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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣶⠶⠶⠶⠶⠶⠶⠶⠟⠛⠛⢻⡆
⠀⢀⣴⠶⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⣦⣀⡀
⢰⡟⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⡀⠀⢀⣀⡀⠀⢀⣤⡀⠀⢠⣤⠀⠀⢴⡦⠀⠀⠶⠶⠀⠀⠶⠗⠀⠐⠛⠃⠀⢸⣧⠀⣿⠙⣷
⣾⠃⠀⠀⢀⣤⠀⠀⠛⠋⠀⠈⠉⠁⠀⠈⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠈⣿⠀⣿⠀⣿
⣿⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⣿⠀⣿⠀⣿
⣿⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣿⠀⣿⠀⣿
⣿⡄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⣿⠀⣿⠀⣿
⢸⡇⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣿⠀⣿⡆⣿
⢸⡇⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣿⠄⢸⡇⣿⡇
⢸⡇⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⢻⡇⢸⡇⢸⡇
⢸⣧⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣦⡄⠀⠀⠀⠁⠀⢸⡇⢸⡇⢸⡇
⠀⣿⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⡁⠀⠀⠈⣿⣤⣤⣤⣶⣴⣾⠷⠾⠷⢾⡇
⠀⣿⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠛⠉⠉⢿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⡇
⠀⣿⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⢿⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿⡆⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿
⠀⢸⡇⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠀⠀⢸⣇⠀⠀⠀⢸⣧⣤⣤⣤⣤⣿⣦⣤⣤⣴⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⢿⣄⣤⣤⣾⠋⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿
⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⣿⠀⣿
⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿
⠀⠈⣿⠀⠀⢺⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣿⠀⣿⡇⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⡀⢹⡇⣿
⠀⠀⣿⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⢻⡇⢸⡇⢸
⠀⠀⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇⢸⡇⢸
⠀⠀⣿⡀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⢸⡇⢸⡇⢸
⠀⠀⢻⡇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⡇⢸⡇⢸
⠀⠀⢸⡇⠀⠀⠹⠆⠀⠀⣀⡀⠀⢀⣀⡀⠀⢠⣤⡄⠀⢠⣤⠄⠀⠴⠶⠀⠀⠶⠶⠀⠀⠶⠗⠀⠀⠛⠃⠀⠘⠛⠋⠀⢸⡇⢸⣷⢸
⠀⠀⢸⣇⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣧⠀⣿⠸
⠀⠀⠘⣿⣦⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢉⣁⣀⣿
⠀⠀⠀⠹⣿⣿⣭⣉⣉⣉⣉⣉⣉⣉⣁⣠⣤⣤⣤⣤⣤⣤⣴⡶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠛⠛⢛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣉⣉⣠
⠀⠀⠀⠀⠈⠻⢦⣭⣍⣉⣉⣭⣭⣭⣭⣭⣤⠶⠶⠶⠶⠶⠶⠶⠶⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣶⣶⣶⠶⣶⠶⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⠀⢀
⠀⢸⠃⠛⡶⢻⠆⠀⠰⢗⠶⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⡀⢻⣿⠀⢰⠟⠀⠀⠀⠀⠀⠀⢠⣦⣤⠀⠀⠀⠀⠀⡂⡄⡀⣄⣴⣶⣾⣶⢾⡄
⠀⠸⠀⢠⠈⠡⠤⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢿⡿⠏⠋⠀⠀⠀⠀⠀⠀⠻⠂⠃⠸⠿⠿⠀⠀⠀⠀⠀⠇⠄⠃⠍⠉⢨⡐⠓⠈⠁
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽
⠀⠀⠀⠀⠀⠰⠶⠄⠀⠀⠀⠀⠠⡂⠤⢐⠄⣤⠀⠀⠀⠈⢠⣄⣠⣾⡆⠀⢆⠈⢆⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣺⣿⣳⣦⢀⣠⣠⣦⣼⡇
⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠂⣀⣡⢸⣾⣿⣿⣿⡇⠀⣀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠐⠀⢾⢺⣿⣿⣿⣻⣿⣿⠛⣹⣿⡇
⠀⠸⠛⠇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠇⢶⠉⠗⠈⠋⠀⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣝⣟⡡⠟⠈⠁⠀⠽⣿⠁
⠀⠀⠀⢠⠀⢡⣬⣁⣩⣤⣁⣀⠠⣀⣤⣴⠃⠇⠀⠀⠀⠁⠁⠁⠉⠛⠀⠀⣴⠶⠶⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⣉⢀⠀⠁⠙⠁⠁⠈⠁⠀⠈⠉⠉⠀⠀⠈⢘⣀⣛⡛⠛⠃⢘⠀⠀⣿⣿⠃⠀⠀⢀⡀⠀⣷⣤⣄⠀⡰⠖⠀⣀⣤⣾⡇⠰⠆⠘⠃⠸⠂
⠀⠰⠿⣷⣼⡂⠢⠓⠀⠐⠀⠂⡁⠐⠐⠂⠀⠀⠀⠀⣨⣅⢡⡆⠀⢻⠃⠠⠟⠛⠀⠀⢴⡇⠀⠀⠈⠛⠛⠦⠿⠶⠶⠿⠛⠋⠀⠀⢀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠈⠇⠐⠇⠑
⠀⠀⢤⣄⣁⠈⠉⠑⠃⠀⠀⠀⠀⠠⠀⣾⣇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⢀⣴⠀⠆
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣒⠁⠀⠠⣰⣿⣿⣃⠤⢰⡤⠖⠿⠒⠶⠄⠀⠀⠁⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠠⠀⢀⣀⠀⣼⣟⣀⣆⡀
⠀⠀⢠⠞⠙⠳⡄⠀⠈⢹⣿⡋⠁⠀⠈⠙⠋⠂⢀⠀⠀⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠹⡿⠠⠄⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠸⠀⠒⢃⠌⣀⡀⠈⣿⠀⠀⠀⠀⢀⢀⠀⠠⠐⠂⠚⠚⠒⠚⠂⡀⠀⠀⠀⠀⠀⠀⠀⠊⠝⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣰⣾⣿⣦⣶⠀⠀⢴⠂⢶⠀⣀⠀⠠⢄⣀⣀⣰⢿⣶⡇⠀⠀⠀⠀⠀⣷⡂⠀⠁⠀⠀⠀⠀⠀⠺⠀⠀⠀⢠⡀⠀⠀⢸⠿⠇
⠀⠀⠀⢠⣹⡆⢸⣯⢤⣈⣿⡠⣄⠀⠡⢆⠘⠏⠠⣤⡜⠻⢯⣠⠼⠃⠀⠀⠈⠀⠀⠘⠃⢬⠗⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⢐⠀⣘⠀⣀⠰⡒⠀⠀⠀⠉⠀⠈⣯⣿⡀⣀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢦⣀
⠀⠀⠻⠛⠛⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⢻⣽⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠈⠚⠧⡤⠿⠎⠀⣤
⠀⠀⢀⠤⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⡸⠛⠟⠀⠀⠈⠀⠀⠀⣀⠀⡀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡰⣿⢶⠄⠀⠀⠀⠀⠀⠀⠘⠎⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠈⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⢪⣷⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⡼⠋⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⢀⠜⠀⠞⠁
⠀⣀⠤⠀⠀⠀⠀⠀⠠⠶⠶⠶⠦⠤⠤⠤⠤⠤⠤⣄⠀⣠⠤⠤⠤⠤⠤⠤⠤⠀⠴⠶⠶⠶⠶⠶⠦⠤⠄⢤⡀⠀⢀⡼⠁⠀⣠⠊
⢸⠁⣀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠤⠤⠄⠄⠬⣿⠥⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⢤⡤⠀⢹⣠⡞⠀⢀⡴⠁
⢸⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡇⡼⠋⠀⣠⠟⣠
⢸⡀⡟⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠄⠀⠤⣤⣤⠤⠊⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠭⠭⢹⣇⡟⠁⢀⡼⣣⠞⠃
⢸⡇⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠔⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣰⠋⠀⢠⠟⢱⡅
⢸⡇⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢀⣈⠁⠀⠈⠉⠉⠉⢡⡽⠁⠀⣴⢿
⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⣼⠁⢸
⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡴⠁⠀⣠⠊⣿⠀⢸
⢸⡇⠘⠷⠀⠀⢉⣉⣉⣁⣀⣀⡀⠀⠀⠀⠀⠈⠁⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠁⠀⡰⠁⠀⠛⠀⢸
⢸⡇⠘⢤⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⣠⠟⠁⠀⢸⣉⠀⢹
⢸⡇⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠋⠁⠀⠀⠀⢸⣟⠀⠹
⢸⡇⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣾
⢸⡇⢠⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⣿
⠘⡇⠀⢿⡄⠀⠀⠠⠤⠤⠤⠄⠴⠶⠀⠶⠶⠀⠐⠲⣾⡴⠶⠀⠰⠶⠄⠀⠤⠤⠠⠤⠤⠤⡤⠤⠤⠼⠗⠀⠈
⠀⠙⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠞⠁⠙⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠖⠃




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠐⠂⠀⠀⠐⠀⠐⠐⠀⠀⠀⠀⠀⠐⠀⠒⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠂⠀⣀⣀⣀⣀⣠⣀⣀⡀⠀⠰⠀⡆⠀⢠⣀⣀⣄⣀⣄⣀⣄⠀⠰⠀⡆⣀⣠⣀⣀⣀⣀⣀⣠⣀⣠⡐
⠀⠔⠈⡂⠀⠀⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢤⠀⠇⠀⠀⠈⠀⠀⠀⠁⠁⠀⠀⢰⠀⡧⠈⠈⠀⠀⠁⠀⠀⠀⠈⠀⠠⠀⠀⢐⠁⠢
⠀⠀⠀⠧⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠼⠐⠁
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠋⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠙
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠌⠁⣠⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣄⠈⠡
⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠃⠀⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣳⡀⠘
⠀⠀⠀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍
⠀⠀⡀⠗⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢀
⠀⠀⠀⡖⠁⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠈⢲
⠀⠐⠀⡀⠀⠀⡀⠘⠑⠒⠖⠗⠒⠺⠺⠚⠂⠈⠀⡇⠀⠀⠛⠚⠒⠗⠓⠚⠀⠀⢈⠀⣇⠀⠓⠗⠋⠲⠚⠺⠒⠓⠃⠀⠀⠀⢀⠀⠂
⠀⢀⠀⠇⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠸
⠀⠀⠀⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⣄⣄⣤⢄⣤⡤⡄⠀⠈⠀⠈⢢
⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠈⠀⠠
⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⢶⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡉⡶⠁
⠀⠀⠀⢉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡉⠀⠰
⠀⠀⠀⠛⠈⠀⠄⠄⠄⠤⠤⠀⠀⠀⠀⠀⠀⠴⠀⠃⠄⠤⠤⠤⠀⠤⠤⠀⠀⠀⠴⠀⠣⠀⠠⠤⠤⠤⠤⠄⠀⠀⠀⠠⠀⠁⠛

⣿⣟⢽⣟⠻⡿⢛⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣤⣤⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣠⣤⢄⣴⡦⠀⠀⠀⢸⣿⣿⣿⡿⠛⠻⠿⠿⠿⢿⣿⣿⣿⡿⠿⠿⠿⠟⣿⠋⢹⣿⣿⣿
⠀⠀⠀⣿⡇⣤⣤⣀⣉⡉⠛⡾⠋⠉⠉⠀⠀⢠⣾⡟⠁⠀⠀⠀⢸⣿⣿⣿⠁⣾⡛⠿⠿⠶⣶⣦⢩⣶⣶⣾⣿⣿⡟⠁⣠⣿⣿⣿⣿
⠀⠀⠀⣿⡇⠤⠤⢤⣄⣉⠁⣧⠀⠀⠀⠀⣤⣄⢉⣶⠀⠀⠀⠀⢸⣿⣿⣿⠀⣿⣛⣛⣛⠿⠶⣿⢸⣿⣿⣿⣿⠛⠿⡶⢻⣿⣿⣿⣿
⠀⠀⠀⣿⡇⠶⠦⠤⢤⣌⡁⣗⠀⠀⠀⢼⡟⠁⢸⣿⠀⠀⠀⠀⢸⣿⣿⣿⠀⣿⣭⣛⣛⣛⠿⣿⢸⣿⣿⣿⡃⣠⣾⡇⢸⣿⣿⣿⣿
⠀⠀⠀⣿⡇⠒⠒⠲⠦⢤⡀⡯⠀⠰⣿⠇⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⣿⣿⠀⣿⣭⣭⣝⣛⡻⢿⢸⣿⡏⠐⣹⣿⣿⡇⢸⣿⣿⣿⣿
⠀⠀⠀⣿⡇⠛⠓⠒⠲⢤⡀⣇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⣿⣿⠀⢿⣶⣭⣭⣝⣛⣿⢸⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿
⠀⠀⠀⠿⠿⠿⠿⢷⣶⣦⣤⣯⣤⣴⣶⠾⠿⠿⠿⠿⠀⠀⠀⠀⢸⣿⣿⣿⣀⣀⣀⣀⣉⠉⠛⠛⠘⠛⠋⢉⣁⣀⣀⣀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠘⠚⠛⠃⠁⠀⠘⠷⠶⠶⠶⠆⠀⠀⠀⠀⠨⡇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠖⠃⠀⠀⠀⠀⠋⠙⠀⠀⠀⢀⠄⠀⠀⠀⠁⠀⠠⢉⠑⠒⠲⠂⠀⠀⠀⠠⠤⠒⠒⠒⠀⠈⠉⢩⣥⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣀⡀⢀⣀⣀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⢶
⠀⢀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠐⠁⠐⠀⠀⠀⠀⠀⠃⠀⠀⠀⠸⠀⠀⠀⠠⠧
⠀⢸⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠇⠀⠀⠀⠰
⠀⠘⡄⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⠂⠀⠀⠙⠃⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠇⠠⠤⠤⠒⠒⠒⠂⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢢⠄⠸⠄⠀⠀⣀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⡶⠀⠀⡇⠀⠀⠀⠀⠀⠀⡆⠀⠀⠙⣦⡀⠀⠀⠀⢀⡀
⠀⢠⣤⢀⡀⢀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢀⢀⣀⣇⡀⠀⠀⠈⢻⣄⡤⠖⠋⠁
⣀⣀⣀⣀⣀⣋⣁⣀⣀⣀⣀⣀⣒⣂⣒⣒⣲⣒⣒⣒⣀⣀⣀⣈⣁⣀⣀⣈⣁⣀⣁⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡍⠀⠀⠉⠉⠉⠉⠐⠒⠒⠒⠶⠦⠤⠤⢄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡠⠤⠼⠼⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠭⢍⡉⠒⠒⠒⠲⠦⠤⠤⡀
⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⠄⠀⠀⠀⠀⠀⠀⡜⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢠⣧⠃
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠀⠀⠀⠀⠀⠀⡎⡇
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣿⠀⠀⠀⠀⠀⢰⢹⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⡆⡌
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⠀⠀⠀⠀⢠⢧⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠬⡬⠭⠤⠤⠥⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢗⡛⠀⠀⠀⠀⠸⡜
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠤⠤⠬⠤⠬⠥⠤⠤⠴⠶⠦⠴⠖⠶⠦⠬⠥⠤⠀⠀⠀⠀⠀⢸⢸⣧⠀⠀⠀⠀⢇⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣰⡲⣄⠶⡤⠤⠤⠤⠴⠶⠶⠶⠧⠤⠦⠤⠤⠤⠄⠀⠀⠀⠀⢸⢸⡟⠀⠀⠀⢰⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⣀⣺⣒⣒⣶⣦⣀⣀⣤⣄⣠⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⢸⢻⡇⠀⠀⠀⡮⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⢰⣿
⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣿⡟⠀⠀⢼⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡇⠀⠀⡧⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡇⠀⢰⣽
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣼⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣿⡅⢀⣳⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⡇⢸⡙
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣿⡇⡄⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⢻⠁
⠀⠀⠀⠀⠸⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣸⣿⣿⡜
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⡴⠶⠶⠿⠛⠛⠋⠉⠉⠁⠀⢀⣤⡀⠀⠀⠀⢀⣴⣶⣤⣄⣀⠉⠙⠛⠿⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠒⢚⣋⣿⣿⣭⣭⣭⣤⣤⣤⣤⣤⣤⣶⣶⣦⣴⣶⣶⣶⣾⠭⣭⣿⣷⣿⣿⣿⣷⣤⣭⣿⣿⣿⣶⣶⣾⣿⣭⣿⢷⡄⠀⡄
⠀⠀⠀⠀⢀⣬⣽⣿⣿⣿⣟⣿⡍⠉⠉⠉⢉⣉⣉⣀⣀⣀⣀⣀⣠⣴⣾⣯⣤⣤⣽⣽⣽⣿⣿⣥⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢰⡇
⠀⠀⠀⠤⣼⣿⣿⣿⣿⣿⣿⣯⠀⠀⠐⠈⠂⠉⣻⣿⡾⠋⣿⣤⣽⣿⣶⣖⢱⣶⣿⣿⡿⢿⠿⣿⣿⢧⠀⠟⣹⣏⢹⣿⣿⡏⠀⣾⣇
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡌⠀⠐⠋⠀⠀⠀⣽⠏⣤⣾⡀⠿⣿⣏⣀⣿⠻⠻⣧⡀⠀⢼⠀⡟⠉⡴⢊⣌⣹⣿⣿⣿⣿⠃⣾⣿⠛⠂
⠀⠀⠀⠈⠉⠙⠛⠻⠿⢿⣿⡇⠑⠀⠀⠀⠀⢀⡸⠛⣧⡬⣿⠐⢿⣿⣿⣿⠷⢶⠤⢽⠧⠮⣪⣥⣞⠿⢿⣿⡿⠟⣿⣿⡟⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⠖⢶⣾⣷⡎⣉⠻⠹⣻⣿⣿⣿⣍⣸⣿⣦⡞⢋⡤⠀⢀⠧⣤⡀⣿⣿⢿⣿⣶⣿⣿⠁⡼⠃⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠛⣿⠇⢸⣿⣿⡝⠛⣧⡀⠉⣿⠉⢺⣿⣿⣿⣭⣀⣼⢃⡿⠙⡇⢸⡟⠀⠀⠀⠙⢻⣽⡟⠀⠁
⠀⠀⣾⣶⢤⣤⣤⣄⣀⠀⣿⡃⣁⢺⣿⣿⣟⣿⣅⣈⠙⢆⡽⠀⠹⠟⠑⢻⣿⢿⣧⣿⣇⣐⣧⠷⠁⠀⠀⠀⠀⢘⣿⡇⠀⠀⠀⠀⢰⣾⣷⣦
⠀⢸⣿⡟⣾⣿⣿⣿⡟⢰⣿⣾⣿⣻⣿⣿⡏⠛⠿⠋⣙⡞⠑⢶⡶⠶⠶⠿⣧⣴⣿⠁⠿⠟⡭⣠⣄⣀⡀⣠⣄⣼⡟⠀⠀⠀⠐⠀⠘⢿⣿⡟
⠀⣿⣿⢱⣿⣿⣿⣿⢇⠈⢹⡿⢛⣷⣦⢹⣶⣶⠾⢟⡋⣤⢠⠨⣉⠳⣤⣶⡿⢿⣬⢲⠳⣞⣷⢼⣯⢿⢿⠟⢿⣿⣿⠀⠀⠀⠀⠀⠀⠘⠛⠁⣰
⠀⠈⠉⠛⠛⣿⡛⠛⠀⠀⢸⣿⣿⠟⢻⡗⢾⣯⣰⣿⠄⣴⡈⠷⣌⢧⣸⣟⣃⣜⠁⣼⣷⣿⡿⠲⣧⠉⣿⠀⣹⠿⣿⠀⠀⠀⠀⠀⠀⠙⠃⢸⣿
⠀⠀⠀⠀⢀⣿⣧⣼⡀⠀⢸⣿⣾⢟⡺⣖⢻⣿⡝⣧⣀⣿⠇⠀⠿⢾⣟⣿⣱⡏⢲⣿⣿⣿⣶⢶⣌⢷⠉⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣀⣸⣿⣿⣿⣷⣾⡿⠉⣾⣿⡟⠃⠂⣀⡈⠀⠛⣿⢛⣿⣿⣿⣿⡯⢀⡤⣌⣿⢉⢹⣟⠳⢶
⠀⠀⠀⢰⣾⣻⣿⣯⣽⣿⣿⣿⣿⣿⣿⣦⠽⢂⣘⣻⡇⠒⠤⠘⠟⢉⣀⣿⣿⣿⣿⣿⣿⡛⠃⢠⡟⣸⠋⢣⠉⠻⣦⣶⣤⡀
⠀⠀⠀⢸⡿⢿⣿⣿⠻⣿⡿⠛⠋⠉⠉⠉⠈⠉⠉⢿⣿⣿⣿⣿⡿⠤⠽⠟⠻⠯⠿⠇⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠦
⠀⠀⠀⠸⠿⠿⠿⠟⠋⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣄⣉⣉⣉⣉⡹⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠓⠲⠶⠶⢤⣴⣿⣿⣿⣿⣿⣿⣿⣀⡀⠒⠒⠠⠬⠻⠿⠿⢿⣿⠿⠛⠉

⠛⠻⢿⣿⣿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿
⠀⠀⠀⠀⢹⣿⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿
⠀⠀⠀⠀⢉⠿⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⢈⣀⣩⣤⣦⣶⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢿⣿⣿⣿⣻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⣿⣿⣿⢿⣿⠀⠀⠀⣾⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⣿⣿⠿⣾⣿⣀⣠⡾⠚⣿⠙⠦⠄⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠿⢳⡄⠀⠻⠀⠠⠀⠀⠃⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⣾⡿⠀⢀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟⠣⠄⠄⢀⣿⠁⣀⠆⢁⡘⠀⢶⡄⠀⢀⠀⣀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⣿⣿⣿⣻⣟⣹⡆⠀⠈⢾⣏⠰⠋⠀⢬⠁⢀⠜⣿⠐⠋⠀⣬⠛⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⣿⣿⣿⡟⢿⣷⠀⠀⠘⣿⠀⠀⠀⢸⡷⠛⠀⢸⡇⢀⡜⠀⡀⠈⢷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⣿⣿⢿⡞⣼⣿⡀⠀⠀⣿⡠⢀⣤⣿⠁⠀⠀⢈⣷⡆⠰⠉⠁⠐⢏⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠈⣿⣿⣿⡏⡸⢻⣿⣇⠀⠀⣿⢡⣾⣿⣏⠀⠀⣠⡾⠛⠐⠂⠲⠶⠶⣼⣿⣡⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣿⣧⣟⣿⣷⣼⣿⡿⣆⢸⣿⠃⣸⠘⣟⠀⣰⠋⠀⠀⠀⢠⣴⠚⠉⠈⣷⠆⢹⣿⡿⣿⣷⢶⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣿⣯⣉⣹⣿⣿⢿⣇⢸⣿⠃⢠⠟⢀⡏⣴⡇⠀⠀⠀⠻⣿⠟⠷⡤⢀⡟⠀⠘⣀⠃⠉⡼⣿⡏⠉⠻⡌⠋⠘⠗⢠⣶⠂⢀⠜⢀⡯⠄
⠀⠀⠀⠰⣿⣿⣭⣧⠟⣉⠈⠙⣾⣿⠀⠀⠀⢀⣼⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢄⡀⠁⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢠⡉⠀⠲⠀⡀
⠀⠀⠀⠀⣿⣿⣿⣷⣄⠻⠇⣀⣻⣿⠀⠀⠀⠘⠋⢉⣰⠟⠀⣰⡯⠤⠄⠀⠀⠀⠀⢀⠓⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⢀⠇⠀⡀⠀⣌⠀⠐⠀⠙⡧
⠀⠀⠀⢠⣿⣯⣿⣿⣿⡶⣾⡋⣿⣇⡾⣂⣴⠔⢠⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣠⠄⢀⡀⠀⠀⠀⠀⢀⢠⠊⢠⣀⠷⠀⠹⠄⢀⠠⠄⠀⠀⢠
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠃⣿⣟⣥⠟⣡⣶⡋⣁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⡁⠈⠀⠉⢹⣆⢈⢠⡀⠀⠀⠀⠸⠀⠀⢀⠉⠀⠀⢀⡷⠛⣤⡤⠀⠀⢸
⠀⠀⠀⢸⣿⣿⣿⣽⣿⣿⣿⣾⣿⣿⣧⡾⢋⣉⣤⣬⡿⢿⡷⠀⠀⢤⣤⣄⣾⣿⠙⠂⠀⠒⠀⠈⠻⢾⡇⠀⠀⢀⡀⠀⠀⡘⠀⣠⢰⠟⠁⠀⠀⣴⠀⠀⢸
⠀⠀⠀⢸⣿⡿⢻⡿⣿⣿⣿⠿⣿⣿⣿⣿⠻⢿⡟⠛⠳⣄⡳⢀⠀⠀⣿⡟⡟⠁⠀⣁⠀⠀⠀⠀⠀⠀⠈⢶⡠⣘⡄⠀⢰⡇⠀⡡⠞⠓⠒⠶⠉⡉⠀⠀⢸
⠀⠀⠀⠘⣛⣷⣿⣿⣿⠿⠁⢰⡟⠛⣶⡌⢣⡈⠻⠀⣠⠦⠿⠿⣷⣦⣸⣿⣯⡋⠀⣀⣀⡰⠖⠋⠀⠀⠀⠀⠀⠹⢧⠀⠈⠀⠰⠁⠀⠀⠀⠀⠀⠙⠀⠀⢨
⠀⠀⠀⠀⠛⠋⠁⠿⠛⠀⢀⣿⠃⠀⠀⠙⠳⢾⡃⠉⠲⡦⠀⠀⠀⠉⠻⢿⣤⣤⣾⣿⠥⠒⠚⠛⠋⠀⣀⣀⣤⡀⢤⡄⣸⠆⠁⢰⠀⠀⠀⠀⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠐⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡽⠷⠖⠒⠐⠶⠚⠉⠉⠀⡄⠁⢀⣠⡜⠻⠆⣤⠛⢣⡤⠤⠄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⣀⠀⠀⠀⠀⠄⢀⣨⡀⠶⣿⣯⠀⠀⣀⠦⠸⠇⠀⣄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠙⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠙⠲⠂⠾⠷⠾⠛⠋⠀⡾⠉⠀⠀⠀⡀⠀⠀⠀⠀⠙⠶⠂⠶⠀⠀⠐
⠀⠀⠀⠇⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⡀⠀⠈⠳⣄⡽⠂⠀⠀⠀⢾⠉⠉⠐⠆⠀⠀⠰⠀⣀⠀⠀⠘
⠀⠀⢰⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢡⣄⠀⠀⡿⠓⠂⠂⠠⢤⣀⡹⢦⡀⠀⠀⠐⠀⠒⠀⠀⠀⠸
⠀⠀⢸⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠢⣄⠉⢀⡾⠁⢠⣤⠠⠀⠀⠉⠒⢠⣀⢈⡇⠀⠀⣀⢠⡄⠀⢠
⠀⠀⢸⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡈⢷⣌⣁⣤⠄⢀⡀⠀⠀⠀⠀⠀⣠⢟⢃⣀⡙⠉⠀⡀⠀⢸
⠀⠀⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣦⣛⠁⠀⡀⠈⠙⠶⠁⠀⠐⠚⡷⢈⠈⠈⠁⠈⠉⠁⠀⢸
⠀⠀⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗⠩⢄⡀⡀⠀⠠⡆⠀⢸
⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⢸⡄⠈⢱⣷⠒⢤⠀⠀⢸
⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣿⡀⠀⣮⠈⠁⠀⢸⠀⠛⣤⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⢀⡏⠀⠀⠀⣤⠀⠀⢹⠇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠧⠀⡀⢠⠆⠀⠀⠈⠀⠀⠈
⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⡶⣿⠸⡦⠀⣠⠀⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠟⣴⠁⠆⠸⡇⠓⠀⠰
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠷⢬⣯⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣏⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⡀
⠀⠀⠀⣰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⢉⢠⣤⡑⡄
⠀⠀⠀⣿⠀⠑⠒⠠⠤⠤⠤⠄⠐⠒⠒⠒⠉⡉⠁⡀⢠⡦⣾⣉⣣⣍⡀⠘⢆
⠀⠀⠀⠸⡄⠀⠤⠄⠀⣶⣄⢰⣆⢲⡟⠉⡛⠻⣄⢹⡿⣿⡌⣭⣉⣉⡿⠀⠈⢦⡀
⠀⠀⢠⡤⢷⡀⠘⠀⢀⡘⣿⡳⣽⣧⡹⣿⣉⣀⣬⠛⠿⠈⢙⣫⣤⠲⠌⢃⣀⢤⡙⢄⡀
⠀⠀⢸⣧⡀⢳⡀⠙⢒⣟⠈⢻⡦⠙⣛⣚⣛⣭⠴⠒⢁⡀⢀⡤⣔⣒⡭⠵⢒⣋⠁⠀⠙⠢⣄⡀
⠀⠀⠈⢿⣷⡀⠻⣄⠀⠰⠶⠟⢒⣊⡉⠉⡴⠖⠚⠋⠁⠙⢄⠉⠱⠶⡾⠟⣛⣤⣴⠮⣴⡶⠒⠩⠭⠉⢙⠒⢤
⠀⠀⠀⠀⠻⣿⣄⠘⢦⡀⠲⠭⠿⠾⠅⣀⠘⠦⣌⠁⠄⠀⠀⠓⢤⣈⠙⠟⡭⠶⠞⠋⠉⠀⠀⠀⠀⠀⠈⢱⠀⡇
⠀⠀⠀⠀⠀⠘⢿⣦⠀⠻⣄⠀⠈⢢⣻⣻⠴⠦⠈⢧⡀⠀⠀⣀⡤⠽⠛⢀⡈⠳⡄⠀⠀⠀⠀⢀⡠⣤⣶⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠘⢧⡀⠀⠀⢶⡶⣿⠷⣖⡬⠓⢉⣁⣤⡴⢾⣋⠄⠀⢸⣄⣠⣶⡾⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠝⠶⣄⡈⠛⣪⣝⡳⠬⣛⣩⠭⣶⡿⣟⣿⡽⠴⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⠀⠀⢠⢉⢷⡀⠀⢨⣻⣥⣶⡿⠿⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠈⠀⣷⢀⡽⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠖⠋⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣴⣀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⣀⣤⣴⣶⣿⣶⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣶⣥⣖⣠⣀⠄⣀⣀⣤⣤⣶⣼⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠔⠚⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢠⠀⠀⡠⠀⠠⠀⢸⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡈⠀⠀⣅⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠣⢀⣀⡀⠉⠀⢀⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⠉⠀⠉⠉⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⢀⠔⠀⠀⠀⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠸⡀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢱⠂⠀⠀⠤⠀⣀⠉⢀⡌⣿⣿⣿⣿⣿⣇⣿⣿⣿⡟⠁⠀⠱⡀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡏⠀⠁⠀⢹⣿⣿⣿⣿⣿⢿⣿⣿⣿⣦⠀⢀⠆⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⢸⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⡷⢄⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠛⠛⠛⠛⠛⠓⠠⢄⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠃⠤⠄⠠⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⡇⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⡇⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠛⠛⠛⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⢀⡀⠀⠀⠀⠀⠀⣀⠠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⣀⠠⠄⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⢠⣊⢁⢀⠠⠄⣂⡉⠀⠘⠉⠙⠓⠶⣤⣠⣶⠿⠋⠁⢸⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢄⠀⡖⠂⠂⠈⠉⠀⠙⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠈⡀⠀⠰⠂⠀⠀⠁⡆⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢸⠀⢘⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⢨⠀⡇⠀⠀⠀⠀⢀⠀⠇⠀⣚⡀⢰⠶⠶⣶⣶⢶⢰
⠀⠀⠇⠀⠀⠀⠠⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠁⠀⠰⠀⠀⠀⠀⢠⠀⠴⠀⢸⣏⣿⣽⣭⡏⢸
⠀⠀⠰⠀⠀⡀⠀⠀⢠⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⡸⠀⢸⠀⠀⠀⠀⠀⢀⣤⠄⠀⠀⠀⠀⢻⠀⢰⠀⠀⠀⠀⠀⠀⠸⠀⢠⠁⠀⠁⠈
⠀⠀⠀⢃⢀⡘⠀⠄⠒⠈⠀⠁⠂⠠⠀⣀⠀⣂⠠⠐⠉⠀⠀⠘⠓⠋⠒⠒⠒⠚⠛⠂⠀⠀⠙⠓⠛⠀⢸⣤⣠⣄⣀⣤⣤⣤⠀⢘⣀⣀⣀⣀⣀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠉
⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡔⠒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⢀⠊⠁⠀⠀⠀⠀⢠⠅⡀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡀⣀⠀⠀⠐⡀⠀⠀⠀⠈⠀⠀⠉⠁⠂
⠀⠀⢠⠂⠀⠀⠀⠀⠈⠥⡄⢸⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠐⠀⠘⡀⠀⠁⠀⠙⠓⠋⠀⠀⢡
⠀⢠⣋⣀⣀⡀⠀⠀⠦⡞⢠⠆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠈⠀⠘⠁
⠀⢻⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠸⠤⠴⠦⠭⠬⠵⠷⠬⠭⠉⠰⠤⠼⠀⠀⣇⣀⠀⠀⠀⠀⠀⠀⡅⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣘
⠀⠀⠉⠉⠉⠉⠉⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠉⠟⠟⠛⠀⠀⠰⠯⠭⠭⠥⠤⠉⠩⠏⠀⠙⠛⠿⠉⠁
⠀⠀⣀⠄⠒⠈⠁⠀⠀⠀⢈⠱⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⣿⢵⣐⡀⡤⠀⣂⠠⠔⠲⣏⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣶⣄⠀⠀⠀⠀⢠⠶⠶⠴⠒⠒⠲⢦⠀⠀⡴⠒⠒⠒⠂⢤⠠
⠀⠀⡧⣄⠀⠉⢛⠉⢀⣠⠔⡈⢙⡀⠀⠀⠀⠀⢰⣿⡇⠀⢸⠀⡎⠇⢠⢹⠀⣰⡀⢾⣿⣿⡆⠀⠀⠀⢸⠤⠀⠀⠀⠀⠀⠋⡆⠀⡆⠀⠀⠀⠀⠀⠈⠃
⢀⣸⡓⠠⠍⠲⡾⠚⠁⢀⡠⢖⠉⢳⠀⠀⠀⠀⣿⣿⡇⠀⣸⠀⢃⡆⠘⣼⠀⡘⡗⢀⣿⣿⣿⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠛⠃⠀⢀⠀⠀⠀⠉⠉⠀⠘⡀
⠈⡇⠂⠍⠐⠀⠀⣡⠔⠉⠀⠀⡠⠚⠀⠀⠀⠀⢿⣿⣿⣿⡟⠛⠛⠛⠛⠛⣻⢻⣿⣿⣿⣿⡏⠀⠀⠀⢸⠖⠀⠀⠀⠀⠀⠐⡃⠀⠘⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠑⠤⢀⠀⠀⠀⠠⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⣠⣀⣄⣄⣤⣹⣸⣿⣿⡿⠟⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⣸⠁⠀⠀⠶⢊⢠⠐⠒⠒⢓⡚
⠀⠀⠀⠀⠈⠐⠢⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⡖⠂⡆⠀⣠⡁⢀⠈⡇⠠⠂⠐⠀⠀⢀⠀⣦⠀⠀⠀⠀⠀⢀⠠⡤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠠⢄⠀⠀⢀⠔⠖⠂⠀⠀⠀⠰⣀
⠀⠠⠁⠀⡏⢩⠘⠁⠀⠀⡇⠀⠂⠐⠇⠀⠋⠀⠁⠀⠇⠰⠀⠀⢨⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣸⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡥⠇
⠀⠀⠘⡗⠃⠀⠐⠂⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠽⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢖⡀⡳
⠀⠀⠨⠅⠅⠀⢠⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠓⠂
⠀⠨⠐⠃⡂⠀⢀⠀⠀⠀⡃⠀⡄⢀⠀⠀⢠⠀⡄⠀⠀⠀⠀⠀⠀⢸⠀⢀⡀⠀⠀⠁⠀⠀⡆⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⢈⡁
⠀⠈⣁⢀⣇⢸⣰⡀⠀⢈⡇⢀⡁⢀⡤⠀⣄⠀⣆⢀⡆⢐⠀⠀⠀⠈⠫⠥⠭⠭⠭⢽⢹⠿⠁⠀⠀⠩⣥⣤⡀⠀⠀⠐⣰⠆⠀⠘⣤⣤⣤⣤⣤⣤⣤⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⢀⣄⣀⣀⣀⡀⣠⣤⣤⡄⡀⣀⣀⣀
⠀⢸⢿⣿⡿⣋⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣰⢤
⠀⠈⠈⠹⣿⠿⠛⠛⠻⣿⣿⣿⣿⠟⣿⠀⠀⠀⢰⢬⣿⣿⣶⣶⣶⡀
⠀⠀⠀⠰⣿⡆⠀⠀⢀⣿⠛⠛⡟⠀⠐⠀⠀⠀⠀⠀⠧⠏⠙⠉⠉⠁
⠀⢸⠷⢾⣿⣿⣶⣰⠿⡟⠁⠀⢱⡀⢀
⠀⠀⢀⣵⣿⡿⠿⣿⠀⣾⣶⣿⡿⢧⣿
⠀⢰⣿⣿⣿⣷⣸⣿⣾⣏⣻⣿⣿⡆⠻
⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠
⠀⠀⢀⣤⣿⣿⣿⣿⣿⠀⣙⣿⣿⣿⣿
⠀⠸⠿⠿⠿⠋⠻⠿⠿⠛⠟⠿⠿⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠚⠀⠀⠰⠀⠈⠲⠈⠐⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠉⠀⠀⠀⠀⠀⠉⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠶⠤⠤⠤⠤⠴⠮⠶⢤⠀⠀⠀⢠⠱⠀⠀⠀⠀⢠⠀⠀⠀⡈⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠤⠤⡔⠂⠒⢦⠀⠒⢺⡀⠀⡄⠉⠀⠀⠀⢠⠀⠒⡀⢠⠀⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⠒⠒⠳⣄⣀⠜⠒⠋⠉⡇⠀⡄⠀⠀⠰⠄⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠤⠤⠖⠒⠒⠒⠒⠒⠒⠃⢀⠀⠀⠉⣀⠀⠐⠀⠀⠚⣀⠀⠄⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡄⠁⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠀⠀⠀⠀⠠⠀⠀⠀⡀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠉⠀⠀⠀⠠⠄⠛⠀⠰⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠉⠒⠂⠀⢀⡼⠤⠀⠀⠀⢀⡨⠅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⣠⠴⠟⠒⠒⡂⠋⠈⠁⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣿⣀⣴⠞⠋⠀⠀⠀⠀⠀⠑⠒⠃⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠅⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⣠⣄⣰⠏⠀⠀⠀⠀⠘⣧⣰⠻⡶⣄⠀⠀⠀⠀⠀⠀⠀⣠⠟⢦⡤⢶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣧⢀⣴⣿⠋⠳⡀⠀⠀⠀⢠⡞⢉⣯⠻⡄⠀⠀⣠⠶⠦⠞⠃⠸⣟⠎⠳⢄⠀⠀⠀⠀⣴⠋⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠙⣦⣤⡴⠋⢰⡿⣌⣢⠈⠳⢮⣁⣀⣀⠀⠀⠀⠹⡗⢀⣀⣵⢶⣷⡾⠁⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⣠⠴⣿⠁⠀⠀⠀⠘⢧⣀⡈⠑⠀⠀⠀⠀⠈⠑⠄⠀⠀⠉⠉⠁⠀⠸⣅⠙⠲⠦⢤⡴⢺⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡀⢀⣿⡄⠀⠀⠀⠀⣤⣻⡍⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣹⠇⢸⡄⠀⠀⡞⣰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇⢸⣿⣇⠀⠀⠀⠀⠸⣟⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠀⢠⠘⣆⣿⣇⠀⡼⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⠀⢹⠀⠈⠙⠢⠀⣤⠀⠀⠀⢀⣿⣷⠀⠀⠀⣿⡆⢸⣿⣿⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⢄⠀⠀⣾⡇⠸⡀⠀⠀⠀⢠⣿⡆⠀⠀⢸⣿⣿⡂⠀⢠⣿⡇⢸⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣦⣿⣦⣿⣷⠀⠙⢂⠀⠀⢸⣿⣇⠀⠀⣼⣿⣿⣇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡷⢦⡴⢄⣀⣿⣿⣿⣀⣀⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣷⣾⣧⣴⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠟⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⡀⠘⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠁⠀⠀⠀⠀⢠⠀⠀⠘⠦⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠊⠸⠈⠃⠀⠀⠀⠀⢀⠰⠀⠀⠀⠠⡀⠀⠀⠠⠀⠀⡀⠀⡀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⠀⠀⠀⠀⢀⠐⠒⠀⠀⠀⠀⠀⠔⡀⠀⡠⠀⠁⠀⠀⠲⠀⠀⠀⠀⣀⠀⠉⠀⠀⠀⠀⠐⡀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠠⠈⠀⠈⠢⠔⠊⠁⠀⠀⡀⠄⠂⠀⠀⢎⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠄⠀⠀⠀⢀⠇⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠒⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠤⠀⠀⣀⡠⠄⠀⠈⠁⠀⠀⠀⠀⠀⠠⠀⠉⠐⠢⢀⠀⢧
⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠐⠂⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠸⡆
⠀⠀⠀⠀⠀⣎⠠⠂⠁⠀⡀⠀⠀⠀⠀⠈⠁⠀⠡⢄⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⣀⣤⣄⣠⣬⣭⣷
⠀⠀⠀⠀⠀⡏⠡⠀⣀⣀⣀⣤⣄⣀⠀⠀⣠⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣇⠐⠋⠉⠉⠉⠀⠀⠀⣀⣀⠀⠀⠀⣀⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⡿⠿⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡟⠛⠳⢦⣠⣄⡀⠴⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⠀⠀⠀⠀⣤⣾⣦⡐⠛⠉⢁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢷⣦⡀⠀⠈⠁⠀⣀⣤⣾⣷⣦⣤⣄⣀⣀⣤⣦⣴⣶⣾⣷⣷⣶⣶⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠻⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢿⠀⠄⠺⠦⠴⠶⠴⠶⣤⣤⡤⣤⣤⣤⣤⣠⣤⣄⣀⣀⣀⣈⣉⣉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⠛⢻⠖⠊⠈⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣭⣬⣭⣭⢽⠶⠶⠶⠦⢤⠄⢀⣀⠀⠀⠀⠈⠉⠛⠛⢛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⡤⣿⠶⠶⣦⣄⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠠⠅⠀⠀⠀⠀⠀⠀⠉⠙⢩⣙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢴⣶⡒⠶⣿⡙⠻⣿⣍⠉⠻⣿⡙⢿⡖⢶⡶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠺⢿⣭⣉⡛⠷⢮⣝⡲⢬⣙⠳⢤⡉⠳⢤⡈⠳⣄⠹⡌⢧⢹⢸⢇⡟⣹⠖⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⢯⣛⡿⠷⠶⢤⠤⠭⡙⠳⠦⣬⣙⠳⢮⣝⡲⠬⡓⠤⣙⠢⠈⠃⠈⠀⠈⠈⠈⠐⠡⢞⡡⠞⣋⣷⢤⡀
⠀⠀⠀⠀⢀⣴⣿⠿⠀⠀⠈⢲⡀⠀⣰⠃⠀⣨⠉⠓⣤⣌⣙⣒⠦⢍⡓⠤⢁⠀⠁⠀⠀⢠⠞⠉⠉⠉⠳⡄⠀⠐⠈⠥⠴⠶⢛⣳⣄
⠀⠀⢀⣴⣫⠔⠢⣅⣠⡞⠉⠳⢄⡀⠉⠀⠠⠉⠁⢀⣼⠷⢻⣿⣷⢶⡌⠉⠂⠀⢀⡴⢲⡇⠀⠀⠀⠀⠀⢸⣀⣠⠞⠛⣯⡩⠭⠭⠭⣷⣄
⠀⢠⣾⣿⣶⣾⣿⣿⣿⣷⣄⠰⠞⣉⡦⠴⠞⣳⠖⠋⠁⠀⠘⣿⡏⠀⠈⠓⣲⡯⣍⡀⣨⢷⠴⠚⠓⢦⣤⣾⣾⡃⠀⣼⡿⠻⢏⣹⢷⠚⠙⢦
⢠⣿⣿⣿⣿⣿⣇⣤⡤⣉⡛⠳⢿⡀⣀⣴⣞⠁⢀⠀⠀⠀⣩⣿⡿⠀⢀⡟⠙⠲⡀⠉⠳⢤⣴⡶⠾⡿⠟⡻⠉⠀⠀⣘⣷⠀⠀⢙⣷⣯⣴⣾⣇
⣼⣿⣿⣿⣿⣷⠟⣽⣻⣯⣿⣷⣶⡉⠛⢿⣿⣿⣶⡄⠀⣴⣿⠓⠀⣠⣿⣤⣤⣠⠞⢀⣀⢤⣬⣻⣮⣄⠈⠀⠀⢀⣠⣿⣿⣤⣾⣿⡿⣿⣟⣿⣿
⣿⣿⡟⣷⡿⣿⢻⡼⣧⣷⣿⣶⣿⣿⣶⣄⣌⡙⠻⢿⣼⣧⣤⣤⣾⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⢬⣿⣿⢿⡟⣿⣾⣿⣿⣻⣯⣿⣿
⣿⣿⢿⣿⣷⣞⣚⣿⣿⣭⣿⣿⣿⢻⣿⣶⡬⠝⠳⣤⣈⣙⠻⢿⣿⣿⣿⣟⡷⠾⠿⠛⢛⣋⣭⣭⣷⣶⣾⣷⣿⣿⠿⣾⣝⣗⣯⣷⣿⣿⢾⣿⣿
⢹⣿⣾⣿⣷⣮⣿⣧⣿⣭⣻⣏⡽⣻⣿⣿⣿⣷⣄⡙⠻⣿⡷⢯⣤⡿⠉⣿⣀⡀⠝⠻⠿⠷⠶⠯⢿⣧⢾⣛⣿⣶⡿⣿⣿⣿⣷⣿⣿⣹⣿⣿⣿
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡟⣿⣆⣤⣍⣳⣯⡳⣄⣈⠙⠛⠻⠶⣬⣵⣾⣿⠿⠷⠿⠯⠿⢶⣿⣟⣿⣞⢻⡿⣽⣿⣿⣿⠃
⠀⠀⠹⣿⡽⢿⣿⣿⣾⣿⣿⣿⣿⣿⣯⣗⣺⠭⣿⡷⣷⣿⣾⣳⣾⣿⣦⣽⡷⠀⠀⠀⠈⠹⣿⣿⣿⣯⠭⠽⢿⣟⣒⣶⣭⣝⣛⣿⣿⣿⡿⠁
⠀⠀⠀⠈⠻⣷⣾⣷⣺⣭⣿⣿⣿⣿⣭⢟⡻⢿⡲⢭⣓⢪⣽⣿⡿⠟⠋⠁⠀⠀⠀⡀⠀⠀⢸⣿⣛⣛⣯⣭⡿⢾⣛⣛⣿⣭⣭⣭⡽⠋
⠀⠀⠀⠀⠀⠈⠛⢿⣞⣛⡻⣍⣯⣭⣝⣿⣮⡓⣭⣷⡾⠟⠋⠁⠀⠀⠤⠀⠀⠀⡀⠀⠀⢀⣼⡿⣟⣻⠽⢗⣛⣭⠭⠴⣶⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣿⣽⣹⣯⣷⡿⣿⣭⠁⠄⠀⠀⠀⠒⡤⠀⠁⠀⠴⠀⣠⡶⣛⠽⣛⣭⠶⣚⡭⢵⣲⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣯⣶⣶⣖⢐⣠⣶⠖⠔⠐⠀⢢⣾⣷⣻⣵⣟⣯⣶⠿⠗⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠲⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣧⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⣠⣷⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣦⣆⣾⡄⠀⠀⠀⠀⠀⠀⠀⢠⡀⢶⣿⣷⣠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣟⠛⠛⠃⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣓⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠉⣻⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠒⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⢴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢅⣀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢤⣄⠀⠀⠀⠀⠀⠀⠀⢀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢲⣀⣀⠀⠠⡤⣽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠀⠠⠀⠀⠀⠘⢿⠟⠳⣿⣷⣤⠐⠀⠀⠀⠀⠀⠀⢆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⢀⣤⣶⣶⣶⣶⣶⣶⣦⣿⣿⣿⣧⣴⡦⠀⠀⠀⢃⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠶⡿⣯⠿⠿⠟⠙⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⢂⡴⠀⠀⠀⠀⠀⠀⣀⣤⡦⠀⠉⠀⠁⠀⠀⠀⠀⢠⡘⠀⣲⣀⠂⣳⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠳⠀⠠⠀⣂⣠⠾⢩⣽⣏⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣠⠛⠋⣠⡶⣯⠿⠊⠀⠀⠀⡀⠀⠒⠀⠀⠀⢠⣀⣿⣾⢟⢀⣨⡈⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣷⡾⠁⣤⣞⣿⡏⠁⠀⣀⠀⠚⠀⠀⣠⣼⣶⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠰⠛⣥⣾⣿⢾⣿⣿⣶⣴⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠺⠿⠃⣴⣿⣿⣁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣧⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢻⠒⢀⣼⣿⣿⡿⠛⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⢖⠐⢦⡄⠀⣼⣿⡿⢿⡶⣻⣚⣿⣿⣿⣯⡙⢿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⢣⣾⢿⣧⣾⣇⣿⣇⣿⣿⣿⣿⣟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⣾⣿⡏⣿⡁⢹⣿⡟⠈⣿⣿⣿⣿⣯⠙⣻⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣯⣽⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣟⣁⣙⣃⣘⣛⣻⣜⣛⣿⣿⣿⣿⣿⣛⣿⣿⣟⣛⣻⣿⣿⣛⣛⣩⣤⣾⣿⣿⣿⣿⣿⣟⣛⣙⣿⣿⣿⣿⣿⣟⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠀⠠⠄⠀⠀⠀⢤⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⣠⠾⠛⠃⠄⣀
⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡴⠟⠉⠀⢀⣾⣿⣿⣿⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⣿⣴⣷⣧⢀⢠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠰⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠉⠉⠃
⣿⣿⣿⣿⣿⣿⣷⣿⣠⡀⣦⣷⢠⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠰⠶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣼⣧⣧⣼⣰⣇⢠⡀⣿⡆⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡆⣠⢸⣷⣿⣦
⣿⣿⣿⣿⢿⢿⡿⠿⠛⠛⠟⠿⠛⠋⠛⠉⠉⠋⠉⠉⠋⠈⠉⠛⠁⠀⠋⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⡀⣸⣧⡆⢠⢀⣿⣾⣷⣿⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⣾⡧⢀⡀⣸⡇⣰⣠⣧⣴⣰⣿⣰⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⢸⣿⣿⣾⣧⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣽⣗⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣸⣷⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡿⠻⣿⣿⢿⡿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠃⠀⠀⠈⠁⠈⠀⠁⠈⠈⠉⠈⠉⠙⠛⠋⠉⠉⠛⠻⠟⠿⢻⡿⣿⠿
⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⢠
⣿⠃⢹⣿⣿⣿⣿⣿⡃⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣸⡇
⡟⠀⢐⣻⣿⣿⣿⣿⣷⣼⣿⣧⡀⢀⣦⠀⠀⠀⢰⣿⠄⠀⠀⠀⠀⠀⢸⣿⣯
⠉⠀⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣸⣿⡄⠀⢰⣾⣿⡇⠀⢰⡆⠀⠀⣿⣿⣿⡁⠀⠀⣾⠄⠀⠀⠀⢰⡧
⠂⠀⠴⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢺⣿⣿⣶⣤⣾⣿⠀⣘⣿⣿⣿⣶⠆⢸⣿⡇⠀⠀⠀⣿⣟
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣷⣘⣿⣿⣿⡟⢻⣿⣿⣦⣽⣿⣿⣿⣿⡥⣿⡿⠿⠀⠀⣽⣿⣿⠇
⠀⠀⠀⠀⣰⣿⣿⣿⣿⠃⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⣶⣤⣿⣿⣿⣷⡤⠀⠀⢰⡇
⠀⠀⠀⣤⣽⣿⣿⣿⡧⠀⠀⢽⣿⣿⣿⡿⢻⣿⣿⠦⠀⠠⢽⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠠⢽⣿⣿⣿⣿⣿⣧⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠤⠼⣿⡟⠓⠀⠀⠙⢿⣿⣿⡃⠀⣿⡟⠃⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⠀⠙⢿⣿⣿⣿⣿⣷⡖⣸⣿⣿⡅⠀⢀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠒⠚⣉⠁⠀⠀⢤⣿⣿⡟⠀⠀⠹⡏⠁⠀⠀⣤⣿⣿⡿⠁⢿⣿⣿⣏⠁⠀⠀⢤⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢁⣾⣇⠀⣸⣿⣟
⠀⠀⠀⠀⠠⠴⠒⠀⠀⠀⠀⠐⠚⠛⠂⠀⠀⠀⠀⠀⠀⠀⠐⠚⠻⠃⠀⠘⣿⣟⠀⠀⠀⠀⠐⢺⠻⣿⣿⠁⢻⣿⣿⡟⢻⣿⣽⣿⣧⣠⣿⣿⣿⣃⣾⠀⠀⣴
⠀⠀⠀⠀⠤⠦⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⢒⣿⠁⠀⢈⣿⣿⠀⠈⣿⠙⣿⣿⣿⠇⢸⣿⣿⣿⡆⢰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠉⣉⠅⠀⠀⠀⠙⠀⠀⠀⠋⠀⣿⣿⡿⠀⠈⣿⣿⣿⣷⣾⣿⣿

⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⡿⣿⣿⣷⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠹
⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣠⣿⣿⡟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⡀
⣼⠟⢤⠀⠀⠀⠀⠀⣀⣀⣠⣴⣽⣿⣿⣿⠟⠻⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⡟⢠⣯⡄⠀⠀⠘⠛⠻⠿⠿⣧⣀⣿⣿⣦⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡿⢿⢿⣄
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⣿⣿⣟⡉⢛⣉⡏⡀⣀⣸⣿⣿⡶⠿⠒
⠀⣾⡻⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⠃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣻⣿⣄⡄⠀⠀⠀⠀⠀⢸⡏⣿⣿⠟⢿⣿⠿⢿⣿⣿⣿⡀⢀⣠⣶⡀⣠⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣀⣄
⡟⠀⠙⠃⠀⠀⠀⠀⠀⢸⡿⢯⣿⣤⣀⣻⣆⣠⠿⣿⠟⣿⣯⣨⣿⣹⡟⣹⣿⣿⢿⣋⡀⣄⣽⢳⣻⡿⠁⠀⠀⠀⠀⣼⣇
⠅⠀⢰⡀⠀⠀⠀⠀⠀⠈⢄⣈⣭⣿⣿⣿⣣⡿⠷⣯⣟⣯⣅⠉⣻⣿⣾⣿⡿⠿⣫⡻⣿⠿⢻⣿⣼⣧⠀⠀⠀⠀⠀⠈⢿⣧⡀
⡟⠀⢺⡇⠀⠀⠀⠀⠀⠀⢘⣿⡿⡿⡼⡟⢶⣅⣄⣀⣿⣾⣿⣷⣿⣿⣿⣍⣴⣶⡿⠏⣥⣤⢿⣷⠖⢯⠀⠀⠀⠀⠀⠀⠈⢻⣷
⠁⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢗⡋⢱⣐⣯⣶⣽⣿⠿⠿⣻⣿⣿⣭⣔⡻⠿⢿⣾⣷⣚⣺⣛⣸⣗⡋⣻⡖⠀⠀⠀⠀⠀⠀⠘⢿
⠀⣿⣿⡇⢠⡄⠀⠀⠀⠾⣷⡿⠷⠛⢫⣽⠋⠡⡶⡖⠀⠛⠿⣿⣿⣿⣥⣤⡀⠈⠉⢻⣿⣿⣿⣿⣵⣾⣷⠄⠀⠀⠀⠀⠀⠀⠈
⢸⣿⡿⣰⣿⡇⠀⠀⠀⠈⠛⠿⣿⠿⣿⣯⣤⣤⡍⢵⣶⣿⣿⣿⣿⣿⣿⡿⠟⠙⣛⠋⠛⠋⢻⣾⡿⠟⠛
⣿⣿⣧⣿⣿⠃⠀⠀⠀⠀⢀⠀⠈⠙⢿⣿⣿⣻⣶⣿⣿⣿⣿⡿⠋⠁⠀⣠⣤⣤⣤⡼⣿⣿⡫⢋⣤⣀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠐⢿⠇⡀⠀⢸⣿⢿⣽⣿⣿⣿⣿⢿⣿⣶⡶⣾⠟⡓⣿⣫⣿⣿⠋⢀⣈⠛⠃⠀⠀⠀⠀⠀⠀⢰⣾⡇
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣀⡀⠉⠀⠈⣿⠐⠙⣿⣥⣿⣇⣦⢻⠻⠐⡿⣾⣿⣧⣼⢿⣿⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠁⠀⣈⠀⢿⣰⣿⣿⣿⣿⣿⣷⣼⠀⣰⣿⣿⣿⣿⠀⢘⣿⣤⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣴⣦⡉⠙⣿⣿⣿⢿⣿⣿⠀⣿⣿⡿⠛⠃⠀⢘⣿⠙⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⡏⢀⣶⣦⠀⠀⠀⠀⠈⠀⠀⠻⠟⠁⠀⣿⣿⣿⠀⢻⣿⣶⣿⠏⠀⠀⡀⠀⢸⣿⠀⢀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣝
⣿⣿⣿⣷⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣿⣿⣿⠀⢸⡟⠟⠡⣿⣿⡟⠀⠀⠘⠁⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠛⠿⠿⠀⠘⠇⠀⠀⠉⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄⠒⠒⠒⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡩⠛⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⣀⢤⡈⠂⠀⠀⠀⠀⠀⠀⠀⢀⣞⡤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⡁⢸⠀⢧⠀⠀⠀⠀⠀⢀⡴⢋⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢇⠘⡆⠀⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡘⡤⠒⠢⣄⠀⠀⠀⠀⣿⠀⢀⠀⠀⠀⠀⠐⠲⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢸⣀⠀⢀⡼⠦⣄⡀⠀⢻⠀⠎⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠋⡍⢱⡀⠀⠀⠉⠓⠘⣧⠀⠀⣿⣧⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⡠⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢳⠀⢧⠀⠀⠀⠀⠀⠈⠳⣄⠈⠛⠀⠀⠀⠀⠀⠀⠀⢀⠴⠾⣤⢀⡴⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⣆⠸⡄⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⡀⠀⣀⡠⠔⠊⠉⣉⣩⠕⠋⠀⠈⢫⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠘⠀⢧⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⢧⠀⠀⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢻⠀⢇⠘⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠲⠎⡇⢀⣀⡤⠖⠋⠉⠉⠛⠒⠲⠤⠤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⢙⣦
⠀⠀⠀⠲⠶⠶⠶⣄⡠⠴⠖⠒⠒⠒⠒⠒⠂⠳⣤⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⠭⠤⢤⣶⠤⠀⢀⣀⣘⣀⠤⠤⠖⠒⠚⠉⠁
⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⢀⠜⡩⠋⠙⠒⠶⠶⠶⠤⠦⠤⠤⠤⠤⠄
⠀⠀⠠⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⢉⡥⠊
⠀⠀⠀⠀⠉⠙⠒⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⣄⣀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠒⠊⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠙⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠒⠒⠒⠒⠒⠂⠒⢲⣤⣶⣾⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠉⠓⢄⡀⠀⠀⠀⢀⠔⠉⠀⣠⣀⣀⣀⣄⣤⡶⢿⣻⡽⠾⢏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⢀⠀⠈⠲⢄⣰⠋⣀⡠⠞⢣⣭⡿⣟⣫⡵⠞⣻⠁⠀⠀⠀⠙⠦⡀
⠀⠀⠀⠀⠀⣀⡤⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠚⠉⠠⠞⠁⣀⠄⠀⢰⠏⢠⣾⣶⣿⣿⠿⡛⡹⣇⢨⢎⡄⠀⠀⠀⠀⠀⠀⠈⠲⣄⡀
⠀⠰⣤⡐⠊⠁⠀⠀⠀⠀⢀⣀⡄⠛⠁⣀⠄⠋⢁⡤⠂⠁⢀⠤⠈⠁⢀⣠⣼⡿⢛⡛⠛⠓⠤⢴⣁⣠⣣⡎⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢸⣽⣽⣷
⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠘⠉⠀⠐⠃⠀⡠⠀⠁⠀⠤⠂⠀⡀⠐⠞⠛⠋⣏⠧⣤⣘⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠠⠄⠈⢀⡴⠛⠀⡀⠐⠊⠀⠀⠀⠀⠀⠈⠉⠉⠚⠛⠛⠒⠒⠶⠦⢤⣤⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠒⠉⣀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⣖⠒⠚⠛⠒⠲⣦⣿⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⠀⠀⢻⣾⣀⣧⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠗⠓⠐⠐⠁⠃⠛⠑⠘⠘⠀⠘⠋⠚⠂⠛⠀⠞⠀⠺⠊⠘⠊⠛⠁⠂⠀⠚⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣿⣿⣿⣶⣶⣶⡄⠘⠛⠀⣠⣶⣶⣿⣷⣾⣷⢦⣶⠆⠈⠙⠛⠷⠶⠄
⠀⠀⠀⠀⢸⡿⠊⠉⠉⠙⢻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢰⣦⠾⠿⠛⠃⠀⠀⠀⠀⠐⠓
⠀⠀⠀⠀⣶⡿⠀⠀⠀⢀⣾⠛⠋⠻⣿⣿⣤⣄⣀⣀⡰⠖⠀⠀⣀⣀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢀⣤⣶⣤⣴⣿⠋⠀⠀⠀⠈⢻⣿⣿⣿⣿⡇⠀⠀⠘⢿⣿⣿⣿⠀⠀⠈⠻⣿⣶⣷⡆
⠀⠀⠀⠀⣻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠙⢿⣿⠏⠀⠀⠀⠀⠀⠻⡿⠁⠀⠀⠀⠀⠀⠙⢿⣄⡀⠀⣠⣤
⠀⠀⠀⠀⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿
⠀⠀⠀⠀⠸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠤⢀⡀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠀⠀⠲⣦⠀⠀⠀⠀⠸⠆⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠋⠀⠀⠀⠋

⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠧⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠘⠛⠛⠀⠂⠙⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠂⠀⠾⠐⠀⠆⠀⠀⠀⡠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣸⢿⣿⣷⠔⠛⣁⣴⢾⣿⣻⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⣿⡿⢟⣩⣴⣶⣿⣿⣷⡿⠛⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⡿⠝⣻⣿⣿⣿⣿⣿⠿⣿⠉⠏⠁⣀⠄⡀
⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠁⠉⠛⢻⢿⣻⠋⠁⠀⠀⠀⠀⠀⠠⠛
⣿⣷⣤⡌⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⡉⠀⠁⠀⠙⠛⠁⣀⣤⣈⠀⠻⠿⢿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⢆⠀⠀⠈⠀⠞
⣿⣿⣿⡏⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⠀⠋⠀⡤⠦⠀⣀⣀⣀⣀⡈⠑⠀⠰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣟⣾⣿⣙⣿⣿⡿⠛⠙⠃⠀⠷⡆⠂⣸⣆⡋⠉⢸⠁⠸⠀⠈⣶⣦⡛⡄
⣿⣿⣿⡏⡀⠀⢸⣿⣿⣿⣿⣿⣿⠿⢿⠿⡋⢀⣴⣶⣄⠀⣞⣿⠂⢿⣿⠃⠆⣤⠀⠀⠀⢸⣿⣿⡇⠁⠀⠀⠀⢀⣴⣿⣦⠄
⣿⣿⣿⣿⣿⣦⢸⣿⣿⣏⣹⣿⣿⣶⡉⠈⡁⣸⣇⣰⡧⠀⠻⠿⠀⠀⡝⠂⠘⢿⠀⠀⠀⠘⢻⠿⠇⠀⠀⠀⠀⠈⠥⡈⢛⣀⡀⢠⣀
⣿⢿⣿⣿⣿⣿⣼⣿⡏⠈⣷⠘⢋⣍⠩⡛⣿⣿⣿⣷⡄⠀⠀⢀⠀⠀⠁⠀⠀⠈⣶⣤⠄⠀⣈⠀⠀⠀⠀⣀⣀⣤⠚⠀⠈⡏⣿⠷⠌⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠙⠀⣼⣿⡄⠲⣌⣿⣿⣿⣷⡴⠢⠔⡅⠀⢸⣿⣏⡄⢿⣿⠀⠀⠘⠛⠻⢿⣿⠿⠟⢣⠀⠀⠀⠀⣯
⣿⣿⣿⣿⣿⣿⠿⢻⣬⡉⣷⣘⣻⠛⣿⣶⠈⣹⣿⣿⣿⣆⠀⠀⡇⠀⢸⣿⣽⣿⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⠀⠀⢀⣿⡄
⣿⣿⣿⣟⡟⡟⢳⠀⠉⠁⠴⠋⣩⣿⣧⣿⣿⢿⣿⡯⠀⢈⢛⣲⠿⡄⢠⣿⣿⣿⣿⣿⣦⣤⣀⣀⣀⡀⠀⣀⢤⠋⠀⠀⢠⣾⣟⠁⢠⠆⠀⠀⣀⡀
⣿⣿⣿⣿⣿⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠈⡀⢀⠿⢀⣼⡇⠀⢾⣿⠿⣿⣿⣿⣿⣧⣿⣿⡋⡄⢺⡇⢸⣷⣶⣾⣿⠟⢃⣠⣡⢤⣤⣤⣼⣧
⣿⣿⣿⣿⣿⠋⢹⣹⣛⣻⣿⣿⣿⠁⣀⣀⢀⣀⣰⡶⠀⠛⠛⠛⠃⠀⠘⡿⠘⠿⠛⠉⠉⠁⠸⠟⠃⠀⠙⣷⣾⡄⡏⢿⡇⣰⡟⢍⡞⣛⣛⣛⣻⣛⣿⣶⣤
⣿⣿⣅⣤⡀⠒⠛⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣦⣴⣿⣗⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⢘⣿⣿⡿⢸⣧⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡇⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⢠⠂⠀⣴⡇⣿⣿⣿⠿⠓⠀⠀⠀⠀⠀⠀⢀⣀⣠⡿⠍⠉⣴⣾⣿⢽⡇⢸⣻⡇⣿⣿⢿⣿⣿⣿⣿
⣿⣿⡇⣿⣟⣛⣛⣝⠃⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⡾⢀⣷⣿⠃⢩⣿⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⣿⣫⣍⣀⣠⣿⣤⣇⣿⣿⢻⣿⣿⣿⣿
⣿⣿⣧⢸⣯⠉⣁⡈⠁⠁⢈⣠⡤⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⣿⣿⣿⣽⡭⠟⠂⠀⠀⠀⠀⠀⠀⢀⡀⠉⢁⡈⠉⠉⠈⠉⠈⠉⠉⠉⠉⠉⢹⣿⣿⠿
⣿⣿⣿⣿⣿⣿⡯⣥⣤⣾⡋⠀⠀⣀⡀⠀⠀⠀⠒⠂⠀⣻⠿⢋⡔⠊⠉⠉⠙⠃⢀⢀⣀⣤⣶⣿⣿⡾⡿⠭⢟⠉⠛⠒⠀⠀⠀⠀⠀⠀⠀⢢⢤⣬⣽⣿⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣎⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⣻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣛⣷⣴⣝⢧⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⢻⡉⠈⠉⠙⠻⣦⣷⣌⣩⡉⠉⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢋⣠⣶⣿⣿⡶⡖⠀⠀⠀⠈⠙⠻⠿⣄⡀⠈⢦⡀
⠀⠀⠀⢀⣀⣀⣠⣤⡤⠤⠤⠴⠚⠃⠀⠀⠀⢚⡏⠉⠉⠙⠛⠇⢠⣄⣀⠀⠀⠀⠀⠀⠙⠲⣆⠈⡂
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣤⣿⣧⡀⠀⠀⠀⠀⠈⠛⢿⣿⣴⠆⣤⡄⡀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠂⠀⠀⠀⠀⠀⠀⠙⠿⣿⠷⢬⣄⡀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣶⠀⠀⠀⠀⠠⣀⡀⠀⠀⠀⠀⠈⠓⠦⣌⣻⣶⣄⠀⠐⠉⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⢿⣿⣷⣄⣀⣀⠁⠘⢻⠿⣷⣅⣀⢀⡀⠀⠀⠀⠉⠻⣿⣷⣦⣄⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢐⣀⣈⡍⠻⠿⠸⣇⣆⣦⠀⠈⢹⠛⢿⣾⣡⣀⣀⡀⠀⠏⠉⢻⣌⠁
⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣦⣿⡇⡾⢹⣿⣇⣀⠀⠀⠘⡿⣿⣿⣤⣦⣠⠴⣾⣿⣿⣿⣿⣿⣶⡶⠶⠛⢷⣄
⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣭⣿⣻⣷⡶⢠⠀⠉⠿⡼⢻⢿⣿⣿⣿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠙⢷⣤⡀
⠀⠀⠀⠿⠿⠿⠿⠟⣻⣟⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢆
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠉⠙⠛⠿⢿⣿⠿⠿⠿⢿⣿⡿⠿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


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

⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⢠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢰⠀⠉⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠢⣃⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⢠⣼⣿⡿⠿⠂⠀⢀⡾⢁⠔⠀⢀⣠⡴⠖⠛⠛
⠀⠀⠀⢸⣿⡟⠻⠀⠀⣠⣿⣿⣥⡶⣚⣩⡥⠖⠛⠉⠉⠁
⠀⠀⠀⠈⢻⣿⠀⠀⠀⠈⠙⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡀
⠀⠀⠀⠀⠀⣸⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠠⢩⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡛⠿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣹⡆⠀⠀⠀⠀⠀⢐⣾⣯⣷⠀⠀⠀⠀⠀⠀⣲⣿⣽⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠃⠀⠀⠀⠀⠀⢸⣿⡿⠋⠀⠀⠀⠀⠀⠀⣿⣿⠟⠁
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣤⣶⣶⣿⣿⣆⠀⠀⠀⢠⣶⣶⣿⣿⣷⡀⠀⠀⠀⢶⣿⡿⢿⣿⣆
⠀⠀⠀⠀⠀⣭⣭⣩⣭⡆⠀⠀⠀⠀⢀⣴⣿⣷⣾⣿⣿⡄⠀⠀⣴⣾⣿⣶⣿⣿⡇⠀⠀⣠⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⣤⣤⣾⡿⠿⣿⣿⣿⡿⢡⣤⣼⣿⠟⢿⣿⣿⣿⠇⢴⣶⣿⠿⠻⣿⣿⣿⠟
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠉⠉⠀⠀⠈⠉⠉⠀⠀⠀⠉⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⡀
⠀⡀⡇⠀⠀⡀⣼⡀⠀⢠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐
⢰⡇⣿⣿⠀⣧⣿⡇⢰⢾⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠈⠛⠿⣿⣀⣿⣿⡇⣼⣾⡆⣷⢀⡇⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢿⡧⠇⢄⡀
⠀⠀⠀⠀⠓⠉⠛⠓⠿⠿⢯⣿⣤⣧⣾⣠⡆⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⠇⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠇⠀⠀⢳⠀⢸⠄⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠉⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠑⠄⠀⡬⡆⢀⠂⢠⠈
⠀⠀⠆⠠⠒⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⢀⠀⠀⠒⠣⣀⡤⡀⣸⣭⠆⠠
⠐⡆⢰⣛⣒⡀⣤⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢘⠌⢨⠅⣀⠹⣷⣥⣕⠂⢓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⢀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣦⣾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⡀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣆⠀⣠⡄⠀⢀⣴⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢀⣾⣿⣶⣿⣿⣿⣿⣿⣆⣴⣿⣆
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣻⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⠿⠿⢟⣫⣶⣿⣿⣿⢭⣽⣛⡝⠂⠩⣭⣛⡛⠻⠿⠿⠿⢟⣛⣫⣵⣿⠟⣿⡿⢟⡪⣔⣮⠳⢍⡙⠛⢫⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣑⣃⣀⣠⠄⣐⣬⣽⢟⣋⡭⠷⠦⢶⣢⣤⡉⢔⣲⣭⣿⣿⣷⣮⣭⣿⣿⣿⣯⣙⣶⠾⠫⢾⣿⣿⣭⣖⣿⡶⠿⢛⣛⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⠙⠋⣉⣉⣭⣭⣅⣛⠻⢭⣛⣫⡴⢮⣽⠷⠾⠿⠿⠶⡾⢿⣿⠿⣭⣷⣶⣮⠭⠽⠿⠛⠛⣒⣙⣫⡵⠾⠛⣉⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣼⠻⠷⠶⢖⡫⠽⣛⣴⣬⠭⣿⣛⠤⣶⠾⢝⣟⣭⣝⣿⣿⣾⣷⣽⣍⣋⣙⣀⣈⣊⣛⠽⣵⣒⠴⢞⣋⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣩⣭⣭⣭⡭⢽⣓⠶⢿⣻⣶⣬⡵⢞⣛⣯⡥⢶⣿⣥⠯⠽⣟⣿⣶⠭⢭⣭⡉⣉⣩⣭⠭⢴⣞⡯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢭⣿⣶⣢⢌⣉⣑⡒⠒⠂⢘⣫⢭⠒⣉⣽⣖⣛⣩⣭⣽⣭⣭⣽⣛⠤⠮⣌⡕⠀⠚⠯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣭⣛⡻⠿⣙⣻⣤⣤⠈⠶⣯⣝⡻⣛⠭⣑⣻⣧⣶⣾⣿⡿⣛⡿⢾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠴⢿⣤⣞⠿⠯⠉⣭⣯⣿⡶⠾⠧⣬⣽⣿⢥⣼⣟⡻⠴⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠟⠛⢭⡴⣬⠴⠾⠿⠟⠟⠋⠛⠈⠁




⣶⣶⣶⣶⣶⡾⢶⣶⣶⡾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣾⢿⡾⠶⡶⠶⢾⣶⡾⠾⠾⠶⠶⢾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣀⠤⠴⠖⠛⠉⠉⠙⠛⠓⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢀⠜⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⣀⣄⡈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣃⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠇⠀⣞⠁⢧⠀⢧⠀⠀⠀⠀⠀⠀⠀⣠⠖⡻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠸⡆⠘⡆⠀⠀⠀⠀⠀⠚⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢳⠀⣙⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠓⢤⠋⠀⠈⠙⡆⠀⠀⠀⠀⢸⠀⢠⠆⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣦⣄⡤⠔⠉⠓⠦⣄⡀⠸⡆⠚⠀⢀⣄⠀⠀⢰⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢒⠁⣆⠀⣇⠀⠀⠀⠀⠉⠀⢻⡄⠀⠘⣿⣷⠀⠈⢿⡿⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢸⡀⢸⡄⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠛⢶⠄⣠⠞⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢧⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀⠀⠀⠀⣀⡤⠔⠊⠉⠉⣉⡥⠚⠁⠀⠈⢯⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠘⠆⠸⡆⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠷⠶⠶⠒⠚⠉⠁⠀⠀⠀⠀⠀⠈⣧⠀⠀⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢻⠀⢰⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠉⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⣇⣈⣃⠼⡇⠀⠀⠀⣀⣠⠔⠲⠤⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣘⢦⡉⢰⠠⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠒⠶⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⣀⣀⡠⠤⠞
⠈⠉⠉⢉⣩⠿⠒⠚⠉⠉⠉⠉⠉⠉⠉⠉⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠒⠛⠺⣯⣥⣤⠤⠤⠤⠴⠒⠒⠋⠉⠉
⠀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡁⠀⠀⠀⠀⣀⠤⢃⠔⠁⠉⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠒
⢾⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢉⡠⠞⠁
⠀⠀⠉⠓⠲⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠖⠒⠒⠒⠚⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⣀⡀⠀⠀⣠⠖⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣳⣞⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠉⠙⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠘⠀⢿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣆⣤⣤⠐⣲⡿⠁⠁⠀⠀⠀⠀⠀⠈⠻⣷⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠠⢠⣀⣀⠀⢸⣷⣮⣿⡯⣹⠋⠀⢀⢠⣤⠀⠀⠀⠀⢀⠀⠀⠀⠈⣿⣾⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠢⡤⡘⣿⣿⣷⣌⢻⣿⣿⢽⢿⣤⣌⠁⣈⡿⠆⠀⠀⠀⢨⠁⠀⠀⢠⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠫⣫⡿⣿⣿⣷⣿⣿⡅⠁⠉⠉⠉⠉⠉⠉⠁⠀⠈⢸⠳⢴⢃⡘⠈⠻⠋⠑⠉⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣻⣿⣿⠟⠉⠓⠒⣲⠖⠒⠒⠒⠒⠲⡖⠒⠚⠢⡀⠙⠷⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡁⠀⢀⡴⠞⡀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿
⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⢄⣀⢝⢶⣫⣤⣾⣿⣶⣤⣤⢀⠀⢀⠀⢀⠙⢦⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿
⣿⣿⣿⣿⣿⣿⡇⠄⣩⣵⣶⢻⣻⣷⣾⢣⣿⣿⣿⣷⣷⣷⣷⣾⡏⣽⣿⣶⡆⣴⠀⡦⢤⡄⢠⡶⠀⠀⠀⡀⠀⠘⢿
⣿⣿⣿⣿⣿⣿⣿⣛⠿⠟⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠾⠁⠺⠦⠻⣅⠠⠔⠈⠀⠀⠀⠸
⣿⣿⣿⣿⣿⣿⣿⣿⡧⠤⠽⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠋⠉⠉⠉⠉⠉⠙⠒⠢⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⡘⢡⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠐⠀⠀⠡⡀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠀⠀⠀⠃⠀⠀⠀⠀⠠⠀⠀⠀⠈⠄⠀⠀⠀⠀⢂⠀⠀⠀⠀⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢘⠦⠀⠀⠀⡑⡄⠀⠠⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⢹⢄⡃⠀⢠⣶⣶⣦⠘⠀⠀⠀⠀⠀⠀⣴⣶⣶⡄⢱⠈⣆⡎⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠸⡀⢀⠀⢆⠀⠐⠜⣿⣏⡿⠀⠀⠀⢀⠀⠀⠐⢿⣹⣿⠃⠀⠀⡘⠀⠐⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠄⠁⠈⢆⡀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡰⠉⠀⢠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣠⠜⠒⠤⣀⠀⠁⠂⠀⠀⠀⠀⠀⠀⢀⠠⠐⠣⢄⡀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢀⡞⠁⠀⠀⠀⠀⠈⠁⠄⣀⣀⡀⠀⠈⢀⠎⠀⠀⠀⠀⠹⣄⠀⢳
⠀⠀⠀⠀⠀⠀⢀⠜⠀⣀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠣⡀
⠀⠀⠀⠀⠀⠒⠓⠒⠉⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠙⠒⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠸⡆
⠤⠤⠤⠤⠤⠤⠤⠤⡎⠀⠀⠀⠀⠀⠀⠳⠒⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⢷⠤⠤⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⢰⠀⠀⡀
⠀⠀⠀⠀⢠⣮⡉⠉⠁⠈⠣⡀⠀⠀⠀⠀⠀⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠸⠁
⠀⠀⠀⠀⠠⡷⠄⠀⠀⠀⠀⠐⢄⡀⢀⠖⢎⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⡟⠂⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠢⠞⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⡄⣄⢂⠑
⠀⠀⠀⠀⠘⣟⡀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣨⠤⠾⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡧⢸⠂
⠀⠀⠀⠀⠀⣍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈
⠀⠀⠀⠀⠀⡯⠄
⠀⠀⠀⠀⠀⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⠀⢠⣀
⠀⠀⠀⠀⠀⣏⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠤⢀⣀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⣇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠀⠀⠀⠀⠀⠹⠉
⠀⠀⠀⠀⠀⢗⡂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠉⠉⠃⠀⠙⠁⠒⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠒⠤⡀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢀⢸⠇⠀⢠⠃
⠀⠀⠀⠀⠀⢀⣀⡤⠖⠠⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠐⠦⣄⣀⠀⠀⠀⠀⢱⣸⡌⡠⠒⠁
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠠⠄⣀⡈⠁⠐⠢⠒⠀⢉⡀⠤⠄⠐⠂⠀⠁⢹⡅⠀⠀⠀⣼⣄⡽
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠒⠠⠄⠉⡁⠀⠀⠥⠤⠐⠂⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⡇⠄⠇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⡉⠐⠒⠀⠠⠄⠀⠀⠐⠒⠂⢉⣁⡀⠀⠀⠀⢸⡇⠀⠀⡾⠐⡘
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⢁⣁⠐⠂⠀⠀⢈⣈⡉⠄⠀⠀⠀⠀⠀⠘⡇⠀⢠⠇⢆⠃
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠒⠂⠠⠬⠀⢰⠀⠠⠀⠒⠀⠈⠉⠀⠀⠀⠀⡇⠀⡼⠈⡈
⠀⠀⠀⠀⠀⣿⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠓⡳⠁
⠀⠀⠀⠀⠀⣿⣀⣴⢤⣤⣀⣀⡀⠀⠀⠀⢸⠀⠀⠀⠀⣀⣀⣠⣤⠴⣄⡀⡇⠀⠖⠁
⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠲⠞⠶⠖⠟⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠡⠀⠀⢀⠀⠀⠀⠀⠀⠀⠔⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀
⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣿⣷⣤⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣮⣩⡄⠀⣥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⢳⠀⠀⠠⠀⡀⠀⠀⣤⣀⠈⢸⠘⢻⠛⠟⠛
⠀⠠⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠐⠀⠀⠱⡄⢸⣿⡿⠟⢻⡆⠖
⢲⣾⣻⠯⢀⣀⡀⢀⣄⠀⠀⢀⠄⠂⠀⢠⠀⠀⠀⠐⡄⢠⡈⠑⠄⣀⠀⢹⢨⣤⣤⡈⢻⠐⣿⣷⣦
⠘⡇⢨⠀⠫⠀⠀⢹⣿⠀⠀⣠⠀⠀⠴⠋⠀⣾⠀⠀⡅⠀⠁⠀⠀⠻⣤⡀⢢⠀⠙⠑⢾⣠⠐⠈⠁
⠠⠶⠾⠶⠴⣤⡄⠉⡉⠀⠘⣿⠗⠀⠈⠀⣤⣤⣤⣦⡥⣤⣤⣤⣥⣴⣴⣶⣶⣷⣶⣶⣼⡷⠶⠛⢻⣯
⢰⣦⢞⣀⣠⡀⡀⣀⣀⣒⣸⣿⣧⣤⣤⡤⠬⣿⠭⢭⠽⠟⢻⣿⣿⣿⣻⣿⣿⣿⡟⡙⣹⡄⢠⡀⣾⣿
⣀⡶⢚⣻⢲⣿⠌⣱⣿⠟⣿⡿⠿⠿⠃⠀⠀⠠⠴⠫⣷⣖⠶⢿⢿⣿⣽⠟⠉⠈⢰⡟⣿⡥⣿⣄⣿⣿⠄
⠡⠺⠯⠖⠁⠀⠀⠿⠷⠏⠁⠀⠐⠂⢚⡁⢀⣉⡉⠉⡉⠙⠛⠿⠓⡬⣁⠉⠃⠀⣷⡀⣿⣗⠥⣸⣿⣿⣦
⠀⠀⢐⣒⣀⠀⠀⠀⠀⠀⠒⠀⠠⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠸⠦⠉⠀⠀⢠⣿⠀⣼⠁⠀⣿⣿⣿⣧
⢴⣾⠏⣽⡿⢿⡶⣴⣏⣿⠦⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⢟⡃⢹⠅⠈⣻⡿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠠⢀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⢅⠀⠐⠀⠀⠀⡀⠀⠔⢀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠢⠀⠀⠀⠀⠀⢀⠘⢀⠀⠀⠀⠐⠁⠠⡀⠀⠄⠐⠠⢄⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⠄⠀⣀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠑⣀⠌⠀⠀⠀⠈⢀⣤⣴⣶⣶⣿⡿⣷⣶⡶⠀⠀⢤⣀
⠀⠀⠀⠀⠀⠴⠶⠿⠟⠻⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣷⡿⠷⠿⠿⠿⠷⠤⠄⠀⠀⠈⠉⠉⠉⠉⠉⠀⠈⠉⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⡀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠈⠀⠀⠀⠀⠈⠀⠉⠀⠀⡐⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⡀⠀⢀⠀⠀⠀⢀⣀⠀⢀⠀⠀⠀⣀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⢸⡏⠙⣷⢸⣷⠀⡖⣿⠁⢠⣿⡇⠀⡦⠉⣿⠁⣾⡇⠀⣸⣼⣿⠉⢷⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣷⠀⣿⠈⣿⣴⠇⣿⠀⢸⣿⣿⢸⠇⠀⣿⠀⣿⡇⢀⡇⣿⣿⣠⡿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⣿⢸⣿⢀⡟⠀⢻⣿⠀⣿⠖⢸⡏⢻⣿⠀⠀⣿⠀⢻⣧⡾⠁⢿⣿⠻⡄⢸⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠈⠋⠉⠀⠀⠈⠉⠀⠙⠃⠈⠁⠈⠁⠀⠀⠉⠀⠈⠉⠁⠀⠈⠁⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣀⣀⣀⣀⠀⠔⢢⠀⠀⣀⠔⠂⠀⡃⠀⠠⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⠈⠑⠑⠀⠁⠊⠁⠀⠊⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠋⠉⠉⠉⠓⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠒⠀⠀⠄⣀⢀⡀⠀⣀⢀⠤⠂⠀⠀⢹⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠴⠋⠀⠁⠀⠀⠀⠁⠬⡀⠀⢐⡀⠀⠈⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠰⠋⠀⠁⠄⢀⠄⠀⠀⡀⠀⠀⡆⠀⠀⡇
⠀⠀⠀⠀⠀⢇⠀⠀⢠⣔⣊⣤⠴⠋⠀⠀⠀⠀⠀⢠⡄⠀⡀⠂⣀⠀⠀⠉⠀⣸⠃⠀⢸⠃
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠈⠹⡀⢠⡄⠀⠀⢀⡀⠐⠤⠠⠔⠀⠂⠉⠀⠠⠀⠀⠀⡀⢠⠏
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠉⣿⠀⠀⠀⠨⡑⢄⠀⢦⡀⠀⣀⠀⠀⠀⠀⠀⠀⡟⢋
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⢸⡄⠀⠀⠀⢿⣶⣅⡂⠈⠉⠀⠤⠂⠀⠀⠀⠀⡗⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⡈⢳⡀⠀⠀⠈⠫⠉⠛⠟⠂⡈⠁⠀⠀⠀⠀⣰⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠶⣤⣀⠀⠀⠀⠀⠀⠉⠀⠀⣀⣠⠞⠁⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠨⢇⡉⠉⠛⠓⠒⢲⠶⠟⡋⠥⣤⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢩⠁⠀⠈⢣
⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⡆
⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⢆
⠀⠀⠀⢠⡏⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠹⠉⠉⠉⠑⠢⣄⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠶⢶⠋
⠀⠀⠀⢀⠃⠀⠀⠀⠀⡰⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸
⠀⠀⠀⡎⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠈⠀⠰⡀⠀⠀⠀⠀⠀⠀⣿⠀⡆
⠀⠀⢸⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⠘⠆⠀⠑⡄⠀⠀⠀⠀⠀⠏⠀⠹⠄
⠀⠀⠈⠲⠤⣀⠀⠀⠀⠀⠀⠉⠒⠢⠤⣠⠀⠘⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⢤⣄⣀⣀⠀⠀⠀⠀⠉⠓⠒⠉⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠹⡀⠀⠀⠀⠈⠓⠒⣊⣡⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⡀⣠⠩⠏⢬⠛⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⣀⢀⣀⡀⢀⣀⡀⡀⢀⣀⣀⣀⢀⣀⡀⡀⠀⣀⠀⢀⢀⡀⢀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣓⡦⢀⠏⢰⣃⣠⢧⠷⡖⠁⡿⠋⢀⡞⢀⣟⡒⣸⣁⢠⣇⢀⡞⡜⢻⡞⣞⣐⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⣤⣭⣽⠑⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⣼⠀⠀⠉⢢⠊⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⣖⣉⣙⡘⠻⠿⠛⢠⠀⠀⠀⢸⠀⠨⢥⢲⡬⡇
⠀⠀⠀⠀⠀⣯⠀⠀⠀⢸⠀⠀⠀⣸⡆⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡟⠙⠉⠉⠀⠀⠀⠀⠸⠀⠸⠄⣼⠤⠇
⠀⠀⠀⠀⠘⠙⠿⢵⣦⣬⣤⣖⠽⠋⠓⠀⠀⠀⠀⠀⢀⠠⠜⠂⠚⠤⠄⡀⠀⠀⠀⠀⠀⠀⠑⠶⠒⡒⠒⠒⠊⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⣀⡤⢴⣴⠾⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠢⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠂⠒⡄
⠀⠀⠀⠀⠀⠸⡄⣠⠣⣄⣼⣋⡉⠀⠀⠀⠀⠀⠀⠀⡠⠔⢇⠀⠀⠱⡸⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢠⠃⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠑⠁⠤⠤⠠⠄⢓⠈⠀⠀⠀⠀⠀⠀⢧⣀⣨⠦⠤⠄⠃⠀⠀⠀⠀⠀⣿⣿⣽⠀⠀⠰⠂⠀⠀⡞⠚⡄
⠀⠀⠀⠀⠀⠀⠀⠢⠄⡀⠀⠐⢃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡣⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⣀⣞⠂⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣧⣰⡣⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠆⠀⠰⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⢤⡀
⠀⠀⠀⠀⠀⢀⣴⠆⠈⠀⠉⠉⠀⢠⣤⡄⠀⠀⠀⠀⠀⡈⢂⠃⠆⢁⢂⡀⠀⠀⠀⠀⠹⡏⢐⠣⠄⡼⠃⠀⠐⢄
⠀⠀⠀⢀⣴⠟⠁⠀⠀⢔⠉⠆⠀⢸⣿⡇⠀⠀⠀⠀⣼⡁⠀⠑⡎⠁⠀⣿⠀⠀⠀⠀⠀⠩⠅⠋⠐⠂⢐⢁⡂⠼⠀⠄
⠀⠀⠀⠀⠁⠀⠀⠀⠖⠚⠛⠒⢢⠀⠀⠀⠀⠀⠀⠀⣿⣤⣀⠀⡇⢀⣠⣿⡇⠀⠀⠀⠀⠀⠢⣄⠤⠤⡜⠡⢈⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠧⠤⠗⠊⠀⠀⠀⠀⠀⠀⠀⠁⠉⠙⠻⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠒⠒

⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⣀⣀⣀⣀⡀⣀⡀⢀⣀⣀⣀⣀⣀⣀⡀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⢀⡀⡀⣀⣀⡀⢀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠠⣤⣼⣢⠞⠀⠀⠘⢦⣸⠃⠀⠀⠀⠹⡄⠀⡄⠀⣠⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣇⣀⣀⣻⡀⠈⢿⡀⠀⢀⣠⡖⢷⡄⢻⠞⠃⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢦⣀⠈⡏⠉⣩⠍⠻⣄⠀⣷⠞⠋⠳⢤⠀⢳⣈⡆⠀⠀⢸⣀⣠⠀⠀⠠⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠓⣷⠚⠁⠀⠀⠙⢾⡇⠀⡀⠀⣄⠀⠈⣿⣀⣤⠶⠛⠙⠀⠁⠀⠀⢀⡤⠚⠻⣝⢩⠻⡏⣏⡙⠟⢦⣀⠀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢼⠄⠀⠀⠀⠀⣿⡤⠞⠁⠀⠘⣇⣸⠋⠁⠖⡄⠀⡃⠀⠀⠀⡴⢋⣀⣀⠀⠈⢻⣴⣬⣤⣧⣷⣧⣽⣦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣸⣇⣤⣦⣶⣠⣏⣀⣀⣠⣶⠀⣸⠏⠀⡄⠡⣧⣤⢶⡤⣄⣼⡇⣯⢉⢹⡇⠀⠀⡇⣶⢶⠲⣆⠀⠀⣷⣤⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣡⣬⣤⣤⣤⣤⣤⣥⣄⣉⣉⠉⠉⠛⠛⠛⠛⠛⢳⠲⠶⠷⠦⠼⣯⣍⣇⣹⡈⠀⡇⠀⠀⡇⠿⠾⠴⠟⠀⠀⣿⣉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠛⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠙⢓⠲⠷⠦⣤⣤⣀⡀⠀⣻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡶⠶⠶⠶⠴⠦⠶⠤⠤⣤⣤⣄⣤⣀⣀⡀⠠⢀⠀⠀⠈⠉⠙⠻⢦⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠶⠤⣤⣄⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡤⠤⠤⠤⢤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣶⡿⢳⡄⠀⠀⠀⠀⠀⠀⠀⠳⣌⠠⠀⠀⠀⢀⠀⠈⠉⠉⠉⠙⠛⡓⠲⠦⠤⢤⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⢀⡞⠳⣬⡛⣛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⢶⣶⣶⠓⠚⡛⠓⠓⠺⠶⡶⢴⣤⣄⡀⠀⢂⠌⡃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⡀⠃⠀⢸⡳⣄⡞⣉⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡌⠁⣀⡀⠁⡀⢀⠈⡀⣵⠟⢳⡄⢍⡓⠦⣄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣞⣯⣹⢦⡀⠈⢇⣸⢉⡽⠋⠀⠀⢀⣀⣠⣤⣤⣤⡄⠀⠠⠀⠙⠲⢤⣄⠠⠀⣀⠀⣀⠙⢳⡟⠂⠠⡸⠁⢨⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⡈⢹⡃⡴⠛⣿⣾⡟⣭⡄⢀⡤⢾⡏⠸⣇⡼⠿⡿⠁⠀⠀⠁⠀⢰⡟⠻⡟⢦⣠⣄⢀⡀⣹⡏⣈⡴⢦⣴⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠉⠛⣟⠁⡴⢻⠍⣿⡿⣠⢾⡇⢸⣧⣞⡉⣩⠟⠁⠀⠀⠀⠀⠀⠈⡿⠛⠁⠈⣧⡶⣄⣤⣼⣏⣉⠷⣴⠛⠖⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⡙⠲⢿⠞⠁⡼⠀⢸⣠⢿⣠⠿⢯⣤⡤⠛⠁⠀⠄⠀⠀⡀⠀⠀⢰⣇⣤⡀⠀⢸⣿⠉⡄⢼⡇⢉⣨⣏⢉⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠙⠲⢬⣇⠼⡁⠘⢿⡿⠿⠛⠛⠉⠁⠀⠀⢐⠀⠀⣻⡟⠛⢦⠀⣸⢿⣬⡿⠀⢸⣿⠤⠴⢾⡇⣤⠼⣯⢠⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢴⣤⣿⣾⠷⣦⣄⣻⣠⣤⢀⣄⡀⠂⠈⡆⠈⠳⢹⣷⣊⡼⠀⢻⠀⣿⠀⠀⢸⢿⠄⠰⢸⡧⠄⠄⡧⠤⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠈⠹⡇⠀⠀⠉⠻⣏⣿⠟⠋⢠⠀⢀⡗⠀⢠⠏⠠⢻⠀⠀⢸⠀⢸⠀⠀⢸⢸⠖⠒⢺⡗⠶⠒⡷⠲⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠈⠀⠀⠀⣹⡀⠀⠀⠀⣷⠀⠀⣴⣀⣿⡿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣄⣠⠶⠛⠉⠁⠀⡀⠀⣿⣷⢠⡿⣿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠁⠸⣇⠀⣿⣿⡾⢡⡏⡿⠈⠙⢄⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠞⡄⠀⢀⠔⠀⠐⠂⠐
⡄⠀⠀⠀⠀⠀⠀⠹⣷⣿⣿⢁⣿⣿⠃⠀⠁⠈⢈⡀⢠⣤⣤⣤⣧⡔⠁⢸⠋⡴⠁⡀⢸⡇
⠂⠀⠀⠀⠀⠄⠀⠀⠈⣿⢾⣼⣻⡏⣠⣴⡆⣪⣶⣗⠉⣉⠛⣯⣝⠳⢄⣾⢉⣁⣠⢀⠏⠈⠐⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⣹⣿⣿⣿⣿⣿⡟⠟⠛⠛⠒⠛⠛⢯⡛⠓⣾⢿⣍⣀⣠⠟⠀⡀⠀⠀⠀⠉⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⡟⠁⢸⡀⠀⠀⠀⠀⠀⠹⢾⣃⠘⠓⡾⠉⠳⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠋⠀⠁⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⢈⡄⠀⠀⠀⠀⠘⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣄⡀⠀⠀⠀⠀⡀⣼⣿⣿⣿⠀⠀⠀⢸⠁
⠛⠒⠀⠘⠀⠼⢹⣿⣿⣿⡇⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢤⠤⠤⠴⠤⠤⠤⠤⠤⠤⠄
⡀⠀⢠⠀⢀⡀⣾⣿⣿⣿⠀⡀⢠⠈⠀⠀⠀⠋⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂
⠙⣶⠞⢣⠸⡇⡸⠋⠉⠙⣨⣡⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣆⠀⠀⢠⠶⠆⡒⣂⠀⢀⠀⣔⢉⠰
⡷⠛⣿⡾⢻⣶⣧⡄⣠⠃⡟⢁⡶⠂⠀⢠⡐⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢉⠚⠀⠌⠘⠛⠉⠀⠀⠀⠀⠈⠉⠀⠉⠀⡐⠋⠀⠀⠉⠘⠻⡟⠛⠂
⣁⣤⣯⣧⣾⣿⣾⡖⣹⣰⢴⠋⠀⠀⠀⠀⢙⣆⠙⣄⣀⣀⣀⣀⢀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⣈⣡⣄⣈⢀⣀⣀⣄⣀⡀⢀⣄⣀⠀⣀
⠉⡁⣿⢹⠗⣿⣷⣧⣽⣿⡏⣉⡔⠁⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠂⠀⠤⢭⣝⣯⡙⠛⣛⠛⠛⠛⠧⠷
⠀⢰⢻⢸⡄⣿⠂⣿⢟⣸⠔⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠘⡇⠸⣿⠃⠈⣯⣌⠃⠖⡾
⠲⣶⣼⣆⢻⢠⣾⡟⣿⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠
⣼⣾⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⡠⠆
⣽⠿⠿⢿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠔⠁⠈
⠀⠀⠀⠘⣋⢩⣿⡛⣯⠈⠙⢢⣴⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⢀⡂⢠⠮⠂⣴⠤⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⡀⠀⠐⢒⠉⢤⣼⣿⣾⣷⣽⣿⡯⠭⢭⠹⣿⣿⡂⢀⣴⣦⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⡜⡩⠋⠀⠀⠿⠤⠄⠠⠀⠘⠀⠀⠀⢸⣻⡅⢀⣀⠺⠄
⡄⠔⠀⠀⠀⡀⢭⣥⣥⣨⣿⣷⣤⡶⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢀⣄⣠⢆⣀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢡
⣄⣸⢀⣶⣭⣀⡨⠙⣻⣿⣯⣿⣇⣠⣼⣶⣾⠋⢻⠿⠿⢟⣿⣻⣿⡿⢿⣿⣿⣿⠋⣩⡿⠉⠀⠀⡀⢀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠁
⣯⢿⣿⣿⣿⣿⣿⣿⣾⠿⠻⠛⠙⣿⣾⣿⡿⣿⣿⠤⣴⢿⣭⣭⣿⣿⣿⣿⣿⣿⠀⠟⢿⣧⣤⠛⠉⠙⠺⠇⠀⠀⢈⣤⣄⡀⠀⠀⠀⠀⠀⢀⡀
⣟⣿⣿⣿⢿⣿⠋⡭⣄⠀⣀⣀⣼⣿⣿⣍⣟⠉⣃⡾⢣⣴⣶⠶⣶⡿⣿⣿⡏⠉⠉⠀⠀⠻⡇⠀⠠⣤⣤⢆⠀⠀⣸⠄⠈⠁⠀⠀⠀⠀⠀⠀⠈⠲⣤⣄⣀⣀
⣿⣿⣿⣷⠿⠁⢀⣀⣙⣿⡟⠋⠉⠙⣿⣿⣿⣿⣿⣾⡿⠛⠛⠛⠻⣿⣿⣿⣠⣤⣤⣤⣀⣀⣼⣶⣾⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠙⠛⠿⠿
⣻⣿⣟⣱⣟⠻⣧⡄⠀⠀⢈⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡶⠒⠒⢛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡀⣠⣤⣤⣤⣤⣴⣶⣟⠋⠉⠀⢄⣤⣄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⠔⡾⠻⢦⣱⠈⢰⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢻⠶⡚⣉⣉⣉⡉⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢸⣿⣿⣿⣶⠀⠀⢀⡀⠀⠀⠀⠀⢰⣖⣷⠁⢘⣿⣿⣿⣿⣷⣂⣼⡿
⠀⠀⠀⠀⢠⣾⣿⣷⡆⠀⠈⣿⣿⡿⢟⣥⣶⣿⣧⣀⠀⠀⠤⠦⠄⠹⣄⠙⠿⢿⣿⣿⣯⣾⡿⡃
⠀⠀⠀⠀⢶⡟⣿⢏⣵⣠⣾⣿⣟⡁⣺⣿⣿⣿⡛⣣⣶⣀⠀⠀⢀⣠⣌⠉⢱⣨⣼⣿⣿⣿⣧
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡘⣿⣿⣿⢇⣀⠙⠟⠿⠿⠿⠟⠛⣿⣿⣿⣿⡆
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⡟⢫⣼⣿⣷⣦⠀⠀⠀⢰⣾⣿⣿⣿⣿⣷⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠻⠻⢿⣿⣿⣠⡀⠀⠀⣽⡟⠛⣿⡟⠋
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣝⠿⢦⣾⣿⣿⣿⣿⠆⠀⢹⣷⡀⢻⣷⡀
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⡟⢛⣽⡿⠃⣾⣿⣿⣿⣿⣿⠀⠀⠀⣿⣷⠈⣿⣿
⠀⠀⠀⢸⠿⠿⢿⣿⣿⡿⠿⠿⣿⣿⡿⠿⡿⠇⠘⢯⠹⠠⣿⣿⣿⠿⠹⠿⠄⠀⣠⡾⠿⠀⠼⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⠹⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣇⣦⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣯⣼⣿⣿⡷⣄⡘⣦⠻⠿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢉⢻⢿⣿⣿⣿⣡⡱⣾⡝⢦⡈⠛⠛⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⠈⠫⠳⣄⠻⢿⣿⣝⣾⣿⣷⠙⠲⣴⣷⣦⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣍⣛⠛⠛⠉⣠⣴⣾⣰⣤⡴⠶⣤⣀⠀⠉⠛⠿⠿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣿⣿⣿⣿⡿⡋⣠⣦⡈⡛⠿⣷⣾⣴⣴⣴⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣆⠀⠀⠀⠀⠀⠀⢰⢖⠲⠄⠀⠀⠀⠀⠀⠀⠀⡄⠠
⠀⠀⠀⢀⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢪⡖⠀⠀⠀⠀⠀⠀⠀⠀⢅⡀
⠀⠀⠀⠈⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡
⠀⠀⠀⠰⠄⠠⠤⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠄⠀⠀⠀⠀⠀⠀⠠⠀⠰⠠⠀⠀⠀⠀⠀⠐⠠⠴⠄⠴⠐

⠀⠀⠀⠀⠰⠖⠄
⠀⠀⠀⢀⣤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠺⠿⠀⡮⠆⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡞⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠰⠆⠶⠾⠦⠆
⠀⠀⢀⣀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠈⠀⠀⡀⣀⠀⠀⠀⠀⠛⠒⢦⡀⠀⠀⠀⢠⡄⡀⣄⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠇⠀⠀⢎⡡⢀⠀⢀⢠⠿⢸⡀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⣠⢠⠀⡄⢤⢀⣄⣀⠄⠀⠀⠀⠀⡇
⠀⠀⠀⠐⠦⠿⠃⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠈⠈⠀⠁⠀⠈⠈⠉⠈⠀⠀⢀⡴⠁⠀⠀⠀⡄⢠⡦⠀⡄
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⠓⠲⠖⠊⠓⠲⠖⠚⠙⠒⠶⠒⠒⠲⠖⠒⠋⠀⠀⠀⠀⠾⠿⣿⣿⡿⠿⠆
⠀⠀⠀⢀⠉⣉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⡀⢀
⠀⠀⠉⠈⠀⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⡄⠀⠀⠀⠀⠀⠢⠈⣷⡴⠀⠀⠀⠀⠀⢸⢰⢰⣐⣂⡇
⠀⠀⠀⠀⠐⠖⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠸⠘⠈⠈⠀⠇
⠀⠀⠀⠠⠄⠠⠤⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠠⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⡀⠀⠒
⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠙⠛⡷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀⡤
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⣴⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠙⢿⣧⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⣠⣿⠏⠀⠀⠀⠀⣤⠛⠙⣿⣄⠀⠐⣄⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢰⣶⣤⡤⠀⡀⣠⣿⣿⣿⠋⠀⡀⠀⣠⣴⣞⠁⡀⠀⠀⠙⣳⣶⣿⣿⣿⠷⠦⠀⠸⠇⣷⣤⡄
⠀⠀⠀⠤⠈⣋⣾⣿⣯⣿⣿⣿⣿⡟⣿⣿⣶⠿⢛⡿⠟⠁⢤⣾⣿⣿⣿⣿⣯⠱⠶⢶⣄⠀⢀⠈⠁⠀⠄
⠀⠀⠀⠰⣾⣯⣿⣿⣾⣾⣿⣿⣿⣷⣿⣿⣿⣷⣾⣷⣾⣾⡆⠉⢻⣽⣿⣿⠿⠁⠀⢀⣨⣿⣿⣿⣷⣶⣦⣤⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣾⣟⠙⠿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠉⠙⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⠋
⠀⠀⠀⠀⠀⠀⠻⠿⡿⣿⣿⣿⣿⣿⣏⠙⠻⣿⣿⣿⣿⡛⠀⠂⢀⣀⣠⣬⣯⣭⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣶⣶⡀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⣿⠻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠿⠿⠿⠟⠙⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⣴⠃⠀⠀⠀⡰⠁⠀⠀⠀⠑⡄⠀⠀⠈⣠⣤⣤⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤
⠀⠀⠀⡸⠃⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⡇⠀⠀⠸⣿⣿⣿⣿⡆⠀⠐⠀⠀⠀⠀⠀⠀⠐⠀⢠⣀⣠⣀⠀⠀⠀⠀⠬⣿⣿⣿⡿⠴
⠀⠀⠘⠃⠀⠀⢀⡠⠀⠀⠑⠒⠒⠒⠂⢵⡀⠀⢠⣾⣿⣿⡿⠁⡀⠀⠀⠀⠀⢨⡀⠀⠀⠀⠀⠟⢻⡏⠀⠀⠀⠀⠀⠙⢻⡟⠁⠀⢀⡀
⠀⠀⠀⠐⠀⠒⠁⠀⠀⠀⠠⢀⣀⠀⠀⠈⠁⠀⣿⣿⠏⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⡀⠀⠀⠀⠚⠛⠛⠃
⠀⠀⢀⣀⣀⠈⠀⠀⠀⠀⢀⣾⣿⣿⣦⣄⣀⠀⠙⠧⡆⠀⠀⠀⣠⣤⡄⠀⠁⠀⠀⠀⣴⣿⣿⣿⣿⣷⠠⠀⠠⠀⠀⠀⠀⠀⢀⣠⣴
⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣿⣿⣿⠀⢀⠀⠠⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⠀⡀⠀⣿⣿⣿
⠀⠀⣿⣿⣿⣧⠀⠀⢀⠀⠉⠉⠉⠉⠉⠉⢹⣿⣀⣀⠀⠀⠀⠀⠈⣿⡟⠀⠀⠀⠀⠀⠙⢿⠿⠟⠛⣿⠀⠀⠀⠁⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠻⠿⠿⠇⠀⣠⡟⠀⠀⠀⣀⣤⡀⠀⠻⣣⣿⢿⣷⣄⣀⡀⠀⣿⣇⠀⠀⠀⡤⠖⠲⣤⠞⣶⠛⣿⣄⢠⠤⣤⣶⣦⣄⠀⢹⠞⠋
⠀⠀⠀⠠⠀⡰⢚⣿⢿⣦⡤⠞⢁⣠⡿⢾⣿⣿⠁⢈⠙⢿⡀⢻⣶⣷⣿⠷⠦⢼⡇⢰⣿⣿⠀⣫⣼⣿⠛⣿⣰⣿⠏⢹⣿⠟⢉⣿⣦⡀
⠀⠀⠀⠀⢸⡷⢻⣿⡞⢁⡀⢰⣿⡏⢠⣿⠋⣿⠀⡾⠁⣸⣇⣼⠟⢡⣷⡆⢀⣾⡷⣿⣿⣿⣴⡟⢹⣿⢠⣿⠟⠹⠀⢸⡇⣀⣿⣿⣿⠟
⠀⠀⠀⠀⠈⠳⣄⠙⢷⣿⣷⢿⣿⣠⣿⣿⠃⣸⠀⢠⣾⣿⣿⡃⠀⣿⣿⣷⢿⣿⠀⣨⣿⣿⠋⠀⢸⣿⡿⣿⠀⠀⠀⢸⣿⢿⠟⠷⣄
⠀⠀⠀⣿⣾⠛⠺⠇⢈⣿⡇⢸⣿⣿⠉⣁⣴⡿⠀⣀⣿⣿⣸⠃⢸⣿⢿⠇⢸⡇⠀⣿⣿⡧⠀⠀⠺⣿⠀⣿⠀⡄⣠⣿⡇⠘⠷⠀⣾⣿
⠀⠀⠈⠉⡏⠳⢶⣶⣿⣿⣷⣾⣿⠏⠛⠿⠿⣇⣰⣿⣏⠙⣿⣤⣾⣿⠘⠦⣾⣧⣀⣈⣉⣩⠀⠦⣤⣤⠀⣯⣀⣿⣿⣿⣿⠦⣴⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣷⡹⡀⠀⠀⠀⠈⠻⣿⠟⠶⣿⣿⠛⠉⠀⠀⠈⠉⠉⠛⠛⠻⠶⣶⣿⡿⢤⣿⣿⣿⡟⠋⠁⠀⠈⠛⠛⠋⡇
⠀⠀⠀⠀⣈⣁⠀⠀⢿⣿⣿⣿⣷⡑⡀⠀⠀⣸⣄⡄⠀⠀⠀⠀⡀⢠⡇⣾⣴⠄⠀⠀⠀⠀⠀⠀⠁⠀⢀⣽⣿⣿⣷⡀⠀⠀⡀⢀⠨⠇⢁
⠀⠀⢠⣾⣿⣿⣿⡄⠀⠻⣿⣿⣿⣿⣶⢄⠛⣿⣿⠀⠀⠀⠀⠀⠀⣠⣤⣤⡄⠀⠀⠀⠀⠀⠠⢤⠂⠀⣿⣿⣿⣿⣿⣧⡱⢄⠀⠀⠃⠉⠁
⠀⠀⠈⢿⣿⣿⣿⠃⠀⠀⠙⢿⢿⡿⠋⠊⠘⠁⠈⠀⠊⠀⠀⠀⠀⡇⠀⢀⡋⡐⠀⢀⠀⠀⠀⠀⠀⠀⢻⠿⣿⣿⣿⣿⣿⡷⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠻⣿⡁⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠤⠞⠁⠀⡀⠞⣼⣲⠀⠀⢀⠀⢠⡆⢹⣿⣿⡿⡙⠀⠀⠀⣷
⠀⠀⠀⠀⠀⣹⠇⠀⠀⠀⠀⡤⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠈⠹⢀⠛⠋⣠⣤⣄⠀⠀⢻⡄⢻⡟⠀⠀⠀⢀⡀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠄⠧⠸⠆⠀⠀⣰⠀⠀⠀⢠⣾⣿⣿⣿⣶⣄⠀⠐⠀⠀⠀⠀⢻⣿⡿⠀⠀⠀⠋⠀⠉⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⢀⣴⣆⠀⠀⠁⠚⠊⠀⠒⢻⣿⡿⠓⠂⠘⢿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⣾⠟⠁⠀⠀⠀⠀⠀⠀⢿⢸⡇⡇⣿
⠀⠀⠀⣴⣶⣾⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠋⠀⠛⠀⠀⠀⠀⠙⢿⣿⣿⡿⡡⠁⠀⠀⠀⠀⠋⠀⠠⣤⣰⣇⣀⠀⠀⣠⡠⣴⢀⠂⡆⢀
⠀⠀⠀⠙⣿⣿⣿⣿⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⠔⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠋⢿⠁⠀⠀⠁⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠘⠋⠊

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⣀⠐⢰⠋⠀⠀⠐⠂⠘⡁⠁⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠏⠦⠣⣸⡀⡀⠀⣀⠠⠏⢞⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣕⠒⣶⣶⣶⣦⣶⣶⡖⢓⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⣹⣿⡷⠀⠀⣾⣿⡆⠀⠀⣿⣿⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⣿⣿⡇⠀⠀⠈⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣴⣴⡄⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⡼⠣⣼⠿⠠⣿⣯⠱⣦⠀⠀⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⠷⣴⡚⠇⣴⣺⠳⠌⣥⣮⡻⠞⣦⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣼⢧⣙⣿⢠⡈⡿⣷⠀⠻⣮⡁⢿⡎⡳⠀⣿⣿⡇⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⣡⡈⠟⣡⠘⢿⡿⢟⠸⣧⣧⡗⣌⣷⡻⡄⣿⣿⡇⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢯⡃⣾⡻⣷⠀⣴⣾⣄⠈⢉⣄⠙⣋⢛⠃⣿⣿⡇⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣮⡧⣜⠿⣆⠘⢷⣮⡙⠰⣿⡝⢰⡟⡏⠀⣿⣿⡇⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡴⣌⠛⣋⡸⢶⡽⠏⢦⣼⠻⣾⠋⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⢪⡿⣿⣰⣾⡟⣰⠏⢙⣁⠀⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡏⠀⢳⠀⠈⣷⣾⡄⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⡴⠟⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡿⠀⠀⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡏⠙⣿⡆⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣟⠉⠀⠀⣰⡿⠁⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣽⣷⠀⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇
⠀⠀⠀⠀⣤⣤⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇
⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠈⠁⠄⠀⠀⠀⠐⠉⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢐⠀⢈⠀⠀⠀⠀⡀⠀⢀⠀⠌⠑⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠁⠠⢠⣀⠀⠀⡉⡀⠀⠀⠀⡀⠈⠀⠰⡀⠀⠀⠀⠄⠉⠤⠀⢀⢠⡐⠂⠀⠀⠀⠀⡀⠀⡀⠀⣠⡶⠶⠦⢥⡀⠀⢀⠀⡰
⠀⠀⠀⠀⢀⠀⠀⠘⠁⠀⠀⠈⠀⠆⠀⠀⠤⠀⠀⠀⠘⠀⠐⠄⠀⠘⠀⠐⠁⠀⠙⠁⠀⢣⠀⢰⡄⠀⠸⡟⠁⠀⠀⠀⠀⠹⣆⠀⠘⠅
⠀⠀⠀⠀⢲⠠⡀⠀⡀⠀⠀⠂⠀⠀⠃⠀⠀⠀⠀⠈⠑⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢀⠴⠡⠌⠁⠀⠀⠀⢈⡏⣤⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠈⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠈⠀⠃⠀⠀⠁⠠⢀⠀⠄⠋
⠀⠀⠀⠀⠀⠟⠈⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠐⠀⠀⢢⡢⠀⠀⠀⢏⠀⡁⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⡈⠉⠋⠂⠂⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡇⠀⠀⠓⠀⢸⠋⢠⡊⠀⠀⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠄⠙⡀⡀⢀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠄⠐⠻⠇⠘⠦⠙⠾⠟⠷⣘⠦⠿⢆⠞⠆⠐⣀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⠠⠀⠀⠀⣀⣠⣦⡽⣿⣿⠋⠁⠂
⠀⠀⠀⠄⠀⠀⠰⠄⠀⠡⠁⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣲⠆⠀⠀⠀⠈⠘⡃⠁⠁⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠓⠲⠶⢦⣤⣄⣀⡀⠀⠀⠀⠀⢀⣀⣤⣴⠾⠭⠟⠛⠀⠀⠀⠀⠀⠐⠨⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠁⠀⠀⠀⣷⠴⠖⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣴⠞⠁⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⠿⠿⠛⠛⠛⠛⠿⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⢸⡆⢀⣴⠟⠁⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠄⠀⠀⠀⠀⢸⣷⣟⣁⣤⣤⣤⠴⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋⠀⠀⠀⠀⢀⣤⣶⠿⠛⠛⠛⠛⠛⠛⠳⠦⠀⠀⠀⠀⣠⣾⠟⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠐⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⣰⣿⠋⠀⠀⠀⢀⣤⡤⠶⢤⡀⠀⠀⠀⣠⣾⠟⠁⠀⢀⣀⠀⠀⠀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⢰⣿⠁⠀⠀⢀⣼⠋⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠇⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⢠⡄⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠸⣿⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⣿⡇⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠸⣿⡄⠀⠀⠀⠈⠻⣦⣄⣀⣀⣀⣠⣴⠟⠋⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠈⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁⠀⠀⠀⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠀⠈⠙⠿⣷⣦⣤⣄⣀⣀⣀⣠⣤⣴⣶⠿⠛⠁⠀⠀⠀⠀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠄⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠠⠒⠶⠒⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣯⠀⠀⠀⠀⠀⠀⢀⣵⡼⣶⣿⠀⠀⠀⣤⣤⣀⣀⣀⣄⣀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠒⠲⠶⣾⢟⢭⣴⣶⣿⣏⠁⠒⢶⡟⠀⡫⠅⠉⠛⡶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⢉⠃⣾⣿⠟⠙⠓⠈⠐⢄⢻⡄⠋⠀⠀⠘⠞⣦⡐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠂⢀⣒⣤⣄⡌⢠⡿⠁⠀⠀⠀⠀⠀⢸⣿⣿⡷⡀⢰⠚⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⣉⡠⠤⠐⠛⡭⡙⢀⢎⠎⠀⣦⡀⠀⠀⠀⣼⡟⠛⢻⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢁⠴⣻⣿⠀⠀⠀⠀⢠⠁⣌⠃⠀⠀⠈⢿⣄⠀⠀⠙⠇⢀⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠰⠁⡀⣿⡏⣀⠄⠀⠀⣼⡞⣤⡴⠈⡲⢟⣛⠿⠄⠐⢲⣾⢎
⠀⠀⠀⠀⠀⠀⠀⣿⢀⠇⠀⢸⣿⠏⢁⣀⢠⣤⡿⣿⠩⠔⠁⠀⠉⠉⠀⢢⣄⠀⢧⠸
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠘⣼⣿⠟⠉⠀⠀⠹⢷⡝⣥⠀⠀⠀⢠⣦⠚⠀⠹⣄⠀⠀⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⡀⢠⢊⠄⢀⠀⠀⡅⡟⠂⠀⠀⢹⠏⠀⠀⠀⢻⣷⠀⠀⠀⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⣿⣿⣧⣤⠁⣠⠎⠀⠀⠀⠀⣤⣼⠿⣦⣀⡀⠀⠈⠓⠤⣀⠀⠀⠉⠻⢶⣖⡈⠒⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢝⠻⣧⣾⣧⣄⣀⣀⢀⣀⠿⢏⡀⠈⠓⢮⣳⠶⢤⣀⠀⠈⠁⠒⠢⢤⣈⠻⣿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡉⠀⠀⠀⢘⣶⣦⣴⡞⠁⠀⠀⢀⡞⠀⠀⠀⣽⢧⠀⠀⠀⠀⠀⠀⢸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠐⣹⢿⡇⣿⣷⡄⣀⣴⣿⣡⣤⣴⠤⠉⣸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠟⡘⠻⢿⣿⢀⠏⠈⠓⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠃⠇⠀⠀⠈⠁⠀⠀⢶⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠟⢿⡟⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣠⡾⢧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢰⡿⢄⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⠀⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠠⣷⣤⣤⣤⣦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠉⠩⠁⢀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠐⣀⣡⢴⠶⠿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠒⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡿⣷⢸⠀⠀⠈⠹⠷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⣀⢠⠆⢀⣀⣀⣴⢨⣙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⢀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠀⠛⠛⠛⠛⠋⠉⠉⠉⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠘⠿⠷⢖⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠀⠀⠀⣂⣈⣠⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢤⡾⣉⠺⣿⠟⠉⠑⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠋⠉⢀⡴⢋⣼⣿⣷⡖⣠⠆⡀⠀⠀⠀⠉⠑⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠈⢧⡴⢋⣴⣿⣿⣿⣿⠿⠷⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⡀
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⡼⢋⣴⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⡀
⠀⠀⠀⠀⠀⡰⠁⣠⣠⠋⡴⠟⠁⠰⡰⡄⠀⠀⠀⠀⠀⡄⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⠂⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⢠⠃⢀⡽⠀⠻⡀⠀⠀⠀⢈⣥⣤⣶⣶⣾⣿⣿⠿⣵⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠎⣠⢾⣅⣀⣐⣡⣤⣶⣿⣿⣿⣿⣿⣿⠙⣿⣿⣷⣄⣀⠙⠓⢦⣤⡀⠀⠀⠀⣠⣆⣀⠁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠃
⠀⠀⠀⠀⣿⣷⡿⠉⣿⠛⠋⠛⢿⠿⣿⣿⣿⣿⣿⣆⣿⡿⣿⣿⣿⣿⣷⣶⣾⣷⣾⣴⡾⢻⣿⡇⠀⣸⣿⣿⣿⠀⢻⡏⣿⣿⢹⢸
⠀⠀⢀⣾⠿⠋⠀⠀⠙⠻⣄⠑⠌⡿⠛⠻⣿⡛⠒⠤⣍⠻⢦⣀⠘⠿⣿⣿⢿⣿⡛⠻⠍⢿⣿⠀⠀⣿⣿⣿⣿⣷⣾⣸⣿⣿⠘⠈
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⠚⢧⡄⠀⠘⣧⣤⣤⠼⠀⠀⠉⠙⠛⠲⠖⠛⠛⠉⠛⠛⠚⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣤⣤⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣠⡾⠿⠻⠛⠻⠉⠉⠉⠁⠁⠉⠙⠋⠋⢻⡿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠐⠛⣻⣠⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⡴⠁⡿⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⢀⣀⢀⣀⣒⣶⣶⡦⢀⢀⠀⠀⠀⠀⠘⠛⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⣠⣴⣦⣦⣿⣿⣿⣿⣿⣿⣶⣾⣿⣷⡦⠀⠀⠀⠘⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⢈⣿⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣄⣀⠀⠐⠟⠻⠀⠀⠀⣼⣶⣦⡄⠀⠀⠀⠀⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⢿⢿⣿⣷⣾⢿⣿⣻⡻⠈⠁⠘⠛⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣟⢿⣽⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⣧⣸⠁⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣳⣿⡡⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠏⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣿⢤⡄⠀⠀⠀⠀⢀⣼⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡘⠁⠀⢀⣠⣴⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢿⠷⠶⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⣾⡟⣡⣥⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠈⠀⠉⠁⠙⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢤⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⢶⠀⢠⡈⣉⠉⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠁⠉⠉⠀⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠒⠛⠃⠀⢠⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠈⢭⡿⠃⢀⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣤⣀⣈⣴⣻⣿⣿⣯⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⢲⣾⣿⣿⣿⣷⣿⡌⠙⢽⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠀⡰⣿⣷⣿⣿⣿⣿⣿⣷⠀⢰⣿⣾⣿⣿⣿⣿⣿⣿⡿⢿⡿⣏⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣀⠀⢸⡇⢹⣿⣿⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⣿⣿⣿⡏⠀⢸⣧⡟⠘⣯⠲⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣰⠇⠀⠀⣿⠀⠀⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⣻⣿⣿⣿⣿⣿⣧⣠⣿⡇⠁⠀⠈⠙⠂⠹⣄⡀
⠀⠀⠀⠀⠀⠀⡴⠁⣽⠀⡼⠀⣿⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡁⢣⡀⠀⣄⠀⠈⠪⠙⣦⣀⣀
⠀⠀⠀⠀⠀⣼⡙⠀⡇⢠⡇⠀⢹⣇⣀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠻⠄⠈⠂⠀⣀⣬⠭⠿⠋⢻⡄
⠀⠀⠀⠀⢠⢻⠁⢸⠁⣸⠀⠘⢿⣿⣧⣉⠛⠭⣛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠢⣠⣴⡶⣠⡾⢛⡁⠀⠀⢠⣌⢙⡢⣄
⠀⠀⠀⠀⡜⡿⢀⡏⢠⡏⠀⠀⢸⡟⠿⠻⠿⠶⣦⣌⡑⠂⠤⣈⡉⠛⠛⠿⢿⣿⣿⣿⣿⣿⠁⠈⠙⠛⠻⠿⢧⠵⣒⣋⠳⠦⣶⣦⡙⢯⣿⡇
⠀⠀⠀⢰⠁⣷⣿⠀⠸⠀⣀⡤⢊⣷⠂⠀⠀⢠⣤⡍⢙⣷⣶⣿⣿⣿⣿⣶⣶⣮⣭⣿⣟⡉⠉⠉⠉⠐⠒⠒⠿⣾⠀⠀⠤⠀⣿⣿⣿⣷⣿⡇
⠀⠀⠀⡼⢸⣯⢟⡲⢿⣿⠿⠛⠉⠀⠀⠠⢤⣤⣤⣄⣀⣲⣄⣘⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⠸⡖⠒⢚⢇⣿⣿⣿⣿⡿⠁
⠀⠀⠀⡇⣸⣿⠟⢱⡟⠁⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⣠⣼⣯⣭⣼⣿⣿⣿⠋
⠀⠀⠀⡇⣿⠿⠀⠸⡄⢀⣠⣴⣾⠿⠿⠿⡛⣛⣻⣿⢿⡍⠉⠛⣾⣷⣶⠶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⣾⣻⣿⣿⣿⣿⣿⣿⠇
⠀⠀⣠⣧⣾⣶⣶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠒⠛⠛⠷⠮⠽⠿⢿⣿⣿⣿⣿⡇⣿⣇⣿⠿⠛⠛⠉⠁
⠀⠀⠈⠻⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡁⢀⡀
⠀⠀⠀⠀⠀⠙⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠤⠶⣶⣶⠏⢈⠛⠿⣿⣿⡤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠍⣼⢹⡗⠛⣈⣙⣛⣿⡥⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠞⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣷⠶⠖⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣭⠈⡏⢰⡋⡧⣿⣿⠘⣶⠋⢻⠁⣿⠅⣷⠀⣇⠀⣿⢰⡷⣾⢼⢩⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠉⠀⠀⠈⠀⠀⠀⠈⠀⠉⠁⠈⠁⠉⠁⠉⠀⠀⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢗⣀⡈⢉⣉⣉⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⡇⡠⡌⠨⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢇⡏⡜⢳⢳⠷⠚⢀⠧⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠀⠀⠘⢄⠙⠁⢸⠀⣰⡿⠀⠀⠈⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡉⠀⠀⠀⠀⠀⣔⡌⢃⣀⣘⣘⠁⠀⠀⠀⠀⠀⠀⠈⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠀⠀⠈⠀⢸⡿⠿⣿⠀⣀⣀⣀⣀⣀⣀⣞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡾⠽⠞⠁⠸⣖⢄⠀⠀⠉⠉⠁⠀⡇⠀⠀⠒⠒⠂⠀⠯⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡕⠁⠀⠀⠀⣀⢤⠀⠱⣳⡀⠀⠀⢰⣒⣓⣒⣚⣛⣚⣒⣒⣲⡼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⠀⢸⠉⠏⣉⡴⢈⣦⠀⢹⣧⠀⠀⢀⢼⠀⠀⠘⠢⠭⠴⠃⢀⡀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠀⠘⡖⡞⠓⠤⣼⠁⠀⣸⡟⠀⠀⢸⢸⢩⣿⣿⣿⣿⣿⣽⣯⣯⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣵⣆⠙⠁⠀⠀⠀⠀⠀⠷⠃⢦⣠⢼⣸⣹⣿⣿⣿⣿⣿⣿⣿⡷⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠽⣲⣶⣶⣲⠽⠃⠀⠀⠀⠁⢸⣼⣈⣉⣻⣿⣿⣿⣟⣉⣁⣧⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠒⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠀⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠞⠋⠀⠀⠈⠉⠉⠛⠛⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠠⢤⣦⣀⡀⠀⠀⠀⠀⢘⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⢀⡴⠛⢻⣿⡀⠀⠀⠀⢀⡷⠋⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢀⣠⡤⠹⣿⣀⡠⠞⠉⠀⠀⠀⢀⣨⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢀⣾⠉⠀⣠⣭⣃⠀⠀⠀⠀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⢠⡞⠃⠀⠀⣿⠀⠀⠀⠐⣤⠞⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢀⡴⠋⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠔⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡿⠤⠤⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠈⠉⠉⣭⣽⣿⠀⠀⣀⣀⡄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣀⣀⣀⣐⣒⣿⣿⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣟⠻⠿⠿⠿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠷⠶⠶⠶⡾⢿⠿⠿⠿⠿⠿⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢸⡄⠀⠀⡇⠀⠀⡄⠹⢿⣿⠟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡆⣿⣴⠀⣿⣇⠀⢰⣧⠀⣦⣧⠀⢀⠀⠀⠀⢀⣷⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡇⠀⢀⣿⣿⣿⡴⣋⣠⡕⠺⢿⢀⣿⣿⣤⣿⡄⠀⠀⢸⣿⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⢀⠜⣉⣙⣿⣿⣿⣿⣵⣶⣆⡬⣿⣿⣿⣿⡇⠀⢄⠹⣿⡅⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢛⡙⡟⢁⣚⡟⢫⣾⣿⣿⣿⣿⣿⣿⡇⠙⣿⡟⢻⣿⡤⠀⠀⢹⣿⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⣿⢀⣿⡿⠿⠀⡾⣿⡷⢮⡽⠿⣿⡿⡷⢾⣭⠀⣤⡬⣏⣀⣀⠖⢿⣀⣙⠦⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠃⣚⠋⢘⠃⣾⠆⣼⠇⣿⢁⡚⢀⣡⡿⢀⡖⢸⡟⢰⡿⢠⡟⢠⡟⣰⠆⣿⠁⡣⡮⢶
⠀⠀⠀⠀⠀⠀⠀⢀⠏⢸⠁⠇⢾⣦⣵⣤⡵⣦⡴⠃⠑⠬⠭⠦⠞⠣⠬⠲⠬⣴⣬⢥⣬⣴⣿⠬⠕⠤⠭⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠚⠒⠁⠀⠧⠚⠙⢿⣿⣷⣼⢻⣷⡀⠀⠰⡢⢔⠆⠀⣠⣾⣿⣿⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣬⣻⣦⡈⠠⠄⣠⣾⣿⣿⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣍⢻⣶⣾⡿⣯⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣏⣹⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣠⡾⢛⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⠋⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⢲⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⠀⢹⣾⣿⣿⣷⡄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⡾⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⣼⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⢰⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣽⣿⣿⡀
⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⡀
⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⠿⢿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣿⣿⣧⣀⠉⣨⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣦⣀⣠⡄⠀⠀⡀⢀⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢹⣇⣠⡄⠊⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣎⣉⣉⢛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠟⠿⣿⡿⠿⠿⠟⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢻⡿⠛⠋⢛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠄⠀⠀⠀⠐⢻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡻⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢻⡇⠀⠀⠀⡀⣀⣤⣴⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣥⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠘⡇⣸⣤⣤⣶⣾⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣤⣶⣤⣴⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣯⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⢠⡖⠀⠀⠀⠀⡜⠀⠀⠀⠈⠱⡀⠀⠀⠈⢀⣀⣀⣀⠐⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤
⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⠄⠀⠀⠀⠠⠀⠀⠀⠂⢀⡀⢰⡄⠀⠀⠀⠀⠠⣸⣿⣿⣿⣿⣢⠄
⠀⠀⠠⠟⢀⠤⠂⠀⠀⠀⠑⠤⢤⣤⠤⢄⣣⠀⠀⠀⣹⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠰⡀⢀⠠⠀⠈⢹⠿⣿⠛⠀⠀⠀⠀⠈⠻⢿⡿⠃
⠀⠀⠀⠀⠐⠀⠔⠉⠀⠀⠀⡀⠀⠀⠀⠀⠙⠆⠀⣸⣿⣿⠟⠉⠀⠈⠀⢀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⢤⣾⣷⡆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⣴⣿⣦⣀⠀⠀⠀⢹⣿⣧⠀⠁⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⢀⣤⣶⣿⣿⣶⡀⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀
⠀⠀⣰⣶⣶⣦⠀⠀⠀⢀⠀⣼⣿⣿⣿⣿⣿⣶⣦⡀⠉⠉⠀⠀⠂⢰⣿⣿⣧⠀⠐⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠁⠀⣶⣾⣿⡇
⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⢿⣿⡇⠀⠀⠀⠀⠀⠘⢿⣿⡿⠀⠀⠀⠈⠀⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠄⠔⠀⠀⠀⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣷⠃⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⣸⡿⣴⣶⣄⡀⠀⠀⠀⢸⣿⠅⠀⠀⠀⠀⠂⣀⠙⢛⣉⣁⣸⡇⠀⠀⠀⠀⣀⡀⠀⠀⢿⣿⣿⠇
⠀⠀⠈⠉⠉⠉⢀⣠⣿⡁⠀⠀⣀⡴⠲⣄⣀⣨⣥⡿⠛⣿⣿⡴⢦⡀⢠⠿⣦⡀⠀⢀⡞⠉⣉⣿⡏⢸⠇⣸⣷⣄⡞⢛⣾⣿⣿⣦⣀⣸⣍⡀
⠀⠀⠂⠀⠂⣴⢋⣼⣟⣻⡶⠚⠁⣠⣾⠟⣻⣿⣿⡄⢠⣄⠙⣇⠈⣿⣿⢿⡟⠛⠛⢺⠃⣸⣿⣿⠀⢸⣼⣿⡏⢹⣧⣾⣿⠁⢸⣿⠋⢀⣾⣿⣦
⠀⠀⠀⠀⠀⣿⠛⣿⣿⠋⣠⠀⣸⣿⠇⣰⣿⠉⢻⠀⠸⠃⣠⣿⣤⡿⠋⣸⣿⡇⢰⣿⠟⢿⣿⣿⣴⡿⠁⣿⣇⣼⡿⠃⠸⠀⢾⡇⣴⣿⣿⣿⠟
⠀⠀⠀⠀⠀⢙⢦⡀⠙⣿⣿⡿⢻⣿⣰⣿⣿⠃⣸⠀⠀⣾⣿⡿⣿⠀⢨⣿⣿⡿⢿⡿⠀⣤⣿⣿⠋⠀⠀⣿⣿⢻⡇⠀⠀⠀⣿⡿⢿⠛⠻⢦⣀
⠀⠀⠀⣹⣶⡋⠓⠿⠀⣸⣿⠅⢸⣿⣟⠉⣁⣴⣿⠀⣀⣸⣿⣆⡟⠀⣾⣿⣿⠁⢸⡇⠀⣿⣿⣿⠀⠀⠸⢿⡇⢸⡇⢠⡀⣰⣿⡇⠘⠿⠀⣾⣿⡆
⠀⠀⠈⠉⢿⠙⢶⣶⣾⣿⣿⣷⣿⣿⠏⠻⠿⠿⣿⢠⣿⣿⠉⢻⣧⣴⣿⣿⠙⢦⣾⣧⣀⣈⣉⣹⠀⠲⢤⣤⡆⢸⣇⢸⣿⣿⣿⣿⢦⣤⣶⣿⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⣿⣿⣇⠣⠀⠀⠀⠀⠘⠻⣿⣿⠷⣾⣿⡟⠛⠃⠀⠀⠉⠉⠉⠛⠿⠿⢷⣶⣾⣿⣧⣼⣿⣿⣿⡟⠏⠁⠀⠈⢹⠛⠟⢹
⠀⠀⠀⠀⠀⠂⠀⠀⠸⣿⣿⣿⣟⣦⠣⡀⠀⠀⢰⣆⣀⠀⠀⠀⠉⠀⠀⢰⣆⣦⣶⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢈⣿⠟⢻⣷⢠⠀⠀⠀⠸⠐⣶⠘
⠀⠀⠀⣴⣿⣿⣿⣆⠀⠹⣿⣿⣿⣿⣷⣵⣀⠴⣿⣿⠋⠀⠀⠀⠀⠈⠁⣈⣀⣉⡉⠀⠀⠀⠀⠀⠀⣂⡠⠀⢰⣿⣦⣥⣾⣿⣇⢧⡀⠁⠀⠣⠒⠚⠁
⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠘⢿⣿⣿⣿⠟⡘⠀⠏⠈⠃⠠⠂⢀⠄⠀⠀⡏⠉⢉⡗⠠⠀⠀⠀⠀⠀⠈⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣬
⠀⠀⠀⠈⢿⣿⡿⠁⠀⠀⠀⠀⠻⡿⠅⠊⠀⠀⠀⠀⠀⢑⠀⠄⠀⠄⠀⢇⣀⡸⠃⠊⠀⡴⣢⡄⠀⠀⠀⠀⠀⣿⠘⣟⣿⣿⣿⢟⠃⠀⠀⣷
⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⢶⠕⢻⡾⠀⠀⠀⠂⠀⠻⣆⠹⣾⣿⠣⠃⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠠⡜⢳⠘⣲⠀⠀⠀⢀⠀⠀⠀⠀⢀⣾⣿⣷⣦⡀⠀⠀⢀⠈⠐⠋⠀⢰⣿⣿⡆⠀⠀⢹⡇⠙⠧⠁⠠⠀⠀⠂⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠳⣀⣙⣸⠁⣀⣀⣼⣄⣀⡀⢀⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢈⣿⡿⠁⠀⠀⠀⠀⠀⠀⣀⢠⡄⡤⢤
⠀⠀⠀⠀⠀⣀⣠⣶⣿⣆⠀⠀⠀⠈⠀⠀⠉⣹⠿⣿⠉⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿⡦⠀⠀⢀⠀⣿⠏⠀⠀⠀⢀⠀⠀⠀⠀⠻⠸⠇⠇⠸
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡽⡄⠀⠀⢠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⢿⣿⣿⠏⠜⠀⠀⢀⠀⠀⠈⠀⠀⠲⣦⣿⣦⡄⠀⠠⡶⡌⠾⠰⢨⢐⠠
⠀⠀⠀⠈⢻⣿⡿⡿⠗⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠙⠣⠊⠀⠈⠀⠈⠀⠀⠐⠀⠀⠀⠉⠘⠧
⠀⠀⠀⠀⠀⠛⠈⠀⠀⠐

⡿⢿⠿⡿⠿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣛⣟⣻⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠐⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠂⠤⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠄⡀⠀⠐⡀⠀⠀⠀⠀⠈⠁⢰⠀⠀⡀⠀⠐⠼⠁⠀⠀⠀⠀⢀⠂
⠀⠈⠄⠀⠰⡄⠀⠀⠁⢀⡤⠀⠀⠀⠀⠀⠀⢸⡄⡞⠀⠀⠀⠀⢆⢤⠀⠐⠀⠆⠐
⠀⠀⠀⠀⠀⠁⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢀⣦⠘⢄⠒⠲⠆⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠐⡇⡄⠀⠀⠀⠊⠁⠔⠂⠀⠀⠀⠀⠀⠄⠐⠂⠀⠀⠀⠐⠀⡀⠀⠀⠈
⠀⠀⠀⡀⠸⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣷⠁⠀⠀⠥⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⡆⡀⠠⠀⠀⡀⠀⢀⠖⠀⠊⢀⠁
⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠄⠀⠀⠀⠀⡦⠃⠀⢠⠀⡇⢀⠀⢰⡁⠀⢸⠀⡰⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠱⠠⠀⠀⠘⠁
⠀⠀⠤⠀⡤⠤⠒⡠⠔⠀⠀⠀⠀⠂⠀⡀⢀⡒⢦⠄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠒⠀⠤⠠⠠⣄
⠐⢀⣠⡀⠄⠀⠀⡀⠀⠀⣀⠐⠁⢀⠔⠁⠠⠉⠋⠁⠀⠀⠀⠀⠀⢀⣻⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠐⠰⠲
⠀⠘⠛⠉⠀⠈⠀⠀⠉⠁⠙⠶⠛⠛⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠈⠢⣄
⠀⠀⠀⠀⠀⠤⢠⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠈⠀⠀⠀⠁
⠀⡄⢠⠀⠀⠀⢐⡀⣠⠀⠀⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡾⠋
⠀⠻⠶⠘⡆⠀⠀⡼⠘⠛⠾⠧⠢⠀⠠⠄⠀⠀⠀⠀⠀⠀⡸⠟⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠄⠀⠘⠂⠀⠀⠀⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⠟⠀⢀⢾⣻⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣦⣤⣦⣽⣿⣯⣽⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁⠀⠉⠛⠛⠛⠹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢫⣉⣢⠀⠠⠶⠶⠄⠈⠛⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠀⠀⠁⠀⠈⠒⠃⠀⠀⣠⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣑⡐⠀⠀⠀⠀⡰⢿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣌⠙⠉⠁⢀⣤⠞⢱⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠺⠋⠁⣿⡽⣿⣶⣶⠞⠋⠀⠀⠀⠀⠀⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⡆⠀⠀⡋⠉⢀⡹⠀⠀⠀⣀⠀⠀⠀⢠⠀⠀⠙⠇⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⢀⠇⠀⠀⠀⣰⠏⠀⢀⡀⠀⠘⢇⠀⢠⠃⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠤⠁⢰⠂⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡀⣰⡄⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⢿⡵⡀⣠⡶⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣄⡸⡀⠀⢀⣀⣀⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠶⠶⣬⣻⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣿⣀⠠⠄⠌⢉⡛⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣿⡏⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠜⠁⢀⡀⠢⡀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠄⠀⢠⡟⢀⠀⠂⢈⠠⠠⢀⣈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠈⠉⠢⠾⠳⠀⠀⠀⣠⡁⠀⠊⢻⡌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⢸⡀⠀⢘⣤⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⣁⣄⠁⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣈⣁⣴⣋⣍⣁⣠⣤⣤⣤⣬⣿⣿⣭⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣾⠛⠛⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢸⣧⣼⣤⡶⠶⠶⠟⠛⠛⠛⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⡄⠀⣀⣀⣀⣀⣠⣤⡤⠤⠤⠴⠖⠒⠒⠒⠊⠉⠉⠒⠢⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⡤⠤⠾⠖⠒⢚⡙⠉⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠙⠻⠶⣤⣀
⢀⠤⠤⠴⠶⠒⠒⠙⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢀⠐⡖⠒⠒⢤⠀⢏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤
⡆⣧⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⠀⠀⢠⠀⣴⠴⠶⠶⠆⢶⣄⠀⢸⡀⠀⠘⡇⠀⢸⡄⠀⠸⡄⢻⣠⠴⠚⠀⢸⡤⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠
⢹⢾⡀⠀⠀⢰⣆⠀⠀⠀⣏⠙⢦⠘⣆⠀⡌⠀⢻⣀⣀⣀⠀⢸⡍⢷⡈⣇⠀⠀⢧⠀⠈⢧⡀⢸⠃⢸⡟⢦⡀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁
⠘⣿⣇⠀⠀⢸⠘⣆⠀⠀⢹⡀⢘⡇⠹⡄⡇⠀⢸⡇⠀⠀⠀⠈⡇⠀⠙⢿⠀⠀⠸⠄⠀⠈⠙⠃⠀⠀⠃⠀⠙⠂⠀⠉⠉⠉⠁⠀⠀⠀⡠⠖⠁
⠀⢿⣿⠀⠀⡷⠖⠘⣆⠀⠈⣇⠜⠁⠀⢻⠁⠀⠀⠷⠶⠶⠆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠊
⠀⢸⣿⡀⠀⠃⠀⠀⠙⠀⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⢀⠀⠠⢤⣤⣤⣤⣶⣶⣦⣤⣤⣴⠶⠶⠶⠶⠟⠛⠋
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⡤⠤⣤⣴⣖⣲⠶⣶⠶⠶⠖⠻⠟⠛⠛⠛⠉⠉⠉⠉⠉
⠀⠀⢙⣳⣾⣤⣐⠶⠶⠮⠿⠛⠓⠛⠊⠉⠉⠁⠀⠀⠀⠀⢸⡆⠀⠀⠀⡇
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠳⠀⠀⠀⠸⣤⡆⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠐⡄⢠⠀⠀⠀⢀⡿⠀⠋⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠙⢆⠘⢆⢇⠀⠀⣼⣷⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠣⠘⡞⠀⡈⣿⢿⣀⠞⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡉⠪⠐⠄⢹⠷⣗⣨⠁⣁⠀⠀⠲⠊⠁⠀⠀⢴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠀⠀⠈⠁⠠⠤⠭⠤⠄⠀⠐⠟⠛⠒⠚⠛⠒⠒⠒⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠉⠛⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠄⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⠀⠤⣄⠀⢀⠤⠄⠀⠀⠀⠀⠀⠀⠂⠒⢀⡐⠀⠙⡆⠀⠀⠔⠊⠁
⠀⠀⠀⠀⠀⠀⡤⠋⠀⠀⠀⠀⢀⠁⡔⠁⠀⠀⠀⢂⠀⠔⠁⢀⢄⠀⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⡠⠴⠰
⠀⠀⠀⠀⠀⠜⠀⠀⡔⠂⠈⠉⠁⠸⠀⠀⣠⡀⠀⠀⠟⠀⠀⣎⣸⠅⠀⠈⠀⠀⡇⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⢰⠀⠀⠈⠀⠀⣆⣀⠀⠁⢢⢰⠇⢱⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠢⣀⠀⠑⠠⢄⡀
⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠈⣉⠀⠀⠀⢸⠂⠨⡄⠀⠀⠀⢰⠀⢀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⢑⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢀⠞⠀⠀⠂⠈⠃⠸⠀⠀⡇⠀⠈⡀⢰⠇⠀⠀⠀⠀⠒⠀⠀⢛⠈⠉⠀⠀⠀⠄⠀⠀⠁
⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⠁⠀⠀⠼⠀⠀⠀⠀⠀⡐⠁⠀⠀⠃⢸⠀⠀⠀⡀⠀⠀⠀⠀⠘⣀⣀⡀⣀⣀⢀⡨⠔⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠊⠀⠑⠀⠀⠀⠀⠱⣀⠄⠀⠘⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣤⡄⠀⠀⠀⠀⣀
⠀⠀⠀⢈⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⡉⠉⣉⣩⣭⣭⣥⣥⠀⠀⠀⠀⠀⠀⠀⢤⣤⠄⠀⠀⠀⠀⠀⠀⠀⢤⠤⠴
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⢀⡀
⠀⠀⠀⢰⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠉⠉⠁⠉⠈⠉⠁⠉⠘⠑⠓⠒⠐⠒⠒⠒⠐⠂⠤
⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠼⠀⠀⠀⠀⠠⠠⠤⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⣀⠐
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣁⣀⣀⡀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⣀⡀
⠀⠀⡤⠖⠒⠒⠒⠒⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑
⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠄⠂
⠀⠀⠀⠀⠒⠒⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⡋⢙⠃⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢢⡤⠤⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⡅⠑⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠎
⠀⠀⠀⠀⠀⠀⠆⠀⠁⠠⠴⠀⢀⡀⣀⣀⢀⡀⡀⡄⢠⠠⠤⠠⠤⠤⠤⠤⠐⠪⠄⠀⠀⠈⠀⠀⠀⠀⠀⠉⠉

⠀⣶⣟⡛⠃⠛⢻⡟⠛⢠⡾⠛⢳⣆⢸⣿⠛⣷⡄⠹⣦⣰⡟⠘⠻⣿⠛⠁⣿⣛⣛⠀⣿⡆⠀⠀⣿⠀⠀⠘⢻⡟⠀⣿⣷⡀⣿⠀⣴⠟⠛
⠀⣈⣙⣻⡧⠀⢸⡇⠀⢸⣧⣀⣸⡟⢸⣿⠻⣯⡀⠀⢹⡟⠀⠀⠀⣿⠀⠀⣿⣉⣉⠀⣿⣇⣀⠀⣿⣀⣀⢀⣸⣇⠀⣿⠈⢿⣿⠀⢿⣄⣛
⠀⠉⠉⠉⠀⠀⠈⠁⠀⠀⠈⠉⠉⠀⠈⠉⠀⠈⠁⠀⠈⠁⠀⠀⠀⠉⠀⠀⠉⠉⠉⠀⠉⠉⠉⠀⠉⠉⠉⠈⠉⠉⠀⠉⠀⠀⠉⠀⠀⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣀⣀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣭⠷⠀⠀⠘⠋⠉⠀⠀⠀⠉⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣴⢶⣾⢻⠀⣠⣤⣤⣤⣤⣤⣤⣌⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⡏⣾⢀⡏⠀⢀⣀⡀⠀⠀⢸⡆⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣿⣧⠟⢋⣇⠀⣾⢩⡇⠀⠀⢸⠅⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣿⠀⣿⠏⢠⡏⠙⢻⠇⢸⣧⣾⠛⠋⠀⢠⡇⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠀⠀⢰⠏⠀⣼⣇⣠⠟⠀⣾⠀⠀⠀⠀⣀⣼⠁⠀⠉⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠁⠀⠀⠀⠀⠀⠘⢧⡀⠀⠙⠁⠀⠀⣿⠀⠀⠰⣾⢉⣷⠄⠀⠀⠀⠀⠈⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢸⠆⢠⡶⠋⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀
⠀⠀⠀⠀⠀⢻⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⣦⡀⠙⠆⠀⠀⠀⠚⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠟
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠋⠀⠀⣿⣛⣛⣛⣛⣛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣭⣭⣭⣭⣽⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏
⠀⠀⠀⠀⠀⠀⢀⣠⡤⠾⠛⢲⣦⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠘⠶⠶⠶⠶⠾⠁⠀⠀⣟⠉⠉⠉⠉⢉⣉⡉⡉⣉⡉⠉⠉⢹⡷⣄
⠀⠀⠀⢀⣤⠞⣋⣥⠶⠚⠿⠋⠁⠀⠐⠿⣬⡛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠸⠧⠼⢷⡄
⠀⠀⣠⠟⣡⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠒⠒⠒⠒⠐⠒⠀⠒⠀⠀⠀⠀⢸⡇
⠀⣼⢃⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⡇⠀⠀⠹⣆⢻⡄⠀⠀⠀⠀⠀⠀⣾⡋⠉⢉⡉⠉⡉⠉⠉⢉⠉⡉⠉⠉⠉⠉⡉⢉⣿⣼⣧⡀
⢸⠇⡾⠁⠀⣠⠶⠶⠲⡶⠶⠚⠋⣉⣀⠀⣗⠀⠀⠀⠹⡆⢿⠀⠀⠀⠀⠀⠀⠈⠉⡿⠉⠉⠉⠉⠙⣏⠉⠉⠻⠉⠉⣩⡟⠉⠉⠉⠀⢸⡇
⣿⢸⡇⠀⠀⢼⣛⠀⠀⠇⢀⡀⣠⠟⠟⠀⣟⣻⠄⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⢰⠞⡇⢠⠀⡄⢠⠀⠈⠛⠶⠦⠶⠞⠋⠀⢠⣤⣤⣤⢸⡟
⣿⢸⡇⠀⠀⢺⣍⣀⣀⣴⣏⣙⡁⠀⠀⠀⡟⠉⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⣽⠀⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⡇
⠙⠋⠁⠀⠀⠀⢻⡀⢹⠁⠀⠉⠙⠳⠦⣤⡗⠀⠀⠀⣠⡟⣸⠃⠀⠀⠀⠀⠀⣿⠀⡇⢀⣾⣻⣟⣿⣻⣟⣿⣻⣟⣿⣻⣟⣿⣻⣟⣿⢸⡇
⠀⢸⡷⢦⣤⡀⠘⣧⣼⠇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⢁⡿⢂⣄⠀⠀⠀⣀⣻⠀⡇⢸⣭⣯⣽⣭⣯⣽⣭⣯⣽⣭⣿⣽⣯⣿⣽⣏⢸⡇
⠀⠀⣷⣄⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠸⠋⠀⠀⠙⢦⣀⡴⠋⣽⠀⡇⢰⠿⡾⢷⠿⡾⢷⠾⡾⢷⠿⡾⢷⡿⣾⢷⡿⢸⡇
⠀⠀⠀⠈⠳⣮⣙⠶⣤⣄⣀⣀⣀⣀⣀⣤⠶⢛⣹⠆⠀⠀⠀⠀⠀⠈⠛⠀⠀⣿⠀⡇⠘⠛⠛⠛⣟⣛⣛⣛⣛⣛⣛⣛⣻⡟⠛⠛⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠉⠓⠶⢬⣭⣭⣭⣭⣥⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣤⣧⣤⣤⣤⣤⣭⣭⣭⣭⣭⣭⣭⣭⣉⣠⣀⣀⣠⣼⣧
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⢠⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣷⣦⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣨⡇⠀⠁⢽⣿⣿⣿⣷⣼⣶⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⣦⠄⠙⠿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢦⣾⣿⣿⣿⣿⠇⠀⢀⠀⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣻⣣⣾⣿⣿⣿⣿⣿⠀⠀⠼⡶⢒⣨⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠗⣛⡛⣿⣿⣿⣿⣀⡀⠐⣁⣷⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠸⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⡘⠇⠀⠀⢄⣄⣤⠗⣦⡤⡀⠹⣿⣿⢿⣦⣿⣿⣿⣯⠙⠋⣉⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠋⠀⡖⢶⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢤⣤⣀⣹⠃⠀⠀⠙⢹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢲⣀⢀⣀⢀⠀⣀⣀⣜⣿⣷⠀⡟⢾⡀⠑⢦⠀⣞⠉⠀⢀⣷⡏⠲⣿⣿⣵⠀⠀⠀⠀⠠⡍
⠀⠀⠀⠀⡇⠀⠀⢟⣀⣦⠶⠀⠀⠚⢁⢸⣯⣿⠀⢀⣼⣵⣄⣿⣦⣌⢳⣤⣾⣿⣧⣼⣿⠿⠿⢤⡀⣸⠇
⠀⣀⣂⣶⣏⠀⠀⠿⣿⣿⣇⠀⠀⠐⠚⢻⣿⠿⢿⢿⠿⣿⠿⠿⠟⠋⠛⢻⣿⣿⣿⣷⣥⡆⠀⢀⡻⣏⠀⠠
⢀⣿⣯⣽⣿⣷⣾⣶⣴⣶⣤⡀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣼⣿⣿⣿⣿⣯⣥⣴⢸⣿⣿⡄⠀⢰
⠀⠛⠟⠛⠻⠿⢿⣿⣿⣿⣿⣿⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿⣿⣿⣿⣿⡿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉⠙⠢⢄⡀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠒⠒⠒⠒⠂⠒⣦⣤⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠉⠒⢄⠀⠀⢀⠔⠁⢀⡤⢤⣤⣤⡴⢶⣻⠽⠚⠻⢄
⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⡤⠀⠀⠈⢲⡏⢠⡖⣉⣵⣾⣽⡵⠺⡉⠦⠀⠀⠀⠀⠑⢤⡀
⠀⢀⠀⡠⠔⠊⠁⠀⠀⠀⠀⡤⠄⠀⠠⠔⢃⡠⠄⠉⡀⠔⠁⠀⣮⣴⠾⠿⢟⡋⣹⠒⡹⡠⠕⠀⠀⠀⠀⠀⠀⠀⠉⠳⠀⣦⣤⡄
⠀⠈⠑⠢⣄⠀⠀⠀⠀⠖⠉⠀⠂⠁⡀⠐⠁⣀⠄⠉⢀⢤⠾⠿⠿⣄⡑⠀⢀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣻⣟⣿
⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⢠⠄⠈⣠⠖⠀⠀⠔⠊⠀⠀⠀⠀⠙⠋⠛⠛⠚⠲⠶⠤⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠊⢁⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣖⠒⠓⠒⢶⣾⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠉⠀⠀⠀⠈⢿⣧⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠀⢀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⣦⠸⡦⠀⠀⠃⠚⠃⠀⠰⣉⠆⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠢⡼⠈⢀⣀⣀⠀⠀⠐⠂⠀⠀⠠⠀⠀⠓⠃⡰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⡙⠂⡀⠀⡏⡏⠀⢀⠀⠀⠀⢺⠆⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠃⠀⡀⠀⠀⠑⠬⠶⠋⠀⣀⡀⠀⠀⠀⠈⠀⣴⠄⠀⠀⢁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢍⠛⠁⠀⠟⢀⣤⠀⠀⠀⠉⣰⢻⢳⡠⠖⡆⠀⠀⢈⡀⠈⠁⠀⢚⣁
⠀⠀⠀⠀⠀⠀⠀⢐⣊⠁⢀⠀⡠⠋⡏⢧⠐⠀⡰⠃⡜⠀⢁⡴⠁⠂⣠⠚⡟⣆⠀⡀⠙⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠸⠴⠃⠀⠚⠁⠘⣅⡸⠳⡞⠀⠸⠵⠂⡞⣞⠱⠘⠹⣤⠇⢈⠢⠀⠐⠳⠃
⠀⠀⠀⠀⠀⠀⠀⠐⠺⠄⠀⢼⡧⠀⠈⠁⠀⠘⠄⠀⠀⠀⠘⣌⡆⠑⠀⠈⠀⢾⠦⠀⠚⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣺⣟⠀⢠⣿⡄⠀⢰⠒⠦⠤⢦⡈⠇⣰⣷⡀⠐⣿⣗⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢈⢽⡿⡁⠰⣿⡢⢠⠋⠓⠤⢎⢿⣱⡄⢞⣿⠆⠨⢾⡮⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣻⣏⡂⣪⣿⢝⠀⠭⠍⠑⠚⠉⠥⠀⡻⡿⡅⢐⣽⣝⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠷⠉⢔⣿⡳⠄⣱⣆⢩⠍⣴⡌⠠⢚⣟⡢⠉⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢿⠌⠂⢽⡷⡀⠨⣾⢮⠘⠵⡷⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠌⠀⠈⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢶⡀⠀⠀⠀⢸⠟⠀⠀⢳⣿⣿⣿⣦⡀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⣾⣿⣦⠊⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⡀⠀⢶⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠁⠀⠠⣤⣹⣿⣿⡆⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⢰⡀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⣶⠖⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠁⢀⠀⠠⢈⣷⣄⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣿⣇⠀⠐⣿
⣸⠀⠀⠀⡄⠀⠀⠀⠀⠀⠞⠃⣾⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡧⠀⠀⠀⢀⣼⣿⣿⣷⣄⡈⠙⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣿⣿⠄⢻⣿
⣿⡆⠀⢠⣧⠀⠀⡀⠀⠀⠀⢨⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⢠⣿⠿⠛⣻⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠈⠛⠉⠙⣿⣿⠉⠉⠉⣽⣿⣿⡤⣾⣿
⣿⡗⠀⣸⣿⡀⢰⡇⠀⠀⢀⣸⣿⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⢾⣿⣿⠶⣿⣿
⣿⣿⡀⣻⣿⡃⣼⣧⠀⠀⠠⣽⣿⣿⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣧⠀⠘⢿⣿⣿⡿⣿⣿
⣿⣿⣽⣿⣿⣿⣿⣿⠄⠀⠴⣾⣿⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣧⣤⣶⣾⣿⣿⣿⣿⣿
⣿⣿⣭⣿⣿⣿⣿⣿⡃⢀⣤⣾⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣴⣶⣶⠀⢀⣤⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢻⣷⣍⠙⠷⣾⣿⣿⣏⢻⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣾⣿⣿⣶⣦⡙⠻⢿⣿⣿⣿⣿⣿⢻⣿⣿⣷⣶⣴⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⡿⠥⠤⢤⣭⣟⡻⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⢯⡙⠻⢿⣾⢿⣿⡇⢏⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣮⣑⠢⣝⣧⢹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣦⠀⠻⣧⠹⣿⣿⣿⣿⣿⣷⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣦⠀⢼⣧⢹⣿⣿⢹⣿⣿⡿⣋⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣽⣿⣿⣿⣷⣶⣿⣧⢻⣿⣼⣿⡿⢡⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣦⣙⣧⢻⣿⢇⣄⣼⣟⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣿⣿⣿⣿⣿⣇⢻⣿⡏⢸⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡛⠛⠛⠩⠙⠻⣏⢿⣧⣄⠹⣿⣿⣿⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣭⣿⣿⣿⣿⣿⣶⣿⣎⢿⣿⣴⠏⠀⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣦⣤⣤⣬⣿⣿⡎⢿⣿⣀⠀⣺⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣟⣿⣿⢿⣛⣻⣟⠻⣿⡎⣿⣏⣰⡻⢿⣿⣿⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠂⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⣿⣿⣋⡉⠙⣿⣿⡜⣿⣿⠗⠊⣻⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣈⠉⠛⠋⠓⣦⣄⠀⠀⠀⠀⠀⠐⠛⠛⠿⠿⢷⣶⣦⣌⣻⣿⡼⣿⣶⢻⣿⣿⣿⠃
⠀⠀⠀⠀⣰⣄⣄⢀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢠⡾⠁⠈⠱⠆⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣶⣤⡹⣿⣽⣿⠙⣿⡿⠉
⠀⠀⠀⣰⣿⣿⣿⣧⣶⣠⣤⠆⠀⠀⠀⠀⠀⣠⡿⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣷⣄⣿⣿⣿⣩⠃
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣵⣷⣶⣦⠀⢀⣿⠇⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠈⢹⡧⣽⣧⣿⡷
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣾⠁⠀⢠⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣻⡍
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣱⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⢰⣿⣿⣿⣿⠟⠋⠛⣿⣿⣿⣿⣿⣿⣷⣿⡿⡄⣸⡇⠀⠀⢲⣄⡀⠀⠀⠀⠀⠀⢀⡀⠠⢄⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠘⠿⠿⣿⣿⡀⢘⣅⡿⠁⣄⠌⣿⣿⡿⠋⢑⣡⡿⠇⠀⠀⠀⠈⠹⠷⠆⠀⠀⠐⠲⡶⠙⠋⠠⠀⢀⣍⠳⢤⡄⠀⠀⠀⠈⣄
⠀⠀⠀⠀⠬⣍⣛⡛⠿⢦⣤⣴⣿⣿⡿⡗⠋⠁⠈⠟⠢⣄⡀⠀⠀⠀⠀⠀⠀⢀⡴⠔⠀⠀⣀⡀⠀⠈⠳⢶⣄⠀⠀⠀⡄⠘
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢦⣤⣼⢃⠜⠀⠀⠀⠀⠀⠀⠀⠈⠓⠐⠠⠀⠀⠀⠈⡇⢀⠀⢀⡙⠁⠀⠀⠀⠈⠊⠳⣤⡀⠆⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠠⠀⠀⠀⣰⡄⠀⢰⠟⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠂⠀⠈⠁⠁⠀⠁⠀⠀⠀⣦⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾
⣿⣿⣏⣡⣉⣿⣀⣏⣉⣯⣁⣽⣛⣏⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠠⠴⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠊⠀⠀⢀⠠⢀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠋⣽⣿⣿⣿⣿⣿⠰⡶⠋⢠⠊⠘⠀⡐⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣏⠑⠀⠀⠀⠀⠛⣹⣿⣿⣿⣿⠀⡇⠀⠁⠀⠀⠐⠀⠠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠳⣄⣁⠀⢲⠀⣰⣿⣿⣿⣿⣿⠀⠀⣘⢄⣰⣼⠀⠐⠁⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠀⢿⣶⣶⣾⣿⣿⣿⣿⣿⣿⠀⠀⡏⠁⠉⠉⠃⠀⠀⢀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⣀⠀⠈⠻⠿⣿⣿⢿⣿⣿⣿⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠄⠈⠁⠀⠓⠀⠈⣿⣿⣿⣿⣿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣿⡆⢰⠆⠀⠰⣠⠈⢿⣿⣿⣿⣿⠀⢹⣤⣀⣤⠀⠤⠠⠀⢾⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇⠈⠀⠀⠀⠙⣷⡌⢻⣿⣿⣿⠸⠏⠀⠀⠀⠀⠀⠐⠀⢹⡆⡷⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣏⠀⣶⡀⠀⠀⢀⣿⣧⢸⣿⣿⣿⠠⡄⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣴⣿⣶⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⡹⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠘⣿⣿⣿⣿⣿⣿⠀⠀⠘⣶⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠘⣿⣿⣿⣿⣿⠀⡀⠀⣿⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢿⣿⣿⣿⣿⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⠀⢰⣿⣿⣿⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢿⣿⣿⣿⠀⢀⠠⠾⠆⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣿⣿⣿⠀⡾⠤⣄⣀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⡄⠀⠀⠀⠈⠉⠒⠂⢤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣀⣀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠂⠀⠀⠀⠉⠙⢶⡄⠀⠀⢰⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡡⠀⠀⠀⢻⡄⠀⢸⠀⠀⠀⠀⠳⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⡇⠘⢧⣀⠀⠀⠀⠀⢀⠁⠀⠀⢷⡀⢸⡀⠀⠀⢀⡀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠉⡀⢀⠀⠀⠀⠀⠀⡌⠁⠈⠁⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠈⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⡁⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠑⠀⠀⠂⠀⠀⠀⠀⠀⠁



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

⠀⠀⢴⣶⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⢀⣼⣷⢶⠖⠀⠀⠀⠀⠀⠀⠀⢸⣿⠻⣦⠀⣨⠿⣄⠀⠀⠀⠀⠀⢠⣬⡷⠞⠁⣿⠀⠀⠀⠀⠀⠀⢠⣶⣬⡉
⠀⢸⣝⣿⣧⢶⣄⠀⣼⡇⠀⠀⠀⠀⠿⢷⣼⠿⣥⡀⠈⠀⠀⠀⠀⠀⣾⣏⡛⠋⣼⣿⠀⠀⠀⠀⠀⠀⠘⢿⣿⠛⠶
⠀⠜⠁⠀⠻⣆⠈⣴⣿⡇⠀⠀⠀⠀⣠⠞⠉⢿⣬⣿⡄⠀⠀⠀⠀⠀⢸⡏⠻⣠⣿⣿⠀⠀⠀⠀⠀⠀⢀⣾⢻⡄⡀
⠀⠘⠀⠀⠀⢉⣼⣿⣿⡇⠀⠀⠀⠈⠁⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⢘⣣⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⠃⠘⠃⠁

⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⢄⡀
⠀⠀⠀⠀⣴⣷⣤⣄⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⡠⠊⠁⠀⠀⠀⠀⠀⢘⣛⠀⠀⣛⡃⠀⠀⠀⠀⠀⠀⣀⡀⠀⠙⠐⠠
⠀⢐⡶⣤⣿⣷⣴⡾⠷⠄⠀⠀⠀⠀⠀⣀⡸⠫⣄⣀⠀⠀⠀⠀⣠⣴⣿⣯⣷⣼⣿⣽⣦⡄⠀⠀⠀⠈⢛⣧⡴⠟
⣼⣟⣻⡯⠿⠿⠏⣾⠿⣷⡄⠀⠀⣤⣾⡿⠃⠀⠻⣿⣷⡄⠀⠀⢎⣭⡝⠭⢭⣭⠯⠯⠭⠅⠀⠀⠀⠀⠌⠻⣷⣶⣦⣄
⠈⠛⠛⠁⠀⠀⠀⠛⠿⠛⠀⠀⠀⠙⠋⠀⠀⠀⠀⠈⠛⠁⠀⠀⠘⠋⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠉

⠀⠀⠀⢀⠠⡠⡄⠀⠀⠀⠀⠀⠀⣠⡀⠀⢀⡀
⣴⣔⣊⣹⣞⣓⣼⣦⣤⣄⠀⠀⠀⠉⠱⠀⢸⡧⢄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⡀
⠻⣿⠿⣿⣿⣿⣿⣿⠿⣿⠆⠀⠀⠀⠀⢧⣼⣷⣿⣿⣶⡄⠀⠀⠀⠀⣠⣿⣿⠹⣷⡀⠀⠀⠀⠀⣶⣀⣀⣠⣿⣻⣃⣀⣠
⠀⠻⠶⠟⣁⣀⣈⠻⠶⠟⠀⠀⠀⠀⠀⣿⣿⠉⠀⠈⠉⠀⠀⠀⣀⣼⣿⣿⣿⠀⠙⣿⣄⡀⠀⠀⢙⡛⣛⣛⡛⣛⣛⡛⣛
⠤⠶⠿⠿⠿⠿⠿⠿⠶⠦⠀⠀⠀⠀⠤⠴⠶⠶⠶⠶⠤⠄⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠁⠀⠉⠁⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⢺⠂⠀⣠⣶⠄⠀⠀⠀⣿⣿⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣝⠋⠀⠀⠀⠀⠀⠀⠀⠀⢾⠆⢠
⢀⣀⣤⣤⣿⣷⣾⣭⣄⣀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⣉⠳⠂⠀⠀⠀⠀⣀⣀⣼⣿⣲⣴⣷⣦⡀
⢨⣍⣩⠭⣭⣩⡭⣭⣩⡥⠀⠀⠀⠀⠀⠀⠀⠶⢼⡃⠀⠀⠀⠀⡾⠉⠙⣮⢿⣉⡞⠉⠙⡆⠀⠀⢸⣿⣿⣿⣿⡿⢿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⡄⠀⠀⠻⠤⠴⠋⠈⠁⠳⠤⠴⠃⠀⠀⠈⠉⠉⠉⠉⠁⠈⠉⠉

⠀⠀⠀⠀⠀⠀⢠⣶⠶⠶⠶⣶⠶⢶⡶⠶⢶⡶⠶⣶⠶⠶⣶⠶⢶⡶⠶⢶⡶⠶⣶⠶⠶⠶⣶⡄
⠀⠀⠀⠀⠀⠀⢸⡟⢧⡀⠀⠛⠀⠘⠇⠀⠘⠃⠀⠛⠀⠀⠛⠀⠘⠃⠀⠘⠃⠀⠛⠀⢀⡼⢻⡇
⠀⠀⠀⠀⠀⠀⢸⣇⣈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣁⣸⡇
⠀⠀⠀⠀⠀⠀⢸⣯⣍⠘⣧⠀⠀⣀⣤⡄⠀⢀⣤⣄⠀⠀⢠⣄⠀⠀⣤⠀⠀⠀⠀⣼⠃⣨⣽⡇
⠀⠀⠀⠀⠀⠀⢸⣧⡤⠀⢸⡆⠀⡏⢀⣀⡀⡟⠀⢸⡇⠀⣾⣽⡆⠀⣿⠀⠀⠀⢰⡇⠀⠤⣼⡇
⠀⠀⠀⠀⠀⠀⢸⡷⠶⠀⠀⢿⠀⠻⠦⠾⠃⠻⢤⠾⠁⠸⠃⠀⠻⠀⠿⠶⠶⠀⡿⠀⠀⠶⢾⡇
⠀⠀⠀⠀⠀⠀⢸⡟⠓⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠚⢻⡇
⠀⠀⠀⠀⠀⠀⢸⣟⣋⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢙⣻⡇
⠀⠀⠀⠀⠀⠀⢸⡏⠉⠀⠀⣠⡷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢾⣄⠀⠀⠉⢹⡇
⠀⠀⠀⠀⠀⠀⢸⡇⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠳⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⠤⠴⢶⡄⠀⣀⣀⣀⣀⠀⠀⠀⢀⣠⣴⡞⢙⣧
⠀⠀⠀⠀⢀⣀⣠⣤⡤⠴⠶⠶⠚⠛⠋⠉⠉⠉⠀⠀⠀⢀⣠⡴⠟⠋⠉⠉⠉⢉⣻⠶⠚⢉⣠⡼⠞⠋⠁
⠀⠐⣿⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⢠⣤⡴⠞⢉⣠⡴⠺⣏⠁
⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⢠⣤⠴⠚⠉⣀⣤⡾⠟⠁⠀⠀⠈⢳⡄
⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡴⣞⣉⣀⣤⢶⠿⣟⣿⠃⠀⠀⠀⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⣅⣼⡇⢈⡿⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠿⠿⠛⠋⠀⢿⢯⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠙⠿⠟⠳⠦⣄⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⢯⡉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠈⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣤⡴⠶⠛⠉⠁