Gunk Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡄⠀⠉⠓⠒⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣏⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐⠂⠨⢃⠀⡏⠀⠀⠀⠈⠁⠒⠀⠀⢀⠀⠀⠉⠐⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⢠⡶⠊⠁⠀⠀⢰⡄⠈⠀⠀⠀⠀⠀⠀⠀⠈⠄⣀⠠⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠠⡾⠃⠈⠁⠀⠀⠀⣤⣄⣀⠀⠂⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠀⠠⠈⠀⠀⠂⣉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⠀⠀⣠⠂⡠⠇⠀⠆⠀⠀⠀⠀⢀⣤⣾⣷⣦⡝⢳⣦⣀⠀⠀⠀⠉⠓⠢⢀⡀⠀⠀⠀⠠⠤⣀⠀⡀⠀⡄
⠀⠀⠀⠀⠀⢀⡠⠒⠁⠀⠐⡶⠄⠀⣀⠀⠘⠋⣀⣀⠀⠀⠀⠀⠀⢆⢠⣾⣿⣿⡟⠛⢿⢧⠊⣹⠷⣶⣤⣀⡉⠂⠀⠀⠈⠐⠀⠀⠀⠄⡀⢁⠔⠁⠀⠀⠀⠄
⠀⠀⠀⣠⠖⠉⠰⣆⡀⢤⡂⠠⣶⠄⠀⣉⡁⠀⣸⠉⠁⠀⠀⢾⣁⠝⡸⡿⠟⢿⢿⡶⠄⠀⠙⠣⠶⣁⡌⢻⣿⣦⣄⣀⡀⠀⠉⠐⣀⢀⠁⠈⠀⠀⠟⠀⠀⡠
⠀⢠⡞⢁⠰⠀⠀⣼⢧⡴⢿⠷⢀⣰⠾⡟⠀⡶⢈⣦⠀⢡⣤⣨⡿⢾⡅⣰⠎⣣⠛⢉⡄⠀⠀⠀⠀⠈⠙⢂⣠⠞⣈⠛⢛⣷⣄⡉⠘⠁⠀⠂⠀⢀⠀⠀⣼⠃
⠀⠘⣷⡄⢴⡛⣯⡰⠆⢀⣠⠶⣯⣽⣿⠁⠀⣀⣿⡉⠿⢂⡙⠻⣿⣾⡎⣿⣬⠋⣰⠏⢀⡴⠀⠀⠈⣠⣶⠍⣷⢶⡋⣡⡈⠛⢿⡷⡀⠀⠀⠀⣰⡿⠀⣼⠃
⠀⠀⠈⢿⡌⠛⢁⡡⢴⠁⠛⣾⠏⠈⣩⠄⠀⠸⣿⣷⣼⢡⣵⡶⠆⠙⠻⣿⣿⣸⠧⣰⠏⣴⠃⡀⣠⠊⣱⣴⠁⣼⡅⢹⡇⠀⠀⠀⢀⠞⠁⠀⣿⢃⣼⠃
⠀⠀⠀⠀⢳⡀⠙⣶⡿⠀⢀⡀⠒⠛⣁⣸⣿⣷⣿⣯⣽⣿⡏⠁⠀⠀⠀⠈⢿⣿⡀⠿⠻⠀⠈⡿⠁⡰⠛⣿⠋⢀⡴⠿⠀⠀⠀⠀⢿⡄⠀⡴⢿⡟⠁
⠀⠀⠀⠀⠀⠹⣿⠾⣣⣀⣄⡴⣦⣾⣿⣿⣟⢻⣍⠿⠯⠟⠁⠱⣄⠀⢀⢀⠀⠉⠛⠀⠀⠈⣶⡓⠞⣠⠞⠋⠉⠈⠀⠀⠀⠀⢀⡄⢈⣨⠿⡵⠋
⠀⠀⠀⠀⠀⠀⠈⢿⡙⢋⡐⠿⣿⣿⡿⣿⠟⠉⠉⡔⣠⣿⣶⡶⢨⠏⣁⠈⠁⢻⡞⠋⠁⠰⠘⢃⣨⢻⠀⢀⠠⠂⠀⠀⠀⠀⠘⡟⠋⣳⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡴⣿⣻⣿⠟⠁⠀⠀⢠⣿⣿⣿⣧⣶⣦⣀⣉⡀⠀⢀⣅⠀⠀⠀⠠⠈⢛⣦⡀⠀⠀⠀⠀⠁⠀⠀⠀⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣭⣶⣧⡀⠀⠸⠻⠿⢿⣿⣿⣿⣿⣿⣿⠿⠇⠈⠉⠤⢀⣠⠀⠀⢸⢧⣀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣬⢻⣿⣿⣧⠛⣧⡀⠀⠛⠋⠉⠙⠟⠉⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⠀⠀⠰⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢧⡙⢿⣿⠁⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠙⠎⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒⠀⣠⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡿⠀⠀⠙⠿⣿⣷⣄⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⣴⡄⠀⠀⠀⠀⢨⡿⠻⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠀⠀⠀⠘⠿⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠇⠀⢿⣷⣠⣀⡂⠀⠱⣶⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢴⣾⣴⡇⠀⠙⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⣸⣧⡠⣿⡗⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣅⠅⠀⠀⠀⠙⠻⣿⣦⣄⠀⠀⠀⠀⣿⠏⣁⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⡀⢰⠉⣤⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠟⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⢴⡶⢾⠉⣩⣿⣹⣷⠒⣦⠀⠀⣶⣶⣿⣿⣿⣿⣶⣦⠀⠀⣴⢦⣤⠀⠠⣤⣤⣤⣄⣀⣀⣀⣤⡖⢶⣶⣶⣤⣤⣤⣴⣦⠀⢀⣴⣾⣷⣶⣿⣷⣶⡆
⠀⠀⠀⠀⠉⢨⢀⣿⣿⣿⡥⠚⠁⠀⠀⠈⠛⠋⣿⢟⡟⠛⠉⠀⠀⠈⢺⠁⣠⠔⠛⠛⢻⡿⠛⠛⠿⣿⠀⢸⣿⣿⣿⡋⢛⡿⠋⠀⠙⠙⣿⣿⡿⡟⢿⠋
⠀⠀⠀⠀⠀⠸⠋⠙⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠉⡞⠀⠀⢀⠀⠀⠀⠀⢠⠁⠀⠀⠀⢠⡴⠛⠁⠀⣽⠀⢸⠟⠉⠉⠣⣜⠀⠀⠀⠀⠀⠀⢱⡀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠘⠀⠀⢀⡀⠀⠀⠀⠀⠀⠇⠀⠀⢶⠄⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠈⠦⠆⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⡇⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⡠⠴⣿⡇⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠃⠀⢠⡄⠀⠀⠀⠛⢻
⠀⠀⠀⠀⠀⢠⡞⠹⣄⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡼⠀⠀⡟⠁⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⣼⠀⠀⢹⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢶⣤⣀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠿⡿⠿⠋⠁⠀⠀⠛⠿⢦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠉⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆⠀⠀⠀⣤⠀⠀⠸⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠰⠀⠀⠂⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⣦⠀⠀⢀⢠⠀⠀⠀⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢇⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡄⠀⢈⣬⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⣤⣼⣿⣿⣇⠀⠰⠆⠀⠀⠀⢀⣠⣃⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢸⣦⠀⠀⢰⣶⡄⠀⢠⡀⡀⠀⢼⠁⠀⠀⠉⠀⠀⠀⣠⠹⡆
⠀⠀⠀⢀⣀⣴⣿⡿⠿⢏⠙⢒⡦⠀⠀⠀⣤⣼⣿⣿⣦⣀⠀⠀⠀⢀⡸⠀⢠⣴⣀⣤⣾⣿⣶⣶⣿⣿⣿⣶⣾⣶⣦⣶⣿⣄⠀⠀⠀⢀⣤⣦⣼⣿⣾⣿⣶⡄
⠀⠀⠀⠙⠚⠛⠓⠐⠀⠈⠙⠚⠁⠀⠀⠀⠉⠛⠛⠛⠋⠉⠁⠀⠀⠘⠿⠷⠿⠿⠿⠟⠛⠑⠒⠂⠛⠿⠿⠟⠋⠉⠉⠛⠛⠯⠃⠀⠀⠸⠿⠿⠿⠿⠿⠛⠋

⠀⠀⠀⠀⣀
⠀⢸⣿⣟⣿⡀⠀⢰⠞⠻⢶⡆⠀⠀⠀⠀⠀⠾⠆⠀⠀⠀⠀⠀⣴⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠎⠙⢁⡀⢈⡠⠄
⠀⠸⠟⠛⠛⠂⠀⠈⠷⠞⠉⠀⠀⠀⠀⠘⣻⡷⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠰⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣷⣤⣤⣀⢀⠄⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢘⣿⣿⡏⠀⠀⠀⠀⠀⣠⣿⣥⡄⠀⢈⣭⣝⢻⣆⠀⠀⠀⠀⢾⣟⣉⣀⡴⠊⠉⣙⢿⣅⣀
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠇⠀⠀⠀⠀⠀⠈⠀⢸⡷⣶⣭⣀⡉⢐⢹⣷⣶⣧⣿⣿⣿⣦⡤⠭⠙⠋⠑⠀⠉⣻⣦⣄
⠀⢰⣿⣭⣿⣷⠀⣶⣦⠰⣶⠀⠀⠀⠀⠰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣦⡈⢥⣹⡿⣻⠟⠠⠀⠀⢀⣠⣤⣦⣄⣿⡌⠉
⠀⢸⣿⡟⠛⠿⠆⢹⣽⣣⠙⠂⠀⠀⠀⢨⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡽⢶⣶⣿⣿⡌⢯⣥⣴⣴⣷⣾⣿⠁⠀⠈⠙⠛⠂
⠀⠀⠀⢀⡟⠚⢶⠈⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠹⣿⣽⣷⣽⡿⠿⢿⡿⠆⠁
⠀⠀⠀⠸⠧⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠼⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣤⣤⢤⣠⣤⡽⢷⡏⢴⣄⣀⡦⣼⡦⣽⠿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣥⣤⡄⠀⠀⠀⠀⠀⠀⠀⠲⣶⣯⣿⣿⣞⡷⣯⡝⠳⣬⡀⢴⡿⢂⠋⣿⡽⢾⣴⠟⠛⠁
⠀⠀⠀⠀⠀⠈⠀⠁⠀⠈⠁⠀⠀⠀⠀⢸⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⣽⠿⠟⠅⠛⠀⠿⠻⢐⣿⠟⡈⠁⠈⠀⠼⠆⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣥⡀⢀⣄⣠⢄⣤⣤⣌⢀⣈⣄⡀⢸⣆⠀⣄⣴⠇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠀⠀⠀⢰⣶⣶⠆⠀⠀⠀⠀⠀⠀⠲⣮⡹⣿⣚⡏⣿⢸⣿⣿⣿⡿⡏⢻⣽⣼⣩⣀⣿⡟
⠀⠠⢠⡀⢰⣎⠀⠀⠀⣀⡄⠀⠀⠀⠀⢸⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠈⢻⠛⢿⠁⠀⠈⠛⠏⡼⠃⠻⠊⣿⢿⡯⠿⠘⠇⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡽⣡⠦⡌⣷⢠⡞⠉⣙⣦⡀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠠⣤⣄⡀⢀⡈⢀⣀⠀⠀⡀⢀⣀⣀⡀⠀⠀⠀⠀⣠⣤⣤⡀⠀⢠⣶⢠⡝⣿
⠀⠀⠷⠁⠀⠀⠀⠀⠿⠞⠉⠁⠀⠀⠀⢰⣿⣿⡗⠀⠀⠀⠀⠀⠀⢿⢾⣺⢎⢿⣧⣽⡗⢼⣿⣯⡟⣽⢡⡾⣇⠀⣹⢿⡹⣾⡏⣷⠈⢿⣸⡟⣿
⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⢀⠀⠀⠀⠀⢸⣿⣿⡗⠀⠀⠀⠀⠀⠀⠘⡟⢹⡾⡛⢿⠁⡯⢸⢡⣿⣿⣿⣿⠀⣿⢦⣿⢎⡷⣬⣿⠈⣧⣿⣿⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠇⠘⠀⠀⠀⠀⠈⠁⠙⠁⠀⠁⠀⠁⠈⠁⠈⠉⠀⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠤⠤⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⢀⡴⠶⠚⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⡄⠂
⠀⠀⢀⡤⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠿⠆⠀⠀⠀⠀⠀⠀⣾⠂⠀⠀⠄⢽⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡀⣀⠀⢼⣦⠀⠀⠀⣀⣠⣤⣀
⠀⠀⠀⠀⠠⠀⠀⠀⢀⣠⡆⠀⠀⠀⠀⠀⡱⣯⡏⠀⠀⠀⠀⠘⢿⣆⣀⣠⢠⣿⣿⠀⠀⠀⠀⠀⢀⡄⠀⠀⠰⣥⠖⣶⡷⠿⠘⠛⠃⠛⣿⠉⠈
⠀⠀⢠⣴⣶⣶⣶⡄⠀⣈⡃⠀⠀⠀⠀⠀⠟⠸⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⡆⠀⠀⠰⠲⠄⠐⠀⠀⠀⠉⠀⠈⢠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠄

⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣦⣤⣤⣀⣀
⢀⣤⣶⣿⣿⡿⠿⠿⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠻⠿⠿⣿⣿⣷⣦⣄
⣾⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣿⣿
⢻⣿⣿⣿⣶⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣶⣾⣿⣿⣿
⢸⣿⣿⠉⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣾⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⢹⣿⣿
⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⣿⣿⡄⢀⣀⣤⣤⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣤⣄⣀⠀⣾⣿⡇
⠀⠀⢻⣿⣿⣿⡿⠿⠟⠛⠛⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠛⠛⠿⠿⣿⣿⣿⣿⠁
⠀⠀⢸⣿⣿⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣾⣿⣿
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣅⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠙⠛⠻⣿⣿⣿⣿⣿⣇⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠛⠛⠛⠋⠉⠉⠙⢻⣿⣿⣿⣿⡟⠻⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣶⣦⠀⢠⣶⠖⠚⠛⢛⣿⣿⡇⢠⠙⠇⢸⡿⠿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠰⠸⠃⢠⣿⠿⠿⢾⣿⣿⣿⣷⣄⣸⣷⡀⠘⢿⠟⠀⣰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⠀⢰⣆⢸⡏⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡄⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣟⠸⢐⣿⣿⣧⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠻⡍⠁⠀⠈⣟⣅⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⠀⠀⠀⠀⠀⣼⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠻⠦⠀⣠⣾⢻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡛⣯⣭⣿⣿⠟⣽⣿⣿⣏⣡⣤⣾⡿⠁⠀⣿⣿⣷⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠮⠻⣿⣿⡄⠘⣿⣿⣿⣿⡿⠟⠁⣠⣾⡿⠟⠁⣸⣿⣿⡿⣟⣭⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠁⠀⣀⣀⣘⢿⣿⣦⡌⢿⡇⠀⢀⣤⡾⠟⠉⣀⣴⣾⣿⣻⣵⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⢠⣾⣿⡿⠿⠿⢿⣿⣷⡈⢿⡴⠛⢡⣤⣴⣾⣿⣿⠛⠉⣿⣿⠟⠉⢀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠸⣿⣿⣄⣀⡀⠠⢹⣿⣿⣦⣤⣶⣿⣿⣿⣿⣿⣿⠀⠀⠿⠋⢀⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡄⠀⠙⠻⠿⠟⣁⠇⡘⢻⣿⣿⡿⢻⡇⠈⢻⡇⢸⣿⠀⠀⢀⡀⠈⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣤⣄⣀⣬⣤⣾⡇⢸⣿⣿⡇⢸⡇⢸⣦⡁⢸⣿⠀⠀⣾⣿⣄⠀⠙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⠻⢿⠁⣸⣿⣿⣿⣿⣿⣿⣤⣤⣿⣿⣿⣦⣄⣈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⢿⣿⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣅⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⡀⢀⣀⠀⢸⡇⡀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⡏⢹⡇⣿⢸⡇⢸⠋⡇⢸⣿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠷⢾⡇⠻⠼⠇⠸⠀⠇⠸⠟⠧⠀⠀⠀⠀⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠓⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠁⠀⠀⠀⢀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⢠⡄⢄⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠈⣇⡔⠀⠀⠀⠈⡄⠀⠀⠀⠘⡆⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠳⢦⡀⠀⠀⠘⠤⠁⠀⠀⠙⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣾⣿⣿⣿⡟⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⣀⣤⣤⣴⣶⣶⣿⣿⣿⣿⢛⣟⠻⣿⣿⣇⢿⡿⣱⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⡇⢻⣿⣿⡇⢸⣿⣷⠈⣿⣿⡚⢡⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⣿⣿⣿⣧⠘⠻⣿⡇⡄⢻⣿⣿⠸⣿⣿⣇⢸⣿⣿⡰⣬⡙⢿⣿⠇
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⠙⣿⣿⣿⡇⢰⣌⠁⣿⡌⣿⣿⣧⡙⢿⡟⢸⣿⣿⣷⣿⣿⣷⠌
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⢙⣌⢿⣿⣷⠀⢿⣿⣿⣿⡘⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢸⣿⣦⢻⣿⣦⣸⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠰⡆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⣰⡆⠀⢀⣤⡀⠀⠀⠀⠀⣴⣾⡆⢸⡟⠢⡀⣿⡿⠂⠀⣤⣿⠧⢀⣷⢶⡄⠀⠀⣶⣶⡄
⠀⠀⠀⠀⠀⠀⣿⣠⢦⣿⠀⠀⢺⣿⡃⠀⠀⠀⠸⠿⠛⣧⠸⡧⠀⠀⠙⠻⠆⠀⠀⣿⠀⠀⡇⠈⠿⠀⢸⣯⠙⠛
⠀⠀⠀⠀⠀⠀⠻⠃⠀⠛⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠂
⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⣶⣄⠀⠀⣲⡆⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⢰⣶⠹⡆⠀⠀⠀⢸⡿⢦⣀⢰⣿⡿⠀⠀⢸⡟⠾⣤⣀⠀⠀⠀⠀⠺⠂
⠀⠀⠀⠀⠀⠀⢿⡄⠀⠙⠛⠶⣦⠀⠀⠉⠙⠻⠄⠀⠀⣿⠃⠀⠙⢻⣿⡇⠀⠀⣿⡇⠀⠀⠉⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠘⢿⣄⣀⣀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⢸⡿⠳⣦⡀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠛⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⣄⢀⣀⠠⠤⢀⣀⠔⠒⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⡀⠀⠈⠁⠀⠀⠀⠠⠁⠀⠀⢀⣂⡮⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⣠⡞⣆⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠻⠷⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠙⠛⠃⢑⠆⠀⠀⠀⠀⠈⠓⠄⢀⠀⠀⠼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣄⡀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⡀⠀⠀⠢⡀⠀⢀⠄⠀⢀⠠⠆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠰⠈⠶⠤⢀⠀⠀⠠⠄⠂⠩⢠⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⢡⠀⠀⠈⠀⠀⠀⣀⠀⠐⢸⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⢂⡀⠀⣉⠉⠄⢠⠀⠒⠸⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⢣⠀⠘⠇⠈⠀⠈⠁⠀⠀⠉⠈⢣⠘⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⡆⢀⠀⡄⠀⠒⠑⠔⡆⠀⠀⣀⠀⠁⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡇⠈⠀⠈⠁⠀⠀⠀⠁⠐⠀⠋⠀⠀⠐⢄⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠁⢀⢀⠀⠀⠀⠀⠀⠈⠀⡀⡀⠀⠀⠀⠀⢣⠉⠂⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠠⢄⡀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⡜⠠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⢤⠀⠀⠀⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠰⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡠⠄⠀⠢⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⠣⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢇⡠⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⠤⣀⣀⡀⠀⠀⠈⠈⢀⠒⠤⠨⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠒⠒⠒⠈⠉

⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⣀⣀⣀⣀⣀⣀⣠⡤⠀⣤⣤⣤⣤⣤⣄
⠀⠀⠀⣼⡟⢻⡟⢶⡶⢶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⡶⣶⡖⣿⢻⣿⢻⠇⠛⠀⠘⠛⢻⡟⡿⣟⠛⠀⠀⠘⠋⢻⠛⢹⡁
⠀⠀⠘⢿⠇⠙⢳⣿⣧⢸⡏⣸⣿⡀⠿⠀⣿⡀⡟⠰⡿⠛⢣⣿⣷⠉⣾⠿⡇⠀⠀⢀⠀⠀⢨⠀⠀⠉⣰⠀⠀⢀⡆⢸⠀⠈
⠀⠀⠀⢀⢀⠀⠈⠓⠋⢘⡏⠻⠟⠁⠀⠀⠻⠃⣷⡄⠀⠀⠈⠳⠞⠁⣈⣉⡀⠀⠀⠶⠀⠀⣀⣀⣤⠀⠻⠀⠀⢸⣿
⠀⠀⠀⠀⠈⠀⢀⡀⣀⠀⠁⠀⠰⢆⠀⠀⠀⠈⠻⠃⠀⢰⣶⢿⣿⣿⣿⢿⡿⣿⡇⠀⢴⣿⣿⣿⣿⣿⠿⠿⠂⠈⠁⠀⠀⠸⠄
⠀⠀⠀⠀⡸⢿⣿⣧⣭⣴⣄⢀⣄⣀⣠⣤⣴⣶⣶⣶⡄⠸⢿⡞⣼⢃⡇⡎⣴⣿⠇⠀⢀
⠀⠀⠀⣴⣶⣾⣿⣿⢩⠛⠋⠀⠉⠉⠿⠋⣿⣏⠉⠙⠃⠀⠀⡇⣯⣿⡇⡟⡏⡄⠀⠀⣿⣿⣿⡟⣯⣿⣟⣿⡇
⠀⠀⠘⢯⡿⠏⢿⣛⡷⠀⢀⣤⣐⡂⠀⠘⠿⠋⠀⣤⡀⠀⣰⢣⣿⣻⣧⣧⢇⠁⠀⠀⣿⣏⡇⣿⣷⡿⡿⢻⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠿⠿⡟⠇⠀⠀⠀⠀⠀⠉⠀⠸⠿⠛⠛⠿⢿⣿⣼⣧⠀⠀⣿⡧⣿⡇⡿⣷⣟⢸⡇⠀⠉
⠀⠀⠀⣤⣤⣤⣤⣤⣤⡀⠀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡀⠀⠥⣧⢛⣿⠆⣿⢿⢸⡆⠀⠀⠀⡄
⠀⠀⠀⠘⡟⣿⢻⠙⠏⠀⠀⣿⣿⣿⢿⠟⠿⢻⡿⡶⣹⠀⠀⠈⠛⣻⠟⡿⢿⠛⠁⠀⠀⣿⠀⣏⠀⣿⠀⠘⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠠⣿⢸⠀⠀⠶⠀⠿⡟⠋⢸⠀⠀⢸⢿⣧⠛⠀⠀⠀⠀⠙⠀⡇⠀⠐⠟⠀⠀⠺⠃⠉⠈⠋⡦⢀⡠⣾⣷⠀⠉⡀
⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠸⠋⠀⣾⡄⠀⣿⠈⠉⠀⡀⡀⠀⠀⠀⢠⣷⡀⠀⢀⡀⢀⠶⣮⣠⣶⣿⣳⣾⣿⢿⣷⣦⣾⣿
⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠸⣹⠇⠀⢽⡆⠀⠀⠐⠀⠛⠀⠀⢻⣼⡇⠐⠈⢿⣾⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠁⠀⡠⠚⠉⠉⢢⡤⠒⠉⠁⢢⠀⠀⠀⠀⠀⣀⠠⠤⠀⠀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠳⢄⠀⠀⠊⠀⠀⢀⠁⠰⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠐⢤⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠃⣆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⣾⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣷⣆⠀⠀⠀⠀⠘⠟⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠠⢀⡌⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠈⠈⠃⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⠀⠀⢠⣿⡄⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢸⡷⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢴⠄⠀⠀⠀⠀⠀⠸⣿⣿⡄⠀⠀⠀⠀⠀⣿⣿⣿⡇⢠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⡇⢸⣿⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⢸⣿⣿⣿⠃⣾⣿⣿⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡗⠀⠀⠀⠀⠀⢰⠀⢠⡄⠀⠀⠙⣿⣿⣿⡀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠀⠀⠂⠀⠀⠀⠀⢀⣿⣿⡀⠀⠀⠘⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⠀⠀⠈⠻⣧⠈⠄⠁⡀⠀⠐⠷⠄
⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣄⡀⠀⠘⠂⠀⠢⡈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠛⡀⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠄⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⢃⠀⠸⢆⠀⠀⠀⠀⢀⠀⠀⢱⠠⠀⠀⡠⠤⠀⠁⠀⠀⠐⠀⠄⠀⠁⠀⠀⠀⠀⠀⡀⠀⡘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣾⣭⣿⡶⠶⢤⢀⠀⣀
⠀⠀⠀⣠⠄⠀⠈⡙⣿⠟⢋⣰⡦⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⡀⠀⠀⠈⠁⠀⠀⠐⠄
⠀⠀⣴⠃⣼⠁⡾⠁⠺⠞⠉⣿⠃⣠⠆⢀⠄⠀⠀⠀⠀⠀⣠⠊⢭⢿⣷⣄⠀⠀⠐⠀⠀⠀⠂⠉
⠀⠀⢿⠀⣯⠀⠁⠀⠀⠀⠀⡁⢸⡇⢰⡏⠀⣾⠀⢰⡄⢀⡏⠀⠀⠰⢍⠋⠁⠀⠀⠀⠀⠀⠆⢰⣿⢠
⠀⠀⠈⠳⢿⠀⠀⠀⠀⠀⠀⢠⢸⡇⢸⣇⠀⣿⠀⣿⠀⣾⠃⠀⠀⠀⠀⠉⠀⠐⠀⠀⠀⢈⡅⣿⣿⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣿⣼⣧⣀⣿⡀⢹⡄⢿⡀⠛⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⣿⡏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠟⠛⠛⠓⠨⠍⢹⡾⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⣰⠁⠀⣸⢸⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠛⢇⣀⠀⠈⠁⠀⠀⠀⠀⠠⠚⠁⠀⠀⡟⠘⢿⡇⣿
⢠⣤⣄⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠙⣓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⠀⠀⠀⠛
⢸⣿⣿⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠁⠘⠣⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⣾⠃
⠀⠁⠀⠙⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡤⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢀⣠⣤⣶⣾⣿⣿⣿⡆⠀⠀⢐⠀⠀⠠⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⡿⠿⠛⠓⠂⠀⠈⠀⠀⠀⣦⠀⠀⣄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢀⠄
⢿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢦⣴⢋⡐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⣠⣀⣤⣤⣀⣠⣤⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣾⣿⣦⣤⣤⡄
⠀⠀⠀⠀⠀⠀⣀⡘⢻⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⡄
⠀⢠⣤⠀⣀⠻⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⢀⡄
⠀⡘⢛⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⢀⡀
⢾⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠘⠋
⠈⣰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⣤
⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠉⠛⢿⣿⣟⣿⢻⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡰
⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠈⠛⠲⠀⠀⣻⣿⠿⣿⠿⣷⡿⣿⠿⢿⢷⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠋
⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⡤⣶⣿⣇⠯⣹⢈⠸⣀⣹⣨⣼⢨⡅⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⢠⣤⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠘⠟⠋⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠁
⠀⠀⠀⣤⠿⢿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠻⠃⣀
⠀⠀⠀⠀⠀⠈⠋⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⠟⠑⣾⡆⠘⠁⠘⠆⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⣿⡿⠛⢻⣿⣿⣿⣿⡿⠿⠻⣿⠟⢻⡏⠈⠛⠃⠉⢳⡄⠀⠉⠁⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠉⠉⠛⠋⠀⠀⠀⢾⣿⡇⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣷⣶⣶⣶⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠏⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⣡⡆⢸⣿⣿⣿⡿⢠⡌⠻⠿⠿⢛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣧⠈⢿⣿⡿⠃⣸⣿⣶⣤⣴⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⢀⣴⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣧⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠘⢿⣿⣿⡿⠉⣿⣿⣿⡇⠀⣸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄⠀⠀⠉⠉⠁⠀⢸⣿⣿⡇⠀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡀⠀⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠿⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⣤⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣄⠀⠘⠲⠶⠦⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠁⠀⣸⠇⠀⠀⢠⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⠴⠚⠁⠀⠀⠀⣿⠀⠠⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢰⡇⠀⠀⠀⠀⠀⡞⠁⢀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠈⠳⣄⣀⣷⠀⠀⠀⡀⠀⠙⠶⠊⠙⠋⠀⠀⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡙⢢⣼⣠⠃⠀⠀⠀⠀⠈⠀⠀⠀⣠⠏⠳⡄⠀⠀⠀⠀⠀⣮⠽⠃⢿⣳⢰⣏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢯⠟⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⣸⠿⡗⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠹⣆⡀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣅⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠒⠂⠀⠀⠀⣿⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠀⣠⠖⠚⠢⠤⠤⠤⠤⠤⠤⠤⠔⠒⠉⠁⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢀⡖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠾⠖⠒⠒⠒⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⠀⠀⠀⠀⠸⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠤⠄⠒⠒⠒⠚⠛⠉⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⢤⣤⠤⠴⠚⠋⠷⣤⣤⠴⠓⠦⢤⡤⠴⠛⠓⠒⠒⠚⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⢀⠀⠀⠐⠋⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠂⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄⠀⠀⠀⠠⠲⠀⠀⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡚⣸⣆⡄⠀⢤⣴⠎⠀⣀⣤⠀⣀⡔⢀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠂⠘⣯⡧⣼⠞⠃⠀⣸⣃⣽⣶⡉⡄⠈⠀⡗⣆
⠀⠀⠀⠀⠀⠔⠀⢀⠀⢐⠀⠁⠀⠀⠀⠉⠁⠀⠀⠦⠾⠉⠋⢃⠙⠛⠓⠚⠀⠇⠸
⠀⠀⠀⠐⠂⠀⠈⡀⠀⠀⠀⠀⠀⢀⢀⣷⣤⠀⠀⣸⡃⠚⣆⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⡀⠀⣁⠐⠀⢼⡦⢙⣦⠶⠙⢷⠎⢤⣀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠰⣶⠾⠅⣀⣖⠃⢘⡿⢿⣿⣿⣳⣄⡌⣻⣬⣄⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⡿⢀⣦⡀⡵⠉⠉⠁⠀⠀⠁⠻⢿⣧⣹⣿⣽⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢾⣷⠨⡽⠆⠁⢄⢀⡄⠀⡤⠀⠀⠀⠀⠁⠌⠋⣳⠆⠒⠀⢠⠀⢀⢀
⠀⠀⠆⠀⠀⠀⠀⢰⠀⠀⠀⠂⠑⢼⣿⡷⣴⡋⣁⠀⠀⠀⠀⠀⠀⠸⡎⠉⠀⣄⢴⣷⠺⡂⠈⠄
⠀⠀⠀⠀⠀⠀⠺⠓⠊⢀⡀⢀⡄⣠⢿⡳⡞⢇⠙⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⡙⢰⠈⠁
⠀⠀⠀⠀⢲⣦⣀⡀⠀⠈⠄⢈⡁⠖⡛⠥⢭⣴⠦⢐⣀⠁⠀⠀⠀⢠⡀⠀⠀⠀⠀⢈⠃⠀⡀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠢⠈⠠⠂⠠⠀⠀⠁⠲⠄⠈⠓⠲⡯⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⣏⠉⣿⠀⠀⠀⠀⠀⢠⠞⠛⢣
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠘⠦⡤⠞⠀⢀⠀⠀⠀⠀⢀⡞⢳⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠸⠤⣿⡀⣧⠞⠀⠀⠀⠀⠀⠀⠀⢰⣛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⡿⠀⠀⠀⠈⠳⣬⢶⣯⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⡤⠤⠤⢤⡄
⠀⠀⢀⣀⠤⠴⠒⠒⠲⢤⣀⢠⡴⠶⠶⠒⠒⠒⠲⡆⠀⢸⡇⠀⠀⢹⠀⠀⣠⠴⠚⠋⠉⠛⢲⡴⠏⠉⠉⢳⢘⡋⠉⠉⠉⢹⠀⢠⡟⠃⠀⠀⠸⠃
⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠘⣾⡄⠀⠀⠀⠀⠀⠀⢳⠀⡸⠀⠀⠀⡸⢀⣾⠃⠀⠀⠀⠀⠀⢘⡀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠸⡇⢸⡇⠀⠀⠀⣤
⢸⡆⠀⠀⠀⠀⣤⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⣤⡴⠞⠀⡧⠀⠀⠀⣧⡞⠡⠀⠀⠀⠀⠀⡞⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡃⡄⠀⢀⡇
⡎⢃⠀⠀⠀⠀⠀⠀⠀⣤⣾⡁⣶⠀⠀⠀⠀⢸⣇⣀⣸⠃⠀⠀⠀⣿⠀⢀⡄⠀⠀⠀⠀⠙⠒⠙⡇⠀⠀⠀⣠⢤⡀⠀⠀⠀⣧⠸⣧⣗⣲⡾⠁
⣇⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠈⠉⠉⠏⠀⠀⠀⠀⠉⠁⠈⠉⡂⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡆⡼⡏⠉⣳
⠸⡄⠻⠀⠀⠀⠘⠛⠀⠀⠀⢀⡿⠀⣀⣀⣀⣀⣀⣀⣰⠀⠀⠀⠀⠀⠀⠀⢈⠙⠢⢄⣀⣀⣀⡠⡴⠦⠀⠐⣧⠀⠀⠒⠈⠉⡟⠙⠧⠼⠛⠁
⠀⣧⠰⢇⠀⠀⠀⠀⠀⢀⠰⣻⠁⠉⠉⣉⣉⣉⣠⣿⠏⠁⠈⠉⠉⠉⠉⣩⡟⠶⣤⣀⣀⣀⣀⡴⠶⠶⠶⠶⠋⠳⠶⠚⠛⠛⠁
⠀⠸⣆⡀⠉⠉⠀⠉⣉⣠⠞⠛⠛⠛⠋⠉⠉⠉⠉⠘⠓⠛⠛⠛⠉⠉⠙⠉⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠉⠉⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⣤⡴⠶⠶⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡖⠳⢶⠋⠉⠀⠀⣠⠞⠋⣠⣤⠴⠦⣤⣀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠙⣆⠀⠀⠀⠀⠘⠁⢠⠞⠁⠀⠀⠀⠀⠉⢣⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⡦⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⣶⡆⠀⠀⠀⡇⠈⡷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠉⠁⠀⠀⢀⠇⠀⠀⠀⠈⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⠤⠤⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠐⠀⠀⠀⠀⠤⠀⠴⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⣠⡴⠶⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣀⡠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣀⣀⣀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢻⡄⢠⡾⠉⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡷⠚⠁⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⢸⠀⠀⢀⡔⠛⠒⠦⣄⡀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠑⠀⠀⢀⡀⠀⢰⡂⠀⠀⠀⠀⠀⣼⠀⢠⠏⠀⠀⠀⠀⠀⠙⢦⡀⠀⢠⠁⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⣄⠀⠀⠀⢰⡏⢹⠀⠀⠁⠀⣀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡘⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⡿⢹⡆⠀⠀⣿⠀⢸⠀⠀⠀⣼⠉⢳⡀⠀⡿⠀⢸⠀⠀⠀⠀⣀⣀⣀⣠⣤⣼⣿⠀⠀⠀⠘⠛⠛⠓⠶⢦⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣻⡇⢸⡇⠀⠀⡟⠀⢸⠀⠀⢠⡏⡠⠚⠧⡼⠃⠀⢸⢉⡏⠉⠉⠉⠉⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠠⣤⡤⠞⠁
⠀⠀⠀⠀⠀⢾⣋⠉⠀⠀⠀⠀⠀⠀⠈⠁⠟⢸⡇⠀⢰⡇⠀⠈⢇⣀⡼⠀⠓⠙⠳⠖⠀⠀⢸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⣤⣤⠟⠳⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠉⠛⢶⡄⠀⠀⠀⣠⠤⠤⠦⠀⣇⠀⢸⠓⠢⡀⠈⠉⠀⡤⠤⠀⢀⡴⠀⠀⠸⠦⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠇
⠀⠀⠀⠀⠀⣀⣠⠤⠴⠏⠀⠀⠀⠙⠲⠂⢤⠀⢿⣤⠞⠓⠋⠀⠀⠀⠀⠓⠒⠒⠒⠲⡆⠀⡖⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⣠⣤⡤⣤⣄
⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⣄⠀⠀⠀⠛⢶⡄⠀⠈
⠀⠀⠀⠀⠀⠛⠦⠤⠤⠤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⠀⠀⠀⠾⠤⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠲⠶⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠖⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠒⠶⠶⠦⠤⣤⣤⣀⣀⣀⠀⠀⠀⠀⢀⣠⡿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢀⣀⣀⣀⡈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡄⠀⠀⣤⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⡇⢀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢱⣄⣿⡤⠖⢀⣠⣴⡶⠀⠀⠀⠀⣀⠑⣤⣥⣬⣴⣟⣡⠀⣠⣶⠀⠀⠀⠀⠀⠀⠀⡛⢷⣸⣤⣎⣴⠿⢛⠁
⠀⠀⠀⠀⠀⠀⠰⣶⣌⣹⣿⣿⣷⡖⠉⠁⠀⠀⠀⠀⠀⠀⢉⣑⣿⣿⣿⣿⣿⣗⢚⠉⠀⠀⠀⠀⠀⠘⠓⠮⣹⣿⣿⣿⣿⣖⣈⣬
⠀⠀⠀⠀⠀⠀⠀⠰⠬⣿⣿⣿⣿⣯⣭⣅⣈⡀⠀⠀⠀⢰⡶⠼⣿⣿⣿⣿⣿⡿⣍⣁⡀⠀⠀⢀⣤⣴⠖⢩⡽⢿⣿⣿⣿⣷⣍⠋
⠀⠀⠀⠀⠀⣤⣤⣶⠞⣹⢻⡟⢿⠛⠆⠀⠁⠀⠀⠀⠀⠀⣤⡶⢛⡿⣿⢿⢟⡳⠌⠛⠃⠀⠀⠀⠉⠀⠶⠀⠐⠋⡉⣿⢹⣮⡁
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠁⢾⣇⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠿⠈⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠻⣆⠰⣹⡀⣆⡾⣱⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠸⡇⠀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠃⣌⣷⣿⣿⣿⣿⣷⣒⣭⡶⠄⠀⠀⣤⣄⡀⠈⠣⣀⣧⡜⠁⣀⠀⠾⠆⠀⠀⠀⠀⠺⣦⡀⢄⣧⣸⣀⠖⢀⣤⠶⠟
⠀⠀⠀⠀⠘⠿⣶⡢⣾⣿⣿⣿⣿⣿⣿⣗⣤⡄⠀⠀⠀⠈⣉⠉⠓⢶⣿⣿⣷⣞⡋⠀⠀⠀⠀⠀⢀⣀⣀⠰⣼⣿⣿⣿⣿⣿⡅⠤⠐
⠀⠀⠀⠀⠀⠈⠉⢭⢼⣿⣿⣿⣿⣿⣿⣧⡚⠒⠀⠀⠀⠀⠁⢤⠶⠚⡿⢿⠻⡀⠈⠙⠻⠷⠆⠀⠈⠉⠉⠶⣻⣿⣿⣿⣿⡿⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠰⠋⠝⡟⣻⣏⠹⣄⠙⠃⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⢸⡆⠹⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡝⣿⢻⡻⠝⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠉⠘⠀⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠘⣇⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⢀⢤⡀⠀⠀⠀⠀⢀⣀⠺⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠎⠀⠀⠀⠀⣇⣈⡇⠀⠀⠀⠀⢀⢤
⠀⠀⠀⠀⠰⠭⣆⢼⠝⠀⠀⠀⠀⠀⣔⡦⠀⠀⠀⠀⠀⠀⠁⠀⠾⠆⠀⠀⠸⢤⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡤⠤⢤⠀⠀⠀⣀⣀⣀⡀⣀⣤⣤⠀⡤⠤⠤⡄⠀⣠⠖⠂⢰
⢀⡤⠒⠉⠁⠀⠉⠳⣼⠉⠉⠁⠀⠈⡅⢰⠁⠀⢸⢀⡴⠋⠀⠀⠀⠛⠁⠀⠈⠶⠂⠀⠀⢸⠀⡏⠀⠀⠈
⢸⠀⠀⠀⣤⠀⠀⠀⣿⠀⠀⠀⢠⡤⠇⣸⠀⠀⢸⡞⠁⠀⠀⠀⡔⠋⡀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⡆
⡆⠆⠀⠀⠀⠀⠐⠺⢤⡆⠀⠀⠠⠧⢤⡇⠀⠀⢸⡄⠤⡀⠀⠀⠈⠉⠙⠀⠀⠰⠲⠀⠀⠀⡆⢳⡧⢞⠁
⢣⠸⠀⠀⠀⠦⠀⠀⢈⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠧⢀⠀⠀⠀⣠⡄⠀⣤⠀⠡⠠⢐⠇⣏⡤⠞
⠘⡄⢆⣀⠀⢀⡀⠄⣾⣉⣉⣉⣉⣩⣿⠓⠒⠒⣂⣀⡸⠳⢤⣀⣈⡠⠶⠶⠶⠊⠲⠒⠒⠊
⠀⠑⠦⠤⣤⠤⠴⠊⠉⠉⠁⠀⠉⠀⠈⠉⠉⠁⠀⠉⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⢤⡴⠚⠉⢉⡽⠛⣉⣁⣉⡉⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠙⡆⠀⠀⠀⠋⡰⠋⠁⠀⠀⠉⢂⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠻⠃⠀⠇⠀⠀⠀⠀⡅⠀⠐⠿⠀⠀⢸⠀⠛⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡳⠄⠠⠄⠊⠀⠀⠀⠀⠀⠙⢤⣀⠀⢀⡠⠊⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⠈⢳⡀⠀⢀⡴⠒⠒⠦⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠔⠒⠉⠉⠉⠉⠉⠉⠉⠉⠁⠒⠒⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡴⠁⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡦⠂⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠁⠀⠀⠀⠆⠀⠀⡀⠀⠀⢀⡁⠀⠀⣈⣀⠀⠀⠀⠀⠀⣰⠏⠀⡀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠆⠀⠀⠀⢀⠄⠀⠀⠀⢨⡀⣠⠌⠀⠉⠁⢠⣀⠀⣠⠁⠀⠃⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣤⠀⠀⢀⠞⡇⠀⠁⢀⡀⠀⢸⠀⡁⠀⠀⠀⠀⠀⠈⢆⡏⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣸⠁⡇⠀⢸⠀⡷⠀⠀⠏⢹⡄⡸⠀⣧⣤⣤⡤⠤⠤⠤⠼⠷⠀⠀⠉⠉⠙⢓⡆
⠀⠀⠀⢠⡶⠒⠒⠒⠒⠒⠒⠛⡆⡇⠀⣿⠀⢡⠀⡸⠰⠡⢍⡇⠀⠰⠊⠀⠀⠀⠀⠀⠀⠣⣀⣀⡦⣌⡽⠋
⠀⠀⠀⠀⠉⠓⢦⠀⠀⢀⠤⠤⠁⢇⠀⡗⢢⠀⠉⠁⠤⠤⠤⠆⠀⠰⠀⡄⠀⠀⠀⠀⠀⢀⣸⠇
⠀⠀⠀⢠⠔⠒⠊⠀⠀⠈⠁⠐⡆⠸⠖⠉⠁⠀⠀⠀⠉⠉⠉⠳⠀⠋⠉⠁⠀⠀⠀⠀⠀⠈⢧⣀⡀⠀⠀⢠⣖⠒⠲
⠀⠀⠀⠸⠤⠀⣀⣀⣀⣀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⠀⠀⣼⣧⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠦⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠒⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⢠
⠀⠀⠀⣰⠉⢳⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⠀⢉⡁⠀⠀⠀⠀⠀⠁⠤⠉⠠⠀⠀⠀⠀⠀⠀⠀⠈⠈⠐
⠀⠀⠀⠺⠰⠉⠆⠀⠀⠀⠀⠀⣶⣏⣳⡖⠀⠀⠀⠀⠀⢀⣘⣁⠀⠀⠀⠀⠀⠀⠹⠺⠈⠁⠀⠀⠀⠀⠀⢨⡀⠗⣈⠂
⠀⠀⠀⠐⠿⠗⠃⠀⠀⠀⠀⠀⠁⠙⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈⠺⠏⠁⠀⠀⠀⠀⠀⠈⠁⠀⠁

⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⢘⣏⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡉⣯⠩⠀⠀⠀⠀⠀⠀⡑⣇⣿⡋⠀⠀⠀⠀⠀⣾⠃⢷⡄⠀⠀⠀⠀⠀⢸⠦⠿⠀⠀⠀⠀⠀⠀⠠⠛⡈⠓⠆
⠀⠀⠀⢓⣿⡀⠂⠀⠀⠀⠀⠨⠃⣇⡿⢡⠀⠀⠀⠀⠀⢻⠀⣾⠃⠀⠀⠀⠀⠀⡅⠀⠀⣏⠀⠀⠀⠀⠀⢐⠀⠁⠘⡂
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⢸⣁⡿⠀⠀⠀⠀⠀⠀⠹⣀⡀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠈⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠴⢸⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠰⣤⣤⡔⠢⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⢐⡀⠀⠀⠀⠀⠐⢀⡊⣒⠀⠀⠀⠀⠀⠀⠀⡤⢢⠄
⠀⠀⠀⢐⢠⢰⠂⠀⠀⠀⠀⠀⡜⠀⠂⠀⠀⠀⠀⢀⢀⡻⠻⡟⡀⠀⠀⠀⠀⠀⠈⠀⠃⠔⠀⠀⠀⠀⠀⢀⣄⣀⡀
⠀⠀⠈⢹⡸⢀⠁⠀⠀⠀⠀⠐⠢⢝⢡⠃⠀⠀⠀⠀⠀⢿⡘⣿⠀⠀⠀⠀⠀⠀⠀⡀⢀⠁⠀⠀⠀⠀⠀⠘⢖⣲⠗
⠀⠀⠀⢻⣒⣚⡗⠂⠀⠀⠀⠀⢶⣌⣰⠦⠀⠀⠀⢀⠁⠪⡄⠼⠉⠀⠀⠀⠀⠀⠀⢇⡼⢀
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⡀⢀⠁⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⢀⣄⡄⠀⠀⠀⠀⠀⠀⠀⢨⣡
⠀⠀⠐⡘⠁⠋⠪⠀⠀⠀⠀⠀⢈⠇⣺⠀⠀⠀⠀⠀⠀⠒⠐⠀⠠⠀⠀⠀⠀⠀⠼⠃⠷⠀⠀⠀⠀⠀⠀⠀⠚⠳
⠀⠘⠒⠄⠀⠀⠔⠀⠀⠀⠀⠀⣰⠐⡰⠀⠀⠀⠀⠀⠬⠀⠀⠈⠤⠀⠀⠀⠀⠈⢦⣛⣠⠆⠀⠀⠀⠀⠀⠀⠂⠘⠂
⠀⠈⠈⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠢⢤⠔⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠚⠘⠆⠀⠀⠀⠀⠀⠀⠐⡀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⣶⣤⡴⡇⠀⠀⠀⠀⠀⠘⢡⠛⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡆⣖⠀⠀⠀⠀⠀⠠⠌⠀⠀⠠⠄
⠀⠀⠠⠄⣿⠅⠃⠀⠀⠀⠀⠀⢂⠸⠀⡂⠀⠀⠀⠄⠀⡌⠈⣄⠀⠀⠀⠀⠀⠀⣠⡞⠤⠀⠀⠀⠀⠀⠀⢴⡄⠀⣼
⠀⠀⠀⠶⠊⠛⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠈⠑⠃⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠖⠒⠒⠋⠉⠉⠉⠉⠉⠉⠓⠒⠒⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⠉⠁⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⢄⣀⡀⠀⠀⠀⠀⠀⠉⠙⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⡆⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢀⡤⠤⠖⠋⠁⠀⠀⠹⣄⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⡞⠀⣀⡀⠀⠀⠀⢀⣀⠈⢧⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⡤⡼⠁⢛⣛⠛⢦⠀⠀⠘⣛⠛⢺⡠⢤⠀⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡇⢻⠀⠀⠁⠀⢸⠀⠀⠀⠈⠀⢸⠁⣸⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠘⠛⣆⠀⠀⠀⠙⠒⠀⠀⠀⠀⢸⠗⠃⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢀⣠⠼⢆⠀⠀⠀⠛⠀⠀⠀⣠⠯⣀⡀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⢷⡚⠉⠀⠀⠀⠙⠦⣤⣤⣤⠴⠚⠁⠀⠀⠉⢓⡶⠷⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠉⠓⠦⢤⣀⣀⡤⣤⣤⡤⣤⣀⣀⡠⠴⠚⠉⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢳⡘⣞⢷⠻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢹⣣⡑⣌⢢⠳⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢀⡄⠀⠀⡞⠀⠀⠹⣝⠎⠀⠀⠀⠘⡆⠀⠀⢀⡄⠀⠀⠀⠀⠀⢧
⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⡼⠀⡠⢼⣁⠀⠀⣠⠼⠓⢤⡀⠀⠀⣹⠤⡀⠀⢧⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢠⡷⠊⠀⠀⠈⠙⡿⠀⠀⠀⠀⠸⣟⠉⠀⠀⠈⠲⣜⡆⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⢣⡀⠀⠀⣀⠴⠋⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠳⢤⣀⠀⠀⣠⠇⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⣧⠤⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡤⣤⠤⠤⣼⠁⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠤⠴⠧⣤⠤⠧⢤⡿⣧⡤⠼⠤⠤⢤⡿⠤⠤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⡏⠀⠀⢸⡇⢸⠀⠀⠀⠀⠈⣇⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀⠀⠀⢸⠁⠀⠀⡼⠁⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠸⡄⠀⠀⣇⠀⠀⠀⣠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢾⣀⢀⡼⠁⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠙⣆⣀⣹⠤⠒⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣀⣒⣒⣠⣼⣧⣼⣤⣤⣤⣀⣒⣒⣈⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⣿⣷⣶⣿⣿⣷⡀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⡟⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⣿⣿⣿⣿⠋⠈⣿⡏⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠃⠀⠀⣿⣷⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠏⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⢹⣧⣤⣴⣏⢉⣳⡦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⢹⣧⣤⣴⣯⢉⣓⡦⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣁⡈⢳⡀⠀⠀⠀⠀⢀⡞⠁⣸⣿⢪⣛⣹⣿⣿⣿⡿⢁⡈⠳⡄
⠀⠀⠀⠀⠀⠀⢠⠏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢹⡀⠀⠀⢀⡏⠀⣻⡟⣫⣼⣿⣿⣿⣿⣿⣟⣼⠃⠀⠹⡄
⠀⠀⠀⠀⠀⠀⣼⠐⠿⢿⣿⣿⣿⣿⣿⣿⠟⢿⠋⠉⠀⠀⠀⡇⠀⠀⢸⠀⠿⢿⡇⣿⣿⣿⣿⣿⡟⢿⡇⠉⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠻⠟⣻⣿⡟⠉⣿⠀⢸⡀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠸⠿⢿⣾⡿⠉⣿⠀⢸⡇⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠛⠋⠀⠀⣿⠀⢻⡇⠀⠀⠀⢰⠃⠀⠀⠸⡆⠀⠀⠀⠙⠋⠀⠀⣿⡄⢸⡇⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⢰⣿⡇⠈⠁⠀⠀⣠⠋⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⢀⣿⡇⠈⠁⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⢸⣿⠃⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⢸⣿⠃⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠚⠛⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠛⠒⠋⠉

⣶⣶⢶⢷⣿⠿⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⡶⣶⣶⢶⣶⠶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣤⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣿⣿⡿⡇
⠀⠀⠀⢰⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⡶⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣴⣇
⠀⠀⠀⠀⠙⣿⣿⣿⣽⣿⣶⣤⣀⢀⠰⠃⠘⠆⠀⢀⣤⣶⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⢻⣙⣻⣿⣿⣿⣿⣿⣷⡇⠡⠄⠸⣾⣿⣿⣿⣿⣿⣟⣋⡿⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠉⣵⣿⣿⣿⡽⣛⢵⡂⠙⠋⢐⡪⣝⢯⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠈⠷⠀⠀⡀⢀⣰⡄⠀⣸⢿⣿⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⢰⢹⣿⣾⣿⣿⡏⣆⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡰⠂⠁⣠⣴⣿⠿⠋⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⠶⠭⠭⠷⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⣵⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠸⢿⠿⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⢢⢂
⠀⠀⠀⠈⠖⡼⢀⣃⡤⢠⡄⣀⠀⢄⡘⠃⠘⠃⣀⠀⢀⠠⡄⢰⣔⣇⢧⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢳⡧⠀⡄
⠀⠀⠀⠀⠘⡍⣑⡒⠽⡶⠳⢺⢧⣦⡍⠁⠈⢉⣴⡤⣟⠺⢳⡮⠔⡚⡉⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡸⠮⢷⠢⠌
⠀⠀⠀⠀⠀⠀⠨⠦⠎⠑⠘⠋⢋⣹⡄⠀⠀⢠⣏⠙⠋⠋⠚⠹⠴⠏⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣮⡯⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣉⣴⣤⣤⣤⣶⣦⣉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⡀⠀⠀⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢁⣛⡻⠿⠿⠿⢛⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠔⠀⢠⣀⣼⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢩⣙⣛⣛⣋⡭⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⢻⣛⣩⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢸⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠁⠀⠀⠀⠀⠉⠑⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣼⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⠀⠀⠀⠀⠀⢸⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠒⣀⣀⡀⠀⢀⠀⠉⠁⠀⠀⠀⣀⣀⡀⠺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣶⡟⠛⣷⣧⣄⠀⠀⠀⢀⣤⠾⢿⣿⣷⡆⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠹⣿⣿⣿⣿⣿⡆⠀⠀⣾⣷⣶⣿⣿⡿⢡⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠈⠳⡈⠻⠿⣿⣿⡇⠀⠀⢿⣿⡿⠿⠋⣰⠋
⠀⠀⠀⠀⠀⠀⢰⡉⢧⠀⢸⠀⡇⠀⠀⠀⠹⡆⠀⠀⠀⠠⠄⠤⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⢳⠘⣆⢸⠀⡇⠀⠀⠀⠀⠙⣦⣄⠀⠀⠀⠀⡀⠀⣤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠸⣼⢰⡇⠀⠀⠀⠀⠀⠈⠻⣄⠀⠈⠀⠀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⡠⣼⣆⣩⠼⣄⠀⠀⠀⠀⠀⠀⠀⢹⣷⣤⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠳⡕⠲⣶⠄⠈⡇⠀⠀⠀⠀⠀⠀⣸⠋⠀⠈⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣖⠁⠀⢰⠁⠀⢠⠴⠒⠲⠊⡡⠀⠢⠄⢠⡉⠲⠖⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠸⡀⢠⡇⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢧⢸⡇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⢏⠀⣸⡟⣇⠀⠀⠀⢀⠀⠀⠀⠀⣿⠁⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠈⢠⣿⠁⢹⠀⠀⠀⠀⠀⠀⠀⢸⢿⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⢆⡀⢻⠃⠀⣾⠀⠀⠀⠀⠀⠀⠀⢸⣾⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠥⠏⠀⠀⠻⠤⠤⠤⠤⠤⠤⠤⠼⠻⠤⠤⠇


⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠄⠀⠀⠀⠀⠀⠄⠀⠀⠠⠠⠀⠀⠀⠐⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣤⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠺⡖⣤⣤⡄⠈⣣⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢙⡄⣈⣼⣷⣼⣺⣇⠝⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢘⣷⣿⣿⣿⣿⣿⣿⣧⡀⠀⠸⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣶⣿⠿⣿⠏⠁⣽⠻⣿⣿⣿⣷⡀⠀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠉⠀⠀⠀⠠⠀⠀⠀⠩⠛⠋⠙⠃⠀⠸⣷⠀⠀⠀⠀⡐⠛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⢹⡄⠀⣠⠊⣽⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠌⠀⠀⠀⢀⣠⣄⣀⠀⠀⠀⠀⠉⢀⢠⠆⢈⣀⡀⢈⠛⠚⣡⠞⢹⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⢀⠀⠊⢀⠞⠀⣀⠾⠚⠛⠿⡖⢆⡀⢀⡀⣠⣶⣿⣿⣿⣷⣿⣧⣤⠏⠀⠈⢧⣈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠛⠀⢰⠇⠀⠀⣀⠀⢻⣤⣧⣤⣿⠟⠉⠋⠛⠿⣿⣿⡏⠀⠀⠀⠀⠀⣿⡼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⡆⢀⣀⠀⠀⣴⢺⣿⣿⠋⠡⠔⠁⣀⠀⢉⣿⡇⠀⠀⠀⠀⠈⣿⠃⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢡⡈⠉⠀⢀⣿⡿⡟⣧⣜⠲⣤⡄⢨⣤⢞⣿⡇⠀⠀⠀⠀⠀⢷⣄⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⣠⡴⠋⠑⠒⠊⠘⢷⣿⠟⠻⠦⢠⡶⠛⡫⠀⣿⣄⠀⠀⠀⠀⠀⡞⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⡷⢶⡆⠙⠻⡆⠀⠀⠀⣠⣿⣾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠰⠁⠈⠀⠀⠀⣦⠐⠆⠀⠀⠀⠀⠠⠤⡄⢀⠠⠁⠀⢹⡆⠘⣿⠀⠀⢰⢿⡏⡏⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⢄⠀⠀⠀⢸⠀⠀⢠⣟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⡆⠀⠈⣿⡖⣿⡶⠏⢁⣿⣷⣦⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣳⣿⣡⣠⠚⠃⢸⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⣼⡿⠁⠀⠀⠒⠀⠀⠀⠀⠀⢸⣿⣏⡀⣒⣟⡥⣮⡼⠁⠀⠀⢿⡳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⣼⠀⠀⠈⠁⢀⡤⠊⠀⠀⠀⠀⢠⠄⡈⣙⣿⡇⠉⢩⡄⣾⠂⠀⠀⠀⠀⠹⣯⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠰⣯⠾⢻⡀⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠐⠋⠃⣾⣷⣥⣠⣀⠀⠀⠀⣼⠏⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⡂⢠⣏⣽⡆⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣛⣲⣶⣿⣿⣿⣷⣬⣷⣄⡀⠀⠀⢠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢠⢲⡏⠁⠀⠈⠙⠛⠳⢾⡧⣤⣀⣄⠀⠀⢘⣷⣤⣷⣶⣿⡿⠿⠛⠉⠉⠋⣿⣿⣧⣟⣻⡿⣿⣫⣾⡇
⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠈⣶⣽⢦⣤⣀⣀⣄⠀⠀⠀⠐⠭⡛⢟⣋⣩⣥⠭⠉⠋⠁⢠⣤⣤⣾⣔⣤⡿⣿⣷⣿⣿⣷⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠚⠒⡅⠀⠛⠈⠉⠀⠀⠀⠈⠀⠀⠈⠙⠦⣤⣄⣀⠈⠛⠿⣿⡶⠶⠖⠛⠛⠛⠉⠙⠉⠉⠉⠀⠐⠛⠛⠛⠛⣏⣉⣉⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠈⠋⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⣿⣿⣿⡷⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⢾⣿⣿⣿⣯⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣯⣿⣿⣿⡀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⣉⡀⠀⠀⣿⠿⠿⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠃⠀⠀⣀⣠⣾⣿⣿⣿⢷⣤⡀⠀⠀⠀⠀⠀⠀⢠⣾⢿⣿⣿⣿⣦⣀⠀⠀⠀⣹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡇⣀⣰⣿⣿⡿⣋⠹⡏⣼⢹⣿⡄⠀⠀⠀⠀⣰⡿⣿⣮⡿⠩⣿⣿⣿⣧⣀⣀⣹⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⣻⣿⣿⣟⠁⡁⢙⡀⠄⠠⠎⠞⠛⢦⣾⣆⣼⢋⣇⡱⠀⠀⢰⡋⢈⠙⣿⣿⣿⣯⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣯⣿⣿⠄⠓⠸⠧⠀⠀⠂⣆⡄⠈⣿⣿⣰⣄⡏⡃⡀⠀⠞⠁⠄⠰⠿⣿⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⣶⣶⠲⣶⣾⣶⡷⡖⠒⢦⣿⣟⡻⢿⣶⣶⣶⣻⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣟⠻⢿⣷⣿⢿⡇⢑⠍⣉⠀⣿⡏⠐⣻⣿⠉⣹⠿⣷⣾⣿⢿⡿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡉⢶⣊⠁⠉⠒⠀⡍⠀⠀⠀⢽⡇⠀⠀⢷⣧⠐⠊⠉⠙⣻⣾⣸⣳⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⢷⣟⠲⠍⣀⠖⠀⣤⢀⠀⠀⠀⠸⠃⠀⠀⢀⡆⣤⠑⢟⣈⡹⢟⣿⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠟⢿⣿⣶⢾⣥⢴⣆⠤⠀⠀⠀⠀⠀⠀⠀⠈⢧⣾⢿⣬⣶⣶⣯⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠙⠋⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠙⠏⠉⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣤⠶⠖⠲⢶⡤⠶⠚⠛⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠒⠿⡉⠙⠏⠀⠀⢀⡴⠋⣠⠴⠒⠛⠓⠦⣄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣀⠀⠀⢹⡄⠀⠀⠀⠈⠀⡜⠁⠀⠀⡀⠀⠀⠈⡆⠘⡆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠿⠂⠀⢰⠃⠀⠀⠀⠀⠀⡇⠀⠀⠸⠿⠂⠀⠀⣸⠀⠟⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢳⣀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⣰⠟⠉⠉⠉⠁⠂⠀⢀⠀⢀⠀⠀⠀⠀⠈⠛⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⢻⡄⠀⣠⠞⠋⠁⠀⠈⠻⣆
⠀⠀⠀⠀⠀⢿⠀⢀⠔⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡴⠃⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠈⠳⣬⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⣤⠄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⡆⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠁⠀⡀⠀⠀⠈⠀⠀⠀⠳⠀⠀⠀⠀⠃⠀⠀⡠⠛⠲⢤⣀⠀⠀⠀⠀⢀⡞⠁⢀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⠀⠀⠀⡀⠀⢰⠂⠀⠀⠀⠀⢀⡇⠀⡞⠁⠀⠀⠀⠈⠓⢄⠀⠀⡾⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣤⡀⠀⠀⢠⡞⢹⡀⠀⠁⢀⣀⠀⠀⢸⡇⢠⠁⠀⠀⠀⠀⠀⠀⠀⠳⣠⡇⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢰⠇⣷⠀⠀⣸⠀⢸⠇⠀⠀⡞⠉⣧⠀⢸⠃⢸⡀⢀⣀⣀⣀⣀⣀⣠⣤⡿⡇⠀⠀⠈⠙⠛⠓⠶⢦⡀
⠀⣀⣤⣤⣤⣤⣤⣤⣤⣭⣿⡄⣿⠀⠀⣿⠀⢸⡆⠀⢰⢃⠖⢙⠶⡏⠀⠸⣉⡏⠉⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠀⠰⢦⡶⠚
⠘⢏⣀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢿⠀⠀⡏⠀⠀⢧⣠⠞⠈⠉⠙⠒⡇⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡴⠞⠙⠒⠛⠁
⠀⠀⠈⠑⣦⠀⠀⠀⡴⠒⠒⠃⢸⡀⢀⣏⣙⠄⠀⠀⢠⡖⠲⠤⠴⠇⠀⠀⠓⠒⣦⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⣠⠤⠖⠚⠋⠀⠀⠀⠙⠛⠓⡆⠘⠷⠞⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⣳⠀⢟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⣴⠒⠚⠲⢤
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣄⠀⠀⠉⣳⠀⢀⣸
⠈⠉⠉⠙⠛⠓⠒⠲⠶⠤⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⡤⠟⠀⠀⠈⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠲⠦⠤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣛⣓⣲⣶⣶⣤⣤⣤⣶⣞⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣋⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⠠⢀⣉⠀⠲⢉⡉⠱⠰⠋⡛⠰⠠⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡎⣸⡈⠃⠐⠂⠊⣄⠙⢮⠯⠿⠼⢾⣟⠶⢁⠚⠓⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⣀⣀⢛⠿⠝⠡⠀⠘⠓⠀⠀⠐⢂⠐⡷⠀⠀⠂⠀⠈⠄⢁⣂⠘⠓⢂⣤⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢲⣀⢀⣠⡑⠐⠀⠀⠀⠀⠀⠀⠀⠈⡓⠨⣀⠂⠀⠀⠀⠀⠀⠀⠉⠉⠀⢁⠃⣨⠈⡆⠤
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠏⠲⠁⣠⣄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠥⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠼⠙⠃⠄⡦⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠘⢴⣛⡴⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠴⡆⢵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠊⣅⣴⢄⡇
⠀⠀⠀⠀⠀⠀⡼⢺⠒⢦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨⡠⠷⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⢚⣫⠄⡕
⠀⠀⠀⠀⠀⢃⠣⣞⣳⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠙⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣧⣉⣍⠠
⠀⠀⠀⠀⣉⠈⢀⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⣀⡉⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡄⢙
⠀⠀⠀⡈⢉⣁⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣇⢘⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠾⡋⠐⠂
⠀⠀⢀⢤⠴⣡⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠓⠓⠋⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡈⢈⣌
⠀⠀⠰⡤⠶⠪⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡘⣀⣤⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣄⠈⠁
⠀⠀⠀⢇⣼⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⣌
⠀⠀⣲⠤⡭⣉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠰⠶⢀⢓⠳⠃⡒⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣇⣼⡀⡃
⠀⠀⣅⠴⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣦⠓⣢⠾⠦⣾⢶⠥⠘⣩⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣙⠲⠞⠁
⠀⠀⢈⣛⡚⢻⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡠⠀⠆⡒⣧⢼⢆⡸⢤⠦⠗⠉⡀⢈⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠄⠲⠗
⠀⠀⠀⠐⣦⣚⡢⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢉⠓⢢⢀⠋⠐⠀⠠⢩⠛⡗⢲⠀⠉⠤⠀⢈⠉⡐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠟⢢⢁
⠀⠀⠀⠐⠠⠭⡄⣈⠀⠀⠀⠀⠀⠀⠀⠄⠂⠳⣽⡳⠎⠉⠁⠀⠀⠀⠹⣴⣓⡼⠀⠀⠀⠀⣠⢤⠀⢺⣆⠀⠀⠀⠀⠀⠀⠀⠀⠲⢜⣳⡴⠐
⠀⠀⠀⠀⠐⢳⠄⡿⢅⠀⠀⠀⠀⠀⡄⣚⠓⢈⠀⠀⠀⠀⠀⠀⠀⠨⣁⠄⠉⣀⠀⠀⠀⠀⠀⠀⢠⠁⣄⠈⣀⣄⣀⠀⠀⠀⠄⠁⣨⡁⠂
⠀⠀⠀⠀⠀⢸⡤⠷⣠⠂⢄⡔⢚⠓⡤⠦⠌⠛⠁⠀⠀⠀⠀⠀⠀⠀⣴⠄⢀⡁⠀⠀⠀⠀⠀⠀⠀⠛⠶⠏⡇⣠⡀⣣⠀⠐⠗⢂⠈⡁
⠀⠀⠀⠀⠀⠀⠈⢙⠅⣨⡘⢦⣛⠦⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠆⠈⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠜⠾⠚⢃⣍⠩⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢍⣌⡹⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠆⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢴⠓⢤⠤⠩⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⡥⠛⠒⢆⠠⢀⠀⠀⠀⠀⠀⠀⠀⡤⡬⣅⣉⠀⠀⠀⠀⠀⠀⠀⢀⣀⢧⡸⡦⡴⠐⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣜⣗⡜⢰⠤⠦⣤⣤⣄⠀⡀⣼⣠⡇⣈⠄⠀⠄⠀⢀⣭⡀⣰⢁⣏⠱⡍⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠴⠄⣘⣠⣆⢈⡄⡄⡚⢓⢳⢛⠊⠘⠓⠴⢨⠅⠰⢘⠯⠭⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠓⠉⠄⠘⠙⠀⡄⠛⠃⠠⠐⠺⠊⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠉⠉⠛⠿⡟⠻⡟⡻⠿⠿⢿⣻⠟⠛⢿⠛⠛⠋⠉⠙⠻⣿⣿⠉⠟⠉⠻⣿⠙⣿⣿⠿⢿⣿⣿⡏⢻⡿⣿⠟⠋⠉⣫⠉⠈⢿⢿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡀⠀⠁⠀⠀⠀⠀⠀⠀⣿⢉⡟⠀⠀⠀⠘⣇⢸⡇⠀⠀⠃⣿⢷⡘⢸⠇⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⡿⢸⠇⠀⠀⠀⠀⠘⣾⡅⠀⠀⠀⡇⢈⢡⡏⠀⠀⠀⠀⠀⢹⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠀⣷⡿⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⡇⠈⢸⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⢸⢃⢸⡄
⠀⠀⠀⠀⠀⠀⠠⠀⠀⣼⢸⡄⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⡆⠀⠀⠀⠀⡼⣸⠀⢷⡀
⠀⠀⠀⠀⠀⣰⠇⠀⠠⣻⠈⢣⣸⡀⠀⢰⡀⠀⠀⠀⠀⠀⠀⡀⢰⡀⢠⠀⠀⠀⠀⢀⣴⣏⠀⢻⣄⠀⢀⣴⣁⣿⣧⣀⣳⣤⣀⣆⡀⣀
⢀⣄⣀⣠⣾⣿⣷⣤⣴⣿⣦⣾⣿⣷⣤⣾⣷⣶⣤⣤⣴⣴⣾⣷⣶⣷⣾⣶⣦⣤⣶⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⢿⣿⣿⡿⠟⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠯⠭⠭⠭⠭⠌⠉⠉⠀⠀⠐⠒⠀⠀⠀⠀⠾⠇
⠀⠈⠛⠋⠉⠉⠉⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⢠⣄⠴⣶⢲⡆⢤⣴⣤⠄⣰⢲⡶⢂⣤⠄⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠻⣧⣄⠀⠹⠀⣽⡇⠰⠉⠀⣠⣾⠋⠀⠀⠀⠀⢠⣾⠏⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⢻⣿⡄⠁⠏⢀⠁⣰⡿⠉⠀⠀⠀⠀⠀⠀⢊⣽⣿⣯⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠈⡿⠞⠀⠈⠻⣏⠀⠀⠀⠀⠀⣴⣶⣶⡽⢻⡿⡿⢻⣹⣾⣶⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⠀⠀⠀⣤⣄⡘⢿⣿⠇⠀⢇⠃⠀⣿⣿⠋⣀⣴⡄⠀⠀⠀⠀⠈⠟
⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⡠⠀⠀⢠⠤⠾⠟⠋⠁⠀⠀⠀⠀⠀⠈⠙⠻⠿⢧⣤
⠀⠀⠀⠀⠀⠀⠀⡀⠀⡶⢶⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢶⣦⠆⡀
⠀⠀⠀⠀⠀⠀⢀⣀⡾⣏⡙⣃⢈⡄⠀⠀⠤⠄⢀⠀⠀⡀⢠⣶⡄⢀⡀⠀⠀⡠⠤⠀⠀⠀⢈⣠⣼⣧⣄⡃
⠀⠀⠀⠀⠀⠀⣨⣭⣿⣿⣟⣯⡭⡄⠀⠀⠀⠀⠀⠈⣓⢉⣿⣿⣿⡉⢉⠀⠀⠀⠀⠀⠀⠀⣈⣉⣿⣿⣋⣀⡀
⠀⠀⠀⠀⠀⢀⠃⣾⣿⣿⣿⡇⢇⠰⠀⠀⠈⠙⠱⠋⠁⠞⠈⣿⠉⠣⠀⠙⠟⠛⠁⠀⠀⠎⠀⣿⣿⣿⣿⠁⠰⡄
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣇⢸⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣿⣿⣿⣿⡷⠦
⠀⠀⠀⠀⠀⢸⠁⣿⣿⣿⣿⡟⢸⢈⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⢀⠇⠹⣷⣿⣍⣠⠊⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠈⠉⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠒⠒⠒⠲⢼
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡇⢀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⢿⣿⣿⣷⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠣⠤⣀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠞⠚⠀⠋⠛⠋⠁⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠳⡀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠔⠀⠀⠀⠀⠀⠀⠀⢣⠀⡀⢹⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠇⠀⠀⢠⣿⠆⢸⠋⠉⠊⠀⠀⠀⠀⠈⠙⠲⣄⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠐⠀⢸⡿⢀⠜⠀⠀⠀⠀⣀⠀⣀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣠⣥⣤⣤⡼⢿⣷⣦⡤⠤⠔⠊⢉⠵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⠟⠁⡾⠵⠜⠓⢀⢘⠐⢢⡟⡆⣀⣀⠴⡅⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠒⡞⠺⠶⠷⠯⠴⠿⠶⠖⠛⠉⡎⠉⠀⠀⠀⠙⣄⠀⣀⠤⠒⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠈⣑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠕⠀⠀⡠⠊⠈⠉⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠄⣈⠷⣀⣀⠀⠀⠀⠀⣀⣴⡊⠥⠄⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠢⡤⠤⡤⠤⣠⣀⣀⡀⣸⣴⣿⢹⣷⣼⠄⣀⡀⣠⡠⠤⡠⢦⡤⠆⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠈⠀⠘⣦⣽⣿⣿⣻⣭⡍⡏⠈⡏⢩⢙⡹⠻⠙⣣⠔⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢄⠀⠀⢀⣠⠄⠀⠰⠁
⠀⢂⡛⣯⠛⠘⠫⠝⣿⡛⠅⠣⠀⠓⠘⣛⣚⠠⠛⠈⠋⡏⣻⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⢠⠀⢀⠀⠁
⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠉⠈⠂⠀⠈⠁⠋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠈⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠀⣀⠠⣤⠀⠀⠀⣤⢄⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢰⠁⠀⠀⠀⠉⡀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠴⠀⠂⠀⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠰⡄⠈⠢⠀⠀⠠⠊⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠄⢀⠀⠀⠀⠐⠂⠀⠉⠀⠁⠂⠺⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢀⡤⠀⢄⡀⠸
⠀⠀⠂⢬⠓⠉⠀⠄⠐⢒⡏⠈⠈⠀⢟⠒⠀⠀⠀⠑⢫⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠻⠈⠀⠠⠟⢃⡀
⠀⠀⠀⠀⢣⡖⠢⡍⠙⡻⢃⠀⠀⠀⠚⢛⠁⡥⠒⣦⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠠⠃⣀⠘⠄⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠈⣰⡇⠀⠀⠀⢳⡈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⡀⠀⠀⠀⣸⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⢣⡀⠀⢀⠃⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢠⠠⠀⠀⠀⠀⠀⢄⡄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠁⠇⠀⠀⠀⠀⠀⠰⠈⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⢠⢠⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠤⠀⠀⠘⠠
⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠈⠆⠉⡞⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⣀⠓⠀⢃⣠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠄⠀⠣⡀⠀⣠⠀⠀⠀⡀⠀⠀⠀⠠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⢈⠀⠀⠀⠉⠀⠉⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠁⠀⠎⠁⢱⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠒⠀⠐⢤⠊⠁⠀⠉⠓⠴⡀⠀⠒⣁⠐
⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠁⢴⡃⠀⠀⠀⢘⡆⠌⢀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⡸⠃⠆⠀⠀⠀⠀⠀⢆⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠁⠀⠀⠀⢈⠁⠐⡀
⠀⠀⠀⠀⠀⠀⠜⠁⢂⠘⡀⠀⠀⠀⠘⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠄⠀⠀⠀⠀⠀⢲⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⡤⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠼⠀⠀⠀⠀⠀⠀⠀⡆⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠀⢀⠠⠁⠀⠐

⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣖⡀⣀⣄⢀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⢀⡀⠀⠀⠀⢠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣼⡿⢀⡇⠀⣼⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣻⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣟⣿⡇⠈⠀⠀⣿⣿⣿⣻⣿⡆
⠀⠀⠀⠀⣸⡄⠀⠸⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠄⠀⠀⠀⠀⠀⠀⠀⢈⣿⣧⣿⠀⠀⠀⣸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢠⡟⢢⢀⠀⢀⣠⣤⣠⣀⠀⡤⠘⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠄⢠⣿⣿⣿⣿⣿⣿⢿⡇
⠀⠀⢀⣸⣷⣾⣸⣦⣾⣿⣿⣿⣿⡄⡀⠀⠀⠀⢠⣄⠀⠀⠐⠂⠤⢀⢸⣿⣿⡿⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠰⣚⣛⣙⣿⣻⣿⠿⢿⣥⣌⣉⠹⣿⣟⢲⣀⣀⠠⠙⠿⣦⡄⠀⠀⠀⠸⣿⣿⣧⣣⡿⠿⠛⠛⠛⠛⠿⣿⣿⣧
⠀⣿⠉⡭⠩⣿⣿⣴⣾⠿⢿⣿⣿⣿⣿⣿⠟⠻⢶⡗⢦⣼⣿⡆⠀⠀⠀⢹⣿⣿⣿⠀⣠⣤⠆⠲⣴⣤⣄⣘⣻
⢨⣿⠀⠃⠀⠈⠙⠋⢁⡀⠀⣸⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠄⠀⠀⢸⣿⣿⣿⣾⠋⠀⠀⢰⣿⣿⢿⡿⠿
⢸⣿⠀⠒⠀⢀⡒⠀⠒⠀⠀⢻⠒⠒⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣇⠀⢀⠄⣾⣿⣿⣾⣿⣿
⢸⣿⠀⠀⠀⠘⠂⠀⠀⠀⠀⣾⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⢸⣿⣿⣿⢻⣄⡌⠀⣿⣿⣿⣿⣿⣿
⡿⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⢻⡂⠂⢚⣿⣿⣿⣿⣿⣿⣿⣿⣿⢴⠀⠀⢸⣿⣿⣿⣿⣿⣳⣠⣿⣿⣿⣿⣿⣿
⠘⡿⠀⠁⠀⢠⠄⠀⠀⠀⠀⢼⠅⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⠀⠀⠺⣿⢿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⡿
⢸⣿⠀⠀⠀⠈⠁⠀⠀⠀⠀⢿⡃⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠄⠀⠸⣿⣿⣿⣷⣶⣶⣶⣿⣩⣭⣽⣿⣿
⢸⣿⠀⢀⣀⣁⣀⣀⣀⣀⣠⣿⣍⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣾⣿⣿
⠸⣿⣀⣻⣿⣿⣿⣿⣿⣿⣟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣯⣿⣿⣟
⠀⠉⠛⠛⠻⠿⠿⠿⠿⣿⠯⢤⣤⣼⣿⣿⣿⣿⣿⣛⣯⣭⣿⣿⣿⣿⣶⣿⣿⡿⢿⣿⣿⣿⡿⢿⠿⣯⣿⡉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣬⣭⣭⣿⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣞⣻⣾⡏⣉⡿⢛⣿⣏⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡾⠻⠿⠟⠛⠛⠋⢛⡷⢿⡿⠸⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢨⣿⠶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⢓⣾⣻⢧⣶⣶⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⢹⡿⢭⣿⣄⣿⢿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣿⣾⠇⣿⣴⠏⣿⣾⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡁⢿⣦⣿⣄⣄⡿⢿⣸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡛⢶⣭⣷⣼⢍⣷⣾⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣿⣶⣴⡶⣷⣶⣿⣿⡃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠙⣻⠟⢩⡿⢹⣿⠁⢹⣿⠹⣎⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⡁⣀⣀⣿⣴⣿⠇⣾⡏⠀⠈⣿⡇⠙⠓⢰⡟⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⣿⠋⣿⣦⣾⢿⣆⢼⡟⣿⣧⣼⠗⢼⣷⣟⢻⢌⣛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡾⠻⣿⣴⡿⣿⡏⣸⡿⣾⡿⠿⠋⢻⣦⣼⣇⢻⡏⣿⡟⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⢠⣾⠍⢀⡻⠿⠿⢿⣿⠃⢀⣤⣴⡿⠟⠻⠿⣛⠛⢹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⡿⠀⣾⡏⣾⠛⣷⠀⣿⢠⣷⣶⣶⠗⣿⣄⣿⠐⢿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠉⢀⣿⡇⢀⣿⢡⡿⣶⠋⠀⣿⠀⢹⣿⠀⠀⠘⣿⡟⢰⡟⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣿⡇⣸⡿⢸⡇⢸⡇⢸⣿⠀⢸⣿⠀⠀⢀⣾⢃⣈⢛⣿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠉⣛⣛⠃⢻⡁⠸⠧⣾⡏⠀⢸⣿⠀⢠⡾⠋⠸⠿⠋⢉⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⢸⣿⠉⠀⣿⣟⡛⢠⣶⡄⢸⡆⢶⣾⠿⣶⣾⣧⡾⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣸⣇⠀⠀⣿⡟⠁⢸⣿⣧⢸⡇⠀⣿⠀⠙⢷⡼⠿⣿⡌⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡿⠹⣿⡀⠀⢸⡇⠀⢸⡿⢹⣿⡇⠀⣿⠀⢀⣸⣇⣼⣿⡗⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢷⡄⣤⠉⣻⢋⠹⠷⠶⠺⠇⠘⣿⠃⢠⡿⠘⣛⡋⢸⡏⢹⡃⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⢿⡿⢠⡟⠛⢻⡟⢀⣶⣰⣦⠰⣧⢰⡆⢾⣏⢱⡏⢻⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⣸⡇⠀⣿⠀⢸⡟⢸⡏⠀⢿⣼⡇⠘⣿⡏⣇⢸⣿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣀⣈⣛⠛⢀⡀⠻⡶⠾⠇⠸⣧⡄⣠⡿⠣⣶⠟⠃⠙⠛⠙⠁⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣉⣿⠘⣧⣤⡶⢷⡄⣿⣄⣴⣿⣦⠘⠛⢻⣟⣿⠐⢶⣟⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠶⠶⣭⣭⢠⡾⠿⠶⢾⣅⣨⣽⣥⣤⣿⢀⣤⣶⣿⡧⢸⣟⣉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⠶⠶⠀⣛⣿⣿⡛⠉⠿⣀⣀⣀⠙⠛⠿⠶⠾⣶⢠⡿⠿⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠶⢦⣿⣸⣯⣍⣉⣿⡧⣿⠉⢻⣯⣿⠴⠿⠿⠶⠿⠘⠛⢛⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠶⣦⣨⣥⣭⣭⣭⡤⢦⣴⡶⢴⣧⠐⠶⠶⠶⣖⠘⠻⣟⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠶⠿⢧⣠⠙⢷⡾⠛⠷⠟⢻⣿⠛⠰⠟⠛⠛⣿⣛⡟⢉⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⢰⣿⣵⢞⣃⣄⣰⡟⠛⠛⠿⢃⠺⣦⣠⣾⣏⣩⣵⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⣉⡀⠘⣧⠈⢡⡾⠛⢻⣶⣿⣧⢸⣯⣽⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠀⢿⣤⣤⡾⢻⡟⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣴⣶⣶⣶⠶⠿⠿⣛⣛⣛⠿⣿⣛⣛⣛⣉⣽⣿⣿⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⢾⣿⣛⡭⠵⠾⣛⣋⢷⠴⠿⠗⣚⣻⣭⣝⣛⣀⣹⣯⠭⠿⠭⠟⠛⠛⠛⠛⠛⠋⠉⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡙⠻⣿⣻⣦⠭⠷⠒⠒⠚⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣷⠀⣠⡴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⠋⠁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢣⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠉⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⡤⠤⠤⠖⠒⠋⠉⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣇⣠⣤⡤⠤⠐⠒⠒⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⡤⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠟⠛⠉⠉⠀⠀⣀⠀⠀⣀⠀⠀⠉⠉⠛⠻⢿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠋⠁⢀⣠⣴⣶⣿⣿⣿⣿⠀⠀⣿⣿⣿⣷⣶⣦⣄⡀⠀⠙⠻⣿⣦⣄
⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⢀⣴⣾⡿⠟⠛⠉⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠉⠛⠻⢿⣷⣦⡀⠈⠻⣿⣷⣄
⠀⠀⠀⢀⣴⣿⠏⠁⢀⣴⣿⠟⠉⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⡀⠈⠹⣿⣦⡀
⠀⠀⢀⣾⡿⠃⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠘⢿⣷⡀
⠀⢀⣿⡿⠁⢀⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀⠈⢿⣿⡀
⠀⣾⣿⠁⢀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡀⠈⣿⣷
⢸⣿⡇⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⢸⣿
⣾⣿⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⣿
⣿⡿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿
⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿
⢿⣿⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⢀⣴⠀⠀⣦⡀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⣿
⢸⣿⡇⠀⢹⣿⡄⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⢀⣴⣿⣿⠀⠀⣿⣿⣦⡀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⢠⣿⡏⠀⢸⣿
⠀⢿⣿⡀⠈⢿⣿⡀⠀⠀⢀⣴⣿⠟⠁⢀⣴⣿⠟⣿⣿⠀⠀⣿⣟⠻⣿⣦⡀⠈⠻⣿⣧⡀⠀⠀⢀⣿⡿⠁⢀⣿⡿
⠀⠈⣿⣷⡀⠈⢿⣿⣄⣴⣿⠟⠁⢀⣴⣿⠟⠁⠀⣿⣿⠀⠀⣿⣿⠀⠈⠻⣿⣦⡀⠈⠻⣿⣦⣤⣿⡿⠁⢀⣾⣿⠁
⠀⠀⠈⢿⣷⡄⠀⠻⣿⠟⠁⢀⣴⣿⠟⠁⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠈⠻⣿⣦⡀⠈⠻⣿⠟⠀⢠⣾⡿⠁
⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⣴⣿⣟⠁⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⣻⣿⣦⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠈⠛⢿⣿⣶⣤⣀⡀⠀⣿⣿⠀⠀⣿⣿⠀⢀⣀⣤⣶⣿⡿⠛⠁⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣄⡀⠈⠙⠻⠿⢿⣿⣿⣿⠀⠀⣿⣿⣿⡿⠿⠟⠋⠁⢀⣠⣶⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣶⣤⣄⣀⡀⠀⠉⠀⠀⠉⠀⢀⣀⣠⣤⣶⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠁
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⡆
⠀⠀⠀⠀⠀⡠⠤⠤⠤⠤⠤⡇⠀⢸⠤⠤⠤⠄⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺⠀⠀⡗⠒⠒⠚⠓
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⡝
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁⢀⡜
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⢀⠎
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡠⠋
⠀⠀⠀⠀⠀⠈⠁⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠒⠶⠶⠶⠤⠤⠤⠤⠤⠴⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣤⠴⠶⠒⢶⣤⠶⠚⠛⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠓⠚⢯⡉⠛⠀⠀⢀⡴⠋⣠⠴⠚⠛⠓⠲⣄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⡞⠀⢀⡀⠀⠀⣧⠀⠀⠀⠀⠀⡜⠁⠀⢀⣀⠀⠀⠈⡆⠸⣆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠋⠀⢀⡏⠀⠀⠀⠀⠀⣇⠀⠀⠘⠟⠁⠀⠀⡇⠀⠋⠙⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣳⣄⣀⣀⠤⠊⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⣀⠴⠁⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠈⠀⠀⠄⠀⠄⠀⠀⠀⠀⠉⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⣠⡤⠤⢤⣄
⠀⠀⠀⠀⠀⡟⠀⠀⢀⣀⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀⣀⡀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠈⢷⠀⣠⠞⠁⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⢷⡀⢴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠙⠺⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣸⠞⠁⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⠀⠀⠀⠀⠸⠁⠀⠀⢠⠀⠀⠀⠀⡇⠀⠀⠀⣜⡀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡀⠈⠁⠀⠀⠀⡇⠀⣠⠊⠀⠉⠓⠦⣀⠀⠀⠀⡞⠀⠀⠂⠁⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⢶⡀⠈⠇⠀⠀⠀⠀⢀⡇⠀⠃⠀⠀⠀⠀⠀⠈⠳⡀⢸⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⢳⡀⠀⢀⡏⠀⡇⠀⠀⣰⠶⣄⠀⢸⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡀⠀⠀⠶⣯⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣿⢸⡇⠀⢸⠇⠀⡇⠀⠀⡏⣠⢿⣄⡜⠀⢸⠶⡖⠒⠂⠊⠛⠋⠉⠋⣹⠁⠀⠀⠀⢀⣀⣀⣽⠇
⢰⡞⠋⠉⠉⠉⠉⠉⠉⠉⢻⢸⡇⠀⢸⠀⠀⢧⠀⡸⠁⠓⠲⢬⡇⠀⠘⡛⠁⠀⠀⠀⠀⠀⠀⠀⠈⣤⣀⣨⠥⣤⣼⠟
⠀⠙⠓⢦⡀⠀⠀⢀⣠⠤⠼⠀⡇⠀⣸⠒⡄⠈⠉⠁⡤⢤⣀⣀⡇⠀⠈⠧⠤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⢀⣠⡤⠴⠃⠀⠀⠈⠒⠶⢤⠀⢷⣤⠟⠛⠁⠀⠀⠀⠓⠒⠒⠒⢢⠀⡶⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⢯⡀⠀⠀⠀⠀⠀⣠⠤⠴⢤⣄
⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣄⠀⠀⠙⢲⠄⠀⢨
⠈⠓⠒⠒⠲⠶⠦⠤⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡼⠂⠀⠀⠛⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⠤⢤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣦⠀⠀⢀⠀⡀⠀⣠⣿⣄⠀⢀⠀⠀⠀⠀⣠⡔⠁
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠻⣄⠈⢲⣷⣶⠿⠿⠿⣿⣿⡶⠁⣠⠶⠁⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠂⠉⠉⠁⠉⢉⣛⣿⣿⣶⣤⣄⣀⡈⢓⣿⣿⣿⣧⣰⣤⣿⣿⣿⣛⣁⣤⡴⠶⠟⠻⠿⠍⣉⡀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠁⣀⣤⣎⣁⣈⣹⣿⠟⠙⢻⣿⣿⡿⣿⣿⡿⠋⠻⠍⠲⣤⣤⣤⠷⠦⢄⡀
⠀⠀⠀⠀⠀⠈⠀⠀⢀⠔⠉⠀⠀⡤⢻⠛⠀⠀⠀⠀⢸⣿⣟⣛⣿⣿⣷⡀⠀⠀⠀⠈⢦⠉⠲⣄⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢠⣟⠀⠀⠀⡠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠢⣀⠀⠀⢆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡤⠖⠊⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⡄⠈⠛⠶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⠻⣿⣿⣿⡿⠛⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⢀⡤⠖⠋⠉⢹⠀⠀⠀⢸⠉⠉⠓⢦⣄⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⡴⠋⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⢰⠇⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢷
⠀⠀⠀⠀⠀⣼⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢹⠀⠀⠀⢸⡀⢀⡤⠞⠁⠀⠀⠀⣠⠀⠀⠀⢠⡀⠀⠀⠀⠙⠦⣄⠀⣸⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠈⠷⠋⠀⠀⠀⢀⡴⠚⢹⠀⠀⠀⢸⠙⠲⣄⠀⠀⠀⠈⠳⠏⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⢸⠀⠀⠀⢸⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠘⢧⡀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⣠⠟⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠉⠳⠦⣄⣼⠀⠀⠀⢸⣀⡤⠶⠋⠁⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣀⣀⣀⣀⣀⣀⣀⣠⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠶⠟⠿⠾⠟⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⢀⣤⣤⣤⣤⣤⣤⣴⣶⣶⣾⣿⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠉⠉⠉⣿⡀
⠀⠀⣿⡏⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⡶⣦⣤⡶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣾⠟⠉⢛⣿⣷⣦⡀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢀⣴⡿⢿⣄⠙⢿⣦⡀⠀⠀⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡿⢿⣦⡀⠙⢷⣄⠙⢿⣦⡀⢿⡇
⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠙⢿⣦⡀⠙⢷⣄⠙⢷⣾⡇
⠀⠀⠀⢸⣇⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠙⢿⣦⡀⠙⢷⣄⠙⢷⣄
⠀⠀⠀⠘⢿⣿⡛⠛⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠙⢿⣦⡀⠙⢷⣄⢙⣷⣄
⠀⠀⠀⠀⠀⠉⠻⣦⣀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠙⢿⣦⣀⣽⡿⢋⣽⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣤⣤⣤⣤⣤⣤⣴⣶⣶⡶⠶⠶⠶⠾⠿⠿⢷⣄⣽⡿⢋⣴⠟⠁⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣴⡟⠁⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣶⡾⠟⠁

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿

⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠙⣿⣿⣿⣿⡏⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠸⣿⠟⠀⠀⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠁⠀⠀⠈⣿⣿⣿⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⢰⣿⣿⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⡇⣿⡇⠸⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⢰⣧⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁⠀⠀⠀⣾⣿⢠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢨⣿⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠁⠀⠀⠀⠀⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠻⠋⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠴⠏⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⣠⡀⠀⠀⠀⠀⠀⣠⣤⡀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠟⠀⠀⠀⠀⠀⠷⠞⢁⣌⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣲⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣤⠴⣄⠀⠀⢺⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡝⣉⡀⠀⠀⠀⠀⠘⣧⣼⠇⠀⠀⠀⢸⡇⠀⢸⠂⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞⠈⢯⡽⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠙⠛⠋⠀⠀⠸⡽⠇⠀⠀⠀⢠⡖⢦
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢿⡽⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⢤⣤⣀⠙⠋⠀⠀⠀⠀⣀
⠀⠀⣠⠶⠋⠉⠉⠉⠉⠉⠉⠛⢷⡀⠀⡴⠒⠛⢻⡴⠛⠉⠉⠙⢦⡀⠀⣠⠶⠛⠉⠉⠛⠶⣄⠀⢠⡞⠁⠀⠀⠀⠀⠉⢳⡀⢀⡴⠚⠉⠉⠙⠛⠲⢦
⠀⡴⠃⠀⠴⠀⠀⣀⣀⡀⠀⠀⢸⡇⠀⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⢹⣴⠋⢠⠄⠀⠀⠀⠀⠈⢳⡟⠀⠀⠖⠀⠀⠀⠀⠀⣷⡏⠀⠀⠀⠀⢀⣠⣤⡾
⣸⠁⠀⠀⠀⣴⠛⠉⠉⢻⣴⣆⣼⠁⠀⡇⠠⡄⠀⢠⣤⠀⠀⢀⣤⢿⡇⠀⠈⠀⣀⣀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣠⠤⠴⠒⠙⣆⣀⣀⣀⡀⠈⠛⠓⠦⣄⠀⣴⢲
⡏⠀⠀⠀⢸⠃⠀⠀⠀⠈⠁⠈⠁⠀⠀⣷⠀⠀⠀⢸⣸⠀⠀⠀⢷⣸⠀⠀⠀⢰⠁⢸⠀⠀⠀⠀⣿⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⢳⡀⠀⠀⠈⣧⠈⠋
⣧⠀⠀⠀⢸⡀⠀⠀⡟⠛⠛⠛⠛⠛⠛⠛⢻⠀⠀⠀⠉⠀⠀⠀⠀⢻⡄⠀⠀⠈⠓⠋⠀⠀⠀⢀⣧⣤⣤⣀⠀⠈⠛⠛⢶⣠⡶⠶⢤⣄⣤⠞⠐⣦⠀⠀⣻
⢹⡀⠀⠀⠈⢳⡀⠀⠛⢶⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠶⠤⠷⠀⠀⠀⠀⢵⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡿
⠈⢳⡀⠀⠀⠀⠉⠛⠒⠋⠀⠀⠀⠀⣿⢻⠉⠀⠀⠀⣼⣆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⡏⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠁
⠀⠀⠙⠳⣦⠀⠀⠀⠀⢀⣤⡄⠀⠀⣿⢸⣀⣀⠀⠀⣿⠙⣦⠀⠀⠀⢀⣸⠟⣆⢠⣶⠀⣤⡇⠀⣇⡾⣇⠀⠀⠀⢠⡶⢾⠁⠀⠀⠀⠀⠀⠀⣤⡀⡿
⠀⠀⠀⠀⢸⠀⣀⠀⡀⢸⠀⣷⡆⣴⣿⠈⠉⢹⡇⠀⣿⠀⢹⠀⣤⡄⣿⠉⠀⣿⢸⡏⠷⠿⠇⠀⠀⠀⢻⢰⢷⡀⢸⠁⠘⣇⢸⢳⡄⣾⢧⠀⡏⠛⠁
⠀⠀⠀⠀⢸⢸⡟⠞⢻⢸⡆⠀⣿⢿⠀⠀⠀⠘⣿⣧⣿⠀⢿⠀⡇⠙⠋⠀⠀⢹⢸⠇⠀⠀⠀⠀⠀⠀⠈⠛⠘⡇⣸⠀⠀⠹⣾⠈⠷⠏⠸⣦⡿
⠀⠀⠀⠀⠘⠿⠀⠀⠘⠿⠀⠀⠹⣾⠀⠀⠀⠀⠀⢸⣿⠀⢸⣄⡇⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⢀⡀⣀⢀⣀⠀⢀⣀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢹⠙⣿⣧⣿⢿⠧⠀⢸⣿⣾⡇⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠈⠛⢙⠉⠙⣙⣋⣁⣘⣋⠙⡃⠀⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠏⣿⡏⢻⢿⠁⠀⠀⠀⠀⠃⡇⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⡉⢹⠀⠀⠀⠀⠀⡅⢠⣤⣄⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠋⢸⠀⠀⠀⠀⠀⠃⡿⠿⠿⣧⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⢠⠀⢰⢸⠀⠀⠀⠀⠀⡆⡙⠦⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢿⣇⡀⢸⢸⣿⡻⣶⡄⢶⡇⣆⠝⠫⢝⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⡟⠟⣁⣸⢸⣿⣷⣿⣷⣇⡁⢯⡤⢶⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⢾⣿⣿⢸⣽⡿⣿⣿⢿⡅⢷⣧⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣲⣿⣿⢸⡿⠛⣟⡟⠋⡇⢸⡟⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠉⠩⡹⢸⣷⣿⣟⣇⣽⡇⣸⢘⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡶⢤⣼⢸⣿⣿⣾⣾⣻⡇⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣼⣯⢸⣛⠻⠛⠋⣵⣷⢠⡤⣤⣄⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣿⢸⣿⣷⣇⣀⣀⡽⠾⠷⠟⠺⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⣿⣯⡏⠉⣿⣿⣾⣟⢳⠛⢗⣟⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠙⠧⠀⠇⠸⠼⠧⠞⠦⠄⠷⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡆⠀⢠⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣷⠀⠘⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠈⠻⣿⣿⡇⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⣿⣿⣿⣷⡀⠀⠘⣿⡇⠀⠀⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣤⣼⣿⣷⣤⣤⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠄⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⠤⠴⠿⠿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠺⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢻⣿⣿⣿⡏⠀⠈⠻⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣴⣶⣄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢾⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⢹⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡷⣶⣄⣶⣰⣷⠀⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣼⣿⠿⣿⣿⡇⣿⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡉⣛⣘⡛⢸⡇⣿⠶⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⡄
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣽⣯⣭⡝⢻⣿⡟⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣉⡀⢸⣿⣷⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣋⣉⠉⠁⣼⡿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣿⣿⣿⣿⣿⠿⠿⠟⣿⣩⣿⣦⠶⠎⢰⡆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡏⢿⡇⠀⠀⠀⢿⠉⢸⡿⠾⢂⣿⣿⣿⠈⠉⣿⣭⣽⡟⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠃⠀⠀⠀⠻⠂⣼⣧⠦⣼⠋⣿⠿⢶⢿⣯⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢢⠎⠘⠀⠀⠀⠀⢀⠀⠀⡠⠒⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢄⡤⠁⠘⢄⠀⢇⠀⢀⡠⠤⠀⠺⣄⢀⡜
⠀⠀⠀⠀⠀⠊⠁⠀⠈⠉⠁⠉⠑⠊⢢⡀⢄⢡⠀⠑⢟⣀⣈⣙⡷⠻⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠑⢂⠆⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠈⠒⠁⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠉⠀⠀⠀⢀⠴⠊⠁⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⡀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠹⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡶⠀⠁
⣠⣤⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢋⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢀⣿⠗
⠉⠁⠀⠀⢠⠏⠀⠀⣀⠀⡀⠀⠀⠀⠈⡄⠣⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡸⠱⡀⠀⠀⢀⢼⠔⢒⢼⡀
⠀⠀⠀⠀⣼⠀⠀⠈⠢⠥⠃⠀⠀⠀⠀⢱⠀⠑⣄⠀⠀⠀⠀⠀⠈⡡⠊⠀⠀⠀⠈⠒⠀⠆⢠⢇⣸⣭⠕
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠀⠀⠈⠀⠄⣀⡠⠤⠚⠀⠀⠀⠀⠀⠀⣀⡤⢦⣳⠄⠠⠛⠈
⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⢠⠦⠔⠃⠀⠀⠤⠴⠒⠒⠂⠀⠐⠲⢄⣀⠤⠖⠊⠁
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢠⠃⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣶⣶⣶⡶⣷⠄
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⡞⠚⠫⢉⡉⠉⠁⠀⠀⠀⠀⠒⠂⠀⢀⣀⠜
⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⢸⠀⠀⠀⠀⠀⠉⠉⠀⠀⠖⠐⢂⡨⠟⠁
⠈⠑⠂⠠⢐⢉⣽⢮⠂⠀⢸⠁⠀⢈⡩⠟⠁⠒⡲⠛⢻⡟⣁
⠀⠀⠀⠀⠀⠀⠫⠭⠤⣠⡈⡗⠊⠁⠀⠀⠀⠘⠒⠚⠙⠛⠁⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠄⠀⠀⠀⠀⠀⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⣀⠀⠀⠀⠀⠀⢀⣤⣾⣧⣠⣿⣦⣄⠀⠀⠀⠀⠀⢀⣘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠙⠛⠛⠛⣿⣿⣿⣿⡟⠛⠛⠛⠁⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡄⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⢹⣿⣿⣿⣿⣿⣿⣿⢁⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣌⣻⣿⣿⣿⣿⣿⣣⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠾⠿⠿⠟⠛⠛⠛⠻⣿⣦⡴⢿⣿⣿⣿⣿⡿⢦⣴⣿⠟⠚⠛⠛⠻⠿⠷⠷⠦⣄
⠀⠀⠀⢀⣠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠋⠀⣿⣿⡏⠙⠓⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠋⠀⠀⠀⠀⢠⣿⣿⣧⠀⠀⠀⠀⠙⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⢀⣈⣿⣿⣏⣀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠉⠉⠢
⠀⠀⣀⠠⠤⢄⡀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⡅
⢠⠊⠁⠀⠀⠀⠈⠢⡀⠀⠇⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠙⣄⠸⡀⠀⠀⠀⠀⢀⠁
⢱⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠱⡀⠀⡀⠀⡘
⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠈⣆⠙⢦⠀⣠⠃
⠀⠀⠈⠓⠤⣄⣀⠀⠀⠀⣀⡸⠤⠇⠉⠁⠑⠒⠒⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡽⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣈⠙⢂⠶⠴⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⠀⠸⠁⠋⠈⣋⣈⠹⡄
⠀⠀⠀⠀⠀⠀⠀⡠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠤⠐⢣⠴⢄⣀⣀⢸⠃
⠀⠀⠀⠀⠀⠀⠐⡁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠚⠀⠒⠁⠀⠀⠀⠀⢼⠃
⠀⠀⠀⠀⠀⢀⡔⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⠦⣤⢤⢀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠤⠛⢦⠀⠀⢱
⠀⠀⠀⠀⠀⠈⠲⠀⠀⠀⠀⣀⡄⢶⠒⡎⠒⠤⡖⠲⠤⠞⠉⠁⠀⠀⠀⡱⠀⠀⢉⠆
⠀⠀⠀⠀⠀⠀⠐⢄⣀⣸⡌⢳⡙⣦⡓⢄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⡏⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡹⡿⣌⢲⠦⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⠈⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠃⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡤⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠥⢦⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠈⣷⠈⠳⡀⢠⠖⣉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⡇⠸⡷⡄⠈⠛⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠴⠃⠀⣷⠙⢄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠿⠀⠀⠑⣦⣄⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢀⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠉⠁⠑⢦⠈⠹⡕⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠸⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠈⣿⠄⠀⣼⡏⠉⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠐⠲⢤⠤⠲⢄⠴⠚⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠤⣤⠠⠤⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⣄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡠⠤⠀⠒⠀⠀⠀⠁⠒⠒⠒⠂⠀⠀⡠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡏⠒⠶⠤⠤⠤⠤⠤⠄⠠⠀⠀⠒⡈⣇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠉⠓⠲⠶⠶⠶⠶⠂⠁⠀⡼⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢸⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⢓⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡸⠤⠒⠉⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣨⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢁⣄⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠢⣤⣀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡼⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⡿⣄⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⣠⠁⠀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢐⡄⠀⠀⠀⠀⢱⠈⠳⢄⡀⠀⠀⠀⠀⠀⣀⡀⠈⠁⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⢫⠀⠀⠀⠀⠠⣏⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠤⠀⢠⠀⠀⠀⠀⠀⣧⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠘⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠠⠇⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠰⠷⠂⠀⠘⠀⠀⠀⠀⠀⠘⡆⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⢀⠃⠀⠀⢀⢤⡀⢤⡖⠚⢀⠉⡆⠉
⠀⠀⠀⠀⠀⡎⠐⠐⠐⢀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠇⠀⡸⠀⠀⠀⠀⠀⢧⡀⠀⡘⠀⠀⠀⢠⡀⠊⠄⣟⠡⠖⣂⢀⠀⠐⠢
⠀⠀⠀⢠⠴⠀⠀⠀⠀⠀⠑⠢⢤⣀⡈⠀⠃⢠⡄⠐⢀⡀⠤⠤⣼⡜⠀⠀⠀⠀⠀⠀⠀⠙⡀⣇⠄⠐⠒⠢⠉⠂⠁⠀⠀⠠⡅⣂⡀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠈⠀⠀⠁⠀⣈⠁⠒⠐⠒⠒⢀⣤⣈⣀⣈⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠚⠛⠋⠁⠀⢠⣴⣄⣍⣥⣤⠄⠦⠝⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⠛⠁⠙⠋⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡻⢁⣿⣿⣿⣿⣿⠿⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣿⣿⣿⣿⣿⣿⡍⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣶⣾⣿⣿⣿⣿⣿⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⣿⣿⣿⡿⡋⠐⠀⣠⣤⣄⠐⠂⡠⠀⠀⠄⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣿⠟⠋⠀⠀⢡⣤⣿⡉⠻⣿⣯⣀⣠⣶⣶⣤⡄⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠉⠀⠀⠰⡀⠀⢈⣿⢧⠁⠀⢈⠉⣿⡿⣟⣿⡟⠃⠀⠀⢿⣿⣿⡿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢱⣿⣷⣀⠀⠀⠀⢇⢠⣼⡏⠠⠀⢙⠀⠸⢻⣷⢨⣿⡇⠀⠀⠀⣿⣿⣿⣷⡀⠈⠻⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⢀⣶⣶⣿⡿⠁⠀⠀⠈⡃⢀⠀⣿⣿⣿⠁⠀⠀⣰⣿⣿⣿⣿⣿⠀⡀⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣍⣿⣟⣸⣿⡿⠻⣿⣶⠀⡷⠀⠀⢠⢰⣿⣿⣿⠀⠀⣰⣿⣻⣿⣿⣿⡁⠈⣡⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⡟⠡⠀⢻⣿⠃⣷⣿⣴⣿⣼⣿⣿⡇⠀⠀⠉⠉⠛⣷⣿⠿⡱⠞⠁⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⠿⣿⣿⡗⠀⠀⢾⣿⣿⣿⡟⠉⠹⠏⢹⣿⡗⢀⣀⠀⢸⡷⢄⡠⢆⣀⢀⢰⣿
⠀⠀⠀⠀⠀⠀⠾⢿⡦⣞⣹⣿⣁⠐⠀⣿⣿⣿⣧⣤⣾⣲⣴⣿⣿⡀⠀⠀⠈⢻⡿⠋⠀⢀⠔⠃⣼⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣩⣭⣦⠄⠀⣉⡏⣠⡿⣿⣻⡛⢛⣿⣿⡷⠄⠀⣀⣿⣇⡀⠒⠁⢀⡴⢣⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢿⣿⣾⣷⣾⠻⡈⣧⣿⣿⣧⣷⣿⣿⣿⡀⢀⢠⣿⣁⣀⣀⣠⠿⠛⢋⣿
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣟⠁⠀⢸⣿⣿⠇⠤⠚⣟⣉⣟⣛⢠⡞⢉⠸⣯⣠⣽⣿⢭⣥⠾⠁⣰⣶⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠈⠉⢹⣿⣆⡠⠜⣿⢻⣿⣿⣷⠡⠀⠀⣀⣉⣿⡏⠀⠀⢠⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣷⡗⠀⠀⠙⠛⢿⠋⠀⢰⣾⣿⣿⣭⣿⠉⠀⢚⠉⢹⣿⠁⠀⠀⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⡀⡰⠃
⠀⠀⠹⣿⣿⣦⣀⠀⠀⡰⢆⠀⠀⠀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠙⠻⣁⡐⠙⠆⠓⠈⣇⣀⣀⣼⣇⣀⣈⠧⠀⠠⠐⠀⠀⠀⠀⠀⢠⠃⠀⠀⠁⠀⢀⠀⠤⠀⠠⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠙⢛⢠⣴⠖⠋⠉⠙⠄⠈⠀⡄⠀⠁⠒⠂⠀⠀⠀⠀⠀⢸⠄⠀⠀⠈⠉⠁⠀⠀⠀⡀⣠⣀⠈⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣤⠀⠀⠀⣀⢤⡆⠀⠅⠂⠘⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠽⠍⠙⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠒⠒⠂⠀⠀⠐⠂⢄⡀
⠀⠀⠀⣠⣴⣶⠶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠉⠉⢓⡐⠒⠂⠀⠀⠈⠉⠻⣦
⠀⡴⣾⠿⠿⡶⡤⣄⠈⠀⣖⢲⡖⢲⣴⣤⣄⠀⠀⠀⢀⠀⠴⢶⠋⢻⡀⠀⠀⢨⣿⠁⠀⠀⠀⠀⠀⢀⣻⠄
⠀⠀⣙⠻⡱⣓⡗⠀⠘⢼⠃⣜⠣⡞⠥⠗⢽⣳⠄⠀⠀⠀⠀⠘⢷⠮⢶⣦⣤⣸⣷⣤⣶⣷⣶⣶⡶⠿⠛
⠀⠀⠈⠋⠀⠈⠈⠱⠤⢥⠔⢤⠞⣥⠖⠀⠲⠟⠁⠀⠀⠀⠀⠀⠘⠀⣀⡿⠙⢇⠈⠉⠉⠉⠉⠙⠶⠶⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣴⠠⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠡⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⢀⠀⣰⡓⣐⣄⣶⣷⣰⣄⠀⠙⠻⠶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠘⠳⣄⡀⠀⠀⠀⠀⠈
⠀⢀⡏⣴⣆⡀⣿⣿⣍⣽⣿⠏⡷⢧⣀⡄⢋⠁⠦⠀⠀⠀⠀⠀⣀⣤⣠⣴⣤⠴⠊⢉⣿⢟⡶⣺⢿⣷⣾⠉⠈
⠀⠘⢦⡟⠁⠀⠉⡿⠛⠁⠉⠙⠃⠀⠺⢣⣄⠱⠀⡁⠀⠀⣠⣾⣿⣿⡇⣼⡿⢿⢸⢧⡿⣻⠟⢯⡞⠙⡾
⠀⠀⠀⠑⢰⣶⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⢠⠄⠀⠈⠙⠛⠉⠉⠁⠁⠀⠀⠉⠉⠉⠑⡄⠀⠀⠠⠐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣀⣠⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠂
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣶
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠙⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⡀⣠⡦⠀⠀⠀⠀⠀⠚⢜⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⢀⣴⣿⣿⠁⣰⠀⠐⠘⠀⠐⠰⠀⠡⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢻⣿⣿⠿⣿⣿⠿⠟⠛⠛⠋⠉⠉⠁⠀⠀⠀⠸⣿⠿⠿⠿⠟⠃⠀⠀⠂⠀⠀⣀⡀⠀⠀⠈⠛⠛⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢤⣤⣴⣶⣶⣶⡷⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣷⣤⣄⣉⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡉⠙⠛⠁⠀⠀⠐⠀⠒⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠀⠀⠀⠉⢨⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠨⠭⠩⠤⠄⠀⠀⠁⠀⠀⢀⣀⣠⣤⣤⣶⣶⣤⡶⠖⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣶⣤⣠⣤⣤⣶⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢴⣶⣿⣿⠄⣀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⢽⡿⠏⠐⠟⠀⢹⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠐⡾⡿⠴⠀⠀⠀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⡉⠁⠀⢀⣼⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠉⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡞⡄⠀⠀⠰⣿⣤⡄⠛⣿⡿⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠑⢦⡀⠀⡌⠀⠈⢄⠀⡸⠀⢹⡆⢠⣿⡟⢿⣤⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠙⠦⣽⣄⠀⠈⠂⠑⣿⣿⣾⣴⣿⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⢀⡤⠀⠀⣻⣿⣦⣾⣷⣶⣿⡟⠉⠻⠏
⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⣀⡴⠉⢠⡀⣠⣿⠟⠁⠉⠋⠈⠉⠀⠀⠀⠈⠄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⣴⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣹⣛⣝⣛⣟⣿⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⣨⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⠁⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⢸⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠘⡄⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⡇⠀⢀⣀⡸⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠅⠛⠋⠉⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣀⣠⠤⠄⠒⠘⠛⠉⠉⠉⠉⠉⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠤⣀⣀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠐⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠄⠀⠀⠀⣠⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡄⠀⠸⡆⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⡔⠋⠀⠀⠀⠀⠀⢀⣀⠀⠀⣴⡄⢸⣷⠀⢿⡏⠁⠀⠀⠀⢱⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⡇⠀⠀⣰⣶⣦⠀⠸⣿⡀⠀⢸⣷⠀⣿⡇⠸⣿⣀⣀⡀⠀⠈⡇⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢷⠀⠀⣿⡇⠹⠃⠀⣿⡇⠀⠀⣿⡄⠸⣿⠀⢻⣿⠛⠋⠀⠀⢸⡀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠘⡀⠀⢹⣧⠀⠀⠀⢸⣿⠀⠀⠸⣧⠀⢻⡇⠘⣿⠀⠀⢀⠀⠀⢇⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢣⠀⠘⣿⡀⣴⣶⡀⣿⣇⣀⡄⢻⣦⣼⣿⠀⠻⠿⠿⠿⠃⠀⢸⡄⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠈⡄⠀⢹⣧⣀⣽⡇⠸⡿⠟⠋⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢰⠀⠀⠛⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠤⠶⠶⠖⠒⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢠⠀⠀⠀⠀⠀⣀⡤⠴⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢦⡀⠴⠚⠉⠁⠀⢀⣀⣤⠤⠶⠒⠚⠋⠉⠉⠉⠉⠍⠁⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⣀⡤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠖⣂⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠤⣤⣄⣀⣀⣀⣀⣀⣉⣁⡀⠤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⠶⡶⠶⠶⣶⠶⠶⢶⡶⠶⢶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⡀⠀⡠⡀⠀⠤⠀⠤⠐⠁⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡁⠀⠛⠀⠀⠀⠰⠄⠀⡀⠴⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠉⠀⠠⡀⠀⠀⡀⢄⠀⢰⢏⣠⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡊⠀⠀⠀⢀⠄⠀⢈⠱⠖⢓⠷⢖⣣⡀⡄⠀⠘⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⢐⡤⠐⠁⡠⠊⡡⠒⠉⠇⠀⠈⠄⠀⠈⢢⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢀⣈⠉⠐⠺⡀⡌⠀⠀⠘⠓⠆⠀⠞⠀⠀⠀⢂⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠉⠐⡦⡈⠉⠀⠂⡄⠀⠰⠖⠬⠀⠀⠀⠈⡄⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢁⠈⢢⠀⡀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢄⠀⠀⠀⠑⢌⠓⠓⠏⣐⠂⠠⠄⢀⣀⡀⠠⠜⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠠⡵⠂⠄⡀⠈⠁⠐⠒⠀⠀⠐⠒⠒⠀⠈⠉⢉⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠠⠀⠀⠀⠀⠉⠁⠀⠐⠒⠒⠂⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡐⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣤⠀⢠⣤⣥⣾⣧⣬⣤⡄⠠⣤⠀⡀⠀⢰⠀⠀⠀⡡⠤⠁⠀⠀⠐
⠀⠀⠳⡀⠀⠀⠀⢀⣴⠋⢳⣆⠀⠀⠀⠀⡠⠊⠀⠀⠀⠊⢱⡄⠀⠀⠀⠀⠀⠀⣀⠘⢺⠿⢣⡜⠿⡖⠃⣀⠀⠀⠀⠀⠀⠢⡠⣆⠀⠀⢀⣠
⠀⠀⠀⠀⡘⠂⠂⣈⢻⡄⣼⢋⡀⠀⠐⡂⠀⠀⠀⠀⡇⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢹⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠉⠛⡿⠋⠉⡄⡀
⠀⠀⠀⠀⠀⠈⠁⠉⢁⠸⢀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠑⣄⡳⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡈⠀⠀⠀⠀⢀⢁⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠑⢄⣀⠴⢊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠠⠆⡀⠑⢦⢠⠖⠁⠰⠄⠀⠀⠀⠀⠀⡀⢡⠌⠉⠁⡎⠀⠀⠀⠀⠂⠉⠉⠀⠀⠀⠙⣗⢸⡏⠁⠀⠀⠀⠉⠐⠂⠀⠀⠀⢿⣿⡄
⠀⠀⠀⠀⠀⢀⡡⢬⡇⠀⣿⠤⣅⠀⠀⠀⠀⠀⠀⠈⠴⠈⠹⠁⠶⠒⠁⠀⠀⠈⠂⠀⠀⠀⡀⠀⣯⢸⡃⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠸⣿⣇
⠀⠀⠀⠀⢇⠈⠳⢤⠛⠈⢃⠴⠋⣀⠆⠀⠀⠀⠀⢠⠀⠛⠂⠛⠀⢤⠀⠀⠀⠀⠀⠀⢹⣏⢇⣀⡘⣏⣀⣠⢯⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠙⠂⠂⠀⡉⠀⠒⠒⠁⠀⠀⠀⠀⠀⠄⠈⠓⠐⠘⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠋⢹⡛⢟⠻⠌⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡠⠉⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠂⠒⠲⠤⠐⣄⣧⣥⡔⠡⠤⠒⠂⠀⠀⠀⠀⠀⡆⠀⠀⡥⠀⢀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠈⠀⠀⠀⠀⣶⡐⠙⡆
⠀⠀⠀⠀⠀⠀⠐⢻⠂⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣠⠔⠛⠢⣈⠆⠀⠀⠀⠀⢀⠘⣇⠀⠀⢰⡤⣴⠀⠀⢀⡟⢀⠀⠀⠀⠀⠀⣹⣇⢀⡇
⠀⠀⠀⠀⠀⠀⡠⡖⠋⠙⣦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠤⠦⠤⠏⠀⠀⠀⠀⠀⠀⠀⢼⡄⢀⣻⣿⣿⢒⡀⣼⠁⠀⠀⠀⠀⠀⠀⣿⣷⣾
⠀⠀⠀⠀⠀⠌⠐⢇⠀⢀⠟⠀⠣⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠸⠐⣄⠀⠀⠀⠀⡌⠸⣇⠀⠙⢿⠿⠋⢀⣿⠞⡄⠀⠀⠀⠀⠀⢸⠳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠊⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⢀⢴⡄⠈⠀⠀⠀⠀⠐⢀⠘⡦⠀⠈⠀⠠⡞⢀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⠀⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠐
⠀⠀⠀⠀⠀⠑⠂⡀⠀⡄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠰⠆⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⡽⠤⡧⠬⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠗⢏⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠤⠐⠂⢆⡇⠀⠀⠀⠀⢤⡀⠀⠒⠒⠀⢀⡤
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠹⣦⡾⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠴⠖⠢⠔⢣⠤⠀⠀⠀⢰⠒⠇⢹⠋⠉⠉⠸⠐⡆
⠀⠀⠀⢀⠄⠁⠀⠀⠁⠀⠀⠊⢃⠀⠀⠀⠀⠀⠀⡆⡠⠊⠤⠆⠤⠄⠢⡀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠆⠀⠀⠀⠀⣄⠀⢠⡕⠀⣠
⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⢰⠂⠀⠀⠀⢠⠀⡀⠀⠹⠁⠨⠖⠀⠐⠲⠀⠹⠅⠀⣀⢠⠀⠀⠀⠀⠃⡀⠀⢈⠀⠀⠀⠀⠀⠢⡀⢦⠂⠐⡄⢀⠔
⠀⠀⠀⠀⠀⠀⢙⠒⠶⠚⡘⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠐⠔
⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠛⢦⡀⣦⠀⠀⣀⣀⣀⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⡤⢿⡶⠶⠚⢻⡟⢿⡉⠉⠉⠁⠀⠀⠀⣷
⠀⠀⠀⠀⠀⣀⣀⣤⣤⡤⠤⠶⠶⠖⠚⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠹⣆⢀⡄⠹⣄⢳⡄⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠹⣆⠙⣧⠹⣆⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡘⣧⡈⣷⠘⣧⠀⠀⠀⠀⣇
⢀⣀⣤⠴⠶⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣌⠛⠉⠀⣼⠀⠀⠀⠀⢿
⢿⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠛⠁⠀⠀⠀⠀⢸⡄
⠸⡇⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⢿⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⢿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡆⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⣷⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⢸⡆⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣷⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⢸⡆⠀⣇
⠀⠀⠀⠀⠀⢷⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤
⠀⠀⠀⠀⠀⠸⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡤⠴⢶⣶⡿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⢷⢸⡆⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⡤⠴⠶⠶⠒⠛⠛⠋⠉⢉⣁⣤⠴⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠸⡾⠷⠶⠖⠚⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠖⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠸⡆⢀⣀⣤⡤⠶⠚⠋⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣿⣯⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠀⠀⠀⣤⠀⣠⡆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⢸⡿⣰⡟⠐⠃⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠋⠀⠀⠀⠀⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠁⠀⠀⢈⡇⠀⢀⣀⡤⠴⢤⡀⠀⠀⠀⠀⠀⠀⣀⢀⡴⣶⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠈⢠⡶⠋⢹⠁⠀⠀⢹⡄⢰⠄⣰⣧⢾⡟⣾⢁⡟⠀⣰⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⣰⡟⠀⠀⡿⠀⠀⠀⢸⡇⣾⡼⢹⡟⢸⣵⣿⠋⠀⣰⠏⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⢠⡿⠀⠀⢠⡇⠀⠀⠀⣼⠃⣿⠁⠘⠁⠀⠁⠙⠶⠞⠁⡀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠁⠀⢸⡇⠀⣠⢚⡟⢦⡀⣰⠏⠀⠀⠀⠀⠂⠀⢠⡖⡆⠀⠀⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠈⢿⡞⠁⠘⡃⣠⡿⠋⠀⢰⡆⠀⠀⠀⣠⡈⢳⣷⡟⠀⠀⣼⠇⠀⠈⠁⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⡀⣀⡼⠃⠀⠙⠛⠟⠛⠉⢀⣶⡄⢸⡇⣴⡂⣼⡽⢃⡞⣿⠃⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠙⠛⠋⠀⡄⠀⢂⡀⠀⢀⣄⠉⣿⠀⣿⣾⣿⢠⣿⣠⠞⠀⠻⣷⠿⠋⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣠⠶⣾⠋⠀⣠⡄⠀⣰⠏⣿⠀⣿⠏⣸⡇⡼⢻⡇⠘⠋⠀⠀⠀⠀⠆⠀⠀⠀⠘⣫⡴⠂
⠀⠀⠀⠀⠀⠀⠉⢀⠀⠚⠋⠀⢀⣿⠀⣰⣇⡿⣠⣿⡼⢹⡴⣏⣰⠋⠿⠁⠈⢀⠀⠐⠂⠀⣴⠛⣧⠀⠀⠀⠀⠀⠖⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⢈⡟⠁⠉⢁⣾⡇⠈⠁⠀⠀⠀⠀⣀⠀⠈⠀⢸⡏⠰⠇⣾⢻⡆⠀⠀⠠⠀⠀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⢶⣶⠶⠋⠀⢸⣿⣰⠿⡾⠁⠀⠀⣾⢸⡇⠀⠀⠘⠁⠀⠀⠀⠐⠀⠀⣿⠇⠀⢸⣏⡾⠁⣠⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠛⡋⠀⢀⠀⠀⠀⢻⣼⣧⠶⠛⣻⡇⠀⠀⢀⣠⡄⢠⣿⠀⠀⢸⡏⠀⢀⡿⠀⠀⣿⠅⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢡⠀⠀⠀⠀⣤⣶⢸⣴⢀⣴⠟⠉⠀⢀⡴⠏⣠⡀⢠⡟⣹⡇⣸⣿⠀⠀⣿⣷⣀⣾⠃⠀⢰⡿⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠛⠈⠀⠵⠋⣽⠀⣠⡾⠋⠀⣸⣷⠇⣼⣱⢿⣷⠏⢿⣄⣼⠇⠈⠛⠁⠀⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⡄⠀⡄⠀⢿⡟⠋⠀⠀⠴⣿⢁⣼⠻⠋⠀⠀⠀⠈⠛⠋⠀⣴⣤⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⠀⠀⠀⠀⢸⡇⠐⠂⠀⠀⠙⠟⠁⢠⡶⣤⡴⠛⣦⣀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⣿⣿⠂⠀⣧⠀⠸⡇⠀⠀⣦⣄⡀⣀⣴⠏⠀⠀⠀⠀⠈⠉⠉⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢿⣦⣀⠀⣀⣴⠃⠉⠛⠉⠁⠀⠐⠀⠀⠐⠐⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠁⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⢀⡈⠁


⣿⠿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠿⠿⣶⣌⠻⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⣠⣶⣿⣶⣄⠙⣷⡈⢿⣿⣿⣿⣆⠸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣾⣿⣿⣿⣿⠃⣤⠘⣷⠘⣿⣿⣿⣿⡆⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⡟⢰⣿⡇⢹⡆⢻⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⡇⢸⣿⣷⢸⡇⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⣿⣿⣿⣿⡇⢸⣿⡟⢸⡇⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠘⠛⠛⠛⠛⠃⠘⠛⠃⣼⠃⣾⣿⣿⣿⡿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢉⣭⠭⠤⠤⠭⠁⣰⡟⢰⣿⣿⣿⣿⠃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⡟⢠⡶⠒⠀⣀⣴⠟⣰⣿⣿⣿⡿⠃⠸⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⠿⠿⠛⠁⠾⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣸⣀⡇⢸⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣄⣀⣴⠟⠋⠉⠛⢿⣄⣀⣠⡾⠋⠉⠙⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠉⠉⠀⢀⣠⣤⣀⠀⠈⠉⠁⠀⣀⣤⣄⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⣤⡶⠛⠉⠀⠙⠻⣦⣤⣴⠟⠉⠀⠉⠻⢶⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠖⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠶⠶⠶⢦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⢤⣄⡀
⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠀⠀⣀⣠⣤⡴⠶⠶⠟⠛⠿⢯⣙⠉⠉⠉⠉⠙⠛⠛⠓⠲⠶⠶⢤⣤⣄⣀⡀⠀⠀⠀⠀⠈⠙⠳⢦⣄
⠀⠀⢀⡴⠛⠁⠀⠀⢀⣠⡶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⢦⣤⣀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⣴⠋⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠀⠀⠀⠈⠻⣆
⣸⠃⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠹⣇
⢿⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⢻
⣼⣧⡀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⠶⠶⠶⠶⠶⠒⠒⠒⠒⠲⠶⠶⠶⢶⣶⣾⣦⣤⣤⣀⣀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⣸
⣿⠈⢷⣄⠀⠀⠀⠈⠙⠷⣤⣖⡚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣲⣤⠾⠋⠁⠀⠀⠀⠀⠀⣰⢿
⣿⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠉⠙⠳⠶⢦⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⠖⠛⠉⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⢸
⣿⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠚⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠚⠁⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠉⠻⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠶⠞⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠒⠶⠶⠶⠶⣤⣤⣤⣤⣴⢶⠶⠶⣶⣶⠶⠖⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋
⠀⠀⠀⠉⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡶⠞⠋⠁
⠀⠀⠀⠀⠀⠈⠙⠻⠶⣄⣀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣶⡶⠾⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⣲⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠚⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡄⠀⢀⡠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠶⠒⠒⠛⢛⣋⣉⣉⣉⣉⣉⣉⣙⠛⠛⠒⠲⠶⠤⣤⣀⣀
⣤⠤⢤⣤⣤⡤⢤⣤⣤⠴⠚⢋⣡⠤⠔⠒⠒⠉⠉⠉⠉⠁⠀⠀⣀⡠⢄⣀⠉⠉⠉⠉⠉⠒⠒⠢⠤⣉⡙⠲⢦⣤⣤⠤⣤⢤⣤⣤⢤
⡇⠀⠀⠀⢀⡴⢻⡟⢁⠔⠊⠁⠀⠀⠀⠀⠀⠀⣠⠴⠚⠿⢿⣋⡤⣍⣙⡚⠿⠯⣖⣢⢤⣀⣀⠀⠀⠀⠉⠓⢄⠙⢿⠳⣤⡀
⡇⠀⠀⣰⠋⠀⣾⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠦⣄⣀⠀⠀⠉⠙⠚⢺⣭⣖⣶⣬⢍⣙⠓⠯⢵⣲⡀⠀⠀⠁⣼⡇⠈⢳⡄
⡇⠀⠀⣿⠀⠀⢻⡈⠲⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⠭⠭⠭⠭⠭⠽⠒⠊⠉⠉⠙⠒⠯⠧⣀⣀⢈⣩⡷⠚⢡⠇⠀⢸⣿
⡇⠀⠀⡏⢷⣄⠀⠙⢦⣀⠉⠙⠒⠶⠤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠤⠖⠛⠋⢁⣠⠴⠋⢀⣰⠏⣿
⡇⠀⠀⡇⠀⠙⠳⣦⣄⣈⠙⠒⠦⢤⣄⣀⣀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢀⣀⣀⣤⠤⠶⠚⠉⣀⣤⠶⠋⠁⠀⡿
⡇⠀⠀⠹⣄⠀⠀⠀⠉⠙⠛⠷⢶⣤⣤⣌⣉⣉⣙⠛⠛⠓⠒⠒⠒⠒⠒⠒⠛⠛⢛⣉⣉⣉⣤⣤⠴⠶⠚⠋⠉⠀⠀⠀⢀⡴⠃
⡇⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋
⡇⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠖⠚⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠒⠛⠋⠉⠉
⡇
⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⢠⣿⣼⣾⣇⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠆⢠⣷⢠⣾⡇⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠘⢧⡠⢿⠟⠉⣿⣿⣧⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⢸⣻⣾⣿⡏⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⣶⣦⣴⣶⡶⢉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣁⣤⣿⡿⠁
⠀⠀⠀⠀⠀⠀⢀⡞⠙⣿⣶⡄⣿⡀⢰⢸⡏⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣿⣿⡿⢄
⠀⠀⠀⠀⢀⠴⠛⢀⣴⣿⣿⣿⣿⣧⣼⣦⣷⡄⠉⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠀⠀⣿⣿⡇⠀⡆
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠹⣿⣿⣿⣿⣿⡟⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠰⢷⣤⣿⣿⡿⡆⠁
⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠈⠛⠋⠉⠁⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢰⡈⠻⠿⠟⢠⠇
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡆⢰⠀⡄⠀⣿⠀⠀⠀⢣⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠸⣦⣷⣶⣿⣶⣷⣴⠇⣸⠆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⢤⣤⣄⣀⣀⠀⠈⠦⠠⣄⡀⠙⢦⡟⣋⣙⣿⣿⣿⣿⠶⠃⣀⡴⠰⠀⠉⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⡠⠐⠋⠁⠁⠀⠈⠉⠛⠛⠲⠤⠌⣹⣶⢤⣀⣀⣉⣗⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣀⣀⣠⣤⣖⣋⡩⠭⠛⠛⠒⠀⠐⠀⠂⠈⠑⠢⣄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⠤⠶⠀⠐⠲⠖⠚⠛⢲⠟⠽⠿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⠿⠶⠤⣤⠄⣢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢀⣠⢴⠏⠀⠄⠀⣼⡇⢠⣿⣿⣿⡌⢿⠁⣙⢿⡉⢹⣦⣀⠀⠀⠀⠀⠀⠈⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⡠⠚⠉⠁⡼⢀⡄⠀⠘⠁⣭⣿⣿⣿⣿⣿⣦⡀⠘⠆⢁⢹⡇⠈⠑⠦⡀⠀⠀⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠊⠀⠀⠀⣀⣿⡀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⢜⣶⣇⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠘⣿⡾⢻⡏⢿⣿⣷⣿⣿⣿⣿⣿⠟⢹⣦⣿⣿⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡇⠀⠙⢿⣿⣿⣿⠟⠋⠀⢸⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠗⠀⠀⠀⠈⠉⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢰⣶⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡀⢸⠇⣻⠿⢿⡁⢻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⠿⢛⠿⠦⠄⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠎⠀⠘⠶⣿⣡⣚⡷⠊⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠐⠿⠗⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢀⣤⡶⣾⣿⣿⠶⣦⣄⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠎⠹⠃⣤⡄⠀⢀⣤⡀⠟⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⣘⠀⡀⠀⢼⠆⠀⡀⢸⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡿⣴⢿⣦⢸⢠⣾⠷⡜⣆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡊⠺⣦⠀⠀⠀⢾⡷⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠈⠀⣆⠘⡟⢰⢈⡿⠀⡆
⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠈⠀⠐⠀⠁⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢹⠀⠹⠾⠞⠁⢸⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠷⣝⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢿⣻⣿⣿⡟⣻⣫⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠙⢿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠛⣷⡄⠀⠀⣶⡆⠀⢿⢧⡀⠀⠈⢿⣿⠟⠉⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⡈⠻⣦⡀⢸⣷⠀⠈⣷⣽⣶⣤⣾⣇⣀⣀⡤⢔⣿⠟⠁⢠⣶⠄⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣌⣷⣤⣿⣆⠀⠹⣿⣿⣿⣄⣿⣿⣶⣿⡿⠃⠀⣰⡿⠁⢀⣠⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠀⠀⠈⢻⣧⣤⣼⡿⠛⠋⠀⢀⣾⣿⣥⣶⢛⣩⣤⣶⠶⠶⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠙⠟⠋⠀⠀⠀⠀⠸⣿⣿⣿⣿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⡿⠷⠀⠀⠀⠀⣀⣀⡀⠀⠀⣀⣀⠀⢀⣀⣀⡀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇⠀⠀⠀⠀⣿⣿⣟⠛⢠⣿⣿⣿⡇⢻⣿⣿⣿⣿⣆⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟⠛⠃⠀⠀⠀⠻⠿⠿⠛⠀⠛⠛⠛⠃⠘⠛⠙⠋⠙⠛⠘⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⠆
⠀⠀⠀⠀⠀⢀⣿⡏⢰⣿⣷⣶⣶⠀⠀⠀⠀⣶⣿⣿⣷⡀⣾⣿⣿⣧⢀⣤⣶⣶⡆⢠⣶⣶⣷⡆⢰⣶⣶⣶⡀⢸⣿⣿
⠀⠀⠀⠀⠀⠸⣿⡇⠸⣿⡏⠾⣿⠀⠀⠀⠀⣿⣿⠿⠟⠁⢿⣿⡿⠉⠘⢿⡿⣿⡷⢸⣿⣿⣷⡆⢸⣿⣿⡛⠁⠀⣭⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣠⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡤⠖⠛⠓⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⣠⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢠⠖⠛⠒⠚⠓⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠳⣄⠹⣤⣀⣀⣀⡴⠚⣁⠀⠀⠀⠀⠀⢸⠀⠀⢠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢠⡄⠀⠸⡆⠀⠉⠉⠁⢦⡀⣈⠳⡄⠀⠀⠀⢸⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢳⠀⠀⢹⡀⠀⠀⠀⠈⣇⠸⡆⠹⡆⠀⠀⢸⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⢀⡼⢧⠀⠀⠷⠤⠤⠤⠤⠤⠤⢤⡀⢹⠀⠀⢸⠀⠀⠈⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⣀⠀⢀⡴⠋⠀⠘⢦⣀⣀⣀⡀⢀⣀⣀⣀⣠⡇⣀⣀⡀⣸⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠈⢳⡋⠀⠀⠀⠀⠀⠉⠉⠉⠙⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣉⣉⣉⣀⠀⠀⠀⢀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⠤⠼⠆⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⣿
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠙⠦⠤⠤⠤⠤⠤⠤⠤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠨⠀⠸⠰⠁⢘⠀⠀⠘⠀⠈⠀⠈⠸⠂⠂⠀⠒⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣶⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧⣿⢸⣿⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡏⢛⣁⣤⣍⠛⠋⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢀⡋⢈⡅⣌⠈⢱⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢡⠘⠿⢿⣇⣼⡟⠟⢀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠀⣼⠀⢳⣬⣭⣭⣴⠇⣾⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢁⣴⠀⠿⠀⢀⡉⣭⡍⢀⡀⣡⠀⢰⡆⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⡀⠸⣿⠀⡀⠿⣄⠀⢴⡆⢀⣼⠟⡀⢸⡇⢸⡀⣧⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⡇⢹⡇⢸⣿⠀⣿⡄⠈⢳⡘⢣⡿⢋⢠⡇⢸⡇⠘⠃⣉⣤⡄⢾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡄⣰⠀⣿⠘⣿⢸⣇⢸⣿⠀⣿⡇⢸⡄⢡⠟⡄⡿⠘⢃⣀⣤⣶⣿⠿⣿⡷⠈⡁⢌⣉⢿⣿
⠀⠀⠀⠀⠀⠀⣋⣀⡉⠀⣿⠀⣿⠈⣿⠈⣿⠀⣿⠃⣼⠇⠿⠘⣡⣴⣾⣿⡿⢛⣻⣇⠺⣻⡆⠂⢉⢠⣠⢸⡿
⠀⠀⠀⠀⣠⡾⢋⣥⣤⣤⡌⠀⣿⡃⣿⠀⡿⠀⠟⢀⣩⣴⣶⣿⣿⠛⠻⣿⡀⢿⣿⢿⣄⣩⡀⢁⢈⣈⠈⠄
⠀⠀⠀⠀⢿⢠⡞⢉⣤⡶⠶⠆⠸⠃⠉⢀⣤⣶⣾⠿⢿⡏⢥⢿⣿⠈⣃⣈⣳⣦⣶⣾⡿⠋⠀⠤⠍⠋
⠀⠀⠀⠀⠀⠾⠁⠛⠁⣴⠖⠓⠀⣤⣾⡄⢿⣿⡀⠶⢨⣷⡀⢚⣻⣦⣿⣿⡿⠿⠋⠁⣰⠃⣆⠘
⠀⠀⠀⣀⣤⣶⡿⢿⣶⠄⢐⠋⣠⠄⠠⠥⠠⠍⢓⠈⣿⣿⣿⣿⠟⠛⠋⠁⣠⡀⢰⡄⣿⠀⢠⡆
⣤⣶⣿⣿⣿⠟⠀⣨⣤⣾⣿⣧⡁⢶⡺⣶⢌⡓⠶⠾⠿⠋⠉⠀⢠⠀⣾⠀⣿⠃⢸⡇⢸⡇⢸⠃
⢹⣿⣿⣿⣿⣷⣦⣽⠿⣛⣥⣾⢃⡈⠛⣠⢒⠭⠀⠂⢀⡀⢸⡇⣿⠀⣿⢸⣿⠀⣿⡇⢸⡇⣾
⠀⢿⣿⣿⣿⣿⣿⡁⠈⢠⣾⣿⢸⡇⢰⡄⢠⡀⢨⡀⣼⡇⣸⠃⣿⠀⣿⢸⣿⠀⣿⠀⢸⡇
⠀⠈⢻⣿⣿⠿⢛⣩⣴⡾⠟⢋⣈⣋⠘⠃⢸⡇⠹⢀⣈⠁⠹⠀⢛⣀⣉⡘⠃⢀⣛⠀⡘⠃⣀⠀⢀⣀⣀⡀⢀
⠀⠀⠈⢩⣴⡾⠛⢋⡅⢰⡇⠉⣽⡏⠋⠁⠸⠃⣼⢋⡴⡴⠂⣴⠋⢹⡏⠙⢦⡈⢹⡟⢻⡟⢻⡦⠈⢹⡏⠉⠉
⠀⠀⠀⠀⠀⢺⡇⢸⡇⢸⡇⠰⣿⡟⠋⠀⠀⠸⣧⢺⡇⣧⠸⡇⢀⣼⣧⡀⢸⡇⢾⡇⢸⡇⢸⡇⠐⢾⡟⠙
⠀⠀⠀⠀⢠⠾⣷⠾⣧⠾⠇⣠⠿⠷⣦⠆⠀⠀⠻⣾⣧⣧⠄⠻⢿⣽⣧⣽⠟⢠⣾⠇⢸⡧⠸⣧⢀⡼⠷⢦⡴
⠀⠀⠀⠀⢀⣀⣀⢀⣀⠀⠀⣠⣄⠀⠀⠀⣠⣀⣀⣀⠀⢀⣄⣀⣀⡀⠁⠀⡀⠀⠀⠀⠀⣀⠤⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠈⠉⣿⠀⢹⣧⡀⢹⠀⠀⠀⠀⠁⢻⠏⢹⡇⠉⣿⡋⡉⠁⠀⣠⣿⡀⠀⢠⡟⣡⢶⡖⠈⢹⣟⢉⠉
⠀⠀⠀⠀⠀⢾⣿⠐⢿⠙⣷⣼⠀⠀⠀⠀⢠⣾⡷⣾⠇⠰⣿⠋⠉⠀⠈⣿⣈⣷⡀⢺⡇⣿⢸⠇⠀⢾⡟⠉⠁
⠀⠀⠀⠀⢠⠾⠟⢠⢾⡦⠘⢿⠄⠀⠀⠀⢀⣼⣧⠀⠀⣠⠿⠷⣶⠦⠴⠧⠀⠘⠷⠈⠻⣿⣼⡧⢀⡼⠿⢶⡶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣛⠶⣤⣀⣠⣴⢟⣽⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣮⣭⣶⣿⣿⣿⣿⣿⣦⣀⣀⣀⣀⣀⣀⠀⠈⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣽⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠙⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⠿⠗⠛⠉⠉⠛⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⢆⠀⡿⡄⣰⣳⠀⠀⠀⠀⠀⢀⣀⢀⡴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢳⡧⣿⣟⣻⣿⣦⣀⢠⣾⣭⣭⣝⡳⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⣿⣶⣦⣇⢀⣴⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣫⣭⣶⣽⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⡿⠻⡏⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣴⣩⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢸⠁⠀⠀⢻⠀⠀⠀⢧⠀⠉⠛⠻⣿⠿⣿⡆
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⡞⠀⠀⠀⠸⡆⠀⠀⠈⠳⢄⡀⠀⠈⠀⠈⠇
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⢠⠚⠁⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣶⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⣠⡶⣿⡿⢿⣿⣿⣶⣦⡀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠹⣷⡘⣿⡌⠻⣯⡻⣿⡻⣷⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣿⣷⡘⣿⡄⠹⣷⡘⢿⣎⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠸⣷⠀⠹⣷⠈⢻⣆⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⢻⡇⠀⢻⣇⠀⢿⣆⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⣿⠀⠈⣿⡀⠈⣿⡄⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠿⠆⠀⠻⠇⠀⠹⠇⠀⠸⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠹⢿⣿⣷⣾⣿⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣶⣬⡙⠻⠟⠛⠉⣠⣴⣶⣶⣦⠀⣀⣴⡦
⠀⠀⠀⢀⣀⣀⣀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⠿⠛⠐⠛⠉⣀⣤⣤⣤⣄
⠀⠀⠀⢿⣿⣿⠿⠿⠶⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣁⠀⠀⠐⠶⢾⣿⣿⠇
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⢻⣿⣿⣿⣿⣿⣿⠿⠿⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠈⠉⠉⣉⣡⣴⡆⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⢰⣿⣿⣿⣿⠇⠀⢸⣿⣿⣿⣿⠿⠋⠉⣀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠛⠛⠋⠁
⠀⠀⠀⠈⠉⠁⠘⠋⠀⠀⠀⠈⢩⣽⠄⢠⣶⣿⣿⣿⣶⣤⣬⣝⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠉⠉⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠔⠋⠉⠢⡀⠀⠀⠀⠀⢀⡴⠚⠉⠓⣆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⡌⠀⡀⠀⠀⠹⡄⠀⠀⣠⠋⠀⠀⠀⠀⢸⠀⡟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⡇⠀⠀⠀⢀⠀⢳⠀⢰⠃⠀⡀⠀⠂⠀⡸⠀⣇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢱⠀⠀⠐⠀⠀⢸⡄⡇⠀⠂⠀⠀⠀⢠⠃⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢀⠀⠀⣿⠀⠀⢀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⡤⡌⢣⠀⠀⠀⠀⡇⠀⠀⠀⠂⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢌⢦⠀⠀⢸⢁⠇⠈⢢⠉⠀⢸⡇⠐⠈⠀⢀⡞⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢀⡴⠖⠛⠛⠛⠃⠓⠒⠛⠘⠛⠉⠉⠑⠒⠚⣄⠀⠀⢀⠌⠉⠙⠋⠉⠉⠳⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⢀⡴⠉⠀⠀⢀⣀⠠⠀⠀⠀⡀⠀⡠⠄⠒⠢⢄⠀⠈⢦⡴⠃⠀⠀⠀⠐⡀⠀⢀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠀⠈⠙⡄
⠀⠀⠀⡾⠀⢠⠦⡤⠤⡤⠵⡀⠀⠀⠁⠞⠒⠲⠰⠒⠘⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠸⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠋⠁⠀⠀⠀⠀⠀⠓⠈⠒⠲
⠀⠀⠀⣧⠀⢇⣀⣐⣚⣁⣀⡇⠀⠀⠀⢦⠐⠒⠒⠒⢒⠇⠀⠀⠀⡀⠀⠂⢀⡎⠀⠂⠀⠀⣀⣀⣸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠾⠤
⠀⠀⠀⢧⡀⠈⠢⣤⣠⡤⠊⠀⠀⠀⠀⠈⠒⠤⠤⢒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰⣏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞
⠀⠀⠀⠀⢹⡀⠀⢀⡠⠄⠀⠀⠀⠀⠀⣀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠉⠁⢀⡞⠉⠙⠓⠲⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⢀⡤⠛⠋⠉⢀⣀⣤⣀⣄⠼⠉⠛⠀⠀⠀⣀⣤⣀⡀⠀⠀⠀⠀⣀⣀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⢤⡤
⠀⠀⠀⡼⠀⢀⡔⠋⠉⠳⠤⢤⡄⢀⠀⠀⡤⡶⢋⠀⠀⡏⢹⠀⡆⣷⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⢠⠇⠀⡼⠀⠀⠀⠀⠀⠈⠳⠋⠋⠛⢡⠃⡎⣼⠀⡇⢸⠀⡇⢸⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠋⠁
⠀⢀⡞⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡇⣿⡀⡇⢸⣠⢧⡎⢣⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⡤⢤⠀⠀⠀⠀⢀⡖
⣞⠉⠄⠰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠤⠖⠋

⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢂⢐⠞⠂⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠢⡀⢈⡄⡀⠀⢀⠀⠀⠀⠀⠀⣊⣀⢀⠀⠠
⠀⠘⠓⠀⠀⠈⢐⣁⠀⠀⠀⠀⠀⠀⣲⣏⣷⡄⠀⠀⠀⠀⠀⠀⠀⣀⣦⡾⡉⡹⡓⠒⠁⠀⠀⢠⠆⠈⣹⠟⠷⡄⠊
⠀⠰⢹⡄⠀⠀⠀⠉⡧⠀⠀⠀⠐⠀⢣⢺⠃⡏⠀⠀⠀⠀⠀⠀⠚⠁⢠⣾⢋⡔⠘⠐⠊⠀⢠⡀⡘⠁⣠⡀⣠⠡
⠀⠀⠹⢄⠀⣀⠀⠄⠁⠀⠀⠀⠀⠠⠊⢹⠡⠁⠀⠀⠀⠀⠀⠊⠁⠀⠚⠕⠂⣸⠀⠀⠀⠠⠚⠀⠁⢀⡈⠀⡤
⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠠⠀⡤⡼⠀⠈⠈⠄⠀⠒⠁⠀⠀⠀⠐⠀⠀⠐⠀⠁⠀⠁
⠀⠀⠀⢈⣉⠑⢨⡚⠠⠄⢐⠀⠀⠀⠀⢸⠀⠠⡀⠀⠀⣡⢜⠞⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⡀⢀⣈
⠀⠀⡔⠉⠇⠀⡡⠝⡠⡍⠁⠑⠀⠀⠀⢸⠀⠀⠁⠀⠀⢦⠈⠆⠑⠌⠀⠀⠀⠀⠠⠀⠀⠀⠤⡚⠈⢀⣤⡀⢀⢠⠄
⠀⠘⠀⠀⠀⠤⠔⣾⠸⠁⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠐⠀⠃⠀⠀⠠⠶⡦⠤⠀⠀⡀⠀⠉⠰⢸⠀⣈⣭⣀⢀
⠀⠀⠀⠀⠀⠠⠔⠁⢀⠀⠀⠀⣰⠀⢰⡴⢻⡆⠪⡁⢠⣞⡹⢠⡎⠀⢸⠇⠰⣉⠂⠀⠁⠄⢰⣾⣛⣭⠾⠯⡲⡀
⠀⠀⠀⠀⠀⠂⠀⠰⠁⠀⠀⡄⢿⡠⢺⠀⢸⡠⢄⣸⠾⢧⡤⠚⢧⡤⠾⠤⢄⣨⡇⠀⠀⠀⠁⡇⠡⢶⣺⠁⠇
⠀⠀⠀⠀⠀⠀⢀⠰⣀⠀⠰⠀⠀⠀⢀⠀⡇⡀⢀⢀⢀⠀⡀⡇⢸⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡉⠁
⠀⠀⠈⠂⢀⡲⠊⠌⢶⠈⠀⠈⠀⠀⠸⠄⠇⠃⠘⠹⠰⠛⠆⠧⠸⠘⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⢶⣀⠊
⠀⠁⠀⠉⠀⠁⠂⠀⠼⡁⠂⠒⠀⠈⢠⢤⠤⡤⢀⠀⢀⠀⠀⠀⢀⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠐⠀⢽⣿⡅
⠀⡀⠄⠂⠁⠀⠀⢠⠀⠈⠄⠀⠀⠀⠀⠀⠒⢩⣥⣄⣩⠇⠀⠁⠞⢦⠼⠧⣄⣤⠤⢄⡀⠀⠀⠀⢀⢚⣘⡶⡀
⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⡀⠠⣎⣠⠀⠑⡌⠉⠆⠀⠀⠀⠁⠀⠀⠀⠀⠐⠄⠀⠀⠁⡠⠃⢰⣶⡺⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⢁⣆⠀⠀⠀⠀⠀⠀⠀⠠⡄⢠⣴⣆⣠⠂⠀⠀⠀⠀⢠⢼⡖⣄⣈⠳⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠠⡤⠀⡏⠊⠀⠀⠀⣠⡄⠀⠀⠀⠈⣋⠉⠉⡇⢀⡀⠀⠄⣀⣀⣧⠙⢦⣌⣀⡐⣁⠀⠠
⠀⠸⠁⠀⠀⠀⠀⠀⠀⠠⠜⠁⠀⠁⠀⠘⢢⣀⡯⠂⠀⠀⠈⠰⢼⠀⠠⡎⠁⠀⠀⠀⠀⠀⠸⠇⠴⠉⠁⠙⠈⠂
⠀⠀⠐⠦⠄⡀⠀⡀⠑⢢⡀⠀⠀⠀⠀⡐⡻⣿⡅⠐⠀⠀⠀⠂⢂⠀⠀⠙⡤⠀⡀⠀⠀⠀⠠⠤⠤⢠⣧
⠀⠀⠀⠀⠀⠈⠁⠈⠉⠢⣉⠄⠀⠤⠀⠸⡴⠋⠀⠀⠀⠀⢀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠂⡀⠸⠍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠈

⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠤⠤⣤⣄⣀⡀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠈⠂⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⢤⣤⣾⣿⣿⣿⣶⣍⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢤⡀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⣛⣩⣭⣭⣭⣽⣟⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣷⣼⣦⠀⠀⠀⠀⣴⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⣤⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠉⠉⠉⠀⠀⡟⠛⠉⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠀⠀⠉⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢧⠉⠛⠿⠿⠿⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢨⠁⠀⠀⠀⠀⠀⠀⠀⡄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣴⣬⡀⠂⢀⠀⢀⣠⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢟⣿⣉⣤⣈⣐⠓⣊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣬⡁⠼⠅⠀⠀⠀⠀⠀⠀⢀⠒⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⢦⠄⣈⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠸⢾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣱⠏⠡⢤⣘⢿⣇⢦⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⢀⣀⣴⣭⠑⠈⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠑⢆⣶⣿⢟⣥⠆⢠⡜⠯⣧⡀⠀⠀⠀⠀⠐⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣠⣞⣋⠕⡿⣫⡶⠀⢁⠀⢻⣿⣦⣠⣦⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡏⢛⣿⡿⣋⡾⠁⠈⠀⣿⣿⡿⣿⣿⠏⠀⠀⢀⠀⠀⠀⠀⠀⢠⣶⡀
⠀⠀⠀⠀⠀⠀⡀⠙⣴⠛⣿⣿⢿⣟⢩⣵⡟⡧⡐⠓⢺⣿⣟⣀⡈⠉⠀⣠⣴⣿⡷⠀⠀⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⣇⠀⢸⢚⣿⣿⣷⣷⡶⡁⢂⣀⣽⣿⣷⣿⣿⣿⣷⢤⡾⠛⠻⠁⠀⡀⠀⠀⠀⢠⡾⠻
⠀⠀⠀⠀⠀⠀⠸⠀⣗⣿⣿⣿⡿⣿⣟⣛⠉⣰⢿⡿⡿⢿⡟⣻⣽⠇⠀⠢⠁⠀⠀⠆⢾⣷⣾⣿⡟⠃
⠀⠀⠀⣤⡄⠀⢀⠀⠛⢿⣿⡿⠁⣾⠻⣟⡟⣋⣽⢲⡧⢈⣿⣿⣯⣄⣠⣾⣿⣶⣆⢴⣾⣿⠟⠁
⠀⠀⠀⠢⡆⠀⢘⠀⢀⣾⣾⣇⣴⠋⠀⢻⡆⠁⣿⢷⣾⣿⣿⣿⣿⣿⣧⣴⡌⢿⢏⣮⣷⣿⣁⣤⡀
⠀⢴⣴⣶⢦⡀⠁⠀⣾⡧⣾⣿⠋⠀⠀⢹⡇⢀⡿⠀⠻⣿⣟⣿⣟⣿⣭⣍⣨⣹⣻⡿⣷⠾⠏⠙⠃⠀⣀⣨⢶
⣠⣾⣿⠟⣼⣧⣀⣰⣋⣠⣿⣿⡆⠀⠀⠈⣧⣼⣇⠀⠀⠘⢿⣿⣿⣿⡿⢿⣿⣿⣿⣼⠿⣄⣸⣿⡗⡂⣤⡅
⣿⣿⣉⣻⣯⣼⣫⣿⣿⣿⣿⣯⣿⡄⢀⡠⠿⣿⣿⣷⣆⠀⠈⢿⣿⣿⣿⣷⣶⣽⢷⣿⣷⡉⠉⠽⣿⠟⠛⠛⠉
⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣏⢐⣾⣷⣿⣿⣿⠓⠄⢀⣿⣅⣀⡾⣿⡟⠛⢿⣿⣿⣶⣾⣿⡛⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢦⣙⣻⣿⣿⣿⣿⣿⣶⡿⢤⣌⣹⣿⣿⣶⢀⣶⠞⡉⢁⠀⠀⠀⠈⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠿⠿⠛⠋⠁⠴⠂⠀⣠⣄⣄⣀⡀⠀⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⣴⣶⣿⡆⠀⠁⠀⠀⠉⠉⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠹⣿⣦
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷
⠀⠀⠀⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⠀⠀⠀⣿⣿
⠀⠀⢀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣾⣿⠇⠀⠀⢻⣿⣇
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⡄⠀⠀⠀⢿⣿⡄
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠈⠛⠁⠀⠀⠀⠈⣿⣿⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇
⠀⠀⠀⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⠉⠉⠉⠉⠁
⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣷⣦⡀
⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣤⣀⠀⠙⢿⣿⣦⡀
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣀⣀⣀⣀⡀⠀⠀⣀⣀⠀⠈⠙⠻⢿⣷⣄⠀⠙⢿⣷⡄
⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⣿⣿⠀⠀⠿⠿⣿⣷⣦⣄⠀⠙⣿⣷⡀⠈⢿⣿⡄
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⠀⠀⠀⠀⠀⠈⠻⣿⣧⠀⠘⣿⣷⠀⠘⣿⣷
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⢹⣿⡆⠀⣿⣿
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⠀⢸⣿
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠘⠛⠃⠀⢸⣿
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢠⣤⡄⠀⢸⣿
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⠀⢸⣿⠂
⠀⠀⠀⠀⠸⠿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠸⠿⠇⠀⠸⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢦⡀
⠀⠀⣀⣀⣀⣀⣀⣀⣠⣞⠉⣴⣛⣀⠀⠀⠀⠀⠀⠀⣠⣴⣗⣶⣶⣦⣴⣏⢁⣾⣯⣄⡀⠀⠀⠀⠀⠀⣤⠾⠽⠷⠶⠶⠶⣏⢁⠶⠷⣦
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⢸⢹⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⣿⡟
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠳⡖⠒⠒⣶⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣻⢏⠛⠛⢻⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠈⠢⡄⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣯⣀⣳⣄⣸⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠸⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠻⠶⠶⠶⠶⠶⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠒⠒⠲⠶⠟⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠯⠿⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⣤⢤⣄⣀⣔⢒⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣠⣄⣀⡏⠉⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢷⡆
⠀⠀⡏⠉⠉⠉⠉⡍⠉⡏⠉⠉⠉⡷⡄⠀⠀⠀⠀⠀⡇⠀⠀⣧⣤⡏⠉⢹⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⣶⠒⠒⠒⠺⡇⢸⡗⠒⠒⠒⢶
⠀⠀⡇⠀⠀⠀⠀⠑⠒⠁⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⡇⠘⠛⠒⠀⠀⠀⢸⡇⠀⣧⣤⣄⠀⠀⠀⠀⠀⣿⠀⢠⣤⣤⣏⣻⣥⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠒⠒⠒⠒⠒⠒⢸⡷⠶⠇⠀⣸⠀⠀⠀⠀⠀⣿⠀⠤⠤⠤⠤⠤⠤⠤⠤⠀⢸
⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⡇⠐⠒⠒⠒⠒⠃⢸⡉⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠰⠶⠶⠶⠶⣶⣶⣶⣀⣸
⠀⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣠⣠⣤⣼⠉⢉⣉⡁⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣠⠞⠁
⠀⠉⢿⣤⣤⠤⠤⠤⣤⣤⣤⡤⣤⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠉⠉⠉⠉⠁⢻⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠶⠶⠖⠚⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠈⠻⠷⣶⣤⣀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈⠉⣳⣤⠹⠿⠭⣤⣁⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⣆⣁⡴⡇⠀⣀⣴⣿⣆⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠐⣿⣥⣾⣷⣶⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⢀⣉⣉⢹⣿⣿⣿⡟⠁⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉⠉⠛⠓⠦⢤⡀⠀⠀⠘⣇⠀⢀⣾⡛⠉⠀⢼⣷⡘⠿⢿⡛⠋⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⠄⠀⠀⠙⢆⣾⠛⠻⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⣀⣀⣠⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⣼⠁⠀⣾⣿⣻⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⣤⠴⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠙⠿⠆⣰⠃⢠⣼⡿⠟⠻⣦⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠛⠛⠻⢿⣿⣷⣦⣤⣾⣧⣿⣿⣋⠀⠀⠀⠀⠉⠉⠛⠛⠛⠳⢦⣄⣀⢀⡀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣾⣿⢿⣟⠉⠻⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠚⠒⠒⠒⠒⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⢺⣿⡜⠋⠋⢿⣿⣿⡆⣿⣧⠀⠀⠀⠈⠉⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡋⠔⠟⢿⣷⣄⡈⠻⣿⡇⠘⢿⣇⠀⠀⠀⠀⠀⠈⠳⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠿⣑⡀⠀⠀⢽⣿⣿⣿⣿⡇⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠉⢽⣶⣦⣀⣻⣿⡿⡛⠀⠀⠀⢻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢳⡀⠀⠨⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣤⣈⣛⡻⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠾⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⡈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢄⡀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠶⠶⠶⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠔⠋⠁
⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⠤⣄⣀⠀⠀⠹⡄⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢈⡙⠿⣯⣉⠉⠛⢉⡉⠟⠛⠻⠿⣿⣶⣄⣽⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⠖⠉⠉⠉⠁⠐
⠀⠀⠈⠻⡄⠀⠙⠻⢶⣤⣀⡀⠀⠀⠀⠀⣈⣹⣿⠇⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⣾⣁⡀⢤⠤⣤⣤⣀⣀⢀
⠀⠀⠀⠀⠙⢆⠀⠀⠀⠈⠙⠛⠿⠿⠟⠛⠉⠁⣴⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢙⡛⠻⠶⣶⣟⢿⡛⢿⣿⣷⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡈⣧⠀⠀⠀⠙⠷⣤⣉⠀⠀⠀⠀⣠⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦⣀⣀⠀⣀⣠⣴⣾⠿⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢷⣄⠀⠀⠀⠀⠉⠛⠶⠶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠫⠾⠿⡿⠿⠋⠁⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢿⣷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⠻⣿⣿⣷⣶⡦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠙⠋⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃⠐⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣠⡶⠟⠛⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢠⣾⠉⠀⠀⠀⠀⢸⡟⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡇⠀⠀⠀⠀⠀⠈⣧⠈⠙⠳⢦⣄⡀⠀⠀⠀⣰⡿⣤⣉⣁⣀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠈⣋⠙⣿⡷⠶⠟⠀⠀⠉⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣠⡶⠏⣡⠀⠀⢹⣇⠀⠀⠀⢹⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠁⠁⣊⡡⠀⠀⠀⠻⣄⠀⠀⣘⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠹⣶⠛⠉⠳⠞⠋⠙⠰⣶⣶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠙⢷⡀⠀⠀⣠⡶⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⢀⢀⣀⠀⠀⢹⡇⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⢸⡇⢠⡾⠋⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠀⠀⠐⠁⠀⠀⠙⠃⠼⢿⣡⣿⢃⣀⠤⠤⠄⠀⠀⠀⠀⣿⡟⠁⠀⠀⠀⠀⠀⣿⢀⡤⠴⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⣿⡀⠀⢠⡔⠒⠐⠒⠀⢀⠀⠉⠉⠉⡠⠆⠒⠂⢠⠀⠀⠀⠟⠋⠉⠀⠀⠀⠁⠀⠙⠋⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⡒⠂⠁⠹⣧⠀⢨⠀⠀⠀⠀⠀⠀⠀⠢⡄⠀⢃⠀⠀⠀⠀⠀⢱⡀⠀⢰⡏⠀⠀⠀⢠⠀⠀⠀⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠈⠉⠛⠦⢤⣿⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⣻⠀⠸⡄⠀⠀⠀⠀⢨⡅⠀⠈⢃⠀⠀⠀⠠⠆⠀⢠⡎⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠳⣄⠀⠀⠀⣀⠔⠁⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠘⢇⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠿⠿⠿⠿⠻⡿⠟⣽⠛⢿⠟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣛⣻⣟⣛⣿⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿

⠀⠀⢠⡀⠀⠀⠀⠀⡶
⠀⠀⠈⣇⠀⡼⣇⢰⠇⡟⠋⠀⠀⠀⠀⠀⣴⠞⠛⠂⢠⠞⠛⠛⢷⡀⣼⢧⣠⢷⡀⠀⢺⠟⠛
⠀⠀⠀⢹⡞⠁⢹⡾⠀⡟⠛⠀⠀⠀⠀⢸⡁⠀⠀⠀⡏⠀⠀⠀⣸⠇⡟⠈⠋⠀⢳⡀⢸⠗⠒
⠀⠀⠀⠀⠁⠀⠈⠁⠀⠟⠛⠛⠀⠀⠀⠀⠙⠲⠶⠀⠙⠶⠶⠚⠁⢰⠃⠀⠀⠀⠀⠇⣸⠀⢀⣀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀
⠀⠀⢰⡆⣠⡀⠀⢿⠀⠀⠀⠀⣾⠛⣦⠀⣶⠶⠆⠀⣀⡀⠀⠀⠀⣀⣀⠀⣤⣤⣀⠀⠀⠀⠀⠀⠻⠟
⠀⠀⢸⡇⣿⠹⣄⢸⠀⠀⠀⠀⢻⠀⣸⠀⣿⣤⠄⣰⠋⠹⣆⣀⡟⠁⠀⠀⣿⣤
⠀⠀⠸⡇⡿⠀⠘⢿⠄⠀⠀⠀⢻⡛⠁⠀⣇⠀⢰⡟⠛⠋⣿⠈⣇⠀⠀⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠛⠛⠘⠁⠀⠀⠙⠀⠈⠙⠛⠁⠛⠛⠋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⣿⣿⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⢩⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣅⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟⠁

⣿⣿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⡀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⢠⡄⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡟⠀⠀⠀⠀⠀⠀⢀⠤⠶⠶⡂⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠈⢻⣿
⣿⣿⣧⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⠀⠀⠀⣽⣿
⣿⣿⡟⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠈⣂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢻⣿
⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡄⠁⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⢾⣿
⣿⣿⣿⣆⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⢠⠐⡀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠂⠄⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⢀⣾⣿

⡻⢶⠀⡃⠀⠀⣤⡤⠀⣀⠀⠀⡀⠠⢰⣶⡆⠄⢀⠀⠀⠀⣀⣤⡤⠀⠀⠀⠀⠠⣀⡤⣤⣀⡤⠀⠀⠀⠀⠀⠀⠀⢸⠀⡾
⡿⣿⣉⣁⠀⠀⠀⠊⠌⣉⠙⠯⠙⢛⣿⣿⣿⡒⠫⠖⠒⠀⠀⠀⠀⠀⠀⣀⣀⠴⣿⡀⢁⣿⡦⣀⡀⡀⠀⠀⠀⠀⣈⣉⣿
⣿⣿⢡⠸⡀⠀⠀⠶⣶⣤⡴⠖⢂⡾⠻⣿⡟⢷⡐⠒⢦⡤⣶⡦⠀⠀⠀⠈⠁⣼⢩⢿⡟⡉⢷⡈⠁⠀⠀⠀⠀⢠⠃⣾⣿
⣿⣿⢸⡄⠁⠀⠀⠀⠀⠉⠀⠀⠘⠁⠀⣿⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⣸⠟⠄⣸⣃⠠⠺⣧⠀⠀⠀⠀⠔⠀⢀⣿⣿
⣿⣿⢸⡏⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⠃⠈⢫⡄⢂⠑⢿⡇⠀⠀⠀⠀⢼⠛⣿⣿
⡟⠃⡼⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⡀⠁⣀⣼⣆⡀⠀⢈⢿⠀⠀⠀⠀⠘⡄⢿⣯
⠿⠋⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣡⡴⠊⠁⠈⠁⠈⠂⢠⣀⡄⠀⠀⠀⠊⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠁⠀⠁⠐⠀⢀⢀⣤⣶⣄⣠⠄⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠠⡷⢶⠊⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣿⣿⠦⠀⠀⢀
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡾⣟⡻⣦⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟
⣿⣄⣀⣄⣀⣀⠠⠴⠖⠂⠀⠀⣘⣛⣷⣿⡷⣯⣙⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠁⠀⠀⠰⠒⠂⢀⡤⣄⣠⣄⣃
⣿⣿⠉⠈⡁⠀⠀⠀⠀⠀⠀⢰⠁⣿⣿⣿⣿⣏⢧⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠙⢒⡈⣽
⡟⠘⢇⠀⠙⣻⡽⠟⠓⠀⠔⠀⢰⣿⣿⣿⣿⡿⢸⡀⠁⠀⠀⠀⠀⠀⠀⠀⢄⠀⢤⣠⣄⣤⠀⡠⠀⠘⠻⠿⡶⠋⡉⣼⠋
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢻⣿⣿⣿⣿⣷⢸⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⣿⣿⠁⢴⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⣿⣽⣿⡟⠃⡸⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⣧⣾⣿⣿⣷⡼
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠈⠙⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣼⣿⣿⣧⣤⣤
⡇⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢂⠄⠀⢀⡎⠀⢸⣿⣿⣿⣿⡇⠀⢱⡀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣶⣤⠖⠈⠀⠀⠀⠀⠀⠀⠁⢀⣤⣿⣿⣿⣿⣿⣿⣦⡀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⣿⣻⡟⠂⠀⠠⠄⠀⠀⠀⠀⠀⠈⡄⢿⣿⣿⣿⣿⡿⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠌⠻⠿⡿⠟⠡⡆
⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⠀⠀⠀⠠⣄
⢾⠊⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⢀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣴⣶⣶⣾⣶⣶⣶⣶⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀⠀⠀⠉⠛⠿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⠟⠉⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄⠀⠀⠀⠀⠈⠛⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠋⠀⠀⣠⣾⣿⠿⠿⢿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣰⣿⣿⡟⠁⠀⣰⣾⣿⠋⠀⠀⠀⠀⣼⣿⠿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡀
⠀⠀⠀⠀⣰⣿⣿⠏⠀⠀⣼⣿⢿⣿⣦⣀⣀⣀⣴⣿⠏⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣇
⠀⠀⠀⢠⣿⣿⠏⠀⠀⢸⣿⠇⠀⠙⠛⠿⠿⠟⠛⠁⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⢀⣾⣿⡏⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⣼⣿⣿⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢻⣿⣷⣦⣤⣴⣶⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⢸⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠸⣿⣿⠉⢻⣿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⣿⣿⡇⠀⠉⠛⠿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣧
⠀⠀⢻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⢻⣿⡆
⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠀⣸⣿
⠀⠀⠀⠘⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡿⠃⠀⢰⣿⡿
⠀⠀⠀⠀⠈⠻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠋⠀⠀⠀⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠋⠁⠀⣀⡀⠀⠀⠘⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣷⣶⣤⣄⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣠⣴⣶⣾⠿⠟⠉⠁⠀⠀⠀⠀⠻⢇⣀⡀⠀⠀⠙⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣠⣤⣴⣶⣶⣶⣶⣄⡀⠈⠛⠁⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠻⠿⢿⣿⡿⠿⠛⠛⠛⠛⢻⣿⣷⡄⠀⠀⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡇⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡇⠀⠀⠀⠀⢀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣴⣾⣿⣿⠿⠛⠉⠀⠀⠀⠀⢀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣾⣿⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⡿⠿⠿⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⣶⣿⣿⡿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⣤⡄⠀⠀⠀⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠸⠿⠇⠀⢿⣿⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠿⠿⢿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣦⣤⣤⣤⣶⣶⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠟⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣷⡄⠀⠀⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⢀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⠀⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠟⠉⠉⠉⠉⠉⠉⠻⣿⣿⡀⠀⠀⠀⣠⣿⡿⠃⠀⠀⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⠀⣠⣶⣿⠟⠁⠀⠀⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⢠⣶⣾⡶⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢀⣼⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠻⣿⣿⣤⣄⣀⣠⣴⣾⣿⠿⠛⠁⠀⠀⠀⠀⣀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣶⣶⣦⣶⣶⣶⣿⣿⣿⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢩⣿⣿⡿⠉⠀⠀⣹⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠏⠁⠀⠀⠀⢿⣿⣿⣦⣶⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⢠⣄⠀⠀⢸⣿⣿⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄⠘⣿⣷⠀⠀⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠃⠀⣿⣿⣧⠀⠈⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠃⠀⢰⣿⣿⣿⡄⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠀⠀⣠⣿⡿⠻⣿⣿⣦⣴⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⣠⣼⣿⡿⠁⠀⠈⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⢷⡀
⠀⠀⣀⢤⡲⠟⠳⣦⡀⠀⠀⠘⣇
⠴⠮⠓⠉⠀⠀⠀⠈⢳⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⡤⠤⠿⠒⠒⠙⠒⠦⡀
⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⣴⣀⠀⠀⠀⢱
⠀⠀⠀⠀⢀⠜⠀⠀⠀⠘⢿⡿⠁⠀⠀⠈⠲⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⣀⣠⠞⠁⠈⠒⠒⠢⢄⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⣇⠤⡄⢀⠤⠤⢲⢋⡼⢠⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠈⠐⠛⠁⠀⡰⢁⡾⢡⠃⠀⠀⠉⠲⣄⣄⣀⣀⣀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡔⢡⢻⢁⠎⠀⠀⠀⠀⡸⡜⠀⠀⠀⠀⠀⠱⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⢀⣠⠞⡴⢣⢃⡎⠀⠀⠀⠀⡰⡹⠀⠀⠀⠀⠀⠀⠀⠱⡘⡍⠒⠦⠄⣀⢀⣀⣀⠠⡊
⠀⠀⠀⠀⠘⠓⠒⠛⣠⠇⡼⠀⠀⠀⠀⠾⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⡐⢄⡀⠀⠈⠓⠬⠷⠓⣝⡢⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠃⠀⠀⠀⠀⠀⠈⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡆⠀⠀⣀⣀⣀⠀⠀⢸⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠷⠊⠉⠀⠀⠀⠈⠑⠺⡀⠁
⠀⠀⠀⠈⠫⢔⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣒⠍⠋
⠀⠀⠀⠀⠀⠀⠈⠢⡑⢄⠀⠀⠀⠀⠀⣠⢶⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡷⢤⠀⠀⠀⠀⠀⣠⢖⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡑⣄⠀⠀⠀⢻⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣾⠃⠀⠀⢀⡜⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠇⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠀⠀⠀⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⠀⠀⠀⠈⠲⠤⠤⠄⢄⠤⠄⠤⣀⣀⣀⣀⠠⠎⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⡰⠤⣀⠀⠀⠀⠀⠀⡠⠚⠉⠁⠉⠑⢦⠀⠀⠀⠀⠀⢀⠠⠰⣀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠙⠢⡀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⡠⠊⠁⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠄⡏⢙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢉⡄⠤⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡄⠀⠠⠠⠄⣀⠀⠀⠀⠈⡇⠀⠀⠀⠐⠒⠒⡊⠀⠀⠀⢀⡀⠤⠀⠐⠀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠸⠤⣀⡀⠀⠀⠉⠒⡔⠉⡧⠤⠤⠤⠤⠄⠤⡋⠑⠔⠊⠁⠀⠀⣀⠠⠸⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⡎⠀⢀⠃⠀⠀⠈⠉⠁⠀⠈⠑⠲⠇⠀⠀⠀⠀⠀⠀⠷⠐⠉⠀⠉⠉⠁⠀⠀⠀⢃⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠠⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠚⠢⣀⡀⣀⡠⠚⢢⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠠
⠀⠀⠀⠀⠀⠀⡼⡠⠋⠀⣀⠤⠀⠐⠂⠈⠉⣀⡀⢬⡦⠄⠇⠀⠀⠃⠤⢮⡤⢀⣀⠉⠁⠒⠂⠠⢄⡀⠈⠣⡼⣄
⠀⠀⠀⠀⡠⣪⠎⠀⠀⡜⠑⢤⠔⠐⠂⠉⠁⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠈⠁⠐⠐⢤⠖⠙⡄⠀⠈⠪⡢⡀
⠀⠀⡠⡪⠚⠁⠀⠀⢰⠀⠀⠸⠀⠀⠀⠀⡜⠁⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠈⢢⠀⠀⠀⠸⠀⠀⠰⠀⠀⠀⠈⠪⡢⡀
⠴⠞⠊⠀⠀⠀⠀⠀⠰⠀⠀⠇⠀⠀⠀⢰⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠘⡆⠀⠀⠀⡆⠀⠘⠀⠀⠀⠀⠀⠈⠙⠲
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢹⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠁⢀⠆⠀⠀⠀⠀⢸⠠⠤⠤⠄⠀⠀⠶⠶⠶⠀⠀⠀⠤⠤⠤⢠⡇⠀⠀⠀⠀⢆⠀⢁
⠀⠀⠀⠀⠀⠀⠀⡸⠤⡌⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠈⡄⠸⡀
⠀⠀⠀⠀⠀⠀⡰⠁⡸⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠸⡀⠱⡀
⠀⠀⠀⠀⢀⡔⢁⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣔⠒⠂⠀⠀⠀⠀⠐⠒⠢⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠘⠄
⠀⠀⠀⣠⢊⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠄⠤⠤⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠌⠢⡀
⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⢀⣴⣿⣷⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⡀⠀⢠⣾⣿⡿⠋⠀⠀⢻⣿⣦⡀⠀⠀⠀⢠⣾⣿⡟⢸⠀⠀⠀⠀⢨⣧⣀
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠙⢿⣿⣦⣄⣰⣿⣿⠟⠀⢸⡀⠀⠀⢰⣿⣿⣿⡇
⠀⠀⠀⠀⢈⣷⣦⡀⠀⠀⠀⠀⣸⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠀⠀⠸⠃⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⢀⣼⣿⡿⠛⠛⢿⣿⣶⣄⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢰⣿⣿⣿⡇
⠀⠀⠀⠀⢹⣿⣿⣿⡀⢠⣾⣿⠏⠀⠀⠀⠈⠻⣿⡿⠀⠀⠀⠀⠀⢠⣿⣿⠋⠛⢿⣿⣿⣷⡀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣷⣿⣿⠋⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠀⠀⠉⠻⠿⠃⠀⠀⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠘⣿⣿⣿⡉⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠹⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⠙⢿⣿⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⢿⡿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠰⠀⠀⠀⠀⠀⠀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠴⠂⠀⠀⠀⠀⠀⠸⡀⠀⠀⠠⣀⢤⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢱⠃⠀⠀⢀⡐⠁⣀⠤⠒⠘⡞⠁⠀⠀⣇⣀⠆⠀⠀⢀⡠⠒⠉⠀⠀⢱
⠀⠀⠀⠠⢦⠀⠀⠀⠀⢇⡀⣀⠔⢊⠩⠿⡏⠀⠠⢤⠄⠑⠦⡤⠊⡩⠋⣱⠄⠒⠉⠀⠀⠀⠀⣠⠃⡴⠞⠁⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠚⣉⠀⠀⠓⠾⣄⠀⠸⢤⣋⡠⠴⠳⣆⣀⣀⡀⢤⣐⡨⠔⠚⠣⡀⠀⠀⠀⠀⠘⠤⠠⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⠁⠀⠀⠀⠀⠙⢆⠀⢀⡁⢀⣴⡆⠀⠀⢀⡠⠖⠁⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠾⠂⢎⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⡀⠀⠀⠰⡰⠀⠀⠀⢀⠤⣄⢀
⠀⠀⠀⠀⣠⢬⡗⣎⣠⠤⠤⠊⠃⠀⠉⠐⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣐⠀⠙⢶⣄⢀⠀⠀⠀⠀⢈⠂⠠⠄
⠀⠀⠀⠀⢯⣉⣻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣁⣀⠀⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⡋⠻⠷⠿⠫⢥⢾⣕⣒⣒⢒⠒⢒⢒⣒⣒⣾⠡⠮⠢
⠀⠀⣀⠀⢈⣿⡿⡟⡆⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠀⠀⠀⠀⠈⠙⠳⢦⣄⡀⠀⠀⢸⠀⠀⠐⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⡎⠘⢷⣺⡿⠁⡅⠈⠄⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠁⠈⠁⠘⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤
⠀⠀⠀⠀⠠⡇⠀⢀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠐⠚⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣽⠲⢋
⠀⠀⠀⠀⠀⢀⡀⠀⣆⠀⠈⠻⠮⣄⠀⠀⠀⠀⠀⠁⠢⠠⠄⠀⠄⠤⠊⠉⠒⠦⠤⣤⠤⣤⠀⡤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠑⠁
⠀⠀⠀⠀⢀⣀⢴⠉⠁⠀⠀⢀⠇⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠁⠠⢀⠔⠁⢰⠾⡀⠀⠀⠀⠳⢤⣴⣀⡑⠈⠑⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠈⠒⠒⠁⠀⠑⠤⠀⠀⠠⠴⠶⠯⢝⡛⠲⠶⣤⣄⣈⠉⢒⠲⠦⠖⢒⡒⠢⢠⠴⠓⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠴⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⣚⠛⠭⠿⠿⠶⢦⡤⠄⠄⠴⠁⠀⠀⠀⠀⠀⠉⠁⠐⠒⠒⠒⠀⠐⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠐⠒⠒⠾⢗⣒⣒⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⢶⣾⣿⣿⣿⣿⣿⡍⠉⠉⠉⢩⣿⣿⣿⣿⣿⣷⡶⠖⠶⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣅⣀⡀⠀⠀⣉⣻⣿⠿⠿⠛⠛⠒⠒⠒⠛⠛⠛⠿⠿⢿⣉⠀⠀⣀⣀⣤⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣯⣩⣤⣶⣶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣦⣤⣼⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⠙⠓⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠉⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢀⡴⣧⡀⠀⠀⣬⠛⠀⠀⢿⣷⣢⠀⠀⠀⠀⠀⡀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣀⡴⠊⠠⡏⠹⠓⡤⠹⣧⠀⠀⢀⡉⠉⠑⣄⠀⠀⡐⣿⣦⡀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣠⠹⠀⢠⠀⠁⢤⡀⠀⠀⠻⠀⠀⠈⢻⡂⠀⠈⡗⠋⡀⠙⠑⠛⢦⡀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠲⡉⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⠀⠀⢀⠘⠿⠀⠀⠀⠀⡘⠆⠀⢀⠀⠓⠄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⢰⣿⡀⠀⣾⣧⠀⢠⡄⠀⣰⣷⣀⡀⣾⣇⠀⢠⡄⡀⣰⣷⡀⠀⣾⣇⠀⢠⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⣿⣿⣷⣾⣿⣿⡄⣾⣿⢀⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⡇⢸⣿⣿⡀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠁⠀⠙⣿⣿⣿⣿⣿⣿⣭⣿⣿⣤⣿⣿⠻⠟⣿⡿⠀⢻⡿⣿⣿⣿⣿⣿⣿⣿⠟⣴⣃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢲⡌⠻⣿⣿⡿⠿⠟⠛⠉⠁⠈⠟⠀⠀⠹⠃⠀⠀⠀⠀⠘⠏⠈⢉⠝⣡⢦⠄⠙⠳⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⡄⠈⢠⡶⣮⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⣥⣄⠈⠳⣄⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⡼⠁⣰⣗⣤⠉⣓⡲⠤⠤⢄⣀⣀⡀⠤⠤⠔⣒⠋⢵⡖⠀⠘⣿⣷⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣰⣿⣿⠏⢠⣏⠉⠀⠀⣶⠀⣦⣶⠀⠀⠸⣏⡷⠈⢿⣃⡀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢼⣅⠀⠀⠀⣿⠀⡿⣿⠀⠀⠀⢻⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠋⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⢄⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⠀⣠⠞⠁⠀⣀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⣠⣶⣿⡏⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣌⢳⣤⣇⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠠⣄⢀⣠⠤⣄⠈⢧⣠⣿⣿⡿⠁⢀⠄⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⢿⣿⣿⡷⣾⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠘⣆⢾⣿⣿⠶⠶⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⢶⠋⠹⡏⢧⣈⣛⠛⠋
⠀⠀⠀⠀⢀⡴⠋⠀⠉⣛⣲⣼⠿⠽⠶⠒⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⢦⠉⠉⠙⠆
⠀⠀⡠⠔⠈⠀⠀⢠⣾⡟⡹⣿⢳⡀⠀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡴⠛⠀⠓⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⡀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠿⠿⠋⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠊⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢾⣿⡷⡆⠀⠀⢓⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⠈⠿⢿⣷⡲⣤⠏⢀⠤⢤⣀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣤⠋⠀⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠒⠋⠉⠉⠉⣠⠟⣻⣶⠿⢿⣤⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣼⠁⡴⠃⣿⣤⣼⣿⡇⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⡇⠀⠈⠛⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀
⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠹⣧⠀⣽⠶⣄⠀⠀⢀⡴⢾⡅⢠⡿⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣇⣸⠀⡀
⠀⠀⠀⠀⠀⠈⢱⣿⣷⡊⠀⠀⠀⠀⠘⣇⠨⠳⣬⣦⣤⣮⡔⠫⢀⡟⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⣤⣸⣇⣤
⠀⠀⠀⠀⠀⠐⢹⣿⣿⡙⠄⠀⠀⠀⠀⠈⠙⠋⠉⢹⣿⠉⠉⠛⠉⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡿⣿⠃⠀⠀⠀⠀⠀⠀⠱⠀⢀⠘⡟⠀⠀⠴⠃⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠴⣿⣿⡦
⠀⠀⠀⠀⠀⠀⢸⣧⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣰⣶⡆⠀⠀⠀⠻⠿⣿⣿⣿⣶⣶⡎⣳⣶⣶⣿⣿⣿⠿⠋
⠀⠀⠘⣾⢿⣟⣶⣤⣠⣞⣿⣿⡾⠃⠀⠀⠀⠀⣻⠿⣟⠀⠀⠀⠀⠀⢀⣨⣽⣿⡿⢿⠿⣿⣿⣭⣅⡀
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡉⠀⠀⠀⠀⠀⠀⢿⣿⡿⠀⠀⠀⠀⠘⠛⠛⠉⠉⠀⢸⠀⠀⠉⠙⠛⠛⠀⠀⢠⣾⣧⡄
⠀⠀⠀⠀⠈⠛⠛⠁⠘⠛⠛⠁⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⡀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⣛⡻⣿⣦⡀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⠻⠽⢿⣿⣶⣞⣳⣶⣿⡿⠮⠞⠁
⠀⠀⠀⠀⠀⠀⠐⢬⠕⠀⠀⠀⠀⠀⠀⠾⠛⠋⠀⣀⡀⠈⠛⠿⠆⠀⠀⠀⠀⠀⠸⣿⡗⠀⠀⠀⠀⠀⠺⠿⠿⢻⡟⠿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢷⣦⣄⠀⠀⠀⢀⣴⣶⢿⡷
⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⢿⣿⣏⠈⠣⠀⠰⠃⢨⣿⣿⠃⠀⠀⠀⠠⢀⠰⢆⠀⠄
⠀⠀⠀⠀⠀⢀⣶⣿⣿⣆⠀⠀⠀⠀⠈⣻⣷⡶⠒⣿⡗⠲⣾⣷⠋⠀⠀⠀⠀⠄⣾⡆⣸⡇⠠⠀⠀⠀⠀⠀⣠⣽⣯⣄
⠀⠀⠀⠀⠀⣼⣿⡛⣛⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠙⢀⠀⠈⣿⡇⠀⠀⠀⠀⠀⢿⣿⣿⡧⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢀⣿⠏⣭⡭⢹⣿⠀⠀⠀⠀⢸⣿⡷⠃⠀⠀⠶⣿⣟⠀⠀⠀⠀⠀⠀⠘⠿⠿⠁⠀⠀⠀⠀⠘⠋⠁⠈⠉⠈⠙⠃
⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠁⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠈⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠳⠶⠶⠶⠶⠶⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⣀⣤⡴⠿⠿⠛⠛⠛⠛⠻⠿⢶⣦⣤⣤⡶⠖⠲⠶⠶⠶⠦⡤⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⣠⡶⠟⠉⠁⠀⠀⡀⠀⠀⠒⢦⡀⠀⠀⠀⠉⠛⠿⣦⣄⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠋⠀⠀⠀⠀⠴⣚⠅⠀⠀⢠⣤⠁⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⢀⣈⣭⣤⣤⣤⣄⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⣾⠇⠀⠀⠀⠀⠀⣠⡴⠶⠶⠶⢏⠉⠁⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢠⣿⠀⠀⠀⠀⠀⠸⣯⣀⠀⠀⢀⠀⠀⠀⠀⠀⣦⣠⡿⠋⠀⠀⠀⠀⠀⠤⣿⡷⠶⠶⠦⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣀⣀⠀⠀⠀⠀⠈⠙⠛⠻⠾⠷⠶⠶⠶⠿⠛⠉⣠⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠚⠋⠉⠉⢿⡄⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⣀⠀⠀⢀⡤⣼⠁⠀⠀⠀⠀⠀⢠⣾⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⠄⠀⠀⡀⠃⠀⠳⣤⠴⠋⠁⠱⣄⡞⠀⠘⣆⠀⠀⠀⠀⣠⣿⣏⠀⠀⠉⠓⠤⡀
⠀⠀⠀⣠⠤⠶⠶⠤⣄⡀⠀⠀⠀⠀⣠⠞⠻⣦⡀⠀⠙⠂⠀⠀⠀⠀⠠⠲⠦⠀⠀⠀⠀⠈⠑⢀⣤⡾⠿⡅⠙⣧
⠀⢠⠏⠁⠀⠀⠀⠀⠈⠹⣄⠀⠀⠐⠁⠀⢰⡟⠙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠋⠀⠀⠘⢦⠘⣧
⠀⣿⠀⢠⠖⠒⠒⢦⡀⠀⢹⡄⠀⠀⠀⢀⡿⠀⠀⠀⠀⠈⠉⠂⠀⠀⠠⠴⠶⠖⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠸⡇
⠀⢿⠀⣿⠀⠀⠀⠀⣿⠀⢸⠇⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⡄⠸⡄⠀⠀⣸⠃⢀⡟⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⡇
⠀⠈⡇⠀⢻⡀⠀⣇⠀⠈⠳⢤⣀⠀⢸⡇⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠘⣧
⠀⠀⡇⠀⠀⢷⠀⠈⠓⠲⡄⠀⠈⡇⣿⠀⠀⠀⠀⠀⠙⠶⣤⣀⠀⢀⣤⠤⢤⣀⣀⡀⠀⠀⠀⠀⣀⣴⠟⠁⠀⢰⠀⠀⣿
⢀⡼⠁⠀⠀⢸⠀⠀⢠⠞⠁⡠⠚⠁⣿⠀⠀⠰⡄⠀⠀⠀⠈⠉⠛⠻⠁⠀⠈⣿⠈⠙⠶⠶⠶⠛⠉⠀⠀⠀⢀⡟⠀⠀⣿
⣿⡀⠀⠀⣠⠟⠀⠀⢻⡀⠀⠳⠤⣤⡏⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠔⢻⡤⠄⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⣿
⠈⠛⠛⠛⠁⠀⠀⠀⠈⠓⠦⢤⣤⣤⣧⠀⠀⠀⠀⠈⠻⢦⣀⣀⠀⠀⢀⣀⣠⣾⣁⡀⠀⠀⠀⠀⣀⣤⠞⠋⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⠉⠻⣄⣀⣠⡿⠹⣏⣉⣀⡴⠒⠚⠛⠉⠁⠀⠀⠀⠀⢀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢷⣶⣤⣤⣄⣀⣀⣈⣉⣉⣀⣀⣈⣉⣉⣀⣀⣀⣀⣤⣤⣤⣶⠶⠿⠛⠉⠁