Laugh Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢴⣶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠛⠛⠛⠙⠛⠓⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠤⠴⠖⠒⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⠤⣤⣤⣤⣤⠶⢚⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠄⠒⠢⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠞⠉⠉⠢⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠲⣄⠀⣿⠀⠀⠀⠀⠀⠀⢀⣀⣐⡲⢦⡀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠈⠱⡈⣧⡏⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠓⠕⠀⠀⠀⠀⠀⢸⠗⠛⠓⠆⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢀⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠐⠋⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠛⠆⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⠶⠾⣇⠀⠀⠀⠀⠀⠀⠀⢸⣟⡛⠛⠛⠒⠲⠦⢤⣀⡀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⢀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠘⣿⠿⣿⣷⣶⠶⠤⠤⠤⣽⡇⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡽⠋⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠹⣆⠀⠙⢯⠀⠀⠀⢀⡞⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠖⠋⠙⢦⡀⠀⠀⠀⠀⠀⠈⠳⠦⣼⣧⣤⠶⠋⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠶⠶⠶⠒⠚⠋⠁


⡿⠿⠿⠿⠿⢻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠤⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡆
⠀⠀⢰⡉⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⣱
⠀⢀⠀⠉⠉⠉⠉⠀⠀⢀⣤⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠈⠉⠉⠉⠁⢀
⠀⠣⠄⠒⣀⣤⡄⠀⠀⠉⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠙⠂⠀⠀⠀⢰⠀⠐⠂⠤⠃
⠀⠀⡴⠛⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⠉⠓⠲⠤⠤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⠒⠒⠤⠤⠤⠤⠤⠀⠀⠀⠀⠠⠤⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⠀⠀⠀⠀⣀⠤⠤⣄⣠⠤⠚⠶⣄⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⡎⠀⠀⠀⠀⠈⢂⣀⣤⠞⠉⣾⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠚⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠒⠛⠛⠉⠉⠉⠛⠛⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠭⠅⠀⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠛⠓⠶⣄⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⢀⡴⠋⢀⠠⠀⠀⠀⣤⠞⣻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⡾⠀⡐⠁⠀⠀⠠⢴⠃⣰⠃⠀⠀⠀⠀⡀⠤⠀⠒⠒⠂⠤⣀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⡇⡘⠀⠀⠀⠀⡇⢻⢰⠇⠀⢀⠔⠊⠁⠀⠀⠀⣀⣠⠤⠶⠶⠶⢦⡀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠂⠤⠀⣀⢸⡀
⠀⠀⠀⠀⢷⡄⠀⠀⠀⠀⠁⠈⣿⡠⠊⠁⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢦⣄⣀⣀⣠⡤⠤⠤⢤⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠐⠀⢸⡀⠀⡠⠐⠀⣹⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠂⠀⠀⢹⡆⢸⠃
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠈⣷⠉⠀⠀⠀⣿⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡾
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⡀⠀⠀⣿⣆⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⣼⣰⠃
⣴⢿⠃⠀⠀⣠⠃⠀⠀⡰⠀⠀⢠⡇⠙⣦⠴⠦⢼⡆⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠒⠊⠁⠀⢻⡃
⣇⠈⠙⠛⠋⠁⢀⡠⠊⠀⠀⢀⡞⠀⢸⠁⠄⠠⠀⠀⠀⠀⠀⠀⠀⠒⠚⠁⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠘⢷⡀⠀⠀⠉⠀⠀⠀⢀⣠⠟⠀⠀⠸⣆⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⣿
⠀⠀⠉⠓⠲⠶⠶⠒⠛⠉⠀⠀⠀⠀⠀⠈⠉⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣤⣤⣤⣶⣶⣿⣿⡇⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠿⡿⢛⡿⠁⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠒⠊⠉⠀⢀⣠⡞⡉⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⡤⠦⣄⣀⣀⣀⣀⣀⣀⣤⠤⡶⠚⢻⡉⠉⢻⠓⠒⠒⠒⠒⠒⠉⡉⡙⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠀⠀⢀⡙⠉⠉⠉⠃⠠⠤⠚⠀⠀⠈⣧⣀⣼⠤⠤⠤⠤⠶⠤⣀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⢬⣅⣾⠋⠀⣀⠀⡄⠐⠒⠒⠒⡆⠀⢸⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠳⠤⠞⠘⠗⢻⡄⠀⠀⡰⠁⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⣤⠤⠤⣄⠀⢠⣭⠏⠉⠉⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠙⠒⠃⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠒⠒⠐⣶⠈⠉⢹⡏⠉⠉⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡀⢸⣇⢀⡀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠁⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠠⠀⠉⠛⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣠⡀⠀⠀⠰⠓⠀⠀⠀⠀⠀⢿⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⢈⡃⣠⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⡔⢒⣀⣶⡆⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠈⠦⣈⡿⠟⠀⠀⠀⢀⠜⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠀⠀⣠⣤⣶⠁⠀⠀⣉⢩⠋⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠠⡎⢀⡤⠖⠁⠈⡄⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠤⠈⠙⢶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢁⠀⠀⠀⠀⣄⣀⠀⢀⡀⠄⠀⠁⠀⢈⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⢸⡀⡠⢶⣶⣌⣁⠉⠄⠀⠒⠀⢾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣀⡠⢀⣀⣸⣿⡟⠁⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠂⠀⠠⠤⠤⠤⠀⠀⠒⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⢨⠉⠉⠉⠉⠉⠁⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⡆⠀⠀⠀⠀⠀⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣀⡇⠀⠀⠀⠀⠀⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢠⡄⠀⣴⣶⡝⠋⡇⠀⠀⠀⠀⠀⢀⠿⣶⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠰⠤⠟⠛⠛⠛⠁⠀⠀⠀⠠⠖⢅⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⡀⠗

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⠿⢿⠿⣉⠿⢿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣟⢛⣿⣛⣛⢛⣿⣿⡛⣛⢛⣛⢛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠚⠉⠋⠰⢿⣿⠟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠫⡉⢃⠀⠠⠶⠮⡛⢛⣛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠻⠡⠧⠜⠀⣶⠉⡆⢀⠘⣿⢻⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣤⣶⣦⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠠⣤⣤⣤⣀⠀⠉⠈⠀⣿⣶⡿⢿⣯⣴⡚⠉⠙⠛⠿⣿⣿⣦⡀⠀⠀⣿⣼⣿⣿⣌⠟⠛⢿⣿⣧⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢸⡷⣶⣮⣽⡷⠓⠀⠈⣹⢟⡝⢢⠀⡀⠀⠀⣀⡈⠉⠙⠻⣿⡴⢛⠙⢂⡀⠀⠞⠛⠦⢀⣿⣛⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⡇⠀⠸⣇⣀⡤⠊⠀⠀⢠⠞⠁⠉⠗⢦⠒⠁⠀⣠⣌⠻⣆⠀⢀⡞⣰⡅⡑⢨⠇⢀⣖⠒⡄⠈⣿⢻⡏⣏⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣹⡄⠀⠈⠁⠀⠀⠀⣰⠃⢸⠖⠦⣄⠈⠀⠀⠺⢗⡨⠃⠙⢰⡿⠀⠈⠉⡔⡁⠀⠘⠛⠚⠀⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠉⠳⢤⣀⠀⠀⣸⠁⢀⣾⣿⣷⣄⠑⣄⠐⡄⠀⠁⠀⠀⣠⡇⢴⣃⣤⣥⣤⣄⣀⣢⡤⠤⠄⠘⠛⡿⠛⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠳⠀⡇⠀⢼⠀⠒⠛⢻⣷⠾⠛⠛⠆⠀⠀⠸⣿⡇⠀⢻⣶⣶⠶⠶⡾⠋⠀⠀⠀⠀⢈⠇⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⡀⣇⠀⠈⠛⠚⠋⠉⠀⠀⠀⠀⠀⡀⢀⡖⠊⣧⠀⠸⣟⠓⣂⠟⠀⠀⠀⠀⠀⢀⣿⡧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠱⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣀⣠⠖⡿⣦⠀⠈⠉⠁⠀⠀⠀⠀⠀⣴⠻⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠉⠀⠀⠧⠃⠛⢦⣀⠀⠀⠀⠀⠀⡼⠁⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⣠⠴⠚⢉⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣯⣷⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣷⠖⢟⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠑⠓⠻⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣿⣿⡏⠁⠀⢻⣿⡆⢀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣇⠀⠀⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢛⣿⡿⠿⠆⢀⢀⠹⠛⠛⠛⠛⠛⠤⠤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠦⡀⠀⠉⠻⢿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠀⡠⠚⠓⠠⠤⠀⣀⣀⣀⡤⠤⠔⠒⠋⢁⠀⠉⠀⠀⠀⠀⠙⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⡿⠁⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣏⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠛⠻⠇⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢋⡁⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⢄⠠⢄⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⣤⣤⣤⣴⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡵⢫⡀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⠓⠾⢀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠖⠋⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠻⣦⠀⠀⠀⠀⠀⠀⠈⢯⠀⠹⠏⠀⣨⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠈⠢⣄⣠⠔⠃⠀⠀⠀⠀⠀⠀⠀⣀⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⣀⠀⠀⠐⠦⠤⠂⠀⠀⠀⢀⣀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠨⠭⠷⠶⣄⠀⠀⢲⢏⠉⠉
⠀⠀⠀⣠⠤⠶⠒⢒⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⡅⠀⠀⠀⠀⠂⠀⠫⢍⣉⠥⠎⠁⠢⣀
⠀⠀⠀⠈⠑⠚⠩⡗⠬⠄⠀⡦⠄⢀⡀⠀⠀⠀⠀⠂⠀⡇⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠁⠀⠀⠀⠉⠒⠂⠤⣤⠀⣇⣠⢎⣀⠤⠄⠒⠒⠛⠛⠓⠦⠀⠀⢂⠀⡠⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣠⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⣠⠾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⡀⠀⢟⠁⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠇⠀⠀⠀⠀⠀⣀⠠⠇⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦⣈⡄⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠃⠀⠀⢀⡠⠒⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠳⣀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⢀⢤⠊⠁⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠻⣿⣇⠀⠱⡄⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠈⠀⠀⢡⠀⠀⣀⣀⣀⣇⣀⠤⠔⠒⠒⠒⠀⠀⠐⠒⠒⠲⠤⣄⣸⠀⠀⠹⣿⡄⠀⣏⣀⠀⢄⣪⣄⣲⠄
⠀⠀⠀⠀⠀⠀⠀⢈⣿⡧⠐⠀⠀⠉⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⠄⣹⣧⠜⠁⠀⠉⠑⠊⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠶⠿⠿⠛⠛⠛⠛⠿⠿⠶⠶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠉⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠛⢿⣦⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣉⣛⣻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣟⣛⣉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⡇⠀⠀⠀⠀⠀⣠⣤⠀⠀⣠⣶⠿⠿⠛⠛⠛⠿⢿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠛⠛⠛⠿⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⢀⣴⠿⠋⣿⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⣼⣷⣤⡀
⣿⠀⢀⣾⠟⠁⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠻⣦⣀
⣿⡇⢸⡇⠀⠀⠀⠀⢸⡇⠀⣴⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⠀⢰⡿⠀⠀⠀⠈⢻⡆⢸
⢸⣇⠈⠻⣦⣄⣀⣤⡿⠁⠀⢻⡇⠈⠙⠛⠷⢶⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⡶⠾⠛⠋⠁⣸⡏⠀⠸⣷⡀⠀⠀⠀⣸⡇⣸
⠈⣿⡀⠀⠈⠉⠉⠁⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠘⠻⠶⠶⠿⠋⢀⣿
⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡍⠉⠛⠛⠛⠻⠿⠶⠶⠶⠶⠶⠶⠶⠾⠿⠛⠛⠛⠋⢉⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⢠⣴⠶⣶⣤⣀⣤⡶⠶⣦⡄⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣿⠁⠀⠀⠈⠉⠁⠀⠀⠈⣿⣀⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣦⣤⣤⣤⣤⣤⣴⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠶⣶⣶⣶⣤⣤⣶⣶⣶⠶⠾⠛⠛⠉
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣶⣤⣶⣿⣿⣿⣶⣤⣶⣿⣶⣄
⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣶⣦⡀
⠀⠀⠀⠀⠀⢠⣿⡿⠋⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⢠⣶⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆
⠀⠀⣠⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⡄
⢸⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠙⣿⣿⣿⣿⡿⠁
⣸⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠀⠸⣿⣿⡇
⢿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⣿⠁
⢸⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⢻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣴⠶⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⣦⠀⠻⠻⢶⡄
⠀⣨⡿⠿⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢸⡇
⣼⠏⠰⠶⢦⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠟⠁
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠈⠻⣦⣴⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣄⠀⠀⠀⢀⣿
⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣤⣀⣀⡀⠀⠀⠀⠀⢀⣀⣠⡼⠋⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣭⣿⣻⣛⣿⠋⠉⠁⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣉⡛⠛⠛⢿⡏⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡀⠉⠁⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠶⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠻⠶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⠾⠛⠉⠛⢻⡶⢶⣤⣤⣄⣀⣀⣴⠟⠁⠀⣶⠀⠀⢸⡟⠛⠛⠻⢶
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⣿⡀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣭⣽⠿⠟⠁⠀⢀⠀⠀⣼⡇⠀⠀⠀⠉⠉⠉⠉⠀⢀⣰⡿⠶⠶⣿⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠸⣧⣀⣤⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⡀⠀⠀⣽⠟⠛⠛⠛
⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⢰⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠛⠛⠻⠿⠿⠿⠟⢿⣦⣀⣟⠉⠉⠛⣻⡏
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⠶⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣀⡄⠀⠈⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠉⠁⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⣠⣾⣇⣀⣀⣤⣄⣀⣀⣀⣀⣀⣀⣄⣠⣄⣠⣀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣿⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⣤⣤⡤⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡏⣹⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⡞⢻⣶⣶⣶⣶⣿⠇⢸⣷⣶⣶⣶⣶⠟⢱⠖⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠋⠙⡷⠦⠿⠖⠃⠀⠀⠀⢹⡇⢸⡇⠀⠀⠀⠘⠶⠷⠴⡿⠋⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠲⠶⠶⠚⠲⠶⠶⠶⠶⠒⠶⢾⣧⣾⡗⠶⠶⠶⠒⠶⠶⠖⠳⠶⠖⠞⣿⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣥⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇
⡇
⡇
⡇
⡇
⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄⡄⣤
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠿⠀⠋⠻⠻⠀⠀⠀⣆⡟⣰⡇⣰⡆⢰
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠻⠛⢷⠿⢇⡋
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠑⢤
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠈⡅
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⡇⠀⠀⠀⠀⡄⠀⢀⢀⡀⢠⡄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⣀⣀⣀
⡇⠀⠀⠀⠀⠳⠄⠘⠨⠇⠘⠙⠜⠋⠀⠄⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠧⠐⠟⠳⠞⣧⡟⠙
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⡂⠐⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂
⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠒⠚⠛⠉⠉⠉⠉⠉⠉⠛⣷
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⡆
⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠖⠒⠒⠒⠦⠤⢄⣀⠘⣧
⠀⠀⠀⢰⡏⠀⠀⠀⠀⣀⡤⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢦⠀⠈⠳⣿
⠀⣠⡾⠋⠀⠀⠀⣠⠞⠉⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⣴⣯⡄⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢀⡟⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇
⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠤⠶⠃⠀⢸⡇
⠈⣷⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠹⣧⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⢹⣆⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢻⣆⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⣹⡆⠀⠀⠀⣿
⠀⠀⠀⠀⣹⣦⣤⣤⡾⠃⠀⠀⠀⠀⠀⢿⣷⣶⣦⣶⣶⣶⣾⣿⣿⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠰⣯⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⠁⠀⠀⣾⠁
⠀⠀⠀⠀⠙⠷⠶⣷⡀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣋⣀⣈⣥⠴⠋⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡯⠤⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠿⠶⣤⣤⣤⣤⣤⣤⣤⠶⠶⠚⠛⠋⠉⠹⣦⠀⠀⠈⡟⠛⠛⠋⠓⠒⠒⠒⠒⠊⠉⠀⡀⠉⢉⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣠⡴⠷⠶⠶⠶⠶⠶⠶⠶⠦⠶⣄⡈⠁⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡂⠤⢤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⢻⣿⠁⠀⠀⠀⡴⢶⡀⣀⣤⡀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠾⠿⠤⠤⠤⠞⠁⠈⠉⢁⣤⣇⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⣄⣤⣄⣠⡤⣄⠀⠀⠀⢤⣤⡼⠃⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠈⠙⠛⠲⠦⠼⠃⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣴⣶⠞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡿⠻⠷⠶⠶⠶⠶⠿⠟⠛⠛⠋⠉⠉⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⠀⠀⠀⠀⠀⢰⠊⠀⠀⠀⠀⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣉⣉⣉⣉⣹⣍⣉⣉⣉⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡏⠉⣿⠉⠉⠙⡏⠉⢹⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⣷⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠤⣿⠀⠀⠀⣿⠤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣷⣴⣿⠀⠀⠀⣿⣶⣾⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡀⠀⢰⣿⣿⣿⣿⣿⣷⣄
⣿⣿⣿⠿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣤⣤⣤⡀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠷⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠿⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷
⠀⠀⠀⢰⠚⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠑⡄
⠀⠀⠀⠈⠳⠤⠤⠤⠶⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⠤⠤⠞⠁
⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⢿⣿⠿⠿⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⢶⣤⡀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠠⠤⡀
⠀⠘⠶⠖⠛⢀⣀⣤⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⡾⠀⠐⠓⠒⠤⠃
⠀⠀⠀⣠⠾⢿⡍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠘⠁⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡉⠛⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠈⠉⠙⠛⠲⠶⠦⠤⠤⣤⣤⣤⣤⣤⠤⠤⠶⣶⡶⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠓⠒⠲⠦⠤⠤⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢀⡤⠴⠶⠶⠦⣤⡴⠚⠉⠙⠳⣦⣤⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⣰⠋⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⣤⠾⠋⢠⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠤⣤⣤⡤⠴⠶⠚⠛⠉⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠄⠄⠀⠀⠄⠀⠀⠀⠄⠂⠤⠤⠂⠐⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⢖⣶⣟⣃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠈⠉⠉⠉⠉⠉⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠈⠀⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠤⣤⣀⣀⣀⣀⣤⠴⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠄⠐⠒⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠰⠋⠉⠑⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⢉⠙⢦⡀⣿⠀⠀⠀⠀⠀⢀⣠⣤⣝⡲⣄⠀⠀⠀⠀⠀⢀⡴⣂⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠘⢆⣷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠟⠋⠉⠑⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⣠⠾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠁⢀⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠈⠁⠈⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠉⠉⢳⠀⠀⠀⠀⠀⠀⠀⣿⣶⣤⣄⣈⣉⠉⠛⠲⢦⡄⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣶⠂⠈⢧⠀⠀⠀⠀⠀⠀⠸⣍⠙⠻⣿⠉⠉⠉⠉⡿⠁⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢤⣀⣠⠞⢳⡀⠀⠀⠀⠀⠀⠘⠦⣄⣘⡆⢀⣠⠞⠁⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⢤⣤⡤⠤⠴⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⣆⠀⠀⠐⠄⠀⠀⠀⠀⣠⡆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠈⠀⠀⠀⠠⣄⣀⣀⣤⣿⠁⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⡀⠀⠀⠀⠈⠛⠛⠛⠁⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠒⠦⢄⡀⠀⠀⠀⣠⢊⡈⡑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠰⠦⣀⠈⠐⡄⠀⠘⠦⡅⠰⣔⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⠀⠀⠀⡈⠉⠀⠀⡀⠠⠐⠈⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣄⠀⠇⠀⠀⠄⢠⡄⠀⠘⠀⠀⠀⠙⢢⡀⠂⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢈⠔⢉⠀⠀⠀⠀⢀⠄⠊⣀⢀⡀⠄⠀⠀⠂⡀⡀⠄⠤⡑⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⠴⠷⠤⠤⢄⡤⢠⠀⠊⠀⠀⠀⠤⠤⢒⡉⠀⠂⠀⠀⠀⠀⠐⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⠀⢐⣃⠀⠀⡥⡵⠀⣀⠀⠀⠀⠀⢀⣩⠀⠔⢊⠀⠀⠀⢠⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠂⠉⠉⡄⠸⠤⠤⡄⠈⠑⠒⠒⠋⠁⠣⣭⡤⠔⣀⠔⢊⣇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣄⠀⢠⡀⠀⢠⠀⠀⠂⠠⡭⠉⠈⠀⠞⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⣾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣶⣦⣬⣹⣯⣴⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠈⣾⣽⠿⠿⠟⠋⠀⢈⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠐⣿⠅⠀⠀⢸⠀⠐⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⡄⠀⠀⠈⠀⠀⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠀⣾⡇⠀⠀⢠⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡈⢹⡇⠀⠀⣸⠀⣸⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣴⣁⣼⣇⢀⡤⠿⣶⡇⠀⣤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⠄⠀⠠⡆⣈⡹⠉⠉⣉⣹⣿⡟⠂⠀⣄⠀⠀⡀⠸⠟⣻⢀⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⢠⡀⠀⠃⠀⠀⠀⢛⣛⣳⣀⣤⣿⡿⠟⠿⠷⠾⠿⠀⢿⣥⡤⠶⠖⠋⠈⠁⠀⠀⠀⠀⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⣠⢀⡀⢀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠤⠄⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠨⠋⠁⠀⢀⡀⠤⠀⠀⡠⠊⠁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡠⠤⢄⠀⠀⠀⠈⠀⠤⠒⠁⠀⠀⣀⡀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠋⠉⠉⠂⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⡠⠔⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠠⢄⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⡀⠀⠰⠀⠀⠀⢰⠁⠀⡀⣠⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⢡
⠀⠀⠀⠀⡠⠀⠃⠀⠀⠀⢸⣀⠀⠶⠿⠀⠀⠀⠀⠀⠀⠀⠀⠨⣾⡇⠀⠘⠀⠀⠀⠀⠠⠔⠒⠅
⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢀⠀⠘⡆⠀⠀⠠⢖⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠦⠄⠀⠀⢠⡟⠀⠘
⠀⠀⠀⠀⢂⠀⢀⠀⠀⠀⠸⣷⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⣶⣿⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠈⠀⢂⠀⠀⠀⢻⣷⣶⣦⣭⣩⣍⣉⣉⣩⣍⣁⣠⣦⣶⣿⠇⠀⠀⠀⡐⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠻⣿⡿⠿⠛⠛⢿⠿⠿⠟⠿⢿⣿⡿⠋⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⣀⠤⠋⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⠠⣀⠀⠀⠈⠉⠀⠂⠈⠉⠀⠀⠀⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠒⠂⠤⠤⠤⠤⠤⠀⠒⠈

⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⢹⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⣿⣿⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⣦⣾⣿⣏⠉⢹⣿⣿⣿⣿⣿⣿⣷⣶⣶⣄⡀
⣾⢃⣆⠀⠀⠀⣠⣤⣤⣴⡾⠻⣿⣿⠟⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠃⢸⣿⠀⠀⠀⠈⠉⠙⠋⢻⣶⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠙⢻⣧⣤⣀⣀⡀
⠀⣸⡿⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⠋⠁⠈⠙⢿⣿⣶⣤⣶⣶⣴⣶⣾⣿⣿⠋⠉
⣴⣿⡃⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⢀⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀
⡿⠙⠻⠃⠀⠀⠀⠀⠀⣧⣿⣯⣀⠹⢧⠀⢹⣿⣿⣷⣴⢛⣿⣳⡿⣿⣿⣿⡿⠋⠙⣻⣏⢿⡿⠀⠀⠀⠀⢘⣟
⠣⠀⢰⠀⠀⠀⠀⠀⠀⠇⠈⣛⣿⣿⣿⣿⡷⣾⣵⣟⡍⠙⣿⣿⣵⣿⣿⢷⡻⣷⣿⢿⣾⣹⡏⠀⠀⠀⠀⠈⢿⣧
⡗⠀⢺⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⡿⣬⣀⡀⢻⣿⣿⣶⣿⣿⣿⣏⣤⣾⡟⢫⣁⣾⣧⠼⣇⠀⠀⠀⠀⠀⠀⢻⣷
⠁⢰⣾⠀⠀⠀⠀⠀⠀⠀⢿⡓⣱⣸⣥⣬⣽⡿⠿⢻⣿⣿⣏⣒⡻⠿⣿⣿⣖⣺⣟⣺⣗⡓⣿⠆⠀⠀⠀⠀⠀⠈⢿
⢀⣿⣿⠀⣤⠀⠀⠀⠺⣷⣿⠟⠛⣽⡿⠉⠴⡶⠃⠘⢛⣿⣿⣿⣥⣤⢀⡉⠙⣿⣿⣿⣿⣷⣿⣶
⢸⣿⡇⣾⣿⠀⠀⠀⠀⠙⠻⢿⡻⣿⣷⣶⡮⢱⣾⣿⣿⣿⣿⡿⠿⠟⠃⢹⡛⢉⣁⣙⣾⠿⠛⠉⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⡟⠀⠀⠀⠀⣠⡀⠀⠉⢻⣿⣯⣿⣿⣿⣿⣿⣟⠃⠀⣀⣶⢶⣶⣾⣻⣿⠏⢡⣶⡄⠀⠀⠀⠀⠀⠀⢸⡄
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠛⠃⡤⠀⢸⡏⠻⣿⣿⣿⡿⠛⣿⣿⢛⣿⣩⣽⠏⣾⣿⡏⠠⠤⢉⠀⠀⠀⠀⠀⠀⠘⣿⣧
⣿⣿⣿⡏⠀⠀⠀⠀⠀⠰⠄⠀⢠⢸⡇⢀⣼⣷⣿⣷⣾⢸⠉⢨⣿⣿⣿⡿⠋⣿⡃⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡠⠖⣈⡣⠿⢿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⠃⠀⣿⠿⣿⠇⠀⢀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⡇⢀⣀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⢸⣿⣿⡏⢿⣿⣄⣿⡟⠁⠀⠀⠀⣿⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⡻
⣿⣿⣿⣧⣾⣿⡷⠀⠀⠀⠀⠀⠀⠈⣰⠀⢸⣿⣿⡇⢸⣿⡿⢛⣶⣶⡖⠀⠀⠟⠠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢿⠀⠸⢿⣿⡇⠸⡇⠀⠈⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠦⣀⡆⠀⢀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠈⠊⠹⠀⠚⠆⠈⠒⠲⢾⠤⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠃⠸⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠧⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢁⣀⡀⠄⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⡠⠲⢄⢀⡽⡖⢲⣒⠎⠉⠳⠴⢚⣇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⡖⢣⣟⢋⣉⠉⣉⡉⡩⣑⢤⠎⠀⠀⢣⣌⣀⣀⡀⠀⠀⣀⡉⠢⠥⢿⢀⡄⠀⣀⠀⠀⢰⡎⠀⠀⠀⡀⠀⢰⠒⣒⢺⠁⠀⡷
⠀⠀⠀⡇⠀⠉⠘⠤⠃⢳⣛⣓⡚⢸⠀⠀⠀⠑⢯⣛⣩⠃⠀⠀⢯⣛⣩⠇⠈⠩⡤⣺⠍⠃⠀⡜⠁⠀⡴⠒⠿⢳⠃⠠⠄⠀⣹⠋
⠀⠀⠀⡇⠀⠈⢉⣉⣉⣀⣀⡀⠀⢸⠀⠀⠀⢻⡀⠘⠡⣖⠀⠀⠀⠙⠃⠀⠀⠀⠀⡋⣠⠃⠀⠑⡄⠀⡇⢰⣲⣻⣶⡶⠶⡤⠏
⠀⠀⠀⠠⠡⠇⠐⠒⠀⠒⡉⠁⠀⠸⠀⠀⠀⠸⡁⠈⠀⠈⠀⠀⠀⠁⠈⠁⠀⠀⡜⢉⡄⠀⠰⣮⡀⠀⡇⢸⣾⣺⣿⡇⠀⡇⢀⣴⢳⠤⢴⢢⣤
⠀⠀⠀⢰⠨⠅⠨⠗⠒⠒⠀⠀⠁⠀⣇⣀⠀⠀⠱⡀⠀⠀⢠⣤⡄⠀⠀⠀⣠⠔⠉⣏⠙⠒⠚⠁⠁⠀⡇⠘⡟⠚⠛⣃⠀⡇⠸⠘⠻⡤⠤⠍⢸
⠀⠀⠀⢸⠸⡽⠀⠒⠒⠐⠛⠀⠀⠀⢳⡜⡉⢱⠀⠈⠒⠤⠤⠤⠤⠴⠒⣏⠀⠀⢀⣼⠑⢆⡀⠀⠀⠀⠙⠒⠛⠒⠚⣿⠚⠁⢰⡀⠨⠤⣀⡀⢸
⠀⠀⠀⢸⠀⣽⠀⠘⠛⠀⠐⠂⡀⢀⣨⢋⣀⠀⡆⠀⠀⠀⠀⠀⠀⢀⡞⠩⠶⠮⠗⡀⠀⠀⠱⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠨⠁⠐⠂⢸
⠀⠀⠀⠈⠒⠒⠒⠶⠒⠒⠚⠛⠉⠹⠤⠖⠒⠊⠁⠀⠀⠀⠀⠀⠀⢸⢘⠀⢠⠒⢺⣷⠼⡆⠀⠱⡖⠁⣘⢆⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠒⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡀⠀⠀⠀⠀⢀⡰⠇⢸⠀⢸⢰⢉⣍⠓⣳⠀⠀⣱⣾⣩⣂⢻⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣣⢴⡖⣤⡲⠉⠀⠀⣸⠀⢸⡆⠠⣭⠀⢻⠀⠀⠙⡇⠀⠈⠇⡧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡎⠀⠘⠳⣇⣼⣠⠤⢊⠏⡀⢸⠃⣓⠆⠰⢿⡆⠀⠀⡇⠀⢸⠛⣁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠈⠢⣤⡦⠊⠀⠀⠠⡋⣀⠈⠙⠤⣍⡦⢵⣪⣇⠠⠀⡇⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡝⢩⠉⠙⡷⡶⠳⢶⡃⠀⣹⣛⠷⣴⠏⠀⠻⣓⡒⠛⢛⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠂⢀⠸⢃⣧⠀⠀⢋⠉⠅⡌⠳⡌⠑⢤⡀⠈⠓⠚⠁⠀⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠰⡤⢄⠀⠀⠀⢙⠂⠃⠀⠈⡀⠀⠑⣄⠀⠀⠀⠀⢧⣀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠄⠀⠀⣰⠀⢸⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠙⢄⢸⡀⠉⠀⠂⠀⢠⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⠦⣄⠀⠈⢆⡧⠤⢄⡸⡀⠘⠆⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠑⣄⠁⢀⠜⠛⠛⠓⢄⠑⣄⠀⠀⢀⣀⡀⠀⠀⠀⠈⡏⣝⠂⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠶⠶⠖⠚⠒⠚⠦⠤⠤⠴⠚⠉⠀⠑⠲⠤⠤⠤⠤⠤⠴⠊⠓⠲⠒⠁

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

⠀⠀⠀⠀⢰⡄⠀⣤⠀⣰⣿⣀⡀
⠀⠀⠀⠀⠈⠳⠤⠝⢲⡿⠉⣀⠿⢂⡀
⠀⠀⠀⠀⠠⣾⡿⢠⣿⠃⠐⠋⢀⢽⡇⠀⢰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠿⠄⠻⠧⠄⠀⠀⠣⠾⠃⠀⠸⣇⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣾⣿⠧⠀⠒⠒⠒⠋⠉⠓⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣀⣀⠀⠀⣈⣑⡒⠀⠘⠏⡇⠀⠀⢀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢣⠞⢁⣼⢶⣾⠟⢩⣿⣿⠂⣰⣫⣿⠆⣼⣿⠃⠀⣰⣿⠂⢀⣾⡆⠀⠀⢀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢋⡞⣽⡟⣿⢏⡴⣿⡿⢃⡴⢳⣿⠃⣸⣿⠃⡠⣾⡿⠁⣠⣾⡟⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠈⠛⠋⠀⠙⠋⠀⠙⠷⠋⠀⠈⠓⠊⠉⠛⠊⠀⠘⠃⢊⣿⠏
⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⣀⠤⢒⣾⠏⠒⢄⡀⠀⠀⢀⣤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⣰⡿⠀⠀⣿⣠⣄⡀⡞⠁⣠⣾⠃⠀⠀⠀⠳⣄⠀⠸⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⣀⣠⣿⠓⢀⣿⠆⠀⠘⣃⡙⠛⠋⣵⡞⢓⠀⠀⠀⠘⢦⣀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⡠⢾⣏⣴⣿⢋⣿⠗⣿⠰⣿⠒⡡⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠀⠀⠀⠻⠃⠈⠉⣸⣏⠜⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⠚⠛⠉⠉⠉⠉⠛⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠂⠀⠀⠒⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠉⠓⣤⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⢀⡞⠁⠀⠂⠁⠀⣱⠏⡴⠃⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⢸⠁⠎⠀⠀⠀⠊⡏⣸⠃⠀⢀⠤⠂⠈⠁⠀⠀⣀⣈⣐⣠⡀⠀⠀⠀⠀⠀⠁⠂⠄⣀⠀⠀⠀⢻
⠀⠀⠀⢸⡌⠀⠀⠀⠀⠀⠻⡇⡠⠂⠁⠀⠀⢀⡠⠖⠋⠉⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡄
⠀⠀⠀⠀⢻⡀⠀⠀⠀⡆⠠⡏⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣀⣀⣀⣀⣠⣤⣤⣀⠀⢸⡇
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠂⠀⣇⠀⡠⠐⠂⣿⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠘⠒⠀⠀⢸⡇⢸⠁
⠀⠀⠀⠀⠀⢸⠀⠀⠀⡆⠀⢹⡎⠀⠀⠀⣇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡾
⢀⣀⠀⠀⠀⢸⠀⠀⠀⠁⠀⢸⢷⡀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡢⠀⢸⡽⠁
⡞⠻⣄⣠⡴⠃⠀⢀⠈⠀⠀⡞⠀⡷⠋⠙⠛⠂⠀⠀⠀⠀⣀⣠⠆⠀⠀⠀⠠⣄⠀⠀⠈⠉⠀⠀⠘⡇
⢳⡀⠀⠀⠤⠀⠂⠀⠀⢀⡞⠁⠸⡇⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⠀⠀⠀⠀⣿
⠀⠙⠦⢤⣤⣤⣤⠤⠖⠋⠀⠀⠀⠙⠲⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣀⣠⣤⣴⣾⡆⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⢿⡿⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠇⠖⠒⠋⠀⢀⣤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⠴⠦⢄⣀⣀⣀⣀⣤⡤⠤⠖⠛⢻⠉⢹⡗⠒⠒⠒⠒⠊⢉⠉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡯⢀⠀⢠⠁⠀⠀⠠⠤⠤⠒⠀⠀⠈⣧⣼⠦⠶⠒⠒⠒⠢⢄⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠒⢦⠏⢀⡴⣄⣀⣄⠀⠠⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⠈⠉⠋⠀⠉⠴⡇⠈⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣉⣙⣲⣌⣿⣁⣀⣤⠤⠤⠴⢶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣍⠉⠀⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣈⣹⣏⣁⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⠚⠒⢺⡗⠓⠚⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣸⣇⣀⣰⣇⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣗⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⢠⡄⠀⣠⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠀⠀⠀⠘⡿⠷⠀⠀⠀⠈⠁⠀⠉⠀⣶⠀⡟⠀⠆⡀⠀⠸⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠈⠃⠀⠀⠀⠀⢀⡀⠒⢀⠀⠀⠀⡄⠀⠛⠀⠀⠀⠁⠁⠀⡤⠀⠀⠒⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣶⠆⠀⠀⠀⠀⠀⠀⢰⡆⠀⠈⢁⡒⠈⠀⠀⠀⠀⠀⠀⠀⣤⣼⣦⡄⠀⠀⡴⣠⠀⠰⠟⠁⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠃⢴⣶⡶⠶⠂⠀⠀⠁⠀⠀⠈⠁⢰⣷⣰⣦⢠⡞⠀⠀⡼⠻⡏⠀⠀⠘⠱⠃⠀⠀⠀⠀⢀⣄⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠰⠀⠀⣠⠟⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠟⠁⠘⠋⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠘⣬⠃⠀⣠⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠘⠃⠀⠾⢧⡤⠀⠀⣸⣿⠟⠳⣆⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠲⠌⠀⠀⡀⠀⠀⣰⠟⠁⠀⢠⣿⡿⠀⣠⡟⠀⣾⠋⢻⠀⠿⠟⠛⢹⣿⠋⣩⡍⠉⠉⠀⣀⡀⠀⠀⠘⠃⠀⠀⢠⣾⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⠀⠁⠀⠀⠀⣼⣿⡿⠿⠋⠀⠀⠈⢹⠟⢛⣿⠆⢠⣿⡏⢠⣿⠃⠀⣠⢾⣿⠁⠀⣀⡀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠰⣶⣾⣷⡤⠄⠀⠘⠃⠀⠀⣿⣿⠁⣴⢏⣿⠏⣰⠋⢠⣿⠏⢠⣿⣿⠀⢸⣯⠴⠋⣀⣿⣿⠶⠛⠋⠀⠀⠰⠀⠀⠉⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠟⠹⠏⠀⢀⣀⠀⠀⠀⢸⣿⣿⠀⠿⠋⠻⠞⠁⢠⣿⠏⠀⣾⣿⡇⠀⠀⢀⣴⠟⣻⣿⠃⠀⠀⢠⢀⠀⠀⠀⠀⡀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⡀⠰⠆⠀⢶⣿⡛⠉⠀⠀⠀⢸⣿⡏⠀⢚⣛⣛⡀⠀⣸⣿⠀⠀⣿⣿⠁⠀⢰⣟⠁⢠⣿⠏⠀⠀⠐⠂⡌⠀⠀⢹⣾⣷⠶
⠀⠀⠀⠀⠀⠈⡋⠰⠆⠀⣠⡾⠟⠃⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠈⠁⠀⠀⠘⠿⠶⠿⠋⠀⠀⠀⢀⡀⠀⠀⠀⠘⠙⠇
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠉⠀⢀⠀⠀⠀⠙⠀⢀⣀⣤⣴⣶⡶⠶⠶⠶⢶⣦⠀⠀⠙⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⣾⡤⢶⡇⠀⠀⠀⠐⠖⠒
⠀⠀⠀⠀⠀⠠⠶⠶⠓⠀⠀⠀⠈⡄⠀⠀⠀⡾⠟⠛⢉⣿⠇⢀⡄⠀⠀⢠⣶⠀⠀⣴⢻⣿⠇⢀⣤⣶⠶⡄⠀⠀⠀⠀⣾⣤⠞⠃⠀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⣈⠁⠀⣠⣿⣿⢂⡾⢁⣿⡟⣰⣿⡟⢁⡴⠃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠿⠀⠀⠀⠀⠒⠀⠂⠀⢠⣿⡇⢠⣿⢀⡞⣿⡿⢻⡿⠁⣸⣿⠃⣿⡿⠛⠋⠀⢀⡤⠀⠀⠀⠀⠂⣈
⠀⠀⠀⠀⠀⠀⠀⢤⠄⠴⠀⠀⣀⣴⠆⠀⠀⠀⠀⢸⣿⠀⠘⣿⠟⢰⣿⡇⠀⠀⢀⣿⣿⠀⣿⣧⡀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠘⠛⠓⣶⠆⠀⠀⠀⠸⠿⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⡇⠀⠈⠛⠛⠉⠀⠀⠀⠐⣾⣷⠂⣠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠸⠿⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀⠀⠉⠁⠀⠀⠀⢠⡄⠀⠀⢀⠀⠁⠈⠀⣩⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣡⠆⠀⠀⡀⣀⠀⠀⠀⠖⠀⠀⣠⡄⠀⣀⣀⠈⠁⠀⠀⠀⠀⠀⠀⢀⣴⣄⢀⡴⠀⠈⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠿⡿⠃⠀⠀⠀⠀⠞⠁⠷⠚⠁⣿⠀⢀⠀⠰⣶⣿⡤⠀⠸⠃⠹⠞⠀⠀⠰⣌⠳⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣰⠆⠀⠀⠀⡰⠀⠤⠀⠘⠀⠀⠀⠀⠋⠻⠀⠀⡤⡐⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⠉⣠⣦⡀⠀⠀⠀⢀⠀⡀⠀⢀⡀⠀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠻⠀⠀⠀⠀⠘⠀⠁⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠐⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠀⠢⣄
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠢⡀
⠀⠀⠀⠀⢀⡠⢴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢰⠀⠀⠀⢰⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠤⣀
⠀⠀⠀⢰⠋⠀⠀⣀⠀⠀⠀⡆⠀⣀⣀⠀⢸⠀⣤⠤⡀⡆⠀⠀⡆⢠⠤⡀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠸⠀⠀⠀⡇⠀⣦⠀⢡⠀⣯⣽⠃⢈⡠⡇⠀⡇⠆⠀⠀⠇⡿⠀⠉⣈⢀⡞⠓⣦⢱⠁⣀⡀⠀⡇⠀⠀⢸
⠀⠀⠀⢠⠰⡄⣳⡄⢸⠟⢧⢸⡀⡿⠈⠧⠀⡇⠓⠴⢃⣷⠀⢰⣆⠧⠴⠂⡇⢸⠁⢠⡇⣸⠀⣿⣹⠇⠀⣄⡀⢨
⠀⠀⠀⠀⢂⢫⠈⢷⠸⠀⠈⢹⣗⡁⢀⣀⡼⠷⠶⠖⠛⠘⡇⡟⠘⠓⠶⠶⠿⣎⣙⢋⣐⣿⠼⠋⡏⡘⣸⣅⢉⠂
⠀⠀⠀⠀⠀⠁⠀⢀⣰⠤⠤⠞⠋⠉⠉⠏⠀⠀⠀⠀⠀⢠⠇⢧⠀⠀⠀⠀⠀⠈⠏⠉⠉⠙⠦⠤⢵⣑⢷⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠣⠤⠤⠴⠤⠖⠋⠀⠈⠓⠦⠤⠤⠤⠤⠃⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠖⠒⢲⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⢀⠜⠉⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣠⠖⠋⠉⠙⠢⣼
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⡠⠒⠈⠁⠀⠀⠈⠙⠲⠁⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢹⡤⠤⢷⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣄⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⡄⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⣴⣾⣿⣿⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣋⠉⠈⠁⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⣀⡀⠀⠀⣀⣀⣀⠤⠖⠚⡅⠠⡤⠤⠤⠄⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢈⠉⠉⠉⠀⠀⠀⠀⠀⢹⣤⡧⠤⠤⠤⠤⣈⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠲⣞⢀⡴⣄⣀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣄⡈⣉⠀⢀⢲⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠤⠵⠶⠗⠒⠒⠒⠛⢩⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠠⣤⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢶⠶⠶⣿⠶⠶⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣆⣀⣿⣀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⢀⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⣠⡾⠛⠉⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣠⡾⠋⠀⠀⢀⣀⣀⣀⠙⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣴⡟⠁⠀⣤⣾⣿⣿⣿⣿⡷⠀⠀⠉⠛⠳⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣼⠏⠀⠀⠈⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⣿⡛⠿⢶⡿⠛⠛⠛⠛⠻⠿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠘⢷⣀⠀⢀⣀⣄⠀⠀⢀⣠⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠈⠙⠛⠛⠋⠁⢀⣴⠟⠁⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠈⠋⢠⡏⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠻⣦⣀⠀⢀⣀⡀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⢀⣠⣤⣤⣾⠁
⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣴⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠞⢛⣉⠀⠈⠹⣷
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠐⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣷⣄⠀⠀⠀⠀⠀⠀⣴⡟⠉⠛⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠘⣯⡉⠙⠛⠛⠷⠶⠶⠿⠿⠟⠛⠛⣉⣽⣿⠈⠁⠀⠀⢀⡀⠀⢸⣧⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢿⠙⠳⠶⣤⣤⣤⣤⣤⣴⣶⣶⠟⠋⠉⣿⠀⠀⠀⢀⣿⡇⠐⠛⠁⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣰⠏⠀⢠⡟⠁⠀⠀⣸⡟⠀⠀⠀⣾⠿⣷⣄⣀⣠⣴⠟⠃
⠀⠁⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⢻⡄⠀⠀⢰⡏⠀⠀⠈⠀⣀⣤⡾⠟⠀⠀⢀⣾⠟⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠙⠶⢦⣼⣷⡶⠶⠾⠛⠋⠉⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣤⣄⣀⡀⠀⠀⠀⣀⣀⣤⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣤⠓⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⣦⡄⠀⠀⠀⢀⣤⣄⠀⣀⣤⣿⣆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣙⠗⠛⣠⣄⣠⣶⡄⠀⢰⡟⠉⠀⢶⣄⣼⠇⣶⡄⠀⣿⡁⣿⠿⢿⡍⠀⠙⠛⣷
⠀⠀⠀⠀⠀⠀⣤⣶⠿⢻⡟⢋⣩⡿⢿⡇⠀⠈⠃⠀⠀⢿⡿⢋⣠⣿⣿⡶⣿⣿⣟⣀⣈⣿⡄⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠈⠻⣿⡟⢛⣿⣩⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⠈⢩⣿⠈⠿⠘⠿⠛⠋⣩⣌⠁⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⢿⣻⣹⡿⠀⠀⠀⠀⠀⠀⣀⣉⡀⠈⡋⠀⠀⢠⣄⠀⣾⠏⢁⣀⣤⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡙⣻⡇⠀⠀⣠⣴⡶⢿⣟⠉⣥⡀⢹⣆⣀⣼⣿⠀⣿⡾⠟⠛⢉⡄⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠿⠃⠀⠀⠈⠀⠀⠈⣿⡀⢸⣧⠈⣿⠛⠉⠹⡧⠈⠻⠶⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⣿⠀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣴⣶⢶⣄⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠓⠀⠻⠋⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣶⣶⡾⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠿⠟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡄
⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⢠⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡄
⠀⠀⠀⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆
⠀⠀⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆
⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠈⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠁⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣏⣤⣤⣤⣄⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣠⣤⣤⣤⣹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠛⠛⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠛⠛⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣶⣦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣶⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿
⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣯⣙⠛⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⣋⣽⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⣴⣶⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏
⠀⠀⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏
⠀⠀⠀⠀⠘⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠃
⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣶⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣿⣆⠀⠉⡒⠤⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⣾⣿⣿⡄⠀⢸⣿⣶⣶⣾⠀⠀⠀⣠⠴⠂⠐⢦⡀
⠀⠀⠀⠀⠀⢠⠊⠒⣒⣒⣒⣒⣒⣤⡀⠀⠀⠀⡜⢰⣿⣿⣿⣧⠀⠸⣿⣿⣿⡏⠀⣠⠞⠁⣠⣀⣀⠀⣷⡀
⠀⠀⠀⠀⠀⢁⣠⡀⠀⠙⠿⣿⣿⣿⣿⣦⡀⢠⠃⠈⢿⣿⣿⣿⣦⠀⢻⣿⣿⢃⡜⢁⣴⣿⣿⣿⣿⣷⣌⢿⣦⡀
⠀⠀⠀⠀⢀⣼⣿⣿⣦⠀⠀⢿⣿⣿⣿⣿⣿⡞⠀⣠⣾⣿⣿⣿⣿⣷⡌⢿⣿⣯⣦⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⢷⡀
⠀⠀⠀⠀⣨⣿⡿⠋⠉⡆⠀⠈⢻⣿⣿⣿⣿⠁⠀⠀⠠⣾⣿⣿⣿⣿⡷⠠⠿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠻⢿⣿⡘⣷
⠀⠀⠀⠀⣿⠏⠀⠀⠀⠑⣄⠀⠀⠸⢿⡿⠋⠀⠀⠀⠐⠺⠿⣿⣿⡏⠀⠀⠀⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢹⡇⡟
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠘⢤⡀⠀⠀⠀⠀⠀⠀⠀⠴⢶⣶⣦⣬⣷⣤⣤⣴⣿⡿⢟⣿⡟⠀⠀⠀⠀⠀⠀⠘⠛⠣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢀⡤⢒⣭⠷⠒⠒⠀⠀⠈⠙⠛⠛⠛⠛⠛⠿⣷⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠛⣁⣤⠒⠋⠉⠈⠀⣈⣉⠙⣻⣿⣿⣶⣄⡙⢻⡀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣴⣿⣿⡿⠦⠄⠀⠀⠀⠩⠴⠛⣻⣿⣿⣿⣿⣿⣾⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⢿⡏⢉⣶⠷⣶⣤⡀⠀⢸⣟⣿⣿⣿⣿⣿⣿⡿⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢹⣤⠁⠘⠣⠌⠉⣿⣿⣿⠈⢿⣿⣥⡾⣿⡿⣿⣷⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠈⠉⠙⠀⢸⣿⡄⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣹⡆⠀⠀⠀⠀⣀⣤⠀⠀⢸⣿⣿⣄⢀⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⡀⢀⠀⡾⠉⠙⠶⣤⣼⠿⢟⣻⣧⣿⣿⣧⣶⣶⣶⣾⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⣇⠸⡄⢿⣿⣟⢓⠒⠒⣛⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠰⡌⠋⣿⠈⣻⡟⠉⢹⣿⣿⣿⣿⣿⣿⠯⣉⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠈⠆⠈⢢⠸⠃⠀⢸⣿⣿⣿⣿⣿⡯⣉⠐⠊⠄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠘⡄⠀⢀⣿⣿⣿⡿⢟⠪⡉⠂⠁⠀⠀⠐⠄⠬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠙⡒⢛⣡⣿⠟⠀⠀⠑⠈⠀⠀⠀⠀⠢⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⡀⢰⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡵⠀⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢣⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠀⣤⠐⡃⠸⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁

⠀⠀⠌⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠐⠂⠙⢄
⠀⠰⠀⠀⠀⠀⠉⠐⠒⠀⠒⠂⠀⠤⠤⠤⠀⠀⠀⠀⠀⠀⠤⠤⠤⢄⡀⠀⠒⠠⠐⠒⠒⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⡠⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡶⠁
⠀⠀⠈⠿⡁⠲⢤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⣀⡤⠶⡎⠁⢰⡄
⠀⠀⠀⢸⠃⠀⢠⠂⠀⠈⠁⠐⠂⠤⠄⣀⠠⠀⠀⠀⠄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⠤⡐⠂⠁⠀⠀⠀⡁
⠀⠀⠀⠸⣣⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠱⠄⠀⠀⢇⠀⠀⠀⠀⠀⡠⠂⠀⠂⠀⠀⠀⠀⠀⢹⡟⣼⡂
⠀⠀⠀⠛⠁⠀⠿⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠃⠀⠀⠀⠀⠈⢦⠀⠈⢦⠀⢀⡠⠞⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠙⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠉⠉⠑⠠⠀⠀⠃⣀⠄⠲⠒⠲⠀⠷⣀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠡⠠⠏⢁⠔⠚⠂⠀⠒⠠⣍⠃⡀⢀⠄⠄⠀⠦⢄⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⡇⠰⠁⠀⠀⠀⠀⠠⠀⠀⢠⡗⠁⠀⠀⠀⠀⠀⢣⠀⢀⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠓⠄⠀⠀⠀⠀⢠⢁⠆⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⠘⠂⠀⢶⡶⠶⠶⠶⠾⠁⣠⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠠⠀⣄⡐⠉⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡑⠤⡀⠀⠀⠀⢀⠔⠉⠀⠀⠀⢀⣠⣞⠁⠀⠀⠀⠞⢋⡵⡋⠖⣇
⠀⠀⠀⠠⢀⠀⠀⠀⡄⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠣⣀⠀⠋⢘⡣⢑⢦⠀⣀⠽⠢⣀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠲⠄⠁⠀⠀⠀⠘⢃⠠⡀⠀⠀⠴⠀⠁⠀⡀⠀⠀⢠⠉⠑⢄⢃⠉⠃⠈⠁⠀⠀⣸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢺⠀⡜⠉⠋⠉⠋⠓⠙⢹⠇⠐⠀⠠⠂⠀⠒⠀⠂⠐⠂⠈⢋⣠⣀⡀⣷⠒⢋⠠⡀⠠⠂⠉⢱⡠⣄
⠀⠀⠠⠀⠀⠀⠀⢸⡐⠅⠠⣂⠀⡠⠀⠀⠨⠜⠃⠀⠀⠑⠀⠀⠐⠃⠀⠰⠀⠀⠀⠉⠛⠁⢽⣉⠋⢝⣺⠛⠛⢦⠂⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡣⠈⠇⠀⡀⠉⠀⠐⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈⠈⢱⠀⠸⣆⠀⠮⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⢀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡌⡀⠒⢺⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠱⠆⠊⢩⠺⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⠤⠀⠉⠂⠤⢀⠀⠀⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡆⢀⠸⡀⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡠⠓⠦⣄⡀⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠐⠠⢀⡉⠐⠂⠠⠤⠤⠤⠄⠒⠈⢀⣠⠔⠊⠀⠀⠀⠀⠉⠓⠆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡇⠀⠀⠀⠀⢸⠉⠙⠓⠒⠒⠒⠒⠂⢩⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣀⠠⠄⠐⠒⠃⠃⠈⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠤⠄⠒⠉⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠃⠀⠀⢠⠄⠀⠀⠼⠀⠀⠀⠐⢀⠀⠀⠄⠀⠀⠀⠑⠖⠃
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⢧⠀⠀⠀⠈⢂⠀⠀⠀⠀⠄
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠱⠤⠀⠤⠂⠀⠀⠙⠦⣤⡄⠞⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣾⣷⣿⣶⡷⢾⣾⣿⣿⣿⣿⣿⣷⣶⣿⣾⣾⣿⣿⣿⣿⣶⣶⣶⣿⣷⣿⣾⣷⣶⣶⣶⣾⢿⡶⠾⠿⢶⣶⡶⢿⣾⠿⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⠿⠛⠿⠿⢿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⢿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣏⠗⠀⢀⣀⣦⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⠜⠇⢰⣿⠿⣟⣛⣻⣿⣿⣿⣿⡶⢀⣴⣶⣶⣤⣀⣀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣯⣿⣿⡇⠐⠿⠦⠶⠿⣉⣉⣉⣛⢛⠻⠏⠀⠈⢿⣿⣿⣿⣿⣿⣷⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠭⣍⣙⣿⣻⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⣀⣾⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠸⣶⢾⣶⠀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣷⠀⠀⢠⠄⠀⠀⣄⠱⠘⢛⠋⠉⠁⠈⠀⠀⡀⢀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⠴⠶⣷⣦⣤⣤⣄⣀⣀⠀⠀⠀⠀⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⡄⠀⠀⠘⠷⣦⣀⠉⠙⢛⣿⠻⠷⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠙⠛⠛⠋⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⣾⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠁⠘⢿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡄⠀⠀⠄⠀⣀⣄⡴⢺⡆
⠀⠀⠀⠀⢀⢀⣀⣤⣤⣤⣴⣾⣿⣿⣯⣿⡀⠀⠈⠻⣿⣿⢿⣿⣿⣿⡟⠛⣢⣤⣙⠛⠛⠟⠋⠀⣾⣿⠀⢰⣦⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠘⠫⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠻⣿⡟⢿⣿⣷⣾⣿⣿⡻⣷⡄⡀⠀⠀⠘⠏⠀⠀⢻⣿⣿⣿⣿⣻⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⠄⠹⠚⠉⠛⢟⠫⠃⠉⠉⠉⠁⠒⠀⠈⣠⣶⡒⠀⢻⣿⣿⣏⡝⡟⠻⣺⠏
⠀⠀⠀⠀⠀⠀⠈⡟⣿⣿⣿⣿⣿⣿⣿⠿⣿⠀⠀⢀⠎⠻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡴⠾⠿⠿⠷⠀⠀⣿⣽⣯⡾⣴⣶⣿⣄
⠀⠀⠀⠀⠀⠀⠰⡌⡻⣛⢧⣻⣿⣿⣿⡶⢺⡆⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠯⠋⣑⡠⡎
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠿⠷⠨⠺⢿⣿⣿⣼⣧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣦⢢⣧⠆⠋⢞
⠀⠀⠀⠀⠀⠀⠀⢼⡛⠓⣝⣋⡻⣦⣿⣏⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣽⣬⣇⣌⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⡩⢏⡛⠪⢷⣯⢔⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⡉⠴⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⠅⡢⠁⠉⢹⠊⢛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣹⠇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣛⣉⣀⣀⣀⣀⡀⠈⠙⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠛⠋⠁⢀⣀⣤⠶⠞⠒⠈⠉⠉⠉⠉⣉⡉⠉⠉⠉⠛⠛⠷⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⢀⣀⣀⣀⣉⡙⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⠛⠳⣆⠀⠀⠀⠀⠸⠀⠀⠀⣠⣴⡿⠟⠛⠛⠛⠛⠿⠿⣦⣍⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⣴⣿⣋⡴⠞⠋⠙⠛⠒⠀⠀⠀⠉⠓⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠜⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⣠⣴⡿⠟⠛⣉⣉⣙⣛⣿⡿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⡀⠀⠹⣿⣆
⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⢠⣾⡿⠋⢀⡴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠀⠁⠀⠀⠘⢿⣧
⠀⠀⢠⡟⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡇⠀⠀⠀⠀⠀⠈⢿⣧
⠀⢀⡟⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠈⣿⣧
⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣷⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻
⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟
⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠶⠶⠶⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⠶⣤⣤⣤⣄⣀⣀⣀⣀⣀⣠⣤⣤⣤⠐⠾⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠉⠻⠟⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠂⠊⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡴⠊⠀⠉⠰⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠈⠓⢀⡀⠀⠀⠀⠀⢀⡜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡏⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢠⠀⡀⠀⠀⠀⠀⠀⡀⠈⠂⠀⠀⠀⠄⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡾⠁⠀⠀⠀⠐⠚⠹⠆⠀⠀⠀⣠⣄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠙⢃⠀⠀⠀⠀⣶⣄⠒⠂⢠⡀⠁⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⢀⠟⡀⠀⠀⠀⠹⡙⢿⡆⠒⠚⡇⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡉⠀⠘⢤⡀⠀⠀⠑⠰⠧⠄⠂⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠊⠀⠀⠠⡁⠀⠀⠀⠽⠶⠦⢤⣤⣤⣤⡐⠋
⠀⠀⠀⠀⠀⢀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠈⠢⢀⣀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠦⠴⠤⠤⠤⠤⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⢸⠁
⠀⠀⠀⢀⠸⣄⠀⠀⠀⣠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠂⢤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⡠⠊⠀⢀⠗⠐⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠰⠁⣀⡔⠁⠀⠀⠀⠃⠀⠀⠀⠀⢀⣤⠄⠀⣀⠀⠀⠀⢠
⡌⠀⠀⠈⠀⠀⠀⠙⠛⠋⠁⠀⠀⠜⠁⠉⠻⣿⣿⠆⠀⠀⠈
⠓⠀⠀⠒⠒⠂⠀⠀⡞⠉⠉⠁⠀⠈⠁⠀⠚⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢡⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⡆⢰⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠒⠒⠶⠒⠀⠘⠲⠒⠒⠒⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠘⠀⠀⠀⠀⠀⡆⠘⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠋⠀⠀⠀⠀⠀⠀⠀⠇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⣸⡆⠀⠀⠀⠀⠀⠀⠀⣶⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣏⠑⢄⡀⠀⠀⠀⠀⢠⣌⠙⢩⣶⣶⠆⠀⢴⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⠎⠙⢂⠀⠀⠀⠀⠉⠛⠛⠿⠻⢤⣤⣌⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⣀⣀⡚
⣿⣿⣿⣿⠿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣀⣀⣀⣠⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠲⣆⠀⠈⣷⠀⢰⡏⠀⢀⡴⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⠀⣏⣠⣤⠿⠶⠾⠧⣤⣈⡇⠀⠀⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⢳⡄⠀⠉⣻⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣾⡋⠁⠀⣴⠋⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠛⢦⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠞⠋⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠛⠲⢦⡟⠀⠀⣠⠖⠒⢦⡄⠀⠀⠀⠀⣴⠖⠒⢦⡀⠀⠘⣧⠶⠚⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⡾⠀⠀⢰⣇⡤⠦⣤⣹⠀⠀⠀⢸⣡⡤⠤⣄⣷⠀⠀⠸⡆⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢻⡇⠀⠀⠈⠋⠀⠀⠀⠛⠀⠀⠀⠘⠃⠀⠀⠈⠋⠀⠀⠀⣿⠋⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣠⣼⡇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡿⣤⡀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢷⠀⠀⠈⢿⡛⠲⠶⢤⣤⣤⣤⣤⠤⠶⠚⣻⠏⠀⠀⢰⠇⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠤⠶⠛⢧⠀⠀⠈⠻⣶⠦⠤⣤⣤⣤⡤⠤⢶⡾⠋⠀⠀⣠⠟⠳⠦⠤⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⢀⡴⠞⠳⣄⠀⠀⠈⠓⠶⢤⣤⣤⠴⠖⠋⠀⠀⢀⡴⠛⠶⣄⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡼⠁⠀⣠⡼⠷⢦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠿⣤⡀⠀⠹⣤⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡀⠀⠀⣏⠉⠛⣿⠒⢻⡟⠋⢉⡇⠀⠀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠋⠀⢠⡟⠀⠘⣧⠀⠈⠳⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈



⣟⣿⣿⣿⣻⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣻⣟⣿⣿⣛⣛⣿⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠙⠉⠰⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠰⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠂⠤⢀⣀⣀⣀⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⣠⠆⠀⠀⠀⠀⢤⡀⠀⠀⠀⡁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠐⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡋⢁⠀⠀⠀⢰⣆⡒⠀⠄⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⡜⢄⠀⠀⠈⢍⠻⠃⠀⡙⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠤⠀⠀⠐⣄⡀⠀⠉⠓⢈⡠⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠁⠀⠀⠀⠐⠤⢀⠀⡀⠉⠉⠉⠛⠁⠐⠒⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡸⢆⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠒⠂⠤⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁
⠀⠀⠀⠀⠀⢠⠊⠀⡠⠋⠉⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢠⠁⠀⠶⠄⠀⠀⠆⠀⠀⢀⡠⠾⢧⣤⣦⠀⠀⠀⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠠⠤⠤⠄⠀⠒⠒⠂⠠⠴⣿⡛⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠀⠀⠀⠀⠐⠒⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⠤⢀⡀⠄⠸⢀⠀⠀⢤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢠⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⠀⠀⠀⠀⢸⡄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢀⡇⠀⠀⠀⠀⠀⢀⣄⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠃⢄⠀⠀⠀⠀⢠⡉⠛⣵⣶⠄⠀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡤⠈⠑⠄⠀⠀⠈⠉⠛⠛⠛⠦⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡟⣿⠿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠛⠛⠲⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⣄⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠘⣿⣿⡄⠀⠀⠀⠀⠀⣿⣧⢀⣤⠤⢤⡈⠛⠛⠻⠷⠶⠶⢶⣿⣿⣦⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⣹⣿⠈⠛⠒⠒⢻⡖⠒⢶⣄⠀⠀⢸⣿⡟⠀⠀⠀⠉⠉⠉⠙⠛⠛⠷⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⢈⡇⠀⢸⣿⡇⠀⣾⣿⠁⠀⢀⣤⡄⠀⠀⠀⣀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⢸⠃⠀⢸⣿⠃⢠⣿⡟⠀⠀⣾⡟⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣠⡿⢃⣴⠗⣶⡿⠀⢸⠀⠀⣿⡿⠀⢸⣿⠇⠀⣸⡿⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⢀⡴⠟⢁⣾⠏⢠⣿⠇⠀⠺⠀⢰⣿⠇⠀⣿⣿⠀⠀⣿⠇⠀⢀⢾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡷⠋⠀⠀⣼⡏⠀⣼⣿⠀⠀⡇⠀⣼⣿⠀⠀⣿⣿⠀⠀⢿⡀⣠⢫⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⣿⠀⢰⠻⣿⠀⣸⠁⠀⣿⡇⠀⠀⠹⣿⠀⠀⠈⣉⣡⣿⡿⠤⠤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⢿⣀⠆⠀⠙⠶⠃⠀⠀⠻⡇⠀⠀⢀⡠⠔⠊⠉⣠⣿⠏⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠀⢀⣀⣥⣴⣶⣤⡀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣴⣾⡿⠟⠛⠉⠁⠀⠁⢸⣿⠀⠀⡎⠀⠀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⠁⠀⠀⢰⣶⠄⠀⠀⣿⣿⠀⠀⠙⠛⠉⠁⠀⠀⢠⣶⣿⠟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠛⠉⣿⣿⣿⡆⠀⢀⣘⠃⠀⠀⢸⣿⡏⠀⣠⡀⢠⢺⣿⠂⣰⣿⠟⢁⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠛⠁⠀⠀⠀⢹⣿⣿⡇⠀⣾⣿⠇⠀⠀⣼⣿⠇⡞⣼⡵⢁⣿⠃⢰⣿⡯⠔⠋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⡟⠀⠀⢀⣿⣿⡾⠀⠿⠁⣼⡏⠀⣼⡿⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⣿⡇⠀⠀⡜⣿⣿⠃⠀⠀⠀⣿⠁⠀⠸⣷⡀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠸⣇⠀⡰⠀⠙⠿⠀⠀⠀⠀⠙⠁⠀⠀⠈⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡄⠀⠉⠛⠁⢀⣀⣠⣤⠴⠶⠒⠒⠛⠋⠋⠉⠉⠉⠉⠛⠛⠛⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠒⠛⢛⣛⣋⣙⣛⣓⣒⡶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢋⣤⡶⠞⠛⠋⠉⠉⠀⠉⠉⠉⠛⠛⠾⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢀⠄⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢿⡄
⠀⠀⠀⢀⣤⡶⠛⠑⠲⠦⣞⠀⠀⠀⢠⡇⠀⠀⠀⠀⢀⠤⠤⠤⢤⡀⠀⠀⠀⠀⠀⡎⠉⠉⠉⢹⠀⢸⡇
⠀⢢⡶⠛⠁⠀⠀⠀⠀⠀⠈⠑⠒⠲⠞⠉⠉⠛⠓⠒⠺⣀⣀⣀⣠⠷⠦⠠⠤⠤⠤⢇⣀⣀⣀⡼⠒⠚⢧⣄
⠀⠈⠛⠶⠶⢶⣶⣶⡦⠤⡦⠤⣶⠤⢤⣦⠤⢶⠤⡴⠦⠤⣦⠤⣤⣤⡦⠤⣤⡦⠤⣦⣦⣶⡦⠴⡶⠶⠞⠫⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠟⠛⠛⠋⠉⠙⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠋⠙⠛⠛⠻⠿⢶⣤⡀
⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠈⠙⠷⣄
⠀⠀⠀⠀⠀⠋⠀⠀⣠⡴⠛⠉⠁⠀⠀⠈⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠁⠀⠀⠈⠉⠛⢦⣄⠀⠈
⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢀⣤⣶⣶⣶⣄⠀⠀⠀⠀⠈⡄⠀⠀⢠⠃⠀⠀⠀⠀⣠⣶⣶⣶⣤⡀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⣾⠁⠀⢹⣿⣿⣧⠀⠀⠀⠀⠇⠀⠀⢸⠀⠀⠀⠀⣸⡏⠀⢘⣿⣿⣷⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⣷⣴⣿⣿⣿⣿⠀⠀⠀⠀⡀⠀⠀⢸⠀⠀⠀⠀⢿⣷⣶⣾⣿⣿⣿⠃⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠤⣤⡤⠤⠖⠋⠁⠀⠀⠀⡴⠋⠳⠄⠀⠀⠀⠈⠙⠲⠤⢤⣤⠤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢿⡶⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⢖⡿⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣔⠋⠍⣙⠓⠒⠒⠲⠒⠒⢒⡚⠋⠉⠙⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣄⣀⣀⣉⣀⣀⣀⣤⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⣤⣾⡿⣷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⣦⣀
⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢼⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠰⣶⣾⣿⣿⣿⣧
⠀⠀⠀⠀⢺⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠀⣀⣀⣀⣀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⡀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⢠⣾⡿⠽⠿⣿⣿⣿⣿⠀⠀⠀⢠⣿⣿⣿⣿⣿⣶⣬⣽⣿⣇⠀⣿⣿⠟⣩⠙⠇
⠀⠀⠀⠀⣰⢋⠛⣿⣿⡇⠀⣿⣿⣶⣻⣯⣛⣿⣿⣿⣆⠀⠀⢸⣿⣿⣻⣭⣿⣿⣿⡿⠿⠭⠤⢿⣿⡼⠁⡸
⠀⠀⠀⠀⠸⡌⠳⣼⣿⡷⠒⠋⠛⠻⡿⠿⠿⣟⣛⠿⠛⠀⠀⠘⣿⣿⣧⣿⢟⣃⣰⠟⠀⠀⣷⣿⣿⡇⠀⡇
⠀⠀⠀⠀⠀⢻⠀⣿⡿⠁⠀⠀⠀⠀⠉⠻⠛⠋⠀⠀⠀⠀⠀⠀⣿⣿⡿⠛⠛⠛⠉⠀⠀⠀⢸⣿⣿⡇⢀⡇
⠀⠀⠀⠀⠀⢸⡀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇⡼⠁
⠀⠀⠀⠀⠀⠀⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠒⣄⠀⠀⠀⣸⡿⢻⣿⣶⣄⠀⠀⠀⠀⣼⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠙⠦⣶⠀⠀⠀⠀⢀⣠⡴⠖⠁⠳⠴⣮⣙⣶⣾⣿⠷⠞⠀⠉⠛⢷⡀⠀⣸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢾⡏⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⢨⡇⣰⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣳⡄⠸⠃⡾⣤⣤⡴⣶⣶⣶⣤⣤⣶⣶⣷⠶⢤⣼⣾⢧⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⠀⠁⠹⣷⣤⣀⡁⠉⠹⠉⠉⢁⣠⣴⣾⠛⠋⣾⣷⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠙⠉⣿⣿⢿⣾⣿⣿⣿⡿⢻⠏⠀⣼⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠘⢧⡀⠉⢋⣩⡿⠃⠁⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠑⠶⣤⣤⣤⣴⣦⡄⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠘⠛⠛⠛⠋⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⢀⣀⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠲⠶⠶⠾⠟⠛⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣲⣯⣍⣩⣭⣍⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠛⠛⠉⠉⠉⠙⠛⠛⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⢁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⣠⣾⠟⠋⠙⠻⣦⠀⠀⠀⠀⠀⠀⣴⠟⠋⠙⠻⣷⣄⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠛⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠛⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢠⣶⡿⠟⠋⠉⠙⠻⣦⡀⠀⠀⠀⠀⢀⣴⠟⠛⠉⠙⠻⢿⣶⡄⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠈⢻⣿⣆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣇⠀⢀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⡀⠀⣸
⠀⠀⠀⠀⣿⣴⢋⣤⠴⠦⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠤⠤⣤⡙⣦⣿
⠀⠀⠀⠀⠸⣷⠏⠀⠀⠀⢠⡟⠀⠉⠙⠓⢶⠦⢤⣤⣤⣀⣀⣤⣤⡤⠴⡶⠚⠋⠉⠀⢻⡄⠀⠀⠀⠹⣾⠇
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⡾⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢷⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⢳⣄⠀⣸⠃⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠘⣇⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⣄⡀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⠀⠀⠀⢀⣠⣼⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⡉⠛⠲⠶⣧⣤⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣤⣼⠶⠶⠛⢉⣥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣤⣄⣀⣀⣠⣤⣤⠴⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠋⠉⠁⠀⠀⠉⠉⠙⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠃⠀⠀⠀⡠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢨⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣁⠀⠀⠀⠀⣀⣀⢀⣀⣤⢖⣀⡤⢶⣯⡶⣀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠇⠀⣠⠖⠉⣛⣩⡁⠀⠉⠉⠀⡤⠶⠬⣍⠘⢶⡄⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣱⡰⣊⠥⠤⠜⠀⠀⠀⠀⠒⠒⠂⢤⣱⠘⣧⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣟⣛⢁⡤⠤⠤⠀⠀⠀⠀⠠⠤⠤⣄⠙⡀⣻⡼⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢉⣟⣩⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡄⢹⣠⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣞⡏⠁⠀⠀⠀⠀⣀⣀⣀⣀⣤⠀⠀⢀⠀⠁⢸⢿⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣻⠀⠀⣴⢤⣀⣀⣉⣉⣠⣤⠴⠖⣻⠷⠀⣸⣉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⡆⠀⠹⡖⠦⣤⠤⣤⠤⠶⢶⠋⡿⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢳⡤⣧⣤⣤⣤⡤⠼⣻⠇⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⠙⠒⠒⠒⠒⠚⠋⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡶⢤⣀⣀⣀⣠⠤⠖⠋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠒⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠒⡶⡶⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠀⢸⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢷⠇⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⢣⡑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢫⠞⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣝⡲⠤⣤⣀⣀⣤⠤⢖⣫⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠿⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷
⠀⠀⠀⢠⠖⠒⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠒⢆
⠀⠀⠀⠘⠦⣤⣤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣤⣤⡤⠏
⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣀⡀
⠀⠸⢤⠤⠔⠂⠀⣀⣀⠀⠀⠀⠈⠉⠀⠈⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠂⠉⠙⠂⠀⠀⠀⠀⠀⢸⠀⠀⠒⠤⢄⠼
⠀⠀⠀⢀⣴⢾⡟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⢰⠟⠁⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡛⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠉⠛⠲⠶⢤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢤⣄⣀⣀⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠈⠉⠉⠉⠙⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀⣠⣽⣧⣄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⢠⠔⠋⠉⠉⠉⠉⠛⢏⠀⠀⠀⠈⣻⡾⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣏⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠟⠋⠀⠸⠿⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢯⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⡿⣿⡷⣶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣷⠾⣷⣿⣷⣿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣿⣿⣶⣿⣿⣿⣿⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⡏⢸⣿⣿⣿⣿⡧⡯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡼⠿⡟⢻⡿⠉⢿⠉⢉⣻⣟⠛⠿⢧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣭⡀⣰⡆⢸⡇⠀⠃⠀⠸⡆⠸⣿⣿⣷⠀⡆⢀⡍⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣻⡇⢹⣿⠸⡇⢸⡀⢣⠀⡇⢀⣤⡟⢻⠀⣇⢸⣿⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣧⢸⡿⠀⢻⡀⣷⣼⡄⢷⠈⠛⠳⣾⡀⣿⣄⠿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⢛⡛⠿⣿⠛⠛⢿⡿⠾⠷⠖⣛⣥⣽⢿⣿⣷⠾⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⠿⠃⢀⡟⠀⠀⠈⣇⠀⣰⡆⢠⡄⢀⣦⡄⠛⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡀⢲⡾⢿⡇⠀⠛⠀⢿⠀⠛⢀⣼⡇⢸⣿⣿⡆⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⢿⡋⣰⠿⠧⣜⠅⢤⠸⠀⣄⢺⣿⡇⠈⣿⣿⣧⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠙⠶⠞⠛⠷⠟⠳⠼⠷⢶⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠃

⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠒⠒⠚⠛⠉⠉⠉⠉⠉⠙⠛⠓⠚⠛⠛⠛⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡎⠡⡗⣘⠃⠀⠖⡂⠀⡀⠋⣉⠑⡔⢕⢒⠝⠀⠸
⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠑⠒⢒⠀⡁⡆⡔⡙⠜⡠⡂⠧⢅⠆⢸⢠⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⠬⠥⠚⠁⠉⠁⠈⠈⠁⠉⠀⠉⠀⠈⠉⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠶⣀⠓⠀⡔⡒⢠⠖⠒⣄⠤⠤⢄⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠁⣅⢡⠈⡱⡃⠀⣅⠀⡆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠦⠅⠤⠀⠤⠂⠼⠧⠑⠼⢇⣠⠔⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢿⠀⢀⠴⠚⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠐⢲⠃
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠸⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢥⣒⡠⠤⠄⠀⠀⢀⣀⣀⣀⣀⣐⡒⠒⠒⠀⠀⠀⠀⠤⢠⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠒⠶⠦⠤⠤⠤⠤⠤⣬⣭⣭⣭⣭⣭⣭⠽⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠛⣀⢴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⢠⡀⠀⠉⠰⠷⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠋⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣄⠀⠀⠀⠄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⢂⣁⠉⠀⠀⠀⠀⠀⢀⠈⠁⠺⢦⡀⠀⠀⠀⠀⠀⠐⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠋⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠹⣀⠀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠖⠙⠛⠀⠀⠀⠀⠋⠁⢦⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⢀⣀⠰⠆⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠺⠆⢠⣤⣤⣤⣤⠄⠚⠛⠀⠁⠀⣤⠺⠃⠀⠀⠀⠀⠶⠚⠀⡀⠓⡄⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠆⠀⠀⣀⣀⣀⣀⣠⣤⡄⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⡇⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⠀⠀⠀⢸⠀⠀⠈⢻⣿⣿⣿⣿⣿⡇⠀⣀⠟⠀⠀⠀⠀⠀⠀⠘⣦⣤⡠⠖⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⢸⠀⠀⠀⠌⠉⠙⠋⠉⢙⣁⣤⠋⠀⠀⠀⠀⠀⠀⠀⣀⠟⠁⠀⠀⠀⠘⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⢶⣄⠃⠤⠤⠶⠀⢀⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠙⠢⣤⣤⡤⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠖⠐⠃⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠁⠛⠁⠰⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⠶⢀⣀⣀⠀⠀⠀⠀⠀⣀⣀⡀⢴⡾⠋⠀⠀⠃
⣀⠀⢀⣤⠾⠛⢷⣄⣾⠃⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠒⠒⠀⠀⢀⡌⠙⠉⠀⠐⠛⠓⠂⠀⠈⣡⡞⠡⡆
⠉⠛⠀⠈⠠⣤⠄⠙⠋⣠⠀⠙⠋⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⣄⠀⠀⠀⢀⡠⠔⠋⠀⠀⠀⠀⠤⢀⣀⣀⡠⠾⠋⡀⢶⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠰⡦⢀⡙⠳⠦⣠⣄⣀⣀⣀⣀⣀⡀⠐⠶⠶⠖⠲⠶⠾⣧⣀⡤⠖⠉⠀⠀⢀⡆⠀⠀⠙⠆⢀⣁⢉⣀⠀⢂⡹⠂⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⡀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣠⠖⠀⢠⡶⢠⠜⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠠⠤⠐⠂⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡠⠚⠁⣀⡴⠋⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⣀⠴⠉⢀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠲⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠂⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠀⠈⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⢀⣀⡀⠀⠒⠖⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⠂⠠⠤⠤⠤⠤⠤⠀⠀⠀⠐⠛⠛⠛⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠀⠀⠀⠀⠀⠀⢠⠋⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀⡼⠂⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⠤⣤⣤⡄⢠⠌⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⢀⠈⠀⠀⠀⠀⠀⠀⠀⠠⣧⣀⣤⠤⠤⢤⣤⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠰⢦⠀⠀⠀⠀⠀⠀⠀⢻⠀⠄⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠈⠉⠀⠐⠦⣄⠐⠒⠒⠀⠀⠉⠀⠀⡀⠚⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⣀⣀⣦⢀⣤⡾⠀⠀⠀⠀⠀⠀⠀⢃⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡠⠤⠴⠶⠶⠶⠟⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠟⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠁⠶⣄⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⣀⣀⣀⣀⠀⠀⠀⠤⠤⠄⠘⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⡟⠛⠉⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣇⣠⣄⣠⣤⣀⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⡀
⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⣼⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠋⠉⠉⠉⠉⠙⠛⠛⠿⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠿⠟⠛⠛⠋⠉⠉⠉⠙⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿
⠻⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⡿



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




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⣿⣿⣷⣶⣶⣶⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣶⣶⣶⣶⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣯⡙⠛⠻⠿⢿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⠿⠿⠟⠛⢉⣽⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢀⣠⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⠿⠿⠛⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⣄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⣰⣿⠟⠁⠀⣿⣿⡆⠀⣠⣴⡶⢲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣴⡿⠃⠀⠀⢀⣿⣿⠁⢰⣿⡟⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⢀⣼⣿⠇⠀⢺⣿⡇⠀⢰⣿⢀⣤⠄⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢇⣀⠀⣠⣾⡿⠃⠀⠀⠘⣿⣷⡀⣾⠇⠸⢿⣧⣤⣿⣷⠾⠛⠋⠉⠁⠈⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠉⠛⣩⣵⡾⠛⠛⠇⠀⠈⠛⣿⡿⣤⣤⡄⠀⣿⣿⠇⠀⢀⣠⡄⠀⢀⡀⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⣿⣿⠀⠀⢰⣿⡟⠀⢀⣾⣿⠀⠀⣰⡟⣰⣿⠏⠀⢸⣿⡿⠀⠀⣾⣿⠃⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⢿⣿⠀⣿⣿⠀⠀⢸⣿⣇⣀⡼⣿⣿⣀⣴⠟⢰⣿⡟⠀⠀⣿⣿⡇⠀⢠⣿⡟⠀⢀⣿⣿⠀⠈⣿⣿⠄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠿⠋⠀⠻⠿⠃⠀⠀⠉⠉⠉⠀⠀⠉⠉⠁⠀⢸⣿⡇⠀⣸⣿⣿⠀⠀⢸⠿⠷⠤⠚⣿⣿⢀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄⡀⠀⠀⠀⢀⣀⣀⣤⣤⡶⠶⠶⠶⠶⠶⢶⣦⣄⠘⢿⣷⠾⠛⣿⣿⠀⠀⡾⢀⡴⠖⢺⣿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⣿⣿⣿⠋⠁⠀⠀⠀⣀⣀⡀⠀⠈⣿⣿⡇⠀⠀⠀⠀⢿⣿⣄⡼⠁⣿⣁⣴⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⠀⠸⠿⠿⠃⠀⠘⠛⠋⣠⠞⣿⣷⡄⠈⠙⠋⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡏⠀⠀⠀⠀⣤⣶⡆⠀⠀⣀⣀⠀⣼⠋⠀⣿⣿⡇⠀⣀⣀⠀⠀⣴⣾⣿⠟⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⠀⠀⠀⣼⣿⡿⠀⠀⣰⣿⣿⢸⠇⠀⢸⣿⣿⣠⠟⣽⣿⡇⣼⣿⣿⠃⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣿⣿⠇⠀⠀⣿⣿⣯⡟⠀⠀⣾⣿⡿⠁⢰⣿⣿⠀⣿⣿⡧⠤⠚⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣿⣿⡄⣀⡴⠛⠿⠟⠀⠀⠀⠻⠟⠃⠀⣾⣿⡇⠀⠹⣿⣿⣄⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⢀⡼⠃⠉⠉⠉⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⣿⣿⣧⠤⣤⡀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠤⠴⠋⠀⠀⠀⠀⠀⢸⣿⠃⣼⡿⠀⠀⠀⢀⣠⠴⠻⣿⣷⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⣤⣤⡤⠴⠞⠋⠁⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠛⠛⠉⠉⠉⠉⠛⠛⠒⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠒⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠉⠓⠢⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣠⣤⣄⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣼⣿⣿⣿⣧⢸⠀⠀⠀⠀⠀⢀⡠⠤⠶⠶⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣿⣿⣿⣿⣿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠹⣿⣿⣿⣟⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠉⠛⣩⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢰⣶⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⡄⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠘⣿⣿⣦⣤⣈⣉⠉⠙⠛⠓⠒⠒⠒⠛⠛⣉⣉⣩⣥⣼⣿⡿⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠈⠉⠉⠛⠋⠉⠉⠀⠀⠀⠀⠀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⠤⣤⣤⣀⣀⣀⣀⣤⣤⠤⠶⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣠⣶⣿⣿⣿⣦⡀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡏⠀⠀⢀⣴⣿⣿⣿⣶⣄⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣿⣿⣿⣿⣿⣿⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣀⣀⣀⣀⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⠿⣿⣿⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢛⣀⣀⣿⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⢁⣾⣿⣿⣿⣿⣿⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠠⣿⣟⠻⠋⢿⣿⣿⠀⢸⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⣠⣾⣿⣿⣟⣀⡀⣾⣿⣿⠀⢘⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⣠⡶⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⡈⠻⣧⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⢀⣧⠉⠀⠛⣻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠙⠙⠉⠀⠀⠀⠀⢀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠢⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢁⣴⣿⡯⣤⣄⣸⣿⣿⣿⣿⡿⣿⣿⣿⣅⠀⠀⠀⠠⠤⠀⠀⠀⣉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⡀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⡉⠁⢛⣹⣿⡄⢄⣤⡶⠀⠀⠀⢀⣴⡏
⠀⠀⠀⠀⠀⠀⠠⣄⢘⡇⠀⠀⢀⠀⠀⢀⢠⢠⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡿⠛⣿⣿⣿⣡⣶⣼⣿⣿⣇⣸⡿⠁⢠⣶⠿⡿⠋⢀
⠀⠀⠀⠀⠀⠀⠀⢈⡿⠀⠀⠀⠈⢠⠠⣿⣿⣾⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠻⠿⢿⣿⣿⣿⣿⣿⠛⠉⣀⠐⠋⠀⠀⣼⣾⠏
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠰⣆⢶⣸⡇⢿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣥⣿⣿⣾⣅⣠⣶⣶⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣷⡈⢦⡙⢿⣿⣿⣌⠻⡏⢀⠀⢀⣠⣤⣤⣤⣄⠀⠀⠀⠀⢀⣤⣤⣤⣀⣀⠀⠓⢠⣿⣿⣿⣿⣿⣿⣿⠿⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣯⣿⣿⣿⣷⡀⠈⢠⠟⠛⡅⠀⢀⣽⡀⠀⠀⠀⠘⠛⠙⠉⣿⠟⢷⡄⠸⣿⣿⣿⣟⣏⡁⣤⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡇⠿⣿⣿⣿⣿⡇⢠⠏⣠⣴⠿⢾⣿⡿⠂⠀⠀⢀⢎⠷⣻⠿⣿⣤⣀⠷⠀⢙⣿⣿⣿⡿⣟⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣷⣾⣿⣿⣿⠃⠛⠻⢿⣁⣴⣄⣱⣄⡄⠀⠀⠈⠀⠾⣡⣶⣦⡽⠿⢇⡀⠀⣿⣿⣿⢿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢠⣬⡉⠻⣿⠀⠀⠀⠀⠈⣿⠟⠉⠉⠀⠀⠀⠀⠚⠃⢸⡿⠁⠀⠀⠀⠀⢀⠙⠋⣠⣶⠀⣉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⢻⣙⣦⣿⠃⠀⠀⠀⠀⠀⢀⠤⠄⠀⠀⠀⠀⠀⢤⡈⠀⠀⠀⠀⠀⢀⠿⢀⣾⡏⠐⠬⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⠀⠀⠀⣠⠴⠿⢄⣶⣤⣀⣀⣴⣶⣸⡷⣤⡄⠀⠀⢠⡏⠀⢘⣡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠙⢃⠀⠚⠁⠀⠀⠀⠀⠈⠹⠋⠁⠈⠁⠀⠀⠙⠀⠀⠃⠀⠀⣾⠏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣽⡇⠘⠀⠸⣷⣶⣖⠰⠶⠶⠒⠒⠲⠶⠤⣤⣶⡞⠀⠐⠀⠀⣰⡿⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠻⣻⣿⣿⣶⣶⣶⣶⣶⣶⣾⣿⠋⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠉⢿⣿⠛⢿⣿⡿⠿⢿⣿⠃⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣇⠀⠀⠈⠻⣦⣀⣀⣀⣤⡾⠉⠀⠀⠀⢀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣇⠀⠀⠐⢤⣈⣉⣩⣴⠃⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⣀⣀⡄⠀⣀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠋⠉




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡡⠠⣟⣋⡛⣛⣛⣛⡙⢋⣙⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⢉⡗⡤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⣿⣿⡷⡄
⠀⠀⢠⣤⣀⡀⠀⠀⠀⠀⠀⠁⠂⠀⠀⣼⠟⠛⠛⣿⣿⣾⣿⠁⠈⠢⡀⠀⢀⣀⣴⣿⣿⣿⡳⣭⡉⠔⠇⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄⠀⠀⠀⠸⣼⡷⠀⢀⣿⣿⠟⠛⢻⡴⣀⣤⣟⠓⣻⡿⣿⡟⣟⠻⣷⣿⣿⣆⠀⣾⠈⡇⠀⠀⠀⢀⣀⡤⠔⠒⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⢸⣿⣿⣷⣆⣸⣾⡯⠚⡟⠉⢉⠓⢼⣿⣾⣆⡈⠻⣿⣿⣦⣬⣍⣀⣀⣤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⡿⣷⡂⠀⢀⣼⣿⣿⣿⣿⣿⣿⡿⠚⣋⣀⣈⡑⠦⣿⣿⣿⣵⣷⣼⣿⣿⣿⣟⣷⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡄⠠⣤⣄⣽⣶⣾⣿⣿⣿⣿⢃⣿⡟⣴⡊⢘⣛⣛⢿⣶⣝⣿⣿⣿⣿⣿⣿⣯⣿⡿⠃⡄⠀⣻⠇
⠀⠀⠀⠀⠀⠀⢀⡀⣚⣷⠶⣾⡗⢙⢻⣿⣿⣿⣿⡇⣠⣾⣿⣹⣿⢡⡾⢿⠏⣧⣿⣿⣿⣿⣿⣿⣿⣟⢿⡟⠁⠈⣧⣴⣿⣀
⠀⠀⠀⠀⢠⣴⣿⣛⢫⣜⣤⠃⣠⠚⣉⠿⣻⣽⣶⣣⣿⣿⣿⠘⣿⣜⣧⣾⣿⣯⣼⣿⣿⣿⣻⣿⣿⣿⠿⣴⣾⣿⣽⣿⣿⣯⣤⣤⣄
⠀⠀⠀⠀⢴⣉⣻⣳⠿⣼⣍⡇⠘⣉⣮⣧⣿⣿⣿⣯⣽⣾⣿⣷⣙⢿⣿⣯⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣓⣿⣯⣿⣆⣿⣾⣃
⠀⠀⠀⣾⢢⣅⣗⠻⠀⠻⢦⣙⠢⠻⣷⣿⣯⣿⣿⢿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣟⣯⣽⡿⠃⣿⣟⢻⡦
⠀⠀⢀⣻⣆⣤⠄⡛⢶⠶⣤⡀⠘⠷⢼⣿⣾⣿⣿⣦⣝⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⣭⣿⡿⣿⡿⣽⢷⣧
⠀⠠⠗⠛⠀⣓⣀⣀⠀⠀⠰⠝⠟⢦⡘⢿⣿⣿⢿⣾⡟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣟⣷⣿⣭⣿⣿⣿⣆
⠀⠀⠈⡉⣌⢻⣭⣿⣿⡓⢶⣬⣠⢊⡹⣾⣿⣧⣼⣟⡀⠀⠾⡏⠙⠛⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠟⠛⠉⠉⠈⠙⠒⠋
⠀⠀⠀⠙⠄⠈⠻⣿⣿⠿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣳⠦⣤⣤⣀⣘⣛⣛⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠠⠀⠩⠋⠀⠈⢹⣿⣿⣿⣿⣿⣦⣼⣿⢿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣻⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢹⣿⣿⡿⢿⣿⣿⢿⣿⣾⣿⡴⠻⣿⣅⠀⠀⠀⣈⣿⠟⢇⣨⣶⣿⣿⣿⣿⣿⣿⣿⣽⣟⣿⣿⣾⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠠⠇⠀⠀⠁⣠⣿⡿⠃⣼⣶⣿⣿⣿⣿⡿⣿⣅⣀⡈⢻⣷⣤⣾⡟⠁⣠⣼⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⡿⠫⡞⣠⣿⣿⣿⣿⣿⡏⢀⠻⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡆
⠀⠀⠀⠀⠀⠈⠻⢯⡙⠿⣁⣈⣣⣿⣿⣿⡌⣿⠏⣰⣿⣷⠹⣏⣱⣾⣋⣿⣿⣿⣿⣿⢏⣾⣿⣤⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢀⠀⣈⡄⠱⠈⠀⢿⣿⣿⣗⠰⣿⣿⣷⣿⡦⣻⣈⣧⣽⣿⣭⣽⡻⠟⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⣆⠀⣴⡄⢀⣀⣀⠀⣨⠈⢀⠀⢄⢈⣿⣿⡉⠖⠀⣾⡿⢻⣷⣿⣿⡿⠛⠉⠉⣻⣿⣦⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗
⣿⣿⣿⣿⣿⣿⢻⡷⣜⠢⡷⠄⢸⡈⠌⢿⡆⠤⠀⠙⣿⣽⣿⣿⣭⡿⠻⠿⡊⢏⣤⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣩⣧⣴⣧⣶⣼⣛⣶⣤
⣿⣿⣿⣿⣿⣟⣿⣿⠈⣷⣤⠀⡌⢻⢈⣿⣿⣿⣷⣤⠉⢻⣿⣿⣬⡉⡅⠀⠑⣼⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠏⠁
⢿⣿⣿⣿⣿⣿⣿⡀⠀⠈⢿⣾⣿⡄⠒⠩⢟⣿⣿⡻⠛⢻⡿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡥⠟
⠸⣿⣿⣿⣿⣿⣏⠑⢄⠀⠀⠃⠙⣿⣟⢦⡈⣎⡉⠈⠖⢟⡉⠲⣽⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠿⠇
⠀⢿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠈⢷⣮⡁⢾⣿⡆⠀⠚⢷⣠⡌⣹⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣁⣤⣈⠉⣿⣿⡿⠃
⠀⢸⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢌⠻⢷⣆⣻⣇⠀⢴⠆⠙⢳⡏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠛⢠⣿⣿⣿⡇⠀⠀⢀
⠀⠘⣿⣿⣿⣿⣿⣿⣍⠃⠀⢀⠀⠠⠀⠀⢫⡸⣿⣿⣦⣄⣀⢀⠄⠘⡀⠋⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢸⣿⣿⣿⠇⣠⠆⣠⡇
⠀⠀⢿⣿⣿⣿⣿⣿⣟⠁⠂⠀⠀⠀⠀⠡⠠⣶⢀⣄⣀⣈⠉⣀⢀⣶⢻⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠓⠏⢸⣿⣿⣿⣷⣧⣤⣿⠇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠐⠒⣺⣿⣍⣯⣿⢿⡤⠾⣟⣮⢽⣿⣿⣿⡟⢿⠛⠋⠉⠀⠀⠀⠀⠀⣾⣿⣿⠛⣛⣛⡟⠁
⠀⠀⠀⣿⣿⡏⣿⣿⣿⣷⠀⠀⠀⠀⠀⠈⠀⠈⢉⠀⢳⣿⣼⣷⣤⣮⣿⣻⣿⣿⣿⠧⣼⡇⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⢹⣿⣷⣿⣿⣿⣿⡧⡀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠿⣿⣶⣿⣿⠿⣧⢻⣤⣤⡀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⠿⠁
⠀⠀⠀⠘⣿⣿⣿⢿⣿⣿⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢫⢛⣾⣿⣿⣿⡿⢾⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡟⠁⣀⣤⡶⠾⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⢶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣿⡴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⣿⣟⣉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢴⣾⣟⡛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢈⣻⣆
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣥⣦⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠘⢷⡀⠀⠀⠸⣿⢿⠄
⠀⠀⠀⠀⠀⠀⠀⢉⣼⠏⠁⠀⠀⠀⣿⣇⠀⠀⣼⠿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠈⢷⡀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⢸⡇⠙⣧⣀⣿⠀⠻⣄⠀⠸⣷⣄⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠙⣷⣄⢸⡇⠀⢰⣦⣷
⠀⠀⠀⠀⠀⠀⢰⣿⣴⠶⣶⠀⢲⣼⡇⠀⠈⠙⠛⠀⠀⠙⢷⣄⣻⡝⢦⣄⠹⣿⣦⣄⡀⠀⠀⠀⠀⠈⢿⣿⡏⠀⠀⣿⠛
⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⣿⠀⠈⣟⠃⠀⠀⣀⣀⣀⣀⠀⠀⠈⠛⠛⠀⠙⣷⣿⣶⣿⣿⣿⡶⠶⣶⣤⣼⡷⢠⣶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⡀⣿⠀⢴⡿⠿⠿⠿⣿⡷⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⣿⣿⣦⢸⡇⠀⣀⣬⡿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣀⣻⣿⠀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠀⢸⣧⡾⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢛⣷⣿⠀⠀⠾⠋⢿⣿⡟⠙⠂⠀⣶⠀⠀⠐⠛⠹⣯⣿⠙⠿⠀⠀⢸⣯⠶⠆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⡇⣽⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⡄⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⠷⢸⡄⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⡇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⣸⡇⠀⠀⢰⣶⣶⣶⣶⣶⣶⡏⢉⣿⣶⣶⣶⡶⢶⡶⠀⠀⢸⡇⠉⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⠀⠀⣿⣷⣦⣤⣬⣉⣉⣀⣀⣉⣩⣤⣤⠴⢾⣿⠆⢀⣿⠷⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⢹⣷⣤⡀⠉⠉⠉⠉⠁⠀⠀⢤⡴⣿⠇⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠙⠲⠦⢤⣤⣤⣤⣤⣤⠤⠶⠚⠉⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⣴⠤⣤⡀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⣼⢶⡇⠀⠸⣷⣶⠀⣠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠋⢠⡀⠀⠀⢁⠘⠟⣹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⠿⣆⣰⠟⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⠶⠶⠶⠶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄
⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⢠⣿⠋⠉⠙⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠉⠙⣿⡄⠀⠀⠀⠀⠀⢻⣇
⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⣿⡛⠿⢶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣴⡶⠿⢛⣿⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⢻⣷⣶⣤⣄⣈⡉⠉⠛⠛⠛⠛⠛⠛⠉⢉⣁⣠⣤⣶⣾⡟⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠈⢻⣦⠈⠉⠛⠛⠛⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠁⣴⡟⠁⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠹⣷⣄⡀⣀⣤⣤⣶⣶⣶⣶⣤⣤⣀⢀⣠⣾⠏⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠙⠻⣿⣯⣤⣀⣀⣀⣀⣤⣽⣿⠟⠋⠀⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠶⠶⠶⠶⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣐⣤⣼⠠⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣾⣿⣿⠿⠟⠿⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⡟⣃⠠⠄⠀⠀⠀⢙⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣷⣿⣷⡴⠒⣧⣦⡰⡄⠰⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⠻⠛⣯⠟⡦⣄⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣿⣿⣿⣿⣿⢀⣿⣿⠷⢊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣞⣷⣹⣿⣿⣿⣼⣿⣷⣾⡋⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣓⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⣿⣿⣿⣿⡿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣟⡟⡫⠅⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣭⣛⡿⢦⣝⣄⢸⡇⢀⣮⠠⠄⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣤⣍⣀⣡⣾⣿⣷⣄⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡎⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣛⡛⠛⣻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⢠⣗⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡖⣫⡟⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠙⠻⠷⣦⣤⣀⣄⠠⠀⢘⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢉⣉⣧⣀⡠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡭⠛⢷⡀⢠⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡆⠀⠈⢳⠞⢀⣿⣿⣿⣿⣯⠈⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄⠀⣞⡰⣇⣉⠛⣛⣫⡵⠚⣩⠄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡷⠒⡋⠋⠁⠄⣠⡬⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣏⣁⡄⠀⠀⠈⠀⠤⠉⠉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⢀⣴⣶⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⡦⢙⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡋⣴⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⠎⠻⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢀⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⡀
⢠⣿⡟⠁⠀⢀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠈⢻⣿
⣾⣿⠁⠀⠀⣾⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢿⣿⡷⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⠀⠀⠈⣿
⣿⣿⡄⠀⠀⠻⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢼⡿⠃⠀⠀⠀⣀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠟⠀⠀⢠⣿
⢹⣿⣷⡄⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠸⠿⢛⣤⣤⣴⣶⣶⣶⡿⠟⠁⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢠⣾⣿
⠀⣿⣿⣿⣶⣦⣤⣀⣀⣀⣀⣀⣠⣴⣾⣿⠏⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣦⣄⣀⣀⣀⣀⣀⣤⣴⣶⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⠿⠿⣿⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⣼⡟⠁⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⢿⣿⠿⠿⣿⣿⣿⣿⣿⣿
⠀⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠘⠛⢻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟
⠀⠈⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣼⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠃
⠀⠀⣙⣛⣻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢰⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣟⣛⣋
⠀⢰⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡆
⠀⣿⣿⣿⠏⠙⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠋⠹⣿⣿⣿
⢸⣿⣿⣿⠀⠀⠀⠻⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠀⠀⠀⣿⣿⣿
⠸⣿⣿⣇⠀⠀⠀⠀⠈⢻⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⠁⠀⠀⠀⠀⣸⣿⣿
⠀⢻⣿⡿⠀⠀⠀⠀⠀⠀⠙⢿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⢾⣿⡿⠋⠀⠀⠀⠀⠀⠀⢿⣿⡟
⡀⠀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣿⣿⠟⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢠⠄⡤⠠⠁⠑⠂⠑⠈⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠂⠀⠀⣀⡀⢀⡀⠴⡀⢊⡠⠀⣀⠀⠦⠰⡌⠂⠐⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠈⠀⣤⣤⣡⣬⡟⢻⣿⠩⣷⡁⣰⣧⡄⢲⣊⢠⣮⡄⢀⡴⠠⢤⢆⡒⠉⠈⢠⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡉⡀⡆⠢⡖⢻⣡⡻⠞⢉⡀⣁⣠⣠⣿⣾⡉⢓⡨⠗⢠⣁⣿⣯⣗⣈⠿⠂⡈⣶⠞⠠⢎⠁⡘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢔⣠⢲⣯⣤⣤⣗⣾⣹⡏⢱⢆⡨⠾⢟⣛⣻⣿⣽⣻⣾⣿⣶⣿⣿⣿⣿⡿⣦⣻⣶⣏⣍⣰⣤⢷⣘⢈⣸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⠓⢴⣿⣿⣷⣾⣿⣿⣿⣖⣋⣵⣿⣶⣤⣽⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣶⣷⣽⠉⣻⡀⢹⡝⣾⣿⣿⣦⣨⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠐⣰⢷⢂⣠⡿⢿⣷⣼⢻⣿⣿⣿⣹⠿⣿⣿⣻⣿⣿⣷⣿⡿⠉⠀⠀⠀⠀⠈⣿⢻⢿⣿⣿⣿⣻⣷⣿⣼⣿⣿⣿⣿⡿⢸⢀
⠀⠀⠀⠀⠀⠀⠀⢘⣹⣿⣿⢏⣇⣴⢀⣥⣿⣿⣿⢏⣿⣿⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠢⠀⠙⡉⢻⣾⣮⡝⡛⣷⣼⣿⣿⣿⣿⣿⣿⡟⢣⠸⠁
⠀⠀⠀⠀⠀⠀⠀⡋⣿⡿⣿⣸⢥⣼⣿⣿⣿⣿⣿⣿⣿⣼⣷⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⣶⣶⢄⣿⠈⣷⢟⣻⣼⡿⣿⣿⣿⡿⣿⣿⣿⣿⠧⠄⢠
⠀⠀⠀⠀⠀⠀⠈⠂⢻⣿⠟⡵⣸⠟⣩⣾⣿⣿⡿⣿⣿⣿⣿⣿⢿⣿⣛⣿⡿⠻⣿⣦⣿⠶⣦⣼⠟⢀⡿⢿⣟⣯⣶⡵⠊⠙⣷⣿⣿⣿⣿⣇⡶⡐
⠀⠀⠀⠀⠀⠰⠤⡤⡀⣿⢸⡷⢁⣾⣿⣿⣿⣿⣷⣿⣾⣿⣟⣿⣂⣨⠏⢀⣈⣇⡙⣿⣿⣿⣿⣽⠓⣾⣿⣿⣿⣿⡶⠁⡀⢺⣿⣿⣿⣿⣿⣿⣌
⠀⠀⠀⠀⢀⠀⠐⠀⠷⣿⣿⣴⡟⣩⡶⠿⣿⣿⣿⣿⣿⣿⣿⣻⢿⡏⣶⣯⣿⣏⢳⡌⢿⣿⣿⡟⢰⡷⠷⢾⣿⡬⠁⣼⣯⣻⣿⣿⣿⣿⣿⣯⣧⡀⢂
⠀⠀⠀⠀⠈⢀⣼⡄⣾⣿⡿⢿⢧⣴⣶⣾⣿⣿⣿⡷⣭⡿⢃⣾⣿⣿⢿⣿⡟⢿⣧⠻⡎⢿⡗⡇⢸⣿⣿⣷⡖⢀⣾⣿⣿⢸⣿⡟⣿⣿⣿⣿⠙⠓
⠀⠀⠀⠀⠀⣃⠙⠀⣼⢋⣼⡟⣫⣭⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⢷⣿⣿⣿⣽⣳⢾⣥⣟⡈⠿⣥⣿⡯⢻⡟⢠⣿⣦⣬⡽⢻⣿⢟⣷⡼⣻⣿⡓⠓
⠀⠀⠀⠀⠀⢃⠨⣰⡃⡼⢉⡴⠋⣠⣾⢗⣭⣶⣿⡟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣭⣛⡛⣶⣬⣿⡁⣰⣯⣭⣟⣛⢧⣴⣿⣿⣿⣷⣾⣿⠣⠔
⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⢋⣴⣾⢟⣵⣿⣿⣿⣿⠞⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⢷⣶⣿⣟⡳⢿⣿⣻⣦⣽⣿⣿⣿⣿⣹⣿⠴⠁
⠀⠀⠀⠀⠀⠐⠸⣿⠖⣰⡿⢟⡩⣣⣿⣿⣿⣿⣿⣼⢯⣼⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⢹⡄⢻⣏⣭⣿⣷⣞⣺⡟⢿⣿⢿⣯⡯⣿⠻⠈
⠀⠀⠀⠀⠀⠘⢀⣿⡴⣻⠚⣡⠾⣛⣻⠫⢹⡿⣟⡻⢿⣿⢿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⢫⣵⢷⡄⠙⣋⢤⡀⠙⢿⡇⠜⢻⣿⣿⣃⡽⠐⣍
⠀⠀⠀⠀⠀⠀⢂⣽⣰⣷⣿⣷⠿⠋⣀⡤⠶⠚⠿⢻⣌⠛⢦⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣙⠆⣙⢦⡀⢸⣇⣠⣼⣟⢼⣤⡿⣷⣿⠓⢨⠅
⠀⠀⠀⠀⠀⠘⢸⣿⣿⣿⣿⣶⣞⣫⡤⠭⣯⣛⠿⢳⣿⡿⢿⡛⠾⠿⠿⠟⠋⠙⠙⠋⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉⠛⠻⠟⠶⠛⠉⠘⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⠶⠶⠶⠶⠶⠶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⣠⣴⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣦⣄⠈⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢠⣾⡟⠁⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠈⢻⣷⡄⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣠⣾⣿⠟⠛⠛⠛⠿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠛⠛⠻⣿⣷⣄⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠈⢻⣿⣧⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⢀⡤⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⢤⡀⢸⡇
⠀⠀⠀⠀⠘⣷⡟⣠⠶⠛⠛⠓⢶⡶⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⢶⡶⠚⠛⠛⠶⣄⢻⣾⠃
⠀⠀⠀⠀⠀⢻⣿⠃⠀⠀⠀⠀⣼⠃⠀⠀⠉⠙⢻⡖⠶⠶⢤⣤⣤⣤⣤⡤⠶⠶⢺⡟⠉⠉⠀⠀⠘⣧⠀⠀⠀⠀⠘⣿⡟
⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⢠⡟⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⢻⡄⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣾⡀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢀⣷⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣍⠛⠶⢤⣄⣠⡟⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢻⣄⣠⡤⠶⠛⣩⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠉⠉⠛⠛⠒⠶⠶⠶⠾⠷⠶⠶⠶⠒⠛⠛⠉⠉⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠶⠶⠶⠶⠶⠶⠶⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡠⠤⡀⠀⠀⠀⠀⠀⢸⠁⢀⠀⠈⠀⡀⠀⢰⠂⠪⠀⠘⠀⢟⡽⢖⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢉⡤⠦⡤⠒⣤⠰⢤⡀⠀⡴⠋⢀⠒⢨⠗⠈⠉⡉⠉⣕⡳⡙⡦⠀⠀⣷⠀⡴⠋⠀⠀⠉⠉⠙⠢⢉⢴⣻⣇⡀
⠀⠀⠀⣀⠁⡡⣾⠁⠉⠀⠀⣁⣀⣀⡃⠛⢸⠿⣄⠐⠐⠲⠙⠀⠀⠀⠀⠀⠀⠈⠙⣍⠲⠄⠟⡞⢀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠰⢚⠄
⠀⠀⠈⢣⠔⠧⠸⣃⠔⠊⠉⠀⠀⠀⠈⠙⢦⠰⣋⣱⣤⡇⠀⢀⠀⣀⣀⠠⢤⠂⠀⢹⠷⠶⢱⡇⢀⠀⠑⠒⠦⠤⠠⠀⠀⠀⢓⣻⠆
⠀⢠⠂⠘⠀⢀⠜⠁⡁⢄⡒⢈⣉⡉⠉⢱⠀⠁⣀⠈⡏⢱⠀⠈⠳⣁⠀⡀⠊⠀⠀⡞⠀⠀⠀⣧⠘⢄⠀⢀⡠⠔⠁⠀⣠⠞⠏
⠀⠈⠉⢠⠚⠻⠀⢺⡉⠀⠀⠀⠀⠀⢐⡼⢀⣇⠙⠋⢀⠜⠣⣄⣀⠈⠁⢀⣀⣤⡞⠣⣀⡴⠊⠀⠑⠢⠬⠥⠤⡄⠒⠉⠁⢄
⠀⠀⠀⠸⠤⠟⣠⡀⠉⠓⠒⠒⠒⠀⠀⣠⠞⠻⡑⢶⢁⡆⣸⣃⣸⣁⣀⣀⣽⣈⡷⢆⠘⢇⢰⠀⠀⠀⠭⠀⣰⡇⠀⠀⢠⣼
⠀⠀⠀⠀⠠⠂⠀⠉⠓⠲⠶⠤⡔⠒⠉⠀⠀⠀⢳⡔⣻⣥⠀⠁⠊⠀⠀⠀⠙⠒⠁⢸⠀⠈⣿⠀⢄⠠⢴⡺⣀⠀⠀⡠⠞⢻
⠀⠀⢠⠆⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠁⠈⢢⡀⢻⠀⠂⠐⢸⠃⠀⠸⣶⠊⠻⣝⣾⠃⠀⠙⣮⡶⠚⣒⣲⠇
⠀⢀⠋⠉⡘⡦⢤⠒⠒⠋⠋⠉⢹⡋⠉⣏⠉⡛⣿⠳⢄⣀⣼⠘⡆⠁⠒⠒⠉⢸⡉⠎⢹⢤⣀⡵⠙⠦⠤⠚⠁⠀⠲⠤⣼
⠀⠈⠂⠔⠘⠁⡌⠀⠀⠀⠀⢠⠈⠄⠤⢼⠄⡇⢹⠢⣕⣚⣋⠜⠁⢤⣤⠤⠀⠀⠙⠲⠁⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⢸⠀⡆⢰⠸⠀⠇⠸⡀⠀⡿⠤⠤⠤⣼⠀⣣⠤⠤⠐⠚⢢⣠⢤⢤⡀⣸⡀⢀⣀⣀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠧⣼⠠⣄⣦⠾⢶⡷⡼⢲⣶⠞⣾⠁⠘⡖⠒⠶⠒⠾⠀⢳⠔⠲⠖⠒⡾⠥⣄⡀⢽⢉⡿⣡⣬⣬⣿⡒⠋
⠀⠀⠀⠀⠀⠀⠠⡳⠤⢀⣀⣁⠼⣾⡳⠤⠤⢔⡛⠀⠸⣓⢒⣒⣒⠿⠀⢸⣓⣒⡒⠘⡷⠑⠒⠒⠚⠘⠧⠤⠤⠤⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠻⠿⠿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠛⣉⣥⡄⠰⣶⣸⡏⠀⠀⠀⠛⣿⠿⠂⣠⡆⠈⣙⠛⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⢋⣤⣀⠘⠛⣿⡄⠀⠙⣿⡅⠀⠀⠀⢰⣷⠀⢠⣿⠁⣠⣿⡟⢠⣬⡙⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠛⠁⠘⣿⣍⣿⣄⠀⠸⣧⡀⠀⠻⠇⣀⣀⣀⡸⠟⠀⢼⡏⣰⡟⢿⣶⣿⡿⠁⢠⣌⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠠⣶⣦⣄⠈⢿⣏⠹⠷⣀⣭⣶⣾⡿⠿⠿⠟⠿⠿⠿⣿⣷⣶⣭⡐⠗⣡⡟⢁⣴⣿⡛⠷⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢁⣤⣀⠹⣧⣽⠿⡶⣋⣴⣿⠟⠋⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣦⡐⢿⣏⠙⠃⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⢠⣿⣉⣿⠇⠘⠿⢠⣾⡿⠋⠀⠀⠀⣀⡀⠀⢴⣿⣿⠂⢰⣾⡦⠀⠀⠀⠀⠀⠙⢿⣷⡍⠀⠀⠀⠀⠀⠀⠈⣻⣷⣄⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠈⠙⠿⣦⠄⣴⣿⠋⠀⠀⠀⠀⠘⠿⠓⠀⠘⠉⠙⠀⠈⠉⠁⢀⣀⣀⣀⣠⣤⣤⣽⣿⣶⣶⣶⣾⣿⠿⠿⠿⠟⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⢀⣀⣀⣀⣠⣤⣤⣤⣶⣶⣶⣶⣿⣿⠿⠿⠿⠟⠛⠛⠛⠉⠉⠉⢉⣁⣀⠀⠀⣤⣤⣄⠀⣶⣶⣿⢿⡇
⠀⠀⠀⠀⢀⣠⣿⢇⣤⣄⣤⣤⣶⣶⣾⣿⣿⡿⠿⠿⠿⠿⠛⠛⠛⠉⠉⠉⠉⣀⣀⠀⠀⠀⣤⣶⣶⣶⣶⡄⣿⣿⠀⢹⣿⣿⣆⢠⣿⣿⣿⠀⢻⣿⣇⣀⣀
⠀⠀⠿⠿⠿⠿⠛⠛⠛⠛⠉⠉⠉⠁⣀⣀⣤⣤⣄⠀⢶⣶⣶⣾⣿⡟⣿⣧⢸⣿⡏⠀⠀⠀⠉⠙⣿⣿⠀⠀⢻⣿⡆⢸⣿⣿⣿⣸⣿⢻⣿⡀⢸⣿⡿⠿⠟
⠀⠀⢠⣤⣴⣶⣦⡄⠀⣾⣿⣷⡀⠀⢻⣿⡟⢻⣿⡇⠈⠉⣿⣿⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⡇⠀⣿⣇⢻⣿⣿⢸⣿⡇⠈⣿⣷⣀⣤⡄
⠀⠀⢸⣿⡏⠉⣿⣿⠀⣿⡟⣿⣇⠀⢸⣿⣧⣼⣿⡃⠀⠀⣿⣿⡀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠈⣿⣷⠀⣿⣿⠈⣿⣿⠈⣿⡧⠀⠿⠿⠿⠛⠃
⠀⠀⠀⣿⣿⣶⣿⠟⠀⣿⣇⣸⣿⡆⠈⣿⣿⠉⣻⣿⠀⠀⢸⣿⡇⠀⠀⠀⣿⡿⡀⠀⠀⠀⠀⠀⠸⠷⠇⠀⠀⠛⠛⠀⠉⠉⣀⠈⣀⣀⣤⣤⣤⣴⣶⣶⣶⣷
⠀⠀⠀⣿⣿⡍⠀⠀⢀⣿⡿⠟⢿⣿⡀⣿⣿⠀⢹⣯⡇⠀⠸⠿⠇⠀⠀⠀⠙⠉⠁⣀⣀⣀⣠⣤⣤⣤⣶⣶⣶⣶⣾⣿⣿⡿⠿⠟⠛⠛⠛⠋⢹⣿⠉
⠀⠀⠀⢸⣿⡇⠀⠀⠘⠿⠃⠀⠘⠋⠁⢉⣉⣀⣀⣤⣤⣤⣤⣶⣶⣶⣾⣿⡿⠿⠿⠿⠋⠛⠛⠛⠉⠉⠉⠉⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⣤⣤⣤⣤⣶⣶⣶⣶⣿⡿⠿⠿⠿⣿⣟⠛⠛⠉⠉⠉⠉⠀⢀⣀⡀⠀⣄⣠⡄⠀⢴⣶⠄⠀⠀⠀⠀⣴⣿⠏⣰⡗⢠⡀⠀⠀⠀⣾⡿
⠀⠀⠀⠐⠛⠋⠉⠙⣿⣧⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⠀⠺⠿⠇⠠⢿⣿⠶⠀⠉⠉⠀⠀⢀⣤⣾⡟⢁⣌⠛⢷⣿⣁⣴⢆⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⢀⣴⡿⣦⡈⠻⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⣀⣤⣶⡿⠟⣁⣄⡹⣿⣷⣄⠈⠛⢫⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⣴⡿⠻⣶⠿⢡⣶⣦⠉⠛⠿⠿⣿⣶⣶⣶⣶⣶⣶⣿⠿⠟⢛⣉⢰⣦⠙⢿⣿⢾⣷⡹⢗⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣬⡀⠀⢀⣴⣟⣿⠃⢰⡿⢳⣦⢠⣤⣬⣭⣭⢀⣤⠀⠀⠿⢿⣏⠀⠻⣧⡀⠻⣷⢉⣵⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣝⠉⢽⡏⢀⣿⠿⣿⠋⠀⣾⡏⠀⢻⣿⡏⠀⠀⠀⠈⣿⡆⠀⠹⢗⣠⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣶⣤⣜⡛⠰⠿⠀⠀⣿⠇⠀⠀⣿⡇⠀⠀⠀⠀⢘⡋⣴⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣷⣶⣶⣭⣤⣤⣤⣬⣵⣶⣶⣾⡿⠏⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠚⠫⣍⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢹⡄⠀⠈⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢲⡄⠤⠞⠃⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⣤⣤⠶⠶⠶⠛⠛⢷⡶⠾⠂⠀⠀⠀⠀⠙⡠⢠⠂⣠⡬⠷⠤⠖⠒⠒⠋⠉⠉⠉⠀⢸
⠀⠀⠀⠀⢠⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡏⡝⠗⠀⠘⠙⠇⠛⢸⡤⠤⠄⠒⠐⠒⠙⢆⢡⢘⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠙⢿⣧⠀⡀⠀⠀⡀⠠⠤⠤⠶⢳⣤⠀⠀⠀⠂⣀⠐⡜⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠸⣿⣧⣾⣿⣷⣍⠢⡀⠃⠀⠀⠀⢀⣸⡇⢣⣀⠀⠀⠒⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣴⠴⢿⣿⣿⣄⣙⠿⣑⢤⡑⢦⡴⠒⡟⠀⢷⠀⠈⠙⠛⢲⡎⠳⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢠⡇⠀⠀⠀⣈⣹⠿⢦⣌⠙⠻⣦⡬⠒⠋⠹⣎⢣⣀⠀⢀⡿⠁⠀⣿⣿⣿⡟⣯⠁⠒⠲⣦⡤⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠁⣠⠴⠛⠉⠀⣀⣀⣨⡽⠒⠁⠀⠀⠀⠁⠚⢷⣿⣭⢟⣴⣶⣶⣿⣿⣿⡇⢸⠀⠀⠀⠀⠙⣿⠀⠹⡌⠉⠉⠓⡖⠶⠦⠤⣤⣀⣸⣀⣀
⠀⢀⣾⠿⠷⠒⠛⠛⠉⠉⢹⠁⠀⠀⠀⠀⣀⡴⠤⠤⠬⢿⡇⣾⣿⣿⣿⣿⣿⣿⠃⢸⠀⠀⠀⠀⠀⢻⠀⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠉⠁
⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⣤⡶⠶⠛⠛⣷⣄⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⠀⢸⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⢀⡇
⣿⠀⠀⠀⠀⠀⠀⠀⢰⣈⣁⣿⡏⠀⠀⠀⠀⠈⣻⡳⣦⣿⣿⣿⡛⢿⣿⣿⣿⡇⠀⢸⠓⠶⠤⣤⣀⡀⠀⠀⠀⠀⢀⡼⠁
⠹⣦⡀⠀⠒⠶⠾⣿⣿⣿⠿⣿⡇⠀⠀⠀⠀⠀⡿⠹⣮⡫⡙⠁⠙⢦⣙⡿⠿⡶⠶⢾⣄⠀⠀⠀⠈⠉⠙⠛⠒⠒⠋
⠀⠀⠙⠛⠒⠚⠛⠋⠁⠀⠀⣿⡇⠀⠀⠀⠤⠊⠁⠀⢸⣿⣌⠢⣀⡀⠈⠀⠀⠈⢦⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠞⢹⢿⢷⣌⠀⠀⠀⣠⡀⠘⡇⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡔⠀⡀⠀⠀⠀⢀⠏⠈⣿⠀⡇⢰⠁⣸⡶⠷⠤⠼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢰⡇⢰⡇⢠⡆⠀⠈⠀⠀⣿⡇⣇⠘⢶⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣾⢁⣿⠇⣾⠇⠀⢀⣧⢠⣿⣷⡀⠀⠀⡙⠛⠛⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣿⣿⣾⣟⣀⣠⣾⣿⣿⣿⣿⣿⣶⣮⣤⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⣠⢀⣀⡀⠉⠉⠉⢉⣩⣽⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣱⣿⣿⣿⣷⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⣀⣠⣀⣄⣄⣀⣀⣄⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠛⠻⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠉⠉⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠋⠉⠉⠛⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠙⠷⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠉⠛⠛⠛⠛⠶⠶⠶⠶⠶⠶⠦⣤⣤⣤⣤⡤⠶⠶⠶⠶⠶⠶⠛⠛⠛⠋⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠘⣿⠛⠶⠶⢶⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣴⠶⠶⢻⡟⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⡀⠀⠀⠀⠀⠀⠈⠛⠷⢦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠶⠟⠋⠁⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠶⠶⠶⠶⠞⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣁⠠⠤⠤⠖⠒⠒⠒⠒⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⡄⠀⠀⣞⢳⡀⠀⠀⠀⠂⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡆⠀⣠⡶⢲⡄⠀⢠⣶⣶⣦⣄⣤⡀⠈⢸⠧⣀⡀⠀⣰⣿⣿⠃⠀⢀⣤⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⡼⠋⣀⣼⡿⣻⣿⠟⢀⣴⣿⣿⠃⢠⡟⣴⣿⠇⢰⣿⡿⠁⠀⣠⣾⡿⠃⠀⣠⣿⡟⠀⠀⠀⣀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⣠⠞⣹⣿⢳⣿⠃⡴⢫⣿⡿⠃⣠⠟⣼⣿⠋⢀⣿⣿⠁⢀⢼⣿⡟⠁⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⠀⠹⠾⠿⠃⠘⠿⠚⠁⠘⢿⣥⠾⠃⠀⠹⠧⠴⠋⠻⠧⠔⠁⠘⢿⣄⠴⣻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣿⢏⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⢀⡴⠚⠉⣴⣿⠃⠀⠈⠳⣄⠀⠀⠀⢠⡾⠛⠉⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⢘⣻⡥⠴⢤⣄⠀⠠⡏⠀⢀⣾⡿⠁⠀⠀⠀⠀⠘⢧⡀⠀⠙⠇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⠀⠀⠀⢀⣿⣿⣡⠤⠒⣿⣥⠀⠀⠀⠠⣿⠇⠀⠙⠛⠛⠉⢀⣴⣶⠖⠲⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠖⣾⣿⠃⠀⠀⣼⣿⡟⠀⣠⣤⣀⡴⣶⣦⢀⣀⡀⢀⣾⣿⠃⢀⠜⠁⠀⠀⠀⠈⠙⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⣠⠞⣿⡟⣀⢼⣿⣿⠋⣸⣿⡿⢫⣿⡟⠸⣿⡿⠊⢁⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡤⠞⠁⠀⠈⠉⠁⣾⣿⠃⠀⠻⠟⠁⣼⣿⠁⡰⠻⠧⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢰⣿⣯⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋

⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢷⠶⠒⠂⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠈⢀⣀⡀⡈⢉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢦⣬⣙⢻⣆⡼⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢤⣹⣿⢿⡏⠹⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣥⣿⡇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡴⣳⣿⣿⣿⣿⣿⣧⡾⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢛⣟⣻⣷⣻⣿⣿⣿⣿⣿⣿⢖⡴⠀⣠⡄⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢂⠼⢿⣿⣿⣥⢹⠛⢛⣿⢻⣵⣹⣿⣿⣏⠙⠢⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣴⡦⢂⣼⣻⠏⠁⣼⣟⢻⣿⢸⣿⣿⡿⣿⣿⣏⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠿⡿⠟⠁⠠⣾⣵⣿⣿⣿⣿⡇⠲⣾⣿⡏⢁⣠⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣻⣮⠠⠄⣅⡀⠉⠉⠛⣿⣿⣿⣴⣄⣿⣿⣿⣾⣿⣿⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣠⣿⣿⣿⣧⡀⠀⠀⠈⠀⢴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡄
⠀⠀⠀⠀⠀⢀⣠⣇⡀⠀⢀⣉⣹⣿⣿⣿⡿⠃⠱⠀⠈⠴⣢⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⣿⣿⣿⣷
⠀⠀⣲⣷⣤⣿⣿⣧⡾⠿⠛⠛⠉⠉⠉⠉⠀⠀⠐⠀⠀⠒⠿⢿⣿⣿⣿⣿⣿⣿⡿⢷⠑⠀⠀⠘⢿⣿⣯⣄
⠀⠐⢩⣻⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠀⠁⠸⠛⠋⣿⣿⣿⡛⡽⠂⠀⠀⠀⠀⢀⡈⢻⣷⣿⣿⣓⡂
⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠈⠙⣋⡹⠿⠄⠀⠀⠀⠀⠀⠒⠶⢿⣍⠋⠑⣀⠉⢹⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠲⢾⣍⡀⠙⠃⡀⣁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⠀⠀⠀⠀⠀⠀⠨⢘⣿⣶⣏⢁⡀⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠘⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠶⠞⠛⠛⠛⠛⠛⠛⠳⠶⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀
⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⣴⠏⠀⠀⠀⠀⢀⣤⣶⠿⠿⠿⠿⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠿⠿⠿⠿⣶⣤⠀⠀⠀⠀⠀⠹⣦
⠀⠀⣼⠋⠀⠀⠀⠀⠀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⠀⠀⠀⠀⠀⠙⣧
⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠟
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⢹⡆⠀⠀⠀⠀⠀⠀⠀⣾⠶⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠶⣷⠀⠀⠀⠀⠀⠀⠀⢰
⠘⣷⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠈⠉⠛⠛⠳⠶⠶⢶⣤⣤⣤⣤⣤⣤⡶⠶⠶⠾⠛⠛⠉⠁⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠹⡇⠀⠀⠀⠀⠀⠀⠀⢿⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⠟⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠈⢻⣌⠉⠙⠛⠛⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠛⠛⠉⢉⣼⠏⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠹⣷⡶⠶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡶⢶⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣦⣤⣤⣤⣤⣤⡴⠶⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⣶⣶⣦⣤⣤⣴⣶⡶⠶⠛⠛⠉⠁
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢠⡄⠀⣀⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠺⡿⠧⠀⠀⠀⠘⠁⠀⠉⠀⣴⠀⣿⠀⠤⣀⠀⢐⣿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠛⠀⠀⠀⠀⠁⢀⠀⠦⢀⠀⠀⠀⡄⠀⠿⠀⠁⠀⠃⠉⠀⢀⡈⠀⠐⠂⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⡴⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠛⠀⠆⠉⠀⠀⠀⠀⠀⠀⠀⣀⣠⣆⣀⠀⠀⢠⢄⡄⠀⣴⠋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡟⠷⢀⣤⣤⡴⡦⠀⠀⠘⠃⠀⠀⠘⠃⢰⡆⢠⣦⢀⣴⠂⠀⣸⠿⣿⠉⠀⠀⠋⠾⠁⠀⠀⠀⠀⢁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⠉⣻⠇⠀⠀⠀⠀⠀⠐⠂⠀⠀⠸⡿⠋⠹⠞⠁⠀⠀⠈⠀⠈⠀⠰⠆⠀⠀⠀⢆⡒⠀⠀⣸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠶⠀⢠⣾⣁⡀⠀⠀⣶⣶⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠟⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣍⠃⠀⠀⠀⠀⠀⣴⠟⠁⠀⢰⣿⡟⠀⢸⡗⠀⣴⠶⣦⠀⣾⣿⠿⢿⣿⡟⢛⣛⠛⠻⠃⠀⠀⠀⠀⠀⠴⠀⠀⠀⢠⣞⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠘⠃⠀⠀⠀⣿⣿⣧⣶⠟⠁⠀⠻⢶⡾⠤⣤⡄⢀⣾⡿⠀⣾⡟⠀⠀⣠⣾⡗⠀⠀⠀⠀⠀⠀⠀⠉⣹⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣠⣾⣀⣀⠀⠀⡤⠀⠀⢰⣿⣿⠉⣠⢶⣶⠆⢠⠞⢀⣾⡟⠁⣼⣿⠃⢸⣿⣁⡴⠚⢱⣿⣧⣴⠶⠗⠀⠀⣀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣽⠿⣿⠋⠁⠀⠀⠀⠀⠀⣸⣿⡏⢰⣿⠞⣿⡴⠋⢀⣾⡟⠀⣸⣿⡟⠀⠈⠉⢉⣤⣶⣿⡿⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠲⠒⠂
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢉⣤⣴⠶⠆⠀⠀⠀⣿⣿⡇⠀⣤⠤⠬⠀⠀⣸⣿⠁⠀⣿⣿⠇⠀⠀⣴⠟⠁⣸⣿⠃⠀⠀⣈⠁⠖⠀⠠⣦⣴⣆⣀
⠀⠀⠀⠀⠀⠀⠶⢈⣁⠀⠈⢛⣷⣦⠀⠀⠀⠀⠛⠛⠃⠀⠒⠛⠛⠂⠀⢿⡿⠀⠀⠛⠛⠀⠀⠐⣿⣄⣴⣿⠃⠀⠀⠀⠀⠙⠀⠀⠀⡾⣿⠋⠁
⠀⠀⠀⠀⠀⠀⡦⠈⠁⠀⠺⠟⠉⠀⠀⠀⠀⡆⠀⠀⠀⠀⣀⣀⣄⣠⣄⣀⣀⡀⠀⠀⢴⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⢠⡿⢀⣠⠀⠀⠀⠈⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣤⣶⡾⠿⣿⡟⠉⠉⠉⠉⠙⣛⠀⠀⢀⣴⣿⡿⠀⠀⣀⣀⡀⠀⠀⠘⠛⠉⡟⢀⣤⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⢀⠀⠀⠘⠀⠀⠀⠘⠋⠁⠀⣼⡿⠀⢶⠆⠀⠀⣼⣿⡀⢠⡞⣹⣿⠃⣰⣾⡟⠉⣿⠀⠀⠀⠀⠘⠟⠋⠁⠀⢨⣽⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⢠⣦⠀⣠⣾⣿⣿⣥⡟⢠⣿⡏⣸⣿⣏⣤⠞⠁⠀⠀⠘⠀⠀⢀⡀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠀⠀⠀⢀⠈⠁⠈⠀⠀⣼⣿⠀⢸⣟⣴⢋⣿⡇⠘⠋⠀⣼⣿⠁⣿⣿⠁⠀⠀⣠⡾⠀⠀⠀⠀⠈⠀⠆⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠘⢃⡘⠃⠀⣠⣾⣋⡁⠀⠀⠀⠀⣿⡟⠀⠈⠛⠁⢸⣿⠃⠀⠀⢀⣿⣿⠀⠻⣿⣤⣤⠾⠋⠀⠀⣀⣤⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠉⣹⣏⣠⡄⠀⠀⠛⠃⠀⠀⠀⠀⠺⠿⠀⠀⠀⠸⠿⠇⠀⠀⠀⠉⠀⠀⠀⠀⠀⢹⠿⠏⢠⣾⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠄⠀⠀⠀⢠⣞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⢤⣤⣄⠀⠀⠘⠃⠀⣠⠶⡆⣀⡤⣆⠀⠀⠀⠀⠀⣀⠀⠀⢀⡾⢷⢀⡾⠀⠀⠀⣀⠀⠀⢀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠻⠋⠀⠀⠀⠀⠘⠃⠐⠛⠁⠀⣿⠀⠠⠀⠘⣿⣿⠶⠀⠘⠃⠈⠛⠀⠀⠀⠳⠌⠃⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⢀⠀⢀⡋⠐⠆⠀⠈⠀⠀⠀⠀⠉⠙⠀⠀⠶⣈⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⣴⣦⠄⠀⠀⠀⢠⠀⡄⠀⢠⡄⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣟⣻⣽⣭⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠉⠉⠙⠟⠋⠋⠉⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢀⣀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡇⠰⠌⢷⠀⢘⡉⠓⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡇⠀⠠⠆⠠⠀⠀⢠⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠐⣒⠂⠈⠔⡩⠺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣟⠻⢧⣀⠀⠀⣨⡌⣜⣴⡌⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣿⠿⢟⢿⡿⢸⠀⠪⡉⠉⢱⡏⠓⢉⠋⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⢋⠁⠈⠁⠀⢀⠆⠘⠃⢸⡀⢀⣼⣦⡉⢿⡆⠃⠀⠘⡉⠛⠛⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢦⡀⠸⡀⠀⠀⠀⡎⠀⠀⠀⠈⡇⢊⣽⣯⠁⢸⣿⡆⢀⠄⡘⢂⠀⠀⢀⠈⣀
⠀⠀⠀⠀⠀⠀⢀⣾⢇⠨⣅⡀⡇⢀⡜⠒⠚⠓⠂⠀⠀⢧⣼⣿⣿⣇⣿⣿⣓⠀⣰⠁⠀⠣⡀⠂⣠⡞
⠀⠀⠀⠀⠀⣴⠛⠣⣄⡀⠀⢻⣶⡀⢿⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣿⣿⣇⠀⠈⣆⠀⠀⠙⢞⢫⢇
⠀⠀⠀⠀⢠⡏⠠⡀⠀⠛⢯⠻⣾⠃⠈⢷⡀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠈⢿⣿⣆⠀⠈⠀⠂⠀⠘⣇⠀⡀
⠀⠀⠀⢠⠾⣃⣤⣬⣵⣒⣦⣾⣿⠀⠀⣿⢳⡀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠈⢿⣿⣦⡄⠀⠀⠀⠀⢨⠳⡌⡄
⠀⠀⣰⣵⣯⠚⠲⢦⣌⢉⣼⣿⣿⠀⢸⡇⠀⢳⡀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠈⢿⡿⣟⢦⠀⠀⠀⠈⡇⢀⡟⡄
⠀⠀⠛⢋⠊⠀⣀⠄⠈⠛⢿⣿⠷⣢⣿⡀⠀⠀⢳⠀⠀⠀⠘⣿⠀⠀⠀⠀⢠⣿⡷⠛⣧⡀⠀⠀⠀⠘⠼⣱⠃
⠶⠶⢷⣀⢠⠞⠁⡠⠔⣪⣾⠁⢦⠈⠲⡎⢶⣶⣶⣷⣶⣶⣶⣿⣶⣶⣶⣶⣿⡿⠿⠷⣼⣟⣳⣤⠀⣀⡼⠷⠶⠶⠶⠶⠶⠶⠶
⠀⠀⠈⠛⠿⢶⣦⣵⡼⣙⣏⣘⡲⠀⠐⠤⣻⡿⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠒⠦⣄⡀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⡀⠤⠝⠛⠀⠉⠮⠥⠄⠐⠉⠀⠁⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣽⠶
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢢⣄⣠⠔⠰⠪⠛⠒⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⠀⠀⠀⠀⣀⣠⠤⠤⠐⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⢀⣾⣿⣿⡿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠋⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⢸⣿⣿⡿⠁⠀⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠇⠈⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠈⣉⣿⣿⣿⠇⠀⠀⠀⠀⠿⠿⣥⣤⡀⠀⠘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠈⠁⠈⠀⢀⣀⣀⣀⣀⣀⣤⣤⣀⣀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡆⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣆⠀⠀⠙⢿⣟⠉⠁⠀⠀⠀⠀⢈⣿⡿⠁⠀⣠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣀⠀⠀⠙⠻⣶⣤⣤⣤⣴⠿⠋⠀⣤⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣷⣦⣤⣀⣀⣉⢉⣉⣀⣤⣴⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢻⠛⡏⠉⠁⠀⠀⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⡶⠿⠷⠶⠶⢶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⢿⣻⣿⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⣋⣵⣾⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣩⣾⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢃⣼⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⣰⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⣸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⣸⡟⢰⣿⣿⣿⣿⣿⡟⠀⠀⢀⣤⣶⣶⣶⣤⡀⢸⡆⠀⠀⣞⠁⣠⣴⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⢀⣿⠁⣾⣿⣿⣿⣿⣯⣥⣄⢠⣾⠿⠟⠛⠛⠿⣿⡄⠁⠀⠀⠃⣼⡿⠿⠛⠛⠿⢿⣧⢀⣠⡀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⢸⡏⢰⣿⣿⣿⣿⣿⣷⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⣼⡇⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⠶⠚⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⡄⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⢸⣇⠈⣿⣿⣿⣿⣿⣿⠀⢀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣄⠀⠀⢻⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠘⣿⡀⣿⣿⣿⣿⣿⣿⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣧⠸⣿⣿⣿⣿⣿⢿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠹⣿⣿⣿⣿⡾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠹⣿⣿⣿⣿⣮⡛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡈⢿⣿⣿⣿⣷⡄⠀⠀⠀⠑⠒⠒⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣙⢿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣬⡻⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣾⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⢶⣶⠶⠶⠾⠟⠛⠋⠉

⠀⠀⠻⣿⣿⣆⠀⠘⣿⣿⡄⠀⢻⣿⡇⠀⢹⣿⡇⠀⣿⣿⠁⠀⣿⣿⠃⠀⣼⣿⡟⠀⢀⣼⣿⡿⠃⠀⢀⣴⣿⣿⡿⠋⠀⠀⢀
⣷⣄⠀⠘⢿⣿⣦⠀⠘⣿⣿⡀⠈⣿⣷⠀⢸⣿⡇⠀⣿⣿⠀⢰⣿⡏⠀⣰⣿⡟⠀⢠⣾⣿⠟⠀⢀⣴⣿⣿⠟⠉⠀⠀⣠⣴⣿
⢿⣿⣧⡀⠀⣹⣿⣷⣦⡈⢿⣷⠀⠸⣿⡄⠀⣿⡇⠀⣿⡏⠀⣾⡿⠀⢰⣿⠏⠀⣰⣿⡿⡁⠀⣠⣿⣿⣿⠇⠀⢀⣴⣾⣿⣿⠿
⠀⠙⢿⣿⣦⣹⣿⣿⣿⣿⡈⢿⣧⠀⢻⣧⠀⢿⡇⠀⣿⡇⢠⣿⠃⢠⣿⣏⣠⣶⣿⣿⣦⣷⣾⣿⡟⣿⣿⣧⣶⣿⣿⣿⣿⣷⣶
⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣷⠎⢿⣆⠈⣿⡀⢸⡇⠀⣿⠀⣼⡏⢀⣾⠏⢈⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣦⠈⢿⡆⢸⣇⠸⣇⢠⣿⢀⣿⠁⣾⠏⢠⣾⠛⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣷⣾⣿⣤⣿⠀⣿⢸⡇⣸⠇⣼⠏⣰⡟⢁⣴⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣀
⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⢳⠈⡇⠛⢸⠁⡟⣰⢋⡼⢋⣴⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀⠁⠀⠀⠀⠀⠀⠁⠞⡡⠞⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠁
⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣯⣤⣤⣿⣿⣿⡿⢻⣿⣿⣿⡀
⣀⣴⣿⣿⣿⣿⣻⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠈⠩⢯⣩⣉⣉⣽⣿⣿⣿⣿⡟⠀⠈⣿⣿⣿⣷⡀
⠿⣿⣿⣿⠟⠉⣉⣙⣿⣿⣿⣷⣦⡀⠈⠙⠛⠻⠉⠀⠀⠀⠀⠀⠀⠀⠒⠶⣤⣬⣉⣿⣿⣿⠿⠻⠿⠛⠛⠛⠿⢿⡄⠻⣆
⣤⣿⣷⣿⠿⠟⠛⢉⣈⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢤⣙⠳⢦⣤⣉⣿⣿⠻⢿⣶⣦⣤⣀⣀⠀⠈⠃⠀⠙⣷⣄
⠛⠉⣽⣁⣤⣶⠿⠛⠉⣸⠿⠛⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⢦⣌⠻⢦⣌⠙⠻⢷⣦⣤⣀⠉⠉⠙⠛⠛⠿⢷⣖⣦⣌⣿
⡶⢾⣟⣉⡥⠀⣠⣾⠆⠀⠀⠀⠀⠙⣿⣇⠀⠀⠀⠀⠀⢠⠐⣆⠉⠀⠙⠷⣄⠙⠿⢶⣄⠈⠙⠻⢿⣶⣦⣄⣀⡀⡀⠀⠉⠙⠛
⣰⣿⡿⠋⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠘⠁⠙⠀⠀⠀⠀⠙⠀⠀⠀⠀⠁⠀⠀⠀⠙⣿⣿⣿⣿⣿⣶⣄⣰
⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣄⣤⣤⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⢹⣧⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣿⣿⣦⣈⢹⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⡀⢀⠀⣹⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿
⣿⣏⣁⡁⠁⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠉⠁⠀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠂⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⣿⡿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⢽⡿⢿⢿⠿⢿⠿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠙⠃⠀⠀⠉⠛⠻⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡿⠋⠙⣿⣿
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠿⠋⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠋⠀⠀⠀⠀⠉⠀⠀⠀⠘⢿⣿
⢀⣀⠀⠀⠀⠀⣀⠤⠤⠠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢏⣉⡇⠀⣠⠞⠁⠀⣀⣀⠀⠈⠑⢆
⠈⠉⠀⢠⠃⠀⣴⠛⠉⠉⠙⣦⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠆⠀⠀⢀⡀⢠⡄
⠀⠀⠀⢸⠀⠀⣯⠀⠀⠀⠀⣸⠇⠀⡇⠀⠀⠀⠀⠈⣅⠗⠀⢦⠰⡂⢻⠀⢢⠆⠀⠸⠀⠇⠉⠠⠣⠘⠆
⠀⠀⠀⠀⠳⡀⠙⠷⣤⣤⡴⠋⢀⡜⠀⠀⠀⠀⠀⠀⠁⠀⠙⠈⠀⠀⠈⠀⠘
⠀⠀⠀⠀⠀⠉⠢⠤⣀⣀⠠⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣏⡀⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠘⠻⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠁⠀⠀⠀⠠⣤⡖⢀⡀⠀⢀⡄⣴⣦⡀⠀⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⣄⣀⠀⠁⠛⠘⠃⠀⠨⠄⠻⠛⠁⣀⣤⣶⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⡶⠦⠤⠤⠶⠶⠾⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣿⡿⠟⢛⡉⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⢹⣿⡟⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣬⣿⡇⠀⠈⢡⡀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⣾⣿⣷⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡀⢀⠄⠀⠀⠀⠘⠀⠀⠀⠀⠤⢀⣼⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⠿⠿⠻⣿⣿⣿⡶⣄⣀⠀⠀⣤⠀⠀⢀⣀⡴⢻⣿⣿⠿⠋⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠉⠛⣇⣀⣉⣙⣗⠒⣲⣏⣉⣀⣀⡼⠋⠀⠀⠀⢀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠠⠤⠤⢄⣀⠀⢴⣏⠉⠀⠀⠉⢻⠉⠀⠀⠀⠉⠀⢀⡠⠖⠋⠉⠀⠀⠀⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣧⠀⠀⠀⠀⢀⡼⠓⢾⣿⣄⣀⣀⣀⣼⣄⣤⣤⡤⠶⣏⠉⠳⡄⠀⠀⠀⠀⠀⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⢠⠏⢀⡴⠁⠀⢉⣧⠠⠴⠶⢀⠰⣏⠁⠀⠈⣧⠀⢹⡄⠀⣀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⠤⡿⡷⠶⠶⢾⡇⣀⣉⣉⣉⣁⣈⣉⡛⠋⢹⡛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠧⠤⠤⠤⠤⠤⠤⠤⠴⠶⠒⠒⠒⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⣀⣀⣀⣀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣟⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠓
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠀⠀⠈⠙⠻⢿⣯⡙⠻⠿⣿⣿⡻⢿⣿⣿⡿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠉⠀⠈⠉⠛⠈⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⢀⣤⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠶⢦⡀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡁
⠀⠀⠀⠀⠀⠀⡠⠿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⠔⠸
⠀⠀⠀⠀⠀⠀⠀⠰⢌⠣⠀⠀⠀⢀⡤⠶⠦⢄⠀⠀⠀⠀⠀⠀⢀⣤⠴⠦⣄⠀⠀⠀⢼⠁⠀⠸
⠀⠀⠀⠀⠀⠀⢃⠀⠀⠂⡅⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⡆⠀⣀⠂
⠀⠀⠀⠀⠀⠀⠈⠢⣀⡠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⢾⣭⣔⣒⣂⣭⣿⠇⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠈⠓⠤⠉⠿⠟⠃⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠄⣀⠀⢀⠤⠒⠈

⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣟⣿⣟