⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⣤⣶⣒⣛⣓⣚⣛⣲⣶⡤⣌⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣟⣯⣎⣩⣾⢿⣟⡻⣭⣟⡷⣝⡾⣷⢰⡙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⡷⡟⣻⣭⣼⡶⠿⠟⢽⣛⣺⣿⣿⠨⣹⡌⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠻⣟⣉⣼⣿⠽⠿⠿⣷⣿⣋⣐⣒⣒⣲⣽⣽⡠⣻⢷⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢗⣼⣿⣭⠶⠂⠀⠀⣠⣤⣽⣿⣯⠍⠭⠍⠛⠿⣷⣝⡆⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠵⠒⠚⣛⡋⠛⠓⠒⠊⢉⣙⣛⣛⣛⢦⣜⢛⡿⣗⢮⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢴⡶⣟⣯⢉⣁⢀⣀⣀⣤⣭⣭⣭⡤⠀⣀⣨⣽⣷⣾⣿⣿⣶⣦⣝⣮⣿⣿⣟⡒⣶⠤⣤⡄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣍⣛⣙⣯⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣯⣍⣭⣽⣿⣿⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠙⠛⢿⡿⡿⠀⠈⠛⠒⠉⠁⠀⠈⠛⠟⠛⣿⣿⡿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⢹⣿⡇⠀⠀⠀⠀⠀⠀⣸⣃⣀⣀⠳⢦⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢿⡿⠿⠿⡿⠿⠁⠀⠀⠀⠀⠀⢀⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄⠀⠀⠀⠀⢀⡴⠚⠁⠀⠓⠶⣤⡀⠀⠀⠀⢀⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢿⣄⠀⠀⠀⣫⣶⠿⠿⠿⠿⠷⠶⢯⣤⠀⢀⣾⢱⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠘⣿⢷⡀⠀⠀⠀⠐⠶⠶⠠⠀⠀⠀⢀⣴⣿⣏⣾⣿⠇⠀⠰⣯⣍⣉⠉⠓⠲⢤⣼⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠸⣶⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⣾⡾⣿⡿⠀⠀⠀⠀⠉⠉⣿⣶⡕⠀⠙⠛⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⣏⢿⣦⣤⣴⡄⠀⠀⠀⣠⣾⣿⣿⣿⡿⣱⣿⢿⣿⣷⠶⠶⡒⠋⣉⠻⢿⣤⢀⠈⣾⡁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⢶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⠀⠉⠛⢿⣿⣷⣶⣾⣿⣿⡿⠛⠉⢠⣿⡟⠀⠟⣁⡀⠈⠉⠳⢽⡶⠋⠉⠉⠀⠈⢧
⠀⠀⠀⡀⠀⠀⢀⣠⡶⠛⢿⣿⣍⡟⠉⠻⣧⠀⠀⠀⠀⠐⠀⠀⠀⠑⢂⣐⣋⡴⠛⢿⣷⡀⠘⢦⡉⠛⠛⠛⠋⠁⠀⠀⢀⣾⡟⡁⡀⠈⠉⣉⠳⣦⡀⠀⠉⠃⠀⠀⠀⠀
⠀⠤⠀⠀⠀⠀⠿⣿⣤⣤⠞⠉⠙⠃⠀⢀⣿⠠⠀⢀⢀⡤⢤⣄⢊⡶⣫⡍⢹⣷⡀⠘⣿⣿⣄⠀⠘⠲⠄⠀⠀⠀⠀⠀⣼⣟⣠⣿⣿⣷⣿⣿⠟⠁⠙⠂⠀⠀⢀⠀⠀⠀
⠀⠀⠥⠀⣰⣖⣛⣿⡟⠁⠀⡠⠷⠒⠀⢸⢿⣄⣤⡶⠋⣴⠃⡟⢿⣄⣧⡳⣌⠉⠻⠶⠛⢿⣿⣷⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀
⠀⠀⠴⠉⢙⣡⡿⠋⠀⠀⢀⣀⡤⠖⠚⠁⠀⠙⢿⣁⣸⡟⢰⣧⡄⠘⣮⣭⣻⠀⠀⠀⠀⠀⠙⠻⣟⣦⣤⣤⣔⣴⣿⠿⠛⠋⠁⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠤⣤⣤⣤⣤⣿⣇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡀⠈⣹⢇⡟⠈⠻⣇⡊⠯⣽⠀⠀⠀⠀⢀⣠⡴⢊⣽⣿⢿⡿⡋⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠷
⠀⢀⡀⠀⡀⣶⡾⣧⡄⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣾⠀⠀⠀⠀⠙⠒⠺⡄⣠⣴⣾⣻⣧⡀⡈⢻⠷⢯⡀⠅⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠘⠃⠀⡇⣼⡅⣇⠻⣄⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠳⢟⡴⠋⠀⠀⠙⢦⡀⢀⣴⣿⣿⣿⣿⣿⣿⣿⢻⡦⠀⠀⠀⠀⠀⠀⠀⣘
⠀⠀⠀⠀⡇⢿⡇⠸⡄⠈⠁⠀⠀⣻⣿⣿⠃⠨⠙⢿⣿⣟⣻⣿⣿⣦⡀⡀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⢎⡟⢦⡀⠙⢿⣿⣿⣿⣿⣿⣿⣟⣡⣴⠿⣆⠀⠀⠀⢀⣴⡿⠏
⠀⠀⡇⠀⡇⣤⡄⢠⡟⡆⡄⠀⠀⢸⣿⣇⢹⡆⠀⠀⢜⣹⣿⠟⣻⣿⣿⣷⠀⠀⠀⠀⣤⡏⠀⠀⢞⢁⢞⡴⡻⣣⣆⡀⠙⢿⣿⣿⡿⠋⠉⠉⡀⢘⣿⣶⣤⣶⣿⣿⣿
⠀⠀⡀⠀⡇⣿⡇⢸⡀⠀⢹⡆⠀⠈⣿⣿⣿⣿⣄⡴⠋⠑⠋⢹⡟⣹⣿⣿⣦⡀⣤⣾⣷⣬⠒⡀⠀⠛⠫⣺⢟⣕⢙⠟⣦⠀⢹⣿⢋⣥⣶⡯⠟⠉⢿⣿⣿⣿⣿⣿
⠀⠀⡆⠀⡇⣉⡁⢠⡇⠀⠈⠻⣦⣴⢿⣿⣿⣿⠋⠀⠒⣤⠀⠀⠉⣙⡁⠙⢿⣿⣯⣭⡾⣿⠟⢮⡓⢤⡑⢅⡻⡿⠋⣼⣷⣷⣿⣧⣯⠟⠋⠀⣠⠶⣿⡏⠉⠉⠀⠀⠉
⠀⠀⡇⢸⡇⣻⠽⣾⠀⠀⠀⠀⠀⢀⡿⣿⡿⣧⠀⢠⡾⠋⣳⡴⣊⡥⠖⠁⠈⢻⣿⣿⣿⣿⣧⣾⣿⣦⡑⢦⡙⢮⢺⣿⣿⣿⣿⠡⠀⣠⠮⡭⣵⣿⣿⣇⠀⠀⠀⠀⠀⠀
⠀⢠⣷⣾⣷⣿⣾⣿⠀⠀⠀⠀⠀⠀⢷⣹⡇⣿⣧⡉⠀⡼⣯⡇⠑⠁⠄⠀⣈⣤⣽⣿⣷⣝⠻⣿⣿⣿⣿⣦⣙⡾⣯⣿⣿⣿⣿⡷⠚⠁⠀⠀⠀⠀⠉⢹⡄⠀⠀⠀⠀⠀
⠀⠈⠉⢀⠄⠀⠀⢈⠀⠀⠀⠀⠀⠀⢾⣿⡇⢸⣿⢿⣄⣽⡯⣷⡆⣠⣴⠿⢝⣿⣿⣿⣿⣿⡇⠈⠻⣿⣿⡿⠋⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀