Spidermanlogo Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢳⢱⢰⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢧⡸⠂⠋⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⠁⠀⠐⠣⢤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⣠⡤⠒⠚⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡤⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⠄⠶⢻⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠗⠀⡽⠀⣰⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠶⣾⡁⠀⠀⡆⣤⠾⣫⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⣷⡀⢠⡟⣾⠟⠁⣀⡤⠴⠖⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣾⢿⡀⣿⠃⣠⠞⣕⠈⣠⢀⠀⣀⣄⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⣶⣄⣀⡀⠀⠨⡏⢸⣷⢸⣶⠃⢊⠁⣴⡿⢈⣿⣿⣿⢆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠘⢿⣍⡛⢳⡄⠻⠛⣿⠺⢃⣴⣏⣾⡟⣴⣮⢹⣿⡏⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢸⡇⢀⠀⠤⢽⣿⣟⡀⠀⣠⣇⣠⣿⣿⣿⣿⢴⣿⡿⣼⣿⣗⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣉⠀⠉⠛⠛⠛⢾⡟⠁⡸⠗⠀⠳⣌⣉⡛⢛⢿⣸⣿⣿⣟⣉⡇⠉⠙⢛⣻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣀⣀⣿⣇⡀⢷⣦⣄⣸⠀⠀⣿⠾⣷⡶⠊⠉⢛⣿⠀⠈⢻⣿⢿⠹⠁⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢾⠻⣿⣯⠱⣄⣈⡻⣿⣦⡧⢘⢾⠋⠀⠀⠘⡟⠛⠓⡄⠀⢿⠻⣀⡀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⢄⡹⢿⣿⣿⣿⣽⣟⣷⣼⠀⠀⠀⣤⠴⠃⠀⠀⠘⠢⣜⢀⡄⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⣙⠶⣽⣻⡿⠁⢯⣉⣹⣄⣠⠴⠟⠀⠈⠳⣄⠀⠀⠈⣿⠁⢠⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠮⣭⣄⣠⠖⠛⠛⠉⢀⣤⡴⠶⣄⠀⠙⠷⣤⣀⣇⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢶⡶⠚⠉⠯⣉⠛⠻⣦⣀⣄⣠⡽⠛⠳⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⡄⠀⠹⡄⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⣄⣀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⢾⠛⠛⠻⠿⢶⣤⣀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠛⠛⡷⠶⣤⣀
⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠳⡀⠀⠀⠀⠈⠙⢷⣄⠀⣠⡾⠋⠀⠀⠀⠀⢀⠞⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⢠⡿⠋⠀⢠⡄⠀⠀⠀⠙⣄⢀⣠⠤⠚⠉⠹⡿⠋⠉⠲⠤⣄⣀⣠⠋⠀⠀⠀⣰⡄⠀⠈⢿⡄
⠀⠀⢀⣿⠳⢄⡀⣾⣿⣶⣤⣄⣠⠚⢧⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⡼⠙⣦⣠⣴⡿⢻⣇⢀⡠⠞⣿⡀
⠀⠀⢸⡇⠀⠀⠈⣿⣿⡿⣿⣿⣿⣷⣦⣳⡀⠀⠀⠀⡇⠀⠀⠀⢀⣾⣶⣿⣿⣿⣿⣷⣾⣿⠁⠀⠀⢸⣇
⠀⠀⢸⡇⠀⠀⠀⣿⣿⡇⠀⠀⠈⠉⠛⢿⣷⣤⠴⠚⡟⠓⠂⣤⣿⠟⠛⠉⠁⠀⠀⢸⣿⣿⠀⠀⠀⠈⣿
⠀⠀⢸⡇⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⡇⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⢰⡿
⠀⠀⠸⣿⠉⠁⠀⢻⣿⢻⡄⠀⠀⠀⠀⠀⠀⠀⠙⣦⣧⡜⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠈⠉⠉⣽⠇
⠀⠀⠀⠹⣧⠀⠀⠀⣿⡀⠻⣄⠀⠀⠀⠀⠀⠀⣴⠛⠶⡻⣄⠀⠀⠀⠀⠀⠀⣰⠟⡸⢻⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠹⣧⡀⢀⣧⡨⠦⢬⣿⣶⣶⣶⣶⠿⠋⡞⠀⢫⠙⢷⣦⣤⣤⣖⣋⣡⠼⠅⢸⡀⠀⣴⠏
⠀⠀⠀⠀⠀⠈⢿⣍⠉⠲⡀⠀⠀⣠⠿⠥⢤⡀⢠⠁⠀⠀⡆⠀⡨⠽⢏⠉⠉⠀⢀⡔⠉⢨⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⢹⣀⠜⠁⠀⠀⠀⠈⡟⠊⠉⠐⢺⡂⠀⠀⠀⠑⢄⠀⡎⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣯⡉⠑⠦⡀⠀⣰⠀⠀⠀⠀⠀⢧⠀⠀⣀⡠⢬⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣄⠈⢦⡧⠔⠒⠒⠒⠂⠼⡦⠊⣡⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣾⣀⠀⠀⠀⠀⠀⣠⣽⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣠⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⡿⣿⡿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣻⠋⢉⣀⣀⡀⠀⢸⠀⠀⣀⣀⣈⠉⢻⣻⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⠉⢠⠌⡝⣩⠉⠉⠉⠉⠉⢻⠋⠉⠉⠉⠉⢩⡙⡏⢤⠈⠙⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠏⠀⠀⢀⡿⢠⢃⡏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⣇⢣⠸⣇⠀⠀⠈⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣫⡏⠀⠀⠀⣼⡇⢸⣼⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢿⣼⠀⣿⡄⠀⠀⠈⣯⣻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁⡟⠉⠉⠉⢹⣿⠦⣼⣿⣀⣀⣀⡀⠀⠀⢸⠀⠀⠀⣀⣀⣀⣸⣿⡤⢾⣿⠉⠉⠉⠙⣇⠙⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⡜⠀⠀⠀⠀⣾⡏⠀⣿⡿⠀⠀⠀⠈⠉⠓⢾⠖⠋⠉⠀⠀⠀⠸⣿⡇⠈⣿⡆⠀⠀⠀⠘⡄⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⢰⠁⠀⠀⠀⢰⣿⠇⠀⡿⠷⠶⣶⣶⣄⠀⡀⢸⠀⠀⢀⣴⣶⣶⠶⠿⡇⠀⢿⣷⠀⠀⠀⠀⢱⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⡟⠀⠀⠀⠀⢺⣿⣶⣶⣷⣶⣶⣶⣦⣿⣿⣷⣾⣶⣿⣿⣧⣶⣶⣶⣶⣷⣶⣾⣿⠂⠀⠀⠀⠘⡇⠀⢻⣿⡇
⠀⠀⠀⠀⠀⢰⣿⣿⣄⢰⠃⠀⠀⠀⠀⠀⠙⠋⢸⡇⢀⣀⣀⣩⣿⣿⣿⣿⣿⣿⣿⣏⣀⣀⣀⠀⣿⠈⠙⠁⠀⠀⠀⠀⠀⢷⢀⣼⣿⣷
⠀⠀⠀⠀⠀⢸⣿⡇⠈⣻⠀⠢⠤⠤⣀⣀⠀⠀⢸⣴⣿⠟⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⡉⠛⢿⣷⣽⠀⠀⠀⣀⡠⠤⠤⠀⢸⡋⠀⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣇⠀⡇⠀⠀⠀⠀⠀⠈⢓⣶⣿⡿⠁⣠⣾⡿⠋⢡⣿⣿⣿⣧⠉⠻⣿⣦⡀⠹⣿⣷⣖⠉⠀⠀⠀⠀⠀⠀⡇⢀⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⠀⡇⠀⠀⠀⠀⢀⣴⣿⡿⢻⣣⣿⡿⠋⠒⠒⣾⣿⣿⣿⣿⠖⠒⠊⠻⣿⣯⣻⠻⣿⣷⣄⠀⠀⠀⠀⠀⡇⢸⣿⡟
⠀⠀⠀⠀⠀⠀⢿⣿⣴⠁⠀⠀⠀⢰⣿⡿⠋⠀⢸⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠈⠻⣿⣷⠀⠀⠀⠀⢸⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⠀⠀⢸⣿⠁⠀⠀⢸⢸⣿⠀⠀⠀⠀⠈⢿⣿⣿⠏⠀⠀⠀⠀⢸⣿⢸⠀⠀⠀⢹⣿⠀⠀⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠘⣿⠀⠀⠀⢸⠈⣿⠀⠀⠀⠀⠀⠘⢿⠟⠀⠀⠀⠀⠀⢸⡏⢸⠀⠀⠀⢸⡟⠀⠀⠀⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⢿⡄⠀⠀⢸⠀⢿⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⠇⢸⠀⠀⠀⣼⠇⠀⠀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⠲⢼⣧⡀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⣠⣿⠴⢲⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣦⣿⡏⠳⢼⣀⣈⣷⣀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣰⣏⣀⣸⠴⠋⣿⣣⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣄⣸⡅⠀⢻⡇⠉⠑⠲⢄⣸⣀⠴⠒⠉⠁⣿⠁⠀⣿⣀⣼⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣶⣤⣿⣄⣀⡀⠀⢹⠁⠀⢀⣀⣼⣧⣴⣾⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⠂⠀⠈⠔⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠮⠀⠀⠠⠀⠀⢀⡀⠊⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⣬⣁⠔⠂⠁⣢⣴⣦⣶⢵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠈⢻⣿⣿⠁⠀⢀⣿⢹
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢣⠻⠦⢴⠞⠃⠛⠳⠶⠞⠃⣞⡀
⠀⠀⠀⠀⠀⡏⠀⡑⢄⠀⠀⠀⠀⠀⠈⢢⡀⠂⢈⠁⠈⡀⠈⢀⣴⠩⠁⠈⠕⠢⡄⣀⡀
⠀⠀⠀⠀⠀⠈⠢⣄⢂⠕⠚⠶⠢⡴⡒⠪⠝⠢⠤⣄⡠⡤⠒⣋⠀⠄⠀⠀⡀⠀⠀⠌⠈⠓⡢⣤⠤⣀
⠀⠀⠀⠀⠀⠀⢠⠮⢁⡔⡛⢢⠐⢱⠀⠀⠀⠄⠀⠄⠀⡼⢤⡶⡆⠁⠈⡄⠐⠊⠉⠒⠢⠬⡀⠂⢌⡐⢓⡤⡀
⠀⠀⠀⠀⡠⢂⡧⠄⢸⢀⠀⢸⡀⠸⠒⠂⠀⡀⠀⠠⠠⣿⣿⣿⠷⠊⣠⠗⠒⢄⠀⠀⠀⠀⢸⠉⠁⠉⠓⠲⠿
⠀⠀⠀⡎⠒⣀⠞⠦⠼⢀⣠⠈⠒⠂⠀⠀⠀⠙⡀⠄⠂⠘⠊⠋⡱⢻⠁⠀⠀⠀⢃⠀⠀⠀⠈⠚
⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠏⡊⡰⠀⠀⠀⠀⠀⣷⢰⢸⠨⢠⠞⠀⡆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠔⠁⠀⣀⠤⠒⠊⠉⠉⠒⠪⠏⠀⠀⡄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⡇⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⣠⢾⠀⠁⠀⠀⡇⢍⢁⠃⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⢀⣀⢠⣤⡖⠉⡀⢸⠀⠀⠀⠀⡇⠀⠰⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣀⠠⠔⢊⠝⢚⠀⠄⢀⡌⠀⠀⠀⠀⠇⢀⠨⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠤⠖⠒⠊⠀⠀⠀⠀⠈⢦⡉⠠⠀⠋⡱⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠤⣂⣀⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⢳⡄⠀⣠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⢿⡄⣷⢀⡿⢀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⠱⣼⡇⢹⣾⠇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣛⢻⣷⡈⢁⠓⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣹⠀⣠⠁⣰⠾⢧⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣎⠒⣤⠘⠅⠀⣀⣈⣩⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠛⣥⠚⢠⠘⣠⣶⣤⠟⠉⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⣉⠴⡂⢀⡐⢃⣼⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠡⠎⠁⣠⡞⣯⠰⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⣁⡜⣡⣶⣿⢋⣴⣿⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠉⠃⣴⠟⠁⠈⠉⠀⠿⢋⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠗⠁⡞⣹⠟⠀⢀⣼⡇⣸⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠁⠀⠈⠀⣘⠷⠃⠀⣴⣿⡿⢡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠛⠛⠛⠛⠻⢿⣅⡄⠀⠉⠂⡀⣿⡀⢲⣿⠿⣋⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠦⠀⠀⠀⠈⠃⠹⣧⡐⠆⠀⠙⣿⡇⣹⣶⣿⠿⠃⠀⠀⠀⠀⢀⣠⣤⡶⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣤⠘⢶⣿⢣⣿⠛⠁⠀⠀⣠⡶⠟⢋⡟⠋⠀⠀⠀⠀⠈⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡤⠂⠀⠀⠀⠀⠛⠀⠀⠈⠁⣹⣿⣿⡆⢘⣷⣾⠃⠀⢀⣴⡿⢋⡴⠚⠋⠀⡀⢀⠀⠀⠀⢀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡆⠀⠀⢤⠄⠀⠀⠀⣀⡀⢶⣿⡏⢻⣿⠘⡻⣧⠀⣠⣾⠋⢀⣿⠆⠀⣠⣾⡇⢼⣅⣴⣿⣿⣷⣌⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠹⣧⢹⣆⡀⠈⠢⣤⠀⠁⠀⢸⣿⠀⢘⣿⣷⡇⢿⣴⡿⢁⠞⠉⡀⠀⣴⣿⣿⡇⢘⣿⣿⣿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠙⠛⣿⣿⣿⠿⣿⣶⣶⣄⠃⢸⣷⣤⣾⣿⣧⣥⢸⡿⠁⢁⣴⡿⢁⣼⣿⡿⠉⣤⠸⡏⢿⣿⣿⡿⢃⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠖⠈⠡⠝⢻⣿⣦⣄⠈⠙⣻⡿⠈⢻⡛⠉⢸⣧⢨⡿⡅⣰⣿⣿⣣⣾⣿⡟⢠⣾⣿⣧⠀⣿⣿⣿⡷⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠛⢀⠘⢁⣝⢿⣿⣿⣿⣿⠛⠂⠀⠀⠂⣼⢧⡙⢢⣿⣿⣿⣿⣿⣿⣿⣁⣾⣿⣿⣿⢀⣿⣿⣿⣴⢀⡿
⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⢿⡟⣁⣤⣌⠔⣻⠟⠛⢿⡛⢿⣤⣤⣁⣀⣀⣴⣟⣸⡗⣿⣿⣿⣿⣿⣿⣿⣿⡛⢿⣿⣿⡏⣾⣿⣿⣿⣣⡾⠁
⠀⣠⣾⠛⠉⠀⠀⠀⠀⠈⠉⠻⠟⠻⠿⠿⣷⣾⣿⢿⠄⢀⣼⣿⡶⠤⠴⢿⣆⠀⣍⠛⣿⠿⠟⢿⡏⠱⢸⣿⣿⣿⣿⡿⠋⠸⣟⠂⣍⠛⠾⠿⠿⢿⣿⠟⠁
⢰⡿⠁⠀⠀⢰⣿⣦⠀⠀⢱⣄⠀⠀⠀⠀⠀⣿⠃⡆⠀⢼⣅⢀⣀⡀⠀⠀⣹⠷⢶⣶⣶⣤⣼⠛⠛⠛⢿⣿⣿⣿⣿⣶⡿⢶⡟⢤⠀⢈⠁⣰⣶⠿⠃
⣿⡇⣴⠀⢀⣸⣿⣿⣀⠘⠤⣿⣿⣶⣦⡀⠀⣿⠠⠆⠀⣻⠿⠿⠿⣿⣿⣿⠶⠂⠀⠈⠛⣿⣿⠀⠐⠀⠀⠙⣿⣛⣭⣿⠳⠿⣁⠃⠤⢁⡼⠟⠁
⢹⣧⡹⣿⣿⣿⣿⣯⡹⡇⡀⠀⠙⠻⣿⣿⣿⣿⡀⣸⡇⠘⣿⡆⣸⣿⠋⠁⠐⠉⢠⣤⡴⣿⣿⣷⡐⠀⠀⠀⢹⣿⢦⣿⠐⠠⢄⠉⣴⠟
⠀⠙⠿⣮⠻⣿⣿⣿⣧⡀⢻⣶⣦⣤⣈⠻⣿⣿⣷⣼⡿⢀⣸⠻⣿⠁⠄⠀⠀⠀⠀⢠⣿⠉⡉⠀⣙⣷⡀⠀⠈⣿⠀⠻⣦⣃⡆⣸⡏
⠀⠀⠀⣿⡀⡀⠉⠻⣿⣿⣾⣿⣿⣿⣿⣷⢹⣿⣿⣿⣧⢨⣏⠀⠀⠘⠄⣀⠀⡎⢀⣨⡿⠀⠀⠀⠉⠉⠷⣆⡀⣽⠄⢀⡈⠉⣙⢻⣇
⠀⠀⠀⠹⣷⡘⣦⣴⣌⠻⣿⣿⣿⣿⣿⣿⣿⢿⣦⣭⣿⣾⣿⠃⢀⠀⠀⠈⣸⣿⠛⠉⢠⣀⠀⠀⠀⠀⠀⠈⠻⣧⣄⣰⣿⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠘⢿⣮⡛⠻⣷⣦⣙⠻⣿⣿⣿⠋⠸⣿⡉⠀⠈⣿⡄⠹⢄⣥⣠⣿⣿⠀⠀⠀⠛⣷⣄⠀⠀⠀⠀⠀⠀⢹⣿⠋⠀⢠⣄⣼⠃
⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣍⠿⣿⠶⢟⡁⠘⠤⠈⠻⣾⣷⣿⣿⡶⠛⠉⠉⠀⠀⢲⡀⠀⠀⠈⠻⣷⡀⠀⢠⠆⠀⢿⡟⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣶⣶⣤⣀⣃⣴⠾⠛⠛⠋⠉⠡⠀⠀⣀⣤⣶⠾⢾⣿⣤⠀⠀⠀⠙⣿⣶⣤⡔⢁⢼⡇⢀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣇⠀⠀⠀⢀⣀⣠⣶⣾⣿⠛⠿⢶⣶⣬⣿⣦⠀⢀⡴⢁⡈⢉⠛⢻⣾⢿⣇⡀⠂⣈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⣿⡛⢋⡑⠂⡈⠓⢦⣄⡀⠀⠙⠿⣷⣯⣴⣯⣴⣥⣦⡟⠁⠀⠉⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡐⠶⠉⠠⠄⠀⠙⣦⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠛⣆⡀⠗⢄⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣈⣡⡀⠀⠀⢻⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠌⠉⠀⠋⠉⠀⠀⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣋⠤⠚⠁⠀⠰⠀⠀⢀⠴⠈⠁⠈⠉⠳⠶⢤⡀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠛⠛⠛⠛⠛⠳⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠾⣃⠀⠉⠀⠀⣀⣀⡀⠀⠀⠈⠀⠀⠈⠉⠀⢀⣀⣠⡉⣦⠀⠀⠰⢆⣴⠏⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⢳⣆
⠀⠀⠀⠀⠀⠀⢠⡈⠀⠈⠀⠀⠻⣿⣿⣿⣿⣶⣦⣄⣀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠶⡾⠃⢀⡀⠀⠀⠀⠀⠉⠀⠀⠀⣀⠀⢻⠆
⠀⠀⠀⠀⠀⢠⠈⢳⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡸⢇⠀⠀⠀⠀⠙⢿⣿⣿⣿⠁⠀⠙⣻⣿⣿⣿⣿⣿⣿⡀⠀⠀⠛⠁⠀⠰⣿⣿⣦⡀⠀⠀⠀⠠⠄⠀⡀⠀⠀⢀⡀⠃
⠀⠀⠀⢀⡘⠃⠈⠀⠀⠀⠀⠂⢠⣴⠟⠁⠀⠀⠀⠙⠷⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣿⣿⡿⣷⣦⡉⠀⠀⠀⠈⠁⣤⣴⣿⣷
⠀⠀⢀⡜⢳⡄⠀⠀⠀⠶⣠⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠚⢿⡄⠀⠀⠀⠀⢰⡟⢿⡆⠈⠙⢿⣆⠀⢀⣴⡿⠟⢛⣿⣿⣀
⠀⠀⣨⡁⠀⠁⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠈⢠⡾⠿⣦⣤⣴⡿⢠⣿⡉⠀⣠⣾⣯⡙⠿⠷⣦⡀
⠀⢀⡯⠀⠂⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠛⠀⠀⣤⠀⠐⠃⠈⡇⠀⡈⠙⠉⣀⠀⢨⠿⠟⠋⠀⣿⡆⠀⠀⠈⢿⣦
⠀⠻⠃⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣴⠆⠀⠀⠀⠀⠀⡀⢰⡎⠁⠀⣀⠉⠀⠀⠀⠰⢄⣼⠇⠀⠘⠀⠀⠈⢹⡇
⠠⠆⠀⠀⠀⠠⠀⠀⠉⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠈⠁⠀⠀⣄⡀⠙⠀⠀⠋⠀⣠⠜⠁⠀⠶⠀⠀⠀⠀⠈⢷⡄
⠷⠀⠘⠂⠀⢀⠀⢀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠏⠀⠀⠀⠰⠆⢻⣿⣶⣦⣴⣶⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠂⣀⠃⠀⠀⠀⢀⣸⠿⣄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠹⢿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠘⣷⡄
⠀⠈⠒⠀⢠⣤⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⠀⠤⠀⠀⢁⣠⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣴⡆⠀⠀⢳⣾⣿⣄⠀⠁⠠⠤⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡄⠰⣆⢹⣿⣿⡶⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀⠉⠻⣿⣿⠀⠀⠀⠙⠀⠀⠀⠀⠀⢀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠿⣧⡀⠀⠸⠿⠓⠒⣀⠀⠀⠀⠀⠀⠀⠀⣿⣆⢀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣹⣿⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠉⣹⣿⣿⠄⠀⠀⠀⠀⢠⡄⠁⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠛⣿⣷⡀⠀⠱⠄⠀⠀⠀⠀⠀⣴⣷⣶⣿⣿⠟⠀⠀⣴⣆⠀⠀⠀⠀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⢧⠀⠃⠈⡉⠀⠀⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⢀⣴⣿⣿⠀⠀⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⣸⣿⣷⣀⠘⠖⠀⠈⠂⠀⠀⠀⠺⠿⠿⣛⡟⠀⢰⣿⣿⣿⣿⠀⠀⢀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠰⣿⣿⣿⣿⣿⣷⣄⠀⠘⠂⠀⠀⠀⠂⢀⣽⡇⠀⣿⣿⣿⣿⡟⠀⠀⠈⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣦⣄⣀⣤⣴⣿⣿⣿⡇⠀⣿⣿⣿⣿⡇⠀⠀⠀⣤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣿⣿⣿⣿⠀⠀⠀⢠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠘⣿⡿⡿⠂⠀⠆⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠈⠀⣇⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢀⣴⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⣿⣿⣿⣽⣿⣿⡟⠁⣠⣴⣿⠀⠀⠿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⠀⠀⠀⠼⢿⣿⣿⣿⠏⢠⣾⣿⣿⣿⠀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡘⠀⠀⠀⠤⠜⢿⣿⠏⢠⣿⣿⣿⣿⡏⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⡇⠀⠀⣹⠀⠀⣾⣿⠶⣾⣿⣿⣿⣿⠃⠀⠴⠿⡇⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠁⠀⠀⠙⣦⠀⠘⣿⠀⠈⠉⠛⠛⠉⠀⠐⠂⢰⣷⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢼⣿⠀⠀⠰⠀⠀⠀⠀⠀⢠⣿⣿⡄⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣾⠃⢸⡟⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠈⢿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⣙⠄⠸⣏⠀⠀⠈⠀⠀⠀⢠⣾⣿⣿⣿⣧⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠈⠻⣿⣿⠟⠁⠀⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠻⡀⠸⠏⠀⠐⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠙⡶⠀⠀⠀⠀⠀⠈⣼⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⣀⣀⠀⠀⠀⠀⠀⢰⠞⠁⠀⠀⠀⠀⢀⣤⣀⣀⠐⠀⠀⠀⠀⠀⠐⠚⠿⣿⣿⣿⣿⣿⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢿⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⠴⠛⠉⠀⠀⠀⠀⠀⠀⠄⠀⢠⣤⠂⠀⠙⠻⠿⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⡀⠀⣶⣶⣶⣄⣿⠀⠀⠰⠦⠴⠶⠋⠻⡆⠀⠀⠀⡀⠀⢀⣀⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣄⡘⠷⠀⠈⠉⠉⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠉⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣴⣿⣿⡄⠀⣶⣶⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣄⠋⢠⡄⠻⢷⡘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⣼⣿⣿⣿⣿⠀⢘⠁⣼⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠘⢿⡆⠈⢿⠀⠘⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣾⣿⣿⣿⣿⢀⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡞⠀⠀⢀⠀⣦⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⣹⣿⡿⠿⠿⠿⣿⡿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⣸⣇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣀⠠⠶⠉⠁⠀⠀⠀⠀⣠⡀⠈⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠘⣦⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠀⠀⡶⠀⠀⠀⠀⠀⠈⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⢀⡀⠀⠐⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠀⠀⠀⢀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠐⠃⠀⠀⠀⠉⣄⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢀⡀⠀⠶⠄⠉⠒⠋⣀⣴⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠘⠀⠀⠀⠈⢁⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠘⢻⣿⣿⣿⣿⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡀⣼⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡄⡀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠑⡀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣦⣼⢿⣿⢿⣤⣶⣶⣶⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠲⢀⢾⣷⣿⡇⡑⠃⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⢀⣼⡇⠀⠀⠋⠛⠀⠀⠀⠈⠹⣿⣿⣷⣷⣿⣿⡟⠉⠀⠀⠀⠘⠉
⠀⠀⠀⠀⠠⡖⠁⡇⢻⡇⡇⠈⢲⠀⠀⠀⠀⠀⣧⠈⠓⣦⣤⡶⠋⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣻⣿⣿⠁
⠀⠀⠀⠀⠀⠇⠀⠃⠈⠀⡇⠀⠘⠀⠀⠀⠀⠀⠟⢦⠴⠋⠙⠳⢤⠞⠅⠀⠀⠀⠀⢀⣀⣀⡀⠀⠘⣿⣿⣿⣿⠇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠂⠀⠀⠀⡰⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣦⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⡆
⠀⣀⣀⡀⠀⠀⠀⣀⡀⡀⡀⠠⠄⡀⣀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⡍⠉⣹⣿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⡄⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣴⣿⠋
⠀⠀⠀⠀⠀⠀⢯⣰⣆⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⣤⠰⠄⠴⠁⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠎⠀⣺⣤⡀⠀⠀⠠⠄⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠛⢿⣿⣶⣤⣀⠀⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⢔⠈⠀⠘⠍⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣷⣶⣶⣤⣿⣦⣿⣧⡀
⠀⠀⠈⠉⠀⠀⠀⠀⡗⠀⠤⠊⠀⠀⠀⠀⠀⠀⢐⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⡍⠻⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⡿⣿⣿⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⣿⣿⣿⣿⣿⣿⣿⣷⣬⣵⡿⠃⠀⠀⠉⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠃⠀⠙⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠬⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠊⠀⢇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠟⠛⠿⢷⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠻⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠁⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⢸⣿⣀⣀⣀⣀⣠⡀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⣀⣸⣿⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠙⠉⣉⣩⣭⣍⣹⣶⣧⣤⣤⣷⣾⣉⣭⣭⣉⡉⠙⠉⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣶⠶⠟⠛⠉⣉⣉⣉⣻⣿⣿⣿⣿⣿⣿⣋⣉⣉⡉⠉⠛⠷⢶⣦⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣠⣾⠟⠋⣉⣽⢿⣿⣿⣿⣿⢿⣍⡉⠙⢿⣦⡀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠀⣤⣾⠟⠁⣾⣿⣿⣿⣿⡆⠙⢿⣦⡄⠉⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠉⠀⠀⠀⣿⣷⠀⠀⣿⣿⣿⣿⣿⡇⠀⢠⣿⡇⠀⠀⠈⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠁⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⣿⣿⣿⠁⠀⢸⣿⡇⠀⠀⠀⠀⠀⠙⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠈⣿⣿⣿⡏⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠘


⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣷⣶⣴⣷⣶⣿⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⡿⠿⠟⠛⠛⠉⠉⠉⢹⡏⠉⠉⠙⠛⠛⠿⠿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠿⠛⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢩⡿⠻⢿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠟⠋⠁⠀⠀⠀⢿⡄⠀⠀⠀⠀⢀⣀⣤⣾⡿⣦⣤⣀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠉⠛⢿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠛⠁⣠⡆⠀⠀⠀⢀⣼⣿⡿⠿⠟⠛⠛⠉⠁⢸⡇⠀⠉⠙⠛⠿⠶⢶⣾⣷⣄⠀⠀⠀⠠⣄⠙⠻⣿⣿⣦
⠀⠀⠀⠀⠀⢀⣴⣿⣿⠏⠻⣷⣾⣿⠁⠀⣀⣴⠟⠁⠸⣷⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣼⡏⠈⠛⢷⣄⡀⠀⣿⣷⣤⡾⢿⣿⣷⣄
⠀⠀⠀⠀⢠⣾⣿⡟⠁⢀⣴⣿⣿⣿⣶⠾⠛⠁⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⣸⣇⡀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠙⠻⢶⣿⣿⣿⣦⠀⠙⣿⣿⣦
⠀⠀⠀⣠⣿⣿⠏⠀⢀⣾⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣀⣿⣶⠶⠶⠾⠛⢻⡟⠛⠻⠶⠶⢶⣿⣅⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⡀⠈⢿⣿⣷⡀
⠀⠀⢰⣿⣿⡃⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣤⡾⠛⠉⢿⡄⠀⠀⠀⢸⡇⠀⠀⠀⢀⣿⠃⠙⠷⣦⣄⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢻⣿⣷⡀
⠀⢠⣿⣿⠟⠛⠷⣾⣿⣿⣿⣿⠛⠿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠘⣿⣀⣀⣠⣼⣧⣄⣀⣀⣾⠇⠀⠀⠀⢨⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣧⡶⠾⢿⣿⣧
⠀⣾⣿⡟⠀⠀⢰⣿⣿⣿⣿⠃⠀⠀⠀⠙⠻⣿⣿⣿⣷⡀⢀⣴⠿⣿⠉⠉⢹⡏⠉⠉⣿⠿⣦⣀⢀⣴⣿⣿⣿⡿⠛⠁⠀⠀⠘⣿⣿⣿⣿⡄⠀⠘⣿⣿⡆
⢰⣿⣿⠁⠀⠀⣼⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡁⠀⢹⣇⠀⢸⡇⠀⣰⡟⠀⠈⣹⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣧⠀⠀⢹⣿⣿
⣸⣿⡟⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⡄⠀⢻⣦⣼⣧⣠⡿⠁⢀⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠘⣿⣿
⣿⣿⣷⣤⣤⣤⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣶⠟⠋⠉⠉⠙⠻⣶⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⣶⠶⣿⣿
⣿⣿⡇⠀⠀⠉⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⣤⠀⠀⣤⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡏⠀⠀⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿
⢹⣿⣧⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠶⢶⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⢀⣿⣿
⠸⣿⣿⡀⣀⣤⣼⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠀⠸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣤⣄⣸⣿⣿
⠀⢿⣿⣿⠋⠉⠀⠹⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣯⣤⣤⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠏⠀⠈⢉⣿⣿⠇
⠀⠘⣿⣿⡆⠀⠀⠀⠹⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⡿⣿⠋⠀⠀⢹⡟⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠏⠀⠀⠀⣾⣿⡟
⠀⠀⠹⣿⣿⡄⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢱⡿⠀⠀⠀⠘⣿⠘⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⡿⠁⠀⠀⠀⣼⣿⡿⠁
⠀⠀⠀⠹⣿⣿⣆⠀⠀⣠⡾⠛⠛⠿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠻⣾⡿⠟⠛⠛⠻⣿⡾⠛⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⢷⣄⡀⢀⣼⣿⡿⠁
⠀⠀⠀⠀⠘⢿⣿⣶⡾⠋⠀⠀⠀⠀⠀⠙⢿⣍⣰⡟⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢿⣍⣹⡟⠉⠁⠀⠀⠀⠀⠙⣻⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣿⡿⣶⣤⣄⡀⠀⢸⡟⠀⠀⠀⠀⠀⠘⣿⠀⠀⣀⣤⣤⣼⣿⣟⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⡀⠀⠀⢀⣾⠋⠀⠀⠀⠉⠛⠷⣾⣧⣴⣶⣶⣶⣶⣤⣿⡶⠟⠋⠉⠀⠀⠀⢻⣦⠀⠀⠀⣀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣦⣤⡿⠃⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠹⣧⣤⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣶⣤⣀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣷⣶⣤⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣿⣦⣴⣶⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⣿⢯⣉⣉⠉⠉⣉⣉⡽⣿⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⣠⠎⢸⠀⠀⠀⠉⠉⠀⠀⠀⡏⠳⣄⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡡⠔⠊⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠑⠦⣝⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⡿⢆⠀⠀⠀⠀⢀⣿⠲⠤⠤⠤⠤⠖⣿⡀⠀⠀⠀⠀⣰⣿⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢁⡇⠘⣦⠤⠤⠔⠋⢸⠀⠀⠀⠀⠀⠀⡇⠙⠢⠤⠤⣴⠃⢸⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⡼⠀⠀⢸⣆⠀⠀⠀⠈⣇⠀⠀⠀⠀⣸⠀⠀⠀⠀⢰⡇⠀⠀⢇⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢡⡀⠀⡸⠈⢆⠀⠀⣰⢯⠉⠑⠊⠉⡿⣄⠀⠀⣰⠃⣇⠀⢀⡜⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⣷⡴⠃⠀⠈⣖⠊⠁⢸⠀⠀⠀⢀⡇⠈⠙⣲⠃⠀⠘⢦⣾⣿⠟⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⣿⡏⢿⣦⠀⢀⡟⢆⠀⢨⡷⠤⠤⢾⡀⠀⣰⢻⡀⠀⣴⡟⢹⣿⢘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⠀⠙⢷⣏⠀⠈⣖⠉⢣⠀⠀⡎⠉⣲⠁⠀⣹⡾⠋⠀⢸⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⢸⣧⠀⠀⠀⠙⢷⣎⠈⣦⢾⠒⢲⠧⣴⠋⣱⡾⠋⠀⠀⠀⣼⡇⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣮⣻⣧⡀⠀⠀⠀⠙⢿⣮⣮⠦⠾⣴⣵⡿⠋⠀⠀⠀⢀⣼⣟⣵⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠱⡝⠻⢶⣤⣤⣴⣶⠟⢿⣭⣹⡿⠻⣶⣦⣤⣤⣶⠟⢫⠎⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠘⡄⠀⢱⢀⡔⠁⠀⡎⠀⠀⢣⠀⠘⢦⡀⡏⠀⢰⠃⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⢱⠀⡼⠣⢤⡀⡸⠀⠀⠀⠈⡇⢀⡤⠽⢧⠀⡏⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⣎⠀⠀⠀⢹⠧⠒⠒⠒⠒⠼⡏⠀⠀⢀⣱⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡙⢆⠀⡞⠀⠀⠀⠀⠀⠀⢧⠀⡴⢋⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣌⣶⣁⠤⠤⠴⠤⠤⠤⣈⣾⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣤⣀⣀⣀⣀⣤⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠁

⣿⣿⢛⢿⠛⠟⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠶⢤⣄⣀
⠀⠀⠀⢀⣤⡶⠟⠛⠉⠀⠀⣀⣀⣤⣤⡴⠶⠖⠚⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠒⠲⠶⢦⣤⣄⣀⣀⠀⠀⠉⠛⠷⣶⣤⡀
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢹⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⡏⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠈⢿⣦⣤⣤⣤⣤⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⢶⣦⣤⣤⣤⣤⣴⡿⠁⠀⠀⠀⠀⢰⡿⠁
⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣦⣄⡀⠀⠙⢷⣄⢰⣆⡀⢀⣰⡄⣠⡾⠋⠀⣀⣠⣴⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣀⣀⣤⣴⠶⠟⠋⠉⠀⠀⠈⠉⠙⠻⠷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠾⠛⠋⠉⠀⠀⠀⠉⠙⠻⠶⣦⣤⣀⣀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠉⠀⠀⠀⠀⠀⢀⣠⣶⣶⣶⣶⡶⢿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢶⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠉⢿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣠⡶⠟⠉⠀⣀⣀⣤⣶⡿⢛⣿⣿⣿⣿⣿⣿⡻⢷⣶⣤⣀⣀⠈⠙⠻⢦⣄⡀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⠁⠀⠀⢀⣼⡟⠋⠉⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⠉⠙⢿⣦⡀⠀⠀⠉⠛⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⠄⠀⠀⠀⠀⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⡜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⡄⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⢦⡇⣦⠇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠗⠱⠅⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡇⡬⠚⢁⣍⣿⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⠕⢪⢂⡵⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢫⣾⣿⣷⡵⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢋⣶⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣿⢢⣾⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠶⢴⣟⡉⠰⠀⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠐⠂⠀⠘⢙⣧⡂⣈⣿⣿⣿⠟⠁⢀⣀⣀⣤⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠦⠀⠂⣿⣿⡌⣿⡿⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⡀⠖⠀⠀⠄⢸⡿⢹⣯⡿⡇⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠘⢻⣿⡶⣾⣧⡔⢸⣇⣼⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⠑⡽⣷⣦⣬⣿⡏⠙⣁⣿⢼⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢀⣠⣤⣤⣄⡀⢀⣀⡀⢸⣟⣤⣃⣺⣿⣿⣿⣥⣆⣀⣴⣿⣷⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⠃
⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡃⣼⣿⣿⣿⣿⣔⣋⣛⣫⡿⣬⣿⣿⣿⣿⣿⡏⡊⠩⢛⣻⠟⠁
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠠⡀⣿⣿⣿⣿⡿⠂⠉⢻⣿⠐⠀⠙⣿⣿⣿⠾⣑⠬⣱⠞⠁
⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣧⠘⣿⡿⠁⠀⠁⠚⡿⠟⠳⣤⠐⠸⡗⢿⡬⡒⡾⠁
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣹⡀⠁⠂⣄⣖⣨⠗⠀⠈⠙⢧⣠⠧⢈⡙⠛⢧
⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⢻⡟⠛⢻⡇⢆⣀⣸⡯⠀⠰⣦⡀⠀⠀⠘⠳⣾⠃⢀⣠⠇
⠀⠀⠀⠀⠉⠻⣿⣿⣿⡛⢍⠢⣙⣶⣾⡷⠞⠋⠉⣀⣆⠀⠈⢻⣄⡠⢂⢼⡇⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠲⣟⠉⠀⢀⣈⣤⡶⠿⣭⣽⣷⡀⢀⠍⠛⠷⣾⣷⡀⠈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢿⡫⠂⠜⠳⠤⡀⠙⠻⠷⠾⣶⠾⠃⠈⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⠰⡁⠤⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣵⣄⣀⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠖⠚⠛⢻⠛⠛⠓⢶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠁⢰⡀⠤⠐⢸⠑⠠⠀⣰⠀⠁⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠃⠀⢀⠔⢺⠀⠀⠀⢸⠀⠀⠀⢸⠓⢄⡀⠀⠁⠻⣦
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⡠⢗⠈⠀⠀⢘⡀⠠⠔⢸⠐⠤⠀⣘⠀⠀⠈⢐⠧⡀⠈⢷
⠀⠀⠀⠀⠀⠀⢀⣟⡖⠈⠀⠈⣀⡀⠔⠉⡄⠀⠀⢸⠀⠀⠀⡈⠑⠤⡀⡈⠀⠈⠐⣾⣇
⠀⠀⠀⠀⠀⠀⣸⣽⣿⣆⠀⠐⠙⡀⠀⠀⢠⣀⠠⢺⠢⢀⣠⠁⠀⠀⡘⠑⠀⢀⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡀⠀⠐⡄⠔⠈⡄⠀⢸⠀⠀⡈⠐⠄⡔⠀⠀⣰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⡟⠙⢿⣿⣶⣎⠈⢄⠀⣠⠄⢺⠂⢤⡀⢀⠌⢈⣶⣾⣿⠟⠙⣿⣿⣇
⠀⠀⠀⠀⠀⢸⡿⣿⡇⠀⠀⠙⠻⣿⣷⣤⠫⡀⣆⢸⢀⡄⠨⢣⣴⣿⡿⠟⠁⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣇⣿⣷⠀⠀⠀⠀⠈⠛⢿⣷⣞⡝⣸⡐⣙⣴⣿⠟⠋⠀⠀⠀⠀⢠⣿⣧⡟
⠀⠀⠀⠀⠀⠀⣿⢽⣿⡆⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣼⣿⢿⡇
⠀⠀⠀⠀⠀⠀⣿⠄⢿⣿⣄⠀⠀⠀⠀⣠⣿⣟⢝⢻⢛⢝⣿⣧⡀⠀⠀⠀⢀⣼⣿⠇⣼⡇
⠀⠀⠀⠀⠀⠀⢹⣦⠀⠻⣿⣷⣶⣶⣿⢿⡋⠈⠟⢸⠐⠏⠑⣙⢿⣷⣶⣶⣿⡿⠋⢠⣼⠁
⠀⠀⠀⠀⠀⠀⠸⡆⠑⡄⢈⠟⠟⡉⡠⠊⠉⡮⢄⣸⣀⠬⡎⠉⠢⡈⠉⠟⢟⠁⡔⠁⡾
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠸⠧⡀⠀⠜⠓⠄⡰⠀⠀⢸⠀⠀⠰⡀⠔⠚⠄⠀⡠⠽⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠈⣎⠀⠀⢀⠉⠁⠂⢼⠄⠂⠉⢁⠀⠀⢈⡎⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣕⢤⠀⠈⠂⣌⡀⠀⠀⢸⠀⠀⠀⣈⡄⠊⠀⢠⢔⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡁⠢⣀⠃⠀⠈⠐⢼⠔⠈⠀⠀⢃⡠⠂⣹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣈⡃⠒⠠⢀⢸⢀⠠⠄⠂⢋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⣀⣹⣁⣀⣠⠴⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⡶⠾⠿⢿⡿⠿⠿⠿⠿⠿⢶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠋⠉⠀⠀⠀⠀⢀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣶⣤⡀
⠀⠀⠀⠀⠀⢀⣤⣾⡟⠉⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⣽⠉⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⣷⣤⡀
⠀⠀⠀⠀⣴⡿⠋⠈⣷⣤⣤⠤⠴⠶⠚⠋⠉⠀⠀⠀⣿⠀⠀⠀⠈⠉⠛⠲⠦⣤⣤⣤⣀⣀⣀⣼⠁⠀⠙⢿⣦⡀
⠀⠀⠀⣴⡿⠀⢀⡾⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢦⠀⠀⠀⠙⢿⣆
⠀⠀⢰⣿⠁⢀⡾⠁⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠈⢳⡄⠀⠀⠘⣿⡀
⠀⢀⣿⠇⢀⡾⠁⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠙⣦⠀⠀⢹⣷
⠀⣸⡟⣠⠟⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠈⢳⣾⡆⣿⡆
⠀⣿⣷⡋⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⣠⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣼⣷
⢸⣿⢸⣿⣆⠀⠀⠀⠀⠀⠀⠹⣄⣀⣀⣠⣤⠶⠚⠉⡏⠙⠳⠶⣤⣄⣀⣀⣀⣀⣀⡞⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠈⣿
⣼⡇⢸⣿⣿⣷⡀⠀⠀⠀⢀⡾⢻⡍⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠉⠉⠉⡽⠻⣆⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠞⢻
⣿⣇⣸⣿⣿⣿⣿⣦⣀⡴⠋⠀⠀⢳⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠈⠳⣄⣴⣿⣿⣿⣿⣿⣿⣿⠀⢸
⣿⣯⠹⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸
⣿⡿⡆⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⢳⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⠏⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⠀⢸
⣿⡇⢿⣿⣿⣿⡀⠙⢿⣿⣿⣿⣿⣦⡀⠀⠻⣄⠀⠀⠈⣇⠀⠀⠀⠀⢠⠏⠀⢀⣼⣿⣿⣿⣿⡿⠋⠁⠀⢸⣿⣿⣿⠶⢺
⢸⣇⠸⣿⣿⣿⡇⠀⠀⠙⢿⣿⣿⣿⣿⣶⣄⢹⣦⠴⠖⢿⠒⠲⠶⢶⣿⣀⣴⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⣾⣿⣿⡇⠀⢸
⠸⣿⠀⣿⣿⣿⣿⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣝⢧⡀⢸⡇⠀⢠⡟⣼⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⢠⣿⣿⣿⠁⠀⣼
⠀⣿⡆⢻⢻⣿⣿⣧⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣧⣳⣤⣷⣤⣟⣼⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠏⠀⠀⣿
⠀⢻⣇⢸⣇⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢉⣿⣿⣿⣿⣿⡟⠉⠙⣻⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡟⣳⠶⣴⣿
⠀⠸⣿⠞⢿⡈⢿⣿⣿⣿⣷⣦⣤⣤⣶⣿⣿⣿⣿⡿⠟⣿⣶⣾⡟⢿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⡟⣴⠃⠀⣼⡏
⠀⠀⣿⡇⠈⢷⡀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢠⡏⠀⠈⣧⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⣼⠃⠀⢀⣿⠃
⠀⠀⢸⣷⠀⠈⢷⠀⠀⠀⠉⠛⢿⡛⢻⡿⠋⠀⠀⠀⡾⠀⠀⠀⢹⡆⠀⠀⠙⠳⣄⣸⠋⠁⠀⠀⠀⠀⣼⠃⠀⠀⣸⡿
⠀⠀⠀⣿⡆⠀⠘⣧⠀⠀⠀⠀⢀⡿⢯⣄⡀⠀⠀⢸⠇⠀⠀⠀⠀⣷⠀⠀⢀⣤⠞⠳⣤⡀⠀⠀⠀⣰⠇⠀⠀⢀⣿⠃
⠀⠀⠀⠸⣿⡀⠀⢹⡆⠀⢀⡴⠋⠀⠀⠈⠙⠳⣤⡟⢀⣀⣀⣀⡀⢸⣦⠞⠋⠀⠀⠀⠀⠙⢦⣄⢠⡟⠀⠀⠀⣾⡏
⠀⠀⠀⠀⢻⣧⠀⠀⣷⡴⠋⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠉⠉⠉⠙⢻⡀⠀⠀⠀⠀⠀⠀⠀⢈⡿⢧⣀⠀⣰⡿
⠀⠀⠀⠀⠈⢿⣆⣠⠟⠷⣄⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠉⢳⣿⠃
⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠈⠳⣄⡀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢀⡴⠋⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠈⠻⣦⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢠⠟⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠈⢳⣴⡇⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡘⣷⠏⠀⠀⠀⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⣹⠛⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠹⡄⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠠⠀⠀⠘⠆
⠀⠀⠀⠀⣀⢀⡠⠤⠐⠺⠁⠀⠁⠀⠠⠄⣀⠀⠀⠀⠀⢘⣀⠃⠀⠀⠀⠀⢃⣈⠀⠀⠀⠀⢀⡀⠤⠀⠀⠁⠀⠹⠒⠠⠤⣀⢀⡀
⠀⠀⢀⠄⠖⡩⠀⢀⣈⣃⠀⠀⠀⠀⠀⠋⠀⠀⠀⠄⠐⣺⣿⠲⣆⠈⢀⡶⠞⣿⡂⠈⠄⠀⠀⠈⠃⠀⠀⠀⠀⢀⣋⣀⡀⠈⡁⠦⢄
⠀⠀⠀⠉⢃⣄⠠⠂⠀⠀⠁⠒⠢⡀⠀⠀⠀⠀⠀⠀⠄⠐⠻⣄⣸⠦⢾⣀⡼⠓⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠂⠁⠀⠀⠢⢀⣄⠋⡁
⠀⠀⠠⠀⠁⠀⢂⡀⠀⠀⠀⠀⠀⠈⠢⢤⡈⠁⠀⠀⠀⠠⠑⡀⠁⠀⠀⠉⡐⠁⠀⠀⠀⠀⠁⢠⠤⠊⠀⠀⠀⠀⠀⠀⣁⠂⠀⠁⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠀⠀⡄⠀⠀⠃⠢⠀⠠⠊⠁⠀⠀⠄⠀⢠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠄⢤⣴⣆⡤⢠⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠄⠀⡀⢨⣹⣿⡁⢀⠀⠂⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠘⠀⠀⠀⠀⠀⣀⣀⣀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠄⡀⠄⣀⡀⠉⠀⠈⠁⠀⠀⡀⠀⢄⡀⠄⠒⠈⠁⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠌⠀⠈⠀⠈⠀⠀⡁⢀⠄⠈⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠁⠌⠁⠈⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠌⠦⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠎⠄⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⡠⠌⡀⠁⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠁⡀⠤⣀⠠⠐⠁⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠔⢀⡄⠉⠘⠂⠀⠀⠀⢀⠚⠈⠉⣄⠐⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⣯⣫⣑⣻⡇
⠀⣠⡞⢟⣿⠟⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⡄⢄⠀⠀⠌⡄⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⡿⣿⡿⢿⡇
⢰⣯⣭⣭⣭⣯⣭⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠈⣦⠎⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣦⣦⣼⣼⡇
⠘⣿⣿⣟⠃⣻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⣛⠟⣿⡇
⠀⠈⠻⠬⣭⡥⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢁⣤⠇⡁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣷⣷⣾⡿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⠿⠖⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠟⢻⣿⣷⣶⣶⣶⡚⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠁⠀⣰⣟⢿⣿⣿⣿⡏⠃⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣀⣠⣴⣾⣿⣿⣷⣿⣿⣿⣷⡠⣀⠀⠀⠀⠀⠀⠀⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⢃⣈⣹⣿⣿⣿⣿⣿⡿⠿⢿⣇⣸⠇⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠸⣻⣿⣿⣿⣿⣿⣽⣿⣿⠟⠁⣁⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⢸⣾⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⢸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠸⢿⣿⣿⢿⣿⣿⣿⣿⣿⣤⣦⣼⣃⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣶⣿⣿⠃⢠⣿⣿⠿⢿⣿⣿⡿⣶⡾⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⣾⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣼⡿⣿⣷⣔⣦⣰⣄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣠⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠈⠻⣿⣿⣿⣿⣦⣄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢿⣿⣇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⠄⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⣽⣿⣯⡈⢙⣿⣿⣿⣿⣿⡟⢻⣿⣾⡇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢸⡎⢻⣿⡇⣾⣿⣿⣿⣿⠏⠀⢸⣿⣿⠇⢸⣿⡀⠀⠀⢠⠀⠀⠀⠀⠀⢠⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⢺⡇⢸⣿⠁⣿⣿⣿⣿⣯⠀⢠⣿⣿⣿⡆⢺⣿⣧⠀⠀⢸⡄⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⣚⡛⠂⣿⣿⣿⣿⡘⣇⢻⣿⣿⣿⠀⢸⣿⣟⡆⠙⣻⣷⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣴⣿⣿⣿⣿⣷⢻⣼⣿⣿⣿⢠⣿⣿⣿⣼⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⡙⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠾⠿⠿⠿⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠶⠶⠶⠶⠦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⢰⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣇⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢰⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡜⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠈⢀⡴⠀⠀⠀⠀⠀⠈⣷⠀⢸⣿⣸
⠀⠀⠀⠀⠀⣀⠀⠈⢷⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⢠⡀⠀⠀⠀⠘⠁⠀⠀⠀⠀⢀⣤⣿⣷⣼⣿⣿⠤⠤⣤⣀
⠀⠀⠀⢀⡾⢻⡿⣿⠟⢿⣿⣿⣿⣿⣍⡛⠷⣄⠀⠀⣿⣠⣤⣦⡀⠀⠀⢀⡤⢞⣿⡿⠋⣿⣿⣿⡻⢿⣿⣯⠙⣧
⠀⣰⣾⡿⠁⣿⡟⠀⠀⢸⣿⣿⡙⠿⣿⣿⣷⣿⣷⣤⣾⣿⢾⣿⣿⣦⣴⣯⣶⡿⠋⠀⣰⣿⣿⣿⡇⠀⠙⢿⡄⠹⣶⣤⣤
⢠⠏⡞⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⡒⠤⣈⣹⣿⣿⣿⡷⠀⠈⠁⠌⣀⣿⣿⣿⠿⠂⣾⣿⡿⢻⣿⡇⠀⠀⢸⡇⠀⠀⠂
⠋⠈⠀⠀⠀⣿⠁⠀⠀⢸⡇⢻⣿⣷⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠙⠋⠉⠀⢀⣾⡿⠋⠀⣿⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⡄⣽⣿⡆⢰⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢻⡇⢀⣾⣿⣿⡇⠀⠀⣿⡇
⡀⠀⠀⠀⠀⢹⠀⠀⣀⣿⣿⣿⣿⣿⣷⣼⠀⠀⠀⠀⠀⠠⣤⣴⠂⠁⠀⠀⠘⠀⣿⢣⣾⣿⡏⣿⠃⢀⣾⣿⡇⠀⠀⠀⢰
⣧⠀⠀⠀⠀⣸⡆⣰⣿⢸⣿⣧⠻⣿⣿⣿⡀⠀⠀⠠⠤⠤⠍⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⢸⣿⣷⣿⣿⣿⠃⠀⠀⠀⣮
⣹⣆⣧⠀⠀⣿⣇⣿⣿⣼⣿⣿⣧⣿⣿⣮⣷⣠⣤⣶⡿⠿⠿⠿⠿⠿⠿⠶⠂⢸⣿⣿⣿⠃⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸⣏
⣿⣿⣿⣷⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠋⠁⠀⠀⢀⡀⠀⠀⠀⠀⢀⣾⣿⣿⡟⣰⣿⣿⣿⣿⣿⠇⣴⡇⢠⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣴⠿⢃⣿⣿⣧⣿⣿⣿⣿⣿⣟⣼⡿⢁⣾⣿⣽
⢹⣯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⠁⠀⠀⣸⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⢁⣾⢿⣽⣯
⣶⣿⣽⣿⡽⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣷⠙⣿⣿⣿⣿⣿⣿⠇⠀⡀⢠⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⣿⢿⣿⣧⠙⣿⣿⣽⣿⡟⢰⡑⠀⢸⣿⠟⣿⠀⣿⣿⣿⣿⡿⠟⠋⠉
⠀⠀⠀⠉⠙⠛⠻⣿⣿⣿⣿⣿⣿⣷⣶⣷⣾⣿⣿⣦⣽⣿⣿⣿⣷⣿⣦⣤⣿⣿⣾⣷⣾⣿⣿⡿⠍⠁⠀⠀⠀⣀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣶⣶⣿⣿⡿⠿⠟⠛⠛
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⢀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣾
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠇⠀⠀⠀⠀⠀⠀⢧⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣾⠀⢰⠃⠀⠈⢳⠀⢸⡆⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠬⣽⣿⣷⣦⣶⣿⣿⡭⠤⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⠒⣾⠟⠛⡟⠁⠀⠈⠙⡟⠛⢿⡖⠲⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢸⠀⠀⠹⡄⠀⠀⡼⠁⠀⢸⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠈⡇⠀⠀⠙⠶⠞⠁⠀⢀⡏⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢳⡀⠀⠀⠀⠀⣰⠃⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠳⡄⠀⠀⡴⠃⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⣤⠤⠔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⣤⠈⠙⢲⡄
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠘⣦⠤⠤⠴⠦⡀⠀⠀⠀⠀⠀⠀⣠⠶⠦⠤⠤⣴⠃⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⢸⢧⡀⠀⠀⣀⣠⠤⣤⣈⣲⣤⣄⣠⣤⣞⣁⣤⠤⣄⣀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠈⢿⡶⠋⠁⠀⣠⠤⠬⢽⣿⣿⣿⣿⡯⠥⢤⣄⠀⠈⠙⢶⠟⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣠⠔⠋⢁⡴⠚⠋⣿⣿⣿⣿⠙⠓⢦⡈⠑⠢⣄⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸⡅⠀⠀⣿⣿⣿⣿⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢸⠁⠀⠀⢹⣿⣿⡏⠀⠀⢸⡇⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⡇⠀⠀⠀⢿⡿⠁⠀⠀⢸⠃⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠘⡇⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠸⡄⠀⠀⠀⠀⠀⠀⢠⠃⢠⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⡎⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⢲⠀⣶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠀⣴⠖⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣿⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⢸
⠀⠀⣿⡇⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⠀⣿⠀⠀⠀⣠⠖⠋⠉⠉⠉⠓⢦⡀⠀⠀⠃⣿⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣄⠀⢸
⠀⠀⣿⡇⠀⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⢸⠀⣿⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠆⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⠀⢸
⠀⠀⣿⡇⠀⣿⣿⣿⣴⣶⣿⣿⣿⣷⣦⣿⣿⡇⢸⠀⣿⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⡇⣿⠀⠀⡸⣿⣿⣿⠟⢿⣿⣿⣇⠀⢸
⠀⠀⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⢸⣿⠀⣸⡏⠻⣦⡀⠀⠀⢀⣴⠟⢹⡇⠀⡇⣿⠀⠀⢿⣿⣿⡟⢀⠘⣿⣿⡟⠀⢸
⠀⠀⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⢸⣿⠀⢹⣇⠀⠈⠻⣦⣤⡟⠁⠀⣼⡇⠀⡇⣿⠀⠀⢸⡟⠿⢿⣿⡿⠟⢛⡇⠀⢸
⠀⠀⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⡇⢸⢸⣿⠀⢸⠹⣦⣠⠞⠃⠀⠻⢦⣼⣣⣇⣀⠇⢿⡤⠤⠼⠷⠤⠶⠛⠓⠒⠛⠓⠈⠉
⠀⠀⣿⡇⠀⣿⣿⣄⢀⣨⣿⣿⣿⣤⣤⣴⠿⠧⠼⠘⠿⠶⠞⠓⠛⠛⠛⠋⠉⠉⠉⠉⠁
⠀⠀⠙⠓⠛⠋⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠶⠿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠀⡒⢶⡿⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠖⠛⠁⠀⠐⠀⠀⣉⠀⠰⠂⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣠⣾⢻⣤⠀⣆⠀⠀⠀⠀⠈⠀⠀⠀⠀⢦⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⣡⡞⢡⠇⠀⠀⢠⡿⣤⣀⣠⡴⠖⠦⢤⣀⣠⣿⣆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠋⣠⣾⠋⢠⡇⠀⠳⡆⠈⢷⠈⠉⡴⠒⢀⠐⠦⠀⠀⡟⠀⠙⠖⠀⡀⠱
⠀⠀⠀⠀⠀⢀⣠⠤⠖⠊⠉⠀⢀⣤⣾⡿⠁⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠈⢉⠛⠀⠀⠀⠀⠀⠀⠠⠚⠁
⠀⠀⠀⢀⡞⠉⠀⠀⠀⢀⣠⣶⣿⣿⠏⠀⠀⠀⠀⠀⠈⠳⢤⠀⠀⠉⠀⠀⢉⣭⡝⠂⢀⡀
⠀⠀⢠⡟⠀⠀⠀⠀⠲⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠻⣿⣿⣿⠟⠉⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⣿⡇⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠘⡜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⢱⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢀⠜⡑⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⣼⠀⠀⠀⠀⠀⠀⠀⢸⣅⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⠆


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⢸⣿⣤⣤⣤⣤⣄⢀⠀⠀⡀⣠⣤⣤⣤⣤⣿⡇⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⣀⣀⣤⣤⠶⠶⠶⣽⣿⣿⣿⣿⣯⠶⠶⠦⢤⣤⣀⡀⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⠉⠀⢀⣴⠶⠖⣾⣿⣿⣿⣿⣷⠲⠶⣦⡀⠀⠉⠛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢀⣴⠟⢡⣿⣿⣿⣿⡌⠳⣦⡈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠉⠀⠀⣿⣿⠀⠘⣿⣿⣿⣿⠃⠀⣿⡇⠀⠀⠙⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠀⠀⠀⢹⣿⠀⠀⢿⣿⣿⡿⠀⠀⣿⡇⠀⠀⠀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⣿⣿⠃⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠙⠋⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈

⢀⡶⠖⠒⠲⠶⠀⣴⠖⠒⠒⣶⢀⡖⢠⡖⠒⠒⠲⡆⠀⣰⠖⠒⠒⠒⠆⣰⠖⠒⠒⣶⠆⢠⡶⢲⡟⢲⡆⠀⣰⠶⠒⠒⢶⠄⢀⡶⠖⠒⠲
⣬⣍⣉⣩⣽⠋⣹⠋⠉⠉⠉⠁⡼⠁⣾⣀⣀⣤⡼⠉⢹⣯⣉⣉⣭⡄⢠⡏⠉⠉⢛⡟⢀⡞⠀⠞⠀⡼⠁⢹⠏⠉⠉⢩⡏⠉⡼⠁⠀⠀⣸

⠀⠀⠀⠀⠀⠀⠀⡶⠤⣄⣀⣀⣀⣠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⣀⣀⣀⣀⡠⢤⢶
⠀⠀⠀⠀⠀⠀⢸⠁⠑⢄⠀⠀⠀⠀⠀⢹⠢⡀⠀⠀⠀⠀⢀⢼⡃⠀⠀⠀⠀⠀⠔⠁⢸⡀
⠀⠀⠀⠀⠀⣠⠋⠀⠀⠈⡳⡦⠤⠤⠔⠉⣆⠉⠢⣄⡠⠔⢁⡜⠑⠠⠀⠠⡴⠚⠀⠀⠀⠳⡀
⠀⢀⣀⠴⠊⠁⠀⠀⠀⢀⠇⠈⢦⡀⠀⠀⣰⠈⠒⢹⠁⠈⢁⡇⠀⠀⢀⠔⠀⢠⠀⠀⠀⠀⠈⠓⠤⣀
⠈⠳⢤⠄⣀⠀⠀⠀⣠⠎⠀⠀⠈⠙⡍⠉⠀⡗⠀⢺⠓⠀⡚⠀⠁⡸⢹⠀⠀⠀⠡⡀⠀⠀⠀⡀⢠⡴⠛
⠀⠀⠀⠑⢄⠈⠓⢯⠤⢀⡀⣀⠌⠀⠈⢢⣀⣰⣄⣸⣀⣰⣇⣠⠎⠀⠀⠢⡀⢀⡠⢄⡷⠊⢁⡴⠉
⠀⠀⠀⠀⠀⢣⠀⠀⠱⡄⠈⠳⡒⠠⣤⣴⣿⠟⠿⣿⡿⠟⠿⣿⣦⡤⠔⢢⠋⠁⣠⠋⠀⢀⠎
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠹⡄⠀⠐⠀⠀⢻⡇⠀⠀⠈⠀⠀⠀⢸⡏⠀⢠⡇⠀⢰⠃⠀⠀⡎
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⢠⠀⣀⣼⣷⣄⠀⠀⠀⢀⣴⣿⣧⡀⠘⡀⠀⢾⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⢸⣁⠠⠤⠷⡒⠛⠛⠭⡉⠙⢿⣷⣄⣴⣿⠟⠉⢉⠍⠚⠓⡲⠴⠤⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⠀⠀⠑⢄⠀⠘⡀⣸⠟⢿⡿⠻⡄⠰⠁⠀⡴⠋⠀⢀⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⢂⠀⡶⠡⢄⢸⠀⡠⠘⢆⠀⡞⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢸⠮⠤⣀⠀⢺⠎⠀⣀⠤⠵⡁⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡐⠁⠀⠀⠈⠳⣸⣠⠊⠀⠀⠀⠈⢄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠦⡀⠀⢹⠁⠀⢀⠴⠊⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⢸⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡞
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⣤⣤⣤⣤⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⢹⣟⣻⡏⣏⢩⣻⣟⣿⣟⠉⢹⡿⢿⣿⢻⣿⣿⣟⣿⣉⣿⡟⣿⣿⣛⣯⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠈
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠛⠛⠛⠛⠛⠛⠛⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠃⣀⣀⠻⢿⠟
⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡄⢀
⠀⠐⠀⠀⠀⠠⡦⠀⠐⠀⠀⠀⠀⡤⢶⣶⠀⠃⢙⠋⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠛⠣⠀⠀⠀⠀⢰⣇
⠀⠀⠑⢐⡿⠛⠉⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠉⠈⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠐⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⡬⠄⢤⠒⠀⠁
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠆⢀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠈⠉⠉⠛⣿⣿⣿⣿⣿⠟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⢀⣤⣾⢿⣿⡏⣿⡿⢿⣆⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠛⠁⣼⣿⠗⢿⣿⡜⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠄⠀⠀⠀⠤⠐⠊
⠀⠀⠀⠀⠀⢠⣿⣿⠀⢾⣿⣧⠀⠀⠀⠰⠚⢿⡃⡈⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠔⠒⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠃⠀⠛⠒
⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⣿⠿⣿⣶⠿⣶⣿⢿⠿⡷⡿⢶⢶⡶⢶⠶⢶⠶⠶⠶⠶⢶⠶⠶⢶⣶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⣿⣿⣀⣀⣀⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣀⣀⣀⣸⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠈⠛⠉⢉⣉⣭⣍⡉⢿⣶⣧⣤⣤⣷⣾⠏⣉⣭⣉⣉⠉⠙⠋⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣧⣤⣴⡶⠾⠟⠛⠋⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠉⠛⠛⠿⢶⣶⣤⣼⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⢀⣴⡿⠟⠛⢛⣿⣿⣿⣿⣿⣿⣿⣟⠛⠛⢿⣷⣄⠀⠀⠀⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⣠⣶⠟⢉⣾⣿⣿⣿⣿⣎⠙⢷⣦⡀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁⠀⠀⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⢻⣿⡇⠀⠀⠙⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡿⠛⠁⠀⠀⠀⠀⣿⣿⠇⠀⠈⣿⣿⣿⣿⣿⡟⠀⠀⣿⣿⡇⠀⠀⠀⠀⠙⢿⣷⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⣿⣿⠄⠀⠀⢻⣿⣿⣿⣿⠇⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠉⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠘⣿⣿⣿⡟⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠹⣿⡿⠁⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣿⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣦⡞⠁⠉⠳⠿⠛⠉⠉⠐⢶⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢀⠴⠦⡈⢻⠇⢀⣀⠀⠀⠀⣀⡀⠰⣷⠇⠇⢀⠀⠀⠃⠀⠂⢄⠀⠀⠀⣀⠀⠀⠤⠐⠈⠀⣰
⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠉⠁⠀⠐⠒⢈⠁⠀⠉⠁⠒⢄⡘⡎⢧⠈⠉⠉⠀⠉⠉⠀⣸⠇⡀⠀⠎⠀⠀⠀⠀⠀⠀⠙⢖⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⡰⠁⠀⠀⠀⢀⡠⠔⠒⠒⠒⠈⠉⡽⠀⠀⠀⠀⠐⡄⠱⣝⢎⠀⠀⠠⣀⠄⠀⢀⢈⢾⡀⡀⠀⡰⠀⠠⠂⢀⡄⠂⠀⠳⢤⣄⡀⠀⠀⢠⠏⠀⠀⢀
⠀⠀⢠⠁⠀⠀⢀⠔⠉⠳⠀⠀⠀⠀⠀⠄⣆⠀⠀⠀⠉⠀⠙⠀⠈⢎⢿⠂⠐⠀⠀⡰⠃⣸⢺⣄⡉⠩⡱⠖⠄⠔⡡⠀⠀⠀⠸⠀⠀⠉⠙⠒⠂⠀⠀⠈⡩
⠀⠀⢸⠀⢀⠜⢁⠄⠀⠀⠀⠀⠀⠀⢀⠠⠼⠦⣀⣀⣀⣀⠀⠀⠀⠈⢦⡣⠙⠀⠘⠁⠔⢁⣁⢺⡀⠀⠄⠐⢄⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠈⠔⠁⡀⠀⠀⠀⠀⠉⠤⡀⡎⠁⠀⠴⢆⠀⠀⠘⠇⠀⠀⠀⠀⠀⠙⢧⡀⠀⢘⡴⠋⠁⠘⠁⠀⠈⠀⠈⣳⣓⡆⠀⠐⠁⠀⠀⠀⢀⠠⠔⠊
⠀⠀⠀⠀⠐⠀⠀⠐⠄⠂⣠⠔⣧⠀⠀⠈⠀⠀⠝⣓⣊⡧⠀⠀⠀⠀⠀⠀⠀⠀⠏⠉⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⡈⠉⠒⠢⠄⣄⠀⠓⠠⠤⣀⣀
⠀⠀⠀⠀⠀⡠⠖⠀⠀⢄⠀⠞⣠⡖⠊⠀⠀⠈⠉⠀⠀⡼⢆⠀⠀⠀⠀⠀⠀⠀⣂⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠐⠤⣀⠀⠈⠉⠉⢈⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠠⢁⠀⠀⠀⠀⠀⠀⠀⡜⢁⠜⡵⠢⣄⡀⢄⠠⠤⠻⢄⡀⠀⠀⠀⠀⣀⣴⢿⠘⣄⠀⠀⠀⣀⠴⢉⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⠂⠀⠀⠀⠀⠤⡀⠰⢀⠞⡹⠁⠀⠙⠷⣦⣴⣤⣖⢢⣬⡭⠶⠆⠩⢙⠁⡘⠀⠘⢆⢠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⢠⠆⠀⠘⠀⣬⠎⡀⠀⠀⢀⠀⠀⠈⢳⣬⣉⡠⠤⢒⡀⠄⠀⠀⠑⠤⠤⠄⠁⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠠⠃⠀⠀⡞⠀⠀⡄⠟⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢀⠆⠀⠀⣰⠃⠀⠀⡟⠀⠀⡎⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠀⣠⠞⠈⠀⠀⢰⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⡰⠁⠀⠀⠀⢀⠟⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠘⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⢊⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣈⣈⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠓⠶⠶⠶⠶⠶⠖⣻⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⣆⠉⠛⠓⠛⠛⢻⣧⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠃⠙⢦⣀⠀⠀⠀⣠⡾⢹⡟
⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⠁⠙⣆⠀⠀⠀⠀⢸⡛⣇⠀⠀⠀⠀⣀⣴⣿⢿⡏⠀⠀⠀⠉⢛⣶⡾⠋⠀⢸⡇
⠀⢤⣄⣀⣠⣤⠾⠋⣰⠏⠀⠀⣸⣷⣤⣀⣠⣼⡇⠘⢦⣤⣤⣾⣿⠟⢁⡿⠛⠛⠛⠛⣻⡿⢻⡇⠀⠀⢸⡇
⠀⠀⠈⠙⢷⣤⣤⡾⠋⠀⢀⣴⠏⠘⢧⡈⠁⠸⣿⣦⣤⣿⣿⣋⣡⣴⣿⡁⠀⢀⣴⠞⣿⠁⠈⣧⠀⠀⠘⣧
⠀⠀⠀⠀⠈⣿⢿⡟⠷⣿⡋⠁⠀⠀⠈⢷⡄⠀⣿⣀⣠⣿⠿⣭⣀⡾⠉⣹⣷⣿⣵⣷⣿⣆⠀⠹⣆⠀⠀⢻⡀
⠀⠀⠀⠀⠀⢹⡀⠹⣆⠀⠙⢷⣦⣄⡀⠀⢹⣿⠻⣏⢉⣿⣄⣤⣿⣿⠞⣿⣿⣿⣿⣿⡷⠈⠳⠶⣬⣷⣦⣌⣷⣀⣀
⠀⠀⠀⠀⠀⢸⠇⠀⣿⠀⠀⠈⣷⠈⠙⢳⡿⣭⣷⣿⣼⣧⣼⣿⣯⣀⣀⣬⣷⣽⡿⠶⢷⣶⠶⠟⠛⠛⣹⡿⠛⠉⠁
⠀⠀⠀⠀⢀⣿⣀⣰⣏⣀⣠⣴⣯⣤⣤⣾⣷⠶⠿⣻⣿⣿⣿⣯⣿⡏⠁⢹⡇⠀⠀⢠⡟⠁⠀⠀⠀⣾⠁
⠀⠀⠘⠛⠛⠛⠻⢿⣍⠉⠀⠈⠉⢷⡀⠀⢙⣷⣾⣿⠋⣿⠻⣦⠈⣩⣿⠿⠷⣤⣄⣸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⢸⣧⡴⠟⠁⣸⠟⠳⣿⣴⠾⢿⣿⠀⠀⢀⣴⠞⠛⠛⠛⠲⣶⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢀⣤⢾⣟⡛⢦⣠⡟⠀⠀⢸⡇⠀⠀⠙⢦⣠⡟⠁⠀⠀⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣯⣁⠀⠈⠻⣌⣿⣥⣄⡀⢸⣇⣠⡶⠖⠚⢿⡄⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀⢹⡏⠀⠈⠻⣼⡿⢁⣀⣤⣤⣄⣻⣄⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣼⠟⠛⠶⣄⣿⣷⠟⠉⠀⠀⠈⡉⡻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠘⣿⡿⠀⣠⡴⠛⠋⠉⠉⠛⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠓⢦⡀⢸⣇⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⣰⣿⠏⠀⢀⣠⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⣄⠀⠀⢻⣷⡄
⠀⠀⠀⣠⣾⣿⣏⣠⣶⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣶⣌⢿⣿⣆
⠀⢀⣼⣿⣿⡟⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠘⣿⣿⣧
⠀⣸⣿⣿⣿⠁⠘⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃⠀⢹⣿⣿⣧
⠀⣿⣿⣿⡏⠀⠀⠘⣿⣿⣷⣶⣶⣶⣶⣶⣄⣠⣴⣾⠇⠀⢿⣷⣦⡄⢀⣠⣤⣤⣤⣤⣼⣿⣿⠃⠀⠀⠈⣿⣿⣿⡆
⠀⣿⣿⣿⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢻⣿⣿⣿
⢰⣿⣿⡿⠀⠀⠀⠀⠀⠈⠉⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡈⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿
⢼⣿⣿⣿⣷⣶⣦⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣠⣤⣤⣤⣼⣿⣿⣿
⠀⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣯⣉⠉
⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠋⠁⣿⡿⠋⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠙⣿⣿⡏⠻⢿⣿⣿⣷⣄⡀
⠀⢠⣤⣤⣾⣿⣿⠟⠉⠀⠀⠀⣿⡇⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣿⣿⡇⠀⠀⠙⠿⣿⣿⣿⣦⣤⣄
⠀⢸⣿⣿⡿⠋⠀⠀⠀⠀⠀⢰⣿⣷⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⡇⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿
⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠙⢿⣿⣿⣿⠟⠁⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⢻⣿⣿⡆⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠙⢿⣿⡄⠀⠀⠀⠈⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠃⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠁⠀⠀⠀⢀⣼⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣿⠿⠿⠿⠿⢿⣿⣷⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠉⠉⠛⠿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠉⠻⣿⣷⣄
⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⡀
⠀⠀⢀⣾⡿⠁⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⢿⣷⡀
⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⣸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣇
⢀⣿⣿⣿⣷⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠀⢀⣾⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣦⠀⠀⠀⠐⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠀⠀⣤⣿⣿⣿⣿⣿
⣼⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠆⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠇⠙⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⡄⠀⠈⠁⠀⢠⠀⠀⠀⠐⠀⠀⣠⣾⣿⣿⡿⠋⠘⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠙⢿⣿⣿⣷⣤⡀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⢀⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣦⡀⢀⠀⠆⠈⠁⠐⠀⡀⢀⣴⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡁⠀⠀⠀⠀⢈⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿
⢿⣿⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣤⣴⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿
⢸⣿⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠛⡿⢻⠙⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢋⣿
⢸⣿⡇⠘⣿⣿⣷⡀⠀⠀⠀⠀⢀⣠⣾⣿⠟⡁⠀⠀⠀⠀⢠⡻⣿⣷⣄⡀⠀⠀⠀⠀⢠⣾⣿⣿⠏⢸⣿
⠘⣿⣇⠀⠘⢿⣿⣿⣷⣶⣶⣾⣿⣿⠟⡁⠀⠐⠂⠠⡀⠐⠂⠀⠈⠻⣿⣿⣷⣶⣶⣾⣿⣿⡿⠃⠀⣸⣿
⠀⣿⣿⠠⠀⠀⠻⣿⣿⣿⠿⠟⠋⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠙⠻⠿⣿⣿⣿⠟⠁⠀⠀⣿⣿
⠀⢸⣿⡇⠈⠀⠀⠈⠉⠁⠀⠀⠈⠀⠀⠀⠘⠀⠀⢀⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠁⠁⠄⠀⠀⢰⣿⠇
⠀⠀⢻⣿⡀⠐⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠂⢀⣿⡿
⠀⠀⠈⢿⣷⡁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁
⠀⠀⠀⠈⢿⣷⣄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠹⣿⣷⡄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⢠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠠⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⢀⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣦⣤⣄⣈⣁⣠⣤⣴⣾⡿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⠈⢳⠀⠐⠖⠃⠘⠀⣸⢆⡸⠄⠀⠀⢀⠀⢩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡶⠓⠀⠿⠩⠀⠀⠂⠀⠀⠇⠘⠃⠘⠁⠀⠀⠚⡧⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⡾⠃⠢⠁⢄⣡⢾⡱⠀⠀⠀⠀⣾⠗⠀⢲⠚⠒⠀⠠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁⣲⣤⣾⣻⣄⠙⠧⠀⠀⠀⠀⠈⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡥⡏⠌⠳⠘⠖⠲⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠘⣿⣦⣤⠀⠀⠀⠛⠛⠀⠁⠈⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⣻⣷⣿⡁⢀⠀⠀⠀⠀⠀⣴⠃⠀⣀⣤⢠⡀⠈⠀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⣿⡧⢬⣏⡀⠀⠀⠠⠆⠘⠃⠀⢀⠀⠀⢹⠉⠉⠀⡆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢘⡵⣀⡈⠁⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠚⠈⠙⠶⠇⠐⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠑⣀⠀⠘⣦⣤⣀⣠⠟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⡐⠐⠄⣁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⡄⢫⡀⠛⣃⠆⢀⣠⢤⣷⡆⠰⠎⢀⣀⠀⠀⠀⢀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠶⠻⢦⣾⠄⠀⠹⠤⢴⣤⣾⠻⠟⠀⠁⢨⣿⣧⣤⡾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠻⠿⢯⣼⣤⣎⣰⣄⠈⣿⣿⠀⠠⣆⣠⣿⣿⡉⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠂⠁⠉⠈⠙⢿⣿⣿⣏⠂⢀⠋⢫⠿⠛⠀⣽⠟⠀⠀⠀⠀⠀⢠⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣷⣦⠴⠁⢠⠄⠘⠛⣦⣄⣀⣀⣠⡴⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠁⠀⠉⠰⢿⣯⠹⡅⠀⠀⠈⠓⠂⠂⠀⠈⠙⠿⠳⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠸⠶⢇⠀⠀⠀⠹⣷⣿⣿⠷⠈⠗⠂⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠁⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠛⢻⣄⢿⣿⣿⣂⣀⣀⠀⠀⠄⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠴⣶⣀⣨⣀⣠⠈⠁⠀⠀⢰⣿⣶⠦⣴⠆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢀⣶⠟⠙⠳⢶⣿⣄⠀⠀⢻⣯⣀⣁⣀⣒⡖⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣾⣾⡟⠁⠀⠀⠀⠈⢿⣿⣧⣀⣀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢁⣆⡼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠹⢾⣻⣯⠀⠀⠀⠀⠖⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣦⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠛⢟⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣐⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢁⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⢪⣁⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣽⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠾⠿⣦⣐⣄
⠀⠀⠀⠀⠀⠀⠀⠳⠼⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⢀⢎⠌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⡙⡄⠀⠀⠀⠀⠀⢠⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠁⠀⠀⠀⢀⡠⢪⢎⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢠⡐⢄⠀⠀⠀⠀⠀⠠⡑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢣⢆⠃⠀⢀⡤⢊⡴⠇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⢉⡲⡌⠐⠲⢄⠀⠱⢹⡘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⢱⢇⡊⣠⠔⠡⣠⢏⡇⠃⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠆⣼⣷⢠⠰⣮⡐⠍⣚⡼⣧⠣⡀⠀⠀⢀⠆⢰⠀⢠⢃⣿⠦⠭⢖⡅⣰⠏⠸⡇⠸⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛
⠀⠀⣀⠤⠀⠄⠀⠀⠀⠀⠀⢠⢎⠌⣼⣿⣿⠀⠳⣝⠻⠷⠶⢆⠜⠳⡙⢄⣠⢃⠇⡆⡠⢣⢊⢃⡠⡙⢛⣵⢃⠀⠀⡅⠀⠱⣙⢄⡀⠀⠀⣀⠤⡐⣊⠥⣂⠜
⠀⠀⠀⠙⡀⠐⠠⢄⣀⡠⢖⠡⣫⣾⣿⣿⡿⠀⣰⡜⡟⢛⠺⣉⠾⠇⡝⠲⠥⠚⣧⣛⡴⢃⡎⢾⠿⣂⣴⢡⣿⡄⠀⠀⡀⢐⢌⡳⢭⣶⣩⢶⠟⡡⠒⠋
⠀⠀⠀⠀⠈⠒⠬⣓⡲⢦⣁⠺⠿⠿⢟⣋⠴⣡⣿⠃⠁⣀⢧⢠⣭⡥⢱⡙⢿⡇⢻⢄⠀⡞⣠⡅⣹⢋⣜⢌⠻⣿⣎⡢⡈⠪⢠⣭⡥⠞⠡⢡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡝⠍⠛⢍⣉⠐⣚⢛⠅⠀⣠⣿⢘⡧⠭⠔⡁⢿⠲⠖⢻⠠⣰⠧⡛⣡⡟⠸⣿⣶⣅⣂⠈⢡⡤⢻⢊⣥⣶⢃⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⠜⣧⠹⠻⡉⠠⠤⠞⠋⠁⡼⣡⡌⣆⠀⢈⣇⣡⡧⣐⣼⠰⢀⠏⡀⠀⠈⠉⣉⣭⠞⠋⡄⡏⣸⣿⡇⠆⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣿⡆⠂⣌⠻⡛⠲⠶⢄⡚⠿⠃⡸⢏⠉⣯⢰⠃⠆⠟⣳⢯⠸⠟⣃⣤⠔⡋⡍⠀⢸⣷⢃⢻⣿⡇⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢀⣿⡇⡄⣿⣷⢡⠀⠀⠀⠈⠳⠿⡧⣬⡟⢻⣿⣤⡾⡿⣅⢊⣷⣾⢡⣎⢸⡇⡇⠀⢸⣿⡜⡜⢿⣿⡰⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠀⣼⡟⡐⣰⣿⡿⠀⣠⠆⠀⠀⢀⡇⡁⢸⡀⠰⠈⣿⣧⢀⡟⢉⣁⣷⡀⢫⣀⡃⢳⣀⣘⠿⠷⠜⢮⣙⠳⠐⢝⠦⠀⠤⢄
⠀⠀⠀⠀⣀⡐⠶⢖⣭⠶⢥⣜⣭⠔⠁⠀⠒⠚⠛⢷⡒⢓⠶⡟⠿⡉⢍⠳⣦⣰⢇⣽⠛⠓⢩⣏⢉⢙⡿⣋⢉⠁⠘⠉⠑⠚⠋⡉⠩⣉⣛⣚⣛⣛⠯⢬⣧
⠀⠀⠀⠌⢀⠩⠝⠭⠒⠉⠑⠢⢄⡠⠘⢿⣿⣷⡄⠀⠙⣎⢷⠸⣆⣤⣴⣚⠉⣻⣾⣿⠋⣉⠿⠧⡉⠘⢱⠇⡎⠠⢊⣴⠟⡡⡠⠒⠁⠁
⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠙⣿⣿⠀⠐⡸⢈⣼⢋⠀⠀⠈⣴⣇⣽⣈⣷⡡⠖⠂⠙⠶⣬⡁⣧⢀⣿⣏⡞⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠸⡟⢌⣰⠇⠉⠀⠈⢝⠢⡘⢡⡆⡛⣡⡘⠱⢋⢄⠔⠤⠖⠀⠹⣆⠻⢸⢁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢃⣆⡴⠋⠴⢿⣿⣷⣎⢊⠃⢃⠻⠃⡇⠬⠥⣤⡆⠀⠠⠢⢐⣋⡭⠍⢑⣺⣦⡑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡡⢚⣫⠌⠭⠡⣀⠉⠻⣿⠘⢦⢛⣭⡙⣧⡖⣥⣌⠓⠀⢀⡴⠛⢁⣠⠬⠔⠒⠒⠂⢅⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡩⠈⢚⡵⠋⠀⠀⠀⠀⠀⠑⡄⠘⢰⣃⠬⠙⠃⢻⢸⡿⠧⢰⢠⠏⢄⠔⠉⠀⠀⠀⠀⠀⠀⠀⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⢸⠇⣂⣄⡀⣿⠨⢒⣨⣝⠳⡌⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⡈⢉⠉⠓⢸⢰⣿⡿⢿⣧⠱⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡰⢋⠜⠀⠙⡄⣼⠈⠵⣒⣐⠂⠴⡜⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡼⢡⠃⠀⠀⠀⢱⢸⡆⡞⠀⠀⠈⠑⠬⠄⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠃⠀⠀⠀⠀⠸⢸⢠⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠀⠚
⠀⠀⢀⠀⠀⢀⠀⡀
⣿⡟⣉⣻⠟⠿⠿⠿⠻⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢰⣶⣶⣶⠀⠀⠀⠀⠄⠀⡀⠂⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⡀⢈⠁⠀⠀⠀⠀⠀⠉⠉⠹⣷⣄⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡀⢀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣇⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⣤⡀⣴⠟⠁⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠋⠀⠴⠞⠀⠓⠶⠆⢉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠢⠀⠀⠀⡰⠛⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⢡⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠄⠀⠈⠙⠇⠄⠀⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⡀⠀⣀⢸⡇⠀⠀⠀⠀⠀⠀⢀⠁⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠈⠀⠈⠙⢻⠟⣠⡀⠀⠀⠀⠀⢀⣷⣶⡿⠟⣛⣿⣿⣿⡗⠀⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠙⠧⠀⠀⠀⠀⣠⢋⡴⠋⠑⠒⠊⠀⠀⣾⢟⣩⣶⣿⣿⣿⣿⠿⡃⢀⣠⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⣁⣤⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⣠⣾⣾⣿⡿⠿⣿⣭⡟⣩⣾⠇⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠛⢉⣠⣾⣿⣿⣷⣤⡀⢀⣾⣿⠇⠀⠀⠀⢠⣶⣿⣿⣿⠋⠀⠀⢹⣿⣿⣿⣿⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⣠⣴⣿⣿⣿⣿⣿⣿⣿⡿⣫⣾⣿⡟⠀⠀⠀⣰⣿⣿⣿⣿⠃⠀⠀⠀⠈⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⠾⠋⠀⢀⣼⣿⣿⣿⣿⣿⡿⣿⠏⢉⣴⣿⣿⠏⠀⠀⢀⣴⣿⣿⣿⣿⠃⠈⠁⠀⠀⠀⠈⢿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢣⠞⠁⠀⠀⢀⣾⣿⣿⣿⣿⣿⢫⣾⣿⢣⣾⠏⠈⠁⠀⠀⠀⣼⣿⣿⣿⠟⠁⠀⠀⠀⠀⠄⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⣀⣠⡤⣀⣭⣿⣿⣿⣿⣿⣿⣿⡿⠳⠛⡏⠀⠀⠀⠀⢠⠀⠛⠉⠏⠁⠀⠀⢀⠔⠊
⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠀⠀⡜⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠿⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠋⠁⠀⠀⠀⠉⠫⠭⠭⣛⡲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣋⠕⢪⣗⣉⣓⣒⣀⢀⠀⣶⡆⠀⠀⠈⠑⠪⣛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡫⠊⠁⠀⠈⣿⠟⢿⣶⣿⣟⡀⠉⣸⠀⠀⠀⠀⠀⠈⠑⢝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⠜⠁⠀⠀⠀⠀⣿⡲⣤⣈⣿⡿⢷⢿⣿⠗⠀⠀⠀⠀⠀⠀⠈⠣⡻⡄
⠀⠀⠀⠀⠀⠀⠀⢠⢏⠎⠀⠀⠀⠀⢀⣼⣷⣿⣿⢿⣱⣾⣿⠿⢛⣂⣀⣄⣠⡏⠀⠀⠀⠀⠱⡹⡄
⠀⠀⠀⠀⠀⠀⠀⡟⡜⠀⠀⠀⠀⠀⣾⣧⣼⣿⣿⣿⡿⣿⣷⣶⣯⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢃⢻
⠀⠀⠀⠀⠀⠀⢸⡇⠇⠀⠀⠀⠀⠀⠠⣿⠿⠻⠛⠛⣿⣿⣾⣿⣿⢿⡄⠀⠈⢿⣿⣆⠀⠀⣶⠠⢸⡇
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⣿⢛⡁⣿⣿⡊⢷⡀⠀⠈⣿⣷⣁⣼⡗⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⣼⠟⢸⡇⣿⣿⣳⣌⣿⠞⡒⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠓⠒⠚⠛⡃⠀⢸⣄⢻⣿⣿⡿⠬⣑⡀⠈⢿⣿⣿⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣫⣿⣷⣶⠶⢶⠷⣤⣠⣝⣿⣿⣿⣿⣞⡈⠉⠀⠈⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢻⣿⣷⡯⣷⣌⣺⣿⣿⠿⠿⣿⣿⣿⡿⣁⠀⠀⡀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠤⢾⣿⣿⣧⣿⣿⣸⣿⣿⣀⣠⣿⣿⣿⣏⠀⠉⢀⣵⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡖⣷⣯⣿⣿⣿⣿⡿⡟⠉⠉⠉⢻⣿⣿⠡⢤⣈⢰⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢽⣿⠇⢿⢻⣿⠀⠡⣷⢄⠤⠴⢿⣿⣟⠒⠀⠈⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⢀⠀⠹⡆⠀⢀⣸⣿⣟⣥⣦⣶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠐⣆⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢴⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢹⡀⠈⠉⠛⠛⠛⠒⠚⠛⠛⡟⠀⠀⠀⠀⠀⠀⠀⠀⢹⡛⠛⠓⠒⠛⠛⠛⠉⠉⠀⡼⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠁⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠙⠶⣄⣀
⠀⠀⠘⠶⣞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⣷⣀⠀⠀⠀⠀⠀⠀⣸⠙⢦⡀⠀⠀⡂⠀⢀⣴⠏⣷⠀⠀⠀⠀⠀⠀⣀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⣓⡶⠊
⠀⠀⠀⠀⠹⡷⢤⡀⠀⠀⠀⠀⠀⣀⡴⠋⢻⡝⠓⠲⠶⠖⠚⢿⠀⠀⠙⠳⢴⣧⠶⠋⠁⠀⣿⠛⠲⠶⠖⠚⢋⡿⠉⠳⣄⠀⠀⠀⠀⠀⢀⣤⢾⡟
⠀⠀⠀⠀⠀⢳⡀⠉⠳⣶⡖⠒⠋⠁⠀⠀⠀⠹⣄⠀⠀⠀⠀⣸⣄⣀⠀⠀⢀⡇⠀⠀⣀⣠⣿⡀⠀⠀⠀⣠⠞⠀⠀⠀⠈⠙⠓⢲⣦⡖⠋⠀⣼
⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⡍⠳⢤⣀⠀⠀⠀⣠⠟⢷⡖⠚⠋⠉⣇⠉⠉⠉⠉⡏⠉⠉⠉⢸⠏⠙⠓⢲⡶⠻⣆⠀⠀⠀⢀⡤⠖⠋⡏⠀⠀⢰⡇
⠀⠀⠀⠀⠀⢀⡇⠀⠀⢠⡇⠀⠀⠈⢳⡾⣯⣁⠀⠀⠙⢦⣤⠴⢻⣤⣄⣠⢴⡧⣤⣀⣤⡞⠦⣤⣴⠋⠀⠀⢈⣹⢷⡾⠉⠀⠀⠰⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⠇⠀⠀⣸⠃⠀⠀⠀⠀⡇⠀⠉⠓⠶⣴⣏⠙⠳⣄⣳⣄⠀⠀⡇⠀⢀⣞⣡⠶⠋⣙⣦⡴⠚⠉⠀⢸⡇⠀⠀⠀⠀⢧⠀⠀⠈⡇
⠀⠀⠐⠒⠲⠾⣦⡤⣴⣯⣀⣀⣀⣀⣸⡃⠀⠀⠀⠀⣿⠈⠉⠓⠶⣿⠋⠙⢶⡷⠛⠙⣿⡶⠚⠋⠁⢻⡀⠀⠀⠀⠈⣇⣀⣀⣀⣀⣨⣦⡤⣤⡿⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠈⠉⠙⢿⠉⠉⢻⡋⠛⠻⡟⠙⠉⣹⢧⡀⠀⠀⢀⣴⣏⡉⠙⢉⡟⠋⠛⡿⠋⠉⡽⠋⠉⠉⠁⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠈⣧⠀⠀⢧⠀⢀⣧⣴⠛⣡⡴⠛⢦⡴⠛⠳⣌⠙⢦⣼⡁⠀⢸⠁⠀⢰⠃⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⢸⠀⣀⣼⠶⢟⠁⣨⠾⠧⣀⠀⠀⡇⠀⣀⠬⠷⣇⠀⡹⠶⢾⣀⠀⣾⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢀⣠⠾⠛⠛⢆⠀⢈⣾⣋⠀⠀⠈⠑⢴⣧⠚⠁⠀⠀⢈⣷⡅⠀⡠⠛⠛⠿⢤⣀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡴⠛⠉⠙⢦⡀⠀⠈⣧⡞⠁⠉⠙⠲⣄⠀⠨⡇⠀⣠⠔⠊⠁⠀⠹⣴⠃⠀⠀⡰⠎⠉⠛⢦⣼⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡄⠀⠀⠳⡄⢠⠟⠛⠲⢤⡀⠀⠀⠑⢤⣧⠚⠁⠀⢀⣠⠖⠚⠹⣆⢀⡞⠁⠀⢀⡴⠋⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢳⣏⠀⠀⠀⠀⠙⠢⣄⠀⠈⡇⠀⢀⡴⠋⠀⠀⠀⠀⣘⣿⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⢀⡟⠉⠉⠓⢤⣀⠀⠀⠈⠳⣀⣇⡔⠁⠀⠀⢀⡤⠖⠋⠀⢹⡄⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣅⣀⠀⠀⠀⠈⠳⣄⠀⠀⠙⡏⠀⠀⣠⠞⠉⠀⠀⠀⢀⣠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠈⠓⢦⡀⠀⠀⠀⠑⢄⠀⡇⡠⠚⠁⠀⠀⢀⡠⠞⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠹⡟⠁⠀⠀⢀⡴⠋⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⢢⡀⠀⠀⡇⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⡇⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⢲⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⠒⠒⠒⠈⠉⠀⠀⡰⠁⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠑⠦⣄⣀⠀⠀⡴⠁⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⡜⣽⡇⠀⠀⠀⠀⣩⡿⡄⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢏⠒⠒⠤⠖⠚⠁⢸⠳⡀⠀⠀⠀⠀⢀⡜⣰⢩⠖⠒⠒⠚⢉⡾⠀⢣⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠑⢄⠀⠀⠀⠀⠀⢇⠘⢦⡀⠀⢠⢎⡴⠁⣼⠀⠀⠀⡰⠃⢣⠀⠀⠣⡀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡇⠀⠀⢳⣄⣀⣀⡤⠚⣦⡀⠉⣻⣗⣉⡤⠊⡟⠢⢤⠜⠁⠀⠀⠑⠤⣀⣈⣒⣦⣤⣬⡦⠖⠂
⠀⠀⠀⠀⠀⠀⢀⡴⠃⢠⠃⠀⠀⡸⠈⠳⣄⠀⠀⠹⡌⠉⢸⡗⠤⠤⢴⣅⡠⢟⠀⠀⠀⠀⠀⢀⣀⠤⠒⠋⢁⠎
⠀⠀⠀⠒⠒⠾⣥⣀⣠⠋⠀⣀⠴⠁⠀⠀⠈⠓⢄⣀⠿⡖⠁⣷⢄⣤⣮⠞⠁⠈⢢⣤⠤⠒⠊⡝⠁⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠈⢣⡈⠫⡙⠓⠢⢤⣤⣀⣀⠀⠀⡟⠢⣈⠦⢼⠔⢾⣉⢶⠖⠊⢹⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠘⢆⠀⠀⠙⢆⠀⠈⠙⡏⢑⣻⡄⠀⠀⣸⠒⠸⠦⠤⣤⠷⠤⠤⢤⡧⠤⠤⠤⣤⣵⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⢸⠀⠀⠀⣸⣠⠤⠾⠯⢅⣀⡜⡳⠞⡄⠑⢢⣀⡎⠀⠀⠀⡴⠋⠀⠀⠀⡰⢋⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣀⡾⠴⠒⠉⠉⠉⠲⡀⠀⣠⠛⠑⡗⠒⠾⡴⠋⠈⠑⢤⡀⡸⠀⠀⠀⠀⡼⡇⡜
⠀⠀⠀⠀⠀⠀⠀⠔⠚⠉⠉⠑⠲⢄⡀⠀⠀⠀⢱⣔⣁⠀⠀⡇⠀⠀⢱⠀⠀⢀⡠⠽⠧⡀⠀⠀⢰⠁⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⣠⠞⠤⢄⡙⢶⣃⣀⡀⠀⢇⡰⠁⠀⢀⣀⣈⣢⡀⢾⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣞⣁⣀⠀⠀⠙⢮⡏⠀⠈⠱⣜⡇⡠⠊⠉⠀⠀⣈⣉⣻⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠈⠉⠓⢄⠈⡧⠒⠢⠤⡘⣿⠁⠀⢀⠔⠋⠁⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡇⠀⠀⠀⠈⢻⡆⢀⠎⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠒⠒⠢⢄⡀⢷⡎⠀⠀⠀⢠⠄⣠⢰⠃⡐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⡇⠀⠀⠀⠘⢆⣙⣎⣷⡡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⣎⠯⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠃⡇

⠀⠀⡀⠀⠀⡀⠀⡠⢀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠄⠄⠀⠀⠀⠀⠀⣀⠀⢄
⠀⠀⠀⠀⣄⠀⠔⠎⣀⠰⠀⠈⠁⢐⠀⠌⠀⡀⠀⠈⠙⠿⠩⢝⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠂⠂⠀⠈⠀⠘⠋⠡⠈
⠀⠔⠍⠀⠆⢄⠀⡄⠀⠀⡬⢆⠀⠠⠀⠀⠀⠀⠀⠠⢀⡄⠀⣄⠠⠄⢀⠀⠊⠉⠉⠉⠁⠀⠀⡀⠀⠀⢀⠀⠀⠂
⠀⠀⣄⠀⢀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠈⠙⠀⠈⡁⠀⠉⠀⠀⠀⠀⢄⠐⣀⠀⡀⠀⠈⠁⡀⠀⣤⡖
⠀⠀⠀⠴⠐⠈⠋⠀⠴⠀⠀⠀⠀⠲⡀⠁⠀⠀⠀⠀⠀⠋⠄⠁⠀⠄⠂⠀⠈⠄⠀⠀⠄⠀⠐⠀⠈⠀⠀⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠂⡀⠂⠀⠀⠠⠀⠀⠀⠁⠀⠀⠢⡄⠀⠀⠄⠀⠄⠀⠄⠀⠀⢐⡀⠀⢀⠂⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠈⠀⠄⠀⠲⠳⠆⠀⡠⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂⠀⠀⠀⠂⠀⠀⠈⠁⣄⣀⣠⠀⠁⠀⠤⠀⠰⠆⠁⢬⠀⠀⠂
⠀⠀⠀⠀⠼⠋⠈⠓⡀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⣿⠀⠀⠀⠀⠤⠀⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠠⠀⠸⠿⠀⠿⢇⠀⠀⢘⣴⠀⣤⡆⠈⠀⠃
⠀⠀⠀⠄⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠁⠀⢨⠈⠀⠉⠁⠀⠀⠀⠌
⠀⠐⠄⠀⠠⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠄⠀⠈⠂⠁⠀⠀⡈⠕⠐⠪⠑
⠀⠄⠀⠀⠠⠀⠀⢀⠾⢀⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠰⠆⠤⠰⠀⠀⠠⠀⠀⠄
⠀⠰⠂⠐⠆⠀⠠⠈⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⡞⠀⠀⠀⠀⡀⠆⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢤⡆⠀⠀⣠⠄⠀⢀⠁⠂⠀⠀⠂⠀⡀⡀⠀⠈⠂⡀⠀⠀⠀⠤⣠⠏⠀⠁⠂⠐⣿⢼⡗⠐⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢶⠞⢋⠀⠀⠀⠌⠀⠨⡀⠁⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠾⠁⠀⠈⠀⠀⠂⠀⠐⠀⠂⢀⠠⣦⣦
⠀⠀⠀⠁⠐⠀⢰⡆⢀⠀⣯⠁⠠⣶⠄⠂⠈⠀⠀⣀⠈⠁⣠⠀⠀⠀⠀⢨⠀⠂⠀⡆⠉⠛⠉⠓⠀⠶⢀⣤⣈⣤⡀
⠀⠀⠀⠠⠀⠀⠿⣇⠿⠇⡿⠇⠀⣀⠀⣄⠈⠀⢈⠋⡀⢀⡈⠄⠀⠀⠀⠈⠈⠁⠈⠀⠀⠈⠠⠁⠀⠀⠨⠅⢈⠌⢡
⠀⠀⠄⠀⠀⠀⠒⠈⠀⠀⠐⠂⠀⢁⡀⡐⠀⠀⠀⠁⠉⠀⠀⠁⢀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠁⠀⠀⠈⠀⠀⠁⠁
⠀⠀⠀⢼⣿⠷⣦⡀⠀⠀⠀⠂⠀⠤⣼⡐⠈⠃⠀⠀⠁⠊⠄⠀⠑⠁⢀⡀⡀⠠⠀⠀⠄⠀⠄⠀⠃⠀⠀⠀⠀⠀⠀⢀
⠀⡀⢀⡀⢀⠀⠠⡼⠰⠀⠀⠀⠀⠘⠿⠃⠀⠠⠀⠰⢀⠀⠆⠀⠀⠀⠀⠘⣣⠄⠉⠈⠀⠈⠀⠀⠀⠀⣐⠀⠌⠀⣄⠈
⠀⠙⠛⡛⠛⠀⠀⠀⠀⠀⡀⠂⡤⠄⠂⠀⡀⠀⠀⠀⠀⠀⠀⠘⡛⡃⢀⡴⠃⠀⡀⠀⡠⠘⠘⠀⡆⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠘⠗⠀⠀⢀⠀⠀⠐⠀⠀⠀⢀⠀⠁⡀⠀⠀⣶⠂⠰⣶⠀⠉⠉⠋⠀⠀⠀⠀⠀⡀⠀⡀⠀⡄⠀⢀⠀⠀⢂⡀⠠
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠩⠀⠄⠩⠀⠀⠈⠀⠈⠈⠀⠀⠀⠀⠀⠁⠀⠁⠐⠁⠀⠐⠃⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⠀⠀⡴⢛⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠳⡄⠀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⠏⠀⠀⠀⠀⣸⠃⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢹⡄⠀⠀⠀⠈⢿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⡏⠀⠀⠀⠀⢠⡇⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣷⠀⠀⠀⠀⠈⢧⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡞⠀⠀⠀⠀⠀⡾⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⡇⠀⠀⠀⠀⠘⣆⢳⡄
⠀⠀⠀⠀⠀⠀⠀⣰⠇⣸⠁⠀⠀⠀⠀⣸⠇⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢻⠀⠀⠀⠀⠀⢹⡄⢷⡀
⠀⠀⠀⠀⠀⠀⢠⡟⢠⠇⠀⠀⠀⠀⢀⡟⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⣇⠀⠀⠀⠀⠀⢧⠈⣧
⠀⠀⠀⠀⠀⠀⣾⠀⣼⠀⠀⠀⠀⠀⣼⠁⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣄⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢹⡀⠀⠀⠀⠀⢸⡆⢸⡄
⠀⠀⠀⠀⠀⢠⡇⢠⡇⠀⠀⠀⠀⠀⠙⠳⣤⡀⠈⠛⢦⡀⠀⠀⠀⢠⠟⡇⠀⡟⢧⠀⠀⠀⠀⣠⠖⠉⠀⣠⠴⠚⠁⠀⠀⠀⠀⠀⣧⠀⣧
⠀⠀⠀⠀⠀⢸⠁⠘⠓⠶⠶⠤⢤⣀⣀⠀⠀⠉⠳⢄⡀⠙⢦⡀⢠⡟⠀⠷⠶⠇⠈⣧⠀⣠⠞⠁⣀⡴⠋⠁⠀⢀⣀⣤⠤⠴⠶⠶⠛⠀⢸
⠀⠀⠀⠀⠀⡿⠀⢀⣀⣀⡀⠀⠀⠀⠈⠉⠓⠲⢤⣀⡙⠦⣀⠙⣾⡇⠀⠀⠀⠀⠀⣹⡶⠃⣠⠞⣁⣤⠴⠒⠋⠉⠁⠀⠀⢀⣀⣀⣀⠀⠘⡇
⠀⠀⠀⠀⠸⡷⠚⠉⠉⠉⠉⠉⠉⠛⠛⠒⠶⠦⣤⣈⠉⠓⠾⣷⣌⢷⡀⠀⠀⠀⣰⠏⣰⡾⠗⠋⢁⣀⡤⠴⠖⠚⠛⠉⠉⠉⠉⠉⠉⠛⠲⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣙⣳⣤⣀⠉⠀⠉⠀⠀⠈⠁⠈⢁⣠⣶⣚⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠒⠒⠛⠛⠛⣉⣉⣉⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣍⣉⣉⡙⠛⠛⠓⠒⠲⢦⣄⣀
⠀⢀⣶⣄⣤⣤⠤⠶⠚⠉⠁⢀⣀⡤⠴⠖⠚⢋⡭⠞⠋⢁⣠⠴⢾⠃⠀⠀⠀⠀⠀⢻⡶⢤⣀⠉⠓⠮⣍⡙⠒⠶⠤⣄⣀⠀⠉⠛⠒⠦⠤⣤⣠⣴⡆
⠀⠈⣇⠀⠀⠀⢀⣠⡴⠖⠛⠉⠁⠀⣀⡴⠚⠉⢀⡴⠞⠉⠀⠀⡏⠀⠀⠀⠀⠀⠀⠈⣇⠀⠈⠙⠦⣄⠀⠙⠲⣄⡀⠀⠈⠙⠓⠦⣤⣀⠀⠀⠀⠀⡟
⠀⠀⢿⠀⠀⢸⠋⠁⠀⠀⠀⠀⣠⠞⠁⠀⣠⠴⠋⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠈⠓⣤⠀⠈⠙⢦⡀⠀⠀⠀⠀⠉⢻⠀⠀⢸⡇
⠀⠀⢸⡆⠀⣿⠀⠀⠀⠀⠀⣼⠁⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢹⡄⠀⠀⠀⠀⢸⡇⠀⣾
⠀⠀⠀⣇⠀⢿⠀⠀⠀⠀⠀⢹⡀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢙⡆⠀⣸⠃⠀⠀⠀⠀⢸⠇⢠⡇
⠀⠀⠀⢸⡄⢸⡀⠀⠀⠀⠀⠈⣧⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠇⢠⡏⠀⠀⠀⠀⠀⣸⠀⣸⠁
⠀⠀⠀⠀⢧⠘⣇⠀⠀⠀⠀⠀⠸⡆⠸⡆⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⡾⠀⡾⠀⠀⠀⠀⠀⠀⡟⢀⡇
⠀⠀⠀⠀⠸⡆⢻⡀⠀⠀⠀⠀⠀⢹⡀⢷⡀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⢰⠃⣸⠃⠀⠀⠀⠀⠀⣸⠃⡼
⠀⠀⠀⠀⠀⢻⡈⣧⠀⠀⠀⠀⠀⠀⢳⡈⣧⠀⠀⠀⠀⠀⠀⠀⠘⣆⢰⠖⠚⢶⢀⡞⠀⠀⠀⠀⠀⠀⠀⢠⡟⣰⠃⠀⠀⠀⠀⠀⢠⡏⣰⠃
⠀⠀⠀⠀⠀⠈⢧⠘⣆⠀⠀⠀⠀⠀⠀⢷⡘⣇⠀⠀⠀⠀⠀⠀⠀⠈⠻⠅⠀⠻⠋⠀⠀⠀⠀⠀⠀⠀⢀⡟⣰⠋⠀⠀⠀⠀⠀⢀⡞⢠⠏
⠀⠀⠀⠀⠀⠀⠈⣧⠹⡄⠀⠀⠀⠀⠀⠀⢳⡜⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣰⠃⠀⠀⠀⠀⠀⠀⡼⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠹⡄⠀⠀⠀⠀⠀⠀⠹⣌⢳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⢋⡼⠁⠀⠀⠀⠀⠀⠀⡼⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠹⣆⠀⠀⠀⠀⠀⠀⠈⠳⣬⣳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣯⡴⠏⠀⠀⠀⠀⠀⠀⢀⡾⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣜⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⡵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠟⠋

⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣼⣿
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠆⣰⢾⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⢀⡼⡿⣦⣄⣀⣸⠆⢥⣿⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡿⠄⢀⣠⣶⣿⣿⣿⣦⡀⠀⠀⣿⣦⣠⣴⣴⣦⣾⣿⠁⣸⣷⣆⣾⣿⣻⣿⣆⣾⣿⣿
⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣤⡀⠀⠀⠀⠀⣹⠿⠋⣲⣾⡄⢙⣿⣿⣿⣿⣇⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣯⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⣴⣏⣠⣶⣿⣷⣻⣼⣿⣿⡿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣦⠀⠀⠀⠀⢿⡗⠂⠀⠀⢀⣾⣿⣿⣿⡄⠀⢹⣿⣿⣿⣿⡈⠷⣿⣿⣿⠟⠁⣽⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣯⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣿⣿⣿⣷⣦⣼⡏⢿⣿⣿⣷⣤⣼⣛⡛⠶⣴⣏⣹⡿⣿⣎⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣇⠶⠀⠀⠀⠀⠀⢸⠂⠀⠀⠿⠿⢿⣿⣿⣿⡇⣺⣿⣿⣿⣿⣿⣿⡤⢌⣿⣿⣿⣷⣼⣿⣆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡃⠀⠀⢠⡄⠀⠀⣿⡎⠋⠀⠀⢀⣼⣿⣿⣿⠧⢹⣿⣿⣿⣿⣿⣷⣶⣿⣿⢻⣿⣿⣇⣿⣿⡆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡧⠀⠀⠈⠀⠀⠀⠈⠁⠀⢀⣴⣿⣿⠿⠛⠁⠀⠈⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢠⣿⠇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠃⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⡿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠀⠀⠀⣿⣿⣄⠀⠈⢿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠘⠀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⡟⢹⣿⣿⣇⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠿⣿⣿⣿
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⠀⠀⠀⢹⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⢴⣦⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⡀⠈⠟⠁⠀⠀⠀⠈⢻⣿
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠉⢠⣼⣿⣿⠗⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣿
⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣄⡀⠀⣀⠀⠀⠀⠀⠈⠙⠿⣿⠇⠀⠈⠛⠿⠿⣿⣿⣿⡿⣿⣿⣿⣀⠀⢀⣀⣀⣀⣤⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢖⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣤⠤⠤⠶⠖⠚⠋⠉⣀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣗⠲⠶⠶⠶⢶⠋⢡⠏⢸⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡏⢷⡈⠙⠲⠦⠤⣬⣷⡯⢤⣼⡄⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⣀⡀⠀⠀⠀⠀⠀⣀⣤⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡇⠀⠙⠲⠶⢖⣾⣿⢿⣷⡄⠀⢳⡸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⣿⣍⣛⠛⠛⣉⣡⡾⠙⡟⢦⡀⠀⠀⠀⠀⠀⠀⢀⡾⡅⢸⡷⠶⠒⠒⠛⢉⣼⡏⠀⢷⠙⢦⣄⡛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣇⠈⠳⡍⠉⠉⠉⡼⠳⠶⢿⠖⠻⣄⠀⠀⠀⠀⣠⣏⣀⣙⣾⠳⢤⣀⡀⢠⡾⢹⡛⠲⠾⢧⡀⡬⠟⣻⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢹⢤⣄⣙⣦⣤⠞⠁⠀⢀⣨⣷⠶⠚⠳⢴⣦⡼⠁⠉⢉⡽⣿⣄⠀⠀⣹⣿⡇⠀⠳⣄⠀⠀⠙⢧⣀⣻⡀⠙⢦⣀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⡿⠀⣀⡴⢻⡟⢿⡙⠉⢉⡿⢹⡷⢦⣤⡤⢿⡶⠶⠚⠉⢰⡇⠈⢉⡟⠁⡿⣧⣤⣤⡬⠿⠒⠶⠶⢾⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⢀⣠⠖⢻⠀⣰⠟⠛⠁⠀⡼⠀⠀⢹⣶⣯⠴⠋⠹⣆⠀⠀⣼⡏⠓⠶⠶⡿⣄⡴⢻⡇⠀⢹⡌⠳⢤⣀⣀⣤⡶⠚⣩⠟⢁⡞⠁
⠀⠀⠀⠉⠙⠛⠛⢶⣤⣼⣷⣃⣀⣀⣤⠞⡇⠀⣠⠏⢳⠙⠦⣄⣀⡽⣟⠛⠁⣷⣄⣠⣴⣣⠞⠁⠀⠙⢦⣤⡽⣶⠞⢉⡽⠁⠀⣰⡇⠀⡾⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠙⠻⣙⠛⠲⠶⣷⣞⣁⣀⢸⠀⠀⠘⣿⢤⡈⢧⣀⣻⣠⣴⡟⠳⣤⣤⢴⡞⡟⣡⠞⡏⠀⢸⠳⢦⣀⡏⢻⡀⡇⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠙⣦⣄⡀⠈⠙⢮⡉⠉⠛⣶⣶⣧⡤⣽⡏⠉⠙⠁⠀⣟⣋⣹⡄⣸⠀⠹⣇⠀⠻⣄⠸⡄⠀⠈⡇⠀⢷⣷⠾⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⡿⣇⠉⠳⣄⠈⡇⣠⠞⢁⣟⣨⡷⣴⠛⢦⣄⣠⡾⢭⣉⡽⠛⠉⡭⠟⢛⣋⡽⢿⡿⠟⢛⣶⡟⠓⣺⢿⠗⢺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣰⠃⡿⠀⠀⢈⣷⡿⠿⣾⡿⠯⢽⣶⣼⣴⠋⡾⠉⢳⡞⠉⢳⣤⣸⣡⣖⣋⣁⣰⠋⠀⣰⠏⢸⣧⠞⣱⠏⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣃⣼⣷⢖⠛⠛⠛⠲⠦⣄⠉⠓⢦⣠⡟⠋⢻⡷⠒⠲⣧⡼⢉⡏⠙⠻⣍⠀⢸⠃⣀⣠⡏⠀⢠⠏⣰⠋⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠚⠉⠉⠙⠛⠿⣍⡉⢳⡶⢦⣄⢻⣄⡼⠋⠉⠙⢺⡇⣠⠖⠛⣇⠈⡷⠶⠶⢮⣷⣿⡚⠁⢸⠁⢀⡟⢠⠏⠀⠸⣶⣲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⠷⢶⠚⣻⣟⡓⠒⠲⣄⢸⣴⣃⣀⣀⣹⣾⠁⣴⣿⣭⣀⣀⡙⢦⣸⡀⢸⠁⡟⠀⠀⢰⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣠⠟⠉⠹⡍⠓⢦⡈⢿⡯⠵⠤⣌⡉⢻⡞⠀⠀⣀⡤⣬⠿⠿⠿⣧⣸⣸⢣⠀⡤⢸⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠷⠦⢤⣄⣹⠀⠀⠙⣾⠀⣀⣤⡤⠿⣜⡇⣠⠞⠁⢠⡏⠀⣀⣠⣤⡽⢿⡘⣌⣷⡟⣷⢏⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠉⠓⢦⡀⢸⡿⠷⢶⡤⣄⡘⣿⡇⣠⣤⣸⡷⠚⠉⠀⠀⠀⠀⠙⢨⢿⣟⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⡇⣠⢞⣀⣤⡽⣾⣿⠁⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⠉⣯⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⣉⣀⣠⠤⣬⣿⡞⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠁⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠈⠉⠳⢤⡙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠙⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇

⠀⠀⢤⠀⠀⠀⢤⡀
⠀⠀⠀⠑⠄⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⢯⣏⣶⡀
⠀⠀⠀⢂⠈⣤⠀⠀⠒⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⢾⢼⣳⣿⣦⣤⣀
⠀⠀⠀⠀⠳⢿⣵⣄⡀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣻⣿⣿⣷⣟⣿⣟⣿⣽⣧⡀
⠀⠀⠀⠀⠀⠀⠁⢯⢯⡟⣢⠼⣦⣀⢀⠀⠀⠀⠀⠀⢠⣴⠑⣼⣿⣿⣿⣿⣿⣟⣹⣾⣷⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠑⠚⡼⣛⣻⣭⡟⠻⢿⡻⠦⣼⣿⡙⣧⣿⣽⣿⣿⣿⣇⣿⣿⣿⣧⣾⣽⣭⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡹⢿⣇⣽⣷⣾⣷⣤⣜⣿⣻⣿⣿⣿⣿⡿⣿⣽⣿⣿⣿⡿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢧⡈⠻⣿⣿⣿⢿⣧⢿⣿⣿⣿⣷⣾⣿⣿⣿⣟⣾⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠁⡹⣿⣷⣹⡚⢿⣿⣟⣾⣿⣵⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠙⢿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠉⠙⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⣀⣀⠉⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣅⠀⠘⣿⣯⣿⠉⢛⣋⢱⣿⣿⠇⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⣸⣽⣧⡾⠿⠋⣿⣿⣿⢀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠄⠀⣼⣿⣿⣷⠏⢳⣄⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⣿⣿⣿⣹⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠀⠀⠀⡀⡀⡀⠀⡀⠀⢀⡀⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⢀⣠⡶⠋⠉⠀⠀⠀⠀⠀⠀⠁⢀⠁⢘⢸⡇⠀⠀⠈⠙⢶⣄⡀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣥⢀⡤⣀⣈⡉⢉⣛⡓⠶⠤⣄⣛⠃⣚⣤⣾⣘⠈⢀⣡⡤⠶⢚⡛⠉⣉⠁⢀⣀⣠⣬⠿⣦
⠀⠀⠀⠀⠀⠀⠀⡐⠀⣶⡔⠂⣲⡄⠀⢲⣤⡴⣾⣬⠉⢩⣭⣷⡏⠉⣿⡏⠉⠛⠟⠁⠙⣷⣶⠋⠩⠉⠳⢶⣦⣄⠀⢠⣤⣶⡆⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠘⣶⣶⣶⣾⡟⣿⣶⣶⣶⣾⡋⢹⣶⣾⠉⡁⣷⣶⣿⣷⣶⣿⣾⣷⣶⣤⣶⣶⡶⢷⣶⡾⡿⢿⣶⡾⡉⢛⣷⣶⣶⣶⣶
⠀⠀⠀⠀⢀⣠⡔⠀⣠⣶⡆⠀⢠⡀⠉⠷⠀⢠⠁⢀⡄⠉⢢⡃⠀⣴⣾⠀⢠⣄⠈⢡⠀⢡⠀⠀⢡⠸⠀⠀⠀⢸⡇⠀⠘⣿⡆⠀⢻⡄⠀⣧⣄⡀
⠀⣀⣤⡶⠛⠉⡀⠀⢻⣾⠃⠀⠛⠁⣠⡆⠀⣼⠀⢸⣿⠀⠀⠁⠀⠃⢼⠀⠈⠋⢀⣾⠀⢺⠀⠀⠀⠟⠀⠀⠀⣼⠃⢰⡄⠸⣧⠀⠀⠑⠀⠸⠉⠛⢷⣤⣀
⠈⠻⣿⣷⡄⠀⢣⠀⠈⡿⠀⢠⣶⠿⠛⠁⠀⡌⠀⢸⡏⠀⣠⠀⠀⡾⢟⠀⢰⣆⠀⠙⡷⢿⠀⠘⡄⠀⠀⡆⠀⢻⠀⠘⡇⠀⢿⡀⠀⡆⠀⠀⠇⢠⣾⣿⠟
⠀⠀⠈⠙⢿⣦⡸⠀⢀⠁⠀⣼⡇⠀⠐⠀⢠⡇⠀⠈⠀⣰⣿⠀⠀⠓⠊⠀⠠⠹⡀⠀⠇⢸⠀⠀⣿⡀⢸⡇⠀⠸⠀⠀⣤⡀⠘⣇⠀⢻⡄⠀⠸⡿⠛⠁
⠀⠀⠀⠀⠖⠉⠁⢀⣾⠀⢀⣿⠻⢦⣤⣤⣼⣦⣴⣶⠿⠋⠸⠶⠶⠶⠿⠷⠿⠀⠿⠶⠷⠸⠷⢶⡿⣷⡾⣿⣤⣤⣤⣀⣿⣇⡀⠘⠀⠈⣿⡀⠀⢇
⠀⠀⠀⢸⣤⣤⣶⠿⠻⠿⠿⢏⡀⠀⠉⠙⠧⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠽⠛⠉⠁⢛⣻⠿⠿⠶⠿⣷⣤⣼
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠒⠀⠀⠉⠐⠒⢒⠒⠀⠀⠈⠁⠀⠀⠈⠉⠀⠀⠒⠒⠒⠂⠉⠁⠀⠐⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⡐⠆⠀⠀⠀⠀⠀⠀⠀⠶⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠰⠀⠣⡀⠀⠀⣠⢠⡀⠀⠀⡠⠂⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠠⡀⠘⠀⢰⡠⠀⠀⠀⠄⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠀⠁⠀⠀⠀⢈⣑⠜⠀⠀⠘⢖⣉⠀⠀⠀⠀⠀⠀⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠂⠀⠀⡀⠔⠁⡀⠀⠀⢠⡀⠰⢄⡀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠⠀⠀⠀⠄⢀⠄⠀⠀⠆⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠁⠂⠀⠀⠀⠃⠀⢀⡠⠁⠀⠀⠀⠢⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠪⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠤⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠑⠢⣄⠀⠀⠀⠀⢠⡤⠐⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⡩⢇⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣿⣿⣷⣄⠀⣭⣿⣷⣿⢧⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢀⣾⣿⣿⣟⡂⠈⠛⠒⠿⣞⣿⣌⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡟⡼⣿⣿⣿⢿⠀⠀⠀⠀⠀⠸⣿⣿⣷⡙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡙⢋⣿⢠⡄⠛⠈⠻⢿⣶⣦⣤⠐⠂⠠⢿⣿⣿⡗⣜⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣫⡀⠹⣻⣶⡅⠀⠀⢀⠿⠶⠶⣿⣿⣿⣿⣿⣿⣿⡷⣻⣏⠁
⠀⠀⠀⠀⠀⣠⡖⣄⣻⣿⣧⣀⠈⠩⡷⣃⣴⣿⣶⠖⢠⣭⣿⠷⠶⢿⡿⠿⠷⠼⠿
⠀⠀⠀⠀⣞⣫⢿⣯⣿⢾⣿⣿⣧⣀⢀⣿⣟⡉⠀⢠⣾⣶⠛⢶⣶⠿⠟⠋
⠀⠀⠀⠀⠹⣖⣮⣿⡷⠟⠹⣿⣿⠿⠤⣭⢿⣿⠀⣾⢟⣹⣷⣸⡇⣠⣤⣤
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⢸⣾⣿⣿⣶⣿⣼⣿⠀⠁⠀⠔⣻⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠉⠁⠉⠁⠉⠀⠀⠀⠀⠑⠻⢿⣿⡿⡿⣿⡟⢉⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣀⠱⣖⠠⠤⢴⣦⠀⢠⣾⣿⣽⡿⣿⣿⠋⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⢿⣷⠈⢳⣶⡀⢀⣄⣾⣿⣿⡏⠀⣿⣿⠀⣆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠻⠷⠘⠿⠇⠈⢻⣿⣿⣿⣧⢀⡿⣿⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢠⡟⢲⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡛⢛⣼⡏
⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⡟
⠀⠀⠀⠀⠀⣰⢟⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⣟⡛⢛⣿⣿⣿⣿⣯⢸⣿⡇⡿
⠀⠀⠀⠀⣼⡿⠁⠀⠀⣠⣶⣷⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠁⢼⣿⢱⡇
⠀⠀⠀⣸⣏⣀⣀⣠⣾⠛⠋⠁⠶⢶⣾⣿⡿⠟⠋⢁⣾⡿⠃⡀⠀⢀⣿⡟⢸⡇
⠀⠀⢰⣿⡿⠿⠿⠋⠁⠀⠀⠀⠀⠀⢹⣿⡷⠾⠿⠟⣋⠁⠾⠿⢦⣼⣿⠁⣿⡇
⠀⠀⠸⣿⣅⠀⢦⣀⣀⣠⣶⡿⠷⣛⣩⣴⣶⣿⣿⠿⠿⣿⣿⣿⣾⣿⡏⠀⣿
⠀⠀⠀⠙⠻⢿⣿⠿⠿⠿⠿⠿⠿⠷⠚⠉⠁⠀⠀⠀⠀⠀⠈⠉⠻⣿⣃⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⡶⠤⣤⣀⣀⡀⠀⠀⠀⢀⣀⣀⣤⡤⣶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠙⢦⡀⠀⠀⣳⠀⠀⢀⡟⠀⠀⢀⡴⠃⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠁⠀⢀⠎⠻⡉⠉⠉⢧⢀⡞⠉⠙⣫⠟⢣⠀⠀⠘⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣗⠦⣤⡊⠀⢀⡟⢷⠶⠞⣿⠦⠖⡾⢻⡀⠀⣳⣤⠔⣻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠘⡍⢻⡧⣄⣸⣷⣤⣯⣤⣾⣄⡠⢿⠋⣹⠁⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡇⢀⡇⢀⣿⣿⠛⠛⢹⣿⣿⡀⢸⠀⢻⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠷⣞⠻⢯⣙⠻⣍⢻⣷⣤⣿⠏⡭⠛⣉⠽⠛⣳⠾⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠈⢦⢸⠯⣌⣿⣩⢿⣁⡞⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⣸⠛⠦⡈⡿⣡⠴⠻⣄⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢧⣀⠀⠘⣿⠃⢀⣠⡼⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣿⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⠍⠑⠲⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠒⠛⠛⢭⣥⣤⣄⡀
⠀⠀⣠⣤⣦⣶⣿⣿⣿⡿⣧⣶⣾⣦⡀⠀⠀⠉⠉⠓⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⣴⣶⣦⣿⢿⣿⣿⣷⣲⣦⣤⡀
⣤⠞⣽⣿⣿⣿⠿⠶⠒⢛⣛⣩⠀⠙⢿⣷⣿⣷⣦⡄⣴⣿⣿⣿⣦⢀⣴⣶⣿⣷⣾⠟⠁⢨⣭⣝⣛⡓⠒⢿⣿⣿⣿⡝⢶
⣤⢾⣿⣿⡟⣱⡶⠚⠋⠉⠀⢸⣆⣀⣀⠈⣩⣯⣽⡈⢿⣿⣿⣿⠟⢈⣯⣭⣯⡉⢀⣤⣄⣼⡇⠀⠉⠉⠛⢧⠙⣿⠿⣿⢦
⢡⡟⠁⣾⠐⠁⠀⠀⠀⢀⣠⠞⠋⠉⢀⡞⣵⣿⣿⣧⡀⢙⣿⠋⠀⣾⣿⣿⣧⣹⣆⠀⠀⠉⠻⣦⣀⠀⠀⠀⠀⠸⡆⠙⣧
⣾⣤⣤⣿⠀⠀⢀⣤⠶⠋⠁⠀⠀⠀⢸⣿⣿⠉⣷⣙⢿⣿⣿⣿⣿⢏⣼⠉⢻⣿⣿⠀⠀⠀⠀⠀⠙⠻⢦⣄⡀⠀⡿⠤⢾
⡏⠀⠀⢹⣴⡾⠋⠁⠀⠀⢠⣿⣿⣿⣶⣿⡁⠀⣿⣿⡷⣿⣿⣯⣶⣿⣿⡄⢈⣿⣧⣾⣿⣿⣦⢀⣀⠀⠀⠉⠻⢷⡇
⡇⠀⠀⢸⡏⠀⠀⢀⡴⠋⢸⣿⣿⣿⣿⣿⡟⣸⣿⣿⣷⣶⣿⣶⣾⣿⣿⣧⢸⣿⣿⣿⣿⣿⡿⠀⠙⢷⣄⠀⠀⢨⡇
⣧⠀⠀⢸⣧⣤⣾⠋⠀⠀⣀⢻⢹⣿⣿⠟⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠋⣿⣿⡏⡿⢁⠀⠀⠀⠙⢿⣶⣼⡇
⢿⠀⠀⢸⣿⠟⠁⠀⣠⡾⠟⠁⢈⣿⣿⡇⠉⠛⠿⠿⠿⠛⠛⠛⠿⠿⠿⠛⠉⢸⣾⣿⡀⠀⠈⠙⢦⣀⠀⠀⠙⢿⡇
⠸⠀⢀⡾⠁⠀⢀⡴⠋⠀⠀⠀⠘⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠃⠀⠀⠀⠀⠈⠳⡄⠀⠀⠳⡄
⠀⢠⠏⠀⠀⠠⠊⠀⠀⠀⠀⠀⢀⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⡀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠙⣄
⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢟⣽⣶⣿⠿⣶⣄⠀⠀⠀⠀⢀⣴⣶⠿⣷⣶⣭⡻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣅⠀⠈⠉⠁⠀⠀⠀⠈⠉⠁⠀⣸⣿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠹⣟⠳⠶⠤⠀⠀⠀⠀⠀⠴⠶⢞⣿⠏⣸⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠄⠀⢀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠆⠀⠠⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⡁⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠀⠀⠀⠀⣠⠶⣿⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢌⡄⣀⣼⠗⠀⣿⠇⠐⡾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣽⢃⢿⣶⡶⠜⠀⣼⡤⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢌⠁⠄⠈⢲⡾⠹⠠⠸⡐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣤⠝⠋⠀⠀⠀⠀⠁⠲⠈⠐⣤⡀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡒⠝⠉⠀⠄⠀⠄⠀⠘⠀⠀⠀⠉⠀⠀⠠⠀⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠐⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠊⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡃⠑⠂⠀⠀⠀⣹⣏⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠏⡇⠀⠀⠀⠀⠙⠋⠀⢀⠀⠀⠀⠀⢀⠀⣠⣿⣦⣈⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡼⡇⠀⡀⠐⣢⣿⣤⣀⡂⠀⢀⠀⢀⣤⣾⣿⣿⣾⡃⠀⠀⢀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡃⢧⣠⣶⣿⡿⠛⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⢳⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡌⡈⣿⣿⡅⠄⢆⠀⠀⠀⡄⣼⣿⢟⣿⡿⠋⠀⠀⠈⡇⠠⠀⠈⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢻⣿⡟⠓⠛⠈⡉⠉⣹⣿⣿⢿⣿⠃⠀⠀⠀⠀⠈⠢⣢⣀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢃⣡⢸⣿⣿⣆⢥⣠⠤⢶⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢀⠔⢁⠈⣠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⢸⣿⣿⠉⣾⠀⠀⣸⣿⣿⣿⣿⠇⠀⠀⠀⠀⢀⠎⠖⠀⠐⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠆⠸⢿⣿⠀⠁⠀⢀⣿⣿⡿⠛⠩⢆⠀⠀⠀⢀⠎⠂⠀⠀⠣⣎⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⣸⣻⢷⡀⡠⠜⠋⢁⣠⣴⣾⣿⡷⢀⢠⠊⠈⢀⡀⠤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢨⣿⣔⠉⢐⣠⣾⣿⣿⣿⣿⡷⡟⢀⠨⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡖⠀⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡤⠒⡬⣴⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⠉⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⣾⣿⠁⠀⢀⣴⣿⣿⣦⡀⠀⠀⣿⣷⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⣿⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⣿⣿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⣿⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⣿⣿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⠹⣿⣧⣀⣀⣽⣿⣿⣯⣀⣀⣼⣿⠏⠀⢀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⠿⢿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣛⡿⠿⠿⠿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠛⠉⣿⣿⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⣿⣿⠉⠛⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⠀⢻⣿⡀⠀⢿⣿⣿⣿⣿⣿⣿⡿⠀⢀⣿⡟⠀⠀⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠘⣿⡇⠀⠘⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⠃⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠘⣿⠀⠀⠙⣿⣿⣿⣿⠋⠀⠀⣿⠃⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠘⡇⠀⠀⠈⠙⠋⠁⠀⠀⢸⠃⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠶⠿⠟⠛⠛⠛⠿⠶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠾⠛⣿⣅⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⢀⣴⡾⢻⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡟⢷⣦⣀⠙⠻⣦⣄
⠀⠀⠀⠀⠀⢠⣾⠻⣷⣠⡾⠟⠁⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠈⠻⢷⣄⣾⠿⣷⡄
⠀⠀⠀⠀⣰⡟⠁⣠⣿⡏⠀⠀⠀⢀⣴⣿⠷⠿⠿⠿⠛⠻⠿⠿⠿⠾⣿⣦⡀⠀⠀⠀⢹⣿⣄⠈⢿⣆
⠀⠀⠀⣰⡟⠀⣰⡿⠹⣷⠀⢀⣴⡿⠋⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠙⢿⣦⡀⠀⣾⠏⢿⣆⠀⢻⣆
⠀⠀⢠⣿⣧⣰⡟⠀⠀⢿⣷⡟⠋⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠙⢿⣶⡟⠀⠈⢿⣆⣼⣿⡀
⠀⠀⢸⡟⢹⣿⠁⠀⢀⣾⣿⡄⠀⠀⠀⠘⣿⣀⣀⣀⣀⣀⣀⣀⣀⣿⠃⠀⠀⠀⢀⣿⣷⡀⠀⠀⣿⡏⢹⡇
⠀⠀⣾⡇⢸⣿⡇⢀⣾⠃⠸⣧⠀⣀⣴⡿⣿⡏⠉⠛⠛⠛⠛⠋⢹⣿⢷⣦⣀⠀⣾⠇⠘⣷⡀⢸⣿⡇⢸⣷
⠀⠀⣿⠀⣿⡟⣿⣾⠃⠀⠀⣻⣿⠟⠁⠀⢸⣇⠀⠀⠀⠀⠀⠀⣼⡇⠀⠈⠻⣿⣟⠀⠀⠘⣷⣿⢻⣿⠈⣿
⠀⠀⣿⠀⣿⠁⣿⡃⠀⠀⣰⡿⢿⡆⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢰⡿⢻⣆⠀⠀⢘⣿⠈⣿⠀⣿
⠀⠀⣿⣷⣿⢀⣿⣧⠀⣴⡟⠀⠈⣿⡄⠀⣠⣿⣷⢶⠶⡶⠶⣾⣿⣄⠀⢠⣿⠃⠀⢻⣦⠀⣼⣿⡀⣿⣾⣿
⠀⠀⣿⢸⣿⣾⣿⠹⣿⡟⠀⠀⠀⢘⣷⡾⠛⠹⣿⠀⠀⠀⠀⣾⠏⠛⢷⣾⡃⠀⠀⠀⢻⣿⠏⣿⣷⣿⡇⣿
⠀⠀⣿⡀⣿⢹⡏⠀⠙⣷⡄⠀⣠⡿⠛⣷⡀⠀⢻⡆⠀⠀⢸⡟⠀⢀⣾⠛⢿⣦⠀⢠⣾⠋⠀⢸⡏⣿⢠⣿
⠀⠀⢿⡇⣿⢸⣇⠀⠀⠈⢿⣾⠋⠀⠀⠘⣿⣤⣼⣿⠿⠿⣿⣧⣤⣾⠃⠀⠀⠙⣿⡿⠁⠀⠀⢾⡇⣿⢸⣿
⠀⠀⢸⣷⣿⡚⣿⠀⠀⠀⠀⠙⢷⣄⠀⢀⣾⢿⣅⢻⡇⢸⡟⢨⡿⣷⡀⠀⣠⡾⠋⠀⠀⠀⠀⣿⣓⣿⣾⡇
⠀⠀⠸⣿⢹⡿⢿⡆⠀⠀⠀⠀⠀⠙⢷⣿⠁⠈⢻⣾⣿⣿⣷⡟⠁⢈⣿⡾⠋⠀⠀⠀⠀⠀⢰⡿⢿⡏⣿⠇
⠀⠀⠀⣿⡌⣿⡈⣿⡄⠀⠀⠀⠀⠀⠀⠙⢻⣦⣤⣽⣿⣿⣯⣤⣴⡿⠋⠀⠀⠀⠀⠀⠀⢠⣿⢁⣿⢡⣿
⠀⠀⠀⢸⣧⢹⣇⢈⣿⡄⠀⠀⠀⠀⢀⣴⡿⢋⣴⣿⠟⢻⣿⣦⡙⢿⣦⡀⠀⠀⠀⠀⣠⣿⡃⣸⡏⣼⡇
⠀⠀⠀⠀⣿⣴⣿⡟⠋⢿⣦⣤⣴⣾⡟⢻⣶⡟⣿⠏⠀⠀⠹⣿⢻⣶⡟⢻⣶⣦⣤⣴⡿⠙⢻⣿⣦⣿
⠀⠀⠀⠀⠘⣷⡈⢿⡄⠈⢿⣍⠀⢸⣿⡟⠙⣿⣿⣦⣤⣤⣤⣿⣿⠋⢻⣿⡇⠀⢩⡿⠁⣠⡿⢁⣿⠃
⠀⠀⠀⠀⠀⠹⣧⠈⢿⣆⠈⣿⣶⠟⠙⢿⣄⣿⠁⠀⠀⠀⠀⠈⣿⣠⡿⠋⠻⣶⣿⠁⣰⡿⠁⣼⠏
⠀⠀⠀⠀⠀⠀⢻⣇⠀⣿⣿⠋⠻⣧⡀⠈⢻⣿⣶⣶⣶⣶⣶⣶⣿⡟⠁⢀⣾⠟⠙⣿⣿⠁⣼⡟
⠀⠀⠀⠀⠀⠀⠀⢻⣶⡟⠙⣷⡄⠈⢻⣦⣾⡇⠀⠀⠀⠀⠀⠀⢸⣷⣴⠟⠁⢀⣾⠋⢻⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠈⢻⣦⡀⢹⡿⠿⠷⠿⠾⠷⠾⠿⠿⢿⡏⠀⣴⡟⠁⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠙⣷⣾⣇⣀⣀⣀⣀⣀⣀⣀⣀⣸⣷⣾⠋⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⢀⣾⠏⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠹⣷⡀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣤⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⣤⣾⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠦⣀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⣰⣿⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⡍⠐⠠⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⣀⡤⠂⣴⠃⣏⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠱⡄⠀⠈⠑⠚⢉⣧⡀⠀⠀⠀⢠⠀⠀⠀⣠⢻⢫⠉⠁⠀⡼⠋⠀⢻⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⡀⠈⢷⣤⣤⡤⠊⠀⠹⢦⣀⣀⣷⠀⠀⡰⢣⠇⠀⠁⢂⡾⠁⠀⡄⠘⡀⡀⠑⣤⡀⠀⠀⣀⣤⠖⠂⠉
⠀⠀⠀⠀⠑⠢⣄⣀⡠⠒⠁⠀⠀⠀⠀⡠⠁⠀⠸⡇⢣⠉⠢⠤⠸⡄⠈⠁⢸⡗⠊⢠⡟⠠⠤⣴⠛⣇⠀⠀⠘⢄⠐⢄⢀⠀⠉⣹⠟⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⠦⣤⠴⠒⠋⠁⠀⡀⠀⠀⠀⢧⡀⢀⡤⢻⣤⣀⣤⣏⠀⣸⠀⠀⣰⠃⠀⠈⠢⠄⡀⠀⠑⣢⣽⠟⠋⠁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠙⢶⡶⠶⠉⠀⠀⠀⠀⡾⠉⢥⠀⠠⢷⠀⢸⡀⠉⡏⠑⢺⠋⠄⠀⠀⠀⠀⣀⠴⠋⢸⠃⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠲⡤⢤⣄⣁⠈⠀⠀⠈⣷⠴⢿⠃⣿⠑⢺⣧⣠⠏⠀⠈⠀⣀⡴⢛⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠈⠉⠲⣤⣖⡉⢳⣼⣆⣇⣠⣦⣼⠉⢣⣤⡖⡻⠋⠀⣼⠀⠇⠀⠈⡄⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⡘⠀⠀⠀⠘⢠⠃⠀⠀⢠⡹⡙⣿⠀⡙⣿⣿⡉⢻⡖⠚⢻⡁⢱⠀⠀⢸⡀⠀⠀⠀⠙⣄⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠊⠀⢀⡴⠁⣀⣀⠤⢶⡏⢠⣄⣤⣾⡶⠒⠿⣄⡇⣻⠛⣴⠾⠴⣶⠾⠷⢶⡷⠖⢒⠻⠗⠢⠤⠴⠞⠛⡲⠦⠬⣶⠤⠴⠤
⠀⠀⠀⠠⠤⠤⠚⠉⠈⠉⠛⠉⡀⠀⠀⠀⠀⠀⠙⢦⠀⠙⣦⠘⣀⡠⠟⠳⣧⣿⡷⠶⢢⣿⠁⢠⡞⠀⢠⠁⡠⠂⠁⢀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠈⡇⠀⡼⠟⠉⠙⢢⣴⢛⡏⢣⡔⣉⠤⠐⢤⣁⠀⢷⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣰⡡⠊⠑⠒⢦⡀⡾⠛⢻⡗⠉⡟⠁⢀⠔⠀⣉⣉⣺⣇⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰⠞⠉⠀⠀⠈⢢⠀⢱⠀⠀⢸⠀⣀⣀⡰⠃⠰⠊⠉⠴⠶⠞⠷⣤⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠟⠒⠒⠠⠄⠀⠀⠀⢧⡸⠖⠒⣾⢡⠄⠊⢻⠄⣰⠂⠀⠀⠠⠴⠒⠋⠉⠙⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⡀⠀⢸⡏⠀⠀⠐⡄⠁⠠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠯⠉⠀⢨⡇⡠⠔⠒⠧⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣀⠀⠀⠘⡇⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⢀⡇⢀⡤⠤⠤⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠘⣷⠋⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⣀⣤⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⠴⣿⣿⡿⠿⣿⣿⣿⣿⣷⣤⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠉⣋⠁⢰⣿⣿⠟⢀⡾⢹⣿⣿⣿⣿⣿⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠘⣦⣿⡤⢾⣿⣿⣤⣾⣡⣾⣴⢿⣿⣿⢋⣽⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢻⣿⡆⢸⣿⡏⣼⠫⢫⡟⢉⣿⣿⡇⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠙⣿⢧⣹⣿⡼⣿⣶⣿⣶⣯⣴⣿⣿⠿⠁⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⠀⣿⣍⢳⣿⣿⣿⣷⣿⡿⠋⠁⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠈⠻⢿⣿⣿⣿⣿⡉⠀⠀⠀⠀⣰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠂⣠⣀⣴⣿⣿⣿⡿⠻⣿⠆⣤⣴⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠧⣜⡿⠋⢹⣇⠘⣿⡀⢳⣶⣿⣭⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣿⠓⢳⣼⣇⣷⣼⣿⡉⠉⢿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠦⣤⣿⠁⠀⣧⡹⣧⣴⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡌⣿⡗⠈⠉⣯⡽⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣿⣾⡇⠀⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢚⣿⢟⢨⣿⣽⣿⣿⣿⣿⣿⢿⣿⢻⣿⣿⣿⣿⣿⡿⢋⣉⡛⠦⣄⠀⠒⠁⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⠈⠁⠀⣼⣯⣍⣹⣿⣿⣿⣏⣸⠇⣿⣷⣾⣿⣿⣿⣿⣿⡿⠉⠀⢀⣠⣴⣶⣼⣧⣶⣶⠶⠤⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⠧⠀⢀⣌⡀⠀⠀⠃⣾⠛⣿⣿⠟⠻⢿⣿⠿⣿⣿⡛⢻⣿⠛⠋⠀⣠⣞⢿⣿⣿⣿⣿⣿⠟⠁⠀⢀⣼⣿
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⠃⢀⣿⠒⠄⠀⠁⠀⣈⣿⢁⡃⠘⠡⣄⣿⣿⣶⠶⠟⢁⣄⣸⣷⣿⣿⣟⣿⣿⠟⠁⠀⠀⣠⣾⣿⣿
⠀⠀⠀⠀⠊⠉⠳⠀⢄⠀⠄⠀⠀⠀⠈⠲⣄⢰⡄⢀⠈⠙⠋⣛⡆⠀⠁⠹⠇⢉⣀⠀⠪⡿⣷⡤⡿⠗⠛⠉⠀⠀⣀⣠⣾⣿⣿⣿⣿
⠀⠀⠀⠇⠀⠀⠀⠀⢀⠓⠀⠀⠀⠀⡀⠀⠈⠻⠀⠀⠏⢰⡄⠀⠀⠀⠀⠀⠾⠇⣿⣶⣿⣿⠿⠋⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⠉⠙⠂⠢⣤⠃⠀⠀⠀⣶⣤⣿⢴⣄⡀⠀⠐⠲⠼⠓⠓⠀⠀⠀⢤⡶⣶⣤⣤⣤⣤⢤⣶⣶⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣿
⠀⠀⢰⠀⢠⠀⠀⠉⢉⣿⣶⣦⣻⣿⣿⣀⣈⣿⠁⣀⣦⢰⠇⢀⡀⢀⠀⣉⣉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣘⢠⣟⡀⠀⣠⠿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠞⠛⠛⢁⣴⠟⠁⢿⣷⡌⢻⣷⣦⣄⣉⡉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⡰⠉⡿⠉⣹⠂⣥⠀⢿⣿⣿⣿⣝⠋⠁⠀⣀⡠⠴⠞⠛⠁⠐⡁⢸⣯⣛⡀⠙⢻⣿⣩⡿⠶⣤⡀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢀⡃⣸⠃⢠⣿⡆⣼⡇⢸⣿⣿⣿⣷⡄⣰⣧⣽⡶⠶⠆⠀⣰⠾⠿⣿⣿⠻⢿⣦⡀⠉⠁⠀⣰⣿⣷⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⢿⡟⠛⣿⣿⡇⠸⠃⢸⣿⣿⣿⣿⣿⣿⣿⣸⠗⠀⠀⠀⣿⠀⠀⣿⣿⠀⣀⣈⠙⢦⡀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⢸⡇⢰⡏⣼⡇⠸⠀⣼⣿⣿⠟⣿⣿⡋⠟⠁⠀⠀⢰⣶⣾⣿⣿⣿⣿⣿⣭⣉⠉⠻⣿⠆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢘⣼⠷⣾⣏⣿⣷⠀⠀⣿⣿⠋⠀⢹⣿⡅⠀⠀⢀⣼⣿⣿⣿⣿⠟⢙⣿⣿⣿⣿⣿⣶⣿⣴⡆⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡿⣿⠀⣿⠿⣿⣿⡇⠀⡿⠁⠀⠀⠸⣿⣿⢧⠀⣶⣿⡿⠷⣾⣿⣤⣽⣿⣿⣿⣿⣭⣍⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣦⣿⣆⠹⣿⣿⢸⣿⡀⠀⠀⢠⣿⣿⡄⡀⣿⣿⣅⢀⣀⣿⠉⣿⣿⣛⠩⢈⠉⠉⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡿⢋⣾⣿⣿⣿⣿
⣻⣿⣀⣿⣏⣻⣿⣿⣿⣿⠃⠀⠀⢸⣿⣿⡷⣿⢿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣾⣿⣿⣿⣿⣿
⠿⣿⠉⣿⡏⢩⣿⣿⣿⡿⠀⠀⠀⢸⣿⣿⣿⣼⣇⣿⣿⣿⡿⣿⠟⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⢻⣿⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⠿⢿⣯⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⢿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠋⠁⠀⠈⢿⣆⠀⠀⠀⠀⠀⣀⣠⣾⣿⣦⣄⣀⠀⠀⠀⠀⢀⣾⠇⠀⠈⠙⠿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⣡⡖⠀⠀⠀⢀⣼⣿⡶⠾⠿⠿⠟⠛⠉⢹⣿⠈⠙⠛⠿⠷⢶⣶⣾⣿⣄⠀⠀⠀⠠⣌⠻⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠻⣷⣾⣿⡇⠀⢀⣴⡿⠋⠹⣷⡀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣸⡟⠈⠻⣷⣄⡀⠀⣿⣷⣤⣿⣿⣿⣿⣄
⠀⠀⠀⠀⢠⣾⣿⣿⠏⠀⣰⣿⣿⣿⣷⣾⠟⠋⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠈⠙⠿⣶⣿⣿⣿⣇⠈⠻⣿⣿⣧⡀
⠀⠀⠀⣠⣿⣿⡿⠁⠀⣼⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢀⣿⣷⣶⡶⠾⠿⢻⣿⠻⠿⠷⣶⣶⣿⣇⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣧⡀⠙⣿⣿⣷⡀
⠀⠀⢰⣿⣿⡟⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣤⣾⠟⠋⢿⡆⠀⠀⠀⢸⣿⠀⠀⠀⢀⣾⠏⠛⢿⣦⣄⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⠀⠘⣿⣿⣿⡀
⠀⢠⣿⣿⡿⠛⠿⣶⣿⣿⣿⣿⠟⠿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠘⣿⣄⣀⣠⣼⣿⣤⣀⣀⣾⡏⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣧⣶⠿⣿⣿⣷
⠀⣼⣿⣿⠃⠀⢀⣿⣿⣿⣿⠏⠀⠀⠀⠙⠻⣿⣿⣿⣷⣄⢀⣤⣿⣿⡛⠋⢹⣿⠉⠛⣿⣿⣦⣄⢀⣴⣿⣿⣿⡿⠟⠉⠀⠀⠸⣿⣿⣿⣿⡄⠀⠸⣿⣿⣇
⢰⣿⣿⡏⠀⠀⢸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⡉⠀⢹⣧⠀⢸⣿⠀⣰⡿⠀⠈⣻⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣇⠀⠀⣿⣿⣿
⣸⣿⣿⠃⠀⠀⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣄⠀⢻⣧⣼⣿⣤⣿⠁⢀⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠀⠀⢸⣿⣿
⣿⣿⣿⣦⣤⣤⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣶⡿⠋⠉⠉⠙⠻⣷⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣶⣶⣾⣿⣿
⣿⣿⣿⠀⠈⠉⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⣤⠀⠀⣤⣴⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⢿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡟⠀⠀⢹⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⢠⣿⣿
⢹⣿⣿⡄⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⢶⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡏⠀⠀⢸⣿⣿
⠸⣿⣿⣧⣀⣤⣾⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠁⠘⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣦⣤⣿⣿⣿
⠀⢿⣿⣿⡏⠉⠀⠘⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣤⣤⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠇⠀⠈⢹⣿⣿⡏
⠀⠈⣿⣿⣷⡀⠀⠀⠘⢿⣿⣿⣿⣦⣀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⡿⣿⡏⠀⠀⢹⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠋⠀⠀⢠⣿⣿⡿
⠀⠀⠘⣿⣿⣷⡀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢡⣿⠀⣀⣀⢘⣿⡈⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⡟⠁⠀⠀⢠⣾⣿⣿⠁
⠀⠀⠀⠙⣿⣿⣷⡀⢀⣠⣾⠟⠙⠻⢿⣿⣿⣿⣿⣿⣿⠛⠋⠉⠻⣿⡿⠛⠛⠛⠛⢿⡿⠋⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⠿⠛⠙⢿⣦⡀⣠⣿⣿⡿⠁
⠀⠀⠀⠀⠘⢿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠈⢻⣧⣴⡿⠁⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠈⢿⣏⣽⡿⠁⠀⠀⠀⠀⠀⢉⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠀⠀⠀⢀⣿⡿⠷⣶⣤⣄⡀⢸⣿⠀⠀⠀⠀⠀⠈⣿⡀⢀⣠⣤⣶⡾⢿⣿⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⡀⠀⢠⣾⠏⠀⠀⠀⠉⠙⠻⣿⣷⡶⠶⠶⠶⠶⣶⣿⡿⠛⠋⠁⠀⠀⠀⠻⣷⡀⠀⢀⣴⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣿⡃⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣶⣦⣄⣀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⣀⣠⣴⣾⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣷⣿⣷⣤⣤⣤⣤⣤⣤⣤⣴⣿⣷⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⠉⠉⠉⠑⠲⠤⣶⣿⣿⣿⣿⣿⣿⡿⠿⡆⠀⠀⢀⣤⠴⠖⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⢀⠰⡄⢸⣿⡟⠁⠀⠀⠀⠀⠀⠱⡀⡴⠋⠀⠀⠀⠀⠀⢙⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⠀⣀⡴⠶⠚⠚⠒⢠⣾⣿⡧⠀⠠⣶⡆⠀⢲⡎⣿⠃⠀⣀⠤⠒⠉⢉⣉⢭⣭⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⣀⡼⠚⠁⢐⣂⠀⠀⣠⡘⡏⣽⠳⣤⣀⡠⠔⠊⠭⠭⠽⣤⣾⡟⠀⢠⣭⠀⢰⡞⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢀⢰⡿⢤⣄⣀⣈⡭⠴⠖⠛⠀⢿⣌⠀⠀⢹⠀⠀⠀⠒⠆⠀⣿⣯⠓⠦⣤⠔⠊⠁⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢙⡇⠀⠈⢹⡁⠀⠠⠖⣒⠒⢲⡿⡟⢦⡟⠀⠈⠉⢉⣉⣩⡇⢳⣤⡀⡘⠀⠴⠊⠤⢨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠛⡶⣄⣸⠀⠀⠈⠉⠩⣝⡾⠀⡇⠀⠈⠑⠲⠤⢠⡤⠎⠀⡴⡇⠙⠣⣄⡀⠀⠒⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠁⠈⠛⠦⠤⣄⣀⣀⣾⡗⢾⢧⠀⠀⠀⠀⠀⢸⣧⡴⢺⠀⠙⣆⠀⠀⠈⠙⡟⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠉⠳⡀⠀⠀⠀⠀⠹⣏⣀⠸⡄⢫⠳⣄⡀⠀⠀⣸⠏⢹⡹⡗⠦⣌⠳⣄⣀⣠⡗⢀⣹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠈⢻⣦⣤⣌⠢⡀⠀⠀⠀⢳⠘⡟⠻⣄⠳⡀⠉⠛⠛⠉⠀⠀⢧⢱⠀⠈⢳⣤⣣⣜⡴⠋⠁⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢸⡇⠀⠈⠳⣌⠳⢤⣀⣸⠀⢹⡀⠈⠳⡱⣀⣀⣀⣀⣀⣀⠼⠀⡇⠀⠀⢳⠘⣟⠀⠀⠀⠀⠀⡺⡆
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢀⠀⠀⠙⢆⠀⠀⠘⣆⠀⠈⠉⠀⣠⣧⣠⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠈⡆⣿⠀⠀⠀⠀⠀⡇⠇
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠳⣄⠀⠘⣆⠀⠀⠀⠹⠿⠁⢣⠀⠆⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢱⠹⠀⠀⠀⠀⠀⣿⠸
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⡠⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠢⡀
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⡁⠢⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢒⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡦⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⢤⢀⠀⠀⠀⠀⠀⠀⢓⠀⠙⢦⠀⣠⠀⣠⠬⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠸⠦⣄⣾⣋⣉⣙⢦⡴⠶⠃⠀⠀⠀⠀⠀⠠⠀⡠⠂⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠼⡞⢿⣿⡟⣸⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡼⠅⠀⣀⡝⣶⣽⡾⢣⣄⠀⠙⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢸⠍⠠⠈⠉⠁⠀⡞⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠃⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣷⡄⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿⠟⡽⠿⠟⠛⡇⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣷⣶⣷⣿⡿⠃⠀⢸
⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠀⠀⠀⡸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣽⡆⠀⠀⢛⡿⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⣾⠁⡠⠰⡶⠖⢛⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⠗⠚⠀⢀⡾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⣧⠀⠀⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣧⡤⠄⠈⡏⠉⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣇⢀⣼⣧⣰⣗⠻⠉⠿⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡿⠃⣻⠁⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢷⡀⣠⠀⠀⠐⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⣸⠿⢿⣦⡀⢱⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠚⠓⠲⣤⣈⠻⣌⣇⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⠤⠤⠭⠿⢺⣿⠢⡤⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠸⠧⢤

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⠤⠠⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠊⠉⠀⡠⠁⠀⠐⠦⢀⣀⠀⣰⠦⠀⠀⠀⠀⠀⡴⠂⠈⠂⢦⡀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⢀⠀⢀⠀⠈⠀⠀⠀⠀⠘⣦⣄⣀⠀⠀⣄⠀⡄⠀⣼⠁⠀⠀⠀⠀⠑⠆⠀⠀⠀⢠⡆
⠀⠈⢁⡶⠚⠁⠀⠀⠀⠀⡀⠀⠀⢰⣄⠀⠀⠀⢀⠉⠁⡈⠀⠀⠃⠀⠈⠀⠀⠀⢀⣀⣠⠈⢧⠀⡄⠀⢹
⠀⠆⠈⠁⠀⠁⠀⠰⠞⠁⠀⠀⠀⠀⠈⢀⢀⠀⠀⡉⠁⠈⠀⠀⣖⠀⠀⠀⠀⠀⠉⠙⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠤⣴⠊⡏⠀⠉⣽⣿⣧⠀⡀⡜⠐⢢⠄⠀⠀⠀⠀⠀⠀⠀⢠⠼
⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⢆⣧⣤⣶⣿⣿⣿⣶⣟⣣⣠⠜⠀⠀⠀⠀⠗⠀⠀⢀⠀⠀⠀⠄
⠀⠀⢀⠀⣀⠤⠐⠛⠲⠤⡀⠈⠛⠀⠉⠢⣳⣿⣿⣿⣿⣿⣿⣷⡀⠆⠀⠐⠂⠀⠀⠀⠀⠀⣼⠀⠀⡀
⠀⠀⠈⠟⠓⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢤⡏⢿⣿⣿⣿⣿⡇⠉⠓⠒⠒⠀⠀⡀⠀⠀⠀⣇
⠀⠀⠀⠀⠐⠂⡀⠀⠀⠀⠀⡄⠀⠲⠂⠄⠈⠁⠄⠻⡿⠋⢨⢯⠀⠀⠀⠀⠀⠂⠉⠀⠀⢀⣘⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠈⠀⠀⠀⠀⡄⠠⠀⠀⠀⢀⠐⠊⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠈⠁⠀⠀⣇⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⡆⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠢⣄⣠⠄⠂⠐⠳⣀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠃⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⢳⣸⡴⠊⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢨⡏⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢴⠒⠚⠋⠉⠉⠉⠉⠓⠒⠲⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠉⠑⠒⠒⠒⠀⠐⠒⡒⠞⠋⠉⡉⠛⠦⣄⡀
⠀⠀⣠⢚⣉⣀⡐⠒⢦⢀⣀⡀⢀⡴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⠘⠄⠀⠀⠙⠦⣄
⠀⠀⢿⣡⠤⣿⠇⠄⣽⣾⠃⢡⡞⢡⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠈⢳⡄
⠀⠀⢸⡅⠰⢟⠒⠒⠻⡟⠰⡿⠀⡘⠀⠀⠆⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠙⣆
⣠⠤⣤⢿⠀⢀⣱⣌⠀⠉⢳⠃⢀⠧⣀⡀⠀⠀⣀⣠⠤⠾⢃⠀⠀⠀⠀⠀⠀⡀⢰⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠘⣧
⢇⠀⣇⢘⠀⡞⠀⠈⣣⡀⣾⣶⡿⠀⠀⠈⠉⠉⠉⠀⡰⠁⠀⢣⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⣠⢼⡆
⠈⠉⠉⠉⠒⢧⣀⡘⠁⠘⣿⣿⣷⡀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⡝⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠉⠻⣄⢰⣿⣿⣿⣷⡀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠑⠦⣀⠀⠀⠀⠀⣀⡴⠞⠋⠁⠀⢠⡇⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡿⣿⣿⣿⡀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣲⣶⠟⠉⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⢹⣿⣿⡗⠒⢹⢳⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⢸⠀⠀⠀⠀⢀⣀⠀⠸⡇⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢿⣿⣧⠃⡎⠀⢙⢦⣀⠀⠀⠀⢀⣰⣏⣥⣤⣤⣤⣶⣤⣴⣶⣿⣿⠀⠀⢇⣾⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⣸⣿⡏⠀⡇⠀⠋⢀⠌⣳⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⡏⠉⠛⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⣿⣿⡇⢸⠀⠀⠀⢃⣼⣿⣿⣿⡿⠟⠋⠉⠉⠀⠀⠀⠀⣽⣿⣿⡟⠀⢠⡿⣶⣿⠇⢹⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣶⣿⣿⣾⠸⠄⢀⣄⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠒⣲⣿⡀⢹⡷⣄⣈⣩⠛⠻⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡛⠿⡟⠋⢡⠀⢀⣸⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠋⢀⡴⠋⠀⠙⢦⣇⣘⣿⠏⠀⢀⣀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠰⣀⠎⠈⡏⠀⠹⣿⣿⣦⣤⣀⣀⣀⣀⣠⣴⣾⣿⡿⠋⣡⣴⠟⠓⠦⣄⠀⢀⡟⠁⠙⡆⢺⣿⠏⢠⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡟⠠⠄⣃⡴⠒⠚⠛⠿⣿⣿⣿⣿⣿⣿⠿⠛⣡⣴⣿⠟⠁⠀⠀⠀⠈⢳⣼⣃⣀⣀⡇⠈⢿⡆⠈⢧⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⢙⠀⠀⠀⠀⠀⠀⣈⣑⣤⣤⣿⣶⣿⣿⢿⠁⠀⠀⠀⠀⠀⠀⢠⡟⠉⠀⢈⡇⠀⠈⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⡛⠛⠛⠛⠋⠉⠉⠉⠙⢿⣿⣿⣍⠉⡷⠶⠤⢤⣀⠀⠀⠀⣼⠁⠀⠀⢸⡇⠀⡴⠂⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠉⠙⠛⠛⠷⠶⢶⣤⣤⣀⣈⣻⣿⠙⢧⣇⠀⠀⠀⠈⠙⢦⣼⣃⣀⠀⢀⣾⠷⠞⣀⢀⡴⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⢀⠀⠠⠄⠀⠀⠀⠀⠀⠉⠉⢹⡇⠀⠈⠻⣆⠀⠀⠀⢠⡾⠉⠉⢉⣽⠞⢉⣋⣩⡥⠴⠧⠼⣧⣄⡀
⠀⣤⣄⠀⠀⠀⠀⠀⠀⣀⡴⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠈⢳⣄⣴⣋⣤⠴⢺⡿⣣⡶⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⡀
⠀⣏⠙⢿⠦⣄⣀⣴⠚⠭⠒⠉⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⣰⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⣿⠀⠈⢷⡌⠙⢦⡀⠀⠀⢀⡠⠄⠂⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡃⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⣿⠀⠀⠈⢻⡆⠀⠀⡀⠉⠀⠰⡄⠀⠀⢸⡆⠀⠀⠀⣠⠄⠀⠀⢀⡼⢧⡁⠀⠀⠀⠀⠀⢠⡾⠃⢰⣿⣿⡁⠀⠀⠀⠀⠊⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⢹⡆⠀⠀⠈⢷⠀⢸⡅⠀⠀⠀⣁⡤⠒⠀⡇⠀⣠⠞⠛⠋⠙⢉⣿⢁⠎⠙⢦⣀⠀⠀⠀⠛⠁⢠⣿⠛⠳⣍⠓⠀⠀⠀⠀⠈
⠀⠈⣷⠀⠀⠀⢸⣇⣈⣷⠔⠒⠉⢹⠀⠀⣀⡴⠟⠁⠀⠀⠀⢠⡞⢁⠟⠳⣤⣤⠋⠙⢛⡷⠒⢲⡿⠁⠀⠀⣨⣿⣦⡀
⠀⠀⠘⣧⠀⠀⠀⣯⢀⡏⠀⠀⢀⣸⡶⠟⠉⠀⠀⠀⠀⣀⣠⠟⣰⡏⠀⠀⡼⠉⠲⢤⠾⠄⢠⡞⠁⠀⠀⣰⠿⠜⠙⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⢧⣀⠀⡿⢘⣧⡴⠞⠋⠁⠀⠀⠀⠀⠀⢠⡟⠉⠏⡴⠃⠘⠒⣾⣀⣀⣰⠃⠀⣠⠟⠀⠀⠀⣰⡏⠀⡇⠸⡀⢈⡿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⢰
⠀⠀⠀⠀⠀⠈⠓⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⠟⠲⢤⡀⡼⠁⠈⢩⠋⢒⣾⠋⠀⠀⠀⢰⣟⡵⠊⠑⣴⣀⡞⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣼⠂⣄⠀⣸⠋⠑⢲⠷⣤⡟⠁⠀⠀⠀⠀⣿⠋⠑⠀⡴⢋⡝⢷⡄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⣠⣍⡉⣈⣩⡴⠋⠀⠀⠀⠀⠀⠀⠿⠤⠴⠟⠛⠉⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⢙⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⣶⣤⣤⣤⡶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣴⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢀⣴⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡝⣿⣪⣿⣿⣿⣿⣻⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠓⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣾⡗⣶⣿⣿⣿⣿⣳⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠁⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣴⣴⣶⣶⣶⢼⣦⣦⣿⣧⣦⣤⣦⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⡙⠉⢗⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢋⠍⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⣀⡤⠖⢋⠠⠊⠀⠀⠀⠳⡤⡈⠓⠦⣄⠀⠀⠀⠀⠀⣠⠴⠊⠁⠀⣰⠏⠀⠀⠈⠑⠦⣄⣀⡀⣀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠧⣀⠀⠁⠀⠀⠀⠀⡠⠊⠘⢮⡂⢄⡈⠙⠲⢄⡶⠊⠁⠀⠀⢀⡴⠋⠀⢀⠀⠀⠀⠀⠀⢈⡝⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡆⠈⠑⠶⢄⡐⠀⠀⠀⠀⠀⠈⠡⡄⠈⠁⠐⠚⠃⠀⢀⠠⠂⣰⡃⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠀⠀⠀⠰⢄⠀⠐⢲⠃⠈⠀⢠⠞⡀⠹⣄⠀⠀⣀⠴⠊⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡌⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠲⢆⡀⠀⠈⢧⠀⠠⡃⢀⡰⠉⠄⠀⢀⡠⠶⠊⡅⠀⠀⠀⢠⠀⠀⠄⠈⢧⡀⢀
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠌⠀⠐⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⠒⣘⣁⠋⣀⡥⠶⠛⠉⠀⠀⠠⡀⠀⠀⠀⠀⣂⣀⡤⠄⢺⠏⠁
⠀⠀⠀⠀⠀⠤⣴⣥⢎⣀⣠⣀⣠⣂⣀⣠⣠⣀⡤⠤⠤⠤⠆⠔⠶⠒⢒⠻⣉⡹⢍⡐⠀⠒⠘⠖⠀⠒⠚⠓⠚⠉⠉⠁⡈⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⢸⢻⡢⡀⠁⡀⠀⠁⡈⢆⠀⠀⠀⠀⠀⠀⠀⢀⠠⠖⠉⣐⠘⡅⢦⡈⠉⠒⠰⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⡌⠀⢳⡑⡀⠀⠀⠀⠀⠀⠆⠀⠈⠀⣀⠴⠚⠁⠀⠀⠰⠃⢨⣣⠀⠓⣆⠀⠀⠀⢀⠐⠁⠉⠒⠠⠴⢠⢦⡄⣀⣈⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠃⠀⠈⣇⠰⠀⠈⠀⠀⠀⣠⠤⠞⠉⠀⠀⠀⠀⠀⣠⡆⠀⢀⠀⢦⠀⠈⠱⣄⠀⠀⠀⠀⠀⠀⠠⠀⡠⠊⣠⠖⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣀⡠⠶⠞⡋⠀⡀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⢘⡀⠨⡄⠀⠈⠀⠑⢤⠀⠀⠀⠀⠁⠔⢀⡞⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠛⠦⣄⡀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠎⠀⠁⠀⢼⠀⠀⣡⠀⠀⠀⠀⠀⠑⢦⣀⠁⠀⠀⡞⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠄⠂⠀⣰⠏⠀⠀⠀⠀⢾⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠉⠓⢮⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⣴⠁⠀⢀⠀⠀⠀⢾⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⣀⠴⠞⠛⠳⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡜⠀⠀⠀⠀⠈⠀⠀⠸⠀⠀⠉⠀⠘⡆⠀⢀⡴⠋⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢛⠉⠉⠓⠦⣁⠢⡀⠀⠾⠀⠀⠀⠀⣀⡬⢦⠋⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢣⡀⠀⡃⠀⠀⡴⠊⠁⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⣡⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⢀⣄⡀⣀⣀⢀⢀⣤⡿⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠀⠀⠉⠛⠻⠿⠛⠁⠀⠘⣷⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⡇⠀⢀⠀⠀⢀⡀⠀⠀⣀⠠⢻⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠸⠻⠷⠆⢰⠶⠟⠷⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠤⠂⢀⡄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⢉⠁⠒⢄⠉⠈⠂⢤⠖⠋⡇⠀⠀⠀⠁⠈⠀⠀⠀⢸⠙⠒⠤⠐⠁⠉⡠⠚⠉⡉⠕⠒⠠⢍⠓⡄⢀⠀⠤⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⣄⡠⠴⠦⢤⡀⠈⠂⠀⠉⠢⡀⠀⠰⣄⣠⡀⠀⠀⢢⡔⠂⠀⢀⡌⣠⠆⠀⢀⠔⠈⠀⠐⠈⢀⡠⠴⠦⢄⣱⠀⠇⣀⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠁⣠⠶⠖⠒⢺⠿⢮⣂⡀⠀⠪⣑⠄⡈⢦⢻⢂⠀⠀⠀⠂⡰⠻⡰⢃⠠⣊⠕⠀⢀⣰⣗⣫⣥⠤⠤⣀⠀⠈⠀⠀⠀⠀⢶⣤⠠⠼⣧
⠀⠀⠀⠀⠀⠀⢸⣠⠎⠀⠀⠀⠜⠁⠀⠀⠀⠉⠑⠒⠬⣕⣮⣿⣷⣷⠄⠉⠛⠁⣤⣾⣷⣥⣺⡵⠞⠛⠻⣉⠉⠀⠀⢀⡠⠴⠚⠒⠒⠴⢄⡠⢠⠄⠆⠀⠘⡆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡠⠀⠐⠂⠀⣀⣤⣤⣤⣤⣄⣀⠀⠉⠉⠉⠛⠻⡛⢛⡟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠈⠳⣄⡶⠋⠀⠀⠀⠀⠀⠀⢸⠁⡆⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠈⠁⠀⠉⢀⡤⠁⠀⠀⠀⠀⠀⠉⠙⢶⣄⠐⠀⠀⢹⡏⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣷⡄⢠⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⣰⠟⠳⢧⡞⠁⠀⠀⠀⠀⠀⠦⡀⠀⠀⠀⠀⠈⢷⡀⠈⢺⡗⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢹⣿⢥⡤⠈⠳⢄⣀⣀⡠⢼⡀⠀⣄⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⡰⢃⠀⠁⠿⠀⠀⠠⣀⠀⠓⠀⠀⠈⠀⠄⠀⠀⠀⠀⢳⠀⢸⣇⠀⠀⠀⡐⠀⠀⠀⠀⠀⢀⣴⡿⣿⣆⠉⠀⠀⠀⠀⠀⠀⠀⠑⠦⠈⠀⠀⠀⢴⠇
⠀⠀⠀⠀⠘⠀⠘⠤⢔⠇⠀⠀⠐⠉⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⢸⣧⣾⣷⣄⡀⠊⠀⠀⠀⠀⠀⣀⣴⡟⠁⣿⢸⠳⡦⣤⡤⠀⠬⢽⡦⣄⣀⢀⣀⣀⣀⠎
⠀⠀⠀⠀⠣⣄⠀⢠⠿⠦⠰⠀⠀⠀⢠⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⢸⠇⠘⠃⠉⠉⠒⢒⡒⠒⠋⠉⠁⣸⠁⢰⠇⢸⠀⠀⣿⡆⠀⠀⠘⡇
⠀⠀⠀⠁⠠⡤⠋⠀⠀⠄⠀⡐⠀⠀⢠⣄⠀⠀⠀⠈⢄⠀⠀⠀⠀⣾⣀⣰⣆⣀⡀⠀⠀⠁⠀⢠⠀⢀⣄⡴⠃⠀⠘⠀⠀⢿⣷⠀⠀⠀⣿
⠀⠀⠀⢇⠀⠀⠀⠀⠀⠐⢧⣀⠔⠁⠀⣀⡅⠀⠀⠀⠀⠢⠀⢀⡞⠁⠀⢸⡇⠀⠉⠁⠀⠀⠀⠐⢉⣿⠟⠀⠀⠀⡀⠀⠀⠸⢹⡆⠀⠀⢸⡄
⠀⠀⠀⠀⢳⣄⠀⠀⠀⢐⠄⠀⠀⠀⣠⠞⠁⠀⣠⠀⣀⣠⠖⢫⣀⣀⣀⢼⡧⢀⣀⣀⡀⠂⢈⠀⠀⣿⠀⠀⠀⠀⣧⠀⠀⠀⠈⡇⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠻⣅⣀⣬⣦⢤⣤⠋⠀⠀⣤⠘⣿⠋⠉⠙⠆⢰⡀⠀⠀⢸⡇⠀⠀⢀⡇⠠⠋⠀⠀⢿⠀⢰⠀⠀⣾⠀⠀⠀⠀⡇⠀⠀⠈⡇
⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⣿⠈⠀⢸⠀⣿⠀⠀⠀⠀⢠⠋⠛⠋⢹⡏⠙⠛⠉⡇⠀⠀⠀⠀⣾⠀⡎⠀⠀⣿⠀⠀⠀⠀⠇⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⣠⡶⠁⠀⠀⠀⠀⣀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⡀⠀⠀⠀⠀⠰⣦⡀
⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⣀⡤⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡶⣄⡀⠀⠘⢿⣦⡀
⠀⠀⠀⠀⣴⣿⠏⠀⣴⠞⢩⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⠙⢶⡄⠈⢿⣷⡄
⠀⠀⠀⣼⢿⡏⠀⠀⢻⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠁⠀⠈⣿⢿⡄
⠀⠀⣼⢇⡿⠀⠀⠀⠘⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢠⡟⠀⠀⠀⠸⣇⢻⡄
⠀⢰⡏⣸⠇⠀⠀⠀⠀⢹⡆⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⣾⠁⠀⠀⠀⠀⢿⡈⣷
⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⢿⡘⣷⣀⣀⣠⣤⣤⣤⣤⣄⠀⠀⣠⠶⢻⡏⠀⠘⣿⠳⢦⡀⠀⢀⣤⣤⣤⣤⣤⣀⣀⣰⡟⣸⠇⠀⠀⠀⠀⠀⢸⡇⠸⣇
⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠈⣷⠈⢉⣉⣁⣀⣀⣠⣄⠈⠳⣼⣧⠀⣿⣤⣄⣤⣼⡇⢠⣿⡴⠋⢀⣠⣀⣀⣈⣉⣉⠉⢰⡏⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⣾⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠉⢉⣩⣤⣈⣷⡄⠈⠛⠀⠁⠀⠈⠀⠀⠁⠘⠋⠀⣴⣏⣡⣬⣉⠉⠉⠉⠙⠛⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸
⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸
⣿⠀⠀⠻⠶⢦⣤⣤⣀⡀⠀⣠⡴⠛⢁⣠⡴⣞⣻⡷⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⣿⣳⡶⣤⣀⠙⠳⣤⡀⠀⣀⣠⣤⣤⠶⠾⠃⠀⢸
⠸⣦⣤⣀⣀⡀⠀⠀⠉⠉⠛⢉⣠⠶⢋⣥⠞⠋⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⡈⠙⢦⣍⠳⢦⣀⠙⠛⠉⠁⠀⢀⣀⣀⣠⣤⡿
⠀⠀⠀⠈⠉⠉⠛⠳⠦⣤⡴⠛⢁⣴⠟⢁⣤⠾⣫⠞⢉⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣈⠙⢮⡻⢦⣄⠙⠷⣄⠉⠳⣤⡤⠶⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⣠⡶⢛⡵⠞⣁⣴⠟⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡙⢷⣄⡙⢷⣍⠳⣦⡈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢋⣴⠟⠉⢰⡏⠀⣾⠋⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠈⢻⡆⠈⣷⠈⠙⢶⣌⠻⢦⡀
⠀⠀⠀⠀⠀⢀⣴⠟⣡⡶⠋⠁⠀⠀⢸⡇⢸⡟⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠘⣿⠀⣿⠂⠀⠀⠉⠳⣦⡙⢷⣄
⠀⣴⠶⠶⠾⠋⣡⡾⠋⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠈⠻⣦⡈⠻⠶⠶⢶⡄
⠀⢿⡀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⣸⠇
⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⢸⡇⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣿
⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣿⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣴⠟⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠘⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡟
⠀⠀⢻⡄⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣼⠇
⠀⠀⠈⣿⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⡏
⠀⠀⠀⠘⢷⡄⠸⣧⠀⠀⠀⠀⠀⠀⢸⣇⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢀⣿⠀⠀⠀⠀⠀⠀⢠⡿⠀⣴⠟
⠀⠀⠀⠀⠈⠻⣦⡹⣆⠀⠀⠀⠀⠀⠀⣿⡄⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣼⠇⠀⠀⠀⠀⠀⢀⡾⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⡀⠀⠀⠀⠀⠘⣷⡀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⣰⡟⠀⠀⠀⠀⠀⣠⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠘⣷⡀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⣰⡟⠀⠀⠀⠀⠀⠐⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⡀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄⠀⠀⠀⢀⣼⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢠⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣰⠃⠀⠀⠀⠀⠀⠹⣦⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⣼⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣧⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⣰⣿⠇⠀⠀⠀⡼⠟⠛⠛⠻⢷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⠿⠿⠿⢿⣧⠀⠀⠀⢿⣷⡀
⠀⠀⠀⠀⢠⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡹⣿⣤⣴⡎⢀⣄⡀⣹⣦⣴⣿⠏⠀⠀⠀⠀⠀⠈⠁⠀⢀⣨⣿⣷⡀
⠀⠀⠀⢠⣿⠿⠿⠿⠿⣿⣿⣿⣿⡿⠿⠿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣷
⠀⠀⠀⠛⠁⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⣀⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣀⠀⠀⠀⠈⠉⠛⠛⠉⠀⠀⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠛⣻⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡾⠟⠋⠁⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠉⠻⢿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠋⠁⠀⣠⣴⣾⠿⠋⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠌⠻⣷⣦⣄⡀⠈⠛⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣶⣾⣿⠟⠋⠁⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠙⠻⣿⣷⣶⡆⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡏⠀⠠⠄⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⡩⠙⢗⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢋⠍⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⣀⡤⠖⠋⠠⠊⠀⠀⠀⢳⡄⡈⠓⠦⣄⡀⠀⠀⠀⠀⣠⠴⠊⠁⠀⣰⠏⠀⠀⠈⠑⠦⣄⣀⡀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠦⣀⠀⠁⠀⠀⠀⢀⡠⠊⠘⢮⡐⢄⡀⠙⠲⢤⡶⠊⠁⠀⠀⠀⡴⠋⠢⠀⠀⠀⠀⠀⠀⢈⡝⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇⠈⠑⠲⢄⡐⠈⠁⠀⠀⠀⠈⠡⡄⠈⠁⠐⠚⠆⠀⢀⠠⠊⣰⡃⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠀⠀⠀⠘⢄⠀⠀⢰⠃⠀⠀⢀⠞⡀⠹⣄⠀⠀⣀⡴⠊⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡌⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠲⢆⡀⠀⠈⢧⠀⠠⡃⢀⡰⠉⠄⠐⢀⡠⠶⠊⡁⠀⠀⠀⢰⠀⠀⠄⠈⢧⡀⣀
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠌⠀⠐⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⠒⣈⣁⠋⣀⡥⠶⠛⠉⠀⠀⠰⡀⠀⠀⠀⠀⣂⣀⡤⠤⢺⠏⠁
⠀⠀⠀⠀⠀⠤⣴⣥⢆⣀⣠⣁⣀⣀⣀⣠⣠⣀⡤⠤⠤⠤⠦⠐⠶⠒⢒⠻⣉⡹⢍⡘⠀⠒⠘⠖⠀⠒⠚⠓⠚⠉⠉⠁⡈⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⢸⢻⡢⡀⠀⠀⠀⠁⡈⢆⠀⠀⠀⠀⠀⠀⠀⢀⠰⠖⠉⣐⢙⡁⢦⡈⠙⠒⠠⠄⢄⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⡌⠀⢳⡱⡀⠀⠀⠀⠀⠈⠆⠀⠈⢀⣀⠴⠊⠁⠀⠀⠰⠃⢨⣣⠀⠓⣆⠀⠀⠀⢀⠐⠁⠉⠐⠠⠴⢠⢤⡀⡀⣈⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠃⠀⠈⣇⠐⠀⠀⠀⠀⠀⣠⠤⠞⠉⠀⠀⠀⠀⠀⣠⡆⠀⢀⠀⢦⠀⠈⠱⣄⠠⠀⠀⠀⠀⠀⠀⠂⡠⠊⣠⠖⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣀⡠⠶⠞⡋⠀⡀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⢘⡀⠠⣄⠀⠀⠀⠑⢤⠀⠀⠀⠀⠁⠔⢀⡞⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠛⠢⣄⡀⠀⠀⢄⠀⠀⠀⠀⠀⢀⠎⠀⠁⠀⢾⠀⠀⢡⠀⠀⠀⠀⠀⠑⢦⣀⠁⠀⠀⡞⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠄⠂⠀⣰⠏⠀⠈⠁⠀⢾⠀⠀⠀⢃⠀⠀⠀⠀⠁⠀⠉⠒⢦⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⣴⠁⠠⢀⡀⠀⠀⢾⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⣀⠴⠞⢛⠷⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡜⠀⠀⠀⠀⠈⠀⠀⢸⠀⠀⠉⠀⠘⡆⠀⢀⡴⠋⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢛⠉⠉⠓⠦⣁⠢⠀⠀⠾⠀⠀⠀⠀⣀⡬⢦⠋⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢣⡀⠀⡃⠀⠀⡴⠊⠁⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⣡⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⣀⢀⣼⣄⣱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⡛⣀⣠⣄⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢠⠈⢻⣤⡼⢃⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠑⢺⣯⡽⢛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡿⣀⢶⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣿⣶⣄⣅⣶⢶⡉⠛⠶⣀⣴⠒⠂⡀
⠀⠀⠀⠀⠀⠀⢀⠤⠒⢒⠿⢡⢬⠏⣈⣉⣍⣩⣄⠰⢖⠊⠁⠈⠓⣤⠛
⠀⠀⠀⠀⠀⠀⣌⠀⢠⡌⢠⠀⠘⠀⡇⠀⡁⠀⠥⠀⣈⣤⡔⠈⠀⠙⣤⠳⡄
⠀⠀⠀⠀⠀⠀⢠⠔⠺⠁⡏⢹⡗⠒⠗⠀⡖⠒⠺⡟⠁⠀⠀⠉⠉⠈⠉⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠳⠀⢀⡇⢰⠀⠀⡂⢀⠈⠀⠀⠀⢀⠴⡀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠘⠀⠀⢀⡀⠁⠘⡏⢹⠁⠉⡏⡙⠀⠀⠀⢀⠎⠀⠈⢆⠀⠆⠀⠀⡄
⠀⠀⠀⢀⠜⠁⠀⢠⠘⠁⢃⣰⣇⣸⣀⢀⣇⡅⠀⠀⢀⠁⠀⠀⠀⠀⢻⠀⠀⠀⠡
⠀⠀⠀⠃⠀⠀⠀⡈⠀⠸⠨⠈⠇⠸⠀⠀⡇⠀⠀⠀⡆⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠂
⠀⠀⢸⡖⠰⠆⡤⠁⠀⠀⣦⠘⠗⠺⠒⠒⡗⣀⡠⣤⡅⠀⠀⠀⠀⠀⠀⠃⠀⢀⣀⢀
⠀⠀⢸⢧⢶⡔⠁⠀⠀⠀⣿⢠⡅⢘⠀⣀⣅⢸⡇⣾⡆⠀⠀⠀⠀⠀⠀⠀⢷⣉⠁⢹
⠀⢀⣿⣌⡆⠀⠀⠀⠀⠀⠋⠲⢇⣙⠁⡠⠅⠚⠁⠙⡡⠀⠀⠀⠀⠀⠀⠀⠈⡟⠉⠛
⠀⢨⣿⣈⣷⣄⠀⠀⠀⠸⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠡⣦⠀⠀⠀⠀⠀⠀⣀⢳⠺⢶⡀
⠀⠈⣟⢠⣼⡿⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⡿⠳⣬⣠⣼⣇
⠀⠀⠈⠋⠊⠁⠀⠀⢀⠁⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠻⣦⣁⣌⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠇⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣸⠀⠀⠀⢰⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣇⣒⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡖⠶⠈⣃⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣟⢉⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣃⠙⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⡟⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠶⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⢻⡇
⠀⠀⠀⠀⠀⠀⢀⡠⠖⣼⣠⠟⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠋⣿⡄
⠀⠀⠀⢀⣴⠶⠥⣠⠞⠀⣤⡒⣺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢴⠸⠗⢤⡀
⠀⠀⠀⠙⠥⠤⠐⠛⠂⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡦⣆⡈⣦⡝⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⡀⠀⠑⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈
⣀⣀⡀⠀⢀⡀⢀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⠀⠀⣀⣀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⠿⠿⠿⠿⢿⣿⡿⠿⠿⠿⠿⠿⣿
⣿⣶⣷⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢄⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡀⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠠⠀⠊⠀⢉⡑⠂⠙⣀⠀⠐⠃⠀⣄⠀⡒⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠀⠐⠀⠀⢈⠁⠀⠂⠀⠀⠈⡆⠀⣄⢠⠀⠀⠀⠂⠀⢰⠷⠖⠀⠃
⠀⠀⡀⠄⠈⡜⠀⠀⠀⠀⠐⠊⠀⠠⠀⠀⠀⠶⠀⠁⠀⠘⠀⠀⠀⠀⠌⠁⠈⠉⠃⠀⠀⠀⠀⢀⣨⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⢈⠁
⠀⠐⠘⠂⠀⠁⠰⢄⠠⡆⠰⠀⠰⠂⠣⠀⠀⠃⠀⠀⠠⠀⢀⡀⠘⠃⣾⠀⠀⣶⠄⠓⠀⠤⠀⠈⠂⠀⠂⠀⢰⠁⠀⢀⣠⠀⠀⡀⠠⠀⠄
⠀⠀⢠⢄⠀⠀⡀⠈⠀⡠⠀⠀⡠⡀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠘⠂⠈⢈⠀⠀⠈⠀⠀⠀⠀⠀⢠⡰⠒⣀⠀⢀⡂⠀⠀⠁⢀⠀⠡⢴⡶⠀⠀⠀⠌
⠀⠀⠀⠀⣴⠀⠄⠻⠋⠤⣰⡄⠈⠀⠀⠀⠰⣆⠀⠉⠀⠀⠀⠀⠀⠀⠘⠃⡐⠃⠀⢰⢡⠂⠀⠀⢡⠁⠩⠁⢈⠀⠀⠰⠀⠀⠑⠀⠀⠀⠀⠂⠀⠄⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⡈⠀⠀⠀⠈⠂⠀⠈⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⣀⠐⠀⠀⠀⠀⠀⠠⠀⠉⠀⠀⡀⠀⠀⠤
⠀⠀⠀⠈⠠⠀⣀⡰⣀⡀⠀⠠⠄⠈⠀⠰⠀⠀⠀⠂⠀⠀⣭⠂⠀⠀⠉⠁⠈⠉⢀⠈⠁⠀⠀⢄⠺⠀⠀⠀⠆⢠⠀⠀⡀⠀⠀⠨⠀⠠⠀⠄
⠀⠀⠐⠐⠆⠀⡉⡃⢉⡁⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⢹⣤⣀⣴⡇⠀⠀⠀⠖⠁⠀⠻⠃⠀⠐⠂⠀⠀⠈⠠
⠀⠀⠀⠀⠀⡼⠋⠀⢈⠛⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⡀⠀⠇⠀⡀⠀⢠⠁⠀⠀⠀⠀⠀⢸⣿⠛⣿⡇⠀⠀⠠⠀⡠⠀⠀⠤⡀⠀⠀⠄⠀⠁⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉⢀⠀⠀⠀⢀⠃⠀⠀⠈⠀⠀⠀⠀⡄⠂⠀⠀⣀⠀⠀⣿⡿⠀⢿⣷⠀⠀⠀⠀⣆⣠⢀⣀⡔⠀⢡⠀⠏
⠀⠠⠊⠀⡂⠠⠁⠄⠀⠀⠘⠀⠐⠐⠀⠀⠈⠀⠁⠀⠀⠠⠀⠀⠄⠀⠀⠀⠰⠀⠀⠄⠀⠀⠀⠌⠠⠄⠀⠠⠤⠡⠀⠀⠀⠛⠓⠘⠓⠇⠀⠀⠀⠀⡐⠆
⠀⠀⠐⡜⠄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠐⠀⠂⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⢀⠠⢀⠀⠀⠀⠀⣇⡀⢀⢀⡄⠀⠀⠂⠀⠋
⠀⠀⠑⠌⠈⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⢀⠀⠐⠄⠀⠀⠒⠀⠀⠀⠃⡂⠀⠀⠐⠠⠮⠁⠌⠱⠄⠁⠀⠀⠀⡀
⠀⠠⠀⠀⠀⠀⠄⠀⠠⠀⠼⠇⠀⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⠰⠈⠀⠀⠠⠀⠜⠠⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠐⢀⡏⢀⡅⢸⡄⠐⠀⠠⠀⠀⠠
⠀⠀⢤⠀⠠⡄⠀⠀⠀⠨⢀⡀⠃⡀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣼⠃⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠀⠀⢀⠀⠀⠀⠀⠀⣂⡀⠀⠀⢐⡀⠀⡀⠂⡀⠀⠠⡄⠀⠀⠀⠀⠀⠛⡃⠈⣵⠀⠠⠄⡠⡄⣴⡆⣲⣆⠀⡄⠐⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡁⠀⣡⡿⠃⠀⡀⠆⣀⠀⠀⠄⢠⠀⠀⠀⠐⠂⠀⠀⠀⠐⠄⠀⠀⠀⠘⣣⡞⠁⠀⡀⠀⠀⠁⠺⠏⠙⠋⡀
⠀⠀⠠⠈⠠⠄⠀⢄⡀⠈⠻⠟⠋⣀⡀⠀⠐⠒⠀⠀⠐⡢⠀⠀⠂⢴⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈⠀⠁⠀⡀⠀⠀⠌
⠀⠀⠀⠁⠁⠐⠂⠀⣸⡇⠀⠀⢸⣆⠰⠀⣴⡦⠀⠂⠈⠁⠀⠀⡀⢀⢉⡄⣀⠀⠀⠀⡈⠀⢀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠄⠀⠠⠄⠀⠠⠀⠀⠠
⠀⠀⠀⠀⢀⡀⠀⣰⣿⡅⣼⣧⢸⣿⣦⠀⠈⠀⠀⡀⠀⠂⠀⠰⠟⠀⠀⠀⠹⡀⠀⠀⠐⠀⢐⠀⠘⢀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⢀⢉⠉⠈⡁⢁
⠀⠀⠈⡂⠀⠀⠀⠀⣘⠒⠃⠈⠒⣄⠀⠀⠸⠂⠀⢻⡀⠀⠀⠠⡀⠹⡄⢩⠂⠅⠀⠀⠀⠠⠀⠠⠠⠌⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠘⠌⢅⠒⠍⠼
⠀⠀⠈⠁⢠⣀⡀⠈⠈⠁⠀⠀⠀⠁⠁⠀⠀⠴⣀⠬⠀⡤⡀⠀⠠⡀⠀⣀⠄⠀⢀⢦⠀⠀⠀⠄⠀⠀⠀⠠⠀⠀⠰⠀⠀⢀⠰⠀⠠⠀⠀⡄⠈⠤
⠀⠀⠀⠀⢾⡿⠿⢿⢦⡀⠀⠀⠀⠐⠂⠀⠰⢶⣧⡚⠀⠑⠀⢀⠠⠈⠄⠁⠄⡀⠔⠁⠀⠀⡀⣀⠀⠐⡀⠀⠐⠂⡀⢄⠀⠘⠀⡄⡀⠂⠐⠂⠀⠂⠀⠄
⠀⠀⠀⢀⠀⠁⠀⠀⢀⣸⠠⠆⠀⠀⠀⠀⠀⠻⣯⠃⠀⠀⡄⠀⢸⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠛⣤⠀⠉⠀⠁⠀⠘⠀⠀⠀⠀⢁⠠⡄⢠⡁⠀⡔⡄⠁
⠀⠸⢴⡯⢗⡾⠇⠀⠀⠀⠀⠀⠀⠠⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⠀⠀⠰⠶⡶⠀⠁⢠⡾⠁⠀⠀⠀⠀⠀⢰⠰⡀⣀⠀⠀⠉⠁⢀⡀⠀⠉
⠀⠀⠀⣔⣆⠀⠀⠀⠀⠀⠀⠆⡘⠀⠀⠋⠉⠀⠠⡅⠂⡀⠀⠀⣀⡀⠀⠀⢰⠀⢶⣬⣤⣴⠏⠀⠀⠈⠁⠀⢐⢑⠀⠀⠂⠘⠀⠀⢀⠃⠀⢰⠀⠂⠀⠐
⠀⠀⠀⢘⠃⠀⠀⠀⠀⠀⠀⢈⡀⢂⠄⢰⠀⠄⠡⠀⠎⡀⠀⠀⣿⡇⠀⢹⣿⠀⠀⠀⠀⠀⠠⡈⡔⠀⠀⠀⠔⠁⢠⡄⠀⢼⠀⠀⡰⠰⠀⠀⡤⠄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠂⠐⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⠁⠀⠀⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠄⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠨⠀⠀⣠⣿⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣦⠀⠀⣠⠖⠁⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢰⠏⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠈⢧⣀⣴⣿⣿⣤⣄⢀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠈⢃⠀⠀⠀⣹⣿⣿⣿⣿⣿⣟⡀⠈⠁⠘⠛⢲⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡴⠂⠁⠀⠀⢀⣴⣿⡷⠶⣶⣿⣿⣿⡿⠋⠉⠉⣤⢾⠀⠰⠶⠿⠿⠗⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡠⠶⡀⠰⢶⣶⣾⡿⣿⠏⠻⠂⡿⢋⢛⣉⠁⡄⣤⠀⠀⠀⣶⢐⢶⣷⠶⣄⠀⢳⡐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡞⢀⣴⠎⢀⣴⡄⠰⠞⠋⠁⠀⠀⣸⢁⠇⢰⣷⢀⣸⢼⡄⠀⣼⡟⠀⡌⢿⣷⡀⠑⢮⠳⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣶⣾⢃⣤⣾⢛⠉⠑⢄⠀⠀⠀⠀⠿⢘⣧⣾⣿⣿⣉⠉⠁⣼⣿⠃⠀⠈⠢⠙⠿⠆⠀⢷⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠊⠁⠙⢻⣿⣿⣷⣦⣬⣦⣤⡀⠈⡏⢱⠋⣿⣿⡇⣆⣱⢸⣿⠁⠀⠀⠀⠀⠠⣿⡇⣰⡎⡄⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢸⣿⣿⣿⡌⠙⠛⡟⠓⠖⢷⢸⡀⣿⣿⠁⡇⡟⠼⡟⠀⠀⠀⠀⠈⠀⠉⢛⡿⢃⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⣇⠠⠀⠸⣿⣷⣿⣇⣀⣀⣩⣤⡴⠾⠛⠿⢿⣿⢾⢷⣷⡏⠀⠀⠀⠀⠀⠀⠒⣷⡜⢀⡞⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢷⠷⠾⣿⠛⢿⣿⣿⣿⣿⣯⣄⣀⣀⣀⣾⣃⣀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⢛⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠘⠠⣌⢉⣆⠈⣿⣿⣿⣿⣟⡉⠉⢛⣉⣿⠉⣩⣿⠇⠀⠀⠀⠀⠀⡀⠀⠀⣴⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠀⢻⣧⣷⣤⡿⠿⠿⣿⣿⡟⠛⠛⣛⣟⢛⣻⠛⢀⣉⣠⣤⡴⢪⡄⣀⣸⠟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢂⡐⡄⠁⠀⠁⣿⡃⢀⣾⣉⣽⣿⣷⣾⣿⣷⣾⣻⣯⣿⣿⣾⣿⡿⣿⣆⡐⠱⠤⡄⣼⠛⠢⠀⠀⡀
⠀⠀⠀⠀⠠⠀⣠⠀⡠⠞⠁⠘⣷⠀⠀⠀⠘⠂⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣽⣯⣉⣿⣾⡿⢿⡕⠛⣇⢴⣼⣄⡀⠀⡀⠈
⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⣠⣼⣿⣷⡀⠀⠀⠀⢿⣿⣿⣿⠿⠿⠿⠿⠿⠿⢿⣴⣪⣉⠻⢿⣏⣰⡚⠀⣰⠙⠀⣫⣨⣿⣿⣿⣶⡆
⠀⠀⠀⠀⣸⡟⣠⣀⣴⣶⣿⣿⣿⣿⣿⡄⠀⠀⢘⡿⢋⡡⠀⠀⠀⠀⠒⠀⠈⠉⠿⣿⣛⣗⣩⣻⣿⣷⣿⠿⠿⠿⠿⣫⣿⣿⠟
⠀⠀⢀⣴⡿⣻⣿⣿⠟⠋⠁⠻⢿⣻⣿⣿⡏⠀⠈⣱⠣⣀⣀⡀⠀⠀⠀⠀⠀⠀⠐⡻⢿⣿⣿⡈⠉⠙⠻⠛⠷⣿⣿⠏⠉
⠠⠀⡿⠋⠀⠙⠉⡀⢀⣠⣴⣾⣿⣿⣿⣿⠁⣶⣟⠸⣿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⡀⢹⣿⣷⡤⢀⠀⠀⠀⠉
⠀⠀⢠⣴⣦⣤⠀⣿⣿⣿⣿⠿⠛⠋⢃⡼⣄⢹⢂⣥⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠀⠀⠐
⠀⠀⢨⣻⠿⠿⠡⠞⠋⠁⠀⠀⠀⢤⡺⠂⠙⡾⢃⠍⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈
⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⡇⢚⠃⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙

⢠⣶⣶⣿⠿⠿⠿⠿⠛⠛⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠚⠛⠻⠿⠿⠿⢿⣷⣶⣦
⠸⣿⡇⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⣿⡿
⠀⣿⣇⠀⠀⠀⠀⠀⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠉⠉⠉⠉⠉⠉⢹⣿⡇⠀⠀⠀⠀⢀⣿⡇
⠀⠸⣿⡀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⣸⡿
⠀⠀⢻⣇⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠈⣿⡄⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⢿⣧⣤⣤⣤⣴⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣤⣤⣤⣤⣿⠇⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠘⠛⠉⠉⠀⣀⣠⣄⡉⠻⢦⣄⣴⣧⣄⣤⣷⣄⣤⠾⠋⣁⣤⣄⡀⠈⠉⠙⠛⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⢀⣠⣤⡶⠟⠛⠋⠛⠛⠷⠶⣿⣿⣿⣿⣿⣿⣿⡷⠶⠟⠛⠉⠛⠛⠷⣦⣤⣀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠙⣿⣷⣿⠟⠋⠁⠀⢀⣴⣾⣶⠶⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠷⢶⣾⣶⣄⠀⠀⠉⠛⢿⣷⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⢀⣤⡾⠟⠉⣠⣤⣶⣶⡿⢿⣿⣿⣿⣿⣿⡿⠿⣷⣶⣦⣤⡈⠙⠿⣦⣄⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠟⠉⠀⣠⣾⡿⠁⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠹⣿⣦⡀⠈⠙⢿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢹⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠻⠀⠀⠀⡀⠛⣿⣄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣧⠆
⡄⠀⠀⠀⠠⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⡴⠀⠀⠄⠀⡀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⣿⣷⣄⡀⠀⠀⠰⡌⢻⣧⠀⠀⠀⢀⣾⠃⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣠⣴⣶
⣿⣿⣿⣿⣷⣦⣄⡘⠖⢹⡆⠀⢠⠞⠃⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣏
⣿⣿⣿⣿⣾⣝⣿⢿⡷⠆⣧⢠⠀⠀⠁⠀⠀⠀⣄⢻⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⣠⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠉⠈⣿⣧⠤⡆⠸⣿⠋⡀⠉⠉⠟⠁⠀⠀⠀⠀⠀⠀⢄⡀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠋⠀⠶⠀⠀⠙⠻⡇⠀⠀⢻⠀⠀⠀⣱⣦⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⠾⣿⣧⣲⣟⠃⣸⠃⠀⠀⢀⡄⠁⣸⠶⣬⣧⡤⠾⣿⣿⡶⢿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠈
⣿⣾⣿⣿⣿⣿⣯⣴⣦⣰⣶⣶⣾⣷⣏⣧⠀⣿⡌⠻⠗⠉⡄⣸⡂⠀⠈⠉⢉⠟⠛⡻⡿⢿⠿⣿⣿⣿⣿⣿⣭⣼⡿
⣿⣿⣿⣿⣿⣿⣿⣿⡥⣹⣿⣿⣿⣽⢿⢧⣴⣿⣿⠀⠀⢦⡹⣿⣷⣴⠇⠐⠁⣀⣾⢾⣿⣃⣀⣀⣀⠉⠉⠛⠛⠋⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣟⡞⣸⣿⣿⣿⡀⣴⡈⢷⣭⣿⣷⣶⣾⣿⣵⣿⣿⣿⣿⣿⠿⡛⢷⣆⠀⠀⠀⠀⠘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣿⣿⠿⠿⠿⢿⣧⣼⣿⣹⣿⣿⣿⣿⣿⣿⣿⢟⣻⣵⣶⡞⣰⠏⠀⠀⠀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣤⣤⣀⣐⣢⡴⠭⠉⢹⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣤⣶⣤⣬⣤⣤⠦⠀⠱⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣯⣍⡉⠉⠉⠀⠀⣉⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢹⠿⣿⣿⣆⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿⣋⠀⠀⠒⢬⣄⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣾⠀⠀⠉⠀⠐
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣥⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠑⢸⣿⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣽⠏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢣⠂⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣿⣿⣿⣿⡿⠟⢁⡟⠀⣿⡏⠀⠀⠀⢠⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠘⠛⠿⣿⣶⡄⣸⠁⠀⠹⠁⠀⠀⠀⣼
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⡏⠀⠀⠀⠀⠉⢻⠏⠀⠀⠄⠀⠀⠀⢸⣸⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠈⢻⣿⣿⠃⠀⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢃⡄⠀⣶⣦⡀⠀⠀⢸⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⣿⡏⠀⠀⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠈⠀⠀⠛⠛⠛⠀⠀⠀⠻⡇
⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠰⣿⣿⠀⣼⣿⣿⣿⡇⠈⠙⢿⣦⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣴⣿⣿⣿⣿⡇⠀⠀⠀⠙⠟⢮⠆
⠀⠈⠛⣿⣿⣿⣿⣿⠟⠉⢀⣤⡀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣄⡞⡇⠀⠀⠀⠀⠀⠀⠀⣀⡷
⠄⠀⠈⠉⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠈⢿⡟⢹⣿⣿⡏⠀⠀⠀⠀⠀⢠⣿⣧⡗⠀⠀⠀⠀⠀⠀⠀⠉⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⣸⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠
⣤⣶⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⠀⠀⠀⢶⣤⣤⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⡤⢶⣻⡭⠿⠟⠓⢲⣭⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢈⡿⣿⡛⠛⠛⠛⠒⠲⠦⣤⣄⣀⣀⣀⠀⢸⡇⠀⠀⠀⣀⡴⠛⠁⠀⠈⠀⢀⣠⣴⣾⡿⠃⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣸⠃⢸⡿⢷⡦⣄⣀⠀⠀⠀⠈⠉⠻⢯⣝⣻⡇⣀⡴⠞⠁⠀⣀⣀⣤⠴⠚⢋⡵⠋⣼⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣰⠃⠀⠀⡇⠈⠳⣆⡉⠙⠓⠲⠶⠶⠶⠶⠶⣿⣟⠛⠒⠚⠛⠋⠉⠁⠀⣀⡴⠋⠀⢰⠇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡴⠃⠀⠀⢀⡇⠀⠀⢸⣿⣶⠤⣤⣤⣤⠤⠶⠋⢹⣿⡷⠦⢤⣤⡤⠤⣶⢿⡏⠀⠀⠀⣾⣇⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⣸⠃⠀⢀⡾⠘⡏⠳⣄⢀⡀⣀⠀⢀⣼⣷⣽⣄⠀⠀⣠⣾⠃⢸⡇⠀⠀⠀⡇⢿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣰⣷⠃⠀⢀⡞⠁⠀⡇⠀⠘⡿⣿⠿⠖⠋⢸⡇⠈⠛⣻⢾⠃⡾⠀⢸⡇⠀⠀⠀⡇⣾⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣸⠟⠁⠀⠀⢸⠀⢀⡼⠃⠀⣰⠇⠈⠳⣄⣤⣼⣟⡿⣿⠁⢸⣰⡇⠀⠈⣇⠀⠀⠀⢷⣿⠀⢀⡏
⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⡟⠻⣿⣶⣖⠒⠒⠚⠳⠿⣷⠶⠶⣯⡤⠤⠤⣿⠙⢾⠋⢀⣿⣀⣈⣿⣧⡀⠀⠹⡄⠀⠀⠸⣿⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠹⡆⠹⣿⣄⠀⠀⠀⠘⢧⡀⠀⢳⡀⠀⣻⣧⡀⢀⣾⡟⠉⢉⡽⠋⠉⠙⣻⡿⠒⠒⠶⣿⡿⠧⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⣿⠀⠘⡎⢧⠀⠀⠀⠈⣧⠀⢸⣧⣞⡁⠀⣳⠋⠙⢦⣀⡞⠀⠀⢠⣾⡿⠁⠀⣠⠞⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⠀⠀⢹⡘⣇⠀⠀⢀⣸⣤⠞⠉⠙⢿⡳⣿⠀⢀⣤⠿⣧⠀⢀⣿⡟⠀⢠⡞⠁⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠸⡇⣿⠀⠀⢸⡿⠻⣄⠀⠀⠀⢷⣟⡴⠋⠀⠀⠈⠳⣼⣿⠀⢠⠏⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⣧⣿⢀⣴⠋⠀⠀⠈⢧⡀⠀⢸⡿⠁⠀⠀⣠⠴⠚⠋⢻⣄⡿⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⢳⡀⢹⣿⣏⡈⠳⢦⡀⠀⠈⢷⠀⢸⡇⢀⣴⠛⠁⠀⠀⣀⣠⣽⣧⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣹⣏⡀⠀⠉⠳⣄⠻⣄⠀⠘⣇⢸⢠⠞⠀⢀⣤⠖⠋⠁⣀⢠⠏⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠈⠙⠳⢤⡀⠘⢧⡈⢧⠀⢹⣿⠏⢀⡴⠋⠀⠀⠀⢘⡷⠋⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⢦⣌⢳⡈⢧⢸⡟⣴⠋⠀⠀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡹⣾⣿⡟⢁⣠⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠔⣦⣷⣶⣶⡶⡼⠄
⠀⠀⠀⠀⠀⠀⡀⢀⠴⢯⢹⡄⠀⠀⠀⠀⠀⠐⡀⠀⠘⣿⣿⣿⣷⣿⡆⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⢻⣇⣀⣼⣾⡆⠀⠀⠀⢰⣷⣾⡇⠀⠀⢸⣿⣿⡿⠟⡀⠀⠀⠀⠀⢀⣠⣿⣿⣶⢄
⠀⠀⠀⠀⠀⠀⣀⣷⣴⣟⣿⡧⠀⠀⠀⠈⣿⣿⣿⣤⣤⣾⣿⡯⠀⣴⡇⠀⠀⢀⢐⣈⣻⣿⡟⡯⡴⠂
⠀⠀⠀⠀⢀⣾⣿⣿⣷⣿⡟⠛⠛⠃⠀⢠⣾⢿⣿⣿⣿⣿⣿⣷⣿⣿⡇⠀⠀⠀⠉⢉⣿⣹⡇⢳
⠀⠀⠀⠘⠻⠿⢿⣿⣿⣿⣿⣅⣿⣿⠀⢈⣽⣿⣿⣿⣯⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠀⠀⠬⠙⠿⢿⣿⣿⣿⣽⣿⠏⠀⠀⠀⠀⠀⠀⠈⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⠀⠀⠇⡠⠞⣵⡽⣾⢿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⡤⠤⠤⢤⣀⠀⢀⣠⠤⠤⢤⣼⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣤⠴⠖⢚⣛⣺⣿⣿⣿⣿⣚⣛⠒⠶⢤⣼⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡴⠋⣩⡟⣻⣿⣿⡻⣯⡙⠳⣄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠀⢸⡏⠀⣿⣿⣿⡇⠈⣿⠀⠈⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⢸⡇⠀⢹⣿⣿⠁⠀⣿⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⢸⡇⠀⠈⢿⠏⠀⠀⣿⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⡞