SONIC Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢠⡖⠶⠖⠖⠶⠶⠲⠲⠖⠶⠒⠶⠒⠒⠶⠶⠶⠶⠶⠲⠖⠖⠶⠆⠂⠀⠁⠀⠈⠁⠀⠐⠶⠶⠖⠒⠶⠲⠶⠶⠶⠖⠲⢶⡄
⠀⠀⢸⡇⠠⣤⣀⣀⠀⣠⣦⣄⣤⣤⣠⣼⠃⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠸⡷⠿⠿⠟⠛⠟⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠰⢠⣶⣤⡀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⡄⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⢠⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⣴⠇⠀⡀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠈⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠀⠸⠀⣬⡍⠀⠰⠤⠠⣄⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠄
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠤⠀⠤⢈⡀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⠤⠴⠊⠀⠠
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⡀⠈⠀⠀⡇⠀⡀⠁⠀⠀⠀⠀⠀⠈⠀⢀⠈⠁⠀⠀⠀⠀⠀⠂⠀⢰⡄
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⢀⠀⠛⠋⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡆⠈⠀⠲⢤⠸⠃⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠈⠁⠀⠐⡶⠀⠀⠀⢌⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⢐⠁⠀⠀⠀⠀⠠⡤⠀⠀⠀⡀⠠⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢰⡆⠀⠀⠀⠈⠀⠀⠀⠀⠸⣄⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠉⠉⠐⢄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠐⡄⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠟⡒⠀⠂⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠃⠀⠀⢄⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢄⡤⠦⣤⢼⡦⡤⢦⡤⠶⡁⡴⠶⠦⢤⡤⡤⠶⢦⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⢤⠀⡆⠀⠀⠃⡇⠈⢰⣦⡅⠙⠀⣿⠈⠇⡇⠘⢋⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠃⠀⠀⠀⠀⠀⣏⠑⢠⡄⠃⣤⢸⡀⡀⢰⡈⠃⣄⣀⠂⣿⢠⡆⣇⠘⣣⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡇⢹⢐⡀⢴⠀⡆⠹⢡⣆⡇⢰⡆⢸⠀⠋⣤⠘⠁⣆⣹⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⢸⡅⠾⠀⠇⣠⠠⠆⡅⠰⠆⢠⠀⡄⠿⢠⡀⠆⢹⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠚⠃⠒⠒⠊⠉⠑⠊⠉⠐⠊⠙⠒⠋⠒⠉⠙⠒⠁⢸⡇
⠀⠀⠸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⠇
⠀⠀⠀⠈⠁⠉⠀⠀⠈⠀⠁⠈⠀⠉⠀⠀⠁⠈⠁⠈⠀⠀⠉⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣧⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣸⣿⣿⣿⠟⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡾⣿⣏⡈⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢙⡀⢁⣭⣿⡿⠃⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⣹⣆⠀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡋⠁⠈⠙⢢⣄⣀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⡿⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⣀⣉⠀⠀⠀⢠⣄⣬⣀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠘⠛⠁⠀⠀⠀⢈⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣠⣴⣶⣾⣿⣿⣦⠡⠀⠀⠀⠀⠈⠀⠀⣞⠀⠀⠀⠈⠻⣿⣧⠀⠀⠀⠀⠀⠑⢄
⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠁⠀⠀⢀⡤⠋⡇⠀⠀⠀⠀⠙⢿⣧⠀⠀⠀⠀⠀⠈⠣⡀⠀⣠⣾⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⡒⠒⠒⠉⠀⣼⡃⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⢀⠀⣀⣾⣿⡿
⣶⣟⡻⠿⣿⣿⣿⣿⣿⣧⣙⢂⣀⣠⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃
⣿⣿⣿⣾⣥⠯⠙⠛⠛⠛⠛⠛⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠿⣏
⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⢀⠈⡄
⠀⠀⢀⣀⣠⣤⣤⣤⣤⣶⣦⣦⣤⣤⣿⡁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡴⠻⡧⠇
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠐⡆⣀⣒⣊⣢⡀⠀⠀⠀⠀⠀⠀⠀⡇⢠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢹⣿⣷⣯⣿⣾⣿⣿⡟⠀⡀⠀⠀⡇⢸
⢄⠀⠀⢹⡟⠙⠛⠻⢿⣿⣿⣿⡿⠿⠟⠻⣿⡄⠀⠀⠀⢿⣿⣿⣿⠋⢁⣀⠙⠘⠀⠀⣰⡇⠄
⣄⠙⠒⠞⢧⡀⠀⠀⡠⠈⠉⠀⠀⢀⣠⣴⣿⡇⠀⠀⠀⠘⣿⣿⡏⢠⣿⣏⠀⠀⠠⠈⣸⠇
⠈⠳⢾⡇⠀⠙⡶⢊⣀⣤⣤⣶⣾⣿⣿⣿⣿⣷⣤⣤⣤⣴⣿⣿⡇⠘⣿⣿⡄⠀⠀⠀⠀⢠⣤⣤⡄
⢦⣄⡀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠿⣀⠈⠙⠁⠀⠀⠠
⠀⣹⡷⣴⡟⠛⠋⠉⠉⠻⢙⣻⣿⣿⣿⣿⣿⣿⠟⣥⠌⠀⠀⠀⠀⣧
⠀⣽⠁⠀⠈⢻⣦⡀⠀⠀⠛⠿⣿⣇⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠉⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢀⡍⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⣿⣽⣿⡻⣿⡀⠀⠀⠀⠀⠀⢀⢴⡇⡀⠄⠀⠁⠀⠀⠀⠉⠉⠀⠐⢰⡾⠋
⣾⠃⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⣿⣿⣿⣿⣿⡄⠀⠀⢀⡠⠂⠡⠈⠁⢀⣠⣴⣾⣶⣶⣦⣤⣤⣀⣰⣿
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣿⣿⣿⣿⣿⣿⣤⣴⠿⠐⠁⣀⣴⣾⣿⡿⠛⠉⠈⠉⠛⠻⢿⣿⣿⣿⠀⢀
⡇⠀⠀⠀⠀⠐⢦⣄⣀⣀⡀⢀⣿⡯⠟⠛⠻⣿⡿⢃⣠⣴⣿⠿⠟⠉⠀⠸⡄⠀⠀⠀⠀⠀⠀⢈⢹⣷⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣇⠀⠀⠀⢸⣿⣿⡿⠋⣁⠀⠀⣀⠀⠀⢹⡄⠀⠀⠀⢴⣷⣤⣿⣿⣿⣿⡄
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⡄⠀⠀⠀⢹⡁⠙⠀⣿⣴⣿⣿⡀⠀⠀⢳⡀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿
⣦⣤⣑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⢳⡀⠀⠈⢻⣿⣿⣷⣦⣤⣈⣧⣤⣤⣶⣿⠟⠈⣿⣿⣿⣿
⣿⣿⣿⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠱⡄⠀⠀⠙⢿⣿⡟⣹⠿⠿⠿⠿⠛⠁⣠⣾⡿⠛⠋⢉
⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠘⢦⡀⠀⠀⠙⠃⠱⢦⣤⣀⣀⠄⣮⡿⣿⣡⣤⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⣴⡀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠙⢦⡀⠀⠀⠘⢾⣿⣿⠿⠟⠓⠊⢻⣿⣫⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠙⠲⣤⣄⣀⡁⢀⣤⣶⣤⡀⠀⠈⠙⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠘⠻⣿⣶⣤⣤⣼⣿⣿⣿⣿⣿⣿⡄⠙⣿⣿⣿⣿⣿⣿
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⣿⣿⣶⣄⡀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⠙⣿⠟⠛⣍

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣄⣀⣀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿⣿⣿⣿⣿⣿⠇⠀⠀⢀⡿⣦⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡏⠀⠀⠀⠈⣷⣾⡆⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠤⠐⠀⠀⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⠀⠘⣿⠿⣿⠟⠀⠀⠀⠀⠀⠹⣿⠃⠰⠟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠁⠀⠀⢀⠀⠀⠀⢿⣿⣤⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠈⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⣠⠉⠀⠀⠀⠀⠀⠉⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿
⠀⠀⢀⠀⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠀⠀⠰⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⡶⠤⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⡤⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣤⣀⡀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⠃⠀⠀⠀⠰⠿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠿⡧⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠏⠉⠻⠏⠉⠀⣶⡀⠀⢀⡀⠀⠀⠀⠀⠘⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⠊⠀⠀⠀⠀⠀⠀⠈⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣠⣾⡇⠀⠀⠀⠀⠐⠀⠀⢠⣤⢠⡄⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣠⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡝⣧⡀⠀⢠⡤⠀⡚⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⠈⣿⣷⣶⠐⠻⠿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣧⠀⢹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠃⠀⢿⣿⣿⡆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⣀⠀⠀⠂⠀⠘⠻⠿⠿⠤⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠉⠛⢦⣄⣤⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣿⣿⣦⠀⠀⠀⠀⠀⢻⣿⣿⣿⣶⣤⣶⣶⠟⠋⠁⠀⠀⠀⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢀⣀⣤⣌⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣷⣬⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠶⠿⠛⠋⠙⢿⣿⡇⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣄⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠉⠙⠛⢛⠛⠿⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠒⠂⠤⢀⡀
⠀⠀⠀⠀⠀⠴⣂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠤⢀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⡀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠢⣅⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣂⡠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⢀⠈⠂
⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡜⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⣧⣿⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠘⣿⠇⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⡆
⠀⠐⠐⠂⠀⠉⠉⠀⠀⠉⠉⢑⠆⠀⠀⠀⠀⠰⠉⠉⠙⠂⢀⡀⠀⠀⠀⠀⠀⢀⠀⢸⣋⡤⢾⣦⠀⠀⠀⠀⠀⡀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢃⠀⠠⢶⠀⠀⠈⠁⠒⠒⠒⠉⠀⠀⠸⠿⠶⠟⠁⠀⠀⠀⠀⠈⡗⢁⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠑⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⣠⠀⢆⣀⡸⠀⠐⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣢⣄⡀⠀⠀⠀⠀⠀⢀⡀⠀⠂⠀⠀⠀⢠⠴⠀⠀⠊⠉⠅⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠉⠀⣠⠍⠀⢻⣉⠝⠛⠀⠀⠉⠛⠆⢄⠀⠀⠀⠀⣄⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁⠀⠀⠀⠀⠀⠈⠀⣀⠤⢪⠏⠀⠀⠀⠀⠀⠀⠀⢢⠈⠐⠒⠁⢈⠷⠄⡠⠖⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣾⠉⠀⢰⠋⠣⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣆⠀⢱⠀⠏⠋⠀⠠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⠸⠗⠁⠀⠀⡘⠀⠀⠈⡆⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠂⠤⣀⣀⠔⠁⠀⠀⠀⡖⠒⣒⣲⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠟⠠⠄⠀⣀⢼⠁⠈⠃⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠈⡀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠃⠀⠄⠀⠀⠀⣧⠀⣻⠆⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠠⠜⠓⠀⣃⠀⠉⠁⠀⠀⠀⠈⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠀⠀⠀⠀⣠⠊⠘⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⠀⢀⣀⠀⠀⣠⠄⠀⠠⠄⠀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠈⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠘⠀⠀⠈⠀⡀⠀⠀⠀⢀⣀⣸⣄⣘⣂⣈⣁⣀⣀⣀⣀⣐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠤⠊⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⣀⠠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⠛⠋⠉⠉⠁⠀⡠⣄⡀⠀⠈⠉⠙⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢦⣀⠀⠀⠀⠀⠀⣸⠐⠀⠨⣑⢦⡀⠀⠀⠀⠈⠙⠢⣄⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⡇⠂⣀⠤⠚⠃⠀⠀⠀⠀⠀⠀⠀⠈⠳⣎⠉⠉⠉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠬⠗⠀⠀⠃⠈⠁⠀⢀⠔⠊⠉⠑⠢⣄⠀⠀⠀⠀⠈⢳⡀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢀⡀⠀⠈⢆⠀⠀⠀⠀⢈⡙⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⡇⠀⠀⠸⡆⠀⠀⢠⠏⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⢳⡀⢀⡏⠀⣸⠁⠀⠀⠀⠀⠀⣀⡤⣄
⠀⠀⠀⠀⠀⠀⢠⣞⣤⡤⠤⠤⠤⠤⢤⣤⡀⠀⠀⠀⢀⡤⠦⢦⣀⠘⠋⠀⠀⠀⠀⠀⠉⣿⠃⣠⡇⠀⠀⠀⠀⢀⡜⠁⣀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⣟⠀⢀⣤⠈⠳⠦⣤⣤⠤⠖⠚⠉⣷⣯⣿⠟⠀⢠⠞⠛⣮⠔⣚⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠹⣆⠀⠘⠢⢄⣀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⣊⠰⣀⢹⣎⠭⢽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢈⣓⣦⣄⡀⠀⠀⠀⠀⣀⣠⠴⠋⠀⠀⢀⠼⠉⠀⠈⠻⣁⠠⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⣠⠴⠚⠋⢉⣽⠯⠖⣦⡩⠛⠙⠉⠉⠻⣦⣄⡀⠀⢀⣿⠀⠀⠀⠤⣤⠤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠚⠉⠀⠀⢀⠔⠋⣠⠴⠚⣙⠀⠀⠀⠀⠀⠀⠈⣆⡉⠛⢉⣠⠗⠤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⢸⠲⣄⡸⠉⢳⠀⠀⠀⠀⠀⠀⣹⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡜⢣⡸⠀⠈⡇⠀⠛⢤⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠔⠷⣀⡀⠀⣰⠇⠀⠀⢸⠇⠀⢠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⡿⢥⠀⢀⣰⢻⠒⣆⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢲⡏⠁⡏⠀⢸⡄⠸⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⣼⠀⢸⣅⠀⣠⣇⠀⣻⠍⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀⠀⠳⠤⠞⠀⣹⡅⠉⠉⠁⠀⠈⢹⣦⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠁⠀⠀⠀⠀⢘⡇⡏⠀⠀⠀⣀⣴⠋⠀⠀⠀⠀⠈⠉⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠫⡙⠓⢶⠶⠒⣯⠀⣹⠓⠚⠉⠁⢸⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠘⢆⠀⠂⠀⠘⡇⣏⠀⠀⠀⠀⠈⣀⣀⣤⡴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠸⠀⢐⣀⡴⠏⠻⠤⠶⠒⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⣀⣤⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡤⡆⠐⠒⣤⢠⡒⠄⠐⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠏⠀⠀⣟⡆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠏⠀⠀⠀⢻⡇⠠⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠿⢶⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⠛⠿⣿⣿⣿⣿⣿
⠀⠀⢀⠀⠐⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠉⠻⣿⣿
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠐⠻⣷⣶⣶⣿⣿⠦⠉⠙⠛⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠂⠀⠀⣠⡿⠀⠀⢀⡟⠀⠀⠀⠈⢿⣿⣿⣿⣶⣤⡀⠀⠀⠈⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠻⠟⠃⢤⡁⠀⠀⠀⠀⠘⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⡀⠀⠀⠀⠀⢨⣥⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠳⣄⠀⠄⣰⡉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⢻⣿⡇⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡀⠀⠈⠙⢶⣶⣤⣀⣀⣀⣤⠤⠖⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣆⠀⠀⠈⣿⣿⣿⡿⠋⠀⠀⠀⢀⣀⣙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢀⣀⣠⣿⣿⣿⠃⠀⠀⢀⣶⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠈⠻⢦⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⢀⣠⡤⢄⣀⣠⣤⣴⠶⠛⠉⠀⢠⣄⣀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⣟⣩⣶⣿⠿⢟⣫⣥⣴⣶⣿⣿⣿⣿⣿⣿⣿⡿⠿⠤⠄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⢌⠛⠉⠀⠺⡿⠿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠁⠀⠀⠛⠻⣦⠄⠀⠈⠁
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣀⣀⣀⣼⠁⠀⠐⠀⠀⠀⠀⢀⣄⣀⣀⣀⣀⡉⠉⠛⠲⣤⣀⠀⠀⠀⠀⠂⠘⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⠀⠀⢀⠀⠀⡀⣸⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡈⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⡀⠀⠠⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢶⠃⠀⠀⢀⠀⠈⠀⣀⣈⣉⣙⢶⣄⡀⠀⠀⠀⠈⠉⠉⠙⠛⠦⢄
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠒⠶⠖⢉⣴⠀⠀⠀⢻⣿⣿⣿⠿⣿⠄⠀⠀⠀⠂⠀⢄⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⣴⣃⣤⠔⠂⠀⠹⣧⠀⠀⠀⠋⠙⣿⣷⡄⠀⠀⠀⠀⠀⠀⠘⠷⠆
⠀⠀⠀⠤⠀⠀⠄⠀⠀⠀⠀⠀⠀⣄⣀⣻⡿⠁⠀⠤⠤⠤⠴⠀⠀⢀⠀⠨⡋⠩⠿⠶⡄⠀⠀⠀⠀⠀⠀⠙⢠
⠀⠀⠀⠀⠀⠀⠘⢋⠠⠂⠀⢀⣠⠟⠉⠉⠀⠐⠔⣶⣦⣉⣍⠓⠀⠘⠛⠒⠀⠀⡸⢀⣰⣖⠚⠃
⠀⠀⠀⠀⠀⠀⡠⢋⠀⢠⠞⣽⠟⠀⠀⠀⠀⠀⠀⠙⢿⠟⣹⣿⡆⠀⠀⢀⡠⠖⠐⠿⡉⠑⠢⠆⠀⠂⠈⠒
⠀⠀⠀⢀⠴⠚⣓⠿⠖⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⣬⣭⣾⣷⠀⢰⡉⠀⠀⠰⡄⠘⠗
⠀⢀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⡀⠀⠸⡿⣟⠋⠂⠀⣷⡄⠀⠐⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠰
⠀⠈⠀⠒⠄⠀⠁⠀⠀⠀⠀⡄⠀⠀⠀⠉⠾⣿⣯⣁⣀⠉⠄⢠⣀⠀⠶⠿⢿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢤⡄⠉⠛⠿⢿⣶⣶⣧⡀⠀⠀⣻⣿⣿⣄⠀⢠⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⣶⣬⡛⢿⡄⠦⠀⡸⣿⣿⣿⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠠⣾⣿⣿⣷⣦⣀⡀⢁⣸⣿⣿⣷⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡄⣠⡄⠀⣠⣈⠻⢿⣿⣿⣿⣿⣷⣮⣉⠙⣻⠀⣴⠀⠀⠀⠀⡄
⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠇⠤⠉⡉⠀⠀⠙⠻⠿⣿⣿⣿⣿⠿⠃⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⠀⠒⠂⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠐⠒⠒⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠁⢀⡀⠠⠦⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠓
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠉⡱⠞⠁⠀⠀⠀⠀⠀⢀⡠⠐⠒⠠⣄⠀⠠⠎⡁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠂⠉
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⠔⠉⠀⣀⣀⡀⠈⠃⠀⠀⠉⠂⣄⢀⠀⢰⠀⠀⠀⠀⠀⠀⡠⠔⠉
⠀⠀⠀⠀⠀⠈⡄⠀⢀⡐⠀⠀⠀⠀⠀⠀⠀⡠⠃⡴⠊⠁⠀⠀⠀⠉⠢⡀⠀⠀⠀⠈⠞⠀⢸⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⢡⠎⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠒⠠⢄
⠀⠀⠀⠀⠀⠀⡌⠀⠆⡀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄
⠀⠀⠀⠀⠀⠀⣧⠀⠀⢡⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣄
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢃⡀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣷⡀⠀⣠⠂⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠈⣿⡟⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⢿⡏⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠰⣿⣿⣿⣷⣄⡈⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠟⠂⠁⠀⠙⡄⠀⠀⠀⠋⠀⠀⣀⣀⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⡿⠀⠀⠈⠁⠲⠀⣀⣀⡀⠀⠐⠀⠉⠀⠀⠒⢢⡄⠀⠀⡃⠀⠀⢠⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠅⠀⢀⠃⠀⢀⠂⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠋⠀⠀⠀⡠⠂⠀⡠⠊⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⣀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠒⠒⢻⠓⢒⡤⠄⠒⠀⢤⡀⠒⠈⠁⠀⠞⠁⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠰⠋⠀⡠⠀⠠⣀⣴⠃⠀⡠⠂⠉⠈⠉⠁⠀⠐⠦⣀⠀⠀⡰⠃⠀⠀⠄⣀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡈⢈⠀⣄⠀⡀⠀⠀⠙⢧⡀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠎⠑⡀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠛⠁⠙⠦⡀⠀⠀⠈⠙⡀⠀⠀⠀⠀⡄⠀⢀⠀⡄⠀⢸⠀⠈⢆⠀⠀⠀⠀⠀⠈⢂⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠒⣤⠀⠀⠂⠙⠃⠄⠀⠀⠜⢳⠆⠀⠀⠀⠀⠘⠀⢀⠀⡜⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠰⣀⠀⣀⢄⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠇⢀⡼⠠⠞⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⡰⠒⠉⠀⠀⣠⠄⠒⠀⠀⠤⣀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⢠⡀⠠⡄⠐⠊⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠒⠤⠤⢄⠀⠀⠸⠑⠤⣀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠈⡀⠀⠀⡆⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇⠀⠀⣇⠀⠀⢃⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠀⡀⠀⠃⠀⠀⡏⠀⠒⣼⡀⠀⠘⡤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠉⠉⠉⠈⡉⠆⠉⠀⠇⠀⠀⢣⡔⠁⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢸⠀⠀⠐⠒⠛⠂⠈⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠓⣄⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠣⡀
⠀⠀⠀⠀⠀⠀⢀⠐⠊⠉⠠⡀⠀⠉⠳⠒⠂⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠠⠚⠁⠀⠀⠀⠀⠈⡆⠀⠀⠆⠀⠀⠀⠘⣤⢀⡀⠀⠀⠀⠀⢀⣀⣠⣄⠊⢆
⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢰⠀⠀⠀⢀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠙⠶⣄
⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⡰⠂⠀⢀⠔⠀⠀⠒⢠⡀⠉⡀
⢎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⣸⠤⠤⠒⠹⠀⠀⠠⠀⠀⡐⠁⠀⠀⠀⠀⠀⠙⢄⠃
⠀⠀⠉⠙⠉⠉⠉⠉⠉⠉⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠑⣄⡆⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠆⡄⠀⠀⢀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣁⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣿⡋⣿⣿⣿⣿⠿⠿⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣿⡟⠛⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⢹⠃⠳⢸⣻⣟⣿⠙⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢠⡟⠙⣿⠁⠀⢈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠖⠛⠲⢤⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⠀⠀⠀⠀⠙⠖⡦⠀⠀⠀⢀⣀⣶⣿⣿⣿⣿⣿⣿⣿⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡖⠳⣶⣶⠄⠀⠀⠀⢶⣶⢒⣦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠁⠀⠀⠀⠀⣀⣤⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⠛⠛⠉⠉⠉⠉⠉⠛⠓⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠧⠤⣄⣀⢀⣀⡤⠴⠒⠒⠂⠀⠀⠐⠀⠀⠀⠈⣉⠳⠴⠒⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⣽⡂⡲
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠉⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠰⠯⠤⠤⣤⠤⠤⠤⠤⠤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⡀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢁⡀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠝⠁⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⡸
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠢⢄⣀⣿⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠓⠛⢉⡭⣍⣿⣀⣀⣠⠴⠲⡤⠔⠒⠲⢤⡀⠀⣼⠀⠀⠐⠀⠀⣨⠟⠛⣛⡴⠚⢓⠦⣄
⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⣏⠀⠈⠁⠀⠀⠑⡄⠀⢹⡤⠖⠛⠢⢌⠆⠈⠳⡶⢖⡚⠉⠀⡴⠋⢅⢀⡴⠁⣠⠈⢧
⠀⠀⠀⠀⢀⠴⢋⠤⠂⠘⡧⠀⠀⠀⠀⠀⠀⢰⢀⣬⠷⡄⠀⠀⠀⠀⠀⠀⡟⠢⣍⣉⣉⣇⠠⣀⠿⢤⠊⣁⠔⣸
⠀⠀⢀⡴⡡⠊⠀⠀⠀⠀⢹⢦⣀⣀⣀⡤⠶⣯⠭⠤⠔⠁⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠉⠉⢦⣀⡬⠞⠛⠥⠞⠁
⠀⠀⡞⡌⠀⠀⠐⢄⢀⠞⠁⡀⠀⠀⣨⠃⠀⠀⠉⠁⡲⠂⠀⠀⠲⢏⠁
⠀⢰⣧⠁⠀⠀⢀⡤⢿⡀⡜⠀⢀⣴⠋⠓⢦⣀⣠⠞⠁⣠⠞⠲⣄⠀⠙⢦⡀
⠀⢸⣿⠀⠀⢠⠏⠀⠀⠉⠧⠴⠋⠈⠙⠲⢤⣀⣀⣠⠞⠁⠀⠀⠈⠳⣄⠀⠹⣄
⠀⠘⣧⣂⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠳⣄⡤⠶⡄
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡦⠚⠁⠀⠀⢳⣄⣀⣤⡴⠖⠚⠉⠉⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⠀⢀⣠⠴⠋⠉⠀⢀⡤⠒⢆⠀⣼⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣾⠉⠀⠀⠀⠀⢸⠁⣂⡱⢈⣯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⠀⠀⠀⣀⡠⠵⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠈⣉⣩⣤⣶⠋⠡⢀⢀⢤⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠈⠀⠈⠈⠈⠉⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⠚⠋⠉⠉⠉⠉⢉⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠴⠡⢄⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠲⢄⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣤⠤⠤⠬⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢶⣫⡏⠁⠉⠀⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣔⠉⠉⠓⢄⣀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠜
⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠈⢣⣨⠦⠘⠆⠀⠀⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣎⣁⡀
⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠸⠧⣛⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠌⣫⡉⡹⠂⠂⠀⠈⠑⢄
⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠓⠒⠊⠉⠀⣀⡤⢤⡀⠀⠀⠀⠀⠀⢳⠐⠒⠠⢧⠀⡼⠀⠀⠀⠀⠀⠀⢈⢣
⠀⠀⠀⠈⠉⠳⢄⣀⠀⠠⠐⠀⠄⣠⠤⠔⠒⠚⠉⠁⠀⠀⠙⢤⡀⠀⠀⠀⠈⠖⢒⡸⢛⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠢⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⣀⡀⠈⠣⣄⣸⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⢠⠒⠊⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⢀⠤⠒⠉⠀⠉⠢⢄⡀⠁⢀⡤⢀⢠⡄⠀⠀⠀⠀⠀⢸
⠀⠀⢀⡋⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠱⠀⠀⠀⠀⢀⡟
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⠦⢄⡀⡔⠒⠈⠉⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢀⡴⠁
⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀⠂⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠉
⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡇
⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠋
⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠈⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠒⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⣀⡀⠀⠀⣀⣀⣤⣤⠶⠒⠀⢀⡀⠀⠀⠘⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⣶⡿⠛⣡⣴⣿⣿⡿⠟⢋⣀⣤⣤⣴⣾⣿⣿⣶⣶⣤⣤⣒
⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠀⠘⢦⣿⠿⠛⣩⣴⣶⣿⣿⣿⣿⣿⣿⣿⠿⠟⡛⠿⠟⠋⠉⣩⣅⡀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠐⠂⠁⠀⠀⠙⠛⠛⠛⠛⠻⠿⢯⣍⣀⠀⠀⠀⠑⠀⠀⠀⠀⠙⢻⡇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⣤⣤⣤⣿⠀⠀⠈⠀⠀⠀⠀⠀⢠⣦⣤⣤⣤⣤⣤⣀⡀⠉⠙⠶⠤⣄⠀⠀⠀⠀⠉⠀⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠀⠀⡀⠀⡀⠀⠀⠤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡈⠉⠂⠄
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢀⠀⡇⠀⠀⠄⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠤⠇⠀⠀⠀⠂⠀⠊⠁⢀⣈⣉⣙⠳⣦⣀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠻⢦⣄
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠻⠦⢤⣤⠔⢋⣴⠀⠀⠈⢻⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠠⠄⠀⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠐⢈⡠⠀⠙⢿⡄⠀⠀⠀⠻⠿⢿⣿⣮⠉⠀⠀⠀⠀⠀⠀⠀⢹⣆⣀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⣿⣾⡟⠉⠀⠀⠀⠈⢛⠀⠀⠀⠀⠀⢰⡿⠺⣷⣀⠀⠀⠀⠀⠀⠀⠀⠉⢩⡄
⠀⠀⠀⠘⠁⠀⠈⢠⣤⠀⠀⠀⠀⠀⠰⣦⣶⡾⠛⢀⠀⣈⣉⣉⣭⣭⣀⠀⠀⣆⣀⡑⠄⠘⢩⠉⠓⣀⠀⣤⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠁⠀⣀⣤⡾⠁⠀⠀⠀⠀⠑⢝⣿⣷⣴⢶⣌⠀⠀⠉⠉⠀⠀⢀⠞⣠⣴⣛⠉⠉⠁⠀⠀⢀⣀⠐
⠀⠀⠀⠀⠀⠀⣠⡞⣁⠄⢠⠞⢱⡿⠁⠀⠀⠀⠀⠀⠀⠀⠙⠿⢋⣼⡿⣧⠀⠀⢀⣤⠔⠉⠈⠋⢢⠀⠉⠺⠆⠀⠁⠀⠉
⠀⠀⠀⠀⡤⠞⠋⣫⠿⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣬⣥⣾⣿⡄⠀⣦⠀⠀⠀⠘⠦⠀⠻
⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣄⠀⠀⠻⡿⢿⠫⠲⠀⢸⣧⡀⠀⠐⠂⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠈⠁⠰⠦⠀⠐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠑⢶⣿⣿⣛⠁⠐⢋⡁⢀⡀⠀⢴⣾⣿⣿⣶⡄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣀⡈⠛⠻⢿⣿⣶⣤⣨⣝⠁⠀⠀⠐⣾⣿⣷⡀⠀⠈⠀⠶⠗
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠄⠀⢤⣉⠛⢿⣿⡃⢀⡀⠈⢿⣿⣿⣿⠆⠀⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⣿⣿⣶⣬⡓⠄⡀⠰⠌⣿⣿⣿⣦⠀⠀⡀⠈
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠆⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣦⣄⡒⠾⢿⣿⣿⡇
⠀⠀⠐⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⢴⣷⠀⢀⣶⡦⠉⠻⣿⣿⣿⣿⣿⣿⣷⣦⣄⣸⠁⢰⡗⠀⠀⠀⠀⠶
⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠐⠆⠠⠄⠀⠀⠈⠝⠛⠿⢿⣿⣿⠿⠿⠃⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠉⠙⠛⠲⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢀⡶⣆⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣤⠶⠋⠀⣿⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠒⠛⠛⠛⠛⠃⠚⠉⠀⠀⠀⠀⡿⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢧⣤⣀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠉⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⡦⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢀⣾⠂⠈⣷⠀⠀⠀⠀⠀⢀⣀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⠀⠀⢸⡏⠀⠀⢸⡄⠀⠀⣠⠎⠁⢻⡷⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⣏⠛⣷⣄⣳⠀⠀⣿⠇⠀⠀⠈⠳⣶⠿⠁⠀⠀⢸⣧⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢿⡞⢥⣬⣽⠳⣄⣿⡀⢀⣀⠠⠤⣤⡀⢀⡀⠀⣼⣟⣾⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢿⣯⢱⣮⣭⣍⠉⠁⠀⠐⢿⣿⣯⡿⢞⣛⣭⠹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣀⣠⣤⣤⣤⣤⣄⡀⠀⠈⡆⠘⣿⣿⣸⡏⠙⣷⣄⡀⠀⠀⠈⣉⣴⣿⡿⠃⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠞⠋⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⣴⠀⢻⣿⣿⣧⣠⣿⢈⣿⠓⣶⢋⣿⡾⢡⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠘⣿⣿⣿⣿⣿⣾⣷⣾⣿⣾⡿⢱⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⢣⠀⠀⢀⠀⠀⠀⠀⢸⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠃⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠘⢿⡄⠀⠸⡀⠀⠀⠀⠈⡇⢸⣿⢿⣿⣿⣿⣿⣿⣿⢃⠎⠳⣄⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠘⡇⠀⠀⣧⣴⣶⠶⠾⣷⢸⡏⣸⢿⣿⣿⣿⣽⡏⡞⠀⠀⠈⠉⠁
⠀⠀⠀⣀⣤⣤⠶⠶⠞⠛⠚⣷⠀⠀⠀⡄⠀⠀⢣⠀⠀⢹⢻⡛⠛⠛⣿⢸⡿⠁⣾⣿⢻⣆⡿⢰
⠀⣴⠞⠉⠁⠀⢀⣀⣤⣤⣤⣽⡇⠀⠀⣝⢦⣀⠘⡄⠀⠘⡼⠙⠶⣤⣽⢸⣿⣶⡇⣿⣸⣿⢃⣷⡀
⢸⡇⡠⠀⢠⣾⣿⣿⣿⣿⣿⠟⠛⢳⣞⠙⠿⣝⡛⢇⠀⠀⢣⡴⠶⠚⠻⣆⡈⠛⠻⠟⠛⣡⡞⢨⡇
⢹⡆⠳⠄⠈⠛⠿⣿⣿⣏⣁⢀⠀⠈⣿⣿⠀⠈⣿⢸⠀⠀⠈⣧⣴⡄⠀⢿⠙⠲⣶⡶⠚⠙⣧⠞⠁
⠈⠳⣬⡑⠀⠀⠀⠀⡈⣩⢉⢻⣆⠀⣸⣿⣀⣀⡿⣾⡇⠀⠀⢸⡆⢷⣤⣾⣧⣤⢿⡧⠀⠀⣿
⠀⠀⠈⠙⠛⠶⠦⣤⣤⣄⣀⣁⣉⣛⣋⣠⣽⣿⣥⣼⡇⠀⠀⣸⠷⣶⡾⠋⢙⡻⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡍⠉⢩⠟⣽⠋⢀⣠⣤⣽⠃⠀⢰⡇⢰⢱⡇⢰⣿⡿⠟⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣿⢸⠇⠀⣿⣿⠏⠁⠀⢀⡿⠻⣾⣸⡇⠘⣿⣇⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣿⣿⣆⠀⠙⠿⠷⠶⠖⣻⠃⠀⠈⠻⢷⣄⣙⣻⣛⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠲⠶⠶⠖⠛⠁⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠖⠚⠛⠛⠛⠛⠛⠛⠒⠲⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣭⣁⣀⠀⠀⠀⣀⣠⠤⠤⠤⠀⠀⠀⠠⠀⠀⠀⠉⠛⠲⣤⡤⠴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡿⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠐⣿⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠤⠋⠰⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣞⣉⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣩⠋⠉⠉⠉⠉⠉⠙⠛⠛⠒⠒⣶⡦⠀⠀⠀⠀⠀⠀⠀⢀⠎⠉⠙⢦⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠥⢤⡤⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣼⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⣀⣀⣀⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠉⠒⠦⣤⣿⡊⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣉⣡⠤⢤⣼⣡⣤⣠⠴⠋⢳⡔⠚⠉⠛⠦⣄⠀⢸⡀⠀⠀⠡⠀⣀⡼⠋⢛⣫⠔⠋⢉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⢩⣇⡀⠀⠀⠀⠀⠈⢣⠀⠀⢳⡴⠚⠉⠲⠬⠕⠀⢙⢲⠶⣚⠉⠁⠀⡴⠋⢦⣀⡠⠋⢀⠄⠙⡆
⠀⠀⠀⠀⢀⣠⠞⣁⠔⠂⠘⡦⠀⠀⠀⠀⠀⠀⠀⡇⣀⣼⠦⡄⠀⠀⠀⠀⢀⠀⢸⠓⢤⣉⣉⡙⡇⠢⣀⡽⢦⠔⢁⡠⢂⡏
⠀⠀⠀⣠⢎⠕⠉⠀⠀⠀⠀⢳⣄⣀⣀⣀⣀⡤⣾⡭⠥⠤⠜⠁⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠉⠉⠙⢦⡀⣠⠶⠽⠯⠤⠊
⠀⠀⣰⢣⠃⠀⠀⠀⠀⢀⠔⠁⠀⠈⠉⢉⡇⠀⠀⠉⠑⠒⣤⠀⠀⠀⠠⣖⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⡏⡗⠀⠀⠀⠀⣠⣏⠀⡰⠁⠀⣠⠞⠳⣤⡀⠀⣠⠞⠁⢀⡵⢤⡀⠈⠓⢄
⠀⢰⡏⡇⠀⠀⢠⠞⠁⠘⠲⣇⣀⡴⠛⠲⢤⣀⠉⠉⠁⢀⡴⠋⠀⠀⠙⢦⣀⠈⠳⣄
⠀⠘⡇⡇⠀⣰⠃⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠙⠒⠚⠋⠀⠀⠀⠀⠀⠀⠘⢦⡀⠈⠳⡄⠀⢀⣀
⠀⠀⠙⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢀⡽⠛⠉⠨⣇⠀⠀⠀⠀⣀⣀⣤⠤⠤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⠀⠀⠀⢀⡬⠷⠖⠚⠛⢉⡀⠀⠀⢠⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⣀⡤⠞⠋⠀⠈⢀⡠⠖⣉⠈⢆⡰⣳⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⠘⣄⠫⠔⣃⣼⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣄⣀⣀⣀⡠⠤⢚⣲⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⣀⣀⡤⢴⡞⡏⠄⢀⡄⡴⠀⠀⢠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠀⠁⠀⠁⠀⠉⠁⠀⠈

⣀⣤⣴⣶⣾⣿⣿⣷⡉⢁⠀⠀⠀⠀⠀⠉⠀⠀⢸⠇⠀⠀⠀⠀⠹⣿⣿⣧⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠂⠀⠜⠀⠀⠀⠀⠀⡸⢳⠀⠀⠀⠀⠀⠈⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢀⣴⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢤⣀⣀⣀⣀⡴⠛⠀⣼⠁⠀⠀⠀⠀⠀⠀⠙⢿⣧⠀⠀⠀⠀⠀⠀⠀⠨⠂⢀⣴⣿⣿
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⣦⡀⠉⠁⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣴⣀⣴⣿⣿⣿⡏
⣿⣿⣙⡻⡻⢿⣿⣿⣿⣿⣿⣷⣤⣙⣀⣀⣤⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿
⣿⣿⣿⣷⣾⣥⣿⠍⠿⠛⠛⠛⠛⠛⠻⣿⣿⢿⡟⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⣻⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡇⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠁⠀⠀⢳
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠈⢻⡷⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⢀⣴⣦⢸⡆
⠀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⠁⠀⠀⠀⠐⡀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⢻⡘⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢻⣀⣤⡥⢥⡸⠦⣀⡀⣀⣀⡀⠀⠀⠀⠀⢸⡇⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢿⣿⣿⣧⣾⣿⣾⣿⣿⣿⠃⢠⠀⠀⠀⢸⡇⡟
⣄⠉⠉⠉⣻⡿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣷⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠛⠉⠉⠙⠂⠸⠀⠀⢀⣾⠁⠇
⠈⠃⢤⣀⣿⡇⠀⠀⠀⠀⠉⡙⠿⠛⠉⠁⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⢹⣿⣿⣿⠇⢠⣾⣟⠃⠀⢠⠀⠠⢚⣿⠄
⠳⣤⠀⣨⠀⠘⢦⡀⢀⡴⠋⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⡀⠀⠀⠉⠀⠘⢋
⠀⠈⠛⢿⠁⠀⠀⣹⢋⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⡀⠘⢿⣿⣷⠀⠀⠀⠀⠀⠀⠶⠶⠾⠇
⢷⣤⣀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠉⠛⢧⡀⠀⠉⠁⠀⠀⡀⠠
⠀⢹⣿⣧⣴⠿⠟⠛⠛⠉⠙⢿⡛⢛⣿⣿⣿⣿⣿⣿⣿⣿⡿⢫⣄⠁⠀⠀⠀⠀⢸⣇
⠀⢸⣟⠀⠉⠑⢶⣦⠀⠀⠀⠀⣵⣿⣿⣿⡏⢿⣿⣿⣿⡟⠀⠉⠀⠀⠀⠀⠀⠐⠾⡿
⠀⢿⠇⠀⠀⠀⠈⠻⣿⣄⠀⠀⠀⠀⠉⣿⢹⣟⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠠⠄⠀⠀⠂⠤⠤⠤⠀⣀⡀⢀⣠⡾⠋⠉
⢰⣎⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠀⣿⣾⣻⣯⣝⣿⠀⠀⠀⠀⠀⠀⠀⡠⠜⢿⡄⠄⠈⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄⠀⣿⣿⣿⣿⣟⣿⡄⠀⠀⠀⣀⠄⢋⠠⠒⠀⠀⢀⣠⣶⣾⣿⣿⣿⣷⣶⣶⣤⣤⣰⣿⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣤⣿⣿⣿⣿⣿⣾⣿⣤⣤⣾⠿⠒⠁⠀⣠⣴⣾⣿⣿⠿⠋⠁⠀⠉⠙⠻⢿⣿⣿⣿⣿⡇⠀⢀⠀⢠
⡏⠀⠀⠀⠀⠀⠠⣤⣀⠀⠀⠀⠀⠀⠘⣿⣟⣿⠿⠿⣿⣿⣿⡿⠁⢀⣤⣾⣿⣿⠿⠛⠉⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡇⠀⣸⠀⢈
⠃⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣶⣶⣶⣶⣿⡏⠀⠀⠀⠀⢿⣯⣴⣾⣿⡿⠟⠉⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⢠⣄⠀⢦⣿⣿⣿⣿⡀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⠀⠀⠀⠀⠘⣿⡿⠿⣁⠀⢴⠀⣀⣠⣤⠀⠀⠀⢻⡄⠀⠀⠀⠘⣿⣿⣷⣿⣿⣿⣿⣿⣧⣠
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣇⠀⠀⠀⠀⠘⣇⠀⠈⠀⣿⣿⣿⣿⣿⡄⠀⠀⠈⢷⡀⠀⠀⠀⠈⣛⢻⠻⣿⣿⣿⣿⣿⣿
⣷⣤⣌⢣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠹⣄⠀⠀⠈⠻⣿⣿⣿⣷⣶⣤⣄⣸⣧⣤⣤⣴⣿⣿⡟⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠈⢦⠀⠀⠀⠈⢻⣿⣿⣿⢛⣿⣿⣿⣿⣿⡿⠟⠁⢀⣴⣿⣿⠛⠛⠋⢉
⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⣀⠠⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠙⢿⣷⠸⣅⡏⠁⠀⠀⠀⣀⠴⣿⣿⡿⠀⢀⣰⣶⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⢀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢠⣍⣿⡿⣴⣖⣛⣴⡿⣫⣽⣷⣻⡿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣶⣦⣰⣾⠄⠀⠀⠀⠀⠙⣿⣧⣄⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⢿⡿⣿⠛⠛⠉⠉⠀⠙⢿⣯⣾⣿⡿⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⡀⠀⠀⠀⠉⠓⣦⣤⣄⣀⡀⠀⣰⣶⣶⣦⡄⡀⠀⠈⠙⢻⣿⣿
⣿⣿⣿⣿⣿⢿⣿⣿⣿⣷⣾⣷⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣤⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣯⠁⢹⣷⣾⣿⣿⣿⣿⣿⣏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡿⣿⣿⣿⢿⡛⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠉⠉⠉⠉⠀⠒⠂⠠⠀⡀
⠀⠀⠀⠀⠀⠶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀⠀⠀⠀⠀⡀⠈⠀⠈⠐⣌⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢃⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠦⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈
⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡎⣿⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡿⠀⠀⠀⠀⢀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⢁⡀⠀⠤⠄⠀⠠⠤⠄⣀⠀⠀⠀⠀⠀⡠⠤⢴⡀⠈⠛⠁⠀⠀⠀⠀⠀⠀⣾⡏⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⣀⠈⠀⠠⠄⣀⣀⡀⠠⠀⠐⣫⣞⣫⣿⠂⠀⠀⠀⠀⣄⠠⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠙⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠀⠀⠀⠀⡀⠀⢫⠔⢂⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⡐⠃⢈⠶⠞⠁⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠒⠲⡊⢗⠲⢦⡤⠀⠀⠤⢶⡀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁⠀⠀⠀⠠⠚⠁⢀⡼⡷⠁⠀⠀⠀⠀⠀⠱⡑⠀⠀⠠⠊⢣⡀⠀⣀⠨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⡔⠀⢠⠔⠈⢁⠴⢅⠀⠀⠀⠀⠀⠀⠀⠰⠤⠄⠠⠐⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢧⠀⢹⠁⠲⠜⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠋⣧⠞⠀⠀⠰⠀⠀⠑⡄⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠈⠣⢀⡀⠀⡠⠃⠀⠀⠀⠧⠤⢤⣤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠽⠢⣀⠀⢀⢴⠃⠈⠏⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠆⠀⡀⠀⠀⠀⣇⠀⣧⠄⠈⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠘⠻⠤⠴⠓⠀⡃⠀⠋⠉⠀⠀⠀⠉⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠄⢠⠂⠀⠀⠀⠀⢀⡔⠁⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⣀⠀⠀⣠⠄⠀⠠⠀⠠⠄⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠈⠄⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⣄⠀⢀⣀⣀⣠⣧⣤⣉⣀⣀⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠔⠋⠀⠈⠉⠉⠉⠉⠉⠙⠛⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⡀⢀⡀⠤⠄⠒⠉

⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⡄⠐⠀⠉⠁⠈⠉⠉⠐⠀⢤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤
⠀⠀⠀⡇⠀⣀⠀⠀⠀⠀⠀⣠⣀⢀⡀⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⡇⠀⢸⣛⣿⣗⣶⣾⣿⣽⣻⣽⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠘⠉⠀⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡄⢁⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⠀⠀⠀⠃⣾⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠖⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡄⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠒⠐⠢⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠀⠀⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡦⠀⠀⠀⢀⣼⠀⠀⡀⠀⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠐⠃⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢛⠀⠀⡆⢀⣚⠋⠀⢴⡤⠀⣀⠀⠀⣻
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⠀⠀⠀⡀⠀⠈⡀⠀⠉
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠐⢉⠀⠀⠀⠃⠀⠀⠀⠀⠆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⢀⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠠⠀⠀⠈⠁⠈⠉⠲⠂⠉⡇⠀⠀⠦⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠤⠐⠀⠀⠀⠉⠉⣀⠀⠐⢁
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⠀⠀⣇⣀⠼⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠎⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠃⢠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠷⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠀⣶⠀⠀⠈⠓⢂⠸⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠄⠈⠀⠀⠂⢶⠀⠀⠀⠠⢌⡄⢠⠁⠀⠀⢀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡊⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠈⡀⠤⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⡦⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣇⠀⠀⠀⠀⡠⠆⠀⠀⠄⠀⠀⠀⠀⠁⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠒⢤⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠒⠒⢠⡀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⡙⠀⠀⠀⠀⠀⠀⠈⡂⠀⠀⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠋⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠘⠁⡍⠀⠁⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠊⠀⠀⠀⠢⢀⠀⣠⠴⠊⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⣠⠤⢤⡠⢼⡤⡤⢤⡠⠴⢎⢀⠤⠤⠤⢤⡤⢤⠤⠤⡄⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣇⠰⣤⡁⢰⠀⠁⠘⠇⡇⢘⠁⣦⣸⠈⢳⠀⣶⠘⡇⢰⠰⠾⡁⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⡏⠳⠈⡀⠸⠀⡄⢶⠀⠇⢰⡀⠏⢩⠀⢸⠀⣿⠀⡄⢸⠀⢾⡁⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡏⢰⠀⡏⢀⣈⢁⡙⣶⠋⡈⢳⠀⣉⠋⣟⢠⠟⣈⢱⡞⣀⠸⡇⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠀⡇⢈⣻⢸⡇⠈⠈⡓⢺⠀⣙⡄⠛⢈⠀⣿⠀⠀⣛⠺⡇⢻
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣸⣀⣇⣈⣋⣈⣡⠾⣌⣃⣼⣀⣙⣅⣿⣈⣤⣉⡰⢦⣙⣨⠃⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠀⠉⠀⠀⠉⠀⠀⣾
⠀⠀⠀⠷⢤⠤⠤⠤⠤⢤⠤⠤⠤⡤⠤⡤⠤⠤⠤⢤⡤⠤⠤⠤⠤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠾
⠀⠀⠀⠈⠉⠀⠉⠀⠀⠉⠁⠉⠀⠁⠈⠉⠀⠀⠁⠈⠁⠈⠉⠀⠈⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣖⠚⠉⠉⠁⠀⠀⢀⠔⠤⢄⡀⠀⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⠀⠀⠀⠸⠀⠀⠐⣍⠂⠀⠀⠀⠀⠈⠳⣄⡠⠤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⠄⠀⠇⠰⠊⠁⢀⠠⠀⠤⣀⠀⠀⠀⠀⠱⡄⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠈⢣⠀⠀⠀⠀⠘⢲⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⢇⠀⠀⢠⠋⣿
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠘⣄⣠⡇⢀⡇⠀⠀⠀⠀⢀⠤⢤⡀
⠀⠀⠀⠀⠀⠰⠧⠒⠒⠒⠒⠒⠒⡦⠀⠀⠀⣰⠚⠚⠢⣌⠁⠀⠀⠀⣀⣀⣿⡡⣼⡄⠀⢀⣀⡰⢃⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⢷⠀⠐⢆⠈⠙⠓⠚⠉⠁⠀⠛⣿⠋⠁⢰⢁⠀⢟⠭⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠈⠳⢤⡀⠁⠂⠀⠀⠀⠀⣠⠴⠁⠀⢀⡞⠀⠳⣞⠋⠉⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢀⣠⠴⠒⢒⣿⠯⢭⣙⠖⠒⠒⠚⢯⣁⠀⠀⢠⡇⠀⠀⠀⣈⡡⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠴⠊⠁⠀⢀⠔⢋⡥⠖⣫⡁⠀⠀⠀⠀⠀⢣⣉⠛⣉⡼⠢⠴⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡀⢺⠒⣤⠃⢹⠀⠀⠀⠀⠀⢸⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣸⠱⠞⠀⢰⠀⠈⢳⡄⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠓⣦⣴⠋⠀⢀⣼⣁⡀⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠫⠭⡴⠊⡜⠀⢧⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⡇⢀⣇⠀⣸⡄⢸⡏⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⠘⠲⠞⢀⣿⡁⠉⠉⠀⠈⣹⢤⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣀⡀⠀⠀⣨⠇⣇⣀⣀⡤⠾⠁⠀⠀⠀⠀⠀⠉⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠑⢌⠉⠋⠉⢳⡀⡏⠉⠀⠀⠸⠀⠰⠀⢀⣠⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠆⠘⠀⢀⡷⣇⣀⣠⣤⠴⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⢀⣀⣤⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡀⠠⢀⢀⢤⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠈⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠒⠂⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠒⠒⠂⠠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠆⠋⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠶⠴⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠰⠆⡀
⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⣀⡀⢀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⢀⠤⠐⠒⠄⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⡠⠔⠊⠉⠐⢦⠀⠚⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⢀⠞⢀⡀⠰⠤⠀⣀⠁⠀⠀⠀⠑⣄⠆⢰⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠰⢀⠄⠂⠀⠀⠀⠀⠀⠀⠀⢁⠔⠁⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠈⠀⠸⠀⠀⠀⠀⠮⢀⡀
⠀⠀⠀⠀⠀⠀⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄
⠀⠀⠀⠀⠀⢸⠐⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠸⠇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⣧⡀⠀⡠⠁⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⡟⠈⠁⠀⠀⠀⠀⠀⠀⠘⢿⡏⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⢾⣿⣿⣷⣤⡈⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠓⠈⠁⠈⢢⠀⠀⠀⠃⠀⠀⣀⣀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡿⠀⠀⠈⠐⠂⠠⠤⠤⠀⠒⠂⠉⠀⠐⠒⣤⠀⠀⢸⠀⠀⠰⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠌⠀⢀⠃⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠁⠀⠀⡠⠊⠀⡀⠃⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠑⢂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⣤⠶⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⠀⠈⠁⣀⠜⠃⠀⣀⣻⣄⣀⠀⣀⠘⠁⠀⠀⣅⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠐⠊⠁⠀⠊⠀⠡⢾⠃⠀⠎⠁⠀⠀⠀⠀⠀⠈⠑⠢⣀⠔⠁⠀⠈⠁⠒⢄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠂⠵⠄⢦⡀⠀⠈⠓⠒⡄⠀⠀⠀⠀⠀⢀⠀⢀⠀⠈⡄⠑⡄⠀⠀⠀⠀⠉⢆⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢈⠆⢀⣀⠀⢀⡈⠦⢀⠀⠀⣘⢦⡄⠀⠀⠃⠀⠇⠈⠀⠀⠇⠀⠘⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⡄⠁⠀⡀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠆⢀⣆⣀⠖⠈⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠃⠉⠑⣄⠀⠀⠀⠀⠀⠀⠀⢁⠖⠚⠀⠈⡠⠀⠂⠀⠤⣀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⡄⣀⣀⣀⠀⠔⠁⠀⠀⠀⠘⡅⠀⠀⠀⠀⠀⠑⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠐⠢⠤⢤⠀⠀⠸⠁⠄⣈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠈⡄⠀⠀⡄⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠀⠀⢸⣀⠀⠡⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠀⢄⣀⣀⣀⣸⠀⣈⠲⡆⠀⠘⡖⢢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⡔⠉⠀⠀⣀⠀⠠⠱⠂⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⡠⠴⠈⠀⠓⠤⣀⣀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠱
⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠳⡀⠀⢡⠀⠀⠀⠉⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢎
⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢡⠀⠀⡀⠀⠀⠀⠘⠑⠂⠀⠀⡴⠂⠐⠂⠉⠠⢌⣆
⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠇⠀⠀⠀⡇⠀⠀⢠⠎⠀⢀⠄⠀⠠⢄⡀⠙⡀
⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⣦⠠⠀⠚⡇⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠙⢄⠃
⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠀⠈⠁⠈⠀⠀⠀⠀⠘⢄⡄⠀⢠⠁⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢄⡀⠀⠌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠶⠶⠶⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠉⠙⠛⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⢀⣴⣦⠈⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⢀⣠⠞⠁⣿⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣄⣠⣿⠃⣠⠶⠚⠋⠁⠀⠀⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠞⠛⠉⠉⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠳⣤⣤⣀⠀⢀⣴⠋⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠟⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠉⣀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⣸⡟⠀⠀⢹⡀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢀⣿⠁⠀⠀⠈⣧⠀⠀⡄⢠⠞⠁⠀⢿⡿⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⢰⡟⠻⢿⣄⠘⣧⠀⠀⢸⣿⠀⠀⠀⠀⠻⣶⣶⡿⠋⠀⠀⠀⢸⣇⣷⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣰⡞⣋⣛⠿⣦⡀⢸⣿⠀⠀⠀⠀⢀⣀⠁⠀⠀⠀⠀⠀⣼⡿⣽⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠈⢻⣿⡿⣌⡛⠾⠿⠤⠶⠖⠈⠉⢹⣿⣶⣴⢲⣄⣴⣿⡷⢿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⣿⣷⣸⠟⣶⣶⣦⡀⠀⠀⠀⠈⠻⢿⣿⠿⢋⣥⣴⡶⠂⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣀⣀⣤⡴⠶⠶⠶⠶⣦⣄⡀⠀⠀⠷⠀⢹⣿⣿⣆⣿⠀⠙⡿⢦⣀⠀⠀⠀⠀⣠⡾⣿⣿⠟⢁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠈⣿⣿⣿⣿⣄⣼⣇⠀⣽⠛⢳⡞⠋⣽⣴⠏⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⢸⣿⣿⣿⣿⣿⣿⣾⣿⣤⣾⣷⣾⣿⢋⡼⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡞⠁⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠠⣄⣷⠀⠀⠀⡇⠀⠀⠀⠀⠀⢿⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⣆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⢣⠀⠀⠀⠀⠀⣸⡇⢸⣿⠟⣿⣿⣿⣿⣿⢿⣿⡟⣠⠃⠘⠳⢤⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣦⣾⣷⠶⠾⢿⡇⢸⡏⢠⠻⣿⣿⣿⡿⣼⣿⢡⡏⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⢀⣠⣤⣤⡶⠶⠶⠿⠶⢿⡆⠀⠀⠀⡄⠀⠀⠈⢣⠀⠀⠀⣿⢿⡛⠛⠛⠻⣿⢸⣿⠏⢰⣿⣿⢿⡇⣿⡏⡎
⠀⢀⣤⡾⠛⠋⠁⠀⠀⠀⣀⣀⣀⣘⣿⠀⠀⠀⡘⢦⡀⠀⢸⡀⠀⠀⠏⣸⠻⣦⣀⡀⣿⢸⣿⣤⣼⠃⣿⢸⣿⡿⢠⣇
⢠⡿⠉⡀⠁⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣧⣤⣠⡿⣦⡽⢶⣴⣧⠀⠀⠸⡏⢀⣈⣙⣿⣿⠈⠻⣿⣿⣾⣿⣾⠿⢡⡿⢹⡇
⢸⡇⢠⡁⠀⢸⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⢹⣧⣤⡈⠻⠶⣤⢹⡄⠀⠀⠹⡟⠉⡉⠀⠙⣷⣤⡀⠉⠉⠉⢀⡴⣿⠀⣸⡇
⢸⣧⠈⡳⠀⠀⠉⠛⠿⢿⣿⣥⣄⣀⣀⠀⠀⢿⣾⡇⠀⠀⣿⠸⡇⠀⠀⠀⢿⡶⣷⠀⠀⢻⡀⠉⢻⣿⡟⠋⠀⢻⡿⠋
⠀⠻⢦⣍⡒⠀⠂⠀⠀⠀⡄⣬⠉⡙⣿⣀⠀⢀⣿⣧⡀⣀⡿⢶⣷⠀⠀⠀⠸⣧⠹⣧⣤⣞⣷⣤⡼⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠉⠛⠿⢶⣤⣄⣀⣁⡀⠀⠑⠈⠙⠛⠛⠃⣈⣿⣯⣅⣠⣿⠀⠀⠀⢰⡿⣤⣶⣶⠛⠛⠻⢿⣻⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⡟⠛⠛⢛⡿⢿⡿⠛⠉⠉⠉⢻⡿⠀⠀⢠⡟⠀⡼⣹⠇⢀⣾⣿⣦⡿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⡞⢠⡟⠀⢀⣶⣿⡿⠟⠁⠀⠀⣾⣷⣠⡇⣿⠀⢸⣿⣿⠁⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠈⣇⣾⠃⠀⢸⣿⣿⡁⠀⠀⢀⣴⡏⠈⠻⣷⣿⡄⠀⣿⣿⣦⣠⣤⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣿⣿⣇⡀⠀⠙⠛⠛⠓⠛⣉⡿⠁⠀⠀⠈⠛⢷⣤⣈⣙⣻⣥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠓⠦⠴⠶⠶⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⡀⠀⠀⠀⠀⡠⠔⠂⠒⠈⠉⠀⠀⣀⢀⡀⣀⢀⣀⢀⡀
⠂⠀⠀⠀⠀⣀⠄⠄⠂⢀⣀⣨⠤⠥⡄⣐⣂⠂⢐⡐⠀⠄⠀⠀⠀⠀⢠⠀⠀⠀⠈
⠀⠀⠀⠠⡎⠀⠀⢠⠚⠉⢠⠔⣪⣥⣤⣤⠄⠀⠀⠀⠀⢢⠀⠈⠀⢀⣠⠲⠃⢈⠀⠨
⠀⠀⢠⡄⠘⠒⠀⠈⠀⠀⠈⠈⠁⠋⠲⠲⠌⠂⠀⠀⠐⠁⢐⠀⠈⠁⠀⢀⠔⠀⡸⠃
⠀⠀⠀⠉⠂⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠴⠂⠂⠁⢠⡢⠀⠁
⠀⠀⢀⠄⡌⠀⠈⠁⠉⠉⠉⠁⠒⠒⠒⠒⠛⠉⠉⠁⠀⠀⢀⠀⠠⠐⢁⡜
⠀⠀⠀⠂⠡⡀⠈⠠⠄⣀⣀⢀⡀⢀⣀⡀⠀⠀⠀⠈⠀⠐⠂⠀⠀⠄⣈⠃
⠀⠀⠀⠀⠀⠈⠂⠕⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⣀⠤⠀⠀⣀⡎⡁⠀⠀⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣄⠀⠀⠀⠀⠀⠀⢀⣀⡠⠦⢴⠞⢋⣴⡁⠀⠀⡀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠑⢶⣤⡄⡠⠤⠤⠄⢒⡁⠀⡠⢆⢅⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⣶⣚⣛⣓⠊⠴⢋⠆⡅⠀⠀⠀⡄⠂⣀⣤⠀⠀⠦⠀⠀⠖⠋⠐⠦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠁⠈⠹⣷⠶⠖⣷⠷⠊⡁⠀⠀⠀⠁⠀⢀⠘⠃⢀⡀⢷⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢰⣄⠀⠀⠉⠀⠐⠟⠒⢂⡺⢽⣻⡁⢀⠀⣀⠁⠀⠐⡃⠀⠠⡠⢀⠀⠂⠀⠀⠁⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠋⠉⠀⠀⠂⢘⣾⣷⣶⡦⠝⣫⠁⠋⠐⢃⠤⢰⠁⢀⡄⢠⣦⣦⠈⠂⣈⠁⡀⠤⠀⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠃⠀⠄⠀⡀⡀⢠⣾⣴⣶⠮⢿⣴⠃⠀⠀⡠⠁⠄⠌⠀⠈⠠⠄⠋⠉⡠⠔⠄⠀⢳⠠⠄⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠒⠠⠀⠃⠀⠀⢳⡭⠤⠄⢰⣺⡵⠁⠁⠀⡁⠠⠁⠈⠴⠐⠁⣰⠃⠀⠁⠒⠤⠀⠀⣇⠠⠄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠃⠒⢠⡀⠆⣭⠸⣿⡿⣛⣭⡿⣃⡶⠌⢀⠀⠈⠀⢁⠂⠀⠉⠉⠲⢄⠀⣀⣀⣈⠀⢀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣄⣰⣛⣿⠟⠀⠗⠉⢶⣿⣋⠙⠀⠚⠃⠀⠀⢠⠀⠰⣤⠢⠴⠦⠔⠊⠁⠀⢀⠀⠔⣸⠃⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠛⠺⡷⠢⣤⣀⠀⠈⣩⡥⠤⠌⡀⠀⠠⡤⠤⠦⡤⠐⣬⣄⠀⢀⢄⣀⡀⣤⠠⢞⣃⠬⠒⠓⡡⠶⡀

⠀⠀⢘⣆⢠⣾⠆⠛⠋⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠠⣾⣆⣀⣀⠀⠀⠀⠀⠸⠿⠃
⠀⠀⠻⠛⠀⠀⠀⠀⠀⠀⠀⠀⡞⣷⠀⠀⠀⢀⡤⡞⠿⠗⠀⠀⢀⡾⠿⠂⡼⠿⠆⠀⠀⣾⠁⠀⠀⠈⠙⠿⠟⠁⠀⠀⠀⠀⣀
⠀⠀⠀⢀⡴⣦⠀⠰⣞⡇⠀⠀⣵⠏⠀⠀⠀⠈⠙⠁⣤⠀⠀⠀⡜⠁⠀⡼⠁⠀⠀⠀⠠⡞⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣹
⠀⠸⣿⠞⠚⠁⠀⠈⠃⠁⢀⡞⣽⠶⢶⡄⠀⢀⣤⣴⠃⠀⠀⠀⠉⠙⠚⠁⣀⠀⠀⠀⠀⠀⠀⡤⢄⣰⣩⠆⠀⠀⢠⣶⣦⠀⢰⣿
⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠸⣆⠧⢁⣸⠇⠀⠈⠙⠃⠀⡠⠤⡀⠀⠀⢀⣴⣏⠀⠀⢀⡤⣄⠀⠙⠊⠉⠀⠀⠀⣠⠏⠈⠁⠀⠈⠁
⠀⠀⠀⠯⠵⠫⠖⠃⠀⠀⠀⠈⣙⠛⠁⠀⢀⠀⠀⠀⠀⢻⠉⠀⠀⠀⠻⣴⠋⠀⠀⣸⣼⠃⠀⢀⣶⠀⠀⠀⣀⠀⢀⣴⣄⠀⢠⣖⡦
⡀⠀⠀⣀⠀⣀⡴⠛⢦⡀⠀⠀⠉⠋⠀⢠⠏⠉⢹⠃⠀⠸⠀⠀⢀⣀⠞⠁⠀⣰⢞⣽⣄⡀⠀⢘⣧⠀⠀⣼⠻⠃⡼⠉⠁⠀⠸⡇
⠃⠀⠀⠻⠟⢡⣤⣠⠜⠻⣄⠀⠀⠀⣀⡼⢀⣄⡟⠀⠀⠀⣠⣄⠻⡟⠀⠀⠰⡇⠏⠀⢨⡇⠀⠘⠀⠀⠰⢿⣗⣺⠃⠀⣼⠃⠀⠃⠠
⠃⠀⠀⠀⠀⠈⠛⡁⠀⣠⠞⠁⠀⠘⠿⠃⠈⠛⠁⠀⣴⡆⠻⣅⠀⣹⣆⠀⠀⢙⠲⠶⠛⠁⠀⡤⣤⠀⠀⢠⣄⠀⠀⣸⣧⠀⠀⠀⢰
⠀⣠⣖⠁⠀⠀⠘⠿⠋⢁⡀⢠⢞⡆⠀⠀⠀⠀⠀⠀⠹⣄⢀⡼⢾⡥⠞⠁⠀⠙⠓⠁⠀⡔⢠⡟⠁⠀⠀⢸⡸⠃⠀⠈⠉
⠈⠁⠋⣀⣀⠀⠀⠀⠀⠸⠿⠋⠉⠀⠀⠀⣠⢾⡅⠀⠀⠈⠯⠖⠋⠀⠀⠀⠀⢀⠀⠀⠀⠈⣉⠀⠀⠀⠰⣉⠇⠀⢠⠏⠙⢲⠆
⢀⡀⠈⠓⢮⣇⠀⠀⠀⠀⢀⣠⣴⣦⠀⠀⠀⠉⠀⢠⡤⠤⣤⠀⠀⠀⣀⣤⡞⠉⢳⡀⠀⠸⡯⠭⢽⠀⠀⢀⣤⣴⠋⣀⣠⠏
⣻⠏⠀⠀⠸⡿⠃⠀⠀⠐⠋⠀⠈⠉⢠⣶⡄⠀⠀⢸⠋⠉⣿⠀⠀⢺⡁⠀⣳⡀⢀⣷⡄⠀⣷⣦⢸⡇⠀⠀⠉⠁⠈⠛⠋⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⠀⢠⣶⡆⢸⠉⠀⢀⣤⡿⠰⣿⠇⠀⠀⠀⢳⡐⠿⠃⠈⠛⠁⠀⢀⠀⠀⠿⠗⠀⠀⠀⠀⡞⣇⠀⠀⠉⠋
⠀⠀⡀⠀⠀⠀⠀⠘⡷⢄⠀⢸⡥⠤⣾⠀⠀⠀⠉⠁⠀⠀⣤⣆⠄⠀⠿⠗⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢀⣤⡀⢈⠽⡄
⠀⣷⣿⠄⠀⠀⠀⠀⠛⠃⠀⠈⠛⠉⠉⠀⠀⠀⢀⣀⠀⠘⠱⠋⠀⠀⣤⡄⠀⠺⣝⣦⠀⢠⣶⡇⠀⢠⣶⡄⢸⠋⠁⠈⠊⠁⠀⣄
⠀⠉⠈⠀⠀⠀⢠⡒⡄⠀⠀⠀⢠⠖⣛⣛⡶⠾⡥⠾⠃⠀⠀⢠⣦⠀⡟⠁⠀⠀⠀⣿⡆⠀⠉⠀⠀⢸⣉⣤⣿⠀⠀⠀⠀⠀⣿⠋
⠀⠀⢀⡞⠀⠀⢸⡇⠀⠀⢀⣀⢿⠈⠁⢛⣷⠀⠀⠀⡄⠀⠀⢸⡁⠀⢻⠀⠀⠀⠀⠈⠀⠀⣤⣄⠀⠘⠒⠒⠋⠀⠀⠀⠀⠟⠙
⠐⣿⡟⠀⠀⠀⠘⠃⠀⠀⠈⠫⠘⠛⠲⠛⠁⠀⠀⣼⣥⡀⠀⠘⣧⠴⠞⠀⠀⣀⡀⠀⠀⠀⣿⣉⠀⠀⠀⢀⡄⠀⠀⢠⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠛⠒⡆⠀⠀⠀⠀⠀⠀⠀⠸⢧⡞⠀⢠⣦⡀⠀⠀⠀⡴⠛⠛⠀⠀⠀⠸⠜⠀⠀⣰⡟⠁⠀⠀⣾⠀⠀⡇
⢠⣶⡄⠀⠀⠀⢰⠇⣀⣸⠃⠀⠀⢀⣀⠸⣿⠃⠀⠞⠀⠀⣀⡯⠇⠀⠀⠘⠁⠀⠀⠰⣖⢦⠀⠀⠀⠸⠉⠃⠀⠀⠀⢿⣶⠀⣇
⠋⠉⠁⠀⠀⠿⠟⠀⠻⠛⠀⣠⣤⠸⣟⠀⠘⣦⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⢀
⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠙⣏⠀⢈⣶⣚⡭⠇⠀⢠⠤⣄⡴⣻⠆⠀⢰⠯⣟⣷⠀⠈⠛⡁⠀⠀⢸⡭⠃⠀⠀⢀⡞⠓⢦⣤⣀⠈
⠀⢀⡤⠖⢿⠉⠈⢧⠀⠀⠀⠀⠈⢿⣭⠞⠉⠀⣀⠀⠀⠉⠁⠉⠁⢀⣀⡏⣠⣤⡏⠀⠀⣿⣿⠀⠀⠸⡇⠀⢠⣄⡾⢀⣄⡾⠀⢉⡟
⣀⣈⣧⣀⣈⣧⣀⣨⣇⣀⣀⣀⣀⣀⣀⣀⣀⣼⣻⣇⣀⣀⣀⣀⣀⣻⣟⣀⣈⣉⣀⣀⣀⣘⣀⣀⣀⣀⣁⣀⣈⣛⣁⣘⣛⣁⣀⣼⣁
⣿⣿⠋⡛⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠇⠀⠀⣼⣇⠀⠰⠀⠖⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⢑⠚⠀⠀⠀⠀⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡀⠈⠀⠀⢀⣠⣠⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠘⠃⢀⠀⠀⠰⣴⣶⣼⣷⣷⣿⡻⣿⣯⠀⠂⠶⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠠⠄⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦⠀⠀⠀⡀⣴⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠻⠆⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⠟⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆

⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣝⢿⣿⢮⠻⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢮⡑⠌⠳⡀⠙⡄⢻⣿⠸⠟⣿⣿⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠿⠯⣍⡙⠻⢮⡂⠀⠀⠀⠀⠀⢻⠀⠀⢿⡟⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣟⣛⡛⠛⠒⠢⠄⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢻⢫⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣛⣫⣭⣭⡥⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢣⠞⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣛⡫⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡄⣤⢄⠀⠀⠀⠀⠁⣠⡾⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡠⣀⣤⠔⠂⠀⠀⠀⠀⠀⢸⡏⡟⠻⠜⠧⠘⠋⠀⠀⠀⠀⠐⣉⣴⠿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣯⣾⠟⣉⠀⠀⠀⠀⠀⠀⡄⢠⣶⣶⡇⢸⡇⠀⠀⠀⠀⠀⠠⠔⠋⠉⠚
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡾⠋⢀⠀⠀⠀⠀⠀⠑⠊⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣪⣭
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⡴⢣⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⣛⣛⣯⣭
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣣⡧⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠐⠢⠤⣤⣤⣬⣽⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣼⣷⠀⠀⣧⠀⠀⠀⠀⠀⠨⡳⣦⣬⣙⣲⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⣿⣿⣴⡄⣿⣧⠘⣄⠈⢦⡐⢌⡳⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣷⣼⣦⡳⣿⣷⣝⣾⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠋⠉⢹⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣮⡻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠋⠀⠀⣠⣾⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠻⣿⣿⡌⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠁⢀⣀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣯⣠⣶⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡁⢈⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠛⣿⣿⣿⠟⠁
⠀⠀⠀⢀⣠⣤⣤⣤⣤⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⣿⣷⣴⣿⡿⠋
⠀⢀⣾⣿⡿⠿⠟⠿⢿⣿⣿⣿⠀⠀⠀⠀⣀⣠⣤⣄⡀⠀⣠⣾⣿⣿⣧⣀⣼⣿⡿⠋
⠀⠸⣿⣿⣄⡀⠀⠀⣀⣼⣿⡟⠀⢀⣴⣿⡿⠿⠿⠿⣿⣿⣿⡟⠁⢙⣿⣿⡿⠋
⠀⠀⠙⠿⣿⣿⣿⣿⣿⡿⠟⠁⢠⣿⣿⠋⠀⠀⣠⣾⣿⠟⢿⣿⣶⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣶⣾⣿⡆
⠀⠀⠀⠀⠀⠈⠉⠁⣀⣠⣤⣴⣿⡿⠃⢀⣠⣾⣿⡟⠁⢀⣴⣿⣿⣿⡀⠀⠀⣴⣶⣾⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⣿⣿⡇
⠀⠀⠀⠀⢀⣤⣶⣿⡿⠿⠟⠛⠉⠀⣴⣿⡿⠛⢿⣿⣶⣿⡿⠋⢸⣿⣷⠀⠀⣿⣿⡏⠉⢉⣀⣀⣤⣤⣤⣶⣶⣾⣿⣿⡇
⠀⠀⢀⣴⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⣠⣿⡿⠋⠀⠀⣸⣿⡏⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠉⣿⣿⡇
⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠟⠁⣀⣤⣾⣿⠟⠀⠀⠀⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⢸⣿⡏⠀⠀⠀⠀⠰⣿⣶⣀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠛⠁⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠘⣿⣷⠀⠀⠀⠀⠀⠈⠻⣿⣶⡄⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣿⣿⡇
⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠈⠻⠟⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⠀⣀⣀⣀⡀⣿⣿⡇⠀⠀⠀⢀⣾⣿⡿⠿⠿⢿⣿⣿⡇
⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠸⣿⣿⣄⣀⣀⣠⣿⣿⠇
⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⠀⠀⢺⣿⣯⡁⠀⠀⣉⣿⣿⠇⠀⠀⠀⠀⠙⠻⢿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣶⣾⣿⡿⠟⠁⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣁⣁⣀⡀⠠⠄⠐⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠒⠖⠒⢉⠙⠛⠉⠥⢤⣤⣤⣤⣤⣤⡤⠤⣤⣤⣤⣀⣉⣙⣛⡛⠛⠛⠓⠲⠦⠤⢄⣀
⠀⠀⢀⠤⠒⠀⡀⠄⠂⠠⠐⠊⡉⢍⣐⣀⣠⡤⠤⠶⠖⢂⣀⣀⣀⣀⣀⣀⣀⣠⣀⣤⣉⣉⣙⣓⠒⠢⠤⢄⣀⠀⠉⠑
⢀⠔⠁⣠⠖⠁⠀⢈⣀⣤⣶⠿⢛⣋⣭⡤⠶⠶⠞⠛⠛⣉⣉⣀⣠⣄⣀⣀⣀⣀⡀⠀⠀⠈⠉⠉⠉⠙⠓⠒⠈⠉⠓⠶⢄
⡌⠀⠌⠁⢀⣤⢞⡿⣹⣥⡶⠟⢋⣡⣴⣶⠶⢖⣛⣫⠭⠉⠉⠀⠀⠀⠀⠀⠙⠛⠓⠒⠒⠒⠲⠶⠤⠤⣤⣀⣀
⠀⠀⢀⡶⢋⣸⠿⢋⠕⣫⡴⠞⠛⠉⢁⣀⠤⠔⣒⠒⠒⠒⠒⠒⠐⠀⠉⣉⣉⣉⣉⡉⠉⠉⠋⠐⠒⠀⠀⠀⠉⠙⠓⠦⣄
⠀⢀⣟⣴⠟⢁⢔⡵⢟⡩⡲⠼⢒⠩⠁⣀⣈⣤⣤⣤⣤⠤⠶⠶⠶⠶⠦⠉⠉⠉⠉⠉⠉⠉⠛⠓⠲⠤⢄⡀⠀⠀⠀⠀⠈
⠀⠈⡟⠁⣰⣻⢏⠔⠁⢀⣤⣶⡷⠞⢋⡉⠭⠴⠒⠒⠀⠀⠀⣉⣁⡀⠀⠀⠀⠀⠀⣤⣤⣄⣀⡀⠀⠀⠀⠉⠲⡄
⠀⠈⠃⠀⢱⠇⠀⢀⣴⣿⠟⢉⡤⣐⠄⡂⠤⠀⣀⣀⣀⣤⡤⠤⣤⡤⢤⡤⠤⠤⠤⠀⡀⠀⠀⠈⠑⠢⡀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⢠⡟⡐⣡⢖⡡⢊⣠⣤⡶⠞⢛⣩⡥⠴⠶⠒⠒⠒⠒⠒⠦⠤⣀⡀
⠀⠀⠀⠀⠀⠀⡿⠀⠜⠁⣡⢶⠟⢋⣡⠴⠚⣛⠿⠟⠋⠉⢉⣁⣉⣉⣉⣀⣀⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢀⡾⠁⡠⠊⠁⣢⡼⣆⣠⣤⣶⡾⣿⣿⠿⠿⠶⠦⠤⣄⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣡⠶⣫⡵⢟⣫⣭⣽⣶⣤⣤⣤⡤⠤⠤⠤⠄⠀⠀⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣠⠞⣵⡿⣫⢖⣵⣿⡿⢟⣯⠏⣩⠤⠶⢒⣂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢏⡾⡏⢐⡵⢋⡕⡡⡺⠋⣠⣮⠴⠚⣛⣩⣵⣞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠡⠀⡾⠁⠎⠈⡐⣡⡞⡫⢂⣴⣾⡿⠋⢉⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣀⠀⡇⠀⠀⢠⡾⢻⢊⡴⡫⠋⣩⣴⠿⣫⠭⠉⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠸⡇⢆⡟⡐⣱⡿⢋⠕⣪⣴⣾⣯⡭⣭⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠁⠀⠠⢊⣾⢟⢟⢫⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡾⠐⢋⣴⠿⣋⣤⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣧⠞⡻⢛⣩⡥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠇⢸⣶⢟⣥⣶⡛⠋⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣴⡿⢫⣥⡶⠚⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠃⣭⣤⣶⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣥⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⣵⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡁

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠄⠈⠀⠈⠠⠀⠄⠠⠀⠀⠀⠀⢀⣤⡀⡀⠠⠀⠀⠀⠀⡤⡀
⠀⠀⠀⠀⠀⠀⢢⡀⠈⠁⡀⠀⠈⠀⠠⡌⡙⢿⡛⢿⡀⢙⣷⠀⢰⣦⡼⢷⣀
⠀⠀⠐⡆⠀⠀⢨⠁⠀⠀⠀⣦⣦⣄⡀⠀⠀⠂⢥⠰⠌⡛⣁⠀⠼⠏⣾⡿⠛⠉⠀⠞
⠀⠀⠀⢁⠀⠀⠀⠀⠀⢀⠰⠫⢻⢻⠛⢦⡣⠀⠀⠀⢡⠁⠀⠀⠀⢆⠋⠀⡀⠔⢐⣶⣤⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠡⡄⠎⠁⠀⡀⠸⠦⠀⠀⡄⠈⢃⠀⠂⢁⢣⠂⠁⠂⠀⣀
⠀⠀⠀⠂⣈⠀⠂⠀⠀⠀⠀⠀⠀⠢⠀⠘⠀⡀⠄⠋⠢⢠⠸⡦⠜⠀⠀⠀⠐⠃⠄⠀⠀⠈⣡⠆⢔⠀⠀⠂
⠀⠀⠀⡾⠁⠀⣴⡏⠐⠂⢤⣤⣀⣂⡙⠀⠴⣤⡈⠠⢖⠚⡠⠅⢂⠄⠀⠂⠨⠄⠄⣠⠔⢾⡇⢴⠂⠐
⠀⠀⠀⠀⢶⣤⣿⡇⠀⠀⠈⠛⢻⠋⠀⠄⠀⡀⠈⣂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠒⢺⠇
⠀⠀⠀⢀⣰⣿⡏⠀⡓⠊⠐⢀⣈⠀⠠⠎⠀⠀⠨⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠐⠀⠄⣁⣴⡿
⠀⠀⠀⠈⡿⠋⠀⠁⠀⠀⠀⢀⠉⢠⡄⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣀⠀⠀⠀⠀⠀⠙⠂⠰⢿⣽⠇
⠀⠀⠀⢷⠔⠀⢀⣤⣶⡎⣤⠜⠁⠀⢀⠄⣀⡂⣀⠀⠀⠒⠀⠀⠀⡀⠀⢀⠉⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠨⠄⠀⢸⣏⣌⣁⠙⢶⠀⠀⢸⠀⢇⠀⠀⠀⣠⠧⠀⠀⠐⢃⠀⠘⠠⣁⡀
⠀⠀⠐⠁⠀⠀⡀⠉⣱⠏⠀⠰⠀⠀⠈⡇⠈⠃⠌⠁⣀⠀⠀⠀⠐⣠⢦⡤⠚⠻⢟⡀⠖⡤⠀⠀⡠
⠀⠀⠀⠀⠐⢾⣶⣴⣿⠁⠀⠀⠀⣤⣴⣷⣄⡀⠀⠀⠅⠀⢂⣠⠶⢹⣥⣂⢐⡘⠃⠁⠾⠷⢂⣠⡀
⠀⠀⠀⠶⠀⠀⠻⣿⣃⣀⣀⣀⠈⢨⠟⠉⠉⠍⠈⠀⠸⢂⠯⠀⠩⠀⡟⠟⠅⠈⢢⠲⢟⠛⠋⠉⠀⣠⠀⡀
⠀⠀⠐⠀⢰⠀⠀⠈⡿⠏⠛⠉⠀⢀⡀⠀⡘⢆⠛⠠⠠⢹⣥⢤⠴⣎⠁⠈⢃⡔⠁⠁⡀⣘⡂⣴⡞⠛
⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⣰⣾⣿⣗⠀⠄⠈⠦⠤⠀⣸⢕⠥⡊⠀⠑⡤⠋⣀⣤⠅⣬⣭⠃⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡈⡻⠷⠠⣤⣤⣤⣾⡃⠁⢀⠄⠠⡀⡏⠀⠀⠋⠀⠈⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠛⠛⠉⠢⠁⡡⠠⢴⣶⣶⡞
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣉⣁⣀⣀⡀⠀⠐⢶⣶⣶⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣄⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠩⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢿⣿⣷⣦⣀⠉⠪⡳⣤⡈⠳⣕⢄⠙⢿⣧⡀⠳⡱⡱⠱⢱⡱⢻⡀⠀⠇⡇⠘⡇⠀⢸⠀⠸⠈⢀⡟⢌⣎⠎⢎⢎⠞⢀⣼⡽⡿⡫⣪⠞⢁⣴⢞⠕⠋⣀⣴⣾⡿
⣄⡀⠉⠙⠿⣗⣤⡀⠙⠝⢦⡈⠢⡑⢀⠙⣟⣄⠑⠄⠀⠁⠳⠈⢇⠀⢠⢻⠀⡇⠀⠘⠀⡆⠀⣸⠁⠜⠈⠀⠠⠊⡠⣻⠫⠊⠈⠜⢁⡴⠫⠃⢁⣤⣺⠿⠋⣁⣴
⢈⠙⠳⢤⣀⠈⠙⠪⣖⠀⠀⠙⠲⣄⠀⠀⠈⠻⡢⡈⠀⠀⠀⠁⠘⡀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠃⠘⠀⠀⠀⢁⢔⠝⠁⠀⠀⣠⠴⠋⠀⠠⣲⠕⠋⠁⣀⣬⣾⡿
⢄⡀⠀⠀⠈⠛⠦⣄⡀⠉⠂⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠐⠈⠀⡠⠔⠪⠛⠋⠁⢀
⣦⣌⣉⠂⠀⠀⠀⠀⠉⠒⠤⡀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⣈⣥
⠀⠀⠉⠙⠓⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠚⠉⣁⣠
⠷⠶⢶⣤⣄⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⣚⣋⡭⠥
⣉⣀⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠈⠁⠀⠀⢀⣀
⠓⠶⠶⠭⠥⠄⠂⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠉⠁
⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤
⠶⠶⢶⣒⡒⠬⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⣀⣀
⠉⠉⠀⠀⠀⢀⡀⠀⣀⣀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉
⣶⣖⣾⣿⡯⠷⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢶
⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢄⡀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠋⠁⠀⠀⢀⡤⠖⠋⠀⡀⠄⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣦⡀⠀⠀⠢⢄⠀⠀⠂⢄⡀⢀
⢁⣠⣴⡚⠉⢀⣠⢖⠏⠂⠀⣠⠖⠋⠀⠀⢀⣴⠕⠀⠀⠀⠀⡄⢠⠁⠀⢸⠀⠀⠀⠀⡃⠀⠈⡄⢁⠀⠀⠀⠀⠨⣦⡀⠀⠈⠻⣶⣄⠀⠀⠹⡲⣄⡀⠈⢓⠯⣷
⡿⠟⢉⣠⣶⡯⠛⢁⢠⣪⠞⢁⠔⡡⠀⣰⡻⠫⡢⠂⠀⠀⡼⢀⡏⠀⠐⣼⡄⡇⠀⢠⣇⠂⠀⢹⡀⢣⠀⠀⠀⢔⠝⣽⣕⢄⠈⢮⡫⣷⣕⡄⡈⠛⢽⣶⣤⡉⠻
⣤⣾⡿⠟⠉⣠⢔⡵⠛⢁⡴⡻⠊⣠⣾⡟⡕⡵⡱⠀⠀⡼⡱⣼⠁⡀⡆⡇⣿⡇⠀⢸⢸⢰⢀⠈⣧⢎⢧⣆⢆⠈⢮⢪⢻⣷⣷⣄⠙⢮⡻⣟⢮⡢⣄⠉⠻⢿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⠿⠿⠿⠿⢿⡿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⢟⣋⠉⠀⠀⠀⠀⠀⠀⡀⠀⠠⣍⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⣾⣿⣿⣿⣏⠀⣈⣭⣤⡟⢧⣤⣤⣀⣀⡀⡀⠈⠁⢽⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣿⣿⠿⠋⠈⠀⠀⠀⠀⠀⢠⡌⠹⣟⣿⣷⣤⣤⣾⣄⡃⢉⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⡄⠸⠟⢿⣿⣿⣿⣷⣿⣼⣿⡿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠁⠀⣈⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣼⣋⣴⣿⣿⣿⣷⣟⣃⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⣰⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠜⢓⣱⢿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠃⠀⠀⣀⠀⠀⠋⠀⠀⠘⠂⠀⠀⠀⠀⠈⠀⠀⠤⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⡀⢺⣿⠄⠀⣼⠏⠈⠀⠀⠐⠇⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⣠⣴⣿⡟⠉⢻⣿⣿⣿⣿⢿⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⡆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢠⣾⣿⣿⣿⣿⣶⣼⣿⣿⣿⣟⣿⣿⣧⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠰⣤⠾⣿⡿⠀⠁⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣾⣿⣿⣿⣃⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢶⣬⡆⣿⣀⣤⣶⣾⣿⣷⣦⠀⢤⣶⣶⣾⣾⣿⣧⣤⣄⣠⡄⡻⣿⣿⣿⣿⣿⣣⣿⢳⣿⠟⣠⣾⣿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣿⣿⣿⣿⣿⣿⣿⠟⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣸⣿⣿⣿⣿⣿⣧⣿⢇⣼⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⣾⣿⣿⣿⣿⣯⣶⣄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣏⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢾⣿⡏⢿⣿⡿⠟⣿⣿⣿⣿⡟⢧⣿⣿⣿⣿⣿⣿⣿⣿⠏⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⣤⣤⠀⣤⣿⣿⣿⣿⣷⣴⢃⠀⣉⣉⣙⣛⡉⢁⢂⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣽⣿⣿⣿⣽⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⣿⠇⠘⠛⠃⢿⠿⠃⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡤⠀⠀⠒⠀⠐⠾⣷⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⣠⠀⡀⣿⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⠟⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣁⣈⣀⣹⢙⡟⣿⣽⣯⣿⣿⣿⣿⣿⣿⣿⣿⡋⠉⠿⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠟⠛⢹⠁⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠸⠆⠀⢠⣤⣤⣤⡄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⠿⠛⢏⠀⠠⠀⠀⠐⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠛⠀⠀⠢⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢰⣮⡥⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⢖⡎⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⣳⡚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡮⠭⢍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠭⠭⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣭⣭⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀⠩⠭⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠔⣛⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠬⠽⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠒⣎⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡶⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠭⣟⡉

⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣼⣷⣭⣿⣷⣵⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⠿⠿⠛⠛⠛⠻⠿⢿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⢸⣿⠀⢀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠀⢀⣄⠀⣿⡇⢀⣀⠀⣶⡆⢸⣿⠀⣿⡇⢀⣄⠀⢸⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⢸⣿⠀⣿⡇⢸⣿⠀⣿⡇⢸⣿⠀⣿⡇⢸⣿⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⢸⣿⠀⣿⡇⢸⣿⠀⣿⡇⢸⣿⠀⣿⡇⢸⣿⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠀⠘⠛⠀⣿⡇⠘⠛⠀⢿⠇⢸⣿⠀⣿⡇⠘⠛⠀⢸⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⢸⣿⠀⠉⠁⠀⠀⠀⢸⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠘⠻⠛⠉⠉




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⡛⣛⣛⡛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠤⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⢀⠀⠆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠛⠁⠈⠛⢛⠂⢀⡀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⢀⣠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠶⣄⠀⠀⠀⠈⢧⣤⠾⠛⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠸⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠀⠈
⠀⠀⢀⠀⠀⠀⠀⠀⢰⠞⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠈⠀⠀⠄⠀⠈⠀⠀⠘⠋⠉⠉⠩⠿⠶⠦⠤⢤⣀⣀⣀⣴⣄⡀⠀⠀⡀⠀⠀⠘⠙⡄⣠⣴⠃⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣻⣿⣦⡀⠀⢄⠀⠀⠀⠀⢠⣿⣬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⠀⡀⠀⠀⠀⡎⠁⠀⠈⠛⢷⣦⣤⠦⣀⣠⠄⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠊⠀⠆⠀⠺⠀⠀⠀⠀⠀⠂⢹⠃⠀⢸⡇⠁⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⠀⠀⠀⠀⠀⢠⠎⠀⠀⣸⡇⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⢈⠢⠐⢦⣀⡀⢀⡠⠊⠀⠀⣠⡟⠻⠿⠛⠛⠁⠒⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡞⠀⢠⠀⠀⡰⠛⠉⠉⠉⠀⠀⢀⣶⣟⠀⠀⠀⠀⠠⠀⠀⢰⠀⠀⠀⠀⠀⠀⣀⠀⠀⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣿⠀⢀⣧⡀⣀⡀⣀⣠⠤⠚⠉⠉⠀⠀⠀⠀⠰⠷⠀⠘⠣⠄⠀⠀⡴⠋⠀⠀⠀⠊⣦⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⢿⣄⢘⠋⠉⠉⠉⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠐⠊⠀⠀⠈⠠⠖⢶⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢁⣠⠴⠚⠋⠀⠀⠀⠀⠀⠀⠴⠀⣾⠐⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠌⠅⠀⠀⠀⢀⣀⠶⠀⠐⡁⢺⠑⠻⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠛⢋⣅⠀⡛⠈⠁⣘⡦⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⣠⢔⡎⢹⣇⡤⠔⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠗⠀⠐⠀⠀⠀⣿⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⢾⣅⣈⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠠⠆⠀⠀⢀⠈⠃⠀⠈⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄⠀⠀⠀⠀⠈⠠⠀⠀⣀⣀⣀⣀⣤⠶⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠉⠩⠉⠁⢘⠑⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢠⡔⠁⠀⣤⠀⠀⠀⠀⠀⢄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⡀⢀⡼⠷⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠒⠒⠒⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⠀⠀⠙⢷⠀⠀⠀⠀⠀⠀⠀⠀⢷⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⣀⣤⣶⠾⠿⣷⣶⣾⣇⣀⣤⣴⡶⠶⠶⠶⠶⠶⢶⣤⣄⣀
⠀⠀⠀⢠⣾⠟⢉⣠⣴⣶⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣶⣄
⠀⠀⠀⣿⣥⣶⣿⣿⣿⣿⣿⣿⡿⠁⣀⣤⣶⣶⡿⠿⠿⠿⢿⣶⣶⣤⣀⠀⠀⠙⢷⣦
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠿⠋⣠⣾⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣷⣄⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠉⣹⡟⠉⠀⠀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀⠀⠈⢿⡄
⢀⣀⡏⠂⠀⢰⡟⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠈⣿⡄
⠿⠿⠃⠀⠀⣾⠃⠀⠀⢀⣿⡇⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠸⣿⡆⠀⠀⢸⣧⣴⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣧⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣦⣤⣿⡇⠀⠀⢸⣿⡿⣿
⠀⠀⠀⠀⠀⣿⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠀⠀⣿⣶⣤⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡄⠀⣾⠏⠀⣿⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⢻⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⢹⢹⣿⣷⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⢸⡟⠀⢠⠇⠀⠀⢀⢜⣥
⠀⠀⠀⠀⠀⠘⣿⡀⠸⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣾⡾⠟⠋⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⢸⣇⣴⠋⠀⠀⠀⠘⢌⣡
⠀⠀⠀⠀⠀⠀⠘⣷⡀⠙⠻⢿⣿⣿⣿⣿⠀⠀⠙⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⢋⣴⠿⣿⠁⠀⠀⠀⠀⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⣰⡟⠁⠀⣹⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠀⢀⣾⡿⣿⠛⠻⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⣀⡀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠾⠋⢹⣇⠀⠘⣿⠀⢹⡀⠀⠀⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠁⢠⡄⠀⠀⢿⣄⠀⠈⠁⠸⣇⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠀⠀⠀⠀⠙⠿⢶⣦⣤⣿⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⢾⢳⡄⠀⠀⠀⠀⣠⣤⣄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠋⠀⠀⠀⠀⢰⣿⠿⠟⣸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠴⠶⠶⠶⠶⠶⢤⣤⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⣛⣛⣋⣩⣍⣉⣙⣛⠻⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣤⣾⣿⣿⣷⣿⣿⣴⣶⣶⣶⣤⣤⡤⢟⣿⡆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠉⠀⠀⠈⠉⠒⠛⠋⠉⠛⠛⠛⢛⣡⠄⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠹⢿⣛⣛⣛⣛⣛⣛⣛⣛⣉⣭⣴⣶⣿⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣦⠤⠤⠤⠤⣤⣤⠤⠤⠶⠖⠛⠉⠁⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠐⠢⠬⢭⣭⣭⣭⣭⣭⣭⣿⠿⣿⡏⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣽⣛⣓⠒⠶⠶⠒⠒⠒⠂⣀⡀⠐⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢑⠒⠶⠶⠶⠶⠶⠶⠾⢟⣻⣧⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣉⣉⣉⣉⣉⣉⣡⠤⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢭⣛⣛⣛⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠦⠭⠭⠭⠥⠄⠀⠀⠀⣽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡿⢿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⣒⠒⠂⠀⠀⢀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣽⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠒⠒⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡑⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣓⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠒⠒⠒⠓⠒⠒⠒⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⡀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⠉⠙⠋⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⢀⡤⠖⠛⠛⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⢀⣀⠀⢀⡴⠚⠉⣺⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⢀⡤⠞⠁⠀⠉⠺⠀⢀⡜⠁⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⢀⡿⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣹⡄⢧⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⣧
⢸⡄⠀⠀⠀⠀⠈⠀⠀⠸⣄⠘⠿⠿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡗⠲⢤⣄⠀⠀⠀⢹
⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⣀⣤⠊⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠈⠙⢦⡀⢸
⠀⠀⠙⠒⢤⣄⠂⠀⠀⠀⠠⣇⠀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷⣦⣄⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠙⣼
⠀⠀⠀⠀⠀⠈⠣⣄⣠⢤⡀⠈⠉⠉⣀⠀⠀⠀⠀⠀⠀⠀⡰⢉⡳⣦⠀⠈⢻⠳⣄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠋⢹⡄⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⢠⡏⠀⠈⢧⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠈⢧⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⡀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠘⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢳⡀⠀⢀⠀⢴⣳⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠘⣧⠀⠘⡏⠒⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠃⠀⠙⠇⠀⠸⡄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⠤⠤⠤⠄⠒⠒⠊⠉⠁⠈⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⡀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⢀⠀⠀⡀⡀⠀⡀⢀⠀⢀⡀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠒⠊⠁

⡀⠀⠀⠀⣠⠤⡾⠿⠻⠿⠧⠤⠤⢤⣀⣀⡖
⢱⣖⠋⠻⣅⠐⠈⠀⢠⠔⠒⠒⠒⠦⠤⠤⣉⣉⠓⠲⠤⢬⣑⡒⠢⠄⣀
⢸⡉⠳⣄⡈⠓⠦⣄⣀⠀⠀⠀⠈⠉⠒⠒⠒⠬⠍⣛⠒⠦⣤⣉⡙⠒⠦⣄⣀⠀⠀⠀⡀
⠀⠑⢦⣀⠉⠓⠦⣄⡈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠈⠉⠓⠲⠤⣌⡙⠒⠤⣀⠁
⠠⣀⠎⠉⠙⠒⠤⢔⡉⠛⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⣉⠒⢤⡀⠀⠀⠠
⠀⠈⢷⠒⠤⢀⡀⠀⠀⠉⠐⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣈⡐⠒⠥⠤⣒⣂⡈⠙⣦⠈⠁⠸⡄
⠀⠀⢈⠑⢦⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠲⠦⠤⠭⠭⠥⠤⠤⠤⢳⡀
⠀⠀⡀⢠⡆⠉⠙⠲⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠒⠠⠤⢄⣀⣀⡀⠀⠀⠀⢂⣠⠾⠁
⠀⠀⠱⣧⣀⠁⠂⠄⠀⠀⠀⠉⠉⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⢤⣄⣀⣉⣉⣙⣏⣉⠄
⠀⠀⠀⠘⢯⡙⠓⠶⠤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⣄⣀⣀⣀⣀⣀⣉⣭⠽⠋
⠀⠀⢈⠀⠀⠉⠳⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀⣀⣀⣀⣈⣉⣉⣩⡧⠄
⠀⠀⢺⡀⠘⣦⡃⠀⠈⠉⠛⠓⠶⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣉⣉⣭⢽⡋
⠀⠀⠀⠙⢦⣘⡝⠓⠶⠦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠉⠉⠉⢉⣡⠴⣺⣃⠀⣴
⠀⠀⠀⠀⠀⠈⠙⢷⡶⠤⢤⣄⣀⣀⢠⡄⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠚⢛⣉⡤⠞⠁⢈⣽⣇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣠⠴⠋⣠⣯⣀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣄⡈⠉⠉⠉⠉⠉⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠈⢁⡴⠚⠁⠀⣨⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢯⣛⣛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢚⡥⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⠛⢻⡛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠊⢀⡀⠂⠠⠴⢛⡷⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⢾⣧⣄⣀⡀⠀⠀⢀⣀⣀⡀⠀⠀⠁⠀⠰⠚⢉⣠⡴⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣻⠛⠉⠉⠀⠀⠀⠠⠄⠲⠶⠛⠋⢁⣴⣣⣄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣄⣀⣀⡰⠄⠀⠀⠀⢀⡀⣠⡭⠛⠁⣈⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢉⡹⠋⠉⠁⠀⠀⠀⠀⠉⢀⠠⠖⠛⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢶⣉⡉⠁⠀⠀⡀⠀⠀⠠⠖⠛⢧⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⠉⠁⠀⠀⠀⠁⠀⠀⢀⣠⣤⣴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠛⠒⡀⠀⠀⠀⢈⣩⣥⠖⠋⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣹⣗⡒⠠⠁⠀⠀⠀⢀⣠⣬⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢭⣈⠉⠀⠀⠐⣀⣐⠣⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠉⠀⠈⠙⠓⠲⠦⢼⡧⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡌⠷⠀⠀⠀⠀⣀⢳⡖⠚⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠄⠈⠁⠀⠐⣶⠂⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠭⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⣑⠶⠃⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣯⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣖⣿⣿⣿⣿⣗⣲⡶⠦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢶⣯⡷⢿⣛⣭⣽⣿⣿⣿⣿⣿⣿⣿⣶⣬⣉⠛⠲⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣯⣿⢟⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣄⡀⠀⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⡾⣫⣾⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠉⠻⣝⣿⣄⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣵⢟⣵⣿⣿⣿⣿⡿⣫⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣽⢫⣾⣿⣿⣿⣿⢏⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣻⣷⡀⢀⠀⠀⢀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣾⢣⣿⣿⣿⣿⣿⣣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⢱⣞⡵⣫⣆⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⢏⣿⣿⣿⣿⣿⣱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣇⢿⣿⣿⣿⣆⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⣾⣿⣿⣿⣿⢧⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣸⣿⣿⣿⣿⡾⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⢳⣿⣿⣿⣿⡏⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣟⣿⣿⣿⣿⣧⢿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣼⣿⣿⣿⣿⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⣿⣟⣛⣟⣻⣸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣿⣿⣿⣿⡿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡜⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣿⣿⣿⣿⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⣷⣿⣲⣿⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢡⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢿⣿⣿⣿⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡟⣸
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡜⠀⠉⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⣷⣀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠉⢁⢀⣠⣤⠭⠿⡿⣿⣿⣿⣿⣿⣿⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄⢀⣤⣤⣶⢞⣿⣿⡿⠟⠗⠿⠙⡦⣄⠀⠀⠀⣠⡾⠁⠠⣤⣰⡾⠋⢁⣠⣤⡾⣼⣿⣿⣿⣿⣿⣿⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⢿⣿⢻⣿⡿⠁⢠⣤⣶⣤⣄⡀⠋⢷⡀⣼⠋⢡⣀⣀⡜⠉⢀⣰⣿⣿⡟⢙⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣿⡏⣸⡏⣿⣿⡇⢀⣿⣿⣿⣿⣿⣿⣧⣼⣿⢃⣤⣤⣽⡏⠀⣠⣿⣽⣿⠉⣀⣼⣿⣿⣿⣋⣉⣿⣿⣍⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢠⣿⠀⣿⡇⣿⣿⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⣿⣿⠏⠀⣼⣿⣿⠟⠁⣴⣿⣿⣻⣿⣿⣿⣿⣿⣿⣷⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⢀⣿⡇⣿⣿⣧⠀⢿⣿⣿⣿⣿⣿⣿⣿⢈⣻⣿⣿⣤⣼⣿⣿⠋⢀⣾⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣼⣿⣸⣿⣿⡸⣿⣿⣦⡈⢿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⡇⠀⣼⣿⣿⣾⣴⣿⣿⣿⣿⣿⣿⣿⣿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣿⣿⣿⣿⣿⣧⠙⣿⣿⣷⣤⠛⢿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣠⣿⣿⣿⣿⡏⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢻⣿⣿⣯⣿⣿⣷⡟⢿⣿⣿⣿⣶⣯⣽⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⣽⡟⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣿⠟⢿⣦⣘⠿⢿⣿⣿⣿⣿⣿⣷⡙⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣔⣼⠟⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣏⠛⢿⣷⣄⡙⠻⢿⣶⣭⣽⣯⣷⠾⠋⠻⣿⣷⡈⢻⣿⣷⡌⢿⣿⣿⣿⣿⣿⣿⣿⡿⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠿⠿⠿⠒⠒⠚⠋⠉⠀⠀⠀⠀⠀⠉⠙⠲⢽⣿⣿⣶⣿⣿⣿⣿⣿⣿⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣤⣀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⠟⠀⢿⡇⠀⠀⠀⠀⠠⣤⣤⣶⣦⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣟⣋⠁⠀⠀⣿⡇⠀⠀⠀⠘⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠋⠀⠉⠛⢛⡛⠻⠳⣦⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠛⠋⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠿⣿⡉⠆⢀⣠⡶⢛⡥⣌⡁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣻⣿⣤⣾⡟⡝⠉⠀⠀⠹⣆⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣐⠆⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿⡿⠀⠀⠀⢀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢷⣄⠀⣸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟⣡⠇⠀⠀⢸⡄⠀⡿⡇⠀⢠⣶⡅⠀⠀⠀⠀⡀⠀⢠⡉⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⠋⠀⠀⠀⠸⣧⠘⠁⠀⠀⣹⡿⠋⠀⠐⠙⠿⣿⣿⣶⣤⣶⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠿⣧⣀⡀⠀⣀⡤⠦⠄⠒⠊⠁⣀⣀⣤⣤⣤⣀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠹⠿⠟⠁⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣷⣮⣤⣼⣿⣿⡯⠉⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⡶⠿⣄⣀⠀⡀⠀⠀⠀⠀⢀⣴⣗⠄⣀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠉⠻⣤⣈⠙⣿⠛⣻⣿⠇⠲⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⣷⣾⣿⣿⠀⠀⠈⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠈⢿⣿⣿⣇⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢸⣿⡉⠻⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣶⣇⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⣤⣶⣾⣿⣿⣶⡀⠀⠀⠀⢸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠟⠻⠿⠿⠿⠿⠇⠀⠀⠀⠈⠛⠿⠃⠀⠀⢠⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠻⢿⣿⣿⣶⣤⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣷⣼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢠⣶⣿⠯⠷⠦⠤⢤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡿⠻⣶⣮⣽⣿⣷⣶⣶⣭⠍⠛⠛⠿⠷⣶⣤⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⡈⠉⠛⠿⠿⣿⣿⣷⣶⣶⣤⣤⣤⣀⣈⣉⣿⡦⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⡀⠀⠀⠀⠀⠲⢤⣤⣬⣉⣉⣉⣉⣉⣉⣻⣿⣿⣷⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣄⡀⠀⠀⠀⣤⣤⣤⣭⣭⣭⣿⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣉⠀⠀⠀⣻⣿⣿⣷⣾⣿⣿⣿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣉⠁⠀⠀⣹⣿⣿⣿⣿⡷⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣉⡁⠀⠀⢤⣭⣭⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡍⠀⠀⣈⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣗⣂⠀⠈⠭⣭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣥⣤⡄⠘⢻⣿⣿⣤⣄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⠙⠿⢿⣥⠀⠸⢿⣿⣿⣆⡈⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⣿⣗⡘⢿⣿⣿⣿⣤⡁⠀⠀⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠂⠀⢰⣦⠈⠀⠀⠸⠿⣦⠉⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠐⠀⠀⠙⠋⠀⡀⠀⢴⡄⠉⣧⢸⣿⣿⣿⠁⠸⣿⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠈⠀⠀⠀⢈⡀⢀⡙⠀⣿⣾⣿⣿⠏⠁⠀⠀⡀⠈⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⢀⠀⠀⢤⣤⣿⣿⣷⢰⣿⣿⡿⣣⣀⠀⠀⠀⠡⡄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠄⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⠶⠶⠶⢶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠋⣉⣁⣤⣤⣤⣶⣶⣶⣦⣤⣤⣈⡉⠛⠷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⣉⣤⣴⡾⠿⠛⣋⣉⣭⣥⣤⣤⣤⣤⣬⣍⡙⢿⣦⡀⠈⠃
⠀⠀⠀⠀⠀⠀⣠⡿⢋⣴⣾⣿⣿⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣧⠀⢀
⠀⠀⠀⠀⠀⢠⡏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣋⣴⣿⢿⡏⢀⡾
⠀⠀⠀⠀⠀⠈⠀⠀⠻⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠟⣛⣋⣭⣵⣶⣿⡿⢛⣵⣿⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠰⣄⠀⠈⠻⣯⣝⣛⠻⠿⠿⠿⣿⣿⠿⠿⠿⠛⢋⣡⣴⣿⣿⠟⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣸⣻⣿⣿⣷⣶⣶⣤⣤⣤⣶⣶⣾⣿⣿⣿⡟⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣋⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⠦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠁⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⡤⢤⣿⣿⣿⣿⣿⠥⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣰⣿⣿⠟⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠟⠓⠒⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳

⠀⠀⠀⠀⡅⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠢⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⢴⣤⡈⡇⢸⡇⣿⠋⠁
⠀⠀⠀⠀⠀⠠⡀⠐⠀⠈⠃⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⢍⡉⣈⢹⡏⢈⡷⢸⡄⠉⠳⠀⠃⠈⠓
⠀⠀⠀⠀⡄⠠⠄⠀⠀⠀⡉⢈⡀⠐⠀⠰⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠀⠉⠉⠀⠀⢄⠀⡄⢠⠀⠆⠲⠀⠋
⠀⠀⠀⠀⣵⣶⡀⠁⠀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠀⠀⠀⠀⠀⠂⠆⠐⠆⠀⠃⠉⠀⠀⠀⠈⠀⢲⠄
⠀⠀⠀⠀⣿⠿⣢⣤⣤⣴⡆⠁⠀⠀⡀⠠⠀⠄⠀⡀⠀⢈⠀⠉⢕⠃⠀⠆⠀⡤⢶⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⠷⢿⡀⢀⠘⠃⠀⣄⠂⠄⢀⣀⣦⢤⣤⣦⣤⣤⣀⣠⡁⢸⡇⠀⠀⢀⠜⣹⣷⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠉⠀⠀⠴⠆⠀⣀⠗⠉⣁⣤⣶⣿⣿⠟⣫⡍⠉⠉⡁⠀⣿⢲⡴⠋⣰⠟⠁
⠀⠀⠀⠀⣿⢿⣙⢻⣿⣶⠂⠂⠀⣠⠔⠋⠁⣠⣾⣿⣿⣿⣿⣿⠉⢡⣵⠞⠁⡵⠀⠘⠋⠀⡔⠋⠓⢦⣠⣤⠄
⠀⠀⠀⠀⣏⣸⣵⣾⠁⠈⢀⣴⣿⠟⢠⡾⠋⡤⢹⣿⣿⡿⠋⢨⣿⠿⠥⠤⠜⠁⠀⠀⠀⢈⣭⣤⠤⠴⠾⠋
⠀⠀⠀⠀⣿⣻⠉⠉⠀⣴⣿⣿⢋⣤⣿⣄⢀⡴⠛⡿⠋⠣⢶⡀⠙⠿⢤⡤⣶⣀⢤⣀⣀⣀⠙⢤⠠⢄⡀⠀⠀⢀
⠀⠀⠀⠀⢓⡏⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣟⣀⠄⠁⠀⢰⡟⠉⣀⡀⠀⠀⠻⠀⢀⣚⠟⢃⣙⣢⣖⠀⠉⠳⣄⠀⠃
⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⢏⣿⣿⣿⣿⣿⣿⢣⠀⠀⡈⣿⠃⠀⠁⠀⢀⠀⠀⠀⣿⠳⢶⣿⣿⣿⣿⣿⣷⣦⣈
⠀⠀⠀⠀⣿⠃⠀⠘⠛⠛⠘⢿⣿⣿⣿⡿⣁⢀⣼⠆⠸⣿⣷⣀⠀⠀⠀⠒⠀⢰⣯⣄⣀⡀⠉⠃⠀⠀⠀⠈⠙⠷⣄⣠
⠀⠀⠀⠀⢿⡄⠀⠀⠀⠆⠀⢨⠿⠟⠻⠻⡷⠿⣿⣦⡀⠈⢤⡽⠀⠀⠀⡄⢠⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⢰⣿⡿⢿⣿⣦
⠀⠀⠀⠀⣿⣇⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠿⣦⡀⢈⠀⠁⠀⠀⣿⣿⣿⣯⡉⠛⠻⢿⣿⣧⡀⠀⠀⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠀⣿⡼⡀⠀⠛⣰⡏⢀⢠⡔⠀⠀⠀⠀⠰⢋⢠⠈⠻⣦⡳⠄⠀⣠⣿⣿⠛⠻⢷⡄⠀⠀⠉⠛⣷⡄⢠⣿⣿⣻⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣷⡄⣰⡿⠗⠚⠃⠐⠀⠀⢀⣠⣧⡌⣰⣦⡶⠚⠿⣶⣎⠁⠈⠻⠇⠀⠀⠉⠀⠀⠀⠀⠘⣇⢸⣿⣿⣻⣿⣿⠇
⠀⠀⠀⠀⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⡉⠘⠛⠛⠉⠀⠀⣼⠃⠉⠑⢠⠐⢶⣶⢸⣷⢀⠀⠀⠀⠀⠃⣿⣿⣟⣿⣿⣿
⠀⠀⠀⠀⠈⢻⣿⣿⣿⣦⣤⣤⣤⣤⣐⠖⢰⡆⣐⠀⣤⡀⢠⣼⣷⣶⣶⠇⠀⣤⣾⣿⡇⣿⣠⣦⠒⡀⠐⣸⣿⣿⣿⣿⣿⡇⣀⡀
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣠⡮⣙⣿⣯⣯⣿⣿⣾⣿⣿⣿⣿⣠⣶⡟⠙⠛⠈⣿⣷⣿⣿⣷⣞⣿⣿⣿⣹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠙⢿⣿⣿⣿⣿⣿⣤⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡀⣦⣼⢠⣿⣿⣿⠿⢿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⣿⣿⣿⠟⠁⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⣦⣉⠛⠡⠤⣤⣶⣧⣴⠀⣀⣤⣄⣀⢤⠈⠙⢻⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⡀⢠⡀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣶⣾⣿⣿⣶⣤⣈⠻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⠤⣌⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⢰⠁⣘⠄⣀⠛⠄⡀⠂⠌⠀⢀⣀⣀⣀⠀⠤⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠌⡘⢀⡏⠔⠂⠚⠂⠉⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠌⡰⠤⠌⠴⠃⣘⠴⡄⠀⡤⣤⠒⣶⣒⢆⣜⣡⣌⡹⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠐⢠⠁⡊⠀⠀⠀⠀⠀⣯⣱⡇⠀⠁⢉⣉⣉⢉⡁⢤⢄⠀⢤⠐⡄
⠀⠀⠀⠀⠀⠀⠀⣀⠂⠌⣑⠦⠄⠀⠒⠚⢶⢯⡇⠀⣭⡿⣼⣛⠾⣽⢃⠎⣐⠢⠌⡁⠀⠉
⠀⠀⠀⠀⠀⠈⡐⠠⡎⠁⠀⣀⡤⣴⡖⣶⣸⢿⡇⠀⣿⣽⣳⣯⣿⢣⣏⡲⢠⠋⠴⠅⠀⡦⢀
⠀⠀⠀⠀⠀⠀⢀⠂⠴⠀⠀⠯⠓⣋⣩⣭⢯⣿⡇⠀⣿⣿⣟⡾⣭⣛⡾⣱⠃⡌⣑⠇⠀⡵
⠀⠀⠀⠀⠀⡀⠂⠌⡰⢇⠀⢰⢏⡟⣼⣟⡿⣽⡇⠀⣿⣿⣿⣽⢧⣟⡳⢧⣝⢰⣁⠄⠀⣇⠂
⠀⠀⠀⠀⠀⢀⠀⠠⢁⠺⡀⠀⢿⡽⠗⠛⠛⠋⠁⠀⣿⣟⡿⣯⣟⡶⠋⠁⠀⠀⠀⠀⠀⡏⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠂⡘⢃⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⡾⣹⠿⣼⠀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠠⠀⠐⠠⢙⡀⠀⡆⠀⠀⠀⠀⠀⢀⡟⡳⢯⡟⠰⢦⡀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣱⠀⠸⡤⢤⣀⡠⢴⠛⠤⢁⠒⡬⢁⠆⡌⠃⠎⡐⠈⠉⠐
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠉⠀⠀⠀⢇⠂⠬⣑⠂⡌⠑⠎⡜⠠⢌⡐⠠⠉⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⢸⡌⠠⢀⡘⠠⢁⠒⠠⠁⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⡜⠀⠁⠀⠀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⢀⣾⣄⣀⡀⠀⠀⢀⣀⣀⣀⣠⣿⣧⠀⠀⠀⠀⠠⠾⣿⠧
⠀⠀⠀⠘⢿⣿⣿⣏⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣦⣄⣀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣄⣤⡄⠀⠀⣺⣿⠆
⠀⠀⠀⠀⠘⠟⠉⠙⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣇
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠠⠿⠟⠛⠻⣿⣿⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⠋⠉⠁⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠘⠻⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡶⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⢠⣿⣦⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣤⣴⣾⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠰⣶⣶⣾⣿⣿⣿⣶⣶⡤⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣾⣿⠃⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⣿⠟⠋⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠿⠿⣿⣿⠀⠀⠀⠀⠀⠀⠐⠛⠿⢿⣿⣿⣿⠿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣾⣦
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃
⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣰⣧⡀
⠀⠹⢿⡏⠀⠀⠀⠨⢿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠶⣾⣿⣿⣿⡿⠃⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⡀⣠⣾⣿⣷⠀⠀⠀⠀⠀⢸⣿⠿⣿⣇⠀⠀⠰⣶⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠾⠿⢿
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⡿⠉⠁⠀⠀⠀⠀⠀⠀⠰⠿⠿⠛⠛⠻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀
⠀⠀⠀⠀⢸⣿⣿⣦⣠⣤⣤⡄⠀⠀⠀⣤⣤⣿⣿⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣦⣶⣶⣶⠆
⢀⣤⣴⣶⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠈⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⢀
⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠛⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⡇⠀⠀⢲⣶⣿⣤⡀
⠀⠀⠀⠀⣿⣿⣿⠿⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⣠⡄⠀⠀⠀⣀⡀⠀⠀⠀⢸⣿⣿⣿⠿⠿⣿⣿⡀⠀⠸⠛⢿⠁
⠀⠀⠀⠀⡿⠋⠁⠀⠀⠀⠈⠀⢀⣴⡀⠀⠀⠀⠀⣠⣿⣿⣿⣧⡀⠀⠘⠟⠃⠀⠀⠀⠸⡿⠋⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣾⣿⣧⣀⡀⠀⠈⠉⠙⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⣆
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣾⣿⣿⣷⣶⡶⠀⠀⠼⣿⠇
⠀⠀⠀⠀⠰⣾⣿⡶⠄⠀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⣠⣦⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⡏⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠛⠉⠀⠈⠙⠇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡷⠆⠀⠀⠀⢀⣿⠟⠛⠻⠿⠀⠀⣠⣤⣿⣦⣀⡀
⠀⠀⠀⠀⢀⣰⣆⡀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠿⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠿⠿⠇⠀⠀⠀⢈⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⠉⠁⠀⠀⠀⡠⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠐⠚⠒⠢⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠢⣄⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢴⠖⠻⠛⢭⣍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠆⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⢀⡤⠀⠤⢄⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡼
⠀⠀⠀⢀⠼⠃⠀⠀⠈⠉⢆⢘⡄⢰⡄⠀⠀⠀⠀⠀⠀⠄⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣔⠋
⠀⠀⠜⠉⠀⠀⠀⠀⠀⠄⠬⣍⡒⠦⣀⡀⠀⠀⠀⠀⠀⠊⠀⠀⠀⢀⠀⠀⢀⣀⣙⡍⢲⠆⠭⠈⠉⠢⡀
⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠒⠒⠋⠉⢀⣠⢤⡀⠀⠀⠀⠀⠀⢖⠒⠢⠽⡄⣡⠇⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠉⠉⠢⣀⡀⠀⠀⠀⠄⡤⠤⠔⠒⠊⠁⠀⠀⠙⠢⣀⠀⠀⠀⠘⠖⣒⠞⡫⠃⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⣀⣤⡀⠈⠣⣄⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇
⠀⠀⠀⡴⠉⠉⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⣠⠔⠋⠁⠀⠈⠓⠤⣀⣀⠴⠦⠤⢦⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⢀⣀⡴⠋⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢰⠁
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡎⠁⠀⠀⠀⢠⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⣠⡴⠃
⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⢣
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠃
⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠐⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠒⠒⠒⠒⠊

⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀
⠀⠀⠀⠀⠀⠀⠃⠀⢀⡤⠴⣶⡶⢾⣿⣿⣿⣿⣿⠿⠛⠛⠛⢻⣟⣻⡿⢶⣶⣦⣄⠀⠀⠼⠃⣀⣀⣀
⠀⠀⠀⢀⣀⡀⠆⠀⢸⣷⣶⣤⣤⣭⣭⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣶⡿⠿⢿⣛⣫⡆⠀⠀⠀⠤⣤⡉⠛⡆
⠀⠀⠀⡏⢡⣾⡛⠀⠈⢻⣿⠿⠿⠿⠿⠭⠭⢭⣭⣭⣭⣭⣠⣤⣿⣋⣉⣉⣩⣭⣵⡶⢀⣀⣤⠤⠾⡛⠀⢤⡀
⠀⠒⡀⠀⠀⢈⡉⠛⠛⠛⢶⣶⣶⣒⣒⡒⠀⠀⠀⠀⠠⣤⣤⣤⣤⣶⣶⣶⣿⣿⣿⡷⠀⠀⠀⠀⢀⣹⡇⠐⠟
⠀⠀⠀⠀⣀⠻⢤⣤⣤⣀⣘⣿⣿⣿⠿⠍⠉⠩⠤⠤⠤⠤⠬⠿⢿⣿⣿⣿⣿⣿⡿⠤⠤⠶⠖⠚⠋⠉
⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠻⣿⣯⣭⣍⣉⣑⠶⠶⠤⠤⠤⠤⣬⣭⣵⣶⣾⡟⢁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣴⠒⣤⡀⠛⣻⣿⣷⠒⢒⣲⣶⣤⣤⣤⣤⣶⣶⣿⣿⣻⣯⠁⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣁⡉⠀⠸⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣭⣉⣛⣿⣿⣿⣷⣖⠛⠛⠋⠀⠈⠡⠶⠿⠞⣛⡋⠀⢀⣉⣷⠘⡦⠀⠀⠀⠀⠰⡿⠉⠞⠁⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠿⣿⣿⣯⢭⠝⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣏⡉⣐⣶⠀⠀⠀⠀⠀⠀⠙⠂⣴⠏⣠⠄⢈⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡗⠂⠈⠿⠿⢿⣿⣒⣶⣶⣖⠛⠛⡻⠿⠿⣟⣛⣿⣿⡁⠀⢀⣀⡀⠀⠀⠀⠘⠁⡴⠋⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠒⠂⠀⠀⠬⠿⠿⠿⠷⣶⣶⣶⣾⣋⣙⡛⠟⠯⡿⣀⠒⠚⠋⢠⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣠⣤⣀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠉⢙⠛⠻⠿⢷⣶⣾⣿⣭⠿⣦⣤⠶⠟⠀⠀⠀⠉⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠘⠉⠀⣹⣆⣤⠤⠴⠖⠀⠀⠀⠀⢤⡘⠦⠤⢤⠀⠰⠾⠿⢿⡿⢿⡧⠀⠀⠪⠅⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠀⣤⠀⠀⠉⣥⠶⠖⠛⠁⠀⣤⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⣷⣿⡏⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣾⣿⡀⠀⢀⣄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡶⠀⠰⠰⣇⣼⣿⣋⠀⠸⠀⠀⠦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠉⠉⠀⠁

⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⡟⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠞⡇
⠀⠀⠀⠀⢤⠈⣄⠑⢄⠀⠀⠀⠀⠀⠀⠀⢀⠐⠤⠁⠀⠀⠀⠈⢆⠀⠀⠀⢀⠎⠀⠀⠀⠀⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡌⢠
⠀⠀⠀⠀⠸⡀⠈⢦⡀⠑⠂⠀⠀⢀⡤⢃⠈⡐⠄⠀⠀⠀⠀⠀⠈⢧⠀⢠⠏⠀⠀⠀⠀⠀⠀⠐⡀⢀⠣⣄⠀⠀⠀⠐⠁⣠⠎⠀⡸
⠀⠀⠀⠀⠀⢇⠀⠀⠑⢄⠀⠀⠀⠈⠇⠀⢀⣤⣴⠶⠶⠀⠀⡀⠀⠀⠱⠁⠀⠀⡀⠈⠰⠶⠶⣤⣄⠀⠀⠛⠀⠀⠀⢀⠔⠁⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠑⠀⠀⠀⠀⠀⢠⠛⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠐⡌⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠆
⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠈⢨⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⢨⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠈⡿⢦⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⢰⣄⡟⠀⢠⠀⠘⣇⣰⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⣠⢾⡍
⠀⠀⠀⠑⡙⢦⡲⢄⡀⠀⣀⣀⡄⠀⠀⠀⠀⢦⡀⠀⠀⠀⠸⢿⠇⠀⣿⡇⠀⢿⠿⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⣄⣀⡀⠀⣀⠴⣢⠚⡑⠁
⠀⠀⠀⠀⠘⣄⠉⠂⢌⣉⠉⠀⠒⠀⠀⠀⠀⠀⠙⢗⠢⠀⣀⢢⣀⡀⢿⠇⣀⣠⡂⠀⠠⢒⠟⠁⠀⠀⠀⠀⠐⠃⠈⢉⣉⠄⠊⢁⡜
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠈⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⣓⡤⢄⣀⡀⠀⠀⠀⠀⣀⣀⠤⣔⡁⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠁⡀⠀⠑⢍⠁⢀⡀⢋⠕⠃⠀⡀⠩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠀⠀⠀⠜⡠⠀⠘⢉⡉⠀⢀⠆⢂⠀⠈⣉⠛⠁⠠⡐⠄⠂⠀⠀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢠⡁⠀⠀⠀⠔⠀⠠⠀⠠⠀⠐⠌⠀⠀⠀⢡⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢀⡟⠀⠀⠳⡀⠂⡴⠣⡄⡤⠳⡄⠂⡰⠃⠀⠘⡇⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠸⠀⠀⠀⠀⠙⡄⠀⠀⢡⠁⠀⠀⠜⠁⢀⠀⠀⠰⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡁⠹⡄⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠘⢀⣤⠏⠀⠀⠀⡼⠀⣥⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢁⡌⠙⠰⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠠⠶⠐⠙⣄⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠴⡄⠀⠀⠀⠀⠘⠀⠀⡠⡀⠀⠰⠀⠀⠀⠀⠀⡴⠞⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⠀⠈⠂⠌⠀⠀⠀⠀⠀⠀⠰⠀⠰⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⠀⠀⢠⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠀⠀⠠⠄⠀⠀⠀⠀⠀⢀⠃⠀⠀⢃⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣠⡆
⠀⠀⠀⣠⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡈⢀⡀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⣁⠈
⠀⠰⠾⢯⡼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠸⠄⣤⠀⡁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡁⢠⡄⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠁⠀⡘⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠻⡛⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡻⢫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠓⡒⠚⢢⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠚⠒⡒⠓⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡀⠀⠄⠀⠸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠸⠀⠀⠄⠀⣚
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⠴⠶⠶⠶⠶⠶⠶⠶⠶⢤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⠞⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠳⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⣡⡴⠶⠶⣦⣌⠻⣦⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣀⣤⣀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠐⠟⠉⣸⣧
⠀⠀⠀⠀⢀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣼⠋⠀⢹⡆⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⣠⡞⠉⢿⡀
⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡏⠀⠀⢸⡇⠀⠀⠀⢹⡄⠀⠀⠀⠀⣰⠏⠀⠀⢸⡇
⠀⢀⣾⠃⣀⣀⣀⣀⣀⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢸⡇⠀⢠⡿⠀⠀⠀⠀⠈⢷⡀⣀⣠⡾⣿⠀⠀⠀⣸⠇
⠀⠘⠛⠛⠋⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⢛⣷⠆⠀⠀⠀⠀⠀⣠⡾⠛⠛⠛⠻⢿⣤⡛⠶⠟⠁⠀⠀⠀⠀⠀⠈⠙⢻⣇⣠⣿⡶⢦⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠰⣏⠀⠀⢀⣠⣤⣀⠈⠙⠳⢦⣤⣀⣀⣀⣠⣤⠶⠶⢿⡟⠁⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠋⠙⣧⡉⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠈⢷⣄⣀⣀⣀⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠈⠙⠶⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣉⣷⣦⣀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣠⣴⠶⠞⠛⠋⠉⠉⣩⡴⠟⠋⠉⠉⢙⣿⠳⠶⣦⣤⣤⣤⣤⣤⠶⣾⡛⠿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⡶⠟⠋⠁⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⢀⣤⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠈⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⢀⣤⣶⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣁⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠻⠷⠾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠹⣏⠉⠻⣶⠶⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⣀⡿⠀⠀⢸⡇⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠙⢷⣄⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠉⠛⠉⠀⠀⢀⣾⠁⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣀⣠⣴⠟⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⡾⠃⠀⢀⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣏⠙⣧⡀⠀⠀⠀⣠⣿⣤⣤⣤⣴⡶⠛⢿⣶⠶⠟⢋⡁⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣜⣃⣠⣴⠞⠋⣰⡏⠀⠀⠘⣷⡀⠈⢿⡷⢿⣿⣥⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⡏⠀⢠⡿⠀⠀⠀⠀⠘⣇⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢿⡇⠀⢸⣧⡀⠀⠀⠀⣠⣿⠀⠀⣿⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢾⣇⣀⣸⣧⢿⡄⠀⣾⣯⣿⣀⣠⣿⠖⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠉⠉⠁⢸⡇⠀⣿⡀⠉⠉⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⣄⡀⠀⠀⠀⠀⣠⣾⡁⠀⢈⣷⣤⣀⣀⣀⣀⣤⡾⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡋⠙⢷⣌⠛⠛⠛⠛⠛⠉⠘⣷⠀⣼⠃⠈⠉⠉⠉⠉⠁⣠⠞⠋⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠉⠻⣦⠀⠹⣧⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⣾⠃⠀⣴⠟⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠘⣧⠀⢹⡆⠀⠀⠀⢀⣠⠟⠀⠻⣦⡀⠀⠀⠀⢸⡇⠀⣼⠇⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣿⠀⢘⣧⣠⡴⠾⠛⠁⠀⠀⠀⠈⠙⠳⢶⣤⣿⡀⢠⡿⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠶⠶⠶⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠶⠶⠶⠶⠞⠛⠁

⠀⠀⢀⠀⢀⠀⠀⣀⠀⠀⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⡀⠀⠀⠀⡄
⠀⠀⠀⢀⠄⠀⡀⠈⠀⠠⠀⠁⢁⠰⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⣾⣷⡖⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣭⠀⠀⠀⠀⠀⠢⣤⣥⡤⠂
⠀⠉⠠⠀⠀⠀⠀⠀⠁⠀⠤⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠈⣻⣿⣾⠟⠁⢠⡀⠀⠂⠐⠆
⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠠⣤⢿⡤⠄⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠚⠀⠠⠟⠋⠛⢇⠀⠙⠛⠀⠠
⠀⠀⠈⠀⣄⠀⠀⠁⠀⣣⢬⡇⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠞⠙⠇⠀⠀⠀⠀⠀⣀⣀⣠⠃⢧⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠠⣤⠞⠘⠦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠀⠀⠉⠲⡔⠀⠀⣠⠊⠀⠀⠴⠆⢠⡀⠀⠀⠀⠀⠄⠀⠄
⠀⠀⢨⣃⣤⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠠⠈⠀⠀⢸⣥⠖⠦⣸⡀⠀⠀⣤⣤⣾⣧⣤⡤
⠀⠀⢈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣰⢇⠀⠀⠐⠐⠀⠐⠀⠀⠀⠈⠀⠀⠀⠈⠡⠀⠀⠀⣹⣿⣿⡏
⠀⠀⠈⠀⠄⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣤⣤⣾⣿⣏⣤⣤⣄⡄⠀⠀⢶⣶⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠋⠁⠈⠓⠀⠀⠈⣇
⠀⠀⠀⠐⠀⢀⡾⣡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢻⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠠⣄⠀⠋⠀⠐⠦⡤⠀⠀⠀⠀⠈⠛⠻⣦⣯⣤⠔⠂
⠀⠀⠲⢶⠶⢋⡴⠟⣻⣶⣦⠶⠂⠀⠀⠀⠀⠀⠱⡄⠀⠀⢸⣿⡿⠿⣿⣷⠀⠀⢀⣀⣼⣧⣀⡀⠈⠁⠀⠀⢠⡀⠀⠀⢤⣦⡤⠀⢀⡿⠁⢷⡀
⠀⠀⡀⠈⢶⡿⢁⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠛⠁⠀⠀⠈⠙⠂⠀⠀⢹⣿⢿⡏⠀⠀⠖⠉⠀⠐⠇⠀⠀⠘⠉⠃⠀⠈⠀⠀⠀⠉
⠀⠘⠋⠀⢨⢶⠿⠿⣿⣿⡀⠀⠀⠀⠀⠀⠐⠒⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠊⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣷⠂⢀⣀⡰⠛⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠄⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠀⢀⡞⢧⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠙⡆⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠂
⠀⠀⠀⠀⣸⣧⠀⠀⠀⠀⠀⠀⠠⣤⠤⠶⠞⢀⠈⠳⠦⠤⣤⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠸⠛⠉⠛⠇⠠⠀⠀⠀⢀⡀⠀⠀⠀⡀⢀⠀⢀⡄
⠀⠲⣶⣶⣿⣿⣷⣶⠖⠀⠀⠄⠀⠈⠳⡄⠘⣿⣿⠂⡴⠊⠀⠀⣤⠀⠀⠰⣶⣶⡿⠧⢀⡄⠀⠀⠀⣀⣴⣆⣀⠀⠀⠁⠀⠀⠀⠀⣀⡤⠞⠙⠤⢀⡀
⠀⠀⠈⣿⣿⣿⣿⠁⠀⠂⠀⠀⠐⠂⠀⡇⠀⣁⣉⠀⢧⠀⠐⣾⣿⡶⠂⠀⠈⣹⣮⣶⡏⠀⡀⠀⠀⢙⣿⣿⡁⠀⠀⠀⠀⠀⠐⠀⠈⢲⠀⠀⢰⠋
⠀⠀⠸⠛⠁⠈⠛⠇⠀⠀⠂⠀⠃⠀⠸⠗⠉⠀⠈⠙⠻⠀⠀⠋⠉⠓⠀⠀⠀⠉⠀⠈⠛⠀⠀⠀⠀⠈⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠟⠊⠙⠺⠄

⠀⣀⡜⣆⣀⠀⠀⠀⠀⠀⢠⠧⠀⠀⠀⠀⠀⠀⠀⢖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣰⠄⠀⠀⠀⠀⠀⠀⢠⡆
⠙⣤⠁⠡⡜⠁⠀⠀⠈⠳⠀⠀⣩⠋⠀⠀⠀⠉⢛⡽⡻⠋⠀⠀⠀⠀⢩⠰⠆⠅⠁⠀⠀⠀⠑⠀⠀⡈⠉⠀⠀⠀⠘⢻⡭⠶⠉⠁⠀⠀⠈⠻⡭⠀⠈
⠀⠷⠊⠑⠇⠀⠀⠀⠀⠸⠊⠁⠺⠀⠀⠀⠀⠀⠋⠀⠘⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠊⠀⠘⠀⠀⠀⠀⠀⠛⠁⠙⠂⠀⠀⠀⠀⠘⠃⠈⠁

⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⢀⡴⢦⡀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠠⢢
⠀⢖⢣⢗⠆⠀⠀⠀⠀⠛⣟⠙⡴⠂⠀⠀⠀⠒⠀⠀⠀⠒⠀⠀⠀⠘⣭⠁⠈⣭⠃⠀⠀⠀⠒⠆⠉⠲⠂⠀⠀⠀⠐⠖⠈⠐⠛⠂⠀⠀⠘⢭⠁⠀⢩
⠀⠘⠁⠑⠀⠀⠀⠀⠀⠸⠃⠛⠆⠀⠀⠀⠀⠀⠶⠀⠸⠀⠀⠀⠀⠀⠻⠖⠲⠟⠀⠀⠀⠀⠀⠟⠈⠳⠀⠀⠀⠀⠀⠴⠃⠸⠇⠀⠀⠀⠀⠠⠕⠒⠭

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⢀⠄⡄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢠⡄
⠐⢖⣚⣓⠖⠀⠀⠀⠀⢶⣾⣿⣶⠆⠀⠀⠀⢈⠉⠀⠈⡡⠀⠀⠀⠠⣄⠁⠀⠒⠂⠀⠀⠀⠠⣤⠻⡤⠀⠀⠀⠀⠰⣖⠋⢓⡶⠀⠀⠀⠰⢦⠈⠀⠒
⠀⠘⠛⠛⠀⠀⠀⠀⠀⠰⠿⠻⠧⠀⠀⠀⠀⠀⠡⠤⠄⠃⠀⠀⠀⠀⢀⠤⠀⢣⠀⠀⠀⠀⠀⠛⠛⢃⠀⠀⠀⠀⠀⠺⠖⠾⠆⠀⠀⠀⠀⢠⠔⠀⢷

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢠⣄⠊⠂⣤⠄⠀⠀⠠⡆⠀⠈⠀⡦⠀⠀⠀⢤⠐⠁⠓⣠⠀⠀⠀⢰⣟⠃⠘⢻⡆⠀⠀⠀⢤⡀⠃⢤⠄⠀⠀⠀⠤⠀⠀⣀⠠⠄⠀⠀⠠⣤⠘⠑⢤
⠀⣸⠴⢮⡅⠀⠀⠀⠀⢰⡀⠀⣀⠀⠀⠀⠀⠀⣇⠤⣘⡅⠀⠀⠀⠘⢿⣤⣤⡿⠃⠀⠀⠀⠀⡤⠀⢦⠀⠀⠀⠀⠀⡁⢲⣿⡗⠀⠀⠀⠀⢨⡤⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⣀⡀⡀⠀⠀⠀⠀⣀⡠⠧⣀⡀⠀⠀⠀⣀⠴⠋⠢⢀⠀⠀⠀⢀⣀⣰⣀⠀⠀⠀⠀⢀⡠⠞⠀⠳⢆⠀⠀⠀⣀⠠⠇⠤⢤⠀⠀⠀⠀⠚⠁⠁⠙
⠀⠠⠛⢏⠀⠀⠀⠀⠀⠈⡧⢠⡍⠀⠀⠀⠀⠀⢆⣀⣰⠃⠀⠀⠀⠀⢹⣿⣷⡄⠀⠀⠀⠀⢹⠀⣀⠐⡏⠀⠀⠀⠈⠓⣠⠐⡆⠀⠀⠀⠘⢆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉

⣀⠤⠎⠣⠤⡀⠀⠀⠀⣀⣰⢧⣀⡀⠀⠀⠀⢀⣀⣾⣄⣀⠀⠀⠀⠀⠤⠎⠱⠤⠀⠀⠀⠀⣀⠀⡗⢀⡀⠀⠀⠀⣀⠠⠏⠡⢀⡀⠀⠀⠀⡀⢠⢇⣀
⠘⡆⠀⠀⡖⠁⠀⠀⠀⠙⣆⡀⣞⠁⠀⠀⠀⠈⣿⣥⣿⠋⠀⠀⠀⠈⡢⠀⠀⢔⠁⠀⠀⠀⠉⣃⠀⣍⠁⠀⠀⠀⠉⣷⢀⡀⡞⠁⠀⠀⠀⠙⢊⡺⡍
⠀⠓⠉⠑⠂⠀⠀⠀⠀⠘⠉⠈⠉⠀⠀⠀⠀⠀⠋⠁⠉⠀⠀⠀⠀⠀⠙⠋⠉⠊⠀⠀⠀⠀⠀⠉⠀⠙⠀⠀⠀⠀⠀⠛⠉⠙⠃⠀⠀⠀⠀⠘⠉⠀⠉

⠀⠀⡀⣀⠀⠀⠀⠀⠀⠐⠀⡀⠐⠀⠀⠀⠀⠀⢀⠆⢀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⡰⢳⣀
⠀⠨⠀⠈⡄⠀⠀⠀⢠⠀⣻⢿⠃⡠⠀⠀⠀⠑⠄⠀⠀⠌⠀⠀⠀⠈⢫⠀⢈⡝⠁⠀⠀⠀⠘⣿⣿⡿⠃⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠈⢣⠰⠷⢠

⠀⠀⠀⠀⠀⠀⠀⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⠳⣄⠀⠀⠀⠀⠀⠀⠈⢦⣀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠘⣷⡄⠀⠀⠐⠶⠀⠈⢏⠛⢦⡀⢸⣧⣀⠀⠀⠀⢰⡀⣦⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⢹⣿⡆⢀⣀⠀⠤⠤⠼⠿⠶⣇⢸⣎⠻⣆⠀⠀⢸⢷⡸⡿⢦⡀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣿⣿⠀⡀⠤⠄⠒⠒⠘⡖⠚⠺⡿⢿⣿⠀⡀⢸⠀⢻⢃⠀⠃⢸⢿⣄⢀⡀⢀⡀
⠶⠦⣶⣶⠞⠛⠛⠓⠢⢀⡴⠊⢸⣿⣿⡉⢀⠤⠾⠂⠉⣁⣩⡅⠈⣇⠀⢻⢽⣿⣿⠀⠚⠈⣏⠉⢹⣀⢹⡆⠀⠛⢸⡆
⠀⠀⠀⠈⠉⠓⠢⠤⣠⣋⠀⡠⢺⣿⣿⠛⢧⣴⣶⣾⣥⣈⠿⠁⠒⡧⠠⠉⡐⠄⣺⡐⢨⣾⡷⠀⢺⠂⠘⣥⣐⠛⣻⠴⣆
⠀⠀⠀⠀⢶⠆⠀⢰⢣⣿⣿⣇⠸⣿⣇⢀⣿⡟⢧⡞⠙⢻⡆⠸⠿⠷⠀⠀⠀⢁⣼⠀⠉⡩⢐⣬⣽⢴⣶⣍⣿⠛⠁⠀⠈⢷⡀
⠀⠀⠀⠀⠀⢀⣤⣇⠀⢻⠿⠟⡼⠙⢿⣧⡙⡿⠥⠈⠲⣼⠃⠐⢾⡿⠀⠀⠘⠛⠋⠐⠍⡀⠿⠿⠋⠀⠉⠁⢠⡄⠠⣦⣤⠖⠁⢴
⠀⠀⠀⠀⠀⢸⣿⣿⠏⣻⠶⢤⣇⣀⢸⠉⠻⠷⠖⠒⠋⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠜⢵⣶⠎⠈⢀
⠀⠀⠀⠀⠀⣼⣿⣿⠀⣿⠀⠀⡇⠉⢙⠓⠶⣤⣀⠀⣶⣷⣀⠟⠀⠀⠀⢀⡀⣀⢀⡀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⡞⢉⡟
⠀⠀⠀⠀⠀⣿⣿⡏⠀⢸⢀⡴⣷⣶⣼⡀⠰⣿⣿⡆⠀⠉⠀⠀⠀⠀⠀⢸⡇⣿⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣧⠉⡄
⠀⠀⠀⠀⢠⣿⣿⣧⠀⠈⢿⣴⣶⡀⠙⣷⠀⠙⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣸⣿⢿⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢀⡀
⠀⠀⠀⠀⢸⣿⣿⠘⡆⠀⠘⡄⠀⢱⡀⠘⢆⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⣶⣶⣞⢲⡖⠀⠀⠀⠀⠀⢠⡆⢀⣀⡿⣶⡆⣾⡏
⠀⠀⠀⠀⠈⢉⡉⠙⠛⣖⠒⠻⡦⠤⠷⣤⣌⣢⠀⠸⣤⡀⣀⠀⠀⠀⠀⢸⡇⢿⡿⣿⢩⠿⣧⠀⠀⠀⠀⠀⠈⠁⡘⠻⢿⣤⠁⠉
⠀⠀⠀⠀⠀⠛⠃⢀⣀⡈⢶⣦⠘⣆⠀⢿⣿⡷⠑⣔⠿⠳⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢆⠌⠲⡄⠠⡤⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠨⠿⠶⠈⠻⣃⡠⠷⡌⢉⡱⣶⣾⡧⡀⠈⣢⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠖⠙⣦⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡟⠢⣀⡼⠫⡀⢈⣻⣁⠈⣿⢋⣉⠫⡉⠀⠀⠀⣴⠇⠀⢰⠗⣼⠇⠜⣿⡠⠀⣨⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⣿⣧⠖⠉⢢⡠⠚⠻⣿⠿⠓⢷⠾⠛⢄⠀⠑⢄⠀⠀⡄⠀⠀⢢⠜⣄⡠⠊⢣⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⣼⠟⢁⣤⡊⠁⠈⠢⡀⢈⡷⣶⣾⣷⢤⡀⠑⠢⣠⣷⢆⡘⡤⢊⠔⠱⡈⢣⣾⡗⠙⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⢺⣧⠹⣷⡄⣀⠴⠊⠓⢄⠈⠙⠯⡀⠈⠢⣀⠹⠫⡀⢉⢘⣅⣠⡾⠋⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠼⠋⠀⠀⠀⠀⢻⣧⡜⣿⣄⠀⠀⠀⠀⣙⡦⡖⣿⣿⣷⡌⠑⢧⣈⢔⣴⡿⠋⠀⠀⠰⣤⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡘⢿⣦⣠⠔⠊⢁⣀⠈⠛⢍⣩⠗⢆⡘⣷⡿⠋⠀⠀⠘⠃⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡷⠚⠉⠀⠀⣤⣈⠙⠋⢠⣾⠛⠢⡀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⡇⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⠼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀
⠀⠲⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠖
⠀⠀⠀⠙⠻⣿⣿⣿⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⢀⣠⣾⣷⣄⡀⠀⠀⠀⠀⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⠀⣠⣴⣿⣿⠟⠻⣿⣿⣦⣄⠀⠀⠀⠀⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⢀⣴⣾⣿⠿⠋⠀⠀⠀⠀⠙⠿⣿⣷⣦⡀⠀⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢏⣠⣾⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣄⡸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠉⠑⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠒⢦⡀⠈⢁⡔⢁⢍⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢨⣀⡠⠶⠖⠚⣛⣛⣉⣉⣉⢭⡽⠃
⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⣇⠐⡩⢦⡘⠒⣡⡈⠢⡀⡰⣡⠔⣒⣒⢠⠃⠀⠀⠀⠀⠀⠰⠯⣥⠶⡖⠉⠙⢉⠁⠀⠈⢸⡄⠀⡇
⠀⠀⠀⠀⣶⡋⢻⡆⠙⠲⠤⢤⠔⡡⣊⡴⣛⣳⣄⠉⢶⠄⠈⠳⡇⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠠⠈⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠘⢎⡴⣫⣾⢟⣵⣿⣦⡀⠀⠀⡤⠜⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠠⠄⣿⠀⡇⠀⠀⢸⠃⠀⠀⠀⠃⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣏⢔⣹⠿⣋⣴⡷⡄⠀⠘⢆⠀⠻⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢹⡀⢸⣀⣀⣰⣦⠦⠤⠚⠛⠉⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠈⠻⡥⣪⢟⣭⡾⢟⣦⠀⠀⢳⣀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠰⣟⣉⣉⣉⠥⠤⠤⠒⠒⠚⠋⠻⣧⣤⣤⣤
⠀⠀⠀⡠⠂⠀⠀⠐⠄⠄⠀⠾⠿⠀⠀⠀⠘⢷⠟⡫⣴⡿⡫⣣⡀⠀⠉⠀⢱⣀⡀⠀⠤⠠⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⢀⢀⠠⠇⠀⠀⠀⠀⠀⣀⠤⠤⠒⠒⠈⠉⡂⠀⠻⣊⢕⡪⣞⣭⣷⣂⠀⠀⠀⢷⢓⠲⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⣿⣿
⡎⠀⠀⠀⠀⠀⠀⠀⠸⣴⡀⠀⠀⠀⠀⢠⠇⠤⠤⢽⣯⡪⢟⣫⣶⣷⠂⠠⣶⢸⠀⠀⠀⢔⣢⠒⣼⡂⣀⠀⠀⢀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⡇
⠈⠀⠀⢒⢀⡀⠀⠀⠀⠈⠙⢶⣤⡀⠤⡼⢦⠤⣤⣃⠘⣾⣟⠟⣋⢼⡧⠤⠒⠉⠀⢰⠒⣤⠿⣋⡡⠄⠈⢷⣾⣷⣄⠈⠉⠏⠀⠙⢯⡁
⠀⠀⠀⠈⠁⠙⠦⣄⣀⠀⣰⡪⠛⢧⢠⠇⠈⠲⠃⠈⠉⢙⣶⣿⣿⡉⠀⠀⢀⡤⠒⠉⠱⡜⢧⠀⠐⠂⢀⡼⠈⢿⣿⣆⠎⠀⠀⠀⠀⠘⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡹⠒⠒⠒⠢⣤⠀⣾⣿⣿⠋⣡⠴⠊⠁⠀⠀⠀⢀⣵⢈⡧⢴⠊⡿⣦⠀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠘⠂⠤⠼⠁⠀⠀⠀⠀⢹⠀⢻⡿⠗⠉⠀⠀⠀⣀⠴⠒⠉⠉⠈⢋⠀⠛⠃⠈⠁⠀⠀⡟⣠⡆⠀⠀⠀⠀⠀⡟⠫⠯⠃
⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠒⠉⠀⠀⢀⡤⠖⠛⠁⠀⠀⠀⣠⣴⣶⣦⣄⠀⠀⠀⠀⠰⣟⠄⠃⠀⠀⠀⠀⠀⠃⠀⢀⣤
⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⢰⠃⠀⠀⣠⠴⠚⠉⠓⠦⡄⡀⠀⠀⢰⣿⢋⠽⢻⣾⣧⡀⠀⣀⠀⠄⠀⠈⠀⠀⠀⠀⢸⣶⣤⡀⠉
⠀⠀⠀⠀⡄⠀⣀⡀⠀⣬⠀⠀⠹⡆⠀⠀⠀⠈⠳⣍⣉⠀⠀⠀⠀⠀⢄⠈⠳⠀⠀⠸⣷⣝⠊⣠⣿⡴⠋⠓⢤⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡷
⠀⠀⠀⠀⣷⠈⣿⣇⠀⠙⠧⣄⠼⠀⠀⠀⠀⠀⠀⠈⠙⣿⣷⣶⣶⣬⣉⠀⠀⠃⠀⠀⠈⠓⠶⠿⠿⠅⠀⠰⠆⣉⠢⣄⠀⠀⠀⠀⢠⡿⠿⠟⠁
⠀⠀⠀⠀⢸⡆⢹⣜⡄⠀⠀⠀⠀⠀⠀⢀⠔⠉⢱⣶⣿⣿⣿⣿⣿⠟⠉⠛⢷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠈⠳⢄⠀⠀⡜
⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⢀⣿⠿⠿⠻⣿⣿⣄⣀⣀⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠄⠀⠒⠒⠾⢷⣈⠀⢀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠒⠉⣀⡤⠔⠊⠁⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣄⣠⣤⡶⠖⠉⠉⠀⠁⠢⣀⠀⠀⠠⡖⠒⠢⣝⢆⠀⠀⢱⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡀⠀⠀⠰⠋⠈⠀⠙⢦⠀⠘⣿⣿⣿⣿⣿⣿⣿⠟⣉⣀⣀⠤⠤⠀⢠⣤⣈⡑⢦⡔⠋⠀⠀⠀⠀⢣⠀⠀⠀⠠⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡄⠀⢣⠀⠀⠀⠀⡀⡇⠀⠘⢿⣿⣿⠿⠋⠀⠀⠹⡄⠀⡰⠁⠀⢸⡿⠋⠁⠀⡍⠓⠢⠤⠤⠴⠋⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠑⠤⠤⠤⠜⠁⠀⠀⠈⢟⠁⣀⠤⠤⠤⣄⠘⣴⠃⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠠⡄⠀⠀⠀⢀⠏⠒⠲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⡠⠤⡄⠀⠀⠈⢎⠃⠒⠒⠒⠊⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠓⠤⠂⠀⣀⣠⣼⠀⠾⠭⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣤⣤⣴⣾⡿⠿⠿⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣹⣛⣩⣙⣙⣟⣩⣟⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢻⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⠁
⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣆⡀⠀⠂⠀⣠⠀⠀⠀⠀⠀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢀
⡏⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠂⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠽⢿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠿
⠀⠀⠀⢰⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠈⠁⠀⠀⠀⣴⣀⡀⠀⠀⠀⠀⠀⠀⠀⢺⢿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿
⡀⠀⠀⠀⠠⠀⠀⠀⢹⡿⣿⡁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⢻⡿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰⣶⠀⠀⠀⢴⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⢴⡦⠀⠀⠠⣼⣷⠄
⠀⠀⠀⠤⣽⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠃⠀⠀⠀⢀⠀⠀⠀⠰⣾⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠃
⠃⠀⠀⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠟⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐
⢀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠠⣦⠄⠀⠀⢸⣿⡷⠄⠀⠀⠀⢀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⢠⡦⠀⠀⠀⣻⣿⡶
⠿⡗⠀⠀⠀⠟⠻⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠃⠀⠀⠀⠸⢿⠂⠀⠀⠨⠟⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡦
⠀⠿⠆⠀⠀⠀⠀⠂⠀⠀⠛⠙⠂⠀⢴⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠀⠠⣼⣧⠄
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢀⡀
⡦⠀⠀⠀⠀⠈⠛⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⢲⣿⠄⠀⠀⠀⠀⠘⠃⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⣶
⠁⠀⠀⢀⣠⡀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⢀⣤⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠘⠻⠇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⠄⠀⠀⠀⠀⠀⠀⠚⠻⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣶⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⡖⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠙⠀⠀⢀⣄⠀⠀⠀⢀⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠙⠀⠀⢠⣄⠀⠀⠀⢠⣄⡀
⠀⠀⠀⠀⢰⣀⡀⠀⠐⠀⠀⠀⠀⠈⠉⠀⠀⠀⠹⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⡿⠿⠄
⡦⠀⠀⠈⠝⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠈⠟⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢠⣴⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢤⣴⣄
⣴⡄⠀⠀⠐⣾⣷⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠚⠛⠏⠀⠀⠀⢠⣴⡀⠀⠀⠰⣿⣗⠀⠀⠀⠀⠀⠀⠘⠻⠁⠀⠀⠛⠻⠇
⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣀
⣀⣶⣄⣀⣀⣀⣀⣀⣀⣈⣿⣿⣄⣀⣀⣄⣀⣀⣀⣀⣀⣀⣈⣀⣀⣠⣦⣀⣀⣀⣀⣀⣀⣀⣈⣿⣯⣀⣀⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣷⣷⣿⡟⠁⣠⡶⣿⣿⣿⣿⣮⣷⡍⠻⣿⣶⠶⣖⣻⣻⣛⣰⡆⠦⣬⢈⡛⢲⢤⡙⠑⠆
⣿⣿⠏⢀⠞⡙⠂⠘⢹⣿⣿⣿⣿⣿⣿⣮⣻⣿⠁⣇⠉⠈⠹⠙⢋⠗⡶⣍⡑⠦⠛
⣿⡟⠀⡞⠰⠀⠀⠖⢙⠟⠛⢯⡿⣿⣿⣿⣿⣿⠿⠛⠿⠛⠙⠟⡶⢦⣤⣀⠛⠓⠲⢄⠀⠀⠀⠀⠀⢀⣤
⢭⠇⠀⣧⠁⠀⢻⠎⠁⠀⠀⣾⡿⢿⣾⣿⢿⣿⣯⡙⠿⢦⡘⠷⣷⣿⣭⣇⠀⠀⠀⠀⢣⠀⢀⣠⢾⣿⣽
⣧⣾⡀⢲⡀⡐⠀⠀⢰⣿⡆⢀⣴⣾⣿⣕⢌⣿⠟⣮⡢⡈⠟⢶⣽⠿⣿⣿⣿⣤⠀⠀⢠⠐⢿⣿⣿⣿⣿
⣷⣿⣇⠀⠛⠀⣠⡤⡀⠉⣰⠟⣵⣯⣙⣿⢿⣟⠻⣿⣺⡯⡢⣈⠻⣷⣜⠿⣿⣿⣷⣤⠜⠃⣀⢿⣿⣿⣿
⣿⣿⣿⣷⠀⠀⠻⠷⠇⣼⡇⣽⣿⣿⣻⣼⡻⣿⣷⣝⣻⡮⡪⡻⣅⣍⠙⢳⣞⠻⣿⣿⡄⠀⠉⣈⣿⣿⣿
⣿⣿⣿⡇⣠⡖⢦⠀⣸⠿⠗⣿⡿⠛⠛⣿⡿⠿⢿⣟⣛⣿⣛⠊⠪⡳⠕⢌⠹⣷⣮⡉⠙⢄⠀⢀⣸⣿⣿
⣿⣿⣿⠀⠻⣿⠿⠃⣿⠃⠘⣿⣁⣴⣾⡟⢂⣴⡌⣿⠿⠿⢿⣯⣦⡁⠀⠈⠳⣌⠿⡿⠀⠀⠀⠀⢼⣿⣿
⠿⣿⣿⠀⠀⡠⢄⠀⢿⡀⠀⢻⣿⣿⣿⣷⣿⣿⣷⢸⠀⠀⢀⣿⣿⣼⡢⡙⣆⠈⠀⠀⠀⠀⠀⠀⡀⣿⣿
⣷⡿⢿⡄⠸⣷⡾⠂⠸⣧⠀⠈⣿⣿⣦⢿⣿⣿⣿⣤⣷⣤⣾⣿⣿⣿⣿⣮⡺⣷⠄⠀⠀⠀⠀⠀⠑⢎⡛
⡟⣷⣾⣇⠀⠀⣠⠤⠀⢻⣇⠊⠈⢿⣿⣷⣿⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢮⠀⠀⠀⠀⠀⠀⣇⣀⠑
⣟⡿⣿⣿⡄⠐⣿⣷⣶⠀⠁⠀⢀⡀⠀⠙⠿⣿⣾⣿⣿⣿⣄⣀⣽⣿⣿⣷⣤⣼⡇⠀⠀⠀⠀⣨⡃⡝⢴
⣿⣿⣿⣿⣿⣄⠈⠉⠁⠀⢀⣾⣯⣀⣡⠀⠀⠀⢀⣩⡿⠿⢿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⢀⡪⣻⣮⡓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡼⢆⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠳⠛⣀⠀⠀⠀⠀⡾⠁⠘⣧⠀⠀⠀⠀⡠⠻⢲⠀⡄
⠀⠀⠀⢠⡤⠤⢤⣀⣀⡀⠀⠇⢸⣄⠲⠿⡻⣶⣤⣸⡷⠀⠀⢿⣆⣤⣶⠋⠓⠴⣠⠀⠀⠀⢀⣀⣠⡤⠤⣤⠄
⠀⠀⠀⠈⣷⢇⣂⡢⠄⠉⠉⠙⠺⣿⣦⠐⠃⠘⢿⣿⡇⠀⠀⢸⣿⡿⠁⠂⠀⣴⡿⠓⠉⠉⠉⡠⢔⣒⡙⡿
⠀⠀⠀⠀⢻⢰⡸⠙⠳⣦⡲⣄⠀⢀⠃⠤⡀⠀⠀⠙⠃⠀⠀⠉⠋⠀⠀⢀⡴⢛⠀⢀⣤⣶⡴⠞⢫⢧⢷⠇
⠀⠀⠀⠀⠈⣇⢧⢂⠀⠈⢿⣾⣷⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣫⣿⡟⠀⠀⣎⡎⡞
⠀⠀⠀⠀⠀⠈⢮⢧⢢⠀⠘⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⢊⣿⣿⠀⢀⢎⢎⡞
⠀⠀⠀⠀⢀⣴⣾⣷⣢⢳⣴⣙⠗⠀⠀⠀⠀⠀⠛⢻⠀⠈⠁⠀⢙⠂⠀⠀⠀⠀⠀⠟⣻⣲⢯⣮⣾⣷⣦⡀
⠀⠀⠀⣠⠿⠿⣿⡿⢿⠛⠛⢒⣠⣤⣄⠀⠀⠠⠄⠀⠁⠀⠀⠈⠁⠐⠒⠀⠀⣠⣴⣤⠒⠛⠛⠿⣿⣟⠿⠻⢦
⠀⠀⠀⠑⢢⠜⢁⠀⣘⣿⡿⠿⠟⠻⠿⣿⣔⣢⡤⠀⡀⠀⠀⠀⠀⢂⣤⣤⣿⠿⠿⠿⠿⣿⣧⣀⡀⡌⢣⡔⠈
⠀⠀⠀⢠⠎⠀⣠⣿⡛⣻⣄⠀⠀⠀⠀⠀⠉⠳⣖⣀⣾⢀⡀⣳⣠⣾⠿⠉⠀⠀⠀⠀⠀⣠⠟⢛⣷⣀⠀⠱⡄
⠀⠀⢠⢃⣤⣤⣄⡈⢷⡅⠙⢆⠀⠀⠀⢠⠀⠀⠈⢹⣿⣬⣥⣿⠋⠁⠠⠀⠀⠀⠀⢀⡴⠋⢠⡿⠁⣠⣦⣤⡘⡄
⠀⠀⠈⢹⡿⣿⣞⣷⣾⢿⣤⣤⣙⣦⣀⠰⣦⡄⠀⣈⠛⠛⠛⢻⡀⠀⢠⡾⠀⣀⠴⣋⣤⣴⡿⢧⣞⡻⣿⢿⡏⠁
⠀⠀⠀⠘⠁⣽⣿⡿⢧⣿⡅⠀⠈⠉⠒⠭⣍⡁⠚⠉⢀⣖⣶⡀⠙⠂⠉⣃⡭⠒⠋⠉⠀⢨⣿⣞⣿⣿⣷⠀⠃
⠀⠀⠀⠀⣷⣿⣿⣷⣤⣈⣻⣿⣦⡄⠀⠀⠀⠀⠀⠀⠸⣿⡿⠁⠀⠀⠀⠀⠀⠀⢀⣴⣾⣏⣁⣆⣰⣿⣿⣾
⠀⠀⠀⠀⡏⢸⣿⣿⣿⢿⢿⣿⡿⣷⣆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣰⣾⣿⣿⣿⣿⣿⣿⣿⡇⠹⠆
⠀⠀⠀⠀⠀⠨⢻⣿⣿⣷⣿⠋⠀⠀⠙⠷⢮⣝⣒⠶⠶⠶⠶⠿⠶⢒⣪⡵⠾⠋⠁⠀⠙⣧⣿⣷⢹⡟⡄
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡆⠀⠀⠀⠀⠈⠈⠩⠿⠛⢷⡶⠿⠻⠛⠍⠂⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⢻⡟⢹⠹⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⠿⡏⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠃⢸⡄⠉⠈⢿⣷⣦⣄⣀⠀⠀⠀⢸⡇⠀⠀⠀⣀⣤⣴⣾⠟⠉⠀⢠⡇⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡦⢰⣾⣿⣿⠟⠉⠙⠳⢦⣴⣇⡼⠟⠋⠉⠻⣿⣿⣶⡆⡇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣼⡿⠛⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠻⢿⣡⠟⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⣷⡄⠀⠀⠀⠀⠀⡠⣀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⡀⠀⢀⠘⠓⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⣤⣄⣀⣀⣀⣀⣘⠀⢸⡇⠀⠀⠀⢀⠜⢉⣿⡷⠀⣿⣿
⣿⣿⣿⠀⠚⠋⠀⠀⠀⡤⠀⢀⣠⠴⠒⠚⠉⢉⣡⣶⣾⣿⣿⠟⠛⠛⠋⠉⠛⠀⠘⣷⣦⣄⡴⠋⢀⣿⠟⠁⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠁⢀⣴⣿⣿⣿⣿⣿⣯⠤⠒⠉⠀⣠⡄⢸⠀⠀⢻⣴⠟⠀⢠⡾⣧⡀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⣠⣶⠟⠁⢀⣠⠶⢻⣿⣿⣿⣿⣿⡿⠁⢀⣠⣶⠞⠋⠀⣼⠃⠀⠀⠀⠀⣰⣏⠀⠀⠙⠳⢒⣿⣿
⣿⣿⣿⠐⠂⣠⣾⣿⠟⢀⣾⠟⢁⡴⠈⣿⣿⣿⡿⠋⠀⢤⣿⡟⠁⠠⠤⠞⠁⠀⠀⠀⠀⠀⣬⣤⣴⠶⠶⠾⠟⣿⣿
⣿⣿⣿⣠⣾⣿⣿⠁⣠⣾⣇⠀⠀⣠⠞⢻⡿⠋⠂⠀⡄⡎⠉⣿⣦⣀⣤⣤⣀⢀⣀⠀⠀⣀⠈⠳⣄⠠⣀⡀⠀⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⡾⠃⢀⠘⠁⠀⠀⣴⡇⠇⠠⠀⠀⠈⠀⣻⡏⠁⠈⣿⣿⡟⠓⠶⣌⣑⠄⠉⠳⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡴⠃⠀⠀⠠⣼⣿⠀⢂⠴⠒⠀⠀⠀⠈⠑⠀⣶⡶⣉⣠⣶⣶⣶⣿⣶⣤⣄⣿⣿
⣿⣿⣿⣿⢇⣾⣿⣿⣿⣿⣿⣿⡿⠠⠇⠀⠀⣄⣿⡇⠀⠈⠀⢀⠀⠠⠤⠀⠀⢸⡿⠙⠿⠿⣿⣿⡿⠿⠿⠿⢿⣿⣿
⣿⣿⣿⠛⠘⢿⣿⣿⣿⣿⣿⠟⡀⠀⣰⡷⡄⠸⣿⣿⡆⢰⠀⠀⠀⠀⠁⠀⢀⣿⣥⣤⡀⠀⠀⠉⠃⠀⠀⠀⠀⣿⣿
⣿⣿⣿⡄⠀⠀⢹⣿⡿⠻⡿⣿⣷⣾⣿⣷⣀⠀⠈⢓⡉⣫⠀⠀⠀⠀⣀⢀⣼⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠉⠈⠀⠀⠀⠈⠁⠀⠉⠛⢿⣷⣄⠈⠉⠃⠀⢤⠀⠀⠁⣼⣿⣿⣟⠻⠿⠿⣿⣿⣿⣿⣦⡀⠀⣿⣿
⣿⣿⣿⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀⠉⠻⣷⣄⢠⡀⠀⡀⠀⠀⣽⣿⣿⣿⣷⣄⠀⠀⠉⠛⠿⢿⣷⡄⣿⣿
⣿⣿⣿⣸⡇⠀⡀⣠⡄⠃⠀⠀⠀⢦⠀⠏⢀⢰⡆⠘⢿⣷⡝⠖⠁⢀⡴⣿⣿⡏⠉⠙⠻⢦⠀⠀⠀⠀⠀⠙⣿⣿⣿
⣿⣿⣿⠿⠷⠂⠋⠀⠀⠀⠀⠀⠀⠠⣿⣀⠏⣼⣦⣴⡶⠛⢿⣦⣶⡀⠀⠀⠻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠫⠿⠾⠿⠛⠁⠀⠀⣰⡟⠉⠛⠂⠀⠀⢀⣠⣤⠰⣦⡀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⢀⠀⠘⠁⠈⣽⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠂⠀⠀⠀⠈⠀⠘⠃⠀⠀⠀⣾⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⣼⣾⣿⣿⡇⣿⡇⠀⠗⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠉⠠⢶⣿⠉⠙⠻⡇⢿⣧⣤⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠘⠃⡀⠉⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣄⣸⡇⣿⣿⣷⠃⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣸⣿⡿⠁⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠗⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣸⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿

⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣷⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠂⠉⠁
⡀⠀⠀⠀⠀⠀⣀⣤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠻⠿⠛⠋
⣇⠀⠀⠀⠠⠾⢿⢛⣩⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠋
⣿⣿⠖⠀⠀⠀⠸⡿⢻⡟⣿⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣿⣿⣿⠟⠁
⣷⣦⣤⣀⡀⠀⠘⠒⢻⣧⣿⣤⡞⠋⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⣿⡿⠋
⣿⣿⣿⣿⣿⣏⣀⣀⡙⣿⣿⣿⣄⣤⣿⣥⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡟⠉⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣢⣼⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣴⣤⣤⣤⣤⣤⣤⣄⣤⣾⣦⣤⣄⣀⣴⣿⣿⣿⣤⣀⣿⣀⣀⣰⣾⣿⣂⣀⣀⣤⣶⣾⣦⣰⣾⣇⣸⡃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡀
⠀⠀⠀⢤⣴⣧⠄⠀⡀⠀⠀⠀⠀⠀⢀⣠⡴⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣀⣳⣾⡅
⠀⠀⠀⠘⠉⢃⣴⣿⣿⣿⣦⣤⡶⠞⠋⠁⠀⠀⢀⠀⠆⠀⠂⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⣔⣾⡯⢤⡿⡝⠛
⠀⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠘⠄⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⣷⠿⣿⢾⡇⠙⠁⠀⢰
⠀⢿⣿⣿⣿⣿⡿⠛⠿⠿⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠀⠀⠀⣀⣀⠔⠈⠈⠐⡌⠧⠉⠁⠀⠀⠀⡠⠃
⠀⠀⠛⠿⠿⠋⠀⠀⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⣣⣤⡀⠸⠟⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⣰
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣰⠾⣶⡶⢶⠿⣶⢶⡶⠶⢾⡿⢼⡿⠛⢷⡄⠀⠀⣠⡎⠀⠀⠀⠀⠀⢈⠠⠠⠃
⠀⠀⠀⠀⠁⠀⠀⠀⡄⠀⠀⠀⠀⣿⢀⡈⡀⣸⢠⣧⠈⠱⣯⣼⡇⢸⢁⡿⠛⠃⠀⠛⠋⠀⢀⡀⠀⠀⠀⠠⠒⠒⢢
⠀⣀⣠⡄⠀⠀⡀⠀⠀⠀⠐⠂⠀⣿⢸⣿⡇⢹⠈⠋⢸⣷⣄⠙⡇⢸⡀⢿⢶⡄⠀⠀⠀⠀⠈⠡⠀⠘⠁⠠⠟⠒⢲
⠀⠘⠿⢁⡄⢦⠙⠀⠀⠀⠀⠀⠀⢿⡼⠛⠷⠾⢷⣴⠟⢷⣤⠾⢷⡾⠷⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡭⣅
⠀⢀⠔⢤⠙⠀⡀⠀⠀⠀⢰⡆⣾⠀⠀⢰⡆⣶⢠⣶⠀⠀⣶⠀⡷⠀⠀⣶⠀⣶⢰⡇⠀⠀⠀⠠⠀⠀⠀⠀⠔⠠⢬
⠀⠈⠙⠀⠀⠀⠀⡠⠀⠀⠸⠇⠻⠀⠀⠸⠇⠿⠘⠟⠀⠀⠻⠐⠇⠀⠀⠛⠀⠛⠘⠃⠀⠀⠀⠀⣠⣄⠀⠠⠖⠒⢦
⢰⠆⠀⠀⠐⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠠⠧⠤⠤
⢀⡚⠀⠖⠄⠀⡄⠀⠀⠀⢀⣠⣄⣠⣾⣶⠀⢠⠀⠀⢀⠀⠰⠶⠶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣄⣈⣤
⠀⠉⠉⠲⠒⡑⣉⠤⠖⠦⢺⣿⣿⣿⣟⡟⢀⣾⣿⡖⠈⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢡⣙⢻⢒⡀⠀⠀⠙⠿⣿⠏⠀⠀⠈⠹⠀⠀⠀⠀⠛⠀⡿⠀⠀⠀⢀⢀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠒⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⡶⠺⠿⠛⠟⠓⡲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠫⠃⠁⠀⠀⣀⣠⣤⣤⣈⡲⡈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡡⠂⠀⠀⣠⠶⠋⠉⠀⠀⠀⠈⠙⢶⡄⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⠊⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠡⠃⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠣⠃⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠻⢶⡴⠿⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠱⢺⠋⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⠁⡞⠀⢠⣎⠉⠙⢦⠈⢷⡀⠀⠀⣀⣤⠤⠶⠶⢼⠇⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢼⠀⣇⠀⢸⠀⠱⡀⠀⢣⠈⣧⠴⠋⠁⢀⣠⠤⠔⣮⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⢸⠀⠘⡄⠀⢱⠀⢀⠖⠁⢀⡤⠚⠁⠀⠀⡔⠀⠀⠁⢢⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⡆⠈⢆⠀⠱⡀⠈⡴⠃⢀⡴⠁⠀⣀⠔⠂⠉⠀⣀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠸⡄⠈⢆⠀⠈⢺⠁⢠⠋⠀⢠⠊⠀⡠⢔⡩⠠⠄⢑⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣙⢦⠈⠣⡀⢂⢀⠇⠀⢰⠃⠀⠌⡐⠁⠀⢀⣤⢂⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣬⣽⣾⡀⠀⢸⠀⠀⡀⠑⠤⡔⣇⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣤⣤⠬⠽⠖⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠁
⢀⣀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⢀⡀
⠀⠈⠙⠲⢤⣀⠀⠀⠀⠠⣄⡉⠲⢄⠙⠢⣄
⢄⡀⠀⠀⠀⠈⠑⠢⣄⡀⠀⠉⠲⢤⡉⠂⠈⠑⢄
⠤⣄⣁⢦⣤⣀⠀⠀⠈⠙⠳⠦⣄⠀⠈⠓⠄⢀⡀
⠀⠀⠀⠉⠓⠛⠙⠒⠦⢄⡀⢀⣈⠙⠻⠿⣶⣤⣉⠲⢄⡀⢀⡀
⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⢄⣀⡈⠉⠒⠢⢄⡉⠛⠷⢦⣈⠐⠉⠢⢄
⠤⠬⢍⣉⡙⠛⠓⠶⠦⠤⣤⣀⡉⠉⠒⠢⠤⠀⠀⠐⢤⣌⡑⠢⠑⠢⢁
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣀
⠉⠉⠉⠉⠉⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⣒
⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠒⠒⠒⠂⠐⠒⠒⠒⠩⠭⠭⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⣀⠀⠄
⠤⠤⠤⠶⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠐⢊⣉⡉⠤⠀⠈⠉⠀⢀⠤⠔⠂
⠔⠒⣒⣀⡀⠄⠀⠀⠀⠀⠀⠀⣀⣀⣤⠽⠟⠒⠉
⠉⠉⠁⠀⠀⠀⠀⢀⡴⠖⠛⠉⠉⠁
⠀⣀⡤⠴⠒⣊⡩⠥⠒⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡠⠔⡀
⠀⣤⡶⠞⠋⠁⠀⠀⠀⠀⠀⣀⡤⠒⠉⢀⡠⠖⣩⠔⠋
⠋⠁⠀⠀⠀⠀⠀⠀⠀⠒⠋⠁⠀⠀⠀⠀⠐⠊⠁
⠀⠀⢀⡠
⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠇⡟⢳⢼⠋⢸⡏⠁⡟⢻⠀⢸⠂⢸⡇⣿⣸⢸⡋⠱⣏⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣙⡆⡟⠋⢸⡉⢸⡏⠁⣇⣸⠀⢸⡀⢸⡇⡏⣿⢸⡋⢠⣝⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠈⠉⠈⠉⠁⠉⠉⠀⠈⠉⠉⠁⠁⠈⠈⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⢒⠶⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠀⡠⠊⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠁⠀⠀⠀⠀⠚⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⠤⠘⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠒⠒⡲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢰⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⢠⠊⠉⠉⠀⠀⠈⠢⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⢀⠆⠀⠀⠀⠀⠀⠀⠀⣁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⠔⠒⠉⠉⠉⠑⢄
⢈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣠⠃⠀⠀⠀⠀⠀⠀⠀⢱
⠈⢓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠈⠢⠄⠤⠤⠠⢄⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠑⠠⡄⠀⠀⠈⠁⠀⢠⣿⡇⠈⠁⠀⠀⠀⣀⠄⠒⠉⠁⠁⠂⠉⠑⠤⠠⠄⣀⡠⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡴⠀⢣⠀⠘⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⠜⠁⢀⡼⠀⢐⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠈⠀⠈⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⡤⠤⢈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠄⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⡀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⣀⣉⣀⣀⡠⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⣒⣒⣊⣩⣭⣭⣭⣭⣍⣉⣐⣒⠒⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⣤⣄⣀⣠⣴⣺⣵⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣶⣦⣭⣙⡲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣶⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣯⠿⠶⢯⣝⠻⣿⣿⣿⣦⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣮⢻⣿⣿⣿⣿⣿⡵⠋⠁⠀⠀⠀⠈⢷⡹⣿⣿⣿⣷⣾⣡⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠘⣾⣿⣿⣿⣯⠞⠀⠀⢀⡄⠀⠀⠀⠈⣧⢿⣿⣿⣿⣿⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠟⣿⡿⣻⠃⠀⠀⠀⣸⢻⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⡷⠋⠀⠀⠀⠀⢿⣾⠁⠀⠀⠀⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⢷⣤⡹⠃⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡿⠏⠉⠑⠦⣄⡀⠀⠀⠀⠀⢀⣠⠴⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢳
⠀⠀⠀⢠⠞⠉⢭⡑⣲⠖⠲⢄⠀⠀⠀⢳⡀⠀⠀⠀⠀⠈⠙⠛⠛⠋⢩⣥⣤⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠇
⠀⠀⣴⢿⡀⢀⡴⠋⠁⠀⠀⠈⠀⠀⠀⠀⠳⣄⠀⠀⠀⢠⣄⣀⣀⡤⠞⠁⠈⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠉⠙⠻⢿⣿⣿
⠀⠀⣿⢄⡉⢻⣄⠀⣠⠇⡠⠊⢑⣦⣄⡀⠀⠈⠓⢦⣄⡀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠸⣄⣈⣉⣨⡟⠁⡜⠀⠀⠀⢀⣠⣿⠀⠀⢀⣠⠜⠛⠛⠳⢶⣶⡞⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠳⣤⡠⠼⣁⢸⡇⠀⠀⢠⡏⠉⠛⠛⠉⢩⠃⠀⠀⠀⠀⠀⠙⣿⣶⣶⣤⣤⣀⠀⠈⠻⡿⠟⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠾⣄⠘⣹⠗⠒⠒⢺⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⠻⣿⣿⣝⠲⠎⠀⠀⠉⣠⠜⠛⠉⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡀⠀⠉⠙⠺⣤⠀⢀⡼⠁⠀⠀⠀⠀⠀⡀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣄⣀⣀⣤⣶⣿⣿⣿⣿⠇⠀⠀⠀⠀⠘⠦⣼⠁⠀⠀⠀⠀⠀⢢⢸⠆
⣾⣿⣿⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣠⣤⡿⢿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⢃⠀⠠⡄⠀⠀⢧⣠⠟
⠙⣷⡙⢿⣿⣿⣿⣿⠗⠢⢤⡤⢾⠁⢠⣼⣿⣯⡴⠿⠿⠿⣿⣿⣟⠉⠁⠉⣛⣛⢓⣀⣤⣾⣿⣷⣦⠙⠛⠅⣀⣠⠞⠋
⠀⠈⢻⣄⠙⣿⠟⠁⠀⢀⣴⠀⠈⠂⠈⢿⣿⢻⡄⠀⠀⠀⠹⣿⣿⣷⣤⣎⣁⠈⠛⠀⠈⠻⣿⣿⣿⣷⡀
⠀⠀⠀⠙⢷⠀⠀⠀⣠⣾⣿⣧⠀⠀⠀⣀⠉⠀⡇⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣇⠀⠀⠀⠀⢻⣿⣿⡿⠿⢦
⠀⠀⠀⠀⠈⠓⢄⣤⣙⠿⣿⣿⣿⣦⣤⣬⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠙⠛⠀⠀⠀⠀⡾⠋⠁⠀⠀⣀⣧
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣽⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣀⡤⠶⠤⢾⠀⢀⣠⣶⣿⣿⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣇⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠴⠶⠶⠖⠒⠒⠶⠶⠶⠦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠉⠛⠲⢤⣄⡀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⢤⣀⣤⠴⠖⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠜⠛⠋⢉⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢸⣏⢉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠐⠐⠉⠀⠙⣅
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣋⣀⡀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⡹⠋⠉⠉⠉⠉⠉⠉⠛⠛⠛⠓⠲⠶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠉⠳⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⣁⣀⣄⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢻⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⣾⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠼⣀⡀⠀⠸⣿⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⡤⠤⠤⠶⠶⢶⡾⠁⠀⠀⠀⣠⣀⠀⢀⣠⣤⣄⠀⠀⠀⠀⣰⠋⠀⠀⠀⠈⠑⠚⢣⣤⣤⣶⠄⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠛⢳⠾⠷⠶⠶⢶⠛⠁⠙⣮⠉⠀⣀⣈⠙⠦⣄⠀⢷⡀⠀⠀⠘⣀⣠⠴⠋⠀⣠⡴⠞⠁⢀⡴⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⣸⠇⡀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠸⡷⠚⠉⠈⠙⠢⠼⠀⠀⡩⢻⠻⢭⣉⠀⢀⣠⡟⠁⠲⣄⡠⠎⠀⣠⠆⠈⣧
⠀⠀⠀⠀⠀⢀⣤⠞⢁⠤⠒⠁⢹⠤⡀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⡿⢦⡀⠀⠀⠀⠀⠀⠰⠁⢸⠓⢦⣈⣉⡉⢹⡄⢢⣀⣸⠧⡤⠚⢁⡠⠂⡼
⠀⠀⠀⠀⣰⢏⡵⠊⠁⠀⠀⠀⠘⢷⡠⠄⠀⠀⠀⣀⣠⣶⣛⣋⣀⣠⠞⠀⠀⠀⠀⠀⠀⠁⣠⠋⠀⠀⠈⠉⠉⠉⠻⣄⠈⠀⣲⣬⣶⣋⣠⠞⠁
⠀⠀⠀⡾⡡⠋⠀⠠⠀⠀⠀⢀⡴⠋⠉⠙⠛⠛⢻⡍⠀⠉⠓⠲⠤⢄⣀⠀⠀⠀⢀⣠⠥⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋
⠀⠀⢸⢣⡃⠀⠀⠀⠀⠑⢠⠎⠀⢠⠎⠀⠀⢀⣾⣀⠀⠀⠀⠀⢀⡴⠃⠀⣄⠀⠀⠙⠢⣄
⠀⠀⣿⢼⠀⠁⠀⠀⣠⡴⠻⣆⣠⠏⠀⢀⣴⣏⠀⠉⠳⢦⣤⡴⠋⠀⢀⡴⠋⠙⢦⣀⠀⠈⠳⣄
⠀⠀⣿⢸⠀⠀⠀⣴⠋⠀⠀⠀⠙⣦⡴⠟⠁⠈⠛⠶⣤⣀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠉⠳⣄⠀⠘⢷⡀
⠀⠀⢿⢸⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠙⢦⡀⢀⣀⣄
⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⢀⣨⠟⠋⠁⠨⣇⠀⠀⠀⠀⠀⣀⣠⣤⠤⠴⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⣨⣷⠴⠶⠚⠛⠉⠁⠀⠀⠀⡼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢀⣠⡴⠛⠁⠀⠆⠀⣠⠴⠛⠙⡄⢀⡼⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣞⠋⠁⠀⠀⠀⠀⠀⡏⠡⠒⢉⠆⢘⣯⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢘⣆⣉⣥⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡓⠂⠤⠤⠤⠒⢊⣩⡤⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣤⡤⠶⢺⣿⢱⠐⢐⠠⡆⣾⡱⠀⠂⣆⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠀⠒⠒⠀⠀⠀⠀⠒⠒⠀⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠉⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⣀⠠⠐⠒⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⠤⡀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⢠⠂⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠐⡄
⠀⠀⠀⠀⠀⢀⠃⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠐⡀
⠀⠀⠀⠀⠀⡈⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢀
⠀⠀⠀⠀⢠⠁⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⡄
⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢠⠀⡆⠠⠐⠉⠀⠱⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠎⠀⠉⠂⠄⢰⠀⡈
⠀⠀⠀⠀⠈⡄⢸⠑⡀⠀⠀⠀⠐⡄⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢠⠂⠀⠀⠀⢀⠊⡇⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠈⢄⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⡠⠁⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠈⡀⠄⠀⠈⢆⠀⠀⠀⠈⢂⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⡐⠁⠀⠀⠀⡰⠁⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠇⠈⠄⠀⠈⢢⠀⠀⠀⠀⠣⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⡜⠀⠀⠀⠀⡔⠁⠀⠠⠁⠸
⠀⠀⠀⠀⠀⠀⠘⡄⠈⢆⠀⠀⠡⡀⠀⠀⠀⠱⡈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⠎⠀⠀⠀⢀⠌⠀⠀⠠⠁⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⢆⠀⠀⠱⡀⠀⠀⠀⠐⡌⢢⠀⠀⠀⠀⠀⠀⠀⠀⡔⢠⠂⠀⠀⠀⢀⠊⠀⠀⡰⠁⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠢⠀⠀⠘⠄⠀⠀⠀⠘⢄⠡⡀⠀⠀⠀⠀⢀⠌⠠⠁⠀⠀⠀⠠⠂⠀⢀⠔⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠑⡀⠀⠈⢆⠀⠀⠀⠈⢂⢡⠀⠀⠀⠀⡌⡐⠁⠀⠀⠀⡰⠁⠀⢀⠊⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⢈⢆⠀⠀⢢⠀⠀⠀⢀⠏⠀⠀⠀⠀⠹⡀⠀⠀⠀⡔⠀⠀⣰⠁⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠒⠤⠵⠐⠊⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠮⠤⠒⠉