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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⣁⣤⣤⣤⣄⡉⠓⢦⡀⠀⢀⡠⠖⠉⣁⣤⡤⣤⣀⡉⠳⣄
⠀⢠⡤⠤⠤⠤⠤⣄⠀⡴⠁⡰⠋⠁⠀⠀⠀⠀⠙⠲⣄⠙⠶⠋⣀⡴⠋⠁⠀⠀⠀⠀⠙⢦⡈⢳⡀
⠀⠀⠙⠢⡄⠀⢠⡏⣸⠁⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢳
⠀⠀⢀⡴⠃⠀⣿⠙⠳⠤⠧⢤⣤⣤⣀⣀⣀⣤⣤⠤⠤⠤⠤⠤⠴⠒⠒⠒⠛⠉⠉⠉⠉⠉⠉⠙⠚⢦⡀
⠀⠀⠛⠒⠒⠒⠺⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡟⠉⣩⠟
⠀⠀⠀⠀⠀⠀⠀⠓⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠤⠤⠖⠒⠒⠛⠉⠉⠉⡽⠛⣲⣄⢠⠇⢰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡍⠙⢏⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢀⡼⠁⣿⣁⣀⣀⣉⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠋⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⢁⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡈⠑⠦⣄⠀⣀⠴⠚⢁⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⣀⠉⢁⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣎⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣟⣿⠿⠁⠀⠀⠀⠁⠢⠄⠠⢀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠛⠛⠀⠀⠀⠀⠀⠀⠂⢦⡀⠀⠀⠀⣴⣿⣿⡿⠉⠂⠁⠀⠀⣀⠄⠀⠀⠀⠀⠠⠀⢃
⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣮⣆⣾⣿⡿⠁⠀⠀⢀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⣼⣿⣿⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⣿⠿⠃⣀⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⢃
⠀⠀⠀⠀⢠⣿⣿⣿⣿⠱⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣳⡼
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⡄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠟⠉⠀⠀⠀⠀⢻⠽⠃
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⢀⣄⣤⠴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⢹⡛⠻⠿⠿⠿⠿⠟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡀
⠀⠀⠀⠀⠹⣤⢤⡀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠛⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢨
⠀⠀⠀⠀⠀⠹⠦⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠋⠁⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡠⠁
⠀⠀⠀⠀⠀⠀⠣⡀⠉⢠⠀⠀⠀⢀⣀⣤⣶⡟⠛⠁⠀⠀⠀⠀⢀⡠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢲⣶⣾⣿⣿⣿⣿⣇⠀⠀⠀⢀⡠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣌⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣯⠾⠐⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢌⡛⠿⢟⣛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠟⠋⠉⠁⢚⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣽⣿⠢⠁⠀⠀⠀⠀⠀⢀⣠⣤⣾⠟⠋⠁⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣄⠀⠀⠀⠀⣠⣶⣿⣿⡯⠀⠀⠀⠀⠀⢀⣤⣾⠟⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠭⠯⠄⠀⠻⢿⣿⣿⣿⣦⡄⢀⣠⣴⣶⣿⠟⡉⠐⠲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⡿⠋⠙⠯⠁⣀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠠⠀⠀⢲⡀⢧⠀⢻⡄⢸⡄⠰⡄⣇⠀⡄⠀⡀⠀⣠⠀⠈⠁⠀⡞⠀⣰⠃⢀⠀⠀⡀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⣀⠀⠳⡄⢳⡀⢳⡀⠇⠀⡇⢹⠀⡇⠀⡇⠀⡿⠀⡔⠀⣸⠁⢰⠇⢠⠏⣠⠎⢠⡞⠁⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠈⢧⡈⢦⡀⠙⠄⢳⡀⢳⡀⠀⠀⢸⡀⠇⠀⡇⠀⠇⠀⡇⠀⠀⠀⠏⢠⠏⡴⠋⣴⠋⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠳⣄⠳⡄⠳⡄⠀⠤⠁⠀⠃⠀⠐⠘⠃⢀⠀⣄⠀⢀⣤⣤⡶⠶⢶⣤⣄⡈⠁⠞⣡⠀⠀⣠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⡈⠳⣜⠦⠉⣠⣴⡾⠿⠛⠻⠿⣦⣄⠀⢀⣴⠾⠋⠁⠀⠀⠀⠀⠀⠙⢻⣦⡀⠀⠴⠋⣡⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠳⣝⠳⡌⣠⣾⠟⠁⠴⠀⠀⠐⠂⠀⠻⣦⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⣷⡀⠰⠋⠁⠤⠴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠳⣌⠃⣰⣿⢋⡴⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢹⣧⠀⠁⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢦⡀⠦⠈⢠⣿⡇⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⠶⠶⠶⣶⣶⠲⠶⠾⣿⣄⡀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠳⣦⡀⢸⣿⡀⡿⠀⠀⠀⠀⠀⣀⣠⣤⠶⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠫⢻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⡟⠿⢼⣿⣧⣤⡴⠶⠶⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣼⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠰⣦⡷⢸⡇⠷⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⣿⠓⢒⡲⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⡇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⠿⠿⠿⣿⣿⠖⢶⣦⣵⡿⠰⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⡇⣲⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⡞⠛⠉⠁⠀⠀⠈⠉⠁⢀⣴⡿⠋⠀⠸⣿⣿⣦⣤⣌⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠙⠻⠧⣤⣤⣤⠶⠾⠶⢿⣿⡙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⡀⠀⢀⠀⠀⠈⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡄⣠⠄⣠⠀⡀⢀⠆⠀⠙⠿⣦⡀⠀⠀⠀⠀⠀⠄⣠⣴⠿⠋⡀⠠⡄⠓⠀⠈⢦⡀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡴⠁⡴⠁⠼⢡⠏⢠⠂⡠⠀⠈⢻⣦⠀⠀⣠⣴⠿⠛⠁⢀⠀⠹⣆⠙⣦⠀⢀⡀⠙⢦⡀⠀⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢠⡞⠁⠀⣰⠃⢠⠏⠀⠀⢀⡇⠀⠙⢷⡾⠋⠁⢠⠀⠆⠈⢧⠀⠘⣦⠈⢷⡀⠹⣦⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣠⠏⠀⠀⣴⠃⢠⠏⢀⡄⠀⣸⠁⢀⡄⠀⣄⠀⡀⢸⡄⠀⠀⠘⣧⠁⠘⢷⡀⠻⣄⠈⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⣰⠀⡼⠃⢠⡏⠀⣼⠁⢀⡏⠀⣸⠁⠀⠁⠀⢣⠀⣧⠀⣦⠀⠸⣆⠀⡈⢳⡄⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⠃⠈⠁⢠⡟⠀⠀⡿⠀⢸⠃⠀⠏⠀⠀⣶⠀⢸⠀⢹⡄⢹⡄⠀⠙⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠞⠛⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⡀⠀⢠⡶⠛⢁⠤⠔⠒⠒⠒⠂⠤⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⡿⠟⠛⠛⠻⣿⣿⣶⠏⠀⠈⠁⣀⠤⠔⠒⠒⠒⠦⢄⡀⢳
⠀⠀⠀⠀⣀⣤⣶⠶⠾⠶⠶⣶⣤⣀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠠⢹⣿⠃⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠉⣸⠇⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⢠⡾⠋⠁⠤⠂⠀⠀⠀⠀⢨⣙⣧⣼⠃⠀⠀⠀⠀⠀⠀⠀⣹⡏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡟⣀⠴⠚⠋⠉⠉⠉⠉⠉⠙⠛⢷⣦⡀
⠀⢠⡟⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄
⠀⣾⢠⢁⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⡇⢸⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠄⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⣇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⣿⡇
⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠁⠀⠈⠉⠛⠧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠋⣉⣤⣤⣄⡀⠀⠙⢿⣆⠀⠀⠀⢀⣿⠃
⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⢀⣤⣶⣶⣦⣤⣈⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢁⣠⡶⠿⠛⠛⠛⠟⣷⠀⠀⣸⡿⠀⠀⡜⣼⡟
⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⡿⠁⠀⠠⣸⣿⠛⠳⢮⣳⡀⠀⠀⠀⠀⡠⢊⡠⠒⠉⠀⠀⠀⠀⢀⣠⣾⣟⣤⣾⠟⣁⣄⣾⣵⡿
⠀⠀⠀⠀⠀⠙⢿⣿⣷⣷⣶⣶⠶⠶⠶⠿⣧⡀⠈⠿⣖⣢⢉⣭⣿⡆⠀⠀⠈⠻⣶⣶⣶⣶⣦⡤⠤⠤⠤⠤⠴⠶⢾⣿⣿⣿⡿⣿⣿⣯⣭⣭⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠳⣼⡉⠉⠉⠉⠉⠉⠉⠛⠷⣦⣬⣉⣉⣛⣛⣋⡨⠝⠛⠻⠿⢤⠞⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣇⠀⠀⠀⠀⢾⠀⠀⠀⠈⢑⡲⠤⠤⣀⣀⣀⣀⣀⡀⢶⣫⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣻⣿⣆⠀⠀⠀⠸⣿⣄⠀⠀⠀⠈⠉⠉⢓⣶⣾⣿⣭⣟⠻⢯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⠀⠀⠀⠀⠀⠀⠀⢼⣿⣾⡿⣆⠀⠀⠀⢻⢻⡄⠀⠀⠀⠀⠐⢉⣮⠿⠛⠁⠉⢳⡄⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠄⠀⠀⠀⠀⢸⣿⠋⠀⠹⣧⠀⠀⡜⢸⡇⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠸⠇⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⠀⠀⠀⣾⠃⠀⠀⠀⠸⣇⠀⢅⡼⢁⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⢼⡇⠀⠀⠀⠀⠀⢹⡇⠈⡱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠃⠀⠀⠀⠀⠀⠘⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⢀⡀⠀⢀⣀
⠀⠀⠐⠓⠐⠲⣄⡈⢳⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢠⠀⣋⣁⡸⠀⠀⠀⢀⡴⠚⠉⠁⠀⠀⠈⠉⠲⡄⠀⢀⣴⠾⠟⠛⠋⠉⠉⠙⠳⢄
⠀⠀⠀⠀⢸⣀⠀⡈⢉⡆⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⢀⡀
⠀⠀⠀⠀⢀⠌⠋⠁⢁⠇⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⡠⠤⠔⠒⠊⠉⠑⠞⠉⠉⠉⠀⠀⢉⣹⣦
⠀⠀⠀⠀⢸⠀⠈⠉⠉⠓⣿⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⢻⡟
⠀⠀⠀⠀⠸⠤⣤⣀⣀⡀⢸⣇⣀⡤⠴⠒⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣺⡏
⠀⠀⠀⠀⠀⠀⠀⢈⠟⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⣠⣤⣤⠤⠤⠴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠚⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠒⠒⠒⠒⠋⠁⠉⠉⢉⣿⠟⠠⠴⠒⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢾⡵⣀⠀⠀⠀⣀⡤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⡖⠒⠂⣀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡷⣡⠴⠋⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⡔⠃⠴⠾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠁⠦⠀⡀⢨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⢰⡠⣤⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠁⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⠿⠿⠿⠟⠛⠛⠋⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣫⣛⣩⣛⣟⣹⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢹⠀⠀⠀⢠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠈⡇⣀⠔⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢏⠲⣄⠀⠀⠀⠀⠀⡴⡅⡸⢀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⡏⠁⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡈⢧⡇⢹⣄⠀⠀⡼⠀⢹⡿⢃⣾⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⠙⢧⢸⣹⠖⠲⡟⡶⠋⠱⠯⠷⢦⡀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣳⡦⠤⣿⡇⠀⠀⠹⡇⠀⢀⠀⠀⢀⠇⢀⣠⠔⢋⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡓⠦⣄⡀⠉⢻⠋⠉⠀⠀⠰⠀⢻⣀⡆⣀⡴⠛⣿⢉⡴⠚⣩⠟⣄⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠉⠓⠮⣷⣤⣤⣤⣠⣬⡏⠌⣻⠅⠀⠀⠈⢧⣀⠔⠁⠀⠘⣏
⠀⠀⠀⠀⠀⠀⢀⠀⢸⠓⢤⠽⠶⠞⢉⡞⠉⠀⢀⠀⠒⣿⠛⠏⠳⢄⣀⡤⠚⠁⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠘⠛⢍⠉⠉⠙⠳⣾⡀⠀⠀⠘⢮⣀⠀⠀⢀⡰⠧⠴⠾⠶⠮⣅⡀⠀⠀⠀⠀⢀⡴⠁⣀⡤⠖⠒⠒⠒⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠈⠳⢤⡞⠉⠉⠳⣄⣰⣠⢋⠘⣦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⡠⠖⠁⡠⠚⠁⢠⠀⠀⠀⠀⠀⠀⠀⣙⣆⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢼⣋⣿⡟⣶⠁⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠑⣤⠎⠀⠀⠀⢀⣀⠤⠔⠒⠋⠉⠉⠀⠀⠉⠂⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⣶⠞⠉⡟⢹⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠋⢉⡀⢠⡖⣤⠀⣀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡃⠀⡀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠚⠉⢀⠀⣦⠈⣯⠄⠀⢻⠣⡄⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⡠⢊⠉⣿⠋⡙⢦⣼⣛⣇⣀⣀⢀⣀⣀⡤⠤⠖⠚⠉⣁⡀⠰⢾⡁⠨⢷⠛⣆⠘⠖⣂⣈⠤⠤⠤⠒⠒⠢⡤⢼⣤⢤⠖⡲⣷
⠀⠀⠀⠀⠀⠀⡼⠥⠤⠴⡇⢀⡇⣸⠈⢏⠉⠉⠉⠉⠀⠀⢀⣄⣠⡆⢸⢻⠹⡆⠈⠧⠀⢘⡠⠴⠚⢉⣁⡀⢠⡖⠢⠴⠶⣴⠟⢩⣃⠼⠛⣲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣼⡰⠃⠀⠘⣧⡀⠀⠀⢠⡄⠀⣯⠉⢷⠈⠳⠛⣁⡤⠖⠋⠁⠀⠀⠀⣇⠀⠙⢾⡂⠀⠀⣰⠟⠒⡺⠛⢒⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⢻⠁⠀⠀⠀⠀⠀⢉⣀⡠⠤⠒⠋⠁⠀⠀⠀⠀⠀⢀⠖⠉⠀⠠⢬⠷⠺⣞⡁⠄⠀⠙⣏⠩⠉⠓⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⡻⠯⣍⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣲⠶⠒⣻⢌⣡⡟⠦⣤⣤⠴⠛⠛⠛⠃⣠⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠀⠀⠉⠒⢦⣄⡀⠀⠀⠀⠀⠀⣠⠴⢾⣁⠀⠀⠀⣸⠀⠉⠀⠀⢹⠓⠒⠒⢲⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⢀⡴⠋⠀⠉⠲⣄⠀⢠⠞⠁⣠⠊⢉⠷⣀⣠⠛⣄⢀⣴⡚⠉⠀⡤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠈⠳⠃⠀⢰⡇⣠⠀⢀⡏⢹⠀⠈⡏⠀⢹⢤⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢀⡄⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⢁⡴⠛⢆⠈⡀⡸⠓⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⡴⠛⡇⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠉⠀⠀⠈⠳⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠉⠀⠀⣧⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠋⠋⠉⠉⠛⠓⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠖⠒⠉⠉⠉⠉⠛⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⢳⣦⣀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠈⢧⠉⠛⠲⠤⣼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣰⠃
⠀⠈⣧⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠛⠉⠉⠁⠀⠀⠈⠉⠙⠓⢦⣀⣰⠃
⠀⠀⣸⠂⠀⠀⠀⠀⣮⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢌⢳⡄
⠀⡴⠁⠀⠀⠀⠀⠀⣧⡈⠓⠦⣤⣀⣀⣀⣀⣀⣀⣀⣤⠴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀⢹⡇
⢾⡁⠀⠀⠀⠀⠀⠀⣇⠈⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢸⡗⠦⣄⡀
⠀⠉⠓⠤⣄⣀⡀⢀⡇⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡇⠀⠀⠙⢲
⠀⠀⠀⠀⠀⠈⠉⠙⡏⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠒⢚⡿⠛⢲⠒⠦⢤⣈⠐⢼⡇⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠁⠀⠀⢀⡴⠋⠀⠀⢸⠀⠀⠀⠈⠓⢾⡇⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠚⠋⠁⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⠞⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠒⠒⠚⠋⠉⠁⠙⠲⣄⡀⠀⠀⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⢸⣤⠴⠶⠿⠧⠤⣤⣀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣿
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾
⣿⣿⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠂⠀⠉⠉⠉⠁⠀⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠀⠉⠉⠉⠁⠀⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠘⡉⠂⠤⠀⠀⢀⣀⣠⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠰⡀⠀⢠⠔⠉⠀⠀⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠄⠐⠂⠒⠀⠂⠀⠀⠀⠀⠋⠒⠂⠤⣀
⠀⡰⠁⠀⠧⣀⠀⠀⠀⡁⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⠀⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢅⡆
⡔⠀⠀⠀⡗⠄⠉⠐⠒⠦⠤⠤⠱⠤⠤⠤⠤⠀⠒⠒⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢄⣃
⠈⠑⠤⢀⡗⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⡅⠈⠐⠤
⠀⠀⠀⠀⡟⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠠⡤⠄⠒⠒⠒⠒⠒⠀⠤⠤⢈⡐⢤⡇⠀⠀⡰
⠀⠀⠀⠀⢏⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠤⠄⠒⠀⠀⠉⠁⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡇⠀⡜
⠀⠀⠀⠀⠀⠈⠐⠀⠤⠄⢀⣀⣀⣀⣀⠀⠀⠤⠤⠀⠶⡒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⡤⠊⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⠑⠢⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠉⠀⣀⡠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠛⠛⠛⠻⣶⣤⣀⣀⣠⠔⠀⠁⠀⠐⠒⠛⠋⣩⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠶⢦⣤⣀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣯⢻⣧⣤⣴⠶⠶⠶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠉⠓⢿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⢀⣤⡶⠾⠿⠿⠶⢶⣤⣀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣤⣄⣀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⡄⠀⠀⣴⠋⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠉⠻⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⠞⠁⠀⠀⠀⢨⠇
⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠟⠁⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⣠⣶⡿⠁
⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⢀⣠⣶⠟⣩⠞
⠀⠀⠀⠀⠀⠐⣿⣿⣷⣄⣀⡀⠀⠀⠀⠀⣀⣠⠤⠖⠊⠁⠀⠀⢀⣠⠴⠚⣹⡿⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢝⡛⠛⠛⠒⠛⠛⠉⠉⠀⠀⢀⣠⡤⠖⠛⠉⠀⠀⣰⠟⣲⠏
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠻⢆⣀⣀⣀⣤⠴⠶⠒⠋⠉⠀⠀⠀⠀⠀⢀⣴⠏⠀⡿
⠀⠀⠀⠀⠀⢀⠔⣡⠄⠀⢘⣿⡿⠿⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠛⠻⠿⠧
⠀⠀⠠⠀⠀⢡⡴⠋⠀⣴⠟⠁⠀⠀⠀⠉⠻⢦⡀⠀⠀⠀⡠⠊
⠀⠀⣀⡠⠖⠋⣀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢱⣤⠔⠊
⠀⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡦⡄⠀⠀⠀⡟⠀⡠⣞⢁⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢠⠅⢧⠈⣆⠀⠀⣧⣞⠀⣹⠟⢚⡇⠀⠔⠲⣔⠒⢄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠘⣏⠢⣄⡏⠑⣾⢄⡸⣼⠉⠓⠊⢩⢿⣴⡥⢒⠏⠀⠀⠀⠀⠘⡆⠀⢱⡀⣸⠙⣆
⠀⠀⠀⠀⠀⠀⠀⣤⢄⢸⢋⢢⠀⡼⣏⡙⠛⢦⣇⠀⠙⢿⡄⣠⢠⠞⠉⠀⢿⡁⠀⠀⠀⠀⠀⢰⠃⠀⠀⠖⠁⠀⣼
⠀⠀⠀⠀⠀⠀⡠⠬⠨⠼⡆⢆⣿⡀⣓⠙⢮⡟⠁⠀⠐⢦⢛⡙⢧⠔⠀⠀⠀⡷⠀⠀⠀⢀⡴⠃⠁⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⣀⡠⠤⣤⠞⠊⠀⣿⠉⠓⢠⠑⣒⠶⠋⠩⢳⡔⢯⠙⠛⢮⣋⠵⢄⡤⠒⠁⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠚⠥⣀⢀⠾⠓⠦⣴⠷⡦⠴⠾⢴⣥⠴⠚⠓⠚⠓⠢⢄⣀⣠⣿⡕⣢⡦⠖⠚⠉⠉⠙⠒⢶⡋⠀⣀⠀⣀⠤⠤⡀
⠀⠀⠀⠀⠀⣀⢠⡤⠭⣦⠀⣠⠟⠙⢣⡠⢴⠞⠁⠀⠀⠁⠂⠀⠀⠀⠉⠢⣀⡴⠋⠀⠀⢁⠀⠀⠀⠀⠀⠱⡞⢸⠞⠁⢀⡀⢹
⠀⠀⢀⠔⠛⣤⡤⢬⢖⡾⠋⢹⣤⢤⣎⠉⠉⠉⠉⠉⠑⠒⠦⢤⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡮⡶⠀⡏⠃⡜
⠀⠀⠺⠮⢩⣋⠴⢧⣮⣤⢶⠃⡎⡸⣏⣆⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠓⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣱⣇⠀⠈⠉
⠀⠀⠀⠀⠀⠀⣰⣉⡩⠿⣸⢰⢡⡟⠋⢸⠇⠀⠀⠀⠈⠀⢰⠏⢛⠇⢀⢴⢄⠀⠉⠉⠓⠒⠒⠒⠒⠒⠒⠚⢉⡼⠁⠛⡦⠒⢲⠆
⠀⠀⠀⠀⠀⠀⢋⠜⣇⠞⢹⣟⡥⠾⠷⢾⣇⣀⣤⡤⠤⢤⣌⣀⡘⠀⢧⣧⠞⢠⡟⠶⡖⢀⠀⠀⠀⠀⠀⣰⡟⣝⡷⠚⢻⠔⠋⣑⠆
⠀⠀⠀⠀⠀⠀⠛⠈⠰⣎⡉⠉⣳⠀⠀⣉⡷⡞⣷⣄⠀⠀⠀⠀⠉⠒⠢⢄⣀⠈⠀⠘⠀⠀⠀⠀⣀⣠⠞⢙⡟⠟⠓⡦⠼⣋⠉⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡝⢃⣽⠖⢚⡏⢀⠀⢸⠳⠽⠢⢄⣀⠀⠀⠀⠀⠀⠉⠉⠛⠒⣒⡶⠟⣿⡃⠀⠈⢻⢓⡟⢋⡷⠢⢌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣅⣿⠥⠞⢺⠁⠹⢩⠓⠋⠀⠀⠀⠀⠈⠙⠲⢤⡀⠀⠀⣠⡖⠋⠝⢑⡟⠃⠉⠻⡞⠻⢽⣯⡉⣻⣭⠽⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⢢⣎⣸⣥⢤⡀⢸⣇⢀⡴⢶⣛⠒⠲⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⣠⠤⣯⣠⡎⢳⡠⡻⢛⠹⡉⠀⢸⡌⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣋⡷⠚⢱⠁⣠⠀⠉⠲⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠘⠦


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣧⢺⣯⣍⣭⣭⣍⣭⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠇⠀⠀⢸⠀⠀⠈⠀⠀⢀⠆⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠱⡀⠀⢸⠀⠀⣼⢠⠀⡇⠐⢀⡎⠀⠀⡠⠂⠀⠀⠀⠠
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣠⡬⠶⠾⠶⠳⠶⠶⣧⣆⣿⠀⢸⣡⡧⠾⠶⠷⠶⢥⣄⣀⡠⠊
⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⡀⠄
⠙⢦⡀⠀⠐⢄⠑⠤⣄⣀⠀⠀⠀⠀⠀⢨⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡀
⠀⠈⠢⣅⠒⠠⠥⠦⠀⠈⠉⣡⣶⢀⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡺⣧⠤
⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⢰⡏⣩⣤⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⢿⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢼⡇⢹⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠖⠋⠁⠀⠀⠀⠈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢸⡆⠈⠉⠛⠛⠶⠦⠤⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣂⡿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⢶⣿⡉⣀⡩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠙⠋⣲⠆⢠⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⠠⠄⠀⠀⠂⠈⠉⠁⠀⢀⣴⡿⠶⢴⡿⠿⠖⠒⠶⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠀⠐⢡⠜⢙⢿⣯⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠡⡅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠔⣡⠎⡐⢹⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⠁⠑⢄⠁⠀⠒⠂⠐⠂⠒⠠⡀⠙⡆⠨⢶⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⡴⠁⠀⢠⠃⠌⢹⣆⠀⠀⠀⠀⢀⣠⢾⠋⠀⠱⡀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠈⢄⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⢀⠇⠀⠀⡆⢹⠀⢀⣤⠖⡏⠁⠈⢇⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⢠⠁⠘⠁⡇⠀⠀⠁⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠐



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠛⠓⠶⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠒⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠀⠀⠀⠀⠀⢠⡴⠛⠁⠀⠀⠀⠀⢀⡀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⢀⣴⠋⠀⠀⠀⠀⣠⠴⣋⡥⠖⠀⠀⢹⣦⡀
⠀⠀⠀⠀⠀⢠⡞⣁⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⡼⠁⠀⠀⠀⢀⡞⡡⢞⣥⠖⣋⣥⠤⣼⡏⠳⡄
⠀⠀⠀⠀⢠⡿⠋⢹⡇⠀⠀⠈⠛⢦⡀⣄⠀⠀⠀⠀⠀⠀⠈⢿⡃⠀⠀⠀⣠⢏⢞⡵⢋⡴⠋⠁⠀⠀⢸⠇⠀⠹⡄
⠀⠀⠀⢠⡟⠀⠀⠀⠹⣄⠀⠀⠀⠀⠹⣮⡳⣌⠳⣄⢠⡀⠀⠈⢷⠀⠀⠐⡡⢫⠞⡴⠋⠀⠀⠀⠀⣠⠟⠀⠀⠀⢹⡄
⠀⠀⠀⢸⠃⠰⡀⠀⠀⠈⠳⣄⠀⠀⠀⠈⠳⣌⢧⡙⢦⡙⣦⠀⠈⢧⡀⠀⡱⢋⡞⠁⠀⠀⠀⣠⠞⠁⠀⠀⢠⠀⠈⡇
⠀⠀⠀⢸⡇⣇⢳⡄⠀⠀⠀⠈⠳⣄⠀⠀⠀⠙⢦⡻⣌⠳⣌⠳⡄⠀⢳⡀⣠⠏⠀⠀⠀⣠⠞⠁⠀⠀⠀⢠⡟⡜⢸⡇
⠀⠀⠀⠀⢻⡜⢦⡙⢦⡀⠀⠀⠀⠈⠳⢦⡀⠀⠈⠻⠦⠤⠤⠤⠶⠶⠒⠛⠁⠀⢀⡴⠞⠁⠀⠀⠀⢀⡴⢋⡾⢡⡟
⠀⠀⠀⠀⠀⠹⣮⡻⣦⡙⢦⣄⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⣀⡴⢋⡴⢋⣴⠟
⠀⠀⠀⠀⠀⠀⠈⠛⢮⡙⢦⣉⠳⢤⡀⠀⠀⠀⠀⠙⠳⣄⡀⠀⢀⣠⠖⠋⠀⠀⠀⠀⢀⣤⠞⣩⡶⢋⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣍⠳⣤⡙⠶⣄⠀⠀⠀⠀⠀⠙⠶⣏⠀⠀⠀⠀⠀⣠⠶⣋⡴⠛⣡⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣙⠶⣌⠳⢦⡀⠀⠀⠀⠀⠈⠑⢦⣀⡴⢞⣡⠞⣉⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣌⡳⢦⡉⠳⣄⡀⠀⠀⠀⠀⠉⠳⣏⣡⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠛⢦⣝⠳⣌⠙⢦⣀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠈⣳⣮⡙⠦⠙⠓⠀⠀⠀⠀⠀⠀⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣠⠊⣠⠞⣡⢞⣹⢶⣜⢳⣌⠳⣌⠲⣄⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⣠⠞⣡⢞⡥⢞⡵⠛⠁⠀⠈⠳⣌⠳⣌⠳⣌⠣⡀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⣀⣤⡶⠀⣠⢎⡴⢋⡴⠋⠀⠀⠀⠀⠀⠀⠈⠳⣌⠳⣌⠳⣄⠀⣶⣤⣀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⣴⣋⡴⠞⠋⠁⢸⠃⠘⡡⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠳⣌⠂⢹⡀⠉⠛⠶⣤⣻⣄
⠀⠀⠀⠀⠼⠛⠁⠀⠀⠀⠀⡟⢀⠞⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⢣⠈⣇⠀⠀⠀⠀⠉⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠛⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠋⠁⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠟⠛⠛⠋⠉⠙⠛⠛⠳⠶⣤⣀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠛⠋⠉⠙⠛⠛⠛⠿⢿⣍⣀⠀⠀⠀⠀⣠⣿⡁
⠀⠀⢀⣴⠾⢛⣛⣛⣻⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣦⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⡀⠀⠀⠉⠛⢶⣤⡀
⠀⠀⢺⣅⠀⠘⠿⠿⣿⣤⣤⣤⣤⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⡀⠀⠈⠹⣦⠀⢀⣠⣤⠾⠟
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠶⠾⣿⣯⣉⡀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢷⡄
⠀⣠⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠐⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠈⠙⠛⠳⣶⣶⣶⣶⣦⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣠⣤⣿⡟⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣀⡀⠈⠻⣦
⠀⠀⢸⡇⠸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢂⣴⠏
⠀⠀⠈⠻⢶⣌⠉⠛⠷⢶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣟⣻⣭⠶⠛⠁
⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣤⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⢀⣼⠋⠉⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠶⣶⣟⡁
⠀⠀⠀⠀⠀⢿⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦
⠀⠀⠀⠀⠀⠀⠙⠻⠶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢙⣛⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠉⠀⠀⠀⠀⠙⠻⣿⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣶⣤⡀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣥⣤⣤⣤⣤⣤⠀⠀⠀⠀⠈⠙⠷⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣯⣭⣭⣭⣭⣭⣭⣤⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⡏⠀⠀⠀⣀⣤⡶⠶⠾⠛⠷⣤⡀⢀⣤⠾⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠛⠉⢿⣆⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⣿⡇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⣍⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠁⠀⣀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠈⠳⣄⠀⠀⢀⡴⠚⠁⠀⢀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠀⠈⣣⡴⠋⠀⠀⣠⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⡀
⠀⠀⠀⠀⢀⣴⡿⠉⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⡆⢀⣴⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⣷⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣼⡿⠃⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠻⣦⡀⠀⠀⠀⠀⠐⠁
⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⢀⡞⠁⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠈⢻⣄
⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⡜⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⡼⠁⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⢀⡼⠁⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠋⠲⣄⡀⠀⠀⣠⠟⠀⠀⣠⡟
⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠀⠀⠀⠀⠙⢢⡴⠋⠀⠀⣰⡏
⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⣰⠏⣅⠀⢀⣼⠟
⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⢿⣠⣿⠋
⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠷⠶⠶⠾⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⣾⡟⠁
⠀⠀⠀⠈⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⣴⣿⠋
⠀⠀⠀⠀⠀⠹⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣯⡳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠖⠋⠁⠀⠀⣀⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣬⣉⡛⠓⠶⠶⠶⠶⠶⠶⠶⠖⠛⠋⠉⠀⠀⠀⢀⣠⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣶⣤⣤⣄⣀⣀⣀⣀⣀⣤⣤⣤⣴⡾⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠌⣧⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⡧⠀⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣜⠮⠁⠀⠀⢀⣴⠖⢀⡠⠐⢀⠀⣄⡴⡀⠀⠀⠀⣷⠀⠀⠀⠀⡄⡀⠀⠠⢀⠀⠶⣄⡀⠀⠀⠈⢕⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⡁⠐⢁⣄⠀⠀⠄⢉⠈⠀⠁⠀⠀⣿⡀⠖⠀⠉⠀⠓⠀⠀⠀⠈⠢⡈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠛⢦⣌⠡⠘⠁⠡⣀⡄⠆⠒⢙⡂⠀⣿⡓⠃⠓⠬⣤⣀⠎⠀⠐⢏⠦⣠⠏⡈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠔⢁⣀⠀⡙⢦⣐⡤⠃⡡⡔⢀⠧⠌⠃⠀⣿⡗⠣⠽⡒⢠⣉⠒⢤⠃⣠⣞⠁⠀⠈⢄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡌⠞⠀⢈⡢⠙⢷⣾⡔⠉⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠉⠢⠓⣠⡾⢟⠁⠀⠺⢃⡀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠀⢑⠞⢡⣂⠄⠙⢿⣦⣀⠄⠂⠀⠀⣿⣿⠐⢠⠀⠀⣠⣾⢯⠁⡈⢖⡁⠀⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢄⡄⠉⠒⠌⢠⣁⠃⠀⠀⠀⠙⢿⣷⣄⠀⠀⣿⣿⠀⠀⣠⣾⡟⠁⠀⠩⣑⡈⢇⠒⠉⠀⡀⠀⠀⢀⡀
⣀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠈⠀⡖⠂⠀⢨⠁⣦⡄⠀⠀⢠⠁⠀⠀⠙⢿⣷⣶⠋⠁⣠⣾⡿⠃⠐⠀⠀⠀⢃⣠⢨⡅⠀⢩⣣⠀⡄⢸⡇
⣉⣽⠿⠆⠀⠀⣀⣀⣀⣿⣠⣤⣤⣬⣥⣴⣾⣶⣶⣷⣶⣾⣿⣿⣿⣿⣄⠀⠙⢿⣠⣾⣿⠏⠀⠀⠀⠀⠀⠀⠈⠈⠀⠋⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⢰⠶⡶
⣉⣻⣶⠆⠀⠀⠀⠀⠀⣶⠀⠀⠠⢤⠀⠀⢀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠉⣧⡀⠈⠻⣿⣿⣿⣿⡿⠿⢿⣿⢿⡟⠛⠻⠿⠛⠛⢙⡏⠉⠉⠉⠁⠀⠻⠀⠇
⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠠⠀⡒⣃⠈⠨⡆⢢⢥⠀⠀⠈⢆⣠⣿⠿⠁⢀⣴⠻⣿⣧⡄⠀⢀⠐⠀⠀⠀⣆⡌⠸⠅⠁⡰⡒⠀⠃⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠹⡌⢤⢆⠀⢠⣾⠟⠁⡀⠰⣿⣿⠀⠈⠛⢿⣦⡀⠀⠀⢀⢌⡔⠠⠏⠐⠢⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢚⣥⠀⠒⠱⢌⢲⣷⠟⠁⠀⠀⠁⠘⣿⣿⠀⠀⠂⠀⠙⠿⣦⡀⠢⠄⡠⠉⠂⣀⠐⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⢠⠰⢀⠔⣺⠟⣁⠠⣑⠦⢀⠀⠀⣿⣿⠀⠀⠀⢀⡀⢔⡉⠿⣦⡈⠑⢄⠠⣪⠊⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠐⢄⠀⣠⠞⣁⠚⠁⢓⠤⡁⠚⠩⠱⣿⣿⠀⠸⠈⠝⠐⣈⢴⠚⠈⠛⢦⣀⠀⠀⡐⢁⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡴⢋⠘⢠⠐⠀⠎⠀⠀⠋⠋⠲⢸⣿⠀⠒⠂⠋⠹⠈⠀⠳⠰⣊⠓⠈⡷⣌⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠛⢷⣄⠑⠀⡀⠀⠸⢱⡄⡠⠀⠂⣸⣿⠀⡀⠁⢠⠄⢶⠰⠄⠀⡀⠄⢉⣠⠾⠋⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡞⡠⡀⠀⠀⠈⠛⠀⠀⠈⠐⠀⠄⢀⡀⠈⠚⣿⠀⠃⠀⢀⡁⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⢀⠤⡙⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣈⣉⣿⢈⣉⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣍⣻⠆
⣿⣿⣿⣿⣿⣿⠿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠿⡿⠿⢿⣿⠿⠿⠿⠿⣿⠿⠿⢿

⠀⠀⠀⠀⠀⠀⠠⠄⠀⢀⣤⣶⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⡠⠒⠉⣉⠭⠝⠳⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⡠⠊⠀⠀⣀⡀⠀⠈⠳⡀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠟⠛⢻⣿⣿⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠰⠁⢀⠔⠛⢢⠀⠑⡀⠀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡏⠀⠀⣼⣿⣿⣿⣿⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢠⠃⠀⡞⠀⠀⡸⠀⢀⠃⠀⡇
⠀⠀⠀⠀⠀⠁⣼⣿⣿⣿⣿⡟⠀⠀⢰⣿⣿⣿⣿⡿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡸⠁⢀⠎⠀⡸⠀⠀⢠⠃⠀⡘⠀⣸⣁
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠁⠀⢠⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣃⢀⣜⠀⢠⠃⠀⢀⠏⠀⢰⠁⢠⠇
⠀⠀⠀⠀⢠⠏⠀⣼⣿⣿⠇⠀⠀⣾⣿⣿⣿⣿⡏⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⢠⠇⠀⠰⠀⢑⡏⠀⠀⡞⠀⢀⠇⠀⡞⠀⠀⠀⠘
⠀⢰⠀⢀⡏⠀⣰⣿⣿⡟⠀⠀⣼⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡎⠀⢀⠃⠀⡼⠀⠀⡸⠓⠒⡾⣄⡸
⢇⠈⠄⠸⢶⣶⣿⣿⡿⠁⠀⣰⢿⣿⣿⣿⣿⠁⣀⠀⠀⠀⠀⠀⠀⠀⢢⠈⠆⠘⢆⠒⠚⢄⢰⠃⠀⢰⠃⠀⠰⠁⢠⠃⢀
⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⣯⣿⣿⣿⣿⠇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⢧⠤⠤⢇⠀⡎⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠨⡇⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣠⠜⠳⣄⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⠋⠀⣤⣳⡌⠳⣄⡏⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠂⢀⡠⠟⢷⡄⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡖⠉⠀⠀⠀⠙⢦⡀⠈⠳⣄
⠀⠀⠀⠀⣀⡤⠖⠛⡿⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠙⢦⡀⠘⢦
⠀⠀⠀⠀⡇⠀⠀⠀⢷⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠈⠳⣄⠈⢳⡀
⠀⠀⠀⠀⡇⣀⡤⢞⠋⢢⡀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⢧⡀⢧
⠀⠀⠀⠀⢻⡉⠰⠯⢤⡀⠙⢦⡾⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠙⢼⡆
⠀⠀⠀⠀⠀⠙⢄⠀⠋⣠⠖⠋⠙⢄⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠈⠳⡏⠀⠀⠀⠀⠈⠳⣄⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡺⢹⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢸⠇⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠈⠳⣄⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢠⠞⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠳⣤⣀⣀⣀⣀⠤⠚⠁⡰⠋⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠈⠓⢦⣄⡀⠈⠉⠉⢀⣠⡴⠋⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠉⠉⠁⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠶⠶⠶⠶⠒⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠶⠒⠒⠚⠉⠉⠉⠉⠉⠉⠙⠒⠒⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠊⠉⠀⠀⠀⠈⠉⠑⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢇
⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡄
⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣄⠤⠔⣇
⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢀⣀⡤⠴⠒⠚⢉⣹⠀⠀⠀⢻
⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⠙⢿⣄⡀⠀⣀⣀⣸⣤⣅⣀⣀⢸
⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡴⡹⠻⢦⣌⣩⡿⠉⠉⠉⠉
⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣟⠜⠁⠀⠀⣽⣟⠢⣄⡠⠤⠖⠚
⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢻⣀⣠⠤⠾⠛⠊⠙⠺⢕⣦⡀
⠀⠀⠀⠀⠀⠘⡆⠀⢀⡀⠀⢀⣀⡤⢽⡗⠚⠉⠁⣀⣸⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦
⠀⠀⠀⠀⠀⠀⠸⣤⡿⠛⠉⠁⠀⠀⢸⢹⣤⡶⠿⣟⢇⢹⡀
⠀⠀⠀⠀⠀⢀⣴⠿⣕⠢⣀⠀⠀⠀⣘⣘⣄⣀⣀⠘⡌⡎⢇
⠀⠀⠀⠀⠀⠚⠃⠀⢻⡙⠺⢷⣟⡛⠛⠉⠉⢉⣁⣠⠼⢿⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣀⣨⡿⠶⠒⠋⠉⠀⠀⠀⠈⢷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠶⠶⠲⠶⠶⢶⣶⡶⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠈⢧⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢀⡖⠋⣠⡞⠁⠀⠀⠈⠉⠻⣆⠀⠀⠀⠀⠘⡇⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢠⠏⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⣷⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣾⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣀⣀⣧⡤⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠓⠲⠦⣤⣼⡤⠶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠋⠉⠉⢹⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⢀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢀⣀⣸⣦⡴⠇⠀⠀⠀⠀⠀⠀⠀⠘⠷⢦⣤⣄⣀⣸⣡⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⢀⣠⠶⣦⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⣿⣄⣠⡄
⠀⠀⠀⠀⢛⣿⢀⣽⠗⠀⣼⢿⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⠘⢷⡄⠀⢸⣧⣌⠹⡇⠀⣶⢶⡆
⠀⠀⠀⣠⣾⠿⠋⠀⠀⣼⣯⡄⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣠⡾⠋⠀⠈⠁⠘⣧⣧⠀⢷⠶⣽⣆
⠀⠀⠘⠛⠁⠀⠀⠀⠀⠈⢸⢇⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠞⠉⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠈⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠿⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣁⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠒⠲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⢀⣴⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠒⠲⡆⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠋⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠉⠛⠛⠛⠛⠋⠁⠀⠙⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⡀⠀⠀⠀⠀⢀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠖⠒⠒⠒⠶⠶⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣄⣀
⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⣀
⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠒⠒⠒⠛⠓⠒⠶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣨⠟⠲⢤⣀
⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣄⡀⣠⠞⠁⠀⠀⠀⠈⢻⡄
⠀⢸⠁⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠷⢤⡀⠀⠀⢀⡴⠋⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠸⣦⠶⠒⠒⠒⠚⠛⠛⠛⠛⠒⠲⠶⠤⣤⣄⣀⠀⠀⠀⠀⠀⢸⠇⠀⠙⢦⡴⠋⠀⠀⡇
⠀⡿⣆⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⢤⣼⡀⠀⠀⢰⠀⠀⠀⢸⠇
⠀⣧⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣼⢀⣠⠴⠋
⠀⠸⡄⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠹⣄⠀⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣄⣀
⠀⠀⠀⠈⠦⣀⠀⠀⠉⠳⢤⣀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠉⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡝⠛⠬⠍⣉⡐⠒⠒⠒⠒⠒⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠈⠙⠲⠤⣀⡴⠋⠀⠀⠀⠀⠀⠀⣠⠞⢻⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⡶⠶⠒⠉⢀⣀⡤⠤⠂⠒⣊⡩⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠀⢸⠏⠙⠒⠤⣀⣠⠔⠊⠁⠀⣾⠀⠀⠀⠀⠀⠀⣀⠔⠋⠁⠀⠀⠀⠘⠛⠻⠭⠭⣉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀⢸⡄⠀⠀⠀⠀⡇⠀⠀⠀⢀⡇⠀⠀⠀⠀⡴⠟⠡⠤⠤⠤⠤⠤⠤⠤⠤⢄⣀⣀⡀⠀⠉⠉⠒⠲⠦⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠷⣄⡀⠀⠀⣿⠀⣠⠖⠉⠀⠀⠀⠀⠀⠙⠂⠉⠉⠉⠉⠉⠉⠀⠒⠒⠒⠒⠒⠚⠩⠽⠯⠭⠭⠗⣒⣒⣛⣛⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠲⢶⠞⠁







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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡶⠶⠿⠿⠛⠛⠛⠛⠛⠛⠻⠶⠶⢶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠋⠉⠁⠀⠀⠀⠈⠉⠛⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠿⠟⠛⠛⠛⠛⠛⠻⢷⣦⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠤⠀⠀⠀⠀⠀⢀⣤⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻
⠀⢠⡿⠉⠛⠓⠲⠦⢤⣄⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣰⠃⢸
⢠⡿⠁⠀⠀⠀⠀⠀⠀⢀⡿⣇⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⢸
⣿⡟⠶⣤⣀⡀⠀⠀⢀⠞⠁⠘⣆⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⣸
⠸⣧⠀⠀⠈⠉⠓⠲⣿⠀⠀⠀⢹⡄⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⣿
⠀⢻⡄⠀⠀⠀⠀⠀⠸⡆⠀⠀⠈⣧⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⣾⠇
⠀⠈⣷⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠙⠓⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠸⣇⠀⠀⠀⠀⠀⠈⣇⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢦⣀⠀⠀⠀⠀⠉⠛⠲⢤⣀⡀⢀⡠⠊⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠸⣤⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠉⠳⣦⡀⠀⠀⠀⠀⠀⢈⣽⡟⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠛⠿⠛⠛⠛⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⠶⣄⠀⣠⠴⠋⠀⣇⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⣿⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣿⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀⠀⠀⣧⠀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⣿⡾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠠⠚⡀⠀⢀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠤⠀⠀⠈⠒⠀⠀⠐⠂⠀⠀⠰⡀⠠⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠉⠁⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⣨⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢉⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⠁⠀⠀⠀⠀⠀⠈⢄⠀⣀⠴⠊⢉⣠⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⡋⠁⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠋⠁⢠⣥⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠉⠳⣄⠀⠀⠀⠀⠀⠁⠠⡀
⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⢀⠀⠘
⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⢀⡴⠞⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠘⠁
⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⡴⠋⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⢀⡎⣸⠃⠀⠀⠠⠄⠀⠃
⢀⡟⠀⠀⠀⠀⠀⠀⠀⢰⠃⣰⠞⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⡞⢰⡏⠀⠀⠀⠠
⢸⡇⠀⠀⠀⠀⠀⠀⠀⡜⣰⠋⠀⠀⠀⠀⠀⠀⢠⠀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⢀⣟⢠⡟⠀⠀⠊⠉⠈⠁
⣾⣇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡠⠄⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⢀⡞⢹⣾⠃⠀⠰
⣿⢿⡀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⢀⣼⠏⠀⠀⠀⠀⠀⠀⢠⠞⠀⣼⠃
⢻⠘⢷⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡟⠁⠀⠀⠀⠀⠀⠀⣰⠋⠀⡾⠃
⠸⣦⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⢀⣼⠁⢀⡾⠁
⠀⢻⡄⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⠶⠶⠶⠶⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠨⢠⡟
⠀⠀⢻⡄⠀⠘⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣼⠏
⠀⠀⠀⠹⣆⠀⠀⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⡼⠃
⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⠀⣤⠟
⠀⠀⠀⠀⠀⠀⠙⠷⣄⡀⠀⠀⠀⠉⠓⠶⢤⣤⣄⣀⣠⣤⠤⠖⠛⠁⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⣤⣤⣤⣤⣤⣤⣤⣤⠶⠚⠉
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⠶⠿⠿⠿⠿⠿⠿⠿⢶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠉⠉⠀⠀⠀⠀⠉⠉⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣀⡀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣠⣴⡿⠟⠛⠉⠉⠉⠉⠙⠻⣶⣄⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢀⣴⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡘⣇⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⣾⢻
⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣰⠇⢸
⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢸
⠀⠀⠀⣸⣏⣀⣀⣀⡀⠀⠀⠀⣾⠁⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣼
⠀⠀⠀⣿⠁⠀⠀⠉⠉⠉⠉⢹⣿⡄⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢀⣿
⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⣸⠁⢻⡀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢸⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠳⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡃⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣿⡏
⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠉⠳⠦⣄⡀⠀⢀⡼⠃⠀⠀⠀⠀⣾⡟
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠉⢳⡞⠁⠀⠀⠀⠀⣼⡟
⠀⠀⢿⣏⠉⠉⠉⠉⠉⠉⠙⡆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⣿⠀⠀⠀⠀⣼⡟
⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠘⣄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⣿⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠘⡆⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠓⠦⢄⡀⠀⠀⠀⡰⠋⠀⠀⠀⢹⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠻⢷⣦⣤⣤⣤⣤⣤⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠈⠐⢦⡞⠀⠀⠀⠀⠀⢸⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⣧⠀⠀⠀⣠⡾⠋
⠀⠀⠀⢸⡄⠀⠀⢀⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣀⠀⠀⠀⢻⠀⢀⣼⠟
⠀⠀⢠⡞⠀⠀⠀⢿⡀⠀⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣄⣸⣤⡿⠁
⠀⠀⣾⠁⠀⠀⠀⣼⠇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋
⠀⣰⠟⠀⠀⠀⠀⣹⠇⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⢠⡄
⠀⡿⠀⠀⠀⠀⠘⣯⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⢠⡟⠀⠀⠀⢠⠂
⠘⠁⠀⠀⠀⠀⣼⠃⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣿⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⣰⠏⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⡾

⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⡀⠀⠀⠐⠰⠜⡐⠀⠀⠠⠚⠃⠀⠀⠀⠀⣠⣞⡡⢪⣾⠭⡁⠠⣀⠈⠁⠀⠀⠀⠀⠀⠒⠒⠦⠄
⠀⠀⠀⠀⠃⡀⡄⠀⠀⢈⠷⠀⢀⣤⡀⡄⠀⠀⠀⢿⢡⢀⣻⠻⣷⣤⣌⣙⣀⠆
⠀⠀⠀⠀⠀⣸⡡⠀⠀⠘⠀⢰⠇⠉⠋⠀⠀⠀⠀⢾⠢⢈⣽⠂⢘⣯⣿⣟⣉⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠃⠀⠀⠀⠀⠀⠀⠀⠛⢿⡧⠰⠻⢃⣤⣤⠍⠀⠀⠀⠀⠀⠘⠲⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠋⠀⠀⠀⠀⠀⠀⣀⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠠⢂⠀⣂⠀⠀⠀⠀⠘⠷⠓⠁⠀⢠⣰⣩⣶⡶⠋⠁⢲⡆⢰⡔⣖⢒⠂
⠀⠀⠀⠀⠀⠀⠀⠸⢀⡿⠀⠠⠒⠒⠀⠀⠀⠀⢸⣝⣽⠃⠸⢧⣠⣤⡶⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⢡⣿⣆⣼⡟⠩⠍⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠈⠉⠁⠉⢻⠲⠶⠛⠁⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠠
⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠁⡐⠂⠐⠀⠀⠀⠀⠀⠡⡂
⠀⠀⠀⠀⠀⠃⠀⠀⠸⢭⣀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠀⣀⡤⢴⡆⠀⠀⢀⠄⠰⠠⠀⣄
⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠄⠉⠉⣒⢦⡀⠀⠀⠀⠀⢸⠀⢰⠋⠡⠀⣀⠀⠈⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢠⡎⠁⠀⣽⠈⠑⠂⠒⠄⣸⡠⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠐⡇⢈⢸⠁⠀⠀⢀⠀⠀⠈⠀⠀⠓⠒⠤⢄⣀⠀⡀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠉⠉⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡉⠀⠀⠀⠀⠠⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠐⠒⠉⡐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠠⡠⡔⠈⠈⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠋
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠃⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠈⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣇⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⣦⣀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠃⠀⠙⢷⣄⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠰⠶⢦⣤⡀⠀⠀⠙⣧⡀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠙⢷⣄⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀⠀⠈⣷⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠻⣦⡀
⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠈⠻⣦⡀
⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢷⣄⠀⠀⠀⠀⠀⠈⣻
⣾⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠙⢷⣄⠀⠀⣠⡾⠋
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠙⣷⡾⠋
⢻⡄⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠈⠻⣦⡀⠀⠀⣠⡾⠋⠀⠀⠀⠙⢷⣤⣀⣀⣀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠈⠻⣦⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢶⣶⣤⣤⣴⣶⠞⠛⠁
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠒⠒⠊⠉⠉⠉⠙⠒⠒⠒⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⡫⡡⢢⢔⠴⠄⠴⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠛⢟⠳⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢫⡤⣾⠞⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⡻⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢀⠜⠙⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣡⠙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⣠⡶⡟⢯⣹⣍⣯⣹⣝⣟⣶⣦⣀⠀⠀⠀⠀⠀⠀⠰⣣⣻⣗⣻⡄
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢳⡼⠟⠉⠉⠀⠀⠀⠀⠈⠉⠉⠛⢷⣄⠀⢀⠀⣠⣾⠞⡡⢻⣅⢿⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢦⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⢫⣾⠟⣡⢾⠜⣩⣯⡳⣷
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠐⣡⠔⡵⣣⠞⣠⣷⡙⣾
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⢌⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢘⡥⢋⠜⡡⢊⡴⣯⠑⣼
⠀⠀⠀⠀⠀⠀⢸⢀⡤⡠⢤⠔⡠⢂⢦⣞⢢⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣈⢞⡡⢊⡴⡫⢞⣧⠱⣿
⠀⠀⠀⠀⠀⠀⢸⠀⣊⠔⡡⢊⢔⡵⡡⢾⠣⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢗⡵⣊⢔⡡⢊⡴⣻⣄⢣⡝
⠀⠀⠀⠀⠀⠀⠸⡇⡡⢋⠔⡡⣫⠞⡡⣻⡳⣝⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣩⢖⡵⢊⠔⡫⣚⡟⡌⣾⠇
⠀⠀⠀⠀⠀⠀⠀⣯⠐⡡⢊⠔⡵⢫⢞⣽⣟⢬⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡵⣫⢞⡵⣪⢞⣽⡙⡸⡞
⠀⠀⠀⠀⠀⠀⠀⢸⡛⠛⠛⠛⠛⠛⠛⠛⡟⠲⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⠚⠓⠛⠚⠛⠛⡿⣧⣱⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⡇⢔⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡇⢀⣽
⠀⠀⠀⠀⣀⠴⠂⠀⢇⠀⠀⠤⠦⠌⢦⠀⡧⢹⣇⣀⡀⢢⡀⠀⠀⠀⠀⠀⣀⣀⠀⢸⠀⠀⢠⣠⣠⡄⢸⠁⡂⠏
⠀⠀⣰⡯⠀⢠⣞⠙⢺⡿⡶⠶⠶⠶⣶⣖⡿⢺⣿⡶⠛⢠⡷⠀⠀⠀⠀⢠⡏⣸⢇⣸⣤⣤⣤⣥⣤⣤⡼⣦⣫⣇⡀⠀⠢⣀
⠒⣩⠤⢄⣀⡼⣻⣿⣿⡇⣸⠾⣾⠶⣶⣶⣗⣾⢸⠀⠀⠀⠙⣄⠀⠀⠀⠸⢿⡟⠿⠾⣻⣯⡷⣼⢿⣯⣽⣿⡧⠦⠶⡼⠃⠈⠓⢄⡀
⠀⠙⢲⣶⣡⡼⠏⣿⠿⣌⣷⣿⠛⣻⣿⢻⣽⣻⣓⡲⡞⢉⠷⡄⠀⠀⢀⠔⠙⠶⠛⢹⣏⡛⣿⠟⣿⡟⣾⣿⣷⣆⣾⣹⠆⠀⠀⢀⡽⠂
⠀⠀⠀⠛⠋⢀⡤⠜⣓⣼⣯⡝⣿⡙⣿⡈⡇⠈⠉⠉⢷⣾⣠⠃⠀⠐⢫⠶⠀⠀⠀⡴⣻⠛⠻⢠⣿⣡⣟⣼⣷⡿⠻⠛⠘⠃⠀⠈⠙⠲⢦⣀
⠀⠀⠀⠀⠐⠆⠛⠛⠉⠉⠀⠑⠛⢣⢻⣷⡾⠀⠀⠀⠀⠈⠁⠀⠀⢴⠟⠀⢴⢠⣞⡼⠁⠀⣷⣾⠋⠛⠈⠙⣧⠳⡀⠀⠀⠀⠀⠀⠀⠀⠐⣢⣝
⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⢴⣶⡀⠀⠀⠀⠀⠐⣻⠆⠀⠀⠀⠈⠀⠀⠀⠀⣀⡀⠀⠀⠀⠘⢷⡷⠀⠋⠀⣀⣄⠀⠀⠈⠛⣄
⠀⠀⠀⠀⡴⣩⠇⠀⠀⠀⠀⠀⢀⡠⡤⣄⠀⠈⣧⢱⣀⠀⠀⠀⠈⠁⠀⢠⣤⠀⢴⢯⣿⣧⠀⠳⣟⢆⠀⠀⠀⠀⣠⢤⡤⣶⣻⣬⣇⠀⠀⠀⠈
⠀⢀⡤⡜⣼⠃⠀⠀⢠⡾⣉⣿⡷⠓⠋⠁⠀⠠⣟⣧⣯⣷⠀⠀⠀⠀⠀⠈⠛⠛⠙⠻⠿⢋⣀⣀⠽⣚⣿⣄⡀⠀⠙⠛⠉⠉⠉⠛⠉⠔⠤
⠀⠸⣱⣿⣽⠀⠀⠀⠘⢆⣿⠟⠀⠀⠀⠀⠀⠀⠛⠐⠛⠁⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠜⠳⣞⡷⠛⣹⠛⣣⡜⣦
⠀⠀⠈⠉⡃⠀⠀⠀⠀⠀⢰⠒⣦⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠟⣅⠀⠘⠻⠟⠛⠡⢖⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠛⠒⠒⠛⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁
⠀⠀⠀⣠⡤⢤⣀⠀⡀⠀⠀⠀⠍⠀⠀⢀⡀⣀⡤⢖⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⡔⣪⡼⠃
⠀⢀⡄⢧⣴⡾⠛⠿⠿⠇⠀⠀⠀⠀⡞⢧⣹⡷⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣾⡏⠁⠀⠀⠀⠀⣖⣒⣀⠤⣤⠖⢦⡀
⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠟⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢻⣷⣏⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⣀⠀⢀⡀⠀⡀⠀⢀⠀⢀⡀⠀⡀⠀⢀⡀⠀⢀⡀⢀⠀⢀⡀⠀⣀⠀⢀⡀⠀⣀⠀⢀⡤⢀⡤⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠕⢊⡥⠚⣩⠔⢋⡤⠚⣁⠴⢊⡥⢚⡡⠖⣉⠴⢊⡡⠒⢉⡤⠚⣡⠔⢉⠴⢋⡤⠚⣡⠖⢉⠔⠊⣡⠖⣡⠔⢁⡴⢋⡴⣫⢖⡵⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠚⠁⠐⠋⠀⠚⠁⠐⠉⠰⠊⠡⠔⠉⠠⠞⠁⠴⠋⡠⠚⢡⠔⠋⠴⠊⠠⠞⠁⠰⠊⠀⠚⠁⠚⠁⠐⠉⠀⠁