Ruminate Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣶⣤⣀⣴⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡃⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⠃⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠀⠉


⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠈⠁⣾⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠰⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠰⠀⠘⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⡄⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠁⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠟⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠉⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⢠⣖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠄⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠶⠄⠀⡀⠀⣀⡴⠁⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠛⠉⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⡄
⠀⠀⢸
⠀⢀⠇⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠒⠢⢤⣀
⠀⡌⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠈⣆⣀⣀
⠀⡇⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⡄⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⣄
⠀⢳⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠈⠢⡘⡄⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⢸⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠘⢎⠳⢄⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿
⠀⢸⠀⠀⠀⠀⠁⠁⠀⠈⠛⠲⢤⣤⣤⣤⣤⡈⢣⡈⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠉⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡀⠴⠦⣀⣀⣠⣀⡠⠔⠋
⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣀⣀⣀⠔⠁
⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⢀⣀⠤⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣥⣬⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⠤⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢀⣠⣾⣿⡿⠿⠂⢀⣀⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠙⠂⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣠⣮⣾⣿⠿⠋⠉⢤⣰⣿⣿⣿⣿⣿⣷⣄⣀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣠⣼⣯⣿⣶⣿⣿⣿⣻⣃⣴⡶⠷⠶⣾⣿⣿⣿⣽⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠶⠶⠶⣾⣿⣿⡿⠿⠿⢿⠟⠿⠿⠛⠉⠁⢨⡀⠀⠀⢀⣯⣽⣿⣿⣻⣿⣯⣿⣿⣿⣿⣆
⢠⠓⠀⠀⠀⠀⢶⣰⣾⣶⣶⣿⡞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⢇⠓⠀⢘⣥⣿⣿⣿⣯⡻⣿⢸⣿⣽⣿⣿⡆
⠀⢀⠀⠀⠀⠀⢨⣿⣿⣿⣿⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠘⢦⣄⣀⣠⣿⣿⣶⡘⢧⣽⢿⣿⣿⣿⣿⡇
⠀⠸⠏⠀⢹⠒⢿⣿⣿⣿⢇⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⢘⠁⣱⣿⣿⣯⣥⣤⣈⢻⠼⠶⠶⣶⣽⣿⣿⣆
⠀⠀⠀⢀⠀⠀⢘⣿⣿⣿⠫⡇⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠠⣰⣿⣿⣿⣫⡔⠋⠿⣷⣤⡶⢶⣶⣽⣿⣿⣿
⠀⠒⣤⣴⣶⣶⣴⣿⣿⣿⡇⢷⠀⠀⠀⣠⣴⣾⣶⡄⠀⠀⠀⣀⢀⣀⢘⡆⠀⠀⢀⡠⡀⢠⣿⣿⣿⣿⡟⣠⣾⣧⡑⠹⣿⠿⣿⣿⣿⡟⠛
⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣧⣼⡿⣷⣿⡟⣿⣿⣿⣿⣿⣶⣶⣿⣿⣶⣾⣷⣤⢲⣿⢹⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷⢠⣿⣷⣮⣿⣿⣿
⠀⠉⠁⢙⣓⡻⣤⡀⠈⠉⠉⠛⣿⠟⠉⠛⡹⡋⠛⠛⢿⠟⠟⠻⠛⠿⠿⠿⡛⠘⢻⢘⡿⢿⠟⠧⠿⠿⣿⣿⣿⡿⠿⢿⡿⠟⠟⠟⠿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣴⣶⣷⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣿⣿⣇⣻⣿⣿⣷⣿⣿⣿⢿⣿⣿⣿⢿⡿⠿⠶⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠹⠟⠛⠉⠉⠛⠙⠛⠛⠉⠉⠩⠆⠉⠉⠉⠉⠁⠀⠁⠁⠀⠀⠀⠀⠀⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢶⠀⠆⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠁⠆⠈⠀⠀⠀⠀⠂⢀⠀⠀⠉⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠠⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢀⠀⠒⠀⠀⠀⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠐⠂⠐⠨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡄⠁⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣂⣦⣆⣀⣆⣶⣷⣿⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠂⠀⠂⠀⠀⠀⢀⠂⠀⣤⡀⢀⡀⠐⠒⣾⠛⢻⣋⠉⠙⠻⠿⣏⠹⣏⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⢀⠀⠀⢂⠈⠀⠀⠘⣤⡴⢶⣶⡿⠋⢳⠟⠾⢩⣴⣶⣌⡻⠋⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠐⠁⠌⠀⠂⠈⠁⠀⢼⠿⠁⠴⣿⣿⣷⡾⣧⠰⠀⠈⠛⠛⠋⠉⠰⢦⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⢐⢂⡀⠀⠴⠀⠐⠁⠄⠓⠄⢀⠀⡀⠀⢀⣀⣿⠷⡻⠷⢀⠰⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠌⠣⠙⠃⠨⠿⣝⢦⡰⣄⡿⠢⡄⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄
⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⢰⠀⠀⠘⠐⠁⠈⠀⡐⠀⠈⣂⠐⠌⠠⡄⠈⢿⡀⠓⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄
⠀⠀⠀⠀⡰⠁⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠐⠀⠀⠁⠀⠄⢢⠀⠏⠶⠈⡰⣆⣄⠉⠲⡀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⠀⢸
⠀⠀⠀⠠⠁⠀⠀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠂⠈⠀⡀⠀⠈⠂⡀⠁⠙⠸⡉⡳⠄⡶⡇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢪⣵⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠈⠀⠀⠃⠀⠀⠠⠀⠔⠓⠈⡄⢀⢮⡙⢻⣿⡿⣿⣿⠟⠒⠒⠢⠀⠀⣈⣰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢀⣉⣃⢠⡜⣿⣷⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠁⠈⠀⠠⠿⠏⡃⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄⠠⠞⠟⠖⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡦⠀⠀⠂⠈⠹⢆⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠐⠐⢦⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⣤⣴⣦⣴⣤⣤⠺⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⡿⣿⢿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⢶⣶⣶⠶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣾⡿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣧⣥⣬⣴⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣇⣀⣠⡀⣤⢰⣶⠶⣲⡶⣠⣄⠠⣤⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠧⢻⣆⠲⣿⡿⣱⣿⡸⣿⡶⢡⡿⠱⠷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣸⣷⣙⢷⣬⣐⣛⣛⣃⣭⡶⢟⣵⣿⣰⣽⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⢱⣮⣭⣍⢋⣭⣥⡦⠙⣻⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣥⣰⣽⣿⣷⣷⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠉⠈⠉⠉⠉⠉⢙⣿⣿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⠿⢿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⡿⠿⢿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡅⢨⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣯⠀⣨⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣿⡻⠟⣟⣿⣿⡏⠉⠉⠉⠙⠻⠿⠆
⠀⠀⠀⠀⠘⠉⢵⡦⢵⣤⣦⡤⢀⣶⣿⣿⣿⣿⣿⣿⣿⡏⠓⠂⠘⢿⣿⣿⣶⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣈⣟⠧⠘⠋⠈⠛⠻⢿⣿⣿⣿⣷⡀⢀⠀⠈⠛⢻⣿⣿⣿⣿⡆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣈⣉⣍⠀⣀⣤⣶⣶⣾⣍⣿⣿⣿⣿⡇⢨⣧⠀⠀⠀⠿⣋⣿⡟⠃⣿⢷
⠀⠀⠀⠀⠀⠀⠀⠘⢭⠬⠗⠊⠉⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣼⣿⡷⢶⣮⠤⡾⢻⡇⠀⠹⠈
⠀⠀⠀⠀⠀⠚⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠉⠻⠛⠀⠁⠀⠀⠀⠀⢀⡞⠁⡼⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣴⣾⣾⣦⣄⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠋⣿⣿⣿⠛⢋⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⣹⣿⣧⣴⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣿⣿⣿⣿⣿⣫⡴⠋⢈⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡀⣿⠿⢻⠉⠱⠤⡟⠾⠛⣈⣹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⣠⡀⠀⢀⣄⢠⣾⡅⢐⠀⠁⡠⢾⣇
⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠈⣧⠀⡀⠚⠁⠐⠋⠀⠀⠸⠋⠈⠀⠈⢋⣠⣆⣻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠀⢠⣤⣤⣰⣿⢿⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣛⠿⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣦⣴⣴⣻⣿⡁⢬⠀⠀⣀⠀⠀⢠⢴⠀⠀⢠⣴⣿⣿⣭⣭⢿⣛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⣿⣿⣯⣠⣅⠀⡝⠁⣶⡟⠚⢿⣧⣾⣿⣿⣿⡿⢏⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣶⣿⣼⣿⣷⣿⣿⣿⣷⣬⣷⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣯⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⡽⣥⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣯⢀⣼⣇⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣣⣼⣿⡟⢻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⣬⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠁
⠀⠀⢤⣤⣴⣾⣷⣶⣶⣾⣿⣿⠀⠀⠀⢠⣤⣤⣴⣶⣶⣦⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣀⣀⣀⣤⣶⣿⣿
⠀⠀⠀⠈⢿⣿⣿⠿⣿⣿⣿⣏⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣆⢀⣎⠄⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⡿⠿⢷⣤⡀
⠀⠀⠀⠀⢸⢿⠃⠀⠀⢸⡏⠈⢳⠀⠀⠀⠀⡾⡟⠁⠀⠀⢠⢿⠁⠈⢻⣿⠇⠀⠀⠀⣠⢾⠿⠛⠁⠀⠀⠀⠀⠀⠀⠈⠛⢷⠄
⠀⠀⠀⠀⠀⡎⡆⠀⠀⢸⠀⠔⠁⠀⠀⠀⠀⣇⢱⠀⠀⢀⠏⠀⡇⠀⠘⠏⠀⠀⢠⠔⡡⠃
⠀⠀⠀⠀⠀⠇⠸⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠋⠘⠀⠀⠘⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣵⣔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣤⡀
⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠈⣿⡷⠶⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠋⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⣾⣿⠉⠁
⠀⠀⠀⠀⠀⠈⣿⣶⣦⣤⣤⣴⣿⣇⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣶⣶⣶⣶⣿⣿⡇⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣠⢤⣾⣿⠿⠿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠿⠿⠿⢿⣿⣿⣷⡆⠀⠀⠀⠀⠘⣿⣿⠛⠛⠛⣿⠏
⠀⠒⣊⠔⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⣀⠴⠋⠏⠉⠀⠀⠀⠀⠀⠀⠹⠋⠀⠀⠀⠀⠀⠀⡟⡇⠀⠀⠀⣿
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡧⠀⠁⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⣠⡆⠀⠀⠀⠀⣄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡀⣠⣶⣶⣶⣾⣿⣿⣿⣿⣇⣼⣿⣿⣿⣿⣿⣦⣤⣀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢹⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⠿⠿⠟⠋
⠀⠀⠀⠀⣠⣶⣶⣆⠀⠀⠀⠘⠻⠿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣿⠏
⠀⠀⠀⠀⢿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠙
⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣴⠀⠋⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣠⡀⠀⣴⣆⡤
⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⣰⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡶⠾⢶⣿⣿⠏
⠀⠀⠀⠀⠀⠘⠛⠛⠋⠀⠀⡀⠀⠀⠀⣿⣿⣿⠿⠛⠟⠻⠛⢿⣿⣿⣿⢿⣿⡿⣿⡟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠔⠀⠀⠀⠹⠿⠁⠀⠰⣦⣤⣀⠀⠜⢿⣿⣷⡿⣶⠏⠀⣿⣿⢸⣿⣾⣿⠟⢿⣽⠿⣿⣿⢋⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣱⠖⠦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣸⠋⠉⠀⠀⠀⠘⠿⠛⠻⠋⠉⠁⠈⠆⠡⢈⠈⢀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⣠⣴⡖⣾⡿⡡⠀⠂⢠⡀⠀⠀⠀⠀⠀⡠⠄⠀⡄⠠⢀⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⣹⠷⣄⣤⣀⣀⣈⣛⣉⣀⣠⣅⣈⣈⠤⠗⠀⠀⠀⠠⣿⣷⠾⠦⠄⠞⡻⠃
⠀⠀⠁⠀⠀⠀⠀⢰⣿⣷⡄⣀⣀⡠⢴⠟⠉⠉⠛⠛⠫⠭⠝⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠙⡷⠆⠰⡎
⠀⠂⠀⠀⠀⠀⢠⣿⣿⡟⠀⠉⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢁⠀⠀⡇
⠈⠺⠾⠀⠀⢀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⢸⡀⢀⠇
⠀⠀⠀⠀⠀⠸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠰⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣓⣼⣄
⠀⠀⠀⠀⠲⣶⣾⡟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⡄⠀⠀⠙⠛⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⣠⣶⢆⢶⣦⣀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡯⠐
⠀⠀⠀⠄⠊⠩⢋⠛⡏⠍⠍⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢀
⠀⠀⠀⠘⠷⠶⠖⠰⠷⠶⠟⠁⠀⠈⡀⣠⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⠛⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⠀⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣃⣉⡠⠀⠀⠸⣿⠆⠀⠀⠀⠘⠛⠀⠀⠈⠋⠀⣤⠩
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠍⠁⢀⡀⠾⠿⠀⣴⡆⠠
⠀⢀⠀⠀⠀⠀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠄⠛⠟⠀⣶⠆⠈⠀⠂
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⡁⠐⠶⠶⠆⣀⣀⠀⠀⠀⠈⢨⡀⣬⠑⠀⠁
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠈⠉⠃⠀⠀⠈⠉⠉⠀⠀⠀⠀⠠⠄⠄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠠⠀⠰⠆⠀⠁⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠾⠂⠀⠶⠷⠃

⢀⡆⠈⢦⣄⣀
⢸⡇⠀⠀⠈⠉⠙⠶⣄
⠸⣇⠀⠀⠀⠀⣀⡴⠾⠷⠶⠶⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤
⠀⠈⠓⠦⣀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣀⡤⠤⠤⠤⠶⠞⠉⣼⢳⠂
⠀⠀⢠⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⣀⣤⡾⢫⡿
⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⣠⡶⠛⠋⠉⢠⡿⠁
⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⣠⡿⠁
⠀⠈⣷⠀⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⠛⠉
⠀⣠⠏⠀⠀⠀⠀⠀⠳⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢄⡀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣄
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⠉⠉⠉⠉⠉⠉⠃⠤⠤⠴⠶⠖⠛⠉⠀⠀⠀⠀⠀⠈⠉⠛⠶⢤⣄⠀⠀⠀⠀⣠⣤⠎⠁⠀⢸
⢇⢠⡀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢠⡤⠴⠋⠀⠀⠀⠀⢸
⠀⠀⢻⢯⣭⠽⣋⡄⠀⣀⣀⠀⠀⢠⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠓⠒⠋⠁⠀⠀⠀⠉⠉⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢀⣤⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠈⢃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠓⢦⣄⡀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⢀⣀⣤⠼⢿⣄⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠠⠈⠉⠛⠒⠂⢿⡀⠀⠀⠀⠀⠀⠀⣸⡇⣀⣤⡤⠶⠾⠛⠉⠀⠀⠀⠹⢦⡀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢀⡟⠉⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⡿⠀⠙⠆⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢠⡇⠀⢸⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⢻⠀⠀⠀⠀⢰⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⡇⠀⣿⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⣈⠀⠀⠀⠀⣼⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡤⠔⠋⢀⡆⠀⠀⢠⡇⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣸⡇⠀⡇⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣤⠼⠁⠀⠀⢸⠁⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠴⠾⠏⠀⠀⢃⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠘⠤⡠⠠⠊⠀⠀⠀⢨⠈⠛⠛⠋⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠉⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣠⠇⡠⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠤⡀⠀⢀⡀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠿
⠀⠀⢠⣤⡀⠀⠀⠀⠀⣠⣤⠠⠐⠈⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⢿⣤⣼⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡠⠁⠀⠀⠀⠰⣿⣿⠉⠙⠓⠀⠀⠀⠀⠀⣴⡞⠿⠃
⠀⠀⠀⢀⣴⣿⣿⣦⠀⠀⠀⣠⣾⣷⠀⠀⠀⠱⡄⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠈⣽⣿⡟⠛⠀⠀⢸⣿⣿⡿⠇⠀⠀⠀⠀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡿⠇⠀⠀⠀⠀⢹⡷
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢺⣿⣿⠀⠀⠀⠀⠀⣾⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠛⠙⠃⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠠⠓⠛⠀⣴⡄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⡄⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⠛⠋
⠀⠀⠀⠀⠀⠄⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢦⡀
⠀⠀⠀⠀⣨⣾⣿⡷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣠⡀⣴⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣷⣿⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠿⠿⠿⠞⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⣿⣿⣿⣿⠃
⠀⠀⠀⠀⢿⣿⣿⣿⡏⠀⠈⢷⡀⠀⠀⡠⠐⠀⠀⠀⢸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⠀⠀⠀⠀⠀⡀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣟⣷⣤⠶⠛⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠠⠀⣀⣀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⢠⣿⣿⡏⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣉⡛⠿⣶⣄⠰
⠀⠀⠀⠀⠀⠙⠋⣠⣄⠀⠛⠣⣨⡍⠙⠛⢀⣤⡄⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠇⠀⠀⠀⠂
⠀⠀⠀⠀⣰⣿⣦⠙⠋⣴⣿⣆⠙⢁⣼⣷⣌⠛⢁⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠉⠙⢋⣭⡛⢆⠉⠒⠐⠈
⠀⠀⠀⠀⣿⣿⣿⡀⢀⣿⣿⣿⠀⢸⣿⣿⣧⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣻⣅⣀⣀⣀⣈⣻⣃⣨⣟
⠀⠀⠀⢰⣿⣿⣿⣧⣼⣿⣿⣿⣦⣼⣿⣿⣿⣤⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣽⣿⣿⣿⣿⣯⣤⣾
⠀⠀⠀⢸⢻⣿⣟⢻⡟⣿⣿⡟⣿⡟⣿⣿⡟⣿⡟⣿⣿⢻⡄⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠯⠭⠭⠽⠿⢿⣿
⠀⠀⠀⠈⠈⠉⠁⠈⠁⠉⠉⠁⠁⠁⠉⠉⠁⠉⠀⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄⡤⢤⣀⡀⠀⠐⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢬⣙⠺⣄⡤⠈⠓⣄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣹⡇⠹⢷⣬⣦⣿⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⠶⠶⡴⠊⠉⠀⠀⠀⠀⠖⣾⣇⠀⡄
⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡄⠀⢻⣿⣧
⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀⡇⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠹⣿⣦⣄⡀⡀⣠⣆⠀⠀⠀⠀⠀⠀⠀⢨⣾⠀⣆⣶⣶⣤⣼⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣮⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣄⡀⠀⠀⠀⣤⣜⣿⣿⣿⡟⠉⠙⢻⢛⡛⠛⡛⣛⡻⡋⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣷⣦⣵⣾⣿⣿⣿⠿⠋⠀⠀⠀⢸⢭⠭⠉⢻⠻⣿⡏⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸⡉⠀⠀⠘⢸⡿⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠠⠀⠀⠾⠀⢀⡴⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⠀⠐⠒⠉⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠛⢯⣴⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⣀⣀⣰⡟⢻⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⣟⠛⣩⣷⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠘⣿⣿⣠⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢉⣇⣙⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣰⣬⣽⣸⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⣩⣽⣿⣿⣿⣿⠁⠹⣿⣿⣿⣿⣿⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠓⣾⡟⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⠟⠿⣷⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣒⢰⣿⡟⠀⣹⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡙⣿⣿⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣹⣾⡟⠀⠀⣿⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣽⡗⣿⣿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠸⣿⣿⣿⣆⡸⠃⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⠀⣠⣾⣿⠁⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣿⣿⣿⣿⠁⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⢀⡨⣇⣿⡿⠐⠒⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢠⠴⣿⣿⣿⣿
⠀⠀⠀⠐⡶⠀⠠⣭⣴⣄⣀⣸⡆⠀⣤⣤⣀⣤⢀⡀⠀⠈⠀⠹⣿⠶⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⣀⣴⣿⣿⣿⣿
⣄⣦⡄⣠⣥⣤⣴⣿⡀⠀⣹⣿⣷⣿⣿⣿⣿⣿⠁⣹⣿⣧⣴⣿⣿⣧⠐⠋⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣿⣟⣿⣿⣿⣿⣿
⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣄⣴⡖⣙⣿⡿⠿⠿⠛⠿⠿⠿⢿⡿⠿⠿⠛⠉⣹⣿⣿⣿⣿⣿⣿⣿
⡿⡿⢻⣿⢺⡟⣿⣿⠋⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⠏⠀⣀⣤⣤⣤⣴⣾⣷⣶⣦⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿
⣶⣶⣼⣿⣿⣤⣾⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⡿⢡⣿⣿⡿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣱⣿⣿⠟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣹⣿⣿⣟⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⢹⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣿⣿⣿⣿⣷⣄⣀⢀⢀⠉⠉⠖⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣶⣠⣾⣿⣿⣿⣯
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠇⠡⡠⡠⢀⡤⡤⣀
⠀⠀⠀⠀⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⣙⠣⣬⡓⠀⠁⠙⠷⣤⡀
⠀⠀⠀⡬⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠈⣷⠆⠁⠀⣄⢸⣦
⠀⠀⠠⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⡀⠀⠀⠀⠀⠀⢰⣿⠇⠐⠾⢶⠄⢠⣼⣿⡇
⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠻⢶⣦⡶⠊⠉⠀⠀⠀⠀⠀⢠⠒⢂⢽
⠀⠀⠘⣬⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠒⣻⡀
⠀⠀⠀⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⣽⣿⣿
⠀⠀⠀⠀⠹⣯⣇⡀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠂⢀⢰⢘⠀⠀⡏⠀⠀⣾⢻⣿⣿
⠀⠀⠀⠀⠀⠈⢻⣗⡧⣄⡀⣠⢀⣠⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡁⠄⣀⢶⣞⣼⡐⣠⣿⣐⣴⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⢱⣶⣤⣿⣪⣿⣦⣿⣿⣿⣿⣿⣿⣿⡛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠭⣿⣿⣿⡾⢿⡿⡿⣿⠿⡿⠿⣟⡯⡎⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡡⣄⠀⢀⡀⠀⠀⢝⣤⣤⣿⣿⣿⡿⠁⠀⠀⠀⣿⣟⠏⠟⠛⡁⠃⢹⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣷⣶⣴⣶⣶⣾⣯⣾⣿⠿⠋⠀⠀⠀⠀⢠⡟⢓⠀⠀⠀⠀⠀⢸⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠘⡇⠆⠀⠀⠀⠇⠀⣾⡷⠀⠀⠀⠀⠀⠀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢀⠟⠁⠀⠀⢀⠄⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠠⠀⠀⠀⠼⠀⠀⣠⠔⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠐⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⢀⣤⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣽⣽⣯⣬⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣯⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡿⠋⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⠉⠉⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠈⠁⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣆⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡷⠀⠀⠠⣴⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣶⣤⣀⣤⣾⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠿⢿⣿⣿⣿⣿⣿⣦⣠⠤⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠈⠛⢿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⢤⣀⠀⠀⠉⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⡵⠚⡛⠲⣯⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠘⢀⣠⠷⠶⢾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠠⠄⠀⠤⠄⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡜⣄⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣨⠁⠠⠞⡅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠋⢠⣀⡀⠙⠦⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⢻⡀⠀⡏⣿⣧⠀⠀⣼⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⡄⠀⠓⢰⠇⣿⠸⠀⠚⠁⢀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⡗⠀⣠⠉⢀⡏⠀⠀⣀⠀⢸⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣀⡼⠟⠋⠁⣠⠞⠀⠱⣄⠈⠳⢾⣧⡀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⣠⡴⠛⠁⠀⠀⠀⠈⠳⣦⣀⠈⠙⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠶⠛⠉⠰⡀⠀⠀⠀⠀⠀⠀⣴⠙⠳⠶⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⡀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⠛⠛⠒⠒⠀⠀⠨⣽⣷⣶⣶⠶⠶⠾⠿⠛⠛⠛⠛⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠀⠈⠉⠙⠛⠻⠶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣉⠛⠻⠶⢶⣤⣤⣄⣀⣀⣀⣤⠤⢴⣒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠠⢤⣤⣤⣴⡶⠾⠟⠛⠉⠉⠀⠀⠀⠈⠙⠛⠳⠶⠶⠶⠶⠶⠞⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⡀⠀⠀⠀⣴⢃⡼⣄⢳⡄⠀⠀⠀⣠⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⣙⠦⡀⣸⢃⠎⠀⠈⢆⢻⡀⡠⠞⣡⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢸⡇⡏⠈⠁⢸⣇⠀⢠⣄⣤⠀⢀⣿⠀⠉⠈⡇⣿⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣛⡲⢦⣸⡀⠀⠀⠀⠸⠀⠀⣿⣧⣿⡇⠀⠘⠀⠀⠀⠀⣽⣤⠶⣚⡋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢻⠉⠀⠈⣷⢄⠀⠀⠀⠀⠀⢹⣿⣿⠁⠀⠀⠀⠀⢀⢴⡏⠀⠈⢹⠃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠸⠀⠀⠀⠘⢆⠁⠀⣠⣤⣴⣾⣧⣿⣶⣤⣤⡀⠀⢁⠞⠀⠀⠀⠸⢸⠃
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣈⢧⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⢏⣀⣀⣀⡀
⠀⠀⠐⢾⡛⢉⣩⡭⠭⠍⠉⠙⢦⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⡄⠀⠀⠀⢠⠞⠉⠉⠭⠭⣭⡉⠙⣻⠖
⠀⠀⠀⠀⠙⢦⡙⢆⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⠏⢸⣿⣿⣿⣿⣿⣿⠈⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⢀⠞⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⣰⣿⣿⣿⣿⣿⣿⣉⡀⢸⣿⣿⣧⣿⣿⣿⠀⣈⣙⣿⣿⣿⣿⣿⣷⡀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢈⣳⣤⣉⡁⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣉⣡⣴⣋
⠀⠀⠀⠀⢀⣴⣞⡉⠴⣄⠀⠁⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠁⢀⡴⠌⣙⣶⣄
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢤⣀⡀⠀⠀⠀⠀⠈⠉⠻⠿⠟⠛⠿⠿⠿⠿⠉⠁⠀⠀⠀⠀⣀⣠⠴⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠚⠋⠉⠁

⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⣠⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣽⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣕⣤⣄⠀⠀⣀⣉⡄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⢿⣿⡿⠟⠋⠉⠁
⠀⠀⠀⠀⢸⣦⠄⡀⡄⢀⣤⣿⡆⠑⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠳⣦⡙
⠀⠀⠀⠀⠀⣿⠶⣍⡏⠀⢼⣿⡷⣠⠛⣥⡐⡤⢶⣺⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⣦⠀⠀⠀⠀⠀⠂⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⣿⠃⢽⡘⢣⣿⣿⣷⡘⣧⣮⠷⠘⣝⢿⣿⣷⣂⣤⡤⠄⠘⣤⠀⢀⡚⠂⠺⠅⠀⠀⠀⠀⠀⠄⠀⠀⠃
⠀⠀⠀⠀⠀⢻⣷⣬⣹⣞⣿⣿⣿⣿⣍⣋⣦⢳⣞⣿⣿⣿⣿⣿⣧⣾⣿⣿⣷⣿⣿⣶⣄⣦⡀⠀⠀⠀⠐⠄⠀⢀
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⡄⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⢯⣸⣚⢀⡠⠄
⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⢿⣿⣿⡏⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⡷⢬⣿⢿⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⢸⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⣤⣡⣞
⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⢸⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣌⣼⠇
⠀⠀⠀⠐⠂⣼⣋⠀⢤⠄⣘⣿⣷⣐⢲⣦⣤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢤⣄⣻⣟⣱⣇⡀⣉⡙⢻⡿⢿⣿⣷⣾⣷⣿⣿⡴⡠⢰⡂⡀⠀⠀⠀⠀⢸⠇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢬⣴⣤⣶⡎
⠀⠀⠀⠀⠉⠛⠛⠘⠃⠉⠀⠀⠁⠉⠛⠋⠛⠛⠻⢿⣿⣻⣶⣭⣄⣶⣇⠀⢢⠄⣾⡀⢸⣿⢻⠀⠂⠒⠦⠎⠳⠎⠛⠻⠋⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠙⠟⠛⠛⣷⣶⣼⣠⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠐⠒⠀⠀⠐⠒⠒⣆⣀⣀⡠⠤⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠐⠒⠀⠀⠀⠤⢄⣀⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠀⠒

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠁⠀⠀⠉⠛⠛⠛⠛⠛⠛⠉⢉⡙⢩⡍⢿⣿⣿⣟⠏⠈⠀⠀⣠⣿⣝⠻⢍⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣄⣲⠄⠀⠀⠀⠀⣀⣶⣶⣦⣤⣤⣄⡴⠼⣯⠿⠄⠰⠓⠀⠛⠛⠛⠑⠐⣿⠟⠻⢿⣿⣿⣿⣟⠛⣀⡈⢙⠻⠟⠛⣿⣿
⣉⣉⣙⣁⡀⠀⠀⠀⠀⠉⠻⠿⣿⣿⠛⠿⣿⣶⡂⠀⠀⢠⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⣀⢀⣀⣀⣐⣻⣿
⠈⠉⠉⠉⠉⠋⠉⠉⠀⠀⠈⠀⠈⠉⠀⠀⠀⠈⠉⠀⠀⢿⣿⣿⣿⠆⠀⠀⠁⠀⠀⠈⠀⠐⠂⠚⠛⠛⠋⠉⠉⢉⣉⣉⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣀⡀⠀⢸⣿⠿⣿⣿⣿⣿⣿⢻⣿⣀⣀⣀⣴⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡐⠀⢀⡀
⠀⠀⠀⠀⠀⠰⡇⠤⠦⠤⠾⢿⣿⣃⡀⠀⠄
⠀⠀⠀⠀⠂⠛⠗⠀⠀⠀⢴⣾⣿⣛⣡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠏⠀⠀⣀⢀⢀⠀⠀⣀⣀⢀⠀⣀⠀⠀⢀⢀⠀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠰⠭⠐⢠⡴⣞⣿⣿⠗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠐⠲⢫⣾⣿⣿⡟⡴⠡⡔⣢⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠳⠀⢐⠺⠛⠏⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡌⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⣬⣁⡄⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⣺⣧⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣤⣄⡤⠤⠆⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⣺⣇⠀⠀⠀⠀⠲⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡤⠶⡦⠤⢊⠄⢀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⢠⢀⡀⣢⣿⠋⠀⠀⠀⣀⢤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣽⣿⣋⠶⢠⣾⡀⠀⠀⠀⢰⣝⣰⣤⣴⣤⣿⣶⣽⣟⣻⠀⠀⠀⢀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣶⣶⣿⣿⢤⠀⠀⠲⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢴⠿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠛⠛⠯⡄⣀⣿⡿⠛⠛⠛⠛⠛⠹⣿⣿⣿⡿⠋⠻⣧⣀⣹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠁⠀⠀⠀⠉⣨⣿⠁⠀⠀⠀⠀⠀⠀⢻⣿⠏⠀⠀⠀⠀⠛⢻⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠇⠀⠀⠀⠀⠀⠸⡟⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠈⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⠃⠀⠀⠀⢠⣀⣼⠃⠀⠀⠀⠀⠀⣰⣿⠇⠀⡀⠀⠀⠀⣄⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⠳⣿⣿⣦⣤⠐⡶⣿⠿⢿⠴⠶⠿⠿⠿⠿⠿⠿⠴⠿⠻⠶⣷⢴⡿⡿⠶⠃⠚⢺⣆⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠒⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠈⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢀⠴⡀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⣀⣤⠤⠒⠊⠁⠀⠱⡄⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣰⡋⢈⣉⡁⢀⠀⢈⣉⡁⢨⣷⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢻⡇⠀⠀⠀⣼⠀⠀⠀⠀⢸⠜⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠱⡀⠀⠀⢀⡀⠀⠀⢠⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣠⠔⠛⢢⣄⣀⣀⣠⡖⠛⠢⣄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⡁⠀⠈⠙⢦⠀⠀⡴⠋⠀⠀⢈⣂⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠒⠤⣀⢣⡜⣀⠄⠒⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢠⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⡄⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⢠⠏⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠹⡄⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠠⢄⡀⠀⠀⠀⣰⣋⣠⠧⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠼⣄⣙⣆⠀⠀⠀⢀⡠⠄⢷⡀
⠀⠀⠀⠀⠀⠀⢠⡯⣄⡀⠀⠙⡆⣠⠞⠉⠁⠀⠀⠀⠀⠉⠒⠢⠤⠤⠤⠤⠔⠒⠉⠀⠀⠀⠀⠈⠉⠳⣄⢰⠋⠀⢀⣠⢽⡄
⠀⠀⠀⠀⠀⣰⠋⠀⠀⠈⢣⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡜⠁⠀⠀⠙⣆
⠀⠀⠀⠀⢠⡇⡎⡜⢠⡆⢸⠀⠒⠒⠒⠒⠒⠒⠒⠚⠛⠓⠒⠒⠤⣀⠤⠒⠒⠚⠛⠛⠓⠒⠒⠒⠒⠒⠒⠂⠀⡇⢰⡄⢣⢱⢸⡄
⠀⠀⠀⠀⠘⢷⠀⠃⡏⢧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠒⠲⣄⡤⠔⠤⢄⡀⠀⠀⠀⠀⠀⠀⠘⡼⢱⠘⢀⡼⠃
⠀⠀⠀⠀⠀⠈⠛⢶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠙⠲⢄⣀⠀⠀⠀⠀⠀⠈⠒⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣆⣀⣤⡤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠐
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠘⠉⠒⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣤⣗⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⢠⢈⠁⠀⡀⠐⠁
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠉⠃⠀⠀⠘⠗⡽⠻⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠈⠁
⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⣄⠀⠀⠘⠷⠀⠛⠃⡈⣇⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠐⠀⢠⣄
⠀⠀⠀⠀⠀⠀⢢⡆⣔⡚⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠃⢤⠄⢀⠊⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢸⡟⡅⠀⠀⠀⠀⠀⠀⠀⠀⠞⢤⡷⠢⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣼⠏⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⣇⠖⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⡇
⠀⠀⠀⠀⠀⠀⢨⡠⠀⠠⣤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⣷⣯⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠈⣗⣀⠇⣈⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⢻⣿⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⢽⣁⢸⠄⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⠀⠮⣽⣿⡂⠀⡀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⣿⣰⠾⣛⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡔⡌⢊⢧⣾⣿⠃⠀⠀⣀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣽⢼⣥⡌⠐⠄⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⡀⠀⠘⣴⣥⣊⣿⣿⣿⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣿⣿⣟⣵⢂⡀⠂⠀⠀⠀⠀⠀⡘⣿⣿⢳⣿⣷⣧⣯⣽⣿⣿⣿⢃⡀⣀⠈⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⠟⣿⣿⣿⣀⠀⣒⡄⢾⡖⢁⣙⣷⣈⣿⢿⠹⣯⣿⣿⣿⣿⣿⡿⠶⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢰⣿⠋⢷⠈⠉⢀⣾⣧⡔⠀⠈⠀⠀⠀⠀⠸⠍⠟⢿⣿⣟⠇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠻⠀⠸⡀⠀⠀⣹⡏⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠒⣿⠙⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡞⠀⠀⠹⠦⢘⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡞⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠰⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢰⠃⠀⠀⢠⠀⠠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⣤⣶⠀⢀⡁⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣼⠀⠀⠀⡘⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠸⢑⣲⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠶⠾⠁⠀⠀⠠⢭⣴⣾⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡽⠂⠀⠸⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⠀⠀⠀⠀⠀⠀⠈⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⢻⠀⠀⠀⠀⠀⠀⢰⢻⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⢻⣿⣿⣿⣿⣿⣿⢀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⡈⡇⠀⠀⠀⠀⠀⢸⣸⠃⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠙⠁⠀⠀⠀⠀⠀⠘⠃⠀⣠⠃
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⣤⡀
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠉⠉⠉⠉⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉⠉⠉⠛⠒⠒⠒⠒⠒⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⢸⠀⠀⠀⠀⣀⣀⡀⢀⣠⣾⣿⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣠⣤⣀⡀
⠀⠀⠈⠀⠀⢠⣿⡏⠀⠀⠄⠀⠈⠉⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣠⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠖⡴⢀⣤⠤⠀⠀⠀⠀⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣅⣤⣈⡀
⠀⠀⠀⠀⠀⢸⣿⡟⠂⢰⠀⠀⣬⠙⠷⢿⣿⡤⠀⠀⢀⠀⢹⠀⠀⠀⢨⣿⣿⠿⠟⠛⣿⢿⣿⣿⣯⣿⣍⣩⠟⠛⠛⠛⠛⠃
⠀⠀⢀⣤⣤⣼⣿⣿⣠⣼⠷⠶⣼⣷⢆⡀⢈⣷⣆⠀⣰⠆⢠⠁⠀⠐⠿⠿⠛⠋⠉⠉⠛⠚⠇⠉
⠀⠀⠀⠀⠉⠉⠉⠋⠀⠀⠀⠀⠀⠈⠉⠩⠙⠓⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣠⠤⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡄⣀⣠⣤⣴⣶⣶⣶⣿⣿⡛⠀⠀⠀⠀⠀⢀⣼⣿⣿⣧⣠⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣾⣿⠿⣿⣿⣿⣇⣼⢿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⣀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠐
⠀⠀⢀⡆⠀⠀⢹⣣⣦⣌⢿⡏⣿⣿⡟⣿⢻⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⠏⠀⣠⡄
⠀⠀⠌⢻⡧⠁⠐⣯⣽⣛⣿⠿⠿⠿⠟⠛⣷⣿⣿⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⣿⡏⠉⠉⠉⠉⠁⢸⣿⣷⠀⠀⠃
⠀⠀⠁⠘⠛⠀⢸⣿⠹⣧⡟⠷⠾⣿⣿⡿⠿⣷⣿⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡤⡤⠶⠴⠤⠤⠬⣿⣼⣧⠤
⠀⠀⠀⠀⠀⠘⠬⠏⠀⠹⣁⠀⠀⠀⠀⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠛⠓⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣁⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⡀⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣽⣿⣧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⣾⣿⡇⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠸⣿⡟⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠹⣿⠃⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣶⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣾⣿⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⣿⣿⣿⣿⠛⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⣿⣿⣿⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⣿⣦⣿⣿⣿⣿⣧⣿⣷⣶⡆⠀⠀⠀⠀⠀⢀⣀⡀⠀⣿⣿⣿⣿⣿⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⣿⣿⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⣿⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢻⣿⣿⣿⡟⣷⡄
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣿⣿⣿⠁⠙⢧⡄⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⣿⣿⣿⣿⡆⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢧⣼⣿⣿⣿⣧⡼⠿
⠀⠀⠀⠀⠀⢰⣯⣄⣀⣰⣿⣿⣿⣧⢀⣠⣼⡆⠀⠀⣀⣤⠴⣿⣍⣀⢰⣿⣿⣿⣿⣇⣀⣩⣿⠦⣤⣄⠀⠀⠀⢠⣤⣄⡀⣾⣿⣿⣿⣇⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⠈⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠁⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⣿⣿⣿⡟⢿⣇
⠀⠀⠀⠀⠀⠀⢀⣰⠿⠀⣿⣿⣿⡁⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡹⣿⣿⣿⣿⢏⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣥⣿⣿⣿⣿⣧⣼⡿⠄
⠀⠀⠀⠀⡴⢶⣯⣅⣀⣠⣿⣿⣿⣇⣀⣨⣽⣶⢦⠄⠀⠀⠀⣤⣤⣌⣷⣿⣿⣿⣿⣾⣡⣤⣤⠀⠀⠀⠀⠀⠀⣶⣶⣤⣀⣿⣿⣿⣿⣯⣠⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⠿⠿⠋⠁⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠶⠶⣶⠶⢶⠶⡶⣶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠚⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣡⣤⡄⡀⢠⡤⢄⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠃⠀⠐⠀⢀⠀⠀⠀⠀⡟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡇⠈⠉⠀⠸⠀⠈⠉⠀⣧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⣀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⢢⣀⣀⣀⣤⠚⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⠀⠀⠙⢦⠀⢠⠞⠁⠀⢀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠘⠲⢄⣳⣃⠴⠚⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢰⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⣰⡇⠀⠀⡀⠀⠀⢸⠀⠀⠀⣀⠀⠀⣷⡀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⣰⢿⢀⡤⠊⠁⠉⠉⠹⠉⠉⠁⠉⠢⣀⢸⢷⡀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠧⢄⡀⠀⠀⠀⣰⣃⣾⣼⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣸⣸⣇⣳⡀⠀⠀⠀⣀⠤⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠥⣀⠀⠙⣄⡠⠚⠉⠁⠀⠀⠈⠑⠒⠦⠤⠼⠤⠤⠖⠒⠉⠀⠀⠀⠉⠙⠦⣀⡜⠁⢀⡠⠵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠱⡔⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡑⡴⠁⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣜⢰⢁⡶⢠⠇⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⠢⣤⠒⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢧⠰⡄⢱⠘⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡈⣠⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠶⠔⠂⠁⠲⢄⠀⠀⠀⠀⠀⠈⠒⢡⡀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⡆⠀⠀⠀⠀⠀⠙⠒⠤⣀⠀⠀⠀⠀⢉⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠶⠶⠶⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠿⠤⠤⠷⠤⠤⠴⠶⠶⠶⠦⠤⠴⠷⠶⠖⠒⠉



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

⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⢀⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠹⣯⠄⠀⠀⠀⠀⣼⣟⡁⠀⠀⠀⠀⠀⠀⢀⣠⠄⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣿⠀⠀⠀⠀⠀⣿⡷⢧⡁⠙⠉⠀⠉⠛⠀⠈⠁⢴⣂⠀⠀⠄⠴⠄
⠀⠀⠀⠀⠀⠀⡈⢺⢿⣧⡀⣤⠀⣾⠛⠆⠂⠓⠀⠀⠀⠀⢡⠀⠀⠀⡐⠨⢣⡙⠂⠁⠠⡄⡄
⠀⠀⠀⠀⠀⠀⠀⢿⡾⣟⣷⣬⣾⡧⠈⠀⠀⠀⠀⠄⠀⠀⠀⠁⠨⠁⠊⠀⠀⠹⣴⠁⡔⡀⠉
⠀⠀⠀⠀⠀⠀⠁⢄⡏⡀⣡⣿⢫⠁⠀⢠⢀⡀⠀⠀⠀⠊⠀⠀⢀⠀⠂⠀⠀⠀⠉⣿⢉⣿⣷⡆
⠀⠀⠀⠀⠀⠐⠀⠰⠄⠛⣈⡟⠀⠀⠀⠨⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⢥⡄⠀⡐⠀⢸⣿⡏⣿⡅
⠀⠀⠀⠀⠀⠀⠀⡃⣤⠊⠀⢰⡀⠐⣢⢈⠀⣗⠀⠀⠄⣼⡀⠀⠰⢀⢂⡩⣿⡇⠘⡆⢙⣿⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠸⢠⢋⣤⡯⣅⣦⢹⡌⠙⡾⢆⠀⣴⣘⣳⢦⢳⣼⣾⣷⣼⢿⢵⡶⡿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⢹⡧⣮⡷⣷⣬⡯⣷⣤⡱⠳⢾⣿⡿⣿⣿⣿⣿⣿⣿⣷⢿⣯⣿⣿⢅⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣿⣿⣿⣿⣿⣿⣿⣷⡯⣯⣿⣿⣿⣿⣞⣿⣿⣿⢿⣿⣿⣟⣿⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⡿⣿⣿⠿⠿⠛⠋⠛⠛⠋⠉⠉⣿⡏⠁⠈⢟⣭⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡟⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠼⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁

⠀⠀⠀⠀⡀⠴⠀⠘⠉⠋⠱⡢
⠀⠀⢀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠑⠐⠄⢀⣠⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢄⡠⡠⠤⠒⠄⠤⠄⠀⡀
⠀⠀⡂⠀⠀⠀⠀⢤⠀⠤⡀⠀⠀⠠⠈⠚⠐⡀⢀⠀⢉⠉⠉⠈⠉⠁⢈⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠂⢄⡀
⠀⠀⠋⠀⠀⠉⠁⠈⠳⣦⡷⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈
⡔⠁⠀⠀⠀⠀⠀⣠⡞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣆
⢧⣠⣂⠤⠒⠐⡉⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢳⣼
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡻
⠀⠀⠀⠀⠀⠀⠀⠈⢄⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⠀⠀⠀⠀⠀⠀⡄⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡇⠀⡀⠀⢀⠀⠀⠀⠠⠄⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⡀⡀⠀⠀⠀⢰⣸⡃⢀⣀⢀⠀⠀⡄⠀⠂⢐⣇⠤⠛⠢⡀⠰⠀⠁⠀⠀⢢⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢴⡆⠀⣠⢋⢁⣭⠁⠀⠀⠁⠈⠀⠀⠉⠁⠀⠀⠀⠀⠀⠑⠀⢀⡀⢀⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢨⡇⠈⢅⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡇⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠸⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢸⡇⠀⠀⣇⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠷⠞⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⣺⣀⠟
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⢀⣴⠀⠀⣄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢺⡀⠀⢸⢹⣈⡃⣀⣷⠀⠨⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣜⠷⣟⡬⣃⣼⣿⣿⣿⣻⣽⢿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡯⣭⢞⢽⢳⣷⣿⣿⣿⣿⣿⡿⣿⣽⡀⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣧⢬⣾⣿⣿⠿⠿⠛⢿⣿⣿⣦⣿⣽⡄⠀⠀⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠉⠉⠻⣿⣿⣿⠃⢀⠀⠀⠀⢈⠛⠿⣿⣿⣿⣦⣇⡐⣀⢡⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⢴⣂⡈⠁⠠⣿⣿⠁⠀⠈⠉⠀⠈⠁⠀⠀⠈⣿⣿⣿⣻⡟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⣿⠉⠉⠁⠀⢾⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣧⢉⣹⠈⢠⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡧⠂⠾⠿⢠⣾⣿⣿⣆⠀⠀⠀⠀⠀⢀⣼⣿⣿⡁⣨⡀⠘⠇⠀⠳⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠀⠀⠀⠀⠹⢟⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣷⣤⣠⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⢈⣧⡄⠀⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⢹⡏⠩⠙⢳⡄⡨⢁⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⠈⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⠈⢌⠸⠋⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡛⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⢁⠐⣿⣿⣿⣤⡄⠀⠀⠀⠁⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠑⡀⠀⠀⠀⠈⠀⣿⣿⣿⣿⣿⠀⠀⠄⠀⠠⠂⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠄⠀⠀⠀⠐⡄⠀⠀⢻⣿⣿⣿⣿⠇⠘⠀⠠⠀⠠⠂⠀⠐⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠑⠀⠀⠁⢀⠀⠢⠀⠈⠀⠀⠀⠀⠈⢿⣿⣿⠏⠀⠀⠠⠁⡐⢀⠔⠁⠀⠔⠁⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢀⠈⠂⢀⠑⢄⠀⠄⠈⠂⠁⠀⠀⣸⣿⣿⣄⠀⠀⠡⠂⠀⢁⠀⠊⠀⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⡀⠀⡑⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠊⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠩⠀⠀⠈⠀⠀⢀⠀⠠⠄
⠀⠀⠀⠀⠀⠄⠀⢀⣀⠀⠀⠁⠀⠀⠀⠤⠈⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣐⠀⠈⠁⠄⠤⠄
⠀⠀⠀⠀⠠⠤⠄⠠⠤⠤⠤⠤⠀⠄⠠⠤⠀⠀⠀⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⡟⣻⣿⡆⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠐⠒⠂⢀⣼⣿⡟⠀⢸⣿⣿⣿⣿⣿⣿⠁⢸⣿⣧⡀⠀⠂⠠⠀⡉⡁⠀⠀⠀⠀⠒⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠉⠁⠀⢀⣴⣿⣿⠟⠁⠀⢠⣿⣿⣿⣿⣿⣯⠀⠈⢿⣿⣷⡄⠁⠂⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣤⣿⡿⠋⠁⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠙⠻⣿⣦⣤⣴⡀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠀⠀⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠈⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡿⠀⠀⠂⢄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠄⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠛⠛⠳⢦⣄⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠛⠋⠙⠓⠟⠁⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠖⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗
⠀⠀⠀⠀⠀⠀⣀⣠⣤⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⠶⠤⣤⣄⡀
⠀⠀⠀⣠⠶⠋⠉⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠈⠉⠳⢦⡀
⠀⢀⡾⠁⠀⠀⠀⠀⠀⣀⣏⠀⠀⠀⠀⠀⠀⠀⠀⣀⡷⣄⡀⠀⠀⣀⣤⠶⣄⣀⠀⠀⢀⣠⢶⡀⠀⠀⠀⣀⣀⠀⠀⣠⡾⠓⠲⠤⣄⡀⠀⠀⠙⣦
⢠⡟⠀⠀⢀⣠⠖⠛⠉⠉⠙⣦⣀⣀⣀⡴⠛⠛⠛⠉⠀⠈⠉⠛⠋⠉⠀⠀⠀⠉⠙⠉⠉⠁⠀⠉⠛⠒⠛⠉⠉⠛⠛⣿⠀⠀⠈⠒⠄⠙⢦⡀⠀⠘
⣾⠀⠀⡴⠋⠐⢀⠀⠐⠀⢸⠏⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⣆
⣿⠀⡼⠁⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣠⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣄⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠘⡄
⠙⠳⠷⣤⣤⣤⣤⠶⠞⠛⣿⠀⠀⠀⠀⢰⡟⠙⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠁⢹⡇⠀⠀⠀⠀⣿⠙⠓⠶⢤⣤⣤⣤⡴⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⣿⣶⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⣰⣶⣿⣶⡆⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⢀⡟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠙⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⠾⠋⠳⣤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠳⠶⠦⣤⣤⣤⣤⣤⣤⡤⠶⠶⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠔⠋⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡄⠘⢦⡀⠀⠀⠀⠀⠀⠐⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡴⠂⠀⣰⡇⠀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣤⣤⣤⣤⠾⠋⢻⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⡟⠛⢦⣤⣀⣀⣤⣤⡀⠉⠉⢀⣠⠞⠒⣿⠋⠀⠀⠀⠀⣷⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢨⡇⠈⢹⡟⠛⠋⠁⠀⠀⣿⠀⠀⠀⠀⠀⢸⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⣧⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⠀⠀⠀⠀⢀⣸⣿⣿⡿⠇⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⢀⣸⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⢐⠃⠁⠴⠁⠀⡌⠘⠲⠴⡆⣀⠀⠀⠀⠠⠄⠀⠀⠀⠤⠀⠀⠀⠠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⣌⡚⠂⠈⠀⠀⠀⠀⠀⠂⠠⠀⢩⡶⣤⣀⠀⠈⠠⢴⠐⠐⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠛⡿⠀⠁⠀⠀⠀⠀⠀⣠⣶⣶⣤⡌⠀⠀⠈⠀⠁⣀⣰⣠⣄⣀⣀⡀⡀
⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁⠀⠀⠀⠀⢤⠑⡻⣯⣿⣿⡟⠛⠛⠿⠯⠙⠲
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠌⡇⠘⢫⣺⡋
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠨⠈⠀⠀⠀⠠⣼⣿⣒⠇
⠀⠀⠀⠀⠰⠒⠒⠒⠂⠀⠀⠀⠀⠀⢀⣀⢂⣀⣄⠓⠁⠀⠀⠀⠀⠀⠀⠉⠂⠛⠆
⠀⠀⠀⠀⠀⠑⠒⠤⠦⣦⣘⣤⣾⣶⣿⣿⣿⣿⣇⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⣿⣿⣿⣿⣿⣿⣤⣘⠆⠀⠈⠀⠀⠀⠀⢠⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢻⣿⣿⣿⢿⡏⠀⢙⡄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠕⢙⣿⢿⠽⠯⠀⠰⠀⠘⠗⠀⠀⠸⢔⠀⠀⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢪⢘⡟⡽⠒⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⡄⢋⠀⠆⡠⠀⠠⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠀⠀⠀⠀⠀⠘⠀⡈⠀⠰⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠇⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⢤⠴⠖⠒⠲⢤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⠀⢀⡠⡠⠤⣄⡠⢀⠀⠀⠉⢢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠐⣄⡀⠀⠀⢠⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⣤⠀⠀⠀⠉⢧⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠆⢀⡴⠀⣀⡼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⠔⠙⠲⠖⠲⠶⠋⠙⢲⡀⠀⠀⠀⠀⠀⠀⠙⠚⠙⣹⠉⠉⣰⠏⠁⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠞⠁⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠴⢤⣀⣀⡤⠦⠶⠞⠉⠙⠋⠁⠀⠀⠘⢦⡀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡩⢄⣀⣀⡀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠈⠳⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⢤⣤⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠘⡗⠀⠤⠄⢂⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⣷⠀⠀⠀⠀⠀⠀⣿⠶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⢀⣠⣀⡀⠀⠀⠈⠻⠟⠋⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠎⠀⠀⠀⠀⠀⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⡟⠲⡞⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠉⠙⠳⠶⢦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠶⠚⠻⣅
⠀⠀⠀⠀⠀⠀⠀⢨⡗⠀⠀⠀⣿⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠶⣤⢾⡇⠀⢀⡤⠄⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣥⣠⠞⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡔⠀⣇⣀⡀⠀⠀⣴⠂⠀⠀⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢦⡼⠁⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠙⠲⠦⠞⠁⠀⠀⠀⠀⠀⢀⡄⠀⢀⣀⠀⠀⠀⢀⠀⡼⠀⢾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠋⠁⠀⠀⠀⠘⠒⠃⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢳⣾⣁⣠⡄⠀⠘⠳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⠁⠙⠶⢶⡶⢦⣄⣀⣈⡷⢦⣤⡴⠞⠳⣶⣤⣤⡾⠻⠾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⡼⣆⠀⠀⠉⣸⢻⡄⠀⠀⠀⠠⡿⣏⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠇⠘⣿⣿⣿⡿⠀⢷⣶⣶⣶⣾⠃⢻⣷⣶⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠀⠀⠈⠉⠉⠀⠀⠈⠿⠿⠿⠏⠀⠀⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣷⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⢻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⡏⠩⠿⠿⠿⠿⠄⠀⠀⠀⠠⠶⠒⠒⠶⠄⢸⡿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣄⠘⡀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢇⣠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣷⣄⠀⠀⠀⠉⠍⠍⠀⠀⠀⣠⣾⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⢀⠈⢉⠛⠛⠛⠷⣤⡀⠀⠀⠀⢀⣤⠞⠛⠛⠉⠁⠀⠀⠀⠀⠀⢀⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠘⡌⠱⡏⢣⠋⢣⠀⠀⠀⠈⠙⠻⠶⠟⠋⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠉⠓⢄⠀⢨⠀⠸⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⠀⢸⣿⣵⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢓⡂⢀⣿⣻⣦⡗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⠲⢦⣄⡈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣂⣉⣉⣁⠟⠫⣸⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠙⢸⣈⠔⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠸⠀⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠁⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢀⣤⣤⡌⠙⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⢠⣀⣠⠀⡄⠀⣄⣀⡄⢹⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣇⠀⠀⠀⠀⠇⠀⠀⠀⠀⢸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣆⠀⠀⠢⠤⠴⠂⠀⣠⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⣏⣀⣀⣀⣹⡿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡏⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣸⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡆⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣧⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⢸⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢻⣸⠇⠀⠀⠀⠀⠀⠀⠀⢸⣠⠇⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣷⡀⠀⠀⠙⢦⣀⢀⣀⣀⣀⠀⢀⡼⠋⠀⠀⣠⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠀⠀⠙⢦⣀⠀⠀⢉⣙⠒⣄⠉⠛⠉⠀⠀⣀⡴⠃⠈⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠙⠓⠲⠦⣄⣈⠛⠒⠯⣌⣉⣁⣀⣠⡤⠶⢚⣁⣠⠤⠴⠖⠒⠂⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣄⡀⢀⣠⠴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠷⠶⠤⠤⠤⠶⠶⠖⠚⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡤⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣾⣿⡿⢻⣷⠈⢿⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⢺⡀⠀⠀⣿⠟⠁⠘⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⢸⡇⠀⣀⠀⠀⡀⠸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣀⠀⠀⠠⣿⣿⣿⡿⠀⠸⢇⣀⣿⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠆⠀⠀⢿⣿⣿⣧⢰⡿⠞⠛⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣤⣶⠀⢸⣿⣿⣇⠘⠁⣠⠖⠈⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣿⣿⣿⡤⠀⠻⣿⣿⣶⣶⣧⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣴⣿⣿⣿⣧⣿⣸⣤⣾⠿⠋⠉⠁⣾⠿⠿⠃⠀⣠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣧⣼⡟⠃⣿⠇⠐⠟⠃⠀⠀⢀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⠈⢻⣿⣷⣾⣄⢁⡀⠲⠀⠿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣇⠀⣱⡟⠃⠉⠉⢙⣤⠜⢺⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣶⡿⠷⠾⣷⣆⣹⣧⣼⣿⣿⣿⡿⠿⠟⠛⠃
⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡿⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠘⠟⠙⢣⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀⠀⠀⢸⡇⢠⣾⡅
⠀⠳⣄⡀⢸⣿⡀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⡀⢸⣿⣾⠟⠁⢀⡄⠀⣠⡆
⠀⠀⣿⡇⠘⣿⡁⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣀⣽⣧⡴⠞⠋⢀⣶⡟
⠀⠀⠉⢁⡴⠈⠉⠀⠀⠁⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠻⣿⣿⣿⣿⡟⠉⠁⣠⠶⣠⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠑⣄⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⢿⣆⠈⠻⣿⣿⣷⣮⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠻⣶⠃⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠀⣿⡇⠀⣹⠿⠟⠋⠉⠀⠀⡀⠀⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠘⠃⠀⠀⣶⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡜⢷⢀⣿⠀⠀⢠⡤⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣼⣿⢁⣴⡗⠉⠀⢀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⡀⠉⠀⠀⠀⠠⣤⡀⠀⢸⣿⣿⠉⢿⣿⣯⡘⠻⣧⣴⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣼⡿⣨⢻⣶⣌⣿⠃⠀⠀⠹⠁⣠⡾⢛⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣶⣿⣿⣠⣆⠀⠀⠿⠄⢹⣿⡏⠀⠆⠀⠀⣴⣟⣵⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣆⡀⠀⣤⡀⢻⣿⠀⢺⡆⠀⣼⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠻⣿⠉⠈⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⡟⠉⠉⠉⠉⠀⠹⣿⡾⣿⣇⠈⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣇⢰⣦⣤⣄⠀⠈⠁⢸⣿⣤⣿⠇⠀⠀⢀⣤⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⣰⠁⣀⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡌⢻⣿⣿⣷⣤⡀⠈⠻⣿⠇⢀⡀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡇⢰⡿⠻⣿⣿⡿⢡⠀⣿⢠⡜⢣⣾⠏⠀⠀⠀⠀⠀⢠⣄⡀⠀⠐⠀⠀⠺⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⠜⠁⠀⠘⢿⠇⢸⡄⠸⣿⣧⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⠈⠁⠀⣴⠁⠈⢠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣄⠀⠀⢠⠆⠀⢀⣤⡀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡌⠙⢧⡀⢰⣈⣷⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀⣾⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣶⣼⣿⠊⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠟⠀⠀⠀⠺⣿⡄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣏⠈⣿⣿⣇⣀⣤⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⡀⠈⣤⠀⢀⣴⡀⣿⣧⠁⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢸⣿⡀⢃⣼⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠛⢷⣄⡹⡀⢸⣿⠇⢻⣿⠀⠀⠀⠀⠀⠀⠀⠠⠿⠁⠀⠘⡷⠗⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣶⣤⠀⠀⠀⣉⠛⣿⠈⠻⠀⢈⡟⠀⠀⠀⠀⠿⠃⢠⠀⣀⠀⠀⠾⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣶⣤⣘⠿⣤⡄⠀⠀⠸⠟⠀⠀⠀⠀⠀⡂⠁⢠⣿⡆⠘⣀⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠿⣿⣷⣽⣿⣴⡀⢰⡆⠀⠀⠀⢠⣾⣿⣿⠀⢻⣿⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠈⠻⣿⣿⣿⠀⢸⡇⢠⡆⢠⡿⠋⠈⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠈⠻⣿⣤⣤⡆⢀⣶⠏⠀⣶⣾⣿⡀⠀⠀⠀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡄⠀⠀⠈⠙⣿⠁⠈⣡⡄⠘⣿⡇⢀⠁⣴⣾⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄⠀⠀⠀⠸⡀⢀⣿⡇⢸⣿⣇⡈⣡⡭⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⠀⠀⢰⣧⠘⣿⡇⠈⢻⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⣿⣿⣿⣄⠀⢸⣿⠀⠸⠇⠀⠏⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⢿⣿⣿⣿⣿⢈⣿⠀⠀⠀⠀⢸⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⢻⣿⣇⠈⠸⠿⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣌⢹⣿⣆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢣⣼⣿⡀⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣟⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢈⠃⠀⠰
⠀⠀⠀⠀⠀⢠⠀⣠⣴⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢰⣿⠛⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠄⢺⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⡴⠊⠁
⠀⠀⠀⠀⣰⠏⠀⠆⠀⠀⠀⠀⠀⠀⠀⣠⠾⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⣤⡿⠏
⠀⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠨⣗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡦⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋
⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⢀⡈⠁⠀⠀⠀⠀⠀⣼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠈⠉⠀⠀⠀⠀⢡⣴⡟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠰⣀⣿⠿⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⣦⣿⣯⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠹⢷⣦⣄⡁
⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧
⠀⠀⠀⠀⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠻⣧
⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠘⠓⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡤⣶⣤⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠁
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢂⠐⣋⢙⠊⠍⢉⠒⠢⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠼⠐⠠⠄⢁⠀⠈⠠⠈⠡⠑⠄⢄⡀⢐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⡃⠦⠁⠢⡀⡀⡀⠁⢀⠂⠑⠂⢀⢇⡀⢀⣀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⡁⠠⣂⡀⡐⠁⠀⠴⡊⠉⡱⡁⠄⠀⢰⡂⣭⣁⣧⢢⣽⡦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⢈⠀⠀⠢⠐⠸⣄⠁⣀⣁⠠⠪⡰⠈⢀⣟⡾⠛⠋⠁⠀⠀⠻⣏⠍⠃
⠀⠀⠀⠀⠀⠀⠀⢀⢔⠯⠊⠐⠁⢒⡤⠂⣈⢕⣪⣭⣭⣽⢘⣷⠥⠁⠀⠀⠀⠀⠐⠂⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠐⠀⠒⡡⠂⡟⠑⡞⠛⠿⣿⣿⣿⣿⠣⢾⣏⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⢠⢠⡀⡠⢁⠄⠒⣀⠥⠘⠦⢄⣀⣀⣹⣿⡟⣽⠁⠈⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⢰⠋⠁⠀⠰⠁⡎⠨⠐⠩⡀⠀⠽⣧⡼⢛⣯⣄⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠓⢄
⠀⠀⠀⢠⠫⡀⢈⠰⠀⡖⡠⠀⠀⠆⢈⠐⠠⢃⢴⡡⢁⠼⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⡼⡣⠃⠀⠐⡨⠥⠆⡁⠴⠐⠀⡸⢸⠘⠃⡇⢒⠑⠨⠟⣧⡀⠂⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠘⠿⠳
⠀⠀⢸⡀⠔⡂⠈⡀⢀⠂⢠⡀⡰⠈⡠⠀⠑⠚⣈⠁⠂⠌⠎⠀⠁⠛⡓⣤⣄⣀⣀⣀⣰⠤⠤⢀⣉⣒⣒⣲⠋
⠀⢀⡼⠠⢀⠄⠅⡪⡃⣘⠢⡁⠄⢠⢈⠴⢡⠜⡼⠨⠀⠀⠀⠀⢠⡁⠂⢙⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠁⠁⢀⡀⠈⠬⡘⡇⠸⢀⠁⡀⡊⠀⢀⢱⠎⠐⢀⢊⡀⠀⠐⡀⢩⠠⠀⣿⡀
⢸⠀⠈⠀⠀⢲⠁⡁⢨⢈⢸⠌⢀⠔⡐⢐⠀⣩⠔⡚⢀⠤⢂⢡⠨⢰⠀⢀⢿⢳
⡎⢢⠜⡰⢈⠬⠃⡼⢀⠃⡂⡳⠘⡔⠈⠆⠠⢃⡨⠬⣁⠄⠀⠀⠀⠀⠀⢌⡇⣯
⢇⠘⢨⠰⡈⠢⠈⡂⢇⠘⠀⢇⠨⠀⠃⠀⠰⠒⠜⡡⠀⠀⢠⢒⠡⢀⠌⢨⢡⣸
⡞⠰⡀⡇⠈⠃⡀⠑⢂⠹⠘⢄⠐⡔⠸⠀⢂⡈⢺⠰⠔⢸⠈⠀⢠⠀⡢⡀⠀⠻⠂
⣱⣁⣓⣘⣐⣀⣑⣈⣀⣃⣘⣀⣃⣀⣀⣀⣁⣘⣀⣀⣀⣘⣀⣐⣀⣁⣈⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⢚⠩⠴⠲⠬⣝⠲⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⡡⠎⠉⠁⠀⠀⠀⠀⠀⠉⠹⢦⣙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢋⡞⠀⠀⠀⠀⢀⠤⢀⣀⣀⡀⡤⣀⣄⠀⠀⠀⢻⡜⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⢳⣀⡀⡸⠀⠀⠀⠁⠀⠀⠈⠁⠀⠈⢧⣠⣠⡿⢂⠠⢀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⢨⡇⠀⣠⡤⣄⠀⠀⠀⠀⠀⠀⣠⢤⣄⠀⢸⡇⠀⠈⠀⢑⡄⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢌⠀⠀⠀⣀⠐⢂⠀⠙⠿⠋⠀⠀⠀⠀⠀⠀⠙⠻⠋⠀⡼⠮⣀⠀⠀⠈⡡⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠬⢭⡤⢞⠀⠈⠄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣰⠃⡄⡳⢤⠬⠥⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢿⠀⠀⠈⢆⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡰⠃⠀⠀⡿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢊⠧⡀⠀⠀⠉⠲⢤⣄⣀⣀⣠⡤⠖⠋⠀⠀⢠⠾⡃⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡸⠀⠉⠢⠤⢤⡄⠀⠀⠉⠉⠀⠀⢀⡦⣤⠴⠋⠀⣟⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡜⡅⢀⠀⠀⠀⠀⠑⠶⢄⠀⠀⣠⠶⠖⠀⠀⠀⠀⡀⢸⠃⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢧⠈⠀⠀⠂⠀⡀⠀⠀⠑⠚⠁⠀⢄⠀⠄⠀⠠⠉⡬⢃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣐⣲⠄⠂⠄⡀⢀⠀⠀⠀⠀⠀⠀⢀⢀⣠⠞⠢⣖⡓⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣏⢳⠆⠀⡀⠀⣀⠀⠀⠀⠀⠀⡈⠀⣀⡀⢶⢋⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⠳⠋⠋⠒⠋⠱⢖⣒⣳⠊⠙⠒⢻⢹⣏⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠰⢠⠀⠀⠀⢸⣿⢹⣿⠀⠀⠀⢸⡼⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⣈⡄⣠⡀⢸⢏⣿⢹⡄⢀⣄⣸⣦⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠭⠵⠦⣝⣋⡼⠙⢦⣛⣫⠶⠬⠭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⡠⢆⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢀⣠⣔⡒⠂⠉⣀⣈⢦⡀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⢋⠀⠤⠄⡀⠀⠠⠤⢠⣹⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠈⢻⠀⠀⠀⠃⠀⠀⠀⡸⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠵⢄⠀⠙⠁⣀⠴⢅⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘⠁⠀⠺⣍⠀⢈⠝⠂⠀⢹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠀⠀⠄⡈⣦⣎⡀⠄⠀⠈⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠈⢹⠉⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢠⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⢀⠟⡏⡠⠞⠉⠉⠉⢹⠉⠉⠉⠙⢤⡈⡏⢇⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠤⣀⠀⠀⠀⣠⣏⣨⣧⣄⡀⠀⠀⠀⢸⠀⠀⠀⠀⣀⣇⣧⣈⣧⡀⠀⠀⢀⡠⠼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠤⡀⠈⢣⣀⠔⠉⠉⠀⠀⠀⠉⠑⠒⠠⠬⠤⠐⠒⠉⠁⠀⠀⠈⠉⠑⢄⣀⠁⠀⡀⠼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⠀⠀⠈⠲⠋⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠲⠊⠀⠀⢈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⡌⢰⠃⠆⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠒⢤⡊⠉⠉⠉⠀⠉⠉⠉⠉⠉⠁⠀⠀⠆⢳⠈⡄⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣇⠎⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠉⡗⠚⠉⠉⠲⢄⠀⠀⠀⠀⠀⠘⠉⢆⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠓⠦⡄⠀⠀⢀⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠱⡆⠠⣀⣀⣠⠤⢶⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠛⣡⠛⣿⡻⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈⠀⠈⣧⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢰⣿⣦⣄⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣩⣝⣿⣿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⡖⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠐⠀⠄⣁⠘⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠆⠀⠀⠀⠀⠅⠀⠀⡄⠀⠀⠀⠀⠂⠠⠀⡖⠀⠁⠀⢀⡀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢧⠄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠂⢰⡁⠅⠀⡄⠀⣅⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠁⣿⠃⠀⠀⠀⠈⠀⠀⢘⣥⠀⠀⠀⠀⢀⠀⠀⠀⣩⣹⠾⣯⠵⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⣠⣿⠃⢄⠀⠀⠀⠀⠀⣈⡁⠀⠀⠀⠀⠈⢀⣰⣾⣥⣯⣦⡶⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣾⣿⣖⣦⣤⠁⠀⠐⠺⢫⣿⣀⠀⠠⠀⠀⢸⣥⡟⠾⢾⡝⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣼⡿⠛⠉⠀⠀⠙⣶⣾⣧⣹⣿⣿⣀⠀⣠⣠⣿⣿⣾⣟⡈⢛⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⢩⡇⠀⠀⠀⠀⠀⠘⡟⠻⣏⠀⠀⠙⣶⣿⣦⣈⡁⠉⠻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢀⣄⠀⠀⠀⠀⠻⠀⠀⠀⠈⠛⢻⠟⢿⠀⢘⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠘⡇⢠⡀⢬⡀⠀⢀⠀⠐⠀⠀⠐⠀⢠⠠⠈⠀⣾⠀⠊⡎
⠀⠀⠀⠀⠀⠀⣤⠀⢄⠙⣾⣷⣾⣿⣿⣿⣿⠷⣤⡁⢾⣿⣿⣶⣿⠋⣶⢀⡤⢸⣿⣷⣦⡦
⠀⠀⠀⠀⠀⠀⠉⠐⠊⠀⠉⠙⢿⠟⢯⣿⢺⣅⣿⢿⡸⠋⠙⣯⣽⣯⣝⣟⣧⣄⣿⣿⡿⣷⢷⡶⢳⠔⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠂⠛⠘⠙⠃⠉⠻⠟⢿⠟⠳⠗⠏⠁⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣾⣥⣭⣯⣬⣵⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠚⠙⠒⢤⡤⠖⠒⠒⠂⣤⠦⡤⠚⠉⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠉⠀⠹⢦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠉⠉⠗⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠋⠀⠁⠀⠀⡤⢷⠈⠉⠑⣄⠠⢀⡀⠀⢠⠔⢧⠄⠀⠀⠈⠠⠞⠳⢄⢀⡀⠤⣄
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢊⣁⠒⠀⠀⠀⠀⠀⠀⠴⠇⣠⠶⠛⠲⠾⢄⣠⠤⠦⠤⣤⢂⡰⠶⠾⣍⠀⠀⢈⠆
⠀⠀⠀⠀⠀⠀⣸⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠘⠂⢱⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣀⣠⡞⠁⠀⠀⠀⠈⢫⠀⠀⠀⠀⢩⠉⠁
⠀⠠⠀⠒⢢⣎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠘⡆⠀⠀⠶⠀⠀⢸⠘⠀⠀⠀⡘
⡠⠓⠀⠀⢠⠋⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⡗⢄⣀⢀⣀⠴⠋⠓⠐⠀⠊⠱⣄
⢣⡀⠀⠀⠘⣄⡄⠀⠀⡰⢡⡘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⡙⡆⠀⠀⢀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠸⢄⣀⣀⡠⠿⠀⠀⢰⠀⠫⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⡴⡷⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⢹⣁⡀⠀⠀⠈⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢇⡇⢀⠤⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠸
⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠆⠘⠏
⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠾⡀⠀⠀⠀⠀⡠⠊⣡⣄⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⡇
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⢇⠀⠛⠈⠀⠀⠀⠀⠀⠈⢩⣁⠀⠀⢣⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠔⠀⠀⣸
⠀⠀⠀⠀⠀⠳⣄⣮⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢘⠂⠀⠀⢣⡀⠀⠀⠀⠀⠉⠉⠉⡏⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠁⣰⢻⣠⠀⡽⢦⣀⠀⠀⠀⠀⣠⠧⡤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢲⠶⡟⢄⣀⠀⣀⡴⠶⣞⠁⢠⢠⠀⠀⠰⠒⡄⠀⠀⣏⣀⡠⡾⠓⠒⠚⠁⠈⠙⠉⠀⠀⠈⠙⠒⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠇⠀⢸⢹⠀⠀⠀⠈⠑⠛⠛⠤⢤⠤⠟⠳⡖⡞⠁⢠⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡇⠀⢸⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⡀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠸⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⢸⠀⠸⣴⢮⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢀⢰⠀⠀⡆⣬⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠇⠀⠻⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡜⠳⠧⠀⠹⠇⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⢠⣤⣶⣿⣿⣿⡷⣿⣷⣦⣤⡀
⠀⠀⠀⠀⢼⣿⣿⣂⣀⡁⠙⣿⡞⠻⢥⣤⣀⣀⠀⢀⠀⠀⣀⢀⣠⠴⢶⢳⠋⠉⠁⠀⠐⠀⢤⡀
⠀⠀⠀⠀⢸⠋⠛⣿⠟⢃⣠⣴⢿⢊⠽⠹⢉⠁⢔⣋⠜⢋⠍⠂⠀⠀⡀⠒⠄⠀⠀⠀⠀⠬⠀⡺⣦
⠀⠀⠀⠀⢸⠀⠀⢀⣶⣿⣿⣿⠗⠵⠄⠐⠀⠀⠨⠇⠀⠁⠀⠀⠀⠀⢃⢠⠀⠂⣀⠐⠀⠀⠂⢩⠐⣱⡄
⠀⠀⠀⠀⢿⣶⣶⡿⢻⣿⣿⣻⣟⠀⢂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⡆⢸⠅⠀⠀⠀⠀⢉⢄⡤⢚⣿⡄
⠀⠀⠀⠀⠈⠉⠉⠀⠀⢻⣿⣿⣯⠋⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂⠦⡉⢠⢦⠀⠀⠀⠄⢚⣳⣠⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠸⣧⣲⣉⣸⠇⠀⠀⠀⠂⡲⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣆⠀⠀⠂⠀⠀⠠⠀⢀⣴⣾⣾⣿⣿⣿⣿⡷⢰⡄⠀⠳⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣿⡅⡣⡖⢠⣅⠒⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢢⠀⢹⣿⣷⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣷⣢⣾⣿⣷⣿⣿⣿⡿⠿⠿⠿⠙⢿⣿⣿⣿⣾⣷⠀⢷⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⣿⣿⡟⠟⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⢿⣿⣏⢻⣾⠿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣼⣿⠀⠀⠀⢤⠤⠤⠤⠄⠠⠄⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⢀⡀
⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢧⡏⡟⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢲⣮⣿⣿⠀⠀⠀⣀
⠀⠀⠀⠀⠈⠀⠈⠑⠲⢶⣶⣤⣴⣿⢷⣺⡟⠁⠀⠒⡲⠖⠒⠲⠶⠶⣶⣿⣿⣿⣿⣯⣿⣴⡷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠉⠁⠛⠙⠥⠀⠀⠀⠀⠀⠒⠒⠂⠀⠉⠒⠒⠒⠚⠛⠛⠃

⠀⠀⠀⠀⣀
⠀⢠⣼⣄⣀⡀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⣤
⠀⠀⠻⣿⣿⣷⣬⣿⢀⠀⠐⣆⢘⣿⣿⡿⠋
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣧⣴⣿⣿⡿⣿⣤⡀
⠀⠀⠀⣹⣾⣬⣿⣿⣿⣿⣿⣿⠇⣴⣿⣿⣿⣷⡢⠤⢥⠀⠀⠀⠀⡀⠤⠤⢤⠤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣷⣊⣽⣿⣿⣿⣿⣿⣶⠇⣀⣰⣶⣴⣶⣳⡆⣼⣷⣶⣷⣖⠾⠆⠀⢖⠀⠀⠀⠀⣶⣶⣿⣾⣿⣿⣯⢀⡀⠀⣄
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠟⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡟⠛⣿⠁⠿⠨⠟⢰⡤⡲⣾⣾⣬⣿⣿⣿⣿⣿⣿⣿⣿⣷⡢⠘⢳⡀
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⡟⣿⢻⡇⠘⠀⠤⠶⣾⡄⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣥⠀⠆⢷
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡟⢻⣿⣿⣿⣿⣿⣿⣷⡏⣀⡇⢀⡆⠒⡂⡀⢠⠟⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⡇
⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⢿⣿⣿⣿⣿⣿⣿⣯⣆⣿⣰⣬⣿⣽⡟⠲⣆⢡⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢈⣷⣾⣯⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⡿⣿⡿⣿⣗⠊⠰⡟⢿⣟⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣵⠏⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢏⠿⣩⠇⠀⠰⠐⡾⠝⢻⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣯⣤⣤⠄⠀⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣒⢠⠠⠀⠀⠂⣡⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣷⣶⡶⢺⣤⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡤⠀⠀⣀⢻⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣷⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣷⣿⣶⣯⢟⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠏⠀⠀⠀⠉⠉⠉⠉⠛⠛⠿⠿⠿⣿⠿⠿⠿⠛⠋⠁⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠹⣿⣿⣿⣟⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣭⣿⣿⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣼⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣿⣷⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⢩⠿⣿⣟⣿⣿⣿⣽⣿⣿⣿⣿⣷⢻⣿⣿⡿⠿⡿⠿⢛⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣾⣌⣿⣿⣿⣟⠻⣿⣿⣿⣿⣿⣋⢹⣿⡝⢷⣾⣽⣏⣷⣶⣿⣿⣿⣽⣛⠿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡖⠚⠉⠉⠒⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡁⢀⠀⠀⠀⠀⠀⠀⠉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠚⠵⠛⠠⣤⡄⠀⠑⠠⠀⠀⠀⠀⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠉⠁⠢⡽
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠦⠤⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢠⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠢⣀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⢢
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⢀⠠⠐⠁⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣶⠶⢤⣤⣤⠤⢤⣤⣤⠤⢤⣤⡤⢴⣦⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣦⣸⣿⠀⠀⠀⠀⠀⢀⣶⣿⣧⣸⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⣿⣿⣧⠀⠀⠀⠀⠘⠻⠿⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠇⠀⠀⠀⠀⡠⣦⣀⣠⠃⢀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⡃
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢰⡾⠁⠀⠀⠀⢀⡏⠀⢸⡉⠉⠳⣿⠳⣶⣦⠄⠀⠀⠀⠀⠀⠈⢷⣤⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢠⠟⠉⠀⠛⠀⢰⡏⠁⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠉⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⡆⠉⠉⢿⣧
⠀⠀⠀⠀⠸⠀⠀⠀⡀⠠⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠃⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⢸⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⡾⢹⡇⢰⡞⠀⠀⠀⠀⠀⠀⠀⣠⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠃⠠⠣⠾⠀⠀⠀⢀⠀⢀⢴⣶⠟⡇
⠀⠀⠀⠘⣆⠀⢀⣀⣀⡤⡄⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⡁⠿⡦⠉
⠀⠀⠀⠀⡟⢱⡆⠀⠀⠀⢸⢻⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠉⠈⠋⠂⠂⠀⠙⠉⠀⠀⠀⠀⢠⠎⢹⡇⣶⠶⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡴⢴⡴⢦⡤⠶⠦⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠿⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠀⠀⠇⢀⣧⠀⠀⠈⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⣈⡻⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⢈⣉⠁⡄⢈⣉⠉⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⠀⠀⠃⠀⠀⢀⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⣄⠈⠉⢁⣠⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⢿⣁⣀⣘⠿⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⢷⠀⠀⠀⠀⠀⠀⢠⢇⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠘⡀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⣀⠇⠀⠀⠀⠀⠀⢸⡸⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠽⡄⠀⠙⢤⣀⣀⣀⡀⣀⠜⠁⠀⣼⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⣀⣀⡈⠢⢄⣀⡬⠭⠆⠈⠀⣀⡤⠊⠀⣀⣉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠑⠲⠤⣉⡓⠒⣉⠭⠕⠒⠋⠉⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄⡀⠀⠀⠀⠀⠀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠓⠒⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣿⣿⣿⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠁⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⡛⢿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⠚⠙⠃⠀⠀⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⡟⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⣿⣿⣿⠀⠙⠚⠁⣸⠀⠙⠚⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣁⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣇⠀⠀⢀⣀⣀⠄⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣄⣼⣿⣿⣿⣿⣿⣿⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢿⣏⠀⢀⡠⢟⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉⢹⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠈⠉⠙⠋⠉⠀⠀⠉⠻⢿⣿⣿⣾⣿⣦⣛⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠻⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢿⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⡿⡀⠀⠀⢸⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠃⣇⠀⠀⠀⠀⠀⠀⠀⢸⠃⡇⠀⠀⢸⠀⠀⠀⠀⠈⠉⠉⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⡀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢧⡝⠀⠀⠀⠀⠀⠀⠀⢸⣴⠃⠀⠀⡟
⠀⠀⠀⠀⠀⠀⣀⣤⠴⠷⡄⠀⠀⠙⢦⣄⡤⠤⢤⣀⣠⠞⠁⠀⢀⡼⢧⣀⡀
⠀⠀⠀⢀⡴⠋⠉⣀⣀⠀⠙⠲⣄⣀⣠⠤⢍⠵⠀⠀⠀⠀⣀⡴⠋⠀⠀⢈⠙⠳⣄
⠀⠀⢠⠏⠀⠀⠀⠀⠉⠙⠛⠲⢤⣌⡉⠓⠶⠶⠶⠖⢚⣋⡥⠴⠒⠛⠋⠉⠁⠀⠈⣧
⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠙⠲⢤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣄⣀⠀⢀⣀⣀⣀⣠⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠛
⠀⠀⠀⠀⠀⠀⢀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠴⣦⢴⡴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠪⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠂⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠋⠄⡿⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀
⠀⠀⠀⠂⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠀⠀⠀⣴⣿⠛⣉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⡦⣦⣴⡾⠛⢛⠛⠻⡏⢈⡙⣷⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡟⢁⣀⣰⣏⢀⡘⠛⠙⢁⣹⠏⠀⢸⢞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡔⠓⣄⣭⣼⣿⠃⢸⡄⠀⠈⠉⠀⠶⠤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢻⣿⣿⠇⠈⠃⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣿⣧⣀⣀⡁⠐⣶⠑
⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡘⣻⠿⣿⣿⣝⣛⠁⠄
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⠓⠾⢶⡅
⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠉⠀⠀⠀⠁
⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠉⠁
⠀⠀⠀⠀⠈⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⢹⡔⠲⠀⠀⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢨⣧⠀⠀⠀⠀⢀⣾⣇⠀⠑⠀⠂⠀⠀⠀⠀⢄⢘⣿⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠁⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠘⡇⠱⢄⠀⠀⢬⣹⣟⣶⡄⢸⣦⠀⠀⠀⠂⠘⣾⣿⣿⠀⠀⠀⠄⠙⠳⢡⣍⣷⡀⢀⠠⠀⢠⠆
⠀⠀⠀⠀⠀⠀⢹⣧⢀⣀⣤⠟⢹⣿⣿⣦⣼⣟⣧⡄⣮⣵⣙⢿⣿⣿⣷⣆⣴⠓⠿⣿⣿⣿⣿⠁⠤⠄⠐
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠁⠀⢸⣿⣯⡏⠀⠈⠉⠉⠛⠛⠛⠉⠉⠉⠙⠻⠿⠀⡀⠙⠛⢹⡟
⠀⠀⠀⠀⠀⠀⠸⠹⣿⠀⠀⠀⠈⢿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢸⣇⣤
⠀⠀⠀⠀⠀⠀⠀⡃⣿⠀⠀⠀⠀⠘⢏⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⡏⠃
⠀⠀⠀⠀⠀⠀⠀⡃⣿⠀⠀⠀⠀⠀⠈⠎⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠸⡏⠉⠀⠀⠀⠀⠀⠀⣷⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠐⢾⡿⠤⣤⣀⣀⠀⠀⠀⠾⠿⢇⣀⣀⣀⠀⠀⠀⠀⠀⠀⣃⣀⣀⠀⠙⣂⢀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣛⣃⣀⣈⣛⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣋⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡀⠀⠀⠀⠀⢀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⢿⣆⠀⢸⣆⠘⠋⣀⠀⠀⠀⣷⡆⢀⠀⢠⣄⠀⠰⠀⠀⠀⢀⠘⠇⠀⠀⠀⢠⣄⠁⣀⠀⠀⠠⣤⣁⣼⠂⠀⠀⢀⠛⢀⠂
⠀⠀⠈⠻⠆⠈⡛⠀⠾⠋⠀⠀⠀⠈⠁⡿⠁⠀⠁⠲⠀⠀⠀⢀⡘⢧⡶⠂⠀⠀⠀⣉⠾⠋⠀⠀⠀⣬⡙⠁⣀⠀⠀⠈⣈⠺⠈
⠀⠀⢶⣦⡀⣰⠆⢀⣴⡶⠀⠀⠀⠀⠃⠀⠀⠀⠰⡀⢀⡀⠀⠈⣿⣆⣴⡇⠀⠀⠈⢻⣧⡶⠀⠀⠀⠉⠛⣿⡟⠀⠀⠀⠉⠲
⠀⠀⠀⠉⠀⡿⠀⠈⠁⠀⠀⠀⠀⠶⠶⠂⠀⠀⠀⡹⠈⠁⠀⠀⠈⠉⠋⠀⠀⠀⠸⣇⣡⡄⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⢈⡙⡔⠗
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠈⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⢀⡀⠀⠀⠉⠀⠀⠀⠰⣶⡄⢠⡄⠀⠰⢾⣶⢄⢿⠄⠀⠀⠸⣶⣾⣯⡔⠀⠀⢀⡀⠀⡄⣧⡀⠀⢰⣿⠀⠀⡀⠀⠀⢠⡄⠀⠀⡀
⠀⠀⢈⡄⠈⠀⠃⣀⠀⠀⢠⣌⠃⢸⠇⢀⡀⠀⠺⠃⠙⢇⣴⡆⠀⠀⠁⠀⠀⠀⠰⣌⢰⣧⡿⠆⠀⢠⣜⣡⠀⠀⠙⠳⠀⠘⠃⠀⢚⡁
⠀⠀⢀⠀⡄⠀⠀⣨⠀⢰⣬⡛⠀⡆⢰⣿⠃⠀⠀⠀⠶⢌⣾⠀⠀⠀⣠⠀⠀⠀⢳⣶⡧⠍⠀⠀⠀⢦⡟⢋⠀⠀⢠⣤⣦⣶⣤⠛⠉⠁
⠀⠀⠉⠀⠀⠐⠀⠀⠀⠘⠿⣷⠀⠄⠈⣁⣀⠀⠀⠀⠀⠀⠁⠀⠐⢶⣿⣶⠆⠀⢸⠃⠀⠀⠀⠀⠠⣤⣷⣿⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠐⠛⠛⠛⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠉⠉⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣁⠤⠤⠤⠤⠤⠤⠤⢄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢳⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡇⠀⠒⠆⠀⠀⠀⠀⠖⠆⠀⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠒⠒⠀⡆⠀⠐⠒⠂⠀⠇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣧⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡀⠀⠀⠂⠂⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠙⣦⢄⣀⣀⢴⡊⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣇⠀⠀⠈⠙⢄⠀⢀⠜⠉⠀⠀⢀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠲⢤⡈⢦⠏⣠⠴⠂⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣴⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢳⡄⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠙⠢⡄⠀⠀⢀⣾⠥⠖⠓⠒⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠒⠓⠶⢽⣆⠀⠀⠀⡤⠚⠉⠹⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠟⠒⠤⣀⠀⢸⣀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠒⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⣸⠀⢀⡤⠔⠚⢧⡀
⠀⠀⠀⠀⠀⠀⣼⢃⠀⠀⠀⠈⢶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠎⠀⠀⠀⢀⢹⡄
⠀⠀⠀⠀⠀⢰⡇⡏⣸⠁⣴⠂⡜⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⠒⠦⣤⠖⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠘⡄⢲⡄⢹⡈⡆⣷
⠀⠀⠀⠀⠀⠘⢷⡀⡇⢸⠙⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⠶⢤⣄⡤⠴⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠱⠜⢹⠀⡇⣰⠟
⠀⠀⠀⠀⠀⠀⠀⢻⡓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢸⠁⠀⠀⠀⠀⠈⠓⢤⣀⡀⠀⠀⠀⠀⠀⠈⠒⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⠀⠀⣀⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠄⠤⠀⠂⠈⠉⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠠⢠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠂
⠀⠀⡠⠊⢀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠒⠆⠀⠘⠘⠃⡀⠰⠂⠐⢄⢀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠈
⠀⡈⠀⡀⠀⡀⠄⠉⠀⠀⠀⠠⠁⠀⠀⠀⠁⠢⢸⢂⠤⠄⠀⠀⠀⠀⢤⢶⡜⠄⠄
⠀⠘⠦⠐⠁⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⡂⢖⣤⢠⡿⠄⠀⠁⣧⢀⡁⠩⠴⠄⠠⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⣠⢢⢠⠀⣀⡀⢈⠆⣠⠇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠸⡁⠀⠀⣫⢠⠁⠀⠀⠀⠀⠀⠀⠀⢀⢸⠆⠤⠤⠤⢴⡀⠌
⠀⣀⠀⠄⠀⠤⠄⠀⠤⠄⠀⡔⣆⠒⣵⡔⣰⣵⣌⠀⠒⠂⠈⠁⠁⠈⠓⠤⢼⡾⠐⠒⠂⡼⢸⠂⠐⠒⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⢠⢇⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠃⠀⠀⠶⠥⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣬⣭⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⢹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⣀⠺⣿⣿⣿⣿⣿⣿⣿⢇⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣷⣬⡻⠿⣿⠿⣋⣵⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣌⣙⠺⠙⠿⠿⠟⠋⣁⣡⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣶⣌⣡⣶⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉




⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠠⠀⠀⠀⠀⠀⠰⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢢⠀⠀⠀⠤⠄⠀⢀⠀⠠⠀⠀⠐
⠀⠀⠀⠀⠀⠠⠒⠒⠿⣷⣤⡉⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⡖⠔⠠⠀⢀⡤⠔
⠀⠀⠀⠀⠀⠀⠀⠉⢠⡅⡀⢿⢀⡡⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⣀⠀⠈⢽⠛⢻⡟⠀⠀⠉⠂⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠠⠀⠀⠀⢀⣀⡀
⠀⠀⠐⠂⠰⠾⠗⢾⡶⢅⢔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠐⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⠄⢄⡠⢂⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠱⣔⠀⢀⠀⡀⢀⠄⠀⠀⢠⠀⡄⠄⠀⠀⣇⠀⢀⢀⠀⠀⠀⠈⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡜⣥⣀⣀⠉⠀⠈⡀⠰⣰⠈⠀⠁⢰⢰⠀⠎⢠⠸⠘⡤⠃⢠⠰⡀⠆⠠⠀⠘⠀⡆⠀⠀⠀⠀⢸⡄⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠿⣁⣧⣇⠐⡀⢠⠂⢁⠄⠀⡞⡏⠋⠐⢁⡾⢐⠀⡉⠆⡅⡀⣏⠄⡀⠀⠀⠀⠁⠀⠀⠀⠀⣼⠃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⡭⡴⣷⣧⠐⣀⠀⠈⠂⢘⠁⠗⣀⠀⡤⡇⢙⢸⠀⣢⡽⢰⠙⡞⠃⠀⡄⠀⡆⠆⠀⠀⢀⡟⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⣧⣡⣽⣰⡧⠴⠀⠇⢾⣆⠀⢹⣷⡸⠃⠊⠸⢀⣟⡃⢰⡯⠄⢠⠀⠰⠀⠈⡸⠀⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣿⣿⢆⡀⠀⠀⢹⣧⢣⡢⢸⣟⢡⢂⠚⣼⡿⢢⣵⣴⣷⣶⣷⡖⡉⢠⢣⠂⡇⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏⣿⡧⠀⠀⢰⠏⠉⠙⠷⠾⣯⣮⣿⣾⣿⠷⠟⠛⠉⠀⠀⠙⢿⣿⡘⣺⠢⢄⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⣷⡄⣼⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡙⢦⡼⣺⠓⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣹⠁⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡟⢯⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢠⠟⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡇⠘⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠈⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢀⡜⢀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⢤⠚⠀⠀⡟⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠀⠞⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠏⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⣤⣴⡔⠦⠑⢔⣠⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠉⠋⠠⠀⠛⣿⡿⠋⠀⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀
⠀⠀⠀⠀⢉⣠⠆⠀⠲⣤⣄⡀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠂⠀⠄⠠⠀⠤⠤⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠙⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⢨⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢾⣤⡃⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧⣝⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⡀⣌⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡗⠀⠀⠀⠀⠀⠀⠒⢹⣈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣏⡠⠂⠀⠀⠀⠀⠀⠀⠐⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢏⠞⣧⡀⠀⠀⠀⢀⢰⣼⣥⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢎⠶⠂⠀⠀⠀⠀⠀⣤⠾⣷⠼⣦⡀⣶⢯⡼⣆⢤⣼⣼⣟⣻⣷⣦⡀⠀⠀⠀⠈⣓⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⣿⡅⠀⠀⢀⠀⢠⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠠⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡃⠀⠀⢸⣿⣿⡿⠀⠈⠋⠉⠛⠛⠻⠿⠿⠿⣿⠿⠟⠈⠈⠛⢿⣿⣦⡀⠈⠹⢿⣰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⡄⠀⢱⣾⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣽⠃⠀⠈⢹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣼⠁⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡔⣰⣶⣤⣄⣤⣄⡄⣠⣶⢿⢶⣿⣿⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⢀⣴⣶⣥⣶⣴⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠈⠘⠋⠋⠘⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠛⠉⠁⠈⠈⠁⠉⠁

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

⣶⣶⣶⣶⣶⡶⠶⠶⠶⠶⠄
⠛⠛⠋⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣤⣤⣴⣿⣸⣿⣿⣿⣟⢻⣧⣤⣴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣶⣶⣶⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣦⣤⡀⠀⠀⣀⣤⣶⣶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣦⣿⣿⣿⣿⡄⣰⣿⣿⣿⣥⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡏⣹⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣀⠻⣿⣿⣿⣿⣷⡌⣿⣿⣿⣿⡇⠻⠿⠿⠿⠋⢴⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢠⣿⡇⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣷⡜⣿⣿⣿⣿⡿⠗
⠀⠀⠀⠀⣾⣿⣧⡹⢿⣿⣿⣿⣧⠻⢿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⢏⣶⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣶⣶⣶⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠿⠿⠿⢿⣿⣿⣟⣛⣩⣴⣿⣿⣿⡇
⠀⠀⠀⠀⢹⣿⡿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⣷⡜⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠈⢩⣴⣿⣿⣿⣿⣎⡻⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⡿⣩⣍⢋⣴⣷⣆⢀⣤⡄
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⢋⣶⣿⣿⣿⡆⣿⣿⢿⣿⣿⣿⠛⢿⢡⣿⡏⣾⣿⣿⣿⣇⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧⣾⣿⣿⣿⣿⡇⣩⣵⣿⣿⣿⡇⣿⣦⡘⢿⡁⣿⢟⣉⠻⣿⢸⡿⣠⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⠃⠹⣿⣿⣿⣿⡆⣿⣿⣿⣦⡀⢡⣾⣿⣷⡉⣠⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠈⠙⠛⢉⣁⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣌⡛⠿⣿⣿⣿⠿⣿⣿⡿⠿⣋⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠶⠶⠒⠀⠒⠶⠶⠿⠟⠋



⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣿⣛⣛⣛⣿⣛

⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⣀⡀⠀⡀⠀⠀⣀
⠟⠛⠿⠟⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠯⠭⠉⠤⠀⠤⢼⣿⣿⣤⠭⣿⣭⣭⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣷
⣄⣀⠀⣀⢀⣀⠀⠀⠀⢀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣸⣿⣇⠠⢤⣄⠀⠀⠠⠤⠨⠿⠉⠁⠉⠈⢁⠀⠁⣀⣀⣈⣉
⣧⣼⣿⡿⠚⠿⢿⣿⣶⣾⣿⣿⣿⣿⣿⣷⣶⠶⠂⣠⣼⣟⣿⣿⣿⣷⣾⣿⣿⣶⣶⣷⣶⣖⣠⣶⣤⣤⣼⣿⣶⣿⣿⣯⣿
⠉⠉⢀⣤⠶⣾⣶⡞⠛⠛⠛⠛⠛⠛⠛⠛⣁⣶⣿⣿⣿⣿⡿⣿⠿⠿⠿⠿⠛⣛⣛⡻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿
⠀⢀⣾⣿⣴⣿⣿⣟⣿⣤⡀⠀⠀⠀⠀⢀⣽⣿⣶⣿⣿⣿⣿⡿⠀⠀⢀⣴⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⡀⠀⣀⡄
⠀⠀⠛⢋⣉⣤⣈⣉⣹⣿⣷⣾⣿⣦⣴⣿⣿⣿⠿⠿⠛⠋⠁⢠⣄⠰⣏⣽⣿⣿⣿⣿⠛⣷⣶⣤⠴⠒⠉⠉
⠀⠀⢐⣷⣿⢋⣹⣿⣿⠉⣉⣉⠁⠀⠀⠀⠀⠂⢠⣀⣠⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣿⣿⠋⠀⣀⠀⠀⠀⣀⣴⣾⣤
⠀⠐⠛⠛⠿⢿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⣀⣀⣩⣙⣿⣿⣿⣿⠿⠛⠉⠉⡉⢉⠉⣉⠛⠃⠀⣼⢿⣷⣾⣾⣿⣿⣿⣿
⠀⢀⣀⣠⡀⠀⠉⠉⠋⠀⠀⠀⠀⢀⣀⣶⣿⣿⣿⢿⣿⣿⣄⣀⡀⢀⠴⢢⣿⣿⣿⣄⡀⣼⣿⣿⡟⣿⣿⣿⠿⠿⠛⠁
⠆⣿⣿⡟⠁⠀⠀⢀⠀⠀⠀⠀⢼⣿⣿⣿⣿⠁⠀⠀⠘⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠟⠁⠀⠀⠠⠄⠀⢀
⠀⠀⠀⠀⢀⣀⡀⢉⣶⣿⣿⣿⣦⣀⣸⣿⣿⣄⠀⠤⠄⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡦⠀⠀⠀⣀⣴⣾
⠀⠀⠁⠺⣿⣿⡷⠞⠛⠛⠛⢿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠠⡄⡀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢈⣩⣿⣿
⠀⠀⠀⠀⣡⣿⣿⡆⠀⠀⠀⠙⠛⠛⠻⢿⣟⢁⣸⣶⡆⠀⠀⠀⣈⣦⣀⣤⣀⡀⠻⣿⣿⣿⣿⠁⠀⠀⠀⣾⣿⣹⣿⣿⣿
⠀⠀⠀⢠⣿⣿⣿⣷⡄⠀⠀⠀⣴⣶⡶⣦⣤⣿⣿⣟⠃⠀⠀⠀⠈⠁⠀⠉⠙⠃⠀⠈⠛⢿⣏⡀⠀⠀⢀⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⡄⣠⣶⣶⣾⣈⠙⠉⠉⠉⢹⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣀⡤⣤⡄⠀⡀⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣿⡧⠙⢿⣿⣿⡅⠀⠀⠀⠀⢸⣿⣿⣧⡀⠀⢀⣀⠀⣠⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣼⣿⣧⡄⣼⣌⣻⣿⠀⠀⣀⣠⣼⣿⣿⣿⣿⣤⣤⣴⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣰⠦⠂⣸⣏⣿⣿⣿⣿⣿⣻⣿⣇⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠀⣶⣻⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣇⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠂⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉
⠀⠀⠀⢌⠎⠆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰
⠀⠀⠀⠀⠁⠀⢸⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣷⣣⠂
⠀⠀⠀⠀⠀⠀⠸⠀⠐⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠄⠜⢂⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡥
⠀⠀⠀⠀⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⢀⡀⠉⣀⣼⢏⡴⠒
⠀⠀⠀⠀⢡⠈⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡌⠀⠆⠀⠀⠀⠀⠀⢀⡡⣼⣿⣷⡈
⠀⠀⠀⠀⠀⢣⡉⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠠⠐⠀⠀⠀⠀⢀⣵⡿⠉⡄
⠀⠀⠀⠀⠀⠀⠳⡀⢣⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⢹⣟⡿
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⡁⠈
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠄⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣠⢆⣉⡀⠀⢉⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣟⡇⠹⢯⡿⣏⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠀⠆⠔⠲⠄⠤⠤⠤⠤⠦⠠⠤⠤⠴⠾⠋⠀⠳⡄⠀⠀⢄⠈⠀⠈⢂
⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣐⣴⣴⠧⠀⠨⠍
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡇
⠀⠀⠀⠀⠀⣾⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣻⡇
⠀⠀⠀⠀⠀⣷⠀⠀⢀⡀⠀⠀⢀⠈⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠨⣽⠇
⠀⠀⠀⠀⠀⣿⠅⠀⠘⣷⠀⠀⡮⠔⣐⠀⡀⠐⡄⡆⠀⠀⠀⠄⠀⡄⡀⠀⠀⠐⠀⢉⣿
⠀⠀⠀⠀⠀⢻⣿⠄⠀⣿⣷⣿⣺⣤⣽⣷⡿⣿⣿⡧⡀⠀⠀⡰⣞⣷⣿⣱⣕⣆⣤⡼⠃
⠀⠀⠀⠀⠀⢸⡇⠷⢂⣻⡿⣿⣿⣿⣿⣿⣿⠿⠿⠷⣊⠡⠔⣒⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠸⣧⣼⣾⠟⠀⢸⣿⡿⠃⠀⠀⠀⠀⠀⢹⣿⣷⡟⠀⠀⠻⣿⡿⠃
⠀⠀⠀⠀⠀⠀⣿⢹⠁⠀⠀⢸⡷⡇⠀⠀⠀⠀⠀⠀⠀⠧⣾⠃⠀⠀⠀⢹⡿
⠀⠀⠀⠀⠀⠀⢸⠸⡀⠀⠀⠀⣇⢁⠀⠀⠀⠀⠀⠀⠀⡰⠘⠀⠀⠀⠀⠸⠃⡄
⠀⠀⠀⠀⠀⠀⠸⣦⠓⠀⠀⠀⢾⣌⠀⠀⠀⠀⠀⠀⠀⣅⢠⠀⠀⠀⠀⠰⡆⠡
⠀⠀⠀⠀⣀⣀⣤⣽⣿⣦⣄⢠⣼⣿⣷⢤⡶⠾⠿⢶⣦⣽⣿⣦⣶⢴⣶⣶⣽⣿⣧⠶⡴⢦⣠⡄
⠀⠀⠀⠀⠉⠉⠁