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⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⠶⠶⠶⠶⠶⠶⠶⠿⠿⠛⠛⠛⠛⠛⠛⢛⣋⣉⣉⣉⣉⣉⣉⣉⣥⡌
⠀⠀⠀⠀⠀⢸⣦⣀⡀⡈⠙⢿⣿⣯⣭⣤⣤⣤⣤⣤⣤⡴⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠛⠛⠛⢛⣛⣛⣛⣛⣉⣉⣉⣉⣉⣭⣭⣭⣭⠁
⠀⠀⠀⠀⠀⠰⠆⠀⠀⠇⣤⠀⠙⢿⣿⣷⡶⠶⠶⠶⠶⠶⠶⠖⠒⢛⣛⣛⣛⣛⣛⣛⣋⣉⣉⣩⣭⣭⣭⣭⣭⣭⡤⠤⠤⠤⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣛⣛⣛⣛⣩⣭⣭⣭⣭⣭⣭⡭⡭⠭⠤⠤⠶⠶⠶⣶⣶⣶⣶⣒⣒⣒⣒⣛⣛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠄⢉⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣭⣭⣭⣭⣭⣭⣽⣿⣿⣷⣾⣶⣶⣶⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠃⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠭⠭⠿⠭⠭⠭⠭⠭⠭⠭⠽⠿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠘⠽⢿⣿⣿⣿⣿⣿⣿⣭⣭⣿⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣭⢭⣭⠭⣭⣭⣭⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⠿⣿⣿⠿⡿⠭⠭⣭⠍⠉⣉⣉⣉⣉⣙⣛⣛⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⢿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣷⣿⣿⡟⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠸⠿⠭⠭⣩⠿⠉⠉⠿⣿⡅⠀⡀⠀⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡤⢢⠜⠁⢠⠄⣤⠈⣙⡆⠀⡇⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣸⠸⡧⠀⢀⣀⠀⣹⡇⠀⠄⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠠⠎⠉⢀⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⢶⣿⣭⡥⢭⡭⠉⠉⠉⣿⣛⣛⣶⠀⡀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣠⣀⣀⣤⠰⠶⠾⠛⠛⠛⠓⠲⠶⠶⠶⠶⠂⠀⠀⠀⠀⠸⣯⣠⠉⢈⢹⡇⢀⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣓⣒⣛⣧⡁⠘⠈⠙⠉⠉⠉⠙⠛⠋⠉⠋⠉⡋⠋⠀⠀⠀⠀⠀⠀⠀⡏⢁⡜⢁⣸⠁⠀⢙⠀⢀
⠀⠀⠀⠀⠀⠠⠓⠂⠀⠀⠐⡤⠤⠭⠭⠭⠭⠭⠭⠭⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⢄⠀⠂
⠀⠀⠀⠀⠀⠓⠂⠀⠀⠀⠑⠛⠒⠒⠒⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⣰⠿⢳⣷⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⣿⢲⣬⠧⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡰⢰⠆⠀⠠⣌⡈⠙⠋⠀⣹⣾⣦⣶⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣿⣷⠛⠀⠀⠀⠈⠛⠷⠂⣾⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢟⠟⠀⠀⠀⠀⠀⠀⢠⣿⡌⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⣿⣍⡀⠊⠀⠀⠀⠀⠀⢸⣿⣣
⠀⠀⠀⠀⠀⠀⠀⠀⡞⢨⣿⣿⠉⠀⠀⠀⠀⣶⣾⣶⣾⣿⠿
⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⣿⠉⠀⠀⠀⠀⣴⢿⠟⢛⣿⡷⢸⡋⠾⠆
⠀⠀⠀⠀⠀⠀⠀⣸⢀⣿⠏⡄⠀⠀⠀⣰⣟⣈⣠⠜⢋⣴⣾⣄
⠀⠀⠀⠀⠀⠀⢀⡏⣸⠿⢺⡇⢀⣴⠾⣿⡇⠀⣀⣀⠈⢤⠾⢿⣦
⠀⠀⠀⠀⠀⠀⢸⣷⡇⢰⣄⢿⠟⠁⢰⠟⣀⣠⣤⡠⠤⠒⠶⣛⣟⠳⣤
⠀⠀⠀⠀⠀⠀⢾⣿⣷⣿⣿⠀⣀⡴⢷⣾⠀⢉⣠⠀⢀⡤⠞⢋⣥⣁⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⡝⠉⠀⠘⣿⣦⠉⢠⡾⢭⡴⠞⢋⣵⡌⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢢⡜⢦⣀⠚⢥⡴⣾⠟⣥⠤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠁⠖⢛⣵⣿⣿⣧⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡯⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠠⠀⠀⠘⢧⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⣶⡦⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⡿⢙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟⠓⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⠟⡩⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠡⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣴⣾⣿⣌⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣸⣿⡇⠘⠁⢘⣶⣾⣳⣾⣷⣾⣿⣭⣭⣭⣭⣭⣥⣶⣿⣿⣿⣿⣯⣿⣷⡆
⠀⠀⠀⠀⠀⠀⣶⣾⣷⣎⠉⠛⠻⠿⠿⢷⣶⣿⣤⠭⠏⠚⠛⠛⠛⠉⠉⠉⠉⠉⠐⠶⠤⠀⠙⢿⣿⢿⣿⣦⣀⠠⠄
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⠐⢀⡀⢀⡀⠀⢀⣀⣀⣀⣀⣒⣀⣀⣀⣀⣃⣀⣉⣛⣃⡀⠀⠀⠀⠈⠓⣹⣿⣿⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠁⠀⠀⠈⠱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⣿⣷⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⣿⡀⠀⠀⠀⣰⢿⠉⢳⡄⠀⠉⠉⠉⢻⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠟⢻⡏⠀⢿⣷⡀⢀⡾⢷⡞⠀⢿⡁⠀⠀⠀⢠⠏⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠁⢀⣀⣼⣦⡀⠸⡟⢿⡾⠃⣿⠃⠀⣠⠃⠀⠀⠀⡟⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡽⠶⠛⣏⣧⣬⠇⠀⣧⢸⠃⢸⣿⠀⢠⠏⠀⠀⠀⢸⡇⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣁⣤⣶⡞⠛⠋⠹⣆⠀⢸⣼⠀⣸⡇⠀⡞⠀⠀⠀⠀⢸⠁⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢿⣿⣿⣿⣿⠏⠀⠸⣧⠀⠀⠀⠹⣆⠀⣿⠀⣿⡇⡸⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣏⣿⠀⠀⠀⢿⡀⠀⠀⠀⠹⣆⢹⡆⣿⣷⠇⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠁⠈⢹⠃⠀⠀⠀⢸⣇⠀⠀⠀⠀⠹⣶⣇⣿⡿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠋⠉⠀⠀⠀⡾⠀⠀⠀⠀⢸⠻⡄⠀⠀⠀⠀⠙⣿⣿⠇⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠉⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⢸⠀⣧⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣟⠁⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⣸⠀⢻⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⢀⡞
⠀⠀⠀⢀⣤⣾⣿⠿⢿⣿⠿⠂⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⢸⠁⠀⢺⡄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⡾
⠀⠀⣶⣻⠿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣳⣦⣽⡆⠀⠀⠀⢠⣏⣀⡀⠀⠀⢀⣾⣷
⠀⠀⠈⠀⠀⠀⠈⠉⠙⠛⠷⢤⣄⣀⠀⢸⣿⡿⠛⠉⠉⠉⠛⠻⣶⡶⢦⣄⣀⣬⣭⣭⣍⣿⣿⣿⣷⣤⣤⠾⠛⠻⢿⣦⣀⣾⠋⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⢿⡅⢀⣀⡀⣀⣠⣉⣙⣿⣿⣦⣀⣀⣼⣿⣿⠃⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⢿⣿⣯⣉⣁⢿⣤⢧⣦⣡⣽⣿⠟⠛⠉⠉⠁⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠁⠉⠁⠀⢸⠏⠈⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⡿⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⡿⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣀⣤⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⡷⣶⡶⠾⠋⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢻⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⠀⠸⡄⠀⠀⠀⠀⠀⠀⢸⢹⡆⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⣇⠀⠀⠀⠀⠀⠀⢸⠀⣧⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⡀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠿⠂⠀⠀⠀⠀⠀⠸⠇⠀⠻⠀⠀⠸⠇⠀⠀⠀⠀⠀⠛⠀⢿⡄⠀⠀⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⢠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡀⠁⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠧⠤⢤
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡆⠀⠂⢀⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠁⠈⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⡞⢤⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡀⡀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⡁⠀⢀⡁⠈⠀⠀⠠⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡤⠰⠀⠀⠀⠀⠠⠂⠉⠠
⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠑⠀⠈⡀⠠⢀⡀⢠⣄⠄⠐⢁⡀⢈⠇⠀⠠⣃⠀⡀⠀⠐⠀⠀⠀⠰⠄⠀⠀⠀⢶⢂
⠀⠠⠀⠁⠙⠀⡀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠓⠈⠆⠀⠂⠀⠄⠀⠈⠉⠉⢻⠅⠀⠀⠀⠈⠀⠑⠀⠀⠀⠂⢠⠁⠀⠂
⠀⠀⢄⠀⠀⠀⠀⠂⠀⠀⡈⠀⠀⢀⠄⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣄⠀⢀⠀⠐
⠀⠀⡀⢀⡀⠘⡤⠄⠠⠄⠀⡀⠤⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠖⠔⠐⠀⠀⡡⠈⠀⠀⠀⠦⠀⠀⠒⠀⠀⣀⠏⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠉⠠⠄⡄⠁⠀⠏⠀⠀⠀⠀⠀⠀⠒⠪⠂⠀⠀⠀⠀⢀⠄⢠⡀⠀⠀⠀⢀⠀⠀⠐⠀⠀⠀⡆⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠨⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠖⠒⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠉⠉⠀⣀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡍⠛⠶⠦⠴⠖⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣈⡢⡀⠀⢀⠤⠔⠊⠉⠀⣄⠤⢴⡷⠆⢯⠌⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢠⡽⡗⠒⠺⡄⠀⢉⡁⢀⣀⡀⠀⠄⡂⠈⠀⠐⠀⠀⠀⢉⡁⢇⣴⣿⣂⢤⣰⣎⠸⠛⠁⠄⣁⠈⠈⠃⠀⠀⢸⣧
⠀⠀⠀⢀⣀⡿⡁⠻⡄⠀⠸⡀⠀⢺⣿⣿⣷⠀⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⡼⠉⢈⡀⠈⢊⠩⠄⠠⠄⠀⠀⠀⢸⡏⡆
⠀⠀⠀⢸⡍⢻⢱⣧⢳⠀⠘⣧⠀⠀⢿⣿⣽⣦⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⢿⣿⣿⣿⡿⡄⠨⠒⠉⠀⠴⠃⠀⠒⠀⠀⠀⢸⣿⡘⡄
⠀⠀⠀⠈⣧⠸⣧⣻⣇⢇⠀⢸⣧⠀⠈⠑⢉⠵⡀⠀⢡⡀⠠⠄⠊⠦⠀⠀⠸⣿⣿⠋⡰⣷⠀⣀⠠⠀⠠⠒⠐⠒⠂⠠⣤⠈⣿⣧⠘⡄
⠀⠀⠀⠀⢻⡄⠿⣯⣧⡞⡆⠀⣿⣧⠀⠁⢠⠀⠁⠀⡈⠀⢀⠀⠀⠀⠀⠀⠀⢻⣿⢷⡀⢻⡆⠡⠊⠁⠠⠀⠀⠀⠀⠀⡄⠀⠘⣿⣇⢹
⠀⠀⠀⠀⠈⣷⠀⣾⣾⣿⡸⡄⢸⡽⣆⠀⠛⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣄⣸⣷⠀⠈⣀⠀⠀⠐⠒⠶⠤⣄⠀⠀⠈⣿⢸⡄
⠀⠀⠀⠀⠀⢹⣆⢹⣿⡉⣿⡇⠀⠻⣿⡄⠐⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣿⣿⣿⣷⢿⠁⠚⢁⣠⣤⣤⣤⣤⣤⣀⣀⣀⣀⣿⣆⣷⡀
⠀⠀⠀⠀⠀⠈⣿⠀⠿⣷⢸⡹⡆⠀⢁⣿⡠⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣩⣴⣾⣯⣽⣿⣦⣭⣭⣭⣉⣛⠻⢾⣿⣧⣵⡹⡄
⠀⠀⠀⠀⠀⠀⠸⡇⠀⢻⣯⡇⣇⠀⠸⣏⡉⠒⢄⠀⣼⣤⣀⢀⣀⡴⣶⢿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣯⣟⣛⠻⢿⣻⡿⢿⣷⣶⣿⣌⢦
⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⣷⣹⢸⠀⠀⠐⠂⠁⠈⢦⣿⣿⣿⣿⣿⣿⣾⣿⣿⣋⢻⣟⣿⣿⡿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⣶⣭⣿⣶⣮⣭⣭⡤⠷
⠀⠀⠀⠀⠀⠀⠀⠸⣤⠀⢻⣻⡀⣇⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣷⣿⡿⠟⠛⣿⡹⠿⢿⣿⡿⣿⡇⠀⠀⠈⠉⠙⠛⠲⠶⣮⣽⣻⡛⠿⠿⣿⣶⣾⣶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠘⣯⣧⣩⣭⣽⣿⣿⣿⣿⢽⣿⡿⠿⠋⠉⠀⠀⠀⠈⠙⠻⠷⠷⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣶⣤⣭⣓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠑⠚⠻⠿⢿⣛⣭⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠶⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠋⠉⠀⠀⠉⠁⠐⠆⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠲⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠹⣆⡀⠀⠀⡀⢀⢀⣠⡠⠴⠄⠐⠒⠒⠒⠒⠰⢶⣤⡀⠀⢀⠄⣠⠔⠁⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠙⠳⢶⡾⡋⠩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠞⠓⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⢀⣤⣤⣄⡀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣠⣿⡿⠿⠃⠉⠉⠀⠀⠙⠋⠙⠻⠿⣤⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣼⣿⡟⠁⠉⢡⣴⣶⢄⠀⠀⠀⣊⢤⣶⡶⢤⠀⠈⣿⠗⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠐⠁⢤⣈⠿⠧⠄⠀⠒⠛⠛⠉⠀⠀⠀⠁⠉⠉⠉⠉⠀⠤⠁⠶⢆⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣢⠎⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⡞⢭⠑⠢⡀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢠⡀⠀⠀⢀⣀⣴⣷⣤⣿⣤⣴⣷⣾⣧⣀⠀⠀⢀⣴⣀⣠⠿⠆⠀⠀⠀⠀⠀⣼⣶⡤⠔⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⠿⠿⡭⢿⣿⣟⣽⣗⡿⢿⣼⣿⣿⣿⣿⣷⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⢄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠙⣍⡂⠤⠀⣊⡽⠁⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢀⡤⡄⠩⠀⣀⠀⠉⠙⠳⢄⡀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⠋⠀⠀⠀⠀⠀⠈⠑⡄⠀⠈⠳⣦⣀⠀⠀⠀⠘⠦⠀⠤⠒⣠⠕⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠰⣌⠫⡒⠢⠤⠀⠀⣠⡴⠗⣥⣂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡳⠀⠀⠀⢩⠐⠬⠁⠒⠀⢲⣿⣧⣅⣂⠩⢛⡒⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠎⠀⠀⡅
⠀⠀⠀⠀⢀⠄⠐⠀⣿⡀⠀⠀⠀⣠⠴⠚⣉⡩⠍⠿⣄⠀⠀⠈⠁⠀⠀⠉⠙⠛⠿⠿⡿⣿⣿⣶⣮⣕⡀⠨⢕⡒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠃⡎⠀⠉⠻⣷⣅⣤⠛⣡⠴⠋⠁⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡽⠁⣾⣝⣿⣿⣿⣶⣤⣁⡂⠬⢐⡠⢄⣠⢀⠀⠀⠀⠀⠀⠀⠰⠇
⠀⠀⠀⠀⠳⡔⠢⡀⠀⠀⢹⣁⠾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠉⠁⠀⠈⠉⠉⠀⠒⠋⠀⠀⠁⠀⠀⠈⣈⢉⡉⠙⠻⣷⣶⣬⣐⠪⢝⡳⢴⣔⠠⠐⠐⣾
⠀⠀⠀⠄⠀⠀⠑⠶⠶⠂⣭⡽⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣨⠍⢉⣋⣿⢻⠿⣶⣦⣅⡛⠩⣗⢼⡇
⠀⠀⠀⠢⢀⠠⢤⠖⠙⠋⠀⠉⠊⠝⡢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⠀⠀⠀⠀⢤⣤⣤⡄⠉⣲⠤⢚⠷⠒⠊⡩⠛⠿⢷⣦⣽⣿
⠀⠀⠀⠀⠀⠐⠡⠈⠀⠁⠀⠀⠀⠀⠀⠉⠙⠳⠲⣤⢦⡀⠀⠀⠀⠀⠀⠀⣀⣤⢶⠛⠋⠊⠉⠛⠙⠛⠗⠦⣌⣫⠑⢄⠀⠀⢁⡠⠔⠈⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⠙⠓⠺⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⢯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣑⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⢄⣀⡀⠀⠀⠀⠀⠀⡀⠀⢀⣀⣀⣀⣠⣤⣤⢴⢶⠶⣮⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠈⠀⠀⠈⠀⠀⠀⠁⠁⠋⠓⠉⡳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠱

⠀⠀⢠⣄
⠀⠀⠟⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⣀⠀⠦⠄⠀⠤⠀⠤⠄⢀⡀⣀
⠀⢼⡦⠀⣰⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠗⠒⢂⣀⣉⣀⣙⣓⣒⠒⠂⢄⠒⠀⠂⠄⠠⠤⠤⠄
⠀⣠⣠⣿⣿⣿⣿⣿⣇⠀⢀⡂⠀⢐⠀⠑⣿⠛⡏⢻⡟⢸⣿⠋⣿⡏⢽⡍⠂⡀⠀⣀⠈⢉⠉
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣃⠈⠁⠀⠈⠀⢀⣉⡀⠁⢈⠁⢈⢹⠀⣿⠇⠙⠀⠀⠃⠀⠉⣀⣘⡀⠸⠀⣀⣀⣀⣀⣤
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⠁⠁⢨⠀⢸⠞⠀⡇⠀⢸⠀⠀⣶⠀⣲⣿⣿⣷⣶⣶⣿⣾⣷⠀⣠⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣷⣴⣯⣿⣿⣧⣶⣶⣶⣶⣊⡒⠀⢓⠂⠘⠂⣠⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⢸⣧⣼⡗⢋⡴⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠇
⠀⠀⠀⠀⠀⢤⡤⣤⡤⣶⣄⢤⣤⡶⢶⣷⣾⣿⣛⣛⣛⣷⣿⣿⣿⣿⣿⣈⡉⣹⣟⠿⠛⠋⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠜⣤⠁⠀⢀⢈⡩⠉⠽⠿⣽⣿⣿⣿⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠈⠀⠁⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⢀⠀⡄⡄⠀⠄⠀⠀⠐⠆⠀⢀⡖⡄⠀⠀⢄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡲⡀⠰⡁⢊⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⢐⠄⠀⠀⠈⠀⠐⠩⠋⠀⠤⡄
⠀⠀⠀⠀⠀⣀⠠⡀⢀⠈⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⢠⠀⡀⡪⠢⢤⡤⡀
⠀⠀⠀⠀⡀⢨⢛⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣾⣾⣿⣿⣯⣅⣦⣀⣡⡙⠉⡊⠀⢼⣿⢆⣠⡄
⠀⠀⠀⠉⡪⠖⠁⡁⠀⠀⠀⠁⡀⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣦⣤⡛⠱⠛⢙⢇⡀
⠀⠐⠯⢊⠀⠀⠢⠀⠂⣀⣴⣿⣶⣿⠿⠿⠟⠛⠛⠋⠉⠉⠉⠉⠉⠉⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣦⠐⡖⡻⣛⡂
⠀⣆⠆⠀⠀⠀⠀⢀⣼⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⡇⡍⣭⡰⢤
⢀⣩⠅⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⠶⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠠⠀⠦⠼
⠰⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠛⣉⣠⣤⣴⣶⣶⠶⣶⣶⣶⣶⡦⣤⣉⠙⢿⣿⣿⣿⣿⡗⠀⠀⠀⠔⣺
⢀⣈⣂⠀⠀⠀⠀⣀⣤⣤⠄⠒⠋⣁⣤⠶⠛⢉⣿⣿⣿⣿⣧⣀⣿⣿⣿⣿⣷⠀⠙⢿⣦⡉⠻⢿⠿⠁⠀⠀⠀⢀⡉
⠰⢶⣤⠀⠠⠤⠀⠀⠀⠤⠤⣶⠏⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠘⢻⣿⣦⣀⣀⠀⠀⠀⠀⠈⠤
⠀⣜⡒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣿⣧⣼⣿⠇⠀⠤⠀⠀⠢
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣀⠤⠞⢋⣠⡁⠀⠀⠀⠀⠀⡠⠈
⠀⠀⠱⠮⠢⡰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠟⠛⠓⠒⠋⣉⣤⣴⣾⠟⠋⠁⠀⠀⠀⠠⡠⠐⠃
⠀⠀⠀⠐⢿⠿⢪⣀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⣶⣶⣶⣿⣿⠟⠟⠋⠉⠀⠀⠀⠀⠀⠀⢀⣄⠥
⠀⠀⠀⠀⠀⠀⠽⡵⠫⣤⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡐⣀⢑⢕⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⡵⠁⡔⢄⢆⠀⠀⠎⠀⠀⠀⠱⠀⠀⠴⢀⡀⢂⢐⣀⡔⡡⢎⢞⡏⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠊⠀⢰⡼⠎⠀⠀⠠⠀⠧⠀⡆⠀⠤⠰⠉⠉⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠈⢉⡁⣀⣀⣀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⢀⢠⡤⢀⠀⣠⡤⢄⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣂⢙⡣⠤⠭⠥⠠⠀⠐⠉⢀⣀⠚⠂⠐⠀⢁⣄⠀⢀⠠⠀⣤⠰⡄⢀⣶⣼⣯⠙⡆⣄⠞⠉⠢⣷⠸⡼⠓⢺⡇⠏⠀⠈⡇⠛
⠀⠀⠀⠀⠀⠀⠀⠌⡴⠉⠀⠀⢸⠀⡇⠀⡐⠖⢐⠀⠾⡦⣀⠐⡒⣤⢿⡀⢰⠊⠷⠋⠓⣿⠘⡁⠀⠀⢰⡇⢸⢀⠀⠀⣹⠀⡇⠀⠠⡇⢘⠠⠀⠇⠐
⠀⠀⠀⠀⠀⣢⠀⢸⠁⠀⠀⢀⠼⠀⠗⠖⠀⠀⠀⠙⢀⣿⢁⢚⢢⠁⢠⠁⢸⠀⠈⠀⠀⣼⠀⡆⠀⠀⢈⣿⠀⡀⡀⡀⠼⠀⠱⠤⠒⠃⠈⠒⠚⠁
⠀⠀⠀⠀⠀⠏⠂⠸⢄⢀⣀⠀⠂⠁⠈⠀⠀⠀⠀⠀⢠⢬⠈⠈⠸⠀⡸⡄⠘⣄⣀⡀⠠⠘⠀⠑⠒⢃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠈⠒⠆⠠⠀⣄⠀⠀⢀⠀⡀⣠⠀⢀⠀⠀⡨⠸⣄⡁⠀⠒⡒⠣⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠐⠒⠤⠤⠤⠴⠦⠔⠂⣀⣀⣀⣀⢀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠲⣄⠐⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡄⢤⡀⠀⠀⠀⠀⢀⡀⢀⡤⠴⠖⠒⠒⠲⠏⠁⠐⡀⢠⣾⡷⣟⡷⣿⡀⣐⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠤⣤⢀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡞⠘⢳⢡⠀⠀⠀⠨⠀⠀⠈⠁⠀⠐⠀⠀⠠⠴⠲⠺⠧⠞⠯⠷⣬⣿⣿⢿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠘⠤⠤⠸⠀⠠⠄⠀⠀⠀⠀⣄⡁⢈⠀⠀⡐⠀⢰⠅⠘⠈⢠⢐⡀⡀⡈⠐⠶⠲⠛⠛⠋⠛⠛⠋⠛⠑⠑⠊⠒⠐⠋⠊⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠶⠻⠿⣀⣀⣀⣀⣀⣀⣀⢰⣯⠤⠠⠄⠠⠒⠧⣅⣘⡤⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠉⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⢸⣿⣷⣄⡀⣀⣤⣤⣼⣿⠀⠂⢠⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣤⠄⢈⣹⣿⣿⣿⣿⣿⣿⣿⡿⣶⣿⣿⣿⣿⣿⣶⡄⢀⣴⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣼⣿⠇⠀⠈⠈⣿⡟⣿⣿⣿⣿⣿⡇⠀⠙⣩⣈⠙⠛⠻⣿⣾⣿⣿⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣿⠿⠿⢷⡄⠀⠉⠀⠘⢿⣿⣿⣿⣿⡀⢌⡛⣋⠸⣿⠆⡃⢻⡿⢻⣌⣡⡉
⠀⠀⠀⠀⠀⠀⣤⣶⣾⣿⣿⣷⣋⣤⡀⢴⡶⢿⣾⣿⣿⠓⠈⠀⠀⠉⠉⠃⠸⢿⣯⣷⣶⣾⣿⣿⣷⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⢿⣋⣠⣤⣾⣿⠉⢉⡴⢦⣠⠜⠛⠻⠇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠻⠟⣀⠉⡛⢿⣴⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠈⠿⢟⣿⡍⢸⣠⣿⣿⣿⠃⠀⠀⠀⠀⠀⠘⠀⠀⢂⣴⣶⣿⣷⣾⣿⣷⠀⠀⠀⢹⣿⡏⠁⠉⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣻⡭⣵⣿⡿⠿⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠛⠛⠙⣿⡿⠙⢀⡀⠀⣿⡏⠑⠂⠀⠒⠁
⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣍⠀⠀⠈⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⠠⠄⣹⣷⣿⣿⣿⣿⡿⠗
⠀⠀⠀⠀⠀⠘⠟⠉⠙⢿⣿⡟⠀⢀⣀⣼⣿⣧⠀⠀⣾⣿⠿⢷⣀⡀⠉⠀⠀⠀⠈⡟⣀⠈⢉⢸⡿⣦⣀⣀⡤
⠀⠀⠀⠀⠀⠀⢠⠀⠉⠀⠻⠀⢰⣿⣿⣿⠋⠉⡲⠼⡏⠀⠀⠀⢹⣿⣶⣿⣿⣿⣷⣧⣠⢦⣈⡼⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠘⠀⠂⠀⠀⠐⢾⣿⣿⡟⠐⠀⣙⣛⣉⣁⣀⠉⣈⣿⠀⠀⢹⣿⣿⣿⠟⠿⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢹⣿⡏⠀⢠⣭⣭⣽⣿⣿⣿⣻⣿⣧⡀⠀⢻⡟⠉⠀⠀⠈⠁⠀⠀⠀⠉⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣤⣤⣴⡶⠆⠒⠂⣤⠄⠛⠛⠿⣿⣇⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⠀⣽⡀⢠⣶⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠐⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⡄⠰⠍⠱⠉⡤⠤⠤⠤⠤⠤⠤⠤⡄⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⠆⠠⠀⠀⠀⠠⢘⠀⠐⠠⠀⠀⠀⠀⠀⠀⡄⠀⠀⠰⠂⠀⠘⠀⠀⡴⠂⠂
⠀⠀⠀⡰⠊⠀⠀⠀⠄⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠈⠲⠀⠀⣆⠀⠀⠖⠁
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠀⡆⠀⠢⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⣀⣀⡀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠈⢠⠀⠀⠀⠀⠀⢀⡎⠀⡇⠀⠂⠂⠐⠀⡆⠐⠀⠂⠂⠀⠀⠀⠀⠀⠀⡇⠠⠤⠄⠀⠀⠀⠧
⠀⠀⠀⠀⠠⡗⠐⠒⠈⠀⠀⡇⢀⠄⠄⠠⠀⠄⠠⠀⠄⠄⠀⡀⠀⠀⢀⡀⡇⢰⠀⠘
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⡿⠀⠀⠀⠀⠀⠀⠸⡏⠀⠀⠀⠀⠀⠀⠇⠈⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠈⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡠⠤⠤⠤⠤⠤⠤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠺⠯⠭⣥⣀⣀⡀⠀⠤⠄⠒⠂⠈⠉⠉⣹⢄⡉⠉⠓⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⢩⣝⣧⡤⢀⣀⡀⠀⢸⠀⠀⣀⣠⣤⣤⠶⠾⠝⠃⠀⠁⠀⠀⠀⣀⣀⣉⣑⣢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⠒⣒⡻⡥⠤⣄⡀⠀⠀⠀⠈⠉⠙⠒⠪⠭⠭⠭⠖⠊⠉⠁⣀⣀⡤⠤⠖⠒⠛⠋⠉⠉⠉⠉⠉⠁⠈⠉⢣
⠀⠀⠀⠀⠀⢀⠔⠋⣇⠀⠀⠀⠀⠉⠉⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠛⠉⠁⠀⠀⢀⡀⢤⡀⢀⣀⣠⠤⠤⡄⡔⠒⢲⠸⡄
⠀⠀⠀⠀⢰⠋⠀⠀⠈⠉⠓⠒⠶⠤⢄⣠⡿⠀⠀⠀⠀⢀⡤⠖⠛⠉⠀⠀⠀⠀⣀⢄⢀⣠⣤⣓⣶⣿⣿⠉⢩⣿⠀⠀⡇⡇⠀⢸⠀⡇
⠀⠀⠀⠀⣼⠓⠶⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⣀⡀⢯⣶⡿⠛⣿⠉⠀⢸⣿⡿⠀⢸⣿⣀⣠⡇⠧⠴⠞⢰⡇
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠈⠉⠉⠓⠲⠤⣤⣀⡀⠀⠀⡇⢀⣀⡀⡴⠚⠉⠀⡇⢘⣿⠀⠀⢹⡇⢀⣸⣿⠘⠒⠋⠀⠉⠀⢀⢀⣀⠤⢼⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⡟⠉⠀⡇⢇⠀⠀⠀⢸⠸⣿⣄⢤⣺⣉⡉⠀⣿⣀⣀⠤⠒⢒⣉⡩⠤⠖⠒⢸⠁
⠀⠀⠀⠀⠘⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸⢸⡀⣀⠠⠚⠀⣿⠀⢸⢹⣿⡇⠀⣿⡥⠔⡶⠛⠳⣄⣀⠤⠔⠒⣺
⠀⠀⠀⠀⠀⢧⠈⠉⠓⠲⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠉⠁⠀⢀⡠⣿⠀⠈⠋⡇⠀⠀⣿⣀⣼⠠⡞⣦⢸⣤⠴⠖⠚⠁
⠀⠀⠀⠀⠀⠸⡭⢷⣄⠀⠀⠀⠈⠉⠛⠒⠦⢤⣄⡀⢸⡄⣀⡸⢆⣠⠔⢊⡡⠔⣿⡇⠀⠀⢠⠀⠀⣻⣥⣼⣄⣑⣋⠾
⠀⠀⠀⠀⠀⠀⣇⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣹⣏⡥⠒⣉⢴⡮⠷⢤⡀⣿⡇⠀⣀⡴⠖⠋⠁⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⢹⠟⠥⣀⡴⠤⣀⡀⠀⠀⠀⠀⡞⡩⠻⡇⣿⠊⢠⠋⢠⠶⡄⢹⡽⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠓⠤⢄⣿⣷⡆⢨⣹⠲⣄⠈⣟⢇⣀⡇⣿⢀⠼⡄⠸⣤⠇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠬⣏⣘⠂⣇⠈⠉⠛⠓⣿⠋⢁⡠⠷⠤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠑⠒⠦⠿⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⡀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⠻⣿⣿⣿⣿⠿⠿⠛⣨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣋⣉⡹⢦⣿⡄⠛⠫⣿⣼⠇⠀⣾⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣾⣿⣿⠽⢿⣿⣿⣶⣆⣀⡁⢀⣼⣿⣿⣿⣷⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣧⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠓⢼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⣻⣿⣷⡄⠀⠙⠿⣿⣿⣿⣿⣿⢿⣯⣳⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⣿⣿⣿⢋⠘⠹⣿⣿⣦⣠⣀⣠⡉⢉⣉⣿⣏⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠻⠟⠀⠠⠄⠛⣹⡿⣿⣿⣿⣷⣿⣿⣟⣦⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣵⣿⡝⠿⣿⣿⣯⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠋⠁⢸⠿⠿⢿⣿⠿⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡃⠀⢤⣨⡄⠰⠛⠁⠀⠈⠠⢀⣠⡀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠎⢈⣿⣿⠶⣶⣿⣷⠀⢄⣀⠀⠀⠀⠀⠈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠲⠀⠀⠀⠂⠀⠀⠀⠈⠁⢿⣿⣿⣷⣤⣬⣿⣿⣾⣿⣿⢧⡄⠀⠀⠈⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⣿⣿⣿⣿⣿⣿⣿⣟⣿⢷⣽⡿⡀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⠀⢸⣿⣿⣿⣿⣿⢿⠟⠊⠁⠀⠙⠰⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢨⣤⣈⡿⠁⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣼⡽⠃⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠼⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣷⠂⠀⢀⡑⠀⠀⠀⠄⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⡅⡁⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠖⠛⠋⠘⠒⠶⡤⠆⠀⠘⢻⣾⣥⣿⣆⠠⡀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⢀⣐⣄⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣷⣶⣤⣀⡀⠚⠓⠲⣄⠠⣿⣾⣿⣿⣿⣆⠡⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⣤⠀⠀⠀⣤⡄⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠙⠉⠙⠛⠛⠛⢿⣦⡻⠆⠀⠀⠀⢀⣴⣶⣶⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠤⠀⠾⠛⠛⠉

⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⡎⠀⠄⠄⠤⠄⠀⠀⠤⠤⠤⠤⢤⠄⠤⠤⠤⠤⠤⠤⡄⠀⠠⠠⠄⠤⠄⢄⢠⠤⠤⠤⠤⢄⣱⡀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⡇⠀⣀⣀⣀⣄⣀⣀⣀⡀⠀⠀⢨⠀⠀⠀⣀⠀⠀⠀⡄⠀⠀⠀⡀⠀⠀⡸⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⢠
⠀⠀⡇⢀⡀⠀⠀⠀⢀⣴⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⢀⠀⠀⠀⠩
⠀⠀⠣⠤⠤⠤⠤⠤⣿⡁⠀⣹⡧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⣿⠁⢈⣿⠤⠤⠤⠤⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠋



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⣴⣤⣼⠏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠈⠠⡀⠶⠀⠀⡄⠀⠀⠀⠀⠀⠀⠘⢷⣴⡤⠀⠀⢠⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⡔⣤⠀⠆⢠⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠢⠀⣒⠀⠀⠑⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡏⠙⡇⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢂⣰⣶⡿⠃⢾⣈⠉⠀⠀⠀⠀⠀⠻⠀⠀⠀⡄⠀⢹⣇⠔⠡⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡟⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣂⣐⠁⠸⡉⢀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠈⠀⠀⡐⢁⣤⢝⣷⡀⡤⣌⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡼⠫⡆⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⢾⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠿⠿⠛⠛⠛⠻⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣠⠀⠀⠀⠀⠀⠰⠖⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⠤⠀⠄⠀⣤⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣸⠏⠀⠀⠀⠀⠀⢀⣴⡆⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⣭⣄⣤⣤⣼⣶⣿⣾⣷⡯⠄⢐⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⡾⢠⣤⠁⠀⠀⠀⠀⠀⣈⡾⠿⡃⠀⠀⠀⠈⠋⠼⠍
⠀⠀⠀⠀⠀⠀⢀⣐⢮⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁⠙⠀⠀⣠⡀⠀⠀⠈⠃⠀⠀⠀⠀⠹⡧⠉⠿⠄⠀⠀⠀⠀⠘⠁⢀⣠⣅
⠀⠀⠀⠀⠀⠀⠰⣿⠿⠟⠋⠙⠏⠉⣹⣇⡀⠀⣤⠀⠀⠀⠻⡇⠁⠀⠀⠀⠀⠀⢸⡶⢀⡏⠰⠄⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣧⣄⠀⠀⠀⢰⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠛⣯⡿⡀⠈⠀⠀⠀⠀⢳⡀⠀⠀⠀⢄⠀⠘⡆⢸⠷⣄⠀⠀⠀⠐⠀⠀⠀⠞⠛⠉⠁⠈⣿⡷⣶⣷⢻⣷⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠇⢸⡏⠑⠀⠀⠀⠀⡠⣄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠾⠀⠘⢣⡀⠀⠀⠀⣤⠀⠀⢀⣠⣶⣿⣿⣇⣿⣿⢀⠀⠉⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢧⠀⡀⠀⢠⠀⠀⠀⠀⢳⣦⡄⢀⣀⡀⠀⠀⡷⠄⠀⠀⢱⡄⠀⣠⡜⠃⠴⠿⠋⠉⢹⡾⣸⣿⡟⢸⣷⡄
⠀⠀⠀⠀⠀⠀⢀⠤⠶⠀⠀⢸⡁⠀⠁⢈⡾⠃⠀⢀⠀⠀⠀⠀⠈⢿⣆⠒⠉⠛⠂⠀⠇⢀⠈⡀⠇⢷⡀⢻⠀⠀⢀⣤⣶⣿⣽⢧⣿⣄⠀⠾⢿⣿⡆
⠀⠀⠀⠀⠀⠰⢿⣿⣦⣵⠛⠉⣿⣶⡅⠘⣆⠄⠀⠈⢽⠗⢀⠀⠀⠈⢯⠆⠀⢿⣶⡄⠥⠀⠁⠁⡀⠈⣷⠀⠀⠐⣿⠉⠉⢹⡿⣸⢀⡻⢐⣆⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠴⠀⠃⠀⢩⠀⣯⣄⣤⣴⣏⣀⣤⣤⠀⢀⠸⢻⡴⣐⠂⡄⢠⣤⡀⡜⡠⠀⣿⡇⠀⠀⣩⣴⣿⣿⢡⣿⡄⠁⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⡌⠾⣶⣍⡻⡿⢧⠉⠀⠿⠿⣮⡳⣿⢸⣋⣁⣌⡠⠆⢹⣷⠀⠛⡏⠀⣼⢃⣿⠿⡟⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠆⠀⡄⢻⠀⡴⠈⢉⣝⢷⠦⣴⣆⢠⡄⠈⢳⣮⡀⣿⣿⠟⠟⠋⢸⣿⣀⣴⣷⣿⢏⣾⣿⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠆⠀⡁⣰⠎⢿⣷⡵⣬⣿⣾⠟⠁⣿⣿⡇⢻⣿⣶⣄⣄⣸⣿⣿⣧⣿⢯⣿⣿⡿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⠀⠀⠇⢹⣿⡻⣧⣄⠸⣿⠋⠀⠸⡹⣿⣿⣿⣿⢸⣿⣿⣫⣿⡇⠈⠡⠇⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣳⣿⠉⢿⣾⣿⣷⣿⣶⠦⠂⣷⢻⣿⣿⣏⣿⡿⣹⡟⠉⢁⠐⠁⠀⠀⠀⢀⠀⣨⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠙⠃⠀⠸⣿⣷⣝⣿⡀⠀⠈⢸⡏⣿⡿⣸⣟⣿⠋⠀⡠⠊⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⣿⠏⢻⣎⢿⣟⡀⠈⣿⢸⣿⣿⠟⡃⠔⠁⠀⠀⠀⠀⠀⠀⠀⢼⣿
⢰⣷⡄⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢌⡀⠸⠿⣷⡽⣧⠀⣿⠜⣫⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠈⠀⠀⠀⠀⠀⠀⠄⢸⠦⣀⣀⣠⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣉⢢⣄⣈⣿⣝⡼⣣⣾⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣠⡀⠘⠛⣰⡀⠀⠀⠀⠀⠀⢲⣶⣬⣿⣿⣿⡿⠿⠋⠉⠉⡷⠞⡾⡿⠁⠀⠈⠻⠿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠰⣦⡄⠀⣾⠟⠁⠄⠉⠙⠀⠀⠀⢾⣷⣾⡿⣿⣿⠉⠟⠀⠀⠀⠀⢠⣇⠀⢻⡧⠀⠀⠀⠀⠀⠤⠈⡉⠳⠶⣾⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠙⣤⠀⣧⡆⠈⣆⠀⠁⠀⢰⣶⣿⡟⠻⡧⠀⠈⠀⠀⠀⠀⠀⠀⠸⠀⢀⣾⠇⠀⠀⠀⣰⣄⠀⠀⠘⠀⠀⣈⠉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⢺⠁⠀⠀⠀⠀⠀⢸⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣻⠀⠀⠀⠀⢿⡏⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣶⡦⣊⣻⣿⣆⣾⡆⣠⣤⣧⣀
⠀⠀⠀⠀⠀⠀⣶⣾⠟⠛⠻⢿⣿⣿⠋⠉⢉⣟⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢠⣶⣿⣧⣠⣴⣴⣾⣿⣿⣶⣶⣌⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣯⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢻⣿⣤⣿⣿⣷⣄⣤⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢴⣦⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢀⣠⣾⣿⣧⠠
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣠⣶⣿⣿⣿⣿⣿⢀⠄
⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉
⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡀
⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠈⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⡴⠂
⠀⠀⠀⠀⠀⠀⠉⠙⠿⢿⣿⣿⡿⠟⠋⠀⠀⠀⠉⠻⣿⣿⣿⡿⢿⠟⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣤⣦⣤⣄⣀⠈⢯⡁⠀⢸⠀⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣵⣄⠀⡇⣼⡏
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣷⠏
⠀⠀⠀⠀⠀⠀⣼⣿⠟⠋⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⡿⠿⠛⠛⠋⠉⠙⠛⠻⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠉⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢦⣶⣶⣶⠀⠀⣴⣿⣶⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⠘⣿⣿⡿⣰⣦⢻⣿⠿⠃⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣀⣀⡀⠀⡿⢿⠄⢀⣀⣀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢹⣯⡄⠤⠤⠄⣹⡏⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⠓⠾⠷⠞⠋
⠀⠀⠀⠀⠀⡀⠄⠒⠋⠁⠀⠀⠉⠒⠤⡀⠓⢄⠀⠀⠀⠀⡀⠔⠀⠀⠤⠒⠂⠀⠀⠀⠀⠀⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠉⠉⠁⠀⠀⠊
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢠⠀⠀⠀⠀⠀⠀⢠⢦⣶⣶⣶⠄⢰⣶⡆⠀⠀⠀⠀⢸⣦⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠸⣿⣿⡿⢠⣾⢿⠇⠀⠀⠀⠠⡠⣿⠇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠠⡀⠀⠠⣤⣤⣁⣀⣀⠀⠀⣿⠿⣂⡈⠀⠀⠀⢀⠅⠀⠀⣑⣂⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠐⠢⠤⠼⡏⠉⢹⡅⢀⣄⠠⠄⢤⠀⠀⠀⠀⣀⣀⣀⣸⡇⠉⠉⠋⠈⢀⠀⠀⢀⡀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⠙⠁⠲⠶⠶⠾⠀⠀⠀⠀⢲⠷⠷⠛⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⢈⠀⣀⠠⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠿⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣦⡀⠀⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣹⣿⣿⡿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣽⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⢤⣤⣄⠀⠀⣀⣸⣿⣿⣿⣿⣤⠀⠀⠀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣦⣤⣈⣴⣿⡿⢿⣿⠟⢻⣿⠿⣿⣿⣧⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢿⣿⣿⣿⣿⡛⠛⣛⣫⣤⣶⠁⠀⢿⣿⣿⣿⣿⡿⠻⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⢸⣿⣿⣷⡿⠛⣛⠛⢻⠇⢁⣀⢀⣈⣿⣿⣿⣿⡇⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠸⣿⣿⣿⠗⠖⠉⠉⣸⠃⣿⡉⠈⣉⣉⣩⣿⣿⣇⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣿⠛⣿⡦⠞⠛⠁⠈⠀⣿⣇⢀⡀⠈⠛⠛⣿⣿⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢸⣿⣿⣧⣤⡞⠛⠁⢾⣆⣸⣿⡦⠀⠀⠛⠢⣬⣻⠄⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⣾⣷⣟⣉⣸⠷⠀⢂⣼⣿⣹⣿⡆⠀⠀⠻⠦⣈⣿⡄⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢹⣿⣾⣽⡿⠀⠀⠀⣨⣿⣿⣯⣄⡀⠀⠲⢦⣹⣿⠃⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠸⣿⡿⠯⠀⢀⡘⠛⣽⣿⣿⣿⣝⠛⡀⠰⠆⠀⠿⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠸⠷⠶⠴⠛⠛⢉⣉⣿⣿⣟⠉⠙⠛⠢⢤⡴⠆⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⢀⣀⣄⣀⠀⠀⠈⣻⣿⣧⡟⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢻⡏⠀⠀⠀⣴⣿⣿⠿⢿⣷⣄⠈⣿⣿⣿⣿⣠⣶⣿⣿⣷⡆⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⠀⠰⣟⠻⠟⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⠟⠁⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⡶⠤⠛⢿⣿⣿⣿⣿⠿⠻⠿⣿⡿⠆⠀⠀⠀⠀⠀⢸⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣾⣷⣾⡿⢷⣴⣿⡎⢹⣿⣿⣿⠶⣤⣾⣿⣷⣾⢆⣀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⡄⠀⠀⠈⣿⣿⣿⡁⠀⢿⣿⣇⣰⣿⠿⠋⠀⢻⣿⡏⠈⢿⣾⣿⡆⠀⢰⣾⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡇⠀⠀⠀⠘⣿⣿⣿⠀⠈⠿⠿⠋⠀⠀⠀⠘⠿⠋⠀⠀⣾⣿⡿⠁⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠇⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣠⣿⣿⡿
⠀⠀⠀⠀⣤⠟⠁⣿⠈⠙⣿⡄⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠃⢠⣿⡇
⠀⠀⠀⠈⠁⠀⣾⠉⣿⡇⢸⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⣿⡟
⠀⠀⠀⠀⠀⢠⠛⣤⠘⠃⢸⠃⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢰⣿⠃
⠀⠀⠀⠀⠀⠙⠀⢹⠀⠸⠇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡏⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⡇⠀⠀⠀⠀⠀⣠⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⣿⡇⠀⠀⠀⠀⢀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⢀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⣤⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⣿⠿⠿⠿⠿⠿⠿⠿⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠉⠉⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡇⢀⣴⢀⣿⣿⣿⣿⣆⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⣿⣿⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠉⢹⣿⣿⣿⠄⠉⠛⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠈⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⢴⣶⣿⡇⠀⠠⠆⠄⠤⠀⡀⣾⣤⣤⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠋⢸⣏⠶⠒⠀⣦⡀⠠⢽⣿⠃⢉⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠈⠛⠁⠀⠀⠀⠀⠀⠙⠛⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢄⣀⣠⣄⣀⣀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⢰⣌⠉⣉⡶⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⢠⡈⠉⢉⣠⠤⠖
⠀⠀⠀⠀⠀⢀⣠⣄⢀⡀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠶⢂⡉⠀⠈⠀⡶⠞⠛⠦⣄⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣠⣀⡀⣀⡀
⠀⢠⡆⠀⠩⠓⠲⠶⠶⢖⣒⠂⠉⠛⢦⣄⣀⣀⡤⢊⣀⠐⠶⠦⠖⠿⠛⠋⠛⠴⠴⠶⠂⣀⠁⢤⣀⢀⣠⣴⡚⠉⠀⣒⡲⠶⠶⢶⠚⠯⠁⢢⡄
⠀⠀⢡⡀⣀⡀⠀⢀⣠⢤⡌⢉⡲⠞⢛⡦⣍⣙⠓⠻⠤⠴⠖⠒⣧⣼⠶⠶⢲⡶⠒⠶⠶⠴⠷⠚⢛⣉⢴⡛⠷⠖⡋⢉⣤⣄⡀⠀⢀⣀⢀⡌
⢀⡴⠆⠉⠉⠙⢶⠉⠀⠀⢈⠁⠀⣠⠋⠰⠉⡙⠳⣄⠀⠀⠶⠞⠉⠉⠛⠛⠉⠉⠳⠦⠀⠀⣠⠞⠟⠉⠦⠙⣆⠀⠈⡉⠀⠀⠉⡷⠋⠉⠉⠱⢤
⢸⡇⠀⠀⠀⠀⢸⠋⠀⠰⠁⣠⠴⣿⡀⣇⠤⠗⢒⠉⠙⠲⣆⡀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⢙⡂⠰⠤⣰⠂⣿⠣⣄⠈⠆⠀⠘⠏⠀⠀⠀⠀⢀
⠈⢧⠀⠀⠀⣸⢿⠀⠀⠀⢰⢃⡤⠜⣷⣈⠲⣶⠀⠲⠴⠀⠈⢱⡀⠀⠀⠀⠀⢀⡞⠁⠀⠶⠖⠒⣸⠶⢃⣼⠣⣤⡘⡆⠀⠀⠀⡿⣦⠀⠀⠀⡼
⠀⠀⡆⠀⠸⠃⠀⠳⣄⠀⡎⢾⡵⠟⣀⠭⡷⠢⣍⣉⠻⠉⠛⠀⠷⠤⠴⠦⠤⡾⠀⠙⠁⡎⢉⣉⠥⢾⡍⣀⠻⢮⡷⢺⠀⢠⠜⠁⠘⠇⠀⢠⠁
⠀⠀⠇⠀⠀⠀⠀⠀⠘⠀⣷⣌⠷⣎⡁⠔⠛⢂⡀⠀⠛⠃⠀⢈⣷⠀⠀⠀⠀⣾⡁⠀⠘⠛⠀⢀⡐⠚⠆⢈⣡⡾⢃⣾⠀⠇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢰⠀⠀⠀⠀⢰⣷⠸⡙⢷⣌⡓⠾⠉⠀⠀⠠⢤⣤⡌⠛⣁⠀⠀⠀⠀⢈⠛⢡⣄⡠⢄⠀⠀⠉⠶⠞⣡⡴⢋⠃⣾⡄⠀⠀⠀⠀⡆
⠀⠀⠀⠸⠀⡇⠀⠀⣼⢹⡆⢱⡊⣉⠿⢲⣄⠉⠙⠛⢂⣁⠰⣦⣥⠀⠀⠀⠀⣼⡴⠆⣈⡙⠛⠛⠉⣠⡴⠿⣍⠑⡮⢡⡏⢷⠀⠀⠠⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⣽⣦⠙⢶⡞⠁⠀⠠⠶⠞⢋⣉⣡⡶⢣⡄⠀⠀⢠⡜⢶⣍⣉⠙⠲⠶⠅⠀⠉⢓⡴⠋⣠⣿⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢛⡀⢓⣄⡉⠛⠁⠴⣀⣈⠠⠔⣉⣴⠇⣀⠀⠀⢀⠸⣦⣉⠂⠄⣁⣐⠤⠌⠛⠋⣠⡚⢁⡻⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡜⢻⣄⠙⣁⠀⠙⢡⢤⣠⣩⡅⠖⢫⡾⠳⣿⠇⠀⠀⠸⣿⢎⢷⡝⠂⢀⣉⣀⣤⡄⠛⠠⡈⠋⢀⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡼⠉⢶⣌⠒⢮⣀⡔⠋⠁⣀⠴⣋⡞⡲⠃⠀⢀⡀⠀⠘⢞⠳⣝⠦⣀⠈⠙⠢⣀⡱⠖⣡⡴⠋⢧⠄⠀⠀⢰⡄
⠀⠀⠀⠀⢀⡄⠀⠀⠀⢰⡙⠇⢈⠿⠦⣀⠐⠀⣀⠀⠘⣩⠟⡀⢟⣴⠉⠙⢦⡹⢀⠻⣍⠃⠄⣀⠐⢂⣈⠴⠾⡉⠸⢏⡜⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⢳⡤⠑⢤⡔⢉⣉⠛⢁⣠⠞⡵⢺⠏⣼⠧⠶⠶⠤⣦⠰⡗⢬⡳⢄⡈⠛⣉⠙⢢⡤⠚⣡⠞⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⠇⠀⠀⠀⢸⡌⢦⠙⢠⡈⡹⠉⠋⣉⠔⠛⣰⢃⡼⡇⠀⢀⡀⠀⠘⢣⡙⣆⠛⠦⢉⠑⠈⢋⢁⡔⠛⣤⢃⡇⠀⠀⠀⠘⠇
⠀⠀⡀⠀⢰⠀⠀⠀⠀⣾⠟⢄⠳⣅⡸⠛⠛⠋⠀⣠⠞⠁⣼⠃⣾⠋⠉⠉⠉⣱⠈⣷⠈⠳⣄⠀⠙⠛⠛⠇⣈⠞⣡⠞⣷⠀⠀⠀⠀⡆⠀⢀
⠀⠰⡇⠀⣼⠀⠀⠀⠀⡟⡆⢎⡱⢌⠳⠮⠤⠴⠚⠁⠀⣸⠇⠀⣏⢀⣤⣤⣀⣻⠀⠙⣇⠀⠈⠓⠤⠤⠴⠖⠁⢎⡱⢠⣿⠀⠀⠀⠀⣷⠀⠸⠆
⠀⡀⠀⠀⣿⠀⠀⠀⠀⢇⣿⡌⢣⡜⠓⠄⡴⠖⣲⠀⣰⠋⠀⠔⢟⠉⠀⠀⠈⢻⠀⠆⠘⣆⠀⣦⠐⠦⠠⠚⢃⡜⢡⣯⡸⠀⠀⠀⠀⣿⠀⠀⣠
⠀⠃⠀⠀⢻⠀⠀⠀⠀⠈⣼⢉⣤⠳⣤⣦⠴⠊⣡⣾⠃⠀⠀⠀⣤⠀⠀⣀⣀⣸⡀⢀⠀⠈⣷⣈⠑⠦⢴⣤⠞⢠⡉⣯⠁⠀⠀⠀⠀⡟⠀⠀⠉
⣸⠀⠀⠀⠀⠰⡀⠀⠀⠀⠛⢾⡱⣁⠠⡤⠶⠛⠁⠀⠀⠀⠁⠀⠏⠀⠉⠋⠉⣠⠏⠠⠀⠀⠀⠉⠛⡶⠤⡤⣘⢌⣧⠻⠀⠀⠀⢀⠄⠀⠀⠀⠐
⣇⠀⠀⠀⠀⢤⡸⠀⠀⠀⠘⡈⢻⣎⣉⣉⡁⢀⠂⠀⠀⠀⠀⠀⣄⣀⣠⣤⣀⣈⠀⠀⠀⠀⠀⠀⡀⠀⣈⣉⣡⡞⢁⠃⠀⠀⠀⢎⡤
⠸⣧⣤⣦⣄⣸⡇⠀⠀⠀⠀⠹⣦⡙⠦⢞⡡⠃⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠋⠉⠀⠀⠀⠀⠀⠀⠑⢌⡳⠤⢛⣴⠏⠀⠀⠀⠀⢸⣇⣠⣴⣄⣼
⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣾⣷⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣾⣷⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣹⣷⣦⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⡀⠀⠀⠀⠀⠀⡔⠀⠀⡀
⠀⠀⠠⣤⣤⣀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⣿⣦⣄⣷⡄⠀⠀⢀⣼⣃⣤⣾⠇
⠀⠀⠀⠀⠉⠙⠻⢿⣷⣦⣤⣨⠹⠿⠛⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣿⢿⣿⣾⣷⣾⣿⣟⡿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣿⢧⡀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⢯⣿⣿⣿⠚⣍
⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⡷⠉⠘⢿⣿⡷⣍⣿⣦⣤⣤⣤⣤⣤⡄⠘⣃⢹⣿⣿⡿⢘⡛
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣦⡀⢸⣿⣷⣬⣬⣿⣿⠻⠿⣿⣿⡇⠈⠉⣾⠰⠰⣜⡆⠁
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⢿⣿⢻⣿⣿⡇⠀⠀⠈⠀⠀⢼⣿⠀⠀⢻⣷
⠀⠀⠀⠀⣾⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣾⣿⣿⣿⣿⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣠⠤⠤⠤⢦⢤⢤⡤⣄
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠀⠀⠀⠀⢠⣷⡾⠶⢚⣚⣒⠺⢧⣿⡠
⠀⠀⠀⢿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⢿⣿⣷⡄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠹⣿⣮⣿⣿⣿⣿⢿⣼⣿⠋
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⠹⣿⢃⣾⣿⣿⣆
⠀⠀⠀⠀⠈⠿⠛⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡷⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⣴⣿⡌⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠻⢿⢫⣾⣯⣿⣿⣎⢿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⡴⠉⠉⠙⠛⠋⠉⠁⢧
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠙⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠙⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣏⣻⡏⠀⠀⠀⢹⣿⣷⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⠿⣿⣷⡀⠀⠀⣸⣿⡇⠀⢶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣇⠀⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠿⠿⣿⠿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⣄⡀⢤⣄⠀⠀⠀⠀⠀⣾⣿⣿⡿⢿⣿⡿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⢻⣿⣶⣾⣷⡀⠘⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⡿⠆⢻⣿⣿⣿⣷⣀⣶⣶⣶⣶⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⢿⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣥⣾⣇⠀⠀⠀⠀⠀⠀⢠⣀⠀⣀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣉⣹⣿⣧⣀⠀⠀⠀⠲⠀⠈⢹⣿⡿⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⢿⣿⣿⣿⡁⠀⠀⠀⠀⢰⣿⣿⣧⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢣⣤⣾⣿⣿⣿⣇⣠⣴⣿⣶⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣻⣿⣿⣿⣿⣿⣿⠿⠛⠛⠻⠿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠋⠉⣽⣿⣿⣿⣦⣀⢀⠀⣀⣼⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣶⣛⣭⣿⡍⠛⠿⠿⠟⠛⠛⠙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣶⣴⠆⢻⣩⣽⠿⣿⣿⣷⣦⣤⣄⣀⣤⣤⣶⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡆⠈⢿⣯⣤⣾⣿⣿⠟⠛⠛⠛⠉⠙⠀⠀⢈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠹⠃⠀⠘⠻⣿⣿⣿⣿⣶⣶⣶⣶⣴⣶⣶⣾⠿⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⠟⠿⠛⠛⠋⠁⠀⢀⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠟⠁⠀⠀⠈⠷⢠⣤⣴⣁⣼⣿⣿⣿⣿⠄⡐⢀⢀⣠⣴⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⢀⡤⠈⢙⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣴⡏⣿⡏
⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⠺⣥⣤⣄⣀⣤⣿⣿⣿⣿⡏⠉⢉⣿⣿⢿⣿⠇
⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢻⠿⣿⣿⣷⣿⣿⣿⣷⣾⣿⠿⠷⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣟⠀⡾⢩⣿⣿⠿⠿⣿⣿⣄⡀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠙⠳⢾⣷⣤⣴⣿⣿⣿⣿⣿⣿⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⡆⠀⠐⠠⠝⠛⢿⣿⣿⣿⢿⣧⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⣠⡆⠀⠀⠀⣀⣉⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢨⠉⠛⠷⢾⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣄⣀⡀⠀⠀⠈⣻⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⢘⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⠉⠀⠉⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣁⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⣾⣿⣿⠇⠀⣀⡀⠀⠀⠀⠀⠀⣈⠂
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣸⣿⠟⠃⣴⣿⣿⣿⣷⣤⠀⢰⣿⣿⡀
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠏⠀⠀⣿⣿⣿⠿⠿⠃⠀⠈⣻⠹⠁
⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠍⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠟⡵⠋
⠀⠀⠀⠀⠀⠀⠀⣟⠻⡄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⣷⣾⢿⠿⠓⡄⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⣿⠗⠲⢤⣄⣄⡤⡔⢲⢢⠈⡈
⠀⠀⠀⠀⠀⠀⠀⢠⠒⠮⣓⣤⡀⠀⠀⠀⢀⣧⠀⠀⠀⠉⠀⠂⠀⢠⢦⢷⢷⡜⠛⠊⣄
⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠉⠛⠀⠀⢣⠈⡟⣆⠀⠀⠀⠀⠀⠁⠈⠈⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠙⠛⠒⠮⢆⡇⠹⡶⣤⣤⠤⢀⣀⠀⠰⠀⠀⠀⠀⠀⢁⡀
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣷⠀⢻⡼⠿⣿⡃⣼⠅⡑⠒⠤⠤⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢻⣿⠶⢹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣶⣤⣶⣿⣿⣿⣿⣿⣿⡛⠂⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢀⣠⣧⣴⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⣰⡟⠉⠙⣷⣦⡀⠀⢠⣿⣿⣿⣿⣿⣿⣿⢟⣿⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⣿⣈⠓⠐⣿⣿⣿⣷⣿⣽⣿⣿⣿⢿⣩⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠘⢿⣷⣶⣿⣿⣿⡿⠽⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⢿⣿⣿⠿⠿⢻⣿⡿⢁⣀⡁⠀⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣷⣶⣤⠚⠿⠿⣿⣿⣿⣷⡜⢃⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⡄⠀⠈⣹⣿⠟⠀⠸⣧⣀⠀⠀⠹⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣏⢹⣏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠿⢿⠉⠈⠁⠀⠀⠲⢶⣤⣤⢾⣿⠟⣴⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⡂⠒⠒⠢⠜⠿⠇⠀⠀⡼⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣤⣤⢤⢄⣀⣀⠀⠀⠂⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠦⣤⣀⡀⣀⣀⣠⣶⠿⣾⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⣿⣿⣿⣿⠖⣾⠟⠋⣰⣾⣷⡗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣨⡽⣿⢻⣿⣿⣟⣵⣟⣴⣿⣿⣿⢡⡇⢄⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⣀⣈⣿⣿⣹⡇⣿⣷⣿⣿⣿⣾⣿⣿⡜⣿⢣⣿⣿⣧⣬⡭⠁⣒⠂⢄
⠀⠀⠀⠀⢀⣴⣿⣾⣿⣷⣿⡏⠙⢿⠿⢿⣿⣸⣿⣿⣿⣿⣿⣿⣁⣷⡀⢸⣿⡿⠙⠁⢰⣿⣿⣷⣆⠑⠄
⠀⠀⠀⠀⠀⠉⢾⣿⣿⡏⣠⣀⠀⣠⣶⣦⣿⣷⣝⣿⣿⣿⣿⣿⣿⣿⣿⣾⢟⣡⣴⣶⣄⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⢯⡷⡎⣿⡽⣿⡿⠛⡛⢿⡇⠹⠋⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠉⠀⣿⣿⣿⣿⡇⢸⣷⣶⣿⣿⣿⣆⢻⣿⠚⠃⣠⣤⣉⠙⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣦⣙⠻⠿⣿⠿⠻⣛⣡⣴⣿⣿⣿⣿⣠⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣯⠉⠲⢿⣟⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠮⠅⠁⠀⠀⠈⠀⠉⠙⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠁⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣌⠁⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⠃⠘⣧⠈⠛⣛⢀⡀⢐⣒⠂⠀⢤⣤⠀⠀⠤⢶⡴⢦⣄⣙⣼⠃
⠀⠀⠀⠀⠀⢀⠀⠀⢠⣿⢞⠳⠄⠀⡀⠐⡀⠀⠲⠖⠀⡄⠀⠈⠀⠘⠿⠾⢿⠁⠾⠁⣄⣀⠀⠀⠀⠀⢀⠨⠶⣾⡷⠚⠋⠁
⠀⠀⠀⠀⠀⠈⠻⣶⣾⣿⣡⣴⠾⠋⢉⡽⠋⠀⠲⠀⠀⠀⣈⡀⠀⠀⠶⠶⣶⣶⠆⣤⡀⠉⠛⠶⢄⠾⠃⢀⣛⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠀⠀⠈⠁⢀⡀⣠⡤⠶⢈⡁⠁⠀⠀⣀⣠⣶⣄⣈⠙⠛⢰⠀⢶⣄⠀⠀⠠⣿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⣀⢶⠉⢅⠄⢀⠀⠈⠇⠀⠀⠀⠈⠉⠉⠉⠻⠟⢡⣶⠀⣀⠹⡆⠀⠀⠀⠺⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠘⠁⠀⣴⠊⠉⠆⣧⡄⢠⠀⠆⠀⠀⠀⠐⠲⠶⢿⣶⣦⣄⡈⠀⠁⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣒⡀⣴⢫⡴⢿⣿⡼⢃⡜⠀⡄⠀⠀⠀⣈⠑⠲⠦⣄⠈⠙⠁⢶⣶⠀⣶⠀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⣤⠌⠷⣙⣂⣀⠀⠀⣤⠞⠁⠀⠀⠀⠻⠿⣶⣶⣵⡖⠲⢶⣦⡌⢀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⣨⣉⠙⠻⠂⢁⡀⠀⠀⠀⠀⣤⣤⡀⠉⢙⡻⠟⠂⣀⡀⠛⠿⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠻⢧⡀⠰⠖⠀⠧⠄⠂⠀⠀⢠⡝⠛⠿⢷⣬⣀⠀⠈⠙⠿⡷⠀⣀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⡰⠀⠀⠸⠟⠂⠀⠀⠈⠙⠻⠶⠀⠀⠀⠈⠻⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠀⠀⠀⠀⣀⠀⠀⠀⠠⠆⠘⢿⡆⠀⢠⡄⠀⠀⠀⣀⠈⠁⠀⠘⠻⠆⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⢀⣀⣀⣤⠀⠀⠀⠈⠀⠀⠈⠀⠤⠆⠀⠀⠀⠈⠻⣦⠃⠀⠀⠐⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠓⠒⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⠈⠀⠀⠠⠈⠃⣀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦⣶⡀⠀⠀⠀⠀⠀⠀⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⢾⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⠐⢦⡀⠀⠀⣀⠀⣿⣌⢀⣇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠀⠀⠀⠘⢦⡀⠀⢪⠛⢀⠀⣽⡤⠴⠒⢚⣿⣯⣥⠬⠉⠭⢷⣶⣾⠾⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠈⠁⠀⠸⠆⢨⣽⣿⣶⠐⣔⣶⣾⣄⣀⠰⢶⣶⣶⣬⡽⣿⢿⣶⣽⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⠷⠀⠀⠀⠀⣠⣶⡶⣤⡄⠨⡿⢃⠈⠷⣌⡻⣿⣛⣤⣼⣟⡿⠿⠅⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⠀⠀⣤⣠⠋⠀⠀⣩⡄⠐⣷⠀⡄⠀⠈⠳⣮⡙⢿⣌⣀⠁⠀⠈⢀⠈⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠄⠈⠻⠀⠀⠀⠁⠴⠆⣿⠐⣻⠀⠀⠀⠀⠙⢶⣌⠛⠶⣍⣄⡾⡚⠲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄⠀⠳⡀⠀⠀⠀⠀⠀⣯⣭⠿⠁⠀⠀⠀⠀⠀⠉⠛⢷⣄⣨⡉⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠻⣤⣤⣌⡀⠀⡀⠀⠰⣇⣄⠀⢇⣠⡀⠀⠀⠀⠐⠀⠳⣯⣟⠷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠙⡁⠀⠀⣿⠒⢻⣭⠀⡘⡟⠳⡀⢠⠀⠀⠀⠀⠘⢻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠁⢀⣤⠟⠀⠘⡛⠀⣿⠁⠀⠀⢶⣀⣄⠀⢸⣦⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠚⠃⠀⠀⠀⠀⠀⡟⠀⠀⠀⠈⠻⠛⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠒⠋⠉⠉⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⣦⣀⣀⣀⡀
⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⢠⢞⠭⢝⢢⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⣖⠋⢻⡉⡇⠀⢨⢨⠇⠀⣷⡀
⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⢹⡸⣄⡼⣻⠃⠀⠀⠀⠈⢣⠀⠀⠀⡴⠹⡄⠀⢣⣧⠀⢸⣼⠀⢰⠇⠙⢦
⠀⠀⠀⢻⠀⠀⠀⠀⠀⣀⣀⣉⡓⠚⠁⠀⠀⠀⠀⠀⠈⣇⠀⡼⠁⠀⠙⠒⠋⠙⠒⠋⠀⠉⠉⠀⠀⠈⣇
⠀⠀⠀⠀⢣⣰⠋⠉⠉⠉⠀⠈⠉⠉⠉⠒⠲⢤⣀⠀⢠⠏⢠⣇⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣠⠤⠴⠒⠋⢳
⠀⠀⠀⠀⣠⠛⠶⠤⠤⠤⢤⣄⣀⣀⣀⣀⡀⠀⣸⡴⠋⠀⢇⣀⠀⠈⠉⠉⠉⠉⠉⠉⠀⢀⣀⣀⡤⠤⢶⠋
⠀⠀⠀⢰⢃⣄⣀⡤⠔⢚⠉⠀⠀⠀⠀⠀⠙⡏⠉⣇⠀⠀⠀⢹⠉⢉⡟⠉⠛⠉⠉⠉⠉⢉⠁⠈⠳⠤⢼⠆
⠀⠀⠀⢸⡏⠀⠀⠀⢠⣦⠀⠀⠀⠀⣉⠀⠀⢧⠖⣾⠀⠀⠀⢸⠋⠉⠀⠀⠾⠆⠀⠀⠀⠘⠃⠀⠀⠀⢈⡇
⠀⠀⠀⢸⠙⣟⠀⠀⠀⠁⠀⡀⠀⠀⠋⠀⠀⢀⣄⡟⠀⠀⠀⠸⣄⡀⠀⠀⠀⠀⠀⢒⣀⡀⠀⠀⠀⢰⠋
⠀⠀⠀⡏⠀⠸⡄⠀⠀⢠⠤⠬⢤⡀⠀⠀⠀⡸⢻⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠸⣍⣉⣉⠟⠀⠀⢀⡏
⠀⠀⠀⠈⠓⠲⠿⢄⠀⠀⠑⠒⠚⠁⠀⢀⡜⣁⡸⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⠀⠉⠁⠀⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⠤⠤⠤⡴⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⠶⠤⠶⣶⠋
⠀⠀⠀⠀⠀⢀⠔⠒⢒⢿⣄⠀⣠⣷⡒⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⣟⡿⣄⠀⢀⡿⡟⡟⠲⡄
⠀⠀⠀⢀⠔⠁⠀⠀⠾⠚⢻⠺⡟⠓⠷⠀⠀⠙⢦⡀⠀⠀⠀⠀⢀⡞⠁⠀⡏⡇⠈⠉⠉⠀⡇⡇⠀⠘⢆
⠀⢀⡴⠁⠀⢀⡀⣴⣶⣦⣼⠉⣧⣶⣶⡆⢀⡀⠀⠙⢦⠀⠀⢠⠏⠀⢀⡤⠿⠓⠒⠒⠒⠒⠻⠧⣄⠀⠈⢧
⢠⠎⠀⢀⡴⢻⡇⠧⢭⡼⢿⠉⡿⠬⣥⠇⢸⠙⢦⠀⠀⠳⣠⠋⠀⣰⡏⠀⠀⡶⠶⢶⠶⠶⡆⠀⠈⣷⡀⠀⢧
⡏⠀⠀⣎⠀⢸⡇⠀⠀⠀⢸⠒⡇⠀⠀⠀⢸⠀⠀⡗⠀⠀⡏⠀⢰⠇⡇⠀⠀⡏⠉⠉⠉⠉⡇⠀⠀⡇⣷⠀⠈
⠹⡄⠀⠈⢦⣸⡇⠀⠀⠀⢸⠒⡇⠀⠀⠀⢸⢀⡜⠁⠀⣠⡇⠀⢸⠀⡇⠀⠀⣇⣀⣀⣀⣠⡇⠀⠀⡇⢸
⠀⠘⢦⣀⣀⣙⣇⣀⣀⣀⣸⣤⣇⣀⣀⣀⣜⣋⣀⣀⠔⠁⣇⣀⣸⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇⢸⣀⣀
⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄⠀⠤⠤⠤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣀⡠⠤⠤⠤⠤⠤⠀⢰⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠋⣝⣇
⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢹⢠⡄⠀⡄⠄
⠀⠀⠀⠀⣤⣤⠯⠤⠵⠤⠀⠀⠀⠐⠛⠛⠛⠳⠀⠀⠀⠀⢸⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⠾⠸⠧⠰⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠤⠶⠒⠒⠚⠶⠶⠄⠀⡴⠀⠀⠀⠀⢰⠀⠀⢰⣶⣿⡄⣄⡼⢴⠀⠀⠀⠀⠀⠀⠁⠐⢷⡾⠇⠂⠀⠛⠒⠚⡃⠀⢤⡤⠤⡤⠤⠠⣤⣤⣤⠍⡋⠁
⠀⠀⠀⠓⠂⠓⠒⠒⠒⠃⠀⠰⠃⠀⠀⠀⠀⢸⢂⡄⣸⡛⣿⠃⠛⠿⠬⠂⠀⣀⢀⠀⠀⠀⠀⡱⣣⡜⡇⢰⠚⡞⠞⢱⢠⣾⠿⠿⠛⢛⠃⠀⠀⠀⠓⠓⢠
⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠰⣯⢹⣿⠀⠀⠠⠐⡒⣖⢒⠢⡆⢴⠬⡇⣿⡇⠛⠀⠇⢹⡇⡟⠿⢾⠘⠋⠂⠀⣒⢐⠀⠀⠀⢀⡀⢠⡞
⠀⠀⠀⡇⢀⡿⢀⣂⣤⣤⢀⣼⣸⣤⡔⠚⢁⣘⡀⢻⡸⣿⠀⠀⠀⠘⡏⣻⠛⠟⠟⠻⠛⡃⠻⣇⡠⠀⠴⠤⠶⠗⠚⡛⠃⠀⠀⠀⣷⣼⣆⣶⢲⢲⣖⣶⠂
⠀⣤⣼⣿⣷⢾⣿⠿⠟⡄⡄⠔⠞⠛⠛⠗⠒⡝⢻⡿⠄⠨⢀⣀⣚⣻⣩⠉⠈⠁⠹⠈⠁⠇⠀⢀⠋⠓⠒⠒⠒⠘⠛⡥⢖⢠⣀⣺⡅⠈⠀⠩⠰⣤⠦⠶⠂
⠀⠧⠃⠽⠷⠶⠶⢲⡖⠶⠲⠂⠒⠒⡂⠀⠊⠛⢃⣠⣆⣘⡇⠀⠀⣛⡓⡚⣁⠀⠀⢀⠀⠠⡤⠀⠅⠀⠀⠀⠘⠒⠒⠒⠘⠀⠀⣀⣀⠠⠤⢶⠦⠤⠐⠐
⠀⠉⠈⠈⠉⠉⠁⠀⠁⠁⠬⠥⠤⢤⣍⣺⠐⠀⣻⠀⠀⠀⡄⠀⠀⡡⡗⠄⢟⠸⡏⡸⠄⣀⠐⠒⡆⠀⠐⠠⠀⠄⠄⠄⠠⡟⠛⠿⡗⠖⠒⣥⣤⣤⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠠⠭⠈⢀⠉⡀⠀⡀⠐⠂⠀⠂⠋⢲⣾⠬⠁⠀⠀⠀⠙⡋⡷⠀⠠⢠⠦⣆⣆⠀⠰⡗⠀⠂⡀⠀⠠⠶⣶⠶⠶⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠠⠇⠀⠂⠹⢶⠛⢋⠀⠴⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠐⠠⡄⠀⠐⠊⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⡆⠀⠀⠀⠀⠀⠀⢤⣤⠤⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⢸⣿⡟⠉⢻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⢯⣽⣧⣤⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣺⢦⣤⣤⡿⢉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡋⠉⢩⣿⣿⣿⣿⣶⡝⠛⠳⢦⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⡠⠛⢻⡿⠿⣿⣿⡇⢀⣠⠞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣰⡿⠁⣀⣨⣛⣴⣻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣽⣿⣹⣿⣟⣋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠁⠀⠘⠻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡆⠀⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣯⡁⠀⣭⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠈⠀⠀⠈⣿⣷⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣾⣿⣿⣷⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠿⠛⠋⠉⠁⣠⣄⠈⠉⠙⠛⠿⢿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠛⠉⠀⠀⠀⠀⢀⣶⣾⣿⣿⣷⣶⡀⠀⠀⠀⠀⠉⠛⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠛⠿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠙⠋⠀⠈⠻⢿⣿⣶⣦⣤⣤⣴⣶⣿⡿⠟⠁⠀⠙⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⠿⠿⢿⣿⣿⣿⣶⣶⣿⣿⣿⡿⠿⠿⢿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠁⠀⠀⠀⠀⠻⣿⣧⠉⠉⣼⣿⠟⠀⠀⠀⠀⠈⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⣶⣦⠀⠀⠀⠀⠀⠈⠿⠿⠁⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣀⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⡀⢀⡎⠀⠀⠈⠛⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠒⠒⠦⠔⠒⠉⠁⠀⢈⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠨⠤⢤⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠼⠆⠀⠀⠀⠀⡜⣡⡀⠀⠀⠘⢦⣤⣤⠤⠤⠤⠤⣀⠀⢀⣀⣀⣀⣀⠔⠁⠀⠀⠀⠀⠤⠶⢤⠤⠒⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠢⢀⡀⡀⠀⠀⠀⢸⣎⣠⡷⠀⠀⠀⠐⢇⠹⠖⢒⠒⠒⠲⠟⠷⡄⢀⠤⣤⢤⠀⡄⠀⣀⣸⣀⡀⠀⠀⠀⠀⠀⠓⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠀⠈⠃⠺⣁⠀⠀⠀⢰⢸⢠⠘⠛⡃⠀⠃⠇⢰⠀⣿⡶⠝⠛⠊⢠⡭⠤⠄⠐⠞⢷⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⣴⠖⠈⠉⠹⠉⠉⠹⣗⠛⠘⠂⣶⠼⢴⣇⠗⠒⠒⢦⣼⡀⣏⡏⡍⡇⢸⠑⠉⠉⠑⠒⠒⠁
⠀⠀⠀⠀⠀⠀⢠⣊⣉⣉⡉⡯⠤⠤⠀⠶⠒⠒⠒⠒⠒⠒⠒⠒⢺⠋⠓⡟⢉⡑⣒⡒⢒⠒⡎⡠⢤⢸⢃⠀⣗⣇⣸⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⠀⢀⠔⠛⠓⠒⠒⠒⢲⢸⢳⢘⠉⡇⣎⠣⠀⣇⣁⣛⣇⣸⡧⣧⣇⣯⠅⡇⢧⡸⢸⢩⡥⠁⠀⠀⠀⠀⡷⠒⢦⡀
⠀⠀⠀⠀⡐⠉⠉⢺⠉⡤⣔⢲⠶⡖⢲⣺⢨⢩⠠⠭⡅⡼⠳⡄⡇⠀⠀⠀⠀⠀⠀⠀⢻⡆⡇⠀⠀⢸⠴⢄⡀⠀⢀⣀⡀⢱⠀⡌⢳
⠀⠀⠀⠀⢱⠀⠀⠘⡀⡇⠸⢸⠀⡇⠈⣷⢨⣸⢠⠠⠇⠧⠤⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢷⡇⣀⣤⣚⡳⠲⣌⠀⣭⠤⠤⠭⣴⠁⢸
⠀⠀⠀⡔⠁⠀⠀⠀⣇⢓⣚⡀⠓⠃⠀⣿⢰⠒⢰⣒⢰⠒⡆⢰⢒⡒⢒⠒⣒⠒⣒⠒⢒⢚⡟⠀⠀⠀⠙⢤⠀⢭⡇⣞⡲⣎⢿⠀⢸
⠀⠀⠀⠳⠤⢄⠀⠀⢸⠘⡄⣥⣏⣉⡆⣿⠸⠤⠀⠚⠈⣒⣃⢸⢩⡹⡏⡧⣇⣿⣉⠇⢘⢸⢇⠀⠀⠀⠀⠘⡄⢠⢓⡨⠝⠓⠾⢄⣸⡀
⠀⠀⠀⠀⠀⠀⠳⣤⠼⡇⠧⠼⠹⠤⠇⡇⡶⣋⡉⠥⠭⠤⠜⠘⠠⢤⡼⢢⡠⢢⡠⢦⣸⢾⠶⢷⢦⠶⠚⠯⢔⡋⢁⣀⣀⣀⡀⠀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢃⠀⡶⢲⡄⠀⡇⠇⡖⠒⡦⡖⠲⠄⡇⠒⠊⠙⠊⠉⠈⠙⠂⠟⠚⠀⠘⣇⠀⠀⠀⠀⣿⠤⠤⠤⠤⠤⠤⠭⢹⢸
⠀⠀⠀⠀⠀⠀⠀⠈⡀⠸⠀⢰⢸⡇⠀⠃⠂⠙⠒⠃⡇⢸⠀⡇⡠⣠⠊⠉⢦⡤⡤⡤⡤⡄⠀⠀⢮⣀⢀⠴⡆⢹⣒⣒⣒⣒⣒⣒⣒⣸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣇⣘⡼⠧⠤⠆⠄⠀⠀⠀⠁⠈⠀⠛⠀⠀⠀⠀⠸⠇⠇⠄⠂⠡⠀⠦⠠⠘⠀⠀⠃⠘⠃⠤⠤⠤⠤⠤⠠⠜⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⡄⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢘⣿⠀⠀⣠⣿⣤⣤⣤⡄⠀⠀⠀⣼⡇⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⣸⠇⣀⠀⢹⡏⠀⠀⠉⠁⠀⠀⠀⣿⡇⠀⣠⠀⠀⣿⡶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠐⠋⢠⡟⢀⣿⠀⢰⡟⠀⠀⢻⣇⠀⣿⡇⠀⣿⠀⠀⣿⠀⢀⣴⠞⢛⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⢠⡶⣾⠃⣼⠇⠀⠹⣧⣤⡴⠟⣿⠀⢽⡇⠀⣿⡄⠀⣿⣄⣿⠛⠛⠛⠉⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⢠⡾⠁⠋⢰⡟⠀⠀⢀⣨⡅⠀⠀⣿⠀⢸⡇⠀⢙⣁⣠⣬⡼⣿⣄⣀⣀⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⣀⣴⠟⠀⠀⠀⣾⢁⣠⡶⠟⠉⠀⣠⡾⠃⠀⢸⣷⠞⠛⠉⠉⠁⠀⠈⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋⠁⠀⠀⠀⠀⠛⠛⠉⠸⢷⠶⠞⠋⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠐⠑
⠀⠀⠀⠀⠐⢶⣶⣶⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠊⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⣠⠄⡀⠀⡾⢹⣿⣿⢻⡇⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠊⠁⠀⠇⠘⣟⣾⣻⡇⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠓⠒⠀⠀⠀⠀⠀⣟⣿⣻⡗⠀⠀⠀⢛⠀⠛⠛⡟⢿⣻⡗⢺⠆⣷⠓⡇
⡟⡗⠒⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢙⠀⠁⢈⡇⠉⢘⡃⢸⠀⡋⠀⡇
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⡏⠬⢡⠄⠀⣿⣿⣿⡇⠀⠀⠀⢨⠀⡴⢄⠀⢸⠀⠅⢸⠀⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⡀⠀⠀⠀⠀⣤⡆⠀⢲⣶⠂⠀⠀⠀⠀⣟⣿⣻⡇⠀⠀⠀⢸⠀⣿⢋⣃⣻⡇⠆⢰⠀⠀⠀⡆
⠀⠠⠄⠤⠀⠀⠶⠀⠆⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣲⣶⣾⡇⣶⠰⠶⠶⠀⠀⠀⠀⣯⣿⣯⡇⠀⠀⠀⢐⢀⢶⣿⣿⡐⠳⢂⢰⠀⠂⠀⡇
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣼⣿⣿⡧⣿⡆⠀⠀⠀⠀⠆⠀⣿⣿⣿⡇⠀⠀⠀⢸⣸⡞⠓⢹⡇⠅⢸⢻⠀⣁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣈⣽⣛⡇⢿⡗⡆⠀⠀⠀⠁⠀⣿⣿⣿⡇⠀⠀⠀⢨⣸⣇⠤⣀⠁⠂⠨⢸⠀⠄⠀⡇
⠀⠀⠀⠀⠀⢀⣠⣤⢴⣲⠿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣌⡛⢙⣏⡇⢼⣿⣙⡆⠀⠀⠄⠀⣯⠿⣽⡇⠀⠀⠀⠸⠸⠽⠀⠈⠂⢰⠎⠸⠀⠤⠀⠇
⢠⢠⡖⢲⡅⠘⠛⠉⠀⠂⠒⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸⠿⣿⡿⠿⠃⢾⡟⢻⢸⣆⠀⠀⠐⣿⣿⣿⡇⠀⠀⠀⠀⢀⠸⠀⠀⢰⣥
⡇⠐⠃⠉⠁⠐⢶⣶⣴⣤⣤⣤⠀⠀⢠⠴⠂⠀⠀⠀⠀⠀⠀⠈⠉⠁⠁⠉⠀⠀⠘⠘⣿⠄⠸⡏⣦⠀⠈⣿⣿⣿⡇
⠈⢀⠀⠀⠀⠀⠀⠉⢿⣛⠩⠇⠀⠀⠠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⡄⢡⣧⠸⡇⣿⠂⠐⣿⣿⣿⡇⠀⢀⣀⠀⣀⣀⣀⣀⣀⣀⢀⣀⠀⠀⠀⣀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⣀⠀⠀⢰⠂⢀⣠⣴⢶⣿⣭⣿⠀⠀⠀⠠⠘⣼⢆⠃⢿⠀⠘⣷⣟⡾⡧⠀⢸⡇⠀⢠⣿⣿⡷⣿⣿⣾⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⠠⠀⠓⠲⠤⠀⠀⠀⠈⠁⠀⠀⠀⠹⢭⢭⠽⢭⡟⣭⡍⣯⠀⠀⠀⢀⠀⢹⢹⡇⢸⠀⠐⣿⣿⣷⡇⠀⢸⡇⠸⠉⡇⣿⡇⠿⣿⣿⣿⠀⠀⠀⢸
⡆⠀⠃⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣀⣉⣉⣉⣿⣏⡇⠀⠀⠀⠀⡆⠈⠸⠇⣜⠀⠰⣿⣿⣿⡇⠀⢸⡇⠀⠀⣫⣿⡇⢸⣿⣿⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠚⠛⠛⠛⠟⠛⠳⡇⠀⠀⠀⠀⢁⠀⠀⠀⣿⠀⠐⣯⢿⣽⡇⠀⢸⣇⠀⠀⠑⣿⡇⠞⢻⢾⣿⠀⠀⠀⢸
⠀⠀⢀⡀⠔⠊⠀⠀⠀⠀⠰⠤⠀⠀⣀⣌⢔⡚⢂⡘⣂⣒⣐⣀⣓⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⣿⠂⠰⣿⠶⡛⡇⠀⢸⡏⠂⠀⠀⣿⡇⠀⠄⢸⣿⠀⠀⠀⢸
⠀⡠⠋⠀⠀⠀⢀⠀⠀⣀⠵⠖⠛⠁⠀⠀⠀⠀⠀⡤⢤⠤⠤⢤⡤⠤⢴⠀⠀⠀⠀⠀⠀⠇⠀⠀⠉⠀⢠⣿⣶⣻⡇⠀⢸⡇⠀⠀⠀⣿⡧⡁⠀⢸⣿⠀⠀⠀⢸
⠳⡱⣄⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠐⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⢁⠀⢸⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢠⣟⣷⡛⡇⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⢸
⠀⠙⠘⠀⠀⠀⣀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠐⠀⣲⣶⣶⣶⣶⣾⣧⣶⠀⠀⠀⡂⠀⠀⢸⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡷⢀⡀⢾⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⠛⠻⠿⣿⣿⣿⡿⠿⠛⢋⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⢉⣁⣤⣿⣿⣿⣿⣿⣿⡀⠀⠀⣸⣿⣿⣿⣿⣿⣧⣄⣉⠙⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⡿⢛⣛⣛⣛⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣷⣜⠻⣿⢏⣠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠿⠿⠿⠟⠛⠋⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⠖⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠈⠁⠉⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⢦
⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⢠⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⡾⢶⠆⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⣦⣼⣷⠀⢀⣀⣤⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣤⡀⣀⠀⠀⣸⡏⣰⡄⢰
⠀⠀⠀⠀⢸⡇⢀⣻⢭⡿⠀⠈⠋⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢻⣷⡟⠃⢸
⠀⠀⠀⠀⢸⡟⠛⣿⡟⠁⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣶⣍⠛⠛⡁⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⡀⠀⣀⣤⣶⣶⣶⣶⣶⣶⣦⣄⡀⠙⠂⠈⢿⡟⠀⢹
⠀⠀⠀⠀⢸⡇⣶⣿⣁⠄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢂⢀⣿⡄⢸
⠀⠀⠀⠀⠈⣧⣿⡇⠯⣸⡟⣿⣻⢯⡟⠷⠿⠿⢿⣿⡿⣿⣿⣇⠀⠀⠀⠠⠄⠀⠗⠒⢄⣀⠀⣸⣿⣿⡿⠟⣻⡛⣟⡿⣿⣿⣿⣿⣿⣧⢼⡾⠙⣇⠨
⠀⠀⠀⠀⠀⢸⣿⢉⡆⣼⣿⣿⢿⣶⣾⠳⠀⠚⠃⣨⣠⣬⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢲⣿⣟⣯⣤⣶⠎⠀⠉⠑⠠⢾⡿⣿⣿⣷⠀⠋⠀⣿⡇
⠀⠀⠀⠀⠀⠈⡿⢸⠃⣿⣸⡷⣨⣿⣅⠀⢠⠈⠰⢉⠻⣿⣿⡇⠀⠀⠀⣀⣄⠀⣤⣄⡀⠀⠈⢿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠈⢳⣾⢿⣿⡄⠀⠀⢹⡇
⠀⠀⠀⠀⠀⣰⠁⢼⡁⣿⣯⣷⢹⡿⢿⣆⢰⣧⡂⢤⡁⣻⡿⠇⠀⢠⣾⣿⣿⣿⣿⣿⣧⠀⡀⠸⣿⣿⡟⠩⢀⣤⣔⣀⣀⣠⣼⣇⣹⣿⠇⠀⠀⠸⡇
⠀⠀⠀⠀⢠⡟⢰⡟⠓⠈⠻⣿⣾⣥⣤⣿⣿⣷⠶⠾⠛⠋⠀⡀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡖⢦⡈⠉⠛⠻⠿⠷⣮⣭⣿⣧⣿⣿⠿⠃⢀⡀⠀⠀⢻⡆
⠀⠀⠀⠀⢸⣧⣾⠁⠀⠀⠒⠀⠈⠁⠀⠐⠀⠁⠀⠀⠀⠸⠿⠀⢸⣿⣿⣿⡿⣿⣽⣿⣿⣿⡇⢰⡏⠐⠒⠀⠀⠀⢀⠛⢨⣍⣉⠀⠀⢰⡟⠤⠀⠀⢶⡄
⠀⠀⠀⠀⢸⡇⣧⡀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⣾⣿⣿⣿⡝⣿⣿⠻⣿⣿⣧⡄⠉⠀⠀⠀⠛⣿⠉⠛⠀⠈⠙⢦⣄⠐⣿⡂⠀⠀⣼⡇
⠀⠀⠀⠀⠈⠳⣌⢻⣶⣦⡔⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢼⣟⡉⣹⣿⣿⣿⣿⣾⣿⣿⣿⣷⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠙⠀⠘⣧⣀⣴⡽⠃
⠀⠀⠀⠀⠀⠀⠙⢶⣬⣿⣋⣀⣀⡀⠀⢲⣶⣦⣀⠀⠀⢀⡀⠘⢿⣿⣿⣿⣿⠇⣿⣿⣿⣿⡿⣯⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣶⣶⣾⣷⣶⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣦⣄⠈⠻⣯⡳⣴⠿⡟⠀⠀⠈⠉⠉⢹⡖⠀⠉⠉⠉⠀⠁⠸⠃⠀⣾⡿⠛⠽⢦⣿⣿⣿⣿⣿⣿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡻⠿⣿⣿⣿⣧⠀⡙⢿⡃⣔⠃⢀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⢸⣶⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠂⢺⣿⣿⣿⣧⡿⣾⡟⣿⠁⡞⠀⣶⠀⢀⡀⠀⢀⠀⠀⡀⠀⠀⡀⠀⠀⠳⠀⣿⣿⣿⣿⣿⣻⣏⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⣿⣿⣿⢿⣼⣿⣿⣿⣼⣇⡀⣿⣄⢸⣇⣄⢸⣆⣀⣣⣤⣤⣧⣤⣄⣀⣀⣽⠟⠉⣻⡏⣹⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠺⢻⣷⡄⠉⢻⠉⢇⠉⠈⣿⠁⠈⣿⠁⠈⣿⠁⠉⡿⠁⢨⡏⠀⣿⢻⢿⡇⠀⣠⣿⢁⡆⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠰⡆⠉⠻⣦⣙⣀⠻⢧⠀⡏⠀⢠⡏⠀⢸⡇⠀⠀⠁⠀⠀⣧⠀⣿⢸⣞⣃⣴⢻⡿⢸⠇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢠⡇⠀⠀⠈⠛⠟⡻⣿⢴⣷⣴⣿⣧⣶⣾⣷⣶⣶⣿⣶⣶⣿⣦⣿⠿⡟⠉⠙⠋⠀⢸⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⡇⠀⠀⠀⠀⠀⢁⡀⠀⣿⡄⢿⡍⢿⣥⠹⡏⢻⠁⣿⡏⢹⣿⢻⡼⠿⠀⠀⠀⢠⠈⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣃⠸⡇⠀⠀⠀⠀⠀⠈⠙⠖⢹⣤⣬⣧⠾⢯⣬⠥⠼⠠⡿⠷⣬⠿⢬⠁⠀⠀⠀⠀⡘⠀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣀⠀⢀⣤⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⣰⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣶⣶⣶⣶⠶⡤⢄⣀⣀⣀⡀⢀⣠⠴⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠉⠀⠀⠈⠀⠁⠀⠉⠀⠉⠀⠁⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⢠⡄⠀⣧⣤⢦⡈⠲⠀⠀⠀⢠⣀⠀⠀⠀⠐⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠚⠀⠀⡇⠀⡠
⠀⢸⡿⠀⠋⠈⠐⠛⠁⠀⠀⡆⠸⠛⠒⠄⠀⠀⠀⠁⠌⠀⠀⠀⠀⠀⠀⠰⠃⠀⣭⡏⠀⠀⢠⡴⠛⠀⢀
⠀⠈⠃⠀⠀⢹⠀⠀⠀⢘⡓⠂⢸⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠱⡂⣃⡀⠀⠴⠈⠀⣠⠔⠁
⠤⠤⡃⠀⣀⣠⣀⠀⠀⢸⡇⠀⠸⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠊⠁⠛⠂⠁⠀⠀⠊⣀
⠐⡀⠀⠈⠙⠠⣼⠁⢠⢸⡇⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠀⣠⣤⡎⢙⣄⣀
⠀⡃⠀⠀⠀⠀⠀⠀⠺⠄⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂⠄⡄⠀⠝⠓⠈⠋⠁
⠤⠧⠀⠀⠰⠄⠀⠠⣠⢀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⡀⠒⠂⠁⡠⢊⣀⣠⠤⢴
⠒⠀⠠⠤⠀⠀⠠⢤⣤⣄⣀⣀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣁⣠⠀⠀⠒⠚⢯⢥⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⢀⣏⠈⠇⠋⠉⡟⠋⠐⡈⣹⣛⡆⡃⠈⠀⠀⢀⣀⣀⡉⠀⣨⡇⠀⢀⣀⠀⢰
⠀⠀⠀⠀⠀⠀⠘⣧⠀⡄⣤⣠⡧⠀⠴⠀⣹⡽⠗⠛⠛⠛⣾⡏⣰⠉⢹⠃⠉⠉⠉⠉⠉⠉⠛⢄⡀
⠀⠀⠀⠀⠀⡀⣀⠛⢲⠉⠁⠀⡁⠀⡠⠁⠁⠀⠀⠀⠀⠀⢯⣠⠉⠃⠾⠄⠀⠀⠀⠀⠀⠀⠀⠈⠧⠉⠩
⠀⡴⢾⣿⣆⠈⠁⠀⠀⡀⠀⡠⠝⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣠⡀⣄⣀⡠⠀⠤⠖⡪⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⣠⢤⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⢠
⠄⠂⠐⠂⠀⣀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋⠀⠃⠀⠀⠀⠄⠆⠀⠀⠀⠀⠀⠡
⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠠⠐⠀⠒⡄⠀⠀⠀⠀⠀⠀⠀⣠⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢅⠀⠀⢀⣿⡀⠀⠀⠀⠀⢀⣾⣿⠋⠙⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠑⠺⣛⠿⣷⡀⠀⠀⠀⢸⣿⡿⠋⢩⣽⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⣝⢿⣄⠀⠀⠘⠿⡟⣁⢫⣤⣤⠛⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⢯⡻⡇⡀⠀⠀⡇⠀⠈⠃⠀⡀⠛⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢰⣶⠛⠀⣿⣄⠀⢠⠀⠤⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢀⡜⠣⡿⠂⠀⠢⣤⣴⡶⠚⠀⡷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⢸⠀⠀⠀⠀⠀⢀⣿⣿⠁⢠⡴⠁⣿⣿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡆⠀⣀⣤⣶⣿⣿⠉⣷⣯⠀⢀⣿⡿⣾⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢧⣀⡤⣷⣾⣿⣿⣿⣻⢿⣧⣼⣇⠀⣼⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣧⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠙⠛⠛⠛⠛⠛⠛

⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⢏⠷⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⡀
⣼⠉⠀⠙⡆⣠⠤⠤⡀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠿⣶⠇⠀⠀⠀⠀⠀⠀⠀⣾⣅⢀⣿⡀⢸⣇
⠙⢦⣤⡜⠱⡇⠀⠀⣹⠀⠀⠀⠀⠀⠀⣀⡀⢀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣌⣛⣋⣄⡀⠀⠀⠀⠀⠀⠀⣿⢸⢸⣿⡇⣿⣿⡅
⠀⠈⢻⣁⢰⣿⠆⠚⠁⠀⠀⠀⠀⠀⢸⣿⣧⣽⣼⣻⡇⠀⠀⠀⠀⠀⠀⢸⣓⡲⠿⠶⣚⣳⠀⠀⠀⠀⠀⠀⣿⠼⠸⣿⠇⢻⡿
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⢀⣤⡶⠖⢶⣤⣀⠀⠀⠀⠀⠀⠠⣴⣶⣶⣶⣶⣶⣦⣶⠀⠀⠀⢀⡀⠘⠚⠋⠓⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠘⢻⣿⣷⣤⣼⣿⡿⠃⠀⠀⠀⠀⢙⣿⣿⡟⠙⠋⠉⠈⠁⠀⠀⠀⠠⠿⣿⣿⣿⣿⣿⠿⠄⠀⠀⠀⠀⣀⣀⣤⡖⡉⣈⣇⣈⣙⣶⣤
⠀⠠⣬⣭⣭⣭⣥⠄⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠿⣟⣛⣻⠿⣾⡃⠀⠀⠀⢲⣾⡿⣿⡄⠶⢀⣿⣿⣿⠿⣿
⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢀⣀⣤⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣯⣿⣿⣽⡟⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣶⣾⣧
⠀⢸⣿⠿⠋⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣼⡇⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣹
⠀⢸⣿⣇⣠⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠠⣼⣿⣭⣭⣯⣯⣿⣧⠄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠻⢿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠐⢻⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣾⣀⣀⣀⠀⠀⠀⠀⠀⠠⡤⢤⠤⢤⠤⡤⢤⡤
⢠⣤⣤⣄⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢸⢀⣇⡀⡇
⠈⠉⠉⢹⣿⣿⣿⣿⠖⠀⠀⠀⠀⠀⣼⣿⠟⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢸⢸⣹⡇⡇
⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⣤⣤⣭⣤⠀⠀⠙⠳⢦⣄⠀⠀⠉⠉⠉⠉⣥⣿⣤⡉⠉⠉⠉⠀⠀⠀⠀⡇⢸⠀⢸⠈⡏⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠞⠁⢀⠀⣈⡨⢷⣤⠄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠴⠢⠶⠀⢲⠟⠸⡄⠁⠐⠧⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡠⠞⠁⠀⠀⠀⠀⠲⠀⠂⠘⠁⠀⡀⢁⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠒⢍⣁⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠂⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠒⢜⠂⣀⠀⢠⣿⣿⣷⣦⣿
⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡶⠈⠀⣠⣴⣌⠳⠀⠀⠀⡉⢀⠸⣿⣿⣿⡟⠛
⠀⠀⠀⠀⠀⠀⠁⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡄⡀⣀⠻⣿⡿⠀⠀⢀⡸⣰⣼⡆⠀⠉⠉⠁⢹
⠀⠀⠀⠀⣼⠀⠜⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠋⠓⠈⠓⠀⠀⡠⠀⠉⠀⠉⠈⠁⠀⠠⢀⠉
⠀⠀⠀⠀⠈⠢⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⠐⣠⠄⠄⠤⠓⠟⠈⠄⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠺⠉⠁⣀⡠⢼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⢠⠘⠀⠀⠰⢤⠀⢀⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣀⣀⣀⡀⢤⠀⢸⡄⠀⢀⡀⠃⠀⣆⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠈⠤⠀⢺⡇⠀⠈⠆⠀⢦⡄⠀⡧⠤⢶⣧⣀⣰⠿⠏⠀⠀⠀⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣀⣤⢸⠿⡆⠀⠐⣦⠶⠇⠀⡗⡀⠾⠇⠈⢀⡄⠀⠀⠀⠀⠀⣘⠁⠀⠄⠘⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠉⠀⠛⢇⣄⠀⠀⣇⠀⡂⠀⢂⠀⢰⡃⠀⣀⠙⠤⡀⠁⡀⣛⠃⠐⢀⣴⠶⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⡈⠏⠀⠀⡏⠁⣠⠀⢠⠀⢠⡇⠨⠿⣄⠐⠒⠀⡒⠀⣰⠂⠹⠁⠀⠀⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡆⠀⠀⣿⠉⢉⠀⠈⠁⢤⠄⠐⡇⠈⠏⣉⠅⠁⠀⠀⠐⠀⠀⠈⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣤⠈⠃⠀⠀⠈⠀⢘⠀⠘⠀⣉⠤⢚⠈⠰⠒⠁⠀⠚⠁⠀⢠⡄⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⢻⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠚⣿⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⡏⠀⠀⠀⠀⠀⠀⠀⠘⡎⠀⡠⠀⠀⢀⡠⠂⢘⠉⠹⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡞⠀⠀⠀⠀⠀⠀⠀⠂⡤⠟⠋⠀⡀⠀⠁⠀⠀⠸⠅⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡄⠂⠀⠀⠀⠀⠀⠀⠀⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣼⡐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠇⢹⣇⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⣿⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠈⠿⢤⣶⣾⣿⣿⣶⣶⡖⠲⢴⣶⣿⣿⣶⣦⡤⠇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢻⢸⣿⣿⣿⣿⣿⠿⣃⣘⠘⣿⣿⣿⣿⣿⡇⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠈⠶⠝⠛⠉⠉⢒⣰⣿⣿⡖⠦⠍⠛⠉⠋⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⢀⠀⠀⠀⠀⠈⢻⣿⣿⡇⠀⠀⠀⢀⣀⡄⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢿⡏⢷⠀⠀⠀⠍⠩⠀⠀⠀⣾⢹⡿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢳⣜⠒⠢⠤⠤⠤⠤⠶⢖⢣⡞⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⠈⠱⡄⠀⠀⣸⠀⠀⠛⠉⠑⠒⠀⠢⡒⠋⠙⠚⠁⠀⠇⠀⠀⠀⠔⠉⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠱⡀⠀⠘⠧⣀⠀⠀⠀⠂⠤⠤⠀⠀⠀⢀⣀⠔⠁⠀⠀⠚⠀⠰⢦⡉⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣶⣄⠀⠀⠑⠄⡀⠀⠀⠑⢦⡀⠀⠀⠀⠀⢀⡤⠊⠀⢀⠠⠊⠀⠀⠀⠀⠈⠃⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠠⡀⠀⠀⠐⠄⡀⠀⠈⠉⠉⠉⠉⠁⠀⡠⠔⠁⠀⠀⠀⡠⠄⠒⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠑⠀⡀⠀⠀⡠⠔⠉⠀⠀⢀⠠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⣀⠀⠀⠁⠃⠀⡀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠉⠢⢄⡀⠀⠀⠁⠂⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠈⠀⠀⢀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠦⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣎⠁⠈⠁⠀⠀⠀⠀⢀⠠⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⡀⠀⠀⠀⠀⠀⡀⠈⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⡶⠂⢠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠐⠇⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣤⣄⢀⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡀⢼⠀⠀⠀⠀⠀⠀⠀⠰⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⣐⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠠⡧
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡒⡇⠚⣴⣶⠁⠀⠉⠉⠉⢘⣯⣽⠀⠉⠉⠉⠃⢀⢀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠇⠐⣾⣿⡇⠀⠀⠀⠀⠀⠓⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠂⠉⠀⢠⢄⠀⠀⠀⠀⠁⠀⢸⠀⠀⠀⠀⠀⡂⠀⠀⢀⠀⠀⢀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢸⢠⠉⠀⠀⠀⠀⢀⢸⠀⠀⠀⠀⠀⠂⠀⠀⢺⠀⠀⠈⠀⠘⠀⠀⢨⠀⠀⠀⠒⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠐⠁⠈⠐⡄⡅⣇⣴
⠀⠀⠀⠀⠀⠀⠐⠘⠂⢀⣀⠀⣆⣀⣈⠄⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⠇⠋⠽⠁
⠀⠀⠀⠀⠀⠀⣦⣄⣤⡄⢠⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠄⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠁⠈⢀⠁⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸
⠀⠀⢀⡠⠀⢠⣦⢀⠉⡉⠿⠝⠀⠀⠀⠁⡠⠠⡴⣀⣠⢈⢩⡀⢳⠎⠶⠙⠀⠀⢛⣖⠐⢂⠲⠮⠏⠋⠏⠓⣸⠀⠰⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠖
⠀⡖⣀⣀⠀⣾⣁⠀⢀⣽⡧⢤⡞⠁⠀⠀⠁⠉⠙⠈⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠴⠀⠀⠀⡠⠂⠘⠀⠗⢆
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠋⠁⠐⠒⠒⠒⠒⠒⠒⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠈⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⠿⠟⠋⣿⣿⣿⣿⣿⣿⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⠀⠘⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡏⠀⠀⠀⠘⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⡶⠀⠀⠀⢠⣶⣶⠆⣿⣿⣿⠁⣶⣶⣶⡆⠘⢿⣿⣷⡄⣶⣦⠀⠀⠀⠰⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠃⠀⠀⠀⣼⣿⣿⢸⣿⣿⡇⠀⣿⣿⣿⡇⠀⠈⢻⣿⣿⣌⢿⡄⠀⠀⠀⢻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠃⠀⠀⠀⢠⣿⣿⣿⢸⣿⡿⠀⠀⣿⣿⣿⡇⠀⠀⠀⠹⣿⣿⢎⣧⠀⠀⠀⠀⢻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡏⠀⠀⠀⠀⣾⣿⣿⣿⠀⠈⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⢰⣶⣾⣿⡆⠀⠀⠀⠈⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠀⠀⠀⠀⠐⠛⠛⠛⠛⠀⠀⠀⠀⠘⠛⠛⠛⠛⠀⠀⠀⠀⠘⠛⠛⠛⠓⠀⠀⠀⠀⠘⠛⠛⠛⠓



⣿⠿⠿⠿⡿⠿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣛⣻⣛⣿⣿⣟⣟⣻⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢿⣿⣿⡿⣿⣿⣿⢿⣿⢿⡿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣪⠞⣡⡞⢡⡟⢠⣿⠀⣿⠀⣿⡈⢷⡈⢷⡝⢻⣕⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣽⠟⢁⣾⠋⣰⡿⠁⣾⡇⢀⣿⠀⢻⣧⠈⣿⡄⠹⣦⡘⠷⣝⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢞⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡳⣄
⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡙⢦⡀
⠀⠀⠀⠀⣠⠞⣫⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣝⠶⣄
⠀⢀⣴⠞⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠳⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠉⠀⠘⡿⢻⡏⠀⠉⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⣀⠁⠀⢢⡀⠀⢠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣠⣄⣼⣿⣿⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡛⠉⢹⣟⣻⣿⣿⣿⠛⠛⢻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⢷⣇⠸⣿⣿⣿⣿⡏⢠⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⡿⠿⡿⡿⣟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣍⣟⣛⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⣭⣿⣭⣼⣓⣀
⠀⢠⣶⣢⣚⣉⣛⡋⠛⠢⢄⡀⠀⢀⡴⢿⣭⣉⡋⣭⣯⣛⣋⣹⡿⢦⡀⠀⢀⡠⠔⠛⢛⣛⣛⣓⣔⣦⡄
⠀⠀⣛⣛⠿⣟⣭⣝⣿⣴⣾⣭⣛⠲⣶⣶⣶⣆⣙⡠⠜⣁⢰⣶⣴⣶⠶⣛⣭⣶⣴⣿⣫⣽⣻⠿⣛⣛
⠀⣾⣿⣿⣧⣹⣿⣻⡿⡋⢼⠿⣟⣀⣬⡿⠳⣿⣿⣿⣿⣿⣿⡾⢿⣥⣀⣻⡿⡧⢹⢿⣟⣿⣏⣼⣿⣿⣷
⠀⢻⣿⡿⠁⣿⣿⡿⡻⣳⢌⠣⣍⡤⠤⠤⠦⡜⣿⣿⣿⣿⢣⡤⠤⠤⣄⣉⡜⡡⣞⢯⣿⣿⣿⠈⢿⣿⡟
⠀⢸⣿⡇⠀⠈⢹⡧⣇⢵⣺⡷⢈⣹⣷⠒⢾⣿⢨⣥⣬⡅⣿⡷⠔⣾⣏⡁⢾⣓⡮⣸⢼⡟⠁⠀⢸⣿⡇
⠀⠈⣿⡇⠀⠀⠘⢳⢈⡳⣝⡛⠉⠁⠀⠀⢀⣩⣾⣿⣿⣷⣍⡀⠀⠀⠈⠉⢛⣫⢞⣁⡞⠃⠀⠀⢸⣿⠁
⠀⠀⣿⡇⠀⠀⢰⠘⣎⢿⣢⠝⠳⢶⣼⠿⠋⡙⣹⣿⣿⣇⢋⠙⠾⣦⣶⠞⠫⣔⡮⣳⠃⡆⠀⠀⢸⣿
⠀⢰⣿⡇⠀⠀⠘⡄⣼⢷⣄⣀⣤⠶⠒⠋⣉⠔⣹⣿⣿⣏⠢⣉⠙⠲⠶⣤⣀⣠⡾⣧⢠⠃⠀⠀⢸⣿⡆
⠀⢸⣿⡇⠀⠀⢠⠻⣥⠶⠋⠛⢿⠶⠶⠛⢛⠔⢉⣿⣿⡉⢢⡛⠛⠶⠴⡿⠛⠙⠶⣨⠟⡄⠀⠀⢸⣿⡇
⠀⢸⣿⡇⠀⠀⠘⡴⠙⡳⣔⣟⠁⢀⣀⡴⠣⣤⢾⣿⢿⡷⣤⠜⢦⣀⠀⠈⣻⣢⢞⠋⢮⠃⠀⠀⢸⣿⡇
⠀⠘⣿⡇⠀⠀⠀⡝⢾⡋⠈⣙⡻⠛⠉⣀⠞⢡⢊⡿⢷⡵⡌⠳⡀⠉⠛⢻⣋⠁⢙⣷⢫⠀⠀⠀⢸⣿⠃
⠀⠀⣿⡇⠀⠀⠀⢹⡀⣽⢾⣭⣤⡴⠞⠁⣹⠋⢾⣿⣷⣖⠹⣍⠈⠳⢤⣤⣭⡷⣯⢀⡏⠀⠀⠀⢸⣿
⠀⢰⣿⡇⠀⠀⠀⢠⡿⣍⢠⡴⠀⠀⣀⣼⠇⠤⢴⣶⣾⡶⠴⠸⣦⡀⠀⠀⢦⡄⣩⢿⡄⠀⠀⠀⢸⣿⡆
⠀⢸⣿⠀⠀⠀⠀⡌⢇⣼⠓⠶⠶⡾⣿⡟⠀⠀⢙⣭⣭⣇⢀⠀⢻⣿⢷⠶⠶⠞⣧⡰⢡⠁⠀⠀⠈⣿⡇
⠀⣿⣿⡄⠀⠀⠀⠁⡼⢧⡀⢀⣺⡵⠋⠀⠀⠀⠼⠿⠿⠿⠀⠀⠀⠙⢮⣗⠄⢀⡼⢧⠈⠀⠀⠀⢠⣿⣿
⢰⣿⣿⣿⣄⠀⠀⠀⡇⢄⡹⠉⣡⡾⠀⠀⠈⠁⢻⣿⣾⣟⠙⠋⠀⠈⢷⣌⠙⠏⡠⢺⠀⠀⠀⣠⣿⣿⣿
⠘⢿⠿⣿⠍⠀⠀⠀⠙⣆⡭⢉⡿⠁⠀⠀⠀⠀⠉⠉⠍⠉⠀⠀⠀⠀⠈⢿⡉⢭⣰⠋⠀⠀⠀⢩⣿⠿⡿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠦⣄⠔⠒⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⣀⠤⠔⠒⠒⡤⠒⠒⠂⠀⠀⠈⠀⠀⠀⠈⡆⡀
⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⡀⣀⠀⠀⠐⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠁⡈⠃⠀⠀⠀⢀⠤⣀⡠⠤⠤⣀
⠀⠀⠀⠀⠀⠨⣀⣀⡀⠀⠀⠰⡇⣸⡄⠀⠀⠀⡏⠳⠠⠠⠤⠬⠿⢯⠉⣉⣁⢄⢠⡀⠀⣀⣇⡀⠈⠀⠀⠀⠜⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠂⠀⠓⠻⡄⠀⠀⣸⣸⣀⣘⢋⡁⠛⠟⢠⢸⣷⠎⠛⠉⢴⠅⣄⣒⡚⢷⠀⣀⡀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠄⠐⠀⠒⠟⠉⠁⠙⢀⣈⢹⣍⣉⣡⣿⣒⣾⡜⠉⣑⠶⠾⣸⠼⢯⡇⢈⠉⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⢰⠗⠶⢠⣼⣃⣉⣉⡏⣉⡉⡥⡄⣠⡀⢈⢫⣬⠁⢠⣤⢶⢠⠌⡇⡎⡇⠨⠤⡷⠷⠒⠒⢲⡀
⠀⠀⠀⢀⠤⠾⢤⠮⢤⠤⣤⡤⢤⡅⢣⡃⣳⡂⣳⣃⢸⠒⠚⠛⠚⠙⠉⢺⣅⡇⠉⠁⣖⡋⡇⠀⠀⠀⠘⡁⢹⢇
⠀⠀⠀⠸⡄⠀⢸⢸⠉⡇⡇⡃⢸⣿⡇⣿⣁⡇⣇⣸⢸⠀⠀⠀⠀⠀⠀⠀⢫⡇⢀⣠⠳⠥⣗⠒⣖⣀⣒⣃⡆⢸⠂
⠀⠀⢀⠔⠀⠀⠈⡖⢃⡇⠓⠊⠀⣧⡜⠃⡄⠀⠲⠄⠸⠒⠒⠒⣒⢒⡒⠒⡞⡇⠉⠉⠉⢦⡈⢱⡇⣒⢶⣺⠇⢸
⠀⠀⠘⠢⠤⡀⠀⢣⢩⢨⣾⣉⡆⡇⠠⠆⠐⠈⢒⡀⠶⡌⢃⡰⡌⠹⠠⠀⠠⠇⠀⠀⠀⢀⡧⢠⣓⠩⠛⠒⠦⢼⡀
⠀⠀⠀⠀⠀⠙⢶⠚⡘⠒⠙⠒⠃⣇⠨⣄⡘⠐⠒⠂⣀⠒⣄⣑⣄⣲⣖⣜⣺⠃⢻⡒⠊⠙⠣⣍⠤⠤⠤⠤⠀⠀⣈⡇
⠀⠀⠀⠀⠀⠀⢸⠀⡇⠸⠉⡇⠀⡏⠸⣁⣹⡏⢹⠀⠁⠉⢀⠤⡀⠀⠈⠈⠉⠀⠈⡆⠀⠀⣀⢾⣓⣒⣒⣒⡒⢒⣹⡇
⠀⠀⠀⠀⠀⠀⠘⠀⢸⠀⣇⡇⢀⣅⠃⠀⠀⠇⠘⠠⠆⠙⠃⠀⢻⣿⢹⢹⢹⣀⣀⣙⡖⠊⠉⠸⣖⣒⣒⣒⣒⣒⣺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠁⠀⢀⣀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⣿⠀⠀⣸⣯⣤⣤⡀⠀⠀⢀⣷⠀⠀⠀⠀⣿⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⢠⡟⣠⠀⣿⠃⢀⡉⠁⢀⠀⢸⣿⠀⣠⠀⢰⡿⠶⠶⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠉⢠⡟⢰⡏⠀⣿⠁⠀⣻⣇⠈⣿⠀⣿⠀⢸⣇⢠⣴⣟⣭⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢀⡾⢿⠃⡿⠁⠀⠻⠷⠛⠉⣿⠀⣿⠀⢻⠇⠈⠿⣾⡁⠉⠀⠀⣠⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⡞⠁⠀⣸⠇⢀⣠⠶⠃⠀⣰⠏⠀⣿⣠⡴⠶⠞⠛⠙⠷⠶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⠞⠋⠀⠀⠀⠿⠶⠋⢷⣤⡴⠞⠁⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⠴⣪⣵⣦⣄⣠⣀⠀⠠⣄⣠⣴⣦⣍⠢⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⣫⣴⣦⣽⣟⣋⠉⠻⣿⣯⣴⣤⣝⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠠⢠⠎⢩⠨⠑⠳⣤⡤⣤⣶⡟⡁⠀⠠⠀⢕⢌⢀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣡⠎⠠⠈⠒⢷⢤⡤⣤⣶⡏⡀⠀⠄⠀⠑⠌⢿⣿⣿⣿
⠀⠀⠀⠀⠰⠃⠐⣀⠤⠔⠒⠛⠛⠉⠙⠛⠓⠢⠤⣄⡂⠀⠂⠀⠀⠀⠀⣿⣿⣿⡿⠑⠁⠐⣠⠤⠔⠒⠛⠛⠉⠛⠛⠓⠢⠤⣄⡀⠈⠈⢿⣿⣿
⠀⠀⠀⢠⡛⠖⠉⠀⠀⠀⣀⣀⣤⣤⣤⣀⣀⡀⠀⠀⠈⠲⠛⣂⠀⠀⠀⣿⣿⣿⣇⠛⠆⠉⠀⠀⢀⣀⣀⣤⣤⣄⣀⣀⡀⠀⠀⠉⠲⠛⣸⣿⣿
⠀⠀⠀⠈⣷⣄⣀⠴⠚⠉⠁⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⣀⣾⡇⠂⠀⠀⣿⣿⣿⢸⣷⣀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⣀⣾⠃⣿⣿
⠀⠀⠐⣦⣰⠏⠁⠀⠀⣠⣤⠶⠒⠒⠒⠶⢦⣄⠀⠀⠀⠙⢇⣤⠆⠀⠀⣿⣿⡿⣤⡸⠋⠀⠀⠀⣠⣴⠶⠒⠒⠒⠶⣤⣄⠀⠀⠈⠙⢆⣤⢻⣿
⠀⠀⢰⠈⡄⠀⠀⠰⣿⠋⠀⠀⠀⣴⠀⠀⠀⠙⢿⡆⠀⠀⢠⡉⡶⠀⠀⣿⣿⢷⢉⡀⠀⠀⠰⡿⠋⠀⠀⠀⣶⠀⠀⠀⠙⣿⠆⠀⠀⢠⠁⡾⣿
⠀⠀⣄⠎⢰⣤⣀⣀⠇⠠⣄⡀⠰⡇⣡⡤⣤⠄⠈⣅⣀⣤⡆⠙⣀⠀⠀⣿⣿⣄⠋⢱⣤⣀⣈⠁⠠⣄⡀⠸⡇⣦⣤⣤⠄⠘⣀⣀⣤⡆⠑⣠⣿
⠀⠀⠈⢷⠞⣿⣿⣿⡀⠀⠀⢙⢛⣄⢶⡎⠁⠀⠀⣿⣿⣯⡟⢶⠋⠀⠀⣿⣿⠙⣶⢻⣽⣿⣿⠀⠀⠈⠙⢋⡄⢶⡎⠀⠀⠀⣿⣿⣿⡛⣶⠋⣿
⠀⠀⠀⢼⣾⣿⠛⢿⣧⡀⠀⡴⠛⠉⠑⢷⠀⠀⣰⣿⠟⣿⣧⣷⠀⠀⠀⣿⣿⣧⣼⣾⣿⠛⣿⣧⠀⠀⡴⠛⠉⠚⢧⠀⠀⣼⡿⠛⣿⣷⣧⣼⣿
⠀⠀⠀⠈⣻⣿⠀⠀⠙⠷⣤⣀⡀⠀⠀⣀⣤⠾⠛⠁⠀⣿⣟⠃⠀⠀⠀⣿⣿⣿⡝⣹⣿⠀⠈⠛⠷⣤⣀⠀⠀⠀⣀⣤⠾⠛⠁⠀⣿⣟⢸⣿⣿
⠀⠀⠀⠐⠿⣜⣦⣀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⢀⣴⢋⢼⠆⠀⠀⠀⣿⣿⣿⣷⡷⡙⣦⡀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⢀⣴⢋⢾⣾⣿⣿
⠀⠀⠀⠀⠀⡽⢝⢻⡷⣶⠤⣤⣀⣀⣀⣠⡤⢴⢾⣿⣫⢾⠌⠀⠀⠀⠀⣿⣿⣿⣿⣧⡿⣝⣿⡷⡶⢤⣤⣀⣀⣀⣤⠤⢶⢾⡿⡯⢿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠐⡌⠯⣳⣿⡇⠀⠈⠉⠀⠀⢰⣿⡿⡹⢃⠆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡈⢏⣳⣿⡆⠀⠈⠉⠀⠀⢸⣿⣞⡽⢁⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠁⠰⣝⠻⣤⣀⣀⣀⣤⠿⣫⡖⡀⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆⠰⣝⠻⣤⣀⣀⣀⣤⠟⣫⠆⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠠⠤⣿⣧⣿⡦⡄⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠠⠴⣿⣴⣿⠥⢌⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠿⠅⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣿⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣻⣛⣛⣟⣛⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽⣿⣿⣽⣿⣿⣽⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠁⠉⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡁⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠘⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⢠⡄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⢛⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣧⣤⠀⠀⠀⠀⣠⣿⣿⣿⣧⠀⠀⠀⣤⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠃⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⣿⣿⣁⠀⠀⠀⠐⠟⢻⣿⣿⣟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢈⠀⠀⠀⠀⠶⠰⠆⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠋⠀⠀⠀⠠⠶⠿⠿⠿⠿⠆⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠉⠀⠀⠀⠀⣶⠀⣤⠀⠰⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣠⠀⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠸⠆⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣷⡿⠁⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣶⡆⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⢠⣦⡆⣤⣴⣴⣶⠰⡆⠀⠀⠀⠀⠀⠰⡆⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠉⠁⠘⠋⠰⠆⠦⠀⠀⠴⠳⠧⠀⠀⠰⠢⠄⠀⠀⠉⠉⢀⡁⠀⠈⠉⠀⢠⡄⠀⠀⠀⠀⠀⢀⣀⠄⠀⢘⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⣀⣀⠀⢀⡀⢀⡀⢀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠀⠀⠀⠀⣀⣀⣠⠄⠀⠀⠀⠀⢛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠈⠁⠈⠀⠈⠉⠀⠿⠿⠓⠀⠀⠀⠀⠀⠀⠀⠋⠉⢁⡀⠀⠀⠈⠙⠉⠀⠀⠀⠀⣇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠆⠀⠀⠰⠆⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠈⠁⠀⠀⠛⠁⠙⠃⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⢀⠀⠀⠘⢋⣍⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢀⡄⣤⣤⢀⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠻⠿⡿⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⢻⣿⣿⣄⢀⣌⣤⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠛⠋⠙⠉⠋⠉⠛⠁⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡛⠀⣿⣿⣿⡼⠿⡏⠃⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⢠⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⢁⣽⣿⡇⠀⠁⠁⠈⠀⠁⠀⣤⣤⡤⢤⠄⠀⠀⠀⠀⠉⠀⠀⠄⠀⠀⠀⠉⠀⠀⠰⠟⠙⠁⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡋⠛⠀⠛⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠐⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⠀⢰⣿⠟⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣠⡆⠀⠀⠀⠀⡀⣀⠀⠀⠠⠟⠀⠀⠀⠀⠀⣄⠀⠀⠀⠈⠿⣿⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⢀⣾⠿⠃⠀⠀⠀⢡⡏⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠃⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠀⠀⠀⠀⠛⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⣴⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡠⠤⠀⠀⠀⠀⠀⠀⠠⠤⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿
⠀⠀⡠⣨⠚⠻⣿⡚⢒⣀⣵⠖⠋⠛⠂⠑⠀⠀⠀⠀⣿⣿⣿⣿⠟⡱⠛⠟⣿⠓⢂⣠⣿⠛⠙⠛⠪⡻⣿⣿⣿
⠀⠀⠜⠡⣀⡡⠤⠟⠛⠛⠛⠢⠤⢅⡁⠀⠀⠀⠀⠀⣿⣿⣿⠏⠎⠀⣀⠥⠤⠛⠛⠛⠛⠠⠤⣀⡠⠀⠙⣿⣿
⠀⣘⠓⠈⠀⢀⣀⣠⡤⠤⢤⣄⣀⠀⠈⠑⢚⡆⠀⠀⣿⣿⣿⡚⠒⠉⠀⢀⣀⡠⠤⠤⣤⣀⣀⠀⠈⠒⢒⣿⣿
⢀⠸⣧⡤⠚⠉⠀⠀⣀⣀⣀⠀⠈⠉⠲⢤⡾⢃⠀⠀⣿⣿⡇⠿⣦⠔⠊⠉⠀⢀⣀⣀⡀⠀⠈⠙⠢⣴⠿⢸⣿
⢈⠗⠋⠀⢀⣴⠞⠋⠉⡉⠉⠙⢶⣤⠀⠀⠑⠛⡀⠀⣿⣿⡙⠛⠁⠀⣠⣴⠟⠉⠉⡉⠉⠛⢶⣄⠀⠈⠓⢋⣿
⣈⠞⣄⡀⠈⠃⢀⡀⢰⡇⣀⣀⡀⠹⠀⣀⡜⠢⡁⠀⣿⣿⡡⠚⣄⡀⠸⠁⣀⠀⢼⢁⣀⣀⠀⢓⠀⣀⡜⠎⣽
⠙⡶⢫⣿⣿⠀⠀⠙⢓⡜⣵⠉⠀⢸⣿⣿⡟⡾⠃⠀⣿⣿⠳⡞⣿⣿⣇⠀⠈⠛⣢⡨⡔⠁⠀⢸⣿⣿⠷⡞⣻
⠀⢷⣿⡏⢿⣧⡀⠰⠋⠈⠘⠆⢀⣾⠟⢹⣷⡗⠀⠀⣿⣿⣶⣷⣿⠙⢿⣄⠀⠞⠉⠈⠻⠀⣠⡿⠋⢿⣿⢶⣿
⠀⢠⡿⣇⠀⠈⠛⠶⠤⠤⠤⠖⠋⠁⠀⣸⢻⡀⠀⠀⣿⣿⣿⣸⢿⡀⠀⠉⠓⠦⠤⠤⠴⠞⠋⠀⢀⡿⣧⣼⣿
⠀⠈⢹⡿⣷⣤⣀⣀⠀⠀⠀⣀⣠⣤⡾⣳⡉⠀⠀⠀⣿⣿⣿⣿⣳⡿⣦⣄⣀⡀⠀⠀⢀⣀⣠⣴⡿⣾⣽⣿⣿
⠀⠀⠀⠌⢟⢿⣷⠈⠉⠉⠉⢁⣾⣜⡟⡀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡘⣟⣾⣆⠉⠉⠉⠉⢰⣷⣟⠏⣾⣿⣿⣿
⠀⠀⠀⠀⠡⣲⠟⠧⣤⢤⣤⠞⠻⡆⠈⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣄⡾⠛⢦⣤⢤⡤⠟⢷⣠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠢⢖⠻⣾⠟⡦⠈⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣤⡞⢿⡾⢓⣦⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣟⣛⣛⣙⣻⣟⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣭⣥⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠈⠙⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠏⣿⣿⣿⣿⡇⠀⠀⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⠟⠋⠁⠀⢸⣿⣿⣿⣷⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣰⣿⡿⢋⣀⣀⢻⣿⣧⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⠀⠀⣿⣿⢧⣿⣿⠃⢸⣿⣿⠈⠻⣿⣿⣿⣿⡇⠀⠀⠈⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⠀⠀⣸⣿⣏⣿⣿⠇⠀⣼⣿⣿⡄⠀⠙⢿⣿⣾⢿⡀⠀⠀⠈⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠀⠀⠀⢀⣿⣿⣞⠿⠋⠀⠀⣿⣿⣿⡇⠀⠀⠈⢻⣟⣼⣇⠀⠀⠀⠘⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠀⠀⠀⠀⣼⣿⣿⡏⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠈⣿⣿⣿⡄⠀⠀⠀⠘⣿⣿⣷⡀




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⠛⢿⠛⡟⢿⡟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣿⡟⢹⣿
⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠁⠀⠀⠀⣠⣖⣠⣿⣻⣿⣿⣿⣟⣖⠠⣄
⠀⠀⠀⠀⠀⢸⣿⣿⣟⣛⣓⡒⠒⢒⠒⣶⠶⣶⣤⣬⣤⣤⣾⠿⠿⠾⠻⡅⠀⢀⡀⢠⣤⣤⡤⠤⠤⣤⣤⠤⣼⣷⣶⣤⡄
⠀⠀⠀⠀⠀⠈⢩⣿⣿⣿⣿⣿⣤⣾⣿⣿⣟⣋⣉⣿⣿⣏⣅⠀⠀⠀⠀⣬⡒⠛⠻⣿⡿⣿⣧⣤⣿⣶⣦⣶⢿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⣾⣿⡿⠃⠁⢹⣿⣿⣿⣿⣿⣿⣍⣀⠶⠶⠛⠆⠀⠀⠘⡟⢻⢛⣫⣥⣾⣿⡿⣻⣿⣿⣿⣿⡀⢻
⠀⠀⠀⠀⠀⠀⠈⠻⣧⠀⠀⢼⣿⣿⣿⣿⣻⡋⠉⠉⠙⠋⢿⣷⠄⠀⠀⣞⠻⢿⣿⠿⠛⣉⣴⣿⣿⣿⣿⣿⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣴⣿⣿⣝⣿⣿⣿⣷⣶⣌⠁⠀⣉⡁⠀⠀⠀⠈⠰⣈⣤⣶⣿⣿⣿⠟⣸⣿⣿⠏⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣾⣿⣿⣯⣿⡋⠁⢀⣀⣀⣼⠇⠀⠀⠀⢺⢦⣬⡿⠿⠟⢉⣠⣾⣿⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢿⣾⡿⣿⣿⣿⣿⠷⠀⠀⠀⣤⡀⠀⠀⢠⡖⠀⠂⢴⣶⣿⣿⣿⡿⠋⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣸⣿⣿⣿⣟⣋⠀⣀⠤⠂⠀⠛⠁⠀⠀⠀⢈⢑⡂⠀⠙⠛⢋⣁⣴⣿⣤⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣯⡉⠛⠿⣟⣛⠿⠀⠀⠀⠀⣶⠂⠀⠀⠀⠈⣀⠁⠀⠛⠿⢿⣿⣿⠀⢉⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⡿⠿⣯⣽⣉⠀⠀⣠⣴⡾⢀⡴⣠⣆⣀⠁⠐⣌⡻⠶⠤⠀⠀⢉⡠⢴⣧⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢿⣤⣀⣀⣉⢿⡿⠇⠀⠀⡤⠈⣾⣿⣏⣙⡿⣧⠈⠀⠀⠐⠲⣾⣅⡀⠀⠈⣩⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⡿⠿⣿⣿⡠⠀⠀⠂⠘⣡⣤⣿⣿⠿⢿⣇⣀⠀⠐⢌⠲⢤⠀⠙⢉⣠⣞⡁⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣦⣤⡴⠾⠓⠒⢂⣴⠾⢋⣹⣿⣿⣿⣺⡟⠛⢷⣄⠀⠑⠐⠒⢾⣧⣄⠙⢛⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣟⠛⠷⠶⡤⡔⠋⣁⣴⠚⠫⣿⣿⣅⣸⣧⣀⠀⠙⠷⣦⡀⢀⠠⣩⡤⢴⣿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣙⠛⠿⢐⠞⢡⡾⠋⠁⣠⣀⣿⣿⡿⢿⡇⠈⠻⣆⠀⠈⠻⣦⠀⠀⠀⢀⣉⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁⠀⠈⠀⠀⠀⠈⢹⣿⣿⣷⢶⡟⠙⠀⠈⠧⠀⠀⠉⠈⠑⠄⠩⠥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣿⣿⡟⠙⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⡉⢈⡿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢄⡀⡠⠠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⣄⡠⠄⢼⠀⠀⠀⠉⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠈⠁⢈⠁⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢢
⠀⠀⠀⠀⠀⠀⠔⠒⠋⠀⠀⠀⢀⡢⠒⡄⠀⠀⠈⣲⠲⣒⠒⠒⠒⠒⣤⡄⠀⠀⠀⠐⠁⠀⠀⠀⠀⠂⠉⠑⠊⠁⠈⢙⣆⡀
⠀⠀⠀⠀⠀⠀⠑⠒⠲⣀⢀⠀⠈⣇⣺⠋⠀⠀⠀⠈⡆⠃⣤⡄⠀⡍⡩⢉⢁⣔⡙⢯⣇⠜⡠⠀⠒⠒⣢⣀⠀⠀⠀⠈⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣒⡠⠤⠠⡶⠷⠦⣄⠰⢠⡄⣁⢸⢰⡟⡍⣀⣀⡀⡏⠡⢦⠤⡄⢹⠤⠄⠦⠤⠀⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⢰⢉⣉⣉⣩⣉⡀⠠⠤⠥⠤⠤⢽⠦⠤⡴⠻⠭⠭⠯⠤⢤⠂⣉⠓⡾⠀⡅⡇⢸
⠀⠀⠀⠀⠀⠀⡏⠉⢉⠱⠯⠤⠤⠀⠅⡤⠄⠐⢲⢠⠢⠀⠘⠠⢶⡀⢸⣆⣋⠂⣒⢸⢸⢈⠀⡗⡃⠋⠉⠉⠉⠉⡅⠀⡀
⠀⠀⠀⠀⠔⠒⠓⡞⢃⣰⠶⣤⡶⢶⠾⣙⣂⢸⢚⢈⠿⡄⡌⠉⠉⠉⠉⠀⠉⠉⢧⢸⠀⠁⠀⣏⡅⡆⠀⠀⠀⠀⠇⠀⠛⡄
⠀⠀⠀⠀⠆⠀⠀⠇⢰⠈⠀⡀⡇⠘⣀⢇⡨⣀⢸⢸⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⠀⣀⣔⡳⠤⣍⠀⣉⣉⣉⣁⡔⠀⡏
⠀⠀⠀⠔⠁⠀⠀⢸⠘⣒⡂⠓⠃⠀⣿⠔⠒⢰⠂⠔⢲⠀⡖⡐⢒⠒⣒⢒⣒⠒⠲⢻⠂⠀⠀⠘⢤⠀⢡⡇⣒⡲⡲⠆⠀⡇
⠀⠀⠀⠣⠤⣄⠀⠈⡄⣧⢩⣎⣉⡆⡿⠣⠄⠘⠂⢐⣊⡀⢿⣩⢇⠡⣇⣽⡨⠀⡀⡾⡄⠀⠀⠀⠀⡆⢠⢓⡨⠝⠒⠦⢄⣇
⠀⠀⠀⠀⠀⠈⠢⡄⢣⠹⠴⠸⠤⠇⡄⡾⡉⠩⠃⠧⠔⠀⢠⠢⡜⢲⡄⢢⡔⢆⡷⡓⠚⡖⠶⠛⠻⢤⡉⡀⣀⣀⡀⣀⣀⣀⡙
⠀⠀⠀⠀⠀⠀⠀⠇⠘⡄⠸⠙⡆⠀⡇⠀⡏⢉⣶⠒⡒⢸⠈⠊⠈⣉⠈⠁⠈⠹⠋⠃⠀⢻⡀⠀⠀⠀⣿⠂⠲⠶⠀⠀⠠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠸⠀⡇⠀⠁⠇⠉⠉⢸⠀⡇⢸⡀⠠⠊⠀⢳⡒⠒⠖⠖⠄⠀⠐⠢⡄⠖⡏⢸⠭⠭⠭⠭⠭⠨⠥⠅⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠑⠤⠗⠑⠒⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠃⠀⠀⠀⠋⠉⠀⠉⠀⠀⠁⠀⠉⠉⠀⠒⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⢷⡄⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢀⡶⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣸⠇⠀⠰⣿⠛⠛⠿⠆⠀⠀⠀⣿⠀⠀⠀⠀⢸⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⢰⠟⢰⡇⢰⡏⠀⣴⠀⠀⣤⡄⠀⣿⠀⢰⡇⠀⢸⡏⠉⢉⣤⣴⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢀⡄⣿⢀⣾⠀⢸⣧⡀⣠⡼⣷⠀⣿⠀⢸⣇⠀⢸⡇⣰⠿⠷⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀⡾⠹⠇⣸⠇⠀⠀⠙⡛⠉⠀⣿⠀⣿⠀⠈⠟⠀⣘⣃⣿⡀⠀⠀⣀⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⠟⠁⠀⢠⡟⢀⣠⡴⠛⠁⢀⣴⠏⠀⣿⣤⠶⠛⠛⠋⠁⠘⠻⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠛⠁⠀⠀⠀⠘⠿⠛⠁⢷⣤⠶⠛⠁⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⢀⣀⣀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⢸⣿⣿⣿⣧⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠘⢿⣿⠿⠃⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⢠⣾⣿⣇⠀⠀⠀⠀⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⢠⣿⣿⣿⡿⢷⣤⡀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠙⠛⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠉⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⣠⣤⡄⠀⣠⣿⣇⠀⢹⣿⣤⡀⠀⣀⣀⡀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⣴⣾⣿⠏⠀⣰⣿⣿⡟⠀⠀⣿⣿⣷⡀⠈⢿⣿⣦⣄⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⣿⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⡄
⠀⠀⠀⠀⢀⠀⠏⠀⠘⠎⠁⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⡊⠀⡂⢀⡇⠈⠂⠀⠀⠀⠀⠀⡠⠄⠒⠈⠁⡠⠤⠒⠃
⠀⠰⠀⠀⠀⠀⠀⠰⠂⠂⠰⠰⠀⠀⠀⠀⠀⠈⠆⠀⠀⢸⠒⠈⠉⠀⢀⠀⠀⠈⠂⠀⠈⢃⠀⢰⢆⡎
⠀⠀⠁⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠈⠈⠀⠰⠍⡀⠀⠀⠈⠁⠂⢄⡀⠀⢀⠀⠀⠈⠀⠰⡍⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠑⠀⠀⠙⠁⠀⠀⠀⠀⢆⢀⠷⠀⠀⠀⠀⠁
⠀⠀⡠⠂⠄⡀⢂⠔⠀⠁⡺⠀⠀⠀⠀⠀⠀⠀⠁⠊⠀⠀⠀⣀⡀⠐⠁⠨⠂⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠲⠄
⠀⠘⠁⠦⡰⠸⡉⠀⠀⢄⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠐⠀⠀⠀⠉⠀⠰⡍⣈⠉⠈⠀⠀⢰⡐⡀
⠀⠈⠂⠉⠀⠀⣁⡈⡤⠋⠀⠀⠴⣀⢄⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⢠⠀⡡
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣴⡡⠊⠀⠻⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡰⠊⠀⠀⡎⠠⠁⠇
⠀⠀⠀⠀⠀⠈⢠⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢠⡀⠀⠀⠐⠀⡄⠀⠀⢀⡠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠘⢓⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀
⠀⢀⠀⠀⠑⠀⠀⠐⠈⠀⠰⠊⠛⢤⡤⠠⢀⠀⣠⠴⠲⣄⠀⠀⢠⣀⡃⣀⣨⡀⠀⠀⠁⢁⠑⠤⠤⠤⠊⠏⡞⡇
⠀⠘⠤⠠⢀⡡⠀⠀⠀⠀⠰⠄⠠⠊⢁⠀⠀⡹⠃⠀⠀⠀⠀⠀⡄⠀⠀⢀⣠⡄⠀⠀⠃⠉⠀⠀⠀⠀⠀⠉⢸
⠀⠀⠀⠀⠀⠀⠂⠑⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⡀⠰⢖⠂⠀⠀⠀⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠉⠀⠄⠀⢀⠀⡀⠀⠀⠀⣀⡀⢀⠠⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠
⠀⠀⠀⡀⡔⠂⣳⠤⡀⠀⠥⠀⠀⠀⠀⠘⠀⠁⠀⠀⢸⢷⡀⠀⡀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⣈⠂
⠀⠀⡔⠑⠈⠩⠀⠂⠁⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠘⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣦⣴⣿⣷⣍⣉⡙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⡀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢱
⠀⠀⠀⠀⠀⢸⡏⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡛⢻⣿⣿⣽⣿⣦⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣇⣿⣿⢿⣿⣿⣿⣿⣿⣛⣯⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠇
⠀⠀⠀⠀⠀⠀⡟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡟⠻⣿⣿⡿⠞⠋
⠀⠀⠀⠀⠀⠀⠹⣨⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⠁⠀⠉⡀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠟⠛⠋⣿⣿⣿⣿⣿⠇⣠⣶⣼⠇
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣟⠿⣿⣿⣷⣶⣶⣿⣿⣿⣿⠅⠀⢰⣿⣿⣩⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣤⣌⡛⠿⣿⠿⠟⠋⠉⠀⠀⠸⠟⠃⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠉⠉⠁⠙⣿⣿⣿⣿⣿⣿⣿⣿⡏⢳⣶⡾⠓⣲⣦⣀⣀⠀⠀⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣇⠈⠡⡴⢠⣿⠻⣿⠻⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⣿⣿⣿⡆⠀⠘⠘⠛⠀⠉⠛⠗⠓⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣶⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠞⠂⠀⠀⠀⠀⠀⡀⢷⡀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⠋⣻⢿⡟⠑⣤⣤⣀⠀⠀⠀⠈⠓⠺⡇⠀⠀⠀⠀⠀⢛⣨⣇⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⣾⠀⠀⠀⠀⠀⠉⠲⠦⣤⣀⣠⠜⠀⠀⠀⠈⠁⠘⢯⠉⡇
⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⡤⠈⠀⢲⣦⠌⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡵⣧⡖
⠀⠀⠀⠀⠀⣠⣿⠋⣴⣿⣷⡌⠀⠀⠁⠀⠀⠀⠛⣢⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠈⡇⢸⡇⠀⣼⠂⡒
⠀⠀⠀⠀⡼⠟⢀⣤⣿⣿⣿⣿⡄⠀⠰⣿⣿⣶⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠠⣧⣾⡷⢸⡟⣸⠇
⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⣿⣆⠀⣿⣿⠟⠉⢙⣟⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⢘⣳⠀⣸⣿⣿⣿⣟⡗⣿⠇
⠀⠀⠀⠀⠀⠀⣽⣷⣿⣿⣿⣿⣿⣿⣷⡈⠁⠀⠀⢸⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣩⡿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢸⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣷⣦⣍⣉⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣸⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢷⣈⣻⡟⠛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢾⡟⠉⣷⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣶⣄⠀⠁⠘⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡟⠇⠀⣼⣿⣿⣿⣶⣤⣄⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠋⠙⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⠋⠛⠉⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⡿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣥⣤⣥⣴⣤⣼⣧⣴⣿⣥⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⠴⢾⡿⠿⡿⠿⣿⠿⢿⡿⠿⡿⠿⣿⠿⢿⡷⠦⡀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⢁⡴⠋⢀⡼⠁⢠⡟⠀⢸⡇⠀⢻⡀⠈⢦⡀⠑⢄⠈⠳⢦⡀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣷⣶⣿⣷⣶⣿⣷⣶⣿⣷⣶⣾⣷⣶⣾⣷⣶⣾⣷⣶⣾⣿⣶⣾⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⡀⠀⣄⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⣹⣿⡿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡏⠹⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢷⡶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢶⣶⣶⣶⣶⣥⠙⠋⡿⣱⠶⣴⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠐⢻⣿⣿⣷⡃⠬⠐⠅⠩⠞⣻⣿⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⣿⣿⣾⡭⠀⢉⣑⣛⣩⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣋⠻⠞⣁⢠⡖⠠⣍⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣌⠀⠀⢿⡆⣥⣀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠏⠐⠒⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⠁⠠⠖⣻⡗⠃⢬⣦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣇⠀⠀⠀⠀⠉⠀⠀⠠⣼⡯⠀⠀⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢩⡄
⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⠦⠂⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠳⡀⠀⢰⣿⣿⣶⡿⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⡚⢿⣿⣿⠷⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢾⡏⢳⣽⡆⢸⣷⣌⠂
⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⢻⡿⠈⠻⠋⠀⠀⠘⠃⠘⡀
⠀⠄⠠⠌⣽⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⣴⡆⠀⠀⠀⠀⢠⣾⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢸⠆⠀⠀⠀⠀⠈⢿⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⡄⠀⠀⠀⠀⠀⠀⠈⣿⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡘⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠴⠶⠶⠶⠶⠚⠞⠻⠒⠒⠒⠛⠓⠒⠒⠒⠛⠿⠿⠿⠶⠿⠂

⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠒⠶⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠛⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⢰⣲⡄⡴⠲⡄⣦⠀⢰⢠⠔⠂⣴⣒⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠘⠁⠀⠓⠒⠃⠛⠒⠘⠈⠓⠂⠛⠒⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠉⣉⣉⡉⠉⢉⣉⣉⠉⠉⣉⣉⡉⠉⢉⣉⣉⠉⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣟⠉⠉⣹⢸⡏⠉⠉⡇⣾⠉⠉⢹⢰⡏⠉⠉⣷⠀⢸
⠀⠀⡴⠒⠒⠒⠒⠒⠒⢺⡇⠀⠉⠉⠉⠉⠀⠉⠉⠉⠁⠈⠉⠉⠉⠀⠉⠉⠉⠁⠀⢸⡤⠤⠤⠤⠤⠤⢤⡄
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⡴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⢀⣧⠴⢶⣶⣾⣿⣿⣶⣶⣶⣶⠿⣤⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⢀⣠⠼⢛⣩⠴⠚⠉⠉⠉⠉⣿⠉⠁⠉⠉⠙⠲⣍⡳⣄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⣰⠖⠒⠋⠁⠀⠉⠛⠲⠦⠤⠤⠤⠤⠿⠤⠤⠤⠤⠤⠤⢿⣹⡏⠙⠛⠒⠾⠦⠤⣤⣄⡀⠀⠀⣿
⠀⠀⡇⠀⡏⠀⠀⠀⣀⣀⣀⠀⠀⢠⣢⢠⠔⢤⢠⠀⣤⣠⠔⢠⣄⠀⢸⠀⠀⠀⣀⣀⣀⠀⠰⣧⣽⡇⠀⣿
⣀⣀⣇⣀⡇⠀⢀⡞⣩⠤⣍⢳⡄⠘⠀⠈⠓⠋⠘⠂⠉⠈⠒⠈⠓⠀⢸⠀⢠⡞⣡⠤⡍⢷⠀⠀⢀⣇⣀⣿⣀
⠀⠀⠀⠀⠛⠛⠛⣇⠻⠤⠟⣸⠗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⠺⣇⠳⠴⠏⣼⠒⠒⠋
⣤⣤⣤⣤⣤⣤⣤⣬⣿⣶⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣶⣯⣥⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡧⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⢦⡩⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠿⡀⠈⠛⠀⠀⠖⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣟⢲⣮⣇⠀⠀⠙⣫⣿⣾⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡓⢽⡫⣧⠀⠀⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⢿⣿⡧⣼⣿⣬⡺⡿⡟⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣩⣯⢹⣿⣿⣿⠿⣴⣧⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣷⣾⢿⣿⣿⣿⣽⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣬⡿⣟⣷⣿⣿⣛⣹⡟⣻⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⡟⠻⠟⢿⣿⠿⣣⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠸⣻⣿⣟⡟⠙⢻⡁⠠⠀⠀⢹⢆⣽⣽⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣿⣻⣤⣀⣸⠲⠿⠄⢳⣦⠘⣿⣭⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣿⣾⣤⣶⣩⣄⠼⠃⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠛⡿⢿⣿⣿⠿⣿⡙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠠⠤⠄⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣼⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⣿⡀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⣿⣇⠸⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⠀⢿⣿⠤⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣿⣿⠇⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣿⡀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠘⡿⢤⣴⣶⣿⣿⣿⣿⣿⣷⣶⡿⠖⠳⣴⣲⣿⣿⣿⣿⣿⣷⣦⣤⢼⠁⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⢿⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⡆⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⠄⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢸⡅⢿⣿⣿⣿⣿⣿⠿⠛⠙⣱⡆⣶⠀⡘⠻⢿⣿⣿⣿⣿⣿⠇⡀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠹⠶⠯⠍⠉⠀⠒⠒⢡⢦⣿⣿⣿⣇⠙⠶⠤⠈⠉⠉⠉⠐⠊⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⡀⠀⠀⠀⠀⠀⠒⠀⠻⢺⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣼⣦⣶⣤⢤⡀⠀⠀⠀⠈⢿⡟⢿⠟⠀⠀⠀⠀⣠⣴⣶⣾⣦⣤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠿⣿⣏⠈⡧⠀⠀⠀⠀⠐⠆⠐⠀⠀⠀⠀⠰⡏⢀⡿⠿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢻⣆⠻⠶⠤⢤⣤⡤⣄⡠⣤⣤⣤⡤⠞⣃⣾⠁⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢻⣟⠲⢦⣠⢀⡀⢈⠀⣀⣠⢠⠶⢎⣸⠇⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠃⠐⠢⢅⢁⣉⢹⠙⣉⡀⠄⠒⠁⠋⠊⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣦⡀⠀⠀⠀⠀⠐⠢⠤⣤⡤⠄⠂⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣀⣀⣀⣀⣀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠉


⠀⠁⠀⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢹⣿⣿⣿⠿⠿⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⡻⣿⣿⣆⢴⣶⣶⣶⠄⠀⣴⣶⣶⣦⠀⠠⣶⣶⣶⣦⠀⠀⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⣫⣾⠙⣿⣿⣎⢿⣿⡟⠀⢠⣿⣿⣿⣿⡄⠀⢻⣿⣿⣿⣇⠀⠘⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣺⣿⣛⣽⣿⠃⠀⣨⣻⣿⣺⣿⠃⠀⣸⣿⣿⣿⣿⣧⠀⠘⣿⣿⣿⣿⣆⠀⠘⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠃⠀⢰⣿⣿⣿⣿⡏⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⢹⣿⣿⣿⣿⡆⠀⠘⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠃⠀⢠⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠈⣿⣿⣿⣿⣿⡄⠀⠘⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣦⣤⡤⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⣀⠴⢺⠀⠀⠀⠀⢸⡟⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢰⠊⠁⠀⢸⠀⠀⠀⠀⢸⡇⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⢸⢧⡀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠖⠁⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠙⠲⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠙⢷⣄⡀⠀⠀⠈⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⣠⠃⠙⢦⡀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⣰⠃⠀⠀⠀⠈⢢⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⡰⠃⠀⠀⠀⠀⠀⠘⡆⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⢤⡄⠀⠀⡜⠁⠀⠀⡴⠁⢠⠤⡤⡄⠀⠀⠀⢳⠀⠀⠀⢧⢠⡤⡤⢤⡀
⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⡸⠁⠀⢸⠀⠀⠀⡼⠁⠀⡏⠀⠀⢻⠀⠀⠀⠈⡇⠀⠀⢸⡌⢆⠀⠀⠘⢦
⠀⠀⠀⠀⣠⠞⠀⠀⠀⣰⠃⠀⠀⠈⠓⠒⠚⠀⠀⢠⠇⠀⠀⠸⡄⠀⠀⠀⠘⠢⠴⠊⠀⠘⣆⠀⠀⠀⠱⡄
⠀⠀⢀⡴⠁⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠈⢦⡀
⠀⠀⠚⠒⠒⠒⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠠⣀⠂⠒⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢀⣀⡀⠄⠀⠒⠲⠄⠂⠚⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣀⢨⠀⠀⠀⠀⠐⡀⠀⠀⠸⣄⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠩⠀⠀⠀⠀⡀⢀⣀⠠⠤⢄⡀
⠀⠀⠀⠀⠀⠨⣀⣀⠀⠀⠀⢰⡆⣨⠄⠀⠀⢸⠋⠳⠀⠀⠀⠈⠻⠯⠉⠁⡠⠄⢠⠀⠀⢀⡀⠀⠀⠀⠀⠀⠂⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠈⠃⠫⡄⠀⠀⣀⣀⡂⡋⣃⠈⠛⠚⡄⣸⡖⠙⠋⠁⠮⠀⣐⣒⠛⡆⢀⣀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠄⠀⠀⢐⠏⠁⠀⠁⠀⠈⢻⣌⣀⣠⣗⡓⠧⠊⠉⡀⠒⠦⡠⠜⢠⠀⡏⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠰⠓⠂⣠⣞⣉⣁⣈⡏⣈⠀⠤⡄⣠⠀⢨⠀⡌⡃⡤⢠⠀⠠⠌⠀⡎⡆⡥⠄⠖⠂⠓⠒⣆⣀⡀
⠀⠀⠀⢀⠄⠲⠤⠎⠤⠤⢠⠤⠤⠆⠲⠀⣒⡃⣓⣀⠘⠀⠁⠓⠉⠈⠑⢻⡀⠀⠁⠁⣒⡁⠀⠀⠀⠀⢠⠀⠹⡄
⠀⠀⠀⠘⡀⠀⠸⢸⠈⡆⡀⡇⢸⠠⡁⣇⣀⠆⣇⡘⠀⠀⠀⠀⠀⠀⠀⠈⢥⠀⣀⡀⠢⠬⡁⠀⣂⢀⠈⣀⠁⡗
⠀⠀⢀⠀⠀⠀⠀⡀⠒⡃⠓⠃⢸⣧⡆⡆⠖⢰⢲⠀⡗⡒⢒⠒⣒⢒⡒⢲⢺⠎⠀⠈⠙⢄⠈⢽⢰⡲⣖⢾⠀⡇
⠀⠀⠈⠢⠤⡀⠀⢃⢣⢸⢬⢨⠀⠏⠣⢃⡒⣘⠊⡀⠫⠼⢁⠃⡠⠤⠅⠘⠸⢥⠀⠀⠀⣐⠆⣜⠀⠄⠓⠚⠤⢅⡀
⠀⠀⠀⠀⠀⠉⠖⠘⡘⠒⣘⠒⠀⢀⠣⡀⡂⠑⠀⠁⢠⢲⣎⡳⣄⡦⣳⢸⣛⠈⣟⠚⠉⠛⠪⡠⠤⠤⠤⠤⠤⠠⠘
⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠈⠀⠀⡆⠨⠀⠍⠁⠀⠀⠀⠀⡀⢤⠀⡀⠀⡇⠀⠀⠘⠄⠀⢀⡀⢯⠁⠀⣐⣒⣂⡀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⣄⠀⣀⣀⠀⠀⠀⠂⠀⠰⠆⠙⠀⠀⢫⣏⠉⠹⢁⢀⣀⠑⠂⠁⠀⠲⣀⣀⠀⠀⠀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⣀⣤⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠈⣿⠀⢀⣼⣧⣤⣤⡀⠀⠀⢸⡇⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣰⠇⣤⠈⡿⠀⣀⠀⠀⣀⠀⢸⡇⠀⣤⠀⢸⡷⠶⠖⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠉⢰⡇⣸⠇⢰⡏⠀⢀⣿⡆⢸⡇⠀⣿⠀⢸⡇⣠⣾⣭⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⢠⡟⠿⢠⡟⠀⠈⠛⠛⠋⢸⡇⢸⡇⠀⠿⠀⠘⠃⣿⠀⠀⠀⣀⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣰⠏⠀⠀⣾⠁⣀⡴⠞⠃⢀⡼⠃⢸⣧⡴⠶⠟⠛⠉⠙⠷⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠖⠛⠁⠀⠀⠀⠻⠚⠉⢷⣤⠶⠋⠀⠀⠘⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠾⠛⠛⢛⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⢰⣿⣯⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⢾⣶⣷⡾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⣻⣿⠉⠉⢁⣼⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣏⢉⣠⣿⣿⣿⡟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⣛⣏⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠠⣶⣿⣿⠄⠀⠀⢀⣀⣀⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠙⣿⣤⣯⣀⣀⠒⣶⡶⠞⠉⠀⠁⣤⠸⠷⣿⣅⣀⣁⣤⠙⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠻⣿⢿⣿⣙⢿⠆⢀⣤⠆⠀⣧⠀⣀⠀⢹⣿⣿⣿⠿⠷⢤⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣍⣻⠿⣦⣤⠀⠂⠀⢩⠀⠩⣶⠂⣉⡿⢿⠇⠀⠀⢨⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠾⠿⢿⣦⣤⠤⠀⠆⠀⢶⣟⠁⠶⠿⣛⣻⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠿⣦⣬⣤⠀⠀⢠⡀⢿⣯⣀⣤⣾⣿⣿⣾⡇⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠶⠮⢏⡁⠤⣀⢿⣿⣧⠉⡛⠠⣼⡏⠙⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢳⣤⣴⠋⠁⠐⠁⠀⢛⣿⣏⣅⠀⣨⡿⠛⣿⡌⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣤⠜⠉⣀⠴⠂⠠⣟⣿⠄⢈⠉⠀⠰⠛⣿⡄⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠚⢁⣴⠟⠁⠐⠂⠀⣿⣿⡗⠈⠙⠃⠀⠛⠛⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⣽⣍⠀⠀⠀⠀⠀⠀⢰⡄⠀⠈⠻⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⠾⣿⠿⢷⣄⣸⣏⣙⣿⡶⠛⠀⠀⠀⢠⡞⠁⠀⠀⠀⠈⢿⣿⡆⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠺⢿⣷⣄⠀⠀⣉⣽⣿⣿⣷⣶⣤⠀⠀⣄⠀⠀⠀⠀⠀⠀⠈⠋⣵⡆⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠛⣿⠶⠛⠛⢻⡟⠛⠛⠁⠀⠀⡀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠁⢼⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠸⠿⢀⠀⠀⠀⠀⠉⠁⠀⠀⣄⡑⣶⣿⠷⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⢠⣤⣾⣶⡄⠠⡄⠀⣴⡄⠀⠙⠿⠀⠈⠀⣾⡟⠀⠀⠀⠀⠀⢡⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⠀⠀⠀⠀⠀⣿⠁⠀⠀⠁⠀⠀⠀⠈⠻⢷⣴⠆⠀⠰⣶⣿⠀⠀⠀⠀⠀⠀⠀⢳⣄⣿⡆⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢿⡀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣄⠀⠀⠀⠰⠋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠛⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⣙⡿⡉⠻⣦⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠃⠀⢸⡇⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣉⠀⣽⣄⠈⢷⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⠟⠃⠸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢹⡟⣿⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣿⣄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣼⣿⣿⠆⠀⠀⠀⠀⠀⠀⠟⠛⠛⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⢿⡧⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡶⣶⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⢀⣌⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢿⡆⠀⠀⠀⠀⠀⠸⣿⣦⣿⡇⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣾⣿⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣧⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠠⣿⣿⠻⣿⠿⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⡿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡗⠐⠶⢶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⠀⢢⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⡆⠈⠀⢻⣿⡿⠿⠿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣻⡟⠋⠉⣀⣼⡗⠤⠐⣿⣿⣷⣤⡀⢈⣿⣿⣧⡀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣾⣿⣿⡛⢀⣀⣀⣬⣥⣭⣿⣿⣿⣿⣿⣯⣤⣄⢹⣿⣿⣍⣀⣀⣀⣉⣉⣽⣴⣷⠾⠿⠿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢻⣷⣦⠜⣿⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿⡿⠄⢠⣿⣿⣿⣿⣿⣿⣫⣿⣿⣿⣿⣶⣶⢟⣉⣤⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⢿⡿⢹⣿⣿⡿⠿⢿⣿⠟⠿⠿⠿⢿⣿⡇⣀⠀⢙⡿⠟⠛⠛⢶⣦⣿⣿⣟⣻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡰⢟⣧⠈⣿⣿⡿⢶⣶⡄⠉⢉⣶⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣾⣷⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣝⠛⠻⠾⣿⣿⣿⢿⣿⣿⣿⣿⣿⠙⠛⢿⣿⣿⡛⣩⣽⢶⣿⣿⣿⠿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⡿⠷⣤⣄⠀⠀⢶⣿⠯⢸⣿⣿⣁⣀⢨⣿⣟⣛⡻⠿⠛⢿⣿⣟⣷⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠻⣿⣦⣤⣾⣿⣾⣷⣿⣿⣿⣿⣿⣿⠉⠉⢈⣿⣿⣿⠓⠂⠀⠻⢿⣏⣻⣿⠋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣶⣤⠀⡉⢙⠋⣯⣽⣿⠿⢷⣶⣿⣿⣷⡄⢸⣿⠿⠛⢷⣾⣿⣷⣶⣿⡋⠀⢠⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠻⢏⣡⣤⣀⣼⣥⣭⣤⣦⡴⠛⢿⣿⣿⠟⠀⠙⣿⣿⣧⣤⣬⣉⡉⢿⣠⣠⣾⣿⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⣰⣄⡉⠉⠛⠛⣿⣋⣭⡀⠰⣶⣿⣿⣿⠗⠀⠀⣯⡵⠻⣿⢻⣉⣽⣿⣿⣿⣿⣇⡀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠟⠉⠁⠀⣓⣟⣩⣤⣴⣾⡛⣿⣿⣯⠀⠀⢸⣿⣿⣶⣤⣙⣛⡻⠟⠛⠉⠙⢩⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⡀⠉⠲⠶⣿⣿⣿⣿⠟⠁⣴⣿⢿⣿⣿⣶⣦⣾⣿⣿⣯⡉⠻⠗⣬⠑⣤⣤⣾⣿⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣃⣸⣷⣦⢠⣾⣿⣟⣩⣤⣶⣟⣁⣾⣿⡏⠀⣿⣿⣿⣿⣿⣉⠁⠂⢠⣽⣿⣿⣿⠿⣦⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡟⠻⠏⠀⢀⣴⣿⣿⠿⣋⣭⣿⠟⢹⣿⣿⣤⣿⣿⣇⠹⣿⣿⣧⣄⣀⠉⠋⡀⢀⣼⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠦⠀⠀⢹⣿⣧⠜⠒⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⠿⣿⣿⣯⣾⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢀⣾⠿⠊⠉⠀⢤⢠⣿⠿⠋⠁⢠⣿⠛⢁⣿⣿⣇⠀⠙⢿⣿⣿⣷⣦⡀⠙⠛⠛⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣀⡀⢠⣤⣾⠛⠁⢀⣤⣴⣿⡏⠀⢈⣹⣿⣿⣷⣦⠀⠈⠻⣿⣏⡿⣶⣤⣴⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣉⢹⣽⣿⡿⣡⣴⣶⠟⠋⠉⢈⡿⠃⠙⣿⣿⣏⡀⠀⠀⠀⠀⠘⢿⣿⣿⠿⡾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠾⠻⣿⡿⠟⠉⠀⠀⠀⠸⢿⣇⣀⣀⡙⣿⡿⠋⠀⠀⠙⠀⢀⣆⡙⢿⣾⣟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡟⠉⠀⠀⠀⠀⠀⠀⣀⣼⠛⠋⠻⣿⣿⣧⣄⠀⠀⠀⠀⠀⠹⠇⠀⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣤⣀⡀⠉⢨⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⣤⣼⠏⠉⠁⣶⣾⣿⣤⡀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⣿⣿⠿⡠⢄⣀⠀⠉⣷⣀⠀⠀⡀⠈⣿⠏⠁⣀⣤⣶⡟⠋⠁⠀⠐⢺⣴⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⢿⠉⠈⠉⢰⣷⠈⢻⣷⣶⣿⠛⠛⢶⣷⣿⣿⣿⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⢸⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⢹⣥⣾⣿⣿⡏⠀⠀⠀⠙⢻⣿⣷⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢽⡾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣄⠀⠀⠀⠀⢈⣿⣿⡇⠉⠻⠶⠶⢶⣶⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⠆⠀⢀⣰⣿⣿⣿⣿⣶⣤⡀⠀⠀⢀⣉⣤⣿⣏⣿⣿⣆⣀⠀⠀⠀⠀⠀⣴⡟⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠟⠁⠀⠘⣿⣿⡏⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡟⣻⣿⣿⣿⣿⣿⣿⠃⠀⢀⡱⣤⡉⣩⣿⣿⣿⣿⣿⡏⠀⠈⣿⢄⣄⣤⣦⠈⢻⣷⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⢿⡿⢿⣿⣿⣧⡀⠀⢤⠀⠀⠀⣹⡷⣍⣩⡿⠀⠀⣴⠟⠀⠙⠁⢻⣶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢳⠀⠀⠀⠉⡟⢻⣷⣦⣀⠙⢦⣴⣿⣿⣿⠋⠀⣠⠾⠋⠀⠀⠀⢀⡿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢹⣾⡟⠙⠛⢷⣤⣿⣿⣿⣿⣤⠞⠉⢀⡀⠀⣠⡴⣀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⠛⠀⠀⣠⣉⣙⣿⣿⣋⣁⣤⡴⠞⠿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠛⢿⣿⡄⠀⢀⡰⠋⠙⠛⣿⣿⣿⣿⣧⠀⠀⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠙⠿⣶⡞⠁⠀⠴⠟⠛⠛⢿⡉⠛⠳⡶⠛⠉⠁⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⢀⣤⣤⡤⠈⠀⠀⠀⠀⠀⠻⠶⠀⠀⠀⣠⣿⣁⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⠉⠁⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠻⢿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇⢸⣶⣤⣉⠛⠿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠛⠀⣾⣿⣿⣿⣿⣦⠀⠉⠛⠿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⢀⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠋⣠⣾⣇
⠀⠀⠀⠀⠀⣴⣶⣶⣶⡆⠀⢀⣴⣶⣶⠄⣰⣿⣿⣿⠟⠁⠸⣿⣿⣿⡆⢰⣶⡄⠀⠀⣶⣶⣶⣶⡄
⠀⠀⠀⠀⣼⣿⣿⣿⡟⠀⠀⣸⣿⣿⡇⣴⣿⣿⡿⠃⣰⣿⣆⠘⢿⣿⣿⡀⢿⣿⡀⠀⠘⣿⣿⣿⣿⡄
⠀⠀⢀⣾⣿⣿⣿⡿⠁⠀⢠⣿⣿⣿⣇⠻⠿⠋⢠⣾⣿⣿⣿⡄⠈⠻⠿⠇⢸⣿⣧⠀⠀⠹⣿⣿⣿⣿⣆
⠀⢀⣾⣿⣿⣿⣿⠃⠀⠀⣼⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⣶⣶⣿⣿⣿⡄⠀⠀⢻⣿⣿⣿⣿⣆
⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠈⠈⠀⠈⠉⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡽⣁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣰⠃⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠈⡇⠀⠅⠀⠀⠀⠀⠾⡇⠀⠀⠀⠀⠀⠀⠤⠄⠀⠐⡒⠀⢠⠄⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣾⠃⠀⣟⢹⠉⠉⠁⢳⡆⠀⠁⠀⢀⠀⠐⢶⢲⣀⢠⣠⠤⡴⠤⠀⢁⣀⣀⢠⡞
⣀⣤⡀⢠⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⢱⠛⡶⣿⣿⣿⠶⠄⠀⢨⡇⠀⠀⠀⠬⠀⡄⠀⣨⠄⠀⠁⠈⠀⠀⠀⠀⠀⠸⠉⠉
⠈⢷⠔⢸⠉⡆⠀⢴⠀⢶⡅⠀⠀⠀⠘⠀⣿⣿⡟⣿⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⣀⣀⣀⣀
⢘⠙⠁⢸⠀⡇⠀⠸⢀⠋⢑⠀⠈⣧⣄⠠⣿⠀⡇⢸⠀⠀⠀⢈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⢹⡁⢠⢿
⠀⠈⡀⣸⠀⠇⠤⢀⠸⠇⠀⠀⠘⣿⠌⠀⢄⠀⣇⢸⠀⠀⠀⠠⣿⠀⢢⣀⠀⠀⠀⠠⠀⢸⡆⢰⠸⠿⠷⠾
⣼⣭⣵⡞⣧⣆⣗⣢⠀⠀⢰⣧⡀⡿⡶⠞⠦⠠⣌⣸⠂⢀⣣⡼⣿⠁⢩⠘⠀⠀⠀⠀⠀⢘⢤⢘⣛⠃⠀⠐⠀⠀⠀⠀⡀
⠭⠀⠤⠒⣘⢛⠛⠻⠀⠀⣄⢾⢦⠄⠆⠀⠀⠈⡄⣘⡂⠀⠀⠀⢘⠟⣿⠢⣤⠀⣓⣢⠔⠉⠀⢨⣀⢀⡀⠀⠀⢰⠀⡇⡇
⣿⣼⣿⡇⣿⢸⣿⣷⣴⣶⣼⣿⣿⣷⡇⡇⢀⣿⣝⣿⡃⣂⡀⣔⣛⣃⡻⠒⡶⠀⣄⣼⠿⢠⣶⣆⠼⣬⡏⣀⠀⢸⠀⡇⡇
⣯⡜⣩⣥⣭⣿⠿⠿⠿⣿⠭⣽⠿⠿⡿⠿⠿⣿⠿⣛⢟⠿⠿⢯⣿⠿⠿⣿⡟⢆⣴⣶⣮⣷⣶⣽⣿⠛⠛⢸⢏⡸⠒⣧⣷
⠀⣠⡁⠁⠁⠁⡠⢄⡤⠷⠶⠞⠛⠛⣴⣦⣆⣴⣴⣭⠈⠀⠉⠉⠙⠛⠛⠛⠉⠈⠉⠉⠉⠉⠉⠉⠙⠉⣿⣥⠿⠿⠆⠣⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠛⠛⠛⠛⢡⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣴⣶⠀⠀⣶⣦⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠁⢀⡀⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠺⠶⠶⠆⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⠄⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠐⠃⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣶⣾⣯⣤⣿⣧⣼⣿⣤⣭⣶⣶⣤⣤⣤⣤⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⣤⣤⣤⣤⣼⣭⣼⣿⠭⣿⠉⠉⣿⠭⣽⣧⣼⣧⣤⣤⣤⣤⣼⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣻⣿⣿⠀⢠⣿⢿⣿⣦⠤⠤⣿⣿⡤⠤⣤⣿⠿⣿⣄⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠹⣶⣟⠻⣶⣶⠦⠴⣶⠀⠀⣶⠦⠴⣷⣶⠟⣻⣶⠏⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⣟⣻⣶⣼⣅⡀⠰⣿⠀⠀⣿⠆⢀⣨⣧⣾⣟⣿⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡿⢿⣿⣁⣤⣤⠲⢿⣄⣠⣿⠆⢈⣀⣈⣿⡟⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠘⠛⣦⣤⣈⣉⣡⣶⠟⣿⣿⣿⠗⢠⣤⣤⣤⣤⠟⢃⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⠘⠿⣤⠀⠈⣿⣿⠅⠀⢿⣯⠙⠷⣼⣯⣉⣁⣀⡿⠇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢰⣦⣀⣴⡿⠏⠀⠠⠀⣾⣷⠀⠤⠘⣿⣿⣿⣟⣷⡆⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠃⠀⢠⣾⠇⠘⠇⢀⡁⠤⠀⣿⣿⠀⠤⠌⡙⢿⣿⠙⢷⡎⠀⠸⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠛⠀⣿⠀⠀⠱⠶⠉⠀⠉⠁⠀⠀⣾⣿⠀⠀⠀⠑⠆⠙⠷⠀⠀⠀⣶⠀⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢣⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡜⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣿⠃⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⢠⣿⣿⠄⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠘⣿⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢠⡌⢠⡌⠀⠀⠀⢀⡼⠋⠀⠀⠙⢣⣤⣤⡿⢿⣤⣤⣤⡞⠁⠀⠀⠛⠀⠀⠀⠀⢡⡄⢡⡄
⠀⠀⠀⠀⠀⠀⠀⣹⠃⣿⠇⠀⠀⠀⠀⢧⣀⠀⠀⢀⡸⠿⣉⠱⠾⠋⣠⢿⡇⠀⠀⢀⣰⠃⠀⠀⠀⠸⣿⠘⣿
⠀⠀⠀⠀⠀⠀⣸⡇⢰⡏⠀⠀⠀⠀⠀⠀⠙⢦⡀⠸⣇⠀⠿⣀⠀⣼⠁⢀⡅⠀⣠⠞⠁⠀⠀⠀⠀⠀⢹⡆⢸⡇
⠀⠀⠀⠀⠀⢠⣿⠁⢺⠃⠀⠀⠀⠀⢀⣄⠀⣨⣿⣄⣼⡇⠀⠻⣷⠉⠀⣨⣧⣶⣯⡀⢀⡀⠀⠀⠀⠀⠘⢇⠈⣷⡄
⠀⠀⠀⠀⢀⣼⣿⣾⡇⠀⠀⠀⠀⠀⢈⠙⠛⠹⠟⢿⡁⣿⣦⣄⣀⣤⣶⣭⠉⠛⢯⡕⠋⣿⠀⠀⠀⠀⠀⢸⣷⣽⣧⡄
⠀⠀⠀⢀⣴⣿⣿⡏⠀⠀⠀⠀⠀⠀⠸⡇⠀⢇⠀⠈⠠⠤⡍⠛⠛⠉⢥⠌⠃⠀⢸⡇⢀⠛⠀⠀⠀⠀⠀⠀⢹⣿⣿⣦⡀
⠀⠀⣤⠘⠻⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠁⣠⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣸⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⠿⠛⣤
⠀⣴⠙⢿⣷⡀⢿⡆⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠻⠀⠀⠀⠀⠀⠀⠀⢠⡿⢂⣼⡿⠋⣤
⠂⠈⢳⡄⢹⡇⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⡏⢠⡞⢁
⠀⠳⡀⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⢀⠟
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⠇⠀⠀⠀⠀⠀⠀⢠⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⡀⠀⠀⠀⠀⠀⢸⡇⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣤⣘⣧⠀⠀⠀⠀⣀⢷⣤⣤⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠉⢙⡿⣿⠆⠀⠀⠀⢻⣿⡏⢙⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠿⠿⠿⣶⠀⠀⠀⠀⢠⣿⠿⠿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⢴⠆⠀⠀⠀⠀⠀⢱⡄⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢱⣆⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⠀⠀⠀⠀⠀⢀⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⣠⠀⠀⠀⠀⢘⣇⢘⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⠀⠀⠀⠀⢸⣿⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣅⠀⠀⠀⠀⢀⣽⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯⡟⣿⠟⠀⠀⠀⠀⠘⢻⣿⢿⣵⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣧⡿⠃⠀⠀⠀⠀⠀⠀⠀⠙⢟⣽⣿⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠙⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⠿⠟⠻⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⢠⡖⠋⠉⢠⡶⡆⠉⠙⠳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠉⢹⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠈⠻⣤⣤⣬⣭⣥⣤⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣛⣛⣛⣛⣛⣛⠋
⠀⠀⠀⠀⠀⠀⢀⡤⣿⠲⠶⠶⠶⠖⢻⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣭⣿⣛⣛⣯⡥⣠⡀
⠀⠀⠀⠀⠀⠀⠈⠓⢿⠀⠀⠀⠀⠀⣼⠚⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣿⣿⣿⣿⣿⡇⠉⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣈⡷⣤⣀⣠⣼⣁⣰⠶⣴⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡻⠿⠿⣛⡀⢀⣶⣤⡞
⠀⠀⠀⠀⢀⡤⠖⠛⣧⣀⡴⠛⢧⣀⣼⠛⠳⢾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦⡙⢿⠟⣵⣜⢿⠟⣡⣮⣝⡁
⠀⠀⠀⢠⡏⠀⠀⠀⢸⡉⠁⠀⠀⠉⡟⢀⡶⠒⠛⠲⣦⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⣷⣾⣿⣿⣷⡇⡿⢏⣭⣭⣤⣄
⠀⠀⠀⢸⡇⠀⣿⠀⠘⡇⣸⠋⢷⢰⡇⢸⠀⠀⠀⣤⢸⠀⠀⠀⠀⠀⠀⣿⣿⡟⣿⣿⢸⡟⣴⡎⣿⢹⡇⣿⣿⣿⡿⣿⡇
⠀⠀⠀⢸⡇⠀⣿⠀⠀⣷⡏⠀⠸⣿⠀⢸⠀⠳⡾⠁⢸⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡜⢳⣿⣿⠘⣼⡇⣿⣯⡛⣱⣿⡇
⠀⠀⠀⠘⠗⠒⠛⠒⠒⠛⠒⠒⠒⠛⠖⢾⡀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠛⠛⠓⠛⠛⠃⠚⠛⠛⠂⠛⠃⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡽⢯⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠋⠀⢀⣤⣤⡄⠀⠀⠀⠀⠀⠙⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠀⠀⠀⢺⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠉⣩⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⡀
⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠛⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡀
⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⢿⣿⣿⣿⣿⡇⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡀
⣔⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠛⠋⠉⠀⠘⣿⣿⣿⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿
⠫⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾
⠀⠈⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁
⠀⠀⠀⠈⠛⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⠀⠹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠈⠛⣷⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠛⣷⣄⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠻⠿⠇⠀⠀⠀⠀⠀⠈⣿⠿⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣳⣞⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠢⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠒⠀⠉⡉⠉⠉⢉⠉⢀⠐⠀⡀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠐⡆⠀⠀⠀⠀⠀⠂
⠀⠆⡄⠀⡀⡀⠀⠼⣀⢀⢠⠤⠅⠀⠤⠀⠆⠀⠠⠀⠄⠀⠀⠀⠀⠀⠃⠀⠀⡇⡏⡉⠉⡉⡉⢉⠉⡉⣉⢙⣄
⠈⡁⡀⠠⣄⠈⠀⠀⠤⠤⣸⠀⡀⣀⣀⣰⣀⣰⡀⡆⣠⡀⣀⣀⣀⣩⣀⣀⣀⣰⡀⢀⣀⡇⡀⠀⠂⠀⢀⣿
⢀⠁⠀⢀⠈⠀⠀⠴⣠⡠⢝⢠⠀⠄⢤⠀⠄⠀⠄⠠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠄⠤⢄⠀⠀⠀⡀⠐
⠀⠁⠀⠀⠂⡖⢰⡄⠀⠐⢸⠈⠀⠆⠼⠀⠀⠀⠀⣀⠀⠠⠠⠀⠀⠀⠀⠀⠀⢀⠃⠚⠓⠀⠀⠘⠀⠤⠄⡇⠄
⠀⠅⠀⠀⠂⠐⠐⠀⠀⠚⢸⠚⠀⡀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠒⠀⢰⠶⣦⢸⠀⠁⠉⠤⠀⢠⠊⠈⡃⡇
⠀⠀⠈⠀⢸⠎⠀⠀⠀⢈⢸⠀⡇⠃⠐⠀⠢⠤⠄⡄⠂⠒⠂⠀⠀⠊⠘⠒⠚⡿⠀⠈⣈⠀⡈⡁⢀⠀⢀⡇
⠀⠀⠀⢈⣉⡈⠉⡀⠀⢀⠀⡀⠀⠊⠉⠀⡇⠉⡽⠁⡆⠀⠂⠀⠉⠉⠂⠉⢢⡟⠀⠄⠜⠀⠄⠆⠠⣀⠜⡇
⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠃⠄⠀⠀⡄⠀⣠⠀⣩⡅⠁⠀⠀⣀⠀⠈⠁⡇⠐⠀⠀⠀⠀⠁⠀⠀⠱⡇
⠀⢀⠡⠀⠀⠄⠀⠠⠄⠀⠠⠤⡄⠀⠀⠀⠁⠀⠚⡀⣅⣀⣀⣠⣤⠜⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠢⠈⠀⠀⠀⡀⠀⠀⡄⠠⠀⠀⣾⠒⡖⠢⡀⢀⣀⡀⠀⠄⢶⣂⢴⠂⣡⡀⠀⣀⠀⠀⠀⠀⠀⠀⢲⠠⡀⡀
⠀⠤⠀⣠⠊⠉⠁⠆⠙⠒⢰⠒⠚⠁⠀⢈⡵⠉⣶⠉⡲⣈⣉⢸⠀⠜⠰⠼⡄⠘⠀⠀⠀⠀⠠⠀⢸⣠⠜⠙⠏
⠒⠄⠀⠚⠄⠀⢠⡄⠀⠀⡼⠀⡄⠜⡇⢰⠐⠀⠀⠀⢀⠁⢐⠈⠙⢇⡆⠀⠠⡠⡘⠉⠟⠀⣷⣆⡸⢫⠀⠀⣦
⠀⠱⠒⠠⠉⠀⠀⠀⠀⠀⠈⠂⠉⠀⠀⠚⠑⠀⠁⠂⠁⡑⠀⠄⠴⢁⠔⠂⠒⠀⠠⠀⢢⠅⣸⠃⠀⠀⠂⢸⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⡙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠁⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠔⠁⠀⠒⠀⠀⠀⢈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⢠⠂⠀⠀⠀⠀⠀⠈⢦⠁⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠄⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⣼⠇⠀⠀⠀⠔⠳⣶⡄⠀⠀⠀⢻⡄⠂⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢀⣾⠏⠀⠀⠀⠐⢶⣾⣿⠃⠀⠀⠀⠀⠹⣄⠂⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣠⣿⠋⠀⠀⠀⠀⢀⣴⣶⣾⣿⣷⣄⠀⠀⠀⠙⣦⠁⡈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢁⡤⣲⠳⠁⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⡿⢿⣧⠀⠀⠀⠘⣧⠁⡌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⡠⠋⡴⠁⠀⠀⠀⠀⠀⡾⠟⠉⢸⣿⣿⣿⡧⠘⣿⠀⠀⠀⠀⠘⣷⡀⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠊⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣟⠀⠉⠀⠀⠀⠀⠀⠈⣷⡀⠀⢻⣄
⠀⠀⠀⠀⠀⢀⡾⢁⡠⢪⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⢻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠈⠿⡄⠀⠹⣆
⠀⠀⠀⠀⢠⠞⡤⠋⡴⠃⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⡀⣼⣿⠋⠀⣀⣙⢿⣧⡀⣀⣀⣀⠀⠀⠀⠀⠈⣆⠀⠹⣧
⠀⠀⠀⣰⠏⡨⠂⡜⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣸⣿⣿⠀⠐⣿⣿⣿⡿⣷⣾⡏⠙⢧⠀⠀⠀⠀⠈⢶⠁⠘⣧
⠀⠀⣰⠃⠈⢡⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠐⣿⣿⣿⣿⠀⠀⡟⠁⠀⠁⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠘⣧
⠀⣼⠃⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⢧
⣼⠋⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈
⣿⢠⠂⠈⠧⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠏⢠⠀⠄
⠘⢦⣀⡀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠊⠁⠈⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⡴
⠀⠀⠈⠉⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠖⠒⠲⠶⠶⢶⣶⣶⣶⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⡏⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣇⣿⣿⣿