Trolling Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣦⣤⣤⣀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣟⢿⣷⣄⠀⠀⢀⣿⣿⣸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣶⣶⣶⣤⣘⣿⣿⣿⣆⠻⣿⣷⠀⣼⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⢀⣀⣨⣿⣿⣿⣿⡻⣿⣿⣿⣿⡄⠁⢻⣿⣿⠟⣿⣿⣿⡇⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⣠⣾⣿⣿⣿⣿⡿⢿⣧⡙⠈⠻⡞⣿⣧⡀⢸⠟⠋⢰⣿⣿⡿⠀⣀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠛⣉⣀⣀⣀⡀⠀⢙⢿⣆⢄⠀⢸⣿⣇⠇⡀⣰⣿⣿⠿⣿⣿⣿⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣦⣷⣿⣾⣆⢸⣿⣿⣼⣷⣿⣿⣷⡿⠋⣭⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣿⣿⣷⡍⢰⣾⠟⠛⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠛⠻⣷⣆⢠⣾⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠛⠛⢿⣿⣿⣾⡧⠄⠀⠀⠀⠈⠻⠙⣿⣿⠏⠞⠁⠀⠀⠀⠠⢼⣿⣾⣿⡿⠛⠛⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠼⠛⣷⡄⢹⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏⢠⣾⡛⠶⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠞⣿⣷⡄⢿⣇⠀⠛⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠀⢸⡿⢠⣾⣿⠳⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⢿⠋⣻⣶⠿⠂⠀⣼⡉⣿⣶⣤⣤⣼⣧⣤⣤⣶⣿⠋⣧⠀⠀⠻⣶⣟⠙⡿⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣈⣿⣟⠁⠀⠈⠉⠛⠳⢶⡾⠿⠯⠈⠁⠽⠿⢷⡶⠞⠛⠉⠁⠀⠈⢻⣿⣃⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣧⠀⢀⣠⣤⠤⠀⠻⡀⢤⣄⠀⠀⣠⣤⢀⠟⠀⠤⣤⣄⡀⠀⣜⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣄⡞⠻⣇⠀⠀⠀⠀⠀⠙⠆⠰⠏⠁⠀⠀⠀⠀⣸⠟⢻⣀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣇⢀⠹⡶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢶⡏⡀⣸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡆⢻⣾⠉⠙⣶⣶⣶⣶⣶⣶⠋⠉⣷⡾⢠⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠜⢿⣿⡿⣿⡌⢿⣤⣔⣋⣩⣥⣬⣍⣙⣢⣤⡿⢁⣿⢿⣿⡿⠃⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⢸⣷⣄⣈⠉⠁⠀⠀⠀⠀⠈⠉⣁⣠⣾⡟⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣤⣶⣶⣤⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠟⢿⣿⣿⣿⣿⡿⢻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠦⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠉⠀⠀⠀⠂⠉⠉⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⠓⡖⠓⠒⠒⢶⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠐⢤⠶⡀⠀⠀⠀⢀⣄⠀⠀⠀⡀⠘⢦⠔⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠔⠁⠀⠈⠒⢤⠴⠏⠠⠑⢒⠊⠀⠀⠀⠑⣮⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡿⠋⠀⠀⠀⠀⡔⢁⣶⡆⠀⣶⣦⠑⡀⠀⠀⠀⠈⠻⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠀⠀⠀⠀⡀⠀⢇⠈⠿⠇⠀⠿⠟⢠⠃⠀⡀⠀⠀⠀⠙⡇⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠑⢎⡀⠀⠀⠑⠂⠤⠤⠔⠊⠁⠀⠀⣨⠆⠁⠀⠀⢸⠀⠀⠀⠀⣀⣠⠃⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢸⣿⣷⣶⣤⣤⣤⣤⣤⣤⣴⣶⣿⣿⠀⠀⠀⠀⢠⠀⠀⠀⠐⡄⠀⢔⢁⠘
⠀⠀⠀⠀⢀⡤⠒⠒⠒⠴⠙⣄⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢀⠎⠱⠄⠂⠐⠁⢀⣀⠩⠔
⢀⡠⠤⠴⠋⢀⠤⠤⣤⠀⠀⠈⠳⢄⡀⠀⠙⠻⢿⣿⠉⠉⠉⠉⢻⣿⠿⠋⠀⠀⣀⠔⠃⠀⠀⠠⣀⡠⠔⠋
⢏⡠⢴⠄⠀⡹⠀⢀⠃⠀⠀⠀⠀⠀⠈⠉⠒⠒⠠⠤⠤⠤⠤⠤⠤⠄⠐⠒⠈⠉⠀⠀⠀⠀⠀⠀⢇
⠀⠉⠁⠰⠤⠇⠀⡬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⢸⠄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠥⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⡆⠰⠆⠀⠀⠄⠀⠀⡀⠀⡰⠊⠁⠀⠀⠀⠀⠀⠉⠲⡀⠀⡀⠀⠀⠄⠀⠀⠖⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠒⠂⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠒⠂⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠤⠤⠄⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡖⠤⠢⡄⣀⣀⢤⡀⠀⡠⢄⣀⡀⡤⠢⠔⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡇⠀⠀⠀⠈⠊⠀⠀⠀⠀⡇⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠉⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠈⠙⢢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⠀⠀⣴⣦⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣲⣦⣴⠷⠒⠂⣹⣿⣿⠟⠛⢿⣿⡟⠉⠛⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⢿⡀⠀⢴⣿⣿⠁⠄⣒⡀⠙⠃⠀⠀⠈⢽⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⢢⠀⠁⠀⠘⣿⡟⠀⢴⣷⣿⠀⠀⠀⠀⠘⠻⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠰⣿⣶⡜⣀⣠⡆⠀⠹⡃⠀⠀⢈⠀⠀⠠⠀⠈⠐⠂⠀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣷⡄⢠⡻⣿⣿⣯⣿⣷⠀⠀⠀⠀⠀⠀⢸⠂⠀⣀⡀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⠟⠢⠈⠻⠟⠋⠻⠹⠃⠐⠢⣤⣤⣶⡿⠷⣾⣿⡿⢤⣀⣀⣀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⠋⠀⠀⡆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢨⡗⣸⣿⠛⡆⢀⠠⣽⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⠀⠀⠀⠙⠢⠤⠶⠳⠀⡄⠀⠀⠀⠀⠠⢿⣿⠃⠚⠛⠛⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠇⠀⠀⠀⠀⠛⠀⢄⠀⠀⠈⠰⠟⣴⣤⠀⠀⠀⢼⣿⠀⢾⠟⠉⢀⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠐⠒⢄⠁⠁⠀⠀⠀⠈⢿⣿⣦⡀⢸⣟⠀⠀⢀⠰⠋⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠑⠂⠀⠀⠀⢀⠤⠙⠿⣿⢿⣿⣦⡾⠁
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⢀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢸⣿⡿
⠀⠀⠀⠀⠀⠀⢠⠋⢁⠀⠀⢠⣿⡟⠛⢧⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣠⡞⠹⡇
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡌⠋⠁⠀⠈⠙⠲⠤⠀⠀⣀⠠⠔⠛⠢⠖⠛⣿
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣸⠘⣆
⠀⠀⠀⠀⠀⠁⠀⠀⠀⡰⣿⣷⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡿⠀⣿⠳⢤
⠀⠀⠀⠀⢰⠂⠀⠀⠀⠄⠘⣧⡶⢟⡧⢑⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠜⢁⢤⡇⠀⠀⠑⢄
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠐⠀⣿⢷⣿⢁⣾⡗⡶⣷⡖⢂⠒⠀⠛⢀⠈⢷⠈⣾⡁⠠⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠣⡀⠀⠀⠠⠀⢘⠋⡤⠺⠷⣭⣆⣭⢹⠀⢶⠀⠀⠀⠀⠸⠋⢹⠈⠁⠒⠄⡰⡀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠘⣄⡀⠒⣂⣰⣿⣿⡿⢀⠰⠆⢠⠄⢀⡎⠀⣾⠀⠀⠀⠀⡟⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠸⡏⠉⠉⠉⢩⣿⣃⡤⠴⠒⣂⡴⠛⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠛⡄
⠀⠀⠀⠀⠀⠀⠀⣠⠛⠀⠀⠀⠀⠐⡀⠀⠀⡏⠙⠋⠓⠒⠋⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠣⡀⠀⡇⠀⠀⠀⠀⠀⢠⠃⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⡴⠠⡀⠀⠀⠄⠑⡀
⠀⠀⠀⠀⡀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠂⡅⡐⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⡷⠠⡼⠀⠀⠀⣠⠁
⠀⠀⠀⢸⢀⠼⠀⠀⠀⡀⠀⡀⠀⡀⢰⠀⠓⢘⡆⠀⠀⠀⠀⢈⠀⠀⠀⠀⠢⢄⣀⡐⠄⣀⠔⠛⠲⠀⢀⠀⡐⡀
⠀⠀⠀⢘⠀⠀⡠⠆⠈⠀⠀⠁⠰⠀⢨⠆⡤⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠓⢦⡤⢴⡦⡴⠉
⠀⠀⠀⠈⢲⠊⠀⢀⠔⠃⢀⡤⠀⢠⠃⠀⠀⠢⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠢⢌⠱⢄⡺
⠀⠀⠀⠀⠀⠉⠙⠤⡤⢖⠋⠀⢠⠃⠀⠀⠀⠀⣱⠀⠀⠀⠣⣀⢀⣠⣠⣄⠀⠀⢀⢀⡄⠀⢀⡴⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⢗⡺⢅⡀⠀⠀⠤⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⢅⣀⣘⠀⢣⡀⠐⠲⠄⠠⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣠⣈⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣠⣦⠀⣀⡤⠒⠈⠉⠉⠉⠉⠉⠁⠐⠢⣄⡀⢠⣦⣀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠎⠒⢄⣀⣼⡿⠋⠀⣨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡀⠈⠻⣿⣄⡀⠔⠊⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠢⡙⣿⣤⣤⣀⢉⡛⣷⣄⠀⡀⠀⠀⡀⢀⣴⡟⣋⢁⣠⣤⣼⡟⡡⠂⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠜⠞⠀⠀⠀⠀⠈⠙⠿⢦⠗⠒⠒⢧⠾⠟⠉⠀⠀⠀⠀⠘⢖⠴⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⡔⠉⠀⠀⠀⠀⠈⠒⠛⠡⠃⠀⠀⠀⠀⠣⠙⠓⠂⠀⠀⠀⠀⠈⠑⣄⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡀⠀⠀⣴⡡⠄⡴⠁⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⠑⡤⠤⣱⠄⠀⠀⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡿⠀⠀⠀⠓⢤⡓⢈⣳⠀⠀⠀⠀⠀⠁⠀⢠⣋⠐⣣⠔⠃⠀⠀⠸⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⡿⠁⠀⠀⠀⠀⠠⢌⣀⡘⡄⠀⠀⠀⠀⠀⠀⡜⣀⣨⡄⠀⠀⠀⠀⠀⠹⣿⣿⣇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣿⣿⣿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⣀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢿⣿⣿⣇⣀⣤
⠀⠀⢀⣀⣸⣿⣿⡿⠛⠛⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠛⠿⣿⣿⣟⣀⣀
⠀⠰⣿⡿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠾⠿⠶⠤⣤⣀⣀⠀⠑⠒⠒⠒⠒⠉⢀⣀⣠⠤⠴⠾⠿⠶⠧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠉⠻⣿⡷
⠀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⠏
⠀⠀⠀⠀⠀⠀⠀⢠⠮⠥⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠥⠤⠤⠤⠤⠤⠤⠤⠬⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠭⢧
⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⢠⠇⠀⠀⣀⡀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⣀⡀⠀⠈⢧
⡘⠀⢠⠎⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠈⢦⠀⠘
⡇⠀⠏⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠠⣀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⢸⡰
⠀⠙⡆⠀⢀⣆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⡞⠁
⠀⠜⠳⣤⠞⠻⠦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠾⠛⢦⡴⠛⠄
⡞⠁⠁⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠁⠀⠀⠀⠙
⠑⠂⠒⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣀⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠔⠆⠀⠀⠄⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⢈⣂⡤⠤⠄⠈⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡔⡂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠐⠥⠀⠀⠀⠀⠀⠀⠃⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠀⢠⡀⠀⠀⠀⠐⠀⠀⢀⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠈⠃⠈⠑⣶⡘⠶⠦⠳⠀⠀⡀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠀⠌⠀⠀⠀⠀⠀⠀⠀⠐⢅⣠⠤⠤⠀⣀⠀⠈⠛⣂⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠈⠣⣄⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⢌⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠀⠁⠁⡀⠀⠢⡄⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⢦⠉⠁⠀⠳⣄⠑⢄⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠁⠀⢀⡠⠊⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠆⠠⠑⠐⠪⣕⠒⠬⢁⣒⠤⢀⡠⠊⠀⠀⢀⠠⠐⠀⡐⠀⠀⠀⠁⠀⠀⢀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠙⠂⢀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠰
⠀⠀⠀⠑⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢸⡅⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⠀⢄⠀⠀⠀⠒⠂⠀⠇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⢸⠀⠀⠀⠁⠉⠓⠐⠄⠠⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⢸⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢸⠈⠹⠀⠠⡀⠀⡄⢀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⠀⠉⠀⠀⠈⠀⠈⠱⠔⠣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡇⢠⡇⠀⢀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡆⢀⢀⡇⣸⣃⣀⡌⠀⠀⡸⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠋⠉⠈⠈⠇⡞⠀⠈⠁⢠⡴⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⣇⡀⡄⠀⠀⣦⠀⣀⡼⠴⠀⣠⠞⣱⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠱⠜⠛⠀⠀⠂⠋⠉⠀⠀⠀⠀⠑⠀⣼⠏⣾⣗⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢀⡌⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢡⣷⣿⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⠿⢼⡗⠀⠀⠄⡄⠀⠀⠀⠘⠀⠀⠀⠠⣄⡀⠤⠀⠀⠀⠙⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣆⢾⠃⠀⡈⡄⠈⠐⠊⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠈⢦⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠉⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡿⠁⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⢀⣿⡷⠖⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠻⠈⠛⢷⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣻⣿⣿⣿⣻⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣀⡾⠃⣀⠀⠀⠀⠀⠀⢀⡀⠻⣆⡀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨⠗⠂⠉⢿⣤⠄⠤⣼⠏⠁⠒⠿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠂⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣼⡿⠿⠃⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣤⣄⣀
⠀⠀⠀⠀⠀⡴⠚⠋⠁⠉⠁⠀⠀⠤⠶⠄⢄⣀⠀⠀⠀⠀⣀⠄⠴⠦⠄⠀⠀⠉⠁⠉⠛⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢶⡀⠀⠀⠀⠀⠈⠽⢿⣿⣿⠿⠍⠀⠀⠀⠀⠀⣰⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠛⣿⡟⠇⠀⠀⢀⠀⠀⠊⠀⠀⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⣴⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡄
⠀⠀⢘⣿⣿⠄⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⢼⣿⣟
⠀⠀⠀⢿⡯⠀⠀⠀⠐⠀⠀⠀⠉⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠁⠀⠀⠀⠀⠀⠀⢨⣿⠃
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠄⠀⠀⠀⠀⠀⣈⠀⠀⢀⠀⠀⠀⠈⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠀⣀⣤⣤⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣤⣤⣄⡀⠘⠀⠀⠀⠁
⠀⠀⠀⠀⠠⠀⠤⠴⠿⠻⠿⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠿⠻⠷⠤⠄⠠
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠂⣠⣀⣠⣿⣷⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⣠⣴⣿⣿⣿⣿⡿⣻⣽⣿⣿⣽⣷⣄⢠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⣿⢿⣿⣿⠟⣿⡍⠉⣜⣿⠟⣿⣿⣿⢿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣿⣿⣿⣿⣷⣴⣿⣿⣾⣿⣿⣷⣾⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⣿⣿⡿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⠟⠉⠛⠛⢿⣿⣿⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⢄⢻⣿⡿⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠻⣿⡟⡠⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢶⡄⢻⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣿⡟⢠⡶⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⣧⠀⣿⣯⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣼⣟⠀⣾⡏⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⢹⣷⡟⠟⢹⣿⡻⠿⣷⣦⣶⣦⣴⣶⣶⣾⠿⣛⣿⠯⠉⢻⣾⡏⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣴⣿⡀⠀⠀⠈⠛⣛⡿⠛⠋⠉⠀⠘⠛⢯⣛⡛⠉⠀⠀⢀⣿⣦⣼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣥⣤⣾⣿⣿⠟⠀⡴⠛⠉⠧⢿⣶⣤⣤⣶⡿⠼⠛⠙⢲⡀⠹⣿⣿⣷⣤⣬⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡟⢀⡼⠀⠔⣂⡀⠀⠀⠈⠉⠀⠀⠀⣐⠢⡀⢷⡀⢻⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⡀⣺⠇⣰⣿⠿⡝⢻⠛⠛⠛⠻⠛⡟⢻⣿⣆⠘⣿⠀⣭⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢹⣿⣿⣿⣿⣿⡟⡔⣿⡀⠿⠿⠶⠛⠛⢋⣙⣛⡙⠛⠓⠾⠾⠿⢀⣿⡳⢹⣿⣿⣿⣿⣻⣿⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⣽⣿⣿⣶⣯⣶⣾⣿⣿⣿⣿⡟⢿⣿⣶⣭⣷⣿⣿⣿⣤⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠇⢡⠈⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣸⣿⠀⢠⣧⠀⣸⣿⢿⣻⣿⣿⣿⣿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢫⢿⣿⣿⡟⢹⣷⣿⣇⣿⣿⣿⣿⡆⠀⣿⠀⣿⣿⣾⡏⢹⣿⣿⡿⡹⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡇⠀⠹⣯⢻⣿⣿⣿⣿⣿⢠⣿⣿⡿⣻⠟⠀⢸⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠈⠈⠻⣿⣿⣿⣿⣿⣿⠟⠁⠉⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡆⡆⣀⡀⢀⡀⠀⢀⡀⣀⠛⠓⢯⣛⣋⣉⣁⣀⣀⣠⣤⢶⣦⠤⣤⡀⣀⣤
⠀⠀⠀⠀⣀⠄⣀⣤⣼⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣴⣧⣽⣿⣿⣷⣿⣮⣿⣿⣿⣿⢿⣿⣽⣿⣿⠛⠛⠉⡛⣿⡿⣤⣴⠀⠀⠀⠀⠂
⠀⠀⢀⡔⣡⣾⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿⢿⣿⣿⡟⠀⠀⢈⠁⣿⣿⣿⣿⣶⣶⣤
⠀⠐⢫⣿⣾⣿⣿⣿⣿⣿⣿⣿⣣⣏⠿⢯⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⠁⣰⣿⣿⣿⣿⣶⡄⢀⠀⣿⣿⣿⣿⣿⣿⣿⣰⣤⣀
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠛⢿⡴⠋⠀⠘⢦⣄⡀⣨⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣾⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⣠⣾⣿⣿⣿⣿⣷⡀⠴⣶⠝⠁⠀⠀⠐⣏⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄
⠀⣀⡎⢿⣿⣿⣿⣻⠋⠀⢀⣀⡀⣠⡀⠀⣀⠨⠰⢦⡀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠟⣟⣿⣻⣿⣚⣷⠂
⠀⠛⣿⠾⢛⣿⡿⠋⠀⠉⠈⠳⢤⣿⣿⣦⣀⣤⣶⣞⢻⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢙⣉⣿⣿⣿⣿⣿⣿⣿⣏⠙⠂⠤⣿⣿⣿⣿⣿⣿⠁
⠀⢀⣧⣴⣿⣿⠁⠀⠀⠀⠀⠂⠈⢿⣿⢿⣿⣿⣿⣿⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣸⣿⣿⣿⣿⣟⣠⡴⣶
⠀⠈⢹⣿⣿⡟⠀⡀⠀⠀⣀⠀⠆⠈⣧⠉⠈⢿⡟⠫⣣⡀⢺⣿⣿⣉⣉⣀⡀⢸⢾⣿⣿⣿⣿⣿⡿⣿⡿⠙⣩⣴⣿⣾⣿⠋⠈⢻⣿⣿⣷⣦⣴
⠀⠀⣺⣿⣿⡇⠀⠀⡠⠊⠉⠓⠒⠶⣎⠀⣤⣤⣤⣤⣿⠁⢰⣿⣿⣿⣿⣿⣿⣶⣿⣿⣟⡰⠻⠛⠉⠁⢠⣾⡿⣹⡿⠿⢿⡇⠀⠀⢻⣿⣿⣿⣿
⠀⣬⢿⣿⣿⣧⣿⢿⠤⠤⣀⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣉⣉⡉⢛⣟⢁⣶⣿⣿⣦⣶⣿⠏⢠⣿⣀⣠⣤⡀⠀⠀⢸⣿⣿⠛⡟
⠀⣠⣿⣿⣿⠋⠀⠀⢀⣶⣿⣧⣤⣾⣿⣋⠀⢹⣿⣿⣷⣿⣿⣿⣿⣻⣿⡉⢉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⢸⣿⡿⠉⠈⣷⣷⣾⣿⣿⣿⢀
⢀⡯⣽⣿⣿⠀⠀⣀⣼⣿⣿⣿⡿⣿⣿⠿⠛⠛⢿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⠁⠘⢿⡗⠀⢠⡿⠛⠙⣿⣿⠇⢰
⠈⠀⠀⠀⣻⣿⣿⡿⠋⠀⣩⣼⣿⣿⠁⠀⠀⠀⠘⣧⣼⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⡿⠀⠋⠉⠀⠀⢸⠁⢰⠟⠃⠀⠀⡽⣏⢀⣿⠇
⠀⠀⠀⢶⣿⣿⣿⣷⡀⠘⠏⠽⠿⣿⣷⡀⠀⠀⢠⠽⡿⠁⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⣿⡟⠛⠁⠀⠀⠀⠀⠀⣾⢀⠛⠀⢐⣠⡼⠗⠻⠺⠉⠁
⠀⠈⠁⠙⠛⠛⠛⠻⠻⠶⠦⠤⠴⠂⠛⠳⠂⠀⠀⠀⠀⠀⢠⡶⠦⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠈⠈

⣀⣀⣀⢀⣀⡀⡀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢄⠀⣰⡾⠂⠒⠀⠀⠀⠀⠀⠀⠀⠈⠖⠐⢷⣆⠀⡀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣤⡀⠛⣳⣄⢀⠀⠀⡀⣠⣞⠛⢀⣤⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠋⠀⠀⠀⠉⠟⠛⠁⠉⠛⠛⠉⠀⠀⠀⠘⢔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⠀⠀⣀⠀⡠⠀⠂⠀⠀⠀⠀⠑⠀⢄⠀⣀⠀⠀⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⠇⠀⠘⠢⠃⢘⡀⠀⠀⠀⠀⢀⡃⠘⠄⠃⠀⠸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣾⣿⡏⠀⠀⠀⠀⠒⠂⠣⡀⠀⠀⠀⠜⠀⠒⠀⠀⠀⠀⠸⣿⣷⣶⡤
⠀⠀⠀⠀⣀⣰⣶⡾⠿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⢷⣶⣆⣀

⠀⠀⠀⣺⡟⠉⠉⠀⠀⠀⠀⠀⠀⠐⠒⠚⠓⠒⠢⠤⠀⠀⠀⠀⠀⠀⠤⠴⠒⠚⠓⠒⠂⠀⠀⠀⠀⠀⠀⠉⠉⢻⣗
⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣠⣄⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣿⣿⣿⣿⣿⡛⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠠⠃⠀⠀⠀⠔⠏⠈⠓⢄⠀⠀⠀⠀⠀⠀⠈⢩⣿⣿⣿⣿⡍⠃⠀⠀⠀⠀⠀⠀⠠⠚⠁⠹⠢⠀⠀⠀⠘⠄
⠀⠀⠀⡄⠀⠀⠀⠀⠄⠐⠀⠀⠀⠁⠂⠄⠀⠀⠀⠀⠀⠋⢿⡿⠙⠀⠀⠀⠀⠀⠠⠐⠈⠀⠀⠠⠂⠀⠀⠀⠀⠀⢠
⠀⢀⠐⠀⠀⠀⠀⠐⠢⡀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⢀⠐⠂⠀⠀⠀⠀⠂⡀
⠀⣰⣄⣀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠰⢇⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⡸⠆⠀⠀⢀⠁⠀⠀⠀⠀⠀⣀⣀⣆
⢴⣿⣿⠅⠀⠀⠀⠀⠀⠀⡇⠀⠀⠢⠀⠈⠁⠀⠀⠁⠀⣀⣀⢀⡀⠀⠨⠀⠀⠈⠁⠀⠐⠀⠀⢸⠅⠀⠀⠀⠀⠀⠨⣿⣿
⠸⣿⣿⡒⠀⠀⠀⠀⠀⡀⠁⠀⢴⡀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢀⣦⠀⠀⢅⠀⠀⠀⠀⠀⢒⣿⣿
⠀⢻⣿⡄⠀⠀⠀⠀⠊⠀⠀⠀⠀⠐⠀⡀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⠀⠂⠀⠀⠀⠀⠑⠀⠀⠀⠀⢠⣿⡿
⠀⠀⠛⠀⠀⠀⠀⠂⠒⠠⠀⠀⠀⠀⠀⠀⠀⠘⡀⣠⠀⠀⠀⠀⠀⠀⣄⢀⡃⠀⠀⠀⠀⠀⠀⢀⠄⠒⠀⠀⠀⠀⠀⠛
⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠐⠀⠐⠀⠊⠉⠁⠀⡠⠀⠀⢄⠀⠈⠉⠁⠀⠂⠠⠂⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⢸⠀⠀⠀⡄⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⢠⠀⠀⠀⡁
⠀⠈⠀⠀⠀⠌⡂⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢐⠡⠀⠀⢀⠁
⠀⠀⠀⠀⣀⠀⠈⠀⠀⢀⡀⢀⣀⡄⠀⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢠⣀⡀⢀⡀⠀⠐⠁⠀⣀
⠀⠀⠀⠀⡀⡀⠀⣠⣶⣿⣿⢿⠿⡇⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⠿⡿⣿⣿⣶⣄⠀⠀⢀
⠀⠀⠀⡁⠀⠉⠉⠉⠋⠈⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠁⠁⠘⠉⠉⠉⠀⢈⠂
⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤
⣿⣿⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠾⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠩⢉⠉⠉⠉⠉⠛⢦⡄
⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠈⡁⠀⣀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⣀⠨⡑⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⢀⡟⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⣀⣀⠀⠈⠢⠑⠄⠀⠀⠀⠹⣧
⠀⠀⠀⣠⡾⠃⠀⠀⠀⣠⣴⣶⣶⣤⣄⡀⠀⠀⠀⠀⣠⠶⠛⣉⣿⣿⣿⣿⣦⡀⢀⡀⠀⠀⠀⣘⣷⣄
⠀⠀⣼⠫⢐⠅⣀⣀⡀⠈⠉⠉⠉⠛⠻⣿⠛⠀⠀⠈⠻⠶⠛⠉⠉⢯⣍⡉⠉⢁⣠⣤⠶⠶⢶⣄⠀⢛⢷⡄
⠀⠀⣿⠀⠀⠘⠉⢉⠹⠷⠶⠆⠀⢀⣴⠏⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠉⠙⠛⠉⠉⣠⣼⣆⠀⠘⣧⠀⠀⣿
⠀⠀⢻⡧⢌⠂⢠⣿⡄⠀⢀⠠⠾⢿⡅⠀⠀⠀⣤⣤⣌⢙⡇⠀⠀⢀⣀⣤⣴⡶⠛⠉⣠⡟⠛⢣⡿⢀⢀⣿
⠀⠀⠀⢻⡄⠀⢸⣿⢻⣶⣤⣀⣀⠀⠙⠷⠖⣀⣀⣀⣠⣬⣷⡶⠛⠋⠉⢁⣾⣤⡴⣾⡿⠁⠀⠀⢐⣡⡾⠁
⠀⠀⠀⢸⡇⠀⢿⣿⣼⣇⢸⡟⠉⠙⢻⡟⠋⠉⣿⠉⠁⢀⣈⣷⣤⣶⡿⠟⣿⠁⣴⠏⠀⠀⠀⢀⡾⠋
⠀⠀⠀⢸⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢻⡏⠁⠀⢀⣼⠟⠁⠀⠀⠀⢀⡾⠁
⠀⠀⠀⢸⡇⠀⠈⢿⣿⠙⣯⠙⣿⠛⠻⣟⠉⠉⢹⡇⠀⠀⠀⢈⣷⣤⠾⠋⠁⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⢸⡇⠀⠀⠀⠙⠛⠻⠶⠿⠷⠤⠿⠦⠤⠼⠷⠶⠖⠛⠛⢉⠠⠔⡀⠄⢀⣡⡶⠛⠁
⠀⠀⠀⢸⡇⠀⢄⠀⠑⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⣂⣡⡴⠞⠋⠁
⠀⠀⠀⠸⣧⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠘⠳⣦⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠶⠶⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⣶⡤⢀⠀⣀⢤⡶⢐⣢⣴⣴⣦⣔⣲⣶⠶⡶⠶⡶⣴⣲⣶⣶⣖⣶⣶⣶⡶⠶⣶⣾⣿⣾⣶⣶⣶⣴⣷⡷⣷⣾⣷⣶⣮⣦
⡷⢾⣿⣢⠁⠸⡙⢪⡌⠣⠙⠰⠋⠙⠉⢾⣟⠳⠝⡛⡵⡲⠶⠨⣛⠻⣻⡿⢾⣷⡾⣉⣀⡀⢿⣿⣿⣷⣄⣿⣳⣿⡻⡟⠟
⣿⡷⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠚⠁⢀⡉⠁⠂⠀⠀⠂⠸⠆⣸⣌⠻⢿⠟⡬⠛⣟⠊⠄⠙⢿⡻⠥⣼⠃
⣏⡁⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠠⠤⣖⣾⣷⣾⣿⣿⡿⣧⣷⣿
⠿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠊⣻⣿⣿⠴⣯⣼⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⢀⣀⡀⠀⢀⣀⣠⡤⢤⣤⣤⡤⠤⣪⣹⣿⣿⣿⣿⣿⡿⢉⣼⢀
⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣨⢉⣾⣿⣳⣀⣴⣾⠁⠀⠁⠀⠒⠒⠒
⣿⣿⣻⣿⣿⡷⠶⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠛⠃⠿⠿⣿⣿⣿⣿⠀⠀⠉⠩⠉⠛⠍
⡿⣿⣿⣿⣭⣴⣾⣷⠙⢋⢻⣷⣦⣄⣈⠀⠀⠀⠛⠛⠻⣿⣶⣄⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠷⠀⢀⠀⠀⠛⠀⠉
⣿⣹⣿⣿⣸⡆⠘⠛⢳⣘⠛⣿⣸⣿⣿⣿⣶⣄⣛⠓⠦⠈⠙⠛⠛⠿⣶⣶⡟⠛⠃
⣿⠹⢿⣿⢲⠾⠀⢀⠻⢯⡘⡿⠌⠻⠾⠙⡛⣷⡻⠕⠶⠈⠻⠟⠰⣶⣤⠈⢺⣦⣀⡀⠀⢠⡾⡿
⣿⣧⠈⣁⡀⣀⠱⣷⡂⠀⠀⠀⣠⠀⠀⠀⠀⠀⠁⠀⣀⠀⢠⣤⣶⠒⠐⡎⠹⠉⠋⠁⣀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀
⣿⣭⣷⣼⡟⠁⠀⠀⠀⠲⢼⡿⣄⣀⣀⡤⣾⠛⠂⠄⣉⣁⣠⣴⣾⣷⣾⣦⣶⣶⣤⣤⣝⣻⣦⣾⣿⣶⣄⣿⣿⣙⠙⠄
⣿⣿⣿⣿⣿⣿⣷⣤⣀⡀⠘⣓⠶⠯⢽⣷⣟⣛⣻⣦⣾⣻⣿⣿⣿⣿⣾⠟⡋⠛⠿⠿⣿⣿⣿⣝⣰⣷⣾⣿⠿⣿⣿⠒
⣟⡻⢿⣿⣿⡿⠁⢸⢟⣻⠿⣯⣟⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⣃⡄⠀⠀⣌⣽⣿⠿⢿⣗⣟⣫⠛⢉⠂⠀⡠
⣿⣥⣭⣿⣿⣾⣷⣤⣬⣽⣿⣬⣽⣿⣿⣿⣿⣿⣿⣽⣭⣿⣯⣭⣿⣯⣿⣿⣿⣧⣤⣽⣧⣿⣤⣷⣼⣿⣧⣾⣷⣤⣤⣬⣥
⣿⣿⣉⣩⣍⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠉⠑⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⢅⠀⠀⢀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠣⠤⠂⠀⡄⠁⢠⠀⠀⠼⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡠⠐⢢⠀⠈⠁⠀⠀⠀⣀⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠓⠠⠔⠀⠀⠀⠊⠑⢀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠒⠂⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⣀⠀⠠⠞⠁⠀⢀⡉⠢⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⡞⠀⠀⠀⠀⢀⠀⠀⡁⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⡝⠋⠉⠉⠉⠉⠙⡍⠋⠉⠉⠻⡿⡁
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢠⠎⠀⠀⡏⠒⢤⡴⢇⠀⠀⠀⠀⠃⢐⡇⠘⠀⠀⠀⡄⠘⢦⡤⠒⢱
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⡰⠃⠀⠀⠀⢣⡠⠊⠀⠈⠢⢄⣀⣀⣴⣋⣈⠢⠤⠤⠚⠀⠀⠀⠑⣄⡎
⠀⠀⠀⠀⠀⢠⠁⠀⠀⢀⠞⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⢠⠔⢉⣀⠀⠀⣀⣁⠢⡄⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⡜⠀⠀⡠⠃⠀⠀⠐⣿⡿⠃⠀⠀⠀⠀⠀⠀⠇⠀⣿⣿⡆⠀⣿⣿⡄⢸⠀⠀⠀⠀⠀⠀⠘⢿⣿⠂
⠀⠀⠀⠀⡰⠁⢀⡔⠁⠀⠀⠀⠈⡝⠁⠀⠀⠀⠀⠀⠀⠀⢣⡀⠻⠟⠁⠀⠻⠟⢀⡜⠀⠀⠀⠀⠀⠀⠀⠈⢫
⠀⠀⠀⣰⠁⢀⠎⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠤⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⣠⡏⢈⠱⠊⠲⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢰⠈⣆⠠⠄⠐⠒⠂⠀⠀⠀⠒⠂⠀⠄⣐⠈⡆⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠖⠑⠎⡀⢱⣀
⣸⢄⠀⠙⠖⠀⢰⣀⣀⡠⠤⣀⠾⡀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⢄⡀⠀⠀⢀⠷⣤⠤⢄⣀⣀⡆⠀⢲⠋⠀⡨
⢣⣀⣈⡁⢀⣀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⣀⡀⣹⣁⣀
⠸⢥⠎⠀⠁⠀⠉⠒⠠⢤⠆⠀⠀⠀⠈⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠄⠰⡤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠠⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠂⠤⠤⠤⠤⠤⠤⠤⠐⠐⠒⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣴⣦⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡧⠄⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠬⠥⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠠⢴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠠⠀⠀⠀⣀⠈⠉⠉⠁⠀⠒⡶⠊⠁⠀⠀⠀⠀⠀⠀⠈⠑⢶⠀⠀⠈⠉⠉⠉⣀⠀⠀⠀⠤⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠀⠠⠤⠤⠀⢀⣀⣀⡉⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠉⣀⣀⣀⠀⠤⠤⠄⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠀⠀⠀⠀⠀⠀⣀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡦⠤⠴⢄⡀⠀⠀⣠⡀⠀⠀⢀⣄⡀⠀⣀⣠⠤⠤⢴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠉⠉⠀⠈⠢⠖⠁⠀⠉⠉⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠒⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⣰⣀⣄⣀⣀⣀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣀⣀⣀⣆⣉⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠩⡕⠦⠤⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠄⠀⠀⠀⠀⠀⠀⠙⠆⠀⣀⣀⡀⠈⡡⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⢀⡠⠐⠀⠀⠀⡠⠀⠀⠉⠉⠉⠁⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠻⠠⠂⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠑⠒⠂⠀⠀⠀⠀⠀⠀⠁⢀⠠⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⠀⠀⠀⠀⠀⡌⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠑⠀⠊⠑⣦⠤⣤⠤⣴⠢⢦⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⢲⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢴⣉⠱⣬⢻⡷⠆⠀⠄⠙⠆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⠀⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⠀⢘⢦⡉⠹⠏⠁⡀⠤⣀⠀⠀⠑⠒⠰⡄⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠤⠤⢠⡏⠀⡀⣀⠂⠈⢁⢩⣥⡰⠋⠀⠀⠈⠆⠀⠐⢒⡀⠑⠒⡀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⠢⡄⠰⡥⠖⠦⠍⠙⠋⠁⠀⠀⠀⠘⠁⠀⠀⠀⠀⢀⡀⠔⠀⠀⠀⢀⠄⠀⠙⢖⠀⠀⠀⠈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠁⠀⠀⠈⢦⡉⠁⠐⢆⠀⢹⡀⠀⠀⠀⠀⠀⣀⠄⠒⡩⠀⠀⠀⢀⡠⠔⡁⠀⠀⠀⠀⢢⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣀⣀⠀⢳⡄⣀⣠⠄⠤⢌⡓⠶⢦⣙⣦⠙⠢⠀⠀⡠⠊⠀⠀⠀⢀⡀⡴⠂⡱⠁⠀⠃⠀⠁⠀⠀⠀⡆⠀⢀
⠀⠀⠀⠀⠀⡌⠁⠈⠃⠉⠘⠂⡀⢉⠂⠐⠍⠲⣄⠀⠉⠙⠲⠄⠈⠀⠀⠀⠀⠈⠁⠀⠀⠀⢳⠄⠀⠀⠀⠀⠀⢀⠀⢸⠆
⠀⠀⠀⠀⠀⢹⡀⠀⠈⠀⠙⠏⠉⠉⠉⠉⠁⠀⠈⠑⢦⡀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡀⠐
⠀⠀⠀⠀⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⠀⠀⠈⠀⠀⠀⠀⠀⠐⠀⢀⠂⢀⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡈⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢸⠀⠀⠀⠙⠂⠒⢦⡀⢄⠀⠀⠸⠠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠣⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠈⠀⢸⠠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⢸⠀⠠⡇⢀⠴⣆⢀⠴⡀⣠⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸⠀⡄⠉⠀⠀⠈⠁⠀⠉⠰⠄⠳⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⡐⠁⠒⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣉⢁⡁⠈⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⢐⠒⠠⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⡆⠀⠄⠀⠀⡲⣄⠀⠀⠀⠠⠀⠃⠀⠀⡔⢀⠀⠀⠁⡀⠑⠀⡇
⠀⢸⠀⠀⠈⠃⠰⠤⠄⠠⠤⠀⠀⠠⢀⢺⠀⠀⢳⡰⣖⣦⠇⣸⠆⠈⡇⠀⠀⠀⢤⠀⠈⠘⠂⠀⠄⠀⢈⡟⡇
⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⣇⢳⣦⠨⠟⡹⣿⣶⡏⣲⢦⣰⣀⡠⡞⠁⢀⡀⠀⠀⢠⠀⠠⡞⠀⠁
⠀⠘⠁⣷⠈⠢⠠⠀⠀⠀⢀⠀⡀⠈⠈⠉⢻⡿⠀⠀⠀⠋⠚⣻⣽⡗⣤⣿⡦⠴⠓⠊⢁⠶⠆⢀⠀⠈⡇⠀⡇
⠀⠘⠀⢘⡶⠄⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⠚⣧⣠⠄⠐⠂⠀⢹⣾⣷⣿⢿⣓⣙⡲⢒⣥⠜⠁⠀⠀⢘⠇⠀⠃
⠀⠀⠀⠈⡷⠐⠒⠄⠤⢀⣀⠠⠦⠴⣇⠘⠂⠻⣦⡀⠀⢠⠄⠸⡾⣿⣃⠈⡠⠋⠓⠻⣶⡀⠄⣀⠀⡎⠀⠀⡇
⠀⠰⠈⠀⡟⠀⠀⢀⣤⠒⠏⡀⠲⣀⠀⠰⡂⠀⢹⡉⠀⢀⠀⠀⣧⡟⠀⠈⡤⠀⠀⠀⢸⠛⣧⡀⢈⠧⠀⠄⡇
⠀⠈⠀⠀⡇⠀⠠⡾⠀⠀⠀⠉⢦⣱⡀⢠⡀⠀⢘⠏⠀⠀⠀⠀⢉⣷⣤⡑⢄⣘⣁⠰⣾⣠⣽⡗⠚⠀⠀⠀⠇
⠀⠸⠀⡄⠈⡆⢠⠃⠀⢀⡤⠦⠄⠘⢷⠀⡙⢦⡀⢠⣤⣄⡀⠀⠈⡿⠝⢤⡀⢹⣤⡍⠁⡰⢿⠁⠆⠀⠀⡀
⠀⠈⠀⢄⠀⢡⢺⡤⠐⠩⠴⢂⡀⠀⢠⣴⠇⠀⠙⡿⠁⠘⡦⠖⠀⠀⠀⠸⢋⡴⡃⠘⠁⠀⠈⢻⣄⠀⡄⠀⠇
⠀⢸⠀⠆⣀⠘⣾⠀⠀⠀⠀⠋⠽⠗⠁⠘⣿⠀⠀⠐⠠⠷⠋⠄⠀⠀⠀⠀⢸⠃⠈⢦⠀⠀⠀⠀⢷⠀⠈⠁⡄
⠀⢀⠐⠚⣡⠾⢇⣀⡀⠀⠀⠇⠀⠀⠀⠀⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⡸⢧⣀⣄⣠⣾⣆⡀⠀⡃
⠀⢈⠀⢶⠀⣿⠀⣴⢉⣷⣼⡇⠀⢀⣰⣮⣍⣠⡁⢲⣀⢀⣀⠀⠀⢀⠴⢋⡟⠀⠀⡃⠘⣷⠈⠁⠉⣬⠃⠀⡇
⠀⠨⠐⠈⠳⣅⠀⠈⣇⣿⠋⠑⢀⡤⠞⠉⠉⠉⠰⣿⠏⣹⣥⣾⡛⣷⢲⡏⡉⠀⡜⡔⠲⣿⢦⣀⠐⡻⠀⠀⡇
⠀⠸⠘⣧⠀⡜⠀⠰⣿⠃⠂⣠⠾⣤⢯⣭⠁⡞⠊⠚⠋⠛⠙⣢⣽⡏⠈⢿⢿⡤⢻⣿⢚⣿⠈⠙⣶⡇⠀⠀⡀
⠀⢀⠀⡌⠻⠁⢀⠀⠑⣾⡏⢧⠀⠀⠀⠀⠣⢶⣆⠀⠀⠃⠀⠙⢿⡁⠀⣺⡧⢹⣆⠋⣻⡟⠀⠒⠛⣣⡄⠀⠃
⠀⠀⠀⠈⠳⡁⠈⢲⢠⣸⡇⠀⡄⠀⠀⠀⢀⣀⣡⡀⠀⠀⠀⠀⡠⢋⡠⣼⠃⠀⢀⣷⡉⢀⣄⡠⠀⣼⠃⠀⠂
⠀⠀⠀⠰⢄⠙⠛⠖⠃⢀⣧⠀⣿⡀⠀⠈⡽⢏⠁⠀⡀⠀⠀⢀⡇⣦⡄⢸⣄⠛⣘⡾⠳⢨⡅⢽⣠⠏⠀⠀⠇
⠀⠀⠀⣀⠀⠺⡠⢀⠻⠀⣼⠀⢸⡗⣀⣿⠄⠤⠀⠀⢃⠸⢢⠎⢳⣯⡝⠻⣁⣤⢈⡇⠘⢀⢫⡌⠣⣀⠀⠀⡃
⠀⢐⠒⠀⡀⠁⢠⣄⣁⣀⢻⣧⢼⠇⠋⣿⠛⣶⠀⠀⠘⢂⠀⠀⠀⠳⣄⠀⠛⠀⢿⡀⡀⠀⠨⣤⠤⢯⠀⠀⡄
⠀⢠⠰⠖⠛⣉⠄⣤⠉⠉⢻⠇⡈⠀⠀⠉⡶⠘⠇⣤⠀⠀⠉⠀⠀⠀⢿⡆⢀⣀⡼⢿⣿⠆⠀⠈⡆⠹⣴⠀⠂
⠀⠐⠀⠀⠘⠛⠊⡉⠀⣜⠛⠉⠉⠻⠉⠋⣻⠛⠿⣿⡀⠀⠤⠤⠆⠀⢸⣇⣈⣁⠁⠘⢫⡆⠲⢶⣹⣦⣷⢩⡆
⠀⠐⠀⠀⠀⠀⠄⣠⡄⠈⠛⢓⡖⠚⠛⠛⡉⠛⠀⡉⢳⣦⡄⠀⠀⠐⠎⠁⣛⢹⡄⠀⠀⠀⡈⢨⣴⣿⠋⡿⠃
⠀⠀⠀⠀⠀⠀⠚⠶⠶⠖⠀⠠⠀⢮⠑⠠⠀⠀⠚⠀⠈⠛⣷⢖⠖⠐⠀⢨⣿⠠⣧⡀⢀⡀⢀⡿⡎⠉⠈⠀⡄
⠀⢠⢤⣤⠄⠀⠂⠀⣤⡀⠡⠀⠀⠼⠛⠀⠀⠀⠀⠀⣤⡖⠀⢛⠀⠀⠀⡁⠹⣇⠀⢷⣟⠃⢾⡀⠘⠀⠂⠀⠃
⠀⠘⠈⠁⠀⠀⠀⠠⠟⡀⠀⠄⠀⠐⠀⠀⠠⠈⡀⣈⠛⠙⠃⠈⢣⡄⠀⠃⣤⡟⠃⠈⠉⠻⣈⡟⢢⡀
⠀⠀⠀⢀⠔⠈⣁⠂⠀⠀⠐⠀⠀⠈⠂⢀⡀⠚⢣⣿⠃⠀⠀⠀⠀⠈⠁⢉⡿⠤⠶⠀⢤⣄⢶⣃⣬⠀⣀⡄
⠀⠀⠀⠸⠀⣀⢛⣱⣆⢀⣄⠶⠀⠀⢀⠈⠉⠁⠁⠙⢀⠀⠀⠁⠀⠀⠀⢻⣤⣀⣀⣠⣠⣬⠿⠸⠉⠉⠛⠓⠆
⠀⠠⠚⠳⠀⢻⣜⠻⠛⠓⢘⣵⣤⢄⢄⠐⠀⠀⠁⠔⠀⡌⠽⠀⠀⠁⠀⠀⠀⠃⠀⠀⠀⠀⢤⠰⠞⠁⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢠⣀⠀⠀⠀⠓⠀⠀⠀⣢⡀⠀⢀⣤⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⢀⣀⣏⡁⠀⠀⠀⠀⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡴⠂⢴⠁⠀⣀⣤⡷⠄⠐⠓⠖⠉⢈⡉⠉⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⡥⠦⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠲⠶⠚⠳⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⢀⡀⢀⣤⣤⣴⣄⣀⣀⣀⠀⢀⠰⡆⠀⠀⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠂⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⡀⠀⠀⠰⠖⠢⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢀⣼⣿⣿⣿⡿⠟⠿⠛⠿⢿⣿⣿⣿⣿⣤⠀⠀⢂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠈⢹⣿⣿⣿⡿⠁⠁⠀⠀⠀⠀⠛⠿⣿⣿⣿⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣾⣿⣿⠉⠠⣆⠀⠀⠀⠀⢀⣀⠄⠈⢻⣿⠏⠀⠛⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠈⣿⣿⣿⡅⠀⠘⡷⠤⢀⠠⢿⣏⣿⡿⠂⢻⡄⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠻⣿⣿⣿⡿⡄⠀⢣⡀⠐⢀⣼⣿⠻⢧⣤⣀⠃⢸⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠘⣿⠿⠁⠀⣠⣿⣿⣶⣾⣿⣟⠀⢸⣿⠿⠀⣾⠁⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠰⠉⢀⣠⣼⣿⣿⣿⣿⣿⣿⣟⡀⢸⣿⠀⢀⠀⢀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⠀⢠⠂⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠀⠋⠀⠈⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⣀⣿⡿⠋⢼⣿⣿⣿⣿⣉⡛⢫⡍⠀⢡⡀⠀⠱⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢟⡤⣠⠮⠛⠋⠀⠀⠀⠛⠿⡿⢏⠹⣏⠀⠀⢀⡸⠃⠀⢤⠤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠁⠀⠗⠛⢂⠀⠀⣀⣀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣖⣒⣶⣿⣿⡿⡟⣋⡉⠉⠶⠶⢶⣤⣀⣀⡀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡏⠀⠀⢸⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠘⠻⣿⣿⣿⣿⡇⠀⠀⢸⣿⢸⣿⠀⠀⠀⠀⠀⠀⢀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣯⠻⣷⣇⡀⠀⠙⠻⢿⡇⠀⠀⢸⣿⢸⣿⠀⠀⠀⣀⣤⣶⣿⣿⠿⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣌⠻⢿⣦⡀⠀⠀⠀⠀⠀⠘⠛⣼⣿⣶⣶⣿⣿⣿⡿⠟⢁⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣝⣻⣷⣤⣠⣤⣤⣤⣤⠛⠛⠛⢛⡿⠟⠋⡠⠔⣋⣥⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠿⠿⠿⠿⠿⠿⠿⠯⣒⣋⣉⣥⠄⣠⣦⡸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣾⣿⢿⣿⣮⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣠⣿⣦⡀⢀⣼⣧⠀⣿⠃⠙⢿⡿⠏⠈⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣏⠛⠛⠻⣿⣿⠇⠀⣿⣧⠀⠀⣤⣤⣶⠖⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠟⢻⠟⠀⠀⣿⣿⣆⣠⣅⠀⠀⣰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠀⠀⠀⠀⢸⢻⣿⣿⣿⣿⡄⢰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣧⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⢈⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠛⠛⠿⠿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣩⣴⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠟⠋

⣿⣏⢹⣟⠻⣿⢛⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⢠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣟⣯⣵⣿⣿⡟⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⠟⣠⡿⠃
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠸⣿⣍⡛⠛⠛⣉⣤⠾⠋⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠸⣿⣦⣤⣀⣤⣿⣟⠛⠛⠉⠉⠀⢠⡄⠀⠀⠀⠀⠀⣻⣷⣤⣠⣤⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠰⣽⣿⣿⣿⣿⣿⣿⣄⣴⠀⠀⠀⢸⡇⠀⠀⠀⣦⣠⣿⣿⣿⣿⣿⣿⣯⠆
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⣿⣿⣆⠀⠀⠾⠷⠀⠀⣰⣿⣿⢿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠐⣮⣿⣿⣿⣿⣿⢸⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⡧⣿⣿⣿⣿⣿⣵⠂
⠀⠀⠀⠀⠀⠀⠀⠉⣛⣿⣿⣿⣧⡀⢻⣿⢿⡻⢟⣿⣽⡻⢟⡿⣿⡟⢀⣼⣿⣿⣿⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣦⣁⡀⠀⣿⣿⣿⣿⠀⢀⣈⣴⣿⣿⣿⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢛⡿⠟⢻⣿⣃⠙⠻⠟⠋⣘⣿⡟⠻⢿⡛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⢳⣌⡙⠛⠛⢋⣡⡞⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠌⠙⠓⠚⠉⠠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠁⠀⠀⠄⠒⠀⠀⠈⠃⠐
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠈⠀⢀⢀⣀⠈⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠄⠀⢸⠞⠉⢆⠀⠀⠀⢀⠉⠀⢀⠘⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⡀⢀⠈⠙⠖⡋⠘⠂⠂⠀⠚⢂⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠄⠉⠐⡀⠚⡅⠛⠐⠧⡤⠀⢶⠀⠆⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠸⠖⠌⠀⠸⠧⡇⠀⡀⠠⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢀⠷⡀⡁⠞⠀⠀⠀⠀⢀⠀⠆⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠀⠐⡄⠀⡐⠂⠄⠈⠙⢱⠾⠶⣁⠀⠀⠒⣈⠦⡲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⠘⠛⠲⣄⢸⡆⢠⡄⣵⢀⡔⠋⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⠀⠀⠈⠀⠀⠀⠀⠄⢸⠀⠀⠀⠀⠈⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⠀⠠⠀⠀⠀⠀⢀⠊⠘⡁⡀⠀⠀⢀⣸⠀⠂⠀⠀⠀⠀⠀⣀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⠸⠷⠈⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣠⠞⠀⠀⠀⢱⠈⠑⠔⠀⡆⢥⡀⠀⠀⠀⠻⣄⠀⠀⡾⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠙⢄⠁⠀⠀⠀⠄⠀⠀⠀⠀⠁⢀⢧⠀⠀⠀⠀⠈⠀⠀⠙⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠟⠴⡆⣿⢳⡄⠧⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢸⢀⣬⠀⠀⠀⢿⠠⠃⢰⠀⢀⠀⠀⠀⠐⠶⠄⠀⠠⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⣁⠛⢢⡀⠀⠸⣧⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣍⡄⠀⠳⠀⠀⡿⣇⢰⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣧⡟⢰⡗⠀⣰⠇⣇⣿⠀⢀⣀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⠁⠀⠀⠈⠀⢀⣤⡠⠞⡩⠀⡞⠀⡐⠛⣠⣿⠟⠀⠸⣁⠉⣠⡀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠂⠂⠀⠴⠦⡈⠀⠄⠀⠊⢀⣼⠁⠐⣇⠀⢼⡸⣟⡄⠁⠀⠀⢨⣄⠠⠀⠀⢤⡀
⠀⡀⠀⠀⠀⠀⠀⠘⣃⡀⠀⢄⡑⠆⠀⠑⠲⡆⠰⠶⠋⠉⠃⢰⡿⠄⠀⠓⠿⠄⠒⠀⡴⠕⠘⠧⠀⣀⠀⠀⠀⢀⡀
⠀⠅⠀⠀⠀⡠⠄⠀⠈⠀⠘⠉⣔⠊⡇⠀⠀⠱⠀⢢⡀⠁⠸⠸⠫⢶⡀⢢⣴⠲⠶⢏⡁⠀⠈⠦⠘⠋⡈⢰⠐⠊
⠀⠂⠀⠀⠀⠀⠤⠀⠀⠠⠀⠀⠈⡠⠇⣘⠂⢀⡲⡦⢄⢠⡴⢐⠁⢺⠀⣠⠀⠀⠐⢀⠱⠀⠁⠀⠆⠀⠐⢣⣄⠀⡁
⠀⠀⠀⠀⠀⠀⠘⡂⠀⠘⠑⠄⠘⣠⠠⣀⡼⠏⠀⠀⠀⡈⢤⢀⣀⠈⠻⠑⡀⢠⠀⢀⠠⠃⠂⠉⠤⡷⠄⠀⠀⠈⠉
⠀⠃⠀⠀⠀⠈⠒⠓⠖⣠⡄⠐⠀⡠⠏⠈⠀⠚⠀⠀⠀⠈⠠⠠⠈⡼⠀⢈⠲⢈⡕⠠⠆⠠⡄⠹⠏⠀⠉⠁⠠⠐⠐⠒
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠃⠀⠘⠂⠀⠀⠀⠀⠀⠚⠒⠈⠁⠀⠁⠀⠉
⠀⠛

⣿⠏⠉⠁⠀⠀⠀⠀⠠⠀⠀⣛⡋⢿⡿⢿⣿⣿⣿⡿⠛⣛⣁⣼⣿⣿⣿⣿⣿⡷⠀⣿⣿⣿⣿⠃⡞⣿⠯⢻⢿⠾⣿⠿⠿⢿⠿⠇⣿⢻⣿⣿⠏⣾⣯
⡟⠀⠀⢠⣶⣾⣆⢿⣄⠀⠘⢟⣣⡝⢣⣾⣿⣿⡏⢰⣿⣿⣿⠛⠉⠉⠁⠀⠀⠀⢀⣿⣿⣿⣿⡇⢿⣷⣶⠄⡉⠀⠀⠀⠀⠀⠈⠴⣡⠔⢛⣥⢖⡣⠬
⣧⠀⢄⠹⠿⠿⠟⠘⠃⠈⠀⠙⠿⢃⣼⢼⣿⣿⡅⣿⡯⡉⠀⢠⣶⠀⠀⠃⠀⠀⠈⠿⠿⠻⢿⣿⢀⣥⡏⢧⠀⡀⣤⢀⡀⠀⠸⣵⠇⠚⠁⠀⠉⠲⣀
⣿⣦⠄⣉⣒⡂⠀⠀⠀⢀⡀⠀⣸⣛⠉⠸⣿⣿⣧⠺⣹⠍⡄⠀⠀⠐⠀⠀⠄⠂⣠⣴⣶⣦⡀⠙⠘⡿⠃⠀⢀⠼⣷⠷⡷⠀⠀⠦⡀⠀⠉⡠⠐⢃⡄
⣿⣿⣷⣌⢠⠀⠰⡆⠺⠿⠿⠛⠋⣶⣌⡠⠈⠊⣿⣶⣝⠺⠁⠀⠀⠀⠀⠀⡑⡄⢿⣿⣿⣿⣧⡀⠀⠓⠓⠀⠈⠚⠛⣷⣼⢶⣤⠂⠈⡀⠀⠶⠂
⣿⣿⣽⡛⣴⠟⣶⣧⣤⣶⡤⢤⣤⡼⣿⡅⠉⠒⡂⠿⡧⠀⢠⣄⣀⠀⠀⠀⠀⠀⢀⠁⣀⣤⣤⡥⣖⠀⠖⠟⣃⠀⠀⢸⡿⠖⢎⣡⣰⣹⣭⣹⣠⠄
⣿⣿⣿⠁⡅⠀⠀⠀⠉⠉⣹⡟⣿⣾⡟⠀⠀⠀⠘⠃⠁⢀⣿⣿⠿⠃⠀⠀⠀⢀⠈⠙⠽⠻⠛⠟⠃⠀⢐⠺⣜⢧⠶⠞⡕⣦⠆⠄⠺⠟⣛⣿⢿⢣⡅
⣿⣿⣿⣆⠓⢄⠀⠀⡄⢲⣨⠃⢙⣒⡇⢤⣤⣴⣤⡤⢈⣧⣃⠀⠀⠀⠀⠀⠀⡀⠐⠂⠀⠀⠀⠀⠀⠀⠸⠃⡉⢦⡞⠓⠀⠀⠀⠀⠘⠃⠛⠧⠜⢚⠂
⣿⣿⡿⠏⠴⢀⡐⠼⠷⠘⢁⣴⣏⡹⠇⠠⡄⠈⣤⣤⡈⠉⣍⠀⠶⠦⣲⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠙⠀⠀⠀⠀⠀⠀⡀⠀⡠⠐⢊⣂
⣿⣿⣾⠂⠈⠺⣷⣶⡜⠻⣿⣿⣿⠃⣾⠄⢹⡛⠟⠁⣻⣳⢉⣤⠑⣠⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐⠂⠂⢀⣄⢻⣷⡝
⣿⣿⣿⣶⣆⠀⣿⣹⡆⠀⠸⢟⣾⣦⠛⠰⣆⣀⠐⢛⣫⢳⣾⣿⡆⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠀⠐⠀⠀⠀⠀⣾⠁⢠⠀⡶⠂⣂⣨⣿⣿⣿⣿
⣿⣿⣿⡿⠛⠀⡅⣿⡇⠀⠀⠉⠛⠉⢀⡈⣿⣿⡇⢮⣦⣿⣾⠟⡃⠾⢷⠀⢀⣀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠻⠦⡤⠀⢿⣿⣿⣿⣿⣿
⣿⣿⠿⠀⠀⠀⠈⠶⣰⣷⡐⣢⢠⡔⣮⣷⣤⣩⡙⢂⣀⡁⠧⠸⠿⢶⡆⢸⣿⣿⣷⣤⣾⣞⣠⣴⢿⣿⡗⠀⠃⠀⠀⣀⣶⡆⠈⠀⠀⢸⢿⣿⣿⣿⣿
⣿⣿⠄⠀⠀⢈⠀⢀⡿⢟⣣⡿⠜⠃⠉⠙⠛⠨⢑⠸⠷⠍⢶⠉⣾⡀⠀⠀⠙⠛⠛⠉⣡⡆⠘⠿⠿⠛⠁⠈⣻⣿⡁⠀⠈⠀⠀⠀⢀⡄⠀⢻⣿⣿⣿
⣿⣇⠀⠀⠀⣠⡄⡯⠀⠉⠛⡴⣀⠀⣄⠀⢀⠀⣼⠢⠰⢐⢂⢠⡞⡥⣤⢠⠄⠀⠀⠰⠋⠙⠀⠀⠀⣀⡀⣠⣿⣿⣿⣿⠽⠀⢀⣼⣿⡗⠀⣸⣿⣿⣿
⣿⣿⣤⠛⠓⠋⠠⣴⣷⠀⡸⣆⠛⣠⠟⠊⠻⡁⢘⠁⠀⠨⡘⣾⣷⣿⣿⣿⣖⣭⠀⣄⢠⢀⣠⡞⣿⣿⠻⠨⣿⠿⠿⠁⠀⣠⣿⣿⡟⢀⣾⣿⣿⣿⣿
⣿⣿⣿⡆⠀⠀⠆⢭⠙⠻⠇⡀⡉⠣⠰⢈⣤⡀⠀⠀⠀⠘⣿⢿⡿⠎⠁⠖⠈⠉⠀⠉⠉⠀⠉⠙⠛⠋⢀⣼⣿⣿⡗⠀⢠⣿⣿⣿⣅⠘⠿⣻⣿⣿⣿
⣿⣿⣿⣿⣄⡀⠀⠄⢿⣿⠞⠀⠀⠑⣾⣿⠸⢇⣀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⢀⡄⢀⣄⢠⣤⠀⢠⣶⣿⣿⣿⣿⡃⠀⢸⣿⣿⣿⣿⡟⣈⣀⢄⢹⣿
⣿⣿⣿⣿⣟⣳⣦⣤⣀⢠⡔⠖⣀⣽⡈⣱⣽⣿⣿⣷⠀⠀⠀⠀⠀⠀⢀⣀⣶⣿⠻⠾⠼⣈⣏⠜⣿⣿⣿⣿⣿⣿⣧⠀⠈⠻⡿⠿⠛⣱⢟⣿⣾⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣂⣤⣴⣶⣶⣿⣿⣶⣶⣦⣄⡀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⣰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⣠⡆⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠹⣿⣿⣿⡿⠟⠛⠛⣿⡿⠻⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢿⣿⣧⠀⠀⠀⠈⠉⠛⠇⠙⠋⠁⠀⠀⠀⠙⠋⠁⠀⠀⡙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠀⠀⠀⠛⠿⢿⣿⣀⡄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠟⠿⠿⠋⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢸⡿⠿⠃
⠀⠀⠀⠀⠀⢀⡀⢿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠈⣿⣾⣿⣿⣿⣿⣿⣿⣁⣤⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⣀⡆⢠⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡏⠉⠙⢯⣠⢸⣿⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡁⠀⠀⢸⡃⠈⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣆⠀⠀⣄⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣀⣀⣠⡎⠓⠀⢸⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣾⣿⠀⠀⠀⠚⠗⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠀⠁⠀⠀⣾⠿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⣠⠞⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣟⢙⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⣀⣠⡤⠶⠋⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣛⣿⣿⠟⠳⢤⣄⣀⣀⣀⣀⡠⠖⠉⠙⠲⢶⡿⠿⠋⠉⢉⡡⠤⠶⠒⠉⠁⢨⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⢨⠋⠈⠈⠀⠀⠀⠀⠀⠀⠙⢦⡀⠾⢯⣤⣄⣠⣤⣌⡀⠪⣝⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠻⣿⣭⣿⡾⠛⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣿⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⡇⠀⠇⠀⠀⠀⠀⠀⠘⢦⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠋⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢶⣄⠀⠀⢀⣴⡄⠀⠀⢀⣴⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡠⠍⠛⠯⠀⢿⠇⠨⠟⠋⠥⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠋⠁⣤⡦⠀⠈⠀⠠⡦⡄⠉⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠁⠀⠮⠴⠀⠀⠀⠠⠬⠆⠀⠉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠛⠛⡛⠂⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠚⡛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠁⠀⢀⠀⠈⢳⣶⣦⣶⣶⠋⠀⢀⠀⠀⠡⠒⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⢀⠀⠀⠈⢹⠉⠀⠀⢀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠦⣀⠀⢑⠐⠂⢩⢐⠂⠒⠁⢀⡠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⠄⣄⠀⢈⠀⠘⠘⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠂⠀⢸⣿⣶⣅⣴⣾⣿⠀⠀⠚⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣀⣈⠀⠸⠿⠛⠁⠙⠻⠿⠀⢈⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠙⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠄⠁⠈⣆⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠈⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣀⠆⠀⠀⠀⠀⠀⡐⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠟⠁⠀⢀⡠⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⡇⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠎⠀⠀⢀⡄⠀⠀⠀⠀⠀⢆⠀⠰⣆⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠁⠀⠆⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⡀⠀⠀⢈⡀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠉⠀⠀⠀⢀⡆⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠇⠀⠀⠀⠁⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⣀⣴⠿⠟⠃⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⡠⢤⣀⠀⢿⣧
⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠙⠒⠀⠚⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠙⡀⠈⣿⡆
⠀⠀⢀⣾⠏⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠀⠀⠇⠀⠸⣿⡀⣴⣶⡄
⠀⠀⣼⡇⠀⣤⡶⠟⠛⠛⠛⠛⢿⡛⠛⠿⠿⢷⣶⣤⣍⣙⠲⠦⣀⠀⠀⠀⠀⠀⠀⠈⣤⡀⠀⠀⡇⠀⠀⠹⣿⠏⢸⣧
⠀⠀⣿⡀⠀⢉⠧⣤⣄⣀⣀⣀⣀⣉⣑⣶⣦⣤⣤⣬⣽⡿⠿⢶⣤⣍⡐⠀⠀⠀⠀⠀⠈⠙⠃⠈⠀⠀⠀⢰⡿⠀⠸⣿
⠀⠀⠹⣷⣄⠹⣦⣀⣈⣙⣩⣿⣿⠟⠋⠁⠀⠀⢀⠀⠀⢀⡀⢀⠀⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⢐⢻⣇
⠀⠀⠀⠈⠙⠻⢾⣿⣿⣾⣟⣁⣤⢔⣢⣴⣦⣼⣷⣴⡶⠞⠛⢿⣰⣄⣇⣴⠀⠉⠙⠻⣶⣤⣀⣁⠀⠀⢰⡟⡔⠂⠌⠤⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣿⣿⣟⣥⣈⣩⣿⠇⠀⠀⠀⠀⠈⠋⠛⠻⠟⢷⣷⣤⣶⣤⣿⢯⣙⣛⣿⡟⠁⠃⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⣿⠋⠛⠛⣿⡟⢀⣾⣧⠀⠀⢀⡀⠀⠀⣷⢸⣷⣬⣴⣾⣿⠶⣮⣽⡿⠁⠄⠀⠀⠀⠄⣰⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⢼⡇⠀⢠⣿⠀⣼⣿⣿⠀⠀⣾⣿⡆⠀⢟⢸⣿⠷⣿⣿⣿⣿⠿⠛⠀⠀⠀⢀⠀⠀⠀⠈⠒⢁⡼⠃
⠀⠀⠀⠀⠀⠀⣀⡴⣏⠀⢀⠐⢸⣧⠀⣼⠇⠀⣿⣿⠇⠀⢸⣿⣿⠀⠀⠈⠈⣿⠀⠀⠀⢾⣇⡀⠀⠀⠀⠀⠘⠂⠀⣀⣀⣴⠾⣅
⠀⠀⠀⢀⣴⠞⠋⠐⠁⠀⠀⠀⠀⣿⣀⣿⠀⠀⠟⠋⠀⠀⠸⠿⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⣹⡟⠛⠶⠶⠶⠶⠾⠟⠛⠉⠠⣀⣿⠇
⠀⠀⠀⢾⣥⡄⠒⠚⠛⠛⠠⠤⡀⠹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠈⣿⣏⣤⣀⢀⣰⣄⣦⣀⣶⠶⠛⠁
⠀⠀⠀⠀⠛⠿⢶⣶⣤⣦⣦⣴⣄⣰⣿⠁⠀⢀⡀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⢸⣧⣠⣤⣤⣄⣈⡉⠉⠛⢻⡏⠛⢻⣿⠁
⠀⠀⠀⠀⠀⠀⣾⠁⢨⣿⠉⠉⠉⠙⣿⠀⢰⣿⣿⣿⣷⣤⣾⡆⠀⠀⠀⠀⠀⡼⠛⠉⠉⠉⠉⠙⠛⠻⢿⠿⠃⠀⢀⣿
⠀⠀⠀⠀⠀⢸⣿⠀⢸⣷⣀⣀⣀⢀⡟⠠⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⢿⣦⣀⣉⡉⠉⠉⠛⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡶⠶⠶⠶⠿⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⡀⠀⠁⠘⣿⣿⠿⠿⠋⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠰⢃⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⣿⡟⠀⠀⠀⠛⠹⠿⣿⣿⡄⠀⠀⠀⠀⠀⢸⣧⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣿⠀⠀⠹⠃⠀⠀⠀⠀⠀⠀⠻⢿⣷⠀⠀⣀⡀⠀⠸⣿⠋⡰⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣆⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⡏⠀⠀⠈⠀⠀⠈⣧⠘⠇⢘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⢏⣹⠏⠰⣟⠳⣶⣤⡀⠀⠀⠀⣼⠋⣷⡀⠀⣠⣤⣀⣄⢀⣴⢟⣷⡄⠀⠉⣳
⠀⠀⠀⠀⠀⠀⠀⠰⡋⢰⠈⠓⣀⣄⡽⠆⠀⠈⢷⡄⢸⠿⣿⠀⢸⣇⣴⠋⢹⡏⢹⡟⣿⠈⣼⠃⠲⠆⠙⢷⡀
⠀⠀⠀⠀⣠⡄⣀⣤⡉⠁⡀⠈⠁⠊⣱⡀⠀⠀⣼⠻⣿⡇⠙⠀⠺⠛⠇⠀⣿⠁⠘⠐⠇⠀⠀⠀⢢⡀⠀⠸⠇
⠀⠀⠀⠀⠈⠤⠉⠁⠉⠁⠀⠀⠀⠤⠟⠁⠀⠀⠄⠀⠀⢀⠄⢀⠠⠄⠀⠀⠠⠤⠀⠐⢿⡂⠀⠂⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠿⡟⠿⢿⠟⠛⡿⠿⠿⢿⠟⡿⣿⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣷⣼⣦⣾⣧⣶⣿⣶⣶⣿⣧⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⢛⣿⣯⣥⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⡴⠊⠙⠋⠉⠉⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡷
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢤⣄⣀⣀⣀⣀⣀⡤⣶⠉
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠠⠐⠒⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠚⠁⠉⠂⢹⠃
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⢉⠳⡀⡇⠀⠀⠀⠀⠀⠤⠤⠭⣕⣆⠀⠀⠀⠀⢀⣴⣂⣀⠀⣿
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢣⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠉⠀⠈⠁⡇
⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⡴⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢤⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡤⠴⡇⠀⠀⠀⠀⠀⠀⢴⣖⠒⠒⠶⠤⢤⣀⣀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣀⡀⢹⡄⠀⠀⠀⠀⠀⠸⡿⠿⣷⣶⠶⠤⠤⠭⡿⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣋⠀⢀⣱⣄⠀⠀⠀⠀⠀⠹⣄⠈⢻⡀⠀⢀⡼⠁⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠢⣄⠀⠀⠀⠀⠈⠙⠒⠓⠒⠉⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⣀⣀⣠⡤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠀⠉⠉⠁⠒⠢⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠦⠀⠙⢆⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠣⢄⡀⡰⠁⡴⠁⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠾⡀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡏⢀⡵⣕⠀⠀⠀⠀⠈⢢⣀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡏
⠀⠁⠢⠀⠀⠒⠄⡀⢀⠔⠀⠈⠊⠐⣈⡻⠃⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⢰⠁
⠀⠈⠉⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⢀⠌⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⡇⠀⠀⠀⠀⢸
⠀⢰⠁⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⡠⢂⡔⠛⠃⠄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠈⡇⠀⠀⠀⠀⢸
⠀⠘⠦⣀⡀⢀⡠⠃⠀⠀⠀⢀⠤⣚⡐⢛⣤⠤⣀⡀⠀⠈⠁⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢹⠀⠀⢧⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠞⠁⠄⠀⠣⠄⢠⠤⠊⠉⠓⠢⢤⠴⠃⠀⠀⠀⠀⢀⣀⡤⡞⠁⠀⣸⠀⠀⠈⠳⢤⡀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠀⠀⠀⠀⢰⠀⠀⡀⠀⠀⠀⠀⠀⠀⡤⠒⢉⣉⡀⠔⠀⡠⠔⠁⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠐⠁⠀⠀⠀⠀⡸⠀⠀⡱⠦⠴⡄⠀⠀⢰⡆⠠⣥⠀⢀⡀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡇
⠀⠀⠀⠀⡄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣀⣀⣇⣀⣀⣧⠀⠀⠀⡇⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⢸
⠀⠀⠀⠐⠲⢄⣄⣠⢶⠊⠉⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡁⠀⣀⣀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⢀⣄⠀⣰⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡀⠀⠀⣰⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣄⠀⠀⣰⣷⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣤⣤⣄⣀⡀⠀⢹⣿⣿⣿⠙⢿⣿⣿⡟⠙⣿⣿⣦⣤⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠻⠏⠀⠀⠈⢿⠿⠻⣿⣿⡿⠟⢩⣿⣿⣧⣄⡀
⠀⠀⠀⢀⣀⣤⣤⣴⣾⣿⣿⡆⠈⠙⠻⢿⣿⠈⠙⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⣼⣿⡟
⠀⠀⢀⣠⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣶⣤
⣴⣾⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⢠⣄⣀⣀⣀⣀⠀⢀⡀⠀⠀⠀⠉⠉⠉⠛⠻⠷⠆⠀⠀⠀⣠⣴⣾⣿⠿⠿⠛
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠘⢛⣉⣭⣽⣝⣛⡋⠁⠀⠀⠀⠀⣴⣶⣿⣷⣶⣤⠀⠀⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣻⣷⠀⠿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠙⠿⢿⣿⣿⠿⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣾⡿⠿⣿⣿⣿⡿⠿⠿⠿⠿⠿⢿⣷⡀⠀⢀⣾⡿⠛⠛⠻⠿⠿⠿⣿⣿⣿⠿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⣿⡟⠀⢰⣿⣿⡟⠀⠀⡀⢲⣦⠀⠈⣿⣿⣿⣿⡟⠀⠀⢀⠒⣶⡄⠀⢸⣿⣿⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣇⠀⠀⠻⣿⠟⠀⢀⣿⠟⠉⢻⣧⠈⠀⠸⢿⡿⠇⠀⢸⣿⣿⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⣹⣿⢿⣦⣀⠀⠀⢀⣠⣾⠟⠀⠀⠀⠻⣷⣄⡀⠀⠀⣀⣴⣿⣿⠀⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡇⠀⠉⠛⠻⠟⠛⠋⠁⠐⢦⣤⠆⠀⠈⠙⠛⠛⠛⠛⠉⠀⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⣴⠾⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⠂⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⡀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣾⡿⠋

⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠁⠠⠤⠀⠀⠶⠶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠑⡀⠀⠠⠫⠹⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠙⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡎⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠃⠀⡙⢿⣿⠀⠀⠀⢀⣀⠰⠄⠚⠻⡃⠀⠈⠁⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣀⠀⠀⡀⢀⠄⠀⢾⡿⠢⡟⠀⠐⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠚⠀⠀⠈⠀⢰⡇⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠣⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣀⠀⢀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠍⠁⠀⠀⠀⠀⠀⠐⠀⠀⢠⠀⢨⠓⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⣃⠄⠁⠀⠀⠱⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⡠⢤⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠔⠀⠀⠀⢁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡧

⠀⣼⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠄⠀⢀⡀⢀⠀⡀⠀⠐⠂⢀⢀⡀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠀⠃⠐⠀⠀⠀⠀⠀⠙⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⡀⠀⠀⠂⠘⠂⠒⠒⠊⠓⠘⠀⠃⠚⠈⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣦⠀⠠⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⡃⠀⠀⠀⠀⠀⠿⠇⠠⠭⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡷⢶⢤⣤⣿⣼⠃⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠈⢀⠂⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠠⣦⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠳⠆⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢶⠖⠒⠒⠂⠒⠆⠀⡀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠒⠠⠋⢻⠀⠀⠀⠀⠀⡒⠺⠋⠂⠰⣶⢤⣀⣶⣶⢚⣶⣛⣦⣤⣄⡄⠀⠙⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠂⠀⠈⠐⠒⠤⣀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣊⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣘⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣪⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢡⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢀⠠⢤⡴⣂⠤⠐⠒⠒⠒⠂⠤⠀⠀⠘⠊⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⠀⢠⠞⠊⠀⠀⠀⠀⠀⣀⡀⠀⠀⠐⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡠⠈⡀⠰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠤⠈⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⡄⠈⢦⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠉⠑⢄⠀⡄⡠⠒⠒⢄⢀⣀⡀⠀⠀⠂⠀⠀⠀⠃⢸⠀⠀⣀⣈⡆⡎⠀⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠷⡀⠘⡄⢿⣧⠴⠦⠀⢧⠀⠀⠀⠀⠒⠛⠛⠛⠿⠀⠣⡎⢝⢀⡁⢨⠊⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⣹⠀⢰⠀⠀⠃⡜⠀⢀⡤⠔⠒⢠⠄⠀⠀⠀⢀⡿⠑⠂⠀⠀⣀⠼⢁⠎⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢂⡑⠋⠈⠳⣊⣀⠣⠀⢣⠀⠸⣄⠀⠘⠁⠀⠀⠠⠁⠀⠀⠀⠀⢀⡔⠁⡠⠊⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠑⠢⠤⣉⠋⠀⠀⠀⣀⠀⠀⡀⠤⠶⣭⠶⠊⠁⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠒⠊⠀⠀⠰⠁⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡑⠦⢄⣀⣀⡀⠤⠴⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠈⢏⠁⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡇⠐⢢⡄⢸⡤⡔⠒⢰⡴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⢪⣀⠉⡆⣸⡅⠀⠉⣧⠠⠼⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠉⠀⠀⠈⠳⡄⠉⠉⣽⠃⠐⣍⠗⡲⠶⠛⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬⣒⠲⠠⠤⠴⠗⣒⠬⠋⠀⠀⢟⢼⣀⠀⠀⠀⠀⡨⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠑⠢⠍⡉⣉⡉⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠴⠶⠒⠶⠚⠰⠶⠖⠒⠂⠀⠀⠀⠘⠞⠧⠅⠂⠔⠒

⣀⣀⣀⠀⠀⠀⠀⠈⢳⣦⣄⡀
⠈⠛⢿⣿⣿⣷⣶⣦⣤⣹⣿⣿⣦⡀⢸⣧
⠀⠀⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢶⣤⣀
⠀⢰⣿⣷⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⣀⣀⣀⠀⢻⣿⣿⣶⣄
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⣀⣀⣤
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⡿⠃
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠘⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠸⣿⣿⡿⠋⠁⠀⠀⠈⠛⠿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⣠⣿⣿⣷⣾⣿⣶⣄⡀⠀⢀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⡇⠈⠛⣿⣿⣿⣿⣿⣿⡟
⠀⢀⣾⣿⣿⣿⠟⠙⠻⣿⣿⣿⣷⡏⠀⡇⠀⣶⣿⣿⣿⣿⡿⠟⠉⠀⠙⠃⠀⠺⣿⣿⣿⣿⣿⡿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠃⠀⢈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡁
⠀⠘⣿⣿⣿⡅⠀⠀⠀⢩⣿⣿⡟⠿⠿⠿⠿⠿⠟⠛⠛⢩⣿⣿⣆⠀⠀⠀⢸⣿⠸⡿⠋⣉⡻⣿⣆
⠀⠀⠹⣿⣿⣧⠀⠀⠀⢸⣿⣿⡇⠀⢀⡆⠀⠰⣧⠀⠀⠸⣿⣿⡿⠀⠀⠀⣾⡏⠀⣠⡾⠋⠉⢻⣿
⠀⠀⠀⠹⣿⣿⣦⡀⠀⠀⠉⠉⠀⢠⡾⠃⠀⠀⠹⣧⡀⠀⠈⠉⠀⠀⢀⣴⡿⠁⠀⠈⠃⠀⢀⣾⡟
⠀⠀⠀⠀⠈⠻⣿⣿⠶⣶⣶⠶⠖⠰⢦⣤⣤⣤⡶⠈⠛⠷⣶⣤⣴⣶⠿⠋⢁⣶⣤⣀⣀⣤⣾⡿⠁
⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠀⢸⣿⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠙⠀⣾⣿
⠀⠀⠀⠀⠀⠈⣿⣧⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠓⠀⠀⠀⢀⣠⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣷⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⢿⣿⡆⣀⣴⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⡿⠛⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠲⠀⠀⠈⠀⠀⠀⠀⠀⠳⠀⢨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡀⠄⢀⣤⠀⠀⠀⠀⣴⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠣⣈⡿⠏⠀⠀⢀⠔⢻⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠀⢀⡤⢶⠁⠀⠈⡍⠉⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠊⠀⠔⠊⠀⠠⠀⠈⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⡉⠡⠀⠈⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣇⢀⣤⣤⣛⠁⠢⠡⠄⠠⣶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣀⠅⠀⣽⣿⠋⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⢀⣀⣀⡀⠀⠤⠐⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠐⢐⠒⠒⠒⠒⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⠀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⡴⡇⠀⠀⠀⠀⠀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠈⢁⣘⣿⡷⠴⠇⠀⠀⠀⣠⠀⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠰⠀⠘⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣹⣛⣛⣹⣋⣎⣛⣟⣋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣭⣿⣭⣭⣽⣿⣭⣭⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠋⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⠿⠿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢸⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⣼⠀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣟⠋⠀⠀⠀⠀⠀⢠⡇⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠛⠻⣏⠀⠀⠘⢶⣄⣠⡞⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠘⣧⠀⠀⠀⣜⣿⣇⢰⣿⡟⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠛⣿⢿⣿⠋⠉⠙⣿⣿⣷⣿⣧⠀⢀⡼⠋⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣯⠀⠏⠁⠨⠁⢰⣶⣿⣿⣿⣿⣿⣿⣶⣼⣷⣴⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣴⠀⠉⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢠⣿⡿⡿⠁⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡖⣦⢄⣻⡇⠀⠙⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠏⠳⡀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⢿⣿⣿⡿⣿⠿⠟⠛⠷⣶⣦⣄⠀⠀⣿⣿⣿⣿⣿⣿⡇⠇
⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠈⠢⡀⠀⠀⠀⢠⣿⣿⣿⡽⢛⡟⣯⣿⣧⡀⠊⠀⠈⢹⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢰⢦⣄⡀⠀⠀⠈⠂⡀⠀⣼⣿⣿⣿⣤⣀⢠⣿⣿⣿⡛⠷⣾⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⡴⣾⡇⠀⠉⠳⠄⠀⠀⠈⢦⣿⣿⣿⣿⣿⣿⡅⠀⠀⢙⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠈⠻⠿⢿⣷⣄⡀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣤⣤⣾⣿⡿⠿⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠲⢦⡤⣀⣉⡻⢷⣦⣤⡀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣡⡄⠀⠀⠀⠀⠀⠉⣻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣷⣄⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢀⣙⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡄⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⢻⣿⣿⣿⣿⣿⣧⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⢻⣿⣷⣿⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡟⢻⣽⣟⡿⣿⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠟⠉⠉⠉⠀⠀⠀⠙⠛⠿⠛⠙⠛⠋
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⢀⣴⣾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣷⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠉⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⢀⣬⣤⠄⠀⠀⠀⠴⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⣀⣴⣿⣿⣶⣦⣄⡀⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⢿⣟⠛⠛⠉⡉⣠⡆⣁⣾⡟⠛⣛⠉⢻⣿⢿⣿⡝⢄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢱⡷⠿⣿⣏⢉⣿⣇⡜⠻⣷⣬⣯⢞⣿⣿⣦⡙⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠛⠿⣿⣼⣦⠻⣿⠿⠿⠵⠇⢀⣿⣿⣿⣮⣿⣿⣿⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣶⠶⢂⣠⠿⣿⣿⣿⠉⠃⠀⠾⣷⡶⣶⣿⣿⣿⡿⡿⡿⢿⣿⠿⢿⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⠁⠀⠀⠀⠀⢰⣿⠯⡴⠴⠶⣾⡿⠁⣿⡿⣿⡿⢧⣰⡉⠈⠛⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⡄⠀⢀⠄⠸⣿⢿⠗⠛⠉⢉⣰⣾⣯⡴⣋⡵⢞⣽⣿⣷⣶⢻⣿⡿⠶⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⢸⣄⣀⣹⣦⣤⣤⣶⣿⣿⡿⠟⣫⣥⣶⢿⣻⢿⣿⡟⠛⠁⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡇⢀⣀⣼⠯⠝⣻⠟⠿⢿⡭⢭⠴⠒⣫⣾⠿⣛⣿⠟⢠⢾⠃⠀⠀⠀⢻⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⢏⠈⣷⡿⡏⠀⢻⡲⡍⢽⣯⣾⣏⣤⣾⣿⣏⡄⡄⢠⡗⠁⠀⡀⢤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⡿⠹⣦⣤⣐⣦⡍⣹⣬⣿⣿⣿⣿⣿⡿⣿⣽⣇⣀⣸⠁⢠⣾⣿⣶⣶⠀⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣯⣝⢿⣾⠿⠛⣙⣋⣬⣿⡿⣿⣿⡿⢩⣾⣿⣟⡿⠙⣫⣴⣿⣿⣿⠟⢹⣿⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠛⣯⣄⢙⢦⡟⣻⡝⠫⣿⣿⣿⣿⣴⣿⠿⠟⠀⠀⣞⡿⠋⠉⡉⢁⣴⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠋⠘⣦⣬⡅⠈⠀⠈⣉⣵⣇⢸⣿⣿⣿⡙⠀⠀⠀⠀⣸⣿⣆⡠⠂⣰⣿⣿⣿⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢨⡿⠱⠂⠠⢾⣭⣭⡿⢿⣭⣿⡿⠛⠂⠀⢠⢚⣿⣿⠟⣡⣾⠿⠃⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠁⢀⣤⣾⠉⠁⠀⣶⣿⢱⣿⣶⣆⣾⣿⡿⢃⣾⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢁⡦⠞⠙⠁⢠⢠⣿⠟⠉⣩⣿⣿⣿⣿⢏⣴⣿⠿⠿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠁⢀⣤⣶⣾⡏⡀⠀⠐⠉⢻⠟⣡⢞⣿⣿⣧⡀⠀⠉⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣮⡁⠽⣉⣹⣏⠁⠀⡀⢠⣴⣦⠄⣼⡷⣿⣿⣿⣿⣷⠀⠀⠀⠀⠹⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⡤⠉⠓⢻⡿⡿⢀⣿⣵⣿⣿⠏⠀⢸⡇⣿⣿⡟⢿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠘⣧⣼⠋⣹⠃⠀⣼⠇⣽⣉⡋⣠⡾⠏⣷⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣠⢟⣀⡴⡅⣼⣿⣿⣿⡋⢡⣤⣿⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠈⠉⠉⠉⠉⠈⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⡀⡀⠀⠐⠀⠀⠀⠀⢄⣠⠀⠀⠀⠀⠀⠈⠁⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠠
⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠆⠂⠀⠐⠄⣀⠀⣀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠠⠀⠀⢀⠀⠘⠖⠀⠀⠀⠰⠐⡂⠐
⠀⠀⠀⠀⠀⠀⠤⡌⠀⠀⠀⠀⠀⠀⠷⠀⠀⢠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢈⡄⠀⠐⠈⠒⠉
⠀⠀⠀⠢⠂⠀⠀⡄⣀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⡠⠀⠀⠤⠩⠀⠀⠀⢄⠀⠀⠀⠀⡠⠀⡀⠀⠁⠀⠀⠀⠀⠠⠈⢁
⠀⠠⡀⠀⠀⠀⢀⠀⢈⠁⠀⠀⠀⠈⠀⣤⠄⠀⠘⠤⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠀⠀⠂⠈⠀⣀⣁⠈⠀⠀⠘⠁
⠹⠆⠀⠀⠐⠀⣁⣠⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢨⣤⠠⣄⡖⠢⠀⣀⠀⡀⠀⠈⠹⢆⡀⠀⠀⡀⡐⠂⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡦⢀⠄⠀⠀⠀⠀⠙⠁⠐⠀⠀⠆⠈⠀⠁⠀⠀⠀⠠⠴⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⡂
⠀⠀⠀⠀⠁⠀⢀⣄⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠳⡂⠀⠠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠚⠛⢄⡀⣀⠀⠛
⠀⢘⢀⠀⠀⡼⡏⠁⠀⠀⡀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠄⡀⠐⠂⠀⡓⢠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠦⠄
⠀⠦⠀⠡⠔⠀⠸⠀⠀⠀⠀⠀⠀⢦⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⠀⠈⠁⠗⠀⢦⡀
⠀⠠⣄⠓⠀⣆⠸⢧⢀⣀⣀⣀⠸⠏⠁⠀⠀⠀⠁⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠈⠢⠄⠐⠿⠀⠀⢠⠀⠀⠀⠀⠰⡦⠃
⠀⠀⠈⠁⠈⠁⠀⠤⠴⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⣜⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠴⠆⠀⠰⠦⠆⠈⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣰⢁⡁⠀⣰⣆⠀⢰⡖⢀⡀⣀⠀⣸

⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡴⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠛
⣤⠤⠦⠤⢤⡀⠸⡧⣾⠀⠀⣀⣀⡀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢸⢸⡇⠀⠸⡳⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⢹⡄
⣿⠀⠀⠀⠀⠙⣦⡟⢳⣠⠞⠁⣤⡏⢀⡔⠋⠁⢸⣷⠶⣦⣰⠲⣆⣞⡉⢩⡿⢾⠈⣛⣷⣰⠋⢻⣤⠖⢶⠖⠒⢦⠀⣠⠴⠟⠉⢻⠀⡇⢰
⣿⠀⠸⠟⠀⢠⠙⠃⢸⠛⠦⠤⡄⠈⣾⡀⠻⠃⠀⣿⡀⠉⠁⠀⡏⡴⠚⠀⢙⠸⡄⠳⠶⣤⡀⠈⡇⡄⢰⢲⡀⠸⡏⢧⠀⠛⠀⢸⣠⣟⣿
⢿⣀⣀⣀⡤⢋⠀⠧⠼⢃⡏⠉⠁⣀⢇⠙⠚⡆⠀⡇⠙⠦⡤⠞⠀⠓⠒⠒⢋⡀⠙⠒⠚⠃⠙⠉⣀⠉⢉⣈⠙⢛⣁⣌⠉⡇⠀⠘⡏⢣⡼
⡼⠮⠭⠤⠔⠋⠲⠶⠞⠦⠭⠭⠭⠕⠋⠙⠒⣇⠀⢻⠓⠦⠤⠴⠋⠑⠒⠒⠋⠙⠒⠖⠊⠉⠙⠋⠉⠋⠋⠈⠉⠉⠀⢸⠀⡇⠀⠀⡟⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠤⣼⣁⡇⠀⢸⠃
⣷⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠙⠶⠋⠀⣼⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡾⣏⠀⠈⠉⠀⢀⣾⡀
⣿⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⢠⠟⣗⠦⠤⠤⢞⣿⡀⣠⢄⠀⠈⢠⣄⠀⢀⠀⠀⢀⢦⣀⠀⡄⠉⢀⣀⣈⣃⣈⣉⣃⣒⣾⠿⢹⡇
⢸⡀⡠⣄⠀⢸⠁⡇⠀⢰⠻⡄⢸⠀⠀⠉⡟⠋⢉⡇⠓⠃⠸⡆⠄⡏⠈⠓⠋⠹⡆⢸⠀⠀⠀⣿⠀⢸⠉⠈⠉⢸⡇⠀⣿⢹⡀⢨⠇
⠀⠉⠀⢸⡀⣸⠀⢹⣀⡟⠀⠳⠎⠀⠀⠀⢳⠀⢰⠁⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⡇⢸⠀⠀⠀⢸⡀⡸⠀⠀⠀⠘⣇⠀⡏⠀⠉⠉
⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠎⠀⠀⠀⠀⠳⡴⠁⠀⠀⠀⠀⠧⠏⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡟⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⣧⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⠀⠀⡇⠳⢴⡮⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⣷⡴⠋⡤⠀⢀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣠⣌⠑⢤⡇⠀⠀⣷⣄⡈⠿⠛⢲⣦⠤⢀⣀⡤⢤⣶⠚⠛⠿⣁⣴⡏⠀⠀⢥⠔⠉⣠⡀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠏⢙⣧⡀⠀⢀⡞⠋⠈⠻⣿⠿⠛⠁⠀⠈⠉⠀⠀⠈⠻⢿⡿⠛⢁⠛⢦⠀⠀⣠⣞⠉⢳⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⠀⡞⠉⢻⣆⡎⠀⠀⠢⠀⠤⠔⠀⡀⠄⠀⠀⠀⠀⣀⠐⠠⠄⠠⠌⠀⠈⣇⣼⠛⠙⣧⠸⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢧⡀⠁⠀⠙⢦⡀⠀⣠⠴⠊⠁⠀⠘⠂⠀⠐⠂⠀⠉⠒⢤⡀⠀⣠⠜⠁⠀⠁⣠⠟⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⣹⡆⠀⠀⠀⠉⠀⠁⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠤⣤⡄⠀⠰⠁⠀⠀⠀⢼⡃⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⠀⣀⠔⡄⠀⠀⠀⠀⡇⠀⠖⠒⢢⠤⡤⢤⠖⠒⠂⠀⡇⠀⠀⠀⠀⡤⢄⡀⠀⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠸⠀⠀⠀⠀⣿⣶⣶⣦⣼⣀⣇⣠⣦⣴⣶⣶⡇⠀⠀⠀⠐⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⠀⠈⡏⠙⠿⠿⠟⠋⠟⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠹⣄⠠⣇⠀⠀⠀⠀⢀⣷⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠂⠈⠑⠂⠤⠢⠤⠶⠌⠒⠉⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠈⠉⣟⠉⠀⠀⠀⠀⢀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⠀⠀⠀⠿⠀⠀⢀⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⡀⠀⣀⠴⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣴⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠛⢻
⣀⣠⣤⣤⣀⡀⠀⢸⣇⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣷⠀⠀⢸⠧⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢹⡆⠀⢸
⢿⠀⠀⠀⠀⠉⢳⣈⣷⣾⡀⣠⠞⠋⠙⡗⠀⢀⠴⠞⠛⢳⣀⣤⣄⢠⣤⣤⢀⡖⠛⠛⣶⣸⠇⠻⢤⡄⢸⠷⢿⡀⣠⣤⣄⡤⢤⣀⠀⠀⢀⣠⡖⠚⣷⠈⡇⠀⣿
⢸⠀⠀⣖⡄⠀⠈⠿⡇⠈⡷⣇⠀⠘⠛⠳⣤⡏⢠⣤⠀⢸⣿⠈⠻⠼⠀⢸⠞⣓⣢⠈⠳⡸⡇⢴⣒⡓⠛⠀⠈⡏⡇⠈⢁⡀⠀⢹⡶⢾⠉⠡⡆⠀⢻⠀⣧⣤⡏
⢸⠀⠀⠉⠀⠀⡼⠀⡇⠀⡇⢈⣍⣩⠆⠀⡟⢧⣀⣁⠀⢸⡹⣆⠀⠀⠀⡜⢸⣁⡀⣀⡠⠃⢣⡀⠉⣹⠿⣄⣀⠇⠧⢤⠇⢹⣀⣈⡇⠘⠦⣤⠀⠀⢸⣶⡧⢾⡅
⢸⡤⣤⠤⠴⠋⣠⠀⠙⠛⡁⢾⣀⣀⣀⡴⢣⣄⠀⢹⠀⠈⣇⡈⠛⠛⠋⢀⣀⡈⠉⠉⣠⣄⠀⠉⠉⣩⣀⣀⣀⣤⣀⣀⡰⢦⣀⣠⠤⠶⡄⢸⠀⠀⢸⣇⣨⠟⠁
⡞⠓⠲⠶⠒⠋⠈⠓⠚⠛⠓⠶⠶⠶⠖⠚⠁⠈⠙⢻⡆⠀⣿⠙⠲⠶⠖⠋⠀⠉⠉⠉⠁⠈⠙⠓⠋⠁⠈⠉⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⢸⡏⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⡇⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢤⡇⡼⠀⠀⢸⠁
⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⠶⠾⠁⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⢠⢞⡏⠀⠀⠉⠀⠀⢀⣿
⢻⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠀⢠⠿⡷⢤⣀⣀⣠⢞⣏⠀⢀⣄⠀⠀⠃⢀⡀⠀⠀⠁⠀⠀⣠⡄⢠⢤⡀⠐⠀⡀⢀⡘⢦⣙⠒⠦⠤⠤⣒⡿⢹⡄
⢸⠀⠀⡀⠀⠀⢰⠛⡆⠀⠀⡴⣄⠀⢸⠀⠉⠒⣦⠥⠶⢻⠉⢧⡞⠈⡆⡀⢰⠏⠙⣤⡼⠶⡆⠀⡟⠈⠁⠀⢷⠀⠀⡟⠚⠙⠚⢹⡏⠛⢻⠻⡇⠀⢸⡇
⠘⠧⠞⠉⡆⠀⣟⠀⢧⠀⢸⠀⢸⣀⡿⠀⠀⠀⢹⠀⠀⢸⠀⠀⠀⠀⣧⠀⢸⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⢸⡇⠀⢸⠃⠙⠦⠼⠁
⠀⠀⠀⠀⢳⣀⡇⠀⠘⠳⠞⠀⠀⠉⠁⠀⠀⠀⠘⣇⠀⡾⠀⠀⠀⠀⢻⠀⢸⠀⠀⠀⠀⠀⣟⢀⡇⠀⠀⠀⠘⣦⡼⠁⠀⠀⠀⠀⠳⣤⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠒⠒⠋⠉⠉⠉⠉⠉⠉⢹⡆
⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣷
⠀⠀⠀⣠⠴⠋⠀⠀⠀⢀⣀⣤⠴⠖⠒⠛⠋⠉⠉⠉⢙⠓⠲⢤⣹⡀
⠀⢀⣠⠟⠀⠀⠀⣠⠖⠋⠁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠹⡇
⣠⣟⠁⠀⠀⢠⡞⠁⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢉⡟⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⡁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡤⠜⠂⠀⡇
⠘⣧⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠹⣆⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠆⠀⠀⠀⠀⠀⠙⣦
⠀⠀⢹⡄⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸
⠀⠀⠀⠻⣄⠀⠈⣻⠄⠀⠀⠀⠀⣤⣄⣀⣀⣀⣀⣠⣤⣴⣾⡄⠀⠀⡾
⠀⠀⠀⢰⠏⠉⠛⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢰⡇
⠀⠀⠀⠘⠷⡤⣦⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠋⠙⣋⡡⠟⠁⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⠧⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠛⠲⣤⣤⡤⠴⠶⠶⠚⠛⠉⠉⠈⢹⣆⠀⠀⡏⠉⠉⠉⠉⠉⠐⠒⠉⠁⢀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⠞⠋⠉⠉⠉⠉⠉⠉⠁⠛⠶⣤⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠈⡷⠤⣤⡞⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⣿⣁⣀⡀⡴⠃⠱⠖⢊⡦⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣀⣀⣀⣀⡀⠀⠀⢀⣉⣙⠆⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠈⠒⠒⠤⣩⠇⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡴⣶⠞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠰⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⠛⠛⡟⠛⠛⡿⠛⢻⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⡇⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⢼⡇⠀⠀⣿⠤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⡇⠀⠀⣿⣿⣿⣷⣤
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣧⣤⣤⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠋⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⠢⡤⠖⠉⠉⠙⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠴⢅⡀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⣔⠲⣄⠂⠉⠉⠀⠉⠉⠒⢄⡔⠋⠀⠀⠀⢀⠤⠒⠑⠬⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠁⠈⢁⡢⠤⠄⣀⠀⠀⡀⢀⡤⠒⠂⠁⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⣱⠊⠉⠫⣠⠤⠤⣄⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡁⠠⠤⢰⣍⠉⠉⠁⠀⠀⠀⠀⠀⣼⡻⠭⢯⡉⣏⢩⡆⠀⣀⡴⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣽⡋⢭⣭⣺⣀⠀⠀⠀⠀⣀⣀⣤⠯⣤⠶⢖⣛⣻⡄⢉⡩⠥⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢿⣉⣉⣁⣤⠭⣽⠋⠉⠉⢫⡏⠀⠀⠈⢏⠉⠀⢰⡇⠸⣄⠀⢐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠓⢮⠽⠖⠒⠛⡇⠀⠀⠀⣀⠀⠀⠤⠐⢺⠀⠀⢸⡇⢠⠤⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣽⠀⠀⠀⠀⡗⠀⠁⠁⢀⢆⠀⠀⢀⠃⠀⠀⠘⡇⠈⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠈⠢⠤⠔⠃⠀⠉⠈⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⢘⡧⠤⠦⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠉⠒⠤⠤⠀⠀⠤⠤⠐⠉⠀⠀⠀⢨⡗⠒⠒⢦⡀⠙⢄
⠀⠀⠀⠀⠀⣀⠤⠤⠼⠧⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢹⠀⠈⠓⢤⡀
⠀⠀⠀⡠⠊⢀⠔⠒⢰⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⡜⠀⡰⡦⡀⠑
⠀⠀⢰⠁⢰⠃⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠑⠊⠁⠳⠝⠒
⠀⡴⠋⠀⠈⡆⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡇
⡜⠀⡴⡦⣀⣙⡆⠀⢸⡇⠀⣿⠀⠀⠀⢀⡀⢀⠀⡄⡆⠀⠄⠪⠀⢒⣂⣠⡼⠇
⠳⠴⠵⠃⠀⠀⠀⠀⠈⢧⠀⣿⠀⠀⠀⠐⢃⣘⣪⡥⠤⠶⠒⠛⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣿⣶⣶⣿⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣬⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣠⣶⣾⣿⣿⣿⣷⣶⣄
⢠⣾⣿⣿⠿⣿⣿⣿⢿⣿⣿⣷⡄
⣿⣿⣿⡁⠀⢀⣿⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⣿⣿⣿⣿⣶⣿⣿⣷⢶⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠟⣋⣥⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠘⢿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⣾⡿⢋⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠙⠿⢿⣿⣿⣿⣿⡿⣿⣿⡆⠀⠀⠀⠀⢰⣿⡟⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠀⠀⠀⠀⣿⣿⠀⠿⠿⢛⣛⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠿⠟⣫⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣤⣤⣀⡛⠉⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠟⠛⠛⠛⠻⢿⣿⣆⠀⠀⣴⣿⠿⠛⠛⠛⠛⠻⣿⣷⡀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠠⣶⡦⠀⣿⣿⣄⣸⣿⡏⠀⠀⠀⣶⣶⠀⢘⣿⣧⣀⣀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⣀⣀⣀⣈⣀⣠⣿⡿⠉⠉⢿⣷⣄⣀⣀⣈⣁⣠⣾⣿⠋⠉⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠿⠿⠿⠿⠿⢋⣁⣤⣤⣬⡛⠿⠿⠿⠿⠿⠿⠛⠁⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡀⠀⠀⣠⣾⣿⠿⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⣸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⠿⢿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⢿⣿⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣿⣿⣿⣿⣿⣠⣿⣿⣇⣀⣿⣿⣿⣿⣿⣿⣿⣿⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⣤⣤⠀⠀⣠⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⢀⣀⠀⠀⠀⣀⣰⣾⢿⣷⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⢸⡟⢻⣷⣴⣄⣻⣿⣆⠀⢠⣿⠛⣿⡄⣿⣿⠀⠀⣀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⣿⣿⢿⣿⣿⣿⣿⡿⣿⣦⣾⡏⠀⣿⣇⣿⡿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣷⣿⡇⠈⣿⡇⢻⣷⠀⠘⣿⠇⠀⠀⢹⣿⡿⠿⠇⠀⠰⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⣿⠿⢷⣤⣴⣶⣬⡟⢃⣶⡿⠃⠀⢸⣿⠀⠀⢸⣿⣤⣾⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣏⣀⣾⡟⠉⠉⢻⣿⣾⡟⠀⠀⠀⠀⣤⣄⣀⣾⡿⢿⣿⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠋⠋⢿⣧⣀⣀⣼⣿⠻⣿⣄⣀⣀⣤⣨⣿⣿⣿⣥⡄⣈⣉⣹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡉⠀⢠⣤⣽⣿⣟⣿⣥⣶⣾⠛⠛⣿⣯⣿⠟⢿⣿⠛⣋⣿⡛⠛⡏
⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠀⢸⣿⣇⣹⣿⢿⣿⣀⣀⠀⢀⣿⡋⣿⡇⢸⣿⣾⣟⠋⡀⠀⠁
⠀⠀⠀⠀⠀⠐⣿⣇⠀⠀⢀⢹⣿⣿⣿⣁⠸⣿⡟⠉⠀⢸⣿⠿⣿⣧⢸⣿⠘⣿⣿⠛⢀⠀⠤
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠿⣿⡇⠙⢿⡷⣿⣷⣿⠿⣿⡯⠀⢚⣿⣼⣿⠀⠘⣿⡿⠟⠁

⠀⠀⠀⠀⠀⣠⣄⠀⣠⣄⠀⣠⣤⣀⠀⠀⠀⢠⣄⣄⠀⠀⢀⣤⣀⣄⢀⣄⣤⣤⡀⣤⣄⢀⣤⡀⣤⣤⣀⡀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⢸⡿⣷⣼⡇⣾⡟⠈⣿⡇⠀⠀⢸⣟⣿⡆⢠⣿⠋⢙⣋⠀⣿⣧⠍⠁⣿⣿⣦⣿⠀⢸⡇⠘⣿⠀⣸⣏⣿⡆
⠀⠀⠀⠀⠀⢼⡧⠙⢿⠇⠹⢿⣴⡿⠁⠀⢠⣿⠍⣽⣷⠌⠻⣦⠾⡿⠠⣿⣧⣴⠆⣿⡄⠻⣿⠀⢼⣧⡼⠟⢠⣿⠍⣽⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡄⢸⣧⡈⣿⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣾⣄⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢶⢃⡀⠳⢋⣨⡤⢖⣠⣴⣴⣿⣿⣷⢷⣄⣠⣴⠇⠀⠀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠸⣥⠌⣋⡴⢞⡋⠡⢰⡿⣟⣽⠀⣿⣿⣿⣿⣿⣿⡿⠰⢦⣄⠀⠀⠀⠀⣦⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⢿⠛⣠⠞⣫⠖⢿⣿⣷⢸⡀⣙⣿⣷⣿⣿⣿⣿⣿⣭⣀⠀⢀⠖⠒⢄⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⠷⢃⡾⠣⠾⣿⣷⣿⡿⢿⢆⣿⣿⣿⣿⣿⣿⣿⡤⠀⠀⠛⣿⣇⠰⠞⠀⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡓⠾⢠⡟⣰⣷⣶⣿⡿⣿⣷⣾⢸⣿⣿⡟⠉⠛⣿⠻⢿⣶⣦⣔⣀⣙⣲⢤⠀⢤⡄⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠶⢯⢁⡿⣠⣼⣿⣿⣿⣤⣾⣿⣿⢸⡿⢻⠟⠒⠈⢀⡀⠀⠀⠈⠙⡿⣿⣿⡃⠸⠊⣿⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠸⠦⡖⣸⢃⣿⣿⣯⣈⣿⣿⣿⣇⣸⡜⣧⠝⡃⠀⢷⣶⣝⠆⠀⣀⣤⣇⣀⣀⣀⣀⢀⡇⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠾⠧⠄⣿⢸⣏⣹⣿⣿⣿⣉⣿⣿⣿⣏⠈⡇⠀⠀⠀⠀⠈⠀⠞⢓⡶⠄⠹⠿⠿⠿⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢦⠤⠇⣿⢸⣿⣿⡟⠙⣿⣿⣿⡏⢻⣿⢰⠁⡀⠀⠀⢀⡀⠀⠀⡸⣁⣀⣀⣀⣀⣀⣸⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢰⣛⡆⢻⡘⠿⠿⠿⠿⠿⠿⠿⠿⠿⢣⠇⠀⠘⢦⡀⠘⠿⡣⠀⠾⠿⠿⠿⠿⠿⠏⡏⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢤⣶⡘⣧⢀⣀⣀⣀⣀⣀⣀⣀⢀⡾⡶⠚⠉⠉⢉⠙⠛⠒⠢⠤⠤⠤⠤⠤⢀⣀⣇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣩⠥⠹⣆⠻⠿⠿⠿⠿⠟⣱⢋⡞⠀⠀⠀⠀⠀⠀⠠⠠⠒⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠙⣦⠀⣀⣀⣀⢠⠇⣼⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⣠⠌⠳⣌⠻⠿⢸⡀⢻⠀⠠⠀⠀⠀⣠⠤⢤⣤⠤⠤⣤⣤⡴⠶⠒⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡲⠓⡌⠳⢦⣀⠀⠈⠳⣄⠀⠀⠈⠁⠀⡸⠏⠀⠀⠈⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢩⣆⡍⠃⠀⠀⠈⠓⠒⠶⠒⠂⠀⠀⠀⠀⠀⠀⠀⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⢠⡶⢸⢀⡿⣾⢄⡇
⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⣀⢀⣀⡀⢀⣀⡀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⢀⡀⢀⣀⡀⣀⣀⠀⢀⣀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⢸⣇⣉⢸⣀⡿⢸⡷⣇⢸⡷⣇⠈⣷⠁⠀⠀⠀⠀⠀⠀⠀⣿⣸⡇⢻⣼⠁⣯⢸⡇⣿⢾⡁⣸⢽⣇⢸⡷⣇
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠈⠁⠉⠈⠁⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠁⠀⠈⠉⠀⠉⠈⠁⠉⠈⠉⠈⠁⠉⠁

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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⠹⣿⡿⠿⣿⢿⡿⣿⢿⡿⡇⣿⣿⡿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠋⠀⠉⠛⠋⠁⢀⠐⠠⠀⣿⣿⣿⣿⣿⣿⣷⡘⡇⣿⣠⣘⣿⣤⣹⣇⡿⢈⠿⣇⢿⠇⣿⣿⣿⣿⣿⣿⠋⢀⡀⢈⡄⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠘⠳⠖⠀⢿⣿⣿⣿⣿⣷⣿⣶⡇⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⢿⣷⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠛⣿⣿⣿⣿⣷⣿⣿⠿⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿
⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡿⠁⠠⣄⢀⡤⠘⢿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠃⠀⠲⠴⠶⠞⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣷⣀⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢀⣀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⣠⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣰⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⡉⢉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⡄⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⡄⣀⠹⣿⣿⣿⡿⠿⠿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠤⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣠⣴⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠈⠙⠿⠁⠠⠤⠤⠀⠙⠛⠉⠀⠀⢰⣿⣿⣿
⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⠙⢿⣿⡏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿
⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣆⠀⠙⠛⠀⠙⢿⡇⠈⠻⣿⣿⣿⣿⣿⣿⣿⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡟⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⠭⠉⢻⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⠌⣸⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣶⣶⣦⡄⠀⠀⠀⠀⢀⣶⣷⣶⣶⣿⣿⣿⣿⣿⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣴⣶⣦⡀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡤⢀⢀⡘⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠠⡥⣤⠈⣿⠿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣀⢀⢻⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠶⡤⠀⢹⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⣿⠉⠀⠈⠻⠿⠀⣉⣍⣀⡙⠋⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡛⠉⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢠⣄⠙⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣦⣾⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣿⣿⣿⣟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣾⣿⣿⣿
⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡄⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣅⣀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⣀⣠⣤⣦⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣃⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⡟⠻⠛⡛⡛⠛⠛⠛⢻⡟⠛⡟⡟⠟⡛⣿⠻⣿⠛⠛⠛⠛⢻⡟⡟⡟⠛⡛⣛⢿⠛⡟⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠙⢿
⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠛⠚⠛⠓⠛⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⢀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠂⠀⠀⠀⠀⠐⠂⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⠍⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠫⢉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⡇
⠀⠀⡇⠀⠀⠀⠈⢂⠀⠀⠀⡠⠁⠀⠂⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⢀⠈⠁⡃⢠⠎⠉⠉⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠲⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠑⠄⡀⠀⠂⡴⣶⠐⡀⠀⠀⠔⡭⣅⠢⠀⡤⠁⠀⠠⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆⠛⠛⠄⠁⠀⠀⠢⡳⢛⡴⠁⠁⠀⠀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⡜⠀⠰⢓⠆⠀⢰⠦⠀⠀⠀⡾⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠀⠀⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠑⡀⠂⠋⠀⠀⠋⠀⠀⣀⡞⠁⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡌⡀⠈⠉⣁⡀⠤⠚⠁⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠐⠁⡄⢂⠀⠀⠀⡀⡀⠀⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠈⠁⠀⠈⠁⠉⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠱⣄⠀⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⠳⣤⣠⠴⠻⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡀⠀⠀⠀⠆⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢯⣁⠀⠀⠀⠐⠠⠄⣀⠀⠀⠀⠀⠘⠲⠤⠶⢲⡟⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⢠⠟⠘⣇⠀⠀⢀⣀⣤⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⠀⠀⠀⠘⢲⡀⠀⠀⠀⠀⠀⡰⡟⠀⠀⠀⠀⠀⠀⠀⡼⠛⣇⠀⠀⡴⠃⠀⠀⠈⠉⠉⣩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠣⠀⠀⠀⠀⠐⢰⠁⠀⠀⠀⠀⠀⠀⡜⠁⠁⠈⠓⠊⠀⠀⠐⠠⠐⠈⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠁⣀⡀⠀⠀⡼⠊⠋⠉⠲⣄⢀⡞⠀⡘⠀⠀⠀⢀⠠⠂⠓⠠⠄⣀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⠤⢄⣀⠀⢠⡏⠁⠈⠹⣼⠀⠀⠀⠀⠀⢘⣿⠎⠉⠑⣦⠐⠈⠀⠀⠀⣀⠤⠔⠖⠚⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠾⠧⣀⣀⠼⠉⠣⣀⠀⠀⣠⠞⠹⡀⠀⢀⡸⣄⠀⠀⠀⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠈⠉⠉⠀⠈⠱⣖⢤⣸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⢀⡤⠔⠤⡀⠀⠀⠀⠀⠀⣠⠤⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⣀⡤⠚⠁⠀⠀⠀⠘⢦⡀⢀⡠⠞⠁⠀⠀⠀⠹⣄⠀⠀⡤⠤⢤⡀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⠒⠉⠉⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⠊⠀⠀⠀⠈⠓⠤⠼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⣾⡦⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⣤⡀⠀⠀⠀⠿⠂⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢠⣄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠉⠁⠀⠀⠀⠈⠋⠀⠀⠀⣀⡀⠈⠋⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠛⠀⠀⠀⠀⠀⣴⣆⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣠⡀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⣤⠤⠴⠧⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠙⠛⠉⠈


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⢻⡿⢿⡛⣿⣩⢛⢿⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣛⣛⣛⣿⣿⣿⣟⣛⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⣶⣽⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣃⣼⣿⣿⣯⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠸⠿⢾⣿⣿⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣿⣿⣿⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣤⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣽⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣝⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣽⣯⣿⣇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠶⣶⠖⠒⠛⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⡞⠋⠉⠉⠓⠦⣄⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣁⣠⠞⠀⠀⠀⠀⠀⠀⠈⠙⠲⣾⡉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣰⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢰⡇⠀⠀⠀⢀⡴⠶⠶⢤⣀⠀⢀⣀⣀⣀⣀⡟⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⡾⠀⠀⠀⢰⡏⠀⠀⣴⣚⠉⠉⠻⢧⡀⠀⠈⠉⠳⣄⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⠁⠀⠀⣼⠁⠀⠀⠀⠈⢷⡀⢸⡇⠈⠙⢦⠀⠈⢷⠀⠀⠀⠀⠘⡆⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠹⡄⠀⠀⠀⠀⠈⢷⠀⠓⠦⣤⠞⠀⠀⣸⠀⠀⠀⠀⢀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡟⠲⢤⣀⡀⠀⠈⣧⣀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⡞⢀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣘⡇⠀⠀⠈⠉⠛⢲⡇⠈⠙⠛⠋⠉⠀⠀⠀⠀⠀⣸⠋⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⠀⠀⠀⠀⠈⠳⣤⣀⡀⠀⠀⠀⢀⣠⠶⠚⠁⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢈⣩⠷⠒⠒⠋⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠈⠙⣳⠖⠛⠻⣍⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⢻⠀⠀⢠⡟⠀⠀⠀⠈⠉⠓⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⡟⢸⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⣾⠃⣀⡤⠶⠒⠛⢛⣋⣩⡽⢿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣷⣸⣾⢗⣋⡤⠴⠖⠚⠋⠉⢁⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⡿⢟⣉⣠⣤⡤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠾⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣛⣙⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣴⡟⠉⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠚⠉⠁⠀⠁⠈⠁⠐⠛⠉⠀⠀⠀⠜⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⡀⠀⠀⠀⢀⣤⠜⠁⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⣇⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠈⣿⣦⡀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⢀⡤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠃⠉⣿⣦⡀⠀⠀⠀⣀⣠⣾⣿⣿⠃⠀⠀⠀⠀⠹⣷⠀⢀⣠⠾⠃⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠸⣿⣿⣷⡆⠀⠈⠻⠿⠿⠋⠀⠀⠀⠀⠀⣰⡿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠙⠛⠛⠁⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⣤⣀⣄⣀⣶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⢤⣀⣀⣠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⣾⡟⠉⠁⠀⡟⠁
⠀⢀⡴⠒⠀⠐⠒⠒⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠿⠷⣶⣴⠇
⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠐⠲⠿⢶⡄
⢸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣀⣀⣀⣤⡶⠀⠀⢀⡔⠋⠛
⠀⠉⠉⢣⠀⠀⣤⡀⠀⠀⠀⢶⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠛⠫⠙⠋⠀⠀⠀⠀⡞
⠀⠀⠀⠈⢧⣀⣽⡧⣄⠀⠀⢸⣿⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁
⠀⠀⠀⠀⠀⠉⠉⠁⠹⣷⣦⣾⡟⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠧⠬⣯⣤⣦⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⢀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣴⡟⠉⠙⠷⢤⣤⡶⠶⠟⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠁⠦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⡘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢂
⠀⠀⠀⠀⠀⠠⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠂⠐⠄
⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠗⣃⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⢠⣁⠀⠠⣀⠀⢠⡔⣫⣼⣟⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠙⠯⠣⢾⠶⠋⢯⠋⠹⢏⠀⢠⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢶⠆⠀⠀⡄⣀⠆⠀⣴⠂⠘⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲
⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠒⠀⠀⠠⠀⠀⠀⢠⠂⠀⠀⠀⠀⠠⣼⠲⠇⠀⠀⠀⠑
⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠸⠀⠀⠀⠬⣀⣤⢈⡆⠀⢀⠆⠀⠈⠀⠀⠀⡀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠐
⠈⢿⠀⠈⠁⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⣧⠈⠻⣄⠡⠴⠖⠶⠋⡞⠁⠀⣼⢰⠀⠀⠀⠂⠀⠀⠀⠙⡇⠀⠀⠁
⢸⠸⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⣧⡀⡟⢤⣀⣈⡀⡤⢳⣀⣰⡇⢸⡐⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠨⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣧⡿⠟⠛⠃⠁⠀⠀⠀⠁⠀⠈⣿⡟⡹⠾⢿⡿⠂⠁⠈⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠐⠀⠀⠀⠑⠂⠀⠀⠋⠁⠀⠀⠀⠐⠂⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠠⠀⠂⡩⠀⠀⠀⠀⠀⠄⠀⠀⠁⢀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠢⠀⠀⠀⠃⠀⠀⠀⠀⠁
⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣿⡿⢿⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄
⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⣠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀
⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⣰⣿⠇⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈⢿⣷⡀
⢠⣿⡏⠀⠀⠀⠀⠀⠀⠛⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠃⠀⠀⠀⠀⠀⠈⣿⣧
⣸⣿⠁⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣶⣦⣄⠀⠀⠀⠀⠀⢹⣿
⣿⣿⠀⠀⠀⠀⠀⠘⠛⠉⠉⠉⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠉⠙⠛⠀⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⢀⣤⣄⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠻⣿⣶⣿⠟⢿⣷⣾⡿⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⠁⠀⠀⠉⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠙⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁
⠀⠀⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣶⣦⣤⣀⣀⡀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣾⡿⠟⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣛⣛⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⣀⣀⠀⣀⡀
⠀⠀⠀⠀⣀⣀⡠⠖⠉⠀⠀⠀⠀⠑⠎⠁⠀⠀⠀⠈⠓⢤⣀⣀
⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⣠⢄⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⢸⠒⠤⢀⣀⠤⠔⠊⠁⠀⠙⠢⠤⢀⣀⠤⠔⠁
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡒⢤⠀⠀⢀⠔⠈⠉⠒⠈⠉⠒⢄⠀⠀⢠⠄⠠⡀
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡎⠮⠜⣀⠔⠁⠀⠀⠀⣠⠀⠀⠀⠀⠑⢄⠐⠼⠌⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠈⠀⠉⠀⠀⠀⠀⢀⠔⠁⠳⡄⠀⠀⠀⠀⠉⠒⠊⡰
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⣀⡀⠤⠒⠁⠀⠀⠀⠈⠑⠤⢀⣀⣀⠠⠔⠁
⠀⠀⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⢦⠔⠒⠢⣄
⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⣀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⡇⡇⠀⠀⡎⡔⠃⠀⡠⠃⠀⠀⠀⢠⠏⢆⠀⠀⠀⠀⢣⡀⠐⠓⡌⡆
⠀⠀⠀⠀⠀⡇⡇⠀⠀⢇⠉⠒⠊⠀⠀⠀⠀⡰⠃⠀⠈⠣⡀⠀⠀⠀⠈⠒⠊⢁⠃
⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠑⠠⠤⠠⠤⠔⠋⠀⠀⠀⠀⠀⠈⠓⠤⠄⠠⠤⠔⠁⠀⠀⣀⠀⡀⡀⢀⡀
⠀⠀⠀⠀⠀⠧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡠⠂⠀⠀⠢⡀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠄⠀⠀⠀⠀⠀⠈⠄⢸
⠀⠀⠀⠀⠀⣄⣀⣀⡠⠔⠒⠁⠀⠀⠀⠀⠈⠑⠢⠄⣀⣀⣀⠄⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠘⠦⣀⡀⠀⢀⣀⠤⠚⠑⠦⣄⣀⠀⠀⣀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠛⠣⢿⣿⢲
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡎⠌⢡⠀⢒⠲⠟⣫⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠃⠑⠂⠸⠇⠅⣤⣹⣯⡿⢀⣀⡠⣄⣤⣀⡀⠀⠀⢠⡾⣿⡾⢻⣦⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⣯⣍⣒⣤⡤⠀⠛⣿⣽⢛⠛⠀⠀⠀⠙⠻⢿⣷⣠⠾⠟⠛⠙⠯⢄⠙⡿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⠠⡇⠀⢧⣈⡩⠋⠀⢀⠞⠁⠻⢬⠔⠀⢀⣘⢳⡄⠀⡼⣵⡘⠀⡇⢀⡔⠄⠘⡿⣻⢛⣿⣇
⠀⠀⠀⠀⠀⠀⢧⣀⣱⣀⠀⠀⠀⠀⣠⠃⢸⣒⠢⡈⠀⠀⠻⡀⠀⠁⣿⠁⡈⠱⠈⠀⠘⠓⠁⠀⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠑⣦⡀⢠⠃⢠⠟⢻⣷⣌⣢⣰⣄⠀⠀⣤⡿⠘⣷⣒⣒⣒⣢⠗⠒⠂⠈⢹⠋⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⢸⠀⠘⠤⠤⠔⠋⠉⠀⠀⠀⠀⡬⢷⠀⢸⠿⠯⡽⠃⠀⠀⠀⢀⣾⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠎⢆⠀⠀⠀⠀⠀⠀⠀⠀⣇⣈⡠⢾⣆⠀⠑⠋⠀⠀⠀⠀⣠⢻⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠈⠣⡀⠀⠀⠀⢀⣠⠞⠉⠁⠀⠰⠋⠑⢦⣀⠀⠀⠀⣰⠃⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢶⣴⠊⢉⡴⠂⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⣀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠻⠿⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠯⠿⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠒⠒⠂⠀⠀⠐⠒⠒⠒⠤⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠉⠉⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⡆⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠶⠶⠶⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⠶⠶⠶⢦⣤⡀⠀⠀⠀⠀⣾
⠘⣇⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣅⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⢠⡟
⠀⢻⡄⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠶⠶⠶⠶⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠶⠶⠶⠶⠞⠋⠁⠀⠀⠀⣸⠃
⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠶⠶⢶⣤⣤⣀⣀⣀⣀⣀⣤⡴⠶⠶⠖⠶⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⣧⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣸⠇⠉⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⢀⣠⣴⠶⠶⠚⠛⠛⠛⠛⠛⠛⠶⢶⣤⡀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣾⣋⣁⣀⣀⣤⣤⣴⠶⠶⠶⣤⣤⣀⣀⡀⠙⣷⣄⣤⡾⠋⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠞⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣓⣒⣦⣔⣶⣤⣴⣶⣞⣛⣛⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠐⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁⠀⠀⠀⠀⠈⠙⠢⣄⠀⠀⢀⣀⡀⠀⠀⠀⡀⢰⡅
⠀⠀⠀⠀⡀⠢⡉⠓⠆⠉⠉⠀⠰⣦⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢳⢊⡅⠀⣀⠤⠒⢊⣰⡿⠁
⠀⠀⠀⠀⠐⣦⣉⠳⠤⠄⠤⢄⠀⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡖⠋⡠⣔⡺⠛⡩
⠀⠀⠀⠀⠀⠻⠿⠿⠶⠦⠦⢄⢠⣜⣿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣴⣋⣥⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⣾⣦⠙⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠸⣿⣿⠛⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡙⠇⣠⣶⣦⣀⠀⠀⠀⠀⠀⢠⣿⣿⣶⡆⠘⢃⡞⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⣧⠀⣿⣿⣙⣿⣻⣶⣶⣶⣾⣟⣻⣯⣿⣇⠀⢼⢆⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠻⠀⠘⠛⠿⠟⣹⣋⣼⣿⣿⣯⠛⢿⣿⠻⠀⡾⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡆⠀⢀⡈⡴⢿⣿⣿⣿⣿⡿⠳⠖⠂⠄⢰⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠐⢄⣀⡾⡽⡿⡿⡿⠷⣦⡤⠂⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢹⣇⣒⡀⡂⠂⣰⡿⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⢝⣛⣛⡛⡛⠱⠃⠀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⣄⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣾⣿⣦⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠉⠉⠀⠀⠀⠀⠈⠉⠑⠢⢤⡀⠀⢹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⡡⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⣧⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠎⠀⠀⠀⠀⠀⠀⣀⣀⡠⠀⢀⣀⠤⠤⠤⢄⣀⡀⠀⠀⠀⠳⠔⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⡠⠔⠊⠉⣠⣫⠔⠊⠁⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀⠄⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢀⠴⠏⣠⡤⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠠⠶⣦⠀⠀⡼⠁⠀⠀⢣⠀⠀⠀⠀⠀⠀⠦⣀⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡴⠁⡸⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢸⠇⠀⠀⢀⡼⢄⡀⠀⠀⠀⣠⢤⡈⢳⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠰⢋⡥⠤⠈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⠀⢸⠀⠀⠀⠸⡄⠀⠙⣄⣄⣠⡇⠀⢹⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠞⠙⠲⡄⠀⡇⢀⡴⠒⠲⣄⠀⢀⡀⠀⠀⠀⡔⠉⠀⠀⠈⢳⠸⡆⠀⠀⠀⠈⢢⢰⠁⠀⡝⠃⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⣄⠀⠹⣀⣱⢺⣀⣀⣀⡈⢼⠁⠉⠆⠀⠀⠤⠶⠶⠶⢶⣾⠄⠳⣄⠊⡟⠉⠛⠘⠢⠞⠁⠀⣰⠃
⠀⠀⠀⡇⢀⠔⠒⢲⠃⠀⡝⠀⠀⡇⢀⠟⠋⠁⠀⢀⣈⠁⠀⠀⠀⠀⠀⠀⢠⣤⢄⡀⠀⠀⠏⠀⢳⡖⢲⠀⠳⠀⢠⠞⠁
⠀⠰⡀⣇⢸⡀⣠⠼⣄⢀⡇⢠⠀⠀⢸⠀⠀⡴⠋⠁⠀⡨⠝⠓⠲⠤⣤⠤⠖⠁⠀⠀⠀⠀⡰⠖⠋⣠⠃⠀⠀⠀⣾
⠀⠀⠳⡈⢦⠈⠀⠀⠈⠯⣤⡤⠑⠀⠘⣄⠀⠙⢤⣀⠈⠀⢀⣀⣠⠔⠁⠀⠀⠀⠀⠀⢀⡴⠁⢀⠜⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠈⠙⠲⠤⣬⣉⣉⣉⣁⣀⣀⠠⠀⢠⡤⠖⠚⣧⡶⠖⠉⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠈⠒⠒⠉⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠺⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡓⠤⣀⠀⠀⠀⢀⣀⡤⠔⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⠦⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⢢⡤⠄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣤⣄⡀⠀⡾⢷⠀⠀⠀⣀⣀⣘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢬⣇⠀⠉⣶⠇⢸⣖⣟⠉⠉⡽⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣧⣀⡈⣳⢀⣧⡀⠀⠀⢹⡤⠔⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⠁⠀⠀⠈⠙⢦⠀⠈⠉⢩⡟⠁⠰⡋⠿⢄⣶⠶⠿⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡓⠤⢀⣀⣀⠀⣠⣇⠠⢔⡻⠃⠀⠀⣿⣠⡋⠀⠀⠀⠀⠀⠀⡳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠢⠤⠤⠔⠒⠂⠉⠀⠀⠀⠀⠘⠪⣝⡒⠤⠤⠤⠴⠚⣁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠂⠒⠀⠀⠀⠀⠰⠀⠶⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠐⠀⠒⠰⠂⠀⠀⠀⠆

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠀⢲⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠿⠛⠋⠉⠀⠀⠈⠉⠙⠛⠿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⢿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠀⠀⠙⢿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⣄⢀⠀⠀⠀⠀⠔⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡀⠀⠀⠈⠀⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣦⡀⠀⠀⢠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⣄⣠⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣷⣮⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣽⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠆⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠀⠃⠀⠀⠀⠊⢀⢀⠄⠰⡆⢀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠐⠛⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠢⠂⠀⡀⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⣠⠔⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⣀⡠⠀⠀⠀⠀⠂⠀⠀⠀⠀⡌⠀⠀⠁⠀⠀⠀⠀⢀⡴⠇
⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⠃⠐⠧⠒⠒⠈⠀⠈⠒⠂⠚⠀⠄⠄⠐⠈⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠁⠀⡈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣀⠤⠀⠤⠄⠠⠤⠤⠄⠀⣀⡀⠀⠀⠀⠀⣀⡠⠔⠀⠀⠀⠀⠀⠉⠓⢄⡀⡀⠀⠀⠀⠀⠴⢠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠀⡒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠆⠵⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⢦⡽⠉⠑⠰⠤⡀⠀⠔⠃⣀⡤⠀⠀⠀⡘⢀⠞⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠙⢠⠀⠠⠃⠉⢠⡶⢀⠀⠀⠠⢲⠌⣠⡁⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⡀⠛⠒⠏⠠⡘⢀⠀⠀⣀⢶⣡⠧⠄⠎⡡⢡⢰⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠂⠈⣴⣴⣶⣿⣇⡸⣷⠀⠠⢀⣀⠀⠀⠂⡞⣨⠞⠥⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣠⠔⠂⠀⠉⠁⡜⠿⠴⢟⠇⠀⠃⠀⠀⢀⠈⠉⠃⢌⢁
⠀⠀⠀⠀⠀⠀⢠⠀⠇⡴⠗⠴⠶⠀⢀⠀⠁⠀⠈⠁⠀⢀⠖⠀⠀⠈⣗⠀⠖⠀⡈⠑⠠⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠈⠘⢠⠤⠤⠄⢠⡰⠁⠀⠐⡦⢰⡀⠄⠋⠀⠀⠀⢠⡷⠠⠄⠀⢁⠀⣼⠐⣧⠗⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠀⠀⠘⠌⢤⡀⢀⠄⢪⠌⢔⠆⠀⠘⠃⡏⠀⠀⢀⡀⠶⠀⠈⠐⣧⠔⠉⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⡴⢀⠀⣩⠀⠀⠨⣿⢍⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠁⡒⠋⠑⠒⠪⠷⠈⠀⠀⢀⠃⠀⠀⠀⠀⠀⠦⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠀⠀⠀⠀⠠⠀⠀⠘⢦⠀⠀⠀⢀⡜⠈⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢈⡀⢢⠀⠀⡅⠀⠀⠀⢀⠾⢷⡖⠀⠀⠘⡆⠀⢀⠰⠀⠠⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠰⠂⠀⠀⡰⠃⠀⠀⠘⡇⠀⠀⡀⠀⠈⠁⠀⠀⠀⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢀⠀⠀⡀⠄⠀⡀⢀⠃⠀⠀⠀⠀⢸⠔⠋⠉⠑⠢⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠚⠝⠂⠉⣱⡄⠀⠀⠀⠸⠤⠄⢀⣄⡀⠀⠠⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⡈⠀⢀⡴⠊⠉⠁⡈⠀⠀⠀⡀⢀⠀⠀⠀⠈⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠧⠔⠉⠁⠀⠀⠀⠫⠀⠀⠀⠀⠀⠁⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠀⡀⠔⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠛⠉⠓⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢹⢀⣀⠀⠀⣀⣠⡤⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠤⠇⠀⠀⠀⠀⠈⠀⠀⠙⠊⠁⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠞⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠢⣤⠴⠋⠉⠙⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠀⠀⠀⠀⣀⠀⣀⡴⠃⢋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣆⣀⣠⡖⠚⠁⠉⠉⣠⠞⠛⠁⠀⠀⠀⠀⠀⠘⠛⠓⠆⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠚⠁⢀⡤⠶⠦⣄⡀⠀⢀⡴⠒⠒⠦⣄⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠉⠲⡏⠀⠀⠀⠀⠈⠳⡄⠀⠀⠹⣄⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢀⣹⣔⣦⠀⠀⠀⢀⡇⠀⠀⠀⠘⣶⠹⣋⡭⠭⠉⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠘⢻⣏⡁⠀⢀⣠⠞⢁⡤⠶⢤⡀⠘⣯⡽⠆⢾⣉⣌⡟
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣸⠀⠀⠀⠀⠀⠀⢀⣨⠦⠤⢤⣄⣀⠀⢠⠞⠀⠉⠉⠉⠀⢠⠏⠀⠀⢈⡇⠀⠘⣦⠔⠋⡡⠋
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠩⢍⣳⣌⣉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢧⠀⠀⠀⠀⠀⠀⠂⢠⢀⡠⠞⠀⠀⠀⢿⠀⡎
⠀⠀⠀⠀⠀⠸⡆⠀⢀⡈⢳⣦⣸⠷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡆⢹⡄
⠀⠀⠀⠀⠀⠀⠹⡄⣸⠶⠯⣄⣀⣠⡤⠴⠖⢦⡀⠀⠀⠠⠚⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣸⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡇⣗⠀⢀⣤⠤⠤⠤⠴⠦⠂⠛⠓⠤⣄⡀⠀⠀⠉⠓⠶⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⣆⡴⠋⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⢸⠀⣟⣀⣸⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣄⣀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠋⢻⡉⠁⠙⢆⠈⠳⣞⠉
⠀⠀⠀⠀⠀⠀⢸⣀⣈⣉⣉⣉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⣶⠦⠤⠤⢤⣤⣶⣧⡀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⣯⣇⣀⣠⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠦⢤⣀⠀⢀⡟⠛⠋⠉⠉⠁⠀⠀⠳⣤⣀⣀⣤⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣦⡀⠐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠈⠙⠒⠲⠦⣤⣤⣤⣄⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡜⠀⠀⠀⠀⠀⣀⣾⣰⠃⠀⢰⠃
⠀⠀⠀⠀⠀⠀⡞⠉⠀⡹⢍⢙⢓⣦⡤⣄⣉⣠⣏⣉⣁⣠⣾⢤⣤⠤⣀
⠀⠀⠀⠀⠀⠠⣷⣤⣀⣧⣠⣭⢭⣝⣶⠿⠛⠛⢶⣤⣮⡤⠤⠚⣦⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠈⠈⠉⠋⠋⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⠷⡖⠯⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠋⠀⠀⠀⢄⠓⠒⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⣀⣀⣀⣠⣀⣠⣄⣀⣀⣀⣀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠃⢀⠀⠀⠀⡊⠀⠀⠀⠀⡀⠈⠎⠁⠀⠀⠉⠗⣄⠀⠀⢀⡀⠰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⣯⠉⢳⠞⢣⡀⠀⠀⠀⠁⣠⢆⠈⠀⠀⢀⠆⠈⠳⡞⠉⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡔⠃⠀⠀⠙⠲⠤⣤⠞⠥⠤⠉⣒⠚⠁⠀⠀⠀⠈⢦⣇⣀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⡰⠋⣠⣦⡀⠀⣴⣦⠑⢄⠀⠀⠀⠀⠀⠙⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡟⠁⠀⠀⠀⠀⠀⠀⡇⠀⣿⣿⡇⠀⣿⣿⠇⢸⠀⠀⠀⠀⠀⠀⠈⢻⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡈⠋⠀⠀⠙⢋⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀
⠀⠀⡞⠐⠲⠋⢱⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣭⣽⣿⣿⣯⣭⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠎⠙⡶⠋⢳
⢸⠁⠉⠢⢸⠂⠀⢣⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡔⠀⠐⡃⠔⠉⠈
⡞⠁⠀⠂⠈⠀⠀⠈⠁⠂⠤⢀⣀⣠⠛⢇⠀⠀⠀⠀⢠⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡀⠀⠀⠀⠀⡸⠙⢆⢀⡠⠄⠒⠈⠁⠀⠀⠁⠀⠀⠈
⠈⠐⠒⠒⠒⠒⠑⠲⢤⣀⠀⠀⠀⠁⠀⠈⠳⣄⠀⠀⢸⣿⣀⣹⣿⣿⣁⣀⣈⣁⣀⣘⣿⣿⣏⣀⣿⡇⠀⠀⣠⠞⠁⠀⠈⠀⠀⠀⣀⡤⠖⠊⠒⠖⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢲⡆⠀⠀⠀⠀⠈⠙⠲⠬⣉⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⠥⠖⠋⠀⠀⠀⠀⠀⢸⡖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⣿⣿⡯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠐⠐⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⣀⣀⠤⠖⠒⠒⠒⠒⠲⠦⣄⣀⡀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠂⠂⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠘⠃⠀⠀⠰⠀⠀⠀⠤⠀⢀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠤⠀⠀⠀⠖⠀⠀⠀⠃⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡈⠁⠀⠁⠀⠀⠐⠒⠂⠀⠈⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠇⠀⠐⠒⠂⠂⠀⠉⠀⠀⢁⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⠤⠄⠀⠤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠲⢄⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡠⠖⠛⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⢹⠒⠤⠤⠚⠢⡀⣀⠴⠓⠤⠤⠔⡞⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡇⠀⠀⢸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⡙⢢⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣶⣶⣶⣶⣶⣶⣮⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣶⣶⣶⣶⣶⣷⣾⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣖⣦⠀⣀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⡀⠈⣾⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢻⣿⣿⣿⣿⣿⣿⣿⡶⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠏⠀⣀⣉⣈⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣇⣀⡀⢻⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠍⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⣿⣿⣿⣿⣿⣿⣿⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⢀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣄⣀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡾⡿⡇⠀⣿⣿⣿⣾⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣿⣧⣶⠏⢸⣿⣿⣿⣕⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⣷⣿⡇⠀⣿⣿⣿⣞⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⠋⠉⠁⠀⠀⢹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⠀⠀⣭⣿⣯⣭
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣧⣤⠀⣠⣶⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡟⠁⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⣦⣄⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠯⠿⠿⠀⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠀⠽⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠁⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠀⠀⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢠⢞⣭⣝⢦⠀⡴⣫⣭⡑⡄⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡔⣫⣭⡳⡄⢠⢞⣭⡝⢦⠘⡇
⠀⠀⠀⢠⡖⠲⢤⣀⣸⠃⢹⣌⠻⢟⡼⠀⢧⡻⠟⣁⡇⠘⣇⡤⠖⢛⣲⠀⠀⠀⠀⡴⠒⢦⣄⣀⡇⠘⣇⡻⠿⣣⠇⢸⣘⠿⢋⡼⠀⢿⣠⠴⠊⣳⡄
⠀⠀⠀⠈⠛⡇⠀⠈⣿⠀⠀⠉⠉⠉⠠⣶⠄⠉⠉⠉⠀⠀⣿⠀⡀⣸⠁⠀⠀⠀⠀⠙⣿⠀⠈⢹⡇⠀⠈⠉⢉⠥⠤⢤⢭⡉⠉⠀⠀⢸⡅⢀⢀⡏
⠀⠀⠀⠀⠀⢳⣼⠀⣿⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⢻⠀⣿⠋⠀⠀⠀⠀⠀⠀⠸⣦⡇⢸⡇⠀⠀⠀⠈⠛⣺⣞⠋⠁⠀⠀⠀⢸⡇⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠘⢦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⣿⣶⣶⣶⣦⣄⡀⠀⠀⠀⢀⣀⣤⣤⣤⣀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⣶⣤⣀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠉⠙⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠉⠉⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠋⠀⠀⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢠⢞⣭⣕⢆⠀⡴⣪⣭⡓⡄⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡔⣫⣭⣢⡀⢠⢞⣭⡝⢦⠘⡇
⠀⠀⠀⢠⡖⠢⢤⣀⢸⠃⢻⣘⠿⢟⡸⠀⢇⡻⠿⢁⡇⠘⣇⡤⠖⠛⣶⠀⠀⠀⠀⡴⠒⢤⣄⣀⡇⠘⣆⠻⠿⣣⠇⢸⡘⠿⢋⣼⠀⢿⣠⠴⠚⣳⡄
⠀⠀⠀⠈⠛⡇⠀⠉⣻⠀⠀⠉⠛⣩⡀⠀⢈⣝⠛⠉⠀⠀⣿⠀⡀⢸⠉⠀⠀⠀⠀⠙⣷⠀⠈⢹⡇⠀⠈⠙⠋⣥⣤⣤⣍⠛⠉⠁⠀⢸⡅⢀⢀⡏
⠀⠀⠀⠀⠀⢷⣼⠀⣿⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⢿⠀⣷⠋⠀⠀⠀⠀⠀⠀⠹⣦⡇⢸⡇⠀⠀⠀⠀⢻⡟⢛⡏⠀⠀⠀⠀⢸⡇⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠘⣦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣼⠃⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⣿⣶⣶⣶⣤⣄⡀⠀⠀⠀⢀⣀⣠⣤⣀⣀⣀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⣦⣤⣀⠀⠀⠀⠀⣀⣀⣤⣄⣀⣀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠉⠙⠛⠓⠛⠋⠉⠉⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠙⠛⠓⠚⠛⠉⠉⠉⠉⠛⠉⠁

⠀⠀⣿⣿⣿⣿⣿⣿⣟⣏⣋⣍⣏⣋⣋⣻⣻⣿⣛⣿⣏⣿⣋⣋⣉⣿⣟⣟⣿⣿⣟⣿⣻⣿⣿⣿⣿⣿⡟⠉⠉

⠀⠈⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁
⠀⠀⠀⠀⠀⠀⠐⠠⢀⡀⡀⠀⢀⣀⡀⠤⠂⠁⠀⠀⠀⠁⠒⠀⠤⠤⠤⠀⢒⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⡹⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⣎⠦⠤⡀⠀⠚⠑⡬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣄⠀⠀⠀⠀⢀⡀
⠀⠰⠀⠤⠤⠒⢒⠯⠭⠭⡀⠀⣠⠀⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⡄⠀⢀⠁⠐⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠒⡖⠁⠀⡌⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠈⠒⠁⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠇⢀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⠉⠀⠀⣧⠀⠴⠀⡀⢠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠐⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡐⠋⠣⣤⠴⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⢎⠰⠀⠀⠀⠀⣀⣣⠜
⠀⢀⠀⠀⡀⡀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢈⡁⠉⠁⠑⠊⠀⠀⠀⠀⠀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⣿⢀⣀⠀⠀⢀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠼⠇⠀⠀⠀⠀⠉⠉⠉⠓⠚⠉⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠔⠀⠐⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⣀⡤⠊⠀⠒⢌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡆⠀⠀⣠⠤⠞⠁⠀⠉⣠⠶⠛⠀⠀⠀⠀⠀⠀⠰⠶⠤⣄⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢋⡟⠁⠀⠀⠀⠠⠞⠁⢀⣀⣀⣀⠀⠀⠀⢀⣠⠤⢤⣀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠉⠢⣠⠊⠀⠀⠀⠀⠙⢦⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠈⢆⣠⣄⠀⠀⠀⠈⡇⠀⠀⠀⠹⣤⢔⡶⢒⣒⠒⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠸⢿⠞⠋⠀⠀⠀⡰⠁⢀⣀⡀⠀⠹⣦⢻⣍⣠⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢀⡞⠑⠂⠤⠖⠋⢀⡴⠋⠀⠙⡆⠀⠹⡏⣀⠼⠂⡰⠂
⠀⠀⠀⠀⠀⢠⠒⠀⠀⠒⠒⠲⢶⡒⠒⠶⢇⠀⠀⠀⠀⠀⠐⠚⠉⠁⠈⠈⠐⠢⢤⡚⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⣰⠃⠀⠀⢿⠁⠀⠊
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡉⠛⣟⣲⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠒⠋⠁⠀⠀⠀⢸⡄⠡
⠀⠀⠀⠀⠀⠀⠁⠀⠠⣀⣀⠉⠒⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢣
⠀⠀⠀⠀⠀⠀⠀⡇⢠⠁⠀⠀⠒⠒⠂⠉⠀⠀⠀⣄⠀⠀⠀⠀⠉⠒⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠐⠀⣨⡆
⠀⠀⠀⠀⠀⠀⢀⡇⢸⠀⢠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⢤⣀⠀⠀⠈⠉⠛⠲⠶⠤⢤⣄⣀⣀⡀⠀⡤⠤⠐⢶⠉⠂⣌⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⢸⡇⠸⠤⠼⠧⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⠳⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠘⠇⠠⠤⠤⡤⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⣏⣉⣉⣭⠭⠽⠛⠻⣄⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠧⠴⢲⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠙⠲⢤⣄⡰⠃⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠶⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠻⠄⠀⠳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠉⠉⠓⢂⡶⢶⠦⠤⢤⠾⠁
⠀⠀⠀⠀⠀⠀⠀⣀⡤⣤⣤⣼⡀⠀⠀⠀⠀⢠⣼⣥⣏⠀⠀⡞
⠀⠀⠀⠀⠀⠀⡞⠁⠀⡨⠳⣀⣉⣳⡶⠦⣤⣤⡾⢤⣬⣤⠼⠷⣶⠶⠦⣄
⠀⠀⠀⠀⠀⠀⣟⣦⣤⣷⣤⣤⣶⣒⡾⠿⠛⠉⠻⢶⣾⣗⡶⠖⠱⡄⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠷⠿⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠏⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⢀⡴⠂⣤⣿⠟⠁⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⣴⠟⢡⣾⣿⠋⣠⣾⡿⠁⠀⠀⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣰⣿⡿⢣⣾⣿⠟⠀⠀⠀⠀⢸⣿⣧⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣿⣿⣿⣿⡿⡁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⣿⣿⣿⣿⡗⡰⠀⠠⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣫⣿⣿⣿⣷⣇⣴⢁⣰⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⠟⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣥⣤⣾⣿⣿⠟⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠿⠛⠿⢶⣤⣄⡀⠙⢿⣿⣿⣅⣘⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣍⣀⡀⠀⠀⠀⠈⠙⢿⣷⣮⠻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⣿⠟⠿⠷⣿⣤⡐⠺⠅⠘⠋⠀⠀⠀⣠⣾⣿⣤⣤⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣧⠀⣾⡀⠛⣀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠋⣠⣾⣿⣉⣩⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠛⠷⠾⠿⠿⠉⠉⠀⠀⠀⠀⠀⠀⢠⣿⣶⢾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠐⣿⢷⣶⣦⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈⣿⣦⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⣿⠷⣿⣿⣽⡟⠋⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⢿⣿⣷⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣝⢿⣿⠿⠿⣿⡧⠀⠀⠀⠀⢀⣤⠀⠀⠀⠁⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠉⢸⣧⣈⣛⣠⣴⣿⡄⠀⠀⠀⠀⢼⣿⠀⠺⠀⠀⣰⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠿⣦⡀⣬⠈⣿⠛⠛⢡⠉⠁⢀⣀⣤⣶⣾⣷⣿⣆⠀⣰⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣮⡀⢿⠀⠀⠀⠀⢻⣿⢻⣿⣿⣿⣿⠉⣿⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠰⣦⣙⠿⣿⣿⣿⣿⣄⠈⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣰⠀⠀⣤⡀⠟⠻⣶⣿⣷⣈⠉⠀⠻⢺⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⠻⣷⠈⣄⠈⠀⢙⣿⡷⣄⣼⠂⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣻⠀⠙⢷⣿⣷⡄⢹⡇⠱⠿⡟⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠈⠇⠀⠘⠏⢿⣷⣼⡇⠀⠹⠇⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠐⠆⠀⠀⢁⣸⣏⠿⠀⠀⣦⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠹⡄⠀⠈⢻⣇⠀⠀⠀⣿⣧⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠆⢰⣧⠀⠀⠀⢻⣷⡀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣴⣿⣿⡇⠀⢀⡿⣻⡇⣠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠏⠁⠀⢸⣷⡟⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣡⠀⠀⠀⢸⣿⡷⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⠀⠀⢠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡆⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⡷⠀⢉⣠⣽⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠠⢀⡀⠀⠀⠀⠀⠀⣀⣤⡴⢶⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣦⣤⣔⠋⣁⣽⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣁⣒⣦⣄⣴⣿⣿⣶⡟⠩⣩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢸⡇⠴⠮⣿⣬⡹⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⢀⣤⣶⣶⠿⡟⠛⠉⠉⣯⣻⣿⣯⣽⣿⣻⡰⢻⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⠃⣿⢓⡏⠀⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⢀⣴⣿⣿⡿⠓⠀⠽⠖⢁⣴⣿⣿⣿⣿⣿⣿⣿⣧⣿⣬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣥⠀⠉⠁⠀⠀⠙⢿⣿⡟⣹⣿⣿⣦⡄
⠀⠀⣾⣿⣿⡏⠦⢠⣠⡴⢶⠿⣿⣿⣿⡏⠀⠙⠛⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣿⣷⠀⠀⠀⠀⠐⣻⣶⣴⣿⡿⠋⠙⣁⣀
⠀⣼⣿⣿⣿⢠⠀⣾⣿⣆⣐⣾⡿⢿⡻⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠁⢀⣠⣴⣴⣷⠿⢹⣿⠏⠀⣠⣾⠿⠹
⠀⠸⣿⣿⣿⡌⠀⠿⢿⣿⠋⠉⠁⢏⣩⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠋⠀⠀⠈⠈⠀⣰⣿
⠀⠀⠙⣿⣿⣵⢦⣿⣼⣿⡄⠀⠀⠈⠛⠄⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⡀⠀⠀⢀⡧⠰⣽⡃⠶⠆⡰⢿⡏
⠀⠀⠀⣟⣿⣹⣾⣿⣿⣿⣷⠀⠀⠀⠀⠀⠃⠄⠀⠑⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⣤⣾⠃⠀⠀⠀⠀⠩⠘⣦⣼⣿⣰⣿⡅⠈
⠀⠀⣠⣮⣿⡏⢿⣿⣿⣿⣿⣷⣶⣷⣶⣶⣤⣤⣈⣀⠲⣥⣤⣄⣀⡀⠀⠀⠀⠀⠠⠤⣄⣀⡀⠀⠀⠀⠂⠀⣄⣐⠇⠀⢠⣄⠀⠀⢴⠒⠛⠛⠀⠑⢟⢷⡄⢻⣦
⠀⢸⣿⣿⣿⣿⣜⠻⠿⠿⣿⣿⣿⣿⣿⣟⣭⣟⣽⡟⠛⠛⠷⠤⠯⣽⡿⠃⣀⣤⣶⣶⣿⣿⠁⠀⠀⠀⠈⠙⢿⡿⠀⢘⡏⠸⣳⡦⡞⠁⠀⠀⢀⢳⡄⢻⣿⣯⣿
⠀⠀⠐⠶⣴⠲⠿⠻⠶⣻⡿⠉⠛⠛⠛⠛⠿⠿⠿⣧⣤⣤⣄⡀⠀⠀⠀⠈⠁⠀⠀⠡⠙⣿⡀⠀⠀⠀⠀⠀⠀⢠⠿⠖⠀⣮⡿⠈⠅⠀⠀⠈⠤⠆⠙⠁⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⣤⡀⠀⠀⠀⠀⠀⠘⠻⠃⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠐⠄⠀⠀⢀⡀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉







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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠣⠀⢀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⠀⢀⠄⠀⠀⠈⠁⠀⡀⠁⠐⡀⠀⠁⠀⠐⠧⠀⠈⠁⠢⣄
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠆⠀⡠⠃⠀⠀⠘⠀⠀⡆⠀⠀⠀⠁⢀⡀⠈⠢⠀⠀⠸⠀⠀⠀⠀⢀⠈
⠀⢸⡄⠀⠀⠀⠀⠣⠠⠊⠀⠘⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢡⠀⠀⠀⢰⠀⠀⠀⠱
⠀⠘⠄⠀⠰⢄⡠⠃⠀⠠⢂⠘⠀⠐⠀⠀⡀⠀⢠⠀⠀⢀⠀⢠⠄⠀⠀⡇⠸⠀⠀⠀⡌⠀⠀⠀⠠
⠀⠀⠀⡄⠀⢨⠀⠀⠀⣠⠀⠸⠀⠀⠀⠀⠉⠀⠘⡁⠀⠀⠀⠀⠀⠀⢠⡆⢀⡄⠀⠸⡀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠁⠄⢠⠜⠀⠑⠠⡄⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠨⡠⠞⠀⠄⠀⠀⠇⠀⠀⠙⢠
⠀⠀⠀⠀⠀⠀⠀⠊⠄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡎⢀⠐⠊⡆⠑⠄⡁⠀⡆⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢃⠸⠁⡆⠀⠀⠠⢁⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⠔⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⣈⡃⠀⠀⠄⠀⠂⠀⠼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠑⠊⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⡀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⢀⡀⠀⠐⠀⡇⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠆⠀⢻⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠘⠀⠀⠒⠊⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡄⣠⡀⠀⠀⠀⠀⡀⠑⠲⠴⠒⠁⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠁⠀⠁⠈⠐⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⢀⠄⠀⢄⠀⠀⠀⠀⠀⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠉⠢⢄⣀⣀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠘⡀⠀⠀⠄⠁⢠⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠀⡐⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⡄⠀⠀⠀⠀⠡⣬⠀⠸⡉⠀⡜⠀⠀⠀⠀⠀⠀⡠⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢀⡜
⠳⠄⠀⠜⠁⠐⠀⡅⠉⠉⠈⠀⠀⠀⠀⠀⡠⠊⡜⠓⠂⠤⢤⠊⠉⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⢠⡄⠠⠤⠔⠈⣠⣾⣤⣀⡀⠀⠀⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠈⢦⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠤⠤⠐⠋⠁⠙⠁⡐⠀⢹⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⡄⠀⠀⠀⠑⢤⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠀⠀⠘⢦⣀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⣀⠵⠀⠀⢀⡤⠊⣀⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠂⠤⠤⠤⠄⣀⡨⠀⢨⠁⠒⠚⡄⠱⡤⠤⠄⠒⠊⡡⠐⠂⠉⠀⠀⢀⠔⠋⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⠄⢀⠀⢠⠤⠤⠀⣠⠖⠒⠚⠒⠒⠢⢼⡄⠱⡤⣄⣄⣀⡟⠉⠉⠒⠒⠞⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢾⣀⣼⣀⡸⠀⠀⠀⡇⠀⡄⠀⠀⡀⠀⣄⠈⠂⠘⢄⠀⠸⣀⡆⠀⡄⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⣆⣀⣀⡧⠤⠇⠈⠒⠢⠚⠀⠀⠀⠈⠉⠙⠒⠚⠁
⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢸⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢭⡙⠲⣾⠀⠀⢸⡀⣧⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡘⠓⠶⣧⣦⠙⠆⢦⠈⢷⣌⠳⢾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠉⣁⠀⠀⠀⠀⠛⠂⠈⠳⢤⣈⡉⠀⠀⠙⠷⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⢀⡴⠋⠁⠠⠒⠉⠉⠉⠁⠀⣀⡤⠖⢒⣠⡤⠤⠄⠀⠀⠉⢙⡛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⣰⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠐⠒⠒⠛⠛⠙⠛⠓⠂⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⡰⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡉⠲⣄
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⡇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣆
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢻⡀
⠀⠀⠀⠀⠀⠀⡖⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢸⡇
⠀⠀⠀⠀⠀⣸⠁⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢸⡇
⠀⠀⠀⠀⠀⣿⡄⠀⢸⠁⠀⠀⠀⠂⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⠀⢀⡼⣡⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢢⡀⣀⢸⡇
⠀⠀⠀⠀⠀⣿⠃⠀⢸⡆⠀⠀⠀⢀⣤⣤⣤⣤⣤⠴⠖⠒⢾⡆⠉⠛⠋⠛⠁⠀⠀⣄⣀⣠⡴⢻⣿⣷⣦⣄⣀⠀⢹⡏⠘⡇
⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠠⠟⠀⠀⠠⠤⠤⠔⡂⠔⠈⠁⠀⠀⡀⢠⡀⠀⠀⠈⣹⠟⠓⠾⠿⣿⣧⣄⣉⣷⠘⡇⢰⠇
⠀⣠⡴⠶⠶⣿⣤⣄⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡇⠀⢠⠊⠀⠲⢤⣤⣤⣤⡄⠀⠀⠀⢸⣇⡾
⡾⠁⣀⣀⣀⠀⠈⣿⠟⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣠⡴⠂⠀⠀⠠⠋⠀⠀⠀⠸⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠟⠛⠛⠳⢦
⠇⠚⠉⠉⠙⢷⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠈⢀⣤⡶⠶⣤
⠀⡀⠀⠀⠘⠻⣿⢀⠀⠀⢀⠄⠀⠀⠀⠀⣰⠟⠁⠸⣆⠘⠿⢷⡄⠀⠀⠀⠀⠀⣀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠷⠂⠀⠀⢀
⠀⠹⣆⢀⠀⠀⢿⢸⡀⢠⡏⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⢀⣀⠚⠋⠉⠉⠘⣿⠀⠀⠀⠀⠀⢸⠀⢀⡼⠀⠀⣀⠔⠁
⠀⠀⠈⠻⣆⠀⠼⢸⡇⣾⠀⠀⠀⠐⠁⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠀⡇⠀⠋⠀⠀⠉
⠀⠀⠀⠀⢹⡄⠀⠈⣿⡇⠀⠀⠀⠀⠀⢀⡴⠟⠛⠹⣄⣴⠉⠳⣄⡙⠒⠤⠤⡀⠀⠀⠀⠀⡟⠀⠀⠀⢸⣾⠃⢀⡤
⠀⠀⠀⠀⠀⠙⠶⠶⢻⡇⠀⠀⠀⣠⠞⠁⠀⠀⣀⣤⡴⠿⣄⠀⠸⡟⠒⠤⣤⣤⣀⠀⠀⢸⠇⠀⠀⠀⣾⡟⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢠⡞⠁⠀⠀⡴⠛⠉⠀⠀⠀⠀⠉⠀⠃⠀⠞⠙⢿⣿⡆⢀⡟⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡿⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⢴⣻⠖⡼⠁⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠁⠀⠀⠈⠁⢀⡼⠁⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠞⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣤⣤⣤⣤⠶⠛⠉
⣿⣿⣿⣿⣷⡾⢿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣾⣿⣿⣿⣷⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣾⣿⡿⡿⡿⠿⠿⣿⣷⠿⠿⣿⠿⢿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⣻⠿⠷⠝⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣛⣭⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣿⣿⣿⠿⠟⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⣀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣛⣷⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠙⣱⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆
⠀⠀⠘⡏⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢋⢸⡇
⠀⠘⢿⠛⢹⠀⠉⢓⠲⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⠙⢿⠢⣄⠰⡿⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠋⢁⠔⠁⣼
⠀⠀⠈⢧⠈⣇⠀⠀⠈⠁⠒⠢⠬⢍⣛⠲⠤⣄⣠⣤⣈⣧⠈⠙⢧⠀⠉⠙⠛⠛⠛⠛⠛⠶⠶⢦⣤⡀⠀⠀⣠⡤⠶⠚⠋⠉⡀⠤⠐⠊⠀⠀⠀⣿
⠀⠀⠀⠈⣇⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢬⡙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠲⠀⠉⠐⢋⠁⠀⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠸⡆⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠹⡄⠣⡀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠙⢆⠑⢄⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢀⠃⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠑⢄⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠊⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡑⠦⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⢠⠎⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⠚⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠰⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠛⠛⠻⠿⠿⠿⠿⠿⠀⠛⠃⠀⠀⠀⠀⠀⠛⠛⠻⠿⠿⠿⠿⠿⠀⠛⠃⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⡀
⠀⠀⠀⠀⠀⢤⡴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠙⠓⠦⢤⣀⡀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠛⠿⠿⠟⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⣀⣀⣉⣳⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠴⢾⣭⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠠⠄⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠄⣀⣀⣀⠀⠀⠀⠀⠀⢀⣙⣻⣭⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠦⣄⡀⠀⠀⠐⠂⠉⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⠀⠀⠈⠉⠉⠒⢒⡿⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⡠⠔⠊⢕⠒⠢⠤⣀⣀⠀⠀⣀⡀⠤⠴⠂⠀⠉⠓⠦⣀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣔⠉⠀⠀⠀⠀⠉⠉⠉⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⣤⣀⣀⣀⣀⣠⡤⠴⠖⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣤⡤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣯⡍⠉⠉⠉⠙⠂⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⣴⣮⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⣨⠟⠁⠀⣠⠖⢻⠉⠉⠙⠒⠲⠤⠤⠤⠤⢤⡀⠀⣼
⠀⠀⠀⢀⣾⣵⠆⠀⡞⠁⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡴⠃
⠀⠀⠀⢘⡝⠁⠀⠰⡇⠀⠈⣠⡶⠶⠶⠴⡄⠀⠀⢠⣤⣤⠀⣇
⠀⠀⢠⣾⣴⠂⠀⢀⡷⠀⠀⠋⡠⠔⠢⢄⠀⠀⠀⣈⠤⣈⠓⡿
⠀⠀⠘⣱⡇⢀⡀⢸⠀⠀⠀⡞⠀⠀⠀⠀⢣⢀⡞⠀⠀⠈⢷⡇
⠀⠀⡞⡡⠒⠂⣼⡎⣇⠀⠀⣇⠈⣀⡤⢀⡼⢸⣀⡤⣈⠁⡼⠓⠲⢤⣀⡤⠤⢄⡀
⠀⠰⡏⢷⠁⣎⠉⠓⠛⠀⠀⠈⠙⠛⠒⠊⠀⠹⡀⠀⢢⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠦⣄
⠀⠀⢳⣄⣈⣃⠀⠐⣫⠔⣒⣢⠀⠀⠀⡀⠀⠀⡇⢀⣮⠤⢤⡀⠀⢠⠇⠀⠀⠀⢀⠀⠀⠀⠘⠦⠤⢤⡀
⠀⠀⢸⣤⠀⣸⠀⠀⢠⡾⠓⣌⠑⡄⠸⣀⡀⢠⠃⡞⠓⠒⢤⣸⢀⠎⠀⠀⠀⢠⠋⠀⠀⠀⡴⠀⠀⠀⣧
⠀⠀⠈⠹⡄⡏⠀⠀⣾⣤⢼⠋⠉⢻⣦⣤⠬⣭⡤⢷⣤⣀⣀⣥⡟⠀⠀⠀⠀⡏⠀⠀⠀⡸⠁⠀⢀⠀⡟
⠀⠀⠀⠀⢷⡇⠀⠀⢸⠀⠀⠓⠶⢯⡀⢈⣟⠉⠈⣯⠃⠀⠀⡼⠀⠀⠀⠀⡸⠀⠀⠀⢰⠃⠀⢀⣏⣰⠇
⠀⠀⠀⠀⠀⠹⡄⠀⠈⢧⡖⠢⣄⣠⢍⠉⢉⠛⠛⡏⠀⠀⠀⣧⡀⠀⠀⣰⠃⠀⠀⢠⠏⠀⢠⠟⢩⠏
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠑⠶⣿⣄⣠⢿⣉⣿⠷⠳⡄⠀⠀⠈⠛⢍⡉⠉⠒⠒⠲⣅⣀⡴⠃⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠈⠉⠉⠉⢁⣠⠔⠛⢦⠀⠀⠀⠀⣹⢤⣀⠀⠀⠀⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠶⠒⠚⠉⠀⠀⠀⠈⠳⣄⣀⣠⠃⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣽⠊⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡿⠫⠂⠀⠀⣀⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣃⡾⣥⠶⢁⠇⠀⠻⠿⣧⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⢋⣹⣿⣏⣋⣴⡀⢀⣀⣦⣤⢻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⣅⣾⣷⣾⣽⡛⢿⣿⣿⡋⠍⠙⣾⡿⡀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⡟⠁⠙⢏⠉⢿⣿⣿⣏⣽⣿⡞⣿⣿⣿⣏⣐⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣦⣿⣿⣿⣇⠴⠂⢸⠀⡽⣿⣿⣿⣿⣄⣴⣿⣿⣿⣷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⢿⣿⣿⣿⣿⡀⠄⠘⢷⣅⣿⣿⠿⡿⣛⠿⣿⡿⡿⠋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣷⡼⢻⣿⣿⣿⣷⠴⡀⠈⢯⣿⣇⡀⢉⣶⣶⣿⣿⠏⡀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣿⠆⠈⣿⣿⡿⣿⠀⠘⠂⠈⣿⡍⣁⣾⣻⣿⣿⣿⡟⠸⣿⡅
⠀⠀⣤⡄⠀⠀⠀⣿⠋⣸⡏⠈⣴⡎⠀⣿⣿⣿⢆⡄⠀⠀⣿⣿⣿⠿⣿⣿⣿⡏⠀⠀⠁⢰⡀
⠀⠀⣻⣥⡆⣴⡺⠁⠀⢻⣇⠀⡞⢳⣶⣿⣿⣿⣼⣧⣄⠀⡘⣧⣿⣷⣴⣿⡏⠀⠀⢰⠂⠀⠁
⠀⠀⠁⠘⢹⡿⠃⠀⠀⠘⣿⣦⣿⣾⣿⣻⣿⣿⣯⡿⠙⠀⠀⠈⢻⣿⣼⠃⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⠷⣿⣿⡟⣿⣟⢠⠀⠀⠀⠀⠀⢀⣭⡀⠀⠈⠁⠂⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠿⢿⠗⣻⡹⠀⠘⠋⢿⡃⠄⠀⠀⠀⠸⣟⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⡀⣼⣿⣯⠁⠀⠀⠁⠀⠘⢖⣄⠀⠀⠀⣠⡄⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⣿⣻⡏⠀⠀⠀⠀⠀⠀⠀⣪⡀⠀⠀⠀⠈⢦⡀⠀⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠛⠛⠀⠀⠀⠀⠀⠀⢸⣿⣟⣠⠖⠆⠀⠈⣷⣾⡿⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢠⠃⠀⣆⢀⡀⣤⡤⠴⠒⠚⠟⠿⣿⣶⣿⣦⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠆⠻⡟⠘⠁⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠄⢄⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠉⠀⠀⠀⠀⠈⠓⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⣀⠠⢤⣀⡀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠴⠊⠁⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢲⠒⠄⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠐⢄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠀⠀⠀⠉⠁⠀⠲⢄⢀⠞
⠀⠀⠀⠀⠀⠀⠀⡈⢀⠴⠀⠀⠡⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢣⠀⠀⠈⣀⣀⡀⠼⡀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠔⢪⡁⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡠⢄⠁⠀⠀⠀⠀⠀⠀⠘⠁⠂⢢⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠈⣤⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⣄⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢸⡀
⠀⡐⠂⠀⠂⢴⠂⠉⠀⠀⠀⠘⠿⠋⠀⢸⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⣀⡇
⠀⡇⠰⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⡀⠀⠀⠀⠀⠀⠀⠤⠴⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⡐⠁
⠀⠙⠒⢳⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⣰⡿⠋⡅⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠈⠓⠲⠤⠤⠶⠒⠻⣁⡠⠊⠉⠳⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠂⠀⡐⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⣧⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⢿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠥⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠀⠀⠀⡜⣸⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⢀⡴⠋⣠⠀⠀⠀⡜⢀⠇⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠞⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣠⠴⢋⠔⠊⠀⠀⠀⡘⡠⠎⠀⠀⠀⡴⠁
⠀⠀⠀⢀⢀⣀⣀⣀⠠⠄⠒⠂⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡉⠑⢄⡀⠀⠀⠀⠀⠘⡅⠀⠀⠙⠢⠴⣅⠴⡴⠋⠀⠀⠀⠀⢸⠁
⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠓⠚⠉⠉⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠸⢆⡠⠀⠀⡀⣠⠏
⠀⠀⠸⢅⣀⣤⠤⣤⠤⠤⠠⠤⠔⠒⠒⠚⠛⠉⢉⠭⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⣀⡀⠀⢀⡀⣀⡤⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⡄⢤⠄⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⠸⠇⠳⠁⠘⠀⠑⠥⠆⠚⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⠙⠀⣁⣀⣀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠀⠴⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠃⠀⠀⠀⠀⠀⠉⠉⢦⠀⢀⠴⠋⠁⠀⠀⠀⠀⠀⠹⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⠉⠑⣀
⠀⠀⠀⠀⠀⠀⠒⠁⠀⡀⠀⠀⠀⠀⠂⠀⠀⠐⠎⠙⠂⠈⢃⠈⠁⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠢
⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠉⠀⠀⠀⠀⠤⠀⢄⡈⣶⠤⢿⣀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠠⡗⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠞⠋⠀⠀⠙⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠨⣇
⠀⠀⠀⠀⠀⢈⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⠀⠃⢻⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒
⠀⠀⠀⠀⠀⠐⣂⠀⠀⠀⠀⠀⠀⠀⠀⠠⢞⣿⡷⠫⠽⠶⢮⣝⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡂
⠀⠀⠀⠀⠀⠀⢺⡀⠀⠀⠀⠀⠀⠀⠠⠚⠉⢰⠀⠒⣉⣙⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⡰⠅
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⢈⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠎
⠀⠀⠀⠀⠀⠀⠀⠈⢒⡀⠀⠀⠀⠀⠄⠀⠋⠀⠘⠄⡀⠈⡀⠈⠑⢀⡀⠀⠀⠀⠀⠀⣔⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠈⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠⣦⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣎⠈⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠻⢱⡤⢠⠆⠀⢀⠀⡀⣀⠀⠀⡄⢰⢿⠀⠣⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠞⠷⠓⠅⠇⠀⠃⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒⠒⠒⠒⠒⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠐⠚⠋⠁⠀⠒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣤⠠⠄⢀⡈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⢀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣰⠃⡖⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢱⢀⠞⠁⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠱⠒⠉⢱⡨⠀⠀⠀⠀⣠⡀⠀⠘⡆⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠰⠄⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠄⠀⠀⠀⠀⠀⡀⠀⠠⠄⠀⠠⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠁⠂⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡄⠀⢀⠀⠂⠀⠤⢀⣀⡄⢄⣸⠛⣯⡲⢂
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠔⡁⠀⠉⠱⠇⠀⢀⣀⠈⠀⠀⠈⠓⠈⠁⠁⡀⠈⣀⠀⠈⡀⠠⠠⠀⡄
⠀⠀⠀⠀⠀⢸⠀⢠⡰⠀⠀⠀⠂⠄⠀⠀⠀⠀⠘⠈⠁⠀⠀⠀⠀⠀⠀⠘⠀⠙⢠⡄⠁⠀⠀⣀⠈
⠀⠀⠀⠀⠀⠀⠱⡌⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠔⠀⠀⠢⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠂⢀⠒⠁⠀⠀⠀⠀⢠⠀⠀⢀⠀⢤⢀⢲⠀⠠⠀⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠀⠁⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠰⢸⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⠀⠀⣀⡀⢙⢚⢀⣀⢘⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⢙⢛⡟⠛⠛⠿⣛⠛⠻⢿⣉⣐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⣿⣆⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⢶⡶⢾⣿⣶⣶⣴⡴⣢⣤⡂
⣿⣿⣷⡄⣀⡄⠀⢻⣿⣄⠀⠀⢂⠀⠀⠀⡀⢠⣦⣤⣤⣼⣴⠀⠖⠂⠀⣠⡯⢻⠀⣠⣶⣿⣿⣯⢿⣻⣿⣿⡷
⣿⣿⣿⣿⣿⡇⢀⠉⢹⡿⣷⡀⠳⢆⠰⠀⢠⠈⡌⣀⢽⣿⢧⣞⢀⢖⣼⣿⣷⣮⣾⣿⣾⢟⣿⣿⣶⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣷⣿⣷⡄⢡⣷⣿⣤⣀⣧⠀⢸⠀⢶⠛⠦⡭⠤⠟⢹⡟⠋⠁⣀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⢛⣛⠻⣿⠸⢻⣮⣧⣸⣆⣹⣀⣀⣇⣄⣴⣿⣇⠹⢀⣙⡃⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠀⣜⠓⢃⣿⠠⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⠟⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠀⠀⠨⠁⣿⣿⣧⣷⣿⣿⠛⢽⡇⠀⠀⠹⣿⣿⣿⡟⠉⠀⠙⠤⠀⠀⠀⠢⢄⡀⠙⢿⣿⣿⣿
⣿⣿⣿⡟⢋⠙⠀⠀⠀⠀⢈⠉⣉⡉⠙⠿⠄⣻⡧⠀⠀⠀⢫⣿⡟⠁⠀⡀⠀⠀⠀⣤⡀⠀⠀⠀⠀⣦⠙⣿⣿
⣿⣿⡏⠀⠛⠦⡀⠀⠀⠀⢘⣷⡜⣿⣸⣿⣿⡟⡟⢠⢀⡼⠞⠋⠀⣠⣾⡇⠀⠀⠀⠐⠻⠀⠀⠀⡔⠋⠀⣿⣿
⣿⣿⣷⢀⡄⣀⠀⠀⠀⠀⠀⢿⣿⣿⣇⡟⠛⣆⣥⠄⢀⠀⣠⣤⢸⡅⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⢶⡆⣿⣿
⣿⣿⣿⢸⡇⣹⡇⢄⢀⠀⠀⠘⣿⣿⣿⣷⡿⠿⠏⠞⢸⣽⣿⣿⣄⣇⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⣿⢻⣿
⣿⣿⡏⢸⣧⣿⠃⠈⠈⠀⠀⢰⣌⢿⣿⣿⡯⠭⠭⠭⠜⣿⣿⣿⣿⢱⣿⡟⢰⠀⠀⢀⠀⠀⢀⣺⣿⡄⣿⣿⣿
⣿⣿⡇⢸⣿⡟⠀⡤⢀⠀⢐⢄⠙⢿⣿⣿⣶⡶⠶⠶⠶⠿⢿⣿⣿⣱⡿⢀⠞⠀⠀⠀⢀⠤⣄⢻⣿⡇⣿⣿⣿
⣿⣿⣇⢸⣿⣧⠀⣳⢸⠀⠈⢧⠱⠶⣿⡏⠛⠛⠒⠒⠒⢒⣂⣙⡿⢛⡕⢉⠠⠤⡄⡀⠃⠀⠈⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣦⣻⡿⠀⣿⠈⠣⠀⠸⡄⠀⢹⣿⣉⣉⣉⣉⠛⠛⢩⡿⠇⠈⠀⠁⠀⠈⠐⠰⠀⠀⢴⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠃⠀⡇⠀⠀⠀⠀⠙⢦⡀⠙⠻⠿⢿⣿⣾⠎⠀⠀⠀⠀⠀⠁⠀⠀⠐⠄⢁⣴⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⡄⠀⢳⠀⠀⠀⠂⠀⠀⠉⠓⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠟⢉⣴⣿⣿⣶⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⢀⡴⠊⠀⢨⠛⣿⢿⣿⣟⠀⠙⣿⣿⣿⣿
⠠⠈⠀⠈⢛⡻⠋⠢⣄⠀⠙⠢⣄⠀⠀⠀⠀⠀⠻⢦⣄⣀⠀⠀⣰⠛⠀⠀⠀⠀⢇⠘⢦⡙⠿⣥⣀⠘⢿⡟⠛
⠀⠠⠔⠚⠉⠀⠰⠃⠈⠳⠀⠑⠬⡛⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠁⡌⠣⠄⠙⠶⢈⡙⠳⢦⡀
⠀⠀⠂⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠙⠖⠔⠀⠀⠄⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣬⣍⣂