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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠸⠀⠀⡏⠉⡆⠀⠀⠀⠀⣠⠀⢀⡔⠒⢢⡀⠀⠀⠔⡆
⠀⠀⠀⠀⠀⠀⠀⢈⠣⣄⣘⠤⠴⠃⡴⠁⠀⠀⠀⢰⢈⡦⠊⠀⠀⠀⢘⡤⠈⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠸⡀⢀⡏⠀⠀⡜⠀⠀⠀⠀⢀⠼⢏⡀⠀⠀⠀⠀⠸⡀⢠⠎⠀⠄⠀⠀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡀⠀⢰⠁⠀⠀⢀⣶⠁⠀⠀⢣⠀⠀⠀⠀⠀⠉⠁⠱⡀⠀⠁⢀⡎⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⢸⠀⠀⣠⣿⡿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠞⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠘⡀⢰⠁⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠁⠄⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠱⠉⠒⠒⠒⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠖⠂⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⢱⠤⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠚⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠠⠄⢀⠀⠀⠀⠀⠈⠑⠢⡀⠈⠌⠹⡗⢒⣲⣶⠉⠀⠀⠀⠀⡀⠀⠀⠀⢸⠃⠀⠀⡀⠀⡐⡹⠴⠑⠒⢦⡂
⠀⠀⠀⠀⠀⠀⠦⠂⠉⠀⠀⠈⠑⠂⠌⡒⠄⢀⢈⣿⠿⡿⠀⠀⡸⠀⢠⠁⠀⠀⠀⢸⠀⣄⡬⠔⠋⠁⠀⠀⠀⠀⣸⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠥⢤⣉⡡⠀⣲⣿⠃⠀⡰⠁⢀⡎⠀⠀⠀⠀⡰⠚⠁⠀⠀⠀⣀⠤⡄⠀⢰⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⣀⠀⠈⠉⠘⠛⠁⠀⡰⠁⠀⡜⠀⠀⠀⠀⠘⠀⠀⠀⠀⡠⠊⠁⠀⢣⠀⠸⠜⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣍⠂⠀⠀⠀⣀⠜⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠙⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢤⡠⠚⠁⠀⠀⡜⠀⠀⣀⣀⡀⠤⠄⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣰⠋⠓⠢⠒⠉⠀⠀⠀⢀⠎⡎⠈⡆⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡋⠀⡇⠀⠃⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⠊⠉⠁⠀⡀⡸⠀⠀⠈⠉⠉⡳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢅⣀⡀⠤⠚⠁⠤⠤⠤⠔⠁⠣⢄⣀⡀⠀⠀⢀⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⠶⠶⠶⠞⠶⠖⢶⢄⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠏⠉⡁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠉⠹⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠋⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠐⠉⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠈⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡋⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⢠⠀⠀⠙⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠡⠀⠀⠀⠰⢎⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰⡄⠁⠀⢰⡎⠀⠀⠀⠀⠁⢀⠀⠀⠀⢹⣦⠀⠠⠛⣿⣦
⠀⠀⠀⠀⠀⠀⢀⣼⣿⠁⠀⠀⠀⢀⣿⠈⣄⠀⠀⣤⠀⠀⠀⠀⢀⣈⣷⡜⠀⣾⣧⣄⣀⣀⣦⢰⡘⣧⣴⣴⣿⣧⠠⠆⠀⠘⣿⣄
⠀⠀⠀⠀⠀⠀⣸⠟⠋⠀⠀⠀⠄⢼⣿⡿⢿⠿⡶⢦⡔⠾⠲⣶⠿⢿⣿⠁⠀⢿⡿⠻⠟⠛⠛⡿⡟⠉⠉⠉⣼⣿⠀⠀⠀⠀⠋⣿⡆
⠀⠀⠀⠀⠀⣸⡟⠂⠀⠠⠀⠀⠀⠈⢿⣷⠀⠀⠀⠈⡇⠀⠀⠁⢀⣾⠁⢀⡀⠈⣿⡄⠀⠀⠀⠃⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠸⣷⡀
⠀⠀⠀⠀⢠⡿⢧⠀⠀⠀⠀⠀⠀⠀⠉⣿⢷⣤⣀⣀⣅⣀⣀⣴⡟⠟⠀⠈⠀⠀⠃⣿⣦⣤⣄⣀⣠⣤⡾⠛⠊⠀⠀⠀⠀⠀⠉⢄⣸⡇
⠀⠀⠀⠀⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠋⢏⢿⡿⣿⣯⠀⠀⠀⢰⡀⠀⠀⠀⠛⠀⠉⢭⢿⠏⢻⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⢹⣿
⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠃⠀⠈⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠀⠀⠲⣿⡆
⠀⠀⠀⣾⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡌⠀⣠⣾⣀⣀⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⢀⠀⠀⢠⣿⣧
⠀⠀⢈⡿⠃⠀⠀⠀⠀⠀⠠⠀⡄⠀⠀⠀⠀⠀⢀⣀⣀⣶⣾⣷⣴⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢠⣼⣿
⠀⠀⣿⠃⠀⡀⠴⠁⠀⠀⠀⠀⠀⢀⠀⠀⢰⣦⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠛⠻⢿⣿⣶⡿⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⡄
⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠄⠀⣠⢿⣿⣤⣄⡻⢿⠏⠹⢻⡿⠿⣿⣿⣷⣶⣦⣤⣤⡴⣾⢻⣷⠈⣿⠰⣶⣧⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⣿⣟
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⣿⡟⠹⠃⡀⠙⠀⠈⠙⠘⢀⡏⠉⠙⣻⠏⠁⠀⠀⠈⠁⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⠆
⠀⢼⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⢰⠃⠘⠀⠀⠀⠀⠜⠀⠀⣼⠁⠀⢠⠀⠀⡇⠀⠰⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣿⡆
⠀⣾⡟⢀⡉⠀⠀⠀⠀⠀⠄⠀⠀⣼⣿⡄⡄⠀⠀⡈⢀⡒⠀⠠⠀⠀⠠⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⣸⣿⠕⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣈⣿⡇
⠀⣿⡁⣼⠟⠀⠀⡀⠀⠀⠀⠀⢦⣽⣿⣿⠃⢀⠘⠃⠈⠀⠀⠀⢀⣾⣷⣿⣖⣀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣀⠀⠀⠀⠀⠀⠙⠀⢠⡀⢎⠁⣿⡇
⠈⣿⡟⠉⠠⠀⠀⠀⠀⠀⠘⢀⠀⢾⣿⡿⠀⠈⠀⠀⠀⠀⠀⣤⣿⣿⣿⢷⣿⣿⣿⣤⠀⠀⠀⠀⠀⢿⣿⣇⠀⠀⠀⠀⠀⠶⠰⠄⠀⡘⢉⣿⡇
⠀⣽⣯⡖⠀⠄⠀⠀⠀⠀⠀⠀⠀⣹⣿⣷⣀⠄⠀⠀⠀⠀⣰⢈⣿⠋⠀⠀⠀⠹⢿⣵⡏⡀⠀⠀⠀⢸⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣯⣹⡇
⠐⢻⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⠃⠀⡀⠀⠐⢠⣿⣿⠋⠀⠀⠀⠀⠀⠘⣿⣗⣠⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⡀⠀⠀⠀⠈⢎⣽⣿⡇
⠀⢸⣿⣿⣃⡶⠀⠁⠀⠀⠀⠀⠀⢈⣿⣿⡗⠀⠁⠀⠀⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⣿⣿⠆⠀⠀⠀⠋⠀⠀⠀⠀⣴⠷⣽⡇
⠀⢸⣿⣿⠁⡇⣀⠈⠀⠀⠀⠀⠀⠈⢹⣯⡇⠈⠀⠀⡄⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⣟⡁⠀⠀⠀⠀⣿⡇⠆⠀⠀⠃⠀⠀⢀⠀⣰⣬⠶⣿⡇
⠀⠸⣿⣷⣞⢁⡈⠰⡀⠂⡀⠀⠀⠀⠘⣿⣧⠀⠀⠐⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣃⠀⠀⠀⠀⣹⡗⣀⡀⠀⠀⠀⢠⠀⣠⣧⠘⣇⣿⠃
⠀⠀⣿⣿⣿⣼⡟⣶⠱⣤⢿⡀⢀⢀⣿⣿⣿⠀⠀⠀⣾⣬⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠈⠁⣿⡟⣏⣣⠤⡆⢀⣠⠘⣽⡟⣷⣾⣿
⠀⠀⢸⣿⣿⣧⣼⣿⣷⣬⣷⣾⣾⣿⠿⠛⠿⡿⠃⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⣿⣿⣿⣫⡚⣁⣬⢁⣾⣿⣷⣿⣷⣿
⠀⠀⠘⠻⣿⣿⡿⠛⢻⡟⠋⠛⠉⠀⠀⠀⠀⠀⠁⢀⠐⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⢀⡿⠋⠙⠿⠿⠿⠿⠛⢿⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠈⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡋⡄⢀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠇
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠅⣺⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡤⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠀⠙⠀⠀⠸⠁⠀⡀
⠀⠀⠀⠀⠀⠈⣷⠀⢀⠀⠀⠀⡀⠀⠠⡄⠀⢸⣯⣽⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣷⠁⣳⡆⠀⠀⠐⠋⠀⠰⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠻⣄⠀⡀⠀⠀⠀⠀⠀⠱⠀⣸⢿⣿⣿⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣌⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠈⢈⣿⣷⣶⣾⣷⣶⣾⣿⣷⣾⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣯⢐⣁⣤⣤⣆⠀⢀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⣀⡴⠟⠉⠉⠉⠁⠈⠁⠎⠁⠁⠉⠉⠉⠿⠛⠛⠀⠀⠀⠀⠀⠀⣠⡿⠿⠛⠟⠙⠻⠛⠛⠟⠻⡟⢿⣿⠻⡿⣿
⠀⠀⠀⣀⠴⠛⠁⡠⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⣆⠀⠀⢠⡿⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠐⠋⠀⢀⣠⡄⠐⠛⠉⠁⠈⢢⡀⠀⠉⠃⠀⢸⣇⠀⢀⣾⠻⢦⣄⡀⠀⠀⠠⣦⡀⠰⠄⠈⠣⠀⠈⢢⡀
⠢⣀⠀⠀⠀⠀⢀⡄⠁⠀⠀⠀⠀⠀⣠⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠂⠀⠈⢿⡄⠀⢻⣄⠀⠉⠛⢾⣧⣀⠈⠙⢦⡀⠑⡄⠀⢦⠀⢳⡀
⠀⠉⠀⣀⠠⡖⠀⠀⢠⣶⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⠘⠀⠀⠀⠀⠈⠻⣦⣄⣈⣷⠀⠀⠀⠀⠉⠻⣶⠀⠙⠆⠐⠀⠠⣆⢈⣧
⠀⠘⠉⠀⢰⠁⢀⡰⢟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢻⣄⠀⣾⣀⣤⠀⣿⣄⡽
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣀⣼⣤⠿⠀⠋

⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⠂⡀
⠀⠀⠀⠀⠀⠀⠀⡀⢀⡄⠀⠂⢄⠀⠀⠀⡠⠐⠀⠀⠀⡀⠀⠀⢀⠀⠁⠂⠄⡀⠀⢀⠄⠀⢀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠢⢀⢀⠖⠀⠀⠀⠠⠂⠁⠀⠀⠀⠀⠈⠢⠀⠀⠀⠘⢄⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠁⠀⠀⣠⣤⣀⠀⠀⠀⠁⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠐⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⢸⠀⠀⠀⠀⡀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑⠄⠂⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠁⠀⠀⠀⠀⠀⠘⠄⠀⠀⠨⡄
⠀⠀⠀⡠⠠⠀⠀⠀⠀⠀⠯⠸⠀⢈⠁⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⢢⢀⡒⠃⠀⠀⠀⠀⠀⠒⢄
⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠘⠜⠀⠀⠀⠀⠌⠀⠀⡘⡀⢀
⡀⠁⠈⠘⢄⠀⠀⠀⠀⡠⠄⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠐⠠⢄⠀⠀⠀⢀⡴⠉
⠣⡀⠀⠀⠀⠁⠀⠁⠁⡀⠔⠁⡀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠈⠢⢄⠀⠀⠀⠀⠀⢀⣀⡠
⠐⠈⠉⠁⢀⠠⠀⡀⠊⠀⢀⠠⠀⡀⠀⠀⠀⠑⢠⢀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⠀⠀⢠⠀⠄⠀⠀⠁⡢⠄⠀⢌
⠀⠀⠀⠐⠃⠀⠀⠀⠁⠀⠀⠃⠀⠁⠀⠀⠀⠀⠆⠀⢀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠆⠀⠈⠀⠀⢀⠈⡃⠐⠒
⠀⠀⠀⠀⠁⠉⠐⠒⠠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠒⠚⠈⠨
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠄⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠄⠠⢄⡀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠐⠐⠘⠁⠀⠼⠀⢈⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢐⠀⢸⠄⠀⠉⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠢⠠⠈⠀⠀⠀⠠⠀⠀⠁⠂⠠⠀⠀⣀⡀⢀⣀⡀⠀⠀⠄⢖⠁⠠⠀⠀⠀⠀⠈⠢⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠡⠀⠐⠲⠆⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⣰⠴⠀⠠⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣄⡀⠀⢀⣀⠀⠐⣆⠀⠀⠀⠀⡄⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠟⠀⠈⢲⠶⠾⠙⠚⠇⠉⠛⠷⢶⠟⢦⣄⡴⠾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠃⠀⠀⡠⢤⣤⣤⣄⠀⠀⠀⠀⠀⢀⡠⢤⣤⣤⡀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⡜⠀⠘⢿⢻⣿⣧⠀⠀⠀⢀⠊⠀⠻⣟⣻⣿⡆⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢇⠰⣶⣿⣿⣿⡿⠀⠀⠀⠐⡄⠰⣶⣿⣿⣿⡇⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠈⣿⢀⠀⠀⠈⠂⠬⠭⠭⠚⠁⠀⢠⣄⠀⠈⠢⠬⠭⠽⠋⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠎⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠀⠀⠣⡟
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠸⣿⡉⢱⡒⠒⢲⣶⣶⣶⣶⣶⡖⠒⢹⡅⣸⡇⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⡀⠀⠀⢻⣷⣿⣷⣤⣿⣿⣿⣿⣿⣿⣷⣤⣿⣿⡿⠁⠀⠠⡆⠀⠀⠳⡀
⠀⠀⠀⠀⠀⢠⠃⢀⠀⢀⡇⠀⠀⠀⠙⠿⣿⣯⠟⢿⠁⠀⢀⡝⢻⣽⢿⠟⠁⠀⠀⠀⣇⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠙⠼⠦⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠈⠁⠀⠀⠀⠀⠀⢹⠧⠾⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣸⣠⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣫⣉⣩⡀⠀⢸⣄⣀⣀⣠⣾⠀⠀⠀⣩⣉⣍⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠷⠾⠶⠾⠶⢾⣿⣿⣿⣿⣿⢿⠷⠾⠷⠾⠶⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⣶⣶⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣴⣿⣿⣿⣶⣤⣤⣤⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣦⡀
⠀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣃⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣟⡀
⠈⠉⠀⠀⠈⠉⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠈⠛⠋⠙⠛⢿⣿⣿⣿⣿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟

⠀⣀⡀⠀⡴⡛⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢆⠀⠀⠀⠀⠀⣠⢶⡄
⠸⡉⢧⡠⠗⠛⠉⠁⠀⠀⠀⣠⠴⠛⠓⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡄⢀⡴⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠈⣇⠀⠀⠀⢠⡏⠀⡇
⠀⠉⢻⠀⠀⠀⡞⢹⡆⠀⢼⡁⢈⡥⠖⡃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢰⠇⠀⢹⡟⠀⢸⣴⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢿⣠⣤⡤⠼⠃⠀⣇
⠀⠀⢸⠀⢠⠶⢬⡈⢀⣀⣨⣤⣄⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⣰⠋⠙⠶⠚⠀⠀⠈⠀⠀⠈⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⢸⠀⢾⠀⠀⠙⢯⣹⣇⡼⡆⣾⣨⠇⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢴⠋⣶⣊⡶⠀⠠⡆⢸⣷⡇
⠀⠀⢸⠀⢻⡄⠀⠀⢀⣤⡈⣀⠙⣄⠀⣤⡄⠀⠀⠀⠀⠀⢀⡇⣠⠔⠒⣒⠊⢳⠆⢠⡖⡴⢒⡦⣼⣀⡀⠀⠀⠀⠀⠀⠀⢸⠐⣋⡷⣆⣶⡖⢰⣖⣹⣏⣹⡃
⠀⠀⢸⡄⠀⠙⠂⠶⠯⢼⣷⣼⣯⠼⠾⠴⡟⠀⠀⠀⠀⠀⢸⡇⠙⠓⢲⣟⢒⡉⢀⣬⡉⢉⣭⡀⣿⠃⣻⣀⡀⠀⠀⠀⠀⢸⡆⠙⠂⠚⠣⠼⠧⠼⠥⠿⠖⠳⡄
⠀⣿⣦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣖⡽⠀⠀⠀⢸⠇⠈⠛⠛⠛⠛⠛⠛⠒⠛⠛⠒⠛⠛⠉⠉⠉⠹⡆⠀⠀⠀⠀⣧⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⡴⠁
⠀⠈⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡞⠁⠀⠀⠀⠀⠀⢺⣄⣄⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⣜⣥⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⡃
⠀⠀⠀⠀⠀⠈⠙⡆⢰⠒⢲⢶⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⣯⡌⠉⠉⢹⣀⡟⠛⠛⢇⢿⠻⣼⠛⠻⠇⠀⠀⠀⠀⠀⠳⣀⣶⠶⠶⠶⠶⢶⢞⠇
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⣠⢴⡆⠀⢀⡤⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⡰⠁⢌⡇
⠀⠀⠀⠀⣼⠁⢸⣃⡞⠉⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⢡⣤⠈⢳⡤⠷⠶⠾⠤⣄⡠⢠
⠀⠀⠀⠀⣿⠀⠈⠋⠀⠈⠓⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⣠⡴⠁⠀⠀⠀⠀⠀⠀⠉⠺⣃⣆
⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⣠⢤⠤⡙⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⣧⠰⠽⢤⣎⣵⣔⣶⢲⡤⢻⡜
⠀⢀⡏⠀⣖⢳⣾⢛⣦⠰⣅⣙⣡⠇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠉⠛⠓⢻⠆
⠀⢸⠁⢀⡈⢉⣍⠉⣁⣄⢀⣄⠀⣠⣸⠇⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⢸⠀⠪⢵⣃⣸⣞⣁⣸⣏⣸⣟⣠⣿⣤⠀⠀⠀⠀⠀⠀⠀⢻⡘⣆⠀⠀⢀⡇⠀⠀⡜⣦⠀⠀⠀⠀⠘⢇⠀⠀⢀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⣸⠆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡾⠤⡄⠀⠀⠀⠀⠀⠈⣧⠈⠳⠤⠞⠁⠀⢀⡇⣿⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⡞⢁⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠓⠒⠋⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⢀⣠⠞⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠉⠉⠀⠀⠑⡆⢀⣀⣀⣀⣀⢤⠀⢞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⣇⠈⣦
⠀⠀⠀⠀⠸⣥⡟⠀⠀⠀⠀⠈⠣⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⣸⠛⠒⣤⢤⣀⣀⣀⣀⣀⠤⠴⠾⣇⠀⡵⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠘⢧⡤⠇⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⢠⣄⠀⣠⡴⠟⠻⠶⢦⡤⠴⢦⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⢀⣖⣣⣖⠲⠒⣚⢛⣙⠛⠛⠛⠛⠒⠶⣶⢞
⠀⠀⠘⢮⡛⠁⣠⠤⣄⠀⠸⣅⣋⡼⢧⣀⡠⢶⡼⠉⣷⣄⠀⠀⠀⢀⡴⠒⠓⢶⠖⠛⢻⠏⢀⡀⠀⢹⡆⠀⠀⠀⠀⠉⠀⡸⠀⣞⠹⠗⣈⡇⢰⡏⠴⢛⡆⢸⠂
⠀⠀⠀⢸⡇⠀⢧⣩⠆⠀⠀⣠⠶⣤⣎⣹⡧⠼⠧⠴⠃⠸⡇⠀⠀⢸⡁⠾⠇⣸⣄⠀⢸⡄⠙⠁⢀⣼⠇⠀⠀⠀⠀⠀⠀⡇⠀⢈⣙⣉⣁⣀⣀⣙⣓⣋⣠⠼
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠛⠒⠚⣡⣼⣠⡄⢉⣷⠚⠋⠁⠀⠀⠀⠀⠀⠀⢸⠇⠀⠫⠿⢥⠿⠬⠿⣼⠿⠟
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⣇⣹⣇⣸⣇⡼⠀⠀⠀⠀⠀⠀⠀⣠⢖⣉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⢤
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠀⠀⠀⠀⠀⠀⡞⠉⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠻⠋⣽⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⣾⠇
⠀⠀⠀⠀⠈⠓⠖⡶⢶⡄⢰⠖⠚⢲⡆⡞⠛⣻⢻⠏⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⡇⡇⢸⠄⢸⠀⠀⠸⣇⡇⠀⡇⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⣤⣤⡸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡞⠒⠒⢲⡖⡟⠁
⣀⣀⣀⣀⣀⣀⣻⣁⣈⣷⣋⣀⣀⣀⣉⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣷⣟⣀⣳⣗⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣁⣀⣀⣈⣿⣃⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢾⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣴⠿⠿⠿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⢿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢦⣄⣀⣽⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣛⣿⣿⣿⣿⣧⠀⠀⠀⢀⡀⠀⠀⠀⠙⠓⠺⠛⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢻⣿⣧⠀⢰⣿⡃⠀⠀⠀⠀⠀⠀⢸⣷⣹⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⣿⣿⣧⣶⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠙⠿⣿⣿⣿⣿⣿⣿⡿⠟⠁⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢰⡇⠀⠉⢻⣿⠋⠁⠀⣾⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⣦⣄⣡⣤⣾⣿⣿⣿⣦⣤⣈⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠁⣿⠃⠉⠘⡿⢿⡿⣿⡿⢿⡟⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⡟⠀⠀⠀⠀⠈⠀⠘⠁⢸⠃⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⢠⠀⣠⠀⣸⣇⠀⣠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣿⣶⣿⣶⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠘⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣮⣽⣿⠇⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠞⠺⠿⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠐⣢⡤⡤⢖⡀⣤⠊⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⡠⠚⠐⠈⠀⠀⠀⠀⡨⠃⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⢘⠣⠈⠀⠤⡠⢤⡀⠀⢸⠁⠀⠀⣨⠓⡜⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⡍⠤⠀⣴⣿⣿⣷⠀⠀⠀⠙⠂⠈⠑⡀⢼⡆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⣼⣷⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠓⢚⢱⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠻⠏⠀⠻⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢡⣄⠘⢿⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠒⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠘⡇⣹⣄⢇⢀⠵⠠⢄
⠀⣰⠒⢮⠉⠢⡀⠀⠙⠒⠂⠤⠤⠤⠤⠄⠠⠤⠤⡤⢤⣴⣶⣶⣿⠃⣎⢈⡇⢳⣇⠀⣹⢌⠒⡠⠞⢢
⣠⠧⣤⠼⠃⡴⠛⠙⡆⠄⣀⠴⠀⠀⠀⢆⣀⣀⡄⡇⢸⡿⢿⠟⢩⠀⠈⠉⢀⡉⡞⠉⠁⠘⠀⠓⢀⡰
⢇⠀⣨⠇⠀⠳⠊⡜⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⢠⠁⡘⠀⠈⣠⠆⠀⠀⠀⣠⠞⠀⠀⠀⠀⡖⠒⠉
⠈⠓⠒⠆⠂⠀⠀⠘⣄⠀⠀⠀⡀⠀⠈⠑⠒⠰⠆⠴⠠⠔⢊⡥⠀⠀⢀⠔⠁⠀⠀⠀⠀⢰⠃
⠀⠀⠀⢠⠀⠀⠀⠀⠈⢦⣠⢎⠀⠤⢄⣀⠀⠀⠀⠀⠒⠊⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠘⣄⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠈⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢱⢸
⠀⠀⠀⠀⠈⢆⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣀⣐⢏⢱
⠀⠀⠀⠀⠀⠘⣦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠧⠜⠘⡾⣀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠰⣒⡆⠀⣠⠇⢵⣈
⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠦⠒⢇⢳
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠢⠖⠀⡸⣖⢤⡀
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠣⡄⢀⠈⠉⠦⣿⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣤⠅⠘⠛⢀⠤⢈⠢⢄⣇⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠸⣧⠇⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠠⠄⠈⠈⢉⠐⣄⡏⠉⠳⠬⣇⣠⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⣀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡞⠉⠀⠑⠄⠀⠀⠀⠀⠤⢠⠒⢳⠑⠂⠂⠈⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠛⠛⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠤⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠀⠉⠀⣀⠤⠽⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠑⠲⠤⠤⠤⠴⠶⠴⠶⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠐⠄⠂⠀⠔⣉⠔⣢⠄⠒⠬⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣨⠐⠒⠒⣪⡗⠲⠅⢤⡀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠇⠉⠓⠁⢆⡮⠤⠤⠤⠔⠒⠋⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡀⠀⡏⠳⣄⠀⡠⢆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠒⢄⣘⣠⣱⣮⣤⣦⣬⣎⣥⣘⡠⠞⢹⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⢒⣾⠼⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⢼⡤⠒⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣮⠔⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⢧⡎⠤⠴⡆
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠐⠒⢄⠀⠀⠀⠀⡇⠀⠀⠻⣎⣰⠥⡆
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣤⠀⠀⡇⠀⠀⢰⡇⠀⠀⢠⡌⢳⣠⣃⢴⠀⠀⠀⠀⢀⡠⢴⡄
⠀⠀⠀⠀⠀⠀⢸⠃⠀⡔⠒⠐⠢⡀⠀⠀⠀⠀⠀⠣⡀⠀⠀⢀⠜⠁⢀⡴⠫⠥⠀⠀⠀⢰⣶⠙⢾⣼⠔⣆⡀⢠⠬⢧⡟⡇
⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⣴⠄⠀⡁⠀⠀⠀⠀⠀⠀⠈⠉⠉⢁⣠⠶⢉⠀⠀⠀⠀⠀⠀⠀⢩⣤⡛⣙⡳⣯⣬⣮⣴⠞⡅⡃
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠣⠄⠠⠔⠁⠀⠀⠀⠀⠀⣀⣠⢴⠚⠋⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠛⢡⠛⣡⣷⣤⠁⢻⡃⠀⡇
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⣀⣀⡤⡴⢖⢛⠉⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⢈⣛⡋⣤⡝⠁⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠉⠉⠙⠭⠁⠑⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠈⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣄⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣀⣠⠞⣰⠶⠒⠲⠶⣤⡀⠀⢀⣀⣠⣤⠤⢤⡀⠀⡟⠁⠀⢈⡇⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⡼⠁⠀⠀⠀⠀⢈⡇⣼⠋⠁⠀⠀⠀⢀⡷⠀⢹⡀⠠⣯⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣎⣒⣛⣀⡿⠀⠀⠀⠀⣶⣋⣡⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠤⠄⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠠⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣀⣀⣀⣀⣴⠾⠳⠞⢋⣉⣥⣴⣶⣤⣉⠛⢶⣤⡀⠀⢀⣴⠾⠛⠻⠷⢶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠟⢛⣉⣉⣥⣤⣤⣌⣉⣉⡉⢉⣡⣶⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣆⡘⢷⡶⠟⣡⣶⣿⣶⣶⣄⠛⣷
⠀⠀⠀⠀⠀⠀⢠⡾⠋⠤⠶⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢉⣉⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⠛⣩⣤⣴⣾⣿⣿⣿⣿⣿⣿⡇⠻⣦⡀
⠀⠀⠀⠀⠀⣰⣿⣤⠶⠶⠳⠶⣤⣤⡈⢻⣿⣿⣿⣿⣿⣿⣇⢸⣏⣸⣧⢸⣿⣿⣿⣿⣿⡿⠋⣈⣛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⡀⢾⡅
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⣇⠘⣿⣿⣿⣿⣿⠋⣡⣼⡿⢿⡁⢸⣿⣿⣿⣿⣿⡇⣾⡏⠙⠻⠶⠶⣦⠘⣿⣿⣿⣿⣿⢁⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠗⢈⣿⣿⣿⣿⠐⢿⡄⠀⠈⢿⡄⢹⣿⣿⣿⣿⣇⠻⣇⠀⠀⠀⢸⡏⢠⣿⣿⣿⡟⢃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧⢈⣿⣿⣿⣿⡗⣸⡗⠀⠀⣾⡃⣼⣿⣿⣿⣿⡿⢠⡿⠀⠀⠀⢸⡇⠸⣿⣿⣿⠃⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣾⣿⣿⣿⡏⢰⡟⠀⠀⠀⢹⣧⢸⣿⣿⣿⣿⣿⠘⣷⠀⠀⣴⡟⢡⣾⣿⣿⡏⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢰⣿⣿⣿⣿⣷⠘⣷⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⠇⣻⡆⠀⠘⣷⡄⢻⣿⣿⡇⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡈⢿⣿⣿⣿⣿⠀⣿⠀⠀⠀⢾⣧⡈⣿⣿⣿⣿⡇⢸⡏⠀⠀⠀⣿⠀⣿⣿⣿⣷⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡈⣿⣿⣿⣿⣇⠹⣧⠀⠀⠀⣿⠀⣿⣿⣿⣿⡁⣿⠁⠀⠀⠀⢻⡄⠻⣿⣿⡏⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠸⣿⣿⣿⣿⡦⢹⡷⠀⢠⣿⠰⣿⣿⣿⣿⣇⠹⣧⠀⠀⠀⠈⣿⠀⣽⣿⣇⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⡿⢡⡾⠁⠀⠀⢻⣆⢹⣿⣿⣿⣿⡆⣿⡀⠀⠀⠀⣿⠀⣿⣿⣿⣆⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢠⣿⣿⣿⡧⢸⡷⠀⠀⠀⣰⡟⢠⣿⣿⣿⡟⢡⡿⠀⠀⠀⠀⢻⡆⢹⣿⣿⣿⢈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣧⡄⣽⣿⣿⣧⠘⣷⡀⠀⠸⣿⠀⣼⣿⣿⡟⢠⡿⠁⠀⠀⠀⠀⢸⣷⢸⣿⣿⠇⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⢁⣿⠀⠀⣸⡏⢰⣿⣿⡿⢡⡿⠁⠀⠀⠀⠀⠀⢸⣿⢸⣿⣇⠸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢁⣾⣿⣿⡇⣸⠇⠀⢀⡿⢀⣿⣿⣿⣧⠸⣧⠀⠀⠀⠀⠀⠀⢸⡇⠘⣿⣿⡆⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡁⢾⣿⣿⣿⢀⣿⠀⠀⠘⣧⡈⢻⣿⣿⣯⠸⣯⠀⠀⠀⠀⠀⢀⣼⠇⣸⣿⡏⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⠀⣿⣿⣿⠘⣿⠀⠀⠀⣾⠃⣼⣿⣿⣿⢸⣏⠀⠀⠀⠀⠀⠸⣧⠐⣿⣿⣧⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠆⣹⣿⣿⡆⢻⡆⠀⠀⢻⡇⢹⣿⣿⡏⣠⡿⠂⠀⠀⠀⠀⣠⡿⠁⣾⣿⡏⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢹⣿⣿⠄⣿⡀⠀⠀⠸⣿⢠⣿⣿⡇⣼⠇⠀⠀⠀⠀⠀⢿⡀⢾⣿⣿⣇⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣷⡘⣿⠀⠀⠀⢼⡏⢰⣿⣿⡄⢿⡀⠀⠀⠀⠀⠀⢘⣷⠈⣿⡿⢋⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢹⡿⢰⡟⠀⠀⠀⠈⣿⠈⣿⣿⣷⡌⢻⡄⠀⠀⠀⠀⢸⣇⠸⣿⠃⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠘⣿⠘⣷⠀⠀⠀⢸⡏⢰⣿⣿⣿⢡⣾⠁⠀⠀⠀⠀⠀⣿⠈⣿⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠹⡇⢻⡆⠀⠀⠈⢻⣄⢹⣿⡟⢸⡇⠀⠀⠀⠀⠀⠀⣿⣀⣡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠘⠇⢹⡇⠀⠀⠀⠀⣿⠀⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠸⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠾⠇⠀⠀⠀⢠⡿⢀⡿⠃⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠘⢡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⢼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢹⣷⡀⠀⣀⡴⠖⠛⠉⠉⠉⡉⠛⠶⣤⡀⢀⣠⣾⡟⠀⡠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠻⣷⣾⠋⢀⠖⠉⠉⠀⠀⠀⠀⠀⠈⠙⢿⣿⡕⠁⡸⣱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣆⠁⠘⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠹⣿⣔⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⢦⠈⢦⡉⢻⣿⡄⠀⠀⠀⢀⣠⢤⡀⠀⠘⠀⠙⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢦⡑⠀⢈⣿⡿⢠⠞⠋⣉⣉⣀⣀⠈⠘⢉⠉⠢⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡌⠻⣶⣿⡿⢁⠁⣴⣿⣿⣿⣿⡟⡁⢈⢿⣿⣶⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡁⠀⣈⠉⠀⢺⠀⣿⣿⣿⣿⠟⠉⣼⣿⡾⣿⣿⣿⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣴⣿⠿⣶⣶⣦⣈⣀⣀⡠⠄⢰⣿⢿⡇⡙⠿⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⣿⣿⣿⣧⠀⠀⠀⠀⠁⠀⠁⢹⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⠀⠙⢇⠉⢻⣦⣜⣠⣠⡃⣰⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⣷⣶⣤⡀⠃⠸⠁⡟⢹⠉⡏⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⢀⣀⠀⠀⣀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡾⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡖⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⢛⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢬⢷⡌⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⠴⣛⡵⢋⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠧⢈⠻⣵⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⣔⣋⠀⠺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣡⠞⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢧⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⡜⠁⡼⠁
⠀⠀⠀⠀⠀⠀⡔⢾⢹⡦⢤⡀⠀⠘⡆⠀⡇⠀⠀⠀⣀⠀⠀⣸⠀⢰⠃⠀⠀⠀⢀⡴⢦⣤
⠀⠀⠀⠀⠀⢺⣙⣀⠁⠰⡞⠀⠀⠀⣧⠀⠉⠒⠋⠉⠉⠉⠛⠁⠀⠸⣄⠀⠀⠀⢯⠁⢉⣘⡧
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠈⠲⢄⣰⠃⠀⣀⣀⣀⣀⣀⣠⠔⢲⠒⢤⠀⢣⡤⠖⠋⠀⡼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⢀⠎⠙⠥⠳⠜⠓⠚⠑⠉⠀⠀⠇⠀⣀⡤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⡀⠀⢠⠔⠒⢴⠋⠉⢢⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠳⡀⢸⡀⢀⡀⣠⣠⠺⣄⠜⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠙⠾⢭⠯⠜⠓⠛⠚⠁⠀⠰⢯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠓⠀⢀⣀⣀⣀⣀⣠⠴⠶⢤⣀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢠⠏⠈⠉⠉⠀⠀⠀⠀⠀⠈⢳⡀⠀⢱⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢌⡳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⡉⠉⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠤⠴⠖⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣤⣤⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⢿⣿⣿⡟⢻⣿⣿⡿⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠘⣿⣿⠄⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠀⠀⣿⣿⠃⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠇⠀⢿⣿⡄⠀⢽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢀⣿⡟⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠘⣿⡇⠀⠀⢼⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⣿⡇⠀⠀⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢿⡇⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠈⢠⡄⠾⠃
⠀⠀⠀⠀⠀⣏⡣⣤⡯⠑⡎⣽⢸⡏⡖⢩⡒⠼⡇⢎⣻⡾⡿⠛⠉⠛⠓⡏⣟⡿⢺⢹⢻⡕
⠀⠀⠀⠀⠀⠿⠑⠚⠀⣠⠘⢮⣸⠗⠁⠿⠀⠱⠇⢤⠴⡡⠃⠐⢤⠀⠀⠧⠧⠴⠿⠄⠑⣕⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢨⢤⣤⠈⢩⠤⡤⠄⠀⠤⣤⠄⢉⠤⡤⡄⠀⣤⡤⣄⠠⣤⢤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠛⠀⠘⠋⠒⠃⠀⠒⠛⠃⠈⠒⠒⠓⠀⠋⠓⠊⠀⠘⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠐⠒⠃⠀⠀⠒⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠱⣄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣀⡤⠂⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢄⣀⣀⡀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⣠⠋⠑⢢⡎⣩⢙⠉⠒⠒⠠⠤⢄⣀⡀⠀⠀⠀⣀⠄⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠨⣆⠀⠀⢳⣀⢻⡾⠳⠼⠛⣠⠒⠉⠙⢦⡀⠀⠉⠉⠉⠉⠁⢀⡞
⠀⠀⠀⠀⠀⢠⠖⠒⠄⠲⣀⠀⠀⣀⣸⡄⠀⠀⢈⠛⣡⠀⣤⠰⡇⢾⣯⠀⠀⡇⠀⠀⠀⢀⡠⠖⠉
⠀⠀⢠⠖⠉⢯⡀⢀⡈⢢⡀⠉⠉⠁⠀⢷⠀⡀⠘⢀⡠⠤⠤⢄⡛⢦⣄⣤⣼⠅⢀⡤⠾⢉⡀⠀⠠⠀⠀⣀⣤⡀
⠀⢠⠃⠀⠀⠰⡍⠉⠀⠀⠙⢆⠀⠀⠀⢸⢠⠀⡴⢋⢀⣀⣀⣸⡗⣄⠀⠀⠀⢠⠻⢤⣀⣀⡠⠔⠁⡔⠁⠀⠀⢠⠄
⠀⠸⣄⠚⣖⣠⢇⡀⢘⡶⡄⢸⠀⠀⠀⢸⣬⣾⣿⣿⣿⣿⣿⣿⣷⣾⡄⠀⠀⡿⠀⠀⠀⠀⠀⠀⢰⠁⠀⠐⠒⠛⢆
⠀⠀⠈⢠⠉⠑⠇⠑⠋⠀⡇⠘⡀⠀⠀⣿⡿⠇⠀⠀⠠⠤⢄⣹⡟⠻⢷⡆⢠⠻⣄⠀⠀⠀⢀⠔⠁⢀⠄⣠⠖⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠱⣀⣀⡟⠁⠀⠀⠈⠛⠒⠂⠀⠉⠒⢦⡇⡼⢦⡀⠉⠉⠉⠀⢀⡴⢧⠀⠓⠤⡤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠈⣼⠁⢀⡴⢦⠀⠀⢀⡤⢄⡀⠀⢸⡇⠱⡀⠈⠙⠒⠒⠋⠉⠀⠀⠉⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡨⠿⠛⠁⣠⠞⠀⢸⠀⠀⡞⠀⠀⠙⡆⠈⠐⠆⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢀⡤⠖⠋⠁⠀⠀⠸⠀⢠⠃⠀⠀⠀⠈⢓⡀⠀⢇⡀⠘⢃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡏⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠈⡇⠀⢸⡙⡄⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠸⡄⠀⠀⠀⠀⠀⡇⠀⠘⡇⢣⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⢧⠀⠀⠀⠀⠀⡇⠀⢀⡇⠘⡆⠈⢆
⠀⠀⠀⣀⣤⠤⣄⣀⡼⠀⢀⣏⣀⠀⠀⠀⠀⠀⢹⡀⠀⠘⡆⠀⠀⠀⠀⣧⠀⢸⡇⠀⠘⣆⠈⠣⡄⡀⠤⠤⠤⠤⣄
⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⣀⠤⠤⠤⠧⠀⠀⢻⣀⣀⣀⡤⠼⠀⠈⠧⢄⣀⡼⠃⠀⠀⠀⠀⠀⢀⠀⣀⡇
⠀⠈⠳⠶⠚⠲⢤⣤⣤⠶⠤⠤⠶⠋⢰⡅⠀⢠⡄⠀⠀⠀⠐⢦⠀⣀⣷⢀⡗⠀⠀⠀⠐⣽⠄⠤⠦⣀⣮⠴⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠙⠲⠴⠧⠤⠞⠉⠁⠈⠉⠳⢤⣀⣄⣤⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣠⣤⣴⣶⣦⣀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠿⣿⣿⣿⣿⣿⠟⠛⠛⢿⣿⣿⣿⡿⢂⣤⣤⣶⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠛⠁⠀⠀⢸⣿⣿⣿⠁⠀⠈⠉⠉⢻⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⠀⠀⠀⠀⠈⣿⣿⣿⡀⠀⠀⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⠀⠀⠀⠀⠀⣻⣿⣿⠅⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡆⠀⠀⠀⠀⢻⣿⣿⡄⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡗⠀⠀⠀⠀⠘⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⣸⣿⠏⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⢸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣿⣿⠆⠀⠀⠀⠀⠀⢸⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⣀⠀⠀⠀⢀⡀⠀⢀⠀⢰⡀⠀⠀⢀⡀⠀⠀⡦⠀⠀⢀⣀⡀⠀⣀⠀⣀⣀⣀⠀⢀⡀⣀⡀⠀⣀⣀⣀⡀
⠀⠀⣿⢷⡀⡰⢋⡟⠀⣿⡉⣽⠉⢻⠂⢸⠛⢦⣄⡇⠀⠻⠯⠭⣭⡈⠁⠈⡿⠀⠉⠀⣿⡅⡌⠃⢸⣯⢤⣾⠃
⠀⠀⣿⠈⠛⠁⠘⣿⠀⠘⢷⣾⡤⠟⠀⢸⠀⠀⠙⡧⠀⣀⠀⣰⡟⠀⠀⠸⡗⠀⠀⠀⢿⣄⣀⣤⠸⡿⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠀⠀⢠⣄⢠⡀⠀⠀⣤⡤⠀⠀⢠⡤⣤⠀⠀⢠⡤⢤⠀⠀⠠⣄⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠀⠀⠘⠈⠙⠃⠀⠀⠛⠒⠀⠀⠘⠉⠛⠀⠀⠘⠒⠚⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢆⠀⠀⠀⠀⠀⠰⣷⣄⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣠⣾⠀⠀⠀⠀⠀⠀⡰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡜⢣⣄⠀⣠⠴⠒⢧⡬⢻⡍⠉⠁⠀⠀⠀⠀⠀⠈⢩⡛⠥⡼⠓⠦⡀⠀⣠⡞⢇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣠⠇⡞⡹⠁⠀⠀⠘⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠁⠀⠀⠘⡏⢧⠸⣈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣁⡇⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠉⠉⠙⢢⡀⠀⠀⠀⠀⠀⠀⠀⢱⣸⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⣿⣿⡄⠀⢱⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠉⠉⠀⠀⡜⠀⠀⠀⠀⠀⠀⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠤⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢸⠀⠀⠀⠀⠀⠀⠈⠑⠦⢄⣀⣀⣀⣀⣀⠤⡖⡏⠀⠀⠀⠀⠀⢸⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⢸⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⡟⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⢀⠏⣧⣀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠉⠁⠀⠙⠲⠤⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠔⠋⠀⠈⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠁⠀⠀⠈⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠁



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣰⣶⣿⣶⣴⣰⢻⣵⣶⣀⡀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣦⠀⠀⠀⠀⢤⣬⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠿⢧⡀⠠⠀⠀⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏⠁⠀⢁⡤⠞⠿⠿⠿⣿⡿⡿⢿⣿⠃⠀⠀⠀⠀⠀⠈⠓⢤⡀⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣤⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⠏⠀⠀⣀⡤⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⢄⣀⠀⠀⠈⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢠⠞⠁⠀⠀⠀⠀⠈⠳⡀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠈⠳⡄⠀⠸⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠁⢀⡇⠀⠀⣴⣶⣶⣦⡀⠀⢱⠀⠀⡏⠀⢀⣴⣶⣶⣦⠀⠀⢹⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠈⡇⠀⠈⣿⣿⣧⣤⡇⠀⣸⠀⠀⡇⠀⠘⣿⣿⣧⣼⡇⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⠀⠘⢄⠀⠪⠙⠛⠟⢁⡰⠃⢀⠀⠙⢄⠀⠨⠛⠛⠋⠀⡠⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡀⠀⠀⠀⠙⢒⣒⣒⣒⣉⠀⠀⠉⠁⣀⣀⣙⣒⣶⡒⠒⠉⠀⠀⠀⢀⣿⢳
⠀⠀⠀⠀⠀⠀⠀⢸⠉⣿⣇⠀⠀⠀⠀⠘⣷⣿⣏⢉⣽⣿⣿⣿⣍⢀⣼⣿⣶⣿⠀⠀⠀⠀⠀⣸⡇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢹⣿⣆⠀⠀⠀⠀⠈⠻⢿⣿⣿⠋⠻⣿⣿⣿⣿⣉⡽⠋⠀⠀⠀⠀⢰⣿⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠾⠿⠿⠙⣾⣤⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⣴⠛⠿⠷⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⢮⣶⣤⣀⢀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣦⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⣿⣿⣿⢷⡿⣾⡷⣶⠶⣿⣿⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⠿⠐⢿⠀⠀⠈⠁⠀⠀⡟⠉⠧⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣴⣿⣾⣁⣀⣀⣼⣶⣶⣶⣶⣶⣾⣧⣀⣀⣨⣿⣿⣤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⡀⣄⣀⡀⣠⣰⡢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣝⢿⣷⣿⡅⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡻⣷⣽⣿⣿⣮⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣘⠛⠻⢿⣿⣿⣿⣿⣿⣮⣻⣿⣿⣿⣿⣩⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡀⠀⠀⣿⡟⠃⠀⠀⠻⣿⣿⣯⡤⠻⣷⡀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡳⣦⣴⣀⣴⢛⣛⡳⣦⡙⣿⡀⢱⣄⡽⣿⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣟⣼⣧⠘⠻⢃⣿⣿⢹⣷⠛⠛⡄⠘⠆⠳⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⠁⠙⠛⠛⢻⣿⣿⣴⣿⣧⡤⠛⠠⠀⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣤⣄⣄⣠⣬⣵⣾⣻⣿⣿⡗⠀⠱⡀⠅⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣶⣭⣿⣿⣿⣿⣭⣭⡿⣿⣷⠀⢠⠽⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢌⢿⣿⣿⣿⣷⣿⣈⢉⢈⣦⣷⣾⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡙⢦⣈⠛⢿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠈⠙⠛⠻⠿⠇⠀⠂⠀⠀⠺⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠎⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡎⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢧⠀⠀⡀⠄⠚⠓⠒⠒⠂⢄⠀⠀⠀⠀⠀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠑⠪⠄⠀⠘⠷⣦⡀⠀⠀⢑⣤⡀⠀⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠉⠙⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠒⠠⠀⠁⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠁⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠒⠒⠂⢠⣽⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⢀⠄⠤⠀⠁⢊⠄⠊⠁⠉⠉⠈⡺⠃⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠉⠂⢄⠀⠀⠀⠀⣸⡦⠀⡀
⠇⢀⣠⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⡟⠓⠠⢀⠑⢄
⠀⠉⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠐⢄⠠
⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠈⠄⠂
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠔⠀⠻⡄
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠈⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠤⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣒⣤⣄⣀⣀⣀⣀⣈⣐⣂⣀⣤⣀⣀⣀⣂⣁⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠⠉⠢⢀⠀⠀⠀⠀⢀⠠⠐⠂⠁⠁⠀⠀⠀⠀⠉⠐⠠⢄⠀⠀⠀⢀⡠⠊⠀⢡⡀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁⠂⠄⡀⠑⠢⡠⠊⠁⠀⠀⡠⠐⠈⠀⠀⠀⠈⠐⢄⠀⠀⠈⠂⢔⠁⢀⠄⠊⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡰⠊⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠈⠂⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⠑⠈⢀⠀⠀⠀⢠⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠰⠀⠀⠀⠐⠀⠂⠀⠐⢧
⠀⠀⠀⠠⠐⠠⡀⠀⠀⠀⠀⡘⠤⢃⠀⠁⠃⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠰⢀⢁⠉⡀⠀⠀⠀⢀⠔⠀⢢
⠀⠀⠀⢁⠀⠀⠈⠄⠀⠀⠀⠃⠀⠅⠅⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠀⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠔⠀⠇⠀⠀⢀⠃⠀⠀⠘⢀⠀⡀
⠐⠁⠈⠐⢦⡀⠀⢰⠀⠀⢀⡤⠐⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠑⠠⢄⠀⠀⢸⠀⠀⡴⠋⠀⠀⠈
⢇⠀⠀⠀⠀⠙⠀⠈⠀⠈⠁⡠⠂⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠢⡀⠈⠁⠁⠀⠈⠀⠀⠀⠀⡠
⢀⠁⠖⠂⠤⠄⠀⠀⢀⠔⠊⠠⠀⡈⠀⠀⠀⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠈⡀⢀⠁⠑⠤⡀⢀⣀⠒⠐⠛⠁⠂
⢀⠀⠀⣀⣠⠂⠀⠈⠑⠠⡀⠈⡤⠁⠰⠀⠀⠀⠀⠀⠘⠁⠒⠠⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠘⠀⠐⡌⠀⡠⠐⠉⠀⠀⢱⠦⠄⣀⡠
⠀⠈⠀⠀⠄⠂⠄⣀⡀⠀⠈⠒⠁⠀⡀⠀⠀⠀⠀⠀⠃⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠊⠀⢀⣀⣠⠴⠡⠂
⠀⠀⠀⠀⢀⠂⠉⠉⠉⠉⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠐⡀⠀⠀⢀⡠⠔⠀⡄⠰⡑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⢠⠀⠒⢤⡀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠉⢁⠋⠀⠀⡸⡀⠀⡁⢀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢘⠀⢀⢆⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠢⠠⠔⠀⠀⠁⠀⠈⠀⢀⠉⠒⠠⠄⠀⢀⣀⠀⢀⣀⣀⠀⠀⠤⠰⡊⠀⢁⠀⠀⠈⠁⠀⠑⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⡀⠐⠤⣴⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣈⣠⠐⢂⡒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠁⠀⠀⠀⠀⣀⡀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠁⠀⠀⠀⠈⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⡀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠢⠀⣀⡀⠌

⠀⠀⠀⠀⠀⠀⣀⢀⣄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠲⠂⢀
⠀⠀⠀⠀⢠⡤⠋⠉⠈⠋⠙⠃⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠸⢻⡇⠀⠀⠀⠀⠀⠀⢠⡖⢊⠐⣦⠀⠀⠀⠡⢸⣷⠄⡎⡂
⠀⠀⠀⣴⠊⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⡧⡀⠀⢀⡸⡌⠀⠀⠀⠀⠀⠀⠀⠰⣆⠚⢃⢼⠀⠀⠀⠢⡙⠉⡚⠵⠁
⠀⠀⣠⠋⠀⠀⠛⠀⠛⠀⠘⠀⠀⠀⠸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢉⠀⠠⢀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠣⡈⢣⡀⠀⠀⡇⢸⠁
⠀⠀⢹⡁⠀⠀⠀⢠⣄⣀⡄⠀⠀⠀⢀⡕⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁⠠⠷⠈⡆⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠱⣄⣠⠁⡇
⠀⠀⢈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡅⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠑⠀⠀⠊⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠂⠈⠀⠀⠙⢦
⠀⠀⠈⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⣁⣋⣀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⣀⠀⢀⣈⡶⢣⣀
⠀⠀⠀⠀⠘⠓⣤⣄⣠⣀⣀⣤⣜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠙⠉⢣⠎⠉⠀⠀⠀⠈⢂⡀
⠀⠀⠀⠀⠀⠰⠿⡿⡇⠀⠻⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠆⠠⠀⡌⠸⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠤⠤⠤⠒⠂⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⣁⡀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⢤⣦⡀⠀⠀⠀⢠⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠉⠙⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠰⣀⡓⠀⠂
⠀⠀⠀⠀⠠⠘⠃⠓⠒⠒⠒⠚⠚⠂⠀⠀⠀⠀⠀⠀⢀⡔⠚⠁⠀⠀⡆⠀⢠⡄⠀⠀⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠙⠃⠀⠄
⠀⠀⠀⠀⠀⠐⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢰⡆⠀⢠⠀⡠⠄⠀⠀⢰⠀⠀⡶⡀⢣⠀⠀⠀⠀⠀⠀⠀⠈⠑⢀⠴⢀
⠀⠀⠀⠀⠄⠀⠀⠀⠻⠿⠟⠀⠀⠀⢠⠀⠀⠀⠀⠸⠀⢸⢃⠀⠸⣾⣿⣾⣷⣿⣾⠀⠀⡇⠇⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁
⠀⠀⠀⠜⡆⠀⠠⠄⠄⠠⠄⠀⢀⠀⠛⡂⠀⠀⠀⢰⣤⣼⠈⢦⠀⠙⠸⠃⠙⠘⠃⢀⡖⣹⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠈⠀⢄⠀⢀⡀
⠀⠀⠀⢈⣄⠀⠀⠀⠀⠤⠤⠔⠃⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⣠⣄⠀⢰⠋⠀⠈⠉⠁⠀⠀⠀⠀⠀⠾⠷⡔⠁⣀⠀⢀⡀⠉⠚⠃
⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇⢸⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠉⠁⡀
⠀⠀⠀⠀⠀⠀⡀⠄⠡⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠀⡇⢸⡀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⡀⢀⠐
⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠁⠀⠶⠂
⠀⠀⠀⠀⠀⠀⠲⣄⡀⢀⣀⡸⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣤⡀
⠀⠀⠀⠀⠀⢀⡖⣀⠀⠀⢀⣄⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡵⠖⠂⠀⠒⠣⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠃⣀
⠀⠀⠀⠀⠀⣦⠀⡋⠀⠀⢀⣉⠠⢃⠀⣀⢤⡀⠀⠀⠀⣠⣤⡠⠏⣴⣶⠄⣴⣻⡓⣜⢂⢰⡻⠄⠀⠀⠀⠀⢠⡄⠀⠠⣶⠀⠀⢰⡖⠀⠀⠙⡄
⠀⢼⠃⡀⠐⠀⠀⠻⣾⣿⠿⠏⠀⠈⠁⡤⠠⠇⠀⠀⠀⠈⢤⣧⠀⠛⠛⠉⠿⣽⣧⠏⢸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢡
⠀⠈⠋⠁⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣶⣶⣦⣿⡄⠀⡼⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠙⠋⠿⠛⠁⡼⠁⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⢋⡀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠙⠦⡀⠀⡠⠚⢡⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠨⠀⢰⠂⠰⡄⠸⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠰⡀⠈⠚⠁⢰⣈⣆⠀⠀⠀⠀⠀⠀⠀⠳⡤⡠⣀⢄⡤⠠⠤⢄⠠⣤⠄⠴
⠀⠀⠀⠀⠀⠁⠀⠀⠘⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡴⠔⠂⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⢠⠾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⢁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠖⠉⣀⣀⠘⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⠰⠐⠤⠠⠤⢀⡈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⡵⣏⡇⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠠⠄⠀⠲⠂⠰⠆⠐⠖⠁⡤⠈⠂⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣈⣉⣴⣶⣤
⠀⠀⠀⠀⠀⡀⢀⣄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣯⣝⣛⣫⣹⡾⢋⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣶⢾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠤⡴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠉⠙⠉⠉⢠⠉⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠚⡵⠋
⠀⠀⠀⠀⠀⠀⣼⠆⣶⠆⢺⡎⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠶⣉⡰⠂⡐⠏⡁
⠀⠀⠀⠀⠀⠀⠉⠀⠙⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠃⠘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣿⣿

⢀⣤⠀⣔⣃⡠⠤⠤⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡀⠀⠀⠀⡴⢦
⠘⢬⡎⠉⢀⡄⡀⠀⠔⠩⢍⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠙⣦⠎⢱⣠⣄⠀⠀⠀⠀⠀⠀⠀⡎⠀⢱⠀⠀⢰⠇⢸
⠀⠀⡀⢠⢬⡀⠁⣀⣃⣈⣁⣀⠀⠀⠀⠀⠀⢀⡖⠢⠴⠃⠀⠋⠀⠘⠁⢸⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠉⠉⠉⠀⠈⡆
⠀⠀⡇⣷⠀⠙⠫⠿⡼⣇⠧⠟⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⢠⡎⠰⠰⠆⠰⣂⣒⡏
⠀⠀⡇⠙⢄⣀⡀⣲⡐⡤⣂⠔⣆⠀⠀⠀⠀⣘⠰⣍⣱⣀⠝⠀⠯⠮⠝⢺⡴⡄⠀⠀⠀⠀⢸⠘⢞⣣⠛⣤⢣⠟⣜⣧
⠰⣆⡇⠀⠀⠀⠉⠉⠉⠉⠁⠀⣇⣴⠆⠀⠀⡏⠐⠲⠿⠶⠷⠮⠷⠾⠽⠞⠒⠛⠳⡄⠀⠀⠸⡄⠀⠀⠀⣀⣁⣁⣀⡠⠁
⠀⠉⠑⠤⣀⡀⠀⠀⠀⠀⣀⠴⠁⠀⠀⠀⠀⣧⡤⢤⡀⣀⣀⡀⢀⣀⣀⢀⣀⢀⣀⡝⠀⠀⠤⢵⠀⠀⠈⠉⠉⠀⢠⠅
⠀⠀⠀⠀⠀⠡⡌⠉⡽⠏⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠚⠀⠀⠣⠏⠉⠙⠼⠙⠞⠉⠃⠀⠀⠀⠀⠣⠜⠀⠀⠈⠻⠍
⠀⠀⠀⠀⣀⡄⠀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡐⡄
⠀⠀⠀⡼⠁⣁⠐⢁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢡⠍⢲⠼⠶⠞⠁⢠
⠀⠀⢀⠗⠀⠉⠀⠈⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⡀⠀⠀⠀⠑⠶⣾⢁⣄⢀⡄⢀⣀⡙⡽⠄
⠀⢠⠃⡤⢤⡦⡄⢰⠱⢋⠎⡆⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⡞⠁⠀⠀⠀⠈⢢⡀⠀⠀⠀⢸⠈⠉⠉⠙⠛⠣⠯⢼
⠀⡎⢀⡉⣉⠉⣁⡀⣩⣁⢄⡇⠀⠀⠀⠀⠀⢰⢻⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠇⠘⠚⠤⠿⠤⠿⠴⠯⠜⢳⠀⠀⠀⠀⠀⠈⡎⢧⡀⢀⠇⠀⠘⡗⠀⠀⠀⠘⠦⡠⠤⠃⠀⠀⠀⠀⠀⠀⢰⠃
⡰⢋⣀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢉⠆⠀⠀⠀⠀⠙⢄⡉⠁⠀⣠⠇⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠑⠉⠈⢲⠄⣀⣀⡀⢠⡀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⡉⡦
⠀⠀⠀⠘⠜⠁⠀⠀⠀⠑⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠌⠙⡖⢢⠤⠤⠀⠒⠒⠛⢄⠜
⠀⠀⡀⠀⢀⣴⣧⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠉⠉⠀⠀⢀⡄⡀⢠⠀⠤⠤⠤⠤⠤⣄⣄⣠
⠀⠈⢯⠒⢁⡤⡄⠀⢏⣂⠶⣀⡠⢤⠎⢳⡀⠀⠀⢀⠔⠒⢤⠖⢲⠎⣀⠉⢱⡀⠀⠀⠘⠋⠸⢠⡪⠏⣱⠀⡖⠶⠢⢸⠈
⠀⠀⠀⡇⠘⠤⠂⠀⣠⢲⡶⠭⠧⠚⠒⠁⢸⠀⠀⢺⡘⣃⠴⡄⠘⢄⣉⣠⠜⠁⠀⠀⠀⠀⡇⢀⢩⡭⣄⢤⢈⣭⡥⠚
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⡞⣷⢢⡜⡏⠀⠀⠀⠀⠀⢀⠴⠃⠀⠉⠑⠉⠋⢹⣉
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞⠀⠀⠀⠀⡰⠋⠉⠉⢹⠃⠀⠀⠀⠀⠀⠷⢹⠀⠀⠀⠀⠀⠀⢸⠳⡇
⠀⠀⠀⠈⢙⢲⢲⠀⡎⠈⢳⢸⠉⡏⡟⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⡀⠀⠀⣀⡼
⠀⠀⠀⠀⠸⡍⠸⠀⠃⠀⠸⠎⠀⠥⠀⠀⠀⠀⠀⠀⠀⠀⠱⣲⠖⢜⡄⠀⠀⠀⠀⠀⠀⠀⠀⠵⠉⠉⠹⡍
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠶⠶⠖⠛⠒⠲⠆⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠛⠉⠁⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠋⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡿⢛⠁⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠂⠀⣠⠃⠀⠀⠀⠂⠀⠀⠀⠘⣦⠀⢈⠿⣿⡀
⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⣰⡆⣀⠀⠀⡄⠀⠀⠀⠀⡈⣷⡰⢤⣾⣄⡀⡀⣠⢠⠸⣀⣤⣰⣿⠀⡄⠀⠻⣿⡀
⠀⠀⠀⠀⠀⢰⡿⠋⠀⠀⠠⠀⣿⣾⢿⠿⡦⢧⠰⠶⢤⠾⢿⣿⠃⠐⣿⡟⠿⠛⠛⡾⠟⠉⠉⢉⣿⡆⠀⠀⠀⠙⣷
⠀⠀⠀⠀⢠⣿⠓⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⢸⠁⠀⠈⢀⣼⠇⢠⠀⢙⣷⡀⠀⠀⠇⠀⠀⢀⡼⡋⠀⠀⠀⠀⠀⠽⣇
⠀⠀⠀⢀⣾⠷⠀⠀⠀⠀⠀⠀⠈⢹⡛⣦⣤⣤⣤⣤⡴⠟⠋⠀⠈⠀⠈⢨⡿⢶⣤⣤⣴⣶⠟⠁⠁⠀⠀⠀⠀⠡⢠⣿
⠀⠀⠀⣸⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡥⢻⠟⠷⠆⠀⠀⠰⠄⠀⠀⠀⠁⠀⠒⠛⠃⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠄⠀⣰⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠈⢹⣧
⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣀⣿⢀⣼⣿⣶⣾⣶⣤⣤⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠁⠰⠄⠀⠙⣿⡀
⠀⢠⣾⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣴⣶⣾⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣷⣴⣥⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡇
⠀⢰⡷⠀⠈⠀⠀⠀⠀⠀⠀⣰⣷⣦⡈⢿⣿⠛⠿⣿⠿⣿⣿⣯⣤⣄⣠⣤⣴⡶⣾⠉⣟⢡⣴⡄⠀⠀⠀⠀⠀⠄⠀⠀⢸⣿
⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⠓⠇⡀⠂⠀⠛⠂⠋⡍⠙⠛⠿⠏⠁⠈⠀⠉⠀⠁⢺⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡆
⠀⣾⣧⠀⡀⠀⠀⠀⠀⠀⠐⡿⣿⡿⠀⠰⠃⠀⠀⠀⠀⠘⠁⠀⣼⠀⠀⠂⠀⢾⠀⠐⠀⢺⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⣹⣇
⢠⣿⠃⢨⡀⠀⠀⠀⠀⠀⢀⢱⣿⣧⡀⠀⢠⡈⠀⠃⠀⠆⠀⠐⢁⠐⠀⠀⠀⠀⠀⠀⠀⣼⣧⠃⠀⠀⠀⠀⣄⠀⠀⠀⣠⢿⡇
⢀⣿⡠⠞⠁⠀⠀⠀⠀⣀⠈⢲⣿⣿⠃⠀⠈⠁⠀⠀⠀⣀⣼⣿⣻⣧⣦⡀⠀⠀⠀⠀⠀⣿⣿⠄⠀⠀⠀⠀⠀⠀⠰⠈⢀⣻⡇
⠈⣿⡅⠀⠃⠀⠀⠀⠀⠈⠂⠈⣿⣟⠃⠀⠀⠀⠀⠀⠐⢾⡿⠟⠛⠻⣿⣿⣢⠄⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠁⠘⠀⠐⣀⢾⡇
⢠⣿⣿⡁⠈⠀⠀⠀⠀⠀⠀⠰⣿⣿⡗⠀⠀⠀⠀⢰⣧⡿⠁⠀⠀⠀⠈⢿⣯⠄⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⡀⢂⢾⣺⡇
⠀⢸⣿⡡⢀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡧⠀⠃⠀⠀⣻⣿⠃⠀⠀⠀⠀⠀⠘⣿⡞⠀⠀⠀⠀⣿⣧⠀⠀⠀⠖⠀⡀⠀⢀⣸⢻⣇
⠀⢸⡿⡟⢹⢀⠀⠀⠀⠀⠀⠀⠙⣻⣷⠀⠀⢀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⣿⠱⠀⠀⠂⠀⠀⠀⠀⣌⡃⣽⡟
⠀⢸⣿⣧⠊⡈⠀⠀⠀⡀⠀⠀⠀⢿⣧⠀⠀⠈⠀⣽⡇⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⣠⡍⢷⢹⡇
⠀⠘⣿⣾⣧⣏⢲⠐⣌⣣⠀⠀⣔⣻⣿⡀⠀⠠⢶⣼⡃⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠁⣿⢻⣛⡄⡆⢀⡠⢘⣽⢻⣾⣿⡃
⠀⠀⣿⣿⣯⣼⣿⣿⣶⣷⣾⣾⠟⠋⠻⠿⡁⠀⢘⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠘⣿⣿⣿⣅⣡⣴⣰⣮⣷⣿⣷⣾⠃
⠀⠀⠈⠻⣿⡿⠋⠙⠏⠉⠁⠀⠀⠀⠀⠀⠀⠠⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⠀⠀⡤⠋⠈⠙⠛⠛⠛⠉⠛⠙⢿⣿⠇
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠖⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡘⠚⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⣀⠀⢰⡜⣧⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠆⢆⡀⠁⠀⠴⠀⢀⡀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠸⣄⠈⠀⠀⠈⠀⠀⠹⡄⢸⣿⣿⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠉⢀⣿⣶⣶⣷⣦⣴⣷⣧⣾⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣟⠒⣀⣀⣠⠀⢀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⣀⡴⠞⠉⠉⠉⠀⠉⠼⠉⠉⠉⠉⠙⠟⠛⠂⠀⠀⠀⠀⠀⣴⠿⠟⠻⠋⠻⠛⠛⠛⢿⠻⣿⠻⠿⣧
⠀⠀⠀⠔⠚⠁⠐⠂⢀⡤⠀⢀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⣤⠀⠀⢸⠇⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠈⠣⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⡴⠆⠈⠉⠀⠀⠐⠄⠀⠀⠀⠀⣼⡇⠀⣾⠋⠳⢦⣀⣀⠀⠲⣤⡘⠃⡈⠓⠀⠈⢦
⠑⡦⠀⠀⢀⡠⠂⠀⣀⠀⠀⠀⡴⠂⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⢿⣄⠈⠷⡄⠀⠉⠛⠳⣤⣀⠙⢦⠈⢂⠈⢂⠈⣧
⠐⠠⠄⠒⠛⠀⠀⣼⠛⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠙⠳⠦⠟⠀⠀⠀⠀⠀⢻⡀⠀⣁⠀⠀⢸⡄⢻⡄
⠀⠀⠀⠀⠰⠂⠋⠈⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⣛⢘⡇⢸⠛⠋

⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢠⡄⢀⢄⢠⡀⣠⡀⠀⠀⠀⠀⣀⣤⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣀⣀
⠀⠀⠀⣠⡾⠛⠉⠛⢷⡀⠀⠀⠀⠐⠷⣤⠶⠿⢿⣧⡀⠀⠀⠀⢀⣈⣿⣿⡾⣾⣿⠯⢄⠀⠀⠀⢰⣟⠀⣦⠀⣹⠇⠀⠀⠀⠀⠀⣀⣠⠞⢿⢻⡿⢷⣶⠃
⠀⠀⣰⡟⠘⠓⠘⠓⣈⢿⠀⠀⠀⠀⢸⡇⢰⣧⡄⣻⣿⠀⠀⠀⠸⣖⣽⡇⠶⢌⣿⣯⢎⠀⠀⠀⣼⠿⠶⠶⠞⢋⢠⣄⠀⢰⡿⣿⢉⣿⡾⣾⣉⡷⣾⠗⢶⣤⣀
⢾⣇⣿⢸⣷⣿⣾⣿⡿⢸⣇⡶⠀⠀⠘⣿⣦⣤⣴⣿⣿⡇⠀⢀⡳⣾⢛⠛⠷⢛⡛⣿⣏⡤⠀⢸⡇⠀⢀⣀⣀⠙⣿⠉⠀⠀⠀⠀⢸⣿⣿⣿⣽⣿⡟⠀⠐⠏⠿
⠀⢸⣿⣦⣬⣤⣥⣤⣴⣿⡏⠀⠀⠀⠀⣿⣿⣿⠠⠆⢹⡇⠀⠈⢻⣿⢐⣄⢀⡈⣁⣿⣿⠁⠀⠈⠻⣿⡿⠿⣿⡟⠁⠀⠀⠀⠀⠀⣿⠟⠉⠉⠻⣿⡇
⠀⠘⠿⢿⡿⢿⡿⣿⠿⣿⠇⠀⠀⠀⣰⣿⣬⣿⣷⣶⣿⣷⡀⠀⢸⡿⣿⠿⡿⣿⡟⣿⣿⠀⠀⠀⠀⢸⡿⠿⢿⡇⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠈⠁⠈⠀⠉⠀⠉⠁⠀⠀⠐⠋⠙⠋⠙⠉⠙⠁⠈⠁⠀⠘⠃⠟⠘⠃⠙⠁⠋⠘⠀⠀⠀⠒⠛⠀⠀⠀⠛⠦⠀⠀⠀⠐⠚⠃⠀⠀⠀⠀⠀⠹⠶⠄
⠀⢀⣠⣤⣄⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡄
⠀⢺⣿⣿⠿⡆⠀⢠⡾⢿⣿⣿⡄⠀⠀⢼⣗⣠⣤⣤⡽⠛⠀⠀⠀⠀⠀⢸⣧⣀⠀⠀⠀⠀⣀⣿⠀⠀⠀⣰⠛⠛⠛⠦⣄⠀⠀⠀⠐⣧⣤⣶⣶⣦⣼⠇
⠀⠀⠘⠁⢀⣽⠶⠾⣄⠀⠉⠁⠀⠀⠀⢀⣾⢟⠛⡛⠿⣦⠀⠀⠀⠀⠀⠀⠈⢻⣿⢿⣿⡿⠛⠋⠀⠀⠀⡟⠐⠂⠰⢠⡏⠀⠀⠀⠀⢸⣿⣐⣂⡿⣿⡀
⠀⠀⠀⠀⣾⠁⢴⡄⢘⣧⠀⠀⠀⠀⢠⣿⣧⣈⠁⢉⣠⣿⣧⠀⠀⠀⠀⠀⠷⣾⠰⠆⢸⣤⠇⠀⠀⠀⢸⣷⣤⣤⣀⣼⠁⠀⠀⠀⠀⠘⡷⢩⢻⣖⣿⣧
⠀⠀⠀⠀⠹⢷⣤⣤⡾⠃⠀⠀⠀⢠⣿⣿⣿⣿⣟⣛⣿⣿⣿⡆⠀⠀⠀⣠⣴⡿⠛⠺⣿⣦⡀⠀⠀⠀⡌⠀⣿⢹⡏⢹⠀⠀⠀⠸⣇⣠⡟⡲⣎⢍⣿⣿⢀⣠⠇
⠀⠀⠀⠀⠀⡴⠁⠙⠦⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠸⠇⢸⣷⣿⣷⡇⠸⠃⠀⠀⣰⣀⢠⡇⢸⡇⢸⠀⠀⠀⠀⠀⠸⣷⣶⡿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⢸⠟⠛⠻⣇⠀⠀⠀⠀⠓⠀⢸⡇⢸⠀⠹⠷⠀⠀⠀⠀⣶⣴⣤⣴⣿⣿⡟
⠀⠀⠀⠀⢀⣼⠿⠻⢷⣶⠃⠀⠀⠀⠠⠏⠩⠏⠹⠏⠹⠍⠳⠄⠀⠀⠀⠀⠛⠀⠀⠀⠹⠀⠀⠀⠀⠀⣀⣤⠇⠸⣤⡀⠀⠀⠀⠀⠀⠈⣛⣿⠿⣟⠋
⠀⠀⠀⠀⢾⡇⠸⡶⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⢀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⢈⣿⠶⢶⡾⠋⠀⠀⠀⠀⠀⠀⠸⣏⣀⣀⡾⠃⠀⠀⠀⠀⠀⢰⢀⣠⣤⣾⣄⠀⠀⠀⠀⣆⠉⢀⣠⣤⣤⣀⣀⡤⠋⢀⠄⢄⠀⠠⣉⠌⠀⣥⣂⠅
⠀⠀⠀⠀⣿⡁⠶⠄⣿⠀⠀⠀⢠⡆⠀⣠⣾⠟⠛⢿⡗⠀⣤⠀⠀⠀⠀⠀⠿⣟⢡⠙⣯⣂⠀⣀⠀⠈⢻⣿⠉⠆⠆⢹⣿⠀⠀⠘⠈⠺⣄⢰⡇⢠⡞⠁
⠀⠀⠀⠀⢨⡿⠒⢾⡇⠀⠀⢀⡘⢷⣴⣿⣯⠐⠷⢀⣷⡴⠋⢀⠀⠠⣤⠘⣵⣿⠶⢻⢿⣿⡴⠋⠀⠀⡜⠛⡶⠶⢶⠾⢏⠀⠀⠀⠀⠀⢸⣾⣧⣿⡀
⠀⠀⠀⠀⢸⣧⣛⣼⠇⠀⠀⠘⢿⣄⣼⣿⠿⣷⣶⣿⠿⣷⡶⠿⠁⠀⠉⠉⣿⡇⠃⠀⡄⣿⡇⠀⠀⢘⡁⠀⡁⠀⠈⠀⢨⡆⠀⠀⠀⠀⢀⣯⣐⣨⣧⡀
⠀⢤⣶⡂⠹⣏⣊⣿⠐⣶⣦⠄⠀⠀⠹⣿⣄⡈⠈⣁⣼⣿⠇⠀⠀⠀⠀⢠⣿⣤⣥⣤⣴⣾⡇⠀⠀⠈⠁⠐⡇⠀⢾⠀⠸⠃⠀⠀⣴⠟⠛⣿⠟⠉⠉⠙⢷⡄
⠀⠈⢀⣀⡜⠉⠉⠙⣆⣀⠉⠀⠀⠀⠀⠙⠿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⣾⠟⣿⠟⣿⡟⢻⡇⠀⠀⠀⢀⣰⣧⠀⢸⣧⣀⠀⠀⠀⠻⡀⠀⢿⡀⠀⠀⠀⡸⠃
⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠈⠀⠀⠀⠀⠈⠉⠁⠀⠈⠉⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀
⠀⠀⠀⣹⣥⣤⣤⣼⡋⠀⠀⠀⠀⠀⠀⢀⣄⢀⣴⣧⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣄⣀⣀⡀⠀⠀⢠⣤⠤⠒⠒⠒⡒⠶⣷⠀⠀⠀⠀⠀⢉⣤⣤⣤⣤⡍
⢶⣄⣾⣿⡿⠿⢿⣿⢷⣤⠦⠀⠀⠀⣠⣈⣿⡿⢿⡿⢿⣿⣀⠀⠀⠀⢸⡏⢍⣩⣭⣩⠏⣿⠀⠀⢸⣧⣀⠻⠀⠘⣛⣠⡿⠀⠀⠀⠀⣰⡟⠩⠄⠦⠙⣿⣆
⠀⣾⡱⢨⣇⠻⢢⣇⣂⣿⠀⠀⠀⠀⢙⣿⡏⠊⢵⣫⢕⢿⡟⠀⠀⠀⠈⡇⢾⣿⣿⣿⠀⡏⠀⠀⠀⠉⠙⠋⠻⡟⠉⠁⠀⠀⠀⠀⠀⣿⣇⠒⠀⠀⣠⣿⣿
⠀⠙⢿⡿⠿⢻⣿⣿⣿⣿⠀⠀⠀⠛⣿⣿⣮⣃⣞⠭⣪⣾⣿⠃⠀⠀⠀⢧⣈⣉⣩⣬⡴⠇⠀⠀⠀⢠⣤⡄⠀⡇⠀⠺⡟⠀⠀⢠⡞⠻⣿⣿⣿⣿⣿⣿⠏⠙⠂
⠀⠰⣶⣾⣷⣿⣿⣿⡿⠉⢢⣀⠀⠀⣿⣿⣿⣿⣿⠿⢟⣿⣿⠀⠀⠀⠀⠀⢸⣍⣥⣼⠀⠀⠀⠀⠀⠈⠉⣄⢠⣿⣄⠴⠁⠀⠀⠀⠀⠀⠈⢻⠟⣿⠛⣧
⠀⠀⣈⡹⠛⠛⢋⡁⠀⠀⠁⠉⠀⢠⣿⣿⣿⣶⣤⣶⣿⣿⣿⣆⠀⠀⠀⠀⠈⣿⠛⢻⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸⠀⠘⡄
⠀⠀⠈⠁⠀⠀⠘⠁⠀⠀⠀⠀⠀⠺⠟⠛⠛⠿⠿⠛⠻⠿⠛⠛⠃⠀⠀⠀⠾⠉⠿⠳⠛⠂⠀⠀⠀⠀⠀⣠⠛⠛⠋⢦⡀⠀⠀⠀⠀⠀⠶⠗⢀⣸⣄⠀⠿⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⣀⡸⠦⠤⠤⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠈⢂⠀⠀⠀⠀⠈⢳⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣀⣀⡤⠖⠁⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠾⠿⢤⣀⡜⠛⡛⢟⠲⠦⢤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢰⡇⠀⣠⣄⣿⣄⢺⣧⡾⠀⣀⡤⢤⣄⡉⠉⠛⠓⠶⢤⣤⣤⡤⠖⠋⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠲⠤⠄⠒⠺⡆⠀⠀⠈⠳⣤⣙⣿⠃⠈⠛⠉⠀⡞⢁⣀⠀⠈⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠘⠢⡀⠱⣄⠀⠀⠀⠀⣀⣻⡄⠀⠀⠀⢀⠎⣡⣤⠀⣤⡀⢺⡇⢾⣯⡴⠀⠀⣹⠀⠀⠀⠀⠀⢀⣠⠞⠋
⠀⠀⠀⢀⡤⠒⠲⣏⠀⠀⠀⠙⣄⠀⠉⠙⠋⠉⠉⠈⣷⠀⠀⠀⠾⠄⠀⣀⣀⣀⣀⠸⠳⣄⠉⠀⢀⣴⢏⠀⠀⣀⣴⡞⠉⠀⡀⠀⢀
⠀⠀⢠⠏⠀⠀⠀⣙⠶⠶⠞⠀⠈⠳⣄⠀⠀⠀⠀⠀⢻⠀⡖⠀⢀⡴⠊⠁⠀⠀⠈⣷⣄⠉⠛⠛⠛⠒⠁⢠⡞⠁⠀⠈⠉⠉⢀⡴⠃⢀⡴⠒⠉⠉⠓
⠀⠀⡏⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⢸⢠⠇⣠⣏⣤⣤⣤⣴⣤⣤⣿⡈⢳⡀⠀⠀⠀⢠⡏⠉⠓⠶⠶⠶⠒⠋⠀⢀⠎⠀⠀⡀⠀⣠⠟
⠀⠀⠳⣔⠋⣹⣃⣰⢃⡀⠀⢹⠟⣆⠀⢸⠀⠀⠀⠀⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⢠⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠙⠛⠉⠳⡄
⠀⠀⠀⠈⠀⡏⠉⠣⣯⠩⠶⠋⠀⢸⠀⢸⠀⠀⠀⠀⣿⣿⡯⠤⠀⠠⠤⢄⣈⡉⠉⣻⠟⠻⣿⡆⡆⢀⡿⣀⠀⠀⠀⠀⠀⠀⣠⠞⠀⢀⣀⠀⣀⡶⠀⠀⢸
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠐⠀⠀⠀⡞⠀⠀⢇⠀⠀⢠⡿⡏⠀⠀⠀⠀⠤⣤⣄⣉⠙⠛⠦⢄⡈⢻⡇⢸⣇⠈⠓⠲⠤⠤⠒⠈⠁⠀⢀⡎⢀⠸⣇⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠳⠤⣼⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⣿⠘⠷⣤⣀⡀⢀⣀⣀⣠⠶⠋⠹⣄⡀⠉⣩⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⣠⡟⠀⠀⣴⠟⢳⡀⠀⠀⢀⡶⠲⣄⠀⠀⠀⣿⠁⠘⣆⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣿⠾⠋⠀⢀⡾⠁⠀⢸⠇⠀⠀⡞⠀⠀⠈⠳⣄⠀⠁⠀⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠁⢀⣠⡤⠶⠋⠀⠀⠀⣸⠀⠀⢸⠃⠀⠀⠀⠀⠈⢷⡀⠀⠘⢦⠀⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⡏⠀⠀⣾⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠘⡷⣄⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣿⠘⡄⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢺⠀⢳⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠘⡄⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠹⡄⠀⠹⣄
⠀⠀⠀⣀⡤⠶⠶⠶⠶⠤⣤⡏⠀⠀⡼⠤⣤⡀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⣧⠀⠀⠀⠀⠀⠘⣇⠀⠀⣿⠀⠀⠀⠹⣦⠀⠈⠷⠤⠄⠒⠒⠒⠒⠒⠲⣤
⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢀⣤⠤⣤⣤⡿⠀⠀⠀⢸⣄⣀⣀⣀⣠⠤⠿⠆⠀⠹⣤⣄⣀⢀⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⡇
⠀⠀⠻⣄⣤⣼⠷⣄⡀⠀⠀⣀⣶⣄⣀⣸⣤⡾⠁⣰⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⣤⡀⠉⠉⢳⡄⠀⣦⠀⠀⠀⠀⠨⣍⢣⡀⢀⣤⡀⠀⣴⢀⣠⠞⠒⠚⠋
⠀⠀⠀⠀⠁⠀⠀⠈⠉⠛⠛⠉⠁⠈⠉⠉⠀⠀⠀⠘⢦⣤⡶⠟⢦⣀⠀⢠⠄⠀⠈⣷⠤⠶⠛⠦⠴⣏⠀⠀⠀⠀⠀⢈⡟⠉⠉⠁⠉⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠚⠛⠁⠀⠀⠀⠀⠀⠈⠓⠶⠴⠶⠶⠚⠁

⠀⠀⠀⠀⠀⠀⢀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣄⠀⠀⡀⠀⠀⣠⡦⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⣰⠏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣾⠁⠸⣆⡼⢳⣤⠞⠁⢹⠀⡀⢠⠞⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢦⡀
⠀⠀⠀⠀⣰⠏⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠸⣯⠀⠀⣿⠃⠀⠟⠀⠀⣾⠾⣷⡎⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢻⡄
⠀⠀⠀⠀⡟⠀⠀⠀⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⢉⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠻⠁⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠈⠹⡄
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⢀⣠⡴⠶⠛⠛⠉⠉⠁⠀⠀⠀⠀⠈⠉⠙⠛⠶⠶⣤⣼⡁⠀⠀⠀⠀⠀⠀⣠⡴⠞⠃⠀⠀⠀⣇
⠀⠀⠀⠀⣿⠀⠈⠀⠀⠀⠠⠀⠀⣠⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢦⣄⣤⠶⠋⠁⠀⠈⠀⠀⠀⠀⡽
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢀⠀⠀⠀⠐⠀⢰⠇
⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣄⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⡶⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⡤⢴⡶⠂⠘⠛⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠠⠀⠀⢹⠉⠉⠙⣿⠛⠛⠛⣿⠟⠛⠛⢻⡟⠛⠉⢹⣏⠉⠉⠙⣿⡀⠀⠀⡇⠀⠀⠀⣉⠀⢈⠀⣿
⠀⠀⠀⠀⠀⠀⠈⡆⠐⠀⠐⠀⠀⢸⣄⣀⡴⠛⠶⠶⠶⠛⠦⣤⣤⡾⠻⣤⣀⣴⠟⠲⠶⠚⠁⠙⠓⠛⠁⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⢻⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢧⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢸⡇⠀⠀⠀⠉⠙⠛⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢤⡤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠀⠀⠀⠀⠀⠉⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠙⣦
⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢧
⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⢳⡀
⢠⡟⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠳⡄
⡜⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢹
⡇⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⢸⡀
⢳⡀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠙⢦⣀⠀⠀⣠⠞⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠈⠉⠉⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣯⠙⢦⣀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠉⠳⠦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡌⠙⠢⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠉⠛⠲⠶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠶⠖⠛⠉⠁⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠉⠉⠉⠉⠉⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⣀⣀⣤⠴⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠶⠶⠶⠶⠛⠁

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠿⢧⠀⣠⣶⣿⣷⣶⣄⣸⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣧⡀⠀⠸⠽⠇⠀⢴⣫⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠻⣿⠿⣿⢾⣿⠿⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⡦⠐⠂⠐⠂⠠⣎⡟⠀⠀⠀⠀⢠⢤⡀⡠⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⡤⣄
⠀⠀⠀⠀⠀⠀⠠⠀⠀⣠⠆⣀⠤⢤⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠘⠠⠃⠀⡀⠇⠉⢆⠀⠀⠀⠘⠦⡼⠁⠀⠀⠀⠊⣡⣄⠉⢢⠀⠀⠀⢸⠗⠁
⠀⠀⠀⠀⠀⠀⠇⠀⠀⡇⠘⠿⠃⢀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡘⠀⢀⢁⠚⠄⠘⠀⢉⣀⠀⠈⡆⠀⠀⣀⣀⡁⠀⠀⠀⠀⡀⠻⠞⠁⡘⠀⠀⠀⢠⣄⣀
⠀⠀⠀⠀⠀⡠⠀⠠⠖⠈⡁⠒⠀⡡⠒⠦⠀⠈⢆⠀⠀⠀⠀⠀⡇⠰⠥⣤⠸⠀⠀⢰⠧⣤⠼⠀⠁⠀⠀⠿⠷⢇⠀⠡⢄⠀⠀⠉⠉⠁⢀⡠⠊⠀⡼⠔⠋
⠀⠀⠀⢠⠊⠀⠀⠠⡀⠠⠀⣉⣁⠠⠀⡠⠀⢀⠀⢣⠀⠀⠀⡐⠀⠀⠁⡀⠡⠠⡤⠠⠁⠀⠀⠀⠸⡀⠀⠀⠀⠈⠢⣀⠀⠈⠑⠛⠓⠈⠉⠀⡠⠊
⠀⠀⠀⠃⠒⠉⠀⠀⠈⠐⠂⠐⠀⠐⠊⠀⠀⢸⠁⠊⠀⠀⠈⢀⡀⠀⠀⠇⠲⠀⠇⠸⠀⠀⠀⠀⣀⠘⡀⠀⠀⠀⠀⠀⠈⢱⣶⠒⠒⣶⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠒⠄⠀⠀⠀⢸⠀⠀⠀⠀⠉⠀⡇⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⢀⠁⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⣄⠤⢤⣼⠀⠀⠀⠰⣄⡄⣠⡴⠀⠀⠀⠀⠀⠀⢁⠀⠀⣾⣿⣿⣿⣿⣿⡆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠀⠀⠻⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡄⠚⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠃⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠉⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢲⣿⣿⣻⣿⣿⡇⠀⣱⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠄⢠⣌⠀⡀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⠛⠛⠛⢉⠋⠀⠈⠤⢧⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡂⠀⠀⠆⡙⠊⠠⠁⠀⡘⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠉⠀⠈⠑⢄⠀⠀⠀⠀⢈⣁⣈⡁⠀⠀⠀⡎⢴⡌⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠚⠑⣄⠀⠀⠠⠠⠄⠀⡠⠁⠀⠀⣾⣄⠀⠠⠊⠀⠀⠀⠀⠀⠀⢰⠉⠉⢀⣶⠀⠀⠀⠀⠀⠀⡠⠄⡀⠐⠒⠂⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠠⠤⠄⠁⡂⠀⠐⠲⠉⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⣶⣶⣿⣿⠀⠀⠀⠀⠀⠄⠠⡆⢸⠀⢀⠤⠀⠄⠀⢸
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠂⠀⡴⢦⠀⠈⡀⠀⠀⢁⠀⠀⠀⠀⠈⡿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠃⠀⠀⠀⢀⡀⠈⠁⠀⠀⠀⠠⣘⢛⠀⠀⢸⢀
⠀⠀⠀⠀⣴⢻⠀⠀⠀⢄⠀⠛⠛⠀⠠⠀⠀⠀⠸⢦⡀⠀⠀⢰⠁⠀⠀⠤⠤⠀⠀⠀⠠⠀⠀⠄⠀⠸⠀⠀⠀⠰⢁⠆⠀⠐⠀⠆⠴⠤⠀⠀⠀⠀⡚⡈⡄
⠀⠀⠀⢸⡇⠘⡀⠀⢀⡀⠈⠉⠉⠁⣀⠔⠀⠀⡈⢸⡇⠀⠀⠀⠀⠀⡀⠺⠇⡸⠀⠀⠀⠺⠇⡰⠀⠀⡇⠀⠀⠀⠠⠐⢄⣀⠀⠀⠀⠀⠀⠀⠀⡠⣡⠂⠁
⠀⠀⠀⢸⡗⠀⠱⡀⠀⠈⠛⠘⠘⠉⠀⠀⢀⠔⠀⠾⠇⠀⠀⢀⠀⠀⠈⠀⠀⠤⡤⢄⠦⠁⠈⠀⠀⠀⡇⠀⠀⠰⠶⠋⠀⠀⠀⠩⠏⠈⠭⠉⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⠤⠤⠤⢴⡂⠁⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠘⠡⠙⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⢴⡾⠇⠀⠶⠷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠂⠀⠘⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣬⣉⠁⠉⢨⡙⣠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠴⠶⠦⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣽⣶⣄⣀
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠉⠉⡶⢤⣀⡰⠋⠀⠀⠀⠙⢯⡻⣦⣀⡀⣀⣀⣀⡀
⠀⠀⠀⢠⠞⣠⠖⠋⠉⠉⠉⠙⠓⠶⣄⡀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⣳⡌⢻⡙⢯⠀⠉⠉⠻⣦⣄⡀
⠀⠀⢠⢏⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⡀⢧⠈⢣⠀⠀⠀⣷⠀⠙⠳⣄
⠀⠀⣾⡟⠉⠙⡆⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣧⠸⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⢷⡀
⠀⢠⢟⠀⠀⢠⣿⣿⣿⣿⣿⣷⠀⠀⡠⡄⠐⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠞⠉⣿⠀⠂⠀⠀⠀⠀⠇⢀⠀⠀⠀⠀⠙⣦⡀
⠀⡜⠈⠒⠒⣿⣿⣿⣿⣿⣿⣿⡇⠁⠈⠈⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢩⠏⠀⠀⣻⡇⠀⡆⠀⠀⠀⠀⠸⠀⠀⠀⠠⡀⢩⠳⡄
⠀⡇⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣾⠀⣸⠁⠐⠁⣀⠴⠧⣄⠀⠀⠀⠁⠸⠀⢻
⠀⢷⠀⠀⠀⣿⣿⣿⣿⣿⣉⣿⠁⠀⠀⠀⠀⢣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⢀⡟⣰⠟⠋⠉⠉⠁⠀⠀⠈⣯⠁⠀⡇⠀⠀⢸⡆
⠀⠸⡆⠀⠀⠘⠿⣿⣿⣿⠟⠡⢳⣿⡆⣠⣧⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⡧⠀⠀⢸⠃
⠀⠀⢻⣆⠀⡐⠀⠀⠀⠀⠀⠀⠈⢉⣴⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⡇⢠⠂⡞
⠀⠀⠀⠻⣝⠦⣄⣀⣀⣀⣠⣤⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠐⠀⠀⡼⠁
⠀⠀⠀⠀⠈⢷⣄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢁⡴⠋⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠁⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠰⠀⠀⠀⠀⠀⠀⢀⣧⣀⠤⣀⠀⠀⣠⡴⠋⢀⡔⠋⠁⠈⠉⠛⠲⠶⢶⣖⣚⠉⠁⠀⢔⣠⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣄⣀⣀⡀⠀⢀⣀⣀⣤⡤⠿⠛⠁⠀⢀⡎⠀⣴⡯⢤⣄⣀⠤⠀⢀⣀⣤⣤⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣸⠀⣸⠁⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣌⠳⢤⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⢦⡀⠀⠀⠀⠀⣀⣤⣄⠀⢀⣀⠀⣀⣤⣤⠀⠀⠀⠀⠀⣀⡴⠞⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⠛⢳⡴⠚⠉⠀⠛⠛⠉⠉⠉⠉⠀⠈⠓⠳⠞⠳⠛⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢆⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠸⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠞⠉⠉⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠑⣆⠀⠀⠀⠀⣼⣧⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠚⠛⠋⠉⠉⠉⠀⠀⠸⡄⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠁⢀⡟⠀⠀⠀⠞⠁⠀⠀⠈⠙⠳⣤⡀
⠀⠀⠀⠀⠀⢀⡤⠞⠋⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠛⠋⣁⡴⠒⣻⠋⠉⣟⠛⢲⡖⠶⣌⡙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⡀⠈⠻⣦⡀
⠀⠀⠀⠀⡴⠋⠀⠀⢀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠑⠖⠋⠓⠊⠉⠓⠋⠣⣤⠋⠙⢄⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⢀⡞⠁⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠠⠶⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠘⣧
⠀⠀⡾⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣠⠖⠛⠉⠙⠲⠞⠉⠉⠙⠢⡀⠀⡼⠀⠀⠀⠀⠀⠐⠦⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⣧
⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⣀⣀⡤⠴⠒⠲⠦⠤⠤⠿⠚⠁⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⢸⡄
⠀⠘⠛⢳⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠈⠙⠶⣄⠀⠀⠀⢠⣄⣰⠦⣬⡇
⠀⠀⠀⠘⠟⠛⣇⠀⣀⣤⡄⣀⡴⠛⠉⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠈⢧⡀⠀⡎⠘⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠛⠋⠀⠀⢀⠏⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⠃⠀⠀⠀⠘⢧⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠶⠦⠤⠤⠤⠤⠤⣤⣤⣤⣬⡽⠛⠛⠛⠛⠛⠛⠛⢻⣉⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠒⠒⠒⠚⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣋⡤⠐⣀⣠⣔⣾⣿⣿⡦⣥⡀⠉⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠘⢃⡴⣾⣽⣿⣿⣿⣿⣿⣿⣻⢿⣷⣤⣦⣵⡔⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠰⠃⠐⣩⣾⣿⡿⠛⠛⠋⠸⠋⠍⢛⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣮⡞⠠⢟⣴⣿⢿⣋⡙⡀⡀⠀⠀⣤⡀⣀⡁⢨⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣏⣴⣳⣿⣿⣿⣿⣛⣛⣿⠇⠀⠐⠛⣿⣟⣛⣶⣾⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⣿⣿⣿⣿⣿⣿⣿⢋⠉⠈⢁⣤⣤⣤⣀⠀⠈⠙⢫⣙⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⣿⣿⣿⣿⣿⣿⣿⣷⠆⠀⠰⢒⣿⣻⣿⣁⡀⠀⠄⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢰⣿⠟⢻⠉⡛⡿⣦⣰⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣷⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢓⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⡄⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢀⣾⠛⢋⣿⣿⣿⣿⣿⣿⣟⣿⣻⣟⣻⣿⣟⣿⣿⣿⣍⣿⣿⣿⣿⣿⡘⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⡆⣼⣿⣿⣿⣿⣿⣿⡝⢻⡼⣯⣭⢵⡿⢿⣿⣿⣾⢻⣿⣿⣿⣿⣷⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣼⣿⠿⠿⣿⣿⡿⣿⣿⣧⣠⣀⠀⣄⠀⢀⣼⣿⣿⡷⢸⣿⣿⣿⣿⣿⣟⡿⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢙⡏⠀⠐⢻⠟⠀⢹⣿⣿⣿⣿⣿⣷⣾⣿⣽⣿⣿⡇⠈⠻⣿⣿⣿⠉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡤⠀⣸⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⣤⠀⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣼⡇⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠹⣿⡟⠀⠿⣿⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣠⣤⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣋⣹⣯⣿⣿⡏⣽⠷⢦⢋⣠⣄⠐⠀⠻⡌⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣕⠛⣛⣛⣉⣩⣿⣿⣿⣿⣿⣿⣶⣆⣄⢸⣿⣷⣦⡳⣍⢲⢷
⠀⠀⠀⠀⠀⢀⣀⣶⠀⡿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣷⡳⠟⢿⣿⣿⣿⣷⢸⢧⠤
⠀⠀⠀⠀⠀⢳⣿⣿⣟⣴⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣻⣿⣿⣿⣿⡄⢴⣄
⠀⠀⠀⠀⢠⡏⣿⡿⠛⠁⠞⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢪⢁⢳
⠀⠀⠀⠀⣾⣹⣿⠁⢸⣅⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢦⠙⣎⡃
⠀⠀⠀⢠⣿⡿⠃⠀⢸⢇⣀⣿⡿⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢡⣳⣾⡄
⠀⠀⡠⣼⣿⣷⣀⡔⠃⣺⣿⣿⠇⢯⣿⣿⣿⣿⣿⡿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢻⣿⣿⢯⢣⢿⢻⡆
⠀⠀⣆⣿⣿⡿⢣⠀⠀⣿⣿⡏⠀⢠⡉⢙⣿⣿⢟⡰⢋⣼⣿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣯⡀⠌⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠑⣦⣀⣤⡿⠿⠒⠒⠉⠉⠉⠁⠀⠐⠒⠲⣶⣄⡀⠀⡠⠋⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⡠⠔⢎⠁⠄⡠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠭⠏⡙⠣⣄⠀⠀⠀⡜
⠀⠀⠀⡖⢆⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⡜⠀⠈⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠑⢠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣀⠤⣀⣱⡀⢂⣠⠤⡀⠀⠀⠀⠜⣉⠒⠐⠊⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⡰⠁⡇
⠃⠄⠀⢈⡁⠈⢡⡔⠁⠀⠀⢠⠊⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠈⠆⠀⠀⠀⢸⡉⠲⠁⠾⠐⠊⢑
⠀⠀⠰⣉⠴⠛⢦⠣⡀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠈⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠄⠀⠠⠐⠁⠀⠀⡴⠀⠀⠈⡄⠀⠀⠀⡹⡠⢄⠒⢇⠉
⠀⠀⠀⠀⠀⠀⠀⠑⢕⠀⠀⠀⠀⢈⣦⠀⠀⠀⠀⠀⠀⠀⢀⡀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⢀⠀⠀⠈⠔⠁⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠀⠀⢸⣄⣑⣤⡀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣁⡸⠀⠀⠀⠀⠘⡄⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢰⠀⠀⠀⠈⠒⠤⣄⣀⣈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⡀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠡⣄⣉⠍⠉⠉⠉⠉⠉⠙⠤⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠠⠂⠐⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠢⠔⠀⠀⠀⡀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢓⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢢⠀⠘⣏⣱⠔⠋⠉⠉⠉⠉⠑⠫⠀⠀⠀⠀⠑⢈⡧⠨⠖⠊⠉⢹⠀⠀⠀⠰⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⠀⠀⠀⠙⠂⠤⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠁⠀⠀⠀⢀⠤⠄⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠊⠻
⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⢠⠖⠉⠀⠀⠠⠠⠄⠀⠀⠀⠾⠀⠐⠠⠐⠁⠈⠳⠄⣀⠀⡀⣀⣀⡠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈⡧
⠀⠀⠀⠀⠈⠒⠚⠛⠀⠒⠒⠒⠒⠒⠒⠤⠤⠤⠤⠤⠴⠒⠒⠒⠂⠤⠤⠤⠤⠤⢤⣄⣀⣀⠠⠤⠤⠒⠒⠋⠒⠈⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠀⠀⠠⠀⠀⠠⠀⠀⠀⠂⠀⠀⠀⠠⠀⠀⠀⠀⠀⠰⠀⠂⠀⠀⠀⠀⠄⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡀⢀⠏⠳⣄⠀⡰⢆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠒⢄⣘⣀⣱⣮⣤⣦⣬⣎⣥⣘⡠⠞⢹⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⢒⣾⠼⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠳⢼⡤⠒⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣮⢔⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⢧⡎⠤⠴⡆
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠐⠒⢄⠀⠀⠀⠀⡇⠀⠀⠻⣎⣰⡡⡆
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⣤⠀⠀⡇⠀⠀⢰⡇⠀⠀⢠⡌⢳⣠⣃⠴⠀⠀⠀⠀⢀⡠⢴⡄
⠀⠀⠀⠀⠀⠀⢸⠃⢀⡔⠂⠐⠢⡀⠀⠀⠀⠀⠀⠣⡀⠀⠀⢀⠜⠁⢀⡴⠫⠵⠀⠀⠀⢰⣶⠙⢾⣼⠔⣆⣄⢠⠬⣧⡟⡇
⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⣴⠄⠀⡇⠀⠀⠀⠀⠀⠀⠈⠙⠉⢁⣠⠶⢉⠀⠀⠀⠀⠀⠀⠀⢩⣤⡛⣙⡳⢯⣬⣮⣴⠞⡅⡃
⠀⠀⠀⠀⠀⠀⢸⡀⠈⠣⠄⠠⠔⠁⠀⠀⠀⠀⠀⣀⣠⢴⠚⠋⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠛⢡⠛⣡⣷⣤⠁⢻⡃⠀⡇
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⣀⣀⡤⡴⢖⢛⠉⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⢈⣛⡋⣤⡝⠁⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠉⠉⠙⢭⠁⠑⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠿⠃⠈⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣠⠞⣰⠶⠒⠲⠦⣤⡀⠀⢀⣀⣀⣤⠤⢤⡀⠀⡞⠁⠀⢈⡇⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⡼⠁⠀⠀⠀⠀⢈⡇⣼⠋⠉⠀⠀⠀⢀⡷⠀⢹⡀⠠⣯⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣎⣒⣛⣀⡿⠀⠀⠀⠀⣶⣋⣡⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠒⠃⠈⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⢤⣤⡀⠀⠀⠀⣴⣷⠂⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠋⠉⠑⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠵⣒⠆⠑
⠀⠀⠀⠀⠄⠋⠛⠀⠀⠀⠓⠓⢄⠀⠀⠀⠀⠀⡠⠖⠋⠀⠀⠆⠀⠠⠄⠀⠙⠦⢄⠀⠀⠀⠀⠀⠀⠀⢄⠀⠛⠁⠠
⠀⠀⠀⠈⠀⠉⠀⣠⣤⡄⠀⠁⢸⠀⠀⠀⠀⢸⠁⣾⠀⢰⠂⡀⢀⢀⡐⡆⠀⡶⡄⢣⠀⠀⠀⠀⠀⠀⠀⠑⡠⢆⠅
⠀⠀⠀⡐⠀⠀⠀⠙⠛⠃⠀⠀⢈⣢⠀⠀⠀⢸⠈⡟⡆⠈⣿⢿⡿⡿⣿⠇⢀⣇⠇⡎⠀⠀⠀⠀⠀⠀⠀⠘⠤⢑⠄
⠀⠀⠘⡜⠀⠁⠀⠀⠀⠉⣉⠄⢈⣮⠀⠀⠀⠻⠶⠇⠙⢄⠈⠈⠀⠃⠉⣠⠎⠸⡶⠇⠀⠀⠀⠀⢀⡄⢀⠔⠈⠁⠑⢀⣠⡄
⠀⠀⠀⠛⡆⠀⠀⠀⠉⠉⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⡞⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠇⠐⠦⠤⠶⠀⠁⠁
⠀⠀⠀⠀⠁⠀⡆⢔⠀⢰⢰⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠀⡇⡇⠐⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠊⠅⠘⠀⠐⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠃⠙⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠁⢣⠤
⠀⠀⠀⠀⠀⠰⡄⢀⣀⡀⡶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⡼⣀⡀⠀⢨⣤⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡵⠖⠒⠒⠺⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠈⠃⣄
⠀⠀⠀⠀⢰⠀⣉⢀⣀⣈⡅⠈⣄⡀⢤⡀⠀⠀⢠⡤⣤⠎⣶⣶⣄⣾⣭⢳⢳⣸⡻⠀⠀⠀⠀⠀⡎⠀⠀⠶⠀⠀⡶⠂⠀⠉⡄
⠀⢺⣉⣍⡃⠀⠙⠿⠿⠛⠡⠀⢀⠔⠒⠃⠀⠀⠀⠹⣿⠀⠙⠋⣈⢳⡽⠞⢈⠏⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠸⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⣷⢿⡾⣿⠀⡼⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⡀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⢈⢦⡈⠉⠉⢁⡼⡀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠳⠀⢠⠤⢤⠀⣜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⢂⠈⠢⠔⠁⡜⡇⠀⠀⠀⠀⠀⠀⢤⢀⠀⠀⡀⡀⠀⢀⢀⠀⠀⣈
⠀⠀⠀⠀⠰⠭⠤⠼⠀⠸⠤⠤⠕⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠂⠀⠀⠀⠑⠛⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠉⠈⠁⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡑⠁⠀⠀⠀⣑⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⠖⢊⣉⡉⠛⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠾⠀⠀⠀⡄⠀⢀⠀⡄⢀⠀⣀⡄⠀⠀⣷⣣⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣤⣀⣤⣼⠈⠛⠈⠁⠈⠙⠁⢧⣤⣶⠟⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣽⡿⠋⢀⣀⠀⠀⠀⠀⢀⣀⠀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⢡⣾⠁⠀⠉⢢⢠⣾⠁⠀⠉⠂⠀⠀⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠻⣿⣄⢰⡷⡸⢹⡿⣄⢾⡆⡜⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⠗⣤⣝⣓⣒⣋⣁⣀⣙⣓⣒⣫⣤⣆⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡀⣿⣿⣦⣿⣦⣿⣼⣿⣼⣷⣾⣿⡿⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⢻⣿⣧⠻⣽⠻⡿⢿⡿⢿⣿⡿⣿⣛⠟⠁⠀⢀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⣸⣿⣿⠗⠬⠉⠛⠚⠓⠒⠛⠛⠉⠁⠀⠀⠀⢸⡇⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⢨⣿⡆⠀⠀⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⢘⡁
⠀⠀⠀⠀⠀⠀⠀⠸⢻⣷⣠⣤⣸⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢟⣴⣀⠎⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠯⠀⠙⣸⡿⠿⠿⣷⡀⠀⠀⠀⠀⠀⠀⣠⠄⠰⢌⠃⠠⠿⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢹⢿⣿⣶⣶⣴⣷⣿⣿⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡽⠷⠄⠀⡂⠀⠈⠉⠉⠉⠀⠈⣿⡧⠄⠰⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣿⣶⣷⣾⣿⣶⣷⣶⣶⡶⢶⢶⣶⣾⣿⣿⣿⣿⣿⣷⣦⠤

⠀⢠⡖⢦⡴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣶⠛⢳
⠀⣸⡇⠘⠡⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠇⠀⠠⣯⡀
⣴⠋⠀⣴⠶⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠞⠻⣦⠈⠹
⢿⡀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣸⢷⣄⠀⠀⠀⢀⡿⢷⡀⠀⠀⣰⣆⠀⠀⠀⠀⠀⠀⠀⣽
⠈⠻⣦⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⢠⠟⠻⣦⡀⣰⠏⠀⠉⢳⣄⣰⠟⠀⠈⢻⡄⢀⡟⠙⣧⠀⠀⠀⠀⣠⡾⠃⣠⠞
⠀⠀⣼⠀⠀⢀⡇⠀⠀⣴⣦⡀⢠⡏⠀⠀⠈⠻⠃⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⣷⠟⠁⠀⠸⣇⠀⠀⠀⣿⠀⠀⣏
⢀⡾⠁⢀⡴⠛⠀⠀⣼⠃⠈⠙⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⣦⡀⠙⢷
⢸⡃⠀⢸⡇⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⣆⠀⠀⠈⢻
⠘⣧⠀⠀⢷⣀⡶⠾⠆⠀⠀⠀⠀⣠⠴⡶⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⣠⢖⡶⠶⢦⡀⠀⠀⠀⢠⣯⡄⠀⠀⣼
⠀⠘⠷⣄⡀⠙⠷⠀⠀⠀⠀⠀⡼⠁⠸⣅⣀⣀⡼⡆⠀⠀⠀⠀⠀⢸⠁⠘⢦⣀⣤⢿⡄⠀⠀⠈⢹⣇⣠⡾⠃⢀⣼
⠀⠀⠀⠈⣹⡷⠀⠀⠀⠀⣀⠀⣇⠀⠀⠈⠉⠉⣰⠇⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢀⡜⡀⠀⠀⠀⠈⠉⢁⣠⡴⠛⠁
⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠘⣆⠘⠲⢤⡤⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠛⠩⠴⠃⠀⠀⠀⠐⢿⡋⠁
⠀⠀⠀⠀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⠀⢀⣀⣀⣀⣀⡀⠀⡤⠶⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣷⠄
⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠸⠤⠞⠉⠉⠁⠈⠳⠞⠁⠀⢀⡇⠀⠀⠀⠀⠀⠀⠐⢿⣍
⠀⠀⠀⠀⠀⠈⠛⠶⣤⠀⠀⠀⠀⠀⠈⢷⣄⣀⡴⠒⠓⠲⠞⠛⠳⣦⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠒⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠻⢶⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⢶⠇⠀⠀⣴⣤⠀⠀⠀⠀⣤⡀⠀⠀⠀⣶⣄⠀⠀⢻⡍⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢸⠇⢺⡀⢀⡴⠛⠉⠻⡆⠀⣰⠏⠹⣇⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠈⢷⡟⠀⠀⠀⠀⠿⠟⠁⠀⠀⢿⠀⠀⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠶⢦⣿⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠛⠋⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠈⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢲⡀⢀⡤⠖⠊⠉⠉⠉⠒⠢⣄⠀⡠⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠉⠉⠀⠀⠀⣀⣀⣀⣀⣀⠀⠉⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢀⠔⠊⠉⠀⠀⠀⠀⠀⠙⣆⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢰⠃⣀⣀⠀⠀⢀⠔⠋⠉⠳⡜⡆⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡇⡎⣀⠀⢱⠀⢸⡀⠲⠂⢀⠇⢹⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢰⠀⠑⠬⠤⠾⣄⢀⢝⠒⢒⣁⣀⠈⡇⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢸⠀⢀⢤⠤⢴⢥⢤⡪⠴⣅⠼⠈⢷⡇⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⡜⣴⠁⠀⠑⠃⠀⠀⠀⠀⠀⣀⣀⣾⢣⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⣼⠃⠹⣗⢄⣀⢄⢀⠴⠒⠒⣦⢗⡵⠁⢸⠀⠀⠀⠙⢤⣀⣀⣴⣧⡀
⠀⠀⠀⢀⣦⣀⡠⠞⠁⠀⢀⡼⢹⠀⠀⠈⠑⠪⠽⠿⠶⠾⠭⠒⠁⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⢹⠅⠀⠀⠀⢀⡤⠊⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠑⠤⣀⣀⡠⠔⠋
⠀⠀⠀⠈⠑⠒⠒⠋⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⢰⢾⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠈⠦⣀⠀⠀⣠⠊⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡾⠏⠀⠀⠻⠶⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠲⡄⠀⠀⠀⢀⡖⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⢠⠃⠀⠀⠀⠀⢑⡆⡞⢀⣀⣤⠤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⣒⣒⣒⣚⠃⠑⠒⠒⠚⠉⠉⠀⠉⠉⠀⣀⣠⣤⣀⡀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣠⠖⠉⠁⠀⠀⠈⠙⠲⣄⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣀⣀⠉⠳⡄⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣸⠁⠀⠀⠀⡤⢶⣤⡀⠀⠈⣧⠀⠀⠀⠀⢀⡏⠀⠀⣾⣉⣿⣷⠀⢻⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢻⠀⠀⠀⠸⣷⣾⣿⡇⠀⠀⣿⠀⠀⢀⡀⠘⣇⠀⠀⠛⠿⠿⠋⢀⡟⠀⡿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⢆⠀⠀⠀⠉⠛⠉⠀⠀⣠⠇⠀⢰⡏⠉⢢⡈⠢⣄⣀⣀⣀⡴⠋⢀⡼⠁⢸⠿⢷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠓⠦⣤⣄⣠⣤⠴⠞⠁⠀⠀⠀⢧⠶⠗⠃⣠⡄⠀⠁⠀⣠⠶⠋⠀⢠⠏⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣄⠀⠀⠀⠀⠀⣤⠤⣤⣤⠤⡤⢶⢋⢿⡇⠀⢠⡞⠁⠀⣠⠴⠁⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠘⣧⢉⡀⠒⢈⡁⢌⡿⠀⠀⠀⠙⠛⠉⠁⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠀⡀⠀⠀⠀⠀⠀⠈⠻⠴⠧⠾⠼⠋⠀⠀⠀⠀⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⡴⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⡞⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠻⠋⡇⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣤⣼⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡛⠶⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠚⢉⡇⠀⢀⡤⠦⣄⠀⠀⠀⠀⠀⠀⠀⣠⢄⠀⢿⠉⠳⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠖⠋⠀⠀⢸⠀⠀⡴⠋⠳⡄⠀⡇⠀⠳⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠞⠁⠀⠀⠙⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⣺⠟⠋
⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⢠⣴⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢠⠶⠂
⠀⠀⠀⠀⠀⠀⠀⣀⣱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⢀⣴⡿⠛⠇⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⣠⣾⣿⡛⠿⣆⢀⡀⠀⠀⠀⠴⣾⠟⠁⠀⢠⣾⠋⠀⠀⠀⢀⣤⠀⠀⣠⡦⢛⡴⠾⠒⡲⠿⠟
⠀⠀⠐⠊⠉⠀⠀⠈⠙⢤⡉⢻⣧⡀⠀⠀⢰⣿⠀⠀⣴⣿⠃⠀⠀⠀⣠⡾⢃⡤⢚⣡⠶⠋⣠⠀⠀⠀⠀⢀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠹⣷⡀⠠⢸⣿⢠⡾⣿⣿⠀⠀⣠⡾⢋⡔⢉⡴⠋⠁⠀⠀⠉⠉⠀⢀⣸⠋⠀⣀⣠⣤⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠈⠻⣦⣸⣿⡇⠘⣿⣿⣴⣿⣿⠷⢋⣴⣿⡗⠀⠀⠀⠀⣠⣤⡾⠟⠋⣉⣭⣽⡿⠿⠛⠛⠛⠛⠛⠛⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢄⠘⢿⣿⡇⣰⣿⣿⣿⣿⣿⣶⣿⡿⢻⣶⣟⣩⣶⠟⢋⣡⣴⣾⠟⢛⢋⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣸⣧⡈⣿⣷⣿⣿⡟⣹⣿⣿⣿⣿⣟⡉⠻⢿⣿⣶⣾⣿⡟⠁⣀⡀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣟⣿⣚⠛⣿⣿⣿⣿⣿⠿⣷⣬⣻⣾⠛⠤⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡾⢿⣿⣿⣿⣿⣿⣿⣿⣾⡻⣿⣿⣿⣷⣼⣯⣤⣃⣱⣄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣝⡍⠻⣮⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣣⠶⠉⠁⠀⢠⣬⣿⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣬⡶⣿⣽⣿⣿⣿⣿⠟⡿⠛⠋⢹⡿⣽⣿⣿⣯⣀⣄⠀⢠⣤⡿⢛⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡷⢀⡿⣦⢼⣿⣏⣿⡿⣿⡈⠓⠀⢀⣼⠃⣿⣿⢟⡟⣩⢿⣶⣶⣿⣃⠈⣟⡢⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⠷⡢⣄⣿⡅⠛⣋⠿⠏⢻⡿⣿⣷⣾⣿⣿⣸⠇⡄⣻⠺⠯⣸⡿⠀⢿⢿⣤⡏⢀⣠⣿⣶⠂
⠀⠀⠀⠀⠀⠀⠀⢐⣖⠳⢶⣟⢻⣿⠀⠀⠀⠀⠀⣆⢷⣿⣿⣿⣿⣿⣿⠰⣼⡁⢠⣦⣇⣄⠀⠀⠶⣹⣿⡿⠁⣀⡛⢋⣀
⠀⠀⠀⠀⠀⠀⢂⣿⠉⠀⠀⢙⣿⠁⠀⠀⡼⠆⣰⣿⣯⣿⣿⣿⣿⢿⡧⢻⣿⣷⣿⡟⠋⠘⢗⠀⠘⡹⢛⣷⡟⠋⢀⠠⣝⠁
⠀⠀⠀⠀⠀⠀⢈⣍⠀⠠⠴⡿⠏⠀⠀⠀⠆⠀⠆⢀⣿⣯⣿⣿⣷⣿⢢⣿⣿⣯⡟⢱⡆⠀⠘⣧⠀⠁⠈⣿⢿⣿⣶⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣏⣀⠰⠶⣷⡄⠀⠘⡗⢀⠀⢀⣼⣿⣿⣿⣽⣉⣷⡸⢿⢹⣿⡿⢻⡟⠀⠀⠘⣧⠀⠀⠀⠀⣼⣟⠁⠈⣿⡀
⠀⠀⠀⣠⠀⠀⠂⣿⣥⡤⠴⣷⠀⠀⡾⠀⢉⡠⠂⢸⣿⣿⣿⣿⣾⣿⣷⠶⣾⣿⠃⠚⠋⠁⠀⠀⠘⠓⠤⠀⠀⠙⡿⠲⢏⡽
⠀⠀⢠⡏⠀⠀⠀⠃⢸⣷⢀⣇⠀⣰⠃⠀⠉⢀⡴⣿⣿⣯⣻⣿⡏⠠⠷⠰⣿⠿⠀⠀⠀⢀⡀⠀⢠⡄⠀⠀⠀⠀⠀⠄⢸⣧⣆⠀⠀⠀⢀
⠀⠀⢸⡇⠀⠀⠀⠀⠈⣿⡎⠚⠂⠀⠀⢰⠋⠈⢠⣿⣿⣼⣿⣿⡀⠒⠃⣠⡅⢘⠃⠀⠀⠾⠀⠈⢀⠓⠲⣀⡆⠀⠀⢾⣼⣿⣿⡗⢤⣶⣿⠃
⠀⠀⢻⡇⠀⠀⠀⠀⠀⣈⢻⣠⡀⢠⣀⣀⠀⠀⠐⡯⠞⠛⠶⠂⠘⠙⠋⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠐⠉⠠⣼⢧⡼⠁⠛⡋⠁⢰⠋
⠀⠀⢸⣧⠀⠀⠀⠀⠀⠁⠏⠀⠆⢸⠟⣿⡆⢦⣌⡱⠂⢸⣤⡶⠒⢶⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣶⣴⡆⣠⢶⢁⡈
⠀⠀⠈⠻⠀⠀⠀⠀⢠⡗⠠⡄⠳⠈⢴⠌⣿⠃⠈⠉⡼⠀⠙⠳⣄⠀⠉⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⣮⡿⢟⣷⣠⡿⠾⠛
⠀⠀⠀⠰⡄⠀⠀⠀⠘⢿⣶⣦⠈⣷⣠⠏⠋⠀⢠⡾⣄⣀⣀⣠⣟⠲⢶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠟⠋⣀⣀⢽⡀
⠀⠀⠀⠀⠀⠀⠆⣄⠀⡬⠭⣿⡤⠛⠁⠀⠀⠀⠉⣠⡾⠿⡋⢉⣀⣠⣌⣀⠀⠀⠀⠀⠀⠀⠀⠐⠏⠉⠀⠀⣸⡿⠧⠀⠇
⠀⠀⠀⠀⠀⠀⠘⠛⠀⡷⠀⠈⠁⠀⠀⠀⠀⠀⠀⣝⣀⣼⣿⣿⠟⠛⠉⣹⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠉⢙⠀⠀⣈⡦⠛⠁⣠⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢿⣟⠻⠋⣠⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣤⡾⡿⢓⣮⣀⣁⡀⠀⠀⠀⠀⠀⠀⡛⠒⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠉⣡⣄⠚⢿⡷⢻⣅⠲⠀⠀⢮⡉⠻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠠⠀⠀⠈⠉⢹⡶⠆⣠⠄⣉⠉⠐⠻⠀⠂⠄⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠁⠀⣀⣤⣀⣶⣶⣤⣔⣚⣉⡈⠋⠤⣶⡶⢟⣣⣬⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢲⣄⠀⠀⠀⣠⡾⢛⣿⣿⣿⣤⣍⡛⠛⢿⣿⣿⡷⢿⣟⣿⡿⢛⡿⠟⢙⣛⠀⠀⠒⠒⡐⠒⠾⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⠂⢀⣤⣴⣤⣴⣶⣦⢉⣡⣾⣿⠋⠉⠛⢿⣿⣷⣤⣬⣭⣽⣶⣾⣿⣶⡄⠃⣦⣴⡟⠉⠒⠦⣾⣷⣶⣶⣶⣶⡖⠊
⠀⠀⠀⠀⠀⠀⠀⠚⠟⠛⠟⠀⠠⠀⣤⣿⠛⠉⠉⠿⣿⠿⠟⠉⠓⠶⠄⡁⠚⢿⠿⠟⠛⠉⣤⡾⠃⠚⠛⠛⠛⠓⢦⣤⣬⣿⡟⠁⠠⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠁⠀⠀⢉⠙⠂⡀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⠀⠀⠄⠀⠀⠠⠀⠀⠱⡤⠶⠚⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡴⠒⠂⠉⠀⠲⠴⠇⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠠⠀⣠⣤⣤⣠⢀⣐⣦⡶⠚⠻⢦⠜⣿⣶⠾⣿⠷⠞⠛⣛⣀⢂⣤⢠⣀⣤⡴⠊⠀⠀⠀⠀⠀⠀⠀⠞⠀⢰⣧⣶⣀⣠⣶⣆⡸⢃⠀⠀⢀⡤⠰⡄
⠀⠀⣠⣄⣠⣼⡿⠏⠀⡉⣭⣦⠑⠋⠀⢀⣤⣄⢀⡈⢻⡌⡁⠘⠓⠛⢩⣿⢈⣩⡼⡷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⢿⣿⣿⣿⣿⣿⣿⣷⡘⡀⠀⡇⠀⠀⣇
⠒⠢⣹⣿⣦⡾⢿⣗⣩⡅⠛⢁⣀⡰⡄⠉⣛⡁⢾⣿⣿⣅⢉⣉⣼⠞⣴⠿⢏⠹⠀⣤⠀⠀⠀⠀⠀⡴⣯⢱⣠⠀⡇⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠹⡄⠀⠈
⠶⠟⢹⣇⣹⣷⠾⠏⣉⡅⢄⠿⠿⢢⣶⡞⠻⣷⣿⡟⠛⢼⠇⠈⢻⢀⣶⠺⠦⠵⢶⣿⡄⣼⣿⣦⠀⢉⣶⢴⡷⠊⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⠄⢠⡀
⣷⠏⠀⠙⣋⣻⣴⡜⢿⠿⡘⢳⠄⠉⠻⣷⠚⣿⠉⢁⣈⣶⠸⣿⣦⠀⠿⡆⢠⡄⠚⢁⠀⣉⣤⣿⣏⠋⣉⢹⡇⠀⢀⠎⠸⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⢸⠟⣛⠻
⣡⣼⠿⠟⢙⣿⠿⠿⣴⣿⣤⡤⠐⠦⠄⠙⢦⡈⠈⠿⠛⠉⣦⠄⢠⣤⣶⡎⣤⠀⣆⡉⠩⠁⢽⢠⣾⡆⠽⣶⣇⠲⠈⠀⠀⠟⢡⣿⡿⢿⣿⠉⠈⠀⠀⣾⣿⣿⠟
⠿⠿⢷⣬⡿⠁⢠⣤⡄⢤⣥⣤⣶⣶⣶⡾⠛⢿⠆⠀⠤⠈⢻⡗⠸⣿⢃⣤⣮⡁⣟⣥⠀⠀⡀⠈⢋⢀⣤⡶⠚⡆⠦⡀⠀⠀⠈⠙⠃⢸⡿⠃⠀⠀⣡⣿⠏⠰⢶
⢳⠆⠀⠉⠐⢂⠛⠟⣁⡼⢿⡟⠉⠙⠿⣇⣀⣿⠶⠀⠀⠀⠉⢱⡜⠁⠈⠉⠙⠡⣼⠿⣧⡀⠀⠀⠀⠻⡿⠃⠀⠃⠀⢨⠀⠤⣀⠀⠀⠀⠀⢀⠠⠊⠙⣧⣴⣿⣶
⣤⢀⣠⡴⠞⠙⣿⣦⡁⣠⣄⢠⣠⣀⣐⡛⠻⡏⠀⠀⣀⣀⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠻⡟⠀⠀⠀⣤⠄⠀⠀⠀⠀⠁⠀⣠⣾⣛⣻⣶⣎⠁⠀⠀⡀⠘⠩⢋⣽
⣿⣦⣤⡀⢋⣴⣿⡟⠁⣼⣿⡈⠉⠣⠀⠀⣴⣧⣤⣶⣴⣶⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠲⡀⢠⡷⣎⣠⠀⣄⠀⣀⣼⣿⣿⣿⣿⣿⣿⡄⠀⠰⠁⠀⠀⣠⣾
⠀⠙⣿⡏⢨⣿⠉⠁⠃⠻⣿⡏⠠⡀⠀⣴⣿⣿⣿⡿⠿⠥⠴⠀⠀⠀⠀⠀⠀⠀⣀⣰⢾⣟⠀⠘⡇⣠⣴⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣴⣯⣍⠉
⣠⡴⠊⣥⡀⠁⢀⡀⣀⣀⣨⣿⣤⣿⣾⣿⡿⠁⠀⠀⠀⠐⠂⠁⠀⢀⢐⡀⢴⣾⣿⣿⣟⣭⣤⣶⣿⣿⡿⠟⢁⢀⡠⠄⠀⠉⠛⢿⣿⣿⣿⣿⣦⣤⡾⣿⣿⣿⠇
⠁⣠⣌⢹⣿⣶⡈⠙⠛⣉⣤⡀⠐⢿⣯⠀⣰⡀⠀⠠⠤⣤⣄⡈⡻⢿⣿⣿⣾⣿⣿⣼⣿⣿⣿⣿⡿⠋⠀⢀⠘⠈⠧⠤⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠇⠻⠿⠛
⣷⡟⠉⠙⢿⣇⣤⣼⡿⠋⠉⠙⠗⠚⠷⢿⣿⣷⠀⠀⠀⣿⠃⡞⠀⢯⠈⠘⣫⡿⢿⣿⣿⣿⣿⡟⠀⠀⢠⠄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢿⣷⣦⣄⣀
⣽⠇⠀⠀⠀⢀⢤⡈⠁⢠⣾⣷⡈⣿⠶⣿⠟⠛⠷⡀⣠⢷⠀⠙⠄⠀⠐⠺⠿⠿⠀⣹⣿⣿⣿⡇⠀⠀⠈⠒⠀⠀⠀⠀⠀⠀⡀⠀⢰⣄⠀⢰⣦⡀⢙⣿⣿⣿⣿
⠋⡀⣀⣠⣶⠟⠈⢀⣄⠹⣿⠟⠀⠀⠀⣰⡕⣶⣶⣿⣧⠈⠓⠤⠤⠤⡀⠀⠀⢀⣴⣿⣿⣿⣿⡇⢰⣷⣄⠀⣀⣀⠀⠀⣶⣿⣿⣦⣸⣿⣤⣼⣿⣿⠿⢿⣿⣿⡿
⢈⣿⣬⡀⠠⠶⠈⠉⠙⢦⠀⢀⡀⢺⣿⣿⡇⢈⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⠈⢿⣿⣠⣿⣿⣧⣤⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⠆⠘⢿⡋
⠻⣿⣿⠿⠂⠸⣷⣶⣶⣶⢠⣀⠁⣀⠉⢉⣀⣃⣴⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠙⠿⠿⠛⠛⠛⣙⡆⠀⠻⣿⣿⣿⡟⢻⡟⢿⣿⣿⠉⠃⠻⠟⠀⠀⠀⠀⠀⣿⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢰⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢣⠷⣀⠀⠀⢀⡠⢼⡋⣱⢖⡛⠛⠉⠉⠉⠉⠉⠉⠉⢛⡷⣏⢹⠷⢤⡀⠀⠀⣀⠾⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⡰⢩⠗⡞⠁⠀⠀⠻⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠟⠀⠀⠈⢶⠺⡙⡆⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡾⢸⠁⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠒⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣧⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡟⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢀⣴⣦⡀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⠻⠟⠁⠀⢸⠀⠀⠀⠀⠀⠀⠠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⠀⣀⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡇⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠂⠀⠀⠀⠀⠀⡏⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠒⠒⠒⠉⢣⠎⠀⠀⠀⠀⠀⠀⡇⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⢷⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⡠⣼⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠞⠘⠦⣄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠉⠓⠒⠲⡶⠶⠦⢤⠤⠤⠤⠤⡤⠤⠤⢶⠔⠒⠒⠋⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣼⠀⠀⠀⠀⣇⣀⣀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⠁⠀⠀⠉⠉⠙⠛⠳⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠠⠀⠀⠀⠀⠀⠀⢀⣤⠼⠿⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠋⢹⠇⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⣠⣀⣀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠀⢸⡀⠀⣠⠖⢛⣿⣿⡍⠉⠉⢳⡅⠀⠉⠉⠙⠳⢤⣀⣿⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⢀⡴⣋⣀⣤⣀⣀⠀⠀⠀⠀⢙⣾⣷⣴⣿⣿⣿⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣹⠃⠀⢀⣤⠴⠶⠶⣤⣄⢹⡄
⠀⠀⢰⠿⠊⠁⠀⠀⠀⠈⡿⣠⣤⣀⣾⣿⣿⣿⣿⣿⡟⠀⠀⠀⢀⡇⠀⠀⠀⢠⠦⠞⠁⠀⣴⡏⠀⠀⠀⠀⠀⠉⠳⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⢠⠄⠠⡌⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠙⠛⠛⠉⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⣀⡴⠋⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠹⠦⠤⠤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⣤⣀⠀⢠⣾⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣠⡶⠦⣤⠟⠉⠙⠛⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠗⠀⠀⠀⢀⣀⣀⠀⠙⣿⣿⣿⣧⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠉⠉⠙⠳⢿⣿⣿⠇⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢀⡤⢤⣄⣠⡾⠟⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⣠⡼⠇⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠇⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⡀⠀⠀⠐⢿⣆⣀⣀⣀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠀⠀⠀⠀⣀⣀⡤⠞⠁⠀⢀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠛⠋⠉⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠶⢤⣄⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⣀⣀⣀⣤⣤⣤⣾⠀⠀⠀⠀⣠⠖⠛⠛⠒⠒⠋⠉⠁⠀⠀⠀⠀⠈⢳⡄⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢸⠁⠀⠀⠀⠀⠸⣇⠀⠀⠀⠻⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠘⢧⡀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠉⠉⠉⠛⠲⣄⠀⠀⠀⠀⠀⠀⢸⣆⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⡉⠳⡄⠀⠀⠀⠀⠉⠛⠛⠛⠛⣻⡆⠀⠈⣧⠀⠀⠀⠀⠀⢾⣥⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠇⢀⡟⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⢀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠖⠚⠋⠁

⠀⠀⣀⠀⢀⡠⠤⡤⠄⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⣀⠀⠀⠀⠀⡀
⠀⠀⠈⡄⡁⢀⠀⠀⠀⠀⠀⣈⠄⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⢁⠀⠀⢀⡈⠄⠀⠠⠄
⠀⠀⡐⠀⠀⠀⣍⠑⠂⠉⣎⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠄⠀⠀⣘⠀⠀⠀⠐⠂⠐⠂
⠀⠀⢇⠀⠀⠀⠐⠀⠀⠀⠁⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠉⠀⠱⡦⡀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀
⠀⠀⠀⠓⠒⠠⡤⠀⠠⠠⠤⠐⠁⠀⣀⡀⣀⣀⠀⠀⠀⠀⡀⠤⠒⠒⠂⠧⡀⠀⠀⠀⠀⠀⠄⣴⢁⠆⠀⠀⠀⠀⠀⡰⠀⠀⠈⠉⡂
⠀⠀⠀⠁⠀⠀⡇⢀⠆⠀⠐⢢⠂⠁⠀⠀⠀⠀⠉⠢⣀⠞⠉⠈⠀⠀⠠⢀⠈⢂⠀⠑⠤⠤⠒⡩⣥⠃⠳⠊⢓⠖⡖⡅⢡⣄⣀⠃⡇
⠀⠀⢸⠀⠀⠀⡇⢰⠀⠀⡇⡈⠀⠀⠀⠀⡄⠀⢤⠀⢹⠀⢀⠀⠀⠀⢀⡀⢀⠀⠆⢀⠴⣎⡹⠤⠀⠂⠀⠠⠄⣠⠀⣄⠉⠉⠁⠀⠂
⠀⠀⢸⠀⠀⠀⣠⠜⡨⢁⠀⠣⣢⠀⠀⠀⠀⠀⠄⠀⠸⠀⠉⠀⠀⠀⠈⠁⠀⠀⢛⠹⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⢀⠃
⠀⠀⢸⠀⠀⠀⠩⠀⠀⠉⡀⠀⣆⠠⠄⠀⠀⠀⠠⠀⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢀⡀⡄⠀⢀⠋⠆
⠀⠀⠐⠂⠀⠀⠀⡆⠀⢘⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠲⢋⠀⠀⠀⠀⡌⠀⠅⠀⡄⠀⠀⡀⠀⠀⠀⠠⢄⡼⠤⠤⠤⠵⠏⡐⡎
⠀⠀⠀⡅⠀⠀⠀⠘⡀⢈⡃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢘⠀⣶⠦⠐⣇⠜⠁⢈⡕⠃⡌⠀⠀⡆⠨⠒⠠⠃⠀⠀⠀⠀⠀⠈⠱
⠀⠀⢠⢰⣀⠠⠤⠀⠋⠁⢡⣀⡄⠀⠀⡀⠀⣴⡎⠛⡰⠉⠉⠁⠀⠀⠀⢀⠔⠋⠀⠀⣈⠦⣸⢦⢁⠀⡀⣰⢠⡇⡄⢰⠀⣄⠀⡀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⣤⣼⠀⠘⣿⠶⢾⠿⠛⠃⠰⠁⡀⠀⠀⠀⣠⢰⣁⠀⠀⠀⠀⠛⠀⠀⠰⢸⠸⡇⠙⡌⢣⢧⡜⣼⣿⣰⠇
⠀⠀⠸⠀⠘⠃⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠀⠀⠙⠈⠛⢻⣶⣄⠀⠀⠀⠀⠀⠎⠃⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠸⠀⠀⢀⠤⠒⠉⠁⠀⠁⠉⠒⢄⠀⠀⠀⠀⡇⠀⠀⠠⠀⡀⢂⠀⠀⣿⣿⣿⣷⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⢄⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠀⠀⠀⡃⠀⠈⠀⠀⠀⠀⠀⢰⣿⡟⠹⢛⠃⠀⠀⠰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠒⠒⠓⠒⠠⢄⡀⠀⠠⢶⣻⣿⣧⣀⠎⠀⠀⠀⡀⢠⡀⠀⠂⠄⠠⠐⠈⠀⠀⢠⠀⡄
⠀⠀⠰⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⣀⠀⠀⠀⠙⠄⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠇⠈⡧⠐⠂⠉⠒⠀⠈⠐⠠⣜⠀⡇
⠀⠀⠃⠀⠀⠀⠀⠿⣼⣇⣴⣀⡄⣠⠀⡌⠀⠉⠀⠃⠉⠢⡀⠀⠞⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠃
⠀⢀⠀⠀⠈⠆⠀⠀⠈⠙⣿⣿⣿⡏⡸⡆⠀⠃⠀⠀⠄⠀⠈⠲⣆⠠⠆⠒⠒⠊⠉⠁⠉⠁⠉⠓⠅⣔⠢⠀⠀⢠⡀⠀⠀⠀⠙⢺
⠀⢨⠀⠈⢆⣰⠀⠀⠀⠀⠈⠻⡿⢀⡆⠁⠀⠀⠀⠀⠀⡀⡘⠁⠀⠀⠀⢀⡀⠄⠀⡀⢀⠀⠀⡀⠐⠋⡐⠀⠀⠘⠁⠄⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡎⠈⠁⠈⠀⠀⠈⠀⢁⠀⠀⡀⠈⠀⠀⢰⠜⠦⠹⠊⠎⠎⢱⠁⠀⢤⣤⣤⠤⠴⠂⠠⠀⠀⡄
⠀⢸⠀⠀⠀⠀⠀⠈⠈⠂⠁⠀⣰⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢀⠔⠁⠀⣀⠀⢶⠀⢰⠈⠤⢐⠀⠀⠀⠀⠀⠀⡃⢠⠁
⠀⠈⡀⢀⠃⠒⠀⠀⠀⠒⡀⡰⠁⡄⡤⠤⠤⠄⠠⢄⠜⠀⡀⠄⣀⣤⡁⠀⠒⣁⣄⠀⠀⠀⢀⡬⢤⡂⠀⠐⠖⠦⠀⠀⣉⡉⠨⠆
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣘⣿⡾⣯⣳⣔⣄⡞⣿⡶⣿⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⠀⠀⠀⠀⠂⢠⣀⣹⣿⣾⣿⣿⣷⣿⣿⣿⣹⡭⠿⠿⠷⣯⡀⠀⠠⠀⠀⠰⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠿⠋⠀⠰⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠈⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⣠⠞⠉⠁⠁⠉⠀⠈⠛⠛⠙⠉⢻⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣴⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠏⠀⠀⠀⣀⡤⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣄⣀⠀⠀⠀⠈⢯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠉⠳⡄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠘⣇⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣼⠁⠀⠀⢀⣠⣤⣄⡀⠀⠀⠸⡆⠀⠀⣰⠃⠀⠀⢀⣤⣤⣄⡀⠀⠀⠈⣇⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠐⣿⣿⣿⠟⠻⡆⠀⠀⣷⠀⠀⣟⠀⠀⠔⣿⣿⣿⠛⠻⡄⠀⠀⢸⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡇⠀⠀⢧⠀⠀⠀⢿⣿⣿⣶⣴⡟⠀⢀⡏⠀⠀⢻⠀⠀⡄⢿⣿⣿⣦⣾⡇⠀⠀⡾⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡇⠀⠀⠈⢧⡀⠈⠢⢙⡛⠛⠟⠁⣠⠞⠀⠀⠀⠈⢳⡀⠈⠢⢙⢛⠻⠋⠀⢀⡜⠁⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⠀⠀⠀⠉⠳⠤⠤⠤⠤⠴⠚⠁⠀⠐⠛⠂⠀⠀⠉⠓⠦⢤⡤⠤⠴⠚⠉⠀⠀⠀⠀⣸⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣿⣿⠀⠀⠀⠀⠀⠀⠰⡟⣶⣶⠒⠒⠒⣶⣶⣶⣶⡖⠒⠒⣻⣿⣏⣛⣆⠀⠀⠀⠀⠀⠀⢀⣿⠃⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣤⣾⣿⣿⣿⣿⣷⣤⣴⣿⡿⢿⣿⡟⠀⠀⠀⠀⠀⠀⣼⣏⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣽⣿⣦⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⡏⠀⠙⣿⣿⣿⣿⣟⣠⠴⠛⠀⠀⠀⠀⠀⠀⣰⣿⣁⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢷⡿⠿⠙⣶⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⣼⠋⠿⢿⡾⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣔⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠈⠁⠙⠻⢻⣿⣶⣤⡄⣲⣤⣠⣄⣄⣀⣀⣤⣬⣴⣶⠿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⡿⠻⣿⠛⠟⠛⠿⣿⠻⠟⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⣶⠾⠃⠈⢻⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠹⢶⣶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣼⣿⣾⣁⣀⣀⣀⣼⣶⣶⣶⣶⣶⣶⣶⣾⣇⣀⣀⣀⣸⣮⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠶⣤⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢩⠆⢘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠨⠻⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠲⠶⠒⢲⠶⡒⠲⠾⣕⠶⠶⣄⡸⡿⠛⠚⣉⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣶⣶⢢⣶⢤⣤⠬⢉⣉⠡⠤⠄⣒⢚⣉⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⣤⣿⣿⣿⣿⣿⣿⡶⣿⠐⠾⠿⠛⣈⡁⢈⣡⣼⡇
⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠰⣾⣿⣿⣿⣿⣷⣄⣀⣀⣴⣤⠰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⢹⣹⣦⠀⠙⠋⠉⠀⠀⠙⠁⠀⠸⣼⡿⣻⠏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠈⡀⠀⢼⡙⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡟⢰⣿⡀⡏⣧
⠀⠀⠀⠀⠀⠀⠀⣿⡄⢃⠉⢎⣤⡿⢶⡓⠒⠢⢤⠴⠶⠟⠉⠙⢦⣾⠃⠈⠓⣽
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠈⢾⡇⠈⡝⠀⠀⢰⠈⠙⣣⠀⠀⢈⡁⢤⡀⢈⠙⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡀⠀⡐⠀⢀⡂⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡄⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⠃⠀⡇⠀⢸⠇⠀⠀⢠⠀⠀⢸⠀⠰⠀⠀⡀⢻
⠀⠀⠀⠀⠀⠀⠀⢿⢠⡆⠀⣸⡏⠀⢠⣿⠀⢸⡀⠀⠀⣼⠀⠀⣼⡄⢰⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⣼⣸⠇⢠⡿⠀⣠⡿⡿⠀⠠⠀⠀⡰⠁⠀⠀⠈⢻⡌⡄⢀⣧⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢹⣿⢃⣾⠃⣴⡿⢁⣵⠀⢸⡤⣴⣧⡴⠄⠀⢰⣄⢻⡇⠸⣿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⠟⢀⣿⣿⠀⣸⣄⠀⠀⣀⣷⠀⣽⣿⣦⡗⢀⡟⣿⣧
⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣏⣤⢼⣿⣙⠶⢿⡛⠀⠀⠻⠟⠄⣡⣿⣻⣧⢸⣷⣿⣿
⠀⠀⠀⠀⠀⠀⢿⡋⢀⠉⢿⡿⡷⠀⣾⣿⣿⣿⡇⠀⠀⢨⡽⣶⣿⠀⡨⢻⣾⣿⣿⡿⡆
⠀⠀⠀⠀⠀⠀⠸⣧⣙⣦⠘⢻⣿⣐⣯⢿⣇⡴⢹⣿⣿⡏⣉⢻⣿⣦⡐⠛⡗⠙⢿⣶⡇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡍⠀⢸⣿⣍⣁⣸⣿⡉⢨⣏⣸⣠⣴⣟⡄⠀⣑⡞⠀⠻⢻⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣷⣷⣰⣆⣿⣟⠻⣟⣻⣿⣿⡟⠾⡭⢘⣷⣤⣄⣹⡧⣀⣴⠟⣫⣿⠁
⠀⠀⠀⠀⠀⠀⡀⢀⣹⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣯⣀⢀⠸⣇⢠⣁⣾⣿⣛⡁⠀⠃⠘⡇
⠀⠀⠀⠀⠀⠀⠻⣯⣀⣼⢹⣿⡇⣷⣸⡇⢻⣿⠹⡿⢿⣻⣿⠿⢟⡛⢻⣼⣿⣿⣆⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠙⣿⠏⠀⢹⣿⡻⣌⠛⠻⠱⢠⠊⠀⡟⢉⢄⣤⣿⢿⣿⣿⣷⣾⢰⠀⠀⠙⡂
⠀⠀⠀⠀⠀⠀⢀⣤⣿⠂⠀⢘⣿⢳⡌⢳⣶⠄⢠⣄⡄⢠⣴⡟⠉⣥⣾⠋⣴⡟⣹⢼⢢⣀⣠⣠
⠀⠀⠀⠀⠰⣿⣿⠋⠩⣀⣠⣬⣿⢜⣿⣎⠙⠁⠉⠈⠁⠘⠛⠃⠰⠟⠇⢸⡿⠃⣘⣸⠀⢙⣧⣟
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢹⠀⠙⠒⠿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⢀⣿⡿⠀⠘⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣶⡶⠤⠤⠤⠤⣔⣶⠿⠓⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⢀⣀⣤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠐⡂⡀
⠀⠀⠀⠀⡶⠪⠉⠀⠁⠈⠁⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⡆⠀⠀⠀⠸⠻⠀⠀⠀⠀⠀⠀⢰⠋⣉⠻⡄⠀⠀⡰⣼⡿⢸⢰
⠀⠀⢐⠏⠀⢀⡀⣀⠀⣀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠈⢛⡦⠠⠤⢼⡀⠀⠀⠀⠀⠀⠀⠸⢤⣩⠐⣇⠀⠀⠑⡌⢨⠕⠁
⠀⠀⣮⠀⠀⠈⠁⠈⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⢀⠌⠠⢂⣅⠢⠙⠄⠀⠀⠀⠀⠀⠀⠀⠈⠣⡈⢦⠀⠀⠃⡌
⠀⠀⢷⠀⠀⠀⠈⠓⠚⠁⠀⠀⠀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠓⠚⠀⢧⡀
⠀⠀⠰⣆⠐⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠰⠖⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⠀⠀⠢⡱⡀
⠀⠀⠀⠉⠶⢄⣀⣀⢀⣀⣤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⡃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠳⠚⢧⡞⠋⠁⠈⠁⢢
⠀⠀⠀⠀⠀⣾⣿⡇⠉⠿⠿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⢠⠀⡁⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠤⠤⠔⠒⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣁⣀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠲⣶⣄⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠉⠓⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⣫⡕⠀⠂
⠀⠀⠀⢠⠀⡁⠉⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⡴⢋⠁⠀⠀⠽⠀⠸⠆⠀⠀⠉⠒⣄⠀⠀⠀⠀⠀⠀⠐⢀⠈⠁⢀⠂
⠀⠀⠀⠀⠰⠀⠀⢰⣶⡆⠀⠀⢀⠀⠀⠀⠀⢸⠀⢿⠀⠰⣁⣤⣠⣄⣄⣥⠀⢸⢃⢘⠀⠀⠀⠀⠀⠀⠀⠀⠐⣋⠔
⠀⠀⠀⣌⠀⣀⠀⠈⣉⡁⠀⠀⢠⢦⠀⠀⠀⢸⠀⡏⢦⠀⠻⢻⡟⢿⣻⠇⢀⡼⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⢀
⠀⠀⠈⣎⠀⠀⠀⠀⣀⣀⡤⠂⠠⠶⠀⠀⠀⠙⠛⠃⠈⠳⣄⠀⢀⡀⠀⣠⠊⠘⠳⠛⠀⠀⠀⠀⢀⣶⣀⠔⠁⠀⠀⠀⠤⠶
⠀⠀⠀⠈⣇⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⡋⣽⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠀⠙⠒⠚⠁⡀
⠀⠀⠀⠀⠀⠀⠇⠂⠀⠘⠸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⡇⢸⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⢰
⠀⠀⠀⠀⠀⠉⠀⠈⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠁⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠁⠘⠖
⠀⠀⠀⠀⠀⠘⠆⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠠⠦⠀⣀
⠀⠀⠀⠀⢀⠜⣤⠀⠀⠠⡶⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠛⠁⠈⣙⣳⡅⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⢠⡀
⠀⢀⡀⠀⡘⠂⣤⣠⣤⣤⡆⠈⢠⠀⠲⡆⠀⠀⠠⢶⢦⠋⣺⣿⠢⣾⣿⢽⡏⣮⡟⠃⠀⠀⠀⠀⠸⠀⠀⠿⠁⠀⠸⠏⠀⠀⠰
⠀⠻⠤⠆⡁⠀⠈⠛⠛⠉⠀⠀⠀⠘⠁⠁⠀⠀⠀⠈⢻⡀⠀⣈⣀⣙⣿⠋⠀⡏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢻⣟⣿⣿⠀⡼⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⢠⡻⢄⠀⠀⣠⠾⡁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢐⠀⢠⠄⢢⠀⠮⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⠀⠑⠖⠁⣀⢡⡀⠀⠀⠀⠀⠀⠰⢆⣀⡀⣀⣀⣀⣀⣀⣀⣀⣠⠄
⠀⠀⠀⠀⠘⠀⠀⠘⠀⠘⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠁⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠖⠂⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⠀⠀⠀⠀⢀⡰⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠎⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠉⡤⢤⠈⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠐⣴⡂⠂⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠐⢒⡊⢀⠓⢒⣠⠀⠠⣄⠀⠀⠀⠀⠀⠀⡎⠀⠈⢾⡷⠃⡀⢸
⠀⠀⠀⠀⠀⡀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠀⠈⠁⠈⠁⠈⠁⠐⠀⠈⠀⠀⠀⠀⠀⠀⣷⣾⠷⣦⡶⢿⣿⡾
⠀⠀⠀⠀⢀⠀⢷⣄⣀⣠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠙⠻⠾⠾⠟⢉⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠉⠉⠡⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠨⠴⡵⠂
⠀⠀⠀⠀⠀⢰⡆⣤⠄⣴⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣕⡠⠄⠰⡚
⠀⠀⠀⠀⠀⠘⠁⠙⠀⠙⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠀⠑⠊⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⠶⠶⠿⠿⠿⠿⣿⠿⢶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠉⠛⠻⢿⣶⣤⣄⡀
⠀⠀⢀⣠⣴⣾⠿⠛⠉⠁⠀⢻⣆⠀⠀⠀⣀⣤⡶⠶⠟⠛⠛⣿⠀⠀⠀⣀⣀⣀⠀⠀⠀⠈⠙⣿⣿⣷⣤⡀
⢀⣴⡿⠟⠋⠀⠀⠀⣠⣴⡾⠛⢻⡆⠀⠞⢹⡇⠀⠀⠀⠀⠀⢻⡀⠀⠀⢸⠋⠛⠛⠛⠶⣦⣴⣿⠃⠈⠙⠻⣷⣄⡀
⠻⣿⡄⠀⠀⠀⢠⣾⡿⠋⠀⠀⠈⢷⠀⠀⠀⠷⠶⠾⠛⢻⡆⢸⡇⠀⠀⠘⠛⠶⠶⢦⡀⠀⣽⡏⠂⠀⢀⣴⡾⠿⠃
⠀⠈⠻⣦⠀⠀⠉⠈⣷⠀⠀⠀⠀⠘⣇⠀⠀⢀⣤⡶⠶⠛⠁⠀⣇⠀⠀⢰⡖⠛⠛⠛⠀⢠⡿⠀⠀⠀⣿⠃
⠀⠀⠀⠘⣧⡀⠀⠀⠹⣆⣀⣠⣤⡄⢹⡄⠀⠋⢻⣦⡤⠤⠶⢾⣿⠀⠀⠘⠷⣤⣤⣄⣀⣸⠇⠀⠀⣼⠇
⠀⠀⠀⠀⠘⣧⡀⠀⠀⠉⠉⠀⢈⣿⠈⣷⠀⠀⠀⠀⠀⠀⠀⢠⣿⣄⠀⠀⠀⠀⠀⠉⠻⣿⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⣤⡾⠛⠀⠸⣷⣤⠴⠶⠞⠛⠛⠛⠛⠛⠿⠶⢶⣤⣤⣀⣸⡏⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢻⣦⣀⣀⣠⣤⠶⠞⠛⠛⠛⠛⠛⠛⠛⠶⠶⣤⣄⡉⠙⠿⣧⡀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠹⣦⣴⣾⣿⡿⠿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡈⠙⢦⡀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢉⣄⠀⠀⠀⠈⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⢶⣄⣻⣆⠀⠀⢈⡷
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠻⠿⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⣈⠉⢻⡿⣧⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠙⠁⠀⡇⠘⢿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡘⣆⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢀⡸⡟
⠀⠀⠀⠀⠀⠀⣠⣾⠃⠈⠑⠒⠤⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠤⠔⠊⠔⣿
⠀⠀⠀⠀⠀⢸⡏⠉⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣀
⠀⠀⠀⠀⠀⠸⣧⠈⢷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡉⠉⣷
⠀⠀⠀⠀⠀⠀⠙⣧⠀⢏⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⡟⠁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠘⣷⠈⢧⡀⠙⠛⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠛⠁⣠⠞⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠳⡀⢀⠀⠀⠀⠀⠈⠉⠉⠑⠒⠒⠊⠉⠉⠉⠀⠀⠀⢀⠀⡠⠉⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⣀⡀⢿⣧⡀⠘⢦⡀⠀⠐⠀⠀⢀⡀⠀⢀⡀⠀⢀⡀⠀⠀⠂⠀⣠⠞⠁⣴⡟⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣰⣿⣿
⠀⠀⠀⠀⠀⢀⣴⡿⣿⣾⣿⣷⡄⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⢀⣾⣿⣤⣴⣾⣿⣿⡀⠀⠀⠀⠀⣴⣿⠿⣿⣧
⠀⠀⠀⢀⣴⣿⠟⠀⠈⢿⣿⣿⣿⣦⣄⠀⠉⠓⠤⢤⣀⣀⣀⣀⣀⡀⠠⠔⠋⠁⢀⣴⣿⣿⣿⡿⠟⠛⠎⣿⡇⠀⠀⠀⣼⣿⠃⠀⠘⢿⣆
⠀⠀⣠⣾⡟⠁⠀⠀⠀⠈⣿⠋⠈⠛⠿⣿⣦⣄⣀⠀⠀⠁⠀⠀⠉⠀⠀⣀⣤⠾⠟⢋⣿⡟⠁⠀⠀⠀⠀⠸⣿⡄⠀⣴⡿⠁⠀⠀⠀⠀⣹
⢀⣼⣿⠋⠀⢀⣼⣷⣀⣴⡿⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣶⠶⠶⠿⠛⠛⠉⠀⠀⠀⢸⡿⠿⢿⣷⣦⡀⠀⠀⢻⣷⣾⠏⠀⠀⠀⠀⢀⣾⡟
⣾⡿⠁⠀⢠⡿⠋⠀⠛⠋⠀⠀⢀⣤⣤⣤⣤⣴⣾⣿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀⣀⠘⣿⣿⡄⠀⠈⣿⠃⠀⠀⠀⠀⣴⡿⠋
⣿⡁⠀⠐⢿⡇⠀⠀⠀⢀⣀⣀⣼⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣿⡏⠉⠉⠉⠛⠛⠛⠛⠻⠿⠿⠿⣿⣿⠀⠀⢹⠄⠀⠀⢠⣾⠟⠁
⣿⠀⠀⠀⢹⡇⠀⠀⣴⡿⠟⠛⠛⠻⠟⠛⠻⠿⠛⢿⡆⠀⠀⠀⢺⣇⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⣠⡿⠃
⣿⠀⠀⠀⣼⡇⢀⣾⣿⣥⣄⠀⠀⠀⠀⠀⢀⣀⣀⣼⡇⠀⠀⠀⣼⡿⠛⣿⡟⠁⠀⠀⠀⠀⠀⣠⡾⢿⣿⡆⠀⠀⠀⣴⡿⠁
⣿⠀⠀⠀⣿⣷⡿⠋⢈⣿⠏⠀⠀⣸⣿⠿⠛⠛⠉⠉⠁⠀⠀⠀⣿⣥⡾⠋⠀⠀⠀⠀⢀⣴⣟⣉⣀⣿⣿⠁⠀⠀⣸⣿⡇
⢿⡆⠀⠀⠈⠀⠀⣠⡿⠋⠀⠀⢀⣿⠃⢀⣶⠛⠛⠛⠛⢿⡇⠀⣿⡟⠀⠀⠀⠀⠒⠛⠋⠉⠙⠛⠛⣿⡿⠀⠀⠀⠟⣸⠁
⢸⣷⣦⣤⣤⣤⣾⣟⣁⡀⠀⢀⣸⣿⡄⠀⠛⠲⠶⠶⠾⠛⠃⢀⣽⣇⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣴⣿⣧⣄⣀⣀⣤⣿
⠈⠉⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣶⣦⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠾⠟⠛⠉⠉⠉⠉⠉⠉⠉⠙⠛⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⣾⢏⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠈⢻⣇
⠀⠀⠀⠀⠀⠀⣸⡏⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⣿⢁⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣷⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⣿⠀⣿⣶⣖⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠒⣂⠀⢹⣿⡰⣣⡦⠀⣼⡇
⠀⠀⠀⠀⠀⠀⣿⠀⢻⠟⠛⠉⣀⣀⣈⢙⠒⠒⠒⠒⡠⠤⠞⣁⡴⢂⡀⣉⣁⠰⢾⡗⣿⣿⠀⣿⠇
⠀⠀⠀⠀⠀⣴⠻⡆⣼⢃⣶⣿⣿⣿⣿⣿⣿⣦⡯⠁⠂⣤⣭⣵⣾⣿⣿⣿⣿⣷⡀⢿⣶⡏⢠⡟⢛⣦
⠀⠀⠀⠀⢰⣿⡄⢻⣿⢿⣿⡿⢛⣉⡙⢿⣿⣿⠅⢤⣠⡹⣿⣿⡿⢛⠛⡳⣝⣿⣷⡘⣿⠁⡼⣠⣿⣹⡆
⠀⠀⠀⠀⣼⠹⣷⡼⢟⣿⣾⣧⣈⣛⣃⣼⡿⣿⡆⣤⡸⣿⠟⢿⣄⣘⡻⣃⣸⣿⢿⢇⡘⠆⢱⣏⠋⣿⠃
⠀⠀⠀⠀⣿⢰⣿⡀⠐⠿⡿⠿⠿⠧⢯⣿⡇⡜⣸⣿⣿⣜⢷⣷⣞⠛⢩⢭⠴⢚⣣⠞⠀⢠⡟⣿⡇⡿
⠀⠀⠀⠀⠹⣎⠻⣧⣀⣀⣀⠄⠉⠉⠁⠉⢀⣾⣿⣿⣿⣿⣧⢻⣟⣆⣈⢉⡙⣉⢩⣤⣶⡿⢱⠟⢱⡇
⠀⠀⠀⠀⠀⠙⣧⠆⢙⢻⣿⢀⡀⢎⠄⠀⠘⠛⠟⢡⠻⠿⠟⠘⠁⣄⣠⣾⡇⣿⣿⣿⡟⡇⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠹⣦⣼⡆⡿⣾⣷⡏⡆⡄⠀⢀⠀⢀⠀⠀⠀⠀⠀⠈⢻⡏⠘⣿⠋⡿⣴⣷⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⢳⣹⣿⡴⠷⣷⠷⣾⠷⡾⢤⡴⢾⡾⣦⣴⣼⡹⢰⣿⣸⢿⡇⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⢹⣿⣷⣤⣿⡀⣸⡀⣇⣸⣠⣼⣄⣴⣤⣿⢹⣴⣿⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⠀⢟⠛⠿⠿⢿⠿⡿⢿⣿⠟⣿⡟⢻⡿⠟⣨⠟⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠰⡌⠻⣶⢧⣼⣠⡄⣰⣿⣀⡦⠵⢊⣴⡞⢁⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡉⠰⡉⠀⠲⣦⣬⣥⣤⣤⡶⣶⡟⠁⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡿⣦⡀⠚⣄⠀⠉⢩⣤⣴⡆⠈⢀⣴⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢾⡜⣠⣶⢰⢠⣤⣠⢴⢾⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠛⣷⠇⠸⡿⠞⠀⠀⠈⠋

⠀⠀⠀⠀⠀⣀⣠⡄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢐⢢
⠀⠀⠀⢠⠖⠉⠀⠈⠀⠉⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⡋⠗⠀⠀⠀⠀⠀⠠⠎⣩⠹⠂⠀⠀⠀⣼⡗⡥
⠀⠀⢰⠇⠀⣤⠀⡄⠀⡀⠀⠈⣣⠀⠀⠀⠀⠀⠀⠀⠈⢋⠦⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠣⠬⡔⢇⠀⠀⠑⡄⠨
⠀⠀⠧⠀⠀⠀⢀⠀⢀⠀⠀⠀⢙⡄⠀⠀⠀⠀⠀⠀⢠⠊⠔⢠⡄⢂⠑⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠣⡀⠀⠁⡀
⠀⠀⠳⠀⠀⠀⠈⠉⠉⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⢀⠀⠢⠀⠠⠊⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠙⠉⠈⠣
⠀⠀⠙⣇⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⣐⣀⡀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢀⡀⢀⢳⠦⣀
⠀⠀⠀⠀⠓⣦⣤⠤⣤⣴⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡘⠉⠳⠊⠁⠀⠀⣈⣂
⠀⠀⠀⠀⠐⠛⠛⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣔⣃⠄⠰⣀⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠈⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡄⠂
⠀⠀⠀⠀⠹⠟⠦⠀⠀⠸⠽⠅⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⡄⠀⢀⠀⠉⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠺⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣠⠀⢀⠀⡁⠀⠈⠀⡀⠀⣤⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⠀⠂⠀⠀⠀⠐⠀⠸⡀⠀⣶⣷⣾⣾⣶⠇⠀⣇⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠠⡳⠀⠒⠒⠂⠒⠀⢀⠀⢫⡀⠀⠀⠰⣤⠆⠰⣄⠉⠚⠀⠃⠋⢠⠖⢭⣤⡁⠀⠀⠀⠀⢀⠀⢀⠄⠂⠐⠄⠀⣄
⠀⠀⠀⠒⠀⠀⠀⠐⠒⠊⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⡖⡆⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⠁⠤⣀⣤⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡇⡇⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠂⠀⠄⠀⠠⠥⠂⠀⠀⠀⠀⠀⠀⠀⠀⠺⠤⠤⠃⠑⠂⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⠁⡠
⠀⠀⠀⠀⠀⢠⡀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⡖⣀⠀⠀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡵⠖⠂⠐⠲⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠑⣀
⠀⢀⠀⠀⠰⠀⣁⣀⣀⣬⠐⢡⠀⠴⣄⠀⠀⠀⣴⢤⠊⣶⣷⢀⣾⣿⣿⢣⣸⠗⠀⠀⠀⠀⢰⠀⠀⠶⠀⠀⠰⡆⠀⠈⡄
⠀⢪⠠⢈⡁⠀⠙⠿⠟⠋⠀⠀⡀⠂⠈⠀⠀⠀⠀⢾⠀⠉⡉⣀⢻⡿⠋⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢧⠀⢷⢿⣿⡷⢀⠎⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠳⣄⠀⠀⡠⢟⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠡⠀⠤⠀⡄⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢨⠠⠀⠑⠎⠀⡈⣄⠀⠀⠀⠀⠀⠀⢆⣀⡀⡀⣀⠀⣀⢀⣀⣀⠠
⠀⠀⠀⠀⠈⠀⠀⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠒⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⢀⠀⠀⢀⡼⠓
⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⡄⣤⡈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠠⢮⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠈⢉⡀⣉⠉⢁⠖⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⣁⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠰⣿⣻⠶⢾⣻⡿⠇
⠀⠀⠀⠀⠀⠈⢻⣤⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠐⡫⠊
⠀⠀⠀⠀⠀⣼⠃⣶⠘⣇⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⠮⠔⠀⠘⢁
⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣌⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢰⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣽⡄⠀⠀⠈⡧⠠⡼⠠⠃⠰⠦⡞⣍⠀⣀⣠⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡿⢃⡠⠴⠢⢄⡀⠀⣀⠤⠖⠤⡀⠀⠑⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣃⡾⡆⠀⢀⡀⢱⣰⣷⠁⢀⣀⠈⡆⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠈⢳⣕⠄⠻⣟⠜⠘⣏⡦⠘⢿⡰⠁⠀⠀⠐⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡗⢱⣶⣭⣭⣉⣁⣀⣀⣈⣭⣭⠭⡴⣶⡆⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⡇⢿⣿⣿⣶⣿⣶⣿⣶⣿⣶⣿⣶⣿⣿⠇⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠾⠉⢻⣿⣿⡘⢯⡟⢻⡿⢿⡿⠿⣿⡿⢿⣟⡻⠏⠀⠀⢀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡏⠀⠀⢠⣿⣿⡿⠢⠉⡛⠓⠚⠓⠒⠛⠛⠋⠉⠀⠀⠀⠀⣿⡄⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡀⠀⠀⢾⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣷⡄⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡾⢿⣇⢀⡀⠘⣿⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣁⡀⡸⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣾⠉⢻⠻⣿⣿⣿⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠹⡏⢉⣿⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠉⠀⠀⣼⣿⠁⠀⠙⣷⣶⣤⣀⣄⣀⣤⣠⣾⡏⠀⠈⠳⠀⠈⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠏⠛⠻⠿⠿⠿⠟⠛⣿⣷⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣂⣚⢛⣉⣀⢘⡄⠀⠀⠀⠀⠀⠀⠀⣼⣿⣫⣄⣸⣦⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠿⠿⠿⠿⠿⠿⠿⠛⠟⠋⠙⠛⠿⠛⠿⠿⠿⠿⠿⠿⠟⠛⠓

⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡟⠀⡠⠉⠙⢿⣿⣿⣿⣿⠿⠛⠉⠉⠀⢀⣀⡀⠀⠉⠙⠻⢿⣿⣿⣿⠟⠋⠀⣡⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣦⣀⠈⡻⠋⠁⠀⠀⡠⠊⠀⠀⠀⠀⠀⠁⢀⠀⠀⠈⠛⢁⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⡐⠀⠀⠀⣀⣀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢄⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⡀⡀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡿⠟⠁⠀⠀⠀⠀⠀⠘⡀⡀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢂⠄⠀⠒⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⢄⠀⠠⢻⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿
⣿⣿⣿⡇⠀⠈⢿⣿⣿⣿⡇⠀⠂⡂⠁⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠊⠰⣁⠀⣿⣿⣿⣿⠁⠀⢀⣿⣿⣿
⡿⠋⠛⢷⡀⠀⠈⣿⣿⣿⣇⠠⠂⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠢⢀⣿⣿⣿⡇⠀⢀⡾⠋⠉⠙
⡇⠀⠀⠀⠈⠂⠀⠛⠛⠋⢁⠔⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⢄⠈⠙⠛⠁⠀⠋⠀⠀⠀⢀
⣿⠶⠦⠤⠤⠀⠀⠀⣠⣶⣷⠀⠈⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠈⡀⢀⣿⣦⣄⠀⢀⠀⠒⠒⠚⠻
⣇⠀⠀⣀⡔⠈⠁⠙⠻⢿⣿⡤⠁⠐⠀⠀⠀⠀⠀⠋⠐⠠⠄⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠈⡀⠐⣸⣿⡿⠋⠁⠀⠈⢦⣄⣀⣠
⣿⣿⣿⣿⣌⣢⣄⣀⡀⠀⠉⠁⠀⠃⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⢀⣀⣤⣴⣮⣼⣿⣿⣿
⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿
⣿⣿⣿⣿⣷⣀⣀⣀⡴⠊⠀⠆⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⢠⠀⠑⠢⣄⣀⣤⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⣼⣆⣠⣿⡟⠁⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢶⣿⣿⣄⣼⣧⡀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⠃⠀⣾⣿⣿⣷⣶⣦⣤⣤⣤⣤⣴⣶⣶⣿⣧⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⡇⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢠⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠐⠀⠀⠈⠙⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠘⠋⠀⠀⠂⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⣠⣴⣾⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣄⡀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⣀⠀⣀⣀⠆⠀⠀⠀⠠⣄⣰⣤⣠⠄⠀⠀⠀⣤⠖⠲⣄⠀⠀⠀⠀⠀⢀⣴⣤⣄⢀
⠀⠀⡼⢫⠀⡌⢳⡀⠀⠀⢹⡟⣭⡙⣷⡄⠀⠀⣶⣿⢫⡛⣿⡴⠀⠀⢸⣧⣘⣡⠟⡀⠀⣠⣤⢶⣾⣽⠿⣭⣿⣀⡀
⢶⣸⠳⣶⣶⣶⠟⣧⡶⠀⠘⣧⣭⣵⢿⣇⠀⢐⣾⠟⠳⢚⢻⣟⡅⠀⣏⠀⣀⣀⢲⡟⠁⠈⠈⢸⣿⣾⣿⣾⠁⠸⠻
⠀⣿⣷⣶⣶⣶⣾⣿⠀⠀⠀⡿⣿⣐⣂⣿⠀⠈⣿⣰⣤⣤⣼⣿⠁⠀⠙⢿⣟⣿⡏⠀⠀⠀⠀⡿⠉⠉⠙⣿
⠀⠈⠙⠋⠋⠛⠙⠃⠀⠀⠼⠿⠻⠟⠿⠛⠂⠀⠏⡏⠽⠹⠹⠿⠀⠀⠠⠼⠉⠙⠧⠀⠀⠀⠤⠇⠀⠀⠀⠸⣄
⠀⣠⣤⡄⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄
⠀⠻⡟⠻⣀⣰⠛⠿⠟⠀⠀⠛⣧⣶⣶⡏⠀⠀⠀⠀⠻⣦⣤⣤⣤⣼⠆⠀⢰⢋⡉⡙⡶⠀⠀⠘⢷⡞⢻⣾⠃
⠀⠀⠀⣾⢩⡍⢳⡀⠀⠀⢀⣾⣕⠂⠂⣹⣆⠀⠀⠀⠰⣼⢩⢙⣃⠄⠀⠀⢼⣄⣀⣠⠇⠀⠀⠀⢸⢗⢾⣽⣇
⠀⠀⠀⠻⣦⣴⠟⠀⠀⠀⣾⣿⣿⣟⣹⣿⣿⡄⠀⢠⠴⣟⣓⢿⢧⠀⠀⠀⠁⣼⢹⠉⠀⠀⠐⠧⣼⠓⣎⣹⣿⣠⠖
⠀⠀⠀⠐⠃⠈⠁⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⢠⣿⣿⣿⠈⠀⠀⠞⠀⡇⢸⢠⣆⠀⠀⠀⣛⡛⣻⣿⣿
⠀⠀⠀⢠⡷⠶⣤⠆⠀⠀⠨⠟⠿⠻⠛⠟⠫⠀⠀⠀⠸⠁⠀⠹⠀⠀⠀⠀⣀⡇⢨⣀⠀⠀⠀⠀⠛⣿⢿⡟⠁
⠀⠀⠀⢿⡐⠗⣸⠇⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠄⠀⠀⠀⠀⢢⠀⠀⠀⠠⡀
⠀⠀⠀⣾⢫⡹⡇⠀⠀⣀⠀⢀⣽⣶⣶⠁⡀⠀⠀⠀⢳⡾⡛⢷⡄⠀⠀⠳⣤⠖⣚⠲⣶⠋⠀⠠⡀⢈⡆⢠⠖
⠀⠀⠀⣹⠶⢾⡃⠀⢀⡻⣤⣿⡁⠦⢸⣤⠏⡀⠀⣄⢠⣿⠶⣿⣷⠜⠃⢀⠟⠶⠤⡴⢏⠀⠀⠀⠈⣾⣧⡇
⠀⠀⢀⣹⡮⣾⡁⠀⠘⠳⢾⡟⠻⠿⠟⣿⡞⠃⠀⠈⢹⡇⠀⠐⢻⡄⠀⢨⠀⡄⠀⡀⢸⡆⠀⢀⣀⣿⣤⠾⣤⡀
⠀⠛⠃⡸⠛⠻⡈⠙⠁⠀⠈⠿⣶⣶⣾⡿⠁⠀⠀⢀⣿⣿⢿⣿⢿⡇⠀⠀⠀⣇⠀⣧⡈⠀⠀⢏⠁⢿⠁⠀⠀⡹
⠀⠀⢈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠁⠉⠀⠁⠈⠀⠀⠀⠉⠉⠀⠉⠉⠀⠀⠀⢀⡀⠁⠀⣀
⣀⢀⣼⣶⣶⣾⡅⢀⠀⠀⠀⢰⣦⣾⣧⣤⠀⠀⠀⢠⡤⠴⠦⢤⣄⠀⢰⡖⢠⡁⢩⡝⣷⠀⠀⠀⢠⡶⠶⠶⣦
⠉⡟⢽⡫⢌⣯⢽⡍⠀⠀⢺⣾⠯⣮⣿⢿⡶⠀⠀⢸⢨⣾⣿⡏⡏⠀⠈⠛⠶⢶⡖⠚⠋⠀⠀⠀⣿⠨⠉⢁⣿⡇
⠀⠻⢿⢟⣿⣿⣿⠃⠀⠘⣿⣿⣭⣞⣽⣿⡿⠀⠀⠘⢤⣭⡭⡤⠃⠀⠀⢠⡤⠀⡇⠐⡗⠀⠠⠎⠻⣿⣿⣿⣿⠋⠃
⠀⠛⢿⣿⡿⠟⠁⠐⠆⠀⣿⣿⣟⣛⣫⣿⣿⠀⠀⠀⠀⣷⣶⡇⠀⠀⠀⠀⠐⣴⣿⡆⠀⠀⠀⠀⠀⢸⠉⡏⢹
⠀⠈⠃⠀⠐⠂⠀⠀⠀⠰⠿⠟⠿⠿⠿⠿⠟⠗⠀⠀⠠⠛⠶⠷⠄⠀⠀⠀⢀⠛⠛⢣⡀⠀⠀⠀⠠⠦⢀⣧⠀⠷⠄

⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢀⡈⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⣈⠀⠀⠀⠀⠀⠀⡖⠀⢲
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⢀⣀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠴⠁⠀⠀⠀⠀⣤⠀⡉⠄⢩
⠀⠀⠀⠀⠀⠀⠀⠀⢦⣠⠄⠀⠨⣀⠄⢀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⡱⠈⠀⠀⠀⠂
⠀⢀⣠⣤⠀⠀⠀⠀⠘⠈⣶⢀⠣⢀⣀⣀⣀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⡄⡠⢄⠐
⠤⠼⠿⣉⠃⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠁⠀⠀⠓⣄⠡⠡
⠐⠀⠘⠂⠀⢺⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠐⠑⠁⠀⠀⠳⠀⠈⠁⠑⠀⠀⠀⡠⠴
⠀⠢⣄⣀⠁⢙⠀⠀⠀⠀⠀⡐⠀⢃⡀⡀⠤⠀⠀⠀⣀⢠⡓⡰⠆⠤⢄⣀⠀⠠⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠠
⠀⠀⠀⠈⠉⠉⠀⠀⠃⠸⠇⠀⠨⠀⠐⠂⠂⠀⠀⠀⡀⠀⠀⠀⠘⠓⠝⢢⠎⠬⠙⠀⠀⠔⠐⢄⠀⠀⠀⠀⡌⠐⢈⠀⢀
⡜⠀⣀⠀⠀⠀⠀⠀⠀⡘⠀⠉⢂⠐⠀⠐⡀⠀⠀⢀⠅⢀⣀⠀⠀⠀⠀⡞⠉⠀⠀⠄⠀⡆⠀⠈⢃⠀⠀⢠⠀⡈⠠⠈
⡐⠀⠨⠇⠀⠀⠀⠀⠀⠖⢎⠁⠀⠀⢑⠤⠴⠀⠈⠈⣱⠏⣈⠉⡂⠀⠰⣦⡐⠀⠈⠀⠀⢉⠀⠀⢩⠀⠀⢸⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡁⢈⠁⠀⠀⠀⠀⠀⠡⢀⠀⠀⠀⠀⡀⠈⠪⠄⠀⠀⠀⠈⠁⠀⡌⠀⠀⠈⡂⠀⠺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠠⠄⠀⠀⠀⢀⣔⠑⠁⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⡀⠲⠠⡀⠀⠀⠀⠁⠄⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡃⠆⠈⠀⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⡁⠀⠀⢀⠀⠀⠀⠀⠈⠓⠂
⣠⡂⡰⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠒⠒⠈⠠⡄⠀⠈⠆
⡄⠁⠒⠒⠑⠀⠀⠀⠀⡠⠂⠩⢰⡙⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠅⡠⠰⡄⠀⠀⠀⠀⠁⠀⠀⡶⢄⣂⠐⢦
⠀⠀⠀⠀⢸⠀⠀⡠⡚⠁⠀⢰⡄⠙⠦⠡⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠉⠄⠐⠀⢠⢀⡀⡤⣧⠀⠀⠘⠐⠡⠀⣇
⠀⠴⠒⢒⡤⠧⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⣶⣢⠀⠀⠀⠀⠀⠀⠰⠄⠤⠤⢀⡡⠪⠇⠈⠁⠀⠀⠈⠠⠁⠡⢈⡀
⠀⠐⠂⠁⡓⠈⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⡀⣉⠤⠛⠃⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢐⠏
⠈⠉⠉⠀⠁⠀⠀⠀⡰⠁⠀⣰⠀⠀⠀⠀⠀⡃⠀⠘⠁⠀⠀⡈⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠢⠀⠀⠀⠀⠀⠀⢈⠈
⠀⠀⠀⠀⠈⠚⠁⠀⠂⢠⠠⠆⠀⠀⠀⠀⢀⣁⣀⣀⡤⠂⠀⡀⠰⠀⠳⢄⠀⠀⠀⠠⠀⣤⠐⠀⠄⠓⣀⠈⠃⠀⠶⠅
⠀⠀⠀⠀⠀⠀⠀⠀⡁⢸⠐⠃⠀⠀⠀⠀⠀⠉⠁⠀⠇⢀⠀⠀⠌⠠⠄⠈⢇⠀⠀⠀⠁⡍⠀⢄⠁⡠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡘⣰⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢠⡄⠀⠀⠀⢁⠨⡄⠀⠀⢐⣀⠠⠂⠀⠠⢀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠀⢱⣊⠀⢀⡠⠀⠸⡄⠀⠀⠈⠀⠀⠠⠀⠤⠊⠉⠁

⠀⠲⡆⢠⠖⢢⠤⠠⡄⣀⡄⠐⠢⡆⠂⠀⢰⠦⣴⡀⠴⢠⡤⠀⠀⠄⡴⠀⣤⡄⠀⢠⡶⠠⠀⡄⠀⠀⠠⠀⡀⠀⠀⠀⠀⢀
⠀⠄⢃⣀⠑⢆⣄⣀⠉⠀⠙⠂⠚⠋⠀⠀⠀⣤⡼⠁⠀⢀⠀⠃⠀⠀⢄⠂⢀⡭⠤⠐⠓⠈⠋⠉⠓⠐⠄⢀⠁⠀⠀⢀⠴⠁⢀⡀
⠀⠀⠀⠀⠀⠾⠋⠉⡿⠐⠆⠀⠰⡌⠉⣀⠀⡉⠥⣤⡀⠀⠀⠐⠃⠀⢀⠔⠉⠀⡀⠄⠤⢒⠲⢀⠀⠀⠂⠀⠑⢀⠄⢁⡀
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⠁⠀⠃⢉⣽⡁⠀⠈⠀⢠⣃⠀⠀⠀⡘⠀⠀⣮⠀⠀⠡⡈⠀⠀⠀⠑⠂⠐⠀⢀⠀⠛⠵⠈⠉⠉
⠐⡇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⣠⡎⡆⣠⠔⠒⠰⠐⠚⠃⠀⢐⡁⠀⠀⠀⠀⠀⠀⠁⠈⡀⠀⠀⠀⢆⠀⠘⠃⡠
⢀⠇⠀⠀⠀⠤⠀⢀⠈⠀⠀⣠⠀⠠⡞⡾⠅⣶⠶⣚⡃⠀⡀⠀⠀⠜⠂⠀⠀⢆⠁⠀⠀⠀⠀⠁⠂⡀⠀⠘⠛⣠⣄⠿⠓⠀⠀⠀⢢
⠸⡄⠀⠀⠀⡲⠒⠋⠐⠇⠀⠈⠴⠒⠀⠴⠶⠶⠄⡀⡀⠤⣄⠀⡇⠀⠀⠀⠀⠈⡄⠈⠉⠀⠀⠀⢈⠀⠀⠂⠀⣉⡻⢿⣆⠀⠀⠀⠟
⢀⡃⠀⠀⠻⡄⠛⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⢀⠀⠀⠙⠒⠚⠛⠁⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⣠⠀⢠⡄⢀⡃⠐⠒⠀⢀⠀⠀⠠
⠀⣣⡄⣠⠐⡆⠀⠀⠀⢠⠀⡆⢠⢘⠀⠀⣾⣿⡙⠉⠢⣄⡀⣀⢀⡆⠄⠤⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⢀⠀⠈⢇⠀⣈
⠀⠈⠀⠀⠀⠁⠀⠀⠠⠈⠀⠘⡀⠀⡵⠔⠇⠙⡄⠶⠈⠀⠈⠻⡄⣣⠨⣂⡀⠀⠀⠀⠘⢠⠀⠀⠰⠀⠀⠀⠠⠀⠀⠀⠀⠀⠼⠘
⠀⠀⠛⠀⠀⠀⠆⠀⠴⠆⠀⠀⠈⠀⠀⠒⠀⠘⠀⠙⠁⠀⢀⠏⠉⠉⢰⠼⢀⠀⠉⠁⠐⡆⠀⠀⠀⠀⠀⠀⠀⣀⢀⡆⠀⠁
⠀⠀⠀⠆⠌⢹⣐⣤⡀⠀⠁⠀⠂⠀⢀⠠⠐⠀⠀⠰⠀⠀⠷⠀⠀⠀⠀⠀⠈⡀⠀⠀⣄⠀⠀⠀⠀⠀⠁⠄⠀⠉⠈
⠀⠀⠀⠀⠐⠚⠒⢍⠙⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠰⠀⠀⠀⠀⠀⠐⢐⠂⠁⠀⢀⡈⠀⠂⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠡⠈⠂⣤⠀⠀⠀⠀⠀⠀⠀⢐⣂⣀⡐⢀⠀⠀⠀⠀⠀⢀⡀⠀⠐⠄⠀⠀⠀⢀⣁⡈⠈⠀⠀⠐⠰⠹⠄⠘⠛⢰⠲⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠉⡀⠀⠀⠀⡀⠁⠀⠀⠀⢹⠁⠉⢀⡳⠀⢈⡔⢂⡄⠐⠀⠐⢀⢀⠀⠀⠀⠠⠀⠀⠀⠀⡐⠓⠀⡀⠀⠀⠀⡂
⠀⠔⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⠂⠀⠉⠁⢀⠄⠀⠀⠀⣬⣤⠤⠐⠂⠀⢄⣤⡬⠀⠀⢬⡄⣤⠤⣀⢤⣤
⠀⠤⣐⣚⡉⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⡁⠀⠀⠀⠀⠀⠀⡀⢀⠀⢀⣶⢀⣸⢀⣠⣄⣰⠄⠸⣷⣝⢺⣅⣀⣽
⠀⢠⡄⠁⠀⠀⣀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠃⠀⠠⠂⠀⠀⠀⠰⠠⠀⠤⠤⠤⠤⠤⠄⠤⢠⣠⠄⠤⠛⡯⠼⠻⠞⠽
⠀⠆⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⢶⣄⠀⠀⠀⡀⠀⠀⠓⠀⡔⠀⡀⠋⠀⢀⣤⡄⠀⠄⠀⠤⣤
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣸⡤⣀⣀⣀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⢁⠀⠙⠂⠈⠀⠠⠋⠀⠠⠊⠁⠀⢉⣠⠂⠀⠉⢁⡄⠀⠀⠀⠘
⠀⡇⠀⠀⠀⠀⠈⡉⡧⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠁⠀⠉⠀⠀⠀⠀⠀⠀⢀⠈⠀⢀⣸⣯⠀⠀⠀⠀⠀⠀⠠⠀⢠
⠀⠦⠀⠄⠒⠀⠈⠁⠁⠀⠀⢀⡀⡀⠀⠀⠠⠆⣴⠀⠁⠲⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⣂⠀⣤⣼⠀⠀⠀⠃⠀⠀⠀⣤⣤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⢐⠴⠷⠖⠂⢸⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⢤⣤⠄⠀⡀⠀⠀⠀⠀⠺⣄⠀⠠
⠀⠀⠁⠂⢁⠀⠀⠀⠀⠀⠀⢀⠭⡀⠀⠀⠀⠖⠀⠘⠀⠙⣴⠱⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⡿⠗⠁⠀⠘⠃⠀⠀⡀⠀⢠⣧⠄
⠀⠴⠦⠤⠾⠁⠀⠰⠦⠶⠶⠄⠀⠈⠒⠖⠊⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠞⠻⠖⠲