Bola Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣏⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣽⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠿⠷⠎⢀⣀⣠⠄⠀⠀⢀⣀⣀⡀⣵⣾⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⡿⡫⠓⠀⠉⠉⠛⠿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⢻⣿⣿⠏⠄⠀⢀⠌⠉⠙⠲⢶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢿⡟⠌⠀⣠⠋⠀⢀⣠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠼⠼⠧⠦⠴⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⣼⣾⣿⣧⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠂⠀⠀⠀⠀⢀⠜⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠱⣦⣄⡀⠀⠈⠩⡀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠛⠛⠋⠉


⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣷⣼⣷⣶⣶⣶⣦⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠖⠺⠿⣿⡿⠿⠿⠛⠓⠶⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠛⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⣄
⠀⠀⠀⠀⢀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⡀
⠀⠀⠀⢠⣿⣿⣇⣀⣤⣤⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣄
⠀⠀⣰⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣆
⠀⢰⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡆
⢀⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿
⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿
⣿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠉⠉
⣿⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⣿⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⢿⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⢀
⢸⡇⠀⠀⣿⣿⣶⣶⣶⣶⣶⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣤⣤⣤⣤⣤⣶⣶⣶⡟⠀⠀⠀⠀⠀⣸
⠀⢿⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⡿
⠀⠘⣧⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣾⠃
⠀⠀⠘⣧⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢀⣾⠃
⠀⠀⠀⠈⢿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢳⡾⠁
⠀⠀⠀⠀⠀⠙⣦⡈⠻⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠈⠻⢦⡈⠙⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠿⠋⠁⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠛⠁⠀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣽⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢋⣀⣤⠶⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⢿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣭⣭⣻⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢞⣭⣷⣦⣀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢰⡄⠀⠀⠀⠀⠀⠙⠿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠟⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠆⠠⡾⠀⢾⠀⠸⡆⠠⣦⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠠⡾⠀⣚⠃⢘⡋⠀⣹⡁⢈⣟⠀⢻⠂⠸⠄⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⣴⠆⢺⠃⠘⡋⠀⢹⡉⠀⣹⠁⠈⣋⠁⠙⡏⠐⢳⠂⢶⠌⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣬⢠⡶⠐⠿⠂⠺⠟⠈⢻⠋⠉⢻⠉⠈⢻⠁⠙⠟⠂⠻⠖⠰⡦⢨⣌⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡊⣨⡁⣴⡄⢴⡷⠀⢾⠗⠐⢿⠗⠒⢿⠗⠒⢿⠖⠲⣿⠆⠴⡧⠠⣦⡀⣥⠙⡒
⠀⠀⠀⠀⠀⠀⠀⠰⠚⠋⣉⠀⣨⡀⣠⣇⣀⣴⣄⣠⣴⣄⢠⣾⣄⢠⣼⣄⣀⣶⣀⣀⣧⡀⣨⡀⢘⡉⠛⠲
⠀⠀⠀⠀⠀⠀⠀⠔⠚⠈⠟⠈⠛⠁⢙⠁⠀⢙⠀⠀⣙⠀⠀⣹⠀⠀⢹⠀⠀⢙⠀⠀⣋⠀⢙⠋⠙⠏⠙⠒⠄
⠀⠀⠀⠀⠀⠀⣠⡤⡶⠰⡷⠲⡿⠒⢿⠗⠒⢿⠓⠚⢿⠓⠚⢿⠓⠘⢿⠗⠒⢿⠗⠚⢿⠂⠺⡗⠲⡷⠰⡦⣦⡄
⠀⠀⠀⠀⠀⠀⣠⣄⣄⣠⣄⢠⣦⢀⣴⣄⢀⣾⣄⢠⣶⣄⢀⣼⡄⢀⣼⣄⢀⣴⣄⢀⣶⡄⣠⣆⢠⣦⢠⣄⣄⡀
⠀⠀⠀⠀⠀⠀⣀⡁⡁⢈⡁⠈⡃⠀⢹⠁⠀⣹⠀⠀⣛⠀⠀⢹⠀⠀⢹⠁⠀⢙⠁⠀⣙⠀⢈⡁⠀⡁⢀⡁⣁⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠙⠟⠙⠟⠒⢻⠗⠐⢿⠓⠒⣿⠗⠒⢿⠗⠒⢿⠗⠒⢿⠗⠒⡿⠃⠺⠟⠙⠟⠙⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠻⠰⡷⠠⣶⠄⣴⣦⢀⣴⣄⣀⣶⣄⣀⣼⣄⣀⣼⣄⣀⣴⣄⢠⣷⠄⣴⡦⢴⠗⠺⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠲⡆⣤⡀⣠⡀⢈⡅⠀⢹⠀⠀⢙⠁⠀⢹⠁⠀⢹⠁⠀⣹⠀⢀⣅⠀⣨⡀⣰⠤⡶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⢨⣄⢘⡁⠉⡟⠉⢻⠟⠘⢻⠓⠒⢿⠓⠚⢿⠓⠘⠿⠁⠙⠏⠀⣉⢀⣄⠰⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢭⡈⣛⠘⠿⠒⠺⡷⠠⢾⡦⠤⣾⠦⠤⣾⠆⠰⡷⠒⢺⠃⢘⠋⣨⠡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢙⡃⠺⡆⠰⡷⠄⢼⡦⠠⣼⡤⢠⣶⠄⢴⠷⠐⠞⠂⡛⠡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠌⢳⡀⢲⠀⠰⡦⠀⢼⠄⠠⡷⠀⠾⠂⣸⠃⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⠈⠷⠀⠺⠂⠸⠇⠈⠋⠀⠁


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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠶⡶⠶⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⠋⠉⠀⠀⢠⠞⠻⣄⠀⠀⠀⠀⠉⠙⠳⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⣀⣀⣀⡴⠋⠀⠀⠈⠳⣤⣀⣀⣀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠯⢤⣤⣴⡚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣛⡶⣶⡒⠚⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⣠⠖⠋⠀⠈⠉⠛⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⠉⠁⠀⠀⠉⠳⢤⡈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⠞⠁⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⡴⠚⠉⠓⢦⡀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣡⠖⠋⠀⠀⠀⠀⠀⠈⠳⢼⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢻⡇⠀⠀⠀⠀⠀⠀⢀⡴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠳⣄⠀⠀⠀⠀⠀⢀⣼⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⢦⡀⠀⠀⢀⡴⠋⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⠳⣄⠀⢀⡴⠋⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⡀⠙⠶⣤⠏⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠈⠳⠞⠀⠀⡏⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠈⢇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣸⠁⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⢘⡆⠀⠀⠀⠀⠀⠘⠲⢤⣀⡀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠳⡀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢓⣦⣤⣶⣛⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠲⠤⠤⡄⠀⠀⠠⡖⠒⠛⠋⠉⠀⠀⠀⠀⠉⠉⠙⠛⢳⠀⠀⠀⣰⠒⠛⢉⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠙⢦⡀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣠⠞⠁⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠙⠦⣜⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⠚⠁⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣌⡙⣶⠤⣄⡀⠀⢀⣠⠴⠒⢹⣇⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠾⠿⠿⠿⠶⠾⠛⠛⠉⠁




⣿⠿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠚⠋⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠸⡇⠀⠉⢉⣉⡩⠿⠿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣿⣴⣾⣟⣀⣀⡀⠀⢸⡆⠉⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠒⠚⠛⠛⠛⠛⠋⠉⠉⠉⠉⢩⠿⣿⡿⢿⡟⠛⠛⠛⠻⠿⣿⣶⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢿⡇⠀⢻⣆⠀⠀⠀⠀⢻⣷⠟⠛⠻⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢸⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠉⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢸⣿⠀⣠⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣤⣾⣿⠛⠋⠻⢿⣿⣿⣿⢿⣿⠷⣦⣄⣀⣠⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠃⢸⣿⠀⠀⠀⠀⢹⡟⠁⢸⣿⠀⠀⠉⢻⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⠀⢸⣿⠀⠀⠀⠀⢸⡇⠀⢸⣟⣀⣤⣶⡿⣿⣿⠋
⠀⠀⠀⠰⠦⠤⠤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣳⣼⣿⣠⣤⣤⣤⣼⣷⣶⣿⠟⠛⠉⣡⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⣹⠟⠻⢭⣽⣿⣿⣿⣿⣿⣤⣴⡾⠟⠁
⠀⠀⠀⠰⠦⠤⣤⣄⣀⣀⣀⣀⣀⣀⣀⡀⠀⢀⣿⠀⣀⣀⣀⣠⣭⣽⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⡿⠛⠉⠉⠉⠛⠒⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢻⣿⠿⠋⠉⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⣼⠃⠀⠀⠀⠀⠀⠀⠹⣦⣤⣤⣤⣤⣤⡞⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⣼⣷⡀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣅⣰⣿⣿⣿⣿⣦⣄⡤⠖⠋⠁⠀⠈⠙⢿⣿⣿⣿⠟⠉⠀⠈⠙⠲⢤⣀⣰⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡟⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⣇⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡌⢿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣮⡏⠁⠀⠀⠙⠦⣄⠀⠀⠀⠀⢀⣴⣿⣄⡀⠀⠀⠀⢀⣠⠞⠉⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠈⠛⢶⣶⣿⣿⣿⣿⣿⣿⣷⣶⠖⠋⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣤⣴⡟⠛⠛⠛⠛⣻⣿⣤⠴⠞⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠾⠛⠋⣩⣿⠟⠉⠀⠀⠀⠀⠉⠉⠙⠛⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⠁⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠉⠉⠛⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀
⠀⠀⠀⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⣸⡟⠙⣿⡄⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢿⣧
⠀⢰⣿⠁⠀⠘⢿⡄⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠈⣿⡆
⠀⣾⡇⠀⠀⠀⠘⣿⡄⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⢸⣷⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠀⠀⠀⠀⠀⠸⣿
⠀⣿⣦⡀⠀⠀⠀⠀⣿⡄⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⡟⠿⣦⣀⠀⠀⢹⡇⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⢻⣇⠀⠈⠙⠻⠶⣼⣧⣀⣀⣀⢘⣿⣀⣀⣀⣀⣀⣀⣿⣧⣤⣴⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠘⣿⡄⠀⠀⠀⠀⠘⣿⠉⠉⠉⠉⠻⣟⠛⠛⠛⠋⠉⠉⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠃
⠀⠀⠹⣷⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠿⠛⣽⡏
⠀⠀⠀⠹⣷⡀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⠿⠿⠛⠛⠋⠉⠀⠀⣼⡟
⠀⠀⠀⠀⠙⣷⣄⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠈⠻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣶⣤⣤⣀⣀⡀⣀⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣨⣭⣿⡿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠒⠒⠒⠒⠦⣤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⠴⠒⠒⠒⠒⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠚⣉⡔⠁⠀⠀⠀⠀⠀⠀⠈⠳⣌⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⣡⠞⠁⠀⠀⠀⠀⠀⠀⠈⢢⣉⠓⢤⡀
⠀⠀⠀⠀⠀⡴⣫⠔⠋⠁⠙⢦⠀⠀⠀⠀⠀⠀⣠⠊⠀⠉⠲⢌⢢⡀⠀⠀⠀⠀⠀⠀⢀⡴⣡⣴⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣦⣝⢦
⠀⠀⠀⢀⡞⢹⠁⠀⠀⠀⠀⢀⠟⠈⠉⠉⠀⠙⣅⠀⠀⠀⠀⠀⢻⠱⡄⠀⠀⠀⠀⢀⠎⣿⣿⣿⣿⣿⣿⣿⠋⠉⠁⠉⠈⠻⣿⣿⣿⣿⣿⣿⡏⢳⡀
⠀⠀⠀⡜⠀⢸⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⢸⠀⢹⡀⠀⠀⢀⡎⠀⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡇⠀⢳
⠀⠀⢰⠇⠀⡸⠒⠒⠒⠲⡇⠀⠀⠀⠀⠀⠀⠀⠀⢈⠶⠒⠒⠒⢾⠀⠀⡇⠀⠀⢸⠁⠀⡿⠛⠛⠛⠻⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠛⠛⠛⢧⠀⠘⡆
⠀⠀⢸⠀⢰⠁⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⢇⠀⡇⠀⠀⢸⠀⣸⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠈⡆⠀⡇
⠀⠀⠘⡖⢯⠀⠀⠀⠀⠀⠀⠈⣦⠤⠤⠤⠤⢤⠃⠀⠀⠀⠀⠀⠀⢸⠒⡇⠀⠀⢸⣶⡇⠀⠀⠀⠀⠀⠀⠘⣤⣤⣤⣤⣤⣴⠁⠀⠀⠀⠀⠀⠀⣹⣶⠇
⠀⠀⠀⢣⠘⣆⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⢠⠋⣸⠁⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠈⢧⠈⢢⣀⣀⣀⣀⡎⠀⠀⠀⠀⠀⠀⠘⣆⣀⣀⣀⡴⠃⡰⠃⠀⠀⠀⠀⠈⢿⣿⣦⣀⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠳⣼⡀⠀⠀⠀⠑⠦⣀⠀⠀⣀⡤⠚⠁⠀⠀⠀⣇⠜⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⠿⠋⠀⠀⠀⢀⣿⠟
⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠙⡏⠁⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠈⢹⠋⠀⠀⠀⠀⣀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠲⠤⠤⠧⠤⠖⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠤⠼⠤⠤⠖⠒⠉⠁

⠀⠀⠐⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂
⠀⠀⠀⠘⠮⠙⠛⠛⠛⢶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠉⠀⠀⢉⠵⠂
⠀⠀⠀⢁⣀⠀⠀⠀⠀⠠⣄⣙⡟⢦⣀⡤⠤⠤⠴⠶⠶⠶⢦⣄⢀⣴⠒⠒⠒⠢⠤⠤⢤⣀⡄⣡⣎⣠⠄⠀⠀⠈⠉⣀⡀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⣀⠀⠀⠀⠀⠀⠈⠁⢿⠉⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⢶⠿⠿⠀⠀⠀⠐⠲⠇⠘⢶⡄⠀⢀⡀⠀⢀⡵⠂⠠⠖⠛⠂⠀⠘⢿⠿⡶⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠉⢿⠀⠀⠀⠋⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⠙⣄⠀⢀⡿⠉
⠀⠀⠀⠀⠀⠀⠈⠁⠀⢤⠂⢱⣼⣦⣤⣤⣴⣦⣄⣰⡀⠀⠀⢺⡿⠀⠀⢀⣆⣠⣴⣦⣤⣤⣴⣥⣮⣤⣿⢢⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣲⠾⣷⣠⠿⠟⣿⣷⣤⣀⣼⣇⣴⣤⣾⣿⠉⢻⡇⠾⠷⣖⠂⠀⠀⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠉⠋⢩⡿⠿⠿⢿⡟⠻⠉⠹⣿⠟⠛⢻⡿⠿⠛⠻⢯⡙⠉⠁⠀⠀⢀⡈
⠀⠀⠀⠀⠀⠖⡋⠀⠈⡏⠀⠀⠀⢀⣰⣉⣀⠀⠒⠶⢦⡀⠀⠀⠁⠀⠀⣠⡤⠤⠄⠀⣀⣀⡀⠀⠀⠂⢹⠀⠀⠙⠒
⠀⠀⠀⠀⠀⢰⢁⠀⠀⠇⠀⠀⠀⠋⣤⠞⠛⢛⠋⠈⢻⣷⣄⣀⢀⣐⣊⣁⣶⡶⠒⡛⠛⠳⢬⢻⠀⠀⠸⠀⠀⡈⡆
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢸⠁⡀⢠⣼⠇⠀⠘⣿⠿⣿⣿⣿⠿⣿⡿⠀⠰⢧⡄⢀⠀⣧⠀⡗⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡸⡖⠁⢸⣧⡁⠂⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⣼⡇⠈⣆⡿⢸
⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠛⠀⢻⠀⢉⡿⣹⠟⠒⠶⠶⣿⣿⣿⣶⣶⡖⣾⡟⣿⡟⠀⣿⠃⠀⠑⡜⣦⣳⢀
⠀⠀⠀⠀⠀⠀⠙⠱⡀⠀⠀⠀⠀⠀⠀⠣⠘⣧⣾⠀⣿⡗⢱⠃⠿⠻⣎⢹⡅⠸⣷⣿⢃⡜⡿⠀⠀⠁⠛⢠⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⣀⠀⠀⠀⠳⣄⠹⣿⡀⣿⣷⡿⠷⣿⠶⢿⣾⣿⢀⣿⢿⣿⠄⠀⠀⣶⡀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡄⠀⠄⢀⠘⠳⡜⡓⢿⣏⠀⢀⠀⠀⠀⣹⡿⣾⢃⠞⡃⢠⠀⢠⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠀⠐⠀⠀⢹⡄⢃⣰⠀⢛⣠⢆⡌⢠⡏⢠⠀⠀⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠦⠍⢁⠀⠒⠉⣚⠻⢾⣣⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⠀⠈⢹⠇⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⢀⣤⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⣀⣀⣀⣠⣤⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢠⣶⣿⣿⣿⣿⣾⣷⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠩⠛⠉⠉⠀⠀⠀⠀⠀⠀⢠⣾⡟⠉⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃

⠀⠀⠀⠀⠀⠀⢸⢹⣿⣶⣶⣶⣤⣶⣤⣼⣧⠀⠀⠀⢶⠦⣤⢤⣼⣷⣶⣶⣦⣤⣤⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠘⣿⠛⠛⠻⣿⡟⠻⠛⠛⠛⠁⠀⠀⢸⡦⠷⢒⣿⣿⠿⠛⠛⣿⣿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⡟⠿⠛⠛⠛⠛⠛⠿⠻⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⣀⣠⣄⣀⣀⣀⣠⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⢀⣼⡿⠃⠀⠀⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠘⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣤⣶⣤⣴⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣾⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠟⢛⣿⡿⠛⠛⠛⠛⣿⣿⠋⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⣿⡟⠀⠀⠀⠀⠀⠻⠛⠀⠀⠀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢈⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣷⣶⣶⣿⣤⣤⣴⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⠟⠉⠉⠉⠉⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⢀⣿⣶⣶⣦⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣾⣀⣤⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠀⠈⠙⠻⠿⠛⠛⠛⠉⠉⠉⣿⡏⠉⠈⠉⣿⠁⠈⠙⠛⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣁⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡄
⠀⠀⠀⠀⠂⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⡀⠒⠠⢔⡀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⣇⡄⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⢁⠠⠀⠤⠤⠤⠤⡀⠀⠄⠀⠤⢈⠀⡄⠀⢀
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢀⣾⣧⠃⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢸⡥⠂
⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠀⠼⣛⡽⠆⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⠀⢤⠀⠀⣱
⠀⠀⠀⠀⠀⠀⠀⣀⡿⠛⠀⠈⠁⠀⠀⠀⣼⠶⠿⠿⣾⣧⡀⠀⢸⣦⠀⠷⠿⣏
⠀⠀⠀⠀⠀⠨⠉⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⡀⠀⠙⢇⠼⣿⣿⡏⠁⠀⠀⠀⠉⠙⠂⠄⡀
⠀⠀⠀⠀⠀⠘⡁⠀⠀⡰⡏⠀⠀⠀⢀⣾⣋⠄⠀⠀⠀⢻⡉⠻⠙⠦⠀⠀⠀⠀⠀⠀⣀⠀⠑⢤⣀
⠀⠀⠀⠀⠀⣸⢀⠀⠀⠀⡷⠀⢸⠀⢸⢋⠴⢶⣦⣸⡄⠁⠀⢀⠀⠀⠉⠉⠢⣄⣠⣸⣯⣤⡈⣱⠋⡆
⠀⠀⠀⠀⠀⠇⠀⡀⠀⠀⠀⠀⡈⡄⣿⡇⢳⣄⣾⠉⢻⡀⠰⡀⠑⡇⠀⠀⠀⠀⠀⠈⠉⠉⢹⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢳⠘⢾⣿⠊⡦⠩⢅⣒⣟⠷⣶⣤⣹⡄⠀⠀⠀⠀⠀⠀⣠⣼⠁⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠆⢀⡀⠀⠈⠈⠙⢻⠶⠛⠁⠙⣆⢰⠓⠂⠉⡛⠷⠤⠤⠔⠚⠉⣙⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⣀⣠⣄⣀⣈⣡⣠⣀⣀⢹⡀⠠⠖⠂⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠣⣀⡴⠶⠤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣶⣶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⣠⣄⣴
⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⡿⢿
⠀⢸⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣾⣿⡀⠀⠰⡆⠀⠀⠀⠀⣠⠀⠙⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⣇⠀⠀⠀⢀⠀⠀⠉⠈⣿⠀⠙⠛⠀⡷⢶⣿⠀⠀⣿⣿⣿⡇⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢹⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠰⠇⠀⠀⡏⠈⠻⠁⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⢰⡇⠀⠀⣀⡀⣿⢀⣀⠘⠛⢻⡛⠛⠁⠀⢀⡇⠀⠄⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇
⠐⢾⡗⠂⠀⠉⠙⡟⠉⠀⠉⠉⣿⠉⠉⠀⠀⢸⠇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡇⠀⠐⣿⠗⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⢠⡧⠀⠀⠀⠀⢿⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢨⡧⠀⠀⢹⠁⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣦⣿⣶⣯⣖⢄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠐⠒⢿⡿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⣿⣿⣿⣟⣿⣿⣿⣷⣇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⢀⣠⣧⣤⣶⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⣿⣿⣷⣿⣿⡿⣿⣿⣿⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⢸⡏⠉⠋⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⢽⣿⣿⣾⣿⣾⣿⡿⠗⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⠙⠟⠿⠿⠟⠟⠈⠀⠀⠀⠀⠀⠀⠙⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠸
⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⡇
⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⣇
⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⣶⣿⣶⠀⠀⠀⣿⡇
⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠃⡠⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠠⠄⠉⣿⠉⠀⠀⠀⢸⠁⠀⠀⠀⣠⠶⢤⡀
⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⢠⡏⠀⠀⣿
⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠸⣿⠆⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⢀⣾⣷⡀⠀⣿
⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⡀⠀⡟⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠉⡿⠿⠁⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡘⢦⡀⢄⣠⡶⠒⠛⠙⢷⣚⠛⠛⠒⣋⡿⠗⠒⡲⢤⣆⢠⠇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢷⣄⠉⠲⢿⡦⠀⠀⠉⠛⠶⠂⠀⠀⠶⠚⠉⠀⠀⠐⣿⠁⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⡌⣻⣷⣄⠔⠀⠀⠀⠐⢦⡍⠀⠀⠀⠉⠀⢀⡀⠀⠀⠲⢄⠀⣿⣧⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣈⣲⣿⣧⣤⣿⠁⢲⣦⠴⣿⢶⣄⠁⠀⠀⢰⠀⢸⢋⣴⡶⢦⡴⠀⣦⣙⣿⡯
⠀⠀⠀⠀⠀⠀⠀⢤⣬⢿⣿⠀⠋⢹⠀⡈⠃⠓⢿⠾⢻⡆⠀⠁⠀⠈⠱⠻⢬⠷⡺⠁⠀⠈⣤⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡿⠶⢦⡞⣰⡷⠈⣲⠟⢤⡀⢠⠤⣄⣀⣠⣄⠠⠀⠙⢄⠰⣧⢫⡁⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢠⣿⠀⢹⢛⣿⢻⣶⣢⠈⠚⠲⡆⢠⢰⠄⠀⢴⣾⣿⠯⢻⡉⣿⡀⢸⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢛⡿⠀⢘⡇⠘⢸⣿⣅⣁⡀⠀⠙⣻⠃⢀⣀⣠⣝⠛⠋⠯⢴⣿⠏⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠂⠹⣦⣾⣁⣀⣼⡿⣿⠉⣙⣛⡛⢛⣛⣋⠙⣿⡏⠙⢻⣶⣶⠏⢩⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠲⢾⠉⣹⠏⡀⣻⡝⡆⡿⠿⢿⡿⠟⣿⠸⣿⣀⢴⣉⢉⡤⠒⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣠⡙⠉⣷⢻⡷⣄⣀⣁⣴⣿⢰⡇⢁⣀⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣴⡟⣾⡇⣛⣛⣛⣃⣿⢷⠷⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⢘⠬⣛⠭⠀⠭⢝⣫⡎⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⢀⡤⠆⠀⠀⠲⠤⠹⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⣁⡀⣀⣄⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡿⠛⠹⢧⡿⠅
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣴⣭⣴⣭⣿⣼⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡟⢛⢻⡇⠀⠀⠀⠀⠀⠀⡁
⠀⠀⣱⡄⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣭⣼⠃⠀⠀⠉⠀⠐⢠⡆
⠀⣥⣤⣿⣯⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣻⣟⠉⠀⠁⣰⡏⢉⣉⣛⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣷⣽⣖⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣴⣲⡤⣯⣟⣧⣶⣿⣿⣍⢥⣀⣨⡇
⠀⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣉⣉⣉⣁⢀⣼⣿⣿⣿⡟⠻⣏⢉⢛⠉⢷⣴⣾⡆
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠟⠛⠀⠀⢀⠀⢶⣶⣷⣾⣿⣶⣼⣍⣉⡙⢿⣿⣿⣿⣟⣛⣿⣿⣿⣁⢠⡏⠉⡇
⠀⡿⣿⣿⣿⣿⡟⠘⣿⣾⣾⣿⣿⣶⣦⣾⣷⣝⣛⠛⠿⠿⣯⣾⣿⣿⣿⣷⣦⡉⠛⢛⣿⣿⡥⢜⠻⣯⡀⢈⡇
⠀⣿⣿⡿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⢦⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣽⣿⣿⣿⣾⣿⣿⡇
⠀⣿⣿⡧⣽⣿⡿⢾⣛⡉⠁⠉⠙⠛⠿⡿⣿⣿⣿⣥⠾⠒⠙⠻⢿⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡇
⠀⣿⣿⣶⣽⣯⣝⠻⡟⠁⠀⣀⠀⣰⡀⠘⣿⣿⣿⣿⣦⣽⣮⣵⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⡷⢛⣿⣿⣿⣟⣯⣥⣤⣶⣿⣿⣿⣷⣤⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⡭⣼⢼⣟⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣆⣤⣽⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣯⠿⣯⣿⣾⣿⢿⣿⣿⣿⡿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣱⣮⣿⣿⣿⣿⠿⣿⣟⣿⣿⡇
⠀⡿⠻⣽⠿⠿⣿⡿⢿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢷⣾⣿⣿⣿⢿⡇
⣀⣈⣁⣉⣉⣉⣉⣉⣈⣋⣀⣀⣀⣈⣉⣉⣉⣉⣉⣉⣉⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡄⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⡛⠛⠉⠁⣀⠀⠀⠀⢀⣴⣷⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⡶⢻⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣶⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡷⠛⠛⡟⣿⣿⠟⠛⢿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣷⣶⣶⡀⠀⠙⠘⠿⢿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣾⣷⣶⣶⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣶⣷⠉⠈⠛⢿⣿⣿⠋⠉⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣘⣿⠿⠛⠁⠀⠀⠀⠈⢿⡇⠀⠀⠀⢠⠉⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡾⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⢺⡇⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⢿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠆⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠁⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠁


⠀⠀⠀⢀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢀⡀
⠀⣀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣿⣿⣽⠇⠀⠑⣄⠀⠀⠀⠸⣟⡋⠀⠀⠀⠀⢦⡀⠀⠀⠀⠋⠚⢃⣀⣾⣿⣶⡀
⢠⣿⡿⠻⣿⢿⣿⠇⠀⠀⠀⠀⠀⠀⠈⢿⡛⠃⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣿⣿⣦⡄⠈⠁⠀⠀⠀⢠⠶⣿⣿⣿⣿⣿⣷
⠈⣟⠀⠀⠁⠀⠍⠤⠀⡄⠀⠀⠀⠀⠀⠀⠈⣰⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⠁⠀⠀⠀⠀⢸⣿⣷⣶⣿⡿⠛⠟⠩
⠀⠉⠀⠀⠀⠀⠀⠴⠾⠆⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠚⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡤⢀⣀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⡄⠀⠀⠈⠉⠉⠓⠒⠂⠀⠉⠉⣁⣀⡄⣴⠋
⠀⠀⠀⠀⠀⢀⠀⠀⠐⠢⠴⠤⣠⣃⡼⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠉⢧⣙⣄⡤⠦⠔⠊⠀⠀⡀
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠠⢄⠀⠙⡙⠀⠀⠈⠢⠈⢄⠠⠄⡠⠀⠖⠁⠀⠠⠋⠋⠀⣠⣴⠖⠃⣰⠃
⠀⠀⠀⠀⠀⠀⠈⠦⣀⠀⢀⣣⣴⣠⣄⣠⠀⠀⠀⠈⠀⠀⠁⠀⠀⢀⣄⠀⢠⡀⠘⠛⠁⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⡀⠛⢛⣦⠀⠀⠹⡏⠀⠀⣴⡛⠛⠀⠀⢿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠍⠁⢺⣿⠓⠦⠴⠳⡖⠙⣿⡷⠈⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⠀⠀⠀⠙⢧⠈⠀⠀⠀⣽⠋⠷⠤⡴⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⡷⢶⣴⡐⠈⠀⠀⠀⢸⠀⠃⠀⠀⠠⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠋⡟⢁⢈⡀⠀⠀⢘⡄⠩⣄⡐⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡶⡻⠁⡬⠥⢤⣀⣘⡡⠀⡾⢵⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣱⢄⡀⠀⠀⠒⣉⢵⡱⠷⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠉⢑⠚⡛⡋⠁⣺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠀⠤⠴⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣀⣀⣤⣤
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣿⡫⣒⣾⣿⡟⠻⣿⣿⣿
⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢨⣿⣷⠉⠉⠛⠷⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⣇⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣾⣿⠇
⠀⠀⠀⠀⠀⠀⢻⠀⢀⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⣿⣿⣿⣿⡟
⠀⣠⡠⣦⣤⠴⢾⡿⠛⠻⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠰⣿⣿⢰⣿⡆⣤⡀⠀⠀⢻⣿⣟⣿⣿⣷⠃
⠀⠈⠃⠉⠀⠀⢰⣵⡆⠀⠀⠁⠀⠀⠀⢰⡆⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠈⠁⢿⡿⠸⠿⠿⠀⠀⣼⣹⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⢘⡧⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⢀⡶⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣻⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣛⣟⠀⠀⠀⠀⠀⣸⣿⣿⠃⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⢸⣚⣿⣷⣿⠽⣴⣀
⠀⠀⠀⠀⠀⠀⠀⠘⣟⡆⠀⠀⠀⠀⠀⠨⣇⠀⠀⠀⠀⣽⣿⠀⠀⠀⠀⢠⣿⣿⡏⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⢠⣳⣿⣿⣿⠋⠈⠙⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⠀⠀⠀⠀⠀⢘⣷⠀⠀⠀⠀⣸⢿⡀⠀⠀⠀⡾⣿⡿⠁⠀⠀⢠⡟⠀⠀⠀⠀⠀⢠⡳⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡄⠀⠀⠀⠀⠠⢿⡄⠀⠀⠀⣹⣧⡀⠀⠀⡜⣽⠈⠁⠀⠀⠀⣼⠃⠀⠀⠀⠀⡠⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⣧⠀⠀⠀⠀⠠⢸⡇⠀⠀⠀⡼⣻⠆⠀⢸⣵⠃⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⡉⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢿⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠰⠿⠆⠀⠸⡟⣦⠀⠀⠀⠀⣾⠁⠀⠀⠀⠠⣔⢯⡳⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠾⡇⠀⠀⠀⠀⣸⠀⠀⠀⠀⢰⣿⡆⠀⠀⠈⠁⠀⠀⠀⢰⡏⠀⠀⠀⠀⣴⣽⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠸⣭⠅⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣾⣶⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠈⢀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⠓⠀⠀⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠲⠤⢄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠄⠀⠀⠀⠀⠀⣔⣠⡀⣣⣠⠤⠤⠠⠀⠀⠐⠂⢀⣤⠔⠂⢑⣝⠗⡂⢤⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⣎⠡⣼⢶⣵⠟⠈⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⢀⠎⣠⠞⡁⢈⡌⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰⣆⣀⣴⠯⡓⡾⠋⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⠻⠂⢡⠎⠐⣏⢡⣣⢄⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣈⠹⢫⡱⠊⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠰⠰⡜⣘⢇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⣠⠄⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢄⠀⠀⠀⢼⠀⣜⢬⣃⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⣀⠴⣈⣰⣥⠀⢄⠀⠀⠀⠀⠀⠀⠀⢀⡄⠊⣴⢭⣶⣄⠀⢘⠀⡅⠻⠄⠀⠀⠀⠀⠀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢰⠁⠀⠀⠀⢈⠵⢎⠡⣾⡆⣣⡞⠀⠀⠀⠀⠀⠀⠀⠠⣷⣭⣾⣴⠙⣮⠙⠈⢣⣠⢶⣁⠀⠀⠀⠀⢠⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⢤⡇⠀⠀⠀⠀⡎⠀⠣⣀⡭⠓⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⠯⣍⣀⣠⠇⠀⠈⡝⢇⠍⠃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢰⠈⣾⠀⠈⢸⠁⠀⠀⠀⠀⠈⠐⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⣈⣃⠀⠀⠀⠈⠁⣐⠆
⠀⠀⠀⠀⠀⣨⠛⢻⠀⠀⢤⠀⠀⠀⢀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⡔⡋⠰
⠀⠀⠀⠀⠀⠘⠶⣦⠠⡃⣾⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⢸⠮⠘⢤
⠀⠀⠀⠀⠀⠀⠀⠈⠀⣽⢸⡄⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠁⡟⢈
⠀⠀⠀⣀⡀⠤⢠⠀⣤⠌⠈⢧⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⠖⡥⠊
⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠴⠛⠛⠛⠛⠳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠀⡔⣭⣚⠧⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠂⠀⠙⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠨⠥⠀⠀⠠⠀⠔⠀⠹⡅⠀⠀⠀⢂⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⡛⣛⠇⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠘⣤⡠⠀⠛⠃⠀⠀⠀⠠⢄⣀⡘⢓⠛⠉⠉⠁⢀⡵⡄⠴⠁⣘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠐⠉⠀⣤⢚⢹⠀⠀⠀⠰⠠⢀⡄⠀⢀⣀⣉⣙⣻⣧⣴⡿⠁⣰⢷⠈⠉⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠂⠀⠉⠥⣀⣮⣄⣀⢠⠖⠀⣁⣠⡴⠛⢋⠀⠐⠋⣗⣄⡀⠰⢇⣠⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠈⠁⡂⡉⠀⠾⠞⠚⠃⠈⠉⠉⠉⠁⠈⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⡀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢾⣿⣿⣿⣧⣀⠀⠀⢠⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣧⣄⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⢀⠠⡈⢻⣦⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡶⠀⠀⠀⠙⠷⣶⣿⣷⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇⠀⠀⡆⠀⢀⠈⣿⣿⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣾⣹⡇⣀⣾⢞⣿⠏⠀⠀⠀⠀⠀⢸⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠉⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣶⣤⣤⣤⣤⡀⠈⠻⠃⠀⢻⣿⣿⣿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠸⣿⣿⣿⡇⠹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢻⣿⣿⡇⠀⢹⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⢻⣿⣿⡇⠀⠀⣿⣿⣿⠀⠀⠀⠀⠠⣤⣄⣤⣴⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠻⣿⣿⣿⣿⣷⣄⠀⠀⠘⣿⣿⡇⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠸⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⡀⠀⠀⠀⢸⡿⡏⠀⣿⠇⠘⣿⡟⢿⣄⠀⠀⢿⣿⡇⠀⠀⣀⣿⣿⣄⡄⡠⠂⠀⠀⠀⠉⠙⠓⠂
⠀⠀⢀⠐⢄⣀⠀⢨⣣⣠⣀⠱⣤⡘⢦⠱⡄⣸⢹⡇⣼⠃⠀⣰⠏⠀⠀⠉⢀⣠⣾⣿⣷⣾⣿⣿⣿⣿⣿⣟⣡⢎⣴⡠⠀⢀⡠
⠀⠀⠀⠐⢦⡽⣷⣤⣿⣿⣿⣷⣽⣿⣌⣧⣳⣿⣼⣵⣷⣄⣴⣃⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣧⣴⣫⣿⡔
⠀⣈⣀⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⣄⣀⡀
⠈⠉⠉⠉⠉⠉⠉⢉⠁⣀⠀⢀⣀⣐⣒⣒⣒⣒⣒⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⢠⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡄
⠀⠀⠈⠉⠿⠿⠿⠿⠿⠿⠿⣉⣉⠉⠉⣉⠉⠉⠉⠉⠉⠉⢹⣿⣶⣶⣶⣿⣿⣷⠀⠠⢀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠉⠁⠉⠛⠻⣉⢋⣿⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣷
⠀⠀⠀⢰⢣⠒⠌⣿⡆⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠢⠼⢄⣠⠄⠤⣀⡀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⢈⡏⠾⡀⣿⡇⠀⠀⠀⠀⠸⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠤⢀⣿⡇
⠀⠀⠀⢸⡇⠀⠇⣿⡇⠀⠀⠀⠀⠐⠀⡄⠀⠀⢀⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠠⠤⠤⠤⢄⠀⢂⣀⡀
⠀⠀⠀⢸⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠋⠹⠿⠯⢭⣭⣭⣭⣭⣭⣿⣗⣶⣶⣧⣤⡬⠈⣞⡏
⠀⠀⠀⠀⡇⢀⠸⣿⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠒⠛⠿⣾⣿⣿⣷⣿⠶⢤⣀
⠀⠀⠀⠀⠅⡀⠀⢻⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠼⢳⠟⠶⣄⠈⢳⡀
⠀⠀⠀⠀⡇⡇⠀⣸⡇⠀⠀⠀⠀⣂⠄⡇⠀⠀⢀⣤⣦⣤⣤⣤⡀⠀⠀⠀⣀⣀⣠⣶⣄⣠⠀⠈⠉⠁⠀⠀⢹⠀⠈⢷⡀
⠀⠀⠀⠀⡇⢇⠀⣿⠁⠀⠀⠀⠀⡁⠀⡇⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣧⣀⠀⠀⠀⠀⠀⠾⠶⠷⠮⠷
⠀⠀⠀⠀⣷⢸⡀⣿⠀⠀⠀⠀⠀⡏⢽⡇⠀⠀⠀⠀⠉⠉⠉⠉⠙⣏⠀⠝⠉⣯⠉⠉⢿⠉
⠀⠀⠀⠀⢻⡈⠁⣿⠀⠀⠀⠀⣖⡃⢸⡿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠋⢻⡆⠀⠈⣇⠀⠀⠀⠀⢀⢠⡾⠛⢳⡄
⠀⠀⠀⠀⢸⠣⡌⡇⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠘⡆⠀⠀⠀⣿⣿⣧⣤⠾⢧⡀
⠀⠀⠀⠀⠸⠀⠠⡇⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⠿⠿⠇⠀⠀⢸⡿⠉⢸⣇⣀⣠⡟
⠀⠀⠀⠀⠀⡇⢘⡇⠀⠀⠀⠀⠀⣿⣙⣿⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣧⠀⢰⡟⠛⠻⣇
⠀⠀⠀⠀⠀⣇⢸⡇⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡾⠷⣤⡶⠋
⠀⠀⠀⠀⠀⡇⢸⠇⠀⠀⠀⠀⠀⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣄⣀⣀⣄⠠⠀⠀⠀⠀⠀⠀⠀⠙⠹⣇⣀⣸⠇
⠀⠀⠀⠀⠀⡇⢸⡃⠀⠀⠀⠀⠐⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⡇⢸⡄⠀⠀⠀⠀⠀⠛⢻⡟⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⠀⠙⢯⣷⣤⡀⠀⠀⠀⠀⠀⣠⣄⣀⣀⣤⡄⡀
⠀⠀⠀⠀⢀⠇⢸⡇⠀⠀⠀⠀⢸⣀⣾⣿⠀⠀⠀⠀⢀⣤⣾⡿⠋⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠻⠋⠉⠉⠻⠟⠃
⠀⠀⠀⠀⠺⠶⠶⠟⠀⠀⠀⠀⠈⠌⠁⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠙⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠘⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⢹⣞⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢠⡄⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⡼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠸⣷⡔⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡜⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠻⣿⣦⣄⠀⠀⠀⠀⢶⡀⠀⠀⠀⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⢹
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⢠⣝⣿⣟⢷⣦⣄⡀⠸⣷⡀⠀⠀⢻⣇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸
⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⢸⣿⡇⠈⠻⢯⣟⡿⣿⣶⣥⣤⣤⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⢸
⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠇⡾
⠀⢿⣷⣄⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⣿⣮⡝⠛⠿⢿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠟⡼⠃
⠀⠈⢳⡝⣦⣀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⣿⢻⡇⠀⠀⠀⡀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢀⣤⠾⣿⣿⣿⣿⣿⠞⠁
⠀⠀⠀⠙⢮⡙⢷⣤⣀⠀⢸⣿⣷⡄⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠹⣷⣤⡀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠙⠷⣬⣽⣿⣾⣿⣿⣿⣶⣤⣤⣾⣿⣿⣿⣄⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⣴⠘⣿⣿⣆⠀⠀⠀⠀⠀⣀⣬⣝⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⡇⠀⢀⣼⡟⠀⢹⣿⣿⣄⣀⣠⣴⣾⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠿⢿⣿⣿⡿⣿⣿⣿⣿⣿⣶⣿⡿⠁⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡎⣿⡻⣷⣹⡟⢋⣀⣀⣴⡿⣵⣿⡿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣞⣛⣿⣿⣿⣿⣾⣟⣿⣾⣿⣵⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣾⣿⣿⣿⣿⣿⣿⣿⣯⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣾⣿⣟⣿⣿⣿⣛⣛⡿⣿⡟⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣽⣾⣿⣿⡿⠟⠛⠿⣮⠻⣿⣿⣿⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⣤⣤⣤⡴⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⣦⣤⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⢡⣤⣤⣤⡜⣿⣿⣿⢫⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⣴⠯⣭⣿⣶⣤⣤
⠀⠀⠀⠀⠈⠶⣤⣠⣤⣤⣤⡴⠶⠚⠋⠁⠀⠀⠀⠀⣰⣿⣿⣿⡟⢱⣿⣿⣿⣀⠀⠀⣀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠖⠛⠉
⢠⣆⠀⠀⠀⢠⡌⠉⢩⣥⠀⠀⣤⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠃⢸⣿⣿⣿⣿⣿⣟⣿⠋⣥⡄⠀⣤⠀⣤⠀⢠⡄⢠⣤⠀⣤⠀⣤⣤⣤⡄⣤⣤⣤⠀⣴⡄
⢸⡿⣧⡀⠀⣿⣿⠀⠸⣇⠀⠀⣿⠀⠀⠀⢀⣴⣿⣿⣿⣿⣯⣴⠀⣾⣿⣿⣿⣿⣿⣿⠃⠀⣿⡇⠀⣿⠀⣿⠀⢸⡇⢸⣿⣆⣿⠀⠉⣿⠉⠁⣿⠋⠉⠀⣿⢿⣆
⢸⡇⣼⠇⢸⡇⢸⡇⠀⣿⠀⠀⣿⠀⠀⢰⢿⣿⢿⣿⣿⣿⣿⣯⣴⣿⣿⣿⣿⣿⣿⡟⠀⠀⢸⡇⢀⣿⠀⣿⠀⢸⡇⢸⡏⢻⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⣿⠀⣿
⢸⣿⡏⠀⣿⠀⠈⣿⠀⣿⣸⡆⣿⠀⠀⣰⣷⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢸⣷⣾⣿⠀⣿⠀⢸⡇⢸⡇⠀⣿⠀⠀⣿⠀⠀⣿⣶⡆⠀⣿⣼⠇
⢸⡏⢿⡄⢻⡆⢰⡟⠀⣿⣿⣷⡿⠀⢀⣿⣿⡏⣿⣿⣿⡿⠃⠀⢻⡿⠽⠿⠛⠋⠀⠀⠀⠀⢸⡏⠀⣿⠀⣿⠀⢸⡇⢸⡇⠀⣿⠀⠀⣿⠀⠀⣿⠉⠀⠀⣿⢿⡄
⢸⡇⣼⠇⠘⣷⣼⠁⠀⣿⡏⣿⡇⠀⢸⣿⡟⣹⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⣿⠀⣸⡇⢸⣧⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⣿⠸⣧
⢸⣿⠋⠀⠀⢻⡏⠀⠀⣿⠃⢸⡇⣠⡿⠋⣼⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⣿⠀⢿⣿⢿⡇⢸⣿⠀⣿⠀⠀⣿⠄⠀⣿⣷⣶⠀⣿⠀⢻

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠐⠒⠒⠒⢒⡽⠿⠻⠟⠃⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠉⠀⣠⣆⣿⣿⣽⣿⣾⡇⢢⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄
⢠⡤⠖⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣶⡟⠁⠈⣷⣾⣷⣿⠆⠀⠘⢿⡀⠀⠀⠀⠀⠀⠀⢀⠆⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣠⣠⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠈⠉⠙⠋⠀⠀⠀⠀⠈⠈⠙⠚⠀⠘⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠤⠀⣀⠀⣐⠛⢖⣒⣓⣚⣓⣒⣲⡦⠴⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⣄⣀⡀⠀⠀⠴⠟⠙⠫⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣠⣀⡀⠀⠶⠟⡥⠉⠩⡍⠉⠉⠉⢉⣅⣴
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣅⠠⠚⠛⠙⣀⣀⣀⣠⣇⣀⣠⣇⣀⣀⣀⣀⣀⣸⣀⣠⣄⣀⣀⣀
⠀⠀⡀⠀⢀⣀⣀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⢀⣤⠗⠁⠀⠀⠀⠈⠻⣷⡄⠀⢀⠐⠂⢀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣴
⡼⡏⠀⠀⠀⠀⢀⠀⠀⠘⢿⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⢲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣝⠊⢂⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣚⡋
⣧⡏⠀⣨⣷⠊⠣⠀⠀⠀⣸⠀⠀⠀⠀⠀⠁⢣⠀⡴⠶⠶⠦⡄⢸⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠉⠻⢤⣧⣴⡁⠀⠀⠀⣱⣤⣧⠾⠋
⠘⣿⣄⠁⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠛⠈⢆⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉
⠀⠈⠛⠿⣶⣶⣶⠷⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠀⠰⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⣕⣀⣀⣠⡔⠁⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠠⡴⠒⠶⠤⠤⢤⣤⡄⣀⢠⣤⣄⣠⣤⣄⣤⣠⣾⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠛⠋⠁
⠀⣐⣠⡠⠤⠶⠶⣏⡿⠿⠿⠛⠛⠛⠻⠿⠛⠛⠛⠛⠛⠩⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀
⠀⠉⠁⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⡀⠀⠀⠀⠀⣾⣿⣬⣹⣿⣟⡒⢾⣶⣶⣾⡿⠚⣛⣟⣉⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣴⡿⠀⠀⣷⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⢾⣬⣌⣉⣤⣴⡇⠈⣉⠀⠀⣀⡀⠀⢠⠀⠀⠀⠀⠈⠋⠳⠶⠖⣺⣿⠀⠀⠀⠀⠀⠙⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠀⠂⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⣀⠭⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⣾⠕⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣴⡋⢉⢿⣿⣦⢠⠀⢀⡀⣰⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣎⣽⣿⣷⣶⣦⠿⣿⡀⠻⠟⠋⠙⣿⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⣿⣿⣫⣾⣿⡍⣝⡷⡄⠀⠀⠀⣜⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣸⣯⣼⣿⣿⠟⠍⢿⣯⣷⣷⠢⣄⣤⣧⣄⣤⢾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⡿⢿⣿⣿⡓⣤⣼⣟⣿⣅⣤⣿⣩⣽⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠳⣄⠲⣿⢿⣿⣦⢬⣿⣍⡻⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠠⢞⣿⡿⠀⠀⠀⠉⣍⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡊⠽⣷⣦⣤⣾⣿⣿⣷⣬⣶⣆⣺⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢉⡟⠛⢹⡿⣏⠀⠈⠙⣿⣿⣿⣿⣿⣯⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⣤⣾⣤⠶⠛⠁⠨⡟⢦⣄⣈⣻⣿⣿⣿⡇⢼⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡴⠦⣄⠱⣌⣠⣿⠿⢿⡟⢿⣿⡙⢿⣿⡇⠀⠀⠀⠀⠀⠀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠁⠒⠰⠿⠷⠿⠟⢤⣤⣶⣼⣿⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢶⡻⠮⣅⣈⣀⣠⣤⣴⣶⡾⠛⠿⠅⠉⣛⣿⣿⡟⠀⠀⠀⠀⠸⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠈⠉⠁⠀⠀⠀⠀⠉⠙⢻⣷⣦⣌⣿⣿⣿⡇⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢟⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣬⣽⣶⣶⣀⣐⣨⣤⣶⣿⣿⡿⠋⣿⣿⡟⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢹⣿⣿⣿⠟⠀⢰⣿⡿⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣀⣀⣿⡿⠁⠀⢀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠀⠀⢀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠋⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠃

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠠⣞⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠈⠁⠀⠛⣀⣠⡤
⠀⠀⠀⠀⠀⠉⠍⠀⠄⡀⠀⢀⡀⠀⠀⠀⠠⠌⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣉⠆⠀⠀⢉⣤⠖⠉⠛⠛
⠀⠀⠀⠀⠀⠒⣂⠛⣦⣮⣀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⢀⠀⠢⠐⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⠻⢛⣾⣿⣯⣠⣫⡕⠂⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢤⠃⠀⢣⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠠⠞⢡⡄⠀⢄⣀⡀⣀⠑
⠀⠀⠀⠀⠀⠀⠀⢫⣿⣿⡾⠛⢻⣏⠠⡀⠀⠔⠀⠀⠀⠀⠀⠐⢀⢀⡀⠀⠀⠀⠀⠁⡴⣦⠀⠀⠀⠀⠀⠀⠀⣀⡶⠉⡟⠧⠎⠙⠉
⠀⠀⠀⠀⠀⠀⠈⠁⣿⣠⣘⣟⡳⠏⠰⠄⠀⠔⣣⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰⠷⡄⣀⠀⢤
⠀⠀⠀⠀⠀⠠⠀⠀⢈⣹⣦⣤⣿⡤⠀⠀⠄⠸⠁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠘⠀⠀⠀⠀⠀⠀⢂⣠⣴⡆⠀⠴⠷⠖⠒⠁
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠹⢟⣳⣘⠻⠎⠀⠲⣄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠉⢀
⠀⠀⠀⠀⠀⠂⠀⠀⠰⠀⠈⠛⣛⣙⣃⠀⠰⢿⡎⠦⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⢣⠀⠀⠀⠀⠀⠐⣉⠛⡉⠹⠂⠀⣸⣈⣀⡄⢠⠀⠀⡀⠀⠤
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠙⣯⣽⠀⠀⡿⣷⡀⢂⠀⠀⠀⡀⠀⠀⠀⠿⠐⠧⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⡼⠃⠐
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠂⠀⠀⠀⢹⡆⠀⠐⠋⠁⠈⠀⠀⠘⡅⠀⠀⠀⡀⠀⡀⠀⡄⠀⢀⡀⢀⣀⠠⠤⠄⠒⣀⠀⡌⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠙⢦⣤⣤⠤⠤⠤⠒⠀⠀⠀⠠⠀⠓⣈⡁⠴⡿⠀⠈⠉⠫⡉⠁⠂⠠⣤⡴⠶⠄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡠⠀⠀⠀⠀⠀⠘⠲⠆⠀⠀⢀⣀⠀⠊⢀⡜⠋⢠⠙⠋⠀⠦⡄⢀⡀⠉⠀⢀⠀⡀⡀⠤⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠁⢈⠀⠢⢉⡙⠏⡯⢉⣐⢀⠘⠄⠙⠒⢤⡀⠈⠁⢀⡀⠄⠂⠀⠀⢀⠀⠀⠁⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠄⠤⠊⠀⠀⠀⠀⠀⢀⠀⠀⢰⡏⣳⣆⣸⣿⠄⠀⠐⠄⠀⠀⠉⠀⠛⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⣉⣩⣠⣿⢀⠀⢀⠄⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠈⠀⠈⢀⡤⠀⠀⠀⠀⠐⢻⡿⠥⢄⣈⠛⠭⠉⠥⠀⢰⠎⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠾⣷⢶⠶⠤⠛⡦⠰⠖⠀⠘⠂⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠦⠀⢀⡀⠐⢁⣠⠶⡄⠀⠐⠒⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠠⠞⠉⠉⠉⠁⠀⡔⣴⡄⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⣿⠉⣷⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⢠⡄⠀⠀⠀⠀⠀⠛⢠⣼⠋⢲⣄⡰⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡼⠀⠀⠀⢀⡀⠀⠀⢘⡃⠀⠀⢿⣧⢈⡀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢈⠁⠀⠀⢸⣏⠀⠀⢀⠀⠈⢣⠀⠀⠀⠂⠀⠀⠀⠀⠘⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⣾⠀⠀⠀⢨⡻⠄⠀⠐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⠀⠈⠀⠀⡿⠀⠀⠀⠘⣵⠆⠀⠀⣿⠀⠘⠠⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠠⠤⠆⠀⠦⠆⠴⠒⠠⠿⠀⡜⠀⠀⠀⠀⣇⠀⠀⠀⠀⢹⠆⠀⠀⢹⠄⠀⠴⠆⠛⠷⠀⠤⠀⠐⠠⠂⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠇⠀⠂⠀⢀⣬⠀⠀⠀⡀⣿⠇⠀⠀⠘⠃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠋⠀⠀⠉⢁⠙⠋⠹⠀⠀⠀⢁⠀⢠⡀⠉⠁⠀⠀⠀⠀⢠⠀⠂⠀⠀⢆
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠘⠀⠾⠀⡀⢀⡆⠘⠆⢨⠧⠀⢀⠀⠀⠰⢜⡧
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⡠⠀⠀⠠⠂⠁⠠⢀⠁⠜⠁⠀⢀⡁⠀⠈⠃
⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⣄⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠇⠀⠄⠀⠀⠆⠀⠋⠀⠈⠡⠀⠘⠉⠀⠀⠀⠀⠠⠀⠄
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠠⠀⠈⠠⠊⠀⠀⠀⠀⠀⠀⠀⣁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢈⡷⠀⠀⠰⣿⡆⠀⠀⠀⠀⠀⠒⠛⠊⢠⡀⠲⠶⢄⠀⠁⠲⠊⠑⢀⡀⠀⣀⠴⠾⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⣿⠆⠀⠀⠸⡅
⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⡺⠃⠀⣼⡟⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠈⠔⠂⠘⠈⡄⠉⠡⢅⠀⠄⠈⢁⣀⡀⠀⠀⢸⣧⡀⠀⠛⠄
⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣠⣤⠘⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠃⢤⣄⡀⠘
⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⢸⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣠⣤⣀⣀⣠⣤⣀⣀⣠⣦⡀⠀⠀⠀⠀⣀⠀⠀⣀⣀⣀⠀⣀⡀⢀⣀⠀⣠⡄⠀⠀⠀⢀⣾⡄
⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⢿⣿⣿⠿⠃⠀⠀⠀⠀⡬⣁⢆⣶⣶⣾⡿⠚⠓⠒⠛⠒⠛⠓⠀⠀⠀⣸⣿⣷
⠀⠀⠀⠀⣸⣿⣿⣧⣘⣀⣼⠀⠀⠀⠀⠀⠀⠀⣾⣾⣾⡿⠿⠻⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣿
⠀⠀⠀⣼⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠤⠤⠤⠤⠴⠿⠧⠀⠀⠀⠘⣿⣷⣀⣀
⠀⠀⢰⡋⠉⠉⠉⠉⢹⣿⣿⣿⢿⡿⠿⣿⠿⣿⣿⣿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠛⠛⠁
⠀⠀⢀⡇⠀⠀⠀⠀⢨⣿⣿⡿⠄⠡⠠⠂⠀⠂⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠂⣷
⠀⠀⠊⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⡂⣿⢴⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡷⠀⠀⠀⠀⢠⡀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⢋⣍⡓⢄⠀⠀⠀⠀⠀⢸⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢜⢿⣟⣼⡀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⢼⣿⣿⣿⢰⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢶⡂⡄⠶⠶⠶⠶⣾⠷⣶⠶⠶⠶⠶⠶⠶⠶⣲⣶⣄⠀⠀⠀⢻⣧⢼⣿⣿⡿⠈⡇⠀⠀⠀⢐⣿⡇
⠀⠀⠀⠛⠓⠓⠲⠆⠛⢣⡊⠁⠑⠚⠓⠚⠃⠐⠋⠁⠀⠀⠀⠀⠀⠈⠻⠾⠿⠿⠁⠀⠀⠀⠀⠀⢸⠏⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣆⣿⡆

⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢿⠆⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⢸⡿⠃⠀⢰
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠀⠀⣸⠆⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⡟
⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢰⣧⡄⠠⠏⠀⠀⠀⠀⠀⢀⣦⠈⢃⡀⠀⠐
⠀⠀⠀⠀⢀⣼⠋⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠉⠐
⠀⠀⠀⢀⣾⠏⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠰⡗
⠀⠀⠀⣸⣿⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠁⠀⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣿⣿⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢸⣿⡄⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏
⠀⠀⢀⣤⣿⣄⢸⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠸⣿⣣⡄⢀⡀⢀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⡀
⠀⠀⢰⣿⣷⣾⡿⠻⠒⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⣷⡆⠀⣄⡀⠀⢸⣿⠀⠀⠀⠀⠀⣣⡁
⠀⠀⠸⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣟⠛⠁⠈⠉⠁⠀⢸⡇⠀⠀⠀⠀⠀⠿⠇
⠀⠀⠀⢿⣻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⣷⡆
⠀⠀⠀⠘⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠹⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢻⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡿⠟
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⡄⢀⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠂⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣿⣷⣦⣤⣤⣤⣴⣾⡇⠀⠀⠀⠀⠀⢰⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⢸⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠐⠒⠂⠈⠁⠉⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠄⠤⠔⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠶⠄⠦⠴⠶⠿⠿⠂⠉⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣿⣿⠀⢀⠂⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃⢸⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠞⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠄⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡐⠁⠄⠉⡀⢢⠄⠤⠚⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⠀⢀⠶⠀⠀⣠⡘⠀⠸⠀⠲⡀⠡⣄⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠘⡴⠂⠈⠀⠉⠁⠛⠒⠊⠳⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠈⣿⡗⣶⡶⠰⠄⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣤⣀⡀⢀⣿⣧⠼⠡⠇⠠⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠛⠛⠿⠿⣿⣿⣿⣶⣤⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⣠⡀⢀⡀⠀⠈⠁⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠉⠁⠀⠐⠂⠐⠀⠀⠀⠀⠀⠀⠀⠲⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠂⠉⠀⠈⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠛⠿⠿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣥⣤⣤⣤⣤⣄⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠤⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠁⣤⣀⡀⢀⠄⠀⠀⠈⠙⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⢻⣿⣿⣤⡀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠈⠙⠢⣄⠀⠐⢦⣄⠀⠀⠀⠦⣤⣀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢗⣀⡐⠤⣤⡀⠰⠄⠀⠇⠁⠈⠻⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠛⠷⠎⡕⢦⣀⠀⠑⠶⠚⠈⠀⠙⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⠄⠀⠀⠀⠀⢯⡀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢠⡀⠈⡀⠀⠀⢠⡀⠈⠑⠄⠈⠛⢤⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠇⠀⢰⣰⡄⠸⡇⢠⡄⢀⠀⠀⠀⠦⠀⠀⢠⣄⠻⣇⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⡄⠀⣞⣇⢀⠆⠘⠇⠀⠀⠀⠀⠠⠤⠎⣼⠏⠀⣿⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣌⢿⣿⣎⣇⢀⣠⠤⠴⢶⣾⣿⣿⠿⡏⠉⣘⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣾⠀⠀⠀⠈⢣⣀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠉⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⢧⡀⠀⠀⣀⣀⣤⣤⣤⣤⣄⣠⣤⣤⣤⣤⣀⣀⠀⠀⢀⡼⠁⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⢈⣿⠞⠋⢉⠀⠀⢀⣠⣤⣨⣅⣤⣄⡀⠀⠀⡉⠙⠳⣿⡁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⠀⣀⣴⠟⢁⣴⢿⣷⠖⠛⠉⢞⣡⢭⡭⣌⡳⠉⠛⠲⣾⡿⣦⡈⠻⣦⡀⠀⣠⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⢋⡴⢻⡇⠀⠀⢀⣘⠛⠛⣩⠤⠤⢍⠛⠛⣃⡀⠀⠀⢸⡟⢦⡙⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠞⠉⣰⣿⣇⡀⠀⠊⢁⡠⠖⠒⠒⠒⠒⠲⢄⡈⠓⠀⢀⣸⣿⣆⠉⠳⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢀⡄⠉⣽⡏⠻⡄⠀⠺⠀⠀⣠⠔⠢⣄⠀⠀⠗⠀⢠⠟⢹⣯⠉⢠⡀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⣾⢇⣾⠁⠀⠗⢢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡔⠲⠀⠈⣷⡼⣷⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⢸⡏⠈⠙⠲⠄⠀⠙⢺⣿⣏⣀⠀⠀⠀⠀⣀⣹⣿⡗⠋⠀⠠⠖⠋⠁⢹⡇⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⢸⣇⣠⡤⡠⢞⣣⣤⠄⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠤⢤⣜⡳⢄⢤⣄⣸⡇⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⣠⣴⡟⣇⠀⢰⣿⣦⡻⢄⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⡠⢟⣴⣿⡆⠀⣘⢻⣦⣄⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣬⣧⠻⠆⢸⣿⣏⠻⢦⡀⠀⢠⡄⠀⠀⢠⡄⠀⢀⡴⠟⣹⣿⡗⠰⢟⣼⣥⣿⣷⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣯⣍⡛⠛⣻⣿⣓⠦⠀⠀⠈⠻⣤⣤⠟⠁⠀⠀⠴⣚⣿⣟⠛⢛⣩⣽⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦⠹⣿⣿⠿⠳⠶⡶⢤⣴⣦⡴⢶⠶⠞⠻⣿⣿⠏⣴⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⣀⣿⣿⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⣿⣿⣀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣆⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⣰⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡏⠖⡀⠀⠀⠀⠀⢀⠲⢹⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣟⣉⣉⣉⣙⣩⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢠⡤⠠⣤⠠⣤⠀⠀⣠⠔⠐⢤⠀⠀⣤⠀⠠⡄⠠⣤⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⣼⠃⠀⠀⠀⠀⠀⣿⠀⢠⠀⠀⣿⠀⡸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⢻⡄⠀⠀⢻⠃⠀⣿⠀⠈⠀⠀⣿⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠐⠛⠀⠀⠙⠒⠄⠚⠀⠀⠛⠒⠒⠃⠐⠛⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠒⠂⠈⠋⡀⠀⢀⠖⠒⠂⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡠⠖⠉⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠊⠀⠀⠀⡈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢨⠠⠄⠀⠀⠀⠀⡌⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠠⠀⠀⠀⠈⠁⠀⠆⠀⠀⠀⠀⡔⠒⡀⠀⠀⠀⠀⡈⠈⡃⠀⠀⠀⠠⠢⠀⠀⡄
⠠⠀⠀⢀⡆⠀⠀⠀⢀⠀⠀⠤⠦⠤⠤⠤⠄⣀⡀⠀⠀⡁⠀⠀⠐⠀⠿⠛⠁⠀⠀⠀⠀⠟⠧⠃⠀⠀⠈⠁⠀⠀⠀⠰
⠀⠀⢀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⡔⠘⠀⠀⠀⠀⠀⠩⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡦⣄
⠀⠀⠀⠀⢀⠎⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠁⠀⠄⠁⠈⢦⠀⢠
⠀⠀⠀⠀⠀⠑⠢⣄⠀⠀⡠⢒⡆⠄⠤⡠⠊⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠤⠄⠀⠠⠀⠀⠈⢢⠁⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠀⠘⠒⠚⠋⡠⢳⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⢇
⠀⠀⠀⡐⡇⠀⠀⠀⠀⠒⡡⠀⠀⢀⠄⡰⠊⠀⠈⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⢠⠀⢈⠐⠀⠀⠈⠄
⠀⠀⠀⠈⠀⡐⠀⢀⡴⠊⠀⠀⢠⠃⠀⢱⠀⠀⠀⠀⠀⠈⠂⣄⠀⠀⠀⠀⠀⠘⠂⠀⠐⠃⠀⠀⠠⠠⡀⠀⡀⠀⠀⡮⠖⠁⠀⠀⡠⠒
⠀⠀⠀⠀⠀⠃⠀⠊⠀⠀⠀⠠⠋⣆⠀⠅⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡠⠄⠒⠉⠀⠀⠀⢈⠆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠔⠁⠀⠈⠢⠀⢇⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠌⠀⠀⡤⡄
⠀⠀⠀⠀⠀⠒⠀⠒⠈⠀⠀⠀⠀⠀⠀⠑⠦⡉⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠴⠀⠀⠐⠤⠊
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠀⠀⠀⠈⠐⠪⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠴⠋⠁⠀⠠
⣀⣀⣀⣡⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣀⣀⣒⣒⣂⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠠⠷⠆⣰⠆⠀⠀⣰⡟⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞
⠀⠀⠀⠠⠄⠀⠀⢙⣯⡑⣤⣥⠃⠐⣤⠈⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⢶⣧⠶⣀⠀⠄⢈⣁⠀⠀⠀⢀⡿⠗
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠑⣟⠿⣆⢳⣿⣷⣄⢸⣧⣄⣀⣀⠀⠀⣀⡀⣠⣶⣿⡷⠟⠀⠀⢀⠀⣄⣸⡿⢷⡄⣠⠋
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢱⣭⢧⡻⠿⡟⠛⢉⣾⣿⣿⣿⣿⣷⣾⣿⣿⣿⣯⣾⠃⠀⣀⣤⣿⡾⠋⣩⠀⠂⢀⡈⠔⠀⠀⣉
⠀⠀⠀⠀⢀⠀⠀⠀⠺⠋⢱⣬⣥⠻⣿⣷⣿⣭⣿⣿⣿⣯⢀⣿⣿⣿⣿⣿⣿⣶⣾⣿⣟⢿⣷⠆⣶⣶⡂⠉⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣻⣿⣿⣟⠏⠁⠄
⠀⠀⠀⠀⢠⣤⣄⡘⢃⣠⠉⣽⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⣀⠈⢁⠀⠀⠈⣹⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⣶⠂⢁⡀
⠀⠀⠀⢨⡀⠘⠆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣷⣈⠁
⠀⠀⠀⠈⠀⠀⡀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣁⣽⡻
⠀⠀⠀⠀⠀⠀⢀⡄⠸⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⣤⣀⡷⠤⣼⠷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣷⡀
⠀⠀⠰⠆⠀⠀⠀⠀⠐⠈⠈⠹⣹⣿⡟⣿⣏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠉⢡⢂⡴⠛⠻⢿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠐⢀⠔⠋⠀⠀⠀⢨⣶⢛⠈⠻⢯⣝⣉⣿⣿⣅⠙⠛⣟⠒⠛⠛⢿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠜⠁⠀⠀⢐⠳⠞⢹⠃⠉⠀⠀⠀⢹⣿⠓⢿⣿⣾⠷⠻⠀⠀⠀⠈⢹⣿⠆
⠀⠀⠀⠀⠀⠘⠃⠀⠀⡰⠃⠀⠀⠀⠀⠘⠁⠲⡟⠶⠀⢀⠀⠒⠀⠉⢀⠸⠟⣧⠀⠀⠀⠀⠀⠀⠀⠉⠢⣔⣼⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡎⠀⠀⠀⠈⠀⠀⢰⡆⠀⠛⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⡀⠀⠀⠀⡄⠀⠀⠀⢠⠀⠀⠀⢀⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⣀⣀⣀⣄⣀⣀⠀⠀⠀⠀⢀⡄⠀⠀⢠⣄⠀⠀⡀
⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⣸⠀⠀⠀⢸⡇⠀⠀⠀⢾⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣿⡇⠀⡇⢸⣿⡆⠀⢿⣄
⠀⠀⢹⣿⠁⠀⠈⣿⠁⠀⠀⣿⡀⠀⠀⣾⣷⠀⠀⠀⣿⡄⠀⠀⢸⣟⠀⢸⠁⠈⣿⠀⠀⠙⠛⢻⡟⠛⠋⠁⠀⠀⢸⣿⢿⣶⣷⡾⣿⡇⠀⠀⢻⣧
⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⠀⣿⡇⠀⠀⠈⠯⠀⢸⠀⠀⠟⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⡟⠙⡏⢹⣿⡗⢸⡖⣿⣿⡇
⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⢻⣿⠀⠀⠀⣿⠀⠀⢀⣀⡀⠀⢸⠀⠀⠀⠠⠄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠻⡇⠀⡇⢸⡿⠃⢸⠛⢻⣿⡇
⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⣿⡆⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⢸⠤⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣄⠀⢠⡄⠀⢰⡄⠀⡇⠀⠀⠀⢸⠀⢸⡿⠁
⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⢿⠀⠀⢸⠉⡇⠀⢸⠀⠀⠀⢶⡶⠀⠀⢸⡇⢸⣿⡟⢻⡟⢻⣿⡇⠀⡇⠀⠀⠀⢸⣠⠞⠁
⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠸⡇⠀⢀⣀⣀⡀⠀⠘⣶⠃⠀⢸⠀⠀⠀⢸⡇⠀⠀⢸⡇⠸⣿⠛⢻⡟⠛⢻⡇⠀⡇⠀⠀⠀⢸⠁
⠀⠀⠘⡿⠀⠀⠀⣿⠀⠀⠀⣿⠇⣀⠀⠀⠁⠈⠉⠉⠉⠙⣦⠀⣿⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠈⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠃⠀⠀⠀⠋⠀⢀⠀⠀⢸⣿⠀⠀⢠⠀⠀⠀⢀⣴⠟⠀⣿⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⢸
⠀⠀⣰⡄⠀⢠⣷⠀⠀⣼⡇⠀⢸⣿⠀⠀⢸⡆⠀⠀⡇⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⢸⡀
⠀⠀⣿⡇⠰⣼⣿⡴⠀⢿⡇⠀⢸⣿⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⢸⡇⠀⢀⣀⣠⡄⣀⡀⢸⡇⠀⠀⠀⠀⠁⠀⠀⠀⠸⣇
⠀⠀⢿⠇⠀⠀⡯⠀⠙⢻⠋⠁⠘⡿⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⢸⡇⠀⢸⣿⣿⣿⣿⡇⢸⡇⢰⣿⣿⣿⣿⣿⠀⠀⠀⠙
⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡿⠀⠀⠈⠀⠀⠀⠀⣷⠀⠀⣿⠀⠀⠀⠀⣀⣀⣀⠀⠀⢸⣿⣿⣿⣿⡇⢸⡇⢸⠟⠛⠉⠁⣿⠀⣀⣈⣿⣀⣰⡆
⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠄⣉⣹⣉⠠⠀⠘⠛⢻⡟⠛⠃⠀⠁⠀⠠⢶⡄⠀⣿⠈⠙⠛⣿⠛⢻⡇
⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢻⠀⢀⣀⣀⣀⠀⠀⠋⢻⠉⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣰⣿⡇⠀⣿⠀⠀⠈⣿
⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⢿⣏⣯⣿⠇⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣼⣿⡿⠀⠀⣿⠀⠀⠀⣿
⠀⠂⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⣰⡆⢸⠀⠀⠛⡟⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣾⣿⡟⠁⠀⠀⣿⠀⠀⠀⣿
⠀⠄⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⢿⡧⢸⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⣼⡿⠋⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠄⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸⡇⠀⠀⣷⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠂⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⡀⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠇⠀⠀⢀⠀⠀⠀⠀⣸⠀⠀⠀⡇⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣿

⠀⢠⡶⣤⣶⡆⠂⢠⣢⡄⠀⠀⠀⠀⠆⠀⠀⣢⣄⢀⡄⢤⣐⡄⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⢸⣇⣨⣿⡟⠀⠎⠰⡦⠐⠀⠅⡈⡏⡥⠀⡟⢿⡋⢹⣶⢂⠱⠂⠘⠇⠀⠀⠀⣠⡀⢀⡀⠀⠀⠀⠀⠁⠀⠀⠀⣀
⠀⢸⣿⣿⣿⣟⡉⣻⣏⠗⠚⣸⢀⡐⣧⡐⠀⡍⢹⣽⣿⢻⣘⣿⢀⠐⠂⠀⠀⣶⣿⣯⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⢸⣿⣿⣿⠁⢀⣏⡙⡂⠀⢸⠦⠤⡿⠇⠂⡁⢸⠤⡿⣻⣿⡏⣀⠀⠀⠀⢰⣿⡟⠀⠁⠀⠀⠀⠀⠀⠀⠀⠸
⠀⢸⣿⣿⣯⠀⠚⠛⠡⠅⠀⢸⡆⢲⣷⡿⡄⠁⣾⣿⣿⣿⡿⠃⠉⠁⠀⠀⣼⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⣴
⠀⣼⣿⢸⣿⢠⣶⣶⡟⡃⠀⢰⠀⢽⣇⡁⡀⢰⢻⡟⠉⢿⣱⠀⠀⠀⠀⠀⡿⠀⡿⠀⢰⣶⣶⠶⠄⢀⣀⢠⡇
⠘⠛⠛⠛⠛⠛⠛⠛⠓⠓⠒⠛⠓⠚⠛⠓⠓⠛⠛⠓⠓⠛⠓⠒⠚⠒⠂⠀⠃⠀⠑⠀⠈⠀⠀⠒⠚⠛⠛⠛⠛⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀
⠀⡄⠀⣴⣦⣀⣀⡀⢀⡠⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠁⠀⣤
⠀⠃⢸⣿⣿⣿⡯⢍⡩⠉⠋⠰⠱⠀⠀⡄⠐⡀⢰⢥⠂⠀⠀⠀⣸⡇⠀⠀⢠⣿⣿⡼⠀⠀⠀⠀⠀⠀⣠⡄⠈⠛
⠀⠁⢸⣿⣿⠀⠀⠉⠍⠠⠱⢾⣷⠤⠀⠄⢠⢀⣴⣶⣦⣤⣤⣴⣿⠷⠄⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⡄
⠀⠀⢺⣿⣿⡃⠈⠈⠷⡬⢀⣄⠖⠀⠀⠀⢫⢘⣿⠿⣿⣿⣿⡟⠁⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠸⡇⢸⡇⠀⢰⡷⠿⡿⠉⠀⣴⣤⡀⠀⢹⡏⠀⠀⢸⠀⡉⠀⠀⠀⠀⢸⡇⢿⠀⣸⡷⢶⣶⠶⠄⠀⠀⠠⢴⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠣⠠⠟⠀⠀⠀⣀⠀⢀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠓⠁
⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠶⠂⠀⠀⡠⠤⡀⠀⠀⠀⠘⠿⠃
⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠉⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠣⠤⠃⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠘⢄⠤⠀⠀⠀⠀⠘⠟⠁⡔⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡆⠀⠀⠀⠀⠘⠃
⠀⠀⢀⣤⡀⠀⠀⢠⡄⠀⠀⠀⠀⠑⠒⠁⠀⡀⠀⠀⢀⡀⠀⠀⢠⣤
⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⡁⣈⠀⠀⠈⠛⠀⠀⠰⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡱⠀⠀⠀⢀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢾⡷⠀⢀⠄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⢀⣀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠜⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣨⠂⠀⠀⠀⠀⠀⠀⠀⢿⠗⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⢿⣿⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠃⠘⢯⠃⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠋⠁⢀⡤⣤⠀⠀⠀⠀⠀⠀⠀⢠⡤⣄⠉⠉⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⣿⠀⠀⠀⣾⣷⠀⠀⠀⣀⡀⠀⠀⢰⣿⡆⠀⠀⢸⡏⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢿⣆⠀⠀⠙⠋⠀⠀⢈⣿⣏⠀⠀⠈⠛⠡⡀⢀⣾⠷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣽⣷⣄⡁⠀⠀⠀⠙⠻⠛⠁⠀⠀⠀⣁⣴⣾⣥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢉⣿⣯⣿⣿⣷⣶⣶⣶⣶⣶⣶⣿⣿⣯⣿⣏⠁⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣤⠤⣠⡟⠉⠉⠉⠉⢉⠉⠉⠉⠉⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠅⢀⣤⣀⠸⣷⠶⠶⣶⡶⠋⠳⣶⠶⠶⢶⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠸⠿⠇⠀⣹⣿⣿⡿⠀⠀⠀⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠦⣤⠼⠟⠻⠿⠟⠁⠀⠀⠀⠀⠹⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠶⠖⠒⠒⠒⠶⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⢉⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⢠⣾⣿⣟⣻⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣥⣾⣿⣿⡾⠧⠀⢾⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⢠⣿⣿⠿⢿⣿⣣⣤⣴⣾⣷⡴⣿⣿⡿⣿⣿⣿⡋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⡄
⠀⠀⠀⠀⠀⣿⣴⣿⣿⣿⣿⣿⣿⣿⠿⠿⠀⠉⣿⣔⣟⣻⣙⡁⠀⠀⠀⠀⢰⡔⠀⠀⠀⠀⠀⠀⣿⠀⢿
⠀⠀⠀⠀⢸⣿⡿⣥⣿⣿⡟⢁⠈⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣷⣤⣀⡀⠀⠀⠻⣄⡀⠀⠀⢀⡤⠋⠀⢸⡇
⠀⠀⠀⠀⢸⠀⠀⠙⠉⠉⢹⣿⣿⣧⣽⣿⣿⣛⣿⠈⢻⣿⣿⣿⣿⣿⡿⠗⠀⠀⠘⠿⣿⣿⠿⠃⠀⠀⠈⡇
⠀⠀⠀⠀⢸⣶⣶⣿⣶⡤⣸⡿⢿⣿⣦⡀⠀⢸⣿⠀⣾⠉⠉⠀⠀⠁⠀⠀⠀⠀⠀⣼⣿⣿⡆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⡏⠛⠻⠿⠷⣿⣿⣖⡉⠻⠷⠀⢸⡟⢸⡇⢀⣀⣀⣀⣀⣀⣤⣶⣶⣾⣿⣿⣿⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣷⠀⠀⠀⠘⢿⡿⣿⣿⣷⠆⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢸⡟⠉⠙⠿⠗⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⢻⣿⡿⠋⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⢸⠀⠀⠀⠀⠀⠀⢸⣿⡟⣻⡿⠋⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣴⣄⣠⠀⠀⢸⣿⡇⠘⣇⠀⠀⠀⠀⠀⣀⠀⡏⣿⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣷⣼⣿⣷⣤⣿⣤⣾⣿⣷⣾⣿⣿⣷⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⢀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⢰⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿
⣾⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⠸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠁
⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⢀⡀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠁⡀⠀⠐⠈⠁⠠⡄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠈
⠀⠠⠈⠁⠐⠚⢷⣤⣀⣤
⠀⠀⠂⠀⠀⠀⣨⣿⢿⠷⣦⣤⡾⠗⠖⠀⣴⢀⣄⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠐⠀⠀⢰⡿⠀⢳⠆⣽⣶⡆⠀⡰⢿⣭⠀⠹⠤⠤⠄⠤⠀⠀⠀⠀⠀⠀⣠⣤⡶
⢠⠐⠁⠄⡀⠀⠀⠀⠠⣾⠷⣤⠈⢛⣀⡀⠉⠀⠀⠀⠀⣀⠀⠀⠀⠈⠀⠀⠀⣾⠛⠿⠄⠀⡀
⠀⠐⠂⠀⠉⠀⠀⠀⠀⣾⣀⠀⠉⠚⢳⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠁⢀⠀⠀⠀⠙⠃
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀
⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣷⣦⡭⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⢀⠾⠒⠑⠀⠙⠳⣦⡀
⣿⣿⣿⣿⢟⣏⣴⣿⣿⣿⣿⣿⣿⣆⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢐⣴⡟⠃⠀⠀⠀⠀⠀⠙⣏⡠
⣿⣿⣿⡿⣻⣽⢿⡿⣿⣿⣿⣿⣿⣃⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⣬⠼⣠⠀⡀⠀⠀⠀⠀⢠⣿⢰
⣿⣿⣿⣿⡿⣿⣟⣿⣿⣿⣿⠟⣻⣿⣿⣽⣿⣿⣿⣿⠀⠀⠀⠀⠁⡂⢘⣔⢀⡀⠀⢀⣤⠊⢜⠘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠁⠀⠀⠌⠘⠈⠀⡯⠥
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠥⢌
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠅⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠐⠀⠸⠀⠂⠐⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠠⢀⠀⠐⠀⠀⢠⢀⠀⠀⠀⠀⠁
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠃⠀⠀⣿⠁⠘⢠⠄⣴⡢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⣤⣴⣄⣺⣣⣖⣷⡀⠈⠀⢴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠀⠄⠀⠈⠉⠑⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢰⣆⣠⣤⢦⡤⠀⠀⠀⢷⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⢂⡀⠚⠳⢾⠋⠀⠀⠀⠀⢸⣿⠎⠁⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⠛⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠇⢺⡘⠇⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣀⣀⣀⣠⠐⠉⠙⣿⣷⡄⠀⠀⡞⠧⣀⠘⢦⠀⠀⠬⠚⠛⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠁⡹⠀⠀⠈⠀⢻⣷⠀⢀⠀⠀⠈⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠁⠳⠢⠆⢁⠀⢺⡟⢠⡏⠀⠀⠀⠀⣡⠋⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⢠⠆⡂⠀⠐⠲⠶⣫⢠⠏⠀⣠⡴⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠁⠀⢀⠀⠀⠀⠠⠄⠀⠀⡰⠋⠀⠀⠀⡠⠂⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⡔⢀⢇⠀⠀⠀⠘⢀⠎⠀⠀⠀⠀⠀⠀⠀⢈⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠝⠒⠀⠈⠁⠸⠃⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠂⠁⠀⠠⢤⣾⣞⡁⠀⢀⣀⡤⠔⠀⢀⡴⠋⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠛⠛⠛⢅⠀⠀⠀⠀⠀⠒⠋⠉⠙⠛⠛⠯⠥⠤⣶⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⣀⡀⠀⠀⠀⠀⠁⢠⠟⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠑⢄⡀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠄⠀⠀⠀⠀⠀⢀⠘⠋⣉⡝⠁⠤⣤⡤⡄⢤⠖⠃
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⠂⠈⠙⠐⠊⠁⠀⠀⠀⠀⣀⠴⠀⣤⠖⠃⢰⣴⣶⣤⠴⡾⣷⠒⠈⣭⣤⠂⠀⠀⢀⠤⠊
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠉⣀⡠⠶⣿⠉⠉⣹⠛⠙⣿⠦⣀⣉⣳⠀⠀⠉⠁⠀⠀⠈
⠀⠀⠀⠈⡀⠀⠀⠀⠀⢀⡤⠤⠚⠉⠉⠉⠉⠀⢀⠈⠁⠀⠀⢸⠳⠀⣻⠦⣼⣇⠀⠈⢉⡟⠛⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⢨⣿⣧⠀⣿⣤⣈⡏⠁⠀⣼⠀⠀⠀⣀⡤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠂⠀⠀⠀⠀⠾⠿⠿⠀⠈⠛⠻⠿⠶⠚⠙⠓⠀⠈⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠠⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂⠀⠀⠐⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠾⣿⡟⠛⠛⠛⠛⠿⠷⢶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟⠋⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⢻⣯⠉⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠋⠁⠀⠀⠀⢀⣴⡿⠿⢶⣶⣶⣤⣤⣀⣀⣿⡆⠀⠀⠈⠹⣷⣄
⠀⠀⠀⠀⠀⣠⡾⢋⣾⠏⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣶⣤⣀⠀⣿⠙⢿⣦
⠀⠀⠀⢀⣾⠏⣰⡿⠁⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣄⡀⠹⣷⡀
⠀⠀⢠⡿⠁⣰⡿⠀⠀⠀⠀⠀⢀⣾⣯⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣸⣧
⠀⢠⡿⠁⢠⡿⠁⠀⠀⠀⠀⢀⣾⠃⠉⠉⠉⠙⠛⠛⠿⠶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧
⢀⣿⠁⠀⣾⠇⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆
⣸⡏⠀⢠⣿⠀⠀⠀⠀⠀⠀⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⡀⠀⠀⠀⠘⣿⣿
⣿⠃⠀⢸⡏⠀⠀⠀⠀⠀⢠⣿⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣄⠀⠀⣿⣿
⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠙⠻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣆⣿⣿
⣿⡄⠀⢸⣧⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢸
⢹⡇⠀⠸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼
⠘⣿⠀⠀⣿⡆⠀⠀⢀⣠⡾⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀⡀⠀⠀⠀⠀⣾⢧⡿
⠀⢹⣇⠀⢸⣷⣤⡾⠟⠉⠀⠀⠀⠀⠉⠛⠿⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⣦⣤⣾⢏⣾⠃
⠀⠀⢻⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠟⠷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢃⣾⠃
⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠉⠙⢻⣷⣦⣤⣀⠀⠀⠀⠀⣠⡿⣡⡾⠁
⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠉⠙⠻⢶⣶⠿⣫⡾⠋
⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⣤⣤⣶⠿⠋⠁⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠀⣰⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣤⣀⡀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⠀⢀⣀⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠶⢶⣾⣿⣷⣶⡶⠶⠶⠾⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠂⠈⠋⡀⠀⢀⠖⠒⠂⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡠⠖⠉⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠎⠀⠀⠀⡈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢨⠠⠄⠀⠀⠀⠀⡈⢀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠂⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠈⠁⠀⡆⠀⠀⠀⠀⡔⡐⡄⠀⠀⠀⠀⡈⠉⡃⠀⠀⠀⠠⠂⠀⠀⡄
⠀⠀⠀⢀⡆⠀⠀⠂⢀⡀⠤⠤⠶⠤⠤⠤⠤⣀⡀⠀⠀⡇⠀⠀⠐⠀⠿⠙⠁⠀⠀⠀⠀⠿⠷⠃⠀⠀⠈⠁⠀⠀⠀⠰
⠀⠀⢀⠀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⡔⠘⠀⠀⠀⠀⠀⠡⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡧⣄
⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⣠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠀⠀⠀⠀⠈⢧⠀⠠
⠀⠀⠀⠀⠀⠙⠂⢄⡀⠀⡠⢒⠆⠀⠤⡠⠊⡇⠀⠼⠋⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠈⠠⠢⠄⠀⠠⠀⠀⠈⢢⠁⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠀⠘⠒⠚⠉⠐⢳⠌⢲⠒⠒⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢇
⠀⠀⠀⣔⡇⠀⡀⠀⠀⠒⡡⠀⠀⢀⠄⡴⠉⠀⠈⠓⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⢀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⢠⠀⢈⠑⠀⠀⠈⠄
⠀⠀⠀⠈⠀⡜⠀⢀⠔⠋⠀⠀⢠⠁⠀⣸⠀⠀⠀⠀⠀⠈⠒⣄⠀⠀⠀⠀⠀⠘⠒⠀⠀⠃⠀⠀⠀⠤⡀⠀⡀⠀⠀⡦⠚⠁⠀⢀⡠⠊
⠀⠀⠀⠀⢠⠅⠀⠋⠀⠀⠀⡠⠋⢆⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡜⠀⡠⠔⠂⠁⠀⠀⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠈⢢⡀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⢀⠎⠀⠀⠔⡆
⠀⠀⠀⠀⠀⠑⠒⠒⠈⠀⠀⠀⠀⠀⠀⠙⠢⡈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠔⠁⠀⠑⠒⠊
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠑⠢⠷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠔⠋⠁⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄⣀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⡄⢸⠁⠀⢀⣀⡀⠀⠀⠀⠀⠀⠁⠂⠑⠈
⠀⠀⠀⠀⠱⢵⣰⣄⣀⡀⢐⣁⣹⠀⢀⣀⣴⣰⠵⠅⠀⠀⠀⠀⠀⠀⠁⠘⠀⡔⠁⠀⠈⡆⠀⠀⢀⠄⢸⠀⠀⡄⡠⡀
⠀⠀⠐⠀⠀⡐⠥⡝⢹⠃⠆⢀⠀⠦⢻⠝⡥⠕⡀⠀⠐⠀⠀⠀⡀⠁⠀⠐⠎⡢⠀⠀⡰⠀⠀⠀⠸⠺⠂⠸⠇⠀⡀⠄
⠀⠀⠀⠀⡧⠀⠀⠈⠒⢻⠀⠼⡀⢸⠓⠊⠀⠀⠂⡅⠀⠀⠀⠘⢀⠀⠀⠀⣯⣆⢡⠎⠀⠀⣀⠀⣰⠺⠀⠀⠁⠀⡃⢄⣀⣀
⠀⠀⠀⠀⠈⢶⠮⢒⡤⠊⢂⠛⢃⠈⠣⡔⠢⢶⠎⠀⠀⠀⠀⠀⠀⠀⠰⠀⣉⡁⠰⠀⠀⠀⠡⡀⠉⠀⠀⢀⢀⠀⠑⠉⢀⠌
⠀⠀⠀⠀⠀⠀⠒⠊⠁⠀⠀⠐⠂⠀⠀⠉⠒⠊⠀⠀⠀⠀⠀⠈⢄⠘⠀⢠⠍⡡⠃⠀⠀⠀⠀⠱⡀⢠⠚⢳⡖⣛⠆⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠊⠀⠀⠀⠀⠀⠀⠀⠑⠀⠊⠉⠀⠁⢠⠊
⠀⠀⠀⠀⣠⠥⠺⢤⡐⣀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⣴⠀⣀⠀⠀⠀⠀⠑⠂⠐⠊
⠀⠀⠀⣻⠁⣒⡥⠒⠛⢠⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⢀⠺⣤⠚⢵⣼⡆
⠀⠀⠀⠋⠍⠁⠈⠀⠐⢸⠀⠀⠀⠰⠄⠀⠀⣀⣀⡀⠀⠀⠶⠀⠀⠀⡄⠘⣜⠀⡜⣀⠅⡂⠀⠀⡐⠄⢡⠤⢤⣊⡠
⠀⠀⠀⠀⠚⠀⠀⠀⠃⡘⠀⠀⠀⠦⠀⡔⢉⠄⠂⢌⠑⡄⠠⠆⠀⠀⢃⠀⠈⠉⠉⠁⡐⠀⠀⠀⠀⠀⢠⣘⣣⠀⠃
⠀⠀⠀⠈⠢⠄⢉⠠⠊⠀⠀⠀⠀⠀⠚⠂⠒⠀⠀⠐⠂⠚⠒⠀⠀⠀⠀⠑⠤⠤⠐⠊⠀⠀⠀⠀⠀⠀⠀⠐⠄⠈

⣿⣿⣿⣿⣿⣿⠟⣣⡤⠄⠐⠞⠻⠶⡤
⣿⣿⣿⣿⣿⣿⣿⡷⠶⠶⠒⠶⠖⠒⠿⠿⠲⠖
⣿⣿⣿⣿⡟⠓⠲⠆⠀⠀⠀⠀⣿⣷⠀⠀⡀
⣿⣿⣿⣿⣁⡀⠀⠀⠀⠀⠀⣼⣿⣯⠀⡴⠁
⣿⣿⣿⣿⠥⢴⣤⠀⠀⠀⠀⢿⣿⣿⣿⠃
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀
⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠁⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⣀⣀⣠⣾⠿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢀⣴⣿⣿⠀⠀⠀⠀⠀⠀⢀⡤⢾⣿⠿⣿⣿⣿⣿⡃⠀⠀⡀⡧⢚⡀
⣿⣿⣿⣿⢀⠀⣀⢀⡀⡀⢀⣿⣹⣷⣿⣿⣿⣽⣠⢀⣀⣄⣄⣠⣫⣀⠞⠁⠀⠀⠀⣈⣽⣷⣤⣴⣿⣿⣤⣤⣄⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣀⣀
⠀⠀⢠⣿⣿⣷⢦⣄⡀
⠀⠀⠀⣿⡟⠻⢷⣬⡙⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⡀
⠀⠀⠀⣿⡇⠀⠀⠈⠙⠷⣯⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠈⠻⣿⣷⣄⡀
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠈⠹⣿⡙⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢰⡄⠈⠻⣿⣿⡆
⠀⠀⠀⢹⡇⠀⠀⠀⠀⢰⣄⠀⢸⣷⡀⢘⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⣸⡇⠀⠀⠈⠻⣿⡆
⠀⠀⠀⠀⢿⡄⠀⠀⠀⢀⣿⣆⠈⢿⣧⠀⠻⣿⣿⣷⣤⣀⣀⣀⣀⣤⣸⣿⣠⡟⠁⠀⢳⡀⠀⢹⣿
⠀⠀⠀⠀⠀⢻⣦⣄⢠⣼⣿⣿⡄⢈⣿⣆⠀⢹⣿⣿⣏⠻⣿⣿⠿⢿⣿⣿⣿⠿⠿⠷⣾⣷⣤⣀⣿⡀
⠀⠀⠀⠀⠀⠀⢯⡻⢿⣿⣿⣿⣿⣿⣿⢏⠀⠈⠀⠙⠿⢷⡈⢻⣶⡿⠶⡷⠶⢦⣴⣶⣤⣄⣀⢉⡉⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠹⢤⡈⠙⠛⠿⢿⣿⣄⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⠈⠉⠛⠛⢦⣠⡄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣓⣶⠶⠿⠿⠿⢛⣷⣶⠖⠀⠀⠀⠀⠀⣸⡟⠋⠙⠋⠉⠂⠈⠲⣤⣄⣀⠀⠉⠛⢦⣀⠙⠿⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠴⢾⣿⣥⡄⠀⠒⢛⣿⣿⡏⠀⠀⠀⠀⠀⠀⢤⣽⣿⢠⡀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣄⠀⢻⣧⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠀⣠⣶⡿⠟⠋⠀⠀⠀⠀⠀⠀⣀⣀⠈⢉⡀⢿⣶⣄⡀⢠⡀⠀⠀⠀⠈⢻⣿⣆⠀⠛⠀⠀⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡧⠀⠾⢻⡟⠁⠀⠀⠀⢀⣴⣿⠃⠒⠺⢿⣿⣿⡿⣶⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢿⣿⡆⠀⠀⠀⠀⠐⣿⡟⡄
⠀⠀⠀⠀⠀⠀⠋⣿⠟⠁⣠⣾⣿⠋⠀⣠⠄⠀⣼⣿⡋⠀⠀⠀⢀⡈⠀⠀⠀⢻⣿⣿⠉⠛⢿⣷⣄⠀⠀⠈⢹⣧⠀⠀⠀⠀⠀⢻⠀⢰
⠀⠀⠀⠀⠀⢀⣼⠏⠀⡰⢫⡿⠁⣠⣾⡏⠀⢸⣿⣿⡇⠀⠀⠀⠘⣿⡆⠀⠀⠘⣿⣿⡄⢀⣈⣿⣿⣷⣦⣀⣼⣧⣤⣄⣀⣠⡆⢰⠀⣾
⠀⠀⠀⠀⠀⢸⣿⠇⠀⢠⡿⠀⠘⣫⣿⡇⠀⢸⡛⣿⣧⠀⠀⠀⠀⠙⠃⠀⠀⠀⠘⢿⣿⣾⣿⣿⣿⣿⠋⠙⠻⣿⠄⠀⠉⠉⢳⣿⣤⡆
⠀⠀⠀⠀⠀⣿⡟⠀⢀⣿⠁⠀⠐⣿⣿⡀⠀⠘⠃⠙⠻⣿⣦⣤⣤⣀⣀⠉⠙⠓⠶⠾⠿⠿⠿⠿⠿⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠻⣃⡇
⠀⠀⠀⠀⠀⠋⠁⠀⢸⣿⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠠⠄⠚⠛⠛⣛⠒⠒⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⡀⠠⣄⡀⠠⣀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⣠⡄⢸⣿⠀⠀⠛⣿⣿⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣦⣤⡀⠀⢠⠈⠹⠟⠳⠄⠀⠉⠳⣤⣈⠙⠦⠈⠳⠆⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠘⢻⣇⢸⣿⡆⠀⢀⣿⣧⡖⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⡿⣿⣷⡀⢧⠀⠀⡀⠀⠀⠀⡀⠈⠻⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢸⣿⡎⣿⣷⠀⠸⣿⡟⠀⣰⢃⣾⣿⣿⣿⣿⣿⣿⣏⠙⢿⡎⠙⣿⣿⡇⢸⠁⠀⣀⠀⠻⣆⠀⠟⠀⢰⣶⣦⣄⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠈⠙⣇⢹⣿⡀⠀⣿⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡆⠈⣿⠀⠈⠙⣿⡐⣦⠀⠙⢦⠀⠙⠀⠀⠀⠘⠛⠻⣿⣿⣷⣶⡶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⣿⡇⠀⠋⠀⣿⣿⣿⣿⣿⠿⠋⠙⣿⠁⣿⠇⠀⠈⠳⡀⠀⠛⣷⡙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠃⠀⠈⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⢀⣿⣿⣿⣿⠃⠀⠀⠰⠃⢀⠟⠀⠀⠀⠀⠣⢄⠀⢸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⢀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⢸⣿⣿⣿⡧⠖⢨⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡿⠛⠛⠻⣦⣄⡀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢪⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠘⣿⡟⠁⠀⠀⣼⠁⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⡀⠐⠋⠉⠛⠿⣷⣶⣶⣾⣿⣧⣄⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢰⡄⠀⢿⣦⠀⠀⢰⣿⣆⢈⡄⠀⢰⡄⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢉⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⢿⡀⠘⣿⣧⣤⡟⠀⠉⠉⡹⢁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠘⢷⡀⠈⠻⠿⣿⣷⣶⣾⡤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣄⡀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠋⠁

⣿⣿⣿⠿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣄⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠿⣶⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣶⡾⣿⠲⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⣿⣿⣿⠿⠟⢛⣭⣽⢂⠀⢠⡶⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢤⣶⣾⣶⠿⣟⣛⣭⣵⣶⣿⣷⣶⠿⠟⠋⢘⣴⣿⡗⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⡿⠿⠛⣋⣸⣥⣴⣶⠈⣿⣿⣿⣿⣿⣿⡀⠄⠀⠀⠘⢿⡈⠐⠀⠀⣟⣂
⠀⠀⢠⣤⣤⣤⣤⣔⡺⢿⡿⠿⣛⣫⣭⣴⣾⣾⣿⣿⣿⣿⣿⣿⠾⠟⠛⠉⠀⣸⣿⡇⠀⠀⠀⠀⣀⣠⡴⢢⠞⠉
⢰⣶⣿⣿⡿⢿⣛⣯⣽⣶⣾⣿⣿⣿⣿⣿⣿⣿⠿⠟⢛⣉⣥⣴⣶⣶⣶⣾⣿⣿⣿⣧⣤⣴⣶⣿⣿⠉⡆⠊
⣸⣿⣿⠖⢋⣩⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⢛⣿⣙⣿⠿⠛⠀⡇
⣿⣿⣿⡄⣾⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡇⠰⠋⢹⣿⣿⣿⢿⣿⠀⠀⠀⠀⢷
⢻⣿⣿⣧⠙⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠀⠀⠘⣿⣿⡷⠀⠀⣾⣿⣿⡷⣾⣿⠀⠀⠀⠀⢸⡀
⠈⢿⣿⣿⣦⣴⣾⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣤⣾⣿⣿⣿⣿⣼⣿⠀⠀⠀⠀⠈⡇
⠀⠈⠻⢿⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠿⠛⢉⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣤⡄⢀⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣤⠈⢁⣀⣠⣤⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢿⣿⣿⣿⣯⣾⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿⣷⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣾⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣶⡤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠙⠳⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⣠⠴⠚⠉⠉⠉⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠉⠐⠤⣄⡀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡈⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡙⢄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣡⣴⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠷⠤⢤⣀⣀⡀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⣀⣀⣤⠴⢿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠉⠉⠛⠛⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠖⠛⠋⠉⠁⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡇
⠀⠀⠀⢸⠈⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⠁
⠀⠀⠀⢸⠀⠀⠙⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠏
⠀⠀⠀⢸⡆⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠘⣇⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⢿⡀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠘⣧⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢠⠁
⠀⠀⠀⠀⠀⠸⣆⠀⠀⢻⣄⣀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⣀⣤⡞⠁⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠹⣆⠀⢸⣿⣿⣿⣿⣶⣶⣤⣤⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣤⣤⣶⣶⣿⣿⣿⣿⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢧⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⠿⠛⠁⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠈⠛⠿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⣶⡶⠶⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣭⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⢀⣦⣤⣀⠀⠀⠀⠀⠀⠀⢰⣶⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣏⣁⣈⣙⣉⣀⣉⡓⣦⡀⢀⣀⣠⣼⣿⢟⣋⣩⣭⣭⣉⣈⡙⣆⠀⠀⠀⠀⣠⡴⠶⠶⢾⡷⠆
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠯⢍⣉⣉⣉⣉⣉⠉⠩⠽⠟⣿⠟⠉⠉⢻⣿⣟⠛⠋⠛⠉⢉⣹⣿⣿⠀⠀⠀⠞⠁⣠⡶⠶⢻⡟⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⣍⣩⡿⠀⠀⠀⠀⣿⠀⠀⠀⢸⠉⠙⠛⠿⠿⢿⠿⠛⠉⢸⠀⠀⠘⠀⠀⠈⠉⠓⢺⡟⠂
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⣀⡀⠀⠀⢀⣀⣠⡟⠀⠀⠀⢸⠀⠀⠀⠀⢀⡜⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⢀⣠⠶⠚⡗⠒⠒⠒⠋⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⣨⡷⠶⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠁⠤⣀⣀⡀⠀⠀⠀⠀⠈⠓⡄⠻⠂⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡜⠋⠀⣇⠀⠀⠀⠙⣦⠀⠀⠀⣠⠔⣚⣻⣏⣶⣄⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠘⠶⣛⢿⣿⠿⠉⠙⠶⢾⡆⠀⢰⠁⠿⠉⢹⡯⠶⠘⡇⠀⠀⠀⠀⠀⠀⣠⢠⡶⠛⠓⣶⣒⣓⠢⣀⣀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⡇⠀⣾⠀⠀⠀⠀⡄⠀⠀⣷⠀⠀⠀⠀⠀⠀⠘⠿⠶⢿⡿⠭⣭⠛⠛⢩⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠸⠀⠀⠀⢀⡇⠀⢀⣿⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠿⠿⠿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⡇⠀⠀⡆⠀⠀⢠⣷⣚⣉⣽⣉⣉⣉⠛⠢⡀⠀⠀⠀⡾⢧⣀⣀⣠⣴⣇⣀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣇⠀⠀⠳⡤⢶⠇⠀⠀⠸⠻⠭⣯⣯⣯⣽⠿⠟⠀⢹⠀⠀⢰⠗⠋⣉⢻⣉⠛⣉⣋⣛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⡴⠖⠁⡠⠤⠀⠀⢠⣠⠤⠊⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⠀⠀⠀⢹⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⡾⠤⠤⠤⠤⠤⠤⠔⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠀⠀⠀⠠⢄⣤⣀⣀⠤⠖⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠘⡄⠀⠀⢸⣇⣀⣀⣤⠾⠁
⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢃⣀⣀⣀⣠⣤⡾⠀⠑⠒⠂⠠⢄⡀⠀⠀⠈⡿⠈⡷⠃⢀⡾⢹⡏⠁
⠀⠀⠀⠀⠈⡟⠳⣄⠀⠀⠀⢠⣶⡿⠟⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⢀⣴⠾⠿⣶⠟⠷⣼⡇⠀⠳⠤⢀⡀
⠀⠀⠀⠀⢀⣷⣤⣼⡆⠀⠀⣾⡏⢠⣶⠾⠿⠿⠿⠿⠿⣶⣤⠶⠖⠒⠒⠒⢻⣇⣀⠶⠶⠦⢄⣽⡇⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠘⠋⠉⠉⢱⠀⠀⣿⡇⠀⠁⠀⠀⡄⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠘⡄⠉⠉⠉⠙⠋⢻⠇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⣿⡇⠀⠀⠀⠀⣧⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠢⠴⠞⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢹⡇⠀⠀⠀⠀⠻⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⠤⣄⠈⠓⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⡿⠃⠀⠀⠀⢀⣩⣤⣤⣼⠏⠀⣠⠤⠒⠒⠒⠀⠀⢀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⢣⣤⣤⣄⣸⡟⠹⠁⠀⠀⠀⠀⢷⣴⣶⣖⣶⣶⢦⣼⡄⠀⢸⡶⠖⠷⢶⡶⣤⣄⣸⡇
⠀⠀⠀⠀⠀⠀⣰⡟⠉⠉⠉⠛⠁⠀⠈⣿⣄⠀⠀⠢⠤⢄⣀⣀⡀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠃⠉⠉⠁⠀⠀⠉⣹⡷
⠀⠀⠀⠀⠀⢰⣏⣤⠶⠛⠓⠶⠛⠓⠒⣿⣮⠷⠀⠀⠀⠀⠀⠀⠈⠉⠑⢦⡀⠀⠀⡇⠀⢸⠀⠀⠀⣀⣤⣴⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠉⠳⠶⠦⠤⠤⠤⠶⠶⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠓⢴⡞⠀⠀⢰⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠈⠙⠂⠀⠘⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠋⠉⠉⠉⠉⣩⡛⢷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠙⣷⣾⣷⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⣉⡙⠿⣿⣭⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣦⣦⣬⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠣⣴⣤⣄⡀⠀⠀⠀⠀⠀⠈⢻⣿⣿⢿⣿⣿⣿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠰⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠹⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⢻⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⢸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⠶⠶⠶⠟⠛⠋⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠷⠆⠠⣤⣘⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡄⠀⠶⠶⢶⠶⠶⠶⠶⠶⠶⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⣠⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣾⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣼⣿⡏⠀⠀⠀⠀⣀⠀⢀⡀⣀⢠⣼⣿⣿⣿⣿⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣤⠀⢸⣿⣧⣀⡀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣾⡟⠀⠀⠀⢸⣿⣿⣿⣿⠄⠉⢉⣩⣍⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿
⠀⠀⢀⠀⠀⠀⢹⣿⠿⢿⣿⣿⣿⡟⠀⠀⠀⠀⢸⣿⣿⡿⣿⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⠿⠿⠿⠿⠿⣿⣿⣿
⠀⠐⡛⠛⠂⢰⡟⠁⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⢸⢇⡟⡇⠇⠀⠀⠀⠀⠀⠀⠉⠁⣒⢉⡀⠀⠐⠚⠂⣿⣿⣿
⠀⠉⠉⣀⣠⣸⣧⣤⣤⣤⣴⣷⣧⣴⣴⣦⣶⣴⣼⣾⣷⣷⣷⣶⣦⣀⣀⠀⣀⡀⣀⢀⢀⣀⢀⠉⣀⣤⣿⣿⣿
⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣳⣤⣤⠄⣀⣠⣤⠴⣶⣶⠶⡒⠒⠒⠶⠆⠀⣀⣤⣴⣶⣿⡁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡷⢖⠛⡩⢶⠃⠿⢟⣫⣽⣶⣶⡜⢟⡴⣛⠽⠻⣟⣭⢾⣵⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⢏⣴⠇⣠⡴⢖⣒⣛⠉⢷⡁⠀⣹⠗⡞⠀⠁⠠⣾⣿⣿⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⢻⣿⡾⢸⢏⡶⢋⣽⣧⠶⠿⢃⡤⢂⣤⣦⣄⠀⢀⣶⣿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠈⡏⡼⠋⠘⣠⣴⣯⣵⣦⣤⣬⡽⠞⠫⠵⢦⣌⣧⠸⣿⢡⣟⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡷⣰⠃⠀⣸⣿⢻⣤⡥⣋⠕⡊⡲⣶⣦⡴⢣⡿⠀⠀⠃⣸⣿⣆⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢋⠄⠁⠀⢰⠋⠀⢀⣤⣬⣤⠄⣽⣇⡴⢂⠀⣯⣜⠏⡀⠀⣿⣿⣹⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⡇⠀⠀⠀⠸⢦⣤⠄⠀⠀⢒⣭⢿⣯⡶⣻⠏⠋⣙⣴⠃⠠⢿⣿⠉⡟⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣗⣻⡿⠃⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣶⠿⠟⢁⠄⠀⠸⣿⣾⠃⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠙⢿⡁⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣲⣶⠶⠞⢉⡀⠂⠆⢻⡏⡇⣿⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣾⠳⣤⣀⣙⣿⣿⡿⣿⣿⣿⣿⡿⢿⣿⡿⣾⢁⡀⣴⣃⣾⠀⣾⣿⢸⡿⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣤⣤⣼⣷⢾⠋⠀⠈⣇⣿⡆⢻⡇⣿⢋⡼⣻⣫⡏⠀⣿⡟⣾⠃⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢏⠁⠙⠅⠈⠀⠀⢀⣸⢇⡏⣸⣿⣷⠏⣸⣩⠟⢀⣾⣟⣽⣫⣦⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣼⢡⣟⡾⠁⢠⡟⣡⣼⠿⣫⣾⢁⣥⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⡼⣡⠟⣋⣤⣴⠿⣛⣩⣴⠟⠫⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⣠⠟⣛⡵⣟⣯⣷⠾⠟⢛⣉⣀⣀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⡡⠂⣴⢃⡾⣫⠾⠛⡉⠀⢀⣶⢛⡍⠉⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⢠⢺⢇⡞⡽⢁⡞⠋⠀⢸⠟⣙⠋⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡌⢸⢸⠣⠇⠈⢧⣀⣀⣾⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠁⢸⢸⡄⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠘⣆⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠈⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣴⠀⣀⣀⣴⣄⣀⡀⢰⡄⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣬⣭⣭⣭⣭⣭⣭⣉⣩⣭⣭⣭⣭⣭⣭⣭⣭⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠿⠛⠋⠉⠉⠉⠉⠛⠛⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠋⠁⠀⠀⣀⣤⣤⣄⣀⡀⠀⠀⠀⠻⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⣴⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⢳⡀⠸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣰⢣⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⣿⣿⣿⣧⢳⠀⠈⠻⣿⣿⠿⠛⠋⠁⠀⠀⢰⢣⣿⣿⣿⣿⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠛⠙⢇⢧⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⢇⠟⠙⠻⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠛⠁⠀⠀⠈⢎⣇⠀⠀⠀⠀⠀⠀⠀⢀⢏⠎⠀⠀⠀⠙⠻⠿⠿⠇
⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡞⣆⠀⠀⠀⠀⠀⢀⡞⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⢤⣀⡀⠀⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⣷⣄⠀⠀⠀⡜⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⣀⣠
⠀⠹⣿⣿⣿⣿⣿⣶⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠱⡹⣿⣷⡆⡼⡱⠁⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣾⣿⣿⣿⣿⡿⠁
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠱⢹⣿⣱⢱⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠈⠉⢹⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢣⠳⢣⠃⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡉⠉
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡿⣷⣶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⣿⢿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣾⣽⣽⣻⣻⣿⢻⢿⢹⢿⣿⣿⡇⣿⣟⣽⣯⣿⣼⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⢿⣧⣿⣿⣿⣾⣿⣿⣿⠿⠿⠿⠛⠛⠉⠁

⠀⠀⣀⣀⣤⣤⣤⣀⣤⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣦⣀⣠⣶⠀⠀⠀⠀⠀⢀⡴⠶⠦⣄⠀⠀⠀⢤⡀⠀⢠⡄⠀⠀⠀⣾⢰⡷⣾⡄
⠀⠀⣿⣿⣿⡿⠿⠛⠋⣽⡿⠋⠹⢿⣿⡏⠉⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⠀⢶⠆⢹⠂⠀⠀⠈⢷⣴⡟⠀⠀⠀⠀⢿⢸⣇⣼⠁
⠀⠀⠋⠉⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣤⡴⠋⠀⠀⠀⣠⠞⠛⠦⡄⠀⠀⠀⢈⣸⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣾⠀⠀⠀⠰⡇⢰⣷⠂⣻⠀⠀⢠⡏⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠒⠂⠀⣀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠘⠀⠀⠀⠀⠙⠦⠤⠖⠁⠀⠀⠀⠳⣄⣠⠼⠁
⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠗⠓⠒⠓⠃
⠀⢸⣿⠿⠟⠛⠉⠁⠀⢀⣾⠟⠁⢻⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠸⣿⡧⠀⠀⠀⠀⠐⠆⠰⠂⠖⠀⠀⠀⢺⣿⡗⠀⠀⠀⠀⠀⠀⣿⣅⣈⣿⣯⣀⣽⣿⣁⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⠇⠀⣷⣌⠀⣡⡾⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠁
⠀⠀⠀⢤⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⢴⡙⢿⠟⣡⡆⠀⣮⡻⡿⢋⣵⠀⠀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡄
⠀⠀⠀⠻⣿⣿⣿⢿⠛⠟⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⢨⠻⣷⡾⢋⡄⠀⣝⢿⣶⠟⣡⠀⠀⠈⢭⣿⣿⣿⣿⣶⢲⢋⣭⢳⢰⣾⣿⣿⣿⣯⠍
⠀⠀⠀⣰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⡀⠀⠘⢷⣮⣴⠟⠁⠀⠻⣷⣥⡾⠋⠀⠀⠀⠀⠰⠿⠿⠛⠋⠉⠲⠔⠊⠊⠛⠻⠿⠷
⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠊⡠⢐⣓⠂⠌⠢⠀⠀⠙⠁⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠰⠰⠆⠀⠰⠰⠄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣶⣿⣸⠡⠶⢹⣼⣷⣶⣶⣿⡿⠋⠁
⠀⠀⠀⠀⣸⡆⠀⠀⠀⠀⠀⠀⠀⢄⠁⠌⡏⠄⠊⠐⠀⠀⠀⠀⠀⢠⣤⣤⣶⣤⣤⠀⠀⠀⠀⠀⠀⠈⠉⠋⠙⠛⠑⠒⠙⠛⠙⠉⠉⠁
⠀⠀⠀⢠⣿⣷⠀⠀⠀⠀⢠⡄⠀⠀⠁⠀⠆⠐⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠉⢻⣿⡇
⠀⠀⠀⡥⡯⢽⠥⠀⠀⣀⣾⣷⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣶⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣾⣿⡆⢀⣀
⠀⠀⠀⣛⣟⣻⡛⠀⠀⣿⡿⣿⢾⠀⠀⠘⣿⣷⣿⣿⣿⣿⣿⣶⡄⠀⠈⠙⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⠈⠉⠋⠉⠀⠻⠂
⠀⠀⠀⠉⠿⠏⠁⠀⠀⠛⠿⠟⠃⠀⠀⠴⠿⠛⠛⠛⠿⠟⠛⠉⠀⢠⡤⠤⠤⠤⢤⡄⠀⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⣴⣄
⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣀⣄⣠⣠⣄⣠⣀⣀⣀⣀⣀⠀⠀⢸⣷⣶⣶⣶⣾⠃⢠⣿⣿⣿⣿⣿⣷⠀⠀⢦⣀⠀⣿⣿⣆
⠀⢸⠿⠿⠿⠿⠿⠿⠿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠀⠀⠈⣿⡏⠀⣹⣿⠀⢸⣿⣿⢿⣿⣿⣯⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡶⠶⠶⠦
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⡿⠟⠀⠀⠻⢿⣿⣿⠿⠃⠀⠀⠞⠋⠁⣿⣿⠏
⠀⢰⡇⠀⣧⠀⣾⠀⢸⡄⢸⡇⢀⡇⠀⣿⠀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋
⠀⢸⣇⢠⣿⠀⣿⡀⣼⡇⢸⡇⢸⣷⠀⣿⠀⣿⡄⠀⣾⠀⢸⠀⢰⡆⠀⡇⠀⣷⠀⢸⠀⢰⡆⢀⡇
⠀⢸⣿⢸⣿⠀⣿⡇⣿⡇⢹⣿⢸⣿⢸⣿⡇⣿⡇⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣶⠀⢰⡆⢰⡆⠀⣶⠀⣶⠀⢰⡆⢰⡆⠀⣶
⠀⢸⣿⢸⣿⢀⣿⡇⣿⡇⣸⣿⢸⣿⢸⣿⡇⣿⡇⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⡀⢸⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿
⠀⠈⠉⠈⠉⠀⠉⠁⠉⠁⠈⠁⠈⠉⠀⠉⠀⠉⠁⠀⠉⠀⠉⠁⠈⠁⠈⠉⠀⠉⠀⠉⠁⠈⠁⠈⠉⠀⠉⠀⠈⠁⠈⠁⠀⠉⠀⠉⠀⠈⠁⠈⠁⠀⠉

⣰⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣶⢶⢶⣾⡷⡶⡶⣶⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚
⠈⠙⢿⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡼⠞⡏⣿⡇⢹⡿⣴⣿⡁⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋
⠀⠀⠀⠈⠻⢿⣷⣤⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇⣇⣇⣿⣿⣸⣇⣿⣿⣷⠀⠀⠀⠀⠀⡠⠊
⠀⠠⣄⡄⡄⢠⠙⠻⣿⣦⣄⡀⠀⠀⢸⣿⣰⣿⢿⣿⣽⣿⣿⣿⣿⣸⣿⠀⠀⢀⠐⠉
⠘⢛⣮⣯⡗⠁⠀⠀⠀⠙⠿⣷⣶⣀⠀⠻⣿⣿⣰⡿⣿⡜⡇⡆⣿⣿⠟⣠⠀⠑
⢀⡸⣷⣟⠉⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣆⡈⠙⠿⣷⣿⣷⡷⠟⢋⡥⠊
⠀⠀⣤⣿⣿⣿⣷⣶⣶⣿⣿⣦⠀⠀⠉⣿⣻⣶⣄⠀⠛⠋⣀⠔⠉⡠⠀⠀⣤⣶⣦⣤⣤⣴⣢⣄⣀⡀
⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⡏⣼⠀⠀⠀⠈⠻⢿⣿⡷⣧⣸⡁⠔⠈⠀⠀⠐⠻⢿⣷⣎⡛⠿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⢹⢻⡏⠁⠈⠻⣿⡿⣄⠀⢀⣀⡤⢀⡽⠻⣿⣟⣿⣶⣄⠀⠀⠀⠀⠀⠈⣿⣿⠿⣿⣾⣿⣿⣷
⠀⠀⠀⠀⢸⠈⡇⠀⠀⠀⢸⠁⢸⠀⠈⢉⠴⡃⠀⠀⠈⠙⠿⡿⠛⠷⣦⡀⠀⠀⠀⠈⠀⠀⠀⠹⡿⣿⣿
⠀⠀⠀⠀⠜⠐⠃⠀⠀⠀⠆⠀⢰⣡⠞⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⢄⠈⢻⣦⠀⠀⠀⠀⠀⣀⣀⣀⣨⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢡⡀⠀⠀⠀⣾⣧⠀⠀⠀⠀⠀⠀⠉⢻⣿⣧⠀⠀⠀⠀⠙⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⢷⣿⣦⡀⠀⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠉⢻⡷⣀⠀⠀⠀⠘⡟
⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠈⢿⣿⣿⣷⣾⣿⣿⣷⡀⠀⢀⣴⡶⠖⠂⣸⣿⣠⣾⣷⣦⡀
⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡷⠁⠀⠀⠀⠉⠉⠻⣿⣿⣿⣮⡢⡀
⢀⣠⣶⠿⣆⡶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣮⡢
⠙⠃⢘⣻⣿⣿⣛⡇⠀⠀⠀⠀⠀⠀⠀⠒⠻⠛⢉⢋⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⡀⠀⣀⣀⣤⣄⣄⣀⣀⣀
⠀⠀⠀⢀⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠋⢉⣉⡀⢀⣈⣉⣉⠛⠛⣉⣤⣤⠀⢩⣍⣙⠛⠶⣦⣄
⠀⠀⠀⣾⠃⢀⣀⡈⠉⠛⠛⠶⢶⣦⡴⠞⠋⠁⠀⢀⡖⠋⠉⠀⣴⠏⣉⠉⠉⣽⣛⣋⣴⠖⡲⠀⠀⢹⡇⠀⠀⠙⢿⣦⣀⣤⣴⠶⠶⠟⠛⣷⡄
⠀⠀⠀⢿⠀⠈⠉⠻⢿⣶⣤⣀⡀⠀⢠⣀⠀⠀⠀⠊⢀⣠⣶⠟⠀⣿⡁⠀⠀⣼⠀⠘⠃⠞⠇⠀⢀⡌⠳⢦⣴⣇⣀⠙⠏⣀⣤⣶⣿⡷⠂⣿⠁
⠀⠀⠀⠸⡇⠀⠀⢰⣿⣿⣿⣿⡏⠀⠠⣬⠳⣄⠀⠙⠛⣩⡿⠷⢿⡟⠃⠀⣈⣵⣀⣀⢀⣐⣶⣾⣋⣭⡙⣾⡷⣄⠸⡆⢀⡉⡻⠟⠉⠀⣸⡏
⠀⠀⠀⠀⢻⡀⠀⠀⣙⣿⡿⠁⠀⠀⣀⠩⣤⣌⠳⡝⣿⣏⣼⣿⣶⣿⡟⠛⣫⡽⠿⠭⠿⠶⠶⢿⡟⠻⠟⢛⣿⢈⡣⠶⣏⠤⠈⠃⠀⣴⡟
⠀⠀⠀⠀⠀⠙⣦⡀⢉⣿⠍⠉⢛⣒⡭⢤⣀⠙⢷⣝⣮⠻⠿⢿⣶⣤⠁⠘⢁⣠⣤⣤⣤⣤⣤⣤⣉⠃⠐⠫⢖⣩⣴⢺⡅⠀⠀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⣸⠷⣾⣯⠥⢤⣘⣛⠃⠐⠾⣿⠶⢭⣝⡓⠀⠴⢤⡀⠀⠀⢻⣿⣭⣤⣤⣤⣤⣼⡟⠀⠀⠪⢭⣤⣖⣲⣶⢒⡂⢺⣷⡀
⠀⠀⠀⠀⠀⢠⠏⣞⠟⢻⡧⣤⣙⣛⣉⣻⠷⠒⠒⠦⢬⣽⣓⠂⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⠟⠉⠀⠀⠀⠀⢁⣉⣀⠡⢽⡭⠁⢺⣿⠃
⠀⠀⠀⠀⠀⣿⡈⠐⠀⣸⡧⢬⣭⣥⠤⢽⣯⠭⠭⢍⣛⣒⡦⠀⠀⠀⠀⠀⣀⣀⣠⣤⣼⠿⢶⣦⣤⣤⡀⠀⣐⠭⣈⠉⢹⣇⠀⠺⣿⡆
⠀⠀⠀⠀⢠⣯⠔⡢⡀⣸⣿⡀⢻⣿⡀⠘⣿⠀⠀⠄⠶⣖⣻⡅⢀⣤⣶⣋⣉⣁⣠⣤⣤⣤⣤⣀⡤⢍⣻⡾⠋⣑⣈⠁⢸⣇⠀⠙⣮⣷
⠀⠀⠀⠀⣼⠡⡪⢞⠁⣼⣿⡆⠸⣿⡇⠀⠻⣦⣄⠀⠉⢻⣿⣿⣿⣿⠀⠀⣿⣄⣧⣦⣦⣤⣤⣼⠀⠀⣿⠇⠀⣹⠃⠀⣿⡗⠀⠀⠩⣻⣧
⠀⠀⠀⠀⣿⢠⢖⡡⠀⣾⣿⠃⢐⣿⣿⣦⡀⠘⢧⠀⠀⠈⢿⣿⠯⣿⡄⠀⡟⡿⡟⡟⣿⢻⣿⢿⡀⢸⣧⣤⠾⠃⠀⢀⣿⡇⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⢿⡶⣫⠖⢀⣿⡇⠀⠈⠸⣿⣿⣷⡀⠈⡇⠀⠀⠼⣿⣢⠙⣿⣄⣿⠉⠧⡇⠟⠈⠸⣸⣷⡏⣼⠁⠀⢀⣦⣿⠿⠃⠀⡀⠀⣜⣿⡇
⠀⠀⠀⠀⢸⡆⠁⠀⢠⣿⡆⣸⡟⡀⠉⢻⣿⡇⠀⢻⡆⠀⢸⣿⣿⣁⢻⣿⣿⠀⠀⠀⠀⠀⢠⢿⡏⣷⠿⢿⣿⣿⡏⠈⠀⠀⢱⣱⣦⣼⣿⠃
⠀⠀⠀⠀⠈⣿⣴⡄⡄⠻⡿⣿⣿⡅⠀⠘⠛⣗⠀⠈⢳⣄⠸⣿⣿⣯⣾⡇⠸⣦⣤⣤⣤⣴⠏⢸⣇⣿⠟⠀⠀⠈⠀⠀⠀⣠⣣⣿⣷⣿⠃
⠀⠀⠀⠀⠀⠹⣿⣧⣷⢠⢠⠡⢻⣦⠀⠀⠀⠀⠀⠀⣈⠛⣶⣾⣿⣿⠏⢧⣀⣉⣥⣤⣭⣏⣀⣸⢻⣿⡆⠀⠀⠀⢠⣀⣰⡟⢳⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢾⣷⣤⣿⣬⢠⢰⠀⢠⢃⢅⣧⣉⢹⡵⠛⣯⣷⣶⠶⠶⣦⣤⣤⡤⣬⣠⢟⡀⢦⢦⢰⣸⢻⣿⣷⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣷⣧⣦⣧⣮⡿⠿⠿⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⣼⢸⡞⡄⢾⣸⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠿⠋⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡟⠻⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⢠⡀⣄⠀⠀⠀⠀⢀⣠⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣿⣿⣦⣼⣶⣴⣾⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡓⠀⠀⢨⡦⠀⠀⠀⠀⣷⠀⠀⡀⣠⣬⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠀⠤⣤⡀⠁⠀⠻⠿⢅⠀⣠⣄⠀⠀⢑⡃⠀⠼⢺⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣴⣿⣥⣿⣷⠟⠙⠃⠖⠀⠱⠀⢸⠉⠀⡄⠀⢈⣥⣄⣄⣰⢿⢷
⠀⠀⢀⡠⠔⠀⢄⡀⡀⠀⢠⣳⣶⣾⣿⣿⣿⣿⡿⢿⣿⣄⣤⠼⡧⠶⠓⠷⢾⣥⡚⠊⠆⠙⠿⠛⠛⠉⠈⠀⠀⡀
⠀⠀⠈⠙⠋⠀⠀⢻⣄⣐⠚⠋⠉⠁⠉⠀⠘⠀⠔⢉⠖⠉⠛⠂⠈⠚⠀⠀⣀⣹⠣⠄⠀⠀⠀⠀⠀⡀⠀⠀⠚⠂
⠀⠀⠀⠤⠴⠒⠒⠨⠾⠛⠿⠆⠀⠤⠀⢠⠴⠖⢀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⢠⣾⣿⣿⣧⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠋⠻⠁⠿⠻⠿⡿⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣿⣆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢾⡿⡿⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⡀⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠚⡿⢰⡤⣦⠸⡓⠄⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠂⠈⠀⠀⠀⣀⣀⡐⢾⠦⢴⣅⠀⣄⠙⢿⠰⣸⠞⢁⡀⢠⣷⠴⣿⠖⣀⣀⠀⠤⠀⠁⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠣⢠⡤⠐⠚⠐⠊⣕⣤⣔⡀⠈⠁⠻⡄⢻⡷⣹⠃⡼⠋⡁⠀⢲⣤⣔⡍⠂⠐⠒⢠⣤⠼⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠖⠂⠶⠄⠀⠈⠑⠂⠛⣮⠿⢷⣾⣿⠈⣹⣿⣶⠿⣭⠟⠂⠒⠉⠀⠠⠶⠂⠚⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠨⠖⣺⡿⣿⡄⣿⢿⣯⡒⢄⠀⠀⠀⠀⡀⠀⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⡘⢀⣴⠂⠀⠀⠀⠀⡽⢠⢹⣷⡿⠥⠹⡄⠀⠀⠀⠀⣾⣀⠘⣤⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢈⡻⠁⠁⠄⠀⠀⠀⠁⠀⡾⢻⢻⠆⠀⠃⠀⠀⢀⠌⠁⠹⡇⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣧⣠⠤⢀⠀⡀⠘⡇⠀⡀⠀⠀⠙⣠⠘⠀⠁⠀⡀⠨⠃⠀⡀⢠⠠⣤⣠⣿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⡻⡧⠘⣾⡣⠒⠄⣀⠐⠀⠉⢲⣀⢤⡭⡥⢆⣴⠉⠁⠰⠀⠀⠕⢦⡻⠊⠰⣿⣙⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢻⣗⠡⠈⠈⣡⠀⣀⠂⠐⠀⠀⠀⢠⡎⢻⠛⡄⠀⠀⢀⠐⠐⢃⠀⢀⡉⠈⡰⢱⡷⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⡈⢹⣶⠳⠄⠨⢦⠙⣇⣰⢠⠌⡑⢳⢤⠟⢦⡔⢀⡀⣤⢰⣀⡎⣴⢮⠠⠞⣳⡟⢁⡄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠿⠶⣌⡓⢼⣦⡀⠀⢿⡻⣾⠓⠙⠋⠀⠀⠀⠈⠻⠗⢻⡾⣻⠆⢀⣀⣦⠘⣣⡴⠿⠧⠞
⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⠀⠈⠩⡈⢁⠙⠳⣌⣛⠻⠆⢃⡤⣾⣋⣲⠦⣀⠋⠟⢙⣡⠴⠛⠁⠉⡄⠁⠐⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠂⠀⠌⢀⡀⠠⢙⣧⠴⣿⣷⣾⣷⣿⣾⣿⡶⣴⣟⠁⠀⠀⠀⠀⠐⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠐⠀⠚⠓⠒⠀⣀⣀⣦⣶⡿⠞⠛⢻⣿⣿⡀⣿⣿⣿⠛⠓⢿⣷⣴⣀⣀⣀⡒⠘⠓⠀⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣴⠾⠛⠛⠛⠛⠛⠛⡻⠛⢷⣄⡀⠀⢍⢻⣿⣽⢻⠀⢀⣠⣴⠟⠛⠛⠛⠛⠛⠛⠛⠻⣶⣄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠛⠉⠁⠀⠐⠂⠀⠂⡸⠁⠈⠅⣸⡿⢯⣴⡿⣿⡻⣦⣽⢿⣯⡠⠍⠀⢳⡐⠀⠀⠂⠀⠉⠉⠙⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣮⡀⡤⠀⠠⠘⠛⠆⠛⢁⠜⣡⣾⠋⣀⡞⠳⠾⣿⡷⠶⢹⣎⠈⢻⣦⡁⢀⠙⠢⠚⠃⠠⠀⠠⡄⣬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠄⣠⠂⢀⠀⢰⡟⠮⠀⠸⠁⠀⠞⠛⢛⠂⠊⠙⠀⢹⠞⣷⠘⣤⠀⢦⡀⠤⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠀⠀⠀⣿⡄⢃⣠⠀⠀⠀⠀⣲⡁⠀⠀⠀⢠⣜⠃⣿⡀⠀⠀⢸⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢷⣤⣭⠭⢤⠀⠆⣿⠃⠀⣤⠼⢭⣥⣾⠟⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠾⠟⠛⠛⠀⠀⠀⠁⠀⠘⠛⠛⠿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⣿⣿⢟⠛⣿⡿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣞⣩⣷⣶⣦⣴⣶⣶⣦⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⠶⣶⣶⣦⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣻⣿⣿⣿⣿⣯⣦⣤⣬⣭⣭⣙⠻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣯⣽⣯⣤⣿⣿⡟⠛⠋⠙⠋⢁⣽⣿⡇⢹⣿⣏⣹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⡤⢿⣹⣿⣿⣿⣿⣿⣤⡀⣠⣶⣮⡛⠛⠁⠀⠈⠻⣿⡿⠉⠀⠀⠀⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣼⣿⣿⣿⢿⣿⣿⣿⠟⠛⠛⢿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠻⢿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠘⢫⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⢿⡟⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣿⣿⣴⣿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣸⣿⣶⣶⣶⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠙⢻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠯⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣦⣤⣀⣠⣀⡀⢀⠀⣤⡶⠀⠀⠀⣀⣀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠖⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣶⣤⠄⠀⠰⢠⢤⣀⠔⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡿⠟⠛⠿⣧⣠⡀⠀⠀⠀⢴⣦⠀⣹⣇⠀⣹⡗⣠⠦⠎⣀⡤⢤⣈⠉⢹⠀⢰⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⣸⡏⠀⠀⠀⠀⠘⣿⠁⠀⠀⢠⠤⡀⡰⢄⠛⣛⠋⢁⡼⠀⡜⠁⠀⠀⠈⢣⠈⢉⠆⠉⣀⣦⣷⣧⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠻⣷⡀⠀⠀⠀⣰⠿⡇⠪⡑⠼⣀⡑⣁⣑⠚⡸⢀⣘⢢⠀⢇⠀⠀⠀⠀⡸⠀⠑⡄⠀⣾⣿⠉⢙⣿⣖
⠀⠀⠀⠀⠀⢀⣲⡾⠷⣾⣘⠿⠛⣷⡶⠻⠿⠚⢍⠙⡕⠉⠀⠀⠀⠀⠙⢎⢡⠞⣻⠤⡌⠑⠲⠖⠊⠁⢸⠉⠀⠀⠹⣿⣷⣿⣟⠃
⠀⠀⠀⠀⠀⢈⣿⣄⣀⣿⡏⠀⢄⡨⣁⡀⠰⢉⡈⡜⠀⠀⠀⠀⠀⠀⠀⠀⡇⡩⠝⠀⠳⠔⠙⢤⠎⢉⣉⣰⣦⣴⡆⢀⠁⠈
⠀⠀⠀⠀⠀⠀⠈⠛⡙⠃⣀⠍⠀⠠⠄⢁⢦⡔⡃⢣⠀⠀⠀⠀⠀⠀⠀⠀⠇⣉⣐⢠⡟⠛⣦⠀⣤⣼⠟⠉⠉⠉⠙⢿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⢨⣀⠑⠤⠂⢠⠈⢁⡼⢀⡣⣀⠀⠀⠀⠀⣠⢞⣈⣳⡄⠀⠻⠛⠃⢤⡾⠇⠀⠀⠀⠀⠀⠈⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠔⠒⢠⣦⠀⠉⢡⠇⡤⡍⢐⣌⢩⠦⣘⡆⠈⠀⠀⠀⠰⠶⠀⣶⡆⠀⠀⠀⠀⠀⢰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠁⠀⠀⠉⠀⠣⠊⠘⠖⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠿⢷⣦⣤⣤⣾⣿⠉⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠛

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡧⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤
⡇⠀⠀⠉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁
⡇⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⢸⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢼⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⡇⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀
⣇⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠾⠷⠦⠄⣄⣀⣘⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣴⣭⣤⣤⣤⣤⣤⣤⣬⣭⣷⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠁⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⢛⣛⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⣼⣿⡄⠀⠀⠀⠀⠀⠀⣴⣶⡆⠀⠀⠀⠈⠋⠁⠈⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣄⡤⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣼⣿⣿⣿⣿⣿⣧⡀⠀⠀⢦⠀⠀⢻⣿⣷⡀⠀⠀⠀⠀⠤⢀⣀⣿⣿⣿⣿
⠀⠀⠀⠀⠘⣿⠁⠀⠀⠀⢜⣏⠀⠀⠀⠀⢀⣴⡆⠀⠀⠀⠀⠸⡿⣿⣻⣿⣯⣩⣭⣥⣤⣀⣀⣸⣧⣀⣾⣿⣿⣷⠀⠀⠀⣠⣤⣤⣌⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣿⣿⣿⣆⠀⠀⢉⣷⣾⣿⣆⡉⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠛⣿⣿⣿⣿⣿⣶⣤⣤⣯⣍⣠⡀⢹⣿⣿⣿
⠀⠀⠀⠀⠈⡿⢿⡿⣿⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢡⠘⠙⠈⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿
⠀⠀⠀⠀⣠⣀⣜⣴⣾⣠⠀⠀⠀⠺⡁⢹⠃⠙⢷⠍⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⣠⣦⣶⣿⣿⣿⣿⣿⣿⣶⣄⠀⣀⣉⣯⡀⣠⣞⣰⣦⣤⢀⡄⢠⣠⣦⣠⣀⣀⣤⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⡠⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣶⣶⣼⣿⣼⣷⣴⣦⣴⣤⣴⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠐⠂⠤⢤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠲⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀⠀⠀⠈⠳⠙⠲⡔⠒⠛⠛⠛⠓⠲⠶⠒⠂⠈⠉⠉⠁⠀⠀⠀⠠⣤⣀⡀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⡀⠀⠀⡷
⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⢒⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⢰⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⢰⠀⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠐⠮⣄⣀⣀⣀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠙⢾⣧⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⣾⠀⣉⠀⠀⣀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢸⡏⠛⣿⣧⡀⠈⠂⠶⠶⣦⣌⡉⠉⠉⠘⠛⠉⢻⠀⣼⣟⠷⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠣⣼⣿⣿⣿⣶⣤⠶⡚⠅⠀⠉⠓⠀⠀⠀⠀⣻⡿⠿⢿⡤⢰⡀⠉⠁
⠀⠀⠀⠀⠀⠀⣰⣣⡴⠀⠀⠀⠀⠀⡰⠁⠀⠠⠖⠛⠛⠓⢦⣄⡛⢶⣤⠄⣉⣀⡀⠉⠀⠀⣠⠖⠓⠢⢄⢀⠀⠠⠣⣄⢠⠚⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⢁⡞⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠁⠐⠋⠐⠒⠀⠼⡆⠴⠤⡀⠀⠀⠀⢀⡜⠁⠀⠈⠑⠀⣁
⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢢⡀⠀⣠⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⢠⡤⠀⠀⠀⠀⠀⠀⢀⠀⠉⣤⡄
⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣧⡉⠂⢿⡀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣠⠂⢸
⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠿⣦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣄⡀⠀⠀⠀⠀⣀⣠⠜⠀⢸
⠀⠀⠀⠀⠀⢸⢘⡇⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣰⠃⢰⣿⣿⣿⣿⣷⣦⣀⣭⠙⣿⠻⠋⠙⠿⠛⠯⠉⢯⠭⠭⠓⢿⡟⠙⠋⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠈⢸⡇⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⢿⠀⢸⣿⡟⠋⠛⢿⣿⣿⣿⣿⣿⠂⠀⣰⠷⣄⠤⠤⠤⠤⣆⠴⡟⠁⠀⠀⢀⡾⢦
⠀⠀⠀⠀⠀⠀⠀⢧⢠⢦⡀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢸⣿⢧⡀⠀⠀⠉⠙⢿⣿⣯⣴⣿⠁⠀⠀⡤⠀⠀⠀⠀⣠⠃⢀⣄⡠⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠈⠳⣄⠀⠀⠀⠀⠀⠀⠙⡆⠘⢻⡄⠱⠾⡤⠲⢀⠀⣿⣿⣿⣿⣀⣀⠢⠀⠀⠀⡠⠀⠄⠐⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠠⢄⠀⢹⡄⠈⢷⣤⣄⣀⣰⣯⠤⠄⠴⠤⠦⣤⡏⣠⠀⢠⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠺⣅⠀⠀⠉⠉⠉⠀⠤⠄⠀⠀⠀⠀⠀⠈⠁⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠞⣧⠚⡧⠊⠑⠸⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠤⠀⠀⠀⠆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⣀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠠⠂⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠎⠶⠂⠈⣠⢢⠤⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠰⠀⠀⠐⣆⡄⢟⣁⢠⡱⠄⠀⠀⡀⠀⠀⠀⠴⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠁⠰⠀⡈⠠⠀⣄⠀⡙⢉⡄⢚⠁⠈⠀⠊⠈⡄⢀⠌⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⢀⠀⠀⠀⠂⠁⠈⢀⠀⢩⡈⠓⣘⠣⠤⠂⠸⠀⠐⠂⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠧⠢⠁⠀⠸⠧⡃⠀⠠⠀⠄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⢠⡖⠰⠶⠀⠁⠶⠁⠀⠐⠀⠀⠀⠠⠄⠁⠀⡂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠐⢄⠀⡀⠰⠄⣀⠄⠒⠓⢤⣵⢶⠉⠁⠀⠴⠀⢢⡤⡴⠋⠀⠀⠀⠀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⢤⣀⠀⣆⠃⠀⠘⣫⠀⣠⠔⠚⠃⠸⠀⠀⠀⠀⠀⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠂⢲⠀⠀⠀⠀⠀⠘⡆⠉⠀⠉⠀⠻⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⡘⢄⡇⠀⠀⠀⠀⠀⠸⠀⡠⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣈⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠡⡤⠀⣠⡔⠰⡼⠂⠀⠀⠀⠀⠀⠀⠀⢤⠈⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⣄⠄⠛⠏⠀⡁⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⢀⠀⡴⡋⠀⠀⠐⢠⠇⠀⠃⠅⠠⡃⠶⡄⠀⠀⠀⠈⠻⡄⠀⠀⣿⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠱⢌⠁⠀⠀⠀⠄⠀⠠⠀⠀⠀⠁⢀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠙⢀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠘⡧⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣽⠧⣤⢆⣶⢶⡄⣻⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠈⡇⢀⡃⠀⠈⠠⣯⢈⡇⢰⠀⠐⠀⠀⠀⠀⠠⣤⣀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⢰⡇⢹⣧⡀⠀⠀⢽⡆⣄⢘⡀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠚⠆⠀⢳⡀⠀⠀⣿⡗⢘⠁⠀⠀⠀⠐⢂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢺⡄⢠⡍⠀⠀⡏⡻⢸⡇⠀⠀⠀⠀⠀⠀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡠⠂⠀⠂⠀⠀⣽⡾⠀⡹⠁⠀⣼⠃⣳⣾⠀⠀⣤⢴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡄⠀⠀⠐⠁⢀⡤⠶⡠⢚⡡⠂⣸⠁⢀⠌⠉⢴⠿⡟⡀⢄⠳⢄⠀⣴⠄⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠐⡀⠢⠟⢢⡄⠀⠄⠀⠊⠀⣠⣇⠀⠘⡆⠀⢺⡀⣟⡳⠀⠀⠀⠀⠰⣆⠐⠀⠀⠠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠚⣃⡀⠀⢤⣈⠶⠀⠁⡑⠲⡆⠰⠖⠎⠁⠙⠀⣴⠿⠄⠀⠑⠾⠥⠑⠂⢀⡮⠕⠘⠧⠀⢀⡂⠀⠀⠀⢀⡀
⠀⢁⠀⠀⠀⢀⣤⠀⠀⠉⠁⠐⠊⢡⠖⢻⠀⠁⠈⠳⠀⢢⡀⠈⠠⡇⠸⢿⣮⡀⢤⣤⠤⠶⢾⡉⠀⠀⠒⢦⠘⠚⢁⠁⡄⠒⠊
⠀⠠⠀⠀⠀⠀⠀⣀⠀⠊⠀⠀⠀⠀⠋⣸⠃⠸⠄⠀⢲⣤⣈⡀⣀⡤⢰⠀⢾⠁⢀⡈⠀⠀⠄⠈⠏⠀⠃⠀⣄⠀⠉⠴⣁⠀⠐⠂
⠀⢠⠀⠀⠀⠀⠀⠤⠀⠀⠠⠆⡀⢀⠋⠀⠀⠁⣰⡞⠈⠉⠈⠁⠋⠀⠉⠀⠘⢴⢻⡁⠀⡀⠃⠀⢰⠄⠅⠲⢄⣦⡀⠀⠉⠉⠶⠓
⠀⣼⠀⠀⠀⠀⠦⣐⣇⡠⠄⠀⢀⠌⠈⢓⠜⠲⠉⡴⠂⢀⠀⠒⠄⠑⠐⠚⢰⠀⠀⢆⠸⠧⡀⢃⡄⠐⠂⠴⣬⠈⠥⠤⠀⠀⡀⢠⣤
⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠀⠀⠴⠉⣁⠀⠈⢠⡄⠀⠀⣄⢀⠌⠈⠔⢃⠀⢰⡊⠠⡾⠄⠫⠀⠈⠧⠀⣁⠀⠀⠀⠐⠁
⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉
⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⡄⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠠⣦⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⢠⠀⠀⠀⠀⢰⡆⠀⠁⠀⠀⠸⢿⠟⠀⠀⠀⢰⣿⡆⠀⠀⠀⠃
⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⢠⣶⡄⠀⢸⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⡇⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠘⠛⠃⠀⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉
⠀⠀⢶⡶⠒⠀⠀⠀⣀⣀⡀⠀⠀⣶⣴⣦⡆⠀⠀⠀⢰⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢰
⠀⠀⢸⡇⠀⠀⠀⠉⠛⡛⠃⠀⠀⠁⢸⠁⠁⠀⠀⠀⣸⡀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢀⣸⣀
⠀⠀⢸⡇⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢹⠈⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢹⠀⠀⠀⢀⣧
⠀⠀⢸⡇⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈
⠀⠀⠸⡇⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢸

⠀⢀⣶⣀⣶⡀⠀⠀⢀⣰⣦⠀⠀⠀⢠⣤⣶⡄⠀⠀⠀⠀⠀⢀⣀
⠀⠘⡿⡿⡿⠃⠀⠐⠉⡿⡿⠀⠀⠀⢸⢠⠟⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⣯⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⢀⣶⣿⣷⣦⠀⠀⣠⣶⠀⠀⠀⠀⣤⣶⣶⣦⡀⠀⠀⡀
⠀⠸⣿⣧⣿⣿⠀⠀⠛⢻⣭⡄⠀⠀⣿⣿⣿⣿⡇⠀⣾⣿⣷⢦⣦⣀⣀
⠀⠀⠉⠛⠋⠁⠀⠀⠈⢹⡿⠁⠀⠀⠙⠿⡿⠛⠀⠀⠿⠿⠿⣼⣏⠙⣯⡙⠓⠲⠦⣤⣀⡀
⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⣷⠀⠀⠙⢶⣄⠀⠀⠀⠉⠙⠻⣶⣦⠄
⠀⠀⠀⣸⣇⠀⠀⠀⢀⣀⡀⠀⠀⢰⣀⣀⣀⣀⡄⠀⠀⢀⡟⠀⣿⠀⠀⠀⠀⠙⢷⣄⠀⠀⣿⣿⠄
⠀⠀⢸⣽⣽⡇⠀⠀⢸⡿⣿⠀⠀⠀⢹⣿⣻⡏⠀⠀⠀⠸⢷⣾⣿⡶⠶⠶⠶⢶⣶⣿⢷⣶⣾⡿⠆
⠀⠀⠈⠿⠟⠀⠀⠀⠘⠿⠟⠀⠀⠀⠈⠿⠿⠃⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠊⠉⠉⠉⠉⠛⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⢤⣄⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⢀⣀⣄⠐⠄⠀⠀⠀⠀⠘⣧⠤⠤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣯⣉⣯⢤⣺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡻⠶⠟⠷⠞⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣌⡀⣈⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣤⣄⢠⡄⠀⠀⠀⠀⠀⠀⠀⢀⠐⢃⠀⠀⠀⠀⠀⠀⠀⢠⣋⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⠀⠀⠀⠀⠀⠀⠈⢁⠉⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⢀⣠⠶⠦⢤⣤⣠⡴⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⣀⡀⣀⠀⢀⡤⠋⠓⢦⣄⠀⠀⠀⠀⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⡏⢻⠒⡖⠲⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣈⢁⡼⠀⡇⠈⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣀⡀⠀⠀⠀⢠⣀⠀⠐⠒⠲⠒⠒⠒⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⡄⠈⠉⠉⠉⠀⠐⠒⠂⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠉⠐⣆⠀⠀⠀⢸⡇⠀⠀⠀⠤⠶⠀⠀⢰⣷⠀⠀⠀⠀⢰⠀⠀⠀⠐⠂⠀⠒⡖⠀⢸⡃⠀⠀⠀⠚⠚⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⡏⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⠀⢸⠀⠀⠀⠐⠃⠀⠀⡇⠀⠸⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡇⠀⠀⠀⢸⠐⣠⡤⢤⣀⣀⣀⣀⣘⣱⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⣗⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠁⠀⢀⣴⣙⣛⣻⣻⣛⣛⣛⣟⣛⣛⣛⣑⣄⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢀⠀⢟⣹⠅⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣶⣴⠶⠤⠤⣤⣄⡀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠋⢀⡀⣹⣆⠰⡶⠶⢶⡿⢿⣿⣿⠋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⣿⣿⣿⡧⠸⣿⣿⡿⢃⣠⠾⠿⠛⣷⣶⣍⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⠋⢰⣟⠁⢠⡞⠋⣡⡾⣶⣵⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣻⣿⣿⡿⠙⣿⡿⠛⢉⣤⡿⢿⣟⠿⠷⡦⠙⢰⣿⣿⡏⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣟⣁⠀⠹⣏⠀⣾⠋⣠⣿⡛⣿⣍⡽⣦⣀⣿⣿⣯⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢟⣿⣿⠟⠀⣴⡟⠁⢸⠇⣰⠿⡟⢧⣙⣻⣷⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣿⣿⡏⠀⣼⣿⡿⠁⠈⢠⣟⠀⠹⣆⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⠁⢹⣿⣷⡏⠀⠘⠛⠛⠛⣿⣶⠋⠩⠭⣉⠛⠻⣯⣝⢛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⡀⡈⠟⣿⣇⣶⢀⣠⠄⠀⠹⣿⣦⡀⠵⠒⡨⠀⢈⡿⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠘⣿⣷⣷⡀⠙⣿⣿⣿⣿⣴⣦⣤⠌⠻⣿⣿⣾⠿⠿⠿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣏⠛⢦⣌⠛⢿⣿⣿⡟⠁⠀⠀⢠⣀⣀⣀⣤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠓⢄⠙⢿⡇⠀⠀⠀⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⡿⠿⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡏⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣄⣀⣠⣼⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠟⠂
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉
⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠙⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣠⣤⣶⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⣀⣀⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣀⣀⣶⣤⣤⣿⣿⣶⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠾⡿⠿⠿⣿⣛⣛⣽⠉⠁⠈⢛⠿⣿⣿⣿⣿⣿⣿⡯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣾⣷⣿⣿⣿⣿⣿⣿⣦⣤⣶⣿⣷⣦⣭⣙⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣳⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡭⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠙⠛⠛⠛⠛⠛⠓⠉⠉


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

⠀⠀⠀⢠⣀⣠⣿⠙⢶⡦⣤⠀⢀⣼⣷⣶⣶⣿⣦⡀⠀⠀⣠⣴⣶⣶⣤⣀⠀⠀⠐⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠃
⠀⠀⠀⢸⣿⣿⠟⠀⠘⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⡁⠀⣾⣿⣿⠟⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠔⠉⠀⢹⣿⡇⠈⠑⢄
⠀⠀⠀⠀⣿⠋⠀⠀⢰⣿⠃⠀⣿⣿⣿⣿⣿⣇⣿⡅⠈⣿⣿⣿⣶⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⢀⣿⡀⠀⠀⠀⠑
⠀⠀⠀⠀⠘⢧⣀⣴⡿⠃⠀⠀⠙⢿⣿⣿⣿⣿⠟⠀⠀⠹⠿⣿⣿⣿⣿⡿⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠙⠋⠋⠉⠀⠀⠀⢸⡇⢀⡇⠀⣤⠀⣶⡆⠀⠀⠀⠀⠀⠀⢨⢉⠁⠀⠀⣜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡤⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾
⠀⠀⠀⠀⢨⠀⠀⠀⡆⠀⠀⠀⠀⠀⢬⡅⠀⢠⣷⣴⠟⢉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠠⢤⡤⠀⢤⣤⠄⠀⣶⠀⠀⢸⠁⠀⠉⡟⠁⠰⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣀⣀⣀⠀⠀⠀⠐⠟⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⢸⠀⠀⠠⡇⠀⠀⠁⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⣀⠀⡀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⣴⣿⣿⣿⠀⠀⣤⣴⣤⡀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠀⠀⠈⠀⠀⠀⡇⠀⠀⣾⣿⣦⣿⡾⠋⢱⠄⣀⣀⠀⣀⣤⡄⠻⠿⢿⣿⣿⠯⠼⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⢀⣴⠇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⡇⠀⠸⢿⣿⠋⣿⠁⣴⡟⠀⣽⡏⠀⠛⢁⡀⠀⠀⠀⣤⠀⠀⠀⠀⠈⠛⣿⡞⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⢸⡿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠈⠛⠸⡿⠀⠋⠀⠀⢈⠁⠀⠰⠿⠁⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠤⣄⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠴⡶⠢⠀⠀⠀⡗⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡏⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢃⡄⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⢔
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁

⣶⡶⠲⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣷⣶⣶⣶⣶⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣹⣷⠀⠀⠸⡦⠀⢿⣿⡷⠆⠀⠀⠀⢻⣧⠀⠀⠀⢀⠐⢻⣷⠄⢠⡀⠛⢀⣤⡆⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠀⠀⠀⠀⢯⠀⠀⢸⠛⠍⠀⠀⠀⠀⠀⢀⡈⠀⠁⠀⠀⠞⣷⡈⠁⣠⣤⡄⢉⡀⠛⠋⣠⣿⡄⠀⠸⠻⠄
⠀⠀⠀⠀⣦⠀⠀⢹⣷⠆⢠⡀⢀⠀⠀⢀⡀⠀⠐⠠⢼⡏⠄⠰⠟⣿⠂⠈⠁⣤⡘⢷⡡⠄⠀⠈⣄⠚⠉⠀⣠⡇⢹⡇⠀⠀⠀⢀⣴⡇⠀⠰⠃
⠀⠀⠀⠀⠁⠀⠀⠛⠙⢣⠀⠁⠈⠃⠀⣿⡻⠀⠀⢰⣶⡿⠀⠀⠀⠐⢷⡇⠀⣿⣷⣌⢣⠀⣀⣤⣾⣦⣤⣼⡿⢷⣼⣷⠀⢠⡖⣀⣿⠇⢀⣀
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠈⣶⠀⠐⠀⠀⠉⠁⠀⠢⡜⠋⠀⠀⠀⠐⣀⠀⡀⠀⢱⣏⢽⣧⣶⠋⠉⠋⢻⣿⣟⣷⠀⣿⣿⡴⢋⣶⣿⡿⢶⣾⡿
⠀⠀⠀⠀⠀⣿⡏⠁⠘⠇⠈⠒⠂⠒⠻⢄⠰⡂⠘⠛⠆⠈⡧⠤⣄⡀⠁⠙⠲⡤⣿⠀⣻⠋⠘⢦⡶⢸⣿⣿⡇⠀⣿⣿⠁⣼⣿⣿⡇⠀⠁
⠀⠀⠀⡄⠀⠙⠓⠀⣀⡀⠀⢠⡄⠀⣰⡾⢎⠁⠀⠀⠀⢷⠄⣴⣿⣧⣀⢰⣾⡛⣹⣶⡏⠀⠀⠂⠀⢸⣿⣿⣇⣠⣿⣿⣰⣿⡿⠟⣻⣿
⠀⠀⠀⠀⠀⠀⣀⠙⠋⠃⠹⣦⠉⠀⠙⠋⢀⠀⠸⣦⣤⣿⠀⢻⣿⣿⣿⡟⠋⠉⣿⣿⠁⢀⡄⠀⢠⣾⠛⠉⢉⣿⣿⡿⢻⣿⡦⢀⣿⣿⢀⣴⡇
⠀⠀⠀⠀⠀⣼⡟⠃⠀⠀⡈⢻⡆⠀⠀⠀⢨⡄⢰⡌⠛⣿⣧⠀⠿⢻⣿⡇⠀⢠⣿⣿⣷⣾⠅⢠⠋⣽⣆⢠⣾⣿⣿⠀⢀⡿⢴⠞⣿⠏⠙⣭⡄
⠀⠀⢀⡀⠀⠘⠃⡀⠸⠒⠃⠀⢃⢰⣾⣿⣦⠑⢼⣷⣴⡾⢿⣧⣶⣾⣿⡀⠀⠀⡿⠋⢹⣿⣷⡏⢠⣿⣿⣿⣿⣾⣯⠀⠊⠀⣠⣿⣿⣶⡄⠉
⠀⠀⠀⠉⢀⠀⣿⡟⠀⠀⠀⠀⠈⠺⡗⡨⠀⠁⢀⠈⢻⡆⠈⢿⣿⡟⣿⣧⠀⣶⠀⠀⠈⣿⣿⡆⠈⣿⣿⣿⣿⣿⡷⢤⣤⣸⣿⣿⣴⣿⣿⡆
⠀⠀⢠⡶⠟⠁⠀⠀⠀⢢⡀⠀⠀⠐⠋⢀⡀⠀⠈⠀⠘⣿⣦⡟⠻⣿⣿⣿⣷⠿⣏⠀⠈⣿⣧⠄⢰⣿⡿⠟⠀⠀⠀⠰⣿⣿⣏⠈⣿⣿⣿⡇⣀
⠀⠀⠉⠀⠀⠀⣠⣤⠀⠀⢱⡀⠀⠀⠐⠞⠓⠂⢶⣀⠾⠋⠙⠳⣤⣾⣿⣿⣿⢀⣼⠛⠹⣿⣿⢣⠜⠋⠀⠀⠙⣦⣶⣿⡿⠿⣧⣶⣿⣿⣿⠛⠋
⠀⠀⠀⣤⣶⡄⠙⠁⠀⠀⡌⠛⠶⣤⠤⠀⠀⠀⠀⡏⣰⡏⠙⠷⠿⣡⢙⡿⣿⠛⠉⠛⡷⢿⣿⣁⠔⠀⠀⣠⣾⣿⣿⣋⣤⣾⡿⢿⣝⠋⠉⢳⡶
⠀⠀⠘⡙⠀⣁⠀⢀⣀⡙⠆⠒⢀⠃⠀⠀⠀⡻⠎⠸⣿⣤⣶⣶⡄⠀⢨⣄⠀⢠⡀⠀⠀⠶⠈⢷⠤⠴⠋⣿⣿⡿⠛⠛⠀⠉⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠁⠀⠀⣀⠀⣭⡙⠓⠂⡀⢀⣤⡄⠀⣽⠇⢰⣿⣿⣿⣿⠇⠀⠈⠛⢠⣟⡿⢀⣾⣿⠀⠐⢠⡆⣰⡞⢿⠉⢷⠊⠹⡝⠛⠛⠛⠏⠉⢻⡄
⠀⠀⠀⠀⣿⡟⠁⠀⠀⠑⠀⠀⠛⢾⢿⣶⠞⣡⣶⣴⣏⣉⠉⠁⣰⣶⠀⠀⠨⣯⣿⣿⣿⣿⣶⡶⢈⣀⠉⠈⣶⡄⠈⠻⣄⠀⠘⣶⣦⢠⡀⠸⣷
⠀⠀⠰⢦⠈⠃⣶⡷⢀⡀⠱⣄⡖⠠⠘⣣⣿⣿⣿⣿⣿⣿⣧⣨⡿⢿⡶⠆⣀⣈⣈⣋⣉⣍⠛⣴⡟⠻⠧⣀⠀⠉⠂⠀⠻⡷⠀⣿⣿⣿⣭⡀⢸⡆
⠀⠀⠀⠠⣦⣤⡴⠁⣨⠃⡷⢌⡑⠂⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⡙⠋⣡⣼⣿⣿⣿⣿⣿⣿⣄⠘⠓⢀⣤⠄⠀⣂⠀⠈⠁⠀⠊⠙⠛⡟⠛⢀⠈⡇
⠀⠀⠀⢀⣈⠙⠀⠀⠀⠈⠁⠀⠉⢩⣿⡁⠙⠻⠿⣿⣿⣉⠀⠐⠈⠎⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠀⣨⣴⠖⠸⣿⣄⣀⡀⢠⡀⠀⠀⠁⠀⠉⠠⠁
⠀⠀⠀⢸⣯⡤⠖⠀⠀⣀⡀⠀⢲⣾⠀⡀⠘⡷⢰⣿⣿⣿⣦⠈⣤⠓⠻⣿⣿⣷⣿⣿⣿⣿⣿⠟⠀⢿⡷⠖⣻⣷⣿⣿⣤⣘⣛⡂⠀⠀⠋⠀⣾
⠀⠀⠀⣾⡟⠃⢀⣤⠐⠈⠙⠉⣹⣧⣶⠀⠀⣴⣿⣿⣿⣿⣿⠸⣿⣤⡄⢉⢉⣉⡛⠻⠿⠛⢁⡰⠀⣸⣿⣿⠿⠛⠛⠛⠛⠛⣿⣿⣿⠂⠀⢀⡎
⠀⠀⠀⠙⠧⠀⠘⠇⠀⠈⡁⢠⣿⣿⠟⣀⢈⠛⠙⢿⣿⣿⡿⣞⢻⣽⣦⣜⠉⣹⢃⣄⡀⡄⢀⣤⠶⢿⣿⣷⡶⢶⣿⣿⡿⠟⡛⠉⠉⠰⠆⢸⡇
⠀⠀⠀⠳⠆⠀⢠⠈⠐⣤⡈⠀⣿⠿⢿⣯⣮⣤⣴⣂⡟⠲⠞⠈⠀⣿⠹⢿⣿⣿⣿⣿⡗⡘⠒⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⢠⣴⣤
⠀⠀⠀⠀⣶⣴⣗⣀⠠⠏⢙⠹⣇⣆⡘⢁⠟⡸⠟⣿⣴⡗⠂⡌⢰⠏⡄⢺⣯⡽⠛⣿⡏⠛⠃⠀⠛⢻⣿⣧⣉⣿⣿⣿⣃⠀⣚⡤⠒⠀⠉⣟
⠀⠀⠀⣾⣈⠿⠋⠁⣤⣦⠀⠰⠞⣿⣿⣿⣶⣿⣴⠃⡼⢻⡟⣠⣧⣸⢁⠛⡏⣀⣾⣿⣶⣋⠁⠞⣃⣉⣉⠻⣿⣿⡟⣿⡯⠙⡿⢷⡗⠀⡰⠟⠛
⠀⠀⠀⠉⠉⣤⡶⠖⠛⢭⢤⣴⣞⡏⢹⣿⢿⣿⡏⢾⣧⣞⣠⠏⢹⡟⢿⣤⣇⣯⣿⣿⣿⣿⢿⣿⡿⢋⡍⣷⣿⣿⣧⣿⣿⣷⣤⡏⢁⣾⠇⣠⣷
⠀⠀⠀⢀⠀⠘⢛⡄⠀⢀⠀⠀⢹⠉⣂⠈⢾⣟⠛⠛⣿⢿⣿⣶⢛⣴⣦⣸⣯⣇⣿⣿⣿⡇⠀⣤⠀⠈⣰⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⡟⠀⠛⠁
⠀⠀⠀⠀⢴⣶⣾⠃⠐⢾⡖⠀⢸⠀⢻⠀⠀⣬⠀⠸⠿⣶⣿⣝⠋⢹⣿⣟⣿⣿⣿⣿⣿⢁⡾⢁⣶⠿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠘⡉⠀⢙⡿
⠀⠀⠀⠀⠈⠉⠁⣤⡈⠀⢸⠖⠘⢒⡈⠑⡒⠧⠀⢃⡀⢀⢀⣼⠀⣾⡋⠛⠋⣿⣿⠛⢇⡾⣠⡎⠁⠀⠀⠈⠙⠋⠉⡉⠉⠀⠀⠰⠎⠉⢷⢀⣦
⠀⠀⠀⠻⠆⠀⠀⢿⣧⠀⠸⢦⠰⠾⣧⡶⠿⣤⢀⠀⠑⣌⠉⢯⡤⢿⡿⣤⡼⢿⣿⡟⠸⣱⠿⠇⠀⠐⠀⠀⡀⠀⠀⠘⠢⠀⠀⠀⠀⣾⠇⠘⠋
⠀⠀⠀⠀⠺⢿⣆⠀⠀⢠⢴⡾⢁⠀⢀⣤⠄⣿⠘⣿⠀⠛⣶⢈⡔⠻⠖⠾⠶⢞⡾⠁⢀⡏⢠⣤⢱⠆⠁⠈⡏⠀⢰⣆⣠⣧⠄⠀⠀⠁⢀⣴⡆
⠀⠀⠀⠀⠀⠸⢻⠀⣠⠃⣸⣇⠙⢃⠈⢁⡄⢤⡴⡇⠹⠶⢩⡏⠁⠀⠘⡓⠀⠈⠐⠆⠈⠀⢿⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⢡⣤⡄⠀⠀⠚⠛⠁
⠀⠀⠀⢾⣤⣦⢀⣨⠷⠂⠀⠙⣶⡿⢶⣦⣄⠈⢻⢃⢄⡀⠙⢁⡉⠀⠈⢀⣴⠾⠀⢀⣧⡀⠈⠉⢐⠀⠀⠠⠀⣷⡄⢀⣀⣰⡿⠛⠃⠘⣷⡆
⠀⠀⠀⠀⠉⢀⣤⠤⣤⠄⢶⠞⢩⢅⡀⢁⡄⠛⠋⠙⠃⠀⡄⠙⠻⠶⠿⡮⢥⢴⠖⠛⠏⠙⠀⢀⡆⠀⠰⠂⠿⡄⠀⣌⠁⠀⠿⢁⣄⠠⡾⠃
⠀⠀⠀⠀⢠⣦⡔⠈⠛⢢⡄⠠⠅⠊⠀⣤⣶⢀⡀⣼⣇⠘⠀⠀⣤⣀⣤⡀⠀⠾⠟⢢⡶⠋⠀⢸⣿⣿⣧⡄⠀⢀⣤⣾⠀⠀⠠⣤⡿
⠀⠀⠀⠀⠀⠻⡶⠂⠀⢿⡄⠀⠀⠀⠀⠈⠛⠛⠁⠘⠛⠃⠀⠀⠛⠁⠚⠁⠀⠀⠀⠘⠛⠁⠀⠀⠉⠛⠛⠁⠀⠀⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡅⠀⢸⠇⠀⠀⠀⠀⠀⠿⣿⠇⠀⠀⠀⢰⠋⠈⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢶⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠈⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣠⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡆⠀⢠⣶⣦⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢦⡀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠘⢧⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⢀⡀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⢀⣀⡀⠄⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠨⣀⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠒⢦⠀⠀⠸⢿⠿⣴⠋⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⡿⠀⠀⠀⠀⠐⠧⠽
⠀⠀⡰⠲⡄⠀⠀⠀⠀⠀⠘⢄⣀⡼⠀⠀⠀⠀⠀⠈⠲⠖⠃
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀
⠀⢠⣶⣶⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡷⠀⠀⠸⣄⣠⠞⠀⠀⠀⣶⣶⡆⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⢠⢮⣶⣤
⠀⠀⠙⠋⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⣠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠛⠋
⠀⠀⠀⡠⠶⢦⡀⠀⠀⡀⠀⠀⠀⣠⣀⠐⣇⡀⣨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣏⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⢧⣀⣠⠇⠀⠀⠀⠀⠀⠸⢿⠿⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡞⠁⠈⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠻⣤⠟
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠸⠿⠀⠀⠀⠘⠲⠞⠁⠀⠀⡜⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⠀⠙⠦⠖⠁⠀⠀⠀⠀⠀⢠⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠕⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢦⡀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠘⠲⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣶⣿⡆⠀⠀⢧⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠈⡆⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠤⠤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⠁⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⠿⠋⠀⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⡼⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⢰⠃⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⡎⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⡀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢿⣿
⠀⠀⢸⡇⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢻
⠀⠀⠀⡇⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢸
⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠈⠻⢿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⡆
⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡿⠿⠿⠛⠋⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⢀⣴⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⡠
⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢀⠂⢀
⠀⠀⢸⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠒⣐⡴⠖⠒⠛⠉⠉⠉⠉⠉⠙⠒⠃⠉⠛⠀⢈⣉⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠈⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡎⠀⢠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢠⠈⠉⠒⠦⣤⣤⣀⣀⡇⠀⠀⠀⠀⠘⡆⠀⢀⡠⠐⠀⣀⡀⠀⠀⣵⡀⢸⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡠⠄⠊
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠢⠤⣀⠀⠀⠉⢻⠀⠀⠀⠀⠀⢁⡴⠋⠠⠒⠉⠁⠀⠀⠀⠀⡷⠾⡷⠾⠛⠛⠶⠖⠒⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠈⠙⠢⣼⠂⠀⠀⢰⣄⣀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠋⠀⠘⠀⠀⠀⢀⣀⣀⣀⡄⠤⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡀⠀⠀⠠⢘⡻⣆⠀⠀⢀⡟⠁⠀⠀⠀⢀⣂⣠⣾⡇⠀⠀⠤⢤⣀⠠⣾⠉⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠒⢩⢴⣿⣦⣴⣾⣇⠀⠀⡀⢀⣾⡏⠉⠹⠋⠀⠀⠀⢠⣨⠛⠦⡉⠀⠀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⠛⠀⠀⠠⡙⢯⠩⠁⢿⣿⣿⡇⡡⠖⠀⠀⠀⢨⣿⡀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣅⠀⠀⠀⠀⠀⣀⣀⠒⣾⠟⠧⠤⠤⠴⠒⠒⠢⠸⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣡⣤⠀⡆⠀⠀⠀⠀⡀⠀⢧⠈⠰⠂⢀⡼⠻⠷⣄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢀⠟⠑⠁⠀⠀⠀⢾⠉⠻⡧⢀⡀⠴⣯⠔⠀⠛⡿⡖⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠏⠀⠀⠀⠀⠸⡃⠐⢹⣉⠷⠐⠛⠒⠒⠟⠁⠁⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡉⣆⠑⠢⠤⠴⠞⠉⠙⠁⠀⡸⠁⠀⢀⣠⠤⠤⠄⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣝⣒⡶⠖⣂⣀⣤⠶⠚⡅⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢏⠉⠁⠀⠙⢦⠁⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠤⣤⣤⡤⠶⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⣀⣀⣀⣀⣀⡀⢀⡀
⠀⠀⠀⠀⠀⣠⠀⠰⣄⠼⢛⠿⣿⣿⣿⢿⣟⢫⡟⡛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡛⠛⣛⢻⡆⢠⣶⣶⡯⣻⣿⣿⡛⣻⣿⣟⢨⣷⣶⣄
⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⢻⢠⡶⠶⠶⢾⡟⠛⣯⣭⣭⣿⣍⢉⡿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠁⣮⣥⣴⣦⡛⠉⠙⣃⡋⠉⠙⣧⣶⣤⣧
⠀⠀⠀⠀⠀⣸⣿⣤⣶⣴⣿⣿⣿⣿⣿⣿⡿⣿⡟⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠻⠮⠿⠛⠻⠧⠾⠋⠈⠉⠁
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡿⢿⠟⢻⣿⠿⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣤
⠀⠀⠀⣴⣿⣿⣿⣿⣿⠿⣿⡀⢿⡀⣠⡏⠀⠀⢀⡀⠒⢢⠐⠀⡐⠢⡀⠀⡤⠒⠢⡄⠀⠀⢸⣆⡀⠀⢀⢿⠧⣀⣀⠼⡿⡀⠀⢀⣰⡇
⠀⠀⣾⣿⣿⣿⣿⣿⠁⠀⠈⠉⠋⠉⠉⠀⠐⠤⢾⡗⠚⢺⠓⠒⡿⠤⡨⣟⣄⡀⢀⣹⠀⢀⣮⠷⢮⡓⣷⠉⠙⡏⢱⠋⠉⣿⢚⠥⠾⡱⡀
⠀⢰⣿⣿⣛⣿⣿⢿⠀⠀⠀⠀⠀⠀⠀⣦⣶⣤⠼⠧⠶⠼⠶⠦⡷⠤⣴⣤⢲⣜⣋⣉⡀⢸⡘⠦⠤⠗⣘⣁⣚⠁⠈⣑⣎⣁⠸⠤⢴⣃⠇
⠀⢸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠻⣿⣿⢾⣿⣿⣸⣿⣇⣿⣿⣿⠟⠉⠉⠉⠉⠁⠀⠉⠛⢺⡋⣯⠕⠛⠒⠒⠛⠫⢍⢙⡟⠚⠉
⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣶⠹⠶⠦⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣃⣛⠒⠠⠤⠤⠄⠒⣋⣘⠆
⠀⠀⣰⠊⠉⠉⠉⠉⠉⠛⢉⠀⠀⠐⢒⣒⠒⠒⠒⠀⠠⠤⠤⠤⠤⠤⠤⠀⣀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⣁⢄
⠀⠀⠺⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⡛⠒⠒⠻⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠒⠲⠶⠾⡶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠶⠷⠀⠀⠀⢀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠀⡠⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡀⠀⣆⠀⠄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠊⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠄⠀⠌⣉⠇⠈⠈⠉⠐⠀⠀⢀⡀⠀⠀⠂⠀⢫⠀⠀⠀⠀⠒⠀⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢲⢄⠰⠘⠂⠀⠠⠄⠈⠁⠄⠁⢀⠬⠋⠀⠀⠀⠀⡒⠀⠧⡄⠀⠀⠀⠀⠄⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠐⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⢀⠃⠰⠀⣰⠂⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⢖⠲⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢲⠀⠀⡀⠠⠀⠻⠀⠀⡄⠓⠁⠀⠀⠂⠁⢀⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠊⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡇⠀⠀⡖⠀⠀⠄⠀⢡⡨⠀⠀⠀⢀⠀⡮⠑⠋⠀⠀⠁⠀⡀⠀⡀⢊⠍⠒⠚⠒⠠⠌⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣠⠇⠀⠀⠀⠀⠀⠤⡀⢀⠄⣸⠀⡆⠀⠁⠈⠀⠀⠀⢠⠄⠠⠃⠠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠑⡌⠀⠀⠀⠀⠄⡇⠈⠃⣼⡄⠀⢠⠀⠙⠐⠀⠀⠀⠆⠁⠀⠀⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡘⢆⠀⢠⠀⠂⠳⢄⡊⢀⠃⠀⡼⠀⠠⢀⡄⠁⠈⠀⣤⢶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠉⠒⠤⠀⠀⢷⠋⠀⠀⡧⠐⠂⣀⠄⠒⠀⣸⠃⠀⣯⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡟⠀⣸⠃⠀⠀⡀⠀⠁⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⢆⢀⠀⠀⠀⠀⣁⣀⣀⣀⠀⠀⠀⢀⠎⠀⠀⠀⠀⣼⠃⠐⣯⡀⠀⠀⡁⠀⡁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡈⠂⢒⠀⠱⠄⣠⠆⠐⠛⠋⠉⠙⠛⠻⢶⣤⣤⣄⣠⡴⠛⠁⠀⠀⢀⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⠃⠄⠘⠀⠀⠀⠀⠀⡀⠈⢳⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠊⢉⠓⠀⠀⡆⠀⠀⠀⠀⠀⠀⠰⠁⠀⠘⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⣠⣴⠶⠞⠛⢛⡇⠀⠀⠀⠀⠀⠀⠐⠂⠃⠀⠁⠀⠀⠁⠉
⠀⠀⠀⠀⠀⠀⡤⠤⢀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠊⡇⠀⠈⡀⠂⠀⠀⠀⠀⠀⢤⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢩⣀⠠⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⠰⠞⢀⠁⠀⠄⠀⠀⠈⠁⠀⠀⠐⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄⠈⠂
⠀⠀⠀⠀⠀⠰⠁⠈⠀⠀⡴⠃⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡤⠴⢦⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⢘⠛⠋⠀⠉⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢄⠀⠀⠐⡀⠀⠆⠀⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠆⠀⠀⠀⠀⠀⠐⣠⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢣⡈⠳⡀⣀⠐⠀⠐⡡⠐⠀⠀⠀⠀⡀⠀⠀⠦⠤⠔⠀⠀⢀⠀⢀⡀⠠⠀⠀⠀⣀⣀
⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠉⠁⠘⠖⠉⠛⠤⠀⠐⠁⢛⠀⠀⠀⠀⠀⠌⠠⢀⣠⠀⠀⠀⡁⠀⠀⠀⠐⠂⠀⠘⠁⠀⢁⠂
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠈⠁⠚⠁⠰⠖⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠁⠔⠂⠀⡄⠀⢠⠇⠒⠀⠈⠰⠀⢀⡀
⠀⢠⠀⠀⠀⠀⠠⠄⠀⠒⠁⠔⢁⠠⢦⠀⠀⠖⠀⠆⠈⠀⠀⡥⠀⠀⠑⠠⠀⠁⠀⠀⠈⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠶⠀⠀⠀⠀⠀⠀⠀⠁⠠⠊⠀⠀⠀⠀⠀⠀⠀⠤⢄⠑⠂⠀⠐⡁⢈⠠⠀⠒⠚⠶⠈⠁⠀⠘⠀⠀⠠⠀⠀⠀⠀⠀⠌
⠀⣶⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠠⠄⠂⠀⠀⠀⠀⠀⠈⠀⢀⠐⠈⠀⠁⠀⠄⠀⠀⠀⠀⠀⠠
⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣤⣁⡂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣾⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⣼⣷⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡤⠔⠊⠉⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣎⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣧⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠓⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠉⠁⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣧⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣤⢶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⢀⣀⣤⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡌⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⢻⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠛⠻⢿⣿⣿⣿⣿⣿⣿⡿⠃⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠄⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢿⡟⠁⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠉

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⢿⣿⡧⠀⠀⠀⠀⠀⠀⣶⣿⣧⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⣠⡟⠈⠁⠀⠀⠀⠀⠀⢀⡜⠙⠁⠀⠀⠀⠀⠀⢐⡾⠅⠀⠀⠀⠀⠀⠀⢤⡼⠃
⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠞⠉

⠀⠀⠀⠀⠀⣀⡴⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⢉⣹⣆⠀⠀⠀⠀⠀⠀⠀⢹⡴⠆⠀⠀⠀⠠⣤⣸⣇⣤⠄
⠀⠀⠀⣶⡋⠀⠀⠀⠀⠀⠀⠀⢰⠾⠀⠀⠀⠀⠀⠀⠠⠖⠉⠀⣟⠀⠀⠀⠀⠙⠁⠀⢀⠇⠀⠀⠀⠀⠀⢰⠟⠳⡄
⠀⠀⠘⠓⠃⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⣤⠂⠀⠀⠀⠀⠀⠀⣠⣷⣾⡄⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠂
⠀⠀⠀⠐⣁⠆⠀⠀⠀⠀⠀⠀⠀⡏⠙⠁⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠲⣯⡿⠂⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠃⡈

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣴⣿⣦⠀⠀⠀⠀⠀⢀⡾⠁⢱⡄⠀⠀⠀⠀⣠⣷⣶⣶⣶⡆⠀⠀⠀⠀⠀⣠⡁⠀⠀⠀⠀⢀⣤⣴⡶⣿⡿⠿
⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⡾⠃⠀⠈⠿⡀⠀⠀⠀⣿⠇⠉⠉⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⠈⣽⣿⢿⢿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⣤⣤⣤⡤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠉⠐⠛⠛⢻⡟⠛⠋⠀⠰⠈⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣴⣾⢿⣶⣄⣬⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣤⣴⣿⣏⣼⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣛⣋⣀⠰⠈⢯⣿⣿⣿⣛⠋⢿⣿⢻⣿⣿⣿⣧
⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣈⠀⠈⠟⠹⠿⢿⣿⣿⡏⣿⣿⣿⣿⡇
⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡻⣿⣿⣿⣿
⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢻⣿⣿⣿⣷⠀⢻⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⣿⡿⠟⠁⠀⢸⣿
⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠀⠀⠀⢀⡇
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⠛⢁⡶⠋⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⡀⠀⠀⠀⠿⣿⡿⠿⠟⠋⠀⢀⣉⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠒⠒⠒⠒⠒⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣿⣿⣷⣶⣿⣿⣷⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⠋⣉⣉⣤⣤⣤⣤⣴⣶⣶⣶⣤⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣵⣷⣾⣿⣿⣿⣿⠏⠙⠿⡿⠯⣽⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠋⠖⠘⠟⠁⠶⣤⡀⠀⠀⣀⠉⢻⢻⣟⣛⣺⣶⣦⡄⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⣟⣆⡀⠐⠀⠀⠀⠁⠀⡀⠀⢈⣀⣁⣈⠀⠀⠀⢈⣹⣿⠀⢀⣾⠟⣿⡆
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣈⣑⡿⡇⠀⠀⠀⠠⣄⠀⠀⠈⣠⠦⠊⡙⠻⠿⠿⢥⣣⣤⣾⡏⢀⣿⣇⡀
⠀⠀⠀⠀⠀⣀⢿⠛⠛⠛⠛⠛⠛⠛⠀⠁⠀⠀⠉⠀⠀⣸⠑⡘⠀⠣⠀⠐⡠⢀⣀⣈⣍⣀⣨⣹⣿⣷
⠀⠀⣴⡮⠛⠛⠚⠒⠂⠀⠀⠀⠋⠀⠀⠀⠐⠂⢀⣬⣹⣧⣁⠀⠀⠀⠈⠉⣤⣯⠊⠉⠛⠛⠛⠛⠛⠁
⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦⣀⠀⠀⠀⠀⢠⣽⣿⣿⣿⣦⡄⠀⣤⢼⣿⣿⢆
⠀⠘⢟⣦⣀⣀⣀⡠⠀⣾⣿⣿⣿⣿⣿⣶⣤⣶⣿⣿⣿⣿⣿⣿⣿⡄⠉⠀⠙⠿⣿⡣⡀
⠀⠀⠀⠉⠛⢟⣄⠀⢉⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠈⢻⣿⡀
⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠚⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢻⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣄⣁⡴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠓⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠶⠖⠒⠀⠀⠈⠒⠀⠐⠲⠒
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣳⣤⡤⠾⠷⢄⠀⠠⡀
⠀⠀⠀⣀⠤⠂⠀⠀⠀⠀⠀⢀⣼⠿⠁⠀⢀⣀⠀⠐⢤⡀⠀⠀⠀⡜⠁⠀⠈⠑⢦⠀⠀⠀⠀⠐
⠀⠐⢯⣀⠀⠀⠀⠀⠀⢀⣠⠞⣇⣀⠀⠀⠀⠈⠓⠦⠤⡙⠂⠀⠀⡄⠀⠀⠀⠀⠀⡑⠠⠤⠤⠐
⠀⠀⠀⠈⠙⠓⠒⠖⠛⠉⠀⠙⠋⠉⠉⠙⢶⡄⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⢰⣆⠠⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠖⠉⣽⣦⡀⠀⠀⢀⣀⣤⣤⣀⣠⣯⣈⡆⠀⠈⢆
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡤⠂⠀⠀⠤⠒⠛⣉⡿⠷⠾⠛⠉⠙⠛⠉⠻⣏⡁⠀⠈⢢⠀⢆
⠀⠀⠀⢠⣠⠊⠀⠀⠀⠀⠀⢣⣀⡀⠀⠀⣠⠊⢁⡤⠖⠒⠀⠀⠀⠤⠄⠀⠉⢙⠢⡀⠺⡀⠘⡄
⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠹⠟⠡⡏⠁⠐⠁⠀⠀⠀⠀⠀⠈⠳⡀⠘⠁⠀⢣⠘⡄⠘⢆⢣
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⠈⢿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠀⠑⢶⣒⣏⠹⠷⢦⣀⠀⠀⠀⣰⣀⠀⢀⣠⠆⠀⠀⠀⠈⣇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡴⢻⡏⠙⠒⠒⠋⠁⠀⠀⢀⣀⡤⠍⠉⠉⠉⠉⠛⠿⣅⠀⠀⣆⠀⠀⢸⡄
⠀⠀⣧⠀⠀⠀⠀⠀⢠⠞⢠⡟⠣⡀⠀⠀⠤⠤⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠁⢠⡇⠀⠀⠀⡇
⠀⠀⠸⣾⠀⠀⠀⠀⠏⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⡛⡇⠀⠀⠀⡷
⠀⠀⠀⢻⡄⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⣀⠔⠈⠀⠀⣠⣴⣦⡀⠀⠀⠈⠂⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠹⣄⠀⠀⠀⢸⡄⠀⣀⣤⣾⠶⠶⠛⠉⠉⠉⠉⠉⠀⠀⠀⠈⠲⣤⡄⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠈⠛⢦⣀⠀⠛⠖⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠆⠀⠈⠻⠿⠃⠀⠐⠂⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⣶⣶⣤⣀⠀⠀⣀⠀⣀⣀⡀⠀⠀⠀⣠⣴⣶⡿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣈⣻⣿⠿⠟⠻⠉⠉⢉⡉⠙⡛⠿⠿⣿⣿⣏⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⡁⠖⠹⡛⠘⠋⢁⡀⠈⠉⢉⣁⠈⠿⠻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠿⠀⠀⢀⡈⣀⡀⠀⠋⢀⣈⣁⠀⠀⠙⠻⢿⡃
⠀⠀⠀⠀⠀⠀⣴⠤⠄⢤⢴⡆⢸⡟⠂⠀⣘⣥⣤⣽⡃⠀⠀⢘⣿⣬⣉⣦⡀⠀⢿⣷⢀
⠀⠀⠀⠀⠀⠀⢸⣶⠀⡄⣸⡁⣾⣇⠀⠀⠀⠀⠀⠛⠗⠀⠀⠈⢿⠁⠀⠀⠀⠀⢸⣿⢻⡿⢛⠒⢿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⣶⢾⠇⠉⠸⣿⣇⠀⠀⢀⡈⠁⠀⠀⠀⢀⠈⡉⠀⠀⠀⢘⣾⡿⢸⣥⠤⠠⣼⠂
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣄⣉⣠⠹⣿⣦⣤⣾⣷⣆⢻⣿⣶⣿⠁⣷⣄⣀⣰⣾⠏⢁⣿⣿⣦⣶⡏
⠀⠀⠀⠀⠀⠀⣸⠛⠁⢹⣿⢿⣿⣀⣈⠙⠿⣿⣿⣿⣿⣻⣿⣿⣾⣿⣿⣿⡿⠋⠰⣾⣷⢹⣿⣯⡇
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠾⠟⠈⠉⠁⠈⠁⠀⠈⠛⢿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠙⠛⠻⣿⣯⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠈⠷⣄⠠⡆⠀⠀⠈⠍⠉⠁⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠭⠒⣪⠟⠒⢺⣀⣈⠛⣣⡰⠆⠖⠂⠋⣃⣈⣴⠏⡨⣤⡆⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⡁⠜⠉⠓⠉⠀⢀⡆⠀⠀⠀⠀⠁⢀⡽⠌⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠛⠻⢟⡻⠿⠿⠛⠛⠯⣉⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠈⠑⢦⣤⣭⣳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡃⠀⠀⠀⠀⠀⣀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠙⢶⣤⣴⣶⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⣸⡏⢁⣴⣿⣿⣿⣿⣿⣿⣷⠶⠖⠛⠋⠉⠉⠻⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢠⣿⣱⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡀⠀⣀⣤⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⠋⠉⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⢿⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡏⠀⠀⠀⠀⠀⠀⠈⣿⠟⠋⠉⠁⠀⠀⠙⢷⣄⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡇⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠙⢆⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠳⣤⣤⣴⣶⣿⣤⡀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⣿⣿⣿⣿⣿⣿⣿⣦⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠖⠒⠉⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⢲⡄⠀⠀⡼⣿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⡠⠞⠉⠉⠉⠉⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠔⠚⠉⠀⠀⠀⢠⠃⠀⡼⣦⣏⢧⠫⠻⣆⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⢒⣒⣛⣷⣣⢃⣿⠀⠀⠀⠀⠀⠀⠀⠈⢷⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⡞⢨⢳⡀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠉⣔⣺⡿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⡚⣧⡀⢀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡚⢻⢲⡲⠲⡲⠶⠶⠲⡻⠁
⠀⠀⠀⠀⠀⢀⠒⣒⣚⢻⣬⣊⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣣⣁⣱⢠⢡⢃⣶⠁
⠀⠀⠀⠀⠀⡽⠋⠉⠁⠀⠈⠉⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣭⠛⠛⢻⣛⠿⠄
⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⢶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠥⠞⠋
⠀⠀⠀⡘⣷⡀⠀⠀⠀⠀⠀⠀⠀⣠⢟⢫⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠁⠀⠀⠀⠉⢻⣦⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⢰⡟⠸⡷⣤⣀⠀⣀⣀⣤⢞⢧⣏⣀⢆⢛⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⢀⣠⣴⡷⠲⢄
⢠⡥⠹⣧⡐⡑⡙⡙⠏⠍⣍⠆⣌⡟⠻⣿⠌⢌⠞⡇⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⡀⠀⠀⠀⠀⢸⡟⠀⠀⢀⣀⣤⣶⠿⠛⠁⠀⠀⠀⠑⣄
⢘⣃⣀⠘⣳⣝⣐⡀⡀⣈⣏⣼⣏⣉⣩⣿⡜⡞⡜⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡄⠀⠀⠹⣷⣶⡾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠐⠠⠤⠀⠉⠛⠿⠿⠿⠯⠽⢦⡴⠒⢺⡷⡐⠠⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠈⠉⠀⠂⠩⡥⡭⠏⠉⠀⠂⠈⠉⣷⡱⡑⣱⣿⣦⣤⣤⣤⣶⣶⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⡹⠳⡶⡤⣤⡤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠠⠀⣻⣄⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡬⢵⣡⡝⡐⡀⡃⣼⠃⣤⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡭⠩⠍⠿⣿⣏⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣛⣊⣽⠿⢶⣿⣼⡟⣼⠇⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠅⠨⢽⣿⢪⠻⣇⠀⠀⠀⠀⢠⡾⡫⡀⡡⢀⠁⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢻⡁⠉⠙⠛⡿⠩⠭⣰⠏⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠐⣿⣣⠑⠹⣆⠀⠀⠀⣾⣴⣷⣥⣵⣧⣡⠆⢌⢿⡄⠀⠀⠀⠀⠀⠀⠀⠙⢤⣀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⣒⣂⠘⣷⣠⠌⠜⠷⣄⣰⠯⠅⣶⠾⠿⠿⢿⣆⣾⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣠⡖⠛⣷⣎⢊⢎⣾⠃⠀⠀⠀⠀⠉⣉⣭⣿⣿⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠛⠋⠁⢠⡞⠉⠙⠃⠒⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⢴⢶⠶⠾⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⢶⡶⠆⢠⠎⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣻⡙⡀⠂⢂⠂⠆⠌⢄⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣾⠂⢀⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⣶⡀⠀⠀⠈⠉⠙⠛⠶⣾⠞⡚⡛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⢂⠹⣆⢎⠎⠙⣦⡀⠀⠀⠀⠀⠀⠀⣀⢴⠞⠋⢡⢃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣥⠴⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡀⠀⠹⢯⡞⠀⠈⠟⣦⡀⠀⣠⡴⠛⡉⠀⡀⣦⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠠⠬⢿⣆⠎⡜⡜⡙⠛⡹⠐⣐⣴⣾⣛⣛⣹⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢙⣾⣜⣜⣆⣠⣤⡷⢛⡉⢭⠤⠕⣿⢃⢟⢳⣤⣀⣀⣀⣀⣤⠴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢭⡠⠀⠚⠛⠛⠉⠙⠋⠉⠈⠙⠛⢛⣛⣿⣧⣧⢡⠩⢉⢉⢁⣦⣾⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠤⠶⠖⠒⠊⠀⠀⠀⠀⠀⠀⢐⣒⣛⣛⣻⣷⠷⠞⠋⣉⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠡⠭⠥⠤⠤⠖⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⡀
⠀⠀⠀⣀⣼⢱⣄⣿⣿⡒⠻⡻⠿⠿⠿⠯⠭⠭⠿⠿⠶⠶⠶⣶⣶⣶⣶⣶⣦⣤⣄⣀⢀⣀⣀⠀⠠⡀⠀⠀⠀⢶⣶⣶⡆
⠀⠀⢸⢿⣇⣶⣴⣶⣶⣾⣿⣿⡿⠛⠛⠛⠛⠟⠿⠿⠾⣿⣶⣿⡾⣄⣀⣀⣀⢀⣀⢙⣶⢷⣾⣷⠀⠉⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠈⢙⣾⠿⠿⠿⢿⡟⠛⠙⣿⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣧⠀⠀⠀⣀⠀⠀⢹⣿⡇
⠀⠀⠀⢻⣿⣴⣶⣶⣶⣿⣿⣿⣿⡿⠸⢸⢸⢸⠹⢹⢓⡿⠶⠶⠿⠟⠒⠒⣶⡄⣶⡂⣾⣿⣿⡿⠀⠀⠀⠉⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠈⠀⠀⠈⠉⠙⠛⠛⠒⠓⠒⠛⠋⠉⠉⠉⠁⠀⠀⠀⡄⠀⠀⠀⠈⠓⢮⢷⣿⣿⡁⠀⠀⢰⣦⣄⠀⠀⠀⢸⣇⡀
⠀⠀⣠⡄⣀⡀⠀⣀⣀⡀⠤⠀⠤⡤⡀⠀⠀⠀⠀⢰⣶⠀⠀⠀⠘⠛⠀⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠋⠻⢿⣿⣦⣤⣾⣿⢣⡖⣀⡀
⠀⠸⣿⣧⣤⣤⣡⣤⣤⣄⣀⣀⣀⡘⡎⠓⢄⡀⠀⣸⣷⣤⡀⠀⠀⠀⢰⣦⠀⠀⢀⣿⣿⣿⣿⠀⠀⣀⠀⠀⢸⣿⣿⣾⣿⢸⡗⠻⡇
⠀⠀⠀⠀⠉⠉⠛⢝⡳⠿⣿⣻⡿⣿⣧⣴⣤⣬⣛⣯⠛⠿⢿⣀⣀⣶⣾⣿⠀⠀⠈⠛⠿⠛⠁⠀⠀⠉⠀⠀⣸⣇⣸⣿⣿⢺⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡉⠻⣿⡿⠿⠿⠿⣿⣿⣿⣷⣶⣶⣤⣤⣏⡙⠒⠤⢄⣀⠀⣀⠀⢠⣴⣶⣿⣿⣿⣿⣿⣹⣿⢹⡇⢸⠃
⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⣀⣀⣁⡀⠀⠀⣀⠀⠀⠀⠈⢿⣿⡯⢍⠿⢿⡟⢬⣷⣶⣦⣭⣝⣎⡆⢿⣿⠿⠿⠛⠻⠛⢻⣿⣸⡧⠼
⠀⠀⢠⣤⣶⡌⠁⢸⣿⣿⠿⡻⣿⣿⡇⠈⢡⣶⣤⡄⠀⠸⣿⡷⢸⠀⠀⣿⠁⡏⠉⠙⠿⣿⠿⠃⠈⠀⠀⠀⠀⠀⠀⢸⣿⣷⠂
⠀⢄⠨⠟⣋⡛⢐⡿⠿⠿⡃⢘⠿⠿⢿⠂⢛⣙⠻⠅⡀⠀⢸⡇⢹⠀⠀⣿⠀⡇⠀⠀⣀⠀⠀⠀⠠⠄⠀⢸⢠⠀⠀⢸⣿⡿
⠀⠸⠶⣆⠿⠿⡏⠁⠀⢠⣶⣶⡄⠀⠈⢹⠿⠿⣰⡶⠇⠀⢸⡇⢸⠀⠀⠉⠉⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣾⢈⡀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠐⠿⠿⠟⠋⠙⠺⠾⠿⠂⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣀⠀⠀⠀⢖⣖⠀⠀⠀⣾⠀⡄⠀⣿⠈⡇⠀⠘⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣒⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⣿⠀⡇⠀⣿⠀⡇⠀⠀⢹⣿⡓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠭⡽⠿⠿⠿⠯⠽⠿⠯⠤⠤⠤⠤⠴⢇⠀⠀⠀⣀⠀⠀⠀⠀⠀⣿⣠⡇⠀⣿⣤⡇⠀⠀⢸⣏
⠀⠀⢰⣶⣶⣶⠶⠾⠿⢿⣿⣿⣯⣭⣭⡼⠿⠿⠿⠿⠿⠿⠿⠿⣿⠏⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢸⡇
⠀⠀⠘⠿⠿⠁⠀⠀⢀⣼⡟⠙⠛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣀⣀⣀⣀⣀⣀⣠⣤⣤⠴⠿⠾⠛⠛⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠝⠛⠓⠀⠀⠺⠛⠻⠛⠻⠛⠟⠛⠟⠩⠉⠀⠂⠘⠛⠛⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠸⠶⣤⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⢷⠃⡏⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⢢⠄⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⢦⠀⠁⣤⠀⠀⣤⣠⠄⠀⠰⠬⠀⠀⠄⠀⢙⡿
⠀⠀⠀⠀⠀⠀⠀⠉⢻⠄⡟⠀⠀⠀⡄⠀⠀⠀⠀⠠⡄⢛⠂⠘⠴⢆⠙⠆⡀⠀⠁⠀⠀⠀⠇⠀⠀⠘⡄⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⡀⠀⠀⠀⠀⠘⠀⡒⠙⠂⠀⠀⠀⠄⠠⡙⠛⠀⠀⠀⠀⠀⠀⠐⡀⠀⠰⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡌⠀⠀⠀⠀⠀⠀⠆⠡⠀⠀⡄⠘⠀⠄⡀⠀⠻⣨⠆⠀⠴⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣀⠁⢀⡄⠀⠀⢂⠀⠀⠀⠁⠂⡀⠀⢈⠆⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⣘⡣⠀⠠⣅⠀⠒⠀⠂⢰⡫⠀⠀⢈⡅⠌⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠠⢂⠈⠁⢀⠀⠑⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⠀⠀⠈⠀⠈⠀⠄⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠀⠀⠟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠅⠀⠀⠄⠀⠂⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⣩⠀⠀⡄⠀⡀⠀⠞⠀⠀⠰⠀⠀⠁⠀⠂⠀⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠒⠀⠀⠰⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡇⠀⠀⠀⠀⠀⠆⠀⠀⠀⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠠⠠⠀⠀⠠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⢀⠀⠀⠀⠀⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣈⠀⠶⠚⠚⠀⡂⠀⠘
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠲⠆⠀⠀⡘⠁⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⣤⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠁⠋⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠊⢠⠁⠀⠀⠀⠀⠂⠀⠀⠐⠰⠄⠀⠀⠀⡀⠄⠀⠀⠈⠀⠀⢀⡠⠁⠂⠀⠀⠀⢰⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠄⠀⠀⡠⠀⠀⠀⠀⠂⠠⠈⠁⠀⠀⠠⠄⠀⠀⠀⠀⠀⠙⠃⡄
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⢄⡀⠀⡐⠀⠀⠄⣀⠴⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁
⠀⠀⠀⠀⡠⠀⠤⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⡘⠀⢀⠈⠘⠧⣄⠀⠘⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠐⢀⡽⣀⠀⠀⠀⠀⣠⡴⠂⣀⠄⠁⠀⢩⠄⢀⡀⠁⠄⠀⠈⠰⣄⡈⢷⡄⠀⠀⠀⣰⠠⢀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⢠⣦⠀⠀⠀⣿⠠⡈⢿⣄⡀⠤⠀⠀⠀⠁⠀⠀⠀⠒⣤⠋⠁⠨⣇⠀⠀⠠⠔⠋⠀⠈⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢃⠀⠀⢹⡀⠈⠀⣈⡗⢂⠀⠀⠀⠀⠀⠀⠠⡀⠒⠖⠀⢸⠇⢀⣴⠎⠁⠀⠀⠀⣀
⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡶⠀⠀⠃⢀⣀⡬⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠑⣀⡔⠋⣁⠐⠀⠀⠀⠀⠀⠀⠁
⠀⡘⠀⠀⠀⠰⠁⠀⠀⠀⠀⡘⠀⠀⠁⢐⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢚⠣⠀⠴⠫⠄
⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠄⠒⠂⠀⠀⠀⠂⠀⢀⠀⣀⠀⠀⠂⠢⠀⠀⠀⠠⠀⢀⠀⠀⠀⢀⠀⠀⠈⠀⣀
⠀⢨⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠉⠀⠂⡃⠀⠀⠀⠀⠀⢀⡅⠂⡱⠀⠄⠀⠀⠐⠑⠀⠀⢀⡀⠈⠑
⠀⢉⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠈⠀⣀⠀⠀⠀⠀⠀⠀⠲⠁⠈⠀⠠⠉⢀⠀⠀⠸⠀⠀⠀⠁
⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣶⣼⣿⣷⣤⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⣠⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣷⠞⠉⠀⠀⠀⠀⣀⣀⣀⣀⡀⣠⣿⣾⣿⡟⠛⠋
⠀⠀⠀⠀⠀⠐⠛⠛⠉⠉⠁⠀⠀⣤⡤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⢸⣿⠟⣽⣿⠋⡙⠻⢿⣿⣿⣿⣿⡅⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢃⣼⠟⠁⠀⠁⠀⢸⣿⣀⣀⣽⡇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⣿⠁⠈⣿⠀⢀⡘⣶⣸⣸⣿⠉⠁⠀⢀⢸⣠⢰⢀
⠀⠀⠀⠀⠀⠀⠀⠐⣿⣼⣷⣾⣷⣧⣾⣶⣿⣦⣤⣿⣿⣿⣿⣿⣴⣶⣦⣿⣾⣯⣿⡏
⠀⠀⠀⠀⠀⠀⠐⠒⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠓⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣾⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⣴⣧⣴⡄⠀⢀⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣾⡿⢿⣿⣧⠀⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣶⣍⠛⠛⠻⢿⣿⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠹⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⣿⣿⣿⣿⡏⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⣴⣿⣷⣤⣄⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠟⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠻⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠉⠉⠁⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⢀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣴⠿⠿⠋⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣿⣤⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⣀⢸⠀⠀⢀⣿⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢻⠟⠉⣿⠉⠙⡿⠁⠀⠀⠀⠀⠀⠀⠀⢿⢿⠖⠀⠨⣿⠅⠀⠀⠺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⣄⠀⡀⠀⣠⡀⠀⢸⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣇⣀⠀⠀⠀⢿
⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⢿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⢸⠀⠀⠀⡟⠀⢠⣤⣿⣤⣀⠀⠀⠀⢹⣿⣿⣿⢏⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠋⠀⣿⠈⠙⠁⠀⢸⠀⠀⠀⡇⠀⠈⠉⣿⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⠂⢺⡗⠂
⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠺⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⢨⠀⠀⠀⣿
⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⢿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠾⠀⠀⠀⣀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠙⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠛⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡾⢀⣴⡿⠋⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⢿⡏⢿⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣀⠉
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠻⢻⣿⠿⠿⠿⠿⠿⠿⠿⠛⠿⠿⠗⠒⠓⠀⠀⠀⠀⠀⢹
⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⠀⠀⣠⣀⣀⡀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣆⢀⠀⠀⠀⠀⠀⣀⡴⠞⣋⣭⠥⠤⠀⠀⠤⠬⢭⣝⡛⠶⣤⡀⠀⠀⠀⠀⠀⠀⣼⣷⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣶⣶⣶⡿⠋⢴⣿⣿⣿⣿⣿⣧⣰⣿⣿⣿⣿⣿⣷⠌⠻⣿⣶⣶⣶⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡝⢿⣿⣿⣿⠏⣠⠞⣿⠿⣛⣩⣭⣽⣿⣿⣯⣭⣭⣙⡻⢿⡛⢦⡈⢿⣿⣿⣿⠿⣣⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠄⠈⠉⠁⠚⠁⠘⠁⠾⠛⠋⣉⣭⣿⣿⣯⣍⡙⠛⠻⡆⠙⠀⠙⠈⠉⠉⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡃⣠⡖⠀⣼⡀⡀⠀⢀⡴⠋⠉⢀⣿⣿⡄⠈⠉⢶⣄⠀⢀⠀⣷⠀⠐⣦⠀⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⡟⣠⡆⢻⣷⣿⢿⡀⠀⢰⡆⠞⢋⡏⢻⠀⡄⠀⠀⣼⢿⣶⣿⢨⣧⡈⢷⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡟⢰⣿⠇⣹⠷⣬⣬⣑⡀⠘⢷⣉⠘⠃⣈⡼⠃⡀⣊⣥⣮⠾⣏⡀⢿⣷⡈⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣽⣿⣿⠃⣿⢿⡆⣟⢷⡽⠢⠉⣿⣿⠛⠙⠙⠸⠙⠛⢿⣿⡍⠔⠯⣾⢟⡄⣿⣿⡇⢸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠋⢡⣿⣿⣿⠘⣿⠈⡿⣿⣶⣴⣞⠃⠈⢙⠉⠡⠀⠀⠌⠉⠚⠉⠀⢻⣶⣴⣿⡾⡏⢸⡇⢸⣿⣿⡿⡆⢸
⠀⠀⠀⠀⠀⢀⠞⠉⠀⠀⠀⠀⣿⣿⣿⡈⣿⣆⣻⣿⣿⢯⣘⢦⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⣡⢏⣬⣿⣿⣷⣃⣿⡇⣼⣿⣿⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣼⣿⣷⣾⣿⣷⣌⡳⠅⠰⣄⡈⠢⠜⢀⣠⠆⠠⢖⣫⣴⣿⣿⣶⣿⣿⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣶⢻⣿⣶⣭⣒⠀⠙⠿⣶⣶⠿⠋⠀⢐⣯⣵⣾⣟⣷⣾⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣾⣿⣯⣼⣿⣿⢶⣬⣤⣤⣿⣿⣤⣤⣤⡶⣾⣿⣿⣬⣿⣷⣶⣿⣿⣿⣿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⠙⣿⣿⣿⣿⣿⣿⣿⡀⣸⣦⣶⣶⣶⣶⣤⣧⢀⣿⣿⣿⣿⣿⣿⣿⠟⠙⠉⠁⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠃⠈⢻⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⠛⠁⣹⠇⠀⢀⣴⣣⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣣⢄⡴⢃⣾⣴⠿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠿⠈⠉⠀⢴⣿⣿⣿⣿⣿⠋⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣙⡷⠲⠶⠺⣟⣼⣿⣿⡿⠃⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠿⢛⣿⣿⡟⠻⢟⣿⡟⠁⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣩⠟⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⣿⣿⣿⡿⠋⠔⠋⠠⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⢐⢠⠄⠀⠀⠀⠀⠀⠆⣰⠂⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠅⠀⠀⢈⠈⢠⡀⠶⠿⠂⢠⡄⣸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢠⡄⠀⠘⠀⠀⠂⠀⠀⠂⠀⠀⡀⢀⠹⠇⠀⠀⠈⣆⣨⣤⠴⠶⢤⣤⡄⠋⠀⠀⠀⠈⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠀⠀⠀⢀⢀⡃⠀⠀⠀⠒⠀⡈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢠⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠄⠈⠉⠁⢀⠀⠀⠀⠈⠉⠉⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠴⢀⠀⠀⠀⠐⡠⡶⠀⠀⢀⡀⠀⠠⠀⢠⡤⣤⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⢱⣴⠆⠈⠁⠀⣀⠸⠡⠀⠀⠀⠛⣰⠾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠙⠠⠀⠁⠀⠈⠀⠀⠀⠀⡔⠈⠇⠀⠀⠀⠀⠄⠀⠀⠀⠀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠃⠈⡴⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⡀⠀⡀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢴⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠄⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⣀⣷⡀⠀⠐⠢⠀⠀⠀⠀⠀⡴⣦⡀⠄⠀⠀⠀⠘⠀⢛⡑⢄⠀⠀⠀⠠⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠈⠉⠓⠀⣀⠀⠀⠀⡀⠀⢰⡿⡁⠀⠀⣠⠀⠀⠤⠛⠱⢀⠑⢄⡀⠄⠈⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⡐⢋⠁⠃⠀⠀⣤⠂⠀⢀⠀⢳⠈⠳⣀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⠄⠀⠃⠀⠀⠧⠀⠀⠀⢄⠀⠠⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡖⠀⠒⠒⠚⠁⠀⠀⠎⠀⠸⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠐⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡀⠀⠂⠀⠠⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⡏⠀⠀⢀⠁⠀⠱⣈⠓⢁⣠⠀⠀⣠⠀⠀⠀⠀⠀⠀⠈⠒⠂⠀⣀⡀⢠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣀⠀⠈⠃⠠⡤⠀⡞⠁⠀⠀⠀⡿⠀⠀⠀⣽⠆⠙⠇⠀⠀⢠⡀⢀⣄⡀⠀⠀⠀⣀⢀⠈⠃
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠸⠀⠀⠇⠀⠷⠁⠀⠀⠀⠀⠀⢶⠀⠀⣠⠇⠀⠀⠀⡄⠀⢸⠟⠉⠃⠀⠀⠀⠀⠀⠀⠤⠄⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠀⠀⠀⠰⠀⢠⠁⠀⠀⠀⣀⢿⠀⢐⣆⣶⣤⣤⡶⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠏⢉⡿⠂⠀⠀⠀⠀⠇⠀⠀⠀⣠⠤⢾⠆⠉⠀⠀⢀⣀⣁⠀⠀⠀⣀⠖⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠠⠀⢠⠴⠋⠀⣼⠃⠀⠀⠀⠀⢰⠀⠀⢀⠀⠥⠂⡠⠟⠋⠀⠀⠊⠀⠉⠀⠀⠀⠁⠀⢀⡄⠃
⠀⠀⠀⠀⠀⠀⠀⣠⠟⢀⡀⠀⠚⣁⠴⡔⠿⠋⠀⠀⠀⠀⠀⡌⢀⠀⠄⠀⠀⣽⠆⣩⡟⠛⠇⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢠⣐⠁⠀⠉⠀⠀⠠⠄⠐⠁⠀⠀⢀⠀⢲⠀⢠⡥⢤⠤⠤⣶⣞⣋⣀⡗⢡⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⣤⡀⢀⡄⠀⢴⡀⠁⠀⠀⠀⡀⢀⡆⡙⠀⣴⡀⠀⣤⣈⠉⢨⣿⠉⠁⠘⡄⠈⠄⠡⡀⠀⠀⠀⠀⠀⠀⠁⢤⠀⠄
⠀⠀⠀⠀⠈⠀⠉⠀⠤⠐⠃⠈⠁⠂⠀⠀⡀⢡⡌⠀⠁⣠⠿⠁⠀⣍⡙⠷⣼⣇⣴⡶⠂⢙⡤⠐⠀⠈⠀⠤⠀⠀⠀⠀⠀⠀⠙⡄⢠
⠀⠀⠀⠀⠀⠀⢤⠀⠢⠐⠈⠰⠴⣤⣄⣠⠄⠙⠳⠀⡉⠁⠀⠐⠓⠀⠈⠂⠀⠉⠹⡆⠀⠢⠄⠀⠀⠀⠦⠀⣀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠰⢀⡒⠀⠀⠀⠄⠀⢠⠀⠀⠀⠀⠔⠛⡓⠆⠀⠀⣠⣴⡣⠖⠁⠀⣥⡀⠀⠘⠀⢲⠄⠀⠀⠃⡀⠀⠀⠀⠀⠰⠁⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠄⠄⠈⡀⠀⠈⡜⢀⣠⠀⠀⣉⣴⠔⠊⠸⣿⠶⠶⢷⣦⡀⠉⠰⠀⠀⠞⣿⡀⠄⠀⠑⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡁⢠⡞⠉⠀⠀⠀⠀⠀⠂⠀⠈⣻⣿⣦⣄⡀⠀⡀⠈⠻⡄⡀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣷⡞⠁⠀⠐⠂⠀⠀⠀⠉⠀⠀⢡⣄⠺⣿⣯⣻⠁⠀⠀⠈⠁⠒⠂⠀⠚⠉⠁
⠀⠀⠀⠀⠀⠠⠄⠀⢀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠨⣿⠀⠦⠠⠀⣄⣀⣀⡀⣀⣀⣀⠸⠷⣾⠛⣻⠀⠀⠀⠀⠀⠀⡀⠀⡄⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠄⢀⡀⠂⣤⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣰⠋⠀⡏⠀⠀⠀⠀⠀⠀⡄⢓⡀⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡄⢘⡧⢤⣀⠀⡄⠀⢦⡀⣠⠀⠀⠀⠀⠀⠀⣠⣿⣤⡧⢠⡾⠀⣀⢀⡀⣄⠾⡁⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠝⠦⠀⠠⠆⠙⠤⠀⣀⠀⠀⠀⡠⢉⣰⠟⠴⠃⠰⠠⠀⠴⠋⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠚⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⣀⣘⣻⣷⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣯⣤⣴⣂⣠⣾⠛⠉⠈⠉⠉⠉⠉⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠉⠉⠀⠘⠛⠋⠁⠀⠀⣠⡇⠀⣰⡾⠛⠛⠦
⠀⠀⠀⠀⠀⠀⠀⢀⣸⣰⣿⠀⠀⡠⠶⠿⠿⠿⢶⣶⣶⠿⠛⢀⣼⣿⣷⠀⢀⣠⣤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠏⠀⠀⣀⣠⣤⣤⣶⣦⣭⣽⣛⡛⠛⢉⣼⠟⠚⠉⠉⡉⠉⠛⠛⠉⠁
⠀⠀⠀⠀⣀⠀⢸⡟⡿⠁⣀⣴⠟⠋⠁⣀⡤⠀⠀⠀⠈⠉⠛⢶⣏⠁⣠⠶⣶⠾⢃⣴⠞⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠸⣷⡌⠇⢀⣼⠋⠀⢀⣴⠞⠉⠀⠀⢀⠄⠀⠀⠀⠀⠉⢿⣤⣾⣁⡴⣿⠇⠀⠀⣦⠀⢀⣾⠋⠉⠉⠃
⠀⠀⠀⠀⠀⢻⣷⢀⡾⠁⢀⡴⠛⠁⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠒⠻⣆⠀⠀⠙⣷⣶⣶⣿⠀⣸⣿⣶⡀
⠀⠀⠀⠈⢻⣶⡝⣿⡇⠀⠊⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣶⠶⢿⣿⣿⣯⣤⠾⠛⢹⡿⠁
⠀⠀⠀⠳⣤⣿⣿⣿⡀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⢀⡴⠃⠀⠘⠳⢶⣯⢹⡇⠀⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⢀⣈⠙⠻⣿⡉⠁⠀⠀⣠⠟⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠉⠻⣷⣽⣗⠀⠀⠠⠋⠀⠀⠀⢠⡿⠁⠀⠉⠛⠳⠶⠦⠄⢀⡿
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣀⠀⠀⠀⠀⢠⡟⢀⡀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣍⡀⠀⠀⠞⠀⠀⠙⠳⠶⣤⣀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣦⣤⣄⣀⣀⣤⣤⡶⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠶⠶⣶⣒⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣯⣿⣭⡻⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⡿⠋⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣇⠀⠀⠀⣀⣀⣀⡀⣀⣀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⣸⣿⣿⣿⣿⣿⣿⣷⣶⣎⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠒⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠐⠂⠤⢄⣀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⢦⣀⠀⡠⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠉⠒⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢎⡟⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⣠⣴⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠤⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡲⠤⡤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⡀⠀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠒⠐⠚⠛⠓⠒⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣞⣾⢁⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⠀⠀⠀⠀⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢻⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⡠⡏⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠘⠃⠀⠃⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⡐⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⡀⠉⠉⠈⠈
⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⣆⢆⡀⢠⡄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⡆⠀⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣷⡄⠈⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡈⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠃⠀⠀⠀⠀⠐⠦⠄⠆
⠀⠀⠀⣀⣠⡀⠀⣀⡀⠈⠉⢀⡠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⡀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠴⠶
⡄⠀⠀⠈⠁⠁⠀⠁⠀⠀⠐⠊⠀⠀⠀⠀⠀⢤⣀⢀⠐⠲⠶⠀⠀⠀⠈⠉⠉⠉⠁⠀⣤⠚⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⢠⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⡦
⠀⠉⠀⡀⣀⣀⠰⠶⠃⠀⠀⡀⠀⠀⠤⢠⡤⠀⠉⠉⠀⠀⠀⠠⠷⠷⠀⠀⠀⠀⢀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠘⠉⠀⣠⠤⡠⠤⠀⠄
⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣴⣶⣦⣤⣤⣤⣤⣤⣤⣾⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣀⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⡿⠯⠁⠀⠀⠀⠀⠈⠉⠻⣿⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠋⠹⢿⣿⠞⣺⣶⡆⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣯⡶⠳⠟⢋⣡⣶⣿⣿⠿⠀⢀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⡟⠁⠀⡾⠿⠿⠿⠛⠁⠀⠀⠈⠀⠀⠀⠇⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖
⠀⠀⠀⠀⠉⠀⠘⠱⠎⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⣴⣾⣶⢦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣴⣿⡏⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢎⣇⢀⣰⣟⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡾⠟⠛⢿⡹⢿⣿⣿⢉⣁⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣯⣥⣤⣬⠠⠍⠘⠏⠹⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢤
⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣦⣄⣠⣤⣀⣀⣠⣿⣿⣿⣿⡛⠿⡿⠛⠉⠙⠛⣿⣟⠀⠀⠲
⠀⢻⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⢿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠘⠿⠄
⠀⠈⣧⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⠻⣿⣿⠿⠉
⠀⠀⠘⣷⡀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣟⣥⡶
⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⡿⠟⠃⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣕⡣⣌⡻⣷⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠉⠓⠿⣾⣧⣄⣀
⠀⠀⠀⠀⠀⠀⢿⢤⣀⠀⠀⠀⠀⠈⠙⠛⠯⣷⡲⠤⣄⡀
⠀⠀⠀⠀⠀⢀⡎⠀⠈⠙⠓⣦⠀⠀⠀⠀⠀⠀⠉⠙⠲⢭⣑⣲⡄
⠀⠀⠀⠀⠀⢸⣄⡀⠀⠀⠀⡟⠑⠻⢶⣦⣄⠀⠀⠀⠀⠀⠈⠉⠓⠾⣴⣢⠄⣠⣀
⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⣼⡃⠀⠀⠀⠀⠉⠑⠢⠄⣀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣝⡲⠤⣀⡀⠀⠀⠀⠀⢩⠖⠰⣒⠢⢤⣀⠀⠀⠀⠈⡙⢓⠶⣭⣓⠢⢤⡀⢀⡠⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣍⠓⣦⠀⠀⠛⢤⣴⣿⡹⠓⢢⣽⡀⡀⠀⡇⡜⠀⡇⡎⡙⠲⠿⣿⣖⠺⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠙⠶⣤⣀⡀⠀⠉⠉⠓⠚⢀⡞⠉⠛⢥⣇⣠⣧⣇⣿⢸⢠⡇⡎⠉⠓⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢸⠃⠈⠉⠙⠓⠲⣼⣆⠀⠉⠲⢄⣻⠍⢻⡀⠺⠿⢼⣸⡇⣧⠀⢰⣇⣀⣨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡟⠀⠀⠀⠀⢀⡜⠀⢈⣽⠀⠀⡞⠈⠀⠀⠙⠒⠲⠚⠛⠯⣟⡤⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣧⡀⠀⠀⠒⠛⠒⠚⠋⢹⡄⠘⣧⠀⠀⣀⡀⠀⠀⠀⠀⢀⣠⣇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢬⣷⣦⣄⣀⠀⠀⠀⠀⢸⡇⣸⡋⠀⣿⣋⣽⠀⠀⢀⡔⣫⣜⣁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣶⠦⣤⡾⠹⣭⠗⠀⠀⠉⠀⠀⠀⡞⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⡆⠀⣇⡀⠀⠀⠀⠀⠀⢸⠇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠇⠀⠀⠀⠀⠀⢸⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣼⠀⠀⠀⠀⠀⠀⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡏⠀⠀⠀⠀⠀⠀⢸⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠁⠀⣴⢛⡷⣄⠀⢸⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣞⡓⠿⣤⣀⠘⠿⠶⠃⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠾⢭⣟⣒⣶⣀⡼⢁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡴⠶⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠷⠶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠿⠋⠉⠀⠀⠑⠲⠤⣄⠀⠀⠀⠀⠀⠀⠀⢀⡤⢾⠟⠒⣻⠙⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠋⠁⣖⠒⠒⠊⠉⠛⢶⡀⠀⠘⣧⠀⠀⠀⠀⣰⠋⠉⠀⠈⠙⠉⠀⠀⠀⠀⠀⠙⠷⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢛⠁⠀⠻⠦⠶⡿⢆⡀⠀⠀⠀⣸⠇⠀⠀⠈⠳⢦⣄⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⢦⡙⢲⡝⢷⣄
⠀⠀⠀⠀⠀⠀⢀⣤⠟⠉⠑⠚⠀⠀⢀⣀⡴⠃⢠⡿⣿⣿⠿⣥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠈⡆⢙⣦⡿⠀⠙⢦⡀
⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠘⢿⣍⠀⢀⡾⠁⠀⠀⠀⠀⠉⠉⠳⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢺⡟⠁⠀⠀⠀⠙⣦
⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠉⠁⣰⠾⢻⣶⡦⣈⢷⡀
⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠒⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣀⣿⠀⠀⠻⠇⠀⠉⢷⡀
⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⡝⠳⠴⠞⠙⢦⣀⡤⢼⣧
⠀⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃⠀⠀⠀⠀⠀⠉⠀⠀⢻⣇
⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⣾⡏⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠇⠀⠀⠀⠻⢦⣼⡀⠀⠀⠀⠀⢀⡤⠙⠲⢦⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣿⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⣟⠀⠀⠀⠀⠉⠋⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣰⠿⠛⢷⡄⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠙⠶⣤⡴⠛⣇⠘⠷⣤⣈⣉⣉⣙⣻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣅⠀⠀⠀⠙⣆⠀⣼
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣀⣀⣀⠀⠙⠂⢦⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢢⠀⠀⠀⠺⣰⣿
⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⣙⡦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠞⠀⠀⠀⠀⣿⡿
⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣦⠉⠐⠒⠒⠶⣤⣀⣀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⡴⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⣧⠘⡄⠀⠀⠀⠀⠀⠀⠀⢻⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⡄⠀⠀⠀⠀⠀⢀⣠⡿⠁⠀⢀⣤⣶⣫⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⣴⠟⠛⣋⣤⣶⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⠷⠶⢶⣶⣿⣤⣤⣾⣷⣾⡿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⡶⠶⠶⠶⠶⢶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⣀⣤⣤⣴⣶⣶⣤⣤⣄⡀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⣠⡶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⡀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⢰⣟⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣸⣷⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣴⡶⠟⠛⠉⠉⠉⠉⠉⠉⠙⠛⠷⢶⣤⣴⠾⠟⠛⠉⠉⠉⠉⠉⠉⣿⡆⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⣠⣴⠟⠋⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⣿⡇⠀⠀⢸⡿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢀⣠⣶⢾⣟⠛⠉⠉⠙⢛⣿⠏⠁⠀⠀⢀⣤⡶⢾⣟⠛⠉⠉⠙⢛⣿⠀⠀⠀⣼⡇⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢀⣴⣿⠉⠀⠀⢿⣆⠀⠀⣰⡿⠁⠀⠀⢠⣶⣿⠁⠀⠀⠙⣷⡀⠀⢀⣾⠇⠀⠀⢠⣿⣄⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⣰⡟⠁⢻⣆⠀⠀⠀⠻⣦⣰⡟⠀⠀⠀⣰⡟⠀⠘⢷⡀⠀⠀⠘⣷⣠⡾⠃⠀⠀⢀⣿⠃⠙⣷⡀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⣰⡟⠀⠀⠀⠹⣧⡀⠀⠀⠙⠻⣧⣄⠀⣸⡏⠀⠀⠀⠈⣿⡀⠀⠀⢹⣟⠀⠀⠀⣠⡿⠁⠀⠀⠘⣷⡀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⢸⡟⠀⠀⢀⣿⠁⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠙⠛⠿⠶⣦⣤⣤⡶⢾⣧⠀⠀⠀⣿⠀⣠⣾⠏⠀⠀⠀⠀⠀⢹⡇⠀⠀⠘⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠛⠻⢶⣦⣤⣤⣤⣤⣤⣴⡟⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⢻⣆⠀⠉⠀⢀⣾⠃⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠈⢻⣆⠀⢀⣾⠋⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠙⢷⣿⠁⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠙⠻⢶⣤⣄⣀⣀⣀⣀⣹⣷⣄⠀⠀⠀⠙⠻⢶⣴⣟⣁⣀⣀⣀⣀⣤⣴⠾⠛⠁⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠈⠉⠉⠛⠛⠉⠉⠉⠛⢷⣄⡀⠀⠀⠀⠈⠉⠉⠛⠛⠋⠉⠉⠀⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣀⡀⠀⠀⠀⠀⠀⣀⣀⣤⡿⠿⠷⣦⣄⣀⡀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿