠔⠊⠁⠈⠀⠀⠀⢸⣆⠀⣿⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿
⠀⠸⣿⠿⠢⡀⣼⣿⢯⡝⣿⣿⣿⣿⣿⠁⡐⠊⠉⠀⣠⠶⠋⠈⠀⠁⠀⠀⠈⠙⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣭⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿
⠀⠀⠹⡗⢤⢏⡿⢵⠂⠀⢌⢿⢿⣿⣿⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⣟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇
⠀⠀⠀⠀⠙⢇⡆⠀⠀⢀⢐⣯⣿⣿⣫⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠀
⠀⠀⠀⠀⠀⠀⠑⢤⢔⣵⡿⣿⣾⡿⡻⣿⣓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢘⢯⣾⣾⣿⢇⣬⠟⢀⠝⠳⣼⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠀
⠀⠀⠀⠀⠀⠀⠀⢀⢎⣙⣿⣿⣛⢷⣤⠀⠑⠀⡨⡙⢾⣕⡢⣠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⣟⣿⣵⣝⠻⣢⣀⠈⠈⠢⠈⠚⠔⡵⡪⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⣠⣀⣲⣤⣄⣀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠸⣿⣿⣿⣿⣿⣾⣯⡪⣻⣧⡄⠀⠀⠤⠀⠀⠙⢳⢍⠦⣀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣼⣓⣒⡻⡿⣿⣿⣿⣿⣷⣦⡀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⣿⣿⣿⣿⣿⣿⣿⣿⣮⣮⡪⢧⡀⠀⠈⠳⣀⣀⡭⡣⣾⣏⣚⢡⡄⡀⠀⠀⠀⡰⠃⠀⠀⠉⠉⠓⢭⡛⢿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣯⣶⣫⣦⣤⣄⣜⠫⣾⠞⠁⠄⠈⣹⡍⣼⡶⡄⡔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣈⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣝⣄⢟⣗⡕⢭⠃⡤⠨⣰⡼⠋⣰⣫⣾⣿⣶⠦⣄⡀⠀⠀⠀⠀⢀⠀⠀⠈⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⠟⠛⠋⠀⠀⣿⣿⣿⣿⣿⡿⣝⢟⣀⠐⣼⢟⣥⣾⡟⠛⣿⣿⣿⣷⣿⣿⣦⣄⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⠛⠙⠿⣯⢻⡜⣹⡗⢿⢃⢾⣾⡿⠿⠾⠿⣿⡿⢟⢿⣿⣿⣯⣙⣄⡀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⡿⠀⠀⠀⠀⠘⠳⡏⠒⣞⣞⣾⠋⠔⢒⢀⢊⢿⣵⣤⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿