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⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⣀⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣼⣭⣵⣾⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣸⣾⣭⣿⣟⣫⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠸⣿⣉⣉⣉⣩⣭⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣷⣾⣧⡄
⠀⠀⠀⣠⣾⠿⢟⠛⡟⠿⣝⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡛⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠽⢗⣒⣽⣆
⠀⠀⣴⡟⠁⠀⢸⢀⣇⡀⠈⢷⡇⠀⠀⠀⠀⠀⠀⠀⠄⠐⣵⡾⠿⣷⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠹⢯⠥⢶⠒⠡⠤⡀
⠀⠀⣿⠀⠀⢠⢻⢿⡇⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⣸⠀⢸⡀⠀⠀⠘⡄⠀⠉
⠀⠀⣿⠀⠀⠣⠚⠎⠑⠃⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠘⣦⡀⠀⠄⠀⠀⠀⠀⠀⠀⠃⠀⡘⣾⣫⡏⢏⡆⠀⠀⢸⠀⠀⢀⡀
⠀⠀⠘⣷⣄⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠄⠀⢀⡿⠀⣠⡤⡜⠒⡈⠻⣦⠀⠀⠀⠀⠐⠂⠀⡆⠀⠈⠷⠙⠃⠈⠁⠀⠀⡜
⠀⠀⠀⠀⠉⠉⠛⠛⠉⠁⠀⡀⠑⠀⠀⠀⠀⠀⠀⢸⡇⠀⠁⠃⠃⠀⠀⠠⢸⠀⠀⠀⠀⠀⠀⡀⠑⣄⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠐⠒
⠀⠈⠀⠀⠄⠀⢀⠀⠀⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⡱⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠒⠒⠒⠈⠁⠀⠀⠐⠤
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣀⡀⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠠⡐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⣇⠀⠀⠀⠀⣠⢇⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠇⠀⢠⠖⢉⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡇⠀⠀⡏⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠃⢠⡀⣇⢸⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣼⢷⡈⠘⡄⠀⠀⡤⢀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⢗⡩⢛⣆⠀⠀⠀⢰⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⢠⣛⠅⣊⠥⢚⠳⡀⠀⠸⠸⡀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⣰⡫⢔⡊⠁⠈⠰⢂⡹⢆⠀⢱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣼⠥⣚⡅⠀⡿⣄⠈⠩⠔⣊⣧⠈⠀⠀⠤⠄
⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢀⡾⢔⡫⠕⠀⢰⠃⠘⣆⠐⢋⡡⠚⣆
⣟⠷⣬⣍⡛⠓⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣒⠭⡒⠉⣠⠃⠀⠀⠘⣆⠠⠖⣋⢽⠀⠀⠀⠒
⠳⢯⣖⡥⣋⠝⣲⢦⣤⣉⡙⠒⠦⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⢸⠫⢒⡉⠀⡴⠁⠀⠀⠀⠀⢸⡄⣋⡤⢾
⠀⠀⠀⠈⠛⠻⠶⣭⣞⠕⣉⠝⡲⢦⣄⣉⡑⠒⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⣚⠁⠀⡜⢀⠀⠀⠀⠀⠀⢸⠃⠡⢒⡿⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠾⣴⣫⢖⡡⢛⡵⣲⠦⣄⣉⠓⠲⠤⣀⡀⠀⠀⠀⠉⠁⠀⣯⠴⠂⢸⠀⠢⠀⠀⠀⠀⢠⠏⠠⢚⣭⠇
⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠷⢮⣥⢊⡥⣋⠝⣲⠦⣄⣉⠙⠲⠤⣄⣀⠀⢹⡫⠀⢺⠀⠩⠄⠀⠀⢠⠇⠰⢚⣥⡞⠀⠀⠀⠐⠂
⠀⠀⠀⠈⠱⠒⡭⠴⣂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⢮⣴⣊⠴⢋⡽⣓⠦⢤⣉⠙⣲⠦⠤⢼⡄⠉⠄⠀⠀⢸⠀⠨⢵⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠭⢒⡣⠴⣂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⢾⣥⡖⡡⣊⡿⢕⢴⢦⡀⡀⠀⠀⠀⠀⠈⠀⢰⠏⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⢚⡭⠖⣂⡤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⢧⡱⠣⢪⣾⢛⢔⢄⡀⠀⠑⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⠥⠒⣊⡥⠄⣀⠄⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⢭⣉⣋⣘⣐⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠴⢚⡥⠔⣀⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠽⢒⡮⢗⡢⢔⡠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠥⠒⣫⠵⣚⠤⣂⡠⢆⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠟⠛⠛⠛⠛⠿⠶⢶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣿⣿⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⣿⣍⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣩⣷⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠁⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣀⣿⣅⣀⣀⣀⣀⣀⣀⣀⣀⣸⣯⣀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⡾⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⢶⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡇⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⠟⢻⣿⠟⠟⠛⢿⡿⠛⠛⠛⠛⠛⠛⠓⠛⠛⠛⠛⠛⠛⠛⢿⡿⠛⠛⠻⣿⠟⠻⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⣼⠇⠀⠀⠀⢸⡇⢰⣿⠀⢸⣷⠀⣾⣧⠀⣾⡇⠀⣿⡆⢸⡇⠀⠀⠀⠸⣇⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⣼⠏⠀⠀⠀⠀⢸⡇⠸⡿⠀⠸⡟⠀⠻⠏⠀⠿⠇⠀⠿⠇⢸⡇⠀⠀⠀⠀⠹⣇⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⣴⠟⠀⠀⢀⣠⡶⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠷⢦⣄⠀⠀⠀⢻⣆⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⣰⡏⠀⠀⢠⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⢻⣆⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⣰⡟⠀⠀⠀⢿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡿⠀⠀⠀⢻⣆⠀⢻⣆
⠀⠀⠀⠀⠀⠀⢠⡟⠀⢠⡟⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⢿⡄⠀⢿⡄
⠀⠀⠀⠀⠀⢀⣾⠁⢠⡿⠁⠀⠀⢀⣾⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠘⣿⡀⠘⣷⡀
⠀⠀⠀⠀⠀⣼⠇⠀⣼⠃⠀⠀⠀⣾⠃⠈⠙⠛⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠸⣧⠀⠸⣇
⠀⠀⠀⠀⢠⡿⠀⢰⡟⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡶⠶⠶⠾⢿⣇⠀⠀⠀⢿⡆⠀⣿⡀
⠀⠀⠀⠀⣾⡇⠀⣿⠁⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⡀⠀⠀⢀⣠⣴⠶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⣿⠀⢸⣧
⠀⠀⠀⢀⣿⠀⢰⡿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣶⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⢿⡄⠀⣿
⠀⠀⠀⢸⡏⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⢿⡆
⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⣿⡄⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⢸⡇⠀⣸⡇
⠀⠀⠀⠈⣿⠀⠸⣧⠀⠀⠀⠀⠘⣷⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⣾⠃⠀⣿
⠀⠀⠀⠀⢿⡆⠀⣿⠀⠀⠀⠀⠀⢻⣶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⣰⡏⠀⠀⠀⠀⢠⣿⠀⢸⡟
⠀⠀⠀⠀⠈⣿⠀⠘⣷⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⢰⡟⠀⠀⠀⠀⠀⡼⠁⠀⡿⠁
⠀⠀⠀⠀⠀⠹⣆⠀⠹⣧⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⡟⠀⠀⠀⠀⢀⣼⠃⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠹⣧⠀⠙⢷⣄⠀⠀⠀⣠⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣄⠀⠀⠀⣠⡾⠃⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠙⠷⣦⣤⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣤⣴⠾⠋⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣄⡀⠀⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠀⠀⢀⣠⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠶⠶⠶⠶⠶⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠶⠶⠶⠶⠶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣀⣤⣼⣿⣦⣶⣶⣤⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴⣤⣴⣶⣤⣤⣶⣦⣦⣴⣿⣦⣤⣄
⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⣿⣇⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣸⣷
⠀⠀⠀⠀⣿⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢈⣿
⠀⠀⠀⢀⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⡀
⠀⣴⠾⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⢷⣄
⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠛⠉⠟⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣠⣤⣤⣤⣤⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⡽⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⣡⡾⢗⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢗⡽⣋⣴⢟⡼⣛⣿⢿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠻⠟⢃⡿⣹⣿⣿⣿⢾⣯⣵⣿⣟⣻⢟⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠠⠋⠀⠉⠉⠉⠰⡟⠩⣠⢟⣽⡫⢞⠽⣻⢟⡿⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠎⠀⡰⠿⡾⢷⡾⢿⣿⣶⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⠖⠩⣶⡽⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣫⣶⢄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⢿⣿⣯⢂⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢿⣯⢿⣿⣷⣻⣿⡿⣣⠄⠀⠀⠀⠀⠀⠀⠠⣺⢟⣻⡿⢛⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠷⠾⠿⡷⢿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠔⠁⠊⠐⠽⣋⣴⣡⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡿⣀⣨⢅⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣠⣶⣿⣿⡾⣷⡴⢖⡠⠀⠀⠀⠀⢠⠿⣿⣿⣿⣿⣽⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣽⣿⣽⣿⣿⣷⠁⠀⠀⠀⠀⢐⣵⣾⣿⣽⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠛⠛⠛⠛⠛⠉⠙⢻⡟⠁⠀⠀⠀⠀⠀⠀⢉⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⢀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢁⣠⣾⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢡⠋⡁⠭⠤⠬⠭⠭⠭⠭⠭⠭⠭⠭⡭⢈⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡇⢀⡀⣀⡀⠀⣀⡀⠤⠠⠄⠀⠠⠄⠤⢀⡀⢀
⠀⠀⠀⢀⠄⠒⡌⠤⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⠉⡉⠬⠥⠶⢤⡀
⠀⠀⠀⡆⠀⡜⡐⢉⡸⢒⠄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⡠⠒⡁⠄⠒⠒⠒⢤⡈⠲⡀
⠀⠀⠀⠇⠀⢢⡕⠁⣠⠊⠀⡅⢀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⠊⡠⢊⠀⠀⡔⠉⠑⡄⠙⣆⠘⡄
⠀⠀⠀⢰⠀⠸⠤⠊⢹⠀⠀⡇⠘⣦⠀⠀⠀⠀⠀⠀⠰⠀⢠⢃⢮⠊⠀⠑⢈⠀⠀⢀⠇⠀⢘⡆⢠
⠀⠀⠀⠀⠑⢄⡀⠀⡜⠀⠀⡇⠀⢸⡀⠀⠀⠀⣔⠠⢅⣀⠃⠎⢨⠀⠀⣠⣁⣑⠐⡡⠊⠁⢮⠁⠈⣤⡀
⠀⠀⠀⠀⠀⠀⠈⠢⡇⠣⠴⠅⠃⢸⠁⠀⠀⢰⠀⠉⠑⠊⠭⡒⡄⢁⠈⠏⠀⠀⣧⡁⠀⠀⣼⠈⢸⠇⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⢀⠟⠀⠀⠀⠈⠀⠀⠀⠀⠀⢱⠀⠀⠱⣄⠀⣠⡃⠱⡀⠊⡘⡄⣸⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢖⠉⠀⠀⠀⠀⠰⣀⣀⠀⣀⣠⣼⠀⣠⡧⠤⡉⡌⠀⠀⠱⠠⡱⢠⠓⠤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠈⡄⢂⠸⡀⠁⠎⠀⠀⠸⢆⠀⣀⡞⡱⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠤⢀⡘⢌⢆⠑⢄⠣⣀⡠⠃⢀⠠⡪⠊⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠢⣀⠑⠒⠒⣚⡃⣁⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢼⣀⡀⣀⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣆⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠿⠯⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣶⣷⣴⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⡿⠿⠛⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⣥⣵⡾⠙⣙⣋⡉⠱⠆⣠⡑⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣷⣿⣿⣳⣿⢸⢿⣿⠟⢰⣦⢹⣿⣦⡙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢿⣿⣿⢳⣿⢿⣿⣦⣦⣾⣇⣿⡇⠻⣿⣷⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡾⣿⣭⣭⡘⢫⣼⠛⢹⡿⠙⢃⡌⠃⣀⡭⠵⠂⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣯⣽⡉⠀⣀⡒⠒⠐⢒⡠⠀⠉⣀⣠⡴⠖⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣯⣿⡛⢻⣛⠛⣿⣯⠛⢛⡚⠉⢩⣜⣿⡤⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿⡟⢨⣯⣿⣿⣿⣿⣯⣧⠈⠀⣾⣿⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⣼⣿⣿⡿⣿⣿⣿⣿⢀⠀⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⡧⢿⣿⣿⡟⢺⠿⣫⢿⢸⠀⡏⡉⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡿⢿⣿⣿⣷⠈⠉⠀⠀⠁⠁⢸⢸⠀⡇⠀⠏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⠁⢀⣿⣽⢿⠠⠤⠀⠀⠤⠔⢸⢸⢠⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡎⢹⡿⡟⠈⠀⠀⠀⠀⠀⠀⠸⢸⠀⠏⢁⣆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣭⡺⢟⣷⢀⣤⢢⡀⠀⣠⣢⡌⢼⠀⠒⣈⠴⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣿⣿⣯⣽⣿⢿⢈⣛⣀⣚⣛⣂⣙⠋⠨⠖⣋⣥⠖⢻⣆⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⣿⣿⣿⣿⡟⣶⠖⣒⣚⣳⣲⣶⣒⡒⠒⠊⣵⣾⣿⣿⣿⢿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⠷⠾⠿⠟⠛⠛⠛⣟⢿⣶⡾⢟⡍⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠰⠿⠋⣁⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⠀⢠⠟⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⣠⣤⡴⢏⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠛⠉⠛⢻⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣴⣦⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣶⣦⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣿⣶⣆⡾⣿⣷⣿⡟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢿⣉⣿⠀⣿⣎⣽⢇⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣀⣚⣉⠛⠛⠛⠛⠁⢘⣯⢍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣉⡾⠁⠀⠀⠀⠀⠀⠀⠙⢦⣙⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣏⡙⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣂⣤⡝⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⢦⣶⠶⠶⢶⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣲⣞⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠛⣿⡇⠀⢀⣄⢓⣎⣉⣉⣉⣹⣏⣙⣋⡉⢉⠀⣠⣼⡟⠃⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣀⣾⡇⠀⣈⢹⡋⢀⣿⣇⣽⣳⠬⡉⢛⣈⡽⢣⡞⢳⡖⣾⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠉⢻⠅⠀⠘⠾⠿⠯⣠⡿⣿⣿⠶⣶⠿⠹⣶⣎⣙⠭⠷⠞⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣤⣄⠀⣀⡀⠀⠀⠀⠀⠀⠂⡛⠲⠇⠀⠀⠀⠀⢠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣿⣿⣷⣉⠀⡄⠀⠀⠀⠀⠈⠁⢸⡇⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣷⣌⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣿⣿⡇⣉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠇⠛⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡆⠀⠀⠀⢸⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⡇⠀⠀⠀⢸⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣿⡿⣤⠚⢣⠀⠀⠀⠀⠀⠀⠀⢿⡇⡇⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⣯⣀⢁⡉⢠⡀⠀⠀⠀⠀⠀⠀⣿⡇⡇⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢸⣿⢿⡿⠐⢻⡄⠀⠀⠀⠀⠀⠀⣿⡇⡅⠀⠀⠀⠘⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⣄⢸⡟⠚⠁⠙⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⡀⠤⢁⣚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠿⣍⢿⡇⠀⣀⠀⣀⠀⠀⠀⠛⢛⠛⣀⣛⠃⡴⠈⣹⠶⣆⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡍⢻⣾⣻⡇⠀⠙⠶⣿⢛⣀⢉⣶⢠⠀⠈⢁⡈⢑⡞⠛⣾⣿⡾⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣆⠙⢿⡇⠀⠀⠀⠘⠂⠀⠉⠙⠛⠶⠶⠋⠀⣉⡵⠚⠚⠉⡀⢀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠾⠓⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⠆⠀⠉⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢠⠀⣠⡴⣷⣠⡀⠀⠀⣲⣀⡀⠀⠀⢀⣤⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⠋⠁⠀⠀⠀⠀⠈⠉⠉⠉⠁⠊⣄⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠤⠶⠖⠒⢛⣛⣛⡋⣉⠉⢳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡏⣡⢶⣤⣶⡄⢾⣽⠃⢹⡇⠸⢷⣿⠀⣾⠋⠉⠉⠛⠳⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⢸⡇⣿⠟⠹⠟⣿⠘⠛⠦⠈⠁⠲⠾⠋⠀⣿⡀⣀⣀⡀⠛⢭⡉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠿⢶⡄⠀⠈⣷⢈⣁⣀⣤⣤⠴⠶⠚⠛⠛⠛⠛⠛⠙⠛⠉⠉⠉⠉⠉⠛⠛⠂⠀⣀⡴⠂⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⡄⢼⠀⣃⠀⢠⣟⣠⠶⠛⠉⠋⠉⠀⠀⠀⠀⠀⠀⢀⣠⡆⠀⠀⠀⢰⣦⣤⡤⢤⣤⣤⢴⣾⣋⢿⠀⠀⠀⣷⡂
⠀⠀⠀⠀⠀⠀⠀⡆⣿⠀⡇⠀⠘⠋⠁⠀⣄⣀⣀⣀⣀⣠⢤⣤⣶⠋⢻⡐⣷⠀⠀⠀⢸⠟⠁⢿⡾⠁⠻⡾⠀⠙⠿⠀⣼⠄⣿⡇⢀⣀⡀
⠀⠀⣠⠤⠔⠒⠒⠿⠿⠖⠷⠤⢤⡀⠀⠀⣾⢼⠃⣿⣀⡿⢸⣿⣼⡇⠈⠿⠉⠁⠀⠀⠀⠀⣶⠈⢁⡄⡀⠀⢀⣤⡄⠀⣼⡎⣿⠇⠘⠿⠃
⠀⠀⣿⠀⠁⠒⣒⠒⠒⠂⠈⣁⡀⠹⡆⠀⠋⠛⠀⠛⠉⠃⠈⠃⠀⠡⣄⠀⣶⠀⣴⠀⠀⡾⢿⠀⠀⣿⡇⠀⠈⡿⢧⡀⢸⡷⠀⢠⡄⠀⠀⣶⡆
⠀⠀⣿⣼⢻⣾⡏⡷⣼⢻⣷⡟⣷⢰⡇⠀⠀⠀⢠⣄⣴⣆⣀⠀⠀⠀⢻⠛⣬⠋⣸⠀⡼⣒⢽⡆⢸⡧⢹⡄⢠⣟⣛⣷⡀⠀⣄⣻⣿⣽⠁
⠀⠀⢸⣧⠈⠉⠛⣷⡿⠚⠋⣀⠿⣾⣁⠀⣀⣠⡤⣿⣿⠗⠯⢤⡀⢀⣬⡀⢈⣭⡄⠀⠉⠉⠉⠀⠻⠷⡴⠷⠈⠛⠋⠉⠀⠀⠀⠯⠟⠊
⠀⢰⢚⡛⠦⠿⠤⠤⠾⠦⠤⠿⠚⠉⠈⢳⠹⣤⠤⢾⣿⠶⠒⢾⠃⠀⣇⣿⣾⣬⣆⠀⠰⡎⣕⠢⢲⣒⢠⡄⢀⡤⢤⣄⢠⡖⠓⢢⠀⠀⣿⡶
⠀⢸⣀⣀⡀⣠⢦⢈⡿⣆⢀⣤⡄⢠⣤⢸⡇⣷⠀⢸⣿⡀⠀⡟⠀⢻⡿⠿⡿⠿⣿⠀⠀⠑⠞⠓⠲⠿⠛⠀⢸⡁⠀⠙⠏⠀⠀⣸⢁⣄⠈
⠀⠈⣿⠉⣷⡟⢸⣿⡇⣿⣼⠃⢧⣿⠈⣿⠃⢹⡀⢸⣿⣇⣰⠇⠀⢸⡷⢀⢀⡻⡇⠀⠃⠀⢀⣴⠖⠲⣤⠀⠈⠳⢤⡀⠀⠀⣰⠏⠘⣿⣿⠋
⠀⠀⣷⣀⡌⡑⠌⠻⢀⢊⣑⣠⢬⡗⢖⣿⠀⠀⡡⠐⣄⠀⣀⣀⠀⠀⠳⠶⠿⠶⠃⠀⢀⣀⣸⠇⠀⠀⢹⣇⡤⢤⡀⠈⠻⣿⠃⠀⠀⠉⣁⡀
⠀⠀⠙⠮⣟⣙⣋⣛⣋⣛⣭⠥⠖⠚⠋⠁⠀⠀⠣⣴⣿⣾⣥⡼⠃⢀⣄⠀⡀⠀⠀⢠⣟⠁⢿⠀⠀⠀⢸⡏⠀⢨⣷⠀⠰⣇⠀⠀⡴⠋⠁⠀⠙
⢀⡼⢦⣄⠀⠀⢀⣁⣀⣤⡄⠀⠀⠀⢗⣳⠀⣾⠙⢹⡏⢹⡏⠁⣳⣺⣿⣿⠿⢢⡾⠉⠈⢳⡼⣆⠀⢠⣿⡶⠛⠙⢷⡄⠀⠈⣷⢸⠃
⢉⣦⣌⣧⠸⣛⠿⠿⠟⠚⠃⢺⡳⢂⣞⣷⡀⢸⡂⢸⡇⢸⡃⠀⣽⢸⣿⢻⡇⢼⠁⠀⠀⢸⣿⠟⠻⣯⣿⠃⠀⠀⢸⡿⠀⢠⡏⠘⢇⡀⠀⣀⡴
⢈⡁⣀⠀⠀⠁⢀⣠⢦⡀⠠⣏⣷⠈⠙⡋⠀⠸⢧⣤⣧⣼⠥⠬⠗⠼⠿⠼⠃⠘⢦⣀⣠⣿⡏⠀⠀⣿⣿⡀⠀⢀⡼⠃⠈⠉⠀⢀⡞⠋⢉⡉⣀
⣸⣋⡻⠄⣀⡴⣉⣹⠛⠃⣀⡤⣤⣾⠁⣣⠀⡀⠀⠀⠀⠀⠀⣀⠴⣤⠀⠀⠀⠀⠀⠛⠃⢸⣇⠀⢰⡟⠀⠹⠟⢋⡤⠄⠠⢄⠀⢸⠀⢠⣸⣿⣿
⢀⡉⠀⠀⠱⢽⠋⠁⠶⣅⣻⠷⠜⠉⠀⢉⡾⢷⡀⠀⠀⡴⢏⣙⣴⣯⣷⢄⠀⠀⠀⠀⠂⢰⠈⣶⠏⢀⠀⡰⢰⠋⠀⠀⠀⠀⢧⠈⢧⠀⠘⠚⠁
⠈⠁⣴⡶⠶⠤⠤⠤⡤⣈⠁⠀⠀⠀⣔⡛⢒⣒⣛⣦⠈⣷⢦⣭⣤⢴⣶⡞⠀⢮⠀⠀⠈⠤⡂⠸⢀⠎⡐⠁⢹⡀⠀⠀⠀⢀⡾⠀⢸⢂⣰⣶⡀
⠀⠀⣿⡀⠟⠂⣀⣤⡾⢻⣧⣀⠀⡸⠇⢌⡡⠶⢤⠾⡄⢻⣻⣿⣸⣸⣿⠁⠀⣾⣗⠀⠀⠀⠙⣆⢂⠜⠀⠀⠈⣳⣤⣤⠴⠟⠁⠠⠋⠀⠙⠉
⠀⠀⠸⣧⣴⡟⠏⢉⣤⠾⣣⠏⠀⢹⡶⡶⢲⢲⣺⢹⠇⠀⠉⠉⠉⠉⢀⠎⠁⠁⠈⠉⠢⢄⠀⢸⡃⠀⠀⠀⡴⠋⠁⡀⠀⢰⣷⠀⠀⣿⡶⠂
⠀⠀⠀⠀⣻⠶⠶⣋⡵⠚⠁⣦⣤⠀⠳⠯⠿⠿⠷⠞⢀⡄⣴⣿⣄⡀⠹⣄⣀⣀⡀⣀⣠⡞⠀⣾⠇⠀⠀⠀⢇⠀⢾⡿⡇⢠⣠⢶⡀⠈⠁
⠀⠀⠀⠀⠷⠖⠋⠁⠀⠀⠒⣟⠯⠂⠀⠀⠀⠀⠀⠀⠀⣼⣻⣟⡞⠁⠀⠈⡇⣏⣯⣇⣿⡇⠀⠃⠐⠀⠀⠀⣸⠂⠈⠀⠀⣼⣦⣸⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣷⣿⣟⣛⣉⣁⣀⣀⣀⣀⣀⣞⣁⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀
⣿⣿⡛⣻⢛⡉⡛⣻⠛⣿⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠶⠚⠛⠛⠛⠛⠛⠛⠓⠶⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄
⠀⠀⠀⠀⠀⠀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⣤⣀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠹⣦
⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠘⣧
⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠸⣇
⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⢻
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠘
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠦⡤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⢤⣤⣤⠴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣳⣶⣶⣶⣶⣞⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣾⣿⡴⠆⣀⣤⣶⣶⣾⠾⢿⣗⠂⢠⣤⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠱⡿⠋⠀⠚⣿⣛⠃⢍⡀⢐⣦⣩⠿⠙⢄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢅⠠⠞⠚⠛⠈⠰⢀⡀⠀⠸⣿⣷⠖⠛⠇⠀⠀⡔⠛⠶⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠁⠀⠀⠀⠀⢀⡤⠞⠉⠙⠢⠸⡁⠀⠀⣀⠰⠀⠀⠀⠈⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⢠⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠞⠋⠳⡄⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠘⠀⠀⢀⠇⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⣤⣼⣧⣤⣼⣤⣄⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣸⣧⡝⠛⣯⣝⢁⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⡔⠀⢀⣿⣤⣬⣿⣟⡈⠈⠱⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⣛⡟⣛⣿⣿⡿⣧⣈⡿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠏⠉⠉⠙⠋⠉⠁⠛⠉⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣺⠀⢀⠀⠂⠀⠀⠀⢠⠀⠀⠘⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣹⢀⠊⠀⠀⠀⠀⠐⠀⢱⡤⠊⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿⠆⠀⠀⠁⠀⠃⠀⠊⠁⠈⠄⠠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⢈⣻⠀⣀⠀⠀⠀⠀⠀⠠⠤⠀⠰⠀⢽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢸⡽⡄⠀⠀⢀⠁⡀⠀⠀⠈⠀⠄⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⣠⠀⠈⣦⣄⣻⣛⣻⣃⣠⡞⠁⠀⠀⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢿⡀⢐⣿⣿⣿⣏⡉⠉⠠⣬⠄⠀⢄⡀⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠆⠉⠉⠉⠛⠋⠉⣤⣤⣠⡶⠀⠘⢿⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣤⣄⣀⣀⣀⣀⣈⣛⣋⣀⣤⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢘⣿⣿⣿⡏⠁⠀⠀⠉⢻⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⠀⠀⠀⠀⢸⣿⣿⠟⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠙⠛⠛⠛⠓⠂⠀⠀⠘⠋⢁⣠⡴⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⣶⣶⣶⣶⣶⣶⣶⡶⢶⣻⣿⠿⠊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠉⠈⠉

⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠶⠶⠀⠀⠀⠀⠀⠀⠸⠿⠧⠀⠀⠀⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠄⢠⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⣠⣴⣶⣶⣄⠀⠀⠀⡀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠶⣀⢢⡈⠀⢀⢀⣀⣀⡈⢡⣼⢧⣀⡹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠈
⠀⣹⣿⣿⣿⠇⣤⣶⣿⣗⢻⣿⠀⠴⠄⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣴⣤⣴⣶⣷⡄⠀⠉⠉⠁⢸⣿⣿⣿⣿⣿⣿⣡⣄⠀⠀⠆
⠀⣿⣿⢿⡟⠀⣙⣣⣿⣿⡆⠃⠀⠀⠀⠀⢰⣿⣿⡟⣌⣥⠀⠀⠀⠀⢿⣿⣿⣿⣟⡁⡆⣼⠀⣤⠈⠛⠿⠃⢰⣧⡹⢿⣿⣇
⢾⣯⣿⣾⣧⣼⣿⣿⣽⡟⠁⣀⣿⠄⠀⠀⠀⠸⠟⠃⣿⣿⠀⢸⣷⡄⢼⠻⠛⢿⣿⢿⡆⠀⠀⠉⠀⠀⠀⠀⠀⡿⠟⠈⢻⣿⡇⢸⡿⣷⣦
⢰⣯⢁⠙⣿⣿⣿⠋⠀⠀⠀⠉⠉⢀⡴⠟⠀⠛⠂⠀⠈⣿⠀⠀⠉⠀⠀⠀⠀⢸⠁⠸⠇⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⣀⡤⠀⠀⠷
⣼⠟⢘⣧⠈⠁⠁⠀⢀⡘⡀⠀⠀⢠⡇⢾⠆⢀⡄⠀⢾⣿⡇⠈⠀⠀⠀⠀⠀⠈⠀⢀⣀⠀⡤⠀⠀⠀⠀⠀⠈⠀⠀⣿⣆⠙⣧⠀⢠⣤⣴
⠀⠀⠛⠛⡀⠀⢠⣾⣿⣿⣧⣴⠇⣾⠀⠸⠇⢠⣿⠃⣾⡟⢡⠀⠀⠀⠀⠀⢀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢻⠆⠀⡔⣿⣿⣿
⢀⣀⠀⢘⣻⣷⣾⡿⠛⠛⠀⠆⣀⠋⡤⠀⠀⠻⠋⠀⠟⢐⡂⠀⠀⠐⣉⣁⡈⢀⠀⢺⣙⠁⠀⠰⣿⡆⠀⠀⠀⠀⠀⠀⠘⠇⠀⢀⣿⣿⣿
⠘⠻⠏⠀⢿⢿⣿⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⢿⣇⡀⣴⣿⣿⣷⣄⣿⣿⣿⣿⡮⠀⠀⠁⠀⠀⠀⠀⠠⢶⡀⠀⠀⠀⠻⢿⣔⠺⣾⣇⠈⠻
⣰⣶⠗⠀⠀⠊⠉⠀⠀⠀⠚⠿⣿⣿⠇⠀⠀⠈⣿⡀⠟⢿⣿⣿⣯⣿⣿⡿⠛⠁⠀⠀⠀⠀⣀⡀⠀⠀⠈⣷⡄⢠⣶⡶⠀⠈⠓⠻⣿⡆
⠈⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠈⠛⠀⠰⢄⣴⡻⢿⣆⢨⡿⠿⠿⣿⣯⡄⢠⣶⠾⠀⢰⠛⠿⢟⣀⣀⣠⣼⠿⠮⠁⡀⣠⣠⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠈⠉⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⣸⡟⠺⠃⠀⠀⠉⠋⠠⣀⡅⠲⠀⢠⣶⠃⢺⣿⠏⠈⣥⠀⠀⠀⠈⠛⠋⢀⡀⠀⠀⠐
⠀⢴⣶⡇⠀⠀⢠⠽⢃⡀⠀⠀⠀⠀⠁⠀⣠⣶⣦⡈⠁⣤⠀⠀⠀⠀⣀⡀⠛⠤⠀⠀⠸⠫⠀⣈⠉⠀⢀⡉⠀⢶⣦⠀⠀⣰⠿⠏⠽⠈⣗
⢰⣾⣿⡿⠀⠀⣶⠀⠈⣡⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⣄⢰⡦⠀⠀⠀⡿⠓⠀⠀⠐⢁⠔⠀⠀⢙⣀⠀⣿⡄⠀⠀⠛⠀⠀⠉⠀⠀⠀⠐⠛
⠈⠉⠀⠀⣀⠀⢸⣶⡿⠃⠀⠀⠀⠀⠰⣤⣝⡛⠁⡈⢧⢸⡇⠀⠀⠀⣴⣾⠀⠀⠀⡖⠀⡀⢸⣿⣯⠁⠙⠿⠲⡷⠀⠀⠀⠀⢳⣤⠀⠀⢠
⠀⠀⠀⢸⣇⠀⢸⡿⠁⢀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⡦⢠⠤⢸⡏⠀⠀⠐⠉⠘⠻⡿⣏⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢀
⠀⢀⣤⣤⣤⡐⠛⠀⠀⠀⠀⣷⠀⠀⠀⠀⠈⠋⡹⠀⠀⠀⠀⠀⠀⢀⣬⢉⣇⠒⠀⠀⠀⠀⠀⣩⠃⣸⣇⠀⢺⣿⣦⠀⠀⠀⠀⠀⠀⠀⠘
⠀⣻⣿⣿⣿⡏⠀⢰⣶⣧⣶⣿⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠐⠻⣥⣌⣬⣥⣦⡀⣿⠛⠂⠀⠸⣟⣹⣿⣷⣿⣿⢃⡀
⠀⣿⣿⣿⡿⠀⠀⠈⣿⣿⣿⣿⡆⠀⠀⠀⢠⣾⣿⣷⡄⣠⠀⠀⡀⢰⣿⣿⣿⣿⣿⣷⣤⣤⣴⠆⠀⠉⠉⠉⠉⠀⢹⣿⣿⡇
⣼⡿⠿⠿⠁⠀⠀⠀⣿⣿⣿⠏⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⣷⠸⣿⣿⣿⣿⠿⠛⠉⠉⠀⠀⠀⢿⣄⡀⠀⠀⢀⣸⣿⣿⣿⣷⣶⣦⣄

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣀⣄⡎⢴⣸⠆⠀⠀⢀⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠋⣴⣹⠀⠀⠀⣠⠤⠤⣄
⠀⢀⠄⠀⠴⠁⡀⣧⠔⢲⠀⠀⠠⠁⢰⠿⢈⡇⠀⠀⢤⠤⡀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠠⡜⠁⠈⡦⠔⣦⠀⠀⠰⠁⢴⠗⢸⡆⠀⠠⡤⢄
⡀⠀⣣⣜⡂⠴⡧⡕⠀⠚⠲⡄⠀⠀⠀⠙⢉⠤⠲⢄⢸⠀⠰⠚⠛⣳⠠⠄⣀⡀⢈⣦⣘⠢⢼⠤⡄⠠⠛⢦⠀⠰⠀⠈⠙⣩⠄⠳⢄⣿⠀⠱⠛⠛⣧⠠⢀
⢱⡼⠁⡀⠈⢣⣑⠉⠉⢲⡊⠀⠀⢉⣷⠀⢏⡐⠗⡆⡟⢠⡖⠀⠰⡁⢀⣀⠈⢣⡞⢁⠠⡉⢆⠃⠉⠙⢦⠊⠀⠀⢉⣇⠀⣇⡰⢳⠈⡇⣰⠎⠀⢾⠁⣀⣀
⡴⡙⠄⠁⠌⡜⠑⠈⢀⡉⠃⣴⠦⣼⢀⠳⠋⢹⠉⠀⢃⣾⠞⠉⣳⠇⢏⠚⣃⢼⠁⠈⠁⢀⠎⠄⠉⣈⡉⠁⡴⠤⡏⣈⠷⡋⣿⠉⠀⢷⡯⠊⠙⣦⠗⣇⠛
⠤⠔⠒⡉⢍⣉⠙⣤⠊⢈⣶⡽⠦⠂⡠⢤⡁⠩⢉⡿⣈⣿⡀⢈⠉⡠⢄⠹⡤⠤⠒⢒⡩⢉⡉⠓⣤⠋⢹⣦⠷⠄⢊⡤⢄⠐⠍⣩⠗⣘⣇⠀⡉⢁⡤⣌⢩
⠓⡒⣤⠀⡉⣴⢩⡾⠀⡎⡹⢦⡬⡀⠳⠵⠃⠬⠙⣦⡠⢒⠙⡾⣼⣜⠛⢀⡝⠒⢂⡌⠀⡍⡖⢡⡾⠀⡏⡳⣦⢅⡘⠮⠞⢁⠍⠳⣄⡴⡪⠳⣷⣼⡘⠋⢀
⠃⠘⠛⢄⠱⡘⠦⡀⢠⢿⡁⢿⡤⢶⠐⣠⡁⢹⠋⢉⠙⠁⢼⢀⡕⠉⣉⡥⢼⠀⠘⠑⣄⢣⡑⢤⡀⣠⢟⠀⢧⡤⡎⠂⣤⡃⣿⠉⣉⠛⠀⣎⢨⠞⢉⣉⠄
⣁⠈⠦⡠⠋⠙⠉⠀⠸⡀⠱⢔⣫⠚⠞⣁⠉⢋⣀⠈⠋⠄⠈⠛⠀⠰⠉⢀⣈⡑⠉⢤⠤⠋⠋⠁⠀⢿⠈⠳⣜⣹⠙⢞⡡⠉⢃⡀⠘⠱⠀⠈⠛⠀⠀⠁⢀
⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢰⡋⣩⡧⠀⡃⠀⠀⠀⢠⠀⢾⡤⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢰⠉⣭⡆⢀⠃⠀⠀⠀⢄⠀⢧
⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠬⠤⠚⠀⠀⠀⠀⠀⠑⠲⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠊⠀⠀⠀⠀⠀⠓⠲

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣦⣤⣀⡀⠀⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣠⣶⡿⠟⠋⠉⠁⠀⠀⠀⠈⠉⠙⠻⢿⣷⣄⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠐⠶⢦⣤⣄⣀⡀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⢀⣀⣠⣤⡶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠃⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠶⠀⠈⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠘⠿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⠛⠉⠁⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣟⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣯⣯⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣭⣯⣽⣿⣿⣯⣭⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠻⣿⣿⣿⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣧⡘⠃⠈⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠁⠀⠐⢧⡄⠀⢉⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣀⣙⣷⣾⣿⣿⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣈⣉⣛⣻⡿⢛⣿⡿⣟⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡉⢉⣿⣧⣤⣤⣀⠀⠀⠈⠀⠀⠈⠓⠽⢤⡠⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠑⠀⣀⠀⠀⠈⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠾⣿⣿⣷⣮⢢⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣠⢼⣿⣶⣄⣀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⡹⢻⡏⢧⢸⣷⣿⣇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠘⣻⣭⣭⣭⣉⡋⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⣼⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⡈⠁⠀⠀⠀⠀⠁⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⡼⢣⣿⡿⠹⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠈⠉⠱⣶⠾⡥⠟⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢾⠦⢀⠀⠄⣀⣔⠀⣀⣀⣀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⡌⠳⡀⠁⢄⣠⠞⢫⠋⢉⡝⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⡷⠀⣉⢶⡉⠀⠀⢨⣤⡊⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⣿⣡⣆⣙⢲⣔⣾⣿⡟⠀⠄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⠸⣻⣿⣯⣿⣿⣯⣿⠟⡠⢊⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢧⣰⡀⠈⠛⠛⠻⠟⠃⢈⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣽⡿⠉⠐⣶⣾⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠴⠾⠿⢿⣿⣿⣿⣿⠋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⣿⡇⠀⠀⠈⣿⣿⣿⣿⣧⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠒⠛⠛⠛⠋⠁


⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠒⠚⠛⠋⠉⠉⠙⠛⠛⠓⠲⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣄
⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⢉⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠈⠳⡄⠀⡠⠂⠁⠀⠈⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠘⣾⠁⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠰⣦⠀⠀⠀⢻⠀⣶⣆⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢻⣿⠃⠀⠀⣸⠀⠻⠏⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⢀⡟⢄⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠘⣧⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠳⢄⡀⠀⣀⡤⠋⠀⠀⠑⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠘⢧⡀⠀⠀⡰⠚⣏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⢸⠁⠈⢻⡀
⠀⠀⠈⠳⡄⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⣸
⠀⠀⠀⠀⠹⣆⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢛⠒⠉⠉⠐
⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⢻⡈⠙⠒⠂⠤⠤⠤⠤⠤⠤⠤⠤⠤⡴⠒⠋⠉⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇⠀⠀⢻⣄⣀⣀⣠
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⢼⡅⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢈⣷⡀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⠸⠧⣀⢀⣀⠀⠀⠀⠀⢀⡿⠿⠿⠷⢶⣶⣶⣶⣶⡶⠿⠷⡀⠀⣀⠠⡀⠉⠓⠒⡟
⠀⠀⠀⠀⠀⣠⣴⡿⠛⢷⡀⠀⠀⠀⠀⠉⠲⢄⣐⠀⠀⠀⠀⣁⣀⡤⠞⠉⠿⠄⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠙⠛⠒⠒⠚⠁
⠀⠀⠀⣠⣾⠟⠁⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠻⣷⡄⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⢙⡿⠖⠶⣤⠾⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠷⣄
⠀⠀⠀⢀⣤⡼⠖⠉⠁⢹⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡼
⠀⠀⡔⠻⣼⠁⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢳
⠀⠀⢳⡀⢸⡀⠀⠀⠀⢘⡷⠶⠦⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠴⠶⠶⠶⠶⡟
⠀⠀⠀⠉⠢⢥⣀⣤⠤⢾⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣯⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣽⠟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠁

⠀⠀⠀⠀⢀⣷⡄⢀⣀⠀⠀⠀⠀⣀⣄⣤⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⠀⠀⢀⣶⠖
⠀⠀⠀⠀⠄⠁⣠⣞⣡⡀⠀⠀⢠⡌⣀⣼⣾⡄⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⡀⣀⣀⣦⠤⢍⠀⠀⠀⠀⠀⢙⣷⢀⣻⣿⣶⣶⣒⣒⣢⡀
⠀⠀⠀⠻⢶⡶⣜⠿⠛⠁⠀⠀⢸⡇⢘⢿⠿⡁⠀⠀⠀⠈⢿⡄⠀⠀⠀⢀⡴⣠⣨⣿⡷⣤⣿⣿⣷⣯⠀⠀⣈⠠⠄⣟⠋⠉⡛⠛⠉⠉⠛
⠀⠀⠀⠛⢶⡁⠀⠀⡄⠀⠀⠀⠈⠻⢦⣬⡷⠃⠀⠀⣲⣼⣾⣑⣶⡶⠄⠀⠀⠟⠉⠻⣀⡐⣸⣟⠻⠀⠀⠀⠉⠆⢸⣿⡀⠈⠻⣷⣄
⠀⠀⠀⠀⠈⠓⢰⡃⠰⢶⣤⡀⠀⠀⠀⠉⠀⠀⠈⢘⡿⢿⡿⣿⡿⠗⠖⠂⠀⠀⠀⠈⠁⠹⠇⠈⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠁⠀⠉⠻⠋⠙⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡿⠿⡂⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠂⠀⠐⠂⠀⠀⠀⢰⣿⠇⠀⠀⠀⠌⣏⢢⠀⠀⡀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⡉⢨⣩⣿⡇⠀⠀⠀⠘⢻⠆⠐
⠀⠀⠐⠠⡀⠶⠉⠀⠀⠀⠀⠀⠀⢙⣵⠀⠀⠀⠂⢿⡄⢧⡸⢟⠂⠀⢐⣿⣰⣦⢠⠀⠀⠀⠀⠈⠁⠀⠈⠻⠃⠀⠀⠀⠀⠀⠶⢡⡄⢰
⠀⠉⠳⠀⠀⠀⠘⠀⠀⠀⠀⣠⣴⢺⢿⠁⠀⠀⠧⣼⣧⠘⡗⢒⠁⠀⠸⣮⣿⣿⣼⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠞⠐⠃⠊
⠀⠀⠀⠀⢁⡘⢀⠀⠀⠀⠀⠿⢿⣿⣶⠀⠀⠀⠸⠦⣿⠀⣁⡶⠀⠀⠀⠚⠙⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡠⠟
⠀⠀⠀⠂⠈⠳⡇⠈⠢⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠙⣿⡇⡿⠆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠘⢿⠏⠀⠀⠀⠀⠀⠘⡿⠀⠆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠹⣼⠟⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⠶⠿⠿⠿⠿⠿⠿⠿⠿⠶⢶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈⠉⠛⠻⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⢀⣴⠟⠁⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄
⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⢃⠐⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠒⠀⠀⠈⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠈⠋⣤⡀⠆⣠⣶⣦⠤⡜⠈⠁⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠈⠁⢸⡃⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⠀⠈⡇⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢀⣇⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠛⠉⢻⠛⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣰⠃⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⡸⠏⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡓⠒⠚⠛⠒⠒⠒⠒⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⠉⠉⠉⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⣀⡀⠀⠀⠀⠀⢴⠛⠉⠉⠉⠉⠙⠛⠆⠀⠀⠀⠀⣀⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠈⠉⠛⠛⠒⠦⠤⢤⣤⣤⣤⣤⡤⠤⠤⠒⠛⠛⠋⠉⠁⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣀⣐⠒⠒⠂⠀⠀⠤⠄⠀⠀⠀⠀⠒⣒⣀⣠⣤⠔⠺⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠈⠉⠉⠓⠒⠒⠒⠒⠒⠒⠒⠒⠋⠉⠉⠉⠀⠀⠀⠀⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡶⠦⣤⣭⣉⣁⠐⠒⠒⠒⠒⠒⠀⠀⣤⣤⣤⡤⠶⠒⠻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⡀⠀⠠⣤⣈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣹⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠑⠶⠦⢤⣤⣬⣭⣉⣉⣉⣉⣭⣤⣤⣤⠤⠶⠶⠚⠉⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⡀⠀⠀⠀⣤⣤⣀⣀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣠⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡉⠉⠙⠲⠶⠶⠶⠤⠤⣤⣤⣤⡤⠤⠴⠶⠶⠛⠋⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠿⠶⠶⠶⠶⠶⠶⠾⠿⢻⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⢀⣤⣴⣶⣶⣤⣄⠀⠀⡠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠒⠂⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠒⠚⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣤⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⠿⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⡆⢸⣿⡇⢰⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠇⠸⠿⠇⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢀⢀⡀⠤⢠⡀⡄⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠈




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⠉⠉⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠸⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢸⠀⣷⣶⣶⣶⣶⣶⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢸⠀⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢠⠃⠀⠉⠛⠿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠲⠤⠤⠤⠤⠤⠤⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠠⡄⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢐⠉⣂⢰⠉⣣⠰⡇⠀⡇⠉⠚⠓⠒⠒⠒⠒⠒⠒⠛⠓⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠭⠥⠤⠭⠥⠤⠗⠒⠃


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⡄⠀⠀⠀⠀⠀⠀⢠⡞⢭⣱⠂⠀⠀⠀⠀⠀⣿⣶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒⢻⡿⠆⠀⠀⠀⠀⢀⡴⢯⣍⡢⠁⠀⠀⠀⠀⠐⠛⠻⣿⠟⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⢇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠⠞⠣⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣶⠃⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⡈⠳⠃⠀⠀⢀⠤⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡛⢟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏⠳⠎⠀⠀⠀⠀⢠⡀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⡋⠁⠀⠀⢎⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣵⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡷⠂⠀⠀⠀⠀⠀⡔⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⢙⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠢⢄⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⡄⠀⠀⠀⠀⠀⣤⠰⠃⠑⣐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡀⠀⠈⠢⣄⡠⢄⡀⠀⠀⠀⠀⠀⠀⠀⣔⠲⠈⢃⠿⠟⠀⠀⠀⠀⠀⢙⣀⠤⢵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠓⢄⠀⠀⢈⠀⠀⢸⠀⠀⠀⢀⡤⢤⠀⠙⣠⠟⠋⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃⠀⠈⠢⡀⠀⠑⠚⠙⢄⣀⠤⡈⠻⢸⡶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⡀⣴⣿⠙⠬⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⣠⣿⣦⡀⠀⠀⠀⠙⡆⡹⠶⠃⠀⠀⠀⢀⠄⠠⡀⠀⠀⠀⠀⠀⠀⣴⠛⠫⣸⡿⠀⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⢀⣼⣿⣿⣿⣿⣦⡀⠰⠿⠟⠣⡀⠀⠀⠀⠀⢂⠀⢀⡄⠀⠀⠀⠀⠘⠺⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⣴⣿⣿⣿⣿⡿⠋⠁⠈⠲⢄⠀⠀⠘⢄⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⣠⣾⣿⣿⣿⠟⠉⠀⠀⠀⣀⣤⣶⣿⣦⣤⣨
⠀⠀⠀⠀⠀⠀⣼⡿⠁⢀⣼⣿⣿⡿⠋⠁⠀⢀⣠⣴⣾⣿⣿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⢰⡟⠀⣴⣿⣿⠟⠁⠀⣠⣴⣾⣿⡿⠟⠋⠉
⠀⠀⠀⠀⢠⡟⣠⣾⡿⠋⣁⣤⣶⡿⠟⠛⠉
⠀⠀⠀⢀⣿⣼⢟⣩⡴⠟⠛⠉
⠀⠀⠀⣾⠿⠛⠉⠁


⣿⣿⣿⣶⣶⡿⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣾⣿⣶⡿⡶⠿⡶⠶⠾⣶⡶⠶⠾⠶⠶⠾⠿⡾⠷⠿⠶⠿⢷

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠯⡀⠀⠀⠀⠴⠚⠛⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣿⣿⣿⣟⣯⣥⣤
⠀⠀⠀⢀⡞⠳⣤⢦⡀⠸⣶⡋⠀⠀⣠⠞⠁⢹⣆⠀⠀⠀⠀⢀⣦⡀⣀⠀⠀⠙⠵⣿⣿⣿⢟⣾⣷⣄
⠀⢀⡴⠋⠀⢢⡈⠻⡀⠀⠀⢈⣦⡞⠁⠀⢀⡏⠸⡆⠀⢀⣴⣿⣯⣻⣯⡀⠀⠀⠀⠘⢟⣵⣿⣿⢧⣿⡄
⠀⠘⢧⡀⠀⠀⣹⠖⠃⢀⡴⣫⠎⠉⢢⣠⠞⠀⢀⡇⠀⠘⢿⣿⣿⡷⠉⠁⠀⢀⡴⠋⠀⠈⢿⣫⣿⣿⠇
⠀⠀⠀⠙⠒⢾⢳⢀⡴⣫⠞⠁⠀⠀⠈⠛⠢⡀⡈⠀⠀⠀⠀⠙⠉⣦⠀⢀⣴⠏⠀⠀⠀⠀⠀⠉⢻⡟
⠀⠀⠀⠀⢀⡞⣯⡻⠞⠁⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⢀⣼⣦⡛⠁
⠀⠀⠀⠀⠈⢧⣛⡵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠕⠁
⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣤⣤⣴⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⡔⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠉⠈⠋⠋⠉⡿⠏⠛⠛⠛⠛⠋⢿⡌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏⠀⢀⣠⣤⣤⣀⣤⣀⠀⠀⠀⠀⠀⢀⣠⣤⣼⣷⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢹⡿⠋⠀⡰⠛⣿⠟⣿⣿⣿⡿⢀⡖⠒⠋⠙⠛⠒⠒⢺⣿⡆⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⢻⡶⠟⠡⣰⡇⣸⡇⡞⣙⡴⠋⠀⠀⠀⠀⠀⠀⠀⡌⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢏⣀⣤⠒⢉⡼⢋⡾⣻⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠈⠉⢉⠐⠉⠉⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠋⢿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣀⣤⡴⣄⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢈⣿⣽⣷⣴⣇⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣡⣶⣾⣿⣿⣿⡍⠸⣷⣦⡕⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠉⣼⣿⣿⣿⣇⣀⠀⢹⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⣿⣿⣿⠟⠃⢀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⢰⣷⣮⣭⣭⣤⣴⣾⠇⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣆⠄⠉⠉⠋⠉⠉⡀⠰⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠐⢀⠀⠀⠀⢀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡄⠀⠠⡀⢠⣼⣶⣶⣶⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⢀⣨⣷⣾⣿⣿⡟⠀⠀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣛⠁⠀⠀⠴⢮⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣀⣀⡀⠠⣾⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣹⡿⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠒⠂⠈⠉⢣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀⣀⣀⠀⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠁⠀⠀⠀⠈⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠀⢄⡀⠀⠀⠀⡏⠀⠤⡀⠀⠀⠀⠠⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢁⣴⣟⠁⠀⠀⠀⠀⢸⠁⠀⠀⠈⠠⠄⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢇⣾⡋⢹⣷⡀⠄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⣿⠏⠀⠉⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⠀⠀⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⡶⠶⡖⠀⠀⠀⠸⡀⢰⣤⣴⣶⣶⣶⣤⡴⣆⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⣿⡏⠁⠀⠁⠀⠀⠀⠀⠙⣶⣿⣿⡟⠉⠈⠉⠘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠋⠟⢁⣈⣓⠃⠀⢠⠄⠀⠀⠀⣸⣭⣶⣶⣶⣶⣶⣶⣤⣜⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡀⠀⠛⠋⠙⠉⠓⠀⠙⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⣿⣿⣿⣍⡃
⠀⠀⠀⠀⢰⣿⣿⣿⡿⠁⠀⢹⣿⣿⣏⠻⢶⡄
⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⡳⠄
⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⢹⣿⡍⢳⣤⠀⣄
⠀⠀⠀⠀⠀⢹⣿⣿⡁⠀⠀⠀⠀⢿⢿⡀⣌⠳⠎⢷⣀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⢀⣾⣿⣿⣧⠀⠀⠀⠀⢸⣷⣥⣼⣷⣬⣶⣿⣿⣶⠶⠋
⠀⠀⠀⠀⢸⣿⣿⠿⣿⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠈⢻⡇⠀⠋⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴
⠀⠀⠀⠀⠀⠀⢸⣿⣶⡦⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢸⠿⠋⠀⠀⢸⣿⣿⠗⠀⢰⣤⣤⣤⠀⢠⣤⣤⣤⠄⠀⣤⣶⣶⠆⠀⠹⣿⣿⡇⠀⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠁⠀⠀⠈⣿⡿⠃⠀⠀⢻⣿⡏⠀⠀⠘⢿⡿⠀⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠛⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⣶⣄
⠀⠀⠀⠀⠀⠀⣰⣶⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠆⠀⠀⠀⠀⡶⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⢸⣄⡀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⣿⠂⠀⠀⠀⣾⣷⠆⠀⠀⠀⠹⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠸⡿⠀⠀⠀⣴⡯⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢰⣿⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢿⠇⠀⠀⢠⣿⡄⠀⠀⢸⣷⡀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢀⣼⣄⠀⠀⠀⣼⣤⠀⠀⠀⣰⣇⠀⠀⠀⢀⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣷⡄⠀⠀⠘⠿⠇⠀⠀⠀⠛⠃⠀⠀⠀⠉⠃⠀⠀⠀⠛⠋⠀⠀⠀⠿⠿⠀⠀⠀⣴⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⡀⠀⠀⠀⠀⠿⠋⠁⠀⣿⣶⡆⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠻⣿⡇
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⠋⠀⠀⢹⡿⠋⠀⢻⣶⡶⠀⢰⣶⣤⠄⠠⣤⣶⠆⠀⢶⣾⡇⠀⠙⢿⡇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠛⠀⠀⠀⢿⠏⠀⠀⠹⡟⠀⠀⠈⠛

⠀⡀
⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠒⠀⠀⠂⠁⠐⢲⣦⠤⢤⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⠔⣸⣭⣍⣰⣇⣀⠀⡯⣀⣀⠀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠄⣾⣯⣾⣧⣦⣤⣍⢻⣿⡿⣾⣿⡿⢛⠃⣸⠂⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⢰⡿⠀⢸⣾⣿⣿⠟⠋⠀⠈⠙⢿⣯⡞⠛⠻⢷⡄⣰⡿⢀⣴⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠉⠀⠈⠿⣿⠁⠀⠀⠀⠀⣠⢊⡝⣀⢀⣶⣸⡑⣫⣾⣿⣟⡥⠚⢁⢮⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠀⠀⣄⣴⠞⣡⣾⣿⣿⣿⣿⣿⣾⣿⣿⡿⠋⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣄⣀⡀⠀⠀⠀⢀⣀⢿⣷⣿⣿⣿⡟⠿⠟⣛⡻⣿⣿⣿⣿⣿⠿⢋⣴⠿⠻⠃⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠓⠒⠀⠠⠽⠿⠚⣶⣶⣿⣿⣾⣿⣟⠛⠛⠋⠉⠉⢀⡴⠟⣀⡤⠆⠀⠀⠀⣀⣀⣀⢀⣴⣶⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠐⣶⣶⣶⣶⡿⠿⠟⢹⣿⡿⣧⡙⠳⢦⣤⣤⣀⡸⢷⡾⠋⠀⠀⠀⠀⣾⣿⠟⠁⠾⠿⠿⠿⠛⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⡉⠙⠻⢦⣚⠂⠀⡀⠠⠆⣠⡿⠛⠴⠿⠿⢦⣤⣄⡀⠀⠀⠀⠉⢠⣤⣄⣀⡤⠊⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠀⢀⠠⣄⣀⣀⡀⣀⣀⣈⣀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠙⣿⣿⣶⢆⣀⣤⠾⠋⠁⠀⠀⣠⣾⣶⠶⠶⢲⣶⠟⠋⠉
⠀⠀⠀⠀⠀⠀⢀⣴⣦⣶⣶⣿⡟⠛⠛⣋⡉⠙⠉⠉⠉⠉⠉⠓⠲⣤⣿⣦⠀⠀⠀⠀⠀⣻⣿⣿⡾⠟⠁⢀⣀⣠⡴⠛⠋⠉⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠉⠉⠉⠙⠿⠿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣤⣠⣠⣰⣿⣿⠯⠶⢶⣶⣿⡥⠄⠴⠶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⣤⣤⣶⣶⣶⣶⡶⣻⠟⠋⠉⠻⣿⠋⠁⠀⠀⠀⡀⠀⠉⠉⠋⠀⠀⠈⣉⣄⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣻⣾⠟⢋⡿⣛⣩⡾⠁⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⢀⣿⡷⠛⠉⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⡤⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢀⣀⣠⣴⣒⣶⣖⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣴⣾⡿⠿⢿⣿⠟⠛⠉⠙⠻⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠒⠀⠤⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠒⢀⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢘⠄⠀⠀⠈⠁⠐⠢⢄⠀⠐⠀⠐⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠚⠃⢱⡄⠀⠀⠀⠀⢀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡠⠄⠀⠀⠀⠀⠀⡀⠀⢁⠀⠀⠀⡀⠇⠀⢀⡀⠀⠀⠀⠀⣨⠆⠀⠀⠀⠀⠳⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢠⡿⠀⠁⢀⠉⢀⣀⠴⠞⠛⢢⡈⠓⠈⡄⢱⡎⠀⢀⡀⠀⠀⠀⢦⡈⢻⡄⠀⠀⠀⢀⡀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠘⠃⡌⠀⣡⠞⠋⠀⠀⠀⠀⠁⠀⠋⠀⠁⠃⠈⠁⠊⠉⠳⠄⠀⠈⢱⡈⢷⣄⡀⠀⠀⠀⠀⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡷⠒⣀⡁⠰⢎⡌⠁⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⠛⠦⠯⡝⣦⡀⣠⠞⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡾⢁⠔⠈⠀⠀⠐⡉⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⣠⣤⣶⢿⡏⠐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⣷⣯⠄⠀⢐⡉⢤⠄⠒⡛⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠂⢀⠀⠀⠠⠀⠈⠻⣆⠀⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠰⣿⡋⠠⡔⠁⠀⠀⠀⠈⡀⠀⠘⠋⠗⣉⣉⡉⢉⡳⡄⠀⢠⡾⠛⠉⣀⣈⣙⠒⠀⢠⠀⠀⠄⡀⠀⠀⠀⠀⠀⢹⠀⠉⠑⠂
⠀⠀⠀⠀⠀⡠⠄⠚⣿⡇⠀⠀⠀⠀⣀⠀⣴⠇⠀⠀⠐⠪⠿⠿⠇⠐⠀⠀⠀⠸⠇⠐⠹⠧⠿⠉⠛⠀⠀⠀⢀⠀⠀⠀⠀⢀⠈⣼⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣎⠦⠀⠀⠫⣄⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠎⠓⠀⠛⠀⠀⠀⠘⠀⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠄⢣⡀⠈⣷⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠔⠀⢈⠀⡠⠃⢀⣼⠃⠀⠠⠀⠀⠸
⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⢸⣐⣴⠇⠀⢻⢾⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⢁⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢰⢃⠎⠀⣴⠋⠁
⠀⠀⠀⠀⠀⠆⠀⠀⢠⡔⠶⠛⠻⠋⠚⠻⠋⢀⠇⠀⠀⠀⠀⢀⡤⠀⠈⠁⢤⡔⠁⠠⠀⠀⠀⠀⠀⠃⠀⠐⠀⠐⠸⡦⢤⣜⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠲⣶⣤⣴⠾⠳⢄⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠞⠃⠀⠀⠀⡄⠐⠒⠿⠋⠙⡆⠀⠈⠀⠀⢸⢠⡖⠺⢋⣉⣉⠝⢳⠀⠀⠀⠀⠀⠀⢸⡆⠀⠤⠤⢤⠤⠀⢈⡙⢦⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⢠⠲⡃⠀⠀⠀⣆⡄⠀⠈⠛⠀⠁⡄⠀⠈⠀⠁⠀⠸⡀⠀⠀⠀⠄⠀⠘⠤⡀⠀⠈⠉⠉⠉⠀⠀⢠⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠘⢆⠀⢀⡣⠀⠀⠀⠈⢥⡀⠀⠀⡀⢠⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⢀⡀⠀⢲⡄⠀⠙⢠⣴⡥⠈⠄⠀⠙⠀⠀⠐⢃⠜⠀⢀⡴⠾⠛⠁⠀⠀⠀⡀⠄⣐⠭⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡛⣀⠈⠱⠀⠁⢱⡀⠀⠀⠈⠁⠀⢸⡧⡾⠂⠸⣀⣤⣿⠶⠊⠁⠀⠀⠀⠀⠁⠘⠀⠃⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉⠁⠀⠀⢸⡟⠁⠀⠀⣀⠄⢠⠎⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠚⣗⠣⠠⠤⠴⠊⠠⠄⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠴⠀⢀⡴⣍⣩⣿⣾⠿⠋⠉⠒⠀⠤⢀⡁⠀⠀⠀⠀⠐⠤⠤⠉⢀⡀⠨⠅⠀⠀⠀⠈⠳⣤⣴⡾⣷⠀⠀⢀⡼
⠀⠀⠀⠀⠀⢰⣶⠁⠠⠤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠘⣤⡀
⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢷⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠻⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⠛⠙⠛⠙⠋⠙⠉⠙⠋⠛⠋⠛⠛⠋⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⢧⣤⢤⡤⣤⡤⢤⡤⠤⠤⣤⡤⢤⣤⢤⡤⣤⡼⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⢁⡴⢋⡴⠋⢠⠎⠀⣸⠃⠀⠀⠘⣇⠀⠳⡄⠙⢦⠛⢦⡈⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣰⠋⢠⡞⠁⢠⠏⠀⠀⡟⠀⠀⠀⠀⢿⠀⠀⢹⡄⠈⢷⡀⠹⣄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⡼⠁⢀⡞⠀⠀⡞⠀⠀⢰⠁⠀⠀⠀⠀⠘⡆⠀⠀⢣⠀⠀⢷⠀⠘⢧⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⡸⠁⠀⡸⠀⠀⢠⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⡄⠀⠀⡇⠀⠈⣇⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢠⠇⠀⠀⡇⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⢸⠀⠀⠸⡀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⡟⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣷⠀⠀⠘⡄⠀⠀⡇⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢻⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⣇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡿⠀⠀⢀⡇⠀⠀⡇⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠸⡄⠀⠈⡇⠀⠀⢸⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⢸⠀⠀⢰⠃⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢣⠀⠀⢣⠀⠀⠸⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠇⠀⠀⡎⠀⢀⡜⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠈⢧⡀⠘⣦⠀⠀⣇⠀⠀⢸⠀⠀⠀⠀⠀⢀⠇⠀⠀⡸⠀⠀⣴⠁⢀⡾⠁⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⢳⡀⠘⣆⠀⠸⡄⠀⠈⡆⠀⠀⠀⠀⣸⠀⠀⢰⠇⠀⣼⠃⢀⠞⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠙⢦⡈⢧⡀⠹⡄⠀⢻⡀⠀⠀⢀⡏⠀⣠⠏⢀⡜⢁⡴⠋⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢍⣢⣽⣦⣙⣦⣘⣧⣀⣀⣼⣁⣴⣣⣴⣯⣖⡩⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢯⡉⠉⠀⠁⠈⠀⠀⠁⠈⠀⠉⠀⡹⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡟⠋⠉⠉⢻⡟⠉⠉⠉⣟⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢸⠇⠀⠀⠀⡧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⢠⡟⢻⡄⠀⠀⡗⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⢻⣛⠀⠀⢠⠟⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡟⠀⠀⠀⠘⠃⠀⠀⠈⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠷⣆⠀⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣯

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣀⣀⣀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⡄⠰⠶⠀⣀⢀⣤⣄⠀⠐⣀⠀⠀⠀⣀⡀⢀⣤⡄⠀⡤⢤⣤⠀⠀⠀⠀⠀⡸⠛
⣶⠀⠀⠀⠀⠀⠉⠉⢁⠀⠀⠀⠀⠀⠀⠙⡿⠁⣾⢶⡏⠀⠀⠈⠟⢑⢘⡫⡅⠀⣥⡽⢤⣿⡷⠀⡀⠀⠀⠀⠰⡿⠃
⠩⣿⣶⠀⠀⢀⣺⡗⡈⡇⠀⠀⠐⠀⠀⣠⣤⡌⠛⣈⡀⠀⠀⠀⡀⡀⠿⣀⠀⠀⡿⡅⢀⠛⠀⠀⠟⠀⣤⣀⡎⢀⠀⠀⢰⠂⣧
⣤⣾⣿⡧⠀⢠⡅⠀⢃⡀⠀⠀⠀⠀⢸⣿⣿⡿⣠⠈⠀⠀⠀⠀⣿⠇⠀⠀⠒⠠⠇⠀⢘⡃⠀⣿⡀⠀⠈⠟⠛⠋⠁⠀⠀⠐⠚
⠉⠉⠀⢀⡀⠀⣧⣾⣿⣿⠀⠀⣠⣤⣤⣛⠋⢁⣾⡄⠀⠀⠀⠀⠑⣊⣤⡀⢰⡇⠀⢀⣿⣿⠧⠙⠿⠀⠀⠀⣀⣺⣿⣤⢄⣀⡠
⠀⠀⠀⣿⣅⠀⣿⣿⣿⠃⣼⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠤⢸⡏⢈⣀⡎⠁⠛⠿⢿⠁⠀⠀⠀⠀⢸⠟⠋⠁⠹⠿⠀⠉⠳
⠀⠀⠀⠉⠻⡞⠉⠙⣿⣶⣿⡟⠉⠉⣝⠋⣩⠇⠀⠀⠀⠀⠀⢠⡀⠉⠛⠉⠁⢀⠀⠀⢠⡀⠀⠀⠀⠀⢈⠛⠘⠷⢠
⠀⠀⠀⠀⠘⠃⠀⠀⠙⠛⠋⠀⠀⠚⠛⠛⠋⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠋⠀⠀⠘⠛⠃⠀⠀⠀⠈⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⠀⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⢸⡏⠀⣠⣤⣤⣤⡀⠀⠀⢰⢴⠀⠀⠀⠹⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠀⣼⣿⣿⣿⡿⠃⠀⠀⠀⣻⠇⠀⡰⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⣿⡿⠋⠀⠀⠀⠀⠀⡛⠀⡜⠁⠀⢀⡤⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣦⣶⣦⡄⠀⠙⠉⠀⠀⠀⠶⠀⠀⠁⠞⠀⣀⠤⠴⠤⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠛⠛⠲⠞⠀⠉⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠢⠶⠌⠀⠀⠀⠀⠀⠂⢠⡤⠠⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠂⠀⢀⠀⠀⠀⠀⠘⢿⡄⠀⣾⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⡾⠁⠀⡉⠀⠀⠈⠁⠀⠀⢰⡻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠿⢿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣠⣦⠀⠀⠛⢛⣛⡿⢷⢄⠀⠀⢠⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠽⠿⠦⠀⠀⠀⢠⣴⣷⡿⠶⢻⡟⠉⠿⢰⠾⣶⡄⠈⠁⠀⠈⠤⠴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡦⣄⠀⢼⠿⢣⣤⣶⣾⣷⠂⠀⠀⠉⠈⠛⠇⠐⠀⣤⠀⠀⣀⣠⣔⣒⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠛⠛⢙⡋⠉⠀⣀⣿⣉⣲⠏⣿⣶⡒⠀⠀⠀⠀⠆⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠠⣤⡶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠉⠉⢁⣹⣿⢿⡿⠁⠀⢀⣠⡤⣶⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠛⢿⣄⣀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢎⣿⠼⢋⣁⣴⣿⣟⣁⡀⠛⠀⠀⡸⠀⠀⣼⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠢⣀⠀⠀⠀⠀⡀⠊⢡⡓⠀⢀⣠⡞⢏⠰⣤⠟⠉⢉⡾⢻⡆⣀⠀⠂⠀⢀⣠⣠⣴⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠄⠀⠀⢐⣀⣠⠟⠀⠀⠀⣀⣀⠘⢀⣿⠀⠀⠀⠀⢸⣿⣿⠆⠻⣦⣼⣿⣄⠉⠁⠉⠙⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠚⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢈⣿⣿⢀⡆⠀⠀⠀⠀⠀⠙⢿⠂⠀⠀⠙⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⡀⠀⠀⣦⣽⣏⡏⠁⣸⠇⠀⢠⣤⣤⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢻⣿⣽⣿⡳⣿⣿⣙⢯⡗⢄⣿⣷⠀⠨⠋⢸⣿⣿⡆⠀⣤⣴⢿⠶⠂⢀⣤⣄⡠⢄⣀
⠀⠀⠀⠀⢀⣴⠶⠿⣛⣛⡷⢠⣤⣤⣬⣥⣿⣿⣇⠈⠻⢿⠆⠳⢸⠁⢸⣧⠀⠀⠀⠿⠏⠀⠀⠾⠤⠌⠰⠿⣿⣿⣥⣿⣟⠉⠙⠂
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠚⠋⠉⠉⠉⢉⣽⣷⡖⠒⠐⠢⠀⠸⡀⠀⢿⣦⠃⢠⠶⠾⣷⣶⣾⣷⣶⣄⠀⠀⠈⠙⠛⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠗⠁⠀⠀⠀⠀⣇⠀⠀⢩⠄⢀⣀⣠⣼⠺⠬⣿⠛⠻⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠉⠀⠀⠀⠀⠀⠀⠠⢰⡀⢠⠋⠀⠙⠛⢿⠿⠿⣷⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡎⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣛⣛⣛⣛⣛⣛⣙⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣩⣿⣯⣭⣍⣩⣭⣿⣿⣿⣛⣻⣛⣛⣛⣿⣍⣛⣛⣛⣿⣛⣿

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡖⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⢰⢲⠀⠀⡠⠀⠈⠀⠀⠀⠸⣉⡝⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡠⢿⠁⠀⠀⠀⠆⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠠⠞⠀⠀⠰⠤⠜⠓⠒⠊⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠋⠈⠀⠀⣠⠾⠭⣙⡆⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠥⢍⣑⡲⡇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡠⠖⠒⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣍⡉⠑⠒⠤⠭⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⢀⡞⠂⢦⣀⡀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢸⠀⢦⡀⠉⠙⠆⡼
⠀⠸⡀⠀⠈⠉⢓⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠗
⠀⠀⠱⣤⠤⠒⠁
⠀⠀⢺⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⡎⠀⡀
⠀⠀⡅⠘⡝⢋⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⠴
⠀⠀⢱⠀⠈⠁
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⠑⠊⠈⡇
⠀⢀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⡠⣄
⠀⠀⠑⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠂
⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⢀⢭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⠀⠠⠦⡀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢑⡠⢴⠎⢹⠁⢸
⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣉⣐⠚⣚⠓⣒⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠈⠲⣼⠀⢸⠀⡘⠀⢴⡄
⠀⠀⠈⠉⠃⠀⡀⢀⡀⠀⠀⡖⠒⠤⠭⠤⠭⠤⢩⡇⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣎⠀⡟
⠀⠀⠀⠀⠀⢺⠈⢁⡇⠀⢀⢇⡠⢤⠄⣀⣀⣀⣀⣅⠀⠀⠀⠁⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⡠⢶⢶⠉⠁⠀⠀⠀⠈⠑⢧⠀⠀⠀⢀
⠀⠀⠁⠀⠀⠀⠑⠋⠀⠀⠈⠉⠉⠉⠐⠋⠚⠓⠛⠊⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⢉⠈⡏⠍⠛⠲⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠀⠀⠀⠠⠗⠀⠀⢠⠄⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠛⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣸⠀⠀⣠⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢿⢀⡾⠋⠀⠀⠀⠈⠙⠻⣷⣴⣦⣤⠿⠛⠉⠉⠀⠉⠙⢷⣿⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣠⠟⠈⣷⠀⠀⠀⠀⠀⠀⠀⡿⢉⡉⢻⡄⠀⠀⠀⠀⠀⠀⣸⠛⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠘⢷⣄⣀⣤⣤⣤⠾⣷⠏⠻⣌⠻⢶⣤⣤⣀⣤⣼⠗⠀⠈⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠈⠉⠁⠀⣠⠞⠋⠀⠀⠙⣧⣀⠀⠀⠉⠉⢀⡀⣀⣀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣄⠈⠿⠳⣶⠶⠂⠀⣰⢿⡳⣶⠶⣰⢶⢶⡶⠍⢳⡄⠀⠈⢻⣿⢿⡟⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠿⠛⠀⠘⢇⢹⡆⡀⢠⡏⠘⡁⠉⠉⠁⠀⠈⠁⠃⠀⠙⣧⣤⣸⣯⠟⠀⠻⢿⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠈⠢⣿⣽⡿⠀⠀⡇⠀⠀⠀⠀⠀⢰⠁⠀⠀⠈⢿⣿⠏⣠⡀⠀⠀⠉⠻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡿⠃⠀⠹⡉⠀⠀⠀⢷⠀⠀⠀⠀⠀⡘⠀⠀⠀⣀⠾⠁⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠱⡄⠀⠀⣼⡏⠉⠙⠲⠛⢿⠀⠀⣼⠏⣶⣦⣀⠀⠈⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠆⢳⠀⠀⢾⠀⠀⠀⠀⠀⣾⠀⢀⡿⠀⠈⢻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⢸⡄⠀⣿⡄⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣿⣧⣤⣠⣶⣺⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⠉⠀⡈⠀⠀⢩⠛⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠀⠀⠀⠁⠀⢠⡾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠈⠁⠀⠀⠈⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣈⠃⠐⠒⠀⠘⣘⣃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣛⠛⢟⣛⠻⣽⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣾⡾⠋⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣤⡄⢠⡶⣶⡀⡤⠠⣶⢺⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⣼⣿⣯⢰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⣿⡅⣶⣾⣷⠶⡿⠂⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠛⠋⠘⠃⠈⠛⠛⠟⠛⠁⠀⠀⠉⠀⠀⠀⠀⢰⣶⠀⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⣦⣶⣶⣄⡀⢀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣸⡇⢻⠿⠏⡀⠀⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠘⠀⠉⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣤⡶⠴⣄⡀⠀⠀⡨⠑⠤⠀⠀⢀⡠⢄⠀⠀⣠⡄⢠⡀⠀⢀⡀⢀⠀⠠⠔⠂⠀⠀⠀⠀⣠⣤⣤⣀
⢸⡟⠁⠤⡄⠄⠅⠀⠸⠠⠴⡄⠐⡃⠈⠐⢒⢾⠞⡁⠤⣄⠙⠂⡀⠶⠓⢿⠁⣬⠤⠵⠒⠀⠸⠁⢠⡈⠉
⢼⠂⣇⡈⠦⡀⠂⠀⠀⠀⠀⠋⠀⠀⡀⠀⠀⢸⣰⢄⠀⠂⠀⢀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⡠⠛⠄
⠸⠦⣁⡸⠀⡊⠀⢖⠀⠂⠀⠀⢦⠀⠀⠠⠂⠀⠙⠋⠀⠈⠀⢈⡄⠀⠀⠀⠐⠀⠀⠀⠀⠆⠀⠦⢰⡆⣀
⠘⣆⢈⠃⠀⠳⠀⡸⠀⣏⡄⠘⠈⠀⠀⠁⠀⠀⠀⡠⠀⠀⠒⠀⠃⠸⠀⠀⠀⠀⣀⣃⢐⠇⣰⡆⢀⠆⣠
⠀⣆⠻⡇⢉⡐⠀⠀⠉⠒⠊⣄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠙⠈⠀⠠⠆⠀⠀⠒⠇⠀⠀⠨⣄⠈⢰⢂
⣤⣼⡀⠁⠋⣀⣤⣤⡀⠀⡄⠈⠂⠀⠀⠀⡄⠐⠄⠁⡜⠁⠂⠠⠀⠀⡀⠀⠀⠜⠂⠀⠠⣠⡀⣈⠑⢁⣼
⠙⢟⢕⣘⣤⣰⣤⣬⣆⣈⣁⣀⣀⣀⣀⣶⣴⣀⣀⣶⢥⣖⣑⣦⣤⣖⣘⠂⠀⠀⢘⣫⣂⣁⣠⣀⣀⣔⢛
⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡸⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠒⠌⠻⡄
⠀⢠⠁⠀⠀⠀⠀⠛⠀⠶⠒⠢⠤⠄⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⣀⡤⠤⣀⡠⠶⠶⠄⠘⠒⠂⠈⠁⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠺⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠤⠄⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡕
⠀⠀⠈⢷⡢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠉⠀⠀⠈⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢊⠹
⠀⠀⠀⠈⣡⠀⠈⠻⠒⠤⠄⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⡀⠀⣀⠤⠒⠊⢻⠀⠀⠸⠇
⠀⠀⠀⠀⣿⠀⡀⠀⡇⠀⠀⠀⠀⠉⠁⠐⠒⠠⠤⠂⡈⠀⠀⠂⣀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠞⠢⠈⠀⠠⠀⠀⠀⢈⡀⣀⢀⡇
⠀⠀⠀⠀⢟⣤⠛⡄⡃⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠀⠋⠀⠀⠀⠈⠶⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⣠⠂⠀⠘⠀⠀⠀⠀⠀⠀⠈⢿⡛⣼⡇
⠀⠀⠀⠸⠏⠁⠀⠵⠁⠀⠀⠀⠀⠀⠔⠂⠀⠉⠂⢰⠂⠀⠀⠀⠀⠄⠑⣄⠐⠈⢧⡀⠀⠀⣠⠞⠁⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠃⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠒⠲⢄⠀⠀⢸⠆⠀⣀⠠⣄⠤⣆⠈⣧⡀⠀⣩⠂⠉⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠱⡀⢸⡤⠞⢁⣠⡤⠄⠤⡄⣉⠚⣆⠀⠀⠀⢀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢁⡜⢁⠔⠉⠀⠀⠀⠀⠀⠈⠳⡄⢶⡠⠚⠀⠁⠀⠉⠙⢦⡀⠀⠀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⢸⠇⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠰
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠓⠄⡀⠀⠀⠀⠀⡆⡸⠀⠀⠄⠀⠀⠉⠁⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢠⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢼⠁⠑⢆⠀⢳⣶⡶⠶⠶⠶⣿⠃⢠⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⢸⠀⣤⣀⠚⠉⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠃⠀⠀⠑⢄⡁⠀⠀⠀⠀⠀⡠⠛⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⡴⠟⢫⠿⢣⠴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⡂⠀⠀⠀⣏⠀⠀⠰⠆⠴⣐⢪⣍⠊⠁⢐⣦⣀⠀⠀⠀⠁⠘⠂⠀⠉
⠀⠐⠂⠈⠉⠐⠢⣀⠀⠰⠀⠀⠀⠀⢪⡗⢄⠀⠀⠀⠀⢀⡀⠜⠑⠀⠀⠀⠂⠓⠶⠤⡀⢰⡦⠴⡦⢣⠴⠋⠁⠈⢳⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢡⠀⠀⣠⢤⡤⢤⣨⣆⠣⣄⣤⠄⢋⠀⠀⠀⠰⢀⠀⠀⡀⠆⡀⠈⢣⡴⠀⠀⠀⡠⣀⢀⡤⠚⠀⡅⢀⠠⠤⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠃⠀⠀⠀⠀⠈⠁⠁⡾⢂⣠⠀⠐⢋⠂⠈⠉⠀⡉⠀⠉⠉⠎⢰⠴⣤⣆⣸⢅⣈⣥⣚⣤⠐⣋⣀⠀⠾⠐⣦⠰
⠀⠀⠀⠄⠀⠀⠀⠀⠀⡇⡲⠀⠰⣀⠀⡀⠆⠀⠀⠰⠙⠃⠀⠀⠈⠁⠀⠀⠈⠛⠀⠀⠃⠀⠀⠀⠀⠹⠁⠠⢷⣉⡩⢆⣭⡍⠙⠻⣶⠁⢀⠤⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣛⠤⠩⠀⠀⠀⠈⠁⠀⢂⠀⢀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⡀⠁⠉⠈⢱⠀⠰⢷⡀⠰⠯⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⣄⡀⠈⠀⠤⠀⠘⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡭⠑⠢⠤⢺⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠠⠖⢂⣝⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠈⢢⡀⠈⠲⠇⢀⡀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢁⠀⠀⡇⠈⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢠⣅⡀⠠⠀⠀⠁⠂⠢⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠸⠀⡇⠑⢄⠈⠑⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠀⢀⡴⠃⠈⠉⠒⠤⣀⠀⠂⠄⠐⢀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠀⠀⠈⠐⠠⢄⣀⣉⠁⠒⠒⠒⠒⠒⠉⢉⣀⡠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣧⣼⠀⠀⠀⠀⠀⠀⡇⠈⠉⠉⠉⠋⠉⠉⠉⠀⠨⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠦⠐⠚⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣀⡀⠤⠔⠒⠈⢉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠇⠀⠀⠀⠀⠁⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⣀⡀⢀⡀⠀⠀⠀⡰⠀⠀⠈⠢⡀⠀⠀⢀⡀⠀⠀⠀⠰⣀⡠⠎
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⣸⡇⠈⠈⠈⠁⢄⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡰⠁⠙⢆⡀⠀⠀⠀⢱⠀⠀⠀⠀⠈
⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⠀⠀⠘⠣⠦⠄⠊⠀⠀⠀⠀⠙⠲⠴⠄⠋⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶
⣿⣭⣭⣽⣭⣭⣭⣬⣍⣿⣩⣍⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣥⣮⣧⣼⣿⣿⣧⣽⣿⣭⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⣵⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿⣿⠅⠀⠀⠀⠀⢀⡀⠂⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⠀⡀⠆⠸⣿⡆⠀⠈⠀⠀⣀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠱⡘⢄⠀⢻⣿⡀⠆⠀⠀⠉⢀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⢎⢂⠀⢻⣧⠀⠀⠀⠀⠠⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡉⠉⠐⠢⢕⡀⠱⣕⢄⠻⣦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠳⠶⣤⣌⡐⠌⠓⠤⡙⠆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠈⠙⠓⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⢁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⡈⡀⡌⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠆⠁⣥⠁⠀⠀⠀⢀⣀⣀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢷⢠⠇⠀⣠⣴⢿⣿⣿⣿⣿⣿⢧⣠⣤⡀⠀⠀⠀⠀⠀⢴⡿⢿⣶⣾⣿⣿⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠈⢸⢠⣾⣿⣿⣿⣿⣿⣻⣷⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢼⠀⢸⣿⣿⣷⣿⣿⣿⣿⡿⠟⢋⣉⠀⠀⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠯⣙⣿⠿⢟⡋⠵⠒⠉⠁⠀⠀⠀⠀⠈⠳⣎⠉⠉⠛⠛⠋⠉⠉⠀⢀⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢼⢸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠈⢷⡐⢄⠀⠀⠀⠀⢀⣮⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡢⠀⠘⣸⡸⡀⠀⠀⠀⠀⢀⡠⣈⠔⠀⠀⡼⠀⢰⠀⠀⠀⢣⠈⢷⡈⠳⠄⢀⣀⡾⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠰⠁⠀⢂⠁⠳⣍⣢⣤⢔⡢⠕⠊⠀⠀⠀⡰⠁⠀⢨⠀⠀⠀⠈⡆⠘⣧⠀⠀⠛⠉⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢀⡀⠄⠈⢄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡜⠁⠀⢸⣴⠀⠀⠀⠀⠰⡀⠸⡆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠓⠋⠉⠉⠉⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⡄⡁⠀⠀⠀⠀⠀⠡⠀⠹⡀⠀⢀⣪⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠡⣀⠠⡈⠳⣄⠀⠀⣀⠠⠔⠊⠀⠀⠀⠀⡇⠇⠀⠀⠀⠀⠀⠀⠁⠀⠱⣶⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢌⡒⢄⡉⠐⠂⠐⠀⠀⠀⠀⢠⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣤⣀⣀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢄⣀⠀⠀⠈⠒⢬⡓⠤⣈⠁⠒⠒⠒⠁⢸⠀⠀⢈⠉⠐⠒⠂⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠤⣀⡈⠑⠢⣑⠢⣄⠀⠀⠀⠣⠔⠉⠑⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣺⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠑⠀⠙⠢⢍⡒⠄⠀⠀⠀⠘⠀⠀⢀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠒⠛⠛⠒⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⢀⣀⣤⠤⠶⠶⠶⠦⠤⣤⣀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠘⣇
⠀⠀⠀⠀⠀⠀⠀⢠⡏⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠘⠄⡠⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠁⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⢀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⣇
⠀⠀⠀⠀⠀⠀⠀⣿⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢹⠘⡇⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⣼⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠸⡆⣷⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⡟⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣼⡶⠋⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢻⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡀⢀⡀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⢀⠀⢀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠄⠀⠔⠀⠀⠀⠀⠠⠄⠰⠄⠀⠀⠀⠁⠠⠃⠠⠃⠀⠀⠀⠀⢆⠀⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠠⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠰⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢀⡂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠤⠐⠂⠚⠒⠂⠤⡀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⣤⠒⢂⠀⢸⡇⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⢀⡐⠊⠾⡱⠳⡄⢸⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢀⠘⢨⠀⠀⠀⠰⠤⠀⢰⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢀⠀⠀⠂⠀⠡⡄⡤⠁⠖⠐⠢⠑⣌⣠⠠⠒⠀⠀⡄⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠈⠀⠁⠀⠈⠀⠀⠁⠉⠉⠁⠀⠁⠀⠀⠁⠀⠀⠁⠀⢾⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠐⠀⣤⠀⠀⠀⢠⡀⠀⠂⢠⡄⠐⠀⣤⠀⠀⠀⢠⡄⠀⠂⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣬⣤⣤⣤⣌⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠈⣤⣤⣤⣤⣥⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⢐⠀⠠⡂⠀⠀⠄⠀⠀⠀⢼⠀⢰⠄⠀⠀⡠⡄⠀⢀⠀⠀⣤⣶⣶⣶⣶⣶⣶⣦⠀⠀⠀⠠⠀⠄⡄⡠⢠
⠀⠈⠀⠂⠄⠀⠀⢘⠀⠀⢁⡉⠀⠨⠵⠀⠣⣀⢧⡤⠋⠐⠀⣧⣾⣤⣥⣼⣼⣦⣿⠃⠂⡀⢀⠄⠀⠠⠐⠀⡔
⠀⡄⡄⠀⠑⡀⠐⠨⠐⠀⠄⠤⠀⡀⡃⠀⠋⡊⠅⠙⠆⠈⢀⠉⠉⠉⠿⠋⠍⠉⢁⡀⡀⠝⢂⠰⠿⠀⠶⠶⡀⠸⠟
⠀⠀⢠⠂⠀⡅⠈⠀⠄⠠⢐⡀⠀⠀⢠⠀⠘⠁⢀⠁⡀⠐⠀⠁⠠⠀⠀⢀⢦⡀⢠⡀⠀⠐⠈⠐⠔⠈⠒⠆⠀⠐
⠀⠈⠔⡀⠁⠆⢀⢀⣃⠔⠀⠄⠀⠀⡚⢀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠓⠀⠘⠀⠀⡯⢘⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠈⠌⠈⠀⠠⠀⠀⠠⠥⠀⠁⣈⠀⠀⠀⠁⠀⠈⠀⠂⠉⠀⠉⢄⠀⠀⠁⢀⢀⠁⡈⢸⠀⠒⠪⠁⠑⠁⠀⠨⠙⡄
⠀⠀⠀⠀⠀⠄⠐⢄⠀⣧⢀⣐⣀⣀⣈⣀⣀⣈⢀⣀⠀⠀⠂⠻⠀⠀⠀⠈⠀⠀⠓⠘⠀⠀⠀⠀⠀⠀⢀⠏⠂⠈
⠈⣀⣀⡁⠂⢠⣶⣷⡆⠈⠀⢣⡰⠁⠂⠀⠈⠢⡠⠁⠑⠤⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠑⠀⠛⡄⢈⠀⠐⠀⠐
⠈⣿⣿⡇⠀⠁⠛⠛⢇⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⣀⠀⠀⠀⠀⠰⠓⠀⠡⣀⠀⠠⠀⠀⠀⠀⡀
⠀⠈⡏⠁⠀⢀⠄⢀⢈⠓⠀⠀⠼⠧⠀⡀⠀⢰⠃⠀⠀⠀⠰⠀⠀⠀⠣⠀⠒⠂⠀⠀⠀⡀⠐⡄⠐⠀⠐⠀⠉⠁
⠀⠄⠃⠂⠀⠈⠀⠀⠂⠀⡠⠀⠀⠀⠀⢀⠀⢈⠀⠃⠀⠉⠈⠈⠀⠃⠀⠀⠀⢠⡀⠀⠐⠀⡀⠈⠀⠀⠐⠈
⢀⡀⠀⣴⠀⢃⠀⠂⡈⠨⠠⠄⠀⠀⠌⠄⡇⠘⣄⠀⠐⠒⠢⠀⠀⠀⡸⠀⠀⠀⠀⠀⠔⠀⠰⡀⠀⢀⠀⠁⠀⠑
⠀⠄⡐⠀⢡⠈⠄⠀⠆⢜⠅⠂⠀⠀⠀⠠⡃⠀⠈⠀⠒⠒⠒⠐⠂⠈⠀⠀⠇⠐⢿⡟⠁⠀⠀⠙⣿⠗⠀⠈⠉⠄
⠀⢠⠁⠀⠀⢃⠀⠁⡀⣤⡁⣰⡀⠀⠂⠈⡀⢠⠄⠀⠀⢠⢀⠐⢀⠀⡁⢀⢄⠀⠈⡀⠀⠀⠀⠀⡏⠄⠄⡂⠉⢨⠐
⠀⠲⠤⠠⠄⠘⠀⢸⠂⠀⠀⠈⠠⠀⢀⠆⠀⠈⠄⣤⠀⠈⠓⠊⠷⠀⠀⠀⠁⢀⠀⠰⠄⡄⡠⠆⠀⠀⠦⠈⠀⠁⠂
⠀⠀⠀⣤⠄⠀⠀⣈⠰⠀⢐⠐⠐⢠⠃⢀⡠⠔⠒⠀⠀⠀⠠⠄⣀⡀⠀⢠⡈⠉⠀⠲⠾⠿⠿⠾⠃⠀⠀⠀⠀⢠⡄
⠀⡰⠠⣤⡌⢂⠀⠈⡁⠀⠀⠀⠀⠈⠉⠀⢀⠤⠀⢀⣀⠤⠄⠀⠀⢀⠀⠘⠁⡋⠉⡖⠂⠀⠒⠖⠒⠒⠂⠀⠁⠈⠁
⠸⠺⠒⡉⡙⠤⠁⠀⠈⡠⠀⠠⠈⠀⠀⠀⡍⠀⠄⢀⡁⢀⠀⡄⠀⠐⠘⠀⠀⠛⠑⠃⠄⠂⠄⠶⠓⠄⠠⠀⠂
⠰⣄⠀⠃⠀⠀⡀⠠⠤⠀⠀⠄⠘⠿⠁⠀⠀⠄⡀⠐⢡⠉⡆⠀⡆⠀⠄⠀⢆⠰⠀⠔⠀⠀⡤⠀⡤⠤⠤⠄
⠀⣀⡐⠂⠈⢁⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⢠⠉⠐⠁⠀⠀⠀⠀⠀⠂⠈⠀⠠⠂⠀⠆⠀⠥⠀⠀⠂
⠀⠀⠀⠀⡄⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⡄⠄⡸⠀⢐⠀⠀⠀⠤⡀⠀⠴⠐⠄⢃⠀⠠⠀⢈⠀⠠⠒
⠀⠆⠀⠄⠀⠀⠃⠀⠀⠠⠂⠁⠀⠘⠁⠙⠓⠒⠃⠃⠀⠂⠈⠁⠀⠒⠀⠑⠐⠃⠃⠐⠂⠐⠁⠊⠒⠂⠀⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣿⡦⡄⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠛⢢⠀⠀⠀⠀⠀⠈⠉⠻⣚⡋⠀⢀⣤⣴⣶⣶⣤⡀⠀⠀⠉⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠐⠤⣄⡀⣀⠀⠀⣰⣿⣿⣇⢀⡀⠀⣤⣶⣶⣶⣤⡀⠥⠀⠰⣿⣿⣿⣿⣿⠏⠠⢄⣀⣶⣾⣿⣷⣦⡀⠀⠀⣶⣶⣷⠀⠘⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠁⠉⠀⠈⢻⣿⠿⢟⢀⠀⠘⠫⣿⣿⣿⣿⡷⠖⠀⠂⠙⠻⠛⠋⠡⠖⠀⠀⣙⠻⣿⣿⣿⠟⠛⠀⠘⢿⣿⣿⠃⢄⠠⣄⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠛⠳⠄⠀⠀⠈⠉⠋⠁⠒⠀⢄⣀⡀⠐⠀⠆⠀⢀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠢⠔⠏⠁⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠳⠀⠀⠀⠀⢠⡿⠏⠀⡄⠀⠀⠁⠘⠷⠀⠀⠀⠀⠐⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⠿⠿⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠿⠿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦
⠀⠀⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⡄
⠀⠀⠀⢠⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄
⠀⠀⢀⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄
⠀⢀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡀
⠀⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧
⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿
⠀⠈⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁
⠀⠀⠈⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁
⠀⠀⠀⠀⢻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠙⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣤⣄⣀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣌⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢁⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣯⡙⠷⣤⣀⠈⠉⠙⠛⠛⠛⠛⠛⠉⢁⣀⣤⠾⢋⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠷⣄⡈⠙⠛⠶⠶⢤⣤⡴⠶⠶⠞⠛⠉⢀⣴⠿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡈⠻⠶⣤⣄⣀⣀⣀⣀⣀⣀⣤⣴⠾⠛⣡⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣤⣄⣈⠉⠉⠉⠉⠉⣉⣁⣤⣴⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⡿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠶⠄⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⣠⣀⣴⠇⠀⠀⠀⢀⠀⣤
⠋⠐⠀⠀⠀⠀⠀⣸⣏⠀⠠⡐⠀⣴⢠⡆⠀⠀⠀⠀⠀⠀⡰⠶⠶⣿⡉⠀⠂⣀⠀⣠⡀⠟⠈⠀⠀⠀⠀⣼⠿⠛⣃
⣀⡄⠀⠀⠀⠀⠐⠉⣉⣀⡀⠀⠀⡆⠘⠀⠀⠀⠃⠀⠀⢸⡇⠀⠀⠸⡇⠀⠀⠀⠁⡙⠀⠀⠀⠀⠀⠀⢰⣧⣴⠿⠛
⢸⡇⠀⠀⠀⠀⠀⣰⣿⣿⡿⠿⣤⣿⠿⢦⣤⣤⣀⠀⠀⠀⠀⠐⠆⣴⡇⠀⠀⠀⢠⣅⣴⠞⠃⠀⠀⠀⢿⣭⡏
⠀⠀⠀⠀⠀⠀⣾⣻⠋⠁⠀⠀⢾⠾⣾⡿⠟⠛⠛⢦⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⣿⡉⠀⠀⠀⠀⠀⠈⠁⡁
⠀⠀⠀⠀⢰⣶⣍⣍⠀⠀⠀⠀⢠⣤⣿⡇⠀⠀⠀⢸⠀⠀⢠⣀⣌⡁⠀⠀⠀⠀⠀⢨⠱⣶⣿⣿⣿⣿⣷⣦⣿⡀
⠐⠃⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠘⠛⠋⠃⠀⠀⠀⠘⠒⠚⠛⠛⠛⠛⠃⠀⠀⠀⠀⠘⠊⠀⠈⠙⠛⠛⠛⠛⠛⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠸⠄⠀⠈⠈⠀⠀⠀⠶⠀⠀⠀⠀⠀⠨⠁⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⣤⢀⠀⠀⠀⠀⠐⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠑⠖⠀⡁⠀⠠⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠁⠂⠀⠁⠀⠂⠤⠄⢀⠀⠂⡀⠈⠀⢀⡀⢀⠚⠀⠔⠁
⠀⠀⠀⠀⠀⠀⢀⡔⠆⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠄⠀⠀⠄⠉⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⡀⠀⠘⠉⠀⠀⠒⠁⠀⠀⢀⣠⠄⠈⠀⠀⠒⢃
⣶⣶⣶⣶⢶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠦⢖⠐⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣘⠤⠴⠚⣥⣴⠀⠀⠀⡀⢤⠶⠖⠋⠉⠁⢀⣠⠤⠴⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⠀⣰⢡⣝⣛⠌⠿⠋⠀⠀⢈⠔⢁⠀⠀⠀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⡘⣉⣵⣿⣯⠰⣾⣦⠐⡴⢃⠀⠂⠈⠀⣸⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡴⣿⠋⠻⡟⠀⠌⠋⡼⠀⠀⠉⠀⢀⠞⢡
⠀⠀⠀⠀⠀⠐⠐⡉⢄⠆⠐⢷⠀⠀⠀⠆⢰⢿⡀⠀⠀⡐⠀⢰⠁⠀⠀⠀⡠⣡⣚⠈⠓⠂
⠀⠀⠀⠀⠰⢀⠈⠸⠁⠀⠈⠆⠹⡠⣀⢰⠀⢼⣷⣀⢠⠁⠀⡆⢀⣀⢔⣫⡄⠿⣿⡇
⠀⠀⠀⣤⡄⠀⢶⡈⠀⠐⠂⠈⠉⠱⠬⠶⠓⠌⡛⣿⡝⢆⣰⣧⢉⣦⣍⠩⢡⣶⡈⠄⠀⠠⢀
⠠⢾⣷⣟⣋⠀⠀⠐⠂⡐⠀⣠⠀⠀⠀⠀⠀⠀⠺⠾⢋⣹⠟⣇⡛⣿⣿⣿⣯⠛⠓⠀⠀⠀⠀⠁
⠀⠀⠰⠿⢿⣿⠆⠀⠀⣠⢛⠁⣋⠀⠐⠀⠀⠀⠀⢠⠙⠛⠃⣿⡄⣿⡿⢻⠿⠀⠀⣤
⠀⠀⠀⠀⠊⠁⠀⡤⢰⠀⠣⣈⢎⡁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠘⠱⢈⠄⠊⠀⠀⠀⢿⢃⣤⠀⣠⣾
⠀⠀⠀⠀⠀⠌⠤⠝⡀⢀⠀⠉⢈⢋⡀⠀⠠⡄⠀⠄⠀⠀⠀⠀⡆⢅⢜⠀⠀⢠⣶⡾⠸⠿⡠⠛⠋
⠀⠀⠀⠀⠁⠠⡀⠐⠨⠙⠊⠐⢨⠀⠁⠈⠢⠀⠀⠀⣀⠀⠀⢈⠼⠘⠠⡁⢰⣾⣍⠀⣰⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠈⠃⠂⠀⠀⠀⢀⠀⢀⠠⢊⠀⢁⣳⣀⢁⢀⠙⠛⢋⣬⣭⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠈⠓⠁⠀⡀⠘⠀⠀⡐⠀⠐⠿⠿⠟⢱⣶⡄⣀⡘⠻⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢁⠉⠹⠀⠀⠠⠀⠀⢠⣾⠌⡛⠇⣿⣿⢰⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠀⠠⢀⣭⡿⠃⠉⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠑⢄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡌⠀⠀⠀⠀⠀⠀⠀⠄⠀⡦
⠀⠀⠀⠀⠀⠀⠀⢀⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠌⠀⠜
⠀⠀⠀⠀⠀⠀⠐⡁⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⢀⠃⠀⠀⠀⠀⠀⠀⠌⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⢀⠖⠀⡰⠆
⢀⠒⢄⠀⠀⠀⠀⠀⠀⠀⠘⢄⠃⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢀⠔⢁⠄⠊
⠢⡀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠈⠂⡄⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⡠⣃⠔⠁
⠀⠀⠑⢄⠈⠢⡀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⠬⠊
⠀⠀⠀⠀⠈⠂⢈⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠙⠲⢤⡀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠱⣄⠀⠀⠀⠀⣠⠔⠁⠀⣀⣤⣴⣦⡀⠀⠀⠀⠴⠿⢿⣿⣦⡀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢀⠜⠁⠀⢀⣾⠟⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠼⠁⣠⠀⠒⠒⠤⡀⠀⡠⠐⠂⠐⠠⡀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠖⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⢀⣀⡈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡃⠀⠀⡴⣦⠀⠀⢸⠀⠀⣴⣶⠀⠀⠘⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⡜⠉⠉⠉⢢⡤⠎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠉⠉⠀⠀⡘⠀⠀⠈⠉⠀⠀⠸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠁⠀⠀⠀⠀⡇⠀⠰⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠾⠛⠁⠀⠈⢄⠀⠀⠀⠀⠠⠉⢄⠀⠀⠀⠀⠐⠁⠀⠲⠶⠄⠀⠀⢸⠀⠀⠀⠘⡅⠀⠑⠤⠐⢇⢀⡰⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡃⠀⠀⠀⠁⠂⠀⠈⠀⠀⠀⠁⠒⠂⠈⠀⠀⠄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠉⠣⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⣤⠆⠀⠀⠀⡏⠀⠀⠀⠀⡰⢁⠞⠢⠤⠤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢹⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⡟⠀⠀⠀⣼⣁⣀⣀⠴⠊⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⣧⡀⠀⢿⣿⣿⣷⣶⣶⣶⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿⡿⠁⠀⢀⣼⣿⣄⠀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣠⠟⠳⣄⠈⢻⡿⠋⠉⠉⠉⠛⠉⠁⠈⠉⠙⢿⣿⣿⡿⠁⠀⡠⠊⠀⠈⠉⠉
⠀⠀⠀⡀⠠⠀⣀⡜⠁⡴⠋⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏⠀⢠⠎
⠀⠰⠉⣀⡀⠀⠈⠁⠎⢀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⢀⠗⠉⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠎⠀⠀⠀⣠⡖⠀⡄⠀⠰⡀⠀⠀⠀⠀⠀⢠⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀
⠘⠦⠤⠾⠇⠆⠀⠱⣄⠀⣈⠀⠀⠀⠀⠀⢸⠀⠀⠉⠁⠀⠒⠀⠀⠀⠀⠀⣀⡀⠀⠬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⣀⠀⠤⠄⠀⠒⠂⠀⠈⠉⣀⣀⠠⠤⠤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠤⠤⠀⠒⠒⠀⠉⠉⣀⣀⡀⠤⠤⠤⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠤⠆⠒⠒⠂⠉⠉⠀⠀⠀⠀⠀⡴⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠶⢄⡀⠀⠀⠀⠀⠀⠀⣀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢷⣶⣶⣾⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡾⢀⣴⣿⡶⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⡟⣿⠠⠍⣻⣿⡇⢸⡟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣿⡿⠀⣾⠅⣠⠟⠘⠓⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣿⡿⢁⡞⢁⡼⠁⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡾⠃⠻⠴⠋⠀⠀⣠⣶⣀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠐⠛⠛⠕⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⣦⡀⠀⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠿⣿⢧⡄⠃⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠿⣾⣿⣶⢶⣦⣶⣶⣶⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⣾⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⡿⢡⣯⣿⣿⣿⢿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠈⠋⣿⡿⣟⢸⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⣹⣿⡿⢄⠚⣫⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⣰⡿⢛⣿⠞⠻⢿⣿⣿⣿⡿⠟⠋⠉⠛⠿⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠐⠀⠀⢾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠐⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠈⣹⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠎⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠈⢇⠀⠀⠀⠀⢠⣤⣶⣿⡇⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⢸⣾⣦⣼⣷⣾⣿⡿⠉⠀⠀⠀⠀⠀⠸⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟⠀⠀⠀⠈⠟⢻⣿⣿⣿⢿⡇⠀⠀⠀⠀⠀⢀⣄⠀⠘⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⠛⠉⠁⠀⢻⡀⠀⠀⠀⠀⣌⣥⢀⣄⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⡄⠀⠀⠀⠀⠀⢀⣶⣛⣿⣷⣄⡀⠀⠀⠀⠀⠸⣷⠄⢠⡀⠀⠀⢴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣽⠟⠁⠀⠀⠀⠀⠀⢸⣏⣨⣻⣿⣿⣿⠀⠀⠀⠀⠀⠶⠚⢏⡀⠀⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⣟⢻⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣏⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡦⢤⣤⣀⡀⣀⡴⠛⠛⠋⠀⠀⠀⠀⡀⠀⢀⣀⠀⠀⠀⠀⢀⡀⠀⢀⢀
⠀⠀⠀⠀⠀⣠⣾⣿⣈⣿⡷⣾⣿⣽⣿⣿⡧⠈⠀⠀⠀⠀⠀⠃⠀⠰⠶⠄⠠⠶⠖⠶⠺⠅⠶⢿⡿
⠀⠀⠀⠀⢰⣿⢿⣭⢋⡀⠐⠀⠧⢢⠙⣿⣿⣿⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠚⢹
⠀⠀⢀⣠⣿⣿⠉⡗⢛⠄⠀⡀⠀⢈⡁⢸⣿⣶⣿⡇⠀⠀⠐⣾⠀⠐⣦⠛⠀⠀⠘⠃⠀⠀⠀⢀⣿
⠀⠠⣿⣿⣿⣟⣶⣧⡀⠄⠚⠡⢀⠀⣡⣾⣿⣿⣟⡇⠀⠀⠀⣿⠀⠀⣿⡇⢠⡀⠀⠀⣠⡄⠀⠸⢻
⠀⠀⡙⠛⢿⣿⢿⣧⡔⠤⣶⢆⣄⡧⠈⢿⣿⠹⠟⡇⠀⠀⡸⡇⢀⠀⣿⡇⠈⠉⠀⠀⠈⠁⠀⠀⠘
⠀⠀⠈⠀⠘⣿⣞⠙⣷⡿⠀⢀⠀⠁⠂⣸⣿⣆⣃⡇⠀⠀⠐⠷⡦⠶⠿⠷⠀⠀⠠⠦⠀⠀⠀⠈⠂
⠀⠀⠀⠀⡀⠹⣮⣗⣻⣧⣀⠐⢂⣇⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤
⠀⠀⠀⠀⣷⡀⠘⢿⣿⣿⣿⣿⣿⣯⣹⠏⢙⢛⣦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⢿⣟⠠⣀⠀⠉⠙⠛⠋⠉⢀⠀⢸⣋⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⣺⣿⠿⣿⡿⣦⢶⣶⡶⢾⣏⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢴
⠀⠀⠀⠀⢩⣥⣆⢿⣷⣴⣽⣟⢲⡄⢉⠀⢰⣾⣷⡆⠀⠀⡖⢂⣶⢰⠀⡆⡖⡶⣦⣖⠠⡆⠀⢀
⠀⠀⠀⠀⢰⣮⣉⣽⣿⣉⣹⣿⣈⣑⣾⠀⢈⣴⡛⡇⠀⠀⠳⡌⣽⢾⡠⡧⡧⢿⠏⣯⠸⠇⠀⠘⣧
⠀⠀⠀⠀⢸⡿⠿⣿⣷⠿⢻⣿⢿⠋⢠⣤⡀⣿⣿⡇⠀⠀⣀⣀⡀⣀⣀⢀⢀⣀⣀⡀⡀⢀⡀⢠⠹
⠀⠀⠀⠀⢸⡇⣿⢿⣿⣷⣿⣷⢾⡦⠀⣸⠳⢻⣏⡁⠀⠀⣟⣿⣿⡏⣿⢸⢸⣉⣈⡀⠉⠘⠃⠀⢰
⠀⠀⠀⠀⢸⣧⣽⣿⣿⣭⣻⠟⢁⣠⡞⠋⢰⡬⡿⠁⠀⠀⠟⠟⠛⠳⠟⠘⠚⠟⠛⠇⠛⠚⠂⢰⣿⡆
⠀⠀⠀⠀⢹⡟⡿⣿⣷⠟⠁⣴⢿⡿⢆⣀⣰⠟⠁⠀⠀⢠⢤⠄⣤⠤⡤⠠⣤⣤⣤⡤⠀⡤⠀⢸⠻⡇
⠀⠀⠀⠀⢱⢶⣿⣻⡟⠀⣼⠷⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠚⠘⠙⠘⠛⠊⠉⠉⠉⠁⠉⠉⠀⢸⣀⡇
⠀⠀⠀⠀⢸⣷⣶⡿⠇⢀⣅⣀⠁⠀⠀⠀⣲⠆⢐⢒⣖⡒⣲⠒⡒⢲⣶⣶⡖⢲⣂⠀⠀⠰⠆⠾⠟⠃
⠀⠀⠀⠀⢸⡟⣃⠀⠰⠾⠿⠋⣀⡀⠀⣀⣀⡈⡀⠀⠈⠁⣀⣀⣀⣀⡀⠀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠸⠟⠻⠷⠖⠤⠶⠾⠿⠿⠦⠬⠿⠦⠥⠾⠶⠤⠬⠷⠿⠿⠷⠦⠼⠿⠿⠿⠿⠦⠼⠿⠿⠇

⣿⣿⠟⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⣴⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⣯⣽⣟⣛⣯⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠄⣠⣾⡿⠉⠈⠉⠀⣼⣷⡒⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣼⠋⣹⣧⣀⣀⣀⣀⣠⡍⢻⡎⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢻⠀⢨⣿⣻⡝⠩⢉⣇⠀⢸⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⠀⠈⠙⠛⠛⠋⠋⠹⡄⢸⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⣏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⡏⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠃⠗⠢⢄⣀⠀⠀⣀⠠⠖⣧⣿⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠇⠀⠀⡈⠙⠋⡀⠀⢰⣿⣿⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣸⢤⡐⣤⡔⣷⡄⠠⡳⣤⡎⣿⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠻⣄⣈⣙⣶⡴⣶⡾⣷⣋⣉⣉⡽⣿⡿⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢾⣿⣿⠿⣧⠞⢳⣾⠋⠻⢿⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⣤⠤⣿⡶⠾⠃⢤⠔⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣞⠉⠻⣄⠀⠀⠀⠤⠾⠒⣋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠠⠴⠖⠒⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣆⠀⠀⡀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⣉⠛⢿⣤⣶⡇⠀⠀⡠⠀⠀⢸⣂⡶⠒⠀⠀⢶⣾⣿⠃⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠤⣤⣀⠀⠐⣾⣿⣿⡿⣆⠸⣿⣿⣶⣄⣷⠀⠀⠀⠻⠿⣤⢞⣿⠸⠟⠂⢀⠛⠋⠁
⠀⠀⠀⣀⣤⣤⣤⡈⠛⠻⠀⠙⢏⣽⣿⣿⣷⣷⣮⡟⣻⡟⢷⣄⠀⠀⠀⢠⣽⠫⠐⠉⠒⣄⠹⣿⡇
⠀⠀⠀⢿⠿⠛⠉⠔⢠⣾⣿⣿⢾⣿⡁⡀⠀⠈⢻⣷⣿⣧⣈⠙⠿⠲⠿⠛⠃⣠⣿⣿⡄⣿⢷⡌⠛⠦
⠀⠀⠀⠀⠀⠀⠀⣤⣄⠸⠿⣿⣾⣿⣾⣿⣷⢀⡄⣿⣿⠟⣣⣆⢀⣠⣤⣀⡈⠻⣿⣿⡇⠹⣴⠇⠀⠐
⠀⠀⠀⠀⠘⠛⣿⣿⠟⠁⣰⣿⣿⣿⢿⣿⣿⣾⣿⡿⠟⢠⣿⠉⡘⣿⣿⣿⣿⡄⠈⠉⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠟⠀⠈⠉⠉⠉⡀⠀⢾⣇⣀⠆⠀⠙⠛⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠛⠁

⠀⠀⠀⠀⢀⣴⡄⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⣠⣦⡀⠀⠀⠀⠀⣀⣶⣄⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⣰⣷⡀
⠀⠀⠀⢠⠾⣿⠷⡄⠀⠀⠀⣴⣿⣿⣄⠀⠀⠀⣤⣿⣿⣧⡄⠀⠀⢠⡿⠛⢿⡆⠀⠀⠀⣠⣿⣿⡀⠀⠀⠀⣴⡛⠙⣳⡄
⠀⠀⠀⢸⠀⢸⠀⡇⠀⠀⠀⡟⠁⠈⣿⠀⠀⠀⡍⡏⠈⡉⡁⠀⠀⢸⡇⢶⢸⡇⠀⠀⠀⠹⠹⠹⠋⠀⠀⠀⣿⡁⠀⣿⡇
⠀⠀⠀⢸⠀⢸⠀⡇⠀⠀⠀⣇⣀⣀⣿⠀⠀⠀⢰⣷⢀⣧⠁⠀⠀⠸⣧⣤⣼⠇⠀⠀⠀⣤⣤⣤⡤⠀⠀⠀⣿⣂⣀⣾⡇
⠀⠀⠀⠘⠛⠛⠛⠓⠀⠀⠀⠋⠉⠉⠙⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠚⠛⠛⠃⠀⠀⠀⠛⠉⠉⠙⠃
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢰⣿⢿⣿⡆⠀⠀⠀⣰⣿⣷⡀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⣿⣿⣟⠀⠀⠀⠀⣠⣾⣦⡀⠀⠀⠀⢾⣿⣿⣿
⠀⠀⠀⢸⡁⠇⢸⡇⠀⠀⠀⢹⠇⢻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⣿⣧⠀⠀⠠⢛⠋⢛⠣⠀⠀⠀⠨⡇⢘⡇
⠀⠀⠀⢸⠁⡄⢹⡇⠀⠀⠀⢸⡇⣸⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⡿⠁⠀⠈⣿⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⣇⣨⡇
⠀⠀⠀⢸⠷⠶⠾⡅⠀⠀⠀⠹⣷⡿⠁⠀⠀⠀⠀⠛⡙⠁⠀⠀⠀⣧⣀⣀⣀⣿⠀⠀⠀⠀⢿⠆⠀⠀⠀⠀⠸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⢛⠻⣛⡛⣻⣟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠖⠚⠛⠛⠛⠲⠶⢤⣄⡀
⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠑⠄⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠐⠒⠦⠄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣠⣤⣀⣤⣄⣠⣤⣤⣀⣀⣀⣀⣤⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢰⡏⢰⡿⡇⢸⢿⡄⢸⣿⠁⣿⣿⠀⢹⡆⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠹⣄⡉⠁⠈⠋⠀⠈⠋⠀⠈⠁⣀⡼⠁⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠈⠙⢷⡄⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⢀⣀⠤⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠐⠦⠤⣀
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠀⢸⡇⠀⠀⠀⢠⡇⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠈⢷⡀⠀⠘⡇⠀⠀⠀⠈⡇⠀⢠⡟⠀⠀⠀⠀⠀⠱⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⣧⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠓⠲⠿⠶⣤⣤⣼⡷⠶⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠳⢦⣤⣤⣤⣤⣤⡴⠾⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣤⣀⣀⣀⣀⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢦⣤⣤⣤⡤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⣴⡿⢶⡀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣶⣜⣧⡀⣓⠋⣈⢟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣶⡄
⠀⠀⠀⢶⣶⣶⣶⡏⣿⣷⣿⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠀⠉⠁
⠀⠀⠀⠸⣿⣿⣿⡇⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣆
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠏⣠⣤⡀
⠀⠀⠀⠀⠘⠛⠿⠿⠿⠿⠟⠛⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡷
⠀⠀⠀⣀⣜⠒⠶⠶⠴⠶⠖⢒⣄⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡙⠟⠁
⠀⠀⠀⠛⠻⠿⠶⠶⠶⠶⠾⠿⠛⠃⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⢿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷
⠀⠀⠀⢦⣄⣀⣀⠀⠀⠀⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠏
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢰⠸⠸⠰
⠀⠀⠀⠀⠀⠈⠂⢀⡠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣾⣾⣾⣾⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡆
⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣶⣶⣶⣶⣶⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠦⢥⠀⣸⣿⡄
⠀⠀⠀⠉⠁⠀⣀⡀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣌⣓⠒⠲⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠘⠛⠃⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠐⢦⣍⣉⣉⡽⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⢀⣛⠛⠃⠀⠀⠀⠠⡄⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⢛⣿⣷⣾⣿⣿⡿⠀⠀⠀⠀⠀⣠⠄⠀⣿⣿⣿⠏
⠛⠛⠃⠀⠀⠃⠀⠈⠀⠀⠀⠀⠀⠙⠿⡏⠀⠈⠀⠀⠀⠀⠀⠉⠻⠿⠛⠋⠁⠀⠀⠀⠠⠀⠈⠀⠀⠛⠋⠁
⡶⠒⢠⠰⣤
⣶⡆⣿⠃
⠉
⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿
⡿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⢻
⣷⡿⠉⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿
⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠘
⠀⣀⣶⣤⣄⣀
⣾⣿⣿⣻⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⡀⠀⣰⣾⣦
⣿⣿⣿⡟⢹⣻⡿⣿⣷⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⠀⠀⠀⠀⣼⣿⣿⡟⠀⠀⠋⠁⡔⠐
⣿⣿⣿⣿⣿⣿⣟⣿⣾⡷⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠤⡀⠀⠀⢰⣿⣿⣿⣿⣷⡀⠀⠀⢿⣿⣿⠃⠀⠀⠀⠀⠀⢤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⢰⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣶⣶⣶⣶⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣼⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿


⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⢰⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣶⣦⣶⣤⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣼⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠀⠀⠈⠉⠁⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢎⣠⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣩⠆
⠀⠀⠀⠀⠀⢠⣬⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⡄
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⢹⢟⠛⠛⡻⠏⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⡿⠿⣿⠿⡆
⠀⠀⠀⠀⢸⡉⣏⠉⢹⠉⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢢⡔⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣇⠀⡇⢸⠁
⠀⠀⠀⠀⠀⣇⢹⠀⢸⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡠⠂⠐⢄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢹⢀⡇⡏
⠀⠀⠀⠀⠀⢹⣼⣤⣾⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠷⠶⠶⠾⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡙⡆
⠀⠀⠀⠀⠀⣠⣼⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣅⡀
⠀⠀⠀⠀⢸⡇⡿⠿⠿⡜⣧⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣼⣿⣇⣧
⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⢸⢿
⠀⠀⠀⠀⢼⡘⣤⣤⣤⡌⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢹⡏⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⣳⣶⣊⡼
⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠠⣾⣷⣼⣧⣾⣷⠄⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣦

⠀⠀⠀⠀⠀⠀⡴⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠲⡄
⠀⠀⠀⠀⠀⠀⢣⣶⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⡁
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠿⠿⣿⠆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣦⣀
⠀⠀⠀⠀⢹⢏⠉⠉⠙⣹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢣⡜⢹⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡛⡟⠛⢿⢻⠋
⠀⠀⠀⠀⠈⡆⠑⣄⠔⢁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣁⣈⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⡟⡼
⠀⠀⠀⠀⠀⢳⡔⠉⢢⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣧⣀⣧⡇
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠠⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡈⠂⢁⣠⣤⣤⣄⡀⠐⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠿⣿⣯⡿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠂⠀⡴⠋⠁⠀⠀⠈⠙⣦⠀⠐⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢋⣴⣿⣿⣿⣷⣦⠹⣿⣯⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣈⠀⢸⠃⠀⢀⡀⢀⡀⠀⢸⡇⠈⡁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⡇⣾⣿⡿⣿⣿⢿⣿⡇⢹⣿⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠠⠀⠘⣇⠀⠀⠀⠈⠀⠀⣼⠃⠀⠄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⠹⣿⣿⣿⣿⣿⣿⠇⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢁⠄⠘⣦⠀⠁⠀⠀⣼⠃⢠⡉⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡾⣿⣧⠹⣿⣿⣿⣿⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠐⠁⢸⡆⢸⡇⢸⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣯⣿⡇⢹⡇⢸⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⢠⡋⢲⠖⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠟⠉⠁⠀⠀⠀⠉⡑⠂⠀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⣸⠓⣴⠂⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⢀⡀⠄⠠⠤⠀⡈⠀⠀⠀⠒⣀⢤⡀⠀⠀⠀⠀⠀⠀⠠⠏⠉⠁⠀⠀⠀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠐⢺⣀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠁⢀⠄⡪⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢱⠎⠀⠀⠀⠌⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢂⠀⠀⠀⢊⠤⠒⢉⠀⠀⠤⠈⠑⢄⠀⠁⢂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⢠⠀⠀⡄⢀⢔⠁⡚⠉⡀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠁
⠀⠀⠀⢀⠀⠀⢀⠄⠂⢘⢟⠛⠒⠐⠄⡅⠀⠀⡂⠄⡀⠀⠀⠁⠀⠀⠀⠀⠀⢀⢊⢔⠡⢒⠅⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⣀⣴
⠀⠀⢀⠊⠀⠔⠁⠎⢔⠕⡱⡪⠂⠡⡂⢀⠀⢈⠔⡨⢂⠄⠀⠔⠁⠞⠝⡐⠀⣾⢐⣡⠎⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⡌⠋
⠀⠀⠆⠀⡌⠀⠀⠀⠀⠈⠀⠁⠁⠊⠢⠘⣆⠠⢊⠔⡡⡢⡢⡀⠌⠚⢈⣩⠁⠛⠶⣅⠨⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⣀⠀⠀⢈⠁⠀⠀⠠⠂
⠀⠀⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⢢⡁⠪⠊⢈⠋⠀⠀⠘⠉⠊⢑⠠⠀⠈⠓⡐⠡⠀⠀⠀⠨⡋⠆⠀⢀⡤⡠⠈⠐⢄⠑⢌⠂
⠀⠀⠀⠀⢃⠀⠀⢂⠀⠀⠀⠀⡊⣂⠕⠊⢅⠠⠄⠾⣷⠃⠀⠀⠀⠀⠀⠀⠁⠂⢕⠀⠀⠘⡎⠴⠀⠀⠅⡂⢀⣠⢖⠎⡁⠀⠄⠀⢡⠀⢂
⠀⠀⠀⠀⠈⡄⠀⠐⡂⠔⡠⢀⢀⠂⠀⠀⠀⠠⠐⠠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡷⠠⠀⠀⠠⠈⠫⣋⠄⡪⢐⢅⠀⠀⠀⡆
⠀⠀⠀⠈⢂⠘⢄⠀⢐⢅⡂⢀⠍⠀⠀⠀⠀⠄⠊⠅⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠔⠅⠀⠀⠈⠀⢱⡨⢐⠕⡠⠂⠀⠀⡆
⠀⠀⠀⠈⠢⢈⠢⡑⠤⡁⠂⢐⢧⠀⠀⠀⠀⠁⡂⠅⣣⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣟⠀⠠⠀⠀⠀⠀⠀⣆⠅⡊⢔⢴⠀⠠⠀⡀
⢀⣀⡰⠂⠀⠂⠀⠀⠀⠀⠁⠈⠘⡤⠀⠀⠀⠀⠐⠡⡹⣆⠀⠀⢂⠄⠠⡀⢀⠄⡠⠀⠀⡼⢑⠡⠊⠀⠀⠀⠀⢀⡡⢊⢔⢅⠕⡠⢃⠔
⠰⠧⡨⠛⠀⠀⠀⠀⠀⠀⠀⠀⢦⡘⢄⠲⢄⠀⠀⠀⠐⠋⠁⢄⠒⡥⠮⠂⠅⡊⠔⣡⣶⣶⣷⠃⠀⠀⠀⡠⢀⣼⡮⢖⣰⡥⠖⠴⠃⣤⣠⡤
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠈⠂⠑⠠⢄⡀⣐⣀⢀⡔⠁⢉⠘⣿⣤⡅⠂⠉⣄⡛⢀⣤⣀⠀⠀⠀⣠⢈⣉⣩⠭⠶⠋⠀⠀⠺⣯⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠇⠎⠀⠐⠀⠉⠃⠁⠀⠀⠁⠩⣤⣤⠤⠴⡠⠂⠀⠀⠠⠐⣋⠉⠴⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠈⡄⠀⡇⠀⠀⠀⠀⡘⠀⠀⠜⠁⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠘⡀⢁⠀⠀⠀⢠⠁⠀⡌⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⢡⠈⠀⠀⠀⡄⠀⠜⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠆⠀⠀⠰⠀⠜⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡈⢰⡆⠂⡌⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⣧⡐⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢰⣆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣼⠿⠿⣿⣤⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁⢀⡀⠀⠈⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠉⠁⠀⠆⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⡏⠀⠀⡤⠀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣴⡄⠀⠀⠀⠹⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠚⢹⡿⠛⠃⠀⠀⠀⠀⠹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠯⠐⠋⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠄⠒⠀⠂⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢣⡀⠰⢶⣾⠛⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣉⡁⣀⢀⣤⣤⣴⠾⠉⠱⢦⣄⣀⠀⣀⠞⠋⠈⣿
⠀⠀⠀⠀⠀⠤⠠⠠⠤⠦⣀⠀⣉⡉⠛⠉⠛⠋⠀⠀⠀⠀⠀⢿⣟⠓⠁⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣾⢾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⣿⣇⡀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠘⢋⣠⣱⣶⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠉⠀⠀⠀⠀⠀⠀⣿⣍⡗⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠄⠈⠉⠉⢉⣉⡀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠉⠀⡀⠀⠀⠀⠀⣸⣿⠛⢠⠀⠀⢀⡴⢉⡶⠄⠀⠀⠐⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⠠⠀⣴⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⢻⡀⣠⠜⠳⢨⡇⠀⠀⠀⠐⢯⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠀⠀⠀⠀⢢⣼⠟⣿⡿⠁⠆⠀⠀⠀⠀⠀⠀⡝⢿⣶⣬⣟⢁⣀⣰⡾⠁⠀⠀⠀⠀⠈⠹⣿⣇
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣧⠀⠀⠀⢰⣿⢃⣼⡿⠀⠐⠀⠀⠀⠀⠀⠀⠀⠻⠋⢼⡻⠿⡿⠛⠋⠁⠀⠀⡆⠀⠀⠀⠀⠙⣿⣄
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡏⠈⠀⠀⢰⣿⣻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠁⠀⠒⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⣿⠆
⠀⠀⠀⠀⠀⠀⠐⢻⣯⡿⠀⠀⠀⠀⢿⣿⢟⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⡀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠈⠓⡀⠀⠀⠀⠘⣇
⠀⣄⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⣶⠾⢋⣾⠗⠀⢀⠀⠀⠀⣄⡄⠀⢠⠀⢀⣼⠶⡇⠀⠀⠀⠀⢀⠀⠰⡀⠈⣆⠀⠀⠄⠰⡄⠀⠸⣆
⠀⠀⠂⠀⠐⠀⢸⣿⣿⠁⠘⢠⣷⠿⠒⣸⣿⠙⠀⢸⠀⠀⠀⠖⠂⠀⠋⠀⣼⠙⠀⠹⠋⠑⠀⠀⠚⠃⠀⠗⠀⠸⡄⠀⠀⠎⠙⢆⠀⢻⡇⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⣿⣿⣏⠀⢀⣾⠇⠀⢀⣿⠋⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠰⣿⣿⠏⠀⣾⣇⢠⠀⣼⠃⡔⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢨⡆⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠈⢁
⠀⠀⠀⠀⠀⣾⣿⡟⠀⣸⣿⢩⡇⣰⡟⢰⠃⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⢾⣿⡇⢠⣿⡏⡾⢀⣿⡀⡈⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⢹⣶⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⣾⣿⣷⣾⣿⢰⢇⣼⠏⠹⡇⠀⠀⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠸⣯⠀⠀⠀⠀⣆⠀⠐⠆
⠀⠀⠀⠀⠀⣹⣿⣿⣿⣧⡿⢰⡿⠀⠀⠇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠙⠟⣛⣟⡛⠁⢸⡇⠀⠀⠀⢸⡏⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣤⠀⠀⠀⠀⢸⠀⠀⠀⣰⣦⡀⠀⠀⢘⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⣰⣿⣿⣿⡏⢠⡎⡁⠀⠀⢠⡿⠃⠀⠀⢰⣅⠀⠀⠀⠀⠀⠀⠀⣼⣧⠋⣶⢶⣀⠀⠀⠄⠀⡄⢻⠈⠉⠲⠀⢀⡤⠀⡀⢤⡂⠈⠃
⠀⠀⠀⢰⡆⣿⣿⠚⢿⣷⣿⡟⣿⡆⠀⠀⡧⣀⡠⣾⡏⠈⣿⠆⣀⡀⠀⠀⢀⣿⣿⠀⢻⠈⢻⠙⣶⡷⠀⣷⠀⢧⠀⠀⠀⠉⢿⠯⠁⠈⠁⠀⠀⠀⣇
⠀⠀⠀⢸⣧⡘⢿⣷⣾⣿⣿⣿⣿⡿⢃⣠⣼⡟⠁⣿⠃⠀⣷⠀⢹⣿⠉⠶⠾⢿⣿⠁⠘⡇⠸⡆⢸⠟⢧⢹⣧⠸⣇⠀⠀⠀⠈⠄
⠀⠀⠀⢸⣿⣷⣌⡙⠻⠿⠿⠟⣋⠁⣾⡟⣿⡇⠀⣿⠀⠀⡏⠀⠀⣿⠀⠀⠀⢺⢻⡇⠀⣧⠀⠁⠀⢧⠈⠛⣿⡄⢹⡀⠀⠀⠘⠃
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⣿⡟⢸⣿⠃⣿⡇⢰⣿⠇⠀⣷⠀⠀⣿⠀⠀⠀⢸⡀⣿⠀⢸⡄⠀⡄⠀⠆⠀⣿⣿⠀⣇⠀⡇⠐⡆⠀⢢
⠀⠀⠀⢸⣿⣿⣿⡿⣸⡟⢀⣾⠁⢸⣿⢸⣿⠀⢸⣷⠀⠀⢩⡀⠀⢹⠆⠀⠂⠈⠀⢿⡀⢸⣿⡄⢻⠀⠀⠀⠸⣿⠄⠸⠀⠱⠀⠑⠀⠘
⠀⠀⢠⣿⣿⡟⠋⠁⠛⠀⣸⡇⠀⣾⠃⣸⡿⠀⣸⣇⡀⠀⣾⡇⠀⢸⠶⠀⢠⠀⠀⡟⠃⠈⡿⠁⢸⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢠⠀⠀⠀⣀⣀⣠⠄
⠀⠀⠈⢻⣿⣿⣦⣤⣄⣀⣟⠀⡸⠃⠀⠛⠀⢀⡿⠛⠃⢀⠙⠁⠀⢰⢀⣀⡘⠀⠀⡟⠃⠀⣇⠀⣠⣤⣀⣴⣶⣾⣿⣶⣾⣷⣶⣿⣿⣶⣿⣿⣿⠟⠁
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣷⣶⣦⣼⣶⣴⣶⣿⣿⣿⣷⣶⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠁
⠀⠀⠀⠀⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠗⠠
⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠛⠛⠛⠛⠉⠛⠛⠛⠛⠛⠛⠛⠛⠙⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⣀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣷⠃⠀⠀⠀⠀⠀⠀⠈⠦⠊⠀⠀⠀⠀⠈⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠛⠂
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡚⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⢀⣞⣒⠒⠸⡄⠀⠀⠀⠠⢄⡀⠀⠀⠀⠠⡀⠐⣑⠀⠀⠀⠀⠀⠀⣀⠤⠤⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢔⠛⠒⠐⠨⠬⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠊⠀⠀⠀⡜⠁⠀⢀⠤⠷⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠊⠓⠒⠚⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠴⠋⠁⡠⠤⢇
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⢀⡀⠀⠘⣯⠀⠉⡆⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠈⢶⡊⠉⠀⠀⡆
⠀⠀⠀⡀⢀⢀⠤⢤⠀⠀⠀⣾⡥⣄⣠⠁⠀⣾⢣⠀⠀⣿⠤⠒⢁⡤⠤⡀⠀⢹⡄⢀⠎⠀⠀⠀⠀⠈⠒⠢⡎
⠀⠀⠀⢆⠀⠁⠀⢰⠀⠀⢀⣿⠀⠀⢸⠀⣼⠧⠬⡇⠀⢻⠀⠀⢸⡇⠀⠘⡄⠀⢿⠊⠀⠀⢠⠤⡤⡄⠀⠈⢺
⠀⠀⠀⠀⠓⢄⡴⠁⠀⠀⠈⠋⠀⠀⠂⢰⡏⠀⠀⢸⠀⠘⠃⠀⢸⡷⠖⠊⠀⠀⣿⠀⠀⠀⠀⠑⠞⠀⠀⠀⠀⡆
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠸⠃⢀⣀⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⢰⡏⠙⢢⠀⠀⣸⠇⠀⣾⠉⠓⡄⠈⢹⣿⠀⢠⡄⠀⢀⢸⡇⠈⠳⡀⠀⠀⠀⣦⠀⠀⢠⠀⠀⡆
⠀⠀⠀⠀⠀⢸⣇⣠⣞⠀⠀⣿⠀⠀⣿⡤⠚⠀⠀⢸⡏⠀⣸⡃⠀⡜⠀⣿⠀⠀⡇⠀⣤⠀⠘⣧⢀⠇⠀⠀⡇
⠀⠈⠑⠒⠀⠸⡇⠀⠈⡇⠠⠿⠀⠀⣿⠈⠢⠀⠀⠘⠇⠀⣿⠉⠉⡇⠀⣿⣀⠜⠀⣼⠋⡆⠀⠸⣟⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⣿⣀⠴⠃⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠛⠀⠰⠇⠀⠈⠀⠀⣼⠋⠉⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠪⠇⠀⠀⠁⠀⠀⠁⠀⠀⠁⠀⠀⠀⢠⢤
⠀⠀⠀⠀⣠⠀⢀⣈⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢱⠀⠀⠀⠀⠀⡖⠒⢳⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠈⠀⠀⠀⡇⢹⠒⡤⢄⡁⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠨⠤⠦⢒⣚⡆⠀⠈⠲⠋
⠀⠀⢀⠀⠀⠀⠀⡇⢸⠀⣡⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠒⣓⣋⠭⠕⠒⢣⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠉⠁⠀⠀⠀⡇⣸⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⢸⡉⢀⣀⣤⠤⣄⣾⠦
⡠⠖⠑⢄⡀⠀⣠⠏⠀⠀⠀⠀⠈⢱⡖⠢⡀⠀⠀⠀⠉⠀⠀⠀⠘⠙⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠒
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡤⠤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠛⠉⠉⠁⠀⠀⠀⠙⢋⣀⠉⠛⣹⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠈⠁⠈⠁⡀⠀⠀⢠⣄⠀⢡⠀⠀⠀⠀⠀⠀⠨⠟⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⢒⠉⠀⠈⡦⢄⡀⠄⠁⠀⠉⠊⠥⠤⢤⣚⡀⠀⣀⠀⡀⢠⡀⡐⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢯⠀⣀⠄⣤⢢⡴⠞⠻⠀⠂⠘⠋⠛⠀⠀⠲⠀⣤⣝⡳⠤⡁⠆⡠⠄⠀⠀⠀⠌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠁⠀⠉⢈⣴⣣⠶⠀⠀⣸⡦⠀⢀⣂⡀⣶⡖⣀⡶⠀⠀⠀⠈⠱⣌⠢⠀⠀⠀⠰⠄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢘⠰⠃⡞⠁⠁⠄⢲⣦⡭⠀⠀⠈⠉⠓⢈⡀⠉⠉⠄⠀⠃⠀⠀⠀⠀⠉⢀⢀⠀⠀⢀⠘⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠈⠀⠈⠀⡜⢀⣤⣄⣤⠀⡿⠿⠻⠧⠀⢀⣀⣾⠃⠀⠀⠆⠀⢠⡦⠀⠘⠃⠰⠈⠋⢸⡃⡀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠎⠀⠀⢀⠘⢰⡇⠈⠀⠈⢉⣤⡁⠀⠀⠀⠀⠁⠀⢠⡄⢡⡀⠀⠖⢀⡄⠀⠀⠀⠀⣷⠈⡀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠉⠀⠀⠠⠤⡇⣿⠀⣼⠗⠀⠉⣩⣵⢤⣤⣚⠀⠀⠄⠀⠀⡄⣳⡀⠀⠀⠀⡐⠂⠀⠗⢾⡆⠂⠀⠀⠀⠂⠸
⠀⠀⠀⠀⠀⠀⠀⢐⠦⠌⠀⠀⠀⠀⠀⠀⠀⠁⣿⠀⠁⠀⠠⠀⠛⠀⠀⣀⠻⢠⡇⠀⠀⠐⠀⠛⠀⠀⢀⣴⡁⠀⠀⢰⡈⡇⠀⠀⢀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⣧⠀⠀⠀⢡⡜⠃⣿⠀⠉⠀⢲⢀⠀⢤⣀⠀⣤⠛⣿⠀⠀⠀⡀⠀⠘⠀⠀⠉⠉⠀⠸⢯⡀⡇⠀⠀⠊⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⣧⠀⢀⠀⠅⡄⣿⠁⠀⠀⠃⣸⣆⢰⠁⠀⠀⠙⣿⢰⡄⠈⠁⣶⣤⠀⠀⠀⠀⠀⠀⠘⣱⡇⠠⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠬⢿⠀⡜⠁⠘⠀⣿⠀⠶⠀⠈⣩⠍⠀⢀⡀⣠⠀⣉⢀⠀⠀⠫⡟⠀⠀⢹⣄⠰⠆⠠⡔⣿⢠⠀⠐⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠁⢈⠙⡄⢿⡄⠀⠀⠀⠁⠀⠀⣈⠁⠀⠄⢹⠸⠀⠋⣠⠀⠀⠀⣈⣙⠀⡀⠀⢸⠃⡌⣃⠰⠆⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡄⠀⠀⠀⢻⡟⠂⠀⢷⢸⡇⢶⠉⠀⠛⠂⣾⢀⠀⠀⣹⣷⠀⠀⠈⠀⠀⠋⡠⠃⢈⣅⠰⠐⢏⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠒⡄⢺⡀⠀⠀⠀⠀⠰⢃⠀⠄⠑⢠⡼⠘⠖⢢⡧⡁⠀⠀⣀⡼⠟⠋⠀⠁⠀⠀⠴⣢⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠘⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠩⣀⠛⠛⣂⡄⠀⠂⢀⠀⣘⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⣀⠀⠀⠈⠙⠓⠶⣠⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣄⠀⣬⠵⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⢢⣤⡄⢀⡄⠑⣤⢬⣤⠶⠓⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⡄⠂⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⢶⠌⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠁⡀⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠂⣰⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡄⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠉⢲⡸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⠤⠰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠻⠀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⡉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣸⠲⣿⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠙⠀⢻⣈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡶⢂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⡐⠀⠇⠀⠀⢄⣿⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠠⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠙⠿⠁⠘⠀⠀⠀⠜⠁⠐⠀⠰⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠛⠀⠀⠴⠷⠦⢀⣀⣀⠤⠊⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣥⣤⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠣⠋⠀⠀⠀⠀⠀⢠⡞⡙⠀⣀⣠⣞⣠⠞
⠀⠀⠀⠀⠀⠀⡠⢶⡄⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⣼⠏⠀⠐⠃⢀⡴⠄⠀⣾⡇⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⢰⠀⢸⡇⣶⡀⢀⣴⠟⡗⣼⠏⣀⠀⠀⠀⢀⡟⠀⣰⡷⠀⣼⣁⣄⢰⣿⢤⡀⢠⡟⠀⠀⠀⠠⣴⡄
⠀⠀⠀⠀⠀⠈⠀⠈⠷⠉⣷⠁⣿⡰⠀⠻⠊⣿⠀⠀⠀⠸⣇⡜⠸⡧⠂⡠⣻⡇⢸⡇⢾⣡⣿⠀⠀⠀⠀⣴⠏⣤⡄⢀⣼⠗⢢⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠹⣆⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠉⠀⠀⢀⠎⠀⣿⠀⠈⠀⠀⠐⣧⡐⠀⠀⠀⠿⠊⣿⠕⣹⢇⡖⣹⠇⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢻⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡏⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⢀⣀⠀⢰⡏⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣼⠃⠀⢠⢦⣀⠊⣿⢀⢾⡇⠀⠀⠀⠘⠤⠟⠁⠀⠀⠀⢠⡞⠁⠀⠀⠀⣰⡟⠍⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣏⠀⢻⠁⢸⠃⠀⡀⠀⠀⠀⠀⢦⣀⠀⢠⡟⠀⠀⠚⠀⣰⡟⠃⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢻⡀⢸⡇⣿⠀⣼⡏⣴⣆⣀⡀⠀⠀⠀⣾⠁⣠⡶⢀⣴⡟⠀⣰⢟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠸⠇⢻⠌⠸⠟⠰⡏⠀⢹⠀⠀⢰⡇⡔⢿⡵⢋⣿⠦⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣿⠀⠀⠀⠀⠘⠿⠁⠀⠀⣼⢃⠇⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠟


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⣠⢞⣍⠓⠦⢄⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⢀⣤⡀⢠⡄⢀⣤⡀⠀⠀⠀⢀⠀⣤⠀⡀⠀⠀⠀⢀⡠⠒⢊⣭⠓⣄
⠀⠀⡼⣡⡘⣿⡇⣾⠆⣙⢦⡀⣄⠹⣆⣿⣸⢇⡀⠀⠴⣶⡝⣷⢸⡗⣿⢡⡶⠢⠀⢠⡹⣆⣿⣰⢏⡤⢀⠔⣡⢸⡇⢸⡿⢡⣌⢧
⠀⢰⢣⣽⢿⡘⣷⢻⢸⢏⣤⠱⣌⢳⣹⣿⣿⡿⠰⠛⢷⣌⣿⢿⢿⡇⡇⡞⣠⠞⢛⠎⢿⣿⣿⣯⡟⣱⠣⣦⣹⣿⡇⣾⢡⠟⣡⡸⡆
⠀⣾⣨⣟⠷⡽⣼⣿⣼⣾⣫⣶⠸⡆⢻⣿⣿⢑⣛⣛⢷⣹⣾⡿⣼⣇⡷⢣⢣⡾⣛⣛⠊⣿⣿⠏⢰⠣⣶⣽⣿⣿⢷⣷⣫⢞⣫⡄⡇
⠀⢿⢨⣭⣓⢮⢿⣿⣿⣿⣿⣷⡆⢻⠀⣝⡃⢘⣭⣭⡳⣽⣿⠣⠻⡿⠃⢠⣯⣟⣭⣭⡃⢸⣿⠀⡏⠶⣮⣻⣿⣿⣿⣟⡵⣫⣭⡄⡇
⠀⠸⡆⣭⣭⣿⣗⠽⣻⢿⣿⣿⡷⢸⠀⢸⡇⠘⣥⠶⣿⣿⣿⠀⣾⣷⠀⣸⣿⣿⠶⣌⠃⢸⡇⠀⡇⢿⣿⣿⣿⣛⣿⣻⣯⣽⡦⢸⠃
⠀⠀⠳⡀⠾⣳⢏⣰⣿⣷⠟⣿⠃⡾⢠⠟⣰⠆⣳⠞⣡⠤⡌⠑⡿⢷⠊⣠⠤⣌⠳⣎⠸⣏⢻⡄⣧⠙⣟⡻⣾⣿⣿⡻⣿⠷⢠⠃
⠀⠀⠀⠑⢄⡃⠸⠔⣿⣹⣮⡥⣾⣷⣯⣾⣥⢤⡿⡌⠷⠖⣸⠀⡇⢀⠰⣇⠲⠞⣰⢿⡠⣬⣷⣿⠾⣷⢭⣿⣃⣏⠢⠃⣉⡴⠁
⠀⠀⠀⠀⠀⠉⠻⣉⡹⡆⢻⢿⡯⢞⡻⣿⠖⢸⣧⠽⣗⣊⠁⡰⢡⡜⣆⠨⣅⡽⢿⢾⡇⠞⡿⣚⡳⣿⡿⡟⣺⣄⡭⠟⠁
⠀⠀⢰⡖⣤⣶⡄⠈⣿⠇⣀⠸⡇⢏⣙⡼⢀⣏⡇⣎⠳⣠⠎⠀⣸⡿⢈⠳⡀⠞⡽⣠⢳⡈⢇⣃⡝⣸⠁⣀⢰⣟⠀⠴⣶⣄⢲⡄
⠰⣶⣜⡇⣿⠁⠀⢀⡁⢀⡟⠀⢙⡢⠤⠔⢊⡞⠗⣩⠞⣡⣙⣿⣿⣇⡿⣡⣌⠲⣅⠿⢱⡑⠂⠴⣚⣁⠀⢿⡀⣀⡀⠀⢸⣿⣼⣡⡾
⢀⡤⢭⣓⣹⡇⢠⣜⡇⣾⣣⡾⣛⣉⡀⠀⠈⢀⠞⠡⠶⣮⣻⣾⣿⣿⣷⢏⣴⠶⠌⢣⡀⠉⠀⣠⣍⣛⢷⡼⣷⣿⣧⡄⣸⣋⣿⣿⣤
⠈⣵⢩⢤⡍⠻⡌⣿⣿⣿⣫⢞⣫⣭⣅⠀⢠⡏⢐⣛⡻⣮⢷⣿⣿⣿⣿⡾⡵⣛⣙⡀⢳⡀⠀⣬⣭⣝⡳⣽⣿⣿⣿⢰⠟⢩⣤⡙⣮
⠀⠹⢬⠴⠃⣤⣧⣀⣹⣿⡗⠭⠷⣤⣅⠀⣾⠀⣭⢭⣽⣿⣿⡿⣻⣝⣿⣿⣿⡭⠶⣭⠀⣧⠀⣩⡷⠦⢭⣎⣾⢿⢻⣟⡦⠹⢤⠴⠋
⠀⠀⠀⠀⣘⣾⣿⡧⢟⡽⠶⢽⡻⠦⠁⢠⢻⡀⢴⣪⢿⣿⠥⣤⣿⣿⡽⠿⣈⣿⣝⠆⢀⢻⡄⠀⡴⢻⡽⠿⢿⡳⢸⣿⡴⣃
⠀⢠⣶⠶⢬⣻⢿⣧⢸⠱⣏⣠⡟⠀⢀⡾⠈⣧⡀⠁⠿⣇⣰⡆⣿⡏⢰⣇⡼⠻⠉⢀⣾⣼⣷⡀⠀⠹⣌⣹⠊⣇⣿⣿⣟⡵⠶⣶⡄
⠀⠘⠻⠂⠀⠈⠁⠂⠙⠷⢤⣤⣤⠴⠋⠟⠾⠳⠑⠦⣤⣀⣤⠴⠃⠘⠦⣤⣤⡤⠖⠛⠏⠾⠹⠛⠦⢤⣤⡤⠾⠿⠟⠛⠁⠀⠐⠛⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆⠐⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠛⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠛⠳⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣀⣉⣀⣀⣀⣀⣀⣀⣉⣀⣠⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⢉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡉⢉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣤⣶⣶⣶⣶⣶⣶⣶⣶⣴⡶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣍⣉⣉⣩⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣸⣯⣽⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⡇⢸⠀⠀⠀⠀⣾⢋⠛⠉⠛⠛⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⣦⠼⠶⠶⠶⠶⠷⠾⠤⠴⠧⠴⠿⠷⠶⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⡤⢴⡄
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⣠⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿
⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⣠⠞⠁⠈⠓⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⡇⢸
⠀⠀⢀⡴⠛⠁⠀⠀⣠⠞⠁⡀⠀⠀⣜⣳⠀⠀⠀⣖⣢⠀⠀⠀⣞⡆⠀⠀⠰⣽⠆⠀⠀⢞⡷⠀⠀⠀⣴⣦⠀⠀⡇⢸
⢀⡴⠋⠀⠀⠀⣠⠞⠁⠀⠺⠽⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⡇⢸
⢻⣄⠀⠀⢰⡊⠀⢠⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣒⡆⠀⡇⢸
⠀⠙⢦⡀⠀⠙⢆⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⡇⢸
⠀⠀⠀⠻⣄⠀⠈⠳⣄⠘⠽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣽⠂⠀⡇⢸
⠀⠀⠀⠀⠘⢧⡀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠙⣆⠀⣛⡆⠀⠀⣞⡆⠀⠀⠀⢔⣲⠀⠀⠀⠸⠽⠆⠀⠀⠺⠽⠀⠀⠀⢺⡙⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠈⢣⡈⠀⠀⡴⠬⠤⠤⠤⠤⠬⠥⠤⠤⠤⠤⠤⠤⠴⠶⠶⠶⠒⠲⠶⠶⠶⠶⠶⠒⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠱⡄⠸⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⡶⠶⠶⠶⠒⠒⠶⠶⠒⠛⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠙⠾⠀⡇⢸⠛⠛⠉⠉⣿⠉⠉⢹⠉⢹⡿⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢠⠁⡯⠀⠀⠀⠀⣿⠀⡄⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣸⠀⡇⠀⠀⠀⠀⡟⠀⡇⠸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⡇⠀⡇⠀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢰⠀⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⡀⢸⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠃⢸⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⠀⠈⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⡄⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⣤⠁⠹⡴⢦⡿⠷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡤⠤⠤⣤⣤⣤⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠄

⠀⠀⠀⠀⢸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⡇⠀⠐⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣷
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⢀⡎⠀⠀⠀⢹⠀⠠⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠤⠠⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣇⠀⠂⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠐⠀⠀⠀⣴⣾⣿⡄⠁⠀⠠⠂⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⢈⠇⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠐⠀⠀⠀⠙⠻⣿⣟⠠⠂⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⡀⠀⠚⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠙⢄⠀⢄⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡀⢄⡀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠉⠛⠛⡿⠉⠀⠀⠈⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⢀⡠⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣰⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⢰⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡆⠀⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠠⠀⠀⠐⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⢠⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⢸⡇⢠⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠁⠀⠀⠀⠀⠠⡀⣈⣦⡆⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠸⣧⠘⣆⢀⠀⠀⠔⣼⠃⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠠⠄⠠⠞⠻⣿⠃⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠪⣳⣦⣭⠷⡾⠋⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⡈⠀⣰⠈⢧⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠰⠗⠊⠁⠀⠈⢧⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⠾⠿⠿⠿⠿⠷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄
⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏
⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⣀⣀⣠⣴⡿⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠨⡆⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⢲⠆⠀⠀⠀⠀⢶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠈⡃⠀⠀⠀⢀⡀⢀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⢸⣿⠘⣧⠰⣷⢀⡼⠁⣀⠀⠀⠀⠀⣀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠀⠀⡀⠀⠀⠈
⠀⠐⠀⢸⣷⠀⣸⠃⠀⠈⢹⣿⡏⠀⠀⠹⡄⢴⡷⢀⡏⠀⠸⣄⢰⡷⠀⠀⠀⠻⣄⢰⡷⢠⡏⠀⠠⣷⠄⢀⡴⠀⢦⣀⠀⣾⡆⣠⠏
⠀⠀⣴⢲⣷⡞⠁⠀⠀⠀⣼⠿⣇⠀⠀⠀⠉⣿⣿⠉⠀⠀⠀⠈⣿⣿⡟⠛⠦⠀⠈⢻⣿⡏⠀⢀⡴⣾⣿⠉⠀⠀⠀⠈⣹⣿⡟⠁
⠀⠀⠘⢾⣿⡇⠀⠀⢀⡾⠁⠀⠘⢧⡀⠀⠀⣿⣿⠀⠀⠀⠀⢠⣿⣿⣇⠀⠀⠀⠀⢸⡿⡇⠀⠈⠇⢸⡿⣆⠀⠀⠀⢠⣿⣿⣿⠀⠃
⠀⠀⠀⢸⣿⡇⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⣿⣿⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⢸⡇⢹⠀⠀⠀⠈⣿⣻⠉
⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠙⠙⠀⠀⠀⠀⠀⠈⠃⠃⠀⠀⠀⠛⠀⠛⠀⠀⠀⠀⠙⠙

⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤
⠀⠀⠀⠀⢀⣨⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠇⠀⠀⢀⣀⠁⠃
⠀⠀⠀⠀⠻⣽⡿⣿⡷⠦⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⢶⡯⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢰⠀⢀⠄
⠀⠀⠀⠀⠀⢸⡗⣦⡀⠀⠁⠀⠀⠀⠀⠐⣿⣿⡾⢿⣄⠀⠈⡁⠤⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠈⠀⣀⡁⠀⠀⢸⡆
⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠀⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠙⢷⠀⢸⣿⣷⣶⣦⣿⠁⠀⠐⠂
⠀⠀⠀⠀⠀⠟⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠰⠟⠀⠀⠀⠀⣼⣧⠀⠀⢠⣿⣧⡀⢈⣽⢷⣤⣭⣿⣿⣿⣁⡀⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⢷⣤⣜⣟⣉⣠⣀⣡⣄⣿⣿⣿⣿⢿⣿⣿⠀⠀⣠⣴⡿
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡸⣿⣿⣷⣷⣦⠰⣷⡶⣶⣶⣶⢶⣤⣿⣷⡶⠿⠛⠁
⠀⠀⠀⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡾⣿⣿⡿⢶⣶⣶⣾⡟⣿⡇⣿⣿⣿⣷
⠀⠀⠀⣽⣅⠀⠀⢀⠀⠀⢀⠀⠀⢨⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡇⣿⣾⣿⣼⣿⣿⣿⡇⣿⡇⣿⣿⣿⣿⣇
⠀⠀⠀⣿⡟⠙⠋⠉⠀⠀⠈⠙⠛⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⡇⣿⡇⣿⢾⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⢹⣿⠻⠇⣿⡇⣿⢸⣿⢸⣿⠀⠈⠀⠀⣿⢸⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠉⠀⠀⠀⠀⠀⢴⠿⠋⠀⠀⣼⡇⠀⠀⠻⠃⠻⢸⣿⢸⣿⠀⠀⠀⠀⣿⠸⣿
⠀⠀⠀⠘⠗⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⢸⡿⢸⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣀⡀⣀⠠⠤
⠀⠀⠀⠀⢀⣴⣆⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢠⣴⣖⣢⣶⣏⠉
⠀⠀⠀⠈⢝⣿⣿⣷⣄⠀⠀⠈⢿⣿⣇⠀⠀⡠⠊⠉⠉⠛⠛⠿⠿⠿⠿⠿⡿⣿⣿⣧⠀⢰⣿⣿⣿⣿⡏⠟
⠀⠀⠀⠀⡿⢻⣿⣿⣿⣷⣤⣄⡀⠙⠛⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠘⠀⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⠀⠀⠀⠀⣀⣠⡄
⠀⠹⣿⣶⣄⣀⡀⠈⠙⠻⠿⠿⠛⠉⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣀⡀⠀⠸⣿⡿⠀⠀⠀⣠⣾⣿⣿⠁
⠀⠀⠉⠛⠿⠿⢿⠟⠃⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⣿⡿⠿⢿⣿⠿⠛⠉⠉⠁⠀⠀⠀⠀⠈⠁⠀⣤⣶⣿⡿⠋⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠴⡤⢄⠀⠀⠀⠁⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⢹⠯⡀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣠⠊⠀⠀⠉⢢⠀⠈⠁
⠀⠀⠀⢣⡠⠊⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠶⠄⠀⠀⣼⠋⠽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣀⣀⠀⠀⠀⠀⠀⣇⢀⠄⠒⠉⣁⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⡊⠭⠄⢳⡀⠀⠀⠀⠀⡠⢄⢤⠀⠀⠀⠀⠀⠘⠃⠀⠃⠀⠀⠀⠀⠹⣀⠠⠒⠊⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡴⠆⠒⠈⣉⣡⡽⠀⠀⠀⠀⠘⢬⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠢⢄⣀⣠⠃
⠀⠀⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠉⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣱
⠀⠀⠀⠀⠀⢀⡏⠀⢠⠁⢀⡄⠀⢹⡀⢈⡆⠀⠀⠀⣆⠀⢀⠀⠀⣶⠒⢢⠀⠈⡟⠀⣤⠄⢄⠐⢲⡖⠒⡀⠀⠀⣾⠑⢂⠀⠀⠀⣄⠀⠀⡀⠀⠈⡄
⠀⠀⠀⠀⠀⢸⡇⠐⠚⠀⣾⢙⡀⢸⡇⠉⢰⠋⠲⡀⠸⣄⠎⠀⠀⣿⠤⢎⡀⢰⡇⠀⣿⡄⠋⠀⢸⡇⢠⡇⠀⡇⢸⠀⠈⠀⢠⡄⠘⣆⠸⠀⠀⠀⡇
⠀⠀⠶⠀⠀⠸⠀⠀⠆⣸⠃⠀⢃⠀⠇⠀⢸⠤⠴⠃⠀⡏⠀⠀⠀⢸⡀⢀⠅⠘⠃⠀⣿⠈⠂⠀⠘⠃⢸⡇⠈⠁⠸⡧⠊⢠⣏⣄⠀⢹⡇⠀⠀⢰
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠿⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠁⠀⡤⡀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠏⠀⠀⠀⠸⠃⠀⠀⡍
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢅⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢦⢤⠀⠀⠀⢇
⠀⢀⠀⠀⠀⠀⢘⠒⡤⢄⠀⠰⠋⠀⠤⠆⠀⠀⠀⠀⠀⠀⠘⠶⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⢸⢸⡀⣀⣠⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠈⠦⠀⠀⠀⢠⡀⡃⠈⣍⡵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⣂⡈⡭⠭⠖⡒⣇⠀⠀⠀⠀⢠⠤⣄⡠⠤⠄
⠀⠀⠀⠀⠀⠀⠘⠃⢰⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⠀⠀⠀⠀⠀⠀⠀⠀⡯⠤⠐⠒⠈⠁⠀⢸⡀⠀⠀⠀⠰⡀⠈⠀⠀⡌⠀⠀⠀⠀⠀⣀
⢀⡠⠤⡀⠀⠀⢀⠖⠉⢀⣤⣀⠀⠀⠀⡤⡀⠀⠀⠀⠑⠢⠼⠮⣀⡄⠀⠀⠀⠀⠀⠀⣰⣤⣤⣖⢮⠽⠮⠛⠋⠀⠀⠀⠀⠑⢄⡠⠎⠀⠀⢀⠀⠀⠎
⠉⠀⠀⠈⠒⠒⠉⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠿⠿⠿⠿⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⡶⠶⠶⢶⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣼⣯⣿⣍⣀⣀⣩⣿⣽⣧⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⡿⣛⣿⣽⣿⡟⣿⣿⣿⣿⡻⣿⣯⣿⣛⢿⢿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣻⢟⣿⡟⣡⣪⢞⣵⡿⣫⢋⣾⡟⡉⢘⢻⣷⡙⣝⢿⣮⡳⣕⣌⢻⣯⡻⣟⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠸⡳⣻⠏⠼⠙⢋⣾⡟⡞⠁⣼⡿⡙⠇⠸⢋⢿⣧⠈⢣⢻⣷⡙⠋⠧⠹⣟⢞⠃⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣸⣷⠏⠀⠀⠀⣾⡿⠀⠀⢰⣿⠁⠀⠀⠀⠀⠈⣿⡆⠀⠀⢿⣷⠀⠀⠀⠹⣞⣇⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣟⡿⠀⠀⠀⢰⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⡆⠀⠀⠀⢿⣻⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣰⢸⣿⣇⡀⢠⠀⢸⣿⡇⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⣸⣿⡇⠀⡄⢀⣼⣿⡇⣄⡴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡼⣿⣿⣿⡿⡇⢸⡏⢸⣿⣿⠘⢱⢿⣿⡇⣴⣸⣇⣦⢸⣿⡿⡎⠇⣿⣿⡇⢱⡏⢸⣻⣿⣿⣿⢷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⢿⣿⣷⡜⢿⡀⣟⣿⣧⠊⣿⢿⣷⢹⢻⡟⡏⣿⡿⣻⠉⣼⣿⡽⢀⡿⢣⣿⣿⡿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡹⣮⣻⣷⣽⡌⢳⡘⣾⣷⣇⠘⣞⣿⡜⣶⣶⢣⣿⣱⠃⣸⣿⣷⢃⡞⢡⣯⣾⣟⣵⢏⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣝⢿⣿⣦⡙⢮⣯⣿⣧⠈⢿⣿⣿⣿⣿⡿⠁⣼⣿⣽⡵⢋⣴⣿⡿⣫⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣮⣿⣿⣦⣝⣷⣽⣷⣄⣻⣿⣿⣟⣠⣾⣯⣾⣫⣴⣿⣿⣵⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⣿⣿⣿⣿⠛⠛⣿⣿⣿⣿⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⣿⣿⣿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢸⣿⣿⡟⢹⣿⣿⡇⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡏⣿⡇⢸⣿⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⣿⣿⠇⠸⣿⣿⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡾⣷⣿⣤⣤⣷⣿⠷⣼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⠋⠙⣶⣶⣄⠀⠀⠀⠀⠀⠀⣠⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⣦⡐⢀⠟⠀⠀⠀⠀⠀⠀⡯⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠘⣇⢂⣸⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⣿⣟⠘⢲⣿⣿⡋⣹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡿⠀⠀⠀⣀⡆⢀⡀⠀⠀⠀⡀⠀⠀⠀⣠⡀⠀⣀⠀⠀⠀⡉⠉⠛⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠑⠋⠐⠁⠀⢉⠀⠂⠉⠁⠀⠉⠐⠈⠉⠀⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠓⠀⠀⠀⠰⠶⠞⠓⠶⠂⠀⠦⠖⠝⠐⠂⠐⠰⠆⠋⠙⠋⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠈⠃⠄⠄⠀⢒⡂⠀⠀⠰⣾⣷⠂⠀⠀⠤⠒⠢⠀⢀⠀⠀⡄⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⡏⢹⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⡇⢸⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡇⢸⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⢾⠋⠀⠨⠙⢷⡀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣶⣴⣶⣾⠇⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⡇⢙⡛⡙⠛⣿⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣿⢸⡇⡇⢰⡿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⢸⠃⡇⢸⣿⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣀⠀⠀⠀⠸⠹⠼⠂⡧⣾⠟⠀⠀⢀⣴⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣏⠉⠉⠉⠉⠉⣉⣉⣹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⠟⠛⠛⠛⠛⢛⣛⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⠛⠛⠛⠛⠛⠛⢛⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⡿⠷⠿⢿⣿⣟⣛⣛⣻⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣯⣭⣭⣽⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⢖⠀⠀⠀⠀⢀⠾⠐⠞⠀⠀⠀⢀⣔⠈⠨⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠘⠉⠀⠀⠀⡰⢅⠙⠊⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣴⠀⠀⠀⠀⠀⠀⣀⠘⢍⠒⠁⠀⠠⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢑⠀⠀⠀⠀⠀⠀⠀⠛⠉⠋⠀⠀⠀⠀⠄⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠛⠀⠠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⡴⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠋⠀⠀⠀⠀⠀⠀⠀⢠⠰⣀⠀⠀⠀⠀⠀⢎⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠃⠀⣀⠀⠀⠀⠀⠀⢀⠛⠄⡤⠀⠀⠀⠀⠀⠀⠐⠣⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢂⠀⠑⠄⣀⠀⠀⠀⠀⠀⠀⠀⡄⠌⢃⠇⠂⠀⠀⠈⠑⡀⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠈⠢⡀⠀⢅⢀⡁⠀⠀⢠⡒⡀⢹⠜⠋⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠌⠀⠀⠈⢄⠀⠀⠈⠲⡄⢆⡥⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢠⢐⠋⢴⠘⠈⠚⠁
⠀⠀⠀⠀⠀⠀⢀⠁⠌⠀⠀⡠⠂⠀⠑⢄⠀⢄⠶⢆⠀⠀⠀⠠⠁⢈⠀⠀⠀⠀⠶⠅⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠌⠀⢀⠌⠀⠀⡠⠐⠁⠈⠢⣀⠀⠡
⠀⠀⠀⠀⠀⠈⠈⢀⠔⠁⢀⠔⠊⠀⢀⠠⠐⠈⢀⠁⠖⠃
⠀⠀⠀⠀⣠⠀⡠⠂⡠⠐⠀⡀⠄⠂⡁⠄⠂⠈
⠀⠀⠀⣠⢃⢌⠄⢊⠄⡀⠅⠐⠈
⠀⠀⢠⣿⣶⠵⠚⠉
⠀⠀⠉⠁

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