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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⢶⣤⣀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠛⠛⠛⠷⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠈⠙⢷⣄⣤⡶⠟⠋⠉⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣤⣴⠶⠶⠶⠾⠻⣿⠃⠀⠀⠀⠀⠀⠀⠈⣿⡁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣴⠾⢷⣶⠿⠛⠛⠶⣤⣀⣤⣤⣶⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⣤⣿⠁⠀⠀⠀⠀⠘⣿⠁⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⣀⣀⣀⣽⣇⠀⠀⠀⠀⠀⢀⣿⢦⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠻⣦⣀⠀⢀⣠⣾⣃⣸⡇⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⢿⣌⠉⠉⠀⠀⢀⣰⣿
⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠶⠶⠛⠉⣿⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⢾⡿⠶⠶⠶⠦
⠀⠀⠶⠾⠿⠛⢻⡟⠛⠛⠁⠀⠀⠀⠀⣴⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠀⠀⠀⠀⠀⠀⢤⣶⣿⣤⣤⣀⡀
⠀⠀⠀⠀⠀⣀⣤⣿⡶⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠛⠳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠈⠁
⠀⠀⠀⠀⠈⠋⠁⠈⢿⡄⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣦⣼⣏
⠀⠀⠀⠀⠀⠀⠀⠀⣨⣷⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠛⠿⢶⣄
⠀⠀⠀⠀⠀⠀⣶⠟⠋⠁⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠛⠛⠋⠉

⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠻⣦⣠⣴⣶⠿⠟⠻⣷⡀
⠀⠀⠀⠀⠀⣾⠏⠀⠀⠈⠉⠛⠿⣶⡾⠿⠿⠿⠿⠿⣿⠋⠀⠀⢀⡀⠈⣿⠉⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣼⠟⣻⡿⠛⠛⠷⣤⣤⡴⠾⢿⣤⡀
⠀⠀⠀⠀⠸⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⢷⣿⠀⠀⠀⠀⢹⣧⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⣤⡴⠶⠿⣦⣀⣀⣠⣾⣥⡿⠀⠀⣼⠇
⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢻⣄⠉⠀⢀⣼⡏
⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⢹⣷
⠀⠀⢀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⡶⠦⡦
⠛⠛⠛⢻⣿⠛⠋⠀⠀⠀⢰⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⠀⠀⠀⠀⠀⢀⣿
⠀⠀⢀⣀⣿⣤⡤⠀⠀⠀⠘⠻⠃⠀⠀⠀⠀⠀⣠⡤⢤⡀⠀⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠘⣿⡿⠶⠶⠆
⠀⠀⠋⠉⠙⢿⣄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⠁
⠀⠀⠀⢀⣠⡾⠿⢿⣥⣴⠶⢶⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠟⠙⠻⠶⠄
⠀⠀⠀⠛⠉⠀⢀⣤⡿⠇⠀⠀⠛⠉⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⢾⡁⠀⠀⣔⢐⡄⠀⣴⡿⣿⠛⠛⠛⠛⠛⠛⢿⡟⠛⠻⣿⠻⣶⣄⡀
⠀⠀⠀⠀⠀⠀⣨⣿⣶⠀⠈⢉⠀⠀⢸⡇⠻⣦⡀⠀⠀⠀⣠⡿⠁⠀⠀⠹⣧⣠⡿⠿⠶⣦⡄
⠀⠀⠀⠀⠀⢠⣿⠁⠻⣦⣤⣾⡷⢶⡟⠁⠀⠈⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⢻⣯⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⢹⣷⡌⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣤⡿⠃⠻⣦⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣃⣀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠛⠛⠛⠛⠛⠿⠿⠟⢿⡿⠿⠿⠿⠿⠟⠛⠛⠛⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣤⣀⣀⣀⣀⣀⣼⣧⣄⣀⣀⣀⣀⣠⣤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⢶⣤⣀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⠛⠷⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠈⠙⢷⣄⣠⣴⠾⠛⠋⠉⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣤⣴⠶⠶⠶⠶⣿⠃⠀⠀⠀⠀⠀⠀⠀⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣠⡶⢶⣦⡾⠿⠶⠶⣤⡀⣀⣤⣤⣤⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⣄⣾⠋⠀⠀⠀⠀⠈⢿⡏⠁⠀⠀⠈⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⠀⠀⠀⠀⣙⣿⠀⠀⠀⠀⠀⠀⢸⣧⣄⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⢻⣆⡀⠀⠀⠀⣠⡿⠁⣿⡆⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⢿⣏⠛⠛⠋⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣤⣤⣤⡾⠛⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣿⣦⣤⣤⣄⡀
⠀⠀⣀⣤⣤⣤⣾⣷⣶⡶⠶⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿⠁⠉⠉⠉⠁
⠀⠀⠈⠉⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⣻⣆⣠⣤⠀⠀⠀⠀⠀⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠈⠻⠟⠁⠀⠀⠀⠀⠀⠈⠙⣿⠟⠛⠛⠶⠆
⠀⠀⠀⠀⠀⣴⠾⠛⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⢀⣈⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣤⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⢿⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠈⠙⠻⠶
⠀⠀⠀⠀⠀⠀⠻⠛⠁⠀⠀⠙⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣴⡶⠾⠛⠉⠁

⠀⠀⠀⠀⠁⠀⠀⢀⠀⠛⠇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡀⠀⠀⠀⡘⠛⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠆
⠀⠀⣰⠶⠂⠀⠀⠀⣦⣦⠆⠀⠀⣈⣀⡀⢀⡀⠀⠘⣿⠇⠆⠀⠀⠀⢻⢻⠂⠀⠀⠙⠈⠙⠁⠀⠀⠐⠀⠉⠀⠀⠀⠀⠐⠿⠆⠀⠀⠠⢀⣤⣄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠉⠂⠀⠀⠀⠀⠈⠀⠀⠀⠠⡀⠈⠳⠀⠀⠀⠀⠈⠁⠀⠀⢀⣡⡄⠰⡆⠀⡀⠀⠉⠀⠀⠀⠀⠀⣀⡉⠀⠀⠀⢀⣈⡃⠀⠀⠀⠀⠸⢇⡾
⠀⠠⠤⠀⠄⠀⠀⠀⠠⡤⠀⠀⠀⠀⠠⢨⠆⠀⠀⠀⠀⠀⠀⠀⠀⣭⢯⡇⠶⠀⠀⠀⠲⠂⠀⠀⢀⣁⠀⠉⠁⠀⠀⠀⠀⠉⠀⠀⠀⠡⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠈⠠⠀⠀⢴⠂⢰⣴⠀⠀⠀⠉⠁⠀⠀⠀⠀⠁⣀⡀⠀⠀⠀⠀⠘⠃⠀⠀⡀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠋⢆
⠀⠈⠀⠀⠀⠀⠀⠈⠂⠚⠁⠀⠀⡰⠒⠄⠀⠀⠄⠀⠖⠸⠇⠀⠀⠀⠹⠟⠀⠀⠀⠀⠶⠂⠀⠀⠀⢰⣶⠈⠀⠀⠀⠀⠀⠐⠀⠀⠀⠑⠀⠀⠀⠐

⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣄⠈⠁⠀⠀⠀⠀⠉⠃⠀⠀⠀⠈⠙⡀⢠⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⡀⠠⢀⡄⢀⣙⣧⠀⠀⢤⣈⡅⠀⠀⠀⠀⠈⠁⠀⡀
⠀⠀⠸⠿⠆⠀⠀⠀⠈⠉⠀⠀⠀⠀⢠⠖⠠⠂⠀⠀⢴⠰⠆⠀⠠⠀⠠⠐⠂⠀⠈⠳⢻⠻⠿⠀⠀⠀⠏⠀⠀⠀⠀⠀⠁⠞⠀⠀⠀⠂⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⡀⠀⠈⢉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠶⠀⠠⠄⠀⠀⠓⠂⠀⠀⠀⠀⠈⠗⠁⠀⠀⠀⠁⠈⠀⠀⠀⠀⠺⠻⠋⠀⠀⠀⠀⠈⠀⠈

⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⣀⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⣼⠀⢳⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⡀⠀⠀⠁⠑⠄
⠀⠀⠘⠛⠃⠀⠀⠀⠼⠷⠀⠀⠀⠀⠙⠟⠃⠀⠀⠀⠀⠒⠀⠀⠀⠀⠛⠟⠁⠀⠀⠀⠁⠈⠀⠀⠀⠰⠾⠄⠀⠀⠀⠐⠾⠈⠉⠀⠀⠈⠀⠀⠀⠈

⠀⠀⢀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠐⠁⠠⣤⠑⣀
⠀⠀⠀⢉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣑⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠃⠀⠀⣤⣅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠠⣄⠀⣤⡄⠀⡤
⠀⠠⠙⠇⠠⠶⠀⠀⠰⠆⠀⠀⠀⠀⠘⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠸⠏⠿⠆⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⠻⠁⠀⠠⠬⠤⠭⠥⠤⠄
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⣿⣿⠿⠻⡿⠛⠿⢿⠿⢻⡿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⡵⠤⠤⢤⡤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠔⡾⠁⠀⡇⠀⠀⠀⠈⠳⣄⠀⠉⠒⢤⡀
⠀⠀⠀⠀⠀⢀⠜⠁⡘⠀⣀⣀⣰⣤⠒⠂⠐⢦⡺⡲⠉⠉⠒⠚⣦⡀
⠀⠀⠀⠀⢠⠋⢀⣠⡷⢩⠃⠀⢰⢻⠀⠀⠀⠀⠙⢄⡴⠛⠲⣤⢞⣁⡀
⠀⠀⠀⠀⣧⠞⠁⠈⢁⠇⠀⠀⠀⡇⡇⠀⠀⢀⠖⠉⢳⡤⠤⣾⣅⠀⠹⡀
⠀⠀⠀⢰⣇⣀⠀⠀⢸⠀⠀⠀⠀⢰⢱⠀⣀⣼⡆⠸⢿⡀⠀⣸⠞⠁⢠⠛⠲⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠉⠲⣶⠔⠒⠊⠉⠙⡨⡟⣶⣅⠈⠒⠚⠉⠉⠀⠙⠒⠋⠀⠀⠀⠉⠉⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢷⡇⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢻⡗⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡏⣻⣆⠀⠀⠀⠀⢾⡲⠀⠀⠀⣤⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠷⠋⠙⡞⠿⢦⣄⡀⠀⠀⠀⠀⠀⠉⠀⢀⣈⡿⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢸⠁⠀⠀⢹⡍⠓⣶⠦⣤⣤⣤⠶⣾⡛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣳⠿⠤⠤⠴⠚⠀⠀⠈⠳⣄⠉⠉⠛⠛⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣦⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠉⠉⠛⠓⢶⠖⠶⢲⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢀⣾⣀⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⣶⣄⣀⣤⣾⡿⠟⠛⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡾⠋⠀⠀⠀⠙⢿⣟⠉⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣤⣤⣤⣴⣶⣾⣿⠿⠿⣿⠃⠀⠀⣠⣴⣦⣴⣿⠶⣶⣤⣤⣶⠾⠿⠿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠉⠙⠛⠛⠉⠁⠀⠀⠀⠠⣿⠀⠀⠰⣿⣀⣿⠃⠀⠀⠀⠙⣿⣤⣀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠙⠛⣿⡀⠀⠀⠀⢀⣿⣉⣿⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣶⣶⡶⠿⠛⠻⠶⣶⠶⣿⡛⠛⠁⠀⣠⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣶⠾⠋⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣿⣷⣶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠛⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠀⣤⣾⣿⣶⣶⠦
⠀⠀⠀⠀⠀⠀⠀⣀⣿⣧⣤⣄⠀⠀⠀⠀⣴⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠁⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠰⠾⠟⠛⠻⣿⡉⠉⠀⠀⠀⠀⢿⣿⠟⠀⠀⠀⠀⠀⠀⣶⠟⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣷⣶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡟⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠙⢿⣦⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠋⠙⠿⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⢿⣷⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣾⠿⠻⣿⣇⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠉⠀⠀⢈⣽⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⡿⠿⠿⠿⣿⠛⠙⢿⣦⠀⠀⠘⢿⣧⡀⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⡋⢁⣼⡟⠁⠀⢻⣧⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠈⢻⣧⡀⠀⠈⢿⣷⣤⣾⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⠈⠻⣷⣾⣿⣶⣶⣦⣄⡙⠿⣶⣤⣤⣶⡿⠛⢀⣠⣴⣶⣶⣿⣷⣤⣶⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⣴⡿⠋⠉⠀⠀⠉⠻⣿⣆⠀⠀⠀⠀⢀⣴⡿⠋⠉⠀⠀⠈⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠇⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⡀⠀⠀⠀⣀⣴⣿⣿⣾⣿⣿⣿⣷⣿⣿⣦⣄⡀⠀⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠙⠻⣦⣀⣠⣴⣶⡿⠿⠿⣿⣦
⠀⠀⠀⠀⠀⠀⣴⡿⠋⠉⠉⠙⠛⠿⣿⣶⣤⣴⣶⣶⣶⣶⣶⣶⣶⡿⠁⠀⠀⠀⠀⠈⢿⡟⠋⠁⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⢠⣿⠁⠀⠀⣠⣶⣶⣤⣼⣷⣤⣄⠀⠀⠀⣀⣹⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠐⣿⣀⣼⠟⠁⠀⠀⠙⢿⡾⠛⠛⠛⠛⢻⣆
⠀⠀⠀⠀⠀⠘⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠙⠻⣿⠀⠀⠀⠀⠀⢸⡿⢶⡄⠀⠀⢀⣿
⠀⠀⠀⠀⠀⢀⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣶⠶⠾⠻⣷⣄⣀⣀⣤⣿⣧⣼⠟⠀⠀⣸⡏
⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠙⢿⣄⠉⠁⠀⢀⣼⡟
⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠶⠟⢻⣷
⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣆
⣠⣤⣤⣤⣿⣧⣤⣤⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠲⠾⠿⣿⡟⠛⠛⠛
⠉⠉⠁⠀⢹⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⣀⣸⣿⣦⣤⠄⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⣤⠤⢤⣄⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠛⣿⡿⠿⠷⠶⠆
⠀⠀⠚⠛⠉⠙⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣿⠃
⠀⠀⠀⠀⠀⢀⣸⣿⣷⣟⠃⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡿⠿⣶⣦⣀
⠀⠀⠀⢠⣶⠟⠋⠁⠈⠻⣿⠏⠁⠈⢻⡶⠶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠿⠋⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠛⠀⠀⣀⠀⠀⠀⠀⣿⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⣶⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢾⣀⡷⠀⠠⣾⡏⣿⡏⠙⠛⠛⠛⠛⠋⠉⣿⠏⠉⠈⢿⣟⠛⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⡟⠀⠀⠀⡀⠀⠀⢸⡷⠸⣷⣄⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠈⢿⣧⣴⡿⠻⠿⢶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠏⠘⢷⣄⣠⣼⡿⢶⣶⡟⠁⠀⠈⠛⠿⠶⠶⠾⠟⠋⠀⠀⠀⠀⠀⠘⣿⣟⠀⠀⠀⠀⣹⡿
⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠉⠉⢹⣿⣄⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⡀⠀⠀⢸⣿⢻⣦⡘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⡾⠟⠃⠀⠙⠷⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣥⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⠛⠛⠛⠛⠛⠻⠿⠿⠟⢿⡿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠉⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⢷⣶⣶⣶⡶⠿⠿⠿⠿⠷⣶⣶⣶⡶⠿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣦⣤⣀⠀⣠⣤⢶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠹⠟⠁⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⡉⠛⣿⣤⠶⠶⣶⡞⠛⠛⠛⠛⠛⠛⢻⣇⣠⣤⡶⠾⠿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣾⠋⠉⠙⠛⢿⡿⠁⠀⠀⠀⠹⣶⠶⠶⠿⠿⠷⠾⠟⠋⠁⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⢀⣠⡼⢧⣴⠶⣦⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣴⠛⠉⠉⣙⡟⠁⠀⠀⠙⣧⠾⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⠃
⠀⠀⠀⠀⠀⢿⠀⠀⢼⣋⣷⣀⠀⢀⣰⠷⢦⣤⠴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠈⢷⡀⠈⠙⢉⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⣿⠛⠳⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⢀⣠⣤⣤⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠶⠾⣿⠟⠓⠓⠂
⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⣀⣰⣿⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠠⠶⠶⢿⡟⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣴⠛⢓⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠋⠙⠓⢠⡶⠛⠉⠉⠙⣷
⠀⠀⠀⠀⠀⠀⢈⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡿⣧⣀⠀⢰⡟⠀⠀⠀⢀⡤⣸⠄
⠀⠀⠀⠀⠠⠞⠋⠉⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠁⠀⠀⠉⠓⢺⡇⢀⣤⣴⠟⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⠶⠿⣿⠿⣥⡀⠀⠀⠀⠀⣠⡾⣋⡴⠶⡥⠶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⢶⣾⠋⣽⠃⠀⢸⣏⠀⠀⠀⣀⣀⣀⣿⠀⠀⠘⣧⣈⡽⠗⠳⢶⡾⣫⡾⠋
⠀⠀⠀⠀⠀⠀⣾⠋⠉⠁⠀⠈⣻⠇⠀⠀⠈⠛⠛⠋⠉⠉⠉⠉⠉⠀⠀⠀⢘⣿⠃⠀⠀⠀⣿⠋⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⣹⠇⠀⠀⠀⢰⡟⠀⠀⠀⠀⢀⣴⠶⢦⡴⠶⢶⣄⠀⠀⠀⣾⠁⠀⠀⠀⢠⡟⠀⠀⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢾⡁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⢷⣤⣠⡴⠟⠀⠀⠀⣀⣿⣉⣿⣄⡀
⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⣀⡼⠃⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⣴⠟⠋⠉⠉⠉⢉⣙⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⠀⢸⣿⣤⡶⠛⠛⠶⠟⠉⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣦⣤⣤⣤⣄⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⡶⠶⠶⠟⢛⣿⠉⠀⠈⠻⣷⠿⠿⠟⠟⠛⠛⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣀⠀⠉⠉⠉⠉⠉⢉⣿⣅⠀⠀⠀⣀⣀⣠⣴⠟⠁⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠷⠾⠿⠟⠛⠻⣛⣛⣛⢛⠛⠋⠉⠀⠀⠀⠀⣀⣀⠀⠨⠷⠶⠶⠶⠶⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⢶⣶⣤⡀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⠿⢷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠋⠀⠀⠀⠉⠻⣷⣶⡶⠿⠛⠋⠉⠁⠀⠹⣷
⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⠿⠿⠿⠿⠟⠛⢻⣿⠁⠀⠀⠀⢀⣀⣀⣨⣿⣤⣀⡀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⢰⣟⣿⠋⣉⡁⠉⠉⠛⢷⣤⣤⣤⡶⣾⣷⣄
⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⢿⣇⢸⣿⣿⠀⢀⣀⠀⢻⡇⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣤⣤⣤⣤⣤⣿⠀⠉⠁⠀⠿⠿⢀⣾⣿⡄⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡉⠀⠀⠙⠿⢷⣴⣄⣰⡾⠿⠷⠟⠁⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣋⣀⠀⠀⠀⠀⠉⠉⠻⢶⣄⣀⣀⣀⣴⣿⣯
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠶⠟⠻⣏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠁⠈⣿⣇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢤⣀⠀⣴⢀⣤⢴⡾⣤⣤⡄⠀⠀⠀⠀⠲⢶⣿⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣀⣀⣀⡀
⠀⣀⣀⣀⣴⣿⣤⣤⣀⠐⢤⣶⠛⠏⠹⠻⠋⣠⠀⠀⠋⢻⡀⠀⠀⠀⠀⠀⣠⠾⣿⣦⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠐⠿⠟⢻⣿⠛⠛⠛⠛
⠈⠋⠉⠉⠉⣿⡏⠉⠁⠠⣾⣟⠁⠀⠙⢦⣾⣁⠀⠀⣌⣿⠛⠂⠀⠀⠀⠈⠁⣀⣽⣮⠅⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⢸⣿⣄⣤⠀⠛⠿⣥⣴⢀⡞⠀⠉⣉⡴⠛⠓⠂⠀⣀⣤⣄⡀⠈⠉⢹⡇⠀⠀⠈⠻⠋⠀⠀⠀⠀⠀⠀⠘⢛⣿⠿⠿⢶⡦
⠀⠀⠀⠲⠿⠛⢻⣿⠁⠀⠀⠘⠉⠉⠙⢳⣾⣴⠋⠓⠀⠀⠀⠸⣏⡀⣈⡿⢂⡀⠈⣷⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⢻⣧⣤⡴⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠙⢶⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣦⣤⣀
⠀⠀⠀⠀⢀⣤⡶⠟⠛⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⠙⠛⠀⠀⠀⠀⠀⠀⣀⣴⡾⠋⠀⠀⠉⠙⠃
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠛⠿⣶⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⠶⠀⣀⣀⣤⣤⣶⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡾⠿⠿⠿⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠟⠛⢻⠟⠛⠛⠳⢦⣀⡞⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⢿⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢀⡿⢶⣶⣶⣶⣶⣦⣤⣤⣴⠾⠟⠛⠉⠉⠙⢿⣆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⣳⣄⠀⠀⠀⡴⠒⠢⡀⠀⠘⠓⠶⢦⡀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⣼⠇⠀⢠⡾⠋⠀⠀⠀⠀⢧⣀⡴⠃⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⢻⡀⠀⢿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⡀⣀⣠⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣿⠁
⠀⠀⠀⠀⠀⠀⣸⡇⠀⠈⠳⠶⠤⠶⠞⢿⠀⠀⠀⠀⢸⡏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯
⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠰⠶⠶⠶⢾⡷⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⢾⡷⠶⠶⠶
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠠⠤⠶⠾⣷⠒⠀⠀⠀⠀⠀⠈⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠁⠀⠀⠀⠀⠐⠲⣿⠷⠶⠤⠄
⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⣀⣤⠶⠿⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⠿⠶⣤⣀
⠀⠀⠈⠉⠀⠀⠀⠉⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠉⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢳⣶⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣶⡟⠛⠉
⠀⠀⠀⠀⠀⠀⣠⣤⢤⡴⣾⠛⠉⢉⣿⠋⠉⠉⢿⡛⠛⠛⠛⠛⠛⠛⢛⡿⠉⠉⠙⣿⡉⠉⠛⣷⢦⡤⢤⣄
⠀⠀⠀⠀⠀⢰⣏⠀⠀⠀⠘⣧⠀⣾⠃⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠘⣷⠀⣼⠃⠀⠀⠀⣹⡆
⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠘⣿⡏⠀⠀⠀⠀⠀⠈⠛⠶⠤⠴⠶⠛⠁⠀⠀⠀⠀⠀⠹⣿⠃⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⠶⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠶⠶⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡛⠛⠛⠻⠶⠶⠶⠶⣶⣶⣦⣤⣤⣴⣶⡶⠶⠶⠶⠶⠟⠛⠛⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣀⣀⣀⣀⣀⣠⣾⣷⣄⣀⣀⣀⣀⣀⣤⡴⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠻⠿⣷⣦⠀⠀⣀⣠⣴⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠿⢿⣶⣶⣤⣀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣾⠟⠁⠀⠀⠀⠈⢻⣿⣿⠿⠛⠉⠁⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠉⠙⠻⢿⣿⠿⠿⠿⠛⠛⢻⣿⠏⠀⠀⠀⢀⣤⣤⡀⣻⣷⡀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⢰⣿⠋⣹⣿⠿⠛⠻⢿⣦⣤⣶⣶⣶⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠈⠿⣾⣿⠁⠀⠀⠀⠀⢹⣿⠁⠀⠀⠈⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣄⣀⣠⣤⣾⣿⠀⠀⠀⠀⠀⣸⡿⢿⡆⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠉⠻⢷⣤⣤⣤⣾⢿⣧⡾⠃⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣧⡀⠀⠀⠀⣠⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢷⣶⠿⠟⠉⠈⣿⡆
⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠾⢿⣿⠿⠿⠿⠿⠂
⠀⠀⢴⡶⠶⠾⢿⣿⡿⠿⠶⠀⠀⠀⠀⢠⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠟⠀⠀⠀⠀⠀⠰⢶⣿⣿⣶⣤⣤⡀
⠀⠀⠀⠀⠀⣀⣤⣼⣿⣶⠶⠂⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠉⠉⠉⠹⣿⣄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⣿⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣟⣁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣾⡿⠋⠙⠛⠿⡦
⠀⠀⠀⠀⠀⠀⢠⡾⠟⣫⣴⡿⠟⠛⠛⠛⠻⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠛⠛⠛⠛⠿⣷⣤
⠀⠀⠀⠀⠀⠀⠈⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷
⠀⠀⠀⠀⠀⣠⣶⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠙⠻⢿⣿⣿⣛⠛⢛⣿⣿⡿⠟⠋⠉⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⣄
⠀⠀⠀⣰⣿⡿⠛⠁⠹⣷⣄⠀⠀⣀⣤⣄⣀⣤⣾⠏⠀⠀⠀⠀⠉⠻⣿⣶⣿⡿⠋⠀⠀⠀⠀⠀⠹⣿⣤⣀⣀⣤⣄⣀⠀⣠⣾⡟⠉⠻⣿⣷⣄
⠀⢀⣾⣿⠏⠀⠀⠀⠀⠙⠻⠿⠿⠟⠋⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠛⠻⠿⠛⠉⠀⠀⠀⠈⠻⣿⣦
⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧
⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⢀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠀⠀⠀⠙⣷⠟⠉⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⣈⢹⣷⣴⠞⠛⠿⣿⠛⠋⠉⠉⠉⠉⠉⠉⣿⣅⣴⣶⠿⠟⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠉⠉⠙⠛⢿⡟⠁⠀⠀⠀⠈⢿⡿⠿⠿⠿⠿⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⢀⣤⣾⣥⣴⠶⣦⡀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣴⠟⠛⠛⠛⣶⠏⠀⠀⠈⠹⣦⣼⠃⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡿
⠀⠀⠀⠀⠀⠘⣇⠀⠀⢰⡾⢿⡀⠀⠀⠀⢠⣿⣀⣀⣀⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀
⠀⠀⠀⠀⠀⠀⢻⣄⠀⠘⠷⠾⢻⡶⠶⠶⠋⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣦⣄⣀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⢰⣿⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣀⣀⣀⡀
⠀⠀⠀⠴⠶⠶⢾⣿⠶⠤⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡀⠀⠀⠀⠘⠋⢹⡿⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠇⠀⠀⠀⢀⣤⣾⣇⣀⠀⠀⠀⢀⣀⣤⣄⣀
⠀⠀⠀⠀⠴⠶⠾⢿⡟⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢠⡞⠉⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠉⠉⠋⢀⡶⠟⠉⠉⠉⠙⣧
⠀⠀⠀⠀⠀⠀⠀⢘⣿⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣾⣿⣄⡀⠀⢰⡟⠁⠀⠀⠀⢠⡄⣿
⠀⠀⠀⠀⠀⢠⡶⠟⠛⠻⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠋⠁⠀⠉⠛⠆⢾⡇⠀⣀⣀⣴⠟⣱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣾⣿⡋⠁⠀⠀⠀⠀⠀⢀⣴⠟⣣⣌⣛⣉⣥⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣴⠟⣿⠟⠉⠙⣿⠛⠛⠛⠛⠋⠉⠉⣿⡏⠁⠘⣿⡉⠛⢷⣤⣤⣄⣀⣴⠟⣡⡾⠋⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠟⠉⠻⢷⣴⡟⠀⠀⠀⣿⣤⣤⣤⣤⣶⡶⠶⠿⠇⠀⠀⠘⢿⣴⡟⠁⠀⠉⢻⣵⠾⠋
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣰⡞
⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢀⣴⠟⠛⠷⠟⠛⠛⣦⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⣠⣿⣤⡀
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠙⠷⣦⣤⡶⠛⠁⠀⠀⢀⣠⣿⣉⣽⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⣠⣼⠏⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⢀⣾⠛⠉⠈⠉⠉⠀⣉⣙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣼⣧⣤⡴⠟⠛⠷⣾⠏⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⣤⣴⠶⠶⣿⠟⠁⠀⠹⢿⣧⣶⣶⣶⢶⠶⠾⢶⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡍⠉⠙⠛⠛⠛⠛⠛⠛⠛⣿⠋⠉⠉⠉⠉⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣤⣤⣤⣤⣤⣴⣾⡿⣷⣶⣶⣶⣶⡶⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⢹⣇⣀⣠⣤⣤⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠙⠒⠒⠒⠰⠦⠤⠤⠤⠤⠤⠤⠤⠴⠖⠂⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠿⠛⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠈⠉⠛⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠈⢿⣦⣴⠶⠾⠿⠛⠻⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠈⠙⠿⠶⠿⠿⠿⠿⠷⢶⣾⠏⠀⠀⢀⣴⠶⣤⣀⣸⣿⣀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⣿⠁⢠⡾⠛⠉⠉⠙⠻⣦⣀⣀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠙⠷⣿⠁⠀⠀⠀⠀⠀⢹⡟⠛⠛⠿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣦⣤⣤⣴⣿⡀⠀⠀⠀⠀⠀⣼⠿⣦⡀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠻⣦⣤⣤⣤⣾⣏⣀⣼⠇⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣇⠀⠉⠉⠁⠀⣠⡿⠃
⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣤⣤⣶⠿⢿⣆
⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⣿⡄
⠰⠶⠶⠶⢾⣿⠶⣦⠄⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣀⣀
⠀⠀⠴⠶⠾⣿⡟⠓⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣆⠀⠀⠀⠀⠀⠘⠛⠛⢹⣿⠋⠙⠛⠛
⠀⠀⠀⠀⠀⢸⣷⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠲⢦⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⢀⠀⣸⡟
⠀⠀⣤⠶⠾⠛⠛⣿⡍⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠳⣤⣤⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⠿⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⢀⡼⠋⣨⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⠟⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡿⠿⣦⣄
⠀⠀⠀⣠⡴⠶⠶⠒⠚⣿⡿⠷⠾⠿⠿⢶⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠟⠉⠀⠀⠈⠙⠷
⠀⠀⠀⣿⣦⣤⣄⣀⣾⠋⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⡛⠻⠿⠷⠶⠶⠶⢶⣶⣶⣶⡶⠶⣶⣾⡟⠛⠋⠁
⠀⠀⠀⠸⣇⠀⠈⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⢶⣦⣤⣄⣀⣀⣀⣀⣤⣴⠾⠛⠉⢻⣄
⠀⠀⠀⠀⢿⡟⠻⡺⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠈⠉⣉⡉⠉⠉⠁⠀⠀⣞⠛⡆⠹⣧⡀
⠀⠀⠀⠀⠘⣧⠺⣭⠆⠙⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⢸⣅⡽⠀⠀⠀⠀⠀⠉⠛⠁⠀⠘⢷⡀
⠀⠀⠀⠀⠀⢹⡆⠀⢾⡿⠀⣸⣟⣿⢶⣤⣄⣀⣀⣀⣠⣴⠟⢀⣄⠀⠀⠀⠀⠀⠀⢠⠶⢦⠀⠀⠀⠀⠀⣈⢻⣆
⠀⠀⠀⠀⢀⣴⣿⡷⠶⠶⢶⣿⣿⠃⠀⢀⣭⣍⠛⠋⠉⠁⠀⢯⣠⠇⠀⠀⠀⠀⠀⠘⠲⠞⠀⠀⠀⠀⢸⣉⣹⠻⣦
⠀⠀⠀⠀⢾⣇⡈⠛⠻⠿⠛⢁⣹⠇⠀⠘⢧⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠉⠛⠛⠛⣿⣿⣿⣿⡆⠀⠀⠀⠀⢀⠠⡟⢙⡆⠀⠀⠀⠀⠙⠶⠎⠀⠀⠀⢀⡖⢲⡄⢠⣀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣤⣤⡶⠟⢷⣍⡉⢀⣀⣴⣷⡀⠀⠀⢀⣤⠿⣦⣀⣑⣚⣵⠟⠙⢻⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠈⠙⠛⠛⠛⠉⠀⠀⠉⢻⡏⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⠶⣶⣶⣶⣶⣤⣤⣤⣶⣶⣶⣶⣶⣶⡶⠾⠿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠙⠷⠞⠳⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠾⠁⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢦⣀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⢤⡴⠏⠻⣦⡀⠀⠀⠀⠀⣰⡟⠁⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⢰⣿⠶⠶⢶⣤⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠋⠛⠳⢦⣄⣀⣀⣼⣷⣤⣤⡟⠁⠀⠀⠀⠘⣷⡾⠛⠋⠉⠉⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⣿⠀⠀⢀⣴⠞⣦⣼⣦⣤⡀⠀⢀⣀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⢷⣼⠋⠀⠀⠈⢻⡟⠛⠋⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣤⡴⢿⡀⠀⠀⢀⡼⢻⡆⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⣟⠙⠋⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡴⠞⠋⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣠⡤⠾⣿⠶⠆⠀⠀⠀⣠⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⠀⠀⠀⠘⠉⢹⡏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠉⠁⠀⠀⢿⣆⣀⠀⠀⠀⠻⡿⠃⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠻⡿⠇⠀⠀⠀⠀⠶⣾⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⢿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⡏⠀⠀⠉
⠀⠀⠀⠀⣠⠶⢦⣀⡴⢶⣬⣿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠻⢶⣄
⠀⠀⠀⠸⣏⠀⠀⠀⠀⠀⢸⡇⠈⠙⠻⢶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠾⠛⠁
⠀⠀⢀⡶⠋⠀⡶⠉⢳⠀⠘⢶⡀⠀⠀⠀⠀⢉⣩⣿⠟⠛⠛⠛⠒⠛⢿⡟⢛⣿⣋⠉⠉⢙⣷⣄
⠀⠀⠸⣧⣀⡀⠙⠒⠋⠀⢀⣸⠇⠀⣀⣤⠞⢫⡿⣹⣿⣷⡄⠀⣀⠀⣸⣷⢿⣿⣿⠃⠀⣿⣿⡿⣷⡀
⠀⠀⠀⠈⠉⢻⣄⣀⣠⠟⠉⠻⣾⠋⠉⠁⢀⡿⠀⠘⠻⠛⠀⣾⣿⣿⣿⠙⣧⠉⠁⣀⡀⠈⠉⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⢸⠃⠀⠀⢸⣿⣿⣦⠀⠀⠀⠈⠛⣹⠇⠀⠘⢷⣾⣿⣿⡆⢀⣴⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡀⠀⣸⡟⠿⠋⢀⣤⣄⠀⣰⠟⠀⠀⠀⠀⠙⢿⣋⠀⠈⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠙⣧⠀⠀⠻⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⣤⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣧⣀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠀⠀⠀⠉⠛⠒⠶⠶⠶⠤⢴⡶⠒⠛⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⢀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠳⠶⠶⠶⠶⠶⠶⠟⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠲⠦⣤⣀⣠⠞⠋⠳⣄⠀⠀⠀⠀⠀⠀⣠⠴⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢀⣹⣇⣀⡀⠀⢹⡶⠶⠶⠶⠶⠞⠁⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠒⠒⢺⠃⠀⠈⢷⠇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠰⠽⢦⣀⡤⠟⠓⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠳⣄⣀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒⠒⢺⡗⠒⠂⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣷⠶⠦⣄⠀⠻⠇⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠈⠉⣿⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠓⢮⣳⡀⠀⠀⠀⠀⠀⠀⠣⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠒⣶⢧⡀
⠀⠀⠀⢀⣀⡀⠀⠀⢀⣀⡀⠀⠀⠀⢿⡀⠀⠀⠀⣿⣄⣀⣀⡴⠒⠲⢦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣷⢾⡁
⠀⠀⠀⡏⠈⠷⠤⠤⠾⠁⢹⠀⠀⠀⠀⠙⢦⣄⣀⡟⠈⠁⠘⡏⠉⠉⠳⣽⢶⡶⠒⢶⡞⠛⠛⠉⠉⠀⠀⠉
⠀⢀⡼⠃⢀⣠⣤⣤⣀⠀⠈⢧⡀⠀⠀⠀⠀⠈⡿⠉⠲⢄⣼⡁⢀⣠⠾⢁⡼⠋⠳⡜⣧
⢀⡾⠁⣰⠋⠶⠀⠶⠀⣽⣆⠈⢷⡀⠀⠀⠀⠀⠻⣤⡤⠞⠻⢯⡿⡷⢶⣙⡧⠤⠤⣷⠘⣧
⣼⠁⠀⡇⠶⠀⢞⣱⠀⣉⢸⠀⠈⢷⠀⠀⡤⢤⡀⡿⠀⢀⡴⢻⠁⠧⠤⣟⠀⠀⠀⡟⠀⠸⡇
⣿⠀⠀⢿⣳⣀⠀⠁⠀⣩⠞⠀⠀⢸⠆⠀⠓⣦⡿⣿⠚⠉⠀⣼⠀⠀⠀⠈⠙⠤⠴⠃⠀⣰⠟⠓⠲⣄
⠙⠦⣤⣤⣁⣉⣉⣉⣩⣤⣤⠤⠶⠛⠛⠛⠛⠉⠀⠙⠷⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⡤⠾⢯⣤⣤⣤⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⠖⠉⠀⣿
⠀⠀⠀⣀⣀⣀⢀⡴⠋⠀⢘⠀⠀⠀⠀⠀⣠⠴⠒⠒⠒⠶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠈⠈⠁⠀⠀⠀⡟
⠀⠀⢸⠀⠀⠀⠉⠀⠀⠀⣸⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⢀⡴⠊⠉⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⠀⠀⢰
⠀⠀⠸⣄⡀⠀⠀⠀⠀⢠⡌⠀⠀⠀⡏⠀⠀⠀⢀⡠⠤⢤⣀⠀⠀⠘⣄⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⣄⣀⡎
⠀⠀⠀⠀⠉⠓⠤⢄⣀⡼⠁⠀⠀⢰⠀⠀⠀⡰⢉⡤⠲⣄⠈⠓⣄⠀⠘⣧⠎⠀⢀⡴⢚⣈⡓⢄⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⡇⠈⠱⢶⣅⡀⠀⠈⢣⠀⠈⡆⢠⡞⠀⢯⣀⡽⠈⢣⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⠉⠀⠀⠀⠇⠀⠀⠈⠒⡁⠀⠀⠀⢣⡀⠸⡇⠀⠀⠰⣟⡆⠀⠘⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⢠⠤⠒⠒⠢⠼⠀⠀⠀⠘⠛⠀⠀⠀⠈⣧⠀⢣⠀⠀⢠⣬⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⠀⣏⠈⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠸⣆⡤⠄⣤⡀⠀⠀⠃⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠀⠀⠘⣇⠈⠉⣖⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⡛⠀⠀⠀⠈⠓⣶⣀⣘⠀⠀⢀⡤⠒⠲⡄⠀⢀⡤⣤
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⣀⣀⣀⢘⡄⠀⠈⠰⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠇⠀⠀⠀⠀⠀⠀⠉⠉⠀⠓⢿⠀⠀⠀⢸⠔⠋⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⣯⠀⢠⠋⢀⣄⣀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠇⢀⡀⠄⣄⠀⠀⣀⠴⠋⠈⠳
⠀⠀⠀⠀⠀⠀⢱⡄⠈⡄⠸⣤⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⣏⠀⠀⢘⡃⠟⠲⠲⡄⠀⢀⠃
⠀⠀⠀⠀⠀⢀⡤⠓⠀⠙⣦⡀⠀⠓⠣⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⡀⠤⠒⡻⠣⠤⠚⠁⠀⠀⠀⡘⢀⡝
⠀⠀⠀⠀⠀⢈⠀⢀⡴⢋⡨⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠐⣶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢸⡏⠳⠤⠞⠁⣼⠁
⠀⠀⠀⠀⠀⠘⡄⠈⢦⠛⣤⡼⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣧⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⢀⡼⠁⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠳⣄⠀⠓⠋⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡀⠀⠀⠀⠀⠙⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠼⢷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢄⡀⠀⠀⠀⡔⢦⠀⠀⠀⠢⢴⣆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠞⠁⠀⠀⠈⠳⠖⠀⠀⠀⢠⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣈⠁⠒⠠⠤⢀⣀⡀⠀⠀⠀⣀⠠⣤⡘⠓⠒⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠜⠉⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠢⠤⣀⣀⠈⠁⠈⠛⢂⡿⠛⠀⠉⠳⡀⠀⠈⢛⠒⠂⠤⠤⢀⣀⣀⣀⣀⣀⣀⡠⠜⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠛⢙⠋⠀⠀⠀⠀⠀⢻⡄⠀⣼⠋⠐⠶⠖⠉⢁⡴⠛⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣦⣼⡀⠀⠀⠀⠀⢀⣤⡗⠾⢄⣀⣀⣀⠀⠀⣾⠀⠀⠀⠀⠈⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠸⡙⠂⠤⢄⣀⡸⠀⠉⠒⠤⣉⣁⣈⣙⣤⢾⡀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⠲⠦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠙⠑⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠈⠒⢤⡤⢤⢤⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⠀⠀⢀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⣽⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⡿⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠐⠒⠒⠒⠀⢀⡈⠤⠄⠀⠙⢤⡀⠀⠀⠀⠀⢰⠁⠈⠉⠭⢉⣉⡉⠀⠒⠒⠒⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠋⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠦⠴⠃⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠶⠢⣤⣄⠀⣀⠈⠉⠒⠶⠤⣤⣤⠀⣀⠀⠀⣀⣀⠀⢀⣀⢠⣤⡤⠲⠖⠉⠁⠀⣀⠀⣠⡤⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠒⠒⠂⠠⠤⠤⠤⠭⠭⠭⠭⠭⠥⠤⠤⠤⠤⠐⠒⠒⠚⠉⠉⢁⠤⠒⢦
⠀⠀⠀⠀⢀⠤⠤⣀⡴⠋⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠒⠦⠔⠉⠀⠀⠠
⠀⠀⠀⠀⣋⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠙⠤⣀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⡀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠈⠙⠂⠤⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠰⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠿⠿⣶⣄⠀⠀⣀⣀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠿⠿⢷⣶⣤⣀⡀⣀⣀⣀⣤⣤⣤⣼⠟⠁⠀⠀⠀⠈⠻⣷⡿⠟⠛⠉⠉⠉⠹⣷
⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠈⠙⠛⠿⠛⠛⠋⠉⠉⣹⡟⠀⠀⠀⢀⣤⣤⣄⣿⣇⣀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢾⡁⣼⡟⠋⠉⠛⢿⣶⣶⠿⠿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢸⣿⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠈⢻⣿⠀⠀⠀⠀⠀⣿⣄⡀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⠶⠾⠛⢿⣦⡀⠀⣀⣼⡏⣹⡇⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⠉⠉⠉⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣦⣤⡶⠟⠉⢿⣧
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⢀⣀⣀⣼⣿⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠿⣿⡿⠿⠻⠟
⠀⠈⠛⠋⠉⢹⣿⡉⠉⠁⠀⠀⠀⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⢀⣿⣧⣴⡦⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠙⠿⠟⠀⠀⠀⠀⠘⠛⣿⡿⠿⠷⠶⠄
⠀⠀⠀⠀⠛⠛⠛⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠿⠻⢷⣦⣄
⠀⠀⠀⠀⠀⣴⠿⠋⠁⠈⠛⢿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⠟⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢷⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠋⠛⠿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠉⠀⠀⠀⠀⠙⣿⣄⣀⣠⣤⣤⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠃⠀⠀⠀⠀⠀⠉⠛⢿⣦⣀⣠⣤⣤⣤⣤⣤⣀⣠⣾⠏⠀⠀⠀⢀⣀⠀⠀⠘⣿⡟⠋⠉⠁⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠉⠉⠉⠉⢙⣿⠃⠀⠀⢀⣾⠟⠛⢻⣦⡶⠿⢷⣦⣄⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠸⣯⡀⣼⡟⠁⠀⠀⠀⠈⠻⣷⣤⣤⣤⣀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠉⢻⣿⠀⠀⠀⠀⠀⠀⠀⢹⡏⠉⠉⠛⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣶⣶⣶⣾⣿⡄⠀⠀⠀⠀⠀⢀⣿⠿⣦⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣤⣤⣴⣿⣃⣀⣼⠇⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣯⠀⠈⠙⠛⠁⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⣀⣀⣤⣾⣿⣇
⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠈⣿⡆
⢀⣤⣤⣤⣤⣼⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠉⠉⠉⠉⢻⣟⠉⠉⠁⠀⠀⠀⠀⢀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣀⣸⣿⣀⣀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣿⣷⣤⣤⣤⣀
⠀⠀⠚⠛⠛⠛⣿⡏⠉⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣿⡇⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⣹⣷⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠙⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣿
⠀⠀⣤⡶⠾⠛⠛⠻⣿⣄⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠘⠷⣤⣤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣿⠟⠻⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣀⠀⠀⣠⡞⠁⣹⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡾⠋⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⢿⣦⣄
⠀⠀⠀⠀⣀⣤⣤⣴⣶⡶⣶⣶⣿⣴⣶⣿⣷⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠋⠁⠀⠈⠛⢷⣦
⠀⠀⠀⢼⣿⣁⠀⠀⠀⣰⡿⠉⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⣶⠶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⠾⠟⠋⠁
⠀⠀⠀⠸⣿⠙⠛⠻⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣤⣈⡉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣩⣿⠿⣷⡀
⠀⠀⠀⠀⢻⣦⣀⣀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡍⠙⠛⠿⠶⢶⣶⣶⠶⠾⠟⠋⠉⣀⣄⡈⢿⣆
⠀⠀⠀⠀⠈⣿⠉⣩⣭⡙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⢠⡴⢦⡀⠀⠀⠀⠀⠀⢯⣀⡽⠀⠻⣧⡀
⠀⠀⠀⠀⠀⠸⣇⠻⠾⢃⣀⠙⠿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠘⠦⠼⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⢿⡄⠀⠻⠽⠀⢰⡟⣹⡿⠷⣦⣤⣄⣀⣀⣠⣴⠿⠁⣠⣤⡀⠀⠀⠀⠀⠀⠀⡴⠛⢶⠀⠀⠀⠀⠀⢀⣈⣿⣆
⠀⠀⠀⠀⠀⣠⡾⣿⡷⠶⠶⠶⣿⣿⡟⠀⠀⢀⣬⣍⠉⠉⠉⠁⠀⠸⣇⣨⠇⠀⠀⠀⠀⠀⠀⠙⠲⠋⠀⠀⠀⠀⠀⣟⣈⡿⠻⣦
⠀⠀⠀⠀⠰⣿⡀⠙⠻⠿⠾⠟⠋⢈⣿⠀⠀⠸⣄⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠈⠛⠻⠶⠶⣶⣶⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⡴⠛⢶⠀⠀⠀⠀⠀⠸⢤⡼⠃⠀⠀⠀⢀⡴⢦⡄⠀⣀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠹⣷⣄⣀⣀⣤⣾⣧⡙⠲⠋⠀⢀⣠⣆⠀⠀⠀⠀⠀⣠⣦⣀⠘⠧⠴⢃⣼⠟⠷⣶⣶⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠙⠉⠉⠀⠈⠛⠷⠶⠶⠟⠋⠙⠷⣦⣤⣴⠾⠋⠈⠛⠿⢶⣾⠛⠁⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣏⣀⣤⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠷⠶⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠛⠻⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠙⣿⣆⣀⣠⣴⣶⡿⠿⠛⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠛⠉⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠉⠉⠙⠛⠿⢿⣶⣤⡀⣀⣀⣤⣤⣤⣴⣶⣿⡏⠀⠀⠀⣴⠟⠛⠻⣶⣤⣾⣿⣶⣤⣀⠀⠀⠀⣀⣸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠟⠛⠋⠉⠉⠉⢹⣿⠁⠀⠀⠀⣿⡀⢀⣾⠟⠉⠀⠀⠀⠉⢻⣿⡿⠟⠛⠛⠿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠈⠛⢿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣧⣀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣄⣀⣀⣠⣼⣿⡀⠀⠀⠀⠀⠀⢠⣿⠋⠹⣷⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠙⢿⣦⣤⣠⣤⣶⠿⣧⣤⣼⠟⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢿⣧⡀⠀⠀⠀⠀⣠⣾⡟
⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⣤⣴⣾⠿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣾⣿⣷⣶⣶⣶⣶⣦
⠀⠀⣀⣀⣤⣤⣿⣷⣤⣤⣤⠀⠀⠀⠀⢠⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠛⠋⠉⠉⠁⢹⣿⡇⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣻⣿⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠉⠙⢷⡄⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⠏⠉⠙⠛⠃
⠀⠀⠀⠀⠶⠿⠛⠛⠻⣿⣄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⣤⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⣀⠀⣠⣄⣴⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⣿⣿⣷⠿⠿⠿⠿⣶⣄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠋⠉⠉⠙⠻⣷⣬⣽⡿⠛⠛⠻⠿⢶⣦⡄
⠀⠀⠀⠀⠀⠀⣴⡾⠛⠁⣰⡿⠋⠀⠀⠀⠀⠀⠈⢻⡿⠿⠿⠿⠿⢷⣦⣄⡀⠀⠀⠀⢀⣤⣶⣾⠿⠿⣿⡏⠀⠀⠀⠀⠀⠀⠈⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⢀⣼⡏⠀⠀⠀⠀⠀⠈⠙⢿⣦⣤⣾⠟⠋⠁⠀⠀⠀⠹⣷⣄⣀⡀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣆⠀⠀⠀⣀⣼⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣦⣄⣀⣤⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠛⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⢿⣧
⠀⠀⠀⠀⣠⣤⣶⠾⠿⠿⠿⠿⠿⢶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⡶⠿⠿⠿⢶⣶⣦⣄⡀
⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠘⣿⣄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠈⢿⡄⠀⠀⡔⠈⠀⠀⢠⡀⢀⡀⠀⠀⡀⠀⣼⡟⠁⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠈⠻⠿⠿⢿⣿⠀⠀⠀⠀⠀⠀⠀⣼⡟⣁⠉⣻⡆⢸⣿⠀⢠⠁⠀⠀⣎⠠⠃⢺⡀⠀⣇⠀⠀⣿⠁⣼⡟⢉⡙⢿⡆⠀⠀⠀⠀⠀⠀⢸⣿⡶⡶⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣦⣤⣤⣶⣿⡀⠀⢿⣆⠛⠻⠛⣀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠘⠻⠿⢋⣼⡇⠀⢰⣷⣤⣀⣀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠙⠿⠶⠿⠿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣾⡿⠿⣷⣶⠿⠋⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣄⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣀⠀⠀⠀⣎⠀⠠⠀⠀⠀⠀⣀⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣀⠈⠉⠀⠀⣀⣤⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠶⢶⣦⣄⠀⠀⠀⢀⣀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⣿⣶⣦⣤⣀⠀⠀⠀⢀⣀⣀⣀⣀⣴⡿⠋⠀⠀⠀⠈⠻⣷⣶⡿⠿⠛⠋⠉⠙⢿⡆
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠉⠙⠻⢿⡿⠿⠟⠛⠛⠛⢻⡿⠀⠀⠀⠀⣀⣀⡀⢹⣿⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡏⢩⣿⠿⠛⠻⢷⣦⣤⣴⣶⣶⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠘⠷⣿⠃⠀⠀⠀⠀⢻⣏⠉⠀⠈⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢿⣷⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣦⣤⣤⡾⣿⣆⠀⠀⠀⢀⣾⠟⣷⠀⠀⠀⣸⡗
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠻⢷⣶⣶⡿⠿⠶⠟⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄⣀⣀⣠⡾⢿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠀⠀⢻⣧
⠀⠀⠀⠀⠀⢰⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣤⣤⣤⣤
⠀⢠⣴⣶⡶⢿⣿⡿⠷⠶⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠉⠉⢹⣿⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣠⣤⣼⣿⣄⣀⣀
⠀⠀⠀⠀⣀⣤⣾⣿⡿⠛⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢠⣶⣿⣷⣦⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢹⣿⠋⠉⠛⠛⠁
⠀⠀⠀⠀⠉⠉⠀⠙⣿⣄⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣶⣿⣯⡀
⠀⠀⠀⠀⠀⠀⢀⣤⡾⠿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠙⠻⠷⡆
⠀⠀⠀⠀⠀⠺⠛⠉⠀⠀⠈⠙⠿⣷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣴⠛⠲⣴⠛⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⢹⡇⠀⡾⠛⣲⠊⠙⣯⠆⢸⡦⣤⣤⠶⠶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣷⠀⣻⣌⣚⣶⠶⠛⠶⠟⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⣦⣀⠘⣿⣶⣿⣶⠏⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠉⠛⢿⣿⣿⣿⡉⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⣾⣿⣿⣿⡆⠀⠈⠁⠀⠠⣄⡄⠀⠐⠗⠀⠀⣺⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠿⣿⣶⣦⣀⣀⣤⣄⡀⠀⠀⢀⣠⡾⣏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡿⣿⢿⣠⡏⠀⣸⠟⠛⠛⠻⠿⣿⡿⠿⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⠸⣿⣷⣾⠏⠀⠀⠀⠀⠀⠸⣇⡤⠞⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢻⣤⣀⢀⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠷⠶⢶⡿⠶⠶⠶⠶⠶⣿⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠺⣿⡀⠀⢸⣇⠀⠀⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⢶⣤⣀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⠛⠷⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠈⠙⢷⣄⣠⣴⠾⠛⠋⠉⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣤⣴⠶⠶⠶⠶⣿⠃⠀⠀⠀⠀⠀⠀⠀⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣠⡶⢶⣦⡾⠿⠶⠶⣤⡀⣀⣤⣤⣤⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⣄⣾⠋⠀⠀⠀⠀⠈⢿⡏⠁⠀⠀⠈⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⠀⠀⠀⠀⣙⣿⠀⠀⠀⠀⠀⠀⢸⣧⣄⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⢻⣆⡀⠀⠀⠀⣠⡿⠁⣿⡆⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⢿⣏⠛⠛⠋⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣤⣤⣤⡾⠛⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣿⣦⣤⣤⣄⡀
⠀⠀⣀⣤⣤⣤⣾⣷⣶⡶⠶⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿⠁⠉⠉⠉⠁
⠀⠀⠈⠉⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⣻⣆⣠⣤⠀⠀⠀⠀⠀⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠈⠻⠟⠁⠀⠀⠀⠀⠀⠈⠙⣿⠟⠛⠛⠶⠆
⠀⠀⠀⠀⠀⣴⠾⠛⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⢀⣈⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣤⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⢿⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠈⠙⠻⠶
⠀⠀⠀⠀⠀⠀⠻⠛⠁⠀⠀⠙⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣴⡶⠾⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠻⣦⣀⣤⣴⠶⠟⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⠶⣶⣤⣄⣀⣠⣤⣤⣴⣾⠋⠀⠀⠀⠈⢻⡋⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠈⠉⠉⠉⠀⠀⢸⡏⠀⠀⢰⠋⣿⠟⠛⠳⣶⡶⠖⠺⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⣻⡇⠀⠀⠀⢸⢦⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠛⠉⠙⠦⣤⣴⠿⠞⠁⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⡴⠞⠉⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣧⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡇⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⠀⠀⠈⠉⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢻⣿⠛⠃⠀⠀⢸⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠚⣿⡿⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣤⣤⣀⠀⠀⠉⠁⠀⠀⠀⠀⣾⣋⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠉⠙⠻⣦⣄⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⢉⣙⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣀⣠⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⡾⠛⠉⣰⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠈⠉⢿⡀⠀⠀⢀⠀⠀⠀⠀⠈⢻⣷⠶⢶⣶⠶⠛⠿⠿⣿⡿⢿⡏⢁⣠⠴⠊⠙⣿⡄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣶⠈⠛⢶⣶⠿⣧⣄⣀⣀⣠⡾⠃⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠿⣯⡤⠤⠤⠤⢽⣷⠤⠘⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⣿⡅⠀⠈⢹⡟⠋⠉⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⣸⣿⠀⠘⠻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠻⠶⠶⠛⣤⣄⣀⠀⠀⠀⠀⠀⠙⠳⠦⣤⣀⣀⣀⣀⣼⠃⠀⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠟⢶⣦⡀⢻⣿⠟⠀⠀⣠⣄⣀⠀⠀⠀⠀⠈⠙⣏⠉⠈⣧⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⡀⠻⣦⣀⠀⠀⠀⢹⣿⡿⠃⠀⢠⣤⣀⡀⢻⡀⠸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠘⣧⠀⠀⠉⠛⠛⠷⣤⡁⠀⠀⠀⠘⣿⠿⠃⠘⣇⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡘⢧⡀⠀⠀⠀⠀⠈⠙⠛⠒⠲⣶⣦⣄⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣽⣦⣄⠀⠀⠀⠀⠀⠀⣠⡿⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⠶⠦⠶⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⢡⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⠢⠄⠀⡆⠀⣀⣀⠀⠀⠀⠀⠀⡰⠚⠓⣄⣀⣠⣄⠀⡆⠀⠤⠊⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⡀⠑⠊⠁⡸⠁⠈⠑⠦⠔⠒⣾⠁⢠⢲⠼⠧⣀⣈⡇⠉⠒⠁⡀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡠⠁⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠹⠤⠤⢧⡀⢀⡼⡄⢈⡆⠀⠀⠡⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠫⣌⡤⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠂⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⢉⡏⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⢻⣧⡀⠀⠀⠀⠀⠀⠨⠔⠀⠀⠀⠁⠀⠠⣼⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠈⠁⠀⠑⢦⠤⣀⣀⠤⠤⠤⢄⡀⣠⠤⣤⡞⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⢤⣀⣠⠞⢰⠉⠀⠀⠀⠀⠀⠈⢿⡄⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠘⠦⠤⣄⢀⣀⠀⠀⣸⡼⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢤⡀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⡏⠒⠤⣄⣀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⢈⡝⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⢀⡤⡄⢸⠀⡇⣀⣀⠀⠀⣆⡔⠀⢲⢰⡆⢰⡆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠘⠸⢝⠇⠾⠸⠇⢧⡼⠀⠀⠟⠯⠗⠸⠸⣴⢸⡤⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣄⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠒⠶⣄⣀⣀⣀⣼⠃⢀⣈⣿⣋⡉⠁⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠉⠉⠉⣏⠀⠻⣾⠉⠀⢹⣟⠛⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠇⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⢦⣤⡾⠿⢃⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣼⡧⠀⠀⠻⠃⠀⠀⣤⢄⠀⠀⠀⢸⡧⠀⠀⢈⣽⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣖⠀⠀⠀⠀⠀⠑⠚⠀⠀⠀⠀⠀⠀⢠⣴⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢻⣶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⡴⠞⠋⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⣾⢁⣾⣯⣭⡽⢟⡉⣿⠉⡹⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠟⢿⠇⠀⠀⠀⢯⡽⠻⣿⣦⠟⠁⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠈⣇⣀⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⢻⣄⣠⣤⡤⠴⠶⠖⠛⠛⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠦⣤⠴⡏⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢤⣤⡤⠴⠶⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄⠀⠀⠀⠀⣠⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⢀⣾⠟⠁⠀⠀⠙⢿⣦⣠⣤⣤⣤⣤⣴⡞⠋⠉⠉⠻⣶⡶⠿⠟⠛⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⢹⡇⠀⣸⡏⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠀⣼⡟⠀⣠⡶⣦⣤⣿⣄⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣷⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⠀⢿⣤⡞⠋⠉⠻⣷⣤⣤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢹⡆⣹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣄⣠⣿⣇⡀⠀⣀⠟⢻⡆⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠛⠿⣿⡛⠶⠞⠃⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣦⣀⠀⠀⠀⠘⢿⣶⣶⣾⣿⣁⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣠⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢾⣏⠀⠀⠙⠳⢦⣤⣤⣄⣿⣿⣿⣿⠉⠀⠀⠀⠀⣠⣬⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⠀⠀⠀⠀⠈⢹⣿⣿⣿⠶⠂⠀⠀⠘⢿⡿⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⢛⣥⣄⠀⠀⠰⠶⣿⡷⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⠶⣶⣦⣼⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢦⡀⠀⠀⠀⠀⠻⠿⠟⠀⠀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠈⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⠋⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡏⠙⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠟⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⣹⣿⣿⣿⠟⣼⠁⠈⠛⢿⣶⣦⣤⣄⣀⣀⣀⡀⣀⣀⣀⣀⣤⣶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⣥⣶⠿⢻⡟⢻⣿⣼⠃⠀⠀⢠⡾⠃⠈⣉⣽⣿⣛⠛⠛⠛⣿⠟⠛⠛⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⢸⡇⢸⡿⣧⣄⣀⣰⡿⠁⢀⡾⣿⣿⣿⣿⣧⠀⠀⢿⡀⠀⠀⠀⣨⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⣷⣄⠉⢻⡿⠁⠀⠸⣷⣬⣿⣯⣭⡏⠀⠀⠈⢷⣀⣴⠞⠉⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠘⣿⠀⣾⠃⠀⠀⠀⠀⢸⣿⡟⠉⠁⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢿⣷⡿⠀⠀⠀⠀⠀⡈⠉⠁⠀⠀⠀⠀⠀⠀⠻⣤⣶⣤⣀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⡷⠶⠶⠶⠶⣾⣷⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⣿⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣤⣼⣿⠁⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠉⣿⣄⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠷⢶⠶⠾⠿⠷⣦⣤⣤⣤⣤⣶⠶⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⠀⠀⠀⠀⣀⣴⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⠀⠀⣴⣿⡿⠋⠉⠙⠻⢿⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣴⣾⡿⠛⠛⢿⣿⣦⣤⣶⣾⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣇⠀⣸⣿⡟⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠿⠿⠿⠿⢿⣿⡿⠋⠀⠀⠀⠀⠹⣿⣿⠛⠉⠁⠀⠀⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⣿⣆⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⢠⡶⠿⣷⣶⣿⣿⣦⡀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢿⣇⣠⡾⠋⠉⠙⢿⣿⣶⣶⣶⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀⠀⢈⣿⣿⡅⠀⠀⠀⣸⣿⣿⣏⠉⠙⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⢀⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠿⣿⣿⣶⣤⣶⣿⣁⣸⡟⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⡛⠛⠉⠀⠀⣰⣿⡟
⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⣼⣿⣄⣸⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⣤⣴⣾⣿⣿⡄
⠀⣠⣾⡿⠿⣿⣦⣄⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠛⠃⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⢸⣿⣧
⢺⣿⣏⠀⠀⠈⠙⠿⣿⣶⣤⣤⣀⣸⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⡀
⠀⠙⢿⣷⣦⡀⠀⠀⠀⠉⠙⠛⠛⠿⣿⣿⣿⣿⡋⠀⠀⠀⠀⠀⠻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡆⠀⠀⠀⠙⢻⣿⣿⠿⠿⣶
⠀⠀⠀⠉⠻⢿⣷⣦⣤⣀⣀⡀⠀⠀⣿⣿⣿⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠿⢷⣄⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇⠀⠀⢀⣄⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⡿⠿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⢹⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⣿⣏⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠋⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠉⠛⠿⣿⣿⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣾⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠟⠋⣁⣽⣿⣿⣿⣿⣿⠁⠀⠀⠀⣠⣿⡿⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣯⣅
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣋⣤⣶⣿⣿⣿⣿⠿⣿⣿⠃⠀⠀⠀⣰⣿⠟⣠⣤⣤⣄⣿⠋⠙⣷⠀⠀⠀⠠⣿⡇⠀⠀⠀⠉⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠋⠁⢸⣿⡇⢰⣿⣿⣷⣦⣤⣼⣿⠏⠀⣿⡁⠀⢹⣿⢶⣼⡿⠟⠻⣦⠀⣿⣧⠀⠀⠀⠀⣠⣼⠿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⠉⣻⣿⠏⠀⠀⠈⣻⣿⢿⣇⣀⣿⣧⣤⣴⠟⠀⠘⢿⣧⣀⣴⡿⠛⠁⠀⠈⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠘⢿⣷⡄⢀⣿⡟⠀⠀⠀⢸⣟⠀⣠⣿⡏⠉⠻⣯⠀⠀⠀⠀⠈⢿⣿⠋⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠈⣿⣷⣸⣿⠁⠀⠀⠀⠈⠛⠿⠟⠉⠻⣶⡾⠟⠀⠀⠀⠀⠀⢺⣿⡀⠀⠀⠀⠀⠀⢀⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣶⣶⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⣀⣠⣿⣇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⢿⣿⣦⣀⡀⠀⠀⠀⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⡿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⣿⣷⣦⡀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠈⠹⣿⣦⣤⣶⣾⣿⠿⠛⠛⠛⠛⢿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠉⠉⠛⠻⢿⣷⣦⣤⣤⣶⣶⣶⣶⣶⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠉⠉⠀⠀⠀⣽⡿⠀⠀⠀⠀⢀⣤⣶⣦⣄⣸⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⣾⡏⠀⣼⡿⠛⠛⠉⠛⢿⣶⣴⣶⣾⣿⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠙⢷⣾⡟⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠈⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⣀⠀⣀⣀⣠⣿⣇⠀⠀⠀⠀⠀⠀⣸⣿⢷⣄⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢘⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠛⠋⠙⢿⣦⣀⡀⠀⣀⣴⣿⣃⣸⡟⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⠋⠉⠉⠉⠀⠀⠀⣠⣾⡟
⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣀⣀⣀⣤⣾⠟⢻⣿⡄
⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠛⠋⠁⠀⠀⣿⣧
⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⢀⣸⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠾⢿⣿⡿⠿⠿⠿⠿
⠶⠿⠿⠿⠿⢿⣿⡟⠛⠛⠂⠀⠀⠀⠀⠀⣠⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣀⣀⡀
⠀⠀⠀⠀⠀⣀⣿⣿⣶⡴⠆⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⢶⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⡟⠛⠛⠛⠛
⠀⠀⠀⠘⠛⠛⠛⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⣴⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⣀⣴⡿⠿⠻⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⢿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⡿⠋⠀⠈⠙⠻⠿⠆
⠀⠀⠀⠀⠀⠾⠟⠉⠀⠀⠀⠈⠙⠻⢿⣷⣦⣤⣀⣤⣤⣶⣾⡿⠛⠉⠀⠀⢀⣽⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⡿⠿⠛⣻⣿⠋⠀⠀⠀⠀⠀⠘⢿⣧⣤⣤⣤⣤⣤⣤⣴⣶⣶⣾⣿⢿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠉⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠛⠛⠛⣻⡿⠉⠉⠉⠻⣿⣆⠀⠉⠛⠿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡏⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣀⣴⡿⠁⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⣨⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⢀⣼⣿⡿⠿⠛⠁⠀⠀⣀⣤⣤⣄⠀⠘⣿⣧⣤⣾⡿⠋⠀⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣿⠿⠋⠀⠀⠀⠀⠀⠀⢰⡟⠁⠈⢻⣇⠀⠸⣿⣟⠁⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠙⢿⣭⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣤⡾⠃⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⣤⣴⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣶⣤⣤⣤⣤⣄⣤⣠⣤⣤⣴⣾⡿⢿⣷⣦⣤⣤⣄⣤⣤⣤⣤⣤⣤⣶⣾⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⠛⠻⠿⣶⣤⣀⣴⡿⠋⠀⠀⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠉⣿⠟⠀⠀⠀⠀⠀⠀⢻⣿⠿⠿⠿⠿⢷⣶⣶⣾⠟⠋⠉⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣀⣀⣀⣤⣾⣿⣦⣶⠿⣶⡀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⣴⡟⠋⠉⠉⠛⣻⡿⠉⠀⠀⠈⢻⣷⣿⠇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢠⣾⢿⣧⠀⠀⠀⠀⣸⣿⣀⡀⠀⢀⣠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠈⢿⣶⡿⢷⣦⣴⣾⠟⠙⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡅
⠀⠀⠀⠀⠀⠀⢠⣿⢿⣦⡀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⣾⡏⠀⠙⠻⠷⠾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⢀⣀⣀⣀⣿⣃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠘⠿⠛⠛⠛⢻⣿⠛⠛⠁⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⣠⣤⣤⣿⣧⣤⣤⣄⣀
⠀⠀⠀⠀⠀⣀⣿⣄⣀⡀⠀⠀⠀⢸⣿⠉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⣿⡇⠀⠀⠀⠈⠉⠉⣿⠉⠉⠉⠉⠋
⠀⠀⠰⡾⠿⠛⢿⡏⠉⠁⠀⠀⠀⠈⠿⣾⠟⠁⠀⠀⠀⠀⢠⣴⣶⣶⣤⠀⠀⠀⠀⠀⠈⠻⣶⠿⠁⠀⠀⢠⣴⣤⣼⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠘⣿⣤⣴⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣼⡟⠉⠙⠛⠃
⠀⠀⠀⠀⢀⣤⣾⠿⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣾⣯⣀
⠀⠀⠀⠀⠛⠋⠀⠀⠀⠙⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠛⠁⠙⠻⢷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣿⣿⢶⣶⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣶⣶⠿⣿⣯⣁⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠉⢀⣾⠟⠁⠉⠹⣿⡉⠉⠉⠉⠉⠉⠉⢩⣿⠏⠀⠈⢿⣆⠀⠉⠛⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣰⡿⢿⣧⣄⠀⢀⣾⠏⠀⠀⠀⠀⠙⢿⣦⣤⣀⣀⣤⣶⠿⠋⠀⠀⠀⠈⢻⣧⠀⢀⣠⣾⠿⣷⣄
⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠈⠛⢿⣾⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢻⣷⠿⠋⠁⠀⠈⢻⣧
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣶⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡶⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⢻⣿⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⣀⣀⣀⣀⣀⣀⣀⣤⣾⣿⣤⣀⣀⣀⣀⣀⣀⣀⣠⣶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⡵⠤⠤⢤⡤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠔⡾⠁⠀⡇⠀⠀⠀⠈⠳⣄⠀⠉⠒⢤⡀
⠀⠀⠀⠀⠀⢀⠜⠁⡘⠀⣀⣀⣰⣤⠒⠂⠐⢦⡺⡲⠉⠉⠒⠚⣦⡀
⠀⠀⠀⠀⢠⠋⢀⣠⡷⢩⠃⠀⢰⢻⠀⠀⠀⠀⠙⢄⡴⠛⠲⣤⢞⣁⡀
⠀⠀⠀⠀⣧⠞⠁⠈⢁⠇⠀⠀⠀⡇⡇⠀⠀⢀⠖⠉⢳⡤⠤⣾⣅⠀⠹⡀
⠀⠀⠀⢰⣇⣀⠀⠀⢸⠀⠀⠀⠀⢰⢱⠀⣀⣼⡆⠸⢿⡀⠀⣸⠞⠁⢠⠛⠲⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠉⠲⣶⠔⠒⠊⠉⠙⡨⡟⣶⣅⠈⠒⠚⠉⠉⠀⠙⠒⠋⠀⠀⠀⠉⠉⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢷⡇⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢻⡗⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡏⣻⣆⠀⠀⠀⠀⢾⡲⠀⠀⠀⣤⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠷⠋⠙⡞⠿⢦⣄⡀⠀⠀⠀⠀⠀⠉⠀⢀⣈⡿⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢸⠁⠀⠀⢹⡍⠓⣶⠦⣤⣤⣤⠶⣾⡛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣳⠿⠤⠤⠴⠚⠀⠀⠈⠳⣄⠉⠉⠛⠛⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣦⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠉⠉⠛⠓⢶⠖⠶⢲⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢀⣾⣀⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠶⠦⠤⠀⠀⠠⠤⠖⠓⠲⠄⠀⠀⠀⠀⠰⠋⠉⠓⠄⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡤⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢀⣠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣢⢤⣈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠒⠛⠉⠀⠀⠙⠲⣤⡟⠓⠲⠲⡄⠀⠀⠠⠶⠦⠖⢆⣀⣤⠶⠋⠁⠀⠈⠙⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠚⠁⠉⠋⠙⠒⠐⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠦⣄⡀⠀⠀⠀⣰⠏⠀⣈⢳⣞⠛⠉⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠙⠛⠛⠛⡟⠀⢾⣹⠟⠉⠙⣶⠶⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠻⠦⡤⠿⣄⣀⣠⣟⡷⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢷⣄⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⢺⡗⠂⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠐⠶⣿⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⢿⣁⠀⠀⠁⠀⠀⠠⣞⣳⠄⠀⠀⠘⠛⠀⠀⠀⢶⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡿⠻⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠈⣽⠿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣴⡶⠛⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣾⣃⣼⠿⠷⠶⠾⣋⣀⣿⣕⡥⣻⣻⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⣿⠀⠀⠀⠀⠳⠾⠙⢿⣴⠟⠁⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣤⣀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠘⢧⣀⣤⣤⡤⠶⠶⠖⠛⠛⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣤⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣤⣤⣤⣤⠴⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠾⠿⠷⣦⣄⠀⠀⣀⣤⣶⣾⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣾⠟⠁⠀⠀⠀⠈⢻⣷⣿⡿⠟⠋⠉⠀⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⢠⣿⡟⠉⠉⠈⠉⠙⠛⠿⣿⣷⣶⣶⣾⣿⣿⣿⣿⢿⠿⢿⣿⠋⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⣠⣾⠿⢿⣶⣾⠿⠿⢶⣦⣄⣀⣠⣤⣤⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣇⢠⣾⠏⠀⠀⠀⠀⠈⢿⣿⠉⠉⠁⠉⠉⠹⣷
⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠈⠛⢿⣿⠀⠀⠀⠀⠀⠀⢸⣿⢿⣶⡀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣤⣀⣤⣤⣾⣿⣦⡀⠀⠀⠀⣠⣿⣇⣠⣿⠇⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠙⠛⠿⠿⠿⣿⡉⠙⠛⠁⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⣀⣤⣾⣿⣷⡀
⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠁⠘⣿⣇
⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣿⣤⣤⣤⣤
⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠛⠛⠛⣿⣿⠉⠉⠉⠉
⣴⣶⡾⠿⠿⣿⣿⠿⠿⠆⠀⠀⠀⠀⠀⣴⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠟⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⠿⢷⣶⡦
⠀⠀⠀⠀⣀⣸⣿⣷⣶⡶⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠛⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠻⠛⠛⠛⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣾⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣶⡿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠙⠛⠿⣶⠄
⠀⠀⠀⠀⠀⣀⣴⣿⠟⠛⠻⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠙⠻⢿⣷⣶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣶⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠿⢻⣿⡿⠿⠿⠿⣿⡿⠿⣿⣿⣿⣿⡿⢿⣿⡿⠿⠿⢿⣿⡿⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⣿⠟⠉⠀⢠⣿⡟⠁⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⣀⣼⡿⠀⠀⠀⠀⠹⣿⣆⠀⠉⠛⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣾⡿⠟⠛⠻⣿⣦⡀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠙⣿⣧⣀⣴⡿⠋⠛⠿⢿⣶⡄
⠀⠀⠀⠀⠀⠀⢸⣿⡅⠀⠀⠀⠈⢻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡏⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠸⣿⣦⡀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⣠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣶⣶⣶⣶⡿⠿⠛⢿⣷⣤⣀⡀⠀⠀⠀⣀⣠⣴⣿⠟⠿⢿⣷⣶⣶⣶⣶⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⡿⠟⠛⠋⠀⠀⠀⠀⠀⠀⠈⠀⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣦⣄⣀⡀⠀⠀⠀⢀⣀⣀⣿⣇⣀⡀⠀⠀⠀⢀⣀⣀⣤⣾⣿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀
⠀⢀⠀⠈⠀⠀⠀⠀⠙⠃⠀⠀⡀⠀⠁⠀⠀⢀⣀⡀⠀⠀⠀⢉⣀⠀⠀⠀⡉⢁⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠒
⠀⠀⠞⠋⠀⠀⠀⠱⠳⠁⠀⠀⠤⠀⠀⠀⠀⠘⠟⠙⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠂⠴⠲

⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠠⠀⠀⠀⠂⡀⢀⡀⠀⠀⠠⠄⠀⠀⠀⠀⠠⠄⠀⠀⠀⠠⠄⠀⠀⠀⠀⣶⢠
⠀⠐⡀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠀⡄⠀⠀⠀⠀⠀⠀⠀⢐⣟⡇⡤⠁⠀⠤⡄⠀⠀⠀⠂⠀⠒⠀⠀⠀⠀⠒⠀⠀⠐⠀⠀⠀⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠁⠀⠀⢰⠆⢰⣤⠀⠀⠈⠉⠄⠀⠀⠀⠀⣀⠀⠀⠀⢀⠘⠃⠀⠀⠀⠈⠡⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠉⠀⠀⠠⠛⠂⠀⠀⠂⠐⠇⠻⠀⠀⠀⠘⠛⠀⠀⠀⠰⠓⠀⠀⠀⠰⠧⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐

⠀⠀⠀⠤⠀⠀⠀⠀⢤⡀⠀⠀⠀⠤⡄⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠠⠀⠀⠀⠀⠠⠄
⠀⠀⣧⣄⠀⠀⠀⠠⠤⠀⠀⠀⠀⣀⠀⠈⠀⠀⡀⢢⠀⠀⠀⠀⠀⠂⡁⢺⣤⣶⣽⠀⠀⠱⡶⠀⠀⠀⠀⠀⣤⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠛⠗⠀⠀⠀⠘⠀⠀⠀⠀⠀⠆⠀⠤⠀⠀⠚⠂⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠋⠀⠀⠀⠀⠁⠀⠁

⠀⠀⠀⢤⡀⠀⠀⠀⣤⠀⠀⠀⠀⡀⣀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠐⣂
⠀⠀⢠⣤⠀⠀⠀⢀⡀⠀⠀⠀⢰⣆⡈⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠄⠀⠀⢀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⠤⡤⠀⠀⠄⠀⠀⠀⠃
⠀⠀⠈⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⣀⠠
⠀⠀⠘⠃⠀⠀⠀⠊⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘⠇⠀⢀⣜⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⢀⡀⢀⣉⠁⣉
⠀⠀⠲⠄⢴⠄⠀⠠⠄⠀⠀⠀⠀⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠆⠀⠀⠿⠹⠆⠀⠀⠀⠠⠀⠀⠀⠀⠰⠺⠂⠀⢠⢥⠬⠭⡤⠅
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣻⡟⡛⢻⡟⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⢦⣤⣀⣀⣀⣀⣀⣠⡞⠉⠉⠻⣶⠖⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠉⠉⠉⠉⠉⡿⠀⠀⡶⣶⡿⠦⣄⣠⣤⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⣀⣹⡏⠀⠀⢸⣧⡀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠙⠶⢶⠛⠚⠁⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠛⠉⣧
⠀⠀⠀⠀⠀⠀⣀⣀⣼⣤⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⠞⠛⠳⣿⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠀⠀⠸⠿⠀⠀⠀⢀⣀⣀⠀⠀⠀⠸⠿⢸⡶⠚⠛⠛⠳⢦⡴⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠻⣇⣠⠀⠀⠀⠀⠀⠀⠘⠣⠞⠀⠀⠀⠀⠀⡟⠀⡄⠀⠀⠀⠀⢳⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⠉⠻⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢷⡀⠠⢴⡀⠒⠀⣸⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠙⠻⣿⡛⠛⠒⣻⡿⠛⢿⣦⠞⠛⠲⢤⣤⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣆⠀⠀⠈⠛⠛⣿⠁⢠⠶⣼⢧⣀⡀⠀⠀⣧⣀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠙⢦⣀⠀⠀⠀⣻⠀⠘⠒⠋⠈⣿⣹⡷⠶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠙⠳⠶⠾⠋⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣥⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠈⠉⠉⢹⣏⠉⠉⠉⠁⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⣀⣀⣴⠟⠹⣦⣀⣀⣀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣄⠀⠀⣠⣴⣶⣿⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠋⠁⠀⠀⠙⢿⣿⠟⠋⠁⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠛⠛⠻⠿⢿⣷⣶⣤⣶⣶⣿⣿⡿⠿⢿⣿⠃⠀⠀⢀⣠⣀⣈⣿⣦⣀⡀⠀⠀⣀⣸⣿⣀
⠀⠀⠀⠀⠁⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠈⠉⠋⠉⠉⠀⠀⠀⠀⢸⡟⠀⠀⢰⣿⠋⣿⡟⠋⠉⠙⠻⣾⡟⠛⠋⠙⠛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠻⣾⣿⠀⠀⠀⠀⠀⣿⡿⣶⡄⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣀⣀⣠⣴⣿⣦⣀⠀⣀⣴⣿⣥⣾⠇⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠉⠛⠛⠛⢿⣄⠉⠀⠀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠋⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣷⣴⣶⣶⡦
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠈⠛⠉⢿⣿
⠀⠀⠀⠀⢀⣀⣠⣿⣧⣄⣀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢀⣤⣿⣷⣤⣤⣄
⠦⠄⠀⠾⠿⠛⠛⢻⣿⠉⠉⠀⠀⠀⠀⢾⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⢹⣿⠏⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢘⣿⣦⣤⡄⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⢸⣏⣉⣹⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣾⣿⣄⣀
⠀⠀⠀⠀⠀⠰⠾⠛⠛⢿⣧⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠉⠉⣩⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠟⠉⠙⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣋⠈⠁⠁⠀⠀⢀⣴⣿⠿⠟⠻⠿⢿⣿⣏⣼⣿⡁⠀⠈⠀⠀⢀⣡⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉⠀⠙⢻⣿⡿⠿⠿⣷⣾⡿⠿⢶⣶⠄⠀⠀⠈⠈⠛⠛⠿⣷⣴⣶⠿⠿⠿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠈⣿⡆⠀⢸⣯⣤⣤⣶⢿⣇⠀⠀⣀⢸⣿⠁⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⣿⡟⠋⠀⠉⠉⠁⡀⠈⠻⣦⣠⡿⣿⡿⠀⠀⠀⠀⠀⣹⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⢹⣧⡀⠀⣴⠀⠀⠛⠀⠀⠈⠉⣀⣿⡁⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠻⣷⣄⣀⣀⣤⣿⠏⠀⠀⠀⢀⡀⠀⠺⠆⠀⠀⠉⠻⣷⣤⣄⣠⣴⡿⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠈⠙⠛⠛⢻⣧⠀⠐⠗⠀⠈⠁⢀⠀⠀⠲⠆⠀⣠⣾⠟⠉⠉⠉⠀⠀⠈⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⣤⡄⠀⠙⠁⠀⣠⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⣠⣴⡿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣶⣿⣟⣉⡀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⣠⣾⡿⠛⠛⠛⠛⠛⠛⠛⠿⢿⣷⣀⠀⠀⠀⠀⢀⣠⣾⡟⠁
⠶⠒⠒⠒⠒⠒⠀⠀⠒⠒⠀⠐⠈⠛⠿⣷⣶⣾⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣶⣾⠿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠈⢿⣄⣤⣴⡶⠿⠛⢻⣧
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠿⠶⣶⣤⣄⡀⠀⠀⠀⠀⠀⢀⣰⡿⠁⠀⢀⣤⣤⣀⠈⣿⣏⡀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠈⠙⠻⣷⡶⠿⠛⠛⠛⣿⡇⠀⠀⣾⡁⢀⣽⠛⠉⠙⠻⣶⣤⣶⠶⢾⣧⣄
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠈⠛⣿⠃⠀⠀⠀⠀⠘⣷⣄⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⢿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣤⣴⣿⣧⡀⠀⠀⢀⣼⠏⢈⡷⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢾⣟⠉⠛⠛⠁⢀⣾⠃
⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣤⣤⡶⠿⣿⡀
⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠹⣷
⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣿⣤⣤⣤⣤
⠀⣀⣤⣤⣼⣿⡶⠶⠄⠀⠀⠀⣾⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⡄⠀⠀⠀⠀⠈⠀⢸⡿
⠀⠉⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠙⠟⠃⠀⠀⠀⠀⠀⠀⣠⠤⢤⡀⠀⠀⠀⠀⠘⢿⡿⠃⠀⠀⠀⠀⠠⢶⣿⡷⢶⣤⡄
⠀⠀⠀⣠⡴⠾⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣠⡟⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀⣀⣼⡟
⠀⠀⠀⠀⠀⠀⣠⡿⣿⣿⡷⠿⠻⠶⣦⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠉⠛⢷⣬⣿⠟⠛⠻⠶⣦⡄
⠀⠀⠀⠀⠰⠟⠉⢠⡿⠁⠀⠀⠀⠀⠈⣿⠛⠛⠛⠻⢶⣤⡀⠀⣠⣴⠾⠟⠛⢿⡁⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⣠⣤⡾⠏⠀⠀⠀⠀⠀⠈⠻⠿⠋⠀⠀⠀⠀⠈⠻⠶⣦⣀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⣾⡟⠷⠶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠙⣿⡄
⠀⠀⢀⣤⡶⠟⠛⠛⠛⠛⠷⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠟⠛⠛⠿⠶⣦⣄
⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠘⣷⡀⠀⡀⠀⠀⢀⠀⣀⠀⠀⡀⢠⡾⠁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠙⠻⠿⣿⠁⠀⠀⣀⠀⢰⡟⣍⣹⡇⢸⡇⠠⠅⠀⠸⠀⠁⠧⠀⠇⠀⣾⡇⣾⣉⣍⣷⠀⠀⡀⠀⠀⢸⣷⠶⠾⠋
⠀⠀⠀⠀⠀⠙⠷⠶⠾⢿⣄⣨⣷⣬⣉⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣌⣉⣥⣿⣀⣸⡿⢶⣴⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠙⢿⣤⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⡀⠀⢸⣀⠀⠀⠀⢀⣠⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣄⡀⢀⣠⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠿⠷⠶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣦⡀⣀⣤⡶⠿⠛⠋⠉⠉⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠈⠉⠛⢷⣦⣀⣀⣠⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠉⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⠀⠀⣀⣀⣀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠛⠋⠁⢹⡇
⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⠶⢾⣿⠾⠶⠶⢶
⠠⠶⠶⠿⠿⢿⡿⠛⠛⠛⠀⠀⠀⠀⠀⣰⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⠇⠀⠀⠀⠀⠀⠀⣤⣤⣿⣇⣀⣀
⠀⠀⠀⠀⠀⣀⣿⣦⣶⠶⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⠶⣤⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠈⣹⡟⠉⠉⠙⠛⠁
⠀⠀⠀⠰⠿⠛⠙⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⣀⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣴⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⡿⢷⣦⣄⡀
⠀⠀⠀⠀⠀⢀⣤⣾⠿⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠈⠙⠻⠆
⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠈⠻⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣶⠶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⢶⣤⣄⡀⠀⣀⣀⣀⣀⣴⠟⠉⠈⠙⣷⡶⠟⠛⠛⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠉⠛⠛⠛⠉⠉⢹⡏⠀⢀⣤⢦⣼⣷⣄⡀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠘⢷⡾⠁⠀⠈⢻⡋⠉⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⠞⢷⣄⣀⣠⣾⣹⠆⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢯⣈⢁⣠⡼⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣿⣤⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣰⠟⠉⠛⣿⡟⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢿⡀⠀⠀⠀⠸⣿⠇⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠐⢿⠇⢻⣤⠖⠛⠛⠓⠶⣤⡴⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠛⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣩⡷⠀⠀⠀⠀⠀⢠⡟⠀⣀⠀⠀⠀⠀⠘⢷⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡆⠀⠉⢠⡄⢀⣄⠀⢸⡷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠙⠻⣶⣦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣤⣴⣏⠉⣻⣦⡈⠙⠓⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠙⢿⣭⣉⣉⣩⡽⠋⠉⠙⣿⠟⠉⠀⠙⠛⣶⠶⢾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⠀⠀⠀⠉⠉⢹⡇⠀⢠⠖⢲⡻⣦⣄⡀⠀⠀⢻⣤⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠉⠻⣦⡀⠀⠀⠀⠈⣿⠀⠈⠓⠚⠀⠹⣯⣹⡷⠶⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠈⠛⠷⣦⣤⡾⠋⠀⠀⠀⠀⠀⠀⢹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢛⣿⠛⠛⠛⠿⠿⠿⣿⡿⠿⠿⠿⠟⠛⢿⡟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⢠⣿⣇⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣤⣤⣤⡶⠟⠁⠙⢷⣤⣤⣤⣤⡶⠏

⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⣐⣺⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣗⣂
⠀⠀⠀⠀⠀⠀⠠⣀⠱⡆⠀⣢⠀⢠⢤⣀⠀⠀⠀⠀⠀⡰⠋⠉⠱⣄⡠⠤⢦⡀⣔⠀⢰⠎⣀⠄
⠀⠀⠀⠀⠀⠀⠼⠆⡄⡨⠋⠀⢠⠇⠀⠀⠙⠲⠒⠒⢺⠁⠠⡞⡶⠛⠢⣄⣀⣇⠀⠙⢅⢠⠰⠧
⠀⠀⠀⠀⠀⠈⢃⡐⠁⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠘⠦⠤⠽⣄⠀⢀⡿⡄⠈⡇⠀⠀⠈⢂⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢯⣉⣡⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢾⠤⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠒⡟⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣦⠀⠀⠈⠋⠀⠀⠀⢠⠂⡄⠀⠀⠘⠟⠀⠀⢲⠗⠤
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠴⠛⠻⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣨⠟⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠸⠤⣄⠀⠀⠀⠀⠈⡏⢓⡶⠖⠚⠉⠙⠓⠢⣤⡞⠉⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡶⠒⠦⠤⠞⠁⣏⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠈⠒⠖⢦⣀⣤⡀⠀⢀⡾⠜⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣾⠙⠢⢤⣀⡀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄⠀⠀⠀⠉⡽⠓⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠴⠊
⠀⠀⠀⠀⠀⠀⠀⣦⢠⠀⣀⡀⢀⡄⢠⠀⠀⠀⠀⠀⣤⢠⠄⠀⣤⢠⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⡟⢻⢸⣇⣟⢸⡇⢸⢰⣿⣷⠀⠀⣿⣯⢶⡄⡆⣼⢁⢸⠉⢠⠄⡶
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⠉⠈⠁⠈⠀⠙⠁⠀⠀⠉⠈⠉⠀⠁⠙⠋⠘⠚⢘⡦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⡠⠼⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠧⢄
⠀⠀⠀⠀⠀⠀⠐⢉⠉⢣⣀⡂⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠲⣄⠀⢀⣀⣀⠀⢰⠄⢐⣠⡜⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⢸⡋⠈⠐⡤⠔⠃⢀⡞⠉⠙⠲⣄⣀⣀⣀⣠⠏⠀⢀⢤⣘⣟⡉⠁⠈⡇⠘⠢⢤⠂⠁⢙⡇
⠀⠀⠀⠀⠀⠀⢰⠝⠁⠷⠐⠁⠀⠀⢸⠀⠀⠀⠀⠀⠁⠀⠀⢸⠀⠀⠣⡜⠁⠀⠙⣶⠒⠳⣄⠀⠈⠂⠾⠈⠩⡆
⠀⠀⠀⠀⠀⠀⠀⠢⠌⠀⠀⠀⠀⠀⣸⠆⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⠳⣄⣀⣴⣏⡇⠀⡼⠀⠀⠀⠀⠡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⡤⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣧⠤⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠒⢺⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢽⡤⠀⠀⠀⠛⠁⠀⠀⠀⢀⠤⡀⠀⠀⠀⢿⠇⠀⠀⠠⣾⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⢉⣩⠷⣤⠀⠀⠀⠀⠀⠀⠀⠘⠒⠋⠀⠀⠀⠀⠀⠀⠐⣶⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⠀⠀⠈⠳⣦⢤⣀⣀⣀⡤⠤⠤⣤⣀⠀⢀⡤⢤⣤⡞⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣄⠀⣀⠀⠀⢀⡟⢠⠞⠉⠀⠀⠀⠀⠀⠈⠙⢿⣇⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠋⠉⠓⠛⠉⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⢆⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠉⠛⠙⠦⠤⠴⣄⡀⣀⡴⠓⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⣿⠙⠦⣄⣀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⠀⠀⠀⠈⠉⠛⡶⠲⠶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠆⣤⠶⡄⢸⡇⢸⠀⣀⣄⡀⠀⠀⣿⡰⠃⠀⢛⠃⣿⠄⢸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠸⠀⢯⠽⡧⢸⠁⢸⠘⣿⣿⠇⠀⠀⣿⢷⣛⡇⢸⠀⡇⡄⢸⢠⠰⡇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠁⠈⠉⢀⣷⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⢶⣤⣀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠈⠉⠻⢷⣦⡀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣾⠋⠀⠀⠀⠈⠙⢷⣄⣠⣶⠿⠛⠉⠁⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠛⠛⠛⠛⠛⢛⣿⠃⠀⠀⠀⠀⢀⣀⡈⣻⣿⣡⣀⡀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⢰⣿⠛⢻⣿⠟⠉⠉⠉⠛⢷⣤⣴⣶⡶⢶⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠻⣦⣿⠃⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⠙⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⣀⠀⠀⣀⣈⣻⣇⠀⠀⠀⠀⠀⢀⣿⠏⢻⣆⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⠉⠙⠷⣦⣤⣤⣴⠿⢿⣶⠾⠋⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢻⣆⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⠀⢀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⢹⣇
⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿
⢰⣶⡶⠾⠿⠿⢻⣿⠛⠋⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⠿⠿⠿⠶⣶⣶
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣀⡀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⣀⣀⠀⢀⣿
⠀⠀⠀⠀⢀⣀⣼⣿⡿⠟⠋⠁⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠙⠛⠿⣾⣿⣄⣀
⠀⠀⠀⠀⠛⠋⠉⠹⣷⡀⠀⠀⣀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⣀⣀⣹⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⡀⠀⠀⣰⡿⠉⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣴⡟⠁
⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠙⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠉⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣴⡶⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⢤⣄⣀⠀⣠⡶⠛⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠈⠙⢻⠏⠀⠀⠀⠈⣷⣤⣤⣤⣤⣤⣤⣴⠟⠁⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⡴⠛⠛⠳⣞⣳⠀⠀⢸⡇⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⣸⡇⠀⠀⠀⣹⣁⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠘⠷⠛⢦⡤⠴⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠳⣄⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣿⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢰⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡇⠀⠀⠠⠤⠤⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⢶⣿⣍⣉⠓⢦⣄⠈⠋⠀⠀⠀⠀⠀⠀⢀⠤⠤⣄⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢠⡏⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠈⠙⢦⣙⡆⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⡿⠧⡄
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⣽⣃⡀⣀⠀⡤⠶⠶⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⣶⡋
⠀⠀⠀⢠⡟⠉⢷⣄⣀⣀⣾⠋⢻⡆⠀⠀⠀⠈⠻⣦⣀⠀⠀⣿⠉⠛⠉⠻⡟⠛⠒⠲⢬⣻⣦⣤⣤⣤⣴⡶⠶⠶⠞⠛⠋⠁⠀⠙⠂
⠀⠀⠀⣼⠇⠀⠀⠉⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠙⠳⣶⠿⣄⡀⠀⢰⠇⠀⠀⠀⣼⠏⢠⡿⢤⣄⢻⡇
⠀⢀⡾⠁⢀⡤⠖⠚⢛⡒⠶⣄⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⣲⣿⢤⣤⡴⠞⢁⣤⡟⠀⠀⠸⡎⢿⡄
⢀⡾⠁⢀⡞⠀⠿⠂⣙⠛⠀⣾⢷⡀⠈⢿⡄⠀⠀⠀⠀⠀⠙⢷⡶⠛⠉⠙⢳⡿⣻⠟⢫⣀⡷⠶⠶⠶⡇⠀⢻⡄
⣼⠃⠀⢸⡇⠿⠂⢸⣉⣱⠀⣉⡈⡇⠀⠈⣷⠀⠀⣠⡶⢤⡀⣿⠀⠀⣀⡴⢻⡇⠹⢤⠼⣟⠀⠀⠀⢀⡇⠀⠀⣿
⣿⠀⠀⢠⡷⢲⡄⠀⠉⠁⠈⢛⡽⠃⠀⠀⢸⡇⠀⠙⢦⣤⣿⣿⠶⠚⠉⠀⢸⠁⠀⠀⠀⠈⠹⢤⣤⡼⠁⠀⢀⣿⠷⠶⢦⡄
⢻⣄⡀⠀⠙⠛⠛⠒⠶⠖⠚⠉⠀⣀⣀⣤⠿⠷⠤⠴⠶⠛⠁⠹⣧⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠁⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠹⣄⣤⠶⠒⠛⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠙⠓⠶⣤⣀⣀⣀⣀⣤⣼⠁⠀⠀⠐⠀⡀⠼⠄⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⣿⠀⠀⠀⠀⡄⠀⠀⠀⠈⣏⠉⠉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⣀⣤⣇⠀⠀⠀⢀⠏⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢲⡁⠀⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠖⠋⠻⣆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣧⣤⣤⣤
⠀⠀⠀⠠⠴⠒⢻⡖⠒⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣀⡀⢀⣠⣷⡤⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⣿⡋⠛⣂
⠀⢠⣌⣏⣇⣾⠱⢊⣷⣴⡦⠤⠦⢤⡀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⡠⠖⠒⠒⢤⣰⡾⣿⣽⣧⡄⠠
⠀⠛⢒⡿⠋⠀⠒⢉⣠⠏⠀⠀⠀⠀⢹⠉⠉⠙⠲⢤⡀⣀⡴⠒⠛⠻⡁⠀⠀⠀⠀⢹⠁⠀⠀⠀⠉⠳⣄⣐
⢀⡞⣉⠁⠀⣀⠀⣤⣹⣄⣀⣀⡴⠒⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠒⠦⣀⣠⢾⡀⠒⡀⠀⠀⠀⢨⡍⣷
⠀⠳⠭⢤⣄⠈⠀⣈⣿⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡑⠏⠰⠂⢀⡶⠚⠁
⠀⠀⠀⣠⠞⠛⢻⡏⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠙⢲⡶⠾⠿⣄
⠀⠀⠀⠻⠤⠤⠤⣷⣈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⣼⢃⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣶⠶⣾⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⡴⠛⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⢧⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠉⠳⣤⣀⣠⠔⠋⠁⠀⠀⠀⠀⠀⣰⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠈⠉⠛⠿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⣤⣤⣤⣀⣀⣀⡀⠀⣾⡏⠀⠀⠀⠙⣿⣆⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠻⠿⣿⡇⠀⠀⠀⠀⢸⣿⣶⡿⠟⠛⠻⢿⣶⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡾⠿⠿⠿⠷⠀⠀⢀⣀⡀⠉⠁⠀⠀⠀⠀⠈⣿⡇⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⣾⠟⠛⢿⡆⠀⠀⠀⠀⣠⣾⡟⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠀⠀⠀⠀⠻⣧⣤⣾⠇⠀⠀⠀⠸⣿⣏⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣶⣶⣦⡄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣾⣷⣄⡀⠀⠀⠀⣸⣿⠀⠀⣿⡇
⠀⣀⣀⣀⣀⡀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⣀⣾⡟⠉⠛⠿⣷⣶⣾⠿⠃⠀⢸⣿
⠀⠛⠛⠛⠛⠻⣿⣿⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣠⣤⣤⣴⣿⣷⣤⣤⠀⠀⠀⠀⠸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠉⠀⠀⠙⣿⡅⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⢠⣤⣤⣼⣿⣀⣀⡀
⠀⠀⠀⠀⠀⣀⣀⣹⣿⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⢶⣦⣄⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⠀⠀⠀⠀⠀⠈⠉⠉⣿⡟⠛⠛⠻⠿⣷
⠀⠀⠀⢠⣾⠿⠛⠛⠛⠋⠉⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⣀⣨⣿⠀⠀⠀⠀⠀⠀⠀⠹⢿⠿⠃⠀⠀⠀⠀⠀⣠⣄⣰⣿⠃
⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠘⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⣿⠿⢷⣶⣤⣀
⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⣠⣾⠟⠙⠻⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⡿⠃⠀⠀⠈⠙⠛
⠀⠀⠀⠘⢿⣦⠀⠀⣠⣾⠟⠁⠀⠀⣠⣾⠟⠻⠿⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠿⣷⣦⡀
⠀⠀⠀⠀⠈⠻⣷⣾⡟⠁⠀⠀⢀⣼⡿⠃⠀⠀⠀⣿⡏⠉⠛⠛⠿⠿⢷⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⡾⠿⠛⠁⠀⠀⠈⠙⢿⡦
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣀⣠⣿⠟⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠉⠉⢻⣿⠻⢿⣯⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣾⠿⠿⣷⣶⣄⡈⢿⣷⣄⣀⠀⢀⣠⣾⡿⠁⠀⠀⠀⠘⣿⡄⠀⢉⣿⣿⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠁⠀⠀⠀⠀⠈⠻⣿⣄⠉⠛⠛⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⣿⣇⣰⣿⠋⠀⠉⠉⠛⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠁⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣀⠀⠀⢀⣠⣾⡟
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠉⠉⠛⠻⠿⠿⣷⣶⣶⣶⣤⣤⣤⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣶⣶⣶⡾⠿⢛⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣶⣶⣤⣤⣤⣤⣶⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠟⠋⠙⠻⣦⣀⣤⣶⣾⠿⠿⢿⣷⡀
⠀⠀⠀⠀⠀⢀⣾⡟⠉⠉⠉⠛⠛⠿⣷⣦⣶⣶⣶⣶⣶⣿⣿⣿⡟⠁⠀⠀⠀⠀⠘⣿⡛⠁⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢠⡿⠀⠀⢀⣴⠶⢶⣤⣾⣷⣤⣄⠀⣀⣀⣸⣿⡀
⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⣿⣠⡿⠁⠀⠀⠀⠙⣿⡋⠉⠉⠉⠛⣷
⠀⠀⠀⠀⠀⢸⣿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠈⢻⣧⠀⠀⠀⠀⢀⣿⠻⣷⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠟⠛⠛⠙⢷⣤⣤⣴⡾⢿⣶⠟⠀⠀⣼⠏
⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⣠⣾⣏
⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠘⣿⡄
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣧⣤⣤⣤
⠶⠾⠿⠿⣿⡿⠿⠖⠀⠀⠀⠀⣤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠈⠉⠉⣿⡏
⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠃⠀⠀⠀⠀⢠⣤⣿⣧⣤⣀⡀
⠀⠀⣤⡶⠾⢿⣟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠉⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠈⢻⣧⣤⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣾⣿⣥⣀
⠀⠀⠀⢀⣤⡶⠟⠛⠻⣷⣶⠟⠛⢷⣄⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠀⠉⠛⠷
⠀⠀⠀⠈⠉⠀⠀⢀⣤⠾⠇⠀⠀⠀⠋⠉⠉⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣶⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⣾⡃⠀⠀⢠⡎⢳⡄⠀⣤⡿⣿⡿⠿⠿⠿⠿⠿⠿⠻⣿⠛⠛⠻⣿⡿⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢈⣿⣶⠆⠀⠉⠉⠀⠀⠈⣷⠸⣷⡀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⢹⣷⣀⣽⡿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠋⢿⣄⣀⣠⣾⣦⣤⡾⠏⠀⠈⠻⢷⣶⣶⡶⠿⠋⠀⠀⠀⠀⠀⢻⣿⠋⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠉⠛⠛⣿⣆⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⠀⣿⡿⣧⡈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣶⠿⠛⠀⠘⠻⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⢿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣀⣀⠀⠀⠀⠀⠀⢀⣸⣧⡀⠀⠀⠀⠀⠀⢀⣀⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⠟⠛⠛⠻⠿⠿⠿⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠿⠿⢿⣶⣦⣤⣀⠀⣠⣿⠟⠉⠀⠉⠻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠉⠛⠿⣿⡿⠃⠀⠀⠀⠀⠀⠘⢿⣷⣶⣶⣶⣶⣦⣤⣤⣀⣀⣴⣾⡿⠟⠛⠉⠉⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣸⣿⠁⢀⣠⣀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠉⠉⠉⠛⠛⠟⠋⠁⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣶⣶⣶⣤⣤⣾⠿⠛⠻⢿⣿⡟⢻⣷⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣾⡿⠉⠀⠀⠀⠈⣹⣿⠃⠀⠀⠀⠀⢹⣿⣿⠟⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⢰⣿⠿⣿⡀⠀⠀⠀⠀⣰⣿⣅⣀⣀⠀⣀⣀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠘⢿⣶⣿⣿⣶⣤⣤⣾⡿⠛⠛⠻⠿⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣦⡀⠀⠀⠈⠁⣠⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠉⠙⢿⣶⣤⣤⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⢀⣀⣤⣤⣤⣤⣶⣿⣧⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠈⠛⠋⠉⠉⠉⠹⣿⡏⠉⠉⠀⠀⠀⠀⠀⣰⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⡄⠀⠀⠀⠀⢴⣶⣶⣿⣿⣶⣶⣶⣦⣤
⠀⠀⠀⠀⠀⢀⣀⣿⣧⣤⣤⡀⠀⠀⠀⢰⣿⡇⢈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⣿⣗⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠉⠉
⠀⠀⠀⣶⡿⠿⠟⢿⣿⠉⠉⠁⠀⠀⠀⠀⠻⣷⣾⠟⠁⠀⠀⠀⠀⠀⢠⣴⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠙⢿⣶⡿⠃⠀⠀⠀⢠⣤⣤⣤⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⣀⣤⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣀⣀⣽⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⣿⠟⠛⠻⠿⠆
⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣦⣴⣿⠋
⠀⠀⠀⠀⢠⣶⡿⠛⠉⠈⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡿⠟⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠛⢿⣷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⡿⠟⠋⠀⠀⠀⠈⠙⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⡿⠿⢿⣿⣷⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣿⣿⣿⠻⠿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠋⠁⠀⣰⣿⠏⠀⠀⠘⣿⣯⠉⠉⠉⠉⠉⠉⠉⠉⣡⣿⠏⠀⠀⠙⣿⣦⠀⠀⠉⠻⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⣿⣤⣀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠈⠻⣷⣦⣄⣀⣀⣀⣤⣾⡿⠋⠀⠀⠀⠀⠈⢿⣧⠀⠀⢀⣤⣿⡿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠈⠙⠿⣷⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣾⡿⠛⠁⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⢈⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣄⣀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣀⣀⣀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣶⣶⣶⣦⣤⣶⣤⣶⣾⠟⠛⢿⣶⣦⣴⣤⣦⣶⣶⣶⣶⣾⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠿⣿⣶⣄⠀⠀⠀⠀⣀⣀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣿⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠈⢿⣷⣴⣾⡿⠿⠛⠋⠉⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠈⠉⠛⠿⣷⣶⣶⣿⡿⠿⠿⠿⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⢀⣴⠟⠻⣶⣼⣿⣷⣦⣄⠀⢀⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠘⣿⣀⣾⠟⠁⠀⠀⠈⠻⣿⡟⠛⠛⠛⠻⢿⣦
⠀⠀⠀⠀⠀⠀⢻⣿⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠈⣻⣿⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢶⣶⡶⠿⠛⢿⣆⡀⠀⠀⢀⣴⡿⢉⣿⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣾⡿⠛⠛⠛⠁⠀⠀⣠⣾⡏
⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣀⣀⣀⣠⣶⠿⢿⣧
⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁⠀⠘⣿⡆
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣀⣀⣀⣀
⢀⣀⣀⣤⣤⣿⣧⣤⣀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠰⠿⠛⢿⣿⠛⠛⠛⠛
⠛⠛⠛⠉⠉⣿⣏⠉⠉⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⢸⣿⣄⣀⡀⠀⠀⠀⠀⠘⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠹⢿⠿⠁⠀⠀⠀⠀⠀⠲⢿⣿⣷⣶⣦⣤⡀
⠀⠀⠀⣤⡶⠾⣿⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⠉⠉⢹⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⣠⣾⡿⢿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠉⠛⠿⢷⡦
⠀⠀⠀⠀⠰⠿⠋⠁⠀⠀⠉⠛⢿⣷⣦⣄⣀⣠⣤⣴⣶⡿⠋⠉⠀⠀⣹⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⡿⠿⠛⣻⡿⠁⠀⠀⠀⠀⠘⢿⣧⣤⣤⣤⣤⣤⣴⣶⣶⣾⡿⢿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⠉⢉⣿⠋⠁⠀⠹⣿⡄⠀⠉⠛⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⢀⣿⣇⣀⣴⠿⠃⠀⠀⠀⠀⠘⣿⣆⠀⠀⣠⣿⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣀⠀⠀⠈⣻⣷⣄⣀⣀⣀⣤⣾⠟⠛⠋⠁⠀⠀⣠⡶⠶⣦⡀⠘⣿⣶⡿⠟⠁⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠛⢿⣿⠿⢿⡿⠟⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⣿⡀⠀⣸⡇⠀⠸⣿⡄⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⣀⣀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⢿⣶⣶⣶⣶⣶⡿⠿⠟⠋⠙⠻⠿⣷⣶⣶⣶⣶⣾⡿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠿⠿⣷⣦⣀⣤⣾⣿⡿⠛⠉⠉⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠈⠙⢿⣿⠛⠁⠀⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣄⣀⡀⠀⠀⢀⣀⣤⣤⣶⣶⣿⣿⣿⡿⠀⠀⠀⠀⣀⣀⣀⣈⣿⣦⣤⣄⡀⠀⣀⣠⣤⣿⣿⣤⣄
⠀⠀⠀⠀⠀⣾⣿⠟⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉⠉⣿⡇⠀⠀⠀⣾⡿⠛⣿⡿⠛⠉⠉⠛⢿⣿⡟⠋⠉⠉⠉⠉⣿⣇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⣿⣧⣼⣿⠀⠀⠀⠀⠀⠀⢻⣿⣶⣦⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠈⠙⢻⣿⡀⠀⠀⠀⠀⢀⣾⣏⣨⣿⠇⠀⠀⣾⡏
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣤⣤⣤⣶⡿⠿⢿⣶⣤⣤⣴⣿⡟⠛⠛⠋⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠈⢿⣿⣴⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠁⠙⢿⣦⣄⣀⣤⣾⠿⣿⣧
⠀⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⢻⣿⡆⠀⠀⢀⣀
⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⡿⠿⠿⠛⠛
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⢿⣿⡀
⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⠀⠀⠀⠀⠀⢀⣀⣼⣿⣧⣤⣤⣤⡀
⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠀⠀⠀⠀⠀⠈⠛⣿⣿⠛⠉⠉⠉
⠀⢀⣀⣤⣴⣿⣿⣶⣶⡶⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣀⣰⣿⡟
⠸⠿⠛⠛⠉⠹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣼⡟⠋⠛⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢹⣿⣦⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⣴⣾⠿⠿⣿⣯⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⢿⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⡟⠀⠀⠀⠀⠉⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠻⢿⣿⣷⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣶⣶⣿⣿⠿⠛⠉⠹⣿⣷⡀⠀⠀⠀⠀⠀⣹⣿
⠀⠀⠀⠀⠀⠀⠰⠿⠋⠀⠀⠀⠀⣈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⢿⣿⠋⠉⠹⣿⣧⠀⠀⠀⠙⣿⣷⡄⠀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⣿⡿⠛⠉⣡⣿⡟⠁⠀⢸⣿⡅⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠙⣿⣷⡀⠀⠀⠈⢿⣿⣆⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢀⣾⡿⠟⠛⠛⢿⣿⣆⣰⣿⣟⣀⣀⣀⠀⠻⣿⣦⣀⣀⣀⣀⣀⣤⣾⡿⠁⠀⢀⣀⣀⣈⣿⣷⣄⢀⣀⣤⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⣹⣿⣿⡿⠿⠿⠿⢿⣿⣶⣄⠙⠛⠿⠿⠿⠟⠛⠁⢀⣴⣾⡿⠿⠿⠿⠿⣿⣿⣿⡟⠛⠉
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢀⣼⣿⠟⠁⠀⠀⠀⠀⠀⠈⠻⣿⣧⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀⢀⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⣸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠇⠀⠀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣶⣶⣾⣿⣷⣶⣾⣿⣿⣦⣀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣶⣶⣶⣶⣿⠿⠟⠋⠉⠉⠀⠀⠀⠀⠉⠉⠉⠛⠻⢿⣿⣶⣶⣾⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠀⠈⠉⠛⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠿⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣤⣤⣤⣤⣤⣄⣀⣀⣿⡇⠀⠀⠀⠹⣿⡄⠀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⢙⣛⣻⣿⡇⠀⠀⠀⠀⢻⣿⠿⠛⠛⠙⠻⣿⡿⠿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠿⠛⠛⠛⠃⠀⢀⣠⣤⡈⠁⠀⠀⠀⠀⠀⣽⡗⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢠⣿⠉⠉⣿⡄⠀⠀⠀⣠⣾⠟⠁⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣀⡀⠀⠀⠀⠀⠻⠷⡾⠟⠀⠀⠀⠀⠻⣷⡄⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣷⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠹⣿⡀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣰⡿⢿⣦⣄⡀⢀⣰⣿⠃⠀⣿⡇
⠠⣶⣶⣤⣤⣤⣿⣏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣀⣠⣾⠟⠁⠀⠉⠛⠿⠿⠛⠁⠀⢸⣿⠁
⠀⠀⠈⠉⠉⠙⣿⡟⠛⠿⠂⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠛⠟⠛⠛⢿⣿⠛⠃⠀⠀⠀⠀⠹⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣷⡄⠀⠀⠀⠀⠐⠿⠿⢿⣿⣷⣶⣶⣤⣤
⠀⠀⠀⠀⣤⣶⠿⠿⣿⠟⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡋⠉⠻⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠉⠉
⠀⠀⠀⣾⡟⠁⠀⠀⠀⠀⠀⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⠿⠛⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠘⠿⣿⣿⣧⣄⡀
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣠⣾⠿⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣼⡟⠁⠉⠛⠿⠷
⠀⠀⠀⠻⣷⡀⠀⠀⣠⣾⡿⠃⠀⠀⣹⣿⢿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⣄
⠀⠀⠀⠀⠙⢿⣦⣴⡿⠋⠀⠀⢀⣼⡿⠁⠀⠀⣹⣿⠛⠿⠿⣶⣶⣤⣤⣤⣀⣀⣀⣀⡀⢀⣀⣀⣀⣠⣤⣶⡿⠟⠉⠀⠉⠻⢿⣦
⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⣠⣿⠏⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠈⠉⢹⣿⠛⠛⠛⣿⡿⢿⣿⣟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣧⣶⣶⣶⣤⣄⠈⢿⣧⣀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⢹⣿⠀⠉⢻⣿⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠛⠉⠀⠀⠀⠉⠻⣷⣆⠙⠻⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠘⣿⣆⣴⡿⠋⠉⠛⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠁⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⡀⠀⠀⢀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠻⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣧⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⢀⣠⣾⣿⠁⠉⠙⠛⠿⠿⠿⣶⣶⣶⣶⣦⣤⣤⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣶⡿⠿⠛⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢩⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢷⣶⣶⣶⣶⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⠀⣴⠟⠛⠳⣦⣀⣀⣀⣀⣠⠖⠒⢦⣠⣤⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠋⢹⡆⢸⡇⠀⠀⠀⠈⠉⠉⠉⣽⠃⣠⢶⣤⣿⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢧⢸⡇⠀⠀⠀⠀⠀⠀⠀⢿⣀⢘⣿⠁⠀⣻⣿⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⣾⠓⠚⢀⡼⠁
⠀⠀⠀⠀⢀⣴⠛⠳⣄⣀⠈⢻⣿⣿⠷⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⠒⠛⣷
⠀⠀⠀⠀⠀⠙⠦⣄⡀⠈⠉⠉⣿⣿⡷⠀⠀⠸⠿⠋⠃⠀⠀⠀⠀⠀⠠⢞⣭⡀⠀⠴⣿⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⡶⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠾⡳⠀⠀⠀⠘⠛⠁⠀⣶⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣾⣿⣿⡟⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⣋⣽⣿⣿⣳⠃⠈⢹⡿⠶⢦⣤⣤⣤⣤⣴⣶⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⢸⡇⣿⣏⣀⣰⠏⢠⣾⣿⣿⣆⠀⢾⠀⠀⢉⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠻⡆⢹⡏⠀⠸⢷⣿⡷⠇⠀⠈⢣⡔⠉⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣿⣾⠀⠀⠀⢈⠛⠁⠀⠀⠀⠸⣄⣄⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡿⠒⠒⠒⡿⠒⠒⠒⠒⠒⠒⠺⡇⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠚⢻⣅⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠶⠦⣤⣀⠀⣠⠖⠛⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠉⢻⠏⠀⠀⠀⠈⣧⣤⣤⣤⣤⣤⣤⡞⠉⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⡴⠋⠉⠳⣏⣳⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠀⠀⡼⣇⠀⠀⠀⣿⣅⣀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡄⠀⠓⠋⢳⠶⠚⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠙⢦⣀⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠒⠒⣿⠒⠲⠄⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣤⣤⣀⡀⠀⠸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠀⠀⠀⠒⠒⣿⠒⠦⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢩⡿⠓⠶⢬⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢰⡉⢑⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⢦⡇⠀⠀⠀⠀⣀⣀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣰⠟⠙⠒
⠀⠀⠀⢀⡤⢤⡄⠀⠀⠀⣤⢤⣄⠀⠀⠀⠘⢷⡀⠀⠀⠀⢾⠶⣦⠤⣤⣏⣉⣉⣙⠳⣄⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⠾⠛⠧⡀
⠀⠀⠀⢸⡇⠀⠻⠖⠒⠚⠃⠀⣿⠀⠀⠀⠀⠀⠙⠶⣤⣀⣼⠀⠀⠀⢸⡇⠀⠀⠉⢳⡿⢻⡟⠛⢻⣿⠉⠉⠉⠁
⠀⠀⣰⠟⠀⢀⣠⣤⣤⣄⡀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⣿⠉⠑⠢⣄⣾⠀⠀⣀⣴⠏⢀⡞⠉⠙⣆⢻⡄
⠀⣴⠃⠀⡴⢩⣦⠀⣶⡆⢉⣳⡄⠈⢷⡄⠀⠀⠀⠀⠀⢻⣄⣀⡴⠾⠿⣭⣿⢯⣴⡞⢻⣃⣀⣀⣸⠀⢻⡄
⣸⠇⠀⢸⢡⣤⠀⣴⠲⡄⠘⠋⢻⠀⠈⢿⡀⠀⠀⣀⣀⠀⣸⠃⠀⠀⣠⣾⡇⢯⡀⣹⡞⠁⠀⠀⢸⠁⠀⢻⡄
⣿⠀⠀⢸⣦⣅⠀⠘⠚⠁⠸⠗⡼⠀⠀⠸⡇⠀⠸⣏⣉⣷⣿⣤⡴⠞⠁⣸⠁⠀⠉⠉⠓⢦⡀⢀⡾⠀⠀⢸⣧⣤⣄⡀
⢿⡀⠀⠘⠳⠾⠦⣤⣤⡤⠴⠊⠀⠀⠀⣰⣧⣀⣀⣨⡿⠋⢻⣇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⣠⡟⠁⠀⠈⢻⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠳⣆⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠋⠁⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢦⣀⣀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣤⣤⣴⠟⠹⣷⡀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⣼⣯⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠶⠶⣶⣤⣀⠀⠀⠀⠀⢻⣆⣀⣀⣸⡿⠋⠀⠀⠈⠹⣧⣀⣤⣶⠿⠟⠛⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠈⠙⠷⣶⡶⠿⠿⠿⠛⢻⡟⠀⠀⠀⢀⣀⡀⠸⣿⠉⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢰⡟⠉⣹⠶⠛⠓⢶⣄⣀⣤⣤⣤⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠈⠳⣾⠃⠀⠀⠀⠀⢹⣏⠉⠉⠈⠉⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣤⡴⢿⣄⠀⠀⠀⢀⡾⢛⣷⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⢶⡟⠛⠛⠁⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣤⡾⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢾⡷⠶⠶⠶⣤
⠀⠀⠀⠀⠀⠀⣠⡶⠾⠛⢻⣿⠙⠋⠀⠀⠀⢀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡆⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⣿⣄⣠⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⢀⡤⠤⣄⠀⠀⠀⠀⠀⠈⠻⠿⠁⠀⠀⠀⠀⠐⢲⣿⠷⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠟⠛⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣾⠇
⠀⠀⠀⠀⠀⣠⣤⣤⣀⣀⣤⣤⣈⣿⣶⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⠿⠷⣦⣄
⠀⠀⠀⠀⢸⡏⠀⠀⠙⠉⠀⠈⢻⡇⠉⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠟⠁⠀⠀⠀⠉⠁
⠀⠀⠀⠀⣸⠷⠀⢀⡤⢤⡀⠀⢼⣅⠀⠀⠀⠈⠉⠛⠻⠷⢶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡶⠶⠿⠿⢿⣍⡀
⠀⠀⠀⣾⠁⠀⠀⢿⣀⣠⠟⠀⠀⢹⡇⠀⠀⠀⠀⣀⣴⣶⣿⣟⣥⣄⠀⠀⠀⠀⠀⣿⡇⣰⣿⣷⣦⠀⢀⣴⣿⣿⣦
⠀⠀⠀⠹⢷⣤⣄⠀⠉⠁⢀⣀⣠⣾⠃⣀⣤⣴⠞⠋⢠⡟⠱⣿⣿⣿⠇⢀⣤⣤⡀⣿⣿⡹⣿⣿⠟⠀⠈⢿⣿⡿⠻⣷
⠀⠀⠀⠀⠀⠀⠹⣆⣀⣀⡼⠃⠀⠙⣿⡏⠀⠀⠀⢠⡿⠀⠀⠙⠛⠋⠀⢿⣿⣿⣿⡟⠘⢷⡀⠀⢀⣀⡀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⢸⡗⠀⠀⠀⢸⣿⣿⣷⡄⠀⠀⠀⠀⠙⠋⣿⠁⠀⠈⢿⣴⣿⣿⣿⠄⠀⣴⣶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⢸⡿⠿⠿⠁⢀⣀⣀⠀⠀⣼⠇⠀⠀⠀⠀⠙⢿⣿⠋⠀⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⡶⠿⣇⠀⠀⠀⣿⣿⣿⣧⡾⠃⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣀⣀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢻⡄⠀⠀⠙⣻⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⢠⡿⣦⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠈⠙⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⠀⠀⠀⠀⠈⠉⠛⠛⠲⠶⠶⠶⢶⡾⠛⠋⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠶⠶⣦⣤⣤⣤⣤⣤⠶⠶⠿⠷⠶⠶⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⠀⠀⢀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠉⠀⠀⠀⠙⢷⡾⠋⠁⠀⠀⠈⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣧⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣇⢀⣤⣤⣤⣴⡶⠶⠶⠶⠶⠶⠶⠶⠶⣿⠀⢀⣠⣤⣴⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠻⠿⢷⣶⣼⣿⠏⠁⠀⠈⠻⣦⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⠿⠛⠉⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠈⣽⠇⢀⣀⡀⠀⠀⠘⣿⡉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣿⣥⣤⣀⣠⡴⠟⠛⠻⣿⡉⢻⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣼⠋⠉⠉⢉⣩⣿⠀⠀⠀⠀⠈⣿⠾⠃⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠁
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⣿⠋⣿⣄⠀⠀⢀⣴⠿⢦⣤⣤⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠙⠛⠛⣹⡟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⢸⡿⢦⣤⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣧⣄⣀⣀⡀
⠀⠀⠀⠶⠶⠶⠶⣿⡷⠶⠆⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠈⠉⠉⣿⡏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡿⠇⠀⠀⠀⢀⣤⣴⣿⣀⡀⠀⠀⠀⢀⣀⣤⣄⣀
⠀⠀⠀⠀⠤⠶⠶⢿⣿⠛⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣴⠋⠙⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠉⠛⠃⣠⡾⠛⠉⠉⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢳⣶⣿⣇⡀⠀⠀⣴⠟⠀⠀⠀⠀⢀⡄⢸⡇
⠀⠀⠀⠀⠀⢀⣤⡶⠟⠻⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋⠀⠉⠛⠷⠄⣿⠀⠀⠀⢀⣠⡿⢁⡿⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠙⠿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣾⠟⠋⠁⠀⠀⠀⠀⠀⢀⣤⡿⣀⡘⠟⢛⣉⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣨⡿⣿⡿⠛⠿⣿⡿⠿⠿⠿⠿⠟⠛⠛⣿⠛⠉⠹⣿⠛⠷⣦⣀⣀⡀⠀⢀⣴⠟⣡⡾⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⡿⠻⢿⣏⣠⡿⠁⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣠⣿⠀⠀⠀⠹⣷⣠⡾⠛⠉⠛⢿⣟⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⢙⣿⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢹⣿⠇⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⢘⣷⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⢀⣤⣶⣤⣤⡶⢶⣦⡀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⢀⣿⠁⠀⠈⠉⠀⠀⠈⣿⠀⠀⠀⠀⢿⣄⡀⠀⣀⣴⡟⠁⠀⠀⠀⢰⡿⠿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠉⠛⢿⡟⠁⠀⠀⠀⣠⣶⠿⠿⣶⠿⠻⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⣄⣠⣼⡟⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⢀⣾⠋⠀⢀⣤⣤⡀⢠⣶⢦⣽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣽⡇⠀⢸⣿⠶⠾⠛⠉⢉⡛⡛⢁⡀⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣶⣶⣤⣤⣤⣤⣤⣤⣤⣀⣤⣤⣤⣤⣤⣤⣤⣶⡶⠶⠾⠟⠛⢛⣿⠏⠀⠀⠀⠛⢿⡿⠻⠟⠟⠛⠛⠛⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠈⠉⠉⠉⠉⠉⠉⠉⣹⣏⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⠋⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢶⣶⣶⣤⣶⣶⣶⡾⠿⠿⠿⠶⠶⠾⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣤⣤⣤⣤⣤⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠑⠒⠒⠲⠦⠤⠤⠤⠤⠤⠒⠒⠒⠒⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⢀⣀⢠⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣤⡀⠴⣾⣾⣷⣿⣏⣿⠝⣸⣿⣿⣿⣿⣇⣤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⡊⠙⠚⣿⣿⣧⡌⣿⣿⠛⠉⠙⢸⣿⡸⢹⠟⡻⢿⡏⠾⠏⠉⠻⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣴⣿⡿⢠⠁⠀⠀⠀⠈⠛⢣⢉⠁⠀⠀⠀⢨⣿⣿⣿⣷⣴⣾⠃⢌⢂⡀⠀⢻⣿⣟⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣰⣶⡆⢛⡓⣈⣄⢉⠀⠂⢀⡁⢈⡄⠳⠆⣤⡀⣛⠉⠉⢻⣿⡯⢰⣿⠆⠈⠀⢠⣾⣿⡟⠀⢉⣦⣤
⠀⠀⠀⠀⠀⠀⠀⣴⠖⠢⣥⣠⣤⣽⣿⣿⣿⣌⣿⣿⡜⣿⠖⣾⣿⣿⣶⣶⣿⢀⣀⣨⣥⣀⠀⣀⣬⣭⣄⡛⢻⠟⠁⠀⠘⢿⣿⣑⡆
⠀⠀⠀⠀⠀⢠⡖⠁⠀⢀⣴⢻⣿⣿⣿⣿⡟⢋⠘⣡⣦⣾⣾⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠂⠃⠀⠀⠈⣿⠟⠀⠐⡀
⠀⠀⠀⠀⢠⣶⢹⡶⣷⣛⠁⣠⠟⠁⣀⣉⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⠿⢿⡇⣿⣿⣿⠿⣿⣿⣿⡛⢡⣶⡍⠇⠀⠀⠀⠀⠀⠁⠀⠀⢀⣴⣤
⠀⠀⠀⢀⣸⡁⣿⣹⣿⠿⣿⠃⢠⡾⠉⠉⠛⢶⣄⣀⣀⣁⠈⠉⠉⣠⣤⣤⡀⠙⠛⠛⠉⠉⠲⣭⠛⠈⢿⣿⣤⣶⣦⠀⠥⠀⠀⠀⠀⠈⣿⣏⠠
⠀⠀⢰⣿⠁⠀⣭⣭⡳⣭⣇⠀⣾⠁⠀⠀⠀⠀⠈⠉⠉⠙⠛⢻⣿⣿⣿⣿⣿⡶⠾⠛⠛⢷⡄⢰⣶⣶⣦⢻⣿⣿⣿⡄⠂⠀⠀⠀⠀⢀⠻⠛
⠀⠀⠀⠻⣤⣾⣿⣿⣿⣶⡯⠀⣿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⢸⡇⠸⣿⣿⣿⠘⠛⠻⣿⣿⠤⠀⠄⠉⠈⠉⠀⠁
⠀⠀⠀⠀⠉⠻⢿⠟⣁⠉⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢘⣠⣿⣦⡰⣤⣵⠋⠠⠀⢀⣁⠀⣄⡀⠡⢾⣷⡀
⠀⠀⠀⠀⠀⡠⠒⠋⠁⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⡿⠁⠘⢛⢿⣿⡇⠀⠀⠀⠀⠀⠈⠉⠚⠛⠃⠀⢿⣿⡇
⠀⢀⣤⡄⠠⠃⠀⠈⠉⠙⣿⣳⣂⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡿⣿⠁⠰⠀⢘⣠⠟⠁⠀⡄⠀⠀⠀⢤⠀⠀⠀⠀⠀⠛⠉
⠀⢈⠙⠁⡇⠀⠀⢀⡤⣴⡏⠁⢹⡆⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⣿⣀⠈⠉⢍⠀⠀⠄⠲⠃⠀⠀⢀⡈⠝⠀⠀⠀⠀⡔⠒⡄
⠰⣿⣿⠀⠣⠀⠀⠋⠀⠀⠁⠀⠀⠷⣤⣀⠀⠀⠀⠀⣴⣶⣦⠀⠀⠀⠸⠿⠃⠀⠀⠈⢩⡟⠉⠉⢁⣸⣧⣀⠀⠌⠀⠀⠀⠀⠑⠢⠀⡀⠀⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⠙⣧⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠛⠲⠀⢸⣿⣿⣿⣿⡆⡠⠨⠁⠀⢀⣴⣿⣿⣇⣛⡻⢾⡇
⢠⡄⠀⠀⠀⠀⠀⢀⣀⣀⡄⠀⠻⣤⡤⠆⣽⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡽⠿⢧⣀⠀⠀⠉⠻⣌⢩⠉⠀⠀⠐⢠⣾⣿⣿⣿⣿⣿⣿⣿⠐
⣼⣷⡄⠀⢀⣴⣾⣿⣿⣿⢧⣄⣀⣀⣠⠾⠋⢀⣀⠉⣉⠙⠛⠛⠛⠚⠛⠛⠋⡁⠀⠀⠀⢁⣀⣀⣀⣀⣈⣋⡀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡏
⣿⣿⣿⣷⣾⣿⣿⣿⣿⣏⠤⠄⠉⠉⠀⠀⠐⠈⢢⠀⠎⢱⠄⠠⠀⡐⠪⠴⣶⣿⠖⠒⠦⡹⣿⢯⠉⠀⠸⢻⢧⣿⣿⣷⣶⣝⢿⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⢀⣹⣿⣿⠃⣾⣿⡷⡀⠑⠒⠁⡄⠀⢼⠁⠀⣤⣼⣇⠴⡄⠀⢱⠙⠶⠶⠑⠖⠀⢰⣿⣿⣿⣿⣿⣦⢿⠿⣿⡿⠁
⣿⣿⣿⣿⣿⠟⢿⣇⠀⠀⠈⠻⠈⣡⣬⣼⡿⣣⡄⠀⠾⠆⡴⣀⢸⣄⣀⣁⣴⣷⣤⣤⣴⣯⠈⢿⣽⠿⡿⢰⣿⣫⣿⣿⣿⣿⣿⣀⡄⠈⠁
⣿⣿⣿⠟⠁⠀⠀⠀⠀⢀⠀⠀⡀⢿⠛⠋⢰⣿⣿⣶⣶⣾⣇⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠐⣠⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⠛⠁⠀⠀⠀⠀⠀⠀⠻⠷⠛⢀⠛⢷⡦⣸⣿⣿⣿⡋⣽⣿⡗⣤⣻⠿⠿⠿⠿⠿⢿⡛⢁⠀⡀⠀⢰⠘⣿⣿⣿⣿⣿⣿⡿⠃
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣏⠠⣀⣀⣀⣴⣿⣦⣈⠉⢻⣿⠛⠉
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣏⣶⣿⣿⢏⣩⣿⣏⣙⢿⣿⣷⣌⢿⡿⠟⠛⣇⣈⡧
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⢿⣏⠀⢸⣿⣿⣿⡷⠀⢈⡿⠃⠤⠷⠒⠊⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⢠⣤⣀⣠⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⢷⠀⣤⠿⡤⠚⢧⡄⢹⣄⣀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⡆⢻⣄⣓⡤⠞⠶⠞⠀⠈⠀⠀⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠳⣤⣈⣿⣾⣷⣏⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣉⣻⣿⣿⡇⠀⠴⠀⠀⠀⠀⠀⣠⠀⠀⣸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣄⡀⠀⠀⠓⠃⠀⠈⠀⣠⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⡇⡼⠉⣻⠿⠿⠿⠷⣶⢶⣾⡋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡇⢿⣷⣶⠏⠀⠀⠀⠀⢻⣤⠴⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⡏⠀⢀⡀⠀⠀⠀⢷⣤⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⣿⠛⠛⣿⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠶⠦⠿⠦⠤⠤⠴⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣤⣄⡀⠀⠀⠀⣀⣀⣀⣠⡿⠋⠁⠈⠙⢷⣤⣶⠾⠛⠛⠛⣷⡀
⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠈⠉⠛⠿⠟⠛⠛⠛⢻⡿⠀⠀⠀⣠⣤⣈⣿⣀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⣇⣼⠟⠉⠉⠻⣶⠾⠿⠿⣿⡄
⠀⠀⠀⠀⠀⢸⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⣀⣀⣨⣿⠀⠀⠀⠀⣿⢦⡀⠀⠈⣿
⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠙⠷⣦⣴⡾⠧⠾⠁⠀⣰⡟
⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⣀⣤⠾⢿⣇
⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⣿⡀
⠀⢀⣀⣤⣼⣿⣤⣤⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠲⠛⣿⠟⠛⠛⠃
⠀⠈⠉⠁⠘⣿⡀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣀⣠⣿⣄⣀⡀
⠀⠀⠀⢠⣴⢿⣿⠛⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⣴⣿⣷⡆⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣽⡏⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢹⣷⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⠿⣦⣄
⠀⠀⠀⠀⢤⠾⠋⠉⠛⢷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠶⣶⣶⣦⣤⣤⣤⣤⣴⣶⡶⠾⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠿⣷⣦⣄⢀⣴⣾⣿⡿⠛⠋⠙⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠙⢻⣿⡿⠋⠁⠀⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣶⣿⣿⠃⠀⠀⠀⠀⣀⣀⠀⢹⣿⣄⣀⣀⠀⠀⢀⣀⣬⣿⣿⣤⣀
⠀⠀⠀⠀⠀⣰⣿⡿⠿⠿⠿⠿⠿⣿⣿⣿⣷⣾⣿⣿⡿⠿⠟⠛⠛⠉⣿⡟⠀⠀⠀⣰⣿⠿⢿⣿⡿⠛⠛⠛⠻⣷⣾⡿⠛⠛⠉⠉⠙⣿⣧
⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢿⣯⣀⣿⡏⠀⠀⠀⠀⠀⠈⣿⣷⣦⣄⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠛⠿⣿⡇⠀⠀⠀⠀⠀⠀⣿⡏⢹⣿⡆⠀⠀⣸⣿
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣀⣀⣀⣠⣴⣿⣿⣦⣀⣀⣀⣠⣾⡿⢿⡿⠟⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠈⣿⣿⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠉⠁⠀⠉⠛⠛⠛⠛⢿⣦⣀⠀⠀⢀⣠⣿⣿⣆
⠀⠀⠀⠀⠀⠘⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛⠉⢿⣿⡆
⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣦⣶⣶⣶⣶
⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡿⠿⢿⣿⡏⠉
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠠⣶⣿⣿⣿⡿⠿⠿⠗
⠀⠀⠀⠀⠀⣿⣿⣄⣀⣀⡀⠀⠀⠀⠀⢀⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⢀⣤⣶⣾⡿⢿⣿⣿⠛⠛⠃⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣿⣿⣀⣀
⠈⠉⠉⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣽⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⡿⠛⠻⠿⢿⣷
⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣏
⠀⠀⠀⠀⠾⠿⠛⠉⠻⣿⣦⣀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠉⠻⠿⢿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⡋⠀⠀⠀⠀⠀⠈⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠉⠛⠿⣿⣿⣷⣶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣶⣶⣾⣿⣿⣿⣟⠋⠉⠀⠙⣿⣿⡄⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠟⠋⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⡿⠿⣿⣿⡿⠿⠿⠿⠿⠿⠟⠛⠛⢛⣿⡏⠀⠀⠻⣿⣦⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣿⣿⡛⠉⢀⣼⣿⠏⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠙⣿⣷⡀⠀⠀⠈⢻⣿⣧⣠⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠛⠛⠙⣿⣿⣤⣾⣿⣧⣤⣤⣀⡀⠙⢿⣷⣤⣀⣀⣀⣠⣤⣾⡿⠋⠀⢀⣀⣤⣤⣼⣿⣷⣄⣀⣤⣴⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢈⣿⣿⣿⠿⠿⠻⠿⠿⣿⣷⣄⠉⠛⠛⠛⠛⠛⠛⠉⢀⣤⣾⣿⠿⠟⠛⠻⠿⣿⣿⣿⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⣰⣿⡿⠋⠀⠀⠀⠀⠀⠀⠈⠹⣿⣷⡀⠀⠀⠀⠀⠀⢠⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⠀⣰⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡏⠀⠀⠀⠀⠀⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⢀⣤⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣦⣤⣤⣤⣴⣾⣿⠿⠛⠛⠋⠉⠉⠉⠉⠉⠙⠛⠛⠿⣿⣿⣶⣦⣤⣶⣾⣿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣠⠠⠠⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠆⠰⠆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⣀⠀⢀⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠈⠐⠉⠀⠈⠋⡀
⠀⠀⠐⣧⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠀⠀⠀⠁⠁⠒⠦⠴⢤⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠐⢿⣾⣿⣿⣿⣿⣿⣿⣾⣿⡶⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢰⣤⢬⣉⡅⡉⠽⠉⠿⠯⠶⠷⠳⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠘⠻⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠤⠽⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡄⠀⠀⠀⡀
⠀⠀⠀⠀⢀⠄⠀⠀⢤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢄⠀⠁⡀⢠
⠀⠀⠀⠐⢪⣷⣻⠯⡑⠀⠳⠟⣓⡃⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠈⠹⠧⠃⡁⠤⠚⡤
⠀⠀⠀⠀⢸⡋⠚⠧⢵⠒⢲⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡜⣒⠙⠉⣀⢐⠸⡆⣀⡚⡀
⠀⠀⠀⠀⠀⣷⠋⠠⢈⣵⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⡟⠻⡶⢄⠻⣯⠑⣾⡟⢘⡟⣾
⠀⠀⠀⠀⠀⠘⣆⠀⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⢛⢯⠟⣾⣠⣿⣯⣽⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠚⠁⠚⠾⠉⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡬⠥⠴⠶⠶⠶⠶⠶⠶⠶⠶⠤⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠉⢀⣴⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢫⡀⣀⣴⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣀⣀⡸⠣⠴⠒⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠉⠛⠻⠯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠞⣛⠩⠀⠀⣀⣀⣄⣀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠭⠭⠅⠐⠒⠂⠉⠁⠀⠀⠀⢸⠁⠀⠀⠀⠙⡆⢂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢀⡇⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⣠⣴⠟⢁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠾⠟⠛⢉⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠳⡀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⣀⣠⠤⣔⠊⠉
⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⠀⠙⢦⡙⢅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠂⢄⠀⠉⠓⢄
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡦⠤⣀⣀⣀⠀⠀⠀⢀⣀⣀⣤⢴⠖⠉⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⢳
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠢⠤⠬⠭⠭⠭⠭⠴⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒⠀⠀⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⠤⣀⣈⠁⠒⠂⠠⠤⠄⠀⠤⠠⠤⠀⠠⠤⠀⠐⠒⠈⢁⣀⣠⠤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉

⠀⢀⡀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠤⠄⡀⠀⠀⠀⢀⡀⠀⠀⠀⡀⣠⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢠⠄
⠀⠘⠀⠀⠀⠀⠀⠈⢄⠀⠰⠁⢀⠀⠉⠆⠀⠀⠌⠠⠊⠠⠀⠟⠊⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠈⠀⠡⣈⢀⡄⠐⠰⣲
⠀⠀⠐⠀⠐⠀⠀⢉⠣⠀⠀⠣⡀⠈⠀⠈⠆⠈⢠⠴⡀⠊⠀⠀⠀⠂⠀⠠⠀⠄⠀⠁⠀⠀⡠⠁⠀⠒⠄⠀⠉⠉⠀⢐
⠀⢀⠀⣋⡀⠀⠀⠀⠈⣀⢀⢀⠈⠒⠈⠄⢀⡴⠀⡯⢚⠁⠀⠀⢠⠐⠂⡐⠠⢀⢀⠀⢀⠜⠀⢣⢄⣀⡀⠀⠀⠀⠤
⠀⠘⠀⠀⡆⠀⡠⠊⡁⠀⠃⠂⢌⡑⢄⠀⠸⠋⠀⢸⡵⠀⠀⢸⠃⠀⢀⡅⡀⠘⠄⠐⠁⢀⠀⠠⣀⡀⠀⡀⠠
⠀⢠⠖⢦⠀⢄⣠⠒⡂⢜⣊⠼⠓⢊⠁⢠⠒⠒⠤⡀⠃⠀⠀⠀⠁⠨⠉⠁⠀⠀⠀⠀⠠⠂⢤⠀⠠⠀⠀⠁⠀⠈
⠀⠀⡆⣈⢂⠀⠀⠀⠀⠒⠒⠒⠈⠁⠀⡌⠀⠀⢰⠀⠰⢠⠀⠀⠂⠆⠀⢀⣴⣶⢻⣟⣳⡆⠀⡀⠀⢀⠀⢀⡰⠉⡘
⠀⠀⡄⠂⢠⠀⠀⠀⢖⠒⠀⠠⠀⢲⠆⠑⠀⠲⠈⠀⠀⠁⠃⠀⠀⠀⠀⢀⢀⠽⠉⠉⠉⠉⠀⠀⡀⠐⠀⠀⠉⠉
⠀⠀⠈⠉⠁⠔⢄⠀⡄⠉⠀⠒⠀⠁⢀⠀⣴⡢⡄⠀⠠⠀⠀⠀⢀⠖⡆⠀⠀⠀⠀⠍⠀⠁⠀⡀⠃⠀⠀⡔⠉⢆⠁⢠
⠀⠦⠀⠀⠈⠀⠀⠀⠀⢀⠀⢈⠈⠡⠏⠓⢉⠠⠣⢶⠠⢅⢤⡤⠀⠀⠀⡀⠈⣁⠀⠈⠈⠀⠂⠀⠀⠀⠨⠀⡠⠊⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠐⠀⠀⠂⠈⢀⣐⣃⡠⠤⠗⢉⠀⠀⠀⠀⠰⠀⠀⡀⠼⢑⠀⠉⡂⠲⢀⠑⢀⡱
⠀⠀⠐⠀⠘⢀⠀⠀⠀⠀⠀⠤⠀⠀⠘⠀⠈⠉⣤⠀⠀⢘⠭⠁⣀⠀⠐⣁⠀⠀⠀⠁⠀⠀⠀⠀⠀⢈⠀⠀⠀⠎
⠀⠐⠂⠀⠀⡠⠄⠀⠀⠊⢟⠀⠀⢠⠶⠛⠉⠐⢹⣶⠄⠈⡠⠀⠀⠀⠀⠈⠀⢠⠄⠀⠀⠀⠀⠀⠀⣡⠀⠀⠘⠀⠂⠄
⠀⠰⠠⠍⢀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⢈⠠⠒⠀⠀⠀⠀⠉⠉⠉⠈⠐⠡⠄⢀⣀⠀⠂
⠀⢠⠀⠀⡀⠄⠈⡆⡕⠀⠈⠀⠐⠀⠀⠠⠒⠃⠀⠀⠀⢰⠀⠀⠀⠌⠀⠀⠄⠠⠏⠀⡀⠀⢀⠠⠀⠀⠙⠇
⠀⠀⠀⡀⢀⡄⠀⢡⠁⢀⡀⠌⠀⡄⠂⠂⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡰⠒⠀⠀⠒⢅⡀⢀⠰
⠀⠠⠐⠊⠸⠈⠆⠈⢀⠡⠀⠀⠦⠠⢖⠀⠀⠀⠈⠁⠀⠈⠀⠐⠀⡄⠀⢠⠂⠉⠀⠀⠀⠈⠀⢀⠜⡀⠀⠀⢻⠧⠂⠁
⠀⠀⠠⠨⠄⠀⠂⠀⠀⠒⠈⠀⠀⠀⠀⠄⠀⠀⠀⠠⠒⠃⠀⠀⣀⠀⢌⢀⠢⠤⠬⠔⡀⠤⠔⢊⠈⠃⡀⠈
⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠈⠠⠂⠀⣀⡀⡔⠒⠒⡆⠀⢢⠀⠐⠀⠀⠁⠀⠈⠀⠠⡈⡀⢄⠀⠠⠀⠈⠀⠀⠀⠀⠐⠁
⠀⢀⡀⠐⠀⠀⠀⠀⠂⠄⠁⠀⠀⠘⠄⠂⢘⣀⣸⠀⠀⡀⠀⠐⠄⢤⠆⠀⠀⠈⠚⠁⠡⠊⠈⠠⠁⠀⠤⠀⠐⠀⠒⢤
⠀⠀⣠⡶⢀⡶⠀⢀⠀⠃⡀⠂⠈⠠⠀⠀⡀⠄⠠⣀⠀⠀⠘⠀⢀⠀⠀⠀⠃⠀⠠⣀⣠⣤⠄⠀⠀⠀⢄⠀⠀⠀⠀⠔
⠀⢸⣆⠅⠊⠀⡀⠁⠀⡂⠀⠀⡀⢊⠀⠌⡠⠊⢉⡉⠠⠀⠈⠠⠊⠀⠀⠀⠁⠀⢀⠜⠈⠡⡀⠀⠀⠀⡈⠀⠈⠁⠁⢠
⠀⠀⠁⠂⠀⠀⠃⠀⠀⠈⠀⠉⠀⠈⠀⠋⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠘⠉⠁

⠀⠀⣀⣀⣤⣤⣤⣶⣶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣶⣤⣤⣤⣤⣀⡀
⠀⣾⣟⡯⠍⠓⠒⠒⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠭⣽⣿⣧
⠀⣿⠈⠉⠻⠓⠒⢶⣶⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢴⣶⣶⣶⡶⠟⠛⠉⢸⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣀⣠⣤⣤⣀⡀
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠟⠋⣉⣉⣉⣙⠻⣷⣄
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣴⠟⠋⠉⠉⠛⣷⡌⢿⣆
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⠃⠀⠀⠀⠀⠀⠘⣿⠸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠃⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿
⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⡟⢸⡿
⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⢀⣼⡟⢠⡿⠃
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⣠⣾⠋⣰⡿⠁
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣠⣴⡾⠋⣡⡾⠋
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⠛⠋⣁⣴⡿⠋
⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⣴⡾⠛⠁
⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠋⠁
⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠈⠻⣶⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⡾⠟⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣽⣭⣬⣽⣭⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠖⠒⠒⠒⠒⠒⠒⠒⠒⠲⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣚⡭⠵⠖⠒⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠓⠲⠤⣍⣓⠲⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠀⠀⢀⣀⣠⣤⠤⠤⠶⠒⠒⠒⠒⠒⠒⠶⠤⢤⣄⡀⠀⠉⠛⢦⡙⢦
⠀⠀⠀⠀⠀⢠⡏⡏⠀⢀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢤⡀⠀⠀⠀⠉⠙⠲⣄⠀⣿⢸⡆
⠀⠀⠀⠀⠀⢸⣇⢣⡞⠉⠀⠀⠀⠀⠀⢀⡤⠴⠒⠒⠒⠋⠀⡤⠤⡄⢹⡀⠀⠀⠀⠀⠀⢀⡿⣡⢾⡇⣀⣤⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⢸⡈⠳⣌⡳⢤⣀⡀⠀⠀⠸⡄⠀⠀⢰⠖⠒⠚⠁⠀⢹⢸⡇⠀⢀⣠⡴⢞⣡⠞⣡⠞⠋⠁⠀⣀⣀⣀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠉⠓⠮⣍⣛⠒⠶⠿⠦⢤⣬⣷⣤⣤⣤⠤⠿⠾⠒⢋⣩⡤⠖⠉⠀⠀⣧⣀⡤⠖⠋⠁⠀⠉⠙⣦⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠶⠶⠶⠶⠶⠶⠶⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢧⣄⣀⣀⣀⣀⡤⠞⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣨⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡀⠀⠈⠉⠉⠁⣀⡠⠔⠋
⠀⠀⠀⠀⠀⣠⠞⠉⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠻⣍⠛⠋⠉⠉⠁
⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⢸⡷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡏⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠙⢦⣌⡙⠲⠦⢤⣄⣀⣀⣀⣀⣀⣀⡤⠤⠖⠚⣋⡥⠞⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⢻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠤⠤⠤⠤⠤⠤⠴⠶⠒⠋⠉⠀⠀⠀⠀⢀⣠⣾⠟
⠀⠀⠀⠀⠀⠙⠪⣝⡲⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢴⣚⡭⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠺⠭⣍⣓⣒⠲⠦⠤⠤⠤⠤⠤⠤⠤⠴⣖⣒⣻⠭⠷⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠈⠀⠀⠦⠀⠀⢘⣺⣄⡠⢀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠄⣠⡁⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠀⠋⠀⠀⠐⠀⠀⠀⠀⠀⢀⠘⠛⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠇⢀⠀⠀⡆⠀⠠⠘⢥⠽⠀⠀⠀⡀
⠀⡀⠀⠐⠀⢠⠀⠒⠈⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠠⠄⠀⠀⠀⠀⢀⡠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⠀⠀⠀⠰⣷⢄⠀⠀⠂⠀⠁⠀⢠⡇
⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣊⢄⠀⠀⠀⠐⢜⢃⡀⢀⣀⣐⣤⡤⢊
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠑⠈⠉⣓⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠈⠀⠀⠀⠈⠆⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠄⠀⠀⢀⡀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⡀⢊⠀⠀⠀⠀⡊
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢈⡠⠀⠀⠀⠉⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠁⠀⠀⠀⠀⠀⠐⠊⠀⠀⠐⡀⠀⠀⡰⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠠⡀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡉⢉⡒⡄⠀⠀⠀⠐⠠⠀⠀⠀⠀⣄⡠⠉
⠀⠀⠀⠀⠀⠤⢤⢤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠠⡄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠺⠀⠀⠈⢶⠀⠀⠈⢱⠂⠐⠒⠒⠋⠁⠄⠀⡇
⠀⠀⠀⠀⠀⢠⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠢⣄⡀⠐⢆⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠈⠃⠀⠘⠀⠀⡠⡧⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠈⠉⠘⣆⡀⠀⢀⠀⠲⡄⠀⠐⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⠄⠼⠀⠀⠀⠀⠀⠀⠀⢌⠢⣄⠐⠀⢀⡴⠤⢛⢽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⠛⠐⠒⠂⠈⡼⠀⠀⠀⠀⡀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⣀⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⣠⠀⠑⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣠⠞⠁
⠀⠃⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠛⠫⠄⠀⠀⠁⠀⠐⠤⢀⣀⣀⡠⠴⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠘⠀⠀⠘⠋⠀⠠⠀⠐⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⣰⠀⠀⠀⠀⣠⠄
⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠶⠶⠦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡀⠀⠀⢠⡀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣦⠠⠾⠷⠶⠶⠿⠶⠄⢠⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⣦⣤⣤⣿⣀⣀⣀⣀⣠⡞⠋⠙⢿⣶⠾⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠈⠙⠛⠛⠉⠉⢹⡟⠀⣠⣴⣬⣿⣄⡀⣀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠻⣾⠃⠀⠈⣿⣯⠉⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠒⠛⠷⣤⣴⡿⠿⠃⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⠾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡧⠤⠤
⠀⠀⠀⠀⠀⠀⠀⣠⠶⢮⡉⠉⢻⡏⠁⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆⠀⠀⢀⣼⣇⣀
⠀⠀⠀⠀⠀⣠⠞⢡⡲⣌⢳⡶⠛⢿⡋⠀⠀⠀⠁⠀⠀⠀⠸⣏⣽⠂⠀⠀⠀⠀⠀⠀⢀⣸⡟⠉⠉
⠀⠀⠀⠀⠘⠧⣜⠳⡝⠎⠁⡇⣠⡾⠿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠟⠛⠷⠄
⠀⠀⠀⠀⠀⠀⠈⠓⠦⠶⠲⣿⢧⡀⠀⠀⠉⣻⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣽⣶⠶⢾⣯⡾⠁⠈⣿⣀⣀⣀⣀⣼⠇⠈⢿⣼⠟⠻⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⣀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣍⣉⠉⠉⢛⣻⣟⡛⠛⠉⣉⣩⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠉⠉⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢠⡏⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣼⠁⠀⡾⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣰⠇⠀⣸⠃⠀⢠⠏⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢠⡟⠀⢰⠇⠀⢠⠏⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠖⣾⠀⢸⠃⠀⡿⠀⢠⡟⠀⢀⡏⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣛⣭⠴⠒⠀⠀⡏⠀⣿⠀⢸⡇⠀⢸⠁⠀⣼⠀⠀⠠⣌⣙⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢾⣿⣶⣞⣋⣥⣤⣿⣤⣿⢠⣼⡇⣤⣼⢠⣤⣿⣀⡀⠒⠦⣭⣛⠆⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⠈⢉⣠⠟⢑⣶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣍⠓⠂⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉⣉⣀⣤⡴⠾⣛⣵⡆⣚⣛⠧⣄⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣝⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⢋⣉⣉⣭⣴⡶⠿⣛⣭⠀⠉⠉⠙⣾⡇⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠃⢸⣟⡳⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣉⣉⣉⣥⣴⡶⠿⣛⡅⠈⠑⣂⡴⣻⡣⢠⡏
⠀⠀⠀⠀⠀⢀⣤⠞⣋⡥⢒⡠⠄⠀⠀⢻⣟⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣉⣉⣉⣩⣭⣴⠶⢟⡋⢡⣞⣛⣽⣾⢟⣵⠟⢶⣄
⠀⠀⠀⠀⣰⠟⣡⡞⢡⡞⢉⡴⠊⡀⠀⠈⢿⣟⠒⠄⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣉⣉⣭⣭⣴⡶⢿⡛⢡⡿⢿⣛⣭⡶⠛⠁⣀⢢⡈⢷⡄
⠀⠀⠀⠀⣿⠀⣿⠀⣿⠀⣿⡀⢾⡀⠀⠀⠀⠻⣯⡒⠀⠀⠀⠀⠀⠀⠀⠈⢉⣉⣭⣥⣤⠴⠶⣞⡛⣱⣿⣿⣿⣿⣿⢿⡌⢳⡈⣆⢳⠈⡇
⠀⠀⠀⠀⠻⣆⠹⣧⣌⠳⢮⣛⠶⣬⣁⠀⠀⠀⠈⠻⣥⣄⣀⠀⠀⠀⠀⠀⠤⠤⠶⢶⣶⣾⣿⣿⣿⣿⣿⢸⣿⡜⣿⢎⡇⢸⠇⡏⢀⡼⠃
⠀⠀⠀⠀⠀⠈⣳⣮⣙⠷⣦⣍⣙⠓⠦⠄⠀⠀⠀⠀⠈⠙⠻⢯⣿⣛⣛⣛⣛⣛⣯⣽⡿⠿⢛⣫⣵⡿⢋⣾⠟⣰⠟⠘⠁⢋⣠⣶⣿⠤
⠀⠀⠀⠀⠀⠀⠀⠨⠽⠿⠶⢮⣭⣙⣓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠽⠷⠶⠶⠶⣚⣛⣋⣭⡵⠾⠛⢁⣈⣡⣤⣶⣾⣿⣿⠿⠭⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣉⣉⣹⣿⣿⣷⣶⣶⣶⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣭⣭⣭⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣛⣉⣉⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠶⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⣛⣛⣛⣛⣛⣛⣛⣛⣻⣟⣛⣛⣛⣛⠛⠛⠛⠛⠛⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢉⣉⣩⣭⣭⣭⣭⣭⣭⣉⣉⡉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣦⣄⠀⠀⠀⠀⢀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⠷⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣤⡴⠾⠛⠉⠁⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠙⠷⠶⠾⠛⠛⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⡿⠿⠋⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⣀⣀⣸⣇⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠾⢿⡟⠻⠿⠶
⠀⠀⠘⠛⠛⠛⢻⡏⠉⠉⠀⠀⠀⠀⣼⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⣧⣀⣠⡀⠀⠀⠀⠙⠿⠃⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⠚⢻⡿⠳⠶⠶
⠀⠀⠀⠀⠠⡶⠟⢻⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣀⣀⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣴⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡿⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⣤⡾⠟⠛⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠈⠉⠛
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠛⠶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠛⠛⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠦⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠠⣶⡄⠀⠀⣧⠶⠛⠛⠉⠉⠉⠙⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⢀⣀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⣀⣀⣀⣀⣀⣠⣴⡾⠛⡅⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠘⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⢿⢻⠛⡟⢻⢻⠛⠙⢹⡇⢀⣷⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⡶⡟⣿⢷⣤⡾⣿⠀⡇⢸⠘⡀⡇⢸⣈⣀⣠⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⣴⢿⠉⡏⢻⡇⣧⢸⢸⡈⡇⢸⣀⣷⡾⠟⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⢀⣠⡴⠶⠶⠶⠶⠾⠿⠟⠛⠁⠀⠀⠀⠘⠃⠀⠀⢠⣴⣿⣾⡄⢇⢸⣧⣼⣼⣤⣇⣧⡾⠛⠁
⠀⠀⢠⡶⠋⠁⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⢸⣿⣧⣼⣼⣿⠁⠈⠙⠛⠋
⠀⢰⡟⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠙⢿⡜⢿⡏⠀⠈⠉
⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⣠⢾⣷⠟⠋
⠀⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⣼⣷⢸⣿
⠀⢸⣷⢹⠛⡿⢿⡿⡷⢶⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣼⡿
⠀⠀⠻⣿⣄⣧⠘⡇⡇⢸⢼⣧⠀⠀⠀⠻⠇⠀⠀⠀⠀⢀⣾⣿⠋
⠀⠀⠀⠈⠙⠛⠿⠿⠿⠾⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠻⡿⣶⣤⣤⣀⣤⣴⡿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⡇⠀⡏⢹⠉⢻⠈⡇⢣⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⣷⣬⣦⣼⣤⡷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠂⠉⠉⠉⠉⠉⠉⠁⠐⠂⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⣷⠀⣶⠀⣾⠛⠁⡆⠀⣶⠀⢠⡾⣷⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣿⠒⣿⠀⣿⠂⠀⣿⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠿⠀⠿⠀⠿⠖⠀⣿⡄⢿⣤⠸⣧⡾⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠀⠤⠀⡠⠤⠤⠤⠤⠤⠤⠤⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⢀⢤⡀⠀⠀⣀⡠⠤⠐⢲⣶⣶⣶⣶⠒⠂⠤⢀⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢮⣤⠑⠖⠉⠀⠀⠀⠀⠈⠈⠉⠈⠁⠀⠀⠀⠀⠈⠑⠤⢎⣤⢡
⠀⠀⠀⠀⠀⠀⢸⠈⠁⠀⠀⠀⠔⠂⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠒⢄⠀⠁⠸⡀
⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⣠⣖⠂⠐⠢⡀⠀⠀⠀⠀⠀⠀⣀⡔⠒⠒⢤⡀⠀⠀⠈⢢
⠀⠀⠀⠀⠜⠢⠤⠀⠀⠠⣾⣿⣿⣷⠀⠀⢡⠀⠀⠀⠀⠀⣼⣿⣿⣦⠀⠀⢘⠄⠀⠤⠤⠷⡀
⠀⠀⠀⠎⠀⠒⠂⠀⠀⠀⠻⣿⣿⣿⠇⠀⡸⠀⠀⠀⠀⠀⢻⣿⣿⣿⠀⠀⠊⠀⠀⠐⠐⠒⠱⡀
⠀⠀⡜⠀⠉⠉⠀⠀⠀⠀⠀⠙⠻⠟⠠⠔⠁⠀⢠⡄⠀⠀⠀⠙⠿⠧⠒⠁⠀⠀⠀⠉⠑⠒⠀⢱
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢞⡱⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⡄⠀⠀⠀⠀⠑⠦⣀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢸
⣧⣤⣤⣤⣤⣤⣤⣬⣵⣶⣷⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣶⣷⣶⣬⣥⣤⣤⣤⣤⣤⣤⣤⣬⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⡿⠯⠥⢤⣱⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣸⡄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⣀⡀⣀⠀⡀⠀⣶⠀⠲⠀⠐⢁⡠⡀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⢲⠀⢷⠀⢸⠀⠀⡆⠀⠀⢨⢸⡁⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡏⠒⠈⠚⠈⠊⠈⠊⠀⠉⠀⠈⠋⠑⠁⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣤⢤⣖⣒⣛⣛⣛⣛⣛⣛⣛⣒⣶⢤⡄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡇⠀⠀⢸⣤⣒⣒⠉⠛⠛⠛⠉⠑⣒⣒⣦⣼
⠀⠀⠀⠀⠀⠀⠀⡇⠠⠭⠍⠀⡇⠀⣠⠖⠉⠉⠲⢤⣤⡤⠖⠋⠁⠀⠀⡇⠀⠀⢸⡀⠀⠉⠉⠉⠉⠉⠉⠉⢉⡁⠀⡾
⠀⠀⠀⠀⠀⠀⠀⡇⠈⠥⠖⠀⡷⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⢀⣒⠒⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢇⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⣧⣄⣀⣀⣀⣀⣀⣀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠧⠤⠶⠶⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠃⠀⠀⠀⠀⠈⠛⠒⠲⠶⠶⠖⠚⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢐⡇⠷⡀⠀⠀⠀⠀⠀⠸⠁⢣⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⣤⣄⣐⣇⠱⡀⢠⣤⣤⡤⢴⣧⣤⣷⠔⠛⣰⠷⠤⢴⣧⣼⡦⠒⢋⡴⠂⢸⠀⢣⠀⣀⣤⡶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢠⣄⠤⠟⠿⣧⣤⣷⢼⣯⡭⠿⣾⣇⠀⣿⡔⠊⠙⠒⠒⣾⡅⢈⣷⣶⣾⣴⠶⣼⡧⣼⡿⠛⢉⡵⠂⠹⣦
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠙⠒⠀⣴⣇⢀⣽⣿⠙⣅⢘⣁⣘⣿⣦⣭⡶⢣⣤⣾⣥⣽⣿⣀⣹⡏⠹⣦⣾⡁⣨⣷⡐⠋⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢰⣧⣬⡿⢦⣿⡀⣸⣿⠿⠛⢻⣍⠉⠃⠈⣿⠉⣉⡭⢍⡙⢛⣧⢀⣿⡶⠟⢛⣾⠵⣆⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⣬⡿⠆⣤⣄⣽⠋⠙⣧⡼⣿⡛⢁⣙⡿⠗⢿⢋⣁⡴⢯⣥⣬⡟⠉⢻⣄⡴⠋⠉⠑⠜⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠙⡷⠞⠙⢦⡀⠈⠉⡙⠳⣤⡼⢿⡉⠀⠀⢠⠋⠙⢶⡉⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣧⠀⢀⡠⠀⠀⠀⠀⠀⢀⣅⡤⠃⠈⠒⢠⡙⠀⠰⡇⢠⠈⠳⠤⣙⢆⠀⣸⣀⠌⠀⠙⠲⢴⡇⠀⠀⠀⠀⠠⣀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢷⡟⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠚⠙⠁⠀⠀⠀⠈⠃⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠹⣷⠛
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣀⣀⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣀⣀
⠀⠀⠉⠉⠛⠛⠻⣿⠶⠶⠦⠀⠀⠀⠀⠀⢙⣿⢙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⢹⡟⠀⠀⠀⠀⠀⠠⠶⠶⢾⡟⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⢀⣤⠶⠟⠻⣟⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠶⢆⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⢻⡿⠻⠷⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠤⠴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣠⡟⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣀⣤⣴⠶⠟⢿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠟⠷⢶⣤⣀
⠀⠀⠀⠠⠞⠋⠁⠀⠀⠀⠀⠙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋⠁⠀⠀⠀⠀⠉⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢷⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣾⢋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠁⠀⠘⢿⣯⡉⠉⠛⠛⠛⠳⠶⠶⠶⠶⠶⠶⠶⠶⠛⠛⠛⠋⠉⣹⣿⠏⠀⠀⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣯⣳⣄⠀⠘⢿⢿⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣾⣿⠋⠀⣠⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠈⠻⢿⣷⡄⠈⠳⠿⠿⠿⠛⠻⠿⢶⣤⡀⣠⣴⠾⠿⠛⠻⠿⠿⠶⠃⢀⣼⣷⠟⠁⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠙⢿⣦⡀⣠⡴⠶⠚⠛⠛⠉⠙⠿⠛⠙⠛⠛⠛⠲⠶⣤⣀⣠⣿⠟⠁⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⢘⣿⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡇⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣰⠶⣦⡀⣠⠶⣶⠁⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢨⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⢠⡾⣿⡖⠶⢦⣄⠀⣠⣷⡶⠙⠿⠛⠲⣾⣇⣀⢀⣤⠶⢂⣿⣿⣆⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣄⣀⣤⣴⣿⡷⠟⠛⠋⠻⣿⣭⣭⣽⣿⣶⣦⣶⣾⣿⣭⣭⣽⡿⠋⠉⠛⢷⣿⣷⣤⣀⣠⣤⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣬⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠛⢿⣯⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠯⡥⣄⠈⢿⡄⠀⠀⠀⢀⣤⣤⠶⠶⠶⣤⡄⠀⠀⠀⢀⣤⡴⠶⠶⢤⣤⣄⡀⠈⠁⠀⣼⠋⢀⡬⢭⢟⣦⣄
⠀⠀⠀⠀⠀⠀⣰⣿⣭⣶⣶⣶⣿⣇⠈⠙⠶⠛⠉⢉⣠⣤⠶⠶⠶⣄⠙⠳⣤⡶⠋⣠⠶⠶⠶⠦⣤⡈⠉⠛⠲⠞⠁⢠⣿⣾⣷⣾⣦⣽⣧
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠿⢿⣿⣏⠷⠶⣶⣶⣾⡿⠿⠛⠛⠿⢶⣌⠳⣄⣀⣠⡴⣫⣴⠿⠛⠛⠲⢿⣿⣶⣖⡶⠾⣫⣿⡿⠿⠟⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠙⠿⣶⣭⣵⡾⠋⠁⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠦⠤⣤⡤⠤⠶⠶⠶⠞⠛⠛⠋⠉⠙⠛⠷⠶⠶⠶⢤⣤⡤⠶⠶⠶⠞⠛⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠿⠿⠿⠿⠿⢿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿
⣿⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣡⠶⠋⡹⠋⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⣠⠞⠂⠀⣼⠁⢠⠀⠀⠀⢈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠛⠀⠀⣴⢃⣠⡦⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠈⠂⠀⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⡤⠚⠀⣠⣦⠰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⢾⠆⠀⠀⠀⠀⠀⠀⠌⠃⠀⠁⠈⠉⢒⡆⠀⠙⣿⠀⡆⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠳⣤⢀⡉⠻⠀⢇⠀⠀⠁⠀⠀⠀⠀⠀⠀⣸⠇⠀⢸⡿⠀⠀⢠⠗⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠱⣤⠦⠈⠢⣀⣀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠏⣰⡿⠚⠁⠰⠟⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⢀⡤⠃⠀⠀⠉⠻⠆⠐⠶⠚⠾⠃⠀⠀⠶⠀⠀⠒⠛⠉⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠠⡟⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠘⣦⡀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⢠⠀⠀⠀⠈⠁⠀⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣅⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿⣧⠀⠀⠶⣦⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⠶⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⠉⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣧⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⠀⠀⠀⠀⠀⠀⠐⠒⠲⠚⠀⠀⠀⠀⠀⠀⢀⣤⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢠⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢤⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠀⠀⠘⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⠀⠀⠀⠘⠀⠀⢸⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠇⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣶⣶⣶⣶⣶⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘






⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣾⣶⣶⣶⣾⣶⣾⣶⣿⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣟⣻⡟⢿⡿⢿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡶⠶⢶⣶⡶⠶⠾⠿⠿⢿⢿⡿⠿⠿⠿⣷⢶⠶⠶⠶⠶⢶⣶⣦⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⠸⡇⠀⠀⠀⢿⢸⠀⠀⠀⠀⠀⣿⢸⡀⠀⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⡏⡇⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣀⣀⣀⣧⣷⣀⣀⣀⣀⣸⣠⣇⣀⣀⣀⣸⣿⣀⣀⣀⣀⣀⣿⣸⣀⣀⣀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠉⠉⠉⠉⠛⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢸⡿⣤⣀⡀⣀⣀⣀⡤⡞⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠳⡄⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣬⡉⠉⠉⠉⣁⣠⣶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠹⣄⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠙⣦⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠈⢧⠈⣿
⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠈⢳⣽⡇
⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⢀⡤⠖⠛⠓⢦⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⢠⣿⡀⠀⢀⡞⠁⠀⠀⠀⣸⠃⠀⠀⠀⡟⠀⠀⠀⠹⣄⠀⠀⠀⣴⠖⠒⢦⡀⠀⠀⠀⣿⠷⠦⣄⠀⠀⢠⠶⠒⢦⡀⠀⣿⡆
⠀⠀⠀⠀⠀⢸⡟⠙⠛⠉⠀⠀⠀⠀⠀⠻⣄⠀⠀⣠⠇⠀⠀⠀⠀⠙⠲⠤⠞⠃⠀⠀⠀⣇⠀⠀⢸⡏⠀⢀⡟⠀⠀⢸⠀⠀⠀⡇⢀⣼⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⣻⡇⠀⢸⣇⠀⣀⡼⠀⠀⠀⠉⠉⣿⠃
⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢀⣿⢿⣆⣴⣾⡆⠀⣀⠀⠀⣿⠇⠀⣿⠀⠀⣀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠘⣿⠈⠛⠁⢸⣇⠀⣩⡀⠀⣿⠀⠀⣿⣴⣿⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠹⠇⠀⠀⠸⣿⠀⢿⠇⠠⡿⠀⠀⣿⠉⠻⣦⡀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣠⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠶⣶⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣿⣀⣤⡶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⣠⣴⣶⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠉⠛⢷⣶⡿⠟⠉⠀⠀⢹⣿⡀
⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀⣀⠀⠀⢀⣀⣤⣤⣴⣶⣾⠏⠀⠀⢀⣀⡀⣻⣦⣀⣀⠀⣀⣠⣿⣧⣄
⠀⠀⠀⢠⣿⠟⠋⠛⠛⠛⠿⢿⣿⠿⠛⠛⠉⠉⠉⣿⠀⠀⢰⡿⢛⣿⠛⠉⠉⠙⣿⣏⠉⠉⠈⠹⣷
⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠈⠿⣾⡇⠀⠀⠀⠀⢸⡟⢻⡆⠀⢀⣿
⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⣀⣤⣾⠿⣦⣄⣠⣴⡿⠿⠟⠁⠀⣼⡏
⠀⠀⠀⠈⣿⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠉⠈⠻⣦⣤⣤⡾⢿⣷
⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣿⣧⣤⣤⣤
⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⢻⣿⠁
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⠀⠀⠀⠀⢠⣼⣿⣤⣤⣤
⠀⠀⠀⢸⣿⣄⣀⡀⠀⠀⠀⢠⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠀⠀⠀⠀⠀⣽⡿
⠰⠾⠟⠛⢿⣟⠉⠁⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⠀⠀⣴⠟⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣷⣤⣄⡀
⠀⠀⠀⠀⢈⣿⣦⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠃⠀⠈⠉⠁
⠀⠀⠀⠾⠛⠉⠻⣷⣄⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠟⠉⠻⠿⣷⣆
⠀⠀⠀⠀⠀⢀⣠⡿⠿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣶⠿⠟⢻⣿⡀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠾⠋⠀⠀⢀⣭⣿⣿⣿⣿⢿⣿⣿⠿⠿⠿⠿⠿⠛⢻⣿⠉⠙⣿⣄⠀⠀⠹⣿⣄⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣏⠁⣼⡟⠁⠀⠸⣧⡀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠘⢿⣆⠀⠀⠙⣿⣦⣾⠟⠁
⠀⠀⠀⠀⠀⣿⡏⠀⠀⠘⣿⣿⣿⣿⣿⣶⣦⣍⠻⠷⠶⠶⠾⠟⠁⣠⣶⣾⠿⢿⣿⣷⣶⠿⠛⠉
⠀⠀⠀⠀⠀⣿⡇⠀⢀⣼⡿⠋⠀⠀⠀⠀⠉⢿⣧⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠻⣷⣄⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠈⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡀⠀⠀⠀⠀⠀⢀⣼⣿⣁⣀⣀⣀⣹⣿⣄⠀⠀⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣦⣤⣠⣤⣴⡿⠿⠛⠛⠛⠛⠛⠛⠛⠿⣷⣦⣤⣤⣤⣾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠸⡇⠹⣆⠀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⢠⣞⣣⠶⣤⠀⠀⠀⠀⠀⠈
⠀⢀⣤⣤⡄⠐⡄⣤⢰⠀⠀⠁⠶⠀⠂⣈⠀⠀⠀⠀⣧⠀⣷⠀⠙⡆⠀⠐⠉⢃⠀⡀⠈⠀⠀⠀⣴⣻⢞⡽⣻⠎⠀⠀⠀⠀⠀⠀⣤⠚⣶
⠀⠀⠀⠀⠀⠀⠓⠃⠘⠀⠀⠈⠛⠋⠛⠉⠀⠀⠀⠀⢹⣁⠀⢂⡼⠃⠀⠀⠒⠊⠉⠀⡂⠀⠀⠈⠐⠫⣼⠟⠁⠀⠀⠀⠀⠀⠀⠈⠿⠵⠋
⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣤⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡄⠀⠀⠀⢀⣤⣤⣠⣦⣤⠀⠀⠀⣀⠐⠄⣈⠀⠀⠀⠀⠀⢠⣄⣠⣄⣤
⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⢀⠄⢠⡒⠦⠀⠀⠀⢠⣤⠙⠃⠀⡼⠂⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⣴⠆⠀⠑⢿⡄⠀⠀⠘⠛⢿⣿⠋⠁
⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⣤⠐⠂⠛⠁⠠⠀⠀⠀⢀⠈⠀⠔⠊⠀⠀⠀⠀⠁⠀⠀⡀⠀⣦⠀⠠⠟⠃⠐⠛⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠈⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠘⠛⠛⠛⠛⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠐⣶⠀⡀⠀⠀⠀⠀⠀⠀⣀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡠⠚⠀⠁⠒⠀⠀⠀⢤⡀⠈⠹⠇⢤⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⣀⡀⠀⠈⠿⠀⠀⠀⠀⠀⠠⠁⠀⠀⡙⣷
⠀⠀⠀⠀⠠⣶⣤⣤⠀⠀⢾⡀⠁⠀⠀⠀⠰⠇⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⣤⠀⠀⠀⠠⠞⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠁
⠀⠀⠀⠀⠀⠿⣮⠗⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠘⠘⠃
⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⢠⠦⠀⠀⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⠀⠉⠛⡂⠀⠀⠀⠀⢰⠀⢃⣠⠖⠀⡀⠀⠀⠄⠀⠰⠀⠀⠀⣴⣶⣶⣶⡶⢤⣀
⠀⠀⠀⠀⠀⢘⡄⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⡣⠀⠠⢤⡄⣤⠖⠉⠀⠀⠀⣅⢠⠄⠀⣀⡀⠀⠀⠀⠳⢄⠀⠀⡴⠖⠋
⠀⠀⠂⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠀⠹⠁⠀⠀⠀⠀⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠡⡄⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢺⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠠⢾⡿⠁⠀⢀⠋⠁⡀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣰⠀⡆⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠐⣄⠁⠠⣀⡀⠀⠀⠀⠀⠀⠀⠐⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠰⠀⠀⠐⣿⡾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠈⠁⠀⠀⠀⠀⢀⠈⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠻⣴⣷⢄⡇
⠀⠀⠀⠀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠄⠀⠉⠑⠀⠄⠀⠀⠀⣀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀⠀⠀⠀⢀⠀⠰⡇⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⣀
⠀⢆⣀⣀⠀⠀⣀⣠⠀⠀⠸⣿⣍⣶⣍⣻⡀⠀⠀⠀⠰⠒⠆⠀⠀⠀⢀⠉⠉⠀⠀⠉⠉⡀⠀⢀⡜⠁⣴⣴⣀⡶⢤⡀⢠⣾⣛⢛⣿
⠀⠀⠙⠛⠛⠛⠛⠃⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⢶⣤⣄⣤⠶⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠠⠀⠃⣜⢁⣸⡧⡴⠃⠀⠁⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⣶
⠀⠀⠀⠘⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⢤⣤⠀⠀⠀⠀⠘⢿⡿⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⡆⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡂⡄⠀⠀⠀⠀⠀⠂⢠⠁⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠄⣀⡆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠑⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠙⠀⠀⠠⠀⠀⠀⠘⠳⠶⠶⠟⠉
⠀⠀⠀⠀⠄⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⢀⠒⠀⢀⡇⠀⠀⠀⠀⠀⠤⢤⠐⠂⠀⠀⠀⣶⣦⠙⢦⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠾⠷⠂⠐⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠉⢠⡜⠛⣅⠀⠀⠉⠉⠉⠀⠀⠃⠀⠀⠀⣷⡾⡿⠻⣆⠀⠀⠀⠀⠀⣾⢀⡀⠟⠀⠀⠀⣤⣤⠀⠀⢀⠀⠀⠀⢀⠸⠋⠙⠉⠙⠁
⠀⠀⠀⠉⠀⠉⠂⠉⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠳⠊⠐⠓⠀⠛⠁⠀⠀⠠⣅⠈⢨⠅⠀⠀⠀⠀⠉⠠⣦⠆⠻⠃⠀⠀⠘⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠓⠂
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢃⠀⠀⠀⠀⠀⠀⢀⣠⣤⡄⠀⠀⠀⢀⠤⠀⠀⣤⠀⠀⠀⠀⠀⣤⣤⡀⠀⡀⠀⠀⠀⠿⡇⠀⢀⣠⠤⢄
⠀⠀⠀⣠⣶⣶⠀⠀⠀⠀⠠⠀⠀⠘⠇⠀⠀⠀⣴⠾⠟⠉⠁⠀⠐⠀⠀⠁⠀⣘⣦⡄⠀⠀⠀⠀⠀⠀⢠⡀⣠⠡⠀⠀⠀⠁⠐⠀⠀⠿⠁⠟
⠀⠀⠀⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠠⠟⠁⠀⠀⢈⣿⡷⠒⠾⠟⠂⠀⠀⠀⠀⠀⡞⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⣶⣶⣶⣶⣶⣶⣦⣶⣴⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠻⠛⠛⣿⠋⢻⠛⢛⡛⠟⣛⣿⢛⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣿⣿⣿⣾⣿⣶⣾⣿⣷⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⠶⠶⠶⠞⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠞⠛⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⢸⡆⠀⡇⠀⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⣾⠀⠀⢸⠀⠀⣿⠀⢸⡇⠀⢸⣀⣀⣧⣤⣧⡤⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⣸⣀⣠⣽⣤⠶⠷⠛⠛⠉⠉⠁⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢷⡖⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠹⣄⠀⠀⠐⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⢀⡼⠃⢀⣀⣤⡤⠴⠶⠚⠛⠉⠁⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⢠⡿⠷⢿⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⢀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠶⠿⣦
⣿⠁⠀⠀⠀⠈⠉⠛⠒⠦⠤⣄⣀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠖⠛⠋⠉⠁⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠦⢤⣀⣘⠆⣀⣀⣤⠤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⢸⡇
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⣴⡄⠰⡟⣷⠞⣹⠆⠸⡇
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⢸⠀⠀⢰⠲⡆⠀⣾⣆⠈⣏⢿⢻⡘⡇⠀⣷⠀⢠⣏⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⡆⠀⢸⠀⠙⢶⠇⠹⡆⢻⢸⣼⡇⣿⣀⣽⠀⣶⣝⢦⡀⢿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢀⠀⢀⡀⢹⣼⡎⣿⣧⣈⣠⣿⣧⡽⠘⠾⠁⢸
⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣸⠀⡟⣧⢘⣿⣦⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⠚⠃⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⠀⠠⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⣿⠀⠀⣒⠦⣭⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⠀⢠⣭⣟⡒⠦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⠀⠈⡇⠀⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⢹⡀⠀⢇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠘⣧⡀⠈⠳⢦⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠞
⠀⠀⠀⠀⠀⠉⠙⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣀⣤⡴⠶⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣤⣄⣀⠀⣸⣤⡶⠞⠋⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠁⠉⢉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢂⠀⢀⠀⠀⠀⠀⠀⠀⢄⠀⣀
⠀⠀⠀⠀⠄⣁⠊⠈⠆⣀⠀⠀⠀⠀⠀⢀⣸⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠠⡂⢀⠘⣸⠃⠆⠀⢀⠀⠁⠙⠉⠈⠙⠙⣿⣥⡀⠀⠒⠀⠄⠀⠀⢀⡀⠔⠛⠀⠀⠠⣀⣀⡀
⠀⠀⠀⠈⢀⡙⣤⡀⠜⠈⠀⠀⠀⢰⠀⠀⠀⠀⠀⢈⣿⡇⠘⢦⡂⡒⢐⠒⠆⠲⡀⡶⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠁⠈⠃⠁⠀⠀⠀⠀⠠⡀⠀⡀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠉⠙⠚⠓⠛⠊⠉⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣻⣄⢀⣠⣿⣿⡟⠀⠀⠀⠒⡀⣀⠀⠀⠀⡇⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠛⠁⠀⠀⠁⠀⠀⠀⡆⠀⢠⠁⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠌⠀⠘⣷⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠙⠈⢀⣤⠃⠀⠀⢇⣀⠀⣀⣀⣦⠆
⠀⠀⠀⠀⡀⠀⠠⡏⠀⠀⠀⠀⢋⡀⠀⠀⠀⠹⣶⣶⣾⡟⣰⠀⠀⠀⠀⠀⣸⠆⠀⠀⠀⠙⠛⠛⠋⠁
⠀⠀⠀⠀⢁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠕⠚⠁⠀⠀⠐⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⢸⠀⢸⠁⠀⠀⠀⠀⡀⠀⣀⣶⠀⠀⡀⣀⢠⡄⢀⠀⠀⠀⠀⣼⠀⠀⣧⡀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠁⢸⠀⠀⠈⠳⠤⡧⡴⠟⠃⠠⠥⠼⡿⠟⠛⠻⣷⡄⠀⠀⣧⠀⠀⠈⠳⢦⣤⣤⡴⠷⠟⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠈⠀⠀⠀⢸⡇⠀⠀⢸⡀
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠇⠀⠀⠀⣀⠀⠐⢄⠀⠀⠀⣠⣠⣼⠇⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⣤⡼⠋⠀⠀⠀⠉⠛⠛⠛⠉⠀⠀⠀⠀⢠⠆⠀⠀⠀⢀⠀⠀⣠⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣤⣶⣤⣤⣤⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⡞⠀⠀⠀⠀⡸⠿⠛⠏⣗⢀⡀
⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⡿⠆⠀⠸⣤⣀⠀⠀⠀⠀⢀⣄⣀⣽⡿
⠀⠀⠀⠀⠀⠈⠙⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠂⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠐⠒⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢁⠈⠀⠀⠒⠂⠈⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠒⠂⠀⠈⠐⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠠⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠢⡀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠶⢺⡀⠀⢁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢂⣹⠀⠀⣏⠉⢵⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⢋⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⠒⠡⠐⢧⠀⠱⣀⡀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⣀⡀⠀⡈⡗⢤⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⡜⢀⢊⠐⠁⠀⠀⠀⠑⢤⡈⠑⠂⠷⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠴⠊⠉⠀⠀⠀⢀⠻⠲⡐⠄⠑⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠆⠀⠀⠀⠀⠀⠀⠀⡼⠉⠑⠂⠠⠀⢤⡀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠡⠁⠀⠀⠀⠀⠀⣠⠞⠀⠃⠀⢨⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢱⡈⠀⠀⠀⠀⠀⣠⠜⠁⠀⠀⠀⠀⠀⠀⠙⢒⠉⠀⠀⠒⠀⠀⠀⠀⠀⠀⢀⣁⠤⠖⣒⠼⠶⠶⠦⠤⠶⠚⠁⠀⠀⠀⠠⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠠⠆⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⣠⠴⠶⠒⢉⣡⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠂⠌⡁⠂⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠉⠀⠀⠔⠊⠁⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠴⠒⢈⠠⢐⡮⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⣀⠀⠀⠀⠈⠉⠁⠒⠒⠀⠤⠤⠤⠤⠤⠤⠠⠤⠤⠶⠶⠾⠛⠛⠓⠊⠉⠉⠠⠀⠂⢀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠠⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠃⠐⠒⠒⠒⠒⠒⠒⠒⠂⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣤⡴⠋⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣸⠀⠀⢻⣏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣧
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣫⣷⣿
⠀⢸⡿⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⡿⢟⣭⣴⢿⣿⢿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⠿⣻⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢽⣶⣿⣿⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⣏⣽⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⢀⣄⣀⠀⠀⢀⣀⡀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⢀⣾⣿⣿⠀⠀⣿⣿⡇⠀⢸⣿⣿⠀⠀⠀⠀⢸⣿⡇⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⠻⣷⡀⣼⡟⣿⣿⠀⠀⣿⣿⡇⠀⢸⣿⣿⠀⠀⠀⠀⢸⣿⣷⣾⣿⡋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣹⣿⣿⣿⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⡄⢻⣶⣿⠁⣿⣿⠀⠀⢿⣿⡇⠀⢸⣿⣇⠀⠀⠀⠀⢸⣿⣿⠛⢿⣿⣄⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⠇⠈⠻⠃⠀⣿⣿⠀⠀⢼⣿⡇⠀⢸⣿⣿⣶⣶⣶⡆⢸⣿⣇⠀⠈⢻⣿⣆⠀⠀⠀⠀⠀⢸⡟⣿⣿⣿⣿⣿⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠁⠀⠈⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢸⣧⣧⣿⣿⣿⣿⣿⣿
⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠙⠉⢉⣭⣍⡙⠿⢶⣶⣶⣶⠶⠋⠉⠻⢿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠯⡭⠤⠤⣤⣀⡘⠻⣿⣇⠀⠈⠛⠛⠋⠠⣿⣆⢠⡀⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⢡⣿⠇⢠⠁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣠⣄⠀⣿⣀⠀⠀⣀⣠⣤⣤⣀⡀⠈⢆⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢻⣿⣿⣿⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡄⣿⠀⠀⠉⠁⠈⠛⠛⠛⠙⣦⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣿⣿⢻⣻⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢡⠟⡀⠀⠀⠀⠀⠀⠀⡧⠞⠀⡿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢻⣿⣾⣿⣿⣿
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣤⣤⣤⣤⣿⣧⣼⣿⣥⣤⣤⣤⣤⣤⣼⣿⣥⣤⣷⣏⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣵⣿⣿⣿⣿⣿
⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠹⣧⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⠀⠀⠀⢠⣾⣿⣷⣶⣶⣶⣶⣿⣿⣿⣷⣶⣴⣶⣾⣿⣿⣿⣦⣤⣴⣾⣿⣿⣿⣶⣤⣤⣤⣶⣿⣷⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⢸⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢸⣿⣿⣿⣿⡿⠋
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⠐⠢⠐⠘⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣄⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠐⠘⠀⠀⠛⠢⠀⠀⠈⠁⠀⠀⠀⠉⢀
⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠄⠀⠀⠀⠘⠘⠀⠈
⠀⠀⠀⠀⠄⠀⣀⣀⠀⠄⠂⠁⠀⠀⠀⠀⠐⠠⢀⡀⢀⠤⠂⠀⠀⠀⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠤⠀⠠⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠂⠠⠄⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠊⠠⡔⠂⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠒⠀⠀⠀⠀⠀⠢⡄⠈⠀⠀⠀⠀⠀⠗⢀⠀⠀⠆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠥⠄⠁⠂⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡆⠀⠁⡀⠀⠘
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁
⠰⢆⠠⠄⠠⠔⠀⠀⠀⠀⢠⠀⠀⠀⠘⡀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐⠲⠂⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠂⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠂⠀⠐⠒⠂⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠦⠀⠀⠀⠀⡀⠀⠀⠀⠈⠐⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠄⠀⠀⠘
⡄⠒⠠⡀⢀⡀⠀⠀⠀⠀⠠⠀⠀⠁⠀⡀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠁
⠃⠀⠀⠈⠀⠉⠀⢀⠁⠀⠠⠂⠀⢀⠂⠉⠁⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢐⡄⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠑⠠⠄⠀⠤⠘⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⡈⠣⡙⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠠⠍⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⣀⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠐⠣⠄⠀⠀⠀⠀⠀⠐⠀⣀⣑⡎⠀⠀⠀⠀⠀⢀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠉⠙⠒⠄⠁⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⣂⣄⣠⠴

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣦⡀⠀⠀⠀⣀⣀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠛⠛⠻⠿⣿⣶⣤⣀⢀⣀⣀⣀⣀⣠⣄⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⡿⠟⠛⠛⠉⠛⣿⣧
⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠟⠛⠛⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⢼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⣤⣤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠸⣿⣷⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡿⠟⠹⣿⡆
⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢿⣷
⣀⣤⣤⣤⣼⣿⣤⣤⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣿⣶⣶⣶⣤
⠉⠉⠉⠉⠹⣿⡏⠉⠉⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡄⠀⠀⠀⠀⠀⠉⠀⣹⣿
⠀⠀⠀⠀⠀⣿⣧⣀⣀⠀⠀⠀⠀⠘⢿⠿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⢠⣤⣿⣧⣀⣀
⠀⠀⠰⠿⠿⠻⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠉⢻⣷⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣽⣿⠋⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠘⣿⣷⣤⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣿⡃
⠀⠀⠀⠀⣀⣴⡾⠟⠻⢿⣷⣄⣠⣴⣾⠿⠿⠿⢿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⠿⠶⣶⣦⣤⡀⢀⣤⣿⠟⠛⠿⣷⣦⡀
⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⣹⣿⠟⠉⠀⠀⠀⠀⠀⠀⠙⢿⣧⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠈⠙⢿⣿⡟⠁⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣶⣶⣤⣤⣤⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠉⠉⠉⠉⠉⠉⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⣦⣤⣶⣿⣿⣶⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⠿⠿⣷⣦⣤⣴⣾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣴⣶⣶⣦⣤⣀⠀⠠⣶⣠⠀⠀⣀⣠⣄⣀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣞⣻⣿⣿⣿⣿⣧⣽⠀⠀⠀⢉⣿⡿⠿⣿⣏⣽⣿⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⡿⢛⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠿⣿⣿⡷⠀⠸⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣤⣴⣿⣿⣧⣶⣼⣿⣿⣿⣽⣶⣿⣽⣿⣿⣽⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⡀⠀⢀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⡴⠀⠀⠰⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡎⠀⠀⠀⢀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠠⠀⡘⢡⡷⢤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢤⡄⢖⠀⣠⣴⣶⣶⣿⡿⢷⣼⣿⣿⣿⣿⣻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠷⢂⢹⣧⣤⣬⠍⠉⠉⠁⠉⠉⠉⠁⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠑⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⠠⠐⠞⠻⣦⡀⠀⠀⠀⠀⠀⢀⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⠷⣤⣍⣓⣶⣶⡤⠴⡋⠁⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣌⠻⣿⣿⣿⣿⣦⣌⣉⣉⣉⣁⡀⠔⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠈⠀⠀⠀⢠⡴⣿⣿⣿⣿⣷⣿⣿⣿⣿⡟⠋⠉⠉⠀⠀⠀⣀⣀⣤⡔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣟⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠈⠉⠁⠀⠈⠁


⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠛⠛⠛⠛⠋⠛⠛⠛⠛⠛⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡋⠓⠐⠒⠲⠤⠖⠒⠲⢤⠴⠒⠒⠒⠊⠉⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠗⠀⠀⠀⠐⠲⠶⠶⠶⠶⠒⠒⠒⠒⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢤⣀⠀⠀⠀⠀⠀⠀⣤⠒⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢱⠀⠀⠀⠀⠀⠀⢰⠋⠁⠀⠀⠉⢧⡀⠀⠀⠀⢨⠁⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠈⢶⣄⠀⠀⢀⡰⠃⠀⠀⠀⠀⠀⠀⠳⢦⣀⡰⠎⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠹⣆
⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⡙⢇⠀⢠⠻⡆⠀⠀⢸⠋⡇⠀⠀⡇⢸⠀⠀⠀⠀⠀⢸⡉⡇⠀⣀⡤⢶⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠸⠃⠈⢷⡎⠀⠻⡀⠀⠘⠀⠇⠀⠀⡇⢸⠀⠀⠀⠀⠀⢸⡆⠰⠞⠁⣠⠏
⡇⠀⠀⠀⠀⠀⠀⠇⣠⢦⡀⢀⣤⡀⢣⡀⠀⡆⢠⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⡇⠀⠠⣞⠁
⡇⠀⠀⠀⠀⠀⢰⢀⡉⠀⠳⠎⠀⢳⠈⢧⠀⣇⢸⠀⠀⡇⠀⡆⠀⠀⠀⠀⠀⡇⢸⣧⠘⠦
⣿⠀⠀⠀⠀⢀⡇⡜⠁⠀⠀⠀⠀⠀⢧⡈⡇⣿⢈⠀⠀⣇⠀⠛⠛⠛⠂⠒⡄⠃⠸⠈⢧⠘⢣
⡇⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠈⠳⠇⠈⠛⠀⠀⠘⠓⠒⠲⠖⠒⠋⠀⠙⠃⠀⠈⠳⠞
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⠄⠀⠀⠀⠐⠒⠒⠲⠶⠶⠶⠶⠆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠂⠀⠈⠉⠉⠉⠉⠉⠉⠁
⡇
⢣
⢸
⢸⡆
⢸⡇
⢸⡇
⢸⡇
⢸⡇
⢸⡇
⠘⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢻⣆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠟
⠀⠀⠻⣷⣶⣤⣤⢤⢤⣤⢤⣤⣤⢤⣤⣤⣤⣤⣤⡤⣤⡤⠤⣤⡤⠤⠤⢤⡤⠲⠲⠖⢶⠖⠒⣩⡟⣻⠟⠁
⠀⠀⠀⠈⢿⣀⣳⣀⠀⠳⠀⠈⠃⠀⠘⠃⠀⠉⠀⠀⠈⠀⠀⠘⠁⠀⠀⠀⠁⠀⠀⠀⠀⢀⣞⣡⡞⠁