Hot water Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⢘⣷⣴⠶⠶⠶⠶⠶⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠖⠛⠛⠛⢻⠀⠀⢰⡟⢛⡄⢠⡉⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⢀⠶⠞⠃⠀⠘⠛⠛⠀⠈⠙⣣⠀⠀⢀⣴⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢄⣸⣄⣀⣠⣄⣀⣀⣀⣤⣤⠴⠿⠗⠛⠋⠁⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠚⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⢀⣰⠟⠁⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀
⠀⣠⡾⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠤⠤⠤⠤⠶⠶⠶⠶⠄⠀⠂⠀⠀⠉⠉⠉⠉⠒⠶⠶⣄⠀⠀⠈⢷⡀
⢠⡟⠀⠀⠀⢀⡠⢤⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠈⢷⡀
⣸⠇⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠛⠀⠀⡀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠸⠀⠀⠀⠘⣇
⣿⠀⠀⠀⠀⡇⠀⣠⡄⠀⠀⠀⠀⠊⠀⠀⡀⠠⡄⠀⠀⠉⠀⣀⠰⢆⠀⠀⠳⡄⠙⠀⠀⠠⣄⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⡇⠀⠀⠋⠀⠀⢀⡀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠄⠀⠀⢀⠀⠀⠀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠠⣄⢤⡀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠀⠀⠀⢠⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠇
⢹⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠢⠌⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⢁⠀⠀⠀⠀⠀⠀⡄
⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠁⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀
⢸⡇⠀⠀⠀⢸⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠃⠀⠀⠀⠀⢸
⠀⣇⠀⠀⠀⠈⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠉⠀⠀⠀⠶⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢿⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⣿⠀⠀⠰⠀⠱⠄⠀⠀⠀⠈⢦⠀⠙⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠸⣧
⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⢸⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠈⠃⠀⠀⠀⠀⠈⠓⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⢸⡆⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠘⣇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠲⠄⠙⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠘⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢹
⠀⠀⡇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⡦⠀⠀⠀⢸⡀⠀⠀⠀⢸
⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠋⢀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠸
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠐⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢸⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢠
⠀⠀⠸⡇⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠞⠁⠀⠀⠀⣸
⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠙⠶⠶⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⠶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⡀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⡇⠀⠀⠀⠀⡆⠀⢸⣆⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⠿⠟⠻⠛⠀⠀⠀⠙⠃⠀⠀⠀⠛⠛⠿⠿⠿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠐⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠒⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠀⠒⠒⠒⠒⠒⠲⠒⠒⠂⠀⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⣠⣴⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡄⠀⠀⠀⠀⠀⠀⠀⣠⡔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣄⡀
⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⠆⠀⠀⠀⠙⠷⣶⣤⣤⣤⣴⠾⠋⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⡀
⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠉⣀⣤⣾⠶⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⣠⣴⡿⠟⠋⠀⠀⣀⠀⠀⠀⠀⠀⠈⢻⣿⣆
⢰⣿⡿⠀⠀⠀⠀⢀⣤⣶⠿⠛⠁⣀⣴⡾⠟⠋⠀⠀⢀⣠⣶⡶⠆⠀⠀⠀⢀⣤⣶⠿⠛⠁⠀⢀⣠⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⢻⣿
⣿⣿⠁⠀⠀⠀⠰⠿⠋⢁⣠⣶⡿⠛⠉⠀⠀⢀⣤⣶⠿⠛⠁⠀⠀⣀⣴⡾⠟⠋⠁⠀⢀⣤⣾⠿⠋⠁⠀⠀⣀⣴⡆⠀⠀⠀⠀⠈⣿
⣿⣿⠀⠀⠀⠀⢀⣴⣾⠿⠋⠁⠀⠀⣀⣴⣾⠟⠋⠁⠀⢀⣠⣶⡿⠟⠉⠀⠀⣠⣴⡿⠟⠋⠀⠀⢀⣠⣴⡿⠟⠋⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠛⠋⠀⠀⢀⣠⣴⡿⠛⠉⠀⠀⣀⣤⣾⠿⠛⠁⠀⢀⣠⣶⠿⠛⠁⠀⠀⢀⣤⣶⠿⠛⠁⠀⠀⣀⣤⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⢀⣤⣾⠿⠛⠁⠀⠀⣀⣴⡾⠟⠋⠁⠀⣀⣤⣾⠿⠋⠁⠀⠀⣀⣴⣾⠟⠋⠁⠀⠀⣠⣴⡾⠟⠋⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠾⠟⠋⠁⠀⢀⣠⣴⠿⠛⠉⠀⢀⣠⣴⡿⠟⠋⠀⠀⢀⣠⣶⡿⠟⠉⠀⠀⢀⣤⣶⠿⠛⠁⠀⣀⣤⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠋⠁⠀⢀⣤⣶⠿⠛⠁⠀⠀⣀⣤⣾⠿⠛⠁⠀⠀⣀⣤⣾⠟⠋⠁⢀⣠⣴⡾⠟⠃⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⢰⡾⠟⠋⠁⠀⣀⣤⣾⠟⠋⠁⠀⠀⣠⣴⡾⠟⠋⠁⠀⢀⣠⣴⡾⠟⠋⠀⢀⣤⣶⠿⠛⠉⠀⣀⣤⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠛⠉⠀⠀⢀⣠⣶⡿⠛⠉⠀⠀⢀⣤⣶⡿⠛⠉⠀⣀⣴⣾⠟⠋⠁⢀⣠⣴⡾⠟⠉⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⢠⣶⠿⠛⠁⠀⠀⣀⣤⣾⠿⠋⠁⠀⠀⣀⣤⣾⠟⠋⠁⢀⣠⣴⡾⠟⠉⠀⢀⣤⣶⡿⠛⠉⠀⣠⣤⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠈⠁⠀⠀⣠⣴⡾⠟⠉⠁⠀⢀⣠⣴⡾⠟⠉⠀⢀⣤⣶⠿⠛⠁⠀⣀⣴⣾⠟⠋⠁⢀⣠⣶⡿⠟⠁⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⢀⣠⣶⡿⠛⠉⠀⠀⢀⣤⣶⠿⠛⠁⠀⣀⣴⣾⠟⠋⠁⠀⣠⣴⡾⠟⠉⠀⢀⣤⣶⠿⠋⠁⠀⣠⣤⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠘⠋⠁⠀⠀⣀⣴⣾⠟⠋⠁⠀⣠⣴⡿⠟⠉⠀⢀⣠⣶⠿⠛⠁⠀⣠⣴⡾⠟⠋⠀⢀⣠⣶⡿⠛⠁⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⢀⣠⣴⡿⠟⠉⠀⣀⣤⣶⠿⠛⠁⠀⣀⣴⡾⠟⠋⠁⢀⣠⣶⡿⠟⠉⠀⣀⣤⣾⠿⠋⠁⢀⣠⣴⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⣴⠿⠛⠁⠀⣀⣴⣾⠟⠋⠀⢀⣠⣴⡿⠟⠉⠀⢀⣤⣾⠿⠋⠁⠀⣠⣴⡾⠟⠋⠀⢀⣤⣶⠿⠋⠁⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⢀⣠⣶⡿⠛⠉⠀⢀⣠⣶⠿⠛⠁⠀⣠⣴⡾⠟⠋⠀⢀⣤⣶⠿⠛⠁⠀⣀⣴⡾⠟⠋⠀⢀⣠⣶⠀⠀⠀⠀⠀⣿
⢹⣿⠀⠀⠀⠀⢴⠿⠛⠁⠀⣀⣴⡾⠟⠋⠀⢀⣠⣶⡿⠛⠉⠀⣀⣴⣾⠟⠋⠁⢀⣠⣶⡿⠟⠉⠀⣀⣤⣾⠿⠋⠁⠀⠀⠀⠀⠀⣿
⢸⣿⡆⠀⠀⠀⠀⢀⣠⣴⡿⠛⠉⠀⣀⣤⣾⠿⠋⠁⢀⣠⣴⡿⠟⠋⠀⢀⣤⣾⠿⠛⠁⠀⣠⣴⡿⠟⠋⠀⢀⣤⣶⠀⠀⠀⠀⢀⣿
⢸⣿⡇⠀⠀⠀⣼⠿⠛⠁⠀⣠⣴⡾⠟⠋⠀⢀⣤⣾⠿⠛⠁⠀⣀⣴⡾⠟⠋⠀⢀⣠⣶⠿⠛⠉⠀⣀⣴⡾⠟⠋⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⢀⣠⣴⠿⠛⠉⠀⣀⣤⣾⠟⠋⠁⢀⣠⣴⡿⠛⠉⠀⣀⣤⣾⠟⠋⠁⢀⣠⣴⡿⠟⠉⠀⣀⣴⡶⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⢾⠿⠋⠁⢀⣠⣴⡿⠟⠉⠀⢀⣤⣾⠿⠛⠁⢀⣠⣴⡿⠟⠋⠀⢀⣤⣶⠿⠛⠁⢀⣠⣴⡿⠛⠉⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⢀⣤⣶⠿⠛⠁⢀⣠⣴⡾⠟⠋⠀⢀⣤⣾⠿⠛⠁⠀⣀⣴⣾⠟⠋⠁⣀⣤⣶⠿⠛⠁⣀⣤⣶⡿⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⢾⠟⠋⠁⢀⣠⣶⡿⠛⠉⠀⣀⣤⣾⠟⠋⠁⢀⣠⣴⡿⠛⠉⠀⣠⣴⡾⠟⠋⠁⣀⣤⣾⠟⠋⠁⣀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⣀⣤⣶⠟⠋⠁⢀⣠⣴⡾⠟⠉⠀⢀⣤⣶⠿⠋⠁⢀⣠⣴⡿⠛⠉⢀⣠⣴⡿⠟⠉⢀⣠⣶⡿⠛⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠼⠟⠉⠀⢀⣤⣶⡿⠛⠁⠀⣠⣴⣾⠟⠉⠀⢀⣤⣶⠿⠋⠁⢀⣤⣶⠿⠛⠁⣀⣤⣶⠿⠋⠁⣀⣤⠀⠀⠀⠀⢸⣿
⢸⣿⣧⠀⠀⠀⠀⢠⣴⣾⠟⠋⠁⢀⣠⣴⡿⠛⠉⠀⣠⣴⣾⠟⠋⠀⣠⣴⣾⠟⠋⠁⣠⣴⣾⠟⠉⠁⣠⣴⡾⠟⠃⠀⠀⠀⠀⣼⣿
⠈⣿⣿⡀⠀⠀⠀⠈⠉⠀⣀⣤⣾⠿⠋⠁⢀⣤⣶⡿⠛⠉⢀⣠⣴⡿⠛⠉⢀⣠⣶⡿⠛⠉⢀⣤⣶⡿⠋⠁⣀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⢻⣿⣧⠀⠀⠀⠀⠀⠘⠟⠉⠀⣀⣤⣾⠿⠋⠁⢀⣤⣶⠿⠋⠁⣀⣤⣾⠿⠋⠁⣠⣴⣾⠿⠋⢁⣠⡶⠟⠁⠀⠀⠀⠀⠀⣼⣿⡟
⠀⠀⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⢠⣴⡾⠟⠋⠀⣠⣴⡾⠟⠋⢀⣠⣴⡿⠟⠉⠀⠠⠶⠛⠉⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿
⠀⠀⠀⠹⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟
⠀⠀⠀⠀⠈⠻⢿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣶⣶⣤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⡀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣶⣿⣿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠶⠶⠶⠶⠶⠶⣤⡾⠛⠓⠶⠶⠶⢦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⣴⠀⠀⢀⡤⠔⠢⠤⠤⠴⠶⠶⢦⠀⠀⢠⡀⠀⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠈⠉⠉⠉⠉⠙⠣⣄⣤⠞⠉⠉⠙⠛⠛⠋⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⢀⣠⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠊⠀⠀⠱⢤⡄⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠙⢦
⠀⢠⡟⠀⠀⠀⢀⣤⢤⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢄⠀⠀⠀⠀⠈⣆
⠀⣾⠁⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠸⡀
⢰⡇⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⣷
⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠆⠀⠀⢸⠀⠀⠀⠀⠀⢹⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠸⡇
⡇⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣷
⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿
⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿
⡇⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿
⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢸⠀⠀⠀⠀⠀⣿
⡇⠀⠀⠀⠀⠀⠇⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢆⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢼
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸
⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⣀⠀⠀⠀⣿⡆⠀⠀⠀⣺
⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠠⣶⠄⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⣿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⣿⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⢸⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣸
⠀⠈⣷⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣿
⠀⠀⠘⢧⣄⠀⠀⠀⠀⠙⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠁⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠓⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠚⠛⠋⠉⠁⠀⠀⠀⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠻⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⠦⡄⠀⠀⢰⠋⠉⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠛⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⢿⠀⠀⣼⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣤⣀⡼⠃⢀⡼⠖⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠰⢯⠀⠀⠀⠀⠀⡞⠛⠛⠛⠒⠒⠲⠶⠤⢤⣄⣀⠀⠀⠀⠻⣄⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣄⠀⠹⠤⠴⠦⠴⠾⣃⣠⣤⠤⢴⣇⠏⠉⠉⠙⠒⠒⠒⠤⠤⠤⡌⢹⠆⣀⣴⣋⣀⣀⣰⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⠀⣈⣳⠀⠀⠀⣠⠖⠋⢁⣀⡤⠤⡾⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⣄⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢉⣡⣤⣤⡾⠁⣠⠖⣋⡁⠀⣰⣣⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣯⠘⣖⣒⡲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣼⠉⢩⡇⡼⠁⠀⠙⠃⢀⡏⡼⠑⠒⠒⠲⢤⣀⣀⣀⡤⠴⡿⢼⠏⣧⢹⠉⠉⣇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢻⣀⣈⣷⢧⣀⠀⠀⠀⣸⢱⢣⡖⣆⠀⠀⠀⠤⠥⠤⠄⢠⡇⣾⣠⣿⣿⣀⣠⠇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠷⣼⠃⠈⠉⠓⠲⠷⠿⠤⣭⣥⣄⣀⣠⣤⣤⠤⠶⠚⠛⠉⠁⠸⡦⠤⠤⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣃⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠈⠉⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⠉⠉⠉⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⣸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡇⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⢰⠛⣯⡟⣷⣦⣤⣀⣠⡄⠀⠀⠀⣶⣀⢀⣀⣤⣶⣞⣿⢸⠻⡄⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⢠⠏⠀⢻⣷⣿⣍⠙⡏⣧⠙⠿⢯⠿⠟⠹⢿⣿⡞⣁⣡⠟⡿⠀⢻⡀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠉⠛⠿⢿⣃⣻⣷⣶⣶⣶⡶⠿⢛⣘⣫⡭⠗⠋⠁⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⢠⣞⣁⣀⣀⣀⣀⣀⣀⣀⣸⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣽⣿⣯⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣿⣀⣀⣀⣀⣀⣀⣀⣀⣳⡄
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣠⠄⢠⡄⢀⡤⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡏⠀⣿⠀⢼⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠻⡄⠙⣦⠘⢧⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⡟⠀⡿⠀⣸⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠋⠀⠘⠁⠘⠃⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣝⠻⠿⠟⣩⣭⣭⣙⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⢛⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⣼⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣘⠿⠿⢋⣼⡏⣴⣮⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⣛⠻⣷⣬⣥⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡘⠿⢃⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⡈⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⡥⠻⠗⠁
⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⢀⠀⡑⢈⡐⠉⢂
⠀⠀⠀⠀⠀⠀⡄⢶⠎⡄⢉⠁⠀⣈⡤⠅⠀⠈⠰⠄⡄
⠀⠀⠀⠀⠀⠀⠈⡡⠀⠇⡐⠓⠞⢉⠄⠀⢦⢄⠄⠄⠠⠄
⠀⠀⠀⢠⠒⢢⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠙⠉⠓⢆⡀⡄⠀⠀⠀⠀⢀⡀⠀⠤⠤⠠⠤⠤⠤⣀
⠀⠀⠐⠁⢠⡮⠠⠚⡠⡂⠀⠀⢧⠀⠀⠀⠀⠀⠀⡂⠠⣇⠜⠀⠀⢀⢼⣡⡦⣤⠴⠤⡦⢤⣠⢤⣼⣦⡀
⠀⠀⠈⠫⠜⢀⠒⢀⠐⠄⠀⣤⠿⠀⠀⠀⠀⠀⢈⢮⠀⡸⠀⠀⢀⠂⠏⠀⠀⠀⢀⠀⡀⠀⠀⠐⠀⢁⢳
⠀⠀⠀⠀⠀⠜⠀⢀⢏⠀⠼⠃⠀⠀⠀⠀⠀⠠⡀⠩⡥⠁⠀⠀⢠⢸⠀⠀⠀⠀⢣⣲⠞⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠣⢀⠉⢘⡁⠀⠀⠀⠀⠄⠠⢀⠀⠈⠢⡐⠄⠀⠀⠀⢄⢇⣠⠐⠊⠉⠚⠛⠒⡤⠄⣀⡱⠁
⠀⠀⠀⠀⠀⠡⠠⠑⠄⠝⡆⠠⠒⡀⠉⠉⣰⡷⢢⣢⠮⢊⣷⡀⢀⢴⠏⠀⠀⠀⠈⠉⠉⠀⠀⠀⠢⠝⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠙⠒⠉⠀⠈⠈⠁⠈⠀⠉⠁⠈⠛⠿⣧⠈⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠂⣴⣶⣼⣶⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠨⢛⠺⠁⠀⠀⠀⢀⣀⢀⠀⣀⣛⢀⣉⣛⣛⣋⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠈⠁⠈⠈⠉⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠐⠂⡀⠀⠄⠐⠐⡈⠀⠀⠀⠀⠀⠀⠀⠘⡘⠄⡟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠑⠢⢄⣀⢂⠀⠌⠊⠌⢄⢀⢄⠀⢂⡢⠡⡀⡸⣡⣷⠚⠁⠀⠀⠢⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣂⣠⣀⠨⠉⠀⠂⠒⠲⠲⠖⠖⠒⢲⢒⣫⣽⠋⠀⠐⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠒⠒⠒⠒⠛⠚⠛⡉⠉⠁⠀⠁⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠁⠢
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠉⠀⠀⢀⡼⡃
⠀⠀⠀⠀⠀⠀⢀⣈⣀⣀⣀⠀⢀⣴⠟⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢢
⠀⠀⠀⢀⢠⣾⣿⠿⠛⠋⠉⠚⠉⢀⣀⡤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢂
⠀⠀⢀⢲⠟⠋⠀⠀⠐⠲⠶⠛⠛⠛⠉⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆
⠀⡔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⠀⠀⣀⠔⠉⠁⠒⡀
⠸⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠛⠛⠛⠉⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠿⢄⠂⠉⠀⠀⠀⠀⣠⡴⠟⢱
⢀⡧⢄⣀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡄⠈⡿⠟⠛⠛⢛⣋⣁⣠⣤⠼⠄
⢸⠁⠀⠀⠀⠀⠀⠈⠉⠁⣠⣌⠐⠒⠂⠈⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠻⠿⠛⠉⠑⠒⠚⠉⠉⠉⠉⠉⠉⠀⠀⢀⠼⢦⣀
⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⢐⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣤⡤⠶⠶⠒⠛⠉⠉⠂⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠘⢟⠶⣕⣀⠤⠄⠒⠤⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣴⠾⡻
⡇⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⢠⣶⡛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⢻⣷⡆⠩⠝⣛⠒⠒⠒⠶⠶⠶⠶⠖⠒⢚⣛⠩⠍⢒⠋
⢹⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⣤⠀⠈⡍⠙⠛⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠛⠉⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⢧⠈⠀⠀⠐⠂⠀⠠⠤⠤⠄⠀⠐⠒⠀⠈⠁⡸
⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠄⠀⠐⠀⠒⠒⠒⠒⠒⠒⠀⠀⠄⡜⠀⠸⡄⠀⠀⠀⠀⠀⢰⣶⡆⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⠀⠐⠒⠂⠀⢀⠀⠀⠀⠐⠒⠀⢨⣇⠀⠐⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠉⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠜⠇⡆⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⢸⠄⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⠇⠀⠀⠀⢱
⠀⠀⠀⢢⢄⠀⠀⠀⠀⠀⠀⠀⢀⠢⠁⠘⣇⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⣜⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠑⢕⡠⣀⠀⢀⠠⡐⠕⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⣧⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠠⠊⠁⠀⠐⠀⠀⠁⢂⠀⠀⠀⢊⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⢠⠠
⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠸⠧⠀⠀⠀⠘⠿⠀⠀⠀⠀⣞⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠈⠐⠠⠀⠀⠀⣀⣀⠀⠀⠀⠄⢃⠁
⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⢀⠁⠀⠒⠀⠀⠀⠒⠐⠈⡀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨
⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣾⣤⣤⣤⣤⣤⣬⣶⣶⣶⣶⣶⣶⣶⣥⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣶⣶⣶⣶⣶⣶⣾⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣦⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠸⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣄⠀⠀⢀⣼⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢰⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣠⣴⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠸⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣄⠀⠈⠙⢿⣿⣿⣆⠈⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠈⣿⣿⡟⠀⠀⠀⢸⣿⣿⣿⠀⠈⠻⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠛⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⢀⠝⠛⠳⠦⣀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠟⠋⠁⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⡔⠁⠀⠀⡘⠀⠀⠀⡄⠀⢠⠀⠀⠀⠈⠀⠈⢢⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⡠⠊⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢃⠀⠀⠑⡀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⡘⠀⠀⠀⢠⠀⠀⠀⢠⠀⠀⠀⠈⡄⠀⠀⠈⢆⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠔⠁⠐⢤⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡌⠀⠀⠀⠀⠆⠀⠀⠀⢀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠃⠀⠀⠀⠀⠸⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⢆⠀⡠⠒⠘⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠇⠀⣀⢀⣘⡀⠀⠀⠀⠀⠀⠱⡄⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⠐⢠⡀⠀⠀⠀⠀⠀⠀⠰⡎⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠈⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁

⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢀⣀⣌⣻⣡⣀⡀⠀⠀⠀⠀⠀⠀⢰⣾⣇⣾⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⡄
⠀⠀⠀⣾⠉⢿⣿⡿⠋⣷⠀⠀⠀⠀⠀⠀⣿⣿⣿⢾⣿⣧⠀⠀⠀⠀⠠⠶⠶⠶⠶⠶⠶⠶⠄⠄⠀⠀⠀⠀⠀⠀⢁⠀⡁
⠀⠀⠀⠹⣷⣶⣿⣶⣾⠏⠀⠀⠀⠀⠀⠐⠇⣼⣶⢨⡄⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⢿⣿⡿⠂
⠀⠀⠀⢠⣦⣿⣿⣯⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣷⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠂⠀⠻⠿⠟⠀⠐
⠀⠀⠀⠈⠱⣿⣿⣷⠍⠁⠀⠀⠀⠀⠀⠀⠘⠿⠟⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠂⠠⠃

⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄
⠀⠀⠀⡤⢄⣰⠿⣆⡠⢤⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣅⠀⠀⠀⠀⠀⠀⢀⠀⢀⣸⣿⣿⣇⣿⡇⠀⠀⠀⠈⣿⣿⣯⠀⢰⣯⣷⡀
⠀⠀⡀⢰⣶⣿⣿⣿⣷⠆⣀⠀⠀⠀⠀⠀⠘⠻⠟⠛⠻⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⣏⣿⡇⠀⠀⠀⠀⢿⣿⣿⠀⣾⣿⣿
⠀⠀⠀⢙⣟⣻⣿⣟⣻⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣽⣿⠉⠁⠉⠁⠀⠀⠀⠀⠘⣿⣿⢀⣿⣿⠏
⠀⠀⠀⠈⠉⠉⠀⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⠿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⣀⣄⡀⢰⡶⡄⢸⣿⣟⠀⠀⠀⠀⢿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⣾⣿⣛⣻⣿⣿⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⠀⣿⣿⣿
⠀⠀⠿⠿⠇⠼⡿⠧⠸⣿⡿⠀⠀⠀⠀⡈⠛⣿⡿⠛⠁⢀⡄⠀⠀⠀⠀⠀⢹⣷⣿⣿⣯⡝⠀⠀⠀⠀⠀⠋⠉⠉⠉⠉⠃⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠈⠋⠀⠀⠀⠀⠀⠀⠻⢿⣿⠟⠁⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠀⠨⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣴⣊⣩⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⢠⡟⠛⠛⣛⣿⣶⡄⠀⠀⠀⠀⠀⠀⣠⡘⡇⠀⡆⠀⠀⠀⠀⠀⠈⣿⣿⣾⣿⣿⣿⣿⣷⡇⠀⠀⠀⠀⢠⠀⠀⣀
⠀⠀⣴⠛⠛⠿⣿⣏⣿⣵⣄⠀⠀⠀⠀⠀⠀⠀⣦⠀⡃⠋⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣤⠲⣖⣲⣿⡖⣲⠖⣤
⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠘⢸⠇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⡠⠄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠑⠂⠁⠀⠈⡰⢖⡁⢈⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠
⠀⠐⠁⠀⠠⠀⠀⠐⠉⠆⠠⠂⠀⠰⠡⠜⠀⠤⠁⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⢃⠀⠱⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠂⡄
⠀⠀⠂⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⡸⢠⠁⠀⠀⠀⠀⠀⠀⠀⢰⠐⠲⠶⠳⠃
⠀⠀⠒⢒⣒⣛⣒⠒⠦⢤⣀⠀⠀⠀⠐⢄⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⡀
⠈⠀⠚⠒⠤⠤⠤⠤⠬⠟⠰⠀⠀⠀⠀⠀⠉⢃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠞⠤⠤⠘
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢨⠄⠀⠌⠆⠀⠀⠀⠓⠄⣀⠀⠀⠀⠀⠀⣀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡆⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⢀⠤⠔⠂⠠⠤⠤⠀⣀⡠⢀⠀⠀⠀⡇⠀⡀⠀⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠡⡡⠒⠒⠺⠭⢉⠁⠒⢌⡂⡅⡀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠉⠉⠉⠉⠉⠂⠜⠐⡌⠀⠀⡃⠀⠇⠄⢲⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣂⡜⠀⠀⠀⠀⡄⡀⢒⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡤⠔⡱⠀⠰⠓⢀⠀⠀⢀⠀⢃
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠔⠣⣀⠠⢣⠀⠀⡀⠀⠀⢀⣡⠀⠀⠀⠀⠀⠀⠀⣠⣉⡉⠀⠀⡇⠀⠈⡀⠀⡀⢀⠀⢂
⠀⠀⠑⠂⠤⠤⠤⠤⠔⠊⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⡅⠠⣘⠏⠄⢘⠀⠀⠑⠦⢤⣀⣀⠤⠎⠀⠀⠌⡇⠀⡇⠆⠀⠀⠀⢀⣀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠇⠀⠀⠉⠀⠀⠑⠓⠂⠠⠤⠤⠀⠀⠀⠀⠚⠉⠀⠀⠈⠢⠄⢀⣀⣀⣀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⠠⠤⠤⢬⣆⠀⠀⠀⠀⠀⢸⠳⠤⠤⠤⠤⠞⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠈⠁⠐⠒⠂⠈⠱⢳⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⣾⣑⣿⣿⣟⡫⠉⠚⣷
⠀⠀⠀⠀⠤⠀⠀⠀⠠⠤⠄⣀⠀⠀⡟⠢⠄⣀⣀⣀⡀⠤⢺⠀⠀⠀⢠⠇⡀⠀⠀⣀⠀⠀⢻⡀⠀⠀⠀⢰⠉⠓⠋⠉⠭⠭⠭⡽⠋
⡐⢁⠠⠄⠀⠀⠀⠀⠀⠠⠄⡀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠣⠀⠀⠀⠀⢀⠀⠘⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠢
⠏⡓⠠⠄⠀⠀⠀⠀⠠⠤⢔⣿⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠌⢰
⠀⣎⠁⠐⠒⠒⠒⠒⠒⠈⢁⡞⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠢⣤⣀⣀⣀⠤⠊⠀⠀⠀⠀⢸⠀⠀⠀⡀⠄⠂⠁⡠⠊
⠀⠈⠛⠲⠄⣀⣀⣀⠠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⢸⢀⠔⠈⠀⠀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢸⡎⠀⠀⢠⠊
⠀⠀⠀⡀⠄⠸⠐⠐⠤⡀⠀⠀⠀⠀⢘⠢⢀⡀⠀⢀⣀⠠⢺⠀⠀⠀⠀⠀⢀⠀⡸⢠⠠⠄⡀⠀⠀⠀⠀⠀⢁⡀⠀⡇⠀⠀⠀⠀⡠
⠀⠀⠀⠡⢀⣀⣀⡀⠔⠃⠀⠀⠀⠀⠈⠓⠦⢤⣀⣀⠠⠴⠋⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⠜⠀⠀⠀⠀⠀⠀⠡⠁⠘⠒⠐⡀⠥⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡡⠐⠀⢱⠂⠂⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⢀⢂⡊⠀⠐⡱⠄⠄⡪⠊⠡⢁
⠀⠀⠀⠀⠀⠀⢠⡀⠒⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⢠⢅⣄⢨⠐⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠀⠀⠀⠤⠤⠤⠤⠤⠤⠔⠖⢶⠖⠛⠉⢸⠢⡂⠀⡐⡈
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣼⠀⠔⡉⠈
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠈⠁
⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢠⠂⠒⠒⠒⠂⡈⠐⠦⢄⡀⠀⣀⠤⠒⠂⠀⠐⠢⡄⡇
⠀⠀⠀⠀⠀⠀⠀⡜⡀⠀⠀⠄⠀⠀⠄⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠏⠃⠀⠰⠀⠀⠀⠈⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻
⠀⠀⠀⠀⠀⠀⠀⢰⢠⠀⠀⠁⠀⠀⠀⡠⠘⠀⢠⠤⠠⠀⠀⠀⠀⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠨⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡄⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠇⡀⠀⠀⠀⡀⠁⡀⠀⠲⡄⠀⠀⠀⠀⠀⠀⢛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠐⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠸⠸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡄⠀⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠜⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡑⠬⣁⣒⠒⠒⠒⢒⣊⠭⠔⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣀⡀⠈⠉⠉⠀⢀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢰⣶⣶⣶⣶⣶⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣐⠦⠒⠒⠒⢴⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆
⠀⠀⢸⡏⣹⢈⡇⣿⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣯⡄⠀⠀⠀⠀⠀⠀⢦⠏⣤⢠⣤⣠⣄⢹⡤⠀⠀⠀⠀⠀⠐⣷⣦⣻⣿⣴⣿
⠀⠀⢸⡟⢻⠙⡟⣿⠿⡍⠀⠀⠀⠀⠚⠲⣶⣤⡰⢶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠲⡆⠚⠛⠋⠘⠙⣰⠖⠀⠀⠀⠀⠀⠀⠙⠿⣷⢿⠿⠃
⠀⠀⢸⡗⣿⣴⡗⣿⠀⠀⠀⠀⠀⠤⠶⢶⣾⣿⣿⣾⣿⣷⡶⠀⠀⠀⠀⠀⠀⠀⠙⠦⢄⣤⡠⠶⠉⠀⠀⠀⠀⠀⠢⢄⣑⣒⣒⣒⣒⣉⡤
⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⢀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⢠⣿⣤⠐⣨⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣦⣄⣰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠰⠶⠶⢶⣿⠶⠶⠶
⠀⠀⠸⠛⠻⠸⢛⣛⠓⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⣿⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀
⠀⠀⠀⣿⡇⠐⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⣽⣿⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣾⣿⣥⣽⣧⣶⣆⠀⠀⠀⠀⢀⣤⡾⠛⠉⠉⣿
⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⠀⡹⠄⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠙⠄⠠⡄⠀⠀⠀⠀⠀⠀⣴⣿⣿⣆
⠀⠀⠠⣶⠟⣻⣦⡀⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⡟⢿⣄⣀⡀⠀⠀⠀⠀⠀⠀⢴⣿⣷⣄⣀⣀⣾⣿⡆⠀⠀⠀⠀⠀⠻⣿⠛⢷⣶⣶⢶⣶
⠀⠠⠶⠦⣽⣻⠿⡿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠐⠓⠚⠋⠹⣇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣖⣒⣒⣒⣚⣀
⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠉⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠆
⠀⠀⢀⣄⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣸⡿⠁⠀⠀⠀⠀⠀⠀⢀⠔⠈⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⣴⣿⣄
⠀⠀⠈⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⢰⣿⡇⣷⠀⠀⠀⠀⠀⢸⣅⣀⣀⣀⣽⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⣿⣿⣿
⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣷⣿⣶⣶⣿⣾⡏⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿
⠀⠀⠀⠘⠿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠠⠞⠛⠛⠛⠛⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⡿⠇⠿⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠂⠀⠀⠈⢀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⣐⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠜⠶⠶⠶⠤⠶⠞⠃
⠀⠀⠀⠀⠉⠉⢁⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⣤⣤⣤⣄⣠⡤⠶⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠴⠲⠄⠂
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠱⠦⢤⣤⣤⣤⣤⡤⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢎⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⣀⠀⣀⣀⡉⣉⣀⣀⠤⠇⠀⡒⠈⠄⠠⠤⠤⠔⠀⠀⠀⠐⠄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠰
⠀⠀⠀⠈⠀⠐⠂⠀⠀⠀⠀⢇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠀⠀⠀⠀⠀⠞⢁⠀⠀⢰⣼⡂⡈⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠁⠀⠀⠀⠀⢰⣴⠓⡀⢈⡥⢃⣡⣆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠔⠡⠰⠀⠌⠨⠨⠘
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠈⠁⠀⠘⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠤⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⢤⠄⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠑⠂⢠⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠈⠓⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⠐⠄⠀⠀⡀⠠⠀⠀⠀⣄⠀⠀⠠⠘⡟⠓
⠀⠀⠀⠀⠀⠀⠐⠒⢂⠆⠈⠀⠀⢀⡀⠀⠀⠠⠂⠈⠀⠀⡄⠀⠁⠲⠁⠀⠀⠀⠀⠀⠁⠒⠂⠄⠁⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢰⡤⠠⠀⠀⠠⠀⠀⢀⡀⠀⠀⠀⠀⠉⠀⢤⣒⠀⠀⠤⠀⠂⠁⠀⡀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢈⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠈⠁⠁⠀⠀⠀⠉⠂⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠁⣼⠃⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠒⠊⠁⠀⣰⢿⣦⣀⣴⡏
⠀⠀⠀⣠⣿⣷⣶⣶⣤⣤⣄⠀⠀⢀⣀⡴⠚⠁⠀⠀⢀⣠⣴⠚⣩⢥⡺⣿⡭⠃
⠀⠀⢰⣿⠿⡿⢿⣿⣿⣿⠏⡴⠋⠉⠁⠀⠀⠀⠀⠀⣰⣿⡿⣋⣴⣿⠟⠋
⠀⠠⢀⣠⣠⣦⣾⣿⣿⠏⢰⣇⠀⠀⠀⠀⠀⠀⣀⡴⣟⣿⡞⠛⠿⠯⠱⡀
⠀⣆⣿⣿⣿⣿⣿⣿⡟⠀⡿⠽⠲⢦⣷⣶⣿⢿⣿⡾⢧⡀⠈⠢⣄⣀⣠⡇
⢠⣸⣿⣿⣿⣿⣿⣿⠃⠀⠐⢒⡶⠾⠛⠉⢀⣿⠿⢷⡄⠙⢦⣄⣀⣨⠇
⠸⢻⣿⣿⣿⣿⣿⣿⣤⡤⠛⠁⠀⣘⣶⣿⣿⣿⣷⣄⡈⢳⣶⡭⠟
⠀⠀⠀⠀⠉⠉⠉⠛⠩⠞⠃⢽⣦⣿⣿⣿⣿⣿⣿⡿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠫⠟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣍⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⢀⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⢠⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⣫⣭⣶⣶⣶⣦⣤⣀
⠀⠉⣵⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⡀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⢟⣵⣿⣿⡿⠿⠿⠿⢿⣿⣿⣷⣄
⠀⠀⠙⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣶⣶⣾⣿⣿⣿⠿⢋⣴⣿⠟⠉⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⡆
⠀⠀⠀⠈⠙⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⢤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣭⠽⠟⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢹⣿⠛⠛⠿⢶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣃⣀⠀⠀⠀⠀⠉⠙⠻⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠞⠁⠈⠉⠉⠛⠲⠦⣤⡀⠀⠀⠙⢿
⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠈⠓⠤⣀⣴⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣧⠀⠀⣸
⠀⠀⠀⠀⠸⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣿⣿⣿⣦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⣾⣫⣥⠴⠞⠉
⠀⣠⣤⡀⠀⠘⢿⣿⣿⠆⢠⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠭⠿⠻⢛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁
⠀⠹⣿⣿⣄⠀⣴⣿⣥⣄⠈⠻⣿⣄
⠀⠀⠈⠛⠋⠀⠈⣻⣿⠿⠛⠊⠉⠹⣆
⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⢀⣴⣿⠇⠘⢧⡀
⠀⠀⣴⠟⠁⠀⠀⠀⠀⠐⠿⠟⠋⣀⣄⠈⠻⣄
⢀⣾⠋⠀⠀⠀⠀⢀⣀⡀⠀⢠⣾⡿⠋⠀⠀⠈⠳⣄
⣼⡇⠀⠀⠀⠀⠀⠘⠿⢿⠿⠈⢹⣶⣦⣄⠀⠀⠀⠈⠳⣄
⣿⣇⠀⠀⠀⠀⠀⠰⣿⣦⣄⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠁
⢹⣿⣆⠀⠀⠀⠀⠀⠈⠻⠿⠇⣰⣿⣷⠄
⠀⠻⣿⣧⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠏
⠀⠀⠈⠻⣿⣷⣄⡀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠈⠙⠿⣿⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣷⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠿⠿⠶⠶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠴⠖⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢡⠈⠠⠠⠊⠀⠀⠀⡀⢀⡄⠀⠀⠀⠀⢄⠠⠀⣂⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⣀⣀⣈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠔⢈⠇⠀⠀⠀⢘⡆⠀⢸⣄⠀⠀⠀⠀⡁⠢⡀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡴⣉⠳⣌⡠⠔⠁⠀⠀⠀⡔⠁⠙⠋⠉⠀⢱⠀⠀⠀⠀⠤⡀⣊⠴⠓⣄⠀⠀⠀⠀⠀⠀⠀⠠⠂⠄⠀⠂⠀⠀⠠⣄⣀⣀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⢑⠎⠀⠀⠀⠀⠀⢘⠉⠉⠁⠉⠉⠉⠉⡂⠀⠀⠀⠀⠈⠡⢔⠩⠈⠀⠀⠠⠀⠀⠰⠂⠠⠀⠀⠂⠀⠀⠀⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢉⡉⠉⠉⠉⠉⡉⡅⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⠀⠤⠀⠂⠀⡆⠐⠀⠀⠀⠀⠀⠈⢠⡌⠀⠘⠁⠌⠃⠀⠀⠀⠀⠀⢀⠂⠀⠀⠖⠀⠀⠀⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠈⢷⠀⠀⠀⠠⠤⠥⠤⠁⠤⠀⠼⠄⠂⠀⠀⠀⡇⠘⠀⢀⡇⠨⠅⢈
⠀⠀⠀⠀⠀⠀⠀⠤⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠣⠄⠄⠀⠒⠚⠒⠒⠒⠂⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠶⠀⠀⠌⠀⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡤⠔⠉⡄⠀⠀⠀⠀⠠⠈⠂⠈⠀⠀⢰⠅⠀⠀⢠⠀⠀⠀⠀⠀⡀⠀⠀⠃⠀⢰⠠⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠟⡠⠔⠊⣁⠴⠀⠀⠀⠐⠁⠈⠀⡤⣃⠀⡀⢀⡀⣀⢀⡀⢀⠀⡀⠀⡁⠀⠀⠀⠀⡀⠠⠄⠘⠀⠐⡀
⠀⠀⠀⠀⠀⠈⠈⠀⠁⠀⠀⠱⠖⠂⠉⠂⠀⠀⡐⠀⢀⠀⠠⠄⠁⠉⠀⠀⠀⠀⠁⠈⠈⠉⠈⠁⠈⠁⠉⠉⠁⠁⠀⠆⠰⡀⠀⠐⡀
⠀⠀⠀⠀⠀⢭⣩⣉⣉⡁⠀⠀⠁⠀⠀⠀⠀⡜⢠⠀⠞⠁⠀⢠⢄⡀⠀⠄⠤⠤⠤⠠⠄⠤⠠⠀⠤⠠⢄⠤⠠⢄⠀⠀⠀⠹⠄⠤⠐⡄
⠀⠀⠀⠀⠀⠲⠤⢤⠠⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠀⠀⠀⠄⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢠⠂⠀⠀⠤⠁⢀⡆⠀⠠⠆⠀⠀⠀⠀⠈⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠘⢤⠀⠡⠀⠀⠘⢂⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠡⠈⠀⠀⠀⢀⠔⠁⠂⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠈⠤⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠈⠀⡠⠊⠀⠀⠀⠀⠀⣀⠀⠀⠸⠁⠀⠀⠀⢰⠀⠀⠀⢼⠀⠀⠀⡀⠀⠀⠈⠣⠀⠈⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡔⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⡄⠀⠀⠘⠀⠀⠀⠀⠀⣀⠀⠇⠀⠀⠀⠀⠐⡄⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⣀⠀⡜⠀⠀⠀⠀⠀⠰⠒⠀⠒⠐⠀⠂⠀⠂⠐⠄⠢⠤⠄⢄⠠⢄⢤⣠⠀⠀⠀⠀⠀⠀⠘⡄⢀⠀⠀⡡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⡀⢀⡜⠀⠀⠀⠀⠀⠀⢠⠤⠠⠤⠤⠠⠠⠤⠄⠀⠀⠄⠤⠠⠄⢤⢠⡤⡘⠀⠀⠀⠀⠀⠀⠀⠐⡤⢀⠉⠐⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣀⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢁⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡎⠀⠅⡠⠂⠀⠐⠃⢀⠀⡆⠈⠠⠁⠘⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠉⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠄⣸⣏⣻⡏⢈⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡖⡦
⠀⠀⠀⠰⢐⠀⠙⠋⡣⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠐⠐⠒⠐
⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢶⣂⣄⠰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⡀⠀⠀⠀⠀⠰⠀⠀⠀⠤⠹⡧⠐⠢⠀⠀⠀⠀⠀⡀⠀⠀⣠⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠡⡉
⠀⢀⠄⠀⠀⠀⠀⠐⠀⢡⡶⢴⡤⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⣀⡤
⠠⠎⡀⠀⠀⠀⠀⠐⠀⠟⣷⡾⠃⡂⠀⠀⠀⠀⠐⣍⠀⠀⠀⠔⠳⠔⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠐⠕⠖⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⡿⠀⠀⡆⠀⠀⠀⠤⠤⠤⠄⣀⣀⣀⡀
⠀⠀⠀⠀⣀⣠⣤⠴⠒⠋⠉⢁⣀⠤⠄⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⣄⣀⠀⠀⢀⣉⠉⠙⠳⢦⣄
⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⣶⡋⠁⠀⠀⠀⠀⠀⠀⢴⡄⠀⠠⠤⠀⠀⠀⠀⠀⠀⠈⠙⣶⠀⠀⠀⠀⠀⠀⣿
⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠳⠶⠶⠶⠦⣤⣤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠛⠛⠁⢀⣀⣀⣤⣴⠞⠋⡀
⠀⢠⣭⠿⣿⠷⠦⠤⢤⣤⣤⣤⣄⣠⣤⣄⠀⢀⣀⣀⣀⣀⣠⣀⣤⠤⠤⠤⠴⠖⠒⠚⠋⠉⠻⠋⠁
⠀⠀⢀⡆⠉⠙⠓⠒⠒⠠⠴⠶⠶⡤⠄⠀⠀⠴⠲⠐⠀⠀⠀⠀⠤⠖⠠⠀⠀⠀⠉⠀⠀⣀⣀⣤⡤⠴⡖⠉
⠀⠀⠈⠇⠐⠒⠶⠤⣤⣤⣤⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⡴⠶⠶⠞⠛⠋⠉⠉⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠃⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣀⠀⢸⠇
⠀⠀⠀⠀⢿⡔⠀⠁⠀⠀⢀⠤⠤⠤⠄⠀⠀⠀⢀⣀⣤⡤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⠁
⠀⠀⠀⠀⢹⡟⠦⢤⠀⢀⡎⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⠀⠉⠐⠈⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠉⠀⠀⠀⣐⠶⠖⠷⠦⣤⢄⣀⣀⣀⣠⣤⣤⣤⣤⣤⡔⠚⠒⠉⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠠⠒⠉⠀⠀⠀⠀⡀⠈⠀⢀⡴⠋⠁⠀⠀⠤⡄⠀⠉⠉⠲⢄⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⢻⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠊⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⣧⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢿⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⡆⠰⠁⠀⢸⡀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠈⠂⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⢀⣀⡀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠤⠐⠂⠉⣁⣤⠌⠒⠒⠲⠖⠛⠁⠀⠀⠀⠐⠀⠀⠄⡁⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣄⡀⠀⠘⠧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢀⣠⠟⠁
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⣤⣤⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠶⠛⠁
⠀⠀⠀⠐⢧⣄⣀⣐⠂⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠳⠶⠶⠶⠖⠒⠒⠊⠋⠉⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣦⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⠄⠰⢂⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢳⣼⣿⣿⣧⣶⣀⣶⣢⣀⣀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠙⣳⡿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣦⣤⣶⣾⣿⣆⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⡄⡀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣴⣾⣆⠀⠀⠀⠀⠀⠀⠀⢀⡄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⢀⠀⠀⣸⣿⣿⣿⣿⡿⠋⠉
⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠈⢀⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣀⣀⣠⣿⣿⣿⣿⣿⣿⡖⠐⠀⠀⠀⠀⣀⣴⣦⠝⠛⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⢻⡿⠿⠿⣿⣿⣟⠛⡿⠋⠉⠀⠈⠳⠶⠍⠉⠁⠀⠀⠀⠀⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⢈⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢸⣷⣀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⣿⣿⡗⢶⣾⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⢹⣿⣿⣯⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠈⣿⣧⠀⢈⠁⠀⠀⠀⠈⣾⣄⠰⡄⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⠈⠀⠀⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣷⡿⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢹⣿⣿⣧⣴⣶⣼⣿⣄⣶⣿⣿⣿⠸⣦⠀⠀⠀⠀⠀⣿⣿⠏⢀⡞⢠⢯⣿⣿⣿⣿⣿⣿⣿⢟⡡⠀⠀⠀⠠⠔⢤⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢿⣷⣤⣀⣠⣶⣿⣿⣷⣞⣷⣏⡾⣻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠢
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⣠⣄⣲⣥⣙⣿⣿⣿⣿⣿⣿⣿⡾⡻⠂⠈⢻⣿⣿⣿⣿⣿⣿⢿⡆⠀⠀⠀⠀⠀⠀⠠⠀⠐
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡋⠀⠀⠊⠁⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⡿⠚⠁⠀⠀⣾⣿⣿⣿⣿⣿⣿⡄⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⡇⠐⣄⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⠄⠀⠀⠊⠹⢿⣿⣿⣿⣿⣿⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣆⠉⠑⢀⡀⠀⠀⠀⠀⠀⠈⠛⢿⡃⠀⠀⣀⣀⣤⠾⢿⣶⣿⠿⠍⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢉⣬⡑⠚⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢉⣉⣉⣹⣿⣿⣆⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠉⣿⣿⣿⢿⣿⠧⢤⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⣿⢸⡇⠀⠀⣹⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⣸⣿⣿⣿⣦⣿⣶⣚⣡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢧⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠂⠀⠉⠉⣩⡉⠉⠉⠉⠉⢉⣭⡉⠉⠁⠀⠒⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢸⣥⣻⠶⠒⠒⠲⣞⣣⡷⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢠⣏⣤⡄⢠⣤⡸⡇⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⣠⣤⣧⡉⠑⠊⣋⣼⣧⣤⡀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠹⠦⠏⠙⠛⠛⠃⠸⠷⠊⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠒⠶⣤⣤⣤⣤⣀⣠⣄⣬⣭⣭⣵⣶⢚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⣿⣿⣿⣿⣿⣿⠟⠛⠛⠿⠏⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⣭⣿⣿⣿⣿⣷⣤⣤⡤⠶⠟⠋



⣿⣿⠛⢿⠛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠩⡲⠄⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⣠⠴⢦⡀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡡⣄⢠⣀⡀⢜⠁⠀⠀⠀⣸⢱⠂⠠⢧⠞⡁⢤⡁⡇
⠀⠀⠀⠀⠠⣤⢤⣠⢀⠀⣠⢦⣀⣠⠢⣀⣠⡿⣀⣑⠐⠂⠋⢷⠃⠀⠀⠀⢸⡈⠀⠀⠀⡔⠁⢸⠁⡇
⠀⠀⠀⠀⠀⠰⢸⣿⢰⠉⠁⢀⠉⠁⡄⠀⡏⠂⠎⣇⠀⠀⠀⢸⡆⠀⠀⠀⠀⣇⠀⠀⠀⡇⠀⡟⢰⠃
⠀⠀⠀⠀⠈⠐⠊⢹⢸⠀⠀⢸⠀⠰⠁⠀⡋⣶⠚⠃⠀⠀⠀⣸⠁⠀⠀⠀⠀⠹⣌⣀⡰⠃⢸⠃⡌
⠀⠀⠀⠀⠀⠀⠀⣾⠈⠀⠀⠀⠀⠀⠀⠀⢠⣋⠀⠀⠀⠀⠞⠁⠀⡴⣦⠄⠀⠀⠈⠉⠀⢀⡏⣰
⠀⠀⠀⠀⠀⠀⢀⡟⠸⠀⠀⠀⠀⣀⠀⠀⠀⠈⡇⠀⠀⠀⠂⠔⠊⠀⡼⠀⠀⠀⠀⠀⠀⡼⢰⠃
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢾⣟⠁⠒⠂⠉⠀⠀⠀⢀⢀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢰⡓⡏
⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠈⢳⠁⠀⠀⠀⠀⠀⠊⠀⠀⠀⢰⡅⠀⠀⠀⠀⠀⡟⢀
⠀⠀⠀⠀⠀⠏⢀⠀⠀⠀⠀⠀⠀⢰⠁⠱⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢸⠁⡆
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢸⠀⠀⠀⠀⡆⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⠓⠒⠂⠀⠀⠀⠀⠀⠀⠄⠀⠈⠑⣄⠀⢀⡇⢸
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠠⣴⠀⠀⠀⠀⠘⠀⠀⡇⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣴⣀⣀⣷⣸⣀
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⣠⣤⠴⠶⠖⠛⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠙⣆
⠀⠀⠐⡄⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⢂⣤⠏
⠀⠀⠀⡇⠀⠀⠀⣤⠞⠋⠁⠀⣀⣠⠤⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⢰⡀⠀⠸⠧⣄⠔⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠘⡏⠓⠒⢄
⠀⠀⡰⠋⠀⢀⠀⠀⣟⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠀⠙⣶⠇⠀⠀⠀⠁⠂⠤⡀
⠀⠰⡇⠀⠀⠉⢀⡞⠁⠀⠉⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠤⣤⣠⣯⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠡⡀⠀⠀⠈⢧⡄⢺⣋⣁⣀⣀⣀⣤⡤⠤⠶⠒⠛⠛⠉⠉⠉⠉⠁⠐⠒⠘⠥⠥⠜⠁⠀⠀⠀⠀⠀⣀⠔

⠀⠀⠀⠀⢀⣤⡚⠁⢈⠻⠷⣄⠀⠀⠘⡛⠅⠀⠀⠀⠀⠀⠀⠀⢱⡄⢠⠀⡄⠃⠀⠀⢻⣿⠂⠀⠀⢸⠀⠆
⠀⠀⠀⠀⢸⡍⠀⠁⠀⠁⠊⢹⠃⠀⢈⣤⡇⢀⡔⠀⠈⠠⠀⠀⠘⢦⣲⡤⠆⠀⠀⠀⢠⣼⡄⠀⠀⢸⠀⠂
⡄⠀⠀⠀⢸⣃⣀⣀⣀⣀⣀⣘⡀⠀⢸⣿⡇⠸⣄⣀⣠⣤⠆⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⣿⡇⠀⠀⠸⠴⠇⢠
⡇⠀⠀⠀⢸⣿⣿⠻⢿⣿⣿⣿⠁⠀⠀⠈⠀⠀⠈⣏⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⢸⣿⣿⠄⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣯⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣇⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⢠⠖⠁⠀⠀⠙⢢⡀⠀⠀⠐⠀⢀⡄⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣠
⣿⣿⣆⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⢻⠀⠀⢰⡴⣯⠀⠀⠀⠈⠉⠀⠀⠀⠀⣰⠋⠋
⣿⣿⣿⠀⠀⠀⠀⠀⢸⣯⣿⠁⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀
⣿⡿⠋⠀⠀⠀⢀⠀⠋⠃⠉⡓⠆⣀⠀⠀⠈⠛⠿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⠂⠀⠀⠙⢿⣿
⠉⠀⠀⠀⠀⠀⠎⢷⠆⠀⠉⠁⢌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠉
⠀⣤⡄⠀⠀⠀⢸⠰⠒⠂⠀⠀⣽⡇⠀⠀⠀⠀⠀⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⢸
⠀⠀⠡⠀⠀⠀⠘⣇⠀⠀⡀⢰⣿⠁⠀⠀⠀⠀⠀⢻⣿⠁⠀⠀⣤⡄⠀⠀⠀⡀⣤⠀⠀⢸⢸
⠀⠀⠀⢠⠀⠀⠀⠛⠿⠿⠿⠿⠛⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠈⠿⣍⣉⣿⠿⠋⠀⠀⢸⢸
⣤⣤⣤⣤⣥⡀⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⢀⡿⠻⢿⣿⡆⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⢀
⠙⠛⠛⠛⢻⣧⠀⠀⠀⢿⣟⣿⠇⠀⠀⠀⠀⣾⠁⠀⠘⣷⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠻⣦⡀⠀⠀⠀⡜
⠀⠀⠀⠀⣬⠿⠀⠀⠀⡾⠋⠹⡆⠀⠀⠀⠀⣿⠀⠀⢠⣼⣿⠇⠀⠀⠀⠀⠀⠀⢀⣴⡃⠀⠈⠈⢳⡄⠀⠀⢆
⣄⣀⣀⣔⠇⠀⠀⠀⠀⡇⠀⠀⡃⠀⠀⠀⠀⠸⣷⣦⣴⣿⡟⠀⠀⣰⢶⣦⠀⠀⢰⡁⠀⠀⠀⢠⣴⡟⠀⠀⠈
⠈⠉⠉⠁⠀⠀⠀⠀⠀⠷⠶⠶⠇⠀⢰⣿⡆⠀⢹⣿⣿⡿⠁⠀⠀⡹⢿⢿⠀⠀⠀⠻⣦⣦⣤⣾⡿⠁
⠀⠀⢰⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢸⢿⡇⠀⠰⠟⠿⠷⠀⠀⢠⡟⣻⣿⡆⠀⠀⠀⠈⠉⠉⠉⠀⢴⢰⡆
⠀⠀⢸⠀⠀⠀⠀⠻⢹⡏⠀⠀⠀⠀⢸⣄⡇⠀⠀⠀⠀⠀⠀⠀⡸⡰⠛⣁⣃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠈⠀⠀⠀⠀⣤⢠⡇⠀⠀⠀⠀⢀⢻⡇⠀⠀⠀⠀⠀⠀⢀⠁⡥⠛⠛⠋⠀⠀⠀⣶⣶⠄⠀⠀⢸⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠗⢲⡀⠀⠀⠀⢤⡤⠤⡦⠦⠄⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣦⡾⠃
⠀⠀⠀⠀⠤⠶⢤⣦⣄⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡯⢁⡖⠀⠀⠀⢀⣀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⡿⠛⠛⡀⠀⠀⠀⢸⣭⢛⣷⣶⠦⢤⣤⡀⡤⠀⠀⠀⠀⠀⢒⡇
⠀⠀⠀⠀⠀⠀⠠⣼⡿⠿⢿⣿⡻⢿⣿⠤⠀⠀⠀⠀⠀⡘⠀⠀⠀⢃⠀⠀⠀⠰⣟⣒⡒⠨⠭⢭⣿⣥⣷⡶⠶⠿⠿⠛⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡯⠐⠈⠃⠀⠀⠀⠀⣐⣃⣀⣀⠀⠘⡄⠀⠀⢸⣍⣉⣉⣙⣛⡛⠻⢿⣤⣤⣴⣤⡶⠶⠿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠚⠀⣸⣿⣿⣿⣿⣭⢿⣿⡄⠀⣠⣿⣷⣖⡰⠶⢶⣾⣏⣛⣉⣡⣤⣤⡶⠶⠞
⠀⠀⣄⠲⡄⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⠀⠀⢻⣷⠀⠀⠈⠉⠉⠉⠛⠛⠛⠓⠛⠋⠉⠁
⠀⠀⠉⠣⡈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣆⣤⣼⡟
⠀⠀⠀⠀⠘⣦⣄⣹⣶⣶⣶⣶⣶⣦⣤⣄⡀⠀⠀⠀⠹⢿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠘⣾⡏⠁⡖⠀⡤⠖⢲⣏⣁⣀⡀
⠀⠀⠀⠀⣸⣿⣿⣿⣟⣿⣿⠛⠛⠛⠛⣻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⠀⠀⣧⡀⠙⠉⠙⣿⣩⡄⢙⣆
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣏⠡⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠀⠀⠀⠨⣹⠀⠀⠀⣿⣿⡟⠉⠉⢷
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⡇⠀⠀⠀⣀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣀⣀⣸⣾⡄⠀⠘⢰⡿⣷⠦⠤⣾
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢹⡆⠀⠀⠘⠀⣿⡆⢠⠇
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠒⠒⠛⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⠞⣯⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⢤⣀⠀⠀⠀⠀⠘⢷⣶⣖⣢⣶⠟⠀⠀⠸⠽⠚⠻⣥⣼
⠀⠀⡀⠀⠀⢀⠀⠀⢀⡻⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⣦⠀⠀⢀⣀⡀
⠀⠀⣍⠀⠒⣤⣶⠶⠞⠓⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣭⣭⣿⣿⣶⣾⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⣿⣶⣤⣿⡄⠀⠀⢠⠄⠀⠀⠀⠀⠈⠉⠀⣠⣶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⣤⣶⣾⣿⣿⣿⣿⣿⣷⣶⣦⣄
⠀⠀⠙⢏⣻⣿⣿⣷⣤⣼⣦⣤⣤⣤⣴⣶⡌⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣛⡋⠀⠀⠀⠀⢿⠛⠛⠛⠛⠛⠛⠻⢿⣿⣿⣿
⠀⠀⠀⠀⠙⢧⠁⠘⡿⠟⠉⠉⠉⠉⠘⠀⠁⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠰⣿⣋⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠙⠻⠿⠿⠿⠿⠿⣿⣿⠿⠿⠟⠋⠀⠀⠀⠙⠶⠤⠤⣤⣤⣤⠶⠿⠋

⠀⠀⠀⠀⣤⠂⣤⣤⠄⣴⣦⡄⢠⣦⣦⣤⣴⡶⠲⣶⢶⣶⡄⣤⣤⡄⢰⣄⣄⠀⣰⠀⠀⠀⠀⣤⠀⠀⠀⢤⡄⠀⠀⠀⣄⡤⠀⠄⠀⣠⢴⡀
⠀⠀⠀⠚⠾⠛⣿⣿⣻⣿⡷⡄⢠⣿⣿⣿⣿⣿⣶⣿⣶⢿⡟⢠⡼⠃⠸⡎⢉⠁⢠⠀⠀⠶⠀⠘⡁⠀⠛⠀⠀⠀⠀⣀⡘⠛⣡⠆⠰⣷⡜⠁
⠀⠀⠀⢀⣴⣧⠀⡿⣻⣿⣖⣅⠀⣽⣿⣿⣿⠋⣿⣿⠉⢈⡵⢿⡅⠀⡴⠠⠤⠙⠃⠀⢀⠂⢂⠀⡀⠀⠖⠀⢀⠀⠘⢇⣽⡆⢁⡤⣾⣝⢷⠂
⠀⠀⠀⢈⡹⠋⠛⠙⠿⣿⣉⣉⠀⢿⣿⢻⣿⠁⣿⣿⣿⣿⡆⢸⡍⠀⢸⠀⠶⠞⠁⠀⠸⠆⠀⠀⠀⢀⠀⠆⠈⠀⢰⡾⢛⣄⠻⢠⠞⠻⣿
⠀⠀⠀⡈⣴⣚⠗⢻⣽⣮⣽⣿⢀⣿⣿⣶⣿⣦⣿⡿⣾⣤⡋⠉⠑⢆⠀⢉⣴⡶⢺⠀⠀⠂⠀⠀⡀⢀⡀⣶⠀⠀⢸⣾⢿⡉⢳⣄⠀⣼⡁
⠀⠀⠀⠛⢤⡿⣶⣶⡾⢋⣡⡁⠠⣶⣿⣿⣿⣿⣿⣶⣿⣯⣟⡎⠛⠀⣷⡉⢁⠸⣤⠀⠀⢠⣤⠀⠁⠀⠁⠀⢀⠀⠈⠧⠀⠉⠈⣞⠋⠐⣃
⠀⠀⠀⣤⡉⠀⠀⣥⣤⣬⡉⠀⠀⡋⠛⠛⣛⠛⣛⣋⠁⣉⠉⠙⠋⠀⠉⠛⠄⠀⠀⠀⢀⡌⣁⠀⠙⠀⠀⠘⣈⡀⢀⡀⢸⡉⢀⢀⠀⢋⣩⡅
⠀⠀⠀⠙⢿⣶⣄⡈⠻⣿⣿⣦⠠⢀⣄⠚⠋⠀⠉⢩⣷⣀⠐⠂⠁⠠⠑⠀⠐⠀⡢⠀⢀⠶⠙⠘⢧⠀⠄⣠⠈⠇⢸⡇⢸⡇⢻⣿⡀⢸⣿⡇
⠀⠀⠸⣷⣦⣽⣿⣿⣦⡈⢻⣟⠀⢉⣿⡟⠰⠟⢳⡌⠙⠉⠀⠈⠡⡄⠐⠀⠀⠀⠀⠀⠛⢠⠞⠛⠀⠴⢎⣁⠀⡀⢸⡇⢸⡇⣾⣿⠀⠈⣿⡏
⠀⠀⢰⣌⠙⢿⣿⣿⣿⣿⣦⡉⢠⡸⠋⡀⠒⣦⠉⣿⣠⠀⠀⢀⣄⣀⢀⣀⣀⡤⡀⠀⠀⠸⠋⠀⠹⡆⣙⠿⠁⣄⢸⡟⢸⡇⣹⣿⠀⢰⣯⣧
⠀⠀⢺⣿⣿⣦⡉⠻⣿⡛⢿⣿⠘⠃⠘⠀⠶⠉⣰⣟⠟⠁⠀⠀⠉⢹⡋⣙⠋⠁⠙⠂⣀⣠⠄⢀⡙⣱⠟⢀⡀⠁⢸⡇⢸⡇⠿⠛⠁⠈⣿⡧
⠀⠀⢴⡌⢻⣿⣿⣄⠈⠻⣦⡉⢠⣬⣿⡆⠀⠙⠛⠃⣠⡀⠀⠠⠄⠀⣀⠸⠂⠀⠀⠀⣠⠀⢘⠟⣡⡱⠀⢈⣃⣄⢀⡅⢸⡇⣀⢠⠀⢀⣿⡧
⠀⠀⠈⠻⢦⠙⠿⠻⠷⠄⠈⠷⠈⠀⣈⠻⠿⠀⡀⠘⠧⠀⠘⠀⠀⠀⠉⠚⠂⠀⢀⠀⠉⠀⡿⠛⠘⠛⢂⡉⠉⡁⢸⠇⠸⠃⠿⠸⠀⠈⠿⡗
⠀⠀⢠⣴⠀⠀⢤⢰⣦⠀⢠⣄⠐⢻⣧⡟⢀⣿⣿⣿⣶⡿⠀⠶⡄⠀⢨⡛⠀⠙⡧⠤⣼⣿⣟⠀⠀⢃⡀⠹⣿⡟⠙⣻⡆⠈⡀⠀⠀⠀⡀
⠀⠀⠰⠄⣀⣀⠠⠼⠉⢥⠈⠋⢠⠈⢉⣾⣛⣿⠿⠛⠁⠀⢘⣆⠈⢷⡆⢰⡄⠣⡆⠀⣾⡽⣿⣷⡀⢺⣿⣿⣿⣿⠚⠋⠄
⠀⠀⢠⡌⣉⠠⠄⠘⢣⡀⠛⠂⢰⣈⣿⣿⣿⣿⣷⣄⡠⣦⠈⠁⣠⠀⣠⣿⠄⢀⠛⡂⢹⣿⣹⣿⣿⣆⠿⢿⣾⣧⠀⠀⠀⠀⠀⠠⠀⠐⣄
⠀⠀⠠⠄⢉⣀⠀⠠⠖⠁⠀⠉⢄⠉⠈⣿⣿⣇⣨⣏⠀⠉⠀⣄⢤⣤⡿⢷⠌⠁⠀⠛⢰⣿⣿⡟⢿⣿⣧⠀⢻⣿⠀⠀⡀⢂⣠⣤⣄⠀⠘
⠀⠀⠠⠂⠼⣿⡂⠀⠀⠀⢡⡷⢼⣿⣾⣿⣟⡋⠉⠻⣿⣦⠀⣼⠟⡉⠝⠀⠀⣣⠌⠆⢀⠹⣯⣿⣾⣿⡋⠀⢻⣯⠀⠀⠀⣾⢁⣀⠙⣷⠐⠂
⠀⠀⠀⡴⠀⠙⠃⠀⠸⡆⠀⠀⠘⢯⡁⠘⠛⣡⣤⣸⣿⡿⠈⠤⠿⠋⠐⢀⠄⠷⠊⠦⠹⠁⠙⠃⠉⠻⣿⣷⣾⡧⠀⠀⠀⠘⠛⠋⠀⠿⡆
⠀⠀⢰⣤⣴⠦⠀⣀⡀⠀⣠⣤⠒⠖⣧⠤⡴⢤⡀⠈⢠⡶⠐⠒⠒⣶⠐⠀⠐⢾⠒⠂⠲⣶⠒⢤⡤⣴⡆⠀⣼⡄⠐⠐⣶⣶⠶⣐⠦⣶⣦⠆
⠀⠀⢸⣿⣷⣤⠀⣹⡷⠸⣿⣿⠀⡾⠋⠀⢇⠙⠁⣠⢾⠇⠀⠉⢏⣽⣀⣰⣄⣼⠀⠀⠀⠈⣡⣄⠙⢛⣁⡀⣿⡟⢺⣛⢙⡿⢆⣡⠙⠋⢾⡆
⠀⠀⢸⣿⣿⣿⣿⠿⣷⣶⣿⣿⠀⠀⠀⠀⠈⠉⠈⣡⠂⠀⣼⣒⡆⠂⢰⠈⠁⠚⢺⡃⢰⡿⠙⠈⣿⣟⣉⠉⡙⠇⣾⣿⠿⠶⠆⣳⣿⣾⢛⡇
⠀⠀⢸⣿⣿⣿⣿⡞⢿⣿⡀⠀⠀⠀⠀⣀⡀⢡⡶⡁⠀⠀⠟⠉⣁⣀⣩⣠⡀⠀⠘⠁⠘⠳⠀⢸⣿⣿⡁⢠⣧⠀⢠⣙⣦⣾⠿⠿⢼⡧⢼⡇
⠀⠀⢸⣿⣿⣏⡀⠀⣾⡿⠁⠀⠀⠀⠒⠀⠐⡏⢀⣁⠀⠀⠐⠂⠀⣶⠈⠀⠀⢶⠖⠂⢀⠀⢀⡤⠛⠛⣤⣨⡄⠀⢹⣈⡟⠛⡳⣯⣿⡧⣄⡁
⠀⠀⢸⣿⠿⣿⣿⣀⠀⣴⡀⠀⠀⠀⠠⠀⠀⢧⣈⡽⠃⠀⠀⠚⣖⣻⡀⢠⡄⣸⠀⠀⠈⢻⣿⠗⠒⠻⣿⠛⠢⠄⠀⣟⣻⡄⣻⠿⢹⣗⠛⡁
⠀⠀⠸⢡⣤⡮⠉⢩⣤⣬⣤⣄⠀⣀⠀⠀⠀⠀⠠⠄⠀⠀⣰⣦⣤⡽⢦⣴⢧⣄⠘⣆⢀⣸⣉⠀⠀⠀⠁⠀⡉⠁⢰⣿⣌⡁⢈⠀⣾⣏
⠀⠀⠀⣩⣼⣷⢨⡦⠤⠸⠎⠹⠀⠨⡀⠀⠀⢎⠀⢐⠁⠄⣻⣟⣿⠿⢦⡿⣿⣩⣷⣍⠸⣿⣿⡇⠀⠀⠀⠀⡀⠀⠸⠿⣿⡆⢻⠧⣸⣿⠇⡀
⠀⠀⢸⣟⢼⠏⣸⣧⠘⠧⡶⢶⠀⠀⣄⠀⠀⠈⠠⠎⠀⠀⣿⣿⡛⣿⣿⢷⣏⢉⣿⣏⢐⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⢀⣠⣛⣂⣾⣿⡏⠘⠋
⠀⠀⠸⢻⡄⢂⡄⠻⠂⠲⢶⡏⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡺⡟⣷⣯⣤⠚⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⢿⣥⢛⡆
⠀⠀⠀⠀⡠⠾⠷⢟⣀⠙⣿⡟⠀⠁⠁⠀⡃⠐⠀⠠⠀⠀⣴⣾⣿⡏⢱⣓⣿⣯⣹⡿⠀⠘⠇⠀⠀⠀⠀⢀⠀⠄⣰⠆⣈⡠⣬⣷⣾⣧⣽⠅
⠀⠀⠐⠞⠳⠿⢾⠿⢟⡘⢾⠿⠀⠉⠀⣨⡠⠀⠀⠾⠀⡀⣩⣼⡟⢙⣿⣬⣿⣯⣭⣴⠀⣴⣿⣦⣼⣧⣠⣎⢀⡀⣿⡰⠿⢀⠛⣿⣿⣦⠁
⠀⠀⠀⠈⠙⠛⠛⠂⠈⠙⠂⠀⢀⣀⠂⠉⠀⠀⠀⣀⣀⡀⢻⠛⠟⠛⠛⠛⢛⡟⢻⣟⠀⠘⣛⡋⠈⠉⠙⢋⠙⡁⠘⠛⠃⡉⠐⠛⠙⢟⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢩⡷⢠⡞⠿⠂⣼⠋⣿⠀⠀⢠⡀⠈⠀⠈⠁⠈⠋⢸⡄⠏⢠⣀⣾⠿⠿⠷⠉⠀⠆⠘⠛⠆⣀⢰⣮⣽⠃
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⢠⡦⠛⠶⠟⣿⣰⠏⠘⠆⢀⡁⢀⡄⠐⠀⠀⠀⢸⡵⣬⣛⣉⡀⢤⣤⠻⣦⠙⣂⠀⠀⢲⠦⡛⠀⠀⠁
⠀⠀⠀⠀⡁⣀⠀⠀⠀⠀⠀⠀⢘⡿⢾⡧⢤⣤⡴⠿⣵⡦⠸⠖⠈⠁⠘⠈⠻⢿⣿⡆⢸⠦⣛⠋⢤⡹⡤⠶⣦⡀⠠⢉⣟⣴⢏⠲⣬⣑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⡟⠃⢠⣏⣠⡿⣧⡀⠀⠀⠀⠂⣀⠄⠀⠀⠀⠀⠸⡆⢞⠛⠎⣷⣝⡀⢻⣏⣀⠈⢈⣱⢦⡴⢠⠏⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡀⢉⣛⠻⣏⠻⣶⠆⡷⠠⢀⠀⠀⠀⢠⠀⠴⠀⠿⠰⣇⣤⣇⠠⣽⡿⠁⣶⡇⣈⡰⢶⠏⠛⠤⢀⡐⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣹⡏⠙⠁⠙⠻⠿⢾⡅⠀⠄⠀⣠⣶⢧⣦⠴⣴⠄⢸⠏⠀⠘⠳⡿⠣⡾⠿⠁⠬⠉⠁⠄⡀⠀⠾⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠠⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠤⠠⠤⠤⠄⠤⠠⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠈⠉⠈⠛⠁⠉⠋⢹⠙⠋⠈⠙⠉⠉⠁⠀⠀⢸⣿⣿⣿⣷⣾⣷⣴⣿⣶⣴⣶⣦⣾⣷⣦⣾⣷⣿⣿⣿
⠀⠀⠀⠀⢠⣦⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠙⢿⣧⣿⢿⣦⣄⣬⡴⠶⢦⣄⣀⠀⠀⠀⠀⢸⣿⣿⣿⣦⡉⠉⢠⣈⠉⠻⠟⣋⣉⣛⠿⢿⣿⣿⣿⣿
⠀⠀⢀⣀⣞⣀⣉⣛⣾⣭⣵⣞⣛⣛⣾⣯⡀⠀⠀⠀⢸⣿⣿⣿⠿⢱⠿⡶⣬⠑⠒⠋⣡⢶⣬⡉⠚⢿⣿⣿⣿
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢸⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢳⡀⠀⠀⠀⠀⢠⠏⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠲⡄⠀⠀⠀⠀⠀⢸⡀⠀⣧⠀⠀⠀⠀⢸⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠹⡄⠀⠀⠀⠀⠈⣇⠀⡟⠀⠀⠀⠀⣿⢀⡞⠀⠀⠀⠀⠀⢀⡴⠚⠉⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⢤⡀⠀⠀⠀⠱⣄⠀⢳⡀⠀⠀⠀⠀⠸⣀⡇⠀⠀⠀⠀⢹⡞⠀⠀⠀⠀⢀⡴⠋⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠙⢆⠀⠀⠀⠈⢦⡀⢇⠀⠀⠀⠀⠀⢿⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⣴⣫⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⢧⡀⠀⠀⠀⠙⠾⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠛⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⣳⡀⠀⠀⠀⢀⡠⠖⠋⠉⠁⠀⠀⠀⠉⠉⠙⠒⠦⣄⡀⠀⠀⠀⠀⠀⣠⠖⠉⣀⣀⡤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⢀⡴⠒⠒⠒⠒⠲⠤⢤⣀⡀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠠⣤⡖⠒⠒⠓⠒⠚⠓⠲⡄
⠀⠀⠀⠀⠀⠈⠓⠲⠶⠶⠒⠒⠒⠋⠉⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠉⠙⠒⠲⠶⠤⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⡄⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠠⣶⠶⠤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠀⣀⠴⠋⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠉⢲
⠀⠀⠀⠀⠀⠀⠀⠘⠲⠶⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠲⠶⠶⠶⠖⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣰⠃⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⡶⡀⠀⠀⠀⠀⠀⢧⠀⠈⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⡴⠁⠀⠀⠀⢠⠏⢸⠀⠀⠀⠀⢸⠃⠙⡄⠀⠀⠀⠀⠈⢳⡀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⣠⠞⠀⠀⠀⠀⢀⡎⠀⢸⠀⠀⠀⠀⢸⠀⠀⢹⡄⠀⠀⠀⠀⠀⠙⠦⣤⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠋⠁⠀⠀⠀⠀⠀⣼⠀⠀⡞⠀⠀⠀⠀⢸⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⡼⠁⠀⠀⠀⠀⠀⢧⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠻⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣯⣤⣤⣤⣤⣤⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠤⠤⠤⢤⣴⣿⣀⠀⠀⠈⠙⠻⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⢩⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡉⠳⠶⠶⠒⠀⠀⠀⣾⡇⠈⣙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣗⣾⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⣶⠶⣾⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⣿⡟⠙⠛⠛⠻⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣋⠙⠶⠶⠶⠶⢾⣿⡷⠈⠉⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣸⡿⠓⠒⠒⢲⣖⠋⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠉⠛⠛⠓⠒⢺⡿⠋⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢰⣿⠁⠀⠀⢸⣿⣿⣧⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⢀⣸⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⠁⣤⣴⣤⡙⠿⠟⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠐⠀⣀⠀⢰⣿⠀⣴⠞⡋⠩⠩⠉⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠸⣿⣿⣿⠃⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠀⣿⡇⣼⠃⠀⠘⠄⠀⢀⡠⠆⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣦⣭⣥⣤⡶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢸⡿⢠⡟⢘⠁⠐⠀⠂⡉⠤⣬⠆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠿⠇⠘⣿⡄⢖⠁⠀⡃⠰⣄⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⣾⣿⣷⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠲⠤⢤⣤⣤⣼⣿⣶⣤⣘⣓⣀⢤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠟⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠖⠚⠿⠟⠛⠋⠙⠛⠿⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠄⠀⠀⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠉⢧⠀⡿⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢉⣛⣛⣛⣿⡿⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠖⠒⠒⢺⣿⠛⢷⣾⣻⣷⠚⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣴⣿⣟⠋⠉⠉⢁⡀⡸⢿⣆
⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣤⣤⣤⣼⣿⡿⠿⢞⣻⣎⠙⣺⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢟⣻⣷⣮⣤⣡⣠⣤⡘⢿⡆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⣸⣽⡟⣿⣩⡹⡏⡩⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠭⣀⣀⣀⣀⣀⣀⣀⠀⠍⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢻⠟⢿⣹⣿⣿⣷⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢻⣧⣾⣤⣿⣼⣷⣤⣹⣤⣺⣿⡏⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢻⣎⡈⠀⠁⠀⢼⣏⠛⠛⠛⢿⣿⡿⢡⠇⠠⠤
⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠈⠉⠉⠉⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠻⢷⣶⣧⣀⣤⣤⣧⣴⠿⢟⣩⠴⠋⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣸⡂⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠟⠀⠀⠀⡈⠉⠛⢿⣿⣿⠟⠋⠉⡀⠀⠀⠈⢷⡆
⠀⠀⠀⠀⠀⠀⠀⣿⢸⣏⠉⠉⠉⠉⠉⠉⠉⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠐⠧⠀⠀⠀⣿⣯⠀⠀⠀⠿⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⣿⣄⠙⠲⠦⠤⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣿⣿⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣶⣶⣶⣶⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣭⣛⣻⣿⣿⣿⣿⣿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⡀⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡖⡆⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠠⠔⠒⠊⡼⠀⠀⠀⠀⠠⠃⠁⠠⢬⠀⠀⠀⠀⠐⠤⢃⠉⠀⠀⠀⠀⠀⠈⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⡊
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠋⢃⡙⠹⠁⠀⠀⠀⠀⡆⠀⠀⠊⠀⠀⠀⠈⠀⠀⡁⠀⠀⠁⠀⠀⠀⠤⠏⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠁
⠀⢠⠂⠀⡢⢀⡂⠀⠀⠀⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⡀⠐⠀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠂
⠀⠀⠢⠘⡅⣠⠎⠀⠀⠀⠀⠂⠀⠄⠂⠀⠀⠀⠀⠀⢡⠈⢠⠃⠀⠀⠀⠀⠈⠉⠰⡴⣦⠀⠀⠀⠀⢀⣄⣅⢠
⠀⠀⠣⠀⠞⠨⠌⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⢀⠘⠃⠛⢀⠀⠀⠀⠀⠀⠐⡿⣗⠋⠀⠀⠀⠀⠈⠉⠋⠈
⠀⠀⠁⠀⠁⠀⠁
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⢒⡄⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠃⠜⠂⠀⠀⠀⠀⠃⠀⠀⢨⣶⣦⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠉⠀⠓⠐⠀⠀⠀⠀⢀⠁
⠀⠀⢠⣷⠼⣧⠀⠀⠀⠀⠂⠀⠜⡮⣖⠃⠀⠀⠀⠀⠄⣀⣀⠄⠀⠀⠀⡀⠠⣀⠀⢀⠄⢀⠀⠀⠐⠃⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈
⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢠⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⢰⠋⢲⣸⠈⢲⠀⠀⠀⠑⢡⡀⠀⠀⠉⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⢀⠶⠒⠲⠖⣤⠀⠀⠀⠀⡈⠈⢀
⠀⠘⠂⠘⠘⠀⠊⠀⠀⠀⠈⠀⠈⠃⠀⠄⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⢐⣾⢧⣀⢀⠔⠁⠀⠀⠀⠠⠇⡤⠧⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠁⠀⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢀⡰⡄⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡠⠤⠖⠉⠀⠠⠜⢤⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠸⣅⠈⠁⢠⠀⠈⠁⣰⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠘⣄⠀⠠⠄⠀⣠⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠋⠀⢗⠂⠐⡿⠀⠙⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠀⠀⠄⣡⣌⠀⠀⠀⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⡴⣹⡠⠊⠉⠉⠙⠉⠉⠉⠑⠄⠏⢆⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⠂⢄⠀⢀⡼⠥⠛⠂⠄⡀⠀⠀⠀⠀⢀⠠⠒⠚⠬⢧⡀⠀⠠⠐⠺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠑⢢⡠⠒⠁⠀⠀⠀⠀⠀⠀⠁⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠒⢄⡔⠈⠙⢆
⠀⠀⠀⠀⠀⠀⠀⢰⡑⡐⣠⢠⠁⠈⠒⠒⠒⠒⠒⠒⠒⠂⢤⠄⠒⠒⠂⠒⠒⠒⠒⠒⠁⠈⡄⣄⢂⢪⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢧⣇⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠖⠂⠁⠢⢄⠀⠀⠀⠀⠘⠁⣼⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣈⣐⣤⣀⣀⠤⠔⠁



⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡠⠖⠊⠉⠉⠙⠒⢤⣀
⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣀⠀⢀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⢹⠀⠀⠀⣿⠉⠛⠦⣄⡀⠀⠀⢈⠟⠋⠛⠦⣤⡤⠚⠉⠀⠀⠀⠉⠙⠲⣤⣀⡀
⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠙⠳⢦⠏⠀⠀⠀⡠⠋⠀⠀⠀⠀⢀⣠⠔⠀⠀⠀⠉⢳
⠀⠀⠸⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⡔⠀⠀⠀⠀⠀⡴⠋⠀⣀⠀⠰⠆⠀⢸
⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠙⢦⡴⠀⠀⠀⠀⢠⠎⠀⠀⠀⠁⠀⢠⡆⠀⡞⠀⠀⠀⠀⠀⠀⠀⢠⣴⡄
⠀⠀⠀⣇⠀⠀⢸⢀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠃⠀⠈⠋⠀⠜⢠⡄⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⣠⠖⠒⣆
⡖⠒⠒⠛⠒⠂⢙⣯⢹⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡎⠀⠐⠃⠀⣠⡄⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠻⡀
⢇⣀⣛⣀⣀⣀⣠⠧⠞⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⢇⠀⠀⠾⠃⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠷⡤⠼⠃
⠀⠉⠉⢹⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣌⣀⣀⣤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⢦⡀⠀⠀⣀
⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠀⠀⠀⣽⠀⠐⠿
⠀⠀⠀⢸⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⣤⡘⢧⣤⡀⠀⠀⣻
⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠁⠀⠀⢹⡄⠀⠛⠒⠚⠁
⠀⠀⠀⠸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣖⡆⠀⠀⣀⣠⠤⣌⠹⣄⣀⡿⠀⢠⣤
⠀⠀⠀⢸⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠈⠀⠸⡄⠀⠁⠀⠀⠀⠉⠀⠀⢠⣤⡄
⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⠞⠓⠚⠁⠀⣶⣳⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢸⡆⠀⠀⣿
⠀⠀⠀⠀⡇⠀⠀⢿
⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⢸⡆
⠀⠀⠀⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⢏⠀⠀⢸⡇
⠀⠀⠀⠀⢸⠀⠀⠘⡇
⠀⠀⢀⡴⠞⠁⠀⠀⠙⣇⠀⣴⠳⡄
⠀⠀⣸⠀⠀⠀⣀⠤⠤⢿⣤⡿⣤⡷⢦⡀
⠀⠀⢹⠀⠀⠀⠻⠤⢤⣤⣥⡄⠰⡷⠞
⠀⠀⠘⣦⣀⡀⠀⠀⢠⡇⠀⣿⢀⡇
⠀⠀⠀⠀⠉⢹⠀⠀⢸⡄⠀⠈⠉
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⢨⡅
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⢠⠞⠛⠇⠀⢸⡇
⠀⠀⠀⠘⣤⣤⣤⣤⣼⡇
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣤⣤⣴⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣤⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⠈⣙⢿⣾⣿⣿⣿⣶
⣿⣟⣴⡏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢒⣿⣿⣿⣿⣿⣿⢿
⣿⣿⣿⡷⣿⡿⢶⣿⣷⣿⣿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠿⠀⠁⡀⠰⣿⣍⢛⢿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢱⡿⢿⣿⣿⣿⣿⣿⣿
⣿⣯⠀⠀⠀⠈⠆⠙⠿⠿⠤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡠⠿⠛⠛⢻⡿⣿
⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⡄⢠⣄⠀⡴⢠⡄⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠃⠀⠀⠒⠂⠚⠉⠁⠂⢀⣿⣿⡆⠀⡏⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠠⠤⠶⠒⢂⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⣀⣠⠤⠤⠤⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠚⠻⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠈⢀⠀⡠⠤⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣀⣀⠈⣭⣿⣥⣤⠀⣀
⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠈⠍⠉⠛⠂⠙⠉⠉⠀⠠⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⢤⣤⣰⣾⠃

⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠂⠠⠀⠀⠀⠀⠀⢀⠀⢤⠄⠂⢀⢀⣄⡀⠀⠂⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⡁⠀⢀⠠⣔⣈⠰⠶⠆⡙⠙⠉⠐⠊⠁⠁⠀⠀⠀⠀⢀⠈⠠⠆⠀⠀⠈⠀⠈⠁
⠀⠀⠀⠀⡀⠀⠀⢉⠀⠐⠉⠀⠔⡫⠅⠈⢀⠠⠆⠀⠈⠉⠀⠀⠉⠀⠀⠓⠠⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠠⠀⠀⠉⠀⠀⠁⠐⠂⠈⠀⠄⡠⠔⠉⠉⠀⠀⠀⠀⠀⢀⡀⠀⢀⡐⠂⠁⠀⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠
⠀⠀⠛⢀⠀⠈⠁⠀⠄⠀⡀⡠⠚⠁⠀⠀⠀⠀⠀⠐⣀⣠⣄⣠⠀⠀⠈⠑⢄⠂⠀⠀⠑⠀⠀⠀⠁⠃
⠀⠀⠀⠈⠀⠀⠀⠊⠀⡀⠊⠀⠀⠀⣠⠔⠀⠀⠐⠈⠀⣠⠉⠉⣛⡾⣄⠄⠀⠑⡀⠀⠀⠠⢀⠀⠀⠀⠀⠘
⠀⠀⠀⠰⡄⠀⡆⠀⡜⠁⠀⠀⠀⠊⠀⣀⠀⢚⠓⠁⠀⢃⢀⠀⣦⠑⢚⠃⡀⠀⠀⠀⠀⠘⠀⠀⠈⠈⠀⠀⠆
⠀⠀⠀⠀⡀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠠⡅⠀⠂⠀⡄⢀⢀⡀⠀⠈⢢⠀⠡⠀⠀⠀⢠⣳⠄⡃⠀⠀⠈⡀
⠀⠀⢱⠀⡁⠀⠱⢸⠀⠀⠀⠒⠀⠀⠀⠁⢸⠀⠠⠥⠀⣐⠘⠈⠀⠀⠞⠁⠁⠀⠀⠀⠙⠁⡇⠀⠀⠒⠀⠀⠜
⠀⠀⠀⠀⠀⠀⡀⠉⠀⠀⡀⠀⢀⠀⠐⠃⠜⢄⠀⠀⠁⠉⠀⠀⢀⠔⠤⠀⡤⠀⡀⠀⠀⠀⠩⢀⠂⠸
⠀⠀⠀⠀⠀⠀⠁⠀⠐⢄⠀⢀⢄⢀⠆⠀⠈⠓⠁⡂⡠⢀⠠⡈⠴⠀⠀⠠⠁⠀⠂⠬⠀⠀⠘
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠢⠈⢧⡅⣄⠠⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠑⡐⠈⡳⣶⣁⠆⠠⠄⠂⠀⠄⠀⠀⠀⠀⠀⠀⡠⠁⠀⠁⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⣤⠁⡀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠙⠗⠀⠌⠀⠀⠀⠀⠀⠀⠀⡄⠈⠑⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⢂⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠃⠠⠀⠀⢀⡁⡄⠀⠀⠀⡀⠠⠆⢂⡀⠁⠀⠀⠀⠀⠀⠀⢀⠐⠁⠒⠀⠄
⠀⠀⠁⠀⠀⠀⢀⡤⠑⠄⠀⠀⠀⠰⠀⠀⢀⠀⠠⢀⡀⣀⣤⡜⠶⠒⠉⣠⠀⡐⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠩⠉⠁⠀⢂⠀⡡⠔⠋⠀⠀⠀⠀⠀⠀⠐⠀⠃⢄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠘⠁⠀⠻⠄⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⠀⠁

⡟⠉⣿⣛⣻⣛⣻⣛⣟⣛⡟⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣻⣟⣻⣛⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⡔⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⠀⠈⢿⠋⠀⠀⢠⠐⠉⢡⡙⠛⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⠀⢠⢀⣷⣱⡵⠃⠀⠘⠀⠁⠀⠀⢈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠙⠦⠿⢿⣿⠁⢴⣆⠀⠀⠠⡙⠆⣸⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⢀⣠⣤⣌⣿⡧⠼⠿⣦⣤⠔⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠉⠁⠀⠀⠀⠀⣠⡿⠉
⠀⠀⠐⢶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⣠⠾⠋
⠀⠀⠀⠀⠙⢿⣿⠷⠶⣤⣀⣀⣠⠔⠠⢤⣤⠀⠄⠀⠈⠻⠷⠶⠾⠛⠁⠀⠀⠀⠀⣴⣶⣶⣦⡀
⠀⠈⢦⡀⠀⠀⠀⠀⠀⠶⣆⠈⢡⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡄
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢌⠢⣄⠀⠀⠀⠀⠀⣰⣿⣦⠤⠤⠤⠄⠀⠀⠀⠀⠀⢀⡌⠟⢻⡀⢹⡟⠘⡆
⠀⠀⠀⠀⠀⣿⣶⣄⡀⢄⠀⠀⠠⣧⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣾⡏⠀⠁⠀⡟
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣶⣿⣶⣿⢇⠰⠆⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢙⠷⠀⠀⣰⠇
⢀⣠⣴⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣾⡟⠉⠉⠉⠛⠛⠦⡀⠀⠀⠀⠀⠀⠀⠊⠀⢀⠎⠀⡆⠀⢸
⠀⠉⠉⠀⠸⠿⠿⠛⢻⣿⣿⠆⣿⣿⣿⣿⣷⠦⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⢠⠞⠉⠀⠰⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⢸⣉⣉⣉⣉⣙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣉⣉⣉⣉⣹⠆
⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠰⣟⣛⣛⣛⣛⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣛⣛⣛⣛⣛⡷⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⢠⡇⠀⠀⠀⢸⡆⠀⠀⣿⠀⠀⠀⠈⡇⠀⣠⠤⠤⠤⠤⢤⡄⠀⠀⡏⠀⠀⠀⢹⡇⠀⠀⢸⠇⠀⠀⠀⢻
⠀⠀⢸⠃⠀⠀⠀⠈⣇⠀⠀⣿⠀⠀⠀⠀⣇⠀⢹⡖⠒⠒⠒⢺⠃⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⣾⠀⠀⠀⠀⠸⡆
⠀⢀⡟⠀⠀⠀⠀⠀⢹⡄⢀⡏⠀⠀⠀⠀⢻⠀⢸⣇⠀⠀⠀⢸⠀⠀⢸⠇⠀⠀⠀⠀⣧⠀⢰⠇⠀⠀⠀⠀⠀⢳⡀
⢀⡾⠀⠀⠀⠀⠀⠀⠀⢻⣼⠁⠀⠀⠀⠀⠘⣇⢸⡿⣇⠀⠀⢸⡆⢠⣾⠀⠀⠀⠀⠀⠸⣤⡏⠀⠀⠀⠀⠀⠀⠈⢷
⣼⠁⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠸⡷⣄⠀⠀⠘⣿⡇⢹⡄⠀⠀⣷⠋⡿⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢹⡈⠳⣄⣸⢻⠃⠀⢷⠀⡴⠃⢀⡇⠀⠀⢀⣠⢶⡟⠀⢻⡀
⡇⠀⠀⠀⠀⠀⠀⠠⣾⣤⣀⠀⠀⠀⢷⠀⠈⠳⡾⠀⠀⠘⠟⠁⠀⢸⣇⡤⠞⠋⣠⠏⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠀⠀⢹⠷⣌⠙⠓⠶⢼⡇⠀⣀⡴⠖⠚⠛⠛⠲⠦⣌⠉⠀⠀⡴⠃⠀⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠳⣄⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠺⠷⠶⠒⠒⢛⣻⡿⠃
⡇⠀⠀⠀⠀⣠⣤⡤⠼⠶⠶⠶⠞⠃⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢀⡤⠖⠋⢸⡇
⡇⠀⠀⠀⠀⠈⠉⠓⢶⠦⣄⣀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠿⢤⣄⣀⢸⡇
⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⡬⠟⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⣀⣀⣉⣙⣳⣶⠄
⡇⠀⠀⠀⠀⠀⠀⣠⠼⠛⠁⠀⣀⣀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠻⣍⠉⠉⠁⢸⡇
⡇⠀⠀⠀⠀⠀⠛⠛⢻⠛⠋⠉⢉⡿⠁⠀⠈⠳⢤⣀⣀⣀⣀⣀⣤⠖⠋⣠⣀⠀⠈⠳⣄⠀⢸⡇
⢷⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠟⠀⣀⣤⣶⠀⠀⠈⡉⠉⠉⢁⣄⠀⠀⢻⡈⠉⠓⠶⢬⣷⣼⡇⠀⠀⠀⠀⠀⠀⢀
⠀⠙⠓⠶⠤⠤⠤⠴⠾⣄⣰⣋⡴⠚⠉⠀⣿⠀⢀⡼⢻⡀⠀⢸⠉⠳⣄⠀⢷⠀⠀⠀⠀⠀⣹⠷⠦⠤⠤⠤⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠧⠤⣤⣤⢤⡇⢠⡞⠀⠀⢧⠀⣼⠀⠀⢸⠷⣜⣧⣤⠤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠯⣄⣀⠀⠘⣆⣿⣠⡴⠋⠀⠈⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⢰⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣷⣄⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠿⠿⠿⠿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣤⣤⣤⣤⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠛⠋⠉⠉⠉⠉⠙⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⡉⠉⢉⣀⣉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⢻⣿⣿⠛⠛⠛⠛⠛⢛⠛⠛⠛⠛⡛⠛⠛⣿⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⢻⣷⡀⠀⠀⣴⡿⠃⠀⢀⣾⠟⠀⣼⡟⠉⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠹⣷⡀⠀⢿⣧⡀⠀⠸⣿⣄⠀⠻⣷⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠙⣿⡄⠀⠈⢻⣷⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣀⣀⣴⡿⠁⠀⠀⣠⣿⠃⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠛⠁⠀⠀⠈⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠠⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠀⠀⠈⠙⣆⠀⠀⠀⠀⠑⠢⡌⡆
⠀⠀⠀⠀⠀⠀⢧⡀⡨⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⢡⢳
⠀⠘⢿⣿⣿⡿⠃⠀⠀⠀⠀⠰⣤⣶⣶⠆⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⢈⣿⡆⢀⡀
⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⢀⡀⠀⣸⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⠀⢻⣀⣰⣼⠇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠰⣿⠀⠀⠀⠀⠀⠸⣿⣿⡟⠀⠀⠀⠀⢸⠏⠈⠻
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠃⢘⣿⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⢸⣄⠀⢠
⠀⠀⠚⠛⠛⠓⠀⠀⠀⠀⠀⠐⠛⠛⠓⠂⠀⠀⠈⠛⠛⠛⠀⠀⠀⠀⠀⠐⠚⠛⠂
⠀⠀⣠⣤⣄
⠀⠀⣿⣿⡏⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⢹⣿⠇⠀⠀⠀⠘⠿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⢠⣤⣤⣤⣤⠀⠀⢠⣶⠤⡤⢶⡆⠀⠀⣠⣤⣴
⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠾⠿⠿⠀⠀⠀⠀⠙⣿⡟⠁⠀⠀⢸⣧⣠⣤⣀⡇⠀⠀⠿⠿⠿
⠀⠀⠙⠛⠃⠀⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠈⠉⠛⠛⠉⠁

⠀⠀⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⢀⣿⣿⣿⣿⣿⣿⠉⢿⡆⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⣷⣶⣶⣶⣾⠗⢲⡀⢀⡀⣀⣈⣉⠉⣀⣠⣤⡀
⠀⢸⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠀⢻⣿⣿⡟⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣀⣸⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⢸⣿⣿⣿⣿⣿⣿⠀⣸⡗⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠙⢿⣿⡿⠋⠉⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⢸⣿⣿⣿⣿⣿⣿⠛⠛⠁⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣹⣿⡇⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠙⠛⠟⠋⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉

⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠤⠤⠤⠼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠂⢐⠐⠐⢰⠂⠒⠂⠆⠂⢒⠂⠂⡖⠒⠀⠀⢂⠒⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⡀⠈⠀⣐⣀⣘⢀⠔⠊⠁⠀⠉⠒⢄⠰⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⠀⣀⢠⠋⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠓⠉⠉⠲⡄⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⢄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠼⠀⣤⠜⢖⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⣀⢀⡀⢰⡲⠣⣤⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⢀⠃⠀⢀⠟⠀⣀⠀⢀⡴⠉⠁⠈⠁⠀⠀⠀⠡⣸⡀⠀⠐⣧⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠂⡊⠀⠎⠀⣴⡁⠀⡰⢁⠉⠁⠀⡄⠀⠀⠀⢰⠀⠀⠀⢠⠉⢢⠀⠹⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠰⠘⡜⠀⢰⡴⡉⠉⠀⠘⡄⠀⢀⡇⠀⠠⢀⠈⢦⣀⣠⠇⠀⡘⡆⠀⢃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠁⡀⠀⡇⠀⢸⠙⢇⠀⠀⠀⠉⠉⠉⠀⠀⠀⢊⠀⠀⠀⠀⠀⢀⠷⡇⠀⢸⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠂⢅⠀⠈⢦⠘⣄⠀⠄⠂⠂⠀⠀⠈⠓⠊⠀⠀⠄⠀⣱⠎⡴⠁⠀⠘⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠈⠀⠂⠀⠘⢄⠀⠀⠑⠯⢳⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡖⠥⠋⠀⠀⡠⠁⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⡀⠠⠀⠂⠀⠐⠈⠢⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢀⠔⠁⠄⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠐⠂⠀⠀⠀⠀⠀⠐⠀⠀⢸⠁⠆⠀⠂⠀⠄⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠐⠀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⡀⠀⠀⠄⠈⠀⠀⠐⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⡀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠈⠀⠀⢨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠇⠀⠀⢹⢀⣤⣤⣤⣀⣀⠀⠀⢤⠉⠘⢀⡴⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⣿⠁⠀⠀⠀⠀⠙⡿⠙⠛⠿⣿⣿⣷⣄⠐⠂⠴⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⣲⣿⣿⠿⠷⠠⡀⠀⠀⠠⣧⠀⠀⠀⠀⠉⢿⣿⣿⠃⠀⠀⠀⢀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠐⡻⠉⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡻⠋⠀⠀⣤⠊⣸⣧⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡷⣌⠀⠀⠙⠉⠛⠃
⠀⠀⠀⠀⠀⠠⣄⡞⡤⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⡰⠀⢦⠆
⠀⠀⠀⠀⠉⠰⢻⠿⢒⠂⢀⡌⠀⡀⠀⠀⠀⠀⠘⠄⡇⠁⢇⢀⠣⡀⠀⠀⠀⠀⢹⣿⡿⠁⠀⠉⠠⡦
⠀⠀⠀⠀⠀⠀⠈⠀⠊⢡⠟⣧⣠⡆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡁
⠀⠀⠀⠀⠀⠀⠀⠁⡌⣩⡿⣯⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⢤⣴⣦⣷⡄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢖⣸⣿⣟⢟⠟⣰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⣦⣔⡈⠙⠉⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢚⣡⣴⡿⠋⠀⢄⣀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠓⢶⠶⣹⡧⣄⡠⠄⠃
⠀⠀⠀⠀⠀⠀⢠⡅⠀⣀⣀⣾⣿⣋⠀⠀⠀⢠⣿⣦⡴⠂⠐⡎⠑⠶⢻⠿⣦⡰⠀⠇⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣶⣦⣴⠟⢹⢁⣹⣄⣠⡷⠶⢦⡤⢷⠞⠳⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⠿⢦⣼⡤⠿⠻⠍⠛⠂⡄⠙⠪⡗⠀⠀⢈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣌⠆⠀⠐⠂⠰⠀⠀⠰⠆

⠀⠀⢸⣋⣉⣉⣉⣉⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣉⣉⣉⣉⣉⡷
⠀⠀⠀⡏⠀⠀⠀⢸⡇⠀⠀⣶⣛⣛⣛⣛⣛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣛⣛⣛⣛⣛⣶⠀⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⡇⠀⠀⠀⠈⡇⠀⠀⢸⡏⠉⠉⠉⢹⠀⠀⣠⣤⣤⣤⣤⣤⡀⠀⠀⡏⠉⠉⠉⢹⡇⠀⠀⢠⡇⠀⠀⠀⢸⡇
⠀⠀⢸⠇⠀⠀⠀⠀⢷⠀⠀⢸⡇⠀⠀⠀⢸⡀⠈⢳⠶⠶⠶⠶⣾⠃⠀⠀⡇⠀⠀⠀⠸⡇⠀⠀⢸⠃⠀⠀⠀⠀⣧
⠀⠀⡾⠀⠀⠀⠀⠀⠸⡆⠀⣸⠀⠀⠀⠀⠘⡇⠀⢸⠀⠀⠀⠀⣿⠀⠀⢸⠇⠀⠀⠀⠀⣧⠀⢀⡟⠀⠀⠀⠀⠀⢸⡄
⠀⣸⠃⠀⠀⠀⠀⠀⠀⠹⣄⡏⠀⠀⠀⠀⠀⢻⡀⣼⣷⡀⠀⠀⢹⠀⢀⣾⠀⠀⠀⠀⠀⢸⡄⡼⠁⠀⠀⠀⠀⠀⠀⢳⡀
⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣶⣄⠀⠀⠈⢧⣿⠈⣧⠀⠀⠘⣧⠟⡿⠀⠀⠀⠀⠀⠀⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⢳
⡟⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠸⡎⠳⣄⠀⣼⣿⠀⠸⡆⠀⣰⠋⢀⡇⠀⠀⠀⢀⣠⣤⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠘
⡇⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⢻⡀⠈⠻⣇⡟⠀⠀⢻⡴⠃⠀⢸⡇⢀⣠⠖⢋⡽⠃⠀⠈⡇
⡇⠀⠀⠀⠀⠀⠀⠀⢻⣯⡓⠶⢤⣄⡈⣧⠀⠀⠈⣁⣠⣤⢤⣤⣄⡀⠸⠟⠋⠀⢀⡞⠁⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠛⢦⡀⠀⠉⠛⢀⡴⠛⠉⠀⠀⠀⠀⠀⠉⠳⣄⠀⣠⣏⣀⣀⣀⣤⣤⣧⣤
⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣙⣦⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡈⠉⠉⠀⣀⡤⠞⡏
⡇⠀⠀⠀⠀⠰⢿⣟⣋⠉⠉⠉⠉⠁⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢀⡴⠚⠉⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⢹⡟⠓⠶⢤⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠉⠛⠲⠦⣤⣇⡀
⡇⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⠖⠋⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣠⣤⣤⠤⠤⠶⡾⠿⠃
⡇⠀⠀⠀⠀⠀⢀⣴⣞⣉⣀⣀⣤⣤⣄⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠙⠷⣄⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠉⠉⢹⡏⠀⠀⢀⡼⠁⠀⠀⠈⠛⠦⢤⣤⣀⣤⡤⠖⠋⠀⣶⢤⣀⡀⠘⠳⣄⠀⡇
⠻⣄⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡞⠁⣀⡴⢾⡇⠀⠀⢀⡀⠀⠀⢠⣄⠀⠀⢹⡄⠈⠉⠛⠲⠾⣷⡇⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠈⠙⠓⠲⠶⠶⠒⠚⢧⣠⣯⡴⠚⠁⠀⢸⡇⠀⣰⠏⢳⠀⠀⣾⠈⠳⣄⠀⢷⠀⠀⠀⠀⠀⣰⠛⠓⠲⠶⠶⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠤⠤⠤⢾⠀⡴⠃⠀⠈⣇⠀⡿⠀⠀⣸⠷⣌⣧⠤⠶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡾⢧⣄⣀⡀⢸⣄⣇⣤⠞⠃⠀⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠻⣦⡶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⢹⣇⠀⠛⠁⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⢾⡇⠈⣷⠀⠀⠀⠀⠙⣷⣶⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣄⣀⣴⡏⠀⠀⠀⠀⢀⡿⢸⠇⠀⠀⠀⢀⣀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢧⠹⡄⠀⠀⠀⢸⡇⣿⠀⢀⣴⣾⣿⣏⣻⣝⣿⠿⣛⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢻⡄⠀⠀⣿⢱⡇⢰⣟⣿⣿⣿⢧⢻⣽⡾⣿⣚⣿⣯⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢻⡄⢠⣿⣼⢁⣿⣷⣿⣿⣿⡞⢯⢷⣛⢩⣭⣷⣾⡖⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣷⣿⡾⠻⣿⢛⡿⣿⣮⣿⠛⣧⢘⣼⣿⣿⣿⠿⣿⣿⠿⣿⣄⡀⢀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣞⣿⡿⢫⣿⣁⠢⠩⣟⢳⠞⣵⡿⣹⣿⣷⡿⢟⡿⣩⡞⣫⢶⡝⢯⠶⣩⠿⡛⣿⣭⣛⣵⣾⡍⢟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⡳⣟⣭⣿⢁⣿⢣⣥⢉⣿⢂⠇⣼⠏⣽⣫⣿⢋⡜⣧⠞⣥⢚⡴⢫⠝⣥⠿⣡⡜⢯⣩⣿⣟⣽⡴⡾⡟⣮⣷
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡼⣱⣟⣾⡧⣜⡏⣗⡮⡙⡙⠾⠽⡟⣼⣳⣿⣋⣏⡸⡏⡶⣹⢎⡸⢇⢿⠙⡤⠹⣧⠚⣇⠴⢻⣿⣷⣻⣿⣱⢾⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣯⣽⣿⣗⣿⣗⣶⣽⡃⢯⣿⣳⣲⢦⣆⣿⣿⣸⠇⣸⣀⢇⡘⢢⡜⢆⣗⠈⡆⠹⡖⢫⣨⣼⡿⣿⣭⣿⣬⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⡷⠟⠛⠛⠛⠛⠻⠷⣧⣽⡒⣬⡼⠟⠉⠉⠉⠻⢶⣦⡟⡴⠣⡜⠳⡜⢻⡆⢡⠜⣧⣾⣥⣾⡷⠿⠷⠷⣷⣯⣀
⠀⠀⠀⠀⢀⣀⣀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠙⠿⣯⣤⣑⣮⣉⣮⣽⡾⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄
⠀⠀⢠⡾⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠷⣦⡀⠀⠀⠀⠀⣠⡾⠋⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠻⣆⣤⣤⣴⡏⠀⠀⢈⡷⠀⠀⠀⢀⣀⣤⣤⢶⡶⠶⠶⢶⣤⣤⣀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠙⣦⣀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⡀⠙⢷⣄⣀⣏⠀⢀⣿⣥⣤⡴⠞⢁⣤⡶⠟⠋⠉⠠⠀⠈⠀⠀⠀⠀⢀⠈⠙⣻⣶⣤⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠉⠻⡟⠛⠛⠉⢉⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⣶⡾⠛⢹⣿⣿⡙⢷⣤⣶⠞⠋⡁⠀⠀⠃⠀⢀⠀⢠⠄⠀⠂⠀⠁⠀⢐⡟
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠂⠀⠀⠀⠀⢁⠀⠀⠈⠀⠀⣿⠋⠀⢀⣾⠋⠙⣷⠀⠻⣧⣀⠀⠄⠀⠐⠂⠀⠈⠀⠀⠀⢀⠄⠀⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢄⠀⠀⠀⠀⠀⠉⠈⣿⣤⣀⣼⠏⠁⠀⢻⣦⠶⠟⠃⠀⠀⠀⢀⡀⠀⠰⠀⠘⠁⠀⡀⢠⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⡀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠨⠙⠫⠀⠀⠀⠰⠀⠂⠂⠀⢈⠁⠀⠠⠤⠀⠄⠐⠐⠁⠀⠁⣀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢈⡀⠈⡀⠀⠰⠀⠀⠀⠀⠀⠆⠀⠂⠘⠁⠘⠀⠀⠉⠀⠀⠀⠀⢀⠀⠀⠄⠀⠒⠀⠘⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣡⣀⠀⡀⠀⠁⠀⠈⠀⠀⠒⠀⠐⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⣀⣀⣤⡴⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠓⠲⠷⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠖⠛⠛⠉⠁


⣿⡟⣛⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡋⡸⢶⡇
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⣡⣈
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢻⡀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠰⠄⠀⣵⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠘⢀⠀⡤⣤⣤⣤⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢼⣿⣇⠀⠀⣰⠻⠟⠻⡇⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⢠⣿⡿⣻⢟⠶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠻⠆⠒⠉⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠐⣸⡿⢟⠁⠙⡇⢿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠠⣀⢿⣠⡀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⢿⣿⣷⣭⣬⣵⣿⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣁⣻⣇⠀⠀⠀⠀⡀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡀⢀⡟
⠀⠀⠀⠀⠀⢾⡂⠈⠉⠉⠉⠩⠅⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣧⣀⡀⢀⣀⣧⣴⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⣿
⠀⠀⠀⠀⠀⠸⣦⣤⣤⣤⣤⣤⣤⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣉⠀⠈⠉⠉⠁⠀⢒⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠑⡠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⡏⠀⠹⣿⣿⣿⠏⠉⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⡐⠿⠻⣄⣀⠐⠒⠚⢀⡀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠉⠉⠉⠁⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠛⠛⠁⠐⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠓⠒⠀⣘⡃⠀⠀⠀⠀⠀⠀⣨⠀⠀⢀⣀⣤⣤⣀⣀⣀⣀⣀⣡
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⡀⣚⠀⠀⠀⠀⠀⠀⢰⣿⣄⣀⣀⣠⣀⣀⣀⣀⣤⣤⣭⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⠿⣿⣣⠄⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉⣿⠃⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣷⠰⠀⠱⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⣿⣁⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⠀⠰⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⣰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⠀⠀⠀⠀⠰⢿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠴⠾⣖⠛⠰⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠰⢿⢼⣧⣤⣠⡈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠂⠈⠀⠐⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⣀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⣀⣀⣀⣤⡤⣦⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀⢀⣀⣠⣀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠀⠀⠀⠀⣠⣿⣿⣿⣿⣶⣶⣶⣿⣧⣤⣿⡄
⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⡥⠀⠀⠀⠀⠉⠉⠍⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠐⠛⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⠍⠉⣛⡛⠛⢛⣛⣋⠡⠀⠀⠀⠀⠀⠈⠉⠉⠉⠋⠀⠀⠈⠁
⠀⠀⠀⠀⠠⠰⣶⣶⣶⣦⣤⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⣀⣀⣀⣠⡄
⠀⠀⠀⣀⣐⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠐⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡇⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠉⠿⠿⢿⣿⣿⣿⣿⡉⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠭⣛⣻⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠠⢶⣶⣶⣶⣶⣶⣾⣿⣷⡆⠀⠀⠀⢸⣷⣶⣤⣤⣤⣤⣤⣴⣶⣿⣷⣶⡇⠀⠀⠀⠀⣠⣴⣶⣿⣷⣶⣶⣄⣠⡄
⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠻⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠭⠭⠭⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠛⠋⠉⠉
⠀⠀⠀⢀⣦⣄⡀⠀⠀⠀⢀⣤⣤⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣠⣦⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠾⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠀⠀⠨⢤⠬⠴⣤⣤⣤⠴⠶⢛⣛⣿⡇
⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠃⠀⠀⠠⠀⠀⣀⣴⣶⣶⣿⣿⣿⡿⠋⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⣰⣿⣿⣷⣶⣶⣆⡀⠀⣠⣿⣷⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⠀⣀
⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠆⠀⠀⠒⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⢿⡟⠀⠀⠀⠀⠼⣿⣿⣿⣿⣿⣿⣿⣯⣤⡄
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢻⣿⣭⣭⡭⠴
⠀⠀⠀⠀⠀⠀⠐⠒⢂⣀⣀⣀⣠⣤⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⡀⠀⠀⢘⣿⣿⣿⣭⣭⣥⣶⣶⣶⣿⣄⠀⠀⠀⠀⢀⣩⣿⣿⣿⣷⣶⣾⣿⡷
⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠈⠋⠉⠛⠛⠿⠿⠟⠛⠛⠛⠒⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢾⡆⣾⣿⣴⣿⢀⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⡀⣿⣦⣿⣷⢸⡷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣷⡹⠟⣼⣫⠟⠁⠀⠀⠈⠻⣿⣯⣽⣿⠟⠀⠀⠀⠈⢿⣝⣧⠻⢏⣿⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⢀⣠⣤⣤⣤⣭⣭⣤⣤⣤⣄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⣠⣜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣣⣄⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣷⣾⣿⣿⠗⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠸⣿⣿⣶⣾⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠟⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠻⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣶⣦⣌⣛⠿⢿⣿⣿⡿⠿⣛⣡⣴⣶⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣥⣚⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⢻⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣝⡻⠿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠒⠉⠙⠛⠿⠿⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠈⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣶⠇⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⣤⠤⠶⠶⠦⠤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠉⠛⠒⠲⠶⠶⠶⠒⠚⠛⠋⠉⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣀⣠⣀⡀⠀⠀⠀⢀⣀⠀⠀⢀⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⢻⠀⢨⠛⣦⣤⣴⠛⣍⠳⢴⡏⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⡾⠀⢸⡀⡇⠀⢸⠀⢸⠀⢸⡇⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⠀⠀⢧⡇⠀⠀⡇⡇⠀⠈⡇⢸⠀⢸⡷⠋⠉⠙⣆⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣤⠞⠁⠀⣿⠀⠀⢸⡇⠀⠀⣧⡇⠀⠀⣧⢸⠀⠠⣧⠀⠀⠀⠈⠉⠈⠉⢻⣄
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⢻⠀⠀⢿⣷⠀⠀⢸⢸⡄⠀⣶⠀⠀⠀⠀⠀⠀⠀⢸⣟
⠀⠀⠀⠀⠀⠀⠀⣨⡇⠀⠀⠀⠀⠀⢀⡿⠀⠀⢸⠸⡀⠀⢸⣿⠀⠀⢸⡀⣇⣀⣽⣄⠀⠀⠀⠀⠀⠀⠀⠉⠻⡄
⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⣿⠟⡗⢲⠾⢶⠷⣶⠾⠿⡶⢲⡟⠛⣯⢉⣧⡼⠃⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠹⢦⡄⠀⠀⠀⠀⠀⠀⠈⠻⠷⢼⣤⣼⣤⣼⣤⣤⣷⠴⠷⠖⠛⠋⠁⠀⠀⠀⠀⠀⠀⡴⠒⠚⠉
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠙⠲⠖⠛⠉⠉⠳⣄⠀⠀⢀⣠⠖⠚⠓⠶⣄⡀⠀⢀⡴⠋⠁⠀⠉⠙⠲⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠉⠛⠉

⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠱⡟⣒⠤⣀⡠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠔⡀⢀⠀⢀⠀⠐⠂
⠀⠀⠀⠙⠀⠀⢱⠋⢀⢔⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢠⠀⡀⠀⠆⠈⠁⠀⠈⠀⠁⠆⠈
⠀⠀⠀⠀⠀⠀⠻⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠸⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠐⠀⢠⢀⡔⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢢⠚⡠⠖⠀⢰⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠄⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠿⣫⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⡤⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⢀⠀⠈⠀⠀⠀⠀⠀⠈⣁⠖⠂⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠍⣉⣉⣁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠐⢤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠐⠂⢰⡞⠐⠀⠀⠐⠀⠀⠁⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠂⠀⠀⠀⠀⠀⠠⡀⣸⠃⠀⠀⠀⠀⠂⢈⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠁⠀⠀⠀⠀⣀⡈⠀⠀⠀⡀⢀⠔⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⡀⠀⠂⠀⠀⠃⠈⢀⠁⠀⠌⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢨⡢⠀⠀⠀⠐⠀⠈⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⡀⠀⠈⠂⡐⡤⠃⠀⠋⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⣉⡀⡇⢀⠀⣈⠄⣀⣀⠜⢁⣀⡀⢘⠀⡶⠄⡀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⡞⠐⢳⠖⡶⠴⠭⡍⠀⠀⠀⠈⠉⣦⣥⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠂⠀⠀⢳⢤⡠⢀⣀⣼⢏⡯⣔⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⡓⠺⠅⢉⡞⠠⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠰⡀⠀⠀⠁⠈⠁⠈⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣄⣠⣶⣶⣷⣶⣶⣦⡀⠀⠀⠙⠒⠀⠂⠁⠤⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣴⡿⠃⡀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠋⠀⠀⢿⠆⣤⢸⣿⣿⡿⣿⣿⣿⡟⠀⠀⠀⠀⣠⣰⣿⣤⡄⠀⠀⠀⠀⠘⣿⣿⣷⣶⡀
⢠⡶⠂⠀⠀⠀⠀⠀⠻⠃⡠⠀⢀⢀⣤⣤⣀⠀⢠⣿⣿⣵⣶⣾⣿⣿⣿⡀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣤⣣⠀⠀⠀⠉⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡁⠉⠛⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢛⠉⠉⢰⣶⣶⣴⣶⣤⠀⣉⠋⠀⠀⠀⠀⠀⣤⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⣴⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠛⢿⣿⣿⣿⡦⡤⢤⣀⡀⠀⠀⠀⢀⣸⣿⣶⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠒⠒⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠏⠉⠉⠉⠉⠉⠁⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠉⠉⠻⣿⣿⣇⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣿⠿⠉⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⣸⠿⠋⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠒⡆⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠾⡀⠀⡞⢍⣳⠀⠀⠀⠀⠀⠀⢰⠒⠒⢦
⠀⠀⠀⠀⠀⡠⠚⠁⡀⠈⠳⢕⠚⣉⣲⡄⠀⠀⠀⠀⢸⠶⠶⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣒⡶⠄
⠀⠀⠀⠀⢸⠁⠉⠁⠈⠉⠉⠉⡍⠀⠀⡧⠀⠀⠀⢀⣎⡀⣀⣘⣦⠀⠀⠀⠀⠀⠀⠀⣟⣿⣿⡇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣇⣀⣀⡇⠀⠀⢀⠎⠀⠀⠀⠒⠈⣧⠀⠀⠀⠀⠀⠀⡗⠒⢚⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⣉⣉⣉⣉⣉⢉⣉⡧⣤⣾⡇⠀⠀⢸⣀⡀⠈⠁⣀⣀⣸⠀⠀⠀⠀⠀⢸⡀⠉⠉⢹⡄⠀⠀⠀⠀⠀⡇⡠⠤⡦
⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠛⠛⡟⠉⢸⠁⠀⠀⢸⠉⠛⠛⠛⠛⣉⣭⡀⠀⠀⠀⠀⡏⠑⠦⠴⠋⢹⡀⠀⠀⠀⣠⠋⠀⠀⠸⡄
⠀⠀⠀⠀⢸⠀⢀⡴⠄⡄⠀⠀⡇⠀⢸⡀⠀⠀⣿⢰⠖⢲⣶⡶⠕⠽⡇⠀⠀⠀⢠⠀⠀⣀⠀⠀⠘⡇⠀⠀⠘⠁⠀⠀⠀⠀⠉⡀
⠀⠀⠀⠀⢸⠀⠈⠛⢚⡁⠀⠀⡇⠀⢸⡇⠀⠀⢻⡰⣄⣠⡽⠖⢶⠆⡇⠀⠀⠀⢸⠀⠀⠉⠀⡄⠀⡇⠀⠀⢸⠀⣀⣀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⢧⡤⠇⠀⠀⡇⢀⢸⡇⠀⠀⢸⠀⠈⠀⠀⠀⣀⣀⡇⠀⠀⠀⣾⠀⠀⢀⠀⠀⢠⡇⠀⠀⢸⡘⠀⠈⢳⠊⠁⡇
⠀⠀⠀⠀⢸⠀⠀⠠⠶⠀⠀⠀⣧⡌⠘⡇⠀⠀⠀⣏⠙⠛⠛⣛⡋⢹⠀⠀⠀⠀⣿⠀⠀⠈⠀⠀⢸⡇⠀⠀⢸⠙⠦⠤⢊⡤⠴⠾
⠀⠀⠀⠀⢸⡆⠀⠐⣲⠀⠀⠀⡿⡀⡄⡇⠀⠀⠀⡏⠐⠔⠀⠉⠀⠘⡆⠀⠀⠀⣿⠀⠀⠀⠀⠂⢸⡇⠀⠀⢸⠀⡠⠚⠁
⠀⠀⠀⠀⠘⡇⢠⠼⠧⠤⢤⠀⣿⢻⠀⡇⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢀⡘⡇⠀⠀⢸⡯⠖⠒⠒⠒⠒⢂
⠀⠀⠀⠀⢠⡇⠸⠀⠀⢀⡸⠀⣧⠘⢀⡇⠀⠀⠸⣄⠀⠀⢀⣀⠀⣸⡟⠀⠀⠀⠸⣀⡀⠀⠀⣀⣠⠇⠀⠀⠘⢇⣀⠀⠀⠀⣀⡜
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⣿⡠⠋⠁⠀⠀⠀⠙⠓⠒⠋⠈⠙⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⣿⣿⠀⠀⣄⠀⠀⢰⣿⣧⠀⠀⠀⠀⠰⣿⡇⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⠀⠀⠚⠛⠋⠁⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣶⡄⠀⠀⠉⠀⠀⢀⣠⠴⠶⠚⠛⠛⠛⠉⠛⠛⠛⠲⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠅⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⣀⠀⠀⠀⡄⠀⠀⠀⢣⣤⣶⣄⣠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠘⣷⣄⢸⣿⡇⢀⡞⠀⠀⠀⠀⠈⣿⣻⣛⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⡠⠤⡿⢿⡀⠀⠀⠀⠈⣮⠺⠋⠙⣿⠀⠀⠀⠀⠀⠀⣹⠈⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠆⠀⠀⢳⠀⠀⠀⠀⢸⡇⠀⠀⢸⡄⠀⠀⠀⠀⠀⡇⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⢸⡁⠀⠀⠀⣇⠀⠀⠀⠀⣰⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡀⢁⠀⡇⠀⠀⢠⣿⠁⠀⠀⠀⢹⠀⠀⠀⠀⠻⣄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⡷⢷⡟⠁⠀⠀⠀⠙⢶⣤⣤⣖⣸⡇⠀⠀⠀⠀⢻⠟⠲⣶⡿⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣧⠀⠀⠀⠀⠀⢸⠟⠀⠹⣯⠀⠀⠀⠀⠀⠘⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣲
⠀⠠⠤⠠⣤⣤⣬⣤⣤⠤⠤⢤⣤⣬⡶⠶⣿⡦⠒⠀⠀⣤⣬⡤⠤⢨⣼⡤⠤⠤⠤⠤⣷⡦⣤⣤⣤⣄⣤⣀⣐⣀⣀⣀⣤⣿⣿⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣴⣿⡏⠀⠀⠀⠙⣿⣿⣶⣿⣿⣤⡄⠀⠀⠘⣿⣦⣼⣿⣦⡍⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠐⠆⠀⠀⠂⠂⠀⠂⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠐⠀⠐⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣿⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣩⠴⠶⠶⢬⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠿⣛⣛⣛⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡼⠁⠀⠀⠀⠀⠙⡆⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⣱⢫⣷⣶⣮⡝⣎⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⠙⠲⡄⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⢟⣥⣥⣝⢿⣿⣿⣿⢸⢸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⡼⠤⠤⠤⠤⠤⢧⡀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⣼⣭⣭⣯⡜⣿⣿⣿⢸⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠚⠁⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⢸⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠅⠨⠨⠈⠅⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⢸⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠒⠐⠂⠒⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣽⣿⣯⣿⣿⣿⣾⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠐⠃⠘⠂⠒⠀⠃⠐⠂⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⠿⠟⠋





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡾⠿⠿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠙⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠟⢁⣤⣤⣤⡈⠻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣧⣄⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠟⠠⢿⣿⣿⣿⣿⡆⢹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⢹⣿⣿⣿⡇⢸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣤⣤⣤⣤⣤⣿⣿⣿⡇⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⡂⡂⢐⢐⠐⡂⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠤⠠⠄⠤⠠⠄⠤⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⣿⢾⡿⡿⣿⣿⣿⣇⣸⣿⠟
⠀⠀⠀⠀⠀⠀⠠⠄⠰⠄⠤⠀⠆⠰⠄⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⣤⣤⣾⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⣿⣿⣿⣿⠟⠛⠛⠋⠁⠀⠈⠳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣻⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣹⣿⣿⣿⣿⣿⠀⣠⢶⡚⠋⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⠛⢿⣿⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⡄⠀⠹⣇⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣦⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠉⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠰⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣷⣤⣄⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠐⠂⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣄⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠰⠟⠛⠛⠙⠛⠛⠛⠟⠿⣿⠿⠃

⠰⢆⠀⠐⣄⠈⢠⠌⠁⢠⡀⠀⢠⠀⠀⠋⠈⠃⠀⠀⠇⠀⠰⠀⠀⠀⠠⠀⠰⠀⠀⠁⣄⠀⣠⠊⠁⢠⠀⠀⡄⠀⠀⠇⠸⠂⠀⠈⠌⠀⠌⠀⠀⠀⡄
⢀⡀⠀⠀⠈⠉⠉⠀⠀⠀⠈⢣⠈⠀⣘⣠⠝⢀⡐⠀⠀⢀⣀⣾⡀⠀⢄⡀⣀⡸⠀⠀⠉⠉⠁⠀⠀⠀⠘⣇⠀⢀⢊⣤⠓⢀⡂⠀⠀⣀⣀⣾⠀⠀⣀
⠀⠀⠀⠀⠀⢠⣦⣴⣄⠀⢄⣃⡐⠀⠀⠬⠧⠼⠩⠀⠀⠊⠉⠉⢵⠀⠀⡆⠐⣲⡒⠄⠀⣰⣤⢶⡄⠀⠈⠀⠀⠀⠀⠌⠥⠭⢸⠀⠠⠋⠉⡩⠇⠀⢠
⠊⣷⠘⡄⠀⠈⡀⢀⠁⠀⠀⣿⠀⠀⠀⠐⢄⠀⡰⠀⠀⣀⠀⠐⢸⠀⠀⢡⠘⣷⢫⡅⠀⠰⠀⢰⠀⠀⠀⡅⠀⠀⠀⠡⡀⠀⠌⠀⠀⡀⠀⠁⠆⠀⢈
⠉⠉⠩⠝⠀⡀⠀⠀⠀⠀⠠⠋⠦⠀⠀⠀⠀⣏⠀⠀⠀⠈⠁⠀⠋⠀⡀⠋⠩⠩⠭⠃⠀⠀⠀⠀⠀⠀⠴⠱⠄⠀⠀⠀⢸⡅⠀⠀⠀⠉⠀⠈⠁⢀⠈
⡄⡀⠀⢀⣀⣕⣁⣀⠀⠀⣔⠒⠒⢓⠄⠀⠛⠒⠃⠠⠤⠬⢤⠀⠠⠠⠇⢠⣀⡀⠀⢀⣀⣂⣀⡀⠀⢠⡖⠒⠚⣢⠀⠀⠓⠘⠀⡤⢤⠬⡄⠀⡄⠤⠂
⡽⠃⠀⠘⡀⠀⠀⣸⠉⠂⠀⣠⠶⠘⠀⣤⠀⣤⠀⢸⢘⡧⡇⠀⠰⢀⡅⢸⠝⠁⠀⠠⡀⠀⢀⡬⠉⠀⠆⡔⡆⠄⠀⡄⠠⡄⠀⢃⣸⣥⡇⠀⢁⢠⡄
⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠘⠈⠀⠇⠀⢠⣀⡼⠀⠈⣹⣟⡇⠀⠀⠟⠂⠎⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠠⠉⠠⠀⠀⣤⣀⡄⠀⢨⣹⣿⠁⠀⠘⠔⠃
⠀⡇⠠⠨⠀⢀⠂⠀⠢⠀⠀⣴⡀⢠⣴⣤⡀⠀⡀⠐⠁⠀⠀⠤⠤⠤⠄⠀⠨⠄⠀⠤⠀⠠⠂⠐⢄⠀⢀⣦⠀⣠⣶⣄⠀⢀⡄⠂⠁⠀⠀⠤⠤⠤⠄
⠀⠠⣀⠔⠀⠘⢀⣀⡨⠃⠀⡋⠁⠘⠿⠷⣤⠟⠞⠁⠀⠀⠀⢳⣄⡀⠃⠀⠀⠢⣀⠄⠀⠣⢀⣀⠘⠀⠀⠉⠀⠻⠿⠷⡰⢻⠎⠀⠀⠀⠀⢣⣄⠘
⠀⠀⡜⢀⠀⠀⠴⠈⠶⠀⠠⠁⠁⠀⠀⠀⠈⡇⠀⠀⠘⠹⠀⠀⠐⠁⠀⠀⠀⢀⠿⡀⠀⠀⠖⠐⠆⠀⠔⠀⢢⠀⠀⠀⠸⠃⠀⠀⠚⠘⠀⠀⠊
⡄⠈⠉⠉⣀⡀⢀⠀⠀⠀⡀⠐⠀⡀⠀⠀⠐⠙⠂⠀⠘⠈⡀⠈⠁⠈⢁⠤⡀⠈⠉⠉⣀⠀⡀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠂⠐⠀⠀⠀⢸⠀⠈⠁⠈⢠
⠌⠆⠀⡊⠀⠀⡀⢱⣀⠀⠑⠒⠀⡇⠀⠖⠒⡆⠀⠀⣆⡀⠱⠀⠀⠀⠆⠤⠰⠄⢠⠈⡀⢀⡈⣠⣀⠀⡖⠒⢀⠀⠰⠒⠒⠆⠀⢠⣁⠀⢁⠀⠀⠠⠅
⠠⠀⠀⠇⠇⢼⠀⢉⣅⠆⠈⠉⠉⠀⠀⡀⠀⡇⠀⠀⡀⢈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠇⠧⠀⢉⣸⠁⠈⠉⠉⠀⠀⡀⢀⠀⠀⢸⠀⡰⢠⠀⠀⠀⠄
⠰⢆⠀⠀⢤⣈⣠⠊⠀⢠⣌⣉⢥⠀⠀⡃⠈⠃⠀⠀⠃⠀⠐⠀⠀⠀⠰⠤⠰⠄⠀⠑⣄⣀⣠⠉⠀⢰⢉⡉⡄⠀⠀⠃⠙⠂⠀⠈⠅⠀⠈⠀⠀⠀⠤
⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⡌⢤⠚⢁⡀⠁⠀⢀⠠⠾⡄⠀⠆⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠘⡥⢐⢀⡊⢤⠋⢀⠊⠀⠀⣀⠠⢿⠀⠀⠄
⣀⣠⣄⣢⠀⢠⠒⠒⡀⠀⠀⣁⠐⠀⠀⠤⠓⠘⠐⠀⠀⠋⠉⠉⢱⠀⠀⣆⢀⣠⣀⡂⠀⣔⠒⢲⡄⠐⢌⣁⠄⠀⠀⠎⠃⠛⠘⠀⢰⠋⠉⡉⠇⠀⢠
⡊⣷⣚⡄⠀⠈⡀⢀⠁⠀⠀⣯⠀⠀⠀⠐⢄⣀⠐⠀⠀⣀⠀⠐⣸⠀⠀⣠⢈⣷⣣⠄⠀⠰⡀⠠⠀⠀⠀⡍⠀⠀⠀⠁⢀⣀⠌⠀⠀⡀⠀⠑⡂⠀⢈
⠁⠁⠉⠙⠀⡀⠀⠀⠀⠀⠰⠁⠆⠀⠀⠀⠀⢏⠀⠀⠀⠈⠉⠉⠁⠀⡀⠋⠈⠈⠉⠃⠀⠀⠀⠀⠀⠀⠰⠼⠆⠀⠀⠀⢸⠅⠀⠀⠀⠉⠉⠉⠁⢠⠈
⣀⡀⠀⢀⠀⠤⠅⢀⠀⠀⡾⠬⠭⢣⠄⠀⠉⠈⠁⠠⠀⠆⢠⠀⠐⠲⠖⢠⣤⡀⠀⠀⠠⠦⠄⢀⠀⠠⠆⠀⠀⠀⠀⠀⠁⠈⠀⡄⢰⠂⡄⠀⡄⠰⠂
⡞⠃⠀⠈⢀⠀⢀⡼⠀⠁⠁⣤⡲⠈⠀⠤⠤⠤⠀⢸⣨⡥⡅⠀⠰⢀⡅⢸⠞⠁⠀⠠⡀⠀⢀⡾⠐⠀⢆⣔⡖⠄⠀⡤⠤⠄⠀⢫⣨⣭⡇⠀⢀⢠⡄
⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠘⠤⠤⠇⠀⠠⠤⠼⠀⠈⣙⣟⡇⠀⠀⠟⠦⠃⠀⠀⠀⠀⠘⠋⠉⠀⠀⠀⠰⠠⠴⠀⠀⠦⠤⠆⠀⢸⣹⣻⠀⠀⠈⠔⠦
⠀⡥⠤⠬⠄⢀⠊⠉⠀⡀⠀⣶⡀⢠⣶⣦⠀⢀⡄⠂⠀⠀⠀⠄⠀⠀⠄⠀⠨⠤⠤⠬⠀⠠⠀⠉⢂⠀⢀⣶⠀⣴⣶⣆⠀⢀⣄⠀⠀⠀⠀⠄
⠀⠐⣤⠄⠀⠘⠄⣀⠌⠁⠀⡍⠅⠘⠛⠳⢤⢛⠞⠀⠀⡀⠀⢱⠆⡀⠁⠀⠀⠂⣤⠂⠀⠡⢄⡀⠜⠀⠀⠉⠀⠙⠛⠳⣰⠻⠊⠀⠀⡀⠀⠳⢆⡈
⠀⢀⠜⣄⠀⠀⠒⠈⠒⠀⡐⠁⠑⠀⠀⠀⠈⡇⠀⠀⠘⢺⠀⠀⠐⠀⠀⠀⠀⢀⠟⣀⠀⠀⠒⠘⠂⠀⠔⠀⠢⠀⠀⠀⢸⡁⠀⠀⠙⢊⠀⠀⠂
⡄⠀⠀⠀⡀⠠⢄⠀⠀⠀⡃⠐⠀⡄⠀⠀⠚⠚⠂⠀⠘⠈⡀⠀⠁⠈⢠⣀⡀⠀⠀⠀⠀⠤⢄⠀⠀⠀⠃⠐⠀⠀⠀⠀⠚⠒⠀⠀⠈⠹⠀⠈⠁⠀⢀
⠐⠆⠀⡊⡀⠀⣀⣱⠄⡀⢑⠀⠀⡇⠀⠇⠉⡇⠀⠀⡇⡀⠐⠀⠀⠀⠆⠀⠘⠂⠠⣉⡀⢀⣀⣡⢄⠀⡇⠈⢀⠀⠘⠉⠘⠃⠀⢠⠀⡀⢁⠀⠀⠰⠂
⣰⣀⣀⣃⣇⣸⣀⣈⣈⣀⣀⣀⣀⣀⣀⣀⣀⣃⣀⣀⣀⣀⣀⣀⣀⣀⣣⣀⣰⣀⣀⣐⣇⣓⣀⣩⣘⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣐⣈⣀⣀⣀⣆
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣠⠤⣤⣤⣤⡀⠀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⢸⣷⡎⠠⣉⢛⣕⠛⠃⠀⠀⠀⢀⣦⣶⣤⡖⣼⣿⡇⠀⠀⠀⠀⠈⢷⡾⠁⠀⠀⠀⠀⢀⠀⢆⡁⠶⣀⠰⢀⡅⠀⣿⣦⣍⠳⡼⣻⣥⠁⢴⣿⠁
⠀⠀⢈⡸⢗⠂⠸⣉⣿⣇⠀⠀⢠⣤⣾⡏⣾⣭⣾⣸⣿⠀⠀⠀⢀⣀⣼⣟⣿⡧⣀⡀⠀⠀⢈⠰⠎⠁⡦⠉⠴⢌⡁⠀⣿⡿⠀⠻⢧⡻⣯⣜⠿⣿⠆
⠀⠀⠹⣷⣾⣉⣼⡏⢛⣽⠆⠀⢸⣇⠻⣫⣬⣭⣽⢿⣭⡀⠀⠀⠈⠻⡟⣙⣳⠿⠟⠁⠀⠀⠘⢠⡜⢃⣤⠛⢠⠈⠃⠀⣿⣍⠶⣶⠈⠳⠆⠉⢠⣿⠂
⠀⠀⣠⠙⣿⣮⠙⠛⣀⢸⠆⠀⢸⣟⣤⡟⠛⢻⡟⣶⣻⠟⠀⠀⠀⠀⠺⣿⣿⠆⠀⠀⠀⠀⠸⠀⡴⠃⣀⠚⢠⠐⠇⠀⣿⣉⣿⢻⣴⠖⠀⣰⡟⣿⡂
⠀⠀⠻⣣⡌⢻⣿⣞⠁⣸⠇⠀⠈⠃⣿⡬⠿⢲⣿⢻⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⢄⡑⠆⣁⠲⢀⠰⡇⠀⣿⡀⢿⡆⠀⢠⣾⠛⠋⣾⡃
⠀⠀⠟⣩⣿⢿⠉⢿⣿⡿⠀⠀⠀⠀⠈⠻⣾⠿⠐⣿⡇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠌⠡⡦⠉⢴⠈⠡⡆⠀⣿⣇⠐⠀⣴⣏⡄⠛⢧⣾⡇
⠀⠀⢼⣿⣟⠀⣰⣤⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢸⣈⣡⣄⢋⣤⣌⣠⡇⠀⣿⡁⠀⠘⠁⣿⡥⠀⠀⣿
⠀⠀⢹⣿⣉⠹⠃⢨⣿⣿⠇⠀⢠⡀⠀⠀⠿⣦⠀⠀⠀⣀⠀⠀⠀⢠⣞⡻⢟⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⡄⠀⢙⣻⣴⡿⠃⠀⣿⠇
⠀⠀⢸⣿⡉⢹⣿⡆⣸⣿⡄⠀⠀⢑⣤⠀⣴⢨⣿⢀⡔⠏⠀⢠⣾⣥⢾⣷⣾⡷⣬⣷⡆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠸⠟⠡⢠⣿⠆⠀⣿
⠀⠀⠈⠛⠓⠈⠛⠛⠉⠋⠀⠀⠠⢼⣿⠇⣼⡾⠟⢸⣟⢦⠀⠈⠻⣅⣟⡽⢻⣟⣩⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠐⠚⠁⠈⠉⠀⠘⠛⠁
⠀⠀⠀⠀⣰⣾⣋⣳⡄⠀⠀⠀⢠⡞⢻⡄⣿⢰⠂⢠⡛⣦⠀⠀⠀⠈⠣⣼⣷⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⣴⣿⣷⣦
⠀⠀⠀⠐⢃⣿⡿⠿⡇⠀⠀⠀⠸⣇⢈⠁⠟⢿⣅⠈⡁⣹⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡀⠀⠀⠀⠘⣷⡿⣿⣿⠂
⠀⠀⠀⠀⠈⠛⠛⠊⠀⠀⠀⠀⠀⠈⡛⠀⠐⠚⡷⠘⢋⠁⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠆⠰⡆⠀⠀⠀⠀⠀⠀⠉⠋⠋⠁
⠀⠀⢠⣄⣤⣶⣄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡤⣤⡀
⠀⠀⣴⣏⠻⡯⣾⣷⠀⠀⠀⢿⣶⠀⠀⠀⢰⡟⢛⡦⠛⢻⡅⠀⠀⠀⣿⡿⣿⣿⠀⠀⠀⠛⢦⡼⢉⣹⡍⠩⣇⡴⠛⠀⠠⠈⠂⡀⠀⣿⡟⢿⣥⢿⠁
⠀⠀⢿⣿⣆⣤⣽⡋⠀⠀⠀⣸⣿⡀⠀⠀⠀⠳⠏⣶⠹⢞⠃⠀⠀⠀⣿⣶⣿⣿⠀⠀⠀⣤⠎⠻⣯⡳⢈⣵⠟⠃⣾⠀⣤⢀⢎⡙⠀⣾⣄⢻⡷⢿⠁
⠀⠀⡸⣿⣿⣿⣾⠃⠀⠀⣾⣿⣿⡿⠀⠀⠀⡹⠗⣖⠚⢿⠅⠀⠀⠀⣿⣼⣿⣿⠀⠀⠀⡅⣠⠋⠌⣳⣞⠁⣝⡆⣹⠀⢀⡹⣮⣥⠀⣾⢋⡷⠛⣽
⠀⠀⠀⠙⡿⠻⣿⠃⠀⢰⣿⢛⣿⣿⡆⠀⢀⠟⠰⣛⠞⣳⠂⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢤⡁⣛⡾⣋⠙⢷⣉⣅⣽⠀⣾⠟⢿⡏⠀⣾⠃⠰⠊⢻⡇
⠀⠀⠉⠾⣿⣶⣿⣇⠀⠀⣿⣿⠿⡻⠇⠀⢸⠃⣛⠧⡛⢹⡇⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣈⡱⣏⠸⠣⠙⢃⡻⢦⣹⠀⣠⡄⣼⣿⠀⣍⠀⣀⠰⣼⡀
⠀⠀⠛⠀⠈⠻⣿⠆⠀⠀⣿⣿⣦⣥⠄⠀⠀⠳⠜⡷⠙⢦⡄⠀⠀⠀⣿⣍⣹⣿⠀⠀⠀⠛⠒⠚⠒⠒⠒⠓⠒⠐⠛⠀⢿⣁⣼⡟⠀⣿⠆⠈⠃⣼⠇
⠀⠀⣰⣄⠀⠼⠿⠀⠀⠀⢿⣿⠟⠋⠃⠀⠠⢙⡓⣄⠸⣻⠄⠀⠀⠀⠿⠏⠹⠯⠀⠀⠀⠠⣿⡴⣷⢷⣴⢿⡷⣴⡳⠀⣨⡿⣯⣶⠀⠛⠀⠤⠄⣹⠃
⠀⠀⢘⣥⣻⡷⣤⡄⠀⠀⣼⣿⣷⣠⡆⠀⢠⡎⠷⣭⠞⢱⡆⠀⠀⠀⢀⣤⣤⡄⠀⠀⠀⠀⡸⢤⣿⣆⢈⣷⣧⠼⡆⠀⣿⣶⡟⣿⠀⣿⣴⣿⠆⢹⡀
⠀⠀⢸⣟⣷⣳⣿⣆⠀⠀⣿⣤⣽⣿⠇⠀⠈⠿⠛⠃⠛⠿⠃⠀⢀⣠⣭⣿⣯⣭⣄⣠⠀⠀⣷⣾⣧⣸⠃⣤⣿⣷⠟⠀⡸⢿⣿⡃⠀⣼⠀⣀⣔⢿⠂
⠀⠀⢰⣿⠾⣳⢿⣄⠀⠀⠛⣿⣟⣿⡀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠈⠁⠴⡹⠟⠦⠈⠁⠀⠀⣮⣿⡷⡁⠁⣸⢿⣷⣆⠀⠘⢂⣽⣿⠀⣿⠿⣳⠿⣿
⠀⠀⠸⣏⣻⡽⣸⠟⠀⠀⠁⢸⣿⠉⠁⠀⠀⢀⣾⣩⣷⡀⠀⠀⢠⣄⢠⠳⠜⡀⣠⡄⠀⠀⢣⠽⢶⡀⠀⣰⠾⢥⠇⠀⠠⠆⠀⠀⠀⣿⢴⣯⡺⣹
⠀⠀⢰⣯⡻⢳⣿⡆⠀⠀⠀⠸⠿⠆⠀⠀⠠⠟⢤⣿⡼⠷⠀⠀⠀⠿⠈⠻⠟⠁⢻⠁⠀⠀⣞⢶⣾⣿⣿⣯⣶⢞⡡⠀⠀⠃⠰⠄⠀⣿⣶⣋⣟⣽⠂
⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣀⡀⢀⣁⣁⣀⣁⣀⠀⣀⣀⠀⠀⣤⣀⣀⣙⣉⣉⣉⠉
⠀⠀⣾⢡⡟⢛⣿⣭⣾⡇⠀⢠⣾⢿⣷⣀⣤⣤⣀⣶⣿⢿⣄⠀⠀⠀⣤⣦⠀⢀⠀⠀⣿⠏⡁⠈⠿⠟⠛⠿⠟⠀⡉⢻⡇⢸⣾⠿⣿⣫⢜⣿⡁⣿
⠀⠀⢿⣟⠝⣳⣼⡯⢹⡇⠀⢸⡇⣠⡞⢉⡀⢀⣘⠻⣆⢀⡿⠀⠀⠘⣿⢿⣧⣤⠀⠀⣯⣘⡿⢃⡴⠿⣿⠷⣦⠹⡿⢸⡇⠸⣣⢜⠻⣷⣾⣫⣻⡎
⠀⠀⣶⣝⣿⣷⣾⣶⠏⣧⠀⣼⣿⡿⢤⣿⣃⣈⣿⣤⢾⡟⣷⠀⠀⠀⠀⠈⠙⠋⠀⠀⡏⠀⠀⣙⠃⠸⠿⠀⢛⣃⢠⢘⡇⠰⣿⣿⡿⢏⡹⣿⠃⣾
⠀⠀⣳⠿⣉⣯⡟⠉⣶⡇⠀⠞⢻⡷⠾⢿⡏⢉⡿⠷⢶⡟⢻⠀⠀⠀⠀⢠⣤⣄⠀⠀⣻⣶⠀⣭⣀⢰⣿⢀⣨⡏⢰⣾⡁⠸⣏⡔⠿⣤⣞⢋⢷⡜
⠀⠀⡿⣻⣯⡱⢝⣿⠿⡇⠀⢡⡾⠛⢶⣤⣷⠾⣧⡴⠟⢻⣆⠀⠀⠀⣤⣿⣿⡿⠀⠀⣿⢁⣴⣌⡉⢀⣥⠈⣉⣴⣄⢻⡅⢠⡝⢮⣶⢟⡹⣷⠋⣵
⠀⠀⡻⠭⣪⡿⡫⢇⣸⠇⠀⠾⠧⠴⠳⠞⠃⠐⠻⠞⠷⠾⠿⠀⠀⠘⠛⠋⠀⠀⠀⠀⠧⠼⡧⠤⠤⠾⠿⠦⠤⠤⠤⠴⠇⠸⣏⡴⣿⣮⣵⢟⣧⡝
⠀⠠⣿⣶⣿⣿⣿⣿⣷⣶⣶⡆⠀⠀⣤⣀⠄⠀⢠⣤⣀⠀⠀⢀⣤⡤⣤⡄⠀⠀⠀⣄⠀⠀⢀⣴⣶⠶⢦⣤⠶⣶⣶⣀⠀⢠⠴⢤⡄⣠⢤⣄⣠⣄
⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢿⣠⡖⣿⢠⣹⠟⠀⠀⠈⣯⡿⣿⡇⠀⠀⣄⠹⠀⠀⢸⠟⣿⠲⣿⢼⠖⣿⠛⣿⠀⠀⠈⠉⠛⠳⡾⠉⣹⠋
⠀⠘⣿⣿⣿⣿⣻⣿⣿⣿⣿⡇⠀⠀⣿⢻⣾⣹⣾⡉⢎⠀⠀⠀⣭⡿⣿⡇⠀⠀⠇⣰⠀⠀⢘⣶⣋⣗⣿⣿⣞⣟⣳⡏⠀⠀⠀⠀⠀⠀⠙⣿⣬⡇
⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢿⣩⢾⣿⢚⣹⢿⠀⠀⠸⣭⣿⣿⡇⠀⠀⣶⢑⠀⠀⢸⠀⣹⡉⣿⣹⡉⣿⠁⣯⠀⠀⠀⠀⠀⠀⢟⣿⠾⡃
⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣾⠹⢿⣭⠷⠿⣦⠀⠀⢘⣛⣷⣿⡇⠀⠀⠛⠸⠀⠀⢈⡿⣩⣯⣿⣿⢯⣭⡽⣇⠀⠀⠀⠀⠀⠀⢊⣿⠛⡇
⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠛⣵⠟⣽⠲⣶⣿⠀⠀⢈⣽⣿⣿⡇⠀⠀⡶⢀⠆⠀⢸⣦⣿⡤⡿⢾⠦⣽⣤⣿⠀⠀⣀⣠⣠⣴⢻⣈⣽⡀
⠀⠠⡿⢿⠿⠿⠿⡿⠿⠿⠿⠇⠀⠀⠹⠶⠋⠉⠱⠈⠁⠀⠀⠈⠛⠛⠛⠃⠀⠀⠀⠉⠀⠀⠈⠹⠿⠦⠛⠛⠦⠿⠿⠉⠀⠘⠦⠛⠋⠛⠛⠋⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⢀⡀⠀⠀⢀⠄⠀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⠀⣿⣿⣿⣿⣿⣿⣿⡇⠸⣿⣿⣆⡀
⠀⠀⠀⢀⡶⣄⡀⠀⣀⣴⣾⣿⡿⠛⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄⠀⠘⠿⣿⣿⣦⣄⠀⠀⣀⡴⡆
⠀⠀⢙⣺⣷⣿⢿⣿⣿⣟⣉⣁⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢀⣉⣙⣿⣿⣿⢿⣷⣷⡊
⠀⠀⠀⠀⠈⠁⢾⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⠆⠉
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢛⣿⣿⣿⣿⣿⣟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟⠁⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠁⠀⠀⠀⠀⠻⠿⠿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣽⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠀⠚⠛⠛⢀⠀⠘⠛⠛⠛⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⠀⠲⠶⠖⠈⡇⠀⠳⠶⠃⢸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡄⠀⠀⠀⠘⠁⠀⠀⠀⠀⣸⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣄⠀⠈⠓⠒⠋⠀⢀⣴⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣥⣀⣀⣀⣬⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠸⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢿⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡿⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠳⣇⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⠁⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠋⢷⡀⠀⠀⠈⠓⠦⣖⣒⠒⠶⣤⠶⠋⠀⠀⢀⡴⠛⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢀⣀⣀⣀⠀⠙⠳⢤⣀⣠⡶⠤⠭⠇⠀⠀⠀⢀⣠⡴⠋⠀⠀⢀⣀⡈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣈⡉⠓⠲⠶⠶⠶⠒⢛⣉⣥⠴⠖⠚⠋⠉⠉⠉⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣤⣤⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠓⠶⠶⠒⠒⠚⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⣀⠀⠀⢀⢀⣀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠙⢲⣶⣿⠋⠀⠀⠀⡀⡠⠾⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠻⣠⢀⢀⠀⡰⠁⠀⣀⠈⢧⠀⠈⠙⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣷⣄⠀⠐⡄⢃⣷⣣⣯⡾⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠛⠺⢆⣹⣼⣿⣿⡟⠀⣰⣆⠀⠀⠀⠀⡈⠻⡄⠀⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠙⣿⣿⠀⠘⢿⡇⠀⠀⠀⠙⠀⠠⠾⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣠⣴⠾⠷⠶⣿⠿⠛⠋⠉⣉⣽⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠲⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⡀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁
⠀⠀⠀⠀⠈⠻⣿⣿⣷⣤⣀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣄⡀⣀⣠⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⡀⠀⠀⠀⠙⠻⠿⠿⠛⠛⠿⠶⠤⣤⣴⠶⠃⠈⠃⢳⡶⠆⠒⠂⠀⠀⠈⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣆⡀
⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠲⣶⡇⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡀⢄
⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⡀⠑⢄⡀⠁⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⠉⠉⢿⡿⠀⣆
⠀⠀⠀⠀⠀⠈⠓⣤⡀⠀⠀⠀⠀⡈⠓⠄⠙⢦⡀⠀⠀⠀⠀⠀⣰⡿⠟⠛⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠈⠀⣸⣷⠀⠸⠇⠀⢘⡆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣤⣈⠲⢄⡐⠀⣀⠲⡇⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠀⢰⣿⣿⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⣶⣿⣶⣼⣿⢃⣀⡶⠀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⢙⢿⡀⠀⠀⢠⡾
⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣑⣂⣠⣾⡿⠿⠿⠿⠿⠿⠿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⡞⠀⢠⠀⠀⠙⡇
⠰⢶⣿⣿⠿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⣸⠃⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠛⠿⠿⠛⠉⠹⣿⡿⠿⠇⢼⣿⣿⣿⣿⣿⡿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠁⠀⠀⠠⠏
⠀⠖⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⣀⣷⣶⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⢋⣠⣤⣖⠰⠂⠒⠒⠓⠀⠀⠀⠀⠠⠀⠀⠀⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣞⠁⢲⠞⢡⡞⠈⠀⠏⠹⠂⠒⠒⠚⣧⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⣤⡉⠉⠓⠀⠖⢠⡀⠀⢰⠎⠀⠒⠀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣨⣿⣷⣿⡀⠀⣀⣼⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣫⣿⠿⢿⡿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⢻⣷⣿⣷⣾⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⣿⡿⠿⠿⠟⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⢰⣿⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⠀⠀⠀⢻⠧⢬⠁⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡰⣋⠻⡀⠀⠀⠀⢸⣧⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⣸⡿⠛⠀⠁⣀⠀⠀⠛⠛⠀⠀⠀⠘⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠄⣸⡇
⠀⠀⠀⠀⣠⠋⠁⢀⣁⣀⣠⣤⣴⣦⣤⣤⣀⣀⣀⠀⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢫⡟⠁⠀⠀⢰⣿⣷⣦⡄
⠀⠀⠀⠀⠸⡶⠛⠉⠉⠉⠂⠀⢸⣿⠯⢀⠈⠉⣻⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢆⠀⣀⠾⣏⠉⠀⠹⣇
⠀⠀⠀⠀⠀⢰⡄⠀⠴⠀⠘⠁⣾⣷⠀⠈⠀⠀⠙⠻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣆⠀⠀⠀⣰⡟⣠⣌⠷⠀⠀⣼⣆
⠀⠀⠀⠀⠀⠀⡧⠂⠀⡀⠀⠈⠛⢿⠓⠄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣶⣾⣿⡟⠁⠙⣦⣶⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢱⣤⠁⠠⡄⢀⣠⣾⡤⠂⠀⣀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠻⣿⣿⣿⣿⠣⡠⠀⠀⠘⠉⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠨⠛⠀⠀⠀⠀⠈⠙⠿⠒⠿⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣿⡏⢧⣶⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠐⠂⠀⢢⣤⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⠚⣿⠀⠀⠀⠀⠀⠂⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢠⡀⢠⡈⠆⠀⠷⠀⠰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠛⣻⣿⣿⣿⣟⡷⠈⡻⠀⢠⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣟⠁⠀⣀⣴⠀⠀⣰⣦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣷⢚⢢⣽⢾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠾⠟⠻⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⢿⣿⣿⡿⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡿⠖⠒⢶⣆⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡝⠻⠀⠀⠀⠀⠈⠗⠀⠘
⠀⠀⠀⢀⠀⠔⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⡿⣿⡷⡴⠆⠐⠂⠀⠀⠘⠿⢹⡄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⡿⠃⠶⠀⢄⠀⠀⠀⠀⢀⡄⠈⠁
⠀⠀⠄⠀⢠⣌⠀⢁⣰⡀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡏⠑⠘⠃⠀⠀⠀⠀⢠⣾⣿⠰⣿⠦⢤
⠀⠀⢀⠀⠀⢻⣿⡿⠿⠋⠉⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠁⠀⠀⠀⠀⢠⣿⣿⣿⠿⣦⡁⠀⠀⠀⠀⠀⠀⠀⠏⠀⢀⣿⠀⢸
⠀⠀⠀⠀⣀⡼⠿⣄⣀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠈⣿⡿⢻⣿⠛⠃⠀⠀⠀⠀⡀⠶⠇⠠⠀⢸⣯⣀⠼
⠀⠀⠀⠀⠈⠁⠀⠈⠙⢿⡆⠀⠀⠀⠀⢸⣿⠃⢀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⣩⠆⠈⣷⣤⣤⡘⠋⠀⠉⠀⠐⣨⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢖⣿⡿⠛⡉⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠁⠈⢹⣷⠀⠀⠀⠀⠐⣿⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⢰⡏⠑⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⢀⡸⣧⡄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣶⡏⠀⢸⣷⣀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⣷⠀⣘⠇⠉⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⣙⣧⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣈⣉⣀⢀⣨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣍⠀⠀⡀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣭⡍⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⢦⡀⠙⢷⡈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡶⠶⠃⠀⠈⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢷⡀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢮⣷⣄⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣨⣭⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠟⠛⠋⠉⠉⠁⠀⢀⣤⣤⣄⡈⠉⠉⠛⠛⠷⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠾⠛⠉⠀⠀⢠⠀⠀⠀⠀⠀⠀⠙⠒⠂⠀⣱⠀⠀⠀⠀⠀⠀⢘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⡀⠀⠀⠀⠀⠈⠒⠤⠤⠤⠴⠖⠒⠒⠚⠋⣁⣀⣠⣤⣶⠶⠟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡛⠷⠶⠶⠶⠶⠶⠶⠶⠶⠾⠟⠛⠛⠛⠋⠉⠉⠀⠀⠀⢰⣏⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠛⠉⣉⡙⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⡾⠟⠛⠛⠷⣦⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⢀⣿⠃⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⠀⠀⠀⣿⠸⣧⣤⣤⣶⠟⠋⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⣿⣤⣬⣥⣤⣤⡶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⢿⡍⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠀⢀⣀⠀⠀⣠⣀⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢷⡶⠋⠙⢧⣀⣴⠟⠙⢷⣄⣿⠙⢷⣄⣿⠉⠛⠛⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠉⠁⠀⠀⠙⠋⠀⠀⠀⢘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⣰⠏⠁⠀⠀⠀⠀⠀⠉⠛⠳⣦⣀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣶⣶⡶⠶⠶⠶⣤⣤⣤⣤⣴⣶⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣴⣶⣿⡽⠿⠛⠛⠛⠋⢉⣉⣉⣛⣛⣛⣛⣛⣻⣿⣯⣭⡴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣛⠳⠶⠶⣤⣤⣤⣄⣤⣤⡶⠞⣋⣉⣤⣴⠶⠾⠟⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠶⠶⠶⠿⠿⠿⠟⠛⠛⠋⠉⠁

⠀⠀⠘⢛⣛⣛⣛⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣛⣛⣛⡛
⠀⠀⠀⣸⣿⣿⣿⣇⠀⠀⠀⠀⠲⠶⠶⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶⠶⠶⠂⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⢀⣿⣿⣿⣿⣿⡄⠀⠀⠀⢸⣿⣿⣿⣿⡄⠀⠀⠈⣭⣭⣭⣭⡍⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⣸⣿⣿⣿⣿⣿⣧⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣇
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣆⠀⢀⣿⣿⣿⣿⣿⣿⡀⠀⠀⠈⣿⣿⣿⣧⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡄
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⣼⣿⡏⠻⣿⣿⣿⣧⠀⠀⣿⠘⣿⣿⣿⡀⠀⡤⢸⣿⣿⣿⣿⣿⣿⡆⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⣿⣷⠸⣮⠻⣿⣿⠇⠀⣿⣇⢻⣿⣿⢃⣾⡇⣾⣿⣿⣿⣿⡿⠟⣿⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣧⢹⣷⣌⠋⣼⢰⣿⣿⡄⠿⢡⣾⣿⠇⣿⣿⡿⢟⣩⡖⣰⣿⣿⡘⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡀⢈⣛⠻⠿⣿⣿⡆⢿⣿⣷⣌⣼⣿⣿⣿⣴⣿⣿⣿⠀⢛⣡⣶⣿⢏⣼⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⡙⢿⣷⣦⣭⣙⣸⣿⡿⢛⣩⣴⣶⣶⣶⣦⣭⡙⠿⣿⣿⡿⢣⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣦⡙⢿⣿⣿⡿⢋⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⢿⣁⣉⣭⣭⣭⣤⡴⠒⣠⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠿⠿⠟⠘⣛⣛⣛⣂⣽⣿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⢿⣿⣿⠿⢋⣥⣾⡇⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣤⣍⡛⠻⠿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣏⡑⠾⢿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣷⣶⠬⢙⣿⣇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣶⣦⣭⣁⡛⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠘⣋⣵⣾⣿⣿⣿⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣿⠛⣋⣩⣭⣭⣭⡅⣴⣶⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠟⠫⠴⠾⠟⢛⣛⣛⡋⣻⣿⣎⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⣿⣿⣷⣌⠻⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣾⣷⢸⣿⣿⣿⠟⣴⣿⣿⣿⣿⣦⣍⣛⡛⠛⢛⣛⣩⣴⣾⣟⢉⣛⠻⢿⣷⣌⠻⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿
⠙⢿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⢋⣾⡿⠟⣫⠀⢸⣿⣿⡿⢿⣿⣿⣿⠛⢿⣿⣿⡌⣿⣿⣶⣦⣭⣑⡈⠁⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠈⠉⠉⠛⠋⠉⠁⠘⠿⠡⠟⣩⣴⣿⣿⠀⣿⣿⠟⣰⡆⣿⣿⡏⢸⣦⡙⢿⣷⡸⣿⣿⣿⣿⣿⡿⠁⠈⠉⠙⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⢀⣿⢏⣼⣿⣿⡘⣿⡇⣿⣿⣿⠀⠙⢧⠙⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠾⣿⣿⣿⣧⢹⠃⣿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⣦⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣄⣿⠀⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⡦⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠐⢛⡿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣿⣯⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠭⠀⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⢀⣾⣿⡦
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠰⣉⢰⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸⣄⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⢐⣶⣶⣍⣻⣿⣿⣿⣷⣖⣦⠀⠀⠀⠀⠀⠐⣻⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠉⠓⣦⣿⡙⠿⠁
⠀⠀⠀⠀⠀⠸⢿⣿⡜⢷⣿⠿⢿⣿⣯⡟⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠏⠸⡍⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠾⠇⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃



⠀⠀⠀⠀⠀⠀⡠⠒⢒⡾⡑⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⣦⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡶⠾⠤⠤
⠀⠀⠀⠀⠀⠀⣷⣤⣌⣛⡋⠀⣰⣇⠠⡀⠀⠀⠀⠀⠀⠀⠀⢸⠛⡝⢿⡟⣉⠻⡄⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣶⣭⣭⣶⡏⠀⠀⠀⠀⠀⠀⠀⠘⣷⣶⣾⣷⣶⣾⠃⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢙⣻⣟⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣯⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠁