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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠀⠀⠀⠀⠠⡇⠀⠀⠠⠤⡄⠀⠀⠀⠈⠀⠀⠀⠀⢰⢲⢄
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠇⠀⠀⠀⣀⠀⠀⠀⠀⡀⠁⠀⠀⠘⠓⠁⠀⠀⠀⢸⠀⠀⠀⠀⡉⢃⠈
⠀⠀⠀⠀⠀⠀⠀⣠⠄⠈⠀⠀⠀⠰⢧⠑⠀⠀⠀⢹⠀⡀⠀⠀⠈⠁⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢀⠀⠀⠀⣠⠀⢺⠽⠒⠲⡀⡀⠀⠀⠀⠒⠀⠀⠀⠘⢧⠳⣀⠀⠀⠀⠀⢀⡜⠠⢀⡀⠀⠀⢀⣀⣀⣀⡴⠋
⡆⠀⠀⠐⢋⡚⠁⠀⠀⠀⠘⣄⢄⠀⠀⠀⠀⠀⢀⡄⠈⠳⢄⡉⣐⣒⣨⠝⠁⠰⠄⠤⠬⠤⠤⠤⠚⠉⣀⣀⣀
⠁⠀⠀⠀⠰⢄⣲⣿⠀⠀⠀⢸⢄⠉⠒⠒⠲⢒⡝⠀⠀⠀⠀⠀⠀⠀⡴⠀⠈⠀⠀⢠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠒⠀⠀⠠⠄
⠀⡘⠦⡀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠈⠉⠀⠉⠁⠀⡰⢄⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⡤⣤⠀⠀⠐
⠀⠙⢦⡈⠐⠂⠀⠄⠐⢊⡵⢃⠀⠐⠈⠀⠀⠀⠀⡳⣄⠑⢄⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠓⠒⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠈⠉⠒⠒⠒⠋⠁⠐⠆⠀⠀⠀⣶⠆⠀⠀⠀⠀⠑⢄⡑⠦⡀⠀⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠁⠀⠐⠂⠠⡀⠀⠑⠦⠉⠀⠀⠀⠁⠀⡅⠀⠀⠀⠀⢰⣉⣉⣳⠋⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠘⢤⡤⠬⢭⡷⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠄⠀⠀⠀⡀⠀⠀⠀⠠⣀⣠⢤⠆⢀⠘⠒⠒⠂⠤⠤⣄⣀⠆⠀⠐⠋⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠃⠀⠀⠀⡤⠬⠦⠃⠀⠛⠒⠲⠤⠤⠤⠤⠄⠀⠀⡴⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠒⢚⡁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡈⣐⠶⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠄⠀⠒⠀⠈⠉⢈⠁⠀⠉⠀⢋⠠⣶⡴⡄
⠀⠀⠀⠀⠀⠀⣀⠠⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⢺⠉⠆⠀⠀⠀⠀⠠⠂⠦⠀⠀⣾⠀⠀⠀⠀⢸⠀⠀⠀⢸⡀
⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⡃⠀⢸⠀⠃⠁⠉⠉⠉⠉⠀⠉⠉⠉⢻⠀⠀⠈⠀⢸⠀⠰⠀⠸⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠊⡇⠀⢘⠀⡀⠀⠀⠀⠀⠀⡁⢀⠀⠀⣘⡀⢀⢀⢀⡘⡀⢀⡀⡄
⠀⠀⢀⠀⠀⠣⡀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⢉⠉⠉⢈⣁⡁⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⣉⢉⣀⢉⠉⢉⣁⠀⠢⡆
⠀⠀⡉⢤⡄⠀⣰⠤⠂⡆⡁⠀⠀⠀⢀⡀⠚⠉⡁⣂⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⢀⠐⠀⠀⢀⠀⠀⠀⠀⠁⠄⠘
⠀⠀⢁⢠⠆⠁⠀⠀⠀⠀⠀⠈⠐⣀⢋⡄⠀⠀⢂⠹⠀⠀⠀⠀⢰⠀⡇⠀⠠⠄⠰⠲⣦⠆⠤⠂⠀⠀⠀⢂⠀⠀⢣⣒⠒⡀⠄⡁⠀⠀⠀⠀⠀⠀⠇
⠀⠀⢈⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠐⡀⢀⣼⣶⣿⠂⠀⢰⡦⠰⡇⠀⠀⠀⠀⢢⠰⡀⠀⠸⠋⠸⢃⠀⠀⠀⠺⡛⠀⠉⠸⠟⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣥⡀⡼⠅⢓⠦⣈⠓⠂⢘⠠⠂⡅⠀⠀⠀⠀⠙⠒⣇⠤⣶⢤⡥⢾⣤⠂⣡⠑⢠⣌⠀⣮⡀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠘⡉⡰⠆⠀⠀⠀⠀⠀⠀⡏⢣⡀⠁⡠⠊⠀⠀⠈⠠⡴⠇⠇⠀⢰⣀⣀⣀⣀⣠⣄⡀⠀⠀⠉⠀⠁⠀⠦⠉⠶⠤⠄⠀⠔⠺⡄⢀⢀⣀⠄⠀⠀⡀
⠀⠀⠀⠈⠀⡃⠀⠀⠀⠀⠀⠀⢠⠀⢨⠁⠀⠀⠀⠀⠂⠺⠀⠊⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢀⣀⠈⢹⠀⠀⠀⠀⢀⠀⠀⣸⠋⢳⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠠⠄⠀⢻⠀⢸⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠸⠃
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠃⠀⢰⡈⣄⠁⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⡈⣰⢠
⠀⠀⠀⠀⠀⠈⢤⠀⠀⡶⡄⠀⢠⠂⠻⡄⠀⣰⡀⠀⡄⢸⠈⠉⠁⢀⠀⡠⠛⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠁⠀⠀⠉⠁⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢰⠀⠀⠀⠀⠃⢠⠀⠃⢈⠀⠀⠀⠀⠀⡅⠀⢸⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡇⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⢸⠀⠀⠀⠳⡀⠤⠀⠀⡁⠀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⣁⠀⠠⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⣗⢂⠦⠼⠦⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠶⡤⢠⠦⢼⡄⡤⡴⡎
⠀⠀⠀⠀⠀⠀⠀⢀⢨⠀⠀⡀⣀⠀⠀⢈⣀⠀⢈⣀⠀⠀⠀⠀⢃⣷⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠳⠐⠀⠀⢀⢻⡧⡇⡀
⠀⠀⠀⠀⠀⠀⢀⠊⠈⢂⠀⠉⠈⠐⠢⡌⠉⠀⡜⠉⠑⢄⡀⠀⠸⢿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠷
⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠉⠀⠀⠈⠉⠉⠁⠉⠉⠉
⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⣶⠒⠒⠒⠒⠒⠒⠒⣶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠐⠒⣰⣶⣶⣦⣄⠀⠒⣶⣾⣶⣄⠀⠀⢐⣶⣶⣷⣄⠀⠀⢀⣶⣶⣶⣾⡟⠀⠀⠒⠒⠒⠀⠀⣿⣤⣶⣾⣶⡒⢠⣦⡀⠀⠒⢲⣾⠀⢀⣤⣶⣶⣶⡐⠂
⠀⠀⠀⣼⡟⠁⠀⢸⣿⠀⣾⡟⠁⠀⢿⡇⢀⣿⠏⠀⠈⣿⡆⢠⣿⠁⠀⠀⢹⡇⠀⢀⣀⣀⣤⡀⠀⣿⠉⠀⠀⠹⣷⠀⢻⣷⡀⢠⣿⠇⠀⣾⠏⣀⣴⡿⠃
⠀⠀⠀⣿⡇⠀⢀⣼⡇⠀⣿⡇⠀⠀⣼⡇⠸⣿⡀⠀⢀⣿⠇⠸⣿⡀⠀⠀⣸⡇⠀⠈⠉⠉⠉⠀⠀⣿⠀⠀⠀⣠⡿⠀⠀⢻⣷⣾⠏⠀⠀⣿⡟⠋⠁⠀⣀
⠀⠀⢀⣘⣿⣿⣟⣻⣇⣀⣘⣿⣶⣾⣟⣀⣀⣙⣿⣶⣿⣋⣀⣀⣙⣿⣶⣾⣿⣿⣀⣀⣀⣀⣀⣀⣀⣿⣷⣶⣾⣟⣁⣀⣀⣀⣿⣏⣀⣀⣀⣙⣿⣶⣶⣾⣛⡀
⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁
⣀⣀⣀⣛⣛⣛⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⢠⡶⣦⠀⠀⠀⡠⠒⠈⠉⠀⠂⠄
⠈⠙⠁⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠱⡄
⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⣀⣞⢡⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⠒⠀⢊⠀⣦⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⡶⢢⠀⠀⠈⠺⠟⢸
⠀⠀⠀⠀⠄⠀⠀⢀⡗⠊⠀⠀⠀⣀⠀⡤⠑
⠀⠀⠀⠀⣾⢐⡶⠋⠀⠱⠤⢀⠀⣹⠛
⠀⠀⣀⢐⢷⢾⠀⠀⢢⡌
⠀⢠⠃⠈⠈⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⡘⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⢴⠂⠠⢄⡀
⠀⠀⠀⢀⠴⡇⠀⠀⠀⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢀⠈⠢⠀⠈⠠
⠀⢠⡶⠁⠀⢱⠀⠀⠀⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣰⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣤⡴⡄⠀⠿⢀⡃⠀⠀⠰
⠀⠉⡶⠀⠀⠀⠆⠀⠀⠛⡟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡳⠝⠏⠣⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠩⠉⠛⠁⠀⣸⠈⠁
⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⣤⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⡇⢶⡴⠂⠀⡏⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⢠⠀⠀⠀⢠⡀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠉⠀⢳⢜⡃⠀⢀⡥⢀⢀⡀⣀⡠⠈
⠀⠀⠀⠈⢄⠀⠀⡺⢧⠀⠀⠀⢓⠇⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⠀⠀⢸⠈⡛⢅⡀⠀⣼⣻⠄⣀
⡁⠀⠀⠀⠈⠀⠀⡄⠀⢠⡤⠖⠉⠒⠋⠁⣉⣁⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⢲⡾⣠⠃⠀⠈⠁⠉⣿⠀⣼
⠀⡏⠒⠄⠤⠤⡤⠇⠀⠀⡷⢤⣖⣗⡞⠉⢛⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢸⡗⠃⠀⠀⠀⠀⠀⣿⠑⢏
⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣧⠀⠻⣿⠁⠀⠨⠾⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⡄
⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⡋⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠀⠀⢀⠀⠀⠀⠀⠀⣄⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠉⠑⠲⢲⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⡄⠀⠀⠈⢣⠀⠀⠀⠀⠘⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠇⠀⠀⠀⠀⠁⠢⢄⠀⠀⠐⡀
⠉⠒⠤⣀⣀⣀⡀⢀⢀⡀⣠⠼⠁⠙⠄⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠈⠀⡇⠀⠀⢸⠀⠀⠈⠀⠀⠈⢢⡀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠤⡄⢤⠀⠀⠤⢀⡀⠀⠠⠜⠁
⠀⠀⠀⢠⠀⡆⠀⠀⢸⠀⠀⢀⠀⠀⠀⠀⠙⠋⣀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠈⢇⠀⢀⠈⡆
⠀⠀⠀⠀⡄⠁⠀⠀⢸⠀⠀⢸⡄⠀⠀⠀⠀⢸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢈⡀⠈⠀⢸
⠀⠀⠀⢠⡃⠄⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⡀⠀⣀⠚⢇⠀⠀⠈
⠀⠀⠀⢠⠹⠳⢄⡀⠘⡀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠈⡞⠁⠀⠈⠀⠀⠀⠃
⠀⠀⠀⠈⠙⠠⠚⢍⠀⠺⢦⡀⢰⠀⠀⠀⠀⡙⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢇⠀⠀⢸⡀⠀⠀⠀⠀⣤⠄⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠃⠘⣆⠀⠀⠀⠛⠂⢬⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⢷⣄⠀⠙⡆⠀⠀⠀⠱⠀⠀⠙⠂⠠⠤
⡤⢤⣤⡤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠶⠶⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠾⠃⠀⠀⠀⠀⢧⣀⠀⣀⡠⢖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⣿⡄⢠⣿⠋⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠛⠛⠛⣿⡄⠀⢠⡄⠀⠀⣿⠇⡾⠃⠺⠋⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⠇⠀⠀⠀⠀⠀⠠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⣠⡶⡄⠀⣠⡶⢳⠀⣠⣾⠋⣽⡟⠀⠀⠀⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣾⡟⢀⣇⣴⣿⣁⡏⣰⡿⠁⣼⣿⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠙⢷⡞⣻⡟⢘⡿⠛⣿⢁⡾⣿⡏⠀⠀⢀⡟⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢤⣶⣶⠾⢷⡆⣿⡇⣠⠞⠠⣿⣥⠟⠀⠀⠻⠟⠀⣿⠃⠀⢀⡾⠁⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠙⠛⣋⣤⡴⠿⠟⠲⢶⡄⠀⠀⠀⣿⡀⣠⡾⠁⠀⠀⠀⠀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢿⣿⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⣤⡾⠋⠁⠀⣰⡇⠀⣰⠇⠀⠀⠀⠈⠛⢋⣀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠘⣿⣆⠀⠀⣠⣴⠟⢁⠀⢀⣾⠋⠀⠀⠀⢠⡿⢀⡼⢋⣾⠀⣠⣿⠀⣴⠟⣹⠇⠀⠀⠘⢫⣴⡶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⠈⠛⠿⠟⠋⠀⠀⠈⠀⢸⡇⠀⠀⠀⠀⣾⠿⣏⠀⣼⣇⡾⣿⡇⣼⣯⠞⠁⢠⡄⢀⠀⠀⢶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⡿⠶⠀⠀⠀⠀⢀⣤⣶⣾⠿⢿⣿⣧⡀⠀⠀⢸⡟⠀⣽⠀⠘⠋⢠⣿⢶⣿⠁⠀⢠⡟⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡆⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⠀⠙⠿⠀⠀⣿⠇⣰⡏⠀⢀⡴⣿⡇⠘⣿⣤⡶⠏⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⢠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢿⡶⠏⠀⣴⠋⢠⡿⠀⢀⢤⡁⠀⠀⠀⠀⣾⠇⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢀⣾⠃⢀⣼⣞⠁⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠘⠃⠀⠀⢷⣤⠾⠃⠀⠘⠗⠛⠂⠀⠀⣼⡟⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⣴⢶⡄⠀⠀⣠⣶⠶⣾⡟⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠠⠶⠿⣿⠆⣠⣾⠏⢠⠇⢀⣾⡇⣸⠃⢀⣼⠟⠁⣼⣿⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢰⡿⢿⣀⣾⠖⣿⠛⣷⣯⠴⣾⡏⢀⡼⢹⡏⠀⠀⢠⡟⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠀⠀⠀⠀⣴⣿⠀⣿⠃⢀⡟⠀⢸⣟⣴⠋⠀⠀⢿⡴⠟⠀⢸⣇⣀⣴⠟⠀⠀⠈⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣤⣤⣴⠾⣿⡏⠀⣿⣴⠟⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⢀⣾⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⡾⢻⠇⣾⠃⣠⣴⠆⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢠⣶⠄⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⣼⠃⢠⡿⠀⣴⡟⠀⢀⣾⣿⣾⠿⠃⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⣸⡏⣰⣿⠇⢠⡿⠀⡰⢫⣿⣿⠀⠀⠀⠀⠀⣴⠟⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⣿⡿⠁⣿⣰⠿⣷⠞⠁⣾⠇⠹⣷⣄⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⣼⣿⣶⣶⠶⠿⠿⠟⠃⠀⠀⠀⠙⠁⠀⠀⠀⠀⠉⠀⠀⠈⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⣠⠀⠀⠀⣰⡟⣾⠀⠷⠀⠀⠀⢠⣶⡄
⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠿⠆⠀⠀⠀⠀⠈⠉

⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⢡⠄⢀⡤⣄⣤⡀⡠⣴⡟⢀⣿⢤⡀⣠⡄⣀⠀⣀⡄⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠘⠷⢚⣹⡆⢿⠴⠻⡷⠋⠓⢿⠆⢸⡧⠖⠁⠻⢶⡿⢶⣋⡀⠠⠍
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣞⡽⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢂⠢⡀⠐⡀⠀⣆⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠰⢠⠀⡆⡂⠠⠙⠀⠈⠀⠉⠸⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⡔⡇⡄
⠀⠀⠀⠄⡀⠀⠂⠌⢠⡄⠒⠣⠄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠤⢄⠀⠄⠃⠇⠃⠀⠀⠀⢀⢀
⠀⠀⠀⠰⠀⠂⠀⠀⠀⠈⠏⠀⢄⠀⠀⠀⠀⠰⠀⡀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⢊
⠀⠀⠀⠀⢁⠀⠀⠀⢠⠊⠀⠀⠀⠢⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠂⠀⠈⠑⠄⠀⠈⠉
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⡜⠀⠀⠈⠘⡄⠀⠀⠀⠀⠀⢀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠸⠀⠀⠀⠆⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡈⠀⠀⠀⡆⡀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢱⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠃⠀⠀⠀⠃⠃⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠂⠀⠀⠀⠃⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠠⡖⠆⠐⠄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠠⠀⠀⠀⠄⠀⠄⠀⠀⠒⠴⠆⠆⠠⠴⠾⠀⠄
⢠⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢀⠀⠀⡀
⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡄⢀⢀⢀⢠⠄⡄⡄
⠀⠀⣶⠀⠀⠀⠀⢄⠀⣠⣀⡠⣠⢤⠠⣠⠤⢄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠙⠀⠀⠀⠀⠒⠛⠃⡏⠓⠛⣘⣛⠛⠘⠛⠀⠀⠀⠀⠀⠀⠀⣠⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠐⢀⣴⠠⠀⣤⠂⠂⢠⣦⠀⣸⠉⠹⠆⢀⣀⡀⠀⣀⣀⠀⢀⣀⣿⢸⣇⣀⡀⣀⠀⣀⢀⣀⣀⢀
⠀⠀⠄⠤⠠⠄⠤⠀⠀⠀⠀⠀⣿⠀⢤⡄⡟⠀⣿⢸⡇⢸⡇⣿⠀⣿⢸⡏⠈⡇⢻⣀⡏⣾⣁⣽⢸⠀⢰⡂⠀⠀⣦
⠀⢰⣆⢼⣄⡤⣠⠄⠀⠀⠀⠀⢿⣀⣸⡇⣷⣀⡿⢸⣇⣸⡇⣿⣀⣿⢸⣇⣠⡇⠘⣿⠁⢻⣀⣠⠀⡖⠔⢀⢈⣀⡆⠀⠁
⠀⠈⠛⠚⠋⠓⠑⠃⠀⠀⡀⡀⠀⠉⡀⠀⡈⡉⡠⠀⠈⠁⠀⠀⠉⠈⠀⠀⠉⠀⣠⡏⠀⠀⠉⠁⠀⠧⠺⠦⠿⠼⠁
⠀⠀⠀⠠⠀⠤⠀⠀⠀⠀⠈⠀⠀⠀⠁⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⣇⡄⠀⠂⠤⡤⠐⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠈⠈⠀⠀⠁⠀⠈⠁⠀⠈⠀⠀⠀⠈⠀⠀⣤⠀⣈⠀⠀⠀⢀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⡂⢀⡄⠀⡀⣈⡈⢁⡠⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣦⣤⣄⣀⠀⠀⢀⣤⠀⣴⠏⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠿⠛⠛⠋⠉⠉⠉⠙⠛⠛⢶⠮⢥⣼⡯⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⢤⡀
⠀⠀⠀⠀⠀⠀⣀⣤⡖⠂⠉⢳⣆⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⣸⡿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⣸⣷
⠀⠀⠀⠀⣠⣾⣿⠏⠀⠀⢀⣾⠟⠀⢰⠛⢷⣄⠀⢀⣀⠀⠀⠀⢀⣴⣿⠿⠲⣾⡟⠀⣿⡇⢀⡤⣤⣸⠏⢘⣷⠀⠀⠀⠀⣸⣿⡏⢀⣼⣿⠃
⠀⠀⠀⢠⣿⣿⡏⠀⠀⠀⠀⣀⡀⢠⣬⠃⢼⣿⡖⣙⠈⣷⡀⢠⣿⡟⠁⠀⢰⣿⠁⢰⣿⡰⠁⠀⣸⡷⠀⣸⡿⠀⠀⣶⡆⣿⣿⠀⠛⠋
⠀⠀⠀⢸⣿⣿⠁⠀⠀⢀⠔⣿⡇⢾⡏⠀⣼⡿⢸⡏⢀⣿⠁⢸⣿⠀⢀⡄⢹⣿⠀⠘⠛⠀⠀⣠⣿⠃⢠⣿⠃⢀⠘⣿⠇⢿⣿
⠀⠀⠀⠸⣿⣿⣦⠴⠒⠁⠀⣿⠇⠘⠧⠴⠟⠁⠀⠁⠚⠁⠀⠈⠛⠖⠋⠀⠈⠿⠆⠀⠀⠑⠚⠛⠁⠀⠸⣿⣔⣁⣰⣿⣀⣈⠻⢧⡤⠄
⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⢲⣿⠷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠉⠀⣼⡏⠉⠛⠿⣷⣶⣶⣤⣤⣤⣤⣤⡾
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣾⡏⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠈⠛⠿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⡇⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢀⣴⣿⠇
⠀⠀⠀⠀⢧⣀⣀⣤⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠾⠿⠟⠁
⠀⠀⠀⠀⠀⠉⠛⠋⠁







⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣬⡳⠀⠀⠀⡄⠀⠀⠀⠸⡝⠂
⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⢀⡖⠄⠀⠀⠰⠀⠀⠀⠀⠄⠀⠀⠀⡇⠀⠀⠀⠀⠂
⠀⠀⠀⢀⡀⢴⡻⠄⠄⠀⠀⠀⠀⠄⠀⠀⠀⣇⠂⠀⠀⠀⠀⢀⡜⠀⡀⠀⠀⠀⠀⠀⣀⠔
⡆⠀⠀⠊⡔⠉⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⡠⠈⠢⢀⣂⣒⡂⠝⠁⠶⠬⠭⠭⠥⠴⠛⠁⢀
⠀⠀⠀⠈⠢⠰⠿⠀⠀⢈⠣⣈⣀⠒⣂⠝⠁⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⣌⠂⢀⠀⠀⠀⢀⡠⢾⠀⠀⠀⠀⠀⠀⡘⠢⡀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⢰⡒⠒⠒⠒⠀⠀⠁
⠀⠈⠑⠠⠌⢉⠁⠤⠒⡁⠀⠀⠀⣀⠀⠀⠈⠑⢌⠢⢄⠀⠀⠀⠀⠀⢠⢀⠀⠀⠀⠀⠀⠉⢉⠀⠀⡀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠁⠀⠄⠀⠀⠑⢄⠱⠀⠀⠀⠀⢸⠠⠀⠀⠀⠤⠶⠶⠾⠀⢰⢁⡇⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠉⠉⠉⠁⠀⠸⢍⡉⢛⡲⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⠀⠀⠀⡀⠀⠀⠀⣀⡠⡴⢀⠐⠒⠒⠀⠤⠠⢀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡞⠀⠀⠀⠶⠚⠒⠀⠈⠉⠐⠒⠒⠀⠐⠀⠀⣆⠡⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣊⠭⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⡀⠀⣴⣾⣿⡿⠿⠿⠟⠿⠿⠿⠿⣿⣿⣶⣦⣀
⠀⠀⠀⢠⣾⠟⠁⠀⣠⡾⠋⠁⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣷⣦⡀
⠀⠀⠀⣿⣿⣦⡀⢠⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦
⠀⠀⠀⠻⣿⡿⠃⠈⠿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀
⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⢀⣴⠟⠛⠷⢀⡶⠟⠻⣦⠀⣴⠿⠛⢷⡄⢸⡟⠛⢶⡄⠀⠘⣿⣷⡀
⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⢸⡏⠠⠶⣤⢸⡇⠀⠀⢸⣿⣿⠀⠀⠀⣷⢸⡇⠀⢘⡧⠀⠀⠹⣿⣧
⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠈⠻⣦⣴⡿⠘⢷⣦⣴⠞⠁⠻⣦⣤⡾⠋⢸⣧⣶⠾⠃⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⢠⣶⣶⣶⣶⣦⡐⣶⣦⡀⠀⠀⣰⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⢿⣿⡇
⠀⠀⠀⠀⢿⣿⡆⠀⠀⠀⠘⣿⡇⠀⠉⣿⡇⠈⢿⣷⡀⣴⣿⠟⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⢸⣿⣿⢿⣿⣯⡀⠀⠀⠻⣿⡿⠁⠀⠠⣿⣿⢿⣿⣿⡆⠀⠀⠀⠿⠿
⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⢸⣿⡇⠀⠀⣹⣧⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⣾⣿⣶⡀⢠⣾⣿⣦
⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠸⢿⣿⣿⠿⠟⠃⠀⠀⠀⡿⠇⠀⠀⠀⣿⠷⠶⢶⠶⠀⠈⠻⢿⣿⠁⠈⠻⢿⣿
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠈⠉⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣶⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣴⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⣿⣿⡿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠀⠐⠀⠀⠀⠁⠀⠈⠀⠀⠀⠘⡄⠶⠶⣆
⠀⠀⠀⠀⠀⡠⠔⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠉⢹⠉⡀⠀⠀⠀⠀⡇⢑⠀⠀⡇⠀⠀⠀⠀⠇⢸⠀⠸
⠀⠀⠀⠀⠐⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢰⡆⠈⡆⠀⠐⡀
⠀⠀⠀⠀⠠⠀⠀⠀⡈⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡃⠔⠀⠐⠠⠄⠀⠤⠤⠤⠶⠸⠤⠤⠾⠦⠦⠦⠆⠴⠴⠦⠦⠄⣠
⠀⠀⠂⢀⠀⠑⣀⠐⡡⠂⠀⠀⠀⠀⣠⠄⠠⠀⠀⠀⠀⠀⠘⠀⠄⠈⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠁⠀⠀⠀⠀⠢⠈⠃
⠀⠀⠄⢩⠔⠀⠁⠀⠁⠀⠐⠄⠀⢋⠀⠀⢈⠹⠀⠀⠀⠀⠀⠸⠀⠀⠠⢠⢤⣤⠀⡀⠈⠀⠉⠂⠀⠀⣀⠤⠀⢈⠠⠀⠀⠀⠀⠁⢠
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡉⢈⠀⣸⡦⣷⠆⠀⢤⡠⣦⠀⠀⠀⠈⣄⣹⠀⠀⠿⠸⠇⢰⠀⠀⢿⠃⠘⠃⡾⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢀⠀⢈⣅⢸⠧⠒⠤⣈⠒⠂⢀⠔⢡⠀⠀⠀⠀⠓⠚⡀⢴⠦⣥⢴⣤⠀⣋⠢⣦⡄⠰⣀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠨⣁⠿⠀⠀⠀⠀⠀⠀⠉⡅⣀⡠⠂⠀⠀⠈⠰⠾⠘⠀⠀⠄⠀⠀⠀⠰⣀⢀⡀⣀⡀⠠⠄⠲⠽⠶⠰⠀⠴⠰⠆⠀⠀⠠⠀⠀⠀⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠱⣀⡀⠀⠀⠀⠀⠇⠉⢀⣁⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣀⠈
⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠄⠄⡇⠀⠀⠀⠀⠀⠀⡏⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⡇
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠃⠂⠗⡄⠋⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡃⢔⠃
⠀⠀⠀⠀⠀⠃⠀⠀⣤⠀⠀⢀⠀⡄⠀⢀⠀⠀⠀⠀⠹⠴⠃⠀⠑⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣤⡤⠂⠀⠻⠼⠃⠁
⠀⠀⠀⠀⠀⠈⠇⠀⡍⠃⠀⠁⠀⠙⠀⠈⢉⠀⡃⠘⠁⠀⠀⠐⠚⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠈⠂⠒⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢦⠀⢀⠀⢀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⡀⢠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⠀⠇⠀⠀⢱⡖⠤⣬⠦⠮⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢤⠀⠤⠤⢨⠦⢼⡥⣤⠶
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⡀⡀⠀⠀⣀⡁⠘⣀⠀⠀⠀⠈⢻⣏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠂⠊⠀⠀⣻⣷⠁
⠀⠀⠀⠀⠀⠀⡌⠀⢢⠀⡁⠀⠀⢲⡀⠁⣾⠀⠁⢠⠀⠀⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⡀⠀⠀⠀⠀⠰⠉⠉⠈⠀⠀⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠑⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠠⢀⠀⠀⠀⢸⠁⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣄⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠁⠢⠀⠻⠀⠀⠀⢸⢤⠓⠀⠀⠀⠀⡆⠀⠀⠀⠀⠘⢷⠊⡇⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⣰⡓⢄⠀⠀⠀⠰⡄⠂⠀⠀⠈⠠⠄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⣠⢴⠁⢀⠀⠀⠀⠀⠘⠹⠀⠆⠀⠀⠀⣧⢰⡀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⢠⠀⠀⠀⢠⣎⢨⡴⠚⠁⠈⢢⠰⠀⠀⠀⠈⠉⠀⠀⠀⠀⠙⢆⠘⠤⣀⠀⠀⢀⣠⣾⠃⠀⠰⠤⠤⠤⠤⠤⠤⣤⡶⠟
⠟⠀⠀⠀⠉⠰⡁⠀⣀⠀⠀⠀⢳⡐⢄⡀⠀⠀⠀⠀⢀⣼⠀⠈⠑⠤⣄⣌⣭⠥⠖⠁⠀⠛⢒⠒⠒⡒⠒⠒⠚⠉⠀⠀⠀⠀⠀⡀
⠆⢄⠀⠀⠈⠳⢌⣺⣿⠀⠀⠀⠈⡷⣄⡈⠑⠒⠒⣚⡽⠃⠀⠀⠀⣀⠀⡀⠀⢰⠊⠁⠀⠀⠀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠐⠠
⠀⣀⠧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠉⠉⠉⠉⠁⠀⢀⢦⡀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢠⠤⡤⣤⡠⣀⡅⠀⠀⠆⢀
⠀⠙⢦⡀⠓⠢⠤⠤⠤⠠⠒⣉⠞⢀⠠⠐⠀⠀⠀⠐⠢⡐⢄⠙⢢⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⠘⠶⠤⠤⣤⡆⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠑⠲⠠⠤⠤⠖⠊⠁⣄⠁⠀⠀⢀⣤⣄⠀⠀⠈⠀⠙⢦⡈⠲⣄⠀⠀⠀⠀⠰⠃⠀⠇⠰⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢠⠂⢠⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠿⠊⠀⠀⣤⢀⠀⠀⠙⠦⣄⠹⠀⠀⠀⠀⡀⠀⡂⢠⠀⠀⠀⠀⣀⣒⣰⣶⡾⠀⠀⡈⠀⡎⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠈⠀⠃⠀⠀⠀⠤⠀⡄⠠⠀⠀⠀⠀⠛⠛⠛⠛⠲⠀⢠⣧⠾⠷⠶⢦⣄⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⣖⠒⡀⠀⠀⠀⠃⠀⠀⠀⠰⠀⠀⠀⠐⠀⠃⢠⣀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣨⠟⠋⠉⠚⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠙⠋⠀⠀⠀⢀⡆⠀⠀⠀⠀⡣⡴⢒⡟⠀⣄⠁⠁⠉⠉⠉⠓⠒⠲⠲⠃⢀⢊⠅⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⡠⡴⡟⠀⠀⠀⠀⠶⠓⠚⠋⠀⠀⠉⠉⠉⠓⠓⠒⠲⠖⠉⠀⠀⢸⠀⢧⡀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠦⠖⠚⢉⣡⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣈⡽⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠁⠉⠠⠒⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠈⠄⠀⠠⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠆⠁⠀⠀⠀⠀⠈⠀⠈⠁⡉⠈⠉⠁⠀⠄
⠀⠀⠀⠀⠀⠡⠀⠉⠀⠀⠀⡈⠀⠈⢂⠀⠀⠀⠀⠀⠠⠄⢀⡀⡀⠀⢠⡀⣀⡀
⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⢁⠰⠨⠙⠀⠈⠉⠀⠀⠠⠃⠬⣥⢥⠀⡈⠄⠠
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠂⠐⠂⠒⠒⠒⠒⠓⠂⠂⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⡀⢰⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢱
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡁⠀⠀⠀⠐⡴⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠰⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠠⣄⠤
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⢠⠂⢀⠂⠀⠀⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⣤⣀⠀⢠⣤⡀⠀⣠⣄⠀⠀⣤⣄⡟⠀⠀⠀⢸⣠⣤⡀⣀⠀⠀⡄⢀⣠⣤
⠀⠀⠀⠀⠀⣾⠁⢸⠇⡟⠀⣿⢰⡏⠈⡇⢸⠁⠈⡇⠰⠶⠆⢸⠁⠈⣷⠘⣆⣼⠃⣾⡴⠞
⠀⠀⠀⠀⠀⠹⠶⢿⠄⠻⠶⠏⠈⠷⠾⠁⠘⠷⠾⠷⠀⠀⠀⠸⠷⠾⠃⠀⣼⠇⠀⠹⠶⠾⠃
⠀⠀⠀⠀⠀⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⣴⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⢀⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠰⣿⣿⡿⠋⠀⢀⣴⡟⠳⣄⠀⣠⣿⠏⠉⣧⠀⢀⣶⣿⠗⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⢀⣤⣼⣷⣄⠀⣴⣿⣿⠀⢸⣿⣼⣿⣿⠀⢀⣿⢠⣿⡿⠁⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠿⠛⠛⣿⡟⣾⣿⡇⠛⢀⣿⢻⣿⡇⠙⠧⣼⡿⣿⣿⠃⣸⣿⣿⡇⠀⢸⣿⡆⢀⣴⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⣼⡿⠀⢿⣿⡀⣠⣾⠏⠸⣿⡇⢀⣼⡿⠁⣿⣿⣼⡟⣿⣿⡇⠀⢸⣿⢃⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣀⣾⡿⠁⠀⠈⠛⠿⠟⠁⠀⠀⠙⠻⠟⠋⠀⠀⠈⠛⠛⠀⠹⣿⣇⣠⣿⠋⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢁⣠⣶⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠉⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣵⡖⠀⣠⣿⣿⠟⠀⠀⠀⢀⣾⣿⡆⠀⢀⣶⠀⢀⣼⣿⠟⠉⣷⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⣼⣿⢁⣼⣿⣿⣃⡀⠀⠀⠀⣾⣿⡟⠀⠀⣼⣿⢀⣿⣿⡃⣠⣾⡿⠀⠀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠸⠃⠀⣸⣿⡟⢿⡿⠟⠛⠛⠻⣷⣄⢰⣿⣿⠀⢀⣼⣿⣿⢸⣿⡟⠻⠿⣋⠁⠀⣠⣬⡉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⡌⣿⣇⣤⡾⢻⣿⡟⠘⣿⣇⣀⣴⣿⠃⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠀⠀⠀⣠⣤⣄⠀⣸⣿⡇⠈⠙⢉⣤⣾⣿⣷⣶⣮⡻⠿⠟⠃⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⢠⣾⣿⠏⠈⢠⣿⣿⠁⢀⣴⠟⢹⣿⣿⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⢠⣿⡟⠁⠀⢀⣾⣿⡟⢠⣾⠏⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⣾⡿⠀⠀⢀⣾⣿⡿⠀⣼⡏⢀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⢰⣿⠇⠀⢠⣾⣿⡟⠀⠀⠹⣧⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣴⣿⡿⠋⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⡶⡶⢶⢶⠶⠶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠛⠉⠉⠉⠙⠛⠒⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢀⠖⠒⠲⣄⠀⠀⠀⠀⡠⠊⠉⠙⠢⡀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⡇⠀⠀⠀⢸⠆⠀⠀⠀⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠘⠦⠤⠤⠊⠀⣴⠀⠀⠣⢄⣀⡠⠞⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠲⢦⣄⣀⠀⠀⠀⠀⠀⠙⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⣠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣉⣠⣀⡉⠉⠛⠶⢦⣤⣠⠶⣆⠀⠀⢯⣤⣆⣴⣄⣴⣀⡞⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠦⡄⠉⢲⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠊⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠀⠀⠀⣠⣀⣀⠤⠴⠖⠚⢋⣉⡤⠴⣄⣈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠄⢀⣠⠴⠒⠊⠉⠀⣠⣤⠤⣭⡀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢀⣤⠤⠤⣄⠀⣀⣀⣀⣾⠋⠉⢧⣀⡼⡶⠚⠛⠋⠉⠙⡇⠀⡀⠀⠙⡞⠙⢆⡰⠋⢳⠖⠒⠶⣄⣀⣸⠋⣹⠃
⠀⠀⠀⠀⣀⣀⣀⡴⠋⠉⠉⠀⡿⠲⡏⠀⠀⠀⠈⠟⠉⠀⠈⠹⡄⢠⣄⠀⠀⢸⠐⣎⣉⠆⠀⡇⠸⠽⠀⢀⡤⣄⠈⠁⣀⡴⠃⠠⣔⠋⠉⡉⠀⡉⠳⣄
⠀⠀⢠⠎⠁⠀⠸⠀⠀⣰⢋⣉⣁⠸⠦⢤⡺⠂⠀⢸⠀⣴⡄⠀⢱⠈⠉⠀⠀⣸⠀⠀⠀⠀⠀⠃⠀⠀⠐⠫⠀⢸⠀⢸⠀⡇⠀⣴⡚⠁⠀⠳⠴⠀⠀⠈⠙⢲⡀
⠀⠀⣸⡆⠀⠀⠰⡀⠀⠙⠾⠂⠀⠀⢠⠤⠦⠤⠴⡃⠀⠈⠀⢀⡼⠦⠤⠤⠚⠁⠀⠀⠀⠀⠀⡀⠘⠗⢀⡠⠃⠸⣄⡔⠸⣇⠀⠀⢈⠀⠀⡖⠢⠀⠀⠀⠀⠀⢷
⠠⠚⠁⠀⠀⠀⠀⠑⠤⣀⣀⡠⡄⠀⠘⡀⠀⠀⠀⠉⠒⠀⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠴⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠘⠦⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠴⠃
⠀⠀⠀⠉⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠳⠤⣄⣀⣠⡤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠙⠦⣤⣄⣀⣠⣤⠤⠒⠉⠀⠀⠀⠈⠳⢤⣀⣀⣀⣀⣀⡤⠴⠚⠉⠀⠀⠈⠳⢤⣄⣀⡀⠀⣀⣀⡤⠔⠋
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⣤⠄⡀⠀⠀⠀⠀⢀⠀⠀⣠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⣰⠀⠀⢠⣾⠁⠈⠆⢀⣾⡟⢃⡀⢰⠀⠀⣴⣾⠋⠉⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⣿⡆⢠⣿⠏⠐⠐⠏⢹⣿⠁⠀⠁⡛⠃⣿⣿⠃⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⢻⣷⠈⣿⡀⢀⠜⠀⠸⣿⡀⣀⠜⠀⠸⣿⡇⢀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⣠⠜⣿⡶⠋⠙⣩⡴⢦⠀⠈⠉⠀⠀⠀⠀⠉⠓⠉⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡡⠊⣿⠁⢀⣾⡟⢀⠜⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠒⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⣿⠀⣾⡿⠂⢁⠀⢠⣿⠀⢰⡆⠀⣴⠟⠉⠀⠀⠀⡴⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⣰⠃⢸⣿⠃⠀⢸⠃⣿⡟⠀⣼⡇⢸⡿⠀⠀⠀⠀⠀⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠈⠀⠀⠘⢿⣄⡴⠃⠀⠻⣧⡤⣿⡁⠘⢷⣤⠔⠁⠀⢾⡩⠿⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠘⡟⠉⠛⢶⣦⣄⣀⣀⣀⣀⣠⣴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣆⠀⠀⠀⣴⣶⠉⡆⠰⡃⣸⠃⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢹⣿⠀⢀⣾⡟⠙⢦⡇⠀⠙⣡⣾⡏⠉⠑⡀⠀⢀⣠⣤⡀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⣀⠼⣿⡦⠸⣿⠀⢀⡞⠉⠁⣾⣿⠃⠈⠒⢲⠒⣸⣿⠋⠉⢻⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢁⠴⣿⡇⠀⠈⠉⠉⠀⠀⠀⢻⣇⠀⢀⡤⠃⠀⣿⠇⠀⢠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣿⠁⢀⡴⢳⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠙⠓⠂⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⢰⡏⢀⣾⠃⠎⠀⣄⠀⢀⡀⠀⠀⠀⢀⣀⠀⠀⣸⡇⠀⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠀⣸⣿⠋⠀⢰⣿⠀⢰⣿⠀⢀⣼⠏⢀⠄⣰⡟⡠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢻⡏⠀⠀⣸⡏⠀⣸⡟⠀⣼⡏⠀⠀⣰⣿⡧⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⡤⠞⠉⠻⠚⠙⠳⠖⠻⠆⠀⠸⠟⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣿⠗⠒⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠟⠛⠋⠉⠁⠀⠹⣆⡀⢀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⡸⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢦⠀⠀⢄⠀⠀⡇⠀⡇⠀⡇⠀⢰⠀⠀⡞⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠐⣆⠈⢆⠀⠸⡀⠀⡇⠀⡇⢀⡇⠀⡇⠀⡼⠀⢠⠂⠀⡠⠋
⠀⠀⠀⠀⠠⡀⠀⠀⣄⠈⢣⡀⠈⢆⠘⡆⠀⢇⠀⢧⠀⡇⢸⠀⢸⠀⣰⠁⣰⠃⢀⡜⠁⢀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠢⣄⠈⠣⡀⠑⣄⠈⢆⠘⡄⠸⡀⢸⠀⡇⣸⠀⡇⢠⠃⡰⠃⣠⠊⢀⠔⠁⢀⡴⠊
⠀⢀⡀⠀⠀⠤⡀⠈⠳⢄⠈⢢⡈⢢⡈⢧⠹⡀⢇⢸⠀⡇⡇⢸⢀⠏⡰⠁⡴⢁⡴⠋⣀⠔⠉⢀⡤⠂
⠀⠀⠉⠒⠤⣀⠈⠑⢤⡀⠑⢤⡙⢦⠑⣌⢧⠱⠘⠈⠀⠁⠃⠃⠞⡰⢡⠎⡠⠋⡠⠞⢁⡤⠚⠁⣀⠤⠒⠉
⠀⠀⠠⠤⣀⡀⠉⠒⠤⣈⠓⢤⡙⠦⡑⠈⣠⠔⠚⠉⠉⠉⠉⠑⠲⢄⡁⠊⡴⢊⡤⠚⣁⡤⠖⠋⢁⣀⡤⠄
⠤⢤⣀⣀⠀⠉⠙⠒⠤⢄⣙⠲⢬⠃⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠘⡡⠖⣋⡡⠤⠒⠊⢉⣀⣀⠤⠤
⠀⢀⣀⣀⣉⠉⠑⠒⠒⠤⠬⢍⠃⢰⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⢐⡩⠥⠴⠒⠚⠉⠉⢀⣀⣀⡀
⠀⠀⠀⠀⠉⠉⠉⢉⣉⣑⣒⣲⣤⣿⣤⣤⣤⣿⣿⣿⣾⣶⣷⣴⣶⣤⣶⣦⣼⣿⣒⣒⡚⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⢰⣿⡄⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⣿⣿⣶⠶⠿⠿⠿⠶⣾⣿⣿⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡾⠛⣁⣤⣴⣶⣶⣶⣦⣤⡉⠙⢶⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣠⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⢿⣀⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣤⣶⣿⣿⣿⣤⣽⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃⣀⣛⣛⣛⣂⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠶⠾⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⣉⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⠍⠉⠉⠉⠁




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣯⣭⣭⣽⣯⣬⣽⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣽⣿⣿⣿⣯⣿⣿⣭⣿⣯⣯⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⢀⠀⡀⡀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡾⠁⣠⢈⣀⢀⣂⢐⣀⡜⠀⡔⠀⠀⠀⠀⠀⠘⠄⡷⠶⠖⠾⠀⠇⠷⢰⠆⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠃⢈⠏⠺⠓⠻⠓⠻⢸⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⣤⠖⢲⣿⠀⢀⣀⣡⠀⢡⡟
⠀⠀⠀⠀⠀⠀⠀⣉⡡⠎⡼⢶⣡⣤⡄⠸⠯⠭⠴⡗⠀⠀⠀⠀⠀⠘⣇⢀⠵⣿⣠⣦⣴⣤⣴⢸⡏⡠⡀
⠀⠀⠀⠀⠀⠀⡞⢁⡄⣐⠒⡁⢞⡠⠔⣰⢃⣉⡀⣰⡄⠀⠀⠀⠀⠀⠈⠋⣸⠏⠈⣵⠈⢁⡉⠙⣛⢸⣇⡘⠃
⠀⠀⠀⠀⠀⠀⠁⠜⠘⠓⠛⠒⠻⠁⠀⠧⠭⢍⣁⠉⠄⠀⠀⠀⠀⠑⠃⢈⡁⠎⠠⠿⠀⠸⠗⣧⠧⠸⠛⠇

⠀⠀⠀⠀⠀⠀⠀⡠⠀⡄⠰⠀⠀⣠⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡤⠀⠀⠀⠀⢠⡄⢠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡇⢲⠶⠱⠇⠎⡇⠈⡽⢜⣰⣠⣠⡄⠒⠀⠀⠀⠀⠁⢸⠀⠤⠶⠀⣶⣦⣴⣾⢠⠀⠦
⠀⠀⠀⠀⠀⠀⢄⠉⠈⠀⠀⠀⠀⠀⠸⠃⠼⢉⡟⠃⠀⠄⠀⠀⠀⢸⡿⣋⡁⣂⢀⣀⠈⢀⣀⣙⣉⢘⣂⡶
⠀⠀⠀⠀⠀⠀⠀⡵⠒⡆⠀⠄⠀⣰⠃⣀⠀⣨⠀⠈⡄⠀⠀⠀⠀⠸⠠⠟⠛⣿⠘⠃⠀⠀⠘⠙⠀⠀⠙⣀
⠀⠀⠀⠀⠀⠐⠀⡇⢺⠺⠱⠇⠯⡇⢸⠉⢠⣤⡔⣼⠀⠀⠀⠀⠀⣰⠁⣀⠀⡋⢀⣼⠀⡏⢈⣁⢀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠒⠀⠀⠘⠐⠊⠉⠁⠈⠁⠀⠀⠀⠀⠙⠈⠉⠀⠁⠈⠙⠀⠁⠈⠈⠈⠈⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡿⣿⠆⣶⣿⢿⣷⡔⠿⠂⣾⣿⡿⠿⢲⣿⣦⢰⣿⡀⣿⣿⣶⣄⢠⣶⡿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⠄⠀⣿⣿⣿⡟⢱⣿⡆⣿⣿⠶⠶⢼⣿⣿⣿⣿⢀⣿⡇⢹⣿⠎⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⣿⠏⠻⣷⠸⣿⡇⣿⣿⣶⣶⡾⣿⡇⢿⣿⠹⣿⣿⣿⠏⢰⣶⣾⡿⠃

⠀⠀⠀⠀⠀⠀⣤⣄⠀⣤⡀⣤⣤⣤⣤⡀⣤⡀⢀⣤⡀⣤⣤⣤⣤⠀⣠⣤⣤⣄⠀⠀⠀⠀⢀⣤⣤⣄⠀⢀⣤⣄⣠⡄⠀⣠⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣦⣿⡇⣿⣿⣉⣉⡸⣿⡇⢸⣿⢻⣿⣿⣉⣉⠸⣿⣯⣽⡿⠀⠀⠀⠀⠻⣿⣍⡉⠀⣿⡛⣿⡘⣿⣤⡿⠁
⠀⠀⠀⠀⠀⠀⣿⡏⣿⣿⡇⣿⣿⣭⣭⡀⢿⣷⣾⡿⢸⣿⣯⣭⣭⢰⣿⡿⣿⣄⠀⠀⠀⠀⣀⣈⣿⣿⢼⣿⠟⣿⣧⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠛⠃⠈⠛⠁⠛⠛⠛⠛⠁⠀⠙⠛⠁⠈⠛⠛⠛⠛⠈⠛⠀⠈⠛⠀⠀⠀⠀⠙⠛⠛⠁⠘⠛⠀⠘⠋⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⠿⠃⣴⣿⢿⣷⡀⣴⣿⢿⣷⡀⣿⣿⣿⣦⢰⣿⣟⣿⣷⠸⣿⠀⣾⡟⣾⣿⠿⠿⠃⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⢀⣶⡄⣿⣇⢸⣿⡗⣿⣇⢸⣿⣷⣿⣇⣸⣿⢻⣿⣿⣿⣅⠀⢹⣿⡟⠀⢿⣿⠟⠛⠃⠸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠿⠁⠙⢿⡿⠟⠀⠹⢿⡿⠟⠸⢿⡿⠿⠋⠘⠿⣷⡿⠿⠀⠘⢿⠇⠀⠺⣿⡿⠿⠗⠠⣶⠄







⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠒⠋⠉⠉⠉⠉⠉⠉⠉⠛⠒⠲⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⢀⣠⣤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⡴⠉⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⢩⠍⡆⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠓⠚⣸⠐⠒⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⢸⣴⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⢀⡇⠀⠀⠀⠀⢀⣄⠀⡏⠀⠀⢨⠇⣰⢟⡄⢀⣴⠆⠀⣀⡀⠀⠀⡏⠀⠀⢸⠀⣴⡦⡤⠀⡤⡤⠀⠀⠘⢆
⠀⠀⠀⢸⠀⠀⠀⠀⠀⢻⡈⢸⠁⢀⡠⠋⠀⠙⢹⠇⠾⠥⠗⠀⠀⠀⡤⠀⡇⠀⣠⠏⠐⠯⢴⠃⣾⢉⣤⠀⠀⠀⠈⣇
⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠙⠛⠚⠉⠀⠀⢾⢁⡞⠀⠀⠀⠀⠀⠀⠀⠳⠼⠷⠚⠁⠀⣠⠀⡞⠀⠈⠉⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠉⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠁⣀⣀⣀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⣻⠀⠀⠀⣠⠶⠟⠉⠙⢷⡄⠀⣠⠞⠉⠙⣷⠀⠀⢠⠞⠉⠀⠉⢳⡄⠀⠀⠀⠀⣿
⠀⠀⠀⢻⠀⠀⠀⠀⠀⢸⠀⠀⢀⣴⠟⠀⠀⣰⠃⠀⠀⠀⠀⢸⡇⢰⡇⠀⠀⣰⠏⠀⠀⢿⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠀⣠⡞⠁⠀⠀⢠⡏⠀⠀⠀⠀⠀⣸⠃⠸⠇⢀⡼⠃⠀⠀⠀⠈⠁⠀⢀⡾⠃⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣰⠏⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⢳⡀⠀⠀⠀⢰⣿⠶⢦⣄⣀⣤⠈⠧⣄⣀⣤⠞⠁⠀⠀⢠⡿⠶⢤⣀⣠⡤⠀⢀⣴⠿⠶⣤⠄⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠳⡀⠀⠀⠛⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠙⠢⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠲⣄⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠤⠤⠖⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠻⣧⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠿⣇⠀⡟⢳⣾⠟⢻
⠀⠀⠀⠀⠀⠀⣠⠀⠀⣿⢀⣇⣸⣿⠀⠘
⠀⠀⠀⠀⠀⠀⣿⠀⠸⠿⠿⠿⠿⠿⢿⡇⠀⢠⣶⣷
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣿⠹⡇⠀⠀⠀⠀⢀⣴⣿⣿⡿⠻⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠙⠛⢛⣦⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣀⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠁⠀⠀⢀⣾⣿⣿⣿⣿⣿⡟⠉⠉⠉⠙⠻⢿⣿⣿⣿⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿⠋⠀⠀⣀⣀⠀⠀⠈⢻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠋⠀⠀⠀⠙⠛⠉⠀⠀⠀⠘⢿⡟⣿⡏
⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡃⠀⠀⠀⢠⣶⣿⣿⣆⠀⠀⣠⣄⠉⢻⡇
⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⣀⠶⠛⠻⠿⢿⣿⣿⠇⠀⠀⠀⠈⠛⣉⣴⠟⠀⢸⣿⡟⢡⣿⡇
⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⣿⣤⣴⣶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⣀⠉⣥⣾⣿⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⣿⣧⣄⣸⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣸⣧⠀⠀⠀⠀⠀⠉⠉⠛⠛⣋⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠻⣶⣤⣤⣾⠟⠁
⠀⣠⠾⢿⣇⠀⠀⠀⠀⢠⣿⢻⣧⡀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡀⠀⠀⠀⠀⠶⣿⣁⠀⠀⠀⠀⠀⠀⣹⠇
⣴⠋⣠⣾⣿⣦⣄⣀⣴⡿⢿⡄⠹⣿⣆⠀⠀⠀⣠⣄⠀⢸⣿⣿⣆⠀⠀⠀⠀⠈⠙⠿⠷⠶⠶⢀⣼⡏
⣿⡐⠻⠿⣿⣿⣿⠋⠉⢿⡌⢻⣆⠈⢿⣷⣀⣾⠉⠉⢻⣿⣇⠘⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠹⣿⡀⠀⠈⢻⣟⣧⠀⠀⠹⣦⡘⢧⡀⠻⢿⣧⣴⣶⣛⣻⣿⣷⣤⡉⢿⣿⣦⣄⣀⣀⣠⣾⠟⠁
⠀⠘⢿⣦⡀⠀⠻⣯⣷⣄⠀⠈⠻⣮⡷⣄⣸⣿⣥⡶⠛⠋⠀⠈⢻⣿⣾⣿⠛⠛⠿⢿⣿⠃
⠀⠀⠀⠙⢿⣦⣄⠉⢿⣿⣷⣄⠀⠉⠳⠉⠹⣿⡃⠀⣠⣤⣶⣿⣿⡿⢿⣿⣦⣀⠀⠈⣧⣀
⠀⠀⠀⠀⠀⠉⠻⣷⣄⠙⠛⡹⣷⣤⣤⠀⠀⢻⣿⣾⡿⠛⢻⡟⠛⠀⠀⢹⣿⠛⢿⣶⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⡏⢻⣿⠁⠀⠀⣾⢻⣿⠀⠀⢸⡟⠀⠀⠀⢸⣿⠀⠀⠈⢻⠁⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣷⣾⣟⠀⠀⠀⣿⠘⡇⠀⠀⠸⣿⠀⠀⠀⣸⣿⡀⠀⠀⣾⠀⣿⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠇⠈⢻⣿⣄⡀⠀⣿⡄⠀⠀⢠⡀⠙⠀⠀⠀⡿⣿⡇⠀⠀⣿⠀⣿⢠⡿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠘⢷⣾⡏⠙⠻⣿⣿⡟⠀⠀⢸⡇⠀⠀⠀⠀⣧⢸⡇⠀⠀⢿⠀⠃⢸⣷⣈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⢨⣿⠇⠀⠀⣿⣿⠃⠀⠀⠘⣷⠀⠀⠀⠀⢿⡼⠇⠀⠀⠈⣦⠀⣸⣿⠟⠛⢿⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⡟⠀⠀⣠⠻⣷⠀⠀⢠⡄⢸⠀⠀⠀⡆⢸⡇⠀⠀⢸⠀⣿⠀⣿⣇⢀⣀⡾⠿
⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⢠⣿⠇⠀⠀⣿⡆⠁⠀⠀⠘⠃⠘⠀⠀⠀⡟⠀⠇⠀⠀⠸⢀⠈⠀⣿⣿⠉⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⣼⣿⠀⠀⢰⡀⠻⠀⠀⢠⡆⢰⡄⠀⠀⣤⠀⡄⠀⠀⢸⡀⣼⠀⠀⣿⣿⠀⢹⡇
⠀⠀⠀⠀⠀⣰⣿⠃⠀⢠⣿⠇⠀⢀⠘⣷⠀⠀⠀⡘⢃⡘⠁⠀⠀⢋⡀⠃⠀⢀⡘⣧⠈⠀⢠⣿⣿⡆⢸⡇
⠀⠀⠀⠀⠀⣿⣟⠀⢀⣿⡏⠀⠀⠈⣷⡏⠀⠀⢸⡧⠸⡇⠀⠀⢾⠀⠟⠀⠀⠸⠇⠹⠁⠀⢸⣿⡿⠃⢸⣷
⠀⠀⠀⠀⠀⠻⣿⣦⣾⣿⠀⠀⠸⣆⡛⠀⠀⢀⣾⣠⣤⣀⠀⣠⡀⣤⠀⠀⢰⡆⢸⠆⠀⠀⣿⡟⠀⢠⡾⠇
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣤⣴⣷⣿⣧⣤⠴⣾⢻⣿⠉⠀⢀⣹⣷⠁⠀⢀⡜⢳⡄⠀⢀⣼⣿⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣧⣤⣿⣿⣿⣿⣷⣿⣿⡏⠀⢤⣼⣿⡎⠁⣠⡿⠟⠃⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠈⠛⠻⣿⣿⣿⠋⠛⠛⠛⠿⠿⢿⣿⣀⣘⣿⣿⣿⣾⣿⡇⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⡟⠉⠀⠀⢹⣧⣾⠏
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠈⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡇⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣸⡇⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣴⡿⠁⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣷⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣿⡿⠁⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⢸⣧⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠸⣇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⠟⣿
⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⢹⡄⠘⢃⠀⠀⠀⠀⠀⠀⢰⡟⠀⠘⣧
⠀⠀⠀⢠⡜⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⣿⡄⠀⠀⠀⠀⠀⡿⠃⠀⠀⢿
⠀⠀⢰⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⣄⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣷
⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⢿⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⢻⣆
⠀⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠐⣿⣯⡙⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠨⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⢸⣿⣿⣦⣌⡙⠿⣶⣄⣀⠀⠀⠀⢠⠉⣿⡄⢹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⡿⠀⠈⠙⠻⠿⣶⣮⣭⣿⡷⠶⣶⡆⠀⠘⣷⠀⢻⣦⣀⣀⣀⡠⠶⠶⠶⠆⠻⣷⡀
⣶⡀⠀⠀⠀⠀⠀⠈⣿⣭⣽⣿⠿⠟⠛⠛⠾⣿⣧⠈⢻⣏⠉⠁⠀⠀⠀⠀⢀⣤⣿⠿⠂⣀⣠⣤⣤⣤⣤⣤⡀
⠿⣿⣦⣤⣤⣤⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣀⡉⠀⣀⣀⣀⣤⣶⡟⠋⠀⠀⣠⣾⠟⠋⠉⠉⠉⠉⢿
⠀⠈⠉⠠⢿⣿⣷⡶⠄⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡘⠻⣿⠛⠛⠛⠛⠁⠈⠻⠶⣶⡾⠋⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠉⠛⠷⠶⠿⣟⣿⣿⣿⣿⣿⡶⠶⠟⠁⠀⢹⣦⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⣠⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠿⠿⠿⠟⠋⣉⣥⣤⡴⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠠⡄⠀⠀⠀⠀⡏⠀⠀⢠⡆⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖
⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢻⡄⠀⠀⠀⡇⠀⠀⣾⠁⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠘⢦⡀⠈⢧⡀⢀⡀⢳⡀⢶⠀⡇⠺⢰⠇⡔⠀⡴⠃⠀⠀⣠⠞⠃⠀⢀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⡀⠀⠀⠙⢦⡀⡑⣄⠱⡄⢣⠸⠀⡇⠆⡌⡼⢁⡞⢠⠜⣠⠞⠁⢀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠢⡙⠢⡈⠣⡀⠀⢀⡀⠇⡀⠈⣠⠋⠔⡡⠊⢁⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⢦⣤⣀⠀⢬⡑⠬⡂⠀⠠⠂⣠⠤⠄⠢⠄⡁⠐⠄⢀⡠⠖⣉⠤⠒⢀⣠⡤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠶⠶⠦⣤⣤⣀⣀⣀⠀⠈⢉⡒⠈⠑⢈⠟⢠⠊⠀⠀⠀⠀⠀⠈⠳⡔⢅⠀⠪⠄⠒⠊⠁⢀⣀⣠⡀⣀⣀⣀⣴⠖⠶⠤⠤⠤⠶⠶⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠈⠉⠉⠉⠑⠒⠒⠤⠌⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⠂⠀⠐⠒⠊⠉⢉⡝⠒⠛⠁⠈⠉
⠠⡤⠶⠒⠒⠒⠚⠋⠉⠀⠀⠀⠉⠉⠳⢄⣀⡉⠉⠉⠀⠐⠀⢇⠀⠀⠀⠀⠀⠀⢀⣀⡧⠬⠥⠤⠬⠭⣱⠖⠋⢀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⠤⣄⣠⠤⠸⠓⠒⠒⠒⠒⠚⠉⠁⠀⠀⠀⢀⣠⠼⠃⣀⠖⠋⠀⠀⠈⠉⢁⣠⠤⠤⠀⠀⠒⠀⠉
⠀⠀⠀⠀⠀⠐⠂⠁⠁⠐⢦⡀⣀⡀⠀⠀⠀⠤⡀⠀⡀⢠⣳⣖⡒⠒⠠⠤⠤⠤⠤⠤⣴⠞⠉⢉⡀⠐⠬⠈⠀⠴⡆⡀⠾⣉⣉⠁
⠀⠀⢀⣄⠀⣤⠄⢤⠄⠀⣠⠀⠤⠤⠬⠭⠥⠴⠦⠤⠤⠤⠴⠶⠿⠖⠂⠀⠀⠀⣤⠼⢃⢰⣀⠉⡤⡀⣀⠠⠀⣠⠔⢁⡽⠄⠈⢫⣉⣙⡤⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢠⡼⣃⣤⠾⠷⡼⠖⠛⠶⢅⣀⣀⣀⣜⣠⣤⣅⣐⣀⣈⣉⣄⣁⢇⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡖⠒⣒⠲⠤⠶⠻⢛⣛⣋⢀⣤⣄⠤⠤⠤⠤⠤⠶⠤⠭⠭⠁⠀⠀⠀⠉⠉⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⣀⣴⠶⠒⠒⠚⠛⠚⠛⠛⠒⣾⣉⡉⠀⠀⠀⠈⣡⣤⣵⣐⠒⣒⣒⣃⡀⠀⠀⠀⠀⠀⠤⠤⠤⠤⡤⠤⠤⠔⠀⠀⠦⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣠⣴⠾⠷⢶⣤⣴⠟⠁⠀⠀⠀⠀⣠⡾⠛⠛⢷⡄⠀⠙⠷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⢾⡇⠀⠀⠀⠀⠀⠀⢀⣿⠶⣤⣄⠀⠀⠹⢦⣤⣤⡾⠿⣤⣀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⣠⠾⠛⠛⠁⢠⣤⠟⠷⣤⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠙⠷⢶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠙⢦⣤⣤⠾⠿⣤⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠛⠙⠛⢶⣤⠶⠛⠋⠛⠷⣦⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⣀⣀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀⣠⣤⣄⣀⡀⠀⠈⠛⠷⠾⠋⠉⠙⠳⠶⠛⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠈⠉⠁⠀⠈⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⡀⠀⠀⡀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠈⠄⠐⠀⠀⠀⠃⠐



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠘⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⠾⣿⠙⠺⠇⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠈⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠛⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣀⠤⠒⠈⠉⠁⠀⠀⠈⠉⠐⠢⢄⡀⢀⣴⠿⠒⠂⠉⠉⠉⠉⠐⠒⠬⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢃⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⡴⠞⠁⠀⠀⠀⠀⠀⠙⠶⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⢉⡉⠛⠲⣄⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠛⠓⠦⣄⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⣼⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢠⣿⣿⣿⡀⣀⠈⣧⠀⠀⠀⠀⢠⠟⣠⣶⣿⠆⠀⠀⠈⢧⠀⠀⠀⠀⠀⡟
⠀⠀⠀⣠⠶⢷⣄⣿⣤⡄⠀⠀⠀⠀⠀⠀⡇⠀⢻⣿⣿⣿⣿⡿⠀⢸⠀⠀⠀⠀⢸⠐⣿⣿⣿⣿⣿⠀⠀⢸⠀⠀⠀⠀⠀⡿
⣴⠛⠙⡇⠀⠀⡟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠳⠀⠈⠙⠻⠿⠛⠁⠀⡚⠀⠀⠀⠀⠘⣇⠙⠿⠿⠛⠃⠀⣀⠞⠀⠀⠀⠀⢰⣇
⣿⠀⠀⣇⠀⠀⣿⠀⠀⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠒⠒⠊⠀⠀⠀⠀⠀⠀⠈⠓⠒⠛⠈⠀⠉⠁⠀⠀⠀⠀⠀⣾⠉
⣿⠀⠀⣿⠀⠀⣿⠀⠀⢹⠉⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⣿⠀⠀⠿⠀⠀⠛⠀⠀⢸⡄⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⣿⠀⠠⠴⠶⠶⠶⣶⠦⠘⠃⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⢸⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⡿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⢸⡆⠀⠀⠀⠀⣯⠀⠀⠀⠀⢸⡇⠀⠉⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋
⠘⣧⣀⣀⣀⣀⣽⣤⣤⣤⣤⠞⠀⠀⠀⠀⠈⣹⠷⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠞⠋⠁
⠀⠀⠉⠉⢻⡍⠀⠀⠀⠈⢷⡀⠀⠀⣀⣤⠾⠋⠀⣿⠀⠉⠉⣹⢿⡛⠛⠛⠛⠉⣹⠇⢹⡆
⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠉⠛⠛⠉⠁⠀⠀⣠⡟⢀⡠⠞⡹⣿⡿⢦⡀⢀⡔⢻⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠙⠷⣄⡀⠀⠀⠀⠀⠀⣀⣤⠞⣻⠁⠈⠀⠀⣿⣾⠁⠀⠙⠋⠀⢸⡄⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉⠀⠀⡿⠀⠀⠀⢸⣿⠙⠂⠀⠀⠀⠀⢸⡇⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣾⣿⣦⣄⠀⠀⠀⠀⢸⣧⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⢾⣿⣿⠟⠀⠀⠀⠀⢸⣿⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣿⡼⠃⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⢴⡶⠶⢦⠀⠀⠀⠀⣿⣶⡆⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡀⠀⠘⣧⠀⠀⢧⠀⠀⠀⣅⣿⣤⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠸⣇⠀⠸⣇⠀⠀⣿⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠿⠂⠀⢻⡄⠀⢻⡄⠀⠘⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡟⠀⠀⠀⠘⣧⠀⠸⣇⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⣀⢀⣀⣠⣞⣉⣀⣀⣀⣀⣸⡇⣀⡀⣿⣀⣀⣀⣈⣻⣦⣄⠀⢀⣀⣀⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⣼⢠⡞⢙⣆⢰⠟⣹⡆⣴⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣼⣿⢸⣇⣸⣧⢿⣀⣿⡷⣯⡴⣿⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣷⣤⣭⣥⣤⣤⣭⣥⣤⣬⣤⣬⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣅⣾⠇⣤⣴⣶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢠⡟⢀⡼⠇⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠉⠛⣦⢠⡟⢀⡟⢠⡏⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⣀⣴⡟⢸⣇⣾⣇⠾⣏⣁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣠⠴⣾⠏⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⢠⢠⡀⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠢⡾⠁⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⣴⡶⡀⠙⡄⠂
⠀⠀⠀⠀⠀⠀⠊⠁⠀⠷⠏⠀⠁⠀⠀⢀⣨⣴⢙⣿⡿⣿⣟⣿⣾⢿⣿⡇⠀⠑⠳⠽⠇⢲⡿⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠙⢯⠠⠑⢶⣶⣬⠭⣤⣥⣭⣤⣼⣿⣿⣣⣬⣛⢿⣿⣇⣐⣀⣖⣚⣿⣷⣾⣷⣿⣿⣦⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠶⣛⣽⣒⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡭⢙⣮⣡⣌⣿⣿⣿⣿⡯⠿⢿⣏⡷⠝⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠉⠁⠀⠈⠉⠋⠉⠋⠉⠚⠋⠉⠙⠛⠋⠋⠉⠁⠉⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⡾⠛⠉⠉⠉⠉⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠤⣾⣏⣠⣤⣤⡄⠀⠀⠀⣀⠹⣷⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⣽⣿⣷⣾⣿⣷⣶⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⡿⠟⠶⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠺⠿⠿⠿⠿⠿⣿⡟⠛⠛⠛⠻⠿⠿⠿⠿⢿⣿⠿⠟⠉⠉⠉⠉⠛⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣷⣄⠀⠀⠀⠀⠀⠀⣠⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠶⠦⠴⠶⠿⠟⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄⠀⠀⠀⠉⠉⠉⠙⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡂⠀⢐⣾⠂⠀⠀⠀⠀⢀⠀⠀⠠⠆⠀⢀⣼⣻⡀⠟⣃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣿⣿⣷⣼⣯⣿⣷⣶⣼⣦⣤⣶⣶⣴⣧⣴⣶⣿⣿⣿⣷⣶⣿⣿⣿⣤⣤⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠀⠁⠈⠉⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣀⣠⠊⠙⢂⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣤⣤⣀⣀⣀⡀⠈⠉⠉⠀⠀⠈⠀⠀⠀⠀⠠⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣉⣷⣶⣄⣀⠀⠀⠀⠀⠴⠀⠀⠠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⢻⣿⣿⣿⣶⡶⢤⡤⠤⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠚⠿⠿⠿⠿⠟⠷


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠛⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠟⠛⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠙⠛⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠈⠿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⢀⣾⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢿⣄⠀⠀⠀⠈⠻⣦⣄
⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⢀⣤⡾⠋⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠉⠻⣦⣀⠀⠀⠀⠙⢷⣤⡀
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⣠⡶⠟⠁⣀⣠⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⣀⠀⠙⢷⣤⡀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⢸⡇⠀⠀⣠⣴⣿⣯⠶⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠶⣬⣿⣦⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠛⠷⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠙⢿⣷⠾⠛⠁
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠛⠛⠻⢶⠶⠶⠶⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣤⣤⣤⣤⣤⣴⡾⠛⠉⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣤⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡶⠶⠟⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⣀⣀⠀⣀⡀⢀⣀⣀⣀⠀⣀⠀⠀⡀⠀⣀⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠿⠟⢸⣿⣛⣿⡇⣙⡁⣿⣿⠛⠛⢹⣿⣷⣸⣿⢸⣿⠿⣿⣆⣾⣿⠻⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠗⠀⣿⣿⣿⣏⠠⣿⡇⣿⣿⣛⣛⢹⣿⠿⣿⣿⢸⣿⣀⣿⡟⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠻⠏⠈⠻⠇⠿⠇⠿⠿⠿⠿⠋⠿⠃⠻⠟⠸⠿⠿⠛⠀⠻⠿⠟⠋

⠀⠀⠀⠀⠀⢰⣿⣦⢸⣷⢰⣿⡿⠿⠇⣾⡇⠀⣾⡆⣾⣿⠿⠿⢀⣾⡿⢿⣷⠀⠀⠀⠀⣶⣿⠿⠆⢠⣾⣷⣴⣷⠀⣼⡷
⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⢸⣿⡷⠶⠆⢻⣿⢰⣿⡇⣿⡿⠶⠶⢸⣿⣿⣿⠋⠀⠀⠀⠀⠙⢿⣶⡄⣼⣧⣼⣷⢹⣷⡟⠁
⠀⠀⠀⠀⠀⠸⣿⠀⢿⡿⠸⣿⣿⣿⡷⠈⠻⣿⠟⠀⣿⣿⣿⣿⠾⡿⠉⠻⡧⠀⠀⠀⠀⢶⣾⡿⠏⢿⠏⠙⣿⠆⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⢀⣤⣶⣦⡀⢀⣴⣶⣤⠀⣶⣶⣦⣄⢠⣶⡶⣶⣄⣴⡆⢀⣴⣄⣶⣶⣶⣶⠀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⢀⣀⢼⣿⠉⣿⣧⣾⣟⢹⣿⡇⣿⡏⢹⣿⢸⣿⣷⣿⡋⠘⣿⣼⡟⢸⣿⣿⣤⣤⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣾⡿⠸⣿⣶⣿⠇⠹⣿⣾⡿⠃⣿⣷⣿⠟⢸⣿⣭⣿⡿⠀⢸⣿⠀⠘⣿⣷⣶⣶⠀⣩⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⠀⠀⠀⠈⠉⠀⠀⠉⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠉⠀⠀⠉⠉⠉⠁⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢸⡟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⢸⠇⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢸⠀⠀⣽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⡇⠀⢀⡇⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⡘⠀⠀⡼⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠐⠁⠀⡰⠁⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠠⡄⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⡔⠁⠀⡼⢹⣿⠇
⠀⠀⠀⠀⢀⡙⠄⠸⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠊⠀⠀⠊⢀⣿⡿
⠀⠀⠀⠀⢀⡙⠆⠰⣳⠂⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠙⠦⠀⢫⠞⣛⡛⠻⣦⡀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠰⢉⡀⠀⠀⠀⠈⢻⣆⠹⣧⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠙⢧⡘⣧⠀⠀⠀⠀⠀⠀⢠⣿⠁⣰⢦⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠈⠳⣽⡄⠀⠀⠀⠀⠀⣿⠃⣰⠃⠘⠁⠘⢉⠽⠛⢛⣳⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⢀⡞⢁⡰⠁⠀⠀⠀⢀⡠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠒⠒⠒⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡀⠀⢀⡀⠀⠀⠀⠀⠀⠰⣋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⡏⠉⠉⢦⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⡼⠀⠀⠀⢀⣰⡏⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠙⢦⠀⠀⠀⠙⠢⠔⠋⠀⠀⠀⢠⡏⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣠⣄⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⣠⣄⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣠⡤⠤⣄⠀⣀⣀⣠⡞⠋⠉⠙⠲⡴⠒⠒⠛⠙⠏⠁⡀⠀⠙⡞⠳⣄⡔⠛⡶⠒⠦⣄⠀⡾⠛⡗
⠀⠀⠀⢀⣠⣄⡴⠋⠉⠉⢨⠗⢾⠁⠀⠀⠈⢛⠉⠀⠉⠣⠀⣤⠀⠀⢨⠰⣋⠽⠀⠀⠀⠿⠀⢀⠤⡄⠈⣀⡤⠛⠠⢖⠋⠉⡇⢠⠙⢦⣀
⠀⠀⣼⠁⠀⠀⠁⠀⣴⢩⠭⠤⠚⠒⠞⠃⠀⠸⠀⣶⠀⠀⡆⠉⠀⠀⡸⠀⠀⠀⠀⢠⢀⣀⠀⢣⡀⣠⠀⡇⣠⠀⠴⣜⡁⠀⢓⠚⠀⠀⠈⠱⣄
⠀⣀⠼⠂⠀⠀⢆⠀⠈⠛⠁⠀⠀⡖⠚⠒⠚⢇⠀⠀⢀⡴⠓⠒⠒⠉⠀⠀⠀⠀⠀⢸⣄⣉⡀⠘⠁⠹⠶⠇⠿⣤⣤⣼⡇⠰⣯⠇⠀⠀⠀⠀⢻
⡞⠁⠀⠀⠀⠀⠀⠑⠲⠖⠒⡄⠀⣡⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠉⠓⠒⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉
⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠦⣄⣀⣀⣀⣀⡤⠞
⠀⠀⠀⠀⠀⠙⠦⣄⣀⣀⣀⣀⠤⠖⠋⠀⠀⠘⠦⣄⡀⠀⠀⣀⣀⡤⠔⠚⠁⠈⠳⢤⣀⡀⠀⠀⠀⣀⣠⠴⠋⠀⠀⠀⠉⠉⠉

⠀⠠⠤⠤⠤⡤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣄⣾⣿⣿⣇⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⣀⡀
⠀⠀⠀⠀⠿⠛⡿⠿⠿⠿⠿⠟⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠺⠀⠐⠉⠘⠛⣿⣿⣿⡟⠙⠁⠀⠠⠀⠀⠀⠀⠀⠀⡹⠯⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠒⠒⠂⠐⠀⠂⠀⠀⠀⠀⠒⠀⠒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠖⠛⠛⠛⠛⠛⠛⠓⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⢀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠟⠁⠀⠀⣀⡤⠖⠛⠉⠉⠉⠉⠉⠙⠓⢦⡀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢤⡼⠀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⢳⣴⣾⣯⠶⠶⠶⠶⠶⠶⠦⣤⣄⡀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⢀⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢶⣄⠀⠀⠀⠀⢻⠀⠀⠀⠀⡀⠀⣸⠻⣆
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢀⣿⡀⢀⣆⠠⣿⣶⢿⣆⠙⣧
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠃⠀⣠⡾⢹⣧⡾⠻⣶⡏⠀⠀⠹⣆⠸⣇
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠰⠛⠋⠀⠀⠉⠀⠀⠀⠀⢻⡄⢿⡀
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠀⠀⠀⠀⠀⠀⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣆⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⣤⣶⣶⠶⠾⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠶⠶⠶⣶⣄⠀⠀⠀⠀⠀⣿⠘⣧
⠀⠀⠀⠀⠀⠀⠈⣷⢀⡴⠋⠁⠀⠀⠀⠀⣠⠶⠚⠉⠉⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠒⠓⠒⠶⣄⠉⠀⠀⠀⠀⠀⣿⣠⡟
⠀⠀⠀⠀⠀⣴⠟⣿⠋⠀⠀⠀⠀⠀⠀⡾⠁⠀⣴⣾⣿⡇⠀⠀⢳⡀⠀⠀⠀⠀⢠⠞⢁⣤⣴⡆⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⢸⡿⣦
⠀⠀⠀⠀⢨⣧⡀⣿⣀⡀⠀⠀⠀⠀⢸⡇⠀⢸⣿⣿⣿⣿⣶⡇⠀⡇⠀⠀⠀⠀⣾⠀⣿⣿⣿⣷⣴⡆⠀⠀⣧⠀⠀⠀⠀⠀⢸⡇⢸
⢀⣤⠤⡼⠋⠉⠻⡟⠋⠙⣷⠀⠀⠀⠘⠇⠀⠈⠻⢿⣿⣿⠿⠁⢠⡇⠀⠀⠀⠀⠸⠀⠻⣿⣿⣿⡿⠁⠀⠀⠃⠀⠀⠀⠀⠀⢸⣇⣼
⣿⠁⠀⣿⠀⠀⢰⡇⠀⠀⣿⠀⠀⠀⠀⠀⠉⠒⠶⠦⢤⣤⣤⣴⠏⠀⠀⠀⠀⠀⠀⠰⣦⣤⣭⠥⠶⠖⠒⠁⠀⠀⠀⠀⠀⠀⣾⠛⠁
⣿⠀⠀⣿⠀⠀⠸⡇⠀⠀⣿⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏
⣿⠀⠀⢿⡇⠀⠀⣷⠀⠀⢸⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⣿⠀⠀⠸⠇⠀⠀⠋⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⣿⠀⢠⣤⣤⡶⠶⢶⣤⡶⠘⠇⠈⣷⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⢺⡀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⢹⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋
⢸⡇⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣿⠀⠈⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠋⠁
⠸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⢨⡟⣷⠶⢦⣤⣤⣀⣀⣀⣀⣤⣤⣤⠶⢶⡟⣿⠁
⠀⠻⠶⠶⣶⠶⠶⠛⠛⢻⣟⠛⠋⠀⠀⠀⠀⣠⠟⠀⣿⠀⠀⠀⠙⠿⣭⣭⣥⠼⠋⠀⠀⢸⡇⠸⡇
⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠙⢦⣄⣀⣤⠶⠟⠁⠀⢸⡟⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⣷⠀⢿⡀
⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣠⣼⡇⠀⠀⠀⢠⣾⣿⣿⣿⣿⣧⠀⠀⠀⣿⡀⠘⣧
⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⣀⠀⠀⣀⢀⣀⠴⠞⠉⢸⠃⠀⠀⠀⠘⡏⠙⠛⠛⡿⠋⠀⠀⠀⢸⡇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠁⠀⠀⠀⠀⣼⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⣧⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠶⠶⠶⠶⠶⠶⠖⠚⠛⠛⠛⠛⠛⠋⢻⡄⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⡶⣤⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣀⣀⣀⣀⣸⣇⣹⣀⢀⠀⠀⠀⣠⣿⣿⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠉⠉⠉⠁⠀⣹⠈⠁⠀⠀⠈⠈⠉⣻⣷⣴⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⢀⣀⣀⣀⣀⣿⣤⣤⣄⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⣉⣹⡏⠀⠀⢸⡇⢈⣟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠉⠙⢿⣄⠀⣼⠟⠋⠉⠛⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⣀⢀⣀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣿⣠⣏⣀⣀⣀⣀⣸⣇⣀⣀⡀⠀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠂⠀⠐⢲⣦⡀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⠀⣀⣀⣤⡖⠂⠀⠑⣲⣆⣀⢀⣀⣀⣀⡀
⠀⠀⣯⣉⣉⣛⠛⠿⡿⠿⢛⣛⣛⣛⣻⣯⣟⣻⣛⣯⣿⡷⠀⢠⣯⣉⣉⣛⠛⠿⡿⠿⣛⣟⣛⣛⣻⣭⣟⣛⣻⣽⣿⡇
⠀⠀⣭⣭⣭⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢠⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠛⠛⠋⠉⠉⠉⢛⠛⠛⠛⠙⠛⢛⡉⠛⠛⠛⠛⠛⠃⠀⠈⠛⠛⠉⠉⠉⠙⣟⠛⠛⠋⠉⠉⢙⠉⠛⠛⠙⠛⠛⠃
⠀⠀⠀⠀⠀⣦⠀⣠⣾⠀⠀⠀⠀⠀⢸⣇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣠⣿⠀⠀⠀⠀⠀⣿⣄⢰⡇
⠀⠀⠀⠀⠀⣹⡞⣫⣾⣇⠠⠤⢤⡀⢸⣯⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢞⣡⣿⣄⠤⠤⣤⡄⢿⣯⣻⣧
⠀⠀⠀⢀⠔⢻⣾⣿⠧⠀⢴⡶⠾⠯⠤⢿⣿⣿⠳⣄⠀⠀⠀⠀⠀⡠⠞⣿⣾⡿⠅⠀⠤⡴⠯⠭⠌⢿⣿⣏⠳⣄
⠀⠀⠖⠁⣴⣿⡿⠁⠀⠀⠀⠠⠤⠤⠾⠯⣿⣿⣷⡎⠳⡄⠀⠠⠊⢠⣶⣿⡿⠁⠀⠀⠀⠀⠀⠀⠻⢯⣿⣿⣷⡌⢳⡄
⠀⠀⣴⣿⡿⣿⡉⠉⠉⣉⢉⡉⢉⢉⠉⣉⣛⣻⣿⣿⣇⡈⠀⠀⣶⣿⣟⣟⡉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣛⣻⣿⣿⣧⡁
⠀⠀⣿⣟⣼⠩⠤⠤⠄⠀⢀⣀⣀⣀⣀⡈⣈⣸⣿⣿⣿⣷⠀⠀⣿⢏⡾⠥⢤⡤⠤⠄⢀⣀⡀⣤⡀⠀⣤⣼⡿⣿⣿⡇
⠀⠀⣟⡾⣉⣀⣀⣀⣀⠀⠀⠐⠀⠀⢀⣤⣄⡍⢭⣿⣿⡿⠀⠠⢏⡟⠁⣀⡀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣤⠭⢭⣽⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⣶⡶⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣞⠙⣷⠀⢸⣆⣿⣿⣿⣡⣼⠇⢈⡁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣿⠶⣦⢻⣿⠈⠉⠉⠁⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣀⣴⠟⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠛⠋⣠⣴⣶⣾⣿⣿⣿⣿⡿⠿⢷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢻⣿⣧⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣀⠂⢸⠀⢠⡤⠈⠛⠀⠀⠀⠁⠈⢫⣤⡄⠀⣸⣿⠏⠀⠈⢿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠸⠛⡣⡞⢄⠀⠀⠀⠀⠐⡂⢘⠀⠀⠀⠀⠀⢀⡿⠃⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠐⢅⣀⡹⠦⠤⣀⣀⠑⠊⢀⡤⠤⠴⣒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⢀⡀⠀⠀⠀⠑⠒⠚⠁⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠢⣄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⣀⣀⣀⣀⣀⣀⠀⣀⠠⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣠⠃⠀⠀⠀⠀⠀⠀⠸⣄⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠛⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⢀⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⢀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⣠⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠀⣀⣴⣿⣿⡿⠃⠀⠀⣠⣤⣄⠀⠀⠀⣴⣶⠶⣄⠀⠀⢀⣠⣤⣄⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠙⠻⡿⠛⠀⠀⣠⣾⡿⠁⢸⡇⢠⣾⣿⠃⠀⣿⡇⢠⣾⣿⠋⣸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢠⣴⣾⣿⣿⣷⣄⣼⣿⢿⡇⠀⣾⣧⣿⡿⣿⡄⢀⣿⣧⣿⣿⠃⢰⣿⣿⡏⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠉⠉⢁⣿⡏⢸⣿⡟⠀⠁⣼⡟⣼⣿⡇⠈⠉⣾⡟⣿⣿⡇⣰⣿⣿⣿⡇⠀⢸⣿⡇⢠⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⢀⣾⡿⠀⠸⣿⣧⣠⣾⡟⠁⠸⣿⣇⣠⣾⡿⠁⢻⣿⣷⣿⠋⣿⣿⡇⠀⣸⣿⢃⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣤⣾⠟⠁⠀⠀⠈⠛⠛⠉⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠉⠉⠁⠀⠻⣿⣇⣴⡿⠃⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣀⣤⣶⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋⠉⢀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢛⣵⡶⠀⢠⣾⣿⡿⠁⠀⠀⠀⢠⣾⣿⡆⠀⠀⣾⡆⠀⣰⣿⡿⠋⢹⣧⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢁⣾⣿⢁⣴⣿⣿⣟⡀⠀⠀⠀⢠⣿⣿⡟⠀⠀⣰⣿⡇⣼⣿⣟⠀⣠⣾⡿⠀⠀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣼⣿⡟⣾⣿⡿⠿⠿⠿⣷⣦⠀⢸⣿⡟⠀⠀⣰⣿⣿⢡⣿⣿⠻⢾⡿⠛⠁⠀⣀⣙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠈⠁⠀⠀⠀⠀⠘⣿⣧⢸⣿⡇⣠⣾⢿⣿⣿⠈⣿⣿⠀⠀⣼⣷⠀⢠⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⣀⣀⠀⢀⣿⣿⠀⠻⠿⠛⢁⣸⣿⣿⣤⣈⡿⣷⣿⡿⠋⠀⠀⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⣠⣾⢟⠫⠃⢸⣿⣿⠀⠀⣠⣾⢿⣿⣿⠏⠉⠛⠓⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠁⠀⣼⣿⠟⠁⠀⢠⣿⣿⡇⢀⣾⡟⠁⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣼⣿⠏⠀⠀⢀⣾⣿⡿⠀⣾⡟⠀⣸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⢠⣿⡏⠀⠀⢠⣾⣿⡿⠁⠀⣿⡇⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢸⣿⠁⠀⣰⣿⣿⠟⠀⠀⠀⠻⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣾⣿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⡀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢀⡤⠤⣀⢀⡀⠀⢀⣀⡀⢸⢉⠇
⠀⠀⠀⠀⠀⢀⠎⠀⢀⣷⠋⠈⠹⠉⠙⠇⣀⠘⠖⡖⢲⠊⠀⠒⢼⠃⡈⠉⠍⠰⠏⠻⣸⠉⠒⣄⢀⣀
⠀⠀⣀⡴⠋⠉⢄⠘⠉⠉⡆⠁⠠⢈⡁⠦⠀⠔⠀⠀⠀⠀⠀⢁⡠⠀⣃⡀⢀⠈⡠⠠⠆⠀⠀⠈⠁⠀⠉⠒⣄
⢠⠊⠀⠀⠀⠀⠀⠸⡠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠈⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠑⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⠀⠀⠀⠀⢀⡠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠓⠒⠶⠖⠒⠒⠉⠁

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⡇⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⡀⣀⡠⠄⠠⠄⠀⠀⠀⠀⠀⠒⠈⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠈⠁⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠂⠄⢀⡀⠀⢀⠀⡆⠀⠀⠀⠂⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⠀⠀⠀⡀⠀⣤⠠⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⡇⠀⠀⢠⡄⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴
⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠘⣆⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⢆⠀⠈⠳⡀⠀⠀⠘⣆⠀⡄⢸⡇⡆⢰⠇⡤⠀⣠⠋⠀⠀⢀⡴⠆⠀⠀⢀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠑⢄⠀⠙⢆⠘⣆⠘⡄⡇⢸⠀⠃⡏⡼⢁⡴⢁⡔⢀⡴⠋⠀⢀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⢠⡑⢌⠠⠣⡈⠂⠁⠀⢸⠀⠀⠐⢡⠎⡠⢋⠔⠋⢀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⡀⠀⠀⢉⠲⢌⠢⠀⠀⢈⠀⠂⣁⣈⣈⠀⠐⢁⠈⠀⢁⡤⠞⢉⡠⠀⠀⣀⣠⠴⠖
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠈⠉⠙⠲⠤⣉⠂⠈⢂⠔⢁⠔⠉⠀⠀⠀⠈⠐⢄⠠⡒⠁⠐⢊⠁⠤⠒⠉⠁⠀⠀⠀⠀⠀⢀⣠⣄⣀⡀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠓⠒⠶⠦⠬⠅⣀⣀⡀⠂⢠⠇⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠈⣀⠀⠤⠴⠖⣒⠿⠥⠖⠛⠓⠋⠀⠀⠉⠉⠉⠁⠀⠈⠉⠁
⠀⣀⣀⣀⣀⣄⣤⠴⠒⠒⠒⠒⠒⠶⢤⡀⠀⠀⠒⠀⠀⠀⠉⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣀⣀⣀⡒⢀⣠⠼⠃
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢤⡀⠀⠀⠈⠀⢣⣀⣤⡤⣄⣀⡤⠴⠚⠉⠁⠉⠉⠀⢠⡞⠁⣀⠴⠀⠐⠂⠖⠂⠀⠀⠀⠀⠀⢠⡄⠠⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠙⠒⠪⣉⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠚⢉⠰⠲⠁⠀⣀⠀⢀⡤⠴⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠒⠂⠠⠤⠄⢈⡓⠁⠐⠀⠓⠻⣲⣭⣀⠉⠉⠉⠉⠉⢩⠟⢁⠀⠀⠀⠀⠉⠀⠀⠈⠓⡂⢬⠖⠲⣤⠤⢤
⠀⠀⠐⠒⠀⠛⠃⠈⠃⠀⠐⠂⠈⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣰⡟⠉⡴⡀⢴⣐⣋⡂⠆⠀⠀⠐⢉⠀⠞⠁⠀⠐⠳⠶⠎⠡⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠚⣁⣀⣜⣫⡶⠛⠉⠙⠋⠁⠉⠉⠳⠶⠶⠶⠦⠖⠛⠒⠲⠶⠶⠶⠶⠶⠧⠿⢦
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⢀⡠⠤⠴⠶⠶⠚⠭⠭⠍⠭⠟⠉⠉⢰⡄⠒⠐⠒⠒⠊⠉⠐⠒⠊⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠰⠾⠋⠉⠉⠁⠀⠀⠀⠀⠉⠹⠦⢤⠀⠀⠀⠀⠐⠶⠞⠺⠭⠥⠥⠄⠠⠀⠀⠀⠀⠀⠀⠘⠛⠓⠒⠒⠒⠛⠋⠀⠀⠙⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠄⠛⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠁⡀⢀⣀⢠⣀⠠
⠀⠀⠀⠀⠈⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢀⡀⢘⠂
⠀⠈⠄⠠⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⢀⡀⣀⠠⠤⠐⠒⠒⠒⠦⠤⣀
⠀⠀⠀⠀⠀⠀⣀⡀⣠⣔⣀⡂⠐⢀⠰⠤⠀⠚⠛⠉⢁⡐⠢⡀⠀⠀⣀⠤⠶⠒⠙⠒⠚⠓⠒⠲⠤⠤⣀
⠀⠀⠄⠂⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠉⠁⠀⠀⢁⣀⠥⠤⣁⣉⡐⠒⠲⠀⡀⣀⡠⠤⠄⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢀⣀⡠⠤⠤⠑⠐⠒⠚⠃⠀⠈⠑⣄⣀⠤⠉⠁⠠⢍⡀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⢀⣀⡀⠠⠄⠀⣀⡀⡀⣈⡆⡴⠒⠉⠉⠀⢄⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠠⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠈⢦⡠⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⢶⡀⢀⡀⠀⣿⢸⠃⠖⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠞⠁⠀⠀⢀⠉⠀⣀⡀⢀⣤⣴⡏⠀⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠈⠃⠀⠀⢀⣴⠋⣇⣾⣣⢣⡞⣡⡿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⢀⣀⣀⢀⡿⢉⡟⣿⣡⠟⢿⡵⢻⠇⠀⣸⢡⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠉⢉⡽⠉⠷⣋⣤⠿⠧⢤⠀⠀⢸⡀⣴⠃⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡈⣿⡀⠀⠀⣠⠞⠀⣠⠞⠉⠀⡾⢀⡼⡀⢀⡌⢉⡥⡀⠀⢠⠞⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠷⡜⠷⠶⠛⠁⠀⢰⠇⠀⠀⣸⢷⡋⣼⡡⣿⢡⡯⠞⢠⠀⠀⢩⡭⠁
⠀⠀⠀⠀⠀⠀⠚⢛⡃⠀⠀⣠⡶⠟⠛⠻⢧⡀⢀⡟⢠⠇⠈⣠⡟⣿⠀⣰⠏⢀⡀
⠀⠀⠀⠀⠀⠀⠰⠟⠁⢀⣾⠋⠀⠀⠀⠀⠤⠁⠸⣧⠎⣠⠞⣽⠁⣨⡋⠁⠀⣸⠇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠂⠀⢀⠀⢠⠀⢠⡃⣰⠃⢰⣯⡀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠉⣡⡄⠀⢀⡤⣤⡟⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠄⠀⢸⡇⠀⠀⠀⠀⠐⢛⡿⣰⣿⢠⣣⣾⣇⢇⣰⠏⣠⣿⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⢀⣾⢱⡏⢸⠋⢸⣣⠏⠉⢿⠔⠁⣯⣀⡼⠁⠈⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠶⠶⢻⡏⠘⠗⠁⠀⠈⠁⠀⠀⣠⡄⣰⠋⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢠⡄⠀⢀⣿⠁⠀⠀⠀⣰⠆⣴⢀⡾⠁⣱⣿⡾⠋⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣸⡏⠀⠀⠀⢠⣯⢾⡇⣼⣣⠎⡿⢧⡀⠀⣠⡞⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣿⣤⠴⠶⠶⠊⠁⠘⢛⠈⠁⠘⠁⠈⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠟⣷⠙⠀⠀⠰⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣾⣿⣿⣿⣴⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⢀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⡿⣿⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠿⠿⠿⠿⢿⣿⣿⢻⣿⡿⢿⣿⠛⢻⣿⣿⣿⠏⡟
⠀⠀⢀⡀⢀⣿⣿⣿⣿⣿⣿⣿⣋⣁⣨⣿⡛⠛⠈⠛⠷⠗⠾⠋⢠⡾⠇⠐⣶⣿⣿⡷⢴⠇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣾⣿⣿⣾⣷⣾⣿⣿⣿⣷⣾⣶⣦
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣸⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈
⠀⠘⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡦
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⠿⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠷⠷⣬⣯⣽⣿⣿⣿⣿⡟⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏⠉⠻⣿⣿⡿⠋⠀⢀⣼⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⠀⠈⠋⠀⢀⣴⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⣄⣀⣴⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠐⠴⡔⣄⠄⠀⡀⠀⠆⠆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠐⠅⢤⡈⠊⠂⠈⢌⢆⢀⠱⡘⢀⠀⠀⠀⠇⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠬⢀⠑⠢⣀⠀⠑⣤⠠⠤⠤⠦⠒⡲⣿⡄⣤⠀⠀⠀⠀⡄⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠭⣑⡒⠤⢄⣭⣿⣈⣽⣍⣡⣤⢶⣖⣺⣿⣻⣿⣿⢹⡾⣶⡇⣰⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠄⣀⡀⠩⠁⠐⠀⠍⠛⠫⠛⠉⢋⠙⠉⠛⠙⠍⠉⢣⣿⣿⣿⢠⡏⠀⠀⠀⡠⣪⡰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⠉⠐⠒⠤⠄⢀⡀⠐⢂⠤⢄⣀⠀⠠⣤⣤⣜⣿⣿⣿⣾⡄⢀⣠⡿⣛⡡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣉⠉⢁⡀⠀⠒⠂⠠⠄⠀⠀⠀⢈⡙⠒⢞⠴⣶⣿⣗⢊⡿⡋⠊⠁⠈⠻⣿⣿⣷⡞⠅⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⠀⠀⠬⠄⠀⢀⣴⣰⣶⡒⠒⠒⠚⠛⠻⢿⣿⡟⠀⠀⠀⠀⠐⢸⣿⣿⣯⡉⠁⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⢀⡀⢀⠀⢀⣠⣤⠬⣑⣣⣶⣤⣭⣒⡁⠑⣮⢍⣭⣿⣷⡣⡀⠀⢀⣠⣷⢛⣯⣿⠟⣁⣄⣮⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠊⣉⠰⣚⡶⠛⠺⣯⠟⠻⠿⠟⢩⡿⢶⣭⠿⣾⣿⠟⣿⣶⣿⣷⣿⣯⣿⡶⣾⣿⣿⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠠⠒⣾⣡⣾⣿⣿⣝⣥⣦⣲⣤⣶⣴⣆⣦⣽⣶⣶⣤⣄⡻⢏⣻⣴⠕⣿⣿⣿⣿⣿⣿⣍⢢⡪⡪⣑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡹⠿⣛⠽⠛⠟⡻⠟⠛⠛⡩⢟⠻⣻⣿⣻⣿⣿⠿⢛⡟⡱⠋⠀⠈⣟⡿⡹⡟⢮⡿⢧⡘⠊⠪⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠩⠂⠬⡒⢕⠪⣐⠌⠀⢂⢵⣿⠿⣻⢵⠟⢁⢤⢎⠂⠡⠀⠀⠀⠙⢻⠀⠈⠈⠵⣜⢵⡔⢕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡩⡒⠅⢢⡐⢝⣡⡤⣨⣾⣿⡧⡊⡤⡡⣢⣪⢫⢆⢠⢂⢃⢂⢀⡄⡄⠠⢂⠀⠀⢌⢦⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⣺⣵⣾⣿⣿⣾⣿⣿⣿⣿⡿⢾⡷⢷⣧⣯⣎⣞⣾⣼⣾⣾⣿⣿⣿⣿⣿⣿⣿⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⠶⠾⠦⠄⠀⠛⢉⠉⣙⢛⠿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⣿⣿⣿⣿⣟⣉⣉⡤⠀⠀⠒⡨⠑⣩⠐⡻⠻⢷⣮⣟⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⢻⣿⣧⣬⣉⣛⠷⠔⣂⠍⢐⠩⠒⠍⠊⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠈⠑⠂⠉⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠔⠒⠀⠠⠒⠢⠀⠀⠒⢼⠀⠀⢸⠐⠂⡀⠀⠀⠄⡀⠒⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠠⠊⠈⠠⠀⠀⠀⠀⠤⠀⠀⠄⠾⠀⠀⠸⠄⠀⠀⠐⠌⠀⠡⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣫⣭⣶⣶⣶⣦⣭⣙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠢⠀⢀⢀⡿⠚⠁⠈⠉⠉⠙⠋⠛⠻⢿⣿⣿⣿⣷⡙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠘⠀⣿⠀⠀⢀⣅⠀⠀⣀⠀⠀⠀⢸⣿⣿⣿⡏⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⣀⣿⡀⠀⠈⠁⠀⠀⠙⠁⠀⠀⠘⢿⣿⡿⠇⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠀⠙⡄⠀⢀⠀⠀⣠⡄⠀⠀⠀⡀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠹⡄⠀⠗⠛⠉⠀⠀⠠⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠹⣾⠓⠦⠤⢤⡤⠴⠶⣦⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠹⠤⣄⣀⣀⣀⣀⡤⠟⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠲⡄⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣇⡀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡇⠠⠈⠙⢙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠄⠐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡔⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⣰⠛⣆⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⢀⡴⠃⠀⠙⢦⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⣧⠀⠀⠀⢠⡞⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⣗⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠉⠁⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠁⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⠀⠀⠀⠀⢀⣀⣠⠇⠀⠀⠀⠀⠀⠸⣄⣀⡀⠀⠀⠀⠀⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠉⠉⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠠⣄⣤⣄⣠⣄⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠂⣀⠀⠁⠀⠁⢈⣙⠳⡄⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡆⠀⠀⢀⡀⠀⢀⡀⠈⠁⢙⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⣀⠠⠀⠈⠉⠀⠀⠀⠀⠉⠙⠂⠀⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠈⠛⠀⠀⠀⢀⡄⠀⠰⠀⠀⠀⠀⣆⠸⠒⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⢲⠀⠀⠀⠀⠈⣇⠀⠀⡇⠀⠀⠀⠀⢀⣀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⢦⠤⠃⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠲⢦⣄⣀⠀⠀⠀⢀⣀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⢤⡀⠀⠀⠀⣀⠴⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣄⠀⠈⢩⣭⣍⠀⢀⡹⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠈⠙⠂⠻⣿⡿⠛⠁⠀⠀⠈⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⣄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡄⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⢸⡄⠀⠀⠀⠀⢸⣧⠀⠀⠀⢠⠻⡀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⡌⡇⠀⠀⠀⠀⠘⢸⠀⠀⠀⢸⡆⢳⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⢰⠁⡇⠀⠀⠀⠀⣿⢚⡇⠀⠀⠀⢳⠈⣇⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⡎⢀⠃⠀⠀⠀⠀⣿⠈⣧⠀⠀⠀⠘⣆⠸⡄⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⡸⠀⡸⠀⠀⠀⠀⠀⡗⡉⢼⡀⠀⠀⠀⠸⡄⢱⡄⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣥⡄⠀⢰⠁⢀⠃⠀⠀⠀⠀⢠⡗⡊⠈⡇⠀⠀⠀⠀⢳⠀⢧⠀⠐⠗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠈⠀⢰⠇⠀⡘⠀⠀⠀⠀⠀⢸⢋⠴⠊⣧⠀⠀⠀⠀⠈⣇⠘⣆⠀⠀⠀⠙⢦⡀
⡠⡔⢳⠎⠵⡄⠀⠀⢀⠴⠊⠀⠀⠀⠀⣠⠋⠀⢠⠁⠀⠀⠀⠀⠀⢸⠠⢂⡡⢼⡀⠀⠀⠀⠀⢸⡆⠈⢷⣄⠀⠀⠀⠙⠲⢄⡀⠀⢀⡒⢠⠉⡎
⡇⡃⠀⠀⢀⢣⡄⠺⢅⠀⠀⠀⢀⡠⠊⠁⠀⠀⡎⠀⠀⠀⠀⠀⠀⡾⢂⠡⠐⣘⠇⠀⠀⠀⠀⠈⣇⠀⠀⠙⠳⣄⠀⠀⠀⠊⠙⣶⢡⠀⠈⡇
⢣⠃⠆⠈⠘⡟⢙⣆⡈⣗⣠⠔⠉⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠻⣇⠔⢋⡼⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠳⣤⡖⢀⣊⡈⠹⠀⠀⠁⢁
⠈⠳⠤⣄⢠⡔⠉⠉⠉⠻⣄⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⢦⡜⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⣠⠔⠃⠉⠉⠉⠓⣄⠠⠒⠉
⠀⠀⠀⠀⢰⢀⠀⠀⠀⠀⣤⣙⠲⣄⡀⠀⠀⠈⣃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣀⠤⠒⢉⡠⠞⠓⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠘⢯⡀⠀⠀⠀⠀⠈⠓⠦⣍⡓⠦⣤⡿⠿⣿⣶⣶⣶⣶⣶⣶⡖⠲⣶⣶⣾⣿⣿⠿⠻⠗⢊⡡⠔⠊⠁⠀⠀⠀⠀⠀⠀⠰⠠⠁
⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢥⣄⣀⠨⠭⢉⣉⡙⠛⢛⢛⠉⠉⠖⢁⣠⠴⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠤⠤⣀⣈⠠⠄⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡛⢶⣄⣀⠀⠉⠛⢦⣄⣀⣈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⢸⠇⠀⠀⢀⠀⣤⡔⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠆⠼⣛⣻⣿⣿⣿⠿⣏⠁⠙⣦⡀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠉⠀⠀⡠⠛⠷⣶⣶⣭⠑⠆⣈⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⣭⣍⣙⡲⠬⠭⠒⠉⠉⠉⠉⠉⠓⠦⠤⠐⠒⠧⠤⠤⣤⣀⡤⠊⠉⠙⢦⣽⣿⣿⡛⠀⣡⠊
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢀⡈⣉⣀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿