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⠀⠠⠀⠀⠢⠀⠀⠀⠀⠀⢠⢊⠩⡍⠆
⠰⠔⠁⠸⢠⠀⠀⡈⠁⠀⠈⠁⢀⠆⠄⢠⣒⠒⢀⡄⠀⣀⣀⢀⣀⣀⣀⣀⠀⠀⠀⠀⢎⠙⡄
⠀⠀⡰⢡⠃⠸⠤⡅⣬⠽⠆⢠⢊⠌⠀⢠⠀⠤⢠⠚⢉⡴⠋⠁⠀⠀⠀⠈⠳⣄⠀⠀⠘⡄⢱⣀⣀
⢀⠞⡔⠁⠀⠀⠀⠧⠇⠀⠠⠡⠎⢀⠀⠀⠀⠀⡇⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢰⠙⣼⠇⠈⠱⣤⡀
⠙⠒⠠⠬⢭⠇⠀⠀⠀⠀⠉⠒⠒⠂⠤⠇⠀⠀⠹⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢸⠀⠈⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⣸⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡤⠴⠶⠒⠒⠒⠶⠾⣯⡀⠀⢀⡏⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣎⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⢒⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣠⠔⠋⢰⡎⠉⠋⠉⠉⠉⠙⠙⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⢀⡎⠀⠀⠀⠀⠐⠦⠀⠠⠤⠤⠤⣤⠃⠀⠀⠀⠘⣇⠀⠀⠀⢘⣇⣀⣀⣀⣀⣀⣀⣀⣎
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠙⣇⠀⠈⢦⡞⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠒⠛⠛⠛⠛⠛⠛⢦⠀⢸⠁⠀⠀⠀⠀⠀⢤⡇⠀⠀⠀⠀⠀⠀⣼⠤⠄⠀⠀⢸⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠚⠁⢸⡠⠊⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢧⣸⡁⠀⠀⠀⠀⠀⡠⠻⠤⠠⠤⠔⠚⠣⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠙⠶⣤⣀⣀⣀⣀⣤⡶⠛⠁⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢨⡏⠁⠀⢨⡇⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⠀⠀⡀⠘⡇⠀⠀⢸⠃⠀⠀⠀⢠⡀⢹⡀

⠀⠀⡄⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⡠⣄⠒⣆⠀⠀⠀⢀⡴⠒⡢⡄
⠀⢴⢡⣶⢹⡆⠀⠰⣶⣶⡶⠆⠀⢠⣴⣶⠀⠀⠘⢧⣢⣀⡴⠃⠀⢀⡠⢻⡛⠠⠷⢤⡄⠀⢾⢬⠁⡱⡹
⠀⠈⠉⠈⠉⠀⠀⠀⠀⠀⠀⢀⣀⣿⣩⠉⡇⠀⠀⠀⣀⣀⣀⠀⠀⠈⡽⣬⣁⢸⡏⠉⠁⠀⠈⠓⠒⠚⠁
⠀⢀⣦⣼⢾⣴⡀⠀⠀⣴⣮⣛⣋⣠⣧⣾⡇⠀⢠⠎⣥⠀⣤⠙⡄⠀⠙⠺⠥⠜⠀⠀⠀⠀⡠⣖⢶⡰⡄
⠀⣲⣿⣿⣷⣼⣗⠀⠀⢩⣍⣹⣭⣤⡌⣽⡁⠀⠸⣄⠉⡷⠩⡤⠃⠀⠀⢀⣴⣶⣤⣀⡀⠀⢿⢻⢙⢏⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠈⠡⠇⠀⠉⠁⠀⠀⠘⠿⠿⠿⠃⠀⠀⡤⡾⣿⢷⣿⡿⠃⠀⠀⠹⣾⠏
⠀⣠⠒⠉⠉⠑⢢⠀⠀⣀⢤⣤⣶⣦⡤⢄⡀⠀⢀⡀⢠⣤⠀⠀⠀⠀⠉⠉⠛⠛⠃⠀⠀⠀⡠⢤⡀
⠀⣇⢴⣏⣸⡧⢨⠃⠰⣗⣿⣿⣿⡿⣿⣣⠇⠀⠹⣷⣼⣯⣬⠐⠀⢠⢀⣄⢄⣀⠀⠀⠀⠾⡗⡚⡿⠂
⠀⠘⡆⡈⣁⢰⠁⠀⠀⠀⢸⠋⣽⡙⡟⠀⠀⠤⠾⣿⣿⣧⠶⠃⠀⠈⡹⠿⠨⠽⠗⠀⣴⠯⠥⡠⠭⢶
⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠿⠿⠿⠿⠿⠿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣦⡇⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡤⣤⣀⡀⠀⠈⠻⣿⡇⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣀⡀⠙⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⠲⣄⠙⠇⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠈⠉⠳⢦⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣾⣄⣸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⠀⠀⠀⠀⠁⠀⠉⠉⠉⠐⠒⠲⠤⣄⡀⠀⠀⠀⢰⡟⠀⠹⢧⣼⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠦⣄⠺⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠉⢤⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⢀⣤⣶⣶⣶⢶⣶⣤⡀⠀⠀⠀⠙⠳⢤⣀⣀⠀⠀⠀⠀⣀⣠⡴⠚⠁⠀⠀⠀⠙⠳⣤⣀⣀⠀⠀⢀⣀⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠘⠛⠁⠘⣿⡄⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠙⠻⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣤⣴⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⢠⡟⠀⠉⠛⠛⠷⠶⢶⣤⣤⣤⣤⣤⣤⣤⣴⢶⡶⠶⠶⠛⠃⠈⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⡾⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⡶⠶⠶⣶⠶⠶⠾⠷⠶⠶⣶⠶⠶⠶⠶⠶⠷⠶⣶⠶⠶⠶⠶⠶⠿⠷⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⣿⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⡧⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠷⠶⠶⠶⠶⠶⠶⠾⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣐⡠⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠦⠽⣛⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡎⢵⢀⠖⠉⠉⠉⠛⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⣴⠀⡸
⠀⠀⠀⠀⠀⠀⢀⡗⠉⠉⡏⠢⠖⠉⣳⠀⢨⡇⠀⠀⢀⢒⡤⡚⢆⠀⠀⠀⡠⠊⠀⠰⠥⠇⢠⠇
⠀⠀⠀⠀⠀⢰⡊⢀⡄⠈⢹⡀⠀⡠⠃⢀⡜⠀⡏⠉⠉⠉⠑⠅⠊⢹⠀⠈⠙⠒⠦⢤⡀⠀⠈⢱⠂
⠀⠀⠀⠀⠀⠀⢛⡉⢸⡀⠀⣷⠊⠀⠀⠚⠒⢲⡗⠒⢒⡶⠀⢀⡡⠟⢀⣠⣄⢔⠀⣸⠁⢰⠋⠉
⠀⠀⠀⠀⠀⣐⣊⢡⡀⢇⠀⠸⡤⠤⠤⠤⠤⠼⠇⢠⠎⠀⣠⠋⣠⠞⠛⣉⡋⠚⢦⡈⠙⠋
⠀⠀⢀⠔⠉⢉⣙⣁⡏⠘⠶⠒⠃⠀⠀⠀⠀⠀⢠⠏⠀⢰⠃⣰⠁⢠⠋⠀⠈⢦⠀⢳
⠀⢀⠏⠀⠰⠯⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣀⣀⡏⠀⣏⣀⣿⡐⠀⠀⢸⠀⢸
⠀⢸⠀⠀⡠⠒⢤⠀⢱⡀⠄⡠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠻⠴⢕⡾⠒⡶⠋⠀⣸⠀⠀⣀⠄⠀⢀
⠀⢸⡀⠀⢇⣀⡼⢠⡏⠉⠪⠣⠝⣹⠂⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⣞⣓⡒⣁⣠⠴⡵⠊⠉⠱⢴⡇⡙
⠀⠀⠳⢄⣀⣀⣀⡤⠋⡩⠛⠀⡴⠁⠀⢀⣋⣆⣙⡀⠀⠀⢸⣀⣈⠉⠙⢯⠉⠀⢸⠀⢠⡋⠉⢳⠀⢹
⠀⠀⠀⠀⠈⠉⠀⠀⢸⣀⣀⠀⠉⢢⡔⠉⡩⠭⡉⠱⡄⠀⢀⠀⢀⡇⠀⠀⡇⠀⠘⢆⡀⠙⠒⠁⠔⡿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣨⠇⠀⢀⣇⠀⠓⠒⠁⡠⠃⠀⣏⠉⠉⠀⢀⡴⢡⠛⠒⠤⠟⠁⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⣀⡠⣾⠊⢁⣠⢤⡈⠑⢦⠀⠈⠉⠛⠉⠁⠀⠈⠒⠤⠤⡤⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⣇⠀⠸⣄⣠⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢄⣀⣀⡠⠔⠃

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠠⠀⢀⠀⠀⠀⠀⡀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠀⠀⠀⡐⠀⠀⠠⠀⠀⠐⣂⣀⡀⠀⢰⠃⢀⠈⡀⠀⣾⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⣍⢉⡉⣉⢡⠀⠀⠀⠀⢀⠀⠀⠰⠛⠛⠻⠋⠀⠘⠰⠤⠰⠀⠀⠀⠀⠀⠂⢀⣀⡀
⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠒⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠃⠀⠀⢀⡀⠀⠀⠀⠌⠉⠈⠉⠅⠀⠀⠀⣤⡀⢠⡄⠀⠀⠀⡄⠀⡏
⠀⠀⠀⠈⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠢⢨⠨⠀⠀⢠⡄⠀⢠⡄⠁⠘⠉⢀⡴⠃⠀⠐⠄⠄⠐⠀⠀⠐⠀⠀⠈⠀⢀⠐⠀⢀⠁⠀⡇
⠀⠠⡀⢆⡰⢄⢤⣠⣠⣀⠀⠀⠀⠀⠀⢀⡂⠀⠀⡀⠀⢂⠀⠀⣁⠀⠀⡐⠀⣠⠀⠀⡀⠠⡀⠒⠀⢈⡀⠀⠀⠈⠁⠉⠉⠋⢁⠀⠀⠀⠀⠁
⠀⠁⠁⠈⠁⠉⠀⠀⠀⠈⠀⠀⠴⠀⠀⠐⠁⠀⠀⡈⠀⠀⠀⠰⠀⠠⠄⠀⠀⠀⠀⠀⠀⠒⠡⠠⣠⡄⠜⠀⡠⠂⠈⠉⠁⠀⠂⠠⠀⠀⠐
⠀⣀⡀⠈⢲⣤⡴⠂⠐⠀⠀⠀⠐⡀⠀⠀⠀⢄⠃⡀⢀⠀⠠⢄⠀⠀⠤⠀⢀⡀⠈⠁⠀⠀⢰⠀⡀⢘⠀⢀⠀⢸⢰⡧⢸⠀⡿⡄⢁⠀⣄
⠀⠀⢉⣉⣀⣩⣤⡀⣀⡇⠀⠂⠀⢀⣂⠀⠐⠂⠀⢰⡆⠐⠀⠐⠀⠀⠀⠒⠀⠈⠁⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⢈⠀⢀⠄⠂⠀⢸⠂
⠀⠰⠋⠉⠉⠉⠀⠉⠉⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⠃⢀⣴⢲⡀⠀⢹⠋⠹⢶⡺⠓⢽⠄⢀⡄⠀⡀⡀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⢁⢀⡀⠀⠀⠀⠀⠀⠀⡎⠉⠈⣳⠀⣀⡀⠙⠁⠈⠛⠶⠽⢁⣀⠰⠥⠀⠀⢔⡁⠅⠀⠥⠉⠦⠀⠀⠀⠀⠌⠄⡀
⠀⣀⣄⣀⣀⣈⣈⣁⣀⣀⡀⣤⠀⠀⠁⠒⠂⠁⠀⠛⠸⠗⠒⠒⢦⠀⠀⠈⠤⢤⠈⠃⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠙⡆⠀⢾⠉⠀⠀⠀⡂
⠀⠈⡀⢀⢀⠀⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠊⠁⢈⣁⠀⠒⠀⠀⠐⢊⡀⢀⡉⡀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠅
⠀⠀⠑⠀⠋⠀⠈⠐⠤⠀⠀⡃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠖⢀⡁⠀⠀⠀⠀⠀⠆⠁⠰⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠠⠄⠀⢤⡄⠀⠀⠀⡂⠀⡐⢤⠀⠉⠀⠱⠠⡀⠐⠀⡐⠀⠠⠄⠀⠀⠀⠂⠀⠀⠀⠂⠀⠘⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢁⡈⠁
⠀⠀⠀⠀⣀⢀⡀⠀⠐⠀⠀⠒⠀⠊⠀⠀⠢⣤⠂⠀⠀⠊⠀⠁⠀⠉⠀⢰⠃⠋⠋⡆⠀⣠⡴⠆⠈⡀⠁⠀⠀⠀⠀⠐⠐⠡⠀⠀⠀⠈⠁
⠀⠊⠀⠡⣈⣉⣁⠆⠈⠂⠀⠀⠀⡀⠁⠀⠀⡟⠀⠠⠄⠄⠀⠀⠠⠄⠀⢫⣥⣀⣥⠏⠰⡏⠀⠄⠀⠀⢢⠔⠀⠀⢠⣂⣁⠉⢄⠀⠀⠀⠀⢌
⠀⢀⠀⠀⣿⣷⡏⠀⢀⢄⢀⠀⠀⡀⠠⠄⠀⠃⠀⣤⠀⢠⠀⠀⠀⢩⠀⢠⡤⢭⡤⠀⢠⠉⢂⡙⠂⠰⠏⠀⠀⣞⢱⣷⡀⣾⣖⣹⡴⠂⠀⣡⡀
⠀⢨⠅⠀⢸⣵⠇⠀⠁⠀⢀⣔⣠⡀⠀⠈⠀⠀⢁⠀⠈⠀⠀⠀⢠⡀⠀⠂⠀⠀⠈⠀⢿⣡⡝⢉⣵⣀⠀⣀⡀⠉⣙⠯⡛⣨⣛⡼⢡⣄⠀⢿⣅
⠀⠰⠂⢀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠋⠀⢠⢼⣷⡇⠔⠀⠀⠁⢼⠛⠃⠘⠛⡨⠀⢀⡴⠞⠁⢱⠀⠁⠊⠀⠡⠘⠛⠃⠌⠀⠑⠈⠀⣠⡽
⠀⠀⠠⠀⡀⢃⠀⠀⢀⠐⡈⢃⠙⢠⠀⠀⠀⠀⠀⣾⣿⣷⠀⣠⡄⠀⠀⠠⠄⠀⠄⡀⠀⠿⠀⠀⠀⢨⠀⠀⢀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠛⢄
⠀⢰⣁⣀⣈⣀⣃⠀⠉⠛⠈⠃⠉⠉⠉⠀⠀⠀⠀⣿⡏⣿⠀⠉⠁⠀⠋⠀⠀⠀⠀⢳⠀⠀⠀⠀⡄⠸⠀⠀⠈⠐⠀⠀⠈⠀⠀⠀⡃⠀⠀⣠⡭
⠀⠀⠀⠀⣀⣀⣀⡀⡄⠐⣤⢀⡔⠁⠀⠠⢀⠀⡀⠛⡃⠛⠂⡀⠀⠀⠉⡆⠒⠐⠡⠋⠀⠀⠀⠀⠀⠐⠀⠀⠀⣠⣷⡆⠀⣼⣼⡀⠓⠀⠰⠏
⠀⠀⠛⣿⣿⠿⣿⠛⠁⠀⠘⠛⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⢀⡰⠀⡂⠀⠀⠈⠀⠀⠂⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠁⢉⠒⠀⡟⠀⠀⠀⠹⣢⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⣰⠁⠰⠀⠀⠀⠐⡂⠀⢀⣀⠀⠀⠤⠄⠀⠀⡀⣀⡀⡀⠀⠀⠀⣀⠠⡀⠀⠈
⠀⠀⠀⠐⠀⠈⠊⠀⢀⠂⢦⠀⠀⠀⠃⠀⡀⠀⠀⢀⠁⠰⡀⠀⠈⢀⡄⠐⠀⠀⠃⠲⠀⠈⠙⠓⣤⡀⣀⣄⣤⣴⠴⢦⡀⠀⠀⡉⡀⠀⢣
⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠈⠀⠀⠀⠈⠁⠀⠈⠀⠀⠁⠀⠈⠀⠀⠀⠀⠘⠊⠀⠈⠀⠀⠀⠙⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠰⠀⠋⠀⡀⠀⠀⠀⠀⠦⠀⠛⠀⠀⠆
⠀⠀⠀⠀⠀⢰⡢⠄⠦⠠⢖⡆⠀⠄⠀⠀⠀⣖⢦⠰⠆⠴⣲⠀⠀⠀⠀⠄⢰⡲⠄⠶⢠⢖⡆⠀⠀⠀⠀⠀⢔⠦⠰⡆⠤⣲
⠀⠀⠀⠀⢀⣐⠈⠄⠀⠬⠀⣂⠀⠀⠀⠀⣀⡂⠠⠀⠀⠄⢐⡀⠀⠀⠀⠀⣐⠈⠄⠀⠠⠀⣂⡀⠀⠀⠀⢀⡊⠠⠄⠀⠆⠘⣀
⠀⠀⠀⠀⠀⠀⠀⠰⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠿⠒⠿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠿⠒⠿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠺⠇

⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠰⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠙
⠀⠀⠀⠀⠀⢠⢄⡀⣀⢀⡤⡄⠀⠀⠀⠀⠀⡤⣀⢀⡀⣀⢤⠀⠀⠀⠀⠀⢀⢤⡀⣠⢀⡠⡄⠀⠀⠀⠀⠀⡠⣄⢀⡄⣀⣤
⠀⠀⠀⠀⠀⠨⠁⡄⠁⢠⠘⠅⠀⠀⠀⠀⠀⠀⢈⠀⠀⠄⠈⠀⠀⠀⠀⠀⠈⠃⠀⠈⠀⠀⠃⠀⠀⠀⠀⠀⠝⠀⡀⠁⡄⠸
⠀⠀⠀⠀⠈⠁⠀⢁⣀⡄⠀⠈⠀⠀⠀⠀⠈⠀⠀⣀⣠⠀⠀⠁⠀⠀⠀⠀⠁⠀⢠⣀⡀⠀⠈⠁⠀⠀⠀⠈⠁⠀⣄⣀⠄⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠧⠄
⠀⠀⠀⠀⠀⠀⠀⣶⠤⣶⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⢴⡆⠀⠀⠀⠀⠀⠀⠀⠀⣶⠤⣶⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁

⠀⠀⠀⠀⠀⠠⢄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠆⠄⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⢀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠘⠂⠉⣈
⠀⠀⠀⠀⠀⠘⠍⡂⠋⢘⠩⠃⠀⠀⠀⠀⠀⠫⢑⠘⠁⠊⠽⠀⠀⠀⠀⠀⠘⠏⠂⠙⠐⠩⠇⠀⠀⠀⠀⠀⠫⠙⠊⠃⠊⠽
⠀⠀⠀⠀⠐⠢⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠰⠀⠈⠀⠀⠂⠠⠆⠀⠀⠀⠀⠢⠀⠒⠀⠐⠀⠴⠂⠀⠀⠀⠐⠤⠀⡀⠀⡀⠠⠖
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣛⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡋⠁
⠀⠀⠀⠀⠀⠀⠀⣤⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠋⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠙⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠴⠀⠀⠀⠀⢊⡀⠀⠀⣭⡁⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣤⠀⠀⠀⠀⢜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⡀⠀⢀⣀⠀⠠⡀⠚⠋⠀⣀⠙⠁⣠⢀⡀⠀⠀⠀⠀⠣⣶⡂⠉⣩⠄⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢼⡄⠰⣶⠀⠀⠶⠀⠄⢠⠏⠀⠀⣚⡡⠋⠁⠀⠀⠀⠀⠀⠨⠄⠀⠈⢢⡀⠀⠀⠀⡇⢸
⠀⠀⠀⣶⠀⢀⣀⠀⠀⠀⢀⠀⠁⢰⠂⠀⠀⡄⠀⠂⠈⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠛⠢⠤⠔⠂⠀⠰⡓⠀⡰⢠⠏⠁
⠀⠀⠀⡇⠀⢵⡄⠀⢀⢦⡀⠀⠲⣄⠄⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⡠⠓⠐⠴⠁⠀⠀⠠⠤⠀⠀⠀⠀⠈⢈⡴⠃⢀⣀⠀⠀⠰⠇
⠀⠀⠀⠈⠂⠁⠀⠀⠝⠃⠀⠀⠀⠀⡏⠁⠀⠀⡅⠀⣠⡑⠒⠊⠉⠀⠀⠶⠘⣉⡀⠀⢰⡶⢠⡀⢀⠀⡔⠉⠀⠀⢩⣦⠀⠀⣤⡙⠆
⠀⠀⠀⠀⠀⠐⠋⠀⡀⠒⠤⠔⠛⠀⠀⠀⢀⡴⠃⠘⠁⠉⠀⠀⠀⠀⠀⠂⡌⠻⠱⡄⠀⢀⣾⡀⠘⠶⠁⠀⠀⠀⣈⡠⠀⠀⠋⠅⢡
⠀⠀⠀⢀⠤⢦⡀⠈⠀⠀⠀⠐⠒⠒⠊⢉⣁⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠺⢄⣤⡀⠀⠀⠘⠛⠂⠀⠁⠀⢠⠋⠒⠁
⠀⠈⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⢰⡦⠀⢀⣝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡦⠀⠀⠉⠢⣄⠔⣂⢀⢘⡲⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⣀⠀⠐⠁⠁⠀⠚⠋⡆⠀⢀⠤⠆⠀⠰⠤⠀⡆⠀⢹⡀⠀⠀⠀⠀⠐⠍⠀⢨⠟⠀⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⠀⠀⠀⠀⠀⠀⠀⡟⠒⠠⡸⠗⠐⢘⣷⠄⡗⠀⠚⠀⠀⣀⡀⠐⠒⠒⠀⠈⣠⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⡇⢁⠀⠀⢰⠃⡀⠀⠀⣤⣤⣤⠒⠁⡇⠀⠀⠀⠀⠈⠉⠘⠁⠀⢠⠞⠁⠑⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⢸⠀⠈⠙⠃⡇⠀⠀⠟⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠁⠀⠀⢀⣠⠇⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠈⠉⠁

⠀⠀⠀⡀⠠⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⣀⠀⢀⠀⠀⠀⠀⠤⡄⡀
⠀⠀⠈⢄⢓⠒⡒⣢⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠛⠃⠀⠀⠁⠀⠀⢠⠀⠀⠈⠄
⠀⠀⣀⠾⠾⠇⠷⠿⠀⠀⣄⠀⠂⠀⠀⠀⠘⠀⠀⠀⢀⢈⠘⠀⠉⠁⡆⠀⢂⠀⠀⠈⠨⠁⠄⠀⠰⠀⠆⠄
⠀⠀⠀⣠⡀⠀⣀⡀⠀⠀⠋⠀⠐⠚⠋⠀⠂⣀⠀⠀⠀⠀⡀⠤⠀⠀⠀⠐
⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⡂⠀⠀⠀⡀⡀⠈⡀⠂⠀⠐⠀⠈⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠅⠀⠀⠉⠄⠀⡀⠀⠘⠁⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⢠⠀⠄⠀⠠⠀⠀⠐⠢⡤⠀⠀⠹⠿
⠀⠀⠀⠤⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠁⠁⠈⠀⠀⠈⠀⠃⣰⠂
⠀⠀⠀⡀⣀⡀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠀⢀⠀⠀⠀⠀⡄
⠀⠀⠈⠉⠙⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡌⠀⠁⠀⠀⠀⠤⠢⠄
⠀⢀⠀⠀⠀⠀⠀⠀⠐⠤⠈⠀⠀⠀⠁⠀⠀⢹⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠖⠀⠀⠶⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⡂⠀⠚⠛⠋⢀⠇⢰⢛⣻⣟⡻⡆⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⡀⠀⠁⠀⢀⣦⡄⢤⡆⠀⠀⠀⢀⣥⢠⡀⠀⣤⣄⠀⢈⢠⠿⢿⡺⠁⡸⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠀⠀⠈⠈⠀⠈⠁⣴⡈⠀⠀⠔⠀⠀⠀⠐⠆
⠀⠀⡀⠀⠀⠀⠀⠊⠀⠐⠘⠂⠀⠠⠀⠂⠀⠀⠀⠀⠀⠘⠀⠀⠀⣀⠀⠀⠈⡀⠀⡀⠀⠀⠈
⠀⠰⠤⠤⠀⠀⠀⠠⠀⢰⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⠛⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠰⠀⠀⢈⣶⣿
⠀⠀⠀⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠐⠟⣰⠿⠂⠁
⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⢀⣀⣀⡀⠀⠀⣀⣀⣀⣀⠀⠀⣀⠀⢀⢀⣀⠀⡀⣀⠀⣀⣀⠀⠀⣀⡀⡀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠌⣉⠉⠉⠉⠉⢉⣁⠤⠔⠊⠁⠀⠀⠀⠀⠀⠀⢀⠠⠤⣉⠑⠢⣀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣤⠶⠶⠶⠦⠬⠭⠥⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠷⠶⠶⠮⠷⠴⠾⠷⠴⣤⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡷⠀⠀⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡞⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⢣⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⣀⣄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⠟⡇⠀⠰⡙⠟⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⡜⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⡀⣀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⡰⢹
⠀⠀⠀⠀⠀⠀⣷⠃⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⠁⢸
⠀⠀⠀⠀⠀⠀⡇⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠎⢀⢾
⠀⠀⠀⠀⠀⠀⣇⠃⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⡘⢸
⠀⠀⠀⠀⠀⠀⡏⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡿⣆⠀⠀⠀⢸⡇⢻⡄⠀⢠⡞⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡇⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣷⠼⢧⡀⠀⠘⠓⠚⠁⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⢃⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠓⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠘⣼
⠀⠀⠀⠀⠀⠀⡇⠀⠀⣺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠀⠀⢹
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⣖⠒⠒⠒⠒⢒⠶⠒⠒⢳⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢣⡀⠀⢀⡠⠔⠒⠈⠉⠉⠐⠒⠢⣄⡐⠠⣀⠀⠀⠀⠀⠑⠢⣀⡀⠈⠉⠀⠉⠀⠀⢀⡠⠊⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠉⠉⠉⠉⠉⠁⠁⠈⠉⠀⠉⠉⠉⠉⠉⠉⠁⠈⠁⠉⠉⠉⠉⠉⠉⠉⠁⠁⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠒⠂⠀⠀⠤⠤⠠⠭⠭⠤⠤⠀⠀⣀⣀⣀⣀⡀⠀⣉⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣶⣤⣴⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣤⣴⣾⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⣠⣴⣾⡿⠿⢿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⠿⠿⣿⣶⣤⡀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢠⣾⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠹⣿⣿⣿⣿⠏⠀⠀⠀⠀⠻⠏⠀⠀⣴⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣶⡄⠀⠈⠿⠇⠀⠀⠀⠈⢿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠙⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣤⣤⣌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣤⣤⣌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣟⠁
⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠛⠁⠀⣴⣿⠏⠀⠈⢻⣿⡄⠀⠘⠻⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⣤⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣷⣦⣄⠀⠀⢻⣿⣿⣿⣿⣿⣿⠃⠀⠀⣀⣤⣤⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⣼⣿⠟⠉⠉⠉⢻⣿⣧⡀⠀⠉⠛⠛⠛⠋⠁⠀⢀⣾⣿⠿⠛⠛⠻⣿⣷⡀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⣿⣿⠀⠸⣿⣧⡀⠙⢿⣿⣶⣤⣄⣀⣀⣠⣤⣶⣿⡿⠋⢀⣴⣶⠀⢸⣿⡇⠀⠀⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇⠀⠀⠀⠀⠹⣿⣷⡀⠘⢿⣿⣦⣀⠈⠛⠻⠿⠿⠿⠿⠟⠋⠁⣀⣴⣿⡿⠁⢀⣾⣿⠇⠀⠀⠀⢀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⠀⠀⠀⠀⠙⢿⣿⣦⡀⠉⠻⢿⣿⣶⣦⣤⣤⣤⣤⣴⣶⣿⡿⠟⠁⣀⣴⣿⡿⠁⠀⠀⠀⢀⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣧⡀⠀⠀⠀⠀⠙⢿⣿⣷⣤⣀⡈⠉⠛⠛⠛⠛⠛⠛⠉⢁⣀⣴⣾⣿⠟⠋⠀⠀⠀⠀⣠⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⣠⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣷⣶⣶⣤⣤⣤⣤⣤⣶⣶⣶⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⣿⣿⠿⠻⠿⠛⠿⢿⠿⢻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣢⠀⠀⠀⠀⠀⠀⠀⢠⠒⡀⠀⠒⠆⠀⠀⠀⠀⠠⠀⠂⠀⠂
⠀⠀⠀⠀⠀⢀⡀⣴⠀⠄⠀⠀⠀⡀⠀⠠⠀⡦⠀⠀⠀⠀⣄⠜⢠⠀⠂⠀⠀⡁⠘⡉⠂⠀⠀⠀⠀⠈⠀⠡⠄⣠⢀⠐⠃⠀⠀⠀⡄⢡⡀⠀⠐⢄⠁
⠀⠀⠀⠀⠀⠁⠀⢀⠁⠀⠃⠂⢐⠀⠐⠤⠜⠠⠉⠆⠀⠐⠀⠤⠽⠀⠀⠀⠈⠠⣀⢀⢋⠈⠒⠀⠀⠂⠖⢀⣸⠁⡀⠸⠀⠀⠀⡎⠀⢄⡼⢁⠂⡘
⠀⠀⠀⠀⠀⠀⠄⠀⠈⠲⠀⠀⠀⠂⠀⠀⠁⢒⠡⠀⠀⠘⠄⠰⠆⠤⠀⠀⠈⠢⡀⠀⠙⢺⠔⠀⠀⠀⢄⠀⠈⠘⡲⠃⠀⠀⠀⠐⠄⣀⠀⠉⡈⠁
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠰⣅⢀⠀⠀⠀⠀⠀⠙⠈⡀⠐⠀⠀⠀⠀⠀⢉⣀⠀⠁⠂⠀⠀⠀⢱⠔⠰⠂⣀⡀⢾⠆⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⣀⠀⡈⠤⢀⠀⣿
⠀⠀⠀⠀⠐⡙⠎⠀⠀⠀⠀⠀⠀⠄⠀⢠⠀⠈⠀⠀⠀⠀⠁⠈⠤⠋⠄⠀⠀⢠⠀⠲⢁⡔⡡⠇⠀⠀⠀⢩⠏⠁⠀⠀⠀⠑⠀⠀⠀⡰⠧⢘⡧⢢⡀
⠀⠀⠀⠀⠀⠠⢀⠀⠀⠈⠀⠀⠨⠂⠀⢐⠀⠀⠀⠀⠀⠘⠐⠁⠁⠀⠀⠀⠀⠈⠢⣀⠀⡀⠄⠂⠐⠀⠀⠈⢒⢀⢁⡀⣔⡔⠀⠀⠀⢄⠀⠈⢠⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠈⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⢀⡀⢀⠀⡀⠀⡀⢀⠀⡄⠀⣀⡤⠤⣄⣀⣀⣤⣀⡤
⠀⠀⣹⠄⢸⠀⡇⢀⠁⣸⠀⡇⠴⠃⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⠠⣀⣀⡰⠈
⠀⠀⠃⠀⠀⠐⠃⠘⠀⠓⠰⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢁⠄⠀⠠⠀⡀⠀⠀⣇⣀⡇
⠀⢀⠂⠀⠀⠈⠘⠀⠀⠐⠀⠑⢠⡄⠀⠂⠀⠐⠂⢀⡄⠀⣤⢠⣦⣰⣄⣶⠀⡖⠁⠄⠀⠀⠰⠁⠠⠈⠀⠀⠈⠡
⠀⢸⠀⠀⠒⠀⠰⢰⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠠⣀⣀⠈⢀⠠⠀⢠⠀⠀⠀⡀
⠀⠀⠀⠀⠐⠦⠆⠄⠀⠈⠃⠀⠀⠀⠀⢠⡄⢰⠀⠀⢀⡄⢀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠐⠀⠋⡠⠀⢸⠀⠘⠂⠃
⠀⠀⠀⠀⠈⠇⠁⠄⠀⠠⠀⠀⠀⢀⡀⠀⣈⠀⠈⠀⠁⠀⣀⣀⣀⣀⣈⣀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⣀⠇⠀⠀⣆⡀⠀⠀⠄⠀⠀⣹⠀⠷⢤⠤⢤⣠⠤⣄⢠⠄⠀⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡄
⠀⠀⡀⠀⡅⠠⠀⠄⠀⠀⠀⠀⠀⡀⢻⡀⠀⢸⢠⣄⠁⣤⢸⡎⡇⢸⣿⠆⡏⠁⠴⠶⠰⠶⠀⠀⠀⠀⠀⠀⠘⣉⠃
⠀⣀⠀⠀⠁⡀⠀⠁⠀⠀⡦⠠⠤⡇⠀⠫⡄⢸⢸⠟⠈⠟⣸⢁⡇⢸⣿⠀⡇⠀⠄
⠀⠚⠀⠄⠠⠀⠠⠀⠄⠠⠧⠼⠤⠇⠀⠀⠷⠘⠒⠋⢳⠖⠁⠘⠒⠒⠻⢾⠃⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐
⠀⠄⢂⣀⣁⢰⠆⢀⠀⠀⠀⡀⢴⢢⡶⢒⠋⡙⠲⣶⡖⠚⣛⠲⠔⠒⠖⣒⣶⢲⢴⡖⣶⡆⢀⠀⠂⢀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠈⠃⠀⢶⡌⠀⠀⠈⢇⠃⡾⢸⣿⠀⢿⡇⠀⢛⣥⣜⢷⡆⢻⣿⢸⠀⡇⡖⠧⠀⠄⠀⠄⢰⢤⢴⣷
⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡘⢃⣆⠸⢇⡀⣷⠸⡙⠇⣁⠿⢿⠸⢤⡇⡇⣰⣶⣶⠀⣀⠀⣮⣮⠿
⠀⠀⢤⠆⠀⠀⠀⠀⠀⢸⣿⡿⠀⠘⠛⠃⢉⠁⠈⠟⠋⠟⠟⠉⢻⡟⢉⡉⣉⠉⠉⠟⠃⠸⣾⡟⢼⠏⠀⣽⠢⣴⡤
⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠈⠉⠉⠉⠀⢀⠀⠀⠀⠀⡀⠀⠸⢄⡰⢂⣀⣄⣀⠀⠚⠁⢲⢶⣾⠛⠀⢻⠃
⠀⠀⡄⡶⠀⠀⠒⠂⠐⠦⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠈⡂⠀⠤⠝⠁⠀⠀⠴⠿⠿⠿⠿⠇⠀⠀⡀⡘⢁⠀⠀⠘
⠀⠀⠀⠂⢠⡄⠀⠠⡄⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣀⡀⡀⠀⠈⡀⢠⠐⠡⢀⠀⠀⠀⠠⡄⠤⠂
⠀⠀⠀⠄⠈⠤⠀⠤⠀⠀⠀⠀⠐⠀⠐⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⢀⠀⠸⠀⢤⠀⠀⠀⠁⠁⠀⠀⡆
⠀⠀⠀⠁⠀⢀⣀⡠⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠂⠀⠀⠀⠀⠸⠀⠈⢠⣶⠀⠀⣶⣄⠀⠇
⠀⠀⠄⠐⠐⠀⠀⠀⠂⠂⠀⠀⠀⠇⠀⠀⠀⠀⡄⠀⠀⠰⠀⠀⡀⠀⢸⢀⡖⢰⠀⣖⢠⡔⠶⢢⠠⣠⢠⠴⠖⢢
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠛⠀⠀⠛⠂⠀⠀⠘⠀⠉⠈⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣴⠀⠀⠀⠀⠀⢀⡴⠛⢶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡿⠀⠀⠀⠀⠀⠈⠱⠴⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠒⣻⠆⠀⠀⠀⠀⣿⣷⣄⠀⠀⠀⠀⠀⢸⠃⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠳⡄⣠⠹⠄⠀⠀⠀⠀⣿⡿⣿⣷⣄⡀⠀⠀⢸⠀⢸⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⣿⡇⠈⠻⣿⣿⣶⣤⣘⣀⣀⣠⣴⣿⣧⡀⠀⠀⣀⣠⣴⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠉⠛⠿⠿⠿⠿⠛⠉⠙⠿⣿⣿⡿⠿⢻⣿⡿⠁⠀⢀⣀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣀⣀⣀⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢿⡉⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⢻⣿⡟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠⠶⠚⠛⠛⠛⠳⠦⣤⣼⡇⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠙⣦⡾⡁⠀⢀⣠⣤⣀⠀⢀⣼⠏⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠁⠀⢀⣼⣿⣤⣤⣀⡀⢰⣿⣛⢧⡀⠀⠀⠀⡿⠀⠀⠹⣿⡄⣾⠟⢋⣩⡿⠿⠞⠛⢻⣿⡆⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⣶⣶⣾⡿⠛⠉⠀⠀⠀⠈⠙⠻⣿⣯⠢⠻⣦⠀⢰⡇⠀⠀⠀⢻⣾⣏⠀⢸⡇⠀⠀⠀⠀⢸⣿⣷⣀⣀⣿⠙⠃
⠀⠀⠰⡦⠴⢣⠀⠀⠀⠀⠀⠀⠈⣻⣿⡠⠀⢀⣤⣤⣤⣀⠀⠀⠈⢿⡥⣀⠈⢳⣼⠃⠀⠀⠀⠀⢻⡟⢦⣬⣷⡾⠟⠀⠀⣸⠋⣿⠁⠀⢸⣦⣤⣄⣀⡀
⠀⠀⠠⡾⢄⣠⠆⠀⠀⠀⠀⠀⠀⣿⣿⡗⣼⡟⠉⠉⠉⠻⣧⠀⠀⠘⣿⣥⡀⠀⠉⠀⠀⣼⡄⠀⠈⣿⡀⠀⠀⠀⠀⣀⣼⢏⣤⣿⣶⣶⣾⡏⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣠⣾⣿⣿⣯⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⢾⣿⣄⠀⠀⠀⡟⢿⡀⠀⢘⣿⣶⣶⡾⠿⠛⢡⡟⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⢧⣀⠀⠀⣠⠞⠀⠀⢠⣿⡿⣿⠻⣷⣄⣼⡇⠘⠿⠿⠛⠛⠀⠠⡀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠒⠀⢶
⠀⠀⠀⠀⠐⠒⠛⠋⠉⠉⠉⠉⣉⠿⢿⣿⣿⣮⠉⠛⠋⠁⠀⠀⢠⣾⣿⣿⣿⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⢐⣤⠞⢶⡒⠒⠚⠛⠆⠀⠀⠀⠈⠑⣤⠞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⠞⠛⢿⣷⣥⣤⣀⣤⣤⣾⢿⣿⣍⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⡎⠈⢳⣄⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⢀⡴⠋⠀⠀⢀⠾⠋⠙⠛⠉⠉⠀⠀⠈⢿⣦⠀⠀⠀⢀⡴⠞⠉⠉⠉⠓⢦⣇⠀⠀⠙⣦⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⣰⠟⠁⣀⣤⠶⠶⢄⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠜⡀⡠⠀⠀⠀⠀⠀⠸⡇⣰⠏⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⠀⠀⣨⠗⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠗⠚⢦⡇⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠄
⠀⠀⠀⠀⣤⡀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠀⠀⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢄
⠀⠀⠀⡁⠈⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠖⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁
⠀⠀⡠⠠⢠⣄⠄⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠠⣀⢀⣠⢠⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡆⠀⣤⢠⡄⣀⣀⢰
⠀⠀⠈⠈⠈⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠉⠀⠈⠁⠁


⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠌⠓⠀⠢⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⡀
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢴⠦⠀⠡⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⡆⠀⡐⠶
⠀⠀⠀⠀⠠⣤⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠛⠸⠸⠄⠋⠛⠘⠖⠵⠢⠸⠀⠀⠀⠀⠀⠀⠘⠸⠘⠷⠉⠘⠂⠔⠡⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⠘⠓⠛⠫⠜⠓⠨


⠀⠀⠀⢀⡀⠀⢠⡀
⠀⠀⢀⢀⣤⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡁
⠀⠀⠈⠢⠐⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⡴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢜⠃⠀⠃⠛
⠀⠀⠀⠀⢩⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡇
⠀⠀⠀⠀⠘⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣀⣛⣘⣘⣃⣓⣂⣘⣀⣛⣀⣀⣀⣀⣀⣀⣀⣇⣛⣜⣺⣋⣇⣬⣈⣠⣛⣛⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣑⣈⣘⣂⣃⣀⣀⣀⣀
⣿⣿⣋⢙⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⣄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⣰⣿⣷⢀⡀⠀⠺⠗⠀⢀⣾⡿⠃⠀⠰⣇⣨⠓⢀⣴⣿⣷⣦⣾⣛⠀⠀⣷⢀⣤⣬⣧⡀⠀⠀⠀⠀⣿⣿⣷⡀
⠀⠀⣴⣼⣷⣿⣿⠇⠀⠀⠀⢠⣼⣿⣷⣶⣦⡀⠁⠈⣠⣼⣿⣿⣿⣿⣯⠉⠀⠀⠈⣿⣿⣿⣿⣿⣦⡀⠀⢀⣿⣿⣿⣷⠦
⠀⠀⠙⢻⣿⣿⣧⣴⡦⠀⢠⣿⡟⣿⣷⣈⣿⣷⠀⠀⠛⠙⠿⠛⡟⠃⠀⠀⠰⠆⠀⢸⣿⣿⣿⣿⣿⣷⠀⠈⠉⠋⠛⠉
⠀⠀⠸⣿⣿⣿⣿⡟⠁⠀⠘⠛⠃⠙⠿⠿⠿⠋⠀⠀⠀⠀⠀⣸⢷⣶⡶⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠀⢀⣦⡀
⠀⠀⢠⣤⣿⣿⣿⣿⣷⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢾⣳⣦⣼⠃⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣻⣿⣿⣧⠀⠋⠀⣶⣶⡆
⠀⠀⠀⠙⠻⠿⠿⠟⠁⠀⢀⣸⣿⣿⣤⣀⠀⠀⠀⠀⠈⠉⢷⡀⠀⠀⣴⡄⢀⣾⡿⣿⣿⣿⣿⣿⣿⡟⢿⣷⠀⠀⣿⣿⡇
⠀⠘⠃⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣇⠀⠀⣀⠀⠀⠈⣷⠀⠀⠈⠀⠀⠉⠀⣿⣿⣿⣿⣿⣿⠃⠀⠉⠀⠀⣿⣿⣿
⠀⢀⣾⣿⣿⣦⠀⠘⠃⠸⢿⠟⠁⠀⠈⢻⣯⠀⠀⠉⠀⠀⣠⡿⣾⣛⡦⠀⠀⠀⠀⣿⡏⠉⢙⣿⠇⠀⠀⠀⠀⣀⣿⣿⡟
⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣴⠋⠀⠈⠁⠀⠀⠀⠀⢶⡿⠇⠀⢾⠿⠃⠀⠀⠀⣿⣿⣿⠿⠁
⠀⠀⢿⣿⣿⠟⠀⠀⣲⠶⣇⡀⠀⣠⣾⣿⣿⣷⡄⠀⠀⠟⠀⠀⠀⢠⣆⣴⡄⠀⠀⠀⣰⣶⣀⣠⣶⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⠀⠈⠳⠴⡏⠀⢀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣶⢸⣿⣿⣿⣆⠀⢠⣿⣿⣿⣿⣿⣇⠀⠀⢠⡟⢙⡶
⠀⠀⠀⠀⣄⣺⣇⣠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⣠⡀⠀⢸⣿⣾⣿⣿⣿⡿⠀⠀⢿⣿⣿⣿⣿⠇⠀⠀⠀⠋⠙⠃
⠀⠀⢠⣴⣿⣿⣿⣿⣶⡏⠀⠀⢠⣶⡆⠀⠀⠀⠀⠀⠙⠋⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠉⠉⠉⠀⢀⣴⣿⣿⣶⠀⢀⡀
⠀⠀⠀⠚⣿⣿⣿⣿⡯⠀⠀⢠⣿⣿⣇⠀⠀⠀⠶⠀⠀⠀⡀⠀⠻⣿⣿⣿⣿⡇⠀⣠⣴⣶⡶⠂⠀⢸⣿⣿⣿⣿⡇⠈⠁
⠀⠀⠀⢰⠟⠋⠋⠻⠇⠀⠀⠻⣿⣿⣿⡷⠀⠀⠀⠀⣸⠛⢣⡄⠀⠀⠀⠀⠀⠀⣼⣿⣿⡏⠀⠀⢠⣿⣿⣿⣿⣿⣷
⠀⠀⠀⢀⣠⣀⠀⣀⠀⠀⠀⠀⣾⡿⠋⠀⢠⣤⣄⠀⠘⠚⠇⠀⠀⠀⠀⠀⣤⠀⢿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣶⣶⣶⣶⣶⣤⡀⠀⠈⠻⢿⡆⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢼⡦⠀⠀⣾⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣠⠟⠀⠀⠀⠀⠀⠀⠀⣼⣷⣾⣿⣿⣶⣿⡇
⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠈⠁⠀⣠⣿⣿⡇⠀⠀⠀⠙⠿⠿⠿⠟⠀⠀⠀⠀⣶⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠙⢿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⢠⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠶⠀⢸⣿⣷⣶⡀⠀⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢀⡟⠛⠻⠛⠁⠀⠀⠀⠀⣿⣿⣿⠃⢀⣾⣿⣿⣄⠀⣼⣿⣿⣷⡄⠀⠀⠈⣿⣿⣿⡇⠀⠀⢰⣿⠀⠀⢸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠈⠉⠁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⠶⢾⣛⣋⣭⠻⣮⣷⣄⡤⣀⣠⣞⣤⠾⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⢡⣴⠞⣏⢻⣭⣉⡇⠈⠻⣿⣿⣿⡿⠟⢡⢿⡷⢦⣽⡛⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⡏⠘⣯⣾⢏⣴⣭⡛⣿⣄⣀⣼⣿⣇⠀⢀⣾⡷⠿⠏⣾⡙⢷⡌⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣶⣿⣿⠇⠀⢸⣯⣾⠇⣌⡻⣾⣝⣩⠿⠿⣯⠛⣟⣵⡾⠛⣷⢹⣷⠞⠁⣿⣿⠁
⠀⠀⠀⠀⠀⢠⣿⢛⣛⠛⡛⠛⠛⢾⣷⣶⣛⣹⡿⣾⠋⠻⣾⣍⢡⠂⠂⠈⢛⣿⡵⢾⡆⣿⣸⣿⠀⠀⣿⣿⣿⣂
⠀⠀⠀⠀⢀⣾⢣⡾⣫⣹⣦⣤⣤⡟⣿⠋⡉⠉⢡⣿⣄⠤⣼⣿⣿⣧⣽⣾⣿⣏⡀⢠⣿⠻⣧⣭⣿⢾⡷⠿⠿⣿⣿⣭⣤⣄
⠀⠀⠀⢠⣾⢣⡿⣙⠛⣫⣭⣭⣽⣶⠟⠐⢤⡘⣿⡷⣿⣛⢫⣿⣷⣿⣿⣿⣟⠷⣽⣞⣿⣠⠂⠌⢹⡟⣷⣀⣀⣠⠴⣶⡈⣿⡀
⠀⠀⠀⣿⡃⣿⡕⣿⡏⢿⡀⣶⠶⠿⠿⣿⣻⣿⣫⣾⣷⣿⢟⡿⣿⣿⣿⣿⣷⣾⣿⣟⢿⣥⣴⣈⠘⣷⣽⣿⣛⡻⣾⣮⣿⣸⣇
⠀⠀⠀⣌⣷⣌⠉⠛⢿⣮⢿⡞⣷⣄⣰⣻⡟⣿⣿⣿⣿⣿⣾⣿⣮⣿⣵⣿⡏⣿⣿⣿⡷⣝⣯⡿⠿⠷⢿⣭⢉⣿⢆⣶⢻⣧⢿⣆
⠀⠀⠀⠈⣻⣿⣦⡀⣰⣏⣼⢏⠘⣿⣿⣿⣾⣾⡿⣿⣿⣿⠯⠿⠷⠡⢿⣻⣿⣟⣛⣿⣿⢋⢿⣹⣀⣠⡾⣵⡿⣣⣾⣯⡼⠟⣸⡿
⠀⠀⠀⠈⣻⣿⡿⠿⣯⣿⡇⢞⢈⣉⣿⣿⣿⣿⣿⣾⣿⡻⠫⢆⣥⣮⢡⡰⡾⣿⣿⣻⣿⣿⡿⣿⣿⠏⠸⣏⢺⡏⠀⠀⣴⣾⡧
⠀⠀⠠⢺⣿⠟⠁⠀⣹⡏⣻⡄⣸⣿⣿⣾⣿⣿⣟⣿⣿⣾⠔⢊⡻⢿⠱⣠⣜⣿⡻⣿⣿⣿⣿⣿⣉⠁⡧⢸⣿⣿⡶⢾⣿⣏⡀
⠀⠀⣴⡏⣴⣖⢶⡿⢫⣾⢟⡾⠛⠹⡛⣿⣭⣷⣯⣽⣿⣿⣦⣶⢁⢭⣆⣬⣿⠿⠿⣿⡿⣿⣿⣟⣿⡀⣱⠟⣻⠏⠉⢿⣿⣏
⠀⠀⠹⣧⢻⣷⠿⢱⣿⡁⢻⠶⣤⣤⣿⣿⣗⢿⣿⣿⣿⢸⣿⢿⣿⢿⣿⡿⣿⣿⣿⣿⣷⣼⣋⠏⠙⢷⡻⣯⡻⣦⣄⣀⠻⣿⠓
⠀⠀⠀⢻⡞⣷⣺⢷⣭⣝⣻⡿⢷⡍⠹⠟⢻⣿⣻⣿⡿⣿⣷⣿⣿⣷⣿⣷⣿⣿⡿⣿⣽⣿⣷⣦⣤⣬⠷⠙⣷⣹⣯⢝⣷⢨⣷
⠀⠀⠀⠸⣿⡙⡷⠶⠋⠉⠉⣷⣼⣇⠒⠄⠻⣿⣿⡿⢶⣿⡽⢿⡿⢿⣿⣂⣿⣿⢾⣿⡉⠋⡅⣴⠾⣿⡿⢟⣛⣤⣙⣾⢣⡿⠃
⠀⠀⠀⠀⠙⠛⣓⣶⣶⣶⣤⢞⡿⣿⣛⢻⣆⣿⠁⠉⣹⣯⡿⡟⢻⣿⣻⡿⠖⠻⣿⠃⣃⣘⣠⣿⣼⠛⠛⠻⣏⣥⣾⢣⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠉⢹⣻⣿⣿⠀⠀⣿⣾⣿⢹⡤⢞⣻⣅⡨⢠⡨⢈⣙⢷⣄⢰⡯⣾⢏⣩⣿⢾⡧⣤⣤⣤⣬⣽⣥⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢀⡴⢿⡇⣿⣤⡴⢟⣯⣍⣳⣶⣶⢋⣛⢷⣎⠛⢸⡿⣿⡇⠀⢠⣷⣿⢿⠛⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⠺⣯⣝⡿⣨⣵⣶⡿⠉⠈⢹⣿⡟⠋⠻⣷⣝⡻⠿⣱⡿⢳⡄⣼⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢦⣽⡛⢷⣦⣴⢃⣠⣾⣿⣾⣷⣤⠀⢿⣩⢟⣧⣻⣵⠾⢃⣿⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣶⡶⠟⡿⠛⠙⠟⠙⣿⠷⣜⣛⣛⣭⣽⠶⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣽⡇⢹⡃⣸⢿⡇⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣹⡇⠉⠁⣿⣼⣧⠐⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⠿⠗⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠶⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⢸⣿⣛⠃⢦⡆⣾⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢈⡛⢹⡇⠞⠇⣿⢻⣧⢨⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠘⠿⠾⠃⠀⠀⠻⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠛⠿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡗⠀⠀⣴⢶⡄⢀⡀⣠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠇⠀⠀⣿⣬⡅⣸⡇⣿⣧⣭⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄⠀⠀⠛⠛⠛⢻⣿⣿⣿⠀⠀⠀⣿⣼⡇⠀⠀⢿⣧⣿⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣺⣿⣿⡏⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⡀⠘⠓⠒⠚⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠺⠿⠿⠷⠶⠾⠿⠿⠷⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣶⣦⣍⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⢋⣤⣶⣶⣤⡙⢿⣿⣿⣿⣇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⠸⣿⡿⠿⠛⢛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣆⢻⣿⣿⣿⣿⡟⢠⣿⣿⣿⣿⣷⣦⣶⣿⣿⣿⣿⣿⣦⡙⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣶⣌⣙⣋⣩⣴⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⣿⣿⣿⣿⣿⣿
⠀⠀⠠⡿⠿⠿⢿⡿⠿⠿⠿⣿⡿⠿⢿⣿⣿⠏⣥⣶⣶⣦⡍⠛⣩⣴⣶⣶⣬⡙⢿⣿⣿⣷⣼⣿⣿⣿⣿⣿
⠀⠀⠀⣴⣿⣿⡆⢰⣿⣿⣷⢠⣶⣶⣶⡌⣤⣶⣶⣌⢻⣿⣿⢰⣿⣿⣿⣿⣿⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⣿⣿⣿⡇⣾⣿⣿⣿⢸⣿⣿⣿⡇⣿⣿⣿⡿⢸⣿⣿⣼⣿⣿⣿⣿⣿⣿⢁⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⢀⣭⣭⣭⡄⣬⣭⣭⣥⢠⣭⣭⣭⠀⣭⣭⣭⡅⣿⣿⣿⣿⣿⢉⣛⣛⣋⣡⣾⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⢸⣿⣿⣿⠁⣿⣿⣿⣿⢸⣿⣿⣿⠀⣿⣿⣿⡇⣸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⣤⣍⣉⣩⣤⣭⣉⣉⣥⣬⣉⣋⣩⣤⣍⣉⣉⣴⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⠀⠀⣠⡤⠖⠛⠉⠉⠙⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠖⠚⠛⣷⡀⢰⠏⠀⠀⠀⠀⢹⣀⡼⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⣠⡶⠦⣤⣀⡀
⠀⠀⢀⣀⡤⣄⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣀⡼⢳⣼⠀⠀⠀⠀⠀⠀⢻⠀⠀⣿⠀⠀⠐⣟⡆⠀⠀⠀⠀⣿⢁⡇⠀⠀⠈⠉⢹⡆⢀⣤⣀⡀
⣶⠚⠉⠀⠀⢋⠓⢦⣄⠀⠀⢻⠀⠀⠀⠀⠏⠀⠀⠀⢻⠀⠀⢰⢦⠀⠀⠈⣇⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⢸⠃⠀⠀⠀⠀⠘⣇⡏⠀⠀⠉
⠘⡆⠀⠀⠀⠘⡄⠀⠈⠳⣄⡸⡆⠀⠀⠀⠘⠲⡄⠀⢸⠀⠀⠈⠉⠀⠀⠀⢸⡄⡏⠀⠀⠀⠀⠀⠀⢿⠁⠀⠀⡸⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⣸
⠀⢹⡄⠀⠀⠀⠰⡀⠀⠀⠀⠙⣷⠀⠀⠀⠀⢰⠁⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⢷⠀⠀⠀⠀⠳⠴⠀⠂⠉⢹⡆⠀⠀⠀⠀⠉⠉⢿⠀⠀⠀⢀⣦⠀⢀⣀⡀⢿⠀⠀⠀⡟⢆⠀⠀⠘⡆⢸⠇⠀⠀⢠⡀⠀⠀⠀⠀⢀⡞
⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⢸⡀⠀⢀⡸⠈⡶⠁⠠⠌⢿⠤⠤⠤⠇⠀⠲⠤⠤⢽⣾⠀⠀⠀⡇⠱⡀⠀⠀⠀⣸⠃
⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢀⣀⠸⠷⠤⠤⠒⠒⠒⢉⡉⠁⠀⢀⣤⣧⡘⠠⢊⣼⠦⣤⣤⣀⡶⣄⣀⣀⣤⠀⠈⠉⠚⢀⠀⢦⣀⠀⢀⡏
⠀⠀⠀⠀⢳⡀⢀⡠⠤⠒⠋⠉⠀⠀⣐⣶⠤⠶⠶⠶⠚⠙⠛⠋⠉⠁⠈⣯⠀⠸⡇⠀⠀⠀⠈⠀⠀⠈⠉⠉⠛⠛⠓⠶⠟⠳⣄⣈⣉⡿⠃
⠀⠀⠀⠀⠈⠻⢥⣀⣠⡤⠶⠒⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠈⠷⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠰⠂⡀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡖⠁⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⢷⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⡞⠈⣇⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢀⠇⠀⠸⡄⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⢀⠇⠀⠀⡸⠀⠀⠀⢱⠀⠀⠀⡇⠀⠀⠀⠀⢀⡠⠒⠉⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠥⣌⣙⠒⠦⠤⢀⡼⠀⠀⠀⡇⠀⠀⠀⠀⠇⠀⠀⠸⡤⠴⠖⢋⣁⡴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠦⡄⠀⠀⢘⡛⠛⠛⠛⢋⢙⡀⠀⠀⢢⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⢣⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⢀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡤⠐⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⢀⣀
⠀⠀⠀⠀⢄⡀⠦⠨⢆⠐⡀⠀⢀⠌⠲⠀⠀⠐⡂⠀⡀⡂⠠⠄⠉⠠⠀⠠⠄⠠⢀⣜⣋⠺⡇⠸⠋⡵⢆
⠀⠀⠠⠢⠊⢠⢰⣖⣦⠙⡈⡀⠈⠀⠀⠠⠄⠀⢯⡇⠀⠀⠀⣾⢰⡃⠀⠀⡂⠀⣬⡝⠋⠢⢠⣀⡈⠠⡄
⠀⠀⢰⠚⡀⠢⡘⠻⠋⢀⢅⠐⠀⣀⠀⠒⠂⠀⢀⠂⢀⠀⢀⠀⠀⠀⠀⠀⠁⠀⢁⣄⣀⣰⡇⠈⠶⠊
⠀⠀⠀⠨⣑⠤⢀⠁⡄⢢⠔⠀⠠⡘⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠉⠁⠈⠀⠀⠀⠀⠖
⠀⠀⠀⠰⢦⡁⢘⠂⡪⡀⠠⠀⠀⠀⠀⢀⣄⠰⡅⠀⠀⠀⣄⠀⠀⠀⠀⡁⢒⠀⣘⢂⡀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢰⠺⣋⡀⠀⠔⠃⠂⢀⢊⠑⡀⠀⠀⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⢂⡁⠏⠁⠉⠃⡁⠠⠀⠂⠈⣤
⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠨⡂⢞⠕⠀⠀⠀⣀⡈⡀⠀⠀⠀⡀⣄⠀⠀⡓⣄⠐⢀⠔⠁⠀⠀⠀⠐⠤⣀⠦
⠀⠀⠀⠄⠀⠐⠁⠀⠀⠀⢄⣨⠌⣤⡄⠀⠀⠇⠻⠁⠀⠀⠀⠛⠉⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⢀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⢀⡄⡄⢄⠀⠀⠉⠀⠈⠀⢠⣄⠀⠀⠀⠀⢀⢀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣀⡮⢣⢇⣂⡌⡁⠀⠀⠠⠂⠀⡀⠀⠀⠀⠠⠊⢬⡄⣬⢂⣀⠀⠀⠀⠐⠀⠀⢀⡍⠉
⠀⠀⠀⠙⢮⠞⢹⣎⣙⠊⠀⠀⡀⠀⠀⠀⠀⣀⠀⠨⠀⠘⠋⠛⢸⡨⠃⠀⠀⠀⠀⠐⠌⠃⠀⠠⠇
⠀⠀⠀⠀⠂⠠⠀⠀⢀⢀⢀⠀⠀⠄⠽⠿⠇⠀⠀⠀⠅⠀⠂⠀⠢⠀⢀⠀⠀⠈⠀⠀⠀⠉⠉⠀⠀⠀⠈
⠀⠀⠀⠘⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠐⠂⠄⠀⠀⠀⠓⠊⠛⠊⠃⠀⠂⠒⠀⠀⠀⠀⠂⠀⠀⠐

⠀⢀⣀⣀⣠⣀⠀⠀⣀⡀⢀⡀⠀⠠⢄⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠈⠉⠉⠉⠃⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⠓⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠒⠀⠀⠀⠇⠀⠒⢀⠀⠀⠀⠠
⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠑⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⢠⠋⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠀⠀⢀⡀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢀⣄
⠀⠠⡀⠀⠀⠘⠛⠻⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠒⠀⠀⠀⠁⠙⠘⢁⠐⠀⠐⠀⠀⠓⠃⠺⠿⠛⠋
⠀⠀⠹⢒⠤⠄⠀⠀⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⠀⠀⡀⣈⠁⠃⡀⢀⢀⠀⠠⠀⠀⠀⠄
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠁⠀⠈⠀⡢⠀⠆⠐⠐⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⢀⠀⠀⠀⠀⠆⠀⠀⠁⠀⠀⠀⠈⠀⠈⢡⠀⠀⢠⠀⠀⠀⠀⠀⠂
⠀⠀⢀⣠⡀⠀⠙⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠄⠀⠐⠂⠰⠀⠀⠀⠈⣀⠀⠀⠀⠈⡊⠉⠉
⠀⠙⠋⠉⠁⠀⠀⠀⠀⠀⢠⣷⣄⠀⠀⠀⣠⣄⠀⠀⠀⠐⠘⠂⠀⣾⣻⡷⠀⠀⠀⠐⠆⠀⠀⠂⠀⠀⣤
⢀⠐⠈⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠐⠀⠊⠀⠀⠀⠈⠉⠀⠀⠀⠠⠀⠀⠀⠀⡀
⠈⠐⢢⠀⢀⠀⠀⠤⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⠈⠘⠀⠀⠀⠀⠀⠀⠅⠀⠠
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⠂⠈⠀⠀⠀⠀⠀⠈⠛⠛⢁⠀⠀⠀⠀⠀⠀⢠⠲⡄⠀⠐⠀⠀⠀⠀⠀⢀⠒⢲
⠀⠀⠐⠀⠐⠈⡈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⣉⣈⠉⠀⣀⣴⣆⣀⠀⠀⠉⠀⠀⠂⠐⠂⣀⣤⡀⠀⢀
⠀⠀⢀⢀⢀⡀⠀⠀⢀⠠⠸⠐⠐⠀⠀⠀⠀⠀⢸⠯⡟⡧⠀⣿⣿⣿⣿⣿⣶⣶⠀⠁⠀⠀⢸⣿⣿⣿

⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠐
⠀⠀⠀⠀⡶⠒⠀⢰⠀⠀⠀⠀⠀⠀⡄⠀⠐⠒⠦⠖⢀⣤⠇⠀⠀⠀⠀⣀⣠⡖⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⢀⠄⠀⠀⠀⠀⠀⡇⠀⠐⠀⠀⠠⠋⠁⠀⠀⠀⠀⠀⠀⠀⢁⠀⠛⡁
⠀⠀⠀⡀⠙⠀⠀⠸⠀⠀⠠⠃⠀⠀⡆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡾
⠀⠀⠀⠈⠉⠉⠉⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠄⠀⠀⢀⣀⠈⠳⡄⠀⠑⠂⠤⠚⠁
⠀⠀⠀⠐⠀⠀⠀⠀⠀⡇⠀⠀⣀⣠⡁⠀⠀⢀⠀⠀⠐⠦⠾⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠐⠚⠀⠀⠃⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠞⠂⢈⠛
⠀⠀⠀⠀⠤⠤⠤⠀⠀⡤⠇⠀⠀⠀⠀⠀⣀⡤⠾⠀⠀⡀⠀⠁⠀⢀⡆⠀⠀⠀⠀⠠⠤⠴⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠠⠴⠆⠀⠀⠀⠀⠀⢀⣉⡀⠀⡀⠀⠈⠉⠉⠉⡉⠁⠀⡀⠀⠀⢠⡂⠐⠂⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⢁⠀⠀⡀⠘⠅⠉⠀⡽⠀⠀⠀⠀⠲⠁⠂⢀⡇⠀⠀⠀⠙⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⠀⢸⠀⠀⠙⠢⠤⠴⠐⠁⠀⠀⠀⠀⢀⢀⡀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠢⠄⠤⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠤⠔⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⣼⣿⣿⣿⣿⣆⢀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⣩⣴⣶⣶⣶⣬⣙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢃⣾⣿⣿⠏⣤⠻⣿⣿⣧⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠏⢸⣿⣿⡇⣈⣉⣁⢹⣿⣿⡇⣿⣿⣿⣿⣿⣿⡿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⣿⣿⣿⢸⣿⡿⢡⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣇⣿⣿⣿⣸⢟⣵⣿⡿⠻⣿⣿⣿⠟⠀⢀⣤⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠃⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣛⡋⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⡀⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⡘⢳⣶⡛⢡⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⡘⣿⢡⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣷⡀⣿⣿⣿⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⡇⢻⣿⣿⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠛⠛⠛⠛⠃⠘⠛⠛⠋⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠒⠒⠚⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠈⡀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⣿
⠀⠀⣤⡤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⣤⠀⠀⠀⠀⠀⢀⣤⠤⠤⠤⠴⠧⠤⠶⠀⠀⠀⠀⠻⠤⠤⠤⠴⠶⠶⣦
⣠⠞⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢠⠞⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⢿⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⡆⠀⠀⠀⣏⠀⠸⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇
⢸⡆⠀⠧⠤⠤⠤⠴⠖⠒⠒⠒⠒⠒⠚⠛⠛⣿⠃⠀⠀⠀⢹⡀⠀⠉⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⡏⠉⠉⠉⠉⣩⠟⠁
⠀⢷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠞⠁⠀⠀⠀⠀⠈⠧⠤⠤⠤⠤⠴⢦⠀⠀⢸⠀⠀⠀⠀⢷⠖⠒⠒⠛⠁
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠓⠚⠛⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠶⠶⠶⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠲⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠙⠳⣤⡀⠀⠀⠀⠀⢀⡞⠁⠀⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠈⢙⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠙⠲⣄⠀⣠⠏⠀⠀⠀⣠⢾⠁
⠀⠀⠀⠀⢀⣀⣀⣀⣙⣶⣾⣯⣤⡤⠤⠴⠶⠶⠶⠶⠶⣆⠀⠀⠀⠀⠀⠀⡼⠉⠓⠦⣀⠀⠀⠀⠈⠙⠃⠀⠀⢀⡜⠁⣼
⠀⠀⣾⡏⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠘⠷⣄⠀⠀⠈⠓⢤⠀⠀⠀⠀⠀⢰⣋⡠⠞⠁
⠀⡼⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⡴⠃⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀
⢸⡁⠀⠹⠤⠤⠤⠤⠤⠤⠴⠶⠒⠒⠒⠒⠚⠋⠉⠉⣩⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⢠⠞⠲⢤⣀⠐⠀⠀⠀⠉⢹
⠈⣧⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⠤⠤⠴⠶⠶⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⡰⠃⣀⡀⠀⠈⠑⠦⣀⠀⡠⣿
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⢀⣤⠞⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠢⡀⢀⡜⠁⡼⠋⠙⠲⣤⡀⠀⠈⣹⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠢⢤⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠈⠋⢀⡞⠁⠀⠀⠀⠀⠉⠓⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⢀⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠈⡟⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣧⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠶⠶⠶⠤⢴⡷⠀⠀⠀⠀⠸⠾⠷⠶⠶⠶⢶⣄
⠀⠀⠀⠀⣴⡶⠶⠶⠶⠶⠶⠛⠛⠛⠛⠛⠛⠋⠉⠉⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⣆
⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣠⣤⣤⣤⣤⡄⠀⠀⠀⠀⢰⠶⠶⠶⠶⠶⢋⣰⡇
⠀⠀⠀⠀⠀⣷⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠶⠶⠶⠶⠚⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣶⣶⠶⢶⡇⠀⠀⠀⠀⢸⠀⣾⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠹⣧⣤⣀⣠⣤⣤⣤⣤⣤⣶⣶⡶⠶⠶⠶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡶⠶⠶⠶⠞⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠛⠛⠛⠛⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠷⣤⡀⠀⠀⣠⡾⠋⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠈⠙⢶⡴⠋⠀⠀⠀⠀⠀⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⢀⡟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⡶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣞⣡⡾⠋
⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⡶⠶⠶⠶⠶⠶⠶⠶⠿⠛⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢙⣷
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣼⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⢀⡾⢁⣙⠳⢤⡀⠀⠀⢀⡞⣿
⠀⠀⠀⠀⠀⠀⢻⣶⣛⣛⣋⣩⣭⣭⣭⣥⣤⣤⣤⣤⣶⡶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⠀⠀⠀⠀⣰⠏⣴⠟⠛⠷⣦⡉⠳⠶⢋⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⣉⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣟⢦⣤⠞⢁⣾⠋⠀⠀⠀⠈⠻⢶⡶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠉⢙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⠒⢛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠒⠒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠁⠉⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⡠⠒⣶⠀⠘⡄⠀⠀⢀⠄⠊⠢⡀⣠⡶⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠉⢉⢤⠾⠋⠄⡰⠁⠀⠀⢿⣦⡀⠀⢈⠁⠀⠀⡠⠂
⠀⠀⠀⠀⠀⢠⡤⠤⠛⠃⠀⠀⠀⠈⠈⠩⠍⡇⠀⠀⠀⢺⡇⠀⠀⢂⣀⠀⠤⠄⠠⠇⠀⠀⠀⠀⣰⣕⠁⢀⡴⠊⠀⠀⠀⠀⠀⣨⡟⢀⠀⠀⠀⢪⡀
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠐⠀⠀⠀⠀⠂⠀⢀⡇⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢙⣣⠤⠌⠀⠀⠀⠀⠀⠠⢾⡁⠀⢀⠤⣦⡀⠀⠑⡄
⠀⠀⠀⠀⠀⠹⣧⡠⠤⣶⡆⠀⠁⢸⠉⠉⠁⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⡀⠈⠀⠜⠀⠀⠀⠀⠻⣁⡂⠆⠀⠀⠀⠀⠀⠀⠀⠙⠒⠁⠀⠈⠛⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠈⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣴⠖⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠈⢣
⠀⠀⠀⠀⠀⢰⡟⠹⢑⡄⣴⣻⠋⠐⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⣸⡇⠀⣶⠶⠒⡄⠀⠀⠀⠀⣿⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠑⠒⢋⣴⡷⠁⠰⢀⠜⠀⠀⠀⠀⠀⠀⢀⣄⡤⢤⠴⠶⠒⠚⠛⠛⠛⠓⡄⠀⠸⢇⣀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⢸⡇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢠⢎⠞⠀⠀⢠⠊⢀⡀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣼⠟⠀⣠⡐⠑⡄⠀⠀⠀⠀⠀⠈⣷⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡴⡳⠋⠀⠁⡔⠁⠠⡟⠭⠳⡄⠀⠀⠀⠀⠙⠒⠒⠒⠂⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠲⠊⠈⠻⠒⠁⠀⠀⠀⠀⠀⠀⠙⣒⣒⠁
⠀⠀⠀⠀⠀⠀⠻⡁⠀⢤⠎⠀⠀⠀⠙⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣈⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠁
⠀⠀⠀⢀⣠⣤⠤⢀⠀⠀⠀⠀⠀⠀⡴⡒⠀⠀⢠⣴⡂⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠙⢆⠀⠀⠀⣾⡏⠁⠂⠤⢀
⠀⠀⢠⢛⠋⠀⣠⡀⠙⡆⠀⠀⠀⢳⡇⠀⡁⠀⠀⠉⢹⡆⢸⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠄⣠⠖⠁⠀⠀⠀⠈⠁⠒⠤⢔⡀⠀⠑⢐⠄
⠀⠀⣇⡎⠀⠌⣸⣿⠀⢸⠀⠀⢀⢾⠋⢐⠁⠀⠀⠀⢅⡄⠸⢄⠀⠀⠀⠀⢠⣾⠟⠁⠀⢠⣋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⡮⠟⢁⡴⠋
⠀⠀⣿⡇⠸⣴⡿⠃⢀⡎⠀⠀⠀⢀⣣⠘⡆⠀⠀⠀⠜⠟⠀⡎⠀⠀⠀⠀⠀⠉⠓⠦⢅⣀⠈⢉⠓⠶⢤⠀⠠⣴⡿⠛⠙⢀⡴⠋
⠀⠀⠈⠳⣄⣀⣀⡤⠊⠀⠀⠀⠀⢸⡟⠰⠇⠀⠀⣤⣤⠖⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣤⠞⠀⠀⠘⢆⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠠⠤⠃⠀⠙⠃⠥⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⢀⡠⠖⠀⠀⠀⢠
⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠃⠀⠀⠀⠋⠁⠀⠈⠀⠀⠀⡰⠃⢀⣈⠀⢀
⠀⠁⢀⢀⡀⢠⡶⣤⢹⡆⠰⠒⢨⢀⢁⢡⡂⠦⣤⣤⢀⡂⠢⠀⠀⠀⡄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⡼⠀⣾
⠀⠀⠎⠀⠱⡀⠙⠋⢸⡇⣀⡁⠀⠀⠀⠀⠀⠀⠉⠈⠈⠁⠀⠀⠄⠀⠁⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⡣⢴⠇
⠀⡆⠀⠀⠀⢇⡠⠂⢸⡇⢿⣿⠀⡀⠈⠀⠀⠀⢀⠀⢰⡁⠀⢠⡄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠀⠀⠸⣁⡜
⠐⡃⠀⠀⠀⠞⠄⠀⠈⠃⠀⠠⠌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⡈⠤⠀⠀⠠
⠀⠆⠀⠀⠘⠀⠑⠘⠀⠰⠂⠰⠀⠰⠀⠀⠀⠀⠀⢀⣄⣀⣀⠀⠠⠀⠀⠀⠀⠈⠀⠐⢠⠐⠃⠰⢍⣁⠀⠒⠈
⠀⠄⠀⠂⡀⠀⠀⠀⠀⠀⡀⠐⠁⠀⠐⠀⠀⠀⠀⠙⠛⠟⠛⠛⢁⠀⠀⠠⡄⠆⠀⠀⣀⠒⠀⢌⢀⠒⠠⠀⠁
⠀⠀⠀⠀⠣⠄⠀⠀⠄⠉⠀⠀⣀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠄⠀⠀⠁⠀⠀⣠⠀⠠⠄⡁⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠄⠀⠀⠀⠠⠀⠀⢀⠀⢰⠋⡆⡿⠀⣼⠉⠧⡏⠙⢲⣶⡄⢸⢻⠀⣸⢻⡆⣀⠄⠠⠂⠀⠐⠃
⢀⡈⠀⠀⠀⠀⠀⠀⢠⡏⠀⣿⣼⡇⡇⠀⡇⢲⡆⡧⠄⢨⡿⣅⢸⣏⠀⡿⠶⡇⠠⠆⠀⠴⠀⠐⡁⠄
⠘⠁⠀⠐⠇⠐⠀⠀⢀⠁⢸⡇⢸⠇⠧⠤⢻⡼⢹⣇⣀⢸⡇⣸⢸⠈⣧⡇⠀⠷⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠
⠀⣐⣀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠠⠀⠀⠀⡀⠀⠀⠀⢉⣀⣉⣅⠈⠀⠀⠀⠀⠀⠀⠂⠀⠃⠂⠀⠀⠁⠀⣤⠤
⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⡏⠘⠁⠀⠀⠀⡀⠒⠀⠀⠀⢠⠀⠇⠈⠀⠀⣈⡉
⠀⠀⠀⠀⠀⠞⠆⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠒⠂⣄⣰⣇⡠⠀⡀⠐⣀⣀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣈⡉
⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⠑⠃⠀⠀⠀⠀⠀⠀⠀⡀⣈⡁
⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠐⠃⠀⣠⠖⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠖⡳⢶⣤⡤⠁⠀⠀⠀⠀⣄⣀
⠀⠑⠄⠀⠀⠀⡄⣀⠄⡈⠁⠀⠈⠀⠀⢠⠀⠁⠈⠀⠀⢀⠀⠀⢀⡀⠀⠀⢀⡼⠁⡎⡀⠀⢡⠀⠀⠸⠀⠤⣅
⠀⡄⠀⠀⡀⠀⠘⠄⠚⠀⢀⠀⠀⠀⠀⠁⠈⠀⢀⠤⠤⣬⠀⠀⠎⠀⠔⠈⠉⠀⠘⠛⠁⠀⠀⢆⢀⠂⠀⠢⢄
⠀⠀⠀⠰⠀⠀⠆⠈⢄⠀⡈⠀⠈⠀⠀⠀⠀⠄⠀⣠⠞⠁⠀⠀⠀⠀⠐⠃⠐⢀⠔⠂⠀⠀⠀⠀⠁⠀⠀⠠
⣀⣀⣈⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣂⣀⣀⣈⣀⣊⣙⣲⣛⣀⣀⣃⣀⣐⣃⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣈
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⢷⣄⠀⠀⠀⠀⣠⠞⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⢷⣄⣠⠞⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⠟⢧⣄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢶⡶⠶⠶⠶⠶⠶⢾⡷⠶⣶⠾⠷⠶⠶⠿⢷⡶⠶⣿⠶⠶⠶⠶⠶⠶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⢸⣧⠞⠁⠀⠀⠀⠀⠀⠀⠙⢶⣿⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣠⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢶⡀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠿⣧⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁⠀⠀⠀⠀⢸⡏⠻⣦⡀⠀⠀⠀⠀⢀⡴⠋⣿⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⢻⡟⠛⠛⠻⣿⡛⢛⡿⠛⠛⠛⣿⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢈⣿⣏⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣴⠟⠁⠈⠳⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣴⠟⠁⠀⠀⠀⠀⠈⠳⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠖⣂⣀⣀⣀⣀⣀⣐⠢⠀⠀⠐⠒⠖⠂⠀⠀⠀⠀⠄⢀⡂⠠⠄⠀⠀⢰⢈⠉⠭⠭⢉⠒⠊⡉⠭⠭⠍⢰⠀⠀⠀⠀⠙⢮⢄⠀⠀⣀⠀⠀⣀⠀⡇
⠀⠀⡀⣇⣀⣀⣂⣀⣀⡸⠘⠀⠀⠘⡉⠉⠃⠀⠀⠀⠁⠜⣁⣀⠑⡌⠀⠀⢐⠐⢒⣀⡒⠒⢨⠀⠒⠒⠀⠒⢸⠀⠀⠀⠀⡠⣄⡙⢦⡄⠈⠲⡀⢸⠀⡇
⠀⠀⠇⠖⣤⠒⣤⠢⡔⢢⠘⠀⠀⠃⡇⠘⠀⠀⠀⠈⠀⢆⠃⠀⠠⠃⠁⠀⠠⢈⠥⠤⠤⢌⠸⠀⠀⠀⠀⠀⢸⠀⠀⠀⣸⠑⢢⠇⠀⠙⢖⡀⠈⠋⠀⡇
⠀⠀⡆⠟⣿⠛⣽⠛⡝⢫⢘⠀⠀⢂⡇⢸⡀⠀⠀⠀⠁⠀⡶⣖⡇⠈⠀⠀⠈⠐⠂⠐⠒⠒⠘⠀⠒⠀⠐⠒⢸⠀⠀⣴⠏⣴⠏⠀⠀⠀⠀⠙⢵⡀⠀⡇
⠀⠀⠇⡛⣿⠛⣿⢛⡇⢸⠠⠀⠀⠘⢭⡭⠃⠀⠀⠀⠀⠀⠛⠒⠁⠀⠀⠤⠘⠮⠍⠭⠭⠉⠾⠏⠭⠩⠭⠭⠜⠀⠸⣟⣾⠏⢀⢠⠤⠤⠠⣄⠀⠙⢇⡇
⠀⠀⠣⠭⠤⠭⠤⠥⠬⠥⠞⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠜⣌⡣⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠉⠀⡔⠃⢀⠤⠤⢤⡀⠙⢆⠀⠁
⠀⢀⣴⠖⡖⠸⠰⢒⢲⣦⠀⠀⠀⡠⠒⠉⣻⡯⢋⢓⣄⠀⠀⠀⠀⠀⡧⣭⢴⠀⠀⠀⠀⠀⠀⡜⢉⢣⠀⠀⠀⠀⠀⠀⣨⠀⢰⠁⠀⠀⠀⢙⡄⠘⡆
⠀⢸⠉⠉⠉⠁⠉⠉⠉⠩⠀⠀⡜⠀⠀⠀⡿⣗⣕⣕⣝⡆⠀⠀⠀⡸⣠⠍⣆⢧⠀⠀⠀⠀⡸⠀⢨⠀⢣⠀⠀⠀⠀⡏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡇⠀⠀⠀⠙⠉⠉⠉⢙⠀⠀⠀⢠⢡⠃⠀⠘⡌⠆⠀⠀⡰⠁⠀⠘⠀⠀⢣⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⣄⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠖⠋⠀⠀⠀⠈⠺⠀⠐⠻⢿⣯⣏⣿⡽⠾⠂⠀⠀⠀⠀⣤⠄⢠⠀⢶⠌⢠⢠
⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠑⠒⠂⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠘⠧⠠⠄⡀⠀⣤⠤⠜⠀⠀⠀⠀⠀⠀⠉⠉⡉⣱⠀⠀⠘⠃⠈⠃⠸⠋⠀⠁⠃⠀⠀⠀⠀⠀⠁⠖⠒⠒⡒⠒⠒⢢⢹⠀⠀⠀⣀⠤⠦⣼⢒⢶⣄
⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢸⠁⠀⠇⠾⠀⠀⠀⠀⢀⣤⠤⡤⢄⡀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠂⠚⢸⠀⢀⠞⠀⠀⠀⣿⢕⢕⠝⣧
⠀⠈⡎⢁⠤⣁⠤⣉⠤⡌⠁⡇⠀⠀⠀⢰⠀⢠⠉⡇⠀⠀⢀⠚⠁⣀⠤⠤⣀⠈⠳⡄⠀⠀⠀⠀⠐⣁⢿⣨⢯⣹⢏⡹⢸⠀⢸⠀⠀⠀⠀⠿⠷⠿⠿⡞
⠀⠀⡏⠻⡠⠻⠄⠟⠄⠏⠙⡇⠀⠀⠀⢈⠀⢸⠠⡇⠀⠀⣌⠀⡜⠀⠀⠀⠀⢱⠀⢙⠀⠀⠀⠀⠸⣥⠿⣬⢯⣼⡇⢹⢸⠀⠘⡄⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⣇⣰⠓⣦⠚⣦⠚⣆⣨⡇⠀⠀⠀⠋⠀⠘⢰⠃⠀⠉⠁⠀⠉⠉⠉⠉⠉⠉⠀⠈⠉⠀⠀⠀⡘⠤⠿⠬⠧⠼⠧⠜⣸⠀⠀⠈⠳⠤⠀⠀⠤⠚⠁
⠀⢀⠣⡘⢒⡉⢒⣉⠒⣁⠜⡇⠀⠀⠀⡆⠀⡆⣹⠀⠀⠉⠉⠁⠉⠉⠉⠉⠈⠉⠁⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢁⣀⣀⣀⡀
⠀⠀⢩⡉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢠⠾⣥⣁⣇⡏⡯⠽⡆⠀⠀⣋⣙⣋⡄
⠀⠀⠀⠘⡦⡀⠀⠀⠀⠀⠀⢀⠀⡠⠬⠄⡀⡀⠀⠀⠒⣉⠭⠤⠍⡁⡒⢈⠉⠤⠭⢉⡂⠀⠀⢀⠞⣘⢢⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢡⠀⡌⡃
⠀⠀⠀⠀⠀⠹⢢⡀⠀⠀⠀⡀⢸⢠⠒⢦⢹⢀⠀⠀⠀⠐⠂⠀⠒⠂⡁⠐⠒⠀⠀⠒⠆⠀⠀⠈⡮⣊⢴⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⡇⡁
⠀⠀⠀⠀⡷⢄⠀⠩⢢⡀⠀⠠⠌⢆⠀⢠⠋⠄⠀⠀⠄⡉⠤⠤⠭⠁⠃⠐⠀⠐⠀⠂⠀⠀⠀⣸⣠⣹⣄⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣸⣀⣇⠃
⠀⠀⠀⠀⣇⣀⣑⡄⠀⠊⢦⡀⠀⢸⡻⢹⠀⠀⠀⠀⠀⠐⠒⠒⠒⠂⡃⠐⠒⠒⠒⠂⠆⠀⢰⢡⠃⠀⠰⡘⡄⠀⠘⢇⣀⣉⡀⢈⣁⡠⠇⠀⠀⠑⠶⠊
⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠬⠼⠦⠀⠉⠁⠀⠀⠀⠀⠻⠭⠭⠉⠩⠅⠿⠨⠍⠉⠭⠡⠟⠀⠗⠃⠀⠀⠀⠑⠧⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⣼⡄
⠀⠀⠀⢀⠤⠐⢶⣟⢶⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡔⠒⠒⢌⠄⠀⠀⠀⡜⢸⠈⡄
⠀⠀⢠⠃⠀⠀⣿⣷⣿⣛⣧⠀⠀⠀⠛⠀⠉⠀⠛⠁⠐⠘⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⣀⢀⡀⣀⡀⠀⠤⢸⠠⠋⠱⢨⠠⠀⠀⡜⠀⢸⠀⠘⡄
⠀⠀⢸⠀⠀⠀⠉⠉⠉⢙⠇⠀⠀⠀⣠⢲⠆⠤⠤⠤⠄⠒⠐⡄⠀⠀⠀⡇⠥⠤⠤⠥⠤⠤⠡⠄⠀⠀⠐⡇⠀⠐⠂⢱⣀⣠⠁⠂⠀⣼⣀⣀⣸⢀⣀⣘⡄
⠀⠀⠀⠳⣄⠀⠀⢀⡠⠊⠀⠀⠀⠀⠙⢼⢂⠂⢒⡒⡂⣠⡀⠇⠀⠀⠀⠧⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠐⠃⠀⠀⠀⠸⢝⡻⠀⠀⠀⠙⠛⠓⠷⠖⠒⠛⠁
⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠻⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄⠀⠀⠀⡇
⠀⠀⠀⠀⣠⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⡄⠀⠀⠀⢀⡴⠖⠒⠒⠛⠒⠃⠀⠀⠀⠳⠒⠒⠒⠒⢲⡄
⠀⠀⠀⡞⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠎⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⢳⠀⢇⣀⣀⣀⡀⠤⠤⠤⠤⠤⠤⢤⡇⠀⠀⠘⡆⠀⠓⠂⠉⠉⠉⠙⠀⠀⠀⢠⠖⠒⠒⠒⣲⠏
⠀⠀⠀⠸⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠎⠀⠀⠀⠀⠳⠤⠤⠤⠤⢤⠀⠀⡇⠀⠀⠈⡶⠶⠒⠚⠁
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢃⠀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⣈⣉⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢇⡀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠙⠲⣄⠀⠀⠀⢠⠞⠀⠈⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⣀⠽⠋⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⢀⡧⣀⠀⠀⠈⠙⠦⣴⠋⠀⠀⡠⢺⠁
⠀⠀⠀⠀⢠⡴⠖⠒⠒⠒⠛⠛⠛⠉⠉⠉⠉⠉⠉⠙⡇⠀⠀⠀⠀⢼⡀⠀⠑⠢⣄⠀⠀⠀⠀⢀⠞⣡⠞
⠀⠀⠀⢀⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠉⠲⣄⣠⠎⠀⠀⠀⠀⠈⠛⠧⣄⡀
⠀⠀⠀⢾⠀⠘⠦⠤⠤⠤⠒⠒⠒⠒⠒⠒⠚⠉⢉⡽⠋⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⡰⠒⠢⢄⡀⠀⠀⠉⣹
⠀⠀⠀⠘⣦⠤⠤⠤⠤⠤⠤⠶⠖⢒⣶⡚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠⣏⠀⠀⢀⠞⣠⢤⣀⠀⠉⠲⢤⣔⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢏⣀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠙⠦⠊⡰⠃⠀⠈⠓⠦⣄⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁

⠀⠀⣤⠀⣀⣀⠀⠀⢠⡄⠀⠀⠀⠀⡞⣩⣭⣭⣍⡍⠆⠀⠀⠀⠀⢠⠶⠶⠶⠶⡄⣚⡇⠀⠀⠀⠀⢰⠒⠒⠒⠒⠒⡆⠀⠀⠀⠀⠀⢰⢫⣩⢩⢩⢳
⠀⠀⣿⢉⣛⣛⣉⣉⢹⡇⠀⠀⠀⠀⡇⣉⣉⢉⢉⡁⠀⠀⠀⠀⠀⢸⠀⣒⣒⠀⡇⢹⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠘⢸⢸⢸
⠀⠀⣿⣨⣭⣉⣩⣭⣸⡇⠀⠀⠀⠀⡇⠒⢲⢰⢸⡇⠀⠀⠀⠀⠀⢸⢀⣂⣀⡀⡇⡼⡇⠀⠀⠀⠀⣸⢻⠛⠛⡟⡏⢇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠙⠀⠁⠀⠀⠉⠘⠃⠀⠀⠀⠀⠳⠭⠬⠬⠬⠥⠆⠀⠀⠀⠀⠸⠦⠤⠤⠤⠇⠘⠀⠀⠀⠀⠀⠿⠿⠧⠤⠼⠿⠿⠀⠀⠀⠀⠀⠘⠦⠤⠥⠤⠞


⠀⠀⠀⣠⠖⠛⠓⢦⡄⠀⠀⠀⠀⠀⢀⡴⠚⠛⠳⣤⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⠶⣄⠀⠀⠀⠀⠀⠀⡴⠞⢛⠓⢦⡀⠀⠀⠀⠀⠀⢠⡴⠚⠛⠲⣄
⠀⠀⢸⡇⢤⣤⣤⡄⢻⠀⠀⠀⠀⠀⣾⠀⣤⣿⣤⠈⡇⠀⠀⠀⠀⢠⡏⠀⣷⣞⠀⢹⡄⠀⠀⠀⠀⢸⠁⣤⣭⣤⠀⣷⠀⠀⠀⠀⠀⡟⠐⠒⠒⠒⢸⡆
⠀⠀⠈⢧⣀⠀⢀⣠⠟⠀⠀⠀⠀⠀⠹⣆⡀⠛⢀⡼⠃⠀⠀⠀⠀⠀⢳⣌⠁⠈⣠⡞⠀⠀⠀⠀⠀⠘⢦⡀⠚⢀⣰⠏⠀⠀⠀⠀⠀⠻⣌⠉⠉⣉⡼⠁
⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⡂⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⡆
⠀⠀⢸⣿⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣉⣙⣿⣧⠀⠀⠀⠀⠀⠀⠀⢴⠟⣧⠀⠀⠀⠀⠀⠀⢰⡇⢀⡶⠳⣆⠀⡇
⠀⠀⢸⣿⣤⣽⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠠⡌⠈⠹⣧⠀⠀⠀⠀⠀⠸⣇⣀⣻⣞⣃⣀⡇
⠀⠀⠀⠁⠁⠁⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⡤⢤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢰⢋⠓⢸⠐⢉⢳⠀⠀⠀⠀⠀⠰⣦⡖⠐⣆⢠⠆⠀⠀⠀⠀⠀⢀⣿⣍⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⣡⠟⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⠉⠁
⠀⠀⢸⠀⣱⣸⣰⠃⢘⠀⠀⠀⠀⠀⠰⠛⠧⠀⢹⡞⠀⠀⠀⠀⠀⠀⠀⣿⠀⢄⣠⠄⠀⠀⠀⠀⠀⠀⢹⣶⠋⠀⠉⠁⠀⠀⠀⠀⠀⠉⢳⣼⠁
⠀⠀⠈⠣⣃⢸⠐⡲⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡼⢳⡀⠀⠀⠀⠀⠀⢠⡾⠉⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⣠⠖⣛⣛⣲⣄⠀⠀⡖⣉⣉⣓⢲⡀⠀⢸⣷⣄⣠⣦⠀⠀⢀⣠⠶⠶⣶⡄⠀⠀⠀⢠⡆⠀⠀⠀⠰⣿⣇⠀⠀⣶⣶⣶⠀⢰⣿⡆
⠀⠀⣼⣱⠋⠁⠀⠙⣿⡄⢸⣿⣿⣿⣯⣼⢧⣄⠈⢿⣿⣿⣿⠀⣰⡟⠁⠀⠀⠈⢿⠀⠀⣤⣾⣷⣶⣦⡀⠀⢹⣿⡀⠀⠈⠛⣁⣀⠸⠿⠃
⠀⢰⡿⣁⠄⣒⡒⠠⣸⠇⢸⣿⣿⣿⣿⣿⣤⠟⠀⠈⢻⡿⠃⣼⠏⠀⠀⠀⠀⠀⢸⠇⠘⢿⣿⣛⠛⠛⠁⠀⢴⡍⣞⣄⠀⢸⡿⠛⠿⠿⢿⣿⠆
⠀⣸⠁⠁⣿⣩⣟⢠⣿⠀⠹⠿⣿⣿⡿⠁⠀⢀⢀⡤⣄⠀⠀⡏⠀⠀⠀⣀⡀⠀⣸⠀⢀⣄⣹⠛⣿⣷⠀⠀⢸⡛⠉⣿⠀⣼⡇⢰⣶⠂⢠⣿
⠀⢹⣠⠀⣿⠤⡗⢸⡇⠀⠀⠀⣀⣤⣴⣶⣾⡿⣏⣿⢿⡆⠀⡇⣰⣿⣀⣿⣿⢀⡏⠀⠘⠿⢿⠟⠛⡛⠀⠀⠀⢻⡭⠷⠂⣿⡁⠘⠛⠂⢸⡏
⠀⢸⠃⢢⡜⢹⡀⣾⠁⠀⣞⣷⣙⣛⣯⣽⢶⠻⢉⢲⡄⢿⡀⠁⣿⡿⢻⣿⣿⠈⢀⣀⠀⠁⠸⠃⠈⠃⠀⠀⠀⠀⠀⠀⠀⠿⠿⠷⠶⠾⠿⠃
⠀⢸⠀⡏⣤⠀⣷⡏⠀⠀⢹⣿⢉⡝⣥⡴⣤⢿⠘⠉⠀⣈⡧⠀⣿⠁⠈⠿⠁⠀⣿⢿⠇⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣠⣾⢦⡀
⠀⣿⢸⣧⡿⢰⣿⢀⣶⣦⡀⢿⡆⠉⠉⢉⣡⣬⠶⠞⠛⠋⠁⢀⣀⣀⣀⣀⣀⣀⡀⠁⠀⠘⠿⠿⠿⠿⠿⠟⠛⠛⣯⠁⢀⣀⣀⠘⢻⣷⣿⣧
⠀⣿⠈⡟⣳⢻⠇⢸⣿⡿⣧⠈⠿⠶⠛⠋⠁⠀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡐⠗⠀⠀⠀⠀⠀⠀⠰⠿⣿⠿⠶⠆
⠀⠸⣧⣙⣡⠟⠀⢸⠁⠀⢹⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠠⣶⣶⣶⣿⠦⠀⢠⣄⢀⣤⢀⣶
⠀⢀⠈⠉⠉⢠⡄⠘⣧⣤⣼⡆⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠿⠿⠿⠿⠇⠀⠀⣿⣿⣿⣿⡟
⠀⢸⡗⢻⣟⣉⣷⠀⣿⣿⣿⡇⢰⣿⣿⡏⣿⢹⠏⣿⡿⠋⠀⣿⠉⡄⠈⣿⠁⠴⠀⠸⡟⢉⡉⢻⡇⣿⣿⡆⢸⣶⣶⣿⡇⠀⠀⠹⠿⠿⠛⠁
⠀⠘⢷⣾⣽⠿⠋⠀⠈⠉⠁⠀⢿⣿⣿⣿⣾⣿⣿⣾⣷⡆⢀⣿⣤⣤⠀⣿⣷⣤⡆⢰⣇⠈⠁⣼⣷⣿⣿⣿⠈⣛⣛⣛⡁⠀⢿⣦⠀⠀⢀⣴⠄
⠀⠀⢀⣤⣤⡀⠀⢠⣶⣶⣦⡀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⡋⢁⣰⣿⣿⡋⢀⣼⣿⣿⣿⣿⣿⣿⣿⡟⠀⢻⣿⣿⡇⢀⣾⣿⣿⣿⣿⡟
⠀⠠⢾⣿⣿⡿⠒⠻⣿⣿⣿⡏⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠈⠉⠉⠀⠙⠉⠉⠛⠻⠿⠇
⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠉⣠⠖⠛⠲⡄⠉⠙⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢁⠀⠀⢀⣠⠴⠖⠚⠛⠛⠛⠛⠒⢦⡀
⠀⠀⣤⣶⣾⣿⣿⣿⡀⠀⠀⣧⢤⣤⡴⢻⣤⡀⠀⢀⣴⠆⠀⠀⠈⠉⠉⠉⠭⣍⠉⠁⠀⠀⣀⣀⣴⣽⣷⣾⣿⣶⣛⣛⣛⣛⣛⠛⠛⠓⢿⣧
⠀⠀⠈⢨⡽⢿⣿⡝⠇⠀⠸⡿⠤⠤⠖⠚⣽⠁⠀⣼⣯⣤⠄⠀⠀⠀⣀⣀⣖⠋⠀⠀⠀⠘⢄⢹⣟⡛⣏⣛⣿⣟⣉⣉⣉⣉⣛⡛⠛⠿⣿⣯
⠀⠀⣠⡞⠀⢼⣿⣷⣄⠀⠀⠙⢦⣀⣠⡴⠋⠀⠐⠛⢻⡏⠀⠠⡾⠿⠿⠛⠻⢿⣷⣦⡀⠀⣨⠿⣿⢿⣿⣿⠉⠉⠁⠀⠀⠉⠉⠉⠓⠲⢿⡇
⠀⠀⠉⠀⠀⣼⣿⣿⣿⣧⠀⠀⠀⠈⣡⣤⣤⣤⣤⡀⠚⠀⢀⣀⣤⣤⣴⡶⣶⣤⠈⠙⠛⠀⠳⠚⠁⠀⠉⠚⠃⣴⣶⠀⠀⠀⢰⡄
⠀⠀⠀⢀⣼⠟⠛⠛⢿⣿⠀⠀⠀⠶⠷⣶⣭⣽⣿⣁⢠⡾⡟⡽⣅⣤⠐⣚⠇⢹⡄⠀⠀⠀⣤⣤⣤⣄⠀⠀⠼⢿⣿⡟⣀⠀⣾⠗⠛⠻⣆
⠀⠀⠀⢸⡇⠀⠀⠀⠘⣿⠀⠀⠀⣶⣶⣶⣶⣬⡍⠁⣿⢨⣻⡷⢃⣭⣤⠤⡀⠸⡇⠀⢠⣾⣿⣿⣿⣿⣇⠀⠀⠸⣿⡿⠁⠀⣿⢀⣠⡀⢻
⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡆⠀⠀⢹⠟⠉⠛⢿⠁⠀⣧⠘⠘⣧⣿⡋⣴⡝⣾⡆⣷⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⢰⠏⠀⠀⠀⣿⠀⠋⠁⢸⡇
⠀⠀⠀⠸⣷⠀⠀⠀⠀⣿⡇⠀⠀⢸⣷⣴⣶⡏⠀⠀⣿⠀⠀⢸⠳⣓⠤⢖⡟⢁⡟⠀⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⣿⡄⠀⢀⣸⡇
⠀⠀⠀⠀⣿⣶⣶⣶⣾⠿⠇⠀⠀⢸⣿⣿⡿⠀⠀⠀⠸⣧⣀⣸⣄⣤⡭⠷⠶⠛⠁⠀⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠸⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉
⡿⠿⠿⠿⠿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣻⣿⣻⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⣀⢀⠀⣄⡐⠀⢀⢀⢀⠢⡐⠂⠀⠙⠙⠉⠈⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠚⠀⠁⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠒⠠⡀⠀⠐⠁⠢⠀⠀⠀⠁⠋
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠚⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⠀⠠⠀⠈⢠⡄
⠀⠀⠀⠀⠀⠀⢈⠂⢀⠀⠀⠂⠐⠈⠀⠀⠀⠀⡀⠀⠀⡆⢀⠢⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⢸⡇⡀⠀⠈⣤
⠀⠀⠀⠀⡀⠸⠮⠀⠀⠁⠢⢄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠸⠠⠊⡀⠀⠀⠘⠆⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠄⢸⡇⢰⣶⠀⣿
⠀⠁⠀⠀⠁⠃⠀⠀⣀⣤⠄⠀⠑⢄⡴⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠈⠉⠉⠈⠉⡉⠉⢩⠉⢩⡍⠉⡏⠉⠃⠐
⠀⢠⣂⡀⠀⠀⠀⠀⠛⠁⠀⠀⡰⠋⡠⠒⡀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠉⠠⣉⠕⠊⠝⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⠊⠀⢈⠁⠥⠐⠂⠃⠀⠀⢀⡡⠤⠔⠒⠓⠋⠀⣀⠀⠀⢀⠀⠰⠢⠀⠒⠀⠁⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢁⢊⡀⠀⠁⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠰⠉⠀⠀⡀⠀⠐⠀⠆⠀⠀⠠⠀⡀
⠀⠀⡠⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠘⠀⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣀⠀⠀⠀⠦⣀⠘⠀⠀⠀⠀⠀⠀⠀⢠⢹⣿⣷⡄
⠀⠀⠃⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠰⣉⠀⠁⠈⣱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠈⠻⣿⠄
⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠻⠶⣲⠴⠟⠁⠀⢀⠀⠀⠀⠀⠀⠀⢀⠆⠑⠒⣀⣤⣧⡄
⠀⠀⠀⡀⠀⠀⠂⣀⠀⠃⠂⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠐⠫⡨⠆⠀⠀⠀⠀⠀⠸⠄⠨⠜⠉⠛⠛⠁
⠀⠀⠰⠀⠐⠚⠀⠀⠀⠀⡀⡆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⢀⠀⠈⠃⠀⠀⠀⠀⠀⠰⠱⠀⠀⠀⠀⢀⡀⠀⢠⠀⣀⠀⠀⡀⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠀⠀⠀⠄⡒⠇⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⠘⠀⠙⠃⠐⠂⠇⠘⠘⠃
⠀⠀⢠⠂⠀⠀⠀⠀⢸⠀⠀⠃⡀⠀⡜⢀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡰⢢⠕⠂⠀⠀⠀⠀⠀⠀⢀⡈⠀⡈⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠈⠒⠀⠀⠀⠀⠈⢸⠀⣦⠆⠀⡏⠀⠑⠀⣴⣁⣀⣩⠆⠀⡠⢊⠔⠁⠀⢀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠈⡂⠀⠀⠀⠐⠀⢀⣠⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠑⠛⠀⠀⠁⠀⠀⠀⠈⠉⠉⠉⣀⠀⡰⠁⠀⠀⢠⢠⠀⠀⠀⠀⠀⠁⠀⠀⢀⠔⠁⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⠀⠀⠀⠘⡀⡄⠀⠀⠀⠀⠀⣮⣤⣼⠂⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⣀⢀⠀⡠⡀⢠⡄⡄⠀⡄⢠⡄⠀⠀⠀⠀⠈⠱⠃⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠉⠀⠀⠀⠉⠀⠀⠀⢀⡡⠶⠂⡀⢀⠀⠀⠀⠀⠀⠀⡀⡀⠤⢀⢄⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⠀⠐⠈⠀⡠⠔⠁⠁⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠓⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢀⠠⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠸
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠰⠊⠉⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐⠀⠐⠀⠀⠀⠈⢀⠈⡄⠀⠀⠀⠀⠀⠠⠈⠀⡀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⢒⠒⢢⡖⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⡐⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣠⡶⣶⣲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰
⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠁⠀⠀⠻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸
⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠚⠙⡄⠀⠀⠀⠀⠀⠀⢸⠀⢰⠉⢙⡀⠀⢠⠞⠀⠀⠀⠀⣶⣶⣶⣶⣶⠆⠀⠁
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⠈⠀⠸⡄⠀⠀⠀⠀⠀⢸⣠⢁⡞⠛⠀⠉⠀⡀⠀⠀⠴⠲⢄⠈⠀⠀⣃⠄⠀⠰⡄
⠀⠀⠀⢠⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢣⡀⠀⢳⠀⠀⠀⢠⠐⠉⠉⠛⡀⠀⢀⠔⣨⠴⠶⠆⣶⡴⣎⠀⠀⠀⣡⡇⠀⠾⠎⣄
⠀⠀⠀⠸⡀⠀⠀⠀⠀⣀⠀⡄⢂⠐⠚⢓⢂⠁⠀⠷⠛⢠⠲⣄⠀⡁⠀⡀⠄⢆⡙⠀⢂⠞⢁⣴⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡴⢠⡌⡆
⠀⠀⠀⠀⠁⠀⠀⠀⠘⠑⠀⠀⠀⠀⠀⠉⠈⠈⠈⠀⠀⠀⠑⢌⠣⡙⠢⣄⣤⠾⡳⢣⠏⢀⠎⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠇⢆⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠶⠬⢀⣀⣨⢟⡎⢢⡙⢦⣿⣫⠟⣠⠋⢀⣎⡠⣄⠀⠈⢻⣢⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠠⠀⠸⠓⠿⠓⠀⠀⠀⠀⠀⢸⡇⢞⣇⠄⠵⡌⠻⣡⠞⠁⠀⠘⡇⣀⣸⠁⠈⠈⠁⠀⠀⠀⠀⠠⠄⣰⠀⠈
⠀⠀⠀⠌⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠩⠅⣸⠷⠋⠈⠁⠈⠉⠳⢍⡛⠚⠛⣋⣴⣿⣟⣳⣲⣆⣶⣖⣖⣖⣖⣖⠾⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠤⠤⡀⠀⠀⠀⢀⣀⣀⢈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠀⠀⠈⠀⠁
⠠⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⠴⢠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠀⠄
⣇⠀⠀⠀⣠⣀⡀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣰⢾⣟⡓⣦⠀⠀⠀⠀⠀⠀⠀⠀⢉⡱⠯⠴⠁⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⢂⠚
⠀⠉⠈⠁⠉⢳⠉⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠺⠞⣀⣿⡼⠀⠀⣀⣀⣀⣀⡠⢃⣀⠃⢀⣄⠀⢠⢸⠤⡄⠀⠀⠀⠀⠀⢡⠏⠄
⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠿⠏⠀⠀⠈⠂⠠⠤⠄⡀⢭⠭⠬⢭⣀⡱⢆⠻⠂⡄⠀⠀⠀⠀⠰⢃⢮⠠
⠀⠀⠀⠀⠈⢳⠀⠄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠠⣾⠆⠀⠀⠀⠀⠀⠀⠀⠠⠆⣉⠀⠀⠁⠀⠈⠹⠛⠁⠀⠀⠀⠔⡁⡔⢡
⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠤⠤⠤⢤⠤⣤⢤⣤⣤⢤⣤⣤⣤⡎⠝⠉⣧⣀⣂⡅⠀⠀⠀⠀⢀⢌⡴⠁⠁
⠀⠀⠀⠀⠈⠳⠄⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⡀⣠⠠⠄⠠⠀⢸⣶⡾⠶⢏⡀⣈⣀⡀⡀⠀⢴⣴⡗⡴
⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠀⣰⣿⢋⡞⠁
⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠱⠋
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠢
⠀⠀⠀⠀⠈⢦⡠⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠄
⠀⠀⠀⠀⠀⠀⠙⠦⠥⠬⠤⠤⠤⠥⠄⠰⠶⠶⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢂⠔⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠃⠌⠔⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠠⠂⠀⠁⠂⢊⠌⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⣠⣄⣄⠀⠀⠀⠠⢊⠔⠀⠐⠔⡁⣀⡀⠈⡡⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠀⠐⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠉⠁
⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠈⡀⠐⠠⠂⠀⠔⡠⠂⠀⢀⢀⠔⡠⠀⠄⠀⡀⠀⠀⠀⠀⢀⠀⡀⢀⠀⡠⢀⠄⠀⠀⡠⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⠤⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⣀⠀⣀⣂⣄⠀⡈⠀⣀⠀⠀⠀⣄⠀⠀⠀⠀⠁⡘⠃⠀⢀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠠⠉⠏⠥⠈⠉⠀⠀⠀⡃⠀⠉⠠⠄⠁⠀⠀⠀⠀⠀⠠⡏⠠⠔⠁
⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠕⢂⠀⠀⠀⠀⠀⠀⢀⠄⠄⠀⠀⠀⠀⠀⠀⠒⠒
⠀⠀⠀⠀⠀⠀⢈⡁⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠈⠀⡀⠂⠀⠄⠠⠀⠃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀
⠘⠓⢾⡇⠀⠄⠀⠰⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠡⠀⠂⠀⠀⠀⠐⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠠⠐⠀⢀⠀⠀⢠⡥⠀⢀
⠃⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠈⠀⡡⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇⠐⠂⠄
⠀⠀⠂⠀⠀⡠⠊⠀⠀⠀⠀⠘⢘⠀⠀⠀⠀⠀⠀⠀⠁⢤⠀⣀⠀⠀⠀⣀⠀⠀⠠⠒⠁⡀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠒⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠊⠀⠐⠀⠀⠀⠀⠁
⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠈⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠸⠀⠅⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠴⠶⠶⠄⡂⠀⠂⠶⠀⠀⠈⠁⠙⠂
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠃⠴⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠠⠄⠙⠢⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡉⠀⢀⣀
⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠠⠀⡐⠚⡃⢀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢨⠻
⠀⠀⠀⠀⣼⠇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠠⠉⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠃
⠀⣀⡀⠀⠀⠀⠀⡀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠀⠀⢀⠂⡀
⠀⠙⠟⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠴⠠⠒⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠈⠈⠀⠀⠀⠇⠘⠡⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢠⠋⠀⠀⠀⠈⠀⠀⠜⠁⠀⠠⠀⠐⠀⠀⠀⠂⠀⡠⠠⠀⠁⠻⠀⠑⡀
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠰
⠀⠀⡀⠀⣀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠁⠀⠀⢠⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⣿⣿⣿⢛⣿⣿⣿⣿⣿⣿⣋⣁⣆⡀⣀
⠀⢳⠀⠀⠀⠈⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⣿⣿⣏⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣾⣿⣿⣼⣯⣟⣿⣽⣿⣿⣿⣿⢿⢫⣿⣷⣾⣷⣶⣶⠄
⠢⡈⠒⠀⡀⠀⠀⠀⠀⠠⣀⡎⠀⠀⠈⠁⠂⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣯⣙⣙⢳⣇⢹⣿⣿⣿⣿⠟⠃
⢀⡀⠄⠋⠃⠐⠃⠠⠀⢄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⡉⠀⠛⠛⠫⠙⠉⠻⠛⠿⠛⠋⠛⠟⢉⡏⠉
⠈⠟⠀⠀⠰⡀⠁⠠⠑⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠒⢀⣀⠄⠓⠀⠀⡀⠀⠀⢠
⠀⣷⠂⠀⠀⡇⠀⠀⢀⡐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠃⠀⠀⠉⠓⠂⠀⠠⠄⠀⠀⠐⠂
⠀⠉⠁⠀⠀⠀⠀⠀⠂⠐⠀⠂⠉⢀⠐⠒⠀⠀⠀⣀⡀⢀⠀⠀⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣴⠷⢦⣴⠏⠉⠉⠙⢦⡀
⠀⠀⠀⠀⣤⣀⠀⠀⣀⣤⠶⠛⠉⠉⢛⡿⠃⠀⠀⠉⠀⠀⠀⠀⣸⡇
⠀⠀⢀⡼⠋⠙⠳⠛⠉⠀⠀⠀⠀⠠⣿⣅⣀⡀⠀⠀⠀⠀⠀⠛⢯⣄
⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⡆⢀⣠⣤⣄⣀⡼⠃
⠀⠈⠙⠶⣤⣀⠀⠀⠀⠀⠀⣀⣤⡤⠤⠶⠶⠶⠞⢡⠟⠁⠀⠈⠟⠁
⠀⠀⠀⠀⠀⠉⠛⠛⠒⠒⠛⠁⠉⠳⢦⣀⠀⠀⠀⢺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⠀⢙⣿⠃⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⠁⠀⣠⠞⠛⠛⣦⡴⠋⢙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠉⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠙⣷⣠⠶⠻⣦⠀⠀⠀⠀⠀⠈⠻⣶⠄⠀⠀⠀⢀⣽⣦⣀⣀⣸⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠁⢀⡴⠾⠇⠀⠀⠀⠀⠀⢾⣅⠀⠀⠀⠀⢺⡁⠀⢻⡍⠁⢀⣼⠃
⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠸⡇⠀⠀⢻⡴⠋⠹⣆
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠻⣄⣀⣠⡾⣄⠀⠀⠀⠀⠀⠈⠳⣄⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠈⠉⠉⢀⣹⠆⠀⠀⠀⠀⠀⠀⠈⠛⣿⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡾⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠤⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⢄
⠀⠙⠋⠂⠀⠀⠀⠀⠈⠰⢄⠀⠐⠄⠀⠀⠀⠀⠀⠐⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠲⣤⣐⣀⣀⣀⠀⠀⠀⠀⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠠⠸⠀⠀⠀⠀⠀⠀⠄
⠀⢀⣀⠀⠀⠀⠀⠀⠀⠢⡊⠀⠀⠓⢄⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠁⠒⠈⡁⠀⠀⠈
⠀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠴⠦⢄⠀⠀⠀⣀⡀⡀⠀⠠⠤⠠⠤
⠀⠈⠀⠈⠀⠀⠀⣀⡀⣀⠀⠀⠀⠀⠂⠤⠄⢀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢄⡀
⠀⠐⠀⠒⠃⠈⠁⠀⡀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠉⠀⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠂⠈⠓⠒⠉⠉⠀⠀⠠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠲⣤⠀⢠⡄⠈⠢
⠂⠀⠀⡀⢀⡀⠀⠀⠀⠀⠈⠁⠁⠈⠑⠋⠂⠊⠒⠀⠀⠐⠀⠀⠀⠀⠀⠀⠳⠀⠀⠜⡀⢀⠳⢀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠐⠲⠦⠀⠀⠀⠀⠀⠀⠄⡀⠀⠐⠀⠂⠀⠀⠀⠀⡂⠘⠀⠀⠀⠀⠈
⠕⠀⠀⠄⠁⠀⠀⠀⡀⠀⠀⠀⠀⠢⠀⠀⠃⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠠
⠀⠐⢆⠀⠀⠀⠀⠀⠐⠂⠀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢊
⠀⠀⠀⠑⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠄⠀⢣⠄
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⡠⠃⠀⠀⠀⡐⠀⠀⢀⡀⠀⠉⠉⠙⠀⠀⠘⠀⠈⠆
⠀⠄⠀⠀⢀⠀⠀⢠⢀⡀⠀⠀⠈⠰⢤⡀⠈⠀⠀⠐⠀⠀⠀⢀⠉⠈⠓⠒⠛⠢⠤⢌⠀⠀⠀⠁⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠂⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠰⠱⠁⠀⠈⠙⠦⠤
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈⠀⠀⠉⠐
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠃⢀⠀⠀⡂⠁⠀⠈⡀⠅⠀⠀⠀⠀⠀⠛
⠀⠀⠉⢀⠈⠉⡁⠀⢉⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠆⠀⠀⠐⠀⠈⠤⠀⠀⠊⠀⠀⠀⠀⠀⡀⡅
⠐⡘⠄⠀⠀⠀⠙⡀⠀⠀⠀⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠶⢂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠻⡁⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⠀⠀⠀⠂⢁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠐⢠⠦⢸⡀⠀⠀⠀⠋
⠀⠘⠁⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡢⣀⠈⠁
⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⠓⣾⠎⠀⠀⠀⠠⠈⠉⠉⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⡄⢠⡆
⡌⠀⡀⠀⠀⠴⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⡀⠀⠠
⠀⠀⠈⠀⠀⠈⠀⣼⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀
⡆⠀⠀⢀⡄⠠⠀⠃⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⣂⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠀⠀⠉⠀⠀⢀⠒
⠀⠀⠈⠀⠀⠀⠀⠀⢀⡀⢠⠀⠀⠀⠀⠀⢀⠀⠈⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⡀⠀⢄⠀⠄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢠⣀⣀⣀⣀⠀⢀⣀⠈⠁⠀⠀⠀⠀⠀⠸⠀⠀⡀
⠀⡀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⢤⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠎⠐⠂
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⠄⢈⢤⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢐⠈⠀⠀⠀⠀⠀⠂⠂⠂
⠀⠀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠀⠠⡆⠀⠀⠀⠈⠈⠉⠃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⡆⠂⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠐⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠬⠀⠀⠀⠀⡀
⠀⠀⠈⡁⠀⢀⠀⠀⠠⠄⠐⠀⣠⠀⡀⠀⠞⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⢄
⠐⠀⠁⠢⣤⠋⠀⠀⠀⢠⠸⠈⠉⠀⠩⠄⠀⠐⠐⠚⠀⠀⠒⠀⡀⠀⢀⠘⢩⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡠⠀⠀⢀⠀⠀⠑⡄
⠀⠀⢀⢀⡄⡀⠀⠀⠀⢰⡀⠀⠄⢃⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠈⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠂⠄⠠⠀⠀⠈
⠰⠀⠃⠀⠀⠀⠄⠠⠂⠀⠁⠀⠀⠀⢃⠀⠀⠀⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠆⠀⠀⠂⠀⠃⠀⣠⠄
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⢡⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⡀
⠀⠶⢤⣦⠂⠀⠀⠀⡈⠐⠲⠒⠂⢸⠀⠀⠀⠀⠁⠀⡠⠀⢀⡄⡀⢀⠉⠀⠠⠄⠀⠀⠀⠀⠒⠎⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⢠⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠁⠀⠐⠀⠀⠀⠀⢀⠄
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠂⠀⠀⠄⠀⠉⠐⠀⠀⠓⠀⣧⡄⠀⠀⠀⠀⠀⠀⠀⣀
⡠⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⠐⠀⣠⠀⣀⠀⠀⠀⠀⠩⠕

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠱⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠢⡀⠀⠀⠀⠀⠋⠀⠡
⠈⠁⠉⠉⠉⡆⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢸⠀⠈⠢⠀⠀⢀⣷⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⡀⠞⠈⠄⠀⠀⣰⠉⠀⠈⠄⠀⢘⡃⠀⠀⠀⠀⢸⣯
⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢂⠀⠀⠀⠀⢯⡀⠀⢀⠀⢀⡀⠑⠀⠀⠀⠀⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣁⣀⣀⣀⣲⣤
⣿⣿⣩⣩⣍⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⠔⠒⠒⠊⠉⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠐⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠀⠀⠀⠂⠀⠀⠀⣿⢢⣄⣀⡀
⠀⠀⠀⠀⠀⠀⡏⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⣿⢸⣼⣿⡇
⠀⠀⠀⠀⠀⢸⠃⡟⠀⠀⠰⠄⠀⠀⠀⡇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠒⠈⠀⠀⠃⠀⠀⠀⣿⠸⣿⣿⡇
⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠐⠀⠀⠇⠰⠂⠀⠁⠀⠀⡀⠀⠀⡄⠀⠀⠀⠂⠀⠀⠀⣿⢀⣼⣿⡇
⠀⠀⠀⠀⠀⢸⢠⡇⠀⠀⠛⠀⠀⠀⢀⠀⡀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠀⣿⡇
⠀⠀⠀⠀⠀⡟⠸⠇⠀⠠⠆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠂⡠⢀⠀⠀⠀⠀⠀⣿⢀⠀⣿⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡄⠀⠀⠀⠀⠀⠀⠠⠂⡐⢀⠂⠀⣄⢀⠀⢠⡟⢸⢠⣿⡇
⠀⠀⠀⠀⠀⡇⢀⠀⠀⣠⡤⠀⠀⠀⢰⡿⠀⡖⢰⠀⠀⠀⢠⠄⠀⠈⠁⠀⠘⠁⠀⠀⢸⡇⣼⢘⣿⡇
⠀⠀⠀⠀⠀⡇⢸⠀⠈⠁⠂⠀⠀⢠⡿⠀⠀⢷⠈⣹⠀⠀⠨⠀⠀⠈⠐⠠⠀⠀⠀⠀⢸⡇⡋⣼⣿
⠀⠀⠀⠀⠀⡇⢸⠀⡀⠀⣄⠀⠀⣼⠁⠀⠀⠸⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠆⠀⢸⡇⡇⣿⣿
⠀⠀⠀⠀⠀⡇⣾⠀⠁⣀⠂⠐⠻⠃⠀⣀⠀⠀⢰⠋⠉⡆⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢸⡇⡇⣿⣿
⠀⠀⠀⠀⠀⡇⠛⠀⠀⠀⠀⣸⠏⠀⠿⠷⠀⠀⠈⠂⠀⠁⠇⠀⠀⢀⠔⠩⡇⡠⣁⠀⢸⡇⡇⣿⣿
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣠⡎⠀⠀⢀⢀⣤⡆⣴⢠⢤⡰⠤⣀⠰⠁⠀⠀⠔⠁⠾⠀⢸⡇⡇⣿⣿
⠀⠀⠀⠀⢸⠇⢠⠀⠀⢘⡟⠀⣴⣷⢻⣾⣿⡇⡏⣿⢸⢃⡠⠊⠀⠀⠀⣀⣀⡠⠂⠀⢸⡇⡇⢸⣿
⠀⠀⠀⠀⢸⠀⣾⠀⢤⡾⠁⡸⠟⣛⣸⣿⣿⠇⢿⣿⢸⣨⣶⠀⠀⠐⢻⠀⢸⠀⠀⠀⣸⡇⡇⢸⡇
⠀⠀⠀⠀⢸⠀⡏⠀⢀⠤⡀⠃⠀⠀⣠⢄⣹⣿⢘⡩⠮⣬⠒⡆⠀⠀⡾⠀⢬⠀⠀⠀⣿⡇⣇⣼⡇
⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⡄⠀⠐⢸⣿⣾⢡⣿⠜⡬⠿⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⡇
⠀⠀⠀⠀⢸⠀⡇⠀⠀⠰⠐⠁⠀⠀⠈⠉⠁⠉⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣶⠶⢿⠁⣿⣿⡇
⠀⠀⠀⠀⢸⠀⠇⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣴⠶⠶⠟⠻⡿⠿⡟⣞⠿⠯⠽⣦⣾⣶⡘⢦⣿⣿⡇
⠀⠀⠀⠀⠘⣴⣖⡺⢿⣿⣿⣻⣻⣝⣯⡧⠽⠦⠤⣄⣻⣿⠶⠤⠤⠐⠒⣊⣉⣩⡿⠥⠴⣀⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠪⡻⣾⡉⠡⠤⠄⠒⢒⣂⣉⣉⣁⣭⣄⣀⣀⣤⣤⣼⣷⣾⠷⠶⠟⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣿⡿⠿⠿⠿⠟⠛⠛⠛⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣲⣦⣼⣶⣶⣿⣥⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠞⠿⢿⣿⣿⣿⣿⣿⡿⢿⣿⢉⡉⢹⣿⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡏⣡⡔⡶⠆⠈⢹⣿⣿⣯⡐⠀⣁⢠⡤⣈⠁⠈⢻⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣠⡶⢀⠀⢄⡁⠀⣿⡟⠀⢀⡚⢿⠈⠡⠟⢃⠀⠿⣿⡉⢿⠟⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠛⠿⠀⡘⠋⣐⡀⠁⠚⠛⠓⠄⠘⣭⡤⠀⠘⠿⣿⠀⠀⠀⢁⠤⡄⠙⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠛⠛⢀⣴⣷⠉⠸⣿⡟⠀⢀⣤⢰⡶⢂⡀⠀⠿⠸⠗⠀⠀⢈⣀⡈⠛⠃⠰⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠈⢠⣿⣿⡿⢀⡈⢻⡇⠠⠿⠎⠀⠁⢛⣛⡀⠐⠂⣦⡄⠀⠛⠛⢧⣾⣷⣤⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣀⠃⢸⣿⡿⠀⠻⠿⠀⠀⠀⣿⢃⡄⢤⡙⢿⠃⢰⡇⠀⠤⠀⠀⠀⠈⠙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠈⠵⠊⢠⣶⣶⣾⡳⠆⠀⠘⠿⠘⠛⠀⠂⢀⠀⠈⣀⠀⠀⠶⠀⠰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠖⠀⠀⠀⠻⠿⠿⠟⠃⠀⡂⠒⠖⢠⡦⣾⣿⣯⠐⠀⠙⠀⠺⠀⢀⣤⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠸⠀⢾⡦⠈⡃⠀⠠⣶⣥⡜⠁⠴⠐⡂⠀⢻⣿⣿⣿⣷⣤⣤⣦⣤⣴⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣁⡀⠠⠄⠤⠈⠁⢰⣾⣿⣿⣧⡄⠙⠂⠉⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣠⣦⡄⣰⣿⣾⣿⣿⣿⣿⣷⣶⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠻⠿⠿⠏⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⠀⠠⢄⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠁⠀⠀⠀⠀⠔⠒⡆
⠀⠀⠀⠀⠀⢀⡀⣀⠆⠀⠀⠀⠀⠀⣀⣀⡄⠀⠄⠀⠉⠁⠀⣉⡒⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⣄⠀⡄⠄⣦⢲⣄⠀⠀⠀⣅⡀
⠀⠀⠀⠀⡆⢂⡾⠁⠀⠀⡄⠀⣼⡟⡿⣿⣟⣿⠀⠂⠓⠎⢀⢁⡁⢸⠁⢠⠐⠁⠀⣤⢰⢀⣤⠖⠇⠛⠓⠁⠈⠁⠀⠀⠀⠀⠀⠀⠐⢶⠠⣤
⠀⠀⠀⠐⠀⣼⡃⠀⠀⠀⠃⢀⢿⣷⣤⡙⢯⡻⠧⡌⠄⠂⠈⠓⠁⢰⠀⠀⠀⠀⢠⡞⠘⠘⠈⢀⠠⠀⠀⠀⠐⠈⠀⠁⠀⠀⠀⠀⠀⠀⣿⠈
⠀⠀⠀⠙⢰⠋⠀⠀⠀⠀⠀⠨⠼⢿⣿⣿⣆⠀⢀⠀⡀⠀⢘⠧⡀⢸⠀⠸⠐⠂⠀⠀⠀⠈⠀⠐⠀⠀⡀⠡⠀⠄⠄⠂⠀⠀⠈⠂⠀⠀⣿
⠀⠀⠀⠉⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⠀⠉⡆⡈⣿⠤⠘⠀⢩⢉⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡌⠑⠊
⠀⠀⠀⠐⠆⢄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠉⠋⣄⠀⠀⢰⡏⠀⠉⠀⣠⣀⠀⠀⠀⠀⠋⠀⣻⣆⠀⠀⠀⠀⠀⠀⢀⠀⠇
⠀⠀⠀⣀⢠⠊⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡃⠟⠀⠀⠘⡇⠀⠁⣰⡃⠀⠀⣰⣷⣀⣠⣾⣿⣿⡇⠀⠀⠀⠀⢰⢸⠀⡆
⠀⠀⠀⡁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣿⣭⡿⠛⢿⠉⢻⡇⠀⠀⡀⠀⢸⠀⠀⠃⠀⡄
⠀⠀⠀⠁⡀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠊⠀⠀⠀⠀⠀⠀⠀⣿⡿⠻⠁⢀⡿⠁⠀⠀⠀⠸⠃⠀⠀⢠⠀⠈⠀⠀⠀⠀⡇
⠀⠀⠀⡀⠁⡀⢠⠀⢰⠀⠀⠀⠁⠀⡀⠀⠐⠀⠄⠀⠁⠊⠀⠀⠀⠀⠀⢰⠇⠀⠀⠘⠇⠀⠀⠈⢉⠶⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⡀⠀⠀⡇
⠀⠀⠀⡨⠜⠁⢈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⢀⠠⠠⠀⠀⠀⠐⠀⡇⠀⠀⡇
⠀⡀⠀⠃⠀⠀⡆⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⡴⣀⡄⢀⠀⠈⠰⠢⠆⠀⠀⠀⠀⠀⠀⡈⠄⠄⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⡇
⠀⠄⠀⠄⠀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠙⠆⢨⠂⠐⢲⠐⠃⠼⠀⠀⠂⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⡇⠀⠀⠄
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠂⠂⠁⠀⠀⡄
⠀⠄⢸⠤⠀⠖⠥⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠈⠉⡁⠀⠚⠼⣶⣶⣤⣘⢀⣠⣤⡶⠶⠦⠄⠀⠀⠀⠀⠀⠀⠀⠐⠤⠤⠴⠂⠀⠀⢂⡌⢀⣠
⠀⡀⠈⠁⠀⠀⠙⠂⠀⢡⡄⠄⠀⠀⠀⠀⣄⣀⣀⢀⡬⠩⠛⠋⠻⣿⡿⠿⠿⣿⠿⠛⠛⢛⠫⣉⠀⣀⣀⣀⣀⠉⠉⠉⡁⡭⠉⠁⠚⠉⠀⠀⢊
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠒⠓⠛⠉⠀⠀⠙⠁⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠈⠿⠃⠀⢰⣿⣿⡆⠀⠀⠀⠀⢀⠀⠀⠘⡆⠀⢀⡀⠐⣶⡂⠀⢴⡆⠀⠀⠶⣾⣿⡤⠿⣷⢶⠀⠛⢃⡘⠁⠀⠠⢽⣿⣿⣿⣿
⠀⠀⠀⢴⣶⣶⣶⣶⣿⡿⢟⠋⠀⠀⢀⢶⠞⡷⣦⡈⠀⡤⠀⢀⣴⣾⣿⡇⠸⣀⣿⣿⣸⢟⣼⣿⠀⢀⣾⠃⠉⡆⢨⢸⣿⣿⣿⣿
⠀⢀⡤⠄⠙⢒⠒⠲⢲⡚⣋⠀⠀⠤⢼⢹⠓⣓⣷⣘⣄⡀⠀⠈⠉⠙⠏⠀⣀⣉⠉⠉⠉⠉⠉⠉⢠⣿⡏⣠⠞⠀⠐⠸⠿⣿⣿⡿
⠀⠀⢀⡜⢁⡎⠰⠂⣀⢳⡈⠀⠀⠀⡈⠺⣤⣄⣲⠟⠀⠀⠀⠀⢀⣤⣦⠀⠙⠟⢀⣤⣤⠀⠀⢠⣿⡟⠘⠁⣀⠐⠶⠀⢀⣰⣦⣀⡀
⠀⠀⠀⠀⡾⠀⠀⠘⠛⠀⢧⠀⠀⣀⣀⣀⡀⢢⡀⠀⠡⣤⣴⠂⣿⣶⣝⡷⢠⢖⣻⣿⣿⠀⢠⣿⠟⢀⡀⠀⢛⣀⣠⣦⠈⠉⡉⠩⠁
⠀⠀⠀⢰⠃⢸⣿⡿⠃⠀⠈⠇⠀⠿⣿⣿⣿⠾⠓⠒⠀⠉⠉⠀⣿⣿⣿⣧⣿⣿⣿⣿⣿⠀⡾⠋⠀⠸⣿⣿⣿⣿⣿⣇⠀⠀⢠⣤
⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⢠⣦⣤⣦⣄⡈⠁⠀⠀⢀⣠⢾⣱⠀⣿⣿⣿⢸⣿⣿⣿⣿⡇⠀⠁⠀⠀⢸⣿⡏⠉⠉⣻⣿⠆⢠⣿⠁
⠀⠀⡰⠁⠆⣻⣿⣷⡄⠀⣿⠀⠀⠀⠉⡇⠀⣠⣞⡍⠂⠊⠁⠘⠿⣿⣿⣸⣿⣿⣿⠿⠃⠠⣾⣿⡶⠀⠙⣿⠿⠿⢿⡏⠀⣾⡏⠘⠟
⠀⠀⢧⣠⢾⡵⣿⣿⡇⠀⡿⠀⠀⠀⢸⣠⡻⠞⠉⠀⣠⣾⣿⣷⣆⠈⠙⠙⠛⠋⠁⣄⡀⢀⣉⠋⢀⠞⣦⠉⠀⠀⠀⠀⢰⣿⠀⣤
⠀⠀⠀⠙⠙⠛⠛⠉⠀⠀⣷⣤⣤⣤⣼⠀⠀⠀⣀⣰⣿⣿⣿⣿⣿⣦⣤⣤⣦⣤⡀⠙⠁⠑⢄⢈⣿⠶⣚⡃⢈⡆⠀⢀⣿⠇⣸⠃
⠀⠀⡤⡁⡄⢤⣃⠀⠀⠀⠀⡉⠉⠉⠉⠀⢰⣿⣿⠟⠛⢿⣿⣿⣿⣿⣿⣿⠿⢿⣿⠀⠀⣠⠜⢻⢅⣓⣨⠷⣉⠀⠀⣼⡟⢠⣬
⠀⠀⣶⣶⣾⡿⠛⣇⠀⠀⣴⠿⠻⢧⡤⠀⠈⣿⣧⡘⠃⣸⡏⢠⡄⢻⣼⡇⠰⣿⣿⠀⠸⡀⠤⠜⡞⠉⢲⢄⣘⡅⢸⡿⠁⣿⣿
⠀⠀⢻⠁⠀⠀⣀⣿⠀⠘⣿⣄⢀⣸⡟⠀⣴⣿⣿⠃⣀⠻⣇⠀⠀⣼⣿⣿⠆⠘⣿⣷⣄⠀⠀⠀⠱⣀⠎⠀⠈⠀⠼⠀⣠⡶⣶⡀
⠀⠀⠘⠿⠾⠛⠛⠁⠀⠀⢘⡙⢛⡋⠀⢸⣿⣿⣿⡀⠉⣠⣿⣿⣿⣿⣟⢁⣠⣾⣿⣿⡿⠀⢀⣠⣴⣶⣶⣶⣶⣶⡄⢠⣿⡟⣿⡇
⠀⠀⠀⢀⣤⣤⣤⣀⠀⢠⣿⠇⢸⣿⡄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⢠⣾⡿⣿⣍⣭⣵⣿⡿⠟⠁⠀⢿⣿⣿⠇
⠀⠀⣰⠏⣅⠉⢻⣿⣧⠉⠈⠀⠈⠏⠙⠀⠀⠉⠛⠻⣿⣿⣿⣿⣿⡟⠁⣠⣴⣶⣦⣄⠈⠛⠃⠙⠛⠛⠋⠉⠀⢀⡤⢤⡈⠋⠁
⠀⠀⣿⣄⠛⠀⣰⣿⣿⠃⢠⣤⢀⡀⠐⡶⠄⠀⢰⣶⡎⠻⢿⠿⠋⠀⢰⣿⣿⣿⣿⣿⡆⠀⢠⣰⣀⣦⣾⡾⠃⠈⠛⠛⠋⠻⠭⠷
⠀⠀⠙⠿⣿⣿⣿⠿⠋⠀⢠⣤⠘⠋⢁⣦⡄⠀⢸⣿⠃⠀⢀⠤⡄⠤⡈⢿⣿⣿⣿⡿⠁⢲⣾⣿⣿⣿⣽⠖⠀⠸⠿⠃⣰⣶⡆
⠀⠀⢶⠆⣀⣀⣀⠀⠀⠟⠆⣛⣸⣟⣊⠉⠁⠀⢸⡏⠀⡼⠓⠢⣷⢔⣛⡄⠻⡟⠉⠀⠀⠀⠙⠛⠛⠛⠋⠀⠀⣠⡤⣾⣿⣿⣿⣤⡄
⠀⠀⢀⣾⣿⣿⣿⣷⡀⠀⠀⣿⣿⣿⣿⠀⠀⣀⣼⣇⠀⢷⣈⠍⡿⢧⡠⠀⠀⡷⠐⣷⣿⠀⠀⢰⡔⠉⢇⡀⠀⣿⠀⠙⠛⠛⠛⠀⡇
⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢿⣿⣿⡏⠀⢸⡿⠛⢿⡇⠀⠑⠺⠥⠒⠁⠀⢰⠁⠀⠀⠉⠀⣰⡶⠶⠤⣽⡃⠀⣿⠀⠀⠀⠀⠀⢠⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⠈⠉⠀⣴⣿⣇⣀⣼⣧⡀⢠⣤⡤⢤⣀⣀⣠⠶⠶⣶⠀⠀⣿⠀⠸⡀⢘⡇⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⠿⠃⢤⡄⠹⣿⣛⣛⣿⡿⠃⠈⢯⡀⢀⣼⠉⠳⣄⣠⠟⠀⠀⠿⣦⣤⣤⣾⠃⠀⣿⣤⣤⣤⣤⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠠⠐⠐⠠⠄⠠⠀⠖⠠⠄⠀⠀⠀⠀⠀⠠⠀⠀⠠⠠⠄⡠⠐⠂⢄⠂⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠢⠤⠒⠀⠐⠤⢠⠠
⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡄⠂⠀⡀⢀⠈⠁⢀⢀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⢀⢀⠢⠀⢀⣀⠀⣀⡀⠐⠐⠄⢀
⠀⠀⠀⠀⡈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⣠⠚⠓⢄⠀⠀⠐⡜⠁⠀⠀⠀⠀⠀⠀⠀⠿⠀⢠⠋⢀⣤⣄⠙⠄⠄⡰⠆
⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠠⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⡘⠆⢀⠣⣀⣀⠜⢁⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡰⡀⠸⠿⠟⢀⡄⡔⠁
⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠁⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠢⠀⠈⠁⠀⢰⠊⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⣉⡙⠒⠀⠒⠋⢈⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠢⡀⠀⠰⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠰⣄⠒⣁⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠐⠀⠀⠐⠊⠀⠂⠂⠐⠐⠀⠓⠂⠀⠀⠐⠀⠀⠀⠂⠐⡒⠒⠀⢂⠀⠀⢀⠂⠀⠒⡐⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠒⠛⠆⠂⢀⡔⠛⠚⠷⠄
⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠸⣌⠀⠀⠀⠀⣵⡆⠀⠀⠀⠈⣸
⠀⠀⠀⠐⠈⠄⠀⢀⣠⠚⠓⣠⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑⠀⠀⢠⣤⠆⠀⠐⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⢤⣤⠤⠋⠳⠤⣤⠔⠁
⠀⠀⠀⠐⠀⠈⢂⠈⠉⢠⡔⠉⠁⡠⠅⠀⠂⠀⠀⠀⠀⠀⠀⠀⠵⠀⠀⠘⠙⠃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠉⠉⣦⡞⠉⠉⠽⢄
⠀⠀⠀⠀⢀⠀⠀⠠⠀⠈⠠⠀⡀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠡⢀⡄⠀⢀⠠⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡡⠀⠀⠀⡀⠜⢃⢀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡔⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⡈⠀⠀⠈⠁⠀⠒⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣀⡀⠃⡀⢀⠙⣄⡀⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠠⠀⠀⠤⠀⠀⠀⢠⠄⠀⠀⠠⠄⠀⠀⠠⠀⠀⠀⠤⠠⠠⠤⠄⠀⠠⠥⠄⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄⠀⠀⠀⠀⡠⠤⡀
⠀⠀⠀⢠⠀⠀⢀⠄⠀⠘⡀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⢀⢡⡐⠆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠉⡀⢀⡀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠂⠀⢄⠀⠂⢅⢀⠍⠐⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢊⢧⡤⠑⠃⢀
⠀⠀⠀⠀⠀⠀⠰⡀⠈⢀⠀⠁⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠐⠃⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡐⠂⠈

⠀⠀⠀⡠⠤⠤⠄⠀⠒⠒⠒⠒⠒⠒⠒⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠒⠒⠒⠒⠂⠠⢤⡀
⠀⠀⢀⠁⠀⠔⠒⠒⠒⠒⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠐⠒⠲⡄⠈⣵⡄
⠀⠀⢸⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡧⡇
⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⢰⢭
⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢸⢸
⠀⠀⢸⠀⠀⡇⠀⠀⠀⢀⡠⠤⠒⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠉⠑⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⠸
⠀⠀⢸⠀⠀⡇⠀⠀⡔⠁⣠⠤⢆⡀⠐⡃⠀⠀⠀⠀⠀⢀⠊⢀⠴⠒⠢⡀⠀⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠲⡀
⠀⠀⢸⠀⠀⡇⠀⡜⠀⡜⠁⠀⣰⠀⢸⠃⠀⠀⠀⠀⠀⠱⢄⠊⠀⠀⢀⠇⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆⡇
⠀⠀⢸⠀⠀⡇⠀⠓⠚⠁⠀⡰⠁⡰⠃⡠⢢⡀⡠⠢⡀⠀⠀⠀⠀⡠⠃⢠⠋⠀⠀⣋⠙⡁⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⡇
⠀⠀⢸⠀⠀⡇⠀⠀⠀⢀⠞⠁⡴⠁⠀⠑⡄⠈⠀⡔⠋⠀⠀⡠⠊⢀⠔⠁⠀⠀⠀⡭⢉⣉⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠸⠀⠀⠇⠀⠀⡰⠃⢀⡜⠀⠀⠀⢼⢁⠤⢄⣹⠆⣠⠊⠀⡴⠁⠀⠀⠀⠀⠀⢦⣀⣤⡨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⡄⠀⡰⠀⢀⠜⠀⠀⠀⠀⠀⠁⠀⠀⠁⠰⠁⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡇⡇
⠀⠀⠀⡇⠀⡇⠀⡇⠀⠈⠀⠊⠀⠙⠁⠐⠆⠀⠀⠀⡇⠀⠈⠉⠉⠙⠂⠀⠒⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⡇
⠀⠀⠀⡇⠀⡇⠀⠙⠀⠈⠁⠀⠁⠀⠈⠁⠉⠀⠀⠀⠘⠃⠀⠂⠉⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⡇
⠀⠀⠀⠇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⡇
⠀⠀⠀⠶⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠇⢀⠣⡇
⠀⠀⠀⠸⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠑
⠀⠀⠀⠀⠈⠓⠛⠩⠯⠥⠴⠰⠆⠀⠖⣲⣀⣂⣀⣂⣐⣂⣐⣂⣐⣂⣐⣂⣀⣂⣐⣂⣀⣒⣒⢒⠀⠲⠀⠴⠨⠯⠭⠭⠭⠥⠐⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠈⠉⠉⠉⠉⠒⠲⢤⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠒⠒⠒⠒⠦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⣄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣀
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⢤⠐⠀⠀⠠⢀⠀⠀⠀⠒⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠤⠤
⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡰⣀⠠⠢⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠒⢦⡀⡢⡀⠀⠀⠀⠠⠁⣀⠀⢀
⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⣤⠀⠀⠀⠀⠈⠀⠙⢦⠠⡀⠈⠀⠀⢀⠈⠉⡀
⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠠⡀⠀⠑⠂⠀⢀⠌⠀⠀⠀⠆⠀⠀⠈⠉⠀⠐⠿⠀⠀⠀⠀⠈⡆⠘⠀⠀⠀⠠⡀⠠⠃
⠀⠀⠀⠀⠉⠑⠒⠒⠒⠂⠉⢦⣀⣀⠀⠀⠀⠀⢀⠀⠳⣄⠀⠀⣀⠄⠀⠀⠀⠀⡀⠐⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⠈⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠙⠒⠲⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣣⣄⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣾⣿⠃⠀⠉⠒⠂⠀⠀⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢸⡅⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⣼⣷⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⢢⠀⠀⠀⠀⡀⠊⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢣⠀⠀⠈⠀⢀⠠⠀⠂⣿⣿⣿⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠀⠂⠁⠀⠀⠀⠀⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠎⠳⢄⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠑⢄⠀⠀⡀⠠⢄⠀⠈⠓⢤⡀
⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣴⣮⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣦⣤⣴⣦⣤⣤⣤⣤⣽⣦⣤⣤⣧⣤⣤⣤⣤
⣿⣿⡏⠉⠋⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡇⠈⠐⢃⠖⠉⠑⣆⠄⣡⠊⠉⢣⠀⠘⡚⠊⠛⠀⡀⠉⠀⢴⡲⠀⢃⣀⠀⠀⣄⠁⣠⡄⠂⣤⠃⠀⣠⡈
⠇⡼⠗⠸⣠⣶⣦⡘⠴⠃⣤⢶⣿⢶⡆⢀⠁⡜⠾⠀⠘⠀⠀⠀⠂⠘⠚⠀⠀⡏⠳⠁⢱⡎⢹⢞⠇⠓⠃
⠠⣠⣠⡀⢣⠉⡉⠁⠀⡀⠬⣹⣋⣉⣀⣀⣤⡀⠀⣠⣴⢻⠀⢾⡽⠀⡁⠠⣟⣷⢦⢀⡴⣶⣦⢸⠀⣴⣈
⠁⠻⠻⠋⠉⠳⡀⠀⠓⠁⢠⠋⡉⢉⠉⠉⠗⢉⠞⡵⢃⡎⠀⣀⣀⣀⣀⠀⠀⣯⢡⡤⠈⢩⠀⢸⠀⡉⠏
⠀⠀⠀⠀⠘⠠⠘⢦⡀⢀⠏⠀⠐⠀⠀⠀⠀⣜⡞⣡⠎⠀⠀⠉⠁⠁⠈⠀⠀⢸⠀⠀⠀⠀⠀⡇⠠⠀⢠
⣠⣇⣰⠃⡤⠀⠄⢀⠙⠊⠀⠎⠻⠉⠞⠁⠀⠙⠋⡅⣔⡦⠀⢼⣱⠀⠀⢀⡀⠀⢳⣄⣀⣀⣼⡆⠐⠠⡀
⣤⣤⢤⠽⢦⠀⠀⠊⠁⠀⢀⠴⠤⡄⢠⢄⡀⣀⣀⠂⠠⢀⢤⡀⠀⠀⠘⠙⢠⡆⠘⡍⣙⠇⢈⡁⣀⢠⣟
⠬⡩⠞⣠⠋⢰⢰⢀⡤⠒⠁⠀⠀⠘⣇⣀⠈⠀⢨⠆⠁⠣⠤⠃⠀⢸⠳⡄⠸⠇⡼⣰⣡⢞⡽⠃⠄⠀⠱
⠓⢒⠚⢁⠉⠈⠀⢸⠀⠀⠀⣤⣰⣏⠁⠈⣉⣳⡋⠉⠉⡦⠀⠀⠁⠀⠓⠛⢠⠞⡴⢃⣉⠥⢄⡀⠶⠲⠃
⠒⠈⠠⠈⣀⣀⡀⠘⢄⡀⠀⠙⡿⢚⣃⠘⠿⠃⡇⠀⠀⢧⠀⠀⠀⠀⡠⢞⡡⠊⠀⠛⠶⠒⠚⣡⢤⣶⢶
⡹⠀⠀⡈⠙⠉⣡⢴⠄⣨⠽⠛⠙⢮⣁⣈⣠⠞⠑⡖⠶⠋⠀⢀⠀⠀⠈⠉⠀⠄⠐⡀⠁⣴⣟⣛⡥⠖⠋
⠀⠀⠀⠄⠀⠸⠵⠊⠀⢧⡀⠀⠀⢡⠃⠀⠀⠀⢀⡸⠀⠂⠄⣠⣴⡋⠉⢉⡆⠀⣀⠁⠐⣀⡄⠀⣀⠀⠘
⠀⣀⠀⠀⠁⢠⠶⠖⠶⠀⠑⠤⣀⠼⣄⡠⠤⠤⠊⠀⠁⠀⠺⠛⠒⠒⠚⠉⣀⠀⡀⠀⡤⡀⠁⡼⠈⠟⠷
⢌⠋⠤⡀⠁⠀⠀⠀⢠⡴⡦⢶⠀⠂⠀⣀⢠⠀⡄⣠⠀⢰⢠⠈⠫⣍⠒⢤⠈⠀⠁⢸⠠⠈⠲⠁⠀⢀⡜
⠊⡤⡌⠁⠀⠠⠀⠀⢸⠀⠀⡏⣀⣡⠀⠈⠉⠉⠈⠁⠀⠈⢀⠀⠀⠈⠙⠊⠀⠡⠅⣼⢠⠺⢦⠀⣠⠊⡀
⠀⠓⢁⣀⡀⠐⢀⣀⢸⠀⠀⠉⠁⢸⠀⠀⣤⡀⠀⠀⡴⢳⡈⠀⡴⢷⠠⢀⣠⡤⠀⣿⠃⡰⡤⡙⠁⠄⠐
⠶⠔⣞⡀⢸⣰⠃⢸⠇⣇⡴⠻⡀⡏⠠⡀⠘⡎⢢⡼⠀⠀⢇⡞⠀⢸⡴⢋⡟⢀⡀
⣀⡘⢫⡿⠆⠁⢶⣺⣦⠉⢰⢠⢻⠃⠈⠁⡀⢹⡀⣥⣤⣤⣌⠀⣤⣮⣥⣜⠀⠙⠊⢀⠤⡀⢠⣊⠝⠁⣀
⣶⠟⠃⠳⡈⢋⡜⠀⠄⠀⠈⠀⡀⠀⡀⠤⡇⠀⣇⣙⣒⣛⣊⣢⣉⠚⡷⠚⠃⢰⣲⠈⠉⠀⠀⢠⣶⣯⣥
⡀⠀⡀⢀⠘⠋⠀⠂⠐⠀⠀⠀⠀⠂⠀⠀⢀⢀⠀⠀⠀⠀⠀⠠⠀⠉⣀⠀⠀⠀⡀⢠⠀⠀⠄⠠⠀⠉⢉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢪⠟
⠀⠀⠀⠀⠀⠀⠀⠀⡰⢻⣿⠀⠀⠀⡾⠒⣿⡆⠀⣠⣄⣀⣐⣶⣶⣤⡀⠀⠀⣰⣷⣃⣀⣀⢿⣇⣀
⠀⠀⠀⠀⠀⠀⢀⡜⣰⣿⣿⠀⠀⠀⢠⡾⠋⠀⠀⠉⠉⢉⢽⣿⣿⠋⠀⠀⠀⠙⠛⠛⠛⢻⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠈⠛⠋⡿⣿⡄⠀⠲⠿⠷⠶⠂⠀⠀⠀⠈⠓⢺⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠚⠁
⠀⣀⣀⣀⣀⣀⠀⠀⠀⡇⣿⡇⠀⣠⣤⣶⣤⡀⠀⣟⠻⣦⣠⣼⣿⠟⠀⠀⢰⡊⠉⢻⣷
⠀⣤⣯⣠⡀⢀⠀⠀⠀⠑⠾⠃⢰⢿⣡⣤⣬⡀⠀⠀⠉⠓⠛⠉⠁⠀⠀⠀⠈⣛⣶⣿⣧⡀⠀⠀⣀⣤⣤⣀
⠀⠈⠁⠀⠈⢣⡅⠀⠀⠀⠀⠀⠈⠾⣧⣀⡼⠇⠘⠒⠒⠒⠒⠶⡼⠃⠀⠀⣺⡁⠀⣹⣿⡟⢠⣮⠚⠛⠻⣿⣷
⠀⠘⠳⠶⠶⠛⠁⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⡰⠁⠠⠀⠀⠈⠉⠛⠉⠁⠀⢸⣇⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡞⣿⠋⠀⠙⣿⣿⣧⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⣠⠤⣤⡀⠀⠀⠀⠙⠿⣿⡿⠟⠃
⠀⠀⠀⣀⣤⣿⣿⣄⣀⠀⠻⣿⣷⣶⣾⠿⣿⣿⠀⠀⠰⠞⠁⠀⠀⠀⠀⠈⠋⣠⠞⠁
⠀⠀⠀⠉⠉⣿⣿⠙⠉⢀⣤⣄⡉⠉⢁⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠻⠷⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡀⠳⢟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⡀⠀⠀⠀⠉⠉⡀
⠀⠘⠚⡳⣟⠛⠃⠀⠛⠓⠒⠛⠛⠃⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢳⣞⣽⣿⠃⠀⢘⠛⣛⣛⣛
⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠙⠛⠃⠀⠈⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⡟⠃⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠂⠒⠀⠂⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣴⣿⣿⣿⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣶⣿⣦
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣿⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣤⣼⣿⣿⣿⣿⣿⣿⣦⣤
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⢻⣿⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠛⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣤⡂⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠓
⠀⠀⠀⠀⢀⣽⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠴⠶⠶⠒⠒⠒⠛⠛⠛⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠶⠖⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⣤⠶⠶⠦⣄⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⣠⢴⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠒⣦⠀⠀⣰⣯⠗⠲⢄⠸⡆⠀⠙⠋⠉⠳⡌⢧⠀⠀⠀⠈⣧
⠀⠀⠀⣰⠃⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣡⡆⢹⠀⠀⢧⠿⠀⠀⢸⢠⡇⠀⢠⣄⣀⡴⠃⡼⠀⠀⠀⠀⠸⡆
⠀⠀⢰⡇⢠⡄⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⡇⢸⠀⠀⠀⠀⠀⢀⡜⣼⠁⠀⠈⠙⠓⣦⠸⡅⠀⠀⠀⠀⠀⢻
⠀⠀⡾⢠⠂⠱⡀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣏⣘⣆⠀⠀⣶⠚⢉⣚⣋⣻⠀⢸⣗⣚⣁⡴⠃⠀⠀⠀⠀⠀⠈⣇
⠀⠀⣇⡇⠀⣠⠔⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⢹⡗⠊⠀⡜⡆⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⢧⠀⡸⠀⢱⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⢸⡄⠉⠐⠒⠋⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⢰⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢵⣲⣇⡾⢸⢀⡶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢹⡀⠀⠀⠰⡏⢻⠀⣀⣄⠀⠀⠀⠀⠀⠀⠘⠒⠒⢁⡇⡇⠸⣜⣿⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠈⣇⠀⢀⣤⠟⠈⢋⣡⡽⠀⢰⠶⠦⠤⣄⣀⠀⠀⠸⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠸⡄⠈⢧⠴⢶⠈⣇⠀⠀⠸⠦⢤⣤⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⢤⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢳⠀⠀⠀⠘⠖⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠴⠒⠚⠋⠉⠁⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠈⢇⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⠤⠤⠴⠶⠒⠒⠛⠋⠉⠉⠀⠀⠀⣀⣀⣀⡀⠤⠤⠤⠶⠖⠒⠒⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⢸⡤⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⠴⠖⠒⠒⠉⠉⢉⣀⣀⡠⠤⠤⠤⠶⢒⣒⣺⣭⠽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢰⠃⠀⡤⠴⠖⠒⠒⠒⣋⣉⣉⣉⣀⣠⠤⠤⠴⠒⠒⣒⣉⣩⣭⠤⠴⠶⠖⠒⠒⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⣾⠀⠀⠀⠀⠀⠀⢉⣉⣁⣀⣤⡤⠤⠴⠶⠒⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠒⠛⠛⠙⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⡤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⡛⠛⠛⠛⠛⠋⠉⠉⠉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣄⣠⠟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⡤⠶⠾⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠞⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⢀⡆⠀⠀⠀⠘⠁⡼⠀⠀⠀⠶⠒⠒⠀⠀⠀⠾⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡿⠀⠀⠀⠀⠀⠀⠀⠰⢻⠁⠀⠀⠐⢻⡏⠉⠀⢀⣠⣞⡁⠀⠀⠘⠛⠉⠉⠀⣀⣀⡤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⣾⣄⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠁⣸⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⡄⠀⠀⠀⣀⣀⡀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⣾⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⢋⣽⠀⠀⠀⡤⠴⠦⠀⠀⠀⢀⡼⠁⠀⠀⢀⣉⣠⡀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠋⢩⠇⠀⠀⠀⠀⠀⠀⣠⡶⠿⢤⠄⠀⠀⠀⠀⠀⠀⠀⠤⠾⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣼
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡿⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣾⠀⢀⣇⣀⣀⣀⣀⣀⣤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠖⠒⠒⠒⠒⠒⢒⣚⣛⣛⣛⣛⣋⣉⡉⠉⠙⠓⠒⠒⠒⠶⠶⠇⠀⣼⠁
⠀⠀⠀⠀⠀⠀⣿⣴⣋⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠠⠤⠤⠤⠤⠤⠤⠶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡷⠤⠤⠤⠤⠤⠤⠶⠶⠒⠚⣻⡆
⠀⠀⠀⣠⠶⠛⠉⢉⡭⠭⠭⠭⠭⠭⠭⠤⠤⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁
⠀⠀⠸⣧⣀⣀⣀⣈⣓⣒⣒⣒⣒⣒⣛⣛⣛⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⠤⠤⠤⠤⠤⠤⠶⠶⠶⠶⠶⠒⠒⠒⠒⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡿⠛⠋⠉⠉⠉⠙⠛⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠏⠀⠀⢀⣠⣤⣄⡀⠀⠀⠙⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡏⠀⠀⢠⣿⣿⣿⣿⣿⠀⠀⠀⢻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⣰⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣴⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣠⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠿⠛⠛⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠂⠀⠐⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠘⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠂⡀⠀⣄⠸⡀⢂⡀⠀⠠⠄⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠂⠀⠁⠀⠀⠀⠀⠔⠐⠒⠀⠈⡄⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠈⠈⠂⠀⠀⠀⠀⠀⡱⠀⡀⠱⠀⠀⠀⠀⠀⠠⠓⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠂⠁⠒⠪⠁⢢⠀⠀⠀⠤⠄⠨⡁⢀⠇⠀⠀⠀⢀⡠⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠒⠒⠈⢀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠁⠢⠁⠀⠈⠁⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⢊⠀⣈⠁⠀⠠⣶⡄⠀⢀⠀⠀⡀⠀⠠⣶⠄⠀⠠⠀⠀⠑⠤⠤⡀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠐⠈⠀⢃⠁⠀⠀⠀⠀⠀⠰⡀⢀⠀⠀⠀⠁⠀⠀⠀⠃⢐⠁⡂⣀⠄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⡁⠀⡠⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠂⣀⠄⠠⠀⢀⠠⡄⠀⠀⠀⠀⠒⢄⠀⠈⢡⠀⠀⠀⠀⠀⠉⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠠⠀⠔⠂⡄⢀⠀⡆⠠⠁⠀⠀⠀⠀⠈⠂⠇⠀⠜⠀⠀⠀⠀⠀⠔⠈⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⠀⡀⠦⠤⠂⠀⠁⠁⢒⠐⠀⠀⠀⢀⠀⠀⠠⠄⢰⠁⠈⠖⠀⠘⠖⡁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢈⠂⠀⠀⢇⠀⡠⠓⠒⠁⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⡌⠀⠀⠉⡆⠐⠔⠃⠀⠠⠓
⠀⠀⠀⠀⠀⠀⠀⠁⢀⠄⠙⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⠀⠀⠈⢙⠀⢀⠤⠄⠂⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠁⠈⡄⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⣆⠔⠀⠀⠀⠈⠐⠁⠒⠄⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠂⣀⢀⠂⠉⠢⠔⠒⢂⠀⠀⠀⠀⠀⠢⠄⠀⡖⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠄⡀⠀⠀⠀⠀⠀⢀⠤⠐⢢⠀⡦⢀⠉⠠⢤⠊⡤⢀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢄⠈⠀⠀⠀⠸⡀⢀⠤⠀⢀⠀⠐⠄⣱⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⡀⠀⠀⠀⠀⠀⠀⠈⡠⠢⢀⡀⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠄⠤⠄⠒⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⠀⠢⡀⠀⢀⡠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠐⠠⠈⠀⠀⠀⠀⠒⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠠⠄⠀⣠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂
⠀⠀⠀⠀⢀⢀⡤⠀⠀⠀⠀⣀⣤⣤⣦⣀⠀⠀⣀⡀⠠⢦⠁⡀⠀⢀⠄⠀⠀⠀⠀⠀⣠⣆⡠⠄⣰⠊⠶⠀⠀⠀⠄⠄
⠀⠀⠀⠘⢋⡌⠀⠀⠀⠀⠀⣿⡙⢻⣿⣿⡀⠁⠉⠀⢄⡲⠀⠀⠐⠀⠀⠀⡆⠁⠶⠊⠉⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠣⡝⠇
⠀⠀⠀⠀⡸⠇⠀⠀⠀⠀⢀⣿⣿⣷⡌⠋⠉⠀⠁⠀⢤⠀⠀⠀⢠⠀⠀⠙⠀⠀⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠂⠀⠀⠂⠀⢲⣞⠀⠀⢐⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠤
⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠁⠛⢃⠀⠀⢈⡏⠀⠂⠀⣀⡀⠀⠀⠀⠏⠙⣦⡀
⠀⠀⠀⡀⡴⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠸⠀⠀⠘⠃⠈⠀⣞⠉⠀⢰⣦⣀⣠⣾⣿⣷⠀⠀⠀⠈⠀⠠
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢸⣿⣷⣶⠿⣯⡿⠛⡿⠙⣿⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠰⠈⠁⠀⠀⠀⠀⠀⠸⣿⠛⠇⢀⣿⠁⠀⠀⠀⠏
⠀⠀⠀⠀⣀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠂⢰⠀⠀⠀⠙⠀⠀⠀⢠⠛
⠀⠀⠐⠂⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒
⠀⠀⠀⠀⠀⢸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠑⣜⠔⡆⠀⣀⢈⠄⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢨⡀⢠⢐⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠒⠒⠀⢠⣦⣤⣄⡀⠀⣀⣠⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⠀⠀⢀⡂⢈
⠀⠀⠘⠀⠀⠐⠒⠀⢠⡀⡀⠀⠀⠀⠀⠀⠀⢉⠉⠵⠒⠚⣿⣿⢿⡿⣿⠷⠒⠲⠖⢂⠀⠀⠀⠀⠀⠀⠀⡀⣐⠋⠡⠔⠉⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠚⠉⠀⠀⠛⠀⠀⠈⠉⠉