Pugilist Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣟⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣉⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠶⠉⠉
⠀⠀⠀⠀⠀⣤⠈⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣖⢦⡀⠀⠀⠀⠀⢀⡠⠔⠉
⠀⠀⠀⠀⠀⣿⣷⡀⠀⠰⣤⡀⠀⠀⡀⠀⢀⠹⢿⣿⣿⠟⠛⠲⠶⠤⠟⠋
⠀⠀⠀⠀⠀⠙⣿⣿⣆⣤⠈⢻⣦⠀⠹⡄⠛⠘⢳⠆⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣧⡀⠛⠂⠀⢀⣀⣴⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⡴⣿⠿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡶⠉⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⡀⠀⢀⣠⣴⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣯⣤⣤⣶⣾⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⢿⣿⠏⠀⠀⠙⢿⣿⡉⠀⠠
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⢰⠆⠀⠀⠀⠀⠈⠻⣿⣤⡀⢠
⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣭⡁⢰
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⢀⣿⡟⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⢸⣿⡃⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣀⡀
⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣏
⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄
⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣤⣤⣤⣠⡀
⠀⠀⠀⣰⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠈⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⣴⠀⠀⠀⢀⡞⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣆⠀⢸⣄⠀⠀⠸⡰⠋⡄⠀⢀⡞⡴⠻⠀⠀⠀⠀⡰⠟⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⠀⠀⠙⣦⣄⠈⠆⠀⠉⠓⢄⡰⠁⠀⠘⣀⡜⠁⠀⢀⠀⠀⠀⢀⣠⡖⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠘⠀⠉⠒⠲⠀⢠⣀⠀⡰⠑⢄⣈⠠⠒⡀⢸⡀⠄⠒⠉⠸⠀⠀⠀⠚⢁⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⣤⣀⠀⠀⠈⠙⢶⠦⣤⠤⠄⢢⠂⠈⠀⢤⢂⣠⣉⣠⠒⢄⣀⠄⣄⠈⠈⠠⡄⠠⠠⣇⠀⠀⠀⢀⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣟⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠱⠄⠀⠀⠀⠁⠀⠀⠀⢤⠀⠀⠀⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠈⠀⠤⣾⣇⠀⡤⠀⠂⣀⠀⣤⡼⢣⣤⢀⣀⠐⠀⢤⠀⣸⣷⠴⠀⠁⠘⢇⠒⣢⣤⠤⠄
⠀⠀⠀⠤⠤⠤⠤⠤⠤⠶⠶⠟⠓⠒⣿⠁⠀⠀⠀⠀⢩⡿⢦⡳⠄⠀⠆⠀⠀⠙⠋⠀⠀⠰⠀⠤⢞⡴⢿⡍⠀⠀⠀⠀⠈⣿⠓⠒⠲⠶⠶⠶⠶⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢹⡀⠀⠀⠀⠀⢚⣷⣤⡄⠀⠤⠠⠤⠤⡜⢣⠤⠤⠤⠀⡀⢄⣤⣾⡓⠀⠀⠀⠀⢀⡏⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⡠⠛⢻⣅⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣠⡟⠻⢤⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠁⠠⠊⠀⠀⢾⡉⠉⠒⠐⠶⠦⡔⠃⠈⢢⠴⠶⠒⠒⠉⢉⡷⠀⠀⠑⠄⠈⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠒⠒⠻⠧⣀⣀⠀⠀⠀⠀⠁⠢⠄⢀⣀⡤⠓⡄⡈⠛⢤⣀⡀⠠⠔⠘⠀⠀⠀⠀⣀⣀⢼⣟⠒⠒⠤⠄⣀⡀
⠀⠀⠀⠀⠠⠖⠒⠈⠁⣀⠄⠀⠀⠀⠀⢀⡠⠀⢹⠃⠲⠆⠠⠸⡁⠙⢁⠄⡚⢌⢁⠤⠤⢠⠋⠸⠓⠤⠄⣐⣶⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠂⢀⣤⠞⠉⠀⢀⠆⠀⢀⡠⠔⠆⠈⠖⠁⣰⢆⠀⠁⢠⠶⣀⠀⠀⡀⠀⠀⡀⠉⠓⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⢀⣬⡤⣢⠟⠀⠀⢠⠀⣠⢞⡟⠀⠱⣀⠆⠀⠹⡕⢤⣇⠀⠀⠈⢢⡄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣁⡴⠃⠀⠀⠀⢸⡶⠁⢾⡇⠀⠀⠹⠀⢠⠀⢻⣆⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠉⠉⠉⠉⠉⠓⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣄⣠⣤⣠⣤⣤⣤⣀⡤⠆⠀⠀⣤⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢰⠁⠛⠀⠀⠀⠨⠽⠞⠀⠰⠞⠃⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⡐⠉⠉⠉⠉⠉⠙⠲⢤⣀
⠀⠀⠀⠀⣀⣀⣠⠤⠤⠤⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠳⠄⣀⡀⢀⣠⠞⠀⠀⠀⢰⡦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠰⠶⠴⠶⢤⣀⡀
⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⠀⠀⠸⠆⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⡀
⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠲⠶⠶⠂⠀⠈⠀⠀⢸⡏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡀⠀⠀⠠⠒⠒⠀⠀⢀⡠⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠁⠒⠢⠤⣀⡠⠴⠛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠶⠶⠀⠀⠀⠀⠀⡞
⠃⠀⠀⠀⠀⠀⠚⠛⠒⠶⢾⣄⠀⠀⠀⠀⠀⠀⡤⠖⠛⠉⢛⡆⠀⠀⠀⠀⡀⠀⠀⠀⢀⡴⠶⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠼⢇
⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⢈⡴⠒⠉⠉⠉⠓⢦⡄⠀⠐⣧⠴⠞⠛⠛⠓⠶⢦⣄⢻⡀⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠙⣆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠻⣇⣀⣀⣀⣀⣀⠀⠀⣈⣤⣀⣀⣀⠀⠀⠀⢀⠀⠀⠀⠘
⡜⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡈⠉⠉⠉⠻⢏⠙⣌⠩⠉⠉⠛⠢⣀⠘⠀⠀⠀⢠
⣇⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣰⠏⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠛⡴⠉⠀⠃⠀⠀⠀⠈⢳⠀⠀⠀⢨
⢹⣆⠀⠀⠈⠻⠆⠀⠀⣠⡞⣿⠋⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⢃⣴⠀⢠⠀⠀⠀⠀⠀⠁⠀⠐⣧
⠀⠻⣦⠀⠀⠀⠀⠀⡘⠏⠀⡇⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠬⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢸⡽⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠈⣷⡄⠀⠀⠠⠁⠀⠀⢁⡀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⡶⢴⡾⠇⣠⡇⠀⠀⠀⠀⠀⠀⣠⡤⠴
⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠳⠈⢧⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⣠⠏⡠⠖⠛⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣴⠁⠀⠀⠀⠈⠙⠛⠒⠒⠒⠚⠛⠉
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠃⡀⠳⣤⣾⠁⠙⠲⠆⢀⡤⠶⠞⠁⠀⠀⠀⠀⣡⠀⠙⢶⣤⣀⣀⡀⢀⣠⡤⠞⢁⡼⠃
⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠈⣰⡟⠀⠙⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⢿⠶⠶⣤⡀⠉⠉⠉⠉⠀⠀⠐⢛
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠈⠛⢧⣀⡾⠛⠉⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠴⡿⠤⠴⠦⠤⠀⠀⠀⢀⡆⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⣠⡀⠴⠶⠶⠷⠶⠶⠶⠶⠦⣤⣤⣬⣅⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠠⠐⠒⠒⠒⠒⠒⠒⠢⢤⣤⣤⣀⣀⡀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢳⣄⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠹⡆⠉⠀⡇⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⠛⢲⡶⠤⠶⠶⢶⡶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠴⠄⠀⠀⡶⠀⠐⠁⣰⠏⠀⠀⠀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠹⡇⠀⠀⠀⡽⠤⣤⣄⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⣿⠀⠀⠀⢣⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠐⠛⠳⠲⡟⠀⠀⠀⠈⣷⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠐⢸⡇⠀⠀⠀⣴⡟⠀⠀⠙⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠈⢸⡇⠀⠀⠸⣧⣽⣄⣀⣀⢀⣀⡤⠛⠷⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠻⣄⡀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠒⠒⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⡄⠀⠀⠈⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣴⡀⠀⠀⠀⠀⠀⢉⣳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⣄⣰⡋⠀⠀⢹⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣔⣒⣖⣠⣄⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣾⣿⣿⣿⣿⣽⠂⠀⠀⢠⢄⣤⠙⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⢉⠈⣩⣥⡞⡍⠝⢿⣿⡿⢸⡇⢉⣿⣯⣿⣴⣻⣿⣿⣟⡤⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⣶⡴⠛⢿⣀⣀⣻⡇⢀⣹⠟⠁⠈⠉⠉⣉⠹⢶⣌⢿⣿⣟⣎⠲⣿⣿⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⣲⣶⠘⠛⠛⠛⢁⣠⣴⡆⠀⢀⣾⡏⣽⣿⠄⣾⣿⣷⢬⠛⢳⡝⣧⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡄⠀⡾⠿⠒⠂⠀⣤⡶⠟⠋⠙⠻⢷⣮⣭⡻⠿⡏⣴⣿⡿⢟⣤⣤⣾⠁⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠚⢳⣄⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠈⠙⠛⣥⠆⢈⣭⣝⡄⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣈⢛⣷⣍⣳⣶⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡺⣿⣿⣿⣿⡿⠛⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⣽⢿⣿⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢈⣽⠻⠻⠗⢀⢞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠁⣿⡏⣻⣯⣉⣇⣝⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠋⠒⠟⣩⢄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⢑⢿⣬⣶⣿⣯⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠠⠐⠀⢚⡉⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⢸⢧⣭⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠐⠩⠼⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠿⣸⣿⡏⢿⡿⠿⣿⣿⣿⣥⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠈⢿⠁⠸⡶⠿⠏⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⠅⠀⠀⠀⠀⠉⢻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠷⢶⣾⣿⣗⣒⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⣉⣿⢳⠏⠀⠀⠀⠀⠀⠀⠈⠛⣿⣼⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣿⠀⠀⠈⠛⠛⠋⠹⠟⢿⣿⣆⣤⠄⢀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢾⣷⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣀⡄⠀⠀⠀⠀⠀⠀⠀⢠⠀⢱⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠅⡀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣋⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠓⠄⠀⠀⠀⠀⠀⠀⠰⠦⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢪⢻⡄
⠀⠀⠀⠀⠀⠀⢀⣾⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⣁⠀⠀⢀⠀⠀⠀⠘⠖⣨⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁
⠀⠀⠀⠀⠀⡀⠚⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⢀⣀⣀⣠⣦⣿⣿⡆
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⠌⣟⣛⣛⣛⣛⢛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣦⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣤⣸⣿⣿⣿⡏⢰⣾⣿⣿⣷⣀⣀⣤⣶⣛⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣿⣻⣿⣿⣿⣿⡶⠶⣾⣿⣿⣿⣿⣿⣿⣟⠉⠈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⣿⣿⣿⣛⣋⣍⣼⡇⢈⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡟⠉⠉⢻⣾⣿⣍⢻⣿⣿⠿⣿⣽⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣾⣿⣿⣿⣿⡏⠀⠀⠀⠘⠿⠿⣿⣮⣽⣿⣴⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣽⣿⣽⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢀⣈⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⢊⣿⣿⣿⠿⠛⢻⡿⠿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⢿⣿⣧
⠀⠀⠀⠀⣀⣾⣼⡿⠋⠀⠀⠀⠸⠷⠖⠒⠻⢿⣷⣽⣦⠀⠀⠀⠀⣿⣿⣿⠿⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣰⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡀⠀⠀⠀⣿⣿⣿⠀⠀⠈⠻⠛⣿⣿⣿⣄⣀
⠀⠀⢠⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⣯⣤⣀
⠀⢀⡿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣷⣦⣤⣄
⢰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⢠⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿
⢾⣩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⣀⣀⣀⢀⣴⣟⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣴⣼⣿⣿⣿⣿⣿⣿⣿⣿⣟⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢠⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣟⣧⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢿⣧⣄⣀⡀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠙⠉⠁⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⢿⣆⠈⢶⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⠳⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⢸⣿⠃⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠏⡸⠃⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⣿⣫⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠋⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣟⣋⣁⣠⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠛⣻⣿⣿⣿⣿⣿⣿⣿⣷⡶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠋⠉⢲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠁⠀⠀⠀⠈⠁⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢟⡡⠔⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣾⠿⠛⠀⡀⠀⠀⠀⠀⠀⣀⣡⠤⠤⢶⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣗⣀⣐⣠⠄⢀⠀⠐⠊⢉⡡⠴⠒⠋⠉⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⢋⣤⣒⣁⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠙⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⡠⢀⠠⢂⣁⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠉⠛⠻⢯⣟⠳⠶⣤⣄⣀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣔⡼⠸⠐⡀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣗⠒⠠⠀⠀⠈⠉⠓⠶⠤⢥⣽⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠞⡂⡁⢀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣥⣉⠙⠀⠀⠀⠀⠀⠀⠀⢄⢠⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡜⠸⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⠓⠯⡁⠀⠀⠀⢈⠀⠂⠰⢹⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠁⢪⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠾⠿⣓⠶⢤⣀⠀⠀⠀⠀⣁⣾⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⣨⠠⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠉⠀⠀⠀⠁⠀⠈⣹⠓⠲⠦⢬⢿⣿⣿⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣴⠯⠁⠉⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠈⠓⠺⠽⣿⡋⠈⣇⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢿⡀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⣼⠗⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡇⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⢸⠁⠀⠀⢹⡀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢳⣀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠉⠓⠲⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋
⣰⠃⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁
⡏⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠱⣄⣤⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠰⠒⠋⠉⠁⠉⠉⠒⠒⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠈⠂⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠂⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⠤⢀⣀⢀⣀⣀⡀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠛⠛⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⡄⠲⠄⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡁⠀⢀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣛⡻⢷⠟⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠎⠁⠀⠁⠀⢠⣾⣿⣷⢠⣇⠘⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⡀⠀⢀⣾⣿⣿⣿⣿⢿⣦⡀⢰⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠈⢻⣶⠎⠻⢟⠈⠁⠠⠀⠈⠋⢍⠀⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠁⠀⡀⢨⡭⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⢀⣼⣧⣤⣤⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣯⣛⣿⣷⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣥⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣵⣆⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠛⠛⠿⡿⠃⢿⣿⣿⡅⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⢀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⣄⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⡅
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⢥⠉⡈⢢⠀⠀⢠⡲⡂⢆⠐⡒⢦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢠⡌⣤⡷⠏⠘⡇⠀⠀⠁⡧⠈⠀⠑⢾⣠⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣛⠫⠐⡠⠂⠁⠀⠸⣅⢇⢧⡀⢀⣀⠳⠃⣯⣿⠀⠀⠀⠀⢀⣄⡄⠰⢐⣖⡞⢇
⠀⠀⠀⠀⢀⢤⠤⡄⡤⢀⡀⠀⠈⢧⡡⠤⠐⡿⠁⠀⡆⠀⠀⠙⠶⣷⣛⠩⠭⠭⠉⢻⡏⠀⠀⣴⣶⣿⣸⠃⠀⣎⠿⣃⣿⠟⡦
⠀⢀⣶⣿⣿⣾⣼⣴⣷⠤⠁⢺⠀⠀⢳⣦⠀⣿⡀⢾⠀⠀⠀⠀⠀⠘⣖⢿⣍⡟⣝⡯⣇⠀⠀⢳⣿⣿⣾⣶⣶⣿⣷⣾⣿⣼⠃
⠀⢸⣿⡟⢹⣒⣒⣻⢻⡄⠀⡺⠀⠀⠐⠀⢠⠏⠢⣿⠀⠀⠀⠀⠀⠀⢸⣎⣿⢻⡳⣽⣽⡄⠀⠀⠉⣿⠃⡀⢀⢀⣼⡿⠃⡍
⠀⠈⠻⡆⠘⣶⣶⣾⣾⡟⢩⠁⠀⠀⢠⣀⠏⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⢇⢊⠎⡓⡓⣷⡀⠀⠀⢹⠀⢌⣉⣵⠋⠀⢠⠁
⠀⠀⠀⡇⠀⠀⠂⠀⠀⠑⣜⠀⠀⠀⢸⠃⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⡟⠉⠀⠀⣀⣀⠀⢳⠀⠀⠀⢷⡴⠟⠳⢬⣻⠋
⠀⠀⠀⢹⣶⣤⣀⢀⠤⣠⡋⠀⠀⠀⣾⡀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠸⣧⠒⡏⣹⣦⣦⣠⣹⡄⠀⠀⣾⡑⠠⣀⡀⢹⡄
⠀⠀⠀⠸⣧⡒⠠⢔⣋⣸⠃⠀⠀⠐⣟⢷⢀⣠⡦⠋⠈⡇⠀⢀⡀⠀⠀⣿⠶⠛⠉⠀⢀⠉⠀⡇⠀⢠⡇⠸⣲⢽⡶⠚⡇
⠀⠀⠀⠀⣿⣻⣦⣄⡈⢹⡄⠀⠀⢰⣟⢝⠓⢅⢀⠀⠀⢸⠀⠀⠀⠉⠳⢿⠀⠀⠀⠀⠈⡀⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⠋⠀⠀⠈⠉⡇⠀⠀⢸⡁⠀⠑⢤⡁⠓⢠⣈⡀⠀⠀⠀⠀⠈⠁⠀⠀⢀⡸⠃⠀⡇⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠁⠀⠀⢸⢇⣀⠀⠀⠉⠲⠒⠙⣇⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣢⣀⣀⡀⣤⢤⡴⠞⣿⠀⠀⠀⠀⢀⣀⣀⡤⠤⠤⠄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠴⡧⠦⠿⠞⠓⠉⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠛⠋⠁⠀⡠⢤⢀⠀⠀⠀⠈⠙⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠃⠀⠀⠀⠀⠀⠀⠄⠀⠎⠀⠀⠀⠀⠘⡣⣴⣿⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠂⠀⠀⠀⠀⠀⠑⠒⠦⣀⣄⣀⠀⠀⠀⠀⠋⠀⠀⠀⣀⠀⠀⣀⣀⡄⠀⠀⠀⢴⣿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢤⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠈⠏⠁⡄⠀⠀⠀⠀⠒⠺⠋⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⠃
⠀⠀⠀⠀⠀⠀⠸⡖⠿⢏⠂⠀⣃⢐⡧⡤⠀⠀⡀⠀⠀⠀⠰⡎⠀⠀⠀⠀⢉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢄⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠘⠓⠞⠓⠁⠀⠀⠼⣆⠀⠀⠀⢣⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⣀⣀⣀⡀⣄⠠⣴⣤⣤⠤⠤⠤⠶⠂⠁⠀⠀⠀⠘⠂⣐⣒⡘⠟⠄⠀⠀⠀⠀⠀⠀⠀⡄⢀⢠⣤⣬⣤⠄⠁
⠀⠀⠀⠀⠈⠯⠼⠋⠺⣄⡡⠘⠉⠻⣿⣿⠦⠀⠀⠀⠀⠀⡰⠒⠂⠏⠉⠱⠂⢣⠀⠀⠀⠀⠀⠀⠀⠈⢺⣿⣶⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢹⣟⡀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⢀⠀⢢
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦⡔⠈⠁⢀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⣿⠻⠀⢀⢴
⠀⠀⠀⠉⡁⠀⢐⡀⠀⠀⠀⠀⠀⠠⢿⣿⢛⣃⠐⢮⠁⠹⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀⠀⠀⠐⠀⡜⢓⠗⠙⠃⢠⡌⠀⠁
⠀⠀⣧⣠⠇⠀⣚⡀⠀⠀⠀⠀⡄⠀⢰⣿⣭⣟⡁⣺⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣄⣠⢀⠀⠀⠀⠀⢀
⠀⠀⣬⣭⡁⠀⠢⠀⠀⠀⢀⠼⠀⢐⣰⣹⣿⡿⢟⡦⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠉⠉⠉⠉⢀⠛⠉⠀⠀⠀⢀⠀⣠⣶
⠀⠀⠻⠿⠇⠀⣀⠀⠀⠀⠈⠒⠀⢨⣿⣇⣿⣟⢸⠇⢻⠀⠀⠀⠀⠀⢠⣄⠀⡒⠷⠁⠀⠀⠀⠀⠀⢰⣿⠀⠀⢠⢠⡞⢴⠶⠛⠁
⠀⠀⠀⠲⠆⠀⣒⡂⠀⠀⠀⠀⠈⠁⠚⠛⣹⡧⢸⠁⠘⠀⠀⠀⠀⠀⠀⠙⠳⣍⠃⠀⠀⠀⠀⠀⠀⠘⠛⠁⢠⣇⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠃⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠋⠉⠉⠉⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⡀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠘⢀⠂⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣂⣀⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡷⠀⣀⣨⣦⠉⠉⠉⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠿⣹⣷⣈⡙⠻⣷⣇⡀⠂⣸⠏⡀⣴⣆⣙⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠋⠂⠀⢀⣿⣿⣿⣿⣷⣮⣻⣷⡼⠋⠀⣴⣿⣿⣿⣦⡌⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⢻⣿⣿⣿⣿⣷⣶⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡟⠉⠉⠀⠀⢰⣿⣯⣿⣿⣿⣿⣻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⣰⣿⣿⢷⣿⠏⠀⠀⠀⠀⠀⠹⢿⠿⡟⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠂⠀⠀⠀⢀⣿⣿⣟⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⢻⣿⣿⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣠⠞⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣸⣿⣿⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢀⣴⠞⠁⠀⠀⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⠀⠘⠛⠛⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠞⠁⠀⠀⠀⣠⡿⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⢀⡭⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣟⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢀⣠⣬⣍⣡⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠳⣄⠀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣿⣿⡿⠫⢠⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡍⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠐⠺⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠡⠲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣇⣐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣠⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⣦⣄⣀⣀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣏⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣥⡾⢃⣴⢟⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⠟⣡⠞⠿⣻⣿⣿⠟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠹⠿⢋⣿⣿⡿⣫⣶⠟⠁⠀⠁⠀⠰⠟⢸⠏⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠉⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢀⠀⣀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⡆⠀⠀⠀⠀⢀⡀⠹⣤⠄
⠀⠀⣠⣦⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣤⡀⠀⠛⠿⣟⠁⠀⠀⣀⡖⠉⢀⣾⣾⣿⣶⣦⡀
⠀⠀⡅⢩⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠛⠛⢉⣇⠀⠀⠀⢈⠘⠀⢠⡇⠀⠀⣤⠻⣿⣿⣿⣿⣿⡆
⠀⢀⡄⢈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠡⣄⠀⠈⠋⠉⠀⠀⣠⣎⠀⠀⠈⠁⠀⠠⠈⠀⢻⣷⣿⣿⣿⠁
⠀⠸⣧⣤⣸⣷⠀⠂⠀⠀⠉⠀⠀⠀⠀⠀⠀⢹⣷⣴⣿⣇⣠⠀⣿⣾⣇⠀⠀⠀⠀⠰⠤⡀⠈⠹⠿⠟⠁
⠀⠀⠘⠳⠄⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡟⢹⣿⣧⣈⡿⠛⠠⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠉⠉⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄⠀⡀⢀⣄⣴⢿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢀⣼⣿⣛⣿⠏⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠀⡆⠀⠀⣶⣄⠀⠀⠻⣦⣀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⣼⠻⣿⣿⣿⣿⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⡁⠠⣦⣼⠟⠉⠳⣦⡀⢘⣿⣿⣖⣖⠉⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣮⣬⣥⣿⣿⣿⣿⣏⡀⠀⠀⠀⠀⠀⠀⠹⠖⠻⠛⢺⣮⠛⢷⣀⣿⣿⡆⠀⠀⠈⠙⠿⠿⠟⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⣠⣴⡞⠛⢿⣾⣿⠀⠙⢟⠋⠛⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣘⣿⡿⣿⠃
⠀⠀⠀⠀⠀⠐⠋⣰⠇⠀⠈⢿⣿⠀⠀⠰⣷⣄⡙⢻⣿⣿⣿⣿⣆⣠⠖⠃⠀⠀⠀⠀⠀⠀⠙⣿⢷⣟⡷⢄
⠀⠀⠀⠀⠈⠒⠋⠄⠀⠀⠀⠙⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣭⣉⣚⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢲⣿⡿⠿⢿⡟⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣌⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⣿⣿⣴⣤⣎⣿⣿⣿⣿⣿⡟⢡⣤⣴⡾⠀⣄⣰⠇⠀⠀⣠⣾⡿⣻⣿⣿⣧⡀
⠀⠀⠀⠂⠀⠀⠃⠘⡇⠀⢰⣿⣿⣿⣹⣥⣤⣽⣿⣽⣿⣤⣤⣄⠉⢰⠿⢋⣵⡆⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀⢀
⠀⠠⠀⠀⠀⢀⠀⢰⡀⢀⣼⡿⡷⢁⢨⣿⣿⣎⣼⣿⣿⠟⠉⢹⣶⣾⣷⢟⣻⣿⡿⢛⣋⣏⣾⣿⣿⣿⣿⣿⣧⢰⣦⣤⣶⣤⣤⣶⣄
⠀⠐⠀⠐⡄⣏⠙⠸⢿⣿⣿⠁⣿⠿⣿⣿⣏⣿⣏⣿⠇⠀⢀⣼⡿⠿⣷⣿⣿⣷⣾⣿⣿⣿⣽⣿⣿⢿⣿⣿⣿⡟⢳⣿⣿⣿⣿⣷⣿⠂
⠀⠀⠀⠀⢘⡁⠀⠀⢸⣿⣿⠀⠛⢩⣽⣿⣿⣙⠝⠉⠀⠀⣿⣿⣧⠀⢽⣿⣿⣿⣿⣿⣿⡿⣯⡿⠁⠈⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⡘⢃⣄⣾⣿⡟⠀⢠⣯⣼⣿⣿⠋⠀⠀⠀⣸⣿⣿⣿⡀⢈⣿⣿⣿⣿⣿⣟⣰⣯⠁⠀⢀⣿⣃⠀⠀⠀⠀⣻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢁⠚⢿⣿⣿⣧⠀⡼⣿⣿⣏⣼⡆⠀⣠⡶⠋⢻⣿⣿⡆⠊⣿⣿⣿⣿⣗⡿⢷⢻⡇⣠⣼⠟⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠘⠐⢸⣿⣿⣻⣿⣿⣿⣿⣯⡏⠀⣸⡟⠀⠀⠈⣿⣿⣷⣀⣿⣿⣿⣿⣿⣷⣇⠞⢀⣿⡏⠀⣿⣿⣿⣿⣿⣿⢿⣽⠙
⠀⠀⠀⠀⠀⠠⣸⣿⣿⣿⣿⣿⣿⣿⢯⡟⠁⣰⠋⠀⠀⢀⣀⣏⣉⣙⣻⣿⣿⣿⣿⣿⣟⠿⢃⣼⣿⣐⣂⣿⣿⣿⣿⣿⣛⠿
⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⠟⠋⢁⡼⢀⣼⣇⣤⣾⣿⣿⡿⠟⠛⠋⠉⠀⠉⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣿⣷⣶⣤⣶⣾⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠓
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣀⡄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠠⣀⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠁⠀⠀⠈⠓⠄⣀⠀⠈
⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⢀⣀⠤⠀⠀⠂⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⡐⠱⠒⠀⢠⠀⠂⠤⠀⠐⠒⠂⢠⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⡐⠒⠐⠈⠲⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠔⢻⢴⡤⢄⠀⠀⢀⠴⠄⠀⠁⠀⠀⠘⡁
⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠐⠉⢲⣄⠞⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡜⠦⢴⡏⢰⠀⠈⠀⠀⠀⠰⠄
⠀⠀⠠⢀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⢁⢀⠈⠢⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⡀⢀⠀⠀⠒⠀⡄⡙⠛
⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⡀⠁⠀⢠⡤⣄⠉⠀⠇⠀⠀⠀⠀⢀⠀⡀⠀⢀⣀⠀⡄⣄⠂⠀⠀⠀⡘⣄⠀⠀⠀⠀⢀⣂⣀⣠⣆⠈
⠀⠀⠀⠀⡇⠀⠀⠀⡘⣠⢀⡀⠀⠸⠀⡀⢰⠞⢐⢀⠀⠠⠐⡢⣄⡔⠈⠀⠤⠅⠄⠈⢣⡀⢀⠌⠵⢶⣥⠦⠤⠅⠊⠥⠄⢄⠈⡇
⠀⠀⠀⠠⠁⡀⠀⠀⠀⣴⠈⣀⡤⠒⠂⠡⢭⠴⠿⠬⡌⠀⠀⢠⠀⠀⠄⠀⠀⠀⠀⠀⠒⠁⠀⠑⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠑⠚⠉⢀⠊⠀⢀⠊⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⣠⢀⢀⡀⠀⢀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡀⣤⠔⠀⠞
⠀⠠⠀⠀⠀⠀⠀⢀⠴⠁⠀⠀⠀⡀⠢⡤⢤⡤⢀⡀⠦⠤⠔⠒⠚⠁⠀⠀⠀⠀⠈⠚⠉⢲⠦⠞⠂⠀⠈⠉⠁⠀⠀⠀⠰⠀⡧
⠀⠀⠀⠀⠀⡤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠣⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢀⠄⣀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⠈⠆⠀⠐⠤⠀⠀⡀⠄⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣀⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡆⠐⠂⠉⠁⠀⠐⢶⡶⠶⠾⡆⢀⠄⠀⠀⠀⠀⠄⡀⢘⣄⠀⠀⠀⠀⠀⠀⢠⡄⠒⠛⠤⠀⠠⡄⢨⠟⠓⠒⠃
⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠘⡄⠀⢰⣰⠋⠘⢰⠆⢀⠃⠊⠈⢁⠀⠳⢄⡀⠀⡠⡲⠋⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⢀⢀⠴⠒⠁⠀⠀⠀⠀⠀⢀⠇⠀⠀⠉⠒⠀⠈⠢⣀⠁⠀⠀⠀⠁⢠⠊⠉⠉⠘⣤⡀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠘⠁⠈⢄⠀⠀⢀⡀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⣀⠊⢀⡠⠴⠘⠁⠈⠑⢒⡶⠤⣄⣀⡇
⠀⠄⠀⠀⠀⠀⠨⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢁⠊⠀⠈⠀⠀⠋⠀⠐⠟
⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢀⡀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠈⠛⠉⠃⠁⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⢀⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠂⠀⠀⠒⠤⡀⢀⡀⢀⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠐⠠⠂⠀⠠⠒⠂⠀⠀⠀⠚⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠄⠀⠀⠀⠀⢠⠖⠉⠁⠀⠀⠀⠀⠀⠀⢀⠴⠰⢄⠀⢢⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠛⠷⠚⠀⠀⠀⠀⠁⠀⠀⠀⣀⠜⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠺⣄⠀⡴⠒⠲⠄⠀⠀⠀⢀⠆⡆⠀⡀⡀⠉⠉⠈⠑⢤⠇⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠈⠂⠡⡠⣀⠀⠄⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⢚⠁⠀⠑⢄⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣀⡰⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢄⠀⠈⢠⣄⠀⠀⠣⠀⠀⠀⢨⠀⠬⠀⠀⠀⠀⠁⢸⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠦⠀⠁⠈⠀⢀⡞⠀⠀⠀⠀⠀⠀⠓⠤⠤⠤⢶⠇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠒⣤⡤⠐⢏⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀⢢⡀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠁⠀⠄⡀⠀⠀⢠⠇⠱⡄⠀⠀⠀⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⠈⠉⠩⠖⠲⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠈⠙⠒⠋⠀⠀⠙⢦⡀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⠀⠀⠔⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠒⠆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢉⠁⠀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⡤⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡠⠤⠴⢁⠀⠀⠈⢰⠦⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢆⠈⢴⠒⠒⠒⡄⡇
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⡄⠐⠁⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡠⠊⠉⠀⠀⠚⡀⢇
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠸⠀⠀⠀⣀⣀⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡀⡇⠀⠀⠀⠀⢀⣰⠎
⠀⠀⠀⠀⠀⠈⠉⠐⠂⠀⠀⠀⠀⠒⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⠛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠎⣠⣾⠖⡳⠐⠒⢂⠄⠊⠑⠠⡐⠒⠂⢞⠲⣷⣄⠱⢤⣀
⠀⠀⠀⠀⠄⢠⠖⡁⠀⢁⣰⡿⠋⠀⠀⠀⠀⠀⠋⠉⠉⠙⠀⠀⠀⠀⠁⠙⢿⣦⡀⠀⢈⠒⠄⠠
⠀⠀⠒⠁⡰⠃⠈⣹⠁⣾⠟⢁⠔⠀⠀⠀⠀⠀⢀⢘⡇⡀⠀⠀⠀⠀⠀⠢⠈⠻⣷⠈⢏⠁⠘⢆⠈⠒
⠀⠓⢀⠀⡇⠀⠀⠀⠀⠉⠀⢀⠤⠀⠀⣤⡶⠄⠘⠘⠃⠇⠠⢶⣤⠄⠀⠤⡀⡀⠈⠰⠀⠀⠀⢸⠀⡠⠚
⠀⠆⠐⢀⡟⠀⠀⠀⠀⡀⠀⣠⣂⣀⣬⣯⣿⣇⢐⢸⡆⡆⣸⣿⣽⣥⣄⣐⣄⠀⢀⠀⠂⠀⠀⢻⡀⠃⠰
⠀⠀⠀⠈⣧⣄⠀⠀⢸⠋⡿⣿⣿⣿⣿⣄⣟⣿⡏⣼⣧⢹⣿⣛⣰⣾⣿⣿⣿⢿⠉⡇⠀⠀⣠⣼⠃
⠀⠀⢳⡄⠈⢿⣶⠀⠘⢸⠹⣷⣿⣾⡿⡻⠟⠉⡠⢟⡻⢄⠉⠻⢟⢿⣷⢿⣾⠏⡇⠃⠀⣶⡿⠁⢠⡞
⠀⠀⠀⠹⣆⠀⢻⠀⢸⠘⡔⣯⡽⣻⡟⣁⣤⡶⣦⣆⠰⣴⢶⣤⣈⢻⣟⢯⣿⢢⠃⡇⠀⡿⠁⣰⠏
⠀⠀⠀⠀⠈⠣⣼⠀⣿⣇⢁⢹⣦⠹⠝⢩⢿⢷⣾⣫⣝⣷⡾⣷⡍⠫⠏⣴⡏⡈⣸⣿⠀⣧⠜⠁
⠀⠀⠀⠀⠀⠀⡏⢠⢿⣟⣼⢠⠃⡟⡀⠀⠈⠵⠾⡧⣽⠧⠮⠁⠀⢀⢻⠘⡄⣷⣻⡿⡄⢹
⠀⠀⠀⠀⠀⠀⠘⠁⠘⢿⢿⠠⣰⠃⠟⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠫⠘⣆⠆⣿⣿⠇⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣾⡁⣾⢦⡐⠀⠀⣰⡶⣿⣿⢦⣆⠀⠀⣂⡴⣷⢈⣷⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢑⣸⠯⣾⣔⢲⡾⢋⣀⣄⣠⣀⡙⢷⡖⣠⣷⠽⣇⡋⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠘⢿⣌⠸⣿⡟⡁⠉⠈⠉⠉⠉⠉⢘⢻⣿⠇⢡⣿⠃⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢊⢿⣆⣼⣷⣤⡼⠖⠒⠒⠺⠧⣤⣾⣧⣰⡿⡑⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠈⠿⣿⡿⠃⠄⠀⠀⠀⠀⠠⠘⢿⣿⠿⠁⠀⠀⠀⠀⢡⡀
⠀⠀⠀⠀⠀⢀⡎⠆⠀⠀⠀⠀⠠⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠄⠀⠀⠀⠀⠐⢱⡀
⠀⠀⠀⠀⢠⠏⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡄
⠀⠀⠀⠀⡏⠀⣂⠀⠀⠀⠀⠀⠁⠀⢳⡘⡧⠀⠀⠀⠀⠀⠀⢼⢃⡜⠀⠈⠀⠀⠀⠀⠂⣀⠀⢹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡉⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠉⠀⠀⡲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⢀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⢯⠄⣬⣁⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠽⠀⠀⠠⣀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣴⠀⠀⠀⠀⠀⠒⠻⡷⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⣧⠘⠀⠀⠀⠀⡓⢉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠞⢻⣶⣤⣤⣀⠀⢶⣶⣃⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣧⡀⠁⢀⠀⡘⡗⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣐⢴⢟⠀⢀⣴⣿⣿⣿⣷⣿⣿⣿⣋⠛⠇
⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⡻⣿⣿⣷⣏⣿⣯⢿⠀⠀⠀⠂⠠⣄⠀⠀⡀⠀⣀⡾⠻⠛⠁⠀⠈⣻⠁⠀⢠⠁⠉⠹⢿⣧⡇
⠀⠀⠀⠀⠀⠀⣜⣸⣧⣤⡄⣼⣿⢻⠿⠿⠙⠉⠉⡀⠀⠀⠀⠀⠳⠾⠿⢳⠶⣅⣀⠀⠀⠀⠀⠀⠛⡀⠀⠀⠀⠀⠀⡏⢻⢦⡔⠀⡀
⠀⠀⠀⠀⠀⠈⠋⣍⠻⠏⣻⡛⣿⣟⠰⡄⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠈⠙⠲⠤⣀⠸⠀⠀⠀⠀⠀⠀⠀⠃⢸⡇⠁⠈⣁⡀
⠀⠀⠀⠀⠀⠺⣇⣿⣿⣄⣮⡿⢿⡛⣹⣧⠀⡄⠀⠀⠆⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⢦⡈⠉⠁
⠀⠀⠀⠀⠀⠀⠟⠃⠀⠉⣹⡟⠸⡇⢾⠟⢃⡄⠀⠀⠸⡀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⡇⢡⠀⠁⠀⠀⠠⠠⠠⠀⠰⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣸
⠀⠀⠀⠀⢘⠲⠄⠀⠀⣰⢿⠐⡀⠀⠀⠀⠀⠃⠀⠀⠀⠣⠀⠐⠀⠁⠄⠀⠀⠀⣰⣧⡀⠀⠇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣿⡦⠀⢀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⣰⠦⡄
⠀⠀⠀⠀⠀⣹⣿⡀⣸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣈⠐⠲⣦⣤⣀⣀⣀⣀⣀⣀⣀⡤⠆⠀⠤⠋⠙⠙⢱⡂
⠀⠀⠀⠀⢰⠝⡍⠻⢆⠦⠶⠖⠂⠀⠀⠢⠤⠴⠶⠿⢿⡅⠄⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠃⠀⠀⠐⠀⠀⠴⠂⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠜⠠⠧⠴⠮⠤⠐⠞⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠚⠒⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠴⠚⢫⣍⠀⠀⡇⠀⠀⢀⡀⠉⠻⢶⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠁⠀⠀⠀⠀⠀⢺⠫⠉⠀⠀⠀⠀⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡤⠖⠚⡁⠔⠊⠀⠀⠀⠀⠀⠀⠀⠸⡗⠃⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠉⢁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⣆
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢠⡀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢳⡄⢻⡀
⠀⠀⠀⠀⠀⠀⡿⢷⠀⠀⠀⠈⣧⣿⠁⠀⠀⠀⠀⠀⠀⢰⡀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢸⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢶⣆⣧⠀⢷
⠀⠀⠀⠀⠀⢰⠃⠘⠀⠀⠀⠠⣽⡿⠀⠀⠀⠀⠀⠀⢈⣺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⣿⡿⢠⡿⡆
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢰⣿⠗⠀⠀⠀⠀⠀⠀⢤⣹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣜⣿⡇⡜⢠⡇
⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢜⣿⠋⠀⠀⠀⠀⠀⠀⢤⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢿⡟⠀⣼
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠨⣿⠅⠀⠀⠀⠀⠀⠀⠠⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⢻⣥⡾⢹
⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢐⠸⣇⡀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣌⣿⡟⠀⡼
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢀⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⣹⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⣾⣿⠋⢀⡇
⠀⠀⠀⠀⠀⠈⣿⠶⠄⠀⢀⢸⣯⣀⡀⠀⠀⠀⠀⠀⣀⣼⡄⣀⣀⣀⣀⠀⠀⠠⠴⢖⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣳⣽⣟⠀⣸
⠀⠀⠀⠀⠀⠀⢹⣷⡦⠄⠐⣾⣧⣄⠠⠤⠔⠀⠀⠀⠀⣸⣷⠆⠀⠀⠈⠉⠀⠀⠐⣋⣿⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣠⣌⣿⣿⠯⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠻⣯⣀⣈⣽⣿⣿⣽⣧⣤⠀⢀⣚⣩⣽⣿⣶⣀⣒⣖⣲⠤⣄⣀⣼⣿⡷⣍⡀⠀⠛⠒⠶⠤⠤⠬⣳⣽⣿⣿⠷⠁⢸⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠙⠛⠿⠿⠿⠾⠿⠟⠛⠉⠙⠻⠿⣶⣶⣶⠿⢿⣿⣿⣿⣶⣭⣭⣭⣥⣀⣀⣤⣶⣿⣿⠿⡅⠡⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠉⠀⠀⠀⠈⠁⡈⠀⠙⠏⠉⠙⠛⠿⣿⣿⠙⡇⠀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⣠⠒⠉⠉⠉⠚⠧⣄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣾⡇⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⢤⣾⣴⣯⠆⡀⠀⠀⠀⠀⢀⡏⠀⠀⢣⢤⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣥⢶⠂⡟⠀⠈⠀⠀⡀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣼⡵⢠⡇⢠⣠⣧⣄⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⠷⠿⠿⠿⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⡋⠀⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⡚⠛⠒⠦⠴⢮⣹⣿⣿⣷⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠋⠉⠀⢀⡴⠛⠉⠙⠛⠿⠿⠟⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣀⡤⠖⠛⢷⣤⣄⣤⣀⡀⠀⠀⠈⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⡼⠇⠀⠀⣴⣿⡏⠀⠈⠻⣿⢿⣷⣀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠈⣡⡇⢀⣀⣶⣿⡌⠀⠀⠀⠀⠘⠋⠙⣿⢿⣧⡄⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣴⡿⢃⢿⡿⢻⣿⠯⢶⣄⠢⠀⠀⠀⠀⠀⢸⠈⠛⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⡀⠈⠘⢠⣯⣀⣠⣾⣿⣧⡄⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠶⠖⠀⠁⠀⠀⣾⠘⢿⣿⣿⡟⣿⣯⠀⠀⠀⠀⢀⣼⠀⠀⠀⠠⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠙⠻⣿⣟⠙⠆⡀⠀⢀⠎⢸⡄⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠈⠢⡀⠈⠐⠎⠀⣠⣷⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣳⡶⣶⣦⣀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡄⣰⠏⠀⠀⠀⡀⠀⢤⠄⠐⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣉⣴⣿⣷⣿⣷⣞⠁⢄⣀⣀⣀⡀⠠⠀⡀⠀⠀⠀⠈⠁⠀⠀⠀⠃⠔⠀⡌⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⡟⣿⣻⠇⡼⣿⣿⣿⣦⣄⣿⣿⣿⣿⣿⣷⡾⢀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠠⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⣷⣼⢿⣼⣿⣿⣿⣿⡄⠈⠛⠛⠻⠿⣿⣼⣷⣀⠃⠀⠀⠀⠀⠀⡞⡁⠀⡆⠀⠀⡀⠂⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠸⠀⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠼⠂⠀⢸⠀⠀⠀⢀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠰⣟⢿⠆⢠⣴⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⣀⣸⡀⠀⣀⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⣻⠻⣽⣿⣿⣿⡿⠿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⣀⢠⡾⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠆⠛⠉⢿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⣀⣷⡿⠛⠛⢿⣿⡻⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡼⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⣼⡿⠋⠀⠀⠀⠈⢿⣿⠠
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⣴⠂⠀⠀⠀⠈⣇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⢠⠞⠀⠀⣀⡅⠁⠀⠀⠀⠀⠀⢸⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣘⡛⠀⠈⠲⣄
⠀⠀⠀⠀⡔⠁⠀⣤⡼⠃⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢧⣀⠀⠀⠣
⠀⠀⠀⡘⠀⠀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠃⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄⠀⠱⡀
⠀⠀⠀⠀⠀⣘⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠐⡀
⠀⠀⠘⠃⠴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⢑⡀⠀⠀⠀⠀⠀⠀⠀⠄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠤⠘⠂⡀
⠀⠀⣄⣀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣾⣧⠀⠀⠀⠀⠀⠰⠀⠀⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣤⣤⣾⣧⣦
⠀⣰⡿⠿⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠷⢤⣀⣀⣠⠆⣄⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠿⠋
⠀⢿⣦⣤⣀⣉⣛⡒⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠗⠒⢒⢛⣁⣘⣛⣉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⣀⣼⠟⠁
⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠁⡀⠀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠔⠂⠄⠀⠀⠀⢈⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⡀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⢀⠁⣠⡀
⠀⡠⠄⡀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠋⠀⠀⠀⠀⠠⠤⠠⠆⠀⠄⢀⡀⠐⠦⡠⢈⠀⢉
⠈⠡⠐⠀⠀⡠⠢⠄⠀⠀⢀⠔⠲⠀⠀⠠⠁⡐⠂⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃⠀⠀⠀⢀⡐⢀⡠⠀⠁⠀⠘⠂⠀⠈⠂
⠀⠀⠀⠀⠀⠐⠒⠀⣀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⡀⠀⠆⠀⠀⢀⡄⠀⡀⢀⠄⠐
⠀⠀⠀⠀⠀⠀⠔⠀⠋⠀⠀⠀⠀⠀⠀⠀⡐⠠⣀⠀⠀⠀⠀⠀⠄⠰⠀⠀⠀⠀⠀⠀⠀⣄⢀⠀⡃⠀⠁⠀⠒⠁⠀⠀⠈⠁⠀⠀⡀
⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠢⢄⠀⠀⠀⠀⠀⠦⠀⢤⠀⠀⠀⠀⠁⠐⠒⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠘⠢⡀
⠘⡡⠆⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠙⡲⠄⠐⠂⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠤⠒⠉⢢⡀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠘
⣘⡀⠀⠀⠀⠐⢆⠀⠃⠀⠀⡀⢀⡠⠁⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢤⣀⡀⠀⠀⠀⠀⠀⢀⠔
⠉⠀⠀⠀⠀⠀⠛⠀⠀⠊⠀⠈⡃⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡉⠩⣀⠉⠉⠁⠉
⠰⢇⢀⡠⠀⠀⠴⠀⠈⠄⣀⠔⠁⠐⠒⡬⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⠢⢄⠁⠀⠀⡀
⠀⠀⠈⠀⠈⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⢙⡤⠴⢬⡦
⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⡀⠀⠀⠀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠐⠃⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⠐⠒⣀⡀⠂⢠⠤⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣶⣦⠤⢶⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢱⠰⠄⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠸⠀⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠀⠀⠀⠁⣄⡈⠀⠰⣤⠀⠀⠀⠀⣠⢊⡑⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⢀⠠⢄⠀⠀⠀⡀⡻⠳⡏⠀⢤⣀⠀⠀⠘⠀⠀⠐⡀⢠⠘⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⡇⣄⠠⠤⠆
⠀⣀⠀⡀⢀⣄⡀⠀⠀⡐⠁⠀⠀⠐⢄⣰⣑⠀⠀⡐⡀⠀⢄⡙⠅⠘⠀⠀⢰⠁⠀⡄⠀⠀⠀⠀⣀⠀⣠⠎⠀⠀⠀⠀⠀⢤⠦⠆
⠺⢥⣐⣀⡀⠀⠀⠀⠍⠉⠄⠀⠀⠀⠀⠀⠈⠂⢄⢡⡈⢀⡀⠀⠀⢠⠀⠀⠀⠃⠈⠀⠀⠀⠀⢰⠉⠈⡉⠛⠒⠦⢤⣤⣤⣶⣤⡤⠴⠗⠀⠐⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠋⠀⠀⠊⠀⠐⠐⠃⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣲⡒⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡳⠞⣿⠫⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⣿⣿⣷⣶⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⣠⠔⠒⠚⠷⠒⣀⣹⣿⣿⣿⠧⣆⡀⠤⠀⢀⠀⣀⣀⣀⣀⡤⠐⠂⠀⢀⣴⣤⣶⣤⣀⣿⣻⡿⣠⠁
⠀⠀⠀⢀⣾⠃⠀⠀⢀⡀⠀⠉⠙⣿⣏⣴⣧⣤⠀⢤⣤⠚⠉⠁⠀⢀⣼⣿⣷⣤⣾⠿⠛⠉⠉⠉⠉⠉⠛⠛⠁
⠀⠀⠀⢸⣿⠀⠀⣠⠿⣅⠀⠀⢀⡞⣿⣿⣿⢧⠃⣼⣿⣷⠦⣶⡾⠟⠋⠉⠉⠉
⠀⠀⠀⢀⣿⠀⠀⠋⠀⢹⣿⡤⠷⢿⡿⠛⠛⣷⡿⠁
⠀⠀⠀⢸⣿⣅⠀⠀⠠⠚⠉⠀⣴⣿⣤⣀⡰⠟⠁
⠀⠀⠀⠈⣿⣿⣷⡄⠀⠀⠀⣼⠟⠈⣿⢋
⠀⠀⠀⠀⢸⣿⣿⣷⣴⣤⡴⠛⠃⠀⣿⠎
⠀⠀⠀⠀⢸⣿⣿⡿⠋⠀⣚⡋⢡⢼⠃
⠀⠀⠀⠀⣜⣿⡿⢷⣶⣾⠉⠭⣿⠏
⠀⠀⠀⠊⠈⠉⠉⠛⠓⠚⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣿⣷⣟⣿⣷⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠛⠛⠋⠉⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⡿⠿⠁⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⠁⠀⠰⣶⣶⠦⢴⠿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⠃⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡠⠞⠻⣻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠁⢀⣾⣄⠀⠀⠀⢀⠰⣶⢾⣶⣿⣿⣶⣦⡀⠿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢺⣿⣿⣷⠀⣄⡈⡈⠁⣾⣿⣿⣿⣿⣿⣷⣄⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⢀⠀⠒⠰⠒⠔⣼⡿⠋⣤⣦⣿⣿⣦⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⢟⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⣠⣿⣿⣿⣿⣿⣿⣿⠿⠻⣿⣿⣿⣿⣭⡶⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢀⣤⣿⣿⣿⢿⣿⣿⣿⣗⠀⠀⠀⠉⠉⠁⠸⣿⠹⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⣄⢀⣿⣾⣿⣿⣿⣋⣿⣿⣿⣿⡟⠀⠀⠀⠀⢀⣿⠀⠻⠧⠕⠊⠑⡄⠠
⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⢁⣙⣿⣿⣿⣿⡿⠻⠛⠋⠀⠄⠀⠀⠀⣸⣿⣧⠀⠐⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢁⣾⣿⣿⡟⣼⡟⠁⠀⠀⠀⠀⠀⣀⣠⠔⠁⠙⣿⡷⣴⠀⠀⠀⠀⢈⣦⣘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⠀⠀⠀⠀⠀⠀⠀⠙⢁⡾⣿⣿⣿⣇⣿⡇⣀⣤⡶⠚⣿⣿⠿⠋⠀⠀⠀⠈⠑⢿⡆⠀⠀⠀⠀⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢀⠠⠖⠁⠀⠀⠀⠁⢛⡛⢻⣿⣿⣿⠿⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠠⣄⣤⣄⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠮⠀⠀⠀⠀⠀⠀⣠⡟⢁⡿⠉⠉⠀⠀⠠⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⣀⠀⠀⠀⠀⠤⡍⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠺⠿⠀⣀⣴⣾⠟⢀⣾⠏⠀⠀⠀⢀⠀⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⠐⡳⢿⠛⢠⣾⠟⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⢲⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣴⣷⣮⣥⣶⣒⣖⣶⣶⣮⣿⣤⡦⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢹⣻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣥⣴⣾⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⢡⣈⡁⠘⣿⠿⡀⢻⣿⣿⣿⣿⣿⣿⡿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣟⢿⡀⡀⣿⣿⣷⡸⡆⢰⣷⣿⣿⣿⣿⣿⡝⣿⡷⠈⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣝⢷⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⢿⣿⡈⠁⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣧⢻⣿⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡻⡛⠃⠙⢷⣌⣻⢿⣿⣿⠿⠋⠛⠛⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣶⡶⠂⣀⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⢷⣽⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣷⣴⢿⣻⠿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣆⡅⣬⣿⡄⢹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠚⣿⣿⣿⣿⣿⣬⡿⢍⡹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣷⣶⣦⡀⢀
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠒⠛⠛⠗⢊⢁⡘⣿⣿⣿⣿⡿⠦⡛⢀⣠⠤⠤⢀⠀⠀⠀⣀⣀⣀⣀⣀⡴⠒⠂⠀⠀⢀⣠⣿⣰⣶⣦⣀⣘⡩⢍⣻⡿⢁⡎
⠀⠀⠀⠀⠀⢠⣾⠇⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⡿⢩⣶⣆⡈⠁⠀⡀⣀⠶⠾⠛⠋⠉⠀⠀⢉⣷⣶⣶⣄⣀⣾⣿⡿⠟⠛⠋⠉⠛⠛⠉⠻⠿⠷⠋
⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⣴⠆⠀⠀⠀⠀⠈⣿⣱⣿⣿⣿⡿⣡⠀⣹⣿⣧⣀⠀⠀⣀⣠⣶⣿⠿⠿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠘⣿⡇⠀⠀⢠⣾⠛⣷⣄⠀⠀⢀⡴⣋⢿⣻⣿⣿⢳⠏⣠⣿⡿⠿⠛⠓⠻⠿⠛⠋
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠙⠁⠀⠸⣿⣿⣤⠾⠾⣿⣿⠟⠛⠛⢧⣿⡟⠁
⠀⠀⠀⠀⢠⣿⣟⠄⠀⠀⠀⠀⣠⠿⠛⠁⠀⣴⡮⠁⠀⠀⢀⣾⠟
⠀⠀⠀⠀⠀⣿⣿⣧⣀⠀⠀⠊⠀⠀⠀⢀⣾⡿⢳⣿⣷⠶⠋⠁
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⠀⢀⠂⠀⠀⣾⡟⠁⠈⣿⠇⡸
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣴⣄⣠⡾⠟⠛⠃⠀⣿⡴⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⡿⠋⢉⡴⠶⠒⠀⣼⡏
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡅⠀⠀⣤⠶⢤⣤⣟⡽
⠀⠀⠀⠀⠀⣰⣟⣟⣻⢻⠿⣾⣿⡧⠤⠴⠻⡟⠁
⠀⠀⠀⠀⠜⠁⠉⠈⠋⠻⠹⠳⠶⠟⠾⠛⠛⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠉⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢿⣧⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏⠀⣴⠏⠀⢀⣴⣿⣧⡀
⠀⠀⠀⠀⠺⣿⣷⡀⠀⠀⠀⠈⢿⡆⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⡿⠃⢀⠞⠁⢀⣴⣿⣿⡿⠛⠁
⠀⠀⠰⣦⠀⠈⠻⣿⡄⠀⢀⡀⠈⢷⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠁⠀⠁⢀⣴⣿⡿⠛⠁
⠀⠀⠀⠀⠁⠀⠀⠈⠻⣄⠀⠙⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢤⣄⣀⣀⠀⠀⣀⣀⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⢀⣠⣦
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣩⠶⣒⠂⠘⣿⡋⠉⠉⠉⠀⠀⠜⢋⡤⠜⠛⠶⣄⠀⠀⠀⠀⠀⠀⠤⠚⠋⠉
⠀⠀⠀⠉⠛⠳⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠁⠀⠀⠈⠉⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡀
⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⡫⠅⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠛⠛⠛⠛⠉
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠒⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠐⠲⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠗⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢻⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠟⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠾⣧⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⢾⡀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⣠⠴⠖⠛⠒⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⢀⣀⠀⠀⣿⣷⡀⠀⢠⣤⡤⠴⠚⣠⣾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠒⠚⠋⠉⣁⠄⠀⣻⣿⣷⣦⣤⣤⣤⣶⣿⠟⠁⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⢀⣀⣠⠄⠻⣧⠀⠀⠤⠤⠤⠒⠀⢹⣿⣦⣀⠙⠛⢋⣁⣤⣶⣿⣿⣿⣿⡿⠛⠉⢉⠀⠀⣼⡏
⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠘⠿⣦⣌⠡⠖⣂⣰⣿⣷⣄⠀⠶⠶⠚⢁⣼⣿⣿⣿⡿⠿⢿⣛⣛⣛⡋⠉⠉⠁⠀⠀⠀⠀⢳⣿⡿
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠉⠈⠙⠿⠿⠶⠿⠟⠛⠁⣿⠋⠉⠉⠉⠉⠉⠙⣿⡄⠀⠀⠀⠀⠀⠀⣸⡿⠁
⠀⠀⠀⢀⣀⣤⣤⠶⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⣀⣀⣤⣿⣧⣤⣤⣤⠶⠶⠟⠋⠁⠀⠀⠀⠠⣀
⠀⠀⠀⠘⠛⠉⠀⠀⠀⢀⣠⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⣴⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣄⠀⠀⠸⣷⣄⠀⠀⠙⢷⣦⡀
⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⣼⠇⠀⠀⢠⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⢸⣆⠀⠀⠈⢿⣷⣄⠀⠀⠉
⠀⠀⠀⠰⣾⣿⣿⠟⠀⠀⠀⠺⠿⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠻⣿⣷⣄
⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠘⢿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⡿⢿⠿⠿⡿⡟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣟⣟⣿⣟⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⢸⣷⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠿⢷⠀⠀⠀⠀⠀⠀⠀⠀⠉⡟⠋⠉⠉⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢟⣁⠀⡀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⢀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠰⡟⢲⠃⡎⠈⠁⡇⠈⢡⠀⠀⠀⠀⠀⠀⢀⣀⢠⡏⠁⠀⠀⠀⠀⠹⣉⠂
⠀⠀⠀⠀⠀⠀⢳⡎⠀⠈⠒⠚⠀⠀⠘⣄⡠⠤⠒⠂⣉⠥⠓⢪⠃⠀⠀⠀⠀⠀⠀⢫
⠀⠀⡠⠀⡤⠀⠸⠀⠀⠀⠀⠀⠀⣀⠀⢫⢀⡠⠺⣽⠦⡀⠀⣧⠤⡄⠀⢀⣀⡀⠀⠈⠆
⠀⡜⠀⡜⠀⢠⠃⢀⣤⠤⠤⠤⢤⠏⠙⠺⠋⠀⠀⠀⠑⠪⣑⠢⢄⣹⣻⢹⣨⡽⠀⠀⢰⠀⠀⠀⠀⠀⡄
⢸⠀⣸⠀⠀⢺⢤⣀⠀⠉⠒⠢⣼⠀⠀⠀⠀⠀⢰⣀⣀⣀⣀⠉⠒⠬⡏⠁⣀⠤⠒⢹⢣⠀⠀⠀⠈⢣⠘
⠉⠀⠉⠀⠀⡎⠀⠉⡻⣶⢤⣀⡇⠀⠀⠀⠀⠀⡠⢠⠀⠉⠙⢿⣦⣄⡜⠉⠀⠀⣠⠃⠀⢣⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢸⠁⠀⢀⣈⠉⠓⠢⠤⠤⠤⠤⠒⣋⠝⠈⠀⠀⠀⠀⠹⡿⠀⠀⣠⠔⢁⡴⠃⠸⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠙⡏⠀⠒⠂⠉⠉⠀⠀⠠⠄⠀⠀⠀⠐⣥⠤⠊⣡⠔⠋⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠒⠒⠒⠒⠁⡄⣖⡩⠔⠒⠛⠤⡀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⡌⠉⠑⠒⠒⠤⠤⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⣸
⠀⠀⠀⠸⣤⡤⠄⣠⢦⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠆
⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠛⠛⠛⣻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡦⣰⣶⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⡈⣿⡿⠟⡿⠉⠙⣷⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣤⣠⠄⠀⠀⠚⠉⠙⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠉⠈⠻⣿⡿⠋⠀⠁⠀⠀⠀⢠⡀⠀⠀⣸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⢀⣀⣧⣤⣿⠇⠀⢠⡀⠀⠀⠀⢘⣷⣶⣶⣿⣿⣿⡜⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡅⠠⣽⣿⣿⣏⠀⠀⠸⣿⣄⣀⣠⣾⣿⣿⣿⠻⢿⣿⣷⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣿⣿⣿⣿⣿⢃⡀⠈⠙⠛⠛⠋⡹⣻⣿⣷⣄⡈⠙⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣇⢻⡟⠛⠓⠂⠀⠀⠀⠀⠈⣾⣟⣭⡿⠿⠂⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⡟⣾⣿⠾⠿⠦⠀⠀⠀⠀⢈⣿⣿⡋⠀⠀⢐⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠀⣿⡁⠀⠀⣤⠶⠛⠛⠛⠛⢿⣛⢿⣦⣶⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣿⣷⣤⣸⠃⠀⠀⠀⠀⠀⢀⣭⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣧⡻⣦⣀⣀⣠⣴⠾⣫⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⡟⠉⠉⠉⠘⠿⠿⠻⠵⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡎⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⡯⠀⠀⠀⠀⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣳⡿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣷⠁⠀⠀⠀⡠⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡄⠀⠀⠀⣴⣶⣦⣄⣀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣧⠀⢳⣄⠀⠈⣉⠉⠁⠉⡇
⠀⠀⠀⠀⠀⠀⣀⣤⠖⠢⢤⡴⠒⠒⠒⢊⣡⡾⢯⣿⣧⠀⠙⠀⠊⠙⠢⢤⣼⣁
⠀⠀⠀⠀⢠⣾⣿⡟⠀⠀⠀⠙⣶⣾⣿⢿⡿⠃⢸⣷⣿⣷⣄⠀⠲⣯⣽⣿⡀⣈⣱⡦⢤⣀
⠀⠀⠀⢀⣿⡇⣿⡇⠀⠀⠀⠀⡿⠤⠤⢀⠀⠢⠈⡏⠙⠿⣿⣦⡙⢲⠶⣿⣿⠉⠀⠉⠂⠈⠙⢦⡀
⠀⠀⠀⢸⣿⡇⣿⡇⢠⡄⠀⣸⣁⠀⠀⠀⠈⠐⢤⡁⢀⡤⢞⣻⣽⣟⢤⣿⣿⣦⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⢏⣿⣿⣷⣾⣿⣿⣷⡀⠁⠀⠀⠀⠀⢰⡽⢫⣾⣿⣿⣿⣿⣿⣌⢻⣿⣷⣄⣀⠀⡄⠀⢸⡆
⠀⠀⠀⢀⡾⣿⣿⡟⠉⠉⢻⣿⣿⣶⣄⡀⠀⡹⢋⣴⣿⣷⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣷⣧⣴⣿⣇⣀⣀⡀
⠀⠀⠀⣾⠀⣿⢸⣿⡆⠀⠈⣷⣝⠻⢿⣿⠞⣱⣿⣿⣿⡻⣻⣿⣿⣿⣿⣿⠀⣿⡿⣆⠉⡿⣻⣿⣿⣿⣿⣦⡈⠳⡄
⠀⠀⠀⣿⣤⣿⢸⣿⣧⣀⣤⣿⡻⣿⡷⠋⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⠀⣿⣷⡽⣿⣿⡿⣿⣿⣿⣿⣿⣿⡄⠈⢆
⠀⠀⠀⢿⣿⣿⠀⢻⣿⣿⣿⡿⠋⡁⣠⣦⣻⣿⣿⣿⣿⣿⡿⢉⣽⣿⣛⣋⣠⡿⢋⡟⢿⡟⣾⣿⣿⣿⣿⣿⣿⣿⡄⠈⡆
⠀⠀⠀⢸⣿⠁⠀⠀⠀⠉⠉⣧⣿⣿⣿⣿⣷⡻⢿⡿⢟⣩⣾⡿⠿⠿⠿⠛⠋⠀⢺⠀⠈⢿⣸⣿⣿⣿⣿⣿⣿⣿⣿⡀⢱
⠀⠀⠀⢸⣿⣿⣶⣤⣤⣴⠿⣻⣿⣿⣿⣿⣿⣿⣷⣖⣻⣿⣿⣿⣦⣤⣤⡄⠀⠀⠸⡆⠀⠈⢷⢿⣿⣿⣿⣿⣿⣿⣿⡇⡸
⠀⠀⠀⠈⢻⡿⠿⢟⣋⣵⠾⣻⣽⣿⣿⣿⣿⣿⣟⠛⠋⢻⣿⡛⠛⠉⣿⣠⠆⠀⠀⣧⠀⠀⠀⠳⢭⣛⣻⣯⣭⢭⠖⠊⠁
⠀⠀⠀⠀⠘⣷⣯⣭⣽⠾⠟⠛⣿⣿⢿⣝⡏⢿⣿⣦⣤⣼⣿⣷⠶⣶⣟⣡⠄⠀⢀⡟⠀⠀⠀⠀⠈⠛⠻⠿⠗⠋
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢻⡹⣆⠹⣿⡆⣿⡿⠛⢻⣾⡞⠋⠉⣿⠏⠀⠀⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣌⢳⣼⣇⣿⣷⠀⡸⢟⣁⣀⣤⣿⡤⠖⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⣝⡻⢷⣦⣴⣿⣿⣿⣿⣤⣤⣤⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⢹⣿⣧⡀⠀⠀⣀⠀⠀⠀⠀⠀⣀⣤⠴⢚⠿⠛⠋⠀⠀⠈⠳⡄
⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⢿⣿⣿⣦⣰⣿⠀⠀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠖⠒⠓⠲⢄
⠀⠀⠀⠈⠛⢷⣄⠀⢸⣶⣦⣼⣿⣿⣿⣿⣿⣀⣾⠁⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠈⠳⣤⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢢⣈⢿⡇⠀⠀⠀⢀⣀⣤⡘⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠒⠶⠬⠿⣧⠀⠀⡴⠋⠀⠀⠹⣿⣠⣤⠤⣿⡁
⠀⠀⢀⣈⣉⣙⣻⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⢳⡀⠀⠀⠀⣿⡇⠀⠀⠀⢳⡄
⠀⠀⠈⠙⢿⡟⠛⠛⠛⠛⠛⠋⣹⢿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⠻⣦⣑⣒⣒⣫⡾⠁⠀⠀⢀⠀⣧
⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⣿⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠉⠉⠁⠀⠀⠀⣀⣼⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⣶⡶⠶⠒⠒⠚⠛⠋⠉⠉⠀⠉⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡏⠀⠉⠀⠀⠀⠘⠦⣄⣀⡀⠀⠀⠀⠴⠋⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⣠⣴⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⡤⠔⠀⠀⠔⠋⢸⣿⣶⣤⣀⡀⠀⠀⠀⠀⢀⡆⣀⡼⠁
⠀⠀⠀⠰⠾⠿⢧⣤⣤⣀⡀⠀⠀⠀⡀⠀⠀⠀⠴⠋⠁⠀⠀⠀⠀⠀⠀⡏⢻⠿⠀⠉⠉⠙⠛⠛⡟⠛⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡦⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⠀⠀⠀⠐⠁⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣾⣷⣄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠗⢦⡤⠀⣀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⣀⣀⣀⣀⣤⣤⡤⠤⠤⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠙⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠘⣿⡏⠻⣿⡇⠀⠉⠙⠛⠿⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡤⢷⡄⠀⠀⠀⠀⠀⠰⡤⠿⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠰⠷⢦⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⢀⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠁⢀⠀⠀⣠⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢦⡀⢀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠲⣶⠶⠶⠃⠈⠉⠉⢸⠀⢠⠞⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⢶⠃⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⢦⡀⠀⠀⢰⠋⠋⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⡴⠒⠛⠲⣄⣠⣸⠀⠀⠀⠀⠀⠀⣤⡟⠀⠀⠀⢳⠀⠀⠘⡷⠀⠀⣀⣠⣀⣠⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡴⠖⠃⠘⠛⠉⠛⠲⣌⠃⠀⠀⠀⠀⣀⡤⠞⠁⠀⢀⣠⠤⠞⠁⠀⢠⠇⢠⢞⣁⣀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢧⣀⣤⠤⠖⠛⠉⠀⠀⠀⠀⢸⠃⠀⢀⡴⠒⠋⠀⠛⠉⠉⠉⢹⠂
⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡀
⠀⠀⠀⠀⠀⡾⠉⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠛⠉⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⢷⡀⠀⠀⠘⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠚⠋⠉⠀⠀⠀⢀⣀⣀⣀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢶⣶⠒⡋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢾⠗⣿⡖⡶⢀⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠇⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠛⠘⠒⠉⠉⠉⠉⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣞⣉⣀⣀⣀⣀⣀⣀⣀⣀
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢿⣿⣿⣿⡷⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠛⠿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⢸⣿⣿⡏⡰⠺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠃⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⢠⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣴⣤⣤⣄⠀⠀⠶⠿⠿⠃⠀⠀⠀⠀⠀⠀⠟⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠛⠶⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠈⢷⡀⠰⠀⠠⠈⠀⠀⠀⠀⢀⡴⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠉⠉⠉⠉⠙⠛⢷⣜⠛⠛⠛⠂⠀⠀⣠⠞⠀⠀⢀⣴⠋⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣼⣿⣿⣦
⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⣀⣀⣠⡴⠛⢀⣠⡴⠟⠁⠀⠀⠠⢤⣴⡟⠉⠁⠀⠀⠀⠉⠙⠛⢿⣷⣄⡀
⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⣛⣤⣶⣟⣋⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢶⡀
⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠋⠉⠈⠉⠁⠀⠈⠉⠑⠒⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⢽⠀⠀⠀⠈⠐⢦⣄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠙⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⣾⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⣿⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠉⠳⡀
⢸⣷⣆⠀⠀⠀⠀⠀⠀⣀⣿⣿⣶⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⠷⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⣾⣿⣿⣷⣶⣶⣶⣶⠀⣧⣼⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠠⠀⠈⠻⢦⡀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠹⣆
⣿⠛⢿⣿⣿⣿⣿⡿⣀⡟⠁⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠙⡆
⣿⠻⠦⣀⣉⣉⣡⠔⠋⡇⢠⣿⣿⣿⡿⠛⣷⣄⠀⠀⠀⠀⢀⣼⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠛⠀⠀⠀⠀⠀⡿⠛⠛⠛⠿⠿⣷⣟⣛⠛⠛⠦⣼⡆
⢹⠇⠀⠈⠉⠉⠀⠀⠀⠹⣿⣿⣿⡿⠁⠀⣼⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣤⣤⣶⣿⠟⠁⠆⠀⠀⢠⢀⡜⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⢻⡀
⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⠀⢠⠾⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠐⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⣴⠋⠀⠘⣿⡿⠁⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⢠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠘⣆⠀⠀⠀⢲⡀⠀⠙⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣶⣶⣿⣿⣷⣄
⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠲⢿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠈⢶⡀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⣀⢸⣷⣤⣤⣀⡀⠀⢠⣀⠀⠉⠻⠟⠁⠀⠈⠙⢿⣿⣿⣿⣿⠙⠳⢦⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢣⡄⠀⠀⣿⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡏⠀⠀⠀⠀⠀⠈⠙⠛⠓⠲⠶⠶⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣘⡇⠀⠀⠁⠀⠀⠀⠀⢀⣀⣄⣀⣀⡀⠤⠤⠤⠤⠤⠤⠤⠤⠼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠛⠛⠛⠛⠛⣿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⡇⠀⠀⠀⠀⢸⡃⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠸⡇⠀⠀⠀⠀⠠⣧⠀⠀⣀⣀⡀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠈⠙⢶⣤⣤⣤⡤⠔⠾⠛⠂⠉⠉⠉⠉⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⢿⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⢸⠇⢠⣶⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⡆⢀⡆⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⡄⢀⣿⡏⣨⠁⠀⠀⠀⠀⠀⠀⣠⣴⡾⠋⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠘⠇⠈⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣠⣿⣿⣿⡷⠀⠀⠀⠀⠀⣠⣾⣿⠏⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⣾⣿⡿⠃⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠟⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⡏
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣟⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⡿⠿⠿⠿⢿⠿⣿⣿⣿⡿⠿⢿⡿⠿⣿⣿
⣿⣦⣼⣦⣤⣦⣴⣶⣴⣿⣦⣤⣧⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣶⣶⣾⣾⣿⣿⣿⣿⣷⣶⣶⣷⣾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⢸⡀⣸⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⠀⠀⠙⠊⠁⠁⠚⠁
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⢀⡴⠋⠀⠀⠈⠙⣦⢤⡀⠀⠀⢀⡀⣠⣶⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢸⠉⠉⠉⢹⠤⠊⠀⠀⣠⢾⡇⠀⣼⣠⡇⠀⠀⠸⠗⣿⣿⠇⡙⢿⣿⣷⣄⡀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠠⡏⠳⢤⠇⠀⣹⠀⠀⢿⣿⣜⣫⣶⢸⣿⣿⣿⣿⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣸⠀⠀⠀⣿⠀⠀⠀⠀⠠⡏⠉⠉⠉⠉⡇⠀⠀⢰⣶⣶⣶⣶⢼⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣀⠀⠀⠀⠠⡏⠉⠉⢙⡏⠁⠀⠀⠀⢠⣶⣶⣶⢸⣿⣿⣿⡿⠟⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠙⠒⠒⠒⠋⠈⠉⠙⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢱⠆⢦⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⢷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠹⣿⣿⣿⠃⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠻⠛⠁⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢀⠀⠀⠀⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠂⠋⠀⠈⣧⠀⠀⠀⠠⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⡤⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⡟⠁⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠿⠋⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠁⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣏⠉⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⢈⣧⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣀⠹⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡄⡤⠤⣤⣤⣤⣤⠴⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠄⢼⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⡍⠻⣿⣷⣿⠏
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠁⠀⣀⣀⢿⣿⡘⣟⠻⣦⣄⣀⣀⡤⣀⣀⠀⣀⡴⠿⣿⡄⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠾⢏⣼⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠘⣷⣠⣾⣿⣾⣿⡿⣿⣯⣿⠳⠀⠀⠀⠈⠉⠉⠀⠘⠛⠋⠉⠀⠀⠙⣿⣦⡀⠀⠈⠣⣀⠀⠀⠀⠀⣠⡾⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠇⣿⣿⣿⠁⠀⠀⠈⠉⠒⠊⠀⠒⠂⠀⠀⠀⠀⠀⠈⠻⣿⣶⠀⣠⡟⠀⠀⠀⣘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡏⣠⣿⣿⣿⡀⠀⠀⠒⠒⠒⠃⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣿⣧⡀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣿⣿⣿⣿⣿⣿⣷⣶⣤⣴⣶⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⠰⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣇⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠆⠘⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⢿⣿⣿⣿⣿⣿⣷⡦⠱⣌⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⣿⣿⣿⣿⣿⣿⣿⣃⣨⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠀⢻⣿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠏⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣄⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⣤⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⡷⠈⠁⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⡴⠃⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣴⣟⣼⠃⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣳⡄⠻⣿⣧⡀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⢀⡀⠈⠻⣿⣦⣀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⢷⣤⣀⠉⠁⢀⣿⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠉⣿⣬⣉⠷⣸⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠄⠁⠀⠈⠻⣿⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⣶⣦⣤⣄⣂⣀⣀⣀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⣶⣦⡄⠀⠀⠀⠰⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⠿⣿⣿⣿⣿⣧⡤⢶⣛⣗⣼⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣿⣿⣿⣿⣿⣿⠏⠉⠉⣈⠛⢭⡻⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣩⣽⣿⣾⣿⣿⣿⣿⡯⢿⣷⣤⣾⣿⡿⢋⣿⣿⣿⣿⡿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢋⣾⣿⣿⣿⡿⠃⠀⠀⠉⠛⠿⠿⠛⠿⣏⠟⠻⣿⡰⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣾⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠈⠁⢾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣯⣠⡴⠾⢻⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡏⠛⠉⠛⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄⢹⣿⣿⣯⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠠⣪⣿⡿⠁⠀⠀⠀⠀⢻⣿⣆⡀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠈⠻⠿⠿⣿⣿⣶⣦⣀⣤⡀
⠀⠀⠀⠀⠀⠀⣰⣷⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡀
⠀⠀⠀⠀⢀⣠⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⡄⠀⠀⠀⣰⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇
⠀⠀⠀⢴⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⢤⣶⣿⣿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢛⣿⢏⣿⡏⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠋⠀⠘⢁⣾⡟⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠈⠛⠠⠏⣾⢻⣿⣇
⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠸⣧⣴⣶⣿⡶⠀⠀⡀⠀⠉⢈⡾⣿⡄
⠀⠀⢀⣴⠞⠋⠉⠉⢻⣆⠀⢨⣿⣿⡋⠀⣠⠎⠀⠀⢸⡏⠀⠸⡟⣦⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⢠⡟⠁⠀⠀⠀⢰⡀⣿⠀⠼⢽⡛⢿⣿⠿⠀⠀⠀⢈⣷⠀⠀⠃⠈⣿⠿⠛⠋⠉⠁⠉⠛⠋⠉⠉⠓⢦⡀
⠀⢸⠁⢀⣠⣤⣠⣾⡇⣿⠀⠀⠀⠻⢾⡷⠆⠀⣠⣶⣿⣿⣷⢤⣀⡀⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠙⣆
⠀⢸⡀⣿⣿⡿⠿⣿⣿⠃⠀⠀⠀⠀⠈⠓⠶⢞⣻⠻⣿⣿⣿⡄⠉⠋⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⢸⡆
⠀⠘⣇⢻⡟⠀⠀⠈⡇⠀⣀⣤⣶⠶⠶⢿⡿⠟⠁⠀⠉⠉⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⣈⢿⣷⣄⡀⠀⢾⣧⡀
⠀⠀⠸⡌⢻⡀⣀⣼⠃⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠉⠁⠀⠀⠀⠙⢦⡀
⠀⠀⠀⣿⠛⠛⠋⠸⣷⣿⠟⠋⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢿⣷⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⢸⡄⠀⠀⣀⣿⡏⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⢀⣴⡿⠁⢸⣿⣧⡀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⣿⠿⣿⣿⣿⣇⠀⠀⠀⢰⡿⢻⣧⡀⠀⠀⠀⠀⢀⡼⠿⣿⡛⠶⠶⣶⡿⠛⠉⠀⢀⣾⠛⣿⡟⣷⡶⠶⠞⠛⠛⢻⡆
⠀⠀⠀⠀⣿⠀⠀⠀⢹⣿⣆⠀⣠⣿⠃⠀⠙⢿⣶⣤⣤⠖⠁⠀⠀⠈⠙⠀⠀⠀⠉⠳⣦⡴⠾⠟⠛⣿⡉⠁⠀⠀⠀⣀⣴⣿⣿
⠀⠀⠀⠀⢸⡄⠀⠀⢸⣿⣿⣾⣿⠏⠀⠀⠀⠈⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠘⣷⡀⠀⠒⠋⠉⠁⣰⡿
⠀⠀⠀⠀⠀⠻⣆⠀⢸⣿⣿⡿⠋⠀⠀⠀⠀⠀⠘⣿⡄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⠀⠀⠀⢠⣿⣷⣶⣶⠶⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠻⣦⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡙⠿⢷⣶⣶⣶⠿⠿⠛⢛⣉⣡⣿⣶⣶⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣤⣤⣤⣴⣶⣶⣿⣿⣿⡿⠟⠋⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣧⣬⣙⣿⣿⣟⣛⡛⠉⠁⠀⢀⣀⣠⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡏⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣇⣤⣀⣀⣀⣀⣀⡀⠀⣸⠀⠀⠀⠀⠀⠠⣄⣀⡀⢠⣤⣤⣤⣴⣶⣶⣾⡏⡿⢁⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⠹⡇⣿⡿⠟⠿⠿⠿⠿⠷⠙⠉⢀⣀⣤⣀⣀⣀⠈⠉⠁⠀⠋⠉⠉⠉⠉⣿⣿⣸⡇⣾⣿⠉⠉⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⣿⣼⡇⠀⠀⠀⠀⠀⠀⠀⠺⠿⠛⠻⠿⠿⣿⣿⣿⣶⣦⣤⣤⣄⣀⢀⣿⣿⣿⢠⣿⣿⣶⡞⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣻⣿⣿⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⢇⣼⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠛⢳⣼⣿⢻⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣷⡿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣾⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⣶⣤⣀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⠛⠻⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠙⠻⣿⣿⡿⠃
⠀⠀⠀⠀⣠⣴⣿⡿⠟⠁⠀⠀⠀⠀⠉⠙⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⢀⣤⣾⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⢠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄
⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣻⣿⣿⡿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⢻⣽⡿⠛⠉⠀⠹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠓⠀⠀⠀⢀⣠⣤⡀⠀⠈⠻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠿⠳⣦⠐⠻⠟⠿⠃⢀⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠇⠀⣀⠀⠀⠀⠬⠉⠀⢰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⠧⣴⠖⠁⠀⠀⠀⠀⠀⣸⠛⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠴⣛⣒⠒⠂⠀⠀⠀⣠⠃⠀⠀⠈⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⠀⢀⣨⣷⡏⠉⠀⠀⣠⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠑⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⣡⣶⣿⣿⣿⣿⣿⣿⣶⣶⣶⡿⠋⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣦
⠀⠀⠀⠀⠀⠀⢠⣿⡏⡄⣴⣿⣿⣿⣿⣿⡿⢿⣿⡿⠿⠋⠀⠂⠿⣷⣿⣥⣔⠢⢆⡐⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣼⣿⣿⣿⣿⣿⢻⡇⠀⠀⠀⠀⠀⣠⠀⠀⠈⠛⠿⢿⣿⣷⣈⠲⡀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡿⢣⡟⠀⠀⠀⠀⢰⣿⣿⣷⣶⣀⣄⠀⠈⠹⢿⣿⢷⠐⡄⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣿⣯⡵⠋⠀⠀⠀⠀⠀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣶⢨⣷⣿⣧⠐⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⢻⣿⡟⣾⣿⣿⢷⣿⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿⣿⣿⣿⣿⣿⣿⢃⣿⣿⣿⣿⠃⠁⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⢠⣼⣟⠁⢿⣿⣿⣾⣿⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣬⣝⣛⣩⣴⣿⣿⣿⣿⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣷⠀⡀⠉⢢⡀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠝⠿⢅⣸⣿⣿⠿⢋⣿⣿⣷⣄⠀⠀⠀⠀⡀⣜⠻⣟⠫⠙⢿⣿⣿⣿⣻⠛⠛⣋⣥⣾⣿⣶⡈⠻⢶⣤⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⡏⣹⣿⢿⣶⣖⣂⠀⠀⠀⠈⠓⠤⣈⠻⣿⣿⣿⣬⣿⣿⣿⣿⠟⠉⠀⠀⠀⠈⠙⠦⠀⢿
⠀⠀⡆⠀⡀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠋⠸⡏⠻⠃⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠙⢻⣿⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾
⠀⢸⡆⢸⡇⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠹⣆⡓⠀⠄⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠳⣮⡄⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠈⠿⣶⠶⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⣀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⢉⣿⣶⣤⣀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣼⣿⣿⣿⠉⠛⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣤⡤⣤⣀⣀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣛⣿⡹⣧⣽⣻⣿⣿⣿⣿⣿⣿⣿⣛⣿⣵⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣻⣿⣿⣿⣿⣷⣬⣍⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⡛⠙⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⡟⡟⢿⣿⣇⠫⠿⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠈⠙⠏
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣾⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡄⠀⠀⠀⡠⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢄⠀⠀⡘⠀⠘⡄⠀⡜⠀⠈⡆⠀⢀⠔⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠙⠦⠃⠀⠀⠘⠞⠀⠀⠀⢣⡔⠁⠀⠀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣣⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡠⠔⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⢸⠀⠀⠈⠁⠀⠀⠈⢣⠄⠒⠒⠢⢄⠀⠀⠀⠉⠀⠀⠀⠀⡇⠀⢀⡠⢴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢬⡶⢸⠀⠀⠀⠀⠲⠂⠀⠀⠀⠀⠀⠀⠀⢡⠤⡀⠀⠀⠀⠀⠀⠓⠊⠁⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣸⠀⠀⢀⣀⠀⠀⡠⠀⠀⣶⡄⠀⠀⢀⠀⢁⡀⠄⠀⢀⢀⠀⠀⠀⠰⠤⠐⢒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠛⠉⠈⡅⠀⢸⠈⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⠠⡀⠀⣠⠄⣠⠋
⠀⠀⠀⠀⠀⠀⣤⣀⡀⢩⠀⠀⠀⠙⠛⠋⠀⠀⠀⠑⠦⢤⣠⡤⣤⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⢁⠔⠁⢀⠜⠃
⠀⠀⠀⠀⠀⠀⠈⢢⠈⠉⠀⠀⣔⣦⠀⠀⢠⣒⡄⠀⠀⠀⠀⠀⠈⠦⠖⠋⠱⣀⣠⠔⠉⠀⠠⣎⢀⡴⠁⣴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⠤⡌⠉⡇⠀⠀⠙
⠀⠀⠀⠀⠀⢀⠤⠤⢄⡸⠂⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⠔⢲⠀⠀⠀⡠⢶⠀⠀⠀⠀⠈⠉⢀⠤⣸
⠀⠀⠀⠀⡰⠁⠀⠄⠊⠁⠀⠀⠀⠀⠐⠄⠘⠒⠃⠀⠘⡅⠀⢸⠀⡠⠊⠀⠈⡆⠀⣠⠛⢆⠀⡜
⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠢⡸⠊⠀⠀⠀⠀⠰⡄⠁⠀⠈⠶⠁
⠀⢀⠔⠁⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡉⠀⠀⠁
⡆⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠤⢢⠀⠀⡀⠀⠀⠁
⠱⡈⠀⠀⠀⠀⠀⠀⡠⠔⠊⠀⠀⠀⠀⢇⡜⠀⠁⠒⠒
⠀⠑⡄⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣤⣤⣬⣶⣶⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⢨⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠛⡁⡀⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠈⠉⠒⠒⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡄⣘⠀⡀⡀⠂⠆⠛⢤⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⣡⣄⠀⠀⣰⠊⠁⠈⢦⡀⠀⠀⠈
⠀⠀⠀⠀⠀⡸⠇⣴⣠⠀⠀⢂⡞⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⢆⣰⣈⣀⣀⠀⢘⠀⢰⣷⣾⣿⣷⠀⣠⣄⢀⣀⠀⡀⠀⢀⡀⢀⣄
⠀⠀⠀⠀⠈⠔⣴⢻⣿⠀⣠⠞⠀⠀⢖⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠟⠋⠀⠀⠀⠀⠛⠘⠛⡛⠁⠀⠀⢦⠙⠉⢁⡄⠀⠀⠁⠈⠋
⠀⠀⠀⠐⠀⠰⠁⣠⠗⠐⡿⠀⠈⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞⠀⠀⠀⠀⠀⠀⠅⠴⠰⠒⠒⠠⢶⣴⣛⠆⠀⠂
⠀⠀⠀⠀⠈⠀⠀⠁⠙⢘⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣀⠒⠂⠈⠀⠀⢀⣤⣤⣤⣤⣶⣬⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢠⣀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠙⢷⣄⡀⣰⠿⠋⢀⣛⣛⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⡎⡉⠀⠀⡼⠃⣴⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠰⢿⣏⠀⠀⢠⣶⣿⡿⠋⠙⣟⣿⣿⣿⡇
⠀⠀⠀⠀⠃⠁⠀⠀⠃⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣦⣤⣄⡀⠀⠉⠳⢦⣶⠟⠋⠀⠀⠀⠈⠙⠌⣿⡇
⠀⠀⠀⠀⠀⠈⠁⠀⠁⢀⡀⠀⠀⠀⠀⠀⣠⠴⠀⠀⠀⠀⠈⠩⣿⣿⠛⠙⢖⡢⠄⠀⠙⢄⠀⠀⠀⠀⠀⠀⢾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠀⠀⠀⣀⠈⢹⡏⠀⠀⠀⠙⣷⣦⣤⠀⠑⡄⠀⠀⠀⠺⢵⡻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢠⢠⡞⠀⠀⠀⠀⠰⣿⡿⠛⠀⣶⠃⠀⠀⠀⠘⠜⣭⡖⠛⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠖⠈⠀⠀⠀⠀⠀⠀⢿⠀⠀⡺⠃⠀⠀⠀⠀⠀⠀⠘⢿⡶⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⣲⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⢰⠃⠀⠀⠀⣀⠀⠀⠀⢀⡀⠙⢾⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠤⠶⠦⢥⣤⡶⠖⣦⣄⣀⠀⠀⠢⠦⠿⣡⡟⠿⠿⠳⠿⠷⣂⣤⣀⣭⣦⡠⠉⠀⠀⠑⡖⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠺⡻⣆⣠⣀⣠⣤⠿⠿⠷⠂⠀⠒⠂⠄⠀⠀⣁⣭⣥⠤⠤⠦
⠀⠀⠀⣠⣤⡭⠶⠆⠈⠁⠀⠀⠀⠀⠐⢆⡤⢠⣔⣄⡀
⠀⠀⠀⠤⠄⠖⢀⠐⢡⢭⣿⣶⣀⣰⣿⣦⣬⣻⡿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠀⢈⠉⣛⠛⠿⠿⣿⡿⠟⠛⡛⢿⣿⣯⣭⣯⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠛⠳⠶⠿⠷⠂⠈⠹⠟⠛⠛⠣⢍⣀⣿⣷⣧⣾⠾⠟⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢠⠋⣟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⣶⣶⡆
⠀⠀⠀⠚⠢⣤⣽⣄⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡻⢢⡽⢹⠋⣿⡆
⠀⠀⡀⣼⠀⠀⢧⣿⡆⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣽⣶⣿⠇⢸⣶⠯⢧⡀
⠀⢰⠁⣾⣷⠀⢰⠌⠃⠀⠀⠀⡠⠈⠁⠀⢿⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡜⠀⡿⣛⠛⠓⡎⠀⠀⢠⣷⣤
⠀⠃⠀⣿⠉⠀⢻⠀⠀⠀⡀⢠⣠⡴⠞⠁⢘⡿⠀⣀⠀⠀⠀⣤⡀⠀⢸⣢⣤⣴⡾⡇⠐⠙⣿⣟⣼⣿⣷⠀⢸⡟⢯⠁
⠀⠀⢰⡇⠀⠀⢸⠀⢀⡐⢀⣾⠏⣠⣄⡼⠿⠿⠒⠛⠃⠀⠀⠸⢷⡤⢤⠋⣤⣤⡀⢧⡄⢀⣼⡿⡿⣿⣿⣇⣺⡷⠀⢣⡀
⠀⠠⣴⡖⠀⠲⡞⠀⣇⣱⠟⣡⠶⣿⡏⠀⠀⠀⠀⠀⠀⢀⠰⣿⡼⠃⢸⠃⠃⠋⡄⠀⠀⠈⢻⣷⠀⢸⡏⠉⢻⠁⠀⣾⡇
⠀⠀⡏⠀⠀⠀⡁⢀⠋⣿⣰⢀⣼⣿⡁⠄⠀⠀⠀⠀⠀⠀⣷⣯⣧⠁⣼⡆⠀⠀⠁⢠⠀⠀⠺⣿⡆⢸⠇⠀⠉⠉⠀⣿⣿
⠀⠰⠁⠀⠀⠀⡇⠈⣘⠀⢙⡟⠉⠻⢷⣄⠠⠀⠀⠀⠀⠸⣿⢻⡋⠈⣿⣿⣆⠀⣷⣶⣤⣤⡼⠏⠀⠀⢠⡄⢀⣤⣤⠛⠛
⠛⠓⠁⠀⠀⠚⠙⠌⢁⣰⠋⠀⠀⠀⠈⢓⡖⣿⣿⡏⠀⣸⣷⣯⣵⣄⠺⢿⣿⡄⠀⣿⢩⢙⣧⠀⠀⠈⠎⡇⠈⠛⢿⡄
⠀⠀⠀⠀⠀⢎⣠⠾⠋⠀⠀⠀⠀⠀⣴⣿⠃⠛⠛⠷⣻⠿⣏⣶⣿⣿⣦⠘⣿⡗⢸⣇⣀⠀⡍⠀⠀⠀⠀⡇⠀⠂⢸⠁
⣶⣄⠀⢀⣷⠟⠁⠀⠀⠀⠀⠀⠀⣀⣿⠇⠀⠀⠀⠀⠈⠀⠲⠿⠿⠟⣛⣻⣿⠇⣸⠿⠛⡀⡟⠉⠀⢐⠘⠃⠀⠀⢸⠇
⣿⣯⡭⢼⣧⣴⡶⠦⠤⠤⠤⠀⠠⠿⠿⠷⠦⠤⢤⡤⠄⠀⠀⠀⠀⠀⢀⣤⣴⣶⣧⣤⣀⣝⡇⣐⠒⠛⠛⠛⠛⣧⣼⣛⣛
⣿⣿⣿⣶⣦⡀⣂⡀⢀⠀⠀⠀⠀⣄⡀⢀⡀⣀⠀⣀⡀⠀⠀⡀⠤⠶⠶⣶⣴⣀⣀⠀⠠⡾⠛⠛⡛⠛⢛⠛⠿⠛⢛⠉⡉
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣚⣭⣴⠶⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠛⠛⠹⣿⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⣠⣤⠀⠉⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠒⠀⢀⡞⢥⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣤⡄⠀⠀⠀⠂⣤⢤⣾⣿⠟⠀⠀⠈⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢫⣿⣿⠟⠛⣀⠀⠀⠘⠿⡆⠈⠁⠀⠀⠀⠀⠀⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⣤⣶⣿⡟⠀⠀⠀⠀⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣠⣾⠟⠉⣐⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⣼⢃⣽⣿⣿⠉⣽⣿⠟⡆⠀⢰⣿⠋⢀⣾⣿⣿⣿⡆⢸⣇⠀⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⡿⠟⣻⠟⠁⠀⢻⣄⠸⣿⣠⣾⣿⣿⣿⣿⣿⠸⡿⠀⠀⠀⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⠈⠁⠀⠀⠀⠀⡏⠢⠀⠉⠹⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠙⣿⡽⢿⡿⠓⢄⠀⠀⢠⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠙⠁⠀⠀⠈⢾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢿⣄⠀⠀⠀⣀⢜⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣀⠀⠓⠤⣀⠀⠘⣟⠶⢤⠾⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠻⠒⡷⡆⠰⣶⣶⣼⣿⣶⣿⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⢠⣤⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⡄⠐⣀⠀⠀⠀⠀⠀⢸⠸⠇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠐⠛⠹⠞⠀⠀⠀⠀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠈⣿⡀⢀⣿⠃⠀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠙⠃⠘⠏⢠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠈⠉⠙⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠁⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢠⡀⣠⠀⠀⢸⡟⠛⢦⡀⠀⠀⢀⡴⠛⢻⡃⠀⠀⣄⢀⡄⠀⠀⠀⠙⢦⡀
⠀⠀⠀⢠⣤⣤⣤⣤⡴⠛⠁⠀⠀⠀⠀⠈⢿⠃⠀⠀⣼⠁⢀⣸⠃⠀⠀⠘⣇⡀⠈⣧⠀⠀⠘⡿⠁⠀⠀⠀⠀⠈⠛⢦⣤⣤⣤⣤⡄
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⣸⠋⠉⠉⢳⡀⠀⠀⢀⡞⠉⠉⠙⣇⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⣻⣽⣷⣋⣀⣀⢀⡼⠃⠀⠀⠘⢧⣀⣀⣀⣙⣾⣯⣟⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠸⡇⠀⠀⠀⠉⠉⠹⡇⠀⠀⠀⠀⢸⠏⠉⠉⠀⠀⠀⣸⠇⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠛⠲⠶⢤⣤⡼⠃⠀⠀⠀⠀⠘⢧⣤⡤⠶⠖⠛⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠘⠛⠛⠛⠳⢦⣄⣀⣀⠀⠀⠀⠘⢳⣄⡀⢀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡀⢀⣠⡞⠃⠀⠀⠀⣀⣀⣠⡴⠚⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⣰⡄⢠⣄⠘⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠀⢠⣿⠁⠈⣿⡀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢹⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⣴⠶⡴⢦⡀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣯⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢉⡄⠀⠀⠛⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⢰⣿⣿⡿⢻⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠋⡴⠆⣀⣀⡇⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⣿⣿⣿⠶⢿⡿⢒⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣻⣿⡿⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣙⣯⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣦
⠀⠀⠀⣿⣿⣿⣿⣶⣷⣿⣿⣿⣿⣿⠉⠉⠠⣄⡀⠸⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⣥⣾⣘⣿⣿⣿⣇
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠟⢿⣿⠏⠀⠀⢀⣬⠁⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⣀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣧⣼⣿⣿⣿⣿
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠀⠈⠛⠀⠀⠀⢾⣿⠀⠀⣧⠀⠀⢀⣀⡤⠤⠀⠀⠀⠀⠀⢰⣿⣿⠀⠈⠙⠂⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⣼⣿⠄⠀⠀⠀⣴⣦⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠃⠀⠀⠀⠀⢀⡴⠶⠦⣴⠏⠀⠀⠘⠛⠓⠒⠀⠀⠀⠀⠀⠀⢻⣿⠀⠙⢋⣤⠀⠀⠀⠿⠿⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢀⡀⠈⠓⠒⢦⡟⠀⠀⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⣬⣇⣀⣀⡀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠧⣤⣀⢀⡞⠉⠓⠶⠶⠋⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠠⠤⠤⠶⠀⠈⠧⣽⣤⣈⡉⢀⣹⠶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠈⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠈⠙⠿⠽⣯⣥⠶⠚⢿⣛⡛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠰⣄⠸⡆⠀⠀⠀⠀⠀⢀⣠⠴⢋⡿⠁⠀⠀⠀⠀⠀⠋⠁⠀⢠⠖⠀⠀⢀⡴⠛⢧⣀⣠⡴⠟⠙⣧
⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠈⠻⣇⠀⠀⠀⣀⡤⣾⠁⠀⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⣼⣃⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢦⡀⠀⠈⠛⠚⠋⠁⢀⣨⠟⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⣆⠀⢹⡄
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠉⠳⣤⣀⣠⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢻⡇⠀⠀⠀⠀⠀⠀⢿⡀⠘⣧
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢨⡗⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⣾⡇⠀⠀⠀⠀⠀⢠⡟⠀⣼⠃
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠛⣳⣤⠞⠁⣾⠁⠀⠀⠀⣀⣰⣟⢀⣼⠃
⠀⠀⠀⢸⣇⣀⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠞⠁⠀⠀⠸⣧⣄⣀⣀⣼⡋⠉⣽⣿⣽⡇
⠀⠀⠀⠀⣿⠉⠉⠙⠛⠛⠒⠲⠶⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠉⠉⠙⠋⠁⠀⠀⡇
⠀⠀⠀⠀⣿⠀⠀⢀⡀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⡀⠀⠈⢷⡀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠈⣧⠀⠀⠀⢠⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠈⣷⡀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣿⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠸⡇⠀⠀⠀⣾⣧⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⢀⡇⠀⠀⠀⡿⢸⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣿⣷⠀⠀⢸
⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⣸⠃⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣏⢸⠀⠀⢸
⠀⠀⠀⠀⢸⠃⠀⠀⣼⠁⣿⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣠⡯⠸⣦⣤⣼⠆
⠀⠀⠀⠀⣿⠀⠀⢀⡟⢸⡇⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠇⠀⢹⠀⢸⡄
⠀⠀⠀⠀⣿⣤⣤⣼⠇⢼⣦⣤⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢀⣸⡇⠀⡼⠀⢸⣇
⠀⠀⠀⣰⠛⠛⠛⠻⢶⡾⠷⢦⡾⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠿⣚⣫⣥⡿⡿⠵⠚⣉⡼⠂
⠀⠀⠀⠘⠓⠚⠛⠛⠛⠳⠶⠶⢤⣤⣤⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠀⠿⠤⠶⠚⠋⠁

⠀⠀⠀⠀⡄⠀⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀
⠀⠀⠀⠀⠇⠀⠇⡀⢐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣵⡇
⠀⢀⠀⠀⠀⠀⠇⠀⠸⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠠⠠⠀⠀⠀⠊⠀⠀⠀⢸⣿⣿⡇⢨⠀⣠
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠄⣀⠀⠀⠀⢀⠀⡇⡀⠀⠰⢶⡶⣶⠂⠀⠀⠀⠈⠀⠀⠀⠂⠀⠈⠄⠐⣿⡿⣿⠃⣴⣾⡏⢿⣿⡬⠀⣭⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣅⡀⢠⡄⠀⠀⡍⠃⡀⠀⢘⣿⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢦⡜⠋⠉⠩⠟⢻⠷⣾⣿⡿⢸⠟⡹⠿
⠀⠀⠃⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠐⠀⠁⡟⠈⢻⣿⡿⠿⡟⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠈⠀⠀⢸⣿⡇⠈⠐⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡸⡤⠀⠀⣠⡆⡤⠀⠀⠀⠁⠀⢸⡁⢴⣰⡿⣴⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡄⢀⣤⡴⢄⣠⣾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠠⣿⣷⣶⣿⡉⠀⠰⢇⡀⠀⠀⢸⣿⣿⣿⣷⣘⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣅⣉⠙⠛⣿⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡀⠀⠘⠏⠀⠀⠈⣿⣿⣿⣿⣿⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⢩⠛⠸⠀⣿⡇⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣧⡀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠋⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⣤⣿⠄⣤⣤⣿⢧⣶⣾⢟⡓⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠿⠇⠘⢿⢿⡁⠀⠀⠀⠀⠀⠹⡟⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣤⣿⡿⠿⠛⠛⠙⠋⠑⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢰⠀⠀⠀⢘⠁⠀⠀⠀⠀⠀⣠⠟⠛⠓⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠉⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠠⠾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢰⣆⣀⣀⡀⠤⣀⣀⣀⣀⣀⣀⡔⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⡦⠤⠤⠒⠒⠋⠉⠁⠀⠀⠀⢷⣀⡠⠤⠤⠖⠓⠒⠚⠉⠉⠳⣼⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⣀⣀⡀⠀⠀⠀⢀⣀⡨⡇⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡘⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠉⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠙⢦⡔⠁⠒⠤⢤⣷⡒⠒⠲⢶⣿⣿⣿⣿⣿⣿⡀⠀⠀⠁⠀⣠⣤⣶⠶⠛⣁⡴⠁⠀⠀⠀⠈⠓⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢎⡀⠀⠀⠀⠀⠀⠉⠒⢲⣿⠟⠛⠛⠿⣿⣿⣿⣄⣀⣀⡬⠭⠔⠶⣿⣿⣷⣤⣑⣶⣤⣀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠶⡤⠤⠶⠒⠚⠉⠉⠙⡦⣄⡀⠀⢀⣴⢏⠏⠀⠀⠀⠀⠀⠈⠙⠋⢻⠁⠀⠀⠀⠀⠈⠻⣿⣿⡿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣸⠀⡇⠀⠀⠀⠀⠀⠀⣠⠶⠶⠿⠿⠛⠙⠈⠀⠀⠀⠀⠀⠀⠀⣦⣶⠸⠀⠀⠀⠀⠀⠀⠀⢻⡟⢂⣂⣀
⠀⠀⠀⠈⡞⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⡛⢲⣾⠇⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡈⠀⠉⠉⠉⠉⠙⣻
⠀⠀⠀⢀⡇⡀⠀⠀⠀⠀⢀⣀⡠⠤⡤⠄⣸⣿⠀⠀⠀⠀⠀⠀⠱⣷⣸⡀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⣧⣇⣀⣀⣀⣀⡀⠀⣏
⠀⠉⠉⠉⠀⢻⡿⠛⠛⠉⠉⠀⣠⠞⠛⠛⠯⠙⠂⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⢹⡏⠉⠉⡏⠉⠉⠛⠛⠉⠁
⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⣸⠃⠀⠀⠀⢀⡀⢢⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠈⣷⠤⠤⠧⡤⡀
⠀⠀⠹⠛⠛⠛⠓⠒⠛⠒⢚⣿⠀⠀⠀⠀⠀⠹⣮⣷⡄⠀⠀⠀⠀⠀⠀⠈⣿⣷⡀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⡇⠈⠳⠒⠖⠂
⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠹⣿⣿⣄⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠈⣿⣦⣸⣄⠀⠀⠀⡇
⠀⠒⠛⠒⠒⠀⠀⠐⠒⢲⡿⣿⣿⣆⠀⠀⠀⠀⠀⠙⣿⣿⣦⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⣿⣿⣛⣓⣦⣀⠷⠶⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠈⢿⡝⢷⡄⠀⠀⠀⠀⠈⠻⠟⣷⡄⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⢨⡿⢿⡇
⠀⠀⠀⢠⡶⠶⠒⠚⠉⠁⠑⠂⣬⣿⣎⠻⣦⠀⠀⠀⠀⠀⠀⠈⢿⣦⣀⠀⠀⠀⢀⣿⣷⡤⠄⠀⠀⢀⣾⣿⣛⡶⠶⠀⣀⣤⡾⠃⢸⣷⣶⡶⡖⠂
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⠜⠁⠱⠙⣿⣿⣷⣄⠀⠀⠀⠀⠀⢠⡿⠿⣿⣿⣿⣿⣿⣿⣿⡷⣶⣶⠟⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⢸⣿⠀⢸⡇
⠀⠀⠀⠸⠶⠶⠦⠤⢤⡴⠋⠀⠀⠠⠃⠀⠙⢻⣿⣷⣤⣤⣶⣃⣊⣀⣤⣾⣿⣿⣿⣿⣿⣿⣼⠏⣠⠖⠛⠛⢲⡀⠀⠀⠀⠀⠀⠀⢸⣯⣄⡠⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡠⠂⠀⠀⢀⣴⣿⣿⣿⣿⠿⠛⠉⠉⢫⠀⢻⣿⣿⣿⣿⣿⣿⡏⢸⠃⠀⠀⠀⠀⡇⠀⠀⠀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣾⣴⣶⣶⣾⣿⠿⠟⠋⠁⠀⠀⠀⠀⣀⣤⣧⠼⣿⠧⢤⣉⡉⢻⡇⢸⡄⠀⠀⠀⣰⣃⣤⣶⣟⣯⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢀⣤⣴⠶⠟⠋⠁⠁⠀⡟⠀⠀⠀⠈⠙⢷⣄⣳⣤⣶⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡷⠖⠛⠉⠉⢀⣀⣀⣠⣤⣴⣾⣷⣦⣤⣄⣀⠀⠀⠉⠉⠁⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⢻⠁⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠙⠋⠀⠀⠀⠀⠉⠉⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣍⣬⣍⣴⣭⣭⣍⣴⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠈⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣷⢹⣎⡿⡾⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣒⣀⢨⠿⠋⠹⣿⣿⠟⢿⣻⣿⣿⣯⣿⣿⡗⣽⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⠟⠀⠀⠀⠀⠁⠀⠹⣼⣿⣿⣿⣿⣿⣿⠿⡿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⠏⠁⠀⣀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⣶⡇
⣷⡄⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⢰⣿⣷⣦⣦⣄⡘⣻⣿⡿⣫⣿⣿⣿⣿⣿⣿⡿⣷⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣶⡟⢟⣡
⣟⢃⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⠆⡻⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⢿⣿⣿⣻
⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣭⣼⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢹⡶⠤⣿⣇
⠟⠛⠁⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣛⣯⣾⣿⣿⣿⠿⢟⣛⡏⠉⠉⣥⣤⠀⢩⣭⠛⣛⡛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡀⣀⡤⠖⢛⡟⠙⠓⠶⣄
⠀⠀⠀⠀⠈⠳⠄⣨⡾⢿⣄⣠⡴⣶⣚⣛⣿⡿⠟⠋⢡⡶⢦⣹⣿⠁⠀⣰⣿⣿⡀⣼⣿⣸⡟⢁⣼⢷⡖⢀⣀⠹⢿⣿⣿⣿⣿⣿⠟⠛⠋⠋⣿⣟⡻⢮⣶⣦⣌
⠀⣀⡤⠶⠶⣶⣿⣿⣾⣴⣿⣾⣿⣿⣿⠟⣫⡶⠄⠀⠸⣷⠘⣿⣿⠀⢀⣿⣋⣿⢡⣿⢹⣿⢣⣿⢿⡿⢣⣾⣋⣿⠀⣈⡛⢿⣿⣿⡇⠀⠀⠀⠻⣿⡿⠷⢬⣷⣿
⠞⠉⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⡰⣯⣶⡶⣶⡄⠙⠶⠟⡉⣁⣈⣉⣤⣥⣤⣭⣈⡉⠛⠓⠿⢧⣾⣻⣿⣁⣙⠉⣿⠀⠀⠉⢿⡀⠀⠀⠀⠀⢀⠀⠸⣷⣿
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡟⣡⣬⣙⣿⣷⣠⣶⡿⠃⣠⢴⣿⣿⣿⣿⡿⣛⣛⡛⢿⣿⣿⣿⣷⣶⣦⣌⡙⢿⣿⣁⡀⣴⣄⢀⣤⡈⠿⠀⢀⢠⡀⠀⠳⠐⣿⣿
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣏⣤⣍⣹⣿⣯⣛⠟⣁⣴⣿⣿⣿⣿⣿⣿⣷⣾⣷⣿⣿⣷⣚⡯⢍⣛⠻⢿⣿⣿⣦⡙⠻⣾⢋⣿⡿⢉⣿⠆⠀⢻⣦⠈⢲⣄⡀⣿⣿
⠀⠘⣿⣿⣿⣿⣿⣿⡿⠁⣿⣍⡙⠛⢿⡿⣧⣾⣿⣿⢟⣿⣛⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⡷⠮⣙⢿⣿⣦⣀⠿⠁⢠⣿⣧⠾⠇⢀⣿⣿⡝⠃⢹⣄
⠀⠀⢹⣿⣿⣿⣿⣿⣷⠀⢈⣻⣿⠷⠟⣱⣿⡿⢫⣾⢋⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠰⣄⠻⠿⡿⢷⣝⢿⣿⣆⠸⣿⠁⠀⢀⣴⠞⣹⣿⣷⠀⠀⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡾⢛⣿⡇⣼⣿⣿⢱⠒⢻⣿⠓⠻⢡⣿⣿⢫⣧⣤⢧⣾⣿⣽⣿⣆⣺⣶⣿⣶⣝⢲⣮⡏⢻⣿⣷⡀⣠⣶⣿⣿⣿⣿⣿⣿⡀⢠⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⠑⠛⠿⣿⣿⣿⣿⣦⡳⣿⡻⣆⣠⣿⣿⣿⠻⠿⠋⣿⡉⠛⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡇⣿⣿⣷⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡉⣿⣿⣷⣶⣾⣿⣵⣿⣿⣿⣿⡇⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⡇⣸⣿⣇⠶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⣿⣷⣼⣿⣿⣿⡿⠿⠿⣛⣿⣿⣿⣻⣿⣿⣿⡎⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣽⣾⣿⣿⣿⣷⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⠿⢿⡿⣷⡁⣸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⣿⣷⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣽⣿⣿⣿⣷⣼⣿⣟⠹⣿⣿⣿⣿⣿⣿⣟⠀⠀⣇⠈⠻⠿⢿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣹⠷⠾⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⣼⠷⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⢿⣿⣯⢙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢻⣬⠿⣏⢈⣛⡻⠻⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣃⣟⠋⢹⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠉⠉⠛⠣⠻⣿⣏⣄⣠⢾⣄⣸⠿⢽⡦⠀⠀⣀⡀⢈⣱⠾⣍⠰⠗⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠹⣄⢀⢸⣄⠀⠀⠀⠀⠀⠐⠀⢈⣙⣛⠾⠶⢤⣄⣬⣇⡀⠻⡬⢵⡆⣄⠀⡀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⡀⠉⠀⠻⣇⣰⣀⣀⣠⣿⠤⣯⣽⣟⣹⡄⠀⠈⠉⠙⢿⣴⣿⣧⣁⡁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠉⠉⠉⣥⣰⠐⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣧⣤⣿⣯⡙⠀⣤⡇⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⣮⠻⣶⣤⣤⣶⠶⠛⠁⠀⠻⢷⣇⡐⣦⣤⣀⠀⠀⠀⠀⠉⠙⠿⢁⠀⠈⠙⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠒⢻⠿⠿⣷⣶⠶⠆⠀⠀⠀⠀⠈⠹⣾⣿⣿⣶⠀⠀⠀⠀⠀⠀⠨⢳⣼⣿⣶⣶⣗⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡃⠉⠁⠠⠾⢿⣿⣷⡆⠀⠀⠀⠀⠀⠀⢺⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⡥⠤⠄⠉⠛⡿⠧⢦⠤
⠀⠀⠀⠀⠀⠀⠀⢐⣈⡁⠀⠀⠀⠀⠀⠀⠀⢽⡁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⢂⣤⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⣼⠁⠉⠀⣀⠴⠋⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠒⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠇⠀⠀⠉⠀⣀⠞⠁
⠀⠀⠀⠀⢀⣼⡏⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⡀⠀⠀⠀⠁⢠
⠀⠀⠀⠀⠈⠀⠀⠀⠃⣠⢾⡇⠀⠀⠁⢆⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠈⣷⡦⠀⢀⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠉⠀⠀⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⡿⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⠀⠀⠀⢀⣠⡄⠀⠀⢰⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣉⣁⡠⣴⡟⠿⠃⠀⠀⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠟⠃⠀⢀⠀⠀⠀⠀⠀⡿⡷⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡄⢠⡤⠀⠁⡀⠀⠀⢠⠇⠁⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠘⠿⣿⣶⠫⠭⠉⠁⠀⣰⠏⠀⢠⣦⣶⣤⡦⢀⣭⠛⠳⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⢤⡀⠀⠀⠈⠹⣧⣤⣤⣴⡿⠃⠀⠀⠀⠀⠀⠀⡴⠟⠁⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢁⠒⠠⢬⡇⠀⠀⠀⠈⠉⠉⢀⡴⠉⢹⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⢀⣈⢱⡞⣷⠀⠀⠀⠀⠀⢀⡾⠀⣰⡇⠙⠂⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣤⣬⡟⠀⣿⠀⠀⠀⠀⠀⣸⢡⠿⡟⠻⢧⡀⠈⡇⢸⣾⣦⡀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠉⢘⡀⠀⣿⡄⠀⠀⠀⢠⣿⡾⢆⢈⠿⣄⠁⠐⣇⣸⣿⠞⠓⠄⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⠀⠙⢦⣾⣿⣦⡶⣶⡝⠿⣿⣦⡉⠂⠀⠀⠀⠘⢿⣿⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢻⣿⣏⠉⠀⠀⠈⠻⣿⣷⡦⢠⠄⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣶⣄⠀⠀⢬⣿⡄⠀⠀⠀⠀⠈⠓⠲⢤⣀⠀⠀⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⡟⠘⡇⠉⠉⠀⢀⡀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⣀⣄⣼⠃⠀⢻⢷⠄⠀⠀⠻⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣹⣿⣦⡀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣷⣶⣦⣀⣀⣀⣀⣙⣿⣿⣿⣦⣤⣤⣤⡴⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⢹⣿⡟⡏⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣏⣟⣹⢸⡌⢿⡇⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⡿⠻⣿⣿⣿⣿⣻⡿⣫⣿⣿⠟⡼⣿⠁⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠿⠁⠋⢰⡿⢿⠏⡼⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⠀⠀⠀⢠⣤⡀⠀⠀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⡀⠀⠈⣿⡇⠀⠀⢿⣿⣿⣯⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⠀⠀⠸⣷⡀⠀⣼⣿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⣠⡤⣀⠀⠀⠀⠘⣿⣿⣿⣶⣄⣀⣹⣅⠘⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⢿⣀⠀⢿⡇⡬⠀⠀⢤⣄⠀⠹⣿⣿⢧⣀⠙⣿⣾⣿⣿⣿⣋⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆⠁⠛⠀⠀⠈⣨⣽⣳⣶⣾⣿⣿⣿⣧⣤⣎⣉⠛⠿⠂⠀⠀⠀⡿⠃
⠀⠀⠀⠄⠀⡀⠀⠀⠀⠘⣿⣤⠄⢀⡷⢤⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣯⠙⠀⣼⡀⠀⠀⢀⣷⡆⢘
⠀⠀⠰⠄⠀⠈⠀⠀⠒⢾⣿⣯⡀⠀⣿⣾⣿⣿⣿⠿⢹⣯⣴⡟⣻⣿⣿⣷⣄⠀⢑⣤⡴⣿⠟⠀⠸
⠀⠀⠀⠀⠀⠀⣀⠀⢀⣼⣿⣿⣿⣦⡙⢿⣿⣿⣿⣾⣼⣿⠟⣼⣿⣿⣿⣇⣸⠆⢻⣿⣇⡿
⠀⠳⣶⠷⠀⢿⣿⢸⣆⠈⠋⣹⣏⣿⢿⡆⢹⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣻⣿⣄⠈⢻⣿⣷⣤⡘⠦⣈⣶⣶⠄
⠀⠀⠀⠀⠀⠀⢻⡟⢛⡀⢠⣿⡟⠋⠀⢷⡈⠏⠛⠿⠿⠽⠟⢿⣏⠻⠿⠉⠿⠿⢶⣄⠹⠋⠹⠟⠛⠛⠉
⠀⠀⠶⠀⣠⣶⣾⣿⢾⣿⡿⣿⣆⣀⣀⣀⣹⣆⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⢀⢀⣼⣿⣿⣿⣿⣿⣿⣉⡃⢙⠛⠯⣧⣯⣤⣤⣶⣶⣶⣿⣿⣤⠤⠶⠦⠴⠖⠒⠒⠊
⠀⢶⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣷⠤⠤⠀⠀⠀⠰⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣱⣿⣶⣟⣵⣧⣿⣷⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⣿⣿⣿⠋⠈⠻⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢀⡁⠀⠀⠀⠠⠶⣶⣤⣤⣴⠃⠀⢀⣀⣀⠲⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡜⠀⢸⣇⠀⠀⠀⠀⠀⡠⠀⢸⡯⠀⢠⣿⣿⣿⣷⣦⣌⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠈⢿⡄⠀⠀⠠⠒⠀⣒⣾⣆⡀⣴⣿⣭⣻⢿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⢷⣌⠙⠦⣄⡀⠀⠐⠒⣼⢿⣿⣡⠙⠿⢿⣷⣝⢿⣿⡿
⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠓⠲⠿⡧⣀⠛⣿⣿⠒⣴⣿⣿⣿⣿⣷⢢⣄⣝⡿⣻⣿⡿
⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⡜⢷⣹⡄⣿⣿⣿⣿⣷⣿⣿⣿⣿⣧⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠻⠁⠈⢿⣿⣿⣿⣿⡿⣿⣿⡟⣿⣿⠇
⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠈⠞⣷⣼⣻⣿⣿⣛⠋⢀⠈⠙⠛⠛⠓
⠀⠀⠀⢨⡻⣷⣄⠀⠀⠀⠀⠀⣠⣾⣏⣀⡀⠀⠀⠀⠀⣔⡻⣿⣿⢯⣿⡟⣤⠋⠀⠀⠀⠀⣿
⠀⠀⠀⠘⡆⢟⣿⡧⠀⠀⠀⠈⠚⠁⠸⣿⣿⣶⣦⡴⠎⠛⢿⣷⣷⣿⡟⠀⢻⡀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠹⣼⡜⣧⡀⠀⠀⠀⠀⠀⠀⢹⠟⠋⠁⠀⠀⠀⠀⠀⣹⣯⣀⠀⠀⠁⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⢹⣿⣎⢷⣄⢀⠀⠀⢀⠞⠋⠀⠀⠀⠀⠀⠀⠀⢠⢿⣿⣿⣿⣦⡄⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣬⣒⢦⠀⠀⠀⠀⠀⠀⣀⠀⢠⠏⢸⠀⠈⠉⠉⠛⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠿⣝⠿⡟⠁⢀⣤⣴⣶⠞⢁⡴⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⠈⠡⣄⡀⠈⠛⠋⣡⠞⠋⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠙⠓⠶⠞⠁⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⣾⡿⢿⡿⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢠⡿⠁⠈⣿⡄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⢀⣾⠃⠀⠀⠘⣿⡄⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢀⣾⠇⠀⠀⠀⠀⠘⣿⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⣧⣤⣤⣾⣿⣤⣤⣤⣤⣤⣤⣼⣿⣶⣶⣿⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⢿⡋⢹⣿⠋⠉⠉⠉⠉⠉⠉⠉⠙⣿⣽⠟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠈⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣀⢴⡿⣿⡀⠀⢀⣀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⣀⠐⠐⢿⢆⠉⠢⡼⣟⠻⢿⣷⣶⣶⣶⣾⡿⠟⠋⣠⢿⡟⠈⠱⡿⠂⠂⣀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠋⢠⠁⢉⠳⣄⠧⡁⢸⠃⣻⣇⠀⠀⠈⠉⠁⠀⠀⠀⣰⣟⠘⡃⢀⠸⣠⠞⡉⠈⣄⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠐⡄⢳⡊⠀⠠⣾⢰⠘⡾⠳⢷⣯⡀⠀⠀⠀⠀⠀⠀⢀⣼⡯⠖⢷⠃⡆⣷⠄⠀⢑⡞⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠉⢭⠖⡡⠊⡠⠁⡠⣲⡏⠐⢄⡢⠀⢀⢄⡨⠂⢹⣖⢄⠈⢄⠑⢌⠲⣍⠉⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠙⠒⠈⣠⣜⡵⠕⠃⠀⠀⡨⡪⢕⢁⠀⠀⠈⠯⢧⣡⣄⠁⠒⠋⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⡄⠄⣲⠟⣌⡀⠀⠉⠛⠖⠠⢤⡄⠀⡀⢦⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠘⢃⡵⡆⠀⡿⡆⠈⣿⠲⣤⡀⣠⡞⠃⢪⡑⢷⠈⢧⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠶⣷⠀⣿⢳⣧⢰⡷⢿⡆⣼⠿⣅⠁⠻⣧⠐⢺⡟⣿⠁⣸⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠈⠀⠙⠀⠛⠸⠁⠘⠇⠸⠀⠈⠓⠀⠙⠳⠘⠃⠋⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠢⡀⠢
⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⠰⠢⠄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⠈⠋⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠖⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠠⠒⠒⠠⠄⡀⠀⣝⠐⢆⣤⢄⠀⡄⠀⠀⠃⠄⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⡀⡴⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠁⠀⠐⢎⠛⢄⢠⠱⣠⣱⠐⠛⠀⠀⠃⠀⠀⠀⡠⠐⠤⣹⠀⠀⠀⢠⢔⡁⡀⠈⢀⡀⠸
⠀⠀⠀⠀⠀⠀⠀⠇⠀⡀⠀⠀⠀⠀⡑⠢⣄⠣⠌⠏⢢⣀⢠⠤⠆⠀⣄⢸⠀⠀⣤⣾⣤⠀⠀⠘⢁⢖⡴⠋⠀⠀⢠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⡀⠈⠁⠀⠀⠀⣠⣀⣤⠒⢆⠀⠀⢀⣼⠈⠳⣤⠽⣅⡄⠁⠁⠘⠒⠋⠀⠀⠠⠴⠅⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠀⠀⠀⢀⣀⡄⢨⠞⠛⡛⠛⠀⠀⣌⠱⠀⠀⠁⡲⠄⣀⢀⡤⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠀⠄⡰⠊⠉⣈⠉⠁⠀⠉⠀⠀⠁⠀⠀⠀⠀⢏⡰⠂⠀⠀⠘⢀⡀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣘⠡⢁⡑⠣⠀⢆⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠈⠉⠉⠉⠁⠉⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⢄⠐⣀⡣⠠⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠦⠬⠉⣇⠀⡀⠓⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠉⠉⠉⠈⠉⠀⢰⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⢁⠖⠂⠚⠁⠄⢤⠍⠁⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢃⠄⠀⠀⠀⠀⠀⠚⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⢀⠄⠀⠀⠀⠀⠀⠀⠑⠢⢄⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠀⠐⠒⠀⠀⠀⠀⠀⠄⠤⠬⠙⠂⠀⠀⠀⠀⠛⠻⠆⠀⠚⠀⠀⠀⠀⠀⠀⠀⢠⣒⣊⣀⠀⠬⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣄
⠀⠀⠀⠀⢠⢾⣿⣿⣽⡿⢟⣧⡀
⠀⠀⠀⢸⢿⣿⣿⣿⣶⣺⣏⡛⣿⣦⣄
⠀⢀⡾⠎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⢸⡗⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠲⣷⡘⢿⡡⢙⣛⡋⠉⢿⣿⣿⣿⣿⣿⠏⢼⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠉⢉⣡⠼⣷⣾⣷⠀⣾⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣟⣿⣦⡭⠛⢉⣡⣾⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠉⠁⢠⡆⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠛⢁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡆⠀⢸⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣞⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡄⢀⠻⣿⣿⣿⣯⠈⠛⠛⠛⠛⠋⠉⠀⠀⢀⣀⣉⣉⠈⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠉⢹⣿⣿⣿⣿⠓⠒⠲⠖⠒⠒⠖⠛⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠋⠀⠀⠈⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣍⠄⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⡀⠀⠀⠲⠦⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⢻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠂⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠀⠈⠻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⡀⠀⠀⠀⠈⢋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠃⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣄⣴⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣤⣴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢠⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⢸⠀⢀⡖
⠀⠀⠀⠈⠉⠙⠲⠶⠶⠦⠤⢤⣤⣀⣀⣀⠀⠈⠃⠈⠀⠛⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⠤⠤⠤⠤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢻⡄⠀⣼⠋⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢠⡏
⠤⠴⠶⠒⠓⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣿⡇⢸⠃⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣒⣶⡤⢶⣄⣠⠀⠀⣷⣿⡇⡿⠀⠀⠀⣺⠟⠀⠀⠀⠀⠀⢀⣰⠏⠀⠀⠀⠈⠉⠛⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣈⡀⠀⠀⣿⣿⠁⠓⠶⠶⡾⠃⠀⠀⣀⡴⠚⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢯⣽⠟⠋⠁⣀⠀⠀⠀⢧⣀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⣿⠀⠸⣄⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣿⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠚⠉⠹⠁⠀⠀⠀⠀⠙⠲⠤⠤⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⡤⠒⠊⠉⠁⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀
⠀⠀⠀⠀⢸⡄⠀⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⢀⣾⡇⠀⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠠⠚⠛⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢰⣤⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣧⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⣿⣿⠁⢳⡀
⠀⠀⠀⠀⠀⠙⢀⣀⣀⣐⣿⣧⡀⠀⢀⣤⣴⣿⣷⣄⠀⠀⠀⠄⣾⣿⣿⣦⣤⣤⣤⣤⣴⠟⠃⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠉⠛⠛⠋⠁⠈⠙⠛⠓⠒⠚⠛⠻⢿⣿⡿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⡠⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠗⠐⠒⠒⠒⠒⠒⠒⠒⠒⠉


⣿⡏⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⣠⡖⡦⠶⡄⣀⠀⠀⠀⠀⣀⡤⠶⠛⠛⠛⠛⠛⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⡤⣾⣼⣇⠀⠀⠀⠀⠀⣗⠿⣴⡖⢟⡼⣛⣦⡀⣾⡏⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁⠀⠀⠀⠐⣆⠀⠈⠳⢦⠀⠀⠀⠻⢭⣿⡇⠋⠙⠷⣺⣼⠏⢻⣸⡔⣆⡀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⣄⠀⣿⢀⡀⠀⠈⢧⠀⠀⠀⣴⡏⠙⢶⣖⠉⠁⡾⠀⠀⠙⠗⠈⢳⠀⡴⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣿⠀⠛⢀⣩⡷⠀⠈⢹⣤⠞⠁⠈⢳⡞⠉⠓⢦⡟⣻⠁⠀⠀⠀⠼⢿⣁⣤⡀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣀⣟⠀⠀⠀⠈⠀⣠⠖⢚⡇⠀⠀⣰⠏⠀⠀⣀⠠⣟⠀⠀⠀⠀⠀⠀⠸⠇⠀⣽⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⣿⠉⠉⠀⠀⠀⠀⠀⠃⠀⡼⠁⣠⣞⣁⡀⣾⠛⠛⠳⡜⣿⠓⠦⠀⠀⠀⠀⠀⢺⡅⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⢀⣈⣿⠖⠀⠀⠀⠀⢀⣠⠞⣡⠞⡿⠋⠛⠙⣯⣤⣤⣄⣽⣈⣳⣦⣤⣤⠴⠶⠛⠛⠙⠓⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣏⣉⣽⠷⠒⠒⠓⠲⠿⠶⠾⠷⠶⣧⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣹⡶⠖⠳⠦⠤⢤⡤⢤⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠶⣶⠒⠚⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠿⡄⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠚⠉⠀⠀⠀⠙⣆⠀⠀⣀⣠⡤⠖⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣤⣤⠴⠚⠳⠶⠦⠴⢶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⣀⣠⡤⢿⡿⠋⠉⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢠⠞⠋⠉⠁⠀⡰⠊⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠸⣷⣄⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣽⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢰⡇⠙⠳⣄⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡇⠀⠀⠀⡟⠀⠀⠀⠀⣴⠛⠁
⠀⠀⠀⠀⠀⢀⣤⠟⠁⠀⠀⢀⡼⠁⠀⠀⠈⠳⡄⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⣾⣓⣶⠤⣼⠀⠀⠀⢀⡇⠀⠀⠀⣧⠀⠀⠀⠀⢧
⠀⠀⠀⢀⡴⠋⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠹⣿⣍⠓⠮⠷⢤⣤⡬⠇⠀⠀⠀⠸⡆⠀⠀⠀⠈⢧
⢀⡤⠶⣯⡀⠀⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⢀⠀⠀⠀⠈⠺⢷⣆⢄⡀⢸⠁⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠘⢦
⣾⣇⠀⢈⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣧⠴⢯⣉⣿⠀⠀⠀⠀⠀⠈⠓⠶⠟⠀⠀⠀⠀⠀⠀⠸⣇⣀⣠⣤⠖⠻⣄
⠈⢿⣦⠸⡄⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⡤⠾⠷⠖⢋⣽⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠏⣀⠴⠃⠀⣠⠼⡇
⠀⠈⠻⣷⣿⣟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⠀⣠⢴⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣉⣽⣋⣥⠶⣚⡩⠖⠋
⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠓⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠛⠛⠿⢿⣶⣤⣀⠀⠀⠀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠈⠻⣿⢷⢶⠿⠛⠋⠙⠛⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠉⠙⠛⠿⠛⠛⠻⠃⠀⢀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢀⡆⡀⠀⠀⠀⠀⠀⠀⢰⡄⢼⣿⠸⠃⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⢇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⢀⡴⠒⠒⠒⠒⠾⠛⣋⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⣄⠀⣿⣆⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⡾⠀⠀⠀⠀⠀⢀⣈⢷⣾⣿⣆⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿
⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠈⢿⣿⡘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⢀⡏⠀⡀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠑⡀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠈⣷⡀⠐⢼⣄⠀⠀⠀⠀⠀⠀⠈⢻⣷⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢹⣷⠄⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⢸⣿⠀⠀⠠⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⣿⡆⠀⠀⠨⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⣠⣿⣦⡀⠀⠀⠀⠀⢀⡠⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠸⣿⡄⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⣸⣿⣽⣷⣶⣦⣤⣤⠄⠀⠀⣼⣿⡉⠠⣄⣈⣉⣀⣐⡟⠀⠀⠀⠀⠀⠀⠀⣀⣴⡟⠙⣷
⠀⠀⠀⠀⠙⣿⣄⠀⠀⠈⠛⢷⣶⣤⣤⣦⣤⣾⣿⠿⢿⣿⣿⣷⣶⣂⣀⣤⣾⣿⣿⡷⠆⠈⠉⢛⣻⣿⣿⣦⣀⣉⣛⣛⣻⣿⣿⣷⠒⠀⢹⣇
⠀⠀⠀⠀⠀⠈⠻⢷⣄⠀⠀⠀⠈⠉⠛⠛⠋⠉⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠉⠉⠁⠈⠁⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣆⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡂⠀⠀⠀⠀⠀⠀⠀⠀⢩⣽⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠒⠿⠷⠒⠶⠶⠶⠊⠉⠙⠙⠿⣿⡆⠀⠀⠀⠀⣰⣿⠆⣠⡆⠒⠒⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⣿⡟⠀⣿⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢿⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠘⡆⠀⠀⠀⢀⣴⡟⠀⠀⠀⠀⠐⢲⣶⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣀⣀⣤⣾⣿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡭⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣽⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣆⠀⠀⠀⠾⠿⣷⣦⠤⣴⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢼⡿⣿⡄⠘⣦⠄⢀⣤⠀⠀⢸
⠀⠀⠀⠀⠀⣠⣴⡞⠉⠓⠲⣍⢉⣉⣡⣴⣾⢹⣟⣿⣄⠀⠀⢁⣨⣑⣲⠿⢤⡀
⠀⠀⠀⠀⣾⣿⣿⠇⠀⠀⠀⢹⠟⠛⠛⠿⠁⠈⣿⢿⣿⣷⡀⢝⠻⢿⣷⡤⠴⠿⢗⠲⢦⣀
⠀⠀⠀⢸⢻⢹⣿⠀⢀⠀⠀⣼⠋⠀⠂⠠⢈⠀⠋⠀⠙⣿⣷⣦⣽⠙⣿⣿⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠸⣸⣟⣿⣆⣾⣤⣴⡗⠦⠀⠀⠀⠀⢉⢢⡴⢚⣵⣶⣷⣮⣗⠿⣿⣷⡄⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⣷⣿⣿⡿⠿⢿⣿⣷⣄⡀⠀⠀⢠⡤⢋⣴⣟⣿⣿⣿⣿⣿⣇⠸⣿⣿⣷⣦⣸⠀⣠⡿
⠀⠀⠀⣼⠉⡏⣿⣤⠀⠀⢻⡻⢿⣿⣶⣤⠞⣱⣿⣿⣽⣿⣿⣿⣿⣿⣿⠀⣿⣿⠻⢿⣿⣿⣿⣷⣾⣏⠒⢄⡀
⠀⠀⠀⣿⢠⡇⣿⣿⡄⠀⢸⣿⣶⣭⣽⢯⣾⣿⣿⣿⣧⣺⣿⣿⣿⣿⣿⠀⣿⣞⢷⣼⣵⡿⣻⣿⣿⣿⣷⡄⠑⡄
⠀⠀⠀⣿⣿⡇⠘⣿⣷⣾⣿⡷⠞⠋⣀⣸⣿⣿⣿⣿⣿⣿⠛⣿⢿⣿⡏⢠⣿⢿⢮⣿⣟⣾⣿⣿⣿⣿⣿⣿⡄⠘⡄
⠀⠀⠀⢿⡷⠃⠀⠉⠛⠛⢻⣠⣶⣾⣿⣷⡻⣿⣿⡿⢟⣡⣾⣿⣿⣿⡿⠟⠁⣾⠀⠻⡏⣿⣿⣿⣿⣿⣿⣿⣷⠀⢰
⠀⠀⠀⢸⣿⣦⣄⣀⣀⣤⡾⣿⣿⣿⣿⣿⣿⣮⣭⢶⣿⣿⣷⣀⣀⣀⡀⠀⠀⢻⠀⠀⠹⣽⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠘⣿⣿⣿⠿⢛⣭⡾⣟⣿⣿⣿⣿⣿⣿⠿⠟⢿⣿⠿⠟⠛⣿⣁⠄⠀⠘⡆⠀⠀⠹⢝⣛⣿⣟⣛⡛⡫⠕⠋
⠀⠀⠀⠀⢸⣦⣶⣛⣯⣷⠿⢻⣿⡿⣟⣿⢿⣿⣄⣀⣸⣿⣧⣤⣤⡟⢁⠀⠀⢠⡇⠀⠀⠀⠀⠛⠿⠿⠯⠞⠁
⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠘⡟⣧⠙⣿⡎⣽⡿⠿⢻⣾⡿⠚⠋⣿⠏⠀⠀⠾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢈⡳⣬⣿⢿⣿⠀⣸⣟⣃⣀⣀⣿⣠⠄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣿⣾⣿⣮⡻⣷⣤⣴⣿⣿⣽⣯⣀⣀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⢀⣀
⠀⠀⠉⠛⠶⢶⣤⣤⣤⣀⣤⣤⣤⣴⠶⠾⠟⠛⠻⠷⢶⣤⣤⣤⣴⠶⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡴⠋⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⣿⡞⠁⠀⠀⢀⣤⠄⠀⠀⠘⣷⣀⣤⣤⣴⠋⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⡏⠙⣦⠀⣰⠟⠁⠀⠀⠀⠀⢻⡏⠀⢰⣏⡆⢀⠇⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢘⣿⡃⠀⠀⠀⠀⠀⠀⢸⣇⡀⠀⢉⣿⠛⠛⠀⠀⢸⡷⠶⠶⠶⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡘⠉⠻⣦⣄⠀⠀⠀⠀⣸⡇⠉⠙⠋⢸⡧⠀⠀⠀⣸⡇
⠀⠰⣄⠀⠀⠀⠀⠀⠈⠳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠙⠂⠀⠀⠀⣿⠙⠲⠤⠤⢾⡅⠀⠀⠰⠋
⠀⠀⠙⠳⣦⣀⡀⠀⠀⠀⠀⠉⠛⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⢠⣿⢦⣄⣀⣠⣷⡃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⠶⠶⠾⠛⠋⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣤⣤⣀⣤⡾⠻⢤⣀⣠⣼⢿⣇⣀⣠⣴⣿⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⣠⣴⠏⠉⠉⠉⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠲⢶⣛⣥⢾⣄⣀⣀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣼⣏⢉⣠⣾⣀⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣿⣿⣿⣿⣿⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡿⠿⠛⠛⣻⣿⣿⣿⡟⢿⣿⣿⣿⠛⠿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⠉⠁⠀⠀⠀⢠⣿⣿⣿⡏⠀⠘⣿⣿⣿⡄⠀⠀⠉⠻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠈⢻⣿⣿⣿⡄⠀⢀⣠⣤⡶⢟⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠇⠀⠀⠀⠸⣿⣿⣇⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣯⣴⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡏⠀⠀⠀⠀⠀⢿⣿⣿⡄⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠁⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣭⡴⠾⠟⢻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⢀⣿⣿⣿⠇⢀⣠⣤⡶⠟⢛⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⢛⣩⡥⠔⠊⠉⠋⢉⣿⣿⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣀⡤⢖⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⢀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣿⣿⣶⣿⣋⣁⠀⣸⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⠀⢀⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠻⣿⣿⣿⣿⠿⠋⢀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄⠀⣠⣾⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠐⠋⠁⢀⠀⠀⠻⣟⣿⣿⣦⣠⣼⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣾⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⡤⠞⠋⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢟⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⣿⣿⡏⠀⠈⠉⠛⠛⠓⠛⠳⠶⠲⠶⠚⠛⠋⠉⠙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠿⠻⡿⠛⠿⢿⡿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡀⣀⡄⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠈⠁⠈⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠌⢀⠀⠀⢀⠀⠀⠀⠀⠀⠠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠈⠀⠀⡀⠀⠈⠢⣀⠂⠐⠄⠣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠈⠀⠀⡀⠄⠘⠀⢀⠊⠀⠀⠀⢀⣀⠀⠀⡀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⠀⠀⠰⠁⠀⠴⣤⡤⠀⠀⠀⠐⠉⠀⠑⡀⣸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⢌⠀⢀⡤⠤⣄⠀⠀⠀⢀⠀⠄⠀⠀⠐⠐⠒⠂⢄⡏⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠑⠘⣀⡀⠀⠄⠔⠀⠊⠀⠁⠈⠈⠀⠀⠀⠀⠸⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠐⡋⠀⠑⢄⠀⠀⠀⠀⠀⣀⢀⣀⣀⣀⣠⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠤⢄⠀⠰⣤⡀⠀⠱⠀⠠⢀⡥⠠⠇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠘⡤⠊⠀⠁⣀⡞⠀⠀⠀⠑⠊⠚⠒⠶⠒⢲⠋⠈⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⡀⠀⠀⠀⠀⠈⢳⢶⠀⠻⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠻⢄⡱⢄⣀⡠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠤⣀⣀⠟⠀⢣⡀⠀⠀⠛⣆⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⢘⡍⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⡀⠀⠀⠀⠂⠀⠀⠀⠀⠙⢦⣤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠄⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⢸
⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⢠⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠙⣠⠤⠤⢤⢸
⠀⠀⠀⠀⠊⠀⠀⠀⠑⡄⠀⠈⠀⠀⠈⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⢎⢸⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⡇⠀⠀⠀⠀⢀⡤⠁
⠀⠀⠀⠀⠀⠉⠁⠂⠀⠀⠀⠀⠚⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠃⣀⡀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⡇⠀⣉⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⡀⢀⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢠⣽⠍⢻⠟⢁⠀⠀⢀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠁⠀⠀⠀⠀⢀⣾⣿⣿⣇⢸⣇⠘⠂⠀⠀⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣤⠀⠀⢀⣼⣿⣿⣿⣿⡿⢿⣦⡀⠰⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠘⢷⣦⡜⠿⣿⠉⠛⠉⠙⠀⠙⠻⣘⠁⠻⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠘⠋⠀⠀⠀⠓⠒⠀⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⢀⣶⠈⠋⠉⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⣱⣿⣿⡿⣿⣿⡿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣤⣴⣿⡟⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡾⣸⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⢿⣿⣿⠟⢹⣿⣿⣿⠉⠉⠁⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠃⠀⠀⡠⠁⠀⠈⠉⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠜⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⣀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⢿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⡰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣄⠙⠀⠀⠀⠀⢸⡄⠀⢹⠀⢀⡆⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⣄⠀⠀⠀⣸⠻⡄⠀⠀⣼⣇⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣷⠈⠛⠦⣤⠏⠀⠙⢦⡴⠃⠹⣄⣀⡤⠞⣹⠁⠀⠀⢠⠂
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠁⢦⣤⣤⣏⠰⣷⡤⠤⠔⠉⠲⠤⠤⠒⢄⣈⣁⣤⠤⢯⣀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢠⡥⠖⠒⠋⠁⠀⠀⠀⠈⣏⡍⠙⠲⣤⡴⠚⠉⣍⡏⠀⠀⠀⡀⠉⠒⠒⠤⣄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⡞⠀⠀⠀⢀⠃⠀⠀⠀⢹⠀⠀⠀⠀⡇⠀⠀⠀⠸⡄⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢀⡀
⠀⠀⠈⠛⠲⠾⠿⠶⠶⡏⠀⠀⠀⠀⠑⢴⢇⠀⠀⠀⢸⡆⠀⢀⣴⠋⣳⣄⠀⠀⡇⠀⠀⠀⢀⢷⠔⠁⠀⠀⠀⠈⡷⠿⠤⠤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠾⣤⣀⣀⡼⠷⠒⠋⠀⣿⡁⠈⠓⠲⠷⣄⣀⣠⡾⣎⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡤⣟⣧⣀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⢤⠟⠒⠳⠶⠶⠒⠒⠛⠉⠀⣾⡄⠈⠙⠒⠒⠲⠶⠶⠓⠛⢦⠀⠀⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⢦⣀⣀⠀⣀⣀⣀⣀⡤⣞⠁⢹⠦⣄⣀⣀⣀⡀⢀⣀⣠⠞
⠀⠀⠰⠖⠒⠒⠚⠛⠛⠚⣧⡀⠀⠀⠀⢾⠀⠉⠉⠉⠉⠉⠀⢀⡿⠀⢸⡄⠀⠈⠉⠉⠉⠉⠁⢸⠀⠀⠀⠀⣠⠗⠚⠿⢿⣓⠒⠒⠶
⠀⠀⠀⠀⠀⠀⠛⠁⠀⠁⠀⢉⠶⠤⢤⡬⠷⢴⣤⣤⣤⣤⣴⡋⠓⢋⣁⣉⠳⢤⡤⣤⣤⡴⠴⢯⡤⠤⠐⠚⠤⢄⡠⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢀⠂⠈⡇⠈⣠⣄⡈⠶⠋⣠⣄⠙⠚⢠⡟⠛⠲⣤⣘⣇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠁⠪⠀⠀⣧⠞⠁⡈⢷⠀⣸⠃⠈⠳⣄⢸⡇⠀⠀⠀⡉⠻⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢠⠀⠇⠈⣧⡏⠀⠀⠀⠈⢻⡇⠀⠀⠀⠙⣆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⡀⢀⠀⠀⠀⠁⠀⠀⠀⠋⠀⠀⠀⠸⠁⠀⠀⢸⡀⠀⠁⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏⠉⠉⠳⠞⠉⠙⠓⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡀⢀⣆⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠙⢿⣿⣷⣦⣄⣀⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢶⠚⠉⠀⠀⠀⠀⠀⢈⡹⠟⠋⠉⠙⠦⣄⣀⣀⣄⣦⣀⠀⠀⠀⣠⣾⣯⣀⠟⢄
⠀⠀⠀⠀⠀⠀⢀⡾⣷⣄⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣾⡛⠛⠛⠁⠀⢠⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⢀⠔⠉⠀⠀⠙⣻⣾⠟⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣷⣄⣀⣠⡴⢋⡤⠞⠉⠀⢀⡟⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⣼⣿⡟⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠹⡍⠛⠉⠉⠀⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠃⢀⠀⠀⢠⣿⡟⠁⡄⢳⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠘⠛⠒⠉⢧
⠀⠀⠀⠀⣠⠤⠛⠓⢿⣿⠁⠀⠹⣼⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠉⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⡀⠖⠚⠀⠀⠀⠀⠘⡿⠀⢀⡞⡽⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠
⡰⠁⠀⠀⠀⠀⠀⠢⢵⠃⠀⢸⡄⠁⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⡇⠀⠀⢠⠀⠀⢀⣇⣀⣴⣤⣤⡤⠔⠒⠉
⠙⢆⣀⠀⠀⠀⠀⠀⠸⡄⠀⠈⣧⣀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⢸⠁⠀⠀⡎⠀⠀⡼⠀⠉⠉⠙⠋⠉⡼
⠀⠀⠎⠀⠀⠀⠀⣀⣤⣧⠀⠀⠈⠛⢶⣤⣀⣀⣠⣤⠖⠁⠀⠀⠀⠀⠀⡸⠀⠀⢰⠃⠀⣼⠇⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⣄⠀⣴⣾⠟⠉⢹⠀⠀⠀⠈⠁⠀⠈⠉⠻⢅⡀⠀⠀⠠⢀⣀⡴⠃⠀⢀⡎⠀⣼⡇⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠘⠞⠉⠁⠀⠠⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⣶⣎⠁⠀⢀⣀⡼⢀⡴⠋⣿⣶⠒⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠈⢇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣯⣭⣷⠟⠀⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠙⠒⢒⣒⣶⡤⠵⣄⡀⠀⠀⢀⣿⣿⡿⠿⠛⠁⠈⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⢠⣾⠟⠉⡟⠀⠀⠀⠀⠀⠉⢉⣴⠿⠛⠦⡀⠀⠀⣀⣀⣀⢀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠇⠈⠀⠀⡇⠀⠀⠀⠀⠀⣴⠻⠃⠀⠀⠀⠈⠁⠒⠉⠉⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣶⣶⣶⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠂⠉⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠠⣔⣀⣀⠈⠁⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢈⠀⠏⠑⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⠾⢻⡙⠛⡿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠔⠁⠐⢀⠀⢇⠈⠈⡅⠀⡀⡀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠘⠂⢀⠀⠀⠄⠀⠀⠒⢄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢠⡀⠀⡠⠀⠀⠀⢠⠖⠒⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡅⠀⠀⠀⠀⠀⢘⠀⡎⠀⠀⠀⠤⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣶⠶⠶⡆⠀⠀⠀⠀⢀⠀⠀⠀⠐⢃⡟⠀⢒⠂⢂⣀⠥⠶⡁⠀⠀⠈⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⢠⡶⠾⠛⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠋⠀⠀⡠⠖⠁⠒⠦⣄⠈⢢⡀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⠎⠀⠀⠀⠀⠀⠈⠱⡄⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣃⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠠⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⡇
⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣷⡄⠀⠀⠀⠰⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡇
⠀⠀⠀⢀⣶⣿⡿⠛⠋⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠗⠈⠐⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸
⠀⠀⢠⣿⣿⠏⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠐⡄
⠀⠀⢼⣿⡏⠀⠀⢀⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⠀⠀⣿⣿⣿⡆⠤⠄⠁⠀⠔⠙⠀⠀⠀⠀⠀⠀⢁⡲⢄⡀⣐⠒⢀
⢀⣴⣿⣿⣇⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⢠⣼⠿⣿⣿⡗⠒⠈
⣿⣿⣿⠉⣿⡀⠀⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠁⠀⣿⣿⠇
⠘⢿⣿⠀⠉⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢠⣶⠿⠃
⠀⠈⠿⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⢡⣿⣿⣿⣿⣿⠋⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠟⠉⣼⣿⣿⣿⠃⠱⢹⣿⣿⣿⠙⠿⣿⣿⣿⣿⣄⣀⣀⣤⣤⣶⣶⡿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠋⠀⠀⢠⣿⣿⣿⡏⠀⠀⢸⣿⣿⣿⣆⣀⣬⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠁⠀⠀⠀⣾⣿⣿⡿⠁⠀⣀⣤⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃⠀⠀⠀⣸⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⢀⣀⣼⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⢹⣻⣿⣿⡇⠀⠀⠈⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠁⠀⠀⠀⠀⠈⢣⢻⣿⣿⡀⠀⢰⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠶⢾⣿⠿⠿⣿⣿⣿⣿⣤⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣷⣤⣀⣀⠀⠀⠀⣀⣀⣠⣤⣶⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣿⣿⣿⡏⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⡿⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠒⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢀⠀⠀⠚⡂⠘⠥⠘⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣘⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⢠⡀⢀⡐⠀⠘⠃⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠄⠀⠀⠀⢀⠠⢰⠀⠠⢐⡀⠘⠻⡈⠑⠱⠀⠀⠀⢀⠀⡁⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⢄⡰⠒⠤⢀⠀⡀⠀⠄⠀⠀⠁⠀⠈⠀⠀⠈⡂⢄⠀⠀⠀⢀⢁⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠤⠒⠂⠀⠀⠀⠀⠠⠔⠂⡀⣀⡂⠂⡁⠀⢰⠄⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⡆⠀⠀⠀⠀⠘⠀⠀⠈⠀⠤⠀⠀⠤⠐⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⣀⣀⡀⠆
⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠠⠀⢠⡀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⣤⣤⣔⡤⣤⣤⡀
⢰⠇⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠈⠈⠐⠀⠀⠀⠤⠚⠀⠈⠁⣀⠀⠀⢀⠀⠀⠀⢻⡵⠞⠡⣄⣀⣉⣺⣿⣿⠶⡄
⢀⠀⠀⠀⠀⠤⠂⠀⣱⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡀⠀⠒⡔⠀⠀⣤⠀⣀⡧⠀⠀⠀⠘⢦⠀⢠⠂⠒⠒⡍⠉⡯⡸⠃
⠀⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠈⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠠⠄⠀⠰⢋⠀⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⣀⠊⣁⣴⣦⠀⠀⠀⠈⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢼⡏⠙⠭⠈⠁⠀⠀⠀⠀⢹⠙⠒⠒⠒⠲⢤⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣀⣠⠑⢄⠀⠀⠀⠀⠀⣼⠖⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⠁⠉⠉⠀⠀⠀⢀⣼⠙⠂⡄⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠛⠓⠪⣿⣷⠒⠋⠀⠀⣀⣴⣿⡿⠓⠈⠀⠀⢀⣤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠛⠲⠤
⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠈⣿⣷⣶⣖⠺⢿⠟⠁⠀⠀⠀⠀⠀⢁⣾⣿⣆⢠⣤⣤⣤⣴⡟⠁⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠻⣽⠃⠈⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣦⣭⣿⣿⣿⣀⠀⠀⠀⢀⡴⠉⠁⠀⠀⠀⠈⠙⠒⠤⣀
⠀⠀⠀⠀⡞⠑⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⡟⣿⣿⣿⣿⠋⠙⣏⣛⣲⡴⠋⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡄
⠀⠀⠀⠀⡇⠀⢶⣦⠀⠀⠀⠀⣴⠋⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⢰⡟⠘⣿⡟⠀⠀⠀⠉⢹⠁⠀⣿⡏⠀⣸⣤⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢀⡴⢣⡀⠀⠻⣷⣤⣤⣾⣿⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣼⠏⠁⠀⣨⠀⠈⢹⠁⠀⠀⠀⠀⠘⢦⣸⠏⢀⣴⣿⣿⣿⣷⣧⣷⣧⣄⢀⣸
⠀⠀⠈⠀⣾⣿⣿⠟⠁⢹⢻⣿⡿⠻⣦⡀⠀⣿⣄⣀⠀⢀⣠⣴⠿⠋⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠐⡇⢠⣾⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠏
⠀⠘⣦⡼⠓⠀⠸⠀⠀⠈⣸⠟⠁⠀⠈⣿⠟⠉⡍⠛⠿⠍⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠲⠭⠭⠭⠿⠿⠚⠉⠉⠉⠉⠁
⠀⠀⢻⡧⠀⠀⠈⠀⢀⡼⠁⠀⠀⠀⠀⢻⣄⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠈⠣⣤⠤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⠳⠂⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⠸⣆⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⢀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠈⡀⠀⠀⠁⠀⣠⠀⢀⣠⠴⢫⢼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣱⢲⢽⣤⡤⠤⠾⣓⠋⠉⠀⠘⠙⠈⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠿⠞⠢⠧⠤⠆⢿⠡⠣⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡠⠂⠣⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣧⣀
⠀⠀⠀⠀⡰⡡⡲⡜⡔⠶⠈⢒⠖⠀⠀⠀⠀⢠⡾⢻⡏⢻⣙⣷⣦⠖⠀⠀⠀⠀⢀⡞⢋⡏⢹⣉⣷⡤⠄
⠀⠀⠀⢀⠀⠓⢡⠍⠲⡘⡙⠀⠀⠀⠀⠀⠀⣿⣤⡾⠛⠛⣿⢨⡇⠀⠀⠀⠀⠀⣸⣤⡾⠓⠋⢻⠀⡇
⠀⠀⠀⠋⠢⣁⠣⢄⡀⢅⢡⠃⠀⠀⠀⠀⠈⠙⣿⣧⣀⣀⣼⣿⠁⠀⠀⠀⠀⠈⠉⢿⣇⣀⣀⣨⣿⠁
⠀⠀⠀⠀⠀⠋⠈⠉⠉⠁⠙⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣶⣒⣲⣒⣒⣒⣒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣶⣖⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣜⠛⠣⠾⠿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢷⡆⠺⣴⣭⠍⣥⠀⠀⠀⠀⠀⠀⠀⢸⣧⣼⠀⠀⣐⠐⢠⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣯⠉⢚⠉⠚⢹⡉⠀⣾⠛⠓⠒⣄⠀⠀⠀⠀⣏⠈⡆⠀⣠⢀⡼⢓⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢷⠾⢤⠀⣄⠀⢠⡇⠀⠀⣤⠈⠓⢲⣀⣀⣈⣦⠷⠶⠶⠙⠛⢾⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣼⠞⠂⢚⣱⣀⣠⣼⠋⠀⠬⣿⣤⣸⣏⠀⠈⠉⢙⣆⠀⢀⠀⠀⠈⠀⠈⢿⡶⢄⡀
⠀⠀⠀⠀⠀⠀⣴⠋⠙⠖⠒⣾⣼⣏⣁⠈⠳⢾⡋⠈⠳⣌⠉⠛⠛⠃⠴⢶⡶⠊⠀⠀⠀⣹⠆⠀⢻⡄⠘⢂
⠀⠀⠀⠀⠀⢠⣯⠀⢠⠞⠉⠁⠉⠉⠀⠙⠲⠶⠟⠠⠤⠈⢦⠀⠀⠀⠀⣰⠍⡀⣀⣠⠖⠁⠀⠀⠀⠙⢦⡘⢧⡀
⠀⠀⠀⠀⠀⢸⡇⢰⡟⠀⠀⠀⠀⠀⢀⡤⠆⠀⠀⠀⠀⠀⣈⡇⠀⠀⠀⢹⡈⠉⠁⠀⠀⠀⠀⠀⠈⠀⢠⡇⠳⡙⢦
⠀⠀⠀⠀⠀⢸⡇⠆⢷⡀⢀⠔⠈⠙⣿⣖⡖⠛⠛⠛⠛⠈⠉⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢳⡄⠀⠀⢷
⠀⠀⠀⠀⠀⣾⣹⣇⠀⠙⢦⠤⠴⠟⠉⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⡈⠑⣁⣠⣶⡾⠿⠻⣆⣀⡤⠜
⠀⠀⠀⠀⠀⠀⢻⡟⠄⡀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠔⠛⠉⠂⠀⠀⢀⣠⣼⡧⡀
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠠⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡷⠀⢠⣤⡴⠚⠉⠀⠈⠀⢺
⠀⠀⠀⠀⠀⠀⠀⣽⢯⢠⡀⠀⠀⠀⢀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠁⠀⠀⠂⠀⠀⠀⠀⠎⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡀⡄⠉⠛⠒⢋⠉⣿⠏⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⢷⡇
⠀⠀⠀⠀⠀⠀⠘⣿⠉⠓⠒⠶⠖⠓⠲⠉⢃⢱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡈⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠰⣯⠃⠀⠈⠂⠀⠀⠀⠀⠂⠈⢙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠋⢠
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠜⠳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠆⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠒⠀⠀⠐⠄⠀⠀⢀⣴⣾⣯⣶⡴⡠⣠⣰⣜⠂⠀⠀⠀⠀⠀⢐⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣳⠙⠈⠀⠐⠑⠻⠋⣿⣻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢿⢻⠟⠀⢀⡠⠀⠀⡀⠃⢠⣿⣿⠀⠒⢐⠀⠀⠀⡄⠀⣀⡤⠏
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠂⠞⠀⠀⠀⠀⡨⣨⡄⠀⠀⣀⠀⠀⠂⠡⣾⣿⣿⠀⠀⠁⠀⠀⣀⠟⠯⣁
⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢀⡠⢇⣠⣄⡀⠀⣀⣀⣀⡈⠱⠅⢣⡀⢀⣬⠉⠀⠀⣠⣿⣿⣃⣠⡀⠀⠰⠄⠀⠀⠀⠀⠂⡁
⢠⡄⠀⣀⢐⠀⢠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢫⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣶⡶⠀⠨⠢⣭⡤⠀⠺⣿⣿⣿⠟⠉⠉⠀⠀⠀⠀⠐⠄⠀⢸⡅
⠀⠀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠸⢿⣿⣿⣟⣻⣿⣿⣿⡿⢋⣀⢉⡗⢠⠙⣄⠀⢀⣿⠀⢿⣿⣧⣤⣠⣴⣤⣢⣤⣴⣠⣤⣤⣿⡇

⠀⠀⠀⢸⡑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢹
⠀⠀⢀⣨⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣯⣯⠭⠇
⠀⠀⠈⠒⢶⠻⡟⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡴⣿⠤⣀
⠀⣠⣒⣿⢿⣿⣧⡄⠀⠀⠀⢀⣤⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢟⣦⣴⣤⡤⣴⣾⠽⢲⡷⢓⡨⣿⡷⣄
⠀⣷⡏⣷⢀⡿⢘⡛⠒⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣰⣤⣽⡌⢻⣦
⠀⢳⣤⣿⣿⣟⡾⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣾⣿⡇⠐⢻
⠀⠀⠸⣿⣿⡟⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠋⣹⣿
⠀⠀⠀⣿⣿⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⣿⡟
⣴⠒⠒⢺⣿⡒⠒⠺⣿⣟⠒⠒⠚⠓⠀⠈⠙⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣿⡿⠒⠒⠲⣿⣿⠶⠶⡆
⣿⠀⠀⣸⣃⣧⠀⢀⣇⢏⡀⠢⠀⠀⠐⠀⠁⠀⠀⠀⡂⠁⢒⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠐⠀⢀⣤⣿⡇⠠⠀⢂⣽⢿⠀⠀⡇
⣿⡀⠀⠀⠀⠀⢀⣀⡉⠹⠁⠄⠀⠀⢀⡂⠀⠀⠈⠁⠀⠀⢘⣈⣐⣀⣀⣂⣀⣂⣀⣀⣀⡀⢀⣀⣀⣀⣘⣀⡀⠁⣀⣀⣀⣀⠀⢀⡀⣀⢀⣀⣀⣁⡇
⣿⣿⣿⢿⣿⠿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣤⣶⣵⣶⣦⣼⣤⣾⣧⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣷⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⢹⡆⢰⡏⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠁⠈⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠙⠓⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠋⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢠⣀⡄⠀⢸⡗⠲⡄⠀⠀⢠⠖⢺⡇⠀⢠⣀⡄⠀⠀⠙⢦⡀
⠀⠀⢀⡖⠶⠶⠖⠋⠀⠀⠀⠀⠀⣿⠀⠀⣾⣤⣤⠃⠀⠀⠘⣤⣤⣷⠀⠀⣿⠀⠀⠀⠀⠀⠙⠲⠶⠶⢶⡄
⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⡠⣿⣤⡼⠁⠀⠈⡇⠀⠀⢸⠁⠀⠈⢧⡤⣿⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠐⠋⢰⡋⠉⠙⠛⠒⢾⠁⠀⠀⠈⡷⠒⠛⠋⠉⢙⡆⠙⠂⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠙⠒⠶⢤⣤⠞⠀⠀⠀⠀⠳⣤⡤⠶⠒⠋⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠉⠉⠉⠓⠦⢤⣄⣀⣀⣈⣳⣤⣀⡼⠃⠀⠀⠀⠀⠀⠀⠘⢧⣀⣤⣞⣁⣀⣀⣤⡤⠴⠚⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⡀⢀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⣸⠇⠸⣇⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢿⠀⠀⠈

⠀⠀⠀⠀⡠⡀⠠⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⠀⠢⣴⡋
⠀⠠⡀⠀⡠⠀⠐⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠈⠁⠀⠈⠁
⠠⡈⠀⠀⢠⠀⠒⠈⡁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢄
⠈⡄⠀⠀⠀⡀⢀⣠⣔⡉⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⢉⣻⠇⡠⠃⠈⠢⠀⠠⠒
⠀⠀⠁⢀⠣⡄⠻⢹⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⡠⠁⢨
⠀⠀⠀⠀⠀⠀⣅⢀⡨⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⣡⠓⠂⠚⢄⠄⠀⠀⠀⠀⠄⡈⠖⣀⠀⡆
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠤⠁⣐⡀⢤⡤⡈⣂⢢⡇⠀⠴⢮⣸⣠⡄⠲⠒⠂⠁⡀⠘⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⢨⠀⠅⣂⠀⡈⠑⠛⠋⢡⠤⢀⠔⠀⠀⠉⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠈⠃⠐⠀⠊⠀⠀⠁⠀⢇⠀⠀⠀⡸⠀⠀⠀⠀⠘⠀⠒⠐⠒⢺
⠀⠀⠀⠀⠀⠀⠈⠲⠀⠐⠒⠃⠀⠀⠀⠀⠀⠀⠀⡊⠑⡆⢈⡇⠀⠀⠀⠠⡀⠀⠀⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠱⠁⠀⠐⠄⠄⠀⠀⠀⠀⠀⠉⠈⠈⠉⠀⠀⠀⠀⠀⢈⡓⠒⠲⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢶⠀⠈⢈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡱⠛⠀⠅⢂⠀⠀⠀⠀⠀⠀⢀⠄⠺⡗⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣅⣠⣤⣤⡼⠛⠀⠀⠀⠀⠀⠀⠙⠄⠀⠤⠊


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠉⠉⣹⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠗⣰⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⣿⣿⣿⣥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣆⠽⠟⠋⠛⠀⣨⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠾⣿⣿⣿⣿⠗⠶⠂⠀⠀⠀⠀⠈⠙⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⢀⠈⢻⡟⠁⠀⠀⠀⠀⠀⢲⣄⠀⢀⣿⣿⡏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⢲⣾⣿⡿⠃⠀⠸⣦⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣀⣹⣿⣿⣿⣦⡤⠀⠻⢷⣶⣾⡿⣿⣿⣷⡈⠙⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣟⢻⣿⣤⣦⡀⠀⠀⠀⠀⠨⣺⣿⢿⣷⣤⡈⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⢼⣧⣤⣤⣀⠀⠀⠀⠀⢠⣿⣿⠟⠉⠁⠠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠏⣾⡟⠁⠈⢉⣠⣴⣶⣶⣾⣿⢿⣦⣀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⣸⣇⣀⢠⠏⠁⠀⠀⠀⠀⠘⣶⡬⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣟⢾⣄⠀⠀⢀⣠⣤⢞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠛⠛⠓⠹⣿⣾⣿⣫⣶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢧⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠑⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡿⠿⠿⢻⣿⣿⡿⠻⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠁⠀⠀⢠⣿⣿⡟⠀⠀⢿⣿⣿⠙⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠃⠀⠀⠀⠀⣰⣿⣿⠟⠀⠀⠀⢸⣿⣿⡄⠀⠈⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠁⠀⠀⠀⠀⣼⣿⣿⠏⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠁⠀⠀⠀⢀⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠸⣿⣿⡆⠀⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣯⣤⣄⣤⣤⣤⣿⣿⣿⣥⣤⣤⣤⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠻⣿⣿⡟⠛⢛⣿⣿⡏⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⣼⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠏⠈⠛⢿⣿⣿⣷⣶⣦⣤⣤⣤⣶⣶⣾⣿⣿⣿⠿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠃⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⠿⠿⠛⠛⠋⠉⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣾⣿⠿⠟⠛⢛⣛⠿⣿⣶⣶⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠿⢿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠉⠳⡌⠻⠟⠉⠀⠈⠉⠛⠿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⡿⠋⠀⠀⠐⠲⣄⠈⠏⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⢀⣤⣾⣿⡿⠿⢿⣿⠁⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⣰⣿⡿⠋⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆
⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿⣿⡄
⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⣿⡇
⠀⣿⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⣿⣿⣸⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡆⢸⣿⣧
⢸⣿⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⢸⣿⣿
⢸⣿⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⣼⣿⣿⡀
⢸⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡿⢻⣿⣇
⢸⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⡇⠈⢿⣿⡄
⢸⣿⣇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠸⠀⠀⠸⣿⣷
⠘⣿⣿⡀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡀⠀⠀⠀⠀⢿⣿
⠀⠘⢿⣿⣶⣶⣤⣴⣿⣿⣦⣖⡲⠖⢚⣡⣿⣷⣤⣀⣤⣤⣤⣤⣾⠿⠿⢿⣿⣯⣭⣁⣀⣠⣤⡾⠟⠉⠉⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⡟⢩⡿⠿⢭⣉⡉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣼⠀⠀⠀⠈⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣄⣀⠀⢸⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⡤⢶⣲⣴⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣦⣄⣀⡆⠀⠀⣸⠀⠀⠀⣀⣬⣷⣾⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣦⣤⣥⣴⣶⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠁
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠸⣿⣤⣿⣿⠀⠈⠁⠀⠀⠀⡀⠈
⠀⣴⣾⣿⣿⠃⠀⠀⠀⠀⠀⠐
⡾⠿⠋⠉⠁

⠂
⠿⠗⠂⠀⠀⢀⡀
⠛
⠀⠀⢠⠎⠀⠀⠄⠀⠀⠀⣀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤
⣠⣶⠀⠀⠆⠀⠀⠀⠀⠀⠈⠁⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠋⠁⡀⠀⠀⠀⠀⠀⣤⡆⠀⠀⣾⠃
⠓⠿⠃⣀⠀⠀⠀⠈⡉⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄
⣶⣶⣾⣿⣦⣹⡦⣾⣿⡟⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣯⣿⣿⣿⣹⣧⣿⣿⡿⢿⣄⡄⠀⠀⠀⠀⠀⠀⠀⢀⣄⠈⠁
⣿⣿⣿⣿⣿⡏⢉⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠛⠙⠓⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⠃⣤⠀⠤⢤⣴⡆⠀⣀⣤⣾⣿⡖⠒⢠⣤⡀⢀⣶⣤⣤⣌⢁⠘⠻⠋⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣧⡀⠄⠁⢀⣶⣿⣿⣿⣿⣿⣦⣾⠏⢻⣿⠙⡿⢿⣉⠻⡃⣀⡴⠛⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢿⣿⣿⣧⣰⠿⠈⢿⡟⠻⣿⣿⣿⣿⣿⣷⣿⣿⣿⣦⠘⠛⠀⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣳⣾⣆⠀⠓⢶⡼⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠐⢿⠛⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢀⣤⡀⠛⠛⠻⠛⠛⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⣀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣷⣾⠟⣳⣶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠋
⢽⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⠇⠀⠹⢿⣿⣿⣿⣿⣼⣿⣶⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁
⠀⠈⠛⠓⠟⠋⠁⠙⢿⡛⠻⣿⡀⠀⠀⠈⢿⣿⣿⣯⣤⣿⣷⠻⢟⡋
⠀⠀⣶⠆⠀⠿⠀⠠⠌⠀⠀⠉⠁⢤⡀⠀⠀⠙⠋⠀⠈⢿⣿⠀⠙⣿
⠓⠄⠹⠶⠀⡀⠀⠀⣴⡄⠀⣶⠂⠀⠉⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠋
⠀⢀⠆⠀⠈⠁⠀⠀⢈⡠⠄⠈⠲⠛⠒⠀⠐
⠒⠿⢶⣤⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⢀⣀⡀

⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠆
⠀⠀⠀⠠⣤⣤⣴⣶⣤⣤⣤⣤⣤⣤⣦⣀⣶⣼⣿⣿⡿⠺⠷⠀⠀⠂
⣉⡷⠦⣴⣿⣙⣿⣿⠋⢾⣿⣭⣿⣿⡉⠛⠁⠀⠈⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈
⣉⠳⣶⣆⣸⠿⢿⠿⠃⠀⠉⠉⠉⠁⠀⠀⣀⣀⣀⡀⠀⢤⡔⠋⠉⠉
⠛⠀⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠛⠉⠘⠟⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠾⣿⠿⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣴⣶⣿⢟⣁⣀⣤⡟
⠀⠀⠀⠀⠀⠻⠿⠇⠸⠟⠉⠙⠷⣤⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⣀⠘⠛⢶⣟⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠂⠉⠁⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢱⠆⠀⠈⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠾⠃⠀⠂
⠀⠀⠀⠀⠀⣀⠀⢀⣀⣀⣀⡀⠀⣀⠀⠀⢀⣀⠀⢀⠐⠒⠀⢀⣠⣴⣿⡀⣀⠀⠀⠀⣠⣤⣀⣴⣶⣿⣦⡀⠀⣴⡀⠀⠀⢀
⠀⠀⠀⠀⠼⠉⠀⠉⠁⠉⠡⠤⠀⠾⠁⠀⠋⠁⣠⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⢀⣶⣾⣿⣉⣏⣽⣿⣿⣿⠁⢸⡍⠁
⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣶⣶⣄⣀⠀⠀⣿⣅⣠⣿⠋⠉⣻⣿⣿⠿⠿⠏⣿⠀⣿⡇
⠀⠀⢠⣶⣿⣿⣷⣶⣤⣤⣄⣠⣤⣤⣶⣤⢠⣶⣶⣿⣿⣿⣿⣿⣄⣴⣿⣿⠿⠁⠀⣰⣿⣿⣷⣶⠶⣾⡿⢠⣿⠃
⣿⣶⣾⣿⣿⣿⡉⠉⠉⠉⠁⠙⠋⠉⠉⢉⡟⠉⠉⠉⠉⢻⣿⣿⡿⠛⢻⣿⠀⣠⣼⣿⡏⠀⢸⡆⠀⢾⡇⣸⡏
⠿⠿⣷⣾⣿⣿⣷⣴⣶⣾⣶⣶⣆⣴⠖⣾⣿⣶⣶⣦⣤⣼⡿⠋⢠⡔⢃⣠⣴⠟⠉⢰⡿⢠⣄⠀⢰⡞⠀⣿⡇
⣶⣾⣿⣭⣿⣿⠿⢿⠿⠛⢻⣿⣿⡏⠐⠟⢻⣿⣿⣿⣿⠛⠟⠛⠛⠿⣿⣿⡅⠀⠘⠿⠀⠨⠉⠀⠻⠃⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⣧⣀⣀⣠⣼⣷⣤⡜⠛⣰⣿⡿⠿⣿⠗⢀⠀⢹⣿⣿⡿⠋⠀⠀⠀⠀⠸⣿⣤⠀⠀⢠⣆⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⠛⠛⢻⣿⠟⠁⠀⠀⠀⣤⡀⠀⠀⠈⠁⣤⡾⠀⠀⠀⠀⠀⠛⠛⠳⣤⠀⠀⠀⠀⠘⠀⠀⠈
⠉⣙⣻⣾⣿⡷⢾⣿⣿⣷⠶⠀⣠⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⣿⡻⢿⣼⣿⣤⠈⠉⠀⠀⣀⣼⣿⣷⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠁⠀⣄
⣽⣿⣄⣹⣿⠟⠀⠀⢀⣼⠟⠻⣿⠁⠉⠳⠀⠀⠀⠀⠀⠀⢰⣀⣀⣴⣶⣿⠛⠻⠀⠈⣿⣀⣀⠀⢠⡆
⠿⢧⣿⡿⠁⢀⡀⣠⣿⡷⠒⠐⣿⣷⠀⠀⠀⢀⣀⣠⣤⡴⠟⠛⠉⠉⠛⠿⠀⠀⠀⠀⠀⠀⠉⠀⠘⠁
⣴⣞⣿⣿⠗⠿⣿⣿⣿⣷⣤⣤⣿⣿⣦⠶⠿⠉⠉⠉⠛⠁⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠿
⣿⣿⣿⣿⠀⠀⠀⠈⢿⣶⣿⡿⣇⠀⣿⠀⠀⠀⠀⠀⢠⣄⣠⣤⣶⠖⢿⣆⠀⠀⠀⠀⠀⡶
⠟⠋⢹⣿⡀⢀⣼⣶⣾⣿⡍⢀⠀⠈⠻⣷⠀⠀⠀⠀⠀⢹⣿⠀⣹⠀⠈⠻⠀⠀⡀
⣴⣶⠤⠿⢿⣿⣿⣷⠉⠹⢿⡀⠴⢄⢰⣿⠀⠀⠀⠀⠀⢸⡷⢠⣿⠀⠀⢰⡄⠈⠻⣤⡀⠀⠀⠃
⢀⣠⠾⠶⠾⣿⣯⠉⠻⣦⡀⠙⢶⣄⣼⣿⠀⠀⠀⠀⠀⢈⣤⠾⠋⠠⣀⠐⠀⣀⠀⠈⠃⠀⢠⠄
⠸⣿⣄⣀⣠⣿⣿⠛⠳⠾⢻⣶⣄⠉⠳⢿⡄⠀⢠⡶⢶⣞⠁⠀⠻⠶⣟⠀⠀⣿
⢿⣦⣭⣏⠙⠿⠿⠦⢀⣠⡌⣾⠿⣷⠆⣀⠀⠂⣀⠀⢨⣅⣰⠄⠀⠐⠟⠀⠀⡿⢠⡄⠀⠀⡿
⠀⠀⢸⣿⣷⣄⠀⢀⠸⢏⣹⡿⠐⠛⠀⣹⣶⣤⡀⠙⢿⣥⣀⠀⠀⠀⠀⠀⢀⣀⣸⠇⠀⢠⡷
⣾⠿⠿⢿⣿⢿⣦⠈⠀⠿⠿⠃⠀⣶⡞⠛⢋⠀⠀⠰⠀⠀⠹⡗⠀⣀⠀⠰⣦⡀⠛⠀⠀⡏⠀⠀⠀⢠⣄
⠀⣠⣄⣀⣀⣀⡀⣀⣀⣀⠀⠀⠀⠉⢳⣾⠛⠀⠀⠀⠈⠛⣾⠄⠀⠙⣀⣤⣤⡀⡶⠀⠀⣱⣤⣠⣤⡤⠁⠀⠠⠄
⣿⡏⠉⠛⠛⠋⠉⠉⠉⠛⠋⠈⠙⠛⡼⠿⣤⣀⣀⣀⣀⣤⣴⡟⠀⠀⠉⠛⠻⣿⠄⠀⠈⠛⠻⣷⣿⣷⣄⠀⠀⠀⠀⠀⡀
⣿⡷⠀⣾⣷⣶⣿⣶⣾⣿⣿⠂⣿⣿⣿⠄⠿⠿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠠⠦⠀⠀⣿⠀⠀⠀⠈⠻⣿⣷⣦⡄⠀⠀⠀⠀⠒⠺⠷⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⡖⠒⠒⠒⢒⣚⣻⠷⠶⠖⠒⠘⣶⠒⠒⠒⠒⠒⣾⠓⠂⠀⠀⢀⠀⡠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠒⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢹⣶⣶⡶⠶⠶⠿⠒⠒⠚⢏⡍⠙⠣⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡄⠀⣀⣀⠀⠀⠛⡀⠀⠀⠀⣀⣠⣤⣴⣶⣾⣯⡤⠉⠄⠀⠀⠀⠀⠀⠀⢀⢀⡧⠀⠀⠀⠉⠲⢤⡀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡶⠁⠈⠉⠒⠾⣷⡄⠉⢙⣿⡿⣿⠿⠿⠛⠛⠄⠀⠀⢀⣠⣤⣴⣶⣛⣡⡎⠀⠀⡀⠀⠀⠀⠀⠉⡗
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠠⣗⣦⣄⡀⠀⢠⡞⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠤⠾⠛⠛⢿⣿⣿⣿⣿⣷⣦⣼⣷⣶⣤⣀⣀⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡶⠒⠚⠋⠉⠁⠀⠈⠹⢲⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⡿⢻
⠀⢀⣀⢀⣀⣀⣀⣀⣠⣇⣇⣀⣀⡀⠀⠀⣠⠴⠚⠉⢷⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠘⠏⠉⠉⡘⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠉⡏⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠉⣹⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠸⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡄⠀⡇⠀⢀⠀⠀⠀⠀⠈⠈⠉⡟
⠀⠀⠀⢇⣀⣀⣀⣀⣀⣀⠀⠠⠔⡆⠲⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠇⣠⠈⠲⣼⣄⣀⣀⣀⣀⣀⣀⣇⣀⡀
⠀⠀⠀⠀⢻⡟⠛⠋⠉⠀⠀⠀⠀⠁⠀⠀⢇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠇⠀⠀⢯⠉⠉⢩⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡼⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⣶⠏⠀⠸⠂⠀⠐⠲⢄
⠀⠀⢰⠖⠛⠓⠚⠛⠛⠉⠉⠉⢿⡁⠀⠀⢶⡄⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⢹⡆⠀⠀⠀⠀⠀⢸⠀⠈⠻⠒⡖
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⣧⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⡇⠀⢄⡀⢀⣀⣸⡀⠀⠀⢰⡇
⠀⠐⠛⠒⠒⠂⠀⠀⢲⣶⣶⣋⣀⣽⡆⠀⠀⢸⡆⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⠁⠀⠀⢸⠟⠻⣟⣛⣓⣄⣸⠷⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⢹⡄⠀⠘⣧⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣀⣀⣀⣠⣤⢔⣢⣼⠿⠁⠀⠀⠀⡏⠀⠀⢸⠀⠀⠀⠁
⠀⠀⢤⣤⠤⠴⠒⠚⠉⠑⠄⠄⠀⠀⠀⢻⣄⠀⢿⡆⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⠷⠚⠛⠳⢤⠀⠀⠀⠀⢠⣿⣷⣶⣾⣶⣶⠦⡤⠤
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⣠⠟⠏⠀⠀⠀⠀⢂⠘⣿⡀⠀⠀⠙⢷⣦⣤⣶⣶⣋⣀⣀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⢹⡇
⠀⠀⣠⣤⣤⣤⣀⣀⣠⡴⠁⠀⢸⠀⠀⠀⢀⣼⣄⡹⠷⡄⢠⣤⠄⣈⣩⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⡀⢨⡅
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⠔⠁⢀⣠⠴⠛⠋⠁⠀⠀⢳⠈⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢹⠁⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣥⣤⣎⠁⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⠿⠿⣿⠀⠀⠀⠀⠀⠀⣠⠃⣀⣀⣤⣤⣀⣤⣴⠋⢠⣥⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠙⢤⢴⡶⠿⠿⠟⠛⠛⠛⠈⠉⠉⠈⠃⠀⡿⢦⣴⠀⠀⠐⠺⠭⢍⣉⡉⠛⣿⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠒⠶⠶⠶⠦⡼⠋⠉⠉⠭⠭⠭⠭⠭⠭⠻⠷⠶⠶⠤⠀⠀⠀⠸⠙⠒

⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿