Function Emojis & Symbols

⠀⠀⢀⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⡄
⠀⠀⠘⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠃
⢀⣤⡤⠺⡿⠆⣤⣄⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠺⡿⢢⣤⣄⠀⠀⠀⠀⠀⠀⠀⣠⣠⠄⠛⠟⢠⣤⡀
⣿⣿⣿⣤⢿⡴⠛⠛⢦⡀⠀⠀⠀⠀⠀⣿⣿⣷⣼⣷⢶⡛⠻⢦⡀⠀⠀⠀⠀⢰⣿⣿⣧⣼⣧⠞⡛⠳⣄
⠉⠛⠻⠿⣟⠈⣽⣶⣄⣷⠀⠀⠀⠀⠀⠙⠛⠿⠿⡏⣸⣷⣶⡎⡧⠀⠀⠀⠀⠈⠙⠛⠿⢿⣇⣾⣧⣤⢸
⠀⠀⠀⠀⠹⢦⣙⣛⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣼⣿⣿⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣝⣛⣯⠞
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇
⠀⠀⠀⣻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⠁
⣰⣾⣿⠈⣯⢁⣿⣶⡀⠀⠀⠀⠀⠀⠀⣴⣾⣏⢉⣏⢙⣿⣶⠀⠀⠀⠀⠀⠀⢀⣶⣿⡏⢩⡏⣸⣷⣆
⠿⣿⣿⣿⣿⠋⠀⣨⡝⣆⠀⠀⠀⠀⠀⢿⣿⣿⣾⡿⣩⣤⣬⡹⡄⠀⠀⠀⠀⠸⣿⣿⣿⣿⡞⣩⣶⣈⢳
⠀⠀⠈⠉⢧⡸⡾⠟⢁⡿⠀⠀⠀⠀⠀⠀⠀⠉⠉⣇⢸⡿⣿⢄⡏⠀⠀⠀⠀⠀⠀⠈⠉⠹⣟⠻⣿⠟⣻
⠀⠀⠀⠀⠈⠛⠲⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠞⠉

⠀⠀⠀⣴⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶⡄
⠀⠀⠘⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠇
⢀⣀⣤⠼⣿⠇⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠼⡿⢄⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢿⡿⢠⣄⡀
⣾⣿⣿⣠⣿⡤⠿⠿⣤⡀⠀⠀⠀⠀⠀⣿⣿⣷⣰⣗⣴⣿⣿⣄⠀⠀⠀⠀⠀⢠⣿⣿⣇⣸⢇⣬⣿⢭⣄
⠛⠻⠿⠿⡟⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠛⠻⠿⢿⣟⣏⣸⣉⣟⡇⠀⠀⠀⠀⠈⠛⠿⠿⢧⡟⠁⠀⠀⠙
⠀⠀⠀⠀⠱⣄⡀⣀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⣼⣠⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠉⠉⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠛⠋
⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇
⠀⠀⠀⢻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠁
⣰⣶⣿⠉⣿⠁⠿⠶⠄⠀⠀⠀⠀⠀⠀⣴⣾⣏⢉⣏⢘⣷⣶⠀⠀⠀⠀⠀⠀⢀⣶⣾⡏⢹⡍⣸⣷⣦
⢿⣿⣿⣾⣣⡿⠀⠀⢾⣆⠀⠀⠀⠀⠀⢿⣿⣿⣾⡿⢋⡍⠉⡳⡄⠀⠀⠀⠀⠸⣿⣿⣿⣿⡟⣩⠾⣍⢳
⠀⠀⠉⠉⠉⠁⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠈⠉⠉⣧⣿⣿⣾⣧⡏⠀⠀⠀⠀⠀⠀⠈⠉⠹⡇⠀⡴⠋⣸
⣀⣀⣀⣀⣀⣀⣿⣶⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣿⣟⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣲⣷⣞⣁
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⡀⡀⢀⣤⠤⢤⡄⠀⠀⠀⠀⠀⠀⢀⡠⠔⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣄⣤⣄⠀⠀⠀⠀⠀⠀⠀⣤⡤⢤⣄⠀⣀⣀
⠀⠀⠀⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡖⠒⠒⠒⠒⢳⣦⠀⠀⠀⠀⠀⢀⠞⢀⠔⠁⠈⠻⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄
⠀⠀⢀⣶⠒⠒⠒⠒⠒⢲⡆⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⢈⣔⠁⠀⠀⢀⡴⢫⠀⠀⠀⠀⠀⠀⣶⠒⠒⠒⠒⠒⢲⣆
⠀⠀⠸⣿⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⢂⡜⠉⠀⠀⠀⠀⠀⠈⠻⣵⣄⣴⣋⡠⠊⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁


⠀⠀⠀⢠⠒⠶⢰⠿⠟⠿⡆⠀⠀⠀⠀⠀⠀⡀⡾⠿⠻⢷⢀⠀⠀⠀⠀⠀⠀⠀⣀⢰⠟⠛⠿⡆⡀⠀⠀⠀⠀⠀⠀⡾⠿⠻⢷⠰⠖⢢
⠀⠀⠀⠐⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⡜⠁⡇⠀⠀⢸⠀⢱⠀⠀⠀⠀⠀⢰⠁⢸⠀⠀⠀⡇⠘⡄⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠘
⠀⠀⠀⡟⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⢣⠀⡇⠀⠀⢸⠀⡸⠀⠀⠀⠀⠀⠸⡀⢸⠀⠀⠀⡇⣠⠃⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠸⡇
⠀⠀⠀⢷⠀⠀⠸⣶⣤⣶⠇⠀⠀⠀⠀⠀⠀⠁⢷⣦⣤⡾⠈⠁⠀⠀⠀⠀⠀⠀⠈⠰⣶⣴⣶⠇⠁⠀⠀⠀⠀⠀⠀⢷⣦⣴⡾⠀⠀⢰⠇


⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⢰⣿⢾⣿⣻⢾⣦⠀⠀⠀⠀⠀⠀⢀⡔⠉⣀⣀⠉⠢⡀⠀⠀⠀⠀⠀⠀⡴⢘⠛⠛⠛⡇⢄⠀⠀⠀⠀⠀⠀⣀⡄⠀⢀⠠⠀⢤⡀
⠀⠀⢠⣾⣣⡄⠀⠈⠯⣻⡆⠀⠀⠀⠀⠀⣨⠀⢠⣧⣼⡆⠀⠷⠀⠀⠀⠀⠀⢸⠀⠨⢸⣭⡆⡇⠈⡆⠀⠀⠀⠀⠀⡟⠀⠀⡍⡇⠀⢸⣷
⠀⠀⠀⢹⣷⣕⣄⣐⣴⣿⠀⠀⠀⠀⠀⠀⠘⢄⠘⠿⠿⠇⣠⠂⠀⠀⠀⠀⠀⠘⢆⢰⠘⠛⠃⡇⡴⠀⠀⠀⠀⠀⠀⠻⠦⠤⠵⠥⠤⠼⠏
⠀⠀⠀⠀⠉⠙⠿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠶⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠶⢢⡀⠀⠀⠀⣠⠖⠲⠚⠉⠱⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠁⠀⠀⠀⣹⠀⠀⢀⡇⠀⠀⠈⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⣀⠀⠼⠁⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠙⠢⡤⠤⣄⠀⠀⠈⠓⠲⣄⢀⡠⠤⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠀⠀⢀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⠀⣀⡀⣠⠔⠲⣄⣠⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣀⠀⣀⣀
⣠⠶⢤⢠⠖⠲⡄⠀⠀⠀⠀⠀⡏⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡠⠞⠁⠉⠁⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⢸⡾⣅⠈
⣇⢠⠞⢿⠀⡴⠃⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⡠⠂⠀⣀⠴⠃⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠞⠁⠾⠔
⠈⠛⠃⠀⠙⠃⠀⣀⣤⠤⠤⠤⠴⠷⠶⠖⠲⢦⠀⠀⢀⡠⠃⠀⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠛⠲⠦⠤⠤⠤⠴⢆
⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢻⡉⠉⠀⠀⠀⠀⠘⣇⣤⡄⠀⡀⠀⠀⠀⠀⢠⠏⠀⢰⠂⠤⠤⠤⠤⠀⠈⢆
⠀⠀⠀⠀⠀⡰⢁⠔⢩⠃⠜⠙⠎⡉⢄⠉⠭⢱⠀⠱⣄⠀⠀⠀⢠⡤⠖⠀⠑⠲⡄⠀⢀⠀⢠⠋⠀⠀⢀⠤⠝⢡⣄⣊⡁⠀⠈⢧
⠀⠀⠀⢀⡜⠁⠈⠒⠀⠛⠒⠋⠈⠑⠊⠑⠒⠊⠀⠀⠈⢳⡀⠀⠈⠓⢲⠀⢰⠚⠓⠒⠉⢹⣃⣀⠀⠀⠘⠒⠒⠒⠒⠒⠃⠀⠀⢀⡳
⠀⠀⠀⡾⠤⠤⠤⠤⠤⠤⠠⠠⠄⠀⠒⠒⠒⠂⠉⠉⠉⢹⡀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⢸⡄⠀⠀⠉⠉⠉⠉⠀⠒⠒⠂⠉⠉⠁⢠⡆
⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠘⡂⠀⠀⣀⣀⣀⣀⣀⣠⣤⡤⠤⠤⠤⠴⠶⠶⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠉⠉⠉⠛⠛⠛⠛⠓⠚⠛

⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠔⠠⠄⢄
⠀⠀⠀⠀⠀⠘⠀⠠⠐⠒⠒⠠⢈⠢⡀⠀⠀⠄⠒⠀⠒⠠⡀⠀⠐⢁⠔⠀⠀⠐⠢⡀⠁⢀⠔⠈⠈⠁⠐⢄⠀⠀⢀⠄⠒⠂⠄⢀
⠀⠀⠀⠀⠆⢀⠊⠀⠀⠀⠀⠀⠀⠡⠐⠰⠁⠀⠀⠀⠀⠀⠈⠄⠁⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠡⠠⠁⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⡄⢨⠀⠀⢠⡰⠴⣤⠂⠀⠀⢀⠀⠐⠜⠂⢣⠂⠀⠀⠀⠀⠀⠚⠉⠓⠃⠀⠀⠀⠀⠙⠁⠻⠋⠀⠐⠇⠀⠰⡴⠰⡤
⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠄⠢⠀⠀⠀⠀⡀⢠⠈⢀⠐⠄⢠⠀⠂⠀⢀⢀⠌⢠⠀⠀⠀⠀⡀⠀⠠⡀⠀⡄
⠀⠀⠀⠀⠀⠁⠀⠈⠢⢰⢰⠊⠊⠄⠆⠀⠀⠀⠀⡦⣤⠂⠁⠀⠀⠀⠀⣰⢶⡔⠊⠠⠌⠀⠀⣤⣶⡠⠂⠀⠀⠀⠐⢀⡆⣴⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠨⠜⢻⠀⠀⠀⠀⠀⠐⡬⠵⣛⠀⠀⠀⠀⠀⢈⠔⣋⣱⠀⠀⠀⠀⠀⠔⣚⡇⠀⠀⠀⠀⢀⠥⢖⣛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⣚⣩⠀⠀⠀⠀⠀⠀⢂⣉⡤⠆⠀⠀⠀⠐⠰⠈⣥⡜⠀⠀⠀⠀⠀⣨⡤⣶⠀⠀⠀⠀⠠⡀⢭⣼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠐⠀⠀⠀⠀⠀⠈⠑⢀⠀⠁⠀⠀⠀⠀⠈⠊⣀⠀⠀⠀⠀⡀⢀⠈⡠⣈⠀⠀⠀⠄⢈⢂⠀⡠⠰
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠐⠀⠈⠀⠀⠀⡀⠀⠀⠀⠉⠀⢀⡀⠀⠀⡀⠀⠀⠤⠤⠤⠠⡄⡀⠀⠀⠀⢀⣀⡀⡀⠀⠀⠀⠀⠠⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⣀⣢⠀⠀⠀⠀⠀⠀⠀⠀⠤⠵⡄⠀⠀⠀⠀⠀⠈⠑⢆⠀⠀⠀⠀⠀⠀⠤⠵
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡈⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢤⠀⠀⡃⠀⠀⠀⠀⠀⠀⠉⢹⠄⠀⣀⠀⣠⡀
⠀⠀⠀⠀⠀⠈⠀⡐⠀⠀⠀⢀⠝⠀⠀⠀⠀⠐⢄⠂⡐⠈⠅⠀⠀⠀⠀⠠⢪⣀⠀⢀⢂⠀⠀⡇⢀⠀⢀⠐⠆⠀⠀⢸⠄⡀⠀⠈⠰⡁⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠘⢕⢂⠉⠀⠀⠀⠀⠀⠸⠀⢃⠄⠡⡐⠀⠀⠀⠀⠰⡈⠑⠡⠀⠀⠀⠀⣿⡀⠑⢀⡸⣤⠀⠀⢸⠀⠁⢤⡐⢁⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠁⠔⠁⠀⠀⠀⠀⠀⠃⠢⠄⠑⠁⠀⠀⠀⠀⠀⠀⠁⠒⠂⠀⠀⠀⡇⠀⠤⠢⠈⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⣆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠐⠒⠒⠒⠒⠲⡶⢃⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠀⠀⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠁⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢰⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡆
⠀⠀⠀⠀⠘⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠏
⠀⠀⣠⣤⡖⠺⡟⢲⣦⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⡖⠺⡟⢢⣦⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⡖⢻⡟⢰⣤⣄
⠀⢀⣿⣿⣿⣼⣷⢞⡛⠛⢦⡀⠀⠀⠀⠀⢀⣿⣿⣷⣼⣷⢾⡛⠛⢦⡀⠀⠀⠀⠀⢰⣿⣿⣷⣼⣧⠞⡛⠻⢦
⠀⠀⠉⠛⠻⢻⡇⢈⣽⣶⣄⡇⠀⠀⠀⠀⠀⠉⠛⠻⠿⡇⣸⣿⣷⡎⡧⠀⠀⠀⠀⠈⠉⠛⠻⢻⣇⣼⣷⣦⢈⡇
⠀⠀⠀⠀⠀⠀⠻⣤⣙⣫⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣽⣛⣯⠞⠁

⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡧
⠀⠀⠀⠀⠀⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠁
⠀⠀⣶⣾⣇⢈⡏⢸⣿⣶⠀⠀⠀⠀⠀⠀⠀⣶⣾⣏⢉⡏⢸⣿⣶⠀⠀⠀⠀⠀⠀⠀⣶⣾⡇⢩⡏⢸⣷⣦
⠀⠠⣿⣿⣿⣾⡿⠋⠁⣩⡳⡄⠀⠀⠀⠀⠠⣿⣿⣿⣾⡿⣫⣍⣭⡳⡄⠀⠀⠀⠀⠸⣿⣿⣿⣞⡿⢉⣴⣍⢳⡄
⠀⠀⠀⠈⠉⠙⣇⠲⡾⠋⢀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠙⣇⢼⡿⣿⢄⡟⠀⠀⠀⠀⠀⠀⠈⠉⠹⡇⠻⣿⡿⢣⡇
⠀⠀⠀⠀⠀⠀⠈⠳⠶⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠮⠶⠋

⠀⠀⠀⠀⢀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡄
⠀⠀⠀⠀⠸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⡗
⠀⠀⠀⣀⡀⣽⣿⢅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢽⣿⢅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣿⣿⢀⣀
⠀⠀⣿⣿⣧⢠⣇⣸⠿⣿⣀⠀⠀⠀⠀⠀⠀⣿⣿⣧⢰⡇⣨⣿⣿⡀⠀⠀⠀⠀⠀⢀⣿⣿⣇⢰⡆⣈⣿⣗⡀
⠀⠀⠿⢿⣿⣿⡟⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠿⢿⣿⣿⢾⡋⢸⠘⡟⡆⠀⠀⠀⠀⠘⠿⢿⣿⣏⡾⠋⠀⠈⠙⣧
⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣏⣹⣩⣯⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠟⠁
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢰⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡆
⠀⠀⠀⠀⠘⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠏
⠀⠀⣠⣤⡖⠛⡟⢲⣦⣤⠀⠀⠀⠀⠀⠀⠀⣠⣤⡖⠛⡟⢲⣦⣄⠀⠀⠀⠀⠀⠀⠀⣤⣴⡖⢻⡟⢰⣦⣄
⠀⢀⣿⣿⣿⣼⢣⣦⠀⠀⣤⡀⠀⠀⠀⠀⢀⣿⣿⣿⣼⣧⠞⡛⠛⢦⡀⠀⠀⠀⠀⢰⣿⣿⣷⣼⣧⢞⣛⡛⢦⡀
⠀⠀⠉⠛⠛⠓⠿⠁⠀⠀⠸⠷⠀⠀⠀⠀⠀⠉⠛⠛⠻⡇⣾⣧⣤⣎⡧⠀⠀⠀⠀⠀⠉⠛⠛⢻⡇⠙⣠⠟⢈⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣭⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠀⠀⡴⠃⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠔⠘⣆⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⣼⠀⡞⠁⡰⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠆⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⢸⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢀⢠⠋⠀⠆⣿⢸⠁⠐⠀⠀⢹⡄⠀⣠⢿⡀⠀⠀⣰⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠙⠳⣷⠀⠀⠀⠀⠀⢸⠀⠀⢀⡴⢻⡇⠀⠀⠀⢿⡯⠀⠀⠀⠀⠀⣇⢀⡏⠀⢳⣀⡴⠃⠀⠄⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡜⢹⡇⠐⠂⠀⠀⠀⠀⠀⠀⡞⢀⡴⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣿⣸⠃⠇⠀⠋⠀⠀⠀⠠⠂⣼⠶⣄⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠸⢀⠀⠷⠀⠀⠠⠀⠀⠘⠁⢸⣵⠟⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠿⠃⠀⠀⠀⠀⠀⠘⠀⠀⠀⡠⠆⣼⠞⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⢀⡄⢱
⠀⠀⠀⠀⠀⠀⠀⢧⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⢀⣠⡟⠃⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⡀⠸⡇
⠀⠀⠀⠀⢠⠏⢁⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⣠⠏⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠙⢦
⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⣧
⠀⣸⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠹⡆
⠀⡇⠀⠀⠀⠀⠀⠀⢀⡞⠋⠉⠙⠛⠛⠛⠓⠒⠓⠓⠒⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠒⠒⠒⠒⠶⠶⠶⠶⡄⠀⠀⠀⠀⠀⠀⠀⢻
⢸⠁⠀⠀⠀⠀⠀⠀⣼⠁⠀⢀⣀⠀⣀⣀⠀⣀⣀⠀⠀⠈⣷⠀⠀⠀⠀⠀⣴⠖⢶⠀⠀⠀⠀⠀⣰⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠄⠈⡇
⣸⠀⠀⠀⠀⠀⠀⢠⡏⢰⢏⡝⢻⡾⠛⠻⣶⣫⣿⠙⡆⠀⠸⡆⠀⠀⡶⠶⠟⠀⠸⠶⢶⠀⠀⢰⠏⠀⠀⠸⢯⡍⢉⠽⠃⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⣸⠁⠈⠻⠿⠟⠛⠙⠳⠻⠷⠖⠚⠁⠀⠀⢻⡀⠀⠛⠒⣦⠀⣴⠶⠾⠀⢠⡟⠀⠀⢀⣴⣉⣀⣙⣂⣲⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⡇⠀⠀⠀⠛⠚⠋⠀⠀⠀⣿⢀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠘
⣷⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠓⠒⠒⠒⠒⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣍⣉⣉⣉⣉⣉⣉⣩⣭⣭⣉⣉⣉⣛⣻⠀⠀⠀⠀⠀⠀⢀
⣿⠀⠀⠀⠀⠀⠘⠳⠶⠶⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢸
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠈⣧⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⢹⡆⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⢻⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠈⢷⡀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⡞
⠀⠀⠀⠈⢷⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢀⡴⠋
⠀⠀⠀⠀⠀⠙⢦⣄⠈⠐⠠⢀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠄⠂⢁⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣄⡀⠉⠑⠒⠠⠤⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒⣁⣁⣠⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⠤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⠴⠶⠖⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣤⡶⠾⠛⠛⠛⠷⢶⣤⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠛⣉⣡⣤⣤⣤⣍⡙⠻⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣴⠟⠁⠀⠀⣿⠛⣿⠀⠀⠈⠻⣦⠀⠀⠀⠀⢸⣿⣿⣿⣿⠋⣠⣾⣿⣿⢀⣤⠈⣿⣿⣷⣄⠹⣿⣿⣿⣿
⠀⠀⠀⣾⠃⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠹⣧⠀⠀⠀⢸⣿⣿⣿⠁⣼⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣆⠸⣿⣿⣿
⠀⠀⢰⡟⢰⡾⢻⣿⣿⣿⣿⣿⣿⣿⡟⢷⡆⢿⡆⠀⠀⢸⣿⣿⡏⢸⡏⢉⡍⠉⠁⠀⠀⠀⠉⠉⢩⡉⣿⡀⣿⣿⣿
⠀⠀⠸⣷⠘⠿⠾⠿⠿⣿⣿⣿⠿⠿⠷⠿⠃⣾⠇⠀⠀⢸⣿⣿⣇⠸⣧⣈⣁⣀⣀⠀⠀⠀⣀⣀⣈⣀⣿⠀⣿⣿⣿
⠀⠀⠀⢻⣆⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣰⡟⠀⠀⠀⢸⣿⣿⣿⡄⠻⣟⣿⣿⡿⠀⢀⠀⣿⣿⣿⣿⠏⣰⣿⣿⣿
⠀⠀⠀⠀⠹⣷⣄⠀⠀⢿⣶⡿⠀⠀⣠⡾⠏⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⡉⠿⣿⣿⣈⣋⣠⣿⣿⠟⢋⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⠻⠶⠶⢶⠶⠶⠟⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦⣤⣉⣉⣛⣉⣉⣤⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊

⠀⠀⠀⡄⢠⢤⡄⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⡤⢤⣠⣄⢤⣠⣤⢤⣤⡄⠀⠀⠀⠀⠀⣠⣠⢄⡤⣠⢤⢤⢄⢤⡄
⠀⠀⠀⠈⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉
⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠘⠃⠃⠧⠚⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠋⠓⠜⠲⠓


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⣀⢀⣀⣀⣀⢀⣀⢀⡀⣀⢠⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡈⣉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⠀⣀⠀⠀⡀⢀
⠀⠑⠈⠊⠁⠉⠁⠁⠉⠁⠁⠉⠀⠀⠀⠀⠀⢀⣀⣈⣋⣁⣉⣈⣀⣀⠀⡀⠀⠀⠀⠀⠁⠉⠈⠉⠁⠈⠃⠉⠈⠁
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⣀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⡂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡰⡆⠀⡀⠀⠀⢀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡐⠀⠑⠀⠃⠔⠀⠀⠀⠀⠢⡀⠸⠂⠀⢀⠔
⠀⠀⠀⠀⡀⠠⠀⢀⠀⠀⠀⠀⢀⡀⢠⢉⡠⣠⠀⠀⠀⠀⠀⢠⡤⣀⣩⡄⠄⡀⠀⠀⠀⠀⡠⠀⠐⠂⠀⢀
⠀⠀⠐⠁⠀⡐⡄⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠌⠰⣌⣃⣠⡾⠃⠡
⠀⢰⠀⠚⠐⠓⠃⠈⠂⠐⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⢀⡵⠝⠄
⠀⠀⠄⠁⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⠀⠀⠂⠂⠈
⠀⠀⠀⠐⠠⠀⠠⠔⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠃⢀⣀⡀⠀⢈⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠰⠂⠐⠂⠃⠂⠒⠀⠀⠀⠀⠀⠀⠀⠐⡀⠄⠌⠁⠝⠀⠠⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⠁⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠃⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠐⠈⠀⠈⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⢠⡤⢤⣄⠁⠄
⠀⢀⠁⠀⡀⣠⡏⡙⡆⠠⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠭⠭⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣀⠀⢀⠈
⠀⠘⠀⣝⠀⠨⡟⠈⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠠⠀⠀⠀⠀⠄⠠
⠀⠀⠐⢌⠋⠊⠀⢀⠔⠁⠀⠀⢀⠄⢐⣀⠂⠠⡀⠀⠀⠀⠀⢀⠄⠐⠀⠂⠠⡀⠀⠀⠀⠀⠐⠠
⠀⠀⠤⡤⠤⠥⠤⠠⠤⠄⠀⠀⠁⠰⠻⠻⠿⠂⠈⠄⠀⠀⠠⠀⠒⠒⠒⠒⠒⡎⠀⠀⠀⠀⠒⠐⠂⠐⠒⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢂⠒⣐⠂⠀⡀⠀⠀⢀⠰⣀⣀⡀⠀⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠺⠶⠟⠃⠐⠀⠀⠀⠀⠂⡀⠀⠑⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡀⣀⢀⢁⢀⠀⠀⠀⣀⢀⣈⡀⡀⢀⡁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈⠈⠈⠀⠀⠀⠀⠉⠉⠁
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢠⠞⢉⣉⡉⠉⠑⡄⠀⠀⡔⠉⠉⠉⠉⠁⢆⠀⠀⢰⠊⠉⠉⠉⠉⢲⠀⠀⢀⠞⠉⠉⠉⢉⠑⡄
⠀⠀⢸⣴⣟⣩⣿⣿⡀⡇⠀⠀⡇⠀⠀⣤⠀⡀⢸⠀⠀⢸⠀⢤⢘⢡⡄⢸⠀⠀⢸⡇⠀⣶⣶⣾⡇⡇
⠀⠀⢸⡝⠿⠿⠿⠿⢃⡇⠀⠀⣇⠀⠢⡠⠊⠀⡿⠀⠀⢸⢸⣤⣼⣤⣴⢸⠀⠀⠸⡗⣴⣿⣿⣿⢇⡇
⠀⠀⠀⠙⠛⠛⠛⠓⠋⠀⠀⠀⠈⠛⠛⠛⠓⠋⠁⠀⠀⠈⠙⠛⠛⠒⠒⠉⠀⠀⠀⠙⠛⠚⠛⠒⠋
⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⢰⠃⠀⢠⣤⠀⠈⡆⠀⠀⡎⠀⠀⢀⡀⠀⢳⠀⠀⢸⢁⣤⣴⣤⣤⢹⠀⠀⢠⡋⠀⠀⠀⠠⠈⡆
⠀⠀⢸⣄⣠⣿⣻⣦⠀⡇⠀⠀⡇⢀⣰⣿⡟⠀⣸⠀⠀⢸⢸⣿⣿⡿⣿⢸⠀⠀⢸⣇⢸⣿⠷⠀⡇⡇
⠀⠀⠸⣏⣛⣛⣛⣛⣠⠇⠀⠀⢷⣿⣿⣿⣟⣀⡟⠀⠀⢸⣘⣛⣛⣛⣛⣸⠀⠀⠘⣏⣒⣒⣒⣒⣣⠇
⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⡠⠤⠤⠤⠤⢄⠀⠀⠀⢀⠤⠤⠤⠤⠤⡀⠀⠀⢀⡤⠤⠤⠤⠤⣀⠀⠀⠀⡠⠤⠤⠤⠤⢄
⠀⠀⢸⠁⠖⢀⣤⣄⠀⡇⠀⠀⡇⢠⣾⣦⣿⠀⢰⠀⠀⢸⠀⠴⠶⠚⠂⢸⠀⠀⢰⡇⠀⠒⢲⡄⠀⡇
⠀⠀⢸⡁⣠⣿⣿⣏⠀⡇⠀⠀⡇⢸⣿⣿⣿⡀⣾⠀⠀⢸⠀⣿⣟⣿⡇⢸⠀⠀⢸⡇⠀⠀⣾⣿⡆⡇
⠀⠀⠘⠧⣬⣯⣤⣤⠴⠃⠀⠀⠳⣼⣿⣿⣿⡥⠏⠀⠀⠸⢤⣿⣿⣿⣧⠜⠀⠀⠈⠧⣤⣤⣭⣭⠵⠃

⠀⠀⢀⡔⠒⠒⣒⡒⠢⡀⠀⠀⡠⠒⠒⢒⠒⠒⢄⠀⠀⢠⠖⠒⠒⠒⠒⢤⠀⠀⠀⡔⠒⠒⠒⠒⠢⡀
⠀⠀⢸⣀⣰⣿⣿⣧⡀⡇⠀⠀⡇⡀⢰⣄⠙⡆⢸⠀⠀⢸⠀⠖⢠⣶⣦⢸⠀⠀⢸⡆⠂⣤⣁⠑⠀⡇
⠀⠀⢸⣿⣟⠛⠛⣿⠇⡇⠀⠀⡇⠱⣄⣁⡠⠃⣾⠀⠀⢸⢠⣤⣬⣽⣭⣸⠀⠀⢸⡇⣦⣉⣡⣤⠀⡇
⠀⠀⠈⠓⠶⠶⠶⠶⠚⠁⠀⠀⠙⠶⠶⠶⠶⠖⠃⠀⠀⠘⠲⠶⠶⠶⠶⠊⠀⠀⠀⠓⠶⠶⠶⠶⠚⠁

⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⡆⢰⡖⠋⢻⠉⠉⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡱⢰⢀⠞⠀⢰⢰⡏⢹⡏⠁
⠀⠀⠀⠀⠀⠀⢸⡥⠺⣏⡀⣻⠉⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡠⠲⡞⠛⢷⡄⣼⠟⠻⡶⠢⣀⠀⠀⠀⠀⠀⣼⠳⣼⠿⠴⡏⣾⢹⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⡀⠐⠺⣄⠈⣷⡏⢀⡼⠒⠀⡸⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡇⡀⢈⠙⠂⠈⠁⢸⠀⠉⠀⠒⢉⠀⡀⡟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡛⠄⠇⠱⠄⣧⣀⡠⠰⣾⡶⠢⣄⣠⡇⠴⠂⠇⠜⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⢑⢴⡀⠀⢸⠆⡴⣄⢸⢀⡴⡄⢾⠀⠀⣰⢔⠉⠉⢳⡀⠀⢀⠞⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠦⠤⢬⣤⡱⠔⠃⠀⢀⣽⣸⣸⣅⠀⠀⠓⠴⣡⣬⠤⠤⢾⡇⠀⢸⣀⢼⢶⠹⠷⠊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠲⠤⢸⢁⠑⡄⢨⡀⠈⢩⣽⣭⠉⠀⣨⠀⡘⢁⣹⠠⠴⠺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⠒⢇⠈⠂⡘⠦⠰⡇⠈⢻⠋⠀⡷⠠⠜⠀⠋⢀⠇⢂⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠤⠀⠀⠀⠀⠀⣷⠄⠐⠀⠓⠒⢠⠀⠄⢱⠙⢿⠟⢱⠁⠄⠠⠐⠒⠃⠐⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⢠⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣬⠟⠁⡠⡞⠂⠈⠉⢢⣸⣠⠎⠉⠀⠚⡦⡀⠙⢯⣤⡿⠁⠀⠀⢀⠳⠴⠉⠐⠀⠈⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡒⣚⣀⡑⡐⠚⠉⢸⣿⣿⠈⠙⠒⠐⣁⣘⡒⣺⡿⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⢤⣄⣤⡤⡄⠀⠀⠀⠈⢷⣄⢄⠩⣌⢦⡀⡉⢹⠉⣁⡠⢎⡬⢁⢄⣴⠏⠀⠀⠀⠀⠀⠀⠀⣸⢹⠇⡏⠀⢀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠋⠉⠉⠃⠁⠀⠀⠀⠀⠀⠙⠻⠶⢾⣤⣤⠾⠃⠻⢦⣤⣼⠶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠤⠧⠼⠴⠹⠿⠊⠞⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠀⠀⠺⠀⠠⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠀⠀⠀⠀⠘⠒⠈⠀⡀⠀⡀⢀⡀⣀⠀⣀⡀⠀⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⠀⡄⠲⢌⠀⠠⠰⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠠⠃⠀⠀⠀⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠄⠂⠐⠀⠀⠐⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠁⠀⠈⠀⠁⠀⠀⠀⠀⠀⢴⡞⠋
⠀⠀⠀⠀⠀⠸⠀⠈⠀⠀⠁⠀⠀⠀⠙⠲⣤⡀⠀⠀⠀⠰⠂⡂⢠⡴⢤⣂⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⢀⣴⠋⠉⣿⠀⠟⢩⡷⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢭⡆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⠿⠂⠀⢀⡴⠋⠉⠙⠇⢀⣽⣛⠁⠀⢀⣫⡛⠳⡄⠀⠀⠀⠀⠀⢀⡘⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡾⠛⣰⠞⠳⣿⣾⠋⣰⡆⢺⣅⣀⠀⠛⢦⠀⠀⠀⠀⠛⠈⠠⣤⠒⡄⠂
⠀⠀⠀⠀⠀⠀⠠⠀⠉⠲⢤⡀⠀⠀⠀⢀⡾⠀⠛⠀⣿⠛⠁⠀⢿⣌⣉⣉⠚⣽⠃⢀⡴⢾⡀⠀⠐
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠈⠑⠆⠀⢸⡆⣴⠞⠛⢿⣄⣀⣸⡏⠉⠹⠏⠀⢙⣷⣞⣁⣼⢳⡄⠐⠆
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⢸⣬⣥⡗⢸⠃⠈⣉⣉⠻⣶⠆⣴⠟⣬⠉⠉⣉⡾⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⣤⣴⡏⡟⠀⢾⣱⡆⠙⣛⢦⣀⣀⡉⣿⠇⠀⣠⡞⠁⢠⡏⠉⢰⣏⠀⠀⠀⠂
⠀⠀⠀⠀⣤⠀⢰⣾⣋⢿⣷⠀⠸⣿⡄⠀⠹⣷⠾⣇⣀⣉⡭⠹⣿⡷⠛⠁⡙⢛⡏⠉⠙⠋⣿⠀⠀⢀⡀
⠀⠀⠀⠀⢻⣧⣤⣿⣯⡼⠿⠀⠀⠁⠀⠀⠀⢹⠷⣬⣿⣍⣀⣀⣿⣴⡶⠀⠹⣏⠁⣀⡶⡰⠋⠀⠀⣾⣻⣷⠂⣀⠀⠃⢀⠔
⠀⠀⠀⠀⠘⠛⠛⠉⠁⠀⠀⠀⠈⠃⠘⢲⠀⠈⠓⢶⡄⢩⡭⡵⠏⣩⡿⣤⣤⡿⢋⣭⡴⠃⠀⠠⠀⠙⠛⠃⠀⢉⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠉⢀⡄⠀⢉⠧⠀⠈⠓⠚⠓⠒⠛⠉⠙⠒⠒⠚⠉⢀⣀⠀⢠⡄⢠⢶⣔⡆⢎⠟⠳⣾
⠀⠀⠀⠀⠀⠀⠀⠐⠆⠐⢢⡄⠀⠈⠋⠀⠰⠀⡔⢒⡒⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢭⡏⢻⡎⣿⣵⣅⣀⢻⡌⣿⡀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠱⡄⠀⠀⠀⠀⠈⠙⠉⡹⠀⡅⢀⢀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣄⢻⡇⠿⡿⠘⠷⠙⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠷⠀⠀⠘⠁⠀⠀⠇⠀⢀⠀⣉⡤⠂⠃⠬⠞⡵⠀⠀⠠⠂⠀⠈⠛⠉⠋⠅⠐⠒⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠴⠃⠀⠀⠀⠀⠀⠧⠊⠁⣠⠆⠀⠛⠂⠀⠀⠀⡀⠀⠀⠀⠀⡀⡀⠀⠀⠂⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠘⠀⠜⠁⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠸⠄⠉⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⢔⣲⠶⠖⢛⣡⡶⠖⠒⣂⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⢉⣤⠞⢋⣠⠶⠛⣉⡤⠖⢚⣉⡭⠤⠖⢚⣛⣒⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⣿⡟⠁⣰⠟⢁⡴⠋⣡⡴⠋⣡⠴⠚⣉⡥⠖⢚⣋⣭⠥⠤⠖⠚⢓⣦⡀
⠀⠀⠀⠀⠀⢀⣔⠉⢿⡀⣿⠀⢸⠇⢠⡟⢀⡾⠃⣴⠋⣡⠖⣋⡥⠖⣋⡭⠴⠒⣚⣋⣩⠭⠭⠭⠽⠦⡀
⠀⠀⠀⠀⣰⣏⠙⢦⡈⠻⣿⡀⢸⡄⢸⡇⢸⠃⣼⢁⡞⢡⠞⣡⢖⣫⠵⣒⡫⠭⠴⢒⣒⣒⣚⣛⣛⣛⣛⣆
⠀⠀⠀⡼⢥⣉⠳⢦⣉⠳⢬⡳⣄⠳⣄⠳⡘⢇⢻⢸⢠⢃⢎⠔⠩⡒⠭⠐⠊⠩⠭⠅⣒⣒⡒⠒⠲⠦⠤⠬⢧
⠀⠀⡜⠓⠦⣍⣙⠲⠬⢍⣒⠪⠭⠓⠪⣑⣈⠊⠣⠁⠈⠀⠀⠁⠀⢀⡀⠀⠀⠉⠓⣒⣒⡂⠬⠭⠭⣍⣙⡓⠲⢧
⠀⣸⣍⣙⡓⠲⠬⠭⣍⣒⣒⣊⣁⡀⠀⠀⠈⠁⠂⠐⠐⢊⠷⠂⠀⠀⠀⠀⠀⢀⠄⠤⣒⡢⠭⣍⣓⠲⠤⣍⡙⠒⢇
⢀⠗⠦⠤⠭⣍⣙⣓⣒⣒⣒⣒⣒⠢⠠⠤⣐⠤⣐⠔⡢⡊⡄⡄⡠⡠⡐⢄⠪⣔⠪⣝⠲⢭⡑⠲⣬⡙⠲⢤⣉⠛⢾
⢸⣉⣛⣓⡒⠒⠒⠒⠒⠒⠒⣒⣒⣊⠭⠥⣒⡭⠔⡫⢎⡼⢡⠃⡇⢱⡘⣦⠱⣌⠳⣄⠙⢦⡉⠳⣄⠙⠳⣄⠉⠳⣤
⠸⠤⠤⠭⠭⣭⣭⠭⠭⠭⠭⠤⠖⢒⣊⣩⡥⠖⣋⡴⠋⣰⠋⣰⠇⢸⠇⢸⡆⢸⡆⠘⣧⠀⢻⣆⠈⢷⡄⠈⢷⣄⠈
⢸⣛⣒⣒⣒⣒⣒⣒⣲⣶⡿⠿⠟⠛⣉⣡⠴⠚⣉⡴⠞⢁⡴⠋⢠⡟⠀⣼⠇⢀⣿⠀⢸⡇⠀⢿⡆⠀⢿⡆⠀⢻⣦
⢸⣉⣉⣉⣉⣉⣉⣠⣤⡤⠴⠶⠚⠋⣉⣤⠴⠚⢉⣠⡶⠋⢀⣴⠟⠀⣴⠟⠀⣸⡏⠀⣸⡇⠀⢸⣷⠀⠸⣿⠀⠀⢿
⠈⡏⠉⠉⣉⣉⣁⣀⣤⣤⠴⠶⠚⠋⢉⣠⣤⠶⠛⠁⣠⡴⠟⠁⣠⡾⠋⠀⣴⡟⠀⢠⣿⠃⠀⣸⡟⠀⠀⣿⠀⠀⢸
⠀⢹⡛⠋⠉⠉⠉⣉⣀⣠⣤⡴⠶⠛⠋⠉⣀⣤⡶⠟⠉⢀⣤⡾⠋⠀⣠⣾⠏⠀⣠⣿⠏⠀⢠⣿⠃⠀⢸⣿⠀⢀⡏
⠀⠀⢻⠟⠛⠛⠛⠉⠉⢁⣀⣠⣤⡶⠾⠛⠉⢁⣠⣴⡾⠛⠁⢀⣤⡾⠛⠁⢀⣴⡿⠃⠀⣠⣿⠇⠀⢀⣿⡏⠀⡜
⠀⠀⠀⠳⣶⠶⠶⠿⠛⠛⠉⠉⢀⣀⣤⣴⠾⠟⠋⠁⣀⣤⣶⠟⠋⠀⣀⣴⡿⠋⠀⢀⣼⡿⠁⠀⢠⣾⡟⢀⠞
⠀⠀⠀⠀⠙⢄⣤⣤⣴⡶⠾⠟⠛⠉⠉⢀⣀⣤⣶⠿⠛⠉⠀⣀⣤⣾⠟⠉⠀⢀⣴⡿⠋⠀⠀⣴⣿⠏⡠⠋
⠀⠀⠀⠀⠀⠈⠻⣅⠀⣀⣀⣤⣤⣶⠾⠟⠋⠉⠀⢀⣠⣴⡾⠟⠉⠀⢀⣠⣾⠿⠋⠀⠀⣠⣾⠟⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡋⠉⠁⠀⣀⣠⣤⣶⠿⠟⠋⠁⠀⣀⣤⣶⡿⠛⠁⠀⢀⣴⣿⢟⡥⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠺⢿⣛⠋⠉⠀⠀⣀⣤⣶⡿⠟⠋⠁⠀⢀⣤⣾⡿⠟⠋⠁
⢀⣀⡀⠀⡀⡀⡀⡀⠀⠀⢀⣀⣀⡀⢈⣉⣓⠚⠻⠿⣯⣥⣤⣤⡤⠔⠚⠛⢉⡁⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⠀⠀⡀
⣿⣏⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣽⣧⡀⠀⠀⢀⣼⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠉⠀⣈⣿⣿⣿⣏⣁⠀⠉⠉⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⣀⣤⢶⣿⠟⠋⠁⠀⠀⠈⠙⢿⣿⠶⣤⣄⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣴⠟⠋⠁⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠈⠙⠻⣦⡀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⢀⣴⠟⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠻⣦⡀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣄⣠⡿⠁⠀⠀⠀⠀⠈⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡿⠀⠀⠀⠀⠀⠈⢿⣄⣠⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠙⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠉⠀⠀⠈⠻⡆
⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣠⣴⡿⠁
⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠛⠛⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⠛⠛⠿⢷⣄⡀
⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣄⣤⣤⣤⣄⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⣯⡉⠙⢿⡍⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠛⠷⣦⠀⠀⠀⠀⠀⠀⠀⠛⢻⡿⠛⢛⣿⠟⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠈⠻⣦⡀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⢀⡴⠟⠁⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠈⠛⢶⣄⡀⠀⠀⠹⣇⠀⠀⠀⢀⣼⠇⠀⠀⢀⣠⡶⠛⠁⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠈⠛⠻⠶⣦⣭⣿⣶⣾⣿⣥⣴⠶⠟⠛⠁⢀⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣤⡾⠋⠉⠁⠈⠉⠛⢷⣄⣤⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠟⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠿⢛⣃⣠⡙⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⡴⠛⣁⣤⢶⣦⡾⢋⠰⣦⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢋⣴⠾⢛⡋⠙⣡⣾⣏⠻⣿⣿⣿⣿⣿
⠀⠀⠀⢀⡾⢁⡾⠋⠀⠐⠋⠀⠙⢷⡈⢷⡀⠀⠀⠀⢸⣿⣿⣿⡿⢡⡿⢡⣾⣿⣧⣾⣿⣦⡙⣷⡘⣿⣿⣿⣿
⠀⠀⠀⢸⡇⣼⠁⠀⠀⠀⠀⠀⠀⠈⣧⢸⡇⠀⠀⠀⢸⣿⣿⣿⡇⣿⠁⣿⣿⣿⣿⣿⣿⣿⣷⢸⡇⢸⣿⣿⣿
⠀⠀⠀⢸⡇⢹⡄⠀⠀⠀⠀⠀⠀⢠⡏⢸⡇⠀⠀⠀⢸⣿⣿⣿⡇⢻⡆⢿⣿⣿⣿⣿⣿⣿⡟⣸⡇⣼⣿⣿⣿
⠀⠀⠀⠀⠻⣄⠹⠦⢀⣴⣂⣀⡴⠟⣠⡟⠀⠀⠀⠀⢸⣿⣿⣿⣿⡌⠭⣬⣹⡿⠋⣿⠿⢋⣴⡟⣰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⠃⢴⡏⠉⣉⣉⣤⠾⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣾⡃⢴⣶⡶⠾⠛⣩⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⢺⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢣⣀⣠⣼⠌⠘⠄⢀⠤⠤⢤⠀⠀⣠⠤⢖⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡈⠉⢁⣀⠀⣸⠀⣸⣤⡼⠼⠒⠚⢋⣍⡦⣧⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠤⣝⢦⠀⠀⠀⣀⡩⢵⢶⡌⠢⢘⣥⣤⡀⢀⣀⣨⡴⣚⣿⠇⠿⢐⡇
⠀⠀⠀⠀⠀⣞⡉⠲⡀⠀⠀⠀⠀⣎⣠⠖⠻⢿⣒⠺⣩⣠⣠⣠⠸⢸⢮⡇⠉⠭⣼⡟⡉⣠⡏⢸⣿⣇⣸⣐⣷⣤
⠀⠀⠀⠀⠀⠈⢫⠖⢛⠯⡉⢻⡉⠀⡞⠄⠀⠀⠉⠊⠉⠒⢤⡈⠙⠫⡉⡇⠀⣓⣿⣃⢱⡿⣠⣦⣬⡍⣙⡛⢁⣿⠁
⢎⣙⢦⡀⠀⠀⢸⢸⠤⡷⠷⢌⣧⣡⠭⠿⢷⡀⠀⠰⡆⢤⣦⣈⠣⣄⣌⢇⠀⠀⢿⢻⣿⠃⠈⠛⠻⠿⠶⢋⣾⠛⢆
⠈⠣⣶⠙⠢⣄⢸⠀⠑⠁⡆⡑⡇⢸⠀⡭⡟⢧⣄⣈⠪⣶⡿⢉⣷⡘⣞⣷⣆⠥⣿⣅⠘⡄⣭⡵⠞⠛⠛⣿⠟⢆⠈⣆
⠀⠀⠉⠳⣦⣂⡽⠀⠀⠀⡆⡢⣇⠈⢶⣶⢋⡌⠠⢮⡟⣛⠟⢩⣿⣷⢹⡿⠿⣩⣿⡝⣺⣘⡤⠶⠄⠀⣼⣟⣦⠈⡦⣙⡆
⠀⠀⠀⠀⠈⣹⠃⠀⠀⠀⣇⠂⣿⣇⣹⠛⢶⠁⠨⣼⡿⣿⠰⣿⣿⡾⠋⣿⣶⣿⢿⣿⠃⣻⡽⠧⢶⣶⡿⠿⢲⢋⡇⢒⡇
⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠸⡀⠉⢿⠻⢿⡏⢀⡥⣿⣶⣿⢣⢄⣉⣡⣴⣿⡧⠿⣒⣋⢭⠗⡶⢫⢋⠎⡐⡱⡡⢣⡧⢽⠁
⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠻⣼⠀⢐⣸⣾⣿⣷⡤⢾⣛⡿⡿⣳⢋⠝⡡⢣⢊⠞⡰⡡⢃⢞⠜⡔⡡⢡⠏⢽
⠀⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⢈⣦⢒⣛⡿⡿⡻⠫⢋⢎⢎⠔⡡⢋⠞⡔⡡⢃⢎⠜⡰⢡⢊⠞⡔⡱⣹⠤⢽⠢⣀⡀⠠⠠⠄⣀
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠳⠀⢽⢉⢔⢎⠔⠔⡱⡱⢡⢊⠜⡰⢁⢊⠜⡰⡣⢊⠎⡰⠡⢂⢎⠜⡔⣼⠭⣿⡑⢐⠀⠂⣙⡂⠈⠣⠥
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⢀⠭⡧⢁⢪⢊⢎⠌⡔⡡⢊⠎⡔⡡⢋⢜⡔⡡⢃⠜⡔⡡⢣⢊⢞⣴⡾⢑⣟⡈⠠⢍⣀⡒⠂⣠⠤⠭
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠥⡷⡑⡡⡣⢣⠎⡔⡱⢁⢎⠔⡡⢋⢊⠜⠠⢊⠌⣔⣱⣥⣿⡻⣽⠷⠞⠩⠭⠉⠐⣒⡐⣃⡈⠉⠉
⠀⠀⠃⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠨⢽⠎⢜⠔⡵⢡⢊⠎⡰⡡⢊⡔⡱⢡⢊⣴⢥⣻⡹⠽⠶⠿⠉⢙⢛⣋⣈⣭⣭⣤⠀⠐
⠀⠓⠤⣘⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢽⢨⢊⠎⡔⡡⢃⢎⡔⣡⣮⠼⣶⣻⠽⠾⠭⠭⠥⠭⠭⠭⠉⠁⠒⠒⠒⠂
⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⡀⠀⡀⠽⡡⢣⢎⣼⡵⡗⣞⣽⡽⡶⠏⠩⠭⠥⠦⠤⠭⠭⠭⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣄⡀⠁⠓⡂⣽⣲⣯⣯⣷⣿⣛⣃⣤⣤⠶⢖⣀⣀⣀⠀⠌⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠶⠿⠿⠟⢀⣐⣒⣒⣒⣛⠓⠓⠖⠛⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⠀⣰⠋⠠⡇⠀⠀⠀⠀⠀⠀⠀⣼⢁⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠃⠀⠁⠹⡄⠀⠀⠀⠀⠀⢠⠋⡀⣿⢰⠃⠐⠀⢷⠀⠀⢀⡄⠀⠀⢀⠇⠈⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢤⣜⠀⠀⠀⠀⢹⠄⠀⠀⣠⣶⠃⠀⠁⣿⡏⠀⠀⠀⠘⡆⢀⡏⢳⡀⢀⠞⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡴⢿⠀⠄⠉⠀⠀⠀⠀⢸⠀⡠⠞⠁⠉⠀⠀⠀⠉⠁⠀⡀⠀⠀⡇⢸⠱⠀⠳⠋⠀⠀⠀⢀⣧⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠸⢠⠸⠆⠀⠀⠀⠀⠐⠀⣯⠞⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡿⠃⠀⠀⠀⠀⠘⠀⠀⠈⡤⢸⠾⡇
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⣧
⠀⠀⠀⠀⠀⠀⢧⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡟
⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⣠⠞⢃⡀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⢧
⠀⠀⠀⡰⠁⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⢦⡀
⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⡎⠀⠁⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢧
⢰⠇⠀⠀⠀⠀⠀⣸⠋⠀⠀⠉⠉⠉⠉⠉⠉⠉⠹⡆⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⡞⠋⠉⠉⠉⠙⠛⠛⢳⡀⠀⠀⠀⠀⠀⠈⡆
⠸⠀⠀⠀⠀⠀⢀⡏⢠⠤⣶⣄⣤⣦⣠⢶⡦⡀⠀⢳⠀⠀⢀⣀⣼⠉⣿⣀⡀⠀⢀⡞⠀⠀⣠⠤⠤⢤⡄⠀⠀⢳⡀⠀⠀⠀⠀⠀⢹
⡇⠀⠀⠀⠀⠀⢸⠁⠳⣿⣴⠯⠶⢬⣿⣽⡧⠟⠀⠘⣇⠀⠹⠤⣅⠀⣨⣭⡇⠀⡼⠁⠀⢀⡭⠋⠶⠅⣤⠀⠀⠈⣧⠀⠀⠀⠀⠂⠸
⡇⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠻⠶⠇⠀⠀⣼⠇⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⢸⡆⠀⠀⠀⡄
⡇⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠛⠛⠒⠒⠒⠒⠒⠒⠒⣾⠀⠀⠀⠀⠀⠀⠀⠀⢨⣬⣄⣌⣽⣭⣭⣭⣭⣽⣽⣿⣟⣻⠇⠀⠀⠀⡇⢠
⣿⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡏⢸
⢸⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠈⣧⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠸⡆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠈⠳⣄⡈⠂⠠⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠂⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣤⣀⠈⠁⠒⠀⠐⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣈⣤⡤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠶⠦⠤⠤⣤⣤⣤⣀⣀⣀⣀⣤⡤⠤⠤⠶⠶⠒⠚⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⡿⠿⠿⠛⠛⠛⠛⠻⠿⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠛⠉⠀⢀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣤⣬⣉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠟⠁⠀⣀⣴⣾⣿⡿⠟⠋⠉⠀⣀⣀⣠⣤⣄⣈⣉⠛⠶⣬⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠀⠀⣠⣾⣿⡿⠛⠁⠀⣀⡴⠾⠛⠛⠋⠉⠛⠛⠻⠿⣿⣷⣦⣝⠪⡣⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠃⠀⢀⣼⣿⣿⠋⠀⠀⡠⢚⣥⣶⠶⠿⠿⣿⣷⣶⣦⣄⡀⠀⠙⠻⣿⣿⣮⡳⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁⠀⢀⣾⣿⡿⠁⠀⠀⣪⢞⣉⣤⣤⣤⣤⣀⡀⠀⠙⠻⣿⣿⣦⡀⠀⠈⢻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⣾⣿⣿⠁⠀⠀⣮⠖⠋⠉⠉⠉⠙⠻⣿⣿⣶⡄⠀⠈⢻⣿⣿⣆⠀⠀⢻⣿⣿⣇
⠀⠀⠀⠀⠀⢀⣿⣿⡟⠀⠀⢸⣿⣿⡇⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡆⠀⠀⢻⣿⣿⡄⠀⠀⢿⣿⣿⡀
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠈⣿⣿⣧⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢀⣿⣿⡟⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⢸⣿⣿⡇⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇⠀⠀⣼⣿⣿⠃⠀⠀⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄⠀⠀⢿⣿⣿⡀⠀⠀⢟⠦⣄⣀⣀⣀⣠⣴⣾⣿⠿⠃⠀⢀⣼⣿⣿⠏⠀⠀⣼⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀⠀⠈⢿⣿⣷⡀⠀⠀⢝⢮⣉⠛⠛⠛⠛⠉⠁⢀⣀⣐⣿⡿⠻⠃⠀⢠⣾⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡄⠀⠈⢻⣿⣿⣄⠀⠀⠑⢬⡛⠷⠶⣶⣶⣿⣿⠿⠟⠉⠁⢀⣠⣴⣿⣿⢏⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⠀⠀⠙⢿⣿⣷⣄⡀⠀⠙⠳⢶⣤⣤⣄⣀⣤⣤⣴⣶⣿⡿⠟⣫⢕⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣷⣦⡀⠀⠉⠻⢿⣿⣷⣤⣄⣀⠀⠉⠉⠙⠛⠛⢉⣉⣥⠶⢛⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣤⣀⠀⠈⠙⠛⠻⠿⠿⠿⠿⠿⠿⠛⢛⣉⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣷⣶⣶⣤⣤⣤⣤⣤⣶⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⠿⠛⠛⠛⠻⢿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠿⠛⠛⠛⠻⠿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⣼⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡄
⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠸⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿
⠀⠀⠀⠀⠹⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⣼⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⡄
⠀⠀⠀⢰⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿
⠀⠀⠀⠀⢻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣿⣿⣿⣿⣿⣿⣿⠹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⠀⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⠀⠀⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡏⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⢹⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠹⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠀⠀⠿⠿⢇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣄⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠙⠛⠻⢿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⢿⣿⠇⠀⠀⠀⠀⠀⠈⣿⡿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⠃⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠋⠀⠀⠀⠾⣧⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⠈⠳⠶⠆⠀⠀⠀⠀⢹⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣟⣛⣛⣏⣍⣋⣻⣋⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣿⣯⣽⣿⣿⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣾⠿⠿⣿⣿⡿⠿⢿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠟⠋⣿⣿⠃⠀⣼⣿⠏⠀⣠⣿⡿⠋⠉⣉⣽⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣿⣿⡀⢀⣽⠟⢋⣤⡶⠟⠋⣩⣭⣭⠥⠤⢼⣿⠟⠋⠉⢈⣙⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇⢀⡿⠋⣴⣿⠁⢀⣿⣿⠁⢀⣾⣿⠋⠀⣠⣶⣶⡶⠦⣤⣾⣿⠿⠛⠛⠿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⢘⡟⢡⣿⠁⠐⣿⣿⠀⣨⣿⡿⠟⠉⣀⣠⣴⣶⣿⣿⠦⠄⣤⣤⣬⣁⠀⣠⣤⣴⣾⣿⣆
⠀⠀⠀⠀⠀⠀⣼⣿⣿⢆⣿⠀⢸⣿⣄⢀⠝⢁⣴⣿⠗⠊⣀⣴⣶⣶⣾⠾⠶⠮⣍⡑⠢⣍⡉⠉⠛⢿⠟⠛⠙⠛⢿⣧
⠀⠀⠀⠀⠀⣸⣿⢀⡏⢸⣿⠀⠈⠻⣿⠏⢀⣽⡟⢡⣴⣿⠿⠓⠒⢀⣀⣀⠤⣤⣀⢈⡓⢮⡛⢿⣿⣶⣅⣤⣶⣶⣤⣿⣧
⠀⠀⠀⠀⢰⣿⣿⢺⡇⠸⣿⣷⡄⢰⣏⠀⣾⠋⢐⣿⡿⠂⣀⣴⠶⠾⠷⣿⣿⣷⣿⡿⣿⣦⡙⢦⡁⠀⠙⢿⠛⠛⠛⠿⣿⡆
⠀⠀⠀⠀⣾⡇⡅⢸⣧⡀⠈⠛⡇⣾⡇⢸⠇⢠⣿⠏⣠⠞⣩⣶⣟⣛⡷⠦⣄⡈⠳⡙⣎⢻⣻⡄⢳⣜⣷⣌⢦⣤⣤⣄⣸⣷
⠀⠀⠀⠀⣿⣿⣇⠘⢿⣿⣶⣼⡇⣿⡇⣿⡆⢸⣿⢰⠃⣾⡽⢋⢩⡉⣉⠲⡀⢻⡄⠘⡈⡄⠃⢹⡄⢿⡇⠹⡞⠛⠛⠿⣿⣿
⠀⠀⠀⠀⣿⡋⢿⣄⠀⠉⠛⠻⡇⢸⡇⣿⡇⢸⡇⢈⢸⢸⣧⣞⡟⠛⣿⡆⢹⠀⣿⡆⣧⢹⢠⠀⣿⠈⣿⣆⢳⣤⣄⡀⠘⣿
⠀⠀⠀⠀⣿⣷⠘⢿⣷⣦⣤⣄⢷⡘⣧⠹⣧⡀⢷⡸⣌⢈⢿⣿⣽⣧⣿⢋⣾⠄⣿⡇⡿⣼⢸⡇⢻⡆⡏⢿⠘⠛⢿⣿⣦⣿
⠀⠀⠀⠀⢿⡿⣧⡀⠉⠛⠛⠛⡜⣧⠘⣆⠘⣿⣄⠳⣝⢶⣕⣬⣭⣭⣶⡿⢋⣼⡿⣹⢃⡿⣿⠁⣼⣇⡇⢸⣤⣄⠀⠘⣿⡿
⠀⠀⠀⠀⠸⣷⡝⢿⣶⣤⣤⣴⣿⣌⢳⣌⢷⣌⠻⢷⣌⣛⠮⠭⠍⠉⠁⣒⠵⢋⡴⢃⡾⣱⡿⠀⡿⢸⡇⠙⠻⣿⣷⣄⣿⠇
⠀⠀⠀⠀⠀⢻⣿⣄⠉⠛⠛⠛⠋⠈⠢⣙⢷⣟⡻⣦⣌⣉⡛⠛⠛⣛⣩⣥⠶⢋⡴⢋⣴⡿⠃⣰⡇⣸⣷⣄⠀⠈⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢻⣿⢷⣦⣤⣤⣴⣾⣷⢮⣷⣮⣙⠒⠉⠉⣉⣉⣉⣉⣭⡶⠞⠋⢀⣴⣿⠇⢼⣿⡁⠈⠻⣿⣷⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣷⣍⠛⠛⠛⠋⠁⢀⣼⣿⡊⠉⠐⠒⣶⣾⣿⣟⣛⠉⠀⣲⣽⣟⢕⠀⠚⣿⣿⡄⠀⢹⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⣦⣴⣶⣿⡿⠏⠀⢠⣿⣿⠛⢛⣛⡩⠤⢶⣮⡍⠊⢹⣿⣧⠀⠘⣿⣿⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣯⣍⣉⠁⢀⣀⣴⣿⡿⠃⠀⣸⣿⡇⠀⢸⣿⣷⠀⢨⣿⣿⠀⢰⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⠛⠋⠀⣀⣴⣿⠟⠁⢀⣾⣿⠃⢀⣼⣿⣧⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣥⣤⣶⣿⣿⣷⡶⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢿⡌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢹⣦⠹⣆⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣈⠳⣌⢳⡘⢷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡟⠷⣝⠳⣝⠳⣍⡳⢦⣄⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡶⢡⡶⣌⡳⢬⡓⢮⣝⣓⠶⠶⢶⣒⣒⠲⣜⣎⣙⠿⢽⣿⠋⣹⡶⢋⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟⡿⣰⢤⣝⡲⢭⣓⠦⠭⠭⠭⣝⣲⠽⣝⣳⢂⡿⣡⠞⣡⠞⣩⡴⢛⣡⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢣⡏⡞⡬⣍⣓⠮⣝⣛⣻⠭⣗⣪⡍⡼⣡⢏⡼⣣⠞⣡⠞⣡⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⣼⡹⣻⣱⣨⣝⣒⠶⣒⣭⡇⡜⡼⣱⢃⡞⡴⢣⡞⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⢻⡇⡇⣧⢧⢣⢳⢨⡭⡷⣼⢸⢱⢇⡏⡞⣸⢡⠏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢣⡟⣼⢱⢇⡏⡾⣸⢸⠑⣛⣛⡘⡜⡜⡎⡇⡇⣿⢼⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⣱⢏⡼⣡⢏⡞⡼⢱⢃⣛⠭⢖⡲⠭⣝⣸⡼⣸⣹⢹⢸⢸
⠀⠀⠀⠀⠀⠀⠀⣠⡴⢛⣥⠞⣡⢏⡴⢣⠞⣼⣁⡫⢭⣒⡭⠭⢭⣝⡲⢭⣁⢣⢇⡏⣼⢸
⠀⠀⠀⠀⣤⠶⢛⣡⠶⢋⣵⠞⣡⠟⠴⠯⣌⣛⣶⢭⣑⣒⣛⣛⣓⠲⢭⡓⢮⣙⠺⢸⡵⣿⡀
⠀⠀⠀⠀⠀⠐⠛⠥⣔⡻⡤⢨⣽⣿⣌⠵⣿⡾⣽⣿⣒⣒⣒⠶⢬⣝⠲⣍⡳⣬⡻⠏⡼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠙⠳⢬⡙⢮⡙⢦⡙⢦⡇⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⣦⠹⣦⡱⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⢳⡌⠻⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠹⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡦


⣿⣿⣿⣿⣛⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣟⣿⣿⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠉⢉⡙⡲⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠠⢂⠕⢒⣠⠴⠷⠛⠓⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢠⡴⠚⢉⣀⣀⣀⣀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣰⠋⡠⠞⠉⠁⠀⠈⠉⠙⠶⣄⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢰⣃⠊⠀⠀⠀⢀⠀⠀⠀⠀⠀⠙⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠟⠁⠀⠀⠀⠀⠈⠀⠀⠀⠘⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠀⠀⢀⠀⠀⠀⠲⠶⠂⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣺⣆⠀⠀⠸⡶⠦⠤⠀⢤⣤⠞⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⣦⡀⠀⠈⠒⠤⠤⠐⠁⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠲⣝⡟⣧⠀⠀⠀⢀⣠⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠘⣟⠄⠀⠀⢠⣅⠈⠑⠤⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝⠁⠈⠑⡄⠀⣸⠈⢳⡤⡀⠸⠃⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠂⠛⢆⡀⢀⠛⠣⣴⠃⡴⢬⠷⣈⠷⢀⣀⡠⢸⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠙⠏⠀⠀⠘⡄⠀⠀⡀⠸⡗⡦⠙⠛⣏⡀⠈⠛⢦⠶⠛⠛⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢈⠉⠙⠀⠈⠷⠀⠀⣿⠙⠳⣄⠀⠀⢀⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⠁⢉⠀⠀⡔⠀⠀⢸⡀⠀⢺⡀⠀⠻⣄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠒⠀⣶⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠳⡀⠁⠀⠀⢸⡇⠀⠈⠳⣄⠀⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⢦⡀⢀⡼⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⠀⣿⢀⠃⢐⢀⡂⠀⠃⠆⡆⢺⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠠⠆⠀⣈⠀⠀⠀⠀⠁⠀⠀⠁⠀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢰⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⡠⣶⠞⠁⠀⠀⢸⡄⠀⡉⠀⠀⠀⠀⠀⠛⡟⠃⠀⠀⣿
⢠⢦⠀⠤⠟⣅⠀⠀⣠⠶⣼⠇⢘⡋⠀⠀⠀⠀⠀⠀⡛⠀⠀⠀⡋
⠸⠀⠑⢄⠀⠈⠳⠄⡏⠀⣸⡀⢸⡇⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⡟
⡰⡔⡄⠀⠑⢄⠀⠀⠂⠠⠂⠙⠺⣇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⡇⠷⠃⢀⠤⠀⡙⠀⠀⠀⠒⠒⠒⠚⢳⠀⠀⠀⠀⠀⡇⠀⠀⠀⢣
⠙⢄⠀⠀⣰⠂⣇⣀⣀⣀⣀⣀⣀⣀⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠑⣄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠈⠳⣬⣀⣀⣀⣠⣤⣤⣤⣤⠁⠀⠀⠀⠀⣾⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠉⠉⠉⠉⠉⠁⢹⠓⠲⠶⠖⠒⠋⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠀⠀⠀⠀⠀⠀⢀⡿⣂⠤⣀⡀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣀⡰⠶⠶⠛⠓⠋⠉⠀⠀⠉⠐⠒⠭⠤⣤⣀⣀⣂⡾

⠀⠀⠀⠀⢰⣇⣤⠀⠀⠀⠀⠀⠀⣾⣠⡄⠀⠀⠀⠀⠀⠀⣰⠗⠀⠀⠀⠀⠀⣀⣸⣃⡄⠀⠀⠀⠀⠀⢀⣴⡠⠀⠀⠀⠀⠀⠀⣠⣆
⠀⠀⢀⣴⠟⠉⠁⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⢠⣤⠞⠀⠀⠀⠀⢀⡠⠞⠉⣸⠋⠀⠀⢀⡴⠚⠋⢩⡏⠀⠀⠀⢀⣤⠤⢲⠟⠁
⣠⣴⡟⠁⠀⠀⠀⠀⢀⠠⡾⠋⠀⠀⠀⠀⠀⢠⣶⡿⠉⠀⠀⠀⠀⠀⠉⠀⢀⠜⠁⠀⠀⠀⠈⠀⠀⢀⡞⠀⠀⠀⠀⠋⠀⢀⡾
⠉⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⢀⡴⠋⠳⠄⠀⠀⠀⠀⠀⣠⣶⡖⠀⠀⠀⠀⠀⠲⢶⣄⠀⠀⠀⠀⠀⠀⣴⣦⡀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠀⠐⣅⠁⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⠀⠀⠀⠀⢀⣴⠏⠙⠗⠀⠀⠀⠀⢀⡼⠻⠿⠂⠀⠀⠀⢀⡾⠛⠿
⠀⢠⣦⠀⠀⢈⡗⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠠⣎⢻⣿⠏⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢠⡶⠋
⠐⠀⠀⠑⠴⠛⠁⠀⠀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠹⠞⠁⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠐⠟
⠀⢠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⣀
⠀⢸⣇⣀⣰⡆⠀⠀⠀⢠⡠⣄⠀⠀⠀⠀⠀⠀⠀⢀⡴⢺⡍⠉⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⢀⣴⣾⣿
⠀⢸⡇⠀⢰⡆⠀⠀⢠⣿⠃⣾⣧⣄⣾⠇⠀⢀⡴⠋⣰⠎⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⣠⣾⣿⠟⠋
⠀⢸⡇⠀⠘⠃⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠉⢠⠞⠁⠀⠀⠀⠀⠀⣔⡿⠁⠀⠀⠀⠀⠀⢠⣾⠿⠟⠛⠀⠀⠀⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠐⠦⣄⡀⢰⡆⠀⠀⢀⣾⣛⣛⣿⣷⠀⠀⠀⣤⣤⣤⣤⣤⡄⠀⠀⣾⣿⣿⣶⠶⠆⠀⠀⠀⠀⠀⢀⣤⣿⡷⠀⠀⠀⠻⠿⠿⠿⢳
⢀⣤⣤⣬⣿⣿⡇⠀⠀⣸⣟⠛⣿⣿⣿⠀⠀⣸⡟⠛⠛⢸⡆⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠋⠀⠀⠀⠀⠀⢀⡄⠈
⢿⣶⣶⣶⣿⣿⡇⠀⠀⠉⠉⠉⠉⠉⡏⠀⠀⠻⠀⠀⠀⠸⠇⠀⠀⠀⠀⣿⣷⠆⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⣀⣀⣀⠀⠀⠀⠀⠖⠖⠒⠒⠒⠒⠀⠀⠀⣀⣀⠀⣰⠶⣦⠀⠀⠀⠀⠀⢀⣶⣶⡀⠀⠀⠀⠀⠀⣠⣶⠃⠀⠀⠀⣶⣶⣶⣶⡦
⠀⣰⣿⠿⣿⡆⠀⠀⠀⡦⠤⠤⠤⠤⠤⠀⠀⢸⣏⣹⡇⠙⠶⠛⠀⠀⣠⣶⣷⣾⡟⠟⠁⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠺⠿⣿⠿⠗
⢠⣼⣿⣿⣿⣧⡄⠀⠀⣤⣤⣠⣄⣤⣤⠀⠀⠀⠉⠉⢠⡟⢻⡄⠀⠀⣻⣿⣼⣿⡇⠀⠀⠀⠀⠸⣿⠥⠄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⢀⡀
⣴⡾⠛⢷⡄⠀⠀⠀⢰⣿⠶⠶⠶⠶⠶⠆⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣤⣤⣯⣭⣧⣤⡄⠀⢀⣀⣤⣶⣾⡷⠁⠀⠀⠀⣴⣿⣾⣿⣷
⣿⣦⣡⣾⡃⣀⡀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠛⠚⠛⠛⠛⠛⠚⠁⠀⣿⣭⣿⣿⣯⣽⡇⠀⠉⣙⣿⠿⢍⡁⠀⠀⢀⣴⣿⣿⣿⠿⠟

⠀⣀⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠙⢆⣴⠛⠳⣼
⡞⠉⢹⣦⠏⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⠄⡸⠙⡶⢀⣿
⢣⣐⣏⠘⢆⣺⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠈⠋⠉⢸
⠀⠀⠀⠀⠀⠁⠀⢠⠶⠖⠒⠚⠛⠋⠉⠉⠉⠉⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠈⠉⠉⠙⠛⠛⠛⠋⠻⡄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢠⠏⠀⣀⡀⠀⡀⠀⣀⠀⢀⣀⣀⠈⢳⡀⠀⠀⠀⠀⠀⠀⢠⡴⠶⡄⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢾⣛⣒⡒⠒⠒⡄⠀⠹⡄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢠⠏⢰⠋⠴⡃⠎⢈⠚⣀⠱⣁⠰⠄⡱⠀⠹⣄⠀⠀⠀⠀⡴⠚⠃⠀⠛⢲⡄⠀⠀⠀⣠⠏⠀⠀⠀⣠⠴⠊⢡⣶⣩⣄⠀⠀⠹⣄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡰⠃⠀⠈⠓⠒⠛⠒⠉⠉⠀⠃⠉⠓⠚⠁⠀⠀⠈⣳⡄⠀⠀⠙⠒⣦⠀⡟⠋⠀⠀⠀⠐⡋⠀⡀⠀⣀⡛⠛⠛⠛⠛⠛⠋⠀⠀⣀⣹⣷⠀⠀⠀⠀⢸
⠀⠀⠀⣼⠥⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠒⠒⠒⠒⠊⠉⠉⠉⢹⡇⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠘⡃⠀⠁⠉⠉⠉⠉⠙⠛⠓⠓⠛⠛⠉⠉⠀⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠤⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠶⠶⠋⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢠⡴⢚⣩⣭⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠿⣿⣦⡀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢰⡟⢠⡿⠛⠙⠻⣿⣦⠀⠀⠀⠀⠀⠀⣾⣿⡟⠁⣠⣄⠀⠹⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸⣧⠸⣇⠀⢀⡇⣽⣿⠆⠀⠀⠀⠀⠀⡄⠀⠘⢿⣿⠟⠀⢀⣿⠁
⠀⠀⠀⠀⠘⣿⣟⣿⣿⣿⣿⢟⡿⠃⠀⠀⠀⠀⠀⠈⢿⣷⣌⣉⣩⣴⣿⠟⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⣠⣾⣿⣷⣶⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠛⠄⢻⢰⠟⣀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⣉⣉⠛⢿⣦
⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢶⡤⠄⣠⣤⣆⢌⠜⢋⠁⠀⠀⠀⠀⢀⣿⡇⢠⡿⢛⠻⣿⣮⣿⡇
⠀⠀⠀⠀⢸⣿⣧⣿⣿⣿⣞⢿⣷⡇⠀⠀⠀⠀⠀⢠⣴⠊⡀⠙⠉⣁⣚⠙⠃⠀⠀⠀⠀⠀⣿⣇⠘⠧⠼⠃⣽⣿⣿⠇
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣵⣿⠟⠀⠀⠀⠀⠀⠀⠀⠠⡾⠃⣧⠑⣦⠉⠋⠀⠀⠀⠀⠀⠀⠈⠿⣷⣤⣤⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣤⢄⡀
⠀⠀⠀⠀⢀⣶⣿⣟⣿⡹⣻⣿⣷⡀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⣉⡉⠙⠳⣄⠀⠀⠀⠀⠀⢠⢿⣿⣿⡿⢿⣿⣾⣢
⠀⠀⠀⠀⢸⣿⠷⣻⣿⣿⣿⡴⣿⡷⠀⠀⠀⠀⠀⢰⣿⡇⢠⢞⣉⡙⢷⡀⠸⠀⠀⠀⠀⠀⣿⣾⡷⠁⠀⠈⢸⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣏⣾⣿⡿⣿⣙⣿⡟⠀⠀⠀⠀⠀⠸⣿⣇⠈⠪⠭⠃⣼⡇⠀⠀⠀⠀⠀⠀⢿⣿⣿⣄⢀⣀⣼⣇⢯⠇
⠀⠀⠀⠀⠀⠙⢿⣷⣿⣼⣾⡿⠟⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣶⡾⠏⠀⠀⠀⠀⠀⠀⠀⠈⠻⣽⣻⣿⢿⣿⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠲⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⢂⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣶⣦⡀
⠀⠀⠀⠀⢠⢄⣞⠕⠛⠀⠀⠳⡝⣆⠀⠀⠀⠀⠀⢠⡞⢁⣾⠿⣛⠿⣝⢷⣄⠀⠀⠀⠀⠀⣾⡿⣫⣽⣶⣯⡹⣿⣿⡄
⠀⠀⠀⠀⣾⢾⢼⠀⠀⠀⠀⢸⣻⣿⠀⠀⠀⠀⠀⢸⡇⢸⣏⣾⣿⣷⢸⠈⣿⠀⠀⠀⠀⢸⣿⢳⣿⣯⡂⢈⡿⣸⣿⣷
⠀⠀⠀⠀⠸⣞⢞⠦⣀⣀⡤⣫⢣⡏⠀⠀⠀⠀⠀⠘⣷⡄⠙⠻⠛⠋⠊⣼⡟⠀⠀⠀⠀⠈⢿⡞⣿⣷⣽⣯⣵⣿⣿⠇
⠀⠀⠀⠀⠀⠈⠳⢭⣟⣛⣯⠷⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣶⣶⣶⠿⠋⠀⠀⠀⠀⠀⠀⠈⠻⢮⣛⡿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⣻⣛⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣟⣻⣿⣛⣿⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠚⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡠⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢂⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠸⣄⠃⠀⠃⠀⠘⠂⠆⠢⠍⢐⠂⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠰⡀⠀⠈⢆⠠⠀⠀⠀⠀⡀⡆⠀⠀⡀⡀⠀⠀⠀⡔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠀⠆⠀⢀⠰⠰⠅⠐⠺⠄⢠⠁⠌⠀⠐⠂⢠⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠑⠤⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠙⠢⣀⢀⢀⡴⢄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠀⠐⠀⢉⣉⣀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⡀⣤⠄⢠⢈⠉⠀⡠⠀⠵⠀⠄⠀⠀⠀⠀⠰⠄⠀⠃⠀⠀⣆⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⠚⣋⠉⠁⠀⠑⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠂
⠀⠀⠠⠐⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄⠀⢰⣆⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⢀⠀⠀⠃⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠁⢀⡞⠁⠀⠈⠁⠀⠀⢀
⠀⠀⠀⠐⠤⠄⠀⠀⠀⠀⠀⠀⠀⠉⣷⠊⠁⠀⠀⠈⠉⠀⠀⠀⠀⠰⡾⠃⠉⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⡇
⠺⠶⠶⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠒⠠⠤⠀⠍⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳
⡉⠁⠀⠀⠓⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣐⠦⠴⣆⣀⡀⡀⠀⠀⠈
⠁⠀⠈⠉⠁⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⣅⣐⣊⠉⠙⠐
⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⠀⠀⠉⠉⠁
⠀⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢸⠑⠄
⠓⢷⣜⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⠠⠒⠈⠉⠉
⠀⠀⢛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠐⠂
⠷⠖⠋⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⢀⡄⠀⠀⠀⠀⠀⠠⠖⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢶⡶⠰⠰
⠀⠔⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠉⠀⠠⣪⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⡄⠀⠀⠀⠈⠐⢲⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢠⠀⠀⠀⣄⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠈⠙⠁⠀⡘⠀⠀⢀⡀⠀⠰⠉⠀⠀⠀⠠⣄⡀⠀⢀⠀⠀⢀⠀⠀⢨⠀⠀⠀⠆⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡠⠧⠤⠀⠈⣧⠐⠖
⠀⠀⠰⢦⣀⢄⠁⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠂⠂⠀⠀⢺⡷⢀⠀⠀⠪⣟⠆⠸⠆⠀⠀⠀⢰⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⣲⠶⢓⣠⡀
⣀⣀⣀⣀⣀⣙⣀⣀⣀⣀⣀⣀⣀⣀⣘⣀⣀⣀⣀⣀⣀⣀⣸⣃⣴⣃⣀⣹⣾⣑⣀⣆⣀⣀⣀⣈⣀⣀⣀⣀⣘⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣉⣉⣉⣁⣹⣀⣀⣀
⣿⢻⠿⡿⢿⢛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣛⣛⣻⣿⣟⣛⣿⣻⣛⣛⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠂⠀⠀⠀⣤
⠀⠀⠁⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠃⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠃⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠰⡿⠁⠀⠀⠀⠀⠀⠀⠈⠀⠠⡘⠁⠀⠀⠀⠘⠁
⠀⠀⠐⠠⠀⠀⠈⠀⠀⠀⠀⡤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠁⠀⢀⠑⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡀⠀⠀⠀⠀⠒⠀⣀⠂
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡠⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠
⣈⠀⠀⠀⠀⠐⠆⢀⠐⠄⠀⠀⠀⠀⠀⠘⠖⣠⣀⠀⠀⠡⠄⠔⠀⠀⠀⠀⠀⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠄⠀⠈⠉⠀⠄⠀⠀⠀⠀⠀⠀⠫⠚⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⣤⠰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢈
⠀⠀⠀⠀⠘⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠄
⡠⢀⠄⠀⠀⠀⠀⣰⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⠃⠀⠀⡀
⠣⢀⠜⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠰⣨⡀⠀⠈
⣶⣷⠶⠷⠿⣿⣿⣶⣶⣶⣿⣿⣶⣶⣶⣿⣿⣿⣾⣿⣶⣶⣾⣿⣿⣷⣿⣿⣿⣿⣷⣿⣾⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣾⡿⠿⠿⣿⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠉⠀⠀⣀⣤⣶⣾⠿⠿⠿⠿⠿⢿⣶⣦⣄⡀⠀⠈⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⣠⣴⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣷⣤⡀⠀⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⠀⢀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⠀⠈⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⠹⣿⡄
⠀⠠⣤⣤⣤⣤⣤⣴⣿⠃⠀⢠⣿⣧⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⢻⣷
⠀⠀⠙⢿⣯⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⢉⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠘⣿⡆
⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠀⠀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠙⠻⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⡿⠛⠁⠀⣠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⣄⠀⠀⠉⠛⠿⢿⣶⣶⣶⣶⣶⣾⠿⠟⠋⠁⠀⢀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⢿⣿⣶⣾⣿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⢸⣿⡀⣀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣀⣀⣠⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⣿⣿⡟⠁⠀⠀⠙⣿⣿⠟⠁⠀⠀⠀⣶⠀⠛⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣇⠀⠀⠀⢀⣿⣿⣷⣦⡀⠀⢰⡆⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢻⣿⣷⣶⣶⣿⣿⣇⠀⠀⠀⠀⠀⠀⠶⠀⠶⠀⠀⠀⠀⠠⠐⠒⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣾⠟⠉⢻⣿⠁⠉⠻⠆⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⠀⠀⠀⠀⠈⠋⠀⠀⠀⢀⣀⣿⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⠙⢢⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠋⠀⠀⢿⣦⣀⣀⣤⣶⠿⠟⠛⢻⡟⠛⠻⢿⣶⣄⡀⣠⣾⠟⢰⡇⠀⠀⠀⡸⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⣄⡀⠀⠀⠀⠀⠀⠀⢹⣿⠟⠋⢀⡤⠔⠂⢸⡇⠒⠦⣄⠈⠙⢿⣿⡁⠀⠀⠳⣀⠀⠁⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⢀⣀⠀⠙⢦⠀⠀⠀⠀⣰⣿⠋⢀⡴⢋⠴⠂⠀⢸⡇⠈⠒⢤⡙⢦⠀⠹⣷⡄⠀⠀⠈⠉⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢰⠁⠀⠀⠀⢸⠆⠀⠀⢠⣿⠃⢠⡏⡴⢃⡴⢋⣠⣼⣧⣄⠙⢦⠹⡌⢧⠀⢹⣷⠀⠀⠀⠀⢠⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⢄⡀⢀⣠⠎⣀⣀⣀⣼⡿⠀⣼⢰⡇⡾⢀⣿⠉⠀⠀⠙⣧⠘⡇⢻⠸⡆⠀⣿⣦⣀⣀⣀⠸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠈⠉⠀⠀⠙⠋⠉⢻⣧⠀⠹⠘⠀⠁⣈⣿⣀⠀⠀⣠⣿⡀⠀⠘⠀⠇⠀⣿⠟⠉⠛⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⣀⠀⠀⠀⠀⠀⠀⠘⣿⡄⢀⣠⠶⠛⠉⢬⣙⡛⢛⣋⠄⠙⠳⢦⣄⠀⣸⣿⠀⠀⣠⡄⠘⠿⠇⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣏⠁⠢⣌⠑⠦⢬⣭⣭⡤⠖⢉⡤⠂⠈⣻⡿⠁⠀⠀⠙⠋⠀⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠿⠃⣠⣄⠀⠀⠀⠀⣸⣿⣦⣄⠀⠉⠓⠒⠶⠶⠒⠋⠁⢀⣠⣾⣿⡁⠀⠀⢸⣿⠀⠀⠀⠀⠀⢼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠀⠀⠀⠈⣁⡀⠀⠀⣾⠟⠁⠉⠛⠿⣶⣦⣤⣤⣤⣤⣶⣾⠿⠛⠁⠙⢿⣦⠀⠀⠀⣴⣶⠀⢰⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣀⠀⢀⡀⠛⠁⠀⢀⣀⣀⡀⠀⠀⠀⠀⠉⠉⣿⡏⠉⠁⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠘⠋⠀⠀⢰⠋⡀⠀⠙⡆⠀⠀⠀⠀⠀⢿⡇⠀⠀⠰⣦⣤⣿⣦⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠑⠤⠞⠁⢀⠆⠀⠀⠀⠀⠀⠠⣤⣴⣿⠟⠉⠉⢻⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⠤⠒⠁⢀⣶⣿⣷⣆⠀⠀⢀⣽⣿⡄⠀⠀⣰⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠟⠀⠀⠉⠉⣹⡿⢿⣿⠿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠋⠀⠈⠟⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⡶⠶⣶⣶⣶⢦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⢿⣿⡅⠀⢻⣿⡆⠀⢿⣿⡃⠀⣼⣿⡟⠛⢳⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⣿⣧⣀⠈⠻⣿⣦⡀⠹⢿⣄⠈⢿⣧⠀⢿⣿⠀⢠⣾⣿⠟⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⣀⠈⠙⠻⠷⢶⣤⠍⣙⡷⠶⢝⣛⣂⣙⣷⣌⢿⡄⢸⣿⠃⠀⣴⣾⣿⣷⣄
⠀⠀⠀⠀⠀⢀⣴⣏⠀⠉⠛⠛⢛⡛⣻⡷⠋⣴⣿⠋⠀⣾⣿⠋⠁⣠⣤⣬⣤⣿⡄⢿⠀⢸⣿⠋⠁⠀⣹⣦⡀
⠀⠀⠀⠀⢀⡾⠻⠿⠿⠿⠟⢻⠟⣰⣿⠀⠀⠿⣿⣦⣀⡙⠿⣦⡀⢿⣿⠁⢀⣤⣭⣘⠇⢸⡇⢀⣴⣿⣿⣿⣷⡀
⠀⠀⠀⢀⣾⣷⣤⣤⣶⢶⢣⣿⠀⠹⢿⣷⡶⠶⡶⢖⣭⠿⠓⣊⣝⣦⣙⣆⢸⣿⠛⠛⠳⣼⡇⣼⡟⠁⠀⣀⠈⢷⡀
⠀⠀⠀⣼⠏⠉⢉⣡⣴⡇⢸⣿⣦⣤⣤⠶⣲⠏⠰⠿⠷⢤⣈⡻⢯⡀⢨⣍⡂⢿⠀⣴⣶⣌⠣⡿⢀⣴⣿⣿⣿⣿⣧
⠀⠀⢠⣿⣷⡿⠟⢋⢸⡇⠀⠙⠛⣉⣴⢣⣿⣦⠶⣲⠟⢁⠲⢿⡓⠛⠜⣿⠛⢾⢰⡟⠉⠙⢷⡃⣾⠟⠁⠀⠀⠉⢿⡄
⠀⠀⢸⡏⢀⣠⡶⢻⠘⣿⣶⣶⡾⢋⣾⠀⢉⡴⠡⠟⣿⡁⠘⣿⡟⠜⣷⡹⢰⣦⢹⢃⣴⣶⡄⢿⠃⣴⣾⣿⣿⣷⣼⡇
⠀⠀⢸⣿⠿⠋⣡⢺⡄⠈⠉⠉⣰⢏⢻⣷⡟⣸⣤⡎⠿⣤⣄⡸⢁⡄⠃⢡⠟⠻⣏⡾⠋⠙⢿⢈⡾⠋⠁⠀⠈⠙⣿⡇
⠀⠀⢸⣇⣠⣾⠏⡌⢿⣶⣶⣾⠏⡼⣄⠉⢰⣇⠻⠃⣤⣻⣍⠡⠽⣷⢏⣪⣶⢠⢏⣤⣶⣆⢸⢋⣤⣶⣿⣿⣷⡄⢸⡇
⠀⠀⠘⣿⠟⠁⣼⢻⡄⠙⠛⠁⣰⡇⡙⣷⣿⠇⣗⢤⣽⣌⠫⢥⣖⣉⠉⢩⣯⠴⠋⠉⢹⡿⣼⠟⠉⠀⠀⠈⠻⣿⣾⠃
⠀⠀⠀⢻⣤⣾⠏⢠⡻⣶⣤⣶⣿⠃⣿⢦⣉⠀⣸⣎⠪⣭⠤⣴⣛⡫⠶⠷⠶⢿⣿⡆⣸⣣⡴⠾⠿⢿⣿⣦⠀⣿⡟
⠀⠀⠀⠈⢿⡏⢠⣿⢱⣌⠙⠛⠁⢀⣿⠀⣯⢛⠿⠟⠀⠘⣷⣦⠉⠛⣿⣷⠀⣰⢟⣐⣩⣥⣤⣤⡀⠀⠹⣿⢠⡿⠁
⠀⠀⠀⠀⠈⢷⣿⠇⢸⡏⠳⣶⣶⣾⡿⠀⢸⣧⠻⣟⠚⣛⣛⣉⣤⠶⠛⣥⣾⣭⣭⣍⣉⠉⠙⢿⣿⡆⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠈⠻⣦⣾⡇⢻⣦⣉⡉⠀⢀⣼⣿⠀⠘⣷⡌⠛⣶⣌⠙⠳⣶⣤⡀⠉⠻⣿⣷⠀⢸⣿⢃⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣅⣸⡇⢸⡝⠛⠿⠟⠋⠀⣠⣿⣿⠀⠈⣿⣷⠀⠈⣿⣿⠀⠀⣿⡟⢀⣾⣧⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣧⣈⣿⠈⢷⡒⢶⠾⠿⠛⠁⣀⣼⣿⠟⠀⣰⡿⠏⣠⡾⢋⣴⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢧⣼⣷⡌⢻⣏⠙⢿⣉⠩⣴⣖⠛⢭⣤⣉⣻⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠷⠾⠿⠷⠾⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⣴⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡆⠀⣀⠻⣿⡿⢏⡀⠀⠀⠀⠀⢀⠻⣿⣿⠟⡀⠀⣴⣿⣿⣷⠀⠀⡹⣿⣿⡿⡁
⠀⠀⠀⠀⢀⣴⣶⣬⣙⣋⣥⣶⣦⡀⠀⠀⠀⢀⣘⠿⣿⠿⣃⡺⣿⣿⣶⣾⣿⣿⡆⠀⠀⢰⣿⣿⣶⣶⣿⣿⢆⡻⣿⣿⠟⡠⢿⣿⣶⣶⣾⣿⣧
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⢱⣿⣿⣶⣶⣿⣿⡏⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠈⠛⠛⠛⠋⠁⠀⠀⠀⠈⠙⠛⠛⠛⠛⢸⣿⣿⣿⣿⣿⣿⡇⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉


⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡆⠀⠀⠀⢠⣾⣷⡆⠀⠀⠀⢠⣾⣷⡆
⠀⠀⠀⠀⠀⠀⢠⣬⣿⣿⣥⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⢀⣬⣛⣟⣧⣄⠀⢀⣬⣻⣟⣧⡄⠀⢀⣬⣻⣟⣧⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣋⠉⣉⣿⣟⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣵⣾⣾⣻⣿⣿⡿⣵⣶⣮⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣴⣾⢿⣿⣿⣶⣿⣿⣿⣷⣦⠀⠀⠀⠀⠈⠛⠛⠛⣻⣝⣿⣟⣜⠛⠛⣫⣝⣿⣛⣜⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⣾⣿⣦⠀⣶⣿⣷⡄⢰⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠀⠛⠛⠛⠁⠘⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠁⠀⠙⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠛⠛⠛⠛⠛⠛⠶⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠀⢀⣀⡀⠐⠶⠶⣦⣴⡖⣻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⡔⠁⠜⢋⣀⡠⠶⠾⠛⠛⠉⠉⠀⠙⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢁⣴⡷⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣠⡿⠉⣀⣴⠾⠛⠛⠛⠛⠛⠻⢶⣦⣀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣰⠋⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⡸⢃⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣴⡷⠻⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠴⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠉⢡⣀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠆⠉⠀⠀⠀⠀⣠⡀⠀⠀⠀⠰⠶⠶⠆⠀⢀⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⣴⡄⠀⠀⠀⠀⣶⣤⣀⣀⠉⠀⠈⠀⠀⢁⣸⠁⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⡄⠀⠀⠀⠈⢿⡉⠉⠉⠁⠐⠛⢿⠿⠃⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣛⣦⣀⠀⠀⠀⠘⠓⠀⠤⠄⠚⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢦⡙⠿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠙⠟⢣⠘⣿⠀⠀⠀⠀⠀⢴⠟⠫⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠈⠿⣟⠀⠀⠀⠀⠀⢠⣤⡀⠀⠉⠲⣤⡀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠃⠀⠙⢳⡄⠀⠀⢀⣎⠉⢻⣦⡀⠀⠀⠹⠿⠁⢹⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠀⠀⠀⠀⣄⣹⡆⠀⣼⠟⠀⠀⣿⣍⠳⣤⣿⠀⠀⠈⠀⠉⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠛⠁⠀⠙⣦⡀⠀⣴⠉⠛⢳⣶⠋⢠⡞⠳⣾⠛⢆⣼⠟⢤⡄⣠⡛⠄⣹⡿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠈⠻⢾⠇⠀⠀⠀⠉⣦⠀⠀⠀⠀⠀⠈⣻⠀⣤⣈⠙⠿⢾⡉⠀⠀⠉⠳⣦⣀⣴⠶⠶⠶⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠠⡤⠀⠘⣧⡔⠉⠛⣼⡅⠀⠀⢰⡿⢧⣤⣀⠀⠈⠉⠁⠀⠀⠀⠀⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢄⣀⣀⣀⠀⠀⠀⢹⡇⠀⠀⠘⣷⠀⠈⠙⢷⠀⠀⠀⠀⡰⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠈⡄⠀⠀⢸⡇⠀⠀⠀⢻⡆⠀⠀⢸⡇⠀⠀⠸⣧⣄⠈⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⣀⣀⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠈⢷⡀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠻⣤⠘⠃⠀⠀⠀⢸⡇⠀⠀⠀⠈⠻⣦⣀⣀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⢷⡀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠙⠻⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣷⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠏⠀⣹⡧⠤⡤⠀⠤⠤⢤⡄⠀⠴⢆⡀⢀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣶⠀⠀⢻⡇⠄⠀⠀⠰⠠⣼⡅⠀⠀⠀⠄⠼⠀⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢀⠉⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⣼⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣰⡞⡿⠟⠁⠀⠀⠀⠀⢿⡆⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠀⠀⠀⠀⢸
⠀⣤⡄⠀⡀⣰⡧⣛⠀⠀⠀⠀⢀⣀⣀⡀⣷⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⢸
⠀⡏⠙⣤⠀⠸⡇⠈⣷⣄⠀⠀⣿⠉⠉⢻⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡆
⠀⠃⠀⠀⠑⢄⠀⠀⠀⠘⠷⠶⡟⠀⠀⣿⣄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⢠⣄⣤⡀⠀⠀⠑⢤⠀⠀⠀⠀⣤⣤⣴⠇⠉⠳⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⠀⢸⡇
⡄⠘⢧⡈⠀⠀⠀⠀⠙⢤⡀⠀⣀⣀⣀⣀⣀⣀⣀⣙⣷⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⣧⠐⠟⠁⠀⢠⡶⠀⠀⡈⠁⠀⠀⠀⠉⠁⠉⠉⠉⠉⠈⡿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠉⢦⡀⠀⠀⠀⣸⡆⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠙⣦⡀⠀⠛⠀⠈⠟⠀⠒⠂⠀⠂⠀⠀⠀⠉⠀⢿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠛⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⡿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠋⠉⠉⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠇⠀⠀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠶⠶⠶⠶⠶⠶⠶⠒⠲⠿⢦⣤⣤⣠⣤⣤⣤⡴⠿⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⣸⣿⣥⠒⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⢀⣠⣤⣤⣴⣷⣶⡦⠞⠛⠁⠀⠀⠛⠓⢤⣴⣞⣛⣁⣀⣠⠤⠤⠤⠤⢤⣿⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣶⣶⣶⣦⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣿⣶⣶⣶⣶⣾⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⠿⢿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣀⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠙⢆⣴⠛⠳⣼
⡞⠉⢙⣦⠏⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢲⠄⡸⠙⣦⢀⣿
⢣⣐⣏⠘⢆⣸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠉⢸
⠀⠀⠀⠀⠀⠀⠀⢠⠴⠖⠒⠚⠛⠛⠉⠉⠉⠉⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠈⠉⠉⠙⠛⠛⠛⠛⠻⡄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢠⠏⠀⣀⡀⠀⡀⠀⣀⠀⢀⣀⣀⠈⢳⡀⠀⠀⠀⠀⠀⠀⢠⡴⠶⡄⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢾⣛⣒⡒⠒⠒⡄⠀⠹⡄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢠⠏⢰⠋⠴⡃⠎⢈⠚⣀⠱⣁⠰⠄⡱⠀⠙⣄⠀⠀⠀⠀⡴⠚⠃⠀⠛⢲⡄⠀⠀⠀⢠⠏⠀⠀⠀⠠⠔⠊⠡⠶⠭⡄⠀⠀⠙⣄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡰⠃⠀⠈⠓⠒⠛⠒⠉⠉⠈⠁⠉⠛⠚⠁⠀⠀⠈⣳⡄⠀⠀⠙⠒⣦⠀⡟⠉⠀⠀⠀⢰⣏⣀⣀⣀⡀⠙⠂⠐⠒⠒⠒⠃⠀⠀⢀⣈⢦⠀⠀⠀⠀⢸
⠀⠀⠀⣼⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠒⠒⠒⠒⠂⠉⠉⠉⢹⡇⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠘⡇⠀⠀⠈⠉⠉⠉⠁⠒⠒⠒⠒⠉⠉⠁⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠤⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠶⠶⠋⠀⠀⠀⠀⢸

⠀⠀⠀⠀⣿⡟⢻⡶⢸⡇⠀⢸⣿⢰⣿⡛⠗⢸⣿⢸⣿⣦⢸⣿⢸⣿⢛⡛⢰⣿⡛⠗⢰⣿⡛⠗⠀⠀⢸⡇⠀⢸⡇⢸⣷⣄⢸⡇⢸⡇⠛⢿⡟⠓
⠀⠀⠀⠀⣿⣟⣻⣷⠸⣿⣀⣸⡿⢈⣙⣛⣿⢸⣿⢸⣿⠹⣿⣿⢸⣿⣛⣋⢈⣙⣻⣷⢈⣙⣛⣿⠀⠀⢸⣧⣀⣼⡇⢸⡏⠻⣿⡇⢸⡇⠀⣾⡇
⠀⠀⠀⠀⠉⠉⠉⠁⠀⠈⠉⠉⠀⠈⠉⠉⠁⠈⠉⠈⠉⠀⠈⠁⠈⠉⠉⠉⠈⠉⠉⠁⠈⠉⠉⠁⠀⠀⠀⠉⠉⠉⠀⠈⠁⠀⠉⠁⠈⠁⠀⠈⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠶⠶⠶⠶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠀⢀⣀⣀⠀⠀⠈⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⠀⠀⣿⠉⢹⡧⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⢀⣴⠿⠿⠿⢷⣄⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⣰⡾⢿⣄⠀⢀⣼⠿⣶⡀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⣦⣴⣿⠀⢸⣷⣤⣾⡇⠀⠀⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣤⣠⡄⢻⡇⠀⠀⢠⡿⠋⠉⠉⣿⠿⢿⡏⠉⠉⢻⣧⠀⠀⠀⣿⠁⣠⣶⣀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣾⠿⠟⠛⠋⠉⠀⠈⣿⡄⠀⠀⠁⠀⠀⢀⣿⣤⣼⣇⠀⠀⠀⠉⠀⠀⣼⡏⠀⠈⠉⠛⠛⠿⢶⣶⣇⡀⣀
⠀⠀⠀⠀⠀⠀⢰⣷⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⢰⡟⠉⠉⠉⠙⣷⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣄⣀
⠀⠀⠀⠀⠀⣶⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⡈⠁⠀⠀⠀⠀⠉⠀⠀⠀⠛⠁⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀
⠀⠀⠀⠀⣠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠀⠈⠙⠛⠓⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠂
⠀⠀⠀⠀⠉⢛⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⢰⡿⠉⠀⠈⠻⣧⠀⠀⠀⠀⠀⣠⣤⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⡶⣶⣄⡀⠀⠀⠀⠀⠀⣠⣶⠾⠛⠃
⠀⠀⠀⠀⢸⣇⠀⠀⠀⢠⣿⠶⢶⣶⣤⣾⠋⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⢿⣦⡀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠛⣷⣶⠶⠟⠁⠀⠀⠀⣀⣿⣄⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣽⣷⡾⠋⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⣀⣀⣼⡟⠀⠀⠀⢀⣠⣶⠿⠋⠉⣿⡿⠾⠻⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⢶⣄⣹⡿⠋⣠⡾⠋⣀⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠸⠛⠉⠉⠙⠻⣦⣴⠾⠛⠉⠀⠀⠀⣰⡟⠀⠀⠀⢘⣷⡶⠶⣦⣄⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⡀⠀⣶⡄⠙⢿⣤⡾⠋⣠⣾⠟⢀⣴⠄⢀⣴⠆⢀⣿
⠶⠀⠀⠀⠀⠀⠀⢹⣧⣀⣀⣀⣀⣀⣰⣿⣀⣀⣀⣀⣾⠋⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠻⣦⣈⣻⣶⣴⣯⣠⣼⣿⣡⣴⢿⣧⣴⡟⢁⣴⡿⠃
⣶⡄⠀⠀⠀⠀⠀⣸⣏⡉⠉⠉⠉⢩⡿⠉⠉⠉⠉⠉⢿⣆⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣦⣨⡿⠋⣨⣿⣿⠿⣿⣏⠙⢿⣄⠙⢿⣿⠟⠋
⠈⠻⣦⣤⣤⣤⣾⠏⠉⠛⢷⣦⣤⣿⠁⠀⠀⠀⠀⠀⠀⠙⣻⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⣹⡿⠋⣠⣾⠟⣠⣿⣧⣈⠹⣷⣄⠙⢷⣄⠙⢿⣄⡀
⠀⠀⠀⠉⠉⠁⠙⣷⡄⠀⠀⢈⣿⠛⠷⣦⣄⡀⠀⠀⢀⣀⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠠⡾⠋⠀⡾⠋⣀⣶⠿⣧⡈⠻⣶⣌⠹⢷⡄⠙⠁⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⣨⣿⠟⠛⠿⣷⣄⠀⠀⠉⠻⢷⣶⡟⠋⠉⠻⣷⡀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠛⢁⣴⠟⢻⣦⡈⠻⣶⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠈⣿⣤⣤⣴⣶⡶⣿⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠺⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛⢁⣴⣿⡿⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠙⢷⣤⣤⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠈⠻⣦⡈⠛
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣶⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⡀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⢀⣀⣤⣶⠟⠋⠀⠀⠀⠀⠉⠻⣶⣤⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠷⣶⣦⣶⣶⡿⠛⠛⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢀⡄⣸⢀⠆⣠⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⣆⢸⣸⣧⣧⣞⡴⡡⣊⠤⣊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠠⠠⠔⠀⡂⠅⠂⡐
⠀⠀⠀⠀⠀⠀⠀⠱⣄⢳⣼⣾⣿⣿⣿⣿⣿⣿⣗⠯⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡐⡀⠠⡥⡡⣢⣒⣇⣕⢂⠢⢠⢀
⠀⠀⠀⠀⠀⠀⣈⠳⣮⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠢⢱⢩⣵⣿⣿⣿⣿⣿⣷⡧⠅⠅⠄⡀
⠀⠀⠀⠀⠀⠤⢬⣙⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢨⣲⣿⣿⣿⣿⣿⣿⣿⣿⡗⡀⡁⢄
⠀⠀⠀⠀⠀⠂⠲⠮⢽⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠌⠼⣽⣿⣿⣿⣿⠿⠿⠿⠋⠉⠈⠊
⠀⠀⠀⠀⠀⠀⠈⣙⣛⣻⣿⣿⣿⣷⣤⣄⣀⣠⣤⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡌⠔⡽⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⣛⣿⢿⣿⢿⣿⡿⡿⡟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⢻⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠔⢉⠔⢩⠊⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡐⠉⠇

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⠃⠀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠟⢻⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡯⣿⣻⣶⣶⣾⣿⠂
⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡟⢂⣿⡏⢸⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣿⣿⣿⠟⠃
⠀⢠⣦⣠⣿⣿⣿⣿⡟⠈⠛⠻⢿⣾⣯⣤⣾⣿⣿⣿⣿⣛⡻⢿⣶⣦⣄⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠛⠉
⠀⠈⢿⣿⡏⠙⠻⣿⣿⡄⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣷⣶⣭⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⢠⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⡆
⠀⠀⠈⠉⠀⠀⠀⠙⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠈⢉⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⣈⣽⣿⣷⣿⣿⣏⠀⢀⣀⣤⣤⣴⣶⠾⢿⣿⣿⣿⣏
⠀⠀⠀⠐⠛⠻⠿⠛⠻⢿⣿⣶⣶⣶⡶⢿⣿⣿⡿⠿⠿⠟⠛⠉⠉⠛⠿⣿⣿⣿⡾⠿⠛⠉⣼⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠙⢿⣿⣿⣧⡀
⠀⠀⠀⠐⠲⠶⠶⢦⣴⡿⢿⣿⣿⡷⣦⣶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⢻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣦⠉⠉⠙⠛⠻⠿⢿⣷⣶⣤⣤⣀⣀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⣸⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⡄⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠻⣿⣿⣿⣿⡿⠟⠉
⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠀⠀⠀⢀⣠⠶⠛⠙⠟⠉⠁
⠀⠀⢻⣏⣹⣿⣿⣿⡿⠿⠟⠛⠛⠋⠉⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⣀⣤⠞⠋⠁
⠀⠀⠘⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡖⠀⠀⣴⣷⣶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣤⠖⢉⣴⡶⠦⠤⠤
⠀⠀⠀⠀⢀⡄⠀⠈⣿⣿⠃⠀⣼⣿⡟⠁⠀⣀⣤⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠸⣣⠞⢡⣾⠏⠁⣰⣾⡟⠛⠒⠶⢤
⠀⠀⠀⢠⣿⣿⡀⠀⣿⣿⠀⢸⣿⠋⢀⣤⣾⣿⠿⠿⠛⠃⠀⠀⠀⠀⠀⢀⡀⢻⡟⢰⢟⣴⠋⣡⣶⡟⠋⢉⣉⣙⡻⠿⢶⣤⡀
⠀⠀⠀⠈⠻⣿⣷⡀⠸⣿⡀⢸⠇⣠⡿⠛⢉⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠸⣷⢸⣇⠈⣾⡇⣠⠟⢋⡥⠶⡾⣭⣛⠻⢶⣄⣈⠻⡄
⠀⠀⣶⣄⠀⠈⠻⣷⣄⠙⢧⢸⡠⢋⡴⠞⠛⠛⠛⠻⠿⣿⣿⡇⠀⠀⢰⣤⣦⠸⣿⠀⣿⢣⣿⠔⡵⣛⣿⣛⢦⡈⢣⣄⠹⡿⣷⡘
⠀⠰⣿⣿⣷⣦⣄⡈⠙⠳⠦⢅⠀⠐⢌⡛⠻⢷⣶⣤⣀⠀⠈⠁⠀⠀⢈⠉⢹⣄⢻⡄⢸⠘⢿⡀⠀⡯⣹⢻⡆⣿⡈⣿⡆⣅⠈⣧
⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⢋⣵⢃⡎⣧⠙⣷⣄⠈⠻⣿⣷⣦⡀⠀⠀⠸⠿⣆⠻⡌⢿⣿⣦⡈⠳⣵⣦⡶⠞⢁⣿⠀⢸⡇⢿⡇⠘
⠀⠀⠰⣶⣶⣶⣶⣶⣶⡿⠛⢁⣾⠁⣿⡇⠘⣿⣦⠀⠘⢿⡿⠁⠀⠀⠀⢶⡾⢷⣽⣄⡈⠈⠻⠿⢶⠭⠤⠖⣿⡿⢁⣾⠀⢸⢇⠇
⠀⠀⠀⠹⠿⠛⠛⠉⠁⣀⣴⣿⠃⢀⣿⡇⠀⢹⣿⣇⠀⠈⠁⠀⠀⠀⠀⠀⢠⣶⣍⡉⠛⠿⠿⠷⠖⠶⢶⣛⡋⠀⣹⡿⢠⡿
⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡟⠁⠀⣼⣿⡇⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⡯⢛⡓⠒⢶⡶⠶⠟⢋⣠⠴⣫⠄⠜⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⢀⣼⣿⣿⠁⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠉⣭⠍⣤⡞⢻⠅⠚⠁

⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠚⠛⠒⠒⠦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠶⣶⣤⣍⠙⠲⣤⡀⢀⡀
⠀⠀⠀⠀⢀⣴⠟⠉⣀⣤⠤⠴⠦⠤⢤⣀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣤⣄⠈⠻⣷⡄⠘⣿⡄⠹⣆
⠀⠀⠀⣠⡟⢁⣴⠟⠁⣀⣀⣤⣀⣀⠀⠈⠑⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠞⢉⣀⣀⣀⣀⠉⠻⣷⡄⠹⣿⠀⢸⣿⠀⢿⡆⢀
⠀⠀⢰⡟⢠⡟⢁⣴⠟⠉⠀⡀⠀⠈⠑⠢⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠛⠉⠙⠛⠻⣦⡈⣿⠀⣿⠀⣼⡏⠀⣾⡇⢸⡆
⠀⠀⣿⠁⣿⢁⣾⢁⣴⠟⠉⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢀⣴⣶⠿⠟⠛⠶⢬⡳⡸⣰⢃⣴⠟⢀⣼⡟⠁⣸⡇
⠀⠀⣿⢸⣿⢸⡇⢸⡇⣴⣖⣂⡀⢀⣀⡀⣤⣤⡀⣤⣤⢰⣶⡦⠀⠀⠈⣰⡿⠋⢀⣤⡶⠟⣛⡥⣺⡮⡵⢭⡤⠶⠟⠋⣀⣼⡟⢠⠄
⠀⠀⣿⡆⣿⡜⣿⡸⣷⣝⡿⢟⣣⣿⡿⢣⣿⡿⣸⣿⡟⣼⣿⡇⠀⠀⠀⣿⠁⣰⣿⠋⢠⡾⢋⡾⢱⠱⣍⠳⠶⣶⣶⠿⠟⠋⣠⡾
⠀⠀⠘⣿⡜⣷⣜⢿⣮⣛⣿⠿⢟⣋⣵⣿⡟⣱⣿⡿⣱⣿⡟⠀⠀⠀⠀⢿⠀⣿⠇⢠⣿⠃⢸⡇⠸⣧⠙⠷⣦⣤⣤⣤⣴⡾⠟
⠀⠀⠀⠘⢿⣮⡻⣷⣽⣛⡿⠿⠿⢟⣛⣵⣾⡿⢋⣴⣿⠟⠀⠀⠀⠀⠀⠈⠀⢿⡄⢸⣿⠀⢸⣷⡀⠹⣷⣤⣀⠉⠉⠉⣁⡤⠊
⠀⠀⠀⠀⠀⠙⢿⣾⣽⣛⡿⠿⠿⠿⢟⣛⣭⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⢻⣧⡀⠹⣷⣄⡈⠙⠛⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠻⠦⣌⡙⠛⠛⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠈⠉⢻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣶⣶⣤⣤⣄⣀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⠀⠀⠀⠀⠀⢿⣿⠋⣁⣩⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠈⣿⡿⠛⠋⣹⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣠⣿
⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⠉⢠⡄⠹⣦⠀⠀⠀⣿⣿⣿⣇⢿⣦⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠇⢸⣿⣯⣿⣿⣿⣿⠏⠀⠀⣠⡆
⠀⢠⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢻⣆⠙⢷⡄⠀⠸⣿⣿⣿⡜⣿⡆⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⣼⣿⣿⣿⣿⣿⣷⠀⢠⠞⠉
⠀⠀⠻⣿⡇⠈⢹⣿⣿⣿⣿⣿⠀⠀⠈⢿⣄⡰⣿⡀⠀⢻⣿⣿⣷⣾⣿⣇⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠃⢠⣾⣿⢿⣿⣿⣿⣿⣿⡴⠃
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠈⠻⣷⣿⣧⠀⠈⢿⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⠃⣤⣿⣿⠋⢸⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠘⣿⠛⣷⣀⠸⣿⣿⡟⠛⠻⣷⣿⣿⣿⣿⣿⣿⣿⢣⣾⣿⡿⠃⠀⢸⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⢹⣧⡘⣿⣧⣹⡟⠻⢷⣦⣼⣿⣿⣿⣿⣿⣿⣵⣿⣿⡿⠁⢀⣶⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⢿⣿⣿⣧⠀⠀⠈⢿⣧⠈⢻⣿⡿⢦⣤⡤⣿⣿⣿⣿⣿⡿⣿⣿⣿⠟⢠⣶⣿⣿⡟⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋⠀⠀⠀⠀⠙⠛⠻⠶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣀⠀⠀⠀⣤⣀⡀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀⣿⠉⠛⠷⣦⣀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⠀⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢰⣶⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⣇⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠈⣿⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠙⠻⣦⣤⣀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠟⠋⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠛⠛⠋⠉⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⣶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⣿⣆⣸⣿⣤⣾⣇⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⣿⣷⣿⡟⣿⣿⣿⣿⣿⢿⣷⣴⡿⠁⠀⠀⠀⢀⠀⢠⡀⢠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣹⣿⣿⣿⠟⢣⢻⣉⣩⣽⣧⣿⣿⣿⣿⣾⡟⠰⣦⣼⣷⣾⣷⣿⣶⣿⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣛⠿⣅⣤⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢿⣉⠉⢻⡟⠁⠈⢻⣿⣧⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣿⣾⣿⡿⠟⠛⠉⠹⡍⠉⠉⠛⠻⣿⣿⣿⣯⣷⣾⣿⣦⣤⣾⠿⣿⣷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⣰⣄⣶⣲⢽⣿⣫⣿⣿⠟⠉⢦⣀⣠⠤⢶⣷⠶⠦⣤⣞⠀⠈⠛⢿⣿⣾⣿⣿⡿⠀⠀⠈⣿⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣬⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⠃⠀⣠⠞⠻⣀⠀⠠⡿⠄⠠⣯⠟⠓⣤⢀⣨⢿⣿⣿⣻⡾⢿⡟⢻⣜⠛⠂
⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠑⠲⣴⣇⢀⡀⠋⠀⣀⣅⣄⠴⠉⠢⡀⠘⢻⡀⠀⢻⣿⣿⣷⣾⢷⡞⢻⣧
⠀⠀⠀⠀⠀⠀⠀⢬⣿⣿⣿⣿⣇⣀⣿⣿⣿⣿⡇⠀⢰⡏⠈⠛⠀⢠⣿⣿⡿⣇⣾⣦⠀⠀⠀⠈⣧⠀⠀⣿⣿⣿⣿⣿⣙⣿⠏⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠒⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⢼⣗⣲⡆⠀⣿⣿⣿⣅⣿⣿⣿⠆⠀⠓⠂⣿⠉⠉⣿⣿⣿⡿⡜⡟⠛⠒⡾⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⡽⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⡇⠀⣠⡀⠘⢿⣿⣾⣿⣿⠟⠀⣰⢦⣀⡟⠀⢀⠿⣿⡋⡠⢟⣙⣤⣘⡳⣇⠑⢄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣼⡔⠋⠻⢭⣻⣿⣿⣿⡄⠀⣹⣾⠟⠓⢀⡀⠈⠉⠉⠀⣠⡄⠀⢉⡟⠓⠦⣼⣿⠒⠽⣱⡟⠉⠉⠉⢻⣎⢶⡩⡵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣧⡀⢀⣸⣿⣿⣽⣿⣿⣎⠁⠈⠳⣤⣼⠋⠀⠘⠇⠀⠈⣭⡶⠋⠀⠀⣜⢫⣏⣀⡇⢿⡀⠀⠿⠀⢠⡿⢘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢛⣿⠛⠉⢻⣿⣿⣿⣿⣿⣿⢿⣿⣷⣄⠀⡰⠉⠛⠒⠒⡗⠚⠉⠁⠘⢠⣬⡷⠿⢿⣭⣵⠛⢌⠻⣦⣤⣴⠟⡡⠯⣝⡳⡆
⠀⠀⠀⠀⠀⠀⠀⠘⢛⣿⣄⣀⣰⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⣦⣤⣄⣀⣧⣤⣤⣶⠔⣶⣿⣇⣀⣸⣧⣈⣣⣴⢫⠖⠒⢶⢻⣠⠞⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠛⣹⣿⣿⠏⠀⠉⣿⣿⣾⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⢛⣿⡿⢧⢿⣿⠟⠻⣎⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣼⠿⣷⣤⣴⣿⣿⣷⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣿⣟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⢸⣿⣿⣿⣷⡒⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣛⣛⣿⣾⠂⠉⠉⠙⣿⣿⣿⠉⢹⣿⣿⡇⠈⠉⠒⢦⣄⡀⠀⠀⣰⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣻⣿⣄⠀⢀⣰⣿⣿⣿⣷⣿⣿⣿⣿⣦⣤⣴⣿⣿⣿⣷⣦⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⢿⣿⣿⣿⡟⠋⠉⠻⣿⣿⠏⠁⠈⠙⣿⣿⣿⣿⣌⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢿⣿⣿⡀⠀⠀⠀⣿⣿⡀⠀⠀⢀⣿⣿⣿⠉⠙⠁⠙⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣴⡿⠿⣿⣦⣴⣾⣿⣿⣿⣷⣾⣿⣿⡉⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⡿⠉⣿⡿⠻⣿⡟⢻⣿⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠻⠀⠀⠉

⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄
⠀⠀⠀⠀⣼⣿⡟⠋⠀⠀⠀⠀⠀⠀⡾⠿⠋⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠘⠛⠂
⠀⠀⠀⢀⣿⣿⣧⠀⠀⠀⠀⠀⠀⣠⣷⣤⡀⣀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⣬⣁⣽⡀⠀⠀⠀⢀⣶⣤⣴⣶⣶⠀⠀⣀⣀⣤⡀
⠀⠀⠠⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⣿⣿⣿⡇⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠟⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⣠⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠇⠸⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣁⡀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⢿⣷⢠⡤⠈⢻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠂⠀⢀⡼⠃⣸⣿⡆⠘⣿⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠂⠀⠀⠀⠰⠿⢧⡀⠙⣿⣷⢠⣿⡟⠀⣽⡟⣀⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠾⢿⡿⠿⠇⢿⣿⢸⡟⢹⣿⡟⠁⠈⣡⡀
⠀⠀⠀⠀⠀⠀⠀⡄⠙⠋⠀⢠⡀⠀⠀⢀⣀⠻⣄⣼⡗⠀⠆⠈⠉⠀⠀⢿⣏⡰⠆⠀⠛⣷⣄
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠈⠓⠀⢀⡏⢰⣿⠀⠘⠀⠀⠰⡄⠀⠀⣴⣿⢳⣄⣀⣴⣿⡟
⠀⠀⠀⠀⠂⠁⠀⠀⣀⠀⠀⠂⠀⠀⣠⡜⢡⣿⣯⢸⠇⠀⠀⠀⠀⠀⠚⢋⡙⠂⠈⠻⣿⡋⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢠⣿⡇⣰⣿⠀⢠⡆⠀⠀⢠⡄⠀⠀⢸⡇⠀⣿⣦⣜⠃⠀⢠
⠀⠀⠀⠀⠀⣶⠀⠀⠛⠃⢀⡄⠛⣿⠈⠃⡀⢡⡄⠈⣅⠀⠀⠀⠉⠀⠰⠊⠁⣾⡏⠀⡿⠇
⠀⠀⠀⠀⠀⠁⠀⠀⣴⠚⠋⠀⠀⡸⢶⣾⠃⠸⠃⠀⠈⠀⢠⡄⠀⠀⠀⠀⠀⢻⣷⣶⣆⣶⡄⠀⢠⣤⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢰⣷⡌⠉⢀⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣄⠈⢹⡟⢻⡏⠀⠀⠙⢿⣿⣦⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⣾⠟⠀⠀⢸⡇⠀⠇⢀⡘⠃⠀⠀⠀⠀⣀⣼⡦⠀⣀⠘⢃⠀⢠⠀⠀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠛⣤⠀⠀⣼⠻⠀⠀⣼⣧⡀⠀⠀⠀⠀⠋⠀⠀⠀⠙⢠⣼⡀⠀⣦
⠀⠀⠀⢠⡤⠀⠁⠀⠀⠀⢠⣇⠀⠸⣷⡴⢿⡧⠶⣾⣿⣿⠃⠀⠀⠰⢄⠘⣷⡄⢠⠀⠘⠛⢷⣀⠈⠀⠀⠀⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⢸⡇⠘⡇⠀⠸⡟⠁⠀⠀⠀⢀⣌⠁⠈⠀⢸⡆⠀⢠⡄⠙⠁⣈⡀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⠈⠃⠀⠁⡀⠰⣯⡀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠃⠀⠠⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⣇⠠⡆⠈⠃⠀⠹⠇⠀⠀⠀⠀⠀⠀⡀⠀⠀⣴⡆⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡟⠀⠀⠙⢶⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠙⠁⠀⠀⣴⠀⠀⠸⠀⢠⠄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣶⠀⢸⡿⠀⠀⡀⢶⣤⠀⠀⠀⠀⠀⠀⠘⠃⣾⠀⠀⠀⢸⠇⠀⣶⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⣷⠀⠘⡿⢠⣿⡷⠀⠀⠀⠀⠀⢠⠀⢸⡀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣿⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⠀⠀⠻⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠘⡇⢀⡀⠰⣿⠂⠈⡇⠀⠀⠀⠀⠀⢠⣿⠀⠀⠸⣧⡀⢰⣆⠀⠀⣠⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⢈⡇⠀⡿⠆⠀⡿⠀⠀⠀⠀⠀⠀⠹⠄⠀⣤⣿⣷⡈⠹⣆⠀⠉⠀⠀⠀⠀⠀⠘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢠⣿⠀⠀⠀⡀⠀⣀⣤⠃⠀⠀⠀⠀⠀⡜⠀⠈⠉⠉⢿⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⣿⠆⠀⠸⣿⣦⣿⡏⠀⠄⠀⠀⠀⠀⠀⠂⢰⡄⠀⠻⠃⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠄⠀⠀⠛⡋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢱⡆⠀⣾⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡟⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡏⠀⡆⠀⣤⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠁⡄⠀⠀⠀⢠⡄⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠁⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠟⠁⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⢀⣦⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠂⠛⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠙⠁⠈⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡄⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠘⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⡆⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⢷⡄⠀⢀⠀⠀⡆⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣵⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠰⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⠀⠀⠀⠀⠀⠀⠠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃


⣿⣿⡿⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⠀⠀⣀⠀⢈⣿⠏⣌⠋⣹⠟⢹⣿⠙⠛⠋⡉⢹⡟⢻⣇⠈⡟⠻⠏⢀⣿⠛⢉⣿⠟⢿⠋⠉⠋⠉⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣄⠀⢩⣤⣼⣿⣦⣿⣶⣿⣶⣾⣿⣴⣶⣾⣿⣾⣷⣾⣿⣾⣷⣾⣶⣾⣿⣦⣾⣷⣤⣶⣦⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⣤⣤⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠒⠔⡠⡀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⣤⡀
⠀⣿⣟⣿⣽⣻⣿⡆⠀⠀⠀⠀⠀⢰⢃⠢⡂⢐⢕⠈⠐⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⢿⡀⠀⠀⠀⠀⠀⡄⣼⢡⡖⢗⣦⠹
⠨⣪⣀⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⢹⠐⠁⠢⡔⢔⠄⢄⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣳⠁⠀⠀⠀⠀⠀⡇⠹⣜⣓⣊⡿⢀
⠀⠙⠿⢿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠈⠳⣅⣀⣀⣀⡤⠂⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⡿⡿⠃⠀⠀⠀⠀⠀⠀⠘⢦⣀⣉⣁⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⢠⠄⠉⠀⠁⠈⠁⡀⠀⠀⠀⠀⠀⠀⢠⣾⣻⡿⣟⣶⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣂⣖⣄
⡨⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⣟⢿⡍⠈⣭⡻⣷⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠠⣱⡟⠉⢹⣷⠄
⠑⢄⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠈⢿⣿⣛⣼⢻⣿⠋⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⢻⠶⡞⠏
⠀⠘⠒⠒⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠑⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⣤⣄⡀
⠀⠀⠀⠀⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⠿⢿⣦⡀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠊⠀⠐⠂⠀⠀⠀⠀⠀⢀⡞⠉⢀⣀⣀⠈⢻
⢠⣄⣤⡄⢁⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⡴⣿⣦⢹⡇⠀⠀⠀⠀⠀⢀⠣⠀⠀⠀⠀⠀⢌⠁⠀⠀⠀⠀⢸⠁⢰⡏⡕⠈⣷
⠘⠋⠁⠀⠀⣿⣷⠄⠀⠀⠀⠀⠀⠀⢻⣆⠻⢮⡶⠟⠁⠀⠀⠀⠀⠀⠀⢁⣀⠀⢀⢀⠀⠀⠁⠀⠀⠀⠀⠘⢧⡘⢧⣙⣚⣁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠉⠉⠉


⠀⣀⣴⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⡢⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣀⡀
⢰⣟⣿⠛⠻⣿⣿⣧⠀⠀⠀⠀⠀⠀⡰⣼⣾⣿⣷⣖⠄⠀⠀⠀⠀⠀⠀⣦⠓⠁⠀⠈⢑⢍⠀⠀⠀⠀⠀⠀⠀⡄⠃⠉⠀⠒⢀
⢸⣿⣷⡀⣀⣼⣿⡿⠀⠀⠀⠀⠀⠀⢝⣿⣿⣿⣿⢏⠂⠀⠀⠀⠀⠀⠀⢣⡂⠀⠀⠀⢰⣸⠀⠀⠀⠀⠀⠀⠀⢝⡀⠀⠀⡘
⠀⠻⢿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠒⠫⠟⠝⠀⠀⠀⠀⠀⠀⠀⠀⠑⠪⠥⠔⡪⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠔
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠉⠑⡄⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡗⠠⠄⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡏⠉⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇⠈⠢⡀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠂⠄⠀⠐⠁⠀⠀⠀⠀⢹⠉⠉⠉⠉⠉⠉⡇⠀⠀⠀⠀⠑⡀⠀⠘
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠁⠈⠀⠀⠀⠀⠈⠀⠀⢠⠋⠀⢹⣷⣦⣤⣀⡀⡇⠀⠀⠀⠀⠀⢡
⠱⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠹⣿⣿⡏⠀⠀⠀⠀⠀⠀⠈⡄⠀⠇⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⢡⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⡆⠀⠀⡇⠀⠀⣿⣿⡇⠀⡔⢊⠃⠀⠚⠑⠘⢄⣰⠀⠀⠀⠀⢀⢄⣿⠾⠻⢷⣕
⠀⠀⢃⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⣟⣛⣛⣛⣛⣛⣛⣛⣛⣛⡇⠀⠀⡆⠀⠀⣿⣿⣧⢸⠀⢂⡀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⡌⣿⠁⠀⠀⠀⣿
⠀⠀⠀⢂⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠁⠀⠀⣿⣿⣿⢘⡀⠀⠉⠉⠁⠀⡰⠀⠀⠀⠀⠀⠀⡿⡙⢷⣤⣠⣴⢟
⠀⠀⠀⠈⢆⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢹⣿⣿⠀⢙⠂⠄⠀⡤⠊⠀⠀⠀⠀⠀⠀⢀⠀⡀⠐⡬⠍⠒⠁
⠀⠀⠀⠀⠈⡄⠀⢰⡉⠑⠤⣀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠁⠀⠀⢸⣿⣿⠀⠈⠀⣤⡀⠆⠀⠀⠀⠀⠀⠀⢀⠎⣼⡟⡰⠁
⠀⠀⠀⠀⠀⠘⣀⠇⠈⠑⠢⠤⢈⡁⠒⠢⢄⡀⠙⡄⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⢰⠀⣿⠀⡄⠀⠀⠀⠀⠀⢀⢂⣾⠟⡐⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⢲⣿⣿⣷⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠘⢸⣿⠀⠀⠀⠀⠀⠀⢠⢃⣾⠏⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⠀⡇⠀⠀⠀⠀⠀⢘⠀⠀⠀⠘⣿⣿⡇⠀⢸⡟⢸⠀⠀⠀⠀⡠⢁⣾⠏⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠧⠧⠀⠀⠀⠀⠀⠼⠀⠀⠀⠠⠿⠿⠧⠤⠾⠧⠴⠀⠀⠀⠴⠥⠾⠏⠬⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣬⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠙⠪⡙⠓⠦⣄⣀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠓⠤⣀⠀⠀⠀⠀⠈⠲⣄⠀⠉⠙⠲⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠉⠓⠦⢄⣀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠈⠳⣄⠀⠀⠀⠈⠙⠳⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣀⠀⠀⠀⠀⠙⠲⣄⠀⠈⠳⡄⢀⣀⣠⣤⡴⠾⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣒⣄⠀⠀⣀⣈⡧⠤⢖⣛⡋⡩⣥⡀⠜⢉⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡶⠾⠟⠛⣛⣛⣯⣭⣷⣶⣿⣿⣿⣿⡇⡁⣸⡾⠠⠾⣻⣁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⢀⡤⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣃⣨⣭⠶⠶⣛⣛⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢣⢸⠀⢸⠿⢸⣿⡿⠿⣛⣛⣭⡽⠷⢞⣛⡫⠭⠐⠒⠉⢉⣁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣤⣼⠿⢈⣓⣻⣥⠴⠶⣚⣫⡭⢥⠶⣾⢿⣽⣿⣠⣤⣶⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠰⠞⣛⡫⠭⠖⠒⠉⠉⢀⣇⣧⡿⠖⠛⣋⣭⣿⣿⣿⣿⣿⠟⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣀⣤⣤⠶⣞⢫⠋⠛⠛⠛⠚⢛⣷⣶⣭⣉⡉⠁⢀⣀⣤⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠧⠾⠬⠽⢿⣀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⣠⣀⡀⠀⠀⠙⢷⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡁⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠛⠋⢀⡼⢛⣶⣶⣭⣥⣄⣀⢡⣉⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠸⣿⣿⣰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⢍⣙⡛⠻⠿⣿⣿⣿⣿⠇⠀⣠⢀⡀⠈⠉⠉⠙⠻⣿⣿⣿⣷⣦⣿⣙⣾⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢤⡬⢉⣓⠒⠼⢟⣁⣤⡶⢋⣨⡶⠚⠛⠀⠀⠀⠈⠉⠛⠻⠿⣿⢙⣽⣿⠾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠛⠀⠀⠘⠓⠃⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣴⣭⣶⣭⣿⣵⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢰⣶⣶⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠊⠓⠀⠙⠀⠀⠀⠀⠀⢠⣿
⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⣶
⠀⠋⠃⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏
⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⢠⣤
⠀⠋⠓⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠔⠢⠄⠀⠹⠀⠔⠀⠀⠀⠄⠄⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠟⠀⠀⠀⠀⠀⣀⠀⠀⢀⠠⠀⢀⣠⢀⠀⡀⢀⢀
⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣀⠀⢀⡀⢀⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣁⣀⣛⣁⣚⣋⣘⣂⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣾⣿⠉⠟⢹⣿⣿⣿⣿⣟⣋⣙⣏⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠛⠛⠛⠋⠉⠛⠛⠛⠉
⠀⣤⣤⣤⣤⠀⠀⠀⠀⡀⠀⠀⠀⡤⠀⠠⡄⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠈⣅
⠀⠀⣸⣿⣿⠀⠀⠀⠘⠁⣰⣧⣀⣀⡀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⢓⠀⠀⠁⠂⠀⠀⠀⠀⠂⣀⠀⠀⡀
⠀⠀⠿⠏⠀⠀⠀⠀⠀⠐⠓⠀⠁⠀⠀⠀⠀⠀⠀⠤⠀⠠⠶⠲⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠈⠋⠐⠛⠱⠀⠀⠀⠀⠀⠀⠀⠸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠾⠿⠿⠿⠿⠿⠿⢷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⣠⡾⠛⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⢀⣾⠟⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢀⣿⠃⠀⠀⢠⣿⠃⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⣸⡟⠀⠀⠀⣾⡏⠀⠀⢀⣾⠏⠀⠀⠀⣠⡄⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⣿⡇⠀⠀⢰⣿⠀⠀⠀⣸⡟⠀⠀⠀⣼⡟⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠰⣿⠀⠀⠀⠀⢙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠸⣿⠀⠀⠀⢹⣧⠀⠀⠀⢻⣦⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢹⣧⠀⠀⠀⢿⣇⠀⠀⠈⢿⣆⠀⠀⠀⠙⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠈⣿⡄⠀⠀⠘⣿⡄⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠈⢿⣦⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠙⢷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢶⣶⣶⣶⣶⣶⣶⡶⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠿⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠿⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡃⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⣀⣀⣼⣿⣀⣀⣀⣀⣀⣀⣀⣸⣿⣄⣀⣀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⢻⣿⠛⠛⠛⠛⠛⠛⠛⢻⣿⠟⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠉⠀⠀⢹⣿⠁⠀⠀⠀⠀⠀⠀⢹⣿⠁⠀⠈⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⢀⣼⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⡿⠟⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠉⠀⠀⣀⡠⠤⠔⠦⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠁⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠠⠀⠀⠉⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣊⡉⠉⠠⠠⡀⠀⡤⠀⠀⠉⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⣴⣾⡿⠛⢿⣿⣿⣷⣦⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠁⢀⠇⠀⢃⠈⠁⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠙⠉⠁⠀⡀⠀⠈⠉⠉⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⠀⠁⠀⠀⡃⠀⠀⠈⠀⣿⣿⠀⠀⣐⡀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⡔⠋⠀⡘⠄
⠀⢀⠔⢳⠀⠀⡗⢦⢀⣿⣿⡄⠀⠀⡄⣤⣄⠀⠀⣼⣿⣿⣆⠀⣼⠀⢎⠈⢩⡇⠀⠀⠀⠀⠀⠀⡠⠔⠋⠁⠀⠉⠁⠀⠀⠀⠀⠉⠈⠙⠲⡄⠈⠂⠁⡈
⠀⢼⠀⠀⠦⠔⠃⣸⢾⣿⣿⣿⣦⣄⠁⠀⠁⣴⣾⣿⣿⣿⡟⢰⣿⢀⠎⡜⠉⠀⠀⠀⠀⠀⡰⢋⣀⡀⢀⠔⢢⡀⠀⠀⠀⢠⠆⠀⠀⠀⠀⢨⠀⣘⠈
⠀⠈⠢⠤⢄⠀⠘⣁⠼⠟⠛⡋⠉⠉⠁⠂⠀⠀⠀⢸⠉⠉⠓⣾⣯⢎⠌⠀⠀⠀⠀⠀⠀⡼⠀⡾⠀⠈⠁⠀⠀⠈⠉⠉⡇⢸⠀⠀⠀⢸⠀⡌⢀⡋⠳⡀
⠀⠀⠀⠀⠈⡄⠀⢅⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⣿⢃⠎⠀⠀⠀⠀⠀⠀⡜⠁⠀⠠⠸⡁⢀⠤⠠⡄⠢⠀⠱⢄⠉⠉⠉⢁⠾⢅⡼⠁⢢⢳
⠀⠀⠀⠀⠀⢡⠀⢰⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⡸⠳⠥⠎⠱⡀⠀⠀⠀⠀⢰⢇⠀⢐⣈⠅⠀⠸⣀⣀⠌⠠⠀⣠⠚⠀⠀⠀⢸⠦⠄⠀⠀⠀⡌⡇
⠀⠀⠀⠀⠀⠸⣆⠔⢳⡀⠀⠧⢀⢀⠀⠀⠀⢀⡀⠸⠰⡉⠂⠀⠀⠀⡇⠀⠀⠀⠀⢸⠈⠢⠤⠞⠀⠀⠀⠀⠀⠀⠁⠀⢇⠀⠀⠀⠀⠈⢯⣂⠀⢀⡴⠁⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⠉⠀⢀⠜⠀⠀⠀⠀⠀⢰⠀⠀⠈⠓⠒⠒⠤⠀⢀⡤⠒⠲⠊⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠊⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠐⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣶⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⡖⠐⠒⠒⠒⠌⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡦⠤⠤⠤⠄⣂⡀⡄
⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣷⣬⣤⣤⣤⣤⣤⣤⣤⣤⣍⠰
⠀⠀⠀⠀⠀⠘⢻⠟⠛⠻⡟⠛⠛⠉⠉⢀⠄⠀⠀⠈⢉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⠀⠻⡈⠑⢆⠘⢤⠉⢩⡭⠈⠙⠛⠋⣭⠟⠛⠑⡆
⠀⠀⠀⠀⠀⠀⠈⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢸⠀⠀⠈⡆⢸⠆⠐⠀⠈⠀⠠⡀⣰⠀⠈⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠑⠒⠊⠀⠸⡀⠀⡆⠀⠀⠀⢀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⣍⡄⢀⣀⡀⠠⣇⠀⠀⠀⠀⠀⠀⣸⡀⠀⢀⠜⡇⢀⠋⣂⠀⠀⠀⠀⠀⠈⠑⠒⠐⡄⠀⠀⠘⡤⠘⢖⠂⠠⣤⡤⠤⠐⡮⠤⢄⣀
⠀⠀⠀⠀⠸⠉⠀⣿⠾⠂⡏⠉⠋⠀⠘⠉⠉⢹⠠⠆⢹⠣⠀⠈⡠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣆⢀⠄⢺⣄⠀⠙⠚⠁⠈⠲⠊⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⣿⡇⠀⡟⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠃⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠘⠃⠀⢊⠜⡂⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⡠⠟⠛⠒⣃⠀⠀⠧⠤⠤⠤⠤⠤⠤⠼⠀⠀⢸⣼⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠸⠀⠀⠀⢈⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢳
⠀⠀⠸⠀⠀⠀⣈⣭⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠜⠐⠚⠢⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠲⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠘⡄
⠀⠀⠰⣀⠀⠀⢤⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢍⣀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠰⠀⠀⠀⠀⢸⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣠⡾⢫⡀⠀⠀⣴⢈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⢀⣴⠟⠛⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢀⣤⣴⠟⠻⢇⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⣀⣄⣤⣄⡄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⣹⣮⡀⠟⣄⣠⡀⠹⣦⡾⢡⢄⣀⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠿⠟⡛⠶⢦⣴⠶⠞⢋⠰⢑⡏⠋⠉⠙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⡿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡛⢿⣿⣶⠐⠀⠈⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⡏⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⢹⣿⣿⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢸⣿⣿⡇⠬⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠥⢸⣿⣿⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⢹⡇⢚⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⡓⢸⡏⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠬⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠠⠀⠀⡄⠀⠀⠀⠄⠀⠀⠀⢀⣀⠀⢀⣀⠀⠀⠀⠀⠲⠀⡀
⠀⠀⠰⣀⢲⡆⠀⠀⢠⡄⠒⣦⠀⠀⠀⠲⠶⠀⠀⠁⠀⠀⢀⡰⠆⠀⠀⠈⣡⣤⠀⠘⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉
⠀⠀⣀⢂⣀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠠⠀⠐⠆⠄⠀⠀⠄⠄⠀⠀⠀⠀⠈
⠀⠀⠻⠂⠉⠀⠀⠀⠑⠀⠀⠋⠀⠀⠀⠠⠀⠀⠄⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⠆⠶⠂⠖⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣾⡟⡟⣿⣿⣿⣿⣟⣛⣟⣦⠀⠀⣀⠀⠐⠀⠂⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⣉⠉⠁⠀⠀⠁⢀⣀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠐⡖⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠶
⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠥⠀⠀⠀⠀⠀⠐⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⣒⣚⡛⠓⠒⠤⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣛⣭⠭⠽⢿⣭⣭⣛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⠴⢪⣥⣤⡀⡘⠿⠿⠋⢀⣠⣄⡈⣆⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣫⡵⠚⠛⣿⣧⣀⣠⣼⡿⠿⠿⣷⢹⣿⣿⣿
⠀⠀⠀⠀⡔⣁⣀⠙⠛⠋⠀⣿⠀⠀⠀⠘⠿⠟⠁⣹⠀⠀⠀⢸⣿⣿⣿⣿⣫⡾⠿⣦⣤⣴⣿⠈⣿⣿⣿⣧⣀⣤⣾⠏⣿⣿⣿
⠀⠀⠀⡞⠸⠿⠿⠃⠸⣷⡀⣿⡀⠀⣠⣤⣄⠀⢠⡟⠀⠀⠀⢸⣿⣿⣿⢳⣇⣀⣀⣼⣇⠉⢿⠀⣿⣿⠿⠛⠻⣿⡿⢸⣿⣿⣿
⠀⠀⠀⢧⣤⣴⡶⠿⣷⠈⠀⢻⣧⡀⠛⠛⢋⣰⡟⠁⠀⠀⠀⢸⣿⣿⣿⡘⠛⠋⢉⣀⡙⣷⣿⡄⠸⢿⣤⣤⣴⠟⣠⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠁⠀⠀⢸⠀⠘⠚⠛⢁⣠⣴⠿⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣿⣿⣿⡇⣿⣧⣥⣤⣿⠟⠋⣠⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣶⠿⢟⣫⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣭⣭⣽⣧⣈⣉⣉⣁⡤⠶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠻⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢸⣄⠙⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢳⠀⢀⣀⣀⣀⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠈⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠁⠀⠀⠀⢠⡖⠚⠋⠉⠉⠉⠉⠙⠓⠲⠤⣄⡀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠦⣌⡉⠓⠲⠦⣄⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠑⡆
⠀⠀⠀⠀⠀⠀⢠⠞⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣸⢁⡦⣀
⠀⠀⠀⠀⠀⣰⠃⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⠀⠈⠓⢤⡀
⠀⠀⠀⠀⢰⠃⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢀⣉⡷⠄
⠀⠀⠀⢀⡏⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡤⠶⡄⠀⢹⡀
⠀⠀⠀⢸⠁⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠈⡇
⠀⠀⠀⣼⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣧
⠀⠀⠀⢻⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡟
⠀⠀⠀⢸⡀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⡇
⠀⠀⠀⠈⣇⠀⠘⠶⠒⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣸⠁
⠀⠐⢾⣉⠁⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢠⠇
⠀⠀⠀⠈⠳⢤⡀⠀⡞⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠙⠺⢁⡟⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⠲⠦⣤⣈⡛⠲⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠈⠙⠲⠦⢤⣄⣀⣀⣀⣀⣠⡤⠼⠃⠀⠀⠀⢈⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠉⠉⠉⠁⠀⢧⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠛⠛⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠖⠒⠲⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣠⡶⠛⠛⠶⣄⠀⠀⣀⠈⣷⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⢀⣴⣾⣿⣶⣦⡀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢰⡏⠀⠀⠀⠀⠘⠁⣶⢸⡇⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⠀⣄⢻⡘⣧
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢷⡀⠀⠀⢀⣄⠁⠿⢸⡇⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⣿⣿⣿⣿⣯⠁⠀⡿⢸⡇⣿
⠀⠀⠀⠀⠀⠈⢷⡀⠀⣠⣤⣄⣙⠛⠛⢋⣁⣤⣄⠈⠁⣼⠁⠀⠀⠀⠀⢿⡀⠀⣀⣤⣄⣉⣛⣛⣛⣁⣠⣤⡀⠘⣰⡏
⠀⠀⠀⠀⠀⠀⠈⢷⣴⠏⠀⠈⠉⠙⠋⠉⠉⠀⠘⣧⡼⠃⠀⠀⠀⠀⠀⠈⢷⣴⣏⣉⣛⣛⣹⣿⣿⣿⣿⣿⣿⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⡀⠀⠀⢀⣀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣯⣭⣽⣯⣭⣭⣭⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣿⣿⡿⠿⠟⣛⣫⣭⣴⣶⡿⢋⣝⣛⣛⣋⡙⢿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⠿⠟⣛⣋⣭⣴⣶⣾⣿⣿⣿⡟⣩⣥⣶⣿⣿⣿⣿⣿⣿⣶⣬⣍⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⠿⠟⢛⣋⣭⣤⣶⠾⢛⣋⣩⣭⣙⡛⢿⣿⣿⡿⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣤⣶⡇⢸⣿⣏⣭⣥⣶⣾⣿⣿⣿⣿⡟⣡⣾⣿⣿⣿⣿⣿⣿⣷⣼⡏⣴⣿⣿⣿⣿⠟⣩⣶⣶⣍⠻⣿⣿⣿⣿⣦⢹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⣿⣿⢋⣵⣶⣶⣮⡹⣿⣿⣧⢸⣿⣿⣿⡟⢸⣿⣿⣿⣿⡇⣻⣿⣿⣿⡇⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠻⣿⣿⣿⣿⡇⣿⣿⡇⢾⣿⣿⣿⣿⡇⢹⣿⡇⢾⣿⣿⣿⣿⣌⠻⣿⣿⠟⣡⣿⣿⣿⣿⡷⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣇⢻⣿⣷⡘⠿⣿⣿⠿⣣⣿⣿⣿⣦⢹⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⠏⣴⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⠻⣿⣿⣷⣶⣶⣾⣿⣿⠟⣰⣿⣆⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣰⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣬⣙⡛⠛⠛⣛⣋⣵⣾⣿⣿⣿⣿⣿⣦⡙⢛⣛⣛⡛⢋⣴⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⢠⡀⠀⠀⠀⠀⠴⢶⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣾⣿⣷⣄⠀⠀⠀⠀⠀⣀⣦⠀⠀⠀⠀⡤⢾⡿⠀⠀⡀
⢬⣽⣿⣟⣿⠂⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠍⣐⠃⠀⠀⣼⣿⣿⡿⠿⠃⠀⣀⣤⣿⣿⣿⡄⠀⠀⣾⡁⠀⠀⠀⠀⢄
⠀⠙⠀⠀⠉⠀⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠘⠱⠊⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠻⠛⠙⠉⠁⠀⠀⠀⠛⠃
⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦
⠀⠀⢸⣿⣿⣿⠶⠀⠀⠀⠀⠀⠀⠈⠛⠻⢦⠀⠀⠀⣼⣭⣤⡆⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠖⠲⠒⠖⠀⠀⠀⢸⡿⠂
⠀⠀⠛⠉⠹⠆⠀⠀⢰⢠⠀⣄⡆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⡀⠘⠃⠀⠀⠀⠀⣠
⠀⠀⠀⠀⣠⣶⣶⣦⠈⠉⠉⠀⠁⠀⠀⠀⢰⣶⢾⡷⠆⠀⠀⠈⢠⡆⠀⠀⠀⠀⠀⡀⣀⡠⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⢿⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠂⠘⠃⠄⠀⠀⠀⠀⡋⠉⠙⡆⠀⢀⡔⠈⠐⢄
⠀⠀⠀⠀⠈⠉⣉⣁⠀⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠉⠻⠄⠀⠀⠀⠀⠀⣤⡖⠀⠀⠉⠉⠉⠁⠀⠸⠏⠀⢰⡎
⠀⠀⠀⠀⠀⠀⠹⡿⠓⠀⠀⠀⠀⠀⠙⠻⠟⠃⠀⠀⠀⠀⣀⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠋⠀⠀⠀⠀⠀⢐⡇⠀⠀⠀⠀⠀⠘⠀⠁⠀⠀⠀⠀⠀⠠⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠆⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢴⣴⡄⠀⠀⠀⠙⠆⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡀⠀⠘⠉⠁⠀⠀⠀⢀⣀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⢀⡀⠀⠀⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣠⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⡏
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣿⣹⡇⢸⡇⢸⡇⢾⣏⡓⠘⣿⠁⣿⣧⣸⠀⣿⣛⡁⢾⣍⡓⢰⣏⣙
⠀⠀⠀⠀⠀⠀⠀⠿⠴⠟⠘⠧⠾⠃⠶⠬⠟⠰⠿⠄⠿⠈⠿⠀⠿⠤⠄⠶⠼⠟⠰⠬⠿⠁
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⡀⠀⢀⣀⡀⠀⣀⣀⡀⢀⡀⢀⠀⢀⣀⡀⢀⣀⡀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⣏⡿⠂⣼⣻⡄⢸⣯⣽⠂⢹⡏⠁⣿⢷⣼⡇⣿⠭⠅⢸⣭⡿⠀⠿⣭⡅
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠘⠋⠉⠓⠘⠃⠙⠂⠘⠃⠀⠛⠀⠛⠃⠛⠒⠂⠘⠃⠙⠂⠓⠚⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⣄⠀⠀⠀⠀⠀⠀⡴⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠼⠇⠀⠾⠤⠤⢤⡤⠤⠼⠧⠀⠼⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠶⣄⠀⠀⠀⢸⠀⠀⠀⣠⠶⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣹⠤⣞⡀⣰⠛⠛⠀⠀⠀⣹⠤⣏⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⣠⠶⠃⢸⠀⠘⠁⠀⠀⠃⠘⠦⢴⠀⠀⠘⠁⡀⠈⠃⠀⢸⡇⠓⠦⣄
⠀⠀⠀⢀⡤⠞⠋⠀⠀⠀⢸⠀⠀⢠⡏⠉⣳⠀⠀⢸⠀⠀⢠⡏⠉⢳⠀⠀⢸⡇⠀⠀⠈⠙⠲⣄⡀
⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠋⠛⠛⠋⠛⠛⠛⠁⠀⠉⠛⠛⠛⠁⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠤⣍⣤⠤⣤⠤⠤⠤⢤⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡴⢷⡀⢀⡴⠃⣀
⠀⠀⠀⠀⠀⠀⢀⡼⣡⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠻⡄⠀⠾⡇⠠⠿⠄⢀⡞⠉⢦
⠀⠀⠀⢤⣀⡀⣈⡛⠁⠀⠙⢿⡁⢀⣠⡄⠀⠀⠀⠀⠀⢀⣀⣹⡇⢸⣉⣀⣀⡇⢰⣇⣀⣉⣿⠀⣯⣁⡀
⠀⠀⠀⠀⠀⠙⢧⣟⣦⡀⢀⠲⣌⡯⠀⠀⠀⠀⠀⠀⠀⢸⢰⢾⡇⢸⠶⠶⠲⠇⠸⠗⠶⠶⣿⠀⡷⣦⢹
⣰⠋⠙⣦⠀⠀⠀⠙⢾⣻⠶⠛⠉⠀⠀⡞⠉⢱⡄⠀⠀⢸⢸⠸⠇⢸⠀⢠⠖⢋⣙⠳⣄⠀⠿⠀⡇⣿⢸
⣸⠦⠴⡇⠐⢂⢂⢂⠀⠀⢀⠂⠒⡐⠀⣳⠤⠾⡅⠀⠀⢸⢸⠀⠀⠀⠀⡏⠀⠻⢭⠀⢸⡇⠀⠀⠀⣿⢸
⡇⠰⡆⢻⡀⠀⠈⠈⠀⠀⠈⠈⠀⠁⢰⠇⣴⠀⡇⠀⠀⢸⢸⠀⠒⢐⢀⠻⣄⡙⠋⣀⡞⡁⢒⠐⠀⣿⢸
⡇⠀⢳⣈⣉⣉⣉⣙⣆⣀⣼⣉⣉⣉⣉⣠⠇⠀⡇⠀⠀⢸⠸⣤⣤⣬⣤⣤⣤⣭⣭⣥⣤⣤⣤⣤⣤⡿⢸
⡇⠀⠀⣿⢹⡭⠭⠭⠭⠭⠭⠭⠭⣽⠏⡏⠀⠀⡇⠀⠰⡟⠓⠒⠒⠒⠒⠒⠲⠶⠶⠶⠒⠒⠒⠒⠓⠚⢛
⣷⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣷⣶⣶⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣴⣤⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠉⠉⠉⠐⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠀⣀⡠⠊⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⣀⡪⠔⠒⠂⠀⠈⠁⠒⠒⠠⢄⡀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡊⣤⠄⠀⠀⠀⠁⣀⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠉⠁⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢵⠂⠀⠀⠀⢀⡖⠂⠈⠉⠐⠯⠉⠒⢤⡀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢀⠀⢣⠀⠀⠀⠀⢀⠀⡀⠂⠈⠑⠤⡀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣾⠀⠀⠀⢀⢀⡎⠀⠠⠀⢄⠀⠈⠒⢄⠱⡦⡀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠏⠀⠀⠀⠡⠀⠀⢘⡀⠀⠠⠀⠀⠀⠀⠑⢵⡈⠢⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⡤⠀⠀⠶⠄⠀⠀⠚⠉⠁⠘⠁⠀⠀⢀⡀⠀⠀⠀⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⢄⠀⢀⠃⠀⢰⠀⠀⠄⠀⠀⢠⡄⠀⠀⠀⠀⠀⠈⠙⠢⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠎⠈⡄⣈⠀⠀⠄⠀⠄⠂⠀⢠⠎⠘⢄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠁⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡇⠀⡤⠒⣹⢿⠀⠀⠀⠀⠀⢀⡔⣁⠀⠀⢘⠣⡀⠀⠀⠀⠀⢀⠞⠀⠀⡘⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠤⠊⠀⠀⠀⡅⠀⡖⡈⢹⠒⠢⡀⠀⡀⠴⠌⠁⣨⣙⠀⢸⠀⠈⠐⠢⠄⠒⠁⠀⠀⠀⠃⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠒⠞⢱⡄⢣⠀⠀⠀⠀⠀⠀⢀⠀⣠⠋⠈⠛⠀⠀⠈⠁⠀⠀⢆⠨⠓⠉⠂⠜⠀⠰⠦⠴⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠰⠀⠰⠁⠀⠀⠑⣄⠀⠀⠀⠀⠈⠑⣌⢣⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣎⠀⡆⢀⣤⣤⠤⠤⠤⠤⠄⢀⠀⠀⠀⠀⠰
⠀⠀⠠⠤⢃⠗⡂⠀⠀⠠⠐⠉⠑⠠⣀⠀⠀⠈⠈⠣⣌⡱⠐⡤⠀⠀⡀⠀⢆⠸⠔⠁⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠀⠀⡀
⠀⠀⠀⠑⠊⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡒⠂⠀⠄⢫⠄⢌⡑⠄⡘⠀⠰⢊⠀⠐⠂⠀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠠⠁⠀⠁⠁⠈⠒⠀⠀⠀⢀⠵⣀⣀⣄⣎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⢀⠂⠀⠀⠀⠀⠀⠢⣄⡀⢠⠔⢠⡿⡹⠺⠧⠥⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⢇
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⡙⠋⠀⠀⡈⠀⠀⠀⠀⢀⡠⡁⢢⠀⠀⠀⠀⠀⠤⠑⠂⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠠⢈⣦⠞
⠀⢀⠄⠀⠐⠶⡄⢄⣀⡀⠀⠀⢠⠁⠀⠀⢀⣃⠀⠀⠀⠀⠈⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⠀⠀⠪⠀⠐⠄⠀⠀⠀⣇⡴⠋⠁
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠺⠉⢈⣐⣠⠀⠀⠀⠀⠀⠀⠀⢀⢤⠤⣶⣶⠒⡞⢳⣶⣶⢲⡖⣶⣶⣶⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣽⣿⡀⣷⢞⡉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡜⣿⣿⡇⠷⠸⠿⠷⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠂⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠆⠀⠰⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⢁⣀⣀⣀⣨⣬⡀⠉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⠏⠁⠀⠀⠀⠀⠁⠁

⣶⠒⠛⠓⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠛⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⢻
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸
⡇⠀⠀⠀⢀⡠⠄⠒⢢⣀⡀⠀⣀⢠⡊⠉⠉⠑⠢⢄⠀⠀⠀⢸⠀⠀⠀⠀⢀⣠⡄⠀⠀⡇⠀⠀⠀⣚⠉⠉⠉⠉⠉⠂⠀⠀⠀⠀⢸
⢷⠀⠀⢠⠊⠀⠀⠀⢸⠀⠈⠊⠀⠀⠃⠀⠀⠀⠀⠀⠃⠀⠀⡼⠀⠀⠀⣠⠼⠀⠹⠤⡄⡇⠀⠀⠀⢈⡩⠜⠋⢀⡰⠀⠀⠀⠀⠀⢸
⢸⠀⠀⣾⠀⠀⢰⠺⠁⠀⠀⠀⠀⠀⢠⠀⠸⠂⠀⠀⢠⠀⠀⡇⠀⠀⠀⠳⢦⠀⢰⠒⠃⡇⠀⠀⠀⡆⠀⠀⠈⠉⢙⠀⠀⠀⠀⠀⢸
⠸⡄⠀⠸⡀⠀⠈⠙⠇⠀⡶⡤⠆⠀⠘⠀⠀⠀⢀⡠⠊⠀⠀⡇⠀⠀⠀⠀⠘⠓⠚⠀⢀⡇⠀⠀⠀⠓⠒⠒⠒⠒⠚⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠈⠒⠤⠤⠜⠉⠀⠀⠘⠂⠈⠉⠉⠉⠁⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸
⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⢸
⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠒⠒⠒⠒⠒⠒⠒⠂⠂⠀⠀⠠⠤⢤
⠀⠈⠃⠄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠈⠉⢉⣉⣀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⢀⣀⡼⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢀⠴⠊⠁⠀⠀⠀⢠⠏⠀⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⡎⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠘⡄
⠀⠀⠀⢀⣠⣤⣤⣤⣤⡜⠀⠀⠰⣥⣤⣤⣤⣤⣄⡀⢰⠃⠀⣿⠊⠙⢦⡀⠀⠀⠀⠀⠀⡠⠚⠉⣾⡄⠀⢇
⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⢸⠀⢠⡇⠱⣄⠀⠙⢦⡀⠀⡠⠊⠀⢀⠔⠁⡇⠀⢸
⠀⠀⠀⠈⠒⠒⠒⠒⠒⡆⠀⠀⡖⠒⠒⠒⠒⠒⠚⠁⣼⠀⢸⠀⠀⠀⠓⢄⠀⠙⠊⠀⢀⠔⠁⠀⠀⣇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⡠⠃⠀⠀⠐⢧⡀⠀⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⠀⠀⡠⠊⠀⢀⠔⠳⢄⠀⠙⢦⡀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⣧⠎⠀⢀⠔⠁⠀⠀⠀⠳⣄⠀⠙⣆⡇⠀⢸
⠀⠀⠀⠀⢀⣠⠜⠁⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢯⠓⠒⠁⠀⠀⠀⠀⠀⠀⠈⠑⠒⢻⠃⠀⡎
⠀⠀⡞⠉⠉⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢰⠃
⠀⠀⠳⠤⠤⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠭⣒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⡒⠀⠀⠭⣗⡯⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⡪⠀⠀⠭⢕⡂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠮⠀⠀⠉⠑⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⠭⠭⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠕⠀⠀⢭⠭⠗⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣒⠀⠀⣙⠭⠅⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠀⠐⠯⠅⠀⢀
⠀⠀⠀⠄⠀⠄⠠⠀⠀⠄⠀⠐⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⡮⠀⠀⣒⣢⠤⠀⢐⡀⠀⠀⠀⠀⠀⢀⢀⡀⡀⠤⠠⠄⡀⢀
⠀⠀⠀⠀⠀⠂⠐⠀⠠⠀⢂⠀⠄⠄⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠒⠚⠇⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⡂⣘⠀⠂⡒⠐⠀⠀⠠
⠀⢀⠀⠀⠀⠅⠀⠀⠐⢀⢐⠠⠄⠅⡂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡂⣚⠈⠁⠭⢘⡒⠖⠾⢹⠧⡴⠒⠖⢲⣮⠈⠨⠅⡁⣒⢰⡂⡶⢸⢺⡇⢸⣾⣶⠒⡿
⠁⠐⠀⡀⠀⠂⠐⠂⠈⠀⢐⠨⠅⠅⠀⠀⠉⠀⠀⠀⠀⠂⠀⠨⠀⠄⠒⢐⡂⣒⠐⡆⠄⠤⢀⣀⢀⠀⠀⠳⣿⢸⠉⠩⠭⠽⠈⠁⠭⢸⣷⡇⢸⣥⠛⠿⣿
⠀⠈⠀⠀⠀⠂⠐⠂⠈⢐⠀⠨⠅⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠅⠤⠠⠄⠦⠈⠉⡅⣠⠀⢀⠀⡀⠀⠀⣀⡀⠀⢀⣄⣤⠈⠃⣿⢸⣭⡇⢸⣾⣋⣶⣏
⠄⢈⠀⠀⠀⠄⠈⠁⢀⠐⠀⠨⠁⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⣀⠈⠁⣭⢈⡁⠀⠒⢐⠀⠉⠀⠀⠀⠐⢀⠐⠂⡀⠚⠘⠃⣀⠀⠈⠁⠸⠉⠉⠅⠁
⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠈⠠⠄⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⠐⠂⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠲⠀⠀⠆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠨⠈⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄⠀⠀⠀⠤⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂
⡄⠀⠀⠂⠀⠀⠀⠀⠀⠠⠈⠠⠀⠄⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉⠁⠀⠀⠈⣿⣿⣿⣿⠁⢀⣀⣄⣀⡀⠀⠀⢠⣿⣿⣿⣿⣿⡆⠀⠀⠀⣴⣿⣿⣿⣿⣷⡄
⣿⣿⣧⠀⠀⢀⣴⣿⣿⣿⡆⠀⠀⣀⣤⣄⡀⠀⠈⠛⠛⠁⠀⣿⣿⣿⣿⣷⠀⠀⠸⣿⣿⣿⠿⠛⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⡷
⣿⣿⣿⡀⠀⣿⣿⣿⣿⣿⡁⠠⣾⣿⣿⣿⠇⠀⢀⣀⣀⠀⠀⠻⣿⣿⠏⢁⣀⡀⠀⠈⠉⠁⣠⣤⣄⡀⠀⠈⠻⠿⠿⠛⠛⠛⠁

⣿⣿⣿⡇⠀⠈⠻⣿⣿⣿⠇⠀⠹⣿⣿⣿⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⢀⣿⣿⣿⣷⠀⠀⣼⣿⣿⣿⣿⠄⠀⠀⢀⣠⣤⣄
⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠿⠿⠇⢰⣿⣿⣷⠀⠈⠿⠿⠋⣁⠀⠀⠹⠿⠟⠛⠋⠀⠀⢠⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⣠⣶⣶⡄⠀⣰⣿⣿⡇⠀⣴⣾⣷⠀⢀⣠⡀⠈⠿⠟⢠⣴⣦⡀⠀⣾⣿⣿⡆⠀⢀⣴⣷⣦⡀⠀⠸⣿⣿⢿⣿⣿⠃
⣦⡀⠀⢰⣿⣿⣿⣧⠀⠻⣿⣿⡷⠀⠻⣿⡿⠀⣿⣿⡇⣤⣤⡄⠘⠿⠟⣁⡀⠙⠛⣉⡀⠀⢺⣿⣿⣿⠇⠀⠀⢀⣀⡀
⣿⣧⣀⠈⠻⢿⣿⠿⠁⠀⠀⠉⠀⣠⣄⠀⣠⣤⠈⣉⡀⠻⠟⢰⣶⡄⠸⡿⠟⠀⢼⣿⣿⠆⠀⣠⣤⡀⠀⢀⣾⣿⣿⣿⡆
⣿⣿⣿⠀⠀⠀⠀⠀⢀⣶⣷⡆⢸⣿⣿⠈⠿⡿⠘⢿⠇⣶⡆⣈⠉⣶⠆⠰⣿⠷⠀⣴⣦⡀⢸⣿⣿⣿⠀⠈⠻⠿⢿⣿⠟
⠛⠋⠁⠀⢠⣶⣷⡄⠸⢿⣿⡿⠀⠈⠉⢠⣤⢠⣶⠠⣶⢨⣄⡛⢴⠀⠾⡂⢴⡶⠈⣛⡛⠁⣠⣤⡄⠀⢀⣤⣶⣦⠀⠀⠀⠀⣀
⠀⢀⡀⠀⢸⣿⣿⣿⡇⠀⢀⡀⢰⣿⣧⠙⠛⢁⡍⣴⠰⠦⠬⣝⢢⠩⠌⣏⠰⠶⢘⠻⠇⠈⠛⠿⠟⠀⠸⢿⣿⣿⡇⠀⢠⣿⣿
⣼⣿⣿⣦⣄⠉⠉⠉⠀⠀⣿⣿⣦⠉⣉⠰⣿⡎⣉⠴⢜⡨⠍⠂⠁⢁⢣⠃⡝⢃⡛⠗⠐⢿⣿⠀⢠⣶⣶⡆⠈⠉⠀⠀⠈⠻⠿
⢿⣿⣿⣿⣿⠂⠀⣴⣶⣄⡉⠉⠁⠀⢿⠿⠀⣶⡍⣡⢋⡖⣦⡀⠀⠀⠭⢚⡰⠎⣃⡸⣿⠆⢀⣀⣈⠻⣿⠏⠀⢠⣶⣶⣶⣦
⠈⠙⠛⠉⠁⠀⠀⢻⣿⣿⠟⠀⢾⣿⡶⠀⣶⡦⠰⠶⢘⡊⣇⢆⠪⢭⣫⢩⡵⡶⠙⠃⣠⣄⠛⣿⡿⠀⢀⣀⡀⠘⢿⣿⣿⡿
⣿⣷⠀⠀⢠⣶⣶⣤⣄⠀⢀⣴⣤⣅⠀⣴⣶⡄⠺⠟⠨⡦⠹⢆⡛⠶⢰⡆⣶⡄⣿⡦⠹⠿⠁⣀⡀⠀⣿⣿⣿⡇⠀⠙⠛⠁
⣿⠟⠀⠀⠘⣿⣿⣿⡿⠀⠀⠻⡿⠟⠀⣈⣩⡀⠺⠿⠀⠺⢇⣈⠡⣾⢠⣤⡈⣤⡄⢠⣦⣄⠸⣿⣿⡆⠈⠿⠿⠃⠀⢀⣀⡀
⠁⠀⠀⣀⣀⠈⠉⠁⠀⠀⣶⣦⣶⣄⠀⠻⠿⠃⠐⣿⡿⠀⠻⠟⣠⣤⠈⠋⠐⠿⠟⠘⠿⠟⠀⠛⠛⠁⠀⠀⠀⠀⢰⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣷⠀⠀⠻⣿⡿⠃⠀⣴⣶⣿⡆⠀⣴⣿⡇⠈⠛⠋⣸⣿⣆⢠⣶⣄⠀⣠⣶⣄⠀⢰⣿⣿⣷⣦⠈⢻⣿⣿⣿
⠀⠀⠈⢻⣿⣿⡿⠋⠀⠀⣠⣄⣤⣤⡀⠈⠛⠛⣀⣀⠉⠉⣡⣾⣦⠀⠉⠋⠁⢾⣿⡿⠃⣹⣿⣿⡗⠀⣿⣿⣿⠏⠀⠈⠻⠿⠋
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠸⣿⣿⣿⣿⠃⠀⢰⣿⣿⣿⠀⠀⠻⠿⠟⢠⣾⣷⡄⠀⠀⠀⠀⠈⠛⠉⠀⠀⠈⠛⠁
⠀⢠⣾⣿⣷⣿⣿⣷⠀⠀⠈⠙⠛⠁⣀⠀⠈⠛⠛⠋⣠⣤⣄⠀⠀⣿⣿⣿⠇⢰⣿⣿⣧⠀⠀⣴⣶⣦⡀⠀⠀⣴⣾⣿⣶⣄
⠀⠸⣿⣿⣿⣿⣿⡟⠀⠀⢀⣴⣶⣿⣿⣷⠀⠀⢠⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠇⠈⣿⣿⣿⣿⡆⠀⢿⣿⣿⣿⣿⣷
⠀⠀⠈⠛⢿⠿⠋⠀⠀⠀⠘⢿⣿⣿⣿⡿⠀⠀⠀⠛⠿⠿⠛⠀⣶⣿⣶⡀⠀⠈⠉⠉⠀⠀⠸⣿⣿⣿⠟⠁⠀⢼⣿⣿⣿⣿⠏
⠀⠀⠀⢀⡀⢀⣤⣤⣄⠀⠀⠀⠉⠙⠋⠀⣠⣤⣄⠀⠀⠀⢰⣾⣿⣿⣿⡇⠀⢀⣠⣄⡀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡿⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡙⠀⠀⠀⢄⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⠔⣂⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⡇⠀⢀⣀⡈⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⢠⣁⣨⠟⢙⡇⠀⠀⠀⠀⠀⠀⠄⠀⠀⠉⠐⠐⠻⢉⣹⢠⣾⣭⣽⠟⠱⣦⡀
⠈⠛⠛⠛⢻⡀⠀⠀⠀⠀⠀⠀⠀⣢⣤⠀⠉⠉⠉⠉⣽⢲⡞⠀⢠⡄⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠙⢦⣀⡄⣠⡤⠶⠿⣿⡇⠀⠀⠀⠀⢨⣿⣿⡃⠀⢸⡇⠀⠀⡀⠀⠀⠀⠀⠀⡇⠀⢠⠀⠀⣧⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠤⣍
⠀⠀⠀⠀⠀⠀⠈⠛⢆⠀⠀⠀⠀⠩⡅⠀⠠⠤⢤⣬⣙⣛⡂⠀⠨⣃⣀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢸⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠔⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⡄⠐⠀⢸⠅⠀⠀⠀⠀⠩⣿⡗⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠸⠄⠀⠻⠀⣠⠇⠀⠀⠀⠄⠀⠠⠀⠀⠀⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡀⠌⡆⠀⠀⠀⠀⠩⡽⠁⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⠂⠀⠀⠀⠀⢈⣯⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣂⣦⣀⢤⣄⡀⣲⣀⣐⣦⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢨⡎⣁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢁⣀⡙⠋⠁
⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢠⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢀⠒⠈⠃⡀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣇
⠀⠀⠀⠀⡚⠋⠀⠘⠁⠰⡄⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢘⠀⠀⠀⠀⠂⠀⢠⠀⠀⡀
⠀⠀⠀⠀⠇⠀⠀⠀⠘⠀⢓⠀⠀⠀⠀⠀⠹⢀⠀⢘⡆⠐⠀⠀⠀⠀⠐⠀⢠⠞⠊
⠀⠠⢡⣿⠁⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠁⠀⡈⠉⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⡀⢰⡏⠀⠀⠀⠀⠀⠘⡰⣯⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⢀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠰⢀⣿⠁⠀⠀⠀⠀⠀⠀⠓⢻⡃⠀⠀⠀⠀⠀⡰⠘⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰
⡆⢨⡅⠀⠀⠀⠀⠀⠀⠀⠸⡌⢻⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠀⠇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠐⢀
⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠀⠀⠀⠀⠀⢠⣼⠀⠀⠀⠀⠀⠀⠀⠘⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣦⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢿⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⡌⢳⡈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⠹⣦⠻⣎⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄⠶⣟⠢⣌⡑⢒⣴⠶⢋⣠⡴⠾⠛⢋⣁⣤⣤⣴⣾⠷⠶⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡴⣌⠳⣌⠳⣌⡓⠶⢤⡤⠴⣒⣮⢗⠯⡣⣽⢊⡼⢋⡴⢋⣵⠞⣋⣥⡶⠞⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣱⢏⢛⢮⣙⠲⢭⣛⣛⣻⢭⣻⡺⡖⣵⢠⢏⡞⡴⢋⡴⣋⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡵⣫⢞⠓⠮⢭⣒⡲⣶⣒⣭⡮⢺⣥⡇⠏⡞⡼⣱⢏⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡵⣳⠬⣛⡓⣒⡾⢿⢖⡿⣺⡆⢎⢹⢸⠑⡇⡟⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣹⣽⢿⣏⣭⢤⢶⡒⣪⣄⢇⢎⢧⢧⣣⣧⢻⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢷⢳⢱⠹⣱⣵⣹⢘⡭⣦⣷⣑⣕⣝⢕⣋⢇⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣼⢸⢃⠇⣿⣿⡘⡯⡫⢖⣚⣛⡺⠤⢧⡵⡣⣪⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣴⢃⡟⡼⡸⢸⢙⣿⣻⡪⢕⣒⣒⠮⢍⣓⠦⣵⣱⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢛⣡⠞⣡⢏⡼⣱⠃⢟⣿⡻⢟⡻⠭⠭⢭⣝⡳⢬⡓⢦⣱⢣⢟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⠶⠛⣋⣥⠶⢋⣵⠞⣥⣟⡠⣝⢮⣻⣕⡯⠝⠒⠛⠛⠓⢦⣝⢶⣝⢦⡙⠏⡼⠁
⠀⠀⠐⠛⠿⠿⠿⢛⡛⠉⢩⣤⣴⡶⠛⠉⡰⠾⠿⠤⣌⣓⠮⠝⠚⠙⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡙⢦⡹⣦⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⢷⡘⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⢷⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠘⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⢀⠀⠀⠀⢀⠀⠀⡀
⣿⡟⡉⣻⠿⠿⠿⠿⠿⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠶⣶⡶⢀⣴⠖⠖⢀⡶⠻⣦⢰⣟⣦⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣩⡉⢩⣿⢋⣭⣭⣿⢋⣭⠙⡏⠩⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠰⠿⣦⢸⡷⠶⣦⢸⡇⠀⢾⠀⠉⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣏⣀⠻⡇⢈⣍⡙⣧⢸⣿⡇⣿⣶⣾⣿⣿⣿
⠀⠀⠀⢀⣴⠞⠡⣤⣤⠟⠘⢷⣤⡾⠘⢷⣤⡿⠀⠳⣦⡀⠀⠀⠀⣿⣿⣿⡿⢋⣡⣞⠛⠛⣡⣷⡘⠛⣁⣿⡘⠛⣁⣿⣌⡙⣿⣿⣿
⠀⠀⠀⠘⢧⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⠀⠀⠀⣿⣿⣿⣧⡘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢃⣼⣿⣿
⠀⠀⠀⠀⠀⠉⠛⠓⠶⠶⠶⣿⣷⡦⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣦⣬⣍⣉⣉⠀⠈⢙⣿⣿⣿⣿⣤⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛




⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠾⣿⣿⣶⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⣭⣛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⢿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠚⠛⠃⠀⠀⠈⢺⣿⣷⣶⣶⡆⠉⠀⠀⠘⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⠶⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣴⣾⣦⡛⣡⣌⢛⣴⣷⣦⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⣿⢿⣿⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢸⣿⣧⣿⣿⣿⣿⣿⣿⣼⣿⡇⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⠀⠀⢸⣿⣿⡿⣫⣭⣭⣍⢿⣿⣿⡇⠀⠀⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣫⣝⡻⠿⠀⠀⠈⠛⠛⠃⣫⣭⣛⡻⠘⠛⠛⠁⠀⠀⠿⣫⣝⡻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠃⠀⠀⠀⠀⠀⠀⠻⣿⣿⠟⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⣀⣤⣤⣄⠀⢀⣤⣤⣄⣀⣀⣠⣤⣤⡀⠀⣠⣤⣤⣀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠈⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠈⠁⠈⠀⠈⠉⠉⠉⠉⠁⠉⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡤⢠⣀⡄⣠⡀⡄⣄⣠⢀⡤⣄⣤⠀⢠⠄⣠⢤⣠⣄⣤⢠⣄⡄⢠⡄⣤⣠⢠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠘⠋⠓⠋⠃⠃⠋⠛⠈⠓⠋⠋⠀⠘⠒⠙⠚⠙⠙⠙⠚⠛⠓⠋⠓⠋⠛⠘⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠛⠿⣿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠋⠀⢦⠀⣿⡾⣿⠁
⠀⠀⠀⠀⠀⠀⢰⠒⠒⠒⠒⠶⡆⠐⠂⢀⡀⠀⠀⠀⠀⢰⡀⠛⠀⢠⡄⠀⠀⠛⢳⠏
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⣀⣀⡿⠠⠀⠈⠉⠙⠒⠲⢤⣀⡵⠀⠒⠀⠀⠆⠰⢂⠮
⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢇⠐⠃⠀⠀⢄⠀⠀⠀⠀⠀⠈⠠⠀⠀⠁⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠘⡄⠀⠀⠀⠀⠁⠀⠀⠀⡀⣀⣠⡅⠀⠤⠶⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠰⡔⠀⠀⠀⠀⠀⠀⢀⣠⣶⠜⠣⠊⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠑⣄⠑⡄⠀⡀⣤⣲⣾⡛⠉⠏⠀⠀⠀⠀⠠⠀⠘⡆⠀⠀⠀⠤⡀⠂⠦⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠈⠳⢬⣵⣾⣿⣿⣿⣮⣱⣦⣄⣀⣠⣤⣐⣀⣀⠈⠀⠀⠃⠈⠚⠚⢈⣉⣀⠴⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠘⢂⣶⣈⠁⠀⠀⠀⠸⠉⠉⠉⠉⠉⠉⠉⣩⢿⠀⠀⠀⠈⠉⠉⢹⠛⠓⣶⠉⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣶⣦⣤⣤⠘⡀⠀⠀⠀⠀⠀⢸⣓⢼⠀⠀⠀⠀⠀⠀⠸⠀⠀⣿⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⣀⣀⣠⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆⡇⠀⠀⠀⠀⠀⠀⠁⢸⡆⠀⠀⠀⠀⠀⠀⡄⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⡇⢸⡀⡇⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠐⠃⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⢸⠀⠘⢄⠈⣿⣿⣿⣿⠛⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠋⠀⠀⠀⠀⠀⡄⠀⢸⠀⠀⠀⠀⠀⢸⡇⠠⠆
⠀⠀⠀⢸⠮⠼⡗⣾⣧⣸⣿⣿⣿⡗⢲⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠇⠀⢸⡆⠀⠀⠀⠀⢸⡇⠀⢽
⠀⠀⠀⠀⢣⠀⠀⣿⣿⣿⣿⣿⣿⣧⠈⠙⢿⡀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠆⠀⠰⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠙⠁⠈⠻⣿⣿⣿⣿⣿⠀⠀⠀⠙⠻⣿⣿⣿⣿⡏⠓⢛⢲⡦⠤⠤⢤⣀⣀⣀⣀⣇⡠⠤⢒⠎⡿⣔⡄
⠀⣀⠀⡠⡂⠀⢇⡠⠒⡍⣻⣿⣿⣿⠁⠣⠔⢠⠖⣄⠽⠿⠟⢃⡀⠒⠊⠂⠂⠰⠾⠆⡔⣂⣉⢱⡀⠀⢃⠘⠤
⠀⠑⠌⠁⠀⠀⠀⠀⠀⠈⠁⠈⢩⡁⢀⠖⠐⠧⠄⠉⠈⠐⠀⠃⠉⠐⠁⠀⠀⠀⠐⠀⠈⠀⠈⠩⠃⠠⠩

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠉⢹⠛⠀⢀⡀⠀⢀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢀⣠⣤⣔⣚⠉⠁⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⢀⡠⢔⡞⠋⠁⠀⠀⠉⠓⣆⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡋⠀⠞⠀⠀⠀⣈⠲⣄⠀⠈⣧⠴⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡙⠃⠀⠀⠀⠀⠉⠉⠀⣠⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠤⢤⡀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⡀⣀⣀⣀⡠⠒⢟⠀⠀⠀⠀⠀⠀⠀⠀⠂⣨⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠀⠀⠀⠈⠣⡀⠙⢎⠢⡀⠀⠑⢄⡀⠀⠀⠈⡶⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠙⣄⣀⢡⡈⠢⡀⠀⠙⢦⡀⢠⡄
⠀⠀⠀⠀⠀⠀⡜⠀⠈⠑⠦⠀⠀⠀⠀⠀⠀⠘⡄⠉⠹⡄⠈⢦⣀⡞⠙⣾⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤
⠀⠀⠀⠀⠀⢰⠃⠀⠐⠉⠀⠛⠀⢠⠀⠀⠀⠀⠸⡶⠶⢷⠀⠀⠉⠀⠀⡏⣶⣽⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⠁⣈⠄
⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢃⠀⠈⡆⠀⠀⠀⠀⢱⡇⠘⣆⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⠀⠀⢰⡀⠀⠀⠀⠸⡇⠀⠫⠈⡷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⢀⣤⣾⠯⠀⣠⣤
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⡇⠀⠀⠉⠉⠉⠒⠒⠤⢄⣸⠲⡇⠱⡀⠀⠀⠀⠀⠀⠀⣀⣴⠦⣠⣴⠟⠁⢸⢷⣿⣼⠟⢁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠇⠀⠀⠀⠀⠀⣇⠀⠀⠘⠈⠉⠙⠒⠢⠄⡈⠀⣿⠀⢳⠀⠀⠀⠀⣠⣞⣓⠦⠝⠂⢹⣀⡀⢸⣸⡟⣧⠞⠁
⠀⠀⢀⠀⠀⠀⠤⠀⣤⣤⣤⣤⠏⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⠈⠄⠀⠀⢰⣧⡈⠉⠀⠀⠀⠞⢹⡇⠈⢁⡴⠃
⠀⠀⢸⠀⠀⠛⠒⠉⠀⠀⠀⠘⠦⣀⡠⠴⠒⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢣⣀⣴⠾⢹⠇⠀⠀⠀⢠⠀⠸⠧⠖⠁
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣤⡾⠛⠁⣀⠾⡀⠀⠀⢀⠇
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⢀⡤⠊⢹⠀⠈⠒⠤⣾
⠀⠀⠀⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⢉⡷⠦⠄⣀⠀⠀⠀⠀⢀⣴⠿⠋⠀⣠⠖⠉⠀⠀⠈⢣⣀⢀⡰⠁
⠀⠀⠀⠀⠀⠈⠉⠓⠢⢄⡀⠀⠀⠀⠠⠀⠀⠀⡴⠋⠀⠀⠀⠀⠉⠑⠒⢾⠟⠁⣀⡴⠚⡟⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣗⡦⠤⠤⠼⠒⠲⠧⡀⠀⠀⠀⠀⠀⠀⢰⠘⣤⠚⠁⠀⠀⡇⠀⠀⠀⠀⠠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣀⠤⠐⠊⠉⠉⠀⠉⢓⣶⡄⠀⡄⣾⢸⢀⣏⠁⠒⠲⢤⡷⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⡶⡻⠋⢱⢠⣁⡷⠚⠁⠀⠀⠈⠙⠢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣼⣁⠄⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⢤⡀⠀⢀⡀⠀⠀⢠⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤
⠀⠀⠈⠀⠀⡠⠀⠀⠀⠀⠒⠂⠒⠂⠀⡛⠓⠀⠤⠀⣹⣅⠀⠀⠀⠀⠠⠁⠀⠀⠐⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⢀
⠀⠀⣄⠀⠘⠀⠀⠀⠀⠐⠀⠀⠀⠈⠉⢀⡄⠀⠀⠀⠀⠈⠉⠀⠐⢰⣥⠔⠄⠀⠀⠀⠀⠀⣠⡀⠀⠀⢄
⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠲⠀⠤⡀⠠⢀⣀⠀⠀⠀⠢⡀
⠀⠈⠀⠈⠁⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⢀⠀⠢⠇⠀⠀⠀⠃⠀⠃⢀⠀⠀⠠⢂⢀⠛⠤⠱⠂⠈⠀⠂⠀⠌
⠀⠀⠀⠸⠀⠀⡴⠀⠀⠀⠤⠄⠀⠀⠠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠄⠀⠀⠈⠑⠂⠀⠀⡇⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⡤⢂⠀⠀⠀⠀⠆⠀⡠⢀⠀⠐⠀⠒⠀⠀⠀⠀⣀⣠⣆⠐⠒⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠁⠀⠉⠰⠀⠀⠀⠤⠄⠀⠀⠀⠀⡀⠒⢀⡀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠸
⠀⢀⠀⠀⢀⠀⠀⠀⠻⠀⠀⠸⠂⡐⠀⢠⠀⠆⠐⠤⠄⠐⠒⠀⠀⠰⡖⢀⣃⣀⣀⣀⣀⣀⡴⠁⠀⠀⠀⠀⢠⠄
⠀⠀⢀⠀⠈⠀⢀⠤⠠⠀⠀⠂⠀⠅⠀⠄⠀⠀⠀⠄⠠⠄⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠁⠀⠀⢘
⠀⠐⠀⠈⡉⠀⢄⡀⠀⠀⠀⢤⠀⠀⠀⠀⡀⡀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠃⠀⠀⠀⠀⠀⠈
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⡄⠀⠀⠀⡄⠈⢳⠄⠀⡠⠀⠀⣠⠀⢀⠀⠀⠀⠊⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠐⠀⡇⠀⠀⠀⠀⣛⠢⠀⠀⠀⡄⠀⠀⠀⡇⠀⠀⡇⠀⠰
⠀⠈⠀⠉⠁⠀⠣⢄⠀⠀⠀⠈⠀⠣⠀⠀⠀⠀⠉⠀⠀⠐⢰⠀⠀⠀⠀⠃⠀⠀⠃⠂⠠⠒⠂⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⢀⢤⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠈⢳⢰⡼⡞⡇⠀⠀⠀⠀⢀⡟⠀⠀⠈⠃⢤⢳⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡆⣠⢶
⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⣠⡾⠈⠙⢳⠇⠀⠀⠀⠀⠈⢧⣀⣀⣠⠌⠘⢸⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠁⠸⢸
⠀⠀⠀⠀⠀⢠⠖⠲⠦⠭⠥⠖⠲⣆⠀⠀⠀⠀⠀⣰⠒⠲⠬⠭⠴⠖⠲⡄⠀⠀⠀⠀⢀⣴⣶⣦⣬⣥⣴⣶⣦
⠀⠀⠀⠀⠀⠻⣄⣀⠀⠀⠀⢀⣠⡼⠀⠀⠀⠀⠀⢧⣄⡀⠀⠀⠀⢀⣠⠟⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⢛⡻⣟⡛⢻⣟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⡀⠀⠀⠀⣀
⠀⠐⠛⠛⢻⣿⠁⠀⢷⣤⡿⠀⢸⣧⣼⡧⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠉⠐⠒⠤⢤⣀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⢾⡷⢾⣿⣿⣿⣿⡷
⠀⠀⠙⢿⣿⣿⡆⠀⠀⣿⠀⠀⢻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⠉⠉⠍⠉⠿⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣴⣶⣶⣶⣄
⠀⠀⠀⣾⣿⣿⡇⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⣾⠻⣦⡀⠀⠀⠀⠘⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠙⠋⠛⠿⠿⠿⠋
⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⣰⣿⣷⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠓
⠀⠀⠀⣿⡿⠿⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⢸⣿⡿⢿⣿⡏⠀⠀⣰⢦⠀⠀⠀⠀⣰⢶⡄⠀⣀⣀⣀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠈⣿⣿⣿⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⣿⣾⡇⠘⠛⣿⡟⠛⠀⣿⣿⣿⣿⣿⣿⠀⠿⠻⢿⣿⡿⢦⡀
⠀⠀⠀⣿⡇⠀⠀⠀⢀⠿⠀⠀⠸⣿⡧⠀⠀⣹⣿⣿⣿⠀⠀⠐⣿⢿⡆⠀⠀⠀⣸⣿⡇⠀⠀⣿⡃⠀⠀⠛⠛⢻⡿⠛⠛⠀⠀⠀⢸⣿⠀⠀⠁
⠀⠀⠀⣿⡇⠀⠀⠀⠈⠒⠁⠀⠘⠅⠍⠀⠀⣿⠇⢹⣿⠀⠀⣼⠇⠘⣷⠀⠀⠀⣿⠹⣇⠀⠀⣿⡇⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⠀⢸⣿⠀⠀⣿⠀⠀⢹⡇⠀⢸⡇⠀⢻⠀⠀⣿⡇⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠓⠃⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠸⠈⠁⢸⡿⠀⠀⣿⠀⠀⣼⠀⠀⢸⡅⠀⢸⠁⠀⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠰⠗⠀⠀⠿⠀⠀⠹⠄⠀⠈⠇⠀⡟⠀⠀⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⠇
⠀⢠⣤⣤⣤⣤⣤⣄⠀⠀⢀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⢸⣿
⠀⣽⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⣤⣤⠀⠀⠀⠀⣤⣤⡄⠀⠀⡆⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⣿⣿⣿⣿⣿⣿⡇⠀⠠⣶⠀⠀⠀⠀⠀⠀⠀⣸⢧⠀⠀⠀⡿⢿⠀⠀⠀⠀⢸⣿⠇⠀⠀⡇⠀⠀⣿⠀⠀⣿⠀⠀⠀⡀⠀⠀⠀⠘⠛⠃
⠀⠈⠻⣿⣿⡿⠛⠁⠀⠀⣿⠄⠀⠀⠀⠀⠀⢀⠇⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⡿⠀⠀⣿⠀⠀⠀⡇⠀⠀⢸⠀⢀⠀⢀
⠀⠀⠀⠘⣿⠁⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⣰⣿⡄⠀⠀⡇⠀⠀⢱⡄⠀⣿⠀⠀⠀⡇⠀⠀⢸⠀⢹⡏⣿⠄
⠀⠀⠀⢸⣿⡆⠀⠀⠀⠸⣿⠃⠀⠀⠀⠤⢤⣤⣄⣀⡀⠀⢀⣠⠦⡤⠊⠀⠀⢸⣿⡇⠀⢸⣇⠀⠀⣸⡇⠀⣿⠀⠀⣠⣧⡀⠀⢸⠀⢸⡇⣿
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣽⡿⠿⠷⠿⢤⠲⠤⠤⡄⠀⢸⣿⠁⠀⢸⣿⠀⠀⣿⡇⠀⣿⠀⣼⣿⣿⣿⡆⢸⠀⢸⡇⣿
⠀⠀⠀⠈⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠁⠀⠘⠛⠀⠀⠘⠛⠀⠀⠛⠃⠀⠛⠀⠈⠻⠿⠛
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⢀⣀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠖⠓⣦⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⢀⣠⣤⡄⠀⠀⠀⠀⢀
⣬⣭⣽⣛⣛⣿⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢤⠃⡚⠔⠀⠀⣠⣿⣿⣿⣿⣿⡶⠀⠀⠀⠀⣤⢻⣼⣇⠀⠀⣰⠋⠙⠛⠀⠀⠀⠌
⠈⠛⠂⠈⠉⠉⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⠀⠀⠕⣰⠍⠀⠀⠀⠉⠉⠉⢽⡇⠀⠀⠀⠀⣤⣿⣹⡞⡧⠿⠀⢰⣿⡆⠀⠀⠀⠀⠀⠈⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠁⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⢰⣿⣿⣶⣄⡀⠀⠀⠀⠀⠘⢧⣿⣄⠀⠀⠀⠀⠤⠄⢀⣀⠀⠀⠀⠀⣤⣤⣤⣄⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢠⣼⡋
⠀⠀⢠⣿⣿⢿⡟⠛⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠀⠀⠨⣿⣿⣒⡎⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠈⠉⣍⠁⠁⠀⠀⠀⠀⠺⠏⠁
⠀⠀⠀⠁⠀⠘⠃⠀⠀⠀⠈⡄⣀⢀⣂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⣠⣶⢿⣷⣄⠁⠀⠀⠀⠀⠀⠀⠀⢶⡶⢿⣿⠆⠀⠀⠀⢴⡦⠀⠀⠀⠀⢀⢀⣀⠤⠄⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⣿⣿⣶⣿⡇⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠻⠃⠠⠀⠀⠀⠸⠙⠙⠑⡆⠀⠀⠀⡰⠒⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢶⣇⠀⠀⠀⠀⠀⠀⢀⡀⠘⠒⠐⠒⠃⠀⠀⢰⡿⠀⠀⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⡶⠀⠀⠀⠼⣿⣿⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⢺⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢀⡀⣤⠀⠀⠀⠙⠁⠀⠀⠀⠀⢴⣾⡆⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠐⣷⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣷⣿⣧⠀⠀⠀⠀⢐⣀⠀⠀⠀⠀⠀⣐⢩⠅⠀⠀⠐⠀⠂⠀⠀⠀⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠀⠀⠀⠵⣹⣿⣩⠇⠀⠀⠀⠁⠀⡀⡀⠀⠀⠀⠀⠐⠇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠸⣀⣰⠀⠀⠀⠀⠇⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⣷⣾⡇⠀⠀⠀⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠑⢄⠀⠀⠉⠉⠀⠀⠀⠀⠾⠶⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⡶⠀⠀⠀⠰⠶⠶⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣴⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣟⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⢷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠾⣽⣿⣿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣤⣤⣤⣤⡀⠀⠀⠀⠀⢀⣤⣤⣤⣤⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣧⣤⣤⣤⣤⣼⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣋⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣴⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣦⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢸⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇⣹⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣾⣿⡿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣉⡃⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣉⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣙⣋⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉

⠀⠀⢰⡂⣰⡆⠀⠀⠀⠀⠀⢠⣦⠀⠀⣤⠀⠀⠀⠀⠀⠸⠇⢀⠀⠀⠀⠀⠀⠀⡀⢘⣗⢀⠀⠀⠀⠀⠀⣀⠀⡀⢀⡀
⠀⠀⠹⣯⠹⣯⠁⠀⠀⠀⠀⣾⣧⣧⣯⣾⡆⠀⠀⠀⢰⣿⣿⣟⣧⠀⠀⠀⠀⢀⣆⠉⡉⢐⣆⠀⠀⠀⣠⣿⡟⠉⢻⣧⡀
⠀⠀⠐⠋⠃⠋⠃⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠛⠋⠀⠘⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠁⠉⠀⠀⠈⠁⠁

⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀
⠀⠀⢸⣿⡖⡿⠋⠀⠀⠀⠀⢰⡖⠿⠗⢶⠆⠀⠀⠀⣠⣟⢀⡀⣻⣄⠀⠀⠀⠀⠐⠀⡰⠰⡿⠃⠀⠀⠀⢀⡄⠙⣻⡆
⠀⠀⣴⣾⠞⢿⡷⠀⠀⠀⠐⠿⠻⠟⠿⠏⠿⠀⠀⠀⢿⡿⠀⠀⢿⡟⠀⠀⠀⣾⠗⠈⠈⠀⠀⠀⠀⠀⠀⢻⣍⣺⣿⡇
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋⠋⠁

⠀⠀⠀⠰⣄⡶⠀⠀⠀⠀⠀⣀⠀⠠⢀⢀⡀⠀⠀⠀⠀⣠⣄⣠⡀⣀⠀⠀⠀⣀⣀⠐⠲⢀⢀⠀⠀⠀⠀⠀⢠⣀⢀⡀
⠀⠀⣼⣯⣿⣭⣯⠀⠀⠀⠀⣽⡱⠐⠠⢼⣯⠀⠀⠠⣷⣯⣥⣽⣭⡿⠄⠀⠀⠘⣿⠊⠘⣺⠏⠀⠀⠀⢀⡄⣹⠿⣝⣛⡀
⠀⠀⠻⠇⠿⠛⠿⠁⠀⠀⠈⠋⠛⠋⠚⠛⠛⠀⠀⠀⠀⠛⠙⠙⠋⠀⠀⠀⠀⠐⠻⠃⠀⠻⠓⠀⠀⠀⠐⠓⠚⠂⠛⠛⠁

⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠘⣷⠃⠀⠀⠀⠀⠀⢶⡆⣰⣀⢴⡆⠀⠀⠀⠀⣤⢘⡃⣤⠀⠀⠀⠀⠀⣮⣥⣼⣵⠀⠀⠀⠀⢠⣶⠡⢌⠐⣄
⠀⠀⣸⣏⣟⣝⣇⡀⠀⠀⠰⣿⣿⠁⠉⣷⣷⠀⠀⠀⠘⠋⣽⣯⠙⠂⠀⠀⠀⠀⢻⣿⢸⡟⠀⠀⠀⠀⢸⡏⠭⣘⠐⣿
⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠘⠋⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⢀⠠⢀⡀⠀⠀⠀⠀⠀⢠⠀⡄⠀⢠⠀⠀⠀⠀⢀⡀⠀⠤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⢀⡀⠀⣀
⠀⠀⡄⠰⣟⠇⠑⠀⠀⠀⠀⢿⡿⡿⢿⢾⡇⠀⠀⠀⣴⣿⡋⠂⣾⣆⠀⠀⠀⠀⣠⣤⢻⡿⠃⠀⠀⠀⢀⡾⣅⣀⣹⣦
⠀⠀⠀⠂⠛⠠⠔⠀⠀⠀⠀⠈⠁⠿⠿⠿⠀⠀⠀⠀⠸⡿⠀⠀⢿⠇⠀⠀⠀⠀⠿⠟⠃⠀⠀⠀⠀⠀⠨⠇⠀⠀⠉⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⢿⡷⠀⠰⣿⠇⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⠙⠋⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⠋⠀⢀⣴⡄
⠀⠀⠀⠀⣠⣀⠀⠉⠁⠀⠀⣴⣦⠀⠀⠿⠿⠿⣿⣿⣷⣶⣤⣀⠀⠉⠁⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀
⠀⠀⠀⠀⠛⠋⠀⢀⣴⣦⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣷⣄⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠻⣿⡧⠰⣿⠟
⠀⠀⣶⡆⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄⠀⠠⣶⡆⠀⠀⠀⠀⠀⢀⣤⣄⢠⣤⣄
⠀⠀⠈⠀⠀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁⠈⠻⠟
⠸⡿⠆⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠾⡿
⢀⡀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣧⣤⣤⣀⣀⣀
⠿⠿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⡿⢿⣿⡿⢿⣿⣿⣿⣷⣦⣄
⠀⣀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⣫⣿⡿⠋⠀⢸⣿⡇⠀⠈⢻⣿⣯⠛⢿⣿⣦⡀
⠘⠿⠃⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠋⠀⢰⣿⡟⠁⠀⠀⢸⣿⡇⠀⠀⠀⠹⣿⣧⠀⠉⣻⣿⣦⡀
⠀⠀⣤⡄⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⢿⣿⣶⣿⣿⣀⣀⣀⣀⢸⣿⣇⢀⣀⣀⣀⣽⣿⣷⣿⡿⠿⣿⣷⡀
⠀⠀⠛⠃⠀⠀⠘⢿⣿⣦⡀⠀⠀⠀⠀⠀⣸⣿⠏⠀⠀⠀⣽⣿⠟⠿⠿⠿⠿⣿⣿⣿⠿⠿⠿⠿⠛⣿⣿⠁⠀⠀⠘⣿⣷
⠀⠀⠀⠀⣾⡷⠀⠀⠙⠻⣿⣷⣦⣤⣀⣠⣿⡟⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠉⠛⠻⠿⢿⣿⣇⣀⣀⣀⣠⣿⣏⣀⣀⣀⣀⣀⣸⣿⣇⣀⣀⣀⣀⣀⣸⣿⣇⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣴⣶⠀⠀⣀⣼⣿⡿⠿⠿⠿⢿⣿⡿⠿⠿⠿⠿⠿⢿⣿⡿⠿⠿⠿⠿⠿⢿⣿⡿⠿⠿⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠻⢿⣿⡇⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⠀⣿⣿⡀⠀⢀⣀⣀⣸⣿⣇⣀⣀⣀⠀⠀⣼⣿⠃⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⣀⣠⣼⣿⣿⣿⣿⠿⠿⢿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣤⣀⠀⣼⣿⠏
⠀⠀⠀⠀⢀⣤⣄⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠟⠛⢿⣿⡄⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⣾⣿⠛⠻⢿⣿⣿⠏
⠀⠀⠀⠀⠈⠻⠿⠀⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⡈⢿⣿⣄⠀⠀⢸⣿⡇⠀⠀⢀⣾⣿⠃⣠⣶⣿⠟⠁
⠀⠀⠀⠀⢀⣴⣶⠀⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣿⣿⣷⣄⣸⣿⣇⣠⣴⣿⣿⣷⣿⡿⠛⠁
⠀⠀⠀⠀⠈⠛⠁⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡤⠄⠀⠀⠀⠀⢠⣶⣤⣴⡄⠀⠀⠀⠀⠀⠀⢠⣶⣦⠄⠀⠀⠀⠀⠀⠀⡺⠄⢀⣀⠀⠀⠀⠀⠀⠀⣠⣴⠆⠄⡀
⠀⢀⣴⣯⣯⣍⣦⣆⠈⠀⠀⠀⠀⠘⠻⠿⢛⠁⠀⠀⠀⠀⠀⠀⠨⡀⢈⠁⠀⠀⠀⠀⠀⠸⢧⣄⣻⡿⠃⠀⠀⠀⠀⢰⣿⣱⣛⡆⢠⡄
⠀⠈⠻⠽⠛⠛⠯⠟⠀⠀⠀⠀⠀⠀⠀⢻⣾⠃⠀⠀⠀⠀⠀⢠⣶⣏⣽⣶⡀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡷⠀⠀⠀⠀⠀⠉⠄⢉⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⣠⡄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠒⢤⡀
⠀⠀⠀⠀⣏⣁⣀⡍⠀⠀⠀⠀⢐⣤⣀⠐⣠⣖⣇⠀⠀⠀⠀⠀⢠⣶⡁⡹⠀⠀⠀⠀⠀⠰⠛⠲⣿⠷⣾⡇⠀⠀⠀⠀⣿⣿⡿⣷⡄
⠀⠀⠀⠀⡻⣿⣿⡇⠀⠀⠀⠀⠸⣟⣻⠇⢿⣛⡿⠀⠀⠀⠀⢰⣿⡿⠛⠀⠀⠀⠀⠀⠀⢠⣬⡸⣷⣽⣾⡇⠀⠀⠀⠀⢰⡆⠑⢹⣧⡦
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⣩⣗⣦⣒⡉⢺⡀⠀⠀⠀⠈⢳⣿⠞⠑⢢⡀⠀⠀⠀⠀⣸⠞⣀⣀⠻⣇⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣁
⠀⠀⠿⢿⡟⢻⠗⠈⠀⠀⠀⠀⠀⣤⢿⡿⠿⢯⠇⠀⠀⠀⢰⣿⣧⣛⢛⣼⣿⡄⠀⠀⠀⠀⡀⠀⠀⣴⡿⠀⠀⠀⠀⠀⠂⣉⣠⣿⣿⡆
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠈⠈⠉⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠐⢦⡀⣤⢤⠀⠀⠀⠀⠀⣠⣬⣵⣤⣄⠀⠀⠀⠀⠀⠆⣀⠒⣿⣿⠆⠀⠀⠀⠀⠀⣴⢟⢛⡷⣆⠀⠀⠀⠀⠀⠀⢀⣠⣀⣾⠉
⠀⢰⠿⠳⣶⡿⠿⣿⣿⠀⠀⠀⠀⠘⠛⡻⡿⡛⠀⠀⠀⠀⠀⠐⠧⣀⣾⣷⡄⠀⠀⠀⠀⠸⣧⠆⣓⢖⣿⠀⠀⠀⠀⠠⣾⣿⠯⢸⣛⣿
⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠀⠀⠀⠀⠀⠙⠛⠚⠛⠃⠀⠀⠀⠀⠐⠛⠚⠁⠀⠛⠁


⠀⠀⢠⣤⣁⣈⣥⣤⠀⠀⠀⠀⢀⣤⣲⣖⣢⣄⡀⠀⠀⠀⢀⡶⣟⢻⢶⡄⠀⠀⠀⠀⠀⠠⡄⠠⣿⣷⠀⠀⠀⠀⠀⠀⣠⣭⣄⢀⣠
⠀⠀⢸⣛⡟⣻⣟⢳⠀⠀⠀⠀⠸⢟⡿⠿⠿⣛⠟⠀⠀⠀⢸⣷⢂⡲⣶⣶⠶⠀⠀⠀⠀⢸⡷⣶⣿⣧⣄⠀⠀⠀⠀⠀⢩⣽⣷⣯⣤⡄
⠀⠀⠘⠻⠓⠋⠉⠚⠀⠀⠀⠀⠀⠀⠢⠤⠔⠂⠀⠀⠀⠀⠀⠙⠛⠓⠿⠃⠀⠀⠀⠀⠀⠀⠁⠸⠿⠿⠿⠀⠀⠀⠀⠀⠘⢿⣩⠟⠛⠃

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠘⠀⠀⢠⡄⠀⠈⠀⠀⠀⠀⠰⠛⠁⠀⢀⡀⠀⢰⠉⠙⢿⡧⠀⠀⠄⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠃⠀⢸⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠳⣄⠬⠀⠀⠂⠠⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡀⠠⢀⠀⠀⠀⠀⠀⠠⠂⠀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠐⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠠⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡧⠁⠀⠀⠀⠀⢀⠀⠀⠀⠘⠀⠐⠀⠀⠁⠈⠉⢂⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠀⠈⠀⠀⠀⠀⠠⠀⠊⣀⠀⢠⣶⡄⠀⠀⠀⠀⢀⡠⡀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⢄⠀⠀⠀⠻⠆⠀⠀⠀⠐⠉⠀⠁⠀⠀⠀⠂
⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡄⠀⠀⢰⠦⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠂⠄⠂⠀⢀⠀⠄⠠⠄⣀⠀⠀⠀⠈⠀⢀⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠊
⠀⠀⠀⢀⡠⠚⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠈⠁⢀⡀
⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠠⠟⠀⠀⠀⠀⠐
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⢠⠀⢀⡟⠁⠀⢠⡀⠀⠀⠀⠀⠀⡀⠘
⠀⢻⣀⣠⣄⠀⠤⠆⠀⠀⠀⠎⠀⠀⠀⣐⡤⠌⠀⠀⡇⠀⠈⠀⠀⠂⠀⠐⠄⠀⠀⠐⠀⠁⠀⠀⠀⠀⠄
⠀⠀⠀⠀⢎⠀⠀⠀⠀⡠⠂⠀⠀⠐⠀⠁⠀⠀⠀⢰⠁⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠴⣄⣄⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⢤⡄
⠀⠀⠠⠀⠠⡀⠀⠉⠀⠀⠀⠀⣰⡟⠁⠀⣠⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣼⠏⠀⠴⢞⣲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠘⠄⣀⡨⠃⠀⠰⠊⠲⢤⠀⠀⠀⠐⠀⢠⡄⠀⢀⠀⠀⠈⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠁⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠶⠶⠦⠤⠶⠶⠶⠦⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢀⣠⣤⠤⠤⠶⠶⠶⠶⠶⠤⢤⣄⣀⠈⠙⢦⡀
⢀⡤⠀⢤⣄⠀⠀⣸⢁⡾⠋⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣉⠳⣄⠈⢷⠀⠀⢀⣤⣤⣄
⢨⡀⠀⠀⠈⠷⣰⡇⣜⡶⠋⠉⠉⠉⠙⠆⠀⠀⣴⠋⣉⣉⣭⣍⡳⢸⡇⠘⡦⠔⠋⠀⠀⢘⡇
⠀⠙⠢⠤⢖⣢⡼⣥⡿⠦⠤⠤⠒⠒⠲⠾⠶⠶⠿⠾⠿⠭⠬⢭⡿⠷⢿⣤⣷⣾⣭⣶⣒⠋
⠀⢀⡴⠚⠈⠁⠀⠀⠐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠂⠀⠀⠀⠀⠉⠉⠻⣦
⠀⡾⠀⠀⠐⠤⠉⠡⠖⠄⠀⠀⢀⡀⠂⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢀⠄⠒⠂⠒⠀⢄⠀⠈⡇
⠀⣧⠀⣆⠀⠀⠀⠀⠈⠀⡄⠀⡠⠁⢠⠉⢱⠀⠒⠒⠲⠠⡆⢄⢢⠀⡀⠂⠉⠀⠈⠁⢂⠓⠀⡇
⠀⢿⡀⢃⡀⠀⠀⠀⢄⠆⠀⠢⠀⠃⠘⣀⡘⠀⠆⠤⠄⠐⠆⢚⡌⠠⡇⠀⠀⠀⠀⠀⡘⠄⢠⠃
⠀⠈⢷⡀⠀⠉⠉⠁⠈⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠓⢀⠤⢄⠐⠀⠀⠸
⠀⠀⠈⢣⡀⡤⠴⠖⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠛⠉⠓⠒⠶⠂⠀⢤⡤⣀⣀⡇
⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡐⠒⠀⠀⢀⣀⣀⣀⡒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⡄
⠀⠀⠻⣄⡀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣈⣀⣀⣀⣀⣀⣀⣀⣀⣈⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠉⣟⠛⠛⠙⠉⠉⠉⢹⠁⠉⠉⢹⠉⠉⢹⡏⡏⠉⠉⠉⢹⡟⠛⠛⠛⠓⠶⢶⡖⠁
⠀⠀⠀⠀⠀⠈⠓⠤⣤⣤⡤⡤⠞⠁⠀⠀⢸⠀⠀⢸⡇⡇⠀⠀⠀⠘⢧⣀⣁⠁⠐⠘⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠀⣀⡠⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠃⠐⠶⠄⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⡤⠶⠶⠄⡙⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠁⠀⠀⠀⠀⠈⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⢄⠐⠂⢀⡤⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡘⠓⠀⠛⢁⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⢆⣀⠴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠈⠓⠀⠙⠦⠈⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⣠⠿⡀⠀⠀⠀⠀⠀⡀⠠⢤⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⢸⠁⢀⡸⢶⠀⠀⠀⠀⠀⠀⠸⠷⢮⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⢦⠀⠱⡈⠀⠀⠀⠀⠀⠀⠰⢀⡜⠁⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠳⣄⠻⣆⠀⠀⠀⢀⠀⠜⠉⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⢀⣨⣂⡏⠡⠾⠀⠜⠁⣂⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠘⣥⡀⣙⡇⠀⠀⢀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠇⠀⠀⠀⠀⠀⠀⠀⢬⡁⠘⡁⠀⠀⢩⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⡏⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠷⠶⡿⡷⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢇⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠳⠶⠶⠶⠶⠾⠶⠶⠷⠀⠀⠀⠀⠀⠀⠀⢀⣁⠀⠀⣸⡀⠀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠊⠁⠈⢈⣍⠁⠈⠀⠙⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠒⠒⠈⠀⠈⠉⠒⠲⠒⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠸⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿