Liturgy Emojis & Symbols

⣿⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣶⣶⠶⠶⠶⢶⣶⣦⣤⣤
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠶⢿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⣤⣾
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾⣯⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡇⢹⣿⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣤⣤⣤⣤⣤⣤⣴⣿⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⣄⡀⣰⠖⠛⣿⣶⡄⠀⠀⠀⠀⠀⢀⠀⠛⠛⠛⠛⠛⠛⢻⣿⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣻⠖⣶⣭⡽⢻⡰⡒⣿⣿⡇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⡇⢸⣯⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⠋⠉⡇⠀⠙⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣷⠾⠾⣿
⣿⠀⠀⠀⠀⠀⠀⢀⣾⠋⠠⠀⠴⠆⠀⠈⠓⡄⠀⠀⠀⠀⠀⢠⣿⣤⡤⣴⡇⢀⠴⠿⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⡿⢤⣿⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⣾⠍⠀⠴⠘⣧⠀⠀⠀⠶⠘⣦⠀⠀⠀⠀⠘⠛⣿⣅⢩⣟⣿⣀⡼⣿⠁⠀⠀⠀⠀⠀⠀⢀⣦⠀⢸⠀⠀⢸⡇⢳⢶⣿
⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢨⡇⠀⠀⠀⣠⠾⢿⠛⠋⢋⣡⢿⡇⣿⣆⠀⠀⠀⠀⠀⢠⠞⠿⢳⡌⢳⣴⡏⡇⢸⢸⣿
⣿⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⢷⣤⠴⠟⠀⠀⢀⣀⠀⠀⣷⣧⡄⠘⣿⣧⠀⢀⣾⡄⠸⣆⠀⣸⠀⠀⢟⣧⡇⣼⢸⣿
⢻⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣆⣠⠖⠈⠲⠄⣤⢴⣞⠳⢤⣄⣀⣤⣾⡉⠳⡄⡟⠙⠒⠒⠛⠉⠀⡟⠛⠙⡇⢸⡟⡇⠀⠀⠘⢎⡷⢿⠟⣿
⢸⡄⠀⠀⠀⠀⠀⠀⣀⣤⠟⠒⠚⠓⢀⣠⣴⠾⡷⣬⣷⣦⣄⠉⠦⣿⡦⠀⠀⣷⡀⠀⠀⠀⠀⠀⢣⣀⣠⠇⠀⢷⢻⡀⠀⠀⠀⡟⣲⢦⣿
⢸⡇⠀⠀⠀⠀⠀⢸⡉⠀⡀⣤⡶⠾⣋⠹⣈⠀⠀⠺⡅⣹⡟⡄⠀⢻⣦⠁⠀⠘⡇⠀⠀⠀⠀⠀⠀⢙⣏⠀⠀⠈⢧⡳⣄⠀⠀⣧⡿⠚⣿
⢸⡇⠀⠀⠀⠀⠀⢸⣟⣿⡟⠐⢂⡄⠀⣤⡀⢾⠧⣬⣵⢞⣁⠀⠀⢸⢴⣾⡀⠀⣿⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠙⠮⢽⣾⢹⠃⠀⣾
⢸⡇⠀⠀⠀⠀⠀⢸⡶⠛⡱⣶⣀⣴⡤⣋⣁⠀⣶⠈⡍⠌⠛⡇⠀⠸⣠⣤⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⣽⣦⡀⣀⣀⣀⣴⣯⣌⣉⣧⣿
⢸⡇⠀⠀⠀⠀⠀⠈⣗⠀⠑⠀⠌⢹⡅⠘⠛⢤⣉⢠⣑⠶⣼⡇⠀⠀⡞⠁⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡏⣡⠄⣄⠎⣫⠛⣿⣿
⢸⡇⠀⠀⠀⠀⠀⠀⣏⡠⠀⣴⢂⣿⣃⡴⠆⠻⡇⡊⠀⣆⠀⠃⠀⠀⡷⠽⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠿⢰⢀⡥⣧⠘⣿⣿
⢸⡇⠀⠀⠀⠀⠀⠀⣿⣡⢀⡟⠉⠂⠀⡔⠀⣶⡈⣿⠶⠖⠓⠃⢠⡀⢰⣤⡇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠾⢭⠈⣃⣙⠀⣿⣿
⢸⠃⠀⠀⠀⠀⠀⠀⣿⡘⣴⡆⢸⣦⣤⠶⠞⠋⠁⠀⠀⢀⣀⠴⣿⡁⢈⣭⠀⢰⡇⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣼⠀⣶⠐⢠⠠⠦⢰⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⣿⠹⠋⠁⠈⠁⠀⠀⣀⠤⣶⠋⢡⠄⠀⠸⠉⠃⠘⠛⠀⢸⡇⢀⣀⡤⠖⠛⠉⠀⢀⡤⢤⣿⣀⣏⡰⠶⠷⠦⢘⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⡿⠚⣤⣤⠤⠐⠶⢰⣧⠀⠛⠀⠈⠃⢶⡖⣨⠿⠟⠅⠀⢸⠟⠋⠁⠀⠀⣠⠴⡊⠁⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠉⣿
⢸⠀⠀⠀⠀⠀⠀⠀⡏⢀⡇⠸⠗⠀⠟⠀⡍⠀⠿⠇⠰⠟⢠⡄⢰⡆⢸⠀⠀⣼⠄⠄⠀⠐⠛⡷⠶⠶⠶⠤⠤⠤⢤⣄⣀⣀⣐⣂⠀⠀⣿
⢸⠀⠀⠀⠀⠀⠀⠀⣷⣼⢧⡤⢤⡷⠠⣮⡠⢒⡆⣸⡧⢠⣼⣧⣨⡄⢰⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣿
⢸⠀⠀⠀⠀⠀⠀⠀⣏⢰⡆⢁⣙⢂⣀⠀⠇⠀⢀⡘⢂⡁⠰⠇⠈⠃⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⠀⠀⠀⠀⠀⠀⠀⡿⢴⣆⠸⠇⠘⠛⣀⠶⡀⠛⠃⠘⠃⣴⡆⠸⢀⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⠀⠀⠀⠀⠀⠀⠀⣿⢸⡉⠸⡇⠠⡷⠘⡋⠀⢾⡇⣺⠇⢨⡁⣦⡞⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⢸⠀⠀⠀⠀⠀⠀⢠⣧⣸⡇⠤⢬⣯⠤⠤⢀⣀⠠⣾⡅⣤⣌⣡⣋⡱⠀⠶⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠖⠀⠀⣿
⢸⡀⠀⠀⠀⠀⠀⢸⡟⠙⠊⠋⠙⠆⠀⠉⠁⠉⣁⠳⣀⡀⢰⠞⠛⠁⠐⠆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠲⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠸⡀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡅⡀⠈⣁⠶⠦⠤⢄⣀⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡀⠆⠀⠐⠰⢤⡄⢠⠋⢀⡎⠁⠀⡠⠀⠀⠈⠦⠀⠀⢀⡩⠴⠄⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢀⠀⠀⠀⡔⡀⠈⠑⠤⠀⠀⠀⠀⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤
⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠠⠀⠄⡀⠀⠀⠀⠀⠀⠋⠀⠐⠀⠀⠀⠀⠠⠁⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡀⠀⠄⠚⠖⠀⠀⠀⠀⠀⠠⠂⠰⠁⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⢤⠀⠓
⠀⠀⠀⠀⠀⠆⢀⠀⠀⠠⠆⠀⠀⡀⠂⠀⠀⠀⠸⠀⠠⠀⢠⡀⠂⠘⠓⠀⠆⠀⠄⠒⠀⠀⠀⠀⠸⣄⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠶⠶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⣀⣽⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⣀⣽⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠻⠇⠀⠀⠀⠀⣴⡶⠶⠶⠶⢾⠿⠀⠀⠿⠷⠶⠶⠶⢶⣦⠀⠀⠀⠀⠸⠟⠛⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠛⠛⢻⣷⠀⠀⣾⡟⠛⠛⠛⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⠀⢀⣠⣤⣤⣤⣀⠀⠀⣠⣤⣤⣤⣀⠀⠀⣀⣤⣤⣤⣄⠀⠀⣀⣤⣤⣤⣄⡀⠀⢀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠀⠀⠉⠛⠛⠛⠋⠁⠀⠈⠛⠛⠛⠋⠁⠀⠈⠙⠛⠛⠛⠉⠀⠈⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⣦⣄⣀⣠⣴⠶⠶⣶⣤⣀⣀⣤⡶⠶⢶⣤⣀⣀⣤⡶⠶⢶⣦⣀⣀⣤⣶⠶⢶⣦⣄⣀⣠⣴⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⡀⠀⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠀⢀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⣦⣤⣴⡾⠛⠛⠿⣶⣤⣤⡾⠟⠛⠻⢶⣤⣤⣶⠟⠛⠻⢷⣤⣤⣶⠿⠛⠛⢷⣦⣤⣴⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⢉⠉⠉⡉⠉⢉⠉⢉⠉⠉⡉⠉⣉⠉⢉⠉⠉⡉⠉⠉⠉⢉⠉⠉⡉⠉⣉⠉⢉⠉⠉⡉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⠇⠘⠟⠀⠿⠀⠻⠇⠘⠟⠀⠿⠀⠻⠃⠘⠟⠀⠿⠀⠻⠃⠸⠟⠀⠿⠀⠻⠃⠸⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⢶⣶⣶⣶⠶⣶⣶⣶⡶⠶⠶⣶⡾⠇





⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⠶⠶⠶⠶⣶⡶⠶⠶⣶⠶⣶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⣭⠷⠀⠀⠠⣡⠄⣢⣤⣽⣭⣅⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢿⣿⢲⡏⠀⠀⠀⠈⣷⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⢦⡀⠀⠀⠚⠸⣇⠀⠀⢀⣴⢃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⣵⡶⢉⡲⣝⢷⣄⠀⠱⠈⠙⠛⣉⡭⡋⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⣶⠋⢱⣻⣯⣽⣿⡎⠳⣜⠧⣀⠀⠀⠀⠉⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡏⣿⠇⠀⣾⣿⡶⠲⣿⣸⠀⠈⠳⣌⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣿⠀⢀⣿⠼⠞⠲⣾⣹⠀⠀⠀⡧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡿⠖⣋⡴⠚⠛⠷⣄⡙⠦⣄⠀⣿⡇⠀⠀⠀⣠⣶⠿⠿⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⣡⡴⠛⢁⠀⠀⠀⠀⠈⠛⠲⣌⠓⢯⡇⠀⠀⢰⠉⠀⠀⠀⠀⠙⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⣉⣴⠞⠋⠀⢲⠋⡱⠂⠀⡀⣀⠀⠀⠨⠑⢦⡙⠿⣿⣹⡀⢄⣠⠄⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣡⠞⠋⠀⠀⢀⠀⠈⠀⣷⡾⠶⣤⢤⣍⡓⢄⠀⠀⠈⠲⢄⡓⢵⣮⣙⠓⠀⣠⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢉⣴⡟⠁⠀⣴⠟⠛⠉⠉⠙⢶⡋⠀⠀⠤⠀⠀⢀⠞⠒⠈⠋⢳⣤⡉⠒⢌⡓⢍⢅⣈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⢼⡇⠀⣼⠁⠀⢀⡤⠴⠂⢘⢧⡀⠉⠉⠁⢌⠁⣀⡀⠠⠄⠀⠈⢻⡄⢸⢻⢿⣄⠉⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⢮⡑⠢⢿⡀⠰⠉⠤⠊⢀⡜⢻⣭⢷⣰⣶⣽⡀⠹⢶⣾⣕⡤⠀⢈⠇⡏⣸⢸⡁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡟⢢⣄⢉⡶⠂⠲⣒⡏⣴⣹⣿⣹⣿⡍⣿⣿⡶⠤⣀⣠⡤⠔⠁⠀⣇⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣷⠘⠟⠛⠣⠀⠀⠘⣷⡉⠴⠒⠉⠙⣿⡏⡀⢉⣛⣫⠶⠂⣤⣀⡀⣟⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⠀⠀⠀⢸⠋⠉⡙⠇⠀⠀⠀⠀⠀⢨⣿⠀⢘⣿⡤⠀⠀⣎⠁⠘⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⡆⠀⠀⣆⣀⠛⢧⣸⡦⣀⢀⡀⣰⣯⠾⠃⠈⠈⡈⠀⠘⠷⠂⢀⣿⠄⣉⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠘⡇⠀⠀⠀⣿⣿⣏⡽⢁⠷⠬⢇⠻⣵⡄⢀⣀⣦⣈⠑⠒⢂⢠⡞⠹⡆⣿⢯⣠⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⡽⠉⠛⢦⠎⠀⠀⠈⣦⡽⠒⠺⢏⡈⠺⠿⠯⢽⢊⣹⣾⣧⣜⣯⣹⠖⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⢸⡄⠀⠀⢹⣄⠀⣈⡷⠄⢤⠶⡁⠀⣠⠿⠮⢥⡤⠦⣄⣛⠄⣸⡇⠈⣿⣿⣿⠄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠛⠛⠛⠛⠛⢿⣿⣾⣟⢀⢘⡵⢶⣿⣞⠿⢧⣀⠜⠛⢦⣀⣀⣿⣯⣤⣤⡿⠿⠿⠼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣀⣀⣀⠔⠁⠀⠀⠀⠀⠀⠉⠓⠶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠛⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⣿
⠀⠀⠀⠀⢀⣠⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⣰⣾⣿⡿⠃⠀⠀⢀⡤⠒⠊⠉⠉⠐⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡀⠀⢸⣿⡇
⠀⠀⠸⣿⡏⠀⠀⢶⢁⡀⠴⠒⠛⢉⡉⣉⡁⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢋⢿⣆⠘⣿⣇
⠀⠀⠀⣿⣿⡀⠶⠛⣍⠃⠠⠐⠛⣉⣉⣉⣉⡀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠈⠿⣏⣿⣸⠏⠀⣿⣿
⠀⠀⠀⢻⣿⣇⠀⠈⠉⠀⢠⠖⢛⣉⣩⣥⣤⣅⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢰⠈⢩⠉⠁⠀⣿⣿
⠀⠀⠀⢸⣿⡟⡀⠀⠀⠀⠀⡶⠛⠉⣀⣀⣈⠉⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⢸⢠⠓⢣⠰⠀⢹⣿⣶⣶⣦
⢰⣿⣿⣽⣿⣿⢡⠀⠀⠀⠀⢠⡶⠛⠛⠉⡉⠛⠃⠀⠸⡄⠀⠀⠀⠀⠀⠀⢸⠀⢶⠖⠀⠀⢸⣿⣿⣿⣿
⢸⣿⣿⣿⢻⣿⡆⣆⠀⠀⠀⠀⣤⠶⢛⣋⣩⣛⡓⠀⠀⢣⠀⠀⠀⠀⠀⠀⠸⡀⢸⢀⠀⡀⠈⣿⣿⣿⣿
⠨⣿⣿⡿⠘⣿⣧⠸⡀⠀⠀⠀⠱⠚⠁⠀⠀⠀⠈⠀⠀⠸⡄⠀⠀⠀⠀⠠⡄⠿⢻⠺⠄⡇⠀⣿⣿⣿⣿
⠀⣿⣿⣷⠀⢿⣿⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠕⠢⣄⣧⠀⡀⠀⠀⠀⡇⠀⣸⡀⠀⡗⠀⢽⣿⣿⣿
⠀⣿⣿⣿⡆⠸⣿⣿⠘⡆⠀⢀⡠⣂⠤⣠⣤⣥⣤⣁⣂⣀⣷⣞⣴⣶⣿⣿⡇⢐⡛⣓⠀⣧⣄⣴⣿⣿⣿
⠀⣿⣿⣿⣇⣀⣿⣿⡇⣛⠐⡩⣊⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣻⠀⢸⠃⡿⠟⠉⠹⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣙⣴⣾⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠋⠉⠉⣿⡟⣷⣿⣶⣾⡷⢻⣷⠂⠀⣿⣿
⠀⣿⣿⣿⣿⡇⠀⠻⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣷⠟⠁⠀⠀⣿⣿
⠀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⠿⠀⠀⠀⠀⣿⣿
⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⡤⠐⠐⠁⠊⠐⠀⠐⠀⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣰⡦⠀⢀⠀⢠⡞⠑⣀⣠⠆⠀⠀⠀⠀⠐⠦⣀⡊⠷⡀⢀⠀⠀⡀⠥⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢄⣀⡄⠀⡀⠴⢋⣉⢴⣶⡻⠟⠀⢈⡀⠸⢧⢀⡀⠀⠘⢏⢶⡂⣈⣙⠓⠄⡀⠶⠦⠔⠂
⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠉⢈⡠⢆⣞⣿⣿⡶⠉⠁⠀⣰⣾⣇⡆⣪⢼⣇⣆⡀⠀⠉⢲⣾⣿⣍⡐⠂⣉⠁⠀⠈⠉⠉⣉⣉
⠀⠀⠀⠀⠀⠒⢚⢙⠂⠀⠄⢦⣞⣿⣞⣾⢏⣉⢊⢤⡀⠀⣠⣭⣿⣿⣯⣽⣿⣿⡽⣄⠀⠀⣠⣙⢈⣉⢶⣸⣿⣏⢦⠤⡀⠀⡶⢠⠒
⠀⠀⠂⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⢙⡈⠌⢛⣿⣿⡷⠙⣆⡙⠋⣭⡿⣻⣿⠿⣏⠟⠃⣠⢞⠵⣿⣿⡋⠡⢜⡉⠀⢀⣀⣀⣀⣉⣉⣀⣀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠜⡃⠒⠂⠐⠠⣍⡉⠩⠯⣥⡙⢦⡈⠹⢸⠁⠈⠉⣤⢞⣭⡼⠗⡉⡩⠄⠀⠒⠦⠔⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⠀⠉⠁⠸⢷⡀⣍⠐⠝⢦⡈⢀⡴⠻⠖⣉⢀⡰⠃⠀⠁⠀⠻⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠒⠠⠄⣠⢝⠻⠀⠀⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⢒⡚⢻⡉⢉⡍⢉⡛⢓⡒⠶⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣋⠡⣦⠸⠇⠘⢃⣘⠁⠈⠁⠘⣁⡙⠁⠺⠂⣶⠈⣙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⡁⠶⠆⢙⠀⠐⠀⠈⡀⣀⣀⠀⢀⡀⠀⣠⡀⠀⠀⠐⠠⠀⡋⠰⠇⣈⡓⢦⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠉⢶⡌⠉⠔⠀⢀⣤⣄⢴⡆⣿⢸⣿⡇⣿⣿⣼⣽⢣⣞⡆⢰⣤⠀⠀⠀⠀⠉⢰⡶⠙⢦⡀
⠀⠀⠀⠀⢀⡴⣃⡘⠓⡠⠊⠀⣀⡀⠘⣷⣻⡞⡧⠸⠞⠛⠃⢙⡁⠙⠛⠘⠿⠣⣾⠁⠀⣼⢣⡖⣀⠀⠀⠛⢁⡝⢦⡀
⠀⠀⠀⢠⠎⣤⡌⢁⠊⠀⣠⠜⢯⣳⡄⠈⠉⠀⠀⠀⠀⠀⣤⣼⣧⣤⠀⠀⠀⠀⠀⠀⠺⢵⢏⡾⣫⣆⡀⠀⠈⢡⡤⠳⡄
⠀⠀⢠⠋⠶⠄⡀⠁⢠⣶⠎⡳⡌⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⣿⢟⣵⡄⠀⠀⠰⠖⠹⡄
⠀⢠⠇⠺⠆⠀⠀⡘⠷⣽⡓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢋⣡⣶⣦⠄⠂⠸⠇⠹⡄
⠀⡟⠘⠓⠐⠀⣬⣝⣛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠘⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣩⡵⠞⠁⠀⠛⡃⢳
⢸⠁⡛⠂⢢⡌⣛⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠀⠀⠀⠀⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡛⢻⣇⠀⠘⢛⠈
⡿⢘⡋⠸⠈⠛⠛⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⠀⠀⣀⣀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠟⠙⠀⠀⢙⡁
⡇⢙⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡭⠇⠸⢥⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡁
⡇⢙⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡉
⣧⢉⣁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡁
⢸⡈⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣍⢁
⠀⣇⢉⣤⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠄⣤⡍⡼
⠀⠸⡎⣡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢠⡄⣰⠃
⠀⠀⠹⡄⢴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⣀⠀⠀⢀⠀⡀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠰⠦⣰⠃
⠀⠀⠀⠘⢆⠼⠇⡀⢄⠀⠀⠀⠀⠀⠀⠀⠨⠡⠌⡅⠰⠛⠆⠘⠊⠀⢅⢀⡀⠘⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠗⡴⠃
⠀⠀⠀⠀⠈⠳⡘⠃⣤⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⣤⠙⣣⠞⠁
⠀⠀⠀⠀⠀⠀⠙⢦⡉⠶⠆⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⣀⠰⠶⣈⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⡋⢰⡆⣈⠐⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠰⡆⢙⡡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣙⡁⠿⢠⣦⢀⣍⢁⡀⢀⡀⢀⡀⣩⡀⣴⡄⠿⢈⣋⡤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠤⠬⣥⣈⣃⣘⣃⣈⣡⣬⠥⠴⠒⠛⠉

⠉⠨⢩⠭⣭⡅⣭⠭⠭⠭⡭⢭⠭⣭⠭⠭⡭⡍⠉⡍⠉⡍⠉⡍⢩⠉⡉⠉⡉⢩⡭⡟⠟⠻⡏⢛⡻⡟⠻⠓⠫⠄⠀⠄⠀⠀⡄
⠈⠀⠀⢀⣗⡉⣜⡀⠀⠀⣇⠪⢸⠇⠀⠀⣷⡇⠀⣧⣤⠓⠚⡦⣸⠐⠄⠀⠇⢸⣿⡇⠀⠀⢿⡣⡯⡗⠀⠂⡸⠄⡄⡇⠀⠀⠃
⠐⣔⣼⣽⠩⠉⠀⠉⠒⠐⠉⠓⠉⠒⠀⠂⠉⠉⢺⠋⢀⣠⣤⣁⠀⣧⠘⠊⠈⠛⠉⠁⠉⠁⠉⠉⠉⠉⠈⠉⠁⠀⠀⠀⡀⠀⡂
⠀⠙⣶⢿⠀⠁⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣴⣿⡿⠿⣟⡷⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠇
⠀⣀⠀⢰⠀⢄⡄⡀⠠⠀⡠⣤⠀⢀⡀⠀⠀⠀⠧⠍⣦⠐⠀⣿⠗⠻⠤⠀⠀⠄⠀⠀⠀⣠⡤⠤⢄⣤⣤⣤⡀⠀⠀⠀⠀⠀⡄
⠀⠋⣞⣞⢔⢎⡙⣿⡢⠉⣳⣽⣻⣿⣯⣤⣠⣀⡠⠄⢹⠋⠘⠃⠀⠀⠀⣠⡆⠐⠠⣬⣾⣿⣿⣤⠊⡖⢄⢐⡇⠀⠀⠀⠉⠅⠃
⠀⠆⠾⢿⣮⣞⣓⣹⣻⣴⣯⠈⣹⣿⣿⣿⣿⣿⣗⡀⢄⠀⠀⠀⠀⠀⠀⣼⠀⣰⣿⣿⣷⣾⣿⠶⡻⣙⣦⣯⡅⣇⣁⣀⠘⠃⠃
⡀⡒⣀⣼⡀⠀⠀⠀⠀⠀⠸⠇⠴⣿⣿⣯⣿⣿⣿⣷⣠⠀⠀⠀⠀⠀⠒⣿⣠⣽⣿⣿⣿⣿⡟⢺⡥⡆⠀⠀⠀⠀⠀⢠⠄⡀⡀
⠀⡿⣋⣿⣷⣄⠀⠀⢀⣠⣦⣀⠹⣛⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣩⠯⣌⡼⠤⡀⠀⠀⠠⠮⢴⡟⡇
⠀⣧⣿⠆⠀⢸⣇⠈⣿⡀⠀⢀⠀⠾⢼⢻⣿⣿⣿⣿⠏⠀⠀⠐⠀⠀⢈⣯⣿⣿⣿⣿⠿⡆⠷⢅⡄⠀⢄⣷⠀⢈⠀⠀⢘⡧⡇
⠀⠇⠿⠖⣀⠙⢁⢈⡽⡗⣳⠟⠖⢐⠈⢩⢼⢛⣿⣿⠀⠀⠈⠀⠂⠀⠈⣻⣿⡿⢻⢭⡭⠁⡰⠚⠻⣏⠚⠠⠀⠈⠃⠘⠁⠃⠃
⡆⣧⣀⣐⡜⠖⠖⠀⠞⢾⣿⢀⣀⣸⣦⡁⡠⠉⢷⢫⠀⠐⠋⢀⠈⠉⢠⣿⢣⡯⠍⣠⣬⣶⣿⣀⣀⡿⡵⠐⠀⠀⠀⠀⣀⡀⡆
⡇⠏⢿⠿⠀⠘⢐⣶⠀⠃⢸⡏⣿⠷⣳⣿⣮⣃⣂⢸⠀⠀⠀⠀⠀⠀⠈⢘⡠⠰⣒⣵⡾⣷⢰⢶⣯⡇⠈⠀⡲⠀⠀⠀⠃⠂⠂
⡇⠂⠀⣸⡅⠤⠾⠶⠦⠤⠸⣇⣿⢚⣟⢵⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢸⣾⡿⣿⣷⣾⢿⣝⢿⣻⠧⢀⠒⠒⠓⠀⠀⡇⠀⠁
⠧⠤⠥⠭⠭⠭⠬⢍⣿⡿⢩⡭⣤⢤⣖⡬⣙⢻⣿⣟⣦⣀⠐⠀⠀⢠⡀⠸⣿⣽⢟⢫⣡⣞⣶⣒⣓⣂⡚⠟⠃⠀⠀⠀⠀⠒⠊
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⡯⣉⣌⣥⢽⣿⡿⣻⣬⢻⣿⣧⠀⠀⠂⢠⣧⢘⡗⣳⣾⣯⢉⢁⡀⣑⣸⡛⢏⢿⡖
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠚⠓⠛⠚⠓⠓⢺⡽⣾⠏⢿⣾⠀⠈⣇⠀⣮⡿⣋⢟⣗⡓⠙⠓⠛⠚⠛⠛⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠘⣼⡟⠀⠀⠉⠀⠘⣧⠏⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣷⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⡇⠀⠀⡇⢠⢸⡽⣂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣭⣼⡇⠀⠐⡁⠀⠐⣿⣧⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣟⡇⠀⠘⠇⠀⢸⣟⠗⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠅⢀⠀⠀⢀⣼⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠗⣶⡿⠝⢟⣞⢬⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢬⡿⢀⣀⠼⠤⠝

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢸⣷⣿⣀⠀⢀⣶⡀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⡀⣸⣷⡀⢠⣼⣿⣿⣿⣄⣿⣿⠃⣼⠇⢀⣴⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣘⣿⡽⡷⣽⣻⢿⣿⡟⢹⣿⢹⣿⣿⡟⣼⡏⢀⣾⡿⠁⣠⡶
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⣷⡙⠽⣿⡈⡿⡇⠀⣿⢸⣿⣿⣿⣟⣠⢿⣿⣤⣾⣿⡆⣠⣤
⠀⠀⠀⠀⠀⣤⣄⣈⣻⣿⣧⡙⠌⠳⡄⠙⠄⠐⢿⠀⠸⠀⡛⣼⢻⡿⢣⡿⢟⣿⣿⠿⢋⣜⣡⣤⡀
⠀⠀⠀⠀⠀⠀⠙⣯⣝⠻⣿⡛⢄⠀⠀⠀⠀⠀⣀⣄⣤⣄⣁⠀⠏⠀⢋⡴⠋⠈⢀⣴⡿⢟⣿⣛⡁
⠀⠀⠐⢶⣦⢤⣤⣿⣾⣝⣆⠙⠢⠀⠀⢀⡴⠋⢩⡴⠶⣦⠉⠻⣦⡀⠊⠀⠀⠀⠉⢀⣚⣯⡟⠋⠁
⠀⠀⠀⠀⢿⡿⠷⠶⣭⣍⡛⠓⠤⠀⢠⡟⢀⣀⣸⡇⠀⣿⣀⣀⡈⢷⠀⢀⣠⣔⣾⣽⣿⡍
⠀⠀⢰⣿⣿⣿⣿⣦⡤⠤⠀⢀⡀⠀⢸⡁⣏⠉⠉⠁⠀⠈⠉⣈⣿⢸⡇⠀⠼⠿⢿⣿⣿⣶⣶⣀⣀⣀⡀
⠀⠀⠀⠀⣀⣛⣻⣿⠿⠶⠶⠦⠀⠀⠘⣧⠙⠋⠙⡇⠀⣿⠉⠉⢁⡾⠀⢠⣄⣀⣀⣒⠛⠻⠿⣿⣿⡛⠁
⠀⠀⠀⠈⠋⠹⢿⣿⣷⣄⣀⣠⣤⣤⣀⣈⣷⣄⣀⣷⣤⣿⣀⣠⣞⣁⣀⣀⣀⣀⣉⣉⣙⡛⢭⣍⣉⠁
⠀⠀⠀⢠⣴⠾⢛⣻⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣭⣛⣻⣿⣃
⠀⠀⠀⠀⠀⠀⢛⡭⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣍⡉⠈⠉⠉
⠀⠀⠀⠀⢠⣶⠟⠻⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡁⠉⠛⠂
⠀⠀⠀⠀⠈⠀⠀⠰⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠸⠋⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠈⠙⣷⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠖⠿⣄⠀⠀⢀⡼⠀⠀⠈⣷⢀⣠⡴⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣄⣀⣀⣀⣰⡏⠀⠀⠀⢸⡷⠞⠿⢦⡀⣀⣠⣿⣯⠵⢶⣛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⢸⣿⢦⣤⣤⣟⠀⠀⠀⡼⠋⠉⠉⢻⡄⠀⠀⣹⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣤⣼⣇⠀⠀⠀⣹⠞⠛⠛⢷⡄⠀⠀⢠⣧⣴⣟⠉⠈⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠋⠀⠀⠙⣶⠶⠲⣯⠀⠀⠀⠀⣿⠶⢾⣋⠀⠀⠈⢳⡄⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠶⠷⢿⣆⣠⣤⣼⡇⠀⠀⠘⢳⣤⠴⣾⠁⠀⠀⢹⡆⠀⠀⣸⣧⠴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⢾⠁⠀⠀⢠⡟⠁⠀⠈⣳⡤⠤⣤⣾⡷⠛⠉⠛⢦⣠⠾⠷⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣼⣷⣤⣠⣼⣧⣀⠀⣼⠁⠀⠀⠀⢻⡀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠉⠻⣦⣀⣀⣠⠞⠛⠶⠴⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⢀⣀⣠⡄⠀⣿⠀⠀⠀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢⣤⠶⠛⠉⠁⠀⠀⠀⣿⠀⠀⠀⠀⠉⢁⠞⠧⠀⠀⠀⠀⠀⢀⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠉⠁⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠂⠈⠉⡏⠉⠙⠓⠶⣤⡀⠀⠀⣀⡴⠋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠄⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠙⣦⠚⠁⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣼⠤⠤⠤⠤⠀⢰⡇⠀⢨⠀⠀⣸⣄⣀⣠⣤⠀⠰⠋⠀⠀⠀⣼⠠⡤⣤⠤⢤⣤⡄⠀⠀⠀⠀⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠐⠋⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⣰⠆⠀⠀⠀⠀⠀⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠈⠁⠀⠀⠀⠈⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⡴⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⠚⠉⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣬⣿⣷⣶⣦⣤⣶⢤⣶⣿⣥⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠰⠀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠛⠛⠛⠻⠿⠓⠒⠒⠃⠂⠐⢻⣶⣿⣿⣿⡿⢿⣿⡿⠛⣿⡟⠀⠀⠀⠀⢀⣻⣟⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠍⠉⠁⠀⠀⣰⡟⠁⢸⡟⠀⠀⠀⠀⣠⡟⣻⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⢠⡟⠀⢠⡟⠀⠀⠀⢀⡞⣁⢿⣽⠏⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⡀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⢀⣨⡟⠀⣴⠟⠀⠀⠀⣰⢿⡷⣦⡾⢃⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠚⠀⠀⠀⠀⢻⣦⡀⠁⠀⠀⠀⠀⠀⢍⣠⠤⣴⠟⣡⡾⠁⠀⠀⠐⣰⣻⠾⣷⡏⠀⡡⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡐⣄⢀⣀⣴⣶⣶⠞⣣⡾⠋⠀⠀⠀⠀⣴⡿⠿⠛⣩⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣴⣶⣦⣤⣀⣷⠈⢿⣿⠛⢿⡀⠠⣿⠀⠀⠀⣀⡴⢻⠋⢁⣬⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠉⠀⣠⡼⠋⠀⠚⠉⣹⡇⠀⠸⣿⠔⢾⡇⠀⣿⣤⣶⣋⣁⣀⣸⣦⣇⡈⢈⣷⣾⠋⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⢾⣅⠀⠀⠀⠀⣰⡟⠀⠀⢰⣧⣄⠐⣿⠀⠹⡏⠉⠁⠀⣼⢫⡟⣻⣿⠋⠙⢻⣷⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠁⠀⠀⢀⣤⣶⡶⢛⠁⠀⠀⣠⣿⠁⠀⠀⢸⡏⠀⠀⢽⡄⠀⢻⡆⠀⠀⠘⠿⣿⡿⠿⡿⣾⡿⣿⣦⣾⣤⡀
⠀⠀⠀⠀⠀⠶⠶⠶⠒⠛⠛⠛⠚⠛⠛⠛⠛⠛⠛⠛⠛⣻⣿⠀⠀⠀⣼⠃⠀⡀⢹⡇⠀⢘⣿⣦⣤⣤⣶⣿⣷⣴⣦⣿⣤⣬⣭⣿⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⣛⠛⠛⠛⠛⣛⣛⣻⣿⣤⣨⣽⣿⣤⣽⣟⢉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠹⣿⣤⣿⣟⢻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠻⠛⢿⣷⡶⠃



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⣄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠒⠀⠀⠀⠐⣁⠒⠀⠀⠀⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⣇⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣩⣹⢏⢩⣉⠉⠉⠉⠉⠉⠉⠉⠉⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⡿⢸⣾⣿⣷⣦⠂⠄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣤⠖⠂⠔⢄⠀⠀⡀⠤⠤⢇⣿⣇⠸⣿⣿⣿⣿⣿⣦⠃⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⢠⠰⠀⠀⠁⢆⡀⢸⡻⣿⡄⢿⣿⣿⡏⠋⠉⠛⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠈⠊⣘⠒⣋⠵⢶⣶⠕⠊⠀⠀⢹⢿⣿⣮⣿⣿⡇⠀⠀⠀⠀⢀⠼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣠⡋⠁⠀⢀⣰⠖⠓⢄⠙⢿⣿⣿⣧⡀⠀⢀⠴⠏⣐⠀⡤⠤⠤⠤⢀⡀⡠⠤⠤⠄⠀⣀⣄⣀
⠀⠀⠀⠰⠶⠀⠐⠈⢀⡔⠂⢽⣿⣦⡼⠿⣿⡀⢀⣸⠄⠠⢼⣿⡿⠇⢀⠇⠀⠀⠀⠑⣧⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠈⢻⡆⠉⠒⠤⡀
⠀⢀⠔⠀⠀⠀⠀⠀⢸⣄⣄⣸⠻⢿⡷⣄⣸⠛⠻⣯⡀⠀⣸⢻⣿⠂⡎⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠿⢀⢀⣐⣷⣷⣄
⠀⡈⠀⠀⠀⠀⢀⣖⡉⠙⣿⡡⡢⣨⡿⢿⣿⣷⣤⣿⣿⡛⠣⣼⣿⣄⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣧
⠀⠡⡈⢉⠀⠀⠈⠿⣿⣷⡷⠻⣿⣿⡷⡆⣹⡏⠉⠙⣿⣷⣦⣿⣿⣿⣿⣍⠒⠶⣖⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣚⣛⣛⣛⣛⠛
⠀⠀⠈⠀⠀⠀⠀⠈⠉⠉⠀⠀⠘⠿⢿⡾⣽⣯⣵⣴⡟⡉⠉⣿⣿⣿⣿⣿⣿⣷⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣥⡖⠉⠉⢉⣩⠭⢍⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⣾⣿⣦⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡯

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⢠⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡘⢀
⠀⠀⡤⢌⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⡡⠃
⠀⠀⠑⠢⢽⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠔⠒⢻⡄
⠀⠀⠀⡠⡾⠀⠈⢉⠋⠑⠒⢒⡶⠤⡤⠤⠤⠤⠤⢤⠤⠤⠤⢤⠤⠤⠤⠤⠔⠒⠒⠞⠉⣉⠀⠀⢀⡘⡀⠀⠀⢊⠓⠄
⠀⠀⠈⠐⠁⠀⠀⡘⢐⡆⠀⠈⠂⠀⠇⠀⠀⠀⠀⢸⢀⡀⠀⠐⠃⠀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠑⠁⢃
⠀⠀⠀⠀⠀⠀⠀⡆⡀⠀⢠⣢⠀⢠⠀⠀⠀⠀⠀⢸⠈⠁⠀⡠⠀⠀⡇⠀⠀⠀⠀⠀⡄⣀⠋⠃⠀⣖⠜⡀
⠀⠀⠀⠀⠀⠀⢠⠀⠚⣀⢄⠀⠶⢮⠀⠀⠀⠀⠀⠀⢰⡆⠀⠉⠠⡲⠅⠀⠀⠀⠀⠀⢚⠚⢀⠊⠐⠠⢀⣃
⠀⠀⠀⠀⠀⠀⠸⠔⠈⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠈⠢⢄⣀⠀⠀⠀⠀⠀⠘⡔⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠸⠉⣼⢃⡖⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣆⠂⣞⠠⠤⠤⠤⠤⠤⢀⡀⢸⠀⠃⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢈⠋⡸⡟⡦⠤⠤⠤⠤⠤⠖⢻⣤⣷⣤⣄⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠃⡠⠁⢃⠀⠀⠀⠀⠀⣄⠀⣼⣯⣯⢞⣟⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠱⠀⠸⡀⠀⠀⠀⠈⠁⢀⠛⠻⣯⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣀⢵⡀⠀⠀⠀⠀⡼⠤⢄⣈⣛⣛⠟⠻⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⢨⠀⠘⣷⣄⠀⢀⣾⡥⠶⠒⠻⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡤⣀⠴⡿⡷⠀⠈⢧⡀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣞⣏⢀⠀⡤⠒⠉⠉⠉⠐⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠁⠀⠈⠢⢄⡀⠀⠀⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣄⣀⣠⢐⠀⠀⠀⠀⠈⠐⠐⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂



⢸⣷⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⡀⠐⠇⠀⣦⡀
⠸⣿⠟⠋⠉⠁⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⡀⣄⠀⠀⠈⠙⣿⠃⠀⠘⣁⡀⣠⠶⠿⢶⣄⠀⠀⠀⠀⠀⠀⠈
⠀⠈⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠉⠙⣆⠀⠀⠀⠁⠀⠀⠀⠘⠛⠿⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠁⠀⢀⠒⠂⠀⣠⣷⣽⠆⠀⠀⠀⠀⠐⠂⠺⠂⠀⠀⠐⢂⠀⠀⠀⠀⠀⠀⠐⣂⠠⠀⠀⠀⠀⠀⠀⠤⠤
⢸⣦⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈⠁⣤⣾⣿⠏⠋⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠁⠀⠀⠀⢶⡞⠀⠀⠀⠀⠀⠉⠐
⠸⢿⠇⠀⠀⠙⢿⣿⡇⠀⠀⠀⠀⠀⢻⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐
⢠⣴⣦⣄⠀⠀⠘⠿⢷⣦⣤⡆⠀⠀⠘⠟⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂
⢠⣌⠙⠋⠀⠀⠀⠀⠀⠀⠀⠓⠠⣤⠀⣀⣀⠀⠄⠛⢻⣷⣤⣦⠀⠀⠀⠟⠓⠀⠀⠀⠀⠀⢀⠀⢰⡋⠗⠀⢰⣦⠀⠀⠀⠀⠀⢰⣦
⠈⠛⣿⣦⣼⠄⠀⠀⠀⠀⠀⠀⠀⠛⠈⠩⠀⠀⠀⢀⣿⠉⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠐⠷⠸⡇⠀⢡⠄⠀⣘⣿⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠉⠁⡿⠧⡄⠀⣀⡀⠘⠳⣦⠀⢀⠀⣾⠛⠳⢌⣧⡠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣚⠉⢁⣀⣀⣋⡉⠁⠀⠀⠀⠀⠀⢀
⠀⣀⠀⠀⠐⠲⠾⠛⠙⠛⣿⣦⣁⠀⠎⠀⠹⢶⣶⣎⠈⢁⡄⠀⠀⠀⣠⣤⣀⢶⡶⠀⠲⡄⢸⢿⣀⠈⠿⣿⡿⢷⣀⠀⠀⠀⠀⠀⠘
⢸⣿⣷⡆⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢻⠀⠘⠣⠀⠀⠀⠈⠉⢻⡆⢻⣤⡄⠀⠸⠋⢙⣦⣴⠀⠀⠈⣛
⢸⣿⣿⡇⠀⢠⡀⠀⠀⠀⣼⣿⡏⢴⣦⣤⡀⠀⢀⡈⠀⠀⠐⣧⠀⠀⢠⣤⢾⢷⣿⣿⡁⣠⣄⠀⠈⠏⢸⡗⡄⠀⣉⡀
⢸⣿⣿⡇⠀⠀⣹⣦⣤⣶⣟⠿⣿⣿⣿⣿⣿⣯⣽⠁⠐⠀⣴⣿⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣧⣼⠃⠀⠀⣸⣿⡀⢨⣷⠀⠈⠀⣠⣴
⢸⣿⣿⠇⠀⠀⠛⢿⠉⠀⠈⢡⣄⡉⠛⠁⠈⠉⠉⠀⠀⠀⣩⣿⠀⢰⣿⣿⣿⠟⠛⠛⢿⣿⣿⣿⠀⠀⠀⠈⢭⣭⠛⠀⠀⠀⢰⣿⡇
⠈⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⣿⣇⣀⡀⠀⠀⠀⠀⠀⠀⠙⠿⠀⠘⢿⠏⠁⠀⢀⡀⠀⠀⠉⢻⡆⣰⡆⠀⠀⠉⣦⠀⢠⡄⠈⣿⣷
⢸⣿⣿⣿⣷⣶⣄⠀⠻⣦⣴⣿⣿⣿⣿⣅⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⣤⣴⣆⠁⠀⠠⠀⠜⢠⣿⣿⣦⡀⠀⣿⡄⠘⠃⠀⠙⢿
⢸⣿⣿⣿⣿⣿⣿⣿⡄⠙⣿⣿⣿⣿⣿⣿⣿⢀⠀⣀⣀⣀⣀⣀⠀⠀⠐⠻⠲⣶⠿⣷⡀⠀⡀⢀⣾⣿⡿⢿⣿⣦⣙⡇⠀⠀⠀⠄⠈
⠘⠻⣿⣿⣿⣿⣿⣿⡷⠀⢻⣿⣿⠿⢿⢿⣿⣴⣾⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠴⢉⣀⣤⣭⡀⠉⠋⠰⡦⢠⣿⣿⣷⣶⢢⡀⠀⣤
⠀⠀⠀⠉⠀⢹⣿⣿⠇⠀⠀⠹⠛⠿⠶⠂⣸⣿⣿⣿⣿⠏⡿⠟⠃⠙⠂⠀⠀⠀⠀⠈⠻⠃⠈⠙⠀⠀⠀⣀⣼⣿⣿⣿⣟⠀⠀⠀⣈
⠀⢠⢠⣤⡀⠈⠛⠋⠀⠀⢀⣀⠀⢀⠀⠀⣿⣿⣿⣿⠄⠀⠀⡀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⣀⣤⡉⠛⣿⣿⣿⣿⣿⡀
⢠⣷⠈⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⣼⣷⣦⣼⠿⠛⠗⠆⠀⠀⠁⢀⣠⡀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠿⠟⢁⣾⣇⠀⣄⣀⣨⣿⣷⣶⣦
⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣨⣿⠿⢿⡷⠀⠀⠀⢰⠚⠲⣿⡞⣩⣿⠟⠛⣶⠂⠀⢀⣀⣀⠈⠀⠀⢸⡟⢿⠀⢸⣿⣿⣿⡏⠁⣀
⠀⠉⠻⣿⡆⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⢨⣧⠀⠀⠀⠿⡄⠀⠻⠈⠛⠙⠓⠀⢀⣤⣼⣿⡏⠙⡇⠀⣠⣾⣿⣿⡄⢈⣥⣶⣿⣿⣿⣿
⢈⣡⣀⣼⣷⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠈⠀⠀⣠⣤⣤⣀⠀⢀⣄⠀⣶⣶⣶⡿⢻⣿⣿⣇⣴⡧⠎⢹⣿⣿⣷⢿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣦⣤⣶⣶⣤⣤⣤⣿⣷⣦⣴⣿⣾⣯⣄⠙⢻⣿⣿⣿⡄⣹⣿⡉⠀⢾⣿⣿⣿⣛⠃⠀⢻⡏⣉⢙⡘⢻⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⢻⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡄⠀⠀⢸⠀⠀⠀⡜⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠙⣄⠀⠀⢽⠀⠀⢸⠀⠀⣠⠇⠀⠀⡘⠁⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⡀⠀⠹⣆⠀⠘⣧⠀⣸⠇⢀⣿⠀⢀⡾⠁⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⡀⠀⠈⠳⣤⠀⠸⣧⡀⢿⣄⣹⣆⣸⡇⢠⡾⠁⢀⣴⠋⠀⠀⢀⡤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡀⠘⠷⣦⣽⡿⠛⠛⠛⠛⠛⠻⢿⣧⣼⠏⠁⣀⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠲⠴⢤⣀⣀⠀⠙⠷⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⠟⠉⠀⣀⣤⣤⠖⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠛⠒⠚⠳⠶⠾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠶⠶⠶⠚⠒⠒⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⠾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡷⠶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠁⠀⠀⢀⣠⣴⠿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢶⣤⡀⠀⠀⠉⠉⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠁⣀⣴⠟⢻⣦⣄⡀⠀⠀⠀⠀⣀⣤⣾⠻⢷⣄⠈⠉⠛⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⡴⠋⠁⣠⡿⠉⣿⠛⣿⣿⣿⡟⢿⡍⠛⣷⠀⠙⠳⣆⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⢀⣼⠋⠀⣼⠏⠀⣿⣿⣿⡇⠈⣿⡀⠈⠳⡄⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠞⠀⠀⢠⠟⠀⠈⡇⣿⡟⣧⠀⠈⣇⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⡟⠀⠀⢸⡇⣿⡇⣧⠀⠀⠘⡆⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠘⣡⣿⣧⣜⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣶⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⢀⣴⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠀⠀⠀⠉⠳⡼⣟⢿⠈⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡚⠁⣠⣄⣀⡤⠬⡟⠳⣄⣹⡈⠩⣿⠗⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠠⠆⠀⠉⣤⡀⠘⠁⣹⠀⠱⣄⡀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢶⣶⡄⠈⠛⣇⠐⠖⠀⢀⣠⢄⠀⢰⠁⢀⡠⠚⠃⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠻⠏⡇⠀⠀⠘⣆⠀⠘⢿⣀⡼⢠⡾⠚⡏⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠶⠿⢧⠀⣀⡴⢾⢓⡶⣶⣲⢶⢋⠿⣄⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⡋⠁⣀⡤⣼⣿⠿⢳⡿⢯⡠⠜⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⢀⣀⣈⡧⣄⣀⣠⣴⠟⣠⠴⣇⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠉⠙⠛⠛⠻⡖⠚⠁⠀⢨⠿⠟⠋⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⣰⠋⠂⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣔⠦⢄⣀⠀⠀⠈⠳⢶⠇⠉⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⣼⠟⠒⣶⠦⣄⠙⠲⠤⣄⣀⣀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⣠⠞⠁⠀⠈⢢⡀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠾⠋⠀⠀⠀⠀⡴⠟⢦⡀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⢳⠀⠀⠈⠁⠀⠀⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠘⣆⣀⣤⠤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⡽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠊⠀⢀⡀⠀⠀⠀⠀⠀⣰⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⡀⠀⠀⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣄⣹⠇⠀⠀⠀⠀⠀⠀⠀⢦⣤⣾⢃⣠⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠸⣿⠟⠋⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠛⠛⢋⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⢱⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣤⣤⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⢀⡸⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠚⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⢠⣿⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⣤⣴⣾⣿⡄
⠀⢀⣼⠏⡀⠁⡈⠁⡀⢀⡀⢀⡈⢓⡦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷
⠀⣸⠋⡄⠁⡄⠁⡄⠁⡄⠁⡄⠁⡄⢁⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇
⢀⠃⠄⠀⠄⠀⠄⠠⠄⠠⠄⠠⠄⠠⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿
⢸⠂⠀⠂⠀⠂⠀⠂⠐⠂⠐⠂⠐⠂⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇
⠚⣷⡃⡀⠃⣀⣧⣄⠃⡐⠃⡀⠃⡐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣷⡀
⠀⠈⠙⠷⣶⠟⠁⠉⠿⣅⡀⢁⡀⢁⣈⣥⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠹⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠛⠛⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠟⠁⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠇⠈⠿⠟⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡾⢷⠀⠀⠀⠀⠀⠀⠘⠛⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠄

⠀⠀⠀⠠⠤⣤⣤⣤⣤⣀⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⠤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⡿⣷⣄⠀⠀⠀⠀⠃⡀⠀⢀⣰⣿⣩⣿⣟⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠽⣿⣿⣟⣻⠃⠀⠀⠀⢣⢻⣟⣿⡿⡟⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠟⠆⠀⠀⠀⡼⠸⠫⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣾⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠝⠟⠟⠟⠂





⣶⡶⠶⠶⠶⢶⢶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢂
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⢸⠀⠀⠀⠀⠡⡀
⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠱⡀
⠀⠀⢠⠃⠢⡀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡀⠱⡀⠀⠀⠀⠀⠀⠀⠀⡜⢀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠱⡄
⠀⢠⠃⠀⠀⠙⢦⡀⠀⠐⠧⣄⣄⡴⠋⠀⠀⢳⡀⠀⠀⠀⠀⠀⡼⠀⠀⠙⢄⠀⠀⢀⣸⡀⠀⠀⢀⡴⠋⠀⠹⡀
⠀⡞⠀⠀⠀⠀⣠⠀⠀⠐⠆⠀⠹⣧⠀⠀⠀⠀⢧⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠳⠄⡇⠀⠀⢀⣠⠏⠀⠀⠀⠀⢳
⢰⠁⠤⠤⠤⠴⣧⠀⢸⡤⠦⠢⡀⣿⠦⠤⠤⠀⠸⡆⠀⠀⢀⡏⠀⠀⠀⠀⡶⠒⠒⠃⠀⠀⠘⠁⠀⢠⠀⠀⠀⠈⣇
⣼⠀⠀⠀⡀⠀⠸⠢⠀⠀⠀⢀⠕⠇⠀⢀⠀⠀⠀⣧⠀⠀⢸⠁⠈⠉⠉⠙⣇⣀⣀⡀⠀⠀⢀⣀⣀⣸⠉⠉⠉⠀⢸
⡇⠀⠀⠀⡌⠉⠒⠒⠒⠲⠶⠒⠒⠒⠊⢁⠀⠀⠀⢸⠀⠀⣾⠀⠀⠀⠀⠀⠈⠉⠉⣯⠀⠀⣸⢦
⡇⠀⠀⠀⣱⠀⠒⠒⠲⠶⠖⠒⠒⠀⠀⣎⠀⠀⠀⢸⠀⠀⡏⠀⠀⠀⠀⠀⡠⠂⠀⡷⠀⠀⣽⡀⠳⣄
⡇⠀⢠⠞⠉⣇⠐⠒⠶⠶⠶⠒⠂⠀⣸⠋⠳⣄⠀⢸⡀⠀⡇⠀⠀⠀⣠⠞⠁⠀⡜⠀⣬⡄⠈⢷⠀⠈⠳⡄
⡇⠐⠁⠀⠀⢈⡆⠀⠀⠀⠀⠀⢀⡴⡇⠀⠀⠈⠢⢸⡇⠀⡇⠀⢀⠜⠁⠀⠀⡐⠀⡆⠀⠀⢢⠈⢧⠀⠀⠘⢦⡀
⡇⠀⠀⠀⠀⡜⠀⠀⠰⡶⠶⠖⠋⠀⠹⡄⠀⠀⠀⢸⡇⠀⡇⠐⠁⠀⠀⠀⣰⠃⠀⠉⣈⣁⡉⠀⠘⣇⠀⠀⠀⠙⠄
⡇⠀⠀⠀⡸⠁⠀⢀⠠⠷⠀⠠⢀⠀⠀⢳⡀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⢠⠇⢄⠀⠀⠀⠀⠀⠀⡄⠘⣆
⡇⠀⠀⢰⠃⠀⠀⠘⠀⠠⠆⠄⠘⠀⠀⠀⢣⠀⠀⢸⡇⠀⡇⠀⠀⠀⢀⡏⠀⠈⠄⠈⠉⠁⠀⡸⠀⠀⠹⡄
⠀⠀⢠⠇⠀⠀⠀⠀⢀⠆⠀⣇⠀⠀⠀⠀⠈⢇⠀⢸⡇⠀⡇⠀⠀⠀⡞⠀⠀⠀⠀⠀⠶⠒⠈⠀⠀⠀⠀⢹⡀
⠀⠀⠎⠀⠀⠀⠀⠀⡼⠀⠀⢹⡀⠀⠀⠀⠀⠈⠆⢸⡇⠀⠃⠀⠀⡼⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠈⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⡄⢠⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⢤⠀⠙⠀⠀⣀⣀⣠⣤⠿⠂⡤⠤⠤⠤⠼⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⡜⠀⠀⢣⡀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⢦⡀⠙⠒⠶⠤⠤⠤⠤⠴⠖⠋⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⡀⠀⠀⠒⣃⠄
⣀⣀⣀⣀⣉⣙⣒⣶⣶⣶⣶⣶⣶⣒⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣑⣢⣬⣭⣭⣤⣐⣊⣉⣉⣉⣉⣉⣉
⣿⣿⣯⣿⣯⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣭⣿⣯⣿⣭⣭

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠀⣠⣤⣤⣤⡸⣷⡀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⡟⠁⠀⢹⣷⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣰⡏⢻⣆⠀⢀⣾⠇⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠙⠛⠛⠁⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢁⣴⠶⢶⣦⠀⠀⠀⠀⠀⠀⢹⠉⣭⠀⠀⣶⠄⣴⠶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡿⠁⠀⠀⣿⠆⠀⠀⠀⠀⠀⢸⡄⣿⣤⣶⣿⡀⣿⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠈⣷⣄⣀⣼⠟⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⣿⠀⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠈⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⣀⣤⣴⣦⡄⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠘⣿⠀⢠⣄⠀⣼⡇⣠⡾⠛⠻⣦⠀⣟⢀⣀⣼⡟⢰⡇⠀⠀⠀⣿⠀⠀⠹⣿⡄⣾
⠀⠀⠀⠀⠀⠀⢻⣇⣿⣿⣷⣿⢡⡟⠀⠀⠀⣿⠃⣿⣿⣟⠉⠀⢸⣿⠀⠀⠀⣿⠀⠀⣠⣿⠇⣿
⠀⠀⠀⠀⠀⠀⠸⣿⡟⠘⣿⡿⠈⢿⣤⣤⣾⠏⠀⣿⠀⠻⣷⠄⢸⣿⣤⣤⡀⣿⣴⡾⠟⠃⠀⡛



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠀⠤⠀⠀⠄⠒⢉⣉⠒⠠⠐⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠖⣋⣉⠀⡀⠐⠒⢒⣈⠤⠤⣈⣀⠀⠈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⢁⡀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡖⠠⢀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣴⡦⠔⠀⠑⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠹⡒⠤⢀⣀⣀⣤⠤⣤⣀⡀⠻⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠼⣦⠤⠵⠖⠚⠉⠑⢦⡀⠀⠉⢿⣿⡷⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡟⣠⣤⠘⢆⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣿⣷⠀⠈⠳⣄⡤⠤⣄
⠀⠀⠀⠆⠀⠀⠀⣼⠙⠟⠋⠀⠈⢆⠀⠀⠀⠀⠀⡇⠀⠀⣠⡿⢿⣦⠔⠊⠉⡆⠀⠀⠑⢄⢰
⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠘⣆⠀⡀⠀⠀⢇⠀⠀⢹⠀⠈⡇⠀⠀⠀⡇⠀⢠⣶⠀⢙⣦⡀
⠀⠀⠀⠀⠀⣰⣿⡏⢷⠀⣠⠀⠀⢀⡼⠐⠁⠀⠀⠀⠉⠀⠀⣇⠀⢻⠀⠀⠀⠙⢦⣄⣛⣠⠞⠀⠙⠀⡀
⠀⠀⠀⠀⢠⣿⣿⡇⠘⠋⢀⡤⠒⠁⠀⠀⠀⢀⣠⠤⠤⠄⣀⠈⠀⠘⡄⢀⠀⠀⢀⠈⣿⣿⡀⠀⠀⠀⠑
⠒⠀⠀⠀⣼⠋⠹⣏⡤⢞⣉⡀⠀⠈⠉⢰⡾⠉⠀⠀⣀⠀⠈⠳⡀⠀⠀⡶⠒⣂⣶⣦⣿⣿⠀⠤⡄⠀⢠
⠀⠀⠀⣠⡟⣠⠖⣏⠀⠀⠀⠈⠓⢤⠀⢸⡇⠀⠀⠈⠁⢹⠀⠀⡇⠀⢸⠃⠸⡿⠋⢻⣿⡏⠀⠀⡃⢠⠇
⠀⠀⢐⣧⡏⣴⣆⢹⠀⠀⠀⠀⠀⠀⠳⣄⠙⠦⢄⣀⡠⠞⠀⣰⠃⠀⡿⠀⠀⢻⠀⠀⣿⠁⠀⠀⢃⠊
⢀⡴⠋⠹⣧⡘⠏⢸⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣀⣀⠤⠊⠁⠀⠀⠃⣄⠀⠘⠂⢠⠏⠀⠀⢠⡇
⢸⠀⠀⠀⢹⣿⣿⣾⡇⠀⡀⠠⠤⠤⠴⠒⠒⠒⠒⠂⠀⠈⠉⣉⡒⠦⢧⠈⠀⠀⣠⠞⠀⠀⠀⠈
⢻⠀⢁⡤⢰⣿⣿⣟⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⠼⠌⠀⠀⠀⠙⠀⡴⠋
⢸⠀⡞⠰⣿⠋⢻⡟⠀⠈⠛⠲⠦⠤⢤⣄⣀⠀⢀⣀⣙⣛⣀⡀⠀⢀⡀⢴⠞⠀⠀⠀⠐⠂
⠸⣸⠁⠀⢻⡀⢸⡇⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⣹⣿⣿⠏⠉⠉⠁⠀⠈
⠰⡏⠀⠀⠘⡧⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠏
⠀⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⣀⣀⣈⡿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠉⣉⣩⡵⠞⠁
⣶⣶⣶⣶⢶⣶⣶⣶⣶⢶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⣷⡄⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢿⠋⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⡀⡌⢀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠄⠀⠀⡠⠄⢯⣷⠈⡴⢸⣇⢀⠀⡀⠀⣤⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢣⠈⠋⢠⡇⠸⠟⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠐⠂⠐⠂⠸⡄⡀⠂⠘⣿⡆⢠⢏⡄⠄⠂⠐⠂⠀⠀⠀⠘⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠠⠀⡀⠀⠀⠀⠀⠙⢞⣧⡀⣽⣿⢨⣿⠀⠀⠀⠀⠀⠀⠔⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⢷⠸⢃⡾⢣⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠒⠀⠐⠂⠚⠁⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠉⠉⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠈⠉⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠁⠀⣀⣀⣀⣀⣈⠑⢄⠀⠀⠆⠰⠀⠀⡠⠊⣁⣀⣀⣀⣀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⢻⡈⠉⠉⠉⠙⢦⠃⠀⢀⠀⠀⠜⡴⠋⠉⠉⠉⢉⡟⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠙⢦⣀⠀⠀⣀⡵⠀⠀⠀⠀⢎⣀⠀⠀⣀⡴⠋⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⡀⠒⠰⠶⠖⠒⠂⠀⠂⠀⠀⠐⠀⠐⠒⠲⠶⠆⠒⢀⡄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠒⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⢒⡒⣛⣛⢋⠉⣉⠙⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠉⣶⣶⣴⣦⢸⣿⠃⢹⡇⠘⢾⡟⢠⡟⠉⠉⠉⢛⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠁⢸⡄⣟⠁⠛⠹⠆⠉⠓⢀⣁⣑⣛⣁⣸⣇⣀⣀⣀⣙⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⠀⠛⣻⠂⢀⣸⣧⣤⠤⠶⠶⠚⠛⠉⠉⠉⠉⠉⠉⠉⡁⠀⠀⠀⠀⠉⠉⢁⣀⣴⠊⠀⢀⠴
⠀⠀⠀⠀⠀⠀⢰⢸⡇⢴⠀⢰⣧⠖⠋⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⢺⡇⠀⠀⠀⣿⡷⢲⣒⡶⣶⣻⠛⢆⣿⠀⣀⢸⣯
⠀⠀⠀⠀⢀⣀⣸⣸⣇⣿⠀⠈⠁⠀⠀⡶⣶⠲⡖⢲⢺⣿⣿⡇⠸⣮⢿⠀⠀⠀⠛⢀⡘⠟⠀⠈⠛⠀⠀⠁⠀⣿⢸⣿⢀⣶⣆
⠀⢀⡞⠩⠁⣀⡈⣉⣁⠉⠍⠙⢦⡀⠀⠯⡏⠐⡷⢾⠀⠿⠛⠇⠀⠁⣀⠀⣀⠀⢀⣼⡅⠀⢾⡆⠀⠰⣿⡆⠀⣿⡜⠋⠀⠈⠀⣀⡀
⠀⠀⡇⣤⣤⣴⢶⢠⡶⣄⣶⣦⢈⡇⠀⠀⠀⢀⠀⣤⠀⠀⠀⠀⢷⢴⠻⠔⣇⠀⣼⠽⡇⠀⣟⢳⡀⢀⣯⡿⣄⠙⢁⢠⣿⣆⡆⠘⠁
⠀⠀⢻⣇⠻⠿⢾⣼⠧⠿⠃⣿⣼⠃⠀⠀⢀⣸⣿⣿⣽⣃⠀⠀⠘⠄⠑⣒⠋⠘⠳⠛⠃⢸⣯⢈⣧⠸⠷⠿⠛⠀⠈⢳⣻⠯⠂
⠀⢠⠼⢯⣔⣆⣈⣩⣄⣀⣺⠴⠛⠑⣆⢏⣉⣩⣿⣧⠤⢼⡇⠈⣏⣧⣼⣹⠀⠀⢠⢒⠤⢠⡬⠉⡁⠀⣀⣀⠀⢀⡤⠠⡁⠀⢴⣤⠄
⠀⢸⠈⣀⠀⣠⡀⣰⣦⠀⣠⡀⢠⣄⢹⠈⡇⠀⣿⡇⠀⣸⠀⢺⡿⣿⣿⠿⡿⠀⠜⠬⠗⠠⠽⠖⠁⢸⡁⠈⠳⠏⠀⢀⡇⣀⠈⠁
⠀⠘⡟⢹⣾⠇⣿⣿⢸⣸⡏⢧⣾⠘⡾⠃⣇⠀⣿⣷⣀⡏⠀⠘⣷⠀⢐⢿⡇⠰⠀⠀⣠⡴⠲⣄⠀⠘⠧⣄⠀⠀⢀⡼⠁⣿⣿⠟
⠀⠀⣧⣠⢉⠠⡙⠇⡀⣙⣂⢬⡗⢔⡿⠀⢈⡅⢠⡀⢀⣀⡀⠀⠻⠼⠿⠾⠃⠀⢀⣠⡟⠀⠀⢸⣇⡤⢄⠀⠉⢷⡟⠁⠀⠈⣁⡀
⠀⠀⠛⠬⣏⣙⣙⣋⣹⡥⠽⠖⠚⠉⠀⠀⠘⢦⣾⣷⣯⡴⠃⢀⣤⢀⣀⠀⢀⣰⣋⠸⡇⠀⠀⢸⡏⠀⢸⡇⠀⠸⣅⠀⢠⠞⠁⠀⠉
⣀⠞⢦⡄⣀⠀⣀⣤⡤⡄⠀⡀⠀⢿⡷⠀⡏⠉⣿⠀⣿⠁⢹⣾⣿⣿⠟⣰⠏⠀⠉⣷⢳⣀⣠⣾⡞⠋⠉⢳⡄⠀⢸⡆⡏
⢨⡶⠼⠆⢹⡽⠛⠛⠉⠃⣸⣽⢠⣏⣷⠀⢻⠀⢿⢀⡯⠀⣼⢸⣿⢹⠇⣿⠀⠀⠀⣿⡏⠉⣷⣼⠁⠀⠀⢸⠇⢀⡾⠁⢹⣄⣀⣠⠼
⢠⣄⡄⠀⠀⠀⠴⣿⡆⠀⠧⠿⢀⠌⡄⠀⠘⠲⠺⠞⠓⠒⠋⠘⠛⠚⠀⠘⠦⣤⠞⣿⠀⠀⣿⡿⢄⣀⣠⠟⠀⠀⠀⠀⡾⠉⢠⣤⣤
⠚⢶⠛⠠⠴⣱⠴⢋⡀⣰⣆⢻⡿⠀⡗⣠⡀⠀⠀⠀⣀⢤⡎⣳⣀⠀⠀⠀⠀⢩⠀⡻⣄⣠⠏⡀⠘⢋⡴⠊⠉⠑⢦⠀⢇⠀⠮⣿⠿
⠴⠆⢀⣀⡉⠋⠀⠈⠙⠾⠋⠁⠀⢀⣔⣛⣓⣄⠀⢴⡙⠦⠟⠗⣚⣳⠀⠤⠀⠀⠄⢃⠘⠉⠠⠁⠌⢸⠀⠀⠀⠀⠈⡇⠈⡆⢀⣁
⠀⠀⣿⠁⡭⠉⠉⣉⣷⡆⠀⠀⢠⡗⠢⠗⡒⢚⡇⢸⣸⣿⢻⢫⣿⠇⠀⡿⡄⠀⠀⠱⣄⢀⢀⠊⠀⠸⣄⠀⠀⣀⡾⠁⣠⠇⠺⠿⠂
⠀⠀⣿⣀⣠⣴⠞⠛⣩⡿⡵⠀⢪⣤⣑⣭⣤⡭⣼⠀⠛⠛⠛⠛⠋⡤⠀⠛⠳⠤⡀⠀⠸⡶⠁⠀⠀⣠⠟⠛⠉⠁⣤⡀⠁⣤⣤⡄
⠀⠀⠈⠙⣯⣄⡴⢞⡥⠚⣁⢀⠈⢿⣿⣿⣶⣧⠏⢀⢀⣴⡄⠀⢺⡁⠀⠀⠀⠀⣈⠇⢰⣧⠀⠀⠀⡇⠀⣴⣶⠀⠉⢁⡄⠘⠋
⠀⠀⠀⠸⣯⠵⠚⠉⠀⠠⣻⣟⠄⠀⠀⠀⠀⠀⠀⢹⠾⡿⣿⠃⠀⠹⡟⡟⣿⣿⣿⠀⠸⠁⠀⠀⠀⢸⠀⠈⠈⢀⣽⡁⡷⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣧⣿⣟⣛⣛⣀⣀⣀⣀⣀⣴⣃⣀⣀⣀⣈⣉⣉⣃⣀⣀⣀⣀
⣿⣿⠛⡿⢛⠻⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⣀⣀⣈⣁⣀⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢄⠐⢦⡀⠀⢰⡇⠀⣿⠉⠉⠉⠉⠉⠉⠉⢹⠀⢧⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠉⠳⠟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⢷⡴⠖⠋⢠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⢀⡐⠒⠋⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠛⠒⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠙⠀⣶⠒⠒⠒⠒⠒⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠓⠒⢲⠀⠘
⠀⠀⠀⣤⠤⠶⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰⣤⣀
⠀⠀⠀⠤⣤⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣈⠉
⠀⠀⠀⠀⠀⠈⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠁
⠀⠀⠀⠀⠀⢀⠀⠙⠛⠛⠛⠛⠛⠛⠛⣷⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⠛⠛⠛⠀⠰⡄
⠀⠀⠀⠀⠀⠁⠀⣀⣀⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⣾⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣀⣄⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠉⠉⠧⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⠉⠙⢦⡀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⣄⣶⡐⠨⠁⠈⠑⠄⠁⡤⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⡒⠃⠀⠀⠀⠀⠀⢂⠀⡀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢠⢈⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⢀⡀⠀⠈⠂⠀⠐⠀⠀⡠⠄⠀⠈⡆⠀⠀⢀⠈⠈⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⣀⠠⠪⠀⠀⠀⣠⡁⠀⠀⠈⢑⠀⡀⠠⡇⢀⠔⠁⠱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣐⡄⢰⣦⠀⠀⠘⣿⠀⠀⠀⢀⠢⠀⠆⠘⠁⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠈⠿⠃⠀⠀⠀⠀⠀⠀⡈⠀⡀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠃⠐⠀⢠⠊⢀⠰⠁⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⢠⠀⢉⠠⣥⠒⢁⠌⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠃⢠⠀⠀⢀⣀⠀⠤⠄⠒⠂⠉⠄⣦⣤⡆⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⡕⠂⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠉⠀⠀⠀⠈⠉⠉⠉⡙⠀⠀⠘⠋⠉⢹⠤⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠄⠀⢸⠷⠿⠿⢿⠀⡀⢀⠄⠂⠀⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⢠⣷⢶⣤⣤⣼⡆⡀⠌⠳⠤⢔⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢲⠙⠿⢿⠠⢍⣯⣽⣌⢁⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⠈⠋⠉⠉⠈⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠉⠁⠚⠊⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠒⠀⠁⢢⣀⡀⠀⠀⢡⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡔⠁⠑⣼⠄⠀⠈⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⡶⠃⢀⡇⢀⠥⠀⠊⣡⠤⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⢀⣠⠶⠋⠀⠰⠀⡀⠊⠁⠀⢀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⠉⠀⠀⠀⠀⠀⠁⠈⠈⠀⠉⠄⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⡜
⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⠃
⠀⠀⠀⠀⢸⠀⠀⢸⡀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⡀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⢧⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⡆⠀⠀⠈⢧⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠈⢳⡀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠃⠀⢸⡄
⠀⠀⠀⡀⠀⠀⠙⣆⠀⠀⠀⠙⢦⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠠⠀⠀⣧
⠀⠀⠀⢠⠀⠀⠀⠈⢷⣄⠀⠀⠈⠳⣄⠀⠹⣄⠀⠀⠀⠀⠀⠐⡇⡆⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠉⠳⣤⡀⠀⠈⠑⠀⠈⠳⣦⡀⠀⠀⠀⣇⠸⡀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠈⠻⣦⣄⠀⢻⡀⠳⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠻⢿⣾⣧
⠀⠀⠀⢆⠀⠀⠀⠈⠻⢶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⢶⣤⣀
⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠉⠛⠿⢶⣦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⢀⡾⠁⠀⠘⠟⠸⠿⣶
⠀⠀⠀⠀⠀⠙⠲⣤⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡻⣦⡀⠀⢀⡞⠁⠀⠀⠀⠀⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠶⠶⠶⠤⡤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣜⢷⡔⠋⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠈⠀⠠⠄⠀⠀⣀⣀⣀⡀⠠⠤⠐⠊⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠘⡎⡧⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣤⠶⠞⠋⠁⣀⡴⠂⢀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠖⠚⠉⠀⣀⡽⠖⠛⠉⢀⡠⢋⠀⠀⠘⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠠⠴⠖⠋⠁⠀⠈⣩⠟⠉⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠤⠤⠶⠛⢁⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠠⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠱⡀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⣠⢴⣶⢀⣠⣤⣾⠛⠁
⠀⠈⠢⣤⣀⣀⣀⣀⣀⣠⣄⠤⠤⠐⣂⡤⠆⡀⠀⠀⠠⠤⠠⠤⠔⠚⠛⠉⠁⠀⠛
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣀⣤⡶⢟⣩⠖⢋⠠
⠀⠀⠀⠀⠀⠂⠤⠤⠖⠛⠉⢀⡴⠋⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠀⣠⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠴⠚⡝⠉⠙⠍⠓⢂⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡀⠁⠀⡘⠀⠀⠀⠈⠀⠀⠑⢄⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⢃⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠂⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⣀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠿⠛⠛⡋⢉⣉⣀⣀⣉⣉⣉⡛⠛⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣛⠭⠆⠒⠂⠉⠅⠀⠀⠀⠀⠀⡇⠀⠀⠉⡙⠒⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠠⠀⠀⠀⢠⠀⠀⠀⠀⠀⠃⠀⠀⠠⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠱⡀⠀⠀⠄⠀⠀⠀⢰⠀⠀⠠⠃⢠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢗⢄⠑⠄⠀⠀⠀⠀⠀⠄⠀⡠⢂⠔⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠢⠢⣄⣢⡀⣀⣠⠴⠐⠁⠠⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠈⠁⠀⠀⠰⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠘⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠓⠾⠀⠀⠀⠀⠇⢀⠆
⠀⠀⠀⠀⠀⠰⠀⢄⠀⣀⢀⣀⣀⣀⣀⡀⢀⠄⠰⠘⡤⠇⠀⠀⠀⠀⠘⢾⠀⠀⠀⠒
⠀⠀⠀⠀⢀⠃⠀⡀⠀⠀⠀⠁⠀⠁⠀⠀⠐⡀⠀⠆⢠⡄⠀⡄⢠⡔⢰⡖⠀⠀⣰⣆
⠀⠀⠀⢀⠊⠀⡰⠀⠀⠄⠀⠀⠀⡄⠀⠄⠀⠡⠀⠀⢴⠃⠘⠃⠈⠃⠘⡇⠀⢔⣿⣿⡢
⠀⡠⠔⠁⢀⠔⠀⠀⠜⠀⢀⠀⠀⢀⠀⠘⡀⠀⠑⢀⠀⠑⠆⡀⠀⠀⠀⠡⢂⠊⠃⠁⠁⠐
⠛⠛⠛⠛⠟⠛⠛⠻⠛⠛⠛⠛⠛⢛⠛⠛⠻⠛⠛⠛⠟⠛⡛⠛⠀⢀⠔⢠⠂⠌⠀⠠⢠⠀⡈⠢⡀
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⡠⠀⠁⡠⠁⠀⠀⡀⠀⠀⠂⠈⢄⠈⠂⢄
⠀⠀⠀⠀⢠⠀⠀⠀⣯⠝⠛⢻⣽⠛⠋⢍⡇⠀⠀⠀⠀⢰⣥⣤⣤⣮⣤⣴⣥⣤⣤⣤⣤⣬⣦⣤⣵⣤⣤⣥⣦
⠀⠀⠈⠀⠈⠀⠀⠀⣿⠀⠀⢸⣿⡀⠀⢸⡇⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⢠⠀⢘⠀⠀⡀⠀⡆⠀⢀⠁
⠀⠀⠀⡇⠀⡆⠀⠀⣿⠀⣆⣿⣿⣇⡇⢸⡇⠀⠀⠇⠀⠃⠀⠘⠀⠆⠀⢸⡟⠋⢻⣟⠉⢻⡇⠀⠘⠀⠃
⠀⠀⠀⠀⠀⠃⠀⠀⣿⢀⣿⣿⣿⣿⣧⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠈⡇⣐⣾⣿⣄⢸⠀⠀⡆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼⣿⣿⣿⣿⡽⡎⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣧⣿⣿⣿⣿⣼⠀⠀⠃
⣤⣤⣤⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣤⣤⣤⣶⣾⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣤⣤
⣿⣿⣥⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⠀⠀⠀⠀⠀⢸⢿⡇
⠀⣷⠈⣧⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣍⠲⣄⡀⠀⠘⣿⠁
⢠⢿⣆⡿⣷⠀⠀⠀⠀⠀⢸⡈⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⡷⡄⠙⢦⡀⠈
⡟⠀⠟⢠⣿⠀⠀⠀⠀⠀⠈⢷⡄⣿⠀⠀⠀⠀⠀⣠⠀⠀⣾⣿⡄⠀⢻⣆⠀⣀⠻⣦
⢧⠀⢠⣿⠋⠀⠀⣀⠀⠀⠀⠈⠻⠃⠀⢀⡀⠀⢀⣿⠀⠀⢸⠁⢸⠀⠈⣿⡆⠈⠻⣿⠳⡄
⠈⠳⡿⠁⠀⢀⣼⠃⠀⠀⣀⣠⣤⢶⡾⣿⡄⠀⠘⡟⣧⠀⠀⢣⡾⠀⠀⢻⣿⠀⠀⠈⢷⣽⡄
⠀⠀⠀⠀⣠⢣⢻⠀⠀⠀⠈⢳⡍⢁⠁⠘⡇⠀⠀⢻⡌⢳⡀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠘⣿⣷
⠀⠀⠀⣰⠃⣾⣾⠀⠀⠀⠀⠀⠹⣾⠀⡀⠙⣄⠀⠀⠙⢦⣷⠀⠀⠀⠀⣿⡇⠀⠀⣄⠀⢹⣿⡇
⠀⠀⢰⠇⢠⢿⣿⠀⠀⠀⠀⠀⠀⠘⣧⡇⠀⢈⣳⣄⠀⠀⠹⠇⠀⠀⣸⣿⡇⠀⠀⠸⣷⣼⣿⡇
⠀⢀⣿⠀⠈⢸⣿⣧⠀⠀⠀⠀⠀⠀⠘⣇⠀⡀⠀⠉⠳⣄⠀⠀⠀⢠⠏⣿⠃⠀⠀⠀⣿⠻⢹⠃
⠀⢸⡇⠀⡄⠘⣿⢿⣦⡀⠀⠀⠀⠀⠀⢹⣾⠁⠀⠀⠀⠈⢷⣀⣴⠏⠀⠋⠀⢰⡆⠀⣿⢠⡟⠀⠀⢠⡇
⠀⣿⡇⢰⡇⠀⢻⡘⢿⣷⣄⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⣀⣻⡅⢀⡄⠀⣠⣿⡇⣼⣷⠏⠀⠀⢠⢛⡿
⠀⣿⣧⣾⡇⠀⠀⠃⠈⠻⣿⠻⣶⣤⣀⡀⢹⡄⠀⠀⠀⠀⠈⠙⠻⣿⣥⡾⢋⣺⠿⠛⠛⠢⣄⠀⢿⠞⠁
⠀⢸⡜⡏⢿⠀⠀⠀⠀⣀⠈⠣⠀⠙⠻⢿⡟⠷⠀⠀⠙⢦⡀⠀⠀⠈⠛⠻⠟⣶⣚⠓⢦⣀⠘⡆⠘
⠀⠀⢷⡀⠸⣧⡀⠀⠀⢹⣦⣀⠀⠈⠲⣤⣀⣀⡀⠀⠀⠀⠙⢷⣦⣄⣀⣀⣀⣈⡩⠴⠒⠋⢁⡇
⠀⠀⠈⠻⣆⠙⣷⣄⡀⠈⢿⡻⢷⣦⣄⣈⣙⣿⣿⣿⣿⠿⠿⠿⠛⠛⠛⠛⠫⣄⡀⠀⣀⡠⠞
⠀⠀⠀⠀⠈⠻⢮⣿⣿⣶⣬⣿⣶⠄⢉⣩⡽⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⠉⠁⠀⣀⣀⣀⣘⣛⠻⣍⠳⣄⠀⢀⣀⡠⢾⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣤⣤⣭⣽⠿⣂⡬⠿⠘⡆⣬⣽⣶⣭⣥⡥⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠁⠀⠀⠀⠙⢿⣧⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⣦⡤⣀⣄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠋⠀⢀⠀⠀⢤⡤⣀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠃⠉⠳⣤⣀⠈⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡃⢽⡶⣄⠈⠑⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⡀⠀⠀⠀⠀⠰⠂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠀⠀⠀⠀⠀⢯⡛⠶⣿⣯⣌⣲⣤⣄⠲⣴⡆⣀⡀⢤⣤⠤⠀⣀⡀⠀⡠⠀⠀⠈⣊⣤⠎⠳⠠⠄⠀⠙⠡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠓⠊⢤⣴⡒⠦⣛⣴⣤⣤⣄⡀⠀⠈⠓⣤⣤⣀⠞⠀⠀⡀⠚⢩⡼⠂⠀⠁⠀⣀⠴⢚⠤⣴⠶⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠁⠀⠀⡐⠦⣤⣀⡀⠀⠈⠛⠷⠾⠇⣻⢿⣤⣿⣿⣟⡷⣮⣞⠉⠉⠒⠊⠀⠀⣀⣀⠠⠀⠂⠩⠐⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠆⠠⠄⣸⠍⠙⠻⠻⢏⣹⠷⢋⣙⠛⠛⠁⠀⠀⠠⠖⠋⠉⠀⠀⠀⠒⠀⠁⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⠤⡀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠈⠀⠈⠉⠉⠴⠶⠾⠀⠀⠉⠀⠀⡀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠙⠛⠓⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠊⠳⡈⠳⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠤⠤⠤⠀⠀⠀⠒⠂⠈⠉⠉⠉⠉⠐⣀⠤⠷⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢿⡀⠸⡄⠀⡀⠐⠠⣄⣀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⢫⣖⠴⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠁⠀⣧⣄⣈⣓⠒⠀⠉⠉⠒⠠⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠐⠂⠀⠀⠀⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠃⠀⠤⠄⠉⠉⠉⠈⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠼⠴⢦
⠀⠀⠀⠀⠀⠀⠀⡴⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⣴⢉⡁⠁⣦⣄⠶⡄
⠀⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡂⠀⠀⠠⣀⠋
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠁⠀⠀⠀⠈


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⣝⡛⣋⡙⢻⡟⣫⣿⢻⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠈⣤⡄⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀⢀⣴⡄⠀⠀⠀⠀⠀⣀
⠖⠠⠀⠀⠀⠀⠀⠀⣿⠁⠀⡀⠄⠀⣤⢠⡀⠀⠀⠀⠉⠁⠀⢀⣤⣤⣾⠟⠃⠄⡀⠀⠀⡀⢰⠏⠛⠉⠀⠀⠀⣼⣾⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠟⠛⠇⠀⠉⠀⢠⠃⠟⠁⠀⠀⠆⠀⠀⢰⡇⠀⠀⠙⣧⠀⠀⠉⠓⠘⠏⠀⠀⠀⠀⠀⠀⢰⣏⣠⣶⣿
⢺⡏⠀⠀⠀⠀⠀⠀⣾⣷⣶⣶⣆⢸⣶⣦⡀⢀⠀⠀⠀⠀⠈⠁⠀⡀⢀⣍⠀⠀⠀⠀⡹⠀⣤⠶⠀⠀⠀⠀⣿⢻⡟
⠘⠁⠀⠀⠀⠀⣠⢾⡿⠛⠋⠁⢙⣿⣭⣥⣿⡿⠿⢷⡀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⢻⡿⠇⠀⠀⠀⠀⠀⠻⠘⠃
⠀⠀⠀⠀⢀⣀⡛⠚⠃⠀⠀⠀⠈⠉⢨⣿⠁⠀⠀⠀⢹⠀⠀⠀⠀⠸⠟⠃⠀⠀⠀⠀⠘⠟⣀⣤⣶⣶⣦⣴⣄⡀⣦
⢀⡄⠀⠀⢀⣇⣿⠛⠂⠀⠀⠀⠀⣿⣿⢻⠀⠀⠀⠀⢸⣠⣤⣼⣷⣷⣶⡄⠀⠀⠀⠀⠀⣇⠝⠛⠿⣿⣿⣿⣿⣿⣿⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡦⠀⠀⠀⢀⡴⠒⠲⡄
⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⣤⣀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⣿⠀⠀⠀⢻⠀⢠⣾⠂⠀⢠⣿⣧
⠀⠀⠀⣠⠴⠋⣉⣐⣦⣀⢦⡈⠳⣄⠀⠀⠀⠀⠀⣿⡇⠀⠀⣹⣦⣀⣀⣸⣀⣸⣿⠀⠀⢸⠘⣿⡄⠀⠀⠀⠀⢀⣤⣄⡀⠀⣰⣶⣄
⠀⠀⠀⠈⠓⠦⣄⣩⡟⠋⣽⣫⠽⠋⠉⠉⠉⠉⠉⢹⣿⡴⢺⣿⠀⠀⠀⡼⠀⠈⣿⠀⠀⡞⠀⣿⣧⠴⠋⠉⠉⠉⠛⠛⠉⠉⢹⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⡤⢺⣿⠀⠘⣿⣄⣀⡼⠁⠀⠀⠙⠓⠚⢁⠴⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣿⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢀⠞⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠛⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣻⡄⣤⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠠⠄⡆⠀⢹⡿⡇
⠀⠀⠀⠀⢠⣄⠀⠀⠀⠉⢿⣟⣯⣄⢽⠯⡏⣀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠁⢀⣴⣷⣷⣶⡁⠀⠀⠀⠀⣀⡀
⠀⠀⠀⣤⣛⣵⣦⠀⡀⠀⢨⣿⣿⣿⣸⣧⣌⣿⣧⡄⠀⣿⣿⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣼⣿⣿⢿⣿⣧⠀⠀⡀⠀⢠⣇
⠀⠀⠀⢾⢍⠁⣿⢶⣷⠀⣾⣿⣿⣿⣿⣿⠿⣿⣿⣷⣶⣿⣍⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣿⣿⣿⣿⣿⡟⢠⣶⣦⡀⠶⣿⡇
⠀⠀⠀⡤⠼⢷⣹⣿⣿⣾⣿⡟⢻⣿⢹⣯⣼⣿⣿⣿⣿⣟⡿⢿⣷⠀⠀⠀⠀⢀⠀⠀⠀⣿⠀⠀⣠⡿⠿⢻⠟⠿⠃⢸⡟⢻⣿⣦⣽⡇
⠀⠀⠀⢛⣴⣿⡿⠿⠿⢻⣿⣿⣿⡿⢡⣼⣿⣿⡄⢿⡷⣿⡅⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠛⠀⢁⢘⠀⣼⣾⣽⣿⣿⣟⠛⣿⡃
⠀⠀⠀⢻⣿⣿⠃⡠⢂⢺⡿⢿⣿⣯⣴⣿⣿⡹⠓⠀⠙⠉⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣿⣈⣷⣤⣴⣶⣾⡤⣧⣾⣿⣿⣻⣧⣰⣿⡇
⠀⠀⠀⢸⣿⣿⣴⣿⢈⣾⢣⡾⠀⠛⢿⣿⣿⡇⠀⢠⠀⠈⢿⣣⣀⠀⠀⠀⠠⠒⠚⠻⠟⠘⡿⠛⠛⢛⠙⣿⡇⠿⠿⣿⣿⣿⣿⣿⣻⡷
⠀⠀⠀⣸⡟⣜⣿⣏⢹⡿⢸⡗⢸⢻⣿⣿⣿⡗⠀⣄⣀⠀⠀⠁⡈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠟⠛⣧⣞⣿⣿⢟⣛⣁⠈
⠀⠀⠰⠿⣿⣿⣿⣽⣿⣿⢸⣷⣿⣾⣿⣝⡺⣗⢀⣿⡯⣵⣤⣄⡀⠀⠀⠀⣀⣼⠄⠀⠀⠀⠉⠉⠁⠀⠀⠘⠛⠉⠀⠀⠀⣩⣥⣭⡴⠆
⠀⠀⠀⠀⠈⠝⠂⡯⢭⠴⣿⡿⠿⢿⡕⠛⠿⣚⣿⣿⣿⠶⠀⠀⠀⠉⠱⣶⣾⣟⣀⣴⣶⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢤⢴⡤⠄⠀⢈⡹⡿⠛⠸⠏

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⣩⣬⡉⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣉⣽⣿⣿⣿⣌⡉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢾⣿⣿⣿⣿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⠫⣿⣿⣿⣿⡏⢶⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⢶⣄⠀⠀⠀⠀⣿⣾⣥⣽⣿⣿⣃⣾⡇⠀⠀⠀⢠⣴⣞⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢏⢠⣾⣷⡄⡝⠛⠛⠛⠛⠛⠿⢽⣿⣿⣿⡿⠟⠛⠛⢿⣿⡿⠟⣽⣷⣦⡘⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣃⡀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠹⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠿⣁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡔⢻⣿⣿⠋⢋⣉⣉⣉⣉⣉⠉⢹⣿⣿⣿⡏⢉⣉⣩⣿⣿⣯⣍⢿⣿⡿⢑⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣀⣤⠾⠉⠉⠉⠉⢹⠀⢸⣿⣿⣿⡇⢸⡇⠀⠀⠀⠙⢿⣷⣤⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⡁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣯⣿⣶⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢀⢸⣿⣿⣿⢇⠸⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢠⣶⣿⣿⣿⣿⣷⣦⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠘⠿⢿⣿⣿⣿⡿⠛⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⠈⠻⠿⠟⢉⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠖⠋


⣿⠿⠿⠿⡿⡟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣛⣿⣿⣛⣻⣿⣿⣿⣿⣿

⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣤⣤⣤⣄⣀⣀⣀⣀⣸⣿⣯⡿⠳⠿⠷⢿⣿⣄⣀⣀⣀⣠⣤⣤⣄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸
⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⣆⣀⣀⠀⠀⠄⣀⣀⠀⠀⣀⡠⠀⠀⠀⠀⣠⣀⣀⣀⣀⣀⣸⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸
⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⠛⠛⠻⠿⣶⣶⣒⣚⢻⡏⠉⠀⠀⣰⠟⣛⣛⣫⡭⠷⠛⠛⠛⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸
⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣷⠀⢀⠀⣿⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸
⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣖⠀⣀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸
⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡟⡂⠩⣿⣿⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸
⢸⠀⢸⡇⠀⢀⣀⣤⣄⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠈⠚⢡⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⣿⠀⢸
⢸⠀⢸⡇⣴⣿⣟⣿⡿⠙⣦⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡧⡀⠀⡾⣼⣿⡇⠀⠀⠀⢀⣀⣀⡀⠀⠀⢸⡄⠀⠀⣴⠛⠻⣿⣿⣷⡀⠀⣿⠀⢸
⢸⠀⢸⣿⣿⣾⣿⡿⠐⠓⣹⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠁⣽⠀⢺⡟⣿⠀⠀⠀⣴⠛⠻⣿⣿⣷⣄⢸⠀⠀⢸⣿⠙⠂⢹⣿⣿⡇⠀⣿⠀⢸
⢸⠀⢸⡏⢿⣿⣿⣿⣧⣼⣿⠃⠀⣿⢸⠀⣴⣾⣿⣿⣻⣦⡀⠀⠀⠀⢸⣿⠀⣿⠀⢸⡇⣿⠀⠀⢰⡏⠰⠄⣸⣻⣿⣿⣼⠟⠋⠉⠉⠉⠘⢿⣿⣿⠀⠀⣿⠀⢸
⢸⠀⢸⡇⣠⡿⢿⣿⣿⡏⢳⡄⠀⠺⠈⣼⣿⣿⣷⡟⢩⠍⣷⠀⠀⠀⢸⠸⡆⢻⠀⢸⡇⡿⠀⠀⠸⣿⣷⣾⣿⣿⡿⣿⠁⠀⢀⣀⣤⣄⣀⠀⠈⢿⢦⡄⣿⠀⢸
⢸⠀⢸⣷⠏⠉⠲⣬⠟⠛⠋⠙⠓⠻⣤⡸⣿⣿⣿⣿⣴⣶⡟⠀⠀⠀⢸⠀⣷⣨⣦⣾⠀⡇⠀⠀⣠⣽⣿⣿⠟⠉⣹⠷⣌⣰⠋⣭⣿⣿⣿⣷⠀⠈⡏⣇⣿⠀⢸
⢸⠀⢸⣿⠀⢠⡾⠁⠀⠀⠀⢀⠀⠀⠐⢝⢿⠟⠁⢿⣿⣿⡀⠀⠀⠀⢸⠂⢸⣷⣿⡿⢸⡇⢀⣾⠋⢻⣦⣄⣀⣤⢏⠞⠈⡟⠇⢻⢹⣿⣿⣿⡇⠀⡇⢹⣿⠀⢸
⢸⠀⢸⣿⠀⣾⠃⠀⢀⣴⣿⣟⡻⡿⢦⡀⣣⡻⣶⣶⣷⠟⢿⡄⠀⠀⠸⠦⣤⣭⣉⣡⡼⢃⣾⠇⠀⣆⠻⣿⣿⢫⢏⠀⠀⢿⣠⣶⠀⢻⣿⣟⡄⡰⢷⡄⣿⠀⢸
⢸⠀⢸⣿⢠⣿⡄⠀⢾⣿⣿⣿⢿⠟⠋⣷⣿⣷⡹⡿⠋⠀⠘⣷⡀⣠⠴⠶⠶⠶⠶⠶⠶⠾⣧⣄⠀⠈⠳⣽⣯⣿⣡⡀⠀⣨⠟⣯⣶⠿⠛⢿⣞⠁⠀⣿⣾⠀⢸
⢸⠀⢸⡏⢸⡇⢳⡀⠈⣿⣿⣿⠄⣀⣸⣿⢸⡟⣷⡹⣀⣤⡞⣿⣿⡇⠐⡆⠁⠀⠅⠠⠃⢀⡾⠘⣧⠀⠀⡏⣿⡿⠛⣷⣼⣯⣾⣛⣡⡤⠶⢤⡹⣧⠀⢻⣿⠀⢸
⢸⠀⢸⡇⣿⠀⠀⣹⢖⣩⠵⣭⡲⣜⢻⡏⢸⣧⢻⣷⠏⡍⣼⡽⠻⣧⣌⠻⠿⢿⣿⠷⠾⠿⢇⣴⠏⠀⠰⢰⢿⡷⣾⠙⢿⣿⣿⠟⢉⠀⠀⠰⣽⣿⡆⢸⣿⠀⢸
⢸⠀⢸⡇⣿⠀⣼⣿⠋⠀⠀⠈⠙⢷⣽⡇⢸⣿⣸⣥⠴⠷⠿⠦⢤⣌⡛⠻⣷⣶⣶⣶⡾⠟⢋⣡⡤⠴⠶⠿⢼⣧⡹⣇⣸⡿⣡⡴⠃⠀⠀⠀⢹⣽⣿⢸⣿⠀⢸
⢸⠀⢸⣇⣿⢠⣟⡟⠀⠀⠠⡀⠀⠙⣿⣧⣼⣻⣿⣅⡴⠒⠒⠶⠆⣀⣿⢿⡉⢉⣛⣉⣉⡷⣯⣀⠴⠒⠒⠒⣀⣼⡟⣿⣿⡿⠋⠀⣠⣎⠀⢰⡟⡯⣿⡟⣿⠀⢸
⢸⠀⢸⣿⡇⣸⣿⢱⠀⣗⡄⠙⢦⡀⣼⢻⡇⣯⠋⠻⢿⣿⣿⣿⣿⠿⠋⢸⢏⡟⣿⢯⢻⡀⠙⠻⢿⣿⣿⠿⠟⠿⣽⣿⠘⣧⣠⡼⣡⣿⡆⢸⠀⣿⢿⣿⣿⠀⢸
⢸⠀⢸⣿⡇⡏⣿⢸⡀⢿⣎⠢⣸⠟⠁⣾⢱⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠃⢹⠘⡌⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠈⠁⣠⣿⣿⣷⡘⠀⢹⢸⣿⣿⠀⢸
⢸⠀⢸⣿⢸⡇⣿⠈⡇⢸⣿⣦⣼⡆⣰⢣⡏⣿⠶⢤⣄⣀⡀⠀⠀⠀⠀⢸⡾⡀⠸⡇⠀⣿⠇⠀⠀⠀⢀⣀⣤⠴⣾⣻⣯⢦⣰⣿⡇⣿⢿⣷⢦⣾⣌⣿⣿⠀⢸
⢸⠀⢸⡏⣼⠁⣿⢈⣷⠘⣇⣧⠙⢣⣯⡟⠀⣿⠀⠀⠀⠉⠉⠛⠛⠒⠶⢾⡇⣇⠀⠉⣸⡿⠶⠒⠛⠋⠉⠁⠀⠀⣿⢹⢻⣧⣿⢃⡇⣿⣾⢿⣆⣸⠟⣿⣿⠀⢸
⢸⠀⢸⡇⣿⢀⣿⠘⢿⠀⢿⠘⡆⠟⣿⡀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠘⢳⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⢠⡿⠁⢸⠁⡟⣿⢸⣿⣧⡀⣿⣿⠀⢸
⢸⠀⢸⣇⣿⣼⡟⠀⣸⡄⢸⣇⢻⡀⢹⣇⠀⢸⡷⢤⣤⣀⣀⠀⠀⠀⠀⠀⣿⠞⠛⠁⠀⠀⠀⠀⠀⠀⣀⣠⡤⣼⡟⢸⣿⠃⠀⡿⠀⡇⣿⢸⣿⢸⡿⢿⣿⠀⢸
⢸⠀⢸⣿⠟⠋⡀⡀⣿⡇⠈⢿⡈⣧⠀⣿⡀⣸⡿⣶⢦⣬⣭⣉⣙⣛⣛⣛⡛⠒⢒⠒⣒⣚⣛⣛⣩⣭⢭⣶⡞⢻⡇⣿⡏⠀⢸⠇⢀⡇⣿⢸⣿⢸⣿⣽⣿⠀⢸
⢸⠀⢸⡇⣤⢰⢁⡇⠇⣧⠀⢸⣧⠸⣆⠘⢿⣿⡇⣿⢸⣿⡇⢹⢹⠍⣿⠿⢿⣿⡏⣿⡿⢭⡯⢹⣿⡇⢸⣿⠁⢸⣿⡿⠀⢀⣿⠀⢸⡇⡏⢸⣿⢸⡏⡟⣿⠀⢸
⢸⠀⠸⠷⠿⠾⠿⠷⠶⠿⠶⠶⠿⠶⠿⠶⠶⠿⠷⠿⠾⠿⠷⠿⠾⠿⠷⠶⠾⠿⠷⠿⠶⠶⠿⠾⠿⠷⠾⠿⠶⠾⠿⠶⠶⠾⠷⠶⠾⠷⠷⠾⠿⠾⠷⠷⢾⠀⢸
⢸⡀⢀⣄⣄⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⢤⣤⣤⣤⣤⠤⠤⠤⠤⢤⣤⣤⣤⣤⠤⠀⢸⠀⢸
⠸⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠛⠉⠉⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢀⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢰⠐⡤⡎⠘⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠓⠦⣧⣄⣀⣀⠀⢀⣀⣀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢀⡤⠤⠤⠴⠊⠁⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⡼⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡛⠒⠶⠖⠚⠋⠁⠀⠀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⠶⠶⠶⡏⠤⠤⠤⠤⠔⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⡇⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⢸⡇⠀⠃⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠸⡇⠀⠀⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⢀⣀⠾⠅⠀⠈⣀⣀⠀⣠⣤⣤⡀⠀⠀⢱⠀⠀⠀⠁⠀⠀⢇
⠀⠀⠀⠀⠀⣀⣀⡀⢰⠋⡤⠀⠀⠁⢰⢏⡍⠑⠀⠀⡾⠫⠍⠙⠁⠒⠂⠱⠀⠀⢸⠀⠀⠀⠀⠀⠀⢇⡀
⠀⠀⠀⠀⢸⠋⠌⠁⠈⠀⠀⢀⡴⠒⠀⠀⠀⣰⠏⠉⠀⣠⠶⠲⠦⢀⣤⣤⡀⠀⢸⠀⠀⡌⠁⠀⠀⠀⢳⡀⠀⠀⢀⠆
⠀⠀⠀⠀⠈⠃⠀⣴⢋⠅⠀⢸⡄⠃⠀⣠⡄⢻⣌⠀⠀⠇⠘⠀⠀⢿⠑⠂⠈⠀⡼⠊⠁⠈⠁⠀⠀⠈⠑⢝⢦⡠⠃
⠀⠀⠀⢠⡶⢂⠀⠻⠄⠀⠀⠀⠛⠆⢸⡡⡀⠀⢨⡞⡃⠀⢀⡴⠛⠛⠓⣤⠖⠈⠠⠒⠀⠓⢆⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠘⠧⠀⠀⣠⠄⠀⢠⡶⢛⠀⠀⣑⠀⠀⠸⣇⠁⠀⠘⣇⠁⠀⠀⣿⡰⠄⣠⠀⠀⠰⡨
⠀⠀⠀⠀⠀⠀⠰⣏⠘⠀⠘⢧⣀⠀⠐⣏⠰⠄⠀⢀⡏⢁⠀⢠⡏⠀⡄⠈⡙⠛⢉⣉⣉⣉⡉⠉⠉⠁⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠀⠀⣤⠤⠀⠈⠙⠂⠀⠀⠳⣬⠤⠀⠙⠶⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠂⠰⣭⣠⠄

⠀⠀⢀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠈⡃⢄⠀⠸⣧⠯⠃⠀⠀⢀⣀⣐⠀⠉⠁⠀⠀⠐⠀⠠⢀⠀⠀⠐⠻⡟⠂⠀⠀⠀⠀⠐
⠀⠐⢤⠃⠀⢀⠔⡅⠀⠀⠐⣋⠤⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⢀⣀⣂⣀⠀⣴⠆⠾⠇
⠀⠀⠀⢀⠔⢁⠢⡇⠀⠀⠀⠀⢀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠽⠒⠅
⠀⢀⡔⠁⢌⢢⢳⡇⠀⢹⡃⠀⣸⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠈⠀⠤⠑⡄⠀⠀⠀⠀⣤⡞⠀⠹⠆⠬
⠀⠫⣉⠉⠚⢦⣻⠁⠀⡈⠳⠎⢸⣀⠄⡟⢃⡠⢧⣀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡠⠒⠀⠋⠀⠑⢲⠊
⠀⠀⠈⠉⠓⠒⠛⠀⠀⡷⠴⢶⣺⢅⡜⠀⠀⠹⢪⠋⣹⠒⠶⠤⢤⠴⠶⡊⠟⣄⠴⠄⠈⠁⠲⠉⠀⢠⠄
⠀⠀⢀⣀⣀⡀⢀⣀⠀⠫⠻⣷⣮⡟⢠⢒⡒⢄⠀⠉⠀⠃⠰⠢⠌⠦⠆⠈⠃⠈⠀⠀⠀⢀⡈⠀⠀⠈⠂
⠀⠀⠙⢄⠀⢠⡋⠶⠥⡀⠀⠀⣀⠀⢧⡇⠈⢪⡄⣰⠞⠉⢢⠀⠀⢤⠀⠀⢰⠠⠂⠀⡠⢊⢁⠝⠑⠒
⠀⠀⣀⠀⠑⢼⢢⣔⠀⠉⠳⣄⣈⣯⠚⠿⣄⠌⣷⡟⠀⣀⠎⠀⠒⠈⣁⣠⠤⠒⠂⠀⠀⠁⠃⠔
⠀⡰⢤⡂⠐⣢⠉⠈⢳⡉⣉⠽⡏⠉⡑⠲⢬⡟⢙⣶⣚⣉⠙⢧⠀⣸⠑⠀⡗⠀⢀⠀⠀⡠⠀⠀⠀⠠⡀
⠀⠰⠔⠄⢀⡽⡄⠀⠀⠻⠴⡞⠁⠀⢦⠀⢠⠀⢀⡀⢀⡰⠀⠀⠀⢇⢀⡔⠱⢄⡜⠀⢠⠁⠀⠀⠀⠀⠄
⠀⠀⢀⣀⣸⢀⠇⢀⠀⠀⠀⢹⠀⠀⡆⠀⢸⠀⠀⠀⢠⠀⠀⠀⠀⡨⠉⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢠
⠀⠀⢘⢒⠢⡿⠤⠒⡃⠀⠀⢸⠀⠀⡇⠀⢸⠀⠃⠀⡆⠀⠀⠀⠴⡇⠀⠀⠀⢠⡼⠤⢀⡆⠄⠤⠒⠁
⠀⢀⣸⠂⠈⠑⠎⢸⠀⠀⢿⡛⣏⠀⠉⠉⠉⠘⠒⠒⠃⠀⢠⡤⠐⠀⠤⠶⠂⠀⠀⠀⠙⠂
⠀⠘⠤⣑⣦⣤⣔⡃⠴⠀⡰⠉⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⡇⠃⡰⡟⣭⣕⡄⢯⡟⠀⠀⠀⢀⣸⠙⣳⠆
⠀⠀⠀⠈⣒⣒⡉⠀⠀⠀⠀⠀⠴⡀⠀⠀⠢⡁⢀⡀⢀⣀⢧⣆⠘⠷⢝⠿⠃⢀⠀⠂⠀⡠⠢⡀⢙
⠀⠀⢠⠊⡉⢳⣹⣀⠠⠰⣷⠄⠣⡁⠠⠀⢠⣌⠀⠲⠟⠉⠀⠉⠑⠒⠀⠀⠀⠀⠀⠀⢀⡫⠒⣠⠏
⠀⠀⠈⠦⣑⣊⣁⣀⠤⠒⠉⠀⠀⠈⠢⠘⠋⠈⠀⠈⠀⠈⠀⢠⡀⢢⠘⢑⡀⠀⠀⠀⢏⣋⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠁⠊⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠶⣶⣶⣶⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠉⠙⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠠⠰⡷⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⢀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢀⣠⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣽⣦⣴⣆⡀⠀⣀⡀⠛⢐⣠⣦⣶⣽⣿⣷⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠸⢿⣶⣶⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⠀⠘⠠⡆⡀⠀⢀⣿⣿⣿
⠀⠀⠀⠒⠄⠀⠰⣄⠀⢠⡀⢀⡀⠀⠀⠚⢒⣀⡀⠀⠷⢠⡠⡏⠉⢿
⠀⠀⠠⢤⠤⠄⠀⠈⠀⠀⠓⠈⢳⠘⣶⡀⠈⢩⣭⣄⡢⡸⢳⡇⠀⠸⠀⠀⠀⠀⡀⢠⣤⣄
⠀⠀⠀⠀⠀⠐⠾⠛⢋⣤⣤⡀⣀⡂⠬⣃⡀⠀⠈⣭⣥⣝⣾⠁⠀⢠⣄⣾⣿⢩⣶⣦⣩⣥⡘⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠱⠾⠟⣫⣴⣶⠭⠛⠶⣤⡉⠉⢹⡆⠀⠀⣿⣿⣿⠿⣿⣅⠘⣛⠃⣶⣶⡤⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣡⣶⣿⠿⠛⠉⠳⣾⠀⠀⢠⢸⡇⠙⠇⢻⡿⠘⢿⠿⣨⣿⡄⣩⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⣿⡿⠲⣿⡷⠈⣻⣧⡜⠛⢡⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠻⠿⠿⠿⠟⠛⠛⠉⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠉⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⣤⠤⠤⠤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢰⣶⡆⠈⠉⣠⡧⠖⠒⠒⠲⢤⡀⢀⣀⡤⠻⣄⣀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠄
⠀⠀⠀⠀⠀⢸⣿⡇⢀⠔⠁⠀⠀⠀⠀⠀⠈⡇⠈⠉⠓⣴⠋⠉⠀⢸⠄⠀⠀⠈⣷⠀⠀⠀⠀⠀⢸⠇⠀⠀⢀⡀⠀⠀⣿⠀⠀⢀⠄
⠀⢰⣶⣶⣶⣿⣿⣷⢂⣶⣶⡆⠀⠀⠀⠀⠜⠁⠀⠀⠀⠁⠀⠀⠀⢸⠀⠀⢀⣀⣈⠀⣁⣀⡀⠀⢸⠄⠀⠘⢻⠟⠓⠒⣿⠒⠚⠿⡟
⠀⣈⠭⣉⠉⢹⣿⠃⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⠛⠋⢉⡿⠀⠈⠉⠙⠛⢾⡆⠀⠀⠀⠀⠀⢀⣿⡀
⡎⠀⠀⠀⠉⢸⡟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⣿⢀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⢰⣲⣶⠀⠀⠉⠿⠉
⠑⠤⠴⠀⠀⢸⣇⡇⠀⠀⠀⠀⠀⠀⠀⣴⡻⠃⣿⣿⣿⠘⢿⣦⠀⠀⠀⠀⠘⠛⠛⠙⠛⠛⠂⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢠⣬⣭⣤⣀⣸⣿⣏⣀⣤⣥⣥⣤⠀⠀⠀⠀⢀⠀⠀⢀⣤⣤⣤⣄⣸⣟⣟⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⢿⣿⡿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠂⠂⠀⠸⣿⣿⣿⡿⢾⡿⡷⢾⣿⣿⣧⡇
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠰⣲⣄⠀⣼⣿⣷⠀⢀⣖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⡤⡄
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠈⠛⠷⣿⣿⣿⠸⠝⠁⢀⠠⠀⠀⠀⠀⣀⠀⠀⠀⠀⠤⡀⠀⢸⣿⣿⠀⠀⢀⣀⣀⡇⢧⣀
⠀⣤⣤⣀⣀⡽⣿⣿⣿⢹⣀⣀⣤⡄⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⣿⣿⠀⠀⠈⠓⠒⡆⡖⠒
⠀⣿⣿⣿⣷⣶⣿⣿⣿⣶⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡞⡆⠀⠀⠸⣽⡿⠀⠀⠀⠀⠀⡇⡇
⠀⣿⣿⣿⡿⠿⣿⣿⣿⠿⢿⣿⣿⡇⠀⠀⠀⠀⠉⠉⠉⠀⠀⠘⠃⠓⡗⠃⠓⠀⠀⠀⠪⢭⡀⡨⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡇
⠀⠛⠋⠉⠉⣷⣿⣿⣿⣬⠉⠉⠙⠃⠀⠀⠀⠀⠀⠀⡀⣀⠄⠀⠀⠀⠃⠀⠀⠀⡟⠃⠀⢘⡗⡇⠀⠀⢰⡏⢱⠦⣠⣶⣤⣶
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⡀⡀⠀⠁⢸⠀⢐⢀⠀⠀⠀⠀⠀⠯⠁⠀⠈⠭⠁⠀⠀⠠⠿⠶⠤⠸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣠⠄⠸⣤⣤⣤⣄⠄⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣿⣿⣇⣀⣀⣀
⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣀⣀⣠⡀⢀⣄⣀⣀⡼⠀⠀⠀⠀⠀⢦⣸⣿⢀⡴⠀⠀⠀⠿⠿⠿⠿⣿⣿⡿⠿⠿⠿
⠀⠀⢀⣬⡓⣹⣿⠱⣧⡀⠀⠀⣀⣰⣆⡈⠀⠄⠄⠀⠀⢸⠈⠠⠠⠀⠀⢀⠀⢰⡩⣯⠭⣿⠬⣿⡍⡇⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠘⠛⠿⡟⢻⣿⢛⠿⠟⠃⣀⣉⣹⣏⣁⣀⠀⠀⠀⠀⢸⠀⠀⠀⢠⣆⢠⣄⠀⠉⠼⢻⣿⠘⠧⠉⠁⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠘⠋⢾⣿⠎⠓⠀⠈⠉⢉⣹⡏⠉⠉⠀⠀⠀⠀⢸⠀⠀⠠⢠⡤⢠⡤⠠⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⢹⡟⠂⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⡤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠠⡀⠀⠀⠠⠂⠒⠀⡄⠀⠁⠀⠀⠀⠀⠁⠠⡀⠀⠀⠀⠀⢀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠁⢰⠒⠂⠒⠀⠀⢂
⠀⠀⠀⠀⠀⠀⢐⠉⠐⠀⠀⠀⠀⠠⠁⠀⡀⠀⠐⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⡠⠀⠀⠠⠄⣄⠀⠀⠀⠤⠐⠀⠠⠀⠠⠠⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠄⠀⠃⠀⡐⠁⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠰⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠐⠅⣠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠊⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⠀⠐⠒⠃
⠀⠀⠀⠀⡀⠀⣀⠁⠄⠀⠀⠉⠀⢀⠄⠀⠀⠰⠀⠀⠘⡀⠀⢀⡀⠤⠒⠇⠀⠀⡄⣀⠈⠀⠀⠀⢡⠀⠀⡰⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠂⠀⠀⠀⠀⠑⠀⠂⠁⠀⡄⠀⠀⠀⠑⠂⠈⠀⠀⠀⠀⡀⠀⠀⠁⠀⠃⠒⠀⠃⠀⠂⠀⠀⠀⠀⠀⠀⠂
⠀⠀⡀⠀⠠⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠁⠀⠀⠀⢀⠀⡄⠒⢤⢀⠀⠀⠀⠀⠀⢐⣰
⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠃⠁⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⢈⢉⠣⡀⠇
⠀⠐⢄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠒⠐⠤⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠈⠀⠀⠀⠢⢀⠂⠀⠀⠀⠀⠐⡄⠂⡀⠀⡡⢄⠀⠀⠀⠀⠀⡠⠤⠄⡀
⠀⠀⠀⠈⠀⠐⠁⠀⠀⠰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠑⠰⢉⡈⠄⠐⠀⠀⠀⠀⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠄⠀⢀
⠀⠀⠀⠀⠒⣤⠒⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⡎⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠐⠀⠒⠁
⠀⠀⢄⠈⢢⠦⣔⠁⡰⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠈⠠⠄
⠀⠠⠌⣠⠝⠰⢂⣈⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠁⠐⠁⠀⠀⠀⢀⡠⠣⠀⡀
⠀⠀⠀⠠⠄⠓⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢄⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⢀⠔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⢀⢐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⡀⠀⢢⠀⠀⠀⠈⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠈⠀⠀⠀⠀⠀⢀⠤⠄⡀⠀⠀⠀⠀⠀⠉⠀⠈
⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⢀⠀⢁⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠄
⠀⠀⠀⠐⢄⠀⠁⠀⠀⠀⠀⠢⠀⠀⠠⠊⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠠⠀⠀⠀⠀⡀⠆⠀⠀⠀⠄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠡⠀⠐⠈⠀⠀⠀⠀⠀⠃⠄⠁
⠀⠀⠀⠀⠐⢠⠉⢒⢀⠆⢠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⠀⢀⠠⠔⠠⠰⣅⠀⠀⠀⠀⡐⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⢄⡀⠄
⠀⠀⣀⡀⠁⢠⠄⠐⠢⢀⡈⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⡡⠊⠀⠀⠀⠀⠀⠈⠁⢀⠀⠒⠀⠀⡐⠀⠀⠀⠀⠀⡠⡅⠙⢠⠁⠀⡃⠜⠠
⠀⠈⠄⠈⠖⠀⠀⠀⠀⠈⢃⡀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡅⠀⢀⠂⠉⠁⠢⠀⠀⠈⠀⠀⠀⠜⠀⠀⠀⠀⠀⡀⠴⠐⠦⡀⠆⠬⠁⣀⠈⠰
⠀⢀⠦⠠⢆⠀⠤⠀⠴⠀⡈⠀⠢⠈⠀⠀⠔⠁⠀⠂⠁⠀⢀⠃⠀⠐⢀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢈⠆⠠⠤⢄⡀⡀⢅⡀⠈⠱⠄
⠀⠈⠰⠁⢀⠱⠤⢤⡄⠊⢋⠐⡆⠀⠀⠀⠈⠢⠀⣀⡀⠀⠊⠡⡀⠀⠀⠈⠁⠀⠀⢀⠴⣀⡀⢠⠀⠀⠀⠀⠀⠀⠈⠚⢀⡠⠎⠀⣧⠀⠑⡈⠄
⠀⠀⠀⠀⠁⠼⡀⣤⠈⠦⠜⠀⠀⠀⠀⠥⠂⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠤⠤⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠐⢂⡠⠂⡐⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡿⢿⠿⡿⡿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣻⣿⣿⣿⣻⣟⣿⣛⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢹⠶⠂⠦⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⠀⠒⠷⣦⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⣿⠖⣤⡜⠛⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠈⣟⢦⣽⣎⣒⢠⣈⡉⢄⣀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠫⠆⡠⢄⣀⠀⠈⠶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠓⠪⠋⣙⡯⣭⡹⣠⣉⣀⡉⠉⠉⠣⣀⣁⠀⠔⠁⠄⠠⠞⣡⠅⠀⠀⠀⠀⡮⢄⣠⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠑⠀⠀⠐⢤⣄⡀⠀⠈⠛⠷⠼⡏⣻⣯⣽⣿⣷⣶⣼⡭⠙⠂⠒⠀⠀⠀⢁⡀⠀⠐⠃⠅⠂⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠳⣄⠀⠀⠀⠀⠀⡀⠈⣙⠳⢂⣦⣜⣎⡙⠛⠻⣛⣿⢾⣍⢛⡛⠁⠀⠀⠠⠚⠉⠁⠀⠀⠂⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣷⢤⣀⠀⠀⠀⠀⠀⠙⠒⠀⠈⠀⠀⢈⠐⡓⠺⠛⠃⠙⠁⠠⠤⠀⣀⣤⣅⡀⠀⠈⠀⡘⠋⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⢧⠈⢷⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠓⠒⠒⠂⣀⠈⠉⠉⠀⠀⠈⠀⠀⣤⡐⠛⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠙⢧⠀⡇⠀⢠⣈⠒⢦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠸⡗⠃⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠟⠓⠲⠤⠭⠀⠀⠀⠈⠶⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⢄⡀⣀⡈⠁⠉
⠀⠀⠀⠀⠀⠀⡉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⠍⠉⠀⡈⠝⠀⠁
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠈⠃⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡄⠤⠤⠤⣶⣿⣿⣴⣦⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⢤⠀⠀⠀⠀⠈⠉⠀⢀⣤⣤⣤⣄⠀⠛⠺⢏⣯⢻⡷⣾⣧⡀⠀⠀⠀⢀⣴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⣀⣀⣤⠴⣶⠟⠁⠀⠉⢻⠀⠀⠀⠀⠀⠉⠉⠘⠋⠀⠀⠀⠀⣠⣽⣿⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢡⠞⣁⠀⣏⠀⠀⠀⠀⠀⠀⠀⠠⣤⠶⢀⣤⣴⣤⡀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣎⡁⣴⣾⣶⣶⣽⣿⣴⣶⣶⣄⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⡄⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⢯⣾⣎⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠈⠻⣿⡿⠿⠛⢻⠄⠀⠀⠀⠀⠀⣠⠀⠘⣿⣿⡇⠀⢀⣵⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠉⢸⣿⣿⣷⣯⣽⣯⣿⡟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣴⣶⣿⣶⣦⠘⠻⠃⢰⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣼⣧⣰⣿⣷⣄⡀⠀⠙⠻⠿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⠟⢁⡄⠀⠘⢿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣼⣻⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠛⠿⠿⠿⠟⠃⠀⠀⠀⠀⢰⡄⣶⠀⢠⡀⢰⡖⣤⢰⡀⠀⢰⡒⠀⣄⠀⠐⡖⠂⠀⠀⢰⢏⡏⠀⠀⠀⡼⣼⣄⡀
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠸⠇⠿⠠⠈⠷⠸⠏⠧⠀⠛⠀⠸⠉⠀⠌⠛⠀⠧⠀⠀⢠⡛⡾⠁⣠⣴⢾⣿⡗⣿⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⢠⣾⣿⣿⣿⣆⠀⣄⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⡀⣠⠀⢠⡀⣄⠀⠀⠀⠀⠀⠀⠀⠈⡉⠀⣼⣿⣵⡿⠻⣿⡿⣷⣝⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠹⠿⣿⣿⠿⢟⣼⣿⠆⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠈⠷⢸⡇⣿⢀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣿⠟⠋⠀⠀⠙⠇⠀⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣡⣤⡌⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⣴⣾⣿⣦⣿⣿⣢⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣿⣿⣿⣿⣾⣿⣷⡴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⣶⢁⣤⣀⣿⣿⣿⣿⣧⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣐⣮⣿⡿⠆⢺⣿⣿⣿⣿⢿⣿⣸⣿⣿⣿⣯⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⡏⣾⣿⣿⣿⣿⣿⡏⡿⣿⣿⠇⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⣾⣶⡻⡯⠊⠀⣰⠀⠻⢿⣿⡿⠘⠿⢿⢸⢫⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣧⣈⣿⣿⣿⣿⣟⣡⣿⡿⠁⢀⣠⣴⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⡀⠀⢀⠀⠹⡆⠀⠄⠘⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠘⠚⠋⠈⠉⣉⣴⣾⣿⣿⣿⡟⠁⠀⠀⢠⣀⠀⢲⡀⠀⠀⠀⠀⢀⣿⣿⣇⠀⣟⡇⠀⠁⠴⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠋⠀⠀⠙⠠⠀⠀⠀⠜⠳⣂⠀⠀⣠⣾⣿⣿⣿⠀⢱⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣿⠟⠁⠠⣤⡀⠀⠀⠐⠀⠄⠠⠞⠁⢠⣾⣿⣿⣿⣿⡿⠀⠈⠣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠃⠈⠉⠁⠀⠀⠀⢀⣭⠀⢆⣀⠔⢿⠆⠀⠀⢻⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠿⠁⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠴⠖⠒⠒⠛⠛⠛⠛⠉⠉⠉⠙⠛⣛⣓⣿⣾⣶⠶⣶⡶⠦⠄⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠐⣿⢤⣤⣤⣤⣤⣤⣀⣀⣀⣀⡀⡀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠋⠀⠘⠓⠒⠚⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠙⠻⣿⣿⣿⣿⣥⣀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢿⣿⣿⣿⡟⠏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⢿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡞⢿⣿⣿⣯⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⢸⣿⣿⣿⣯⡤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢉⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⢻⣿⣿⣟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡙⢿⣿⡿⠶⠀⠀⠀⠀⣀⢀⢀⠄⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣬⣻⠛⡷⣀⡠⣥⢄⡉⠥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣟⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡏⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣶⣶⠿⠿⢿⠿⠿⢿⣿⡻⠆⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣫⣿⣹⣽⠷⡀⢰⣆⢀⡼⣭⠓⣢⡔⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣴⢶⣬⡃⢙⣿⣾⣷⣾⣁⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢈⣿⣿⣷⣾⣷⣿⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⢿⡷⢿⡿⠻⡄
⠀⠀⠀⠀⠀⠀⢂⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣾⣿⣾⠗⣮⡓⠛⢷⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⢐⣥⣬⣿⣿⣿⣧⠦⣤⣾⡀⣵⣿⣶⠞⣱⡯⡹⠶⣤⡿⠆⣾⣿⣉⠙⣦⢡⠂⣀⠠⢀⠀⠀⠀⢀
⠀⠀⠀⢈⣽⣿⢿⣿⡿⣫⢖⣲⣿⣿⣯⣿⢿⣯⣾⠟⣯⠀⠄⠈⣧⢹⢛⢌⡻⣎⢁⣨⠶⠶⠶⢤⡙⢤⠀⠁
⠀⠀⠀⢤⡿⣿⣿⣿⠀⠿⣿⣿⡿⢿⣛⣥⡶⣿⣿⡇⠈⠁⣰⢷⣌⠛⠈⡿⣶⣬⣁⡈⠲⠮⣥⡤⠟⢸⣆⣀⣀
⠀⠀⠀⠈⠛⣿⣿⣿⣧⡛⣴⣻⢰⣿⠟⣏⣽⢻⣾⠇⣣⣸⣿⡙⣿⣧⡁⣿⠿⣙⢿⠿⣷⠹⡞⣻⣟⣼⡏⣷
⠀⠀⠀⠀⠀⠽⢿⣿⣿⣿⣮⣙⠈⠻⠤⢿⣷⢾⡟⢦⣻⡓⠛⣯⣍⠼⡷⣮⣓⠶⠤⠤⠋⡀⢛⣴⣾⡿⢡⡟
⠀⠀⠀⠀⠀⠀⠈⠻⣟⠻⢿⣿⣿⣗⣲⣴⡤⡌⢉⣉⣃⣈⢉⣉⣁⢘⡉⡉⢉⢠⣒⣦⣷⣾⠿⠛⣉⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⠬⣍⡛⠛⠿⠃⠿⠻⠶⠣⠿⠾⠷⠿⠾⠿⠽⠟⠛⡚⣋⡩⠤⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠒⠒⠂⠒⠐⠒⠒⠒⠒⠚⠋⠉⠉⠁
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣴⣶⡆⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⣿⣿⣧⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢠⣿⣿⣿⣶⠀⠀⠀⠈⢻⣿⠿⣿⣿⡟⣱⣷⣶⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣾⡿⣿⣿⣿⡄⠀⠀⠀⠀⢁⣤⣻⠁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⢠⣶⣿⡇⠋⠀⢸⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠀⣨⣍⡛⢧⡹⣿⣿⣿⣿⣇⡇⠀⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠐⠻⢿⣿⢿⡏⢸⣿⣿⣿⣷⣄⣸⣿⣿⣿⢹⡇⠀⣸⣦⣽⣿⣿⣿⣿⡟⠿⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⢘⣯⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⡿⢻⣿⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣷⣿⣿⡿⠟⠁⠀⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣯⣥⣷⣾⣧⣀⠀⣠⢽⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣶⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢻⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠘⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣙⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣫⡽⠖⠀⠀⠘⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣰⣿⣍⠳⣜⠿⣿⠿⣟⣛⣛⡿⣿⣿⡿⣫⡵⢋⣥⣶⣿⡧⠀⠀⠀⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣷⣍⡛⢛⣋⣭⣭⣭⣙⠲⠶⠚⣫⣴⣿⡿⠋⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡀
⠀⠀⣠⣤⣴⣿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⡄
⠀⢺⣿⣿⠟⠁⠀⠀⠀⣰⣿⣿⣿⣿⣿⠏⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⢠⣾⡿⠋⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⡿⢰⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⣿⣿⣿
⠘⣿⠇⠀⠀⢀⣴⣶⣾⣿⣿⣿⣿⣿⡇⠈⠟⠛⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⡿⠟⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡰⢄⠈⠀⠀⢀⠀⠀⠅⠑⢶⣦⣄⣀⠀⠀⠀⠀⠀⠈⠀⠀⠰⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⣁⣀⡴⠖⢾⡟⠀⢀⠀⠀⢀⢀⡄
⠀⠀⠀⠃⠦⠈⢱⠚⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠶⠈⠡⠈⢲⣶⠄⣠⠠⣤⡀⢠⡤⣀⣀⣀⢀⠄⣀⣤⠀⠄⠰⡶⠒⠄⠉⠏⠁⢀⠀⠀⠀⠈⠀⠐⠲⠖⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠀⠛⠀⠀⠀⠈⠟⠁⠁⠀⠙⠁⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣈⠻⣤⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣀⡸⠟⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠁⠀⠀⠊⠀⠈⠓⠖⠒⡀⠤⢤⠀⠀⢀⣀⠀⠀⢌⡀⢀⠀⠀⣠⠶⠤⡤⢀⠀⠰⠛⠀⠸⠁⠉⠉⠀⠅⠀⠀⠀⠈⠁⠀⠀⡏⡆
⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠤⠀⠠⡖⢆⠀⠰⣴⡄⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠶⠈⠀⠀⠀⠃⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⣀⠈⠀⠀⠈⠀⠀⠀⢱⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠽⠁⠀⠀⠀⠁⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠘⢶⡄⠀⠀⢀⣀⠀⢀⡈⠰⠆⠀⠀⠀⠁
⠀⠀⢤⣴⠾⠶⠶⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠀⠀⠘⣛⢂⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢺⠂⠄⠈⠁⡀⠈⠈⠒⢀⣀⠀⠰⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⣤⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠁⠈⡀⠀⠀⠈⠁⠀⠀⠆⠁⠀⠐⠦⡈⠉⢠⠀⠀⠀⠉
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠀⠈⠁⠀⠀⠀⢀⡰⠒⠇⠀⠰⠂⠀⡀⠀⢀⠀⠈⡧⢀⠀⠀⣀⠈⠉⠁⠀⠀⠀⣃⡦⠀⢀⠇⠐⠀⠀⠀⠺⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠈⠋⠀⠀⠀⠀⣀⣷⣄⢈⡧⠀⣧⠈⡷⡀⢨⣦⣀⠀⠀⠀⣠⡝⠀⠀⠀⠀⢀⠤⠈⠀⢠⡦⠀⢀⠀⠘⠛⠅
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⢀⠊⠀⠀⠀⠀⠀⠀⠒⠈⠀⠐⠣⡾⢀⡨⠉⡍⢛⡍⣯⠉⡍⡈⢺⠀⠀⠁⠛⠀⠀⠀⠘⡐⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⣇⡘⡄⢀⢁⠉⡍⣀⢠⢀⡁⣸⠀⠀⠀⣀⡀⠀⢀⠟⠀⠀⠀⠰⠆⠘⠇⡀⢀⠀⡀⠌
⠀⠀⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣘⣋⡶⣀⠏⡇⠀⢿⡜⠙⠇⠘⢿⣟⠃⢙⠛⡆⠀⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠆⣁⠘⠀⠀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠉⠀⠸⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡟⠘⠃⢀⠻⠃⢀⠓⠀⠘⠀⠐⠎⠁⠆⠉⠰⡧⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢆⠂⠀⠀⡀⠀⠘⠋⠀⢀⠘⠁⠀⠘⠁⠀⠀⠖⠀⠛⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⢢⣶⠱⠀⢿⠎⠀⣀⢠
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢘⢥⠀⡛⠩⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⣣⣾⠃⠀⠀⠀⠸⢛⠄⠀⠀⠶⠃⠚⠛⢠⣤⣿⠂⠁
⠀⠀⠀⠀⡀⠀⠄⠀⢀⠀⠀⠀⡠⣤⠀⠀⠀⠀⠘⣞⠀⣟⠢⠤⡉⣀⠒⠀⠀⠀⣀⠤⠤⠤⠐⠒⠀⠀⣈⠔⠀⡀⠀⠀⠀⠀⠀⠀⠚⠠⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠇⠀⠘⠛⠁⠀⠀⠀⡀⠀⠉⠀⢨⠙⠲⢤⣄⣀⣀⡀⠀⠀⠀⢀⡀⠠⠄⠒⠋⣀⠄⠨⢁⣀⠀⠀⠀⠀⢀⣀⠂
⠀⠀⠀⢠⣊⣅⡰⢄⡀⠀⠂⠀⠀⠀⠀⠀⠀⠛⢋⠉⠉⠀⠑⠢⠤⠄⢈⣉⣉⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⢀⠀⠀⠉⠀⡀⠀⠀⠀⠄⠀⡄⠠
⠀⠀⠀⠘⠉⠞⠀⠙⢍⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠠⠄⠀⣰⣆⠀⡀⠀⠀⠀⠀⠠⠊⠀⠰⣤⠶⢎⡛⣠⣀⢠⡄⠀⠀⢈⠀⠈⠱⠀⠀⠠⡄⢸⠀⡇⢀⠆
⠀⠀⢠⡄⠀⠀⠀⠀⠀⠘⠌⡆⠀⠀⠈⠀⠀⠀⠀⢹⠀⠀⠀⠘⠳⠦⠴⠁⣁⣀⣴⣄⠀⠐⠀⠀⢬⣻⢨⣻⣯⢹⣶⢃⡢⠀⠠⠀⠁⢨⠀⠀⠀⡇⢸⠀⣇⢸
⠀⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠾⠿⡧⣥⢿⣼⡇⠀⠐⠐⠒⠐⠀⠀⠀⠁⠈⠀⠁⠈
⠀⠀⠀⠀⠠⠤⠀⠛⠠⠤⠤⠀⠀⠈⠑⣄⡀⠐⠀⠉⠁⠀⠁⠘⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠁⢻⠶⠜⠃⠀⠰⣤⠖⡩⡀⠀⢴⣤⠀⠀⠠⠈⠓
⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢑⠑⡇⠀⠀⠁⣀⠀⢀⡤
⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠉⠀⠀⠀⠀⠾⢗⠀⠀⢤⡀⠀⠀⠀⠀⠄⡀⢸⣦⠀⠀⠈⠀⠀⡅⠀⠀⢀⠀⠑⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙⠛⠉⠀⠒⠀⠀⠀⠄⠀⠀⠀⠀⠈⠙⠀⠀⠀⡐⠿⠁⠈⠋⠀⠀⠠⠀⠰⠀⠀⠀⠈⠀⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠄⠀⠀⠠⠠⠄⠤⠀⠄⠀⠠⠀⠴⠤⠀⠐⠀⠀⠙⠿⠂⠀⠀⠀⠀⠀⠀⠀⢂⡟⠁⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠂⠀⡀⠀⠀⠀⠀⠀⠀⢸⠁⢠⢠⠀⠈⡇⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠘⠁⢀⠀⠀⠘⠀⠐⡀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⣇
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠂⠀⠤⠤⠤⠤⠤⠀⠂⠀⠀⠀⡀⢈⣉⣁⣀⣀⣈⠁⠠⠀⠀⠠⠊⠀⠀⣄⠀⠀⠀⣀⠴⠃⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠈⠂⠠⠀⠀⡠⠊⠀⠐⡀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⠀⠀⠀⠀⠰⡀⠤⠤⠤⠤⠀⢀⡆⠀⠀⠁⠀⡄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⢀⠀⠀⢠⠂⠈⠀⠈⠀⠠⠄⠀⠁⠀⠀⠀⠘⠄⠉⠉⠉⠀⢀⡜⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠋
⠈⠀⠀⠀⠉⢹⠀⠸⢄⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠤⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠐⠒⠛⠉
⠀⠀⠀⡔⠀⢸⠀⢸⠄⠳⡄⠀⠀⡀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠦⠤⢤⣀⠀⠑⢤⡀
⠀⠀⠈⠀⠀⠀⠀⠀⠘⡀⠈⠢⠀⡇⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠀⠀⠀⠀⠀⠐⠃⠀⠓⠲⠀⠀⠀⠀⠀⣄⠀⠀⠀⠈⠓⠄
⠀⠀⠀⠰⠀⠀⢠⠀⠀⠱⡀⠀⠀⡇⠀⠀⠀⠀⢀⡆⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠘⣆⠀⠀⠀⠀⠀⡄⠀⠀⠘⠀⠀⠀⠀⠀⠘⢦⣠⣤⣠⠤
⠀⠀⢀⠁⠀⠀⠘⠀⠀⠀⢱⠀⠀⠇⠀⠀⠀⠠⠊⠀⠀⠈⢳⡀⠀⠀⠀⠀⡀⠀⠋⠀⠀⢀⣘⠆⢀⠀⠀⠀⡇
⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⡇⠀⢄⠑⠦⢤⣀⣈⣉⠭⠔⢁⠀⠀⠠⣈⠒⠢⠤⠤⠬⠤⠒⣁⠄⠀⠀⡇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠑⠒⠒⠒⠒⠒⠋⠁⠀⠀⠀⠈⠉⠙⠛⠒⠛⠋⠉⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠇⠂⠀⠰⠤⠆⠈⡠⠂⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠠⢀⣀⡀⠀⠀⠀⠀⠀⠋⠀⠐⠀⢠⠀⠀⠀⠒⣀⠣⡄⠐⠐⠃⠀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠒⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠐⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠤⠀⠀⠀⠀⠀⠘⠀⠺⠈⠣⣄⠓⠌⠙⠦⠙⠂⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠉⠈⠂⠀⠑⠂⠁
⠀⠀⠀⠀⠀⠀⢠⠀⠉⠉⠉⠀⢰⠀⠀⠀⠈⠑⠢⠤⣀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠢⡐⠒⠚⠋⠉⠉⠁⠀⠀⢀⡤⠲⠭⠀⠀⠀⠨⠗⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠐⠀⡆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠈⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⡀⠠⠀⠀⠒⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠂⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰⠁⠀⠀⠒⠂⠈⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡴⠀⡂⣈⠠⠕⠢⣈⠉⡀⠐
⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠓⠌⠀⡈⠁⢈⠉⠁⠀⠀⠀⠀⡠⠄
⣿⣷⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣿⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⡶⠶⠶⠶⢶⣶⣶⠶⣶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⣿⠀⠀⠀⣶⠶⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⣿⠀⠀⢠⠏⠀⠀⠉⠙⠛⢻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠁⠀⠀⢠⡿⠀⣠⠏⠀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⠛⠛⠉⠁⠀⠀⠀⠀⠀⣼⡷⠞⠁⠀⠀⣀⡴⠚⠁⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣖⣋⣁⣠⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠛⠛⠛⠛⠋⠉⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⣄⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⣤⣀⣤⣠⡀⡀⠀⠀⣠⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠶⠾⣿⠿⡟⣿⣿⣿⣀⠞⠁⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠈⠁⡾⠴⠾⢯⡅⠀⢠⠀⠉⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⣰⡇⠀⢀⣾⡆⠀⢰⡀⠙⠿⣦⣤⣀⣠⣤⠶⢋⡁
⠀⠀⠀⠀⠀⣠⢾⣿⠁⣼⡿⠀⠀⡼⣼⡇⠀⣿⣧⠀⠀⢀⡌⠉⠀⣄⠀⠀⣷⣤
⠀⠀⠀⢀⢞⡕⣿⠇⡜⡽⡇⠀⣸⣹⢻⡇⢠⣿⢻⡀⠀⢸⣧⠀⠀⡿⣄⠀⢸⣹⣣⡀
⠀⠀⢀⠏⣯⣷⠏⢸⢻⣿⠃⠀⡇⣏⣾⠇⢸⡸⣦⠃⠀⣏⢿⠄⠀⡇⡟⡆⠈⢿⠿⢳
⠀⠀⠀⠓⠚⠁⠀⠘⠒⠋⠀⠀⠙⠒⠋⠀⠀⠣⠤⠃⢸⢸⣾⠀⠀⡇⢇⣧⠀⠈⠑⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠉⠀⠀⠈⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⢠⠔⠎⠀⠈⠓⠒⠒⠒⠋⠀⠀⠀⠓⠒⠒⠒⠊⠀⠀⠷⠦⡀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣷⣦⡄
⠀⠀⠀⠱⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠇⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠙⠦⠔⠉⠉⠉⠉⢻⠀⠀⠀⡏⠉⠉⠉⠙⠢⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠋⠉⠉⠉⢹⣿⣿⣿⡏⠉⠉⠉⠉⠻⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢲⠀⠀⢰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣴⠖⢀⡴⠀⡶⡄⢰⠀⣦⡀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢲⡀⠀⣷⡀⠀⠹⣆⢸⡀⠀⣇⠏⣼⢸⣃⡇⢸⠃⣼⢁⡶⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠸⣇⠙⣎⢷⠀⣷⠳⡄⠀⠼⠃⠙⠂⠛⠂⠃⠸⠉⠀⡏⢰⠃⡾⢡⢏⡇⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢤⡙⢦⣸⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠡⢿⢋⣴⣺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⣀⣤⣴⣶⣶⣶⣤⣤⣀⣴⣦⣀⣤⣤⣶⣶⣶⣤⣤⣀⡀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠾⠿⠟⠛⠛⠉⠉⠉⠀⠀⠉⠉⠉⠉⠀⠀⠉⠉⠉⠛⠛⠻⠿⠶⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡴⣇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀
⠀⠀⠀⠀⠀⠀⢠⣾⠴⢃⣞⠉⣰⠦⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⡆⢹⣄⠹⣌⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠠⣏⠀⣰⠓⢠⠏⢸⢠⣷⡀⢠⢀⠀⠀⡀⢤⠄⡄⢠⠦⡄⣦⣹⡘⣄⡀⠹⡌⣷⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢿⣠⠏⣼⡴⢃⣿⣾⡀⣸⣇⢸⠀⣧⢸⠀⢷⢸⡈⣧⠀⣹⠆⠙⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠋⠃⠸⠈⠹⠇⡟⢻⢸⠄⢻⠘⠧⠞⠀⠃⠈

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⣴⣶⣾⣿⣿⣿⣿⣿⣿⠏⢿⠀⠉⠙⢿⡆
⠀⠀⠀⠸⣿⡇⣿⣿⡟⠘⠋⠋⠅⠀⠀⠀⠉⠀⠈⡇⠀⠀⢀
⠀⡀⠀⢀⣿⣧⣸⣿⡇⠀⠀⠀⡆⠀⠀⠀⠂⠁⠀⡇⠀⠀⡄⠂⡄
⢆⠀⣠⣾⣿⣿⠏⢻⡇⠀⠀⠀⢀⠀⠀⠀⠀⡠⠀⢧
⠈⠛⠛⠛⠛⠁⠀⠀⢹⣄⢀⠀⠀⢸⣶⣶⡇⠀⠀⡸⠀⢀⠁⠇
⠀⠀⠀⠨⠐⠄⠀⠀⠘⠮⣅⡀⠀⠈⠁⠁⠀⣀⡜⠓⠆⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠄
⠀⠀⠀⠐⠂⠨⡢⠀⠀⡀⡔⠉⠑⠒⠒⠒⠋⠁⠘⡆⠁⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⠀⠀⠛⠀⠠⠖
⠀⠀⠀⠀⠀⠀⠈⠂⠄⠂⠉⢲⠁⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠶⠎⠀⠀⠁⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣀⣤⠤⠤⠶⢲⠾⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠒⠒⠲⣶⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠀⠀⢀⣀⣴⣏⣀⣾⠄⡼⠀⠀⠀⠀⢸⣿⣭⣭⣍⣉⣹⣿⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢑⢦⠞⠉⠉⠁⠀⠀⣠⢣⠎⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠱⡀⠀⠀⠀⠀⠀⠘⠛⠻⠷⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⣵
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣴⠂⢠⣤⣤⣄⣀⠀⠀⣰⣧⣬⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣽⣿⣶⣿⣿⣿⣿⣿⣆⡀⣿⣯⣄⢠⣀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠹⠇⠹⢿⣿⣿⣿⣷⠸⡅⣩⠀⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡉⠀⣘⠃⠸⣿⣿⣿⣿⠤⣧⡙⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣦⣁⡴⠀⠻⣿⣿⣿⣶⠟⣿⠟⢀⣾⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⠻⢻⠯⣤⡟⠀⠀⣀⣈⣋⠀⠀⠀⠘⣇⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⠄⠀⠀⠀⠀⠀⠀⠀⠄⣌⢠⢠⠀⢈⠄⠀⠐⠙⠿⠈⠳⡆⠀⠀⢻⣿⣿⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⡾⠃⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⢇⠀⠀⢸⣿⣿⡆⠀⢌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣰⣤⣞⠰⣧⠚⣷⣄⣄⠀⠀⠃⠀⢀⣲⡏⢳⣿⠀⢸⢸⣿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠈⢿⡟⠁⢠⣿⣿⡿⠀⢰⠄⠀⣨⢷⡅⢈⣻⠀⢸⣾⣿⣿⡇⢠⣟⠃⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⣠⠟⠈⢻⡀⣰⠃⠀⣴⠋⣾⣯⣨⣿⠀⢘⣿⣿⣿⡇⠀⠈⢁⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠻⠶⢀⠈⠃⠋⠐⠈⣧⠚⠉⠛⢿⡿⢀⣾⠛⠉⢹⡇⠐⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠓⠧⣀⣠⡀⡇⠀⢀⠄⠈⡠⠊⠃⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡛⣡⣤⣴⡾⠁⠀⠀⢀⣴⡟⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⢠⣤⣴⠋⠀⠀⠀⠀⠠⠽⠋⠙⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⢀⠀⣿⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⣰⡏⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠟⠀⠀⠀⢠⠀⠀⠀⠀⠱⡀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⣀⢀⡀⠉⠀⣀⠴⠦⠀⠀⠀⠀⠀⠀⠕⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠠⣻⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠻⠇⠀⡟⠃⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣭⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀⠀⣗⡆⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣤⣿⣧⡀⢫⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣷⡛⢃⣺⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⡟⠉⠀⢠⡤⢠⡄⠀⠀⠀⠀⣾⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠛⠻⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢠⠀⠀⢰⠛⠱⡀⠀⠀⠀⠀⠀⠔⠀⢢
⠀⢀⣤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡆⠀⢸⠀⠀⢱⠀⠀⠀⠀⡘⠀⠀⢸
⠀⢸⢁⠀⠠⢌⠳⡄⠀⠀⠀⠀⢀⠤⠒⠉⠉⠈⠉⠑⠠⡀⠸⡴⣚⠧⢀⠼⣴⢤⣸⡀⠀⠀⢀⠃⠀⠀⡌
⠀⠈⢮⡈⠁⠀⣹⣼⣆⠀⠀⠰⠃⠀⡀⠀⠀⠀⠈⠀⠀⠈⢆⠹⡀⢀⠀⣀⠈⠓⠺⡇⠀⠀⢸⠀⠀⢸⠁
⠀⠀⠀⠑⣤⠀⣩⡅⠈⢆⠀⡅⠀⡀⠀⣠⠔⠢⣄⠀⠈⠀⠘⡀⠈⢦⠀⠀⠀⠉⢀⡇⠀⠀⡧⠂⠀⡎
⠀⠀⠀⠀⠈⠻⡻⣠⣤⣼⡄⡇⠀⠁⣰⠃⠀⠀⠈⡆⠀⠄⠀⡇⠀⠀⢏⠀⠀⠀⠨⠇⠀⢸⣣⡠⣰⠁
⠀⠀⠀⠀⠀⠀⠙⡟⠿⠟⢃⢸⠀⠀⠜⢦⡀⠀⡰⠁⠂⠆⢠⠁⠀⠀⢸⣤⣤⣍⣸⠀⠀⣇⣈⣍⡏
⢀⡤⡀⠀⠀⠀⠀⢸⡄⢒⢻⠀⢃⠀⠠⠀⠈⠉⠀⠠⠂⢀⠎⠀⠀⠀⠤⠸⠿⠿⠇⠀⠘⠯⠿⡾⠁
⣏⠀⠈⢂⠀⠀⠀⡸⠁⠀⠉⠀⠀⠳⢄⠀⠀⠀⠀⣀⠔⠁⠀⣀⠴⠚⠫⢲⡵⠃⠀⢀⣤⢬⡉
⠱⡀⠀⠀⡱⣲⢾⡅⠀⢠⠁⠀⠀⠀⠀⠉⠉⠈⢉⠔⠒⠊⠉⠀⠀⠀⣠⠞⠀⠀⢰⠁⢿⡄⢸
⠀⠑⠤⢀⡇⢘⠂⢉⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⠤⠔⠊⠀⠀⠀⠀⠈⠢⠿⠧⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⣠⣚⢢⠀⠀⠀⠀⠀⣷⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⢧⠀⠀⠀⠀⢀⠼⣁⣪⠉⠀⠀⠀⠀⠶⠿⣿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠻⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⡛⢄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣵⠃⠀⠀⠀⠀⠀⠀⣠⣜⠣⠙⠋⠀⠀⠠⠒⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡙⡅⠀⠀⠀⠀⠀⠀⠀⠛⠁⠙⠁⠀⠀⠀⠐⠄⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠀⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣱⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡧⠄⠀⠀⠀⠀⡠⠢⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⠈⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡂⣣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠠⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠢⣀⡀⠀⠀⠠⢄⠚⠈⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⠀⠈⠒⡤⠂⢄⠀⠀⠀⠀⠀⠀⣶⢑⣈⡄⠛⠁⠀⠀⠀⠀⣥⠐⠺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠉⠢⡀⠀⢄⣀⣴⠀⠀⠀⢴⡙⣄⣨⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⢙⣄⠀⠀⠀⠑⢶⡑⣤⡥⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⢠⣾⠉⠨⠿⠁⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⣰⣿⣿⣷⣄⠀⢠⣶⡷⡉⠀⠀⠀⠀⡊⠀⢐⠀⠀⠀⠀⠤⣾⠇⠀⠙⠁
⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⣠⣾⣿⣿⣿⠿⠋⠉⠢⡀⠀⠈⢆⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⣼⡟⠀⢀⣴⣿⣿⡿⠟⠁⠀⠀⣀⣤⣾⣷⣦⣤⠇
⠀⠀⠀⠀⠀⣸⡟⠀⣴⣿⣿⠟⠋⠀⢀⣠⣴⣿⣿⠿⠟⠋⠉
⠀⠀⠀⠀⢰⠟⣠⣾⡿⠛⢁⣠⣴⣾⡿⠟⠛⠉
⠀⠀⠀⢠⢏⣼⠟⣋⣤⡶⠟⠛⠉
⠀⠀⢀⣿⡿⠷⠛⠉
⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡒⠁⠉⣡⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣨⣿⣿⣯⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⡅⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡶⢻⡟⣿⣿⡟⢎⡴⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡜⢿⣯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢴⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠹⠶⡴⣷⣿⣿⣷⠘⢓⣴⣗⡠⡀
⠀⠀⠀⠈⠀⠠⣦⡬⣿⡇⠘⠁⠀⢀⣤⠄⠀⠦⠀⠀⠠⠀⣤⡄⠀⣦⣀⡈⠉⢉⣀⣲⣆⢠⠆⠠⠀⠀⠀⢠⣾⣿⣷⣶⣶⣾⣿⣿⣷⣶⣿⣿⣿⣿⣦
⠀⠀⠀⠰⢶⡴⠿⠟⠘⣷⣤⠤⠀⠿⠷⣦⡄⡠⠀⠀⢀⠒⢶⡆⢻⣿⠆⡀⠀⠀⢼⣿⡿⣲⠦⢆⠀⠀⠀⠀⠻⠿⢋⣭⣉⡉⢻⣿⡏⣽⣿⣿⣿⠟⠟⠃
⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣙⠂⢻⡇⣾⣿⡇⢏⡸⢏⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠘⡇⣿⣿⡇⣿⠃⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠈⣸⡇⣿⣿⡇⣿⡀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠇⣿⣿⠇⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⣸⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡽⠂⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡾⣿⣿⡿⠐
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⠁⣭⡍⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢻⡏⠁
⠀⠀⠀⠀⠀⠀⡠⠊⢩⣿⡦⢹⢰⣾⡭⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⣍⢣⣤⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣼⡇⣠
⠀⠀⠀⡀⢀⣆⡀⡹⠋⢠⢧⣿⣤⠃⠉⢢⣠⣱⣄⡀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠆⠘⠋⠀⡀⢀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠈⢹⡟⠇
⠀⠀⠀⣧⣉⣁⠲⢋⣩⡘⢠⣎⠰⢒⣮⣉⠛⣃⡉⠁⠀⢀⠀⢾⣔⠦⠄⣀⠀⠈⠣⠀⠠⠴⣲⠧⠄⡀⠈⢫⣤⣤⣼⣿⣧⣤⣼⣧⣤⣤⣽⣿⣧⣬⣵
⠀⠀⠀⡇⠘⣻⣶⢉⣹⣿⢆⣹⣷⣭⣿⡉⢿⣙⠃⠀⠀⠀⠀⣶⠞⣡⠀⠸⠆⠀⢠⠶⢦⣬⢳⣶⠀⠜⠁⠈⠧⠀⠜⠛⠂⠀⢸⡇⠀⠀⠀⠛⠷⠤⠭⠇
⠀⠀⠀⠁⠻⡏⠙⠀⠀⠀⡟⠉⠙⡇⠀⢀⡟⠛⠒⠀⠀⠀⠀⠙⠷⠁⠀⠀⢰⡆⠀⠀⠀⠢⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠠⠀⢰⣾⠀⣸⠀⢰⣖⠉⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠆⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠺⡄⣶⣿⡿⠟⡔⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠃⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠐⣿⡆⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⢠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⣼⣿⣷⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣬⣵⡟⢮⣏⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣸⠃⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣧⣠⣼⣯⣅⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣤⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡶⠶⠶⣶⣶⣶⠶⠶⠶⢶⣶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠟⣰⡇
⠀⠀⠀⠀⠀⠀⣰⡟⠀⣿
⠀⠀⠀⠀⠀⢠⣿⠀⠀⣿⡄
⠀⠀⠀⠀⠀⣾⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⣷⡀⠀⠀⣶⣤⣀
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠙⣷⡀⠀⠹⣿⠻⣷⣦⣄
⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠈⢿⣆⠀⢻⣧⡀⠙⠻⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠙⢷⣄⠻⣷⡀⠀⠀⠉⠻⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠙⢷⣜⣿⣄⠀⠀⠀⠀⠉⠛⢿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣄⡀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⣀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣦⣀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄⠘⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣦⣉⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡛⠷⣦⣝⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠙⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣄⠙⠛⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡌⢿⡿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⠀⠈⠙⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣷⡈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⡀⠀⠙⢷⣄⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠷⣦⣄⠙⣷⡀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡌⠙⠷⣌⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆⠀⠈⠻⣄⠀⠈⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠃⠀⠀⠀⠀⠹⣆⠀⠀⠈⠻⠀⠀⣶⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿
⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀⣀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠐⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣿⣿⠿⢿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡳⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣄
⠀⠀⠀⠀⠀⡰⣻⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⡭⠭⠭⠍⠁⠀⠀⠀⠀⠉⠙⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠀⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠐⠈⠉⠉⠉⠉⠉⠙⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠘⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⡭⠈⠻⡟⠛⠈⢻⡄⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠻⣿⣿⣿⣿⣿⣿⠉⠸⠳⡌⠀⠀⠱⠀⠈⠀⠱⡄⠀⠂⠀⠀⠀⠀⠀⠀⢀⣼⠛⢩⠀⠀⢹⠿⠟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢿⣿⣿⣿⣿⠓⠀⠁⢠⠻⣰⠀⠀⢢⣀⠀⡄⢻⡄⠀⠀⠀⠀⠀⠀⢀⣼⠻⡄⠀⡄⠀⠸⡀⠘⣧⠀⠈⠙⢿⣿⣿⣿⣿⣿
⠀⠀⢊⣿⣿⣿⣿⠀⠈⠀⠀⢰⡇⠀⠀⠀⠹⠿⠃⠀⢳⡄⠀⠈⠁⠀⢀⣾⠃⠀⠈⠾⠀⠀⠀⠹⣆⡟⠀⠀⠀⠀⢹⣿⣿⣿⡟
⠀⠀⠈⠹⣿⣿⣿⡤⢤⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠁
⠀⠀⠀⠀⣿⣿⣧⡀⠀⠀⠀⠋⢠⡆⠀⠄⠀⠀⠂⠀⠀⠀⡆⠨⠄⠀⣸⣿⣀⠀⠀⠀⠐⠀⠀⠀⣀⣀⣀⣤⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠈⢿⣿⣧⣾⡀⣀⣤⣾⠁⢀⠀⠀⠀⣀⣀⣀⡀⡇⠀⠀⠀⢸⣿⣷⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠃⠀⠄⠀⠀⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⠋⠁⢐⣀⣈⣥⣀⣒⣀⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⠿⠛⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠉⠋⠁⠈⠉⠿⠛⠿⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀⣀
⠀⠀⠀⠈⠉⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⣶⣶⡜⣿⠿⣛⢿⣿⢟⡻⢿⠋⣿⠿⣻⡷⣿⠿⣛⢿⣿⢟⡻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⢇⣿⢸⣿⡇⡇⢿⣿⢿⡇⣿⢸⣿⡇⣿⢸⣿⢸⡇⣭⣭⢼⡇
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠸⣶⣤⣶⣾⣿⣶⣵⠾⠿⠶⠵⠿⠷⠿⠾⠿⠷⢿⣾⣿⣾⣿⣾⣵⣿⠇
⠀⠀⠀⠀⣹⣯⠀⠀⠀⠀⠀⢶⣶⠀⣶⣶⠀⣀⣀⡀⢀⣴⣆⣠⣶⡀⢀⣀⡀⢀⣀⣀⡀⣀⣀⡀
⠀⠀⠀⢠⢿⣿⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⠘⢟⣸⣿⠀⣿⡇⢸⣿⢠⣿⣇⣿⡄⣿⡟⠋⣿⣌⠃
⠀⠀⠀⠈⠈⠁⠁⠀⠀⠀⠀⢸⢹⡏⣿⡟⢸⣿⢸⣿⠀⣿⡇⢸⣿⢸⣿⡏⢉⠁⣿⡇⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠄⠠⠿⠿⠸⠿⠾⠿⠄⠿⠧⠸⠿⠀⠻⠧⠞⠡⠿⠧⠀⠷⠼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢠⣤⣠⣤⣄⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠉⠉⠉⠉⠀⠁



⣿⣿⢛⢿⢛⠿⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⣄⠀⠠⡤⠀⠀⠀⠀⠀⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠁⠀⠀⠀⠀⠀⠀⢀⣠⡆⣶⣄⡀
⠲⠂⠀⡀⠐⠀⣈⠀⠐⠂⠀⠨⡡⠶⠶⢯⡷⠶⠶⢈⠆⠀⠰⠶⠰⠶⢾⠲⠖⠰⠖⠀⠰⠆⠭⠭⠵⡯⠭⠭⠰
⠀⠀⠀⠋⠀⠘⠑⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠹⠏⡿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⢰⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⢠⣶⣿⣿⣷⣆⠀⠀⠀⠀⣀⢀⣔⡇⣿⣠⡀⣀⠀⠀⠀⠀⡀⣤⡏⣦⡀⠀⠀⠀⠀⠀⠐⡶⣾⣷⢶⠂
⠀⠻⢿⣿⣿⣿⣿⣿⠿⠂⠀⠀⠤⠩⡥⡅⣬⢤⠥⠤⠀⠀⠐⠒⠯⣭⡅⣭⡭⠒⠒⠀⠐⠿⠿⡿⣿⣿⢿⠿⠿
⠀⠀⠀⠛⢿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠈⡇⡟⠁⠀⠀⠀⠀⠀⠀⠀⠊⡇⠏⠂⠀⠀⠀⠀⠀⠐⠛⢻⡟⠛⠂
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤
⠀⠀⠀⠀⡉⡎⡀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⢀⡤⣠⣾⣗⣟⣿⣦⡤⣄⠀⣀⣤⠤⠤⠇⠣⠤⠤⣄⡀⢀⠔⣠⣂⣀⣿⣀⣀⣄⠂⡀⠠⠤⣀⣀⣸⣗⣀⣀⠤
⠈⠒⠙⢿⡭⣯⣿⠟⠑⠋⠀⠀⠉⠉⠉⡇⡜⠉⠉⠉⠀⠀⠁⠈⠑⠀⣿⠀⠈⠉⠀⠀⠈⠉⠁⠈⢸⡇⠁⠈⠉
⠀⠀⠀⠀⠁⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠲⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠵

⠠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⠄
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⡀⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⢰⠇
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣱⠏
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⡿⣋⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢛⣥⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⢿⣯⣭⠭⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣶⣶⣶⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠘⢦⣀⡀⠀⠀⠀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢛⣉⡙⢿⣿⡇⠀⠀⠀⠹⢤⣀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠻⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠺⣿⡿⢸⣿⡇⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⣠⠴⠛⠁⠀⢠⡴⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣶⣶⣿⣿⣧⠀⠀⠀⢠⠞⠋⠀⠀⠀⣾⠁⠀⠀⠀⠀⠹⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠟⣛⡛⢿⣿⣿⠀⠀⠀⠈⠳⠶⢦⠀⠀⠘⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⣴⣿⣿⣿⣷⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢿⣿⣿⣿⣿⢁⣿⣧⠀⣠⣤⣶⣤⣄⡀⢀⣀⣤⣄⣀⣠⣤⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣙⣛⣋⣥⠞⣩⣥⣬⡝⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⣿⠿⠿⠿⠇⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠌⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋

⠀⠀⠀⣄⢠⡄⡴⢦⡀⡄⠠⣄⡤⠀⣠⠶⢠⡴⢦⢠⣄⢠⡄⣴⡄⣴⢠⡄⢠⢠⣄⢠⢠⣄⡴⢦⡀⣦⢠⡄
⣀⣀⣀⣏⣙⣋⣧⣴⣃⣧⣄⣹⣀⣀⣳⣤⣸⣦⣾⣸⣹⣏⣇⣿⣻⣹⣎⣧⣼⣸⣏⣿⣸⣟⣧⣼⣃⣏⣻⣃⣀
⣿⣿⠋⠛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠐
⠀⠀⠀⠀⠀⠠⠌⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠹⠈⠀⠀⠀⠂⠀⠀⠀⠀⠰⠁⠀⠀⠂⠀⠀⠀⠀⢀⠃⠀⠡⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠐⠀⠐⢦⠀⠃⠀⠀⠀⠀⠀⡀⠀⠀⡃⢀⡴⠀⠀⠀⢀⠄⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠠⠄⢐⢄⠀⠑⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡀⠠⠄⠀⠀⢂⠔⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠢⢀⡀⠒⠠⠊⢊⠳⠦⠕⣇⠛⠰⠀⠀⠀⠀⠄⠛⢈⠠⠴⠚⡓⠑⠄⠒⢁⡀⠔⢀⡀
⠀⠀⠀⠀⠰⠏⠙⠃⠉⠄⢠⠈⠖⠀⠀⠀⠀⠠⣞⢆⠀⡄⠀⠀⠈⠀⠐⣣⠄⠀⠀⢀⠀⠰⠂⣈⠠⠙⠴⠋⠽⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⡀⠢⠑⠲⠁⠀⢀⠀⠣⡀⠂⠀⠀⠀⠀⠐⢀⠜⠠⠌⠔⠉⠖⢊⠔⣀⠠⣤⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡀⠋⠃⠈⠉⠀⠀⠐⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠃⠢⠀⠀⠙⠃⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠘⠀⠆⠀⠐⠀⠁⠀⠀⠀⠀⠃⠀⠱⠀⠃⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢤⣤⡄⠀⠀⠀⠀⢠⣤⡤⠀⡀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂


⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠉⠀⠀⠀⠀⠀⠈⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢠⣄⠀⢀⣄⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢠⡤⠀⠀⠀⠀⢳⡿⠀⠸⠿⠀⠀⠀⠀⠐⠒⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠑⠦⣀⠀⠀⠀⠀⠀⢀⡠⠚⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠉⠁⠤⠄⠉⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠶⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠶⢤⡀⠀⠀⠀⠀⣠⠴⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠀⠀⠀⠀⠀⠙⣆⠀⠀⡾⠁⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⡇⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⡀⠀⠀⠀⠀⠀⠀⠸⣆⢀⡞⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠹⡾⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⢀⣀⣄⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣰⠋⠀⠀⠹⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢻⠀⠀⠀⠀⣿⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠙⠦⣄⣀⠀⠀⠀⠀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⠀⠀⠀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢘⠀⠒⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠤⠀⠀⡄⠀⠈⠃⠀⠀⠀⠙⠁⠀⠀⠀⠹⠁⠀⠀⠀⣀⡀
⠀⠐⠀⠀⠀⢰⡄⢦⠀⣰⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⣦⡀⢸⠃⠈⢳⣿⡖⠃⠐⡆⢠⣆⢀⡆⠐⡆⢠⣆⠀⠀⢰⡀⢠⡆⢸⠃⠀⣦⡀⢀⡄⢀⡀⢀⣤⠀⡴
⠀⠀⡴⣿⡖⠃⠀⠀⢸⡿⣇⠀⠀⠈⢺⣷⠋⠀⠀⠈⣾⣷⠒⠢⠀⠙⢺⣷⠁⠀⡠⣿⣶⠋⠀⠀⠙⣲⣿⠞⠁
⠀⠀⠓⣿⡇⠀⠀⣰⠏⠀⠘⢦⠀⠀⢸⣿⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⠓⢸⡿⡄⠀⠀⢠⣿⣿⡇⠈
⠀⠀⠀⣿⡇⠀⠀⠁⠀⠀⠀⠈⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⣸⢃⡗⠀⠀⠀⣿⣿
⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣘⣿⡀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⢀
⠀⠀⠀⠸⢿⡿⣿⠶⠄⠀⠀⠀⠀⢰⣾⣿⣦⠶⠍⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⡄
⠀⠀⠀⠀⢸⡗⣧⠀⠀⠀⠀⠀⠀⠸⣿⣟⠿⣄⠀⠅⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⢀⣠⡀⠀⢀⡗
⠀⠀⠀⠀⣸⠃⣼⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡟⠀⠀⠀⢠⡀⠀⠀⢰⡄⠀⠙⣇⡘⠿⣿⣿⡾⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⣿⢷⡀⠸⣿⡷⢀⢰⡌⢻⣷⣿⣿⣶⣦⠈⠁⠀⣠⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢿⣿⣶⣿⡟⣶⣾⣾⣿⣿⣧⣹⣯⣤⡶⠟⠁
⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⣿⣿⡷⣶⣷⣾⡟⣿⢸⣿⣿⣷
⠀⠀⢠⣧⣀⣀⡀⠀⢀⣀⣀⣸⣧⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡾⣷⣿⣿⣿⡇⣿⢸⣿⣿⣿⣇
⠀⠀⢸⣿⠈⠁⠀⠀⠀⠀⠉⣿⣿⡄⠀⠀⠀⠀⠀⠀⣻⡿⢿⣿⣿⣿⣷⣿⣿⣿⣿⠇⠿⠛⢿⡟⣿⠛
⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠘⢿⠟⠃⠀⠀⠀⠀⣠⡾⠋⠀⣾⠇⠁⢿⠿⣿⣿⣿⣿⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡿⣿⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠙⢣⣿⣆⠀⣰⣄
⠀⠀⢀⡴⠊⠁⣿⡈⠙⠦⣄⡠⠿⢯⣄⣙⣟
⢠⣲⣿⠘⣿⣶⡉⠀⠀⣂⣱⣶⣶⣯⣭⡀⠉⠩⠓⢤
⢸⡿⠋⠀⠙⠛⠇⠐⠋⠉⢭⠉⠏⠱⣿⡷⠀⠀⣠⣦⣥⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠃⠀⠀⢈⣽⣶⡺⠿⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣇⢸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢷⣿⡟⣸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣤⠾⢿⣀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣾⣄⠀⠨⣹⠯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠑⢉⡸⣿⡿⠀⠀⣴⣶⠷⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠘⠀⠂⠀⠀⢀⠀⢘⡀⠀⠀⠈⠁⠙⠙⠆⠠⢠⣶⣖⡗⢟⠛⠓
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠆⠀⠀⠈⠁⠀⠁⠀⢠⢞⠁⠀⠂⠐⠐⠐⡏⠉⠻⣿⣇⣸⣄
⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⢀⠋⠀⡌⠂⠉⠃⡀⠀⣿⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠈⣹⠋⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⢀⠈⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⣠⠞⠀⢀⢀⣠⠔⠁⠀⠀⠀⠁⠩⣿⣿
⠀⠀⢄⣄⡢⡀⠀⠀⠤⡔⠉⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠚⠁⠀⠀⠈⠉
⠲⠝⡿⠁⠀⠀⠁⠀⠀⢀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠁⠀⠠⣾⡷⣾⡀
⣶⣷⣶⣷⣶⣿⣾⣶⣷⣾⣿⣿⣿⣷⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⣙⣀⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⢦⡑⠂⠤⠄⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⡀⠀⣔⢢
⠀⠀⠀⠀⢀⣶⠆⠘⣷⠀⠀⠉⠹⣟⠉⠁⠀⠀⠀⠀⠀⠀⠖⢸⣿⡿⠿⠘⠿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢀⣉⡄⠈⠁
⠀⠀⠀⠀⠸⢱⣤⣴⡆⢀⣴⣄⢘⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡻⠿⢋⢸⣿⣿⣠⣄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠱⠀⠀⢈⢈⡁
⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⡄⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⣾⣿⠈⠉⢙⢹⣿⣤⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⣀⣈⡀⢠⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⠿⠿⠿⠿⠇⠉⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣦⢻⡿⠗⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⣼⢃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⣶⡄⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⣀⣼⠟
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⢱⣾⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⢠⣤⣶⠟⠋⠁⣿⣿⣿⠟
⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠈⢿⠀⠀⣠⣾⣿⣿⠏
⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⢸⣇⣼⣿⣿⣿⡏
⠀⠀⠀⠀⠀⣿⣿⠛⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⠟⠋⠁⣸⣿
⠀⠀⠀⠀⠀⢿⣏⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠈⠙⢶⣤⠀⠀⠀⠀⠀⠀⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢴⡟⠿⠿⣷⡄⢀⣤⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠲⡄⠀⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⢀⣷⡤⠜⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⢠⡇⠀⠀⠀⠃⠀⠀⠄
⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣀⠤⠶⠒⠒⢤⣄⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠠⣤⣀⠀⠀⠀⣰⠏⠀⠀⢀⠊⠀⠀⠀⠀⠀⡴⠀⠈⠢⡀
⠀⠀⠀⠀⠀⢠⣀⣀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⠛⠢⠚⠋⠀⠀⠀⠀⠂⠁⠀⠀⠀⣰⣃⣄⣠⠖⠛⢇
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠂⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⠿⠿⣆⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠤⠤⠄⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣥⣤⣤⣽⡧⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⣿⡏⠉⠉⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡹⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣏⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣟⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠒⠶⠶⠾⠿⠛⠻⠿⠿⠿⠿⠷⠶⠶⠶⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠀⠁⠐⠒⠒⠈⡉⠁⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡡⠊⡠⠔⠂⠁⢀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⢢⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠐⢀⠀⠀⠐⠈⡠⠐⢀⠐⠢⡀⠠⡀⠀⠈⠂⠀⠣⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡀⠠⠀⠀⢁⠠⠒⠁⠀⠀⠈⠢⣀⠀⢄⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢤⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⡀⠀⠀⠀⠀⠀⠀⢡⠤⢤⠀⠒⠀⠉⢀⡀⠀⠀⠀⠀⠀⢀⡀⠁⠂⠌⠁⣀⣀⣄⠀⠀⠀⠐⠰⠌⠸⠡⢗⠝
⠀⠀⠀⠀⠀⠀⣘⡑⠆⠏⢀⡠⡄⠀⠀⠀⠀⡇⠊⣺⠀⠀⠀⠀⠈⠁⠀⠀⠒⠀⠀⠀⠁⠀⠀⠀⠀⣜⠗⠉⠀⠀⠀⡰⢄⠀⠀⠀⢈⠝
⠀⠀⠀⠀⠀⠀⠩⡁⠀⠀⠀⢰⠀⠀⠀⠀⠀⢱⠀⡌⢃⠀⠀⢆⡰⠀⠀⠐⠤⠔⠀⠀⠀⠅⠅⠀⣔⠀⠀⠠⢄⠔⠊⠀⠀⠑⡄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢢⠤⠐⠊⠢⡀⠀⠀⠀⡮⠀⢨⠞⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⣀⡙⢆⠀⠈⢆⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠑⠂⠴⠁⢠⠂⢤⠂⡎⠁⠂⢤⠄⣀⠀⠀⡠⠄⡶⠁⠀⢡⠈⠂⠀⠙⡀⠠⢆⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⢠⠦⣠⠎⠀⠈⡀⠈⢲⠄⣠⠣⣈⠒⠒⣁⠜⣄⡠⠐⢩⠀⢸⠀⠀⠈⣖⠡⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⡊⠁⠚⠀⠀⠀⡇⠀⢸⠀⠘⢄⠙⡀⢠⠋⡠⠃⠀⠀⠘⠀⢨⡀⠔⠉⠀⠢⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠐⠓⡖⠀⢰⠀⠀⡄⠀⠀⠀⠑⡜⠣⠊⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠀⠰⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⢀⠃⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠀⠎⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢣⠎⠤⠀⠀⣀⣀⣀⣀⣇⣁⡀⠀⠀⠤⠤⠴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡼⢄⣀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⢀⣀⡠⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠠⠁⡠⠉⠹⠁⠐⠒⡖⠐⡆⠘⠉⢹⠀⢡⠈⡌⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⡰⠁⡐⠁⢠⠃⢀⠇⠀⡇⠀⡇⠀⡆⠈⠀⠀⠆⠈⠄⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢁⠌⢀⡜⠀⢠⠃⠀⡘⠀⢠⠁⠀⠃⠀⡇⠀⡇⠀⠘⡀⠈⢆⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⡠⠁⢠⠊⠀⡰⠁⠀⢠⠁⠀⢸⠀⢰⠀⠀⡇⠀⢠⠀⠀⢃⠀⠈⢣⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢒⠧⢀⡜⠀⠀⢠⠃⠀⠀⡆⠀⢸⠀⠀⡇⠀⠘⡄⠀⠈⡆⠀⣀⠱⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠁⠀⠀⠈⠁⡲⠃⠤⠀⢠⣁⣀⣈⣀⣀⣇⡀⠀⠧⡄⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⡏⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣡⣄⡠⠤⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠉⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠁⠂⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠙⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢀⡀⢀⡱

⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣴⣧⣼⣼⣷⣾⣼⣼⣥⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣶⣿⣾⣿⣿⣶⣶⣶⣾

⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⢄⠄⠠⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠁⠀⠀⠀⠀⠀⠀⠀⣀⡆⡆⣦⡀
⠴⠄⠀⡁⠠⠄⢈⠀⠀⠦⠀⠀⡚⣶⢶⣶⣛⡶⣶⡶⢒⠀⠀⠴⣖⢠⣴⢽⢠⠄⣲⠦⠀⠠⠄⠬⠭⠥⣧⠭⠭⠄⡦
⠀⠀⠀⠞⠆⠰⠓⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⡇⡇⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠩⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖

⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⣠⣼⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⣨⡂⣇⠀⠀⠀⠀⠀⠀⠀⠀⡀⡾⡄⡀⠀⠀⠀⠀⠀⠀⣀⣸⣾⣀⣀
⠀⣤⣴⣿⣿⣿⣿⣿⣤⣄⠀⠀⠐⢒⣚⣚⠃⣓⣓⡒⠂⠀⠀⣀⣀⠤⠿⠇⠿⠧⣄⣀⠀⢀⡤⣤⣿⣽⣿⣽⣧⣴⣄
⠀⠈⠘⣿⣿⣿⣿⡟⠉⠀⠀⠀⠈⠁⠙⢽⡏⡯⠋⠈⠁⠀⠀⠀⠀⢉⡿⡇⡿⡋⠁⠀⠀⠀⠈⠉⣿⣽⣿⣽⣏⠉⠁
⠀⠀⠀⠀⢩⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣽

⠀⠀⠀⠀⢠⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⣲⠂
⠀⠀⠀⢀⣄⡗⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣠⠖⣴⣿⣏⣯⣷⣷⣄⢦⡀⠠⣴⡖⡒⠚⠀⠒⠒⢲⣦⠀⢠⠈⣴⣤⣤⣿⣤⣤⣤⠀⡄⠀⡖⣦⣤⣤⣿⣤⣤⣔⣲
⠈⠉⠀⠿⡦⡧⣾⠗⠀⠋⠀⠀⠀⠉⠉⢹⠐⠉⠉⠉⠀⠀⠀⠀⠈⠐⠀⣿⠀⠂⠉⠀⠀⠀⠉⠀⠀⠀⣿⠀⠀⠀⠉
⠀⠀⠀⠀⠀⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠰⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠽⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⢀⡀⠀⠘⠉⣤⠀⠀⠀⠀⠀⠀⠄⠀⠄⠂⠀⠀⠀⠀⠀⠄⠀⢸⠀⢀⠀⣀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⢠⠶⣦⡴⠦⠴⠶⠯⠳⠲⠖⢳⠃⠠⠀⡔⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠹⠐⠘⠀⠀⠊⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⡤⠠⠎⠀⠦⠠⡆⠘⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⢻⣱⠤⡋⠃⠀⠙⠃⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠖⠀⢈⢠⠀⢠⡀⠀⡘⢃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠐⡪⢸⡦⠀⣀⡖
⠀⠀⠀⠀⠀⠀⣀⠀⠈⣠⠀⠀⣿⠁⣾⠀⠀⢱⡄⠀⢸⠝⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢀⡑⢺⣀⠠⠊⠀⠐
⠀⠀⠀⠘⠀⠄⠲⠄⠀⣏⠀⠐⠛⠛⡛⠃⠀⢀⣇⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡠⠂⢃
⠀⠀⠐⠛⠀⠂⠈⣿⣞⢙⢢⠖⠀⠈⠀⠘⢂⠫⣭⠙⠗⠂⠀⠀⠀⢅⠄⢰⠘⠀⠱⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠘⠄⢘⠁⠷⢾⣂⣬⣥⣼⣓⡀⠀⠄⠀⠀⠀⠀⣐⠀⠄⢨⢀⠄⠀⠀⠀⠀⠀⠀⠃⢤⠅⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣽⠉⡁⢠⣶⠀⢏⡍⠉⢹⣀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢯⠀⢇⠈⣽⠀⠀⡒⡀⠸⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠦⠂⠈⠀⠳⠇⠈⢔⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣏⡇⠀⢘⣶⣦⠀⠀⠀⠀⢀⠀⠀⡀⠀⡇⠘⡇⠀⠀⠀⡟⠈⡁⠀⡀⠀⡄⡀⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⢁⡄⢸⣷⢁⠣⠀⠀⠈⠛⠀⠀⠀⠀⡇⢀⡧⡇⠸⠀⡇⠎⠃⠃⠃⠁⠁⠁⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠏⠀⠫⡟⣛⣿⢠⣤⠀⠀⠀⠀⠀⠀⠀⠁⠊⠀⠀⠀⠰⠓⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠸⠓⠀⠀⠀⢡⡄⠸⠟⠈⠷⠆⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⣀⠲⡄⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⢀⡤⢄⠀⠀⠀⠀⠀⠀⣴⠛⠁⠀⢀⣤⠔⢈⡀
⠀⠀⠀⠀⠠⣾⡺⢶⣥⣀⣈⣽⣓⠿⣷⡀⠀⠀⠀⠀⢾⢀⠀⠅⠀⠀⠀⣠⡾⠧⠒⣾⠁⣈⠤⠞⢉⣠
⠀⠀⠀⠀⠀⣬⣥⣧⠶⠦⢽⡿⢋⣓⠀⢫⠀⠀⠀⢀⡴⠘⠁⢄⠀⠀⢊⡜⠀⠂⠀⠾⠭⠤⠖⢛⣋⡀
⠀⠀⠀⠀⠀⠀⠔⠉⠃⠀⠹⣻⣽⣣⣦⠄⠳⠤⡤⠎⠀⠀⠀⠈⢦⠤⠊⠀⠀⠀⢀⠀⠞⠯⠭⠭⠁
⣀⣶⣶⣦⣤⣤⣤⣤⠞⠛⠉⢉⣈⣴⢷⢶⣶⣦⣤⡀⠀⠀⠀⠀⣤⢀⣀⢠⠤⣉⡍⣙⣛⠶⢦⣬⣥⣤⣤⣤⣴⣶⣶
⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠚⠉⠀⠁⠟⠟⠉⠈⠃⢀⠀⣤⠰⠋⠛⠛⠛⠓⠚⠛⠊⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢸⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⡠⠃⢰⠂⠖⠐⢄⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠰⠄⠀⠂⠀⠀⠀⠈⠀⠀⠀⠈⠳⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠘⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠇

⠀⠀⠀⠀⡀⣀⡀⠀⡀⠀⠀⠀⠀⢀⡀⠀⡀⠀⢀⣀⡀⣀⠀⢀⢀⡀⢀⠀⡀⠤⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢀⣀⠠⡀⠀⢀
⠀⠀⠀⠀⠐⡤⢃⡞⣡⠞⠁⠀⠰⠲⠵⣰⣸⣇⠈⡖⠉⠶⠥⠜⠤⠀⠹⠈⠆⠅⢀⣜⣀⢭⣘⣅⠂⠪⡁⠎⡆⢶⡔⢂⠶⠀⠄
⠀⠀⠀⠈⠀⡔⣍⣴⢟⠉⡁⠒⣛⠌⡂⢁⠀⡆⡠⢢⢈⣀⣒⢄⣒⠀⠃⡄⡆⠐⢁⣠⠠⢸⠇⢸⠼⠂⡓⣧⠃⡃⢷⡎⢡⠐⡀
⠀⠀⠀⢀⣼⢏⠛⣁⣈⠡⡉⠀⢌⢉⠧⠠⠰⢀⠀⣗⢺⢒⡇⡈⠀⡀⡆⠇⡥⡪⢀⣝⢀⣀⣀⣸⠕⠨⡤⢒⡄⠐⠍⢈⠥⢚⠆
⠀⠀⠀⠠⠔⠀⠉⢎⠻⠦⠈⠞⠥⠵⠚⡡⢋⣂⠑⡖⠂⢀⡔⡐⣉⠴⣐⠞⠐⠮⠘⠓⠈⠛⢉⠋⡀⢀⡃⣈⠡⢄⣁⡌⡠⠁
⠀⠀⠀⠐⡄⡇⢩⢐⢂⠴⠠⠐⢃⠜⢎⡡⢩⠂⠁⠐⠆⡘⠐⡉⢄⡦⡢⡚⠊⠁⡦⣯⢜⡹⣈⣋⡸⣸⠁⡅⡆⠈⣓⡇⡊⢑
⠀⠀⠀⠐⠀⠀⡀⠠⠤⠂⣈⣀⣃⣈⢖⠜⠰⢐⠉⣇⡘⠁⢂⠡⠩⠀⡠⢀⠡⠠⠲⢲⢲⠒⠒⢓⠀⢄⠂⠑⡤⡔⣄⠠⠉⠌⠤
⠀⠀⠀⠀⠂⠀⢡⠁⢠⠌⢀⡀⢀⢩⢙⠘⠄⠑⢁⠊⠄⡐⠠⠊⠀⢂⡒⠚⢀⠠⣀⡋⠐⠆⢐⡮⠂⠙⡂⡁⡇⡀⠈⠂⣀⠘
⠀⠀⠀⢠⠂⢰⠘⡀⣋⢙⡎⡫⠁⢘⠲⡆⠠⡀⠀⡠⣂⠤⠐⠠⠤⠤⠤⢸⠩⢈⠄⠠⠈⢁⠔⠊⡂⢀⡁⡅⡡⡆⠸⠢⠀⡘
⠀⠀⠀⠀⠄⡉⠔⡁⠈⠁⠰⠔⠒⣰⠀⡁⠀⡀⡄⠐⠁⢀⠁⡆⢰⠀⡆⠀⢣⠘⢌⡡⢡⠁⠈⢋⠂⠒⢂⠡⡆⠀⠊⡥⠤
⠀⠀⠀⠀⠻⢈⠀⠃⠜⠠⠢⠌⡆⠂⠀⠃⠤⡀⡠⢖⠀⢸⢃⠀⢀⢰⠠⡗⠀⠁⡆⢰⠁⡒⠃⢀⠃⣛⠁⠂⢂⠉⠐⠐⢒⠘⡅
⠀⠀⠀⣐⣠⠦⣀⣤⠀⡀⢪⢈⠐⠇⠘⡀⠃⣳⢴⠄⢆⠸⢀⠈⣠⠌⡰⠀⠀⠀⠃⠀⠰⠂⢁⡠⡙⠐⠐⠀⠸⠀⠀⢠⠁⠖⡀
⠀⠀⠀⢈⠣⠤⠌⣠⠞⡣⢒⣀⢈⢁⡘⠠⠀⠙⠉⠘⠠⠴⣀⠉⣴⡾⢁⡠⠊⠀⠤⠄⢈⣰⢥⡐⣬⠁⠊⠠⠊⠈⢀⠂⠁⡠⡁
⠀⠀⠀⣨⢴⡊⠠⢈⢴⠈⡍⠀⡇⡇⠉⠁⠿⡘⡞⢡⣥⠬⢽⣀⣉⠥⠰⠂⡓⢇⠖⠰⠋⠞⢀⢉⠀⠀⠁⠐⣄⣠⠊⠀⠁
⠀⠀⠀⡸⠣⢌⠃⢜⢌⣀⡃⠶⡱⠁⠀⠈⠂⡀⢥⡠⡄⡊⠅⢈⠎⠠⠲⠂⠃⠘⠠⢪⢆⠉⠘⠄⠀⠡⣀⠘⠀⠀⠈⢈⠠⠠
⠀⠀⠀⢀⡱⣄⣰⠆⠁⠒⠁⠋⠤⢌⠒⠤⠀⣒⠀⠑⠌⠌⠔⠁⠊⠴⠁⡿⢀⠇⠀⠋⠁⠒⠄⢠⡌⠚⣌⠣⣋⢍⡽⠀⡀⢀
⠀⠀⠀⢨⠤⠦⠠⣆⣡⡮⠂⢀⡐⢂⠀⡮⠐⠁⡂⡀⣑⠘⠬⢁⠄⡊⣡⢈⠀⠀⢤⡀⠡⠊⠃⠆⠢⠒⠚⣰⡹⠠⠬⡁⡐⠚⡄
⠀⠀⠀⠐⠠⠲⣲⡦⡒⣶⠊⢉⢢⠡⠼⠃⠄⠂⡀⠀⡄⡤⠠⢁⢪⠜⡡⠊⠀⠀⢸⠠⠀⣀⠰⠀⡤⠀⡅⠂⢈⡀⠈⢀⠥⡋⡀
⠀⠀⠀⠐⠃⣀⠰⢑⠛⠚⠀⠃⡭⡀⠆⠠⠀⣀⣁⣐⣃⠠⠇⠈⡄⢪⣒⠀⠀⠂⡾⠀⠀⠂⡑⢂⠆⠡⠠⡂⠁⠼⠉⠰⢆⠑⠣
⠀⠀⠀⢠⠅⠐⣀⠌⠉⡆⠐⠊⠅⠤⠦⣰⣥⢀⣠⣤⢠⣿⢐⠲⢰⠀⡀⡆⢀⠰⠘⠀⠀⡔⢁⠎⠈⠓⣁⢜⠌⡄⣩⢫⡍⠈⡕
⠀⠀⠀⠙⡂⢠⣛⠅⡄⡗⠀⠂⢛⢤⢠⠤⡀⠈⠉⢩⡄⠠⠠⡇⠘⠅⠈⠰⠆⠀⡠⠠⠀⡅⠏⠀⢸⢬⠀⣭⠀⢤⢥⢩⠄⣤
⠀⠀⠀⠐⣀⠂⠀⠀⡺⠃⡈⡑⢁⠊⢡⡀⠊⠐⠤⠄⡀⢄⣁⠆⠤⡓⢡⡂⣄⢀⠕⡉⢃⣄⡌⠅⢠⣈⠓⣵⠄⠌⠘⡾⠃⡜⠆
⠀⠀⠀⠀⠀⠁⠂⠉⠀⠈⠀⠘⠁⠁⠀⠒⠂⠉⠉⠉⠁⠊⠀⠙⠈⠁⠐⠀⠐⠈⠀⠀⠁⠀⠁⠀⠈⠁⠉⠉⠁⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢸⡇⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠘⠃⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⣴⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⣤⠀⠀⣼⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣠⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⣤⠶⠶⠶⠶⠶⠶⠶⣄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⢷⣤⡀⠀⣿⠀⢸⡇⠀⣿⠠⣤⣤⣤⣤⣤⠀⣿⠀⣿⠀⢀⣤⡾⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣼⡏⠀⠀⣿⠀⢸⡇⠀⣿⢀⣤⣤⣤⣤⣤⠀⣿⠀⣿⠀⠀⢹⣧⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⣿⢀⣀⣀⣀⣀⣀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⣿⣀⣀⣀⣀⣀⣀⣀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⢠⣤⣤⣤⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⡀⢠⣤




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⠉⠉⠉⠉⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⢠⠖⠒⡆⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢠⠤⠤⠾⠀⠀⠷⠤⠤⡄⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠘⠶⠤⢤⠀⠀⣤⠤⠤⠇⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⡘⠀⠀⡇⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⠢⠦⢤⣀⣀⡀⣇⠀⢀⣇⠀⢀⣀⣀⡠⠴⠗⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠒⠤⢄⣀⣀⣀⠀⠉⠉⠉⠉⠉⠁⢀⣀⣀⠤⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠐⢄⠀⠀⡐
⠀⠀⠀⠘⡀⠁⠀⠉⠀⠄⠀⠀⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠈⠃⠀⠁
⠀⠀⠀⠀⠡⡀⢀⠐⠀⢂⠀⠐⠑⠄⠑⠢⡤⠖⠒⠒⢤⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⢄⠂⡀⠀⢃⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠈⠑⠆⠉⠀⡄⠀⠈⠀⢼⠁⠀⠀⠀⠀⢱⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⣀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠖⠈
⠀⠀⠀⠀⠀⠈⠀⠄⠀⡀⠀⠀⠀⢄⠀⠸⡄⠀⠀⠀⠀⡜⠀⢀⠴⠋⠁⠀⠀⠀⠀⠀⠈⠉⠲⢄⡀⠂⠁⡀⠀⠀⢀⠄⠀⠁⠔
⠀⠀⠀⠀⠀⠑⠂⠐⠆⠀⡄⠋⠀⠀⠀⢀⢹⠒⠒⠒⣿⠀⡰⠃⠀⠀⢀⣀⣀⢀⡎⠈⢱⠀⠀⠀⠙⢆⠈⠁⡤⠀⠖⠠⠐
⠀⠀⠀⠀⠀⠄⠀⠠⠤⠀⠀⠂⠀⠰⠂⠁⢸⠀⠀⠀⣿⢰⠃⠀⠀⠀⠸⣀⡀⠉⠀⠀⠶⢤⣀⠀⠀⠈⣆⠀⠀⠀⠤⠀⠀⠄
⠀⢀⠀⠀⠀⠈⠦⠀⠀⠰⠗⠀⠉⢄⢀⠀⢸⠀⠀⠀⣿⡎⠀⠀⠀⠀⠀⠀⢸⠃⠀⢠⠤⢤⡼⠀⠀⠀⢸⡀⠀⠀⠠⠠⠎⠀⠀⠀⠀⠐⠀⠁⡠
⠀⠀⠑⢀⠀⠀⠀⠀⠀⢀⣤⣤⡤⠄⢔⡶⠛⠲⠶⠦⠿⢇⠀⠀⠀⠀⠀⢀⡏⠀⠀⡏⠀⠀⠀⠀⠀⠀⢸⠂⣤⡤⢤⣀⠀⠀⠀⠀⠀⠀⡀⠊
⡀⠀⠂⠁⠀⠀⠀⠀⠰⠁⠀⠀⠉⠉⠚⠂⠀⠤⠤⠤⠤⢼⣇⣀⣀⣀⣀⣸⣀⣀⣼⠀⠠⠤⠤⠤⠄⠒⠋⠉⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠂⠀⠀⠄⡀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢱
⠀⠀⠀⠀⠀⠀⠈⠓⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠃
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠒⠒⠂⠀⢤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⡄⠀⠒⠒⠒⠈⠁
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣷⡘⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣷⣄⣤⣦⡀⣠⡶⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠿⠿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⡟⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣶⣿⣿⠿⠟⣠⡟⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣼⡏⠈⠀⠀⠀⠀⠄⢰⣿⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣿⡇⠀⠀⣿⡟⠀⢀⠂⠀⡀⠀⢸⣿⠀⠀⠀⠠⣿⠇⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢸⣇⠀⠐⣿⡇⠀⠉⣠⠄⠈⠄⠈⣿⠀⠀⠀⠀⣿⡀⠀⠀⠈⣿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠐⡤⢼⡏⠀⠀⣿⡇⠀⢰⣧⠀⠀⢈⠀⣿⡇⠀⠀⠠⠙⠷⠄⠊⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠢⠁⢸⡇⠀⠀⠹⠧⠄⠘⢷⡀⠀⠌⠀⠸⣷⠀⢀⠃⠀⠀⠀⡀⠠⣿⠂⠊
⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⣀⣾⠃⠀⠀⠀⠀⠀⣀⠀⠁⠀⠀⠀⠀⠙⠛⠁⠀⠀⠐⠁⠀⠀⣿
⠀⣆⠀⣰⡿⢋⠐⠀⢠⠤⠠⠀⠀⠀⠀⠠⣄⠀⠀⠉⠀⠀⠀⠀⠀⢀⠀⢸⣷⠀⢠⣤⠀⠀⣠⡀⠀⠀⠀⡠⠒⣶⠀⠇⣈⠁⠀⣠⠤⠤⠀⠀⠀⢀⠀⢹⡆
⠀⠀⣸⣿⠁⠀⠀⠀⠻⡀⠀⠀⢠⣄⠄⢠⣿⡠⣶⡄⠀⣶⡄⠊⣿⠆⠀⣸⡏⠀⢸⣿⠀⢠⣿⡇⢀⣿⡆⠀⣸⣿⠀⠀⣿⠇⠀⣧⣄⠀⣀⣸⣷⠂⢀⣿⠇
⠀⠀⣿⡇⠀⠀⠀⢰⡦⠀⠁⠋⢹⡟⠀⠘⠟⢠⣿⠁⠀⠿⠇⠀⠛⠀⠀⣿⠃⠀⠚⢿⡠⠈⣿⠀⠈⠻⠀⢀⣿⠇⠀⠐⢿⠠⠀⣿⡇⠀⠌⠘⠋⠀⣸⡟⢀
⠀⠀⠹⣇⡀⠀⠄⠉⠛⠀⠀⠀⠀⠁⠀⠀⠀⠈⠓⠐⠁⠀⠀⠀⠀⠀⠀⣿⢀⠀⠀⠀⠀⠀⠻⠈⠀⠀⠀⠘⣿⡠⠂⠀⠀⠀⠀⠹⠤⠊⠀⠀⠀⠀⢿⡁⢀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢻⣟⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡷⡿⡛⡽⣻⢷⣯⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠲⣽⡶⣯⠠⡴⣾⡭⢚⣿⣿⡿⢻⠉⠀⠀⠀⠈⠀⠀⠀⠀⣝⡿⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡭⢔⡀⡆⣀⠶⣰⡾⢯⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡄⠹⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣞⡜⣭⣺⣺⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢹
⣿⣿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠨⠙⠛⠿⣿⣿⣿⣿⠠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢸
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⢹⣿⣿⣿⡄⠀⠀⠛⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣋⡆⠀⢸
⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢌⢿⣿⣿⣷⡀⠀⠀⣏⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⣿⣿⣿⣧⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠜⣸⣿⣿⣿⣷⡀⠀⠈⠉⠁⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣼⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣦⣄⣀⠀⠀⠀⠀⣀⣈⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠿⠿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠛⠉⠠⡦⠈⠛⢿⣿⣿⣿⣿⣿⣿⡏⠀⠀⣰⠞⠛⢦⠀⠀⣿⠉⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡀⠛⡰⢗⡛⠃⣸⣿⣿⣿⣿⠿⠛⣧⡀⠀⠈⠀⠀⣸⣃⣀⡀⠀⠀⠀⠼⠿⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⡈⠃⠘⠃⣠⣿⠿⠿⣿⠁⣠⣤⡈⠛⠶⠶⢶⡞⠉⢉⣉⠙⣦⠀⠀⠀⠀⠀⠻⢿⣿⠟⠋⣼⣿
⣿⣿⠿⣛⠿⣿⡿⠿⠿⠃⢀⣀⠘⢦⣤⣴⠇⠀⠀⠀⢸⠀⠀⢿⣛⣤⢿⡀⠀⢀⣤⠀⣴⠳⡆⣰⣿⣿⣿
⣿⣿⣨⡺⣱⣿⣷⣤⣴⣾⣿⣿⣿⣶⣤⣤⠴⠂⣀⠀⢘⣧⡀⠀⠀⠀⢀⣽⣟⠉⠀⣀⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠚⠉⡉⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠇⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⣀⡀⠀⢀⣆⠀⢀⣀⡀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⢹⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⠀⠀⠸⠀⢀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠿⠿⣿⠛⣿⣿⡶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻⠿⣿⣍⣉⣉⣉⣉⣙⣛⣻⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡜⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡘⣧⣀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣝⣿⣛⣁⣤⠾⠋
⠀⠀⠀⣀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⡟⢻⡇⢀⡀⠀⠀⢀⣀⠀⠀⠀⡔⡀⢆⠀⠀⠀⡠⠤⠤
⢠⡴⠛⠉⠁⠀⣠⠾⠛⠛⠛⠛⠳⣦⡀⠀⣠⠤⡄⢀⢀⡤⢤⣧⣇⣸⣷⠛⠛⣧⣤⣯⡈⣳⣀⣠⣌⣡⠟⢴⣶⡚⠀⠸⣁
⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠉⣙⣷⣤⣼⣉⣙⡧⣸⠋⠉⠉⣿⡖⣒⣿⣁⣼⡟⠉⢙⣧⣨⣿⣶⣿⣯⣉⠛⢦⣄
⠀⠙⠻⢶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠀⠀⠉⠛⣷⢃⡏⠀⠀⠘⢷⣄⠀⢸⠋⠈⠙⣿⣍⣨⣿⡿⠃⢀⠈⠳⡄⣹
⠀⠀⠀⠀⠈⠙⢻⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣻⣿⣀⡀⠀⠀⠈⢿⣿⣏⢻⣖⣏⠉⠉⡟⠉⢻⡗⠿⣧⣄⣹⣾
⠀⠀⢀⣠⠴⢚⡫⠕⠛⠉⠙⠛⠿⣶⣤⣄⡀⣤⣾⣭⣤⣭⣤⣤⣤⣤⣤⡤⠤⠬⠿⢿⣦⠼⠋⣉⡙⠶⠾⣃⣨⣽⠭⠷⡽
⠀⠐⠛⠳⠶⠭⣅⣀⠒⠤⠄⣀⠀⠀⠈⠉⠛⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿⠿⠿⢛⣈⣩⠤⠴⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣄⡑⠢⠤⣀⠀⠀⠀⠀⠀⡀⠄⠀⢀⣉⡥⠤⠖⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⣦⣄⣊⣠⠤⠶⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⠀⠈⠻⣦⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣦⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣷⣦⣤⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡿⢿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⠀⠀⢸⣿⣿⣿⣿⡟⠛⠛⠛⠛⠃⠀⠈⠛⠛⠛⠛⠛⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⣿⣿⣿⣿⣧⣤⣤⣤⣤⡄⠀⢀⣤⣤⣤⣤⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠏⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠹⣿⣿⠿⠛⠁⣰⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⢀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡏⠀⠀⠀⠀⢀⣴⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢙⣉⣽⣿⣷⣶⣶⣶⣾⣿⣿⣿⡇⠀⠸⠿⠿⠻⠟⠛⠛⢻⡗⠀⣸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢆⠀⠀⠀⠀⠀⠀⠀⠃⢰⣿⣿⣿⣿⣿⣷⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢈⡁⠀⠀⠀⢀⣠⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣷⡀⠈⠛⠛⠋⠀⠈⠿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠷⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⠀⠀⠀⠀⠀⠰⠄⠀⠀⢀⣀⡀⠀⠀⠀⠀⣄⣠⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⠀⠀⠀⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣤⡴⢶⣶⣶⣤⡀⠀⠀⠀⠘⡇⠀⠀⠀⠀⣠⣴⠶⠛⠛⠛⠂⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠀⠀⠀⠀⠈⠙⢿⣧⡀⠀⠀⠹⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⢀⡾⠁⠀⠀⢀⣤⣶⣶⣶⣶⣦⣄⡀
⠀⠀⠉⠀⠀⠀⠀⠀⠉⠁⣀⣴⠾⠟⠛⠻⠷⣦⣄⠀⠀⠀⠘⢷⡀⠀⠀⣸⠁⠀⢀⣴⠟⠋⠁⠀⠀⠀⠈⠉⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠋⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠘⣇⠀⠁⠇⠀⢠⠞⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠳⡀⠀⠸⡀⠀⠀⠀⠊⠀⢀⣴⡾⠿⠿⠿⢷⣦⣀⠀⠀⠀⠀⠀⠀⣀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠿⠷⠶⣤⡀⠀⠘⠄⠀⠃⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠈⠙⠳⣄⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠐⠂⠀⠀⠉⠲⠄⠀⠀⠀⣼⣾⣀⠞⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⣰⣶⡀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠴⠚⠉⠙⠛⢿⣦⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠃⠀⠀⠀⡼⠁⠀⠀⠀⠀⣠⣿⠟⠉⠀⠀⠉⠀⠀⠀⠸⠧⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠹⣧⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠐⠀⠀⠀⠸⠁⢠⣤⠀⠀⢰⡿⠁⠀⠀⠀⢠⣶⠶⠆⠀⠀⠀⠀⠈⠁⠈⠙⣷⡀⠀⠀⠀⠀⢹⡆⠀⢠⣿⣿⡄⠀⠀⠆
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⠁⠀⠀⠀⢠⡟⠁⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⠁⠀⠀⠀⣿⠀⠀⠀⢀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡀⡟⠀⠀⠀⠀⡇⠀⠀⠀⠼⡿⠆⠀⠀⠸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⠀⠀⢀⡀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠁⠀⠀⠀⠀⠁⠀⠀⣠⡄⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠟⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡖⠉⠀⠀⠈⠙⣄⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⠉⠉⠉⠉⠉⠓⠲⢤⡀⠀⠀⠀⢀⡞⠉⠀⠈⠙⢦⠀⠀⠀⠀⣠⠎⠁⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⢸⡀⠀⠀⠀⠀⠈⢧⠀⠀⣰⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡞⠁⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠈⢧⡀⠀⠀⠀⠀⠈⢇⣰⠃⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⡞⠀⠀⠀⠀⠀⠀⢠⣻⡆⠀⣜⣦⠀⠀⠀⠀⢀⣀⠈⣇⠈⠳⡀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢈⡇⢸⠁⠘⡏⠀⠀⠀⠀⠈⠛⠃⠀⠙⠋⠀⠀⠀⠀⠀⠈⠀⢹⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⢀⣀⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⢸⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢰⠏
⠀⢰⠋⠀⠈⣧⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠸⡆⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⣼⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⠀⠀⠀⠸⠶⠟⠀⠀⠀⠀⠀⠀⢸⠇⠀⢳⡀⠀⠀⠈⠲⢄⣀⠀⠀⠀⠀⢀⣠⠔⠋⠀⠀⠀⣰⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠉⠩⠭⠉⠁⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⠤⠤⠤⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⣿⣿⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣼⣿⣿⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠞⠛⠋⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠛⠛⠷⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢



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