Creature Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣾⠥⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡔⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣘⣋⢀⠀⠀⠀⢳⡔⢀⡀⡦⣈⡇⠀⣤⡀⣰⢚⠎⠀⠀⠀⣻⣷⣶⢞⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣿⣦⡑⢂⣤⣴⣯⠀⠛⠁⠈⠀⠀⠀⠈⠃⠘⢦⣴⣶⣾⠿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⢻⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣧⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⡿⠋⢀⡤⠶⠶⠤⣀⠀⠀⠀⣠⠴⠶⠦⢄⡀⠀⠈⠂⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠒⣴⢯⠆⠀⠀⠀⠈⢣⢠⡞⡵⠀⠀⠀⠀⠱⡀⠀⠀⠹⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠃⣿⡸⡇⠀⣴⣶⡄⣸⢺⡇⢇⠀⢰⣶⡆⢰⠇⠀⠀⠀⢈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⢅⡀⠈⢷⣭⣓⣈⣩⡵⠃⠀⠹⢮⣑⣂⣉⡥⠎⠀⠀⠀⠀⠈⢋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠋⣸⣿⣶⡤⣭⣥⣤⣤⣤⣤⣤⣤⣭⣭⡤⢴⡶⣿⣇⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⠄⣿⣿⣿⣷⣴⣿⣦⣼⣿⣤⣾⣷⣴⣿⣶⣾⣿⣿⡟⠀⠀⠀⠐⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡿⠿⢿⣿⣿⣧⠸⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠈⠻⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠟⠁⠀⠈⣿⣿⣿⣆⠙⢾⣅⣹⡟⠙⣿⠏⠉⢻⣿⣋⣹⣿⠴⠋⠀⠄⠀⠀⣴⡀⠀⠀⢃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠀⠀⠀⢀⣿⣿⣿⡿⠑⠦⢀⡉⠉⠉⠉⠉⠉⠉⠉⠉⠑⠀⡀⠔⠂⠀⠀⠀⣿⡇⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⠁⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣄⠀⠀⠘⢟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡃⠀⠘⠀⠜⣿⣿⣿⣟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡗⠀⠀⢀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣿⣿⣆⢲⣤⣤⢻⣿⣿⣿⣿⡁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⣴⣿⣛⣤⣾⢠⠎⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣿⣇⠀⠸⠇⢻⣿⣿⣿⢿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⡃⠻⠀⢀⣼⡿⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠁⠀⠀⢰⣿⣟⠁⠀⠂⠻⣷⣶⣦⣄⡀⣄⡀⢠⣄⣠⣴⣿⠃⠀⠀⠉⢦⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣏⠀⠀⠀⢀⡟⠿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣧⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣿⣦⠄⠀⡘⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠘⣿⣷⣤⠄⠠⣇⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣤⣬⣄⣤⣴⣤⣄⣽⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣿⣧⣶⣶⣦⣼⣷⣆⣣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠛⠻⠛⠋⠙⠛⠛⠟⠛⠛⠛⠿⠿⠿⠿⠿⠟⠛⠛⠛⠋

⠀⠀⠠⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠆⠰⠀⠀⠀⠠⠜⢻⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⢂⠀⠀⠀⢒⠀⠀⢆⠀⠀⠀⠀⠀⠐⠐⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠒⠀⠀⠁⠀⠀⠁
⠀⠰⠤⣤⡐⠀⠂⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠂⠂⠈⠓⠒⠄⠀⠀⡀⠀⠀⠀⠃⠀⠃⠈⢃⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠠⠀⠒⠋⠀⠀⠖⢀⠀⠀⠀⠀⠀⢀⠀⠐⠀⢁
⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢰⡂⠐⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠈⠅⠀⠀⠰⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⢀⠈⠂⠀⠀⡐⠈⠀⠀⠐⠀⠀⠒
⠀⠀⠈⠁⠀⠀⠀⠀⠢⠀⠀⠠⢄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀
⠀⠀⠀⢀⠀⠠⢀⠀⠀⠀⠀⠴⡇⠀⠀⠗⠄⠀⠀⠀⠀⠠⠊⠀⠀⠐⠄⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠚⠃⠀⠈
⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠐⠀⠄⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⢈⠐⠀⠀⠀⠀⠀⠀⠁⠐
⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⠈⠀⠀⠀⠀⠀⠀⠀⣤⡤⠤⣤⠄
⠀⠀⠀⢠⠠⡀⠀⠀⠴⠦⠀⠀⠀⠶⠀⠀⠀⠡⠒⠀⠠⡂⠀⠀⠐⠀⠢⠀⠀⠀⠐⠀⠀⠈
⠀⠠⠄⠄⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣆⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢀⡈⠀⠁⠀⠀⠀⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠠⠈⢄⠀⢉⠀⠄⠑
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠀⠈⢁⠀⠀⠁⠂⠀⠀⠰⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠠⠉⢀⠀⠀⠐⠒⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠂⢰⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠐⠒⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠰⠔⠊⠀⠀⠀⠀⠁⠐⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⠶⠶⠶⠚⠶⠒⠶⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠏⠉⡁⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠈⠳⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠨⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣏⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡷⠎⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣤⠀⠀⠀⢀⡀⠀⠀⠀⢠⡀⠀⠈⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠂⠀⠀⢀⡛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢀⠀⢸⡇⠀⠀⠀⠀⠀⢠⠀⠀⢀⣿⠆⠀⠊⢹⣷⡄
⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⣼⣇⣴⣄⡀⠸⠀⠀⠀⠀⣠⣤⣿⡎⠁⣿⣷⣦⣤⣴⡇⣧⣸⣶⢷⠾⣿⡂⠛⠀⠈⢹⣷⡀
⠀⠀⠀⠀⠀⢀⣿⡁⠀⠀⠀⠐⠀⣿⣿⠛⠛⠻⠙⣦⠛⠉⡿⠉⢹⡿⠀⠀⢻⣏⠙⠉⠉⢹⠛⠁⠀⠀⢰⣿⠇⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⢀⣿⠏⠁⠀⠂⠀⠀⠀⢸⣿⣆⠀⠀⠀⠇⠀⠀⠀⣠⣾⠀⢰⡄⠀⡿⣆⠀⠀⠘⠀⠀⢀⣠⡟⡅⠀⠀⠀⠀⢀⠐⢻⡄
⠀⠀⠀⠀⣼⡟⠃⠀⠀⠀⠀⠀⠀⠀⠘⡏⢿⣶⣤⣶⣤⡶⠾⠏⠁⠀⠀⠁⠀⠀⣿⠛⡷⢶⣶⣾⡿⠋⠀⠀⠀⠀⠀⠁⠀⠂⠼⣷
⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⡿⠛⠻⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠈⠛⠛⠀⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⣈⣿⡄
⠀⠀⠀⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣰⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⣿⡇
⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣰⣇⢀⣾⣿⣶⣶⣶⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⡆⠀⠈⠻⣿
⠀⠀⣼⡏⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣰⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣾⣡⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢻⠂
⠀⢠⣿⠀⠀⠂⠁⠀⠀⠀⠀⠀⢠⣶⣄⡀⢿⣿⡿⢿⣿⣿⣿⣿⣿⣏⣉⡀⢀⣀⣀⣤⣯⡟⢻⡟⣡⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣷
⠀⢸⡯⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⠷⣾⠉⢡⠀⠸⢷⢬⠛⡛⠿⠿⣿⡟⠛⠉⠃⠸⠃⠘⠃⣼⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡀
⠀⣸⣟⡀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣿⣿⠀⢀⡬⠀⠀⠀⠀⠀⢰⠃⠀⢠⠊⠀⢀⠀⢠⡄⠀⠀⠀⣽⣿⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⣽⡿⠀⠒⠀⠀⠀⠀⢀⠀⠀⢏⣿⡟⢀⠈⠀⠀⠄⢠⠀⠀⡀⠀⢀⠛⢀⠀⠀⠀⠈⠁⠀⠀⠀⣿⡟⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⣿⡇
⠀⣿⠀⣸⡷⠀⠀⠀⠀⠀⠀⢠⢘⣿⣿⡆⠀⠀⡷⠀⠃⠀⠀⠀⣤⣤⣴⡤⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠘⠂⠀⡀⠰⠋⣿⡇
⠠⣿⠦⠛⣄⠀⠀⠀⠀⠀⠆⠀⢹⣿⣿⡄⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣮⣿⣿⣦⡀⠀⠀⠀⠀⢨⣿⣿⡁⠀⠀⠀⠀⢠⠀⡄⠁⠀⠖⣿⡇
⠀⣿⣗⡄⠀⡄⠀⠀⠀⠀⠀⠁⠀⣿⣿⡅⠀⠀⠀⠀⠀⢠⠈⣿⠟⠉⠈⠉⠻⣿⣽⠖⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠁⠀⠈⣎⢻⡇
⠰⢿⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⠁⢀⠀⠀⠄⣜⣾⡟⠀⠀⠀⠀⠀⠹⣿⡖⡀⠀⠀⠀⠹⣿⡀⠀⠀⠀⠀⠀⠀⠠⠀⠆⣻⣻⡇
⠀⢸⣿⣵⢀⡄⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⠀⠈⠀⠀⣈⣿⣿⠀⠀⠀⠀⠀⠀⠀⣻⣿⡁⠀⠀⠀⢐⣿⡧⠀⠀⠀⠸⠀⠠⠀⠀⣤⠿⣹⡇
⠀⢸⣿⣿⠈⡇⡀⠀⠀⠀⠀⠀⠀⠙⣛⣿⠀⠀⠀⡀⢽⣿⡏⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁⠀⠀⠀⢨⣿⠱⠀⠀⠐⠀⠀⠀⠀⢠⣍⡢⣿⡇
⠀⢸⣿⣯⡞⢁⠉⠀⠀⢀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠁⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⠀⠈⣿⠆⠀⠀⠀⠀⢀⠀⢀⣼⠈⡇⢿⡇
⠀⠐⣿⣿⣷⣼⠱⡆⢣⣘⣇⠀⡀⣰⢿⣿⡄⠀⠀⣴⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠈⢰⣿⢺⡛⡄⣤⠀⣀⡔⢫⡟⣷⣮⣿⠃
⠀⠀⢸⣿⣿⣩⣿⣿⣾⣽⣶⣶⣾⡿⠟⠻⣿⠓⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡏⠀⠀⠀⢸⣿⣾⣟⣇⢋⣠⠇⣴⣻⣵⣟⣷⣿
⠀⠀⠘⠻⣿⣿⡟⠛⢿⠟⠙⠋⠉⠀⠀⠀⠀⠈⠀⡀⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⣨⠟⠉⠻⠿⠿⠿⠛⠻⠟⠿⣿⣿⡇
⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⢩⡄⣀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇
⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢠⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢇⡏⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣗⢀⢀⠀⠀⠇⠀⢀⡀
⠀⠀⠀⠀⠀⠈⡇⠀⢤⠀⠀⠠⠀⠀⢶⠀⠈⣿⣼⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⡎⢈⡷⠂⠀⠀⠉⠀⠐⠂⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠳⢄⣀⣀⠀⠀⣀⠀⢀⣇⣠⢟⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣼⡅⢀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⣿⠿⡿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣇⣩⣤⣶⣾⣀⣀⣄⣀⣴⠁
⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠙⠉⠉⠁⠀⠀⠀⠀⠀⣼⠛⠛⠉⠋⠈⠛⠉⠙⠉⠻⠙⢿⡏⠛⠻⣆
⠀⠀⠀⠔⠋⠁⠠⠊⠀⢠⠆⠀⢀⡀⠀⠀⠀⠀⣀⣤⡶⠦⡀⠀⠀⢦⡀⠀⢰⡏⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠁⠀⠀⣴⠖⠈⠉⠀⠀⠀⠑⠆⠀⠀⠀⠀⢸⣇⠀⢰⡏⠙⠳⣦⣀⣀⠀⠺⢦⡈⠑⢄⠈⠂⠀⠙⢆
⠐⢦⠀⠀⠀⣀⠜⠀⠀⡀⠀⠀⠀⡴⠂⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⠀⠀⠀⠀⢻⣆⠀⠹⢦⡀⠀⠉⠛⠷⣤⣀⠉⢢⡀⠱⡀⠙⠀⠘⡄
⠐⢀⡠⠴⠘⠇⠀⢀⡼⠃⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠀⠀⠀⠀⠀⠀⠉⠻⢦⡴⠇⠀⠀⠀⠀⠀⢻⡇⠀⢁⠀⠀⠀⣻⡀⢿⡀
⠀⠀⠀⠀⠰⠤⠒⠉⠱⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠄⢺⠗⣾⠁⣻⠓⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠼⠆⠀⠀⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⠀⡀⠀⠀⠀⠈⢣⡀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠱⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡗⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⢴⣾⣛⠛⣲⠶⢦⡤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⣴⠁⢀⡈⢻⡃⢴⣆⣱⠀⠈⣉⣉⠓⠒⠶⠤⣤⣀⣀⣀⣀⡤⠎⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣀⡀⠤⢷⠀⠀⠀⠻⣄⡘⣷⡟⠙⠾⠛⢁⡴⠋⠁⠉⠛⢦⠀⠀⠀⠈⠉⠉⠁⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⡠⠄⠦⡀⠱⣄⠀⠀⠀⠀⠈⣧⠀⠀⠀⠈⠽⢋⣀⠀⣀⡀⢾⡁⣾⣏⡆⠀⢸⡇⠀⠀⠀⠀⠀⣠⡴⠛⠁
⠀⠀⠀⢀⡤⠖⢾⠁⠀⠀⠘⢄⠈⠛⠒⠚⠛⠉⢹⡆⠀⠀⠰⡅⠈⢉⣀⣉⡁⠸⢧⡈⠉⠀⢀⡼⡁⠀⢀⣠⡖⠋⠁
⠀⠀⣰⠋⠀⠀⢈⡳⠶⠖⠀⠈⠳⣄⠀⠀⠀⠀⠀⡇⢰⠂⠀⣠⠞⠉⠀⠀⠈⣷⣄⠙⠛⠛⠛⠊⢁⣴⠋⠀⠈⠉⠉⢀⡴⠁⢀⠔⠊⠉⠙⠂
⠀⠀⡏⠀⣀⡀⠀⢻⠀⠀⢀⡀⠀⠈⡆⠀⠀⠀⠀⡇⡼⢠⣾⣥⣤⣤⣴⣤⣴⣿⡈⢦⠀⠀⠀⠀⡼⠉⠛⠲⠶⠶⠒⠉⠀⢰⠃⠀⢀⠀⢁⡞
⠀⠀⠣⡜⠉⣟⣠⢞⣀⡀⢨⡟⢦⠀⢸⠀⠀⠀⠀⣷⣦⡿⠿⡿⠿⣿⣿⣿⣿⣿⣿⣿⡇⠆⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠈⠙⠉⠙⢦
⠀⠀⠀⠀⢰⠁⠙⠺⡄⠗⠋⠀⢸⠀⢸⠀⠀⠀⢰⣿⡿⠇⠂⠀⠠⠤⢤⣀⣀⣿⠋⠻⣿⣸⠀⣸⠣⡀⠀⠀⠀⠀⢀⡴⠃⠀⣠⠄⣀⡴⠃⠀⠀⡆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⡞⠀⠀⢧⠀⠀⢸⠿⠀⠀⠀⠀⠐⠶⠦⠤⠈⠉⠓⠢⣌⣿⢀⡿⣄⠈⠙⠒⠒⠋⠁⠀⢀⣴⡅⠄⠻⣄⣀⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠙⠒⡿⠀⠀⢀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⣿⠸⡇⠉⠳⠦⣤⣤⣤⠤⠖⠋⠀⠙⠦⣤⡤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣠⡾⠁⠀⣴⠋⠈⡇⠀⠀⣰⠋⠙⢷⣄⠀⠰⠇⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠛⠉⢀⣠⠞⠁⠀⠀⡇⠀⢀⡏⠀⠀⠀⠙⢦⡀⠀⠐⣄⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⣠⡴⠚⠋⠁⠀⠀⠀⢸⠀⠀⢸⠃⠀⠀⠀⠀⠀⢻⡄⠀⠘⣦⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⠙⡄⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⢹⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢐⡇⠘⡆⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢻⡄⠀⢳⡀
⠀⠀⠀⣀⡤⠴⠶⠤⠤⣤⡾⠀⠀⣰⣅⣀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⣧⠀⠀⠀⠀⠀⢸⡀⠀⢸⡃⠀⠀⠹⣄⠀⠙⠦⠤⠠⠒⠒⠒⠒⠦⢤⡀
⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⣠⡤⢄⣀⣼⠄⠀⠀⢸⣀⣀⡀⣀⡤⠼⠧⠀⠘⢧⣄⣀⢀⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡗
⠀⠀⠳⣤⣤⡽⢦⣀⣀⠀⣀⣴⣄⣀⣸⣤⠞⠀⣞⠁⠀⢀⡀⠀⠀⠀⠀⠀⢤⡀⠉⠉⣳⠀⢰⡄⠀⠀⠀⠀⢍⣳⡀⢀⣤⡀⠀⣎⣀⡼⠓⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠈⠁⠀⠀⠀⠹⠦⡴⠞⠳⣄⣀⣀⠂⠀⢈⡷⠶⠚⠛⠶⠺⡄⠀⠀⢀⠀⢀⡟⠈⠉⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠛⠒⠚⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⢄⣀⠀⠤⢒⣀⣠⣤⣤⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡢⣄⢤⣀⣤⣤⣴⣮⣵⣶⢾⣷⣽⣾⣽⣿⣬⣮⣽⡿⠿⣿⣿⣟⡲⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠮⣛⣩⣔⢟⣯⣫⣝⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣿⠾⢿⣿⣿⣿⣿⣿⣿⣢
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣪⠐⢁⣪⣼⣾⡶⠿⠟⠛⠛⣫⢙⢛⡻⣿⡿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢀⢔⣴⣷⣿⣿⡿⡟⢻⠡⡆⢸⢠⠀⣮⡶⣿⣛⢷⣽⡯⣮⣦⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿
⠀⠀⠀⠀⣴⣿⣿⣿⣿⠟⢡⢰⣧⣸⢁⠇⢨⢧⠺⣿⣿⣿⡟⢸⣧⢃⣿⣿⣃⣿⣿⣎⣽⣻⣟⡻⣍⠙⠛⠿⠿⣿⡿⢿⣿⣿⣿
⠀⠀⠀⢰⣿⠟⠋⠋⠁⠀⡌⢷⠙⢦⠺⢠⠈⢸⠀⢹⡿⣥⡴⢏⣠⣾⣿⡿⣻⣿⣿⣿⣿⣟⣁⡈⠀⠀⠦⠬⣶⣾⣍⠾⣹⣿⣿
⠀⠀⠀⠈⠁⠰⣦⣔⢲⣲⠁⣦⠷⢚⣷⢩⣦⣻⡥⣐⣩⣭⡶⢛⣿⡿⢻⣹⡭⢁⣌⣼⣟⣂⣽⣿⣄⡀⠀⠀⢀⢈⠶⠮⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠙⠿⠇⠹⡦⠘⢇⠋⠻⠈⠉⣉⢈⣉⣭⣷⣾⠿⠿⣷⣥⣷⣿⣿⢿⢿⡟⡙⢻⣿⣿⣿⡖⣬⣐⡂⠲⠶⠾⠁
⠀⠀⠀⡶⢀⢄⢦⣬⣆⡸⣡⣀⣠⣶⡁⢖⢓⠻⡟⣟⣿⣽⣧⣾⣿⣿⡯⡼⣏⢎⣏⡖⠐⢥⣄⡿⣿⣿⣷⣄⢔⢄⣀⣀
⠀⠀⠐⢁⣾⢿⣿⣿⣿⣷⣦⣄⣩⠹⡏⠻⣿⠿⢿⣿⡿⣿⡿⢛⠻⠷⠑⡝⡗⡤⠨⠈⡅⠈⠴⢘⠴⣝⣿⣿⣿⣷⡪⠈⠁⠂
⠀⠀⠀⣼⣿⣞⡿⣿⣻⣿⢿⣿⣿⣿⣿⣶⣿⣿⡶⡮⠽⠍⢊⣆⠕⠇⣀⠍⠌⡡⣅⣡⢰⠀⠂⢩⡌⣼⣿⣿⣿⣿⡓⠄
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣠⡹⠄⠿⡃⣜⡀⠀⠟⢇⡡⢃⠂⢠⣎⣽⢯⣿⣿⣿⣿⣛⡲⠄
⠀⠀⢸⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣾⣭⣐⣮⣾⣴⣼⣴⣷⣦⣿⣿⡿⣾⣿⣿⣿⣿⡟
⠀⠀⠈⣿⣿⣻⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⠯⡹⣿⣿⢇⣿⢿⣿⣿⣿⣿⣿⣿⢿⣿⡿⠂
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⣾⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠔⠖⠠⡶⣿
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢣⣿⡿⡟⠋⠤⢤⡀⠀⢠⡄⠘⠢
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⠋⠀⠀⢀⡄⠈⠁⠠⢀⠙⠁
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡄⠀⠈⠉⠙⠛⠛⠛⠻⠟⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡁⠀⠀⠀⠊⠁⢀⡄⠀⠀⠀⡀⠘⠡
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢡⣿⣿⣿⣿⡿⣿⠆⠀⠀⠀⢰⡦⡤⣾⠟⠁⠀⣤⢷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢺⠣⠻⣿⠿⠋⣡⢿⡟⠀⠀⡼⢀⠊⠀⢰⠇⠀⠀⠀⠁⠋⠛⢄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⣴⡿⠃⠀⠀⠀⡀⠀⠀⠠⠾⢃⣀⢀⡀⣠⣴⣶⡿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⡇⠀⠀⠐⠀⠀⠝⠉⢈⣴⣿⣟⣿⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢀⠂⠆⠀⠀⣔⣠⣿⡿⡋⠱⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣾⣿⣿⣾⣾⣾⣾⣿⡏⢯⠊⠀⠀⣿⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠅⢡⠀⠀⠀⠁⢋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣗⣿⡀⠠⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠋⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠿⠟⠉⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⣀⣀⣠⣴⡄⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⣤⣤⣄⠀⢀⠀⠀⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣬⠀⠀⠁
⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⡴⠿⡿⢟⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡄⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢿⣿⣶⣶⣿⣿⣦⡐⠲⠟⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣶⣤⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⠿⣿⣿⣿⣿⣿⣿⡛⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡃⢸⣷⣿⣿⣿⣿⣿⣿⣷⡄⠰⠁
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟⠻⢿⣿⠿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣷⣠⣤⣄⠀⠀⠀⠀⠀⢠⡇⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠋⠀⠀⠙⣿⡇⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢹⣿⣿⣿⣷⡀⠀⠀⠀⠀⡇⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠈⢿⣦⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⣿⣿⡄⠀⠀⠀⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⡄⠀⠀⠀⢹⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⡇⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢸⡇⠀⠀⠀⠠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢻⣿⡿⣿⣿⣿⡄⠀⠀⠀⠈⠻⣿⣿⢿⣿⣿⡿⡏⠚⠏⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⡀⢂⠀⣆⠀⡽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠻⠀⠀⠀⣳⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⣆⡇⢸⠀⣻⠀⣿⢹⡄
⠀⠀⠀⠀⠀⠀⢀⣢⠀⠀⠀⢻⣧⠀⣗⢦⠀⠀⢀⡀⠀⠹⡇⠀⢠⢶⡄⠀⠀⢠⡏⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣞⡿⠁⡟⠀⡿⠀⣿⢸⠃
⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢀⠀⠈⠂⠘⠮⣷⡀⠸⣿⡀⠀⠀⠀⢼⠀⣧⠀⠀⣾⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡼⢁⡼⢃⣰⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠣⣧⠀⠠⣶⣄⡀⠀⣀⠈⠓⠀⠀⠀⠀⠀⠹⠶⠶⠟⠀⠿⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⡴⢊⡽⠛⠛⠛⠯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⠈⠁⠀⠈⠙⠛⠂⠻⣷⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠶⠀⠀⠀⠉⢛⣳⣦⠀⠀⢠⠏⢠⡏⢠⠏⢀⡎⢀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⣀⣀⠀⠀⠀⠀⠈⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡽⠋⠀⠀⣿⠀⢸⡀⣼⠀⢸⠀⢸⠀⡧
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠢⣀⠔⠋⠉⠉⢻⡄⠀⠀⠀⠀⡿⣄⠀⠀⠀⠀⢰⡞⠒⠶⠶⠶⠚⠋⠀⠀⠀⠀⠘⣆⠘⣇⠹⡄⠸⡄⢸⣴⠇
⠀⠀⠀⠀⠀⠀⢸⠁⠈⠀⠀⠁⠀⠀⠀⠀⢠⠇⠀⠀⠀⣸⣵⠟⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⠛⢦⢿⠶⢿⣏⠁
⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⠀⡀⠃⠀⠐⠻⡄⠀⠀⢠⡇⣸⣷⢃⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⡀⠱⡄⣇⠘⡜⣧
⠀⠀⠀⠀⠀⠀⠀⡠⠛⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡾⠛⣧⣼⡞⠀⠀⠀⢸⠁⢀⣴⡄⠀⠀⠀⠀⠀⠀⠀⡏⢰⠀⡇⢠⡇⡟⠀⡇⣼
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⣀⠀⠀⢀⣸⠃⠀⢰⢷⡞⢻⣁⡿⠀⠀⠀⢸⠠⢋⣿⡶⣿⠃⠀⠀⠀⠀⠀⢳⣎⡜⣱⠏⡼⠁⣼⣡⠏
⠀⠀⠀⠀⠀⠀⠹⣄⡀⢀⣠⠞⣽⠓⠲⡚⠁⠀⢀⣟⣠⣿⠞⣿⠀⠀⠀⠀⣾⠀⠈⢉⣼⠃⠀⠀⠀⠀⠀⠀⠀⠙⠿⣥⣾⡵⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⣄⠀⣿⣶⠟⠁⠀⠀⡾⠉⣷⣸⢿⠁⢀⠘⣆⠀⣿⣀⠀⠈⢉⣻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⡇⠀⠀⠀⣸⠳⡟⢹⣥⠾⠀⠘⢦⡈⢷⣯⡉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⢀⣿⡴⢳⣁⣼⣷⠇⠀⠀⠉⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⣇⡄⠀⠀⠀⢀⠀⣀⡀⠀⣀⠀⣠⢶⡦⢶⣦
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⡴⢶⣲⣾⠷⢿⡾⢹⣏⣿⣸⠃⠀⠀⠀⢸⡧⠠⠴⣶⡶⠂⠀⠀⠀⡟⢷⡠⡿⠦⠛⠗⢛⣿⠞⠟⠃⠁⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠛⠿⢭⣍⣹⣿⣹⣷⢻⣏⢿⡼⣧⣶⠛⣯⢹⡇⣹⡇⠀⠀⠀⠸⣷⣶⣿⠵⠶⣶⡶⠀⠀⠉⠀⠀⠀⠀⠀⠀⠸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠷⠿⠷⠿⠶⠦⠤⠴⠶⠶⠶⠒⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠛⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠉⠙⢇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⢓⠦⠤⢵⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡖⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏⠀⠀⠀⠈⢆⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡆⠀⠀⠀⠀⠀⠈⠢⣛⠛⠛⠿⠿⠟⠋⣠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣶⣤⣤⣶⣶⣾⣿⣭⡉⠓⣰⠶⢶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣷⠮⡁⠄⠉⠀⢼⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣟⣫⣬⣝⣚⣷⣆⡲⣈⠻⠽⣿⣷⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⣶⣿⣷⣾⣷⣿⣿⣿⣿⣿⡿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⢽⣿⣿⣿⣻⠋⠻⠿⣿⣿⣿⡿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣧⣾⣿⣿⡿⠏⠀⠀⠀⢸⣿⣿⣽⡇⠘⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡏⠀⠀⠀⠁⠀⠈⠻⣿⣿⣇⠀⠈⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⠀⠀⠘⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣷⡀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⢹⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⣼⣿⠂⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠹⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢠⠛⢢⡀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡖⢤⡀⠇⡈⣦⣊⣰⣀⣹⣜⣁⢸⢀⠴⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣖⠲⢤⣇⡷⠟⠋⠉⠉⠀⠀⠀⠀⠉⠉⠙⠓⠶⣨⡃⠤⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢮⣜⡠⠖⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣰⠁⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⢄⡀⠀⠀⠀⠀⣇⠀⠈⠻⣬⠀⡟⣀
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⡀⠀⠙⡄⠀⠀⢀⡟⠀⠀⠀⡙⢾⡍⡸⢀⡀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⡟⠀⢀⡠⠤⢄⡀⠀⠀⠀⠀⠀⠀⡄⠀⠘⠃⠀⣠⠃⠀⢀⡞⢃⠀⠀⠀⢉⣤⡹⣧⠅⣇⣄⠀⠀⢀⢺⣁⣿⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡎⢀⡀⠀⠱⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠚⠁⣀⠴⠋⠈⠉⠀⠀⠀⠈⠛⠃⢨⡿⣿⣸⠞⢦⡏⣌⣿⠁⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢇⠘⠋⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠚⠉⠐⠀⠀⠀⠀⠀⠀⠀⠐⢿⡷⢰⡷⢩⣛⠻⢛⣿⠌⠁⡇
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠉⠈⠉⠀⠀⠀⣀⣠⣤⠴⠒⢋⡉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣬⣇⠐⢿⣿⠇⠠⣶⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠹⡆⠠⠤⢤⡤⠶⠲⣛⠉⠡⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠀⣶⣶⠘⠿⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠲⣄⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠦⢤⠴⢣⡾⠛⠋⠛⠳⢦⡀⢀⣀⣤⡤⠴⠶⠶⣆⠀⢺⠀⠀⢀⡼⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣴⠏⠀⠀⠀⠀⠀⢨⡇⣼⠁⠀⠀⠀⠀⢀⣼⠀⠈⡇⠀⠺⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣭⣉⣉⣠⡿⠀⠀⠀⠀⢸⣭⣤⡤⠞⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⣶
⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣄
⠀⠀⠀⠀⠀⢺⣿⡇⠀⠀⠀⠀⠀⠠⣤⡄⠀⢠⡄⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⣄⣄⣀⣒⣀⣤⡔⠀⠀⠀⠈⣿⠿⡟⠀⠀⠀⠀⠀⠀⠓⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣷⣶⣶⣄⠀⠀⠀⠀⠱⠶⠆⠀⠀⠀⠀⠀⠛⠛⠛⠋⠀⠀⠀⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢰⠀⡦⣀⠤⠁⠀⠀⠀⠀⠀⠀⣴⣾⣷⡄⡀⢀⢠⣿⣿⣿⣿⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣽⢰⣟⠟⠠⡒⠄⠀⠀⠀⢀⣾⣿⣿⣿⣿⣆⠉⢀⣿⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢾⡀⢻⣿⣿⣿⠃
⠀⠀⠀⢰⣧⣸⣏⣴⣴⣖⠃⠀⠀⠀⢸⣿⣿⣿⣿⡿⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣴⡇⠀⣿⠃⠀⠀⠀⠀⣀⡀⠈⡇⠀⠀⠀⠀⠈⠙⠻⠿⠿⠟
⠀⠀⣴⣿⣤⠟⠘⢿⡈⠀⠀⠀⠀⠀⠈⠁⠉⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⡴⠿⢴⣦⣻⠀⠀⠀⠀⠈⠉⠉⠀⡇
⠀⠀⢧⣿⠀⠀⣠⣞⡧⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢹⠀⠀⠀⣹⡠⣤⣿⡗⡃⠘⡇⠙⠀⠀⢀⠀⣀⣀⣤⢠⣿⣆⣀⣀⣀
⠀⠀⠈⣿⣤⡾⢿⠉⢡⣇⠀⠀⠀⠀⢠⡇⢸⡇⠀⢀⠀⣿⢸⡇⢸⠇⢹⠿⠈⠉⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠤
⠀⠀⠀⣿⡇⠀⣸⠞⠉⠀⠀⠀⠀⠀⢸⣷⢾⡇⣼⣿⡤⣿⣘⠛⠉⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⡀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠙⠉⠉⠀⠀⠀⠀⣦
⠀⠀⠀⢹⣧⡼⠃⠀⠀⠀⣀⡀⠀⠀⢸⡇⢸⡇⡟⠙⠁⠛⠉⠀⠀⠀⢀⠀⠤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿
⠀⠀⠀⣼⡟⠁⠀⠀⠀⠈⠿⠃⠀⠀⢸⡇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣦⡄⠠⣴⡆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠁⠀⠀⣿⡇
⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠁⠆⣁⡆⠀⠀⠀⠀⠛⠀⠀⠻⠀⠀⠘⠀⠀⠀⣿⣿
⠀⠠⠾⠿⠆⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⠻⠛⠻⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿
⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⢠⣴⣶⠀⣴⣦⡄⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⠀⠂⠀⠀⠀⠀⠈⣿⠇
⠀⠀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣷⣶⢶⣆⠀⠀⠀⠉⠁⠿⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠐⡀⢀⣈⡲⢦⣄
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⡀⢉⠉⠉⠙⠶⣿⠳⢦⡄
⠀⠀⠀⠙⣿⡟⣿⠏⣁⡿⠋⠉⠛⣿⡀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠢⣥⡈⠁⠀⠀⠦⣼⣤⣼⡁
⠀⠀⠀⠀⠸⣿⠏⠀⠉⠀⠀⠀⠀⠚⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠹⣄⠀⠀⣷⡀⠀⠓⣦
⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠛⢿⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠇
⠀⠀⠀⠀⠀⢠⣤⣀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣦⣶⣦⣴⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠹⣀⣾⡿⠃
⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢺⣿⣿⣿⣿⣿⡙⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠈⠃⠸⣿⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣇⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠙⠉⠉⠘⠋⠉⠉⠃⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⢸⠀⠀⠀⠀⣠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢛⣳⣄⠀⢈⠆⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣦⡄⣄
⠀⠀⠀⠀⠀⠠⠐⠠⠀⠀⠈⠛⠿⡇⠨⣆⠳⡔⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣽⠿⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠙⢆⠙⠿⣦⣠⣧⣿⠀⢻⡀⢠⣀⢩⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠿⣿⣿⣿⣿⣿⣻⣿⡿⠻⡿⠋⠀⠀⠀⠀⣠⡄⠈⠙
⠀⠀⠀⠀⠀⠀⠀⣸⡷⢄⢨⡟⣡⣿⠀⠈⠲⣷⣛⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢉⡤⣌⢻⣿⣿⣿⣿⣿⠛⠚⠀⠀⠀⠀⠀⢰⡝⠋
⠀⢄⠀⠀⠀⠀⣠⡿⢁⠀⠀⠀⠀⠀⠀⠈⢀⣀⡀⠀⠀⢰⡆⠀⠀⠀⠀⠀⢀⡘⠃⠀⢃⣼⣿⠋⡙⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠷
⠀⠈⠡⠦⠤⣶⠛⠁⢠⠄⠀⣠⣶⠤⠂⠀⢼⡏⠀⠀⠠⣏⠀⠀⠀⠀⠀⠀⠬⠉⠙⠛⠛⠛⠁⠉
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠈⠛⠉⢢⠀⢠⠻⠧⡄⠀⠀⠈⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠓⠀⠀⠑⠤⠤⠤⠀⠑⠆⠀⠊⠐⠒⠒⠁⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠊⠻⣿⣶⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠙⠻⢿⠴
⠀⠀⠀⠀⢀⣤⢰⣦⣄⠀⠀⠀⠀⠀⠀⢀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠰⡇⠀⠀⠀⠈⣇⣠⣞
⠀⠀⠀⠐⠒⠿⣅⣿⣹⣷⠂⠀⠀⠀⠀⢸⣧⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠐⠃⠓⠀⣶⠇⠴⠟⠉⠀⢀⣤⠴⠶⣦⣄⣰⠇
⠀⠀⠀⠀⠀⠀⠙⢿⣻⡾⣆⢾⠙⢖⣀⡈⣧⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠠⠲⡔⢳⠖⠀⡿⢢⡀⢸⣿⣦⣴⣶⣽⢯⠤
⠀⠀⠀⠀⠀⠀⠀⠀⡹⢅⠨⣝⡾⣆⠙⢓⡹⡋⠰⣄⣤⡀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠉⠛⢠⡆⠀⠀⠀⠐⠘⢄⡅⢤⠿⣿⡄⠹⣿⣧⠤
⠀⠀⠀⠀⠀⠀⠀⢀⡅⠀⢱⡌⠓⠾⣻⡟⢳⣿⢿⣇⡿⠉⠠⠀⠀⠀⠀⠀⠀⣿⡇⠀⢁⠀⠀⠀⠀⠀⠀⠀⠸⠖⠀⣶⡾⡏
⠀⠀⠀⠀⠀⠀⠀⡨⠁⠀⠀⢿⣮⡀⠙⠁⡀⠀⠠⠟⠶⢶⣤⠀⠀⠀⠀⠀⠀⠛⠁⠰⠆⠀⠀⠀⢀⣤⣄⡀⠀⠸⠆⠘⢃⡇
⠀⠀⠀⠈⠀⠂⠂⠁⠀⠀⠀⠀⣿⢏⠀⣀⠈⣿⠃⢀⣴⡿⠆⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⢰⡀⠈⠙⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⡳⠀⠘⢇⠀⠈⠹⣧⡀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠛⠀⠈⠀⠀⠔⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⣰⠂⠀⠀⢀⣀⠈⣁⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠒⠢⠀⠁⠈⠤⠙⢢⠤⡀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣼⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣰⢿⣏⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⣤⠀⠥⣤⣠⣿⣵⡶⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣦⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⣧⡖⠰⢿⣿⣻⣿⠟
⠀⠀⠀⠀⠀⠀⠀⣀⣶⣈⣴⢿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⣸⠀⠀⠀⠀⠀⠠⡀⣠⣄⡢⠖⠙⣦⡀⢻⣾⣇
⠀⠀⠀⠀⠀⠀⢠⡗⣿⡏⣼⣿⣿⣶⠀⠀⣠⡴⠀⢈⣡⠔⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡆⠻⢿⣷⠘⡷⣿
⠀⠀⠀⠀⠀⢠⣾⣷⣾⠛⡫⠉⠉⠉⠹⠖⠙⣇⣀⠈⠴⠋⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡍⢶⣾⣿⣾⣿⢠⣷⠏
⠀⠀⠀⠀⠀⣸⣿⢉⡿⢟⣧⣠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⢠⡉⣿⢷⡿⠾⣿⠁⡼⠃
⠀⠀⠀⠀⠀⠟⠁⠈⠳⡿⠻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⢰⠀⢧
⠀⠀⠀⠀⠀⠂⠀⢀⠀⢣⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢃⡰⣤⢀⣀⠴⢲
⠀⠀⠀⠀⢊⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣤⠜⠁
⠀⠀⠸⠦⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⣹⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠁⢠⡀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⣠⡾⠏⠁⠀⠀⠀⠀⣠⠂⠀⠀⠀⡠⠈⠁
⠀⠀⠀⣀⡰⢾⡟⠷⣀⣀⠀⠀⠠⣶⡞⠁⢀⣼⠋⠀⠀⢀⣤⠀⣀⠤⣣⠶⠒⠲⠿⠃
⠀⠀⠈⠁⠀⠀⠈⠢⡈⢿⡄⠀⠀⣿⠀⣠⣿⠃⠀⠀⣠⠟⡡⣊⡵⠚⠁⣄⠀⠀⣀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠙⢦⡁⣿⡖⢿⣿⣀⣴⣾⡵⢊⣴⡃⠀⠀⠀⢀⣠⡴⠧⠠⣴⣾⣧⢤⣶⠦⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡈⢻⣿⡇⣸⣿⣿⣿⣷⣴⡿⢿⣴⣒⣴⠾⣋⣥⣴⠾⠟⣻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢼⣦⣻⣷⣿⡿⣹⣿⣿⣿⣿⣉⠻⣿⣷⣿⢿⣉⠠⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⡛⢷⠞⡻⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢯⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣩⠗⠋⠓⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⡶⣿⣿⣿⣿⡟⠿⠛⠛⣿⢻⣿⣿⣁⣄⠀⣤⡿⢻⠎
⠀⠀⠀⠀⠀⠀⢀⣸⡿⠢⣿⠗⣿⣿⣿⣹⣧⣉⣀⣴⡇⡾⠟⡙⣩⢻⡷⢾⣧⠈⡯⠒⢢
⠀⠀⠀⠀⠀⠀⣈⣏⣡⠵⣾⠃⠁⠈⠁⣹⣿⣿⣿⣿⣿⠇⢠⠃⢉⡞⠃⠈⣙⢿⣤⡶⠛⠯⠄
⠀⠀⠀⠀⠀⢰⡛⠀⠙⢷⡏⠀⢀⠆⢠⣷⣿⣿⣿⡿⣿⢰⣿⣴⣿⠛⠱⡄⠠⡻⢻⣶⠚⠀⢹⠂
⠀⠀⠀⠀⠀⣩⡀⠠⢾⠟⠀⢀⡰⠀⠂⢸⣿⣿⣿⣿⠗⣽⣿⣽⣣⣆⠀⢻⡀⠁⠹⠿⢿⣶⣾⣿
⠀⠀⠀⠀⠀⢹⡔⠈⢻⠇⠀⡸⠡⠄⡰⢻⣿⣿⣿⣸⣧⣛⣹⣿⢉⡧⠄⠀⠻⡀⠀⠀⠾⣇⣀⢽⠄
⠀⠀⣸⠀⠀⠸⢛⣯⢹⠁⢀⠇⠰⠊⣠⣿⣿⣿⣿⡟⢻⢉⣿⡇⠀⠀⠀⠀⢀⡀⠁⠀⠀⠁⠈⣞⡀
⠀⠀⣿⠀⠀⠀⠈⣷⡘⠇⠈⠀⠸⠈⢡⣿⣯⣿⣯⠐⠂⣀⠁⠘⠀⠀⠬⠀⠈⠓⢆⡠⠀⠀⢦⣿⣿⡆⣤⣾⠆
⠀⠈⣇⠀⠀⠀⠀⣬⡿⠄⢰⣤⡄⠀⢈⡗⠋⠓⠀⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⣼⠧⠊⠈⠏⠁⡎
⠀⠀⢿⠀⠀⠀⠀⡌⢁⢠⠸⢉⢿⡜⠲⠎⡀⠳⣏⠉⠣⠆⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣷⡟⢰⣻⠔
⠀⠀⠠⡀⠀⠀⠐⣷⣬⡀⣦⣨⠾⠃⢠⣼⡀⢀⣀⠷⣤⠌⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⣘⣏⠉⠁
⠀⠀⠀⠀⠐⣠⠀⡬⠹⣧⠛⠁⠀⠀⠈⣠⠟⣛⣉⣀⣤⣁⠀⠀⠀⠀⠀⠀⠊⠉⠀⢠⡿⠇⠸
⠀⠀⠀⠀⠀⠉⠀⠗⠀⠀⠀⠀⠀⠀⠀⠧⢴⠿⠛⢋⢁⡼⠀⣀⡤⠀⠀⠀⠀⠀⠀⠤⠒⠆⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⢴⠞⣁⣴⠞⠋⠀⠀⠀⠀⠀⠀⠐⠂⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣤⢾⠻⣅⣀⣀⠀⠀⠀⠀⠀⠒⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⠛⢩⣤⡜⠻⠟⢣⣐⠂⣐⠍⠛⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⣀⣀⠛⢡⠖⢠⣄⣀⣁⣌⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢀⡴⢿⣷⣿⣛⠻⠶⣶⣶⣮⣿⣿⣶⠿⣧⠖⣶⠄⠀⠤⠀⠤⠤⣄
⠀⠀⠀⠀⠀⠀⣀⣴⣀⠠⠴⢥⣶⣶⢉⣡⣿⣟⡙⠻⠿⠶⣬⣭⣽⣶⣿⠷⣄⢣⣤⣟⠓⠒⠶⣷⣴⣶⢶⡖⠂
⠀⠀⠀⠀⠀⠈⠉⠙⠋⠀⠠⠾⠋⠁⠀⠛⠛⠋⠉⠙⠓⠀⠘⠛⠛⠉⠠⠞⠉⠈⠉⠉⠉⠲⠶⠺⣏⡀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⡀⣄⣀⡀⣠⣰⡢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣝⢿⣷⣿⡅⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡻⣷⣽⣿⣿⣮⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣘⠛⠻⢿⣿⣿⣿⣿⣿⣮⣻⣿⣿⣿⣿⣩⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡀⠀⠀⣿⡟⠃⠀⠀⠻⣿⣿⣯⡤⠻⣷⡀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡳⣦⣴⣀⣴⢛⣛⡳⣦⡙⣿⡀⢱⣄⡽⣿⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣟⣼⣧⠘⠻⢃⣿⣿⢹⣷⠛⠛⡄⠘⠆⠳⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⠁⠙⠛⠛⢻⣿⣿⣴⣿⣧⡤⠛⠠⠀⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣤⣄⣄⣠⣬⣵⣾⣻⣿⣿⡗⠀⠱⡀⠅⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣶⣭⣿⣿⣿⣿⣭⣭⡿⣿⣷⠀⢠⠽⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢌⢿⣿⣿⣿⣷⣿⣈⢉⢈⣦⣷⣾⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡙⢦⣈⠛⢿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠈⠙⠛⠻⠿⠇⠀⠂⠀⠀⠺⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠎⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡎⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢧⠀⠀⡀⠄⠚⠓⠒⠒⠂⢄⠀⠀⠀⠀⠀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠑⠪⠄⠀⠘⠷⣦⡀⠀⠀⢑⣤⡀⠀⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠉⠙⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠒⠠⠀⠁⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠁⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠒⠒⠂⢠⣽⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⢀⠄⠤⠀⠁⢊⠄⠊⠁⠉⠉⠈⡺⠃⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠉⠂⢄⠀⠀⠀⠀⣸⡦⠀⡀
⠇⢀⣠⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⡟⠓⠠⢀⠑⢄
⠀⠉⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠐⢄⠠
⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠈⠄⠂
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠔⠀⠻⡄
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠈⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠤⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣒⣤⣄⣀⣀⣀⣀⣈⣐⣂⣀⣤⣀⣀⣀⣂⣁⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠⠉⠢⢀⠀⠀⠀⠀⢀⠠⠐⠂⠁⠁⠀⠀⠀⠀⠉⠐⠠⢄⠀⠀⠀⢀⡠⠊⠀⢡⡀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁⠂⠄⡀⠑⠢⡠⠊⠁⠀⠀⡠⠐⠈⠀⠀⠀⠈⠐⢄⠀⠀⠈⠂⢔⠁⢀⠄⠊⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡰⠊⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠈⠂⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⠑⠈⢀⠀⠀⠀⢠⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠰⠀⠀⠀⠐⠀⠂⠀⠐⢧
⠀⠀⠀⠠⠐⠠⡀⠀⠀⠀⠀⡘⠤⢃⠀⠁⠃⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠰⢀⢁⠉⡀⠀⠀⠀⢀⠔⠀⢢
⠀⠀⠀⢁⠀⠀⠈⠄⠀⠀⠀⠃⠀⠅⠅⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠀⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠔⠀⠇⠀⠀⢀⠃⠀⠀⠘⢀⠀⡀
⠐⠁⠈⠐⢦⡀⠀⢰⠀⠀⢀⡤⠐⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠑⠠⢄⠀⠀⢸⠀⠀⡴⠋⠀⠀⠈
⢇⠀⠀⠀⠀⠙⠀⠈⠀⠈⠁⡠⠂⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠢⡀⠈⠁⠁⠀⠈⠀⠀⠀⠀⡠
⢀⠁⠖⠂⠤⠄⠀⠀⢀⠔⠊⠠⠀⡈⠀⠀⠀⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠈⡀⢀⠁⠑⠤⡀⢀⣀⠒⠐⠛⠁⠂
⢀⠀⠀⣀⣠⠂⠀⠈⠑⠠⡀⠈⡤⠁⠰⠀⠀⠀⠀⠀⠘⠁⠒⠠⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠘⠀⠐⡌⠀⡠⠐⠉⠀⠀⢱⠦⠄⣀⡠
⠀⠈⠀⠀⠄⠂⠄⣀⡀⠀⠈⠒⠁⠀⡀⠀⠀⠀⠀⠀⠃⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠊⠀⢀⣀⣠⠴⠡⠂
⠀⠀⠀⠀⢀⠂⠉⠉⠉⠉⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠐⡀⠀⠀⢀⡠⠔⠀⡄⠰⡑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⢠⠀⠒⢤⡀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠉⢁⠋⠀⠀⡸⡀⠀⡁⢀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢘⠀⢀⢆⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠢⠠⠔⠀⠀⠁⠀⠈⠀⢀⠉⠒⠠⠄⠀⢀⣀⠀⢀⣀⣀⠀⠀⠤⠰⡊⠀⢁⠀⠀⠈⠁⠀⠑⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⡀⠐⠤⣴⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣈⣠⠐⢂⡒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠁⠀⠀⠀⠀⣀⡀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠁⠀⠀⠀⠈⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⡀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠢⠀⣀⡀⠌

⢀⣤⠀⣔⣃⡠⠤⠤⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡀⠀⠀⠀⡴⢦
⠘⢬⡎⠉⢀⡄⡀⠀⠔⠩⢍⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠙⣦⠎⢱⣠⣄⠀⠀⠀⠀⠀⠀⠀⡎⠀⢱⠀⠀⢰⠇⢸
⠀⠀⡀⢠⢬⡀⠁⣀⣃⣈⣁⣀⠀⠀⠀⠀⠀⢀⡖⠢⠴⠃⠀⠋⠀⠘⠁⢸⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠉⠉⠉⠀⠈⡆
⠀⠀⡇⣷⠀⠙⠫⠿⡼⣇⠧⠟⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⢠⡎⠰⠰⠆⠰⣂⣒⡏
⠀⠀⡇⠙⢄⣀⡀⣲⡐⡤⣂⠔⣆⠀⠀⠀⠀⣘⠰⣍⣱⣀⠝⠀⠯⠮⠝⢺⡴⡄⠀⠀⠀⠀⢸⠘⢞⣣⠛⣤⢣⠟⣜⣧
⠰⣆⡇⠀⠀⠀⠉⠉⠉⠉⠁⠀⣇⣴⠆⠀⠀⡏⠐⠲⠿⠶⠷⠮⠷⠾⠽⠞⠒⠛⠳⡄⠀⠀⠸⡄⠀⠀⠀⣀⣁⣁⣀⡠⠁
⠀⠉⠑⠤⣀⡀⠀⠀⠀⠀⣀⠴⠁⠀⠀⠀⠀⣧⡤⢤⡀⣀⣀⡀⢀⣀⣀⢀⣀⢀⣀⡝⠀⠀⠤⢵⠀⠀⠈⠉⠉⠀⢠⠅
⠀⠀⠀⠀⠀⠡⡌⠉⡽⠏⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠚⠀⠀⠣⠏⠉⠙⠼⠙⠞⠉⠃⠀⠀⠀⠀⠣⠜⠀⠀⠈⠻⠍
⠀⠀⠀⠀⣀⡄⠀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡐⡄
⠀⠀⠀⡼⠁⣁⠐⢁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢡⠍⢲⠼⠶⠞⠁⢠
⠀⠀⢀⠗⠀⠉⠀⠈⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⡀⠀⠀⠀⠑⠶⣾⢁⣄⢀⡄⢀⣀⡙⡽⠄
⠀⢠⠃⡤⢤⡦⡄⢰⠱⢋⠎⡆⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⡞⠁⠀⠀⠀⠈⢢⡀⠀⠀⠀⢸⠈⠉⠉⠙⠛⠣⠯⢼
⠀⡎⢀⡉⣉⠉⣁⡀⣩⣁⢄⡇⠀⠀⠀⠀⠀⢰⢻⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠇⠘⠚⠤⠿⠤⠿⠴⠯⠜⢳⠀⠀⠀⠀⠀⠈⡎⢧⡀⢀⠇⠀⠘⡗⠀⠀⠀⠘⠦⡠⠤⠃⠀⠀⠀⠀⠀⠀⢰⠃
⡰⢋⣀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢉⠆⠀⠀⠀⠀⠙⢄⡉⠁⠀⣠⠇⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠑⠉⠈⢲⠄⣀⣀⡀⢠⡀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⡉⡦
⠀⠀⠀⠘⠜⠁⠀⠀⠀⠑⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠌⠙⡖⢢⠤⠤⠀⠒⠒⠛⢄⠜
⠀⠀⡀⠀⢀⣴⣧⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠉⠉⠀⠀⢀⡄⡀⢠⠀⠤⠤⠤⠤⠤⣄⣄⣠
⠀⠈⢯⠒⢁⡤⡄⠀⢏⣂⠶⣀⡠⢤⠎⢳⡀⠀⠀⢀⠔⠒⢤⠖⢲⠎⣀⠉⢱⡀⠀⠀⠘⠋⠸⢠⡪⠏⣱⠀⡖⠶⠢⢸⠈
⠀⠀⠀⡇⠘⠤⠂⠀⣠⢲⡶⠭⠧⠚⠒⠁⢸⠀⠀⢺⡘⣃⠴⡄⠘⢄⣉⣠⠜⠁⠀⠀⠀⠀⡇⢀⢩⡭⣄⢤⢈⣭⡥⠚
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⡞⣷⢢⡜⡏⠀⠀⠀⠀⠀⢀⠴⠃⠀⠉⠑⠉⠋⢹⣉
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞⠀⠀⠀⠀⡰⠋⠉⠉⢹⠃⠀⠀⠀⠀⠀⠷⢹⠀⠀⠀⠀⠀⠀⢸⠳⡇
⠀⠀⠀⠈⢙⢲⢲⠀⡎⠈⢳⢸⠉⡏⡟⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⡀⠀⠀⣀⡼
⠀⠀⠀⠀⠸⡍⠸⠀⠃⠀⠸⠎⠀⠥⠀⠀⠀⠀⠀⠀⠀⠀⠱⣲⠖⢜⡄⠀⠀⠀⠀⠀⠀⠀⠀⠵⠉⠉⠹⡍
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⢀⣄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠲⠂⢀
⠀⠀⠀⠀⢠⡤⠋⠉⠈⠋⠙⠃⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠸⢻⡇⠀⠀⠀⠀⠀⠀⢠⡖⢊⠐⣦⠀⠀⠀⠡⢸⣷⠄⡎⡂
⠀⠀⠀⣴⠊⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⡧⡀⠀⢀⡸⡌⠀⠀⠀⠀⠀⠀⠀⠰⣆⠚⢃⢼⠀⠀⠀⠢⡙⠉⡚⠵⠁
⠀⠀⣠⠋⠀⠀⠛⠀⠛⠀⠘⠀⠀⠀⠸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢉⠀⠠⢀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠣⡈⢣⡀⠀⠀⡇⢸⠁
⠀⠀⢹⡁⠀⠀⠀⢠⣄⣀⡄⠀⠀⠀⢀⡕⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁⠠⠷⠈⡆⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠱⣄⣠⠁⡇
⠀⠀⢈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡅⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠑⠀⠀⠊⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠂⠈⠀⠀⠙⢦
⠀⠀⠈⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⣁⣋⣀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⣀⠀⢀⣈⡶⢣⣀
⠀⠀⠀⠀⠘⠓⣤⣄⣠⣀⣀⣤⣜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠙⠉⢣⠎⠉⠀⠀⠀⠈⢂⡀
⠀⠀⠀⠀⠀⠰⠿⡿⡇⠀⠻⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠆⠠⠀⡌⠸⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠤⠤⠤⠒⠂⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⣁⡀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⢤⣦⡀⠀⠀⠀⢠⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠉⠙⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠰⣀⡓⠀⠂
⠀⠀⠀⠀⠠⠘⠃⠓⠒⠒⠒⠚⠚⠂⠀⠀⠀⠀⠀⠀⢀⡔⠚⠁⠀⠀⡆⠀⢠⡄⠀⠀⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠙⠃⠀⠄
⠀⠀⠀⠀⠀⠐⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢰⡆⠀⢠⠀⡠⠄⠀⠀⢰⠀⠀⡶⡀⢣⠀⠀⠀⠀⠀⠀⠀⠈⠑⢀⠴⢀
⠀⠀⠀⠀⠄⠀⠀⠀⠻⠿⠟⠀⠀⠀⢠⠀⠀⠀⠀⠸⠀⢸⢃⠀⠸⣾⣿⣾⣷⣿⣾⠀⠀⡇⠇⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁
⠀⠀⠀⠜⡆⠀⠠⠄⠄⠠⠄⠀⢀⠀⠛⡂⠀⠀⠀⢰⣤⣼⠈⢦⠀⠙⠸⠃⠙⠘⠃⢀⡖⣹⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠈⠀⢄⠀⢀⡀
⠀⠀⠀⢈⣄⠀⠀⠀⠀⠤⠤⠔⠃⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⣠⣄⠀⢰⠋⠀⠈⠉⠁⠀⠀⠀⠀⠀⠾⠷⡔⠁⣀⠀⢀⡀⠉⠚⠃
⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇⢸⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠉⠁⡀
⠀⠀⠀⠀⠀⠀⡀⠄⠡⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠀⡇⢸⡀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⡀⢀⠐
⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠁⠀⠶⠂
⠀⠀⠀⠀⠀⠀⠲⣄⡀⢀⣀⡸⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣤⡀
⠀⠀⠀⠀⠀⢀⡖⣀⠀⠀⢀⣄⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡵⠖⠂⠀⠒⠣⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠃⣀
⠀⠀⠀⠀⠀⣦⠀⡋⠀⠀⢀⣉⠠⢃⠀⣀⢤⡀⠀⠀⠀⣠⣤⡠⠏⣴⣶⠄⣴⣻⡓⣜⢂⢰⡻⠄⠀⠀⠀⠀⢠⡄⠀⠠⣶⠀⠀⢰⡖⠀⠀⠙⡄
⠀⢼⠃⡀⠐⠀⠀⠻⣾⣿⠿⠏⠀⠈⠁⡤⠠⠇⠀⠀⠀⠈⢤⣧⠀⠛⠛⠉⠿⣽⣧⠏⢸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢡
⠀⠈⠋⠁⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣶⣶⣦⣿⡄⠀⡼⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠙⠋⠿⠛⠁⡼⠁⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⢋⡀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠙⠦⡀⠀⡠⠚⢡⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠨⠀⢰⠂⠰⡄⠸⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠰⡀⠈⠚⠁⢰⣈⣆⠀⠀⠀⠀⠀⠀⠀⠳⡤⡠⣀⢄⡤⠠⠤⢄⠠⣤⠄⠴
⠀⠀⠀⠀⠀⠁⠀⠀⠘⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡴⠔⠂⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⢠⠾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⢁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠖⠉⣀⣀⠘⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⠰⠐⠤⠠⠤⢀⡈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⡵⣏⡇⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠠⠄⠀⠲⠂⠰⠆⠐⠖⠁⡤⠈⠂⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣈⣉⣴⣶⣤
⠀⠀⠀⠀⠀⡀⢀⣄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣯⣝⣛⣫⣹⡾⢋⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣶⢾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠤⡴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠉⠙⠉⠉⢠⠉⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠚⡵⠋
⠀⠀⠀⠀⠀⠀⣼⠆⣶⠆⢺⡎⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠶⣉⡰⠂⡐⠏⡁
⠀⠀⠀⠀⠀⠀⠉⠀⠙⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠃⠘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣿⣿

⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢠⡄⢀⢄⢠⡀⣠⡀⠀⠀⠀⠀⣀⣤⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣀⣀
⠀⠀⠀⣠⡾⠛⠉⠛⢷⡀⠀⠀⠀⠐⠷⣤⠶⠿⢿⣧⡀⠀⠀⠀⢀⣈⣿⣿⡾⣾⣿⠯⢄⠀⠀⠀⢰⣟⠀⣦⠀⣹⠇⠀⠀⠀⠀⠀⣀⣠⠞⢿⢻⡿⢷⣶⠃
⠀⠀⣰⡟⠘⠓⠘⠓⣈⢿⠀⠀⠀⠀⢸⡇⢰⣧⡄⣻⣿⠀⠀⠀⠸⣖⣽⡇⠶⢌⣿⣯⢎⠀⠀⠀⣼⠿⠶⠶⠞⢋⢠⣄⠀⢰⡿⣿⢉⣿⡾⣾⣉⡷⣾⠗⢶⣤⣀
⢾⣇⣿⢸⣷⣿⣾⣿⡿⢸⣇⡶⠀⠀⠘⣿⣦⣤⣴⣿⣿⡇⠀⢀⡳⣾⢛⠛⠷⢛⡛⣿⣏⡤⠀⢸⡇⠀⢀⣀⣀⠙⣿⠉⠀⠀⠀⠀⢸⣿⣿⣿⣽⣿⡟⠀⠐⠏⠿
⠀⢸⣿⣦⣬⣤⣥⣤⣴⣿⡏⠀⠀⠀⠀⣿⣿⣿⠠⠆⢹⡇⠀⠈⢻⣿⢐⣄⢀⡈⣁⣿⣿⠁⠀⠈⠻⣿⡿⠿⣿⡟⠁⠀⠀⠀⠀⠀⣿⠟⠉⠉⠻⣿⡇
⠀⠘⠿⢿⡿⢿⡿⣿⠿⣿⠇⠀⠀⠀⣰⣿⣬⣿⣷⣶⣿⣷⡀⠀⢸⡿⣿⠿⡿⣿⡟⣿⣿⠀⠀⠀⠀⢸⡿⠿⢿⡇⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠈⠁⠈⠀⠉⠀⠉⠁⠀⠀⠐⠋⠙⠋⠙⠉⠙⠁⠈⠁⠀⠘⠃⠟⠘⠃⠙⠁⠋⠘⠀⠀⠀⠒⠛⠀⠀⠀⠛⠦⠀⠀⠀⠐⠚⠃⠀⠀⠀⠀⠀⠹⠶⠄
⠀⢀⣠⣤⣄⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡄
⠀⢺⣿⣿⠿⡆⠀⢠⡾⢿⣿⣿⡄⠀⠀⢼⣗⣠⣤⣤⡽⠛⠀⠀⠀⠀⠀⢸⣧⣀⠀⠀⠀⠀⣀⣿⠀⠀⠀⣰⠛⠛⠛⠦⣄⠀⠀⠀⠐⣧⣤⣶⣶⣦⣼⠇
⠀⠀⠘⠁⢀⣽⠶⠾⣄⠀⠉⠁⠀⠀⠀⢀⣾⢟⠛⡛⠿⣦⠀⠀⠀⠀⠀⠀⠈⢻⣿⢿⣿⡿⠛⠋⠀⠀⠀⡟⠐⠂⠰⢠⡏⠀⠀⠀⠀⢸⣿⣐⣂⡿⣿⡀
⠀⠀⠀⠀⣾⠁⢴⡄⢘⣧⠀⠀⠀⠀⢠⣿⣧⣈⠁⢉⣠⣿⣧⠀⠀⠀⠀⠀⠷⣾⠰⠆⢸⣤⠇⠀⠀⠀⢸⣷⣤⣤⣀⣼⠁⠀⠀⠀⠀⠘⡷⢩⢻⣖⣿⣧
⠀⠀⠀⠀⠹⢷⣤⣤⡾⠃⠀⠀⠀⢠⣿⣿⣿⣿⣟⣛⣿⣿⣿⡆⠀⠀⠀⣠⣴⡿⠛⠺⣿⣦⡀⠀⠀⠀⡌⠀⣿⢹⡏⢹⠀⠀⠀⠸⣇⣠⡟⡲⣎⢍⣿⣿⢀⣠⠇
⠀⠀⠀⠀⠀⡴⠁⠙⠦⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠸⠇⢸⣷⣿⣷⡇⠸⠃⠀⠀⣰⣀⢠⡇⢸⡇⢸⠀⠀⠀⠀⠀⠸⣷⣶⡿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⢸⠟⠛⠻⣇⠀⠀⠀⠀⠓⠀⢸⡇⢸⠀⠹⠷⠀⠀⠀⠀⣶⣴⣤⣴⣿⣿⡟
⠀⠀⠀⠀⢀⣼⠿⠻⢷⣶⠃⠀⠀⠀⠠⠏⠩⠏⠹⠏⠹⠍⠳⠄⠀⠀⠀⠀⠛⠀⠀⠀⠹⠀⠀⠀⠀⠀⣀⣤⠇⠸⣤⡀⠀⠀⠀⠀⠀⠈⣛⣿⠿⣟⠋
⠀⠀⠀⠀⢾⡇⠸⡶⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⢀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⢈⣿⠶⢶⡾⠋⠀⠀⠀⠀⠀⠀⠸⣏⣀⣀⡾⠃⠀⠀⠀⠀⠀⢰⢀⣠⣤⣾⣄⠀⠀⠀⠀⣆⠉⢀⣠⣤⣤⣀⣀⡤⠋⢀⠄⢄⠀⠠⣉⠌⠀⣥⣂⠅
⠀⠀⠀⠀⣿⡁⠶⠄⣿⠀⠀⠀⢠⡆⠀⣠⣾⠟⠛⢿⡗⠀⣤⠀⠀⠀⠀⠀⠿⣟⢡⠙⣯⣂⠀⣀⠀⠈⢻⣿⠉⠆⠆⢹⣿⠀⠀⠘⠈⠺⣄⢰⡇⢠⡞⠁
⠀⠀⠀⠀⢨⡿⠒⢾⡇⠀⠀⢀⡘⢷⣴⣿⣯⠐⠷⢀⣷⡴⠋⢀⠀⠠⣤⠘⣵⣿⠶⢻⢿⣿⡴⠋⠀⠀⡜⠛⡶⠶⢶⠾⢏⠀⠀⠀⠀⠀⢸⣾⣧⣿⡀
⠀⠀⠀⠀⢸⣧⣛⣼⠇⠀⠀⠘⢿⣄⣼⣿⠿⣷⣶⣿⠿⣷⡶⠿⠁⠀⠉⠉⣿⡇⠃⠀⡄⣿⡇⠀⠀⢘⡁⠀⡁⠀⠈⠀⢨⡆⠀⠀⠀⠀⢀⣯⣐⣨⣧⡀
⠀⢤⣶⡂⠹⣏⣊⣿⠐⣶⣦⠄⠀⠀⠹⣿⣄⡈⠈⣁⣼⣿⠇⠀⠀⠀⠀⢠⣿⣤⣥⣤⣴⣾⡇⠀⠀⠈⠁⠐⡇⠀⢾⠀⠸⠃⠀⠀⣴⠟⠛⣿⠟⠉⠉⠙⢷⡄
⠀⠈⢀⣀⡜⠉⠉⠙⣆⣀⠉⠀⠀⠀⠀⠙⠿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⣾⠟⣿⠟⣿⡟⢻⡇⠀⠀⠀⢀⣰⣧⠀⢸⣧⣀⠀⠀⠀⠻⡀⠀⢿⡀⠀⠀⠀⡸⠃
⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠈⠀⠀⠀⠀⠈⠉⠁⠀⠈⠉⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀
⠀⠀⠀⣹⣥⣤⣤⣼⡋⠀⠀⠀⠀⠀⠀⢀⣄⢀⣴⣧⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣄⣀⣀⡀⠀⠀⢠⣤⠤⠒⠒⠒⡒⠶⣷⠀⠀⠀⠀⠀⢉⣤⣤⣤⣤⡍
⢶⣄⣾⣿⡿⠿⢿⣿⢷⣤⠦⠀⠀⠀⣠⣈⣿⡿⢿⡿⢿⣿⣀⠀⠀⠀⢸⡏⢍⣩⣭⣩⠏⣿⠀⠀⢸⣧⣀⠻⠀⠘⣛⣠⡿⠀⠀⠀⠀⣰⡟⠩⠄⠦⠙⣿⣆
⠀⣾⡱⢨⣇⠻⢢⣇⣂⣿⠀⠀⠀⠀⢙⣿⡏⠊⢵⣫⢕⢿⡟⠀⠀⠀⠈⡇⢾⣿⣿⣿⠀⡏⠀⠀⠀⠉⠙⠋⠻⡟⠉⠁⠀⠀⠀⠀⠀⣿⣇⠒⠀⠀⣠⣿⣿
⠀⠙⢿⡿⠿⢻⣿⣿⣿⣿⠀⠀⠀⠛⣿⣿⣮⣃⣞⠭⣪⣾⣿⠃⠀⠀⠀⢧⣈⣉⣩⣬⡴⠇⠀⠀⠀⢠⣤⡄⠀⡇⠀⠺⡟⠀⠀⢠⡞⠻⣿⣿⣿⣿⣿⣿⠏⠙⠂
⠀⠰⣶⣾⣷⣿⣿⣿⡿⠉⢢⣀⠀⠀⣿⣿⣿⣿⣿⠿⢟⣿⣿⠀⠀⠀⠀⠀⢸⣍⣥⣼⠀⠀⠀⠀⠀⠈⠉⣄⢠⣿⣄⠴⠁⠀⠀⠀⠀⠀⠈⢻⠟⣿⠛⣧
⠀⠀⣈⡹⠛⠛⢋⡁⠀⠀⠁⠉⠀⢠⣿⣿⣿⣶⣤⣶⣿⣿⣿⣆⠀⠀⠀⠀⠈⣿⠛⢻⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸⠀⠘⡄
⠀⠀⠈⠁⠀⠀⠘⠁⠀⠀⠀⠀⠀⠺⠟⠛⠛⠿⠿⠛⠻⠿⠛⠛⠃⠀⠀⠀⠾⠉⠿⠳⠛⠂⠀⠀⠀⠀⠀⣠⠛⠛⠋⢦⡀⠀⠀⠀⠀⠀⠶⠗⢀⣸⣄⠀⠿⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢯⠲⣄⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⣀⡤⠴⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⠳⣩⢷⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⠞⢧⣹⠞⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⢃⣀⣙⣲⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣖⣋⡀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡞⠋⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠹⡄⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢠⠏
⠀⠀⠀⠀⠀⠀⠀⢠⢤⡖⢲⢀⣀⡀⠀⠀⢹⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⡏⠀⠀⠀⠀⠀⢀⣄⡀⡀
⠀⠀⠀⠀⠀⠀⢰⠾⡌⠳⠘⢋⣨⠇⠀⠀⠘⡇⠀⠻⣤⣤⠴⠶⠒⠲⠦⣴⠃⠀⢸⡇⠀⠀⠀⠀⡖⠻⠸⠋⣿⡄
⠀⠀⠀⠀⠀⠀⠈⠲⠶⣄⠀⠈⢧⡀⠀⠀⢀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⢀⣽⠆⠀⡴⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠈⠳⢤⡞⠀⠀⣠⣤⢤⡤⢤⡤⠴⣞⠉⡹⠉⠓⡄⠈⠷⠖⠋⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣀⠀⠀⢠⠎⠀⠳⠞⠳⠞⠑⠚⠉⠉⠁⠀⠀⢸⠀⢀⣤⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⠀⠀⠀⢀⡤⢤⣀⡖⠓⠲⡄⠀⠀⡸⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⢇⠀⢠⡇⠀⠀⠘⠁⠀⢀⣹⠀⢠⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⢦⡀⡷⢄⡠⢦⡖⢲⣏⣈⠶⠋⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠉⠓⠶⠓⠋⠉⠉⠉⠁⠀⠀⠀⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⣀⣀⣀⣀⣀⣠⠤⠖⠒⠦⣄⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⡼⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠱⣄⠀⠀⢹⣄⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⣀⠲⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠧⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢤⣤⡼⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣢⠂⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠁

⠀⠀⠀⠃⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⣶⣶⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⢸⣣⣼⣦⣿⠀⠀⠀⠀⠀⠀⠀⠈⠀⠿⡿⢿⠟⠿⠆
⠀⠀⠀⠀⠀⢠⣶⣿⠞⢹⢻⣿⡦⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⣠
⡀⠀⠀⠀⢀⣟⠛⠀⠀⢰⠀⠋⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⢂
⢿⠀⠀⠀⠾⣿⡆⠂⣄⣸⠀⠈⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠓
⣿⡆⠀⠀⢸⣿⣃⠀⠙⠛⠀⠀⣽⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣡⣴⠀⠀⠀⠀⠀⣼⣾⣷⣆⢠
⢿⠃⠀⠀⠀⠼⢿⣶⣶⣖⣦⣼⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠸⣿⠟
⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣶⣶⣶⢰⣶⣶⠎
⠀⠀⣀⡀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠃⠀⢻⡟⠀⠀⠀⠀⠀⠀⠠⣄⠠⣶⠖⣀⠂
⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁⠈⢣⣶⣦
⠀⠀⠀⠀⠠⠶⠄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⠀⠠⣽⢹⣿
⠀⠀⠀⠀⠀⠀⠄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣿⡮⣭
⠀⠀⠀⠀⢀⠀⡀⠀⠄⠇⠀⠀⠀⠀⣀⣀⠔⠒⣒⠒⠠⣄⣄⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠛⠋
⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠈⠙⢏⢀⠀⢐⠀⠀⢸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢰⣿⡷⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⣿⣷⣾⡧
⠀⠀⠀⠶⢤⣤⡤⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠛⠛⠛⣿⣷⣿⠆
⠀⠀⠀⠀⠸⢿⡿⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⣰⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⡏⠀⠀⠀⣿⣿⣧⠄
⠀⠀⠀⡖⢢⣤⣴⠐⡇⠀⠀⠀⠀⠀⠀⠀⠓⠒⠉⠉⠖⠚⠀⠀⠀⠀⠀⢨⣿⣿⡇⠀⣶⠄⢿⣿⣷⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣦⣄⢀⠀⣠⣆⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⡠⠀⠘⣿⣷⣺⣯⣟⣾⣿⣾⣥⣿⣿⣷⣷⠂⠀⠀⠀⠀⣠⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠇⢀⠀⠀⣀⣸⣷⣾⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⠉⠉⠓⠦⣈⠀⠀⠘⢻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣏⠀⠀⢀⡤⠞⠉⠋⠛⠉⠿⠻⠛⠿⢻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⣀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⡿⠁⠀⠀⣀⡤⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⢤⣀⠀⠀⠀⠹⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⣠⠊⠁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢳⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢰⠃⠀⠀⣠⣶⣶⣤⡀⠀⠈⢧⠀⠀⡞⠀⠀⢀⣤⣶⣦⣄⠀⠀⠘⡆⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⠀⠀⠌⣿⣿⣿⡁⣹⠀⠀⣸⠀⠀⡇⠀⢠⢿⣿⣿⡉⢙⡆⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⣆⠀⠰⡙⠿⣿⣿⡿⠀⢠⠇⠀⠀⠱⡀⠈⢜⠿⣿⣿⡿⠃⠀⡸⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡀⠀⠀⠈⠳⢤⣈⡀⠀⣀⣠⠔⠁⠀⣴⡄⠀⠙⠦⣄⣁⠀⢈⣀⡤⠞⠁⠀⠀⢠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⣇⠀⠀⠀⠀⠀⢠⡭⣭⡥⠤⠤⣤⣤⣤⣤⠤⠤⠤⣶⣾⠻⣍⠀⠀⠀⠀⠀⠀⣸⡟⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣻⣿⡄⠀⠀⠀⠀⠀⢿⣿⣿⣄⣠⣿⣿⣿⣿⣧⣀⣼⣿⣿⣿⡿⠀⠀⠀⠀⠀⢠⣿⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢈⣿⣷⡄⠀⠀⠀⠀⠀⠙⠻⢿⣿⣟⠁⠙⣿⣿⣿⣿⣃⡤⠟⠁⠀⠀⠀⠀⢀⣾⣏⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠏⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣴⡟⠹⠻⠾⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⢮⣤⣄⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣦⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠉⢹⣿⣷⣷⣼⣶⣴⣦⣦⣤⣴⣷⣶⡿⠋⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠍⢽⡇⠉⠀⠙⠋⠈⠁⢸⡟⢻⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣹⠛⠀⠀⠀⣧⣤⣤⣤⣤⣤⣤⣼⠀⠀⠀⠻⣽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠓⠒⠒⠿⠿⠿⠿⠿⠿⠿⠛⠛⠗⠒⠒⠚⠛⠛⠛⠋


⠀⠀⣀⠀⢀
⣿⣯⢭⣟⡛⣛⣛⡛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢲⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠈⢄⡀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄⠀⠀⠀⡻⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡄⠁⠀⠁⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠤⠤⣾⣿⣟⣶⣃⠀⠊⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠠
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠆⠑⣤⡟⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡿⢿⣿⠁⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⣤⡏⠀⣼⠇⠀⠀⠀⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠁⢸⣦⠀⠀⠀⣀⠴⠃⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⣀⡄⠘
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠘⠋⠀⣴⣿⠃⠀⠀⠀⠁⠀⠀⠀⠘⡿⢽⡢⢄⡀⠀⣀⡆⠸⣿⣷⣷⣼⠋⠀⠀⠀⠀⡄⠀⠀⢣⠀⠸⠇⠤⣿
⠀⠀⠀⠀⠀⠀⠀⡻⣦⡀⠘⣶⠋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⢹⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠁⠀⠀⠘
⠀⠀⠀⠀⠀⣎⢰⡇⢸⣿⡆⢹⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠁⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⢸⣯⠀⡘⠉⣷⡀⢹⡄⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⡀⠄
⠀⠀⠀⠀⠀⠈⠁⠘⠁⠈⠙⠳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠀⠀⠀⠀⡀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⡀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣰⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⠶⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⣷⣴⡿⠇⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⠒⠶⠶⠶⠶⠾⠑⠛⠁⠀⠀⠀⠠⠒⠛⠹⠉⠃⠁⠙⠁⠀⠀⢢⠄⠋⠉
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣹⣄⠀⠼⣿⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣰⠀⠀⠀⠀⡀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⢸⡦⣤⠀⢠⣤⠿⠛⢿⠀⢹⠀⡁⠀⠀⠀⢀⡌⣠⣦⠀⠂⢸⠤⠀⡀⣴⠆⢀⡴⠀⠀⠀⠀⢠⣿⠀⣰⠟⠦⠀⠀⠔⠃
⠀⠀⠀⢠⠊⢀⡾⠀⠈⠀⠀⠀⠀⠀⠈⣆⢘⣴⠃⠀⠀⢀⣼⣴⠋⡰⠃⠀⠀⠀⠀⠀⠀⠀⠎⠇⠀⠀⠀⠀⠘⠁⣼⠁⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⣼⣠⡏⣇⠀⡂⠀⠀⠀⠀⠀⠀⢻⣸⡟⠀⠀⠀⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠸⣁
⠀⠀⠀⢽⣿⡄⠛⠛⠛⠁⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⢀⣴⣾⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⣥⠞⣠⣴⣿⣿⣿⣿⣿⡿⠟⡀⠀⢶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣤⡀⠈⣳⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣰⣾⣿⣿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⠿⢻⣡⣾⣿⣿⣻⣯⣽⣿⡿⣿⣻⣿⣿⣿⣿⠿⢃⣈⠁
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣶⣾⡿⠿⢿⡿⣿⣿⣿⠐⣎⣿⣿⣯⣴⣬⣦⣶⣾⡏
⠀⠀⠀⠀⠀⠀⢀⢘⡿⠿⠟⢋⣹⣴⣾⣿⡿⠭⠛⢿⣷⡿⠁⠹⣿⣿⣿⣰⣿⣯⡿⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⣿⣿⣿⣿⣿⣿⣧⠀⠀⡼⡌⠀⠀⣸⣿⣿⣿⡯⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⡟⣷⠙⠻⠿⣦⢸⣿⣷⡀⠠⠟⣿⢱⢻⣿⠂⣽⡧⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⡀⣸⣿⣄⢀⣄⠀⣉⠀⢀⣤⠀⢀⣀⡈⠁⣿⣄⣿⡿⢏⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢠⡏⢻⣿⣿⠟⣿⡅⣿⣷⣶⣽⣿⣷⢴⣿⡿⢹⣇⣘⢿⣧⡾⢚⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠐⠻⣤⣷⣿⣿⣴⠘⠿⠾⠿⠛⠫⠉⢉⠛⠿⢖⣻⣾⢿⣼⣟⡱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠻⣿⢿⣿⢻⣦⣶⢦⣄⡙⠃⠋⣰⡤⣾⣿⣣⣽⣷⢚⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣋⣿⣿⣗⢬⡾⣽⣷⢹⠿⣍⣏⣿⢯⡗⣻⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⣛⣓⣺⣽⣟⣙⣻⣏⣥⡿⣶⣿⣧⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⣱⣟⡿⢿⣿⣿⣟⣫⣥⡶⢦⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⢁⣠⣭⣴⣿⣽⣿⣯⡶⠈⡴⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠉

⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⣿⣶⣦⣄
⠀⠉⠉⠉⠛⢿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣽⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⣀⣤⣤⣶⣦⣤⣌⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉⠀⠀⠀⠈⠉⠻⣿⣿⣿⣿⣿⣧
⠾⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⣦⣤⣄⣀⣀⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣟⣛⣛⡻⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠙⢿⣮⣙⠻⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⢿⣿⣿⣿⣧⠀⠀⠀⢀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⢻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣧⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⢸⣿⣿⣆⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣇⢰⣿⣿⣿⣷⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣼⣿⣿⣿⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢰⡆⠀⠈⠙⠻⢿⣿⣿⣿⡿⠟⠋⠁⠀⢰⡆⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⢀⣹⣿⣏⡀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⣿⣿⡄⠀⠈⠁⠀⢀⣠⣴⣿⣿⣿⣿⣿⣶⣄⡀⠀⠘⠃⠀⣠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⣿⠿⣷⣶⣶⠟⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡏⠀⣼⣿⠇⠀⠘⠀⠈⣿⡿⢿⣿⣿⢿⣿⣿⣿⣿⣿⡟⠙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⣿⡇⠀⣿⡿⠀⠀⠀⠀⠀⠟⠀⠘⣿⠁⠀⢿⡿⠁⢸⣿⡇⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠃⠀⠰⣿⠃⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⢠⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣿⡄⠀⠀⠀⢀⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⡀⠀⠀⢰⡇⠀⢠⣿⠀⠀⣸⣿⣿⣀⢀⣰⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣤⣿⣿⣤⣾⣿⣧⣤⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠉⠉⢹⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢻⣿⣿⣿⠀⠀⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣾⣿⣿⣿⡟⢀⣠⠼⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠵⠿⠿⠟⠋⣰⣿⠴⣾⣿⣿⠟
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠁⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣋⡤⠐⣀⣠⣔⣾⣿⣿⡦⣥⡀⠉⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠘⢃⡴⣾⣽⣿⣿⣿⣿⣿⣿⣻⢿⣷⣤⣦⣵⡔⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠰⠃⠐⣩⣾⣿⡿⠛⠛⠋⠸⠋⠍⢛⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣮⡞⠠⢟⣴⣿⢿⣋⡙⡀⡀⠀⠀⣤⡀⣀⡁⢨⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣏⣴⣳⣿⣿⣿⣿⣛⣛⣿⠇⠀⠐⠛⣿⣟⣛⣶⣾⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⣿⣿⣿⣿⣿⣿⣿⢋⠉⠈⢁⣤⣤⣤⣀⠀⠈⠙⢫⣙⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⣿⣿⣿⣿⣿⣿⣿⣷⠆⠀⠰⢒⣿⣻⣿⣁⡀⠀⠄⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢰⣿⠟⢻⠉⡛⡿⣦⣰⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣷⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢓⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⡄⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢀⣾⠛⢋⣿⣿⣿⣿⣿⣿⣟⣿⣻⣟⣻⣿⣟⣿⣿⣿⣍⣿⣿⣿⣿⣿⡘⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⡆⣼⣿⣿⣿⣿⣿⣿⡝⢻⡼⣯⣭⢵⡿⢿⣿⣿⣾⢻⣿⣿⣿⣿⣷⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣼⣿⠿⠿⣿⣿⡿⣿⣿⣧⣠⣀⠀⣄⠀⢀⣼⣿⣿⡷⢸⣿⣿⣿⣿⣿⣟⡿⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢙⡏⠀⠐⢻⠟⠀⢹⣿⣿⣿⣿⣿⣷⣾⣿⣽⣿⣿⡇⠈⠻⣿⣿⣿⠉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡤⠀⣸⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⣤⠀⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣼⡇⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠹⣿⡟⠀⠿⣿⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣠⣤⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣋⣹⣯⣿⣿⡏⣽⠷⢦⢋⣠⣄⠐⠀⠻⡌⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣕⠛⣛⣛⣉⣩⣿⣿⣿⣿⣿⣿⣶⣆⣄⢸⣿⣷⣦⡳⣍⢲⢷
⠀⠀⠀⠀⠀⢀⣀⣶⠀⡿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣷⡳⠟⢿⣿⣿⣿⣷⢸⢧⠤
⠀⠀⠀⠀⠀⢳⣿⣿⣟⣴⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣻⣿⣿⣿⣿⡄⢴⣄
⠀⠀⠀⠀⢠⡏⣿⡿⠛⠁⠞⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢪⢁⢳
⠀⠀⠀⠀⣾⣹⣿⠁⢸⣅⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢦⠙⣎⡃
⠀⠀⠀⢠⣿⡿⠃⠀⢸⢇⣀⣿⡿⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢡⣳⣾⡄
⠀⠀⡠⣼⣿⣷⣀⡔⠃⣺⣿⣿⠇⢯⣿⣿⣿⣿⣿⡿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢻⣿⣿⢯⢣⢿⢻⡆
⠀⠀⣆⣿⣿⡿⢣⠀⠀⣿⣿⡏⠀⢠⡉⢙⣿⣿⢟⡰⢋⣼⣿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣯⡀⠌⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠴⠶⠦⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣽⣶⣄⣀
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠉⠉⡶⢤⣀⡰⠋⠀⠀⠀⠙⢯⡻⣦⣀⡀⣀⣀⣀⡀
⠀⠀⠀⢠⠞⣠⠖⠋⠉⠉⠉⠙⠓⠶⣄⡀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⣳⡌⢻⡙⢯⠀⠉⠉⠻⣦⣄⡀
⠀⠀⢠⢏⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⡀⢧⠈⢣⠀⠀⠀⣷⠀⠙⠳⣄
⠀⠀⣾⡟⠉⠙⡆⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣧⠸⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⢷⡀
⠀⢠⢟⠀⠀⢠⣿⣿⣿⣿⣿⣷⠀⠀⡠⡄⠐⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠞⠉⣿⠀⠂⠀⠀⠀⠀⠇⢀⠀⠀⠀⠀⠙⣦⡀
⠀⡜⠈⠒⠒⣿⣿⣿⣿⣿⣿⣿⡇⠁⠈⠈⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢩⠏⠀⠀⣻⡇⠀⡆⠀⠀⠀⠀⠸⠀⠀⠀⠠⡀⢩⠳⡄
⠀⡇⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣾⠀⣸⠁⠐⠁⣀⠴⠧⣄⠀⠀⠀⠁⠸⠀⢻
⠀⢷⠀⠀⠀⣿⣿⣿⣿⣿⣉⣿⠁⠀⠀⠀⠀⢣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⢀⡟⣰⠟⠋⠉⠉⠁⠀⠀⠈⣯⠁⠀⡇⠀⠀⢸⡆
⠀⠸⡆⠀⠀⠘⠿⣿⣿⣿⠟⠡⢳⣿⡆⣠⣧⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⡧⠀⠀⢸⠃
⠀⠀⢻⣆⠀⡐⠀⠀⠀⠀⠀⠀⠈⢉⣴⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⡇⢠⠂⡞
⠀⠀⠀⠻⣝⠦⣄⣀⣀⣀⣠⣤⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠐⠀⠀⡼⠁
⠀⠀⠀⠀⠈⢷⣄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢁⡴⠋⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠁⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠰⠀⠀⠀⠀⠀⠀⢀⣧⣀⠤⣀⠀⠀⣠⡴⠋⢀⡔⠋⠁⠈⠉⠛⠲⠶⢶⣖⣚⠉⠁⠀⢔⣠⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣄⣀⣀⡀⠀⢀⣀⣀⣤⡤⠿⠛⠁⠀⢀⡎⠀⣴⡯⢤⣄⣀⠤⠀⢀⣀⣤⣤⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣸⠀⣸⠁⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣌⠳⢤⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠞⣶⣢⠀⠀⠀⠀⠀⢰⣢⠔⠒⠒⠒⠒⠒⠲⢖⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢿⠀⠠⠌⠳⣄⠀⠀⠀⠀⢸⠀⠛⢠⣌⡆⠘⠀⠣⣤⠤⢤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⠈⢧⡀⠀⠀⣨⣦⡀⠀⠀⠈⢆⠀⠠⠛⠄⠀⠀⣀⠀⣤⠀⠉⠀⠾⠇⠙⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣶⣿⣿⡉⠳⣄⠹⣦⣴⣿⣾⠇⠀⠀⠀⠀⠙⣶⡖⢒⣌⠀⠉⢀⡀⠀⠛⠃⢠⣄⠛⠃⢸⣦⡀
⠀⠀⠀⠀⢀⣤⠚⠹⡇⠀⢿⠀⠹⡆⠈⠳⡄⠈⠳⡇⠈⠉⠉⠀⠀⠀⠀⠀⠀⢩⡀⣈⠉⢠⡄⠈⢃⡀⠿⠂⣀⡀⢰⡶⠀⡦⣍⣒⠒⣶⣶⡦
⠀⠀⠀⣰⡏⠸⣆⠀⢻⡀⠘⣧⠀⠹⣆⠀⠙⢦⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢱⠉⡷⢠⠊⠉⠉⠉⠉⠑⢻⡁⡎⢑⠸⠃⠀⠀⠈⠉
⠀⠀⢰⠃⡿⣄⡙⢦⡀⢷⡀⠈⣧⣀⠈⠷⣤⢄⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⠀⢰⣼⠀⠀⠀⠀⠀⠀⠀⣷⡇⠸⣶
⠀⠀⠀⢰⣆⢹⢹⡈⡟⠉⠁⠀⠀⠈⢷⡀⡇⠸⢦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣀⣀⣀⣄⣀⣀⡤⣈⣤⡀⢉⡴⠶⢄⡀
⠀⠀⠀⠀⡟⡿⢨⣹⠇⠀⠀⠀⠀⠀⢨⣙⠇⢸⣤⡇⠀⠀⠀⢀⣀⣤⣤⣲⣶⠶⠭⠿⠽⠷⠚⠓⠛⠚⠓⠛⠀⠉⠀⠉⠉⠀⠀⠀⠉⠉⠑⠒⠒⠒⣤
⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⠀⠈⣉⣀⣀⠉⠀⠀⡤⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣋⣹⣉⢭⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠁⠉⠉⠓⢦⡉⠙⠒⠒⠒⠒⠒⠢⠤⣄⡀⠀⣀⠀⢀⠤⠤⠤⠤⠤⠤⠤⣄⠀⣤⡀⢠⠴⠖⠒⠒⠒⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⣀⢀⣀⣀⠀⣀⣀⡀⢀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠱⣼⠉⢦⠏⠀⠀⠀⠀⠀⠀⠀⠘⠴⠁⠣⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠈⢳⠁⢤⢀⠒⠀⣤⠈⡟⠁⢻⠀⠀⠀⢀⡄⢀⡄⠀⠀⠀⠀⠀⠀⠠⡋⠑⠒⠒⠒⠲⢤⡤⢤
⠀⠀⠀⣠⠔⠚⠋⠉⠉⢹⡄⠀⠸⡀⠀⠘⣿⠃⠀⢠⠃⠀⣸⠀⠀⣀⠴⢛⠛⣆⠀⠀⠀⠀⠀⢀⣠⠃⠀⠀⠀⠀⠀⠀⠨⣴⣁⣀⣀
⠀⣰⢿⠁⠀⠀⠀⠀⠀⠀⠳⣀⠀⠑⠦⢄⣀⡡⠴⠋⠀⡰⠃⠀⣞⣉⡀⠀⠀⢸⡿⡄⠀⢠⢊⡉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉⠉⠑⠢⣄⢀⡀
⣰⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⣀⣀⣤⠴⠚⠁⠀⠀⠀⠀⠙⣦⠶⠀⣿⣧⠀⢲⠈⠁⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠏
⠀⠀⢸⢀⡶⡄⢠⡤⠤⠤⠤⠤⠤⣄⠀⠴⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢰⣿⠾⡆⠈⠳⢄⣀⡀⣀⣀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⢸⠀⡇⡿⠀⠀⠀⠀⠀⠀⢹⢸⠀⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⣟⣿⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠘⠞⠀⠳⠃⠀⠀⠀⠀⠀⠀⠘⠞⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢻⠽⡄⠀⠀⠀⢸⠀⠀⣠⡀⠀⢠⠖⠒⠒⠲⡄⠀⢠⡀⠀⢸
⠀⠀⠀⢀⡤⠤⢤⣀⣠⣤⣄⣀⡀⣠⠴⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣴⡼⡗⢶⠀⠀⠀⠸⡀⣀⡇⡇⠀⣸⠀⠀⠀⠀⣇⣀⡾⡇⠀⡸
⠀⠀⣰⠃⠀⢀⠞⠁⠀⠀⠀⠀⠹⡅⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢯⣹⣄⣀⣀⣀⡀⣀⣀⣉⣉⣁⡀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⡇⠀⠀⣜⡀⠀⠀⠀⣀⠀⠀⢸⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⠀⠀⠿⠿⠁⢉⣴⣤⠀⠀⠉⠉⠉⢿⠿⠋⠲⢤
⠀⠀⠹⣄⡀⢫⠁⠀⠀⠈⠉⠁⠀⠘⠤⣀⣀⡠⠎⠉⠛⠲⢤⡀⠀⠀⠀⠀⠀⢸⡿⠃⠀⢠⣤⡀⠈⠛⠋⠀⢰⣿⠆⠀⠀⢰⣿⡆⠈⣇
⠀⠀⢀⣀⠉⠙⣇⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀⠀⠀⠀⠈⣇⠀⠀⠙⠛⠃⠀⢸⣿⠆⠀⠀⠀⢾⡷⠀⠉⠀⡴⢻
⠀⠀⠘⢌⠙⠒⠃⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⣿⢸⢳⠰⡿⠳⠶⠖⠒⠒⠒⠒⠛⠉⢳⢠⢿⢠⠇⢸⡀
⠀⠀⠀⠀⠉⠓⠒⠒⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠇⠀⠀⠀⡇⢸⠘⡇⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢸⢸⢸⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡄⠀⠀⡤⠤⠤⠤⠤⢤⡀⠀⢀⠀⠀⣿⠀⠀⠀⠀⡇⣾⠀⣷⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢻⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⢸⡆⠀⣿⠀⠀⠀⠀⣇⡿⠀⢸⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢹⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡸⢸⡀⢀⠇⠀⠀⠀⠀⠀⢇⠀⡼⡇⠀⡟⠀⠀⠀⠀⢿⡇⠀⠘⣆⡇⠀⠀⠀⠀⠀⠀⠀⠀⢇⡾⢸⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢿⡟⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡴⠔⡛⠻⠷⠋⠀⠀⠀⠘⠧⣀⣀⡶⠁⣀⡠
⠀⠀⠀⠀⠀⠀⠠⠞⠁⠀⠀⣤⠿⠛⠳⣄⡀⠀⠀⠴⣴⠞⠉⢠⠀⠙⠈⠀⣼⡆⠀⠀⠀⢀⡌⠋⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠈⠳⣌⠑⠂⠓⠂⠀⠀⠀⣾⡆⢰⣿⣯⠀⣠⢔⠿⠋⠄⢠⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡠⠀⠀⠀⠀⠀⠀⣌⣷⡦⠋⠀⠔⠃⠀⣿⣷⣾⣿⣿⡎⠀⠁⠀⡠⠐⣋⣴⡿⠛
⠀⠀⠀⠀⢿⣦⣤⡀⢡⢎⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⢀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠂⡀⠀⠴⠾⢋⣀⣄⣀
⠀⠀⠀⠀⠀⠈⠉⢻⣷⣾⠷⠶⣤⡀⠀⢀⣴⡿⠃⠀⣤⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣺⣿⢠⣿⣦⣼⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠈⠳⣬⡁⠀⠀⠈⠻⢾⣿⣏⣀⢀⣸⣿⣧⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⠋⢉⠀⠔
⠀⠀⠀⠀⠀⠀⣀⣶⠀⠈⠙⢶⡀⠀⠀⠀⠀⠉⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢉⣀⣀⠄⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠘⠿⠃⢶⣄⡀⠀⠉⠲⣦⡀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠚⠛⠋⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠈⠳⢶⣠⣄⣙⣷⣿⣿⣿⣿⣿⣿⡿⢿⠿⠟⠁⠀⠀⡀⠀⠀⠀⠀⢀⣤⠾⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠉⠻⢹⣿⣿⣿⣿⡿⠉⠈⠀⠀⡀⠀⢀⣴⣾⡷⠚⠁⠀⠀⠊⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⠛⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣿⠀⠀⠀⠸⣦⣤⣤⣶⣶⣮⣶⣰⣦⣖⣴⣶⣶⣤⣤⣴⠇⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⣿⡀⠀⠀⠀⠈⣻⣿⣯⢻⣎⣿⢹⡟⣿⣱⡟⣿⣿⣟⠁⠀⠀⠀⢀⣿⣿⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⣿⣷⡀⠀⣸⣶⣿⣿⣿⣸⣿⣿⣮⣽⣿⣿⡟⣿⣿⣿⣶⣇⠀⢀⣾⣿⡿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣌⠻⣿⣿⣷⣿⡟⣽⣿⣷⣿⢿⣯⡻⣟⣽⡿⣿⣿⣿⣯⢻⣿⣾⣿⣿⠟⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣾⣿⣿⣿⠀⠘⢿⣿⣿⣼⣿⣻⣿⣿⢧⣿⣿⡿⠃⠀⣿⣿⣿⣷⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣧⣀⡀⠝⢻⣿⡎⢻⡟⢱⣿⡿⠉⢀⣀⣼⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣴⣿⣷⣿⣿⣿⡟⠛⠓⢤⡙⣷⣀⣀⣾⢋⡴⠚⠙⢻⣿⣿⣽⣾⣿⣦⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⢿⢻⣿⣿⠳⣤⣙⣈⣿⣿⣁⣋⣤⢚⣿⣿⡟⡿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⣾⣿⣿⣌⠀⠈⠛⠷⠾⣿⡟⠃⠸⢻⣿⠷⠾⠛⠀⠀⣤⣿⣿⣷⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⣷⣶⣿⣦⡴⣯⣤⡀⢀⣤⣽⢦⣴⣷⣶⣾⡿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠈⢻⣿⣿⣿⠀⣁⣭⣙⣋⣭⣉⠐⣿⣿⣿⡟⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢠⣯⣄⡀⢀⣠⣽⡆⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⣿⣌⢷⣬⣭⣭⣥⡾⣣⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣶⣶⣶⣶⣶⣶⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢼⡗⣦⣀⣀⡤⠀⠠⡎⢠⡾⢳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣿⠛⠋⠁⠀⠀⣀⣽⣁⣸⡉⠀⠀⠀⠀⠀⣀⠀⣤⣶⣶⣄⢀⣀⠀⠀⢠⡿⠿⣿⣷⣄
⠀⠀⠀⠀⣠⣆⢧⢻⣜⣿⣿⣿⠀⠀⠀⢀⡾⠋⣙⠿⡛⢿⣿⡄⠀⠀⠀⠈⣽⣛⡙⣛⢿⣏⠀⠀⠀⢻⠀⠀⢸⣿⣿⣷⡀
⠀⠀⠀⠢⢸⣿⣾⣿⣿⣿⣿⡿⠀⠀⢀⣾⣿⣤⣥⣾⣶⣿⣿⡇⠀⠐⠲⣾⣿⡿⠉⠛⢿⣿⡛⠛⠀⠈⠳⣶⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡇⠀⠀⠸⣿⣿⣿⣏⠉⢹⣿⣿⣿⠀⠀⣰⣿⠏⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠈⢿⣿⣇⣶⣤⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠀⠀⠀⠀⢸⠉⢹⠛⠛⠻⡟⠛⡏⠐⠛⠉⠙⠶⣦⣀⣄⣤⠾⠛⠛⠲⠀⠀⠀⠀⣿⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠿⠉⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢻⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⣠⠶⢤
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⡀⠀⠀⠀⢠⣿⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢻⣿⢳⡄⠀⠀⠀⠀⠀⠰⡀⠀⠀⠆
⠀⠀⠀⠀⠀⢀⣠⣼⠋⠉⠉⣷⣄⠀⠀⠀⠙⠛⠛⠁⠀⣀⡀⠀⠀⠀⠀⠀⢤⣀⣀⣹⣿⣿⣿⣇⣀⣀⣠⠄⠀⠀⣼⣿⣧
⠀⠀⠀⠐⠻⢿⣿⣿⣷⣶⣶⣿⣿⣿⣶⡤⠀⣄⡀⣠⣾⠿⣿⣦⡀⢀⣤⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢠⣟⣯⣿⢿⣽
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠟⠉⠁⠀⠀⠈⢹⣯⣂⣀⣀⣿⣿⡍⠀⢸⣿⣿⡿⠟⢿⡟⣿⠿⢿⣿⣿⡇⠀⢿⣭⣿⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⡇⠀⢸⡟⠁⠀⠀⠈⠇⠏⠀⠀⠈⢻⡇⠀⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠙⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠉⠁⠁⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠋⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣭⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠖⡄⢀⣀⣀⠀⡖⢶⡀⠙⠛⠓⢺⣿⣷⣦⣄⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣷⣦⣀⡀⠀⢀⣀⣀⡄⠈⠙⣥⣿⣿⣿⣦⡝⠉⠀⠀⣤⣤⣼⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⣀⡀⢀⣼⣶⣶⣀⠴⡦⠀⠜⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⣀⣴⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠁
⠀⠀⠀⠈⠉⣽⡟⠻⠻⠛⠟⢷⡀⠀⠈⠀⠉⠙⠋⠉⠂⠁⢠⣾⡏⢹⣿⣿⣿⣿⣿⣿⣿⣿⢛⣿⣦⠀⠀⠀⠀⣤⣦⣴⣄
⠀⠀⠀⣶⣦⣙⣷⣿⣾⣿⣿⢛⣠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⢸⣟⠉⠀⠤⠄⠈⢹⣿⠀⠹⠏⠀⢀⣤⢼⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠈⠻⠿⠛⠋⠉⠉⠛⠻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣶⣶⣶⣶⣿⠇⠀⠀⠀⠀⠈⠁⠈⢯⣀⣸⡟⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙

⠀⠀⠀⠀⠀⠀⠀⣄
⠀⢿⢿⣦⡇⠀⠀⠀⠀⢠⣶⣾⣷⢄⣀⠀⠀⠀⠄⠀⢀⡀⣀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⣏⠀⠀⠀⠠⠦⡐⢿⣾⠋⠀⠈⠙⠻⡀⠀⠀⠀⠀⠐⠖⠚⠻⠟⠛⠇⠀⠀⠀⠈⠀⠀⠀⠰⠶⠶⠶⠦⠑⠺⠾⣷⣮⡂⡀
⠀⠀⠀⠘⠀⠀⠀⠀⠀⣠⢸⣧⣶⣦⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣿⣗
⠀⠀⢦⣈⡤⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣇⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⣄⠀⠀⡆⠘⣿
⠀⠀⠙⠿⠿⠶⠿⣿⣿⣿⡞⢹⣿⣿⣿⣿⡏⠀⠀⠀⠐⣦⣀⣄⡀⠀⠀⠄⢀⡀⠀⢠⠀⠀⠀⠀⠀⣼⣿⡾⡣⠊⢿⣷⣿⣇⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣇⣀⠙⣿⣿⣿⢆⣴⣾⡆⠀⠈⢿⣿⣷⣤⣸⣾⣼⣧⣤⣿⣇⣾⣤⢠⣾⣿⣿⣿⣜⢄⡀⠊⠀⠀⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠙⠛⢿⣻⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⣿⣿⣿⣿⣦⣂⣤⢠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠠⢸⣿⣿⣿⣧⡈⣻⠟⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣃⣀⣈⣛⢿⣿⣿⣿⣿⣿⢿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⣄⣄⢿⣿⣸⣿⣿⣏⣀⣜⣿⣿⣿⣿⣿⣿⣿⣿⡟⡉⢘⡩⠙⠉⢿⣭⣏⣹⡾⣿⡟⠛⢿⣮⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⢭⣶⣿⣿⣿⢧⠉⠁⠀⠈⠉⠉⠛⠛⠻⣿⣿⠟⠁⠀⠀⠀⠀⢸⣿⣿⣿⠀⠈⢳⣤⣿⡿⣀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⣿⣿⣿⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣮⠀⠀⠹⣿⣇⠉⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣡⠀⠀⠀⠀⢹⣿⡷⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠛⠁⠀⠀⠀⠀⠀⣟⠣⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣯⡝⢿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇⡡⠀⠀⠀⠀⠀⠀⠀⠓⠤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠿⠷⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠰⡟⠶⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⡀⣰⣿⣿⣷⣿⡄⠀⠀⠀⢢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⠃⠀⢀⣠⣾⣾⣿⣿⣿⣿⣿⣿⣆⠀⠀⠈⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⣦⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣀⣴⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠈⠛⠻⣿⡿⠛⠻⢿⣿⣿⡿⠻⠛⠻⣿⠟⠛⠁⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣷⣾⡟⣄⣔⠄⡀⠻⡿⢁⠄⢢⣠⡹⣿⣶⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⠟⠋⠱⣿⠹⣦⣈⣀⢀⣁⣴⠏⣻⠦⠉⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⡶⢃⠀⠀⠈⠓⠺⢩⡍⢉⣍⠟⠚⠋⠀⢠⣀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣧⡃⠊⢻⣿⠳⠲⣬⣛⣚⣡⡴⠺⢛⡿⠣⣹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⡄⠂⢻⣿⣤⣿⣷⣿⣿⣿⣸⡿⠱⢠⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⠈⣿⡿⣿⣿⣿⣿⢿⣿⠁⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢹⣿⣿⣿⣿⣿⣧⢸⡃⢿⣿⣿⣿⠀⡏⣰⣿⣿⣿⣿⢻⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠸⣿⡻⢿⣿⣦⠳⠼⣙⣋⡧⠜⣡⣿⣿⢟⣿⠏⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⠙⢿⣷⣤⣷⣴⣤⣾⡿⠋⢡⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠾⠀⠀⠀⡄⠀⢀⠀⡄⢀⠀⣀⡄⠀⠀⣷⣣⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣤⣀⣤⣼⠈⠛⠈⠁⠈⠙⠁⢧⣤⣶⠟⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣽⡿⠋⢀⣀⠀⠀⠀⠀⢀⣀⠀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⢡⣾⠁⠀⠉⢢⢠⣾⠁⠀⠉⠂⠀⠀⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠻⣿⣄⢰⡷⡸⢹⡿⣄⢾⡆⡜⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⠗⣤⣝⣓⣒⣋⣁⣀⣙⣓⣒⣫⣤⣆⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡀⣿⣿⣦⣿⣦⣿⣼⣿⣼⣷⣾⣿⡿⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⢻⣿⣧⠻⣽⠻⡿⢿⡿⢿⣿⡿⣿⣛⠟⠁⠀⢀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⣸⣿⣿⠗⠬⠉⠛⠚⠓⠒⠛⠛⠉⠁⠀⠀⠀⢸⡇⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⢨⣿⡆⠀⠀⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⢘⡁
⠀⠀⠀⠀⠀⠀⠀⠸⢻⣷⣠⣤⣸⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢟⣴⣀⠎⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠯⠀⠙⣸⡿⠿⠿⣷⡀⠀⠀⠀⠀⠀⠀⣠⠄⠰⢌⠃⠠⠿⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢹⢿⣿⣶⣶⣴⣷⣿⣿⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡽⠷⠄⠀⡂⠀⠈⠉⠉⠉⠀⠈⣿⡧⠄⠰⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣿⣶⣷⣾⣿⣶⣷⣶⣶⡶⢶⢶⣶⣾⣿⣿⣿⣿⣿⣷⣦⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠶⣒⣛⣉⣉⠹⣟⣿⣷⣮⣍⢣⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠦⠴⠿⠀⡼⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⠃⡀⠀⠀⠚⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡉⠀⣀⠁⣠⡆⢀⠩⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣿⣿⡿⠀⣞⡶⣿⠏⠸⣿⣏⠴⢶⣭⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣡⣿⠂⠀⡿⠋⠀⠀⠊⠀⠛⢻⡿⢻⣿⣿⣿⢿⣿⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣼⣿⠿⠯⢹⢿⡛⢫⠦⠞⠀⠀⠀⠰⠦⠀⠀⠀⠀⠉⠐⠿⠛⠁⠀⠙⢿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⡿⠆⠠⡎⣾⡇⠀⡶⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠆⢻⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⢀⢱⡝⡥⢠⡁⠀⠀⠀⠀⠀⢀⣤⣶⣤⠀⠀⠀⠀⠀⠚⠋⣀⣀⠘⠋⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠙⢿⣿⣆⠁⠀⠐⠂⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠑⢺⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣶⡞⢛⣛⡛⠿⣎⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠺⠛⠗⡼⠟⣛⢛⠛⣶⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣿⠰⠟⢶⣈⣶⡘⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣰⣏⣰⠞⠷⢸⣧⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠸⡏⣇⠀⠂⢳⡿⠳⠸⣿⣆⡀⠀⠀⠀⡀⠀⠀⢀⣤⡿⠰⠻⣷⠃⠂⢀⡏⡿⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠸⡄⠀⠀⠙⠒⠳⡙⣿⣿⣇⢠⣴⣷⣤⣿⣿⡟⡱⠓⠚⠁⠀⠀⡼⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⠐⢾⣶⣤⣀⣀⣠⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣀⣀⣠⣴⣾⠖⢀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠠⠶⠛⢉⡽⣫⡿⠉⠁⠀⠀⠉⠹⣯⡻⣍⠙⠳⠤⠀⠀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡀⠀⠀⠀⠀⠁⡸⠉⠈⠳⠀⠀⠀⠰⠃⠈⠹⡀⠁⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢶⡳⣦⡀⠀⠀⡄⠀⠘⠓⠛⠀⠀⡄⠀⠀⣤⡶⣳⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠙⠛⠀⠀⣔⣦⠠⠄⠤⢠⣆⡄⠀⠘⠋⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠢⢤⡖⠀⠉⠁⠐⣦⠔⠂⣠⡶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡄⢁⡎⠀⠘⣆⠁⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣄⡀⣀⣼⡟⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠐⠲⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⡀⠀⣼⠃⠘⡀⠀⠀⠀⠀⠀⠀⢀⣶⢟⠛⠻⣦⠀⠀⡄⠀⠀⠈⠒⠈⠀⠀⠀⠀⡀⠤
⠀⠀⠀⢀⡉⠩⣿⣷⠦⢀⠀⠀⠀⡗⠉⠉⠁⠐⠛⠒⠒⠂⠈⠃⠀⠘⣿⡆⠄⣰⣿
⠀⠀⠀⠚⠁⠀⢻⣿⡄⢀⡽⠒⠂⠑⠂⠐⢇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠻⣿⣾⣿⠇⠀⠀⠀⠀⢰⠇⠀⢠⡆⠈
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⢹⡏⠀⠀⠀⠀⠀⠀⠃⠀⠉⠀⠀⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠂⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⢽⡀
⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠠⠌⡈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣤⣤⡄⠀⢀⣾⠁⠈⣷⡄
⠀⠀⠀⠄⠀⠀⠀⢰⡀⠀⣀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣀⠄⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠛⢿⣿⣿⡆⠀⠁⡟⠀⠀⢸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠐⡿⡄⠀⠀⠀⣴⠈⠀⠀⠙⣧⠀⠈⣧⣤⡴⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠐⠀⠀⠘
⠀⠀⠀⠀⠀⠰⠆⠀⠐⡂⠀⠀⠀⠿⣷⣤⡀⠀⠹⠾⠀⠀⠠⣇⠀⢀⠀⣿⡇⠀⠀⠀⠀⠤⣤⡾⠛⠁⠀⠀⠀⠘⠁⠈⠃
⠀⠀⣃⠀⠀⣴⡀⠘⠁⠁⠀⠀⠀⣓⠷⢿⢿⠇⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠠⠐⠒⠤⣄⠀⢰⣄
⠀⠀⢿⣷⣤⣽⣿⡿⣿⡇⠀⠀⠀⠉⠀⠙⠃⠀⠠⠀⠤⣤⣠⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠈⠛⠋⠁⠙⠛⠛⠁⠀⠀⢠⠀⣀⠠⠄⡄⠀⠀⣀⠔⠛⠛⢉⣿⣿⣿⣿⣿⣿⡿⠿⢶⣄⠀⠀⡀⠘⠂⢰⣦⠈⠉⠛⠉
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⢒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠉⢻⡿⠃⠈⠛⠁⠀⠀⠀⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡯⠄⠀⠀⢸⠐⠷⠀⠀⢀⠀⠀⠀⠀⢠⣤⡄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⣀⡀
⠀⠀⠀⠀⠀⣰⣧⣤⠀⠀⠀⠀⠀⠟⠇⠄⠀⠆⢲⠄⢠⣴⣶⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣾⣿⣿⣷⡶⠂
⠀⠀⠀⢀⣼⣿⣧⢰⡆⠀⠀⣐⡼⠀⠀⠱⢠⠀⠀⠀⢸⠃⢌⣻⣷⡆⠈⠀⠀⠰⢦⣀⠀⢠⣠⣄⠀⢰⠁⠀⠀⠈⠿⢻⠁
⠀⠀⣀⣾⢿⣿⣿⣋⠀⠀⠚⠀⠀⠀⡜⠀⠆⠀⠀⠀⠸⠀⠈⢧⠹⡇⠀⠀⠀⠀⢸⠃⠀⠈⣿⠋⠀⡏
⠀⠈⠁⠀⠈⠉⠉⠁⠀⠀⠀⠈⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈
⣶⣶⣶⢶⠶⢶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡶⣶⠶⣶⣶⢶⣶⡶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠘⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⡂⠽⢫⡀⠀⠀⠀⠐⣠⢄⡤⣅⠀⠀⠠⠈⠈⠀⠀⠠
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⣀⡨⠁⠀⠀⡘⠠⡇⣏⣼⣸⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠑⠀⠉⠘⠁⠀⠀⠀⠒⠂⠀⠁

⠀⠀⠠⠄⠀⠀⠂⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⢾⠆⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣀⠰⠠⢶⠀⢇⠀⡁⠀⠀⢲⠠⠀⡃⠆⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠈⠉
⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠈⡦⠄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡆⠀⢀⣾⣿⣿⣿⣟⣂
⠀⠀⠰⠀⠀⡄⠀⠀⠀⠀⠈⠁⠀⠀⠂⠀⠀⠀⠀⠀⠑⡦⠀⠀⠀⠀⠀⣼⣿⠿⠋⠀⠀⣾⣯⣿⣯⣏⣵⠍
⠀⠀⠀⠂⠘⡇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠰⠛⠁⠀⠀⠀⠀⠘⠻⢻⡽⠓⠃⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣹⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠉⠙⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢛⣛⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣶⡶⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣝⣻⣿⣿⣿⣿⣧⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠙⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⢿⣿⣧⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡌⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⢸⣿⣿⣦⣴⣶⣾⣿⣿⣷⣶⣤⣿⣿⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⢈⠙⠿⣿⣿⣿⣿⣿⡿⠟⢉⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⢸⠅⠀⠀⢉⣿⣏⠁⠀⠀⢸⠆⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⣷⣤⣀⣠⣤⣾⣿⣿⣿⣿⣦⣤⣈⣠⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡏⢹⣿⠁⠉⠈⣿⢿⣿⠿⣿⣿⣿⣿⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⡏⠀⠀⠀⠋⠀⠟⠀⠹⠃⢸⡿⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⢰⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⢰⠀⢰⡇⠀⣾⣧⣀⣠⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣿⣷⣿⣿⣾⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⢹⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢾⣿⣿⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣻⣿⡿⢋⣔⣺⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠛⠉⠛⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡀
⠀⠀⠀⠀⠛⢤⣶⣿⣿⣦⣝⠁⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢸⡵⠀⠘⠛⠀⠺⡗
⠀⠀⠀⠀⠀⠀⡩⢙⠹⠉⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠛⢀⠀⠀⠺⢵⠂⠀⠀⢀⣀⠀⠀⠀⢻⠗
⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⡀⠈⡈⠀⡀⠄⠀⣤⡄⠀⠀⠀⠘⠋⠄⠀⠀⢰⣤
⠀⠀⠀⠊⠀⠁⠀⢐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⡀⠁⠉⠁⠀⠀⠀⠉⠄⠀⠀⠠⠄⠐⠁⠀⠁
⠀⠀⠁⠈⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠠⡀⠀⠁⠀⠂⠃⠀⠀⠄⠀⠀⠀⠀⠀⢸⡇⢰⡇
⠀⠀⠀⠀⣀⣀⡈⠀⠀⢄⣀⣠⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠠⠾⠇⠸⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠿⠿⠿⠿⠇⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠃⠰⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠺⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠂⡀⠀⠀⢐⣂⡂⠀⠀⠀⡄⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢂⠀⠀⠀⢸⣦⣀⠀⠀⠀⠀⠀⢈⣁⣶⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠈⠀⠀⠀⠘⣿⡿⠀⠀⠀⠀⠀⢺⣿⠟⠀⠀⢀⠀⠈⠀⠀⠀⠆⠀⢀
⠀⠀⢠⠛⠀⠀⠀⠈⠁⠀⠀⠀⠘⡆⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠰⠀⠀⠒⠒⠂⠀⠀⠃
⠀⠀⢼⠀⢂⠀⠐⠠⠐⠊⠁⠀⠠⡇⠀⠀⠀⠀⠈⢀⣀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠐⠂⢲⠂⠈⠑⠂
⠀⠀⠀⠀⠀⠁⣐⠒⠀⣒⡀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠈⠁⠀⠀⡠⠁⠀⠀⠀⠀⠀⠚⠂⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣒⠂⠐⢂⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣖⣲⣶⣶⣶⣶⡄⢀⣶⣶⣾⣯⣴⣶⣶⣶⣶⣶⣶⣶⣶⠆⡼⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⡿⢟⣵⣿⣿⣿⡿⢿⣏⣭⣭⣿⠟⢻⡿⠿⠛⠛⠀⠈⠁⠚⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡝⢻⣿⢿⣿⡇⠰⣿⣿⣿⣿⠋⠁⢀⣠⠄⢀⣀⠀⠀⠀⣠⣶⣶⣄⣠⣄⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢣⣴⢶⢶⣄⢷⣄⠙⠿⠃⠉⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠸⣿⣿⣿⠟⠀⣻⡷⠀⣸⣷⣿⣿⣿⣿⣿⣿⣶⣦⡀⠀⠀⢀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢹⠏⠁⠀⠀⣛⣥⡶⢰⠂⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣤⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣿⢠⣶⣤⣾⣿⡿⠙⠘⢀⣼⣿⣿⣿⡟⣿⣿⡏⠀⠀⠀⠀⠘⣿⣿⣿⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣯⠈⠻⣯⣿⣿⠿⠛⠛⠻⣶⠳⠶⢾⣿⣧⣀⠀⠀⠀⠀⠘⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡼⠋⣠⠀⣹⡟⠁⢂⣀⡀⠈⠉⠀⠀⠠⠉⠛⠿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢻⡀⢻⣾⠟⠁⢆⠀⠈⠁⠁⠀⠀⠀⠀⠁⠸⠗⡄⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣙⣡⡮⠨⠀⠈⢀⠀⠀⠀⠁⠀⠀⠀⠐⠀⣀⡐⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄⠀⣦⣤⠀⠀⠑⠂⢀⠀⠀⠃⠀⠠⠀⠀⠀⠀⠠⢸⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⣼⡇⠀⢿⣿⠓⢄⠀⠀⠊⠻⠷⣤⣤⡤⠠⣾⣷⣆⠀⠀⠙⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⡇⠀⢸⣿⡔⣆⢀⢀⣀⠀⠀⠀⠀⠀⣤⣼⣿⡘⠂⢠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣤⣠⣼⣿⣥⣬⣿⣿⡛⠱⣤⣤⣠⣶⣿⢩⣿⣧⠄⠁⣬⠸⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⡟⣿⣽⢻⣿⣿⣿⣿⣿⣥⣌⣉⣁⡿⠿⣿⣿⣿⢻⡄⣢⠽⠺⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠙⠛⠿⠒⠛⠁⣿⡟⣿⣟⡉⠀⢈⣿⣿⣿⣿⣿⢻⡇⠀⠂⠭⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢸⣖⣲⡆⢀⣇⢀⡳⠄⠀⢀⣀⠈⠉⠉⠉⠋⠸⠇⡰⣤⣨⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠉⠉⢡⠊⠉⠁⠉⠀⠀⠉⠉⣻⠀⠀⠈⠀⠀⢈⡟⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣥⣤⣤⣄⡀⠀⠀⠀⢠⡀⢽⣷⣶⣶⣶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠉⠙⠛⠏⠉⠏⠱⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡉⣽⠂⠄⠀⠀⠀⠀⠀⠀⣶⢦⣄⠀⠀⠀⠀⡞⠀⠀⠙⠀⠈⠀⠀⠀⠀⠃⠘⠉⠀⢀⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⢩⡎⣰⠀⠀⠀⠀⠀⢠⣿⠛⢾⣷⡄⠀⠀⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠂⠀⠈⢃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠓⢢⠀⠀⠀⠀⡸⣿⡀⠀⠙⣻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠘
⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠐⢏⠀⠈⠂⡀⠀⣸⣁⣼⣷⣄⠀⠘⢛⣷⣄⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠁⠀⠳⢄⠀⠀⠀⠛⠁⠈⠻⢿⣆⠀⢸⡋⠉⠃⠀⠀⢀⣀⣤⣴⣯⡄⠀⠀⠀⠀⠀⠀⠹⡀
⣤⣤⣶⣸⣧⡀⠀⠀⠀⠀⠀⠀⠈⠐⠺⠂⠀⠀⠀⠀⠀⠀⠀⢹⡧⢸⡇⠀⠈⠛⠢⠤⣍⣺⠿⠻⣧⠤⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⢤
⣿⠛⠉⠉⠛⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⡄⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⣼⣣⣶⣶⣿⣿
⡇⠀⠀⠀⠀⣿⠟⠛⠲⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⠀⢠⢆⡀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⢷⣿⡇⠀⠘⡏
⠁⠀⠀⠀⣰⡟⢰⢀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡻⣆⣀⣤⣤⠄⠀⠀⠀⠈⠛⠛⠿⠛⠿⢅⡀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠻⢷⡀
⠀⠀⠀⠀⠏⡴⠿⠞⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣇⠀⠀⠈⠛⠐⠒⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⣀⢀⠀⠀⠘⣇⠀⠀⠀⠀⠀⣀
⠀⠀⢀⠀⠀⠤⠄⠀⠪⠋⠀⠀⠀⠀⠀⣀⣤⡶⠲⣶⠒⢲⠀⠀⠀⠀⣀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣶⣄⣸⣶⣦⣤⠀⠀⠘⠷
⠈⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣵⣿⠿⠿⠇⠀⠀⠺⠄⠀⠀⠀⠀⠉⠛⢶⣞⢿⣻⡷⠶⣶⣤⣄⣀⣀⣤⣤⠶⠦⠄⠀⠀⠀⠀⠰⠻⠟⠋⠉⠙⠻⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠁⠀⠀⢀⡀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠟⠃⠀⠀⠭⣙⠻⢯⣍⡑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣀⡀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠐⠛⠉⠁⠀⠀⠀⡀⠠⣤⡀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠈⢳⡌⠳⢤⣈⣻⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⢈⣿⣄⠒⠉⠀⢀⣀⡤⠀⠒⠋⠀⠀⠀⠀⢀⡞⠀⠀⠈⠙⠆⠀⠀⠀⠀⠀⠀⠀⢀⣸⡄⢷⠀⠀⠉⠉⠛⠻⠿⠿⣿⣿⣷⣄⡀⣀⣤⡤⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠙⣷⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣟⠟⠁⠀⠀⠀⠀⠀⣿
⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠀⠀⡀⠀⠀⠀⡙⢀
⢿⡆⢒⠘⡠⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢢⣀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⣀⣬⠖⡸⠯⠿⣤⣤
⣠⣥⣼⠀⡧⠀⠀⠀⢀⣠⠞⠁⠀⣀⡀⣀⣤⣤⠤⣤⡾⠃⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⣴⣍⠀⠍⠲⣦⡀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠄⢀⡔⠡⠛⠋⣀⠀⢀⡜
⠋⢹⡇⣴⣷⣤⠴⠞⠋⠁⠀⣰⡿⢿⡛⠿⣧⣄⣐⣤⣤⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⢸⠓⢸⡃⠀⠀⠀⠚⠓⠀⠀⠀⠀⠀⢻⠀⢀⡞⠉⣷⠀⣠⠔⠉⠉⠹⠋⠉
⠂⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⠀⡿⠗⠤⠌⢉⡉⠁⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⢋⣉⡠⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢰⡟⣤⡆⠀⠀⢻⣄⣀⢀⡼⠒⢠
⢐⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⢿⣣⣤⣷⣶⣶⣿⣉⣁⢤⠀⠀⠀⠀⢀⡀⢹⡄⠀⠀⠘⣧⣥⡰⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠾⠷⠿⣧⣤⣶⠋⠁⠈⣹⡁⢀⡼
⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⡿⢛⡿⢯⡛⠷⣯⣴⣴⣠⠀⣼⠁⣸⠃⠀⠀⠀⣹⡄⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣤⡀⠀⢙⣿⡠⠀⠐⣿⣉⢤⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠸⣴⡇⢿⡇⠀⠉⣷⣤⣌⡉⣀⣾⡇⢠⠃⠀⠀⠀⠀⢻⣥⢰⡆⠿⠗⠀⠀⠀⠀⠀⢠⡞⠀⠈⣠⣿⣄⠸⣯⣁⣈⡿⠁⠐⣈⠄
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠃⠈⠑⠀⢀⣿⠏⣹⣿⣿⡟⢧⣸⠀⠀⠀⠀⠀⢸⣧⠤⠤⣰⠞⠀⠀⠀⠀⢠⡏⠐⢃⡀⠌⠉⠙⠛⠓⠶⠓⠲⠾⠿⠷⠶
⣹⣷⠄⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣿⣸⠏⠈⠙⣷⠸⡟⠀⠀⠀⠀⠀⠀⢸⣛⡻⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉
⠀⠀⠀⠀⣠⣴⡾⢿⡋⠫⡹⣦⡀⠈⠁⠀⠀⠀⢸⣿⣟⠀⠀⠠⢾⡇⡇⠀⠀⠀⠀⠀⠀⢸⡦⠆⡠⠶⢢⡀
⠀⠀⠀⢰⠟⠀⠃⠈⠁⠀⠄⠠⣤⠀⠀⠀⠀⢠⣿⣿⡟⠡⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠶⡁⢷⣾⡧⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⢤⠀⣀⣀
⠀⠀⠀⢸⠄⡠⠀⠀⡀⠂⠳⡰⠃⠀⠀⠀⠀⠛⢹⡟⠀⠀⠛⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢋⠛⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠀⢀⡀⠀⠀⠉⠘⠶⡀
⠀⠀⠀⢠⢊⣳⣦⡀⡀⣑⣐⣈⣀⡀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⣋⡓⡆⠀⠀⠀⣈⣀⢀⣤⡀⠀⠀⠀⠀⠀⢰⣆⠀⠐⢛⠛⠀⠀⢀⡀⣈⣻⣀
⣿⣿⡿⠻⠿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⣀⣤⣾⣿⡟⢹⡄⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⣛⣭⠿⣿⡟⣽⢿⢁⢸⡀⠀⠈⠲⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣖⠶⣒⠭⠀⠼⠛⢛⣠⡞⠟⠈⢋⣿⡊⠀⠋⠦⡀⠀⢬⡳⠀⠐⠦⣄⡀⡄⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣎⡁⢜⣃⡤⢒⣿⣟⡴⢠⣴⡿⢻⣧⠐⠢⠀⠻⡄⠀⠁⡰⠄⡀⡂⠌⣿⡇
⠀⠀⠀⠀⠀⢀⣿⣿⠿⠋⠿⣋⣥⣴⣿⠟⡴⢡⠏⣅⢰⠏⠈⠧⠀⠈⠳⠀⣴⠠⡇⠠⣿⣧⣶⣿⣷⡀
⠀⠀⠀⠀⢠⣾⠟⢉⢁⣠⡌⢉⣤⡊⠁⠀⣴⠃⢠⡀⢠⡇⡄⢃⡐⠀⡆⢠⣈⣤⣶⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠈⠙⠛⠋⠉⠉⠛⠋⠙⢷⢀⡀⡀⠁⣿⣇⣸⣿⣷⣼⣧⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⣸⡜⣃⣧⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⣦⣉⣸⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⠿⣿⣿⣿⣯⣿⣿⣿⣽⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⡇⣄⣸⣿⣿⣿⣿⣿⣿⣿⡿⠏⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⡿⠋⠀⠹⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣷⣿⣿⡿⠟⠛⠛⠉⠉⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⣿⣿⣿⣿⡿⠁⠀⠀⠀⠈⠻⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣿⡿⠀⠀⣀⣀⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⢠⣽⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠘⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣦⣀⠀⠀⠀⠀⠀⣠⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⢀⣠⣴⣦⣤⣄
⠀⠀⠀⠀⠀⢀⠀⠠⠴⠶⠶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⠿⠿⣿⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣶⠦⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⢀⠀⠀⢻⣿⣿⣿⣿⡟⢁⡴⠟⠿⠿⠛⠛⠿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠰⠀⢀⣼⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⠀⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⡔⠀⠸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢻⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠘⡀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢻⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠈⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡈⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠰⠀⠀⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢃⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⣸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡈⣠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⡔⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠠⠁⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠡⡀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢹⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠘⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣼⣷⣿⣷⣶⣴⣹⣶⣧⣤⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⠆⢀⠀⠀⣰⣶⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠦⣄⠂⠀⠘⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡁⠀⢀⠴⠛⠙⠛⠛⠿⠟⠟⢻⣿⠁⠀⠀⠀⠀⠀⠀⠑⢦⣀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡟⠀⢀⡠⠖⠒⠒⠒⠦⡀⠀⠀⠀⠀⣀⠴⠒⠒⠒⠲⢄⠀⠀⠘⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢠⠎⠀⠀⣀⣀⡀⠀⠈⢆⠀⠀⡴⠁⠀⢀⣀⡀⠀⠀⠱⡄⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣾⠀⠀⢾⣿⣿⠛⣆⠀⢸⠀⠀⡇⠀⢰⣿⣿⠛⢳⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠸⡄⠀⠘⢿⣿⣿⠟⠀⡜⠀⠀⢧⠀⢈⢿⣿⣷⡿⠀⢠⠃⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠘⠢⣄⣀⣀⣀⠤⠊⠀⠴⠄⠀⠓⢤⣁⣀⣀⡠⠔⠁⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⣸⢻⣷⠀⠀⠀⠀⠀⣖⣶⡖⠒⢲⣶⣶⣶⠒⠒⣲⣾⣛⣆⠀⠀⠀⠀⠀⣿⠃⢇
⠀⠀⠀⠀⠀⠀⠀⡇⠘⣿⣇⠀⠀⠀⠀⠙⢿⣿⣶⣿⡿⢿⣿⣷⣶⡿⠛⣿⠇⠀⠀⠀⠀⣸⡟⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢳⣤⣿⡿⣧⣄⠀⠀⠀⠀⠈⠙⠛⠦⠤⠿⠿⠛⠓⠊⠁⠀⠀⠀⢀⣴⢿⣶⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠈⠹⣷⣦⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⠎⠁⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⢹⣿⣶⣦⣷⣦⣦⣤⣤⣶⣶⡿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⠉⢿⠈⠁⠉⠃⠉⢀⡿⠻⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⣷⣋⣀⣀⣸⣶⣶⣶⣶⣶⣾⣇⣀⣀⣹⣽⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⣀⣀⣴⠦⠄⠀⠀⠀⠀⠠⢤⣄⡀⠀⡔⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠔⠊⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⠢⡀⠀⡌
⠀⠀⠋⠀⠀⡀⠀⠀⠀⠔⠄⠠⠂⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⢲
⠃⠀⢉⠀⠉⠜⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠢⠁⠆⠂
⠀⠀⠓⠊⠑⠅⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠐⠀⠀⠀⠁⠢⢹
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡡⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⠀⠀⠈⠐⠠⡄⠀⢀⣀⠀⣀⣀⡀⠠⠔⠊⠁⠀⠀⠀⠀⡀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⢁⠊⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⣀⢄⢀⠠⠄⠀⠀⡀⠀⠀⠀⠘⡀⢀⣀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠅⣀⡄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠇⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⠤⠒⠊⠁⠀⠀⠀⡄⠀⢀⠀⠡⡀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠀⠀⠀⠀⠀⢢⡀
⠀⠀⠀⠑⠐⠒⠀⠒⠂⠒⠠⠤⠄⠀⠤⠐⠂⠀⠤⠤⠍⠁⣀⡀⠀⠠⠤⠐⠒⠀⠒⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠛⠷⣀⠀⠀⢀⠉⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⡜⣻⠀⠀⣹⡆⠀⠐⣀⠘⠃⣐⠃⠀⠀⢠⡆⠀⢱⠒⢛⢳⠦⠐
⠀⠀⠀⠀⠊⠀⠐⢬⣒⢥⡬⣟⠁⠀⠉⠁⠀⠀⠈⠘⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠨⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠊⠑⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠀⠐⠠⠀⠀⠀⠀⠀⠀⢸⣀⠆
⠀⠀⠀⠀⠀⠀⠖⠰⠆⠀⠒⠘⠃⠀⠀⠀⠀⠀⠀⡼⣆⠀⠠⠄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢈⠀⠀⠀⠀⢄⠂⠓⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡴⡒⢃⠈⡐⠀⠀⠀⠠⡄⡄⠀⠁⠀⠀⠀⠀⠀⠄⠎⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⣀⠥⢀⡐⡀⠐⣀⠈⡀⠀⠀⠀⢡⡄⠀⠀⠀⠀⠀⠀⠀⢔⠁⡔⠀⠀⢀
⠀⠀⠀⠀⠀⠒⠒⢀⠃⢐⡈⠀⢩⡀⢡⡄⠀⠀⠀⡆⠀⣤⡆⡀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣀⠀⡘⢃⡄⠈⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠈⡋⠈⠃⠀⡀⠀⠀⠀⠂⠀⠰⠄
⠀⠀⠀⠀⡠⠋⠈⠒⠋⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠐⠙⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠲⠦⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠖⠒⠚⠉⠛⠋⠋⠋⠀⠠⠦⠤⠆⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡏⠁⠀⠈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⢸⠸⢀⢣
⠀⠀⡀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢠⠀⢈⡜⠀⠀⠀⠀⠀⠀⠀⠀⠰⡓⠂
⠀⠀⠀⠄⠐⢀⠀⠄⠀⠀⠀⠀⠀⠀⢠⠄⠂⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⡀⢀⠀⠀⡀⠀⢀⡀⠀⠀⠀⢀⠀⠀⣀⠀⣀⢀⢀⡁
⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⡇⢾⠀⠀⠀⡗⠀⠀⠀⢱⡀⠀⠑⣄⠀⠀⢸⠀⠀⢸⠀⡗⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠁⠀⠁⣀⣀⠁⠀⠀⢈⣀⠀⠈⠀⠁⠈⠀⠁⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣭⡇⠀⢸⣯⣭⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠋
⠀⠀⠀⠀⠀⠠⠎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠻⠃
⠀⠀⠀⠈⠀⠃⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⢀⣀⠀⠀⠁⠈⠃⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡤⠴⠃⠀⠀⠚⠋⠉⠁⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠹⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠄⠀⠀⡾⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣤⣤⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠏⠙⠛⠿⠙⠉⠈⠉⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣦
⠀⠀⠀⠀⠀⠀⠀⢛⣤⣀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⣀⣠⡎⡇
⠀⠀⠀⠀⠀⠀⠀⣟⢋⣿⢿⡟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⢿⣏⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣩⡼⣱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⣄⣻⡿⠁
⠀⠀⠀⠀⠀⢀⡴⠶⢮⣽⣿⢣⢞⡀⠀⠀⣠⣄⣀⣰⣠⣆⣀⣠⠄⠀⠀⢖⢲⠙⣿⡋⢀⣀⡀
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠙⠁⠘⣾⡍⠳⣄⡀⠀⢝⠟⠈⠛⠫⠀⢀⣠⠶⢻⣿⠀⠹⠟⠋⠉⠙⢷⡀
⠀⠀⠀⠀⠀⣿⣦⠔⢱⣮⡈⣳⢼⣿⠚⣦⡉⢳⡒⠀⣀⠒⢒⣟⢉⡗⠶⠛⠊⠤⣠⠖⢲⡤⢤⣼⡇
⠀⠀⠀⠀⢸⠁⢹⣦⣾⠋⢛⣁⠀⠹⡇⠈⢿⣿⠈⣞⠉⢷⣿⣍⣿⣿⠁⢰⢲⡆⢙⡒⢿⣀⣴⣿⠛⡆
⠀⠀⠀⠀⠈⠛⢻⠈⢷⡀⠹⣿⣆⠀⢷⠀⡀⠛⠀⣿⣦⡀⠙⢿⣿⡟⠀⡏⡼⠀⡞⠉⣻⠋⡉⢿⣤⠇
⠀⠀⢠⡖⠦⣀⠈⣇⠈⢳⡄⢳⡹⡄⠸⡆⢿⡀⢀⣿⠻⠟⣆⣘⢩⡇⢸⣿⠃⣀⢹⣶⠃⡴⢻⠂⢹
⠀⠀⠀⠙⢦⠈⠑⢾⡀⣄⠻⣄⠳⣹⠀⠗⠊⠁⠀⠀⠀⠀⠀⠀⠉⠑⢯⡎⣸⠋⣹⠃⠘⠷⠃⣠⠏
⠀⠀⠀⠀⠈⠳⡄⢢⣀⠘⣦⠙⣆⣀⣠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⠙⣶⠃⣼⠃⡴⠖⠁
⠀⠀⠀⠀⠀⠀⠘⢦⡙⣷⠟⠑⠞⠉⡷⠀⠤⣤⠤⠤⠤⠤⠤⢤⣤⠜⠠⡏⠙⢧⣾⡏⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡼⠃⠀⠀⠀⠗⠒⠒⠋⠀⠀⠀⠀⠀⠈⠓⠒⠒⠓⠀⠀⠘⢶⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣄⡀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⡀
⠀⠀⠀⠀⢀⡀⠀⠠⠀⠀⠀⠐⡇⡾⣫⠾⣧⢤⡄⠀⢠⢞⠿⣷⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠒⠛⠋⠉⠓⢡⣠⠁
⠀⠀⠀⠀⠈⣷⡀⠀⢳⡀⠀⣀⢿⡟⠡⡎⠀⡄⡇⠀⢸⡘⠆⠈⠀⠀⠀⠀⠀⠤⠀⠀⠀⠄⠀⠀⠀⠉⢀⣤⠞⠿⠃
⠀⣀⠄⠂⢀⣿⣿⡆⠉⢷⣶⡏⢉⢧⠀⢻⠓⢣⠇⠀⣾⣿⠆⠀⠀⠀⠀⠀⢰⣀⠀⠀⢀⢁⡀⢹⣄⠠⡟
⠸⡿⠀⠀⡏⠉⠈⢷⠀⠀⠈⠻⡌⣧⢳⣀⡴⠃⠀⣰⡿⢿⠂⠀⢠⣶⣶⣶⠇⠈⠂⠀⣸⠉⠉⢐⠨⢧⣀
⠀⠀⠀⠀⠀⠠⠀⠈⠇⠀⢀⣴⡷⠸⠋⠁⠀⢀⡼⣯⣟⡞⠀⠀⠀⠀⢸⡟⠁⠀⠀⢸⣇⠀⢰⡏⠀⠀⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠛⠁⠀⠀⠀⣔⣭⣿⣮⣛⠁⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠉⠳⠚⠷⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⢎⠟⡟⡝⣿⡇⠀⠀⢢⠴⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡍⢲⢊⠂⡓⡬⢷⡩⣇⠀⡠⠿⠅⢾⣙⣿⣇⢀⣀⣀⣀⣤⣤⠴⣆⣀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⢇⡍⡦⢚⣣⠋⠠⢽⡇⠘⠿⠋⠁⠿⡿⢻⢳⣋⢟⣿⢴⢫⢗⡭⡛⢒⠏⠑⠢⠴⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢉⢂⡁⣀⡕⠀⠀⡹⠢⠾⡛⣀⠀⠀⠀⠀⠻⠅⢋⡫⣿⢻⣸⠛⠘⣋⣵⡆⠁
⠀⠀⠀⠀⠀⢀⣶⡦⠀⢀⢋⠁⢑⠀⠀⠀⠀⠇⠃⠀⠀⠑⡁⠀⠀⠀⠀⠀⠁⡰⠟⠀⠛⠂⠀⠈⠁⠍⠂
⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠀⠀⡤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠸⠇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀⢀⠠⠀⠀⠀⠀⡀⢀⣧⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠤⠶⠚⢀⡜⢡⣠⣼⣡⣴⡄⣐⣼⣿⣿⣿⠇⠀⠀⢰
⠀⠀⠀⣦⣴⠎⠁⠐⠖⠀⠀⠀⠀⠀⠀⠠⡀⠀⠹⣆⡽⢭⠉⠹⠿⣿⣿⣿⣿⡿⠁⠀⠀⢠⣾⡀⢠⣤⡷⣄
⢰⠀⠀⢸⡿⠀⢀⠀⠀⠐⣧⣆⠀⠀⠐⢰⣷⠀⣦⠹⢄⠨⠀⠀⢠⣿⠿⣿⣿⣿⣦⣤⣴⣿⣿⣧⣿⣿⠇⠘⡇
⠘⣆⠲⣼⠁⠀⢸⣿⣶⣶⣿⣿⣶⣶⣾⣿⣟⣾⢯⡀⠀⠀⠀⢀⠎⣀⡴⠛⢿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣆⠀⠙⠶⠄
⠀⠈⣿⡇⠀⠐⠛⢿⣿⣿⡿⠿⠿⠛⢹⣿⣻⣿⠀⠀⠀⠀⢰⣴⡞⠋⠀⠀⠘⠿⣿⠀⠀⠀⣿⣿⡏⢻⡋
⠀⠀⠈⠁⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⣸⣿⣹⣿⠀⠀⠀⠀⠘⠻⠟⠀⠀⠀⠀⠠⠙⠆⠀⣰⣿⡏⠀⣸⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠁⠈⠙⠛⠁⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⠁⠀⠀⠉⠉⠙⠛⠳⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠠⠀⠀⠀⠀⠀⠀⢀⣤⠼⠿⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠋⢹⠇⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⣠⣀⣀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠀⢸⡀⠀⣠⠖⢛⣿⣿⡍⠉⠉⢳⡅⠀⠉⠉⠙⠳⢤⣀⣿⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⢀⡴⣋⣀⣤⣀⣀⠀⠀⠀⠀⢙⣾⣷⣴⣿⣿⣿⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣹⠃⠀⢀⣤⠴⠶⠶⣤⣄⢹⡄
⠀⠀⢰⠿⠊⠁⠀⠀⠀⠈⡿⣠⣤⣀⣾⣿⣿⣿⣿⣿⡟⠀⠀⠀⢀⡇⠀⠀⠀⢠⠦⠞⠁⠀⣴⡏⠀⠀⠀⠀⠀⠉⠳⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⢠⠄⠠⡌⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠙⠛⠛⠉⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⣀⡴⠋⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠹⠦⠤⠤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⣤⣀⠀⢠⣾⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣠⡶⠦⣤⠟⠉⠙⠛⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠗⠀⠀⠀⢀⣀⣀⠀⠙⣿⣿⣿⣧⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠉⠉⠙⠳⢿⣿⣿⠇⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢀⡤⢤⣄⣠⡾⠟⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⣠⡼⠇⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠇⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⡀⠀⠀⠐⢿⣆⣀⣀⣀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠀⠀⠀⠀⣀⣀⡤⠞⠁⠀⢀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠛⠋⠉⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠶⢤⣄⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⣀⣀⣀⣤⣤⣤⣾⠀⠀⠀⠀⣠⠖⠛⠛⠒⠒⠋⠉⠁⠀⠀⠀⠀⠈⢳⡄⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢸⠁⠀⠀⠀⠀⠸⣇⠀⠀⠀⠻⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠘⢧⡀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠉⠉⠉⠛⠲⣄⠀⠀⠀⠀⠀⠀⢸⣆⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⡉⠳⡄⠀⠀⠀⠀⠉⠛⠛⠛⠛⣻⡆⠀⠈⣧⠀⠀⠀⠀⠀⢾⣥⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠇⢀⡟⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⢀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠖⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⢀⣀⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠉⠀⣠⣶⢿⣿⠛⣧⣄⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⡇⠀⢰⣿⠁⠀⢹⣿⡏⠹⣷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡶⢊⡉⡉⠑⣦⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢛⣴⣿⠾⣷⡀⠈⠻⣦⣀⣾⣡⣯⣤⣟⠘⣃
⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⠟⠛⢷⣿⣦⣀⠀⠀⠀⠀⣀⡴⠾⠋⢉⡀⢿⡁⠀⠀⠀⠙⢆⠀⠸⣿⢋⡿⠁⣼⠏⢠⡟
⠀⠀⠀⠀⠀⣿⣿⡟⠉⠀⠀⠀⠀⣿⡏⠻⣦⣠⡴⠞⠁⠀⠀⠀⠀⠙⠶⢻⡆⠀⠀⠀⢸⠀⢠⣿⣿⣦⣼⣏⣰⣿⠁
⠀⠀⠀⠀⠀⢩⠿⣄⠀⠀⠀⠀⠀⢻⣧⣤⡴⠋⠁⠀⠀⠀⠀⠛⢢⣴⣷⡜⠃⠊⢀⢰⡏⢀⣾⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠈⠢⡉⢶⡄⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⢈⡿⠀⣼⣿⣿⣿⣿⣿⠟⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡧⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡀⠘⠶⠶⠟⠀⣴⡟⠀⢀⣿⣿⠁⠀⠟⢀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣠⣾⠃⠀⠀⠀⠀⠠⢀⣀⣀⣠⣼⣿⣛⣻⣶⣶⣿⠟⠑⣶⣾⠿⠛⠀⠁⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠋⣽⣦⣄⠀⠀⠀⠀⠊⣉⢙⣻⣿⣿⣿⣿⡿⠟⢃⣴⡾⢏⡁⣤⣤⣤⢀⠀⠴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⣏⡈⢻⡆⠀⡄⢀⡾⠉⢉⣵⣿⡟⢿⣷⡶⢿⣿⣿⣤⣿⣷⡟⢉⣀⡈⠛⠒⠛⢁⡠⠤⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡠⠙⣷⣿⣾⣳⠞⢉⣹⣿⣧⣀⣤⣶⣾⣧⣤⣽⣿⣷⣙⣩⣿⣿⣦⣀⡴⠋⣁⣤⣤⣄⠉⠓⡀
⠀⠀⠀⠀⠀⠠⠂⢀⣐⠶⣦⣶⣾⣿⣿⣿⣧⠘⢦⣹⣯⡙⠛⠛⠛⣩⣉⣙⣛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⡼⠛⠙⠻⣿⣿⣄⠀⡄
⠀⠀⠀⠀⠀⢃⡾⠋⠉⠓⠤⣤⠟⢿⣏⡿⠋⢰⣄⠙⢻⣾⣿⣷⡶⡏⣿⣿⣿⣿⣿⠟⠉⠁⠀⠉⣹⣟⣛⣉⣁⣰⡏⠀⠀⠀⠀⣹⣯⣿⠀⡇
⠀⠀⠀⠀⠀⠘⣀⠀⠀⠀⠀⠀⠀⠸⠿⢿⠃⠀⣿⣧⣾⣿⣥⣻⣿⣼⣿⣿⣿⣿⠃⠀⢠⣶⣾⣿⣿⡉⠉⠉⢻⡟⠉⠙⠿⢶⣾⣷⣿⠿⠀⠃
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⢛⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⠏⠀⢸⣿⣿⣿⣍⠻⢿⣷⣄⠀⠹⣦⣄⣀⡀⠈⠉⠁⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠤⠤⢿⣿⣾⣿⣿⣿⣄⠸⡧⣿⣟⠉⠻⣧⠀⢀⣿⣿⣿⣿⣿⡶⠀⢿⣿⣧⡀⠹⡏⠙⠋⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⣿⣿⡙⢿⣧⡘⠿⠷⠾⠉⢀⣼⣿⡙⠟⠻⣿⣿⠀⠸⣿⣿⣿⠀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡍⢿⣻⡇⠈⢿⣿⣶⢦⣴⣶⣿⣿⣿⣿⡄⠀⣿⡇⠀⢰⣿⣿⣿⣆⣹⣄⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠶⠶⠋⡠⠃⢸⠇⣻⡄⠀⠻⣿⡆⠉⠻⢋⣿⣿⣿⣿⣾⣿⡇⠀⠸⣿⣿⣿⣿⡏⠁⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⣁⠤⠒⠛⠃⣱⠀⣸⠂⣽⠆⠀⢠⣿⣿⡄⠀⠀⠈⠈⢻⣿⣿⠿⣧⡀⠀⢿⣿⣿⠿⢀⣿⠇⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢀⠀⠸⠁⢠⡏⠀⣿⠀⠀⣤⣿⣥⣿⣄⣀⡄⠀⠸⣿⣿⣶⣿⣷⣤⠾⣿⣛⣲⡿⠋⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢣⡀⠈⠠⣇⡠⠎⠀⢰⣿⠖⠋⢁⣀⣉⡀⠉⠙⢷⠀⠀⣿⡿⠿⡟⠉⢀⡀⠈⠹⣿⠀⠠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠑⠚⢅⠛⢷⣤⡔⠋⢁⣴⣿⣿⠋⠉⢷⡔⡀⠠⣄⣸⡟⣡⣶⠃⣴⣾⣿⣧⠀⣿⡷⠒⠚⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⣤⣬⣷⣦⣤⣤⣤⣿⣿⣿⠃⠀⠀⠸⣷⡇⠀⢻⡿⢉⢹⠁⣾⣿⣿⡏⣤⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠉⢙⠉⠛⣿⡟⠀⠀⠀⢠⡿⠆⠀⣿⣧⠘⢰⡄⢿⣿⠿⣷⡿⠛⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⣿⡄⠀⢀⡾⠃⠀⣴⣿⢿⣷⣌⠣⣤⣽⠶⠾⠛⠒⠿⢤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣤⣈⡙⠛⣁⣤⣶⣿⣿⣶⡄⠺⣿⣷⣶⣶⣶⣶⣾⣷⣤⣀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣻⣿⣧⣤⡈⠻⣧⠀⠀⠙⢿⣿⣿⢏⣰⣿⠏⢁⣀⠙⠙⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⡿⠁⠙⣷⣿⡆⢹⡆⠀⠀⠈⠉⠛⠋⢉⣧⢠⣿⣿⠷⣄⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣳⣄⠀⣿⠋⣷⡈⣷⣀⠐⠀⠀⠀⠀⢈⣿⠘⣿⡟⣀⠟⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠏⠀⠀⢿⣿⡀⠙⠻⣿⣶⣤⣦⣴⣿⣧⣾⡿⡟⠁⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⡾⠿⣏⢿⣿⡿⠉⠉⠁⢰⣤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢸⣿⣿⡟⠀⠀⠀⠈⠻⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⢿⡅⠀⠀⢀⣀⠄⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠙⠛⠋⠀⢠⣤⣤⣾⡙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⡾⠛⠉⠛⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠘⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⢿⣿
⣿⣧⣶⣶⣴⣦⣴⣶⣶⣦⣤⣼⣧⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣾⣷⣶⣶⣾⣿

⠀⢀⠀⣀⠀⠀⠀⣠⣤⠀⠀⣠⡄⢀⣠⡀⠀⠀⡤⠠⠄⠀⠠⠀⢀⣠⣄⣀⡀⠀⠀⣀⣀⣀⡀
⠀⠸⠀⠉⡜⠀⠀⠙⠿⠃⠐⠿⠀⠐⠛⢋⣤⣤⣤⣶⣶⣤⣤⣄⣈⠙⠉⠉⠁⠀⠻⢿⣿⣿⡿⠂
⠀⢠⠠⠊⠀⠴⢶⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡈⠁⣀⣀⠀⠀⠀⠀⡀⠄⠀⢸⢰⢺⡇
⠀⡄⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠘⠿⣿⣿⣿⣿⣿⡿⡛⠛⠛⠛⠛⢻⣿⠛⠁⠀⢤⢠⣿⠇⠀⠀⠈⠈⠈⠁
⠀⠈⠑⠀⠀⣾⣖⠀⠀⢠⣤⠀⠀⠉⠉⠀⠀⠀⢡⠄⠀⣀⠀⠀⢠⣶⣆⠀⠈⠻⠀⠀⠀⠈⠛⠀⠀⠀⠀⣶⢠⡄⡆
⠀⢰⣿⣷⡄⠙⠇⠀⠀⠘⠋⢠⠀⠀⢠⠀⠀⠀⠉⠀⠀⢉⠀⠀⠀⠉⠁⢠⣴⡶⠿⡇⠀⠀⠀⠀⣍⡀⠀⠛⠘⠃⠃
⠀⠘⣿⠀⠀⠀⠀⠘⠃⠀⡀⢸⡇⠀⠈⠉⠡⢠⢀⣤⢠⠉⠉⠀⡤⠀⢀⠘⠻⢷⣦⡇⠀⠀⠀⢸⣿⠄⠀⢠⢀⣀⡄
⠀⠀⣿⠀⠀⠀⠀⢸⣦⠀⣿⢸⣷⠀⠑⠀⠀⠀⠄⠀⠀⠀⢀⠠⠈⠀⠈⣀⣀⠀⠀⠀⠀⠀⢠⠀⠋⠀⠀⢸⠸⠯⡇
⠀⠂⠻⠤⢝⣤⡄⠀⣿⣆⣿⣿⡿⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⢿⡇⠀⠐⠄⠀⠀⢰⡀
⠀⠘⢷⣲⣼⡽⠀⠀⠈⠿⣿⠟⢁⣀⡈⠉⠀⠀⡆⢰⢶⡶⠂⠀⠀⡄⠈⠀⠀⠀⣰⣿⣿⣿⣦⠀⣳⠀⠀⢸⣶⣴⡇
⠀⠀⠀⠉⣿⣇⠀⠐⠀⢀⣴⣾⣿⡟⠓⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠰⣿⡄⠀⠈⠈⠉⠁
⠀⠠⡀⠀⣿⣿⣿⣶⣶⣿⣿⣿⣿⠃⠀⠀⢰⣿⣷⡄⠐⠐⠄⠀⠄⠀⠀⠀⠀⣀⣸⣿⣿⣿⣥⣄⢼⠃⠀⢀⢤⣤⡆
⠀⠀⠀⠀⠘⣿⠿⡿⢛⡉⠿⠛⠁⠀⠄⠀⠀⠉⣿⣷⣄⠂⠀⠀⠀⠀⠀⠀⢀⣿⡟⣿⢿⠿⣿⢻⡆⢀⠀⠰⠛⠹⠇
⠀⠀⠀⣶⣤⠈⠓⠁⠋⡀⠀⠀⣴⠀⠘⡶⠀⠸⠏⢹⡏⠀⠀⣿⠘⣿⡄⠙⢛⣱⢇⡇⣿⢸⣇⡀⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠶⠶⠶⠶⠶⠶⠶⢦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠛⠉⣁⣠⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣄⣈⡉⠛⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⢁⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠈⢳⣄
⠀⠀⠀⠀⠀⢀⣼⠋⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠙⣧
⠀⠀⠀⠀⢀⡾⠁⣰⣿⣿⣿⣿⣿⣿⣿⣯⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣽⣿⣿⣿⣿⣿⣿⣿⣆⠈⢷⡀
⠀⠀⠀⠀⣾⠃⣰⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣦⣌⡙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣋⣡⣴⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣆⠘⣷
⠀⠀⠀⣸⠇⢠⣿⣿⣿⣿⣿⣿⣄⠀⠉⠙⠛⠿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠀⣰⣿⣿⣿⣿⣿⣿⡄⠸⣇
⠀⠀⢀⡿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⠀⢿⡀
⠀⠀⢸⡇⢰⣿⣿⡇⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣯⣤⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢸⣿⣿⡆⢸⡇
⠀⠀⢸⡇⢸⣿⣿⡄⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢠⣿⣿⡇⢸⡇
⠀⠀⢸⡇⢸⣿⣿⣷⠀⠀⢀⡙⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⢋⡀⠀⠀⣾⣿⣿⡇⢸⡇
⠀⠀⢸⡇⠸⣿⣿⡞⣧⡴⠟⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⢦⣼⢹⣿⣿⠇⢸⡇
⠀⠀⠈⣷⠀⢿⣿⣧⠈⠀⠀⠘⣧⠀⠀⠀⠀⠀⣴⢿⡄⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢠⡿⣦⠀⠀⠀⠀⠀⣼⠃⠀⠀⠁⣼⣿⡿⠀⣾⠁
⠀⠀⠀⢹⡆⠸⣿⣿⡆⠀⠀⠀⠘⣧⠀⠀⢀⡾⠃⠈⢿⡀⠀⠀⢠⡿⢿⡀⠀⠀⢠⡿⠁⠘⢷⡀⠀⠀⣼⠇⠀⠀⠀⢰⣿⣿⠃⢰⡏
⠀⠀⠀⠀⢿⡀⠹⣿⣿⡄⠀⠀⠀⠘⣧⣀⡾⠁⠀⠀⠘⣷⠀⢠⡿⠁⠈⢷⡄⢀⣾⠁⠀⠀⠈⢷⣀⣼⠃⠀⠀⠀⣰⣿⣿⠏⢀⡿
⠀⠀⠀⠀⠈⢷⡄⠹⣿⣿⣆⠀⠀⠀⠈⠻⠃⠀⠀⠀⠀⠸⣷⠟⠁⠀⠀⠀⠻⣾⠇⠀⠀⠀⠀⠘⠟⠁⠀⠀⠀⣰⣿⣿⠏⢠⡾⠁
⠀⠀⠀⠀⠀⠈⢻⣄⠘⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠃⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠙⢿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠋⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠙⠿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠿⠋⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡈⠙⠻⢿⣿⣿⣷⣶⣶⣤⣤⣤⣤⣤⣤⣶⣶⣾⣿⣿⡿⠟⠋⣁⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⣤⣀⡉⠙⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠋⢉⣀⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⠶⠶⠶⠶⠶⠶⠶⠞⠛⠋⠉

⠀⠀⣀⠀⢀⡠⠤⡤⠄⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⣀⠀⠀⠀⠀⡀
⠀⠀⠈⡄⡁⢀⠀⠀⠀⠀⠀⣈⠄⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⢁⠀⠀⢀⡈⠄⠀⠠⠄
⠀⠀⡐⠀⠀⠀⣍⠑⠂⠉⣎⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠄⠀⠀⣘⠀⠀⠀⠐⠂⠐⠂
⠀⠀⢇⠀⠀⠀⠐⠀⠀⠀⠁⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠉⠀⠱⡦⡀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀
⠀⠀⠀⠓⠒⠠⡤⠀⠠⠠⠤⠐⠁⠀⣀⡀⣀⣀⠀⠀⠀⠀⡀⠤⠒⠒⠂⠧⡀⠀⠀⠀⠀⠀⠄⣴⢁⠆⠀⠀⠀⠀⠀⡰⠀⠀⠈⠉⡂
⠀⠀⠀⠁⠀⠀⡇⢀⠆⠀⠐⢢⠂⠁⠀⠀⠀⠀⠉⠢⣀⠞⠉⠈⠀⠀⠠⢀⠈⢂⠀⠑⠤⠤⠒⡩⣥⠃⠳⠊⢓⠖⡖⡅⢡⣄⣀⠃⡇
⠀⠀⢸⠀⠀⠀⡇⢰⠀⠀⡇⡈⠀⠀⠀⠀⡄⠀⢤⠀⢹⠀⢀⠀⠀⠀⢀⡀⢀⠀⠆⢀⠴⣎⡹⠤⠀⠂⠀⠠⠄⣠⠀⣄⠉⠉⠁⠀⠂
⠀⠀⢸⠀⠀⠀⣠⠜⡨⢁⠀⠣⣢⠀⠀⠀⠀⠀⠄⠀⠸⠀⠉⠀⠀⠀⠈⠁⠀⠀⢛⠹⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⢀⠃
⠀⠀⢸⠀⠀⠀⠩⠀⠀⠉⡀⠀⣆⠠⠄⠀⠀⠀⠠⠀⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢀⡀⡄⠀⢀⠋⠆
⠀⠀⠐⠂⠀⠀⠀⡆⠀⢘⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠲⢋⠀⠀⠀⠀⡌⠀⠅⠀⡄⠀⠀⡀⠀⠀⠀⠠⢄⡼⠤⠤⠤⠵⠏⡐⡎
⠀⠀⠀⡅⠀⠀⠀⠘⡀⢈⡃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢘⠀⣶⠦⠐⣇⠜⠁⢈⡕⠃⡌⠀⠀⡆⠨⠒⠠⠃⠀⠀⠀⠀⠀⠈⠱
⠀⠀⢠⢰⣀⠠⠤⠀⠋⠁⢡⣀⡄⠀⠀⡀⠀⣴⡎⠛⡰⠉⠉⠁⠀⠀⠀⢀⠔⠋⠀⠀⣈⠦⣸⢦⢁⠀⡀⣰⢠⡇⡄⢰⠀⣄⠀⡀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⣤⣼⠀⠘⣿⠶⢾⠿⠛⠃⠰⠁⡀⠀⠀⠀⣠⢰⣁⠀⠀⠀⠀⠛⠀⠀⠰⢸⠸⡇⠙⡌⢣⢧⡜⣼⣿⣰⠇
⠀⠀⠸⠀⠘⠃⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠀⠀⠙⠈⠛⢻⣶⣄⠀⠀⠀⠀⠀⠎⠃⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠸⠀⠀⢀⠤⠒⠉⠁⠀⠁⠉⠒⢄⠀⠀⠀⠀⡇⠀⠀⠠⠀⡀⢂⠀⠀⣿⣿⣿⣷⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⢄⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠀⠀⠀⡃⠀⠈⠀⠀⠀⠀⠀⢰⣿⡟⠹⢛⠃⠀⠀⠰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠒⠒⠓⠒⠠⢄⡀⠀⠠⢶⣻⣿⣧⣀⠎⠀⠀⠀⡀⢠⡀⠀⠂⠄⠠⠐⠈⠀⠀⢠⠀⡄
⠀⠀⠰⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⣀⠀⠀⠀⠙⠄⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠇⠈⡧⠐⠂⠉⠒⠀⠈⠐⠠⣜⠀⡇
⠀⠀⠃⠀⠀⠀⠀⠿⣼⣇⣴⣀⡄⣠⠀⡌⠀⠉⠀⠃⠉⠢⡀⠀⠞⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠃
⠀⢀⠀⠀⠈⠆⠀⠀⠈⠙⣿⣿⣿⡏⡸⡆⠀⠃⠀⠀⠄⠀⠈⠲⣆⠠⠆⠒⠒⠊⠉⠁⠉⠁⠉⠓⠅⣔⠢⠀⠀⢠⡀⠀⠀⠀⠙⢺
⠀⢨⠀⠈⢆⣰⠀⠀⠀⠀⠈⠻⡿⢀⡆⠁⠀⠀⠀⠀⠀⡀⡘⠁⠀⠀⠀⢀⡀⠄⠀⡀⢀⠀⠀⡀⠐⠋⡐⠀⠀⠘⠁⠄⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡎⠈⠁⠈⠀⠀⠈⠀⢁⠀⠀⡀⠈⠀⠀⢰⠜⠦⠹⠊⠎⠎⢱⠁⠀⢤⣤⣤⠤⠴⠂⠠⠀⠀⡄
⠀⢸⠀⠀⠀⠀⠀⠈⠈⠂⠁⠀⣰⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢀⠔⠁⠀⣀⠀⢶⠀⢰⠈⠤⢐⠀⠀⠀⠀⠀⠀⡃⢠⠁
⠀⠈⡀⢀⠃⠒⠀⠀⠀⠒⡀⡰⠁⡄⡤⠤⠤⠄⠠⢄⠜⠀⡀⠄⣀⣤⡁⠀⠒⣁⣄⠀⠀⠀⢀⡬⢤⡂⠀⠐⠖⠦⠀⠀⣉⡉⠨⠆
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡴⠶⠾⠿⣛⣻⣿⡿⠟⠛⠉⠁⠘⠀⠈⠉⠉⠛⠻⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣤⣴⡶⠶⢟⣿⡿⠏⠁⠀⢀⣠⣤⣶⣶⡶⣶⢶⣶⣶⣤⡀⠉⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠶⠶⠚⢻⡿⠛⠀⣀⣴⣾⠟⢛⣉⣤⣤⣦⣤⣤⣤⠈⢙⠿⠿⢷⣄⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠛⣻⡿⠁⣀⣿⠏⣉⣴⣿⣿⣿⣿⠛⣿⠋⣿⣟⢻⣿⣶⣤⡀⠽⢿⣮⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠠⣼⡿⣁⣴⣿⣿⣿⣿⣿⣿⠀⢿⡆⢻⣟⣛⣛⣻⣿⣿⣦⡄⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⣸⣿⢁⣿⣿⣿⣿⣿⣿⣿⣿⡄⢿⡇⢻⣿⣛⣛⣉⣉⣫⣽⣿⡅⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡆⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇⢸⣟⣛⣛⣉⣉⣉⣭⣽⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣸⡇⢸⣿⣛⣛⣛⣿⣿⣿⣿⣿⣄⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⣿⣧⠘⣿⡟⠛⣛⡛⠟⣛⣿⠟⢻⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⠋⣹⣿⡅
⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⠀⠀⠀⠀⢨⣿⣆⠜⢿⣟⣛⣛⣛⣛⣿⡇⣾⡇⢸⣿⣿⣿⣿⣿⣿⣿⢟⣴⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣷⣷⣿⣿⡶⣶⣶⡶⢾⣿⡟⢷⣬⡙⢿⣿⣭⣯⣿⣇⣻⣇⢸⣿⣿⣿⡿⠿⣛⣵⣿⠟⠁⣿
⠀⠀⠀⠀⠀⠀⢹⣧⣾⠏⠹⣷⡿⠻⣷⣿⢿⣧⠀⠙⠻⣷⣮⣍⣛⣻⣿⢿⡿⠿⢿⣿⣫⣴⣾⠿⢯⣐⣂⢰⣿
⠀⠀⠀⠀⠀⠀⠸⠿⠋⠀⠀⠉⠁⠀⠉⣁⣼⣿⠒⠀⡁⠐⠈⢙⣻⣿⣿⡿⠿⣾⣿⣿⣟⡋⢀⡶⣼⢯⡝⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣯⣿⣿⡀⠀⠀⣠⣼⡿⠛⠁⠸⡟⠀⠀⣿⡅⠙⠿⣯⣷⣦⠔⠛⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣇⣴⣾⠟⠉⠀⠀⠀⢠⣧⠀⠀⢹⡇⠀⠀⠀⠙⠻⢶⣶⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠟⠋⠁⠀⠀⠀⠀⠀⢸⡿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠹⣿⠀⢠⣴⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠿⢿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⣁⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⢿⠀⠄⡀⠀⠀⠈⠛⠿⣦⣄⡀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠁⠈⠰⠀⠀⠉⠈⡀⠀⢀⢀⡈⠛⣻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣋⣄⣀⣡⣤⣤⣬⣠⣤⣤⣩⣾⣟⣛⣹⣆⣥⣾⣻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⢻⣿⣛⠛⠛⠛⠛⠟⠛⠫⣿⣿⢿⠿⢟⣻⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣷⣥⣼⣉⣑⣒⣀⣀⣿⣿⡶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣆⠀⠀⠀⢀⣤⣴⣶⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⢻⣿⣿⣿⠀⠀⠀⣿⡿⣛⣶⣿⡷⢾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣿⣿⣽⣿⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⠀⠈⠉⠋⠉⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣘⣿⡾⣯⣳⣔⣄⡞⣿⡶⣿⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⠀⠀⠀⠀⠂⢠⣀⣹⣿⣾⣿⣿⣷⣿⣿⣿⣹⡭⠿⠿⠷⣯⡀⠀⠠⠀⠀⠰⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠿⠋⠀⠰⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠈⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⣠⠞⠉⠁⠁⠉⠀⠈⠛⠛⠙⠉⢻⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣴⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠏⠀⠀⠀⣀⡤⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣄⣀⠀⠀⠀⠈⢯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠉⠳⡄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠘⣇⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣼⠁⠀⠀⢀⣠⣤⣄⡀⠀⠀⠸⡆⠀⠀⣰⠃⠀⠀⢀⣤⣤⣄⡀⠀⠀⠈⣇⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠐⣿⣿⣿⠟⠻⡆⠀⠀⣷⠀⠀⣟⠀⠀⠔⣿⣿⣿⠛⠻⡄⠀⠀⢸⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡇⠀⠀⢧⠀⠀⠀⢿⣿⣿⣶⣴⡟⠀⢀⡏⠀⠀⢻⠀⠀⡄⢿⣿⣿⣦⣾⡇⠀⠀⡾⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡇⠀⠀⠈⢧⡀⠈⠢⢙⡛⠛⠟⠁⣠⠞⠀⠀⠀⠈⢳⡀⠈⠢⢙⢛⠻⠋⠀⢀⡜⠁⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⠀⠀⠀⠉⠳⠤⠤⠤⠤⠴⠚⠁⠀⠐⠛⠂⠀⠀⠉⠓⠦⢤⡤⠤⠴⠚⠉⠀⠀⠀⠀⣸⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣿⣿⠀⠀⠀⠀⠀⠀⠰⡟⣶⣶⠒⠒⠒⣶⣶⣶⣶⡖⠒⠒⣻⣿⣏⣛⣆⠀⠀⠀⠀⠀⠀⢀⣿⠃⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣤⣾⣿⣿⣿⣿⣷⣤⣴⣿⡿⢿⣿⡟⠀⠀⠀⠀⠀⠀⣼⣏⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣽⣿⣦⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⡏⠀⠙⣿⣿⣿⣿⣟⣠⠴⠛⠀⠀⠀⠀⠀⠀⣰⣿⣁⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢷⡿⠿⠙⣶⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⣼⠋⠿⢿⡾⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣔⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠈⠁⠙⠻⢻⣿⣶⣤⡄⣲⣤⣠⣄⣄⣀⣀⣤⣬⣴⣶⠿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⡿⠻⣿⠛⠟⠛⠿⣿⠻⠟⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⣶⠾⠃⠈⢻⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠹⢶⣶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣼⣿⣾⣁⣀⣀⣀⣼⣶⣶⣶⣶⣶⣶⣶⣾⣇⣀⣀⣀⣸⣮⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⡂⠀⠀⠀⠀⡀⣀⠀⠀⠠⠀⡄⠀⠀⠀⠀⠀⠀⠴⠖⠀⠀⠀⠀⠀⢀⢀⡀⠀⢀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⢣⡀⠀⠀⠀⠐⢮⠀⣀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⠑⠀⠀⠐⠙⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠶⢄⡀⠀⠡⠀⠀⠀⢀⢼⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣀⣰⣶⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢂⡀⠒⠀⠰⢏⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⠀⠀⠀⠀⠒⠂⠀⠀⠀⠁⠀⢀⢮⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⡀⢀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡢⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⠁⣅⡾⡄⠤⣤⠀⣀⠒⠄⠀⠀⠀⠈⠉⠑⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⣖⣼⣭⠋⠀⠁⠁⢀⡇⠀⠈⡺⠴⠴⡲⡄⠀⠰⠆⣸⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠳⠖⠸⠿⢧⣸⡀⣟⣀⠀⡇⣃⣄⣠⠾⠛⠻⢞⣄⣀⠀⢄⣀⠀⣀
⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠴⠤⠼⡠⢴⠶⠼⠖⢊⠉⠀⢸⡏⠉⠁⠀⠉⡊⠉⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠘⠋⢀⣀⡈⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠌⣐⡒⠐⠠⠠⠠⠠⠀⠀⡀⣀⣀⣀⠀⠠⠄⠄⠄⢄⡤⠖⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠐⠒⠀⠠⠤⠤⠤⠤⠤⠀⠒⠂⠀⠁

⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠄⡀⠀⠀⠀⠃⠀⠁⠢⠄⡑⠀⠀⠀⠐⠁⠀⠀⠀⡀⠀⠀⠂⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠆⡒⠛⠛⠒⢁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⢡⠀⠀⠀⠀⠀⠱⢤⣀⡠⠊⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⡀⠉⠉⠀⠀⠠⣖⡚⠂
⠀⠀⠀⠀⠀⠀⠘⢄⠀⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⢄⠀⠀⠈⠓⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠠⣉⢈⠄⡀⢀⢉⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⠀⠍
⠀⠀⣦⡀⠄⡤⠀⠀⠈⠀⠇⠳⠩⣳⠀⠄⠂⠀⠀⠈⠁⠀⠀⠀⢀⠀⠀⠘⠢⠄⡀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⡀⠄
⠂⠑⠊⠈⠀⢣⡀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠉⠂⠂⢤⠀⢠⠀⢤⠀⢀⠀⠀⡤⠀⠀⠀⠡⡀⠂
⠀⠀⠀⠀⢀⡄⠁⠀⠀⠀⠀⠀⣔⠋⠀⠐⠰⢤⠀⠀⠀⠀⠀⠀⠀⢱⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠊⠀⢄
⠀⡇⠀⠀⠀⠀⠀⢰⠀⠀⠠⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡉⠂⠤⢄⣀⢐⠀⠀⡄⢀⡀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠉⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢂
⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⢦⡀⠀⢂⠀⠀⢂⠀⠀⠀⡀⠀⠀⠑⠄⠀⠀⠀⠁⠀⣠
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡀⢀⠀⡈⢷⡄⠀⠀⠀⠀⠠⠲⠀⠀⠀⠀⠀⠀⠈⠐⡲⠄⢀⠢⡀⠀⠈⠂⠄⡀⠈⠂
⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣎⠎⣘⢙⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⠑⠒⠀⠄⠀⠀⠌⠀⠀⠀⠀⠈⢄
⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠨⠦⠘⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄
⠐⠂⠀⠀⠱⠦⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣠⡔⡠⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠍⠂
⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠤⠐⢠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠬⡑⠈⠳⠿⠠⠈⠉⠀⠓⡡
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠑⠂⠐⢀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠐⡔⢈⡐⠈⠵⠒⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠄
⠀⠀⠀⠀⠢⡙⣱⠀⠀⠀⠀⢀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠉⠁⢄⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠶⢃⣷⢌⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠙⠢⡀
⠀⠀⠊⢌⣁⠘⠃⠀⠀⠀⣀⠀⠀⠀⠀⠆⠒⠤⠄⠲⠠⠄⠈⢢⣀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠑⡠⠀⠀⢠⠁⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⣄⠐⣄⢀⣀⡀⠀⠂⠀⣴⣧⠷⡀⠀⠀⠀⠢⡉⠀⣀⠠⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣈⣀⣄⣠⣀⣠⣀⣀⣀⣀⣀⣨⣀⣀⣀⣠⣀⣀⣭⣠⣄⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠉⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣴⡎⣠⣴⣾⣯⠖⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠠⣤⣼⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣧⣤⠄⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢹⣠⣤⣤⣝⣿⣿⠿⠿⡿⣿⣿⣿⣿⢿⠿⠿⣿⣿⣫⣤⣤⠄⡏⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡈⠳⠾⣿⣿⠿⠇⠀⠀⠀⠘⠸⠇⠃⠀⠀⠀⠸⠿⣿⣿⠷⠞⢁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣤⣾⣿⡸⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⢿⣷⣤⣶⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣽⣿⠿⣿⣟⠁⠀⠀⠀⡠⡀⠀⢄⠀⠀⠀⠈⣢⡈⠻⣿⣯⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⣭⣿⡿⠂⢹⣿⡿⠳⣄⠀⠘⢳⡞⠃⠀⣠⡞⢿⣿⡏⠐⢿⣿⣍⡙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠋⠁⠀⠻⣷⣄⣋⣷⠶⠾⠷⠶⣾⣙⣡⣾⠟⠀⠈⠙⢿⣿⣟⡂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡞⠀⠀⠀⣀⣈⠛⢶⣷⣦⣴⣾⡾⠛⣁⣀⠀⠀⢀⢳⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⣿⣿⣧⡟⠀⠘⣏⣸⠓⣄⣈⠛⢛⣁⣠⠚⣇⣩⠃⠀⢹⣾⣿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠇⢰⣿⡿⣿⣿⣷⡇⠃⠘⣿⣼⣴⣦⣿⣿⣼⣦⣧⣿⠃⠘⢰⣾⣿⣿⢿⣿⡆⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢹⣿⢿⣿⢡⡀⠀⡇⢻⣿⣿⣿⣿⣿⣿⡟⢸⠀⢀⡌⣿⡿⣿⡏⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠸⣿⣿⣠⠀⠳⠶⠶⠔⠒⠒⠢⠶⠶⠞⠀⣄⣿⣿⠇⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⣿⢷⣄⡀⠀⢠⡀⠀⡄⡆⢀⣠⡾⣿⡿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⠻⣷⣾⣾⣧⣸⣷⣷⣿⠟⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠏⠁

⠀⠀⠀⠀⠀⢀⣀⣤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠐⡂⡀
⠀⠀⠀⠀⡶⠪⠉⠀⠁⠈⠁⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⡆⠀⠀⠀⠸⠻⠀⠀⠀⠀⠀⠀⢰⠋⣉⠻⡄⠀⠀⡰⣼⡿⢸⢰
⠀⠀⢐⠏⠀⢀⡀⣀⠀⣀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠈⢛⡦⠠⠤⢼⡀⠀⠀⠀⠀⠀⠀⠸⢤⣩⠐⣇⠀⠀⠑⡌⢨⠕⠁
⠀⠀⣮⠀⠀⠈⠁⠈⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⢀⠌⠠⢂⣅⠢⠙⠄⠀⠀⠀⠀⠀⠀⠀⠈⠣⡈⢦⠀⠀⠃⡌
⠀⠀⢷⠀⠀⠀⠈⠓⠚⠁⠀⠀⠀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠓⠚⠀⢧⡀
⠀⠀⠰⣆⠐⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠰⠖⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⠀⠀⠢⡱⡀
⠀⠀⠀⠉⠶⢄⣀⣀⢀⣀⣤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⡃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠳⠚⢧⡞⠋⠁⠈⠁⢢
⠀⠀⠀⠀⠀⣾⣿⡇⠉⠿⠿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⢠⠀⡁⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠤⠤⠔⠒⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣁⣀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠲⣶⣄⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠉⠓⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⣫⡕⠀⠂
⠀⠀⠀⢠⠀⡁⠉⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⡴⢋⠁⠀⠀⠽⠀⠸⠆⠀⠀⠉⠒⣄⠀⠀⠀⠀⠀⠀⠐⢀⠈⠁⢀⠂
⠀⠀⠀⠀⠰⠀⠀⢰⣶⡆⠀⠀⢀⠀⠀⠀⠀⢸⠀⢿⠀⠰⣁⣤⣠⣄⣄⣥⠀⢸⢃⢘⠀⠀⠀⠀⠀⠀⠀⠀⠐⣋⠔
⠀⠀⠀⣌⠀⣀⠀⠈⣉⡁⠀⠀⢠⢦⠀⠀⠀⢸⠀⡏⢦⠀⠻⢻⡟⢿⣻⠇⢀⡼⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⢀
⠀⠀⠈⣎⠀⠀⠀⠀⣀⣀⡤⠂⠠⠶⠀⠀⠀⠙⠛⠃⠈⠳⣄⠀⢀⡀⠀⣠⠊⠘⠳⠛⠀⠀⠀⠀⢀⣶⣀⠔⠁⠀⠀⠀⠤⠶
⠀⠀⠀⠈⣇⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⡋⣽⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠀⠙⠒⠚⠁⡀
⠀⠀⠀⠀⠀⠀⠇⠂⠀⠘⠸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⡇⢸⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⢰
⠀⠀⠀⠀⠀⠉⠀⠈⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠁⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠁⠘⠖
⠀⠀⠀⠀⠀⠘⠆⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠠⠦⠀⣀
⠀⠀⠀⠀⢀⠜⣤⠀⠀⠠⡶⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠛⠁⠈⣙⣳⡅⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⢠⡀
⠀⢀⡀⠀⡘⠂⣤⣠⣤⣤⡆⠈⢠⠀⠲⡆⠀⠀⠠⢶⢦⠋⣺⣿⠢⣾⣿⢽⡏⣮⡟⠃⠀⠀⠀⠀⠸⠀⠀⠿⠁⠀⠸⠏⠀⠀⠰
⠀⠻⠤⠆⡁⠀⠈⠛⠛⠉⠀⠀⠀⠘⠁⠁⠀⠀⠀⠈⢻⡀⠀⣈⣀⣙⣿⠋⠀⡏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢻⣟⣿⣿⠀⡼⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⢠⡻⢄⠀⠀⣠⠾⡁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢐⠀⢠⠄⢢⠀⠮⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⠀⠑⠖⠁⣀⢡⡀⠀⠀⠀⠀⠀⠰⢆⣀⡀⣀⣀⣀⣀⣀⣀⣀⣠⠄
⠀⠀⠀⠀⠘⠀⠀⠘⠀⠘⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠁⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠖⠂⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⠀⠀⠀⠀⢀⡰⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠎⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠉⡤⢤⠈⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠐⣴⡂⠂⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠐⢒⡊⢀⠓⢒⣠⠀⠠⣄⠀⠀⠀⠀⠀⠀⡎⠀⠈⢾⡷⠃⡀⢸
⠀⠀⠀⠀⠀⡀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠀⠈⠁⠈⠁⠈⠁⠐⠀⠈⠀⠀⠀⠀⠀⠀⣷⣾⠷⣦⡶⢿⣿⡾
⠀⠀⠀⠀⢀⠀⢷⣄⣀⣠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠙⠻⠾⠾⠟⢉⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠉⠉⠡⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠨⠴⡵⠂
⠀⠀⠀⠀⠀⢰⡆⣤⠄⣴⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣕⡠⠄⠰⡚
⠀⠀⠀⠀⠀⠘⠁⠙⠀⠙⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠀⠑⠊⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣴⠋⠉⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⡇⠀⠀⢻⠀⠘⡆⠀⠀⠀⠀⠀⣠⠀⠀⢠⠖⠉⠑⣄⠀⠀⠀⡰⢓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠧⣀⠀⢳⣄⡠⠟⢀⡼⠁⠀⠀⠀⠀⢸⠂⢧⡴⠃⠀⠀⠀⠀⢳⡠⠊⠐⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠈⢙⡏⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⣸⡖⠁⠀⠀⠀⠀⠀⠀⡎⠁⠀⣰⠃⠀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⢾⠀⠀⠀⡞⠀⠀⠀⠀⠀⡠⠊⠁⠙⢦⠀⠀⠀⠀⠀⠀⠘⠤⠔⢧⡀⠀⠁⢄⠀⠀⣏⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⣠⣾⡆⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⢣⡜⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⡇⠀⠀⣰⣿⡿⠃⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢀⠜⠂⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⡇⠀⡴⠁⠁⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠃⠀⠀⠀⠀⠀⡧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠹⡴⠧⢄⣀⣀⣀⣠⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢸⣀⡀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⠀⡅⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⠀⣀⠀⠀⢯⠒⣤⣄⣀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠉⠀⠀⡀⠀⡀⠄
⠀⠀⠀⠀⠀⠠⠖⠁⢠⠀⠀⠀⢀⠀⠀⠈⠑⢄⡀⠈⠢⠠⢿⡉⣛⣻⣿⡏⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢀⡀⠀⡀⣋⠵⠞⠐⠓⠒⣬⠃
⠀⠀⠀⠀⠀⠀⠀⢠⡡⠒⠈⠉⠀⠈⠉⠑⠢⢄⡙⠢⣀⠀⠀⠙⣿⣿⣿⠇⠀⠀⢸⠃⠀⡘⠁⠀⠀⠀⠀⣸⠀⡀⢀⣇⠴⠚⠉⠀⠀⠀⠀⠀⠀⣏⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠒⡈⠑⠲⠎⠩⠍⣭⣶⡿⠀⠀⢠⠃⠀⢠⠇⠀⠀⠀⠀⠀⣯⡤⠞⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⡜⣃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠉⠛⠺⢶⣉⣦⣿⠿⠁⠀⢠⠇⠀⠀⡞⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⣠⠔⠋⠁⡇⠀⢸⠧⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⡸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠱⡀⠈⠉⣩⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡌⠁⠀⠀⠀⠀⢀⠔⠁⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⣀⣠⠞⠁⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠤⠤⢄⣀⣠⠴⠋⠀⠀⠀⠀⢀⠞⡔⠒⢢⢸⡁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠺⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⣠⠏⢘⡇⠀⢸⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣱⠀⠈⡃⠀⡄⠀⠹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠁⠀⠀⡇⣰⠁⠀⠀⠀⠉⠉⢉⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⣍⠀⠀⢀⣀⠴⠎⠠⢀⣀⣀⣀⠼⠁⢣⣀⠀⠀⠀⠀⠈⠁⠈⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠒⠒⠒⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢿⢿⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣽⢿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧
⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⡴⣾⣿⠛⠉⠁⠈⠻⠋⠀⠀⠀⠀⠀⠀⢠⣀⣀⣸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠉⠁⠈⠉⠉⠀⠀⠀⠀⠈⢀⡤⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠉⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠙⠚⠈⠉⠁⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠂⣀⡘⠛⠂
⠀⠀⠀⠀⠀⠀⣼⣿⣾⠆⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣧⡀⠀⠀⢀⡆
⠀⠀⠀⢀⠀⢰⣿⡟⠁⠀⠀⠀⠉⣆⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣼⠀⣸⣿⣇⣴⡆⠰⣦⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⣠⣿⠇⠀⣀⠀⠀⠀⠀⠀⠀⣾⡟⠠⠀⠠⡷
⠀⠀⠘⣿⣄⠘⣿⣿⣿⣤⣄⣸⣾⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⠟⠀⠀⠀⠀⠀⠀⡈⠱⠆⠀⣶
⠀⠀⠀⣿⣿⣷⡿⣿⣿⣿⡏⣿⣿⣿⠀⠀⠀⠀⠀⠀⡀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣛⠁⠛⠃
⠀⠀⠀⣿⣿⣿⠁⣿⣿⣿⠃⣿⣿⡇⠀⠀⠀⠀⢀⠴⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠋⠈⠐⠠⡃
⠀⠀⠀⠘⠛⠛⠀⠙⠿⠿⠆⠿⠿⠇⠀⠀⠀⢠⡍⢰⣠⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠃⠓⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄⣄⡀⠀⠀⢀⣬⡇⢿⠛⡕⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡄⠛⣀⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⣷⣙⣟⡟⠵⣾⣿⢸⣯⣿⣣⠉⠻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡈⠁⠀⣫⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢘⢦⠹⡜⣿⡟⡼⣟⣿⣿⣽⣻⠯⣩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡀⠁⡀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⢸⣵⣆⢳⢿⣷⢻⢷⣿⠻⡳⠭⣭⠜⠀⠀⠀⠀⠀⢀⣶⠶⣾⣇⠀⠀⠀⠀⠄⠀⠛⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠃⢻⠈⠻⠋⠢⠽⠯⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⢈⣠⣶⢾⡾⠻⣦⡀⠀⠀⠢⠄⠠⠄⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⠀⠀⢠⠆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠗⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⢠⡶⠀⢸⣇⣀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠰⠄⠀⠄⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⢴⠃⠴⢿⠟⣀⠤⠿⠼⢾⡯⠿⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠒⠃⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⢀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣿⣷⣄⠀⠀⠀⠀⢀⣾⣿⣿⣿⣦⣾⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣃⣀
⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣿⣿⣿⣿⠃⠀⠀⣠⣤⣤⡀⠀⠀⠈⢿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢺⣿⣿⣿⣿⡇⠀⠀⢰⣧⣽⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⡍
⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣇⠀⠀⠀⠙⠿⠿⠋⠀⠀⠀⢸⣿⣿⣿⣿⣯⣤
⠀⠀⠀⠀⠀⠀⠀⠠⢼⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⠿⣿⣿⣿⣶⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⠁⠶⠈⣿⣿⡿⣿⡿⢿⣿⠿⣿⠿⣿⠁⠴⠀⢻⣿⠏⠻⣷⣦
⠀⠀⠀⠀⠀⢀⣠⡾⠛⢿⣿⣦⣤⣴⣿⣿⣶⣾⣷⣼⣷⣶⣿⣤⣿⣦⣤⣴⣿⣿⠭⠀⠈⠻⣷⡄
⠀⠀⠀⠀⣠⣿⠏⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠀⠀⠀⠀⠙⣿⡆
⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠋⠀⠘⠃⠙⠉⣿⣿⠹⠉⠉⠟⠉⢹⣿⡇⠋⠀⠉⠀⠀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⢸⣿⣿⣃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣹⣿⣿⡆
⢀⣾⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣶⣄⠀⣴⣶⣿⣿⣷⣶⡄⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣧
⠞⠁⣿⠋⢿⡎⠻⠈⠁⠀⠀⠀⠀⠛⠙⠟⠙⠛⠙⠛⠘⠛⠙⠛⠙⠟⠙⠃⠀⠀⠈⠁⠼⠋⣾⠏⢹⡇⠙

⠀⠀⠀⠀⠀⠀⡶⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣤⣄⣠⣤⣀⣀⣀⣠⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣶⣦⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⣴⣧⠛⣿⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠠⣾⣏⢉⣭⡉⠉⢹⣿⣿⣿⣤⣭⣿⣶
⠀⠀⠀⠀⢸⣿⣷⣄⡟⠀⠀⠀⠰⣲⡆⠀⠈⠙⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⠆⠀⠀⠀⢻⣿⡋⠿⣿⣶⣼⣿⣯⣿⠟⠻⠛⠻⠆
⠀⠀⠀⠀⣸⣿⣿⢿⣁⣤⠀⠀⠐⠁⠐⠀⠀⠀⠀⠀⠘⣿⣿⣧⣿⠛⠛⢻⣿⣿⣿⣿⡟⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⢸⡿⠁
⠀⢀⠤⡴⠛⠷⠿⠋⠙⠁⠀⢘⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⣿⠀⠀⢸⣿⠋⢿⣿⡇⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢨⣑⠀⠀⠀⠀⠀⠀⠀⠶⠟⠁⠚⠋⠀⠀⠚⠋⠀⠐⠞⠃⠀⠀⠀⠀⢯⣇⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢮⡅⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢿⢗⡅⠀⠀⠀⠀⠀⠈⠙⠢⡀⠀⠀⠀⠀⠀⠀⢀⡇⠒⠙⠛⠳⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢾⡱⠎⠀⠀⠀⠀⠀⠀⢸⡄⠀⠙⠄⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠉⡆⠀⠀⢀⣀⣠⣤⣶⣶⣷⣿⡿⠷⠒
⠀⠀⠀⢀⣠⣤⣶⣿⣻⣿⡷⢠⡃⠀⠀⠀⠀⠀⠀⢹⡇⠐⠂⠄⠿⠄⠀⢸⣿⣿⠞⠁⠀⠀⠀⠀⢀⡞⠁⠘⣲⣿⡿⣿⣿⣿⠿⢿⡉⠁
⠀⠀⠀⢺⣿⡿⠏⣷⡛⡝⣿⡎⠷⠀⠀⠀⣴⣶⣤⣼⠁⠀⠀⠆⠀⠀⠀⢸⣯⣬⠶⠖⠃⠀⠀⠀⠀⠛⠿⠿⠛⣩⠟⠉⠀⠀⠀⠀⠉⠉
⠀⠀⠀⢸⣷⠀⠀⠿⠈⠄⢹⠃⠀⠀⠀⢀⣥⣿⣿⣷⡀⠀⠀⠀⠀⠀⠐⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⣿⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⣿⣿⣹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣦
⠀⠀⠀⣤⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⣸⢛⠟⣋⠽⠷⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣤⣤⣾⡗⠀⠁⠀⠀⠀⢀⡀⠀⣴⣿⣿⣿⠋
⠀⠀⠐⡿⢸⠁⠀⠀⠀⠀⠀⡄⠀⠀⠘⠁⢰⣧⠞⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠃⠀⠈⢉⣿⣿⣿⣿⣿⡖⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠙⠛⠂⠀⠀⠀⠀⠀⠀⡘⠻⡟⠋⠁⠀⠀⠈⠛⣇⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⢣⡀⠀⠀⠀⠀⠆⠀⢄⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣠⣴⡾⢷⣜⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠉⠁⠀⠀⠻⡿⠻⣿⣦⣄⣀⣀⣀⣠⣄⣀⠀⠀⢀⣾⣿⣿⣿⣿⡿⠋
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣼⣿⣿⣽⣿⣆⠀⠀⠀⠀⠀⠀⠀⣸⡟⠉⠀⠈⢃⠙⠋⠙⣿⠲⣄⠉⠉⠙⠋⠉
⠀⠀⢹⣿⣿⣿⡿⠈⠻⣿⣿⣿⣼⣿⣿⣿⢿⣏⠋⡟⠋⢿⡆⠀⠀⠀⠐⣿⠿⢿⣤⣤⣤⠾⢧⢦⣰⡿⠀⣿⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀
⠀⠀⣨⣿⣿⡿⠛⠁⠚⠛⠛⠙⡿⣹⡏⠀⠀⠀⠁⠙⠒⠛⠁⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⡅⠀⠋⠀⠀⠀⡶⢿⣿⣽⣿⣷⣿⣿⡄
⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠥⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⠀⠀⠀⠀⠀⠀⠺⡟⠛⠛⠛⠛⢿⠁
⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢀⡇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⠟⠁⠀⠀⠀⢀⡄⢀⣴⢃⣤⡶⠊⠀⠀⠀⠀⠀⠀⠙⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⣾⠀⠀⠀⠀⢠⣿⣷⣿⣿⣿⣿⣥⣴⣶⣿⣯⣄⣀⡀⠀⢸⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠘⠦⣤⡤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⢤⠴⠋⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣤⣤⡾⠋⠀⠹⣿⣿⢿⣿⣿⠿⠛⠛⠛⠻⠿⣦⣤⣤⣶⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡿⠀⠃⠀⠀⠘⢿⡀⠙⢿⡄⠀⠀⠀⠈⠙⢎⢿⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⣿⠃⠀⢾⣌⡁⠀⠀⠁⠀⠀⠉⠀⠀⢈⣤⡇⠈⠘⣿⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡿⢠⣿⠀⠀⠺⢻⡙⢦⣀⢀⡀⣀⡀⣠⡶⢹⡟⠛⠀⠀⣿⡔⢿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠁⢸⣿⡇⠀⠀⠀⠈⠛⠶⡟⣉⠓⢋⣉⣻⠒⠋⠀⠀⠀⠀⣿⣿⠇⠈⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢰⢣⣂⠀⠀⣀⣻⣿⠀⢀⣠⢤⡶⠦⣭⡍⣿⣿⣭⡥⢶⣶⢤⣄⠀⠀⣿⡟⣀⠀⠀⣀⡜⡆
⠀⠀⠀⠀⠀⠀⠀⠿⡿⠁⠀⠀⠈⢻⣿⢠⢸⣿⣄⡇⢀⣇⣀⣸⡀⣈⣃⣀⣇⣰⣿⠇⡞⣿⡟⠁⠀⠀⠈⢿⠷
⠀⠀⠀⠀⠀⠀⠀⣸⠁⣠⠀⠀⣠⠈⢻⡀⠀⣿⢿⣿⢿⣿⣿⣿⣿⣿⣿⢿⡿⣿⡿⠀⢠⡟⠁⣄⠀⠀⣀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⢹⠀⣸⠀⢸⡇⠀⢾⣷⠀⠘⢎⡿⠮⡽⠿⣿⠿⢿⠅⢺⣁⠟⠁⠀⣿⡗⠀⢸⡇⠀⣇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣿⣦⣾⣿⣤⣿⣿⣧⡀⠀⢉⡒⠷⠤⠿⠤⠼⠗⢚⠁⡠⢂⣾⣿⣷⣤⣿⣷⣴⣿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⡟⠋⢸⠿⢿⣷⣤⡐⠉⠛⠒⠒⠒⠚⠉⠀⢊⣴⣿⠿⠿⡇⠙⢹⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠈⢻⣿⣷⣶⣦⣤⣤⣴⣶⣾⣿⡋⠀⠀⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣄⠀⠘⣿⣿⣿⣯⣭⣭⣿⣿⣿⣿⠃⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠟⡓⢲⣿⣄⣈⡉⠉⠉⠉⠁⣸⣿⠗⣒⠺⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠻⠿⠾⠿⠿⠿⠿⠿⠟⠋⠀⠉⠛⠿⠿⠶⠿⠶⠿⠟⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⢀⠀⠀⠀⣴⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⡀⢠⣿⣧⠀⣼⣿⣿⣿⠀⠀⣠⣶⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⡀⣿⣿⣿⡿⢺⣿⣿⡟⠛⠿⠿⠟⣦⣾⣿⡟⢀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣤⣄⡀⣻⣿⣿⣯⠉⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠙⠿⠿⠋⢿⣿⣿⣿⠁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠉⠻⠛⠁⠀⠀⠀⠀⣀⡀⡤⠤⣀⣀⠀⠀⠀⠀⠀⠀⠉⣻⣿⣾⣿⠏⢀⣠⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣸⠿⠟⠁⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠉⠒⢤⡀⠀⠀⠀⠻⣿⣿⢿⣾⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠂⠀⠀⠀⠀⡰⠋⠀⠀⣀⣤⣶⣶⣶⣶⣤⣀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠈⠿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡏⠛⠁⠀⠀⠀⠀⡼⠁⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠈⢧⠀⡀⠀⠀⠀⣴⣾⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠙⢿⠟⠁⠀⠀⠀⠀⢸⢰⡇⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡆⠀⢸⡄⠃⠀⠀⠀⠙⢿⡿⣯⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢸⢸⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣠⣿⣿⠀⠈⡇⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠻⡟⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⠋⠁
⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⣀
⠀⠀⢀⠞⠁⠠⠤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠉⠻⢿⣿⣿⣿⣿⡷⠆⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⢠⠃⠀⡠⠊⠉⠀⢹⠁⠀⠀⠀⠀⠀⠀⠲⡴⠂⠀⠀⠙⠲⠤⠤⢄⡠⠤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⠙⠶⢤⡀⠀⠣
⠀⡏⠀⡸⠁⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠙⣄⠀⢳
⢘⠃⢰⡇⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⢸⠃⢸
⢸⠀⠈⠘⢄⡀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠙⠢⢤⣀⣀⣀⣀⠤⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⢀⡌
⢘⡆⠀⠀⠀⠉⠙⢢⡀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠐⠠⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠃⠀⠀⠀⠀⣀⠤⠒⠋⠀⠀⢠
⠸⠁⠀⠀⠀⣠⢤⣿⣿⡄⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⣼⣇⡀⠀⠀⠀⠀⠘
⡇⠀⢀⣀⣀⡁⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣤⣀⣀⣀⠀⠀⣀⣀⣀⣤⠀⠀⠹⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠉⠱
⢿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⡜⠀⠀⠀⠀⠀⠀⠀⠀⡆⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣸⣷⣤⣦⣤
⠈⠻⣿⣏⠛⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⢫⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠊⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠈⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣤⣄⡀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⣄⣤⣶⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⠟⢻⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠻⣿⣿⠙⢿⠟⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣥⣴⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣭⣬⣭⣤⣭⣽⣭⣼⣯⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢀⠀⡠⠀⠈⠁⠀⠀⢀
⠀⠀⡄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠐⠂⠈⠁⠀⠀⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢂
⠀⠀⠀⠀⠠⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⡄⠐⠀⠀⠀⠀⠀⠀⠀⢀⣁⠀⠀⠂⠐⠠
⠀⠀⠀⠘⠃⠀⢙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⣄⡻⢂⠃
⠀⠀⠀⠀⠀⠀⠀⠂⠐⡀⠀⠀⠀⠐⡁⠄⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢀⠂⠀⠀⠒⠆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄
⠀⠨⡀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⡀⠙⠀⠀⡀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⠐⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠰⠈⠂⠐⠢⠆⠇⠀⠀⠀⠀⢈⡂⢂⠃⠀⣌⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⢂⠆⠐⠰⢫⠄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⣠⠄⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢎⠄⡄⡈⠂⠆⠐⡄
⠀⠀⠀⠀⠀⠏⣀⡄⠰⡿⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠚⢋⡡⠌⢂⠀⠀⠀⠈⠨⠘⠀⠔⠈
⠀⠓⢶⣄⣀⠐⣁⡀⠀⠁⠀⠀⣨⠔⢂⠀⡄⠀⣠⡄⠍⢀⠒⠌⡈⠃⠀⠀⠀⠀⠀⢦⠙⠐⠊
⠀⠀⠀⠀⠀⠀⠙⠂⠀⠐⠠⠞⠋⠀⠀⠀⠀⠀⠊⠁⠀⠀⠁⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⢰⣻⣶⣾⣿⣿⣶⣿⡷⠀⠀⠀⠀⠀⢀⣠⣴⣶⣤⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠹⠀⠍⠀⢀⡀
⠀⠀⣼⣿⣿⡟⢩⡙⣿⣿⣿⠄⠀⠀⠀⢠⣾⣷⣦⢀⣾⣿⣧⠀⠀⠀⡀⠀⠀⠀⠀⣴⣿⡖⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠈⠛⠡⠦⠈⣃⠘⠛⠁
⠀⠀⠾⣻⣿⡆⠀⠀⣾⣿⡷⠆⠀⠀⠀⣼⠛⠛⠋⠆⠙⠛⢻⡆⠀⠀⡂⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠈⠋⣿⡇⠀⠀⣿⡏⠛⠀⠀⠀⠀⣿⣶⣶⣶⣶⣶⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀⡀⣀⣀⣤⣦
⠀⠀⠀⠀⠘⠛⠷⠷⠿⠇⠀⠀⠀⠀⠀⠹⡿⡿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣍⣛⣡⣿⣿⣿⣿⠀⠀⠀⢀⠀⠀⠀⠀⠘⣿⣿⣿⣟⣿⠆
⠀⠀⠀⠀⠒⠶⠶⠶⠖⠂⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠰⠆⠀⠻⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣊⡀⠀⠀⢀⡄⢠⠄⢀⣄⢠⡀⡄⣤⠀⣄⢠⣀⡀⢠⡄⣄⠀⣠⠀⣤⠀⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢣⡀
⠀⠀⠀⠀⠀⠰⣼⣔⣂⠀⠀⠺⠦⠄⠀⠀⠀⠀⠵⠸⠀⠘⠿⠸⠿⠇⠿⠀⠿⠸⠸⠇⠻⡇⠹⠀⠿⠀⠻⠇⠏⠿⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠉⠽⠫⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡄⣀⣠⣄⣠⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣶⢄⠀⠀⠀⢠⣶
⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⣾⣷⣶⣶⣷⠁
⠀⠀⠀⠈⠛⠈⠛⠛⠘⠏⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢠⣶⣬⡿⠼⢿⣧⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⣶⠽⣿⡇
⠀⠀⠀⠀⣀⡀⠀⠀⠀⡀⠀⠀⠀⢿⣧⠀⠀⠀⠀⢸⣿⠁⠀⠀⣰⣦⣽⠆⠀⠤⠀⠘⣯⣤⡆⠀⠀⠀⠀⠀⢶⣶⣆⡀⠀⠐⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⡍⠉⠁⢒⣛⡃⠀⠀⠀⠘⣿⡀⠀⠀⠀⣿⠃⠀⠀⠀⢩⣭⣿⠀⠐⠀⠀⠀⢸⣭⣅⠀⠀⠀⠀⠀⠈⢩⣷⣿⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⢠⣿⠇
⠀⠀⠀⠀⣶⢶⣶⣶⣿⡇⠀⠀⠀⠀⠈⠀⠠⡄⠀⠉⠀⠀⠀⠀⠘⠟⣹⣦⣤⣠⣤⣴⣏⠻⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⣠⠋
⠀⠀⠀⠀⠙⢿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏⣾⠿⣿⠸⠟⠁⠀⠀⠀⠀⣀⣀⠀⠸⠿⠿⠀⠀⣀⣀⣀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠘⠿⠟⠃⠀⣴⣾⣷⣦⠀⠘⠛⠛⠁⠀⠀⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⢀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⡆⠀⠀⠀⢠⠁⠀⠀⠀⢀⣠⣾⣿⣿⣶⣿⣿⣿⣿⣥⣂⠀⠀⠀⡘⡀⠀⠀⠀⢰⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢸⡇⠀⢠⡀⣿⣿⣀⣧⣬⣿⣿⣿⣿⡌⠉⠙⠛⠿⠿⠿⢿⣿⣼⣠⣿⡇⢀⠀⠀⢸⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⣧⡀⢀⣿⣿⣿⣿⣷⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣞⡀⣀⣾⠇⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠙⠛⠙⠉⠉⣠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⡙⠙⠛⠋⠉⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣤⣆⣀⣀⡼⢏⣀⠀⠈⠀⠀⠀⠀⠐⠂⠀⠀⠀⡀⠐⠉⢀⣀⠿⣄⣀⣠⣤⣶⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⠿⢫⣾⣿⣿⣷⣄⠀⠈⠛⣷⣶⣼⣿⣟⣋⠀⠀⣴⣿⣿⣿⣮⠻⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣷⣾⣿⡃⠀⣽⣿⡿⠁⠈⠻⢷⣤⣤⣿⣿⣿⣿⣿⣥⣶⠿⠋⠀⠹⣿⣿⡁⠘⣿⣿⣾⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣻⣿⡟⠉⠀⠀⠹⣿⣷⡀⠻⠆⢀⣽⡿⣟⡛⢛⣛⡻⣿⣇⠀⠾⠃⣰⣿⡿⠀⠀⠈⢻⣿⣟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⣿⠟⠀⠀⠀⠀⠀⠈⠉⠛⠷⠶⣿⠿⣞⣻⣥⣤⣭⣟⣿⣿⣿⠶⠾⠛⠉⠀⠀⠀⠀⠀⠹⣿⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡀⠀⠀⠀⣀⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣄⡀⠀⠀⠀⣿⣷⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⣷⡄⢠⣾⣿⣿⡏⠉⠙⠟⠙⠻⠒⠛⠛⠻⡿⠛⠛⠛⠻⠛⠙⡿⠉⠉⢻⣿⣿⣦⠂⣸⣿⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣿⣿⣷⡸⣿⢿⣿⣇⠀⣠⣾⣄⣾⣦⣄⣀⣀⣧⣀⣀⣤⣼⣇⣰⣧⡀⢀⣾⣿⢿⣿⣰⣿⢿⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⢿⣿⣿⡈⢿⣿⣿⡟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢻⣿⣿⠏⣼⣿⡿⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⠟⣿⣧⠈⡻⣿⠀⠀⠻⡟⠙⠏⠉⠈⡏⠉⠉⡟⠉⢿⠋⠀⢸⡿⡋⣰⣿⠿⠁⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣿⣷⡙⢮⣓⠲⠾⢦⣤⡷⡤⠴⠷⠤⣤⣷⡤⠼⠶⢖⣫⢞⣴⣿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢻⣿⣿⣶⣦⣉⡀⠀⠀⢀⠤⣤⣤⡤⠀⠀⠀⢀⣬⣽⣶⣿⣿⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠹⣿⣿⣭⣄⣀⢀⠀⠀⠀⠉⢀⣀⣀⣤⣿⣿⠏⢹⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠀⠙⠿⣿⡟⢿⡟⢷⣆⣠⣶⢿⡿⢻⣿⠿⠃⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠙⠀⠹⡿⠁⠈⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁