Lunchroom Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣆⠀⠀⣨⠇⡼⠀⠀⠀⠀⠀⠀⠀⢀⡽⢠⠇⠀⠀⣰⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡟⠀⠀⠘⠂⠓⠀⠀⠀⠀⠀⠀⠀⠈⠃⠘⠂⠀⠀⢻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⢠⣶⣶⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡀⠀⠀⠀⠉⠛⠉
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡆⠀⠀⣨⣴⡍⠀⠀⠀⠀⠀⠀⠀⠈⢩⣶⣄⠀⠀⢰⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣮⡻⣿⣶⣿⡿⢋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡙⢿⣿⣶⣿⢟⣥⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣮⡛⠁⠀⠙⠛⠛⠛⠛⢻⡟⠛⠛⠛⠛⠛⠀⠈⢛⣵⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⢀⣀⣀⡀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣀⣀⣀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢸⡇⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣭⣭⣭⡍⠉⠙⣿⣿⡀⠀⢸⡇⠀⢀⣿⣿⠋⠉⢩⣭⣭⣭⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⡇⠀⢸⣿⣿⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⣧⠀⢸⡇⠀⣼⣿⡏⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠘⠿⠟⠀⢸⡇⠀⠻⠿⠃⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠛⠁⠀⠈⠉⠙⢣⠔⢀⢐⣶⣤⣤⢀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣻⣿⣠⣴⣿⣶⣤⣄⡀⠠⡸⠶⢛⣠⣴⠟⠃
⠀⠀⠀⢀⣤⡶⡟⢩⡾⣿⣿⣿⣻⣿⢿⣿⣿⡆⠉⡆⣀⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⣀⣤⡶⢶⣤⣄⣀
⠀⠀⢾⣿⣁⠀⠘⠦⢙⣙⡻⠿⠿⣷⡟⠚⣛⡠⣶⡿⠛⠁⠀⠀⠀⠀⠀⣀⣠⡶⠟⠋⠁⠀⠀⢠⣽⠙⠛⠓⠶⣦⣤⣀⡀
⠀⠀⠀⠈⠉⠛⠲⠶⣧⣿⣿⡿⢟⠛⠛⢟⡿⠞⠉⠀⠀⠀⠀⣀⣤⣶⢿⡿⠁⠒⡤⠆⣰⣢⣤⣈⣉⣐⡞⠀⠀⠀⣈⣽⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠞⠋⠀⠀⠀⠀⢠⠶⠛⠉⢱⠲⣮⡙⠟⡁⠐⠌⠙⡋⠙⠀⡣⠀⢀⣠⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠑⠲⢤⣌⣳⣬⣄⣈⣁⣉⣀⣀⣠⣧⣼⣧⡾⠟⠉
⠀⠀⠀⠀⠀⣀⠠⠄⠴⠶⠟⠛⠉⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢿⣟⣫⠉⢩⣭⠿⠋⠁
⠀⢠⣶⣶⠋⠅⠀⠀⠀⠀⠠⠓⠄⠀⠀⠀⣈⣉⠻⡶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋
⠀⠀⠉⠻⢷⣦⣤⣤⣤⣤⣴⠀⠀⠀⢀⣀⢄⠗⠁⠷⢔⠙⡿⢶⣤⣀
⠀⠀⠀⠀⠀⠈⠛⠿⣦⣄⣸⣯⡙⠋⠉⠩⠁⠀⠰⠑⢚⣠⡞⣠⣮⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣽⣻⣗⠶⠶⣶⡶⣶⣾⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢧⡼⠖⠛⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡴⣠⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⣾⣿⣃⠁⢸⢦⣶⣿⣿⣦⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠇⠀⢾⠀⠀⠀⠀⠀⣰⡟⢉⡀⣾⣿⣍⠄⣰⣿⣿⣟⠠⣴⣾⣯⠉⠻⡏⠀⢀⠀⠀⢀⣼⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⢴⣠⣦⣀⠀⣀⣴⣿⡋⢂⢰⣿⣿⣿⣬⣿⣿⣿⣿⣷⣷⣿⣾⣷⣾⣷⣿⣿⡛⠓⠻⣿⡍⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣿⣿⣿⣿⣏⠙⠐⢶⣶⣿⣿⣯⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠂⢿⣿⡇⠸⣿⡇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡄⠈⢁⠁⣿⣿⡿⡿⠿⢿⣿⣿⡇⠀⢸⣿⣿⣿⣮⡤⠥⢾⣿⣿⣁⠀⠈⣻⣿⡷⢾⣷⣤⣤⣝⣋⣤⣿⣷⠰⠛⣿⣤⣼⡄
⠀⠀⠀⠀⠀⠀⠱⠀⠀⢻⣿⣿⣅⡀⠀⠉⢉⣿⣿⢙⠿⣿⣿⣿⡌⠑⠶⣾⣿⣯⡥⠄⢠⣍⡿⠇⣾⣿⣏⠹⠇⠀⠙⢻⣿⠦⡁⠘⢻⣿⠃
⠀⠀⠀⠀⠀⠀⢀⣷⡖⢶⢻⣿⣿⣿⡄⠐⢺⣧⣾⡀⣰⣿⡻⣿⠻⠂⠀⣄⢹⣿⣿⡦⠈⣽⣁⣐⣿⣿⡿⡇⠀⡀⣤⡈⠙⡇⢰⣿⠏⠉
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣾⡟⠉⠡⢬⣦⣀⡏⢹⣿⣿⣿⠿⠿⣥⠀⠀⣠⣿⣾⣏⣛⠛⣦⡘⢿⣿⣿⣿⣿⣿⡷⠾⠿⠗⠀⠰⣿⡟⡃⢠⡆
⠀⠀⠀⠀⠀⠰⠆⣠⣿⠃⠀⠀⠀⠀⠈⠟⣇⠈⠉⠩⣽⣿⣷⣄⡸⠷⠒⠠⠼⠿⠿⢤⠤⢩⠟⠉⠀⢀⣨⡽⠴⠖⠀⠒⣴⣾⠟⣧⣀⣾⡇
⠀⠀⠀⢀⠀⢀⣴⣿⡟⠀⠀⡴⠰⣤⣄⡀⠈⠲⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⡤⠴⠛⣿⣿⣷⣠⣄⣀⡠⠃
⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⣈⡀⠰⠟⠻⠿⠷⣦⡴⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⣙⢚⣅⠀⠂⠀⣀⡤⠚⠁⠀⢈⣡⣿⣿⣿⣿⠿⣿⠄⢠⡀
⠀⠀⠀⠘⣻⣿⣿⣿⣿⠀⠉⠀⠀⠀⠀⣿⣲⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣭⣭⣭⣥⣤⣤⣤⣶⣻⣿⣇⠀⠀⠀⠈⣾⣿⣾⣿⣿⠀⠀⢠⢞⡇
⠀⠀⠀⠀⠁⣿⣿⣿⣧⡏⠀⠧⠀⡰⠚⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⠿⠿⠿⡜⣿⡆⠀⠰⠾⣿⣿⣿⣿⣿⡠⢤⣼⡿⠁
⠀⠀⠀⠀⠰⢿⣿⣿⣿⣷⡀⠀⠛⣀⠀⣀⢀⣀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⠤⠴⠤⠠⡾⣿⣦⠀⠀⣿⣿⣿⣿⣿⡇⠸⡿⠃
⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠟⠛⠓⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠋⠛⠛⠋⠛⠉⠛⠿⣿⣿⡟⠂⠑⠹⣿⠀⢸⣿⣟⡋⠉⠉⢻⡂⠁
⠀⠀⠀⠀⠀⠀⢉⣉⣙⢛⣷⣶⠒⠒⣶⣶⠶⠶⠤⠤⠤⣤⣀⣀⣀⣀⡀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣴⢦⡴⢚⣇⠀⠉⠉⠀⠀⠀⢶⢳
⠀⠀⠀⠀⠀⠲⣿⣿⣿⣿⣿⣷⢤⣠⢿⡟⠀⣤⣄⣴⣶⠄⠀⠤⠊⠉⠉⠉⠉⣿⠟⠋⢹⡀⠉⠙⠛⠷⠮⠴⠃⠈⠀⠀⠙⢶⣤⣀⣊⣸⣇⡄
⠀⠀⠀⠀⠀⠐⠻⣿⣿⣿⣿⣷⣾⠿⣽⡇⠈⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣟⣉⠕
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⢿⣦⣤⣤⡤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⣽⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣾⣿⣿⣿⡇
⠀⠀⢀⣠⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⣼⣿⣿⣿⣿⣇
⠀⠀⣾⡟⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣧⣿⣿⣿⣿⣿⠗
⠀⢠⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⣶⠄⠀⠀⠠⠿⠻⣿⣿⣿⣿⣿⣿⣿⡆
⠀⣙⣿⣿⢿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣿⣿⣟⣿⣾⣿⡦⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠿⠀⠀⠀⣤⣤⡀⢸⣿⣿⣿⣿⣿⣿⡇
⣼⡿⠏⢻⣷⣿⣿⣿⣷⣄⠀⠀⠀⠀⠹⠾⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠸⠿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣾⣿⡿⠀⠀⠀⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣴⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠻⠿⡿⠿⠿⣿⡿⠟⣿⣿⣿⣿⡀⠀⠀⢸⣿⡿⣿⣶⡆⠀⠀⠀⠀⠀⠘⢿⠿⠿⡿⣿⣿⣿⣿⠃⠀⠀⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⢰⣿⣿⢻⣿⣧⠀⠀⠘⣿⣧⢿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢘⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⣾⣿⡇⠸⢿⣿⣷⣤⡄⣻⣿⣄⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣿⣿⣧⣤⠀⠀⠀⠉⠀⠀⠉⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠈⢳⡀⠀⠀⠀⢳⡄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠘⣦⠀⠀⠀⢀⡴⠋⠁⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⣾⠁⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢰⡋⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣠⡴⠋⠀⠀⠀⠀⠙⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠳⣄⣀⢀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢠⡞⠉⠉⢳⡄⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⢻⠀⠀⠀⠀⣀⣈⣉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠉⢳⡄⠀⢈⡷⠒⢦⡼⠁⠀⠀⠀⠀⠀⠀⠘⢧⡴⠒⢾⣀⠀⢀⡞⠉⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠰⣄⠀⠀⠳⠞⠉⠀⢀⣼⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣧⣀⠀⠈⠳⠞⠁⠀⣠⠆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠈⠳⣄⠀⢀⣤⢾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣷⢤⡀⠀⣠⠞⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠈⠙⡏⠀⠈⠛⠚⠛⠛⠛⠛⡖⢳⠞⠓⠛⠓⠛⠛⠁⠀⢹⡏⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠷⠤⠤⠤⠤⣤⡀⠀⠀⡇⢸⡃⠀⢀⣠⠤⠤⠤⠤⠾⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⡇⢸⡇⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠛⠒⠶⠶⠶⠲⣶⠲⣦⠀⠀⢸⡆⠀⡇⢸⡁⢀⡇⠀⠀⣰⠖⣶⠶⠖⠖⠶⠒⠛⠖⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⡶⠶⠶⠶⠛⠀⢹⡀⠀⠈⡇⠀⡇⢸⠇⢸⠇⠀⠀⡟⠀⠙⠶⠶⠶⢶⡶⠶⠶⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡇⠀⠀⣿⠀⡇⢸⠆⣼⠀⠀⢰⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⠀⡇⢸⠃⡟⠀⠀⢸⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠹⢦⣤⠞⠀⣇⣸⡆⠳⣤⣤⠟⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⡰⠀⢸⠀⠱⡀⠒⠀⠀⠀⠀⠀⠢⠀⠁⠰⠄⢆⢈⠸⠆⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠈⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣒⣒⣒⣤⡼⣟⠥⠔⠒⠒⠒⠒⠒⠤⢤⣼⣧⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠿⣿⠟⠁⠀⠀⠉⠠⣄⠤⠉⠀⠸⠿⠟⢦⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⢠⡏⠀⣠⠖⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡿⠀⠈⢿⢂⡞⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⣸⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠈⠙⠣⣤⣀⣀⡤⠴⠚⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣀⣀⠀⠀⠀⣀⣠⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⢻⡈⠻⡿⠟⢡⡏⠀⠉⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠙⠓⠶⠞⠋⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠈⠙⢦⡀⠀⠀⠀⢰⢒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⡀⢰⣴⢂⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣞⣵⣠⠀⠀⠙⠦⣄⠀⣄⡎
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠄⣧⠾⠛⣯⣴⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠳⢿⣆⡄⠀⠈⡻⣼
⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⢀⠸⣤⠟⠁⠀⠀⢹⣿⣞⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⣀⣀⣀⣠⣽⣧⣤⠶⢳⠉⢧
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⣸⢛⣁⣀⣀⣀⣤⣼⣿⡿⠿⠴⠶⠶⠒⣒⣚⣛⣛⣋⣉⡉⢉⣉⣁⣠⠴⠶⢶⣦⣰⣇⠀⢣⡀
⠀⠀⠀⠀⠀⡼⠀⣠⢛⡿⠿⠯⡍⢉⡷⠒⠚⢛⣻⡟⢧⣌⣉⠉⠉⠉⠉⠉⢻⣅⠀⢹⣨⣭⣭⣉⣙⣻⠀⠀⢹⠻⣷⡞⠳⡀
⠀⠀⠀⠀⣠⠷⣴⠛⣇⠀⠀⢀⡄⢸⣄⠀⠀⠻⣾⣍⣕⡒⠬⣝⣳⡟⣛⣉⣉⣩⠧⣾⡷⣶⣄⠀⢨⠏⠙⢓⣾⡄⣝⡰⢆⡱
⠀⠀⠀⢰⠁⣰⣧⠤⠬⠽⠿⠧⠶⠶⠮⣝⡲⢦⣿⣶⣬⣭⣽⣚⣯⡷⢷⠞⣩⣏⡾⢼⡧⠀⠙⠳⡟⠛⠉⠉⠩⠗⠛⠓⣎⢧
⠀⠀⠀⢸⣞⡿⠁⠀⠀⠀⣀⡤⠖⢦⣶⣀⠙⠳⢬⣍⡉⢉⡭⠭⡉⣛⢛⢋⡷⣿⢰⠋⣿⠷⡀⠀⢱⠀⠀⠀⠀⡄⠀⠀⠸⡜⡆
⠀⠀⠀⣼⡞⠀⢀⡠⠔⡋⣥⣠⣴⡼⠿⠟⠀⠀⠀⠀⡙⣦⠶⠋⠁⠀⠈⠙⠟⢿⢸⡀⢿⣿⣿⣄⣼⣀⣀⣀⠴⡁⢀⣀⣠⠇⢻
⠀⠀⡜⡟⠰⠶⠥⠒⠚⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⣠⡟⠿⣀⢀⣀⠀⠤⠴⢤⡞⣀⣛⣒⣊⣉⣉⣉⣍⣤⠩⠭⠭⠦⠶⠶⠖⠛
⠀⢼⠁⠧⢤⣤⣤⡤⣤⡴⡶⣶⣶⣶⣶⣶⠶⠿⢿⣭⣶⣶⠶⠚⠛⠉⠉⠉⠉⠉⠁⡏
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣗⣫⠀⠀⠀⣠⠖⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠴⠂⠀⠀⣽⣫⠗⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⢚⣁⠀⠀⣿⡥⢞⡀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣩⠆⠀⠀⡎⣷⣫⠄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢗⣋⠀⠀⡇⢸⣥⡖⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⣠⣀⣰⠇⠘⠿⠱⡶⣶⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠻⠀⠀⠀⠀⠀⣧⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢴⠀⠀⠀⠀⠀⣿⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢿⡟⣆⠀⠀⠀⣾⣉⣉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣤⣤⣤⡴⣗⣲⡶⠿⠠⠴⠲⠿⠒⠒⢋⡀⣤⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠉⠙⣆⠀⠀⠀⢤⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢦⠰⣄⠀⠀⠀⢠⠖⠛⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣸⠀⠀⠀⣼⠃⡞⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⡟⠀⠀⠀⡏⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣤⠴⠃⠀⠀⠀⠈⣀⣉⠀⠀⠀⠀⠀⠀⠀⠀⣉⣀⡁⠀⠀⠀⠙⢤⣀⣀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⣄⠀⠀⠀⡏⠉⠉⣧⠀⠀⠀⠀⠀⠀⣾⠉⠉⢹⠀⠀⠀⣠⠤⢬⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢳⣀⡤⠛⠉⢳⠃⠀⠀⠀⠀⠀⠀⠘⣾⠉⠛⢦⣀⡾⠁⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠈⠳⣄⠀⠉⠀⣠⠶⠷⠶⠶⠶⠶⠶⠶⠶⠶⠾⠶⣄⠀⠉⠁⣠⠞⠁⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠳⣶⠋⠙⠦⠤⠤⠤⠤⣤⢤⠤⠤⠤⠤⠴⠋⠙⢶⠞⠁⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠹⣤⣤⣤⣤⣀⠀⠀⣿⢸⠀⠀⢀⣤⣤⣤⣤⡟⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⣿⢸⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠛⠓⠒⠒⠒⢲⠒⣦⠀⠀⣇⠀⣿⢸⠀⢸⠀⠀⢰⠒⡶⠚⠒⠒⠚⠛⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢻⡟⠛⠛⠋⠀⢹⠀⠀⢹⠀⣿⢸⠀⡿⠀⠀⡿⠀⠉⠛⠛⠛⡟⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸⡆⠀⢸⡆⣿⢸⢀⡇⠀⢠⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢧⡀⣰⠇⣿⢸⠘⣇⢀⣸⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⠛⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠀⡄⠠⠀⠀⠀⠀⢀⠀⠠⢀⠀⡀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘⠘⠀⠈⠁⠀⠈⠀⠀⠀⠀⠘⠁⠁⠀⠉⠀⠃

⠀⠀⣀⠀⠀⠀⠀⠀⣀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢫⣽⠏⢿⠩⡏⢭⡍⡍⢭⡭⢝⡏⣽⠋⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⣼⣿⡆⠠⣷⠀⣸⣿⣦⠀⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣶⣾⣾⣧⣽⣷⣷⣬⣶⣾⣧⣿⣶⣮⣾⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣀⣿⢠⡿⠙⣿⠀⣿⣿⡇⢿⠸⡀⣜⢈⠶⠇⣍⠀⡡⡅⠶⣉⠶⢘⣈⢃⢻⣷⠛⡆⢸⠆⢡⠨⢸⠔⢿⣿
⠀⣻⣿⣿⣟⠛⣻⣿⣶⡟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⢿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠙⠛⠿⠿⣿⠿⠿⠛⠋⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠾⠷⠿⠶⠿⠾⠷⠾⠷⠿⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⡀⢒⡒⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠤⠆⠀⠂⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣅⣀⡁⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈

⠀⣴⠀⠀⠀⠀⠀⠀⠈
⢸⣿⢄⣀
⣼⣏⠁⠀⡈
⣹⣿⠀⠐⠁
⢹⣿⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁
⣾⣿⢀⢀⣤⡄
⣿⣿⠨⠟⠷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⣏⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇
⣿⣿⣾⡄⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠨⠀⠀⠀⠀⠀⠀⠀⠀⡆
⣿⣿⠀⠀⡇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⡿⡧
⣿⣿⠀⠀⡇⠀⠀⢀⣀⡐⣀⡄⠀⠀⡄⠀⠀⠐⡠⠀⠴⠂⠀⠀⠂⠀⠠⡀⡀⢤⠀⠤⠀⠀⣼⣼⣶⣞⠀⠀⠡⣷⡗⠰⠀⠀⠁⡀
⣿⣿⠀⠀⠇⢈⣿⡟⢊⠛⣧⣏⣿⣿⣹⣤⠀⠀⢰⠎⠀⢀⣀⡰⠀⠀⠀⡁⠁⠀⠀⠀⢠⠀⣿⠋⢻⣿⣦⢸⣿⣿⣷⡄⠀⠀⠀⣷⠄
⣀⣁⡀⠀⢸⣽⣿⣷⣼⡿⠛⠛⣻⣭⣿⣿⣤⠀⢸⣿⠂⠈⣿⣷⣦⣿⣶⣢⣤⡀⠀⠦⣤⠃⢀⣀⣀⣈⡦⢘⡿⢿⡿⠁⠀⡇⠀⢩⣤
⣿⣿⣿⣾⣞⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡆⡔⡈⢿⣿⣿⣿⣿⣿⠏⠁⢀⣴⣶⣖⣪⣻⠍⣻⡇⢸⣷⣾⣇⢀⣰⣷⣾⣾⣿
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡓⢶⡇⣿⣿⣿⣿⣿⣿⣿⣶⡆⠀⢻⣿⣿⣿⣿⣿⣿⣧⣼⡇⠸⡇⢀⡾⠿⠿⠿⠿
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣀⣿⣿⣿⣹⡿⣿⡿⢿⠧⠼⠿⠿⠿⠛⠟⠛⠛⠋⣻⢁⣸⣧⣤⠤⠄
⣿⣿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢫⣽⠏⢿⠩⡏⢽⣍⡍⢽⠩⣻⢩⡟⢹⡟⣿⣿⣿⣿⣿⣿⣿
⠀⣼⣿⡄⠰⡧⠀⣾⣿⡄⢸⣿⣿⣿⣿⣿⢿⣿⡷⣿⣾⡶⢾⣷⢿⣷⠷⣿⣶⣿⣾⣷⣶⠾⡿⢿⡿⢿⡿⣿⣿
⠀⣿⣿⣷⣰⣧⣸⡏⢹⡇⢸⣿⣿⠸⡼⣇⣨⣁⡸⣇⣭⣀⣱⣕⢗⣕⠷⣁⣌⣄⣿⣎⣱⡨⠇⣡⡆⢕⡦⣻⣿
⠀⣿⣿⣿⣏⣉⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⢿⢿⣿⠿⣿⢿⡿⡿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠉⠙⠛⠻⠿⠛⠛⠉⠁⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠏⣬⠍⢓⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⡏⠉⠠⠌⠇⠀⠀⠀⠀⠀⠀⠀⠉⠈⠃⠀⠈⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠥⠀⠀⠀⠐⠐⣐⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢈⠀⠠⠠⢸⠐⠠⠥⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠖⠒⠚⠀⠀⠀⠀⢈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠄⠀⠀⠌⠀⠀⠐
⠀⢄⡐⢒⣂⠇⠀⠀⠀⠀⠀⣐⣒⣂⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀⢁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⣒⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠐⠀⠀⠎⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠩⠭⠭⠁⣀⡀
⠀⠀⠀⠀⠀⠀⠁⠒⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⣠⡶⠞⠛⠛⠲⣦⡀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⠦⣤⣀
⣼⠋⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠘⣷⠀⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠘⣧⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠙⣧
⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⡟⠀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⣰⠟⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⢸
⠘⣦⡀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠘⠧⠀⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠹⠆⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⢀⣼
⠀⠈⠙⠳⠶⠶⠛⠋⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣤⣤⡴⠟⠁
⠀⠀⠀⠀⠀⣀⣠⣤⣀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢹⡆⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⠀⠀⣠⡞⠉⠁⠀⠉⠻⣆⠀⠀⠀⣿⡴⠶⢦⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣠⡶⠶⣾⡃⠀⠀⢀⣴⠋⠁⠀⠈⠙⢷⡀
⠀⠀⠀⣿⠀⠀⣀⠀⠀⠀⢻⣄⣴⠞⠋⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠙⢷⣄⣸⠇⠀⠀⠀⣀⠀⠈⣧
⠀⠀⠀⣿⠀⠀⠙⢷⣄⠀⠀⠉⠀⠀⠀⣠⣶⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣷⣦⣀⠀⠀⠈⠁⠀⠀⣠⠞⠁⠀⢠⡇
⠀⠀⠀⢻⡀⠀⠀⠀⠙⢷⣄⠀⣀⣴⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠳⣤⡀⢀⣤⠞⠁⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⢻⡏⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⢻⡟⢻⡟⠛⠛⠛⠛⠛⠛⠛⠁⠀⠈⢹⠏⠁⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠸⣇⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⢷⡄⠀⠀⢸⡇⢸⡇⠀⠀⣴⠟⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⡁⢸⡁⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⣴⠶⠶⠾⠿⠷⠶⠶⠶⠶⠶⣶⠶⠶⣆⠀⠀⠀⢿⡀⠀⢸⡇⢸⡇⠀⢸⡇⠀⠀⠀⣴⠶⢶⡶⠶⠶⠶⠶⠶⠾⠿⠷⠶⢶⡄
⠀⠀⢿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⡾⠁⠀⣿⠀⠀⠀⢸⡇⠀⢸⡇⢸⡇⠀⣾⠁⠀⠀⢰⡏⠀⠸⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⢹⡏⠁⠈⠉⠁⠀⠀⠀⢹⡄⠀⠀⠘⣧⠀⢸⡇⢸⡇⠀⣿⠀⠀⠀⢸⠇⠀⠀⠀⠉⠉⠈⠀⣿⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⢸⡇⢸⡇⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢹⡆⢸⡇⢸⡇⣸⠇⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⣀⡼⠃⢸⡇⢸⡇⠸⣆⡀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠀⠀⠸⠷⠾⠃⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻


⢠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢤
⠰⠤⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠧⠼
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⣠⠤⡀⢀⠠⡤⠀⣄⠀⡠⢄⠀⠀⠀⡤⠄⠀⡠⠄⠠⡤⢀⠤⡀⣠⠤⡀⢀⢀⠀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢿⠀⡜⠀⠀⡇⠰⠘⠆⢛⢹⠈⡀⠀⡇⠀⠀⠉⠢⠀⡇⢸⠓⡇⢇⠀⡼⠸⠣⡀⡇
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠤⠀⠀⠒⠂⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⠀⡆⠀⢸⢸⠈
⠀⠀⠀⠀⠀⠀⠀⠐⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⢰⣿⠐⡂⠀⣤⡤⠀⠀⠀⠀⠀⢀⣼⣿⣄⣧⣤⣼⠸⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣆⠀⠀⠀⠀⠀⢠⣶⡿⠛⠁⢸⣷⠀⠀⢀⣀⢸⣿⣤⣤⣤⣬⣥⣤⣤⢀⣀⣀⣸⣿⣿⣿⣁⣙⣉⣀⣐⣒⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⡇⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠈⠙⠒⢠⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡇⠀⠀⠀⠀⣾⣿⣿⡿⠦⠀⠀⠀⠀⣇⡴⣿⣿⣇⣀⠀⠀⠈⠉⢹⣿⡏⠙⠛⠛⠛⠛⠻⠿⢿⣿⣿⡿⠿⠇
⠀⠀⠀⠀⠀⢒⣶⣶⣶⣾⣿⢿⠀⠀⣤⣤⣼⣿⣿⡇⠀⠀⠀⠀⠀⠁⠀⢈⣿⣿⣿⠁⠀⠀⣿⣿⣿⣇⣀⣀⡀⠰⠆⠰⠆⣸⣿⣿⡇
⠀⠀⠀⣤⣤⣤⣿⣿⣿⣿⣿⠃⠀⠀⣿⣿⣿⣿⢻⡟⠀⠀⠀⢸⢰⡾⢿⣿⣿⣿⠿⠀⠀⠀⠿⣿⣿⣬⡏⠉⠁⠀⠀⣈⣩⣿⣿⣿⠇
⠀⠀⠸⠿⠿⠿⢿⣿⣿⡿⠿⠀⠀⠀⠙⠛⢻⡟⢻⠓⠀⠀⠀⢸⢸⡗⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠷⣾⣤⠀⠀⠀⠸⠾⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⢀⣤⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢠⣇⠀⠀⢸⢸⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣇⠀⠀⣾⡏⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⢺⣿⠀⠀⠸⠿⠀⠀⢸⢸⠀⢸
⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠘⣿⣷⠂⢸⣿⣿⠘⣿⡿⢀⠀⣠⠀⠀⠀⠀⢰⣿⣿⡄⠀⢠⢾⣶⣶⣿⠘⠂⠚
⠀⠀⠀⠀⠶⠶⠆⠈⠀⠀⢠⣾⣿⣿⣶⣴⣿⣧⣤⣤⣤⣤⣤⣤⡀⣀⣀⣀⡀⠀⠸⣇⣀⣀⣀⣙⣃⣈⢈⠙⠛⢒⣾
⠀⠀⠀⠀⠒⠀⠁⢀⣁⣀⠀⠉⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠰⠄⠀⠄⠈⠉⠉⣿⡟⠛⠛⣿⡿⠿⠿⠿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⡏⠉⠙⠛⠛⠛⠛⠛⠛⠛⠻⠿⠛

⠀⠀⠀⠀⠤⠬⠭⠭⠭⠭⢭⢵⡄⠀⠀⠀⣀⠀⢀⣀⣀⠀⠀⠀⠰⠋⠉⣱⡄⠀⠀⡰⢢⠜⣬⢣
⠀⠀⠀⢡⡀⠂⢂⠈⡄⡤⠄⢸⡇⠀⠁⠀⢹⠀⡠⠰⠻⢆⠀⢀⣀⡆⠊⣩⠇⠀⠀⣅⠏⢠⣇⡿
⠀⠀⢤⣬⣥⣶⣼⣤⣧⣧⣼⣼⡇⡇⠀⠀⢸⠈⠀⡠⠒⠺⠀⢀⠉⠀⢶⠁⠀⠀⡐⢀⣤⡉⢈⢷
⠀⠀⣀⣀⠀⢠⣿⣿⠷⠂⢸⢸⡇⡇⠠⡄⢸⠈⠢⢆⡠⠆⡀⣸⠀⠀⣰⡷⡄⠀⡇⢸⡈⠀⠸⣼⣤⡄
⠀⠠⠷⠷⠷⣨⠿⠁⡈⠀⢠⣤⠀⠁⠲⣗⢻⠃⠀⢈⢳⡆⢰⠉⣆⠀⠏⣿⡋⠀⢱⠒⢺⢟⠀⠀⠀⣇
⠤⢴⣄⢀⡞⠁⠘⠿⡀⢴⡁⠊⠀⠀⢶⣶⢾⠀⠀⢸⢸⣤⢮⡅⠉⠣⣀⡉⢙⣻⠙⣻⠋⠺⠤⠤⠤⠏⠁⠠⡀
⠀⠘⢿⠻⢹⡄⡀⡄⣷⠞⠠⠤⠶⠶⠶⠶⢤⠳⣄⢀⣠⠾⠬⡜⠀⠀⠀⢹⠉⠩⠩⠷⠶⠶⠶⠶⠶⣤⢤⡤⣬
⠀⠀⢹⡀⠹⣇⢡⣑⣈⣂⣠⡄⣀⠀⠀⠀⡇⠀⠈⠉⢸⡀⠀⢧⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡈⠁⠀⠿⠿⣟⣛⣛⡉⢹⠛⢻⡄⢸⠁⠀⠀⠀⠈⡇⠀⠈⠒⠲⠶⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢠⣽⢀⡴⢆⣆⠀⣀⣙⢦⡠⢼⡇⢸⡇⠸⠀⠀⠀⠀⠀⡇⠀⠐⠤⣀⣀⣸⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠐⠇⡏⠀⠢⠉⡌⠡⠄⢳⣷⢸⣇⢸⣅⡀⠈⠲⡤⢤⡶⠃⠀⢠⠀⣿⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠛⣇⠈⢀⣌⡿⡏⠒⡾⡿⠺⣿⣴⡦⠟⠈⠀⢧⠘⡇⠠⠤⢼⠀⡇⢧⠘⡆⠀⠀⠠⠤⠤⠤⣤⣤⡇
⠀⠀⠈⠢⣌⣀⣃⡨⣞⠝⠁⢴⣚⣷⠀⠀⠈⠂⢾⣀⣋⣀⡢⠘⠀⠧⣀⡇⠹⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠉⠉⠑⠂⠬⠓⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠈⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡤⣤⣤⡀
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⡆⠀⢀⡀⡗⣟⠀⢹
⠀⠀⠀⠀⢠⣾⣯⡀⠀⠀⣠⣿⣦⣄⠀⠀⠀⠀⠀⠀⣴⣶⡌⣻⠇⠀⠀⠀⠀⣀⡀⠀⠀⠏⠀⠀⠀⡇⠀⠻⢷⣿⣿⣇⣸
⠀⠀⠀⠀⢿⣿⣿⣇⠀⢰⣿⣿⣿⠿⡆⠀⠀⢸⠁⣀⣼⣿⣿⣿⠀⠀⠀⠀⠸⣿⡿⡇⠀⠀⠀⠀⠀⣿⣶⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⠷⠀⠸⣿⣿⣿⡇⠁⢠⣤⣤⣤⣿⣿⣿⣿⣇⣀⣀⣀⢀⣴⣿⣿⣧⣄⡀⠀⢠⠤⢧⣼⠿⠟⣿⣿⣿⡿⢶⣶
⠀⠀⠀⠀⠀⡇⣿⣶⠀⠀⠹⠿⠿⠇⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⠛⠛⠛⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⣶⢤⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣶⡆⠀⢠⣿⣿⣿⣿⠋⢀⣶⣤⣄⠀⠉⠉⠉⢙⣿⣿⠛⠛⠻
⠀⠀⠀⠀⠸⣿⣿⣏⣙⡛⠉⣿⢸⡇⢚⣛⣿⣿⣿⣿⣿⣿⣿⠟⠿⠃⠀⠾⢻⣿⣿⣿⡇⠘⣿⣿⣿⠀⠀⠀⠀⠸⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠋⡾⠉⠉⠉⠉⠉⠉⠀⣿⣿⣿⢤⣶⡼⠞⡘⢿⡿⠿⣿⢛⣿⣿⣿⣿⠀⠀⠀⠀⠐⡏⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⣿⢸⣿⠠⠠⠄⠠⠀⢼⣷⠀⣴⡎⠻⡟⠋⠯⠀⠀⠀⠀⠀⠷⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢨⣇⠀⠀⠀⠀⠀⣺⡏⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠈⠏⠁⠀⠀⠀⠲⠿⠇⠀⠈⠟

⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠋⠙⠋⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⣀⣤⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣛⡛⠀⠀⢀⢀⣛⡛⠿⣿⣿⡇⠀⠀⢀⡀⠀⠀⠀⠀⣀⡀⠤⠴⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⣿⣿⣿⣿⣿
⣋⠉⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⢉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠂⠀⠀⠀⠀⠀⣠⠖⠲⣄⠠⣖⠒⠒⠒⢒⡆⠀⠀⠀⡦⠴⠤⣄⢀⠤⣄⣀⢶⡄⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠰⡇⠀⠺⠥⠤⡄⠹⠀⠏⢩⡄⠀⡤⠄⡇⢠⢤⠈⠇⠑⡌⢡⠊⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠀⠀⠀⠀⠀⠙⠛⠒⢲⡀⢈⡀⡇⡆⢣⠱⠴⠃⡄⡆⢸⡤⢀⡆⠀⡇⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣰⡒⢒⣊⡠⠞⠀⠧⠇⠈⠓⠶⠚⠁⠷⠤⠔⠋⠀⠀⠙⠚⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⣇⢸⠩⠀⢰⠋⣉⠆⢰⠋⡉⠳⡠⡖⠒⠲⢤⡀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢅⠀⡆⢸⠀⣉⠘⡌⠀⠋⣴⠃⠀⢘⢢⠀⢣⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⢸⠀⡇⢸⠀⡟⡄⢻⠀⡷⡈⢢⡇⠘⠋⢀⡎⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠈⠳⠃⠘⠚⠁⠘⠋⠓⠃⠙⠊⠱⠶⠖⠋⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⠴⠚⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠒⢶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⣄⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣬⣑⠲⠦⠤⠤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠉⠉⠚⠒⠒⠒⠒⠲⠶⠄⠀⠤⠤⠄⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠶⠶⠤⣤⣤⣤⣉⣉⣉⣉⣉⠉⠉⠉⠉⠉⠛⠒⠒⠒⠒⠒⠒⠶⠶⠾⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠖⠛⠚⠛⠓⠒⠚⠆⠀⠀⠀⠀⢀⠀⠀⠀⠠⣄⠀⢠⡄⠀⢀⡀⠀⣤⢠⢤⡤⠄⢠⠤⢤⠄⣤⠤⠄⢀⠀⢀⡀
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠈⢀⡏⢰⡀⢸⠑⢄⡿⠀⠘⡇⠀⢸⡀⣀⠀⣧⣀⡀⢨⠃⢸⡇⠀⠁
⠀⠀⠀⠀⠀⣾⣛⣧⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠳⣤⠖⠸⠉⠉⠳⠸⠂⠈⠛⠀⠀⠃⠀⠸⠶⠶⠀⠳⠶⠆⠸⠀⠹⠆⠀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⠀⢄
⠀⠀⠀⢀⡀⠀⢀⣀⣀⣀⡴⣀⠀⠠⠤⠤⠴⠦⠤⠤⠤⠤⠤⠔⠢⠀⠀⠀⠀⠀⠚⠁⠀⠒⠒⠒⠒⠓⠒⠒⠒⠒⠒⠛⠒⠒⠒⠒⠒⠋⠒
⠒⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒
⠀⠀⠀⠀⢀⣀⡀⣠⠤⠤⣤⣤⣴⡒⠶⠖⠒⠒⠒⠛⠛⠛⠋⠉⠉⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡛⠛⠛⠛⠒⠒⠒⠒⠒⠶
⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⢦⣄⡀⠀⠀⠀⠀⠀⠸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⡀⠀⢤⣤⣀⠀⣀⠈⠁
⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢨⣧⡄⠈⠉⢉⡃⢹⣷⠜
⠀⠀⠀⠁⠀⠀⠄⢀⠀⠀⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠋⢈
⣀⡀⠀⠀⢀⡠⠔⠁⠀⠀⠈⠆⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠠⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠁⢷⡀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠊⠀⠀⠀⠀⡔⠁⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⢀⣤⣴
⠀⠀⠀⡀⠈⣇⠀⠀⠀⠀⠐⠀⠀⢠⠀⠀⠀⠀⠀⠨⡥⠶⠀⠰⠤⡅⠀⠀⠀⠀⠀⠀⠈⢠⣾⡟⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠺
⠀⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠴⣴⠶⣶⡀⠁⠛⠲⣶⠶⠂⠀⠀⠀⠀⠀⠙⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢥⠑⠄⡀
⠀⠀⠀⣡⠈⠀⠀⠀⠀⠀⣤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠘⢄
⠀⠤⠀⠁⠀⠀⠀⣠⠄⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠈⠁⢀⠀⢀⠉⠀⠀⠀⠀⠀⢀⡀
⣴⣦⡀⠀⠀⡀⠀⠉⠀⠀⠀⢀⠃⠀⣀⠀⠀⠀⠀⣄⠊⠁⠀⠀⠀⠀⣶⠎⠀⠈⠶⢐⠂⠀⠀⣀⣤⣶⠏⠁⠀⠀⠔⠉⠀⠀⠀⠀⢀⡈⠉
⣿⠟⠁⠀⠀⠀⠀⠐⠢⡀⠈⠁⢠⠋⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠈⣁⡤⠚⠁⠀⠹⡀⠀⠀⠘⡄⠀⠀⠀⡄⠀⠀⠀⠠
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠒⢄⠈⠢⡀⠄⠊⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠱⡄⠀⠀⠓
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⡄⠀⠈⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⢠⡀⠈⠁
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⣃⣤⠤⠤⠤⠀⢀⡤⠀⠀⢴⣤⠀⠀⠐⠂⠀⠀⠀⢰⣄⠀⢷⣄⠀⠀⠀⠐
⠀⠀⠀⣄⡀⠀⠀⢂⠀⠰⠁⢠⠃⠀⠀⠀⡶⠀⠚⠉⠀⠀⠼⠉⠀⠀⠀⠀⠀⠠⠄⠀⠤⠀⣵⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⢺⣿⣷⣶⣤⣤
⠀⠀⠀⣿⠹⡿⡆⠈⠃⠀⠘⠁⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡼⠛⠀⠠⢀⣤⣤⣤⡀⢀⣤⣆⣰⣯⣛⣛⠻⠿
⠀⠀⠀⠉⠀⠀⠀⠰⠀⢈⠀⡀⠀⠀⠀⠉⠀⠀⢀⠀⠈⣧⠀⠀⠀⡄⠀⠐⠃⠘⠆⠀⣁⠁⠀⠀⣤⠐⠀⠀⠀⠠⠿⢀⡠⠘⣿⣿⣿⣿⣆
⠀⢀⡘⠀⢸⢇⡈⠀⢀⡈⠁⠠⠄⢀⠃⠀⠀⠀⠀⠳⡄⠂⠀⠑⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠈⠃⠀⣴⠀⠠⠀⡀⠐⠉⠀⠀⢻⣿⣿⣿⣿
⠈⠉⠀⠀⠉⠀⠉⠀⢈⣁⡀⠀⣀⣿⠀⠀⠁⠀⠀⠀⢿⠀⠀⠉⠀⢰⠦⢄⣀⠀⠀⠠⠀⠄⠀⠅⠀⠈⠁⠀⠁⡀⠀⠀⠀⠀⠀⠙⢿⣿⣿
⠀⠄⠀⠀⠀⢀⡀⢀⣼⣿⣿⣾⣿⠟⠀⠀⠀⠀⠀⠀⠈⠀⠀⠶⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠿⠆⠠⠀⣄⣀⣡⣀⣀⣠⠄⠀⠀⠀⠙⠻
⠆⡀⠀⠀⠀⠈⠀⠀⠹⢿⣿⡟⠉⣠⣄⠀⠀⠀⠀⠈⠀⠈⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢺⡄⠀⠂⠂⣰⠦⠀⠀⠀⠀⢹⡏
⠀⠙⠠⠀⢀⠀⢀⠂⠐⠀⢠⠁⠀⠋⠉⠰⠆⠐⠆⠁⠀⢀⠀⢀⠎⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠘⠑⠃⠀⠀⠆⢰⣤⡆⠀⠀⠄⠂⠀⠀⠠⠄⢸⠴⡎⠀⠀⠀⠀⠦⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢦⠀⣸⠀⠀⠀⠀⠀⠀⠁⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢡⠑⠀⠀⢰⠁⠀⠀⠀⠀⠐⢦⠇⠀⠸⡄⠀⠀⠀⠀⡸⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⠿⠿⠿⠿⠿⢿⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⣿⢸⢸⡇⣷⠀⠀⠀⢀⣶⣿⣿⣷⣦⠀⠀⠀⠀⢀⣴⣿⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⣿⢸⢸⡇⣿⠀⠀⢀⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠙⣿⣆
⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⢰⣿⢸⢸⡇⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⢸⣿⣼⣼⣧⣿⡄⠀⠀⢿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⠋⠀⠀⠀⠈⠻⣿⣿⡿⠋⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⠛⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⠀⠀⠀⠀⢰⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠸⣿⣿⣿⠃⠀⠀⠀⠀⠹⣿⣿⡿⠁⠀⠀⠀⠀⠸⣿⣿⣿⠇⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⣶⣶⣶⣶⣶⣶⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⣄⠀⠀⠀⠀⠀⣀⠴⠒⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⠖⢉⣀⠀⠀⠀⠀⠀⠀⠑⢤⣄⡀⠀⢴⣂⡦
⠀⠀⠀⠀⠀⠀⠀⠀⢴⡐⠆⠀⢠⠃⡴⠉⠈⢦⠀⢶⡀⠀⠀⠀⠈⣿⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢰⠃⠀⠀⠀⠑⢬⣯⠲⣄⠀⠀⢹⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡄⠀⣀⠀⠀⠀⠈⡁⠀⢙⠒⣼⠀⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠉⠀⠠⠀⠀⠁⠀⢸⡞⠓⢤⡈⢃
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠢⣄⠀⠀⠀⢀⣠⠔⠣⣀⠀⢀⡧⠃
⠀⠀⣖⣩⡄⢨⣑⣩⣭⣍⡒⡄⠀⡸⠁⠀⠀⠈⠛⠝⠁⠀⠀⠀⠈⢆⠀⠀⢸⣉⣉⣉⣉⣉⣹
⠀⠀⠈⢎⣀⣀⣀⣀⣀⣱⣈⠁⠀⡇⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⡧⠤⠤⠤⠤⠤⢼
⠀⠀⠀⠀⣸⣀⣀⣀⣀⣀⣀⡇⠀⣟⢀⣪⣆⣀⡀⠀⠀⣀⣀⠤⠤⢄⠇⢀⠋⠉⠉⠉⠉⠉⠹
⠀⢀⣠⠚⠒⠒⠒⠒⠒⠒⠒⣇⣀⣻⣲⠫⠝⠀⠈⡉⡉⠤⠹⢯⣂⣸⣀⣀⠗⠀⠀⠀⠀⠀⠰⣄⣀⡀
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠊⠉⠀⠉⠒⠊⠉⠉⠉⠋⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠁⠈⠛⢟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠠⠀⠀⠀⠀⠄⠤⠤⠤⠤⠤⠄⠤⠀⠀⠀⣤⠄⠀⠀⠀⠀⠈⢅⠀⡠⢀⣿⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡅⠀⠀⠉⠉⠁⠀⠀⠁⠀⠀⠈⠉⠉⠉⠉⠀⡇⠇⠀⠀⠀⠀⠀⣠⡶⡂⢁⠿⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣱⢠⠀⠀⠀⠀⠌⢸⠇⣈⡁⠼⠃⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⢰⡖⠛⠛⠛⠛⠋⠏⣁⣰⣛⣉⣉⣉⣉⣉⣹⡇⡈⡀⠀⠀⡘⢀⠁⠀⠀⠀⢨⠘⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠋⠉⠉⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⡥⠅⢅⡀⠠⠁⠈⠄⠀⠀⠀⠀⣾⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠤⠄⠠⣄⡤⠒⠉⠉⠒⢶⠒⠒⠒⣶⠊⠀⠀⠀⠈⣦⢶⣿⣿⡖⠀⠀⣎⣁⠤⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠝⠿⠧⠀⠉⠉⠁⢾⠀⢠⠴⠶⢒⣶⡅⠀⠐⠁⢰⠂⡄⠂⠶⠋⠀⠁⡇⠐⠂⠁⠀⠸
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠤⠀⠀⠄⠁⠀⠒⠂⠀⠀⠀⠈⢶⣇⠀⡀⠄⠸⠧⠏⠽⠦⡻⣦⡀⠼⢆⠟⠭⠭⡝⢩⠭⠭⠭⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢿⣦⡾⠣⡀⠀⠀⠀⠀⠊⡹⠿⡄⠶⣖⢒⣆⡸⢶⣌⣅⣤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⡀⣃⣂⣥⡀⠀⠀⠰⠈⠢⣀⡝⠓⠋⢛⡩⠛⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠲⡛⢄⠘⠨⠙⡀⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢀⠙⢄⠑⣄⡄⠃⠀⠀⣄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠁⠉⠉⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⡀⠟⡤⠤⠀⠀⠀⠰⣀⣀⣠⣁⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠀⠀⠀⠀⠀⠀⡰⢁⠜⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⢨⠀⢠⠀⡄⠀⡔⠓⢴⠈⠀⠤⠠⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠃⡠⠀⠢⢅⠸⢦⠱⡀⠚⠆⠀⢆⣀⣐⢛⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⠀⠀⠀⠀⠑⠀⠀⠈⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⢭⣟⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢤⡤⠤⠄⢀⣀⠀⡸⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠣⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠀⠀⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠂⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠈⡉⠉⠉⠁⠀⠀⠒⠂⠐⠈⠉⠉⠁⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠢⠤⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠔⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠃⠘⠉⠭⠽⠟⠛⠛⠛⠋⠈⠉⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠚⠛⠻⠛⠛⠓⠒⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣠⠤⣄
⠀⠀⠀⠀⠀⢀⡴⢚⠁⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠈⠙⢦⡀⠀⢀⣠⡤⠖⠛⣉⠥⠄⠠⠜⠻⣦⡀
⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣞⠉⠀⠀⠀⠁⠀⠔⠊⠉⠴⠊⠩⡝⢳⣄⡀
⠠⢶⠤⣰⣇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⢣⡺⡤⣿⣌⠙⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣿⣠⢼⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡲⠤⣄⣀
⡠⢀⠙⠺⠻⣷⣶⡤⠤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣩⡾⡃⣀⣚⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣀⣠⣤⣽⣦⣀⠤⢶⢺
⠀⠐⠣⡅⣙⢄⣋⣭⣛⣓⣶⠶⢶⣤⡤⣤⡴⠶⠶⠶⠶⠚⡟⠛⠶⠿⢱⣚⡏⠩⠈⠓⢄⡀⠀⠀⠀⠀⣀⣤⣤⣶⡶⢿⢿⠿⠿⠟⠛⠉⠉⢀⣀⣼⠿
⠀⠀⣾⣷⡔⠀⢨⠃⠂⠘⠑⠷⢶⣧⣲⡖⠃⢂⡵⠷⠒⠛⣓⣲⡒⠄⡽⣿⡀⠀⠀⠀⠀⠙⠢⣀⣴⡟⠛⢿⡟⠀⣠⣲⣷⣲⠶⢶⣒⠚⠿⢍⡳⢤⣤
⠀⠀⡟⠬⡙⠗⠦⣤⣀⠀⠀⣀⣀⠀⠀⣙⡉⠀⢀⠭⠒⣈⠡⠖⠴⠼⢂⣿⡁⢠⢠⡔⠀⠀⢠⣾⡇⠘⡳⡖⠀⡴⣱⡁⢸⡌⠉⢳⢦⣍⡒⠤⣉⣲⠖
⠀⢀⠛⣦⡘⠀⢀⡈⠉⠛⠖⠀⣀⣀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⢀⣠⠎⢱⠙⢦⡸⢃⠄⠀⣾⣿⡇⠀⣷⡉⢙⡴⣝⢧⡀⣷⠀⠈⣶⣄⣹⣲⣇⡰
⠀⠐⡀⠈⠀⠩⠴⠿⠧⠤⠤⢤⣁⣉⣥⣌⡱⣶⡴⠭⢄⣒⣂⠭⠔⠋⠀⡸⠀⠀⠙⢼⠃⢠⡟⣿⣿⠀⠸⣿⡋⠵⢾⣭⣏⣻⡖⠐⡧⣄⣈⣉⣏⣹
⠀⠀⠑⢄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠙⢮⡇⠈⢻⠇⠀⢳⡉⠲⣤⣄⠤⣞⠛⢟⠿⣓⠊
⠀⠀⠀⠀⠈⠉⠒⠛⠛⠉⠁⠲⠶⠶⠶⠦⠤⠤⠄⠀⠐⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠈⢿⠭⡛⡘⣤⡉⢉⠇⠈⠑⢤⣙⡚⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⣈⣆⠱⣸⣌⣄⠉⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢉⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢉⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣃⢸⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠓⡏⠀⠀⠀⠀⢀⡠⢄⢴⠊⠉⢱⡀⡏⠉⠀⣁⢀⠖⠤⣄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠰⡁⠀⠈⢎⡆⠀⠀⢣⠡⠀⠀⢠⡌⠀⠀⡰⡰⠁⠉⢢
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣼⠀⣧⣤⣤⣤⣀⣱⡀⢺⠁⠘⡀⠀⠀⣾⠀⠀⢸⠁⠀⢠⠛⠳⡄⢀⠎
⠀⠀⠀⢰⡉⢖⠉⠉⠀⠀⠀⠘⠤⠃⠀⠀⠀⠉⢁⡣⢉⣇⠀⢇⠀⠀⢹⠀⠀⡍⠀⠀⠸⠀⡸⠀⡎
⠀⠀⠀⢸⠪⠭⢉⣛⣒⣛⠛⠚⠶⠚⠛⣛⣚⣛⡉⠭⠑⡟⡀⢸⠀⠀⢸⠀⠀⠇⠀⠀⡇⠀⠇⣼
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠧⣀⡇⠀⠸⠀⢸⠀⠀⠀⣃⡴⠊⠸
⠀⠀⠀⠀⡇⠀⠀⠀⣼⢷⡀⠀⠀⠀⢠⡾⣦⠀⠀⠀⢸⡀⠀⠈⠉⠒⠓⠒⠚⠒⠒⠉⠁⠀⠀⡇
⠀⠀⠀⠀⢡⠀⠀⠈⠁⠀⢁⣴⣶⣦⡉⠀⠈⠁⠀⠀⡏⡇⠀⠞⠋⠳⠀⠀⠀⠀⠞⠉⠳⠀⢰
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠘⣿⡟⣻⠁⠀⠀⠀⠀⢠⠄⢻⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠛⠁⢀⣀⠀⣀⠀⢼⢀⣸⡀⠀⠀⠀⠀⠻⡿⠀⠀⠀⠀⠀⡇⠀⠀⣀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⡴⠋⠒⡝⡤⠖⢀⣉⣛⠫⠽⠿⣓⠒⠶⣶⣶⣶⣶⣤⣬⡿⣍⡓⠛⣉⣐⡶⣚⠭⠭⠓⢄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡃⠀⢀⡴⠛⢒⡒⢛⡒⠢⠤⠤⠒⢬⠲⣦⣄⣢⣶⠖⠥⠦⠶⠶⠛⠛⠛⠚⠲⢄⠀⠀⢸
⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠉⢲⠋⠀⠀⠰⠃⠈⠁⠺⡄⠘⣀⠘⢣⡄⠰⠂⣰⠂⡤⢸⡇⣨⡃⠘⠁⠀⠀⠀⠱⡖⠁
⠀⠀⠀⠀⠀⠀⢣⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⠀⠀⠀⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠉⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠆⠤⠤⠤⢀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⡀⠤⠤⠄⠒⠒⠊⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠚⠛⠛⠛⠛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠐⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⣴⠟⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠞⠁⠀⠀⠀⠈⠙⠓⠲⠶⠤⠤⣴⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠶⢤⣄⣠⣤⡤⢾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠃⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠆
⠀⠀⠀⢀⣤⠴⠖⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⢤⣄⠀⠀⠀⠀⠀⣟
⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠘⢦
⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠈⢷
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠛
⢸⠃⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠚⠚⠛⠛⠒⡶⠀⠀⣀⣽⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡇⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⡇⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⠶⠚⢛⣛⣦⣀⢀⡀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⣿⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⡏⠀⠀⠈⢉⣭⣭⣭⣽⣶⠿⠶⠶⠶⠶⠿⠶⠶⠶⠶⠿⠶⠶⠶⠶⠶⠶⠶⠶⠾⠷⠶⠶⢶⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠰⠶⠶⠶⠶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⠛⠛⠛⠛⠛⠛⠛⠃⠀⢀⡼⠁
⣈⣧⣤⣀⣀⣀⣀⣀⣀⣀⣀⣧⣀⣀⣀⣘⣛⣛⣛⣻⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣴⣟⣁⣀⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣤⣤⣄⣠⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣼
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠠⠒⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠁⠀⠀⠀⠀⠈⠄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⢀⡀⠘⡄⠀⠀⠀⠀⠀⢀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⠄⠂⠀⠁⠀⠃⠨⠐⠀⠀⠒⠚⡑⠈⢤⠊⠡⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣈⠀⠀⠀⠐⣄⡄⠀⠀⠢⡀⠀⢀⠀⠀⠀⠔⠀⡀⠁⠐⣰⠃⣀⣀⣡⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠈⠉⢁⠃⠠⣌⠉⡀⠠⠀⠈⠂⠤⢠⣤⠤⠔⠊⠀⠐⠊⠉⠉⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⠤⡂⢀⠲⣌⠓⣷⡦⠴⠦⠤⠀⠠⣀⠦⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠙⠊⠡⠄⠀⠀⠀⠈⠁⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡒⣒⢒⣒⣒⣀⣀⣈⡁⠀⠐⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠛⢉⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡵⣾⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠾⠱⠿⠏⢺⣿⣏⣼⠋⢹⣯⡉⣯⡑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣼⢀⠴⠒⠿⠛⢁⣽⣯⠀⠀⠀⡛⠟⠚⢆⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡼⣶⢁⠀⠄⡤⡸⠈⣻⣦⡀⢄⠹⠆⣀⣀⡹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠿⡀⢻⠀⢶⢋⣽⡿⠯⢭⣟⠿⠖⡺⢿⡝⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠠⣿⣿⣤⠾⣸⠏⠀⠐⡭⠛⠛⠋⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠁⠁⠙⠋⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣔⡐⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠁
⠀⠀⠀⠀⠀⢀⡴⠀⠀⠈⠐⠂⠀⠀⠈⠂⢤
⠀⠀⠀⠀⡴⠡⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⡀
⠀⠀⢀⠎⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⢀⠔⠀⠀⠂
⠀⠀⡎⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠰⠃⠐⠀⠀⠀⠀⣠⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠓⢠⠀⠀⠀⠀⠀⠀⠀⠐⡀⡄⠘⠀⠀⠀⠀⠐⠀⢀⠔⠁
⠀⡆⡠⠀⠀⠀⠀⠀⠀⠠⠼⠀⠈⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠢⠤⠄⠀⠀⠀⠈⢁⠂⠀⠀⠀⠴⢀⡔⠁
⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠀⠠⠁⠀⠀⠀⠀⠀⠄
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄⠂⠠⠘⠂⡀⠁⢦⣀⠴⠤⣀⡀⠀⠀⠈⠀⠀⠀⠀⠨⠡⠀⠀⠀⠀⢀
⡄⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠤⠀⡀⠕⡊⠁⠁⠀⠀⠀⢼⠀⠀⠁⠀⠀⠂⠀⠉⠁⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠊⠃⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠠⢤
⠘⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠠⠊⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠐
⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⡠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠘⠒⠀⠀⠄⠀⠀⡀⢀⡠⠊⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠲⠤⠄⢀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢀⣀⣠⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠂
⠀⠀⠀⠀⠀⠀⠉⠑⠒⠢⠤⢤⣄⡠⠄⠀⡤⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠉⠀⠉⠀⠀⠀⣠⠤⠀⠀⠐⠠⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠐⠒⠤⡄⠴⡴⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠞⠣⠒⠗⠒⠀⠀⠀⢄⢠⠤⠄⠤⣤⠲⠈⠀⡠⠶⠈⠑⠀⠤⣥⡔⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣁⣂⣀⠁⠀⠈⠀⠀⠀⠀⠀⢂⠤⣎⣁⠠⠄
⠀⠣⠀⠐⠠⠄⠀⠀⠀⠤⠀⠀⠀⢠⠞⠀⠀⠀⢎⡀⠀⠀⠀⠀⠀⠀⠁⢀⡠⠐⠁⠀⢠⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠤⠄⠊⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠓⠦⢀⡀⠀⡠⠖⡩⠄⠀⠀⠀⢸⡄⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠐⠈⠀⠈⠀⠀⠄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⠭⠔⠉⠀⠀⠀⠀⠀⠘⡇⡞⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠠⠄⢠⡄⠀⠒⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⢀⡀⠀⠀⠀⠄⠠⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⠀⠀⠈⠀⢠⠘⣠⠀⠅⠤⠀⠃
⠀⠀⠀⠒⠂⠀⠀⠀⠀⠠⢀⡌⠀⠈⠀⠸⠁⠐⠀⠀⢀⠁⠉⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⠠⠴⠄⠊⠀⢀⠘⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠠⠃⠀⡜⠚⠀⠁⠂⠀⠀⢄⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠄⠀⣴⠄⢀⡀⠙⠂⡀⢀⡀⠀⠀⠐⠧⡄⠣⡀⠀⠀⠀⠀⠀⠛⡠⠄⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⡀⠀⠰⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠄⢠⣄⣀⠀⠀⠀⢀⠀⠐⠠⠤⠀⠉⠈⠒⠈⠁⠀⠀⠀⠀⠀⣀⡀⠀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠈⠠⢌⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠄⠀⠀⡐⠁⠀⠀⠀⠁⠀⠀⠀⢰⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⠄⠈⠀⢀⠀⠒⠂⠉
⠀⠀⠀⠀⠀⢆⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⠀⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠨⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⢀⠀⡀
⠰⢦⡀⠴⠄⠦⠔⠃⠀⠀⣀⠤⢠⡄⡶⠠⢄⠀⡆⠀⠀⠀⡀⠀⡝⠤⢀⡀⢠⡄⢤⣤⠀⣀⡠⠄⠀⡢⠄⠠⠄⠀⠀⣀⠤⠀⢨⣅⡰
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠐⢠⠀⡥⠀⣶⠀⠀⠀⠀⠀⠀⠀⠁⢀⢈⣛⡀⠀⢠⠀⠀⠘⠡⡤⠀⠁⠀⠕⠀⢀⡀⣤⠀⠀⠂⠁⠁
⠀⠀⠀⠀⠀⠀⠠⠄⠆⠀⠀⠐⠈⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⢩⠁⠀⣠⡀⠑⠓⠐⠆⠀⠀⠀⠀⠈⠉⠀⠉⠀⠂
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⠀⣀⣀⣈⡀⠴⠗⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠤⠀⠀⠀⠠⠀⠀⠃
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⢬⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠁⠀⠈⠁⠉⠀⠦⠤⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠷⠀⠀⠐⡶⠀⡀⠀⠀⣀⡀⠀⠁⠀⠄⠀⠺⢷⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠔⠀⠀⠠⠤⠀⡀⠠
⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠌⠓⠀⠀⠁⠀⠒⠀⠀⠀⠑⠠⠌⠂⠉⠐⠂⠉⠀⠀⠀⠀⠔⠀⠑⠒⠂⢀⣁⡀⠀⠈⠁⡠⠀⠀⠀⠄
⠀⣤⣄⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠊⠆⠀⠰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⡀⠀⠀⠀⢈⠄⠠⠐
⡇⠀⠀⠁⢄⠘⠦⣄⢠⡄⠔⠀⠘⣰⠀⠀⠀⣀⣦⠄⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠡⠀⠀⣤⠀⠀⠀⢠⠁⠀⢠⡏⠀⠀⠀⠀⠀⡘⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠄⡀⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⡀⠘⢇⣀⠀⣀⠀⠔⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⢋⡤⠀⠀⠀⠀⠐⢀⠁⠀⠀⠠⠒⠈⠉⠉⠁⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠀⠀⠀⠇⠀⠀⠀⠀⢀⠄⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠤⡀⠀⢀⠀⠀⠀⡀⠀⡇⠀⠀⠀⠀⣦⠀⠀⠀⢰⠀⠀⠃⢄⣀⡀⢀⣀⡰⠞⠀⡈⠀⠀⠀⠀⠀⢀⠀⠘⠀⠀⠀⠊⠔⣈⠉⠋⣀⠲⠀⠀⡄
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⣊⠤⠖⠶⠤⡑⠀⠀⠡⠀⠀⠀⠀⠄⠀⠀⡐⠀⠀⣀⡀⣀⣀⠈⠁⠈⠣⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠊⠀⠀⠁⠀⠀⠀⠀⠀⠑⠀⠀⠀⠐⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⡧⠼⣀⠀⠀⠀⠀⠀⠆⠀⠴⠀⠀⢀⠤
⠀⠀⠀⠀⠓⠈⠜⢉⠲⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣄⣰⠁⠀⠀⠀⠀⢠⠈⠁⠜⠂⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⢶⠀⠃⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣀⣠⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⣈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠋⠙⠛⠋⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⣤⠀⠀⠀⣤⠤⠤⠤⣤⠤⠄⠤⣤⠠⠤⠤⣤⠤⠤⠤⢤⡤⠤⠤⠤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠦⢾⢟⣶⣲⣚⣛⡓⠠⠒⠿⠶⠶⠾⠿⠷⠶⠾⠟⠲⠶⠶⠻⠿⢶⠴⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠲⢰⡀⢸⢸⠀⠀⠀⠀⡃⠀⠀⡃⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠌⡆⢸⢸⠀⡠⢄⠀⠀⠀⠀⡆⡀⠀⠠⠀⢀⡄⡀⡾⠀⠖⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠅⠚⡑⢸⠸⠂⠀⠀⠀⢀⡀⠀⡇⠀⠀⠠⠀⠈⠀⠀⠁⢀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠠⡞⡃⢸⠘⠀⠀⠀⠀⣠⠟⠀⡷⠥⠄⠀⠀⠀⠰⠁⠀⠀⠈⠈⠁⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠍⠅⢸⢸⠀⠀⠀⠀⡇⠀⠀⠡⠤⡀⠦⠄⡄⠒⢀⠒⠂⠀⠀⠀⠉⠀⠁⢀⡀⠀⠀⠀⣔⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢋⠃⠈⠀⠀⠀⠀⠀⠃⠀⠀⠈⣿⣧⠀⠀⢠⠓⠄⠤⠄⠀⠠⠠⣤⡤⣤⢤⢴⡇⠀⠀⡯⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠟⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣤⣤⣤⡀⠀⠠⠤⠐⠀⣄⠠⠧⠸⠂⢛⣵⣶⣶⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠺⠦⠦⠓⠚⠛⠛⠛⠛⠛⠛⠛⠒⠒⢸⢹⠙⠛⡏⡗⠒⠂⠒⠒⠒⡗⠒⠒⠒⠒⢺⠇⡏⠛⡇⡇⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠘⠐⠂⡇⠇⠀⠀⠀⢄⠐⢓⣢⠀⠀⠀⠘⢸⠃⠀⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⡁⠀⠀⠀⠀⠀⠈⠉⠙⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⡀⠶⠀⠀⠆
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠤⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠒⠲⡖⠀⡀
⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠊⠁⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠖⠓⠒⠒⠒⠿⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⠀⠐⠆⠀⠖⠈⣦⠠⡄⠶⠴⠲⠊⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢄⣀⠜⠁⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡀⠀⢠⠀⠀⡤⠀⢰⡇⠐⠟⢠⠦⡀⠐⠀⠀⠀⠀⠀⠀⠰⠀⠀⡆⠀⡄⡛⠉⠉⠙⠛⠛⡟⠒⠒⠒⠒⠒⠀⠐⠚⡇
⠀⠀⠀⠀⠀⠀⠈⢡⠀⠀⠀⠂⠀⠘⠀⣈⣤⣤⣶⡄⠘⢧⠀⠀⢀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠐⠾⣄⣴⣾⣿⣿⣿⣿⠀⠀⠀⠀⡀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠿⠍⡉⠛⠛⢳⠲⠶⠤⠤⠤⠄⠀⠀⡏
⠀⠀⠃⠀⠀⠀⠀⠀⠀⣿⡔⠉⠙⣿⣿⣿⣿⣿⣿⣇⠐⠂⠈⠁⣀⡠⠂⠑⡀⡀⠀⠀⠀⡀⠃⠀⠘⠀⢘⠁⠀⠀⠀⠻⣦⣄⡘⣄⢂⠀⠀⠀⠀⠀⠀⢃
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠉⢿⡀⠀⠘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⡄⠐⠡⠀⠀⠀⠀⠀⢁⠀⢰⡆⠠⢘⡇⠀⠀⠀⢠⡏⠀⠈⠹⠃⠠⠠⢤⡜⣉⠀⠉
⠀⠀⠀⠀⠀⢶⠀⠀⠀⣀⣘⣧⠀⣴⣿⣿⣿⣿⣿⡟⢂⣠⡀⠀⠀⠀⠀⠄⠂⠀⢠⣴⠀⠈⡀⢸⣇⡔⠋⠀⠒⠒⠂⠈⠀⠀⡀⡀⠀⠀⠀⠤⠀⠀⣀
⠀⠀⠐⡀⠀⠈⠇⠀⠀⠀⠉⠙⡆⣿⣿⣿⣿⣿⣿⢀⡀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⢀⢧⡽⠂⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⠉⠒⠂⠒⠀⠀⠉⣤
⠀⣴⣿⣿⣦⠀⠈⠀⠛⢲⠀⠀⠡⢈⣿⣿⣿⣿⡇⠈⡀⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠆⠀⠀⠂⠀⠒⠀⠀⠀⠀⠀⠈⠁⠒⠀⣤⣀⡀⠀⢹
⠰⣿⣿⣿⣿⣷⠀⡆⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧⡀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⡿⠽
⠀⣿⣿⣿⣿⣿⣿⠀⡸⢃⣠⠀⣾⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡰⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⢣⠃
⠀⠟⠛⠛⠛⢻⣉⢱⣧⣬⠛⠋⠉⠉⠉⠉⠛⠟⣿⣷⣶⣾⣿⣾⣿⣿⣻⣿⣶⣶⣸⣿⠀⡄⠚⢼⠀⠀⠰⠃⢠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⣼
⠀⠠⡄⠡⠀⢀⡛⠉⠘⠙⠀⣀⠀⠀⠀⠀⢀⣀⠉⠋⠛⢻⣿⡶⠒⠛⠻⡿⠿⠶⢿⣿⠀⠘⠀⢸⠷⠠⠃⠘⡎⡐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡏⣞⠁
⠀⠀⠚⠓⢶⣤⣄⣤⣦⡤⠞⠛⢪⡵⡟⠻⢿⣿⣿⣶⡤⠞⠙⡁⠤⠀⠀⠀⠀⠀⠈⠉⠐⡂⠐⠿⠃⠀⠀⠄⠘⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣼⠿⠁
⠀⠀⠀⠰⠹⢿⣟⣿⡿⠓⠈⠑⠀⣠⣄⠀⠤⠦⠀⠈⠚⠓⢀⠀⠀⠀⢠⣢⠄⠀⠶⠖⠂⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⠄⠀⠀⠀⠀⢠⣿⡦⠤⠶
⠀⠀⠀⢢⡀⠏⢡⡏⠵⠀⠠⡄⢀⣉⠀⠀⠀⠀⠀⠀⠀⠉⠏⠀⡀⠀⠀⠐⠂⠀⣼⡟⠀⣤⣄⣀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠓⠶⠆⣁
⠀⠺⠊⠉⣭⡄⠁⠐⠀⠤⡀⠁⠀⠀⠉⠀⠀⠀⠀⢠⣀⢠⣄⠀⠀⢀⣀⠀⠀⠀⠈⠉⠀⠻⢿⣯⡛⠦⣤⡀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠚⠛⢋⣀⣀⡤⢤
⠀⠀⠀⠀⠛⠘⠛⠲⠦⠄⠀⢹⣶⣶⣤⣀⣀⢀⣠⣤⡞⠁⠉⠍⠉⠉⠉⠀⠤⠠⠀⠀⠀⠀⠀⠈⠙⠻⢾⣿⣶⢤⣄⣠⣶⠟⠿⠶⠶⠖⠋⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⠛⣿⠛⠋⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣿⢿⣥⢀⠀⠀⠀⠀⣀⣨⠥⠆⠒⠛
⠀⠀⠀⠀⠀⢤⠤⢄⣀⠩⠁⠒⠀⠀⠀⢀⠏⠐⠀⠀⠀⠀⠓⠒⠒⠒⠒⠀⠒⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠒⠒⠀⠈⠙⠖⠓⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⠀⠀⠀⠀⠀⠤⠄⠠⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠐⠂⠀⠀⠀⠂⠀⠂⠂⠁⠂⠂⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐

⢀⡀⠀⠠⠅⠀⠀⡠⠂⠈⠢⡈⠁⠠⠄⠄⠀⠂⠀⢀⠴⠀⠐⠛⠻⠀⠸⠇⠄⠄⢄⠀⠀⠊⠅⠀⠀⡠⠊⠉⠠⢀⠇⠘⡄⢀⠀⠀⠀⢀⠴⠀⠐⠛⠳⠁⠸⠔⠠
⠀⠀⠈⠀⠐⠀⠂⠁⠀⠀⢀⠜⠀⠀⠖⠀⠨⠀⠠⡀⠀⠀⡠⠊⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠂⠑⠄⠀⢀⠘⠀⠀⠖⠂⠀⠀⠀⠀⠀⠀⠠⠊⠀⡀
⠺⠂⠳⣀⠀⠈⠈⠒⠚⠀⠁⠀⠀⠀⠠⠀⠠⠃⠀⠈⠠⠈⠀⡔⠓⡜⢂⠀⣾⠃⠺⢆⠽⣠⠀⠈⠈⠒⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠔⠂⠔⢂⠀⣺
⠀⠘⠂⠁⠀⠀⡠⠐⠀⠁⠐⠠⠀⠀⠀⠀⠈⠀⠀⠀⡀⢀⠈⠲⢪⡖⡛⠀⢙⠒⠂⠛⠛⠏⠀⠀⠠⠐⠀⠀⠐⠢⠀⠀⠀⠀⠌⠀⠀⠀⢀⠀⠈⠶⢢⠔⡛⠀⢛
⠀⠀⠀⠀⠀⠀⡶⠤⣂⢴⡤⢴⡆⠀⡏⠀⠄⢩⠂⠀⠀⠆⠐⠈⡧⢾⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⡶⠄⣂⢠⡆⢶⡆⠀⡍⠀⠈⢩⠂⢰⠀⠀⠀⠈⡳⢮⠃⠀⠈
⠀⠀⠉⠀⠀⠀⠉⠉⠷⠘⣣⠍⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠆⠀⠐⠂⣀⢀⡀⠁⠀⠀⠉⠀⠀⠀⠉⠉⠳⡜⣫⠝⠁⠀⢠⠀⠀⡀⠀⠀⠑⠀⠆⠀⠐⠂⢀⢀⡀
⠀⢀⠊⠀⠈⢢⠀⠀⠀⠈⠁⠀⠀⠀⠸⠄⠀⠄⠀⠀⢀⣲⠦⠀⠀⡁⠀⠀⡀⣥⠀⠠⠈⠀⠈⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⣀⣶⠦⡀⠰⡉⠀⢀⡀
⠀⢨⠀⠀⡠⠂⢀⢀⣑⠄⢀⣴⠤⠂⠀⠀⠀⣦⣡⡾⠟⠁⠀⡀⠀⠀⠂⠀⠀⡀⠀⢀⡀⠀⡀⠂⠀⢂⡐⠄⢀⡕⠴⠠⠀⠀⠀⣤⣱⡾⠟⠋⠀⠀⠀⠁
⡄⢢⠘⠈⠀⠀⢈⠠⡔⠲⠵⠷⣄⠒⠀⠀⠈⢜⠁⠀⠀⠠⠊⠀⠀⠀⠒⠐⠀⠀⠄⠡⠚⠈⠀⠀⠄⠀⡴⠲⠾⠶⣄⠂⠀⠀⠀⢎⠀⠀⠀⠠⠊⠀⠀⠈⠂
⢀⡁⠀⣀⠀⠀⠈⢐⠂⠠⠄⢂⠄⢀⠀⢠⣂⠀⠀⠀⠀⠀⠘⠀⠇⡈⣄⠀⠀⠀⢀⡀⠀⣀⠀⠀⠐⢐⠂⠠⠄⢂⠄⠀⠀⣘⣀⢀⠀⠀⠀⠀⠸⠁⠇⡘⡀
⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⡯⠇⠀⠀⢀⠀⡰⠂⠺⠀⠀⠠⠀⠌⠢⠰⠁⠄⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⣠⡿⠇⠀⠀⢀⠀⡯⠒⠲⡀
⠀⠃⢀⠀⠀⠀⣠⠄⠒⠀⠲⠀⠐⠂⠁⠂⠈⠁⠐⠀⠀⠀⠁⠀⠀⠀⠀⢸⠀⢀⠀⠀⢀⠀⠃⠀⢠⢄⡒⣀⢶⣄⠐⠂⠑⠐⠈⠁⠐
⠀⠂⠁⠐⠊⠀⠃⠀⠀⠀⠀⠀⠆⠀⠀⡀⠀⠀⡀⡀⡀⠀⠄⢀⡀⠀⠀⢀⠀⠀⠀⠒⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⠀⡀⠀⠄⠀⡄
⠀⠀⠀⠀⠀⠀⡆⣀⠠⠀⠒⣀⠁⠀⠦⢷⠀⠘⡉⠁⡙⠀⠀⠀⠉⠀⠀⠉⠀⣀⠀⠀⠀⠀⠀⠀⣂⣀⠠⠀⠒⡀⠁⠀⢨⠷⠄⠘⠃⠉⠛⠀⠀⠀⠉⠀⠀⠉
⠉⠀⠉⢉⡄⠀⠀⠀⠀⢀⠨⠀⠁⢀⣀⡀⠄⠀⠠⠀⠀⡀⠴⣀⢦⠀⢀⡁⠉⠀⠉⠀⠋⢉⡆⠀⠀⠀⠀⠀⡤⠀⡃⠀⢌⡄⣄⠀⠃⠀⠂⠠⠴⣐⢦⠀⢀⡀⠉
⢉⣑⠈⢛⠄⠀⠀⡠⠊⠈⠐⡈⠃⠀⠖⠀⢀⠂⠀⣀⠖⠐⠒⠋⠺⠀⠰⡃⣀⣄⢢⣄⠨⢋⠖⠀⠀⡀⠊⠈⠀⢀⠃⠐⡔⠀⠀⠀⠀⠀⠔⠐⠒⠉⡳⠁⠘⡏⣑
⠀⠀⠀⠀⠀⠀⠪⠀⠀⠀⢀⠘⠀⠀⠔⠀⠰⠀⠨⡀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢋⠐⠀⠀⢀⠜⠀⠀⠆⠂⢀⠀⠈⡀⠀⠀⡠⠈
⠿⣆⡵⣠⠀⠀⠈⠃⠈⠃⠀⠀⠀⠀⠂⠀⠀⠃⠀⠈⠂⠊⠀⡐⡁⣜⢂⠀⣷⢃⡿⢂⠵⣠⠀⠀⠈⠁⠈⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠈⠂⠂⢀⡜⡁⣄⢂⠀⣾
⠒⠛⠈⠁⠀⠀⡀⠂⠁⠀⠁⠢⠀⠀⢀⣀⣃⣀⠀⠀⠤⠀⡀⠒⠢⢔⡚⠁⠙⠐⠓⠛⠋⠇⠀⠀⠀⠀⠉⠉⠁⠢⠀⠀⢀⣀⣂⣀⠀⠀⠤⠄⡈⠖⠢⢔⡺⠀⢋
⠀⠀⢀⠀⠀⠀⡷⢒⣒⢒⡒⢶⡇⠐⡅⠀⠀⢨⠂⢀⠀⠆⠀⠈⡗⠺⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⣶⢒⣂⢒⡒⢶⡆⠐⠍⠀⠀⢨⠂⢰⠀⠀⢐⠈⡷⢮⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠐⠌⠶⡜⣣⠍⠀⠀⢀⠀⠀⠈⠀⠀⠁⠒⠒⠀⠐⢂⠀⠀⢀⠀⠀⠀⠀⣀⡀⠀⠙⠌⠷⠜⣯⠈⠁⠀⠀⠀⠀⡀⠀⠀⠁⠒⠂⠀⠐⠂⠀⠀⢀
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠁⠀⠀⠀⠰⠀⠀⠆⠀⠀⣐⣶⠶⡄⢠⣁⡀⣀⡀⡤⠀⠠⠀⠀⠀⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⣶⠶⡀⠰⣁⠀⣀⣀
⠀⢠⠀⠀⡀⠂⢀⢀⡠⠄⠠⡔⠴⠂⠀⠀⠀⣤⣦⡿⠟⠁⠀⠈⠀⠀⠂⠀⠀⡀⠀⢠⠀⠀⡠⠂⠀⢀⡈⠄⠨⡔⢤⠢⠀⠀⢠⣔⣤⡿⠟⠁⠀⠄⠀⠁
⡔⢢⠊⠀⠀⠀⠀⠀⠌⠒⣋⠶⡴⠒⠀⠀⠈⢊⠀⠀⢀⠄⠊⢀⡀⢈⠁⠂⠉⠀⣔⠠⠈⠁⠀⠀⠁⠀⡌⠒⣟⠧⢴⠒⠀⠀⠀⢎⠀⠀⠀⠠⠊⣀⡀⢈⠁⠂⠉
⢀⡁⠀⣀⠀⠀⠈⠜⠂⠰⠄⢀⠄⠀⠀⢠⡇⠀⠉⠁⠀⠀⠘⠒⠃⡌⣄⠀⠀⠀⢀⡁⠀⣀⠀⠀⠀⠈⠊⠡⠤⢈⠄⠀⠀⢘⡄⠀⠉⠉⠀⠀⠘⠒⠂⡬⣄
⠁⠈⠂⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣨⠯⠇⠀⠀⢀⠀⡸⠂⠸⠀⠀⠠⠁⡀⠃⠰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠨⡀⠀⠀⣨⡿⠅⠀⠀⢠⠀⡯⠂⠒
⠀⠁⠀⠀⠀⠄⣠⢀⡔⠭⣳⠄⠘⠄⣉⣀⢈⡁⠜⠀⠀⠀⠁⠀⠀⠀⠀⢘⠀⠀⠀⠀⢀⠀⠃⠄⣠⢆⡔⢭⣳⢄⠘⠆⣉⣀⣈⡁⠜⠀⠀⠀⠁
⠀⠂⠀⠐⠀⠀⠁⠂⠀⠀⠀⠐⠇⠀⠀⡀⠂⠀⣀⣀⡀⠀⠄⢠⡀⠀⠀⢰⠀⠀⠀⠂⠀⠑⠀⠀⠡⠀⠀⠀⠀⠈⡆⠀⠀⠠⠂⠀⣀⣀⣀⠀⢀⢠⡀
⠀⣀⠀⠀⠀⠀⣎⠀⠄⠀⠀⢈⠀⠀⠲⢿⠀⠘⠅⠀⠜⠀⠀⠀⠀⠀⠀⢀⠀⡀⡀⣀⠀⠀⠀⠀⣂⠀⠄⠐⠀⣀⠀⠀⢰⡷⠀⠸⠁⠀⠘⠀⠀⠀⠀⠀⠈⢀
⣁⣀⣉⣈⡀⠀⠀⠀⠀⠀⡈⣀⣁⢀⣀⣀⣄⠀⠠⠠⠀⣀⡒⣑⢢⡀⢀⣀⣁⣀⣉⣀⣃⣀⡂⠀⠀⠀⠀⢀⡤⣀⣃⢀⣌⣄⣀⠀⠃⠀⠂⣀⠐⣑⣦⡀⢀⣀⣁
⣿⣿⡿⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⠿⠋⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣹⣦⣄
⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣴⣶⣾⣿⣿⡿⣆
⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠆
⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠁
⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠛⠛⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⡿⠛⠃
⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣄⡀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣄⣀⣀⣀⣀⣈⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣟⢻⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇⢸⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠸⣿⣿⡿⠁⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢯⣿⡿⢿⣿⣮⣿⠻⣷⡼⡛⡙⠈⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠉⠓⠯⣙⡤⠇⠃⠄⠊⢰⠀⠀⠀⠀⣀⣤⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠁⠐⠀⠀⠀⠀⠈⠒⠻⠶⠻⠆⠀⠀⠓⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⡉⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⠀⠃⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠸⡿⣿⣿⢿⣧⣴⠀⠀⠀⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣿⣿⠀⠀⠀⠀⢤⣄⣇⠀⠀⠀⠀⠀⠀⠀⠦⣿⣋⠈⠀⠀⠀⠀⠀⢠⣤⠲⠶⢷⢶⣾⣿⡼⠿⠿⠿⠏⢩⣤⣤
⠛⠛⠀⠀⠀⠀⠀⠙⠛⡃⠀⠀⠀⠀⠀⢸⣿⣿⠿⠀⠀⠀⠀⠀⢀⢸⣧⣤⣆⣿⣿⣽⣆⡇⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⢀⡀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⢀⠀⣀⣀⣤⣠⠴⣷⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣠⣤
⣠⡤⢀⣻⠇⠀⠀⠀⠀⠀⣖⠀⠀⠀⣴⢿⢿⣾⠴⢶⣴⣿⣿⣿⣷⣾⣿⣿⠼⢿⣿⣿⣿⣿⠀⠀⣀⣤⣾⣿⣿
⣾⣧⢉⠄⠃⠀⠀⠀⠀⠐⠋⢠⣶⣶⣶⣶⣶⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⢼⠀⢸⣿⣿⣿⣿⠓⢮⣍⣿⣿⣿⣿
⣭⣸⠻⡛⠒⠓⠛⠐⠛⠾⠿⠿⣿⣷⣿⣾⣴⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣏⣤⣌⣿⣿⣿⣿⠀⠀⠉⢻⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠞⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣌⣛⣦⡤⢤⣀⣀⣀⣀⠀⣀⣀⡀⠀⠀⠉⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣳⣿⠻⢷⣽⠭⣿⠼⣯⣉⠉⠳⣦⠀⠀⠀⠉⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⣿⠉⠻⢦⣉⠲⣄⡀⠀⠉⠓⠒⠛⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣹⠆⠀⠀⠀⠉⠳⣬⡙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⢻⡄⠀⠀⠀⠀⠀⠀⠙⠶⣬⡑⢦⡀⠀⠀⠀⣤⡄⠒⢲⣬⣭⡵⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢾⠛⡛⢻⣿⢦⡀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣌⣩⣭⣵⠶⠞⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠀⣽⠀⢃⠸⡇⠙⢧⣙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠈⠁⣸⡏
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠙⢦⣈⣹⡄⠀⠀⠙⠳⢦⣝⣛⣡⠤⠤⣤⣤⣛⡿⠛⢳⣄
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠙⠉⠉⠀⠀⠀⠙⣦
⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠰⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢢⠀⣿
⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣦⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢿⣅⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣸⣿⠀⢸⡇
⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠘⢿⡿⠃⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠉⠻⣆⠀⠀⠀⠀⠀⠈⢿⣿⠏⠀⢸⡇
⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠉⠉⢻⡿⠛⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠟⠁⠀⠀⠀⠀⠀⠀⢀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⢧⡉⠛⠛⣷⠀⠀⠀⠀⠀⢻⡋⠉⣠⡶⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠀⡎⠙⠷⣾⣧⣦⣀⣤⣀⣀⣌⣷⡞⠋⡆⠰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⠁⠀⠀⠈⠉⠈⠛⠈⠉⠉⠉⠉⠀⠀⠁⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⠻⣷⡶⢤⣀⣀⠀⠀⠀⠀⣀⣠⣷⣶⡿⣋⣀
⠀⠀⠀⠀⠀⠀⣠⡴⠲⠶⠶⣶⠛⠉⠁⠀⠀⠀⠈⠉⠑⢲⡖⠋⠉⠁⠀⠀⠀⠉⠉⠓⣶⠶⠶⠶⢦⣄
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡆⠀⠀⠀⠘⢷⡀
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠈⣷
⠀⠀⢀⡼⠃⠀⠀⠀⠀⠐⢷⡄⠈⢉⣉⣉⣉⡀⠶⢀⡀⠘⠃⢀⡀⠤⣄⣉⣉⣉⡉⠁⢠⣾⠀⠀⠀⠀⠀⠘⢇
⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠛⠛⠿⠿⠿⢿⣿⣿⡿⠿⠿⠿⠟⠛⠿⠿⠿⢾⡇⠀⠀⠀⠀⠀⠀⠘⣦
⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠙⠶⢤⣤⢤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⡤⠶
⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠙⢦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⢉⡏⠁⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠁⠈⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⣴⠿⠓⠉⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠓⠚⠿⣦
⣤⣤⣤⣤⣤⣤⣤⣽⣷⣶⣶⣮⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣵⣶⣶⣾⣭⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡿⠟⠻⠿⠿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣏⣠⣀⣤⣤⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⡆⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠠⢄⠀⠀⠀⢀⠀⠈⠄⠀⠀⠀⠀⠀⢀⠁⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠉⠆
⠀⠀⠀⠈⠙⠤⠄⢠⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠄⠀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠡⠂⠀⠀⠀⠀⣔⠃⠋⠓⡄⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠈⠁⠀⠠⠀⠂⣀⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⡀⢀⢈⠉⠈⠀⠀⠀⠀⡌⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄
⠀⠀⡀⠀⠀⠀⠀⠈⠂⡀⠀⠂⡀⠈⠁⠁⠄⠀⡁⠀⢀⠠⡄⢀⠈⠀⠀⠠⠄⡀⣀⢀⠀⣼⣇⢄⠀⠀⠀⠈⠀⠀⢀⡀⠠⠲⠀⢀⡀⢀⠀⠀⠐⠁⢁
⠀⢰⡀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡘⠀⠀⠁⠐⢁⡄⠀⡀⠈⠑⡽⠋⠉⠀⠀⠀⠀⠀⠀⠀⠁⠈⣤⡠⡤⣤⡴⡀⠈⢤⡀⠔⢀⠠
⠀⠀⣦⣄⡠⠆⠀⠒⠁⠉⠀⠀⠀⠀⠡⠄⠠⡁⠀⠀⠁⠀⠀⠀⢀⠌⠀⠀⢨⡕⠀⠀⡀⢠⠀⠀⠀⠠⠀⠀⠠⠀⠦⠄⠘⣄⢋⠀⠤⠂⠀⠀⠀⠀⠆
⠀⠀⠀⡀⠂⠀⠀⢀⡀⠀⠀⠤⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠐⠀⠄⠀⠀⢀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠠⠄⡀⠀⠀⢘⠀⠀⠀⠀⠀⡊⠀⠀⠀⠀⠰
⠀⠀⠀⠈⠀⠀⠀⠀⠀⡌⠡⠒⢈⠀⠂⠀⠀⠀⠀⠐⠀⢀⣀⠈⠀⠀⠣⠊⠁⠆⠀⠀⠀⡀⠀⠀⠀⠔⠁⠀⠀⠀⠰⠀⠅⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠂⠀⠀⠈⢠⠀⠀⠀⠀⠀⠀⠁⣀⣀⣀⣀⣄⣄⣀⣀⣀⠀⠀⠀⠈⠁⠀⠀⠠⠄⠀⠊⢀⡐⠊⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⢠⣶⣿⣿⣿⠛⢛⡛⡻⡟⣿⣿⣿⣶⡄⠀⠁⠀⠉⠀⠠⠸⡈⣈⠀⠀⠀⠀⠀⠡⠀⠀⠀⠠⡐⠂⠠⠁⠀⠀⠀⠀⠀⠑⠄
⠀⠘⡒⠐⠈⠀⠀⢀⠀⢀⣿⢛⡟⣿⢻⡛⢛⣿⢛⡛⠻⠟⠟⢻⣿⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠃⢀⠀⣂⢀⣀⣀⣀⠀⠀⡀⡂
⠀⠀⢡⠀⡄⠀⠀⠀⠀⠘⣿⢨⠁⢹⡜⣿⢸⣿⢈⡁⡇⠀⠀⡆⣿⠃⠀⠀⠀⠀⠐⠁⠀⠀⠀⠄⡂⠀⠀⡁⠀⠁⠀⠀⠇⠀⠍⠛⠋⠀⠛⠀⠐⠈⠁
⠀⠀⠀⠆⠃⠀⠀⢀⠀⠀⠹⣼⣼⣮⣭⣿⣼⣿⣼⣧⣡⣬⣤⣴⠏⠀⠀⠀⠀⠰⠀⠀⠀⡐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠈⠠⠀⠐⠀⠀⠀⠉⠛⠛⠛⠻⠛⠛⠛⠛⠛⠉⠀⠀⠀⠂⠀⠀⡀⠀⠠⠄⠁⠀⠀⠀⠀⠈⠀⠀⠀⢐
⠀⠀⢅⠀⠀⠀⠂⠀⠒⠠⠀⠀⢠⠐⢀⠀⠀⠠⢀⠀⠠⡀⠀⠀⠀⠀⠄⠀⠀⠠⡈⠀⡀⠔⠀⠄⠀⠀⣀⣀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⢉⠀⠸⠀⣕⠐⠘⠂⠀⠁⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠐⢀⠗⠁⠀⠀⠉⠄⠀⠀⠂⠈⠂⠀⣈⠀⠀⠀⠀⢙
⠀⠀⠀⠀⢀⠀⠁⠈⠀⠀⠀⡀⠀⠐⢀⠀⠀⠘⠕⠀⡀⠀⠀⢘⡂⠀⠀⠆⠀⠀⣀⠀⠀⠀⢱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⠅⠀⠀⠀⠀⠀⠀⠑⠂⠀⠘⠀⠰⣀⣀⠬⠃⢈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠆⠀⠁⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠐⠬⠄⠀⠀⠠⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⡄⠀⠀⠀⠀⠀⠀⢀⠰⠸⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠁⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠢⠀⠀⠀⠀⠒⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⣀
⠀⠖⠃⢀⠄⢀⠀⠒⠑⢀⠀⠀⠠⠀⠀⠀⠃⠁⠦⠀⠐⠀⠅⠀⢸⠀⠁⠈⠀⠂⠀⠐⠀⢴⣊⠖⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠀⠀⠀⢈⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠂⠠⠤⠆⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⠀⠀⠀⠠
⠀⠀⡀⠀⠀⠀⢀⠀⠀⠨⠂⠀⠀⠠⠊⠀⠀⢄⠀⠀⢀⡈⠀⠂⢰⣀⡠⡰⠱⢡⠄⠀⠀⠀⠀⡆⠀⠠⠀⠠⠀⠁⠒⠈⠀⠤⠀⠀⠀⠀⣠⠀⢀⠠
⠀⠈⠀⠀⠀⠄⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠁⠀⠀⠠⢇⡀⠀⢸⢆⡀⠄⢠⠀⠀⠀⠀⠀⠀⠀⡤⠠⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⠀⠀⠀⠐
⠀⠢⠀⠠⠄⠂⠁⠀⠀⡀⠀⠀⡠⠔⠊⠁⠁⠀⠀⠈⠍⠀⡄⠀⠀⠈⠚⠚⠁⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠰⠀⠀⠀⡄⠀⠀⡀⠀⠀⡄⠀⠄
⠀⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠑⠂⢀⠉⠀⠐⠀⠄⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠁⠀⠠⠤⠄⠀⠁⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠓⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⡀⠀⡠⠊⠡⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠒⠂⠂⢡⠀⠈⠁⠀⠀⠀⠈⠁⠁⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠒⠈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠤⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠂⠀⠃⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣀⣀⠀⠀⠀⠀⡠⠒⢄⡀⠄⠒⠄⡀⣀⠠⠀⠄⡀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⢀⢴⠃⡠⠊⠀⠀⠀⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠑⢄⠀⠰⠢
⠀⠀⠀⠀⠀⠀⠀⡈⣨⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⣆⠆
⠀⠀⠀⠀⠀⠀⠰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⢀⣀⣀⣀⡀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢀⣀⣀⣀⡀
⠀⠀⠀⠓⠚⠉⠛⠛⠛⠿⠷⠶⠶⢶⣦⣤⣤⣤⣀⣀⣀⡀⢀⣀⣀⣠⣤⣤⣤⣴⣶⠶⠶⠾⠿⠛⠛⠛⠉⠁⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠉⠀⠀⠉⡏⠉
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡐⠈⠐⠃⢁⡼⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡈⠙⠃⠁
⠇⠀⠀⠀⠀⠠⠴⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠦⠄
⠘⢄⠀⠀⠀⠐⣚⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣓⠂⠀⠀⠀⡠
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠂⠀⠠⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠀⠠⠄⠨⠁⣈⣁⡀⠥⠠⠄⠀⠒⠂⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣤⣀⠀⣄⣀⣈⡀⠀⠁⠀⠈⠀⠀⠀⣈⣀⡤⡰⠂⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠈⠳⢦⣤⣭⣍⣛⣢⡄⠀⣐⣙⣥⡍⣴⠚⠁⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⠑⠀⠀⠉⠀⠈⠁⠈⠳⠋⠉⠁⠀⠀⠀⠙⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡀⠀⠁⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠼⠀⠀⠀⠀⠀⢀⡀⣀⠀⢼⡀⠀⡀⠀⠀⠀⠀⠀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡄⠠⠡⡀⠠⡏⠠⢄⡿⠚⢿⠈⡄⠑⡄⢀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠚⠦⣌⠊⠴⢓⢴⣛⠡⣀⠿⢿⣶⠚⡁⠁⢠⢈⡮⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠒⠶⢖⡛⢷⣾⣷⣄⠀⡈⠳⡤⠤⡤⠠⠄⠉⠓⠤⠤⠤⠤⣚⢃⣡⠿⠖⠂⠊⣁⠤⠓⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⣨⠉⠓⠓⠤⠍⣈⠉⠉⠓⠒⣶⣦⡠⠶⣷⠷⠟⢛⡿⣿⠥⠶⠒⢊⡉⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠐⣚⣭⡍⠀⠀⠀⠀⠀⠀⠀⠉⠉⠺⢭⣀⣁⣁⣠⠐⢉⡡⠁⠀⠀⠀⠀⠀⢛⠒⠂⠠⠀⠁
⠀⠀⠀⡔⠀⢐⡩⢴⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢍⡐⠒⠆⠀⠂
⠀⠀⠀⠈⠄⢡⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⠉⠂⠈
⠀⠀⠀⠀⠀⢺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⠤⠴⢒⡟⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠙⢤⡀⠀⠁⠐⠚⢄⡀⣠⠴⠐⠈⠀⠀⣠⠴⠭⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⣭⣮⢐⣦⣄⢀⢀⣀⠀⡀⣀⣀⣤⣾⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠗⠹⠕⠋⠁⠀⠀⠀⠀⠀⠀⠉⠉⠻⢽⡟⠋

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠁⠀⠁⠀⠀⢀⠠⠤⠀⠀⠀⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⢀⠤⡀⢤⠰⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠦⠤⠐⠀⠈⠀⠠⠀⡀⣤⠀⠤⢤⡌⠛⠂⠀⠤⠆⠉⠉⠁⠠⠤⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠂⠀⠀⠀⠀⠀⣀⣡⡒⠦⠤
⠀⠂⠀⠀⠢⠄⠀⠀⠀⠄⠀⠀⠀⡀⠞⠀⠀⠀⠆⠀⠀⠀⠈⠀⠀⠀⠀⢀⡠⠘⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁
⠀⠀⠀⠐⠀⠀⠀⢀⣀⠀⠠⠄⠊⠁⠀⠀⠀⠀⠀⠁⠒⠦⢀⡀⢀⡤⢊⠡⠀⠀⠀⠀⢠⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠔⠈⠀⠀⠀⠂⠄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠥⠖⠁⠀⠀⠀⠀⠀⠸⡄⠃⠀⠄⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⡤⠄⠤⠀⢤⠄⠀⢂⠀⡰
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠉⠀⠀⠀⡆⠀⠀⠄⠂⢀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠐⠄⠃⡤⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠌⠀⢀⠀⠘⠀⠐⠄⠀⠀⠁⠀⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⡀⠀⠀⠀⠀⠀⠀⡐⠁⢠⡕⠁⠀⠀⠀⠀⢠⡄⠀⠀⠀⠇
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⣀⠂⢀⠀⠙⠂⠀⡀⠀⠀⠀⠑⢧⠘⠀⠀⠀⠀⠀⠀⠐⢁⠤⠀⠀⠘⠀⠠⢀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠄⣀⠀⠈⠀⠀⠀⠀⠢⠤⠀⠈⠀⠂⠈⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠐⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠
⠀⠀⠀⠈⠠⢌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠁⠀⡄⠀⠊⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠦⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠊⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⢀⠀⣀⢀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠘⠲⡀⠶⠄⠦⠐⠃⠀⠀⡀⠀⢠⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢀⠁⠤⠄⠀⠀⠀⠤⠄⠀⢀⡀⠀⢀⠠⠐⠠⠀⠀⠀⠠⠄⠀⠀⢀⠂
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⣁⡀⠠⠄⠄⠀⠈⠠⠄⠈⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⠀⠠⢤⠄⠙⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠤⠀⠀⠠⠤⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠀⠀⠀⠀⠀⣀⠀⠀⠀⢰⠀⠀⠨⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠈⠉⠉⠀⠦⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠐⠖⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠛⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠄⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⠐⠈⠁⠀⠂⠀⠀⠀⠀⠐⠒⠈⣀⣁⠀⠀⠀⡁⠀⠀⠀⠀⠃
⢀⠤⢄⡀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠂⢀⠀⠀⢀⠀⠀⠐⠢⠀⠀⠀⠀⠀⡄
⡇⠀⠀⠈⢂⠘⠢⠄⠀⠠⠒⠀⠈⡠⠀⠀⢀⡄⠆⠆⠀⠙⣂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠰⠀⠂⠀⠀⠀⠀⠁⠂
⠀⢀⠀⠀⠀⠱⡀⠀⠄⠒⠀⠀⢰⠁⠀⢰⠑⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⢂⠈⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⠊
⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣀⠀⢆⣀⣀⡀⠀⠂⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠥⠄⠀⠀⠀⠀⡀⠀⠀⠀⠄⠐⠈⠉⠉⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠁⢀⠀⠀⠀⠠⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠓⢠⠄⠀⠀⣤⠚⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠐⠢⠄⡈⠣⡙⠃⠀⠘
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠔⠒⣉⡐⠒⠤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⣠⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⣤
⠀⠂⠀⠀⠀⡀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⡀⠁⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠶⠾⡄⠀⠁⠀⢂⠐⠃⠐⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉
⠀⠀⠀⠀⠑⠀⠈⣉⠲⠀⠀⠀⠀⠀⠀⢀⡉⠀⠭⠭⣝⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠁⠤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠲⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣑⣤⣤⡤⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⣠⣎⣉⣳⡀⠀⠀⠀⠀⡤⡤⠒⡶⣠⠒⡦⡀
⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠉⢆⠀⠀⠀⠀⢠⠮⠭⠭⠭⠽⠄⠀⠀⠀⣸⠥⠄⠬⠦⠨⣆⠀⠀⠀⣜⠁⣼⡙⣧⢎⢫⣾
⠀⠀⠀⢠⡗⡒⡒⢖⠠⣒⡲⣶⠾⡀⠀⠀⠀⠘⡎⠉⠉⠉⢹⠁⠀⠀⠀⠙⣟⢻⡭⡛⢹⠁⠀⠀⠀⢹⠙⠓⠓⠚⠚⢉⡇
⠀⠀⠀⠀⢹⠒⠾⡉⣘⢿⠓⠞⡟⠀⠀⠀⠀⠀⢣⠀⠀⠀⡌⠀⠀⠀⠀⠀⠘⡟⢤⣺⠃⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠉⠁⠘⠳⣰⠞⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⡁⡝⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠄⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢀⡤⢒⣂⡉⣐⠲⡄⣠⢤⠀⠀⠀⠀⠀⣀⠤⢤⠒⠢⡄⠤⡀⠀⠀⢀⡠⠔⠒⠒⠠⢄⠀⠀⠀⠀⠀⢀⠤⠒⠢⢄
⠀⠀⠀⢠⢋⠖⠧⠎⢀⡠⡫⠞⠹⡆⠇⠀⠀⢠⠞⠁⠀⠀⠣⡀⢣⣀⣈⠆⠀⣎⣀⣀⣀⣀⣀⣠⡕⠀⠀⠀⢠⠋⠀⠀⠀⠀⢣
⠀⠀⠀⢸⠈⠆⠏⠋⠻⠒⡿⢭⢅⠙⠂⠀⠀⡜⠣⡀⠀⠀⢀⡝⠁⠀⠀⠀⠀⠈⡄⢠⠈⡄⠰⢠⠁⠀⠀⠀⠸⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⢆⠳⢖⢐⢢⣀⡰⢋⡞⠀⠀⠀⠀⡗⢄⣈⡶⠒⠁⠀⠀⠀⠀⠀⠀⠀⢡⣘⠀⢃⣃⠎⠀⠀⠀⠀⡼⢑⣤⠴⠊
⠀⠀⠀⠀⠀⠑⠢⠤⠭⠤⠒⠋⠀⠀⠀⠀⠀⢇⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠸⢭⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠠⠄⠀⠀⠀⠀⠀⠀⠀⣔⣒⣢⢠⡆⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⣴⢖⣂⠴⡒⣒⣄⠀⠀⠀⠀⠀⡤⠤⠤⠧⠧⠤⣄⣀⡀⠀⠀⠀⢸⠀⣇⠾⠼⠄⠀⠀⠀⢀⡔⡢⣄⡠⢚⠦⡀
⠀⠀⠀⠀⡎⡡⠒⠉⠒⠒⠒⢝⢦⠀⠀⠀⠀⢧⠀⠀⠀⠀⢀⣯⣗⠅⠀⠀⠀⡸⠀⡧⠋⠈⠣⡄⠀⢠⠊⠀⢱⠈⢳⠸⡄⠈⠆
⠀⠀⠀⢸⡴⠁⠀⠀⠀⠀⠀⠈⢿⠂⠀⠠⡆⠀⠷⠤⠄⠖⠋⠃⡲⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⢸⠀⠸⡀⢀⠇⠀⢸⠀⡇⠀⡄
⠀⠀⠀⠸⠤⠤⠤⠀⠒⠀⠀⠤⠊⠀⠀⠀⠈⠑⠒⠒⠒⠒⠂⠉⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⢸⠀⠀⠑⢤⣃⠴⠛⢄⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠣⠤⠤⠄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣄⡀⠀⠀⠀⠀⠠⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣀
⠀⠀⠀⠀⠀⠀⣀⠀⣞⣵⣵⠗⠺⡄⠀⠀⠀⢠⠃⠘⡀⠀⠀⠀⠀⠀⠀⣠⠊⠱⢼⢌⢷⠀⠀⠀⣠⣀⡠⠤⣄⡀⢀⣀⣀⡀
⠀⠀⠀⢰⣊⠭⣔⡺⣽⠝⠁⠀⠀⡇⠀⠀⠀⡞⣚⠓⠙⡆⠀⠀⠀⠀⣴⠥⠔⠚⠉⢉⡏⠀⠀⢸⠀⠘⠀⣀⠈⠀⠀⠋⠀⢹
⠀⠀⠀⢎⣪⠛⠊⠉⠀⠀⠀⣠⠎⠀⠀⠀⠀⠹⠀⡇⢰⠃⠀⠀⡠⠊⠀⠀⠀⠀⡠⠋⠀⠀⠀⣞⡋⢛⡓⠒⡞⠉⣖⡛⢳⣋⡢
⠀⠀⠀⠈⠡⡀⠀⢀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠸⠋⠁⠸⡄⠀⠀⠣⢄⡀⠀⠀⡔⠁⠀⠀⠀⠀⠘⠬⠅⠈⠫⠬⠍⠤⠬⠥⠜
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⡄⠠⠃⠀⠀⠀⠀⠈⠢⠞
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢠⣛⣟⢅⢀⡠⠂⠄⠀⠉⣹⠋⢍⡩⠙⡏⡩⠀⠀⡏⠉⠙⠋⠉⢹⠀⠀⠀⠀⠀⠀⡸⢇
⠻⠟⠛⢿⠇⠹⠿⢫⠀⠀⠀⠉⠝⢫⠉⠀⠀⠀⠀⡟⠇⡘⠎⠸⢸⠀⠀⢀⣀⡀⠀⡉⢉⡦⣄⠀⠀⠀⡀⣤⣠⣀
⢰⡴⠶⠜⠉⠃⠀⢸⠀⠀⠀⠀⢑⡊⠀⠀⠀⠀⠀⡇⣍⡅⣭⡅⢸⠀⠀⠛⠈⠻⠀⢓⣤⡇⢹⠀⢴⠿⠷⢶⡾⠷⠾
⠀⠀⠀⠀⠀⡧⠤⡾⠀⠀⠀⠀⢅⡨⠀⠀⠀⠀⠀⣇⡦⢔⢦⢆⡸⠀⠀⢴⠤⠦⡤⠿⢴⢿⣽⠀⠘⣷⣅⣦⣤⣴⣶
⠀⢀⣀⡤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀
⢰⠟⣁⢄⡤⣈⣟⡄⠀⢀⠀⢀⣀⣀⡀⣀⡀⠀⠀⠜⢷⣐⣂⡎⠀⠀⠀⡠⠀⣀⢚⢓⠀⠀⢠⠀⠀⠺⠛⠛⠛⠛⠛
⣿⠘⣿⡵⢿⠚⠀⡇⠀⡇⡈⠉⠩⠉⠉⢁⠈⠀⠀⠀⠘⡀⢈⠃⠀⠀⠀⠂⠤⢭⠭⠌⠹⠥⢸⠀⠀⠀⠀⠈⠉⠁
⠨⣽⡙⠂⠘⡀⠼⠃⠀⡇⠃⠉⠉⠩⠉⠘⢀⠀⠀⡠⠐⡿⠛⠂⢄⠀⠀⠀⢱⢸⢈⠃⠀⢹⢸⠀⣀⣤⣀⣀⣀⡀⡘
⠀⠀⠉⠋⠙⠁⠀⠀⠀⠑⠈⠉⠉⠁⠉⠀⠊⠀⠀⠁⠒⠚⠀⠒⠊⠀⠀⠑⠂⠈⠀⠁⠀⠊⠊⠀⠈⠂⠀⠒⠚⠒⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠒⠦⠀⣶⣶⠀⣶⣆⢰⡆⠲⣶⠒⢰⣖⣒⢠⣶⣒⡂⣶⣆⣲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣤⠶⢸⠗⠻⡇⠿⠙⠿⠇⠀⠿⠀⠸⠯⡭⠸⠿⠭⠄⠿⠙⢿⠇

⠀⠀⠀⢠⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⢀⢠⠀⠀⠀⢠⣄⣀⣀⣠⡄
⠔⢀⡴⡴⠚⠒⠃⠄⠀⠠⢦⡠⢄⣀⣀⠀⠀⠀⣤⣈⣉⣨⣡⣭⣡⣦⠀⠀⠒⠏⠉⠉⠹⠗⠂⠀⠀⢨⢴⠦⡶⣴⡇
⢀⢄⠀⠈⠹⠞⠂⡇⠀⣀⣞⣶⣁⣀⣋⣹⡝⠀⣹⣤⣤⣤⣤⣤⣤⣍⠀⠀⡁⠀⡀⠤⠀⢀⠀⠀⠀⢰⢛⣛⢳⡟⠂
⠈⢢⡀⣠⠰⠥⠆⡇⠀⢗⠒⠒⠒⠒⠒⠒⡺⠀⠹⠤⠄⠴⠦⠤⠤⡏⠀⠀⢠⠀⠁⠊⠀⡌⠀⠀⠀⠘⢉⠈⠉⡉⠃
⠑⠚⠓⠛⠒⠒⠂⠂⠀⠈⠒⠒⠒⠒⠒⠒⠁⠀⠀⠓⠒⠒⠒⠒⠚⠀⠀⠀⠈⠒⠒⠒⠒⠁⠀⠀⠀⠈⠒⠒⠒⠑⠃
⠀⢰⡭⣝⣯⣝⠀⠀⠀⠀⣿⣿⣛⣟⣻⣷⠀⠀⠀⣤⠾⠚⢓⠷⣤⠀⠀⢨⡭⠭⠭⠭⠽⡃
⠀⢘⣭⣛⣻⣯⠀⠀⠀⠀⢿⣿⢶⣷⡇⢸⠀⠀⣜⢃⣞⣭⣼⣳⠓⣧⠀⡜⠁⠀⠀⠀⠰⠙⡄⠀⠀⠤⠤⠤⠤⠤⠠
⠀⢸⠹⢭⠌⢹⠀⠀⠀⠀⣿⣿⣿⣯⣿⣿⠀⠀⢻⢼⣯⡓⣹⣿⡆⡟⠀⠆⣶⣁⣀⣁⣉⣈⣵⠀⢸⢸⠍⠌⠅⠍⠨
⠀⠸⣰⣶⣶⣚⠀⠀⠀⠀⣷⣿⣿⣿⣷⣻⠀⠀⠀⠛⢦⣤⣤⡔⠙⠀⠀⣀⣸⣄⣤⣭⣤⣄⡇⠀⠺⣞⣒⣲⣶⣶⣚
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠊⠭⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣴⠞⠉⠉⣛⠿⡖⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⢱⠈⠀⠀⠀⢀⡾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢮⠭⠭⠭⠭⠭⠭⠍⠁⠈⠩⠡⠤⠤⠤⠤⠤⠤⠤⠥⠤⢸⡏⡇⠀⠀⠀⠀⠀⠀⠀⠳⢀⣒⣡⠞⣷⠾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢡⠀⠀⠀⠀⠀⠀⡠⢶⡟⢅⣀⡠⢓⡶⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⢸⠀⠀⠀⠀⠀⡰⠁⡸⢧⣀⣀⡠⢼⠃⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡤⠶⠬⠤⠤⠤⠬⠭⡽⢻⠀⢰⠖⠻⠿⠿⠿⠿⠛⠞⣹⠘⠀⡇⠀⠀⠀⢠⠃⡰⠁⠀⠀⠀⠀⢸⢰⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⠶⠷⠤⠤⠤⠤⣤⣤⣤⡶⠾⠿⠿⠷⠶⠶⠶⠶⠶⠾⠿⠿⠀⡇⠇⠀⠀⠀⡎⢠⢧⠀⠀⠀⠀⠀⠸⣠⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠒⠚⠒⢄⠀⡀⣸⡀⡇⠀⠀⠀⠀⣀⠽⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣸⡄⠀⠒⠒⠒⠒⣲⣤⠖⠉⠁⠀⠀⠉⠻⣏⠉⠉⠉⢻⠏⠀⠀⠀⠀⠀⠈⣧⢹⣿⠿⢿⠗⠀⠈⣉⣇⣠⠤⠊
⠀⠀⣀⣀⣀⣀⣀⣀⣀⢀⣠⠹⠿⢯⣉⣉⠉⠉⣈⡠⣏⠀⢀⡤⡶⠶⠞⡶⣼⠉⠉⢙⠃⠀⡖⠒⣐⠂⠩⠝⡇⠀⠈⢙⠇⠒⠊⠉⠁⠀⢸⡀
⢀⡞⠁⠀⠀⠀⠀⠀⠀⢀⠸⠤⠤⠤⠠⠤⠤⠤⠄⠀⢳⡴⡞⠠⠄⠀⠈⠁⡛⣒⣒⣒⣆⠰⣣⡀⠁⢀⣖⡀⡗⣒⣒⣛⡒⢂⣐⣒⣒⣒⣒⣓⣂
⢸⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠀⠁⠀⠉⠉⠀⠀⠉⠛⢢⣕⣒⣠⡞⠀⠀⠀⠀⠈⠑⠒⡻⣦⣤⡭⠊⣀⢀⣀⣸⠀⢙⣃⡠⠤⠴⠀⣠⠄
⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠙⠽⢲⠛⠈⢦⠀⠀⠀⠀⠀⡜⢠⠚⠙⢽⣆⠙⢷⣤⣬⣧⣞⣙⣲⣶⣤⣭⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠘⢄⠀⠇⢱⡨⣄⠀⠀⠀⠀⠇⠀⠳⢄⣰⠉⠛⠃⠚⣁⠥⠉⠉⠉⠉⠉⠉
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠣⡔⠫⡁⠈⡇⢡⢹⠀⠀⠀⢸⠀⠀⠀⠀⠉⠙⡊⠉
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠈⢦⡈⠢⡅⢸⠀⠇⠀⠀⠀⡆⠀⠀⠀⠀⠀⢣
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣉⣀⣀⣵⣠⡽⢞⠊⠀⠀⠀⠀⡗⠤⠤⠤⠤⠄⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣀⠀⠀⠘⠀⠀⠀⠀⢰⠀⠀⠀⠀⡄⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⢄⡠⠟⣤⢠⠇⠀⠀⠀⠀⠦⣀⣀⣀⣜⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⡇⢀⠀⠀⠀⠀⠀⡴⠁⡸⠁⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡸⠀⠀⡇⢰⠀⠀⢀⠤⡋⢀⠜⠁⠀⡇⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡋⠐⢣⠀⡔⠥⠼⡄⠀⡇⠀⢈⣧⠀⠀⢸⠁⠈⠓⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⣠⠜⠀⠙⠦⡀⠛⢢⠘⣄⠈⠈⡆⠀⠸⢄⣀⣈⣋⣉⠆
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣷⣯⣤⣤⣤⣤⣤⣬⣷⣧⣤⣤⣤⣮⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣤⠃⠘⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⣴⠃⠐⠀⢠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠑⠀⠝⢱⠀⠼⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠍⠱⠀⠼⢉⡀
⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⡀⠀⠀⠀⠀⡴⠋⡏⡩⠦⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠀⠐⠀⠀⠀⡴⠫⡎
⠙⢣⡴⠀⠀⠔⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠆⠅⠜⢃⡶⠀⠀⠔⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠄
⠀⠀⠀⠀⠀⠐⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠰⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠰⠤⠒
⡄⠀⢀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⠀⠐⢷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠐⢷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠖⡼⣿⡆⢊⠀⠀⠀⠌⠄⢉⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣬⣖⣾⣻⡆⠊⠀⠀⠀⠌⠆⢙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣰⠀⠀⠀⠀⠶⢋⠘⠃⢦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡤⠀⠀⠀⠀⠶⠛⠘⠇⢦⡴⠂
⠀⠀⢀⣀⠀⠀⠈⠀⠀⠃⠀⠀⠨⣁⠀⡤⠈⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⣀⠀⠀⠈⠀⠀⠃⠀⠀⠈⣀⠂⠤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢉⠀⠁⠀⠀⠀⠀⠀⠀⠚⠀⠈⠀⠀⡠⠦⡴⠐⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⡁⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⢀⡠⠢⡤⠀⠀⠀⠀⠆
⠀⠀⠈⠛⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠒⠖⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠒⠆⠀⠀⠉⠛⠃
⠀⠀⠀⠀⢀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂
⠀⠀⠀⢀⠉⡰⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⡰⠉⠆
⠀⠀⠀⠀⡶⠃⠘⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⡶⠃⠠⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠜⠹⠀⠼⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠱⠀⠍⠹⠀⠨⢡⡀
⠦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠐⠀⠀⠀⣴⠁⡏⠩⡦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠀⠐⠀⠀⠀⡰⠉⡏
⠙⡃⡶⠀⠀⠔⠒⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠀⠔⠁⠜⠃⡴⠀⠀⠔⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆
⠀⠀⠀⠀⠀⠐⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠐⡤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠒
⡀⠀⢀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⠐⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⠐⠷⠆
⣗⠺⣻⡆⢂⠀⠀⠀⠌⠂⢉⠿⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡷⣊⣖⢾⣹⡖⠊⠀⠀⠀⠈⠂⢉⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣤⠀⠀⠀⠀⠖⢋⠘⠃⢦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⡴⠀⠀⠀⠀⠷⢛⠘⠃⢢⠴
⠀⠀⢀⣀⠀⠀⠀⠀⠀⠃⠀⠀⠈⣁⠐⠤⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⣀⠀⠀⠈⠀⠀⠃⠀⠀⠈⣁⠂⠤⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢉⠀⠁⠀⠀⠀⠀⠀⠀⠚⠀⠉⠀⠀⡰⠦⠶⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠁⠀⠀⠀⠀⠀⠀⠚⠀⠉⠀⠀⡤⠲⡴⠐⠀⠀⠀⠂
⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠒⠆⠈⠀⠉⠛⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠆⠀⠀⠉⠛⠃⠀⠀⢀
⠀⠀⠀⠀⢀⠀⡠⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠅⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠈⠉⠰⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠉⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠃⠀⠊⠀⠀⠈⠀⠐⠀⠓⠀⠀⠀⠂⠐⠀⠀⠀⠐⠈⠒⠒⠀⠁⠁⠘⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠤⠄⠄⠀⡐⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠰⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠤⠐⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⢄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠉⠉⠙⠓⠒⠒⠀⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⠛⠉⠙⠒⠲⢤⡀⠀⠀⠀⢀⣸⠤⠔⢢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⠖⠋⠁⠀⠠⠖⠡⠟⡷⣄⡀
⠀⠀⠀⠀⠀⠐⡒⡾⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠰⡄⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄
⠀⠀⠀⠀⠀⡈⠹⠿⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⢟⣹⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢉⣉⣲⣄⣀⢤⠄
⠀⠀⠀⠀⠀⠀⢁⣮⠆⢽⡍⢿⣖⢶⠶⢶⠒⣒⣚⣹⠉⠞⣳⡽⠉⠈⠢⣀⠀⣀⣤⣴⡶⢛⣟⢛⡛⣉⣁⣤⠞⠃
⠀⠀⠀⠀⠀⠀⠸⢛⠦⢄⣀⠀⠈⠉⠙⡓⠀⣉⠀⠘⠋⡥⢎⡟⢠⢀⠀⠀⣽⠉⢤⡏⠀⡎⢻⠛⢯⣄⡈⢙⣭⠗
⠀⠀⠀⠀⠀⠀⠘⠦⢀⣠⡌⠛⠁⠤⢄⢠⣈⣁⡀⠀⡀⣠⠎⠁⠣⣌⠄⢸⣿⡄⢸⣬⣖⡽⠄⣇⢀⠦⠽⢾⡜
⠀⠀⠀⠀⠀⠀⠐⢄⢀⠈⠉⠁⠈⠀⠈⠁⠉⠉⠉⠁⠈⢀⠄⠁⠀⠈⠢⣥⠙⣧⠈⣟⠬⣉⣁⡟⢴⣟⠓⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠀⠐⠒⠒⠒⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢂⠙⡓⡁⢶⠨⠇⠈⠂⠭⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠱⠘⠏⠇

⣿⣿⠻⢿⢻⠿⡿⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡶⠚⠈⠐⠒⢶⣶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⡖⠒⠒⠒⠒⠒⠒⠒⠓⠒⢲⠒⠒⠒⠒⠒⠒⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⠀⠀⠀⡀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠀⠀⠀⠀⢀⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⢦⣄⠀⠀⠀⠀⠀⣿⣷⣤⣤⣤⣴⣿⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣀⣀⣀⣀⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠙⣇⠀⠀⠀⠀⠉⠛⠛⠟⠻⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⣿⠿⣿⣿⡿⢿⣿⡿⢿⠿⠿⡿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⡄⠀⣤⢤⠀⢠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣤⣤⣄⣠⣤⣤⣀⣤⠀⠄
⠀⠀⠀⠀⠀⠀⠐⣿⣿⠿⢿⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠘⠿⠃⠤⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠮⠁⠀⠀⠴⠦⠉⠀⠀⠤
⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠒⣶⣶⠶⢒⣲⣶⡶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⣸⣻⣿⡀⢄⣸⣷⣶⠀⠀⠀⠀⠀⢰⣶⣶⣶⡶⠾⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⠶⠾⢶⣶⣶⣿⣷⣶⣾⣿⡷⠶⠿⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣨⣿⣟⣗⣺⣸⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠚⠛⠛⠂⠛⣿⣿⡿⡿⡿⣿⠉⠋⠀⠀⠈⢻⠿⠿⢿⣿⣿⣏⣴⣿⣿⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⢻⣿⣧⣼⣾⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⣀⣀⣿⣷⣄⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀⣶⣿⣿⣽⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡘⠛⠛⠛⠛⠸⢿⣿⣿⣿⣿⣿⣿⣿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣁⣟⣛⣛⣛⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢹⣿⣷⢶⣶⡶⣶⣶⢿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⢿⣿⣿⢿⣿⡾⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣤⣄⣤⣿⠿⠿⠿⢄⣤⣤⣤⣤⣤⣄⣼⣿⣿⠘⣿⡟⢸⣿⡀⣿⣿⠀⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⢸⣿⣿⢸⣿⡇⣿⣿⢘⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡆⢠⣜⠿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣸⣿⣿⣿⣿⣀⣿⣿⠿⠛⠉⠀⠈⠛⠻⢿⣿⣸⣿⣿⣾⣿⣇⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⡿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⠋⠠⠄⠛⠋⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠀⠺⠃⠀⠀⠀⠉⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠠⡄⠀⢠⡄⠈⠇⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⢆⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠤⡼⢣⡀⠀⠀⠃⠀⠀⠀⣠⢶⣤⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠰⡎⠀⠀⠙⠢⡤⠒⠂⠉⠉⡤⣣⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢴⣆⠀⠀⠀⠘⠢⠤⣀⢸⠀⠀⢠⠔⠺⣟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣿⣣⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠤⢄⣀⣀⡇⠹⣷⡄⠀⠀⠀⠀⢸⠀⠀⠀⠸⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⡇⠀⠘⢾⣆⠀⠀⠀⢸⠀⠀⠀⠀⢱⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠚⠧⢤⠴⠊⣉⡳⢦⡀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⡀⠀⢻⠀⠀⠀⠀⠀⢠⠀⡒⠤⢽⣾⡃⠘⣄⠀⠀⠀⢀⠇
⠀⠀⠀⢸⠉⡩⠭⠛⠉⠉⠉⠁⠀⠉⠉⠉⠚⢿⡒⠠⠤⠰⠚⠢⠭⠿⠞⠙⠿⠉⠉⠙⠒⠶⣛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠢⣄
⠀⠀⠀⠇⠛⠒⠤⢤⡤⠤⠴⠒⠒⠒⠒⠒⠤⠤⠬⣦⣀⣤⠤⠤⠴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠬⠷⠄⠀⠀⠀⠀⣀⣠⢤⡀⠀⠀⠀⠈⠑⢤⡀
⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣒⡩⠕⠒⠉⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠕⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⢸⠂⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⡒⣒⣒⣒⣒⣒⣛⡇
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣬⣥⣾⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠤⠎⠁⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠴⣶⣄⣀⣀⣀⣠⡄⢀⣠⠞⠁⢴⡄⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠉⠛⠛⠉⠁⠚⠉⠀⠀⠀⠺⣧⠀⠀⠀⢠⣤⡆⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢆⣿⠟⣡⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠀⠀⠀⠀⠨⠋⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠶⢄⠀⠀⠀⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡴⡇⠀⠀⠀⠐⠲⠖⠋⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣧⡀⠀⠀⠀⠀⣰⠁⢐⠢⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⠟⣿⣿⠋⠀⠀⠀⢀⡴⠃⢀⣾⣇⢀⡀⠉⠉⠉⠛⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡎⢧⡀⠀⠀⠀⣀⡴⠋⢀⣴⣿⣿⣿⣿⣗⣴⣦⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣦⣉⠓⠒⠋⠁⠴⠾⠛⠛⠛⠋⠉⠙⣿⣿⣯⣌⡆⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⣀⣠⣤⣴⣶⣷⣾⢿⣿⣿⣿⣿⣶⣆⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡙⠻⡿⠿⢿⣿⡿⠟⠉⢀⣀⣤⣶⣶⣿⠿⠿⠛⠉⡉⠡⠿⢗⣿⣿⣿⣿⣿⡷⠃⣠⠀⠘⢆
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠀⠀⠀⢎⡉⠑⣸⣿⣿⣿⠟⢋⠉⠀⠀⠀⠈⠁⠀⠀⠉⠁⠸⣿⣿⣿⣿⣿⣿⣷⡀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⢻⣽⣿⣿⣛⠛⠛⠛⠛⠋⠙⠈⣿⣿⡟⠘⣿⣿⣯⡿⢷⣾⣯⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠰⠴⠭⠯⣛⣉⡁⠀⠀⠀⠈⢿⣿⣿⡁⢹⣿⣿⣿⣞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣄⠀⠙⣿⣿⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⣀⣠⡼⢖⠀⠀⠀⢸⠏⠻⣇⠠⢿⣿⡿⢛⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⣀⣀⣀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⢀⣠⡯⠗⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣟⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⢁⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡠⠀⠐⣄⡠⠂⠁⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⢀⠂⢀⣀⡐⠂⣰⡄
⠀⠀⣊⠀⠠⣀⡬⢾⡹⠷⢶⣤⣔⣶⣿⠑⠀⠀⠀⡸⠃⠀⠸⠔⢒⢮⣥⢠⡀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠿⠁
⠀⠐⠛⢷⣋⣍⣢⠴⢥⠼⠅⢓⠊⠟⣁⠔⠀⢀⡼⢁⡏⠀⠀⠀⠛⠚⠣⣄⠄⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⠘
⠀⠀⠀⡈⠍⠫⠉⠀⠍⢈⡈⠤⠒⠈⢀⠠⠢⠋⣀⠕⠀⠀⠀⢀⠐⠂⠊⠠⠀⠀⠁⠀⡆⠀⠠⠔⠀⠀⠀⠄⡇
⠀⠀⡀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠾⢋⠁⠔⠋⠀⠀⢀⡀⠀⠀⠀⠀⢠⡐⢂⡀⣦⠀⠠⠀⠄⠠⠤⠤⠴⠊⠁
⠀⡖⠀⠈⠀⠈⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠾⠀⠀⠠⠀⠀⡜⠊⠤⠴⠖⠓⠀⢀⡀⠠⣤⣤⣄⠀⡀
⠀⣍⠁⠀⠀⠀⠀⡅⠀⢂⠂⣀⣀⣀⡬⠄⠀⠀⡽⠀⠀⠀⠀⠀⠀⠀⡖⠂⠀⠂⣀⠙⠀⠨⠈⣽⣉⣭⣿⡿⠇
⠀⡛⡇⠐⠀⠐⠀⢸⠀⠀⠏⢴⣦⣽⡷⡎⠄⠀⠐⠀⠀⠀⢶⠌⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⣀⡤
⠀⠧⡃⠀⠀⠈⢆⡈⡀⠀⠘⡸⠊⠙⠉⠓⠘⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠌⠐⠀⡠⠀⠀⢴⣯
⠀⠀⠰⠀⠀⠀⠀⢃⡇⠀⠀⠀⠀⠀⠀⡀⠠⠀⣐⡠⠬⠤⠾⠀⠀⠀⠚⠀⠌⠈⠃⠑⠠⠄⠀⡏⠕⣠⠬⠑
⠠⢸⣘⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠸⠒⡺⠛⠀⠂⠁⠀⠀⠀⠠⠒⠂⠠⡄⠀⠀⡀⠀⠀⠀⡅⢈⠤⠆⡅⡀
⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠄⠀⠀⡀⠀⠘⣦⠀⡠⠊⠀⠀⢁⠈⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠡⠈⠁⠔⠁
⠐⠨⠛⣃⠵⢑⡀⠄⠀⠁⠀⠀⠀⢱⠀⠀⢸⡆⠀⠀⠀⠀⠊⠆⠀⠀⠀⣀⡄⡀⠀⡄⠀⠀⣀⠤⠤⣄
⠘⠀⠉⠁⠈⢀⠤⢤⠀⠡⠀⣀⡠⠎⠀⠀⠈⣧⡀⠀⠨⠔⠀⠠⠀⠃⠀⠻⡄⢀⣀⠤⠶⠥⠿⠭⢤⣿⡭⡩⡍
⠀⠀⢀⣤⢄⡌⠀⢉⡄⣠⣤⣤⣤⢄⠀⠀⠑⠂⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠄⠀⢲⠠⠀⠄⠀⠀⠠⡧⡐⠘
⢀⠴⣿⢃⣼⣟⣿⣿⠄⢻⠻⢻⡿⣧⠡⡀⠀⡠⠆⢉⢉⠕⠒⠜⠳⢢⠀⢰⠠⢿⣏⠀⢀⢀⠀⠂⠀⠘⡀⢠⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣼⣛⡲⢤⡀
⠀⠀⠀⠀⢠⠖⠋⡍⠉⡉⢉⠙⠳⣄⠀⠀⠀⠀⠀⠀⣴⠶⠾⠷⠶⣦⠀⠀⠀⠀⠀⠀⠀⣼⠀⣿⠀⠀⠀⠀⠀⠀⢀⡴⢚⣩⣭⡇⡿⢟⡙⢦⣹⡄
⠀⠀⠀⠠⣏⣀⣃⣀⣁⣀⣁⣘⣀⣸⡆⠀⠀⠀⠀⢾⣉⣉⣉⣉⣉⣉⡇⠀⠀⠀⠀⠀⢠⡟⠒⠻⡆⠀⠀⠀⠀⢠⡟⣴⣋⣐⢺⡇⣷⡯⢟⣫⣍
⠀⠀⠀⠀⣯⠤⠤⣤⡤⠤⣤⡤⠤⣽⠇⠀⠀⠀⠀⠈⣯⠈⠀⠈⠀⣿⠀⠀⠀⠀⠀⠀⣾⠛⢳⠀⣿⠀⠀⠀⠀⣾⢰⡇⢯⡽⢸⣧⠵⣚⠉⡇⢹
⠀⠀⠀⠀⢳⠶⣶⡒⠒⠒⠒⣶⠶⣾⠁⠀⠀⠀⠀⠀⢻⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢸⡶⠛⠀⡟⠀⠀⠀⠀⢹⡌⣇⢰⣷⡌⠄⠺⠿⣰⢇⡿
⠀⠀⠀⠀⠘⠳⠾⠷⣦⣠⠾⠷⠖⠋⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣇⠀⠀⠀⠀⠀⠻⣌⠳⢭⣴⣏⡷⠞⣣⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠻⠤⠤⠤⠟⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠶⠖⠛⠁
⠀⠀⠀⠀⠀⢀⣀⣠⢴⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⢀⣴⠻⡌⢿⢸⡟⣸⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢘⡎⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡅⠻⣜⣳
⠀⠀⠀⠈⡏⠛⠛⠚⠛⠒⠛⠋⣿⠀⠀⠀⠀⠀⠀⢰⠶⠶⠶⠶⠶⣦⣀⡀⠀⠀⠀⢠⡞⠉⠠⠈⠠⠉⠄⠈⢳⡄⠀⠀⠀⠀⢀⡞⠰⠀⠄⠠⠈⡽
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⣿⢁⡿⠀⠀⠀⠀⢹⡏⠹⡟⠉⢹⡏⢩⡟⠁⠀⠀⠀⠀⣸⣇⠐⠀⢂⣠⠞⠁
⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣠⣤⣼⣧⣤⣤⣤⣤⣿⣭⣤⠀⠀⠀⠀⠀⢷⠀⣿⠀⣸⠁⣼⠁⠀⠀⢀⡤⡼⣃⡼⠓⠛⠉
⠀⠀⠀⠀⠀⠧⠤⠤⠤⠤⠤⠾⠀⠀⠀⠀⠀⠹⢦⣤⣤⣤⣤⣤⣤⣤⣤⠞⠀⠀⠀⠀⠀⠘⠧⠼⠤⠿⠴⠃⠀⠀⠀⠈⠳⣬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣉⣳⣆
⠀⠀⠀⠀⠀⠀⢀⣀⡴⠻⣆⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠻⣄⠀⠀⠀⠀⣾⣁⣈⣹⣄⣙⣦
⠀⠀⠀⠀⢀⣾⣩⣥⣀⣠⣩⣧⠀⠀⠀⠀⢀⡞⠉⠉⢹⠏⠉⠉⡟⠉⠉⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠲⢾⡀⠀⠀⠈⢷⠖⠲⠖⣶⣶⡞
⠀⠀⠀⠀⢻⣥⣄⣠⣤⣀⣤⣬⡇⠀⠀⠀⣼⢶⠶⠶⡶⠶⢶⠶⠶⡶⠶⢶⢾⡆⠀⠀⠀⠀⠀⢀⡴⠋⠳⢤⡀⢨⡇⠀⠀⠀⠘⣷⠖⢋⣡⣼⠁
⠀⠀⠀⠀⢸⡄⠈⡇⠀⣿⠀⣸⠃⠀⠀⠀⣿⣯⣛⣛⣟⣛⣋⣛⣫⣿⣚⣉⣿⡇⠀⠀⠀⢀⡴⠛⠳⣄⠀⠀⠙⣿⠀⠀⠀⠀⠀⢹⡞⢋⣠⡇
⠀⠀⠀⠀⠀⢷⠀⣷⠀⡟⢠⡏⠀⠀⠀⠀⠘⠲⠾⠭⠭⣬⣭⣭⣬⠭⠽⠷⠚⠀⠀⠀⢿⣉⠹⡄⠀⠘⣦⣠⡾⠃⠀⠀⠀⠀⠀⠈⣿⠉⡿
⠀⠀⠀⠀⠀⠘⠓⠛⠒⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⣻⠙⠖⡦⠀⠀⠀⠀⣠⠞⠳⣄⢀⡴⠛⣦⠀⠀⠀⠀⠀⣯⠭⣽⡇⣴⠟⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⡴⣣⣏⡽⠷⣤
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠈⣷⠛⣯⣿⣹⠆⠀⠀⣰⠋⠐⣦⠈⣯⠀⢶⠈⢳⡀⠀⠀⠀⢸⠀⢸⠀⣹⠏⢹⣇⠀⠀⠀⠀⠀⠀⣠⢞⡵⣫⡞⠁⠀⠈⡇
⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡟⠛⢻⡾⠃⠀⠀⠀⢿⠀⣰⠃⠀⣽⢠⡏⠀⢸⠇⠀⠀⢀⡾⠀⠸⣾⠁⠀⠀⢸⡇⠀⠀⢠⡖⣫⡵⣫⡴⠋⠀⠀⢀⡼⠁
⠀⠀⠀⢰⡋⠀⠀⢀⣴⠏⢀⡴⠋⠀⠀⠀⠀⠀⠈⢧⡈⢁⡴⠳⣌⠃⣰⠏⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠸⣏⡽⠛⠁⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠙⢦⣴⠟⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠘⠛⠁⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠉⢳⣄⣀⣀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠙⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠛⠓⠒⠒⠚⠃⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠤⠐⠒⠒⣒⣈⣭⣦⣄
⠀⠀⠀⠀⠀⣀⣀⠤⠤⠐⠒⠒⠈⣉⣉⡠⠤⠔⠒⠒⣂⡉⣉⣥⡤⢬⣾⡤⢳⡀
⠀⢀⠔⠊⠉⢁⣀⠤⠤⠒⢒⣉⣉⣀⡤⢤⣰⡏⣧⡞⢉⣿⣏⠀⠃⢸⡜⣿⡥⡇
⠀⢯⠀⠀⡎⣥⠴⣦⠀⠘⡏⠯⢸⣿⡇⢤⣙⠃⢸⣇⠉⣻⢹⣤⡶⠴⣇⢻⠖⣿
⠀⠘⡆⠀⢇⢹⡆⠛⠒⣻⡷⣄⣼⠀⠗⠛⢈⣉⣉⣬⣭⣵⣶⡶⠟⢻⣿⡸⣯⢾
⠀⠀⢹⠀⠘⡌⠳⣒⣋⣉⣤⣤⣴⣶⣾⣿⣿⠿⣿⡿⠉⣿⣿⡆⠒⣻⣿⡇⣷⣾
⠀⠀⠀⣧⠀⠹⣿⣿⣿⠛⠉⢁⣽⣿⠸⣿⣧⢀⡘⢃⡀⢹⣿⣇⣨⣽⣿⡇⣿⣿
⠀⠀⠀⢸⡔⠀⢻⣿⣿⣿⡇⢻⣿⣿⣤⣿⣿⣼⣿⣿⣷⣾⣿⣿⣿⣿⣿⡇⣯⣿⠀⠀⡠⠖
⠀⠀⠀⠀⢳⠂⠈⣿⣿⣿⣿⣿⣿⣿⣿⠛⠿⣟⠹⣿⠿⣿⣿⣿⣿⣿⠟⠛⠙⠛⣄⣞⣀⣀
⠀⠀⠀⠀⠈⣟⡀⠸⣿⣿⣿⣿⣿⡅⠈⠑⠦⣀⠙⠦⣄⡉⠩⠯⠽⣯⠁⠀⠀⡠⠮⠄⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠸⣇⠄⢹⠛⢛⣉⡉⢹⠀⠀⠀⠈⠙⠢⣄⠉⠲⣄⠚⢲⠷⣀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠉⠉⠙⣯⣷⣶⣿⣿⣠⠀⠀⠀⠀⠀⠀⠙⢢⡈⠳⣌⣹⠗⣂⠀⠀⠀⠀⢀⣰⡿⠿⣟⣻⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⢭⡛⠿⣁⠞⡠⡀⠀⠀⠀⠀⠀⡇⢶⠏⣼⣿⣾⣤⣴⡶⠯⠓⠒⠊⠉⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠙⠺⢭⣛⠥⣄⠀⠀⣀⡸⠧⣞⣺⠭⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡀⠀⠀⠀⠀⠀⠈⠉⠛⣿⠻⠖⠒⠋⠁⠀⠀⠀⠀⠀⠀⢀⠤⢤⣶⠄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠁⡴⢀⡀⠀⠀⠀⠀⠀⢿⡠⠀⠀⠀⣀⡤⢤⣄⠀⠀⠀⠘⠛⢿⣿⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠤⢒⣩⠽⣧⣎⣠⠂⡄⢀⠀⠀⢸⡴⡀⠀⠀⠓⢻⣿⣅⠀⠀⠀⠀⠀⠈⢿⣧⣀⣠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠔⢚⠽⢚⣶⣇⡖⢠⢸⡞⣡⠀⠀⠀⠀⠹⣿⣄⡀⠤⣤⣒⣶⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠽⠻⠿⢾⣵⣡⡧⠴⠐⠒⠊⠻⢿⡿⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⡩⢣⠀⣤⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠓⣀⢦⠠⠤⡏⠇⠰⠰⡅⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠐⢆⡰⢌⡄⠹⡤⠿⠲⠁⣎⡹⠆⠡⠀⢂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⣄⢀⡸⠒⠎⠘⡀⡠⠞⢢⠀⠨⠁⠈⡀⢂⠈⠎⠖⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⢹⠇⣤⠋⢄⠀⠑⢻⠲⠉⢇⠀⠁⠡⠐⠀⣃⣸⠞⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠶⣶⠀⠀⠀⠀⠀⣀⡇⠀⠈⢆⠈⡆⠀⠈⠓⠋⢇⢃⠘⡆⠀⠀⣣⡶⠟⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⣰⠃⠀⠀⠀⠀⢸⡗⠶⠦⢤⣬⣤⣽⣄⣦⣤⣥⣸⣬⣤⡴⣶⠞⠉⠀⠀⠀⠀⠀⡇
⠀⠀⣠⡤⠔⠒⠂⠀⣈⡧⡤⠓⠒⠤⢤⡀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣠⣤⣤⣤⣤⣧⣤⣀⣀
⠀⣼⣏⠀⠀⠀⣠⣾⡿⠀⠀⠀⠀⠀⣀⢟⡆⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠛⠉⠉⠁⢀⠀⠀⠉⠉⠉⠛⠻⢷⣦⣤⡀
⠀⡷⠬⣍⣛⣒⠚⠛⠒⠒⠒⣒⣚⡉⠥⠚⡇⠀⢸⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⢉⡁⡄⠀⠀⠠⠀⠂⠀⠉⠊⠁⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣄
⠀⡏⠀⠀⠈⠉⠉⠉⠉⠉⠁⠁⠀⠀⠈⠀⡇⠀⠸⡆⠀⠀⣀⣴⠾⠋⠁⡠⢰⠄⠉⠀⠂⢂⣠⠤⠖⠒⡈⠁⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠈⢿⣷
⢠⡇⠀⠀⠀⠀⠀⠠⠀⠀⠄⢀⡀⠀⠀⢀⡇⠀⠀⣇⣠⡾⠋⠁⠀⠀⢀⣈⣀⣤⠤⣤⠴⠿⠤⠤⣤⣉⣀⠀⢴⡴⠀⠉⠉⠠⡠⠆⠀⠀⠀⠈⣿
⢸⡇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⡇⢀⣴⠟⠁⠀⢀⠀⠐⠊⠉⠁⣠⠔⠋⠀⠀⡄⠀⣀⠀⠈⠉⠛⠲⣄⠀⣆⠜⠀⠀⠀⠀⠀⠀⣸⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⡀⢸⣷⠟⠁⠀⠀⠺⠈⢀⡠⠂⣤⠞⠁⠀⢀⡀⠉⠀⠠⠊⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⣼⡿⠃
⢸⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠄⡆⠀⣠⠞⠁⠐⠁⠐⠒⠁⢠⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠟⢻⡇
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⣰⡟⠀⠀⠀⠀⠀⠀⠋⠀⡴⠃⠀⠀⠸⠥⠛⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋⢀⣤⡞
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠛⠉⣀⣤⠞⠋⣠⡇
⢸⡇⠀⠀⠁⠀⠀⠀⠀⡜⠀⠀⢨⣿⣦⣄⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡿⠿⠛⠉⣀⣤⠶⠛⠉⢀⣤⠞⠉⡷⢤⡀
⣿⡇⠀⠀⠀⠀⠀⢠⠎⢀⡀⠔⠉⠳⣍⡛⠿⢿⣶⣶⣴⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⡿⠿⠟⠛⢉⣁⣤⠴⠞⠋⠉⠀⣀⣤⠾⠋⠁⣀⡼⠃⠀⣿
⣿⠇⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠸⣇⠙⠓⠦⢤⣈⣉⡉⠉⠉⠉⠉⠉⢉⣉⣁⣠⣤⠴⠖⠚⠋⠁⠀⢀⣠⣤⠶⠛⠉⠀⣀⣴⠞⠋⠀⠀⢠⣿
⣿⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⢦⣄⣀⠀⠉⠉⠉⠛⠛⠛⠛⠉⠉⠉⠀⠀⢀⣀⣤⡴⠶⠛⠋⠉⢀⣀⣤⠶⠛⠉⠀⠀⠀⢀⣴⡿⠁
⣿⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢦⣌⠉⠙⠛⠓⠒⠲⠶⠶⠶⠖⠒⠛⠛⠋⠉⠁⢀⣀⣤⡴⠶⠛⠋⠁⠀⠀⠀⠀⠀⢀⣤⡿⠋
⣿⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⠓⠲⠶⠤⢤⣤⣤⣤⣤⠤⠴⠶⠶⠞⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋
⢿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠁
⠘⣿⣦⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡾⢛⣻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⠿⠛⠁
⠀⢿⣝⡛⠻⠷⠶⠿⠿⠟⢛⣋⣩⡴⠖⠛⠁⠀⠉⠛⠷⠶⣤⣤⣤⣄⣀⣀⣀⣠⣤⣤⣤⣴⠶⠶⠟⠛⠋⠁
⣤⣤⣬⣭⣿⣿⣷⣾⣿⣿⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣭⣏⣼⣉⣯⣩⣿⣹⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠔⠒⠒⠒⠂⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠲⠦⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⠋⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠒⠲⠤⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠒⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣦⣶⣶⣾⠿⠿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠀⣤⣀⣀⣠⣤⣤⣴⣶⣶⡶⠶⠿⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣄⣖⠼⠄
⠀⠀⠀⠀⠀⠀⠀⢐⣤⡀⠀⠀⠀⠈⠛⠲⢦⣄⡀⠀⠀⠀⢸⣿⠆⠀⠀⠈⠁⠀⠀⠀⣀⣀⣠⣤⠤⠴⢖⣚⣟⣛⣻⣭⣭⣷⣾⠟
⠀⠀⠀⠀⠀⠀⠰⠋⠒⠬⣙⠲⢤⣀⠀⠀⠀⠀⠉⠙⠲⢤⣀⣿⣿⣶⣶⣶⢶⣒⣛⣯⣭⣭⣴⠶⠿⠛⠛⠛⠉⠉⠉⠁⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠉⠙⢻⣿⡟⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣤⣷⡤
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣈⠑⠢⢄⡀⠀⠀⠀⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠴⢖⣚⣯⣭⣭⣶⣾⠃
⠀⠀⠀⠀⠀⠀⠈⠐⠦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡈⠑⠠⢄⣻⣿⣯⣀⣀⣤⣤⠤⣶⣞⣿⣯⣿⡶⠾⠟⠛⠋⠉⠁⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⡀⢈⣉⣩⣭⣿⠷⠾⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠒⣛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⢸⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠞⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⢸⣿⢋⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠴⠖⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠼⣿⣿⣄⣠⣤⣶⠶⠖⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠤⣤⠀⠀⠀⠀⠀⠀⠠⣐⠶⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⡤⠢⣄⡀
⠀⠀⠀⠀⡠⠼⢧⣭⣽⠦⢤⡤⢤⠤⠤⠼⣍⢀⣙⡀⠤⢄⠀⠀⢀⡔⠁⣠⠀⠀⠸⣀⣡⠗⡀
⠀⠀⢀⣼⢠⡤⡿⡨⣭⡄⠸⣀⣠⣀⡀⡰⣥⣤⣧⣄⠰⣶⠋⠙⠍⣥⡭⡉⣀⣠⢄⡮⢾⠀⠁
⠀⠐⠛⠘⠛⠙⠉⠉⠉⣿⣫⠀⠠⣽⠏⠉⠉⣉⠉⠉⢉⣩⡔⢾⡉⠉⠉⠉⠉⢹⠾⡴⢽
⠀⠈⠓⠉⠋⠉⠛⠛⠚⠋⠙⠒⠐⠋⠉⠉⠉⠉⠁⠀⠈⠻⠛⠛⠟⠉⠉⠉⠉⠈⢠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡇⣁⠀⠀⠀⠀⣀⣀⣀⣈⣛⣟⡁⠀⠀⠀⠀⠠⠛⠗⠼
⠀⠀⠉⠁⠈⠉⢉⣉⠉⠉⠻⠍⠫⠜⠁⠀⠀⠀⠀⠀⠀⠀⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠁⠄⢵⡀⠀⠀⠀⢀⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣐⠒⠒⠒⠒⠲⡀⢄
⠀⠀⠀⠀⠀⠈⠾⢾⣾⢶⣲⠀⣤⣸⣄⣽⠀⠀⠀⠀⠀⠀⠀⠈⠀⡼⡥⢺⢋⡀⠀⠀⠱⡀⠡⡀
⠀⠀⠀⠀⠀⡌⠀⡦⢱⠻⠞⠀⠠⣦⢻⠟⢣⠀⠀⠀⠀⠀⠀⢀⡠⠺⣳⣾⡾⠃⠀⢀⣀⠑⣠⡋⣰
⠀⠀⠀⠀⣐⠌⠹⠅⠀⠀⠀⠀⠀⠀⠈⡃⢩⠆⠄⠀⠀⠀⠀⠨⠤⠤⠟⠒⢃⢀⠀⠈⢬⣦⣝⡶⢊⠦
⠀⠀⠀⡴⣃⡐⣒⠖⣔⡂⠀⢀⠐⣿⣃⠀⣯⣱⠸⠀⠀⠀⠀⠘⠁⢗⣤⠐⢡⣀⠀⠀⠈⠋⣓⠋⢾⣄⣨
⠀⠀⢰⠋⢈⠩⣥⠎⠀⠀⠀⠘⢹⣲⢼⣣⢆⢾⡀⠀⠀⠀⠀⠈⠤⢼⢦⣾⠿⠛⠀⠀⠀⠀⠱⣶⡚⣭⠇⣃
⠀⢠⠉⠒⠲⠀⠀⠀⠀⠀⠀⠀⠒⠔⠛⢋⡁⠠⡇⠀⠀⠀⠀⠀⢰⣀⢀⢙⡀⠀⠀⠀⠀⠀⠀⠐⠂⠉⢋⠈⠢
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣠⣤⣤⡤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⡏⠓⠒⠯⠝⢻⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠆⠀⠀⠹⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⢉⡐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣾⣿⣥⣤⡀⡥⠀⠀⠀⠀⠀⠀⣐⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢰⠞⡃⠠⡠⢹⣦⣾⣟⣽⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣞⣻⣿⣿⣻⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣯⠙⢋⡿⠣⣮⣉⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣿⣿⣿⣏⡉⠙⠁⠀⠀⠒⠦⠈⠉⠉⠉⠿⣿⣿⣿⣿⡟⣶⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠛⠻⠛⠲⠶⠶⣄⣲⣀⡀⡒⠒⠺⠽⠛⠛⢋⣼⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣛⣉⣤⣶⣶⣮⠙⠉⠉⠉⠉⠉⠉⢉⣭⣶⣶⣤⣀⣾⣿⣿⠏⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡿⣭⣙⡻⣿⣿⣿⣷⡄⠀⠀⢀⣴⣿⢿⡟⣛⣹⣿⣿⣿⣿⢾⣾⡏⠎⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⠆⣈⡴⣋⠿⡝⢁⣙⠳⢶⡟⡇⠈⠀⢽⠋⡛⠞⢻⠁⢀⣰⡗⣵⠐⣯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣷⡔⢡⠗⡂⣨⣍⠳⢮⠀⣀⣚⠋⠘⠀⠀⠀⢡⣴⣿⣉⣸⡏⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣏⡟⠳⠓⠜⠉⠉⠐⠚⠛⠿⢆⠍⣀⣎⣹⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠓⠒⠒⠶⠤⠤⠤⠄⠒⠂⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⢭⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡒⠊⠀⢷⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⢸⣀⠞⠉⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⢤⣀⢳⠀⠀⡏⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡤⠖⠒⠒⠦⠖⠚⠉⠉⠛⠛⠉⠉⠉⠛⠛⠓⡶⠶⣄⡠⠞⠉⠀⠀⠀⠀⠀⠈⠻⡆⢸⣇⣔⣁⡴⠯⣄⣀
⠀⠀⠀⠀⠀⡜⠁⢀⣀⡀⠔⠊⠑⠒⠈⠉⠉⠉⠉⠉⠁⠉⠉⢉⡱⠒⠁⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠐⠋⠀⠀⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠀⠀⠀⠀⣀⠔⢉⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⢰⠇⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠀⠀⠀⠀⠀⣠⠞⠁⠀⠘⡀⠹⡄
⠀⠀⠀⡞⠠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⢀⠀⠀⠀⢀⡠⠚⠁⠰⠈⠉⠐⣁⠔⠁
⠀⠀⢸⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⡠⠀⠀⢀⡠⠊⠁
⠀⢀⡏⠀⠂⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠈⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⡾⠀⡰⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡠⠔⠁⠀⡠⠂⠁⠀⣠⠖⠉⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⢸⠃⢀⠇⠀⡀⠔⠁⠀⠀⠀⠀⡠⠔⠉⠀⠀⠀⢀⠊⠀⢀⡤⠚⠁⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⡟⠀⣇⡠⠊⠀⠀⠀⠀⠀⠀⡔⠁⠀⡀⠒⠒⠠⠞⣀⠔⠉⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⣇⡴⠁⠀⠀⠀⠀⠀⢀⡤⠊⡀⠀⠂⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠁
⠘⠦⡀⠀⠀⢀⠔⠊⢁⠄⠊⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠚⠉⠉
⠀⠀⠈⠲⡎⢀⠀⠀⠁⠀⠀⣀⡴⠊
⠀⠀⠀⠀⠈⠣⣀⢀⣀⠴⠚⠁
⣶⣶⠶⠶⠶⠶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣾⣴⣿⣿⣿⣿⣿⣿⣿⣸⣿⢸⣿⣿⣿⣿⠀⠀⠀⠀⢠⣿⣿⣿⣿⢻⣿⣿⣿⣿⠿⣽⣿⢸⣿⡇
⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡟⠀⠀⠀⠀⣾⣿⠛⡿⢻⣿⣿⣿⣾⣣⡿⣿⣿⢸⣿⡇
⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣼⣿⣿⣭⣿⣿⣿⣿⢞⣽⣿⣿⣿⣿⢸⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣯⣿⣵⣿⣿⣿⣿⢸⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣂⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣟⣟⣻⣿⣿⣿⣿⣿⣿⣿⢸⣿
⢦⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⢯⣽⢚⣿⣽⣖⡛⠛⠉⠛⠻⢻⣿⠿⢿⣿⣭⣿⣿⣿⡿⣿⢸⣿
⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣟⣿⣛⡋⢨⣭⣿⡉⠁⠀⠀⠀⢀⢸⣟⠘⠂⠀⠀⠻⠿⠟⠀⣿⢸⣿
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣄⠛⠋⡉⠛⠛⠛⠛⠛⢻⢟⣋⣳⣄⣀⣀⣀⣀⣀⣀⡇⢸⣿
⠀⠀⠀⢶⠆⠀⣿⣿⣿⣿⣿⣿⣿⠿⠿⢶⡖⣿⠀⠀⠀⠀⠀⠘⠋⠛⠟⢿⣿⢿⡿⣿⣿⣯⡅⢸⣿
⠀⠀⠀⢸⣠⠀⣿⣿⣿⣿⡟⠉⠁⠀⠀⠘⡇⢿⠀⠖⠀⠀⠀⢠⢀⢠⣶⣼⣟⠻⢻⣗⢹⢹⡇⢸⣿
⠀⠀⠀⠘⠂⠀⢹⡿⣿⣿⡅⠀⡀⠀⠀⠀⠇⢹⠀⠀⠀⠀⠀⠀⠸⢸⣿⣿⡇⠀⢀⡟⢹⠟⡇⢸⣿
⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⡄⠀⠀⠀⠀⠀⡀⢸⣇⠀⢀⡼⢸⡇⠀⠈⣿⣿⡇⠀⠀⣷⢈⣀⡄⢸⣿
⠀⠀⠀⠀⠀⠀⢸⡇⢸⠹⠧⠤⠀⠀⠀⠀⠃⢸⣿⣿⣿⣿⣿⣇⣀⣈⣿⣿⣇⠀⢸⣿⢾⣏⡇⢸⣿
⠀⠀⠀⠀⠀⠀⢠⣿⢿⠖⠒⠖⠒⠒⠔⠂⠐⢺⣿⣿⣿⣿⣿⡏⠙⠙⣿⢻⣿⠿⣟⣏⣿⡏⡇⢸⣿
⢠⡆⠀⠀⠀⠀⣿⡇⢰⢸⠋⠉⠀⡉⠁⠁⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⣿⣺⡟⠠⠀⣿⣿⣷⡇⢸⣿
⠘⠁⠀⠀⠀⠀⢸⡇⢸⣸⠀⣀⣄⣀⡆⣠⣤⠀⢹⣿⠀⠀⠀⠀⠀⠀⢀⣘⣓⡦⢀⣿⣿⣿⡇⢸⣿
⣠⡏⠀⠀⠀⠀⢸⣿⣸⣿⢡⣿⣿⣿⣧⣿⣿⡄⠀⢽⣷⢸⡍⠉⠉⠉⠉⠉⠉⠀⠉⠻⢿⣿⡇⢸⣿
⠁⡇⠀⠀⠀⠀⢸⡿⠋⣉⢼⣿⣟⣿⣿⢿⣿⡇⠀⠀⡉⢐⡄⠀⠀⢀⣀⣀⣀⣀⣀⡤⢤⢬⡇⢸⣿
⣿⣷⠀⠀⠀⠀⢸⣷⣰⣶⣲⣶⣐⣶⣾⣯⡽⠇⠀⠀⣿⠘⣿⠀⠂⢥⣤⣤⡤⠶⠶⠶⠶⠾⠇⢸⣿
⣿⣿⡄⡀⠀⠀⣸⣟⠿⣙⣫⣍⣭⢍⠀⠀⠒⢤⠀⠀⢛⠀⢿⡐⣶⠸⣿⠤⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢠⣽⣯⣭⣙⡤⣄⠶⢤⣴⠿⠗⠻⠖⠂⠀⠀⠒⠆⠀⠸⠂⠄⡇⠘⡀⠄⣧⡐⢄⠀⠀⠀⠀⠀⢸⣿
⢸⣿⣿⣿⡉⠠⣀⠨⢤⡤⠂⢤⡐⠄⣐⠀⠤⠀⠀⠀⠀⠐⣾⡇⢀⡁⠀⣿⣿⣆⡱⣄⠀⠀⡀⢸⣿
⣾⣿⡓⣦⣝⢂⣤⡍⣀⡄⣁⡠⢄⡀⠒⠤⠴⢢⡴⠂⡰⢿⣿⣿⠈⢁⠘⣿⠿⠿⠷⠈⠋⣏⡇⢸⣿
⠋⠍⠋⠡⠐⣉⠄⢈⠁⠒⡩⠔⢁⠀⢀⣠⠆⣁⠘⠡⣴⣋⣽⣿⠀⢸⠀⠘⠢⡄⢡⡄⠀⣿⡇⢸⣿
⠤⣈⠀⠰⢄⡀⠚⠤⠖⠛⠰⡎⣉⠻⢫⣉⠚⠠⣈⡛⠳⢮⣽⣟⠀⣴⠀⠀⠀⣧⠈⠇⡇⡿⡇⢸⣿
⢄⡀⠐⠦⣄⠀⠒⢄⣀⡙⠲⣄⡈⠓⢠⣌⡙⣻⢬⣉⡳⠲⣬⡝⠐⣿⡀⢳⠀⢻⠀⠀⠇⣿⡇⢸⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⣿⡿⠻⠟⠋⠀⠙⠿⠻⣿⣷⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢿⣿⠋⠈⠐⠀⠀⠀⠀⠀⠂⠀⠹⣿⠿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠉⠀⠀⠀⠃⠀⠀⠀⠘⠀⠀⠀⠋⢠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⢀⠈⠉⠁⡀⠀⠀⠀⠀⠹⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣤⣶⣿⣄⠀⣠⣷⣦⣤⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣞⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠁⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡛⠛⠛⣛⣛⣛⣛⡆
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣭⣭⣿⣯⣭⣭⣭⣥⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡯⡍⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣖⠒⠁⠘⡇⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⣧⢠⠞⠉⠀⠀⠀⡩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⠸⡄⠀⢸⠃⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠦⢤⡤⠴⠒⠒⠒⠲⠒⠒⠒⠒⠒⠶⠶⣤⣤⣄⢀⡴⠚⠉⠁⠀⠀⠀⠈⠓⢷⠀⣾⢀⠔⢀⣰⣋⡀
⠀⠀⠀⠀⠀⣰⠋⠉⠉⠉⣉⠴⠒⠤⠄⠒⠒⠒⠒⠒⠒⠂⠐⠂⠒⢢⠴⠂⠁⠀⠘⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠽⠋⠉⠁⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⢠⠇⠀⡌⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⣠⠞⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⡞⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠀⠀⠀⠀⠀⡠⠊⠀⠘⡀⠻⡄
⠀⠀⠀⣸⠁⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⠀⠀⢀⡠⠊⢀⠄⠂⠤⠇⢀⠟
⠀⠀⢀⡏⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⡀⠌⠀⠀⡠⠞⠁
⠀⠀⡼⠀⡰⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠠⠊⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⣰⠃⠀⠄⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⣀⠴⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⢀⡏⠀⠎⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⡠⠄⠐⠊⠀⢀⠔⠁⠀⠀⡠⠞⠁⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⣸⠀⡠⠁⢀⠔⠁⠀⠀⠀⠀⠀⡰⠋⠀⠀⣀⡀⠀⡀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁
⡏⠀⡥⠚⠁⠀⠀⠀⠀⠀⢀⠜⠀⠀⡐⠁⠀⠀⠉⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁
⢷⡊⠀⠀⠀⠀⠀⢀⡠⠔⢁⠤⠂⠉⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣀⣀⣀⣀⣀⣠⠤⠖⠋⠁
⠀⠙⢦⡀⠀⡠⠊⠁⡠⠔⠁⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠙⢯⡠⠔⠈⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠙⢤⣀⡠⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠱⣦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠶⠀⠼⠁⠈⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢧⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⡄⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⡋⠂⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠜⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⢀⣯⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⡆⠀⠀⣏⣀⡀⠀⠀⠀⣠⡴⠿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣷⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠉⢩⠿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢀⣒⣚⣯⣷⢶⠶⠞⣻⣿⣟⡉⠉⠁⠀⠈⡄
⠀⠀⠀⣀⣤⣠⢄⣀⠀⢳⣴⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣧⡀⠀⣀⣴⣁
⠀⠀⠈⠀⠈⠐⠋⠚⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡋⣙⣰⡿⠋⠉⠁⠀⠀⠙⠉⠓⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉⠀⠀⠀⠀⠉⠉
⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⢤⠔⠒⠀⢀⡀⠀⠀⠀⠀⠀⠖⠀⠀⠤⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠁⠈⢀⠔⠀⠀⠀⠀⠠⠚⢁⡠⢤⣤⠀⢀⡤⠞⣉⠀⠀⠀⠀⠉⠁⠒⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠉⠀⠀⠀⠀⢰⡃⠀⠀⠀⠀⠀⠀⠐⠃⠀⠛⠀⠠⠊⠀⠞⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣀⠀⠤⠀⠒⠊⠁⠈⠉⠉⠀⠒⠒⠂⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠰⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠗⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠳⠟⠉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣋⠀⣀⠰⠀⠀⠀⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⠋⠉⠉⠱⠂⠈⠁⠃⠈⢷⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄
⠀⠀⠀⠀⢀⣀⠀⠠⠀⢠⡄⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣿⡇⢸⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢸⡇⣼⣻⣟⣻⢩⣿⡤⠌⠿⣆
⠀⠀⠀⣴⣿⣿⣷⣆⠀⠀⢻⠻⣿⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⠉⠀⣷⡀⠀⠀⠀⣰⣄⡀⠀⠀⠀⢸⡇⢸⣿⠋⡏⣻⣿⣇⠀⠀⣿
⠀⠀⠀⢿⣿⣿⣿⣿⠀⠀⡆⠾⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠺⠛⠖⠶⠶⠛⠛⠶⠶⠶⠚⠿⠃⠀⢿⠋⡟⡟⠻⢿⡆⠄⣸
⠀⠀⠀⡸⣿⣿⣭⣿⠀⠀⢠⢰⢾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⡆⢈⡀⠀⠀⡀⠀⠀⠀⢠⠀⡄⠀⠀⠀⠀⠀⠀⠘⠛⡏⡟⢻⡞⡇⢠⡇
⠀⠀⠀⠁⠀⠻⣿⣿⣷⡀⠀⡇⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣇⣜⣸⣧⣧⡙⠀⠸⣶⠇⠀⠀⠀⠀⠀⠀⠀⠛⠛⡿⢳⣶⣟⡏
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣰⡄⢼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠉⣹⢏⠉⣥⣤⡿⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢻⠻⡿⠟
⠀⠀⠀⡄⠀⡐⠃⠀⢻⣿⡷⠅⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣶⡾⠿⠂⠁⠀⠀⠀⠀⠀⠴⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⡾⣿⠁⠁
⠀⠀⡼⠀⠠⠁⠀⠀⢸⣿⡷⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠀⠀⠀⠀⠀⢀⣸⣾⣧⡆
⠀⢰⢀⠀⠀⠀⠀⠀⢸⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠛⣿⠟⠛⠛⠛⠉⠉⢹⣿⠿⡟⡠⠀⠀⠤⣠⠖⢋⡤⠉⠸⢷⡇
⠀⠊⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢿⠀⠀⠀⠀⠀⠀⠀⣿⡄⣿⡇⠀⠩⣰⢄⡟⣠⣴⠏⠴⡻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣦⣤⣤⣤⣤⣤⣤⣤⢶⡏⠀⠀⠀⠀⢿⡇⣸⠀⠀⠀⠀⠀⠀⠀⢿⡇⢿⣿⣷⣤⣄⣀⡈⣉⣠⣡⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢻⡟⠋⠙⠉⣿⡀⠘⣿⠀⠀⠀⠀⢸⡇⢹⠀⠀⠀⠀⠀⠀⠀⢾⡇⢸⣿⢰⣿⠛⠛⠛⠛⠋⢹⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⢸⡇⠀⢀⣀⢸⣇⣠⣿⢀⣤⣤⣤⣼⡗⢺⣷⠦⣴⢖⣀⣠⣄⣾⡧⢾⣿⣼⣿⢀⣠⠄⠂⣤⣸⡗⣿⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⣾⠏⠙⡶⠒⢺⡗⣱⠟⠉⠛⣵⠧⢼⣿⠿⠥⣄⡸⠬⢄⡀⠘⣿⣿⣩⣩⣛⣷⠰⠿⠠⠐⢛⣿⣋⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡷⢛⢛⡟⡙⢠⢶⡿⢿⠒⠋⢛⣁⣠⣾⠟⠀⠤⠾⡷⠆⠴⣷⣿⣟⣋⠼⠛⢁⣹⠆⢰⠶⣿⣏⣉⣿⡧⢄⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠃⠤⠒⠋⢉⠿⠿⠐⠀⠴⣦⡿⡟⠉⠉⣱⣏⣛⣸⣛⣉⣹⠟⢁⠈⠷⠔⠻⠛⠓⠘⠛⠚⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠓⠛⠉⠃⠛⠂⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀
⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠠⠄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠁⢰⡉⠁⢰⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠘⠁⠈⠀⠁⠴⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⢀⠦⠐⠆⠢⢀⡔⠀⠀⠀⠐⠂⠒⠒⠒⠒⠒⠒⠚⠂⠀⠓⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠀⠀⠀⠀⠀⠐⠁⠀⠘⡀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢩⣶⡉⢀⣶⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠀⠠⠼⢃⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣹⣇⡀⣹⣏⡀⣸⡆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣀⣣⣀⣨⣧⡀⣠⢈⡀⢰⠀⣀⡁⢀⣆⡀⢀⣣⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠁⠉⠉⠉⠉⠉⠉⠉⠉⠙⠈⠛⠓⠀⠀⠀⠁⠀⠀⠐⠛⠛⠙⠋⠁⡻⠙⠋⠉⠙⠙⠉⠛⠛⠉⠛⠛⠛⠛⠋⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠰⡆⠀⣆⠂⠀⠀⠀⠤⠀⠤⠤⠆⠀⠀⠀⢀⠀⠐⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡠⠁⠀⣁⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣁⠀⣨⡀⠀⠀⠀⠀⠀⠀⠠⠈⠐⠁⠐⠀⠀⠀⠀⠀⠀⣠⡦⢀⡀⠀⠀⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠆⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡁⠉⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢀⠀⠚⠀⠀⢠⠄⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡰⠀⠀⡀⠠⠀⠀⠀⠤⠙⠃⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠈⠆⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⡀⠀⠠⣀⡀
⠀⠀⠀⠀⠀⠀⠏⠠⠀⠹⠷⠀⡰⠀⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠈⡀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠂⠀⠀⢦⡈⢀⠀⠀⠀⠐
⠀⠀⠀⡄⠀⠀⠈⠀⠀⢠⠀⠎⠀⠐⠀⠀⠚⠋⠀⠉⠁⠁⠛⠃⠉⠈⠁⠈⠛⠋⠀⠉⠉⠈⠛⠃⠀⠉⠉⠈⠛⠁⠉⠈⠁⠉⠀⠀⠀⠀⠀⠈⠈⡃⠁
⠀⠀⠀⠁⠀⠙⠀⠀⠀⢸⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠘
⠀⠀⣀⡆⠀⠀⠀⠀⠀⢀⠀⡅⠀⠴⣈⣉⣉⢉⡉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣙⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠⠀⠀⠀⢠⠶⠟⠈⠛⠰⠛⠓
⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠉⠉⠋⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠁⠈⠀⠉⠀⠉⠀⠃⢈⡀⠀⠀⠁⠈⠀⠀⠈⠁⠀⠀⡇⠀⠒⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⢀⠸⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⣀
⠀⠀⠀⠀⠀⠈⠋⠡⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠀⠁⠀⠉⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠃⠀⢀⠈⠀⠀⠀⠀⠀⠀⢱⢀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡉⠈⠀⠀⠀⠀⠀⠀⠈⠀⠈⣉⡁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠉⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⡤⠤⢤⣤⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢹⠂⠀⠁⢨⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡗⠒⠒⠒⠒⠀⠀⠀⠀⠒⠒⠒⠐⠒⠒⠀⡏⡀⠀⠀⠀⠀⠀⣑⡬⠕⢻⣛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠃⠀⠀⠀⢀⠎⢿⠀⠒⢁⡞⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠴⠤⠤⠤⠤⠴⢲⠀⡴⠂⠒⠒⠒⠰⣾⢇⢠⠀⠀⠀⡌⠐⠁⠈⠉⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢛⠛⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⣃⣀⡀⠀⢰⠀⠓⡀⠀⠀⠀⢢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣈⠀⠀⠀⠀⢀⣀⠔⠊⠚⠒⢤⠤⠤⠤⣤⠊⠀⠀⠀⠑⡜⣾⣷⣷⠖⠀⢲⢉⡀
⠀⣀⠀⠀⠀⠀⠀⢀⠹⠿⠅⠀⠀⠉⠩⡇⢀⡤⠶⠖⣲⡝⠀⠐⠃⢰⢂⡔⠐⠞⡁⠈⢑⠁⠂⠈⠁⠘⡀
⣰⠁⠀⠀⠠⠤⠄⠠⠥⠤⠤⠄⠀⠀⠀⠱⢾⡀⢀⡀⢠⠇⠀⠈⠦⡻⣦⡀⠾⡦⠋⠉⢭⠛⡦⠉⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⡷⡾⠓⡄⠀⠀⠀⢠⠊⠝⢿⡄⠷⣄⣘⣤⣞⣮⣥⣤⣾
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢄⡑⢰⠴⣀⠀⠀⢸⠀⠢⢠⣉⠛⢛⡉⠝⠉⠉⠉⠉⠁
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠑⢎⠢⡀⠃⡄⡆⠀⠈⡀⠀⠀⠀⠸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣂⢀⣱⣈⣧⠇⠁⠀⠀⠧⠀⠀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠒⠂⠓⢸⠀⠀⠀⢈⠖⠰⢺⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⡆⢸⠀⢀⣤⢊⠔⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠀⡃⢐⠃⢲⡀⢂⠀⣽⡀⠀⡏⠈⢱⣄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⣤⣂⣁⣀⣉⣪⣜⣀⣒⣀⣃⣀⣁⣒⣒⣒⣁⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⠀⠐⠒⠶⠖⠲⠆⠀⠀⠰⠒⠦⠶⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⢸⡍⣷⠀⠀⢀⡄⣶⣤⣠⣴⠀⣶⣄⡰⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣇⠀⠀⣽⠀⠀⠘⠃⠁⡉⡉⠀⢀⡀⠁⠀⠀⡆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠐⠋⢀⡇⠸⠀⡖⠀⠏⡖⡸⠂⣾⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠀⠛⠀⠂⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠰⠒⠛
⠀⠀⠀⠀⠀⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⡠⠒⡰⠄⢢⣎⣁⠀⠐⠚⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠶⠶⠒⣒⡒⠲⢤⡀⠀⡤⠀⠀⠀⠀⠀⠀⢀⣠⠖⠻⡄
⡟⠓⠺⣄⠀⢀⣇⣀⣹⡄⠀⢸⣄⠀⠀⠂⠀⠀⠀⠀⢸⣁⠀⠀⠀⠀⠀⢀⠀⠈⢑⡦⢷⡀⣌⠐⢣⡄⢀⣀⡰⠟⢫⣾⠟⠁
⠇⠀⠀⠀⠉⠉⠀⠀⠀⣹⢿⣧⡎⠀⠀⠀⠀⠀⠀⠀⢸⡂⡈⠀⠀⡐⢀⡀⣈⢀⣀⣤⣾⠁⠉⠀⠛⠓⢀⣤⠶⠛⠉
⢰⠀⠀⠀⠄⠀⠀⠀⡐⠉⠀⢧⠀⠀⢐⠀⠀⠀⠀⢴⠉⠻⠽⠿⣭⠿⣯⠽⠛⣛⡻⢟⡟⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀
⠸⠀⠀⠀⠀⠀⠀⠐⢀⣠⠴⠚⠀⠀⠀⠉⠉⠁⠀⠈⠲⢍⣀⣒⣐⣒⣀⡨⠭⠄⠒⠉⠀⠀⠰⠀⠀⠀⠀⠈⠀⠈⠁⡆⠀⠀⡀⠀⠀⠨⢆
⠀⠉⠙⠛⠛⠒⠚⠞⠋⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⢀⡀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⠀⠀⠱⢂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡨⠓⠀⠀⠀⡈⠓⢦⡀⠀⠀⠘⣆⠀⠀⠀⠀⣼⠃⠀⠀⠀⣤⣶⠶⣤⣤
⠀⠀⠀⡀⠀⢀⡄⠀⠀⠀⠀⠰⢄⣨⠁⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠈⠢⡀⠙⢆⠀⠀⠁⣆⠀⠀⠀⡇⠀⠀⠀⠀⠁⢀⠠⠐⡄
⠀⠀⠀⠀⠐⠳⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠤⠀⠀⠀⢀⠎⠀⠀⠐⠦⠀⠀⡀⢀⣨⡦⠠⠇⠀⢀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠈⠀⡁⢄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠄⠀⠀⢀⠀⣠⣴⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡀⢂⣤⠾⠛⠁⠐⠃⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠄⡆
⠀⠀⠀⠀⠀⠀⠄⢙⣣⡄⠀⠀⠈⠀⠉⠀⠀⠀⠀⢸⣀⣴⠟⠋⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠘⠶⠤⠀⠀⠀⠒⠃⠀⠀⠀⠀⠀⠘
⠀⢀⠄⠒⠢⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⡤⠒⠛⠛⠉⠉⠉⠓⠦⡀⠀⠀⠀⠀⡤⢀⢄⡀
⠀⡀⢰⠀⠀⢧⠀⠀⠀⠀⣠⣄⣴⣄⠐⢃⣀⡀⠀⠀⠀⢠⠏⠀⠀⠈⠀⠀⠀⠀⢀⣠⢼⠀⠀⠤⣜⠷⠻⣇⢱⠈⡁⠀⠀⠀⠐⠀⠀⠈⠠
⠀⠄⣸⠀⠀⢸⠀⠀⢠⣾⠏⡠⠊⠀⠀⠀⠀⠉⢶⠀⠀⡟⠀⠤⠀⠀⠀⣠⣤⣶⠿⢳⣻⡀⠀⠀⣿⠄⠀⠟⣸⣄⡿⠃
⠀⡇⣇⠀⠀⣼⠀⠀⢼⡫⠜⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠳⣤⡤⠶⠊⣙⢋⠽⣷⢦⣎⡥⢴⠋⠀⡋⠘⠚⠷⠛⢛⡇⠀⠀⠀⠀⠀⡀⠠⠤
⠀⢶⢸⠀⠀⡇⠀⠀⠀⠉⠓⠲⠤⠤⠤⠤⠤⠀⠠⠃⠀⠀⠈⣼⣄⣸⣭⣴⡶⠿⠋⠃⣱⠎⠀⠀⢸⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢀⠃⠀⠄
⠀⢸⡌⠀⢰⠇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⢀⣀⣈⣉⣀⠤⠒⠊⠁⠀⠀⠀⠀⠃⠀⠀⠠⠓⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠳⣤⠟⠀⠀⠀⠀⠀⠀⠁⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠂⠐⠚⠀⠀⠀⢀⠀⠐⡀
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⠟⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⣠⡌⠢⡀
⠀⠀⠀⢸⠀⠀⠀⠐⠂⠀⠀⠐⠖⠒⠲⡤⠠⢤⡶⠀⠀⢀⠤⣠⠶⠶⠚⠋⠉⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡽⡆⢀⣬⡦
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠀⠀⠂⠀⠀⠁⢨⠀⠀⠀⠀⠀⣀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠘⠦⠂⣀⣒⠀⠉⠝⠃⠚⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣵⣤⣤⣤⣮⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⠤⣽⣋⣛⣛⣛⣛⣋⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠙⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⠀⠀⠈⠙⠂⠉⠛⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⠟⠋⠀⠀⠘⠛⡿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠉⢀⡀⠀⠀⢦⡔⠀⢉⣼⠏⠀⠀⠀⠀⠹⣧⡁⠀⢢⡵⠀⠀⢀⡀⠻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠑⠷⠀⣀⡄⠁⣠⣾⣿⡀⠀⠀⠀⠀⠀⠈⣓⣄⠈⠀⡀⠀⠾⠊⡰⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣶⡤⠀⠐⠚⠁⠈⣿⠧⡀⠀⠀⢀⠔⠁⠀⠈⠐⠂⠀⢤⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣟⣉⡳⣄⠀⠀⠟⠀⠀⠀⠀⣀⢀⡈⠀⢁⠀⠀⡘⠀⢀⡀⣀⠀⠀⠀⠀⠡⠀⠀⣠⢞⣩⣻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣼⠉⢻⠾⣷⣎⣤⣂⠨⠭⠒⡞⡿⠓⢀⢘⠀⠀⡁⡤⠚⢽⢓⢶⡭⣄⣀⣤⣁⣼⠿⠛⢻⣡⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠈⣇⠞⡕⢿⣿⠙⠓⠲⠒⠶⡓⢧⣀⣜⡸⠀⠀⠇⣢⣀⡼⠘⠉⠀⠚⢴⡋⣿⡿⡰⠒⢸⠃⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢧⡻⠸⠸⣿⡆⢀⠀⠀⢀⠶⠁⡇⠀⢠⠀⠀⡄⠀⢸⠸⢁⡀⡀⢀⣈⢱⣿⡇⡘⠌⣼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣷⣶⣿⣷⣂⣤⣵⣶⣾⣿⣿⣶⠌⠀⠀⠡⣶⣿⣿⣶⣶⣦⣭⣌⣸⣿⣴⣾⣉⠁
⠀⠀⠀⠀⠀⠀⠍⠢⣀⡀⡤⠐⠊⠹⣿⣿⠿⠿⠿⠛⠿⣿⡿⠿⣿⠇⠀⠀⠀⠀⠘⣿⠿⠿⣿⠿⠛⠿⠿⠿⣿⣿⡏⠁⠂⢤⢄⣀⠔⠹
⠀⠀⠀⠀⠀⠀⠈⡄⠀⣀⠴⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡉⠀⠀⠀⠀⠀⠀⢠⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠦⢀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠔⠀⠀⠀⠀⠀⠀⠀⠀⠢⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⡀⢉⠛⠒⢯⣿⣲⠤⣀⡀⠘⠻⢯⣉⠉⠉⠁⠀⠀⢤⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣤⢤⣄⣀⣀⣀⣤⣤⣴⡖⠛⠃⠈⠉⠉⠁⠀⠀⠀⢀⣀⡐⢛⣿⢻⠁⡀⠃
⠙⠛⠲⠦⣤⣄⡈⠛⠒⠫⣵⣂⠄⢀⠀⠀⠀⠀⠀⠀⠈⡏⠉⠉⠉⠉⠀⠀⠀⠀⠀⢸⡿⠿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⡆⣰⡟⢸⣧⠃⡰⠁⡤⠃
⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⢄⣈⡉⠑⠦⠽⣂⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⣿⣿⣿⣿⣿⣇⣤⣠⠠⠤⠄⠠⣀⣐⣚⣓⣻⣶⢿⠏⡴⣡⣞⡇⡘
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⢤⣍⣙⠒⠶⣷⡒⠶⠦⠤⠄⠀⠀⣒⣒⣚⣓⣻⣛⣻⣝⣿⣯⣿⣽⡭⠯⠭⠤⠤⠷⠼⠶⠷⢺⡶⠷⠚⢛⡇⠘⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⢀⠈⢹⡟⠋⠉⠉⠉⠉⠉⠉⠉⠍⠉⡉⠉⠁⠀⠀⠀⠀⠀⠀⢠⠖⡶⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⡄⢀⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠨⢅⠀⡀⠈⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠀⠉⠄⠀⡧⠲⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠨⠄⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⢀⡰⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡃
⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠈
⠀⠀⠀⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⠤⠤⠄⡀⢀⡀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣼
⠀⣤⣀⣀⣀⣈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⣁⡄⢹⠄⠀⠀⠀⢁⠉⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⢀⣠⠄⡀⠀⠀⣸⠀⠀⠀⠀⠀⠀⡬⠭⢠⣬⠀⠀⣿
⠀⢹⠆⠀⠼⣿⣶⣒⣲⣶⢦⣤⣤⠴⠒⠚⠭⣿⠀⣾⡆⠀⠀⠀⠀⡐⠀⠀⠀⣠⡟⢲⣷⡖⣖⣿⣿⣿⠆⠁⠀⢠⠇⠀⠀⠀⢠⠀⠀⠂⠀⢸⣾⣆⠤⡏⠀⢠⡟
⠀⠨⣆⡿⣀⣿⣿⢛⣏⡯⢽⣿⣿⢸⡷⠶⣶⣿⣮⣿⣦⣤⣤⣤⣌⣉⣆⠙⡋⣿⡇⢸⣿⠀⠟⠷⣿⣿⡤⢤⠤⡌⠀⠀⠀⠀⠺⣦⡤⠤⣤⣬⡧⡟⢒⣃⣉⣹⠯
⠶⢀⢿⡗⠚⢿⡀⢀⣀⡠⣾⣿⣿⣼⣿⣴⣾⣿⠉⣏⠀⠀⠀⠈⢽⠻⠋⠹⣟⣿⡗⢺⣿⣶⣷⣶⣿⣿⣷⣿⣤⣇⣍⣀⠀⠀⣃⣏⣉⣷⣶⣿⣷⣿⣉⣛⣻⡻⠾
⠀⢀⢸⡯⢭⣿⣯⣭⣦⣾⣿⣿⣿⠿⠿⠛⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠃⠈⠉⠉⠉⠉⡿⢿⠟⠟⢻⠻⠉⠙⠉⠛⠻⣹⢻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣸⣷⣶⣿⣼⣿⣿⣿⣿⡿⣿⠁⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⣟⣾⣏⣿⠉⠁⠐⠒⣶⣯
⠠⠆⣻⠿⠺⠽⣿⠟⠛⠋⠭⢉⡉⠄⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⣸⡙⢸⡿⢿⡯⠉⠛⠛⠛⠛⠻⠿
⠀⠀⠠⠀⣀⠘⠁⠀⡀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⢠⠀⠀⠄⠀⡰⠀⢸⠀⠀⣾⡇⢸⣿⣼⣷⢣
⠀⠀⠀⠀⣀⡠⠒⠁⠀⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠁⠀⡜⠀⣰⠇⠀⠙⠀⠀⠀⠀⢸⡋⣿⣿⣧⢣
⢀⣠⠶⠋⠁⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⡠⠀⠀⢠⠇⠢⠠⠀⣀⡈⠀⠀⠈⠀⠀⠄⠀⠀⠀⠀⠘⡀⢸⢿⣿⣯⣆
⡉⠀⠒⢀⡤⠂⠀⠀⠐⠀⡀⠀⠈⠀⠠⠀⠀⡠⠊⠀⡠⠈⠀⠀⠈⠀⠀⠖⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠉⠑⠃⠐⠨⠤⢤⣀⠀⠀⠀⡆⠀⢸⣿⢿⣷⣶⣶⣶⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⢀⠤⠤⠤⠤⠄⠤⠤⠤⠭⠥⠄⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠈⠈⠀⠀⠀⠀⠀⠐⢠⠀⢠⢠⣄⣴⣶⣾⣟⣧⢖⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⢢⣆⡚⠐⡀⠀⡈⢀⠂⡀⢸⢸⣿⣿⣷⣾⣿⣿⣿⣟⡽⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠈⠦⠀⠍⢰⠀⠎⠀⠄⢸⠙⢽⣿⣿⡿⣿⣿⣿⣯⣾⠲⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠁⠀⢀⠀⠠⢄⠀⢂⠊⠀⠀⠀⠘⠀⠘⠉⠻⠉⠉⠉⠁⠉⠁⠠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠀⠜⠀⠜⠠⠀⠆⠑⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⢀⡀⣀⠈⢆⠀⠠⡀⠀⠂⢸⠠⠂⠀⠀⠀⠐⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⠈⣂⠈⠙⡩⠁⠀⠢⡀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠓⠀⠀⠐⠀⠉⠀⠀⠀⠀⠉⠀⠀⡇⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠄⠤⠤⠀⠀⠀⠖⠠⠤⠤⠀⠀⠤⠃⠀⢄⠀⠀⠀⡈⣑⣀⣀⣂⠁⡀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡄⢠⡄⢰⠀⡠⠂⠀⡄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡀⢆⣀⡇⢰⠀⢶⠀⢣⣀⠀⠇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⡀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⢇⠸⠀⡃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠁⠁⠈⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠀⠒⠒⠒⠒⠂⠰⠷⣭⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⢀⠤⠟⠀⡠⠞⠉⠦⡀⠘⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠷⠾⢎⣡⠶⢶⡉⠀⠀⠀⢀⣈⠶⢼⡏⠲⠶⡦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠘⠀⠠⠐⣀⡀⠀⠀⠀⢠⡴⠆⠠⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠉⣰⠇⠀⢠⣿⣿⣿⠀⠀⠀⢸⣿⣿⣦⠀⠀⢿⡋⠈⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⣽⠀⠀⠀⠙⠛⠁⠠⠀⠠⠀⠙⠛⠁⠀⠀⢸⡇⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠔⠷⠄⢠⡄⠀⠀⠀⠰⣦⠴⠂⠂⢠⡷⠀⠀⠀⠀⣴⠀⠴⠿⠴⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠘⠾⡉⢩⠗⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠓⠤⠤⠤⠤⢤⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡴⠶⠆⠀⠶⠆⠀⠙⠛⠁⢦⡴⠘⠛⠃⠀⠲⠦⠀⠰⠖⢦⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠁⠉⠈⠉⠉⠁⠀⠒⠒⠶⠶⠶⠶⢶⠖⠒⠀⠀⠈⠉⠉⠉⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⢀⡀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠤⠤⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠈⠰⠒⠒⠺⠀⠀⠀⠀⠀⠀⠀⠀⢐⡇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢨⡧⢼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣎
⠀⠀⠀⢀⡠⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⡀⣀⢸⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀
⠀⣀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠂⠀⠀⠞⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢆⡀
⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠤⢤⣤⢤⡤⠤⣤⣤⣤⢠⣤⣀⣀⣠⣤⠤⠤⠤⠤⠤⠄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠠⠤⠤⠤⠤⠤⣤⣄⣀⣀⣤⢠⣤⣤⠤⠀⣤⢤⡤⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣅⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠉⠉⠉⠉⠉⢹⡆⠀⠀⠀⠀⠀⢸⠉⠁⠉⠀⠀⡇⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡤⠀⣀⣀⣀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣀⣀⣀⠀⢤⠈⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣀⣴⡆⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⣦⣀⣴⠂⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢉⡙⠏⡇⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣸⠻⣏⡉⠷⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⢀⡴⢉⠀⢺⡟⡰⠃⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠉⢆⠹⡗⠀⡈⢦⡀⠀⠀⠀⣧⣤⠀⠀⡇
⠀⠀⠀⠀⠀⡀⠐⠂⠀⠁⠀⠀⠀⠀⢠⡍⠀⠀⠑⠊⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠙⠊⠀⠀⠉⡄⠀⠠⠀⠠⡇⠀⡃
⠀⠀⠀⠀⠀⠟⠛⠆⠀⠉⠀⠁⠀⠀⠘⢇⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⡠⠃⠀⠈⠂⠘⠁⢀⠛⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡅
⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶
⣿⣿⣿⠏⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⠀⢀⠠⠀⠴⠀⢸⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣄⣹⣿⣿⣳⡄⠀⠀⠀⠀⠉⠳⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣭⣌⠙⠿⢷⣮⣿⣿⣿⡆⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣾⣶⣿⡿⠋⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠿⠛⡗⡮⠙⠋⢀⣂⠖⠀⠀⠀⠀⠰⣆⣀⣀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡚⠀⠀⠀⠑⠀⠀⠀⠀⢀⡀⠀⠀⠀⠂⢀⣀⡵⢾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣎⠀⠙⢿⢢⡀⠀⠀⠀⠀⠐⠚⠀⠀⠀⠤⢟⣭⣥⣶⣟
⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢙⣦⣀⣳⡙⠢⠌⠁⠀⠤⠀⠀⣘⠯⠈⠙⠀⠉⠉
⠀⠀⠀⠀⠀⢀⠖⠁⠄⠀⠀⠤⣙⢦⡝⠹⣤⣻
⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠈⠑⣝⣆⠹⣟⡤⠀⠒⠒⢲⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⢧⢹⡅⠀⠀⠀⢸⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⢲⢄⣀
⠀⠀⡼⠃⠀⠀⢄⠀⠀⠀⠄⠀⣀⡀⢤⣹⠃⡈⡃⠀⠀⠀⠘⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠑⠤⡀
⠀⡰⠃⢀⠈⠢⢈⢢⡀⠀⠀⣀⣀⣭⣓⠚⢳⣁⣷⣄⠀⠀⠀⡇⠚⣡⣵⡶⢠⠀⠒⠘⢄⠀⠀⠀⠀⠀⠀⢀⠖⠁
⠞⠁⠀⠀⠙⠢⢤⣁⠻⣯⡉⠀⠀⠠⠤⠐⠚⠻⣷⠬⠀⠀⠀⡞⠂⠀⢁⠠⠄⠀⠀⠀⢀⠀⠀⠀⢀⢄⢴⣉⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⢳⣄⠀⠀⠀⠀⠀⠀⢘⡇⣀⣀⣠⣁⣀⠡⣌⡃⠀⠀⠐⠀⠀⠀⠀⣀⠴⢧⣎⡀⢀⡀
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⢞⣰⡿⠀⢠⠀⠀⠀⠀⢫⠢⡄⠳⣄⠀⢀⣢⣛⠆⡀⠀⢠⣄⣤⢀
⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠴⠚⠁⠎⣠⠔⠁⠀⠀⠀⠀⠈⡇⠁⠀⠈⠐⠋⠠⡙⠴⢀⠄⠔⣭⣭⠢
⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠈⠳⢦⣀⠐⠊⣁⣀⣀⣀⣀⣤⡤⠤⠁⠀⠀⠀⠀⠐⠄⠈⠱⡄⠐⠂⠊⠁⠐
⠀⢾⡒⢒⠀⠀⠀⠀⠠⠠⠄⠀⢁⣀⣀⣀⡀⠙⢍⠣⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠢⡀
⢰⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠒⢯⡳⠄⠣⡈⢂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣠⠔⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡓⢦⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⡼⠋⣩⣭⠉⠉⠉⠈⡏⠀⠀⢸⢹
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⢀⣄⡀⠀⠀⠀⠀⠀⣟⠓⢠⠇⠘⠆⢠⡴⢶⣄⠐⠁⠀⠀⢾⠀⠀⠉⠁⠀⣰⡆⠀⡇⠀⠀⣼⢸
⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠤⠈⠇⡝⠀⠀⠀⠀⠀⡏⢀⠎⠀⠀⠀⢟⠰⣟⣹⠀⢀⣀⠀⠈⠳⣄⡉⠀⠀⠛⠁⠀⡇⠀⠀⣿⠸
⠀⠀⠀⠀⠀⠀⠀⢺⠒⠃⠀⠄⣠⠀⠀⠀⠀⠀⢰⡇⠸⡀⠀⠉⠀⠈⠢⠈⢁⠀⠀⠀⠀⠀⠀⠀⠉⢳⡄⢠⣶⣤⠃⠀⠀⣿⢠
⠀⠀⠀⠀⠀⣀⣤⠀⠀⢀⡀⡄⠁⠀⠀⠀⠀⠀⢸⡇⣤⡉⢦⡀⠀⠰⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⣠⡞⠀⠚⠛⢸⡀⠀⠀⡏⢸
⠀⠀⠀⢠⠎⠁⠀⠑⠦⣠⠄⢄⠀⠀⠀⠀⠀⠀⢸⡇⠈⠁⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⡴⠄⠀⢸⠁⠀⠀⡇⢸
⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⢸⡇⡀⠴⠚⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⢸⢸⡇⠀⠀⡇⢸
⠀⠀⠀⢿⠀⠀⠾⠀⠀⠀⠀⢠⢸⠀⠀⠀⠀⠀⢸⡇⠁⠾⠋⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣠⢸⠃⠀⠀⡇⢸
⠀⠀⠀⢈⠇⣠⣶⡀⠀⠀⠀⢈⣈⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⠁⠀⠀⠀⠀⠀⡀⠒⢒⠂⢤⠀⠀⠀⠀⠀⠣⡁⢸⠀⠀⠀⡇⢸
⠀⠀⠀⠀⡟⢉⡁⠈⠙⠒⠢⠌⡉⠀⠀⠀⠀⠀⢸⡇⠴⠃⠀⠀⠀⠀⠀⠀⢸⠆⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⢸
⠀⠀⠀⠀⢹⠀⠈⠉⠀⠒⠂⠤⢄⡀⠑⢲⡄⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⢬⡶⢆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠃⠸
⠀⠀⠀⢀⣿⠂⠀⠀⠀⠀⠀⣿⠠⣼⠐⠟⢁⠀⠘⡀⠒⠦⢤⣄⣀⡀⢀⣠⡶⢿⠾⣕⠒⠋⠈⡄⠀⠀⠤⠴⠖⢿⠀⢀⣀⡇⣇
⠀⠀⢤⣎⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢠⠀⠻⡆⠀⠁⣆⢄⠀⠀⢀⠎⠘⣧⠞⡔⠊⢯⡛⠉⢿⠀⠀⠀⠀⠀⢸⠀⠸⠸⡅⡿
⠀⠀⢀⡿⠛⠶⢦⣤⣀⣀⣐⡞⠀⢧⣄⡀⠴⠀⠀⡄⠐⠋⠀⠑⢄⠨⠀⠀⣿⢷⣄⡶⣦⢉⡷⣋⢄⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠈⡉⠉⣧⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⢦⣀⡀⣠⠂⢹⠒⢫⢹⡟⠤⠘⣽⠈⡆⠀⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⢀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠉⢠⠃⡇⠈⠀⣰⢆⡈⠉⣸⠺⣤⠀⠀⠀⢸⠀⠀⢰⢸⡇
⠀⠀⢸⡇⠀⠂⠀⠀⢸⡃⠀⡟⠀⠀⠀⠀⠀⠀⠀⡷⠤⠤⠤⠔⠒⠂⠸⡄⠘⠢⠤⣽⣎⠥⢔⡁⢰⡟⠀⠤⠤⢾⠀⠀⢸⢸⡇
⠀⠀⠀⣧⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⣸⠱⠖⠒⠖⡖⢲⡉⣷⠛⠀⠀⠀⠀⢸⡀⠀⢸⢸⠁
⠀⠀⠀⣿⠀⠠⠀⠀⣿⠀⣾⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠈⠂⠠⠚⢁⡤⠁⣿⠀⠀⠀⠀⠀⢸⡇⠀⢸⢸
⠀⠀⢀⡿⠀⠀⡀⢀⡏⣀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⠃⢠⠇⠀⠀⠀⠀⠀⢸⡇⠀⠈⠘
⠀⠀⣸⡇⠀⠘⢧⠞⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢰⠏⢠⡟⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⢠
⠀⢀⡿⠀⠀⠀⣶⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠃⣀⣀⡻⠀⣾⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⢸
⠀⢸⡇⠀⠀⢰⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠉⠉⠉⠉⠉⠉⠁⢰⡇⠀⠙⡆⠀⢁⠀⠈⠉⠉⠉⠉⠉⢹⠀⠀⠀⢸
⢀⣸⡇⠀⢠⠏⢺⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠇⠀⠸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⢸
⠀⠻⣇⠀⠏⠀⠾⣤⣤⡤⠀⠀⠀⠒⠒⠒⠒⠒⠀⠛⠛⠛⠛⠒⠒⠒⠂⢀⠃⠀⠘⢀⣴⣖⠀⠛⠛⠛⠛⠛⠛⠈⠳⡄⢸
⠀⢈⡎⢳⠂⠀⠈⣩⣉⠛⡒⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⢶⠖⠀⠤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⢰⠋⡘⡏⠀⠀⠸⣍⣈⣈⣀⠀⠀⠉⠱⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⣫⡆
⡇⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠃
⣼⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠛⠻⠶⣤⡀
⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⢀⡿⠁⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣹⣷⠄
⠀⠀⠀⢸⡇⠀⠀⢀⣠⡟⠉⠉⠉⠉⠉⢹⡏
⠀⠀⠀⠈⣷⠀⠀⢾⣍⡁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠘⣧⡀⠀⢹⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⠻⢦⣼⡇⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⢰⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⣴⠟⠉⠙⠛⠷⢾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⣶⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣷⣀⢀⣀⣀
⠀⠀⠀⢀⡿⠀⠀⣄⠀⠀⠸⣧⡀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠋⠉⠛⣻⠇
⢠⡴⠶⣼⡇⠀⠀⠹⣆⠀⠀⠘⢷⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⣿
⣿⠀⠀⢸⡇⠀⠀⠀⠙⣧⡀⠀⠀⠙⠳⣦⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠘⡇⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠈⠙⠛⠛⠲⠶⠶⣾⡟⠉⠉⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⣿⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢹⡿⠷⣦⣤⣀⣀⣀⡀⠀⢀⣸⣇⣀⣀⣠⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⣀⣠⣿⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠉⠉⠛⢻⡟⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⢸⡿⠶⠶⠶⠶⠶⠶⠖⠺⠿⠶⠶⠶⠶⠶⠶⠾⢷⣶⠶⠶⢂⡔⠒⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠲⠶⠶⠶⠶⢶⡶⠶⠶⠶⠾
⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠘⡿⠺⠶⠖⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⣶⡀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠷⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⣿⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠘⠷⠶⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠶⢾⡇⠀⠀⠀⠀⢻⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢸⣇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⡄⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠸⣧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⣿⠛⠛⠛⠛⠛⠻⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠓⠀⠀⡾⢷⡀⠀⢸⣷⡄⠀⣾⠀⠚⢻⡟⠛⠀⣿⠛⠛⠀⢾⡟⠛⠂⢰⣷⡄⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⣇⣸⣧⠀⢸⡇⢻⣄⣿⠀⠀⢸⡇⠀⠀⣿⠗⠖⠀⣽⡗⠒⠀⢸⡇⠻⣆⣹⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣷⣶⣶⣰⣟⣉⣉⣻⣂⣸⣇⣀⣹⣿⣀⣀⣸⣇⣀⣀⣿⣶⣶⣀⣹⣷⣶⣄⣸⣇⣀⣙⣿⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠟⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣤⣴⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣿⠙⠛⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⠀⣀⡀⠀⣀⣀⡀⣰⢦⣄⠀⠀⢀⡶⠳⣟⠀⢹⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡴⠋⠀⠀⠀⠉⢿⠉⠙⠟⠉⠈⠙⢧⠀⠈⠳⡴⠋⠀⠀⠙⢦⠸⡄⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠳⢼⡇⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣀⠀⠀⢸⡃⠀⠀⠀⠀⠀⣠⠴⣏⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠳⡄⠀⠀⠈⠁⠀⠀⠀⢃⠀⣴⠛⠛⡖⢾⠁⡀⠈⠉⠀⣶⣅⡉⠲⣄⠀⠀⢾⠁⠀⢻⡀⣓⣶⠟
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⣄⣤⠴⢶⡀⠈⣧⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠸⠀⣧⠰⣿⠉⠉⠉⢹⠀⠀⠀⠈⢯⣻⡆⢸⠀⠀⢸⡄⠀⠈⣫⡽⠒⢦
⠀⠀⠀⣠⣄⠀⢺⡉⠀⠀⠙⣄⠀⠀⠙⠚⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⡅⠈⠛⠋⠀⠀⡸⠉⠀⠀⠀⠀⠀⢻⡇⡎⠀⠀⠈⡇⠀⢸⡉⢑⣶⠏
⢀⡴⠚⠁⠘⢦⡀⢳⡄⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄⣀⣸⠓⠦⠔⠙⢧⣀⣀⣀⡤⢾⣁⣠⣤⠟⠦⠴⠊⢉⠁⣧⣤⠤⢤⡿⠀⠈⠉⠉⠁
⢿⡀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⢀⡼⠉⠉⠀⣰⢦⣤⢤⣄⣀⠀⢀⣀⣤⣀⣀⣀⣠⣤⣤⠶⠻⠶⠶⠖⠚⠋
⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡤⡤⠤⠖⠋⣀⡴⠞⠋⠁⠀⠀⠀⠈⠉⠉⠉⠁⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠈⠳⣄⠀⠀⠀⠀⢞⠙⢦⠀⠀⢀⡨⠇⢠⡶⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⢓⡄⢳
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢑⡄⠓⠚⠁⢀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠁⠀⢸⡄⠀⡠⠤⠤⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠳⣄⣠⠔⢋⣤⠞⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⣧⠊⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠳⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣠⠤⠖⠚⠛⠓⠲⣜⡆⠀⡏⠀⡠⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠲⠤⠚⣉⡉⠉⡉⠩⣀⠀⠄⠀⠀⠠⠤⢄⣀⣀⠜⠛⠙⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⣧⠮⠒⠚⠓⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡐⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⢈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⠀⠀⠀⠀⡠⠞⠁⡎⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⡠⠊⢡⠄⠀⠹⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠀⠠⠏⠀⠀⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠠⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠠⠈⠀⣠⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⣠⠞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠈⠁⠀⠔⠁⠀⢀⡴⠊⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⢰⠃⢠⠃⡀⠀⠁⠀⠀⠀⠀⢀⠎⠀⢀⠀⠀⣀⡀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡼⠊⠀⠀⠀⠀⠀⠀⡠⠃⠀⠠⠃⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢏⠀⠀⠀⠀⢀⠠⠒⢉⠠⠊⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠦⠤⠤⠤⠤⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡠⠀⠀⠠⠐⠁⠀⠀⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⠁⠀⠀⣀⡤⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠴⠊⠉⠙⢢⡀⠀⠀⢠⠖⠈⠑⠢⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠁⠀⠀⠀⠀⠉⠓⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣀⡾⠘⠊⠉⠉⠀⠈⠉⠉⠒⠔⢢⠀⠀⠀⠱
⠀⠀⠀⠀⠀⢠⠎⠀⣀⣀⣠⠖⠉⢷⠒⢒⣐⣐⡚⠳⣄⠀⠀⠀⠀⠀⢀⣘⠔⠁⠀⠀⠀⠀⠀⠀⢀⠔⠢⡀⠈⠃⠰⡀⠀⠘
⠀⠀⠀⠀⢀⠇⠰⡏⠀⠉⡤⠦⣀⣸⠉⠀⠀⠁⠀⠙⢍⢆⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⣀⠈⡒⠠⢄⠱⡀
⠀⠀⠀⠀⢸⠀⠀⠙⢄⠜⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⠆⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⢀⡀⢈⡡⠘⠛⠀⠀⡰⠁⡧⠔
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⠀⠑⠚⠒⢤⣀⠀⣀⠀⠀⠀⣔⠃⠀⠀⠀⠀⢸⠙⠉⠉⠉⠉⠁⠍⠀⠀⢸⠀⠑⠈⠀⠀⡇
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡞⠀⠀⠒⠀⠀⠀⠉⣉⠈⡙⠉⠁⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠂⠀⠀⢸⠀⠀⣀⠀⡰⠁
⠀⠀⠀⢠⠃⠀⠀⠀⠀⢠⠃⠀⠀⠂⠀⠀⠀⠠⠌⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠠⣀⡠⠄⠀⠀⠀⠀⠉⡉⠜⠉
⠀⠀⠘⠣⡄⠀⠀⠀⣴⢧⣀⠀⠀⠐⠢⠠⠀⢀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠌⣀⣀⡀⠀⢤⣔⡊
⠀⠀⠀⠀⠙⠚⢢⡼⠀⢀⠑⠭⠶⣦⣒⣂⡬⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠳⠴⠋⡀⠀⠂
⠀⠀⠀⠀⠀⠀⡘⡄⠀⠘⠀⠀⠀⡁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⡆⠀⠀⠁⠰⢄⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠄⠀⠘⠄⠀⢆⠀⠉⠂⠄⣗⠄⣀⠀⠑⠢⢄⣀⠂⠀⠀⢀⠀⠀⠀⠀⠈⠁⠀⢐⠄⠀⠀⠀⠄⠈⠀⢀⠄
⠀⠀⢨⠀⠀⠀⠈⠁⠀⠑⠢⠤⢀⡀⠉⠙⣢⡰⢄⣠⠏⠑⠦⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⢀⡀⠀⠐⠉
⠀⠀⢸⠀⠀⠀⢰⠤⠀⠀⠀⠀⠀⠈⠑⠲⠋⠁⢠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠲⠤⠄⢀⠠⠆
⠀⠀⠈⡄⠀⠀⢸⠁⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠
⠀⠀⠀⠣⠀⠀⠜⠀⠀⠀⠐⠄⠀⣀⡠⠤⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈⠋⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⢂⠉⠄⠀⠠⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⢀⡀⠁⣀⡠⠭⠒⠂⠁⠀⠀⠀⠀⠀⠑⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⠤⢤⣄
⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠒⠲⠤⣄⡀⠀⠉⠑⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⣊⠏⠁⠀⠙⢄⠉⠉⠑⠒⠤⣀
⠀⠀⠀⠀⣀⡴⠷⢞⡉⠁⠀⠀⠀⠀⠀⠉⠳⢄⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⡴⠁⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⢮⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠸⠀⡜⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢃⡤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⡇⡸⠀⠈⠢⢄⠀⠀⠀⠀⠐⢤⠄⣀⣀⣀⣀⣀⠠⡄⠀⠈⠀⠀⠀⠀⣠⠼⠁⠀⠀⠀⡰⠀⠀⢠⡆⠀⣿⠀⠀⠀⠀⠀⠀⢠⡆⠀⠘⢿
⠀⠀⠀⣇⠃⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠃⠀⠀⠀⠀⠀⠀⠘⡀⢰⠀⠀⢀⠎⠱⣼⠀⠀⡴⠸⠇⣀⣀⢾⠀⠀⣿⡀⠀⠀⠀⣀⣀⡠⠧⢤⠀⢾⣼⠉⠑⢄
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣼⠀⢀⠇⡐⠀⣃⠀⢰⠁⠀⠀⠀⠀⠀⠉⠉⠁⠉⠉⠉⠀⠀⠀⠀⠀⢸⠀⢸⡈⠎⠀⠈⡆
⠀⠀⡎⠉⠀⠀⠀⠀⠿⠂⠀⠀⠀⠀⡀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠉⠑⡌⢰⠇⡜⠉⡶⠁⠀⠀⠀⠾⠆⠀⠀⠀⠀⠀⠀⠀⠺⠇⠀⠀⠀⠈⣀⡏⠈⣾⠀⠀⢸
⠀⠀⠑⠂⢄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠚⠁⠸⠀⠘⡆⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠊⠇⢠⠀⢸
⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⡀⣀⠂⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⡀⠀⠀⠀⡇⠀⠈⠢⣀⠀⠀⠀⠀⠀⠐⠲⠒⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡼⠀⣼⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠑⠦⢄⣀⡀⠀⠀⠀⠀⣀⣀⠠⠔⠊⠁⠀⠀⠀⠀⠠⠤⠀⠀⠤⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠉⠀⠀⠀⠐⠓⠐⠿⠤⠌
⠀⠀⠀⠀⠤⢤⣀⡀⢀⡤⢺⠹⡈⠀⠉⠉⠍⣱⣤⠀⠀⢀⣀⡵⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠉⠉⠉⠉⠓⠦⣄⢀⠀⠀⡀
⠀⠀⠀⠀⡇⠀⠀⠉⠙⠓⠦⢷⣦⡭⠭⣴⣾⡋⠤⠉⠊⠉⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⢨⠉⠛⠒⠤⢮⣀⡀⠀⠀⠀⢀⣠⡛⣨⠤⠖⠚⠉⡁
⠀⠀⠀⠨⠁⡆⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⢰⠌⡇⠀⠀⠀⠀⠀⠀⠐⢢⠀⠀⠀⠀⠀⠉⠛⠶⠾⠛⠉⠁⠀⠀⠀⠀⢠⠳
⠀⠀⠀⠀⠳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠀⠀⠀⠀⠀⠀⠀⠰⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡌
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡐⠈⠂⠀⠀⠀⢠⠈⠡⠀⠀⠀⡀
⠀⠀⠀⠀⢠⠓⠒⠒⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠒⠓⢢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢠⣴⠃⠀⠀⡄⠀⠀⡄⠀⠀⢷⠖⠂⠁
⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠌⠑⢶⠤⣦⡖⠉⢰⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠀⠀⠀⡗⠖⠈⡇⠀⠀⢸⡜
⠀⠀⠀⠀⠀⠙⠢⢄⠀⠀⠀⡅⠀⢸⠉⠣⡜⣄⠀⠂⢀⣀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⡀⡀⠀⣃⣄⣤⡇⠀⠀⠺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠥⠠⠮⠤⠄⠋⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠄⠒⠀⠐⠢⠤⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⣤⣴⣿⣿
⠀⠀⠀⢀⣀⢀⣠⡄⠒⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⠛⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠢⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⠰⠙⠙⠂⠀⣀⣀⠀⣀⡀⠀⢀⣀⣘⣿⡛⠻⡿⢏⣥⣶⣶⣶⣤⡀⠀⢀⣤⣾⡶⢋⣵⣾⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠉⠀⠀⣿⡟⠀⠉⢾⠀⠀⠒⠐⣯⠙⠛⠛⠉⣷⡖⠛⠛⠛⠋⠛⠂⠀⢸⣿⣿⣿⠿⠟⠓⢸⣿⣿⣿⠺⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠩⢤⢶⡆⠿⠇⠀⡌⣹⡇⠀⠀⠀⢹⣧⡐⣃⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠁⠘⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢾⠿⠇⠀⠀⠀⢸⣿⢷⣿⣵⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢠⣄
⠀⠀⠀⠀⣤⣀⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣷⣶⣶⠾⠿⣧
⠀⠀⠀⠀⠘⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢋⣿⡟⣵⠾⢭⢸⡆⠀⠀⠀⢤⡄
⠀⠀⠀⠀⠀⠸⣿⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡸⣿⣧⣙⠓⣂⡿⢿⡇⠀⢠⣿
⠀⠀⠀⠀⠀⠀⡿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡋⠉⠿⠿⠀⠌⠉⠁⠈
⠀⠀⠀⠀⠀⠀⢇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡶
⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡇⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠧⠾⠿⠿⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣿⡿⠃⠀⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⠁⠀⠀⠈⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠤⠤⠖⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠂⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣖⢲⡶⣤⡄⠀⠘
⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⣴⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⢰⡇⠀⢀⡀⣿⣸⣄⠀⡇
⠀⠀⠀⠀⠀⣴⣿⣧⠀⠀⠀⣠⣼⣷⣤⣀⠀⢀⢀⣀⡀⠀⢠⣶⣦⡌⣿⠇⠀⠀⠀⠀⢀⣀⡀⠀⠀⠸⠀⠀⠀⢸⡇⠀⠹⢿⣿⣿⣿⣀⡇
⠀⠀⠀⠀⢰⣿⣿⣿⣇⠀⠀⣿⣿⣿⣿⢿⡄⠀⠀⢸⡇⠀⣀⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⢿⡆⠀⠀⠀⠀⠀⢸⣷⣶⣶⣿⣿⣿⣿⣿⡇⠀⢀
⠀⠀⠀⠀⠀⣿⣿⣿⡷⠀⠀⣿⣿⣿⣿⡌⠃⢀⣄⣈⣁⣾⣿⣿⣿⣿⡏⠀⠀⠀⠀⢀⣾⣿⣶⣇⡀⠀⠀⠀⣠⣬⣇⣹⣿⡟⢿⣿⣿⣿⡷⣤⣴
⠀⠀⠀⠀⠀⢸⡏⣿⣤⠀⠀⠙⣿⣿⣿⠁⠀⠈⢉⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⠂⣿⣿⣿⣿⣿⣿⣷⣶⣶⣷⣶⣾⣿⣶⣶⣾⣿⣥⣽⣶⣾⣿
⠀⠀⠀⠀⠀⣼⣾⣿⣿⠀⠀⠀⢀⡀⣀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⣀⡀⠀⠀⣸⣿⣿⣿⣿⡿⠋⢉⣉⣛⠛⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣸⣿⣿⡿⢧⣤⡤⣾⡇⣿⠀⣠⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡟⣿⣿⣿⣿⡀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⢹⣧
⠀⠀⠀⠀⠀⠛⠿⠿⣷⠶⠤⠤⢼⣇⣿⠤⠤⢤⣿⣿⠿⠟⢻⣿⢿⣿⣿⣿⣷⣤⣬⣧⣿⢛⣛⣻⣇⣀⣛⣻⣿⡇⠀⠀⠀⠀⠀⣿⢀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⢿⣿⣤⣭⠄⠀⣿⡿⠉⣿⣭⣽⣿⣿⣿⠇⠀⠀⠀⠀⠀⢻⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⡿⢸⡏⠀⠀⠀⠀⠀⠀⣿⡇⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⢸⣇⠀⠀⠀⠀⠀⢀⣿⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠘⠉⠁⠀⠀⠀⠰⠿⣿⠀⠀⠘⡟

⠀⠀⠀⠀⠀⣀⣀⡤⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⡖⠭⠒⢀⡍⡁⣈⡠⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠒⠃⠢⠀⠒
⠀⠀⣀⠁⠀⠀⠀⠈⠀⠀⠀⢀⡰⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⡐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠂⠀⠖⠀⠀⠊
⠀⠀⠁⠀⠀⠀⠒⠒⠒⢒⣈⣩⠴⠚⡅⠀⠀⠀⠀⠀⠀⡠⠞⠍⠀⠄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠤⠐⠒⠒⢀⣀⡠⠤⠖
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⡀⠠⠔⢻⠀⠀⠀⠀⠀⠀⣼⠁⠀⠐⠀⢠⠊⠁⠀⠀⠀⣠⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀⠀⠀⢀⣠⠄
⠀⠀⢸⠄⠀⡀⠀⠀⠀⠀⢀⣀⡤⢾⠀⠀⠀⠀⠀⠀⠇⠀⠀⢂⠘⠀⠀⠠⠤⠐⢊⠁⠁⡔⠊⡥⠀⠀⠀⠀⠀⠀⢩⠝⣀⢋⠼⡟⢣⠉⡄
⠀⠀⠘⠀⠄⠀⠈⠉⠉⠭⠀⢀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠔⠁⠀⠞⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⢼⣤⣦⣼⣤⢧⢿⠁
⠀⠀⠀⡇⠀⠀⡀⠄⠐⠀⠀⠙⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⢜⣕⢕⡵⡡⡏
⠀⠀⠀⠘⠀⠀⠀⣀⣈⣀⡠⠴⡺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡯⡪⣾⢮⡇
⠀⠀⠀⠀⠀⠀⠒⠀⠠⠄⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⢠⡆⠀⠀⠀⠀⠀⠀⣀⠄⠘⠀⠀⡚⢀⡀⠀⠃⢦⡐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠞⠁⠀⠀⢀⣠⠀⠀⠀⠀⣀⢄⠈⢱⠚⠇⢸⠁⠞⠂⡜⠓⢀⠀⠀⠀⠀⠀⣀⠄⠊⠀⠀⠐
⠀⠀⡠⠖⠂⠒⠉⠈⠁⣄⣀⣠⠞⠁⠘⠉⠀⠀⠀⠀⠁⠈⠃⣰⠸⠀⠟⠲⠀⣼⠇⠀⠈⠀⠀⠀⠀⡸⠤⠠⠄⠤⢀⡀
⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡆⠀⠂⠅⠧⠼⠿⠏⠀⠀⠃⠀⠀⠀⠀⠀⣺⣁⣊⣐⣂⠐⠋⠻⠐⠠⢀⠤⠀⠈⠆
⡌⠀⢀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⠌⢀⡘⡈⠀⠀⠀⠀⢄⣁⣠⠎
⠇⠀⠀⠀⠀⠀⠀⡀⠀⢈⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢨⠆⠀⢀⡈⠉⠀⠒⠃⠐⠂⠐⠀⠜
⠀⠀⠀⠀⠀⣀⣴⡡⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢀⡀⠀⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠐⠀⠤⠌⠀⠉⢐⡋⣘⠁⢀⠀⢀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⣀
⠀⠀⢀⣠⠖⠊⠉⠄⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⢸⠁⠄⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢧⡀⠀⣀⡀
⠀⣠⠋⠀⡈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⠀⠂⠀⠀⡄⠀⡄⠀⠠⠀⠀⢀⡀⠀⠀⠀⢸⠁⠀⠈⠃⠪⡁⢟⣠⣄
⠀⣇⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⣡⠞⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡒⢠⠉⠂⡠⠀⠀⡄⠀⠘⠀⠀⠀⠀⠈⠓⠀⠤⠘⣤⣗⣲⣄
⠈⢙⡁⠶⠄⠤⣤⠤⠤⡬⢄⠒⠚⠋⠀⡰⠁⠀⠀⠀⠀⠀⠀⠠⡔⠀⠀⡁⠂⡠⠊⠀⢀⡰⠃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⡢⡑
⠀⠈⡀⠀⠉⠀⠐⠈⠀⠐⣁⠤⠚⢀⡁⠴⢃⠀⠀⠀⠀⠀⠀⠀⠀⠄⢈⠕⠊⠀⢀⡴⠋⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞
⠀⠘⠀⠘⠀⠤⠶⠒⠒⠒⠀⠚⠩⠉⠂⢁⠇⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢠⣪⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠂⣒⣨⠔⠃

⠀⠀⠀⠀⠀⠀⠀⠐⢢⠀⠈⠀⠀⠀⠀⠀⠀⠐⠒⠛⠗⠊⠀⠀⠀⡯⠥⠠⠄⠢⠴⠇⠀⢀⠂⠁⠀⠀⠀⠀⠀⠠⡅⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣤⢀
⠀⠀⠀⡌⠀⠀⠀⠐⠁⠀⠀⠀⠀⡀⠀⡤⢀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠀⠁
⠀⠀⠀⠐⠄⠀⠚⠀⠀⠤⠀⠀⡰⠆⡅⠠⠃⢡⠀⢀⡞⠉⠀⢀⠀⠙⠄⠀⢠⡤⠀⠀⠄⠠⠤⡀⠀⠰⠭⠜⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠈
⠈⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠈⠮⠀⢰⡂⠰⢀⡀⠀⠀⠰⠀⠀⠀⠀⠀⢀⡕⠀⠐⠀⠀⠘⠀⠂⡀⠀⠺⠀⠐⠈⠣⠐⠂⠀⠀⠀⠀⠠
⠠⠒⠂⠀⠀⠀⠀⢤⢢⣤⡀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠱⠆⣀⣀⡀⠔⠃⠀⠀⠈⠠⠄⢃⠀⠁⠲⠀⠀⠓⠀⠀⠁⠀⠀⠁⠂⠀⠀⠄⠈⠁⢀⡄⠀⠰
⠀⠊⠁⠀⠀⡪⡿⠑⣯⢹⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃⠠⠀⠰⠀⠀⠀⠀⠘⠒⢂⠉⠀⠀⠠⡄⠀⠀⠀⠀⠠⡮⠁⠀⠐
⠀⠀⠂⠀⠀⠀⠖⠴⠛⠁⠀⡄⠀⠀⠀⠰⡖⠀⣀⠀⠰⣃⠂⠠⠀⠐⠀⠀⠀⢀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠊⠀⠀⣄⡐⠀⣚
⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠉⠀⠀⠀⠀⠄⠁⡤⠁⠠⠀⣤⠇⠀⠤⠀⠀⠀⡔⠁⣀⢄⡀⠀⠄⠀⢠⠀⠴⠘⠀⠀⡄⢀⡁⠦⡀⠀⠈⠚⠁⠐⠊⡀
⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⣀⠠⠒⠀⠀⠠⠠⡀⣠⡀⠀⡠⠀⡄⠀⠀⠀⠐⡅⠌⠃⡐⠁⠠⠆⠀⠀⠀⠀⣀⠀⠀⠆⣠⠄⠖⢛⠀⠀⠀⠀⠀⠀⠁
⠀⣀⠀⠀⠀⠀⠀⠀⠠⠂⢉⠀⠢⠀⠀⠀⠙⢺⢃⢹⡏⡀⠇⡏⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⣻⡯⠎⠃⠀⠩⣄⡠⠌⠚⠀⠀⣀⠀⠀⠀⢀
⠁⠀⠉⠀⠡⠀⠀⢠⠂⠄⠒⠀⠀⡃⠀⠀⠀⠀⠐⠀⠀⠊⠀⠙⠈⠀⠀⢀⣀⣀⡀⠀⠀⠀⣴⡿⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⣰⠨⣜⡿⣃⠀⠁⢇
⠀⠒⠀⠈⠀⠀⢠⠀⢰⡀⣀⣀⣸⠀⠀⠀⠀⠀⠤⠄⠠⡀⠀⠀⠀⢀⠞⠁⢀⠀⠈⠑⠀⠀⠑⠛⠒⠂⠀⠀⠀⠀⠀⢀⡀⠀⠈⣇⢀⡹⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡍⠉⠉⡍⠀⠀⠀⣐⢢⢤⣌⢁⠀⠐⠀⠀⢰⡀⠰⣀⠠⠀⠀⡆⠀⠀⣀⠦⣀⠀⢀⡄⡠⢄⠀⣨⢆⠀⠀⠀⢀⠠⠐⠀⠀⢶
⠀⠀⠑⠂⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠹⢮⣄⣞⣯⢋⡿⠀⠀⠀⠳⠆⣀⣀⢀⠔⠁⠀⠀⠰⠂⠂⠀⠨⣧⣤⣝⣤⢤⡜⠀⠀⠀⠀⡠⠤
⠀⠠⠄⠀⠀⡀⠈⢠⠀⠀⠀⠀⡆⢐⠀⠀⠘⣿⡎⢟⣽⠉⠀⠀⡀⠀⠀⠀⠈⠁⠀⠀⣰⡄⠀⡀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠐⡛⠂
⠀⠀⠉⠀⠀⠄⢀⠀⠀⠀⠀⠠⡁⢈⠀⠀⢰⡞⢞⡽⠀⠀⠰⠀⠀⠀⠒⠀⠀⢰⡚⠓⢀⠤⣀⣹⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⢀⠄⠀⠧⠀⠰⠆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠰⣟⡌⠀⠀⠀⠄⠐⠂⠤⢀⡅⠀⠈⠥⠨⠀⠀⠀⠘⠀⠀⠀⠀⠆⣉⠡⠘⢑⠀⠈⢢⠄⠀⠤⢼⠁
⠀⠀⠀⢀⠄⠀⠀⠀⡾⠀⠀⠠⠤⠤⠄⠀⠈⠊⠀⠀⠀⠒⠂⠐⠐⠀⠂⠘⠀⠀⠀⠀⣀⠀⠀⠀⢠⡀⠀⢰⡆⠀⢀⡈⢹⠀⠀⢸⣒⡚⠛⡟
⠈⠀⠀⠈⠀⣠⠀⠘⠃⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠐⠀⠀⠈⠀⠀⠀⠸⠀⠀⠀⣀⠁⠸⠀⠈⠃⠀⠀⠀⠊⠀⠀⠀⠂⠀⠀⠁
⠀⠸⢄⠀⠀⠉⡏⡉⠉⢹⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠄⣀⢑⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⠂⠑⠀⠀⢠
⡀⠂⡘⡂⠀⠀⢠⠀⢀⡞⠀⠀⢀⠈⠀⠒⣈⠀⠘⠄⠀⠄⠀⠀⠠⠀⠃⠀⠃⠙⠀⠛⠀⠀⢴⣉⡤⣉⠂⡧⠀⠀⠀⠓⠠⠓⡀⠀⠀⠀⠒⠂⠀⠀⣎
⡶⡶⢺⠃⠀⠀⠈⠋⠉⠁⠀⠀⠈⠄⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣿⢏⠝⣬⠀⠀⠀⢘⠀⠀⠁⡁⠀⠀⠀⠀⠘⢀⠀⢙
⠊⢶⡍⠀⠀⢀⣠⠤⢦⡤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⠀⢐⣀⠀⠀⢰⡁⠀⠰⠄⠀⠀⠀⠱⣍⣫⡖⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⢘
⠓⠊⠁⠠⣞⠿⠖⠓⠋⠈⠈⠀⠀⠀⠀⡀⠀⠀⠀⢸⣿⣿⣷⡄⠀⠂⠀⢰⣀⣪⠐⠜⢓⠀⠀⠀⠀⠀⠀⢀⠤⣀⡀⠀⡀⠇⠁⠀⠀⠈⠀⠀⠁
⣀⠀⠀⠀⠈⠃⠠⠄⠀⠀⠀⠀⢀⠰⠁⠐⠄⠀⣰⣿⣿⣿⣿⡿⠃⠀⠀⠈⠡⣀⡤⠐⠋⠠⠄⠀⠀⢐⣦⠃⢶⢱⡥⠀⠀⠀⠀⢪⠰⡄⠀⠀⠀⡤
⠛⠂⠀⠀⣀⠀⠀⠀⣠⠀⠀⢰⠢⣔⠈⣉⡃⠀⠙⠿⠟⠉⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠪⠤⠴⠚⠁⠃⠀⠀⠀⢨⡀⠼⡀⠀⠀⠃⢀
⠉⠀⠀⠀⠙⠁⠀⠤⠀⠀⠀⠸⠤⣄⠈⣸⠃⠀⣀⠠⢀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠈⠃⠀⡃⠈⢤⠀⠈⠠⠀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠈⡀⠀⠑⠒
⠀⠒⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈⠁⠀⠠⡅⠁⠈⠷⠀⠀⠀⠀⠀⠰⠗⠂⣀⡀⠀⠀⡉⢄⠰⠖⠂⠀⠀⠀⣰⠶⠂⠑⠀⠀⠑⠒⠀⠁
⠠⠀⠄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⢄⡀⠛⠛⠛⠓⠃⠀⠀⠃⠑⠒⠒⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡇⠸⠤⣧⠀⠀⡆⠘⢵⠶⠟⠸⠿⠆⠗⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⠟⢠⠀⡇⢀⠄⣤⡀⣰⢠⡄⡀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠈⠀⠃⠃⠇⠸⠣⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠤⠾⠁
⠀⠀⠀⠀⢠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⣀
⠀⠔⢂⠶⠘⣆⣁⡀⠔⠒⡄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠴⠶⠖⣒⡒⠒⠤⡀⠀⠤⠒⠀⠀⠀⠀⠀⣠⠖⢻⡀
⡏⠙⠳⣄⣀⣭⠤⣽⣄⣀⡼⡄⠀⠂⠀⠀⠀⠀⣯⠀⡀⠀⠀⠀⠀⠀⠈⢶⠼⡄⢌⢑⣦⠄⠀⠠⠟⣩⠾⠛⠁
⠇⠀⠀⠀⠀⠀⠀⠀⡜⢻⠾⠃⠀⠀⠀⢀⠀⢀⣧⣠⣀⣀⣌⣠⣄⣦⣤⣴⣾⠇⠀⠄⡉⣁⠴⠚⠉
⢰⢰⠀⠀⠀⠀⠀⠈⠀⢀⡇⠀⠀⠄⠀⠜⠀⢧⡈⢙⣛⣚⣛⡓⠋⠍⢝⡪⠟⠀⠀⠀⠀⠠⠤⠤⠤⠀⠀⠀⠀⠀⠋⠢⡀
⠘⠤⢤⣤⣤⣠⣦⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠒⠚⠃⠉⠁⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⢠⠆⠀⠀⠁⠰⡐⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠯⠄⠉⠙⠲⣄⠀⠀⠀⢹⡀⠀⠀⠀⢠⠏⠀⠀⠁⢀⣤⣀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢄⣀⡒⠁⠀⠂⠀⠀⠀⠀⡠⠁⠂⠀⠀⠀⠢⡀⠑⢄⠀⠀⠃⡀⠀⠀⠘⠀⠀⠀⠈⠍⠉⡉⣝
⠀⠀⠀⠀⠲⠅⠀⠀⠀⠀⠀⠉⠁⠀⠀⢄⠀⠀⠀⣰⠁⠀⠀⠆⠀⠀⠀⣈⣦⠄⠆⠀⠀⠁⠀⠆⠀⠀⠀⠀⠀⠀⠐⠄⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠡⠀⠀⢀⢀⣤⣶⠟⠋⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⠃⠀⣠⡴⠚⠉⠀⠘⠀⠀⠀⠀⠀⣸⠀⠀⠀⠠⠀⡆⠀⠀⠀⠀⢰⣼
⠀⠀⠀⠀⠀⠠⠠⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠺⠶⠛⠁⢀⣠⣤⡦⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠁
⠀⠔⢉⠉⢳⠀⠀⠀⠀⠀⢀⡀⢀⡀⡀⠀⠀⠀⠀⣠⠞⠙⠠⠁⠉⠉⠀⠉⢱⡀⠀⣀⣠⣁⣉⣡⡀⡀⠀⠀⠸⣽⣋⣦⣄
⠀⠀⢆⠀⢸⡄⠀⠀⣴⠟⢪⠟⠈⠐⠒⠣⣄⠀⢠⠏⠀⠀⠀⠀⠀⣀⣥⣾⣽⠀⠀⢉⣏⠉⠹⡟⡇⣧⠄⠀⠀⠈⠉⠀⠈
⠀⢰⡃⠀⢸⠃⠀⣺⣛⠔⠁⠀⠀⠀⠀⠀⠈⡀⠸⣄⣀⣤⠔⢲⠟⣿⣥⡴⣚⣡⡆⠀⡇⠢⢴⡽⠿⣏⠀⠀⠀⠀⠀⠀⢀
⠀⡇⡅⠀⢸⠀⠀⠈⠓⠢⠤⠤⠀⣀⠀⠀⠤⠁⠀⠈⣭⣄⣀⣋⣭⣾⡟⠋⢱⡜⠀⠀⢇⠀⠀⠀⢰⠃⠀⠀⠀⠀⠈⠁⡠
⠀⢹⠁⠀⡞⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢅⣀⣉⣉⡠⠴⠒⠉⠀⠀⠀⠘⠀⠀⠀⡟⠀⠀⠀⠀⠀⠉⠉
⠀⠈⠣⡜⠁⠀⠀⠀⠀⠀⠁⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠂⡀
⠀⠀⠀⡇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⡀⣀⣤⣴⠟⠛⠉⣻⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⢶⠈⠢⡀
⠀⠀⠀⡁⠀⠀⠈⠁⠀⠀⡙⠉⠉⠛⠉⠑⡉⠀⠠⠐⠗⠛⠉⠉⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⡿⣼⢄⣶⠟
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠁⠀⠀⠀⠆⣀⣀⡀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠒⣢⡶⠒⠀⠐⠂⠠⠄⢠⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠁⠋⠈⠁⠀⠀⠀⠀⠐⡌⢷⣄⣥⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠱⠄⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⣠⠆⣰⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠠⠀⣦⡄⠀⠀⠀⠀⠀⠀⣀⠤⢚⣡⠞⢸⠳⣼⣄⢄⢀⠀⡀⠀⠀⠀⠀⠀⠄⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡼⣛⣛⡴⠋⠀⠀⠀⠀⠀⠀⠠⠔⠒⣋⣀⠀⠀⠀⠀⠉⠓⠓⠦⢜⢦⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢀⠄⣠⣇⡀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠂⠀⢁⢀⢀⠀⣧⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠞⡡⢊⠏⢸⡇⠋⢾⣿⡆⠀⠀⠀⠀⢰⣶⣶⠦⠀⢈⢰⡾⡄⡏⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠆⡼⡱⠁⠘⢸⠀⠀⠈⠉⠀⠀⠀⠀⠀⠐⠛⠛⠀⠀⠘⢸⠁⢵⠁⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⡄⠇⠄⢂⠀⠀⠀⠀⣄⣀⣀⠀⠀⠀⠀⠀⢀⢆⠟⠊⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢧⠸⣷⢈⢾⡀⢣⣄⠀⠀⠈⠀⠁⠀⠀⠀⢀⠤⢃⠼⡄⢰⡸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⡳⣌⡓⠅⡙⢄⡨⡕⠲⠄⠤⠤⠤⠒⠮⢤⡔⠁⡄⣿⢤⡙⠮⡭⢥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠟⠈⢙⠷⠊⢨⠁⢂⠀⡁⣤⡄⡁⠀⠀⢀⡆⠉⠲⣟⡀⠈⠩⠌⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡁⠀⡀⡘⠀⠀⠁⠀⠨⣅⣠⠵⣚⡍⠵⠒⢒⡠⠳⣾⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⠚⠒⠤⢽⣷⣦⡀⠀⠀⠐⣫⠖⠋⠀⠀⠀⠀⢨⣀⠂⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢈⢶⡀⢄⠀⠈⠙⠻⠖⠛⠊⠀⠀⠀⠀⠀⠀⣔⠘⠏⠢⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠀⠀⡸⠀⠀⠀⠀⢤⣤⣶⠀⠀⠀⠀⠀⠀⠘⣤⢀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡝⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢲⠔⠒⠢⡀⠀⢀⣀⣀⡄⠀⢀⠴⠎⠀⢤⡀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⣠⠃⠀⠀⠀⠘⠆⢌⡛⠉⢡⠊⡰⠀⠀⠀⠀⢹⣄⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⣴⠧⠤⠤⠶⢿⠀⠀⠀⠀⣰⠤⠦⠤⠤⠤⠾⠤⡆⠀⠀⠀⢸⠿⢿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⡄⡀⠀⠈⠄⢸⠆⠐⠀⠀⠸⡀⠀⠀⢀⡓⠐⠒⠒⠒⠀⠀⠂⣧⠀⠀⢀⡸⠤⠤⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠔⠀⣰⣁⣹⣉⣈⣈⢀⢈⡀⠀⠀⠀⣏⠈⠉⢉⣉⣉⣉⣉⡇⠀⠀⠀⣹⣠⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣯⡦⠼⡀⠘⡄⠀⠀⠀⣜⠭⠍⢢⡀⢁⠀⠀⠀⠀⠀⠀⡀⢁⡔⠈⠉⣢⠀⠀⠀⠴⠶⠅
⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠉⠀⣠⣋⣨⡇⣂⡂⠀⠳⡀⠀⠈⣷⠈⠒⠂⢰⢲⣒⡖⢁⡾⠁⠀⢀⣼⢀⢀⡠⠀⠀⢢⣢⠄⠀⢠⣭⡐⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠤⠀⣁⡓⠜⠂⠀⠀⠀⠀⠉⠉⠀⠀⠈⠢⡀⡸⠁⠀⠀⠈⠙⠙⠀⠈⢇⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠬⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠙⠶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠤⢀⢄⢴⣦⣄⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⡤⣧⡦⣬⠶⠒⠻⡛⣿⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠆⠰⣶⡷⡀⠿⠃⢻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⢠⠀⠙⠀⣠⡦⡖⠚⠉⠉⠛⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡞⠛⠁⣜⣶⠲⠒⠚⠋⠈⣇⠇⠀⠀⣠⡖⠛⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⣿⠀⠀⠘⠝⠄⠴⢶⣒⣯⣭⣤⢤⣴⢯⣼⢧⠞
⠀⠀⠀⠀⠀⠀⠀⢰⡷⢫⡗⢻⣖⡄⠀⢀⡤⠞⠋⠉⠁⠀⠀⣠⡿⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⠲⠖⢆
⠀⠀⠀⠀⠀⠀⠀⠸⣅⣸⠉⢻⡏⠀⠀⣾⢥⡒⣆⣠⣭⣴⠾⠋⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢠⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣇⠀⠀⣹⣟⡽⠉⠉⠁⠀⠀⠀⠀⠀⣸⣀⣀⣀⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⡠⠘⠊⢣⣤⡄⡄⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣽⣀⠀⠤⢬⡀⠀⠠⠤⠐⠒⠶⣚⠫⠭⠉⠙⠛⠛⠫⣲⠈⠙⠶⣄⡀⠀⠀⠀⠘⢧⣤⠄⠀⠀⠟⠁
⠀⠀⠐⠒⠚⠛⠉⠉⠉⠉⢀⣠⣼⣷⣄⣀⡤⣿⡀⡀⠀⠀⢦⣘⣣⣤⣤⣤⣤⠤⣶⠞⠁⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠉⡹⠲⠶⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣄⣾⣉⠀⠀⣈⣓⣲⣬⣤⠞⠀⠀⠀⠀⠀⠀⠉⠁⠂⠒⠈⠁⣠⣴⣦⣤⣤⣤⣤⠤⠴⢿⠄⠀⡈⠀⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠛⠋⣀⣀⣀⣀⣤⣤⡤⠴⠶⠶⠒⠛⣻⣿⡁⣫⣟⣙⢋⢻⡷⠒⠛⠀⠀⡏⠙⡇⠀⠘⠇
⠀⠀⢀⣀⣀⣠⣤⣤⣤⠤⠶⠶⠶⢖⣛⣛⣛⣩⡭⠭⢭⠴⠶⠶⠒⠒⠛⠛⣻⣿⣿⣿⣿⣿⣿⣿⢾⣿⣿⣶⣦⠀⠀⠇⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠤⠤⠶⠒⠒⠒⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠿⠋⣿⠙⠤⠾⠀⠶⠆⠛⡧⠙⢿⠀⡴⠀⠈⠐⠚⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣇⣀⠠⠤⡼⠤⠐⠒⠒⠛⠉⠉⠀⠀⠀⢢⡠⠬⠤⠔⣉⠞⠀⠀⠈⠘⣛⣦⡆⠠⢴⠾⠃
⢠⣤⡤⠶⠶⠶⠖⠒⠚⠛⠛⠉⠉⠉⠁⠈⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣴⡖⠛⠛⠁⠈⡛⠛⠛⠁⠀⢈⠙⣲⣤⡈⢀⠀⠀⠈⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢀⡴⣚⡩⡞⠋⠉⢹⡄⠘⢦⣀⠀⠀⡰⠋⠉⠻⣀⢱⡎⠀⢠⡄⡇
⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠐⠒⠒⠒⠒⠒⠒⠂⠀⢸⠃⠀⠀⠻⠤⣭⣳⠼⠓⠋⠉⡙⢿⡞⢛⣶⡤⠀⠐⠂⠉⣁⠞⡇⠀⠙⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠲⢤⣶⣤⣤⣥⣙⣓⣠⣯⣝⡡⠒⠋⠉⣀⣀⡃⠀⠀⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠒⠶⠶⠶⠶⢴⡇⠀⠀⠙⠬⣭⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣀⣀⡀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⢿⡟⠙⢻⢷⣦⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣸⠀⢠⡏⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠄⡾⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣼⠁⣸⠃⢳⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢣⠇⢠⠏⠀⣸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⢀⡟⢠⣚⣡⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⡼⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣆⡉⣿⣷⡀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠵⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣯⣿⢿⣿⣿⣿⡟⠻⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣦⡹⣿⣿⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣦
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢻⣿⣿⡇⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣄⣀⣠⣶⢿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⢠⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⣿⣿⣿⠀⠀⢀⣴⣾⣿⣿⣿⣿⢿⣿⡛⣿⣿⣾⣿⣿⣿⡿⡿⠑⠁
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣇⢹⣿⣿⠀⣴⣿⣿⣿⣿⣩⣍⠷⠶⣋⣴⣿⡿⢛⣽⡟⣻⣭⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣞⣿⣿⣿⣿⣿⣿⡄⢛⣫⣾⣿⣿⣿⣿⠑⣻⣽⣾⣿⣿⢟⣩⢾⣯⣶⣿⣿⣿⣷⡀
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣞⢿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⢣⣿⣿⣿⢟⣯⠾⣫⣵⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣿⢟⣫⠾⣋⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⣿⣿⣿⣿⣿⡿⢋⣴⣯⠶⢋⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠉⣿⣿⣿⡶⠿⢫⣵⣾⣦⢻⣿⣿⣿⣿⣿⣿⡿⠉⣿⣿⣿⣿⣿⣿⣿⣿⢿⣄
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡟⠸⣿⢟⣩⣴⡾⢻⣿⣿⣿⠈⣿⣿⣿⣿⡿⠋⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⣫⣶⣶⣿⣦⢇⣾⣿⣟⣴⣿⣿⣿⣿⡇⢿⣿⣿⠟⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣷⣻⣿⣿⣷⣤⣀⡀
⠀⠀⠀⠀⢻⣿⣿⢿⣿⣭⣽⣾⣿⣿⠟⢿⠋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡍⠙⠛⠛⠛⠛
⠀⠀⠀⠀⣀⣽⣿⣾⣿⣿⣿⣿⠿⠿⣇⣻⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⡇⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣄
⠀⠀⢰⡾⣿⣿⣿⣿⣿⣿⣿⡵⣿⣷⣿⣿⣿⡿⢋⣵⣾⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣆
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣟⣋⣼⣿⣿⣿⣿⡏⣰⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠻⠿⣯⣽⣿⣾⣿⣿⣿⣿⡿⢛⣵⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⡿⢋⣄⢿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠛⠉⠰⠾⠿⠏⠻⠟⠛⠻⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⡟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢍⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⠒⠋⠀⡇⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢻⢀⡔⠋⠉⠀⠀⢉⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠘⡆⠀⢸⡏⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⣀⣠⠤⠶⠶⠦⠤⠶⠒⠒⠦⠤⠤⠤⣤⣀⡀⢀⡤⠖⠋⠉⠀⠀⠀⠉⠳⣿⠀⢸⠀⡠⠊⣀⡞⠁
⠀⠀⠀⠀⠀⢠⠖⠛⠉⠛⠊⣡⠤⠤⠠⠄⠒⠦⠄⠒⠢⠄⠐⠒⠒⠢⣤⡤⠒⠁⠀⠹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⣸⠞⠋⠉⠉⠉⠉⠑⠢⣄
⠀⠀⠀⠀⢀⡞⠀⢰⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⢀⠽⣆⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⡼⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠂⠀⠀⣠⠞⠁⠰⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⢰⠇⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⠀⠀⣠⠚⢁⡠⠄⣀⡇⢀⡇
⠀⠀⠀⡞⠀⡇⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠊⠀⠀⠀⠀⠁⠀⣠⠔⠊⠀⢀⠌⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣸⠁⢠⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⡠⠊⠁⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⢠⡏⠀⡃⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⠀⠔⠀⢀⡠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⡾⠀⢰⠁⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⢀⡀⠤⠚⠁⠀⡠⠊⠀⠀⢀⡴⠋⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⢸⠃⢀⠜⠀⢀⠔⠁⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⡎⠀⠀⣠⠞⠁⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⡿⠀⢺⡠⠊⠁⠀⠀⠀⠀⠀⢀⠎⠀⠀⡄⠒⠐⠒⠔⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠋
⣇⡰⠋⠀⠀⠀⠀⠀⠀⣀⠴⠋⡠⠀⠚⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠘⢦⡀⠀⠀⠀⡠⠔⠊⢁⠔⠉⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠒⠒⠋⠉⠁
⠀⠀⠙⠦⡤⠊⢀⠠⠊⠁⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠘⢾⡁⠀⠀⢀⡤⠖⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡘⣀⠁⠐⠲⠦⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡖⠒⠀⠈⠁⠙⠛⠿⠿⠿⠯⠝⠉⠉⠁⠀⣤⣶
⠀⠀⠀⠰⣄⠀⠀⠂⠀⠀⠀⠀⠀⡄⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠖⠒⠒⠂⣀⠠⠤⠄⠀⢤⠤⢤⡤⠴⠶⢛⡛
⠀⠀⠀⠀⠻⡄⠀⠀⠀⢸⡀⠀⠀⠣⠀⢠⠀⠀⣦⡎⠀⠀⣺⠀⠀⠀⠀⠀⠀⠀⡀⠁
⡀⢀⠀⠀⠀⠂⠀⢠⠀⠀⠁⠀⠀⠀⡄⠀⡄⠀⠈⢠⠄⠀⠋⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠌⡀⠀⠀⠀⠀⡀⠼⣇⠀⠀⡇⠀⠰⠃⡄⠃⠀⠰⣧⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈
⠠⠟⠀⠀⢀⣀⣠⣀⣉⣀⣀⣇⠀⠀⠀⠁⠈⢀⠠⠄⠤⠤⠤⠔⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣑
⢸⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠂⠀⢀⠀⠀⠒⠂⠀⠀⠀⠒⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿
⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⡀⠎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡆
⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⢀⢹⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⡀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠋⠉⠉⠉⢉⠉⠛⠙⣻⠦⠐⡒⠀⠁⠀⠀⠀⢀⠀⠀⠐⣻
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠐⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⠀⠀⡀⠀⠀⠀⣀⡄⢀⠖⢀⡀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠈⣾
⠀⠀⢔⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⣄⠀⡀⠀⢀⣀⣠⣀⠀⠀⢀⠀⠄⢀⡈⡉⠀⠀⠐⠂⠂⠀⣀⡀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⢸⠚⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⠀⠀⠀⠖⠙⠆⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠠⠤⠶⠦⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠛⠳⡶⠲⠶⠓⠔⠂⢀⠀⠀⠖⠒⠂⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠐⠂⠉⠀⠀⠀⠀⠀⡐⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⢀⡀⠀⠀⠀⢀⣠⣤⣶⡒⠒⠛⠉⠭⢍⡀⠐⠂⠂⠄⣀⠀⠀⠀⣀⠤⠄⠀⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⢁⡖⠊⠁⠀⠀⠀⠀⠈⠁⠒⠤⡀⠀⠉⠒⠠⣀⠀⠉⠒⠍⡀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⡠⠄⢒⡤⠊⢁⢻⡑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⠀⠀⠉⠢⢄⡀⠈⠑⠶⡐⠠⡀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠎⠀⠠⠊⠀⠀⢸⠀⠻⣄⠀⠈⠁⠒⠄⣀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠈⠑⠠⢀⡈⠑⢌⡆⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸⠀⡰⠁⠀⠀⢀⠆⠀⠀⢨⠷⡦⣀⠀⠀⠀⠈⠑⠂⠤⣀⡀⠀⠀⠀⠀⠈⠁⠒⠀⢤⣤⠼⠛⡃⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡰⠁⠀⠀⡠⠃⠀⢀⡴⠁⠀⠈⠐⠬⣑⠂⠤⠀⠀⣀⣀⡀⢈⣉⣁⡀⠀⠶⠾⠋⠁⣠⠔⢊⡹⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠀⢀⡠⠊⠀⣀⠔⠊⠀⣀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠄⠤⠄⠀⢀⣀⠀⠀⢤⡄⠀⠒⠋⠁⠈⠓⢳⠄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡶⠋⡴⠂⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠒⠆⠉⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢘⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢠⠃⠀⠀⡸⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠤⢌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⡄⠀⠀⠰
⠀⠀⠀⠀⢀⠂⠀⠀⠰⠁⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠐⡀⠀⠀⢣
⡀⠠⠤⠒⠁⠀⢀⣞⣷⣶⣤⣤⣵⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣼⣤⡤⠴⣶⣟⣆⠀⠀⠑⠤⡀
⠣⠄⡀⡀⠀⠤⡏⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠻⠿⠷⠶⠶⣦⣤⡬⠉⠡⣤⣴⣶⠶⠶⠿⠿⠟⠛⠛⠛⠛⠋⠉⠉⠉⠉⠀⠀⠈⠦⢀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠉⠁⢀⡇
⠀⠀⠀⠀⠀⢀⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣄
⠀⠀⠀⠠⠊⠀⠛⠈⣀⣜⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢸⣄⡉⠘⠂⠉⢢
⠀⠀⠀⢂⠀⠀⠀⠀⠁⣠⡤⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⣄⡀⠁⠀⠀⠀⢠
⠀⠀⠀⠈⢆⠀⠀⠀⠀⠠⠴⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⠤⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠁⠲⠒⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⢲⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⠤⠤⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠂⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠒⠀⠤⠤⠀⡀⢀⡏⠀⠀⠀⠈⣇⠀⡀⠠⠤⠄⠐⠒⠂⠈⠉

⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⢄⠀⠠⢤⡈⣷⡀⠀⣀
⠀⠀⡠⠊⠁⠀⠀⠀⠀⠈⠑⢄⠀⠀⢀⢀⣴⡞⢹⣧⣿⣱⡡⣥⢾⡼⡆⠀⠀⠀⠀⠀⣠⠀⢤⢀⣆
⠀⢰⠁⠀⠀⠠⠀⠀⠄⠀⠀⠈⡆⠀⠈⠢⣇⣧⡞⣼⡟⣸⣸⡕⡯⢁⡯⠃⠀⠀⢀⠈⠀⢱⣶⣿⣷⣀
⠀⢳⣂⡂⠀⠀⠀⠀⠀⣀⣀⣈⢗⡄⠀⠀⢹⠙⠃⠷⠴⠿⠧⠶⠜⢊⠁⠀⢠⠔⠉⠁⠈⢁⣉⣉⣀⠀⠩⠶⢤
⠀⣶⣦⣽⣭⣭⣿⣹⣷⡤⣼⢵⡞⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠖⠢⠖⣄⠖⣤⠒⠤⠲⡔⠶⣾
⠀⢫⣽⡯⠹⢖⣛⣻⣷⣾⣿⢿⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⡻⠶⠶⠶⠶⠖⣖⣒⣒⣲⣟⡇
⠀⠀⠑⠂⠀⠀⠀⠀⠐⠒⠚⠁⠀⠀⠀⠀⠀⠀⢆⣀⣀⣀⣀⣀⡜⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠠⠤⢄⣀⢀⣀⠤⠔⠒⠉⠒⢄
⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⡠⠤⡴⠁⠘⠉⠀⠀⠔⠠⠕⠀⠐⠁⢁⡈⠀⡼⢒⡶
⠀⠀⣀⣔⣊⣁⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠬⠵⠦⠀⠀⠑⢢⣘⣌⣀⣉⣀⢀⣀⣐⣒⣒⣂⣈⣡⣴⡊⣤⠋
⠀⠀⢒⣒⣶⣶⣶⣶⣶⣀⣈⣏⣉⣻⡿⠯⣿⠏⡹⡄⠀⠀⠓⣖⠪⡭⢿⡶⢲⣲⣒⣶⣿⣿⣿⡿⢔⣵⡿⠂
⠀⠀⠁⠐⠚⠒⠚⠒⠒⠂⠀⠀⠉⠀⠈⠈⠁⠀⠉⠁⠀⠀⠀⠈⠁⠒⠀⠉⠉⠓⠂⠐⠒⠓⠒⠉⠉
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⣀⠔⣹⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡔⢣⠃⠀⠀⠀⠀⠀⢰⣍⡠⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡃⢸
⠀⢸⠁⠘⠎⣓⡦⠀⠀⠀⠀⢠⠒⢶⡀⠲⢥⠇⠀⠀⣠⠤⠀⠐⢆⡹⠲⠶⢦⣀⠀⢀⢔⢒⣄⢠⠚⡟⣿⠛⠛
⡀⠮⡝⣉⣉⣩⠖⠳⡀⢠⠋⡀⢁⠉⣈⢢⠜⡄⠀⠀⢿⡓⠒⠒⠒⠒⠒⢒⣚⠏⠀⠰⠧⠄⢉⠎⣣⠿⠉⠘⠂
⠑⠶⠤⣤⢤⠾⣶⡚⠀⠘⣍⠙⠈⢉⡠⢯⠴⢸⠀⠀⠈⣦⣒⣙⣭⣋⡑⠠⡼⠀⠀⢀⠀⣰⣯⣾⡋⣦
⠀⠘⣮⡱⢪⠼⣱⠃⠀⠀⠡⢍⡉⠉⠉⣀⠔⠁⠀⠀⠀⢰⠄⣰⢀⢀⠀⢰⠃⠀⠀⡗⠀⠨⠝⠉⠀⣺⡗⠉⡄
⠀⠀⠘⡾⠿⠾⡾⠀⠀⠀⠀⠀⢀⢙⣉⣀⠀⠀⠀⠀⠀⠀⠄⢽⠈⠤⠀⡏⠀⠀⠀⠹⡉⠛⠛⠓⢈⡽⠧⠕⠁
⠀⠀⠀⠐⡄⢠⠃⠀⠀⠀⠀⢈⡀⠀⠀⢀⡹⠀⠀⠀⠀⠀⢠⣞⡘⢚⣾⠃⠀⠀⢠⡖⠚⠒⠂⠒⠋⠙⢱⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠈⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⡛⢄⠀⠀⠀⢀⡀⠆⢈⡀⡀⠀⣄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠨⠜⠓
⠀⠀⠀⣠⡯⠠⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⡴⠠⡞⠁⠀⠀⠘⠢⢐⠉⢉⡀⣐⠒⠴⠆⣐
⠂⠀⠀⠸⣄⡀⡴⠀⠀⢀⡄⠀⠀⠀⠀⡐⣙⠀⠀⠀⣐⣉⠉⠀⢠⠀⠘⢤⠀⠀⠀⠀⠀⠀⠉⠐⠀⢈⡄⠒⠀⠀⡀⣀
⠐⡄⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠀⠀⠀⠀⡈⡇⡀⠔⠁⠀⢦⡉⠁⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠊⠒⠀⠂
⠀⠈⠀⠠⣤⣤⣴⠴⠋⠁⠀⠀⠀⠀⠀⠀⠶⣉⡄⠐⠤⢂⢡⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢂⠤⠒⠁
⠀⠀⠀⠀⢠⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄⢠⡀⢀⣤⣄⢠⣤⡄⣠⣤⠀⣠⣤⡀⣠⣤⡀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⣿⡇⠘⣧⡉⠀⣿⠀⣿⣀⠀⡏⢸⡇⣿⢸⡇⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⠉⢰⡷⣿⢠⡌⣷⠀⣿⠀⣿⠁⠀⣯⢸⡇⣿⢸⡇⣿⢸
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠐⠴⡀⠅⠠⠄⠀⠀⠀⢂⠀⢀⡤⠁⠀⠀⠀⠀⠀⠈⠀⠈⠁⠈⠀⠉⠉⠀⠉⠀⠉⠀⠀⠉⠉⠀⠈⠉⠀⠉⠉
⠀⠀⣀⢲⡆⠀⠀⠀⠀⠈⠉⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐
⠀⠔⠭⠭⣷
⠀⠉⠒⠛⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡮⠄⢀⣀⠴⠚⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⡀⣀⡈⡔⡆⠚⠰⡤⢤⠀⡀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⢤⡉⠉⠉⠁⢠⠠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢃⠈⠀⠀⢀⠀⠄⡈⢿⠀⠀⠀⠀⠀⠙⠛⠛⠛⠓⠾⠿⢃⣔⡉⠸⠤⡡⢤⡀
⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠀⠂⢀⠀⠘⢀⠄⢚⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⣹⣼⣿⣿⡿⠋
⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⠑⠁⠀⠀⠀⠀⠀⠀⠀⠠⣀⠴⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠉
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⢠⡀⠀⠀⠡⡀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠈⠁⠐⠀⠀⠴⠀⠀⠀⠀⠀⡀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈
⠀⠀⢠⢖⠢⡔⠀⠀⠠⠄⢒⣤⣀⡐⣪⡰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠦⣀⢐⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆
⠀⠀⠸⣄⠈⠛⠓⠀⠈⠲⠄⠀⣀⠄⠚⢀⡆⠀⠀⠀⠀⠀⠀⠀⡀⢉⣸⠘⢢⡌⡐⠺⢏⠀⢀⠀⠈⡦⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⠔
⠀⠀⠘⢤⢀⡀⠀⠀⠔⠀⠀⠀⠀⠀⠠⠠⠄⠀⠀⠀⠀⡀⠼⠄⣀⡀⠀⠈⠁⢉⡈⠓⡢⠊⡡⢶⡄⢂⠶⠀⠀⠀⠉⢄⡀⢀⠀⠄⠊⠁
⠀⠀⠀⠀⠀⠐⠬⣑⠂⠒⠐⢃⡁⠶⠊⠀⠀⠀⠀⠀⠀⠓⠠⠤⠶⢆⣀⠈⠉⢠⠜⠉⠀⠣⠀⠀⠁⠆⠀⣲⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⠶⠤⠤⠤⡆⢠⣀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⣀⣀⣀⣀⣀⣠⡄⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠉⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢀⠀⠀⠀⠀⣀⣸⠀⠀⡀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⣸⠀⢉⠉⠉⠙⡋⠉⠉⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠆⢀⠠⠀⢠⠃⠀⠀⠀⣀⢰⡇⠀⠲⠌
⠀⠀⠀⠀⡀⠈⠀⡇⠀⠀⠃⠐⠀⡀⢀⠀⠀⢀⡡⠷⠓⠚⠉⠉⠉⠉
⠀⠀⠀⡞⢲⡺⣄⢡⠀⢀⠆⢠⡇⠁⠈⣥⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⡀
⠀⠀⠀⣇⢸⡍⠳⣆⣰⠆⣠⠋⣰⡠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢫⡢
⠀⠀⠀⢻⠘⡇⠀⠈⠉⠛⠛⠛⠉⠀⣾⣦⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⡹⢿⣿⣿⣷⣾⣶⣤⣀⣳⣄⣦⣄⣠⡴⡊⢁⣀⣤⣖⣁⣤⠾⠁
⠀⠀⠀⠈⣧⣿⠀⠀⠀⠀⠀⠀⠀⢀⡞⢃⠠⠉⡛⡛⠛⠿⣿⠿⢿⡿⢿⣿⣿⡿⠿⢿⣿⠛⠙⠻⡤
⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⡈⢀⣌⣉⡒⢒⢫⣀⡴⠶⡾⠵⠶⢥⠤⠤⠤⢤⡤⣮⣭⣵⡿⠃
⠀⠀⠀⠀⠀⢹⣇⠀⢀⣀⣀⣀⣀⣈⣍⠙⠿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⢈⠀⢀⣴⣿⣿⡟⢁⠇
⠀⠀⠀⠀⠀⠀⠙⠋⠉⠉⠉⠉⠉⠉⠉⠓⢤⣀⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠋⠉⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠒⠦⠤⠤⠀⣀⣀⡤⠴⠒⠋⠁

⣿⡿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢂⠀⠔⠒⠢⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠂⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢈⠠⠀⠀⠀⠀⠀⠄⡁⠈⠄⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣱⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠚⢄⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠈⠈⠑⠠⢀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣰⡖⣒⣢⡄⠀⢠⣜⣓⢶⡄⢰⠀⠀⠀⡀⠀⢀⣠⣄⠀⠀⠀⣀⣦⣄⠤⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠳⣾⠏⣭⣽⣗⠤⢺⣯⣭⠱⢕⡾⢄⠀⢀⢀⡼⠙⢒⣻⡧⣄⡠⣖⣟⠻⡴⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢁⡎⠹⠑⠦⠿⠿⠃⠀⠈⠿⢏⠴⠋⡇⢀⠒⠁⢀⡟⢄⠻⡿⡗⠀⠈⣿⣿⣁⡜⣧⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠌⠈⢘⠀⠀⠀⠀⠰⠤⠆⠀⠀⠀⠀⠓⠁⠀⠑⠠⠄⠀⠉⠁⠀⠠⠄⠈⠀⠈⠀⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠊⢀⡈⠀⠀⠸⠈⠢⡀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠘⠤⢌⡈⠋⠀⠐⠢⢀⢈⣁⡖⠐⠐⣂⡁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠠⡤⠀⠤⣤⢦⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⡈⠡⢀⣀⡀⠔⠀⠑⠦⡘⠀⠀⠀⠀⠀⣤⠊⠁⠁⠤⠄⠋⠉⠉⣌⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠁⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⡀⠀⠀⢀⠈⠘⠀⠀⠀⠀⠀⠀⠀⡀⡆⡑⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠂⠀⠄⡅⠀⠠⠀⡠⠂⠀⡠⠄⠀⠒⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⢀⠤⠄⠀⠀⠀⠀⠀⠄⢀⠀⠀⠑⠃⠀⠱⠖⠁⠀⢀⠀⠄⠀⠀⠀⠀⠀⠤⢄⠠⠤⣤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠁⠀⣀⣠⠤⠔⠂⠀⠀⠀⠐⠛⠲⠦⢀⠱⢁⠠⠶⠞⠁⠠⠤⠄⣀⠐⠠⢄⣀⡀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⡠⠂⠐⠒⠀⠈⠉⠁⠒⠀⠀⠀⠐⠂⠀⠀⠀⠀⢀⡴⠛⠳⣄⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠉⠙⠒⠒⠂⠀⠦⡀
⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⡟⠀⠀⠀⠨⠗⠒⠒⠒⠒⠒⠂⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠⠐⠦⠀⡀
⠀⠀⠀⠈⠉⠉⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿