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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣤⣶⣲⣆⣴⣲⣆⣴⣒⣦⣀⣶⣦⣠⡶⣦⢠⠖⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠘⡇⠀⠸⡇⠀⠀⣧⠀⠀⣿⠀⠀⠘⣏⠉⠹⣏⠉⢻⡉⠉⠉⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⡤⡤⢀⣀⡀⢀⣀⠀⠈⠀⠈⡁⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏
⠀⠀⠀⠀⠀⣰⠁⠀⠀⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠇
⠀⠀⠀⢠⠏⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟
⠀⠀⢀⡎⠀⠀⠀⠁⠀⠉⠉⠉⠉⠉⠉⠉⠙⠛⠒⠒⠒⠛⠒⠒⠒⠒⠲⠆⠀⠀⢰⣿⣿⠁
⠀⠀⠘⡏⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠓⠒⠒⠒⠲⠶⠶⠶⠶⠤⠤⠤⣤⣤⣾⢷⡇
⠀⠀⠀⠹⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⠤⣤⣤⣤⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡞

⠀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⣀⠀⣀⠀⢀⠀⢀⣀⣀⡀⢀⣀⣀⣀⠀⣀⠀⣀⠀⢀⣀⣀⡀⢀⣀⣀⡀
⠀⠀⢀⠀⣼⡏⢉⡁⢈⡉⠉⣿⠀⣿⠀⣿⡇⣿⡏⢉⡁⢈⡉⢉⣿⠀⣿⠀⣿⠁⣿⡏⢉⡁⢈⡉⢩⣿
⠀⠀⢀⢀⣈⣅⣸⡇⢸⣯⣀⣉⣀⣿⠀⣿⣇⣉⣁⣸⡇⢸⣿⣀⣉⣀⣿⠀⣿⣀⣈⣁⣸⡇⢸⣇⣀⣉⣀
⠀⠀⢠⠄⣉⣏⢉⠁⠈⣉⣉⣉⢉⡉⠀⢉⣉⣉⢉⣉⠁⠈⣉⢉⣉⣉⡉⠀⢉⣉⣉⡉⣉⡁⠈⣉⣉⡉⡉
⠀⠀⢠⠤⢛⡟⢻⡇⢸⣿⠛⠛⠛⣿⠀⣿⡛⠛⠛⢻⡇⢸⣟⠛⠛⠛⣿⠀⣿⡛⠛⠛⢻⡇⢸⡟⠛⠛⠛
⠀⠀⢠⠀⣼⡇⠘⠃⠘⠃⠀⣿⠀⣿⠀⣿⡆⣿⡇⠘⠃⠘⠋⢐⣿⠀⣿⠀⣿⠀⣻⡇⠘⠃⠘⠃⢰⣿
⠀⠀⢠⡀⠚⡿⠛⠃⠘⠻⠛⠛⠀⠛⠀⠛⠃⠛⠛⠛⠃⠘⠛⠛⠛⠀⠛⠀⠛⠀⠻⠛⠛⠃⠘⠛⠛⠟
⠀⠀⠠⡀⣼⡿⠿⠆⠐⠿⠿⣷⠀⣿⠀⣾⠆⣾⡿⠿⠆⠰⠿⠿⣷⠀⣿⠀⣾⠀⣾⡿⠿⠇⠸⠿⠿⣷
⠀⠀⢠⡄⢼⡇⢰⡆⢰⡦⠀⠿⠀⣿⠀⣿⡇⠻⠇⢰⡆⢰⣦⠐⠿⠀⣿⠀⣿⠀⠻⠇⢰⡆⢰⡆⠘⠿
⠀⠀⢀⡤⣶⡶⠾⠇⠸⠿⠶⠶⠶⠿⠀⠿⠷⠶⠶⠾⠇⠸⠿⠶⠶⠶⠿⠀⠿⠶⠶⠶⠾⠇⠸⠷⠶⠶⠶
⠀⠀⢠⣤⣦⡤⣶⡆⢠⣶⣶⣶⣶⣶⠀⣴⣶⣶⣶⣶⡆⢠⣶⣶⣶⣶⣦⠀⣴⣶⣶⣶⣶⡆⢠⣶⣶⣶⣶
⠀⠀⠠⡄⣤⡤⢸⡇⢸⡯⠀⣤⠀⣿⠀⣿⡇⣤⡄⢸⡇⢸⡿⢀⣤⠀⣿⠀⣿⠀⣠⡄⢸⡇⢸⡇⢠⣤
⠀⠀⠀⠄⣼⣧⣤⡄⢠⣤⣤⣿⠀⣿⠀⣿⡇⣿⣧⣤⡄⢠⣤⣤⣿⠀⣿⠀⣿⠀⣿⣧⣤⡄⢠⣤⣤⣿
⠀⠀⠠⠤⣤⣤⣠⡄⢀⣤⣤⣤⠀⣤⠀⣨⡀⣨⣤⣤⡄⢀⣤⣤⣄⠀⣤⠀⣠⡀⣠⣤⣤⡄⢀⣤⣤⣄
⠀⠀⠀⠀⣼⡯⢉⡁⢀⡉⠉⣿⠀⣿⠀⣿⠇⣿⡏⢉⡁⢀⡉⠉⣿⠀⣿⠀⣿⠀⣽⡏⢉⡁⢈⡉⢩⣿
⠀⠀⠀⢀⣈⣥⣸⡇⢸⣿⣀⣉⣀⣿⠀⣿⣇⣈⣁⣸⡇⢸⣿⣀⣉⣀⣿⠀⣿⣄⣉⣁⣸⡇⢸⣇⣀⣉⣀
⠀⠀⠀⣈⣉⣍⢉⠁⠈⣉⣉⣉⣉⣉⠀⢉⣉⣉⣉⣉⠁⠈⣉⢉⡉⣉⡉⠀⢉⣉⣉⣉⢉⠁⠈⣉⣉⣉⡉
⠀⠀⠀⠈⢛⡿⢻⡇⢸⡟⠛⠛⠛⣿⠀⣿⡛⠛⠛⢻⡇⢸⣟⠛⠛⠛⣿⠀⣿⠛⠛⠛⢻⡇⢸⡟⠛⠛⠛
⠀⠀⠀⠄⣼⡇⠘⠃⠘⠋⠀⣿⠀⣿⠀⣿⡆⣿⡇⠘⠃⠘⠋⢈⣿⠀⣿⠀⣿⠀⣿⡇⠘⠃⠘⠃⢸⣿
⠀⠀⢀⡄⠸⡿⠛⠃⠘⠛⠛⠛⠀⠛⠀⠛⠁⠛⠛⠛⠃⠘⠛⠛⠛⠀⠛⠀⠛⠀⠛⠛⠛⠃⠘⠛⠛⠛
⠀⠀⢀⡄⣼⡿⠿⠇⠐⠿⠿⣿⠀⣿⠀⣾⡆⣾⡿⠿⠇⠐⠿⠿⣷⠀⣿⠀⣾⠀⣾⡿⠿⠇⠸⠿⠿⣷
⠀⠀⢀⡄⠼⡇⢰⡆⢰⡖⠀⠿⠀⣿⠀⣿⠇⠻⠇⢰⡆⢰⣶⠀⠿⠀⣿⠀⣿⠀⠻⠇⢰⡆⢰⡆⠘⠿
⠀⠀⢠⡴⣶⡶⠾⠇⠸⠿⠶⠶⠶⠿⠀⠿⠷⠶⠶⠾⠇⠸⠿⠶⠶⠶⠿⠀⠿⠷⠶⠶⠾⠇⠸⠷⠶⠶⠶
⠀⠀⠠⣤⣶⣶⣶⡆⢠⣶⣶⣶⣶⣶⠀⣴⣶⣶⣶⣶⡆⢠⣶⣶⣶⣶⣶⠀⣴⣶⣶⣶⣶⡆⢰⣶⣶⣶⣶
⠀⠀⢠⡄⣠⡄⢸⡇⢸⡿⠀⣤⠀⣿⠀⣿⡇⣤⡄⢸⡇⢸⡿⢀⣤⠀⣿⠀⣿⠀⣤⡄⢸⡇⢸⡇⢀⣤⠀⡄
⠀⠀⠀⠀⠹⠧⠤⠄⠠⠤⠴⠿⠀⠿⠀⠿⠇⠿⠧⠤⠄⠠⠤⠴⠿⠀⠿⠀⠿⠄⠿⠧⠤⠄⠠⠤⠴⠿⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠾⡄⠾⡆⠰⢷⠰⠦⡄⠸⣆⠰⢶⠰⢰⣀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⠞⠁⠶⠃⠰⠞⠰⠖⠁⠶⠃⠰⠊⠰⠞⢸⡇⠀⣼⠃⡝
⠀⠀⠀⢀⣤⣤⡶⢒⢶⠖⠲⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣼⠁⣼⣤⣄⣀
⠀⠀⠀⢸⣿⣿⠉⠉⠉⠉⠉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⡰⠓⠚⣧⢻⡄
⠀⠀⠀⢸⣿⣻⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡰⠁⠀⠀⢸⢸⠇
⠀⠀⠀⢸⡟⢸⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠰⠁⠀⠀⠀⣸⢾
⠀⠀⠀⠘⣿⣾⡄⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣰⠁⠀⠀⠀⠀⣟⣿
⠀⠀⠀⠀⣿⣽⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡠⣿⠀⠀⠀⠀⢀⡏⡏
⠀⠀⠀⠀⣿⢳⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⣾⠀⠀⠀⠀⢸⢧⡇
⠀⠀⠀⠀⢻⣧⣇⠀⠀⠀⠀⠈⣿⣧⡀⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⢸⡷⣿⠀⠀⠀⠀⠀⣿⣦⣝⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⠀⠀⠀⠀⣿⣾
⠀⠀⠀⠀⢸⡿⢹⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡿
⠀⠀⠀⠀⢸⣇⣼⣧⣄⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣟⡇
⡤⠤⠒⠒⠐⠛⠛⠚⠓⠒⠾⠾⠷⠿⠿⠯⠶⠬⠾⠥⠼⠥⠼⠿⠿⠿⠤⠴⣷⣽⣿⡵⠫⡴⠇⠤⠤⠤⢄
⠙⠒⠂⠴⠤⠤⠤⠤⠤⠤⠤⠦⠴⠴⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣄⣠⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠙⠛⣻⠖⠲⠶⢶⠶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⡛⡛⠟⠛⠛⣿⣦⣄⢰⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢰⡋⣹⣧⣯⣿⡇⠀⠀⠁⠠⠂⠀⢀⠀⠀⠀⠈⠀⠁⠀⠈⠀⠀⢀⡗⠘⣿⣿⣿⢿⡷⢶⡄
⠀⠀⠀⠀⠀⠀⢠⠾⠻⡀⠀⠀⢉⣿⡟⣙⣋⡀⠂⠀⠀⠀⠀⠀⠡⢂⡔⡠⢀⢀⠄⠠⢀⠔⠀⢸⠃⢸⡿⣿⣧⣸⡇
⠀⠀⠀⠀⠀⠐⣧⣤⣤⣷⠀⢠⣯⣸⣗⣿⣿⡷⠀⡴⢠⢀⠀⠀⠁⠉⠘⠀⠉⠈⠀⠀⠈⣠⠊⢸⠆⢸⡇⣿⡏⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⣿⡏⠀⠀⠉⣹⣟⠛⣻⡁⠞⠃⠓⠃⠀⠀⠐⠀⠀⠠⠀⢀⠀⣀⢀⠉⡡⢹⠀⣸⡇⣿⡃⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡃⣿⣇⠀⠀⣾⣹⢫⢼⣿⣻⠂⢰⡖⠚⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠙⢷⣄⠟⡀⢹⣷⣿⣇⣿⠃
⠀⠀⠀⠀⠀⢸⣿⣴⣿⣿⠀⠀⠻⢿⣿⢟⣋⣍⡀⣼⡃⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⠈⣧⡴⠀⣾⣿⣿⢸⣿
⠀⠀⠀⠀⠀⢸⣿⢋⣼⣿⠀⠀⣴⢻⡟⢹⣿⣯⡿⢸⡇⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⣿⣿⣾⣿
⠀⠀⠀⠀⠀⢸⣿⡿⢿⣿⠀⠀⠻⢾⣿⠿⢋⣟⡀⢺⡄⠀⠚⠲⠒⠒⠚⠛⠛⠛⠛⠋⠀⢰⣿⣀⣂⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣥⣿⣿⠀⠀⣴⡟⣿⡣⣟⡟⡗⢸⡇⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⡤⠀⠈⣿⡧⠭⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⡗⣿⣿⠀⠀⠈⠻⣿⡷⢋⡩⠆⠸⡇⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣿⣟⠃⠀⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⣿⣞⣿⣿⠀⠀⣲⡿⣿⠄⣿⣻⣿⠆⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⠿⠉⡤⠀⢿⣿⢸⡟
⠀⠀⠀⠀⠀⠀⣿⣯⣻⢾⠀⠀⠻⠷⣷⣶⡿⢛⣋⠀⠑⠹⠿⠶⢶⠒⡒⠒⠛⠟⠛⠋⠉⠋⡡⠀⣮⠀⢹⡿⢸⣧
⠀⠀⠀⠀⠀⠀⣿⣿⢿⢿⠀⠀⠀⣴⣷⡇⣰⣿⣷⣆⠀⠀⠀⠀⠀⡀⢀⠀⠄⠀⠄⠀⠈⠀⠔⠀⢾⠀⢸⣷⢺⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣻⣿⠀⠀⠸⣷⣿⣧⡿⢿⠿⠟⠀⠂⠀⠀⠈⠀⠁⠀⠀⠀⠈⠀⠀⢠⠄⠀⢺⠀⢸⣿⣸⣿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣽⣿⠀⠀⠀⢀⣿⣟⠀⣠⢶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣔⣋⣴⢀⢸⠀⢸⣿⣿⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⠀⠀⠀⣿⣿⣿⣴⡷⡿⠃⠀⡠⠀⣀⣀⢀⡔⢀⠴⠤⣴⠾⣫⡿⠽⠭⠼⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠿⠿⢿⠿⠀⠀⠀⠀⢿⣿⣏⣬⣤⣶⣶⣯⣽⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⢛⡛⠛⠻⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠉⠙⢧⠐⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

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

⠀⢀⠤⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⠒⠒⡆
⢀⠟⣄⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠈⣞⡵⡊⠉⠑⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⢘⢾⡀⠻⡀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⣞⣀⣷⠀⢷⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠈⠉⠉⢳⠙⢿⠑⢕⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⢀⡕⠈⣧⠘⣇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⢸⠄⠀⢸⡆⢹⣄⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠑⠚⢁⣽⡟⢻⡍⠦⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⡞⠁⢳⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠻⣆⡈⣧⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠈⠉⠙⡧⢚⢽⡒⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⡐⠐⣇⠈⢷⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⡅⠀⠘⣆⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠙⡦⢚⣡⠀⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠐⡇⠈⢧⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀⢸⡄⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⡤⠤⠶⠒⠛⠉⠁⠉⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⢻⡈⢎⡦⢤⣒⣒⠒⠈⠉⠉⠉⠋⠉⣉⣉⣀⣤⣤⠤⠶⠶⠚⠛⠛⠛⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡸⢤⣤⣤⣀⣒⡐⠤⠬⠭⠒⠛⠋⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠟⠛⠛⠉⠉⠙⠛⠛⠿⢶⣦⣶⠾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠿⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⣸⣿⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⢠⣿⣯⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⣼⡿⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧
⠀⠀⠀⠀⢰⣿⢣⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣾⡟⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆
⠀⠀⠀⢰⣿⢧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⠀⠀⣸⣿⣾⠁⠀⠀⠀⠀⣀⣀⣤⣤⣤⢤⣤⣤⣀⠀⠀⢸⡇⠀⠀⢀⣀⣤⣤⣤⣴⣦⣤⣤⣤⣤⣀⣀⠀⠤⠀⢹⡏⣿
⠀⠀⠀⣿⣿⡏⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣶⣼⣇⣴⣾⣿⣿⣿⣿⣿⣭⣭⣭⣽⣿⣿⣿⣿⣿⣷⣦⣼⣿⢻⡇
⠀⠀⢠⣿⣿⣷⣿⠿⢟⣛⣉⣉⣭⣤⣤⣤⣤⣤⣭⣭⣽⣿⣿⣿⣿⣿⣯⣭⣤⣤⣦⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⡿⠿⠿⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠷⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠓⠄⢀⡸⠂⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⠀⠀⢢⡘⠋⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠘⠣⠀⣴⠴⠋⠀⠀⠀⡀⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⡃⠐⡀⣠⠟⠀⠀⠀⢀⠤⠊⠀⠑⠄⠀⠀⠀⠀⠀⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠮⢄⠀⢘⡬⠋⠁⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠉⢆⠠⡶⠋⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠰⢏⠁⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠑⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠠⠀⢀⡶⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⢀⡔⠁⠀⢡⣠⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⣠⠞⣇
⠀⠀⡠⠐⠺⡁⢱⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⣠⠞⠁⣠⡿
⢀⡾⠂⠄⢀⡼⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⡠⠂⠀⠀⠀⠀⠀⡠⠚⢁⣴⡾⠋
⠈⠳⣄⠺⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡴⠋⢀⣤⠿⠋
⢾⡢⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣴⠟⠁
⠀⠙⠦⡀⢌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣴⠟⠁
⠀⠀⠀⠈⠳⣕⢌⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⡠⣠⠟⠁
⠀⠀⠀⠀⠀⠈⠳⣕⢌⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⡠⣮⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢕⡄⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢠⡾⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⢔⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡈⠳⢄⡀⠀⠀⡠⠞⢁⢔⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠙⠶⠋⠀⢔⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡤⣠⠖⠁

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢠⠖⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠳⣦⣄
⠀⠀⠀⡟⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠨⣿⣿⣿⣿⣿⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⢀⣿⡇
⠀⠀⠀⡇⠀⢉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⢨⣿⣿⣿⣿⣿⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢸⣿⡇
⠀⠀⠀⡇⠀⢉⡁⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸⣿⡇
⠀⢘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⡇⠀⠉⠁⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸⣿⡇
⠀⢐⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⣿⠀⠉⠁⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⢸⣿
⠀⢐⣿⣿⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⡏⠀⠉⠁⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⣿⣿
⠀⢐⣿⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⡇⠀⠉⠁⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⣿⡏
⠀⢐⣾⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⡏⠀⠉⠉⠀⠀⠛⠚⠛⠛⠛⠛⠛⠓⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⣿⡇
⠀⢐⣶⣿⣶⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⡏⠀⠉⠉⠀⠠⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡥⠀⢠⣿⡇
⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⣻⠇
⠀⠀⠀⠘⢷⣦⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣒⣲⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⡿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⢰⡄⠶⠆⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⢀⡀⡀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠐⠟⠉⠁⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢀⡤⠤⠄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠐⠃⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢠⣄⣀⡀⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢀⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠛⠉⠁⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣶⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣄⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠋⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠠⠿⠉⠀⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⣤⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⡠⠀⣠⢀
⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠛⠐⠂⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠤⠤⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡀⢠⣄⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠢⢀⠤⠺⠃⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠰⢿⣿⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣀⣰⣿⣿⡆⠳⠂⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⢯⣽⡅⠢⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢷⣿⡽⢠⡾⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣶⡧⣤⡀⡆
⠀⠀⠀⠀⠉⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⡾⠁⠸⠇⡀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣜⢿⡄⠐⣠⣌⡅
⠀⠀⠀⠀⠤⠠⠶⠿⠆⠇⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⡀⣴⣆⣠⡀⡀⡀⡙⠿⢿⠛⠋⠀⢠⡖⠀⠋⠀⠀⠀⠀⠀⠸⠀⣀⡀⠀⠠⠆⡐⡿⣶⣷⠿⠄⠈⠆
⠀⠀⠀⠀⢄⡀⠄⣤⡀⣠⡀⠀⠀⠀⠀⢠⣿⡹⠯⠘⠓⢽⠟⠁⠰⢄⠈⢉⣀⡈⠀⠀⢣⣼⡇⠀⠀⠀⠀⠀⠀⢀⣶⢀⡟⠁⠀⠀⡀⢈⣧⣄⣁⣠⡀⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠛⡀⢧⣄⢀⡀⠸⠟⠃⠚⠀⠀⣬⣧⣟⠀⠺⠖⣤⣤⡇⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠐⢾⠋⡠⠁⠀⠀⠀⠙⣾⣏⢡⡄⠈⠢⣠⡆
⠀⠀⠀⠠⠭⠤⠀⠿⠃⢀⡶⠚⠻⣄⠁⠀⠐⠂⠀⡀⢰⡄⠹⠃⠒⠢⢀⣀⢘⡗⠀⢰⠇⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡀⣀⢀⠀⢈⠀⠂⡶⠉⣰⣿⡄
⠀⠀⠀⠀⣠⠀⠀⣤⡀⠘⠣⢤⡾⠛⠇⠀⣴⣖⣿⡷⠄⠀⠀⠀⠀⠀⠀⠅⠘⠃⠀⠀⠀⠀⠀⢀⠚⡅⢹⡄⢀⡄⠀⠀⠀⢀⣙⣏⠀⠈⢺⠞⠁⣰⠏⠉⡃
⠀⠀⠀⠀⠁⢀⠘⢭⡗⢀⢀⡞⡃⠀⠀⠀⠿⣿⣿⡷⡆⠀⠀⣀⠆⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢠⣴⡄⡾⣾⣷⣘⢿⡧⠙⢿⣡⢒⡁⠀⣸⠏⠠⣶⡄
⠀⠀⠀⠐⠛⢻⣿⠿⠃⢀⡴⠞⢘⣷⡧⢀⣺⣿⡿⠀⢖⠀⢠⡇⠀⠘⢷⡀⠀⠀⠘⠀⠀⠀⡶⠄⠀⣺⣿⣧⣿⣿⣿⣿⣿⣦⡀⠸⠇⠻⠇⡸⠁⢀⣤⣍⠁
⠀⠀⠀⠠⣤⢸⣷⣴⣄⡾⠋⣸⠃⠙⣳⣿⣿⡿⠧⠐⢠⣄⢸⡇⠀⠀⠀⣷⡀⠀⣆⠀⠀⠀⠂⠀⠈⠛⢹⣿⣿⣿⡿⢻⣿⣿⣷⣤⠀⠀⣠⢷⣦⡼⠻⠠⠆
⠀⠀⠀⢀⣀⢀⣩⣿⡟⠁⢸⠇⠀⠐⠋⢦⣿⠀⠀⡇⠀⣙⣸⡇⠀⠀⣀⣈⣁⢼⣿⢁⣶⠀⠀⠳⠀⣼⣿⣿⣿⣿⣟⠛⢻⣯⣿⣿⣁⣰⡟⣷⠟⠀⠐⡦⡄
⠀⠀⠀⠈⠛⠂⢾⠿⣧⣀⣅⡀⠀⠀⣠⣤⣶⣶⣤⣴⡟⢿⣿⡷⢀⠐⠿⢟⣧⠀⣃⢸⣿⠀⢠⣤⣾⣛⢿⡿⣿⡿⢿⣤⣾⠘⡻⠚⣿⣿⣿⣿⣄⣠⣾⣤⡀
⠀⠀⠀⠠⡶⣤⣤⣶⣿⠁⢠⣷⣶⠋⢩⣽⡏⠙⠻⢿⣿⣿⣿⠁⠀⠀⠀⢠⣿⠇⠃⣸⣿⣦⡀⠀⢨⠿⠧⠸⠛⠁⠘⣿⠏⠀⢀⣾⣻⣴⣿⣶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠫⣹⡿⠧⢾⣿⡆⢀⣾⣿⠀⠀⠀⠀⢧⠀⣿⠀⠀⠀⠀⠀⢰⡼⠚⣹⠉⣿⠇⠀⠈⠀⠀⠀⠀⠀⠀⠋⠀⠀⠞⣼⡟⠿⣾⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠠⠤⣾⣦⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹
⠀⠀⠀⠈⠉⢻⠋⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢐⣒⣿⣗⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠉⢹⠁⢉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠨⣯⣿⡿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢀⣀⣸⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠸⠿⢿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢠⣤⣼⣦⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠐⠛⢻⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢰⣶⣾⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠈⠉⣿⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣀⣿⣀⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠨⠿⣿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⢀⣀⣿⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠐⠓⣿⠚⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠠⣤⣿⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠈⠙⣿⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⡇
⠀⠀⠀⠀⠀⠹⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⢸⠛⢷⠀⠀⠀⢸⡟⢶⠀⠀⠀⢠⡞⠓⣶⠀⠀⡾⠛⡆⠀⠀⢰⡞⢳⡀⠀⢰⠖⢶⡀⠀⠀⣴⠶⡄
⠀⢀⡶⠲⠶⢾⡀⢸⣤⣤⣤⣼⠃⣸⠤⠤⠤⢴⡇⠀⣿⡤⣤⣿⠀⣧⣤⣤⣼⡇⢸⠧⠤⢼⡇⢸⣧⣤⣤⣽⠀⣧⣤⣤⣤⣤⣀
⠀⢸⡇⠀⠀⠀⣅⡼⠀⠀⠀⢸⣀⣧⠀⠀⠀⠀⣇⠀⣡⠀⠀⢠⣀⡇⠀⠀⠀⣇⣸⠀⠀⠀⡇⣸⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠙
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠘⠒⠃
⠀⢸⠇
⠀⣼
⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠙⢿⣉⣉⣉⣀⣀⣈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⡉⠉⠉⠉⠉⠛⠛⠉⢉⡉⠉⠉⠉⠉⠙⠓⠒⠒⠒⢲⡟
⠀⠀⠙⢯⣄⢡⣉⣉⣉⣉⣉⣉⣉⣉⣉⣒⣒⣒⠒⠂⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⢀⣀⣀⣀⣀⡀⢀⣀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠉⠛⠳⠶⠶⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⠤⠶⠶⠶⠶⠶⠶⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⡮⠟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⡡⠞⠉⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢚⡥⠖⠉⠀⠀⡠⠖⢻⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢞⣡⠖⠉⠀⠀⣠⠔⡫⠴⠣⢸⡀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢞⣩⠖⠋⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠹⡇⢿
⠀⠀⠀⠀⠀⢀⡤⢚⣩⠖⠋⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠒⠃⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⣶
⠀⠀⠀⡠⢞⡡⠖⠋⠀⠀⢀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡔⠀⠢⠀⡄⠀⠀⡆⢠⠔⠂⠲⡄⢰⠒⠒⣄⠰⣿
⠀⠀⣾⣴⡏⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⠉⠒⢢⠀⠋⠀⠐⡇⢸⡀⠀⠀⡇⢸⠀⠐⠁⠀⣿
⠀⠀⢿⣻⡇⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠑⠒⠋⠀⠁⠀⠀⠃⠀⠑⠒⠊⠀⠘⠀⠀⠀⢀⣿
⠀⠀⢸⣻⡇⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣰⡆⠀⣀⣁⠀⣀⣀⣀⡀⠀⣀⣀⢀⣐⣆⣀⣉⣀⣀⣀⡀
⠀⠀⠸⣿⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⣤⣾⠃
⠀⠀⠀⣷⣿⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢸⠯⡟⢩⡟⡿⠁⡿⢹⠏⣽⣩⡏⢉⣯⣽⠁⣽⣽⠏⣹⢿⡟⢈⣿⠉⢸
⠀⠀⠀⢿⣹⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢠⣯⣼⣡⣮⣼⢧⣼⣯⡿⢰⣷⡾⢠⣾⣷⡏⣼⣿⡿⣴⣷⡞⢠⠎⡋⢰⢾
⠀⠀⠀⢸⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡇⢀⠀⠀⡀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⡟⠁⢸⠒⠒⠒⠒⠒⠒⠒⠒⣦
⠀⠀⠀⢸⣿⣿⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡇⠛⠛⠐⠇⡇⠀⠀⡇⠀⠀⠀⠀⢀⠂⠀⠀⡠⠀⢸⠀⠀⡇⠰⢻⠀⠀⠀⠀⠀⠀⢠⢾⡞
⠀⠀⠀⠈⣿⣿⡀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⡥⠠⠄⠀⠀⡇⠀⠸⡇⠀⠀⠀⠀⠁⠐⠀⠌⠀⠀⣼⠀⠀⣧⠀⢸⠀⠀⠀⠀⠀⡴⣣⠋
⠀⠀⠀⠀⣿⣿⡇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣧⠀⠀⠀⠀⡇⠀⢠⡃⠀⠀⠐⣀⣀⢀⣀⣀⣀⣀⣻⡀⠀⣿⣸⣾⢀⠀⠀⢠⢾⡟⠁
⠀⠀⠀⠀⢽⣿⣧⠀⣧⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⣠⠞⡇⢀⡃⠀⠀⠀⠀⢃⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⢞⡵⢡⠟⡴⣫⠎
⠀⠀⠀⠀⢸⡿⣿⠀⢻⡀⠀⠀⠀⢀⡤⠊⢀⣴⠞⠁⢀⣔⣉⣉⠿⠁⠀⢠⡜⠉⠉⢉⡽⠃⠀⢀⠜⠁⢀⡰⢋⡜⠁⠀⠀⡰⠏⡴⢣⢞⡴⠃
⠀⠀⠀⠀⠸⣿⣽⡀⢸⠿⡄⢀⡴⠋⢀⡤⠋⠀⢀⡴⠁⣠⠝⠁⠀⢀⡴⠋⠠⢠⡴⠋⠀⢀⡼⠃⠀⣠⠞⡴⠏⠀⠀⢀⠜⢡⠞⣱⢣⠞⠁
⠀⠀⠀⠀⠀⣿⡿⡇⠈⣇⡴⠋⢀⡴⠋⠀⢀⣔⠁⢀⠔⠃⠀⠀⡴⠋⠀⠀⡠⠛⠀⠀⡤⠊⠉⢡⠞⠁⠘⠀⠀⠀⠐⠋⠐⢁⣜⡵⠃
⠀⠀⠀⠀⠀⢻⡿⣇⠀⠈⢀⡴⠋⠀⣠⠞⠁⢀⠔⠁⠀⠀⡰⠊⠁⠀⣰⠎⠀⠀⢀⣞⣁⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⣻⠟
⠀⠀⠀⠀⠀⢸⣷⣿⢀⣴⠟⠁⣠⣞⡁⣀⡜⣁⣀⠀⣠⡊⣀⡀⣠⣞⡁⠀⢀⡴⠃⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⡿⠁
⠀⠀⠀⠀⠀⠘⠷⢿⣯⣥⢤⣤⣭⣭⡭⣭⣭⣭⣭⣭⡭⣭⣭⣭⣉⣉⣉⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⡴⣣⠏
⠀⠀⠀⠀⠀⠀⠀⠻⠯⠯⢿⠻⢡⣟⣼⣟⢾⣿⣿⣿⣰⣿⣿⣿⣽⣿⣿⣯⣾⣿⣿⣿⣳⣿⣿⣿⣿⣿⣽⣿⣿⠿⠁

⠀⠀⠀⠀⠀⠀⣠⠴⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⣄
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠓⢄
⠀⠀⠀⠀⠀⣿⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠓⣄
⠀⠀⠀⠀⠀⣿⠲⢾⣉⡷⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠂⠀⠳⣄⡀⠀⠈⠑⣄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⣻⣀⣠⠶⣄⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣿⠀⠙⠒⠋⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡠⢾⣷⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⣼⣿⣦⡙⠿⡿⣿⠛⠁
⠀⠀⠀⠀⠀⣿⠒⢺⣅⡷⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣴⣍⠻⣿⣿⣷⣤⠌
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣄⣮⡿⣿⣷⣌⠻⡷⠁
⠀⠀⠀⠀⠀⣿⣀⣠⠶⣄⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡠⣾⣿⡿⣿⣾⡝⠿⠃
⠀⠀⠀⠀⠀⣿⠀⠈⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⣾⡿⣻⣾⣿⣫⠞
⠀⠀⠀⠀⠀⣿⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⣿⣿⢿⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⣾⠒⠾⣉⡷⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⢀⣴⣽⣿⢟⣵⣿⢿⡝⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣟⣵⣿⣿⡿⠋
⠀⠀⠀⠀⠀⣿⣀⣠⠶⣄⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣾⣿⣞⡋⢿⡿⡫⠋⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠈⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡬⠉⠁⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣿⡿⠟⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣻⠒⠺⣥⡷⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢂⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢭⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⡄
⠀⠀⠀⠀⢀⡞⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⣆
⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⣴⣡⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣵

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠤⠤⠶⠶⠶⠒⠒⠒⠒⠚⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⢹⣧⢼⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠏⣤⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣧⡀
⠀⠀⠀⠀⠀⠀⡞⠉⡏⠱⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢷⢹
⠀⠀⠀⠀⠀⠀⠷⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠤⢾⢸
⠀⠀⠀⠀⠀⠀⣠⣈⡁⣘⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢸⢸
⠀⠀⠀⠀⠀⠰⡏⠀⡇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠙⢶⠧⠠⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⡼⠓⣞⠐⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⢸⣼⠃
⠀⠀⠀⠀⠀⠀⠻⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣿⣄⣸⣇
⠀⠀⠀⠀⠀⠀⢀⣀⣛⣚⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠉⠉⠉⠁⠀⠈⠀⠸
⠀⠀⠀⠀⠀⠀⣏⠀⣿⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠙⢳⣿⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣴⣦⡤⢾⣿⣿⠟⠻⡾
⠀⠀⠀⠀⠀⠀⣼⠃⣷⠐⠾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⣧
⠀⠀⠀⠀⠀⠀⠻⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⡄⣛⣒⣻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⣿
⠀⠀⠀⠀⠀⠀⣟⠁⢸⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠙⢲⣾⣄⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡁⠀⡿
⠀⠀⠀⠀⠀⠀⢰⠟⢲⡆⠶⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⢠⠃
⠀⠀⠀⠀⠀⠀⠸⣦⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡋
⠀⠀⠀⠀⠀⠀⠀⢔⣛⣁⣛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⣤⢤⡤⢤⣾⢻⡄
⠀⠀⠀⠀⠀⠀⣼⠉⠉⣯⣭⣤⣤⣤⣤⡶⣶⣶⣶⡲⡾⣟⢿⡻⣯⣹⣽⣍⣪⣋⣱⣽⣮⡶⠼⠶⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠘⢶⣤⣘⣿⣿⣿⡿⠿⠿⠾⠟⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⠤⠒⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣲⠝⠋⢧⠀⠀⡼⠛⠁⠀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣺⠕⠋⢀⡠⢲⡸⠔⠈⢀⠠⠒⠉⠀⠠⡇
⠀⠀⠀⠀⠀⠀⢀⡠⣒⠕⠊⠀⡠⠒⠉⢈⠿⡄⠀⠀⠁⠀⡠⠔⠀⠀⣷
⠀⠀⠀⢀⠠⣂⠕⠊⠀⣠⠔⠋⠀⠀⠀⣼⠤⠟⠀⠠⠐⠁⢀⠠⠂⠀⣿
⠀⢠⣒⠕⠊⠀⡠⠔⠈⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⢀⠄⠊⠁⢀⡄⢸⢸
⠀⢸⣿⠀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠊⠀⢀⠤⠂⠁⢀⠇⢸
⠀⠈⣿⡄⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⠀⠊⠀⡠⠔⠈⡜⠀⢸
⠀⠀⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠉⠒⠦⣄⡁⢀⠤⢼⡁⠀⢸
⠀⠀⢿⣧⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢙⠃⠀⠀⠉⠢⣼
⠀⠀⢸⣿⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⡴⢛⠖⠒⡢⠒⣠⠆⠀⠀⠀⠀⣠⡟
⠀⠀⢸⣿⡄⢃⠀⠀⠀⠀⠀⠀⠀⡠⢾⠀⠀⠀⢀⠎⠀⠀⢀⡰⠊⠴⠁⠠⠞⢡⡞⠁⠀⠀⠀⢀⣼⠏
⠀⠀⠈⣿⡇⢸⠀⠀⠀⠀⠀⡠⠊⣠⢾⠀⠀⢠⠋⠀⢀⡤⠋⡱⠇⣀⠚⢁⣝⡏⠂⢀⡐⡶⣴⡿⠃
⠀⠀⠀⣿⣷⠸⡄⠀⠀⡠⠊⡰⠊⠁⢼⠀⢠⠃⢀⡔⠋⡠⠋⢀⠔⠁⡠⣫⠊⠀⠠⢊⢞⣾⠞
⠀⠀⠀⢹⣿⠀⣟⡴⢋⡴⠋⣀⠀⣁⢾⢀⢫⠔⠉⡠⠏⠀⡰⠃⡠⠊⠘⠁⠀⠔⠑⣡⡿⠃
⠀⠀⠀⢸⣿⡆⢀⡴⠋⡠⠞⣡⠜⠁⢸⠔⠉⢠⠞⠀⢠⠎⢠⠜⠁⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⡿⣧⣋⣀⣘⣓⣒⣒⣒⣐⣓⣒⣒⡓⠀⠐⠓⠔⠁⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠛⠓⠟⠱⠷⠟⠾⠿⠿⠿⠿⠿⠿⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋

⠀⠀⠀⠠⣶⡀⠀⠀⠀⠀⠀⠀⢻⠦⠊⠁⠀⠀⠀⠀⠀⠤⠾⠄⠔⠈⠀⠀⠐⠣⠤⠿⣀⢤⠀⠀⠀⠀⠙⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢒⡀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⢀⡀⠤⠤⢄⡤⢀⠀⠀⠀⠀⠀⢂
⠀⠈⠀⠙⠈⢁⠰⢀⠀⠀⠀⢂⡄⠀⠐⠀⡀⠀⠀⠤⢭⣈⠠⠀⠦⠀⠀⢰⠀⠃⠔⠂⡀⠹⠀⠀⠀⠀⣮⢴⠆
⠀⠀⠑⠂⡤⠈⢠⡀⠀⠀⠀⠀⣂⡄⢀⠰⠃⠀⠀⠀⢰⠀⡒⠄⠀⠀⠀⠀⠀⣴⡀⡠⠇⠀⠀⠀⠀⠀⢈⡈
⠀⠘⠃⠊⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⠀⠀⠀⠀⠀⠀⢐⠣⣄⠀⠀⠀⠀⠰⠉⢃⡉⠋⡕⢂⠀⠀⠠⠀⠮⢠⠀⡤⠀⠀⠀⡀⠒⠀⠢⠄⠀⠀⠀⠀⠁
⢙⠊⠀⠀⠀⠀⠀⠀⠘⠁⠇⠀⠀⠀⠀⠈⢀⡆⠈⢁⠀⠋⠀⠀⠌⠰⠀⠈⡀⠄⠀⠀⠈⠀⣁⡚⢊⠈⠁⠀⢰⡀
⠶⠀⠀⠀⠀⠀⠁⠀⠀⣶⠄⠀⠀⠀⠀⠀⠀⠑⠶⡟⠀⠀⠀⠀⠘⠢⠸⠧⠤⠄⠀⠀⠀⠈⠀⠀⠋⠀⠀⠀⠀⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⢂⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⡀⠤⠤⢤⡴⡀
⠀⠙⠀⠋⠈⢁⠴⠀⠀⠀⠐⢎⡤⠀⠀⠀⠀⠀⠀⠬⢥⢀⠠⠀⠄⠀⠀⠀⠀⢾⢰⡆⠀⠀⠀⠀⢶⢈⠃⠔⢂⠀⠩
⠀⠈⠐⢢⡄⠉⣠⠁⠀⠀⠁⠀⣀⡄⢀⠰⠃⠀⠀⠀⢰⠰⡒⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠠⠀⠀⠀⠀⣴⡀⡠⠇
⠀⠘⠓⠊⠁⠉⠁⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠠
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠄⣂⠀⠀⠀⠀⢐⠢⡉⢉⡑⠋⠴⠀⠀⠀⣄⠂⠀⠀⠀⠀⠀⠀⠐⠂⠴⠂⡆⠠⠄⠀⠀⠀⠔⠂⠐
⠀⠀⠀⠈⠇⠋⠀⠀⠀⠀⠙⠀⢀⠁⢀⡄⠈⠀⠀⠤⠀⡚⢠⠀⠀⢀⠄⠀⠌⡑⠆⠀⢀⠠⠀⠀⠈⠉⢑⢚⣈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣛⡛⠛⠾⠟⠲⠶⠶⠶⠶⠤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠁⠀⠀⠀⠀⠈⠉⠉⠀⠀⠐⠒⠂⠀⠒⠂⠀⠠⠭⠭⠍⠉⠉⠙⡛⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⠀⢀⢠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⢸⢸⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠓⣒⠂⠤⠄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⡉⠐⣀⠀⠂⡀⠉⢉⠀⠒⠂⠈⠭⠅⢐⣒⠀⠠⠤⠀⡀⡀⠀⠀⠀⠀⠀⢈⢸⢠
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠂⠀⠰⢀⡀⠐⡀⠐⠂⠤⠈⡠⠀⡁⠰⣀⠠⠂⠀⡈⠁⢀⠒⠒⠠⠭⠀⢐⣒⠚⢸⣸
⠀⠀⠀⠀⣠⣔⠉⠤⠀⠉⠄⠈⠀⠂⠀⠆⡀⠄⠈⠀⠀⠁⠴⠀⠠⠀⢁⠀⠒⠀⠐⠀⢀⠀⢐⡂⠀⠒⣀⡤⢾⡭⠜
⠀⠀⠀⠀⠧⠤⢈⣉⠐⠒⠒⠡⠤⠁⣐⣀⠀⠀⠀⢀⠀⠀⠠⠂⠠⠀⠀⠄⣀⠁⢐⠀⠐⠂⣁⠤⠒⣉⠼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠒⠠⠤⠄⣈⣉⡁⠒⠒⠠⠤⠅⣀⣀⠤⠀⠠⠀⣀⠤⠒⣉⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠠⠤⠤⣈⣉⢐⣋⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⢀⣿⣡⠀⢰⣶⡄⢰⡖⣴⡀⢰⡯⡄⠀⣶⣆⢠⡖⢴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⣀⣸⡏⢀⣈⣹⠀⣈⣹⡏⢈⣁⣹⠃⣈⣉⡏⢀⣉⣿⢉⣉⣹⠃⣷⡄⠀⠀⡠⡤⠢⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠛⠚⠁⠘⠟⠁⠀⠛⠋⠀⠘⠛⠁⠀⠛⠏⠀⠘⠛⠁⠘⠟⠉⠀⣿⡃⠀⣰⢷⠃⣰⠃
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠤⠤⣤⣤⣤⢤⣼⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣰⣣⠃⢠⡃
⠀⠀⠀⠀⠀⢸⣿⣱⣿⡥⢤⣾⣵⣆⣤⣖⣽⣷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⠃⢠⣟⣝⣽⡟⢛⣶⡀
⠀⠀⠀⠀⠀⢸⣿⡿⣿⠀⠀⠀⠀⠀⠀⠀⠸⣷⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⠃⢀⠎⠍⢉⠙⢻⠜⣹⡇
⠀⠀⠀⠀⠀⢸⣿⡵⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⢀⡞⠀⠀⠀⡂⢸⡏⢾⡇
⠀⠀⠀⠀⠀⢸⣿⡴⢿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⡜⠀⠀⠀⠀⠃⢸⡇⣻⠃
⠀⠀⠀⠀⠀⢸⣿⢁⣸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡞⠀⠀⠀⠀⠀⠄⢸⡶⣿
⠀⠀⠀⠀⠀⢸⣿⣱⣻⡆⠀⠀⠀⠀⠀⠀⠀⣿⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣾⠀⠀⠀⠀⠀⠀⠀⣿⢔⣿
⠀⠀⠀⠀⠀⠈⣿⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠐⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⣿⣪⡏
⠀⠀⠀⠀⠀⠀⣿⢣⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡞⢽⡇⠀⠀⠀⠀⠀⠀⢰⣏⢼⡇
⠀⠀⠀⠀⠀⠀⣿⣿⢧⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠋⠀⣼⡇⠀⠀⠀⠀⠀⠀⢸⡣⣺⠃
⠀⠀⠀⠀⠀⠀⢹⣧⡇⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣗⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡇⠀⠀⠀⠀⠀⠀⣾⣔⣿
⠀⠀⠀⠀⠀⠀⢸⡟⡴⣻⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢧⣀⠸⡏⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠉⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡟⣱⢽⡀⡀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠙⢳⣿⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣠⣀⣍⡁⠀⠀⠀⠀⠀⢨⣿⣾⡏
⠀⠀⠀⠀⠀⠀⢸⣿⢣⡾⡇⢁⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠽⠿⠟⠛⠻⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⢸⡿⣻⠇
⠀⠀⠀⠀⠀⠀⠸⣿⠝⡔⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠌⡰⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣴⣿⡶⣫⡿⡩⠫⣩⢟⣟⢖⣒⡶⡲⠖⡶⢖⢞⢛⢟⡿⣋⢛⣻⢏⠙⣿⣟⣻⣻⡿⡻⢛⢛⣟⣻⢞⣷⣿⠟⣻⢟⠕⣿
⣶⠒⠚⠛⠛⠋⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠛⠓⠓⠓⠓⠛⠒⠒⠛⠓⠒⠚⠻⠿⠶⠿⠾⠴⠵⠿⠭⠴⣯⣽⣥⠾⠥⠮⠾⠧⠤⠴⠤⠤⠤⠤⣤
⠙⠧⢤⢤⣠⣄⣠⣀⣠⣄⣄⣄⣀⣄⣄⣀⣠⣤⣄⣄⣠⣄⣄⣀⣤⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣄⣀⣄⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⢻⣿⡟⠛⠛⠛⠛⠛⠛⣿⣿⠛⠛⠛⠛⠛⠛⢿⣿⠛⠛⠛⠛⠛⠛⢻⣿⡟⠛⠛⠛⠛⠛⠛⣿⡿⠛⠛
⣿⡟⢹⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⣿⡇
⣿⠀⢸⡇⠀⠀⠀⠀⣠⣤⣤⣾⡿⠀⠀⠀⣀⣤⣤⣼⣿⠃⠀⠀⢀⣤⣤⣤⣿⠏⠀⠀⠀⣠⣤⣤⣾⡿⠀⠀⠀⣠⣤⣤⣼⣿⠁
⣿⠀⢸⡇⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠠⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⡯⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⢰⣿⣿⣿⣿⠇
⣿⠀⢸⡇⠀⠀⠀⠀⠛⠿⠿⠋⠀⠀⠀⠀⠙⠿⠿⠛⠀⠀⠀⠀⠘⠻⠿⠟⠁⠀⠀⠀⠈⠻⠿⠟⠃⠀⠀⠀⠀⠛⠿⠿⠋
⣿⠀⢸⡇
⣿⠀⢸⡇
⣿⠀⢸⡇
⣿⠀⢸⡇
⣿⠀⢸⡇
⣿⠀⢸⡇
⣿⠀⢸⡇
⣿⠀⢸⡇
⣿⠀⢸⡇
⣿⠀⢸⡇
⣿⠀⢸⡇
⣿⠀⢸⡇
⣿⠀⢸⡇
⣿⠀⢸⡇
⣿⠀⢸⡇
⣿⠀⢸⡇
⣿⠀⢸⡇
⣿⠀⢸⡇
⣿⠀⢸⡇
⣿⠀⢸⡇
⣿⠀⢸⡇
⣿⠀⢸⡇
⣿⠀⢸⡇
⣿⠀⢸⡇
⣿⠀⢸⡇
⣿⠀⢸⡇
⣿⠀⢸⡇
⣿⠀⢸⡇
⣿⠀⢸⡇
⣿⠀⢸⡇
⣿⠀⣸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⡿⣀⣉⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣠⣤⣾

⠀⠀⠀⠀⢀⣀⣀⣰⣴⣀⣰⣴⣄⣰⣶⣆⣠⣦⣦⣠⣦⣦⣀⣶⣶⣀⣶⣶⣀⣰⣴⣀⣰⣴⣆⣀⡀
⠀⠀⠀⠀⢸⠋⠉⢹⣹⠁⢸⣿⡇⢸⣿⡇⢈⣟⡇⠀⣯⣿⠀⣿⣿⠀⣻⣿⠁⢹⣹⠉⢹⣿⡏⠉⣿⣆
⠀⠀⠀⠀⢸⠀⠀⠿⠿⠃⠸⠿⠇⠸⠿⠇⠸⠿⠟⠘⠿⠿⠀⠿⠿⠀⠿⠿⠀⠻⠿⠃⠸⠿⠇⠀⣿⣿
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⣿⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿
⠀⠀⠀⠀⠻⠯⠭⠭⢭⣭⣭⣭⣭⠭⠭⠭⠭⠽⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠗⠒⠚⠛⠛

⠀⠁

⠀⠀⠀⠀⠀⣴⢾⣿⣙⢉⡽⣻⠟⣉⢟⢟⢛⡟⡻⣛⣻⢟⢟⢿⠟⡒⠒⡲⢚⣻⡿⣻⢛⡛⣻⠟⢛⠖⢒⢶⠶⠶⠶⠶⠶⣤⣤⣄
⠀⠀⠀⠀⢀⣿⣾⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⢛⠓⠛⠚⠾⢯⣤⢾⡇
⠀⠀⠀⠀⢸⣿⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⡧⣸⡇
⠀⠀⠀⠀⠀⣿⣧⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡟⢹⡇
⠀⠀⠀⠀⠀⣿⣇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⢿⠁
⠀⠀⠀⠀⠀⣿⡷⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿
⠀⠀⠀⠀⠀⣿⣷⣺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⡿
⠀⠀⠀⠀⠀⢸⣷⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢜⡇
⠀⠀⠀⠀⠀⢸⡿⣗⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⣸⡇
⠀⠀⠀⠀⠀⢸⣷⢃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣻
⠀⠀⠀⠀⠀⠐⣿⢫⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣿
⠀⠀⠀⠀⠀⠀⣿⢏⣼⡆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⡏
⠀⠀⠀⠀⠀⠀⣿⡞⡰⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⡇
⠀⠀⠀⠀⠀⠀⣿⡜⣱⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡇
⠀⠀⠀⠀⠀⠀⢸⣗⣿⣖⣽⣏⢏⣩⣝⣿⢟⣓⢖⠖⢒⠖⢟⠟⢿⡟⡛⣛⠝⣩⣽⢿⡯⣿⢟⠝⢛⣿⣻⢟⣷⣿⢋⡿⡫⣺⠁
⣾⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠚⠖⠷⠿⠾⠷⠿⠶⠶⠛⠒⠒⠒⠒⠒⠲⣶
⠈⠓⠒⠶⠶⠤⠦⠴⠦⠦⠦⠴⠶⠤⠤⠶⠶⠦⠶⠴⠦⠴⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣄⣀⣀⡾
⣷⣿⡿⠿⠿⠿⣶⣶⣶⣷⣶⣶⣶⣶⣾⣾⣿⣿⣿⣿⣷⣷⣷⣷⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠿⠿⠿⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣠⣀⣤⣀⣄⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠈⠉⠁⠈⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡇⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⢸⣿⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⡇⠀⠀⠀⠀⠀⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⡇⠀⠀⠀⠀⠀⣿⣿⣤⣤⣤⣤⣤⡀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢻⡆⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠲⠶⠶⠶⠆⣿⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⣤⣤⣤⣤⣿⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣏⠙⠛⠋⣻⣿⡏⠀⠀⠀⠀⢀⣼⠇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣆⠀⣼⣿⠟⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⠎⠉⠒⢲⠖⠉⢑⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠁⠛⠀⢠⠇⠀⡴⠁⠀⠀⠈⠉⠓⠒⠠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⠃⠛⠀⢀⡏⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⢛⣄⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⡎⠀⠁⠀⠀⡇⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡜⠘⠃⠀⠀⢰⠁⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡸⣀⠀⠀⠀⢀⠆⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢰⠁⠉⠀⠀⣠⡮⡆⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠟⠗⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢀⡎
⠀⠀⠀⠀⠀⠀⠂⢀⠏⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡼
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢰⠃
⠀⠀⠀⠀⠀⠘⢠⢡⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⠏
⠀⠀⠀⠀⠀⠀⠮⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡞
⠀⠀⠀⠀⠀⠀⠀⡞⠉⠐⠢⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⢄⣀⠀⠀⠉⠑⠒⠢⠤⣄⣀⡀⠀⠀⠀⠀⠀⢠⠃⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠠⠤⣀⣀⡀⠀⠈⠉⠉⠓⠒⠒⠁⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠦⠤⠤⣀⣠⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⡿⠿⠿⠿⠿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀
⠀⠀⠀⠀⠀⢠⣿⠏⠀⢀⣴⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⣾⡟⠀⠀⡾⠁⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⢶⣦⣄⠀⠀⠀⢿⣧
⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⢠⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⢸⣿⡀
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⣦⡀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠛⠿⠋⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⣠⣶⣶⣶⣦⣀⣀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⡀⠠⣿⡏⠀⠀⠙⠿⠛⠛⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠃⠀⢻⣷⡀⠀⠀⠀⠀⠀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣹⣿⣄⠀⣀⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣶⣦⣤⣤⣤⣤⣤⣤⣴⣶⡿⠟⠛⢿⣿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣶⣆⢠⡖⡄⠀⣶⣦⠀⣶⢰⡀⢰⡖⡄⠀⣶⡄⢰⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇⢀⣈⡇⢀⣀⡏⢀⣁⣿⠁⣉⣹⠏⢉⣉⠏⢉⣉⡏⣉⣉⡟⢹⡆⠀⠀⡠⠤⠲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠘⠋⠀⠘⠓⠀⠈⠓⠀⠀⠛⠋⠀⠙⠋⠀⠙⠃⠁⠙⠋⠀⢸⡇⠀⣠⠳⠁⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⡶⠶⢶⠶⠶⠶⢶⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣴⣧⠂⣰⣅⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣴⣿⠒⠲⠿⠾⠶⠾⠵⣿⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⠃⢠⣯⣮⣿⡋⢹⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⣿⠀⠀⠀⠀⠀⠀⠀⣿⣸⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠃⢠⠇⠂⠠⠀⣿⡁⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⣿⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣿⠃⢀⠏⠀⠀⠀⡅⢸⠖⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢹⠀⠀⠀⠀⠀⠀⠀⢸⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⢀⠞⠀⠀⠀⠀⠀⣼⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⣾⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢀⠎⠀⠀⠀⠀⠀⠀⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣻⡆⠀⠀⠀⠀⠀⠀⢸⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⡘⢀⣾⠀⠀⠀⠀⠀⠀⠀⣯⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣼⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢡⢾⣿⠀⠀⠀⠀⠀⠀⢸⠗⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣯⡇⠀⠀⠀⠀⠀⠀⠸⣿⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠋⢀⣿⠀⠀⠀⠀⠀⠀⣸⡃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡇⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⡴⣻⠀⠀⠀⠀⠀⠀⠀⣿⣿⢧⣀⠸⡇⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⣱⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⡡⢻⠀⡀⠀⠀⠀⠀⠀⢻⣿⣄⣙⢷⣷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣸⣿⠀⠀⠀⠀⠀⢰⣯⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢡⢽⡆⠀⠀⠀⠀⠀⠀⠘⠻⠻⠿⠿⠿⠿⠿⠏⠿⠻⠽⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⢸⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢊⣜⠷⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⢗⡿
⠀⠀⠀⣀⣀⣀⣀⣀⣀⣻⣾⣿⣯⣾⣧⣪⣠⣞⣽⣟⣩⣝⣝⣉⣝⣍⣍⣉⣽⣫⣊⣯⣋⣨⣿⣫⢟⡽⡫⠈⢍⠽⠭⣫⣽⡿⢋⡼⡩⢪⡇
⠀⠀⠸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠓⠛⠛⠛⠛⠛⠋⠉⠙⠛⠛⠛⣶
⠀⠀⠀⠈⠛⠛⠓⠛⠒⠶⠾⠷⠶⠶⠶⠶⠶⠶⠓⠒⠓⠚⠓⠒⠒⠒⠒⠶⠶⠶⠶⠴⠶⠶⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠴⠴⠴⠴⠤⣤⣤⣤⣤⣄⣤⣴⠏

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

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠠⠤⠤⠤⠤⠤⠤⠐⠒⠒⠒⠒⠒⠒⠒⠈⠉⠉⠙⡀
⠀⠀⠀⠀⠀⢰⠋⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢂⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠸⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡤⠤⠤⠤⠶⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠐⠈⠉⠉⠉⠉⠁⠀⢀⣀⣀⣀⠤⠄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸⠀⠀⠀⠀⠀⠠⠤⠤⠴⠒⠒⠒⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⡇⠀⠀⠀⢀⣀⣀⣀⡤⠤⠤⠤⠤⠔⠒⠒⠒⠈⠉⠉⠉⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⡠⠒⠉⠉⡭⠭⠥⠄⠀⠀⠀⠀⠀⠀⣀⣀⣀⠠⠤⠴⠶⠶⠒⣒⣛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣧⠤⠤⢶⡇⠀⠀⢸⠬⠭⠭⠭⠭⠤⠐⠒⠒⠒⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢩⣃⠔⢆⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠂⢈⣉⣁⠐⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡴⠞⠋⠉⠉⠉⠙⠳⢦⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⣰⠟⠁⠰⣐⠀⠀⠀⠀⠀⠀⠙⣦⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣰⠏⠀⠀⠀⠀⠀⠐⣉⠂⠀⠀⠀⠘⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢰⠏⠀⣀⠀⠐⠤⠃⠀⣀⠀⠀⠭⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠉⠉⠉⢠⡟⠀⠈⠒⠁⠀⠠⡀⠀⠢⠃⠀⠀⠀⣼⠃⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢀⡿⠀⠀⢀⠤⡄⠀⠒⠁⠀⡤⡄⠀⢀⡼⠁⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⢀⡾⠡⣒⠄⠀⠉⠀⠀⡔⡄⠀⠁⠁⣠⠞⢡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⣆⠆⠀⠈⠀⠀⢀⡴⠋⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⡟⣁⠀⠐⠍⠆⠀⢀⠀⠀⠎⠆⣠⠞⠁⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢀⣴⠋⠀⠦⠃⠀⣀⡀⠀⠣⠃⠀⠀⣴⠋⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣠⡞⠁⠀⠀⡤⡀⠀⠒⠃⠀⡠⡀⢀⡾⠁⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⣴⠋⠀⡔⡄⠀⠑⠁⠀⡄⡄⠀⠑⠁⡼⠁⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⢀⠜⢀⡾⠁⠀⠀⠈⠀⠀⡖⡆⠀⠉⠁⠀⠀⣼⠁⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⡌⢠⡟⠁⢀⠀⠀⢎⠆⠀⠀⠀⠀⢆⡢⠀⣰⠃⣀⣀⣀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠧⠃⠀⢀⡀⠀⠯⠅⠀⠀⠀⢰⠇⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠇⢰⢃⠀⠀⠀⠀⠣⠂⠀⢀⡀⠀⠀⢰⠏⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠪⡢⡀⠀⠀⠀⠀⠀⠑⠊⠀⣰⠋⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡉⠪⢒⠤⠀⢀⠀⣠⡴⠞⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠌⣉⣁⡀⠡⠄⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣴⠛⠉⠛⠳⣤⠀⣠⠞⠛⠉⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣷⠀⠀⣸⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠶⠖⠒⠲⠶⠶⠦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣃⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⣠⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠙⣦
⠀⠀⢠⡞⠁⠀⠀⠀⣠⠞⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠙⠳⡄⠀⠀⠀⠈⢳⡀
⠀⢠⡟⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⢷⡀
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧
⢸⠛⢶⡀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠉⠉⢷⠀⠀⠀⠀⠀⠀⠀⣴⠋⢹⡆
⣾⠀⠀⠙⣦⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⣠⡞⠁⠀⠘⣇
⣿⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠴⠶⠶⠒⠒⠒⠶⠶⠤⢤⣤⣄⣀⣀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣿
⣿⣄⠀⠀⠀⠀⠈⣷⠖⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⢲⡟⠀⠀⠀⠀⢀⡼⣿
⢿⠙⢧⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣴⠎⢰⡇
⠸⡇⠀⠙⢦⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⡞⠁⠀⣼⠃
⠀⢻⡀⠀⠈⠻⣆⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡴⠋⠀⠀⢠⡏
⠀⠈⢷⡀⠀⠀⠈⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⢠⡟
⠀⠀⠈⢳⡄⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠙⣦⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⡀⠀⠀⠉⠛⠳⠶⠦⣤⣤⣤⣤⣤⡤⠶⠶⠛⠋⠁⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠞⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣽⣿⣷⣶⣶⣶⣾⣿⣯⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⡿⢏⡉⠑⢫⣢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⢶⠛⢍⠙⢻⣦⠚⡻⡇⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣲⣯⣭⣛⢍⢹⣢⢳⢉⣿⡘⠻⠃⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⣭⣶⢭⣿⣇⣈⣯⡏⢰⣾⡜⠟⠁⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠾⣭⠲⠶⢌⢸⡤⢯⡇⣾⣷⠀⠉⠀⠀⠀⠀⠀⣠⠔⢤⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⡠⢴⣛⠿⣽⢛⠺⣹⡮⠞⣫⡠⣿⡧⠀⠀⠀⠀⠀⠀⠀⣠⠊⢠⠖⡩⠦⡄⠀⠀⠀⠑⢄
⠀⠀⠀⢀⡠⢞⣚⣛⠣⡟⢿⢻⠖⣿⡇⠘⠛⠃⠀⠀⠀⠀⠀⠀⣀⠴⠊⡡⠟⢄⠉⣠⠞⣁⢤⡀⠀⠀⠀⠑⢄
⠀⠀⣰⣫⡾⠁⠸⠩⣱⡹⢋⣿⡌⠛⠋⠀⠀⠀⠀⠀⠀⢀⡠⢼⣥⠴⢤⡙⢦⠈⠳⣌⠛⣡⠔⠁⠀⠀⠀⠀⠀⠑⣄
⠀⢠⣡⠋⡧⣀⣴⠋⢱⣷⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⡰⢁⡴⠢⣌⠳⡀⠙⢦⣙⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠈⡇⠀⠉⢈⡫⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣍⠲⣧⠘⣆⠀⢘⡆⢳⠀⠀⠉⡤⠒⠒⠒⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠘⡕⢀⠀⠱⡕⣕⢝⢦⠀⠀⠀⠀⢰⠒⠲⠤⣌⣳⡌⠳⣌⣓⣉⡠⠋⠀⡤⢄⠑⢄⠻⣍⠉⢢⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠈⠮⠋⠳⡜⢮⠳⡐⠑⢄⠀⠀⠈⠳⣄⠲⣦⠤⠬⣄⡸⠂⢀⣠⣄⡀⣧⢀⡙⠦⣱⣌⠓⠋⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⢠⡙⢌⢦⠳⡜⢦⡠⣳⡀⠀⠀⠈⠳⣌⢣⠀⠀⠀⠰⡁⢴⣢⠈⣿⠀⢡⠴⢌⣹⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠙⢦⡳⡕⢜⢦⡙⢮⡻⣦⡀⠀⠀⠈⠁⠀⠀⠀⠀⠙⢦⠉⢰⠋⢧⣸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠙⢎⢢⡑⢝⢦⡙⢦⡙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢦⡱⣝⢦⡑⢌⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢮⡳⡙⢦⡳⣝⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢮⡣⡙⢮⡳⣕⣝⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡲⠟⡡⢞⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢮⡳⣝⢮⡻⣷⣝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⢋⠴⣪⢞⡡⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢮⡳⣝⢮⡻⢝⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡺⣙⡥⢚⡥⣊⠔⢋⢔⡩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢝⡪⡑⢝⢦⡑⢮⡻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⢵⠾⢋⡥⢖⡡⠚⡡⢖⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡲⣝⢯⡲⣍⠲⣝⣕⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠿⣫⠔⢁⠔⢉⡴⢋⠔⢋⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⡑⢝⣮⡑⢦⡙⠷⣮⡳⢄⠀⠀⠀⢀⡠⣪⠞⡡⠚⣁⠔⢁⠔⣡⢞⡥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⠮⡳⢌⠲⢌⠙⢷⣭⣢⣖⡥⢊⡴⢋⡤⠊⣡⢖⡡⢊⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢌⡳⣝⠢⡙⠦⣙⠦⡭⠖⡡⠖⣁⠔⢋⠴⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡑⢮⡳⢌⡓⠶⢋⡤⠚⡡⢞⡥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢭⡲⢌⡒⣩⠔⣫⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣍⡥⠚⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠤⠖⠒⠊⠉⠉⠉⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠔⠒⠚⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⠂
⠀⠀⠀⠀⣀⣠⣤⠤⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠶⠖⠒⠂⠉⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀
⠀⠀⡀⣨⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⢀⠤⠤⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⢀⣿⠆⡀
⠀⠀⣧⡏⠉⣿⡆⠀⠒⠂⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⣸⣿⣿⣴⡥⠎⣠
⠀⠀⠸⡿⡄⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠸⠿⣟⡿⢁⣾⡥⢂⡀
⠀⠀⠀⠀⠘⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠐⠋⢷⣩⣿⡿⠯⡤
⠀⠀⠀⠀⠀⠸⡄⠈⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠐⠿⢛⡡⠔⣺⣷⣞⣃
⠀⠀⠀⠀⠀⠀⠹⡄⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠒⣋⣡⣤⣋⡉
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠈⠥⣴⣿⣯⡥
⠀⠀⢠⣤⣿⡿⠀⠀⠹⡷⡀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⢋⡶⠆
⠀⢤⣬⣿⡿⠟⠃⠀⠀⠹⡇⠐⡱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⣿⣷
⠀⠀⣭⣿⣿⣿⣿⠀⠀⢀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠉⠁
⠀⠀⢺⣿⣿⠿⢿⡧⣀⠀⠂⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⢀⣈⡥⠶⢞⡻⣛⣩⠀⠀⢡⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠉⠡⠶⠛⣉⣵⢾⣵⣾⣀⠀⢳⡻⣎⡔⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠐⣋⣭⣴⣯⠟⢛⡥⠀⠀⢀⠙⣜⠄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⡧
⠀⠀⠀⠀⠉⣭⣿⣻⣿⡵⣞⣡⢖⣂⠁⠹⡆⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠒⠈⠉⣺⡟⠁
⠀⠀⠀⠀⠀⠀⣻⡿⢟⣩⠟⣃⣾⣷⡀⠠⡹⡄⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠈⠁⡉⠀⠀⢚⣁⣠⣶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠐⠻⠿⣿⣿⣴⣾⣟⠛⣄⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢐⣀⣀⣨⠤⠴⠒⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠙⠉⠀⠹⣧⠀⠻⣤⢤⡴⠖⠒⠋⠁⠀⠀⠀⣀⣠⣤⡴⠶⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⢸⡏⠀⠀⢀⣀⣤⣤⠶⠞⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠶⠿⠛⠋⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠂⠲⠀⠀⠐⠰⠐⠰⠀⠀⠂⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂

⠀⠀⠀⠀⠀⢀⡠⡤⠤⠀⠀⠂⠒⠂⠀⠀⠀⠀⠈⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⢾⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⢖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠄⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣦
⠀⠀⠀⠀⠐⠀⠈⠀⢧⣀⡤⠤⠤⠤⠤⠀⠒⠒⠂⠀⣰⠊⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠶⠴⠶⠶⠒⠒⠒⠒⠒⠂⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠳⣤⡀⠀⠀⠀⠀⠀⢀⣤⡼⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢐⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⠀⠈⢷⡀⠀⡠⢢⡶⠛⠉⠀⠀⠠⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢨⣈⢃⣃⠀⢀⡠⠤⢄⠀⠀⠀⣔⣒⣶⠶⢾⠀⠀⠀⠙⣿⡵⠋⠀⠀⢀⠄⠀⠀⠀⠁⠂⢄
⠀⠀⡘⢁⢸⠁⠀⢠⡖⠂⠾⠉⡏⡱⠦⢰⣇⣀⣰⡀⠀⢀⡹⡿⠏⠀⢸⠶⡦⠤⠤⡄⠀⠀⢀⠴⡁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⡀
⠀⠀⡟⠀⠚⠀⠀⠀⠛⠉⠃⠀⠓⠃⠂⠚⠁⠈⠩⠴⠂⢰⡀⠀⠀⠀⠀⠀⣘⠀⠈⠃⠀⠀⠈⠢⣈⠑⠤⠀⠀⠀⠀⠀⢀⠄⣀⡑
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⡇⡀⠀⠀⠀⠀⠓⠂⠆⠀⠀⠀⡰⢿⣛⠛⠶⣤⡀⠢⣤⠐⣡⡞⣁⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢸⠀⠀⠰⡇⠀⠀⠀⠆⢸⠀⠀⠀⡀⠀⢀⢧⣄⠉⠛⠶⠤⣍⡓⠶⢞⠝⣰⠋⠐⡀
⠀⠀⣀⣀⣀⠀⠀⠘⠀⠀⢰⣴⠀⠃⠀⠀⡀⠀⣨⠀⠀⠀⠂⠀⣤⣤⡇⢸⢐⣲⠀⡆⠀⠘⢄⠁⠙⢦⣄⠀⠂⠍⡓⠾⠞⢁⡀
⠀⠀⠿⠛⢿⠒⠒⢺⠤⠤⢾⣻⠀⣽⣿⢿⡶⠶⠻⠀⠀⢠⡄⠄⡽⠽⡏⢉⠀⠀⢀⠇⠀⠀⠀⠑⢄⠐⢌⡛⠦⣄⣈⣣⠔⠋⠀⡃
⠀⠀⠑⠀⠚⠀⠐⠚⠐⠂⠁⠈⠀⠛⠀⠊⠁⠀⠉⠁⠀⠉⠁⠀⠉⠈⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈⠢⣈⣫⠅⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢿⠵⠂
⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠙⠳⠄⡈⠉⠙⠲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠂⠀⡇⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡁⠀⠀⡇⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡆⣷⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣯⠀⠀⠀⠀⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠲⠖⠒⠒⠒⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠠⢀⢐⠠⠟⠀⠀⠐⠓⠒⠚⠛⠛⠛⠛⠉⠙⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⡿⠿⠿⠿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣤⡀
⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆
⠀⠀⠀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀
⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⣼⣿⠁⠀⠀⠀⠀⢀⡴⠒⠲⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠇⠀⠀⠀⠀⢻⣿⡀
⢰⣿⠇⠀⠀⠀⠀⠀⣾⡀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣯⣤⣄⡀⠀⠀⠀⠈⣿⣇
⣾⣿⠀⠀⠀⠀⠀⠀⢻⣿⣶⠒⣇⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠋⠉⠛⠿⠀⠀⣀⡀⢹⣿
⣿⡏⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠁⠀⠀⠀⠀⢠⣾⠿⠿⣷⡄⣠⣶⣿⠿⠿⢿⣶⣄⢀⣤⣾⡿⠿⠿⣿⣿⣇
⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣠⣿⣿⣿⠋⠀⠀⠀⠀⠈⢻⣿⠟⠁⠀⠀⠀⠀⠙⣿
⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠻⣿⣏⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣴⣿⠏⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿
⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁
⠀⠀⠀⠻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠙⢿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣷⣶⣶⣶⣶⣾⣿⠿⠿⠛⠋⠁
⣿⣿⠛⢛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⠉⠁⢲⠎⠉⣑⠦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢠⠇⠈⠁⢰⠃⠀⡜⠁⠀⠀⠀⠉⠁⠒⠢⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠇⠛⠀⠀⡇⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣦⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡞⣀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡜⠀⠉⠀⠀⢰⠁⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢱⡄⠀⠀⢠⠃⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢠⣧⡀⠀⠀⠰⠓⣏⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⡎⠈⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡞
⠀⠀⠀⠀⠀⠀⢀⠀⠀⡜⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢠⠇
⠀⠀⠀⠀⠀⢠⠀⢠⠇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡞
⠀⠀⠀⠀⠀⠀⠀⡎⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡸
⠀⠀⠀⠀⠀⠄⡸⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢰⠃
⠀⠀⠀⠀⠀⠀⡇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⠏
⠀⠀⠀⠀⠀⠀⠨⡗⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⡜
⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⡀⠀⠉⠁⠒⠂⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂⠤⢄⣀⡀⠀⠈⠉⠁⠒⠒⠀⠐⠁⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠢⠤⠤⢀⣀⡠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠂⠀⠈⠉⠉⠉⠉⠉⠁⠀⠂⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣶⣦⣄⡀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⣠⣴⣶⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠿⢦⡀⠀⠀⠀⠀⠀⠀⢀⡴⠿⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠁⠀⠉⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⡠⠒⠈⠉⠉⠐⠢⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⢉⠐⠄⡀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⢂⠀⠀⠁⠒⠠⢀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠐⠈
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣷⡀
⠀⠀⠀⠀⣀⣸⣇⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠽⢿⡿⠿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⣀⣨⣇⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠿⢿⡿⠾⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⣀⣰⣧⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠶⠾⡿⠾⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⣤⣤⣧⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠲⠾⡿⠛⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⣤⣤⣷⣤⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠛⢻⡟⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠤⢼⣷⢤⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠛⢻⡟⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠴⢾⣷⢶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠙⢻⡟⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢶⣾⣷⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠉⢹⡏⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⣶⣾⣷⣶⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠉⢹⡏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⣒⣾⣗⣺⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠉⢹⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣸⣄⣀⣀⣀⣤⣤⣀⣤⣀⣤⣤⣤⣄⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠼⣿⡇
⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠸⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⠇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⣼⠇⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠸⣧⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣯⣤⣤⣤⣤⣽⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡧⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠼⡃⢂⣠⠜⠁⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠀⠀⣰⠎⠁⠀⠀⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⢠⡜⠋⠀⠀⢀⠀⠢⡀⠀⠀⠀⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⠉⠠⡰⠖⠋⠀⠀⢀⠔⠁⠀⠀⠀⠐⢄⠀⠀⠀⠀⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠚⠅⢈⣤⠞⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⡏⠐⣀⡼⠋⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠙⠦⣀
⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⣰⠎⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⣤⠁⠀⢡⡔⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢀⡴⢋
⠀⢀⠔⠈⠡⡠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢀⡴⠋⣠⡴
⢰⡋⢀⣄⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠐⠁⠀⠀⠀⢀⠔⠁⣠⡾⠋
⣠⠉⠢⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⣠⠾⠋
⠈⠢⣀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣤⠞⠁
⠀⠀⠈⠳⣄⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣰⠟⠁
⠀⠀⠀⠀⠈⠳⣄⢌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢁⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠑⢶⣌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⡠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠙⠦⡀⠀⠀⠀⠀⢀⡴⠋⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠈⠳⢄⣀⠔⠉⢄⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⣁⢀⡴⠋

⠀⠀⠀⠀⠀⠀⠀⢐⢲⡀⡐⢢⠀⠒⡄⠀⠀⠤⡀⡔⠄⠀⡄⡄⠰⠒⡄⢠⢤⡀⠠⠤⡀⢠⠤⡄
⠀⠀⠀⠀⠀⢰⠀⠩⠼⠁⠱⠼⠹⠤⠏⠈⠙⠤⠉⠥⠝⠈⠥⠏⠡⠤⠏⢹⣨⠁⢅⣠⠇⢹⣀⠏⠉⠉⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢠⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⠀⠐⢠⣂⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠺⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠘⠦⣖⠒⠒⠒⠒⠒⠒⢂⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣛⡓⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⣀⣀⣀⣀⣘⣛⣉⣉⣉⣙⣛⣛⣛⣛⣂⣀⣀⣀⣀⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠿⠟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⢿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣦⡀
⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⣠⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⡀
⠀⠀⣰⣿⡟⠁⢀⣴⣾⣿⣷⣦⣠⣶⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣶⣤⣤⣶⣿⣿⣶⣄⠀⠙⣿⣷⡀
⠀⢠⣿⡟⠀⢀⣿⡿⠋⠀⠉⠻⣿⠟⠁⠀⠙⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠁⠈⠙⢿⡿⠋⠀⠈⠹⣿⣇⠀⠘⣿⣧
⠀⣿⣿⠁⠀⠈⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⢹⣿⡇
⢸⣿⡇⠀⠀⠀⠹⣿⣦⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⣿⣿
⣿⣿⠁⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⢀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⢹⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣿⡿⠋⠀⠀⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⣀⡀⠀⠈⠻⣿⣷⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⣿⣿⡀⠀⣰⣿⣷⡀⠀⣸⣿⡇⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠟⢿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇
⠀⠈⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟
⠀⠀⠀⠀⠀⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣶⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣾⣿⡿⠟⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠛⣿⠛⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⢹⠀⠀⠀⠀⣿⠀⠀⠀⢸⡏⠳⣄
⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠶⠶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠘⢧
⠀⠀⠀⣠⡶⠋⠁⠸⡆⠀⠀⠀⠸⡆⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠘⣧
⠀⠀⣼⢻⡄⠀⠀⠀⢻⡀⠀⠀⠀⢻⡀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣀⣀⣸⣀⣀⣀⣀⣿⣀⣀⣀⣸⣇⣀⣀⣀⣿⡀
⠀⣼⠃⠀⢿⠀⠀⠀⠈⣧⠀⠀⠀⠈⣧⠀⠀⢀⣸⣧⣤⡀⠀⠀⠀⠀⢀⡞⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙
⢠⡏⠀⠀⠘⣇⠀⠀⠀⠸⣆⢀⣀⣤⠼⠖⠛⠉⠁⠀⠀⢹⡄⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠘⡇⠀⠀⠀⢹⣆⣤⠴⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣼⠇⠀⠀⠀⠀⠈⢻⡟⠛⠛⠛⢻⠛⠛⠛⠛⣿⠛⠛⠛⢻⡟⠛⠛⠛⣿⠋
⢀⣿⠴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠞⠋⠙⣇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⡞⠛⠉⠀⠸⡆⠀⠀⠀⣻⣄⣀⠀⠀⠀⣴⠞⠓⠒⠒⠒⠛⠓⠒⠒⠒⠛⠒⠒⠒⠚⠓⠒⠒⠒⠛⠶
⠹⣦⣀⣠⡤⠶⠚⢻⡏⠀⠀⠀⢻⡀⠀⣀⣠⣿⠶⠖⠛⠉⠁⠉⢷⠀⢸⡇
⠀⠀⠉⢿⡀⠀⠀⠀⢿⣀⣠⣤⠾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⣸⠃⠀⠻⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴
⠀⠀⠀⢘⣧⣤⠶⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠚⢻⡅⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠚⠋⠉⢻⡀⠀⠀⠈⢷⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⢷⣀⢀⣀⣠⡴⠶⢾⡋⠉⠀⠈⣧⠀⠀⠀⠈⣧⠀⠀⠀⠘⣇⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠉⠙⣯⠀⠀⠀⠘⣧⠀⠀⠀⠸⡆⠀⠀⠀⠸⡆⠀⠀⠀⢻⡄⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢹⡆⠀⠀⠀⢻⡀⠀⠀⠀⢻⡀⠀⠀⠀⢷⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⢷⡀⠀⠀⠈⣧⠀⠀⠀⠈⣧⠀⠀⠀⠸⣇⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠘⣧⠀⠀⠀⠸⡆⠀⠀⠀⠸⡆⠀⠀⠀⢹⣼⡇⠀⢀⡴⠾⠶⠶⠶⠶⠿⠶⠶⠶⠾⠷⢦⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢹⡆⠀⠀⠀⢻⣄⣤⡴⠶⠛⠛⢳⡄⠀⢿⡇⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢀⣷⡴⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⢻⡀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢸⡇⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢸⡇⠀⠈⠳⢶⠶⠶⠶⠶⣶⠶⠶⠶⢶⡶⠞⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠘⣧⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠛⣯⠀⠸⡇⠀⠀⠀⢸⠀⢀⣠⣤⠿⣤⣄⡀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠙⠶⣤⣴⠖⠛⠋⢹⣆⣠⣄⣀⢸⡆⠀⢹⣦⣀⣀⣸⡶⠋⠁⠀⠀⠀⠈⠛⢾⣇⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢹⡄⢀⡶⠋⠁⠀⠀⠉⠙⠛⠚⠛⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠉⠉⠉⠁⠀⠀⠂⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⣦⡀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠀⠀⠀⢀⣤⡀⠈⠃⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠛⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠘⠃⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠠⠠⡀
⠀⠀⠀⠁⢄⠀⠙⠀⡀⢂⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠔⠁⢀⠂⡀⠜⠁⢀⠄⠁
⠊⠀⠁⠂⠤⢑⠠⡀⠈⠂⢕⢄⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠠⠂⠀⠀⠔⠁⢀⢔⠅⠊⠀⡠⢐⠡⠄⠂⠁⠈
⠀⠀⢀⠀⠀⡀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠑⠄⠀⢠⡀⠀⣀⣀⠀⠠⠐⠀⡠⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⡀⠠⢐⠀⠁
⠣⠀⠀⣀⠀⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠁⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠐⠂⠀⠠⢀⠀⠀⠀⠀⠀⢀⠴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⢄⠀⠀⠀⠀⠀⢀⠠⠀⠀⠒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣬⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣬⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢸⡟⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡟⠿⠇⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⣥⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣼⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣿
⠀⠀⠀⣿⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣿
⠀⠀⠀⣿⠺⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⢀⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⣿
⠀⠀⠀⣿⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠐⠀⠀⠀⣿
⠀⠀⠀⣿⠻⠿⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⢴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣿⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣿⠈⠛⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢰⡇
⠀⠀⠀⣿⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⢸⡇
⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⡏⢩⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡍⢹
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⢧⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⡜
⠀⠀⠀⠀⣠⢚⣡⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣌⡳⣄
⠀⠀⣠⠞⠑⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠊⠳⣄
⠀⠰⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡆
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⢀⣤⡴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠲⠶⠶⠶⠗⠚⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣤⣄⡀
⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⢸⡇⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷
⠀⠀⣛⣻⣿⣻⣿⣿⡇⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⣿⣿
⠀⠀⠈⢹⡏⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⣤⣼⣧⣤⣴⣶⡄⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⣿⣿
⠀⠀⠶⢾⡷⠶⢿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⢸⡇⠀⢀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣿⣿
⠀⠀⣛⣻⣿⣛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠉⢹⡏⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⣠⣼⣧⣤⣤⣴⡄⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠀⠀⠀⢸⣿⣿
⠀⠀⠶⢾⡿⠾⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠄⠤⠀⠤⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠤⠠⠄⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⢸⣿⡇
⠀⠀⣲⣲⣿⣶⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠉⢹⡏⠉⠛⠛⠃⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣼⣿⡇
⠀⠀⣀⣸⣇⣀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⣽⡇
⠀⠀⠾⢿⡿⠿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⣿⣿
⠀⠀⣲⣾⣷⣶⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠉⢹⡏⠉⠛⠛⠃⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⣿⣿
⠀⠀⣀⣸⣇⣀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠽⢿⣿⠿⣿⣿⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣿⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⢰⣿⣿
⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⡏
⠀⠀⠀⠀⢻⣷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⡤⣤⢤⣤⣤⡤⢤⣤⢤⣤⣤⣤⣤⣶⣶⣶⣶⣾⣿⡵⠟
⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠾⠿⠿⠿⠶⠶⠶⠾⠿⠾⠿⠷⠶⠶⠞⠛⠛⠛⠓⠒⠛⠛⠓⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉