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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⡀⠀⢠⡐⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢉⣡⠄⢀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⠂⠀⠰⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠡⠒⠀⠒⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠚⠓⡆⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠢⠀⠀⠂⠂⣠⣴⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡛⣳⠄⠀⠀⠐⠀⠂⠀⠀⠠⢦⠤⠄⠠⠴⠦⣯⠁⠀⠀⡥⠀⠀⠀⠀⢀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠙⠋⠁⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠠⠄⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠠⠈⠀⠠⣵⣝⠟⠀⢨⠐⠉⠁⠀⠁⠁⠀⢀⠈⠁⢀⡠⠆⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠀⠂⠀⠀⠀⠀⠀⣀⠐⠣⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠓⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠤⣤⣄⣀⣀⣀⣀⣀⣀⠀⠀⢀⠀⢀⠀⠀⣀⢙⡀⡁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡠⠀⢒⡁
⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡀⠀⣠⠶⠶⠶⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⠦⣌⠁⢀⣀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⡀⠙⠂
⠀⠀⠀⢀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣿⣿⣿⣿⣿⣿⣿⣦⢀⣀⣀⡀
⠀⠀⠀⠂⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⡆⠀⢿⡄⠀⠀⠀⠀⢀⢀⡀⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢠⠞⣠⡿⠃⠀⠀⠀⠀⢀⡶⢮⣥⠴⢚⣡⠾⠬⣿⣄⣀⡄⣰⣯⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⢡⡞⠁⠀⠀⠀⢀⠀⢀⣾⡿⠃⠲⠊⠀⠃⢠⠀⣌⠃⠸⣄⣸⣿⣿⣿⣿⣿⣿⣿⣶⠙⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠂⢸⡇⠀⠀⠀⠀⠸⣤⢹⡏⢰⣷⠚⠧⡀⠀⡜⢉⣉⣿⠀⢸⣿⡿⠿⠛⣩⣍⠈⢿⣿⣤⣶⣦⠀⠉⠀⠉⠙⣿⣿⣿⣷⣶⡄
⢀⣠⠜⠁⠀⠀⣀⣤⠔⠁⠈⠃⠀⡇⠀⢠⣷⠀⣆⣿⠾⠿⠒⣼⣿⡿⢀⣀⡀⠉⠀⠀⠉⠛⠻⢿⣷⣿⣿⣆⠀⠈⢻⣿⣿⣿⡏
⡟⠀⠀⠀⠀⠀⠙⢧⣞⡁⠂⠀⠀⠛⠛⠘⠛⠀⠀⢠⠀⠀⠀⢻⣿⣧⣿⣿⠈⠀⠀⠀⠀⠀⠛⢷⠀⢀⣿⣟⣀⠀⢈⣿⣿⠟⠁
⣇⠀⠀⠀⠀⡀⠀⠀⠠⡀⠁⣤⠀⠀⠐⣦⢄⣉⠈⣀⠄⣸⢷⠈⠁⠘⠛⢥⡌⠀⠀⣠⣶⡄⢂⡾⠀⣸⣿⣿⣿⣿⡿⣿⣥⡄
⠙⢦⣀⠀⠀⡧⠌⠳⢤⣈⣉⡽⡄⠀⠀⢻⣶⣬⣬⣤⣾⣿⡆⠀⠀⠰⠁⠀⠀⠀⢰⡿⠟⠀⢸⣠⢆⣿⣿⣿⡿⢿⣿⣿⣿⠏
⡀⠀⢹⣤⣠⣿⡷⠶⢀⣀⡄⢀⣹⣆⠀⠈⢿⣿⣿⣿⣿⡏⡏⠀⣷⡀⠀⠀⠠⠀⠀⠉⠑⠢⡾⠉⢸⣿⣿⣿⣥⣾⣿⣿⣷⡄
⡁⠀⠀⠈⢻⣥⡀⠀⠀⠉⠉⠉⠛⠻⢦⡀⠀⠻⢶⣓⡾⠋⣣⠀⢸⣏⣂⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⣿⠿⣿⣿⣽⣾⡿⢹⠇
⢧⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⠒⢶⠲⠻⢦⣄⣀⣀⣠⣾⣿⡀⠸⡿⠿⠿⢶⣶⠏⠈⠀⠄⠀⠀⠀⠾⢠⣿⣿⣿⡟⠀⠁
⠘⢦⠀⠀⠀⠀⠀⢀⠀⣀⠀⢿⠲⡄⢧⠀⠀⠈⠻⢍⠉⠉⠀⢧⠀⠑⠠⠔⠊⠁⠀⠀⠀⣠⣄⣃⣠⣾⣿⣿⠿⠛⠁⠀⠓⠄
⠀⠈⣧⡀⡄⠀⠄⢘⢇⠈⠡⠀⢂⠈⠋⠀⠀⠀⠀⠀⠉⠢⢄⢠⣷⣤⣀⣀⠀⠤⠴⢶⣿⣿⣿⡿⠟⠋⠁⠀⣀⡀⠀⠒⠀⢲
⠀⠴⠈⡇⠰⠄⠘⠘⠌⠄⠠⡡⢈⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣟⠁⡇⠀⢀⡠⠖⠋⠉⠀⠀⠀⠀⠀⣺⣷⠦⠤⠖⠠⠃
⠀⢀⠀⢃⠘⠆⠀⠣⢶⣀⠻⠿⠛⠛⠧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡞⠉⠀⠀⠀⠀⠀⠀⠀⣠⠞⠷⠶⡶⠖⢀⠃
⠀⠀⢄⢀⠁⠂⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠉⠲⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠋⠀⠀⡎⠄⡞⠁
⠀⠀⠸⣎⣀⠀⠀⠀⠀⠀⠙⠢⢄⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⢀⡠⠔⠾⢅⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⢹⡗⠖⠒⠒⣄⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠓⢶⣤⡴⠚⠉⠀⠀⠀⠀⠙⢦⡀⠀⠀⢰⡇
⠀⠀⠀⠀⢸⠇⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠉⠓⢄⠀⠀⠀⠀⠀⣀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣠⣿⡇
⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠑⢤⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⢾⡿⢿⠟⠋⡇
⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⠿⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣀⣠⣤⡴⣶⡿⠻⢿⣿⣯⣍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣤⣄⠀⠀⢰⣿⣶⡀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣧⠀⠈⠉⠻⣿⣦⣄⠀⠀⢰⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣠⣴⣦⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠈⠻⣿⣷⣄⠸⣿⣿⣿⣿⣿⡿⠏⢀⣠⣶⣿⡿⠛⠛⣠⣄⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⢰⣶⣶⣶⣶⡮⠻⢿⣷⣬⣿⣿⣿⣋⣠⣶⣿⡿⠛⠉⠀⠀⢰⣿⣿⡇⠀⠀⢀⣤⣤⣼⣿⣿⣏⣁⡀
⠀⠀⠀⠀⠀⠙⠛⠙⠻⣿⣿⡟⠋⠀⠀⠘⠿⣿⣿⠟⠃⠀⠀⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⣤⣤⣀⣤⣶⣿⣿⣿⣿⣿⣷⣦⣄⠀⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣰⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢹⡏⢹⣿⣿⣿⣿⣿⣿⣿⡟⠛⠿⢿⣿⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠋⢰⣿⣿⣿⣿⣿⣿⣿⣿⠘⢿⣿
⠀⠀⠀⢸⣇⢸⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⣿⡄
⠀⠀⠀⠈⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢻⣷
⠀⠀⠀⣠⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢾⣿⠇
⠀⠀⠀⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢺⣿⣿⠛⠛⠛⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⡆⣼⢿⣿⠛⠉⢹⣿⣿⣿⡿⣿⣿⣿⣿⠀⠀⣸⣿⡏⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠏⠀⠀⣿⣿⣿⣿⡿
⠀⠀⠀⣿⣿⣿⣿⢸⡟⠀⠀⢸⣿⣿⣿⡇⣿⣿⣿⣿⠀⠀⣿⣿⡇⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠙⠋⣿⣿⢸⡇⠀⠀⣼⣿⣿⣿⡇⣿⣿⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⢿⡇⠀⠀⣿⣿⣿⣿⡇⢿⣿⠟⠛⣀⣤⣿⣿⠀⠀⠀⠀⣀⣀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣷⠀⠀⢿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⢻⡟⠀⠁⠀⠀⠉⣽⣿⣄⣀⣸⣿⣧⣤⠉⠉⠉⠉⠀⠀⠀⠈⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⡆⠀⠀⣾⣿⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠈⠁

⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⡘⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠈⠉⠢⡀⠀⠀⠀⠀⠀⠀⢀⠁⢀⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⢀⠀⠘⡀⠀⠀⠀⠀⡠⠊⡠⠁⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠤⠀⠐⠂⢃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠂⠀⠉⠐⠆⠁⢀⠔⠊⡁⠀⢂⠄⡐
⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠐⠒⠠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⡠⠈⠒⠊⡁⠘⠁⡲⠋⢈⢔⡥⠊
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠂⠐⠁⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⡠⠀⠙⠂⠀⠒⠀⡠⡪⠈⢀⠡⠊
⠀⠀⠀⠀⠀⢀⠔⣡⠤⠠⢚⠠⢘⡠⠁⠠⠈⡑⠢⡀⠀⢀⠊⠀⠀⡀⡂⠔⠅⠠⡪⠊⢐⠥⢀⠱
⠀⠀⠀⠀⡰⠁⢈⠀⢀⠨⢄⠪⢐⠨⢐⢢⣜⠀⠀⠈⠀⡆⠀⠀⠀⡈⠐⠁⡨⠊⢂⠔⡡⢊⠄⡂⢡
⠀⠀⠀⡰⢁⠼⢡⠏⢐⠨⡐⠔⡀⠩⡀⠀⠆⠙⠢⡈⠀⡁⠀⠕⠁⠸⢄⠈⡀⢀⠁⠡⢂⠄⡨⢀⠈⠄
⠀⠀⠀⠃⠐⢀⡃⠀⠐⠅⡂⠅⠂⠅⠒⠀⠰⠀⠀⠈⡈⢿⠀⠕⠀⠸⡈⡄⠐⠁⡠⠂⠑⠡⢂⠄⡠⠘⡀
⠀⠀⠀⢆⠀⠀⠙⠢⢌⡁⠀⠀⠀⢀⠐⠠⠀⠀⠀⠀⢇⡜⣆⠀⠈⡀⠉⠉⠁⠚⠒⠴⠋⠩⠗⣕⠨⠀⡇
⠀⠀⠀⠈⢢⡀⠣⠀⠺⡁⠌⠂⠌⢂⠬⣃⢠⠀⠀⠀⠀⠑⠚⠓⠠⢌⡐⢑⠀⠠⡢⢂⢄⠀⠀⣈⢚⡠⢀
⠀⠀⠀⠀⠀⠉⠀⠄⡔⠁⡐⠌⣠⠀⢅⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠚⡄⢉⡁⡈⢉⡉⢁⠁⢨⠀⢀⡠⠒⠒⠲⣄
⠀⠀⠀⠀⠀⠀⢠⠀⡀⢅⠢⢀⠁⡀⢀⠈⢀⠘⠄⠀⠀⠀⠀⠀⠀⠀⢠⠁⠄⠈⠣⢄⠀⡄⠄⡠⠔⠋⠸⠒⡉⠐⠠⡪⠀⠘
⠀⠀⠀⠀⠀⠀⢸⠀⠄⡀⠈⠈⠀⠂⠀⠅⠐⠄⡆⠀⠀⠀⠀⠀⠀⢀⠃⡠⢑⠈⡢⢃⠌⢡⠀⠂⢐⢄⠖⠊⡁⣪⠞⣀⠓
⠀⠀⠀⠀⠀⠀⢈⠀⡐⠀⠁⢀⠀⡆⠀⣡⠐⠀⡇⠀⠀⠀⠀⠀⠀⠸⡀⠂⠕⠀⡢⢘⢄⠘⡄⠀⠑⠁⠠⠪⢊⡡⠊⠀⠓⠁
⠀⠀⠀⠀⠀⠀⠘⠀⠀⢈⡀⢨⠀⠃⠀⠔⡀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢅⠨⢀⠀⠑⡌⠒⠂⠤⠐⠀⠁
⠀⠀⠀⢠⠠⡀⣔⣀⣀⣀⡤⠺⢂⢴⣄⣀⣈⣀⢼⠢⡠⡴⠐⠔⠄⠄⠀⠑⡄⠐⠨⢀⠨⡐⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠈⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠂⠔⠠⠔⡀⠀⠀⡀⠀⠀⡀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠂⠁⠂⢕⠚⢠⠤⣤⠄⠖⠋⠐⠅⠈⠀⠑⠈⠒⠁⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣀⡤⠴⠤⠤⠤⠀⣠⠞⠿⣷⣤⣄⠀⣀⠴⠊⠽⠛⠽⠋⢢⢠⣤⡶⠖⠊⢑⠀⢢⠀⠐⠢⠄⣄
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠁⠀⣴⠃⠀⠀⠀⣈⠛⡼⠀⠀⠁⠐⠂⠛⠂⠸⡀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⠀⠀⠠
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠰⠇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠄⠀⠀⡴⢦⣄⣀⡉
⠀⠀⠀⠀⡔⠒⠶⢤⢤⣤⡴⠀⠀⠀⡀⠀⠀⠀⠼⠃⠀⠀⠀⠀⠀⠀⣴⠛⠛⠋⠁⠀⠀⠀⣼⠁⠀⠀⠈⠙⣆
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠞⠁⠑⢍⠻⣷⣄⠀⠀⠀⠉⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠐⠈⠀⠈⠏⠀⠀⠀⠀⠀⠀⢸⡼⣀⣠⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡍⠁⢠⡌⠉⠀⢀⣴⠶⠚⠟⢋⡁⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠠⠤⣬⠄⠶⡶⠛⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠻⠃⠐⠋⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⠶⠶⣶⠀⠀⣀⣀⣀⣀⣀⣀⣀⠐⠤⣀⠤
⠀⠀⠀⠀⠀⡤⠶⠶⠒⠛⢱⠉⣉⠉⠖⠂⠐⠸⡦⠰⠀⠉⠉⠉⠛⠉⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⢶⠞⠇
⠀⠀⠀⠀⠀⠃⠁⠀⠁⠀⠘⠌⠀⠀⠁⠀⠀⠀⢰⠃⠈⠩⠁⠀⠈⠍⠁⠘⠦⠀⠀⣀⣤⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⠁⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⣀⣀⣠⣀⣤⠖⠀⠀⠶⠤⠤⠖⠁⠀⠀⠠⡀⠀⠀⠁⢀⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣇⠠⣤⡴
⠀⠀⠀⠀⠀⠀⠹⠄⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠋⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠍⠀⠷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠸⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡦⣴⣶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢻⣇⢩⡿⠀⠀⠀⠀⣠⣶⠶⠾⠻⣿⠲⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣻⣶⣄⠀⠀⠀⠀⠀⢠⣢⣾⣟⠀⢹⠀⠀⠀⠾⣍⡿⠀⠀⠀⠙⣿⠤⢄⠐⠄
⠀⠀⠀⠀⠀⠀⠰⡇⡝⠁⠀⠀⠙⣯⣗⠀⠀⠀⠰⠟⢾⡿⣣⠎⠀⠀⠀⠠⣜⣿⡇⠀⠀⠀⠀⣿⢀⣂⣠⣦
⠀⠀⠀⠀⢀⡀⣼⣟⡇⠀⠀⠀⠀⠈⣿⡆⠀⠠⠆⢀⣨⠋⠁⠀⠀⢀⠤⠴⢶⡿⣹⣆⠀⠀⠀⢇⡘⠙⠻⡅
⠀⠀⠀⠀⠈⣰⡿⠉⣿⠀⠀⠀⠀⢠⣿⣧⣤⣶⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠿⠇⠀⠀⠀⠘⠀⠀⠄⠈⢀
⠀⠀⠀⠀⠈⠉⠿⡟⣟⢳⣄⠀⠀⢾⠋⠀⣼⠇⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠘⠦⠤⠊⠀⠀⠀⠀⠀⠨⠃
⠀⠀⠀⢀⡈⠼⣄⢸⠈⢾⡑⡄⣀⠜⢛⠈⠉⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠇
⠀⠀⢀⠀⠀⠀⠸⠇⣠⢸⠇⠉⠁⠀⠀⠀⡄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠨⡉⠁⠀⠀⣂⠐⠊⠁⠀⠀⠀⡀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠉⠀⠁⠙⠁⠂⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠠⠄⠀⠒⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠒⠀⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⣠⡔⠂⠈
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⡇⢏⣿⡀⠀⠀⠀⢀⣠⣤⣴⣶⣦⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢶⢶⣄⡀⠀⠀⠀⠀⠀⢀⣅⣼⡿⠀⢯⠀⠀⠀⣰⣛⣻⠁⠀⠈⠻⣖⣉⠉⠢
⠀⠀⠀⠀⠀⠀⢀⡖⠿⠋⠉⠉⠻⣽⣦⠀⠀⠀⢠⠾⣽⡿⣏⡶⠈⠀⠀⢀⣨⣿⡇⠀⠀⠀⠀⣿⠁⠌⢀⣄
⠀⠀⠀⠀⠀⠀⣠⡷⡇⠐⠀⠀⠀⠙⣿⡄⡀⢀⣀⡀⠘⠟⠋⠁⠀⠀⣀⣠⣥⣿⢣⡀⠀⠀⠀⡏⠴⠶⢿⡋
⠀⠀⠀⠀⢘⢁⣿⠓⣷⠈⠀⠀⠀⢀⣿⣗⢀⣄⠀⠙⣿⠀⠀⠀⠀⠈⢹⡇⠈⢉⣾⡏⠀⠀⠀⣸⣶⠀⡀⠓
⠀⠀⠀⠀⠀⠚⣷⡴⡿⣦⡀⠀⠀⣼⢟⠩⣹⡏⠲⢠⣿⡀⠀⠀⠀⠀⢀⣿⡀⢁⣿⣷⣄⣀⣴⣿⡇⠀⠀⢀⠘⠆
⠀⠀⠀⠀⡰⣴⡌⢱⠛⣼⠣⡀⠀⣰⣾⣶⠙⠐⠀⣸⣿⣷⣦⣤⣤⣴⣾⣿⠃⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠌⠉⠁⢹⡎⠀⢹⣇⠷⣯⣁⠻⢯⡆⠀⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⢀⠒⠂⠀⠀⣴⢻⣽⣿⣷⣼⠿⣋⣼⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠻⢶⡮⣽⣿⡷⠾⠛⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠘⠿⠿⣿⡿⠿⠿⠟⠛⠃⠀⠀⠀⠻⠿⠿⡿⠿⠿⠟⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣠⠔⠐⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠎⡀⠀⠁⠠⡀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⠀⠠⠆⠊⢁⠀⡠⠠⠤⠆⠰⣿⣶⣤⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠤⠀⢀⠱⡀⠐⠀⠐⡜⠀⠀⠀⠀⠀⠈⠉⠀⠴⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢀⣿⣤⣀⣀⢀⠁⠣⢭⢢⠁⠀⡎⠀⠀⠀⠀⠀⠀⠀⢀⠁⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠈⠀⠸⣿⣿⣿⢋⠎⠀⠀⠀⢘⠉⠘⠉⠳⠄⢀⣠⠆⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠈⣉⠴⠚⡄⠀⠀⠐⠈⠓⠠⢀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠠⠀⢀⢤⡀⠠⢀⣀⠀⢀⠇⠀⠀⠀⢀⠄⣀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠉⠈⠁⠈⠀⢀⠜⠁⢀⠎⡠⠐⠂⠀⢀⠀⠔⠋⠁⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⢀⠈⠐⠒⠒⠓⡄⠈⠁⢀⠒⠊⠌⠀⠀⣀⡀⣀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⢢⠀⠀⠀⠀⠀⢂⠀⠉⠊⠀⠠⠊⠀⠀⠀⠐⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⠀⠀⠀⡀⠀⠀⡜⠀⠀⠚⠒⠄⠀⠑⣄⣀⠄⠀⠀⠀⠀⠀⢠⡀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠞⠓⢢⠀⠤⠐⠃⠀⠀⠈⠀⠀⠀⠀⠀⢀⠠⠊⠀⢀⠔⠀⠀⠀⢀⠊⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⢀⠄⠊⠀⠀⠀⠀⠀⢸⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⡁⠀⠔⠀⠀⠀⠉⠉⠁⠀⢀⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠁⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⢀⡡⠀⠀⠘⠊⠀⠀⢀⠔⠒⠂⠉⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡔⠁⠀⠠⢪⣤⣬⡀⠀⠀⠈⠑⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⡸⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⡠⡠⠊⠉⠂⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢁⣁⡠⢴⠀⠀⠀⠀⣀⣀⣀⡠⠂⠀⢀⠔⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁⠀⠀⠈⠒⠠⠄⠀⠀⠀⠀⢀⡠⠶⠑⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠢⢤⡀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠛⠛⠛⠯⠥⡒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⣄⠈⠂⠀⠀⠀⠀⠀⠒⠉⠉⠉⠉⠙⢦⠐⠤⣕⠢⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡠⠆⠀⢄⡀⠀⠀⠙⣎⢳⣠⠀⠠⠀⠀⠀⠀⠐⠦⣀⠀⠀⠈⠀⠀⠈⢳⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠂⢀⡖⠉⢠⠀⠀⠀⢨⠓⢶⠀⠘⡆⣷⡇⢃⠷⣤⣄⠀⠀⠀⠈⢧⠀⠀⠀⠀⣄⠀⣇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⡾⡇⠀⠈⠀⠀⢀⠈⠀⢸⣇⠀⠁⣿⡇⣴⠀⠠⣄⠉⠓⠢⡀⠀⢀⠀⠀⠀⢸⠀⣿⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀⠀⠀⢸⣷⢹⠀⠀⢀⣀⠘⠀⠀⣾⢻⠀⣤⡿⠁⡸⠀⠁⠀⠀⠀⠀⣷⠇⢸⠀⢀⡇⡼⢀⡇⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠉⢯⣧⠀⠀⠈⠁⠀⣰⠃⡟⡰⠟⠀⠀⠑⣄⢀⣀⠀⠀⠀⠈⢠⠏⠀⣸⠃⣠⠞⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠃⡸⠙⠲⠤⠴⡞⠁⠀⠉⠀⠀⠀⠀⠀⠈⡟⠉⠀⠀⠀⠀⠁⡠⠖⢁⣨⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠤⠲⡄⠀⠀⠀⠈⢻⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠋⠉⠃⠀⠀⠀⠀⠸⣯⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⡄⠀⠀⢠⠎⠀⠈⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⡈⢢⡀⠀⠀⠀⢸⡇⡀⠱⡄⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⡼⠀⠀⠀⡏⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢣⠀⠉⠓⠲⠶⠚⠀⡇⠀⢣⠀⠀⠀⠀⠀⠀⢀⠎⠀⡼⠁⠀⠀⡼⠀⢀⠂⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠳⣄⣠⠞⠢⠤⠼⠃⠀⠀⢂⠀⠀⠀⠀⢠⠎⠀⠰⢡⣤⡤⠞⠀⠀⢸⡀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠉⠛⠒⢦⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠠⠏⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢦⠀⠀⡟⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠈⢧⠀⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⣰⠃⠤⠀⢀⠇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢢⡀⢀⣀⣈⡁⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢠⡤⠀⠀⡿
⠀⠀⠀⠀⡼⠁⠀⠀⢀⡞⠀⠘⠀⠀⠀⠀⠀⠀⣀⣀⠤⠄⡄⠀⠀⣷⡠⠀⠀⠈⠒⠤⢩⣀⣀⣀⠀⠀⠀⢀⣀⠀⠈⠀⠀⠉⠀⢀⠃
⠀⠀⠀⢠⠃⠀⠀⠀⡜⠀⠀⢰⠀⠀⠉⠉⢉⣉⡀⠤⠒⠀⡆⠀⡮⠔⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⢢⠀⠀⡰⠃
⠀⠀⠀⡘⠀⠀⠀⡜⠀⠀⠀⢸⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⢰⠀⠈⠛⠁⠒⠊⠉⠁⠒⠤⢄⣀⣀⠀⠀⠀⠀⠀⣀⡀⠤⠳⠺⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⢿⣯⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⠿⢿⣌⠼⠷⠭⣷⣜⣧
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠟⣡⠴⢋⣝⡷⣦⣤⣤⣾⣟⣩⣏⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⢀⣼⡿⢋⣽⡅⠰⢣⡾⠋⠚⠁⠀⠀⢛⣁⠈⠿⠛⣿⠃⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⢶⣄⡀
⠀⠀⠀⠀⠀⣸⠃⠐⠋⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⢰⢿⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣻⢟⣉⡿⠋⣿⠛⠻⠷⣤⡀
⠀⠀⠀⠀⣰⣿⠀⠀⠀⠟⠓⠀⣀⠀⠀⠀⠀⠀⣀⠀⠉⠈⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⢐⣟⡉⠁⠀⠀⠀⢀⣀⡀⠈⢷⡀
⠀⠀⠀⠀⠻⣽⣦⡀⠀⠀⠀⠈⠉⠛⠶⠶⠖⠛⠁⠀⠀⠀⢀⣠⠾⠋⠀⠀⠀⢀⣀⠀⠀⠀⠀⢠⣶⡄⠀⠸⣆⠀⠋⢀⡁⠀⠀⠀⢠⡈⠈⠁⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠈⠛⠶⠦⢤⣤⣤⣤⣄⣀⣀⣤⣴⡶⠶⢿⣯⣤⡤⣤⣀⣀⣀⣿⢹⡀⠀⢀⣀⣸⠇⣷⠀⠀⠙⣦⣄⠈⠙⠓⠶⠚⠛⠁⠀⢀⣴⠏
⠀⠀⠀⠀⠀⢀⣟⢲⣄⣠⣴⣶⣿⠏⠛⠛⠛⠛⠉⠀⣄⠀⠙⣯⡿⣿⣏⣩⣭⣅⠈⠙⢿⣿⠉⠉⠀⣙⠲⢤⣠⣿⠙⢿⣷⠶⠶⠖⠻⣷⣞⠋
⠀⠀⠀⢀⡴⠋⠙⣡⣿⡿⠇⣾⠏⢠⡆⠀⢸⠀⢤⠀⠈⠳⣄⠈⢻⣾⡏⠀⠀⠈⠛⠶⠾⠿⠶⠶⠞⠛⠛⠳⣿⡏⣰⣼⢇⡄⢠⡇⠀⢻⣾⣧
⠀⠀⠀⠘⢧⡴⠞⠉⠹⢿⣿⡏⠀⣾⠁⠀⣿⠀⠈⣧⠀⠀⠘⣦⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡟⣸⠁⢸⠁⠀⡌⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡏⠀⢠⡏⠀⠀⢸⡄⠀⠀⠘⠧⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⡏⠀⡼⠀⠀⡇⢿⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢸⠁⠀⠇⠀⠀⠀⢸⡸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠃⠀⠈⠀⠀⠀⠀⠠⠴⠒⠋⠁⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠛⠀⠀⠀⠀⠀⠀⢸⣧⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡾⢿⣟⣒⠶⠶⠶⠶⠶⠶⠆⣀⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣦⣤⣤⣤⣤⡶⣿⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡆⠀⠉⠛⡛⢿⠿⠿⢿⣿⡋⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⣿⡯⠿⣶⡄⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣗⢢⡀⢹⣿⠛⠛⢿⣏⡁⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⠉⠙⢻⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡄⠀⢸⡟⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣧⣻⡀⠀⢸⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⢶⣿⠇⠀⠀⣿⣟⣛⣲⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⠟⠃⠀⠀⠛⠚⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠀⠀⠀⠀⠉⠉⠉

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

⠀⠀⣶⠛⠛⠛⠳⠶⢦⣤⣀⡀⠀⠀⠀⠀⣀⠤⠒⠒⠒⠒⠒⠒⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠉⠙⠛⢶⣶⣍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣠⣤⡴⠶⠞⠛⠛⠉⠉⠉⠉⣻
⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⡴⠁⠀⠈⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⣠⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣀⠀⠀⠀⠀⢹⣀⣠⡶⠛⠁
⠀⠀⠀⠀⣀⣀⣤⣤⡤⢶⡷⠶⠖⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⢿⣷⣶⠾⡿⠧⣤⣤⣄⣀⡀
⢀⡴⠞⠛⠉⠉⠀⢀⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠛⢶⣷⡀⠀⠀⠈⠉⠙⠛⠶⣤
⢿⣄⠀⠀⢀⣤⠞⠋⠁⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣤⣤⡴⢾⡛⠉⠀⠀⠀⠀⠀⠀⢸⠛⢦⣄⠀⠀⠀⠀⢀⣼
⠀⠉⣻⡾⠿⣥⣤⣀⣀⡇⠀⠸⠓⠢⢄⡀⣀⣠⡾⠶⠛⠋⠁⠀⠀⢹⠄⠀⢀⡠⠔⡆⠀⢸⣀⣀⣹⣷⣴⡶⠚⠋⠁
⠀⢰⡏⠀⠀⠀⠀⠈⢹⣃⣀⣤⣧⣶⣾⢿⣝⡓⠇⠀⠀⠀⠀⠀⢀⣬⡿⢭⣅⠀⠀⡇⠀⢸⠋⠉⠀⠀⠈⠳⣄
⠀⠈⠛⠶⠶⠖⠚⠛⢻⠋⠉⡠⠇⠉⣴⣶⣮⢻⡄⠀⠀⠀⠀⠀⠈⢡⣶⣶⡝⣧⠀⢇⡀⢸⣤⣀⣀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⢿⣷⣿⡾⠁⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⠟⠀⠀⠑⢸⠀⠉⠉⠙⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡀⠀⠀⠀⠉⠉⠀⠀⢀⠤⠤⠤⢄⠀⠀⠈⠉⠁⠀⠀⠀⡰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⢳⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⣨⠇⠀⠀⠀⠀⠀⢸⠉⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⡸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⡄⠀⠀⠀⠀⢀⠔⠋⠉⠉⠲⣄⠀⠀⠀⠀⢀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠘⢦⡀⠀⠀⣎⡠⠒⠒⠲⢄⢸⠀⠀⠀⣠⠎⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⠦⣀⠈⠉⠉⠉⠉⠉⠋⢀⡠⠚⠁⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⡄⠀⠀⡴⠒⠉⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢀⣀⣀⠤⠤⠤⠐⠓⠒⠒⠓⠠⠤⠤⣀⣀⣀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠢⢄⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⡀⠀⢀⠀⡂⠀⠈⡆⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⡔⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡀⢀⠌⠀⠐⡄⠠⢄⠐⢄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠹⢦⡘⢄⠀⠀⠀⢀⠄⠈⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡂⢈⠘⠟⠀⠳⠈⠀⣐⠀⠀⠁⠀⠀⠈⠢⡀⠀⠀⠀⠠⠁⠀⠹⣄⠑⠄⠀⢀⣴⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⠀⣀⠸⡀⢠⣤⡄⢰⡓⠢⠠⡀⢀⠀⠀⠀⠀⠐⠀⠀⡀⠁⠀⠀⢀⠘⢁⡀⠲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠖⠌⢱⠅⣙⣦⣤⢶⡃⡝⠑⠲⣶⡀⠃⠀⠀⠀⠀⠀⠐⠀⠀⠠⢂⣠⡶⠛⠳⣄⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠑⠮⡧⠀⠹⠟⠁⠀⠀⠀⠀⠃⠄⠀⠀⠀⢠⠀⠰⣥⡴⠛⠁⠀⠀⡀⠙⢷⡀⠳
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠀⠉⠡⠀⠀⠀⠀⢀⠀⠀⠿⣦⡑⣄
⠀⠀⠀⠀⠀⠀⡀⠀⡠⠐⠀⠈⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣍⠒⢄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃
⠀⠀⠀⠀⠀⠠⠒⠀⠀⠄⠘⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⡒⠦⡌⡑⠴⡀⠀⠀⣠⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⢀⠀⠀⠈⠛⠷⢦⣀⣀⣀⣀⣤⣀⣀⣄⣀⣀⣀⣀⣀⡀⣠⡠⠒⠁⠀⠀⡠⠐⡀⠀⢠⡀⢰⠀⠀⠛⠖⠐⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠐⠀⠀⠈⠁⠀⢠⠿⣿⡫⠉⠙⠋⠉⠛⠙⢀⠀⢄⠀⠈⠠⠀⠀⢄⠲⡈⠄⢀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠈⣇⠀⠀⠀⠀⠀⠀⠀⢂⠀⠑⠀⠀⠀⠈⠀⠑⠄⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠠⠀⠈⠑⠄⡀⠈⠃⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢄⢠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⡄⡠⠐⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠠⠞⠁⠀⠐⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⠀⣀⡀⠀⠀⠄⠀⠀⢀⠀⠀⣀⣀⠠⠄⠂⠉⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠉⢩⡍⡹⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⢺⠄⠆⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⢠⡀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠒⠒⠠⢀⠀⣤⢤⡄⢸⢤⡀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢁⠀⠁⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⢀⠀⢀⠀⢀⠀⢳⡶⣄⡸⢾⡃⠁⠀⠀⠀⠀⡀⢀⠀⢀⠀⠀⠀⣀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠈⠀⠈⠁⠀⠀⠀⠁⠈⠁⠈⠛⠉⠛⠋⠁⠁⠀⠁⠉⠂⠁⠀⠁⠈⠀⠈⠀⠈⠀⠀⠀⠁

⣞⣟⣉⣉⣉⣉⣉⠉⠉⠉⠙⣶⣶⡄
⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⡏⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⣛⣙⣿⣻⠿⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣟⠋⢹⣏⠉⢉⣿⠳⣮⣛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⣿⠀⣼⠃⣠⠞⠛⣯⡻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠤⠤⠤⠤⠼⠦⠼⠷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠿⠧⠿⠶⠧⠾⣧⣤⡴⠛⣷⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣥⣤⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢦⣄⣀⠀⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⢠⡞⠁⠙⣿⣦⡈⠙⢻⡿⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⡴⠋⠀⠀⠀⠈⢧⣙⣶⢟⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣰⠎⠀⠀⠀⠀⠀⠀⠀⠻⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢿⣿⣿⣿⣿⡿⠋⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠛⠛⠛⠛⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣠⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣼⠀⠀⣿⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠉⠉⠉⠉⠀⠀⢹⡀⢀⣿⠀⠀⠉⠉⠉⠙⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⣤⣄⣀⣀⡀⠀⠀⠈⠉⠉⠁⠀⠀⠀⣀⣀⣀⣤⣴⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉
⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠜⣍⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡀⠀⠀⠀⠈⡞⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠱⠀⡀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⠓⣄⠀⠀⡇⢀⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⢀⡤⠒⣫⠉⡶⠉⣷⠀⠋⠀⠈⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠁⡀⠀⢈⠖⠁⠀⣀⠡⠴⠁⠘⢀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠀⠀⠀⠀⠙⣗⠂⠚⠢⠔⠊⢀⣖⠦⠒⡮⠏⢁⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⡀⠈⠁⠀⠀⠀⣠⡞⠋⠤⡆⠁⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠲⢄⠜⠁⠀⢠⠀⡸⠀⠸⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⢇⣠⡀⠀⡈⢈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠖⢠⠃⢐⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠑⢬⡍⠋⠁⠚⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢤⣄⣈⢹⠂⢴⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢺⣹⠔⠚⠈⠉⠈⠃⠻⠄⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢫⠀⢀⣰⠲⢤⣤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠤⠀⠛⠾⠋⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⢀⡀⠀⠀⠀⢐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢛⡀⠈⢿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡅⠀⠀⣀⠀⠕⠢⠀⡀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠾⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢈⣄⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣛⣛⠃
⠀⠀⠀⠀⠀⠀⡌⣁⣈⡈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⠀⠀⠀⠀⠀⠀⠀⠀⠰⣜⣟⣿⡳⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⣀⣀⠀⠀⠀⠀⠀⠀⠈⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⣿⡷
⠀⠀⠀⠀⠀⠀⠈⣹⣿⡉⣀⣈⠿⠿⠂⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⢟⣇⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠰⢒⣒⣒⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⣍
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠛⠘⣍⣉⣏⠌⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣶⣿
⠀⠀⠀⠀⠀⠀⣧⡉⠐⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠆
⠀⠀⠀⠀⠀⠀⣬⣤⣤⣬⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⠶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠂
⠀⠀⠀⠀⠀⠀⠙⢿⣿⢟⣤⣶⡀⠀⠀⠀⠀⠀⠀⠀⠈⣭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠰⠾⠿⠖⠀⠻⠛⠁⠀⠀⠀⠀⠀⠀⠾⠿⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠘⠿⠇⠀⠀⠀⠀⠀⠀⠀⠠⠾⠷⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠈⢩⣭⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣶⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣭⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠰⣿⣿⡾⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⣰⣃⣠⡄
⠀⠀⠀⠀⠶⣿⡄⢈⣏⠰⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⠀⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣼⣿⣿⡷⢄
⠀⠀⠀⠀⠀⠈⠁⠉⠉⠁⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠙⠋⠁

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⡀⠀⠀⠀⠀⠀⠀⠠⠶⣷⣦⠀⠀⠀⠀⠀⠀⢀⠉⠁⠄⡀⡈⠐⠀⠀⠀⠀⠀⠀⠀⠞⢃⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⢀⠁⠀⡀⠱⡐⠀⠀⠀⠀⠀⠈⢀⢁⠃⡅⠇⠀⢀⠀⠀⠀⠀⠀⠀⡆⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠄⠀⠀⠀⠀⠈⠀⠁⠀⠀⢀⢠⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⣀⠐⠀⠀⠀⠸⢸⠀⠀⠀⡀⢆⠀⠀⠆⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⣇⠁⠀⠀⢡
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡀⢀⠔⡀⠀⠰⠀⠰⠰⢸⠠⢀⠌⠀⠀⡁⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⡠⠶⢤⠤⡔⠀⠀⠀⡀⢃⣀
⠀⠀⠀⠈⠢⡀⠀⠃⠸⢠⢠⠰⠤⠖⠛⠁⠨⣴⣄⡆⢀⠀⠀⠀⠀⠘⡀⢞⠀⠀⠈⠁⠐⠀⠀⠀⠀⠈⠀⠀⠀⣠
⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠈⠀⠀⠀⢀⠄⠀⠀⠀⠀⠘⠀⡆⡆⠀⠀⠃⠀⡉⠐⠠⠀⠀⠀⠀⠀⠀⢰⠀⠀⠐⠁⢠⠄⡀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠗⠲⠶⡄⠀⠀⠀⠃⠃⠀⠀⢘⠰⠀⠀⠰⠀⠀⠀⠀⠀⡄⠈⠈⠀⡀⠀⠆⠀⢠⠃⠀⠀⠄⠤⠔⡔
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡰⢍⠀⠀⠀⠂⠀⠉⠁⠀⠀⠀⠀⠃⠀⣇⡬⠁⠈⠉⠁⠈⠁⠀⠀⠀⣠⡄⣀⣀⣀⣀⠀⠀⢸⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⣀⣀⣀⡀⠔⡀⠀⠀⠀⠀⠉⠀⠈⠉⠙⠀⡠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠰
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠁⠀⠀⠖⠀⢂⢠⠀⠀⠀⠉⠀⠀⠂⠀⠀⠀⠈⠀⢘⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⡿⠛⣿⣿⡏⢹⣿⣿⣿⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣧⣤⣈⢟⠀⣾⣿⡏⢸⣿⣿⠃
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡟⠛⠉⠉⠉⠉⠀⠀⠙⠋⠤⣾⣿⣿⡇⠀⠀⠀⠀⠈
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⠁
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠁⢀⠔⠒⠶⠄⠀⠀⠀⣠⠖⠲⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠟⢧⠀⠀⢠⣶⣒⠀⠀⠀⠈⠠⢖⢶⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣯⣵⣯⣼⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⡄⠀⠈⡆
⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⡽⢻⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠓⠀⠀⣿
⠀⠀⡀⠀⠀⠀⠀⠴⢞⣿⣧⣢⣝⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣰⠞⠛⠻⣿⠀⠀⣿⡦⠤⠤⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣏⣉⣿⣦⣤⠀⠀⠀⠀⢀⣿⣿⣿⣿⡏⠀⣸⠇⠀⠁⠰⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠻⠖⢤⣀⠀⠀⣿⡀⠀⢸⡇⠰⠉⠀⠠⠀⠠⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠾⣿⣿⣿⣿⣿⣿⡿⠀⠐⠀⠈⠁⠐⣄⡘⠷⣤⠾⢣⠀⠂⠀⠀⠀⠀⠀⢐⡰⠄
⠀⠀⠀⠀⢀⣠⡾⠉⠀⠀⡽⢻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢀⣸⠃⠀⠀⠀⠀⢀⡴⠋⠐⠀⠈
⠀⠠⠀⡴⠋⠁⠀⠀⠀⠈⢷⣼⣿⣿⣿⣿⣿⣦⣦⡀⠀⠀⠀⠀⠈⠏⠙⢿⠁⠀⠀⠀⢀⠔⠛⢷⣄
⠀⣡⡼⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⢻⣏⢠⠏⠉⠉⠂⠂⢠⠊⠀⠀⠀⡄⠀⠀⠀⢻⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠂
⣾⠃⠀⠀⣠⠶⠛⠛⠛⠲⠦⣄⠀⠀⠀⠀⠹⡎⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⡀⠀⠀⠘⣇⠀⠀⠀⠘⣧
⠈⢧⠀⢸⡅⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⡜⠀⠀⠀⠀⠀⡘⢆⠀⠀⠀⠀⠘⡇⠀⠀⠀⠘⣆⠀⠀⢠⡏
⠀⠈⠳⡌⣷⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡰⠀⠀⠀⠀⠀⡰⠁⠈⢆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⢆⠀⡞⣷
⠀⠀⠀⠘⢿⡆⠀⠀⠀⠀⠀⠀⠀⢇⡰⠁⠀⠀⠀⠀⢠⠃⠀⠀⠈⢧⡀⠀⢰⠟⡆⠀⠀⠀⠀⠈⢿⠁⠸⡆
⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⢀⠏⠀⠀⠀⢀⡀⢳⣀⠎⠀⣧⠀⠀⠀⠀⠀⠀⠳⡀⠃
⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠤⠤⠴⠖⢻⡇⠀⡏⠀⠉⢻⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢸⠃⠀⠁⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⢈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⢀⠀⠀⡀⠀⢀⡿⡆⣠⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠸⣷⣄⠀⢳⣰⡇⢠⣸⠁⣷⠇⢸⠁⣤⣴⠆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣝⢦⡀⠢⡈⠲⣿⡙⢶⡀⢻⣇⠀⡏⠀⠙⠀⣿⠀⠁⣾⠞⣰⢁⣤⢔⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠢⣌⡃⠙⢦⡈⡦⠈⢅⡄⠀⢧⠉⢸⠀⠀⡤⠀⠀⢰⠆⠏⢀⠀⠚⠁⣽⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣶⣥⣀⣬⠿⠶⠱⣦⠙⠆⠀⠉⠀⠠⠄⠈⠀⠇⠀⠸⠀⠈⡠⠀⠋⢠⠀⢘⡿⠋⣀⠄
⠀⠀⠀⠀⢤⣀⠀⠀⠒⠮⣿⡌⢍⠁⠠⡀⠈⠀⢢⡀⠁⠀⠀⠄⠀⢀⠀⠂⠀⡀⢀⠁⢠⠀⠁⠀⠠⠴⠯⠗⣶⠃
⠀⠀⠀⠀⠀⠈⠙⠳⠦⠍⠒⠦⣄⠁⠀⠉⠀⠠⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣷⡀⠀⣾⡴⠀⠀⠀⠀⠀⠬⣭⣤⠄
⠀⠀⠀⠀⠀⠉⠙⠛⠒⠲⠀⠠⡀⠀⠀⠙⠂⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠠⣄⣿⠳⠀⠻⠁⠀⠀⠀⠀⠐⢻⠯
⠀⠀⠀⣤⣤⣽⠿⠧⣤⠀⠒⠚⠛⠂⠈⠁⣀⠀⠀⠀⠈⠀⠀⠀⠠⣤⣬⣆⠈⠛⠀⠀⢀⣄⠀⠀⢠⣀⢸⡇
⠀⠀⠀⠀⢀⣀⣠⡤⠀⢉⠿⠒⠀⠠⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢙⣦⠀⠀⠀⠀⠀⠋⠥⡀⠀⠀⠪⢸⡇
⠀⠀⠀⠀⠀⠰⠶⣤⣤⡄⠐⣋⡀⠀⡴⠀⠠⠀⠀⠀⠒⠀⠀⠀⠸⣏⠀⠀⠀⠀⠀⠀⠰⠇⡇⠀⢞⢣⢺⠇
⠀⠀⠀⠀⠀⠐⠋⠉⠐⠚⠉⡉⠁⠊⠀⣠⠀⠀⠀⠄⠀⠀⠀⠀⠀⡞⠀⠀⢀⣀⣀⡀⠈⠒⠃⢀⠬⠛⠛⠛⠻⢶⡞⢻⣦
⠀⠀⠀⠀⠒⠚⣋⣴⡾⠀⠚⠇⠀⠚⠀⡄⠀⠆⠀⠀⠂⠀⠀⠀⠈⠃⠀⠀⠀⠹⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁
⠀⠀⠀⠀⠤⠚⢩⣟⣥⡶⢀⣠⣀⣄⡀⡁⠰⠀⠉⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣄⣀⣀⠀⠀⢀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠠⠟⠋⢰⡷⠛⢛⠁⠀⠉⠀⠀⣄⣀⣀⣀⡀⣶⠀⠀⠀⠀⠀⠉⠉⠽⣦⣠⣶⣭⡽⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣴⣤⣤⡼⠶⠏⠈⠉⠉⠁⠙⠳⠶⠶⢶⣦⣀⣠⣤⣾⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠓⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠐⣀⣀⠤⣄⡐⠊⠀⠀⠁⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⢀⠔⠊⢡⠴⢃⡀⢺⡅⠀⣏⠁⣿⠃⠂⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠞⠀⣤⣽⣎⠁⣾⣚⢨⣖⠇⠰⠾⢧⣄⠁⠄⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡔⢱⢄⡨⢟⢯⠉⠰⢒⢩⠛⢀⠀⡏⠧⣽⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⡋⠊⠓⠫⠎⠉⠀⠀⡇⠀⠑⠅⠂⠉⢑⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠙⢷⠀⠉⠋⢩⠉⠒⡉⠒⢚⡉⣥⣉⠜⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⢄⣁⡈⣁⣃⠀⠄⠀⠀⠈⠋⠀⠀⠀⠀⣠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠀⢠⠀⠀⠀⠀⠀⠀⢀⠠⠔⠋⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠘⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⢀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢀⢀⡐⠀⢀⡀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠫⠤⢭⠀⠉⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢲⣤⡸⣀⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢚⠉⢑⠰⠻⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡀⠉⠁⠒⠒⠀⠤⠄⠠⣧⠬⠄⠚⡂⡤⢏⡗⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠠⠄⠐⠂⠒⣿⢸⠀⠀⠠⠒⢧⡟⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡹⣍⢠⠱⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢈⠉⠢⠀⠀⡊⠀⠀⢻⠀⠀⢀⠄⠀⠀⡌⠁⢱⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⣀⠙⣑⠀⣇⠄⢌⠁⡟⠀⠁⠉⢋⡠⠠⠀⢱⢐⡑⢁⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠴⠀⠀⠯⢔⠖⠉⠀⠀⠀⠀⠎⢐⣓⠤⠎⠈⠐⡂⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠔⠹⢀⢀⢠⡎⠀⠀⠀⠀⠀⠀⡄⠑⢄⢑⠀⠀⠘⠈⠢⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠣⡀⠀⠀⠀⢀⠔⠁⡘⠀⢫⠃⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣀⣉⣔⠉⢁⡀⢠⠃⡰⠃⡆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢀⠊⠀⠀⠘⠓⠁⠈⡡⠊⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡑⡈⠢⠔⠁⢰⠀⠀⡔⠁⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢼⠓⢄⣀⡘⢀⡊⣀⠴⠶⣒⠕⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠈⠣⠘⠁⠁⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠠⡀⠀⠀⠀⡩⣀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠠⡤⢷
⠀⠀⠀⠀⠀⠀⠀⠪⡑⢄⣈⣦⡀⡸⠀⠀⠑⢤⣁⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠁⢘⠻⣅⠀⠀⠀⠈⠀⠉⠒⠦⡤⠌⠡⠤⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣼⡦⡀⠀⠦⢄⠀⢀⠁⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⢕⢝⣶⡙⠀⠉⡟⠛⠂⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⠛⠀⠀⠀⢱⣀⣀⣀⣀⣀⣐⣦⣤⣤⣤⠠⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠐⠙⠊⠉⠋⠉⠉⠉⠋⠛⠋⠉⠀⠈⠋⠉⠉⠭⠟⠓⠒⠓⠒⠘⠓⠢⠝⠃

⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⢀⡀⠀⠀⠀⢠⠀⣥⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢰⣧⣄⣷⣶⠀⠀⠀⠀⠀⠁⠐⠂⠜⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣤⠀⠠⣤⣄
⠀⠀⠀⠀⠀⠈⠀⠑⠒⣿⠀⠀⠀⠀⠀⠆⠈⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠛⠀⠀⠀⠀⢰⠀⠀⠀⠀⠹⣶
⠀⠀⠀⠀⠀⢠⠀⠀⠀⣿⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠰⠀⠓⠚⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠐⠦⠈⠾⠃⠀⠀⠀⠀⠀⠣⠀⠠⠾⠃⠀⠀⠀⠸⠆⠀⠶⠃⠀⠀⠀⠀⠠⠄⠠⠶⠀⠀⠀⠀⠀⠘⠦⠀⠀⠀⠴⠟

⠀⠀⠀⠀⣀⡀⠀⢠⣰⢄⡀⠀⠀⠀⡤⠄⠀⠠⠀⢤⠀⠀⢠⠂⠐⠒⠂⢲⡄⠀⠀⣠⡤⢀⠀⣤⣤⣄
⠀⠀⠀⠀⠹⡄⠀⠈⢹⠟⠀⠀⠀⠀⢂⠀⠀⠀⠀⠘⠀⠀⠀⢂⠀⠀⢀⠞⠀⠀⠀⠣⠀⠀⠀⠀⠀⠉⠀⠀⢠⢀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠄⠀⠀⠀⠀⠈⠈⠀⠀⠙⢻⡄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣁⡟
⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⢀⢇⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠸⠋
⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠘⠀⠀⠀⠐⠋

⠀⠀⠀⠀⣀⡀⠤⠄⠀⢀⡀⠀⠀⠀⠀⡀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⡄⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⢏⠀⡀⢀⡀⢻⡇⠀⠀⠀⠘⠓⠛⠛⢛⡗⠀⠀⠀⠂⠀⠈⠀⡶⠀⠀⠀⠀⠃⠀⠀⢹⡇⠀⠀⠀⠐⠗⢒⣆⣰⠛⣿
⠀⠀⠀⠀⠘⡀⠀⣀⢀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢸⠀⠀⠸⡇⠀⠀⠀⠀⢄⡋⢿⢛⡇⣿
⠀⠀⠀⠀⠀⢟⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠁⠀⠀⠀⢀⠐⠀⢀⡀⠀⠀⠀⠀⢸⠀⠀⠐⡇⠀⠀⠀⢀⡁⠀⠒⠊⠀⡁
⠀⠀⠀⠀⠀⠘⠃⠀⠚⠃⠀⠀⠀⠀⠀⠙⠀⠒⠛⠀⠀⠀⠀⠘⠀⠐⠛⠁⠀⠀⠀⠀⠀⠂⠀⠚⠀⠀⠀⠀⠈⠓⠂⠀⠈⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠛⠀⠉⠥⠞⢿⣇⣺⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡯⠟⠁⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠊⠀⠴⡌⠼⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⢶⣦⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠢⠞⠁⣀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣦⣼⣛⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡌⠄⢀⣴⣤⣠⣶⣞⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡹⠉⣻⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣴⡿⣿⣿⣿⣿⣿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣗⠝⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣏⣭⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢯⣿⣿⡿⣟⠳⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣽⠿⣿⠤⢎⣗⣼⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣾⣟⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠛⠬⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣛⣉⣩⡷⠟⠒⠋⡨⠋⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⣿⣸⣿⣶⡝⠁⠁⠉⠛⠒⠀⠀⠀⠀⠀⠐⠚⠋⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⠛⢯⠍⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⠗⣿⣿⣿⣾⣿⣿⣿⢶⠀⠀⠀⠀⠀⢠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢈⡂⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣿⠉⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠻⣿⣿⣿⣿⣿⣗⣐⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄⣘⡀⣨⣁⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣾⣿⣷⡴⡏⣀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⣰⣿⣿⣿⣿⣟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣼⣿⣯⣏⣛⢀⣽⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣽⣿⣿⣅
⠀⠀⠀⠀⠀⠀⢀⡎⣠⣠⢼⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠹⠛⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣰⢋⣀⡀⠇⢸⣿⣿⣿⣿⣿⣿⡟⠑⠉⠉⠿⠂⠀⣀⣀⣀⣤⡴⠚⠁⠀⠀⠀⠻⢿⣿⣿⠿⢿⣿⣿⣷
⠀⠀⠀⣠⣼⣯⣥⣤⣽⣾⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢿⣿⣿⣿
⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠙⠛⠙⣁⣐⣉⣈⣵⣖
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢒⣀⡀⠀⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⣿⡜⣻⣿⣜⣿⣿⣿⣿⢋⣨⡰⢂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⡀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⣻⣿⠟⣽⣿⣏⣿⣿⢿⣷⣿⣿⣿⣻⣾⣏⣣⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣥⣤⣤⣤⣄⣤⣤⣤⡄⠆⠑⡟⠜⣿⣿⣟⣻⣿⣿⣿⣫⣾⣿⣿⣿⠿⣫⣿⣏⣽⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣘⢆⡻⠟⠛⢿⣿⣿⣿⣿⣿⣿⢿⣿⠿⣯⣐⣿⣿⣿⣿⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡱⣽⣿⡇⢌⣹⣿⠿⣟⣻⠻⣦⠿⠻⣿⣿⣿⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡊⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢹⠀⠀⠀⠀⢠⣞⣙⡷⠤⠤⠶⠾⠾⠶⠦⠤⣤⣤⣒⠙⢦⠀⠀⠀⢀⡔⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡼⣧⣠⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠛⠿⣄⡀⢀⡜⣀⣬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠀⠀⡴⠚⢶⠀⠀⢀⣠⣀⠀⠀⠀⠠⠊⠉⠇⠀⠀⠀⠈⠙⢿⣎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⠞⠁⠀⠀⠀⠀⠉⠀⠀⠀⠸⣇⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⡠⠒⢳⡄⠀⡹⠃⠙⣦
⠀⠀⣀⡀⠀⠀⠀⠐⣧⠚⠁⣠⠴⢤⡀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠞⠀⠚⠁⣠⠖⢷⡒⠋⢻⠆
⠀⣼⠁⢹⣀⠀⠀⡜⠁⠀⠸⣥⣀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡴⠋
⠀⠘⠷⠶⢽⣷⡞⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢻⠀⠀⠀⠀⠀⠀⠀⢀⡤⢦⡌⢷⡀⠀⠀⣠⠶⠶⣄
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⡸⠀⠀⠀⠀⠀⠀⠒⢉⡀⢀⡿⠘⣧⢴⣊⣥⣤⣄⡏
⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣿⣷⡀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⢹⡏⠁⠀⠀⠈
⠀⠀⠀⠠⣯⡇⠀⠰⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⣿⣿⠇⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣛⠋⢹⠆
⢀⣀⣀⣀⣙⠁⠀⠀⢳⡈⠉⠛⢶⣶⣦⠤⣤⠄⠀⠈⠈⠙⠒⠦⠤⠤⠤⠤⠖⢉⡤⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⣦⠀⠀⠉⣿⢉⣀
⣎⡀⢈⠥⢼⡆⠀⠀⠀⠳⣄⡀⢈⣛⠯⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠋⠀⠀⠐⢿⣀⢹⠆
⠈⠉⠀⢀⠔⡇⠀⠀⠘⠦⣌⣉⡡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠒⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠘⠦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⡤⠔⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠸⡆⢠⡏⠀⠀⠀⠀⠀⠀⠀⠰⣏⠈⢧⣀⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠜⠀⠀⢠⡀⢹⡶⢤⡀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⡔⠹⡌⠙⠛⠀⠀⢠⡔⠹⣆⢀⡹⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢃⡟⠀⠀⢧⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠳⠤⠿⣆⠀⠀⠀⠀⢹⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⣦⠀⠈⠳⢤⡀⠸⡄
⠀⠀⠀⠀⠀⠀⣦⠰⠽⢦⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣠⠛⣧⠰⠋⠀⠀⠀⢀⡟⠀⠈⠑⢆
⠀⠀⠀⠀⠀⠀⢹⠀⣠⠜⣻⣦⡀⠀⠀⠀⠀⡟⠀⣷⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠦⠴⠃⠀⠀⣠⡞⢥⡀⠸⡆⠀⠀⠀⠀⢸⠀⣤⡀⢰⡈⡇
⠀⠀⠀⠀⠀⣴⠊⠉⢀⡞⠁⠀⢙⡶⣄⡀⠀⠻⠖⠋⠀⠀⠀⠀⢸⣄⣸⠄⠀⠀⠀⠀⠀⠀⣀⣠⠴⠊⠁⢹⡄⠙⡆⢿⠀⠀⠀⠀⢸⡼⠁⠹⣄⡿⡗
⠀⠀⠀⠀⠀⠘⠳⠶⢞⠁⠀⠀⠸⡇⠀⠉⢳⠶⣦⣤⣤⣠⣀⣀⣀⣉⣡⣤⣤⣤⠴⠶⡖⡟⠧⠙⢦⡀⠀⠈⠛⠊⠁⢘⡀⠀⠀⠀⠈⠁⠀⠀⣿⠀⠁
⠀⠀⠀⠀⠀⠀⡞⢀⡎⠀⠀⠀⠀⣷⠀⡴⠋⣼⠁⠀⣰⠁⠜⠁⠹⢡⠀⠘⠒⠂⠀⠀⡇⠙⢶⡄⠀⠙⢦⡀⠀⠀⠀⠘⢧⣄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠇⠈⢶⠲⡄⠀⠀⢻⡘⠷⠤⠟⠀⣰⠃⠀⠀⠀⣇⣸⠀⠀⠀⠀⠀⠀⠙⠒⠚⠁⠀⠀⠀⠙⠳⡶⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⠀⣆⢸⣇⡸⠀⠀⠘⣧⠀⠀⢀⡴⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢹⡄
⠀⠀⠀⠀⣄⠹⠦⣬⣷⡼⠟⠀⠀⠀⣽⣂⠀⠈⠳⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣇⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠳⢦⡀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢉⡵⠞⠛⠶⠲⠦⠟⣠⠼⠦⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠇⠀⠀⠤⠍⡁⠶⠂⠴⣤⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⢼⡟⣥⠶⣄⣀⡴⠒⠦⣤⡤⠛⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠭⠔⠒⢦⣀⣠⠔⠒⢠⡀⠀⢀⡨⣷⣄⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠿⠊⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⢧⣽⠙
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶
⣿⣿⣇⣀⣀⣀⣄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠠⡄⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠀⢀⣀⣀⣠⡀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣼⣿⣦⣀⠀⣿⣿⣿⣿⣿⣯⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣭⣽⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠐⠛⣻⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⠟⢻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠟⠋⡙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣶⠈⠰⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⡀⠀⠂⠀⠰⠒⠚⠦⠹⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⠐⠶⣾⣿⣷⣶⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⣿⣿⣿⣿⣦⡀⠀⠈⠿⠿⠿⠿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣛⣉⣤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡠⡂⠜⢀⠀⠈⢻⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⡄⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠻⢿⣿⣿⣿⣿⠈⠃
⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⢸⠃⠀⢀⡾⢾⣿⠙⠻⣿⣿⣿⠿⠟⠛⠋⠉⢧⠤⣤⡀⠀⠂⠀⠀⠉⠀⠀⢀⡇
⠀⠀⠀⠀⢠⠍⡜⢰⡆⠀⠛⢠⢀⢸⢃⡀⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠈⠃
⠀⠀⠀⠀⠀⠈⣵⠟⣿⠀⠀⠀⠞⡀⢸⡇⠀⢻⣦⡀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠤⠤⠆⠀⡀⠀⢸⡇⠀⠈⡀
⣀⠀⠀⠀⠀⠀⣉⠰⣿⠁⡤⠀⠸⠇⡀⣧⢠⡇⣿⣿⣦⡀⠀⠀⠀⠀⢦⣀⠀⠀⠀⠀⠀⡼⠁⠀⠈⢀⡄⢀⢹
⣿⣿⣷⣦⣄⠀⣿⠀⣿⠀⠀⣤⢠⠀⢸⡇⣼⠁⠈⣿⣿⣿⣷⣄⠀⠀⠀⠈⠉⠉⠉⠀⠀⠆⡰⠁⡄⠈⠑⠋⡋⠀⢀⣀⣤⣤⣤⣤⣶⣶⣿
⣿⣿⣿⣿⠇⣰⠏⣼⣿⡇⢀⢸⠈⣧⠀⡇⠀⠘⠀⠀⠈⠙⢿⣿⣿⣶⣤⣤⣄⣤⡄⡀⠀⠀⣷⢸⡇⠀⢰⡞⣱⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣯⠀⢿⢴⣿⣿⣇⣾⣆⠃⢸⡀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣿⣿⠏⠀⢰⡇⢠⣧⠸⢀⣾⠈⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡄⠀⣼⠃⠙⣻⣦⢹⡀⢸⡄⢹⡀⠀⠀⠀⠀⢀⠀⠀⢰⠂⣿⡟⠁⠀⠀⢰⡇⡸⠀⢀⡾⠁⣠⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣆⠠⣾⠄⢹⡟⢨⣽⡜⡃⠘⣧⠀⠀⠀⢀⠇⠀⡄⢠⡼⠋⠀⠀⠀⡜⡜⢰⡇⢹⡿⠁⢀⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠟⢉⡟⠿⡀⢁⡘⠀⠀⠀⠉⣰⡇⠃⢸⡆⠀⠀⠈⠀⠞⢠⡟⠀⠀⡀⠀⣰⣡⡇⠸⣃⣿⠁⠀⠟⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢿⠃⣼⡀⠀⠀⢰⡆⢀⣿⠀⠀⢠⣿⣿⡆⢀⡇⠀⠀⠀⠀⠀⠤⠀⣀⣼⠁⢰⢃⠋⡷⠀⣿⠇⠀⣶⠀⢸⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢃⣾⣣⡇⠖⠀⠀⣣⠸⣯⡀⠀⠛⠉⢻⣇⠸⣇⡀⢸⡀⠀⠀⠀⢰⣿⡇⠀⠀⢸⠀⣿⠀⣿⠀⣼⡟⡄⢸⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣾⢏⡿⢠⡆⠀⣠⣿⠀⠿⠀⠀⠀⠀⠘⢿⠂⢁⠓⠈⢷⠀⠀⢠⣟⣹⠀⡠⠀⢸⠀⣿⣧⠁⠀⣿⠁⠁⣾⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠎⡘⢁⡾⢃⡀⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠉⣿⣼⡇⢰⢸⠀⣿⣯⠀⠀⢻⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⢀⣤⡟⠁⣼⣧⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢸⠃⣿⡇⠈⣤⢰⣿⣿⠀⠂⡌⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠋
⢠⢋⣴⣾⡿⣽⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⡇⢰⣿⡇⠀⠏⡀⢿⣿⣇⢰⣿⡇⣻⣿⣿⣿⣿⣿⣿⠟
⣱⣿⣿⡟⣱⣿⣿⣿⣿⣤⣤⣠⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⡇⠀⡀⢸⡸⣿⠟⠸⣿⠁⣿⣿⣿⣿⣿⣿⣿⣇
⣿⠟⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣇⠀⠇⣸⣿⣝⠂⣠⣿⡇⢹⣿⣿⣿⣿⣿⣿⠇
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣯⠀⠀⡿⣿⣿⣧⣋⢿⣿⢾⣿⣿⣿⣿⣿⡏

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡖⣀⣤
⠀⠀⠀⠀⠀⠀⢀⣠⠴⣶⠶⢤⣄⠀⠀⠀⠀⠀⠀⢠⡏⢸⠟⣽⣵⠄⠀⠀⠀⠀⠀⠀⣠⠴⠶⡶⠶⣤⡀
⠀⠀⠀⠀⠀⢠⠎⠀⢠⡟⣆⠀⠈⢷⡀⠀⠀⠀⠀⣸⠷⠶⠶⠷⣿⡀⠀⠀⠀⠀⢀⡞⠁⠀⣰⠻⣄⠀⠹⡆⠀⠀⢀⡀⢶⠀⣇⢸⠆⡶
⠀⠀⣀⡀⠀⣿⠀⢀⡞⣀⢈⠳⣄⠘⡇⠀⠀⢀⡞⠁⠀⢠⢀⡀⠀⠹⣆⠀⠀⠀⢸⠷⠶⠞⣁⠀⠈⠳⠶⣿⠀⠀⢮⣹⡾⠛⠋⠙⠛⢷⣞⣉
⠀⠸⣧⣿⣀⢻⣤⠏⢠⣄⣈⣠⠈⣻⡇⠀⠀⢸⡁⠀⡎⠀⠀⠀⣠⠄⣿⠀⠀⠀⢸⡆⢰⣆⠁⠈⣁⡄⢀⡟⠀⠀⢳⡏⠀⠀⠄⠀⠄⠀⢹⡍⠀⠀⣴⠒⣦
⠀⠀⠀⠀⠉⢻⢿⣦⣄⣈⣉⣤⡖⠻⣿⠆⠀⠘⣇⠀⠙⠶⠶⠖⠁⢀⡟⠀⠀⣠⣾⠿⢦⣍⣉⣉⣁⡤⣾⣳⡄⠀⠘⣇⠀⠀⢤⣤⠄⠀⢸⣇⡤⠞⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠈⣧⣄⠀⠀⠀⠀⢘⣷⣤⣄⣀⣤⢴⣯⣀⠀⢻⠏⠁⢀⡴⡟⠉⠉⢹⣦⣈⠿⢿⣦⠀⠙⢦⣄⣀⣀⣀⣠⡞⠉
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠸⡏⠛⢶⣖⣶⠋⣹⠀⠀⠀⠀⠀⣿⠉⠛⣶⠛⡶⠋⢰⠇⠀⠀⠀⢻⡉⠳⣮⣃⣀⡴⠋⡏⠉⠉⠉⠈⢷
⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣀⣀⣀⣿⠀⠀⠉⠁⠀⡟⠀⠀⠀⠀⢀⣻⠀⠀⠙⠛⠁⠀⡼⠀⠀⠀⠀⠘⣧⠀⠀⢧⡼⠁⠀⣇⣀⣀⣀⣠⣼⠇
⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢛⣚⣿⠛⠉⣿⣁⣀⣀⠀⠀⠀⠀⠷⢦⠶⠶⢶⡖⠛⠀⠀⠀⠀⠀⠀⠉⢉⡏⠁⠀⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠸⣧⣹⠇⠀⣯⡷⠀⠀⠀⠀⠀⢰⣋⣉⣹⠇⠀⢯⣥⣀⣽⠇⠀⠀⢠⡴⢾⡄⠀⢸⣗⣲⡄⠀⠀⠀⠀⠀⢾⣽⡇⠀⠀⠙⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⢀⣀⡤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⣰⡛⠋⠈⠋⠙⠃⠘⠛⣧⠀⠀⠀⠀⣠⡴⠶⡶⠶⣤⡀⠀⠀⠀⠀⠀⣰⢶⣄⣶⡆⣠⡄⠀⠀⠀⠀⠀⣏⠙⣧⢼⡁⣹
⠀⠀⠀⠀⠀⢸⡉⢀⡴⠒⠛⠛⠛⠶⣌⠻⡦⠀⠀⡼⠁⠀⣰⢿⡀⠀⠙⣦⠀⠀⠰⣷⣾⠞⠛⠛⠿⣷⣷⡆⠀⠀⠀⢀⡽⠶⠟⠛⠳⣋⣷⣄
⠀⠀⢀⡀⠀⢰⣏⣏⣀⠀⠒⠰⠂⢀⡘⣿⡗⠀⢸⡧⠶⠊⡁⠀⠻⢦⣤⣼⡇⠀⢀⡟⠃⠀⠀⠀⠀⠀⠐⣦⠀⠀⢀⡾⠶⠶⠶⠾⠛⢿⣤⣹⡆⠀⠀⣴⠛⣷
⠀⢰⣏⣹⡀⠀⢸⡇⠉⣿⢲⡖⢺⠋⠀⣻⠀⠀⣸⣇⠀⢦⣀⣈⣁⠀⠀⣸⣁⡀⢸⡁⠀⠐⠂⠀⠃⠀⠀⢸⠇⣤⡴⣦⠀⠀⠟⠀⠛⠀⠈⢹⡇⣠⠞⠻⠶⠟
⠀⠀⠉⠉⠙⠳⢾⣧⡀⠙⠛⠛⠋⠀⣠⠏⠀⠾⠥⠟⠷⣤⣈⣉⣁⣠⠞⠿⠽⠟⠘⣧⠈⠓⠶⠶⠒⠛⢁⡞⠀⠈⠛⠛⣮⡙⠛⠒⠚⢋⣠⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢨⡟⠶⠦⠤⠶⡾⢧⣄⣠⣀⣠⠴⣺⠋⠉⠉⠉⣿⢦⡀⠀⠀⠀⣨⣷⢦⣄⣀⣠⣴⣋⠀⠀⠀⢀⡤⢾⡟⠛⠛⠛⢿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⢿⡀⠘⠷⠋⠀⢠⠇⠀⠀⠀⠀⢿⠀⠙⣶⢲⠞⠁⣿⠀⠀⠀⠀⢻⠙⠳⣞⣿⠉⢀⡞⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠳⡶⠶⠶⢶⡶⠼⠇⠀⠀⠀⢀⣏⣀⣀⣀⣀⣀⢸⡆⠀⠉⠉⠀⠀⠻⣶⠶⠶⣤⡼⠇⠀⠈⠁⠀⣼⣁⣀⣀⣀⣀⣀⣸⠆
⠀⠀⠀⠀⠀⣠⠖⠾⣇⠀⠀⢸⠗⠒⠶⡄⠀⢀⣠⣤⣿⠉⠉⢹⣏⡉⠀⠀⠀⠀⠀⢀⣠⣽⡀⠀⣸⣦⣀⠀⠀⠀⠀⠉⠉⣿⠉⠉⢹⡁
⠀⠀⠀⠀⠀⠻⠶⠶⠋⠀⠀⠘⠷⠤⠶⠃⠀⠘⠷⠶⠛⠀⠀⠺⣯⣽⠂⠀⠀⠀⠀⢿⡤⠶⠋⠈⠷⣤⡽⠇⠀⠀⠀⢰⣏⣹⠇⠰⣏⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠯⠉⠉⣙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠬⠔⠒⠒⠒⠒⠲⢤⣀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⡀⠀⠀⠀⠀⠀⠀⣀⠈⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⣍⠂⠲⠤⠖⠒⢋⡍⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢿⣷⣶⣤⣶⣶⡟⠁⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠈⠛⠿⠿⠟⠋⠀⠀⠀⢠⡏⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢄⡀⠈⠉⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠝⠓⠲⠤⠤⠤⢴⠞⠫⢄⠀⠀⠉⠲⡀⠀⠀⠀⠙⢦
⠀⠀⢀⠤⠐⠒⠒⠒⠤⢄⣀⣀⣀⣠⡤⡾⠁⠀⠀⣀⣠⠴⠚⠁⠀⠀⢠⠇⠀⠀⠀⠙⣆⠀⠀⠀⠀⢳⡄
⡖⢶⠡⣆⣀⡹⢤⣼⣃⡀⢠⠤⠤⠤⠴⠷⠒⠚⠙⠍⠀⠀⠀⠀⢀⡴⠋⢀⠠⠂⠀⠀⠘⡆⠀⠀⠤⠊⢹
⠓⠒⠶⠮⠭⠥⠤⢤⣀⠉⠙⢧⡔⠁⢀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠛⠂⠉⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢸⡗⠺⢿⡹⣄⣀⣀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠸⡇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠈⡗⢬⣃⡇⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊⠁⠀⠀⠀⠀⡇⢰⠀⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⡇⠀⠀⠀⢧⠀⠈⠀⠀⠀⠐⠂⠉⠉⠀⠀⠀⠀⠀⠀⢀⠀⢠⡇⡸⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⣸⢁⡇⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡷⠀⠀⠀⠀⠈⣧⣄⠀⠀⠤⠤⠤⠤⠤⠶⠒⠉⠀⠀⢀⡴⢧⣼⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⢀⡞⠀⠈⠙⠓⠦⢤⣄⣀⣀⣀⣀⣠⣤⠖⠋⠀⠀⠈⠉⠛⠛⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⢀⣠⠤⣤⣇⣇⣠⠤⢤⡏⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⣠⡤⠖⠁
⠀⠀⠀⠀⠀⡜⠁⣜⣆⠀⢀⡞⢁⣤⡀⢻⠒⠾⠦⢤⠼⠤⠼⠶⠤⣤⣄⣀⣀⠤⠤⢄⡈⠉⠉
⠀⠀⠀⠀⠀⢧⠀⠛⢉⡗⢺⠀⢸⡠⠃⢸⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⡔⢢⠀⢷
⠀⠀⠀⠀⠀⠀⠙⠒⠋⠀⠘⣆⠀⠀⣠⠏⠀⠈⠉⠉⠉⠉⠉⠙⠒⢶⠒⢿⠀⠳⠁⢀⡟

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⢀⣽⣿⣅⠀⠀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⠀⢈⣿⣿⣏⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣿⣿⣿⣿⣶⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠙⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⠀⠀⠀⢀⣦⠀⠀⠀⠀⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠲⠀⠀⢹⡀⡼⠀⠀⢀⣼⣘⡆⠀⠀⠀⣠⠞⠁⠀⠀⡆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢀⡀⠈⡏⠳⣄⠀⠰⠃⠀⠹⠀⢀⣴⡁⠀⠀⠀⠀⡷⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⢀⣤⠤⣄⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⢀⣾⠏⠀⠀⡼⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⣠⣿⠃⠀⠀⠀⠁⢠⣿⠀⣀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠛⠀⠀⠀⠀⣀⣀⠀⠉⣹⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣀
⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⢸⣇⡞⠀⣹⡇⢰⣿⢠⠎⣿⠀⣼⣿⠀⠀⢰⡏⠀⠀⣷⠀⣿⠃⠀⣿⠇⡴⢻⡆⢀⣤⡀⠀⣴⡏⢉⣿⡿⠀⠀⢠⣾⠁⠀⢹⠄
⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⣿⠇⢸⣿⠏⠀⡿⠀⣿⡇⠀⠀⢸⣿⣄⡀⠀⠀⣿⠀⢸⣿⡸⠁⢸⣧⠏⢘⡇⢰⣿⠀⠈⢹⡇⠀⠀⢸⣿⣦⡀
⠀⢸⣿⠀⠀⠀⠀⠀⠀⡆⢸⡇⠀⠀⣿⠀⢸⡟⠀⠀⠙⠀⣿⠇⠀⢀⠇⠀⠙⠻⣦⠀⣿⠀⢸⣿⠃⠀⢸⡿⠀⢸⡇⢸⣿⠀⠀⣿⡇⠀⠀⡞⠀⠉⠻⣷⡀
⠀⠸⣿⠀⠀⠀⠀⠀⣰⠃⠘⠇⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⢹⣇⣠⠞⠀⣼⠂⠀⣸⠀⣿⠀⢸⡟⠀⠀⢿⡇⠀⢸⡇⠘⣿⡀⡰⠹⣇⣀⠜⠁⢰⠇⠀⢈⠇
⠀⠀⠙⠦⠄⠠⠴⠚⠁⠀⠀⠀⠀⠀⢹⡀⠈⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠓⠋⠁⠀⢻⡄⠀⠁⠀⠀⠈⠁⠀⠸⡇⠀⠈⠉⠀⠀⠈⠁⠀⠀⠈⠑⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠀⠀⠀⠀⠀⠀⠀⠙⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⣰⡆⠀⠀⠀⣴⠤⣀⠀⠀⠘⢦⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠀⠀⣰⣃⣳⠀⠀⠀⢸⠀⠘⡆⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠰⠋⠉⢹⡄⠀⠀⢸⣀⡴⠃⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠁⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣤⣤⣀⣀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⣿⣿⡇⠈⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⢀⢿⣿⡟⠁⠀⢸⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠳⠄⠀⠀⠀⡀⠠⠐⠊⠁⠃⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⠀⠀⣸⠟⠀⠀⠀⠈⣿⣿⡿⠿⢿⣿⡇
⠀⠀⠀⠀⠈⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢸⣿⣿⣿⠛⠛⠶⠏⠀⠐⠋⠀⠀⠀⠙⠒⣿⣿⡇⢀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⣿⣿⣿⣦⣤⠀⣀⡀⠤⠂⠁⣽⣿⠿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠹⢿⣿⡿⢿⡄⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠈⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠹⣿⡆⢃⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⡠⠙⢷⡘⡀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⣿⡇⠀⠀⠰⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⣿⠁⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⡏⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣏⠆⢾⣋⣙⣛⣋⣉⣙⣉⣋⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⣤⣤⣴⣾⣿⣴⣾⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠠⠒⠈⠦⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣼⣿⣿⠁⠈⠛⠿⠛⠻⠟⠿⠿⣿⡟⠀⠀⠀⣾⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⢸⣿⡏⠒⠓⠀⠀⠀⠀⠛⠂⠀⣿⡁⠀⠀⠀⣿⡏⠀⣀⠀⠉⠛⠛⠛⠛⢻⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡚⢿⠀⢀⠤⠄⠀⠀⠀⠠⠄⠀⡼⠘⠀⠀⢤⡿⢀⣤⡤⠀⠀⠀⢀⡉⠂⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠣⡬⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⢀⠀⠀⠄⣹⠘⠿⠂⠀⠀⠰⣿⠆⠀⣹⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢤⠀⠀⠰⣶⣶⡶⠀⠀⡰⠟⢏⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠳⢀⣀⡀⠀⢀⠀⠊⠀⠄⠎⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠈⠁⠀⠈⠁⠐⠀⠐⡀⠠⠀⣠⣤⣄⣀⣁⣠⡄⠐⢈⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⢀⠘⠂⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠉⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠉⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠢⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠴⢺⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠐⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠄⠂⠀⠀⠀⠆⠀⢲⠀⠀⠀⠀⠀⠑⠂⠠⠐⠀⠀⠐⠄⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠀⠀⠀⠀⠃⠱⠀⠀⠀⠀⠀⠘⠁⠇⠀⠀⠀⠀⠀⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠡⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⢀⡐⠀⢀⡁⠀⠀⠀⠀⠀⣅⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠶⠊⠁⠀⠀⠀⠀⠀⠀⠈⠓⠆⠀⠂⠈⠤⠒⠉⠀⠀⠀⠀⠀⠀⠈⠙⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠣⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡃⠘⢦⡀⠀⣀⠾⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠇⠐⠀⠈⠉⠀⡄⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠈⠄⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣏⡀⠐⠂⠀⠀⠀⣀⠠⠀⠀⠀⠳⠤⠤⢶⡞⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⣠⠋⢫⠀⠀⢀⣀⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠀⠀⠈⢲⡀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣰⠏⢃⠀⢀⠜⠁⠀⠀⠉⠉⣩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠡⠀⠀⠀⠀⢀⣃⣀⣀⠀⠀⠀⡰⠁⠀⠀⠑⠁⠀⢀⠁⠀⠈⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⡋⠀⠀⠀⠀⢀⣠⠤⢄⠀⡞⠁⠁⠀⠙⣆⡜⣁⣄⠀⠀⢀⠀⠂⠀⠁⠐⠠⢈⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠦⢤⣸⠀⠀⠀⡽⣇⠀⠀⠀⠀⡼⡏⠀⠀⢱⠊⠀⠀⠀⡠⠖⠊⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠙⠒⠊⠀⠈⠓⠤⠤⠚⠁⠑⠤⠤⠊⠑⢤⣀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢀⡤⠤⣄⠀⠀⠀⠀⠀⣀⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣀⡠⠖⠁⠀⠀⠈⠳⣄⢀⡠⠞⠁⠀⠀⠈⢦⠀⠀⡠⠤⢄⡀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠛⠉⠉⠁⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⠋⠁⠀⠀⠉⠒⠶⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠸⠿⠀⠀⠀⠀⠀⠈⠀⠀⠀⣴⡄⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⢶⠆⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠾⠂⠀⠀⠀⠀⠶⠄⠀⠀⠰⠆⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⢀⡀⣀⡇⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠸⡁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠁⠙⠓⡲
⠀⠀⠀⠀⠀⠀⠀⠙⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠚⠛⠋⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣽⣵⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣶⣶⣾⣿⣿⣷⣾⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠶⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡔⠂⠐⡄⠀⠀⣿⡣⠄⠙⠀⠈⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⢘⣃⡀⡀⠀⠀⢘⣖⣟⡠⠂⠀⠀⣿⡠⢄⠀⠲⢘⠀⠀⠀⠀⠀⠻⣷⠄⢀⡉⠁⠟
⠀⠀⠀⠀⢸⣥⣅⢛⣄⠀⠈⣼⠋⠀⠀⠀⠀⢸⠟⢢⠒⠂⠈⠀⠀⠀⠀⠀⠀⠘⣷⠄⡒⠊
⠀⠀⠀⠀⠀⣿⡯⠉⢩⡙⠡⠸⠀⠀⠀⠀⠀⢿⣦⠿⠰⡏⢰⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢹⣷⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠿⠓⢄⠘⢸⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠸⠙⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣶⡦⢠⠛⢸⠀⠀⠀⠀⠀⠀⢄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠛⠢⠴⠰⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⢀⣷⡀⠀⡀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡷⠖⠀⠁⣄⠁⠈
⠀⠀⠀⠀⠀⠀⢸⣿⢧⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠐⣠⠶⣌⣶
⠀⠀⠀⠀⠀⠀⢸⣿⣾⣦⠤⠦⣻⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣭⡏⠀⠩⠃
⠀⠀⠀⠀⠀⠀⠘⠛⠿⢀⡀⠀⠀⠀⠀⠀⠀⣯⣁⠌⠂⠒⢙⠀⠀⠀⠀⠀⠀⠈⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⣿⣧⢄⠀⢀⢘⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢁⢨⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠸⠀⠀⡀⠀⠀⠀⠀⣿⠟⠶⠄⠛⠛⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡍⡜⠻⠄
⠀⠀⠀⠀⠠⠔⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⢀⣤⣶⡆⠀⠀⠀⠀⠀⠀⣴⡴⠂⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠸⢀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡆⠀⠀⡼⡿⠿⡇⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠸⢰⣿⡿⣿⡤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⠄⠒⠁⠀⠀⠇⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⣾⠟⠏⠈⢣⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⣤⣯⠤⡀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠈⢄⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⣰⠃
⠀⠀⠀⠀⢀⡙⠻⠿⠷⠾⠿⣫⡄⠀⠀⠀⠀⠀⠀⠈⠲⢼⡷⠀⠀⠀⢠⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡦⢤⠇
⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣁⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠒⠂⠒⠚⠛⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠒⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠘⠢⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⡇⠀⠛⣿⡿⢃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠰⠉⠉⠲⠤⡌⣉⠀⡤⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⣼⡄⠀⠀⠘⠙⠓⠒⠒⣒⣲⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠺⣿⣧⡠⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⢻⡏⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢿⣿⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⣷⣂⡀⠓⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⣿⠧⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⣿⣷⣦⠀⣡⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣥
⠀⠀⠀⠀⠀⢀⣿⣻⣛⡓⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠉⠇⠀⠀⠀⠀⠀⠀⠀⠲⠤⠭⠭⠙⠂⠈⠤⠤⠂
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣠⣤⣤⢤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⡄⠀⢠⣴⣿⣿⣿⣿⣿⣷⣾⣿⣦⡀
⠀⠀⠀⠀⢰⣿⣿⣿⡏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢸⣿⣿⣿⠀⣿⣿⣿⣿⣿⡿⠿⠟⠋⠀⠉⠛⠃⠀⠀⠀⠀⠀⣿⠀⠈⠙⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⡇⣿⡟⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠈⠉⠉⠉⠁⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣼⣿⣿⣿⠿⠼⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠲⡀⠀⠀⠀⠀⠀⠀⠀⡴⣯⠀⠀⠀⠀⠀⠀⠀⠏⢹⣿⣿⡇
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠓⢄⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⢀⠠⡀⠀⠀⠀⠀⠀⠔⠁⢻⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠍⠐⣆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⣗⣾⣀⡤⠠⣔⣁⣀⡀⣈⠢⠀⠀⠀⠀⠀⢀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠐⡀⠀⠈⠑⣤⠀⠀⡀⠀⠀⡀⠀⠆⠉⠀⠐⠁⠀⠀⡠⠀⠈⠂⠀⠈⠁⠎⣀⠠⠚⡛⠛⡀⠜⠚⠷⠺⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠈⠐⠊⠀⡆⠑⢦⠤⣚⣁⣠⠤⠞⢦⣤⡆⠀⠀⠀⠀⠀⠀⢰⠀⣤⠔⠉⠀⠀⠌⠀⠀⠀⠀⠀⠀⠡⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠖⠀⠀⠀⠀⠀⠀⠌⠀⠄⠨⠄⠀⠂⠀⠀⠚⠋⠁⠀⠀⠀⠀⠀⠀⡎⠀⠀⠑⢠⣀⢰⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠠⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⡀⠀⠀⠒⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠐⢦⠒⠂⠒⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠦⠦⠶⠤⠶⢸⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⠂⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⢲⡀⠠⠂⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⠀⠀⠢⢨⡀⠀⠀⠀⠀⠀⣦⣤⣤⣤⣤⡤⠤⢴⡇⢡⡎
⠀⠀⠀⠀⠀⠀⠀⠐⣶⣮⠁⠀⠀⠐⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠨⡇⠀⠀⠀⠀⠀⠇⠀⠀⠀⡹⣄⡀⣾⣷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⡖⠒⠞⠳⠒⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⡇⠀⠀⠀⠀⠀⠃⠀⠀⢈⡃⠈⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⠁⠀⠀⠀⠀⠀⢹⠀⠀⠸⠃⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠀⢸⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢸⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣄⠀⢨⣀⣨⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⢸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⡁⠀⢈⢈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⢰⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠋⠀⠘⠛⠛⠛⠀⠀⠀⠀⠀⠛⠛⠛⠋⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⠉⠻⣦⡀⢠⣶⡄⢀⣠⣤⣄⠀⠀⠀⣠⡴⢶⣄⠀⠀⠀⠤⠴⠶⢶⣤⡄⣴⣦⠀⠀⢠⣶⡄
⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⠈⠁⢸⣿⣇⣾⠋⠀⠙⠃⠀⠰⠋⠀⠀⣿⡆⠀⠀⠀⠀⢀⣼⣿⠃⣿⣿⠀⢀⣾⢿⣿⡀⣀⡄
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⢠⣺⡿⠁⠀⠙⢿⣤⠾⠃⣨⣿⡟⠁
⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⣠⡶⠟⠉⠉⢻⣿⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⢀⡴⠋⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⢀⣶⠀⢸⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⣀⣾⣿⢀⣼⣿⠏⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣠⣾⠏⠀⠘⠿⠏⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠿⠏⠈⠛⠻⠶⠶⠶⠒⠂⠀⠀⠸⣧⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⣀⣀⣀⣤⣤⣀⣴⡆⠀⠀⣾⡏⠷⠀⣴⡖⠒⠂⣶⠀⠀⠀⠀⣰⢠⡄⢰⣶⠀⣾⡄⢀⣶⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠛⠛⠛⠛⠿⣿⣿⡟⢻⡇⠀⠀⢹⣷⡀⠀⣿⠃⠀⠀⣿⡇⢀⡀⢀⡇⢸⡇⢸⡟⡆⣿⡇⣾⡟⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢀⣾⣿⣷⠈⠀⠀⠀⠀⢹⣷⠀⣿⠦⠆⠀⣿⡇⣾⣷⢸⠁⢸⡇⢸⠀⢿⣿⡇⣿⡇⠐⣶
⠀⠀⠀⠀⠀⠀⠠⠶⠶⠶⠶⠶⠶⠶⠿⠿⠿⠿⠶⠶⠦⠀⣆⢀⣿⠀⣿⡄⠀⡀⢹⣷⠻⣿⡟⠀⢸⡇⢸⠀⠸⣿⠁⢻⡇⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠉⠉⠀⠈⠉⢀⣉⣡⣀⡀⠀⠈⠁⠀⠉⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⠈⣷⠀⠀⠀⣠⢶⣲⣖⡶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⢠⡟⠀⠀⣼⢱⣿⣹⣏⣻⣮⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⣰⠟⠀⠀⠀⢻⡸⣿⣹⣯⣽⢏⡿
⠀⠀⠀⠀⠀⠀⢸⣿⡟⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⣠⡾⠋⠀⠀⠀⠀⠀⠙⠶⠭⠭⠷⠋⠁
⠀⠀⠀⠀⠀⠀⣼⣿⡷⠋⠁⠀⠀⢀⣴⣾⡷⠶⣄⠀⠀⠀⠀⠀⢠⣶⣿⡷⢦⣿⣿⠞⠋⠀⠀⢠⣶⣄⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢀⣾⣿⠏⠀⢠⣿⣿⠂⠀⠀⢰⣿⣿⠏⠀⣸⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⢰⣿⣿⡄⠀⠀⣠
⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⣸⣿⡿⠀⠀⣾⣿⣿⠀⠀⠀⣿⣿⡟⠀⢠⡟⣿⣿⠀⠀⠀⢀⣿⣿⡇⠀⠀⣾⣿⣿⡇⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⣰⡿⣿⣧⣀⡾⠃⣿⣿⠀⠀⣼⢿⣿⣇⣠⡟⠀⢻⣿⡆⠀⢀⣾⠙⣿⣇⣀⣼⠃⣿⣿⡇⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⡾⠋⠀⠈⠙⠋⠀⠀⠘⢿⣤⡾⠋⠀⠉⠛⢉⣀⡀⠈⠻⣷⣶⠟⠁⠀⠈⠙⠉⠁⢀⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⢫⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠻⠿⣿⣿⣶⣶⣶⡚⢛⣓⣊⢁⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠶⠾⠿⠟⠛⠛⠉⠁⠉⠙⢯⠀⠀⠈⠙⢢⠀⠀⠀⠀⣾⠃⠀⢀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠔⠋⠀⠀⠀⠀⠙⠶⠶⠿⠟⠁

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⡟⠙⠻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠋⠸⠁⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⠁⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣟⢿⣿⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣢⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣌⠳⣽⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣼⣿⣿⣿⣿⡇⢀⣀⠈⠛⠇⠀⠀⠀⠀⠾⠋⠁⣀⠀⠀⢸⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣢⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⡏⠉⠫⣭⣹⢦⠀⠀⠀⠀⠀⢀⢾⣍⡭⠋⠁⢸⣿⣿⣿⣿⣿⣿⣷⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢛⣛⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⣸⣿⣿⣿⣿⣿⢿⣷⣤⣤⣄⣀⣀⣀
⠀⠀⣀⣤⣤⣴⣶⣾⣿⣿⣿⣿⡿⣿⣿⡹⣿⣿⣿⣿⡿⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡿⣫⠿⣹⣿⣿⣃⣿⣿⣿⣿⣿⣿⣿⠿⢿⠗
⠀⢸⡻⣿⣿⣿⣯⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣷⣦⠤⣐⡦⢤⣠⡾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣤⡀
⠀⠀⠉⠛⠉⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠿⠿⠿⠟⠛⣛⣻⣛⣻⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠚⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⠭⣟⣛⢛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠘⠶⢀
⠑⣄⠀⡆
⠀⠈⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⡀
⠀⠀⠸⡔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣿⣽⣧⠀⠀⠀⣀⣀⣠⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧
⠀⠀⠀⠎⠀⢁⠀⠀⠀⠀⢀⣤⣴⣤⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠄⠀⢸⣿⣿⣿⣿⣷⠀⠀⠀⠀⠈⡻⣿⣿⣿⠇
⠀⠀⠀⢠⠀⠀⠃⠀⠀⠀⣿⣿⣷⣿⣿⡄⠀⠀⠀⠀⠀⠈⢿⣿⣿⠟⠀⠀⠀⠀⠺⠻⣿⣿⣿⠀⠀⠀⡀⠈⠻⣶⣦⣄⣤⣀
⠀⠀⠀⠀⠻⣴⠂⠀⠀⠀⢿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣠⡄⢨⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠠⠀⠀⢳⣆⣶⣿⣿⣏⡽⣿⣷⡀
⠀⠀⠀⠀⠀⠈⢧⠁⠀⠙⠄⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⡘⠘⠇⠈⠘⠛⠻⡛⠛⠒⢶⣥⣦⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⢴⢿⡿⣷
⠀⠀⠀⠀⠀⠀⠀⢷⠄⠀⠺⣶⣤⣶⣤⣦⠼⢿⣆⣞⠀⠃⠀⠀⠀⠀⠀⠈⢃⠀⡀⠀⠀⠀⠉⣠⠛⠁⠀⠈⢿⣿⣿⣿⣶⠾⣷⠙
⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⠠⡀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢷⣄⡀⠀⠈⠁⠀⠀⠀⠆⠈⢿⣿⣿⠢⣿⡯⠆
⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡻⠏⠀⡝⠛⠶⣄⡀⣤⠀⠀⡧⠀⠈⠻⣿⣿⡼⢳⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⠀⠀⠀⠀⣀⣤⣾⡇⠀⠈⢂⣀⣀⣡⡴⠂⠀⡞⠀⠀⢠⡿⠂⣀⠀⠀⠈⠀⠀⣧⡈⠀⠀⠈⠻⡄⣸
⠀⠀⠀⠀⠀⠀⠀⡼⣇⠀⠀⠀⠀⠀⣠⠾⡟⠣⣖⠀⠀⢈⡉⣠⣤⠾⣞⠁⠀⣠⡟⡴⠛⠀⠀⠀⠀⠼⠲⣿⣿⣷⣧⣥⣤⣄⡀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢠⠁⠈⠀⠀⠀⠀⠀⠐⢸⠃⠀⠛⠲⠶⠟⠋⠁⢀⣠⡼⠷⠎⢁⣠⡐⠉⠀⠀⠀⠐⠒⠀⣿⣿⡟⣿⣿⣿⠿⠿⡷⠐⠁
⠀⠀⠀⠀⠀⠀⢸⠔⣡⣤⣤⣶⣶⣤⣤⣀⠃⠀⠀⠀⢀⡤⠞⠙⠛⢐⣀⣉⢋⡸⢈⣉⣀⣀⣀⣀⣀⣀⡀⣿⣨⠍⠋⣉⢽⣿⣿⠘
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢩⣴⡿⣟⠟⠻⠟⠛⢻⣏⠀⡙⢿⣿⣿⣩⣿⣽⠿⣟⠿⢿⣣⡴⠾⢊⣿⣟⠛
⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⠛⠍⢹⣿⡇⢀⠀⠋⠐⠀⠄⢠⠀⠐⢁⡼⠀⢧⣜⢿⣟⢛⡛⠛⠛⣌⡓⢸⣿⣡⣾⡿⠏⠁⢀
⠀⠀⠀⠀⠀⠀⢸⣿⣯⣿⣿⣉⣁⠀⢀⡟⠀⠈⠇⠀⠀⠀⠀⡦⢀⠴⠋⡄⠀⠎⣿⠷⣿⣿⣷⣾⣿⠛⣶⢸⡿⣫⢿⣤⠴⢃⡍
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣻⡷⠈⡇⠀⠀⣇⠀⠀⢖⣰⣿⡃⠠⠞⠁⠀⣷⡾⠰⢻⠇⢰⣿⡟⠟⠉⣹⣿⣿⠏⢓⣀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣻⣿⡆⢻⡇⠀⢶⠀⠀⢈⡀⣀⡀⠾⢻⣏⠀⠀⠀⠀⢉⠶⡿⣷⠾⡇⣻⣽⠗⠀⣿⣿⠏⠐⢻⡷⠛
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣞⠂⢸⣷⡄⢸⠀⠀⠘⢻⠋⠛⠀⡶⢷⣦⡔⠀⠀⠈⠀⡶⠟⠿⠁⠙⠓⢤⠶⣎⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⣿⣿⡄⡅⠀⠰⠈⠂⠘⠀⠰⠂⠀⠀⠀⠀⠀⠀⠧⣀⠰⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢺⣿⠇⠀⠀⠀⢸⣿⠇⡧⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⢲⡿⠀⠀⠀⠀⢸⣿⠀⡇⠀⠀⢹⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⣿⠀⡃⠀⠀⠘⠂⠀⠀⠀⣤⠂⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣇⡄⠀⠀⠀⠀⢿⡄⠁⠀⠀⠀⢠⣦⠀⣀⠙⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣦⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⠀⠀⠀⠀⠘⠊⢀⣠⣤⠀⣶⣿⢸⣿⣢⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⣠⡄⢰⣿⣿⣷⣿⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣈⡇⠀⠀⢸⣿⣏⠀⡛⣶⣿⡏⣸⣿⣯⠶⠆⠀⣿⣿⣣⣦⣼⠄⠀⠰⢶⣿⣿⡿⣭⢻⣿⡿⠴⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠎⠀⠀⠀⠈⢿⣿⣦⣿⠏⠀⠀⠵⠶⠶⠖⠂⠉⠉⠛⢛⡿⠟⠀⠀⠀⠑⠂⠄⠀⠈⠈⠁⠉⠀⠐⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣁⣉⣋⣀⣉⣹⣍⣣⣽⣋⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⢀⠤⠎⠀⠈⠐⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠁⠀⠀⠈⠒⠄⢇⠀⠀⠀⠀⠴⠛⡄⠀⠀⠀⠀⠀⠀⠐⣶⣄
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠠⣄⡄⠀⠀⠀⢈⢌⠟⠀⠀⠀⡀⡠⠛⡄⠀⠀⠀⠀⠀⠀⡹⣿⣿⠛⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠠⡀⠈⠉⠉⢉⡉⠈⢆⣀⣱⣎⡀⣀⠔⠁⠀⠀⠀⠀⠀⡘⠀⡎⠹⣀⣤⣤⣌⣻⡄
⠀⠀⠀⠀⠀⠀⠀⢣⠀⢠⠇⠈⠀⠀⢉⠇⠀⠀⢸⣿⣿⡟⠀⠀⣀⠀⠀⠀⠀⡌⠈⠀⢦⣄⢿⣿⣿⣿⡟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣦⣤⣈⣥⣎⠀⠀⠀⡀⣿⣿⠁⠀⠀⣿⡗⠀⠀⠀⡇⠀⠀⠔⢻⠈⢻⣿⡟⢡⠃
⠀⠀⠀⠀⠀⠀⠀⡰⢍⠛⠿⠛⠙⡛⠛⡄⣀⠔⠁⣿⠇⠀⠀⣰⣿⣇⠀⠀⠀⠈⠑⠄⢀⣸⡤⠤⠝⠚⠁
⠀⠀⠀⠀⠀⠀⡐⠁⠀⡇⠀⠀⠀⡃⠀⠀⠈⠀⢀⠏⠀⠀⢠⢿⣿⡿⡄⠀⠀⠀⣠⡾⠟⠋⡇
⠀⠀⠀⠀⠀⠰⠁⠀⢰⠀⠀⠀⠀⡇⠀⠘⣄⠴⠃⠀⠀⠀⢸⠀⠈⠀⡃⠀⠀⠜⠙⢄⠀⠀⢠
⠀⠀⠀⠀⢠⠃⠀⠀⡎⠀⠀⠀⠊⡇⠀⠀⠃⠀⠀⠀⠀⠀⠀⢷⣶⡞⠈⠾⠀⠀⠀⢸⠀⠀⠈⡆
⠀⠀⠀⠀⡪⠀⠀⡜⠀⠀⠀⠀⠀⡇⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⢀⠎⠀⠀⠀⢣
⠀⠀⠀⣾⣀⣀⣉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣖⣊⣁⡀⠀⠀⠀⠈⢆
⠀⠀⠀⠈⢉⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⠃⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠈⠁⠒⠂⢠⣤⣤⣤⣤⣷⣶⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣶⣶⣶⣶⣶⣾⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠃⢿⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣂⢸⠀⣦⠐⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠻⢶⡷⠙⠛⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣯⣿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣆⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠠⣿⣿⢿⡿⢿⡿⠛⠿⠿⠻⢿⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠄⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⣿⡟⠁⠀⠈⠉⠛⠋⠉⢻⡇⠀⠀⠀⠀⠀⠘⣿⠋⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⠃
⠀⠀⠀⢀⣀⠀⠉⡙⠀⠀⠀⠈⠀⡀⢀⠀⠁⠀⡀⠀⡸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇
⠀⠀⠀⠿⢏⡹⢠⣷⣄⠀⠀⠀⠀⠛⠛⠁⠀⢀⠃⠘⣛⡏⠁⠀⠀⠆⠀⠀⠀⠰⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠃⣀⣀⡀⠁⠀⠀⠀⡫⠉
⠀⠀⠀⠀⠀⠈⠀⢍⠻⢿⣶⣦⡄⠀⣤⣤⣶⠟⡂⠔⠁⢃⠀⠀⠀⠀⠐⠀⠐⠀⠀⣠⠠⠀⢠⡄⠄⡀⢀⠀⠀⠙⠋⠁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡈⠑⠁⠊⠃⢉⠄⠈⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⡤⠊⠀⡠⠊⠀⠀⠁⠒⠒⠁⠐⠲⡄⠠⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢲⡀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠈⣤⠎⠀⠀⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠊⠢⢔⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠈⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⢩⡀⣆⠀⠀⠀⠀⠀⠀⠌⠀⢀⡆⠀⠀⠀⠀⠘⣆⡀
⠀⠀⠀⠀⠀⣀⢩⡏⠀⠀⠀⢀⡈⠉⠁⡅⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠙⢥⡀⠀⠀⠀⡠⠊⠀⡠⠚⠒⠀⠀⠀⠀⠒⠉⠚
⠀⠀⠀⠀⠀⠐⠀⠀⠁⢲⢚⣍⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠉⠀⠒⠂⠈⠀⠀⠉⠒⠒⠂⠀⠀⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢰⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢴⠁⠀⠀⠀⠀⠂⢘⡆⢠⠀⠀⠀⠀⠀⠀⠀⠤⣶⣀⡂⠈⢶⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠁⠀⠀⠀⠀⠈⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠁

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

⡅⣩⢩⣉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠀⡀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠀⢁⠉⠀⢀⡀
⠲⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡠⠆⠉⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠐⠒⠀⠄⠉⠀⠀⠰⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⢄⡊⢡⡅⠨⠁⠈⠀⠀⠀⠘⠀⢀⣄⠀⡀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠁
⠀⠀⢰⠃⣈⠀⠀⠀⠀⠀⠀⠀⣜⢁⠀⠈⠡⠄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠠
⠀⠀⠈⠈⠉⢃⡀⠀⠀⠀⠀⠀⠀⠟⠂⢷⢂⡙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠚
⠀⠀⣄⠀⠀⠙⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠀⠀⡀
⠀⠀⠈⠉⠥⠀⡲⠀⠐⠀⠀⠀⠀⠠⡠⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠈
⠀⠀⠀⠀⠈⢍⡀⠠⠐⠊⢰⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⠀⠚⡃⠶⠶⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉
⠀⠀⠀⡀⠀⠀⢈⡀⢂⠀⠀⣀⠠⠀⠀⠀⠀⠀⠀⠤⠀⠀⠂⠀⠐⢂⠀⠠⠄⡀⠀⠀⠀⢀⡀⠴⠄⢂
⠀⠈⠀⠀⠀⢐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠠⠄⢡⣠⣀⠀⣤⠀⠀⠆⠉⠀⠀⠘⠈⠁⢂⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠁⠀⡠
⠀⠀⠀⠶⣀⣠⢘⠀⡀⢰⠠⡤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⢠⠀⠐
⠀⠈⠀⠐⠀⢀⠹⠌⠷⢞⠀⣉⣷⡁⠠⠴⡤⡀⠆⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⣦⠀⣤⠀⠀⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠤⠀⡁⠝⠀⠀⠈⢤⢄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢀⠃⠀⠀⠀⠉⠉⠀⠈⠉⡀⠀⠈
⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⡀⠄⠀⠈⠀⠁⠀⠘

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⠛⣻⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⢞⣻⣿⡙⠳⣦⡀⠀⠀⠀⠀⠀⠈⠉⠋⢁⣼⠏⠻⣟⠋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠀⢛⣟⠁⣸⣧⣤⣬⣉⣙⠛⢦⡀⠀⠀⢠⣴⡿⠿⣶⣾⣿⣷⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠻⠞⠋⣹⡷⠟⠀⠀⠀⠈⠉⠳⣦⠹⣆⣰⡿⠉⠀⠀⣠⡀⠉⠉⠉⠉⣿⣿⠛⠛⣻⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣤⡀⠀⠈⢻⣿⡟⠀⠀⠀⠀⠙⠁⠀⣤⡄⠀⣿⣿⣷⢴⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⠏⠀⠀⢾⠆⠀⠉⠁⠀⠀⠀⣿⡇⠀⣼⠃⠀⠀⠀⠀⠉⠁⠀⣿⣿⣅⣠⣿⠟⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⢀⡀⠀⠀⠀⢸⡆⠀⣹⣧⡀⠘⣦⣄⣀⣠⡾⠃⠀⠀⠘⢿⣿⠇⠀⣴⣾⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠘⢷⣄⡀⠀⠘⢷⣤⣤⣤⠾⠁⣠⣿⠉⠻⣦⣀⡉⠉⠉⠀⠀⠀⠀⣠⡾⠋⠀⠀⢿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠉⢻⣦⣀⡀⢀⣀⣀⣤⡾⠋⣿⠀⠀⠀⠉⢻⠛⢷⡶⢶⡾⠿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⡟⠋⠛⠻⢿⣯⣍⣁⠀⢠⡿⠷⠶⠶⠶⢾⣇⣼⣷⣫⢆⡄⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠉⠛⠻⠿⢷⣦⣤⣤⣶⠿⠛⠉⠀⠑⠣⠾⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⡾⠏⠀⠀⢘⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣤⣀⣀⣀⣀⣤⡄⠚⠉⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⢤⣴⠞⠻⢭⠽⢿⣦⡀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠈⠉⠉⢀⣀⣀⣤⣀⠀⢤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠠⡭⠏⠀⠹⢿⣄⡀⠀⠀⢀⣼⡿⠛⢿⡛⠛⠛⢋⣿⠟⠁⠈⢻⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠻⢧⡀⢀⣀⣤⣽⠿⢶⣴⡾⠟⠁⣀⣤⡿⢦⡶⠟⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⡁⠀⠀⠈⠉⠉⣀⣀⣀⣤⣤⣴⡶⠛⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠦⠴⠚⠋⠙⠛⠋⠉⠉⠉⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀
⠀⠀⠀⢀⣤⡶⠛⠉⠈⣷⠀⠀⠀⠀⢀⣠⣤⠶⠛⠁⠀⠀⠈⠙⠻⢶⣦⣤⣀⣀⠀⠀⠀⣿⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⢠⡶⠛⠁⣀⣀⣀⣠⠾⣧⣤⣤⣾⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⣾⡿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄
⢰⣏⣀⣀⣠⣿⣿⣤⠿⠲⢤⣀⠀⠉⠙⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆
⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⠙⠻⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣤⡿⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⠛⠷⣦⣤⣀⡀⠀⠀⢸⡇⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣴⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⣿⠉⠉⠛⠛⠿⠿⠇⠀⠀⠘⠛⠿⠿⢿⡿⠿⢿⣿⡿⠛⢻⡟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠸⣿⣷⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⡿⣿⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣀⣯⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣄⣠⣷⣿⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠈⠹⢇⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣉⣭⣼⣿⣧⣴⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⢤⣄⡉⠉⠙⢦⡀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠁⠈⣉⣥⡶⢿⣤⣶⠾⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠦⣤⣠⣿⠀⠀⠀⠀⠘⢷⣤⣀⣠⡾⠛⠉⠁⠀⠀⠀⠀⢀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⢦⣤⣤⣤⠶⠞⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣄⣀⣤⡀
⠀⠀⠀⠀⠀⠀⢠⡾⣋⣭⣶⣶⣿⣶⣄⠀⠀⠀⣠⣾⣿⣿⣿⣷⣶⣶⣤⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣗
⠀⠀⠀⠀⠀⠀⣾⣳⣿⣿⣿⣿⣿⣿⢿⡇⠀⢸⣿⡿⠻⢿⣿⣿⣿⣿⠧⠀⠀⣾⡟⠛⠻⢿⣿⣿⣿⣿⡁
⠀⢠⡶⢤⣰⣾⣿⣿⣏⠙⠉⠉⠈⠉⠀⣣⡄⢸⡏⠠⠀⠀⠀⢃⠈⣟⠀⠀⢠⡾⣠⡄⠀⠀⣐⠂⣼⡟
⠀⠈⠁⠀⠺⣿⣿⣍⡃⠀⠁⠀⠀⠀⠀⠇⠀⠚⡇⢽⡷⠀⠸⠿⠀⣏⣅⠀⠀⠅⠙⠃⠀⠘⠟⠀⡟⠃
⠀⠀⠀⣠⣤⡌⠻⣿⡇⠀⠐⠳⠞⠀⠀⠀⠀⠈⠁⠀⠀⣤⣤⠀⠀⠋⠙⠂
⠀⠀⢸⣿⡿⠿⠦⠀⠁⠀⠰⠂⠰⠀⠀⠀⠁⠀⠀⠀⠠⡀⠠⠀⠀⠀⠀⠀⢠⣤⣤⣤⣶⣀⣤⣤⡀
⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠀⠀⠀⠀⠀⠀⡤⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠿⠿⢿⣿⡿⠿⠿⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠶⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠒⠆⠀⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠓⠒⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⠀⠀⠀⢀⠀⠀⠀⢀⠀⢈⠀⠀⠀⠀⢰⠀⠀⠀⠀⠧⠔⠃⠀⠀⠀⠐⠦
⠀⠀⠀⠀⠀⠀⠀⠈⠘⠁⠀⠀⠀⠀⠙⠂⠀⠀⠊⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠘⠿⠿⠋⢠⣿⣿⣿⣿⣿⣿⣿⣿⡆⠙⠿⠿⠋⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⠿⠿⠋⣰⣿⣿⣿⣿⣿⣿⣿⣿⡄⠻⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⡿⢿⣿⣿⣿⠟⠁⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣿⠀⠀⠀⠀⠛⠿⣿⣿⣿⡿⢿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠃⠈⢿⣿⡿⠀⠀⠀⠀⠀⠀⢿⣿⡿⠃⠘⢿⣿⡟⠀⠀⠀⠀⠀⠀⢿⣿⡿⠀⠸⣿⣿⠏


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡄⠠⠄⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⠐⠓⠘⠁⠘⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⢦⠀⢠⣆⠀⡄⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠇⣮⣼⡄⠸⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠒⠋⠘⠃⠀⠃⠀⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢁⣀⢨⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⡅⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠓⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠐⡄⠠⡀⠁
⠀⠐⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢄⠈⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠈⠄
⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠂⢁⡠⠜⠀⠀⠀⠠⠁⡰⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠐⠂⠤⠐⠈⠁⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠈⠇⠀⠠
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠄⠀⢀⠀⡈⠀⠀⠀⠀⠀⠀⢰⠠⢀⠌⠀⠀⠀⠀⠀⠀⠀⡇⡠⠂
⠐⠀⠁⠀⠌⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁
⡇⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠔⠉
⢠⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠤⠁⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⢤⠊
⠀⠀⠐⠁⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠄⠒⣆⠁⠀⢀⠀⠀⠀⠒⠂⠀⠀⠀⠐⠒⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡨⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠄⠀⢂⠜⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⠀⠎⠀⢀⠀⠀⢠⠌⠲⠀⢐⠀⠨⡀⢰⣀⠉⠀⢀⠄⠠
⠀⠀⠀⠀⠀⠀⡀⢀⠔⢹⠀⠀⠀⠸⠄⢀⠀⢀⠃⠀⠀⠐⠄⠈⠀⠄⢆⠈⡤⠨⠂⡀⠡⠈⢀
⠀⠀⠀⠀⠰⠴⠥⢊⢤⢸⣀⠀⣀⠀⠀⠔⠐⠀⠀⠤⡄⢀⠀⣃⠤⠊⡀⠕⢀⡱⠤⠰⠀⢂⠈
⠀⠀⠀⠀⠀⠀⠦⠁⠲⢐⠄⢠⠊⠀⠀⠐⢲⠀⢠⠊⠀⠣⡘⠂⢀⠈⡠⠈⢣⢀⠁⠈⡄⠈⠀⠠⢆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠀⠀⠀⠈⠇⢱⠀⠀⠑⠀⠂⠁⢤⣒⡀⠀⠈⠁⠠⠄⠄⠋⢣⢀⠁⢤⠀⡀
⠀⠀⠀⠀⠀⠤⡀⠐⢂⢠⠮⣡⠃⠀⠨⠤⠈⠀⠀⠀⠀⢀⠠⡠⠖⠓⠒⠤⡀⠀⢠⠀⠊⠀⢅⡌⠀⠉⡀
⠀⠀⠀⢰⠬⢀⠑⢀⡀⠃⠀⠐⠑⠂⠀⠀⠀⠀⡠⠀⡅⠀⠐⠶⣆⠀⠀⠀⠈⠂⠀⡀⡄⠓⠨⠣⢋⠆⠀⠙
⠀⠀⠀⠀⠑⠁⠣⠂⠱⡀⡐⢂⠀⠀⠀⠀⢠⠊⠀⠜⡁⠀⠱⠺⡋⠀⠀⠀⠀⠐⠎⡀⠈⠀⠀⠒⠀⠈⠋
⠀⠀⠀⠀⠀⢀⢀⠀⠄⠈⠀⠀⠐⠟⠀⠀⠀⢁⠆⠀⢃⠀⠀⢠⠦⠤⠄⠀⠊⠔⢁⠝⢆
⠀⠀⠀⠀⠀⠐⠀⠐⢘⠁⠐⠀⠈⠀⡀⡀⠀⠀⠂⠀⠀⠀⠀⠉⠀⠀⠀⠂⠂⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠓⠁⢁⠀⠒⠌⡲⢸⡀⠦⠔⠀⠀⠀⠀⠀⠀⠀⢀⡀⡠⡤⠄⢠⠄⠋
⠀⠀⠀⠀⠀⠀⠀⠤⠠⢂⢉⠀⠄⠐⠀⠁⠀⠀⠀⢢⡀⠀⡠⠶⠸⠐⠂⠉⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠆⠓⡀⢁⡁⠁⡂⠐⠀⠐⠀⠀⠀⠀⠙⠐⠒⠒⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠒⠈⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠰⠇⠀⠀⠂⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠂⠒⠀⠀⠀⡀⢀⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⢋⡀⠀⡠⡏⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⢫⠕⠟⣸⠚⣆⡳⠐⠊⠁⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠶⠚⠁⠀⠈⠙⠚⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⠟⢓⣀⣴⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣦
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⢀⣾⡿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⡿⣿⣿⣿⣿⡀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠃⢠⣿⣿⣿⣿⣷⣿⣿⣿⣿⣧
⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣱⠏⢠⠏⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣴⠋⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⣾⣿⣿⣿⣣⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⢠⣿⣿⣿⣿⢿⣿⣿⣿⣿⢸⣿⣿⣿⡟⣻⣿⣿⡯⠀⠀⣴⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⢸⣿⣿⣿⣿⣾⣿⣿⡿⣿⣸⣿⣿⡿⠛⠋⠁⠈⠀⠀⠀⠁⠛⠈⢛⣻⣿⣿⣿⣿⣿⣿⣿⠛
⣾⢿⣿⣿⣿⣿⣿⣿⢱⣏⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣧
⢸⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠰⠷⣦⠴⠆⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⢀⣀⣀⣈⡀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣷
⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠘⠚⠋⠉⠉⠉⠛⠛⠛⢦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⠐⠀⠲⠶⠶⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡉⠉⠉⠀⠀⠀⠀⠀⠀⠠⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢆⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡁⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢸⣯⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡷⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣾⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣾⣽⣷⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠙⠛⠛⠻⠟⠛⠿⠿⠿⠿⠿⠛⠿⠿⠃⠀⠀⠀⠀⠀⠀⠛⠛⠛⠻⠛⠛⠛⠛⠛⠛⠙⠛⠛⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⢿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠈⠀⠀⠀⠈⠳⠀⠀⣀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣰⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠛⠻⣿⣿⣇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠸⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣠⣀⡀⠀⠀⠀⠀⠀⡂⠙⠀⠀⠀⠀⠀⠀⠀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠈⡰⣀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⢔⠠⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠴⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠓⠒⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠒⠛

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠔⠲⠏⠉⠉⠍⠐⢢⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠁⡀⠀⠀⠗⠀⠀⠂⠀⡇⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢀⣀⣠⣀⣰⣆⣀⣰⣆⣨⡀⡇⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⡊⠀⢠⡀⠀⠀⠀⠀⠀⢀⣀⠀⠈⠑⡆⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⠈⠳⣄⠀⠉⠀⠀⠀⠀⠀⠈⠉⢀⣠⠞⠑⢆⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡆⠀⠀⠈⠳⣤⣀⠀⢀⣀⣤⠴⠚⠋⠀⠀⠀⠸⡄⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠀⠀⠀⠀⠀⠀⠉⢙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⢀⠀⠀⠀⣠⠞⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢸⠀⠀⢸⠀⢀⠞⠁⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠘⡄⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢸⠀⠀⡘⢠⣿⣿⣿⣿⣿⣿⣿⣆⠰⡄⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢸⠀⠀⡇⢸⣿⡿⠛⠛⠋⠙⢿⣿⠀⡇⠀⠀⠀⠀⠀⡇⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢸⠀⠀⢻⠸⡿⠶⠶⠶⠶⠶⠾⡿⠀⠀⠀⢸⠀⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢸⠀⠀⠘⡄⠻⣄⠀⠀⠀⠀⡴⠃⡀⠀⠀⠘⡆⠀⠀⡏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠈⠛⠶⠶⠾⠗⠋⠀⠀⠀⢀⠇⠀⢠⡇⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⢤⣀⣀⣀⣤⡴⠃⠀⠀⠚⠀⠀⡾⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢀⡼⠉⠉⠈⠳⣄⠀⠀⠀⢀⡼⠃⠀⠀⠀⡇⠀⠀⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡼⢆⡀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠙⠶⠤⠶⢻⠀⠀⠀⠀⠀⡇⠀⠀⠈⢆⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠃⠈⠙⠲⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢹⠀⠀⠀⢸⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢸⡄⠀⢀⡞⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠸⡆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⣧⠔⠉⠀⢀⡞
⠀⡠⠒⠂⠒⠒⠒⠀⠀⢼⠁⠀⠀⠀⠀⣇⢸⠀⠀⢰⡄⠀⠀⠠⠶⠂⠀⠀⠘⡄⠀⠀⣧⠀⠀⠀⠀⠀⠸⡦⠤⠖⠛⠉⣉⠱⡄
⡜⢀⠎⠀⡰⠒⠁⠀⢠⡏⠀⠀⠀⠀⠀⢸⣼⠀⠀⠀⣧⠀⠀⠀⠀⡇⠀⠀⠀⠹⡄⠀⡟⠀⠀⠀⠀⠀⠀⠘⢦⣀⠐⡄⠈⢧⡱
⠙⠺⢤⣤⣇⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢹⠀⠀⠀⠀⣧⠀⠀⠀⠀⠹⣆⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢾⡀
⠀⠀⢀⡴⠚⠉⠁⠀⠀⠀⠀⢀⣠⠤⠤⠏⠀⠀⠀⠀⢸⣦⣤⣤⣤⣷⠤⠀⠀⠀⠀⠈⠛⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙
⠀⠀⠸⣄⠀⠀⠀⠀⠀⣀⡴⠋⡀⠀⠀⢀⠀⠀⠀⢀⠈⠳⡄⠀⡞⠁⠀⢰⠂⠀⠀⠀⡀⠀⠀⡄⠘⣆⠀⠀⠀⠀⠀⣀⣠⡴⠟
⠀⠀⠀⠈⠛⠛⠚⠛⠛⠙⠦⢼⣃⠀⠀⢸⡀⠀⠀⢸⣄⣀⣿⠀⠳⠤⠴⢿⣄⣀⣀⣀⣇⣀⣀⡽⠒⠋⠙⠛⠛⠛⠛⠉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠀⠀⣀⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⠀⢀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠴⠦⠄⠀⠀⠀⠀⠁⠀⡟⠀⠀⠸⠀⠀⠀⠀⠒⠒⠒⠒⠒⠶⠶⠶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⢀⣠⣤⠶⠶⠛⠋⠁⠀⢀⣠⠴⠒⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠶⣤⣄⡀⠀⢀⣀⡀⠀⠉⠛⠻⣦⡀
⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⠀⠠⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⡀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠈⠙⣳⡄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⢶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡶⠶⠶⠞⠋⠀⠀⠀⢀⣀⣤⣶⠿⢃
⠀⢀⠙⠳⣶⣶⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⠶⠶⠾⣿⠟⠛⠉⠀⠀⠈
⠀⠈⠉⢉⠱⠿⢤⣭⣉⣉⣙⠛⢛⣛⢛⠓⠛⠛⠋⠀⠀⣓⠒⣛⠛⠛⠛⠛⠛⠋⣉⠉⠉⠉⠁⢀⡄⠀⠀⠀⠈⠀⠀⠀⢀⣀⡠⠄
⠀⠀⠀⣾⠀⣀⠀⠂⠈⠉⠉⠘⠛⠛⠛⠛⠋⠉⠁⢀⠛⠉⠉⠉⠀⠀⠀⠀⠔⠊⠉⠑⠀⠀⠀⠀⣀⣀⣤⣤⡤⠶⠚⠛⠋⢹
⠀⠀⠀⠸⠀⠈⠙⠛⠓⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠶⠶⠾⠟⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢠⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠘⣧⠀⠀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠶⢀⢸⡇
⠀⠀⠀⠀⠀⣿⣎⠀⠀⠀⠀⠀⢠⠖⠒⠒⠒⠀⠀⠀⠀⠀⣠⡴⠶⠾⠶⢤⣄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⢻⡏⠓⠤⠄⠀⢀⡎⠀⠀⠀⠀⠀⠀⠠⠐⠉⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠈⠀⠀⢐⡒⣤⠶⢷⣦⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⠖⠒⠒⠉⠉⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠠⠖⠋⠁⠀⠀⠀⠀⠈⠃⠌⠉⣉⠝⠋⠉⠉⠉⣉⡉⠉⠙⠛⠒⢤⡀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⡼⠃⠀⠀⠤⠀⠀⠈⠉⢄⡀⠀⠀⠀⠉⠆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⣷⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢻⡆⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡀⠀⠀⡀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠸⠀⠀⠀⣧⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⢤⣤⣤⡤⠐⠂⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠙⠀⠀⠀⢱⣀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⢀⡠⠤⠐⠛⠀⣀⣉⠤⠤⠤⠤⡴⠒⠓⠀⠀⠀⠠⠠⠀⠈⢁⡈⠁⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⠀⠀⠀⠀⢾⣋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠷⣶⣄⣀⠀⠈⠛⠂⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠛⠷⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠶⠛⠋⠉
⠀⠀⠀⠀⠘⠷⣤⣤⣀⣉⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠋⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠦⠤⢠⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣈⠆⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡕⠢⡀⠀⠀⠀⠀⠀⠐⠧⡀⠀⠪⡁⠀⠀⡤⠠⡀⡠⢲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠄⠉⠀⠠⠄⡀⠀⠀⢰⠂⠈⠋⡀⠬⠆⠀⠱⢄⠰⠅⠼⠀⠀⢀⠖⡀⠰
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠊⠹⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠜⠀⠀⠀⠑⠢⣀⠉⠒⠄⠐⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⣌⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡨⠂⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠈⠂⠀⢀⠆⠀⠀⠀⠀⢀⠀⠀⢀⡨⠃⡖⠁⠤⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠀⡄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠁⠠⣎⠀⠀⠤⣀⠀⡔⠀⠐⣏⠀⠀⠈⠢⠀⠀⠀⠀⠀⠃⠀⣨⣇⠄⠀⣀⠼⣀⠀⠀⠀⠉⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⡄⠀⠀⠀⠀⡘⠀⢱⠀⠀⠀⠉⠀⠂⠀⠀⢄⣀⡀⠀⠀⠈⠀⠀⠸⠤⠔⠈⠈⢇⢀⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⠲⡀⢀⢬⠓⡩⠃⠀⠠⣁⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠾⠧⠤⣄⠀⠀⠀⣀⠀⠄⠑⢌⡉⡰⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠳⠎⠀⠑⡑⣄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠑⢄⠀⠈⠀⠈⢹⡠⠌⢃⢠⠢⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⠳⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢠⠃⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠈⡄⣠⠖⠹⣄⡠⠘⡈⢁⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⠀⢳⣀⠜⠙⡄⠀⣀⠀⠀⠀⣴⡁⠀⠀⠀⠀⠀⠀⠂⠘⡄⠀⠀⠀⠀⠀⡰⠉⠢⣄⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⢀⡈⠺⠲⠄⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⢠⠦⠄⠠⠤⠐⠊⠠⠊⢙⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠢⠄⠈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠄⠀⠀⠐⠊⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠆⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠂⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⢀⠖⠄⢠⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢉⢢⣀⠀⢠⠴⠯⠴⠒⠊⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠛⢆⠈⠃⠙⠊⠀⡖⠀⠤⠤⠵⠀⠀⠀⠀⠀⢹⠒⠲⠦⠤⠴⠖⠒⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⣀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠈⠉⠀⡀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠴⠀⠀⠀⠒⠒⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠯⠰⠬⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⣤⣤⠤⠤⠔⠂⢶⢶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠇⣰⠢⡀⢀⠔⠻⠘⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⠉⡰⠪⠾⣠⡾⠹⣦⡙⠒⢺⠀⠀⢱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⣤⣼⡦⠞⠋⠀⠀⠈⠙⠲⠧⠤⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠠⠀⠀⠀⠄⠄⠄⠄⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡰⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠰⠀⠀⠠⠤⠀⣀⡀⠀⠀⠀⣀⣤⣌⠁
⠀⠀⠀⢀⠀⠀⠾⣦⣤⡤⠈⣿⠁⠀⠀⠧⠈⠓⠈⣿⣾⠀⢸⣾⠁⠀⣦⣰⡆⠀⢁⣄⣠⠀⠀⢙⡋⠉
⠀⠀⠀⠈⠁⠀⠀⠘⠟⠀⠰⣿⡆⠀⢸⣿⣿⡇⠀⢻⡿⠆⠀⣿⠀⠀⡿⢽⡇⢀⣿⣿⣿⣷⠀⠯⡿
⠀⠀⠀⠠⠀⠀⠀⠀⠇⠀⠀⠈⠀⠀⠈⠿⠿⠁⠀⠀⠀⠀⠀⠛⠀⠀⠿⠿⠇⠀⠻⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢀⣀⣀⣀⢰⣿⣿⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡇⠀⣾⣿⡇⢸⣿⡏⢡⣤⣤⠉⠠⠖⣶⣿⠆
⠀⠈⠿⡿⠞⢸⣿⣿⠘⠰⣶⡦⠂⢸⠂⢠⡀⠠⡄⣿⣿⣿⡆⠀⣿⣿⡇⠀⣿⠃⢸⣿⣿⠀⠀⣯⡆⡇
⠀⠀⠀⠀⠀⠘⠛⠛⠂⠀⠀⠀⠀⠐⠀⠈⠉⠉⠀⠻⠿⠟⠃⠀⠛⠛⠃⠀⠈⠀⠈⠛⠛⠀⠀⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣀⣼⠟⠰⠶⠶⠶⠆⠦⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢸⡯⠀⠀⠀⠀⠀⠀⢸⡗
⠀⠐⣿⣿⠇⠿⠿⠿⠿⠇⠺⠀⠀⠀⠀⣼⣷⡄⠀⠀⠀⠀⠀⠀⠀⠁⡄⢠⡄⠀⠄⠀⢸⣇
⠀⠀⠈⠁⠀⣿⣿⣿⣿⡷⢸⠇⣤⣤⡄⣿⣿⠀⣶⣶⡂⠀⢰⠀⠀⠀⡇⢸⡇⠀⠀⢀⣯⣹⠀⢤⣤⣤⠄
⠀⠀⠀⠀⠐⠾⠿⠿⠷⠇⠸⠀⠙⠛⠁⣿⣿⠀⠈⠋⠀⠀⠀⠀⠀⠀⡇⠈⠁⠀⠀⢸⣿⣿⡇⠀⠉⠁
⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃⠋⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠛⠛

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣿⣿⣿⣿⡿⣧⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣶⣌⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣀⡀⠀⠀⢠⡀⢸⡆⠀⠀⠀⠀⢰⡿⢿⣿⣿⣿⢿⣿⣿⣎⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⢀⡀⠀⠀⢀⡀⠘⠃⠀⠀⠀⠀⠘⠁⠀⠁⠀⠀⠈⠉⣿⡿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠐⠀⠀⣸⣿⠟⢁⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠴⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁










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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣴⢤⠄⢰⡤⣤⣤⣠⣤⣤⣴⠄⢐⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣬⣶⣾⣿⣿⣦⡀⠀⣠⠔⠂⠐⠒⠠⢄⡄⠀⠀⢀⣀⡀⢠⠀⠒⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⣿⢿⣿⢿⣿⣿⢸⣏⡐⠶⡞⠛⡁⠀⢳⠀⣴⠞⠛⣓⣀⢐⡂⠀⠄⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠻⠁⠀⢀⠀⢊⠁⡎⠀⡙⠀⠁⠈⠑⠙⢹⠋⠀⠃⠀⠀⠈⠈⠇⠘⣿⡟⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠐⡀⠀⢀⠀⠀⡜⢁⡄⠑⡀⠀⣀⡀⢀⠋⠀⠀⠰⡀⠁⠀⠁⠀⢈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡈⢢⢀⡠⠚⡠⠊⠀⠀⠈⠂⠉⠐⠁⠀⠀⠀⠀⠱⡀⢉⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢏⠈⠊⢠⠏⠰⠄⡀⠤⡀⠘⠊⠀⢠⠀⠀⣀⠀⠀⣀⠺⠔⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠈⠁⠆⠊⠠⠧⠀⠝⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠠⡤⠆⠀⠀⠀⠀⠀⢐⡆⠈⠃⠀⠀⠀⠀⢠⠂⣾⣶⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗⢠⡷⠈⠹⠀⠀⠀⠀⠀⠀⠘⡈⢉⠀⣀⠀⠀⠀⠉⠠⠎⠛⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⠁⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠂⡉⠀⠀⢀⠀⢀⣀⣄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠉⠉⠉⠑⠛⠓⠁⠒⠚⠓⠚⠂⠈⠉⠘⠃⠈⠁⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⡴⠗⠀⠀⠀⠰⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠛⠀⠀⠀⠘⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠂⠀⠀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢢⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠁
⠀⠀⠀⠐⠀⠀⠠⠆⠀⠆⠆⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠶⠀⠀⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢠⡦
⠀⠠⢤⠀⠀⠀⠀⠠⠊⠑⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠑⠀⠠⠊⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄
⠀⠀⠈⢢⢀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠂⠉⠑⠂⠤⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⠀⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⢄⠉⠀⠂⠀⠂⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠈⠹⢅⠈⠁⠈⡁⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠤⠀⢢⡀


⠀⠀⠀⠀⠀⢠⣤⠤⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⡀⠀⡀⠈⡑⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠀⠀⠀⠈⠁
⠀⠀⡠⠄⠀⠉⠉⠀⠀⠀⠀⠉⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣁⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡑⣠⣤⣤
⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⢀⠄⠀⠀⠀⠀⠤⠄⠀⡈⠻⣿⠟⢡⡙⠻⠛⠀⠀⠐⠒
⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠁⠈⠁⠀⠀⠀⠀⠀⠈⡄⠀⠈⠀⠀⠂⠁⠈⠁⠈
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠤⠀⠤⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⢠⣄
⠀⠀⠀⢤⣤⣤⣀⡀⠀⠀⠀⣀⣠⣤⠶⡦⠀⠀⠀⠀⠀⣠⠴⠞⠋⢠⠜⠓⠦⠄⡀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⡀⠀⠉
⠀⠀⠀⠐⡀⠈⠛⠛⠛⠿⡛⠻⠁⠀⢀⠇⠀⠀⠀⠀⠸⡀⠐⠲⠂⠀⡀⠒⠂⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠘⣠⡀
⠀⠀⠀⠀⠉⠂⠀⠐⠊⠀⠈⠂⠀⠐⠁⠀⠀⠀⠀⠀⠀⠑⠤⠀⠀⠊⠁⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡰⢀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⡀⠄⢰⠀⠀⠒⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠈
⠀⠀⠀⠀⡰⠯⠤⠤⠘⠀⠔⠒⠐⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡀⠇⠄⠄⠒⠒⠒⠒⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷
⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠒⠒⠶⣄⠀⠀⠀⠀⠀⢀⠴⠒⢺⡖⠶⣄⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⡀⠀⠀⠀⠀⢀⡤⡾⠯⣛⣙⡋⠻⣷⡀
⠀⠀⠀⠀⠀⠀⣠⣿⣀⠀⠀⠀⠀⣿⢳⡀⠀⠀⣰⠃⠀⣠⠏⢷⠀⠈⢷⠀⠀⠀⠀⡴⣿⣀⣀⣀⣸⡟⢷⡀⠀⢰⣿⣩⠖⠋⠉⠉⠙⠳⡜⣷⡀
⠀⠀⠀⣀⡀⠀⣿⠈⠉⢩⠉⢩⠉⠁⠈⡇⠀⠀⣿⣠⠞⠅⠰⠈⢧⡀⢸⠀⠀⠀⢸⣥⠏⢉⡉⡉⠉⢷⣈⣧⠀⠾⣽⡇⣄⠀⠆⠒⢀⠀⢹⣾⠇⣠⣄
⠀⠀⠸⣭⢧⣄⡹⣄⠛⠶⠦⠤⠖⠀⣸⠃⠀⢀⣼⣇⠸⠦⠤⠞⠈⢳⣯⠀⠀⠀⢸⡄⠀⢦⣤⠤⠀⠀⢩⠇⠀⠀⠸⡇⠈⣿⣶⣶⠋⠀⢸⢃⣠⢷⡼
⠀⠀⠀⠀⠀⠈⠙⠿⡷⠦⠤⠤⢶⣏⡁⠀⠀⠐⢋⣨⡿⠲⠶⠖⢾⡙⠯⠁⠀⠀⠀⢙⣦⣄⣀⣀⣠⣴⣏⡀⠀⠀⠀⠙⣦⣀⣀⣀⣠⠴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢸⡇⠙⣿⣻⠞⠉⢰⠃⠀⠀⠀⢸⡟⠛⢲⡶⣦⠞⠉⡟⠀⠈⠀⠀⣿⠈⠙⣷⣲⠖⠋⡿⠈⠉⠁⢻
⠀⠀⠀⠀⠀⠀⠀⣼⣀⡀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⢳⡀⠈⠉⠁⠀⢰⡇⠀⠀⠀⠀⣿⠀⠀⠈⠉⠀⣸⠁⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡟⠉⠙⣿⠉⠀⠀⠀⠀⠀⠘⠓⣾⠛⠛⠛⣿⠛⠁⠀⠀⠀⠀⠈⢹⡏⠉⠙⣿⠁⠀⠀⠀⠀⠀⠧⣤⣤⣤⣤⣤⠿
⠀⠀⠀⠀⠀⢸⣍⣙⡇⠀⠀⣿⡒⢳⠄⠀⠀⠀⢠⡖⢻⡆⠀⠀⣿⣉⡷⠀⠀⠀⣶⠛⢻⠇⠀⠀⢹⠒⢲⡄⠀⠀⣠⠶⢾⡆⢀⡷⠶⣦
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠈⠛⠋⠀⠀⠀⠙⠛⠋⠀⠈⠛⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡶⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢹⣿⡀⠀⣠⣖⡿⠂⠀⠀⠀⣀⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠾⠿⢷⣤⣙⡇⠀⠀⠀⣠⠖⠛⠛⢛⣶⣄⠀⠀⠘⢿⣷⡴⠚⢛⣿⠛⢿⡏⢀⡠⢤⠀⠀⣻⣦⣿⣴⣟⠁
⠀⠀⠀⠀⠀⠀⡴⣫⣤⡶⠖⠹⣇⠻⡅⠀⠀⣼⢁⣀⣤⠴⢿⣄⣘⣇⠀⠀⠀⡿⠶⠒⠋⠘⣧⣀⣻⡿⠛⠉⢀⡾⠁⠀⠀⠀⠉⢷⡄
⠀⢀⡴⢦⠀⠰⡟⢁⡀⠂⠀⠂⡘⠳⣿⠀⠀⣿⠉⠥⠀⠀⠂⠈⠉⡿⠀⠀⠀⣧⠰⣆⡁⢉⡤⠉⣼⠀⠀⠀⢸⡅⠀⠶⠰⠆⠀⠈⣿⠀⣠⣤⡀
⠀⠘⠦⠾⣤⣀⢳⡀⠉⠛⠓⠛⠁⣠⠇⠀⠀⠹⣦⡈⠛⠛⠋⢀⡼⠁⠀⠀⠀⠘⣷⣬⣉⣩⣤⠖⠀⠀⠀⠀⠘⣧⡀⠶⠴⠆⠀⣸⢏⡼⠷⠼⠃
⠀⠀⠀⠀⠀⠉⠛⣿⠳⠶⠶⢶⣿⣁⠀⢀⣀⡴⢺⠏⠛⠛⠛⣿⢦⣤⠤⣤⠖⢋⡏⠀⠀⠈⢿⠳⣄⣀⣀⠀⣠⠼⡟⠲⠶⠖⡿⠛⠉
⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠈⣧⢹⣟⡿⠁⠀⣸⠀⠀⠀⠀⢻⠀⠳⠖⠁⠀⣼⠀⠀⠀⠀⠘⣇⠈⣧⡼⠋⠁⢰⡇⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢀⣏⣀⣀⠀⠀⠀⢸⡆⠉⠀⠀⠀⣿⣀⣀⣀⣀⣼⠇⠀⠀⠀⠀⣇⣀⠀⠀⢀⣀⣘⡆⠀⠀⠀⠀⠸⣷⠦⢤⣤⡿
⠀⠀⠀⠀⠀⠀⠉⣉⣿⠉⢹⠉⠉⠁⠀⠀⠀⠀⣀⣸⠁⠀⣇⣀⡀⠀⠀⠀⠀⣉⣉⣿⠉⠉⢹⣉⡁⠀⠀⠀⣠⣤⣿⠀⠀⣇⣤⣄
⠀⠀⠀⠀⠀⠀⢾⣭⠿⠀⢾⣭⡿⠀⠀⠀⠀⠸⣭⡿⠀⠀⠿⣬⠏⠀⠀⠀⠀⠿⠶⠏⠀⠀⠸⣯⡽⠀⠀⠀⠻⠶⠋⠀⠀⠛⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠶⠒⠶⠶⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠙⠛⠳⠶⣤⣄⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⣀⣀⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⣠⠋⠉⠛⠶⣤⣀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⢏⠉⠁⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⢰⠃⠀⠀⠀⠀⠀⢉⣛⣶⢶⣞⠋⠉⠀⠀⠀⠀⠘⡆⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡷⣄⡀⢀⣠⣴⣞⣋⣁⣀⣀⣉⣛⣶⣤⡀⠀⠀⠀⣱⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠶⢾⣻⡿⠟⠿⣭⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣙⣿⣿⡛⠛⡗⠶⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠀⢀⣠⠶⡟⠁⠠⠄⠀⠀⠹⠷⢦⣤⠀⢀⣤⣤⠶⠿⠋⠀⠠⡝⠳⣧⡀⠀⠀⠀⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⡶⣯⣅⣀⣇⠀⣂⠀⠤⡀⢀⣀⣤⠼⠟⠻⢯⣄⡀⠀⢀⡠⠀⡇⠀⣋⣙⣶⣤⡴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⣉⣋⣩⣿⠿⡿⠿⠿⡟⠒⠒⠒⠒⠒⢻⠿⠿⣿⣿⣛⡟⠉⠋⠉⠀⠉⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⢹⡏⢠⠃⠀⠀⣶⣲⢸⠀⠀⠀⠀⠀⠀⣴⣒⠌⡇⠉⠛⡷⢷⢤⣤⣄⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⠀⠀⠀⠑⠒⠋⠀⢀⣀⣀⠀⠀⠙⠒⠊⠀⠀⠀⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⠲⡄⠀⠀⠀⠀⠀⡋⠀⠀⢹⠀⠀⠀⠀⠀⢰⠒⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⡇⠀⠀⠀⠀⠀⠈⣉⣉⠁⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⡰⢋⣀⡀⠙⠆⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠀⠳⠯⠤⠬⠷⠜⠀⠀⡠⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⢀⡠⠔⠋⠀⠀⡇⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡇⠀⠀⠀⠀⠀⠱⢄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠶⠒⠒⠒⠙⡄⢠⠃⠒⠒⠒⠦⠤⡀⠀⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠱⠆⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠶⠴⠶⠶⠶⠦⠤⠤⠤⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄⠀⠀⠀⠀⠀⠰⢄⡀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠲⠂⠀⠀⠀⠀⠀⣠⠏⡐⠶⠃⠀⠀⠀⢀⡐⠂⠀⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠰⠊⠁⠀⠀⠀⢀⣞⠉⠳⠞⠀⠀⠀⠀⠀⠀⠀⠡⡌⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢬⡂⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢦⡵⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢖⠸⠃⠀⠀⠀⠀⠀⠀⠀⣾⠤⣌⠆⠀⠀⠀⢰⠁⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢥⠊⠀⠀⢔⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣤⠂⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⡔⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⢠⡏⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⢩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢢⣀⡀⠀⠀⠀⢤⡐⠀⠁⡘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⡀⠀⠈⠢⡤⠀⢄⠀⠀⠀⠀⠀⠀⠠⣎⢂⢈⡰⠂⠁⠀⠀⠀⠀⢈⡤⠂⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡐⠉⢆⠀⠀⢄⡀⣠⠀⠀⠀⠀⡖⢢⠀⣩⠜⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡐⠀⠀⠀⠑⡄⠀⠀⠀⠑⢤⡈⢆⢙⡸⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡠⠒⠣⣰⣁⠁⠑⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡐⠁⠀⠀⡠⠊⠈⠢⡀⠀⠀⡠⣅⡜⠉⠀⠀⠀⢀⠊⠈⠲⠀⠀⠀⠀⢀⣠⢉⠀⠐⠗⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠃⡜⠀⠀⢀⠔⠁⠀⠀⢀⠜⠓⢄⠑⠉⠈⢢⠀⠀⠀⠀⠢⠄⠘⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⢀⠃⡜⠀⠀⡰⠃⠀⠀⡠⠖⠁⠀⠀⠀⣁⠦⢀⡀⠡
⠀⠀⠀⠀⠀⠀⡌⠌⠀⡠⠊⠀⢀⠤⠊⠀⠀⢀⠠⠒⠉⠀⣀⠠⠘⠋
⠀⠀⠀⠀⠀⡐⠌⢀⠔⠁⡠⠐⠁⢀⡠⠐⠈⣀⠠⠐⠊⠉
⠀⠀⠀⠀⢰⠏⡠⢃⠴⠊⡀⠄⢂⠡⠔⠒⠉
⠀⠀⠀⢠⣏⣨⢔⣡⠔⠍⠒⠉
⠀⠀⢠⡿⠟⠛⠉


⣿⢿⠿⡿⡿⠛⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣟⣛⣿⣿⣛⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⣀
⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣷⣿⣿⣇⠀⠀⠀⠀⠀⠀⣀⣤⣤⣬⣾⣥⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠸⣿⣿⣿⠿⡿⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠇⢀⣿⣿⠟⠛⠛⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠐⣿⡟⠡⠂⠀⠀⠀⠦⠀⢿⣿⠇⢀⢘⣁⢸⡏⢙⠛⠛⠃⠈⠛⠛⢹⡿⠀⠸⣿⡟⣀⠀⠀⠈⢻⣿⣿⣿⠇
⠀⠀⠀⠀⠐⠻⡇⠀⣼⡆⠀⠀⣼⡷⢸⠛⠀⢠⡇⠀⡉⡇⢿⡆⠀⠀⢰⣧⠀⢨⠟⠂⠰⣟⢤⣆⠀⠀⠀⣄⠙⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠈⠀⠎⠀⠀⡄⠆⠀⠀⢁⠀⠀⡀⠀⡀⠀⠀⣘⠊⠀⠐⢼⠈⠛⠀⠀⠀⠛⠁⢀⡇⠂
⠀⠀⠀⠀⠀⠀⠈⢄⠀⠙⠒⠋⠀⠔⠀⠀⠀⠃⠀⢀⠀⠀⠢⠀⠀⠀⠀⢀⠀⠁⠈⠀⠀⢸⣇⠀⠀⠲⠖⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⢀⣀⠸⠆⠂⡺⢀⡦⠀⠈⢠⠈⠱⠊⠀⠐⢆⠄⠏⣀⠹⡄⣀⠀⠡⢈⠒⠸⢿⠿⣶⠠⠀⡶⠿⣿⡇
⠀⠀⠀⠀⠀⡠⢨⡿⠀⠀⠀⠀⠘⣷⠄⠀⠁⠀⠀⠀⠀⠊⠘⠀⠀⠀⠀⢀⡀⠁⠢⠈⠀⢀⡠⠡⠀⠀⠀⠀⠈⠸⡇
⠀⠀⠀⠀⠀⠂⣾⠇⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣑⣦⡤⠀⠀⠘⠃⠂⠀⠀⠀⠀⢠⠀⡨⡀
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠒⠃⠆⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠸⡆⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⡅⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⡠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡿⠂⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠁⠀⠒⠀⠐⠂⠈⠈⠐⣷
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠘⣿⣿⣿⠏⠘⣿⣿⣿⡿⠀⠀⠀⠀⠀⢀⡀⠀⢀⠈⠂⠀⠀⠀⠄⠀⠀⠀⡼⢶⣦⣤⣤⣤⣤⣴⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠙⠉⠑⠀⠘
⠀⠀⠀⠀⠀⠀⠰⠴⠀⠀⠀⠀⠰⠰⠀⠀⠀⠀⠀⠀⠸⠒⠆⠀⠀⠀⠐⠆⡆⠀⠀⠀⠀⠀⡆⣰⠀⠀⠀⠀⣦⣰
⠀⠀⠀⠀⠀⠀⠰⢘⠀⠀⠀⠀⢐⠠⠀⠀⠀⠀⠀⠀⠨⢠⠀⠀⠀⠀⠀⣄⡁⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⣀⠀⣠⠄⠀⠀⠀⢤⡀⢐⡀⠀⠀⠀⣄⠁⣀⠄⠀⠀⠀⢨⡀⠀⡄⠀⠀⠀⡐⠀⣠⠀⠀⠀⠀⣀⠀⢢
⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠹⠟⠃⠒⠒⠂⠀⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠘⠢⠀⠀⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠓⢦⠀⢀⠔⠀⠀⡀⠀⠄⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⡰⠁⣼⠀⠀⠀⣶⠷⡆⣀⠀⠀⠀⢤⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡽⠶⡀⠤⠄⠀⠀⠉⡌⢃⠿⠀⠀⠐⠒⡐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠊⠀⠀⠀⠠⢀⣀⠠⠐⠐⠈⠉⠉⠖⠒⠋⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢇⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢊⠄⠒⠀⠢⣌⠢⡀⠀⠀⠀⠀⠀⣀⣀⣸⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⣯⣄⣀⠀⠀⠀⠀⠀⡜⢠⠊⠀⢸⠑⠢⢄⣀⠀⡠⠒⡁⠀⣰⢀⣤⠌⣦⠈⠢⡀⠀⠀⠐⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠰⠀⣿⡤⠤⠼⠀⣤⠄⢤⢠⠶⠂⣟⢉⣹⡤⠤⠚⢿⣧⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⡗⠀⠐⡇⣀⣀⢹⠀⣤⡤⠧⡲⠆⢳⠀⠰⠂⠹⣇⠀⠸⠄⠀⠀⠒⢿⡶⠦⠿⠒
⠀⠀⠀⠀⠀⠀⠀⠒⠿⠶⢺⡿⠒⠁⠀⠀⠀⠀⡀⠀⢻⡇⠚⠁⡅⠀⠁⣠⠆⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⡋⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠑⢄⣀⣁⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⠒⠠⠤⢄⣀⣀⡀⠀⠀⠀⠀⠀⣀⡀⠤⠒⠊⠁⠀⠀⠀⠀⠀⠀⠑⢿⠷⣼⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⣧⠞⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠿⠿⡏⠉⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡀⢀⡀⠀⠀⠀⠀⠰⡹⠇⠀⠇⡁⠀⠀⠀⠀⠀⠀⠐⠁⠈⡄⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⠎⠀⠀⢁⠀⢀⣀⡀⠀⠔⠄⠠⠄⠒⠐⠂⢲⠂⠀⠀⢘⠈⠲⣍⠈⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠁⢣⠔⠀⢨⠅⠠⠒⠓⠒⠒⠀⠄⠀⠠⡂⠈⠉⠉⠉⠉⠉⠢⠼⢀⢀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡃⢀⠜⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⠁⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⢿⠈⠟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢦⠐⠈⣀⡇⠃⣠⠄⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠒⠒⠒⠒⠀⠀⠉⠉⠐⠒⠂⠀⠤⠤⠀⠀⠵

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢠⠅⠁⢠⠤⢐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⡐⠉⢢⠢⠰⠈⢀⡠⠓⢌⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠰⠤⠄⠤⠥⠉⠐⣡⠊⠀⠀⡊⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠮⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⢀⠈⠉⣰⣂⡴⡄⢦⠠⠬⠴⡅⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠉⠐⠒⠒⠒⠒⠒⠒⠂⠀⠀⠒⠒⠒⠒⠚⠆⠋⠍⠁⠐⡏⠢⢠⠀⡐⣠
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢄⣇⠠⠔⣉⠔⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠗⠒⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠘⡇⡠⢄⣀⣀⡀⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⣀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡃⡇⠀⠀⠀⢀⠠⠐⠈⠉⠉⠒⠢⠤⠐⠊⠁⠀⠀⠀⠀⠀⢡⠸
⠀⠀⠀⠀⠀⠀⠀⠀⡷⠇⠀⠀⢠⠁⡀⠔⠂⡀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠀⠀⢠⠀⠀⠀⠀⠀⠀⣘⠠⠀⠀⠀⢄⠀⠀⠀⠀⠀⠇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⠀⠀⠀⠡⠀⠀⠀⠀⠠⢹⠀⢠⡀⠀⠀⠑⢄⠀⠀⢰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡇⠀⠀⠀⠑⠀⠀⠁⠀⢸⠀⢸⠀⠀⠀⠀⢸⠀⠀⠘⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢡⠀⠀⠀⠀⢀⠄⠁⠂⠌⣶⡀⠀⠀⠀⠰⢸⠀⠀⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠂⠀⡀⠢⢀⠀⡉⠀⠀⠀⠀⠁⠀⢀⢁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⡆⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠁⠒⠐⠁⠀⠀⠀⠀⠀⠀⢰⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢦⣙⠲⠤⠤⣤⣤⣤⠤⠤⠐⣊⡡⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡈⠉⠒⠒⠒⠂⠒⠒⠂⠉⠁⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠤⠤⠤⠤⠤⠐⠒⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣛⣋⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡄⠀⠀⠀⠀⢀⡴⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢸⡇⢰⡇⠀⠀⠀⠀⠉⠤⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠙⢚⠇⠀⠀⠀⢸⣿⣦⡀⠀⠀⠀⢸⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⣤⠆⠃⠀⠀⠀⢸⣿⠻⣿⣦⣀⠀⢸⠀⡈⠀⣠⡆⠀⠀⠀⠀⠀⣀⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠈⠻⢿⣿⣶⣶⣶⡾⢿⣿⣤⣤⣴⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⢀⣠⣿⠇⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠉⠉⢠⣿⡇⠀⢰⡖⠛⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠻⣿⡟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⠴⠒⠚⠛⠷⢤⣀⣧⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⢹⣷⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡞⠉⢷⡾⡁⢀⣤⣤⣄⡀⢠⣟⢹⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣁⣀⣠⡾⠿⠶⠶⢤⣸⣿⣝⢦⡀⠀⢠⡇⠀⠈⢿⣢⡟⢩⡴⠛⠛⠉⠉⣿⣇⠀⢸⣄⣀
⠀⠀⣀⣀⣄⠀⠀⠀⠀⠉⠻⣿⡟⠁⠀⢀⣀⠀⠀⠈⢻⣿⠀⠹⣆⣸⠁⠀⠀⠘⣿⣧⡀⣷⣠⡤⠀⠀⡿⣿⠖⠚⣇⡀
⠀⠀⢸⣃⢈⠄⠀⠀⠀⠀⢀⣿⣿⢤⡾⠛⠛⠻⣦⠀⠀⢻⣔⡀⠈⠛⠀⢠⡆⠀⠸⣎⠉⠛⠉⠀⢀⡼⢃⣽⣀⣤⣽⠛⠛⠓⠂
⠀⠀⠀⠀⠉⠀⠀⠀⠀⣀⣼⣿⣿⣿⡁⠀⠀⠀⠘⡇⠀⢸⡯⣿⣄⠀⠀⢸⢿⡄⠀⢹⣶⣶⣶⠾⠛⣽⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⢷⣄⣀⣠⠾⠁⠀⣸⣿⣿⠙⢷⣤⡿⠀⠿⠛⠛⠁⠀⢄⠀⠀⠹⣄⠀⠀⠀⠀⡐⠀⠆
⠀⠀⠀⠀⠉⠉⠁⠀⠀⢀⡴⠛⣿⣿⣷⠈⠉⠁⠀⢀⣴⣿⠿⠟⠃⠈⠉⠀⠀⠀⠀⠀⢀⠵⣎⠻⣍⠉⠉⠁⠀⠀⠀⠀⠶⠉
⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋⣠⠞⠁⠙⢻⡿⠶⠶⠿⠛⠙⢿⣆⠀⠀⠀⠀⣠⡤⠤⠤⣄⢸⠀⠘⢦⡈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢠⠞⠁⣀⣠⣄⡀⠙⠆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠖⠀⠀⠀⠀⠀⠀⣿⢰⠏⠀⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⢀⠝⠀⠀⠀⠀⠀⢹⡏⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠻⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⢠⡿⠿⠀⠀⠀⠰⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⡄⠀⠀⣶⠶⠀⠀⠀⠀⠀⠒⠛⠉⠀⠀⠀⠰⠶⢤⡀
⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠠⠀⠀⠀⠁⣀⠔⠀⠀⠀⡀
⢀⠁⠀⠀⠀⠈⠆⢀⠀⠀⠈⠄⠀⠀⠶⡿⣶⣤⣄⣀⣤⣀⣤⣴⣾⠟⠛⢫⠉⠀⡇⠀⠀⠀⣄⠸⢠⡄⠀⠀⢰⠀⠀⠈⠀⠀⠀⠀⠈⠃⠛
⠸⡀⠀⢿⠂⠀⡇⠸⠃⠀⢀⠆⠀⠀⠀⠀⠀⠀⠘⠋⠙⢍⠁⠈⠁⠀⢀⠎⠀⠀⢇⠀⠀⠘⠿⢠⠘⠏⠀⠀⡌⠀⠀⠠⠀⠀⠀⣠⠀⡀⠀⠀⠀⢠
⠀⠣⡀⠀⠀⡠⠃⠄⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⠒⠐⠂⠁⠀⠀⠀⠈⠢⡀⠀⠠⠊⠑⠀⠀⠊⠀⠀⠀⠀⠠⠀⠀⠛⠌⠁⠀⠀⠐
⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠠⠤⣄⠀⠦⣄⣀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⢀⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠢⡀⠀⠀⠠⠞⠉⠁⠀⠀⠀⠀⠁
⠀⢀⣠⡴⠛⠉⢠⡐⠉⠳⢤⣀⠀⠀⠀⠤⠴⢤⣤⣀⠀⠀⠌⠀⠀⠀⠁⡀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠊⠀⠀⠀⠀⠋⠐
⠐⣏⠁⢠⣤⠀⠀⡇⢤⡄⠀⠀⡍⠀⠀⠀⠀⠀⠈⠹⣷⢸⠀⢠⣄⠀⠀⠁⠀⠀⠆⠀⠀⢀⣶⡆⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⢸
⠀⢩⠄⠀⠀⠀⠀⣇⠀⠀⠀⢀⠇⠀⠀⠰⠦⠦⢌⣦⠈⠀⢆⠸⠟⠀⢀⠂⠀⠀⠡⡀⠀⠀⢈⡁⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠃
⠀⠀⠢⣐⠀⢀⠖⠉⠠⠀⠠⠋⠀⠀⠀⠀⠀⠉⠀⠉⠁⠀⠈⠒⠠⠀⠀⠀⠀⠀⠀⠈⠀⠐⠉⠙⠂⠀⠂⠀⠀⠀⠀⠑⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠂⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠀⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⣰⡿⠆⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠘⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡠⠞⠃⣀⠉⠙⠷⢤⣀
⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠆⠀⠀⠀⠀⠰⢡⡴⣦⣤⠘⠁⣴⢷⠶⠐⡀⠀⠀⠄⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢰⡆⢶⠀⠀⠀⢸⠁
⠀⠠⠀⠀⠀⢀⡆⠀⠀⠀⠠⡶⠀⠀⠀⠀⡇⣠⣽⣷⡅⢀⠀⣙⣿⣿⠆⡇⠀⠀⡤⠀⠠⠄⠀⡄⠠⠄⠀⠐⠀⠀⠀⠀⠆⠀⠀⠀⣠⠀⠀⠀⠀⠌
⠀⠀⠃⠀⠀⢀⠐⠀⠀⠀⠀⠁⠀⠀⠀⠀⠰⡀⠉⠋⢁⠊⠡⠉⠈⠁⡐⠁⠀⠀⠐⠀⠀⠀⡸⠉⠀⠀⠀⠆⠀⠀⠀⠀⠀⠂⠀⠀⠀⠑⠄⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠛⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⣴⣿⡆⠀⠀⠀⣶⣄⡀⠀⠀⠀⠀⠀⠰⠟⠃⠀⠀⠰⢤
⠀⢠⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⢠⠀⠀⠉⠭⢙⠆⠀⠀⠀⢈⡥⠴⠂⣄⠀⢀⠬⠍⣇⠀⠀⠀⠀⢀⠴⠀⠰⠄
⠀⠉⠀⠀⠀⠀⠀⠀⠺⠿⠻⠷⠆⠀⠀⠀⡌⢀⢒⡓⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢀⣤⣶⠀⠀⠀⣴⡆⠀⠀⠀⠆⠀⠀⠀⢀⡀⠀⠀⠀⠄
⡆⠀⠀⠲⠶⠀⢐⠀⠀⠶⠶⠒⠀⡆⠀⠀⡇⠉⠙⠉⠀⠄⡆⢰⠋⠀⣰⠆⠀⠈⡇⠀⠀⠘⠿⢸⠂⠀⠀⠉⠑⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠈⠀⠀⠀⠀⠀⠈⠁⠀⠠⠄⠴⠚⠀⠀⠀⠹⡄⠬⡭⠔⣴⣎⠛⠯⢚⡟⠀⠀⠀⢳⡄⠀⠀⢀⡼⢄⡀⢀⡴⠇⠀⠀⠀⢃⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠖⠚⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠂⠰⠋⠀⠈⠙⠉⠀⠀⠀⠀⠀⠈⠓⠂

⠀⠀⡀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣾⣿⣷
⠀⢰⠂⠀⠀⠀⠀⠀⠻⣿⠀⠀⠄⠀⠀⠀⠀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⢰⠆⠀⠀⠀⠀⠐⠰⣶⡿
⠀⠘⢦⡀⠀⠀⠀⢀⡴⠋⠀⠀⠈⢄⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⣀⡈⡸⠃⠀⠈⢧⣀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠉⠻⠶⠞⠋⠀⠀⠀⠀⠀⠈⢳⣾⠟⠀⠀⠀⠀⠀⣀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠻⡿⡟⠁⠀⠀⠀⠀⠉⠛⠶⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⢈⠀⢀⣤⡈⡁⠁⢃⡀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠻⢿⠃⠙⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⡄⠀⠁⠀⣨
⠀⠀⢀⠀⠀⠒⠓⠠⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡿⠿⠿⠦⠾⢿⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠐⠒⠠⢀
⠀⠀⠀⠈⠉⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠁⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠈⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⠀⠀⠀⠀⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⠀⠀⢀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢀⡀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠸⢷⣄
⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢞⠛⢶⠹⡊⡴⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⣷⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⠶⣶⢏⢹⣿⡐⠢⣜⠐⠀⣣⠔⢒⣿⡏⡹⣖⠶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣾⠁
⠀⠀⠀⠀⠘⣿⣏⣾⣺⠁⠙⢽⠻⡟⠑⡼⠂⣰⢿⡟⣧⠙⢿⣦⡑⠂⠘⢊⣴⡾⠋⣾⢛⡿⣆⠐⢤⠊⢹⠿⡯⠋⠉⣗⣷⣻⣿⠃
⠀⠀⠀⠀⠀⠙⣿⣡⣿⣭⣒⡬⢄⡁⠠⣕⣻⠃⢾⠧⠟⣤⣦⡍⠛⢦⡴⠛⢩⣴⣤⠷⢾⣷⠘⢟⡫⠄⢈⡠⣥⣖⣽⣿⣬⣿⠋
⠀⠀⠀⠀⠀⠀⠘⣿⣉⡽⣿⡿⣷⣌⠄⠩⢙⡲⠛⠉⢰⣿⣿⡏⠀⠘⠋⠀⢹⣿⣿⡆⠹⠛⠖⢏⠌⠐⣡⣾⢿⣿⢧⣘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣞⢟⡄⢙⣿⡌⢠⡆⢀⡀⠰⠄⢸⣿⡇⣤⣠⣆⣤⢸⣿⡃⠰⠄⢀⡀⢰⠄⢁⣿⡋⢠⡻⣻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣇⢰⣷⡇⠀⠁⡿⡏⢠⢒⡟⢢⢹⣾⠳⢞⣷⡏⡔⢳⡒⡆⢹⣿⠈⠀⢸⢿⡄⣸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣗⡇⠀⠀⡇⢠⠌⠉⢸⡟⣸⠛⠃⠘⠛⣇⢿⡇⠉⠡⡄⠺⠀⠀⢸⣸⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⡀⠀⢿⣳⣶⣾⣧⣿⡀⠀⠀⢀⣿⣼⣷⣦⣞⡿⠀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠾⣷⣽⣆⣸⢿⢧⡀⢀⡼⡿⣅⢰⣯⣾⠷⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣆⣾⣿⣿⣿⡀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠐⠒⠾⠿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠙⠉⠙⠛⠃⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⡿⠟⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡇⣿⣿⣿⠁⢸⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣿⣥⣼⣿⣿⣶⣾⣿⣧⣤⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠛⠉⢹⣿⢏⣴⣿⡿⠛⠃⠀⠘⣿⣯⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⣿⣿⡜⠿⠟⠁⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠈⠻⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠹⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠠⠂⠀⠀⠄⠀⠐⠠⡀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠂⠐⠀⠀⠤⠀⠠⢄⠀⠀⠀⢀⡀⢀⠀⠈⢆⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠠⠀⡇⠀⠇⠀⠄⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠈⠚⠃⠐⠄⠀⠔⠻⠎⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠤⠠⠄⣠⠄⠀⠠⢤⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢀⠁⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢰⠀⠀⠈⠂⡀⠀⠀⣀⠜⠁⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠋⠉⠀⠀⠀⠈⠀⠀⠀⠀⠙⠉⠛⠁⠀⠚⠃⠀⠀⠒⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠐⣤⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⣠⠈⠀⠰⢀⣄⠒⠈⢧⡀⠀⠀⣤⠀⠉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⢱⡤⠂⠛⠀⠀⠀⠀⠀⠀⠀⠘⡃⠀⠠⡤⠂⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠓⠠⢀⠀⠀⠀⠁⠆⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⣎⡠⠂⠉⠁⠀⡄⠀⡃⠀⠠⡆⠀⠀⢀⡀⠀⠈⠁⣴⠊⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⡠⠜⠀⢃⠀⠀⠈⠀⢀⡀⠀⠠⡄⠋⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠒⠤⠔⠊⠀⠀⠀⠈⠄⠀⠀⠀⠸⡏⠁⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠄⠁⣸⠇⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⡀⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠀⠀⠀⠀⠀⠀⠀⡀⢈⡀⡾⠀⠸⠀⠀⢀⣀⠰⠈⠁⠀⠀⠀⠀⠇⠀⢠⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⠠⣤⠒⠉⠀⠀⠃⠀⠀⢆⡰⠉⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⡠⠤⠤⢀⠀⠀⠀⢠⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢃⠀⠀⡇⠀⠀⢀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠘⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠀⠀⠀⠀⠸⠀⠈⠢⣀⠀⣀⡴⠀⠀⠀⠀⠀⡀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⡰⠦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⣟
⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⣀⣀⠀⠈⠐⠂⠐⠀⠀⠀⠀⠉⠐⠒⣶⡄⠒⠀⡛⠀⢀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠉⠐⠂⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⠀⠁⠀⠀⠘⣄⡀
⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡞⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⢀
⣶⣶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⡱⠆⢾⣋⣉⣉⣉⣉⣉⣉⣉⣙⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⣤⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠈⠉⠙⠛⠷⣶⣴⣚⠉⠀⠀⠀⠀⠀⠀⠉⠓⠆⣀⣤⣴⡶⠶⠿⠛⠛⠿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⡰⠋⠈⠙⠻⠶⣦⣄⡀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⣰⠃⠀⠀⠀⠀⠀⠀⠉⠛⢶⣤⣀⠀⠀⠀⠘⣆⠀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣼⣷⣶⡶⣶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⢶⣶⣤⣿⣿⣏⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⠛⠋⢉⣡⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠾⠛⠛⢾⣆⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⣀⡀⠉⠀⡏⠀⣀⡀⠀⠀⠀⣠⣶⣾⡿⠿⢿⡉⠀⠀⠀⡀⠀⡏⠛⢦⡀⣀⣠⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠙⠛⠻⡇⠀⣫⣉⣷⣶⣾⣟⠉⠀⠀⠀⢠⣷⣴⠞⠋⣷⠀⡷⠾⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢨⡏⢉⡽⠋⣡⣬⣻⡄⠀⠀⠀⠀⠋⣥⣬⣳⡀⣯⡀⣷⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⢸⠀⠀⠻⠿⠟⠁⠀⠀⠀⠀⠀⠻⠿⠟⠀⠀⡟⣏⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠘⢤⡀⠀⠀⠀⠀⡞⠛⠛⠢⠀⠀⠀⠀⢠⡴⠇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠙⠒⠀⠀⠀⠀⠀⠀⢸⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣠⠊⠉⠉⠳⠄⠀⠀⠀⡼⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⢷⣾⡭⠤⣄⠀⠀⣠⠞⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⣰⠛⠒⠒⠢⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⣄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⢶⣤⣀⣴⠟⠳⣶⣤⣤⡶⢃⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠻⠟⠓⠒⠒⠒⠛⠓⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠂⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⢻⡟⠋⠁⠀⠈⠉⠉⠀⡢⠄⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠀⡐⠶⠀⠀⠀⠉⠁⠀⠀⠀⡧⠤⠐⠒⠀⠀⠀⠀⠠⠴⠜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠀⠀⠞⠁⠀⠀⠀⠀⠂⠀⢠⠔⠐⠓⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠂⠀⠀⠀⠘⠀⠀⠀⠀⢸⡆⠀⠒⠺⡆⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠐⠖⡁⠀⢸⠇⣀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠀⠀⠳⠀⠐⢤⡀⠐⠀⢀⣤⠖⠚⠛⠒⠂⠀⠀⠴⠆⠐⠒⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⢸⠁⠉⠉⠒⠒⠒⠒⠒⠈⠉⠉⠁⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⠒⠐⠛⠉⠀⠠⠀⠁⠠⠄⠐⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠄⠀⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⢠⠀⠀⠀⠀⠠⠊⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠒⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠁⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠄⠀⠠⠄⢾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣇⡇⠀⠀⠀⠀⠈⠀⢠⠇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠦⠤⠤⠤⠤⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣿⡇⠀⠀⠀⠀⠂⠀⠁⠀⠀⠘⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠤⠠⠄⠀⢀⡼⠷⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠠⠄⠀⠐⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⢸⠀⠘⠍⠃⠘⠠⠘⡏⠉⠁⠉⠀⠀⣀⠄⠀⠀⡿⡇⠀⠀⠀⡁⠐⠀⠀⠀⠀⡄⢸
⠀⠀⠀⠀⠀⠀⠀⢠⠓⠁⠀⠐⢀⡆⠀⠀⠈⡄⠸⡀⠀⢳⠶⠖⠒⠚⠋⠁⠀⠀⠀⡇⢰⡀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠸
⠀⠀⠀⠀⠀⠀⠀⡎⠇⠐⠀⢀⠞⠀⠀⠀⢠⡇⠀⠇⠀⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠠⠛⠒⠢⠤⠤⠤⠴⠒⠁⢢⡇
⠀⠀⠀⠀⠀⠀⠀⠱⣀⡠⠐⠉⠀⠀⢀⠠⠐⠓⠒⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⣄⣀⣀⣀⣤⠤⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠋⠉