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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠴⠶⠾⠂⠉⠚⠙⠒⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠠⢞⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⡀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⢀⣀⡤⠤⠤⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢞⠁⠀⠚⣉⣁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⢦⠀⢠⠚⠁⢰⠋⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⡀⠀⠀⠀⠀⢞⠰⠹⠸⠆⠼⠆⠀⠸⢄⡘⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢄⣀⠀⠀⡨⠓⠤⣄⣀⢨⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡡⠔⠉⠀⠉⠚⠳⢄⠀⢀⠜⠫⣀⣀⣀⣀⠤⠤⠤⢖⣋⠀⠀⣔⣉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡀⠀⠀⠀⠀⠀⠀⠑⠯⡀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠠⣽⠛⡟⠐⠂⠤⠄⢀⡀
⠀⠀⢀⡠⠄⠐⢲⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠒⠢⠤⣀⡠⠀⠈⠀⢀⠤⢰⠃⡎⠀⠁⠂⠀⢀⡀⠀⠉⠁⠒⠒⠢⠤⢀⡀
⠀⡎⠉⠉⠉⢹⠃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠚⢆⠙⠋⣨⡀⠀⠀⠀⠀⠀⢁⡆⠀⠀⠀⠀⠀⠀⠈⠉⠒⠂⢤⡀
⠀⢡⢀⣀⠠⣱⣄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⣴⠃⣀⠤⠒⠂⠉⠀⠀⠀⠀⣀⡠⠤⠖⠒⠒⠒⠊⠉⡏
⠀⠀⠀⠀⠀⡟⠈⠓⠶⣤⣁⠀⠀⠀⠀⠀⠀⠀⢻⠒⠒⠀⠉⠐⠶⠦⢤⣤⣤⠤⠚⠒⠂⠥⠊⠀⣀⠠⠤⠤⠖⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⣀⡜
⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠉⠙⠲⠶⠤⣤⣤⡤⠼⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠲⠒⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠶⠒⣺⠉⠉
⠀⠀⠀⠘⠁⢼⡧⣤⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡶⠦⠤⠤⠤⣄⣀⣀⠀⡸⢀⣀⣀⣠⠤⠀⠶⠚⠛⠉⠉⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠂⠀⠀⠈⠑⠤⣈⠓⠦⣤⣠⣀⣀⠀⠀⣀⣠⣤⠤⢀⡇⠀⠀⠀⠀⠀⠀⡏⠉⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⢸⠒⠒⢤⣄⡀⠀⠙⠃⠈⠉⠉⠉⠉⠉⠁⠈⡏⠁⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⢸⠀⠀⢸⠀⠈⢱⣒⠒⡖⠒⡖⠖⢲⠒⠒⡞⠃⠀⡟⠒⠒⠶⠤⢤⣀⣀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠃⠀⢸⠀⠀⢸⡧⠀⡇⠀⡧⠀⢸⡇⢐⣇⣦⠴⠋⠉⠙⡄⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⢸⠄⠀⢸⠀⠀⢸⡇⠀⡃⠀⡇⢸⢸⡇⢸⡅⠀⠀⠀⢀⣠⡇⠀⠀⠀⠀⠇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⣼⠄⠀⢸⠀⠀⣼⡄⠀⡇⠀⠿⡼⢸⡇⠘⠟⠿⠟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠘⠒⠒⠋⠀⠀⣿⠂⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣥⣼⠃⠀⠀⠀⠈⠓⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⢀⣠⠤⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠶⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠩⠭⠝⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⢁⣀⠀⠀⠤⠠⣐⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⠄⢻⡍⣙⠛⢛⡋⠉⢻⠀⢀⣴⣾⣿⣿⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠙⠗⠋⠀⡙⠀⠈⢁⠀⣼⢰⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⣦⡀⠑⠒⠒⢁⡴⠃⣿⣿⠉⡉⠀⢙⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠿⣛⣒⣛⠿⠛⢦⣿⡇⠈⠋⡀⠘⠃⠈⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⣀⡤⠖⠒⠦⠤⡴⠊⠹⣿⣿⣄⠁⠤⠘⢀⣴⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠙⢦⡿⣿⠖⣆⠀⠀⣿⢀⡤⠖⢺⡏⠙⠟⠛⢉⡷⢭⣍⡁
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⡞⠀⠀⢀⣯⡀⠀⢸⡋⠀⠀⠘⠶⠋⠳⣤⡼⠁⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠘⠉⠀⣤⠖⢺⠉⠁⣠⠊⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣷
⠀⠀⠀⠀⠀⠀⠈⢧⣀⣀⣤⣾⣁⡀⢘⣶⣶⡁⠀⠲⣿⠀⠀⠀⠀⠀⠀⣴⡄⢸⣷⠛⠉⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢀⣼⣆⢩⠻⣇⠀⣙⣢⢄⡈⠛⠓⢦⠤⡄⢰⡿⠤⣼⡽⠛⠀⢀⡗
⠀⠀⠀⠀⠀⠀⢀⠞⢹⣾⡃⠘⢣⠃⠀⣟⣚⡿⢡⠎⠷⢤⣀⡿⡰⣟⠛⠁⠀⠀⣀⡤⠴⠺⣅
⠀⠀⠀⠀⠀⢠⡿⠐⠚⠒⠒⠠⡧⠄⠒⠚⠛⣲⣃⣎⣎⣎⡛⡑⡝⡿⠛⠲⠖⠺⣅⣀⣠⣤⣌⣧⡀
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠉⠙⠓⠷⠞⠛⠋⠙⣇⣈⣉⣉⣉⣰⠼⣦⠖⠛⠉⠉⠉⠛⠉⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢂⠼⡛⠿⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⡰⠋⣤⠘⣆⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⠀⠀⠀⠏⢠⢸⣀⠀⠀⢢⡘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⠀⠀⠀⠀⡇⢷⢢⠙⢄⠑⠌⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⡇⣀⠀⢠⠀⠸⡟⣆⣇⠈⡆⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⢁⣼⢧⠀⣇⣿⣽⣶⡧⡟⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠉⢻⡷⡿⠿⢿⣶⢿⠟⠉⠉⠉⠹⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⢀⠀⠠⠀⢠⡏⠐⢄⠀⠈⠉⠀⠀⠠⠀⠀⠰⠆⠀⠀⠀⡎⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⢀⠀⡈⠀⢠⢠⠁⢀⣄⠙⣲⣤⣀⡀⠀⠀⡹⠃⠁⢀⠃⠄⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⢫⠀⠸⠒⠋⠀⠢⠈⠒⠀⠁⠔⠀⠀⠀⠀⣼⢀⣀⠀⠀⠀⠀⠈⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⢣⢸⠘⠀⠀⠀⠀⠀⠀⠀⠁⠿⠀⠀⠀⠀⠀⢰⠛⢯⡈⡄⠀⢀⠔⠁⠚⠉⠑⡄
⠀⠀⠀⠀⠀⠀⢀⡴⡀⠀⢀⣄⡘⣢⣽⠃⢀⡀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢰⠃⠀⠀⠙⢦⣀⡁⠀⢀⠀⢀⣐⣯⡄
⠀⠀⠀⠀⠀⠀⠪⠊⠀⠔⠁⠀⠛⠁⠀⠈⠃⠀⠀⠉⠆⠂⠠⠀⠌⠉⠛⠀⠂⠙⠠⠀⠀⠀⣀⠉⠛⠒⠛⢠⡄⠀⠈⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠁⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⢦⡀⠀⠀⠀⠁⠀⠐⢧⡄⠸⡄
⠀⠀⠀⠀⠀⠀⠈⠒⠠⠤⠄⠖⢲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠱⣄⠀⠀⠀⠐⢧⡀⢁⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⢳⡀⠀⠀⠀⠘⠀⣀⠠⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠙⠖⠐⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣎⣁⣤⣤⣤⣐⣶⣒⣒⣒⠀⠀⣄⣰⣀⡤⠖⠒⠒⠀⠤⠤⠤⠤⠠⠤⠥

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⡤⠒⠒⠠⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠼⠀⠒⠋⢔⠈⠢⣄⣀⡀⠀⠀⠀⠀⣀⠀⠀⣀⡤⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⠙⣺⡉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠐⠦⠀⠀⠀⠀⠀⠈⡀⠀⡠⣤⠙⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡖⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠒⠘⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⡠⣒⠄⢐⠁⠐⠁⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠒⠄⣀⡑⠀⠈⠉⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠈⠑⢄⠎⠁⠀⠀⠒⠒⠋⠀⢴⡤⠀⡀
⠀⠀⠀⠀⣀⠄⢀⠆⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⢀⠀⠀⠠⢰⠠⠁⠀⠀⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠁⠈⡄⢐⠆⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠐⠆
⠀⠀⠀⠀⠀⠀⡏⠑⠲⢤⣀⠀⠀⠀⢳⠀⠀⠐⠒⠶⠤⠀⢓⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⣀⣀⠴
⠀⠀⠀⠀⠀⠖⢳⣄⣀⠀⠀⠉⠉⠉⠉⠀⠐⠤⠀⠀⣀⣀⠀⡁⠀⢀⠀⠀⠀⠐⠀⠈⠁⢀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⢑⠲⠦⠤⠤⠤⡶⠂⡄⠀⠀⠀⢸⠉⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⡆⠀⠁⠐⡄⠠⢠⠄⢄⢠⠆⢰⠂⠤⠤⠄⢨⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⡇⢸⠀⢀⡇⠀⠰⠀⣿⢸⠛⠉⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⣇⣈⠀⢸⡅⠀⠘⠆⢸⠘⠒⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠁⠀⠸⣄⡆⠀⠀⠺⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⡠⠄⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⢤⡀⠀⠀⠀⠀⠀⣠⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠒⠒⠂⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠉⠉⠤⣀⣀⠤⠵⠤⠤⠤⢤⡏⣀⢀⠀⠈⠉⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡥⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢎⡑⢂⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣰⠋⢀⣀⣀⣠⡴⠋⠀⢹⡊⣉⣉⣉⣉⣁⣛⢢⡀⠀⠀⠀⠀⠀⠀⣤⡵⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠢⡀⠀⠁⠘⣄⠀⠀⠙⢆
⠀⠀⠀⠀⠀⢠⠃⠠⡏⠀⠈⢡⡤⢤⣀⣀⡏⠁⠀⠀⠀⠀⠈⢫⠱⡄⠀⠀⠀⠀⢨⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⡀⠉⠦⠤⢀⡈⣆⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠘⢆⢀⡞⢠⡀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢤⡄⠀⠀⠈⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡠⢿⠊⠀⠀⢀⠃⢸⣀⠤⠊
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠈⠉⠀⠘⠙⠢⢄⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠆⠀⠀⠀⠀⠙⠒⢲⠒⠦⠤⠴⠒⢚⡓⠚⠉⠀⡜⠘⠤⠔⠁⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡀⠈⠑⠂⠤⠶⠲⠤⡤⠬⠇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠣⡀⣀⠴⣄⠴⠁
⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠈⠀⠀⠀⠀⠀⠆⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⢢⡤⠒⠀⠀⠀⠀⠀⠀⢀⠔⠊
⠀⠀⠀⠙⢶⠀⠀⠀⠀⣠⢞⣤⡀⠀⠀⠀⠓⠤⠄⠀⢀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢀⣀⣀⡀⠠⢠⣔⣂⠁
⠀⠀⠀⠀⠀⠓⠒⢢⣠⠃⠀⠎⠲⢭⠶⣤⣄⠐⢒⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠣⠠⠞⢡⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠐⠀⠀⠀⡏⠄⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠸⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡇⠘⠀⠀⠀⠀⠀⠀⠇⠈⡄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢠⠃⠀⡃⠀⠐⣄⠀⠀⢀⠀⠉⠲⠢⣀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⡔⠈⠁⠉⠐⡄⠘⢄⡀⠀⠈⠒⠼⡉⢢⣄⠀⠀⠉⠂⢤⣄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⠕⢀⠀⢀⠀⠀⠀⠀⢀⣠⠁
⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠉⠐⠦⠄⢀⠀⠀⠘⣂⡜⠦⣠⠞⠀⠙⠢⢄⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⢀⡀⠄⠂⠁
⠀⠀⠀⡇⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠈⠒⠟⠉⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠐⠮⡴⠀⢂⡀⠔
⠀⠀⠀⢳⠀⠀⠀⢸⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⠄⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠀⠀⢀⡀
⠀⠀⠀⢸⡀⠀⠀⡸⠀⠀⠈⠐⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠑⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠅⠀⠀⡀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠊⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣠⠤⠒⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣈⠁⠀⣀⡨⠥⠒⠊⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⢿⠿⡿⢿⢿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⣛⣻⣿⣿⣿⣿⣛⣿

⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡠⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡄⠠⠤⠐⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠛⠛⠛⠛⠛⠓⢦⣄⠀⠀⠀⠀⠀⠀⠈⠁⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢀⡴⠚⣹⢞⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢳⡟⠛⢶⡄⠀⠈⠙⠒⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡞⠀⠀⣯⡾⠀⠀⢀⣀⠀⠀⠀⠀⢠⣄⠀⠀⠸⡆⡇⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢠⠟⡇⠀⣰⠟⣿⣀⣀⣀⣀⣸⡟⢧⠀⠀⡟⢳⡄⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣁⣠⠔⠋⠀⢷⣴⡇⠀⠨⠍⠉⠉⠉⠭⠀⠘⣇⣰⠃⠀⠙⣦⡀⠘⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠸⣄⣷⠀⠀⠛⠀⠀⠀⠛⠀⢸⣇⡼⠀⠀⠀⠀⠉⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠿⢷⣄⣀⠐⠒⠀⣠⡴⠿⠿⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠙⢯⣽⠿⠁⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣾⣄⠀⠀⠀⠀⣴⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠄
⠀⠀⠀⠀⡿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⣿⠀⠀⠀⠀⠀⠀⣰⠏⠈⠙⢦⣼⠧⠿⣌⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠙⢧
⠀⠀⠀⠀⡷⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⣿⠀⠀⠀⠀⠀⢰⡏⠀⠠⠖⠋⠀⠀⠤⣿⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⠈⣧
⠶⠶⠶⠶⡗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡿⠒⠒⠒⠒⠒⠺⣦⣀⣀⡀⢀⣠⣤⣹⠿⡀⣯⣤⣤⣤⣤⠤⠶⠶⠤⠴⠛⠁⠀⠀⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶
⠀⠀⠀⠀⡷⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⡇⠀⠀⠀⠀⠀⠀⠀⠈⣹⠋⠉⠁⠀⠀⠀⠉⠁⠀⠀⢀⡽⢛⠁⠀⠀⠀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⡷⠶⠲⠲⠶⠒⠶⠖⠒⠒⠖⠲⣇⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡵⣢⣄⣠⠞⠁⢷⠶⠦⠤⠤⠤⠤⢤⡄
⠀⠀⠀⠀⠷⠶⠶⠶⠒⠒⠒⠒⠒⠚⠛⠚⠋⠀⠀⠀⢠⠶⠖⠚⠛⠒⠶⠶⢤⣄⣀⣤⠴⠶⠶⠶⠶⠦⢤⣍⡉⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⠶⠶⠶⠶⠿⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣀⣀⣀⣀⣀⡀⠀⠀⣸⠁⠀⢀⣀⣀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⠭⠥⠤⠬⠭⢭⣟⡲⠿⢛⣉⣭⣭⣭⣭⣝⣛⠶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠔⠉⠉⢑⣄⠀⡀⢀⠔⠊⠑⠢⠤⠤⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠈⠙⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⣂⠖⠛⠉⠉⠀⠀⠉⠉⠐⠢⠂⠄⠀⠀⠘
⠀⠀⠀⠀⠀⡰⠃⣀⣀⣠⡔⠉⠙⡒⣈⣈⣁⣀⣓⣄⠀⠀⠀⠀⠀⢠⠼⠞⠁⠀⠀⠀⠀⠀⠀⡔⠐⢄⠀⠑⠐⡄⠀⠘⢢
⠀⠀⠀⠀⢰⠁⠸⡀⠀⢠⠔⠤⣠⠏⠀⠀⠀⠀⠈⠫⣣⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⣄⠑⠂⠠⡘⡄⠀⢀
⠀⠀⠀⠀⠼⠀⠀⠑⠤⠃⡶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡆⠀⠀⠘⠄⣀⣀⣀⠀⠀⣀⣀⣀⠤⠒⡏⠃⠀⠴⠁⢱⠒⠉
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠈⠁⠋⠒⠤⣀⣠⣀⣀⣀⡶⠁⠀⠀⠀⠀⠀⠈⠈⠉⠉⠀⠒⠀⠀⠀⠃⠈⠁⠀⠀⡌
⠀⠀⠀⠀⠇⠀⠀⠀⠀⢀⠁⠀⠈⠁⠀⠀⠀⢠⠀⡈⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠐⠀⠀⠀⢆⣀⡠⣄⡴⠁
⠀⠀⢀⡼⠀⠀⠀⠀⢀⠎⠀⠀⠈⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠲⡤⠒⠀⠀⠀⠀⠀⡠⠂⠁
⠀⠀⠈⠑⡄⢀⠀⢠⠪⣶⡄⢀⠀⠉⠂⢂⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠄⣤⠀⢒⠦⠅
⠀⠀⠀⠀⠈⠁⣷⠃⠀⠁⠉⠊⠻⡓⠤⠔⠉⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢠⠃⠈⠙⠉⠠
⠀⠀⠀⠀⠀⠀⠇⡀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡰⠀⢠⠀⠘⣄⠀⢠⡀⠁⠆⢄⡀⠀⠀⡀⠠⠂⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠠⠀⠀⠉⢠⠈⠢⣀⠀⠉⠚⠭⠶⣄⢀⠈⠐⡦⢄⡀⠀⡐⢀⠀⠀⠀⠀⠀⠀⠜⣀⠀⠀⠂⠀⠀⡠⡀
⠀⠀⢰⠀⠀⠀⠘⠀⠀⠀⠉⠐⠒⠄⣀⡨⠗⠑⠒⠁⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⢀⡈⠁⠀⠀⢀
⠀⠀⢸⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⡀⠼⠆⠀⠀⠀⠈⠀⠈⠉⠄⠂⠉
⠀⠀⠸⡄⠀⠀⠇⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠤
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠐⠊⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⢐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⠀⠀⡀⠠⠀⠀⠤⣒⡪⠅⠒⠀⠀⠀⠀⠐⠀⡀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⠥⠖⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣠⠄⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡀⠈⢀⡀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠤⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠈⠁⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠁⠌⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡈⠁⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡁⠀⠀⠁⠀⠀⢠⠃⠀⠑⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⣀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠆⠄⠀⠀⠀⠀⠒⠐⠒⠐⠀⠀⠂⠰⣰⡦⡐⠈⡉⡆⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⣒⣻⣀⣀⢀⢠⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⡏⠉⠙⠃⣤⡼⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⣿⠁⠀⠉⠓⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢶⠒⡖⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢮⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈⣀⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢡⡾⠤⢄⡀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠾⠁⠀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠘⠛

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⠋⠁⠁⠀⠀⠂⠀⠂
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠄⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡦⠤⠔⠀⠐⠠⠀⢢⠐
⠀⠀⠀⠀⠀⠘⢁⠀⠀⢠⠀⠀⠀⣤⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡄⠀⠀⠴⠀⠀⠐
⠀⠀⠀⠀⠀⠘⡆⠁⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢥⠀⠀⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠣⣀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠰⠀⠀⢀⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢛⡆⠀⢁⡹⠁⢈⡄
⠀⠀⠀⠀⠀⠀⠤⠐⠀⠘⠲⠤⠗⠂⢠⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⢃⠀⠁⠑⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⢁⡠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣐⠀⠀⠀⢠⡄⠀⢤⠀⢀⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡀⡀⠒⠨⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠨⠤⠤⠤⠤⠤⠤⠀⠀⠀⠜⠀⠠⠀⠂⠈⠠⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢉⠁⠒⣀⡖⠒⢂⡉⠃⠀⠀⠐⠛⠀⠀⠀⠁⠀⠀⠁⠀⠈⠛⠓⠂⠘⠀⠀⣀⠰⡖⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣎⣀⡬⠖⠛⠛⢻⠦⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⣤⠼⠛⠛⠿⣭⡀⣡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠂⠀⣀⠀⢸⠐⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⢿⠠⠄⢀⠤⠤⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠀⠈⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢐⠄⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⡇⣠⣆⢸⣇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⠨⣶⡠⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠇⠀⠀⠀⠸⠮⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠌⠠⠀⠀⠸⠿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠒⠂⠐⠒⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠄⡰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢀⡼⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⢠⣉⣀⣀⡰⠁⡊⠀⠀⠀⠀⡰⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠎⠀⠐⢒⡒⣒⠬⠀⠒⠒⠊⠀⠀⢭⣛⡶⠤⡚⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⢀⣔⣅⡶⣶⣷⠀⠀⠀⠀⠀⣼⣱⢶⣬⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡖⠉⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⠛⠁⠀⠒⢠⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠡⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠜⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⢆⠀⠀⠀⠀⢀⣀⣀⡠⠤⣠⣀⠄⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠣⣀⠀⠀⠀⠈⠙⠨⠭⠔⠉⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣀⠀⡤⠤⠡⣦⢀⡀⠀⠀⠀⠀⠀⠀⣀⠄⢚⡤⢄⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣠⠔⠉⠀⠀⠘⡄⠀⠈⠓⢄⡉⠀⠒⠂⠈⢁⠠⠐⠁⠀⡠⠁⠀⠀⠀⠈⢂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠈⠂⢄⡀⠀⢈⠔⠲⢂⡉⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⡆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣨⣄⠠⠊⠈⡚⠀⠈⢦⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡂
⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠸⣧⠈⠆⠀⠈⠚⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢀⣀⢀⡠⠋⠀⠀⠀⠀⠹⣧⠘⡄⠘⠙⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠆
⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⠦⠤⠄⠂⠒⠐⠢⣐⡀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⢀⣠⠖⠒⠚⠙⠑⠛⠒⠶⠶⠶⠶⢤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠢⣙⢦⠀⠐⠒⠒⠒⠒⠚⠒⠒⠒⠒⢲⣿⠋⠉⠛⠛⠛⠣⠤⠶⣶⣆⣶⠶⢛⣉⡽
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⣀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠉⠉⣲⣄⢱⣀⡀⠤⠀⢒⡀⠉⠩⠍⠡⣒⠪⢉⠉⠀⠀⠒⠒⠠⠤⠴⠧⠠⠷⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⢙⠒⠂⠠⠄⠀⠀⠬⠁⠈⠢⡇⠐⠀⢀⠤⠒⠀⠉⠩⣁⣈⠁⣄⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⢀⡠⠄⠭⠀⠀⢧⣀⣤⣤⣐⢊⣁⠤⠶⢉⢣⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⠠⠉⠁⣀⠀⠉⠡⠄⠀⠀⠈⢅⠠⠀⠒⢂⣀⡀⠀⢈⣉⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡭⠥⠒⠒⠶⠶⠶⠦⣬⣥⣀⣀⣠⠤⠒⠉⠑⠀⠀⠀⠀⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣀⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣤⠖⠉⠉⠑⢦⠀⠀⠀⠀⢀⡤⠒⠒⠒⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⠉⠉⠉⠉⠑⠒⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⡀⢔⣒⠤⠴⠒⠚⠓⠚⠒⠒⠾⢤⡔⢠⣀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⢀⡤⣄⠀⣀⣀⣀⣀⣉⣢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⢥⠞⠂⠀⠀⠀⢨⠂
⠀⠀⠀⠀⠀⠀⢀⠞⠁⣀⣤⣤⡤⠞⠁⠀⠘⣏⣉⣠⣤⣤⣤⣌⣉⣳⡀⠀⠀⠀⠀⠀⠀⣠⠤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠈⢢⡀⠀⠉⠉⢦⠀⠀⠈⠱
⠀⠀⠀⠀⠀⠀⡎⠀⠸⡅⠀⠀⢠⡤⠦⣄⣀⡏⠀⠀⠀⠀⠀⠀⠈⢧⡙⣆⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀⡀⢉⠒⠠⢤⡈⢇
⠀⠀⠀⠀⠀⢰⡅⠀⠀⠻⣄⢀⡏⢠⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⡄⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⡥⡟⡇⠀⠀⢀⠆⠸⣄⡠⠞
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠈⠉⠀⢸⠙⠢⣄⣢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡧⠀⠀⠀⠀⠉⠓⡖⢲⠒⠤⠤⠤⠔⢒⡓⠒⠋⠁⠀⡇⠰⠤⠔⠁⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠈⠙⠒⠤⠤⠴⠤⢄⣀⣈⠧⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠈⠁⠀⠀⠀⠀⠀⣄⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘⣄⠀⢀⣤⡀⣠⠃
⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠘⠀⠀⠀⠀⠀⢠⠄⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢄⣀⡰⠄⠀⠀⠀⠀⠀⠈⠉⡩⠔⠉
⠀⠀⠀⠲⣋⡀⠀⠀⠀⠀⢀⣼⡁⠀⠀⠀⠀⠴⣀⠀⡀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣈⠉⠀⠀⠀⠀⠀⣀⡠⠴⠊
⠀⠀⠀⠀⠀⠣⣀⣤⡀⢠⠋⠰⡻⣖⠢⢄⣀⣀⣀⡀⠤⢤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢒⠶⢶⠉⢉⡝⡒⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣽⠚⠀⠀⡇⠀⠉⠉⢿⡒⠦⠤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡛⠀⠀⠉⠁⠀⡆⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⢀⡇⡆⠀⠀⠛⠀⠀⠀⢰⠘⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⢸⠀⠀⠀⢸⠀⠱⢄⡀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠒⠒⢸⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⡀⢀⢀⡟⠀⠈⡄⠀⠀⠣⢄⡀⢸⣄⡀⠀⠋⠒⠤⣀⠀⠀⠤⠀⠎⠉⠀⠀⠀⠀⠀⠀⠀⠓⠂⠤⠀⣸⠠⢀⠀⠀⠀⣀⠔⠀⠀⠀⡀
⠀⠀⠀⡬⠀⠀⠀⠀⢸⠀⠈⢢⣀⠀⠀⠈⠙⠢⠭⠶⡄⠀⡀⠀⠙⢶⠦⣀⠀⠀⢄⠃⣀⠀⠀⠀⠀⠉⠀⠀⠀⠎⢠⣀⡀⠔⠊⠀⠀⠀⡠⢮
⠀⠀⠀⠆⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠓⠒⠤⢄⠀⠀⢭⣵⠚⠢⠴⠋⠀⠈⠑⠦⠀⠀⠈⠀⠀⢀⠀⠀⠀⡀⣠⢴⣃⠀⠀⣀⣤⠴⠊⠁
⠀⠀⠀⢸⠀⠀⠀⠀⣸⠤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢏⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠰⡅⠄⠀⠁⠛⠉⠴⠷⣞⣥⡼⠇
⠀⠀⠀⢸⠀⠀⠀⠀⡏⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣀⠀⡀⠀⢀⡀⠤⠒⠊⠉⠁⠀⠀⠀⠁⠙⢀⣠⡄⠈⠋⠉⠁⠀⣀⣔⠒⠀⡦
⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠈⠒⢄⠠⠀⠀⠀⠀⠀⠀⣀⠠⠜⠓⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⡠⠒⠲⠿⠒⠒⢍
⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠢⠴⠒⢋⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠁⣀⠀⠀⠈⠀⠠⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣠⠤⠒⢈⡀⠤⠀⠀⠰⡀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠄⠀⠀⠀⠁⠂⠄⡀⠒⠈⠀⠀⢒⣋⠥⠔⠂⠉⠀⠀⠀⠄⠀⠀⠐⠄⠀⠀⠀⠄⠁⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣊⡭⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠉⢀⡀⠄⠂⠁
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠒⢤⡀⠀⠀⠀⠀⢀⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠐⠒⠒⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠁⣩⢀⣄⡠⠤⠗⠒⠢⠤⢴⣁⡠⢄⠀⠉⠉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⣀⣄⠀⠈⣀⣀⣈⡓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡥⠚⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢠⠓⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⡼⠁⣀⣤⣤⠤⠞⠁⠀⢯⣁⡀⠠⡤⠄⢉⣑⣄⠀⠀⠀⠀⠀⠀⡤⠼⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠈⢢⡀⠀⠀⠑⡄⠀⠀⠑
⠀⠀⠀⠀⠀⢸⠀⠘⡅⠀⠀⣠⠖⠦⣄⡼⠀⠀⠀⠀⠀⠀⠙⢌⢣⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣀⡄⠈⠒⠀⠤⠸⡀
⠀⠀⠀⠀⠀⡌⠀⠀⠘⢦⡰⠃⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢓⡆⠀⠀⠀⠹⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡠⠔⡋⠃⠀⢀⡜⠀⡧⠤⠚
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠈⠒⠼⠦⣄⡀⠀⠀⠀⠀⠀⠀⡤⠓⠀⠀⠀⠀⠀⢹⠙⠉⠑⠒⠂⠋⠩⠉⠁⠀⠀⡇⠈⠒⠁⠀⠀⡇
⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⢀⠃⠀⠀⠤⠀⠀⠈⠉⠈⠉⠑⡖⠒⠊⠀⠀⠀⠀⠀⠀⢸⠠⠄⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡄⠀⠀⠀⠀⠐⠓⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠓⠤⠔⡻⠖⠁
⠀⠀⠀⡤⠃⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢀⡀⠀⠀⠀⠉⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠑⠚⠉⠀⠀⠀⠀⠀⣀⠔⠋
⠀⠀⠀⠈⠹⡀⢀⠀⢀⠞⠹⢶⠠⢀⡀⠀⠁⠂⣁⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⢐⢤⡆⠒⣸⠧⠄
⠀⠀⠀⠀⠀⠈⠁⢱⠞⠀⠀⠀⠉⠊⢹⡗⠤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⡍⡆⠀⠀⡇⠀⠀⠘⠈⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⠇⠸⠀⠀⢃⠀⠀⠀⡄⠰⢄⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⢠
⠀⠀⠀⠀⠀⠀⠾⠀⠀⢇⠀⠈⠒⠤⡀⡧⢄⠀⠀⠑⠢⢄⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⢤⠀⠀⠀⠀⠠⠐⠀⠀⠠
⠀⠀⠰⠂⠀⠀⠀⠘⠀⠀⠑⠤⢀⡀⠀⠉⠒⠛⠦⡀⣄⠀⢨⠉⠢⢄⠀⠂⠁⠤⠀⠀⠀⠀⠀⠀⠀⢁⡔⠀⠀⠁⠀⢀⠠⠒⠃
⠀⠀⠀⡄⠀⠀⠀⢰⠀⠀⠀⠀⠈⠉⠉⠐⠤⣠⠴⠋⠀⠉⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠍⣀⠀⠠⠐⡈
⠀⠀⠀⡇⠀⠀⠀⢸⠍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⡤⠴⠂⠀⠀⠀⠀⠁⠀⠈⠉⠀⠐⠉
⠀⠀⠀⢷⠀⠀⠀⡼⠀⠈⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡃
⠀⠀⠀⠈⠒⠀⠒⠁⠀⠀⠀⠀⠁⠢⠀⠤⠖⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⢂⠁⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢐⣈⠭⠐⠂⠉⠀⠀⠀⠀⠀⠀⠑⠤⠄⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠠⠄
⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠂⠀⠀⠀⠀⠀⣠⠖⠲⣄⠠⣖⠒⠒⠒⢒⡆⠀⠀⠀⡦⠴⠤⣄⢀⠤⣄⣀⢶⡄⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠰⡇⠀⠺⠥⠤⡄⠹⠀⠏⢩⡄⠀⡤⠄⡇⢠⢤⠈⠇⠑⡌⢡⠊⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠀⠀⠀⠀⠀⠙⠛⠒⢲⡀⢈⡀⡇⡆⢣⠱⠴⠃⡄⡆⢸⡤⢀⡆⠀⡇⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣰⡒⢒⣊⡠⠞⠀⠧⠇⠈⠓⠶⠚⠁⠷⠤⠔⠋⠀⠀⠙⠚⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⣇⢸⠩⠀⢰⠋⣉⠆⢰⠋⡉⠳⡠⡖⠒⠲⢤⡀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢅⠀⡆⢸⠀⣉⠘⡌⠀⠋⣴⠃⠀⢘⢢⠀⢣⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⢸⠀⡇⢸⠀⡟⡄⢻⠀⡷⡈⢢⡇⠘⠋⢀⡎⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠈⠳⠃⠘⠚⠁⠘⠋⠓⠃⠙⠊⠱⠶⠖⠋⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⠴⠚⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠒⢶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⣄⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣬⣑⠲⠦⠤⠤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠉⠉⠚⠒⠒⠒⠒⠲⠶⠄⠀⠤⠤⠄⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠶⠶⠤⣤⣤⣤⣉⣉⣉⣉⣉⠉⠉⠉⠉⠉⠛⠒⠒⠒⠒⠒⠒⠶⠶⠾⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⣄⠀⠀⠀⠀⠀⣀⠴⠒⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⠖⢉⣀⠀⠀⠀⠀⠀⠀⠑⢤⣄⡀⠀⢴⣂⡦
⠀⠀⠀⠀⠀⠀⠀⠀⢴⡐⠆⠀⢠⠃⡴⠉⠈⢦⠀⢶⡀⠀⠀⠀⠈⣿⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢰⠃⠀⠀⠀⠑⢬⣯⠲⣄⠀⠀⢹⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡄⠀⣀⠀⠀⠀⠈⡁⠀⢙⠒⣼⠀⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠉⠀⠠⠀⠀⠁⠀⢸⡞⠓⢤⡈⢃
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠢⣄⠀⠀⠀⢀⣠⠔⠣⣀⠀⢀⡧⠃
⠀⠀⣖⣩⡄⢨⣑⣩⣭⣍⡒⡄⠀⡸⠁⠀⠀⠈⠛⠝⠁⠀⠀⠀⠈⢆⠀⠀⢸⣉⣉⣉⣉⣉⣹
⠀⠀⠈⢎⣀⣀⣀⣀⣀⣱⣈⠁⠀⡇⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⡧⠤⠤⠤⠤⠤⢼
⠀⠀⠀⠀⣸⣀⣀⣀⣀⣀⣀⡇⠀⣟⢀⣪⣆⣀⡀⠀⠀⣀⣀⠤⠤⢄⠇⢀⠋⠉⠉⠉⠉⠉⠹
⠀⢀⣠⠚⠒⠒⠒⠒⠒⠒⠒⣇⣀⣻⣲⠫⠝⠀⠈⡉⡉⠤⠹⢯⣂⣸⣀⣀⠗⠀⠀⠀⠀⠀⠰⣄⣀⡀
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠊⠉⠀⠉⠒⠊⠉⠉⠉⠋⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠠⠒⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠁⠀⠀⠀⠀⠈⠄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⢀⡀⠘⡄⠀⠀⠀⠀⠀⢀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⠄⠂⠀⠁⠀⠃⠨⠐⠀⠀⠒⠚⡑⠈⢤⠊⠡⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣈⠀⠀⠀⠐⣄⡄⠀⠀⠢⡀⠀⢀⠀⠀⠀⠔⠀⡀⠁⠐⣰⠃⣀⣀⣡⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠈⠉⢁⠃⠠⣌⠉⡀⠠⠀⠈⠂⠤⢠⣤⠤⠔⠊⠀⠐⠊⠉⠉⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⠤⡂⢀⠲⣌⠓⣷⡦⠴⠦⠤⠀⠠⣀⠦⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠙⠊⠡⠄⠀⠀⠀⠈⠁⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡒⣒⢒⣒⣒⣀⣀⣈⡁⠀⠐⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⡀⠶⠀⠀⠆
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠤⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠒⠲⡖⠀⡀
⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠊⠁⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠖⠓⠒⠒⠒⠿⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⠀⠐⠆⠀⠖⠈⣦⠠⡄⠶⠴⠲⠊⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢄⣀⠜⠁⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡀⠀⢠⠀⠀⡤⠀⢰⡇⠐⠟⢠⠦⡀⠐⠀⠀⠀⠀⠀⠀⠰⠀⠀⡆⠀⡄⡛⠉⠉⠙⠛⠛⡟⠒⠒⠒⠒⠒⠀⠐⠚⡇
⠀⠀⠀⠀⠀⠀⠈⢡⠀⠀⠀⠂⠀⠘⠀⣈⣤⣤⣶⡄⠘⢧⠀⠀⢀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠐⠾⣄⣴⣾⣿⣿⣿⣿⠀⠀⠀⠀⡀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠿⠍⡉⠛⠛⢳⠲⠶⠤⠤⠤⠄⠀⠀⡏
⠀⠀⠃⠀⠀⠀⠀⠀⠀⣿⡔⠉⠙⣿⣿⣿⣿⣿⣿⣇⠐⠂⠈⠁⣀⡠⠂⠑⡀⡀⠀⠀⠀⡀⠃⠀⠘⠀⢘⠁⠀⠀⠀⠻⣦⣄⡘⣄⢂⠀⠀⠀⠀⠀⠀⢃
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠉⢿⡀⠀⠘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⡄⠐⠡⠀⠀⠀⠀⠀⢁⠀⢰⡆⠠⢘⡇⠀⠀⠀⢠⡏⠀⠈⠹⠃⠠⠠⢤⡜⣉⠀⠉
⠀⠀⠀⠀⠀⢶⠀⠀⠀⣀⣘⣧⠀⣴⣿⣿⣿⣿⣿⡟⢂⣠⡀⠀⠀⠀⠀⠄⠂⠀⢠⣴⠀⠈⡀⢸⣇⡔⠋⠀⠒⠒⠂⠈⠀⠀⡀⡀⠀⠀⠀⠤⠀⠀⣀
⠀⠀⠐⡀⠀⠈⠇⠀⠀⠀⠉⠙⡆⣿⣿⣿⣿⣿⣿⢀⡀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⢀⢧⡽⠂⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⠉⠒⠂⠒⠀⠀⠉⣤
⠀⣴⣿⣿⣦⠀⠈⠀⠛⢲⠀⠀⠡⢈⣿⣿⣿⣿⡇⠈⡀⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠆⠀⠀⠂⠀⠒⠀⠀⠀⠀⠀⠈⠁⠒⠀⣤⣀⡀⠀⢹
⠰⣿⣿⣿⣿⣷⠀⡆⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧⡀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⡿⠽
⠀⣿⣿⣿⣿⣿⣿⠀⡸⢃⣠⠀⣾⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡰⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⢣⠃
⠀⠟⠛⠛⠛⢻⣉⢱⣧⣬⠛⠋⠉⠉⠉⠉⠛⠟⣿⣷⣶⣾⣿⣾⣿⣿⣻⣿⣶⣶⣸⣿⠀⡄⠚⢼⠀⠀⠰⠃⢠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⣼
⠀⠠⡄⠡⠀⢀⡛⠉⠘⠙⠀⣀⠀⠀⠀⠀⢀⣀⠉⠋⠛⢻⣿⡶⠒⠛⠻⡿⠿⠶⢿⣿⠀⠘⠀⢸⠷⠠⠃⠘⡎⡐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡏⣞⠁
⠀⠀⠚⠓⢶⣤⣄⣤⣦⡤⠞⠛⢪⡵⡟⠻⢿⣿⣿⣶⡤⠞⠙⡁⠤⠀⠀⠀⠀⠀⠈⠉⠐⡂⠐⠿⠃⠀⠀⠄⠘⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣼⠿⠁
⠀⠀⠀⠰⠹⢿⣟⣿⡿⠓⠈⠑⠀⣠⣄⠀⠤⠦⠀⠈⠚⠓⢀⠀⠀⠀⢠⣢⠄⠀⠶⠖⠂⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⠄⠀⠀⠀⠀⢠⣿⡦⠤⠶
⠀⠀⠀⢢⡀⠏⢡⡏⠵⠀⠠⡄⢀⣉⠀⠀⠀⠀⠀⠀⠀⠉⠏⠀⡀⠀⠀⠐⠂⠀⣼⡟⠀⣤⣄⣀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠓⠶⠆⣁
⠀⠺⠊⠉⣭⡄⠁⠐⠀⠤⡀⠁⠀⠀⠉⠀⠀⠀⠀⢠⣀⢠⣄⠀⠀⢀⣀⠀⠀⠀⠈⠉⠀⠻⢿⣯⡛⠦⣤⡀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠚⠛⢋⣀⣀⡤⢤
⠀⠀⠀⠀⠛⠘⠛⠲⠦⠄⠀⢹⣶⣶⣤⣀⣀⢀⣠⣤⡞⠁⠉⠍⠉⠉⠉⠀⠤⠠⠀⠀⠀⠀⠀⠈⠙⠻⢾⣿⣶⢤⣄⣠⣶⠟⠿⠶⠶⠖⠋⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⠛⣿⠛⠋⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣿⢿⣥⢀⠀⠀⠀⠀⣀⣨⠥⠆⠒⠛
⠀⠀⠀⠀⠀⢤⠤⢄⣀⠩⠁⠒⠀⠀⠀⢀⠏⠐⠀⠀⠀⠀⠓⠒⠒⠒⠒⠀⠒⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠒⠒⠀⠈⠙⠖⠓⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⠀⠀⠀⠀⠀⠤⠄⠠⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠐⠂⠀⠀⠀⠂⠀⠂⠂⠁⠂⠂⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⠿⠋⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣹⣦⣄
⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣴⣶⣾⣿⣿⡿⣆
⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠆
⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠁
⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠛⠛⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⡿⠛⠃
⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣄⡀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣄⣀⣀⣀⣀⣈⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣟⢻⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇⢸⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠸⣿⣿⡿⠁⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣶⣦⣤⣄⣀⣀⣀⡀⣀⣀⣀⣤⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣄⢘⣿⣷⡍⠶⡌⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣀⡴⠊⠝⢊⡡⠔⠚⠉⠈⠙⢾⠿⣨⣟⢢⣌⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⠴⠊⠁⠀⠀⠉⠁⠀⠀⠀⣠⣴⣀⣀⢻⣙⢫⣶⣯⠱⣦⡈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠛⠛⠓⠦⣜⣫⣶⣮⠸⠿⣇⣼⡷⢀⣈⠛⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡠⣄⠙⣋⡺⣿⢗⣝⡛⣥⣮⡛⢡⣄⡉⠻⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⡿⣗⣉⣋⣾⣿⢝⠻⢟⣦⣽⣿⣧⣮⣩⣾⣷⢏⡱⣶⡄⠙⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠋⠁⢀⣠⣼⣕⡿⠟⠋⠀⠀⢈⡿⠛⠁⠙⢧⠻⡿⣇⣜⡷⢆⡈⠙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠚⠋⢹⡿⠋⠀⠀⠀⠀⠒⠉⠀⠀⣠⠖⠛⠻⣆⣙⠿⣣⣮⣟⠗
⠀⠀⠀⠠⢄⣠⣤⠴⠶⠒⠛⠁⠀⢈⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣤⡿⡿⣟⣭⡛⢵⣶⣦
⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢛⠿⣿⣃⡽⢿⢟⣤⣍⠁
⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠟⠋⠉⢁⣤⣾⣮⡛⠋
⠀⢦⣤⡈⠳⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢶⣿⠇⣙⠿⠋
⠀⠀⠙⢿⣦⠈⢦⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣀⣴⣄⡻⣿⣷⣦⡀
⠀⠀⠀⠀⠹⣷⡄⠙⠲⠂⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠙⠿⢟⡁⠈⠻⠿⠋
⠀⠀⠀⠀⠀⠈⠳⣄⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠙⢿⠶
⠀⠀⠀⠀⠀⠀⠀⢈⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠞⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣌⣛⣦⡤⢤⣀⣀⣀⣀⠀⣀⣀⡀⠀⠀⠉⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣳⣿⠻⢷⣽⠭⣿⠼⣯⣉⠉⠳⣦⠀⠀⠀⠉⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⣿⠉⠻⢦⣉⠲⣄⡀⠀⠉⠓⠒⠛⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣹⠆⠀⠀⠀⠉⠳⣬⡙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⢻⡄⠀⠀⠀⠀⠀⠀⠙⠶⣬⡑⢦⡀⠀⠀⠀⣤⡄⠒⢲⣬⣭⡵⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢾⠛⡛⢻⣿⢦⡀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣌⣩⣭⣵⠶⠞⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠀⣽⠀⢃⠸⡇⠙⢧⣙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠈⠁⣸⡏
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠙⢦⣈⣹⡄⠀⠀⠙⠳⢦⣝⣛⣡⠤⠤⣤⣤⣛⡿⠛⢳⣄
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠙⠉⠉⠀⠀⠀⠙⣦
⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠰⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢢⠀⣿
⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣦⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢿⣅⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣸⣿⠀⢸⡇
⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠘⢿⡿⠃⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠉⠻⣆⠀⠀⠀⠀⠀⠈⢿⣿⠏⠀⢸⡇
⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠉⠉⢻⡿⠛⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠟⠁⠀⠀⠀⠀⠀⠀⢀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⢧⡉⠛⠛⣷⠀⠀⠀⠀⠀⢻⡋⠉⣠⡶⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠀⡎⠙⠷⣾⣧⣦⣀⣤⣀⣀⣌⣷⡞⠋⡆⠰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⠁⠀⠀⠈⠉⠈⠛⠈⠉⠉⠉⠉⠀⠀⠁⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⠻⣷⡶⢤⣀⣀⠀⠀⠀⠀⣀⣠⣷⣶⡿⣋⣀
⠀⠀⠀⠀⠀⠀⣠⡴⠲⠶⠶⣶⠛⠉⠁⠀⠀⠀⠈⠉⠑⢲⡖⠋⠉⠁⠀⠀⠀⠉⠉⠓⣶⠶⠶⠶⢦⣄
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡆⠀⠀⠀⠘⢷⡀
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠈⣷
⠀⠀⢀⡼⠃⠀⠀⠀⠀⠐⢷⡄⠈⢉⣉⣉⣉⡀⠶⢀⡀⠘⠃⢀⡀⠤⣄⣉⣉⣉⡉⠁⢠⣾⠀⠀⠀⠀⠀⠘⢇
⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠛⠛⠿⠿⠿⢿⣿⣿⡿⠿⠿⠿⠟⠛⠿⠿⠿⢾⡇⠀⠀⠀⠀⠀⠀⠘⣦
⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠙⠶⢤⣤⢤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⡤⠶
⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠙⢦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⢉⡏⠁⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠁⠈⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⣴⠿⠓⠉⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠓⠚⠿⣦
⣤⣤⣤⣤⣤⣤⣤⣽⣷⣶⣶⣮⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣵⣶⣶⣾⣭⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡿⠟⠻⠿⠿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣏⣠⣀⣤⣤⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣂⣰⠤⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⢠⠗⠈⠉⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⢀⣀⣀⠀⠀⠤⠤⠤⠤⠀⠐⠒⠲⢲⡀⢀⠀⠄⠂⠀⢸⠊⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡇⠀⠀⠀⠀⠀⠸⡀⣷⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⢀⡜
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠐⠐⠈⠀⠀⠀⠙⢦⣀⡠⠤⠒⠁⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⡆⠀⠀⠀⠀⣀⢀⠀⣀⠀⠀⠀⠀⠀⠀⢏⡇⠀⠊⠁⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⠤⠤⠖⠚⠉⠙⡄
⠀⠀⠀⢡⠀⠀⠀⠀⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⢸⢹⠀⠸⡏⠙⠒⠦⢌⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⡆⢲⡒⠒⠀⠀⠠⣭⣉⠓⠢⢤⡠⠤⠤⠔⠒⠊⢉⠏⠸⡀
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢇⢬⡡⠀⠀⠀⡇⡇⠈⠙⠢⢄⣀⠀⠀⠀⠀⡴⠁⠀⠀⢣
⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠐⠂⠀⠀⠀⠳⠃⠀⠀⠀⠀⠈⠉⠓⠦⠊⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠸⠄⣠⣤⠄⡀⠒⠒⠒⢒⣒⡒⠒⠒⠒⠒⡟⠒⠒⠒⠒⠒⠛⢀⠐⢢⣤⡄⠀⠠⡄⠈⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠐⠒⠚⠓⠛⠛⠒⠓⠓⠒⠒⠒⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⠒⠛⠒⠚⠛⠓⠒⠒⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⡿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡿⠁⠀⢹⣿⣿⡟⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⣤⡴⠋⠉⠘⢈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠅⠀⠀⠀⠀⣿⠉⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢃⡀⠀⠀⠸⠁⢀⠔⠀⣰⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠘⢓⠖⠁⢀⣴⣿⣻⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣰⡿⠛⠉⠀⠀⠙⢷⣼⣦
⠀⠀⠀⠀⢠⣾⣿⣿⣷⣦⣲⣾⣿⣯⣷⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠏⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣯⣿⣽⣿⣿⣿⣽⡄⠀⠀⠈⣓⠂⠁⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⢠⡀⣿⣿⣿⡆
⠀⠀⠀⠀⡏⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣠⠏⢀⠀⠀⠱⠀⢀⡄⣠⣀⡀⠀⠀⠈⣿⡿⠀⠀⠀⠀⠀⠀⣿⡇⡟⣿⣿⣇
⠀⠀⠀⠀⢷⣶⣶⣶⣶⣾⣿⣿⣿⣿⡿⣿⣿⣿⡻⣮⣿⣶⡣⠄⣤⠒⠺⡙⠈⠠⠴⠤⣶⡄⠹⡇⠀⠀⠀⠀⠀⣰⡟⣹⢳⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡵⠀⡇⠀⠀⠀⠀⠒⠤⣄⣘⡆⣰⠁⠀⠀⠀⠀⢠⠋⣰⢯⣿⣿⣿⣿
⠀⠀⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡆⠃⢄⣀⠀⠀⠀⠀⠀⠈⠁⢝⢦⡑⠀⠀⣠⣿⣞⣱⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⡟⠉⠁⠀⠀⠈⠙⢦⡀⠀⠀⠀⢰⠌⠧⠁⠀⣴⣿⣿⣿⣿⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⡀⠀⢀⡜⣿⣿⣿⣿⠟⠿⡟⠈⠘⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢠⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⡿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⠋⠠⠄⠛⠋⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠀⠺⠃⠀⠀⠀⠉⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠠⡄⠀⢠⡄⠈⠇⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⢆⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠤⡼⢣⡀⠀⠀⠃⠀⠀⠀⣠⢶⣤⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠰⡎⠀⠀⠙⠢⡤⠒⠂⠉⠉⡤⣣⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢴⣆⠀⠀⠀⠘⠢⠤⣀⢸⠀⠀⢠⠔⠺⣟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣿⣣⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠤⢄⣀⣀⡇⠹⣷⡄⠀⠀⠀⠀⢸⠀⠀⠀⠸⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⡇⠀⠘⢾⣆⠀⠀⠀⢸⠀⠀⠀⠀⢱⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠚⠧⢤⠴⠊⣉⡳⢦⡀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⡀⠀⢻⠀⠀⠀⠀⠀⢠⠀⡒⠤⢽⣾⡃⠘⣄⠀⠀⠀⢀⠇
⠀⠀⠀⢸⠉⡩⠭⠛⠉⠉⠉⠁⠀⠉⠉⠉⠚⢿⡒⠠⠤⠰⠚⠢⠭⠿⠞⠙⠿⠉⠉⠙⠒⠶⣛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠢⣄
⠀⠀⠀⠇⠛⠒⠤⢤⡤⠤⠴⠒⠒⠒⠒⠒⠤⠤⠬⣦⣀⣤⠤⠤⠴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠬⠷⠄⠀⠀⠀⠀⣀⣠⢤⡀⠀⠀⠀⠈⠑⢤⡀
⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣒⡩⠕⠒⠉⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠕⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⢸⠂⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⡒⣒⣒⣒⣒⣒⣛⡇
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⣀⣠⣤⣶⡆
⠀⠀⠀⠀⢿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠘⠿⠿⠛⠛⠁⠀⠀⠀⠀⠀⠻⠁⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠔⠒⠒⠒⠂⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠲⠦⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⠋⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠒⠲⠤⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠒⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣦⣶⣶⣾⠿⠿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠀⣤⣀⣀⣠⣤⣤⣴⣶⣶⡶⠶⠿⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣄⣖⠼⠄
⠀⠀⠀⠀⠀⠀⠀⢐⣤⡀⠀⠀⠀⠈⠛⠲⢦⣄⡀⠀⠀⠀⢸⣿⠆⠀⠀⠈⠁⠀⠀⠀⣀⣀⣠⣤⠤⠴⢖⣚⣟⣛⣻⣭⣭⣷⣾⠟
⠀⠀⠀⠀⠀⠀⠰⠋⠒⠬⣙⠲⢤⣀⠀⠀⠀⠀⠉⠙⠲⢤⣀⣿⣿⣶⣶⣶⢶⣒⣛⣯⣭⣭⣴⠶⠿⠛⠛⠛⠉⠉⠉⠁⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠉⠙⢻⣿⡟⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣤⣷⡤
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣈⠑⠢⢄⡀⠀⠀⠀⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠴⢖⣚⣯⣭⣭⣶⣾⠃
⠀⠀⠀⠀⠀⠀⠈⠐⠦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡈⠑⠠⢄⣻⣿⣯⣀⣀⣤⣤⠤⣶⣞⣿⣯⣿⡶⠾⠟⠛⠋⠉⠁⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⡀⢈⣉⣩⣭⣿⠷⠾⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠒⣛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⢸⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠞⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⢸⣿⢋⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠴⠖⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠼⣿⣿⣄⣠⣤⣶⠶⠖⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠒⣤⡀⠀⠀⢀⡔⣹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠁⡇⠀⠀⠚⠀⠋⢇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡔⡡⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠑⣀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⣀⠀⠀⣠⣿⣀⠀⢀⡤⠀⠂⠠⡀⠀⠀⢰⡇⠀⠀⣰⠉⠃⠀⠀⠀⠀⠀⠀⡀⠄⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠰⡔⠡⠀⠀⢌⢳⡾⠀⠀⠀⢢⠈⡄⠀⢸⡇⠀⢀⠃⠀⠂⠀⠀⠀⠀⡀⠾⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡇⠃⠀⠀⠘⠇⡇⠘⠀⠀⢈⡇⡅⢀⠘⠃⠀⡎⠀⠸⠀⠀⠀⡠⠂⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡱⢄⣀⠀⠀⡇⠘⢄⣤⠞⢠⠁⢸⠀⠀⣸⠀⠀⡇⠀⠀⠾⠶⠤⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠛⠦⣤⣄⠀⠹⡄⠀⠀⢰⣿⣿⠇⠀⢀⡇⠉⢰⠅⢠⡎⢀⣤⡤⡀⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠂⠀⠀⠡⠶⣧⠀⡇⢸⣿⣿⠀⠀⢸⠀⠀⣸⢠⣿⣱⣿⣿⢏⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⠀⠀⠀⠀⠈⠙⣇⢸⣿⣿⠀⠀⡏⠀⠀⣿⠿⣽⣿⣿⣳⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠒⠿⠶⠶⠤⢶⣤⣀⣿⣸⣿⣏⠀⢀⡁⠀⣸⢏⢾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠉⠈⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡠⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢤⡀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣭⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⠋
⠀⠀⠀⠀⠀⠀⠀⢠⠤⡴⠶⠛⠁⠀⢸⡟⣻⠹⡟⢒⠲⠶⣶⠶⢶⡶⢶⡶⣶⢶⢶⢶⢶⡶⠦⣼⠀⠈⠑⠒⢒⡶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠢⢄⡀⠘⢧⡝⠀⢳⠤⠀⠀⠸⡄⢸⡆⢸⢷⢧⢯⠇⡞⢸⡇⣠⡾⠁⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠈⣿⠀⠈⡇⠂⠀⠀⢳⠀⣷⣜⣞⡟⣼⡸⡁⠀⣿⠋⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠸⡀⡄⠀⠀⠛⢹⠇⣹⣹⠇⡑⠀⠀⠧⣤⡴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡨⠄⠀⠀⠘⡿⢷⣿⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠒⡀⠀⢸⡇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⢡⠄⢸⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡤⠂⢸⡟⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢩⠄⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⢚⠄⠀⣀⡼
⠀⠀⢀⠴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉
⠀⢠⠏⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⢀⣴⡾⡄⠀⢀⡤⢤⡄⢀⣠⠤⠄⠀⣸⣁⣠⠀⠀⠀⢠⣤⣤⣿⠤⠀⣠⠤⣤⠀⠀⢀⣤⠶⣦⠀⣀⣤⠄⢀⣾⣀⠀⠀⣀⣀⠀⢀⣀⠀⠀⣴⠏
⠀⡏⠀⡌⣰⠋⣠⡾⠀⡞⠁⠀⢀⣰⣏⡉⠁⠀⠀⠀⠀⠀⣿⠃⠀⡼⠁⢀⡾⠀⠀⢯⣀⣀⡀⡾⠋⠀⠀⣼⠋⣹⠇⡞⠁⢨⣷⡟⢉⡇⣼⠋
⠀⠳⠜⠁⠱⠞⠋⠓⠀⠙⠒⠚⠋⠈⠀⠉⠁⠀⠀⠀⠀⠀⠻⠖⠋⠙⠒⠋⠀⠀⠛⠛⠛⠛⠛⠙⠒⠛⠉⠛⠀⠛⠀⠙⠚⠋⠘⠓⠋⢰⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⡄⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣠⡀⢧⡤⠒⠉⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢳⣀⡠⠔⠒⠒⢦⡀⠀⠀⠀⢸
⠀⠀⢀⠀⠀⠀⠀⠀⢠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢙⣦⡀⠀⠸⡤⠤⣤⣀⡀
⠀⡔⠃⠓⣠⠀⠀⠀⠈⠒⠊⠀⠀⠀⠀⢠⠔⠂⠀⠒⠢⡀⢀⠤⠐⠒⠀⠐⢢⠔⢋⣠⠽⠆⠀⠀⠀⠀⢀⣨⡆
⠀⠈⠓⢋⠠⠒⠢⣠⠖⠲⠀⠀⠀⠀⠀⠙⢆⠀⠀⣠⡀⡘⢁⠀⠀⠀⢀⡴⠇⢠⡏⣄⢤⠤⠤⠶⠾⢭⣁⣀⣀⠀⠀⠐⠂⠤⢄⣀
⠀⡠⠀⠁⢔⠈⠉⠒⠃⠐⣆⠀⠒⠒⠒⠒⠒⠛⠓⡋⠉⠈⠛⠃⠀⠒⠁⡶⠀⠀⠘⣡⠋⠀⢀⠴⢄⠀⠀⠈⠉⠉⠙⢲⠆⠀⠀⢈⣱
⠈⠁⠒⠂⠀⢬⠐⣠⠀⠀⠀⠙⠒⢤⡤⠤⠄⠒⠉⠁⠀⠀⠀⠀⠀⣠⠞⠁⠀⠎⠉⠈⠒⠚⠁⡠⣈⢢⠀⣴⠀⠀⡰⠃⠀⠀⣠⡞⠁
⠀⠀⠤⣀⣉⡐⠮⢀⠀⠀⠡⠀⠀⠀⠉⠒⢤⣀⠀⠀⣀⣀⠤⠔⠊⠀⠀⠀⣸⠖⢤⣀⣀⠤⠊⠀⠈⢦⣩⠞⢢⠎⠀⠀⢀⣼⠋
⠀⠀⠀⣀⡤⢄⡋⠔⠁⢔⡩⡹⠦⣄⠀⠘⣿⣧⣌⠉⡰⠒⡶⠠⠤⣤⠤⠊⠀⣠⠀⢀⡀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⣠⡿⠁
⣤⠊⠉⠀⢾⡁⠀⠀⠀⣜⡩⢔⢀⠦⢒⠥⠊⠉⠁⠛⢁⠈⠲⢶⣉⠀⠀⢀⣼⡟⠳⠭⣉⠒⢄⠀⠀⠀⡔⠁⠀⠀⣜⠟
⠀⠈⠉⠐⢒⢤⣀⡀⠂⠉⢠⠸⠄⡎⢠⢲⣊⣍⠦⡀⢀⠠⢚⡿⠟⠻⠶⠛⠛⠤⢀⣀⠀⠈⠈⠓⡀⠜⠀⠀⢀⣾⠎
⣴⠦⣒⠪⢽⣼⣶⣗⣦⣅⠂⠝⡒⠁⠈⡭⡌⡗⠂⢋⠅⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡉⠒⢢⣜⣊⠀⠀⠀⢼⡎
⠀⠉⠂⠍⡒⠭⡛⠿⣿⣽⡯⢶⣦⣈⡐⠧⠍⠴⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠵⣀⠀⡿⠈⠉⠓⠺⠞
⠀⠀⠀⠀⠀⠑⠢⢕⡢⢍⠛⢽⡯⢷⠷⣒⣄⣀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠋⢑⠢⣀⠀⣀⣤⢔⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢍⡲⢌⠛⠉⢋⠉⣤⠶⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣽⡪⠔⠉

⠀⠀⠀⠀⠀⡀⢀⢠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡀⡀⢠⢴⠶⢶⣦⡄⠀⡐⡆⠠
⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠘⠉⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡰⠚⠉⠀⠀⣾⣤⣤⣼⣿⠃⢀⢰⠇⡇
⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠬⣄⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢒⠀⣿⣿⣿⣿⢼⣀⣀⣸⠰⡀⠤⠤⠤
⠀⠀⠀⠀⠐⠂⠀⠀⠁⢉⠁⠀⢌⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀
⠀⠀⠂⠀⠀⠀⠀⠀⠨⠷⠀⢀⡀⠒⠠⢀⠄⡀⠀⠈⠉⠉⠀⡀⠐⠒⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢠⠀⠀⠁⠈⠁⠉⠖⣠⠆⠀⠈⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⣀⡀⠀⠀⠀⠀⠇⢀
⠀⠀⡀⠀⣀⠀⠀⠁⠘⢀⠂⢀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⣄⣀⠀⠋⠉⢉⠉⢉⠉⡩⠉⠉⡠⠘⡆
⠀⢀⠋⠀⠋⠀⠀⢀⠀⠀⠀⠁⠠⢤⢤⣇⢘⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠐⢁⡆⠀⢁⠄⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣦⠄⠀⡀⠀⠓⡜⠲⡇⠀⣀⠐⠈⠐⠂⠀⠀⠀⠈⠲⠀⠀⠀⢀⠈⠀⠀⡠⠉⢀⠔⠁⠀⠀⠀⠀⡠⠄
⠀⠀⡀⠄⠀⠀⠀⡁⢄⠀⡀⠀⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⠀⠤⠄⠀⠠⠌⠀⠰⠏⠀⠀⠠⠂⣡⡀⠀⠀⠀⠀⠀⢠⠊⢠
⠀⠀⠀⠀⠀⠠⠎⠁⠀⠈⠙⡁⠀⣀⢀⠂⠀⠀⠆⠚⠀⠀⠀⠒⠀⠀⢀⡁⠀⢀⠤⠤⡄⡄⢀⠈⠐⠫⠄⡀⡀⠔⠁⠀⠐⠆⠈
⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⣦⠀⠓⢀⡆⠀⠰⠆⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠈⠈⠉⢁⡴⠃⠀⠀⠀⠀⠀⠁⢀⠀⠀⠐⡀
⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⢀⠴⠦⣤⣀⡀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠠
⠀⠀⠂⠀⠀⠀⠀⠁⠀⠉⠀⠃⠐⠇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⡒⠈⠉⠒⠢⢤⣄⠔⠁
⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⣀⠉⠉⠁⠉⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠠⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠁⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡀⠠⠄⠐⣂⡀⠀⠀⢀⡀⢀⡠
⠀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠚⡂⠀⠄⠀⣂⠠⠤⠄⠤⠤⠤⠤⠤⠤⠤⠴⠴⠤⠤⠀⠐⠀⠤⠒⠚⠋⣉⣤⡈⠚⠒⠩⠔⠂
⠀⠂⠀⢉⠉⠉⠀⠀⠈⠉⠉⠉⢙⣋⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠉⠹⠿⠍⢁⠀⠀⠉⡰⠂⠛⠋⠀⠀⠂⠀⠁⣀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⣿⡿⠻⠟⠋⠀⠙⠿⠻⣿⣷⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢿⣿⠋⠈⠐⠀⠀⠀⠀⠀⠂⠀⠹⣿⠿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠉⠀⠀⠀⠃⠀⠀⠀⠘⠀⠀⠀⠋⢠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⢀⠈⠉⠁⡀⠀⠀⠀⠀⠹⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣤⣶⣿⣄⠀⣠⣷⣦⣤⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣞⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠁⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡛⠛⠛⣛⣛⣛⣛⡆
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣭⣭⣿⣯⣭⣭⣭⣥⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⢸⠆⠐⠀⢄⠀⠀⠀⠀⠀⠀⠀⠄⠈⣣⠭⠭⠤⠀⠀⠀⠀⠀⢠⡈⢀⠶⡀⠀⠀⠀⠀⢸⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⢀
⡄⠀⠀⠀⠀⠂⠤⠤⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢀⡀⣦⠀⠀⠀⠒⠀⠀⠀⠀⡀⡁⢀⡰⡈
⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠠⠋⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠹⣌⢳⡀⠀⠀⠀⠀⠀⡈⠁⡡⢡⣤⠉
⠄⠂⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠹⠀⠀⠀⠀⠀⠀⠘⠐⠤⠡⠌
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠠⠊⠁⠈⠢⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠈⠠⣀
⠀⠀⠀⡠⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠈⡐⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠃
⠀⠐⠂⡀⠐⠦⠠⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠩⠥⠤⠀⡀⠀⠀⠁⠂⠒⠑⠌⠂⠀⠀⠀⠀⠠⠄⠀⠂⠈⠀⠀⠰
⠀⠀⠈⠁⢠⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠐⠂⠐⠢⠀⠀⡐⠁⠁⠂⠐⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠈⠈⠀⠁⠀⠀⠇⠀⠀⠂⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠁⠀⠄⠀⠀⠃
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠡⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠠⠐⡀⠀⣘⠐⠀⠀⠀⢂⡠⠀⠈⠦⠀⠀⠀⠀⠀⠆⠡⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠐⠀⠀⠈⠁⠒⢆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠙⠒⠨⠇⠀⠀⠐⠀⠀⡄⡀⠄⠂⠂⠀⠄⠀⠀⠀⠀⠉
⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠈⢂⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⡀
⠀⠀⠀⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠈⠀⠤⠈⠠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣴⣄⠀⢀⠠⠐⠀⢄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⠀⠀⠀⠠⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠐⠻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠤⣀⠀⠀⠀⠀⠀⠰⡄⢀⠡⠄⠀⠀⠀⠀⠦
⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⢀⠀⢄⠐⣄⠈⠁⠀⠀⠀⠀⠀⠈⠈⠄
⠀⠀⠀⠁⠢⠀⠀⠀⠀⠀⠀⠀⡐⢠⠆⡤⢀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠈⠆⠀⠀⠀⠀⠘⠄⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⢍⠀⣒⠖
⠀⠀⠀⠀⠁⢡⠀⠀⠀⠀⠀⠠⠀⠀⠠⠁⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣌⠢
⢁⠀⠀⠀⠀⡼⣄⠀⠀⠀⠀⠁⠀⠀⠂⠄⠀⠄⠀⠀⠠⠀⠀⠀⠠⠀⡃⠀⠀⠀⠈⠙⠒⠲⠤⡄⠀⠀⠀⠀⢰⠶⢶⣠⣬⣵⢶⣛⣶⡶⠶⣤⣤⡀⠀⣧⠑
⠀⠈⢐⡤⠉⠁⠀⢑⠀⠀⠐⠀⠀⠀⠁⠀⡘⠀⠀⠀⠰⠒⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⢀⠸⠛⠛⠶⣶⣮⣭⣽⣛⡪⣽⠇⢈⣿⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡟⠀⠀⡛⠀⠀⢚⠀⠹⠉⠉⠉⠛⠋⠉⠛⠛⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠻⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠢⠐⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⠀⠀⠀⠠⠤⠤⢀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⡆⠀⠀⠹⣄⠀⠀⠉⠀⠀⠀⠀⡔⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠒⠒⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⠊⠁⠀⠀⡇⠀⠁⠀⠈⢦⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⡾
⠀⠀⢀⢲⠀⠀⠀⠒⠊⠁⠀⠀⠀⠀⠀⠀⡇⠀⠈⠀⠂⠀⣳⠀⠀⠄⠠⡏⠀⢀⠀⢰⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠤
⠀⢰⣇⣋⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡤⠤⠖⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⡐
⠀⠀⡏⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⠉⣁⠠⠄⠀⠀⠂⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣒⠉⠡⠤⠔⢒⣀⠀⠀⠍⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠒⣒⠀⠁⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠤⠥⠀⠂⠀⢡⠤⠀⠐⠀⢈⠉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢉⡁⠤⠄⢀⠐⢃⣈⡡⠄⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠶⠃⣀⣀⡀⠀⠀⠀⠨⠤⠔⠒⢂⣀⠈⠀⠀⠀⠀⠀⢠⠂⠰⠋⢙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠠⢐⠂⢶⡇⠀⣴⠒⢅⠀⢀⣀⡤⡖⢢⣤⣤⣴⣦⣕⣒⣚⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⡇⣸⣿⢤⣴⣥⣾⣿⡟⠻⡁⢴⢊⣧⣴⢿⡯⢹⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡿⣿⢿⣿⣿⣿⣧⡺⢷⣤⣾⣿⣯⡾⢃⣾⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣧⠘⣿⣿⣟⣉⣽⣿⣷⣯⣭⣭⣥⣶⣿⣿⣿⡿⢿⣟⣿⣙⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⡹⣿⣿⣿⣿⣿⡿⢿⡛⣿⢩⠟⣼⣛⡾⢩⡟⠨⠃⡚⢹⣏⡻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣜⠿⣿⡓⠾⢄⣛⠸⢅⡿⢼⣟⣾⡥⠶⢜⣓⣪⣭⡼⢫⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣮⡻⢷⣤⣖⣁⣚⡛⣭⣤⢶⣶⣿⣿⣿⡿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣽⣿⡿⠿⠿⠛⠛⠋⠉⠉⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⠀⢀⠠⠀⠴⠀⢸⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣄⣹⣿⣿⣳⡄⠀⠀⠀⠀⠉⠳⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣭⣌⠙⠿⢷⣮⣿⣿⣿⡆⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣾⣶⣿⡿⠋⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠿⠛⡗⡮⠙⠋⢀⣂⠖⠀⠀⠀⠀⠰⣆⣀⣀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡚⠀⠀⠀⠑⠀⠀⠀⠀⢀⡀⠀⠀⠀⠂⢀⣀⡵⢾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣎⠀⠙⢿⢢⡀⠀⠀⠀⠀⠐⠚⠀⠀⠀⠤⢟⣭⣥⣶⣟
⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢙⣦⣀⣳⡙⠢⠌⠁⠀⠤⠀⠀⣘⠯⠈⠙⠀⠉⠉
⠀⠀⠀⠀⠀⢀⠖⠁⠄⠀⠀⠤⣙⢦⡝⠹⣤⣻
⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠈⠑⣝⣆⠹⣟⡤⠀⠒⠒⢲⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⢧⢹⡅⠀⠀⠀⢸⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⢲⢄⣀
⠀⠀⡼⠃⠀⠀⢄⠀⠀⠀⠄⠀⣀⡀⢤⣹⠃⡈⡃⠀⠀⠀⠘⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠑⠤⡀
⠀⡰⠃⢀⠈⠢⢈⢢⡀⠀⠀⣀⣀⣭⣓⠚⢳⣁⣷⣄⠀⠀⠀⡇⠚⣡⣵⡶⢠⠀⠒⠘⢄⠀⠀⠀⠀⠀⠀⢀⠖⠁
⠞⠁⠀⠀⠙⠢⢤⣁⠻⣯⡉⠀⠀⠠⠤⠐⠚⠻⣷⠬⠀⠀⠀⡞⠂⠀⢁⠠⠄⠀⠀⠀⢀⠀⠀⠀⢀⢄⢴⣉⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⢳⣄⠀⠀⠀⠀⠀⠀⢘⡇⣀⣀⣠⣁⣀⠡⣌⡃⠀⠀⠐⠀⠀⠀⠀⣀⠴⢧⣎⡀⢀⡀
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⢞⣰⡿⠀⢠⠀⠀⠀⠀⢫⠢⡄⠳⣄⠀⢀⣢⣛⠆⡀⠀⢠⣄⣤⢀
⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠴⠚⠁⠎⣠⠔⠁⠀⠀⠀⠀⠈⡇⠁⠀⠈⠐⠋⠠⡙⠴⢀⠄⠔⣭⣭⠢
⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠈⠳⢦⣀⠐⠊⣁⣀⣀⣀⣀⣤⡤⠤⠁⠀⠀⠀⠀⠐⠄⠈⠱⡄⠐⠂⠊⠁⠐
⠀⢾⡒⢒⠀⠀⠀⠀⠠⠠⠄⠀⢁⣀⣀⣀⡀⠙⢍⠣⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠢⡀
⢰⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠒⢯⡳⠄⠣⡈⢂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠁

⠠⠈⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠄⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠡⠓⠀⠀⠀⠀⠀⠈
⠁⠀⠀⠀⠀⠠⠀⠂⠉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⢄⢠⡢
⠀⡀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠐⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠊⠁⢀⠠⠀⠀⠀⠐⠁
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⢂⢀⠀⠀⠌⠆⢃
⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⠍⠂⠀⠀⠀⠀⡀⠠
⠁⠀⠠⠈⠠⠀⠀⠀⠈⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⣈⡀
⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠁
⠀⣀⡄⠀⠀⠀⠀⠖⠀⠀⠀⠐⠀⡈⠀⠈⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠁
⠈⠐⠀⠠⢔⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⠄⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠈⠀⠈⠂⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠄⠑⠢
⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠄⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠠⣈⠀⠀⠀⠀⠌⠀⠀⡀
⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠊⠂⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡀⠀⠂⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢀⢄⠀⠠⡀⢂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢀⠀⠀⠀⠠⠐⠤
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠠⡁⠀⠀⠔⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠁⠀⢀
⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠂⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠄⠌⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢁⠂⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠈⠐⠈⠀⢁⠠
⠀⠀⠁⠀⠁⠀⠄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠂⢀⠐⠁⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠐⠁⠀⡀⠀⠀⢪⠡⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢐⣠⠄⠫⠉⠠⡒⢄⠀⠀⡠⠀⡀⠀⢀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠈⠀⢀⠀⠠⠒⠉⠀⠔⠀⠀⠀⠈⢠⠁⠁⠀⠀⠀⠈⠁⠄⠔⠀⠀⠀⠢
⠀⠀⠀⠐⠀⠀⠀⠀⠁⠀⠀⠂⠀⠂⠀⠦⡀⠀⠀⠀⠀⠀⢀⠠⠀⡈⠐⠈⠀⠀⡠⠂⠀⠀⡀⠀⠀⠂⠀⢠⡄⠀⢠⡄⠀⠈⠀⠤⢤⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠂⠀⠀⠀⠀⠁⠢⡄⠀⠁⠠⠐⠁⠀⠀⠀⢀⠄⠀⠀⠀⠀⠐⠒⠒⠄⠀⠀⠀⠉⠀⠀⠀⠐⠲⡒⠊
⠀⠀⠀⠐⠀⠀⠀⠂⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⢄⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢀⠀⠀⢀⠊⠉⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠔⠀⠈⠐⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠐⠈⠄⠀⣀⠀⢀⠀⠠⠂
⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⡀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢀⠄⠂⡂⡠⠄⢀⣀⢀⣤⠊⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠁⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠁⠀⠈⣿⠿⠋⠀⠀⢢⣤⡀⠀⠀⠀⠀⢀⣠⡦
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠀⠆⠐⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠦⢤⠶⠟⠋
⠀⠈⠀⠀⠁⠀⠀⡀⠄⠄⠀⠀⠀⠀⠠⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢤⡌⠀⢀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⡀⠀⠀⢸⡗⠂⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠉⠐⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠄⡀⠤⠒⠒⠀⠄⠀⠀⠒⠐⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠱⣄⠀⠀⠀⠀⢀⡀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⢀⠅⠀⠀⠀⠀⠀⠉⠒⠒⠈⠁⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠈⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠈⠂⠒⠠⠐⠂⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠒⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠴⠊⠉⠙⢢⡀⠀⠀⢠⠖⠈⠑⠢⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠁⠀⠀⠀⠀⠉⠓⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣀⡾⠘⠊⠉⠉⠀⠈⠉⠉⠒⠔⢢⠀⠀⠀⠱
⠀⠀⠀⠀⠀⢠⠎⠀⣀⣀⣠⠖⠉⢷⠒⢒⣐⣐⡚⠳⣄⠀⠀⠀⠀⠀⢀⣘⠔⠁⠀⠀⠀⠀⠀⠀⢀⠔⠢⡀⠈⠃⠰⡀⠀⠘
⠀⠀⠀⠀⢀⠇⠰⡏⠀⠉⡤⠦⣀⣸⠉⠀⠀⠁⠀⠙⢍⢆⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⣀⠈⡒⠠⢄⠱⡀
⠀⠀⠀⠀⢸⠀⠀⠙⢄⠜⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⠆⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⢀⡀⢈⡡⠘⠛⠀⠀⡰⠁⡧⠔
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⠀⠑⠚⠒⢤⣀⠀⣀⠀⠀⠀⣔⠃⠀⠀⠀⠀⢸⠙⠉⠉⠉⠉⠁⠍⠀⠀⢸⠀⠑⠈⠀⠀⡇
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡞⠀⠀⠒⠀⠀⠀⠉⣉⠈⡙⠉⠁⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠂⠀⠀⢸⠀⠀⣀⠀⡰⠁
⠀⠀⠀⢠⠃⠀⠀⠀⠀⢠⠃⠀⠀⠂⠀⠀⠀⠠⠌⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠠⣀⡠⠄⠀⠀⠀⠀⠉⡉⠜⠉
⠀⠀⠘⠣⡄⠀⠀⠀⣴⢧⣀⠀⠀⠐⠢⠠⠀⢀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠌⣀⣀⡀⠀⢤⣔⡊
⠀⠀⠀⠀⠙⠚⢢⡼⠀⢀⠑⠭⠶⣦⣒⣂⡬⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠳⠴⠋⡀⠀⠂
⠀⠀⠀⠀⠀⠀⡘⡄⠀⠘⠀⠀⠀⡁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⡆⠀⠀⠁⠰⢄⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠄⠀⠘⠄⠀⢆⠀⠉⠂⠄⣗⠄⣀⠀⠑⠢⢄⣀⠂⠀⠀⢀⠀⠀⠀⠀⠈⠁⠀⢐⠄⠀⠀⠀⠄⠈⠀⢀⠄
⠀⠀⢨⠀⠀⠀⠈⠁⠀⠑⠢⠤⢀⡀⠉⠙⣢⡰⢄⣠⠏⠑⠦⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⢀⡀⠀⠐⠉
⠀⠀⢸⠀⠀⠀⢰⠤⠀⠀⠀⠀⠀⠈⠑⠲⠋⠁⢠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠲⠤⠄⢀⠠⠆
⠀⠀⠈⡄⠀⠀⢸⠁⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠
⠀⠀⠀⠣⠀⠀⠜⠀⠀⠀⠐⠄⠀⣀⡠⠤⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈⠋⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⢂⠉⠄⠀⠠⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⢀⡀⠁⣀⡠⠭⠒⠂⠁⠀⠀⠀⠀⠀⠑⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡿⢡⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠋⠈⢿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⠘⠛⢿⠟⠀⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡅⠀⠀⠀⠀⢰⢾⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⠀⠀⠱⡴⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠘⠢⡀
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣠⠖⠙⠛⠧⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠖⠋⢻⡤⠀⠀⡠⠀⠀⠀⠈⠢⡀
⠀⠀⠈⠉⠉⠁⣀⣠⡴⠞⠉⠀⠀⢀⣠⣴⣾⡿⠧⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⠒⠈⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⣀⠤⠒⢩⣏⣁⠀⠀⢀⣤⣶⡿⠛⣭⡷⡙⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⢿⣷⣦⣿⣿⣿⣷⣾⡿⠋⣑⡬⠛⠉⠟⢽⣮⣍⠃⠀⠀⠀⠀⠀⠐⠒⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠘⠿⢿⣿⣿⡟⠛⣻⡷⣿⣥⣍⠃⠀⠀⠘⡿⠟⠻⠿⢿⣿⣴⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠈⠙⠛⠦⠀⠉⠛⣿⣿⣿⠀⠀⠀⠃⠀⠀⣠⡞⠋⠙⠀⠒⠢⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⣀⠀⠀⠠⠞⠉⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⣾⠿⠛⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠑⠤⢄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣰⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀⠘⠀⠀⠈⠀⠀⡠⠋⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠤⡠⠊⠁⠀⠀⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣉⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⣀⣀⣀⡠⠤⠤⠤⠤⠶⠶⠒⠒⢒⣒⣋⣉⣩⠭⠤⠤⠤⠤⠄⠐⠒⠒⠒⠒⠀⠤⡄⠀⡸
⠀⠀⠀⠀⠹⡀⠀⡠⠤⠖⠒⠒⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢰⠃
⠀⠀⠀⠀⠀⠱⡀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡏
⠀⠀⠀⠀⠀⠀⢳⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸
⠀⠀⠀⠀⠀⠀⠀⢣⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢇⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠱⢤⣀⣀⣀⣤⠤⠤⠤⠤⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠒⠒⠒⠒⠒⠒⠚⠉⠉⠉⠉⠉⣉⣉⣉⣉⣉⣉⣉⠉⠉⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣰⢚⣉⣉⣉⣭⣩⠭⣍⣭⢭⣉⣭⣭⢭⣭⡍⣽⡍⢹⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠙⠒⡊⢉⣉⣈⣭⣭⣨⠭⣤⢤⣤⣠⣴⡆⣖⡶⢸⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡇⢸⣭⣿⢟⡜⡱⢋⣽⣭⠿⡽⣿⡌⢯⢻⠢⡴⢲⠀⡇⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠃⣹⣴⠃⠘⠰⢡⢏⢞⣩⠯⡵⢌⢻⣦⠃⠃⠙⡄⡆⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢳⠇⠀⠀⠀⢠⣣⢋⠞⢉⢅⣾⣧⠉⠀⠀⠀⢸⠃⢇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢸⠀⢸⠀⠀⠀⠀⠈⢁⣞⣤⡬⢵⠦⣄⠀⠀⠀⠀⠀⡇⢸⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣀⡎⠀⠀⠀⠀⢀⠮⠬⠭⠭⠭⠤⠬⢆⠀⠀⠀⠀⢧⠼⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡼⠀⠀⠀⠀⠀⣎⣀⣀⣀⣀⣀⣀⣀⣸⡀⠀⠀⠀⠘⣤⠤⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡽⠁⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⣀⣠⣤⣄⣐⣦⣤⣴⣶⣾⣿⣶⣶⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣀⣴⣦⣤⣴⣴⣶⣾⣿⣷⣾⣿⣿⣿⣿⡿⠿⠿⠿⠟⠛⠛⠛⠋⠉⠉⢹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣴⣶⣾⣶⣾⣿⣿⣿⣿⡿⠿⠿⠿⠟⠛⠛⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⣿⡿⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⠁⢀⣀⣀⣀⣀⣤⡤⠤⢤⡤⠶⠖⠂⠀⠀⠀⠀⣼⣿⣿⡛
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡀⠀⠀⢴⣶⣤⣴⣶⣶⣾⡿⠿⠆⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣸⣿⣿⢳⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡻⣿⣿⣄⠀⠀⠋⠉⠭⠅⠀⠀⢒⣐⣘⣢⣀⣀⣀⡤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢏⡞
⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠝⣿⣿⣦⠀⠀⠀⠘⠚⠊⠉⠁⠈⠉⠁⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣶⣶⣶⣶⣾⣿⣿⣟⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢞⢿⣿⣷⣦⣤⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⣋⣉⣉⡉⠊⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠾⠿⠻⠛⠛⠛⠛⠛⠛⣛⣋⣉⣩⣩⣥⣤⣤⣦⣶⣶⢶⣺⠿⠿⡟⢿⣟⣛⣛⡅⠘⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣧⠁⢰⣶⡶⢾⠽⠿⢿⡟⣻⣋⣛⢫⣭⣭⣬⣽⣷⣶⣶⣶⣿⡿⢿⣿⢿⣿⣛⣿⣇⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡟⠀⠀⣶⣿⢞⣿⠟⣁⣼⠟⣩⣤⣿⣟⣿⣯⣿⣿⣾⣿⣱⡛⠻⣶⣍⡙⢿⣿⠻⣿⠀⠁⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠂⠀⣿⢇⣾⠃⣸⡿⠣⣸⣿⣛⠿⠻⠛⣿⣽⣿⣾⣿⣿⣿⣦⣌⡻⢿⣦⠙⢷⡙⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⢿⠀⠀⠘⢸⠇⢰⡿⠁⣸⡿⠏⣁⣲⣶⣿⣿⣼⣿⣭⣟⡁⡙⠻⣿⣆⠈⠻⡆⠈⠁⠐⣀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⡞⠀⠃⠀⠀⠘⠁⠀⠟⠁⣰⣟⣻⣿⣽⣿⣭⣿⣿⣿⣻⢿⣦⡄⠙⠂⠀⠀⠀⠀⠀⠘⢸⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣆⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⢋⣤⣻⣿⣧⣿⣿⡾⠿⠶⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⡗
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⡧⣿⠀⠀⠀⠀⠀⠀⠀⠸⠁⢀⡾⠛⠛⠛⠋⠉⠉⠑⠚⣶⣤⡄⠳⡄⠀⠀⠀⠀⠀⠀⠸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⢸⡞⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠓⠛⠛⠛⠛⠉⠉⠉⠉⠉⠙⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠟⠀⣦⠀⢠⣼⠁⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⣤⣀⣀⣤⣤⣴⣾⣿⣶⣿⣿⣶⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠛⠛⠛⠛⠛⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣀⣤⣄⣠⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⣠⠠⡉⢉⡉⠉⠉⠉⡉⡉⢉⠉⠉⠉⠉⠉⠉⡉⠑⣤⡶⠤⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠘⠀⠀⠀⠂⠨⢄⢋⢍⢜⢘⠂⠂⠀⠑⠂⠀⠀⠘⠀⠘⢿⣗⡀⠠⠄
⠀⠀⠀⠀⠀⣠⡤⠃⡀⠊⠀⠀⠀⠀⠀⠀⠀⠊⠈⠈⠀⠀⠰⠄⠓⠀⠀⠀⣠⠀⠠⣌⢧⣭⣍⢩⡄
⠀⠀⢀⣤⠶⠾⣥⡾⠲⣄⢀⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠣⡑⣀⢼⠾⠿⣿⡝⠛⠁
⠀⠀⠉⠀⠀⠔⠛⠂⠀⣨⡿⠯⠶⠤⠴⢤⣤⠤⢀⣀⣀⠀⠀⣀⣀⣀⣀⣠⣴⡜⢿⡄⠉⠀⠊⠉⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠰⠁
⠀⠀⠊
⠀⠘⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠋⠁⠀⠀⠀⠀⠀⡀⠀⠁
⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠁⠀⠐⢆
⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣄⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣤⣀⣠⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣏⡁⠀⠈⣹⣷⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠃⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠰⠀⠀⠤⠀⠀⠀⠀⠀⠠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⣤⠀⠆⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⣀⣀⣀⠀⠀⠀⢰⠁⠀⠀⠈⠀⠀⠙⠛⠠⣈⠖⠙⠋⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠉⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈
⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠄
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠀⠀⠀⠀⠀⢀⠀⢴⣆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠋⣀⠦⠀⠀⠀⠉⠀⣀⡀
⠀⠀⠀⠀⠀⠉⠀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⢋⡡⠄⣀⣀⡠⠤⠴⠓⠊⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⣉⣥⠴⠶⠒⠋⠉⠁⠀⣀⣀⣀⣤⠤⠴⠒⠚⠉
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⣁⣐⣂⣉⣤⠤⠤⠴⠒⠒⠚⠉⠉⠉⠀⠀⠀⠀⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠒⠤⣀
⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⢿⣿⡟⠀⠀⢀⠄⠀⠀⠈⠀⠀⡠⠞⠁⠀⠀⠀⠀⠙⠢⢤⣀
⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡠⠊⠀⠀⠀⢙⠃⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⢀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢢
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠠⠃⠀⠂⠀⠀⠀⢀⡌⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠠⠃⠀⠀⢀⠂⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⡜⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⡠⠁⠀⠀⢠⠂⢀⠇⠀⠀⠀⠀⠈⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡫⠫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⡀⠀⠀⠀⠠⠃⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⢟⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⡐⠀⠀⠀⡔⠀⢀⠎⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢄⣀⠀⠀⠀⣏⠜⡡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠐⠀⠀⢀⠌⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢼⡣⠊
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠈⠀⠀⢀⠊⠀⡠⠃⠀⠀⠀⠀⠀⠀⢰⠦⡀⠀⠂⠀⠃
⠀⠀⠀⠀⠀⠀⠔⠀⠤⡀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠰⠄⠀⠀⠀⠰⠁⠢⡀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠋⠁
⠀⠀⠀⠐⠂⠄⠀⠀⠀⠀⠀⠠⠎⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠔⠖⢒⠘⠙⡉⠉⡆
⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢠⡄⠤⠤⠄⠔⠒⠒⠘⠉⠁⠈⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠈⠐⠢⢄⡀⠀⠀⠀⠀⠜⠁⠀⠀⠀⡄⠀⡆⠂⠈⠉⠁⠈⠀⠀⠀⠀⠀⠈⠀⠀⢈⠀⠀⠀⠆⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⡌⠀⠀⠀⠀⠀⠉⠢⡎⠠⠐⠀⠀⠀⠀⢸⣸⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⠤⠴⢶⢒⡞
⠣⡀⠀⠀⠀⠀⠀⢀⠝⠀⠀⠀⠀⠀⠀⠸⢸⡄⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀⣀⢀⠤⠤⠤⢤⢖⢒⡚⠿⠭⠍⠒⠒⠒⠉⠉⠉⠉⠉
⠀⠈⠑⠢⠤⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠘⡇⠒⠒⠉⢈⠍⠈⠉⠡⠠⠤⠐⠒⠒⠂⠉⠉
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⠠⡀⠀⢀⣀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠤⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣸⠀⠀⡀⠀⠀⠀⠀⠠⠖⠒⠚⠛⠉⠉⠁⠀⠀⠀⠀⣀⣀⠤⠤⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡉⣡⠔⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⠤⠐⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠉⠂
⠀⠀⠀⠀⠀⠀⠀⢠⡬⣼⠇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡅⠴⠀⠒
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠐⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠄
⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠙⠃⠀⠀⠃
⠀⠀⣆⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠤⠀⣠⠖⠒⢄
⠀⠀⠈⠓⠈⠀⠁⠀⠀⠀⠀⠀⠈⠂⠠⠂⠀⠀⠀⠀⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⡆⠀⢻⣄⡠⡀⠠⡀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣌⠢⠐⠀⣣⠰⣄⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠘⣿⣿⠀⠘⣿⣏⠀⠀⢈⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⢀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢹⢟⠃⠀⠩⠊⢅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢠⠠⠀⢠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠸⠀⠀⠠⢤⠾⠃⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠘⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠉⠒⠀⠐⠀⠄⠀⠀⠀⠠⠴⠚⠀⠀⠀⠀⠀⠀⠆⠀⠠⢠⣄⠀⡐⠄
⠀⠀⠀⢀⠀⠀⣤⢁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⡍⣭⠤⠂
⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⢠⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⠀⠀⠀⠀⠀⠀⡄⠀⠀⠉⠁⠀⠀⠀⢀⣤⡠⠄⠀⢄
⠀⢀⢀⡄⠁⠀⠐⠀⠀⠀⡠⠃⢸⠀⠀⠀⠀⠀⠠⠤⠖⠒⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣤⠴⠞⠋⠁⢀⣀⡀⢈⡄
⠀⠀⠀⠧⠀⢨⠩⠄⢠⣾⡁⠤⡼⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄⠀⠀⠀⠁⠀⠀⠀⠀⠮⠿⠿⠖⠒⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠶⠖⠀⠀⠀⠀⠀⠀⠲⠶⠆⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⣠⣄⠂⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠛⠟⡀⠀⠤⠀⠈⠻⢃⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠠
⠀⣠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠎⠉⡦⡀⠀⠀⠄⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠠⢄⡀⠀⠀⢀⠀⠀⠀⠀⠒
⠀⠁⠰⠐⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠠⠀⠀⠄⠀⠀⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⢀⠈⠀⠀⠈⠀⠐⠃⠀⡄⠀⠀⠈⢀⠐⠁⠀⠀⡀
⡄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠢⠀⠀⠃⠀⠀⠀⡇
⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⡇
⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡦⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠒
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂⢀⡀⠱⠋⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⡀⠊⠈⠉⠁⠈⠁⠐⠚⠆⠀⠀⠀⠀⠲⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠰⠊⠁⠀⠰
⠀⠀⠀⠀⠀⠄⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀⠀⠀⠀⠀⢀⡤⠆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠔⠀⠈⠆⠀⠀⠀⠃
⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠘

⠀⠀⠀⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠛⠛⠛⠛⠐⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠖
⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀
⠀⠀⠀⣷⣤⣶⣤⣤⣄⠀⠛⠛⠛⠋⠉⣥⣤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠍⠩⠭⠉⠭⠤⠥⠤⠤⠆⠀⠀⡀⠈⠁
⠀⠀⠀⠟⠉⠉⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠃
⠀⠀⠀⠶⠲⠶⠶⠶⠶⠖⠒⠒⠂⠀⠄⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣀⣀⡀⣀⣀⡀⠀⠀⠀⣀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⡦⠤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⢩⣉⣉⣉⣉⡁
⠀⠀⠀⣷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⣤⣄⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⣀⣀⡈⠛⠛⠛⠛⠛⢷⣶⣆⣉⠩⠤⠤⠄
⠀⠀⠀⣦⡤⠠⠄⠀⢀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⡃⠀⠀⠀⠀⠉⠉⠉⣉⣉⣉⣉⣙⣛⣛⣻⣿⣿⣿⣯⣭⣽⣯⠙⠛⠛⠛⠛⠛⠛⠛⢓⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣈⣈⣈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⢀⣀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣀⠴⣯⠉⡠⠀⠀⠀⢄⠉⢿⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠊⠀⠀⠀⠀⠈⠑⠠⠄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠉⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿