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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⣠⡴⠶⠛⠛⠻⠿⣷⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠀⢠⡿⠋⠀⣀⣠⣤⣀⠀⠀⠉⠙⠻⢿⣿⣶⣤⣀⡀⠀⠀⠀⣀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠀⣿⡇⣰⣿⠟⠁⠀⢻⡆⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⢤⡤⠤⠴⠞⠋⠀⠀⠀⠹⣷⣿⡟⠀⠀⣀⡾⠁⠀⠀⠀⠀⣴⠖⠲⣦⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⣀
⠀⠀⠀⣼⣿⠏⠀⢀⣶⡆⠀⢻⡄⠀⣠⣶⡶⢶⣦⢠⣿⣿⠷⠒⠛⠉⣀⣤⣤⣤⡀⢸⣯⣤⠄⢹⣀⢤⣤⠀⡼⠁⢸⣿⢀⡴⠋⣽⣿⢀⣴⣶⠟⣶⡄
⠀⠀⢰⣿⡟⠀⠀⢸⣿⡇⠀⠀⣷⢰⣿⣟⣀⡼⠋⢸⣿⡟⠀⠀⠀⣴⣿⡟⠁⠘⠃⣾⣿⠗⠶⣿⠁⣾⣿⡼⠁⢀⣿⣿⠟⠀⣼⣿⠏⣼⣿⣇⡤⠟⢁⡀
⠀⠀⣿⣿⠃⠀⠀⣼⣿⡇⠀⠀⣿⠘⢿⣧⠀⠀⢀⣽⣿⡇⠀⠀⢀⣿⣿⡇⠀⢀⡼⣿⣿⣀⡼⠃⠰⣿⣿⠃⠀⢸⣿⡏⠀⣼⣿⠏⠀⣻⣿⡇⠀⠀⣸⠇
⠀⠀⠹⣿⡀⢀⡜⠁⠻⠿⠤⠾⠃⠀⠀⠉⠉⠋⠉⠸⣿⣇⠀⢀⡾⠁⠈⠉⠉⠉⠀⠈⠉⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸⣿⡟⠀⢠⡿⠉⠙⠒⠋⠁
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣶⡄⠀⠈⠛⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⢿⣿⡇⣀⡾⠁⠀⠀⢀⣠⣤⣤⣤⣀
⠀⢀⣤⣶⣾⣿⣷⣶⣄⠀⠀⢰⣿⣿⡿⠁⣿⣿⡏⢀⣤⣤⣤⣤⡀⠀⠀⢀⣴⡾⣿⣿⣶⡀⠀⢠⣾⠛⣿⣿⣧⠀⠀⠀⠈⡛⠛⠋⠀⠀⢠⣶⣿⡿⠛⠙⢻⣿⣷
⣰⣿⣿⡟⠁⠈⣿⣿⣿⡄⠀⣿⣿⣿⣃⣾⡿⢏⣴⣿⣿⠏⠀⠙⣿⣆⣰⡿⠃⠀⣿⣿⣿⡇⢠⣿⠃⠀⣿⣿⣿⠀⣠⣾⣿⣿⣿⣆⠀⣰⣿⣿⡿⠁⠀⠀⠀⣿⣿
⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇⢸⣿⣿⣿⠟⠋⠀⣾⣿⣿⣯⣀⣀⣠⣽⣿⠋⠀⠀⢀⣿⣿⣿⢀⣿⠏⠀⢰⣿⣿⣿⣰⣿⠋⢸⣿⣿⣿⢀⣿⣿⣿⠇⠀⠀⠀⢠⣿⣿
⠘⠿⣿⣧⠀⠀⣿⣿⣿⡧⢾⣿⣿⣿⠀⠀⢸⣿⣿⣿⠉⠉⠉⠉⢹⣿⠀⠀⠀⣼⣿⣿⡿⣸⡟⠀⠀⢸⣿⣿⣿⡿⠁⠀⣼⣿⣿⡏⢸⣿⣿⣿⠀⠀⠀⣠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⢸⣿⣿⣿⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣿⡟⠀⠀⠀⣿⣿⣿⣧⣿⠁⠀⠀⢸⣿⣿⣿⠁⠀⢀⣿⣿⣿⠃⢸⣿⣿⣿⣤⠴⠞⠋⣡⡀
⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠈⣿⣿⣿⡀⠀⢸⣿⣿⣧⠀⠀⢀⣼⡿⠃⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⠀⣾⣿⣿⡇⠀⠀⢸⣿⣿⡿⠀⠸⣿⣿⣿⡀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠈⠛⠿⠛⠁⠀⢻⣿⣿⣇⠀⠀⠻⣿⣿⣦⣶⣿⠟⠁⠀⠀⠀⠀⢿⣿⣿⣟⠀⠀⠀⠀⠿⣿⡿⠃⠀⠀⣿⣿⣿⡇⠀⠀⠙⢿⣿⣷⣤⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠀⢀⣀⣤⣭⣭⣉⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣀⣀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠛⠉⠙⢻⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠛⠉⠙⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⢿⣿⣿⡆⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠟⠋⠀⠈⠻⣿⣿⣦⣶⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀⠀⠀⠀⢾⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣷⣶⣤⣤⣤⣤⣤⣶⣶⡿⠿⠛⠋⠁
⣶⣶⡶⠶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⡶⠶⠿⠿⠿⠛⠛⠿⠿⠷⠶⢶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡾⠿⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠿⢷⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠋⠉⠀⠀⠀⠀⣀⠀⢠⣴⣖⠀⠀⠀⢀⣴⡟⠉⠉⠲⠀⣠⣴⠖⠰⣄⡀⠀⢀⠀⠀⠀⠉⠙⠻⠷⣶⣤⣀
⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉⠀⠀⠀⢀⡀⢴⣶⡄⠀⠀⠁⠀⣿⣿⠀⠀⠀⢾⣿⡁⠀⠀⠀⢸⣿⡏⠀⠀⣿⣷⠀⣼⣿⡇⠀⠀⣀⠀⠀⠀⠉⠛⠷⣦⣀
⠀⠀⣠⡴⠞⠋⠁⠀⠀⢀⣠⣄⠀⠘⡄⠘⣿⣷⠔⠚⠃⠀⢹⣿⡄⠀⢀⠘⠿⣧⡀⢀⡀⠘⢿⣇⠀⢰⣿⠟⢰⠉⣿⣇⡠⣺⣿⠟⠀⣤⡀⠀⠀⠈⠙⠳⢦⣄
⠀⠀⠙⣷⡄⠀⢀⣠⣦⡈⠛⣿⣷⣄⡇⠀⠸⣿⣧⣀⠠⠆⠘⠛⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠁⠠⠎⠀⡿⠋⣰⣿⡏⠀⣰⣿⡟⠁⠠⡀⠀⢠⣾⠋
⠀⢀⣠⣬⢿⣆⠈⠙⠻⢿⣷⣼⠉⠻⣧⣤⡄⠉⠁⣀⣠⣤⡴⢶⣾⣿⠉⠉⠀⠀⠀⠀⠉⠉⠛⠓⠶⢦⣤⣄⣀⠀⠀⠐⠻⠏⠀⣼⣿⠟⠤⣀⠀⠈⣰⡿⣥⣄⡀
⣴⣟⠉⠀⠚⠽⣧⡀⠀⠀⠉⠻⠂⠀⣸⣿⡷⠾⠛⠉⠁⠀⠀⣸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⢶⣤⣀⠈⠛⠏⡀⠀⠈⢀⣼⣟⡓⠢⠉⣻
⠉⠙⠳⣦⡀⠀⠙⢷⡀⠀⣀⣤⣶⢿⣿⣿⢷⣄⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⢟⡿⣶⣤⣈⠛⢀⣾⡛⠂⢈⣴⠞⠋
⠀⠀⢀⣾⠋⠀⠀⠀⠻⣾⣯⣉⣈⣠⣿⣏⣀⣙⣦⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⣠⣶⣦⡀⣀⠀⠀⠀⠀⠀⣴⣋⣀⣁⣈⣀⣁⣉⣽⣷⠟⠁⠀⠀⠙⣷⡀
⠀⠀⢸⡇⣀⣤⣤⣤⣤⣤⣤⣉⣉⣹⣿⣋⣉⣽⠿⠃⠀⠀⣾⣿⠁⠀⠀⠀⠀⣤⣦⣾⣿⣿⣿⣿⣿⠄⠀⠀⠘⠿⣯⣉⣉⣉⣉⣉⣉⣤⣤⣤⣤⣤⣤⣀⢸⡇
⠀⠀⠸⠟⠉⠉⠀⠀⠀⠀⠀⠀⠈⣿⡿⠉⠁⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠙⠻⠇
⠀⠀⠀⠀⣤⢠⠆⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⢯⡿⠀⠀⠀⠀⠀⠀⣠⣿⣇⣠⣤⣤⣴⣶⡶⣿⣿⠛⠛⠀⢀⣠⣄⠀⠀⠀⣤⠀⠀⣠⣤⡀⠀⠀⣠⣤⣤⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠠⠄⣧⣾⠗⠀⠀⠸⠿⢿⣿⠟⠛⠋⠉⠁⠀⠀⣿⡇⠀⠀⣰⣿⠃⢹⣇⠀⢸⣿⠀⣼⡿⣿⡇⢀⣾⡿⢻⣿⠀⠀⢀⣾⡿⣿⡆⠀⠀⠘⠁
⠀⠀⠀⡤⣄⡏⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⣿⣿⠀⢸⣿⠀⢸⣿⣼⣿⠁⣿⣧⣿⡟⠀⢸⣿⠀⢀⣿⡟⢠⣿⠃⠀⢀⡀⠀⣀⣠⣤⣄⣀
⠀⠀⠀⠙⠺⣧⠈⣀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠰⣿⡿⢦⣼⣿⠀⣾⣿⣿⡏⠀⣿⣿⡟⠀⠀⣿⣿⠀⢸⣿⣤⡿⠁⠀⢀⡾⠀⢠⣀⠀⠀⠀⠈
⠀⠀⠀⠀⠦⠸⣾⡟⠁⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⢀⣿⡏⠀⣿⣿⣿⠁⠀⢿⡿⠁⠀⠀⣿⡿⠀⢸⣿⡄⠀⠀⢀⣾⠁⠀⢠⡈⠁
⠀⠀⠀⠀⢠⡤⣿⠠⠄⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢿⣷⣼⣿⠃⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠻⣿⣦⣶⡿⠁⣄⠂⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠙⠛⢧⣰⣿⠆⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠙⠛⠁⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠉⠁⠀⠀⠸⡆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠆⠻⣿⢠⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠛⣷⡀⠀⠀⣤⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠰⣗⣿⣦⣸⣿⠋⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⢠⡴⠛⠉⠉⠉⠙⠛⠶⣤⣀⠀⠀⢸⡇⠀⢀⣿⠃⠀⢸⡇⠀⢸⡇⣴⡗⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⠿⣄⠰⠀⣠⡄⠀⠀⢻⣿⣄⠀⢠⡟⠀⢰⣾⠛⠻⣆⠀⠀⠈⠉⠛⠶⠾⣷⡶⠛⠁⠀⠀⠀⣷⣴⠟⠀⠛⠃⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣚⣷⣦⣽⣿⠀⠀⠈⠻⠟⠀⠀⢷⡀⠈⠛⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣤⣴⠾⠛⠛⠶⠶⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠒⠦⠀⠀⠀⠀⠀⠈⠻⢦⣤⣤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠉⠀⠉⠉⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠉⠂⠀⠀⠀⠐⠢⡀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠰⡀⠀⠀⠀⠘⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠙⢦⠀⠀⠙⣤⠀⠀⢀⡤⠒⠉⡛⢦⠀⠀⠤⢄⡀⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢀⣤⣀⠀⠀⠸⣄⠀⠀⠈⣇⠀⠀⠀⠙⢲⡾⠤⠐⠀⠀⠀⢸⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠒⠒⠒⠲⣤⡀
⠀⠀⠐⠁⠀⣿⠀⠀⠀⠈⠉⠒⠀⢻⠀⠀⢆⠀⡾⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠀⠂⠈⠙⠢⢄⣀⡀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⣿⠀⠀⢠⡄⠀⠀⠀⡼⢀⣠⡀⢸⡇⠂⠀⠀⣀⡀⠀⡴⠚⠳⡄⠀⠀⠀⢤⡄⠀⠈⠁⠀⠀⢀⣀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⢠⢃⡏⠀⠅⢸⠁⠀⢠⡎⠀⠁⠀⡷⣯⠤⣿⢲⡆⠀⠀⣇⡠⣦⠀⢠⡞⠉⠉⡇⠀⢠⠇
⠀⠀⠀⠀⠀⢿⠀⠀⠜⢷⣄⠴⠃⠘⣯⣀⠀⠸⣧⣀⡸⣧⣀⣀⠼⣧⡀⣠⠇⢸⡇⠀⢠⡟⠀⡿⠀⢸⣷⠀⠈⠀⢠⠎
⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⢀⠤⠒⠲⡅⠀⠀⠀⠁⠀⠈⠉⠀⠀⠈⠉⠁⠀⠈⠁⠀⠈⠁⠀⠳⠄⠀⠙⠳⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢴⡄⠀⠀⠀⠀⠀⣠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢀⡤⢤⡄⢀⠖⢹⡄⠀⠀⣧⠴⣦⠀⠀⣼⡁⠀⠨⠃⠀⠀⠄
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣿⠀⠀⠲⣄⠘⣶⠧⢿⡅⠀⢸⠇⠀⠀⡏⠀⢻⡀⠀⢿⣍⠉⠀⠀⠀⠘
⠀⠀⠀⠀⢆⡀⠀⠀⠀⠀⠹⣦⠀⠀⢹⡀⢿⣄⡼⠁⠀⠀⠀⠀⠀⠀⠀⠈⣉⣁⣀⠙⠓⠖⠂
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⠃⠀⢸⡇⠀⠀⠀⡰⠊⠛⠲⣄⠀⡠⠂⠁⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠹⣄⠀⢀⠹⡎⠀⠀⠀⢀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠈⠉⠀⢠⡇⠀⠀⠀⢧⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣀⡀⠀⠀⣼⠃⠀⣀⡀⣴⢋⣹⠇⢰⡟⢀⣀⠀⣀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣴⡿⢠⡦⠤⣼⠿⠃⣠⣴⡇⢀⡾⠉⣉⣿⢯⣿⠤⣾⣵⢿⣏⡾⣿⣁⣼⠿⠋⣀⡤
⠀⠀⠀⠀⠀⠀⠀⢀⣞⡾⠩⣧⠟⠀⠀⠙⠒⠋⠁⣿⣰⠿⠷⠛⠁⢿⠞⠁⠰⠿⠁⠻⠋⠀⠈⠉⠙⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁⠀⢠⣶⡄⠀⣶⡄⠀⣠⣿⣷⣤⡀⠀⣤⡀⠀⠀⣠⡆⠀⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣿⡇⢸⣿⠁⠀⢹⣿⠀⣿⣿⣦⣾⣿⡇⠀⣿⣧⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠈⣿⡇⠘⣿⣦⣤⣾⠟⠀⣿⡇⠙⠋⣿⡇⠀⣿⣧⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠉⠉⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠉⠉⠉⠁


⣿⣿⢛⠻⣛⡛⢿⢟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⡴⠁
⠀⠀⠀⣠⠞⠋⠉⠉⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢠⠞⠁
⠀⠀⠀⠈⠁⠀⡠⠀⠀⠀⠀⢳⠀⠀⠀⣀⠀⢠⡟⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⢸⡇⢠⡞⡽⠃⣾⠕⠉⠀⢀⣴⠖⠢⠀⠀⠀⠀⢀⡖⢀⡴⣲⢀⡤⠞⣩⠟⣠⠤⡀
⠀⠀⠀⢠⠏⢀⡴⢳⠀⠀⠀⢸⢇⡿⠋⠀⢴⡏⠀⠀⣰⠟⠁⠀⢀⡴⠒⢰⡄⣾⡴⠋⣰⡿⠋⢠⡾⢣⠞⣁⠔⠁
⠀⠀⢀⡟⣰⠟⠀⠈⠣⣤⡴⠏⠘⣇⠀⠀⢸⡇⠀⡴⣿⠀⢀⣠⣾⠁⣀⡾⢹⡟⠀⠰⠏⠀⣰⡟⠀⣿⠉
⠀⠀⠈⠛⠁⠀⠀⠀⠀⢀⡤⣀⠀⠈⠁⠀⠸⣇⠞⠁⠈⠁⠉⠀⠈⠛⠉⠀⠘⠀⠀⠀⠀⢠⣿⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣼⡴⠋⠀⠀⣀⠀⠀⢀⣴⠟⣦⠀⢰⡄⢀⡴⢺⡷⣠⠞⢉⣼⠏⢀⣴⢶⡆
⠀⠀⠀⠀⠀⠀⠀⡟⠀⢀⡴⢋⡼⠃⣰⠏⠙⣴⠧⢶⡿⣠⠎⢠⣿⠟⠁⢀⣾⠋⢠⣿⡵⠞
⠀⠀⠀⠀⠀⠀⢠⡇⣠⠋⢀⡾⠁⡰⣿⢀⡴⠃⠀⣼⡟⠁⢀⣿⠋⠀⢀⣿⠃⠀⣿⢁⣠⠔
⠀⠀⠀⠀⠀⠀⠀⣷⠃⢀⡾⢁⡜⠀⠙⠉⠀⠀⠀⠋⠀⠀⠼⠃⠀⠀⣼⠇⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣧⠞⠀⠀⠀⠀⠀⠀⠀⢀⡤⣦⠀⠀⠀⠀⠻⠄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠑⡤⠋⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠎

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠤⣀
⠀⠀⠀⠀⢠⡄⠀⠀⠸⣏⢇⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⣆
⠀⠀⠀⢠⣿⠁⣾⢳⠀⠹⣾⡀⢀⣴⠖⠲⡄⣿⡇⣠⠞⠀⢀⣤⣄⠀⠀⠀⣀⠀⢠⣿⢀⠔⣷⢀⠔⣻⡇⢀⣴⠚⢲⠀⠀⣸
⠀⠀⠀⢸⡇⠀⣿⣸⠀⠀⢸⠋⣿⡿⠲⠒⢡⣿⡋⠀⠀⢰⣿⠁⠀⢀⣾⠺⣭⡷⢻⣧⠏⣸⡿⠃⣰⡟⢀⣾⡧⠔⠋⠀⢀⡏
⠀⠀⠀⢸⡇⠀⠸⣿⠀⠀⡸⠀⠸⣿⣄⣠⠎⢻⣇⠀⢀⣿⣧⠀⣠⠛⢿⣀⣠⠃⢸⡟⢀⣿⠁⢠⡟⢠⠎⢻⣆⡀⢀⣠⠎
⠀⠀⠀⠀⠻⢤⠴⠋⠓⠚⠁⠀⠀⠀⠉⠁⠀⠀⠙⠓⠋⠀⠛⠛⠁⠀⠀⠉⠀⠀⠈⠃⠘⠃⠀⠘⠓⠃⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⢼⡀⠀⠀⠀⠀⣼⣿⡄⣰⣿⡆⠀⢸⡿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣸⡇⠈⠰⢦⣸⣦⡴⠆⠁⠀⣿⠻⣷⣿⠹⣿⠀⢸⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠋⢻⡇⢀⢢⠴⣻⡗⠢⡄⡀⢸⡟⠀⠻⠇⠀⢿⡇⢸⣟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠉⠀⠁⢾⠃⠀⠀⠉⣾⡇⠀⠀⠀⠀⢸⣿⢸⣿⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⣤⡄
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠓⠲⣿⡿⢽⡿⠤⠤⣤⣤⣴⣿⣧⢤⣾⠿⢷⣤⠴⠺⣿⠓⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢿⠟⠁⠀⠀⠈⠻⣼⠁⠀⠀⠘⢿⣿⣿⡇⠀⣿⠄⠀⣸⠇⠀⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠻⠇⠀⡿⠔⠛⠁⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢀⠀⠀⢸⣧⠀⠀⠀⠀⠀⣠⠞⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠈⠑⡄
⠀⠀⠀⠀⢠⡟⢰⣷⠈⠻⣇⢠⡶⠒⣄⣿⡠⠞⢀⣤⠄⠀⠀⣀⠀⣾⢡⢶⣆⠔⣺⠂⡵⠒⡆⠀⡇
⠀⠀⠀⠀⢸⡇⠸⣿⠀⠀⡇⢻⡇⠈⡸⣿⠀⠀⣾⠃⢀⣼⡏⠓⠏⣿⠃⣾⠃⣼⢃⣼⡗⠊⠀⣠⠃
⠀⠀⠀⠀⠈⢧⣀⠝⠦⠔⠀⠈⠛⠛⠁⠙⠦⠔⠹⠧⠞⠈⠓⠊⠀⠛⠸⠇⠀⠧⠊⠀⠙⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⠀⢰⡏⠀⠀⣀⡀⠀⣴⠆⢠⣤⠀⣠⡄⠀⣠⠤⡀⠀⣴⡶⣄
⠀⠀⠀⠀⠀⢀⡼⢷⠀⠈⢹⡟⠛⢻⠃⣴⠎⠧⣱⠤⣿⢰⢁⣿⠎⣼⠃⣼⣃⡠⠇⠀⣿⠀⠘⢲⡆
⠀⠀⠀⠀⠀⠻⠷⢬⡇⠀⢸⡇⠀⢿⠀⣿⡀⢀⠎⠀⣿⠃⣸⡏⢰⡇⢠⠻⣇⣀⡄⠀⣿⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠸⠀⠈⠉⠁⠀⠀⠉⠀⠙⠀⠘⠓⠁⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⢀⣠⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢰⡿⠋⠉⠉⣹⡿⠓⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢹⡇⣴⡟⣳⢀⣿⠁⢀⣤⢦⡄⣠⣤⡀⣠⣄⣀⣄⢀⣀⠀⠀⣀
⠀⠀⠀⠀⠀⢠⡶⢿⡄⢰⡇⠈⣿⠛⣿⣟⣡⢾⣿⣀⣿⡃⠘⣡⡷⢤⡿⣿⡟⣹⡿⢫⡿⢠⣾⣋⡗
⠀⠀⠀⠀⠀⠻⠇⢸⣇⣿⣷⣴⠏⣀⣬⣭⠁⠈⠛⠋⠙⠿⠛⠻⠧⠞⠁⠻⠀⡿⠁⣿⣡⢿⣇⡴
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣃⣨⣭⣴⡿⠤⠞⠁⣴⠛⢲⣴⣿⡶⣿⡶⣶⡆⣰⡾⢲⠀⣉⣵⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠉⠀⣿⣵⢾⡧⢰⣟⢋⡿⣸⡏⣰⡟⢰⡟⣸⣿⠛⣩⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠸⣿⡇⣾⣇⡼⠛⠋⠁⠈⠁⠉⠁⠻⠟⠉⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⠂⠀⠀⠀⠐⢢⡀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠘⣄⠀⠀⠰⡀⠀⠀⡀⠃⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠹⡄⠀⠈⠶⣀⠀⣠⠖⠉⠉⠛⠄⠀⠀⠠⢄⡙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠠⠶⣄⠀⠀⠑⢄⣀⣀⢹⠀⠀⠀⠈⣹⠓⠂⠀⠀⠀⠘⣄⠀⠀⠀⠉⠢⣄⠀⠀⠀⠀⠀⠉⠉⠲⢦⡀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠑⢤⡇⠀⠀⠀⠀⠀⠀⢈⣉⠉⠉⠀⠀⠈⠑⠲⠤⠤⠤⠀⠀⠀⠀⠹
⠀⠀⠀⣿⠀⠀⣸⠀⠀⠀⡸⣠⠒⢆⢸⠁⠀⢀⡴⠲⠄⢸⣁⠈⢳⣀⠀⠀⢹⡆⠀⠀⠀⢀⡤⠶⣄
⠀⠀⠀⣏⠀⠀⣿⡀⠀⡰⠃⣿⠀⠀⢸⡀⠀⢸⡀⠀⠀⣾⠙⠊⡹⢸⡇⠀⢸⡧⢉⡇⠀⣿⠀⠀⠎⠀⡸
⠀⠀⠀⠹⡄⠀⠈⠛⠋⢀⣀⣙⡓⠀⠈⠛⠒⠉⠓⠂⠀⠙⠷⠖⠁⠸⠇⠀⠘⠃⠸⣇⠀⠹⢧⣄⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣀⣀⡀⢀⠔⢦⠀⠀⣿⣀⣤⠀⠀⣰⠋⠀⢹⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠠⣄⠐⣧⣎⢷⠁⠀⣿⠀⠀⢿⠁⢸⡇⠀⢿⡓⠂⠁⠀⠀⠌
⠀⠀⢀⠀⠀⠀⠀⠈⣷⡀⠀⢸⡆⢿⣄⡼⠀⠀⠁⠀⠀⠀⠀⠀⠙⣂⡈⠙⠢⠄
⠀⠀⠀⠉⠀⠀⠀⠀⠈⠛⠀⠈⣇⠀⠀⠀⡐⠚⠒⢦⡀⢀⠔⠈⠀⠀⠉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠳⣀⢀⡀⢿⠁⠀⠀⢀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠸⡀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣀⠔⠁⠀⠀⠀⠀⠈⠉
⣿⡷⠻⠗⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣒⣠⣤⣶⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⢀⣀⣀⣄⢠⣾⣿⣿⠟⢋⣴⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⣘⢧⣶⣏⠁⠀⣉⣍⣀⠀⠀⢀⠉⠉⠛⠿⣛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣾⡿⠃⠀⠉⠛⢷⠀⠈⠛⠉⠀⠀⠉⠀⠁⠀⠀⠀⣷⡦⡀⠲⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⠊⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠉⢦⡀⠀⣀
⠀⠀⠀⠀⠀⢠⣀⠀⠉⠀⣼⣶⣶⡦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⣾⣷⡀
⠀⠀⠀⠀⠀⠘⣿⣷⠀⡜⠙⣉⡁⠀⠀⠀⣶⠀⠀⠀⡄⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠱
⠀⠀⠀⠀⠀⠀⠈⠁⢸⢃⣀⡀⠀⠀⠀⠀⣿⠀⣖⠀⣼⣶⢾⢠⠒⣠⢲⡔⡖⣤⡶⠀⠀⠀⠀⠀⠾⢣⣷⡟⠓
⠀⠀⠀⠀⠀⠀⣦⡀⠸⠛⠉⠀⠀⠀⠀⠀⠸⣄⣯⣰⠃⠙⠉⠈⠋⠉⠉⠁⠁⠉⠋⠀⠀⠀⠀⠀⠀⠘⣿⠃
⠀⠀⠀⠀⠀⠀⣿⣷⡤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⣇⣠⠀⢀⣤⣄⣤⣀⠀⠀⠀⠀⠀⢀⣠⣼⣤⡀
⠀⠀⠀⠀⠀⠀⠹⢿⣿⠀⠐⠀⠀⠀⠀⠀⠀⠀⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⠛⣻⠈⠁
⠀⠀⠀⠀⠈⠻⣿⣶⣬⡇⠀⠉⠀⠀⠀⠀⠀⠛⡟⢻⠄⡚⣦⢸⢛⡎⡇⣾⠕⣠⠀⠀⠀⠀⢠⣤⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠉⡻⣿⣷⣴⡇⢰⣤⠀⠀⠀⠀⠃⠈⠁⠙⠁⠘⠈⠁⠃⠉⠛⠉⠀⠀⠀⠀⠛⠋⢀⠾
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠹⣏⠈⠛⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢠⣾⡀⠀⢠⡦
⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠈⠢⡀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⢀⠔⣿⣿⡇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣀⣀⠀⠀⣄⠀⣦⣀⠀⠆⠀⢶⣌⡃⢀⡠⣴⣿⡆⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠁⠉⠿⣿⣷⠍⣀⣙⣿⣷⣀⢠⡤⠛⠋⢁⣼⡿⠛⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⢀⡀⠀⣠⣶⡄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡿⠏


⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣠⠴⢶⠄⣼⠇⣾⠔⢚⡉⠉⠉⠛⠲⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⡏⠀⣠⡤⠀⣿⠘⠁⣼⡏⠉⢠⣾⠙⡂⢠⡬⠙⠻⢶⣦⣤⣀⣴⡄
⠀⠀⠀⠀⠀⢀⣤⣄⠀⢻⣧⡀⠹⡄⣿⡇⠀⣿⡄⠀⣿⣇⠀⣾⡇⢠⢁⣿⠗⣹⡇⢴⡎⠉⠛⡋⠁
⠀⠀⠀⠀⠀⢸⡀⠹⣷⡀⢻⣷⣄⢱⠻⣷⣠⠛⠧⠔⠈⢙⡋⠈⠓⠃⢾⠏⣰⡟⢡⡿⢡⣶⣟⣉⠇
⠀⠀⠀⠀⠀⠈⠛⠀⠘⣿⣌⡏⠙⠋⠀⠀⢀⣀⡤⠖⠊⠉⠈⠙⠒⠦⢤⣀⡈⠀⠿⠵⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢡⣄⠀⢠⣾⠛⢛⡶⣒⠒⣀⡀⠻⠟⢡⣄⢪⣿⠿⡄⠀⠈⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⣿⢠⣿⠀⢸⡆⣿⣇⠀⠀⣾⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠒⢸⣿⢸⣿⠀⢸⡇⣿⢻⣆⢸⢸⣿⢸⣿⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠸⣿⠀⢸⠇⢸⠀⢻⠇⢸⣿⢸⣿⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣭⣤⣬⣭⣤⣬⣭⣥⣤⣭⣤⣤⣤⣬⣭⣭⣭⣭⣍

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠈⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠙⢿⡄⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⣷⠀⣾⠃⠀⢀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠁⠀⣴⡿⠛⠛⠻⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⣩⣉⠛⢿⣧⠀⠀⣼⠏⢀⣾⣿⣃⡄⠉⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⣾⣿⣧⡶⠈⣿⠀⢰⡏⠀⠈⠻⣿⠿⠃⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠈⠛⠛⠁⠀⢹⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢀⣀⣤⡤⠄
⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⢀⡠⠞⠁⠀⣄⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⠛⠉
⠀⠀⠀⠀⠀⠈⠙⠓⠒⢛⣡⡴⠶⠶⠚⠛⠲⠶⠦⣤⣈⠳⣤⣀⣀⣀⣀⣀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⠈⠉⠉⠉⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣶⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣠⣿⡟⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⣀⣀⣀⣠⣤⡾⢟⣁⣀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⢿⣅⠀⠈⠉⠁⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⢀⡴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠏⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠒⣿⠖⠶⢤⡀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠴⠆⠀⣠⡴⠾⣏⠉⠈⠙⣿⣀⣀⡀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⢹⡄⠀⠀⢻⡄⢀⣠⣄⣀⠀⢸⡇⠘⠻⣧⠀⣏⠀⠀⣸⠇⠀⠀⣿⠋⠉⢿⠀⢀⣿⣤⡤⠀⣠⣄⣀⠀⣠⣤⣼⡇
⠀⢸⡇⠀⢀⣼⠃⣼⣥⣤⣽⡆⢸⡇⠀⠀⣿⠀⣿⣓⣚⣁⠀⠀⠀⣿⠀⠀⢸⡆⠀⣿⠀⠀⣸⣧⣤⣼⣷⠏⠀⠈⣿
⠒⠚⠓⠚⠉⠀⠀⠹⣦⣀⡤⢀⣸⣇⠀⣠⣿⣤⡟⠉⠉⠉⠻⣴⠛⠛⠛⣀⣼⣦⠄⠘⠷⡼⠛⢧⣀⡤⠜⢧⣄⡤⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣀⣀⣀⣀⡛⣦⣤⣤⡶⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣤⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠀⠀⠀⠀⠀⠀⢀⡀⢀⡠⠈⠒⠦⠤⠀⣄⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠤⠐⠂⠐⠂⠒⠌⡉⠊⢡⠂⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡈⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠑⢲⡄⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠸⡆⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣤⣄⠈⠑⢄⠀⠀⠀⠀⡰⠜⠿⡀⠀⠀⠀⠀⠀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣈⠫⣂⠜⢀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢂⠀⠤⣥⡼⣠⡀⢀⠴⠤⢤⠏⠂⠀⠀⢀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣺⣑⣛⡧⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠈⠈⠉⠁⠂⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠁⡪⠵⣾⢿⡄⠀⠀⠀⠀⠐⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⢚⣤⣶⣾⠇⠥⣬⡠⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡜⣇⢒⣒⣓⠂⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⣮⠳⡁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠪⠄⠀⠐⠂⠈⠈⠁⠈⠂⠀⠀⠀⢸⢸⡟⣧⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⡄⢷⢻⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠀⠢⠀⠄⠀⠰⠄⠴⡄⠀⢀⠀⠀⠄⢠⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠂⢀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣐⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠌⢠⡀⠀⠂⠀⠰⠀⠀⠀⠀⠈⠣⡤⠀⠀⠀⠀⡄⠳⠒⢀⡀⠀⠉⠱⡦⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠐⠸⡌⠢⠀⠀⠀⠀⠀⢸⡀⠀⠀⢫⠀⢀⣀⡤⢧⡴⠂⠈⢻⢧⡄⠀⢁⡠⠃⢀⠹⢄
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠈⠠⠀⢀⠀⠔⠊⠂⢿⠀⠀⠈⠀⡄⠀⠀⠈⠁⠀⠠⠋⡡⠟⣠⠚⠀⠖⠩⢚⡇
⠀⠀⠀⣀⠀⠉⠀⡄⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠙⠛⠶⣶⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠷⡆
⠀⠀⠈⠀⠀⠀⠈⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⢸⣆⠀⠈⡄⢀⢽⡄⣶⠐⠛⢳⣶⡄⠀⣈⠀⠀⡠⠔⢒⡁
⠀⠌⠀⠀⠠⣄⡀⠈⠱⣴⣶⡢⠄⢀⠀⠐⠖⠛⠁⡀⠀⠣⠀⣧⠀⢀⣼⠅⠋⠈⠩⠈⣅⠢⡈⠃⠀⠀⢀⡼⠛
⠀⡧⣴⠁⠀⣠⠔⠛⠂⠀⠈⢁⡄⠀⠉⠂⠀⢀⣺⡿⠆⠂⢠⣧⣴⣾⡟⠃⠀⠀⠀⢀⣶⠿⡇⠀⠀⣠⠞⠐⠖
⢇⣬⠷⢠⠐⠁⠀⠀⠀⠀⠀⡼⢃⣠⣶⣆⡒⠛⠏⠀⢄⠀⡀⡙⠛⠋⠀⠀⠀⡌⠀⠀⣠⣖⣄⠀⢈⠀⠀⠀⢘
⠀⠅⢴⣇⣀⢀⣀⡀⠀⠀⢰⡦⢋⠀⠉⠉⠁⠀⢠⣇⡌⢃⡻⣿⣿⡁⠀⠀⢤⣠⣤⣴⣟⣀⠵⠂⢤⡉⠲⢰⡾
⠀⠐⣻⡏⠙⡌⠉⠀⠀⠀⡀⠀⠈⠉⠉⠛⠓⢦⡄⣥⠉⣿⡀⠀⠉⠉⠀⠀⠀⠀⠈⠭⠽⡿⡐⣄⡰⠈⢑⠦⠁
⠀⠈⣿⡏⠀⠸⣦⠀⠀⠀⢻⣄⣠⣿⣶⣦⣤⡄⢿⠇⣾⠝⠐⢄⠀⠀⠀⠀⠀⠀⢤⣴⣶⡿⣾⣿⣯⡷⣺⣷⡄
⠀⠀⠍⢿⣀⠔⠹⣆⠀⠀⣶⣾⠷⠶⠿⡿⣿⣿⢂⠼⠋⠀⠀⠀⠁⠀⢠⣾⠷⠖⢟⠙⠌⠁⢈⠛⠒⠂⣀⡀⣹
⠀⠀⠃⢈⣿⣿⣆⣘⣓⠚⠛⠂⠀⠉⠁⠈⡿⠇⠛⠿⣶⡆⠀⠀⠰⠺⣿⠛⢿⢰⡈⢧⡀⠁⣈⡻⣿⣷⠉⠙⠁
⠀⠀⠀⠈⠛⡽⣿⣿⣿⣦⡀⠀⣠⡀⠀⢀⣀⡀⠀⠹⢏⢿⡷⢣⣀⣤⣿⣧⣈⣱⣷⡞⢩⡀⠈⢿⣞⠋
⠀⠀⠀⠀⠀⠘⠣⡠⣾⡙⣿⠋⠁⣿⠛⢛⠟⢽⣦⡀⣻⣘⣿⣧⡟⣻⡿⢿⣿⣍⡥⠤⠌⠓⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣉⠀⡀⠙⠛⠉⢻⣿⢫⠁⢠⡙⣈⣻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢬⣾⣀⠘⢉⡌⠊⠰⠏⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⢀⣴⣶⠆⣴⠟⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠃
⠀⠀⠀⠀⠀⢀⡴⣿⡿⠁⠀⠲⡔⠚⢛⣋⠀⢰⡏⣠⠖⣒⠉⠉⢛⣒⠒⠒⣒⠒⡚⠛⠛⠛⠉⢁⡀
⠀⠀⠀⠀⢠⣿⢰⣿⠁⢠⣶⠀⢡⣰⡏⡹⢠⣿⠘⢂⣾⠋⠃⣠⡾⣉⡧⢾⣯⢺⣿⡔⣿⠇⣼⡟⡹
⠀⠀⠀⠀⢿⣿⢸⡇⠀⢸⡏⠀⡎⠘⠧⠔⠁⠻⠤⠊⠳⠤⠔⠹⠤⠜⠁⠼⠃⠸⠟⠀⠻⠤⠜⠯⠤⠆
⠀⠀⠀⠀⠀⠀⠘⣇⣠⠃⠳⠜⠀⣀⣠⣤⣴⣶⠶⠶⠞⠛⠛⠓⠒⠒⠲⠶⠶⠤⠤⠤⠤⠤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠏⢹⣾⡆⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠁⠀⢠⣿⠑⠌⠛⠿⣿⡿⠇⢿⣿⢏⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡟⡆⢠⢾⠄⠀⠀⢦⠔⠲⠒⠒⠒⠶⠤⣽⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⡆⢸⢫⠈⠦⡼⠋⠀⠀⠠⡀⠀⠐⠂⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠆⠐⡇⢸⢚⡳⣠⠇⠀⣀⣀⣤⡈⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⢁⠸⡍⡁⠚⠀⠀⠈⠀⠴⠗⠀⠀⢘⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡜⠀⠀⢳⡹⣃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡠⡀⡇⣷⠘⣧⠀⠀⠀⢀⣀⣀⣈⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠻⣔⡼⢹⡄⢿⣧⡀⠀⠀⢣⡁⠁⠁⡔⢲⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠘⣆⠹⣷⡘⣿⣿⣦⡀⠀⠉⠉⡱⠀⠈⣇⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢿⡙⠮⣕⠴⣍⠙⢿⣷⡶⠶⠃⡀⠀⢸⠀⠸⡄⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡻⣿⡆⢸⠟⣿⢦⠀⠰⠀⢸⢀⣤⣈⣅⣑⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠂⣿⣷⣾⡀⠸⣄⠀⠀⠀⢸⠈⠛⠟⠛⡛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⣠⣿⡟⠀⠳⠦⠜⠀⠀⠀⢸⡀⢿⣼⣷⣿⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣼⡿⢤⡷⣾⠿⣿⣧⡀⠀⠀⠀⢀⡠⠒⠊⠉⠉⠓⠙⠉⢒⣢⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠘⠇⣀⡠⠴⠂⠈⠉⠀⠀⣀⣀⣀⡒⠒⢉⣼⠙⣇⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⢸⡧⡀⣀⣠⠴⠖⠋⠁⠀⠀⢀⣠⠴⠚⠉⢩⡿⠉⣹⠋⠻⠤⠗⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠾⠛⠉⠁⠀⠀⠀⠀⢀⣤⠖⠋⠥⣤⣤⡴⠛⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠀⠀⠀⠀⢸⡟⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢀⣠⡴⠛⠉⣠⠄⠀⠀⠀⠠⠊⢸⣇⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠷⠒⠲⠚⠉⠁⠀⠴⠛⠁⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠟⠋⠀⠙⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⠿⠀⢑⠂⠀⠤⢤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢷⠀⠀⠈⠀⠀⢃⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⢳⠈⢦⡀⠦⣤⣡⡎⠙⠒⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠀⠀⠈⢆⢀⡝⠓⣯⣤⠃⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⢠⠏⠐⠢⣄⠀⠀⠀⠀⠉⢹⣾⡇⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠀⠀⠀⠀⠀⣾⠀⢻⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠘⡇⠀⠀⣾⣿⣿⣿⣿⣿⠟⡙⠻⣿⡟⢻⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⡀⠀⠀⠀⢸⠀⠀⡇⠀⠀⣼⢿⡿⢛⢏⡤⣪⡿⢤⠙⠃⠸⠿⢛⣒⠦⢤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⣇⠀⠀⡇⣼⢀⡿⢩⠞⢉⠔⣿⠀⠀⣠⠀⡴⠈⣧⠀⢀⠀⢉⣹⣆
⠀⠀⠀⠀⠀⠀⣾⠴⠖⠒⠲⢤⡇⠀⠀⠀⢸⡆⠀⣿⠀⠀⡇⡏⠀⠀⠀⠀⠕⣻⠋⠀⣠⠃⢀⡇⠀⣟⠲⣼⣦⣷⣻⣿⠳⣄
⠀⠀⠀⠀⠀⠀⣷⢀⠤⠶⠤⣄⢡⠀⠀⠀⢸⡇⠀⢹⠀⣀⢇⡋⠀⢀⡤⠶⠋⠉⣖⠋⠳⣄⡞⢄⡀⢸⠀⠈⠁⠙⠷⠃⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⢸⠻⢭⣓⣒⣺⠇⠀⢸⡿⢿⢸⠃⠀⣿⠀⠀⠀⠀⠘⣴⠿⣽⡃⣰⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⡘⠀⠀⠀⠀⠈⢣⣀⡾⠀⢸⡿⠀⠀⣿⠀⠀⠀⠀⠀⠈⠀⠸⣍⣽⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⡇⠀⠀⣿⠀⢀⡀⠀⠀⠀⠀⡼⠁⢹⡀⠐⠒⠶⠒⠂⠀⢱⡀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢸⠘⡇⠀⢰⠃⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⠃⠀⠀⢹⠀⠈⣧⠀⠀⠀⢰⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢸⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢸⠀⠀⠀⢸⡄⢠⡟⢳⡀⠀⣼⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣤⣤⣤⣀⣇
⠀⠀⠀⠠⠤⣤⣾⣤⠇⠀⣾⣀⣀⡀⠀⢀⡇⠀⠀⠀⠀⣿⠀⠀⠀⢸⣇⣼⠀⠀⢻⣀⠇⠀⠀⡼⢻⡇⠀⠀⠀⠀⠀⠀⠀⣿⣁⣤⣤⣄⡈⢻⡀
⠀⠀⠀⠀⣀⣤⡞⠁⢠⡀⢸⡉⠉⠉⠉⠛⠛⠛⠛⠒⠒⢿⠀⠀⠀⢸⠍⣧⣄⣀⣀⡟⢀⣴⠶⠧⣄⣀⣀⣠⣤⠤⠶⠒⠋⠉⠀⠀⠀⠀⠹⡄⡇
⠀⠀⠀⢰⣿⡯⠕⠊⢹⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢀⡾⣰⠃⢀⡼⠛⠁⣾⣵⣶⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠛⠁⠠⣯⣀⣀⣠⠽⠻⣏⠐⡻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈

⠀⠀⠀⣠⠔⠒⠒⠢⢤⡀
⠀⢀⡞⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠊⠉⠉⠙⠲⣄
⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠉⠁⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠑⢤⣀⠀⠀⠀⠀⣠⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⢠⠚⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⣠⠖⠒⢄⠀⠀⠀⠀⠀⠀⠀⠶⠁⠈⠒⠊⠉⠀⠀⠀⢀⡔⠛⠳⣄⠛⠲⣄⠀⠀⠀⠀⢀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⠇⠀⠀⠀⠀⠘⠁⣠⠒⠑⢢⠀⠀⠀⠀⠀⠸⡄⠀⠀⡸⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣓⣶⣉⠀⠀⠀⠀⠀⠀⣠⢧⡀⠀⣸⣇⠀⠀⠀⣀⣤⠬⣭⣭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠠⢄⣹⡓⢤⣀⣀⣠⣤⡏⠀⠉⠉⠀⢹⡦⠖⠊⢱⠇⠀⢠⠎⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠈⢷⠦⣄⣤⠤⢴⡟⠋⠀⠀⠀⡞⣧⠀⠀⡞⠀⡰⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠸⡆⠀⠀⠀⢸⣇⠀⠀⠀⢰⠧⢼⠤⢶⠗⠋⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣷⠀⠀⠀⢸⠘⢦⡀⢠⡏⠀⢸⡇⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢸⡄⠀⠀⢸⠀⠀⠁⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢈⡇⠀⠀⢸⡄⠀⠀⠀⠀⢀⣠⡇⠘⣶⠶⠶⠶⢤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡶⠒⠛⢻⠋⠁⠀⠀⠀⠉⠙⡏⠉⠉⡏⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠂⣾⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣸⠗⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠘⠓⠂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⠃⠀⠛⠋


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

⠀⠀⠀⠀⠈


⠀⠀⡀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠰⢀
⠂⢀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠄⠀⠀⠀⠀⠈⠀⢀⠂⠐⠂⠀⠀⡀⠰
⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠐⠁⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠃⠀⢀⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠂⠈⢁⠔⠁⠀⠀⡀
⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣠⣄⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⠿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⢸⣿⣿⠃⠀⢀⣴⣾⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠙⠿⣿⣷⡀⠀⠀⠀⠀⠙⠁⠀⣰⣿⣿⡿⠿⠿⠿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠋⠀⠀⠀⠀⠀⠉⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⣰⣿⠟⠉⠀⠀⠀⠀⠉⠻⣿⣿⡆⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡆⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⣿⡄
⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠓⡀⠀⠘⣿
⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣶⡇⠀⠀⢹⣇
⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠃⠀⠀⠘⣿
⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠘⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢻⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⡀⠀⣿⡄⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⡀
⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⢸⡇⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠛⠋⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠃⠀⢸⡇⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠋⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⠁
⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁
⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⢰⣿⠏
⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣀⣀⣀⡀⠀⠀⠀⠉⠛⠻⠿⣿⡏
⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠛⠛⠛⠛⠶⠆⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠉⠻⠷⣶⣶⣶⣶⣾⡿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⢻⣿⣿⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⢀⣾⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠛⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⡀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠋⠁⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⣀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣶⣤⣤⣤⣶⡾⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢁⣴⠟⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠋⠉⢀⠆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁⣴⡟⠁⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⣀⣠⣶⡿⠋⠁⠀⣼⡟⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣥⣤⣤⣤⣴⣾⣿⣿⣋⡁⠀⠀⠀⢠⣿⠃⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠈⠉⠙⢿⣶⣤⡀⢸⣿⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣿⡀⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣶⡿⠋



⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣠⣶⣶⡄⢠⣴⣦⡄
⠉⠹⣿⠉⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠀⠀⠘⠛⠃⠀⢀⣴⠟⠛⠛⣧⡀⠀⠈⠉⢿⣇⣠⣤⣤⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣦
⠀⠘⣿⠀⠀⠀⠈⢻⡆⠀⠀⣀⣀⣀⠀⠀⠀⣿⠀⠰⠾⢿⣇⠀⢼⡇⠀⠀⠀⣨⡷⠀⠀⠀⢼⡟⠁⠀⢹⣇⠀⠀⢈⣿⣀⣤⡀⠀⢀⣀⣀⠀⠀⠀⣀⣤⣤⣿
⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⣾⠋⠉⠙⢷⡄⠀⣿⡄⠀⠀⢸⣿⠀⢸⣷⣦⣤⡶⠟⠁⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠘⢻⣿⠋⠉⠀⣼⠏⠉⠙⢻⣄⣾⠋⠉⠉⣿⡇
⠀⠀⣿⡀⠀⣠⣴⠟⠁⠰⣿⠿⠿⠿⠿⠇⠀⣿⡇⠀⠀⠸⣿⠀⠘⣿⣦⣤⣤⣤⣀⠀⠀⠀⣸⣇⣀⠀⠀⣿⠀⠀⠘⣿⠀⠀⣴⣿⡿⠿⠷⠿⢿⡏⠀⠀⠀⢹⡇
⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠻⣦⣤⣴⡶⢀⣠⣿⣇⡀⢀⣴⣿⣤⣾⡏⠀⠀⠀⠈⠙⢷⡞⠛⠛⠛⠛⣠⣤⣿⣶⡆⠀⠙⠷⣶⠟⠹⣷⣤⣠⡶⠌⠿⣦⣤⡶⢾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉⠉⠉⠁⠈⠉⠉⠉⠙⢷⣤⣀⣀⣀⣤⡾⠃⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠛⠉⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠄⠒⠦⠤⢤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠖⣿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⢉⡉⠉⠉⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⡓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢀⡞⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣼⡃⠀⠀⠀⠀⠀⢠⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⡖⠉⠀⠑⠢⠤⠤⠤⣀⣸⠇⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣄⡀⠀⠀⡠⢶⡑⠒⠖⠒⣉⡉⠉⡆⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣀⡀⠀⠀⠀⠀⠀⠀⠈⢹⡴⠣⣷⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠛⠫⠭⠤⠊⠳⢄⡤⠊⠀⣹⡀⣇⣨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡾⠭⠭⡷⢉⡿⢭⣉⠙⠢⣿⠁⠀⢹⣗⡢⠤⣄⣀⣀⣀⡤⢾⠑⡄⠀⠀⠀⠀⠀⠀⠀⢠⠦⠔⠉⠳⠿⣿⡖⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢀⠖⢉⣧⡜⠓⢦⡈⠀⠀⠸⡄⠀⠀⢳⠉⠓⢦⣄⡀⠀⠀⠀⣇⢱⡀⠀⠀⠀⠀⣀⡠⠃⠀⠀⠀⠀⠀⢨⠃⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠩⡇⠀⠀⠀⡇⠸⣼⣷⡀⠀⠀⠀⠀⡇⠀⠀⠀⢇⠀⠀⠙⢍⢳⠀⠀⡌⠓⠺⠿⠿⡉⠉⠁⠀⠀⠀⠀⠀⠀⡴⠃⠀⢀⣀⣤⣤⢽
⠀⠀⠀⠀⠀⠀⠀⡞⠀⢰⡇⠀⠀⢰⠁⡰⠃⢹⣷⡀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠘⡟⠦⣀⡇⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢀⡠⠊⠀⣠⠾⠛⠉⠁⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠣⣀⣸⡇⠀⠀⡸⠋⠀⠀⠀⠁⢱⡀⠀⠀⢸⠀⠀⢠⢸⠀⠀⠀⢠⠇⠀⠀⢣⠀⠀⠀⠀⠀⠑⠦⠤⠴⠒⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⢹⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢇⠀⣿⡸⠀⠀⠀⢸⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡄⡎⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⡆⣿⠇⠀⠀⠀⡼⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠈⡇⡇⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠹⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠹⣇⠀⠀⢰⠃⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏
⠀⡀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⢻⣆⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⡎⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⡜⣀⣀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠶⠟⠋⠁⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⣀⡦⢄⠠⠔⢗⡤⠤⢚⠏⠀⠀⠉⠙⠒⠒⠢⠤⠒⣷⡤⠤⠤⠾⠤⣀⣀⣀⣀⠤⠔⠒⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠋
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⠾⠉⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠲⠒⠒⠒⠒⠒⠒⠚⢧⣀⣀⣀⣀⣀⣀⣀⣀⠤⠴⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⣀⡀⠀⢀⣴⢤⡀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢾⡇⢰⣧⠀⠘⠀⣴⡖⢢⠀⣿⠀⠀⣾⠏⠁⠀⣘⠯⠀⡧⢠⡄⠠⢻⡆⢰⡆⠀⢀⣴⠖⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢸⡧⠈⣿⠀⠇⢸⣿⠀⠈⠰⣿⠀⢰⣿⠀⠀⢸⣿⠀⢠⠃⠈⣷⠀⣿⠇⢸⡇⠀⢸⡟⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻⣧⠀⠘⣿⣀⠀⠀⢿⡀⠈⠻⠦⠀⠀⠿⠤⠃⠀⠀⠛⠀⠉⠀⣾⡀⠀⠘⣧⡠⠂
⠀⠀⠀⠀⠀⠀⢰⠒⢢⠄⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡄⠀⠦⡀⠀⠀⡠⠐⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠰⠀⠠⠀⠀⠀⠀⠠⠄⠀⠀⡤⢖⠐⠒⢄⠀⠀⠄⢀⠇⠈⠢⠊⠀⠀⠀⠀⠀⡠⠔⡴⠒⠄
⠀⠀⠀⠀⠀⠀⠐⠀⠰⠀⠀⠀⠀⠀⠀⠀⡜⠀⡌⠀⠀⠀⠆⢘⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡘⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⢈⠀⢸⠀⠀⠀⠀⢸⠀⢀⡄⠰⠀⠀⠀⠀⡄⢸⠀⢸⠀⠀⠀⠀⠀⢰⠀⠀⠃⠀⡇⠀⠀⠀⠅
⠀⠀⠀⠀⠄⠀⢨⠀⢈⡀⠀⠀⠀⢼⠀⠀⡄⢂⠀⠀⠀⢀⠃⢸⠀⠘⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⠇
⠀⠀⠀⠀⠀⠀⢸⠀⠐⠀⠀⠀⠀⠀⠀⠀⡄⠘⡀⠀⠀⠈⠀⢸⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠱⠀⢱⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠘⡀⠀⡄⠀⠀⠀⠀⠀⠀⠘⢆⡑⣀⠔⠁⠀⢸⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠢⣀⢂⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠧⠀⠰⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠠⠼⠄⠸⠀⠀⠀⠀⠀⢘⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠊⠉⠶⠈⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠑⢄⡠⠊⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣷⣾⣿⣷⣀⣀⣴⣦⡴⢿⣿⠀⠀⠀⠀⠀⠀⢠⣴⣦⠀⠀⣠⣴⣶⣄⠀⠀⢀⣾⣿⣛⣻⣿⡷⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠋⠉⠉⠻⣿⡛⢻⡿⠒⠒⠒⢲⣶⣶⣿⣿⣿⡗⠒⣿⡟⠛⠻⢯⣍⠉⠉⢿⣧⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⠁⠀⠀⠀⠀⠘⢷⣾⠁⠀⠀⠀⠈⠻⣿⣿⣿⣿⠇⠀⢹⡇⠀⠀⣠⡿⠀⠀⠀⢹⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⠀⠀⣸⣇⡤⠞⠋⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⡀⠀⠀⠀⠀⠈⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠲⢄
⠀⠀⠀⠀⠀⠀⠀⣤⡄⢀⠀⢻⡎⣆⠀⠀⠀⠀⠀⠀⣰⡟⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⣸⡟⢠⡿⢃⠀⠻⣿⡀⢠⡶⠒⠲⡀⣿⣁⡠⠊⠀⣠⣤⡄⠀⠀⢀⣀⠀⣰⣿⢠⠞⣿⢠⠖⢹⡗⢀⣴⠛⢳⠄⠀⢸
⠀⠀⠀⠀⠀⠀⣿⠃⠘⣿⡸⠀⠀⢸⠉⣿⡗⠒⠚⢰⣿⠉⠀⠀⣸⡿⠁⠀⣠⣾⠸⢧⡷⢹⣧⠏⣸⡿⠃⣰⡟⢀⣾⡧⠔⠋⠀⢀⡞
⠀⠀⠀⠀⠀⠀⢿⡄⠀⠹⣇⠀⠀⡞⠀⠹⣷⣄⣠⠋⢿⣇⠀⢀⣿⣇⠀⣰⠛⢿⣀⡠⠃⢸⡟⢀⣿⠃⢠⡟⢠⠞⠹⣧⡀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠈⠳⠤⠴⠉⠓⠊⠀⠀⠀⠀⠉⠀⠀⠀⠙⠓⠋⠈⠛⠛⠁⠀⠀⠀⠀⠀⠈⠃⠈⠃⠀⠘⠓⠋⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢶⣿⣤⣀⣀⣿⣇⠀⡀⢰⡟⢢⠀⢀⣿⠃⣠⠊⣿⢀⡔⣻⠇⢀⣴⠋⠉⢣⠀⢰⣿⡟⠙⣇⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡼⠻⣧⠀⠀⠀⣿⠀⠀⠀⣿⠀⣼⡟⠈⠳⢼⠖⢺⣿⢰⠃⣸⣿⠎⢰⡟⠀⣸⣯⠠⠴⠋⠀⠈⣿⡇⠀⠀⣩⡷
⠀⠀⠀⠀⠀⠀⠀⠻⢷⣦⣹⡆⠀⠀⣿⠀⠀⢸⣿⠀⣿⡅⠀⠀⡼⠀⠸⣿⡎⠀⣿⡏⠀⣿⠁⢠⠻⣿⣀⢀⣠⠀⠀⣿⢇⣠⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢹⡄⠀⠀⣿⠀⠈⠛⠒⠚⠀⠀⠀⠿⠀⠘⠿⠁⠀⢿⣠⠏⠀⠈⠉⠉⠁⠀⠰⠛⠋⠁






⣿⢿⠿⠿⠿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣟⣻⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⠀⠀⠸⣭⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢤⣤⣴⣶⠶⠿⠛⠻⠶⢦⣄⡀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⣀⣀⣀⠀⠀⠀⠐⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠻⣧⠀⠒⢤⣀⠰⡀⠉⢻⣆⢀⣤⣴⠟⠉⢠⠂⣀⠠⠉⣹⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠹⣧⠀⠒⠚⠓⢧⣀⠀⢹⣿⢀⡟⢸⣠⠶⠛⠁⠀⣴⡟⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⢀⠘⢆⠀⠀⠀⠙⣷⣄⠐⠂⠀⠛⢦⣸⡏⢸⣠⠞⠋⠀⠂⣠⣾⠟⠀⠀⠀⠀⡴⢡⠀⢈⢧
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠧⠊⡠⠈⡄⠀⠴⠀⠈⣙⣿⣷⣶⣶⣿⣿⡇⢸⣿⣶⣶⣶⣿⣏⣁⠀⠀⠈⠀⢰⠁⡀⠑⡞⠈⡆
⠀⠀⠀⠀⠀⠀⠀⢃⠈⠙⢖⠀⢀⠇⠀⢀⣴⠿⠛⠉⠁⠀⠉⠙⠻⢷⡾⠟⠉⠁⠀⠉⢙⠛⢷⣦⡀⠀⢸⡀⠈⡾⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠢⠬⠷⣏⠀⣴⡟⢁⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠙⢿⣆⢀⡱⠾⠥⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡀⢤⣿⡏⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠈⢳⡀⢻⣧⠤⢀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⢀⣼⡿⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⣿⣤⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⢠⡊⠤⠔⠻⢁⣿⡇⠀⠀⠀⠀⢀⢤⣶⣤⠀⠀⠀⠀⠀⠀⠀⢠⢴⣦⣄⠀⠀⠀⠀⠀⢸⣿⠙⠓⠠⠄⢳⠄
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠁⠈⣿⡇⠀⠀⠀⠀⢺⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⠂⠀⠀⠀⠀⣼⡏⠀⠀⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠹⣷⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⣄⡄⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⣿⠏⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣤⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢻⣦⡹⣄⠀⠀⠀⠀⠀⠀⠛⠶⠞⠃⠀⠀⠀⠀⠀⠀⡰⢃⣼⡟⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣬⡓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢋⣴⡿⠋⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⣤⡶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢿⣿⠀⢸⡇⠀⣠⣶⣿⡆⠀⠀⣴⡄⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣾⠉⠹⡆⠀⠀⢉⣠⠟⠀⣼⣿⡟⣰⠇⠀⠀⠈⠀⠀⠀⠀⢸⣿⡆⢹
⠀⠀⠀⠀⠀⣴⡋⢹⣿⣷⡀⠀⠀⠙⠳⠦⣿⠴⠖⢋⡁⠀⣼⣿⡟⣰⠋⣀⣀⠁⢀⡶⠋⠉⠳⡄⠀⢀⡾⠀⢈⣠⣴⣦⢀⣤⢶⣶⣦⠀⠐
⠀⠀⠀⠄⠀⣿⣷⠈⣿⣿⡇⠀⣶⣦⡀⠀⣿⢠⣾⣟⣹⢷⣿⣿⡟⢡⣾⣿⠋⣷⡜⣷⣾⣷⢀⣻⡴⢻⣦⣴⠏⣹⣿⣿⠋⢀⣾⣿⡏⣠⣴⣦⡀
⠀⠀⠀⠀⠀⠈⠁⠀⣿⣿⣷⠀⢸⣿⡇⢠⡇⢿⣿⡏⠁⣸⣿⡿⠀⣿⣿⡇⠘⠛⣡⣿⣿⡏⠉⣿⠀⣼⣿⠃⢠⣿⡿⠁⢠⣿⣿⡟⣸⣿⡿⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀⣸⣿⣧⣾⠁⠈⠻⠷⠞⢻⣿⡇⠀⠻⣿⣷⡶⠞⠹⣿⣿⠀⣰⠇⠀⢿⣿⠀⠀⠙⠃⢠⣿⣿⠟⠀⢿⣿⠀⣴⡆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡿⠋⠻⠿⠃⠀⠀⠀⠀⠄⠸⣿⣇⠀⣤⣦⡀⠀⠀⠀⠉⠛⠛⣡⠴⠶⢦⠀⠀⠀⠁⠀⣸⣿⡿⠒⢦⠀⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣶⠄⠀⠀⠀⠹⣿⣄⢉⣰⠇⠀⢀⡀⠀⠁⢸⠇⢰⣾⣿⠃⢀⣠⠴⠛⠁⠙⠓⠒⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠉⠀⠀⠀⠈⠁⠀⠀⠘⢧⣤⣭⡤⠖⠋⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣯⣿⣽⣽⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣷⣶⣷⣿⣿⣷⣿⣿⣷⣷⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⡁⠀⢄⢆⠄⡔⠀⡄⠀⠔⠀⡔⢢⠀⣆⠴⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠈⠈⠀⠉⠀⠁⠀⠁⠀⠈⠀⠀⠈⠀⠈
⠀⠀⠀⢀⣤⡶⠶⠒⠒⠒⣾⣟⡡⣄⠀⢠⡿⠋⢱⠀⢀⣴⡦⢊⣷⡤⠚⢻⡆⠀⣠⡶⠛⣳⠀⠀⠀⠀⣀
⠀⠀⠀⠺⠏⠀⠀⠀⠀⣸⡿⠋⣰⡿⠀⣨⡟⠂⠞⠀⣿⡏⢀⣾⠏⠀⢠⣿⠃⢰⡿⠁⠊⢁⣠⠶⠚⠛⠛⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⢰⡿⠁⠀⠙⠃⠀⠀⢀⡉⠀⠈⠁⠀⠀⣾⠇⠀⠘⠳⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⢰⠋⠉⠉⢉⣓⣦⣤⠴⣿⠖⠒⠛⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠹⣆⠙⢦⣀⣀⣀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣎⣴⣤⣼⣁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣾⣿⣿⣿⣿⣴⣴⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣠⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢁⣂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⡟⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣛⣶⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢉⢹⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⣿⣿⠿⣿⢧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠩⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⠿⠏⠙⢡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⣶⢰⡄⠀⡶⣤⠀⢠⡆⢠⡆⡄⢶⣰⡆⣶⠀⣴⢤⢰⡖⡆⠐⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⣿⠀⢻⠀⣇⡿⠀⠸⡇⠸⣇⠇⠸⢻⡇⣿⠀⢿⠸⢸⡏⣇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣤⠀⠀⠀⣠⠀⣠⢠⠀⡄⠀⠀⣀⠀⣄⢠⠀⢀⠀⢀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣆⠀⠀⢸⠀⣿⢸⠀⡗⠀⠀⣿⠀⣿⢸⠀⢻⡄⣿⢸⠎⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠈⠀⠉⠈⠀⠁⠀⠀⠉⠀⠉⠈⠀⠈⠁⠈⠈⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⣩⠠⡄⠤⢲⠀⢲⠀⣲⡦⢸⡆⠀⠀⢀⠄⠚⠦⢀⠄⠲⡆⠀⢰⡄⠀⡤⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⡀⠄⠂⠀⠀⠰⠀⠈⢧⠹⣄⠛⠀⢸⡅⡷⡠⠸⡇⠀⡠⢸⡀⠀⡠⢸⡀⠀⡇⠀⢸⣷⠸⡇⠀⢠⡇⣨⠟
⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠙⠂⠀⠘⢓⢛⣁⣘⣙⣉⣁⣠⠿⣏⣉⣀⣙⣛⣁⣀⠛⠛⠲⠧⠂⠼⣄⣫⠖
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠰⠀⠒⣲⠞⠋⠉⠉⠉⠉⠉⠉⠉⠁⣩⠟⠁⠀⠈⠻⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡁⢀⣠⠄⠒⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡁⡀⠀⢤⠀⠠⣄⢹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠘⠓⠲⢤⣀⠁
⠀⠀⠀⠀⠉⠁⠠⠖⠠⡄⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠥⠤⠤⢼⠄⠠⣀⣀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡔⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠒⠊⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠘⠀⢀⠥⠄⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠦⠤⠴⠾⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⡤⠤⠤⣤⡤⢤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠦⠤⠤⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢰⠀⡇⠀⡖⠆⠀⡀⠀⡟⠳⠀⡀⠀⠀⠀⠀⠄⠁⠀⣾⠛⠉⣉⠉⣿⢻⠀⠀⠀⢸⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⢘⡇⠀⣇⡀⠀⠁⠀⣿⡀⠀⡀⠀⡁⡁⡁⡁⠀⠀⣿⡀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠀⠀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠒⠒⠚⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠒⠒⠚⠒⡆
⠀⠀⠀⠀⠀⠠⡄⠀⠈⠁⠈⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣀⡀⠤⠔⢒⠠⠀⠀⢢
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢰⡇⠰⡦⠠⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠀⠒⢒⣒⠊⠉⠁⠀⠀⠀⠀⠠⠀⠒⠤⠑⡀
⠀⠀⠀⠉⠁⠀⡀⠀⠀⠀⠀⣸⠀⢶⠁⠀⠀⠀⠀⣠⠂⠘⡇⠀⠀⠀⠀⠀⠰⣦⠀⣰⠀⠂⠀⠀⠀⠈⢳⠈⣝⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠁⠈⠆
⠀⠀⠀⣀⢴⡊⠁⠀⠀⠐⠚⢝⡒⠾⠤⠀⠀⠀⠀⢻⡀⢀⡇⠀⠀⠀⠀⠀⠀⣿⡆⢻⡄⠀⠀⠀⠀⠀⢸⡄⣛⣜⠀⠀⣀⠤⠒⠂⠉⠁⣀⠬⠄
⠀⠀⠀⠣⠡⢌⠢⡀⠀⠀⠀⠡⡠⡉⢐⠖⠀⠀⠀⠀⠉⠉⠁⡀⠀⠀⣀⣀⠐⠛⠛⠚⣒⡂⠀⠀⠀⠀⠈⠉⠁⢔⠀⡥⠈⠆⠀⢀⡠⠎⣢⠡⠜
⠀⠀⠀⠀⠀⡘⠀⠀⠀⠄⡀⣀⠄⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⡀⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠒⠻⣿⣿⠟⠂⠀⠀⠀⠀⠀⠀⠺⢿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠐⠚⢻⡿⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃


⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣰⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠚⢿⠟⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⣠⡀⠈⠻⠋⠀⠀⠀⠀⠀⠀⠀⠐⢺⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡷⠀⠀⠀⠀⠀⠀⠀⠺⣿⡿⠂⠐⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⣰⡄⠘⠃⠀⠀⠀⠀⠀⠀⠄⢀⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠤⢾⣿⣿⡶⠄⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⡶⠀⠀⠀⠀⠀⠀⠠⠴⢾⣷⠶⠤
⠀⠀⠀⠀⠀⠁⠀⠹⠏⠀⠀⠀⠀⠀⠀⠀⠰⡦⠈⢻⡿⢁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠇


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡶⠶⢦⣶⣶⣶⠶⣤⡶⠾⣭⣷⢀⡳⢮⣍⣿⠿⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⣭⣋⣻⠛⡶⢾⣗⣛⣧⢼⣏⣱⠶⣶⣙⣛⣯⡝⣛⣋⣬⣿⡃⢶⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⣛⣿⣛⣥⣌⣛⢹⣿⡏⣿⣿⣶⣭⣏⣏⠹⣆⣻⣀⣦⠿⣭⣶⣽⡟⡻⣧⡤⢶⡛⡟⣻⢦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣧⡾⡿⣶⣿⡼⢿⣶⢸⡿⣻⣿⣩⣿⢻⣿⣿⣿⣶⣧⣷⣾⠊⣟⣻⡸⡷⣿⣬⢽⣛⣛⠲⣛⣳⣬⣿⡛⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⢶⣿⡷⢾⣴⣶⣼⣄⣾⢙⡏⢶⣯⢸⣿⢝⡷⣴⡟⣿⣷⠉⣴⠛⠁⣿⢡⡶⠛⡹⣶⣏⢿⡧⢭⡽⢶⣝⣛⣡⣼⡿⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⣊⡿⣿⣿⡹⣧⣾⣭⡟⣿⣛⣿⣇⣿⢻⣦⢷⡞⢿⡆⢸⡞⣧⡟⡟⢹⣬⡟⢣⣿⣷⣾⣘⠷⣭⣥⢴⠾⣯⣵⣦⣽⡷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣟⣓⣶⠿⢟⢛⣿⣭⣭⣏⣶⡿⣤⠿⣦⣽⢾⣟⢫⣬⣿⡏⣴⣛⣷⢹⡖⣿⢀⣽⣿⣳⣟⠳⠶⠶⣾⠋⣿⢛⣋⣿⣼⡇
⠀⠀⠀⠀⠀⠀⢀⡴⡿⣿⣬⣽⣏⣏⡿⢿⣿⣬⡽⣼⣷⣎⣱⠾⣷⣩⡿⢲⣿⡎⣻⣿⡏⣿⣭⡶⡾⠷⢁⣸⣿⣩⣽⢛⣻⡿⡫⢿⣮⠋⣿⢃⣽⠼
⠀⠀⠀⠀⢀⣴⣿⡷⣿⣿⣧⣽⣝⣿⠿⣟⢹⣿⣿⢻⣎⣩⣿⡳⣿⣬⡟⢻⣿⣯⣿⣿⠛⣿⠘⣁⡱⠶⠾⢧⣮⣩⣵⡾⣭⣿⣽⡟⣿⠜⣻⣸⣷⡆
⠀⠀⠀⠀⠀⣿⣿⣖⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣙⣿⣹⣥⣿⣟⢳⣿⢨⣟⠗⢩⡿⢿⣯⣑⣾⣛⣿⣚⣹⣧⠐⣶⠿⣶⣯⣿⠻⣷⢿⡆⢿⣸⡇
⠀⠀⠀⠀⠀⣿⡿⣟⣿⣿⣷⣿⣫⣿⣿⢻⣿⣿⣿⣿⡟⣷⣏⣿⢿⡻⣿⢿⣾⣡⣾⣯⣭⣟⣻⣹⣄⣻⠶⣞⣻⣽⣟⣉⣿⣸⣿⢹⣴⠷⢾⡜⠇
⠀⠀⠀⠀⠀⣿⡿⣿⣿⣗⢺⣿⡿⢽⣿⠾⣿⣋⣻⣿⣿⠖⣽⠟⣽⣇⡯⣿⣿⡽⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣟⣿⢩⣿⢻⢸⣿⡜⢿⣴⡀⠁
⠀⠀⠀⠀⠀⣿⣿⣼⣿⢿⢽⣾⣿⣶⣻⣽⣿⢿⣷⡻⢸⡗⣻⣷⣫⣽⣟⣹⣿⢿⢿⡷⣽⣷⣽⣿⣴⣾⣋⣻⣿⣇⣺⣇⣿⢻⡏⣷⠻⣗⡄
⠀⠀⠀⠀⠀⣿⣻⡽⣿⡿⣿⣧⡿⣿⣏⣟⣿⣾⣿⣿⣾⣿⣺⣴⠟⣿⣾⠋⣿⣿⣿⣿⡭⣻⣿⣽⣯⣹⣿⣷⣏⣹⣧⣛⣹⠿⣶⡍⣾⢻⣇
⠀⠀⠀⠀⠀⣿⢻⣿⣿⢭⣾⣟⣿⣾⣿⡿⣿⣬⣷⣿⣿⣿⣸⣿⣿⣻⣟⣶⣿⣿⣾⣿⣁⣿⡹⢿⣏⢝⡷⢿⣉⣿⣿⠰⣿⡞⢿⣼⡿⠷⠈
⠀⠀⠀⠀⠀⣿⣿⡽⢷⣯⣿⣾⢿⣻⣿⢷⣿⣿⣟⣹⣿⣿⣟⣺⡷⣿⡯⠿⠘⣯⢶⣿⣟⣽⠿⣿⣾⣟⣻⣾⣝⢻⣿⣿⣺⣿⢻⡄⠃
⠀⠀⠀⠀⠀⠻⢿⣷⣾⣿⡟⣿⣿⣻⣿⣿⣿⣿⢿⣻⣿⣿⣿⣻⣷⠉⠁⠀⠀⣷⣸⣻⣿⣻⣶⣾⣽⣯⣻⣿⣻⣽⣿⣿⡏⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠾⠿⣱⣿⡿⢿⣿⣯⣿⣷⠟⠁⠀⠀⠀⠀⠸⣿⣷⣾⣿⣙⣿⣟⣿⡏⣻⡿⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠘⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠉⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣠⠔⡒⡀⢀⡆⠀⠀⢀⢀⡄⢀⠀⠀⠀⠀⠀⣠⠁⢸⡖
⠀⠀⠀⠀⠀⠀⠀⢀⣷⣷⣟⡴⠟⠛⠛⠓⠀⠀⡈⢿⡇⢸⣷⣠⠤⠖⣻⡿⢶⡟⠁⠀⢈
⠀⠀⠀⠀⠀⠀⠀⣾⡿⠋⠁⠀⠀⠀⠀⣀⣀⣈⣉⣸⣿⣌⠿⣿⣿⡿⠋⣠⠟⠈⠢⣄⣉⣤⡑⢐⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⢀⣠⠴⠾⠛⠛⠛⢃⣉⣿⣿⣟⠻⢶⣦⣶⣿⣿⣤⣤⠞⠋⠉⠋⠖⢿⠴⠿⠉⠉
⠀⠀⠀⠀⠀⠀⢀⡝⢦⣴⡿⠲⢶⣿⣇⠴⠀⠖⠉⣽⣿⣷⣦⣿⣿⣿⣿⣿⣧⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣗
⠀⠀⠀⠀⠀⠀⢘⠷⣌⠻⣧⡄⢾⡧⣰⡶⠂⣠⣤⠴⢲⣿⡓⠛⠛⠉⠋⠋⠽⠿⢿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣴⣶⣾⡿⠟⠙
⠀⠀⠀⠀⠀⠀⠘⢄⠙⢦⡈⠙⠾⢿⡋⡷⠿⠇⠀⠀⠀⠀⠙⠳⣖⠒⠒⠒⠛⠓⠒⠦⠭⣉⣛⠛⠛⠛⣛⣉⡉⠉⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠻⣶⣄⡀⠿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣽⣷⣶⣦⣧⣤⣤⣤⠚⠋⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠶⠭⠭⠛⠛⠛⠻⠟⠁⢀⡠⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠂⠐⠒⠂⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣧⣠⣶⡂⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢄⣀⠞⡡⠂⠪⡉⠉⠉⠉⠲⠔⢒⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣑⣾⠃⠐⢧⣘⣁⠇⢀⡔⢉⡙⡆⠊⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢻⣀⠀⠀⢘⡶⠦⠈⠳⠬⠔⠁⠀⢸⠤⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠃⢸⠉⠉⠉⠉⢳⣀⠀⠀⢀⣤⠤⢀⣈⣀⡀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⣀⣀⣀⣤⠊⠉⢀⣀⣈⣧⣄⣀⣀⣀⣈⣉⣋⣙⣮⣑⡾⠁⠀⣨⣦⣤⡄⣀⠀⠈⢧⡀
⠀⠀⠀⡇⠀⣀⠟⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠇⠀⠠⣇⠀⠀⠀⢿⠀⠀⠀⠳⡄
⠀⠀⠀⡇⠀⠓⢾⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢾⣠⠗⠋⠀⠀⠀⢸⠀⠀⠀⠀⢳
⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⣀⡀⠀⢸⠀⠀⠀⠀⠘⡇
⠀⠀⠀⡇⠈⡇⢰⣇⢠⠆⡷⠒⠀⢸⠀⠀⢰⠋⠁⠠⣎⡳⢰⠛⠞⣇⢸⠅⠀⠀⢸⠀⠀⠀⠀⠀⣧
⠀⠀⠀⡇⠀⠙⠏⠘⠚⠀⠙⠒⠀⠘⠉⠀⠈⠓⠈⠀⠈⠀⠈⠀⠀⠀⠈⠉⠀⠀⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⢤⣀⠀⢸⠀⠀⠀⠀⠀⡟
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠈⠙⠺⠒⠒⠀⠀⣰⠃
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⣤⣤⠤⠴⠚⠉


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣤⠊⠀⠑⡲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢆⠎⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣶⡄⠀⠀⠘⠀⠀⢀⠀⠉⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠃⢠⡀⠀⠀⠀⠀⠀⣿⢷⡀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠔⠀⠀⠆
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠓⠀⠀⠀⢀⡿⠘⣷⡀
⠀⠀⠀⣄⡀⠴⠉⢆⡀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠈⠻⢦⣤⣄⣀⣀⣤⣤⣄
⠀⠀⠀⠀⠑⢄⢠⠖⠉⠁⣀⣤⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠛⠉⠀⠀⠀⠐⣶⠤⣠⠤⡾⠁
⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠙⠛⠛⠷⢦⣄⡀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⠤⠤⠤⢽⣄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢿⣀⡿⠁⠀⠀⠀⠐⢤⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠉⠀⡀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠘⣿⠇⠀⠀⠀⠀⠀⠀⠙⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⡀⠀⠀⠀⠀⠿⠀⠀⠀⢸⡆⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠣⡀⠀⠀⠀⠀⠀⠀⠀⣼⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢅⠀⢈⠝⠉⠀⠀⣀⡤⠞⠁⠀⠈⠙⠒⣲⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠃⠀⠀⠀⠈⠉⠛⠲⢄⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠁⠒⠈⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣤⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁⠀⠀⠀⢿⣿⣿⣿⣿⣿⢣⣶⣿⣶⡄⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠁⠸⣿⣿⣿⠇⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⢀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⡾⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠛⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⡇⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢀⡀⡀⣀⡈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢸⠃⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⣼⡴⠒⠋⠉⠉⠁⠁⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡬⠿⢶⣾⣤⡖⢋⡁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠒⠾⠿⢦⣀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠁⠀⠀⠀⠀⠀⠈⠙⠻⣄⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠀⠀⠀⠀⢀⡟⠀⣀⣬⣭⡁⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢸⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣨⣭⣅⠀⠀⠀⠀⠀⠀⢀⣶⣿⠓⣆⠀⠀⢸⠇⠀⣧⣼⣿⣿⠀⠀⠀⠀⠀⠀⣠⣤⣬⡁⠀⢸⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣉⣿⣿⡧⠀⠀⠀⠀⠀⠘⣿⣿⣶⡿⠀⠀⢸⠀⠀⠙⠿⠿⠋⠀⠀⠤⠀⠀⠸⣿⣿⣥⣽⠀⣸⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⠻⠿⠿⠃⠀⠀⠉⠀⢀⡀⠈⠉⠉⠀⠀⠀⢸⡀⠀⠀⠀⠀⠐⣶⠠⠤⢀⣤⠀⠙⠻⠛⠁⢀⠇⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠘⢿⣭⣭⡾⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠙⢿⡿⠟⠁⠀⠀⠀⠀⢀⡞⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⣸⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⣡⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⠄⠀⠀⠀⠀⠀⢀⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⠀⠀⢰⡇⠀⢀⣞⡀⠀⢀⣠⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣦⣷⠀⠈⢹⠵⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠇⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠢⠿⠁⠀⠀⠀⠀⠀⠀⡉
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁
⠀⡄⠀⢠⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠂⠐⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁
⠀⠀⠄⠀⠄⠀⠀⠀⠀⠐⠀⠀⠀⠀⡁⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠
⠀⠀⠐⠀⠩⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⡜⠀⠀⠀⠀⢀⠀⡠⢀⡔⠀⠀⠀⠠⠀⡥⠂⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠃⠇⢰⡀⡄⠇⡀⠳⠏⠃⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡁⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠈⠁⠒⠒⠂⠁⠐
⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠠⠔⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠄⠄⠄⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠀⠀⠀⠀⠀⠅⠀⠀⠐
⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠤⠄⠀⠀⠀⠀⠀⠐⠇
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠋⠓⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠆⠐⠒⠀⠐⠐⠐
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⢨⠉⣙⠉⠉⢭⠉⠉⠉⠉⠍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠇⠈⢷⣀⠉⡿⢁⢀⠄
⠈⠀⠀⢼⣀⠛⠀⠀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⣿⠀⢐⠾⠀⣀
⠀⠈⢦⣄⠉⠶⠄⢢⣹⡉⣦⡔⠁⣼⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣿⢁⣉⠂⡽⢃
⠀⠶⠀⠙⢷⡄⠘⣪⠀⠇⠨⠀⠀⣿⠀⠀⠆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡤⠀⠉⠀⢀⠈⠸⠅⣢
⠀⠰⢆⠀⠀⠹⡆⠁⠈⠀⠄⠀⠀⠈⠃⠠⢻⢓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⡘⣭⠢⡷⡸⠃⡠⠀⠀⠄⠀⠀⠋⢀
⠀⠂⠶⢠⠶⡆⠀⠀⢀⢀⠀⡀⠃⠰⢯⢃⠯⠀⡁⣀⠓⢰⠀⠷⡀⠀⠀⠀⠀⠋⠀⠄⠠⢇⢏⠀⠙⣦⣀⠀⠈⠀⠀⢀⠀⢸⡿
⠈⠓⢦⣄⣀⠀⡆⠀⠀⠀⠈⠠⠟⠁⠲⡍⠀⠠⠋⠁⠀⣿⠀⠀⠁⠀⠀⣄⠈⢳⡄⠀⢀⠀⢹⡀⢠⠇⠀⣠⣤⠀⠡⣤⡆⠀⢀
⠑⢂⡄⢈⠙⠛⠛⢄⠀⢀⡀⠓⠲⢆⠀⢡⠀⠀⢀⠄⠐⠙⡷⠀⠁⠀⠈⠻⣶⣈⡧⢀⡈⠑⣠⠷⢂⡤⠾⠛⠀⠃⢐⠈⢿⠀⠈
⠀⠀⠀⢹⠐⠀⢋⢊⡤⠶⢛⡢⠤⠤⢴⠋⢺⣖⣡⠀⠀⠻⣧⠀⠀⡄⠀⢠⡿⠟⠃⠉⠉⠉⣩⠤⣾⠓⠦⣤⡴⠆⠶⠾⡲⢀
⠀⠀⠀⢸⠀⠀⠞⠃⣤⣶⠇⢀⣠⡄⠊⠗⢛⠉⠇⠀⠀⢀⠘⠂⢀⡀⠀⠊⣀⠀⠀⢈⣋⡀⠀⢰⠀⢡⣄⠀⢡⣤⠀⡈⣤⡉⠃
⠀⠀⠀⢸⠀⠀⠀⠀⠄⠔⠋⠉⠁⠀⠸⠀⠘⢧⡀⠐⠀⠀⠀⠀⢄⠀⠀⢀⡀⣀⣀⣈⡁⠁⠀⠞⠂⠀⣙⠣⣌⢢⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠚⠾⠂⠀⠐⠀⢠⣤⡇⠀⠀⠈⠀⠀⠀⠀⠄⠰⠀⠉⠉⠙⠙⠻⠶⣄⠈⠃⠘⠋⠉⠉⠓⠇⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠀⠀⠠⠤⠴⠛⠛⠉⠒⢀⡀⢠⠂⠀⠁⠀⠈⡄⠀⡀⢀⡀⠀⠀⠀⠀⠈⠀⠈⠂⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢀⡀⠀⠤⠀⠴⡃⠋⠦⡀⠀⡀⢤⠉⠏⠈⢢⠘⡇⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⣟⠀⠈⠀⠘⢧⣸⠀⠘⠒⡠⢷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢰⡆⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⣽⠏⣰⡗⠀⣼⢃⡄⢀⣾⠁⣠⡀⢀⣄⡀⢀⠀⣄⠀⣀⠀⣠⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠋⠀⢰⡏⢰⣿⢁⡼⢱⣟⠼⣾⠃⣾⠋⢳⡟⢠⢧⡿⢡⡿⢰⡏⣼⢏⠟⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠄⠸⠷⠋⠀⠛⠀⠘⠛⠘⠇⠀⠛⠐⠋⠘⠁⠘⠀⠛⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⠀⢀⣀⡀⢀⠀⠀⡀⠀⠀⡀⣀⡀⠀⣀⡀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠀⡄⢀⠀⢀⡀⢀⢀⡀⠀⡀⣠⡄⡀⢀⠀⣠⡀⠀⠀⢠⠀⠀⠀⣀⢀⠀⣠⣀⠀⠀⠀⠀⠀⡇
⠀⠂⠀⢸⠀⠀⠀⠀⠀⠄⠠⠤⠀⠄⠀⠤⠄⠀⠤⠤⠀⠠⠤⠀⠤⠄⠤⠀⠀⠠⠤⠤⠀⠤⠄⠤⠀⠤⠄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠒⠒⠀⠀⠐⠀⠒⠀⠐⠂⠐⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠠⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠄⡇
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠋⠙⠛⠿⣿⠋⠀⣴⠿⠿⠟⠛⠛⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠡⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠏⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⢠⣾⣿⣿⣿⣿⢏⣑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠻⡙⢿⣿⡿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣰⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠁⢠⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢼⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⡑⢦⠀⠀⠀⠀⠀⠛⠉⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠙⠻⣻⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠻⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢻⡿⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠙⠻⠓⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⠙⠃⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣴⣶⣶⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⢀⣠⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡿⢋⣿⣿⡿⠿⠛⠋⠉⠀⣀⣤⣶⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠁⠀⠀⠀⢀⣠⠀⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣠⠀⣷⠀⠀⠀⠀⠀⠘⠛⠃⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⠒⣶⣈⣙⡀⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣦⣤⣤⣤⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⡙⢋⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⣶⡶⠾⠂
⣿⣿⣿⡿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠒⠒⢲⠛⠉⠈⡁⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠰⠶⠾⠿⠟⣛⡉⠉⠀⠛⠉⢉⢧⡀⠈⣙⠃⠙⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠫⠞⠃⠋⠁⠀⠁⠉⠉⠀⠀⠀⠀⠀⠸⠀⠀⠀⠻⠄⢀⠀⠈⢕⠣⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⡐⠁⠋⠁⠀⠀⠀⢈⠔⠀⠈⠀⠀⠀⠀⡄⠀⠀⠠⠀⠀⠄⠀⠈⠀⠀⠐⢬⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠺⢀⡴⠁⠔⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⠀⠀⠀⠘⠂⠀⠘⠀⠀⡐⠂⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⡔⠋⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡣⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⢾⡸⠁⠀⠀⠀⠀⠀⢸⠰⠄⠀⠄⡅⠀⢠⠀⡀⣀⢠⠀⠀⠀⡃⡇⠠⢀⡀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁⠁⠈⠀⠁⠀⠉⠀⠀⠀⠁⠃⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣤
⠀⠀⠀⠀⠀⠀⠀⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠐⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⡀⠀⠀⠀⠀⢀⣄⣤⠀⠄⠀⠀⠀⢄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠆⠀⠀⠀⠴⠤⠴⢤⣠⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⢠⠦⡐⠀⢀⢠⠀⠀⠀⠀⢶⣷⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⣀⠒⠚⣏⣠⠄⣿⣿⣹⡿⣿⣿⢹⣿⢿⣿⡇⢲⣄⣀⣤⡼⠀⠄⠀⠀⣠⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣜⢧⣀⠄⡠⠌⢡⠍⠁⠀⢹⣿⣿⣇⣿⣿⣾⣿⣼⣿⠇⠀⠈⠉⢈⣅⢈⡁⢀⡞⢋⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠙⢶⡀⠀⠶⡀⠀⡀⠸⣿⣿⣿⠃⠀⠀⢿⣿⣿⡀⡄⠀⡀⠀⠀⣀⡠⠟⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣞⢢⡀⠁⣴⣿⣿⣿⠀⠀⠀⢸⣿⣿⡇⡀⣠⠤⠖⣫⡡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠯⠿⠿⠯⠽⠶⠶⠶⠛⠋⠩⠵⠏⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢀⣀⣠⡄⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠘⣎⡿⠁⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡀⢀⡀⠀⣀⡀⠈⠁⠀⠀⠀⠀⢀⣈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⢸⡇⢸⡇⢸⠏⣿⠀⣿⡄⠀⣼⡇⣿⠉⠁⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢸⡷⢾⡇⣾⠀⣿⠀⣿⢷⣸⢻⡇⢿⡦⠄⠀⠀⠈⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢻⣆⣿⢀⡿⠘⠃⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠃⠀⠙⠃⠈⠛⠀⢀⡘⠃⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢙⠆⢀⣰⠿⣠⡴⠚⠁⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣀⠀⠙⠃⢈⣭⡀⣠⡌⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⣷⢰⡃⣾⢡⠇⣾⢋⢼⠋⢱⡏⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁⠀⠀⠙⠁⠀⠈⠁⠈⠉⠘⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢸⡇⢠⡞⣶⠀⣶⡀⠀⣰⡆⣾⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣼⡇⣼⠀⣿⠀⣿⣇⢠⢿⡇⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢸⡇⢿⡄⣿⠀⣿⠸⠟⢸⡇⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠘⠇⠈⠻⠟⠘⠷⠀⠀⠸⠇⠿⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡄⠠⠀⠄⡀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢀⡀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠨⠀⠀⠀⠈⠀⠠⠀⡀⠠⠔⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡄⠀⢀⠀⠀⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣄⠀⠀⠀⠀⡀⠸⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢇⠀⠤⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠈⠃⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠈⠁⠊⠀⠀⠀⠀⠄⠀⢀⣄⠀⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢐⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⠄⢀⠀⠀⠀⠀⠀⢠⠠⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠋⠊⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠆⠀⠠⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⢁
⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⡀⠈⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠈
⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠈⠁⠀⡇⠀⠀⠀⠄⠠⠀⠁⠀⠂⠀⠀⠰⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠘⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠀⠀⠀⠀⠀⢀⠈⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡀⠀⠀⢀⣀⣀⣀⣀⢠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠐⠐⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠔⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠟⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠐⠛⠀⣠⠤⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠘⠀⠀⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆⢀⡠⠟⠓⢶⡠⠤⣀⡀
⠀⠀⠀⠀⢀⣀⣀⣀⠤⠖⢋⣴⣶⣶⣤⣽⠀⢀⠞⠋⠀⢠⣟⡡⠤⣾⠉⢀⣀⡙⠤⠀⠒⠳⠀⠈⠉
⠀⠀⠀⠀⠈⠻⣄⣶⣶⣦⠌⢉⠉⠛⠿⠿⣄⣀⡙⠂⣈⣛⣆⣀⣤⡔⠊⡉⢀⡄⢸⠉⠀⢰⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⣴⠃⠈⢿⠃⡇⢸⣠⣄⣰⢳⡟⠁⢰⠉⠉⡽⠻⠀⡏⢹⠀⡟⠎⣇⢸⢁⠀⢸⢰⣶⣴⣦⢠⠗⠋
⠀⠀⠀⠀⠈⠋⠉⠉⠋⠉⠻⣌⠋⠘⠃⢸⣋⠀⢸⠀⠀⡇⠀⠀⣇⡸⠀⢇⣀⣿⣼⣿⣷⣾⠸⡿⠛⠁⣈⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠆⣀⡨⣍⠛⢚⡿⠦⠼⠭⠤⠽⠾⠴⣶⠒⠋⠉⢹⣿⠿⠿⠟⣋⡠⠐⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣠⠖⠹⠧⠤⣤⡽⠷⠤⠤⣭⡷⠒⠀⠸⠤⠤⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣠⡶⡄⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⢠⡶⠛⠛⠉⠉⠉⣼⡟⠁⡀⠀⢻⡁⠀⣠⡴⠄⠀⣠⡿⠃⠀⢀⣠⢠⡀⠀⣿⡃⠈⠂⢀⣤⡀⢒⣷⡠⣶⡄⠀⣠⡶⠛⡳⠀⠀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⣸⡟⠀⣼⡟⠀⠀⠉⢸⡿⠀⠀⢠⣿⠃⠀⣠⣿⠁⠈⠀⣶⡟⠁⡈⠀⣰⡿⢡⣿⠋⣴⡿⠁⣼⡟⠃⠉⣀⠴⠞⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠄⢿⡇⠀⠀⠀⠈⠉⠀⠀⠸⣯⠄⠀⠈⠛⠀⠀⠀⠻⠤⠊⠀⠀⠉⠁⠀⠁⠀⠛⠀⠀⠘⠃⠒⠈⠁


⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠲⢦⡄⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣿⡿⠃⠀⣾⣿⣶⡀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣦⣄⣀⣣⣀⣁⣠⣤⣾⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠋⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣠⡤⠀⠈⢿⣿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢷⡆⠠⠙⢿⣷⠀⠀⠐⠚⠛⠋⠀⠀⠀⣀⣿⠀⠀⠈⠙⠻⠿⢿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠂⠈⠿⠧⣄⣀⠀⠀⠀⠀⣈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡋⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠟⠛⠛⠉⠙⢷⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣮⠛⠿⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⣀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣶⣾⡿⠿⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣌⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠁⠙

⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣷
⠀⠀⠠⠶⠶⣶⣿⣻⣷⠶⠶⠄⠀⠀⠀⠰⠶⠶⢾⣏⣿⠶⠶⠦⠀⠀⠀⠀⠰⠶⠶⢾⣿⡶⠶⠶⠀⠀⠀⠀⠰⠶⣾⣟⡁⢈⣻⣷⠶⠄
⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⠁
⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣼⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⠀⢀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣸⡇⠀⢀⣀
⠀⠀⠀⣠⣴⠞⠃⠈⠻⢦⣤⡀⠀⠀⠀⠀⠀⣼⡇⠛⠻⣧⡟⠛⠃⠀⠀⢀⣀⣠⣤⡾⠛⣦⣤⣄⣀⡀⠀⠀⠀⠈⠛⣿⡟⢻⣿⠋⠁
⠀⠀⠀⠈⠙⠳⣆⢠⡞⠋⠁⠀⠀⠀⠀⠶⢾⣏⣿⠶⠆⣿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠹⣾⠋⠁⠀⠀⠀⠀⠀⠀⣠⡾⠿⢿⡿⠿⢷⣄
⠀⠀⠀⠀⠀⠀⠹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⢠⡀⠀⣷⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⢀⡔⠀⡀
⠀⠀⠀⠀⠀⣲⡾⠷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣾⣶⡟⠁⠀⠀⠀⠀⠀⠀⠈⢻⣶⣿⣾⠋⠀⠀⠀⠀⠀⠀⠈⠛⢾⣿⠶⣿⡶⠋
⠀⠀⠀⠉⠛⣻⣧⣤⣿⠛⠉⠁⠀⠀⠀⠀⠀⢈⣿⣷⣿⣏⠀⠀⠀⠀⠀⠀⠙⠙⣻⣿⣾⣿⡛⠛⠁⠀⠀⠀⠀⠀⢀⣴⣷⣤⣾⣦⡀
⠀⠀⠀⠀⠀⠁⢸⡏⠈⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠈⠀⠙⠷⠀⠀⠀⠀⠀⠀⠸⠋⠀⣿⠈⠻⠂⠀⠀⠀⠀⠀⠐⠉⡼⠋⠀⠹⣆⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⡀⠀⢰⡆⠀⢀⡀⠀⠀⠀⠀⠀⠰⣦⠀⣴⡀⢠⡶⠀⠀⠀⠀⠀⠀⣀⠀⠀⣆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠉⢀⡾⢻⣄⠘⠁⠀⠀⠀⠀⠀⣤⣤⣴⠏⢷⣤⣤⡀⠀⠀⠀⠀⠀⠙⠷⢰⣿⠀⠞⠁⠀⠀⠀⠀⠀⠀⠛⢀⡀⢀⡐⠛
⠀⠀⠐⠶⠾⣯⡀⠀⣩⡿⠶⠆⠀⠀⠀⣀⡈⢳⡄⠀⠀⣴⠋⣀⠀⠀⠀⠀⠲⠶⠾⣿⣹⡿⠷⠶⠖⠀⠀⠀⠀⠶⠂⢈⡅⢨⡁⠰⠶
⠀⠀⠀⠀⣤⠈⢻⡾⠃⢠⡀⠀⠀⠀⠀⠉⠁⣸⡷⠛⠷⣿⡀⠉⠉⠀⠀⠀⠀⣠⠖⠸⣿⠀⢶⡄⠀⠀⠀⠀⠀⠀⣴⠈⢀⡈⠡⣦
⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢄⣤⠆⠰⡶⠈⣿⡾⡎⢡⡧⡿⡆⠶⠠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡻⢖⣤⠻⠘⠃⠻⢦⣷⡋⠸⣿⣃⢸⡄⣼⣡⣿⣦⡟⣧⢖⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⡴⡆⣌⠩⣷⠘⠋⠾⢧⣤⠨⠝⣂⣈⣽⣤⣙⡃⢘⣿⣿⢆⡴⢟⡷⡐⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⣥⣱⣱⠹⠽⠼⠴⠶⠴⣬⡙⠏⠀⠸⣿⣿⢿⣿⣿⡿⣈⡻⢟⢰⡿⠟⡀⣿⣾⢁⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣤⡾⣦⡘⢵⢯⣿⣱⠘⠚⡚⠚⠺⠗⢼⣿⣷⡿⣻⣿⢷⢹⣿⣿⣟⠘⢫⢾⣿⢿⣯⣿⠠⢉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣛⠛⢓⣛⠀⠶⣌⣻⠀⡎⣿⢻⡬⠅⢸⣿⣿⣧⡻⣾⣾⣾⣿⣿⡟⠀⣿⣯⣟⡽⠛⠉⢠⢬⠘⠂
⠀⠀⠀⠀⠀⠀⢿⣧⣬⡖⢩⡿⠀⠀⣙⣁⣈⣴⡿⠞⢠⣿⠻⠿⠿⣿⣿⣿⣿⣿⡍⣙⢁⣞⠾⣚⠋⡀⠄⠓⣟⢉⡁⣱⡄
⠀⠀⠀⠀⠀⡘⣡⡄⣼⢱⡟⠀⣾⣤⠈⡏⢉⣠⣤⡉⠈⣿⠘⢂⣤⣻⠟⣫⣽⡻⢤⡄⠊⠀⡜⣽⡃⣠⢶⠶⣄⠈⢿⡿⠗
⠀⠀⠀⠀⠀⣀⣿⡇⢻⣿⣇⡼⣿⡿⠀⡟⢈⣯⡴⢇⣷⡇⢀⣇⠋⠛⣸⣿⠋⡇⣿⣧⢾⣾⢱⣿⠁⣇⣏⡠⠏⣤⡈⢠⠶⡀
⠀⠀⠀⠀⠀⠻⡌⠓⣠⣭⣍⡀⠙⠷⠴⢃⠈⠳⠤⠊⣘⠷⠊⠙⠂⢊⡀⠛⠚⡄⠘⠛⠈⣃⠸⡧⠞⣩⡳⠤⠶⢋⡀⡎⢸⡇
⠀⠀⠀⠀⢀⠛⡴⢋⣴⣿⣿⡿⠀⣓⠺⠅⠠⠾⣿⢲⡈⠽⢅⣾⢹⡆⠟⠼⢐⠐⢫⣽⡆⣨⢭⣥⡐⢙⡟⢻⢷⠌⡁⢳⠋
⠀⠀⠀⠀⠈⢸⡇⢸⣿⡙⠋⠥⣸⣹⣧⢾⣿⡆⣿⣼⢃⣀⠞⣿⡾⢠⢲⢄⣭⡰⣿⣸⠱⣇⣾⣩⠟⣸⢣⣏⣞⡈⢏⡇⠶
⠀⠀⠀⠀⠀⠈⠻⢾⣿⣿⣷⣾⣅⠙⣯⠀⠻⠷⠘⠿⠉⠟⠀⢿⣇⡞⠛⠋⢘⣁⢻⣯⡴⠙⠻⠶⠾⡁⠸⣿⠿⢠⡿⣁⠂
⠀⠀⠀⠀⠀⠀⠸⢣⡶⣰⠃⠈⢿⡧⢠⣖⣶⣦⡀⣻⣴⣶⣄⢀⡶⣀⣾⢰⠈⣉⢡⣄⣄⣦⣴⣀⣴⣞⠰⢎⣉⣁⣄⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢻⣄⣠⠿⣡⢼⣿⣶⣽⣿⣯⡍⣿⠏⣠⣞⡫⠵⠋⣠⠿⠿⣿⠏⠻⠙⠂⣍⠀⠺⠿⠥⠳⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⡦⣠⣤⡶⣖⣻⡇⠉⣋⣹⢾⣅⠘⢩⡍⢠⣤⠈⣴⣶⡆⠠⠰⠆⡴⣷⢿⢶⣤⣛⠂⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣼⣿⣿⣴⣤⣟⣿⣾⣿⠀⣂⣴⣿⣏⣾⣵⣟⣴⣾⡦⠀⣧⢻⣼⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⢿⣭⡾⢋⣿⣉⣁⠎⢷⢬⣭⢡⣌⣛⡱⢿⣍⡀⠲⣌⠛⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠙⠿⣟⡇⢸⡿⣿⡿⢹⣽⡇⠙⠻⠗⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣠⣦⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⣰⢿⡿⠀⠀⡴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⢀⡟⢸⠇⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⣸⠃⡟⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢀⡿⣸⠁⠀⠀⠀⢠⡏⠀⠀⢀⣤⠄⠀⠀⢸⠃⠀⠀⠀⠀⣠⣶⠀⠀⠀⢀⣴⣆⠀⠀⠀⣴⣿⢀⣾⣿⠀⠀⢀⣴⣾⡗
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣷⠃⠀⠀⠀⠀⡞⠀⠀⡰⠟⠟⠀⠀⢀⡿⠀⠀⠀⠀⡜⠛⠃⠀⠀⣠⠛⢿⣿⡄⠀⢸⣿⣿⣾⢹⣿⠀⢠⠞⠉⡿⠁
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⡘⠁⠀⡼⢠⠎⠀⠀⡀⣸⠇⠀⠀⠀⣼⠁⠀⠀⠀⢰⡇⠀⠘⣿⠇⠀⣿⢻⣿⡏⢸⡇⢠⡇⢀⡔⠁
⠀⠀⠀⠀⣿⣧⠀⠀⠀⡴⣿⣧⡀⠀⢀⠜⠀⠀⣸⣷⡀⠀⢀⡜⠀⣿⠀⠀⡀⢰⣿⡀⠀⣴⠀⣾⣇⠀⢀⡟⠀⢀⣿⢸⣿⠁⣾⡇⣿⣧⣈⣀⣀⣀
⠀⠀⠀⠀⢿⣿⣷⣶⠟⠀⢻⣿⣿⡾⠋⠀⠀⠀⢻⣿⣿⡿⠋⠀⢸⣿⣷⡞⠁⠸⣿⣿⡿⠃⠀⢿⣿⣿⠏⠀⠀⢸⠇⢸⠏⠀⠟⠀⢿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠛⠁⠀⠀⠀⠉⠉⠁⠀⠀⠈⠉⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⣰⣾⣿⣿⡿⠿⠿⠿⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠿⠿⠿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣶⣶⡶⠶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠀⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⢀⡠⠚⠉⠀⠀⠈⠁⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣱⣀⠀⠰⢿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠶⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠉⠉⠉⠉⠙⠛⣻⣿⣿⣿⣟⠛⠛⠛⠁⠀⡄⢀⣴⣶⣶⣦⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣿⠀⠀⡆⠀⠀⠇⠸⣧⣾⣿⣿⠃⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠙⠻⠿⠟⠁⠀⡰⠁⠀⠀⠹⡀⠉⠛⠛⠁⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢀⡤⠊⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢷⠀⠀⠀⠉⠓⠒⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠁⠀⠀⣽⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⠐⠠⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠄⠂⠁⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠠⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⣀⣀⠠






⣀⣀⣀⢀⡀⣀⣀⣀⢀⣀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠐⠛⠻⠿⢶⠶⢶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠐⢁⠄⠈⠀⠀⠀⠈⠉⠓⠶⠤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠁⠀⠀⠀⠀⢀⠄⡠⠋⠀⠀⠀⠀⣀⢤⡀⠀⠀⠀⠀⠀⢀⠝⠉⠙⠓⠲⠶⠶⠚⠛⠛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⢀⠎⡰⠁⠀⠀⣀⠔⠊⠀⠀⡇⠀⠀⠀⠀⢀⠊⢀⠔⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠎⡸⠁⠀⠀⠀⣇⠀⠀⠀⠀⡷⠀⠀⠀⢠⠃⢠⠋⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠈⠰⠁⠀⠀⠀⠀⢸⠀⣀⠴⠊⠀⠀⠀⢠⠃⢠⠃⠀⠀⠀⠀⠀⠀⢠⢂⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡇⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⡀⠉⠁⠀⠀⠀⠀⢠⠃⢀⠃⠀⠀⠀⠀⠀⠀⣠⢣⡾⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠇⡜⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⢀⠆⠀⠆⠀⠀⠀⠀⠀⠀⠠⠁⣼⠁
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢸⢀⠁⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⡘⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⡇⠘⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⢠⠃⢰⠁⠀⠀⠀⠀⠀⣀⣴⡟⠋
⠀⠀⠀⠀⣠⣾⡏⠁⠀⠀⠀⠀⠀⢠⠃⡆⠀⠀⠀⡠⠚⠉⢧⠀⠀⠀⠀⠀⡘⠀⡇⠀⠀⠀⠀⢀⠞⢱⡏
⠀⠀⠀⣰⡟⠹⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⢰⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠈⣆⠀⢀⠞⠀⠀⠀⠀⢠⠀⠸⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⢿⣇⣀⣠⣤⣤⣈⣳⣄⠀⠀⢸⣄⡄⠀⠀⠀⠙⠖⠋⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠲⠴⣿⣇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠲⠦⢤⣄⣀⣀⡀⠀⠀⠀⠘⡆⡇⠀⠀⠀⠀⠠⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠳⠶⣷⣷⣤⣀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⣦⣀⣴⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣄⣀⢠⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⠟⠹⢿⣿⣧⡉⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⠀⠀⠀⣼⣿⣿⣿⣿⣧⣮⣉⣿⠻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⣿⣿⣿⣿⣦⠀⠀⢻⣿⣿⣿⣷⣿⣴⣿⠟⣳⡯⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢈⣿⣏⠙⠂⢳⣿⣿⣿⠀⢀⣤⣽⣿⣿⣿⣿⡄⠫⣉⣡⣴⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠙⠋⠉⠈⢀⠀⢸⠿⣿⣿⣷⡟⠛⣿⡿⠿⣿⠿⡇⠀⢸⣿⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢠⣤⣶⡆⠀⣿⠋⣶⣬⣟⠋⠀⠀⣇⠠⠀⠀⠀⠀⢠⣿⣴⣾⣿⣿⣿⡭⠀⠀⠐
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣧⠀⣤⡏⠀⠀⢀⣀⣀⢘⣿⣿⡿⠛⠉⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⢃⣾⣿⠟⠀⠀⠀⠀⠀⢤⣾⡟
⠀⠀⠀⠀⠀⠀⠛⠿⠛⠛⠛⠛⠁⣸⡿⠿⠛⠛⠛⠿⠿⠟⠛⠚⠿⠛⠛⠀⠀⠀⡄⡗⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⣷⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠏

⠀⠀⠀⠀⠀⢠⣤⣶⣶⣤⡀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⡀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⣄⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣄⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⣄
⠀⠀⠀⠀⢰⣿⠟⠋⠉⠹⣷⠀⠀⠀⠀⢠⣿⠟⠋⠉⠘⣿⡀⠀⠀⠀⢀⣿⠿⠛⠉⠈⢿⡄⠀⠀⠀⢀⣾⡿⠛⠉⠈⢿⡆⠀⠀⠀⠀⣼⡿⠛⠉⠁⢻⣇
⠀⠀⠀⠀⠟⠀⠄⠀⠀⠆⠘⠇⠀⠀⢀⠸⠁⠤⠀⠀⠄⠘⡇⠀⠀⠀⢸⠁⠤⠀⠀⠤⠈⡇⠀⠀⠀⢸⠃⠠⠀⠀⠰⠈⡇⠀⠀⠀⢰⠋⡠⠀⠀⠠⡀⡻
⠀⠀⠀⢀⣀⠀⢀⡀⣀⠠⠾⠀⠀⠀⡊⢠⡀⢀⡠⣄⠠⠾⠃⠀⠀⠀⠀⠀⠀⡠⢄⠠⠼⠃⠀⠀⠀⠀⠁⠀⣀⢀⡀⠴⠇⠀⠀⠀⠀⠀⠀⣠⢤⡀⠴⠇
⠀⠀⠀⠈⠁⠀⠆⠘⠀⡆⠀⠀⠀⠀⠀⠈⠀⠖⠘⠀⠖⠀⠀⠀⠀⠀⠀⠀⠼⠘⠃⠇⠀⠀⠀⠀⠀⠀⠀⢘⠡⠍⠑⠀⠀⠀⠀⠀⠾⠀⠰⠀⠃⠰⠂
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⢰⠀⠂⠃⠀⠀⠀⠀⠀⠀⢰⠠⠄⡆⠀⠀⠀⠀⠀⠀⠀⠰⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠚⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠳



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣴⣾⣿⡿⢷⣄⠀⠀⠀⠀⠀⣰⣾⣿⡿⢷⣆⠈⠀⠀⢀⠀⣠⣶⣿⣿⢷⣦⠀⠀⠀⠀⠀⢠⣾⣿⣿⠿⣦⡀⠀⠀⠀⠀⢀⣶⣿⣿⠿⣶⡀
⠀⠀⠀⠀⣸⠟⠋⠁⠀⠈⢿⡀⠀⠘⠁⢸⡿⠋⠁⠀⠈⢿⡄⠀⠀⠀⢠⡿⠋⠁⠀⠀⢻⡇⠀⠀⠀⢠⡿⠛⠁⠀⠀⢻⡇⠀⠀⠀⠀⡿⠟⠉⠀⠀⠹⣧
⠀⠀⠀⠀⠃⠀⠃⠀⠀⠃⢨⠃⠀⢰⠒⢫⡠⠂⠀⠀⠂⢈⡗⢲⡄⠀⠸⠀⠒⠀⢀⣶⢀⡇⠀⠀⢴⠺⡀⠘⠀⠀⠘⢀⡵⢲⠄⠀⠘⠁⠔⠀⠀⠘⠄⡝
⠀⠀⠀⠀⠶⠄⡰⢆⠦⠐⠛⠀⠀⠀⠀⠀⠑⢀⢆⠖⠐⠒⠀⠀⠀⠀⠀⠀⢀⢆⡸⢣⠘⠁⠀⠀⠀⠀⠈⠊⠲⡰⡀⠊⠀⠀⠀⠀⠀⠠⣤⠴⡲⢄⡾⠇
⠀⠀⠀⠀⠀⠀⠃⠈⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠘
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠸⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠐⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⠀⠀⠀⠀⠀⠀⠀⢠⡦⠀⠀⠀⠀⠀⠀⠀⢀⣦
⠀⠀⠀⡴⢋⠟⠙⠝⢳⡄⠀⠀⢀⡴⢶⣿⣶⢦⡀⠀⠀⢠⠞⢹⠋⢻⠉⢦⡀⠀⠀⡴⠚⡾⠻⡟⠲⡄⠀⠀⠀⣠⣶⠿⣷⣦⠀⠀⠀⠀⡠⠴⠾⢷⢦⣄
⠀⠀⢸⢱⣼⠀⠀⠀⡇⡷⠀⠀⢸⠀⣾⠀⢸⠀⣿⠀⠀⣿⠀⡎⠀⠈⡇⠈⡇⠀⢸⡇⢸⠀⠀⢱⠀⣻⠀⠀⢰⣷⡈⠀⢱⢻⡇⠀⠀⢸⠐⠀⠀⠈⠈⠈⡆
⠀⠀⠈⠿⣾⣄⣀⣴⡷⠃⠀⠀⠘⢶⣽⣄⣾⣴⠏⠀⠀⠹⣬⣧⣀⣠⣿⡼⠁⠀⠈⢧⣿⣄⣠⣾⣧⠏⠀⠀⠈⢿⣾⣄⣾⡿⠃⠀⠀⠘⢮⣔⣄⣰⣰⠴⠁
⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠈⠁⠉⠀⠀⠀⠀⠀⠀⠈⠉⠈⠀⠀⠀⠀⠀⠀⠈⠈⠉
⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡴⢺⢿⠿⢿⠲⡀⠀⠀⠀⢀⣠⣾⣏⡀⠀⠀⠀⠀⡠⠚⣿⣟⠲⡄⠀⠀⠀⡴⠺⡾⠻⡞⠲⡄⠀⠀⠀⣀⣽⣿⣤⣄⠀⠀⠀⢀⣶⣿⡿⣿⣭⡐⡄
⠀⠀⢰⠁⡏⠈⠃⠘⡆⣳⠀⠀⠀⡞⠁⠁⠈⢹⡀⠀⠀⢰⡁⠸⠀⠘⠀⣹⠀⠀⢸⠁⢸⠀⠀⢱⠀⢻⠀⠀⢰⢣⠰⠉⢱⠉⣇⠀⠀⠘⢧⠱⠂⢸⠰⢹
⠀⠀⠈⢧⣷⣀⢀⣸⣷⠏⠀⠀⠀⠳⣤⣀⣤⠞⠀⠀⠀⠈⢷⣼⣄⣰⣴⠏⠀⠀⠘⢧⣿⣄⣀⣼⣧⠏⠀⠀⠈⢮⣴⣄⣼⡤⠃⠀⠀⠀⢿⣜⣄⣸⣤⠞
⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈⠈⠉⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⢀⣬⣽⢷⣤⣄⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⢀⣀⣼⣇⣀⠀⠀⠀⠀⡴⠒⡾⠻⡖⠲⡄⠀⠀⠀⠀⣀⣸⣃⡀⠀⠀⠀⠀⣠⡤⡾⢿⢤⣄
⠀⠀⠀⡇⠂⠁⠀⠃⠂⡇⠀⠀⢰⠛⠻⠛⠻⠛⢳⠀⠀⢀⡏⠀⠉⠁⠈⣧⠀⠀⢸⠁⢸⠀⠀⢡⠀⢻⠀⠀⢠⠏⠨⠋⠃⠉⣆⠀⠀⢸⢹⢰⠃⠘⠈⡌⡇
⠀⠀⠀⢧⣄⣄⣀⣆⣤⠇⠀⠀⠸⣴⣐⣄⣜⣼⡼⠀⠀⠈⢧⣤⣄⣤⣤⠟⠀⠀⠘⣦⣿⣄⣀⣼⣧⠟⠀⠀⠘⢆⣀⣄⣆⣤⠏⠀⠀⠘⣾⣌⣇⣸⣰⡴⠃
⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠿⠛⡛⠛⣟⠟⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣄⠀⠀⠀⡀⠀⠀⢀⣴⡿⠛⠛⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⣿⠀⢀⣾⡇⠀⣠⡿⣫⣄⠀⢰⣿⢁⣤⣀⡀⢀⣤⡀⢰⣤⠀⢀⠀⠀⡀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⢠⣿⣠⡟⣿⠀⣰⡟⣼⡏⣸⡇⣼⠇⣾⡏⣿⢣⡿⠁⣿⢸⣿⢴⣿⡧⣾⡟⣼⡏⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣾⣿⠏⠀⣿⢠⡟⠀⣿⡟⠋⣼⣟⣰⣿⠀⢀⣾⣇⣾⠇⣿⠃⣼⡟⢀⣿⢡⣿⠛⠋⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⡀⠀⣸⣿⠃⠀⢸⣿⣾⠁⡀⠻⡷⠞⠻⠿⠛⠿⠶⠋⠹⠿⠋⠀⠿⠂⠿⠇⠸⠿⠋⠿⡶⠶⠃
⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠉⣿⡶⠀⠈⣿⡄⢿⡇⠀⠀⠈⠻⠇⢰⣿⠂⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⢠⣿⠃⠀⣰⣿⢃⣴⣤⣀⠀⣤⢤⣄⣼⡏⣠⡄⠀⠀⠰⢦⣶⣿⡿⢿⡟⠛⠻⠿⢷
⠀⠀⠀⠀⣿⡁⠀⠀⠀⢤⣾⣇⣴⠾⠛⢁⣿⠃⣸⡿⣸⡇⠸⢋⣿⣿⣥⠀⡤⠀⠀⠀⣸⡿⢠⡿⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠙⠳⣄⢸⣿⣴⣿⣵⣿⡇⢀⣼⡟⢸⣿⣰⠃⠀⠀⢀⣿⢃⣿⠇⣼⣇⣴⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⣿⣄⣘⣛⣀⣠⣼⠟⠀⠛⠁⢛⣁⣘⣿⠋⠹⠃⠘⠟⠁⠀⠀⠀⢸⣿⠞⢿⣷⠏⠉⠉⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠟⠛⠋⠁⢀⣠⣶⡾⠟⠋⣩⣿⠃⠀⠀⢀⣷⡄⠀⠀⠀⠈⠁⠀⠀⠀⠀⢠⡀⠀⠀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠉⢀⣤⣾⡿⢃⣀⣀⡀⣸⡿⠀⠀⠀⣀⡀⠀⢀⣤⡀⠀⣸⡟⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠘⠿⠟⠋⢀⣾⠃⣿⢃⣿⡷⣾⡇⣼⡏⢹⢠⣿⠉⣿⢠⡿⠁⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣤⣀⡀⣀⣼⣿⠀⠀⣼⡟⢡⣿⣰⣿⢹⣿⣼⣿⣴⣿⣾⣃⡄⢀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡿⠛⢧⠘⢿⣦⠞⠿⠇⢸⡿⠛⠿⠟⠁⠹⠿⠟⠀⣿⠟⠀⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢯⣤⣶⠟