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⠀⠀⠀⠉⢑⠒⠢⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢀⠀⠘⠇⠀⣸⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⢹⠀⠀⢀⣶⢻⡛⠿⣷⡄⠀⠀⠀⠀⢠⡄
⡀⣺⣟⣿⣷⣦⣄⡟⠀⠀⢀⣶⣿⣷⣾⢿⠂⠀⠀⠀⠿⣿⣦⡀⠀⡿⠀⠀⢸⢉⠮⣬⢉⡩⣞⠀⠀⠀⠀⢸⣷
⠉⠻⢍⡀⣹⠞⠋⠀⠀⠀⠀⠛⠟⢋⠞⠁⠀⠀⠀⠀⠀⡐⠚⠭⠓⠇⠀⠀⠈⢿⢢⠆⡼⡤⠋⠀⠀⢀⠲⣶⠜
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠶⠒⠲⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⢰⣇⠀⢀⡏⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⠀⠀⠀⠀⠠⣤⣤⣤⠄⠀⠀⠀⠀⠐⠁⢹⡀⠀⠀⠀⠀⠀⠀⣠⣴
⠀⠎⢙⠒⡼⠀⣠⠀⠀⠀⢰⣿⡀⢀⣤⡄⠀⠀⠀⠀⠀⠜⣿⠯⠀⠀⠀⠀⠀⠀⠀⠂⣇⠀⠀⠀⠀⠀⠛⠋⡏
⠀⠀⠀⡐⠁⠀⠀⠒⠀⠀⠘⠟⠁⠸⡿⠃⠀⠀⠀⠀⠀⢤⣾⡶⠀⠀⠀⠀⠀⠀⠴⠒⡟⠂⠀⠀⠀⠀⢶⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⢰⣋⣷⠀⣤⣄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⢀⡀⠀⠀⠀⠘⣾⠇⠀⠀⠀⠀⢠⣴⣪⣭
⣠⠟⠛⠚⠷⠞⡇⠀⠀⠀⠂⣿⣄⠀⠀⠠⣵⠀⠀⠀⠀⡼⢠⣤⢴⢩⡇⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠇⡇
⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠉⠑⠀⢹⠇⠀⠀⠀⠀⠁⠀⠀⠀⣸⠀⠀⠀⠾⣮⣤⣤⡾⠀⠀⠀⠀⠀⠷⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢰⠖⠒⠒⠒⠒⠒⡆⠀⠀⠀⠀⠀⢸⣿⣻⣟⣿⡇⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⠦⡄
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⣿⡟⢻⣿⡇⠀⠀⠀⠀⢱⣶⣶⣶⣶⣶⡶⢶⡇
⠀⠀⠀⠀⠸⠦⠤⢤⣤⠤⠴⠇⠀⠀⠀⠀⠀⢸⣇⣿⣿⣽⡇⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠷⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠼⠧⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠷⠶⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢄
⠀⠀⠀⠀⠸⠛⠛⠛⡟⠛⠛⠃⠀⠀⠀⠀⠀⣾⣿⡿⢿⣤⣽⡄⠀⠀⠀⠀⠀⣰⠁⣾⣷⡎⣧⡀
⠀⠀⠀⠀⠰⣤⣤⣴⣧⣤⣤⡆⠀⠀⠀⠀⠀⢿⣿⣷⣾⣿⣿⠁⠀⠀⠀⠀⠘⢿⠀⣿⣿⠇⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇


⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠉⠳⡄⠀⠀⠀⠀⠀⠀⡴⠬⣿⣿⣥⣀⡄
⠀⠀⠀⠀⠀⠀⠘⢿⡿⠇⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠀⣷⡄⠀⠀⠀⠀⠀⠧⠤⠿⠿⠿⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠛⠁


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

⠀⣈⣢⣀⣈⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡄
⢸⠉⠉⠉⢹⣿⠀⠀⠀⠀⠀⠀⣯⣭⣭⣭⣭⣷⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⣿⠀⠀⠀⠀⢸⣿⣿⣧⣿⣿
⠸⣤⣤⣤⣼⣽⠀⠀⠀⠀⠀⠀⢿⣾⣿⣿⣤⠇⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠸⣿⣿⠇⠉⠉


⠀⠀⠀⢀⣄
⠀⢀⣴⠟⠁⢑⣄⠀⠀⠀⠀⢰⠛⢛⠛⠛⣿⣳⠀⠀⠀⠀⢠⣤⣄⣰⡞⠛⢲⠀⠀⠀⠀⠀⠀⠀⡿⠷⡄
⢴⠟⠉⢑⣴⠟⠁⠀⠀⠀⠀⢸⢀⣾⣷⣷⡉⢸⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⢸⠀⠀⠀⠀⠀⣀⣀⡇
⠀⠑⢴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠛⠓⠚⠛⠓⠊⠀⠀⠀⠀⠈⠉⠉⠹⠷⠤⠾⠀⠀⠀⠀⠘⠿⠿⠇


⠀⠀⣴⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⡤⠤⠤⡄⡀⡀
⠀⢤⣿⣿⡇⡄⠀⠀⠀⠀⠀⢠⣤⣶⣿⡇⡾⣳⠀⠀⠀⠀⠀⣤⣶⣿⢷⠋⠀⠀⠀⠀⠀⣧⣤⣤⡇⠁⠁
⠀⠘⠾⣿⠕⠁⠀⠀⠀⠀⠀⠘⠛⠿⢿⡗⢽⡝⠀⠀⠀⠀⠀⣻⠟⢿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇
⠀⠀⠒⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⡀
⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⢙⣿⣿⡋⣿⠀⠀⠀⢠⠤⠤⠤⠤⣴⣿⣿
⣿⣭⣿⣿⣿⣯⣿⡇⠀⠀⠀⠐⠤⠤⡤⠤⠼⠀⠀⠀⠀⠀⣿⢿⣿⣿⡿⣿⠀⠀⠀⢸⠀⠸⠆⠀⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣛⣛⣛⣛⣛⣛⣻⣿⡟⢻⣿⡛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⣿⢡⣬⣙⠻⢿⣿⣿⣿⣿⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⣿⢸⣿⣿⣷⣦⠙⣿⣿⣿⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⣿⠸⡿⠟⣉⣴⣾⣿⣿⣿⣿⠷⠞⣛⣡⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⣿⣶⣶⣿⠟⢩⣭⣭⣭⡅⢰⣶⣿⣿⣿⣿⡇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⣿⣿⡟⣠⡆⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡇⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⣿⣿⣧⡙⠇⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡇⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣧⣘⠛⣛⡛⠃⠘⠿⣿⣿⣿⣿⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⢉⣉⣉⣉⣉⣉⡀⢻⣿⡀⠀⠀⠀⠈⠙⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣿⣿⣿⣿⣿⡌⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⢰⣿⣿⠚⢯⢻⡀⠀⠀⠀⠀⠀⠀⢸⡟⡛⣿⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣷⣲⡇
⠀⠀⠀⠀⠀⠀⠸⢿⡞⠦⠾⠿⠀⠀⠀⠀⠀⠀⠀⠸⠷⠵⠟⠀⠀⠀⠀⠀⠀⠀⠾⠿⠻⡿⠿⠷⠀⠀⠀⠀⠀⠀⠀⢧⣤⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⠃⠀⠀⠀⠀⠀⢸⠀⠈⠈⠀⢸⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⢠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⣲⣶⡒⠚⠀⠀⠀⠀⠀⠠⠤⠤⠤⢤⠤⠤⠄⠀⠀⠀⠀⠀⠺⡿⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⢠⡤⣤
⠀⠀⠀⠀⠀⠀⠀⠮⡀⠀⠙⠂⠀⠀⠀⠀⠀⠀⢠⣤⣤⣶⡤⢤⠀⠀⠀⠀⠀⠀⣴⠶⠶⢶⡶⡆⠀⠀⠀⠀⢸⣤⣿⠀⠀⢸⣧⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣠⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣯⡇⠀⠀⠀⠀⢸⣉⣿⣤⣤⣼⣏⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠋⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠁⠀⠀⠀⠀⠈⠉⠉⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⢠⣤⣤⡤⠤⠤⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠰⠆⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⣟⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠰⠆⠀⠀⠀⠀⠀⠀⠀⠿⠛⠛⠻⡇
⠀⠀⠀⠀⠀⠘⠿⠟⠉⢭⡿⠭⠉⠀⠀⠀⠀⠀⠀⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡒⠢⠀⠀⠀⠀⠀⠀⠀⠛⠀⡀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⡼⣿⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⡦⢐⣀⢈⠀⠀⠀⠀⠀⠀⠀⢠⣶⡶⠶⣶⡄⠀⠀⠀⠀⣤⣤⣤⣤⣤⢤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢧⣹⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠈⣴⣷⣾⣦⠁⠀⠀⠀⠰⣏⣛⣛⣛⣟⣹⣿⣫⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⢸⣶⢺⣶⣷⣮⣶⣶⣷⣮⣶⣬⣺⣵⣷⣵⣶⣶⣭⡾⡗⢉⡉⢺⠉⠉⠉⢳⣦⡶⢶⣷⡶⣶⣷⣞⣶⣶⣶
⢸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⣿⡿⣿⣿⠀⢀⡄⠘⠀⠄⠀⣼⣟⣷⣾⣿⠷⠟⠛⠙⠋⠻⠿
⠯⠤⠄⠄⠥⠿⠿⠿⠧⠄⠤⠄⠀⠤⠤⠼⠿⠿⠿⠿⠧⠀⠀⠴⠄⠁⠴⠽⠿⠿⠿⠿⠟⠻⠟⠿⠿⠷⠿
⠀⣆⣀⡀⠀⠀⣠⢤⡠⠀⣤⠀⢠⡄⢀⠀⠀⡀⢀⣴⠶⣦⠀⠀⢤⠂⠀⠀⣀⢀⠀⢠⡤⠐⠂⠀⠒⣤⣤
⠀⡟⠛⠃⠀⠀⠙⠚⠁⠀⠛⠓⠚⠃⠸⠷⡷⡇⠈⠿⡶⠟⠀⠸⠿⠿⠀⠉⠙⠛⠀⠸⠿⠿⠿⠀⠀⠟⠉
⠀⡟⠛⠃⠀⠀⣤⣤⣠⡀⣤⠤⠤⡄⠀⢰⡆⠀⠀⠀⠂⠀⠀⣶⣆⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⡿⠿⠇⠀⠀⠤⠥⠥⠀⢹⡥⢭⠇⠀⢸⡇⠀⠀⠇⠀⠗⠀⣿⡏⠀⠀⠐⠑⠁⠁⢀⣀⢀⡀⠸⠿⠤⠿
⠀⡷⠶⠆⠀⠀⣀⣀⣀⠀⠀⠀⢀⡀⠀⢠⣤⡀⠀⠀⣤⠀⠀⢠⣤⣄⠀⠀⠀⢀⡀⢀⣀⣀⡀⠀⠀⣤
⠀⣷⣶⡆⠀⠀⢿⣿⡿⠇⠀⠴⠋⠀⠰⣶⣭⠁⠀⠀⠛⠀⠀⠈⠀⠁⠀⢠⢴⠾⡇⠸⠯⠭⠯⠀⠀⠉
⠀⣧⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⡀⠀⢀⣥⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⡀⣀
⠀⣧⣤⡄⠀⠀⣴⣄⣶⠀⠛⡟⠿⠇⢸⣿⣿⡿⠀⢸⣿⠀⠀⠲⠿⠀⠀⠀⠘⠂⠀⠘⢿⡿⠟⠀⠧⣤⠜
⠀⣏⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⢀⡀⠀⣀
⠀⣏⣉⡁⠀⠀⠀⠀⢰⠀⠀⣤⠤⠀⠀⠀⡀⡀⠰⡖⠀⣶⠀⠀⠀⣿⡇⠠⠐⣠⠀⢀⣤⣤⣄⢸⠓⠒⢻
⠀⡟⠛⠃⠀⠀⠑⠛⠛⠀⢀⣁⣀⠀⠀⠀⠁⠁⠀⢀⣀⠀⠀⠀⠁⠈⠀⠀⠛⠃⠀⠈⠉⡉⠉⠈⠂⠀⠘
⠀⡟⠛⠃⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⣤⡀⠀⣤⣤⣤⡄⠀⣾⣿⡆
⠀⡿⠿⠇⠀⠀⠁⠛⠀⠀⠘⠂⠘⠀⠀⠀⠀⠀⠀⠘⠒⠃⠀⠀⠛⠃⠀⠙⠛⠛⠁⠀⠘⠛
⠀⡷⠶⠆⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠸⠛⡆⠀⢨⣤⡤⠀⣻⣿⣿⡇⢀⣀⣀⣀⠀⠀⠀⣲
⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠰⠤⠃⠀⠘⠇⠄⠀⠺⠛⠻⠇⠘⠛⠛⠛⠀⠀⠉⠘
⠀⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣤⣤⡄⠀⢰⣿⠀⠀⣠⣤⠀⠀⣀⣤⡄⠀⠀⢀⠲⠀⠀⠀⠀⠄
⠀⠷⠶⠂⠀⠀⠀⠶⠂⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠸⠿⠀⠀⠉⠛⠀⠀⠉⠙⠃⠀⠀⠿⠇

⢠⠤⠤⠤⠤⠤⠄⠄⠀⠀⠀⣤⣤⣤⡀⠀⠀⠀⡠⠄⠀⠀⠀⢄⠀⠀⠀⠀⠤⠤⢤⠀⠀⠀⠀⣠⠤⠤⠤⠤⡀
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⢸⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣿⣿⣿⡁⠀⠀⠀⠧⠤⠤⠤⠤⠼⠀⠀⠀⠀⠋⠉⢹⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠉⠉⠽⠿⠍⠉⠁⠀⠀⠀⠿⠿⠿⠂⠀⠀⠶⠶⠶⠦⠴⠶⠶⠶⠀⠀⠀⠦⠤⠼⠀⠀⠀⠀⠻⠶⠶⠶⠾⠃
⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠰⠋⠀⠀⠈⢳⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⢰⡟⠉⠙⢷⡄
⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⣤⣤⣤⡄⠀⠀⣴⡶⣶⢶⡶⡶⣶⣦⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠸⣦⣀⣀⣾⠃
⠀⠘⠿⠀⠀⠸⠟⠀⠀⠀⠀⠻⣿⠿⠁⠀⠀⢷⣭⣭⣭⣥⣿⣧⡽⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⢠⣬⣿⣯⣤

⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢰⠒⠛⠛⠛⠛⠒⡆⠀⠀⠀⢿⣿⠗⠀⠀⠀⠀⣠⣴⠂⢲⣶⡂
⢀⠴⢿⣯⣨⣿⠷⣄⠀⠀⠀⢸⠟⠿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⣽⣿⣽⠀⠀⠀⠀⣇⣨⣷⣯⣀⡵
⠀⠺⠂⠿⠿⠕⠿⠊⠀⠀⠀⢸⣦⣗⠀⠀⠀⠸⣷⣶⣶⣶⣶⣾⠇⠀⠀⠀⢉⣟⠉⠀⠀⠀⠀⠻⣀⣸⣄⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢀⣤⣠⠀⠀⠀⠀⠀⠀⠤⠀⠒⢲⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠒⡀⠀⢀⠀⠀⣴⣶⣶⣶⣦⡀
⢸⣯⣿⣿⠛⠛⠻⣿⠀⠀⡄⠀⠀⠸⠀⠀⠀⠖⠒⠲⠒⠒⠖⠒⠲⠀⠀⣿⣦⣶⡇⠀⢸⠇⠀⡿⠿⠿⠿⠿⡇
⢸⣿⣿⣿⣀⢀⣠⣿⠀⠀⢧⣤⣤⣴⣄⠀⠀⡀⠀⢀⠀⠀⡀⠀⢀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇
⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠈⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⢀⡠⠔⠂⠁⠀⠀⠀⢠⡶⠶⣦⠀⠀⠀⠀⠀⠀⢀
⠀⣤⣴⣿⡇⡀⡕⡄⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⢿⣠⣆⣆⣴⣠⠀⠀⠀⢸⣧⣀⣼⠀⠀⠀⢠⣤⣾⣿
⠀⠿⣿⣿⡏⢊⢄⡆⠀⠀⢠⣤⡇⣠⣼⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⢸⡏⠍⢻⠀⠀⠀⠸⠿⣿⢿⠃
⠀⠀⠀⠙⠃⠀⠉⠀⠀⠀⠈⠛⠁⠛⠋⠀⠀⠀⠘⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠒⠛⠀⠀⠀⠀⠀⠑⠛
⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣻⣿⣿⣿⣿

⠀⠀⠀⡠⠀⠠⡀⠀⠀⠀⢀⣤⠤⣄⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠄
⠀⠀⢰⡀⠀⠀⣘⠀⠀⠀⣎⣹⣶⣉⡇⠀⠀⢸⣿⢩⡍⣿⡃⠀⠀⠈⠢⡶⠕⠁⠀⠀⠀⠀⣇⣀⡇
⠀⠀⠘⠟⠀⠘⠏⠀⠀⠀⠈⠻⠤⠛⠁⠀⠀⠘⠛⠓⠚⠛⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠿⠼⠇

⠀⠀⢀⣴⣾⣶⣄⠀⠀⠀⣤⣤⣤⣤⡄⠀⠀⠀⣾⠻⢻⣦⠀⠀⠀⢠⣤⣤⣤⣄⠀⠀⠀⡠⡖⡷⣦⡀
⠀⠀⠸⣿⣄⣼⡿⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⠀⣿⠚⠚⣷⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⢇⣇⣇⣇⠇
⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠘⠛⠛⠛⠋⠀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡏⠀⠈⡇⠀⠀⠀⢸⣿⣿⣶⠀⠀⠀⠀⢠⣶⣦⣀⠀⠀⠀⠀⢰⣿⣇⠀⠀⠀⠀⢰⣿⣿⣧
⠀⠀⠀⣷⣶⣶⡇⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠐⠿⠿⠿⠿⠀⠀⠀⠀⠈⢻⠁⠀⠀⠀⠀⠘⣿⣿⡏
⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢸⣟⣻⣛⣿⠀⠀⠀⠀⠀⠈⡎⠀⠀⠀⢀⣞⠁⠈⣳⠀⠀⠀⢸⣻⣟⣟⡷⠀⠀⠀⠸⢿⣿⣦⠄
⠀⠀⠘⠋⠉⠉⠛⠀⠀⠀⠛⠛⠛⠛⠃⠀⠀⠸⠿⠿⠿⠿⠂⠀⠀⠀⠟⠛⠋⠁⠀⠀⠀⠆⠂⠀⠉

⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⡀⠀⠀⠀⡆⠀⠀⢠⡄
⠀⠀⠘⣯⡀⢀⡀⠀⠀⠀⡃⠀⠀⠀⡃⠀⠀⠸⣿⣟⣻⣿⠆⠀⠀⠸⣿⡇⠊⠐⠀⠀⠀⡟⠛⠛⠛⡇
⠀⠀⠀⠈⠙⠛⠛⠀⠀⠀⠈⠉⠋⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠃⠀⠀⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⣠⣦⡄⢀⣴⣤⠀⣠⣶⡄⢀⣴⣦⠀⠀⠀⢸⡇⠀⠀⠀⣿⣟⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠙⠛⠁⠈⠛⠛⠀⠙⠛⠃⠀⠛⠛⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠙⠲⣍⡻⢶⣄⡀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⢸⡇⠀⠀⠀⣿⢀⡴⣋⣤⡿⠛⠉⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⢸⡇⠀⠀⠀⢿⡿⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣀⡀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠈⠉⣿
⠀⠀⠀⣿⣀⡀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠸⣧⣼⠇⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⣿
⠀⠀⠀⠈⠉⠉⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⣄⣀⣼⠇⣠⡾⢦⣀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠛⠶⣦⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⣄⠀⠈⠉⣠⣾⠋⠀⠀⢙⣧⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⠃⠀⢸⡇⠀⠀⠀⣠⡾⢋⣩⠷⣦⡾⠋⠙⠶⠋⠉⠉⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣤⠶⠶⣦⣼⡇⠀⠀⠀⠀⠀⢸⡇⠀⣠⡾⠛⠒⠋⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡿⣱⢫⢍⢢⢻⡇⠀⠀⠀⠀⠀⢸⡇⠐⣯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢿⡙⢬⡩⢞⣼⠃⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠙⠛⠚⠛⠁⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦
⠀⠀⠀⠀⠀⠀⠀⠛⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠟⠃

⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠤⠤⠤⠄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀
⠀⠀⠀⠀⢠⣆⣧⣠⣠⡄⠀⠀⢀⣀⢐⣒⣚⡀⠀⠀⠀⠻⢿⣄⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⢀⠄⠀⠀⠀⠀⢿⣟⣻⣿⣃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠈⢿⠸⠿⠿⠇⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠁⠀⡠⠒⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⡴⠒⠒⢄⠀⠀⠀⢀⣀⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⣠⣷⡄⠀⠀⠀⢠⣾⣆⣼⣷⡄⠀⠀⠀⣴⣶⣴⠿⣦⡄
⠀⠀⠀⠀⣘⠃⠀⠀⢸⡀⠀⠀⢸⣷⣶⣷⡏⠍⡃⠀⠀⠀⢀⣴⣿⢿⡟⠂⠀⠀⢹⣿⡗⢾⣿⠇⠀⠀⢸⣿⣿⣇⠿⣸⡇
⠀⠀⠀⠀⠙⠁⠀⠀⠘⠃⠀⠀⠈⠛⠛⠛⠛⠛⠁⠀⠀⠀⠉⠉⠃⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁


⠀⠀⠀⠀⠀⢀⣤⠀⢠⠀⠀⠀⢀⣀⣠⣀⣀⣀⡀⠀⠀⠀⡄⠀⠀⠀⠀⡄⠀⠀⠀⣾⡿⢿⣷⡀⠀⠀⢀⣠⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠾⣿⣿⠀⢀⠇⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⣇⣤⣄⣴⣿⡇⠀⠀⠸⣿⣄⣠⣯⠃⠀⠀⠸⣷⣶⣿⢱⡦⡃
⠀⠀⠀⠀⠀⠈⠛⠀⠉⠀⠀⠀⠈⠉⠙⠛⠋⠉⠁⠀⠀⠀⠛⠛⠛⠛⠛⠃⠀⠀⠀⠈⠛⠙⠉⠀⠀⠀⠈⠛⠛⠛⠓⠒⠃

⠀⠀⠀⢀⠤⠤⡀⠀⠀⠀⠀⣀⡤⢤⡀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⡄
⠀⠀⢰⠁⠀⠀⠈⠂⠀⠀⢰⠋⣳⣞⠙⡆⠀⠀⢸⣿⢫⡝⣿⡇⠀⠀⠀⠛⢦⣜⠓⠀⠀⠀⠀⢸⠀⠀⡀
⠀⠀⠘⠿⠀⠀⠿⠃⠀⠀⠀⠻⠇⠸⠟⠀⠀⠀⠸⠿⠮⠵⠿⠃⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠸⠏⠽⠇

⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⣴⢶⢦⣄⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⣠⡤⣤⣀
⠀⠀⢰⣿⡋⢙⣿⡆⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⣿⣤⣼⣯⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⢰⢹⠭⠭⠭⠆
⠀⠀⠀⠛⠿⠿⠋⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠷⠤⠤⠿⠀⠀⠀⠸⠿⠿⠿⠿⠇⠀⠀⠀⠛⠯⠿⠋

⠀⠀⠀⣴⠒⠲⣦⠀⠀⠀⠀⣶⣶⣦⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⢠⣴⣦⡀
⠀⠀⠀⣧⣀⢀⣤⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢀⣼⣿⣿⣶⡄⠀⠀⠀⠐⢿⡿⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠻⠿⠿⠟⠀⠀⠀⠀⠻⠿⠿⠟⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠘⠻⠟⠁

⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⡴⢦⡀⠀⠀⠀⠀⡰⡶⣶⣦⡀⠀⠀⠀⣰⣶⣄⡀
⠀⠀⢸⣯⣼⣧⣿⡇⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⢰⣧⣀⣠⣼⡆⠀⠀⠈⣥⣽⣭⣭⠁⠀⠀⢀⠛⡿⡿⣿⠂
⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠘⠛⠛⠛⠛⠃⠀⠀⠀⠘⠉⠉⠀⠀⠀⠀⠘⠉⠁

⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⢠⠒⠒⠒⠲⡄⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣤⠀⠠⠀⠀⠀⢰⡆⠀⠀⢰⡄
⠀⠀⠈⢿⣄⢀⣀⠀⠀⠀⢸⣀⢀⣀⣀⡇⠀⠀⠸⣿⣟⣻⣿⠇⠀⠀⠸⢿⣿⠀⢅⠁⠀⠀⢸⡏⠉⠉⢹⡇
⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠃⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢉⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠖⡀⠒⢃⢀⢀⠘⠂⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⢂⠀⠀⡄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠙⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⢅⠨⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⣀⠀⠀⡠⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠀⠐⠒⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠄⠀⠈⠀⠤⠤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠈⠀⠀⠉⠍⠢⡀⠀⠀⠀⠀⠀⠀⡔⠒⠖⠒⠒⠒⠒⢲⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠐⡂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠂⠀⠀⠤⠀⠐⠂⠙⠀⠀⠀⠀⠀⠀⣉⣉⠀⠄⠀⠀⠀⢸⢉⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡤⠔⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠤⠀⠀⠀⢤⠄⠀⠤⠀⠇⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠂⢸⠠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⠤⠄⠀⠒⠂⠠⢀⠜⠀⠀⠀⠀⠀⠀⡖⠒⠀⠁⠀⠀⠀⢸⠐⢰⠀⠀⠀⠀⠀⠀⠀⡰⠂⢘⠀⠀⠢⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠐⠦⠤⠶⠐⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠄⠑⡴⢉⠄⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠨⠂⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠉⠉⠉⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⡔⠙⣄⣁⡴⠁⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⠼⠦⠤⠤⠤⡀⠀⠀⠀⠀⠀⠀⢸⠀⠄⠀⠄⠠⠄⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠄⠀⠀⠀⠀⡷⠊⡇⠀⠀⠀⠀⠀⢰⠁⠀⠑⠄⠀⠀⢀⠒⡄⠘⡄⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡧⢀⡁⠀⠀⠀⠀⠀⠈⢆⡈⢊⡠⠔⠢⣀⠘⢀⠔⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⠙⠉⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠛⠉⣻⣤⣤⣄⠀⠀⠀⢸⠉⢩⣭⣭⠉⢹⠀⠀⠀⠀⣠⣾⣧⣤⣾⣦⡀⠀⠀⠀⣀⣀⣶⣆⣀⡀
⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⠀⠀⠀⠸⠤⢼⣿⡯⠤⠼⠀⠀⠀⠀⠙⢹⠋⠉⢻⠉⠁⠀⠀⠀⡇⣴⣶⣶⠀⡇
⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣶⣶⣾⠀⠀⠀⠀⠀⢧⣝⣛⣛⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⢠⠤⠤⢤⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢸⣴⣶⣼⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢀⡠⣤⠤⠤⣄
⠀⠀⠀⠸⣤⣤⡼⠀⠀⠀⠀⠀⠈⠉⡏⢸⢉⣭⣭⡀⠀⠀⠀⢸⣿⣿⣻⣿⣿⠀⠀⠀⢀⣶⣾⣶⣿⣀⣀⣿
⠀⠀⠀⠀⠚⠛⠃⠀⠀⠀⠀⠀⠀⠀⠳⠒⠚⠛⠛⠁⠀⠀⠀⠘⠛⠛⠛⠛⠛⠀⠀⠀⠈⠹⠋⠉⠉⠉⠳⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⣼⣿⡏⢩⢭⠅⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣀⣤⣾⠰⠿⠗⠀⠀⠀⠀⣿⣿⣿⣿⣄
⠀⠀⠀⢀⣛⣛⣃⣘⣋⣀⠀⠀⠀⠀⢸⡂⣶⣶⣾⠀⠀⠀⠘⠛⢻⠿⢿⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⣿⠖⠁
⠀⠀⠀⠈⠉⠉⠿⠿⠉⠉⠀⠀⠀⠀⠸⠶⠴⠦⠾⠀⠀⠀⠀⠀⠘⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⣶⣶⡖⣵⣿⣮⣂⠀⠀⠀⠚⡛⠛⠛⠛⠛⠓⠀⠀⠀⠀⢻⣿⡗⠀⠀⠀⠀⠀⢸⣿⣿⠻⣿⣿⡇
⠀⠀⠀⠀⢹⡿⠓⠽⣛⠫⡆⠀⠀⠀⠀⠜⠛⠰⠶⠶⠀⠀⠀⠀⠀⢸⢿⠃⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢰⣶⡟⠚⠓⠛⣧⣶⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠸⠛⠁⠈⡏⠀⠹⠿⠀⠀⠀⢀⠄⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡿⣿⡇⠀⠀⢰⢤⣀⠀⡁⢀⢠⣶⠀⠀⢠⡏⠀⠀⠀⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠁⠀⠀⠀⠴⠯⠶⠱⠮⠷⠀⠀⠀⢀⣀⣀⡀⢸⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣷⣀⠀⠀⣀⣄⣰⡞⣦⠀⠀⠂⠀⠀⠴⠾⠿⠿⠷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠺⡋⣉⡿⢦⡀⠀⠀⠀⠀⠀⢠⣶⣴⠉⠁⠀⠻⣼⣛⣉⣡⣀⣠⡄⠀⠀⡟⠘⡇⢸⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣚⣽⣾⣝⣠⣷⠀⠀⡄⠀⠀⠀⠙⠷⠀⠀⣀⠀⠘⠛⠛⠛⠉⠙⠃⠀⠀⠃⠀⠃⠘⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣘⠞⠝⣞⣠⠏⠀⠀⠀⣠⣄⣠⣄⠀⠀⠀⠀⠀⠀⠈⠈⠈⠀⠀⠀⠀⠈⠀⠀⠁⠈⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠛⠁⠀⠀⠀⠀⠀⠋⠙⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢰⣶⣶⠶⢶⣶⡆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⡆⠀⠀⠀⢸⢸⣿⡁⠶⠀⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⢤⡤⢤⣤⣄⡀⠀⠈⠃⠀⠀⠀⠀⠎⠉⠋⠉⠀⠀⠀⠀⠸⠸⠿⠿⠶⠿⠿⠇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢨⣿⡟⢻⠂⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠇⠿⠿⠀⠀⣠⣄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡀⠤⣖⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡠⣪⣾⣿⣦⠀⠀⠀⢀⣀⢠⡄⠄⠀⠀⣤⣿⣷⢈⣿⡦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⠾⠟⠋⠁⠀⠀⢈⣩⢉⣡⢀⣄⡀⠀⠀⠸⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⣾⢶⣶⢶⣾⢶⡇⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠾⠿⠿⠇




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⢀⠀⠀⡀⠞⠁⠀⠀⠀⠀⠀⠀⣤⡶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠈⣩⠌⠋⠈⠉⠁⠀⠀⠀⠀⠸⣟⣀⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣠⣴⣶⣶⣶⣶⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢟⣛⣛⠿⠿⠟⢣⠀⠀⠀⠰⠰⠀⠆⠶⠰⠀⠆⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠿⢿⣷⡀⠀⠀⠀⠈⢱⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢸⣿⡆⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣈⣥⣿⠇⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠈⢛⣛⢛⡋⠁⠀⠀⠀⠀⢨⢩⠁⡍⣭⢩⠉⡅⣭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡸⣿⣿⢇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⣿⢿⣿⠶⣶⢿⡿⣿⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⣿⣿⣿⣿⣿⡱⢈⣿⣿⣿⣿⣿⣿⡇⡀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡀⠀⠀⣿⣿⣿⡇⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢃⣙⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃⢛⡷⠀⠀⣿⣿⣿⡇⠴⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣬⣤⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣬⣤⣤⣤⣭⣭⣭⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠦⠶⢔⡋⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣛⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣥⡂⠀⠀⠀⢱⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠇⠀⠀⠀⠀⠀⢠⣾⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⠑⠮⣶⣄⠀⡎⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠹⠟⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⣀⡀⠀⠤⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⠍⠀⣀⠀⠀⠀⠀⢱⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠺⡗⠒⢻⡟⢛⣷⡾⠿⠶⠶⠶⣶⣺⣿⣿⡀⠀⠀⣽⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡖⠀⡇⠀⢸⣇⡿⠋⠀⠄⡓⠀⣱⣿⣿⣿⣿⣿⣶⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣤⣧⣤⣼⣿⣆⣀⡄⡊⠯⡐⠌⠙⣿⣿⣿⣿⣿⣿⣿⣿⠋⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⣰⠏⣀⣞⢡⠏⢰⠁⣼⠁⡞⢡⡿⠻⣷⡌⠀⢀⡈⠙⠋⠀⠀⣀⡀⠀⠀⠘⢋⡟⠛⢛⡟⠻⡆⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠉⠉⠑⠛⠒⠓⠴⡇⣼⠀⠀⠘⡷⠌⣂⠈⠛⠀⠀⠀⠛⠁⠀⠀⡀⡸⠀⠀⣸⠁⠀⡷⢠⡆⠀⣦⠤⣀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⠀⣿⠛⡟⣹⡧⣠⠄⠠⡄⠐⡒⢾⣄⣇⠀⠀⡟⠀⠀⡇⠈⠉⠉⠉⣀⡠⡇⢸
⠀⠀⠀⠀⠐⣄⠠⣿⠀⣾⠀⠀⠀⣠⡎⠀⠀⠀⢸⠀⠀⠀⢸⣘⣦⣀⣴⠄⣋⡀⠾⣆⠋⢠⠆⠹⠟⠶⡇⠀⠀⡏⠈⠉⠉⠁⠀⢀⠇⢸
⠀⠀⠀⠀⠀⠈⠓⣿⢀⣿⠀⢀⡔⣹⠀⠀⠀⠀⢸⡆⠀⠀⢸⠉⠀⠀⠀⠉⠉⠉⠁⣟⠀⢨⡂⠀⠀⠀⡴⠀⠀⣷⠤⡤⢤⠖⡚⣻⠀⢺⠂⠀⡀
⠀⠀⠀⠀⠀⠉⣷⡟⣷⣿⠀⣾⢷⣏⠄⠀⠀⠀⢸⡇⠀⠀⢸⣆⣀⣤⡤⠤⠤⠶⠶⡟⠃⣀⠓⠛⠛⠻⡆⠀⢸⡇⠚⠐⠒⠘⠉⡯⠀⢀⣀⠼⠁
⠀⠀⠀⠀⠀⣀⣸⡗⣿⠿⠐⢻⡿⠏⠀⠀⠀⠀⢸⡇⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⢱⡀⠉⠀⠀⠀⢠⣇⣀⣸⡧⠄⠀⠠⠄⣺
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣦⣮⣯⣴⣷⣦⣄⡀⠀⢸⡇⠀⠀⢸⣇⠀⣀⣀⣀⣤⡤⠤⢾⡇⣾⠒⠚⠛⠋⠉⡿⢹⠁⠒⠒⠀⠉
⠀⠀⠀⠀⢸⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀⠉⠳⡾⣷⠖⠚⠛⠉⠉⠉⠀⠀⠀⠀⠀⢘⠦⢽⣠⣤⣤⢲⡶⡇⣼⡀
⠀⠀⠀⠀⠐⣎⢙⠛⠛⢻⠛⠋⢩⣠⠏⠉⠓⢶⡇⣿⠀⢀⣀⣀⣀⠤⣤⣶⣒⣿⡟⣾⠀⢘⣶⢿⡺⣿⣛⣷⣿⣽⣦⣄
⠀⠀⠀⠀⠀⠈⠛⠶⠶⠾⠶⠾⠛⠁⠀⠤⡤⣶⣧⣿⣿⣛⣥⣿⣍⡾⡿⡲⣟⢟⣫⣹⠀⢸⡿⢿⠻⡟⣏⣻⡹⢬⣆⠳⣿⣂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣟⣏⣯⣷⣶⣷⠿⡟⢿⠻⣏⣿⠹⣇⠸⣟⣸⣧⣽⣷⣬⢿⡶⣟⣟⣿⢯⣻⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣭⠧⣽⡞⢷⣏⣘⣷⣤⣿⡶⢿⢟⢻⣻⡻⣝⣎⣮⣷⣷⢿⣾⠿⣿⣻⣯⣽⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠺⣟⣿⢻⡙⣟⣦⣳⣽⣼⢾⣶⠿⣿⡻⣯⢻⣦⣹⣷⣿⣿⡿⠿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⢿⣝⡻⣮⣿⣶⣿⣷⣿⠿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠟⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠛⠛⠛⠛⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⣤⡎⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠹⠁⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄
⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⡤⠤⠤⠤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠇
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡟⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠒
⠀⠀⠀⠀⠀⢀⣀⣙⣛⣛⣛⣛⣛⣛⣛⣛⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢸⡿⠉⢹⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⡏⠉⢿⡇
⠀⠀⠀⠀⠀⣸⡇⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⢸⣇
⠀⠀⠀⠀⠀⣿⠃⠀⣿⠁⠀⠀⠀⠀⠀⠀⢠⣿⠀⣸⡇⠀⢸⡇⠀⠸⣿⠀⠀⠀⠘⣿⠀⠘⣿
⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣿⠃⠀⠸⣷⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⢰⡟⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⣿⠀⠀⢿⡄⠀⠀⠀⢹⡆⠀⢻⡆
⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⠁⢠⡏⠀⠀⠀⢻⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⣸⠁⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⡇⠀⠀⠀⢸⡇⠀⠘⡇⠀⠀⠀⠀⣇⠀⠈⣇
⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⡄⢀⣤⡾⠛⠙⢿⣦⡀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⣿⡏⠉⢹⣷⡿⠋⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦
⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡆⠀⠀⠀⠈⠻⢷⣄⡀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠆⠀⠀⣤⡾⠋⠁⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢉⣿⣅⢀⠀⠀⠀⠀⠀⠙⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⢿⣿⠀⠀⠀⢀⣘⣁⣀⣠⣴⣿⠿⠿⣿⣿⢸⠀⠀⠀⣿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠙⣿⣿⣿⣿⣿⠋⢀⣤⣿⣿⠿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣰⣿⠟⠋⣷⠒⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇⠀⢠⣿⡄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢏⣾⣿⣿⣎⢿⣷⢀⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣸⣿⣿⣿⣿⣸⡇⣿⣿⢟⡹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠇⠿⠿⠿⠿⠿⠇⢇⣭⣶⣿⣿⣮⣭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣇⣿⣿⣿⣿⣿⣇⣧⡹⣿⣿⣿⣿⢏⣼⣿⣯⣭⡍⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⢸⣿⣿⣿⣿⢸⣿⠷⣙⣓⣚⣋⠾⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣙⣛⣃⣛⣛⣛⣃⣉⣚⣛⣙⣛⣛⣋⢺⣷⣝⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡞⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⣼⢹⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⢾⠷⣸⣿⣿⣿⣿⣿⡇⡇⣿⢸⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣧⣿⢸⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⣸⢘⢻⢸⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣈⣀⣈⣉⣁⣉⣉⣁⣀




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠋⠉⠉⠻⣗⠀⠀⠀⠀⣠⣾⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⢠⡠⢤⠀⣿⣠⣂⣀⣀⣿⠀⠀⠀⠀⠻⠛⠛⠻⢿⣿⡇
⠀⠀⠀⠀⠀⢸⠈⠉⡀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠡⠀⠠⠈⠀⠂⠀⠀⠀⠀⠀⠠⠀⠀⣠⠀⠄⠀⡆⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠈⠀⠙⠆⣹⠆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣻⡔⠊⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣷⣶⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠤⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠚⠛⢻⠄⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠒⠀⠀⠀⠤⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⣶⣶⠆⠀⢀⠀⠀⢰⠀⠀⠠⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠀⠀⣀⣀⡀⣠⠂⢀⣀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠈⠁

⠀⠀⠀⠀⠀⡠⠊⠢⡀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢄⠀⠀⠀⠀⢀⣴⣷⣄
⠀⠀⠀⠠⠊⠀⢌⠆⠈⠢⠀⠀⠐⠀⢰⡆⠀⠀⡀⠀⠀⠠⠊⠀⢈⡀⠀⢐⡀⢀⣴⣿⣯⢩⣿⣷⣄
⠀⠀⠀⠀⠀⣸⣀⣺⡀⠀⠀⠀⠀⢰⣏⣹⡆⠀⠀⠀⠀⠀⠁⢰⣂⣸⡀⠈⠀⠀⢸⣿⢱⣶⢹⣿
⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠐⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋
⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⣠⣄
⠀⠀⣸⣿⣛⣻⠀⠀⣀⡀⠀⠀⠀⠀⣿⣿⣿⣿⠉⣽⡉⠑⠀⠀⠀⠀⠀⢻⣿⠇⠀⠀⠀⢿⡿
⠀⠀⠃⠿⠤⠾⠈⣾⠀⣾⠀⠀⠀⠀⠘⢿⣿⣋⣀⣿⣄⣀⠀⠀⠀⠀⣠⣾⣿⣷⡄⢀⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠼⠿⠗⠿⠶⠀⠀⠠⠾⠿⠿⠿⠏⠉⠉⠉⠁⠀⠸⠇⣿⣿⣿⣿⡇⠸⣿⣿⣿⣿⠀⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⡤⡄⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⠀⠐⠄⡀⠀⡀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⡼⠉⠀⠀⠈⠉⡸⠀⠀⠀⠀⠀⠐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠑⣀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠑⠦⠄⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⡄⠂⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⡀⢀⡇⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠘⠄⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠊⠀⠀⠀⠈⠐⢒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⢀⠔⡲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣈⣆⠀⢦⠤⠤⠀⢀⣀⢀⠐⠒⠀⠒⠒⠒⠒⡆⡠⠊⠀⠀⠀⠀⢀⡔⢡⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⢘⠀⠈⡄⠀⡄⠀⡾⣿⠋⠀⠈⡶⠠⠤⠤⠤⠔⠋⠀⠀⠀⠀⢀⡠⠊⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠰⠀⠸⡄⠉⠁⠀⠀⠐⠦⠄⢀⣀⣀⣀⣀⣀⡠⠔⠊⠁⠀⠀⠀⠀⠈⠲⢄⣀⠀⠀⠀⠀⠀⣀⡠⠤⠚⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⢠⠛⠉⠉⣩⠛⠁⠒⠒⢦⠤⠀⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⢉⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠓⠓⠒⠒⠒⠛⠓⠛⠛⠿⠿⠷⠶⠶⠆⢤⢤⠤⠄⠀⠀⠀⠤⠤⠄⠒⠉⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡎⠀⠀⠶⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠳⣀⢀⡰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⣀⣀⢀⡀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⣼⡀⠀⠀⢀⣀⣀⣀⣀⠀⠤⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁




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