Profession Emojis & Symbols

⠀⠀⠀⠀⠀⠀⣠⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⣤⡀⠐
⠀⠀⠀⠀⠀⣈⣩⣭⣅⣚⠃⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⢠⣿⣶⣄⣉⠛⠒⠆⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⢠⡏⢁⠀⠀⢈⢹⡇⠀⠀⡀⠀⡏⢉⠁⠈⢉⢹⠃⠀⠀⠀⢸⠁⡀⠈⠉⡉⡿⠀⠀⠀⠬⡏⡉⠀⠈⡉⢹⠄
⠀⠀⠀⠀⢸⣧⣄⠀⠆⣀⣼⡇⠀⠀⠽⠆⠐⠀⠀⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⠀⠀⠈
⠀⠀⠀⠀⠈⠛⡟⢰⡆⢻⠟⠃⠀⠀⠀⢀⠀⣠⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⢐⠀⢰⠀⠀⠀⠀⠈⢀⠈⣴⣦⣴⣄
⠀⠀⠀⠀⠀⠰⠇⠸⠇⠸⠇⠀⠀⠀⠀⠐⠐⠷⠼⠷⠼⠃⠀⠀⠀⠀⠀⠀⠼⠀⠨⠀⠀⠀⠀⠀⠸⠸⠿⠿⠿⠿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⣠⡀⠀⠀⠀⠀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⢟⣉⣉⡻⢿⡧⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠿⠀⢀⣤⣾⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠰⡏⡙⠛⠛⡋⢳⠀⠀⠀⠀⢸⢋⠉⠉⠁⠈⡇⠀⠀⢸⣿⡟⢋⠉⠉⢀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡆⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⡄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢷⠀⠀⠀⠀⠀⠀⢀⡿⠸⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠸⠿⠿⠛⠋⠻⠀⠀⠀⠀⢠⡞⠛⠋⠛⠛⢶⠀⠀⠀⠀⣾⠿⠿⠛⠋⠹⡇⠀⠀⠀⠀⡖⠛⠋⠙⠛⢲⡀
⠀⠀⠀⠀⠰⣧⠘⠀⡀⠘⢀⣿⠀⠀⠀⠀⠃⠀⢀⡀⠀⠘⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠃⠘⠀⡀⠘⠘
⠀⠀⠀⠀⠀⠀⠀⠦⠠⢆⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡶⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣾⣧⢀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⢺⣿⣭⣿⠂⠀⠀⠀⢠⢶⢀⡆⠀⠁⢰⡄
⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠀⠀⠀⠀⠈⠉⠈⠁⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠚⠂⢀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⡀⠂⠀⠀⠀⡀⠁⠀⠀⢀⠊⠀⠀⠈⠋⠑⠄⠀⢀⠐⠁⠀⠀⠑⠄⠀⠀⠄⠀⠁⠈⠑⣄⠀⠀⠀⠀⠊⠀⠠⠀⠀⠑⠀⠀⢀⠰⡂⠀⠀⡱⠠⡀
⠀⠔⠃⠔⢄⠀⠀⠈⠚⡀⢐⠀⢀⣀⠁⠂⠀⢆⡀⠀⣆⠄⠤⠆⠀⠠⣠⡀⠀⡤⠀⡀⠠⠈⠊⠄⠀⠀⡎⠤⠀⠘⠃⠠⣀⢂⠀⠀⠀⢀⡧⠰
⠀⠐⠒⠀⠀⠀⠉⠉⠒⠃⠀⠐⡁⠁⠀⠀⠀⢸⠀⢠⢠⠀⠀⠀⠀⠀⡃⡅⠀⡁⠀⠀⠀⠉⢈⠁⠀⠘⢈⠁⠀⠀⠀⠀⡇⠌⠀⢌⣹⠀⠀⠀⠀⠃⠁
⠀⠀⠉⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⡄⠀⠀⢀⠄⠀⠀⠀⠠⡀⠀⠀⡐⠀⠀⠀⠡⡴⠀⠀⠄⠁⠀⠀⠘⡀⠠⢀⠀⢀⠔⠀⠊⠀⠀⠝⠀⠀⠀⢀⠊⠁
⠀⠀⠀⠈⠒⠀⢊⡀⠀⠀⠀⠀⣠⠀⠈⢍⡀⠀⠀⠸⠀⠀⡨⠁⠉⢄⠀⠀⠀⠀⠈⠁⡁⠀⠰⡀⠀⠀⠇⢀⠜⠀⠱⢄⠐⠁⠀⠀⠀⢈⠒⠐⣅
⠀⠀⠔⠉⢀⡠⠂⠈⠢⠀⠠⠊⠀⠈⠀⠀⠈⢂⠀⠀⡱⠉⠀⠀⠁⠀⠉⠀⠀⠀⡀⠈⠈⠀⠀⠈⠂⠀⠜⠁⠈⠉⠀⠀⠁⠀⠀⢀⠂⠁⠈⠀⠀⠐
⠀⠘⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⡀⠄⢊⠁⢢⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠉⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⢠⠖⠉⡉⠐⠌⡘⢒⠀⠀⠂⠀⠀⠀⠀⠐⡄⠀⡴⢅⣐⡀⠉⠉⣐⣄⠒⠀⠀⠀⠀⢂⡞⠻⠻⠛⠛⢻⠛⡆⠀⠀⠀⠀⣴⠰⡁⠀⠀⠘⠉⢦
⠀⠀⠀⠀⢾⠇⠀⠀⠈⠂⢀⠀⠚⠲⠄⣀⣀⠲⡆⠀⠄⡊⢀⢀⠔⠄⣀⡀⡆⠀⠀⠀⠀⢸⣇⢀⣀⣀⣀⣘⣀⣟⠀⠀⠀⠐⡄⡤⠃⠀⠀⠐⢤⢰
⠀⠀⠐⢄⡀⠀⠀⠀⠀⠀⠨⣸⠀⠀⠀⠀⠈⡁⠇⠀⠉⢅⡉⠀⠀⠀⠀⢁⡁⠀⠀⠀⠊⠀⠙⠛⠛⠛⠛⠛⠋⠁⠈⢂⠀⠀⠁⢃⠀⠀⠀⠀⠰⠈
⠀⠀⠀⢀⠃⢄⣀⡀⠀⠀⠀⠀⢀⠀⠀⢀⠔⠀⠀⠀⠀⠀⣀⠄⠀⠀⡠⠀⠅⠀⠀⠀⠀⡳⡷⢸⡏⡍⠅⡁⡟⢹⣆⢨⠀⠀⠀⠀⠁⡤⠠⠔
⠀⠀⡠⠈⠀⠀⢇⠀⠀⠀⠀⠀⣀⠋⠡⠡⢄⠀⠀⠀⠀⠀⠀⣠⢍⡩⠤⠀⠀⠀⠀⠀⡀⠀⠃⠐⠁⠂⠐⠂⠃⠋⠈⢸⠀⠀⢀⠔⠚⠡⠤⠘⠒⢀
⠀⢰⣝⣲⠤⣤⠤⣶⠀⠀⠀⠊⠀⠀⠀⠀⠀⠑⡀⠀⠀⢀⠊⠀⠀⠀⠀⠈⢂⠀⠀⠀⠐⠀⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⡆⠴⠀⠀⠀⠀⠰⠄⠇
⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢐⡈⢐⡄⠀⠀⠀⠀⣀⠔⠀⠈⣀⠜⠀⠀⠀⢀⠄⢂⡐⢄⠀⠀⠀⠀⠠⠀⠊⠑⠀⢄⠀⠀⠀⣠⣶⣿⣷⣶⡄⠀⠀⠀⠀⠄⢀⡀⠀⠠⡀
⠀⠀⣔⠁⠀⠀⠈⠦⠀⠀⢈⢀⡄⠀⡠⠈⠀⡦⠀⢀⠁⠀⣐⠂⡀⠑⡄⠀⠀⠀⠀⠀⠂⠀⠀⡆⠀⢸⡿⢛⡻⠿⣿⣿⣦⠀⠀⣄⠀⠀⠀⠀⡠⢅
⠀⢈⠉⣀⡦⠔⢄⡀⠐⠀⠀⠚⡄⣀⠴⡈⠕⢠⠀⢰⣿⠷⠆⠴⡶⣿⡆⠀⢐⢂⠦⠀⠀⠤⢡⣱⠀⠘⠃⠁⠊⠈⠛⠻⠋⠀⠀⠀⢈⠠⠦⠢⣀⢀
⠀⠈⢉⠀⠀⠀⠀⢈⠁⠀⠀⠰⠀⠀⠀⠀⠀⠸⠀⠀⢈⠀⠀⠀⠀⢀⠠⠀⠠⢈⠀⠀⠀⠀⢀⡠⠀⠀⠆⠀⠀⠀⢠⡎⠀⠀⠀⢉⡃⠀⠀⠀⠀⢉
⠀⠀⠈⢀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠁⡀⠀⡀⠂⠀⠀⠀⠀⡀⠀⡀⠂⠀⠀⠀⠈⠀⡀⠀⡀⠄⠀⠀⠀⠀⠂⠀⠄⠈⢀⠀⠀⠠⠴⠠⡀⠀⠀⡠⠝⠖
⠀⠀⠀⠀⡙⠀⠣⡀⠀⠀⠀⡀⠂⣀⣨⠁⠀⠀⠀⠀⢀⠄⠎⠀⠓⠠⠀⠀⠀⢀⠄⠎⠀⠱⠠⣀⠀⠀⠀⢠⠈⡀⠀⠌⠃⠀⠀⠀⠀⢨⡁⢘⠄
⠀⠀⡐⠉⠀⠀⠀⡀⠢⠀⡈⠀⠀⠀⠀⠈⠀⠀⠀⠠⠁⠀⠊⠀⠐⠀⠈⠀⡐⠀⠐⠊⠀⠐⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠊⠀⠀⠁⠀⡀⠄
⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁

⠀⢠⣶⣦⠀⠀⠀⠀⠀⢀⣶⣶⠀⠀⠀⠀⠀⠀⢰⣾⣿⡦⠀⠀⠀⠀⠀⠀⣿⡗⠀⠀⠀⠀⠀⢀⣶⡄⣀
⠀⠈⢿⡏⠀⠀⠀⠀⠀⣘⣿⣿⡁⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⣀⣠⣼⢧⡄⠀⠀⠀⢀⣬⣿⣵⣶
⣴⣿⡶⣾⢿⡄⠀⠀⠀⣿⣿⠿⠿⠶⠀⠀⠀⢠⣾⣿⣿⣷⡄⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⠀⢸⣿⣿⣿⣯

⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠠⣿⡇⠀⠀⠀⠀⠀⠀⢾⣿⠇⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠰⣿⠇⠀⠀⠀⠀⠀⠀⠸⡿
⢠⣤⣿⣧⣄⠀⠀⠀⠀⣤⣶⣿⣦⣤⠀⠀⠀⢠⣶⣿⣿⣶⡆⠀⠀⠀⣶⣿⣿⣿⣶⡄⠀⠀⠀⣴⣾⣿⣶
⠘⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⢺⣿⡆⠀⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢹⣿⠄
⢀⣀⣿⣇⣀⠀⠀⠀⠀⣀⣨⣿⣄⡀⠀⠀⠀⠀⣀⣼⣧⣀⡀⠀⠀⠀⢀⣀⣿⣄⣀⠀⠀⠀⢀⣤⣾⣿⣦
⠞⠿⠿⠿⠟⠇⠀⠀⠐⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠛⠛⠃⠀⠀⠀⠛⠛⠛⠛⠛⠀⠀⠀⠈⠛⠛⠛⠛

⠀⠠⣴⣄⠀⠀⠀⠀⠀⠀⢰⣦⡄⠀⠀⠀⠀⠀⠀⡶⣶⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⣶⣶
⠀⠾⣿⠿⠀⠀⠀⠀⠀⠀⠸⣿⠃⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠀⢻⡿⠁
⢰⣶⣿⣷⣟⠀⠀⠀⢀⣶⣿⣿⣷⣦⠀⠀⠀⢰⢶⣛⣟⡷⣆⠀⠀⠀⣾⣿⣿⡟⡧⠀⠀⠀⢠⣾⣿⣿⣿

⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠐⣿⡇⠀⠀⠀⠀⠀⠐⢿⣿⠗⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢹⣿
⣴⣶⣿⣷⡶⠆⠀⠀⢀⣤⣾⣿⣦⣄⠀⠀⠀⢰⣾⣿⣿⣷⡆⠀⠀⠀⢰⣶⣿⣷⣶⠀⠀⠀⠰⣿⣿⣿⣿
⣿⡿⠻⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠿⠿⢿⣿⣿⣿⣶⣶⣶⣤⣍⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠾⠛⠋⠛⠉⠉⠉⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠉⠉⠉⠉⠙⠛⠛⠻⠿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣤⣄⠀⠀⠀⠀⢀⣤⣶⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⢻⡄⠀⠒⠀⠀⠀⠀⣶⣿⣻⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠻⠿⠿⠃⠀⠀⠀⠀⠀⠁⠸⠿⠿⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣤⡶⠿⣷⡄
⠀⠀⠀⠾⠋⠁⠀⠀⠘⣿⡄⠀⠀⠀⢀⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣀⣤⣼⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⢀⣴⣷⡀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢀⣴⣿⣿⣿⣿⡄⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⣰⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠐⠀⢈⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠙⣿⡟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⡤⠌⠀⠐
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠘⣷⣄⡀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠶⠚⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆⠀⠀⠀⠈⢿⣿⣷⡀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢋⣴⣧⡀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁⣀⣴⣿⣿⣿⣷⢀
⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣿⣿⠟⠁⠀⠀⣴⣾⣿⣿⣿⣿⣿⠉⠋⠆
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠠⠄⠀⠌⠙⠋⠁⢀⣼⣇⡀
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠙⢿⣿⣷⡀
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⣿⣿⣧
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡜⠀⠀⣿⣿⣿
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⢿⣷⣶⣶⠶⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠒⠦⠀⠀⠀⠀⠀⠀⠀⠒⠂⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠒⠂⢀⠀⠀⠀⠀⠀⠀⡤⢄⠀⠀⠀⠀⠀⠀⠀⠈⠉⡀⢄
⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠁⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⡀⠀⠀⠀⠀⠠⠀⠖⠂⠐⠀⠶⠀⠀⠀⣀⠀⢤⠄⡀⠀⠀⠀⠀⠓⠘⠀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠔⠐⠄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠘⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂
⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠘⠂⡄⠀⣠⠀⠃⠀⠀⠘⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⠀⣨⠍⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠆⠀⠀⠀⠰⠁⠀⠀⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠈⠈⠁⠉⠁⠀⠀⠀⠀⠀⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠐⡄
⠀⠀⠀⠀⠈⠏⠒⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠲⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⡄⠀⢀⠄⢠⠀⠀⠀⠀⠀⠁⠀⠀⠃⠐
⠀⠰⠀⠐⠐⠆⠁⠐⠃⠀⠀⠀⢀⠒⠒⠀⠄⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠂⠁⠀⠃⠐⠂⠀⠁⠐⠐⠆⠃⠀⠀⠀⠀⠀⠀⠐⠂⠃⠀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠂⠁⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠁⠠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⠤⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀
⠀⠀⠀⠀⠊⠁⠂⠀⠀⠀⠀⠀⡟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢒⠀⠀⢀⠀⠀⠀⠀⡄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠓⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⡄⠀⢈⠀⠀⠀⠀⠀⡄⢀⠰⠀⣀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠇⠀⠐⠀⠀⠀⠀⠀⠀⢀⠉⢘⠇⣘⠠⡀⠀⠀⠀⠀⠐
⠀⠀⠦⢀⠐⠈⠀⠁⠀⠀⠀⠀⠓⠀⠀⡈⠀⠛⠀⠀⠀⠀⠈⠀⡈⠄⠀⠀⠀⠉⠀⠁⢘⡉⠀⠲⠀⠀⠀⠀⠈⠻⢯⠄⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⡗
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄
⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠲⡿⠿⠿⡷⢢⠀⡆⡆⠀⠀⠀⡹⠿⠿⠏⠀⠀⠀⠀⠀⠀⠈⡿⠿⠳⡆⠀⠀⠀⠀⠀⠀⠰⠟⠿⠿⠇⠀⠀⠀⠀⠀⠀⢲⠉⠈⡖⠂
⠀⠀⠀⠀⠀⢂⠀⡀⠀⠘⢄⣇⠇⠀⠀⠀⠑⡀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠡⠠⢄⠌
⠀⠀⠀⠀⢲⣈⠃⠈⣰⠂⠀⡇⠀⠀⢠⣤⣼⡟⢻⡧⣤⡀⠀⠀⠀⠠⠒⠹⡏⡟⢐⠢⠀⠀⠀⠀⡀⠸⡏⠛⢸⡗⠀⠀⠀⠀⢀⠀⠀⠉⠉⠀⠀⡀
⠀⠀⠀⠀⠘⠛⠛⠛⠛⠐⠀⠃⠀⠀⠘⠛⠛⠃⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠚⠛⠛⠛⠓⠀⠀⠀⠀⠐⠀⠀⠐

⠀⠀⠀⠀⠀⢠⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⣤⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠰⠆
⠀⣀⣠⣤⠀⡛⠛⠛⡃⠀⠀⠀⠀⠀⠀⠀⠘⣛⣛⡃⠀⠀⠀⠀⠀⠈⠹⠟⠛⠿⢩⠁⠀⠀⠀⠀⠀⢀⠏⠛⠻⡀⠀⠀⠀⠀⠀⢰⡶⠒⠒⢲⣆
⠘⡿⠟⢻⡐⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠐⡈⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠈⢁⣀⡘⠀⠀⠀⠀⠀⠆⠸⣷⡀⢀⣼⡏⠀⠒
⠀⠀⠀⢠⢳⡐⡆⣲⢰⢳⠀⠀⠀⠀⢠⣾⣿⣷⣿⣿⣷⡀⠀⠀⠀⣴⣦⡑⣷⣠⣶⣆⠀⠀⠀⠀⡀⣾⡅⣾⢀⣷⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠈⠈⠁⠁⠉⠈⠉⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠃⠀⠀⠘⠛⠛⠛⠛⠛⠋⠛⠀⠀⠀⠀⠓⠛⠛⠛⠛⠛⠒⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⢰⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⡱⣿⣷⡁⠀⠀⠀⠀⠀⠀⠀⢸⠛⠻⡇⠀⠀⠀⠀⠀⠀⠙⡿⣧⡿⡟⠀⠀⠀⠀⠀⠀⠈⣿⣷⣿⠃⠀⠀⠀⠀⠀⠀⢸⠂⠀⡇
⠀⠀⠀⠀⠠⣆⠀⢀⡦⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⢠⡦⠀⠀⠀⠀⠀⠀⠑⡀⢀⠋
⠀⠀⠀⢠⣴⣾⣴⣾⣶⣄⠀⠀⠀⠀⢀⠔⠐⢵⡮⠂⠀⠀⠀⠀⠀⠀⠴⠚⢵⠝⠲⠄⠀⠀⠀⠀⢠⣀⣌⣴⢡⣄⣄⠀⠀⠀⠀⠀⠘⠷⠾⠂⠢
⠀⠀⠀⠸⠿⠿⠻⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠸⠂⠈⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠶⠿⠀⠀⠀⠘⠘⠀⠀⠀⠀⠒⠃

⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⡾⠿⠿⡆⠀⠀⠀⠀⠀⠀⠀⣼⡿⠻⣦⠀⠀⠀⠀⠀⠀⠀⡯⠉⠹⡇⠀⠀⠀⠀⠀⠀⠀⡞⠿⠻⡄⠀⠀⠀⠀⠀⠀⢸⢿⣾⡇
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢶⡀⢀⣾
⠀⠀⠀⠀⠂⠈⠀⠉⠀⠀⠀⠀⠀⠀⢀⣴⡊⢵⡯⢰⣦⡀⠀⠀⠀⠀⢰⡎⢱⠉⣶⠄⠀⠀⠀⠀⣠⣶⣆⣾⢰⣶⣄⠀⠀⠀⠀⠀⠀⠁⠊⢀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘⠿⠿⠾⠷⠿⠿⠃⠀⠀⠀⠀⠸⠷⠾⠶⠿⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠟⠀⠀⠀⠐⠀⠀⠀⠆⠉⠀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣿⣏⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣭⣽⣭⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣴⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣶⣶⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠈⠐⠐⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸⡀⠀⢀⣷⠀⠀⠀⠀⠀⠀⣊⠛⠛⠉⠂⠀⠀⠀⠀⠀⠐⠁⠂⠂⠑⠀⠀⠀⠀⠀⠀⢀⠋⠛⠛⠆
⠀⠀⠀⢀⢁⣀⠴⣦⡷⢄⡀⠀⠀⠀⠀⠀⡀⢼⣶⣿⢇⡀⠀⠀⠀⠔⣱⡦⣶⣴⡦⣄⠀⠀⠀⠀⠀⠀⠐⣲⡆⠀⠄⠀⠀⠀⠀⠀⠀⠀⡄⡄
⠀⠀⠀⢸⢸⣿⠀⢸⠛⢻⡇⠀⠀⠀⠀⢀⣧⠐⡿⠋⣼⡃⠀⠀⠀⠈⠉⢟⠻⡿⠛⣿⠀⠀⠀⠀⠀⠴⠟⣻⡟⠳⠤⠀⠀⠀⠀⠀⠁⠀⢿⠁
⠀⠀⠀⠈⠉⠀⡆⣸⠀⣼⡇⠀⠀⠀⠀⠭⢭⠽⠀⠅⣿⡇⠀⠀⠀⠀⠀⢸⣀⣿⢈⣿⠀⠀⠀⠀⠀⣶⣼⣿⣿⣴⡆⠀⠀⠀⠀⠀⠈⠀⠈⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠶⠋⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠁

⠀⠀⢠⣶⡄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠻⡃⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠓⠒⠚⠀⠀⠀⠀⠀⠀⠀⡴⠟⠻⢧⠀⠀⠀⠀⠀⠀⠠⡾⠿⠟⠛⠄⠀⠀⠀⠀⠀⠀⡼⠟⠛⠿
⠀⠀⠀⠀⠃⠀⠀⣤⡤⠀⠀⠀⠀⠀⠀⠀⢀⣦⣠⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣤⡀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠁⠁⠀⢸⡅⠀⠀⠀⠀⠀⠀⣼⣿⡉⣿⣿⣿⡆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢹⣀⣀⡏
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠘⠟⢻⣿⠟⠛⢉⡃⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠎⠀⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠉⠈⠃⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁⠂


⠀⠀⠀⠀⠀⠀⠀⡴⠞⠶⡄⠀⠀⠀⠀⠀⠀⠶⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠐⢒⣒⡄⠀⠀⠀⠀⠀⠀⣄⠠⠄⢤⠀⠀⠀⠀⠀⠀⠀⡰⠶⠲⢄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣀⣠⡆⠀⠀⠀⠀⠀⠀⠘⣄⡈⣀⡄⠀⠀⠀⠀⠀⢠⢱⢤⡠⢬⠃⠀⠀⠀⠀⠘⠀⠀⡀⢀⠈
⠀⠀⠀⢀⠀⠀⠀⠒⣷⠇⠐⠀⠀⠀⠀⠀⣸⡇⠉⠀⢽⠄⠀⠀⠀⠀⠀⠠⡚⠴⠌⡒⠀⠀⠀⠀⠀⠙⠁⠐⢁⠈⠉⠀⠀⠀⠀⠄⢈⠀⢲⡖
⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠐⡬⢹⢀⣀⠾⡺⠀⠀⠀⠀⠀⢰⠑⣦⣀⡡⠀⠀⠀⠀⠀⢠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠃
⠀⠀⠀⠘⠖⠒⠒⠒⠳⠒⠀⠀⠀⠀⠠⠾⠆⠼⠶⠧⠰⠷⠄⠀⠀⠀⠐⠁⠴⠉⠥⠤⠀⠀⠀⠀⠶⠋⠁

⠀⠀⠀⠀⢠⣶⣿⣟⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣷⡶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⣶⣿⣿⣷⠒
⠀⠀⠀⠀⡬⠿⠿⠿⠿⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⠄⣤
⠀⠀⠀⠀⠹⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⢷⣿⣿⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠟
⠀⠀⠀⠀⣀⣶⣿⣿⣶⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡿⣿⡿⣦⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⡿⣿⣄⣀
⠀⣴⢐⡷⡟⡛⣛⣛⣛⢻⢿⡣⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣯⢳⣴⣴⣫⣿⠿⠃⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣟⣷⣿⣷⣿⣿⣿⣷⣶
⣸⣿⢨⣽⡇⣛⣙⣋⣛⢸⣯⡅⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣾⣿⣿⣯⣿⣿⡟⡑⢺⢿⠒⣒⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠀⠈⠁⠈⠉⠉⠁⠈⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⠉⠁⠉⠁⠉⠀⠀⠈⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢠⣾⣶⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⢾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇⣻⣷
⠀⠀⠀⠀⢨⣿⣭⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣿⣯⣿⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣯⠄
⠀⠀⠀⠀⠛⣿⠿⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣿⣿⣿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠟
⠀⠀⠀⠀⢀⣻⣤⣽⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣆⣽⣫⣅⣻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣺⣿⣿⣦⣀⣀
⠀⣀⣠⣶⣿⡹⣿⣿⢻⣿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠻⣼⢫⡟⢧⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣯⣿⣯⣽⢿⣿⣿⣿⣷
⢸⣿⣿⣿⣻⣽⣷⣾⣭⣿⣿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣼⣿⣿⣾⣿⣿⣾⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣿⣟⣿⣿⣿⣿⣿⣝⣿
⠿⠟⠶⠻⠷⠾⠻⠿⠷⠾⠟⠾⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠟⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠽⠷⠿⠾⠷⠿⠿⠿⠯⠷⠿



⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⣴⣿⣹⣾⡿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⣿⢿⠆
⠀⠀⢀⣀⣟⣋⣿⣿⣛⣟⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣮⣿⣿⣿⣿⣵⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣾⣶⣿⡀
⠀⠀⠈⠻⣿⣿⣿⣿⣿⡷⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⡙⢿⣿⢟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄⠀⠀⠀⢀
⠀⠀⠀⣀⣠⣛⣿⣿⣟⣥⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢲⣾⣿⣿⣿⣻⣷⡞⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣴⡶⣿⣿⣿⣿⣷⢼⣿⣿⣿
⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⣿⡛⢛⣿⣾⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣽⢳⣿⣿⣿⣿⣿⣻⣿⣞⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿
⣿⣿⣇⣈⣉⣀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠈⠀⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⡿⠿⠛⠁⠈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠁⠹⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠐⢤⠂⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢺⠃⠀⡀⠀⣨⢲⣄⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠄⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢿⠀⠀⢉⠀⠁⠈⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠠⠀⠐⠂⠀⣀⡗⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⣠⣷⡄⢀⠀⠀⠀⢸⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢆⠀⠀⢸⠏⠀⢷⠘⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠂⡄⠀⠀⠐
⠀⠀⠀⠓⠒⠒⠒⠂⠀⠀⠒⠐⠓⠚⠂⠘⠘⠓⠀⠚⠘⠒⠒⠒⢺⠒⠒⠒⠒⠂⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠒⠒⠒⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡔⠀⠀⠀⡀⠑⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⠿⠛⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⢠⡄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠉⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⢳⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⡉⠀⠂⠐⠁⠤⠀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⠀⢪⣿⠀⠀⠀⠀⠈⣶⢄
⠀⠀⠀⠀⠀⠀⠐⠉⡀⠐⠀⠀⠀⠐⠀⠠⠈⠀⠀⠁⠀⠢⠀⠀⢸⠀⠀⠀⠀⠀⠄⠀⠀⠀⡏⠀⡀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢠⡾⢷⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠸⠀⠀⢸⠀⠀⠀⠐⠀⠀⢀⠂⠀⣼⠒⡄⠀⠀⡟⠀⠘⡆⠇
⠀⠀⠀⠁⠀⠉⠀⠀⠈⠉⠁⠀⠀⠈⠁⠈⠀⠀⠀⠀⠁⠉⠀⠁⠉⠁⠉⠈⠁⠀⠀⠉⠁⠀⠉⠉⠁⠀⠀⠉⠁⠈⠁⠉⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⣀⠘⡇⠀⠀⢠⡦⠀⠀⠀⠀⢹⣿⠃⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠆
⠀⠀⠀⢠⣟⠶⢻⣿⣷⡄⠀⠀⢀⣬⣯⡄⠀⠀⠀⢀⣼⣯⡀⠀⢠⠃⠀⣷⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢇⣆
⠀⠀⠸⣿⣿⡷⣿⡟⠀⠇⠀⠀⣸⠱⠦⢿⠀⠀⢠⠞⣿⣿⠛⠓⠂⠀⢸⣿⠋⠓⡄⢀⡀⠀⠀⠀⠀⠰⢋⠇⠈⣏⢢
⠀⠀⣾⣿⣿⣷⣿⡇⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⡷⢾⣿⣿⣶⠖⠀⠀⠘⣦⣶⠞
⠀⠀⠙⠻⠟⠛⡟⡇⠀⠀⠀⠀⠘⣤⣤⠅⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⠗⠋⠉⠛⢦⣦⠀⠀⠀⡿⢹⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠃⠙⠀⠀⠀⠀⠀⠃⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠃⠈⠀⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⡀⠀⠀⠀⢀⡴⠂⠀⣠⡀⠀⣿⣇⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠅
⠀⠀⠀⠸⣿⠀⣤⡿⣏⠀⠀⢠⣽⣧⣠⣿⣿⡀⠀⣤⣤⣬⣯⣤⠀⠀⠀⣀⠀⠐⢿⠇⠀⢀⡀⠀⠀⢀⣤⣶⣶⣦⡀⢰
⠀⠀⡴⢿⣿⣿⣽⠆⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠙⣿⣿⣿⣿⡆⠀⠀⢉⣀⡴⣿⣿⠓⠾⠃⠀⢀⣞⣁⣹⣿⡏⠻⣼
⠀⠀⠑⢸⣿⡉⠁⠀⠀⠀⠀⠈⣿⣿⠉⠁⠀⠀⠀⠀⣽⣿⣿⣿⡇⠀⠀⠀⠀⢀⣿⣿⣤⡀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠁
⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠿⣿⣿⡿⠓⠀⠀⠀⢰⣿⠉⠀⣿⣷⡀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⣿⠈⡇⠀⠀⠀⠀⠀⠀⡿⢹⠀⠀⠀⠀⠀⠀⠀⡿⢸⠃⠀⠀⠀⢠⣼⡿⠀⠀⠚⠿⠶⠀⠀⠀⠀⡿⢸⠃
⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠒⠒⠲⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⣁⠀⠀⠀⠀⠐⠦⣄⠀⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠧⠴⠧⣤⣤⣤⣀⠀⠀⠈⠙⢄⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⢶⣶⣶⣤⣤⣤⣉⡙⠓⠦⣶⣄⠀⠈⢦⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠸⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡙⠳⡄⠈⣧⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠚⣷⣀⢸⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣸⠉⠀⠀⠀⠉⠙⠛⠿⢿⠟⣛⠉⣴⣿⣿⠸⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⣿⠒⠋⢉⡛⠒⠦⠀⠀⢸⣿⡟⢰⣿⣿⣟⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠸⣶⢚⣉⣉⡉⠐⠀⠀⢸⣿⠁⢟⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠹⣏⡡⠤⠄⠀⠀⠀⣀⠇⣴⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣀⠀⠀⠀⠘⠳⠶⣶⣶⣿⣿⡏⢸⣿⣿⣿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⢶⣶⣶⣀⣉⣉⣉⣭⣴⣿⣿⡟⠁⣠⠟⠳⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢻⠇⢸⣿⣿⣿⣿⣿⡿⠟⠁⣠⡾⠃⠀⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢀⡏⠀⠸⣿⠿⠟⢛⡽⠿⠿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡋⠀⠀⠀⣼⠀⠀⠀⠀⢠⡞⠋⠀⠀⠀⢨⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣴⣿⡀⠀⠀⣰⠏⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣤⣴⡏⠀⠀⠀⣠⠞⠁⠀⠀⢀⣀⣤⡴⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠿⠷⠶⠶⠛⠛⠋⠉




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣿⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⠿⡿⠛⠛⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠈⢉⣉⣭⡛⠃⠀⠀⠛⣫⣍⣉⠉⠀⢸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠻⣆⠀⠈⠸⠿⠙⢂⣀⣀⠛⠽⠿⠈⠀⢀⡞⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡈⡏⡷⠶⠶⠚⠋⠁⠀⠈⠉⠛⠲⠶⠶⡏⣏⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⡀⠀⠀⠀⠀⠀⢀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠙⠒⠤⠔⠚⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠖⣶⠞⠉⠩⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⠻⡝⠒⠲⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠋⠉⠀⠀⠀⠀⣇⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢹⡀⠀⠀⠀⠀⠈⠉⠓⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠷⢶⣄⠈⢆⣀⠀⠀⠀⠀⠀⢀⣠⠎⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡄⠘⣇⠉⠉⠉⠉⠉⢁⡟⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⠀⠘⡇⠀⠘⡄⠀⠀⠀⢀⡞⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⢿⠀⠀⠸⡄⠀⠀⡼⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢸⡄⠀⠀⠹⡄⡼⠁⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠈⡇⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⡇⠀⠀⣠⣶⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⣄⣀⣀⣀⠞⠓⠒⠚⠁⠀⠀⠀⠀⠀⣠⡤⠛⢵⡄⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹
⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣄⠀⡠⡃⠀⠀⠀⠀⠀⠀⠀⡇
⠉⠓⠲⠤⣤⣀⡀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠉⠊⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣀⣠⡤⠴⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⠊⣉⣉⣙⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣙⣉⡙⠲⠶⠦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣠⡴⠒⠋⠁⠀⠀⠀⠰⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣙⡦⠀⠀⠀⠈⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖
⠀⠀⠀⢀⣠⠴⠋⠁⠀⠀⠀⠀⢀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣀⡀⠀⠀⠀⠀⠀⠈⠙⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⡴⠟⠁⠀⣀⠴⠶⠚⠛⠛⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠓⠶⢦⣀⡀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢸⡇
⢠⡞⠁⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣄⠀⠙⢆⠀⠀⠀⠀⢸⡇
⡘⠁⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠋⠉⠀⠀⠀⠀⠀⠉⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠙⣄⠀⢸⡇⠀⠀⠀⢸⡇
⣻⠀⢸⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⢸⡇
⡇⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⢸⡇
⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⢸⡇
⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡀⡀⠀⠀⠀⢀⡿⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⣿⠀⢸⡇⠀⠀⠀⢸⡇
⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢸⡋⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⣿⠀⠀⠀⣇⠀⠀⣿⠀⢸⡇⠀⠀⠀⢸⡇
⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡏⠀⠀⡿⠀⢸⡇⠀⠀⠀⢸⡇
⡇⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠘⠷⠶⠶⠶⠶⠤⡆⠀⠀⠀⠀⡶⠴⠶⠶⠶⠤⠟⠀⠀⠀⡇⠀⠀⣧⠀⢸⡇⠀⠀⠀⢸⡇
⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣿⠀⢸⡇⠀⠀⠀⢸⡇
⡇⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⢸⡇
⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⣷⣤⣤⣤⣤⠇⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⢸⠁
⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⢸⡀
⡇⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠦⠤⠤⠶⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⢸⡇
⡇⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⢸⡇
⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⡇⠀⠀⠀⢸⡇
⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⡇⠀⠀⠀⢸⡇
⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⢸⡇
⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⢸⡇
⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⢸⡇
⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⢸⡇
⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⢸⡇
⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⢸⡇
⡇⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠇⠀⠀⠀⢸⡇
⡇⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⢸⡇
⢧⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⡼⠀⠀⠀⠀⢸⡇
⠘⣧⠀⠈⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⣴⠃⠀⠀⠀⠀⢸⡇
⠀⠈⠳⣄⠀⠀⠉⠛⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠋⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠙⠳⢦⣤⡀⠀⠈⠙⠳⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠆⠉⠉⠀⢀⣠⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠈⣇⠀⠀⡶⠉⠉⠉⠲⣄⠀⢠⠶⠋⠉⠉⠳⡄⠀⢰⡏⠀⠀⠀⣠⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢀⣀⣼⣀⣀⣇⣀⠀⠀⠀⣿⠀⢸⠂⠀⠀⣀⣠⣇⣀⣸⣀⡀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⢼⡧⠿⠥⠤⠿⢽⡦⠤⠴⣿⠤⢾⡦⠤⠤⣿⠤⠿⠤⠿⠯⣷⠼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⢿⠶⠾⠃⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠤⠤⠤⠤⠾⠀⠀⠀⠀⢷⡈⢷⣄⠀⠿⠤⠤⠤⠤⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡉⠓⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠔⠛⣁⡴⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣴⣶⣶⣶⣶⣶⣶⣦⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴⣛⣉⣀⣀
⣿⣿⣿⡿⠛⠿⠟⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣥⣤⣤⣤⣤⣄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠛⠛⠛⠛⠛⠛⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠤⠤⠤⠤⠤⠤⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺⡇⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡟⢻⠀⠀⠀⠀⢸⡇⠀⠀⢀⡴⠋⠉⠉⣙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⠖⠚⠃⠘⠒⢲⡆⠀⢸⡇⢀⡴⠋⠀⠀⠀⠀⢸⢾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠙⡇⢸⠋⠉⠁⠀⢸⡷⠋⠙⢦⡀⠀⠀⠐⡫⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠓⠚⠀⠀⠀⣰⠋⠀⠀⠀⠀⠙⢦⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠉⠉⠉⠉⠉⠉⢉⡽⠋⠉⠉⠉⠉⢹⠉⠉⠉⠉⠉⠓⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⣀⣀⣀⣀⣀⣸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⢼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⣀⣀⣀⣀⣸⣀⣀⣀⣰⣖⣫⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⣀⢀⠀⠀⣀⢀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠁⠸⠄⠣⠜⠀⠘⠤⠘⠸⠑⠇⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡟⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠈⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡉⠀⠀⠁⢠⣤⡄⠀⠀⠀⠀⡤⣤⡄⠀⠀⠀⢛⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠀⠻⠿⠃⠀⠀⠀⠀⠻⠿⠃⠐⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⣀⠀⠀⠁⠀⣠⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠳⣶⣶⡾⠋⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⠂⠠⢀⡀⠀⡀⠠⠔⢻⠆
⠀⠈⠁⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⠟⠀⠀⠀⠀⠀⠀⠀⠂⠸⡅⠀⠀⣤
⣆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⠀⠀⠀⠀⠀⢂⣀⠙⠁⠀⠀⠀⠈⠀⢀
⠷⠀⠆⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⢉⠐⠂⠰⠖⠉⠆⠀⠃
⠘⢧⠒⠀⢊⡄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠀⠀⠀⠀⠂
⠀⠈⠀⠀⠉⠲⡀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠛⠁
⠀⠀⠀⠀⠀⠀⠻⢤⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢀⠰⠍⠀⠀⠀⠀⠆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠔⠠⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡻⠗⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⢀⠂⡀⣀⠀⡀⣀⡀⣀⡤⠖⠉⢡⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠋⠁⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⣀⣀⠀⠄⣀⣀⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣤⣤⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣠⠀⠐⠄⢀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⡀⠠⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠺⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠁⠀⠀⠀⢈⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⠻⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠶⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢰⡗⠀⢷⡦⢀⣼⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⢸⣃⣠⡼⠂⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⢿⡿⠁⠲⢶⠆⠴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠰⠉⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆⠀⠀⠀⠀⠀⠘⣿⡄⠀⠐⠒⠂⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⡴⣿⡹⠷⢦⡤⢴⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⣀⣴⣇⠘⡳⣶⡀⠀⢸⠃⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⠀⠲⣦⠝⢿⣿⣾⡟⢉⠷⣠⡾⠋⠉⠑⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠑⠉⠁⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⢰⠀⠀⡴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡮⠀⠀⠀⠀⢠⣿⣷⠀⠹⣿⣀⠀⢀⠄⠀⠂⠀⠀⠀⢀⠀⣇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀⣾⡟⠙⠆⠐⣻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡟
⠀⠀⠀⠀⠀⢀⡾⠁⠀⣀⠀⣠⣿⣿⣽⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⢠⢺⣶⣿⠟⠙⠛⠿⠿⠼⠿⠧⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠸⡿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢻⡏⠀⠀⠀⠀⠀⠀⠀⠋⣿
⠀⠀⠀⠀⠀⠙⢿⣿⣧⣀⣀⣤⣤⣤⣴⠾⠃⠀⠀⢼⣧⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡀⣽⣿⣿⣿⡀⠸⡆⠐⣟⠇⠁⠀⠀⠀⠀⠀⠁⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⣸⣿⣿⣝⡏⠀⠣⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡼⢻⣿⡿⣯⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠞⠻⣿⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⠻⣿⣿⡄⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⢻⣿⣿⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⣿⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀
⢠⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣦⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣦⡀
⣿⣿⣿⣿⣿⣿⡿⠛⢉⣉⠙⠻⣿⣿⣿⣿⣿⣿⡆⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣀⣤⣀⣀⣀
⣿⣿⣿⣿⣿⡏⢀⣀⣸⣿⣀⡀⠸⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⡇⠘⠛⢻⣿⠛⠛⢀⣿⣿⣿⣿⣿⡇⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉
⣿⣿⣿⣿⣿⣿⣦⡀⠘⠛⢀⣠⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣴⡤⢤⣤⣀⠀⢰⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢀⣤⣴⣦⣄⡀⠀⣠⣴⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⢁⣰⣆⡈⢿⠀⢸⠀⠀⠀⠀⠀⢀⣾⣿⣿⣟⡁⢈⣹⣿⣿⣿⡆
⠀⠀⠀⣾⣿⣿⣟⣿⣿⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠙⢻⡟⠃⣼⠀⢸⠀⠀⠀⠀⠀⢸⣿⣿⠿⠛⠛⠚⠛⠿⣿⣿⡇
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣶⣾⣿⠀⢸⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇
⠀⠀⠀⠹⣿⣷⣿⢽⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡟⢻⢛⣟⠉⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠸⢜⣛⣀⣀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⢀⣤⣾⣿⣿⠿⠿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣤⣤⣀⢀⣠⣤⣴⣶⣶⣦⡀
⠀⠀⢠⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠈⣿⣿⣿⣿⣿⣿⡗⠀⢠⣤⢀⣀⠀⣀⢀⡀⢀⣀⢀⡀⢀⡀⣠
⠀⢀⣿⣿⠿⠿⠿⠇⠀⠀⠸⠿⠿⠿⣿⣿⡆⠀⠀⠀⠈⠛⠿⢿⣿⣿⠃⠀⢸⣿⣿⠿⠿⠛⠁⠀⢸⣿⠉⢉⣉⠋⠉⠉⠉⠉⠉⠉⠉⠉⢻
⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⣶⣶⠀⠀⣀⡀⠀⢻
⠀⠀⣿⣿⣷⣶⣶⡖⠀⠀⢰⣶⣶⣶⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢘⣿⠀⠀⢛⡿⠀⠀⢿⡧⠀⢺
⠀⠀⠘⢿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢕⣂⠀⠀⠀⠀⠀⠀⠀⢸⣿⠲⠞⠛⠶⠶⠛⠛⠖⠶⠛⠳⠒⢿
⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠻⠛⠚⠐
⣶⣶⠶⠶⠶⢶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣠⣴⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣄⡀⠐
⠀⠀⠀⠀⠀⣈⣩⣭⣅⣚⠃⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⢠⣿⣷⣤⣉⠛⠲⠆⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⢠⡏⢁⠀⠀⢈⢹⡇⠀⠀⡀⠀⡏⢉⠁⠈⢉⢹⠃⠀⠀⠀⢸⠁⡀⠈⠉⡉⡿⠀⠀⠀⠬⡏⡉⠀⠈⡉⢹⠄
⠀⠀⠀⠀⢸⣧⣀⠀⠆⣀⣼⡇⠀⠀⠽⠆⠐⠀⠀⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⠀⠀⠈
⠀⠀⠀⠀⠈⠛⡟⢰⡆⢻⠛⠃⠀⠀⠀⢠⠀⣠⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠐⠂⢰⠀⠀⠀⠀⠈⢀⠈⣴⣦⣴⣄
⠀⠀⠀⠀⠀⠰⠇⠸⠇⠸⠇⠀⠀⠀⠀⠐⠐⠷⠼⠧⠼⠃⠀⠀⠀⠀⠀⠀⠼⠀⠠⠀⠀⠀⠀⠀⠸⠸⠿⠿⠿⠿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⢟⣉⣉⡻⢿⡧⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠿⠀⢀⣤⣾⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠰⡏⡙⠛⠛⡋⢳⠀⠀⠀⠀⢸⢛⠉⠉⠁⠈⠇⠀⠀⢸⣿⡟⢋⠉⠉⢀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⡄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣷⠀⠀⠀⠀⠀⠀⢀⡿⠸⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠸⠿⠿⠛⠋⠻⠀⠀⠀⠀⢠⡞⠛⠋⠛⠛⢲⠀⠀⠀⠀⣾⠿⠟⠛⠋⠹⡇⠀⠀⠀⠀⡾⠛⠋⠙⠛⢲⡀
⠀⠀⠀⠀⠰⣧⠐⠀⡀⠘⢀⣿⠀⠀⠀⠀⠃⠀⢀⡀⠀⠘⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠃⠘⠀⡀⠘⠘
⠀⠀⠀⠀⠀⠀⠀⠶⠠⢆⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡶⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣾⣧⢀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⢺⣿⣬⣿⠂⠀⠀⠀⢠⢶⢀⡆⠀⠁⢰⡄
⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠉⠀⠀⠀⠀⠈⠉⠈⠁⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣤⡄⢠⣤⡌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡍⠉⠭⠉⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⢿⢾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣶⣴⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⡾⢿⡟⠿⣍⠟⣿⡿⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣺⡉⣀⣠⣿⣾⣿⣧⣠⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡏⣤⡤⣤⢠⣴⢤⡄⣿⢻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠿⣿⣟⠿⠃⢋⠑⠛⢚⡛⠛⠃⠸⣸⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣶⣿⡀⢀⡍⢹⣿⠿⠷⣿⣿⣿⣿⣿⣿⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠘⢿⣇⡈⠁⠀⣿⣙⣛⣿⣿⡿⠿⡟⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣄⠀⠀⠁⣶⠖⢿⣾⣿⣻⡍⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣗⣤⣄⣼⣶⣸⠷⣿⣝⣀⠤⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⠀⠉⠉⣡⣶⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⡀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣛⢋⠀⠻⣾⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿⣿⡾⡆⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡏⠇⢹⢹⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⢧⠀⠈⢸⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠀⠀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠖⠒⠚⠉⠉⠉⠉⠉⠉⠉⠛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣩⠄⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⢤⡀⠀⠀⠀⠀⠀⢝⡳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠲⠄⠀⠀⠀⠀⠀⣀⣀⡇⠀⢸⣀⡀⠀⠀⠀⢀⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠬⠍⠀⠀⠀⠀⠀⣿⠈⠁⠀⠈⠀⡿⠀⠀⠀⢀⠥⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠰⡦⠀⠀⠀⠀⠀⠙⠒⡆⢠⡖⠒⠀⠀⠀⠀⠠⢅⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣂⠀⠀⠀⠀⠀⠀⠀⠷⠼⠇⠀⠀⠀⠀⠀⢲⣍⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢁⣀⣀⣀⡤⣤⠤⣤⣤⣤⣤⣀⣀⣀⣀⡀⠰⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢂⠟⢉⡤⢊⡡⠔⣡⠞⡠⠊⠀⠈⠢⣕⡠⣉⠛⢍⡓⠲⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⣫⣖⡯⠒⣩⠔⣊⠔⠈⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣍⠲⣜⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⣡⠔⣋⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢊⡤⠊⠀⠠⠤⠤⠄⣀⡀⣀⠀⢀⣀⢀⣀⣀⣀⣀⠀⠘⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⡏⠉⠡⠈⠀⠀⠀⠀⠢⢀⠶⠓⢶⡄⠄⠀⠀⠄⠐⣌⣭⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⣿⡀⡘⠀⠀⠀⠀⠀⢀⣿⠀⠀⠈⣧⡂⠀⠀⠀⢀⢿⣿⡿⢁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣻⡅⠈⠛⠤⠤⠭⠭⠖⠃⠀⠀⠀⠈⠓⠤⠤⠬⠄⠊⣿⣧⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡄⠀⠀⠀⠀⠀⠀⢀⣠⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡿⠉⠓⠦⠤⠤⠤⠖⠋⢩⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⡶⠲⡖⡺⡻⠋⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⡝⠲⠶⡶⠶⠶⡶⡤⣤⣄⣀
⠀⠀⠀⠀⢀⣤⡖⢿⠋⠁⠂⠈⠀⠀⠀⢀⡞⠁⠀⢸⢣⣄⠀⠀⠀⠀⠀⠀⠀⢈⣿⡇⠘⢦⠀⠀⠀⠀⠀⠀⠳⠹⠹⣻⠳⣤⡀
⠀⠀⠀⡴⢏⣼⠁⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⢸⠀⠋⠹⠓⠒⠶⠒⠊⡉⠉⢰⠃⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠘⡻⣦⡀
⠀⠀⡞⢉⠢⠉⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⡇⠀⡑⢎⡡⠀⣀⡶⠡⠀⡼⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣩⣷⡀
⠀⡸⢍⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣶⣤⣄⡀⠀⣇⠀⢌⠣⠁⣐⣾⠃⠀⢠⠇⠀⠀⢀⣤⡖⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡫⢝⣿
⢠⡓⠢⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡾⠃⠀⢸⠀⠀⠀⠰⣼⢃⠂⡰⡞⠀⣠⡶⠋⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠽⣻⣇
⣸⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠈⡆⠀⠀⣹⢃⠢⡰⣹⠃⠀⠓⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢊⡥⣿
⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⢱⠀⠠⠃⠂⣰⢣⡏⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⣒⣿
⡇⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢟⣦⡀⠀⠀⠈⡆⠀⠀⢰⣿⣾⠀⠀⠀⠀⠀⢀⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡊⢽
⣇⡁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢳⣄⠀⠀⢱⡎⢀⣷⣹⠇⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡊⣹
⡧⠤⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⢷⣄⠀⣇⢸⢣⡟⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢻
⣏⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⠸⣧⡿⠀⠀⢀⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴
⣃⣀⣀⣀⣀⣀⣀⣀⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣟⣓⣒⣒⣚⣒⣒⣖⣒⣒⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤
⣿⣿⡿⠋⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⣀⡉⢛⣤⡽⠭⣍⠛⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣳⠘⢋⢶⣶⣶⡆⠀⠹⢯⠁⢹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠙⠛⣻⠀⠘⠘⠛⠋⠀⠀⠀⢣⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢠⣉⣒⣀⣤⣴⡆⠀⠀⣰⣇⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡙⢦⣤⠶⠋⠀⣠⢾⣿⣿⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠤⢴⡞⠁⢸⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⣀⣤⠤⠔⠆⠳⠶⠍⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠄⠐⣶⣾⠇⠀⢈⣷⠦⣄⡀
⠀⠀⠀⠀⠀⠘⠯⣁⡀⠊⠀⠀⠀⣀⣀⡽⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢁⠄⣠⠞⣿⠏⠀⣠⡟⢹⡄⠀⠉⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠸⡋⠀⡠⠋⠢⣀⠀⠀⠀⠀⠀⣠⠊⠀⣠⢊⢴⠏⣸⣿⣤⣾⠟⠀⡿⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡀⠀⠀⠈⠑⢄⡀⡠⠞⠁⠀⡔⠁⢀⣿⠀⣿⣿⣿⠏⣤⠾⣟⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠙⠆⠀⠀⠀⡆⠀⢸⡇⢰⣿⣿⠏⡼⠡⠀⢸⡦⠀⠳⡄⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⢀⡠⠖⢃⢰⣟⡇⢸⣿⠏⣸⠇⠀⠀⣸⠁⠀⣰⠻⡄⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠤⠒⠉⠀⠀⠘⡛⠿⠃⢸⡟⠠⡿⡄⢀⣰⠏⠀⡴⠃⠀⠘⣄⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢸⡁⠀⠈⠁⠙⠋⢀⡜⠁⠀⠀⠀⡨⠒⠛⠁⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢸⢀⠀⠀⠀⠀⢠⠎⠀⠀⣀⠔⠉⠀⠀⠀⠀⠀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠉⠀⠀⠀⡰⡇⢀⡴⠛⡁⠀⠀⠀⣀⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠦⠁⠀⠀⠀⠀⡘⠉⠣⣀⣑⠶⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⡆⠀⠀⠀⠀⠀⢃⢰⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⡼⢇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠈⢢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⣦⣄⣀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢷⡶⢶⣶⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠛⠛⠛⠻⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡟⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⣀⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠿⠿⠟⣋⣴⣿⣷⡝⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣰⣿⣿⣿⣿⣿⣿⣿⡄⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢻⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠈⢻⣿⣿⣿⣿⣿⠏⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⣍⣛⣛⣛⣅⢀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠛⢛⣘⣿⣿⣿⣿⣏⣘⠛⢉⡁⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⣾⣎⠻⣿⣿⣿⣿⡿⣫⣾⡜⣿⣷⣆⢸⡏⣭⣽⣿⣭⣭⣭⣭⣭⣭⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⢈⣽⣷⡜⢿⣿⢏⣴⣿⡍⢥⣾⣿⣿⢸⣧⡭⢭⣭⣭⣭⣭⣭⣭⣭⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣸⣿⣦⣽⣿⢿⣦⢱⡿⠟⣫⣵⣾⣇⢻⣿⢸⡏⣾⣷⢹⣿⠖⠶⠒⠶⠶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣷⣤⢠⣶⣿⣿⣿⣿⣿⢸⣿⢸⣿⣦⣴⣿⣿⣭⣭⣭⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡿⢸⣿⢸⡿⠿⠶⠶⠶⠶⠶⠶⠶⠶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡘⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⢃⣿⣿⢸⣟⣋⣛⣛⣋⣿⣿⣟⣹⣛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⢸⣿⣿⢸⡷⠶⠶⠶⠶⣶⣶⣶⣶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⢸⣿⣿⢸⣟⠛⢛⣿⣟⣛⣛⣿⡇⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠈⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣤⡄⠀⠀⠀⠀⠀⠀⣠⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⣾⠿⠿⠿⠆⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠸⣿⣦⣀⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠈⠙⠻⣿⡆⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢰⣤⣤⣤⣿⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠉⢹⠏⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠟⠁⠀⠀⠀⣸⢫⣶⣶⡝⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣶⣶⣶⣶⣾⣿⠿⠃⠀⠀⠀⠀⠀⠹⣎⡻⢟⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣄⣠⣤⡴⠞⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢀⡠⠒
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠏⠉⢛⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⠿⠿⢿⠿⠿⣿⣿⣿⡿⠿⣿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⢦⣡⣌⣛⣈⣛⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣭⣯⣯⣭⣭⣽⣯⣽⣽⣭⣿⣭⣭⣽⣯⣭⣭⣭⣿⣭⣭⣭⣭⣭⣿⣿⣿⣿⣯⣯⣽⣯⣭

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣷⣤⣀
⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣾⣿⠿⠛⠉⠙⠛⠿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⢸⣿⡟⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⢻⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣄⡀⢀⣠⣶⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⡿⠟⠋⠁

⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣟⣿⣿⣟⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⢿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠻⠿⢿⡿⠋⠁⠈⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⠀⠔⣛⢻⣶⡄⢰⣿⣛⣳⠀⢸⠗⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠉⠙⠉⠹⠃⠈⠉⠉⠀⠀⣸⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠂⠰⠋⡈⠙⢊⠁⠈⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠤⠤⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⡀⠀⠀⠀⣠⠋⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠻⣧⡽⠿⠶⠒⠂⠁⡀⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠸⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠻⢀⡿⠈⢇⠀⠀⢠⣿⣿⣷⡄⠀⠀⠀⢸⠀⠀⠀⠐⠤⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠃⠸⡇⠀⠈⢆⡠⠛⣿⣿⡏⠻⣀⠆⢀⡿⠤⠼⠀⠀⠀⠀⠈⠐⠒⠒⢄
⠀⠀⠀⠀⢀⠐⠉⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠈⢣⡀⣿⣿⣷⠀⡜⢀⣼⢥⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⡞⠀⠀⠀⡀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠹⣿⣿⣿⡞⢁⡟⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠘⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢘⣇⡠⡀⠀⠹⣿⡟⠀⠸⡇⠀⠀⠀⠘⠇⠀⢠⡇
⠀⠀⠀⣼⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⢴⠿⠋⠀⠘⠄⠠⡏⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⢦⣤⣤⠆⠀⠀⣷⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⢻⣲
⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸
⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢠⡀⠀⣠⡦⣼⢧⡇
⠀⠀⠀⢢⠹⡆⠀⠀⠀⠀⠀⠈⣵⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⡶⡣⠊⡰⡀⠀⠀⠀⢀
⠀⠀⠀⠈⣇⠘⠀⠀⠀⠀⠀⠀⢹⠛⠒⠦⠤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⢀⠈⡰⠁⠀⠀⠀⠈
⠀⠀⠀⠀⢸⡄⠀⠀⠀⣀⣀⣀⣸⣀⠀⠀⠀⠀⢸⠋⠁⠀⠀⠀⠁⠀⠒⠒⠶⠏⠀⠀⠈⠈⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⣈⣭⣿⡶⢶⣤⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠤⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠂⠉⠁⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢢⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠈⢷⡜⣤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣜⡇⠀⢰⣀⡄⠀⠀⠀⡴⢆⢸⣾⡇⢶⣰⠀⠀⢻⣧⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣩⣶⣿⠁⠀⠀⠀⠇⠀⠀⠚⠁⠈⠃⠿⠃⠈⣿⠀⠀⠈⣿⣿⡱⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢊⣽⣾⠃⠁⠀⣴⣶⣦⡀⢸⣷⠀⢺⡆⣄⣤⣤⠉⠀⠀⢀⠛⢛⢷⣎⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠰⣮⣿⣿⠀⠀⠀⢿⡀⠸⣧⢸⣿⠀⣿⡇⢙⣿⠱⣆⠀⠠⠁⠀⠀⣼⣿⣦⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢩⣴⡟⠉⠨⡝⡀⠀⠃⡘⠧⣼⠟⠘⢿⣴⡿⠃⠀⣿⠀⠟⠀⠀⠀⠀⢰⣿⣿⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⣿⣿⠌⠙⠁⢁⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠛⢻⣷⡞⢁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⢁⢀⣵⡟⠀⠀⠀⠀⠀⠐⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢷⣛⡤
⠀⠀⠀⠀⠀⠀⠄⠃⠄⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠔⠁⠀⠐⠀⠁⠄⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣨⡕⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⣀⠂⠀⠀⠀⠀⢢⠁⢀⠀⠀⠐⠂
⠀⠀⠀⠐⠂⠤⢺⣿⣿⣧⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠇⠀⠀⠀⠀⣡⠛⡀
⠀⠀⠀⠀⠀⠈⢫⡽⣿⣿⢀⠀⠀⠀⠀⣿⣿⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠃⠈⠀⠀⠀⠀⣾⣣⡀
⠀⠀⠀⠀⠀⠈⢠⠻⣿⡇⠆⠀⠇⠀⠁⣿⣿⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⢸⡿⠟⠈⠀⠀⠐⢄⠀⠘⡘⣧⠐
⠀⠀⠀⠀⠀⠀⡄⢻⣽⠇⠁⠀⡄⠀⠀⣿⣿⣧⠂⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠨⠀⠀⠀⢀⣤⣶⣿⠆⠀⠀⢹⣧⠃⡀
⠀⠀⠀⠀⠀⠀⠐⢫⣿⠀⠀⢠⡧⡀⠀⣾⣿⣿⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠂⠠⣠⣴⣾⣿⣿⡿⠁⠀⠀⣀⣾⡜⢂
⠀⠀⠀⠀⠀⠀⠘⣨⠿⠓⠠⢸⡇⠀⢀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠋⠄⠄⣤⣾⣿⣷⠛⡀
⠀⠀⠀⠀⠀⠀⣤⣬⠋⣤⢦⡌⣣⣴⣈⣭⣭⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢩⣭⣍⢠⣴⣤⣴⣭⣭⠈⠛
⠀⠀⠀⠀⠀⠀⡿⣾⡅⣿⢾⡇⣿⢹⡏⠉⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⢸⡷⣿⠁⣿⠇⠈
⠀⠀⠀⠀⠀⠀⠛⠛⠁⠛⠚⢃⠙⣛⣣⣄⣛⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⢃⣲⣜⠛⡋⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠛⡉⠷⣛⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢏⠳⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⣈⠙⠂
⠀⠀⠀⠀⠀⢀⢀⡄⡀⢀⠀⢠⣴
⠀⠀⠀⠀⠠⠞⣡⠒⣁⣸⠀⠈⠙⠀⢀
⠀⠀⠀⠀⠴⢞⣡⣿⡏⠄⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠩⠶⢻⠏⠐⠢⠀⢀⣀⡠⠆
⠀⠀⠀⠀⠀⠀⢐⣤⣀⣀⠀⠀⠈⢻⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⣦⡈⠀⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⡀⣸⣿⣿⣿⣯⣿⣿⣿⣿⣦⡀⠉⠻⣿⣷⣶⣦⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣇⠁⢀⣤⣤⠰⢄
⠀⠀⠀⣻⣾⣿⣟⡿⣿⣿⡿⣿⣿⣿⣿⣿⣦⡀⠀⣉⣿⣿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⢉⣿⡭⢍⢅
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣾⣿⣿⣷⣿⣿⣿⣿⣿⠿⠟⠛⠛⠉⠁⠀⠀⠀⠉⠚⠛
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠁
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠛⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣙
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣰⣿⣿⣿⣿⣿⣿⣷⣾⣮⣝⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⡇
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣙⣻⡿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣳⣶⣿⡟⡻⠛⠋⠉⡁
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⢿⣿⣇⡟⠀⢀⣐⢬⠟
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣾⣷⣭⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⣿⡿⢻⣿⣿⣿⠀⠀⠀⠀⠒⠲⠀⠀⠀⠉
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠧⠭⡽⣼⠿⠿⣭⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠
⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣆⣤⣶⣱⡶⢲⣾⠉⣿⠀⠀⠀⠀⠀⠀⣠⠖
⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⣿⣿⣿⡏⣼⣿⡇⠙⣶⡶⠆⡀⠖⢋⡤⠲⠙
⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠉⠉⠉⠁⠉⠋⠙⡁⢻⣿⠇⠀⣨⡴⠚⣡⣶⡟⡧
⠀⣴⣿⣿⣿⣿⣿⣿⠿⠶⡷⢤⣖⣀⠐⡀⠀⢠⡴⠚⠁⠸⣎⡴⠚⣡⣴⣿⠛⠛⠁⠷
⢰⡿⣿⠿⠟⠛⠛⢿⣧⣄⠀⣄⢰⣀⠈⡀⣷⣐⠔⢒⡽⠎⣩⣴⣾⣿⢿⡋⢄⡄
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠚⠛⠛⠓⠉⠁⠖⢉⡴⢚⠋⠁⡅⠙⠆⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⠒⠚⠉⠛⠛⠛⠓⠒⠒⠢⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⡄⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢀⡀⣸⠇⠀⢸⣆⣀⡀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠁⢸⡇⠀⢸⠋⠉⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⣀⣀⣤⣬⠿⠷⣯⣄⡀⢀⣀⣀⡀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠖⠋⣉⡥⠀⠀⠀⠀⠀⠀⢹⡏⠁⠈⠉⠻⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⢀⡤⠚⢁⣀⠀⣠⠄⠀⠀⡼⢸⡇⠰⡄⢠⣄⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢹⢣⠾⠚⢋⣡⡽⠞⢁⣤⢠⠞⣱⠏⢷⡀⠻⣦⠘⣆⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠞⢋⣡⡴⠞⣹⣃⡤⠞⠁⠀⠈⢷⡄⠙⠀⠘⠆⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡶⠋⣡⠤⠴⠿⠉⠀⠀⠀⢀⣠⢤⣝⢦⡘⣧⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣴⡏⠀⠀⠀⣰⣦⠀⠀⠀⠀⠀⠀⢰⣶⠈⠀⠙⣿⡟⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⢹⡇⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⣿⠙⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠠⣷⠀⠀⠀⠀⠀⠀⠀⠀⢐⡆⠀⠀⠀⠀⠀⢸⣧⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠾⣆⠀⠀⠀⢀⡀⠀⠀⠉⠀⢀⠀⠀⠀⠀⡿⠷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠈⠳⣦⣤⣤⠶⠋⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠁⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣦⣤⣤⣤⣤⣴⣾⠋⡾⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢻⡇⠀⠀⠀⠀⣸⡟⣦⡇⠀⣶⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠀⢸⡇⠀⠀⠀⠀⢻⠇⠈⠻⣦⡘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⣿⠁⠀⠀⢘⣇⠀⠀⠀⠀⣿⡀⠀⠀⠘⣟⠻⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⣙⠛⢶⡶⠋⢻⡶⣄⣀⢴⡏⠙⢷⣤⠶⠛⠀⠀⠀⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡀⠀⠀⠀⣾⠁⠀⠈⣷⢾⡵⣾⠇⠀⢀⡽⠃⠀⠀⢀⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠈⠻⢦⡀⠘⣷⢠⡏⢀⡴⠋⣀⣀⣀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡏⠀⠀⠀⠀⠀⠈⠹⣦⡸⣏⡴⢯⡤⠾⠧⢶⣿⢦⡼⣧⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⠀⢸⣷⣿⠂⠿⣟⢸⡇⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡶⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣿⠀⢸⣧⣭⣤⣬⣭⠼⠇⣸⢤⡤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠘⣧⡀⠀⠀⠀⠀⠀⠀⠈⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⢠⡄⢿⠀⠀⠀⠀⠀⠀⠠⣾⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠁⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⠃⣿⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠼⠧⠤⠤⠤⠤⠤⠤⠤⠿⠤⠤⠤⠶⠶⠶⠶⠿⠶⠶⠶⠾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠈⠻⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠈⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣠⣤⡄⠀⠀⠀⣤⣄⠀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⣠⣦⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠙⠛⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠙⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣤⣶⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⢀⣴⡆⢠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁⠀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⠹⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠻⢷⣤⣀⣀⠀⠀⢀⣀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⠀⠀⠀⠀⠈⠙⠛⢻⣶⠊⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡼⠛⠁⢸⣿⣧⣤⣤⣀⠀⠀⠀⠀⠀⣸⡿⣧⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡄⣿⣿⡟⠀⠀⠀⢸⣿⠋⠈⠉⠉⠀⠠⣶⣶⣾⣿⠄⠈⠻⣿⣿⡷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠸⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠉⠀⣴⡆⠀⠀⠀⣿⣿⡇⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⣿⣿⡇⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠋⠀⠀⠀⠈⢃⣄⠀⠀⣿⣿⡇⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⣿⣿⠀⠀⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠟⠀⠀⠀⠀⠀⠸⣿⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⣿⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⣿⣿⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⣿⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣠⣴⣿⣿⣷⣦⡀⠀⠹⠃
⠀⠀⠀⠀⠀⠐⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠟⢿⣿⠙⠻⣿⣦⡀
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠈⠀⠀⠹⣿⣿⣄⣀⡀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⣤⣾⣿⣦⣤⣾⣿⡇⢸
⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⣼⣟⠛⠛⠉⢱⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠛⠉⠛⠛⠉⠙⠻⣿⣿⣿
⠀⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⣄⠈⠻⠿⠀⠘⠃⣨⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠘⣿⣀⣴⡄⠀⠀⠀⠻⣿⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⠇⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃⠀⠀⠀⠀⠿⣏⣀⣴⣦⣄⣼⣿⡇⠀⠀⠀⠀⠀⣰⣿⡿⠇⠀⠀⣾⡀⠀⠀⠀⠀⢠⣿⣿
⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⣿⡟⠈⢻⡇⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⢸⣿⡇⠀⠀⠀⠀⢿⣿⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢩⣿
⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⣶⡇⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣿⡇⠀⠀⠀⠀⠀⢿⣿⣦⣀⡀⣿⣿⡄⠀⠀⢸⣆⣠⣿⣿
⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠀⠀⣴⣾⣿⡿⠛⠃
⣿⡃⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⡇⠀⠀⠙⣿⡿
⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⡇
⠿⠁⠀⠀⠀⠀⠀⠀⠀⠙⣀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆⠀⠀⠀⢸⠃
⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠿⣇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸
⠀⠘⢿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⡄
⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠉⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠙⣀⡀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⠀⣼⣇
⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⢤⣾⣿⠀⠀⠀⠀⢻⡟
⠀⠀⠀⠀⠀⢁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣆⣀⣴⣶⡎⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⠀⣴⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⣶⣾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢠⣤⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣶⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠿⠶⠂⠀⠒⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠛⠓⠀⠀⠀⣀⣤⣤⣤⣤⣄⣀⣀⣀⡀⠀⠸⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠟⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⡸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⣸⣿⠀⠈⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢠⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣸⡆⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣄⠀⠀⠀⠀⢻⡇⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⢸⡇⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠈⠃⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⢻⣄⠀⠀⠀⢸⣷⣤⡀⠀⠀⠈⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⠀⠀⠀⠙⢷⣄⠀⠸⠟⠉⠀⣶⣶⡄⠀⠀⢙⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟⠂⣀⣀⠀⠀⠈⢻⣷⣶⡾⠟⠁⢀⣴⣶⣶⡾⠟⠋⠘⠿⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣤⣴⠛⠛⢀⣤⣤⣴⣿⠟⠁⣤⡄⠈⠉⠉⠉⠁⠀⣤⣤⣤⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠐⠿⠟⠛⠛⠀⠀⠀⠈⣱⡄⠀⠀⠀⠀⠀⠈⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉⠻⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣇⣀⣠⣀⣀⣀⣀⣀⣀⣠⣴⣤⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣠⡀⣄⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠋⠃⠛⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠚⢍⣻⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠂⠺⣧⣱⠴⡷⡀⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠋⠀⢐⣀⣀⣈⣁⣀⠈⠪⡢⡀⠀⠈⠓⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⣏⣉⣉⣉⣉⣩⠆⠀⠈⠪⠇⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣭⣽⡆⠀⠀⠀⠀⠀⠀⠀⡖⡴⠶⠶⡔⡆⣀⣶⣀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢧⠀⣀⠟⠀⠀⠀⠀⠀⠀⠀⡇⢧⣤⣤⠇⡦⣬⣤⣏⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⢯⠢⣉⡗⠢⣠⣄⠀⠀⠀⢸⠠⠮⠼⠿⠄⠂⠿⠤⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡄⠑⢄⠹⡔⠓⠚⠒⡄⠀⠸⣶⣶⣶⣶⣶⣷⣶⠶⡶⡶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢰⠀⠈⣰⣹⡄⠀⣼⣹⡱⠤⡟⡖⠛⠾⠷⠿⡶⠶⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠈⠒⠁⠀⠑⠊⠀⠀⠈⠓⠒⠒⠒⠒⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠋⠉⣘⣿⣇⡀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣏⢀⣯⠵⠶⠶⢭⡅⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣼⡶⣻⠓⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣾⡴⠿⢂⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡏⠑⠳⠚⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠿⣁⣼⠁⣀⣀⡠⠾⣿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠙⣄⠀⢩⣿⣿⣿⡿⠁⠀⠈⠛⣷⢪⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⢵⡀⢈⠀⣼⣿⣿⣿⠃⠀⠀⣾⣷⣣⢀⣛⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣶⣾⣷⣸⣶⣶⣿⣿⠀⣴⣶⣯⡽⠛⠉⠉⢠⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠉⣉⣽⣿⣥⣤⣤⣤⣤⣴⣦⣌⡁⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢡⣜⣿⣿⣿⣿⣿⣿⡟⠛⠛⠉⠉⢁⠀⠀⣠⢄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡈⣉⡻⢻⡟⣿⣿⣧⣄⣀⠛⠒⠒⠾⠿⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣿⣧⣿⣽⣿⣿⣿⣷⣿⣿⣷⡖⢒⣲⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣾⠗⣿⣿⣿⡟⠺⠿⣟⣋⢤⠉⠁⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⣰⣿⠿⣿⣿⡶⣿⠿⠿⠘⠂⠨⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⠓⠂⠀⢰⠋⠀⠀⠀⠀⠳⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢽⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢹⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠛⠛⠛⠛⠁⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⢠⡄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡏⠙⠶⠶⠆⠀⠈⠉⠙⢷⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⡀⠶⠀⠀⠠⢦⠀⠀⠀⠀⠷⣀⠀⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠁⣤⡀⠀⠀⣀⡀⠀⠀⠀⠀⠙⠀⠀⣀⠋
⠀⠀⠀⠀⠀⠀⠀⠐⣯⠀⠈⢻⠁⠀⠀⠛⠁⠀⠀⠀⠀⢀⣤⣼⢿⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠂⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⣽⠖⠘⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢤⣤⣤⡤⣤⣤⣤⡤⠀⠛⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣼⡄⠀⠀⠀⠀⢀⣾⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢶⣿⡏⢻⡷⠴⠶⠀⠉⠙⣿⠀⠀⠙⣷⢴⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⢠⣿⠇⢸⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⣿⡟⠉⠲⢦⡀
⠀⠀⠀⠀⠀⠀⢀⣜⠁⠀⠀⢸⣿⠀⠈⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⣿⡇⠀⠀⢈⡿⢦⡀
⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⡇⠀⣿⡇⠀⢠⡞⠁⠀⠻⡆
⠀⠀⠀⠀⠀⡄⠻⠇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠇⠀⠘⠃⠀⠀⠀⠸⠆
⠀⠀⠀⠀⡀⠋⠀⠀⢀⣴⣿⠿⠿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⠀⢰⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⠇⠀⠀⠀⢸⣿⣵⣶⡦⣼⣿⣇⠀⠀⠀⠀⣴⠀⠀⠀⡟⠀⣸⡿⠀⢸⡇⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⡿⠀⠀⡇⠀⣿⡏⠀⢸⡁⢸⡟⣿⡆⠀⠀⠀⣿⠀⢀⣾⠿⠿⢿⣇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠃⠀⠀⡇⢸⣿⠁⠀⢸⡇⢸⠀⢻⣇⠀⠀⠀⣿⠀⣾⠄⠁⠀⠀⢼⡇⠀⢰⡆⠀⠀⠀⠀⠀⠀⢻
⠀⠀⣷⠀⠀⠰⣇⢸⡿⠀⠀⠘⠁⢸⠃⠘⣿⡀⠀⠀⣿⠀⣸⡶⠟⠈⢻⡆⠁⠀⠀⢱⡄⠀⠀⠀⠀⠀⠀⣶
⠀⣠⠀⠀⠀⢠⡟⢻⣇⠀⠀⠀⠀⣸⠀⠀⣿⡇⠀⠀⣿⠀⣿⠀⠀⠁⠀⠀⠀⠀⠀⠈⢡⡀⠀⠀⠀⠀⠀⠙
⠀⡿⠀⠀⠀⠀⡇⠈⠻⣶⣤⡀⢰⣿⣶⣾⠏⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⣷
⣄⠀⠀⠀⠀⠀⡇⠀⠀⠈⠛⠁⣼⡟⠉⠁⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠙
⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡆
⠟⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠇
⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⣤⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢀⠟
⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢀⠛
⠀⠀⠀⠀⢷⡀⠀⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⢠⡜
⠀⠀⠀⠀⠀⠳⣴⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⡀⠀⠀⢀⣰⠏
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⢃⡈⠓⣦⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⣴⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠓⠀⠼⠀⠀⠀⠀⠀⣸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠛⠛⠛⠛⠛⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠑⠲⠒⠒⠶⠒⢿⠀⠀⠀⠀⣨⠿⠖⠆⣀⣀⣀⡀⢴⡞⠁
⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⣴⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⡴⠂⠁⠀⠀⠀⠀⠀⠀⣠⡖⠛⠋
⠀⠀⠀⠀⠉⠦⠤⠀⠀⢤⡤⠞⠋⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⣤⠞⠁
⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣿⣿⣿⣿⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⡿⠟⠻⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠚⠛⠛⠛⠛⠷⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⣄⣀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⣠⠾⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⣠⣾⣿⣷⣮⣿⣿⣿⣿⣿⣿⠿⠿⠟⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣷⣶⣶⣶⣾⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡧⢻⡏⠀⠀⠐⠛⠲⠄⠀⠀⠀⠀⠀⠀⣽⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣧⠘⡄⠀⠀⠙⠳⣦⣦⠀⠀⣀⠛⠁⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠛⢲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣻⣿⠻⣇⠀⠀⠀⠀⠀⠐⠀⠴⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⣴⠏⠀⣿⣆⠀⠀⠈⠛⠓⠶⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⢻⡄⠀⠸⣿⣷⣄⡀⠉⢉⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠶⠒⠚⠛⠋⠁⣿⠀⠀⠀⢻⡄⠀⠙⢿⣿⣿⠇⢠⡀
⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢿⡀⠀⠀⢈⡏⠀⠘⣿⠶⣤⣄⡀
⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣷⡀⢀⡞⠀⠀⠀⡟⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢂⡟⢰⡏⠉⠙⠛⠙⠛⠛⠛⠛⠛⠓⠓⠓⠓⠚⠛⠛⢻⣿
⠀⠀⠀⠀⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣅⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⣰⠃⣿⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡀
⠀⠀⢠⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⡖⠒⠂⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢘⣷
⠀⠀⢸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣿⣡⣿⡃
⠀⠀⣼⣋⣠⠤⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄⠀⠀⣼⠃⠀⠀⠀⠀⣠⠞⣿⠀⠀⠀⢀⡞⡿⢀⣴⡆⣼⠟⠁⢀⣿⣄
⠀⢠⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣀⠁⠀⣼⠃⠀⠀⠀⠀⡴⠃⢰⠇⠀⠀⣠⡟⢰⡧⠊⡼⢿⣏⡴⠚⣁⣀⡙⢷⡄
⠀⡟⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣿⣷⣴⠏⠀⠀⠀⠀⣼⠁⠀⠘⠷⠒⠛⠉⠠⢏⡤⠊⣠⠟⢿⣿⣛⣉⣉⠀⣀⠙⣦⡀
⣠⠇⠀⠀⠀⠀⣀⠴⣚⡩⠉⠻⢿⣿⣿⣿⠟⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠴⠿⢋⣠⣾⡏⠀⠀⣿⠉⠉⠙⢧⣌⡀⠘⢷⡀
⣿⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⢛⠙⠛⠓⠒⣤⣤⠾⠃⠀⠀⠀⠀⠀⢀⣀⣤⠶⣶⠋⠁⠀⢰⠇⠀⠀⠀⠀⠈⠛⠀⠈⢷
⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⣰⠃⠀⠀⠀⢀⣠⡤⠖⠊⠉⠀⠀⣸⣏⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠙⢷⣤⣀⡀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢰⣯⡴⠶⠒⠛⠛⠳⠶⣶⠶⠶⠶⠶⠛⠉⣿⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠉⠛⠛⠛⠿⠶⣦⣄⣀⣾⠀⠀⠀⠀⣾⢣⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⠹⣆⡀⠀⣀⣀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⠷⠶⣤⣄⣤⠟⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⣠⠞⣿⠀⠀⠈⠛⠋⠉⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣤⣤⣬⣁⡀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⣠⠞⠁⣀⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠠⣶⣬⣍⡛⠓⠀⠀⠀⠀⠀⠀⣱⠀⠀⠁⠀⠈⠁⠀⠘⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣲⣿⣿⣿⣿⣖⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⣠⣫⡾⠋⠀⢫⡝⠀⠙⢷⣝⣄⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⣶⡾⠛⠋⠙⠛⢷⡤⡪⢟⡻⢕⢤⡾⠛⠋⠙⠛⢷⣶⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⢿⣷⡔⡈⢀⢔⡟⣠⣴⣦⣄⢻⡦⡀⢁⢠⣾⡿⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⠀⡐⠩⣱⣲⢿⣾⢏⡥⣖⣲⢬⡹⣷⠿⣖⣎⠍⢐⠀⢹⣄
⠀⠀⠀⠀⠀⠀⠀⡴⣫⡾⢿⣄⢀⢔⣻⣷⣟⠽⠚⢺⡕⢪⡓⠓⢯⣻⣼⣿⣢⡀⣠⡿⢷⣕⢦
⠀⠀⠀⠀⠀⠀⣼⣾⠋⠀⡠⡫⠛⣩⠏⡵⣺⣀⠠⣢⣇⣸⣴⠀⣠⣗⢮⠹⣍⠛⢝⢄⠀⠙⣷⢧
⠀⠀⠀⠶⠶⢾⣿⣿⣏⠶⣿⡱⠰⣿⢸⢹⡱⣉⣉⣹⣿⣿⣏⣁⣉⢎⡏⡇⣿⠆⢎⣿⠶⣹⣿⣿⠷⠶⠶
⠀⠀⠀⠀⠀⠀⢻⢿⣄⠀⠑⣕⣤⣙⣆⡳⢽⠊⠐⠛⡏⢹⠻⠀⠉⡫⢞⣰⣋⣤⣪⠎⠀⣠⡿⡟
⠀⠀⠀⠀⠀⠀⠀⠳⣝⢷⣾⠋⠈⠫⣽⡟⣯⣣⢤⢼⡣⢜⡧⡠⣔⣽⢻⣯⠝⠁⠙⣷⡾⣫⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣏⠀⠅⣐⡹⢽⣶⣿⣎⡓⠿⠭⢚⣱⣿⣶⠯⢇⣂⠈⠀⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣾⡿⢃⢁⠈⠪⣧⠙⠻⠟⠋⣼⠕⠁⡈⠘⢿⣷⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⠿⢷⣤⣄⣀⣤⡾⠳⢕⣮⣵⡪⠞⢷⣤⣀⣠⣤⡾⠿⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠙⢝⢷⣄⠀⣜⣣⠀⣠⡾⣻⠋⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠽⣻⣿⣿⣿⠿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠐⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢸⡏⡉⠉⠉⡉⢻⡆⠀⠀⠀⢿⠀⠀⠀⢈⠙⡇⠀⠀⠀⠀⢾⢁⠉⠉⠛⠛⡇⠀⠀⠀⢸⠋⡅⠀⠀⠉⢹
⠀⠀⠀⠀⠀⣼⣅⠀⠠⠄⢀⣬⡇⠀⠀⠀⠈⠀⠀⠤⠀⠀⠁⠀⠀⠀⠀⠈⠈⣿⣿⣿⠀⠁⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠹⠿⠿⠂⠘⢿⠿⠃⠀⠀⠀⠀⠀⠠⣀⠄⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⢸⢘⡛


⠀⠀⠀⠀⠀⢀⣴⣿⠟⢿⣿⣦⡀⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀
⠀⠀⠀⠀⠀⠿⠛⣋⣤⣌⣙⠻⠿⠀⠀⠀⢰⣿⣿⣿⡿⢿⡄⠀⠀⠀⠀⢀⣠⠤⠤⠤⢄⣀⠀⠀⠀⢰⡿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠸⡇⡄⠀⠀⡀⢸⠀⠀⠀⠀⠚⢉⠉⠀⢀⠀⠇⠀⠀⠀⠀⢸⠀⡀⠀⠀⡀⡿⠀⠀⠀⢸⠀⡀⠈⠉⡍⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⢠⣶⣄⡀⠂⠀⣤⣶⠀⠀⠀⠀⢀⡀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠐⢠⠀⠄⠀⠀⠀⠀⠀⠙⢿⠋⠀⢑⡾⠋⠀⠀⠀⠀⠀⢁⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠸⣷⣶⣾⠇⠀⠀⠀⠀⠀⠸⠀⣦⣤⣴⡇

⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⣀⣤⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣦⣤⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⢿⡆⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢠⣿⣿⣿⣿⢿⣷
⠀⠀⠀⠀⠀⠘⡉⡉⠉⠉⡉⢹⠁⠀⠀⠀⢿⢉⠉⠁⢀⡀⡗⠀⠀⠀⠒⢲⠀⠀⠉⠀⡈⡿⠒⠀⠀⠸⠋⡉⠉⠀⡀⢹
⠀⠀⠀⠀⠀⠀⠡⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⣠⣸⣧⣄⠀⠀⠀⠀⠀⠀⠀⠤⠀⡆⠀⠀⠀⠀⠀⠐⠂⠀⣄⠀⣠⠀⠀⠀⠀⠀⠀⠀⠘⠃⠉
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣤⣿⡆⠀⠀⠀⠀⠀⠈⠇⣶⠻⠛⠟⢺⠀⠀⠀⠀⠀⢸⡀⠐⠀⢸⡄

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣾⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆⢠⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠋⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡋⠈⣿⡆⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣤⣾⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⡿⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⠀⣾⣿⣿⣿⣿⣿⣤⣤⣄⣀⣀⣉⣻⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⣠⣀⣀⠀⠀⣀⣀⡀⠀⣀⣀⣀⡀⣠⣀⣀⡀⢀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⣀⣀⣀⡀
⣿⣿⣿⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⣿⣿⣿⡇⢰⣿⣿⡇⠀⣿⣿⣿⠁⢻⣿⣿⡇⣼⣿⣿⡏⠀⣿⣿⣿⣿⡿⣿⠇⠈⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⠻⣿⣿⣇⠀⢸⣿⣿⣇⣾⣿⣿⣿⣴⣿⣿⡏⠀⠀⢻⣿⣿⣿⣿⡏⠀⠀⣿⣿⣿⣤⣤⣄⠀⠀⣿⣿⣯⣀⣸⣿⣿
⣿⣿⣿⡀⠀⠀⠀⢰⣿⣿⣿⣤⣿⣿⣿⡆⠈⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡇⠀⠀⠘⣿⣿⣿⡿⠀⠀⠀⣿⣿⣿⠛⠛⠛⠀⠈⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣿⣷⣾⣷⠀⣿⣿⣿⠟⠛⢿⣿⣿⣧⠀⢿⣿⣿⣿⡏⢸⣿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⡇⠀⠀⢠⣿⣿⣿⣷⣶⣾⡆⠀⣿⣿⣏⠈⣿⣿⣷
⠉⠉⠉⠉⠉⠉⠀⠉⠙⠋⠀⠀⠀⠉⠉⠉⠀⠈⠉⠉⠉⠁⠀⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠋⠁⠀⠀⠀⠉⠉⠋⠉⠉⠉⠁⠀⠉⠉⠁⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠻⠿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⠟⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⠀⠀⠀⠀⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⠀⠀⠀⠈⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⠀⠀⠀⠀⠙⠿⠿⠿⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣽⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⡿⠿⢿⠿⠿⠿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⣠⠈⢧⣀⣴⣊⡀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⣠⠏⠉⠯⣉⠉⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀⢤⡤⠀⠹⢄⡉⡇
⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠉⠙⠒⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠶⠌⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠓⣖⠁⠀⢀⣶⢏⠷⡄
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠠⡰⠖⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣿⠔⠦⢤⣶⡿⠛⠁⡰⠉⠐⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠐⢇⠀⠀⠘⡆⠀⠀⠀⠀⠀⣠⢶⠋⡇⠀⠸⡇⢀⣾⠛⠒⠒⠉⡠⠀⢀⠤⠈⠳⣄
⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⢀⡀⠀⡾⣀⣀⣠⠶⠤⠚⠹⡼⢸⠇⠀⠀⣿⣼⠃⠀⠀⠀⡔⠠⣞⣡⡴⡶⡶⣿⡆
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢹⠁⠈⠉⠀⠘⢉⣁⣀⣀⣀⣀⣙⡏⠀⠀⢰⢿⡿⠂⠀⠀⡐⡴⢰⠋⢀⠀⠀⢀⢀⡿
⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢸⠛⠲⢦⣀⣀⣤⠴⠶⠬⣭⣉⢻⡇⢀⣰⢏⡾⠃⠀⠀⠌⡜⢠⡏⠔⢁⡠⠔⠉⣼⡇
⠀⠀⢠⡏⠀⠀⠀⢷⠀⠀⢸⣦⣤⢾⡍⣉⡉⠑⣞⠛⠉⠉⢻⠛⠿⠟⡻⠁⠀⠀⢀⠞⡴⠉⠁⠉⠀⠀⢀⣼⢻⡇
⠀⠀⣼⠀⠀⠀⠀⢸⠀⠀⢸⡏⠀⠈⠻⣶⣄⡀⠈⠒⠲⢤⣿⣀⣀⣰⠁⣀⣀⣰⠧⣼⣅⡀⠀⠀⢀⡴⣿⡧⣾⣀⡀
⠀⢠⠇⠀⠀⠀⠀⢸⡄⠀⠈⡇⠀⠀⠀⠀⠀⠙⠦⠤⢼⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⠀⠰⢋⣾⠃⣠⠇⠈⠉⠛⣦
⠀⠸⡧⠄⠀⠭⣤⠰⢇⠐⠂⠙⣄⠀⠀⠀⠀⠀⠀⠀⠈⣿⣤⠤⠤⢤⣤⣀⣀⠀⠀⠀⠀⠀⠉⣠⡾⠁⡼⠁⠀⠀⠀⢰⠇
⠀⢸⠃⠀⢒⠒⠛⣏⠈⠳⣄⠀⠈⢦⡀⠀⠀⣀⣀⡤⠾⣁⠀⢀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⡛⠋⢠⣦⣧⣄⣀⠀⢀⡞
⠀⢸⠈⠉⠉⠀⣀⠈⣆⢀⣨⠷⢦⣀⠙⠶⢋⡉⠁⠀⠠⢌⣑⡪⣭⠽⢒⣒⣒⠲⠄⠐⠊⠁⠀⣠⠎⠈⡇⠀⣹⠛⠛⠁
⠀⠀⠀⠀⠀⠉⣉⣀⣉⣐⡁⠀⠀⠈⠑⠲⠤⣼⡷⢤⠀⠀⠀⠀⣀⣀⡀⠀⠠⠤⠤⠄⢀⣀⡘⠁⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢦⣤⣄⣀⣀⢀⠀⠀⡈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠈⠀⠀⢀⡷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⡶⠚⠁⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣤⡀⠀⢀⣾⡁⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠾⠉⠙⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⢀⣀⣀⣭⣭⣭⣭⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢊⣤⣶⠾⠟⠛⠉⠉⠁⠀⠀⠉⠉⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣑⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣰⡿⠁⠀⠀⠀⠀⠀⢀⡠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠂⣼⡟⠀⠀⠀⠀⠀⢀⣴⣿⣧⣀⡇⠉⢿⢿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡄⢰⡟⠀⠀⠀⠀⠀⡠⠃⠈⠙⣿⣿⣿⣶⣜⣯⣿⣿⣿⣿⣿⣾⣄⡴⠃
⠀⠀⠀⠀⠀⠀⠰⢀⣿⠃⠀⠀⠀⠀⢰⣷⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠛⠛⠿⠿⠛⠟⠁
⠀⠀⠀⠀⠀⠀⡀⢸⡟⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠃⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⡇⠀⠀⣀⡤⢤⣤⣀⡀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠉⠛⠄⠄⢀⡀⠀⢸⣿⣿⣿⣯⡞⠀⠀⠀⠁⢀⣤⣀⠈⠁⠀⠀⣰⡫⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡏⠁⠊⠉⠀⠀⠀⠀⠀⠋⠛⠃⠀⠀⠀⠀⡛⠃⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡿⢿⠿⣧⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣇⣤⣴⣾⣷⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠠⡤⠴⠶⠶⢶⡖⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⢀⠀⠀⠀⠀⠀⠙⢶⣤⠤⣾⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⢿⣿⣿⣿⣿⣿⣿⡇⠀⠁⠢⠄⠀⠀⠀⠀⠉⠉⠁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣠⡊⣁⢀⣠⣤⣤⣿⡇⠀⡀⠀⠀⠀⠀⠐⡆⠄⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣿⣿⣿⣿⣿⣿⢿⠋
⠀⠀⠀⠀⠀⠀⢀⢀⠀⠤⠙⠛⠛⠛⠋⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠳⠐⠢⡄⡀
⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠑⡀⠀⠉⠐⠠⡀
⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠐⠀⠀⠀⠀⠀⠢
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢂⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡠⠐⠈⠀⠀⠀⠑⡀⠀⠀⠀⡀⠌⠀⠉⠐⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⡰⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣦⡆
⠀⠀⠀⠀⠀⠀⢄⣀⠠⠀⠠⢄⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⣈⣁⡀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⡀⠀⠀⠀⠀⢠⡅
⠀⠀⠀⠀⠀⠀⠐⠀⠤⠀⠠⠤⠐⡂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠉⠀⠀⠁⠸⡇
⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡂⠐⠂⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣑⠀⠀⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠂⠈⡁⠆⠀⢩⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⢆⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⣦⠆⠉
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠁⠸⠉⠁⢀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡥⠀⠀⠶⠆
⠀⠀⠀⠀⠀⠈⠀⠀⠠⠄⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠀⠀⠀⠀⠠⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠐⢘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⡀⠀⠀⢐⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠀⠀⢸
⠀⠀⠀⠀⠀⠰⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠐⠂⣀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠈⠑⠤⠀⠠⠠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡃⡤⢤⢚⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠄⢲⡃⡤⢤⢚⡖⠢⡀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠈⡃⣠⡀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠃⡈⢁⠘⠀⠀⢀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡇⠁⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠆⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠤⠅⠤⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠠⠤⠀⡀⠮⠅⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠛⠛⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡴⠡⠤⠤⠤⠄⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢉⡉⠁⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⡄⠂⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠈⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⡤⠀⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⠀⠀⢹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠶⠖⠣
⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠎⠀⢰⠐⡫⢝⠂⠖⠀⠱⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⣤⡄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘⠀⠀⠀⠄⠃⠸⢀⠁⠉⠁⠃⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠘⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⣩⣍⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⣠⣤⣬⣥⣤⣄⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠉⠉⣉⣉⠀⠀⣘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢉⡘⠃⡨⢁⠘⢃⡈⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠛⠋⠉⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠄⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢡⡧⠀⠀⣬⠄⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠐⠲⠔⠂⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠈⠀⠀⠀⠈⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣪⠕⢯⠑⢂⡔⠒⣹⠲⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡀⠀⣤⣤⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⡿⢧⡸⣤⠘⡏⢻⡟⢛⠇⢠⡼⠨⡏⠒⠦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⡦⢄⡡⢴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢸⠀⠀⡇⠳⡄⢹⣼⣿⡜⢀⡜⢀⣴⣣⡄⠀⠸⡄⠀⠀⠀⠀⠀⠀⠘⣿⢿⣿⣀⠀⠀⣨⣷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠘⠀⣰⠁⠀⠘⢦⣻⣿⡡⠋⠀⠈⠻⠿⠃⠀⠀⡇⠀⠀⠀⠀⠀⣴⣿⣹⣿⣿⣿⣷⣾⡿⠿⠿⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠭⠭⠤⠭⠤⠤⠤⠿⠿⠥⠤⠤⠤⠤⠤⠤⠤⠤⠿⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠸⠶⠀⠀⠀⠀⢀⠐⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠘⠀⠂⠀⠤⠀⠄⢀⠀⢀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⡂⢀⠀⠀⠀⠀⠀⢠⠀⡁⠀⠃⠀⠂⠠⠄⠠⠀⢀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⢀⠴⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠐⠀⠤⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡠⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⢡⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡆⠀⠀⡏⠘⣦⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠃⠀⡀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠶⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠂⠀⠀⠀⠀⢈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⢲⣤⠤
⠀⠀⠀⠀⢀⢤⡈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠢⠼⠛⠹⣄⠀⠂
⠀⠀⠀⠠⠽⠊⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠠⠤⡌⠁⠤⠄
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⠠⠤⠜⠈
⠀⠀⠀⠄⠚⠫⠃⠀⠀⠀⠀⠀⠛⠌⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠙⠐⠢⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠋⠉⣉⠭⡭⢍⡉⠉⠙⠛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⠀⠀⠀⢀⠎⡡⠼⠹⢤⡙⡆⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢸⡀⠓⢲⢰⠒⠃⡸⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡀⠀⠀⠀⠳⣄⡸⢼⣀⠴⠁⠀⠀⢠⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠈⠛⠶⠶⣦⣤⣤⣬⣭⣤⣤⠴⠶⠶⠛⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢀⣴⠞⠋⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢠⡟⠁⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⡀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠈⠻⣆⡀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢸⣿⣦⣄⡀⢀⣠⡴⣿⠁⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣤⣤⣴⡇⠀⠈⠙⠋⠁⠀⣿⣤⣤⣤⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠞⢛⣿⡉⠻⣄⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠉⢹⡟⠓⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠀⣠⡾⠋⠉⠙⢷⡜⢷⡄⠀⠀⠀⠀⣠⡞⠁⠀⠀⢸⡇⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠻⣆⠙⣦⡀⢀⡼⠋⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⢹⡆⠈⠻⠟⠁⠀⠀⠀⠀⠀⠘⣇⣠⡴⢦⣄⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡸⠇⣽⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣈⣛⢲⣆⣀⣰⡖⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣋⣁⣀⣀⣀⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡶⠞⠛⠛⠛⠛⠛⠛⠷⠶⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢷⣤⡀
⠀⠀⠀⠀⠀⠀⢸⣶⣄⡀⠀⠀⠀⠀⢀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠈⣯⠉⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⡀
⠀⠀⠀⠀⣶⢦⣤⣨⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦
⠀⠀⠀⠀⠘⣦⡈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⣠⡶⠟⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠳⢶⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⡟⠀⠀⠀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⠀⠀⠻⣇⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠀⡿⠀⠀⠘⠋⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠛⠉⠉⠙⠃⠀⠀⠀⠀⣿⠀⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⣿⠀⡷⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣠⣦⡄⠀⠀⠀⠀⠀⣿⠀⣸⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣿⣸⡇⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠸⣧⣸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠛⠋⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠈⠛⠁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣷⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⠶⠾⣋⣼⣿⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠹⣿⣭⣭⣭⣭⣤⣶⣾⣋⣸⡇⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠉⠷⣤⣤⡟⠉⣁⣠⡿⠋⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠉⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣄⠀⠀⠀⠀⠀⢠⡔⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠀⠀⠀⠀⠀⢹⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣭⣽⣟⠀⠀⠀⠀⠀⢈⣿⣭⣭⠛⢿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⣶⣿⡿⠋⢼⣿⡉⠈⠻⣦⡀⠀⠀⣠⡾⠋⠈⣹⣿⠀⠈⠻⣿⣶⣤⣠⣄⣀⡀
⠀⠀⠀⠀⠀⢀⣤⠞⠛⠛⣿⠋⠉⠉⠉⠁⢀⣼⣿⠁⠀⠘⣿⢷⡀⠀⠈⠻⣤⡼⠋⠀⠀⣰⣿⡟⠀⠀⢠⡟⢿⣯⡈⠉⠉⠉⢻⠟⠛⠛⠳⣤⡀
⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⣿⣿⡀⠀⠀⢿⡘⢷⡀⣠⡾⠛⠳⣦⡀⣼⠏⣸⠇⠀⠀⣾⠇⢸⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⡇⢸⣧⠀⠀⢸⡇⢨⡿⠋⠀⠀⠀⠈⠻⣏⢰⡏⠀⠀⢰⡏⠀⣸⣿⠇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣧
⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⠇⠈⢿⡀⠀⠀⢻⡏⠻⣤⣀⣀⣀⣠⡾⠋⣿⠁⠀⠀⣸⠇⠀⣿⡟⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⣿⣆⠀⢸⡇⠀⠀⠘⣧⠀⣼⠋⠉⠙⢻⡄⢸⠟⠀⠀⢰⡿⠀⢰⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡿⠉⢿⣦⣼⣿⠷⡶⣦⣿⣴⡿⠀⠀⠀⠸⣧⣿⠀⢶⣶⠾⠃⠀⠈⣿⣇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⣿⣶⣿⣛⡿⠷⠶⠶⠛⠛⠛⢿⣛⢯⣿⣿⠁⠀⠀⠀⠀⣿⡇⠀⠀⣿⠇⠀⠀⠀⠙⢿⣷⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣇
⠀⢀⡿⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣏⣿⡆⠀⠀⠀⢠⡿⠀⢠⡾⠃⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡀⠀⠀⣸⠇⣠⡾⠁⢀⣤⡶⠶⠶⣤⣾⡿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡇⠀⢰⡏⣠⡟⠀⣰⡟⠁⠀⠀⠀⠀⠙⣧⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣾⠃⠀⠀⠀⠀⠀⠀⠸⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⣾⣿⠏⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣿⠃⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡏⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠈⠳⣦⣄⣀⣠⣤⠾⠃⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣾⠛⠛⣷⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⢸⢻⣇⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠁⠀⠀⠀⠀⠀⠀⢀⡿⣿⣿⣤⣴⠟⠉⣳⡄⠀⠀⠀⠀⠀⠸⠷⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⢻⡄
⣿⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿⠉⠛⠛⠛⠲⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣇
⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⣿⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡆
⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇
⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⡇
⣿⣀⣀⣀⣀⣀⣀⣸⡟⠀⣿⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣸⡇⠀⢸⣇⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠛⠋⠙⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣄⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄⢴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣫⡭⢭⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣇⣿⠗
⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣰⣯⣵⠆
⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣮⠝⠛⠉⣵⣦⡓⢢
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⢀⣻⣷⣟⣀⠀⢸⣿⡏⠙⠻⢤⣄
⠀⠀⣿⣿⠟⠉⡟⠛⣿⣿⡋⡟⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣇⣤⡈⠣⠤⣤⠾⠥⢤⣴⣿⣿⡏
⠀⠀⠘⣿⣾⠈⣾⢸⠡⡀⠇⡏⠏⡞⡃⡏⠟⢄⠸⢸⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⢀⣤⣤⣶⡾⣿⠟⠇
⠀⠀⠀⠹⣿⣿⣿⣶⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⡿⣿⣿⠙⠆⠀⠀⠉⠟⠓⠶⣶⢶⠖
⠀⠀⠀⠀⠙⣿⣿⣿⣿⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡔⢨⠀⠀⢰⣄⠤⣾⣷⣾
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣽⣿⣯⣿⣿⣿⣿⣯⣿⣷⣯⣿⣿⠧⠋⢀⣤⣤⢄⣦⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⢩⣽⡿⡿⣿⣿⡇⠀⢀⣾⣿⣿⣼⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣧⣉⣶⣤⣵⡗⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⠿⠿⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣝⡅⠘⠋⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣌⠘⢗⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣟⡿⢻⢳⣶⣾⣤⠀⠏⢶⣄
⠀⠀⠀⠀⠀⠀⢠⡞⠹⢤⡈⣾⣇⣾⣿⣿⣿⣿⡇⠀⢸⣿⣷
⠀⠀⠀⠀⠀⢀⣿⣧⠀⣠⠿⣌⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢘⣿⡯⠞⠁⡜⢉⣿⣿⣿⣿⣿⣿⣇⣀⣀⣼⣿⣷⡀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⢠⡄⠊⠀⡠⢊⣥⣟⠡⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⢄⣀⡀⠀⣠⡖⠉⣤⣤⢶⠀⢤⡄
⠀⠀⠀⠀⡟⠁⡀⢊⣴⣿⣿⣁⣀⢸⣻⣿⣿⣿⣿⣿⣿⣁⣿⣿⡿⠋⠀⠟⠟⠁⠀⣤⣤⣴⣦⣾
⠀⠀⠀⠀⠂⢈⣴⣿⣿⠛⣯⡾⠛⡼⢻⣿⣿⡯⢟⣟⣋⣉⣀⣀⠀⡀⠠⢼⢗⣲⣚⣛⣛⣛⡋⠉⠃
⠀⠀⠀⠀⣠⠟⣯⣉⢋⡀⣿⣁⣀⣠⣚⣿⣯⣤⣾⣿⣤⣤⠭⠭⣽⣿⡏⠉⠉⠉⠉
⠀⠀⠀⠀⠙⠠⡛⢻⣿⢹⡏⡉⠉⣽⡏⢻⣿⣃⣿⣿⣿⣿⣷⣶⣄⠙⡇
⠀⠀⠀⠀⠀⠀⠅⠠⣿⣸⡅⠅⠰⣽⣿⣿⣨⣿⣿⡟⣿⣿⣿⣿⡿⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢹⠁⠀⠀⢽⢹⡏⠈⣿⡿⢻⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠰⣇⣿⠀⠀⠀⡄⢸⠁⠀⢼⡇⢸⣿⢸⣿⣿⡇⠸⠁
⠀⠀⠀⠀⠀⠀⠀⢰⡟⣿⠀⠀⠀⡅⠀⠀⠀⠈⠃⢸⡎⠈⡏⠃⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠘⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠇


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⣶⡤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠁⠀⠀⠀⠀⠀⠀⠈⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢁⡶⠛⠛⠛⠛⠚⠛⠉⠙⢳⣄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡾⠋⢀⣭⢭⡀⠉⠳⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢾⡋⣤⢤⡉⣻⢆⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠙⠁⠀⠙⢁⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣿⡙⠓⠲⠶⠚⠋⣹⣦⣤⣄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠋⠉⢉⡿⠁⢸⡆⠙⢳⣤⡤⣴⠞⠉⣾⠀⠙⣧⢸⡟⠓⠶⠶⠶⠶⠒⠛⠉⠙⠛⠒⠶⠶⠶⠶⠚⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠛⢦⣄⠀⢿⢦⣼⣧⣀⣾⣧⢾⡇⠀⣠⠞⢻⡇⢀⡟⠲⠶⠶⠶⠖⠛⠛⠲⠶⠶⠶⠶⢻⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠸⡇⢀⡟⠈⣷⠀⣾⠀⠀⢹⡆⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⢿⣸⠇⠀⢸⣶⠇⠀⢠⡞⠁⢸⡇⠀⡇⠀⠀⠀⣤⠶⠒⠒⠦⠀⣠⠟⠀⢀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠘⣿⠀⠀⠈⣿⠀⣠⠟⠀⠀⢸⡇⠀⡇⠀⠀⣾⠁⣀⠀⢀⣠⠟⠁⠀⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣶⠀⠀⠀⠀⠀⠀⠈⢳⡄⢻⡆⠀⢸⠇⣰⠏⠀⠀⠀⠘⣇⠀⣧⠀⠀⢿⡀⠙⠳⠞⠁⢠⡇⠀⠀⣼⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⢻⣜⣧⠀⡿⣴⠋⠀⠀⠀⠀⠀⢻⡄⠘⣆⠀⠈⠳⢦⣤⣤⠴⠋⠀⠀⣴⠃⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠻⣄⠈⠳⣄⡀⠀⠀⠀⠀⢀⣤⠞⠁⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⣤⡀⠙⠳⣄⣠⠞⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡷⣤⡈⢁⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠘⠃⠈⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⣠⣴⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣷⣿⣶⣦⣄
⠀⠀⠀⣸⡿⠿⠟⠛⠉⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⠟⢻⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠿⢿⣿⣿⢿⣿⡄
⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠛⠁⠁⠀⠀⠀⠋⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣀⠀⠀⢀⣴⣿⣿⠄
⠀⠀⠠⢐⡾⣿⡀⠂⣸⡟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢿⠟⠛⠁⠀⠘⠛⢿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠘⢧⣀⣼⠓⠒
⠀⠀⠁⢸⡇⠹⢷⠰⠟⣡⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠳⣄⡀⠀⣀⣴⠏⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢳⣄⠀⠀⡐⠁⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠈⠉⠀⠄⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠆⠀⡀⠀⠀⠀⡀⠀⠀⠀⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⢠
⠀⠂⢸⠃⠀⠀⠀⠀⠀⠀⠀⡷⠀⠆⠀⠀⠀⠀⠀⠀⡆⢠⠱⠀⠀⠀⠀⠀⠀⠰⢡⠄⠄⠀⠀⠀⠀⠀⠀⠲⠀⢿⣶⣶⣶⣶⣶⣶⣶⡾⡄
⠀⠄⠼⠃⠀⠀⠀⠆⠀⠀⠀⠻⡄⠂⠀⠀⠀⠀⠀⠀⢀⡰⢘⠀⠀⠀⠀⠀⠀⢸⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢺⣿⣿⣿⠀⣿⣿⣿⡟⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢘⠀⠁⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣿⣿⣿⡇
⠀⠀⠀⢻⣿⣿⣿⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣿⣿⣿⡇
⠀⠀⠀⢾⣿⣿⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣿⣿⣿⡇
⠀⢀⣴⣿⣿⣿⣿⢸⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⢰⣦⣬⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡅⣿⣿⣿⣷⣤
⠀⠈⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉


⠀⠀⠀⠀⠻⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⠿⠟⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣿⣿⣿⣿⢿⣾⣷⡀
⠀⠈⢻⡞⠛⠛⠛⠛⠛⠷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠟⠙⠋⠁⠀⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣧⡀⠀⠀⠀⢀⣼⡿⠋
⠀⠀⠀⢀⣤⡀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⢤⡄⠀⡤⢤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⠂⠀⠀⣁⣀
⠀⠀⠀⢸⣇⠙⠐⠋⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠘⢳⣶⠂⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣷⣤⣾⠿⠄⠀⢀
⠀⠃⠀⣿⣻⣿⣿⣿⣏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇⠀⠀⠀⠈
⢀⠀⠀⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠁⠁⠁⠀⠘⣷
⡇⢀⢿⣵⣿⣿⣿⣿⣿⣬⡇⠀⣀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠇⠀⠀⠀⠀⠀⠀⢀⢇⣀⡄⠀⠀⠀⠀⠀⠀⣂⣀⠘⠀⠀⠀⠀⠀⠀⠀⠘⣀
⠅⢨⢸⣿⣿⣿⠛⣿⣿⣿⡇⡅⠉⠀⠀⠀⠀⠀⠀⠀⠉⢹⡆⠀⠀⠀⠀⠀⠀⠀⢬⠉⠁⠀⠀⠀⠀⠀⠀⠈⢉⡅⠀⠀⠀⠀⠀⠀⠀⢨⡍
⠈⠃⢸⣿⣿⣿⠀⣿⣿⣿⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢸⣿⣿⣿⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠈⠉⠉⠉⠈⠁
⠀⠀⢸⣿⣿⣿⢀⣿⣿⣿⡇
⠀⠀⠘⠛⠛⠛⡜⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠄⠠⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣂⣠⠀⣤⣀⣀

⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⢰⣿⣿⡿⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠒⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠿⢿⣿⣶⣄
⠀⠀⠀⠀⣿⣿⠿⠦⠶⢿⣿⣷⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡇⠀⢸⣿⣿⣿⣧
⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⣿⣿⣿⣿⠋⣀⣶⣀⡈⣿⣿⣿⣿⠀⠀⠀⠀⢰⣶⣶⣿⣿⣷⣷⣷⠀⠀⠀⠀⣿⡏⠉⠉⠁⠀⠀⠉⠉⢹⣿⡆
⠀⠀⠀⠀⢠⣤⣴⣾⣷⣦⣤⡄⠀⠀⠀⠀⣿⣿⣿⣿⡈⠛⣿⡛⠃⣿⣿⣿⣿⠀⠀⠀⠀⠈⠉⠉⣿⣿⡏⠉⠉⠀⠀⠀⠀⢿⣿⣶⣶⡆⠀⢰⣶⣶⣾⣿⠁
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⢿⣿⣿⣿⣿⣶⣤⣴⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⣀⣸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠙⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⢰⣦⣤⣤⣤⣠⣄⣤⣤⣤⣤⣄⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⣄⣾⣿⣿⣷⡄
⠀⢸⡇⠀⣴⡄⠀⢀⡀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⡇⢀⣿⡇⠀⣿⡇⠀⢰⣶⠀⢻⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢛⣿⣿⣷⡀⠀⠀⠀⠀⠀⠹⣿⣾⣿⣿⣿⣿⣿⡿⠁
⠀⢸⣧⠤⠾⠥⠤⠼⠧⠤⠴⠯⠤⣾⡇⠀⠀⠀⠀⠀⠀⢸⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣿⣿⣉⣿⣽⣷⡀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡿⠟⠋
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠓⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣀⡀⠀⠀⣀⣀⣤⣄⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠄⠆⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢰⣿⣿⢋⣴⣉⢻⠀⠀⠀⠀⠀⠀⢸⠁⠀⣰⠞⠳⣆⠁
⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⡿⠃⠀⠀⠀⠈⠛⠻⠿⢿⡏⠀⠸⣿⠿⠿⠛⠁⠀⠀⠀⠀⢸⣿⣿⣌⡙⣉⣾⠀⠀⠀⠀⠀⠀⢸⠀⠀⠹⢦⡴⠏
⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠈⠛⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠙⢹⡇⠀⠀⠅⠀⠀⠀⠀⠀⠈⠙⠚⠚⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀
⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⣠⣬⣭⠑⠄⠆⠀⠀⠀⠀⠀⣴⣶⡿⠛⡓⢶⡄
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣹⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢾⡷⢠⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣟⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⡄⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣶⣿⡇
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⢄⡤⠴⠋⠀⡂⠀⠀⠀⠀⠀⠈⢙⡟⡟⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⣀⣀⢤⣄⡤⠤⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣤⣾⣭⣭⣵⣶⣾⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣶⣶⣾⠿⠇
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⢸⣿⣿⠉⠉⠉⠉⠉⢙⣹⠛⠉⠛⠋⠉⢹⣿⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⢀⢠⢈⡈⣀⡀⡄⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠠⣠⠀⠄⡄⢠⠁⠀⠀⢸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠠⢄⢠⠄⠠⠄⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣿⣿⣿⣿⣟⣻⣿⣿⣿⣻⣟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⢷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣟⣿⣿⣿⣿⢿⣻⣾⣿⣿⣟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢀⣠⣴⣠⣤⣶⣴⣶⣤⣴⣶⣾⣿⣽⣿⣯⣿⣿⠿⠿⠿⡿⠿⣿⣿⣿⣿⣼⣶⣶⣶⡶⣦⣤⣤⣶⣤⣤⡺⣶⣦
⠸⠿⠷⠾⠿⢿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠷⠾⠷⠷⠿⠿⠿⠿⠿⠿⠿⠿⡇⠷⢷⣾⠿⠷⠒⢫⠋⠃
⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡧⠤⣾⣿⠤⠤⠤⢼
⠀⠀⢀⣄⣀⣸⣟⣿⣧⣤⣍⣿⣿⣶⣄⣀⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣴⣿⣿⣋⣁⣔⢀⡟⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡿⠺⢷⢾⠤⠤⠤⠤
⠀⠀⠀⠈⠉⢹⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⢿⣿⣿⣿⠛⠉⠉⠉⠉⠉⠉⢿⡏⢱⣾⠉⣾
⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠛⠙⠿⣿⣦⣄⡀⠀⠀⠀⣿⠀⠈⢫⣤⣛⠤⠤⠤⠤
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣀⣴⣾⠿⠋⠁⠀⠀⠀⠀⠀⠙⠻⢿⣶⣤⣀⣿⠀⢰⠀⠐⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣃⣠⣴⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣦⣾⣦⣤⣿⣤⣤⣄⣀⡄
⠀⠀⠀⠀⠀⠸⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠿⢰⣶⣿⣀⣀⣀⠀⡇
⠀⠀⠀⠀⠀⠘⠿⠿⠿⠶⠿⠶⠶⠶⠶⠶⠶⠿⠿⠾⠷⠷⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠴⠿⠿⠿⠿⠿⠤⠧
⣶⣶⢶⢶⡶⠶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⢶⢶⣶⣶⢶⠶⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠⠚⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠑⢀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⡴⠋⢢
⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠈⠁⠀⣠⣶⣄⠀⠀⠀⠀⠀⠑⢄⠀⣰⣾⣿⣷⣄⠀⠀⠀⢰⠀⠀⠀⣠⠞⠁⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣇⠀⠀⠀⠀⠀⠀⠙⢯⡻⣿⣿⣿⣷⠀⠀⠀⢇⢀⡾⠁⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣿⣿⠟⠀⠀⠀⠈⠻⢤⣀⣀⡠⠔⠁

⠀⠀⠀⠀⠀⠀⠒⠰⠀⠀⠀⠀⠀⠀⠀⢠⠖⠲⡄⠀⠀⠀⠀⠀⢠⣴⣶⣦⣀⣴⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣒⣾⣶⣶⣷⣒⡂⠀⠀⠀⣿⣿⣿⡿⢻⣿⣿⣿⣿⠀⠀⢀⠄⠀⠀⠀⢀⣴⡿⠣⡀
⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⠉⠹⠿⣿⣿⡄⠀⠀⢛⣛⣣⣱⣾⣏⢟⣛⡛⠀⠀⠸⣷⣶⣤⣴⣿⣿⣴⣶⠇
⠀⠀⢳⠀⠀⣧⢀⡼⠀⠀⠀⠀⢸⣿⣿⣤⠀⢠⣤⣿⣿⡇⠀⠀⠀⠹⣿⣿⣇⣼⣾⠟⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠳⠤⠼⠋⠀⠀⠀⠀⠀⠸⠿⠿⠿⠷⠾⠿⠿⠿⠇⠀⠀⠀⠀⠈⠙⠿⠋⠁⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠛

⢀⣤⠛⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⣿⣿⡆⣀⡀⠀⠀⠀⠈⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⡆
⠀⠀⠳⣄⠑⣄⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡅⢾⣿⣿⣷⠄⠀⠀⠀⣀⣿⣿⣛⠛⢳⠀⠀⠀⠀⠠⠾⠴⣿⡶⡲⢦⡀
⠀⠀⠀⠀⠛⢄⢱⡀⠀⠀⠀⠀⢀⣬⣿⣿⣏⢾⣿⣿⣤⠀⠀⠀⣾⠉⢘⠲⠛⣷⢺⡀⠀⠀⠀⠀⠀⠀⣯⡪⢈⣼⣃⣀⡀
⠀⠀⠀⠀⠀⠀⠉⠻⢆⠀⠀⠀⠀⠻⠿⢻⣿⣾⡟⠿⠟⠀⠀⠀⢿⡀⠈⠁⢠⡟⣸⡏⢄⠀⠀⠀⠀⠀⠈⠉⠉⡟⡣⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⢿⡄⠈⣿⠀⣿⣧⢰⡟⠀⠀⠀⣿⣛⡛⠈⣿⡟⠀⣿⠛⣻⡆⢸⣿⣆⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣀⣸⡿⠾⣷⠀⢸⣿⡇⢻⣿⠃⠀⠀⠀⣿⠉⠉⢀⣿⣇⠀⣿⠛⢿⣄⢸⡏⠹⠏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠀⠀⠉⠁⠀⠉⠀⠈⠉⠀⠀⠀⠀⠉⠀⠀⠈⠉⠉⠀⠉⠀⠈⠉⠈⠁⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠛⠛⠳⢦⣤⠶⠶⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠤⡼⠋⠀⠀⠀⠀⣴⡀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⣰⣷⠶⠶⢾⣉⣹⠶⠶⣶⣧⠀⠈⠛⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⠏⠘⢧⠀⠀⣿⠁⠀⣰⠏⠈⢧⠀⠀⠀⢼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⢴⠿⠶⠶⠾⣷⠀⣿⠀⢴⡿⠶⠶⠾⣷⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠈⠳⠤⡤⠾⠃⠀⠻⠀⠈⠳⠦⡤⠾⠃⠀⠀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣀⣀⣀⡀⢀⣤⠖⠛⠛⠛⠲⢦⡀⠀⣀⣀⣠⡾⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠀⠈⠉⠉⠉⠁⣾⣡⣤⣤⣤⣤⣤⣤⣿⡀⠉⠉⠁⠀⠉⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣤⣤⣤⣤⣤⣤⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄⡀
⠀⠀⠀⠀⢀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧⠀⠀⠀⠀⠀⠀⠀⣼⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⣀
⠀⠀⠀⠀⠻⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⡼⠛⣧⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠃
⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡇⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⣤⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⢀⣤⣤⣤⣤⣥⣤⣤⣤⣤⣄
⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⡀⠀⠀⠀⠀⠀⢸⡇⠀⠰⠶⠶⠶⠶⠆⠀⠀⣿⠀⣿⠀⠀⠰⠶⠶⠶⠶⠆⠀⢸
⡾⢡⡶⡦⢸⡆⠀⠀⠒⠒⠒⠂⠀⠀⠀⣏⢰⠶⣎⢻⡄⠀⠀⠀⠀⢸⡇⠀⢘⣛⣈⣛⡛⠃⠀⠀⣿⠀⣿⠀⠀⢘⣛⣛⣛⠛⠃⠀⢸
⢻⣜⠳⠶⠋⠀⠀⠀⠴⠀⠆⠰⠀⠀⠀⠙⠳⠶⢋⡼⠁⠀⠀⠀⠀⠸⣇⣀⣈⣉⣉⣉⡁⢀⣀⣠⠟⠀⠻⣀⣀⡈⢉⣉⣉⣀⣀⣀⣸
⠀⠉⢿⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠙⠻⠁⠀⢀⣀⠀⠀⠈⠟⠋⢁⡁
⠀⠀⠀⣿⢻⡟⠛⢻⡟⠛⠛⠛⡟⠛⢻⡟⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠙⡆⠀⠀⠀⡟⠉⢻⠀⠀⠀⢸⠋⠙⡇
⠀⠀⠀⡏⠘⡇⠀⢸⡇⠀⠀⠀⡇⠀⢸⡇⢸⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠷⠾⠧⣄⣀⡤⠿⠶⠿⢤⣀⣠⠼⠷⠾⠧⣄⡀
⠀⠀⠀⡇⢸⡇⠀⢸⡇⠀⠀⠀⣧⠀⠈⡇⢸⡇⠘⢧⡀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠈⠏⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠈⠃
⠀⠀⠀⡇⢸⡇⠀⢸⡇⠀⠀⠀⣿⠀⠀⡇⢸⡇⠀⠀⠻⣆⠀⠀⠀⢠⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻
⠀⠀⢸⡇⢸⡇⠀⢸⡇⠀⠀⠀⣿⠀⠀⡇⢸⡇⠀⠀⠀⠈⢳⡄⠀⢸⡇⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢸
⠀⠀⢸⡇⢸⡇⠀⢸⡇⠀⠀⠀⣿⠀⠀⣇⠘⡇⠀⠀⠀⠀⠀⠙⢦⢸⡇⠀⠀⢸⡉⢈⡇⠀⠀⠠⣏⠀⣻⠀⠀⠀⢼⡁⢈⡇⠀⠀⢸
⠀⠀⢸⡇⢸⡇⠀⢸⡇⠀⠀⠀⣿⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⡴⠞⠛⠛⠳⢤⣠⠴⠛⠛⠛⠦⣄⡴⠞⠛⠛⠷⢤⣀⣸
⣤⣤⣼⣧⣼⣥⣤⣼⣧⣤⣤⣤⣿⣤⣤⣿⣤⣧⣤⣤⣤⣤⣤⣤⣤⣬⣭⣽⣥⣤⣤⣤⣤⣬⣧⣤⣤⣤⣤⣤⣼⣥⣤⣤⣤⣤⣤⣯⣭
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⣏⡌⠀⠀⣠⠂⠀⠀⠀⠉⢛⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢿⠇⠈⠚⠁⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡞⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠋⠁⢿⡇⠀⠙⢦⡀⠀⢤⡀⠀⠀⣰⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠙⠓⠶⣬⣿⡀⠀⠀⠙⢦⡈⠙⠶⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠲⢄⠀⠀⠀⠀⢠⡇⠘⣷⣀⣠⡤⠔⠓⣾⢻⣷⡷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠈⢳⡀⠀⠀⠈⠳⣄⣹⡏⠈⢦⠀⢸⠃⢸⡿⠉⠛⣧⠓⢄
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠈⠙⢿⡀⠈⢧⢸⡇⣿⣧⠀⠀⠈⢧⠀⣇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⣟⣦⡈⢿⣧⡟⢸⠀⠀⠀⠈⢇⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠠⡏⢌⢿⡝⣎⣿⡇⠘⣧⣀⡀⠀⠘⣾
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠙⠒⠋⠀⠈⢿⠁⠀⠀⠉⠁⠘⠓⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢠⢹⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠈⡜⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠘⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⣧⢯⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠲⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⢧⣀⠀⠀⢀⣠⠖⠋⠀⠀⠀⣠⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⢤⣼⡀⢀⡠⠤⠶⠶⠖⣛⠲⢦⣨⣻⠟⣋⣠⣄⢀⠶⡖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡿⠁⠀⠀⠀⠰⠶⣦⣬⠿⠋⢰⡛⡇⢧⠈⡿⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠛⠳⢦⣤⣀⠀⠀⠀⢀⡞⢰⣇⣀⡤⠖⠋⣩⠿⠋⠁⣠⠴⢮⣁⣨⠤⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣄⣸⠉⢻⠃⠀⢠⡴⣋⣁⡤⠖⢻⠯⣦⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣂⣼⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣋⣼⣷⣶⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⠿⠛⠛⠛⠃⡜⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠟⠁⠀⠀⣀⣀⠀⢠⡤⠐⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠉⢠⣴⣞⣿⣿⣿⣶⣿⢡⡌⢻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⠟⠁⣿⣿⣯⣯⣠⣽⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⠉⠀⠀⠀⠉⠙⠋⠘⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⢀⠀⠀⡀⢰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⡀⠀⠀⠀⠀⢀⣼⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧⡀⠈⠿⠷⣦⣤⡤⠈⠋⢹⣿⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡀⢠⣾⣿⣿⡿⠶⣄⣀⠀⠁⣠⠏⠀⣼⣬⠻⢡⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣛⠟⠛⠛⠒⠙⠿⣿⣷⣧⣄⠀⠉⠛⣸⣇⣠⣴⡇⢿⣄⠀⢿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠻⠏⢈⣿⣷⣄⡀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣾⣿⣿⣿⣿⣷⠘⣿⡀⠈⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣮⣻⣿⣦⡀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣧⠸⣧⠀⡜⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⢻⣿⣿⣦⡀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡆⢻⡆⢸⣘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣜⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣿⣦⡀⠀⠀⠀⠈⠻⣟⣿⣿⣿⣿⣿⡼⠟⠀⢻⣿⡿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠈⢶⣿⣿⣾⣿⣷⠀⠁⠘⢿⣿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⣾⣿⣿⣻⣿⣿⣿⣉⣿⣄⠀⠀⠀⠀⠻⣿⣿⣟⢽⣆⠀⠀⠀⠙⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⠹⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠘⣿⣿⣿⣿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⢀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣦⡹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⡿⣿⣼⣦⠀⠀⠀⠈⢿⣿⢏⣿⣿⣿⣿⡀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇⣾⣿⣿⣿⣶⣿⣿⣭⣭⣿⣯⣤⣿⣿⣿⣧⠀⠀⠀⠈⠿⣾⣿⣿⣿⣿⠃⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣿⣿⣿⣿⣿⣦⣀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣧⡀⠀⢀⣿⣿⣿⣿⣿⡿⠃⢸⣦⡀⠀⠀⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡀⠈⠋⠉⠉⠉⠉⠉⠙⠋⠉⢉⠉⣍⠘⣍⣉⢹⣛⠛⠄⠸⣿⣿⣿⣿⣿⠀⠀⢨⠟⠀⠃⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠉⢘⠃⠀⠀⠀⠙⠿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣶⣶⣶⣾⣷⣾⣿⣿⣿⡆⠀⠀⠈⠙⠻⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⠘⠋⢸⡿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣃⣀⣀⣀⣀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣒⣒⣒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠋⠉⠉⠉⠉⣭⡄⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣴⣶⣿⣷⣦⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠈⠉⣿⡏⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⣸⠀⠳⣄⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⣰⠃⠀⠀⠹⣦⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⡼⠃⣀⠤⠤⠀⢹⣧⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⡠⠚⠁⠀⢀⣴⠟⠀⠀⡤⠄⢀⣀⡆⣿⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠃⢸⡇⠀⠀⡜
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⢀⣘⡆⠀⠀⠀⢠⠀⢸⡇⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡎⠀⢸⠁⣠⠃
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢆⠀⠓⢿⠚⠟⠚⠟⠀⡼⠀⢀⢏⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⣠⠜⠀⠀⢀⠎⠀⠀⠐⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠉⠒⠒⠊⠁⠀⠀⢠⠏⢀⡤⠄⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⢀⡶⠒⠒⠘⣆⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⡆⠀⠘⠀⠈⠀⣀⡴⠀⢀⠞
⠀⠀⠀⠀⠀⠀⢺⠀⢰⠓⠀⣹⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⢻⡀⠀⠀⠒⠚⢉⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠈⠣⣈⠀⠒⠋⠀⢀⣠⠞⠁⠀⢀⠀⠀⠀⠀⠙⠒⠒⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠘⠉⠉⠀⠀⠀⠀⠈⠳⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣶⣿⣶⣷⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⠀⠀⠀⢸⣷⣶⣶⣶⡀⠀⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⠀⠀⠀⢠⣴⡶⠷⠿⠟⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢹⡇⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣤⣶⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠘⣿⡎⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⢿⣿⢀⣀⣤⣤⣶⣦⣶⣾⣿⣿⠿⠿⠿⠿⠿⣿⡿⠿⠿⠿⠛⠛⠛⠛⠛⠿⠿⠿⠿⣿⣿⣿⣷⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠘⠿⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠁⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠰⢶⣤⣤⣄⣠⣤⣤⣶⠾⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣤⣀⣀⣀⣀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⣼⡿⣧⡀⠀⠀⣠⡾⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣯⠉⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⢻⣧⠈⢿⣄⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣻⣆⢘⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠷⣤⡀
⠀⠀⠀⠀⠀⠀⣴⡿⠛⠛⠻⡿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣀⣀⠀⠀⠀⠀⠀⠁⣀⠈⠹⣷
⠀⠀⠀⠀⠀⢸⣿⠀⠐⠲⡤⠁⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠈⠻⠛⠃⠀⠀⠀⠀⠎⠛⠀⠀⣿
⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠘⢷⣤⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠰⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⢶⣴⠞⠛⢿⣆⢀⣤⠀⠀⠀⠀⠀⠀⣸⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠶⣤⣴⣏⣀⣀⣀⣀⣠⣤⡿⠟⠁⠀⠀⠀⠀⠀⣼⡟⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠙⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣏⣉⣩⡿⠋⠀⠀⠀⠀⠀⠀⢠⣾⠋⢱⣆⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⡀⠀⢠⠉⢻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠹⣦⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⢀⠀⠀⡐⠀⢹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠋⠀⠀⢿⡀⠀⠘⢷⡄⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠘⠀⢸⠇⠀⡸⠀⠘⢻⣶⣤⣀⠀⠀⠀⠀⠀⠀⣀⣤⡶⢿⡇⠰⠀⢱⡄⠀⢻⡄⠀⠀⢻⡀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢀⠀⠀⡇⠀⡆⠀⠀⡇⠀⡇⢸⡇⠉⠛⠛⠿⠿⠾⠿⠛⠛⠁⠀⠘⣷⡀⠀⠈⠃⠀⠘⠃⠀⠀⠀⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡀⠀⠁⠀⠇⠀⠀⠇⠀⣇⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣀⡀⠆⠀⠀⠀⠀⠀⠀⡅⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⢠⠀⠰⠀⠘⣦⣤⣾⣿⠋⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠙⢿⣦⡀⠠⠆⠀⠀⠰⠀⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⣈⣠⣤⣾⡿⠟⣹⡿⠁⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠙⢿⣦⣄⣀⠐⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠿⠛⠉⠁⠀⠀⣿⠃⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⢻⣟⠿⣷⣶⡟⠁
⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠈⢿⡆⠀⠙⠻⣷⣦⡀
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠙⠿⣷⣄
⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠀⠀⠻⣦⣄⠀⠀⠀⠀⣠⣼⣏⡀⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⣀⣀⣤⣴⠟⠛⠿⠿⠟⠋⠈⠙⠻⠷⢶⣶⣶⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄
⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠏

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠋⠉⠛⠉⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣦⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡲⢖⡢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠕⠀⢭⠢⠀⢀⣿⠋⠀⠀⠈⢎⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠠⠤⢤⣿⡁⠀⠀⠀⢣⠇⣸⡇⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡤⢿⢕⠳⣺⣿⠀⠉⠉⠁⠺⡸⠿⠤⣀⡀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⡏⣉⠉⠓⠺⠦⣷⣥⡪⡇⡯⢚⠔⡠⢀⠈⣹⠿⡆⠀⠈⠉⠁⢒⡛⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⠀⠀⠀⠀⠉⣽⣟⠓⠮⠮⢵⣆⣿⡣⢂⢄⠠⡀⠀⠛⢿⠀⠀⢀⣀⣉⣑⡒⡦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠙⠻⠀⠀⠀⠀⢠⣿⡍⠓⠳⠮⢔⡡⣂⢄⠀⢰⣟⡁⡡⢊⢔⠕⠺
⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⡠⡉⢢⠀⢠⢄⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠈⠉⢓⠳⣞⡯⢊⢔⠕⡩⠆⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⣥⠈⡆⡇⢸⢸⠀⡏⢱⠀⢠⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢟⡡⡡⢊⠔⡁⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣇⢹⣀⠆⡆⢸⢸⠀⠷⢸⠀⠈⠑⠀⡤⢍⡗⡖⢦⠀⣠⣀⠀⠀⡇⣽⢊⠔⡡⢊⠔⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠠⠤⠐⠁⢸⠀⢦⠅⢸⠀⠀⠀⡆⡇⠀⠀⢱⠘⡴⢀⠌⠀⢈⡧⣻⡡⢊⢔⡵⣋⡇
⠀⠀⠀⠀⠀⠀⠸⠀⠂⢀⡀⠀⠀⠀⠀⠑⠤⠤⠜⠀⠀⠘⠁⠇⠀⠀⠘⡆⢠⠎⠀⠀⠀⣗⢽⢔⠕⡡⢎⠅⡇
⠀⠀⠀⠀⠀⠀⠀⡇⣂⢸⠉⡉⠒⡆⢠⢤⣀⠀⠀⠀⠀⠀⠳⠔⠀⠀⠀⡇⠈⡆⠀⠀⠀⠮⡻⣫⢎⠔⡡⣪⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⡀⠩⢍⡃⢸⢠⠬⡉⠲⡀⣤⣀⠀⠀⠀⠀⠀⠙⠲⠇⠀⠀⠀⣗⢝⡗⡡⢪⢞⠕⡇
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⡇⢰⠒⠇⢸⠈⣓⠋⢀⠇⠇⢀⣉⠑⡆⢰⠢⢤⣀⡀⠀⠀⠀⣯⠀⣀⠄⠄⡠⢀⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠘⠄⠑⠼⠀⠀⢸⠀⡍⡇⣇⠀⢻⠘⠃⢭⠁⢸⠀⢴⠤⢬⠆⠀⠀⡗⡰⡶⡠⡢⢂⠄⣦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠐⠊⠳⠮⣤⣀⠀⠀⠈⠑⠃⠹⢼⠄⢸⡘⠯⢝⡁⠀⡆⣨⢉⡆⠀⠀⠀⣭⢻⣯⢎⠔⡡⣊⡷⠭⠭⠭⠭⠍⠁⠒⢒⠒⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⣤⣀⠀⠀⠀⠀⠉⠑⠢⠇⠀⣇⠘⠲⢤⠀⠀⠀⠣⠱⣷⡱⣪⢞⢵⡯⠭⠭⠥⠤⠀⠀⠤⠆⠀⠀⢀⣒⣒⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠢⢤⣀⡀⠀⠀⠀⠀⠉⠒⠦⠂⠀⡄⡗⣽⡫⣻⢷⢵⣹⡯⠭⠭⠤⠤⠬⠭⠭⠭⠍⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠤⣄⣉⠐⠠⢠⡃⡇⣿⣿⡩⣲⣯⡿⠉⠁⠩⠭⠭⠭⠭⠭⠭⠭⠭⠭⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣇⣿⣥⣚⡋⠁⠤⠤⠬⠭⠭⠭⠭⠭⠭⠭⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠛⠀⠀⠒⠻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠤⠹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣀⡀⠤⠴⠤⠏⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠈⣤⣾⣿⡿⠁⢀⣀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡛⠿⠀⠀⠀⠤⠜⠁⡺⢿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠈⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⠀⠀⠀⠀⠀⠀⠀⠀⠾⡀⠙⣿⣶⣄⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⢂⡀⠈⠂⠌⣻⣿⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢷⣦⣤⡋⠀⠀⠈⢷⣄⡀⠀⠀⠈⢙⣷⣿⠶⡴⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃⠀⠈⠙⠿⣷⣦⣄⠀⠹⠛⡴⢦⣠⣾⡿⠷⣾⣿⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⡄⠉⠻⡟⣠⢦⣼⡁⠉⠛⠁⠈⠡⢟⣭⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⠁⠂⢠⠉⡄⠀⠀⠀⠀⠉⠛⣻⣯⣀
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⡀⠀⠀⢠⡶⠀⠀⠀⠀⠀⠒⠚⠉⠐⢻
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣾⡇⠀⠀⢸⣿⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⠉⠀⠀⠀⣾⡏⡀⠀⠀⠀⠆⠀⠀⠀⠤⢺⡄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣼⠿⠋⠉⠀⠀⠀⣴⣿⣿⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⣼⠇⠀⢀⣀⣴⣶⠶⠋⢀⠀⠀⠀⠁⣠⣾⣿⣿⡏⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⢠⣿⠀⢨⣵⣾⠟⠁⠀⠀⠸⣧⠀⠀⠀⢹⣿⣹⣿⡇⠀⠀⠀⠄⠀⠀⠲⢶⣶⣾⡏
⠀⠀⠀⠀⢸⡇⣴⠟⣴⠋⠀⠀⠀⠀⠀⠈⠓⢂⣴⣿⠋⣿⣿⠀⠀⠀⠀⠀⠀⠀⠐⠞⣻⣿⡃
⠀⠀⠀⠀⡿⠃⠀⠘⠃⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣯⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⣿⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡟⠛⣿⣿⣃⣀⡀⠀⠀⠠⠤⠄⠀⢀⣀⡿⠁
⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⣀⣴⣾⣿⣽⣿⡇⠀⠀⠀⣿⡟⠀⣿⠀⠀⠀⠀⠀⢸⣿⣿⢿⡇
⠀⠀⠀⠀⠈⠻⢦⣀⣤⡾⣿⠛⠋⠰⣿⣿⡇⢀⣀⣸⣿⡇⠀⣿⡆⠀⠀⠆⠀⣿⣿⠃⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⣄⠘⣿⡏⡇⠀⠀⣼⡟⠀⠀⣿⡇⠀⠀⠀⠀⢸⡏⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢹⡇⣿⠷⢧⡀⢠⣿⣇⡀⠀⣿⣄⣤⣤⣥⡴⠟⠃⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⣿⣀⣾⠀⢸⣿⠉⠿⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⣶⣿⢻⣏⣴⠇⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⣮⡉⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⠃⣀⠸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡄⠘⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣄⠀⣤⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠜⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠛⢋⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣉⣹⣛⣛⣉⣻⣋⣤⣉⣙⣟⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣯⣤⣭⣥⣤⣬⣿⣿⣯⣭⣭⣭⣭⣭⣿

⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⢀⡺⢗⠀⠀⠀⠀⠀⠀⠀⢐⣿⣆⠀⠀⠀⠀⣀⡻⣃⡀⠀⠀⠀⠀⡻⣆⠀⠀⠀⠀⠀⢿⡗
⠀⠹⠷⠟⣷⣿⣷⡀⠀⠀⠀⠀⢰⣿⣿⡿⣇⠀⠀⢰⣿⣿⣿⣿⡀⢀⣐⣿⣿⣿⠁⠀⠀⣴⡟⢸⡏⢿⡆
⠀⠀⠀⢰⣿⣿⣿⡷⠶⠆⠀⠀⢿⣿⣿⣷⣿⠀⠀⢾⢿⣿⣿⣿⠇⠀⠉⠉⣿⣿⡄⠀⠀⠙⠓⣾⣷⠛⠃
⠀⠀⠀⠘⣿⣿⠀⠴⠆⠆⠀⠀⣼⢸⣿⡟⠀⠀⠀⣿⣾⣿⣿⡀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⡟⣿⡆
⠀⠀⠀⠀⣿⡇⠀⣴⣶⡆⠀⠀⠀⠀⣿⡇⠀⠀⠀⠉⣿⡇⣿⡇⠀⠀⠀⠀⢿⡏⠀⠀⠀⠀⣿⡇⢹⡇
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠈⠁
⠀⠀⢶⡆⠀⠀⠀⠀⢀⣾⡆⠐⢃⣰⣆⣪⡉⠉⢱
⠀⢠⣷⣄⡄⠀⠀⣤⣶⣶⣶⣄⠀⠉⠈⠻⠀⠆⢸⠀⠀⠀⠀⠀⠀⠀⠄
⠀⢸⣿⡿⠿⢟⣀⢻⡾⠿⠿⠏⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⡏⠛⠀⠀⠀⠀⢀⠀⢰⡔⠀⠐⠂
⠀⢺⣿⡇⠀⣸⣿⣻⠀⠀⠀⠀⢹⣿⣿⣿⣇⡈⢻⠘⠁⢠⡄⢀⡀⠀⣇⣀⣠⣤⣤⣴⣿⡇⣸⠷⠶⠶⢦
⠀⠘⣿⡇⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠓⢺⠀⠀⢸⡇⠀⠀⠀⡆⠛⠛⠛⣿⣿⣧⣤⣿
⠀⠀⠿⠃⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠿⡿⠀⠀⠘⠀⠀⠸⠇⠀⠀⠀⡏⠀⠀⠈⠀⠀⠀⠀⠸
⠀⠀⠀⣠⣄
⠀⠀⢀⣻⢟⡀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡇⠀⠀⠀⠀⠀⠀⢸⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿
⠀⢠⣿⣿⣿⢿⡆⢀⣴⣦⡀⠀⢰⣿⠟⠿⣦⠀⠀⠀⠀⢠⣿⣷⠀⠀⢀⣿⠗⠀⠀⠀⠀⢀⣾⠿⣿⢷⡀
⠀⠸⡿⣿⣿⡸⣧⣤⣿⣿⡄⠀⠈⢿⣀⣴⠁⠀⠀⠀⠀⢸⣿⡿⣷⣤⣿⣿⣄⣀⠀⠀⠀⢸⡏⣶⣿⠈
⠀⠀⢾⣿⢻⡇⠀⠀⣿⣿⠓⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⢸⡿⣿⠀⡨⢿⣿⣿⣿⡀⠀⠀⠀⢰⣿⢿⡆
⠀⠀⢸⡇⢸⣿⠀⠀⢸⡇⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⣾⠇⣿⡇⢅⠈⠉⢩⣿⡇⠀⠀⠀⢸⡇⢸⣇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠀⠁⠈⠁⠈⠁⠀⠀⠀⠈⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢒⣛⣉⣉⣙⠛⠒⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⣀⠴⠊⡀⠀⠀⠀⢳⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⡀⢀⣠⡤⠶⠒⠋⠉⠀⠀⠀⠘⣆⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢔⣡⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢁⠖⡠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢠⢃⡼⢱⡇⠀⣀⣠⣀⡀⠀⠀⠀⠀⠀⣠⣤⠄⠀⣿⡇⠇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠇⡼⠁⢸⠇⠈⠘⠿⠎⠁⠀⡀⠀⠀⠀⠙⠛⠀⠀⡇⡇⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢠⠃⡆⠘⡆⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠸⠀⡇⠀⢣⠀⠀⠀⠀⠀⠀⢄⣠⠀⠀⠀⠀⠀⣸⢰⠃⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠈⣆⠀⠀⠀⠀⡀⠀⡀⠀⢄⡀⠀⠀⡏⣼⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢃⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠄⢠⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢀⠈⢆⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡜⢠⠇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡈⢦⠸⡇⠈⠓⠦⠤⠤⠤⠀⠀⢠⡿⢠⣎⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠷⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⡴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡯⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢻⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⡎⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⢋⡄⠀⡏⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠸⣆⠈⠓⠦⣄⡀
⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠸⡇⠀⠁⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠤⠤⠴⢾⠁⠀⠀⢀⠓⠈⠀⠀⠀⠉⠳⢦⣀
⠀⠀⠀⡠⠂⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠄⢀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢄⠁⠀⠀⠀⠈⡆⠀⠀⠀⠈⠃⢀
⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠤⠐⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠈⢧
⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸⣇
⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠑⢄⠀⠀⠀⠀⠈⢧⠀⠀⠀⢠⠃⠀⠀⢀⡴⠊⢄⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠙⣦⠀⠀⠐⠤⡀⠀⠀⠈⡆⠀⠀⡎⠀⢀⠴⠋⠀⠀⠀⠣⡀⠀⡏⠀⠀⠀⡇⠀⠀⠀⢸⡆
⢀⡇⠀⠀⠀⢰⡄⠀⠀⠀⠈⢧⡀⢀⡿⠀⠀⠀⠀⠈⠀⠀⡀⠘⡄⢰⢀⠔⠁⠀⠀⠀⠀⠀⠀⠁⣾⡁⠀⠀⠀⡇⣀⠀⠀⠈⣧
⢸⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢰⡌⠀⠀⠀⠀⠀⠀⠀⠀⡀⡬⠁⠈⠀⠠⠆⣷⠃⠀⠀⠀⠻
⡎⠀⠀⠀⠘⡌⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⠏⠄
⣧⣤⣤⣤⣤⣷⣾⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣶⣦⣤⣤⣤⣤
⣿⣭⣭⣭⣽⣧⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣶⣶⣶⣾⣿⣿⣾⣿⣷⣾⣾

⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⣠⣤⣶⠻⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⠀⣾⡿⠿⠟⠻⣷⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⣰⠿⠿⠙⢦⠀⠀⠀⠀⢀⣿⡿⠿⠟⢿⡆
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⢰⣿⡀⠉⠉⠉⢀⣿⠆⠀⠀⠀⣀⢀⢃⠀⠀⠀⡸⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢳⢲⣶⡚⢃⠀⠀⠀⠀⠀⣸⣿⣶⣶⣶⣿⣦⠀⠀⠀⠘⢄⠈⠀⠁⠀⠉⠀⠁⠀⠀⠀⢠⣾⣧⡄⣤⣿⣆
⠀⠀⠀⠰⠰⠻⠊⠁⠉⡦⠆⠀⠀⠀⠀⠻⣿⣿⣿⣷⣿⠿⠂⠀⠀⠀⠀⠠⠀⢿⣿⣿⠁⠄⠀⠀⠀⠿⣿⣿⣧⣿⣿⡿⠆
⠀⠀⠀⠀⢁⣀⣀⣄⣀⡈⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠘⠆⠀⢴⠀⠀⠂⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣶
⠀⠀⠀⠀⠸⢿⠿⠿⣿⠿⠀⠀⠀⠀⠀⠀⢿⡿⠿⢿⡿⠇⠀⠀⠀⠀⠀⠀⠦⡤⠾⢤⡤⠄⠀⠀⠀⠀⠿⣿⠿⠿⡿⠿⠿



⠀⠀⠀⠀⠀⢀⣄⣠⡀
⠀⠀⠀⠀⠰⢿⣏⣩⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠈⠐⠠
⠀⠀⠀⠰⣆⠂⠀⠀⠀⢰⠇⠀⠀⠀⠀⣸⠿⠲⠒⠂⠻⣇⠀⠀⠀⠀⠀⢤⡀⡀⠀⡀⢠⠄⠀⠀⠀⠀⢄⣤⣔⣀⣱⡀
⠀⠀⠀⠀⢀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠻⢿⣄⠀⠀⠀⣴⠟⠃⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢓⣉⠉⠈⣉⡙⠀⢀⣤
⠀⠀⠀⢀⠀⠙⢿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇⡄⢁⣀⢹⣧⠀⠀⠀⢰⡿⢿⠿⢋⣹⣿⣾⠋⠉
⣠⠶⠲⢿⣿⣛⣛⣟⣛⣿⡇⠀⠀⠀⠰⢁⣤⠲⢄⠀⠐⠈⠀⠀⠀⠀⠈⢹⠾⠷⠶⣿⠿⠛⠀⠀⠀⠸⠁⢌⠻⣶⣟⣟⠋
⢿⣄⣠⣦⣼⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⣀⡉⣙⣡⣀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⣄⣀⣽⣿⣧
⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⣓⡶⠞⢻⠋⠉⠙⠋⠛⠛⠉⠉⣏⣑⣲⣤
⠀⠀⠀⠀⠀⡔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣼⣏⣸⣏⣽⣀⣀⣀⣀⣠⣤⡤⠤⠿⠶⠷⠿⠇
⠀⠀⠀⠀⠐⢓⡒⢲⠒⠒⡖⠒⢺⠒⡟⠛⢹⠛⡛⠓⠒⠒⠒⠛⠛⡇⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠄⠢⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠘⣄⡰⢚⢘⠇⡇⠞⢸⣉⡗⡟⢸⠃⡇⢸⡗⢺⠀⠇⢰⡇⠘⣧⡆⠀⠀⠐⡄⠀⠀⡆⣽⣷⠈⢄⡰⢄⢀⣸⣇⡀
⠀⠀⠀⠀⠀⡗⠿⠾⠿⠿⠿⠿⠿⠾⠷⠿⠿⢿⣿⣿⣷⣾⣶⣿⣿⡇⢿⣹⡆⠀⠀⠀⢈⠒⢛⡧⢸⣻⢾⠾⠯⠙⠫⢽⡇⠃
⠀⠀⠀⠀⠀⡇⢀⠤⢤⣤⣄⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⡀⠀⡇⠰⢾⡷⡀⠀⠀⣀⣠⠎⣸⣛⣙⣈⣢⣀⣀⣀⣠⡇
⠀⠀⠀⠀⠀⡇⠈⣀⣹⡶⣒⣛⣛⡳⠶⠾⠿⠿⣿⣿⣿⣿⣟⡿⠛⡇⠀⢺⡏⢉⠯⡍⠙⠉⠉⠁⠀⠈⠉⠉⠉⠀⣟⣦⢰
⠀⠀⠀⠀⠀⡗⠆⠀⣼⢉⡇⠀⠈⠉⠉⠉⠙⠓⠒⠒⠒⢂⡎⡁⡖⢿⢀⢸⠇⣨⠉⠁⠀⠀⠀⢤⠤⠤⠤⠶⠶⣶⠶⠒⢿⡀
⠀⠀⠀⠀⠀⠙⠿⡤⣿⣿⣿⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⢷⣥⣤⣼⢸⢸⣖⣾⣏⣉⣁⣠⣄⣄⣄⣠⣤⣤⣼⣿⡿⡷⢾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢀⡌⣩⠉⠛⠛⠛⠲⠶⠶⢾⢸⣿⣋⢨⢉⢩⠀⠀⢀⡀⠀⠀⠀⢀⠀⡄⠤⠤⢤⠐⠂⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢆⣷⣫⡏⣙⠚⡤⠤⠞⠁⠀⠀⠀⠈⠈⠣⣀⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠉⠀⠀⡈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⢀⣇⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⣠⠎⠘⢦⡀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠲⣦⠶⠖⠛⠁⠀⠀⠀⠹⣦⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢿⢦⡀⠀⠀⠀⠀⢠⢾⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠘⣿⠛⠲⠤⠴⠞⢻⡞⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠷⣦⡤⠼⠄⠀⠀⠀⠀⠾⠴⣶⡶⣏⣀⠀⢠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣰⠃⠹⣄⠀⠀⠀⠀⠀⢠⠞⠉⢧⠀⠉⠀⢸⢠⡖⠒⠂⠐⠒⠒⠒⠒⠒⠒⣦⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠛⣲⠆⠈⠳⡄⠀⢀⡴⠃⠀⣶⠚⠃⠀⠀⢸⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⡇⠀⠛⠦⠀⠀⠙⣶⠋⠀⣀⡤⠞⠃⢸⡆⠀⢸⢀⡞⠉⢳⡀⠐⣛⣛⣛⣛⣛⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣼⠀⠀⠀⠀⠀⠙⠲⣿⠴⠚⠁⠀⠀⠀⠀⣧⠀⢸⠀⠳⠶⠞⠀⠨⣭⣭⣭⡍⠉⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⡆⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢰⠇⠀⢸⠀⣛⣛⣛⣛⣛⡛⠛⢛⣛⣛⡃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡿⠀⠀⢸⠠⣭⣭⣭⣭⣭⣤⣤⣬⣭⣭⡅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠒⠶⠶⠶⠖⠀⠀⠀⢠⡤⢤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠒⠒⠂⠀⠒⠒⠒⠀⢸⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁



⡿⢿⠿⠿⡿⠻⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢸⣿⣿⣿⣿⣿⣿⣿⣿⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡌⢿⣿⣿⣿⣿⣿⣿⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠉⠛⠻⠛⠋⠁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡌⠳⣤⣀⠀⣀⣠⡾⠋⣼⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡄⠈⠉⣿⠋⠉⠀⣸⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠛⠛⢿⣷⡀⠀⣿⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠘⠃⣸⣿⣧⠀⣿⠀⢠⣿⣿⣿⡟⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡸⣿⣿⣿⣧⣿⢀⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡙⢿⣿⠿⣻⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠷⠶⠾⠿⠿⠿⠿⠿⠿⠇⠸⠿⠿⠿⠃

⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠉⠙⠿⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣠⣰⠛⠛⠛⠓⠶⠶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿
⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⣏⠙⣧⡟⠒⣶⣆⠀⠈⠙⢮⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡱⣿
⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⣙⣹⠀⣇⡇⢰⣧⣼⣆⠀⠀⠈⢇⢷
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⡇⡇⠘⠶⠶⠃⠀⠀⠀⠈⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣛⣛⣦⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢰⡏
⣿⣄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢷⣄⣀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣠
⣿⣿⣾⣆⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⢠⡟⠙⠓⠶⠋⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⣴⣾⣿
⣿⣿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢤⣟⣾⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⢠⣤⣤⣤⣴⣶⣿⣷⣦⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⢛⣭⡻⢿⣿⣿⣿⣿⡷⠀⠀⠘⣿⣿⠟⣫⣭⣭⣭⣍⡻⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣫⣭⣭⣷⠸⠿⠇⣾⣭⣭⣍⠁⠀⠀⠀⠀⣿⢣⣾⣿⣿⣿⡿⣛⠻⣮⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⠀⠀⠀⠀⢰⡇⣿⠟⣙⠿⣫⡾⢫⣾⣿⡌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣮⣙⣛⣛⣛⣛⣛⣛⣛⣋⣵⣶⡄⠀⠀⣸⣧⢻⣷⣝⠿⣫⣶⣿⣿⡿⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠹⣿⣷⣝⠿⣿⣿⣿⠿⢛⣵⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⣿⡿⢿⣿⣿⠇⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣶⣶⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣶⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣙⣛⣋⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣿⣿⣿⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢾⣆⢿⣦⣙⠛⣋⣴⣿⢱⣶⢀⣄⠀⠀⠀⠀⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡘⣿⡜⠟⢡⣶⣌⠻⢇⣿⡏⣾⣿⣷⡀⢀⣀⡛⠟⠰⢶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧⢹⣷⣾⡜⠿⢋⣶⣾⡿⣸⣿⣿⣿⡇⣿⡟⠻⠿⠟⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣇⢻⣿⢸⣿⣧⢻⡿⣱⣿⣿⢸⣿⡇⣿⡇⣿⣿⢘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⣧⡛⢸⣿⣿⢚⣵⣿⣿⣿⢸⣿⡇⣿⡇⣿⣿⣮⣛⣛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⣿⣿⣦⡙⣵⣾⣿⣿⣿⣿⢸⣿⣿⡎⠀⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⢸⠀⠀⡀⠀⠔⢋⢀
⠀⠀⠀⠀⡀⡀⠀⠀⠈⠀⢀⡄⠂⠈⡋⠀⠂⠀⠀⢸⠀⠀⠀⠼⠟⣿⠀⠀⠀⣤⢠⠀⠀⡀⠀⣀
⠀⠀⠀⠀⠁⢤⠀⠀⠀⠀⡎⡗⢀⣼⣄⡀⠀⠀⠀⠀⠀⢠⡄⠁⣸⠃⠀⢰⣏⠢⠈⠀⣴⡆⠘⢿⠇
⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⢀⣿⣾⣿⣿⣇⣀⣀⠀⠀⠀⠀⠄⣤⡿⠇⢀⡲⠈⢀⠈⠙⢋⠀⢀⡈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢸⣿⣿⣿⡏⠅⠀⠀⠀⠸⢦⣤⠀⠀⣶⣗⣸⢡⣡⣤⣤⠆⢉⣿⠇
⠀⠀⠀⠀⠒⢳⣶⠶⠖⠶⠀⠁⣸⣿⣿⣿⣿⣿⡄⠀⠀⣰⣿⣿⠇⠀⣿⡏⢈⣻⣿⣏⣹⠁⢺⣿⡀
⠀⠀⠀⠀⠀⢾⡏⠀⣤⠂⠀⢀⣿⣿⣿⣿⣟⠛⡗⠀⠀⣿⣿⣿⠀⠤⠭⢁⠈⢿⣿⣿⠃⣤⢘⣿⣅⡀
⠀⠀⠀⠀⣤⣼⡿⢺⣹⠀⠀⠀⠙⣿⣿⣿⢇⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⠈⠀⣿⡿⠀⠀⡿⠏
⠀⠀⠀⠀⣿⡟⡞⠋⠟⠀⠀⠀⠀⢿⣿⣿⡛⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⡄⠹⡇⢰⡆⠙⡀
⠀⠀⠀⠀⣿⡇⢳⠀⠀⡀⠀⠀⠀⣼⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠁⠘⠃⠰⠏
⠀⠀⠀⠀⠉⠁⠈⠰⡀⠣⠀⠀⠀⣿⣿⣿⣧⣣⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢿⡃⠈⡿⣇⣠⣤⠄⣿⡤⢀⣿⠇⠀⠀⠀⠀⠈⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠚⠋⠁⠀⠉⠀⠒⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠄⠠⠠⠠⠀⠀⠀⠄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢟⣫⣭⣭⣙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⣱⡿⠋⠉⠉⠛⣿⡌⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠁⠀⠀⠀⠀⢸⣿⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣿⡇⣿⣦⣤⣤⣤⣤⣼⣿⢸⣿⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣭⣭⣭⣭⣿⡇⣿⣭⣭⣭⣭⣭⣽⣿⢸⣿⣭⣭⣭⣽⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠉⠉⠉⠉⣿⣇⣿⠋⠉⠉⠉⠉⢹⣿⣸⣿⠉⠉⠉⠉⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⡿⠛⠻⣷⠀⠀⠀⢠⣿⠛⠛⣿⡄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸⣷⣤⣴⡿⠀⠀⠀⠘⢿⣦⣴⡿⠃⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣀⡀⠉⢁⠀⣀⠀⣀⣀⡀⢉⠉⠀⣀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡏⣿⠀⢹⠀⣿⠈⢹⡟⠁⠈⢳⡾⠋⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⠷⠋⠀⠘⠷⠋⠀⠘⠃⠀⠀⠈⠃⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⣤⣄⠀⣤⣀⠀⢀⣤⣄⠀⣠⣤⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡷⠆⠀⣯⡿⠀⢿⡧⠄⠀⣿⠤⠄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⠃⠀⠀⠏⠻⠀⠻⠶⠆⠀⠿⠶⠆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⣋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢴⣶⡦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠢⠀⠀⠄
⠀⠀⠀⠀⠀⣓⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⣠⡀⢯⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡞⠙⠠⡆⣤⣤⠄
⠀⠀⠀⠀⣭⠙⢈⠡⣿⠀⠀⠀⠀⠀⠀⠀⢸⣏⣿⢀⠠⣦⣶⠀⠀⠀⠀⠀⠀⠀⢳⣴⣤⠀⣴⣶
⠀⠀⠀⠈⠐⠒⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠋
⠀⠀⠀⠀⠐⠀⠐⠂⠂⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠐⠂⠐⠀⠀⠀⠀⠐⠂⠂⠐⠀⠂⠂⠐

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠠⡉⢰⠈⢉⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢟⣛⡻⡀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠆⠀⠤⠄
⠀⠀⠀⠀⠛⠂⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⢀⠰⠯⡸⣿⠇⡿⠀⠀⠀⠀⠀⠀⠀⣦⡀⣾⣻⣆⣠⡀
⠀⠀⠀⠰⣤⠄⠄⠙⠉⠁⠀⠀⠀⠀⠀⠀⠰⣀⠀⣼⣶⡾⠁⠀⠀⠀⠀⠀⠀⠀⠐⡷⠈⠉⢼⣿⡇
⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀


⠀⠀⠀⠀⠀⠄⣶⣶⠄⢀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠦⡄
⠀⠀⠀⠀⠶⠶⣾⣻⡆⢠⠀⠀⠀⠀⠀⠀⠀⡾⠿⡂⢠⡭⠃⠀⠀⠀⠀⠀⠀⠀⢀⠘⢦⣡⣏
⠀⠀⠀⢀⣀⣈⡌⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⢶⣶⡬⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⣿⣯⣧⡀
⠀⠀⠀⠘⠻⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠑⠞⠙⠋⠀⠀⠀⠀⠀⠀⠀⠚⠑⠛⠋⠛⠘⠁
⠀⠀⠠⠤⠄⠠⠀⠄⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢹⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡞⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠐⢦⣀⠂⠀⠀⠀⠀⠀⠀⠰⠏⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⢀⣴⣿⣿⣧⡀⢦⢘⣿⣷⣄⠀⠀⠀⠀⠀⢀⠟⢁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡾⣤⣾⠿⣿⠟⠻⢳⡎⢹⣋⢹⠭⣦⢀⣠⣀⡠⢄⠀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡉⠛⠂⠀⠐⠉⠈⠉⠀⠀⠀⠀⠛⠃⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠂⠠⠤⠄⣀⣀⠀⠐⠒⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡆
⠀⠀⠀⣴⡶⢷⣦⠀⠀⣤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣹⣿⣿
⠀⠀⠀⠙⠁⠋⠁⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠉⡅⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠚⢧⡀⠐⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⣠⣻⣆⠙⠛⠛⠋⠉⠓
⠀⠀⠀⠀⠀⡀⠀⢈⣷⠄⠘⠷⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣠⠿⡟⠛⡄
⢀⠀⢄⡉⠀⢄⡀⠀⠀⠀⠀⢰⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣅⣂⣿⡀⢃
⢠⠀⠀⠈⠓⠦⣌⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⡉⠼⣷⠸
⠈⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣬⣴⣤⠿⣿⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠐⠂⡔⣀⢡⠅
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠴⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢂⠀⢀⠀⠀⢀⠀⠀⠠⢀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⡀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠐⠀⠐⠸⠀⠄⠇⠃⠐⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⠇⠀⠆⠁⠀⠀⠀⠸⠈⠀⠀⠱⠈⠘⠀⠂⠀⠅
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢤⣠⡀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠈⠀⠀⢀⣬⣿⣿⣶⣦⣌⢄
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠈⠛⠋⠙⣿⣯⠉⠀⠣
⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣾⣿⣿⣶⣶⣄
⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⢀⣤⣎⠙⢛⠻⢻⣿⣿⣿⣿⣿⣿⡹⣦⠈⠃
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣠⠚⠻⣿⣿⣶⣤⡘⣯⣿⣿⣿⣿⣿⣵⡃⢀⠄
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢰⣥⣦⣤⣼⣿⣻⡿⠿⠿⠿⠿⢿⣿⣿⠿⢿⠋
⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⢻⠃⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠀⠐⠉⠉⡉⠙⠛⠀⠀⠀⢜⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢢
⠀⠀⠀⠀⠀⠈⠁⣰⣶⣶⣿⠟⢻⢿⡟⠀⠀⠀⠀⠰⠛⣿⠁⠀⠀⠀⠸⠗⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡄⠀⠀⠳⣄⠀⠀⠀⠀⣀⡀⠠⠀⠠⠀⠐⠂⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠠⠦⡀
⠀⠀⠀⠀⠀⠀⢸⠉⡃⢉⣉⣿⣦⣀⠀⠙⠳⢴⠁⠁⠀⠀⠀⠀⠀⠀⠀⠰⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢀⢀⠜⠐⠄
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⢈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠆⠊⠱⠁⠀⡠⢨⠄⠠
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⣄⣸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⡉⠁⠢⡀⠀⠀⠀⠰⠁⠀⣜⠗⠁⢀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡇⠈⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠈⢄⠀⠐⢄⡘⠀⠐⠄⠚⢅⣠⠐⠉
⠀⠀⠀⠀⠀⠀⡞⢀⠘⠏⠉⢉⣉⣉⣿⡇⠀⠀⠈⠁⠒⠀⠠⠄⠀⠀⠄⠂⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⣷⣾⣷⣿⣾⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⢈⡛⠿⠿⠿⡟⠉⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠂⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠁⠐⠠⡀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣗⠠⠀⢀⠀⠠⢔⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠤⠒⠀⠁⠀⠁⡀⠀⠀⠀⠄⠈⠉⠁⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠁⠀⠀⠀⠀⠀⠀⠈⠢⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣤⣤⣤⣀⣀⡀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣀⡭⠟⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣄⡈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡗⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣟⠛⣛⣿⢻⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡼⠃⠀⠀⠀⠀⠀⠀⢀⣴⠟⣁⣹⣄⠀⠀⠉⠻⢿⣾⣿⣿⣿⣿⡿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⢠⡞⣷⣾⣿⣿⡿⠀⠀⠀⠀⠀⠁⠈⠉⠁⠉⠀⠹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⣰⣿⠸⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣤⣿⡄⠀⠀⣀⣀⣀⣤⣀⠀⠀⠀⠀⠀⠀⢀⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟⠑⠒⠛⢉⣭⠉⠙⠻⠇⠀⠀⢀⣴⡾⢛⣉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣿⣿⡋⠁⠀⠀⠀⠤⠷⠶⣿⡿⠆⠢⠀⠀⣾⠿⢶⣶⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠴⢞⣁⣤⣤⣤⣄⣀⠀⠀⠀⠀⢠⣿⣄⣈⣿⡁⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⣿⡇⠙⠛⢻⣿⣏⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⢻⣿⣶⣄⢠⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢼⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣿⣿⠻⢿⣿⣿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⢀⣿⡆⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡎⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠻⡟⢁⡾⠉
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣯⠀⠀⠀⠈⢿⣿⣿⣿⡇⠀⠀⣆⠀⠀⠀⠀⠀⠀⠐⠶⣶⣾⢿⣿⣾⡷⠂⣼⠃
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠈⠉⣹⡇⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠟⠁⣼⠇
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⢀⣿⠀⠀⠀⠀⠈⠻⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣴⣿⠇⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⣶⣤⣄⣀⣤⡴⠏
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣶⣄
⠀⠀⠀⠀⠰⠾⡿⠿⠟⠋⣤⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⣿⣿⣷⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⢳⡀⢻⣿⣿⡇⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⠈⠁⠈⠉⣿⠀⠀⠀⠀⠉⣿⣿⣿⡏⠀⠀⠈⠉⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⢠⡏⠀⠀⠀⠀⢠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠾⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⣠⣶⣿⠆⠐⢦⣤⡀⠀⠈⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢀⣴⣾⣿⡿⠟⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣿⣿⠟⠁⠀⢠⣄⣤⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⢀⡀
⠀⠀⣠⠚⢩⣭⠙⢦⡀⠀⠀⠀⠀⠀⢠⣾⣄⠀⠀⠀⠀⠀⠀⣤⣾⣦⡀⣴⣄⠀⠀⠀⠀⢠⡶⣾⣭⡷⢦⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠰⡇⣶⣾⣿⣶⠆⡇⠀⠀⠀⠀⢀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢛⠃⢀⣽⠋⠀⠀⠀⠀⢸⡇⣀⣶⣀⢸⠀⠀⠀⠀⠀⠀⣿⣽⣝⡇
⠀⠀⠳⣄⠸⠟⢀⡴⠃⠀⠀⠀⠀⠸⣝⢿⣿⠇⠀⠀⠀⠀⠀⠀⠻⠿⠛⠁⠀⠀⠀⠀⠀⢸⡇⠉⠿⠉⢸⠀⠀⠀⠀⠀⠀⣿⣾⣶⡇
⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠛
⠀⠀⠀⠀⠀⡀
⠀⠀⢀⡞⢉⣍⠑⣆⠀⠀⠀⠀⠀⡤⣶⣋⣳⢤⡄⠀⠀⠀⠀⢀⣠⡤⢤⣄⡀⠀⠀⠀⠀⢠⣤⣤⣀⣤⣤⡀⠀⠀⠀⠀⢀⣤⣀⣀⣀⣀
⠀⠀⣼⡆⠻⠟⢣⣾⡄⠀⠀⠀⠀⡇⣒⣒⡒⢸⡇⠀⠀⠀⣰⠋⠀⢾⡷⠀⠙⣆⠀⠀⠀⠻⢃⠻⠙⠿⢿⡇⠀⠀⠀⢠⣾⣤⣿⣿⣅⣽⠆
⠀⠈⢿⡇⠀⠀⠸⠿⠃⠀⠀⠀⠀⡇⣒⣒⡒⢸⡇⠀⠀⠀⠘⠦⠖⠚⠓⠲⠴⠃⠀⠀⠀⠈⠻⣦⣿⡿⠋⠀⠀⠀⠀⠘⠻⠟⠛⠛⠻⠟
⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⣾⠁⠈⢹⡄⠀⠀⠀⠀⠀⢰⠒⠒⠒⣶⠀⠀⠀⠀⠀⠀⣠⠶⠶⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⡂⠀⠀⠀⠀⠀⠀⡤⢞⠛⢛⡲⣄
⠀⠀⠀⣿⣀⣀⣸⡇⠀⠀⠀⠀⠀⢸⢠⣿⡤⣿⠀⠀⠀⠀⢸⣿⡟⢫⡝⢻⣿⡇⠀⠀⠀⠀⠀⣸⠹⡇⠀⠀⠀⠀⠀⢸⠁⣿⣤⣼⡇⢹⡄
⠀⠀⠀⣯⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⠀⠉⠀⣽⠀⠀⠀⠀⢸⣿⣞⢻⡟⣫⣿⡇⠀⠀⠀⠀⠀⡇⣗⣿⠀⠀⠀⠀⠀⠘⣆⠿⠁⠸⢇⡼
⠀⠀⠀⠙⢮⡿⠟⠀⠀⠀⠀⠀⠀⠸⠿⠯⠿⠿⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⢧⣂⡟⠀⠀⠀⠀⠀⠀⠈⠙⠓⠚⠉

⠀⠀⠀⠀⣠⣦⡀⠀⠀⠀⠀⠀⠀⣠⡄⢠⣄⠀⠀⠀⠀⠀⠀⢀⣀⡴⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⣴⡿⢛⣝⠻⣷⡄⠀⠀⠀⠀⢿⠀⢀⡿⠀⠀⠀⠀⠀⠀⢸⢉⡛⠛⠉⡇⠀⠀⠀⠀⠀⠀⡾⠻⡖⠀⠀⠀⠀⢞⠿⣿⣾⣿⣿⣾⣿⣿
⠀⠀⣿⣟⢻⡿⢛⣿⠀⠀⠀⠀⠀⠀⢹⡏⣴⣦⠀⠀⠀⠀⠀⢸⠞⣷⠷⠦⡇⠀⠀⠀⠀⠀⠀⣿⢹⡏⠀⠀⠀⠀⠻⠿⠿⠻⠿⠿⠻⠿⠿
⠀⠀⠛⠛⠛⠚⠛⠛⠀⠀⠀⠀⠀⠀⠸⣇⣨⠇⠀⠀⠀⠀⠀⢸⣤⣬⣤⣤⡇⠀⠀⠀⠀⠀⠠⠿⣼⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠂
⠀⠀⠀⣔⢉⡏⢳⡄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣠⠖⠒⠒⠦⡀
⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⠀⠀⢠⣶⡞⣛⣛⢳⣶⡄⠀⠀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⣼⠁⠀⠀⢺⡷⠹⡆
⠀⠀⠀⠀⣙⣛⡁⠀⠀⠀⠀⠀⠀⣴⠛⠿⠷⣦⠀⠀⠀⠀⠈⢛⣥⠟⠻⣬⡛⠁⠀⠀⠀⠀⢸⣇⡘⣻⠀⠀⠀⠀⠀⢻⣠⣶⣤⠞⠁⢰⠇
⠀⠀⣴⣾⣿⣿⣿⣶⡀⠀⠀⠀⠀⠻⣄⣀⡼⠘⠗⠀⠀⠀⠀⠿⠿⠦⠴⠿⠿⠀⠀⠀⠀⠀⠸⠿⠶⢿⠀⠀⠀⠀⠀⠀⠓⠮⣤⡤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⡀⣾⠂⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⢸⡆⣀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⡟⢸⣏⣩⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⣿⢘⡿⢿⣿⣿⣿⣿
⠀⠀⠀⣀⠸⠇⣀⡒⢂⡤⠖⠛⡉⠍⠭⢙⠓⠦⣄⠒⣂⡀⠿⢀⡀⠀⠀⢸⣿⣿⡿⢿⣀⣿⠯⣭⠿⣛⣭⣶⣶⣶⣶⣮⣝⡻⢯⡭⢿⣇⣸⢿⣿⣿⣿
⠀⠀⠀⠘⠃⠠⠍⡵⠋⠄⣨⣴⣶⣿⣷⣶⣮⡑⠬⠳⡍⠥⠀⠛⠁⠀⠀⢸⣿⣿⣿⣤⣿⣓⡞⣡⣾⡿⠛⠉⠉⠈⠉⠙⠻⣿⣦⡹⣖⣻⣧⣼⣿⣿⣿
⠀⠀⠀⣷⢄⡀⡼⣁⢡⣾⣿⡿⠿⠇⠿⠿⣿⣿⣦⢁⡹⡄⣈⢴⡇⠀⠀⢸⣿⣿⡇⢙⣿⡟⣼⢿⠋⠀⣄⣀⣠⣇⣀⣀⠄⠈⢻⢿⡜⣿⣟⠁⣿⣿⣿
⠀⢀⣀⣸⢸⢲⡇⡀⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡇⠀⣷⢸⢸⣀⣀⠀⢸⣿⡿⠿⣸⢹⢧⣿⡇⠀⠀⠀⠈⠁⡇⠉⠀⠀⠀⠀⣿⣧⣿⢹⡸⠿⣿⣿
⠀⠈⠙⠄⣸⢸⡇⠁⣿⣯⣉⣩⣿⡇⣿⣯⣉⣩⣿⡇⠁⣾⢸⡄⠝⠉⠀⢸⣿⣷⣜⡿⢸⡾⣷⡧⠀⠶⠶⠆⢸⣷⠀⠶⠶⠆⢠⣿⡿⣾⣼⣿⣴⣿⣿
⠀⠀⠀⠀⠈⠃⢧⠉⠘⣿⣿⣟⣉⣀⣈⣙⣿⣿⡟⠈⢡⠇⠚⠀⠀⠀⠀⢸⣿⣿⣿⣿⣮⣧⢴⣶⠄⠀⠠⢴⡾⢿⣶⣦⠀⢀⣾⣿⣿⣿⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠳⣌⠨⡛⠿⣿⣿⣿⡿⠟⡋⢌⡴⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡻⣿⣷⣄⣀⠀⠀⢀⣀⣴⣿⡿⣳⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣅⣐⣐⣐⣀⡥⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⣝⡻⠿⠿⠿⠿⠿⣛⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠂⠀⠀⠀⠀⠀⠀⠙⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠐⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⠉⠃⢐⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠴⠀⢸⡃⠀⠀⠀⠀⠀⠉⠀⠀⣀⡠⠄⠘⠋⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣠⡤⠒⠀⠀⣠⡤⠴⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⡚⠃⠀⡴⠖⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣆⣰⣶⣄⢙⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠻⠻⠛⢿⡿⢠⡄⠀⠀⠃⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠸⠁⠀⠀⠀⢀⣠⠁⠀⠀⢠⣶⣿⡿⣯⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣤⣄⡀⠐⠿⣿⣷⣿⡇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠉⠀⠀⠀⠁⣿⣧⣤⣬⣽⣿⣯⣁⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡐⠆⡇⠀⠀⢀⣼⣳⡖⢀⣰⣶⣶⣥⣤⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡇⠀⡿⢹⡾⣱⣿⣿⣨⡿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⡥⠠⣾⣿⣿⣛⢷⣝⣋⣰⠿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣾⣿⣿⣿⣶⣬⣼⣿⣧⣾⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡁⠀⠈⢷⣿⣿⣿⣿⣿⣿⣿⢻⡏⠁⠀⠀⠘⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⡹⢻⣷⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣾⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣀⢀⡀⠀⠀⠀⣻⣿⡇⠀⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣷⣶⣦⣿⣿⡇⠀⠸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⢿⣿⣿⣿⡯⣿⣷⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣽⣿⣿⠄⠀⠉⢿⡿⣿⡿⠿⣿⣿⣷⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠁⠀⠀⠀⠘⡟⢿⡇⠀⠀⠈⠉⠙⠛⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡴⠞
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡏⢀⠰⣧⡄⢂⣀⣀⠂⣤⡷⢘⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢋⡀⠈⢀⡈⢀⠌⠠⠐⢄⠉⡀⠌⠀⡌⢛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⠀⠀⣰⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠇⠀⠀⠀⠀⠀⠘⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⣾⠏⢘⠀⠀⠀⠀⠀⠀⠀⢐⠋⣿⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⣁⣤⣶⣿⣷⣼⣆⠀⠀⠀⠀⠀⢀⡾⠀⣿⠀⠀⠙⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢠⣿⠋⠀⠀⠈⢻⣿⣦⠀⠀⠀⣠⣾⠇⠀⣿⠇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸⣿⠀⠀⡀⠀⠀⣿⣿⣿⣶⣾⣿⡟⠀⢰⡏⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠂⠀⣴⣿⢹⣿⣿⣿⡿⠁⠀⣾⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⠁⠀⠀⠠⡏⠉⠀⢻⣿⡿⠁⡔⠌⢉⠀⠀⠀⡀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢰⡇⢂⠀⠀⠐⠀⠀⠀⠻⡁⠀⢄⠄⢀⠀⠀⠀⡇⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠀⠈⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠠⠃⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠣⠀⠀⠀⠄

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