Kidney Emojis & Symbols

⠀⠀⠀⠀⠀⠀⣠⣴⠶⠒⠒⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀
⠀⠀⠀⠀⢠⣾⡿⠁⠀⠀⠀⠐⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⠀⠐⠻⣿⡆⠀⢀⣀⡀⠀⠀⠀⠀⢠⣿⣿⣿⣿⢷⣝⢻⣿⣷⡀
⠀⠀⠀⣾⣿⡋⠁⠀⠀⠀⠀⢲⣿⣿⡇⠀⣿⡏⡇⣶⣿⠀⠀⠘⣿⣿⣆⠁⢰⣿⣿⠛⣟⣿⡀
⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⣹⣿⣿⣥⣤⣿⡇⣧⣿⣿⣤⣤⣤⣬⣿⣿⡆⠀⣩⣽⣇⢹⣿⣧
⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⡏⠉⠀⣿⡇⢋⠉⠁⠀⠀⠉⠻⢿⣿⡗⠀⠙⠛⣿⠉⣿⣿⡄
⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⢸⣿⡿⡿⠛⠉⣿⡇⡟⣿⣇⠀⡄⠁⠀⣠⡟⠁⠀⠰⢿⣿⠶⣿⣿⡇
⠀⠀⠀⢻⣿⣿⡄⠀⠀⠀⢘⣿⠃⡇⠀⠀⣿⡇⡇⣿⡏⠀⣿⠀⠀⣿⢠⣶⡆⠀⣤⣿⣀⣻⣿⠇
⠀⠀⠀⠀⢻⣿⣷⣀⠀⠀⣼⣿⠀⠇⠀⠀⣿⡇⡇⣿⡇⠀⠀⠀⠀⠙⣾⣿⣧⣤⣽⠋⣽⣿⡿
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠏⠈⡄⠀⠀⠀⣿⡇⠃⣿⡇⠀⠀⠀⠀⠀⠙⢿⣯⣭⣵⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⡅⠀⠀⠀⣿⡇⡇⠙⠛⠉⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠻⠃⠃
⣿⡿⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠜⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡌⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠆⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠀⠀⠉⠢⡀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⣶⣷⣤⣤⢀⠇⠀⠀⣠⠜⠁⠀⠀⠀⠀⠀⠘⡄⢀⠊⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⣸⣿⡧⡀⠙⡄⢰⠒⢊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠦⠤⠔⣿⣿⡷⠁⠀⠀⠁⠀⠀⡀⠀⠀⠀⣀⣠⡤⠶⡚⠁⠀⠠⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⢿⣿⡇⠀⠀⠀⠀⠀⠀⢲⠁⠃⣈⣠⡴⢚⣿⣯⣏⣶⢡⠻⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢣⣿⣿⣗⠀⠀⠀⠀⠀⠀⠈⠀⠘⢹⡿⠻⠛⣛⣹⡿⠚⢰⣇⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢋⣾⡿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⢿⣛⣛⡓⠁⠩⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣿⡿⠃⠉⡆⠀⠁⢀⣀⣤⣴⣤⣤⣴⣶⣿⣿⣋⣵⣗⡙⣷⣠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⣀⢀⣭⣴⣿⠋⣥⠿⣁⠠⡉⢿⣿⣿⣿⡿⠿⠙⣉⣹⠏⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢡⠎⣡⣾⠋⠊⠁⠊⢀⣄⣘⢦⠀⠀⠻⣿⣿⡉⠉⠛⠁⣀⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢹⣿⢁⣾⡿⠁⠈⢿⣦⡤⣅⣥⡋⠈⠆⠙⣀⣹⣿⣣⡤⠾⠛⡋⢗⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢻⣿⢸⣿⢷⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⣰⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⣼⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠐⣸⣫⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⣼⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡇⠸⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠇⠀⠘⣿⣧⡆⡀⠀⠀⠀⠀⠤⠐⠀⠀⠀⠄⣰⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢶⠀⠀⠈⠛⢿⣵⣄⣀⡀⠀⣀⣀⣠⣦⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠀⠉⠉⠉⠉⠉⠉⠉⠀⠒⠂⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠛⠋⠁⠃⠋⠙⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠛⠒⠗⠺⠃⠖⠺⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠁⠀⡆⠀⠀⠀⠾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠇⠀⠀⠀⠀⠉⠐⠂⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⡈⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⠀⡐⠁⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⢀⡐⠊⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⢨⠐⠀⢀⠖⠀⠀⢠⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⠲⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠐⠒⠂⠉⠀⠀⠀⠀⢀⡀⠄⠂⠉⠀⠀⠀⠀⠀⠀⠀⢠⠂⢀⡀⠔⠈⠀⠀⠤⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠤⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⣀⣀⣑⣄⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⠀⠀⠀⠀⠒⠂⠠⠄⡀⠀⠲⡀⣩⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣈⡲⢀⠌⠀⠀⠀⠀⢰⡦⢶⠸⠦⠲⢰⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⢰⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢠⠀⣤⠲⣤⢠⡄
⠀⠀⠀⠀⠀⠀⠘⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⢀⠔⠂⠈⠉⠉⠉⠉⠑⠢⡀⠀⠀⠀⠀⠀⠉⠀⠒⣾⠡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⡰⠁⠀⡔⠊⠁⠀⠀⠀⠀⠀⠈⠄⠀⠈⡏⠁⠒⢄⠀⠀⠀⠀⠈⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠰⠀⠀⠀⠑⢄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠃⡠⣄⣤⣤⡄⣴⢠⣤⠀⡈⠀⢀⠀⢣⠀⠀⠀⠀⠃⣤⠀⠀⠀⠀⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠸⠀⠉⠀⠀⠹⣄⠀⠀⠀⡰⠀⠀⢰⠇⠀⠑⢄⠀⠀⠀⠛⠐⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠘⠃⠘⠁⠠⡀⡌⠰⠀⠀⠀⠑⣄⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣦⠊⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢦⡴⠆⠶⠴⠰⠆⠦⠇⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁
⣄⣤⢤⣤⢆⣤⢤⠀⠀⢠⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣤⠶⠀⠙⠉⠉⠁⠦⣄⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠂⠀⠒⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢀⡠⣤⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡠⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⠒⠛⠒⠿⠀⠨⠶⢶⣦⣄⡤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢠⣀⡤⠾⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠢⢍⡛⠛⢧⡀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⢀⣀⣴⠶⠛⠉⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⢻⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⣀⡼⠋⠁⠀⠀⠀⠀⠀⠀⠀⠠⠘⣨⠃⠀⠀⢠⡟⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⣤⡇⠀⠈⢱⣟⡀⠀⢻⣤⣄⣀⣀⡀⠆⠀⠀⠀⡌⠷⠀⢀⡍
⠀⠀⠀⠀⠀⠀⠀⡄⢀⡰⠟⠁⠀⠀⠀⠀⠀⠀⠷⡀⠀⠀⠸⣯⡿⣇⠀⠀⠘⠋⠉⠀⠉⠁⢿⣿⣄⡀⠀⠀⠀⠀⠸⠆⣀⡼
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢈⣐⠍⠀⠀⠀⠀⠀⠀⠀⠀⠈⣬⣭⠇⠀⠀⠀⠀⠈⡀⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⢀⡄⠀⠀⢱⣦⡀⠃⠠⡄⠀⠘⢿⡄⠀⠀⠀⠀⠀⠀⢀⡐⠚⣥⡴⢦⠴⠃⠤⠀⠀⡧⠀⠀⠀⠸⡛
⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⢀⠆⠁⠀⠀⠈⠁⠰⣶⣄⠀⠀⡀⢻⣆⡀⢠⣾⣶⡆⠈⠛⠛⠃⠀⠀⣤⠀⠀⠀⠀⣷
⢀⡀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⣾⣛⠷⡄⠀⠀⠀⠀⠈⠻⣷⣶⣤⣴⠌⠛⣿⣿⠟⡘⣧⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠊⠉⠀⠈⡏⠁⠈⠀⠀⠀⠀⠀⠀⠴⠂⢠⣤⣄⡀⠀⠀⠀⠙⠀⠀⠀⢠⣴⣿⣿⠻⠷⣾⠂⠘⠁⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⡄⠀⠀⠀⢈⡍⠛⠛⠻⠷⠶⠶⠂⠀⣠⢾⣿⣿⢻⣶⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⢰⠀⠀⠀⠀⠒⡄⠒⠁⠈⠂⠀⢤⣤⣀⣀⣀⣀⡤⠄⠀⠀⠀⠹⡠⢏⠙⠀⢻⣷⣾⣿⡀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠈⠀⠀⠀⢠⣄⠀⢤⡄⠀⠀⠀⠀⠀⠉⠉⠛⠛⣇⠀⠀⠀⠀⠀⠿⣿⡆⠚⢧⣝⠿⢿⣿⣆⡀⠀⠀⠀⣠⠞
⠀⠀⡆⠀⠀⠀⠀⠉⠠⠂⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠋⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠙⠳⠦⣄⣈⣿⣷⣶⡞⠋⠀⢠⠤
⠀⠀⡇⠀⠀⠀⠀⠰⡷⠆⠀⣤⡤⠀⠀⠾⠛⠋⢉⡙⢳⡆⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠉⠉⠛⠻⠷⣀⠀⠘⠦⠄
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⣀⣀⣀⣀⣸⠃⠀⠀⠀⠠⣷⠈⣿⣿⢸⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠠⢾⣶⣶⣆⠀⠋⠀⠛⠛⠛⠋⠋⠉⠀⠀⠀⡀⠀⠀⢀⣽⡿⠈⠉⠀⠀⠀⢀⣀⠀⢀⣦⣤⡀⠀⠀⠙⢦⡀
⠀⠀⡀⠀⠀⠀⠠⢖⡳⠆⠈⠙⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⣤⡾⠁⠀⠀⣾⡿⠁⠀⢀⣠⣤⡾⣿⣿⣷⠊⠀⠀⠈⠓⢄⠀⠀⠉⢆⡀
⠀⠀⡇⠀⠀⠀⠀⠀⣤⠂⣴⡆⠀⠤⠀⠀⠀⠀⢤⣶⣾⣿⣿⡖⠃⠀⢀⣿⠁⠀⣠⣾⠟⠁⠀⣹⡾⠁⠀⠀⠀⠀⠀⠀⠱⡀⠀⠈⢷⡀
⠀⠀⠇⠀⠀⠀⠀⠀⡏⠈⢶⣿⠛⠀⠀⠀⢀⣴⡇⣠⣿⠿⠿⢷⣦⠀⣾⡇⠐⠓⠉⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⢧
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⢿⡖⡀⠀⠚⠋⠀⠀⠀⠀⢀⣤⣠⣿⣿⣿⣧⢠⣶⣦⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠈⡀
⠀⠀⠈⡆⠀⠀⠀⠀⠀⠉⠀⢣⣾⣿⠛⡄⠀⠀⠀⠀⠀⠀⢸⣿⠟⠁⠈⠙⢷⣎⡛⠲⠦⠸⠟⠀⣀⣴⣦⣄⠤⠄⠀⠀⠀⢸⣇⠀⠀⠀⠁
⠀⠀⠀⠐⡄⠀⠀⠀⠉⣿⠟⢋⣿⠃⠇⢀⠄⠀⠀⠀⣀⠘⠀⠁⠐⠀⠀⠀⠀⠈⠟⠛⣀⣤⣴⣾⣿⣏⡿⠉⠅⠀⠀⠀⠀⢸⡿⡀
⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⡸⠟⢀⡴⠁⠀⠀⠀⠤⣙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠛⠛⣉⡁⠀⠀⠀⠀⠀⠀⣼⠀⢷⡀⠀⠐
⠀⢠⣤⠀⠀⠆⠀⠀⠀⠀⠀⡀⠀⢸⣟⣀⡤⣰⣤⡾⠛⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢽⠀⠻⠋⠹⠂⠄⠀⠀⠀⠀⡟⠀⠸⡇
⠀⠘⠺⠀⠒⠚⢃⠂⠀⠀⠘⠿⡋⠘⡭⠟⠣⣽⠏⠀⠄⡀⠀⢹⡗⢀⠀⠀⠀⠀⠄⠢⠌⠳⠄⠀⣠⡄⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠁⢀⣰⡏⠀⠀⡀⠀⠀⠸⡇⠘⣶⠆⠀⠀⠈⠌⠳⢄⠀⡀⠉⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠈⠩⢄⡈⠀⢁⠀⠠⠀⢻⡄⠀⠚⣠⣦⠀⠀⢳⠈⠆⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠾⠡⠠⣦⠆⡀⠀⠃⢰⡄⠀⠹⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⢠⡤⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠞⠁⠀⠀⠀⠀⠸⠧⠀⠀⠒⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠞⠛⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣬⣽⣿⣿⣿⣷⣿⣿⣿⣷⣯⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡟⠉⠙⠉⠙⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⢻⣿⣿⣿⡟⢹⣿⣿⡿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣏⠉⠁⠀⠘⣿⢩⠛⠀⠘⠁⠀⠀⠉⠴⢿⣿⣷⣤⠀⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠷⠝⠃⠐⣄⡀⣿⡀⠁⠀⢦⠀⠀⠀⠀⢘⣿⡟⠛⢟⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡐⣀⡀⠐⢨⣥⣽⣗⠀⠀⠀⠂⠀⣠⣴⠞⡉⠈⠁⠾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠿⣵⡷⣢⣿⣿⣿⣿⣀⠟⣀⣴⣾⡿⣯⣈⣠⠊⠀⠀⠉⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡦⠀⠈⣽⣿⣿⡿⢷⣽⣿⣿⣿⣿⣿⣿⣄⠤⢀⡀⠀⠀⡀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠈⣿⣿⠗⣠⣤⣿⣿⠏⣵⢄⡻⡟⢻⣿⣿⣿⢿⣋⠀⠠⡀⠀⠀⠀⢀⣩⣿⡇
⠀⠀⠀⠀⡀⠀⠁⠀⠸⢿⣶⣿⣵⡿⠿⢃⡼⠟⠋⡤⣿⡄⢹⠈⣵⣿⣿⣷⣄⣀⠀⠀⢀⡠⠾⠿⣷
⠀⠀⠀⣀⡷⣤⣀⣄⠀⠀⠀⠀⠀⣴⣤⣶⣦⣾⣿⣷⣿⠇⠀⣰⣿⣿⣿⣿⣿⡿⡗⠈⠀⡀⣰⣿⣿⡄
⠀⠀⣴⠛⠾⠿⠿⣿⣿⣿⣷⣷⣾⡅⠺⢋⣉⣴⣿⡿⠃⠀⠀⣿⣝⣿⣿⠟⡁⠀⠐⠀⠀⠀⠀⢚⣿⣧
⠀⠀⢻⣾⣶⣴⣶⣤⣄⣀⣩⡙⢯⡾⣿⡾⢻⣿⠏⠁⠀⢰⣷⣽⡟⠍⠀⠀⠀⠀⠀⠀⠀⠈⠁⠌⢿⣿
⠀⠀⠀⠉⠉⠛⠛⠻⢍⣉⠛⠻⣟⡿⠋⢰⣿⠋⠀⠀⢀⣉⣿⣿⣄⡛⠁⠙⠒⠾⠉⣽⠆⠈⠀⠈⣻⣿⡇
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠻⠒⠉⣀⣴⣿⠃⠀⠀⡀⣀⣽⣿⣿⣾⣿⣶⣏⠀⣝⠻⠷⣦⢤⣀⣤⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠔⢉⣠⣾⣿⡿⠟⢱⠂⠈⣸⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣄⡀⠀⠀⡰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⡜⢡⣴⣾⣿⡟⠋⠀⠀⠀⠀⢀⢿⣿⣮⣿⠿⠿⠿⣿⡿⠿⠛⠿⠿⠿⠿⠶⠤⢼⣿⣿⠀⠒
⠀⠀⠀⠀⢀⠎⠀⢿⣿⣿⣿⣿⣷⡀⠀⠀⣠⣶⣿⣿⣿⣿⡿⣹⠟⡡⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣿⡏
⠀⠀⠀⠀⡼⠀⠀⢰⣿⣿⣿⣿⣿⣿⣄⠀⣿⣿⣿⣿⣿⣿⣷⣿⠦⠁⠀⠀⠀⠀⠀⠀⠀⠀⣒⣿⡿⠁
⠀⠀⠀⠀⠁⠀⢰⣿⣿⣿⣿⣿⣿⡍⣿⣷⣌⠻⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢽⡿⠁
⠀⠀⠀⠐⠀⠀⣼⡏⣿⣟⠛⠉⠋⠓⢛⡻⢻⣶⣿⣿⠜⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⢬⣿⡿⠁
⠀⠀⠀⠸⠀⠀⣿⠇⢸⣿⡄⠀⢠⡀⠋⣀⢭⣋⣻⠇⠀⠈⠹⠯⠍⠙⠛⠿⣿⣶⣶⣷⡶⠏
⠀⠀⠀⠸⠀⠀⣿⠀⠀⢻⣿⡉⡶⠋⢀⠉⠈⣽⡇⠀⠆⠀⠀⠀⠀⠀⠀⠁⢰⣶⣽⡟⠁
⠀⠀⠀⠀⠄⠀⠃⠀⠀⠀⢻⣷⢁⠀⡀⠀⢸⡿⠁⡄⠀⠀⠀⠀⠈⠋⣄⡠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⣿⣄⣾⡇⢸⡇⠀⢀⠀⣀⠀⣔⣿⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⡇⣤⣸⣿⣿⣿⣿⠟⠁⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⡿⢿⡿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠿⠛⠛⢿⣿⣿⣿⣿⣮⡼⣟⣷⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣻⣿⡿⠋⠀⠀⠀⠀⠀⢹⡟⠉⠉⠉⠙⠿⢷⡘⢟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣯⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠈⣵⣿⡆⢴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢽⣿⠿⠁⠀⠀⠀⠀⠀⡀⣀⡒⠸⠇⢀⣀⡀⢀⡀⠘⠛⠛⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣀⡿⠃⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣧⠘⠛⠋⠀⠀⠀⠀⠀⠀⠘⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣼⠃⠀⠀⠀⣀⣴⣾⣷⣾⣿⠁⢹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⠀⢠⣾⡞⢻⣿⡏⠈⠻⢦⣾⠃⢀⣶⣶⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⢻⣿⣿⣋⠿⠋⠹⣧⠀⠀⠀⠀⠛⢿⡿⣿⣦⣄⠰⠶⠖⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⡷⣼⡇⢠⡏⠀⠀⠀⠀⠀⠐⢛⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⠋⣠⡿⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠋⠉⠀⠀⠀⠀⠀⠀⢀⣾⡋⢀⣽⡿⠀⠄⡄⠀⠀⠀⠀⠀⡈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠙⣿⣿⠿⠛⠁⠈⠁⠀⠀⠀⠀⠀⠀⠒⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⡠⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣿⣿⣦⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⣠⠞⠀⣠⣾⣶⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠈⠙⡿⠛⢿⣿⡇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⣼⣿⣿⣿⣿⣿⣻⣿⠃⠀⣶⣄⠀⠀⠀⢘⣿⠀⠀⠉⠀⠉⠄⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣹⣿⣿⠏⠉⢳⣿⢿⣿⣿⡿⠟⢻⡆⢠⣮⣿⠀⠀⠀⠀⠀⠀⠀⢄⡀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠠⡤⠬⡦⠤⣼⣿⣾⣿⣵⣦⠤⢠⣤⢄⡤⠀⣤⣶⣶⣦⣦⡦
⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⣿⣿⣿⣦⡀⠀⠀⢸⡇⢀⣀⣄⣀⣹⣿⣤⣄⣈⣤⣼⣿⠘⡟⣲⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡀⠀⠘⡍⢉⣉⣉⢉⣁⣉⣉⣀⡉⣉⣉⡋⣠⣹⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⠀⠀⠀⠀⠉⠉⠉⢉⡝⠉⠉⠉⠉⣻⣿⡟⣻⣿⡿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣤⣀⠀⠀⢀⣴⣾⣧⣀⣠⣤⣤⡟⣿⣿⣣⣞⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣏⡯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⢿⡿⠿⠛⠛⠉⠉⠈







⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠠⠠⠄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡊⠁⣄⠀⣶⡀⢶⠀⡶⠀⡉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢠⣄⠙⡆⢻⡀⠘⡇⣞⣀⣏⡾⠁⢀⡞⢙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⡀⡀⠉⠓⠛⠒⢧⣠⠋⢀⠤⠀⠄⠘⢾⣁⡟⢀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⠳⠁⠀⠀⡨⢀⠀⢡⠀⡈⡠⠀⠀⠀⠀⣏⡾⠋⣠⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡙⡇⠀⠀⠁⠻⣔⡇⢸⢸⡳⣰⠖⡉⠀⡐⠣⠴⢟⣡⡴⢂⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣙⢻⡀⠀⢷⠚⠎⠑⣾⢪⠆⢠⢀⡴⠊⡀⠀⠀⠀⠙⠖⢋⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠉⠉⣭⠶⢴⡈⠻⠉⢻⠋⡐⡺⣿⠀⠸⠞⠃⠂⠠⠀⢸⣉⡴⠎⣇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢂⠀⠱⢶⣄⠈⣶⠀⠸⣀⠿⠆⠉⠢⡶⠦⠀⠀⢀⣴⠯⡖⠶⠒⢹
⠀⠀⠀⠀⠀⠀⢠⠀⠈⠐⠀⠠⠤⠀⠀⠑⢲⣦⠤⢀⠉⢐⠄⠡⢴⠶⠀⠒⠦⠶⠒⣉⠀⠉⠳⣞⣛⡋⠘⡀
⠀⠀⠀⠀⠀⠀⠈⢡⠦⠄⠤⢀⣀⣀⠈⠉⠛⠁⠀⠀⠢⢀⠲⣄⠈⠒⠂⡞⠣⠉⠉⠁⠁⠀⠀⠳⠮⠍⠀⡇
⠀⠀⠀⠀⠀⠀⠰⡌⡆⠀⢀⡠⠤⠉⠻⠋⠉⠀⠀⠀⠀⠀⡕⠌⠱⣤⡤⡦⠄⡡⠋⢍⠀⠀⠀⣔⣻⣓⢰
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⣀⣀⡝⠢⠀⠙⠍⠛⣶⠦⠤⠤⠬⢴⠺⠴⠭⡥⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠔⣩⠄⣠⠟⣁⡀⢰⢰⣤⡊⡐⠺⠽⠳⠀⠀⢫⣽⣛⠂⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡊⠐⣧⠊⠠⠈⢂⡤⣾⠈⠊⣻⣜⠂⠀⠀⠀⠀⠘⠶⠤⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠓⣻⠇⠀⠀⠠⡿⢠⠏⣬⢠⡁⡈⢧⡐⠄⠂⣠⡓⠦⢄⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠏⣹⠆⠀⠀⠀⠁⡼⠸⡇⠎⠹⠊⠂⢻⣉⠻⣬⣉⠃⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⢈⠡⠶⢓⡶⢤⠖⢆⠀⡑⠀⢈⠀⠀⣼⡘⠷⣬⠉⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⡇⠀⢂⠲⠋⢰⠛⠀⣼⢢⢄⣁⠈⠠⡞⣧⠹⡄⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢧⠀⠀⠑⢄⠋⢀⡽⠋⢸⡇⢹⡀⢸⡄⠘⣂⠰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠉⠢⠥⣀⣘⣀⣈⡡⠤⠔⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠜⠁⠀⠘⠠⡁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠠⢄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡠⠤⠤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠐⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⢤⡀⠀⠀⠙⢕⡄
⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⣀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢰⠀⣸⡀⣷⣦⠚⢦⠈⡜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠐⡆⢀⠀⠀⠀⠀⠀⠀⡇⠄⠀⢱⣼⣿⠛⣣⠤⡞⠀⢰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣶⡦⠲⢷⠐⠈⡑⢄⠀⠀⠀⢀⣴⠖⢂⣢⣉⠀⠀⣬⡉⠙⡇⢰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠟⠠⠀⠇⠀⠁⠚⠁⠀⢀⣾⡵⠞⠁⠙⢋⡇⡖⡷⣿⡙⠀⠆⠌
⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠈⣜⡄⠀⠀⠀⣼⡿⢸⠀⠈⠀⣌⠀⣧⠱⠼⢃⢊⠌
⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠔⠀⠀⡿⡇⠀⠀⠙⣿⠁⢸⠀⠣⡀⠙⠋⠁⢀⣴⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠁⠀⠀⠀⡇⠃⠀⠀⢠⠉⡀⠹⠀⠀⠀⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡌⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⠀⠀⠀⠘⠃⢃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣦⣄⡀⠀⠀⡇⠀⣿⠀⣶⣷⣦⠀⠀⢀⣤⣶⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠛⣿⣿⣿⡄⠀⡇⠀⣽⠀⡇⠀⣿⠀⠀⣼⣿⣿⡉⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣴⣿⡿⠃⠀⢀⣿⣿⣿⡇⠀⡇⠀⢹⠀⡇⠀⣿⠀⠀⢻⣿⣿⣷⠀⠀⠘⢿⣿⣷⡄
⠀⠀⠀⠀⠀⣼⣿⡿⠆⣶⣴⣿⣿⣿⣿⠃⣠⡟⠀⢈⣛⡓⢶⢿⣷⣦⣄⡻⣿⣿⣿⣷⣤⣾⣿⣿⣿⡀
⠀⠀⠀⠀⢸⣿⣿⡗⠄⢻⣿⣿⣿⣿⡧⠞⣩⣄⠀⢸⡇⣿⠛⣷⡖⠿⢿⡿⢯⣿⣿⣿⣿⡀⢸⣿⣿⣷
⠀⠀⠀⠀⣾⣿⣿⡅⠀⢸⣿⣿⣿⣿⠖⠋⠀⢸⠀⢸⡇⣿⠀⢹⣷⠀⠀⠈⣪⣿⣿⣿⣿⣷⣘⣿⣿⣿⠄
⠀⠀⠀⠀⣿⣿⣿⡅⢀⣾⣿⡿⣟⢳⡄⠀⠀⣸⠀⢸⡟⢻⠀⢸⠸⡆⠀⠰⢫⡟⣿⣿⣿⠛⠋⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣟⠁⠾⢿⣿⣿⡌⠆⠃⠀⢠⣿⡴⢻⡇⢸⠀⣼⠀⣷⠀⠃⡎⢸⣿⣿⣯⠀⢀⣿⣿⣿
⠀⠀⠀⠀⢿⣿⣿⡂⠀⢀⣿⣿⣷⠀⠠⠀⣞⣽⠁⢸⠁⣽⠀⣿⠀⣿⠀⠀⡇⠈⣿⣿⣿⣿⣴⣿⣿⠏
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢀⡟⢿⠀⢸⡆⢻⠀⢻⡄⢹⠀⢠⠁⠀⠈⠻⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠁⠀⠀⠀⢸⠃⢸⠀⢸⡇⢸⠀⢸⡇⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡼⠀⢸⠀⢼⡇⢸⠀⣸⠀⢠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢸⠀⢸⡇⡟⠀⡿⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⢸⡇⡇⢰⡇⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡇⠈⣷⡇⠘⡇⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⠃⡼⡿⣸⣧⢳⡀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠉⠐⠛⠛⠂⠓⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣷⣿⣿⠏⣠⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⢠⣤⣿⣿⣿⣿⣏⣡⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢶⣸⣿⣿⡞⡿⣼⠿⢿⠿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⠃⣥⣿⣿⣿⣿⠟⠁⠠⣿⠘⢁⡎⠸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⣿⣿⡦⣄⠆⢸⠆⢸⠇⠠⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⣿⣿⣿⣯⡿⣿⣿⠁⢈⢄⠈⠀⠀⠀⠀⠟⠳⡿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⠿⢃⡀⠒⠚⠀⠀⠀⠀⠄⢠⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣟⠈⠉⠀⠺⠛⠒⠂⠠⢬⡁⢠⣹⣏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣮⣭⡿⢿⣛⡛⠛⠛⠛⠉⠀⠁⠀⠉⠙⠃⢀⠀⠀⠀⠀⠀⠤⠄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣿⣯⠈⠈⠁⢀⠴⠛⠉⠉⠉⣒⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢀⣻⠊⠀⣶⣂⢾⢂⣠⠼⠒⠒⠒⠒⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⢇⠀⣾⠁⠐⣟⣫⣶⣺⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣿⣸⡞⢢⠴⢊⣩⣿⣿⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⣿⣷⣛⠀⠀⢂⣾⣿⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠟⣿⣷⡄⣤⣿⡿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⣱⣾⣿⣿⣿⣼⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠛⠹⠛⠛⠛⠫⠹⠟⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠄⠀⠈⢅⢄
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠛⠿⣷⡄⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⢘⢠⣤⣿⣆
⠀⠀⠀⠀⢀⣾⣿⣏⣈⣿⣿⢶⡄⣼⣿⡄⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠋
⠀⠀⠀⠀⣼⣿⠻⣿⡟⢤⣯⣴⠃⣘⣿⠃⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⣿
⠀⠀⠀⠀⣿⡇⢀⣺⢽⣿⣿⠛⢣⡉⠁⣠⠆⠀⠀⢸⠄⠀⠀⠀⠀⣴⣿⣿⣿⣿⡟
⠀⠀⠀⠀⣿⣿⡿⠯⠠⣾⣯⢀⣰⠾⠋⠁⠀⠀⠀⣚⠀⠀⠀⠀⣀⣿⣿⣟⡛⠙⠂⠀⠀⠐⠒⠂
⠀⠀⠀⠀⢿⣯⣀⣰⡶⠛⣿⣿⣯⣾⣳⣆⠀⠀⠀⢿⡆⠀⠀⠀⠹⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠸⣿⣿⣷⣿⡴⢿⣿⣟⣻⣿⣿⠀⠀⠀⠸⣿⡄⠀⠀⠀⠈⠉⠻⢿⣿⡇
⠀⠀⠀⠀⠀⠹⣿⣿⣉⡟⠻⣿⣏⣹⣿⣿⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⢰⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠙⢿⣷⣶⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠄⠀⡂⠀⠠⢀⠀⠈⠀⠀⠀⠋⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⢀⡀⠀⡁⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠒⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠐⠂⠀⠐⠠⢎⡀⢄⠀⠀⡁⠀⠌⣀⠀⠰⠀⠀⢀⡂⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⣀⢠⡌⠀⠀⠀⠀⠀⠱⡀⠡⡀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠃⠀⠀⢠⠂⠀⠂⠀⠀⠀⠀⠀⠀⠸⢦⡇⠐⠀⠄⠀⠀⣠⠖⠋⡉⠀⣲⣄⠀⡀⢀⠆⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⢩⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡟⠁⠀⢀⡴⠚⡅⠔⠁⠀⠀⠈⠂⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⠈⠀⠰⢨⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⣶⠇⢀⡴⢋⠥⠘⠁⠀⠀⠀⠀⠀⠀⠰⡁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠄⠀⠀⢦⡀⠄⠀⠠⣤⠤⠚⣡⢺⢀⡙⣆⣞⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢠⢀⠀⠅⠠⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠐⠀⠀⠄⠒⠙⢳⣅⣤⠈⠲⣞⠵⠀⠪⢤⠏⣴⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⠀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠁⠀⠀⢹⡍⡂⠸⣄⠀⠴⠀⠙⠏⠋⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠐⠀⡀⠀⢀⡗⣳⡦⣿⠅⠀⠀⠀⠰⠞⠀⢑⣦⠀⠀⠀⠀⠀⠀⠀⢀⣠⠊⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠠⠬⠀⠀⠀⠄⢸⣿⢋⡇⣿⠀⡠⠀⠀⢀⢄⡊⡉⢁⠀⢀⣤⡔⠄⣊⡥⠬⠖⠊⠀⠰⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⣠⣴⣾⠏⢰⢣⡗⢀⠀⠀⠀⠍⠈⠁⣈⡤⠔⠒⠎⢍⣉⠀⠀⠀⢀⠀⠐⠂
⠀⠀⠀⠀⠄⠀⠀⠀⠆⠠⠤⠀⠶⠒⠖⢒⣬⣽⡿⠟⣁⡠⣢⡟⠉⠀⠀⠀⠀⣀⣠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⢨⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣚⣿⣋⣁⣤⠞⢛⡴⠻⠇⠀⡀⠤⠖⠀⣠⢁⣈⠓⠒⢒⣲⣉⡿⣉⣨⣭⡭⢰⠞⠚⢄
⠀⠀⢘⠘⠀⠁⣀⡶⢒⣶⣻⣿⡿⠟⡟⠙⢻⡟⢻⡶⠻⢤⣀⠀⠉⠀⠀⠀⠀⠈⠣⡉⠉⠉⠉⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⡄⠀⡄⠀⢈
⠀⠀⠈⡶⠀⠀⢛⣿⣿⣿⣿⡏⢀⢷⠛⠚⡯⠞⠋⠈⠂⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠈⠉⢈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠁⠀⠁⠀⢈
⠀⠀⠀⠁⠒⠒⠚⠒⠊⠛⠛⠛⠻⣏⣱⢟⡢⢄⠀⢀⡀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⢈⣁⡃⣟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠃⠃⠒⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡖⠛⠁⠁⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠠⢒⢌⠙⠾⡏⠁⠀⠀⠀⠀⠀⠀⠀⠈⣆⠜⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠈⢴⡀⣤⣄⠀⠀⠀⠀⠀⠀⡠⠻⣦⣈⣑⣠⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡿⠀⠀⠀⢀⠄⠠⠐⠈⢻⡉⢺⡋⢄⠀⠀⡄⠀⢗⠀⠘⣤⡈⠉⠲⠆⢭⢧⣤⠠⣤⢤⡐⢂⠐⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠸⠀⢀⠊⠀⠀⠀⠀⠀⢧⠈⣔⠀⠐⠀⠸⠀⢸⡌⠐⡀⣝⠲⡄⠀⢀⠀⢈⢉⠒⠒⠢⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣎⠀⠀⠀⠆⠀⠀⠰⠀⠀⠀⠸⡇⠀⠀⠘⠁⠉⠙⠀⠋⠀⠈⠠⣀⠙⡇⠀⠀⠄⠈⠒⠀⠐⠀⠁⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠙⢆⠀⢀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⠀⠈⠀⢀⠀⠀⠋⠁⠀⠁⡁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⡄⠀⠀⠀⠀⠀⠁⡁⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⠃⠄⡀⠀⡀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢰⠀⠀⠀⠀⠄⠀⢀⠀⠄⠈⢊⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠃⠀⠀⣈⠀⠈⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⡄⠀⠀⠀⠀⠈⠄⠀⢀⠀⠀⡈⠢⢐⠀⠀⠀⠠⠲⢠⠤⢠⣐⠤⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠹⠒⠒⠓⠒⠲⠈⠉⠀⡀⠀⠉⠂⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⢁⠀⠀⡁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠒⠀⠒⠒⠂⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣄⠀⠘⠲⠶⠦⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠁⠀⣸⠇⠀⠀⢠⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⠴⠚⠁⠀⠀⠀⣿⠀⠠⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢰⡇⠀⠀⠀⠀⠀⡞⠁⢀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠈⠳⣄⣀⣷⠀⠀⠀⡀⠀⠙⠶⠊⠙⠋⠀⠀⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡙⢢⣼⣠⠃⠀⠀⠀⠀⠈⠀⠀⠀⣠⠏⠳⡄⠀⠀⠀⠀⠀⣮⠽⠃⢿⣳⢰⣏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢯⠟⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⣸⠿⡗⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠹⣆⡀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣅⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠒⠂⠀⠀⠀⣿⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠀⣠⠖⠚⠢⠤⠤⠤⠤⠤⠤⠤⠔⠒⠉⠁⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢀⡖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠾⠖⠒⠒⠒⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⠀⠀⠀⠀⠸⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠤⠄⠒⠒⠒⠚⠛⠉⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⢤⣤⠤⠴⠚⠋⠷⣤⣤⠴⠓⠦⢤⡤⠴⠛⠓⠒⠒⠚⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⣏⠉⣿⠀⠀⠀⠀⠀⢠⠞⠛⢣
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠘⠦⡤⠞⠀⢀⠀⠀⠀⠀⢀⡞⢳⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠸⠤⣿⡀⣧⠞⠀⠀⠀⠀⠀⠀⠀⢰⣛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⡿⠀⠀⠀⠈⠳⣬⢶⣯⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⡤⠤⠤⢤⡄
⠀⠀⢀⣀⠤⠴⠒⠒⠲⢤⣀⢠⡴⠶⠶⠒⠒⠒⠲⡆⠀⢸⡇⠀⠀⢹⠀⠀⣠⠴⠚⠋⠉⠛⢲⡴⠏⠉⠉⢳⢘⡋⠉⠉⠉⢹⠀⢠⡟⠃⠀⠀⠸⠃
⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠘⣾⡄⠀⠀⠀⠀⠀⠀⢳⠀⡸⠀⠀⠀⡸⢀⣾⠃⠀⠀⠀⠀⠀⢘⡀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠸⡇⢸⡇⠀⠀⠀⣤
⢸⡆⠀⠀⠀⠀⣤⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⣤⡴⠞⠀⡧⠀⠀⠀⣧⡞⠡⠀⠀⠀⠀⠀⡞⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡃⡄⠀⢀⡇
⡎⢃⠀⠀⠀⠀⠀⠀⠀⣤⣾⡁⣶⠀⠀⠀⠀⢸⣇⣀⣸⠃⠀⠀⠀⣿⠀⢀⡄⠀⠀⠀⠀⠙⠒⠙⡇⠀⠀⠀⣠⢤⡀⠀⠀⠀⣧⠸⣧⣗⣲⡾⠁
⣇⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠈⠉⠉⠏⠀⠀⠀⠀⠉⠁⠈⠉⡂⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡆⡼⡏⠉⣳
⠸⡄⠻⠀⠀⠀⠘⠛⠀⠀⠀⢀⡿⠀⣀⣀⣀⣀⣀⣀⣰⠀⠀⠀⠀⠀⠀⠀⢈⠙⠢⢄⣀⣀⣀⡠⡴⠦⠀⠐⣧⠀⠀⠒⠈⠉⡟⠙⠧⠼⠛⠁
⠀⣧⠰⢇⠀⠀⠀⠀⠀⢀⠰⣻⠁⠉⠉⣉⣉⣉⣠⣿⠏⠁⠈⠉⠉⠉⠉⣩⡟⠶⣤⣀⣀⣀⣀⡴⠶⠶⠶⠶⠋⠳⠶⠚⠛⠛⠁
⠀⠸⣆⡀⠉⠉⠀⠉⣉⣠⠞⠛⠛⠛⠋⠉⠉⠉⠉⠘⠓⠛⠛⠛⠉⠉⠙⠉⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠉⠉⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⣤⡴⠶⠶⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡖⠳⢶⠋⠉⠀⠀⣠⠞⠋⣠⣤⠴⠦⣤⣀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠙⣆⠀⠀⠀⠀⠘⠁⢠⠞⠁⠀⠀⠀⠀⠉⢣⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⡦⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⣶⡆⠀⠀⠀⡇⠈⡷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠉⠁⠀⠀⢀⠇⠀⠀⠀⠈⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⠤⠤⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠐⠀⠀⠀⠀⠤⠀⠴⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⣠⡴⠶⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣀⡠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣀⣀⣀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢻⡄⢠⡾⠉⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡷⠚⠁⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⢸⠀⠀⢀⡔⠛⠒⠦⣄⡀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠑⠀⠀⢀⡀⠀⢰⡂⠀⠀⠀⠀⠀⣼⠀⢠⠏⠀⠀⠀⠀⠀⠙⢦⡀⠀⢠⠁⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⣄⠀⠀⠀⢰⡏⢹⠀⠀⠁⠀⣀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡘⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⡿⢹⡆⠀⠀⣿⠀⢸⠀⠀⠀⣼⠉⢳⡀⠀⡿⠀⢸⠀⠀⠀⠀⣀⣀⣀⣠⣤⣼⣿⠀⠀⠀⠘⠛⠛⠓⠶⢦⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣻⡇⢸⡇⠀⠀⡟⠀⢸⠀⠀⢠⡏⡠⠚⠧⡼⠃⠀⢸⢉⡏⠉⠉⠉⠉⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠠⣤⡤⠞⠁
⠀⠀⠀⠀⠀⢾⣋⠉⠀⠀⠀⠀⠀⠀⠈⠁⠟⢸⡇⠀⢰⡇⠀⠈⢇⣀⡼⠀⠓⠙⠳⠖⠀⠀⢸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⣤⣤⠟⠳⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠉⠛⢶⡄⠀⠀⠀⣠⠤⠤⠦⠀⣇⠀⢸⠓⠢⡀⠈⠉⠀⡤⠤⠀⢀⡴⠀⠀⠸⠦⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠇
⠀⠀⠀⠀⠀⣀⣠⠤⠴⠏⠀⠀⠀⠙⠲⠂⢤⠀⢿⣤⠞⠓⠋⠀⠀⠀⠀⠓⠒⠒⠒⠲⡆⠀⡖⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⣠⣤⡤⣤⣄
⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⣄⠀⠀⠀⠛⢶⡄⠀⠈
⠀⠀⠀⠀⠀⠛⠦⠤⠤⠤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⠀⠀⠀⠾⠤⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠲⠶⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠖⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠒⠶⠶⠦⠤⣤⣤⣀⣀⣀⠀⠀⠀⠀⢀⣠⡿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢀⣀⣀⣀⡈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⢀⢤⡀⠀⠀⠀⠀⢀⣀⠺⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠎⠀⠀⠀⠀⣇⣈⡇⠀⠀⠀⠀⢀⢤
⠀⠀⠀⠀⠰⠭⣆⢼⠝⠀⠀⠀⠀⠀⣔⡦⠀⠀⠀⠀⠀⠀⠁⠀⠾⠆⠀⠀⠸⢤⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡤⠤⢤⠀⠀⠀⣀⣀⣀⡀⣀⣤⣤⠀⡤⠤⠤⡄⠀⣠⠖⠂⢰
⢀⡤⠒⠉⠁⠀⠉⠳⣼⠉⠉⠁⠀⠈⡅⢰⠁⠀⢸⢀⡴⠋⠀⠀⠀⠛⠁⠀⠈⠶⠂⠀⠀⢸⠀⡏⠀⠀⠈
⢸⠀⠀⠀⣤⠀⠀⠀⣿⠀⠀⠀⢠⡤⠇⣸⠀⠀⢸⡞⠁⠀⠀⠀⡔⠋⡀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⡆
⡆⠆⠀⠀⠀⠀⠐⠺⢤⡆⠀⠀⠠⠧⢤⡇⠀⠀⢸⡄⠤⡀⠀⠀⠈⠉⠙⠀⠀⠰⠲⠀⠀⠀⡆⢳⡧⢞⠁
⢣⠸⠀⠀⠀⠦⠀⠀⢈⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠧⢀⠀⠀⠀⣠⡄⠀⣤⠀⠡⠠⢐⠇⣏⡤⠞
⠘⡄⢆⣀⠀⢀⡀⠄⣾⣉⣉⣉⣉⣩⣿⠓⠒⠒⣂⣀⡸⠳⢤⣀⣈⡠⠶⠶⠶⠊⠲⠒⠒⠊
⠀⠑⠦⠤⣤⠤⠴⠊⠉⠉⠁⠀⠉⠀⠈⠉⠉⠁⠀⠉⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⢤⡴⠚⠉⢉⡽⠛⣉⣁⣉⡉⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠙⡆⠀⠀⠀⠋⡰⠋⠁⠀⠀⠉⢂⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠻⠃⠀⠇⠀⠀⠀⠀⡅⠀⠐⠿⠀⠀⢸⠀⠛⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡳⠄⠠⠄⠊⠀⠀⠀⠀⠀⠙⢤⣀⠀⢀⡠⠊⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⠈⢳⡀⠀⢀⡴⠒⠒⠦⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠔⠒⠉⠉⠉⠉⠉⠉⠉⠉⠁⠒⠒⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡴⠁⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡦⠂⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠁⠀⠀⠀⠆⠀⠀⡀⠀⠀⢀⡁⠀⠀⣈⣀⠀⠀⠀⠀⠀⣰⠏⠀⡀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠆⠀⠀⠀⢀⠄⠀⠀⠀⢨⡀⣠⠌⠀⠉⠁⢠⣀⠀⣠⠁⠀⠃⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣤⠀⠀⢀⠞⡇⠀⠁⢀⡀⠀⢸⠀⡁⠀⠀⠀⠀⠀⠈⢆⡏⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣸⠁⡇⠀⢸⠀⡷⠀⠀⠏⢹⡄⡸⠀⣧⣤⣤⡤⠤⠤⠤⠼⠷⠀⠀⠉⠉⠙⢓⡆
⠀⠀⠀⢠⡶⠒⠒⠒⠒⠒⠒⠛⡆⡇⠀⣿⠀⢡⠀⡸⠰⠡⢍⡇⠀⠰⠊⠀⠀⠀⠀⠀⠀⠣⣀⣀⡦⣌⡽⠋
⠀⠀⠀⠀⠉⠓⢦⠀⠀⢀⠤⠤⠁⢇⠀⡗⢢⠀⠉⠁⠤⠤⠤⠆⠀⠰⠀⡄⠀⠀⠀⠀⠀⢀⣸⠇
⠀⠀⠀⢠⠔⠒⠊⠀⠀⠈⠁⠐⡆⠸⠖⠉⠁⠀⠀⠀⠉⠉⠉⠳⠀⠋⠉⠁⠀⠀⠀⠀⠀⠈⢧⣀⡀⠀⠀⢠⣖⠒⠲
⠀⠀⠀⠸⠤⠀⣀⣀⣀⣀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⠀⠀⣼⣧⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠦⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠒⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣤⣤⣤⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⠒⠒⠒⠒⢲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠚⢿⡝⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠋⠁⣤⣶⠀⠘⡷⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠄⠀⢸⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡞⡆⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⢰⢡⢱⣦⠲⣄⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠴⣤⡷
⠀⠀⠀⠀⠀⠀⢰⠺⣇⠀⠰⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡆⠀⠀⢀⠆⣞⡜⠁⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇
⠀⠀⠀⠀⠀⠀⠀⢳⡹⣧⡀⠀⠀⢠⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣄⠀⠀⠀⠀⢠⠎⡼⠋⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠑⣜⢿⣦⡀⠈⠁⠀⢰⡷⠀⢀⣦⠀⠀⢰⣆⠀⠀⠈⠉⠀⠀⢀⡴⡡⠊⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣝⢿⡢⣄⠀⠀⠀⠀⠈⠉⠀⠀⠈⠁⠀⠀⠀⠀⣠⠔⣩⠞⢁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⢠⢇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢬⡒⠭⣒⠢⠤⣤⣀⣀⣀⣀⣠⡤⠴⢒⣉⢤⣾⡁⠀⠈⠉⠀⢰⡇⠀⢠⡆⠀⠀⣷⠀⠳⣀⣀⣀⣤⣔⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠬⣁⣒⠒⠀⠀⠐⢒⣒⣉⠥⠰⢞⡢⢍⡒⠤⣀⡀⠀⠀⠀⠈⠁⠀⢀⣀⣤⡶⢿⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠒⠬⢕⣒⢨⣩⠭⠽⣟⡛⣙⣻⠭⠕⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⣾⠛⠛⣿⠲⠷⢾⠿⠶⡷⡗⠾⠷⠿⢶⠾⠷⠾⢿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠶⠿⠿⠶⠿⣷⡾⠶⠿⠾⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⣿⣿⡇⢤⠄⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣦⡀⠀⠀⠀⠘⠃⠀⠀⠀⡤⠤⣀⣶⣶⡦⠀⠀⠀⢀⣠⣤⣤⣤⣤⡀
⠀⠀⠀⣿⣿⡇⠼⠄⠀⠀⠀⠸⢢⠆⠀⠀⠀⠀⠀⠀⠀⠈⠩⠉⠩⢹⡇⠀⣿⠿⠿⠟⠛⠋⠀⠣⣰⣾⣿⣿⣿⣷⠀⠀⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⣿⣿⡷⠤⠴⠤⠤⠤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢸⡇⠀⢠⣶⣾⢸⡀⠀⠀⢠⡎⠉⠉⠉⠉⠉⠀⠀⠀⢀⠄⢠⢰⡆⣿⡇
⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣼⡇⠀⢬⣿⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇
⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⣾⣿⣿⡆⠀⠀⠀⠛⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⣀⣀⣀⣀⣘⣃⣿⡇
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠒⠒⠀⠀⠀⠀⣶⣦⣤⠤⣤⣤⢤⣤⣤⣤⣶⡆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣿⣿⠍⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡓⣿⣟⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠚⠛⠛⠛⠀⠀⠀⠀⣿⣿⣶⡇⠀⠀⢀⣀⣠⣤⠀⠀⠛⠉⠀⠁⠉⠉⠉⠉⠉⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⡶⠆
⠀⠀⠀⠀⠀⢀⣀⠀⣠⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣰⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⢠⣤⣤⣤⣤⡄⠀⠀⠀⢠⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠸⠗⠀⠀⠀⠀⠀⠀⠀⣠⡟⠛⠻⢯⣀⠀⠀⠀⢸⡿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣃⣀⡊⠓⠲⢻⠀⠀⠀⢸⣷⣿⣾⣿⣿
⠀⠀⠀⠀⢈⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⡤⠀⠀⠀⠤⠤⠤⢤⠀⠀⠀⠀⠀⠀⠛⠛⠒⠀⢸⠤⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⢫⠀⠀⠀⠀⠀⢶⣦⣤⣴⡾⠉⠉⠉⠁⠀⠀⠀⠀⠘⠛⠛⠻⠟⠿⠟⢃⡀⠀⠀⠀⠛⠘⠳⠁⠠⢸⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠉⠤⠴⠦⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠷⠴⠛⠁⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠁⢀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠠
⠀⠀⠀⢤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠛⠛⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⡀⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣀⡀⠠⠴⠀⠒⡁⠀⠀⠤⠤⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠐⠒
⠀⡀⠀⠀⠀⠈⠤⠠⠴⠒⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⡀⠉⠁⠀⠀⠀⣤
⠀⢁⠀⠀⠀⠀⠀⢀⡀⠄⠠⠀⠀⢉⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠔⠈⢀⡂⠀⠀⠀⠀⡅⠀⢀⡀⢀⡤⢠⠠⠀⠁
⠀⠈⠀⠬⠀⠈⠀⠃⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠘⡀⠀⠁⠀⠀⠀⠇⠀⢀⣤⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠃⠀⠀⠀⠀⠐⠛⠁⠀⠀⠄⠀⣙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠂⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠂⠀⠀⠀⢛⠉⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠒⢀⡀⠀⣁⠀⣴
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢨⢀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁
⠀⠀⠁⠈⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠄⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠲⠠⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠘⠆⠘⠃⠂⢂⠄⠀⠀⠀⠀⠀⠤⠄⢙⣐⣒⣉⠠⠤⡆⠀⠀⠀⠀⠀⠉⠉⠉⠀⠁⠀⠀⡀⠀⠀⠀⠀⣀⠤⠀⠀⠠⠤⢀
⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠸⡀⢆⢀⣄⠲⠀⢂⠀⡇⠀⠀⠀⠀⠰⢠⠂⠈⠲⠂⠀⠀⠀⠀⠀⢀⠈⠀⣀⡀⠁⡁⣀⣀⢱
⠀⠀⠀⠀⠛⠄⠀⠀⠀⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⢀⠜
⠀⠀⠀⠀⠁⠀⠈⠠⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠒⠒⠒⠒⠒⠁⣀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣤⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠺⡖⣤⣤⡄⠈⣣⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢙⡄⣈⣼⣷⣼⣺⣇⠝⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢘⣷⣿⣿⣿⣿⣿⣿⣧⡀⠀⠸⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣶⣿⠿⣿⠏⠁⣽⠻⣿⣿⣿⣷⡀⠀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠉⠀⠀⠀⠠⠀⠀⠀⠩⠛⠋⠙⠃⠀⠸⣷⠀⠀⠀⠀⡐⠛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⢹⡄⠀⣠⠊⣽⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠌⠀⠀⠀⢀⣠⣄⣀⠀⠀⠀⠀⠉⢀⢠⠆⢈⣀⡀⢈⠛⠚⣡⠞⢹⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⢀⠀⠊⢀⠞⠀⣀⠾⠚⠛⠿⡖⢆⡀⢀⡀⣠⣶⣿⣿⣿⣷⣿⣧⣤⠏⠀⠈⢧⣈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠛⠀⢰⠇⠀⠀⣀⠀⢻⣤⣧⣤⣿⠟⠉⠋⠛⠿⣿⣿⡏⠀⠀⠀⠀⠀⣿⡼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⡆⢀⣀⠀⠀⣴⢺⣿⣿⠋⠡⠔⠁⣀⠀⢉⣿⡇⠀⠀⠀⠀⠈⣿⠃⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢡⡈⠉⠀⢀⣿⡿⡟⣧⣜⠲⣤⡄⢨⣤⢞⣿⡇⠀⠀⠀⠀⠀⢷⣄⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⣠⡴⠋⠑⠒⠊⠘⢷⣿⠟⠻⠦⢠⡶⠛⡫⠀⣿⣄⠀⠀⠀⠀⠀⡞⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⡷⢶⡆⠙⠻⡆⠀⠀⠀⣠⣿⣾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠰⠁⠈⠀⠀⠀⣦⠐⠆⠀⠀⠀⠀⠠⠤⡄⢀⠠⠁⠀⢹⡆⠘⣿⠀⠀⢰⢿⡏⡏⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⢄⠀⠀⠀⢸⠀⠀⢠⣟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⡆⠀⠈⣿⡖⣿⡶⠏⢁⣿⣷⣦⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣳⣿⣡⣠⠚⠃⢸⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⣼⡿⠁⠀⠀⠒⠀⠀⠀⠀⠀⢸⣿⣏⡀⣒⣟⡥⣮⡼⠁⠀⠀⢿⡳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⣼⠀⠀⠈⠁⢀⡤⠊⠀⠀⠀⠀⢠⠄⡈⣙⣿⡇⠉⢩⡄⣾⠂⠀⠀⠀⠀⠹⣯⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠰⣯⠾⢻⡀⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠐⠋⠃⣾⣷⣥⣠⣀⠀⠀⠀⣼⠏⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⡂⢠⣏⣽⡆⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣛⣲⣶⣿⣿⣿⣷⣬⣷⣄⡀⠀⠀⢠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢠⢲⡏⠁⠀⠈⠙⠛⠳⢾⡧⣤⣀⣄⠀⠀⢘⣷⣤⣷⣶⣿⡿⠿⠛⠉⠉⠋⣿⣿⣧⣟⣻⡿⣿⣫⣾⡇
⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠈⣶⣽⢦⣤⣀⣀⣄⠀⠀⠀⠐⠭⡛⢟⣋⣩⣥⠭⠉⠋⠁⢠⣤⣤⣾⣔⣤⡿⣿⣷⣿⣿⣷⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠚⠒⡅⠀⠛⠈⠉⠀⠀⠀⠈⠀⠀⠈⠙⠦⣤⣄⣀⠈⠛⠿⣿⡶⠶⠖⠛⠛⠛⠉⠙⠉⠉⠉⠀⠐⠛⠛⠛⠛⣏⣉⣉⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠈⠋⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢽⣿⣷⣷⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣭⡵⠏⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠉⠀⠐⠀⡀⠀⠀⠀⡠⠂⠀⠁⠁⠉⠐⠲⠁⢀
⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠈⡀⢀⣠⣄⡀⠀⢀⣀⣠⢂⡀
⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⢀⠄⢚⠁⠄⠀⡀⠠⠌⢙⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠤⠤⣤⣤⣀⣀⣀⣤⣤⣠⠤⢴⡆
⠀⠀⠀⠀⢀⠀⠀⠀⠀⡠⢂⠄⠐⠄⠀⠀⠀⡇⠀⠀⠀⠴⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠄⠀⠀⠀⠀⠀⠀⡆⠀⢸
⠀⠀⠀⠀⠈⠀⡠⠂⠈⠠⠁⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⠈⡑⠠⡀⠀⠀⠀⠀⢀⠀⠚⠁⠀⢠⠀⠤⠤⠤⠀⠤⣧⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢀⠠⠉⠃⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠋⢀⠀⠀⠀⠀⢀⠠⠀⢡⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠈⠏⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠔⠁⠀⠀⡎⢀⠎⠀⠀⠀⡄⠀⠈⠁⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠈⠀⠀⠀⠀⣠⠖⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⡔⠁⠀⠀⡄⠸⣣⠎⣠⠃⠀⠀⠰⠷⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠄⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣄⠌⡸⠁⠀⠀⣸⠀⡰⠿⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⢀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠰⠀⠆⠀⠀⢃⡤⠞⠛⠚⠛⡁⠀⠒⠒⠓⠤⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡁⠰⡃⠀⠀⠀⡇⠀⠀⠀⠳⠐⡡⠎⠀⠀⠈⠢⣰⠊⠀⢀⠔⢠⣞⡡⠀⠀⣠⡁⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠁⠀⠤⠀⠀⠁⠀⠠⢄⠀⠁⠀⠐⠂⠀⠀⠀⢟⢀⣼⣡⣴⡿⠥⠒⠀⡐⠙⠁⠀⠀⠀⠀⠁⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠄⠀⣠⡶⠬⠭⠛⠉⠉⠉⠒⢀⠚⠅⠀⠀⠀⠀⠀⠈⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣤⠶⠖⠲⢶⡤⠶⠚⠛⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠒⠿⡉⠙⠏⠀⠀⢀⡴⠋⣠⠴⠒⠛⠓⠦⣄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣀⠀⠀⢹⡄⠀⠀⠀⠈⠀⡜⠁⠀⠀⡀⠀⠀⠈⡆⠘⡆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠿⠂⠀⢰⠃⠀⠀⠀⠀⠀⡇⠀⠀⠸⠿⠂⠀⠀⣸⠀⠟⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢳⣀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⣰⠟⠉⠉⠉⠁⠂⠀⢀⠀⢀⠀⠀⠀⠀⠈⠛⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⢻⡄⠀⣠⠞⠋⠁⠀⠈⠻⣆
⠀⠀⠀⠀⠀⢿⠀⢀⠔⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡴⠃⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠈⠳⣬⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⣤⠄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⡆⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠁⠀⡀⠀⠀⠈⠀⠀⠀⠳⠀⠀⠀⠀⠃⠀⠀⡠⠛⠲⢤⣀⠀⠀⠀⠀⢀⡞⠁⢀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⠀⠀⠀⡀⠀⢰⠂⠀⠀⠀⠀⢀⡇⠀⡞⠁⠀⠀⠀⠈⠓⢄⠀⠀⡾⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣤⡀⠀⠀⢠⡞⢹⡀⠀⠁⢀⣀⠀⠀⢸⡇⢠⠁⠀⠀⠀⠀⠀⠀⠀⠳⣠⡇⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢰⠇⣷⠀⠀⣸⠀⢸⠇⠀⠀⡞⠉⣧⠀⢸⠃⢸⡀⢀⣀⣀⣀⣀⣀⣠⣤⡿⡇⠀⠀⠈⠙⠛⠓⠶⢦⡀
⠀⣀⣤⣤⣤⣤⣤⣤⣤⣭⣿⡄⣿⠀⠀⣿⠀⢸⡆⠀⢰⢃⠖⢙⠶⡏⠀⠸⣉⡏⠉⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠀⠰⢦⡶⠚
⠘⢏⣀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢿⠀⠀⡏⠀⠀⢧⣠⠞⠈⠉⠙⠒⡇⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡴⠞⠙⠒⠛⠁
⠀⠀⠈⠑⣦⠀⠀⠀⡴⠒⠒⠃⢸⡀⢀⣏⣙⠄⠀⠀⢠⡖⠲⠤⠴⠇⠀⠀⠓⠒⣦⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⣠⠤⠖⠚⠋⠀⠀⠀⠙⠛⠓⡆⠘⠷⠞⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⣳⠀⢟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⣴⠒⠚⠲⢤
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣄⠀⠀⠉⣳⠀⢀⣸
⠈⠉⠉⠙⠛⠓⠒⠲⠶⠤⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⡤⠟⠀⠀⠈⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠲⠦⠤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣛⣓⣲⣶⣶⣤⣤⣤⣶⣞⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣋⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠀⠀⠀⠤⢊⠀⠂⠒⠒⠬⡑⢄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⢀⠔⡨⠄⠒⠈⠉⠈⠉⠁⡀⠀⠀⠀⠀⠀⠀⠐⡀⠇⠀⠀⡀⠖⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢠⢊⠔⡩⠂⠡⠀⠀⣠⠖⠄⠀⢹⠀⡄⠀⠀⠠⡀⠀⠀⠈⠀⠀⠱⡀⢢⠁⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⡐⠁⠠⠊⠀⠀⠀⠀⠀⠣⠀⠀⠴⠉⠐⠇⠀⠀⠐⡄⠀⢁⠂⠀⠀⠀⠀⠀⠁⠡⡀⢆⢀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⡈⠀⠁⠀⠀⠀⠐⠀⠘⢇⠐⠈⠀⠀⢀⠀⠐⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⢀⠀⠶⠀⠀⠀⠀⠙⠂⠀⠀⠐⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠇⠘⠀⠀⡛⢉⡖⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⢰⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡹⡷⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠋⡁⠀⠰⢠⠆⠀⠀⠀⠀⠕⠀⠁⠀⠀⠄⠠⠀⡀⢢⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠘⡀
⠀⠀⠰⢣⠀⠀⠀⠀⠀⠀⠀⠉⣀⠠⠾⡦⣤⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠌⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠞⠄⠉⠀⢀⠀⠀⠂⠀⠀⠀⠀⣷⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠐⡀⠀⠀⠀⠁⠂⠄⠀⠀⠀⠀⠀⡈⠠⠀⡀⠄⠈⠀⠀⠀⠀⠀⠀⠑⠂⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⠂⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠒⠂⠁⠀⠀⠀⠀⠂⠂⠀⠂⠀⠄⠐⠀⣤⠀⡆⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⢀⡀⡄⠀⢰⠀⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠄⠊⠀⠀⠀⠉⠣⡀⠀⠡⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠀⠀⠀⠀⠀⠀⠶⠛⠋⠛⠋⠀⢏⠉⢣⡢⡀
⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠘⡣⠈
⠀⠀⠠⠈⠀⠁⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⠀⠀⠀⠀⠀⢜⠁⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡢⢀⠀⣀⡀⠀⠀⠉⠢⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⣴⠛⡏⠉⠷⡀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⢠⢰⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⢀⣤⢴⠆⠈⠁⠒⡭⠄⠀⠀⠚⠉⠀⠀⡀⠀⢀⡡⠀⠦⠌⢛⠡⠂⠀⠘⣄⢀⠾⠁
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡗⠀⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠊
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢄⠀⡀⠄⡧⠀⠀⠀⠠⠀⠀⢸⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢀⢀⠔⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠈⠑⠀⠐⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡜⠃⠀⠀⣀⠄⠈⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠉⠒⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠇⠀⠂⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠚⠀⡠⠁⠀⠰⠇⠐⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣤⠴⠶⠒⢶⣤⠶⠚⠛⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠓⠚⢯⡉⠛⠀⠀⢀⡴⠋⣠⠴⠚⠛⠓⠲⣄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⡞⠀⢀⡀⠀⠀⣧⠀⠀⠀⠀⠀⡜⠁⠀⢀⣀⠀⠀⠈⡆⠸⣆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠋⠀⢀⡏⠀⠀⠀⠀⠀⣇⠀⠀⠘⠟⠁⠀⠀⡇⠀⠋⠙⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣳⣄⣀⣀⠤⠊⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⣀⠴⠁⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠈⠀⠀⠄⠀⠄⠀⠀⠀⠀⠉⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⣠⡤⠤⢤⣄
⠀⠀⠀⠀⠀⡟⠀⠀⢀⣀⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀⣀⡀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠈⢷⠀⣠⠞⠁⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⢷⡀⢴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠙⠺⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣸⠞⠁⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⠀⠀⠀⠀⠸⠁⠀⠀⢠⠀⠀⠀⠀⡇⠀⠀⠀⣜⡀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡀⠈⠁⠀⠀⠀⡇⠀⣠⠊⠀⠉⠓⠦⣀⠀⠀⠀⡞⠀⠀⠂⠁⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⢶⡀⠈⠇⠀⠀⠀⠀⢀⡇⠀⠃⠀⠀⠀⠀⠀⠈⠳⡀⢸⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⢳⡀⠀⢀⡏⠀⡇⠀⠀⣰⠶⣄⠀⢸⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡀⠀⠀⠶⣯⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣿⢸⡇⠀⢸⠇⠀⡇⠀⠀⡏⣠⢿⣄⡜⠀⢸⠶⡖⠒⠂⠊⠛⠋⠉⠋⣹⠁⠀⠀⠀⢀⣀⣀⣽⠇
⢰⡞⠋⠉⠉⠉⠉⠉⠉⠉⢻⢸⡇⠀⢸⠀⠀⢧⠀⡸⠁⠓⠲⢬⡇⠀⠘⡛⠁⠀⠀⠀⠀⠀⠀⠀⠈⣤⣀⣨⠥⣤⣼⠟
⠀⠙⠓⢦⡀⠀⠀⢀⣠⠤⠼⠀⡇⠀⣸⠒⡄⠈⠉⠁⡤⢤⣀⣀⡇⠀⠈⠧⠤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⢀⣠⡤⠴⠃⠀⠀⠈⠒⠶⢤⠀⢷⣤⠟⠛⠁⠀⠀⠀⠓⠒⠒⠒⢢⠀⡶⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⢯⡀⠀⠀⠀⠀⠀⣠⠤⠴⢤⣄
⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣄⠀⠀⠙⢲⠄⠀⢨
⠈⠓⠒⠒⠲⠶⠦⠤⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡼⠂⠀⠀⠛⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⠤⢤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠉⠉⠉

⠀⠀⠀⠰⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⡄⠀⠀⠀⡀⠀⠄⠀⠀⢠⠀⠀⡠⠤⠤⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⢠⠀⠠
⠀⠀⠀⡇⠀⢀⠀⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠠⠠
⠀⠀⠀⠁⠀⠈⠀⠀⠆⠀⠃⠀⢸⠀⠠⠀⢀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⡄⢀⡴⣇⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢠⣴⣾⡀⣆⣠⣦⠖⠀⢸⠀⢠⡄⢒⢫⣴⣷⣿⣅⠀⠀⠀⠀⠀⡀⢦⢎⠃
⠀⠀⢀⣼⢉⣼⣿⠟⠞⢁⣼⣾⡾⢰⠇⠀⠙⢾⠯⡽⣿⡷⢀⢴⢐⣆⡘⣿⣿⡟⡳⠇⠀⠀⠀⠀⡀⡄⠠⠤⠬⠤⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠋⠉⣁⠀⣸⡿⠇⣀⣒⠀⢲⡈⣩⣆⣄⣾⣴⢎⠲⢑⣠⣃⠈⠠⣤⣒⠊⠀⠆⠀⠐⠇
⠀⠀⠀⠀⠀⠀⡂⠀⡀⢡⡄⡠⠨⣱⣏⣯⠁⠀⠀⠀⣀⣶⣿⠦⢅⠀⠄⠋⢩⢤⡯⢤⠀⢉⣀⠆⠤⢤⢲⠆⠂⢀⣴⠂
⠀⠀⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠙⢿⡅⢌⠀⠀⠳⠀⠀⢉⠁⡆⠈⠁⠀⠘⠉⣉⠁⠀⠀⠈⠉⠁⠀⠈⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠓⠘⠑⡞⠒⠓⠻⠘⠛⠐⡶⠀⡟⠉⠉⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠆⠀⠀⠀⢖⢀⣀⡂⣟⠂⡐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⢸⣿⣿⢸⣿⣿⡇⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣷⡄⠀⠈⣿⣿⡄⣿⣿⡇⠀⠀⢠⣾⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⡇⢿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢹⣿⣇⢸⣿⣿⡄⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣉⣀⣀⣀⣼⣿⣿⣝⣛⣿⣃⣀⡀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠨⣭⣭⠭⠭⢹⣿⣿⠉⣭⣭⣭⣭⣍⣉⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣷⣄⠀⠀⠀⢸⣿⣿⠀⣿⣿⡇⠈⠙⣻⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢿⣆⠀⠀⢸⣿⣿⢰⣿⣿⡇⠀⣰⡿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡀⠀⢸⣿⣿⢸⣿⣿⡇⢰⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⡇⠀⣼⣿⣿⢸⣿⣿⡇⣼⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠈⣿⠀⣿⣿⣿⢸⣿⣿⡇⣿⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⠿⠟⠋⠀⠀⠀⣿⠀⣿⣿⣿⣸⣿⣿⠃⣿⠀⠀⠀⠙⠻⠿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⣿⣿⣿⣿⣿⣿⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⣿⣿⣿⢿⣿⣿⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣴⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢿⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⡿⠀⠀⠘⣷⠶⠶⢶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣋⣩⣤⡴⠾⢷⣄⣀⣴⠿⠶⠶⠶⣦⣤⣍⣙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⢷⣤⣤⣀⣀⣉⣉⣁⣀⣀⣀⣀⣤⣤⡾⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠶⠶⣯⣭⣭⣭⣭⣭⣭⣭⣽⠶⠶⢿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠿⣾⣯⣭⣄⣀⣀⣀⣀⣀⣠⣭⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣡⣴⠶⢶⣌⠙⣿⡉⠉⠉⠉⠉⢩⡄⣼⢷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢸⡏⠀⠀⠀⠘⣷⠘⣧⠀⠘⠁⠀⢀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠸⣧⡀⠀⠀⣠⡟⢰⣯⣈⣁⣀⣀⣤⣴⠟⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠀⣈⠛⠛⠛⢋⣴⠿⠿⠿⠿⠿⠿⠷⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⣠⡾⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⡾⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠶⠶⢶⣤⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⣤⠀⠶⠀⣤⡌⠙⣷⡄⠀⠹⣧⠀⠀⠀⠀⠀⢀⣀⣀⣴⣆⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠂⠀⠀⢰⡶⠀⠠⠶⠈⣿⡀⠀⢻⡆⠀⢸⡇⢰⣿⣉⣉⣉⣉⣉⣉⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠶⠀⠀⠀⠾⠷⠄⠀⠰⠆⢸⡇⠀⢸⡇⠀⢸⡇⢸⣿⠉⠉⠉⣉⡉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠠⡆⠀⠀⠀⠀⠀⠠⡄⢀⣾⠃⠀⣼⡇⠀⢸⡇⢸⣿⠀⠀⠀⠛⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠺⠀⢴⠄⠺⠂⣠⡾⠃⠀⣰⡟⠀⠀⢸⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⣶⣤⣴⡶⠟⠋⠀⢀⣴⠟⠀⠀⠀⢸⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣆⠀⠀⠀⢀⣀⣤⡶⠟⠁⠀⠀⠀⠀⢸⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡟⠛⠛⠛⠛⠛⠛⣻⣿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⢀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣤⣤⣤⣤⣴⣾⣿⣤⣤⣤⣤⣾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠁⢸⣇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠴⠏⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⣠⡀⠀⠀⠀⠀⠀⣠⣤⡀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠟⠀⠀⠀⠀⠀⠷⠞⢁⣌⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣲⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣤⠴⣄⠀⠀⢺⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡝⣉⡀⠀⠀⠀⠀⠘⣧⣼⠇⠀⠀⠀⢸⡇⠀⢸⠂⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞⠈⢯⡽⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠙⠛⠋⠀⠀⠸⡽⠇⠀⠀⠀⢠⡖⢦
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢿⡽⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⢤⣤⣀⠙⠋⠀⠀⠀⠀⣀
⠀⠀⣠⠶⠋⠉⠉⠉⠉⠉⠉⠛⢷⡀⠀⡴⠒⠛⢻⡴⠛⠉⠉⠙⢦⡀⠀⣠⠶⠛⠉⠉⠛⠶⣄⠀⢠⡞⠁⠀⠀⠀⠀⠉⢳⡀⢀⡴⠚⠉⠉⠙⠛⠲⢦
⠀⡴⠃⠀⠴⠀⠀⣀⣀⡀⠀⠀⢸⡇⠀⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⢹⣴⠋⢠⠄⠀⠀⠀⠀⠈⢳⡟⠀⠀⠖⠀⠀⠀⠀⠀⣷⡏⠀⠀⠀⠀⢀⣠⣤⡾
⣸⠁⠀⠀⠀⣴⠛⠉⠉⢻⣴⣆⣼⠁⠀⡇⠠⡄⠀⢠⣤⠀⠀⢀⣤⢿⡇⠀⠈⠀⣀⣀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣠⠤⠴⠒⠙⣆⣀⣀⣀⡀⠈⠛⠓⠦⣄⠀⣴⢲
⡏⠀⠀⠀⢸⠃⠀⠀⠀⠈⠁⠈⠁⠀⠀⣷⠀⠀⠀⢸⣸⠀⠀⠀⢷⣸⠀⠀⠀⢰⠁⢸⠀⠀⠀⠀⣿⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⢳⡀⠀⠀⠈⣧⠈⠋
⣧⠀⠀⠀⢸⡀⠀⠀⡟⠛⠛⠛⠛⠛⠛⠛⢻⠀⠀⠀⠉⠀⠀⠀⠀⢻⡄⠀⠀⠈⠓⠋⠀⠀⠀⢀⣧⣤⣤⣀⠀⠈⠛⠛⢶⣠⡶⠶⢤⣄⣤⠞⠐⣦⠀⠀⣻
⢹⡀⠀⠀⠈⢳⡀⠀⠛⢶⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠶⠤⠷⠀⠀⠀⠀⢵⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡿
⠈⢳⡀⠀⠀⠀⠉⠛⠒⠋⠀⠀⠀⠀⣿⢻⠉⠀⠀⠀⣼⣆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⡏⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠁
⠀⠀⠙⠳⣦⠀⠀⠀⠀⢀⣤⡄⠀⠀⣿⢸⣀⣀⠀⠀⣿⠙⣦⠀⠀⠀⢀⣸⠟⣆⢠⣶⠀⣤⡇⠀⣇⡾⣇⠀⠀⠀⢠⡶⢾⠁⠀⠀⠀⠀⠀⠀⣤⡀⡿
⠀⠀⠀⠀⢸⠀⣀⠀⡀⢸⠀⣷⡆⣴⣿⠈⠉⢹⡇⠀⣿⠀⢹⠀⣤⡄⣿⠉⠀⣿⢸⡏⠷⠿⠇⠀⠀⠀⢻⢰⢷⡀⢸⠁⠘⣇⢸⢳⡄⣾⢧⠀⡏⠛⠁
⠀⠀⠀⠀⢸⢸⡟⠞⢻⢸⡆⠀⣿⢿⠀⠀⠀⠘⣿⣧⣿⠀⢿⠀⡇⠙⠋⠀⠀⢹⢸⠇⠀⠀⠀⠀⠀⠀⠈⠛⠘⡇⣸⠀⠀⠹⣾⠈⠷⠏⠸⣦⡿
⠀⠀⠀⠀⠘⠿⠀⠀⠘⠿⠀⠀⠹⣾⠀⠀⠀⠀⠀⢸⣿⠀⢸⣄⡇⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⣴⡆⠀⣶⠀⢠⣦⠀⣴⡄⢠⣶⠀⢰⡆⠀⣶⡄⢠⣦⠀⣴⡄⠀⣶⠀⢰⣦⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⢀⣿⠀⢸⣿⠀⣿⡇⢸⣿⠀⢸⡇⠀⣿⡇⢸⣿⠀⣿⡇⠀⣿⠀⢸⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠿⠇⠘⠿⠀⠸⠏⠀⠿⠇⠸⠿⠀⠸⠇⠀⠿⠇⠸⠿⠀⠹⠇⠀⠿⠃⠸⠿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣋⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠒⠀⠀⠉⠡⠌⡁⠀⣐⠂⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠉⢁⣀⣀⠀⠀⠀⡀⢄⠀⠈⠊⠀⠐⠂⠐⠀⠐⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠈⡱⠊⠓⠠⠀⠤⠂⠁⠀⢎⠀⡠⠔⠀⢀⠀⡀⡀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠁⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⡀⡀⡨⠂⠢⠄⠀⠁⠀⠀⠁⠑⠄⡀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠘⢅⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠐⠠⢄⣁⠀⢬⡄
⠀⠀⠀⠀⠀⠀⣤⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⡅⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠁⢀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠈⠉⠁⠐⠐⠃⠂⠤⢀⢁⡀⢀⠀⠀⠀⠀⠀⠀⢁⠄⠘⠒⡀⠀⢢⡀⠠⠤⡀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠐⠂⢁⣀⡀⠸⡀⠀⠀⠐⠀⠀⠠⠂⠀⠀⠡⠌⠀⠁⠐⠃⠀⠠
⠀⠀⠀⠀⠀⢠⠈⡈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠲⠆⠀⠀⠀⠈⠂⠀⠀⠀⠀⢀⠁⡅
⠀⠀⠀⠀⠀⠀⡀⠠⡈⠄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⡄⠀⠀⠀⠀⠀⡀⠤⡾⠃⠀⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠐⢄⠐⢌⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠂⣐⠣⡀⢀⣀⠐⠒⠊⡘⠁⠀⠸⠄⠀⠀⠀⠉⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠑⠈⡢⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠔⠁⠔⠍⠀⠈⠉⢀⠡⠞⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⢈⠙⠋⠒⠒⠲⠖⠀⣀⣀⣀⣀⡒⠈⠅⠀⠒⠈⡠⠴⠁⠀⠦⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠒⠂⠀⠤⠤⠤⠤⠤⠀⠐⠂⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣶⣦⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣴⣦⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠘⣿⣿⡽⣿⣿⣹⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⣿⡟⣿⣿⣳⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡽⣏⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣿⣿⣿⣳⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣮⢿⣿⡼⣿⣧⣇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣻⣿⡶⣿⣟⡆
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣯⡻⡿⢿⣼⣿⡆⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣟⠿⠧⢟⣾⡇
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣿⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠛⠛⣿⣿⢹⣿⣿⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣿⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⢸⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣿⠈⢿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠈⢻⣿⣧
⠀⠀⠀⠀⠀⠀⣾⣿⠏⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠻⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢢⠎⠘⠀⠀⠀⠀⢀⠀⠀⡠⠒⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢄⡤⠁⠘⢄⠀⢇⠀⢀⡠⠤⠀⠺⣄⢀⡜
⠀⠀⠀⠀⠀⠊⠁⠀⠈⠉⠁⠉⠑⠊⢢⡀⢄⢡⠀⠑⢟⣀⣈⣙⡷⠻⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠑⢂⠆⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠈⠒⠁⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠉⠀⠀⠀⢀⠴⠊⠁⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⡀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠹⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡶⠀⠁
⣠⣤⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢋⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢀⣿⠗
⠉⠁⠀⠀⢠⠏⠀⠀⣀⠀⡀⠀⠀⠀⠈⡄⠣⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡸⠱⡀⠀⠀⢀⢼⠔⢒⢼⡀
⠀⠀⠀⠀⣼⠀⠀⠈⠢⠥⠃⠀⠀⠀⠀⢱⠀⠑⣄⠀⠀⠀⠀⠀⠈⡡⠊⠀⠀⠀⠈⠒⠀⠆⢠⢇⣸⣭⠕
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠀⠀⠈⠀⠄⣀⡠⠤⠚⠀⠀⠀⠀⠀⠀⣀⡤⢦⣳⠄⠠⠛⠈
⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⢠⠦⠔⠃⠀⠀⠤⠴⠒⠒⠂⠀⠐⠲⢄⣀⠤⠖⠊⠁
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢠⠃⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣶⣶⣶⡶⣷⠄
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⡞⠚⠫⢉⡉⠉⠁⠀⠀⠀⠀⠒⠂⠀⢀⣀⠜
⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⢸⠀⠀⠀⠀⠀⠉⠉⠀⠀⠖⠐⢂⡨⠟⠁
⠈⠑⠂⠠⢐⢉⣽⢮⠂⠀⢸⠁⠀⢈⡩⠟⠁⠒⡲⠛⢻⡟⣁
⠀⠀⠀⠀⠀⠀⠫⠭⠤⣠⡈⡗⠊⠁⠀⠀⠀⠘⠒⠚⠙⠛⠁⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠉⠉⠢
⠀⠀⣀⠠⠤⢄⡀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⡅
⢠⠊⠁⠀⠀⠀⠈⠢⡀⠀⠇⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠙⣄⠸⡀⠀⠀⠀⠀⢀⠁
⢱⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠱⡀⠀⡀⠀⡘
⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠈⣆⠙⢦⠀⣠⠃
⠀⠀⠈⠓⠤⣄⣀⠀⠀⠀⣀⡸⠤⠇⠉⠁⠑⠒⠒⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡽⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣈⠙⢂⠶⠴⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⠀⠸⠁⠋⠈⣋⣈⠹⡄
⠀⠀⠀⠀⠀⠀⠀⡠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠤⠐⢣⠴⢄⣀⣀⢸⠃
⠀⠀⠀⠀⠀⠀⠐⡁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠚⠀⠒⠁⠀⠀⠀⠀⢼⠃
⠀⠀⠀⠀⠀⢀⡔⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⠦⣤⢤⢀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠤⠛⢦⠀⠀⢱
⠀⠀⠀⠀⠀⠈⠲⠀⠀⠀⠀⣀⡄⢶⠒⡎⠒⠤⡖⠲⠤⠞⠉⠁⠀⠀⠀⡱⠀⠀⢉⠆
⠀⠀⠀⠀⠀⠀⠐⢄⣀⣸⡌⢳⡙⣦⡓⢄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⡏⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡹⡿⣌⢲⠦⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⠈⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠃⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡤⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠥⢦⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠈⣷⠈⠳⡀⢠⠖⣉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⡇⠸⡷⡄⠈⠛⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠴⠃⠀⣷⠙⢄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠿⠀⠀⠑⣦⣄⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢀⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠤⣀
⠀⠀⠀⠀⠀⢀⡠⢔⠂⠀⠄⠀⠀⠀⠂⠀⠀⠉⠢⠑⡀⠀⠀⠀⣠⡤⠀⠀⠀⠤⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⠄⢁⠪⢀⠀⠠⡀⠉⠑⡆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⡩⣫⣵⣤⣀⣠⣄⡀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠐⠁⠐⠁⠀⠀⠀⠸⠀⠀⠔⠃⢆⠀⠀⢂⠀⢃⠀⠀⢈⢨⠀⠁⠊⢡⢄⣐⠟⠋⠀⠀⠀⡤⠀⠀⠀⠀⠁
⠀⠠⠀⠀⠀⠀⠘⠀⠙⢄⠂⠀⠀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠳⣄⠀⠀⠈⠐⠊⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠑⠄⠀⠀⠀⠀⠀⠀⢀
⠀⡀⠀⠀⠀⠘⢃⠀⡀⠀⠁⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆⠀⠠⠤⠀⠀⠀⡠⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠆⠀⠀⠀⠘⣿⣇⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⢠⢀⡄⠀⡀⠁⠢⢐⠀⠀⠀⠀⠀⠀⠒⠒⠰⠦⠤⠤⣄⡀⠀⠀⠓⠀⠀⠀⠄
⠀⢠⢣⠀⠀⠀⠀⠀⠈⢁⠤⢾⡧⣄⢀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡆
⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠃⠉⠀⠀⠀⠀⠀⠀⠀⡐⠐⠏⠃⠀⢹⠁⢰⠋⢢⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⣀⣀⠠⠄⠀⠀⠀⠀⠀⠀⣤⠀⠀⣄⡀⠀⢸⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⠀⠀⠀⠀⠈⠁⠄⠀⠀⠀⠀⢰⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⢣⡀⠀⡄⠀⠀⠀⠀⠀⠀⢰⣂⣀⠀⢰⠀⢠⡀⡘⠀⠀⠀⠀⢀⠀⠤⠄⠄⠀⠀⠀⠈⠀⠀⠀⠀⢈
⠀⠀⢀⠔⠃⠀⠀⠀⠀⠙⠀⠀⠀⠀⡉⠁⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠮⠁⠀⡐⠀⠀⠀⠀⠀⠀⠒⠀⠀⠸⡀
⠀⠀⢀⠀⠀⠂⢄⢐⣋⠀⠘⠢⡀⠀⠀⢠⡤⠂⠀⠐⡄⠘⠒⡄⠀⠀⢸⠉⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠹⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⢸⡄⠀⠀⠂⠋⣨⢇⣂⡀⠀⠢⡀⢤⠀⠀⣴⠀⠀⠀⠀⡇⠀⠀⢸⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⢠⠑⠂⠀⠤⠴⠇⠀⠀⠈⠐⠄⠈⢀⠈⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⢀⣠⠇⠀⠀⢸⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⣀⣀⣀⠃
⠀⠀⠀⠑⢄⠒⠐⠚⡇⠀⠀⣀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠒⠂⠢⠄⠀⠀⠄⠀⠄
⠀⠀⠀⠀⠈⠓⠤⠀⢀⠺⡤⡤⠟⠀⠀⠀⠀⠀⠀⠀⡀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢈⡓⠀⠀⢀⠀⠒⠉⠉⠉⠛⠃⢦⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠠⡀⠀⠀⠘⠛⠁⠀⣠⡦⠀⠀⠀⠀⢀⠉⠀⠀⠙⣇
⠀⠀⠀⠀⠀⡰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡀⢠⣿⣾⡀⢀⠀⠀⠀⠀⠠⢀⠄⣿
⠀⠀⠀⠀⠀⠣⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠊⠁⠀⠈⣷⠊⢡⡄⠁⠀⠀⠀⠀⠄⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠲⠟⠀⠠⠈⠃⠀⠀⠀⠀⠀⠙⢉⣄⣐⠂⠀⠀⠀⠈⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠠⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠚⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠤⣀
⠀⠀⠀⠀⠀⢀⡠⢔⠂⠠⠀⠀⠀⠉⠂⠀⠀⠈⠢⡑⡀⠀⠀⠀⣠⡤⠀⠀⠀⠤⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⢀⠔⢵⠪⢐⠀⠠⡀⡉⠑⡆⠀⠀⠠⡀⠀⠀⠁⠀⠀⡝⡩⣯⣵⣤⣀⣠⣄⡀⠈⠀⠀⠀⠀⠐⠒
⠀⠀⠐⠁⡐⠁⠀⠀⠀⠸⠀⠂⠔⠃⢆⠀⠀⠢⠀⢃⠀⠀⢨⣸⠀⠁⠋⢡⢄⣘⠿⠋⡀⠀⠀⡠⠀⠀⠀⠀⠁
⠀⠠⠀⠀⠀⠀⠸⠀⠙⣄⠂⠀⠀⡄⠀⡀⠀⠂⠀⠀⠀⠀⠀⠻⣄⠀⠀⠈⠐⠊⠁⠀⠀⠀⠀⠁⠀⠀⡀⠀⠉⠑⠄⠀⠀⠀⠀⠀⠀⢀
⠀⡀⠀⡀⠃⠘⣃⠀⡀⠀⠁⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆⠀⠠⠀⠀⠀⠀⡠⠂⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠁⠀⠇⠀⠀⠁⠚⣿⣇⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⢠⢄⡄⠀⡀⠉⠦⢈⠀⠀⠀⠀⠀⠀⢒⠒⠰⠢⠤⠤⣄⡀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄
⠀⢠⢣⠀⡀⠀⠀⠀⠈⢁⠤⢾⡧⣤⢀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡇
⠀⢈⠀⠀⠀⠂⢀⠀⠀⠀⠀⠈⠑⠃⠉⢀⠄⠀⠈⠀⠀⠀⡐⠐⠋⠁⠈⢹⠁⢰⠋⢢⠀⠀⠀⠈⠢⠀⠠⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⢀⠀⠀⠀⠀⠨⢀⠀⠀⢀⡀⠠⠄⠐⠀⠀⠀⠀⠀⣠⠀⠀⣄⡀⠀⢸⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⠀⠀⠀⠀⠘⠁⠆⠀⠁⠀⠀⢸⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠀⠐⠀⠀⡀⠀⠀⠀⢣⡀⠀⡄⠀⠀⠀⠀⠀⠀⢰⣂⡀⠀⢸⠀⢠⠀⡘⠀⠀⠀⠀⢀⠀⠤⠤⠄⠀⠀⠀⠘⠀⠀⠀⠀⢀⡀
⠀⠀⢀⠔⠃⠀⠀⠀⠀⠙⠄⠀⢀⠀⡉⠁⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠮⠁⠀⡐⠀⠀⠀⠀⠀⠀⠓⠀⠀⢸⡇
⠀⠀⢂⠀⠀⠂⢄⢐⣋⠀⢘⠢⡀⠀⠀⢠⡶⠂⠀⠐⡄⠚⠓⡄⠀⠀⢸⠉⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠹⡅⠘⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠸⡄⠀⠀⠀⢫⣨⢇⣂⡀⠈⠢⡀⢤⠀⠀⢼⠄⠀⠀⠀⡇⠀⠀⢸⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁
⠀⠀⢠⠑⠂⠀⠴⢴⠇⠀⠀⠈⠐⢄⠈⢂⠈⠁⠀⠀⠀⠀⠀⡇⠀⠀⠸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠟⠀⠀⠀⠀⠀⢀⣠⠇⠰⠀⢀⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣀⣀⣀⣀⠃
⠀⠀⠀⠑⣔⠒⠒⠛⡇⠀⠀⢀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠒⠒⠢⠄⠀⠀⠄⠀⠄⠐⠁
⠀⠀⠀⠀⠈⠑⠤⡀⢄⠺⡄⡤⠟⠀⠀⠀⠀⠀⠀⠀⡀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⢈⡓⠀⠀⢀⠄⠒⠉⠉⠙⠛⠇⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠈⠩⠁⠀⠂⠀⡀⠈⠄⠀⠀⠀⠄⠤⡠⠀⠈⠙⠛⠁⠀⣰⡧⠀⠁⠀⠀⢀⠉⠀⠀⠙⣇
⠀⠀⠀⠀⠀⡐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡀⢰⣿⣾⡀⢀⠀⠀⠀⠀⠠⠀⠄⣿
⠀⠀⠀⠀⠀⠣⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠀⠚⠁⠀⠈⣷⠋⢠⡄⠍⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠹⠕⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⡲⠟⠀⠠⠈⠃⠀⠀⠀⠀⠀⠙⢍⣄⣀⠀⠀⠀⠀⠈⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⢰⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠚⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈

⠀⠀⠀⠀⠀⢀⠤⠐⠂⠉⠉⠉⠈⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠀⠀⠈⠉⠐⠂⠄⡀
⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡀
⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⣄⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣠⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⢠⠃⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠘
⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡱⠦⠈⠁⠢⡀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠁⠤⢎⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⢃⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠰⡀⠈⠄⠀⠀⠀⠀⠠⠁⢀⠆⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠈⢄⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠰⠀⠘⠀⠀⠀⠀⠃⠀⠂⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡠
⠀⠈⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠄⠚⠁⠑⡀⠀⠇⠀⠇⠀⠀⠸⠀⠸⠀⢀⠊⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠁
⠀⠀⢃⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠀⢰⠀⠀⡆⠀⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡘
⠀⠀⠈⢄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⠀⢸⠀⠀⡇⠀⡇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡠⠁
⠀⠀⠀⠈⠢⣄⠈⠒⠤⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⢸⠀⢸⠀⠀⡇⠀⡇⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠤⠒⠁⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠉⠒⠀⠤⠄⠀⠤⠄⠂⠁⠀⠀⠀⢸⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⠈⠐⠠⠤⠀⠠⠤⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇⠀⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣶⣾⣤⣤⣷⣶⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣶⣶⣶⣦⣤⣀⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣽⣟⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⠹⣿⣿⣿⣿⣿⡿⢳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣏⣁⠀⠀⣀⠿⣿⣿⣿⣿⣿⠁⢈⣿⣿⣷⡊⠀⠀⠂⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⡀⠀⠀⠘⠿⠿⠿⣿⣿⣿⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⣦⣄⠰⣿⡶⡀⢻⣿⣇⣠⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣶⣶⣶⣿⣿⡏⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣷⡟⢹⣧⠀⢿⣿⢋⣽
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠾⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⣿⡟⠁⠀⠀⠉⠛⠛⢿⣿⣿⣿⣇⠸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠉⣿⠉⠈⣽⣿⣿⣇⣄⢸⣿⠄⡀⠀⠀⠀⠀⠀⢤⣨⣵⣿⣧⣈⣿⣿⠀⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⢠⣿⠀⠀⢼⣿⣿⠁⠈⢹⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠉⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢨⣿⡄⠀⢸⡿⠟⠀⠀⢹⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢹⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠇⠀⠀⠀⠀⢀⣤⣤⡿⣻⡄⠀⢸⣦⣀⠀⣞⠛⠂⠀⠀⠀⢉⣻⡆⠀⠀⠀⠀⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⠈⢿⡏⣾⣿⡇⠀⢸⣿⣿⠷⠿⣿⣶⣶⡶⠶⢤⡽⠀⠀⠀⠀⠀⢸⠿⣿⣇⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠙⣍⢻⡇⠀⢸⣿⣿⠗⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣡⣾⠋⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⢷⠀⠀⠀⠀⠀⠀⠀⢻⣾⣷⠀⠀⢻⣿⠀⠀⣿⣿⡇⢠⡀⣀⣀⢀⣀⣤⣿⣿⡇⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣄⢸⣷⡀⠀⠀⠀⠀⢀⣿⢻⣿⠀⠔⣿⣏⠀⣆⣿⣿⠃⣿⡟⠛⠻⠟⠛⠋⠉⣿⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣗⢡⣿⣿⣿⣶⣴⣶⣶⣿⣿⠘⣿⠀⠀⢹⣿⠀⠋⣿⣿⠀⣿⣷⠀⠀⠀⢠⠤⠞⠃⠀⠀⠘⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣼⣿⠀⠂⢸⣿⠀⠆⣿⣿⠀⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⣾⡟⠋⣀⣠⣤⣴⣿⣿⣿⣿⣿⢸⣿⡆⠀⢠⡿⠀⠀⢻⣿⡀⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⡀⠀⠀⠀⡇⠀⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣷⠀⠈⠁⢀⡄⠘⣿⡇⣿⠛⠿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⢸⣿⣿⡄⢀⣴⠟⠻⣄⠘⣿⣿⡇⠀⣿⣿⣯⣿⣿⣿⣷⡄⠀⠀⠈⢿⡇
⠀⠀⠀⠘⠃⠀⠀⠀⠀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠒⢻⢸⣿⡟⢀⣾⡿⠻⡄⠈⢢⡈⠘⣷⠀⢹⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢸⡇
⠀⠀⠀⠁⠀⠀⠀⠀⢀⠠⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⡇⠛⣀⣼⡿⠁⢀⠀⠛⢄⠙⣆⠻⡆⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠁⢠⣿⡿⢺⠿⣿⣶⣶⣾⣶⣶⣾⣷⠻⣿⣿⡔⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠀⣸⡿⣤⣤⣴⣷⣼⣷⣿⣿⣿⣾⣿⣄⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠀⡟⣹⣏⣤⣉⣭⣌⣽⣁⣉⣉⣭⣏⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣼⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠀⢠⢿⣿⣿⣹⡿⠟⠛⠛⠛⠻⢿⣿⡟⣿⣟⢧⠹⣿⣿⣿⣿⣿⠋⠀⠀⡝⠁⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠃⢠⡟⣼⣿⡟⠁⠀⠀⠀⡀⠀⠀⠀⠉⠓⣼⣿⣦⣳⡈⢿⣿⠟⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⢄⣸⣷⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠏⠁⠀⠀⢀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠋⠀⠀⠀⠀⡀⠎⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢀⢀⡀⠀⠀⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⡷⠀⠀⠀⠀⣀⣠⣶⣿⣿⣿⠿⠟⠛⠋⠉⠉⠉⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠋⠙⢿⣦⣴⣶⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀⠀⠀⠀⢠⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣤⣴⠟⠉⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠻⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⣽⣿⣿⣯⣽⣿⣿⣿⣿⣿⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠉⠉⠀⣿⡇⠀⠀⡇⠀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠀⣿⡇⠀⠀⡇⠀⠀⠘⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠒⠀⣿⡇⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⠀⠀⣿⡇⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠂⢀⣀⣀⣀⣀⠀⣿⡇⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⡇⠀⠀⢧⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠉⠉⠉⠀⣿⡇⠀⠀⠈⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣇⣸⣏⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⡿⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⠶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⣄⣀⣤⡾⡉⢢⣴⣤⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠤⢀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣘⣣⡱⣄
⠀⠀⢀⠠⣾⣿⣿⣿⣿⣦⣇⣝⠛⣿⣿⣿⣿⣟⠟⡟⢿⢿⣿⣿⣿⣿⣆⠀⠀⠀⣾⡿⡻⠍⠴⣈⡉⠣⠀⣝⠻⣿⣿⣿⣿⡿⢛⣋⢹⣿⠟⠇⣿⣿⣇
⠀⠀⢿⠐⠄⠙⠳⣶⡋⡉⠀⡀⠀⠀⠀⠈⠉⠉⠣⢤⡘⠸⠈⡏⡁⢹⣇⡆⠀⢠⣟⡴⡷⣆⡀⠙⠔⢁⡤⠖⠀⣹⣿⢻⣿⢡⢳⡿⠦⠀⠀⠀⠫⠽⠟⣧
⠀⠀⠀⢸⣿⣗⣦⡈⠉⠉⠙⠛⠳⠆⡔⠀⠀⠀⠐⢂⣍⡑⢄⡁⠀⠠⡿⠇⠀⢸⡏⣢⡤⠴⠔⣀⡀⡿⢓⠦⠀⡁⣿⣸⡇⡮⢺⡿⡇⠀⢀⣠⣶⣾⡆⣿
⠀⠀⠀⢸⣿⣿⣎⣯⢶⣀⠄⠀⠀⠀⠆⠀⠖⠀⠁⠀⠈⠀⠂⠙⣆⠀⡇⠀⠀⢸⣧⢻⡊⣠⢂⠍⡑⣀⡀⠒⠴⢧⣿⣿⠜⠀⢸⡟⢵⣾⣷⣿⣿⡿⡡⡿
⠀⠀⠀⠘⣿⣿⣿⡘⡎⡝⡤⡠⠀⢀⣆⠀⠀⠈⠀⠀⠀⠀⠀⠑⢼⣇⡇⠀⠀⠈⢿⣷⣅⡠⠌⠎⢷⠼⠷⢚⣴⣿⠋⠁⠀⢀⠞⠀⠀⣯⣾⣿⢟⣴⡶
⠀⠀⠀⠀⣷⠈⠻⣷⣥⣋⣃⣵⠟⣾⣿⣯⣽⣦⣄⣀⣀⣠⣤⣄⣙⣿⡇⣠⣀⠀⢸⡿⣿⣯⣵⢾⠶⠶⠻⣿⣿⣿⣠⣄⡴⢋⡆⠀⣠⣿⣟⣵⣿⣿⢸
⠀⠀⠀⠀⣿⠀⢀⣨⣿⡿⢿⣿⣿⡿⢿⢿⣻⠛⣿⣛⠿⣿⠹⠟⢿⠿⠏⣹⠏⠀⠸⡗⠀⠀⠘⢷⣤⣀⣴⠊⠿⠛⠻⣖⢶⡜⢷⣴⠟⠣⣿⣿⢻⣷
⠀⠀⠀⢰⣿⣾⣟⣿⣷⢾⣻⡝⠻⠶⠛⠙⠃⠀⠀⠁⠀⢀⣀⣤⡤⢰⡆⣹⣧⡀⠘⣿⣄⠀⠀⠀⢹⡟⣿⣦⣄⣤⡀⣟⠻⠿⠶⠶⠶⠖⠋⢤⣼⣧
⠀⠀⠀⣼⣯⣿⢿⡍⠇⠀⣁⡉⠀⠀⠀⠀⠀⠠⣤⠤⠒⠛⠓⢋⣀⢸⡇⠀⠀⠀⢀⣿⣿⣯⣡⣴⢾⣧⡈⠿⣿⣽⠞⠋⣧⠀⠀⠀⠀⠈⠉⢹⣿⣿
⠀⠀⣾⣿⠿⠿⠼⠕⠀⠼⠟⢀⣀⣠⠴⠾⠛⠋⠁⠤⠦⣿⣿⣷⣦⣾⡷⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⢹⣿⢴⣿⣿⠀⢀⠟⠀⠀⢀⣀⢩⣿⣿⣿⣿
⠀⠀⠉⢑⣾⣟⣛⣛⣛⣛⣉⣩⣠⣔⣤⣦⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠛⠛⠛⢯⣑⠀⡸⠟⠛⠛⠛⠛⠛⠋⠛⠛⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⣀⣀⣀⣀⡤⡤⡴⠒⠛⣉⣉⣙⡓⣶⣀⣀⣀⣀⣠⣤
⠀⠀⠀⠀⢰⣶⣶⣾⣿⡏⠟⠈⠉⠘⠓⣶⣿⢿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⠀⠀⢸⣿⣿⡿⢿⣿⢯⣀⣴⣿⠿⠿⠶⣾⣿⣿⣿⠃⢿⣿⣿⢫⣾⢿⣲⣶⠒
⠀⠀⠀⣰⣿⣶⣿⣯⣿⣶⣶⣶⣶⣶⠶⠿⣿⢾⣷⣿⣿⣿⣿⣹⠏⠀⣿⡿⠀⠀⢸⣿⠋⢃⣼⡇⠍⠻⣿⣿⣦⣴⠀⢉⣹⣿⣿⡏⢸⣿⣱⣿⣶⡟⠊
⠀⠀⠀⡉⣿⣏⣻⣻⣿⣿⡄⠀⠀⠀⠀⢀⣾⢹⣿⢧⣯⣿⣍⣈⠀⠀⣼⠇⠀⠀⣼⡙⢀⡞⣽⣿⣿⣷⣌⠋⠛⠛⠻⠿⠛⠉⣋⣴⡿⢻⣿⣿⣿⣷⣤⣀⡀
⠀⠀⠀⠱⣘⢯⣻⣻⣿⣿⠯⠟⠛⠛⠛⢿⣿⡞⠁⠊⠀⠋⠉⠉⢹⡆⣿⠀⠀⢰⣟⡫⣽⣏⣫⠊⠛⡿⠿⠛⠛⠛⢻⣿⠿⠛⠋⠁⢀⣼⢯⣟⣿⣿⡈⠉
⠀⠀⠀⠰⢿⣆⢹⣿⣿⣿⣉⠀⠀⠀⢠⣾⡏⠀⠰⠞⠀⢀⣄⠀⢸⡇⣾⠀⣠⠼⢷⣾⣷⣿⣷⣷⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠛⠉⢳⣯⣛⣛⡛
⠀⠀⠀⢀⣼⣷⣽⣿⣿⣗⣂⣀⣀⣀⣿⠟⠃⠀⠀⠀⠀⠈⡎⠁⢸⡇⣿⠀⠣⣄⡈⠙⠍⢻⢛⢿⠿⣿⣿⣿⣴⣤⣄⣄⣀⣀⣀⣤⠴⣶⣾⢟⡏
⠀⠀⠀⠈⣿⠻⣏⣿⣿⣟⣒⣛⣻⣿⢏⠀⠀⠀⠀⠀⢀⣡⡴⡞⠿⠟⢻⢄⡀⠈⢻⣷⣶⣤⣄⣀⠀⠚⠋⠙⢝⣿⡝⢿⢯⣭⠯⠒⠿⠀⠸⢿⡇
⠀⠀⠀⠀⣿⠀⣹⣿⣿⣿⠷⠶⢿⣧⣼⣠⣄⢀⢠⡤⠛⠁⠀⠀⠀⠀⡠⠄⡇⠀⢸⡟⣉⣈⣉⡛⠓⠷⣦⣤⣄⡉⠁⠀⠑⣽⣟⠿⡶⣦⣍⣻⡟
⠀⠀⠀⠀⣯⢰⣿⣿⡏⠀⠀⠀⠀⣹⡋⠛⠺⢾⣯⣄⣀⠀⢀⠠⢒⣏⣠⠖⠃⠀⢸⣟⣷⣿⡷⣾⠓⣶⠒⢈⠙⠳⠾⣶⣤⣄⡈⠀⠈⠈⠋⠙⠳⣦
⠀⠀⠀⠀⣿⣦⠹⣿⣧⣀⣀⣶⣿⠃⠀⠑⢦⡀⠉⠛⠻⠿⠤⠚⣋⣹⣿⠀⠀⠀⣾⣋⠵⠾⡧⠾⡇⡉⢹⠋⣦⠄⠄⠠⠄⠉⠙⠷⠶⠦⣤⣄⣤⠼⠃
⠀⠀⠀⢀⣿⣿⣿⣦⣍⡙⢿⣿⣿⣶⡄⠀⠀⣧⣠⣅⣤⣄⣠⣾⣷⣿⡿⠀⠀⠀⠿⣷⣦⣾⣿⣐⣟⣤⣬⣭⣴⣴⣧⣤⣤⣴⠶⣭⣽⣿⣾⣿⠇
⠀⠀⠀⠧⠽⠿⠛⠉⠉⠉⠉⠙⠻⠿⠿⠦⠶⠋⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠀⠀⠙⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠠⢄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠤⠔⠦⠤⣄⡀
⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⡠⣤⠀⠀⠈⠪⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢀⣣⣸⣧⡖⠛⡆⠀⠡
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢠⣶⡿⠊⠸⠀⢀⠀⠀⠀⠀⠀⢀⠀⠄⠀⢥⣽⡟⠛⡤⣴⡃⠀⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⢶⡦⠲⢶⠀⠨⡀⣀⠀⠀⠀⡠⡲⢒⣀⣠⣀⠀⢀⣍⡈⣹⠀⡄⡀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠉⠀⢗⠀⠈⠆⠀⠀⠛⠀⠀⢠⣮⢔⠋⢀⠉⣹⢰⢺⠗⣏⠃⠠⠠⠁
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⠀⠂⠀⠸⠀⠀⠀⡀⣿⠇⡟⠀⠁⢠⡁⢘⡆⠣⠜⠐⡡⠁
⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢙⡿⠀⡇⠀⢆⡀⠉⠁⢀⣠⠬⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠃⠁⠀⠀⢨⢇⠀⠇⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣾⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⣿⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡅⠀⠀⠀⠛⠀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠔⠋⠉⠢⡀⠀⠀⠀⠀⢀⡴⠚⠉⠓⣆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⡌⠀⡀⠀⠀⠹⡄⠀⠀⣠⠋⠀⠀⠀⠀⢸⠀⡟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⡇⠀⠀⠀⢀⠀⢳⠀⢰⠃⠀⡀⠀⠂⠀⡸⠀⣇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢱⠀⠀⠐⠀⠀⢸⡄⡇⠀⠂⠀⠀⠀⢠⠃⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢀⠀⠀⣿⠀⠀⢀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⡤⡌⢣⠀⠀⠀⠀⡇⠀⠀⠀⠂⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢌⢦⠀⠀⢸⢁⠇⠈⢢⠉⠀⢸⡇⠐⠈⠀⢀⡞⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢀⡴⠖⠛⠛⠛⠃⠓⠒⠛⠘⠛⠉⠉⠑⠒⠚⣄⠀⠀⢀⠌⠉⠙⠋⠉⠉⠳⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⢀⡴⠉⠀⠀⢀⣀⠠⠀⠀⠀⡀⠀⡠⠄⠒⠢⢄⠀⠈⢦⡴⠃⠀⠀⠀⠐⡀⠀⢀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠀⠈⠙⡄
⠀⠀⠀⡾⠀⢠⠦⡤⠤⡤⠵⡀⠀⠀⠁⠞⠒⠲⠰⠒⠘⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠸⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠋⠁⠀⠀⠀⠀⠀⠓⠈⠒⠲
⠀⠀⠀⣧⠀⢇⣀⣐⣚⣁⣀⡇⠀⠀⠀⢦⠐⠒⠒⠒⢒⠇⠀⠀⠀⡀⠀⠂⢀⡎⠀⠂⠀⠀⣀⣀⣸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠾⠤
⠀⠀⠀⢧⡀⠈⠢⣤⣠⡤⠊⠀⠀⠀⠀⠈⠒⠤⠤⢒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰⣏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞
⠀⠀⠀⠀⢹⡀⠀⢀⡠⠄⠀⠀⠀⠀⠀⣀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠉⠁⢀⡞⠉⠙⠓⠲⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⢀⡤⠛⠋⠉⢀⣀⣤⣀⣄⠼⠉⠛⠀⠀⠀⣀⣤⣀⡀⠀⠀⠀⠀⣀⣀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⢤⡤
⠀⠀⠀⡼⠀⢀⡔⠋⠉⠳⠤⢤⡄⢀⠀⠀⡤⡶⢋⠀⠀⡏⢹⠀⡆⣷⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⢠⠇⠀⡼⠀⠀⠀⠀⠀⠈⠳⠋⠋⠛⢡⠃⡎⣼⠀⡇⢸⠀⡇⢸⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠋⠁
⠀⢀⡞⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡇⣿⡀⡇⢸⣠⢧⡎⢣⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⡤⢤⠀⠀⠀⠀⢀⡖
⣞⠉⠄⠰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⢰⠤⠤⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⡁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠃
⠀⠀⠀⠀⢀⠄⠐⠁⠀⠀⠀⠈⠂⠢⡀⠀⠀⢠⡄⠠⡰⡍⠀⠀⢀⠠⠀⠒⠀⠀⠤⠤⡄
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠠⣌⡁⠀⠁⡇⡔⢁⠠⢀⣀⣤⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢊⢂⣄⠏⡴⣋⠅⠒⢀⠔⠚⠓⠠⢀
⠀⠀⠀⠀⠉⠋⠉⠉⠉⠉⠉⠉⠉⢹⠙⠀⢀⣠⢤⣤⣄⣷⢔⠉⠅⠃⡰⠁⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⡄⠀⠀⢸⢠⠎⠀⠐⢌⠢⠼⡗⡳⡄⠐⡕⠁⠀⠀⠀⠀⠀⠀⠀⠎⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠰⡀⠀⢸⠇⠀⠀⠀⠂⠈⠐⠀⢊⠱⠊
⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢁⠀⢸⢞⡡⠕⠀⠀⢠⠔⠊⠁⠀⢀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠆⡂
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡈⢢⣪⠮⢥⣀⠀⡰⠁⠀⠀⣠⣶⣿⣿⣿⣿⠁⠀⠀⠀⠀⠠⢀⠁
⠀⠀⠀⠀⠀⠀⠀⢈⡰⠶⠖⢱⣼⣏⣉⠠⠠⠘⢇⠀⠀⢰⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢀⠃⠌
⠀⠀⠀⠀⠋⠁⠀⠀⠈⢹⠟⡩⠇⠱⢄⡑⢄⠀⠈⡄⠀⠸⣿⣿⡿⠟⠁⠀⠀⠀⠀⠠⠃⠌
⠀⠀⠀⠀⢆⠀⠀⠀⠠⢥⠊⠀⢀⠸⠀⠈⠒⠇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣡⠊
⠀⠀⠀⠀⠀⠑⠒⠒⠒⠚⢂⠀⠈⠎⠀⠀⠀⠀⠔⠑⢤⡀⠀⠀⠀⠀⠀⡠⠔⡡⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠠⠤⠄⠒⠁⠀⠀⠀⠉⠙⠂⠤⠭⠀⠒⠉


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣄
⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⢀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦
⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⢀⡴⠋⠀⠒⠀⠐⠒⠬⡑⢦⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⠉⠑⠲⢍⡢⣄⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀
⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢳⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢌⢦⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⣼⣿⠁⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢳⠀⠀⠀⠀⠀⠀⠘⣿⣧
⢰⣿⡇⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢹⣿
⣼⣿⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⣔⣪⠞⠀⠀⠀⠑⢤⣒⠤⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠈⣿
⣿⣿⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⡏⠁⠀⠀⠀⠀⠀⠀⠀⠈⢩⠃⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣿
⣿⣯⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢶⢉⡀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⣈⢳⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡈⢦⣇⠀⠀⠀⠀⠀⠀⠀⢰⣡⠋⡝⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⣿
⢻⣿⠀⠀⠀⠀⢸⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠘⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢀⣿
⠸⣿⡇⠀⠀⠀⠈⡆⢧⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⡿⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡠⢠⠇⠀⠀⠀⠀⣸⣿
⠀⢻⣿⡀⠀⠀⠀⠘⢮⡳⢄⡀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⣀⠤⢚⡡⠋⠀⠀⠀⠀⢀⣿⡏
⠀⠈⢿⣷⡀⠀⠀⠀⠀⠈⠓⠒⠤⠤⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⠤⠒⠚⠉⠀⠀⠀⠀⠀⢀⣾⡿
⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁
⠀⠀⠀⠀⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟
⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠒⠒⠖⠒⠆⠀⠒⠠⠄⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⣀⡀⣀⢠⣠⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠾⢿⣶⣡⣿⣿⠻⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⢶⠟⠻⡇⠉⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡈⠶⠀⠀⠀⠀⠀⠱⢦⣄⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡿⢯⣏⣤⡔⠀⠀⠠
⠀⠀⠀⠀⠐⠂⠀⠀⠀⠺⡷⢘⣿⣖⠄⠀⠀⠀⠀⠀⠛⣷⡀⢀⣀⠀⠀⠈⢹⣦⠀⢹⡞⢶⣭⡶⣀
⠀⠀⠀⠠⠀⠀⠀⢀⣠⣷⣤⡈⠙⠂⠀⠀⠀⠀⡀⠀⠘⣿⣇⢈⣽⡇⣀⣀⣿⡟⠀⠀⠸⢻⣿⣿⣅⠂
⠀⠀⠀⠶⠀⢀⠀⠀⠝⠋⠋⠛⠒⠤⢀⡀⠀⢡⠹⣧⠀⣻⣿⡌⠉⠱⣝⠛⢛⡁⠀⠀⠀⣱⡿⠟⠁⠀⡀
⠀⠀⠐⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⢺⣶⣍⠂⠘⠇⢻⡇⢸⣿⣧⠀⠀⣨⣿⣿⣩⡤⢒⡊⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠻⢷⣤⣄⡀⠀⢀⣿⣿⣷⠀⠠⠆⣰⣄⣿⣿⠀⠀⣡⣿⣿⢋⣴⣿⣷⣶⣤⣤⡞⠁
⠀⢠⠀⠀⠀⠀⠐⢶⡤⠀⠈⠁⠰⠿⣿⠿⢟⠀⣠⣤⡘⠛⠈⣿⡆⢀⠈⣼⢇⣾⣿⣿⣿⣿⡿⠋
⠀⠐⠒⠐⠐⢉⣿⣟⣿⠂⠀⠀⠀⠀⠐⢾⡃⠀⠁⠾⠃⠀⠀⠈⠁⠀⠀⣿⢸⣿⣿⣿⡿⠁⢠
⠀⠂⠀⠐⠒⠒⠒⠛⠿⠟⠀⠀⣒⣲⣖⣿⣟⣓⡶⢤⣄⡀⠀⠀⠀⠀⠀⢿⢸⣿⣿⡟⠁
⠀⡤⠀⠀⠀⣤⣤⣀⣀⠀⠀⠀⠉⠀⠉⠉⣛⠿⢿⣿⣶⣿⣷⣦⡀⠀⠀⠸⣇⢿⣿⡇
⠰⠀⠸⠇⠈⠹⢻⣿⣿⡛⠟⠲⠦⣄⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣦⠀⠀⠙⣎⠿⣧⡀
⢸⠄⠀⠀⠐⢿⣿⣿⡷⠆⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⣠⣬⡛⢿⣿⣿⣷⡀⠀⠈⠓⢮⣝⡀
⢘⠋⠀⠀⠀⢘⣿⣿⣿⣉⡀⠀⠀⠀⠀⠙⣆⠀⠀⠰⣯⣿⠃⠊⠸⡿⣿⡇⠀⠀⠀⠀⠉⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠉⢿⣾⣿⣿⣧⣤⡠⠀⠄⠀⠈⠉⠁⠀⠉⠀⠀⠀⠀⡿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠀⢺⣿⣿⡆⢀⠀⠀⠀⠀⠀⠠⠀⠀⢉⡥⠆⠀⠀⣠⣤⣤⠤⠀⠀⠀⠀⠀⠈⠙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⠟⡠⢠⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⢀⠀⠀⠹⢶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⢀⡜⣡⢾⣄⠀⣠⣾⣿⣿⣶⣄⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣧⡉⠀⠀⠀⠈⠀⠀⡀⠹⠟
⠀⠀⠀⠀⠀⠀⣺⡾⡁⠀⠀⠾⡙⠿⠿⡋⠀⢛⢻⣿⣿⣿⠋⠀⠀⠀⠀⠀⢠⠻⣟⠻⣿⠟⠋⠙⠛⠉⠑⠦⠀⠀⠀⠀⠈⠐⢦⣄
⠀⠀⠀⠀⠀⠀⠋⣡⡷⣢⠀⠀⠉⠉⠉⠀⠀⠀⠘⠛⠉⠀⠀⠐⠀⠀⠀⠀⠀⢳⣝⢦⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠂⠀⠀⠘⢫⣀⠀⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠈⠢⡀⠀⠀⠀⡟⠳⠤⠀⠀⠀⠀⠀⠀⢀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⡟⣄⡑⢂⠀⡀⠠⠀⠀⠐⠡⢾⣿⣿⣿⡄⠀⠀⠈⠙⠛⠿⣄⠀⠀⡈⠙⣿⣿⣦⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠘⠣⠄⠀⠀⠀⠀⠀⠀⠈⠙⣿⡿⠀⠀⠀⢀⣤⣶⣤⣌⠆⠀⣙⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⣿⣿⣎⣼⣿⠿⡿⠟⠳⠂⣽⣟⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠘⠋⠋⠙⠿⠉⠀⠀⠀⠀⠀⠀⠈⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⢧⠚⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⡂⠀⠀⠈⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠳⢶⠣⠿⠀⠀⡐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢄⡀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠶⠶⠶⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠔⠋⠁
⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⠤⣄⣀⠀⠀⠹⡄⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢈⡙⠿⣯⣉⠉⠛⢉⡉⠟⠛⠻⠿⣿⣶⣄⣽⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⠖⠉⠉⠉⠁⠐
⠀⠀⠈⠻⡄⠀⠙⠻⢶⣤⣀⡀⠀⠀⠀⠀⣈⣹⣿⠇⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⣾⣁⡀⢤⠤⣤⣤⣀⣀⢀
⠀⠀⠀⠀⠙⢆⠀⠀⠀⠈⠙⠛⠿⠿⠟⠛⠉⠁⣴⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢙⡛⠻⠶⣶⣟⢿⡛⢿⣿⣷⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡈⣧⠀⠀⠀⠙⠷⣤⣉⠀⠀⠀⠀⣠⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦⣀⣀⠀⣀⣠⣴⣾⠿⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢷⣄⠀⠀⠀⠀⠉⠛⠶⠶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠫⠾⠿⡿⠿⠋⠁⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢿⣷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⠻⣿⣿⣷⣶⡦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠙⠋⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃⠐⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣠⡶⠟⠛⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢠⣾⠉⠀⠀⠀⠀⢸⡟⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡇⠀⠀⠀⠀⠀⠈⣧⠈⠙⠳⢦⣄⡀⠀⠀⠀⣰⡿⣤⣉⣁⣀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠈⣋⠙⣿⡷⠶⠟⠀⠀⠉⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣠⡶⠏⣡⠀⠀⢹⣇⠀⠀⠀⢹⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠁⠁⣊⡡⠀⠀⠀⠻⣄⠀⠀⣘⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠹⣶⠛⠉⠳⠞⠋⠙⠰⣶⣶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠙⢷⡀⠀⠀⣠⡶⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⢀⢀⣀⠀⠀⢹⡇⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⢸⡇⢠⡾⠋⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠀⠀⠐⠁⠀⠀⠙⠃⠼⢿⣡⣿⢃⣀⠤⠤⠄⠀⠀⠀⠀⣿⡟⠁⠀⠀⠀⠀⠀⣿⢀⡤⠴⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⣿⡀⠀⢠⡔⠒⠐⠒⠀⢀⠀⠉⠉⠉⡠⠆⠒⠂⢠⠀⠀⠀⠟⠋⠉⠀⠀⠀⠁⠀⠙⠋⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⡒⠂⠁⠹⣧⠀⢨⠀⠀⠀⠀⠀⠀⠀⠢⡄⠀⢃⠀⠀⠀⠀⠀⢱⡀⠀⢰⡏⠀⠀⠀⢠⠀⠀⠀⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠈⠉⠛⠦⢤⣿⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⣻⠀⠸⡄⠀⠀⠀⠀⢨⡅⠀⠈⢃⠀⠀⠀⠠⠆⠀⢠⡎⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠳⣄⠀⠀⠀⣀⠔⠁⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠘⢇⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠿⠿⠿⠿⠻⡿⠟⣽⠛⢿⠟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣛⣻⣟⣛⣿⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿

⠀⠋⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠑⠀⠀⠶⠂⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠢⡙⢆⡀⠉⣉⡀⡀⠩⠀⠰⠂⠙⠀⡰⣰⠀⠠
⠔⠠⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢄⡀⠀⠀⡆⠀⠀⠀⠀⠠⢈⠉⠀⠀⢨⡁⠀⠀⡙⠢⡀⠀⠀⠀⢀⡔⠔⣠⢮⡼⠁
⠐⠂⣄⠀⠀⠊⠂⡀⠀⡧⠐⢁⡠⡄⡀⠀⠀⠸⣀⣀⠀⠀⠄⠀⠡⠀⡄⢀⠁⠂⠀⠀⠈⠉⡀⠀⠀⠀⠀⠀⠀⠖⠋⠑
⠀⢀⠈⠀⠃⠈⢀⢐⠀⠀⠠⠍⠀⡀⢀⠐⠀⠀⠀⠀⠠⠒⡄⠀⠀⠀⠀⠰⠆⠀⢀⠠⠔⠀⠁⠀⠤⠠⠏⠀⡁⠀⠀⠌⠀⠆⠠
⠀⠀⠀⠐⠈⠀⠁⠨⠀⠀⠀⠀⠀⠶⠅⠠⡨⠀⠀⠀⠀⠀⠠⠂⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⠁⠀⠁
⠒⠂⠀⡀⢈⡁⠀⠀⡠⠂⠢⡀⠀⢀⣔⣽⠣⠁⡠⠒⠀⣈⡀⠄⠀⠀⠈⠀⠊⠁⠀⠐⠀⠀⢀⠀⠀⠤⠀⠔⠁⡇⠀⠀⠀⣀⡘
⠀⠀⡀⣉⢠⢀⠀⠐⠀⠀⠀⠃⢠⡵⠋⠄⠁⠀⠁⠆⠣⠀⠈⠁⠉⠉⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄⠀⠀⠀⡋⠀⠀⠀⠉
⠈⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠱⣞⡖⠀⡀⡀⠐⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⡀⠄⢀⠄⠀⠄⠄⠈⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣒⠒⠂⠀⠀⠀⠀⠀⠌⡀⡟⡁⠀⠀⢈⠆⠀⡀⢁⠁⡀⠀⠀⠀⠐⠊⠀⡀⠀⡀⠀⠐⠂⠜
⠀⠌⠂⡀⠀⠡⠀⠈⠁⠀⠀⠀⠀⠀⠀⠴⠃⡇⠹⠀⠀⠀⠀⠀⠀⢁⠀⡤⢨⠀⠀⠠⣀⠀⠀⣷⡄⠀⠈⠀⠁⠀⠀⠀⠀⠀⣤
⠄⠘⢀⠀⡀⠀⠐⠀⠀⠀⠀⠀⡔⢇⠀⣃⣤⠁⠈⠠⣀⠄⠀⠀⠀⠶⠂⠀⠀⢀⠠⠄⡀⠀⠄⠀⠀⠀⠀⠈⠛⠀⠒⠒⠊⢀⠤
⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠌⠀⠀⠰⠀⣠⠀⠀⠀⠀⠸⠐⠂⠀⠀⠰⠅⡀⡂⢀⢔⠀⠰⠴⠃⡄⠠⠀⠀⠀⠀⠀⢀⠂
⠠⠀⠠⠄⠒⠂⢄⠀⠀⠀⠀⠊⠀⠀⠀⠀⠌⠋⠀⠘⠁⠄⠐⠘⠂⠁⠂⠀⠀⠀⠐⠒⠅⣀⠄⠢⡀⠀⠄⢀⠈⢖⡰⠀⣀⣄
⠀⠅⠰⡂⠢⠄⠀⠅⠀⠀⠰⠀⢀⠔⠁⠀⠀⠤⠤⡞⠁⡀⠀⠀⠀⠀⠀⢃⠁⡄⠀⠠⠊⠀⠀⠀⡗⠀⢀⡌⠀⠉⠀⡀⢀⠀⠂
⠈⠀⠀⠏⠡⠂⠀⠂⠀⣄⡀⠤⠁⠠⢊⡡⡁⠐⠄⠀⠠⣧⠀⠀⠀⠀⢁⡀⢠⠀⠀⠅⠀⠀⢈⠈⠀⠀⢉⠀⠀⠀⠀⡀⠀⠐
⠀⠉⠈⠐⣀⠀⠀⠈⠀⠐⢀⠀⠀⠁⠠⠀⠈⡠⠂⠀⠀⠈⠀⠀⣀⠀⠀⠀⠀⠐⠀⠀⠑⠊⢀⡎⢀⡀⠃⠀⠐⠀⠒⠀⠀⠀⢀
⠄⡀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠉⠀⡁⠀⠀⠂⠀⠀⠀⠧⠔⠂⠀⠀⠀⠑⡀⠀⠂⠈⠈⠠⠀⡁⠀⡁⡀⠀⠀⡙⢈⠁⠀⠀⣠⣁
⢉⠂⡄⢊⠐⠁⠀⠀⠀⠀⠀⣀⠊⢀⡀⠀⢠⡠⡠⠀⠀⠀⠀⠀⠀⠀⢀⠁⠙⡄⢁⠒⠀⠂⠈⢻⣿⠀⠀⠀⠀⠈⠐⠀⠀⠀⠒
⢀⠄⡉⠠⢎⠶⠀⠀⠀⢀⠆⠀⠐⠀⠐⠔⠐⠒⠀⢂⢀⡀⠀⠀⠒⠀⠀⣀⠀⠁⠄⠁⠀⠠⢤⠀⠇⠀⢀⠤
⢈⠀⠆⠀⠆⢺⣡⠧⠔⣸⠀⢀⠀⡀⠆⠀⠀⠀⠀⠸⠀⠉⠀⠀⠒⠉⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠈⠤⠈⢁⢆⡸
⠀⠀⠂⠂⠀⠡⠗⠆⠀⣆⠔⢠⣠⢀⠈⠂⠤⠠⠐⠁⢄⠒⢬⡀⠀⠀⣂⠀⠀⢀⢂⠀⡀⠀⠐⠒⠒⡄⠈⢀⡉⠈⠋⠀⠀⠀⢀
⠂⠄⢀⠆⠀⠀⢀⠀⠀⢀⡴⣲⢏⠂⣠⢠⡄⠐⠀⢤⡠⢀⠀⢈⠓⢊⠰⠁⠂⠡⠈⠈⠀⠀⠀⠀⢸⠀⠀⢉
⠘⠄⠋⠠⠐⠂⠀⠂⠀⠠⠟⠁⠂⠀⠀⠣⡕⣠⠊⠒⢝⡷⡒⠀⠀⠀⡀⠲⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠰⠂⠐⠂⠀⠤⠀⠀⡠
⠀⠀⠁⠀⠀⠀⢀⠐⡀⠀⠂⠄⠀⠀⠀⢐⢻⣅⠅⠀⠀⠙⠙⠂⠀⠀⠁⠀⠀⠀⠈⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠛⠊⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣶⣶⣶⣦⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠿⠟⠛⠋⠉⢉⣤⠄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠿⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⣿⢿⣅⣀⡀⠀⠀⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠈⠙⢿⣦
⠀⠀⠀⠀⠀⢠⣿⢸⢸⠉⡟⢻⠒⣶⠶⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣨⣯⣤⣤⣤⣤⢴⡶⣾⢻⣿
⠀⠀⠀⠀⠀⠈⣿⡾⠼⣀⣇⢸⠀⢸⠀⡇⢸⠀⢸⠀⡏⢸⠁⡇⢸⡅⣿⢸⠀⡇⢸⢠⠃⣿⣼⣿
⠀⠀⠀⠀⠀⠀⢹⣿⠿⢶⣮⣭⣗⣛⡢⠿⠼⣄⣸⣄⣧⣸⣀⣇⣸⣄⣿⣸⣤⠧⠼⢜⣒⣫⣿⡟
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢠⠈⠉⠛⠛⠻⠿⠶⢶⣶⣶⣶⣶⣶⣶⠶⠶⠶⠾⠿⠛⠛⠛⣿⠁
⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠐⣿
⠀⠀⠀⠀⠀⠀⠸⣿⠙⠳⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⠀⠈⠉⠛⠛⠲⠶⠶⠶⠶⠶⠶⠶⠶⢶⡆⠸⠃⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠣⡀⡜⢳⢰⣄⣤⢀⡤⡀⢀⠀⣀⠀⠀⠀⢸⠇⢨⡅⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠒⠻⠋⠿⡏⣿⢸⢸⣤⠟⡾⢀⡧⠀⠀⠀⣾⠀⠘⠂⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢀⠀⠀⠀⠀⠈⠘⠀⠀⠳⠘⠧⠀⠀⠀⣿⠀⢰⠂⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠰⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣧⡜⠛⠛⠲⠦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣼⠏⠉⡉⠙⠒⠦⣄⣿
⠀⠀⠀⠀⠀⠀⠀⢻⡇⠉⢓⠦⢄⣀⡀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⣓⣠⠴⠒⢩⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⠸⠀⠀⠈⠉⠉⠓⠒⠒⠒⠲⠶⠒⠒⠒⠋⠉⣅⠀⠀⢀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣟⡶⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣤⠶⣛⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣶⣭⣭⣛⣛⠛⠛⠛⠛⠛⢛⣛⣋⣭⣭⣶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠿⠿⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢠⡐⠒⠉⠉⠉⠉⠉⠉⠈⠀⠀⠉⠉⠉⠉⠉⠉⠑⠒⢢
⠀⠀⠀⠀⠘⡟⠛⠓⠶⠶⠶⠶⠶⠦⣤⣴⠶⠶⠶⠶⠶⠖⠛⢋⡿⠀⠀⡠⠖⠋⢲⡀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠀⠉⣩⣩⡇⢀⡜⢀⠔⠉⠉
⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣇⡜⣰⣁⣀⣀⠀⠀⠀⣀⣀⣀⡀⢠⠔⠂⠐⠢⠐⠀⠐⠀⡀
⠀⠀⠀⠀⠀⢸⡧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢧⠀⢸⣱⡡⠤⠐⠚⠛⢒⠎⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠸⠿⠀⠀⠀⠀⣠⠄⠒⠒⠒⠒⢒⣳⣾⣅⣀⡀⠠⠤⠄⠺⡀⠀⠀⠀⠀⠠⣀⠀⣀⡴⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠈⠁⢉⢜⡯⠋⣱⠆⠰⣄⠀⠐⠢⣀⠎⠀⠀⠀⠀⠀⠀⠈⡽⠉⠀⠀⠀⠀⠀⡜⡆
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⢀⠜⠁⡆⠀⠇⠙⠦⣀⠸⡣⣀⠀⣀⡤⢉⣛⣋⠴⠦⣄⠠⣀⣠⠜⢁⠇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣴⣦⠀⠀⠀⠀⢠⣏⡤⠚⠁⠀⠀⡇⠀⠰⠀⠀⠈⠑⠳⠬⣉⠩⣴⠣⡙⢌⡾⠀⣺⠑⠂⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⢱⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠃⠙⠪⣍⠁⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⢧⠑⠤⠄⠀⡀⠀⠀⠀⠀⠀⠀⠤⠽⢄⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠈⠆⢠⡌⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠔⠉⠑⠦⢄⣀⠀⠀⠀⣀⡠⠔⠋⠀⠀⠀⠀⠀⠀⣆⡜⣀⠼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁






⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣀⣔⣡⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⡰⡩⢊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢑⠢⡀⠀⡁⠰⡐⢡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢆⠎⠑⢌⢢⡇⢸⢃⡧⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⣁⡀⠀⣀⠱⡱⠸⡞⡐⠪⡓⠄⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠒⠬⡙⣷⢡⡁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠬⣠⠤⠄⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠈⠳⣄⠀⣀⣠⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠙⡯⠗⣚⡁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⠀⢀⡀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⠀⣀⣤⣀⡠⠐⠂⢄⠔⠚⠚⠟⠋⠤⡈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⣠⣶⣿⣿⣿⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣼⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⢑⡖⠋⠀⠀⠀⠟⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢿⣿⣿⡿⠋⠹⣓⠦⢖⣤⡭⣭⠶⣄⢄⣀⠼⡸⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠉⠁⠀⠀⠀⠀⠁⡩⢮⡺⠟⢸⡏⠑⠲⡮⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⢀⠤⢊⠴⠋⠈⠀⡀⣇⠀⠀⠱⡴⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠂⠤⠍⠔⠊⠁⠀⠀⠸⠴⠡⠜⠀⠀⠀⠈⠢⠂

⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣿⣟⣻⣿⣻⣟⣟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠶⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠐⠂⠀⠈⠉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⡀⠀⢀⡠⠤⢄⠀⠀⠀⠀⡇⠠⠄⠀⠈⠁⢀⡠⠈⡇⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣳⣤⣞⠀⠀⠀⢳⠀⠀⠀⠘⢦⣀⠀⠄⠀⠀⢀⡼⠁⠀⠀⠀⢀⡴⠋⣉⣀⣠⣤⣤⣤⣭⣭⣉⡓⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡴⠁⠀⠈⢳⠀⠀⢸⠀⠀⠀⠀⠀⠈⠉⠒⠒⠋⠁⠀⠀⠀⠀⢀⡾⠞⠛⠉⠁⠈⠀⠀⠀⣀⠤⢬⣉⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⡇⠀⠀⠀⢨⣧⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⣀⠤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣀⠜⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡤⢤⣤⣤⣄⣀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴
⠀⠀⠀⣰⡿⠛⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⡫⠉⠐⠒⠈⠐⠒⠂⠈⢍⢻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⡁⠜⣅⢀⠀⡀⢀⢀⠀⡀⣀⢂⡡⣰⠌⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣭⣥⣿⣿⣿⡿
⠀⠀⠀⠈⠳⣄⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⢁⣤⡔⢕⢁⣬⡀⠗⣩⣍⠦⢗⣴⡁⡘⠄⢿⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⣷⣾⣿⣿⠿⣿⠟⠁
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⣟⡵⠋⠀⢙⣒⠋⠈⠓⠞⠁⠉⠛⣋⡈⠙⢶⣔⢸⠀⠀⠀⠀⠀⢀⣴⣿⣖⣋⣩⣯⣿⣷⣾⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠈⣧⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠈⢻⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠙⠛⠻⠟⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠋⠀⠀⠀⠀⢠⠚⡇⠀⠀⠀⣾⣷⠀⠀⠀⣤⠀⠀⠀⣾⡆⠀⠀⠀⣼⠓⡀
⠀⠀⠀⣀⣠⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⢠⠶⣬⣦⢷⠀⠀⣤⠈⠁⣠⣤⡤⠤⢤⣤⣄⠀⠠⡤⠀⢀⣯⣞⡵⢦⠀⠀⠀⠀⠀⣠⣤⣤⣴⣧⠤⣤⡀
⠀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣴⠉⠸⠥⢈⡏⢧⡀⠀⠀⠀⠳⣄⡀⠀⣀⣠⠏⠀⠀⠀⣠⠎⣟⠀⡽⠊⢱⡄⠀⠀⢸⠁⠀⠙⠛⠋⠁⠀⢻
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⠦⣤⣠⠾⢦⡀⠉⠲⢤⣄⣀⣀⠉⠉⢉⣀⣀⣤⠴⠚⠁⢀⠼⢦⣀⣤⠞⠁⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⣸
⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠹⣄⠀⠀⠙⠒⠛⠉⠀⠀⠉⠉⠉⠉⠁⠀⠀⠉⠓⠒⠉⠀⠀⡼⠁⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⣠⠏
⠀⠀⠉⠛⠛⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⣤⠤⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⣀⣠⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⡴⠚⡉⢱⠀⠀⢀⠜⠁⣸⠁⠀⠀⠀⠀⠀⠀⢧⣤⣤⣀⣀⣀⣀⣠⣤⣤⡿⠀⠀⠀⠀⠀⢀⣠⡤⠴⢶⣶⡯⣍⠉⠓⠦⣌⠉⠓⢦⡀
⠀⠀⠀⠀⠀⡜⠘⡆⠱⠀⣡⠖⠁⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠉⣮⡇⠀⠀⠈⡧⠀⠀⣹
⠀⠀⠀⠀⠰⡇⠀⣀⡤⠚⠡⡀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⣠⣠⡾⣄⣀⣄⣠⠇⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⢰⢯⠀⠀⢠⡖⠁⢀⡟⠁
⠀⠀⠀⠀⠠⡏⠉⠁⢄⠙⠆⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣖⣒⣺⠀⣿⣒⣲⡇⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢸⣿⠀⠀⡼⠀⠀⡼
⠀⠀⠀⠀⠀⠙⠦⠤⠤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⢀⣸⠆⣏⡀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠸⣄⣀⣀⣀⣸⡿⢤⡼⠧⠤⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠰⠀⠀⠀⠀⠀⠀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠴⠂⠀⠀⠀⠀⠀⠸⡀⠀⠀⠠⣀⢤⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢱⠃⠀⠀⢀⡐⠁⣀⠤⠒⠘⡞⠁⠀⠀⣇⣀⠆⠀⠀⢀⡠⠒⠉⠀⠀⢱
⠀⠀⠀⠠⢦⠀⠀⠀⠀⢇⡀⣀⠔⢊⠩⠿⡏⠀⠠⢤⠄⠑⠦⡤⠊⡩⠋⣱⠄⠒⠉⠀⠀⠀⠀⣠⠃⡴⠞⠁⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠚⣉⠀⠀⠓⠾⣄⠀⠸⢤⣋⡠⠴⠳⣆⣀⣀⡀⢤⣐⡨⠔⠚⠣⡀⠀⠀⠀⠀⠘⠤⠠⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⠁⠀⠀⠀⠀⠙⢆⠀⢀⡁⢀⣴⡆⠀⠀⢀⡠⠖⠁⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠾⠂⢎⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⡀⠀⠀⠰⡰⠀⠀⠀⢀⠤⣄⢀
⠀⠀⠀⠀⣠⢬⡗⣎⣠⠤⠤⠊⠃⠀⠉⠐⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣐⠀⠙⢶⣄⢀⠀⠀⠀⠀⢈⠂⠠⠄
⠀⠀⠀⠀⢯⣉⣻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣁⣀⠀⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⡋⠻⠷⠿⠫⢥⢾⣕⣒⣒⢒⠒⢒⢒⣒⣒⣾⠡⠮⠢
⠀⠀⣀⠀⢈⣿⡿⡟⡆⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠀⠀⠀⠀⠈⠙⠳⢦⣄⡀⠀⠀⢸⠀⠀⠐⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⡎⠘⢷⣺⡿⠁⡅⠈⠄⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠁⠈⠁⠘⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤
⠀⠀⠀⠀⠠⡇⠀⢀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠐⠚⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣽⠲⢋
⠀⠀⠀⠀⠀⢀⡀⠀⣆⠀⠈⠻⠮⣄⠀⠀⠀⠀⠀⠁⠢⠠⠄⠀⠄⠤⠊⠉⠒⠦⠤⣤⠤⣤⠀⡤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠑⠁
⠀⠀⠀⠀⢀⣀⢴⠉⠁⠀⠀⢀⠇⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠁⠠⢀⠔⠁⢰⠾⡀⠀⠀⠀⠳⢤⣴⣀⡑⠈⠑⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠈⠒⠒⠁⠀⠑⠤⠀⠀⠠⠴⠶⠯⢝⡛⠲⠶⣤⣄⣈⠉⢒⠲⠦⠖⢒⡒⠢⢠⠴⠓⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠴⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⣚⠛⠭⠿⠿⠶⢦⡤⠄⠄⠴⠁⠀⠀⠀⠀⠀⠉⠁⠐⠒⠒⠒⠀⠐⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠐⠒⠒⠾⢗⣒⣒⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠆⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⣴⡦⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣾⣿⣯⣿⣿⣿⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣻⣿⣿⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠿⠣⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣾⣿⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⠀⠀⠀⠈⠳⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣿⣓⣿⣿⣹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⠀⠀⠘⣃⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠠⣤⠀⠠⣤⠀⠠⣤⠀⢠⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⡃⠀⠀⠀⠀⢸⣿⣿⠀⠀⠈⠁⢀⡈⠈⢀⡈⠉⢀⣘⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠠⣴⣶⣶⣶⣾⣷⣶⣦⠀⠀⢸⣿⣿⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⢹⣿⣿⣿⣿⢁⣽⣿⣿⣿⣿⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠀⠀⠀⠈⠿⠿⠿⠿⠏⠑⠒⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⠈⣿⣿⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⢈⣿⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⣸⣯⣉⣩⣿⣸⣿⣿⣿
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠲⠋⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣳⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⣶⠀⠀⠀⠀⢠⣖⡶⠰⠾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣀⠈⠉⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠿⠃⠀⠀⠀⠀⢰⣟⣷⠀⠀⠀⢰⡏⠉⢷⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠟⠸⣟⡆⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠳⠶⠋⠀⠠⣏⡇⠀⠀⠀⢠⠤⡄
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠰⣯⠇⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡈⠛⠃⠀⠀⠀⣀
⠀⠀⢀⡴⠛⠉⠉⠉⠉⠉⠙⢳⡄⢀⡴⠚⠛⣷⠞⠉⠉⠙⢦⡀⢀⡴⠛⠉⠉⠛⠶⣄⠀⡴⠋⠀⠀⠀⠀⠙⢦⠀⡠⠞⠉⠉⠉⠛⠲⡄
⠀⣰⠋⠀⠸⠁⣀⣤⣤⡀⠀⢀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡞⠀⡖⠀⠀⠀⠀⠈⢿⠃⠀⠘⠁⠀⠀⠀⢈⡿⠁⠀⠀⠀⢠⣤⡴⠃
⢠⡏⠀⠀⢀⡾⠉⠀⠀⢷⠷⣼⠃⠀⡇⠐⠀⢀⠶⡄⠀⠠⡞⢹⠀⠀⠀⡠⢤⠀⠀⠀⠘⡆⠀⠀⠀⡔⠒⠋⠉⠱⠤⠤⠤⣤⡀⠉⠉⠲⣄⢸⣿
⢸⡇⠀⠀⢸⠁⠀⠀⣤⣀⣀⣀⣀⣀⣧⣀⠀⠀⠷⠃⠀⠀⠹⣿⠀⠀⠀⣇⡸⠀⠀⠀⢀⡇⠀⠀⠀⠧⣀⣀⡀⠀⡀⠀⠀⠀⣹⠀⠀⠀⢹⡄
⠘⡇⠀⠀⠘⣇⣀⣸⣧⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠈⠁⠀⠀⠀⡼⠻⢿⣶⣄⠀⠀⠀⠹⡏⠉⠉⠛⠋⠁⠨⠃⠀⢸⣧
⠀⣿⡄⠀⠀⠈⠻⠭⡽⠇⠀⠀⠀⣴⢶⠚⠀⠀⢠⣄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢠⡞⢳⠀⠀⠉⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠞⢸⡄
⢰⠃⠙⠶⣄⠀⠀⠀⠀⢠⣄⠀⠀⣿⢸⠀⠀⠀⢸⡟⣆⠀⠀⠀⠀⣶⢦⠀⣤⠀⣸⡇⢸⣠⣦⠀⠀⠀⢀⣤⡔⠁⠀⠀⠀⠀⠀⣀⢸⡇⠀⢸⡇
⡿⠀⠀⠀⢸⠀⡀⠀⠀⣏⢹⣤⢠⣼⠈⠛⢻⠀⢸⠀⢸⠀⣠⡀⡟⠁⢸⢰⡟⠷⠿⠇⠀⠉⠸⣄⣶⡀⢸⠀⢻⡀⡶⡄⣴⢦⠐⡟⠛⠀⠀⠘⡇
⡇⠀⠀⠀⢸⣸⠟⠞⣇⣿⠀⢸⢸⠀⠀⠀⠸⢾⣼⠀⢸⠀⡏⠛⠁⠀⢸⣆⡇⠀⠀⠀⠀⠀⠀⠛⠁⣇⣼⠀⠀⢿⡇⠳⠟⠸⣦⠇⢀⠀⠀⠀⡇
⡇⠀⠀⠀⠈⠟⠀⠀⠙⠃⠀⠘⠿⠀⠀⠀⠀⠀⣿⠄⠸⣦⠇⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⣇
⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢀⡀⠀⣿
⣿⠀⠀⠀⠂⠀⠀⠘⠆⠀⠀⠐⠄⠀⣰⣆⢀⡀⠰⠄⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠈⠃⠀⠀⠀⠉⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢻
⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡇⢸⠞⣧⠀⠀⠀⣴⠀⢠⣦⠀⠀⠀⣤⡀⠀⠀⠀⠀⢀⣿⣀⠞⣧⠀⠀⣤⠀⡄⠀⣴⠀⣀⢀⣄⠀⠀⢸
⢻⠀⠀⠀⠀⠀⡼⢻⠀⠀⢀⣄⠀⢰⠃⠀⠀⣿⠀⠀⢰⡏⡇⡾⠸⡄⠀⢸⠇⢳⡀⣷⣴⠀⢸⠉⠛⠀⢸⠀⠀⢸⠶⠻⠞⢹⡆⠉⢸⢹⡄⠀⢸
⢸⠀⢠⣤⠀⠀⡇⢸⠀⠀⣸⢹⡄⢸⠀⠀⠀⢸⠀⠀⢸⠀⠙⠁⠀⡇⠦⢸⠀⠈⠻⠁⢹⡆⢸⠀⠀⠀⢸⡆⠀⢸⠀⠀⠀⢸⡇⠀⢸⠈⣇⠀⢸
⠘⢷⠏⢸⠀⠀⡇⠈⡇⠀⡁⠀⡇⢸⠀⠀⠀⢸⡄⠀⢸⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠈⡇⢸⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠸⡇⠀⢸⠀⠹⣦⡞
⠀⠀⠀⠀⡆⠀⡇⠀⣧⠀⡇⠀⠹⠟⠀⠀⠀⠈⡇⠀⢸⠀⠀⠀⠀⣷⠀⣿⠀⠀⠀⠀⢀⡇⢸⠀⠀⠀⠀⡇⠀⡏⠀⠀⠀⠀⣇⠀⢸
⠀⠀⠀⠀⢷⣸⠁⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⡾⠀⠀⠀⠀⣿⠀⡻⠀⠀⠀⠀⠸⡇⣸⠀⠀⠀⠀⢻⡼⠁⠀⠀⠀⠀⠘⢦⡟
⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⠃⠀⠀⠀⠀⠸⣆⡇⠀⠀⠀⠀⠀⣷⠏

⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣾⣁⡀⠉⣹⠟⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠉⢹⣏⣼⠏⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣀⡴⠟⠛⠛⠛⠛⠛⠛⠷⣄⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣷⠀⢰⡟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢿⣀⣸⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠉⠁⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣿⡟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿


⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⣇⣀⠗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⠀⠈⠉⢱⠐⠂⢉⠉⢠⢻⠀⢨⣉⡆⢰⠀⢸⣍⡁⠀⠸⣉⠀⢹⡏⠀⣍⡑⢐⠋⠙⠆⢠⡜⠀⣯⡉
⣀⣀⣀⣀⣀⣀⣀⣀⣨⣤⣄⣠⣤⣊⣐⣂⣈⣀⣎⣉⣃⣰⣤⣁⣰⣤⣘⣤⣄⣀⣠⣤⣃⣸⣇⣀⣁⣡⣘⣦⣤⣃⣀⣘⣄⣧⣤⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠻⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣦⣶⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢧⣤⡄⠀⣤⣤⣤⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠇⠀⠀⠀⠘⠛⢻⣿⣟⣛⣻⣿⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛⠛
⠀⠀⠀⠀⠀⣇⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡗⠶⠦⠤⠤⠤⠶⠖⠒⠒⠶⠦⠤⠤⠤⠶⠖⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣭⣭⣛⣛⣛⣛⣋⣭⣭⣭⣭⣛⣛⣛⣛⣋⣭⡅
⠀⠀⠀⠀⠀⢸⡇⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠚⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠿⠙⢒
⠀⠀⠀⠀⠀⠀⢠⠔⠘⠉⡏⠀⠀⢲⠀⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠲⠛⠁⡀⠂⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠀⠀⣥⠀⠀⢈⠉⠀⢡
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠂⠉⠈⢀⠀⠈⠓⠀⠀⣆
⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡔⢠⠲⠨⠒⠉⢒⢦⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠠⠀⠁⢲⠒⠎⠀⠀⠈⠢⠄
⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⡀⡤⠶⡘⣀⣃⠥⡇⢌⠰⡝⢫⠊⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡴⠊⠀⠀⢀⡤⣀⠠⢤⠄⢤⢄⣼⡇
⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠱⠁⡜⠁⠌⡄⠐⠘⠌⠥⢃⠡⣂⠄⠀⠀⠀⠀⠀⠀⣄⠠⠂⠀⠀⠀⠀⢐⡦⠉⠁⠈⣦⠊⠉⠙⢉⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠈⡉⠐⠊⠀⠉⠂⡐⠊⠠⠐⣀⣠⠞⠁⠀⠀⠀⠀⠀⢀⡈⣦⡄⠐⢴⠀⠀⠀⠈⠡⠤⠕⢊⠁⠠⠀⣀⠈⠂⠚⠁⡶⠦⢠⠒⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⢦⠹⡇⡙⡛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠊⠓⠚⠻⠤⠴⠀⠀⠀⠉⠛⠢⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠠⣿⣿⣣⡸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣭⢹⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠖⠊⠉⠉⠉⠉⠑⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⣀⠙⢁⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⢀⠀⣒⠀⢄⠀⠈⡉⠕⡄
⠀⠀⠀⠀⠀⢀⣀⢀⣰⣋⣉⣉⢀⡎⠓⠀⠐⢋⣀⣀⠀⠄⡀⠀⠀⢦⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⡴⠋⣀⠀⢆⠉⠀⠈⠆⢰⡀⠀⢹⠀⢼⡀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠸⢆⠀⠻⢄⠈⠉⠐⠒⣰⠥⢈⣁⠬⠤⢀⡈⢣⠀⠀⠙⠒⢤⢠⡀⠀⡤⠒⠉⠁⠀⠀⠉⠀⠨⢁⠂⢘⠂⣤⡵⠶⢾⣤⠄⢿⡉
⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠢⠔⠀⠂⢠⡀⠀⠤⣈⡂⣀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠈⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡀⣋⡄⠀⠀⡋⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠒⠶⢈⡈⠀⠀⢫⡈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡈⠀⠀⠁⠀⠊⠉⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠀⠀⠀⠀⠘⢅⡈⠀⠈⠓⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢦⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠈⠒⠒⠂⠈⠓⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⡂⠀⢩⠀⠀⠀⠂⠀⠀⢀⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠂⠀⠄⠀⠀⠀⠀⡀⠀⠀⠂⠀⠁⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠸⣠⣀⣀⣀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⡔⢹⡀⠉⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡤⠄⣰⢀⣼⣷⣴⣿⠀⠀⠈⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢀⣈⠛⠻⠁⠁⣸⣿⣆⡂⠀⣰⣷⡾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⢸⠃⢠⡄⢀⣀⠄⢶⣿⢛⣩⣤⡤⠀⠙⢿⡿⠻⣿⣿⠋⠀⣷⠤⠄⠀⠀⠀⠀⠀⠘⠂⠀⡅
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⠀⠈⠀⠚⠢⠿⣤⣼⢹⣿⠶⣡⣼⠟⢁⣴⣿⡇⠘⣷⣻⡌⢻⣶⠒⠛⣧⡠⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣴⢼⢸⣏⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣭⣤⣜⣋⣀⣽⢳⡆⠘⠛⠁⢾⣿⣷⣶⣌⣵⡃⠆⡀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⡈⣿⠛⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⢬⣭⣭⢁⡾⠀⠀⠀⣰⣾⣾⢛⣿⣯⣌⠅⠀⢐⠂⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⣶⡏⢠⣘⠿⣿⣿⣽⡳⣷⡄⠀⡀⠈⠀⣶⠶⠒⣲⣦⠜⣁⠌⠀⠀⠀⢰⣿⣷⣶⣿⣽⣟⣿⣦⠀⠢⠄⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⣿⣧⣾⣽⣻⣿⣿⣿⣿⣯⡿⣷⢈⡀⣄⠘⣿⡟⠋⢡⠔⠁⠀⠀⠀⠀⢼⣿⣿⡛⣶⣾⣿⣿⣿⠛⡶⢤⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⡜⣇⢹⣧⠐⢀⡴⠎⠁⠀⠀⠀⠀⠀⠀⠂⣿⣿⣿⣿⡿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⢿⣿⠞⠋⣠⡾⡍⠂⠀⠀⠀⠀⢨⣷⣶⠀⢡⣹⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠷⠊⠁⣤⣿⡿⠂⠁⠀⠀⠀⠀⠀⠀⠙⠈⢠⣟⠁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠀⢠⣤⣶⡟⠷⠉⠀⠀⠀⠀⠀⠀⣠⡴⢷⣦⣤⣀⣈⣷⣿⡛⣷⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠐⣾⡼⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⢣⣶⣬⣭⢟⣷⣿⣿⣿⣌⠛⠉⠒⠤⡄⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⡠⠁⡄⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡹⣿⣞⢿⣿⣿⣿⣿⣤⣰⠾⣿⣤⣉⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠰⠁⠠⢟⣀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣿⣆⣿⣿⣿⣟⢿⣿⣷⠷⠶⣽⠓⠄⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⢀⠃⠀⠀⠈⢁⣤⣼⣾⣧⡔⠀⠀⠀⠀⠀⠀⠀⣴⣶⡆⠀⠈⣩⣿⣿⣿⣝⡿⠇⠙⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠃
⠀⠀⠀⠀⡌⠀⠀⢀⢾⠟⣿⣿⡟⠛⣿⣥⠀⠀⡀⠀⢰⣿⣿⠋⣠⣤⣀⠛⠛⠸⡏⠟⠀⠀⠠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠘⢻⠏⠀⣿⠟⠀⠄⠙⢿⠀⠿⠏⠀⠀⠉⠁⣹⣿⣿⡻⣿⡶⣤⣴⡄⢳⣶⣄⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⡄⡈⠀⠀⣿⡏⠄⠀⠀⠠⡀⠄⠀⠀⠀⠀⠘⣹⢿⡛⢷⣌⢻⣯⢩⣿⢦⡻⢏⣙⠶⣄⠀⠀⠀⠀⠀⠄⠀⡰⠁
⠀⠀⢀⠀⠀⢸⢣⠁⠀⠀⢹⣇⡀⠀⠀⢀⣀⡀⣰⡀⠀⠀⠀⢻⣎⠻⣗⣾⡿⣿⣦⡙⢷⡤⠈⠈⠣⠈⠓⠀⠀⡡⠞⠁⣸⠃
⠀⠀⢠⠀⠀⢬⡄⠀⠀⠀⠸⣷⣁⠀⠀⣾⡇⠾⣠⣿⣦⡀⢀⡰⠿⠹⣮⠻⣧⣽⣿⣿⣾⣿⣦⠀⠀⠀⠀⢀⠀⠀⢠⣬⠃
⠀⠀⠈⠓⠘⠋⠁⠀⠀⠀⠀⢳⡈⠀⠀⠻⠄⠠⢿⣿⣛⠇⠀⠀⠀⠦⢀⠐⠀⠀⠀⠀⠀⠈⠉⣳⢄⡱⠄⠀⠁⣠⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⡖⠀⢿⣿⠀⣆⡀⠄⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠈⠩⡆⠄⠐⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠈⠀⢨⠋⣶⣿⠱⡆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠈⠆⠈⠐⢿⠂⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⡂⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⢦⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢰⠀⡤⠀⡠⠘⠦⠈⢀⣰⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⣀⢀⣀⣀⣤⠶⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠉⠲⡀
⠀⠀⢀⡠⠤⢄⡀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠗
⢀⠞⠁⠀⠀⠀⠈⠢⡀⠀⢸⠀⠀⠀⠀⠄⠀⠤
⠄⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⡄⠀⠀⠀⠀⠀⡄
⠸⡀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⣄⠀⠀⠀⢰
⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⢢⡀⢀⡇
⠀⠀⠀⠑⠦⢤⣀⡀⠀⢀⣀⣘⠦⠎⠈⠁⠘⠒⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠁⠉⢉⡹⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠒⡲⠖⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⡀⠀⠐⠠⠄⠐⠁⠋⠀⢉⠈⡉⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠚⠄⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠦⠄⠃⡴⠔⣁⢀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⠂⠐⠊⠀⠀⠀⠀⢢⡇
⠀⠀⠀⠀⠀⠀⡴⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⢱⢦⢤⠄⡀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡤⠞⠳⡀⠀⠘⡀
⠀⠀⠀⠀⠀⠈⢳⠆⠀⠀⠀⡀⣄⠲⡒⢲⠰⠶⡖⠒⠲⠖⠉⠁⠀⠀⠀⢸⠄⠀⢈⡅
⠀⠀⠀⠀⠀⠀⠈⢆⣀⣀⣅⠳⣌⢧⣙⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢯⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡙⣿⣮⠓⡾⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠎⠁⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡠⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠶⣄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢻⡄⠙⢄⢀⡔⢉⡉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠇⣣⠈⣿⢆⠀⠉⠃⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠤⠋⠀⢸⠈⠣⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠅⠀⠈⢲⡤⡤⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠠⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⡼⠁

⠀⠀⠀⠉⠢⡀⠐⠀⠑⠀⠀⠀⠀⠀⠀⠀⢄⠰⢦⠀⠀⢀⣀⠀⠀⠁⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠳⠤⡅⠐⠈⠉⠦⡀⠀⠠⣄⢰⠄⣀⠀⢠⡊⢁⠽⠛⠀⠀⠀⠀⠀⠐⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡿
⠀⠀⠀⠀⠀⠀⢦⣆⡀⠀⠀⠀⠀⠀⠀⠂⠀⠰⠲⠆⠀⠀⠀⠀⠈⠀⠀⠘⠛⠂⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢤⠔⠷
⡀⠀⠀⠰⢢⠈⢦⣿⠛⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢢⠀⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁⠀⢀
⣿⠄⡀⠀⠈⡓⣄⢻⣅⡉⢋⠅⠀⠀⠀⠀⠀⠀⠀⠤⢄⠀⠀⠀⠀⠀⠀⠀⡀⠈⢀⠀⠀⠙⠄⠀⠀⠂⠀⠀⠐⠂⠀⡠⠐⢈⠠⣀⡀⠀⠀⠀⠀⠠
⠥⠚⠁⠀⠀⡁⠻⢧⣙⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣄⠀⠀⠀⠀⠀⠀⠰⡄⠈⠁⡆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠐⢢⠀⠀⡔⠶
⠀⠀⠀⠀⠀⠀⠀⠁⠟⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠁⠀⠟⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠱⠀⢀⡴⡎⠉⠉⢷⣤⢤⣀⠀⠀⠀⠀⠀⠁⠉⠀⠀⣆⣀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⡰⠞⡋⠀⠀⠀⠀⡏⢧⣶⢄⠀⣼⣿⡄⠀⠙⢢⠀⠀⠀⠀⠀⠀⠀⣹⠉⠃⠀⠀⠐⠀⣤⣀⡀⠀⠀⠀⠘⠃⠇⠘⠠⠕⠣⠚⠅⠛⠸⠐⠄
⠀⠖⡀⠘⠤⠄⠁⠀⠀⠀⢰⡦⠈⠁⠈⠑⣿⠇⠀⠀⠀⠛⠁⠀⠀⠀⢀⣀⣴⠋⠀⡀⠈
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠻⢦⣶⣶⡾⠋⠀⠀⢠⡼⠃⠀⠀⠀⠀⠈⣉⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⣤⣀⣰⠟⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⡦
⠀⠀⢸⡿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠟⠁
⠀⠀⠈⠁⠀⠛⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠋⠀⠀⠈⠀⢁⠈
⠀⠤⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄⣀⣀⣀⣀⣴⣾⣿⣿⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡀⠀⠀⠀⠆⠀⡸⣄⠀⠢⡀
⠀⠒⠁⠀⠰⡆⠀⠛⠅⠛⠈⠉⢻⡟⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⣰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡤⠀⠤⠄⠀⠄⠀⠀⠂⠈⠀⠈⠁⠀⠀⠀⢃
⢀⣀⣀⣠⠜⠁⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠑⠀⠀⠀⠀⠀⣀⣠⣴⣀⡖⣦⠀⠀⠀⠀⠀⠀⢾
⠀⠉⠀⠀⠀⣀⡀⠠⠀⠀⠀⠈⠛⠿⠗⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣷⣶⠲⠿⢅⣀⣁⣌⡷⠁⠀⠀⢀⠀⢀⡤⠞
⠀⠀⠀⠴⠆⠤⠭⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⡆⠠⣴⡄⠀⣈⡁⠀⠾⡼⢌⡟⢰⡶⣤⣤⣤⠀⠐
⠀⢀⠀⠘⠲⠂⠀⠀⠉⠀⣀⣀⣠⠗⠨⠀⠦⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠽⠀⠀⠀⠀⠀⠈⠃⡜⠀⠩⡝⠦⡀⠀⠀⠀⠤⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠄⠀⠄⠄⠀⠈⠳⠀⠀⠀⢀⡄⢠⡀⠢⠀⠀⡎⠀⠀⣀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠷⠀⠂⠀⠀⢀⠀⢠⠤⠄⠐⠀⠀⠂⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⢯⡥⠀⢀⠆⠀⠑⠀⠠⡀⠈⢢⡁⣶
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢙⠀⠀⠀⠀⠀⠀⣎⠁⠀⠠⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠔⠁⠀⠀⠀⠀⡀⠠⣤⡀⢳⣿
⠀⠀⠀⠀⠀⣀⣈⣀⠒⠒⠁⠀⠀⢀⡄⠠⢀⠀⠀⠲⣚⠀⠇⠀⠀⠘⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣿⡄
⠀⠀⠀⠐⠁⣤⣀⠈⢣⠀⠀⡉⠀⠛⠹⠄⠀⢁⠀⠀⠉⠀⠀⠀⠀⠀⢄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠠⠉
⠀⠀⠀⠀⠠⣿⣧⠆⢀⠀⠀⠃⣀⠀⠀⠀⠀⠌⠀⠐⠙⢄⠀⠀⠀⠀⠀⠣⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠈⠀⠀⠀⠔⠃⠀⠀⠀⠙⠀⠀⡠⠊⠀⠀⣠⣸⠆⠖⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠁⠀⠀⠠⠀⠀⠀⠐⠀⠀⢱⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠠⠤⠚
⠀⠀⠀⠀⠀⠀⠈⠀⠒⢤⣄⡀⠀⠀⠐⠀⠀⠀⠈⠢⣤⣤⡠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠠⠆⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢠⡄⡀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⡆⢀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡤⢤⣀⡇
⠀⠀⠀⠀⠀⠐⠲⣤⡄⠐⠶⠦⠤⣤⣀⡍⠈⠓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣘⣦⠀⠀⠀⠀⡆⠀⠀⢰⠀⢙⠁⠀⠀⠄⠘⠀⠀⡦⠀⠂⠀⠈⠃⠀⠠⣀⣠⡶⠀⠀⠄⠁⠀⠀⠁⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⣍⡳⡄⠀⠀⡇⠀⠀⠈⠀⢘⠀⠀⠀⠀⣠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠂⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣀⢀⡀⡀⢸⠀⠀⠀⠀⠈⠀⢠⢀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⡀⠄
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠠⠄⠀⠀⠀⠁⠈⠈⠀⠀⠘⠀⠀⠂⠀⠀⠀⠈⠀⠀⠃⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠉⠀⠈⠁⠀⠉⢢⠀⠀⠀⠀⠀⡄⡰⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡠⠀⣀
⠀⠀⠀⠀⠋⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢣⣇⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⣂⣀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡄⠀⡀⠀⢸⠀⠀⠆⠀⠀⠈⠺⠏⠁⠄⠀⠀⣠⠤
⠀⠀⠀⠀⠀⡀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠁
⠀⠀⠀⠀⡰⠁⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢐⡀⠀⢰⠂⠀⢀⡀⠀⡄⠀⣀⠀⢀⠌⠀⠀⠀⠀⠐⠀⣠⠊⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠠⠀⠈⠁⠀⠀⣵⠊⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠
⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠠⠀⢸⠀⠀⠀⠀⠀⠀⠈⠃⠀⡰⠁⠀⠈⢀⠀⠀⠀⠀⠀⠠⠀⠂⠃⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⢠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣀⠀⠀⠐⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠤⠊⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠀⠒⠹⢆⠀⢀⡤⢀⠀⠀⡠
⠀⠀⠀⠀⠻⡀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠤⡀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠂⠉⠀⠈⡆⠸⢇⡀⠤⠊⠀⢰⣄⠤⠀⠐
⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠄⠀⠈⠐⠀⢈⠀⠃⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⠀⣀⣀⣠⣴⠄⠀⠀⠀⠀⠀⢴⠈⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣁⡀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠻⠇⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠈⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⢄
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⣸⠀⠆⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠋⠁⠀⠀⠀⠢⠀⠀⠀⠰⢼⠀⠀⠀⠀⠀⠀⠉⠳⣤⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠒⠒⠀⠀⢹⠀⠀⠀⠂⢸⠀⠀⠀⠠⠀⠀⡀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠙⢦⡄⠀⠀⠀⢺⠀⠀⠀⠆⢸⠀⠀⠀⢠⠄⠤⢀⣀⠀⣹⠧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠈⢧⡀⠀⠀⢸⠀⠀⠀⠁⢸⠀⠀⠀⢠⠀⠀⠀⣤⡾⠁⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⣈⡞⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢸⠀⠀⠀⡇⠘⠀⠀⠀⢸⠀⢀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢹⠀⢸⠀⠀⠀⡆⠀⠀⠀⠀⢸⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⢰⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⣀⡎⠀⡄⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠘⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠤⠤⠄⠊⠁⢠⠀⠁⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⡀⢸⠣⣄⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸⠀⠀⠀⡇⢠⠀⠀⠀⢸⠀⠀⡇⢸⠀⠀⠈⠀⠒⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⡇⢸⠀⠀⠀⢸⠀⠀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡄⠀⠀⠀⡇⡘⠀⠀⠀⢸⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠀⠀⠀⠀⢷⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢠⠃⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠌⠀⠀⡠⠀⠀⠀⢃⡀⢄⠀⠀⠘⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡟⠀⠀⠌⠀⠜⡀⠀⠘⠀⠀⠆⠀⠀⢁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡌⠀⡐⠀⢁⠀⠀⠇⠀⠘⡀⠀⠈⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡘⠀⠰⠁⠀⡸⡄⠀⠸⠀⠀⢡⠀⠀⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠔⠁⠀⠃⠀⠔⠀⠐⠄⠀⠃⠀⠀⠢⠀⠐⠃⡠

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠋⠛⠟⠉⡛⠛⣟⠛⣛⡿⣛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠴⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠂⠄⠀⠀⣀⣀⠀⠀⠠⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⢦⣀⠸⠀⠀⠀⣀⠰⠠⠒⠀⠐⠀⠒⠀⠀⠀⠀⠀⠀⡀⠻⠏⢀⠂⠈⢂⠠⠈⢢⣰⠃
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠓⢄⠌⢀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠓⠪⣿⡀
⠀⠀⠀⢨⣈⠑⠀⢰⠂⠚⠧⠀⠀⠀⠑⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁
⠀⠀⠀⠘⠛⢁⠂⠈⢰⡤⠀⠠⠀⠀⣩⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⢡⠀⠀⠄⠀⠤⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠃
⠀⠀⠀⠀⠀⠀⠲⠤⣀⠀⠠⠀⠀⠑⠶⠦⠋⠀⠀⠀⢀⠀⢀⠁⠀⠀⠀⠀⠀⠒⠐⠓⠀⠄⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⢄⣀⠀⣤⣀⣙⠒⠠⢦⠀⠀⠀⠀⠀⠀⠀⠘⠄⠘⠄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒
⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠉⠳⠦⣧⡀⠀⠰⡄⠀⡆⠐⠀⠀⠀⠂⠀⡆⠀⠀⠑⠀⠤⠴⠆⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠟⠀⠀⠁⠀⠀⠀⠀⠀⠀⠇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⠀⠈⠈⠈⠀⡀⠐⡀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⡄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠈⠈⠈⠁⠀⠀⠀⢀⡀⢀⠄⠀⠀⠀⠀⢀⠆⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠷⠗⠀⠀⢀⠄⠀⠀⢀
⠀⠀⠀⡠⢠⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⢀
⠀⠀⠀⠁⠀⠀⠀⠦⡀⠀⠀⠀⠀⠀⠀⠄⠀⠓⠀⢩⡤⠰⠀⠁⠐⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⡠⠀⠀⠀⠀⢀⠄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠌⠀⠁⠀⠀⠀⠀⠀⠂⠃
⠀⠀⠁⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⢴⡀⠀⡀⠀⠀⡀
⠀⠀⠈⢇⢠⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠀⠒⠈⢳⠸⠁⠀⠈⢠⡤⠄
⠀⠀⠀⠀⠈⠀⠀⠀⠐⠄⠀⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⣴⠶⠋⠀⠀⠀⠀⢇⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠓⠋⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢯⣬⠇⠀⠀⠀⠀⠀⠀⠀⠀⠳⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠆⠀⠀⠀⢰⣿⠗⡘⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣠⢤⡀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣶⢉⣄⠀⠀⠀⠀⠿⡼⠃⠀⠀⠸⣇⠀⡿⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠘⠾⠁⠀⠀⠀⠀⠀⣠⣄⠀⠀⠈⠉⠀⠀⠘⠛⠁⠀⠀⠀⣟⡷
⠀⠀⠀⣀⡤⠴⠶⠶⠶⠤⣤⡀⠀⢀⣠⣤⣄⣤⠤⠤⣄⠀⠀⠙⣫⠤⠤⢤⣀⠀⠀⣠⠖⠛⠛⠛⠶⣄⠀⠀⣠⠤⠤⠤⣄⣀
⠀⢀⡞⠁⡠⠀⠀⠀⠀⠀⠀⡇⠀⡏⠀⠀⠋⠀⠀⠀⠈⢳⣤⠞⢁⠀⠀⠀⠈⠳⣴⠇⠀⡤⠀⠀⠀⠘⣦⠞⠁⠀⠀⠀⣀⣸⠇
⢀⡟⠀⠀⢀⡴⠛⠉⢳⣤⣀⡇⠀⡇⢠⠀⢀⣤⠀⠀⣀⠼⡏⠀⠁⢀⣀⠀⠀⠀⢹⠀⠀⠀⢀⡤⠤⠴⢻⡀⠀⣀⠀⠘⠻⠧⣄⠀⡴
⢸⠃⠀⠀⣼⠁⠀⠀⠈⠁⠉⠀⠀⣇⠀⠀⠸⣼⠀⠀⠸⣦⡇⠀⠀⣎⢸⠀⠀⠀⢸⠀⠀⠀⢸⡀⠀⠀⠀⠈⠉⠉⠉⢣⠀⠀⠈⣧⠉
⢸⡆⠀⠀⢻⡀⠀⣾⡉⠉⠉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠈⠋⠀⠀⠀⡼⢶⣦⣄⠀⠈⠉⠑⡶⠚⠒⠶⠖⠃⠰⠀⠀⢸⡄
⠀⣧⠀⠀⠀⠻⢯⣹⡿⠀⠀⠀⢠⣤⠼⠀⠀⢀⡀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢀⡼⣧⠀⠉⠉⠁⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⣠⠏⣧
⢰⠋⠳⢤⡀⠀⠀⠀⢀⣀⠀⠀⢸⢸⠀⠀⠀⢸⢷⡀⠀⠀⠀⢰⢤⠀⣀⠀⢸⡇⢿⣠⡄⠀⠀⠀⣠⣤⠂⠀⠀⠀⠀⠀⡀⢸⠁⠀⢻
⡾⠀⠀⠀⣿⢀⠀⠀⢸⢹⣦⢠⣼⠘⠛⣷⠀⢸⠀⣷⢀⡄⢸⠋⠸⡇⡟⠷⠿⠃⠈⠉⢹⢰⣆⢰⡇⠸⡄⡴⡆⢴⣆⢸⡟⠛⠀⠀⢸
⡇⠀⠀⠀⢸⣾⠓⢻⣸⠀⢸⣿⡀⠀⠀⠸⢷⣾⠀⣏⢸⠙⠋⠀⠀⣇⡇⠀⠀⠀⠀⠀⠈⠋⢸⣸⠇⠀⢹⣧⠹⠟⠸⣶⠇⢀⠀⠀⢸
⡇⠀⠀⠀⠘⡃⠀⠈⠛⠀⠈⠿⠃⠀⠀⠀⢸⣿⠀⢻⡼⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣀⠀⠸
⣿⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠃⠀⡼⢧⢠⡀⠑⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁⠀⢀⡀⢀⣀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠉
⣿⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⡇⠘⠋⣇⠀⠀⢰⢧⢀⡿⡄⠀⠀⡶⣆⠀⡀⠀⠀⢸⢷⡜⠹⡆⠀⢸⣀⣦⣀⣿⠀⠆⢰⣆
⢹⠀⠀⠀⠀⢰⠃⡇⠀⢠⡶⡄⠀⡇⠀⠀⣿⠀⠀⣾⢸⣼⠀⣧⠀⢰⠇⠹⣤⠟⢿⠀⣸⠀⠀⠀⡇⠀⢸⠛⠉⠉⢸⡇⠀⢸⢸
⢸⣀⡞⢣⠀⢸⠀⣇⠀⢸⠀⡇⢀⠁⠀⠀⢹⠀⠀⣿⠀⠀⠀⢸⠈⢸⠀⠀⠉⠀⢸⠀⣿⠀⠀⠀⣿⠀⢸⠀⠀⠀⢸⡇⠀⢸⠘⣇⢀
⠀⠉⠀⢸⠀⢸⠀⢸⠀⢸⠀⢻⣼⠀⠀⠀⢸⠀⠀⡏⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⢸⠀⡃⠀⠀⠀⢻⠀⢸⠀⠀⠀⠘⡇⠀⣸⠀⠈⠛
⠀⠀⠀⠸⡆⣼⠀⠘⢧⠎⠀⠀⠁⠀⠀⠀⠘⡇⠀⡇⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⢸⣀⡟⠀⠀⠀⠀⢳⣀⡇
⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣸⠁⠀⠀⠀⠸⡆⣸⠀⠀⠀⠀⢸⣤⠇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠙