Crystallography Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠒⠤⢀⡀⣀⡀⠀⠀⠀⠀⠀⠀⣁⡀⠀⠀⡄⡑⠂⢀⣀⣀⣀⣀⣀⣀⣄⣀⣤⣠⣤⣤⣤⣤⠴⣶⣦⠄
⠀⠀⠀⠀⠀⠀⠀⣰⠋⢻⡿⠟⠁⠊⠐⣿⢩⡿⡴⢄⠉⡀⠀⠀⠠⠀⠀⠀⠀⣰⣻⣿⣿⣯⣯⣵⣯⣭⣿⣿⣝⣻⡷⠎⠉⠉⣉⠠
⠀⠀⠀⠀⠀⣤⣴⣿⠃⠸⠃⣴⠁⡇⠀⠉⣨⠐⠾⠧⠄⠱⠀⠀⠘⠶⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣽⣛⣭⡤⢼⣿⣷⣮⠛⣦⡀
⠀⠀⠀⢀⣾⣿⣿⣿⣧⠀⢠⡯⠐⠁⠀⠲⠿⡄⠀⠀⠀⡀⠁⣠⣴⡶⠁⠀⢹⣼⣭⠉⠛⣏⠉⢛⣿⣷⡉⣿⣿⠻⡅⠛⢮⡻⣷⣷⡖⠃
⠀⠀⣰⣯⣿⣿⣿⣿⢛⣦⣬⣠⣬⡄⣘⡦⠤⢤⡀⠀⠀⢀⡟⣥⠈⠁⠀⢀⢘⠋⠸⡇⢰⣿⢠⡞⣿⢿⣛⠙⠻⣇⢹⡀⢯⣡⣴⣌
⠀⠀⢸⣿⣽⣿⡻⢃⠛⡿⣿⣿⣿⣿⣟⠠⠀⠀⢴⠆⠀⣠⠿⠀⠀⠀⡀⡈⣯⠈⠰⡇⢠⣏⢸⣷⡜⣧⣬⣣⠡⠻⠀⠂⠈⠉⠠
⠀⣴⣿⣿⢛⡉⢷⠒⢏⢉⡿⣿⣿⣿⣿⣡⣶⡠⢀⣀⣸⣷⣄⠀⠠⡃⡑⠝⣿⣦⠐⡆⢨⠟⠹⣿⢀⠨⢿⣿⣻⠆⢡⡙⢀⢀⠜⢱⡆⠀⣶
⣾⣿⣟⣿⡎⣥⠁⠀⠀⡐⠧⣘⣿⣿⣿⣿⣿⣰⢹⣿⠫⠛⠀⠀⠳⠙⠆⠐⠝⣿⣦⣴⡿⣿⣤⡾⣴⢿⢿⣿⣿⡔⠦⣀⡄⠈⠀⡈⠀⠀⠙
⠈⢿⣿⣿⠁⣐⣠⢷⡶⢿⠀⢛⣿⣹⣿⣟⡉⠁⠘⢛⣬⡄⡠⢼⣄⡠⣠⡆⠸⢿⡼⣿⣧⢿⢏⠁⣿⢣⠆⢸⣝⣻⣿⡿⢀⠀⣠⣿⢠⡆
⠀⠀⢽⡆⠒⠠⢿⣶⣥⣽⣾⡙⢿⣿⡥⠀⠠⡀⡼⠂⢿⡁⠀⠢⢛⠻⣷⡦⡹⣷⣗⣿⢹⡼⡇⠊⣿⡞⢀⠀⣿⣿⠟⡁⠈⢠⣿⣿⣼⣷⣄
⠀⠀⢴⣷⣄⠀⠄⠙⠛⠻⣿⠃⢰⣿⡏⠀⠃⠙⢶⡗⠘⢉⣤⠾⠃⢭⣻⠟⢿⣦⡻⡟⡹⣧⣄⡄⡇⠀⢸⣿⣿⡿⢠⣇⢛⡞⢹⣿⢿⣿⡏
⠀⢀⣼⣿⠛⠟⢳⣄⡀⢼⠃⠦⠉⢻⣿⠂⠠⠀⠚⠀⠠⣽⠒⠀⣀⠾⠛⢷⡀⢿⡙⢼⠆⣿⣿⣏⡀⠀⣾⣻⡟⣿⣻⡿⣾⢡⠙⠉⠀⠈
⠀⠛⣘⠁⠀⠀⣀⣀⣁⠀⠄⡢⣄⡘⠿⠀⣤⠠⠢⠀⡀⢀⢄⣠⣄⣈⢻⣯⠀⠖⠉⠺⠄⠙⣿⡟⠀⢺⡏⣉⢱⣏⡙⠀⠁
⠀⢸⣿⣿⣿⣿⣿⠟⣩⣷⣾⣶⡾⣷⡆⢀⣀⣀⣠⠂⢁⠀⠒⠉⠁⠀⠲⠀⢁⠠⣤⡁⠠⠻⡼⡇⠀⡀⠀⠀⠀⠛⠏
⠀⠘⠙⠻⣿⡿⣿⣶⣼⣿⣿⣾⣿⣾⣧⣴⣿⣤⣿⣿⣏⣸⠀⣀⠀⠄⢀⢃⣀⣄⣀⣨⣹⣖⣹⠖⠀⠁⠐⢋
⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⡶⠶⢿⣿⣿⣿⣿⣿⣿⡿⠿⠧⠳⢾⢿⣿⣿⣿⡿⠿⠛⠛⠋⠉⠀⠀⠀⠢⢶⡖
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣭⣥⣤⣤⣤⣤⣤⣤⣤⣬⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠊⠉⠉⠁⠒⠦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠠
⠀⠀⠀⠀⠀⣠⠞⢁⠠⠀⠈⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⣿⣯⣿⣦⣶⣶⣤⣤⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⠀⡀⠀⠠
⠀⠀⠀⠀⢰⠁⠀⠀⢴⡄⠀⣶⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⢻⣿⣿⣯⣿⣿⡿⢿⣿⠆⠀⠀⠀⠀⠀⠀⠀⢠⡈⣿⣷⣠⣤⣤⣶⣷
⠀⠀⠀⢠⠏⠀⠔⠀⠈⡀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⡿⠟⠳⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣿⠉⣿⣿⣿⣿⣿
⠀⠀⠀⢸⠀⡸⠀⠸⠆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⢿⡿⡿⣿⠏
⠀⠀⠀⠈⠢⡁⠀⠒⠸⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠸⠇
⠀⠀⠀⠀⠀⠘⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⠆⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡉
⠀⠀⠀⠀⠀⠒⠈⠉⠁⠀⠚⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡿⠟⢆
⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠐⡆
⠀⠀⠀⠀⠨⣾⡇⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⢆⠀⠘⡀
⠀⠀⠀⠀⠀⢰⣷⣄⡀⢠⡇⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠈⠙⠻⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣸⣿⣿⣿⢄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣾⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣻⣿⣟⢻⠿⠟⣻⣾⣿⣧⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢸⡀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⣿⣿⣆⣠⡰⣿⣿⣾⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢀⠀⣴⠾⠚⠳⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢿⣶⣾⡿⠿⠿⣿⠿⣿⡿⣿⣿⣿⣿⣉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣿⣿⣧⡀⢉⣼⠃⡤⢒⣠⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢈⣿⣿⣇⠀⠀⣿⣶⣿⣶⢼⣿⣷⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢿⣾⣿⣷⣿⣳⣟⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣅⣠⠀⢻⣿⣿⣿⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣿⣿⣏⣀⣾⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠉⠉⠉⠉⠉⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠛⠛⠛⠶⠶⠶⠶⠶⠶⠤⠤⠤⡤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⡤⣤⣤⣤⣤⣤⡶⢿⣥⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⣤⣄
⠀⠀⠀⠀⠀⠀⣠⣶⠾⠛⣛⡉⠀⣼⠃⢰⣦⡤⠀⠰⠿⠀⠀⢠⡟⠀⢀⣤⡉⠛⠙⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠙⢶⡀
⠀⠀⠀⠀⠀⣰⡟⣡⣄⠘⠟⠃⣼⠏⠀⠞⠻⠇⠀⠀⠀⠀⢰⠏⣠⠄⠀⠋⠁⣠⣾⠟⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⠟⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⣰⡏⢻⣯⣼⠇⣠⣼⠷⠴⠶⠶⠶⠶⠶⠶⠶⠾⢧⡀⠻⣴⠇⣠⠞⢁⡜⠀⣴⡶⣤⡄⠀⡿⠁⠀⢠⡟⠀⠀⠀⠀⠀⠈⠳⣦
⠀⠀⠀⣰⡟⢀⣠⡼⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⣀⡼⠋⢀⡼⠃⢸⣿⠀⣨⣿⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⣰⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⣼⠃⠀⣿⠛⠚⠉⠁⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⢠⣿⠁⠀⠀⠀⠀⣀⠀⠀⠀⢠⡿⢁⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣴⠋⠀⢾⠟⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⣼⡇⠀⠀⠀⠀⢸⡿⠀⣼⡗⠈⠁⣘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⢀⣿⠁⠀⠀⠀⠀⣿⣅⣴⡿⠀⠀⠀⢻⣷⣶⡶⠶⢦⣤⣤⣤⣀⣀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⢸⣿⡀⠀⠀⠀⣤⣌⣡⡿⠁⠀⠀⠀⠀⢹⣏⠻⢦⣄⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⣀⣀⣤⣤⠚⢿⡇
⢸⡟⠻⣦⡀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠙⠷⣤⡀⠀⠀⣀⣤⣴⠶⠟⠋⠉⠀⠀⠀⢸⡇
⢸⡇⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣷⠀⠀⠈⠻⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⢸⡇⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⢀⣠⣤⠾⠋⢉⣿⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⣴⡄⢀⣤⣄⠀⠾⠃
⢸⡇⠀⠀⠀⠀⠀⠈⠻⣦⣴⠾⠛⠉⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣾⣿⣁⣼⣿⠇
⢸⡇⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢰⣿⠀⣶⣶⣾⡿⠀⠀⠘⢿⣶⠗⠀⠛⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣼⡟⣿⡿⢻⡟
⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⣄⠘⣿⠀⣾⠃⢸⡗⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸⡿⠰⠿⠠⡿⠃
⢸⡇⢀⣄⣀⣀⡰⣿⠇⢸⡇⠠⣶⣦⣶⡾⠋⢠⣏⠀⠛⠀⢸⡇⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢸⡇⠀⢹⣿⠉⠁⠀⠀⠸⣧⠀⢰⡿⠀⠀⠀⢰⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢸⡇⠀⣾⠇⠀⠀⠀⠀⠀⣯⠀⣾⠇⠀⠀⠀⢸⣿⢀⣀⣤⠾⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⢸⡇⠀⠛⠀⠀⠀⠀⠀⢠⣿⠀⠁⠀⠀⠀⠀⠸⣿⠋⠉⠀⠀⠀⠈⠙⠳⣦⣄⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⢸⣧⠀⣀⣠⣤⡶⠾⠛⠋⠙⠛⠳⢦⣤⣄⣀⣀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡛⠷⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠘⣿⡟⠋⠉⠀⣴⡶⢶⣆⠰⣿⠆⠀⣀⣈⡉⠉⠳⠶⣦⣼⣧⡀⠀⢰⣶⠀⣰⡆⢀⣿⠃⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠘⣷⡀⠀⢀⣿⡷⠾⠋⠀⠀⠀⠘⣿⠉⠉⠙⠛⠛⠛⠉⠙⠋⠀⣼⠇⣠⣿⠃⠈⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠘⣿⡀⢾⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⣿⠇⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠻⣦⡀
⠀⠀⠀⠈⢻⣦⢤⣤⣤⣤⣤⣠⣤⣀⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠷⢦⣀⡀⠀⢀⡴⠛⠁⠀⠀⠘⢷⣆
⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠙⣻⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣧⣤⣤⣤⣭⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⣾⠃⠀⠀⣤⣤⣤⡀⢠⣶⠆⢀⣷⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⣴⡟⠀⠀⢠⣿⠉⢹⣿⠈⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢰⡆⣰⡦⠀⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢠⡿⠁⠀⠀⣿⠿⠿⠟⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⣸⡟⠀⠟⠐⠛⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⢀⣽⠇⠀⠀⢼⡿⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⣰⡿⠁⠀⠀⠀⠀⠀⢸⠇
⣸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠗⢶⣦⣤⣤⣀⣀⠀⠀⠀⢠⣌⢩⣿⠁⠀⠀⠀⠀⠀⢰⡏
⣿⡇⠈⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣙⣿⣷⠶⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠉⠻⣦⣀⠀⠀⢀⣀⣠⣤⣴⠖⠛⠛⠉⣉⣼⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠉⢻⣿⠋⠉⠉⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠋⠉⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣏⠀⠀⠀⠀⠀⠀⠘⡟⢷⣄⠀⠀⠀⠀⣀⣤⠾⠛⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣀⡀⠀⣀⡀⠀⢰⡇⣰⡷⠀⠀⠀⠀⢸
⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠳⣤⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⣿⡇⣸⣿⡇⠀⠟⠀⠟⠀⠀⠀⠀⠀⢸
⣿⠀⢀⣀⣀⣀⠀⣶⠀⢈⡗⢀⣀⣀⣀⠰⠟⢹⠇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣽⣷⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠈⢹⣿⠋⠀⠉⠀⠀⣷⠀⢹⡟⠉⠀⠀⣹⠀⣠⣄⣀⠸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⣿⢡⣿⠃⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⢀⣿⠃⠀⠀⠀⠀⠀⣿⢀⣾⠃⠀⠀⠀⣿⠀⢈⡿⠉⢱⡆⠀⠀⣶⣶⣶⡄⠰⣿⠇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⠀⠘⠛⠀⠀⠀⠀⠀⠀⣿⠈⠃⠀⠀⠀⠀⣿⠀⣾⠇⠀⢸⡇⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣀⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞
⣿⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠉⠛⠷⢤⣄⠀⣿⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⣿⠀⠀⢀⣠⠾⠋⠁⠀⠀⠀⠀⠀⣴⠆⠈⢙⡿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⣿⣤⡾⠛⠁⠀⠀⢠⣤⠀⢠⡄⠰⠋⠀⡴⠋⢀⣀⣤⣤⠞⠛⠛⠳⠶⣶⣦⣄⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⢻⣅⠀⠀⠀⠀⠀⣼⡏⢀⣿⠃⠀⣠⣾⣴⢶⢉⡉⠀⡀⡀⣀⡀⢀⠀⠀⠈⠉⠙⠛⠶⠶⢦⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠙⣧⡀⠀⠀⠀⠿⠆⣿⡏⠀⠀⠉⠉⠉⠀⠀⠉⠈⠉⠉⠉⠈⠿⢳⣄⠀⠀⠀⠀⢠⣶⠶⣬⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠈⢿⣄⠀⠰⢦⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⢷⡀⠀⠀⣿⣧⡴⠟⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢷⣀⡀⢹⣦⢾⠏⠀⠀⣴⣶⠀⠈⢷⡀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠙⢷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡾⠋⠁⣤⣍⠁⠿⠛⣷⣄⠀⠀⠉⠀⠀⠀⠈⢿⡄⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⡀⠤⣀⡸⠿⢧⣀⠀⠀⠀⠀⠀⢶⣤⠆⣤⠈⠹⣦⠸⣥⡼⢇⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⣴⠟⠁
⠀⠀⠀⠀⠸⢧⣴⠏⠀⠀⠀⠙⢷⣄⣀⣀⣀⡚⣻⣄⣈⣀⣀⣈⣷⣠⣴⠟⠙⠻⢶⣤⣤⣤⣤⣶⠶⠶⠞⠛⠛⠋⠁

⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢠⡏⠑⢀⠀⠀⠀⠀⠀⣀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠃⡀⠐
⠀⠀⠇⡌⠘⠉⠀⠀⠀⠀⢆⠀⢀⢠⠜⠀⠀⠀⢀⡀⠀⠀⠀⠸⠙⠂⡄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠤⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠓⡏⠀⠀⠀⠀⠀⠀⡰⢠⠦⠀⠀⠀⣤⢃⠀⠀⠀⠰⠋⢂⡠⠀⠀⠈⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠁⢷⡀
⠀⠀⠀⣄⠀⣰⠀⠀⠀⠘⠁⠀⠀⠀⣠⡀⠀⠈⠋⢀⠀⠀⢀⠀⠀⠈⠉⠀⠀⠰⠋⠡⠌⠁⡷⠀⠀⠀⠀⠀⠀⠀⠀⡀⢨⠁⠠⠀⠀⠈⠋⠷
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠚⠓⠛⠀⠀⠀⢤⡂⢄⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠐⣀⠆
⠀⠀⠀⠀⠀⠠⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⡉⠀⠀⠀⠈⠂⢂⠁⠀⠀⢠
⠀⠀⠀⠀⠚⡀⠀⢀⡔⠂⠀⠀⠀⡀⠀⢀⠀⠀⠀⠠⠀⢀⠀⣀⠀⠀⠀⢀⠀⣀⢀⠀⠀⢀⠀⠀⠀⢣⠊⢀⡄⠆⠀⠀⠀⠙⠁⠀⢠⡍⢱
⠀⠀⠀⠐⠣⠈⠐⠀⡆⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⡇⢰⠀⠀⢧⠏⢀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠘⣆⠜⠀⠀⠀⠀⠀⠀⠀⠀⠐⣰⣦
⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⡀⠤⡄⠀⠀⠰⠀⠤⠇⠈⠀⠁⠀⢸⠀⠀⠀⢸⡆⠀⠃⠀⠀⠀⠐⠃⠞⠀⠀⠀⣠⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠠⠆⠀⠴⠀⠀⠀⠀⠠⢁⠤⠒⣄⠚⠀⠀⠉⠁⠉⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣸⠟⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠘⠰⠆⢀⡀⠀⠀⠀⠀⠣⠄⠦⠊⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢢⡄⣀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⣆⠀⠈⠀⠀⠀⢀⣀⠀⠀⠈⠑⠦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠉⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠈⣥⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠃⡠⠚⠀⠀⠀⢠⣼⡀⠀⠀⢀⠄⠠⠂⠈⠙
⠀⠀⠀⠀⠀⠀⠠⠻⠃⠠⠈⠁⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠸⠃⣅⠀⠀⠀⠀⠸⠏⠀⠀⠀⠈⠱⠯⠤⢀⠐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠘⠀⠀⠀⠀⠀⠀⠘⠙⡀⠀⠀⠀⡀⠀⠤⣄⠀⠀⠀⠀⠁⣄⡘⠄⠁⠀⠀⠀⠀⠀⠀⠀⢰⠀⠄⠀⠉⠀⠈⠁
⠀⠀⠰⠆⠢⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢻⢸⠁⠀⠀⢀⠒⠠⠄⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⢀⡀⢱⠀⠀⠀⣀⢰⡍⣁⡄⠀⠀⠀⠈⣟⠀⠀⠀⠘⠒⠀⠈⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⠀⠀⠀⠀⠴⠀⠠⣄
⠀⠀⠈⣦⠮⠰⠃⠀⠀⠀⠁⠊⠀⠀⠀⠀⠀⠀⢰⡏⣧⠀⠀⠀⠀⠳⡼⠀⠀⠀⢠⠉⢲⠈⠇⠀⠀⠀⡃⣌⠀⢠⠜⠀⠀⠠⣆⠚⠃⡄
⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠄⢐⠀⠇⠀⠀⠀⢈
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠐⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠈⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠺⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠃
⠀⠀⠀⡀⠲⣿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⣶
⠀⠀⠀⣽⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⢨⡅⢈⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠄⣿
⠀⠀⠀⠁⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠠⠷⠏⠀⠂⠀⢸⡇⢸⠀⠠⣞⡸⠆⠀⠀⠀⠀⠀⠀⠀⠈⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠛⣿⡆⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠘⠿⣷⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⢠⣿⣶⡄⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠀⠀⠀⠀⢰⣿⣿⠇⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⣀⢀⢠⡄⠀⠀⠀⠀⠀⠸⠟
⠀⠀⠀⠎⢷⠀⠀⠀⠀⠈⠁⣤⣀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⠀⢰⡆⠀⠀⠀⢀
⠀⠀⠀⡤⣄⠀⠀⠀⠀⠀⢾⣿⠿⠀⠀⠀⠀⠀⠀⠀⣸⢳⡀⠀⠀⠀⠀⠐⣿⠃⠀⠀⠀⠘⠃⠀⠀⢐⢸⡇
⠀⠀⠀⠑⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⢤⡤⠤⠤⠤⠤⢤⡤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠠⣶⣉⣉⣀⣀⣀⣀⣠⣄⣀⣠⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡴⠖⠛⠍⠳⣄
⠀⠀⠀⠀⠀⣠⠞⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠙⠧⣄⠀⠀⠀⠀⢠⠏⠁⠀⠀⠀⠀⠀⠁⠌⡻⢦
⠀⠀⠀⠀⣰⠛⢶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠂⠀⠀⠀⠀⠈⠳⣄⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠈⣆
⠀⠀⠀⠀⡏⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⢹⡀
⠀⠀⠀⢸⠃⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠀⠈⣇
⠀⠀⠀⡿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⣨⠁⠀⠀⠀⢹⡄
⠀⠀⢰⡇⠀⠀⠀⢸⡀⠀⠀⠀⠀⢀⡐⠀⠀⠑⢆⡀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢀⠐⢁⡀⠀⠀⠀⠘⣧
⠀⠀⣼⠀⠀⠀⠀⠘⡇⠀⠀⠀⠠⠂⠀⠀⠀⠀⢀⠼⠐⠐⠀⠂⠀⠐⠐⠒⢻⠂⠂⠒⠐⠲⡀⠀⠈⡅⠀⠀⠀⢠⢿⡀
⠀⠀⣿⠈⠐⣀⡀⠄⡷⢶⡀⠈⠁⠆⡀⠀⢀⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢦⡀⠀⠀⠀⠈⠰⢀⢠⠀⣀⣰⡏⠸⡇
⠀⠀⢿⠀⠀⠁⠀⠀⣇⠀⠙⠦⣄⠀⠀⠓⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠙⠢⣄⣠⡤⠶⠞⢻⡍⠉⠈⠙⠻⣿
⠀⠀⢸⡄⠀⠀⠀⠀⣿⠀⠀⠀⣨⠗⠒⠚⠓⠶⠶⠶⠤⠶⠤⠤⠴⣏⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⡅⠀⠀⠀⢠⡏
⠀⠀⠈⡇⠀⠀⠀⠀⢹⢀⡤⠚⠁⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⣾
⠀⠀⠀⣷⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⢰⠇
⠀⠀⠀⢸⡀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢀⡟
⠀⠀⠀⠸⡇⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣸⠁
⠀⠀⠀⠀⣧⠀⢰⡏⠀⠀⠀⠀⠀⠀⠠⠈⠁⠆⡀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⣹⠇
⠀⠀⠀⠀⠹⣦⣾⡀⠀⠀⠀⠀⢀⣄⣀⡀⢠⠀⣍⢣⠄⡀⢀⠀⡀⣰⢇⠀⡄⢀⠀⡀⢀⠀⡀⢠⠞⢡⡞⠁
⠀⠀⠀⠀⠀⠈⠓⢿⡢⣄⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠳⣄⡀⠀⠀⠀⠀⣀⣠⠴⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠒⠒⠒⠒⠒⠒⠶⠶⠶⠶⠶⠿⠤⠤⠤⠤⠤⠽⠦⠶⠚⠋⠉

⠙⢺⢀⡴⣋⠴⠋⠀⠀⠀⠀⢠⣏⡵⠞⠦⢤⡇⠀⠀⠀⠀⢸⢗⡿⣄⡼⢁⡞⠁⠀⢠⡧⠤⠷⢾⡉⡆⠀⠀⠀⣠⠦⢤⡀⠀⡠⠀⠀⠀⠈
⣀⠈⠛⠊⠁⠀⠀⠀⠀⠀⠀⠘⢅⡀⠀⣀⡴⢁⡀⠀⠀⠀⢸⣼⠀⡼⢠⠏⣀⣀⡀⠀⠳⣄⣀⣀⡨⠟⠀⠠⡮⣰⠇⠀⠸⡄⢀
⡉⠙⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠄⠀⠀⠀⠀⠀⢸⣋⣿⣷⢣⠎⢰⠇⢹⠀⠠⠀⠈⠉⠀⠀⠀⠀⢳⣼⠳⡄⣠⠃⠀⠀⠀⢀⡴
⠉⡷⠴⣽⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠉⠉⠁⡰⠃⢀⡏⠀⡟⠀⠀⠀⡴⣿⣦⡀⠀⠀⠘⠗⠒⠚⠁⠀⠀⠀⣰⠏⣠
⠞⢁⡴⠃⠀⠀⠀⠄⠀⠀⠀⠀⠀⢠⡞⢛⡟⠲⣄⠀⠀⠀⠀⠀⠸⡟⢛⡟⠳⣴⠃⠀⢀⡜⢰⠀⡇⠱⡄⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⡼⠁
⣠⠞⠀⠀⠀⠈⠀⢀⣤⢄⠀⠀⢰⡏⣀⡞⠓⢦⡈⠓⢦⡀⠀⠀⠀⢻⣼⡤⠚⠁⠀⠀⣼⢤⣼⣀⣿⠴⢻⠀⠀⠀⠀⠀⠀⡴⠋⣰⠋⠀⡴
⠁⠀⠀⠀⠀⠀⣴⠉⡇⠀⣱⣄⠈⠻⣍⠙⢦⡀⠙⠲⣄⠉⠢⣄⠀⠀⠁⠀⠀⠀⠀⠀⣿⠀⣧⠀⣸⢀⣿⣆⠈⠄⠀⠀⣾⣡⠞⠁⣠⠞⢀
⠀⣠⠤⢤⡀⠀⡇⢸⠁⢠⠇⡸⠀⠀⠈⠳⣤⡙⠲⣄⠈⠓⢦⡈⢳⡀⠀⠀⡀⠀⣸⡗⢿⡀⣿⠀⣿⠎⢸⠙⣆⠀⠀⠀⣿⠟⢢⣴⣁⡴⠋
⡴⠃⢠⠃⡿⢸⢀⡷⢔⡾⣤⠃⠀⠀⠀⠀⠀⠙⢦⣈⠳⢤⡀⢹⢿⣧⠀⠀⠀⢠⣇⣧⡠⠟⣾⢰⡗⠤⣸⡤⢾⠀⠀⠀⠿⠞⠛⠋⠉
⣄⣠⠃⠀⡇⣧⣎⣀⣞⡜⠁⠀⢄⠀⠀⡀⠀⠀⠀⠈⠳⢤⣽⣻⣤⣿⠀⠀⠀⠘⡄⠸⡀⠀⣿⢸⡇⠀⡼⠀⡎⠀⠀⠀⠀⢀
⢀⡟⢦⣰⠁⠹⣇⡿⠋⠀⠀⠀⡠⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢧⠀⣿⢸⡇⠀⡇⣸⠁⠀⠀⠀⢀⣀⣠⠤⣤⡄
⣾⠤⠒⠉⠀⠀⠉⠀⠀⠀⠀⣜⠀⠀⢀⡨⣷⠀⠀⠀⠀⣠⡦⡀⠀⡀⠄⠀⠀⠀⠀⢣⡘⣆⣿⠴⢇⢰⣣⠃⠀⠀⠀⡴⠋⢧⣠⠞⠁⡇
⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⡇⢉⡽⣛⣤⠇⠀⠀⠀⡴⠁⡇⠘⣦⠀⠄⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠙⠁⠀⠀⢠⠞⢀⡠⠊⠉⣓⡶⠁⠄
⠀⠀⢠⡶⣻⣽⡋⣿⠀⠀⠀⠘⠋⠉⠉⠀⠀⠀⠀⢸⠓⠲⡗⠋⡿⠀⣠⣴⡀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠘⢶⣋⡤⢖⣫⣅⣀⣀
⠀⡴⠋⡴⠋⢀⡽⣾⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠘⡄⠀⡇⢠⢃⣮⡯⢼⣇⠀⠀⠀⣤⡚⣱⣃⣙⡄⠀⠀⠀⠀⢀⡤⠖⣩⠞⢫⣭⠟
⠞⢡⠞⠁⣠⠏⢠⠟⠀⠀⠀⣸⡛⢗⣢⣄⠀⠀⠀⠀⢣⣀⣇⡞⢸⠸⡆⢸⠸⡄⠀⠀⣇⡼⠉⠀⠀⡇⠀⠀⡠⠚⣁⠤⠚⣇⡰⢫⠏
⣴⠋⢀⡼⠁⣰⠋⠀⠀⠀⠀⣏⣷⠋⠻⡌⢷⡀⠀⠀⠀⠀⠁⠀⡟⢸⠉⢲⡖⡇⠀⠀⠙⠢⠤⠔⠋⠀⠀⢠⡗⠊⠁⣀⡴⢛⡶⠃⠀⠀⣴
⠁⣠⠎⢀⡞⠁⠀⠐⠀⠀⠀⢿⠘⢦⠀⠹⡄⠻⡄⠀⠀⠀⠀⢰⡇⡾⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠾⠥⠴⠋⠀⠀⢀⣾⣡
⢶⠃⣰⠋⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⢧⠀⠹⣆⠹⡄⠀⠀⠀⠀⢳⣇⣀⡼⠞⠁⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠼⠟⠁⠀⠀⠀⣠⠴⣲⢿⣇⠀⠈⢧⡈⢳⡀⠙⣆⠹⣆⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⣤⣔⣒⠒⠉⡗⠦⣤⣄⠒⠒⢲⣄⠀⠀⠠⠐⠀⠀⠙
⠀⠀⠀⠀⡴⢯⣠⠞⢁⠇⠋⠀⠀⠈⢳⡀⢳⡀⣸⠦⢽⡆⠀⡠⠲⡖⢦⣄⠀⠀⠀⢫⡳⢤⡉⢹⠗⢦⣘⣆⠉⠲⣼⣄⠙⢦⡀
⣠⣴⢆⡞⠁⡰⠛⢲⡾⠤⣴⠀⠀⠀⠀⠳⣄⠏⢀⣠⠞⠁⠘⡅⠀⠙⣏⡸⡀⠀⠀⠀⠙⣄⣩⠿⠤⣄⡈⠙⠶⣊⠁⠈⠙⠦⣝⢦⡀
⣠⠃⢸⡆⡼⠁⣠⠋⣀⣞⠁⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠣⣀⣰⣇⣹⠇⠀⠀⠀⠀⠈⠛⠶⢤⣀⣉⣽⣶⣤⣝⡒⠦⢄⣈⣹⠇⠀⢠
⠏⠉⣹⡟⢡⡔⠛⢉⡜⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡯⢄⣸⡄⠈⠉⠛⠷⢶⡏⠀⠀⠀⡼
⠀⣴⡟⡰⠃⠘⣦⣋⣀⡇⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠠⡐⠀⠀⡤⢲⠖⠒⢲⡄⠀⠀⠀⠀⠄⢀⠀⠙⢦⣞⣉⣉⣙⣛⣓⣲⠗⠀⠀⢠⠇
⣴⠏⡸⠁⢀⡴⠁⡠⠊⠀⠀⠀⣀⡠⠤⣶⢦⠀⠀⠀⠁⠀⠀⡜⢀⡎⠀⡰⠋⡇⠀⠀⡴⢶⠤⠤⣀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠸⣠
⡿⠿⠿⠿⡿⡻⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠈⠉⠉⠐⢤
⠀⠀⡐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠠⠑⡄⠀⠀⠀⠀⠀⢡
⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠐⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠈⠂⠈⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠂⣄⠀⠁
⠀⠀⠀⠂⠀⠀⢄⡀⢀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠆⠀⠉⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⠠⢄⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⠂⠄⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄
⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢄⣀⠊⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠉⠀⠀⠀⠈⠑⠢⠀⠀⠀⠀⠀⠀⠀⠠⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠐⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⡀⠀⡀⠀⠀⠀⠀⢀⠶⠄⠤⡀⠀⠀⣀⠘⢃⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠥⠀⠜⠆⢡⠀⠀⠀⢠⠂⡴⠀⠀⠈⠢⡠⠁⠀⢠⠋⡁⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⢄⡀⠀⣀⠃⢘⠓⢸⠀⠀⠀⠃⠀⣠⠃⠀⠈⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠡⡀⠀⡎⠉⡉⠙⡄⠘⠀⢸⢀⡀⢸⠀⠀⠅⠀⠀⠀⠀⠀⠀⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠐⡀⠈⠀⠀⠀⡥⠛⢄⠸⣅⠱⠆⠀⡸⠀⠀⠈⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠘⠄⡾⠀⡾⡇⠂⠀⠰⠀⢈⡠⢶⠑⠀⡀⢠⠌⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡘⠢⠄⡏⠙⡀⠀⠠⠻⢎⠁⡄⡶⢎⣱⠀⢁⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡄⣿⠀⢀⠀⠈⠐⠛⠀⡀⠃⢀⡧⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠲⠒⠓⠊⠋⠉⠉⠉
⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⢸⡇⠀⠀⣦⠖⣴⠀⣶⢠⣖⠂⢺⡗⢠⡖⢲⡄⣿⠀⡇⢠⡖⢲⡄⣴⠒⡆⢰⡴⢂⡶⢲⡆⣶⠖⣦⢸⡷⢲⡄⣶⠀⣶
⠀⠀⠘⢧⣤⠀⠧⠀⠻⡤⢿⠠⣭⠗⠸⣧⠸⢧⠼⠇⢿⠀⢧⠘⢧⡼⠃⠷⡴⣧⠸⠇⠘⢧⠼⠇⢸⣤⠞⠸⠇⠸⠇⠿⡤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠐⠒⠋

⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠒⠛⠋⠉⠉⠉⠉⠹⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⢒⣲⡶⢦⡀
⠀⠀⠀⠀⡠⠂⣩⠗⠒⠒⠒⠒⠒⠒⢓⡒⠚⠓⢶⡖⠒⠒⠒⠒⠒⢲⡶⠒⠋⠁⠀⠈⠠⠙⢦⡀
⠀⠀⣠⣎⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠈⠳⣄⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠈⢂⠙⡆
⠀⠀⡏⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠈⠙⢦⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡚⠀⢹⡄
⠀⢸⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠈⣧
⠀⡾⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⠀⢹⡄
⢰⠇⠀⠀⠀⣧⠀⠀⠀⠀⠀⠠⠜⠀⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⡀⠜⠁⠀⠀⠀⠈⣧
⣼⠀⠀⠀⠀⢸⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⢹⠐⠀⠀⠆⠀⠀⠠⠰⣷⠠⠀⠄⠤⣆⠁⠀⢃⠀⠀⠀⢀⣿
⣿⠈⠠⣀⢀⢸⣧⣆⠀⠙⠠⢀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣄⠀⠀⠀⠈⠃⠄⡈⡀⠀⣠⡞⠘
⣿⠀⠐⠁⠀⢸⡇⠈⠳⣄⠀⠈⠐⠆⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠈⠳⢤⣀⣠⡤⠴⠟⣟⠉⠉⠙⠳
⢸⡄⠀⠀⠀⠀⡇⠀⠀⢀⡽⠖⠖⠓⠶⠶⠶⠤⠤⠤⠤⠴⣏⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣡⠀⠀⠀⢀
⠸⡆⠀⠀⠀⠀⡇⣠⠖⠉⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠞⠁⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣼
⠀⣧⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢠⠇
⠀⢸⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⡾
⠀⢸⡆⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣸⠃
⠀⠀⣇⠀⣸⠁⠀⠀⠀⠀⠀⠀⠂⠀⠁⢤⡀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠆⣡⠏
⠀⠀⠙⢶⡳⣄⠀⠀⠀⣀⠴⠆⠦⠐⠆⠰⠙⠂⠆⠰⠀⠄⣼⡡⠐⠂⠤⠠⠀⠄⠠⠀⠞⣡⠞⠁
⠀⠀⠀⠀⠙⠿⣷⣴⣊⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠙⠦⣄⠀⢀⣀⡤⠴⠒⠋⠁
⣶⣶⠶⠶⣴⣶⣶⣦⣬⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠐⢂⢀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠀⠰⣏⠐
⠀⠀⠀⠀⠀⡶⠦⡘⠳⠄⣀⣁⠀⣀⢐⡀⠀⠀⠀⣠⢊⠃⠌⢢⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⢠⡀⢦⣱⢄⠶⡇
⠀⠀⠀⢠⣀⣀⠀⡤⠃⢀⢨⠎⠂⠂⠐⠀⠀⠀⠀⠉⢃⠑⠋⠀⡇⠘⢀⠀⠀⠀⠠⠋⠳⢀⠀⠀⠀⠀⠄⢠⠁⠂⡄⡀
⠀⠀⠀⠐⠈⢀⡄⠆⡿⠜⠹⠃⠀⠂⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⡖⠁⠀⠀⠀⠁⠀⠄⡀⠀⠀⠀⠀⢢⣸⣄⡌⠂
⠀⠀⠀⠀⠀⠙⠃⠀⠀⡠⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡷⠁⢂⠀⠀⠀⠀⠀⢘⠧⠀⠀⠀⠀⠀⠛⠋
⠀⠀⠀⠀⢠⡰⠄⠤⡬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⡀⡠⠄⠃⠀⠀⠀⠀⠀⠠⠀⠀⢐⠊
⠀⠀⠀⠀⠀⠁⠲⠋⠀⠀⠀⠀⠠⠚⠳⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⡈⡈⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⡀⢀⢀⠄
⠀⠀⠀⠀⣰⢷⡀⠀⠀⠀⠀⠀⢆⠁⠎⡠⠃⠀⠀⠀⠀⠀⠀⢀⡀⡉⢸⢃⠀⡀⡤⠀⠀⠀⠀⠴⣁⠀⠀⠱⣞⡟
⠀⠀⠀⡄⠄⠘⡐⡀⠀⠀⠀⠀⠀⠑⠖⠀⠀⠀⠀⠀⠀⢀⣀⢸⠀⠁⠈⣼⢧⡟⠀⠀⠀⠀⢀⠀⣌⡅⠘⠀⡟⣠⢠⠄
⠀⠀⠀⠠⢤⡟⠄⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠⠘⡠⢊⠍⠉⢃⠀⠀⠀⠀⣄⣠⣙⢂⠂⢠⡿⠟⠃
⠀⠀⠀⠐⡤⢩⠌⠀⠀⠀⠀⠀⢀⡤⡄⠀⠀⠀⠀⢄⢘⡀⠃⢀⡀⠃⡌⠘⢁⠍⠀⠀⠀⠀⠑⣀⢂⢊⠱⠋⠁
⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠆⡤⠀⢠⠀⠆⠀⠀⠐⠻⡼⣴⠨⠇⡎⠠⠀⠂⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠈⠙⠶⣶⣷⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣇⠘⠯⠀⣠⣮⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⣠⢨⠀⣕⣄
⠀⠀⠀⠀⠰⠢⡀⠠⠀⠀⠈⠙⠠⠀⠠⠘⠁⠀⠀⠀⠘⠰⠀⠀⠀⠀⠀⠀⠀⡀⠈⠈⣀⠤⠀⠀⠀⠸⣇⡀⠒⠀⠈
⠀⠀⠀⠀⠄⢒⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡁⣠⡴⡄⠀⠀⠀⠀⠁⢅⠃⢼⡰⠀⠀⠀⠘⣟⡤⡀⠐
⠀⠀⠀⠀⠀⠈⢿⣛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠛⠶⢼⠙⠤⠛⠄⠀⠀⠀⠀⠰⠈⠸⢘⠀⠀⠀⠀⠯⠵⡇
⠀⠀⠀⠀⠀⠀⠘⢯⣷⡄⠀⠀⠀⠀⠀⠀⠀⠐⠏⠘⠄⠈⢄⡀⠀⠆⠀⠀⠀⠀⣂⡄⣴⣍⠀⠀⠀⠀⢦⠤⠑⣤⠆
⠀⠀⠀⠀⠀⠀⠈⠙⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⣈⡂⣾⡵⠊⠀⠀⠀⠀⠀⠈⠲⠞⠁⠀⠀⠀⠀⠀⠙⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠀⠀⠀⢠⣦⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠛⣿⣿⣿⠀⠀⠀⣠⣎⣦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢀⣢⣳⣾⣇⢀⣀⣼⣿⣾⠀⠀⠀⠀⠀⠀⠀⣀⠔⢉⡅⠀⠀⣿⣿⣇⡻⠀⠀⣾⠟⠛⠁⠀⢀⡆⠀⡀⡠⠖⠂⠈⣹
⠀⠀⠀⢀⡄⠀⠀⠀⠀⢠⠀⡠⣾⡧⢹⣿⣿⡿⢸⣿⣿⣾⡆⠀⠀⠀⠀⠀⢠⣤⣴⣿⠃⢠⠐⠛⢿⣦⣼⠒⡞⡅⠀⢀⣠⣴⡾⠀⡐⣩⣅⠀⣠⣾⡟
⠀⠀⢠⣾⣧⡤⠄⠀⠀⣿⠀⣹⣿⣧⡌⣿⣯⣷⠺⣿⢿⠃⡆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣴⣿⠖⢠⣿⣿⣿⠞⢀⣼⣿⣿⡏⠋⠀⠀⣰⣿⣿⡿⣧⡟
⠀⠀⢸⣿⣿⡀⠀⠀⢸⡿⣆⣩⡾⢱⡆⢘⣤⣨⡇⣿⡀⣇⡇⠀⠀⠀⠀⠀⡞⢸⣿⣿⣿⠋⢀⣿⣿⣿⠃⢠⣾⣿⣿⣿⡟⠀⠀⣴⣿⣿⡿⢷⠏
⠀⠀⢸⣿⣿⠡⠠⢄⣾⣿⣿⣿⠃⣿⣧⠆⢿⡿⡧⣿⣧⣄⠃⠀⠀⢀⡀⢠⠃⣾⣿⣿⡁⣰⣿⣿⡿⠃⣰⣿⣿⣿⣿⣿⠓⠂⣼⣿⣿⣿⣶⡎
⠀⠀⢸⣿⣿⠖⠂⣾⣿⣿⡿⡟⢸⣿⣿⡀⠘⣿⣷⢹⣧⡤⢠⣄⣲⡦⠀⠀⢰⣿⢋⣼⣶⣹⡟⡝⠁⣴⣿⣿⣩⣿⣿⣋⣀⣼⣿⢯⣽⢿⡟
⠀⠀⢸⣿⡟⢁⣸⢻⣿⣿⢠⢁⢿⣿⣿⣷⠀⣹⣿⢸⣯⠀⠀⣿⣿⣷⣶⠃⠿⣿⣿⣿⣹⣿⡇⠀⢉⣽⣿⣿⣿⣿⠏⠉⣽⣿⢁⣿⣿⢏⣤⡄
⠀⠀⢸⣿⣷⣰⢏⣿⣿⣳⡏⣼⢸⣿⡏⢹⣷⢿⣿⢸⣿⣷⡀⣿⣿⡿⠇⢰⣶⢿⣿⣿⡿⣉⡤⢞⣿⣿⣿⠟⢸⣿⢃⣼⡿⢡⣿⣿⣿⡟⢫⣅
⠀⠀⣸⣿⣟⠏⣼⣿⣼⣿⣿⣯⠔⣿⢷⣿⣿⢻⣿⡌⣿⣼⡇⣿⣿⣥⠀⠿⣧⣾⣿⡿⣟⣭⣶⠿⣿⣿⣿⠀⣼⣯⣾⡟⢱⣿⣭⣝⡵⢋⢼⡏
⠀⠀⣿⣿⡏⣰⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣸⣿⡇⣿⣿⡧⠛⣉⡟⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠂⣿⣯⡿⣹⣿⣿⣿⣋⢞⣵⡿⠁
⠀⠰⣿⡿⢡⣿⢹⣿⣿⣿⣿⣿⠿⠀⣿⠟⣹⣿⣿⣧⣿⡿⢓⣼⣿⢃⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠀⣿⡟⣿⣿⣿⣿⣿⣱⣿⡿⠃
⠀⣸⣿⣇⣾⣿⣿⣿⣿⣿⣿⣯⣤⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣿⣿⣿⣯⣽⣿⣿⣤⣀
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣭⣿⣿⣿⣿⣯⣷⣦⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣿⣿⣿⣿⣽⣟⣿⣿⣿⣿⣿⣿⠁
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⡸⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠂⡯⠀⠀⠀⠀⡰⠁⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠁⠀⢀⡞⠈⠀⠀⠀⣰⠓⠀⠀⠀⢑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⡾⠁⠀⠀⠀⠘⠃⠀⢀⡠⠐⠉⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⡘⣦⡼⠉⠁⠀⢀⣴⣾⠢⡔⢅⢀⣠⣾⣘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣉⠉⠉⣉⣧⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠛⢦⣹⠟⠀⠀⣠⠜⠉⢉⠃⣸⡅⠘⡯⠟⠛⢉⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⡶⣦⣿⣿⣻⡟⠁⠀⢳⡎⢁⡹⠄
⠀⠀⠀⠀⠀⢠⠜⢣⠀⢠⡟⠀⣠⢞⣒⣒⣰⠃⢠⣯⣷⡀⠁⠀⡀⠁⠠⡆⠀⠀⠀⠀⠀⠀⠀⣞⢧⡥⡇⢻⣿⡟⡠⢠⡄⢸⡉⠙⣧⠘⡆
⠀⠀⠀⢀⠴⠋⢀⡘⢣⠈⣃⡞⠁⠤⠠⠤⡆⠠⠁⣹⣆⢡⠀⡶⠠⠀⠒⣷⠀⠀⠀⠀⠀⠀⡠⢿⣄⣪⡧⠾⡞⢰⡤⢰⠉⢻⡟⠀⡆⢂⢘⣤⠤
⠀⠀⠀⠉⡏⡍⢸⡇⠀⣹⡇⠔⠲⠶⠶⡾⢹⣷⣼⠀⠹⣷⣶⠃⣀⠈⠥⡜⠀⠀⠀⠀⠀⠀⡀⡸⠋⠉⢂⠀⠘⢿⣇⣤⠤⠾⣿⣆⣡⡼⠋⠀⢠⡀
⠀⠀⠀⢸⢙⠢⠆⢰⠎⣇⣀⣠⣤⠐⠁⠉⣇⠈⠛⣤⠔⠛⠯⣀⣀⣠⣴⠁⠀⠀⠀⠀⢰⡎⣹⣷⡄⠀⢳⣦⣀⠜⠋⠀⢀⣠⣤⠟⣿⣵⠂⡠⠀⢻⡗⠄
⠀⠀⠀⡌⠈⣢⡴⠁⠸⢋⡟⠛⠀⠠⡞⢠⣟⡝⢦⡘⠆⠀⣑⣄⠉⠉⠙⡄⠀⠀⠀⠀⢿⡧⡿⣿⣿⣀⠼⠋⠁⠀⢀⡼⢿⣹⡇⠀⣿⠟⠀⠀⠀⣰⠇⢀⡄
⠀⠀⠀⢹⣈⣵⠇⢀⣧⢹⣭⣹⣽⣿⢅⡌⠈⠹⢎⡙⢾⠛⠛⢻⠆⠀⠀⢀⠀⠀⠀⠀⢸⣿⠀⠸⣿⣇⢀⡀⢤⡴⠋⠀⠘⣿⣇⠔⢻⣀⣀⡠⠚⣁⣤⡞⠃
⠀⠀⠀⠈⣥⡏⠀⡌⠈⠲⣶⣶⣶⠀⠀⠙⡷⣀⠀⠹⣼⢀⡠⢧⠀⠀⠀⠘⠄⠀⠀⠀⠈⣿⡄⠀⢫⡏⠀⠀⣸⣷⠀⡠⠚⣻⠁⠀⠸⣿⠇⣸⠉⢸⡇⢇⠄
⠀⠀⠀⠀⢃⠀⣸⠁⠈⢀⡉⠻⡅⠀⠀⠀⢱⢈⠋⢷⣧⣐⣬⣡⣧⡀⢀⣠⡼⠀⠀⠀⠀⢸⡇⢠⣾⠶⣟⣿⣯⠻⠋⠀⣰⡿⠠⣦⣾⠛⢿⡯⠀⠸⣷⠛
⠀⠀⠀⠀⠐⢶⠂⠀⠀⠠⠱⠠⠓⣀⣤⣶⡏⠉⢙⠟⠻⠥⣤⠁⠙⢿⠎⣉⣧⠀⠀⠀⠀⠈⠳⡎⠃⠀⠸⣿⣿⣀⣤⣼⠋⠉⢢⠙⣿⠀⣨⣿⡆⠀⠋
⠀⠀⠀⠀⠀⠘⢧⠀⣀⢣⠇⠛⣾⣧⠀⣤⣆⣰⣯⠀⠀⣀⡴⣤⣀⡀⣸⣿⠷⠀⠀⠀⠀⠀⢀⣄⠐⡠⠊⢾⣿⠀⢿⣿⣦⣶⠋⠀⠘⣶⠛⠋⣡⠜
⠀⠀⠀⠀⠀⠀⠈⢲⣡⠁⢰⠋⢘⣿⣴⠿⣉⢋⣼⣢⣿⣿⠴⠿⢿⡷⠁⠀⠀⠀⠀⠀⠀⠀⢸⢿⠇⠁⡄⠀⣼⠠⣼⠋⣿⣯⢠⣤⡎⢀⣀⢴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⡃⣀⡜⠙⣧⡠⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣗⠙⢆⠸⣿⡴⠋⠀⣿⣿⠟⣀⣸⠟⢀⣾⣇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠙⡅⢸⣆⢹⡏⠀⢰⣿⣟⣈⡥⠘⣡⣿⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⡝⣄⢿⡾⡇⠀⣼⣿⣿⣯⠐⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢿⣼⣷⡇⠀⣿⡃⢠⣿⢫⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣇⡼⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠁
⠀⠀⠀⢠⡶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⣾⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⣿⠁⠀⠀⢻⡞⠋⢸⡆⢰⡇⢰⣏⡛⠈⣻⡏⢀⡾⠋⣿⠀⣿⠀⢸⡇⢠⡟⠙⣷⢀⡾⠋⣿⠀⣷⠞⠃⣾⠋⢻⡆⢺⡞⠙⣷⢸⡿⠛⣿⠀⣿⠀⢸⡇
⠀⠀⠀⠹⣧⣤⡄⢸⡇⠀⢸⣇⡼⣧⢠⣌⣿⠀⢻⣇⠘⣧⣠⢿⡄⢿⣄⢸⣇⠸⣧⣠⡿⠘⣷⣠⢿⡄⣿⠀⠀⢿⣤⣾⡇⢸⣇⣠⠟⢸⡇⠀⣿⠀⢿⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠠⢤⡼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⢢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡰⣷⣄⠀⡀⢃⠀⠀⠀⠀⠀⠀⠀⣰⢉⠀⠈⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠀⠁⢀⠀⠀⢀⡆⠀⠀⠀⠀⠰⠃⣿⡿⠀⠹⣼⡄⠀⠀⠀⠀⠘⣾⠃⡼⡇⠁⢠⡇
⠀⠀⠀⠀⠠⠤⢤⡀⠀⠀⢹⡇⠀⠐⢿⡄⢀⣾⡁⠀⠀⠀⠀⠁⠀⣿⡇⠀⠂⠈⠷⠀⠀⠀⠀⠀⢿⠀⡀⢁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢴⣷⣦⡀⢸⡇⠀⠀⣾⣥⢏⡽⠀⠀⠀⠀⠀⠠⠀⣿⡇⠀⠒⠀⠁⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣼⡃⠀⢀⣿⣣⠞⠀⠀⠀⠀⠀⠀⠀⡆⣿⡇⠀⡄⣰⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⡆⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⣕⠂
⠀⠀⠀⠀⠀⠈⠛⢀⠀⢹⡅⠀⣰⣶⣞⣥⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢂⣠⡇⠀⠀⠀⠀⠀⠰⠀⠀⣦⣤⣭⡿⠇⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠘⠓⠚⠛⠓⠺⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⠆⠘⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠂⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡰⡀
⠀⠀⠀⠀⠀⢰⡄⠀⠈⠀⠠⢴⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠱⣑⣦⠀⠀⠀⠀⠠⡀⠀⠁⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠁
⠀⠀⠀⠀⠀⠈⠁⢹⡆⠀⠀⠸⢀⣆⣄⠀⡀⡁⠀⠀⠀⠀⠀⠷⣄⠀⠀⠀⢠⠁⠘⠀⠀⠀⠀⠀⡆⠀⠀⠀⣿⠒⡗⠀⠀⠀⠀⠀⠀⢨⡄
⠀⠀⠀⠀⠀⠀⣀⠈⣇⠀⠀⠀⢸⣿⡇⠠⢨⠇⠀⠀⠀⠀⠀⠄⠈⠗⣀⣀⡆⠀⡇⠀⠀⠀⠀⢰⡇⠀⠀⠀⡏⢀⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⣿⡄⠘⡀⠀⠀⢸⣿⠀⡼⠋⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⢸⠃⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⢻⡇⠀⠠⠀⣃⣸⣯⣀⠴⠐⣀⣰⠀⠀⠀⠀⠸⠀⠀⠆⠀⣿⡏⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢠⣾
⠀⠀⠀⠀⠀⠀⠘⣿⢣⢠⡒⠁⢸⣿⣿⡇⢠⣿⠟⠀⠀⠀⠀⠀⠣⢀⢠⠀⣿⠃⠀⠀⠀⠀⠀⠀⠧⡀⠐⣲⡷⠞⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠠⠍⠀⠀⣼⣷⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠐⠂⠂⠀⣤⠄⠀⠀⠀⠘⢳⣶⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣽⡇⠀⠀⠀⠀⠉⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣜⡏⠀⠀⣀⡤⢒⣺⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠺⠴⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢠⡟⢸⣏⢘⡴⠃⠀⠀⢀⠄⠁⢈⠐⠁⡂⢺⡆⡀⢀
⠀⠀⢀⣠⣄⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣇⠼⠽⠊⠀⠀⠀⠀⠸⠁⠡⣋⡄⠙⡙⠃⠹⠁⣠
⠀⠀⣟⡉⣿⡀⢬⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⡼⠰⠛⣲⠢⡙⢷⡖⠃
⠀⢠⡌⢠⡘⠇⣘⢳⡎⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠈⠃⢀⡽⠴⣶⠷⡤⠆
⠈⡏⢿⠈⢠⠈⣻⠸⣅⡀⣡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠂⢹⡴⣒⣧⡴⠶⡜⠋
⠀⢧⡼⢶⡞⢂⠖⢤⠳⡆⠿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠓⠁
⠀⠀⡄⠈⠏⡿⡖⣀⢚⠁⠀⠀⠀⠀⢀⠀⡀⠀⠠⠀⠀⠄
⠀⢼⢡⣳⠀⣿⡁⣣⡞⠀⠀⠀⠀⠀⢾⠗⠁⠐⠋⡀⠶⠚⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠈⠳⣿⢰⢻⠞⠁⠀⠀⠀⠀⡔⡀⢈⠀⠁⣁⠘⣇⣠⠠⡆⠀⠀⠀⠀⡴⡾⣿⠃⢑⢠
⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢃⡐⣮⡤⠈⣏⡁⠘⠃⠤⠁⠀⠀⠀⢀⠓⠋⠻⡆⠸⣄⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠐⢚⣻⠓⢌⠻⡏⢉⠀⠀⠀⠀⠰⡾⣧⠘⠂⠀⣾⢸⠃⠀⠅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢁⣠⡜⠸⠷⠛⠦⣲⠀⠀⠀⣇⢸⣠⣽⠀⣸⢀⣯⡅⣿⠁
⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠡⠁⠈⣷⢻⡷⠴⠳⠖⠉⠀⠀⠀⠘⡋⠸⡇⣾⣡⠙⡤⢇⠈
⢀⠀⠌⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⢦⠀⣿⠃⡖⣴
⢿⠎⠀⠐⢣⠠⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⣿⢠⠿⡤⠛⠁
⣈⠀⢁⣁⠸⣇⠠⠀⡄⠀⠀⠀⠀⠀⠀⠀⣀⣰⣤⠰⠀⢀⠀⠀⠀⠀⠀⠈⢻⠎⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀
⣭⠄⠈⣍⡃⠘⠁⠤⠀⠀⠀⠀⠀⠀⠀⠰⣛⠽⣇⠐⣆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃
⢚⣓⠓⣍⠻⡏⡁⠀⠀⠀⠀⠀⠀⠀⢠⣲⡀⣦⠙⢂⣉⢾⠁⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠈⢁⣠⡜⠹⠶⠛⠦⣢⠀⠀⠀⠀⠀⢸⡉⣯⠀⡄⠈⡧⢨⡤⠀⣴⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢠
⠁⠘⡶⣻⡧⠶⠳⠎⠉⠀⠀⠀⠀⠀⠀⢧⠟⢾⣋⡰⢃⣞⢻⡀⠏⠀⠀⠀⢀⠀⠌⠀⠠⠀⠀⡀⠀⠀⠀⠀⠈
⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⢨⣷⣆⡘⣋⠀⠀⠀⠀⠠⣺⠞⠁⠴⢆⢢⡴⢆

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠒⠠⡀⠀⠀⠀⠀⢀⡤⠒⠠⠀⠀⠀⣀⠀⣤⠀⠀⠀⠀⠀⢀⣀⠄⡀
⠀⠀⠀⠀⠈⠀⠀⠀⢰⡆⢀⠀⠀⠀⠀⠀⡀⠠⠀⠈⢀⠀⠈⠆⠀⠠⠀⡞⠠⠀⠈⠀⠀⢉⡉⣠⣅⢠⡄⠀⠀⡔⢎⠀⢸⢖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠅⣾⠀⠀⠀⠀⠀⠀⠍⢠⡌⢡⡄⠀⠀⠀⠻⣬⠑⡯⠃
⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠅⠀⢠⠀⠀⠡⡈⠀⠘⠀⠄⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢁⠉⠀⠀⠀⠈⠀⠀⠀⠒⠊⠀⠀⠀⠀⢀⠐⡀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠃⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠚⡆⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠠⠀⢀⡈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠐⠀⠀⠀⠁⢸⣏⠀⠀⠀⡀⢀⠀⢑⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠆⠐⠤⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⡀⢰⠀⠀⠀⢼⠁⠀⠀⠀⠀⠀⢀⡞⠁⠀⢀⠀⠐⢄⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠆⠀⢀⡁⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠘⠈⠃⠀⡂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠂⠄⠀⠀⠀⠸⠀⠀⠁⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀⠀⠠⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀⡀⠀⠱⣀⢀⣀⣤⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣇⠔⠁
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠐⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡠⠂⠉⣩⠉⠑⠠⡀⠀⠀⠀⠰⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠂⢄⢀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠉⣐⣀⠔⠁⠀⠀⠠⢄⠈⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠔⠉⡑⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⣃⠀⠰⠀⠀⠀⠂⠀⠀⠀⠀⠀⡆⠀⠑⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠃⠃⠂⢀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠊⡀⠀⣀⠈⠳⣄⠀⠰⠌⡤⠂⡐⡄⠀⠀⠛⠀⢠⡷⢾⡇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⠉⠁⠢⠀⡇⠈⠋⠀⠀⠀⠀⠀⠀⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠈⠇⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠁
⠀⠀⠀⢠⡀⠢⡈⠀⠀⠰⠂⠀⠀⠀⠀⠐⢁⡄⠀⠑⡈⠀⡠⠁⠀⠀⠀⠀⠀⢀⠀⢠
⠀⠀⠈⠒⠴⣦⣿⣔⡾⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡷⠈⠀⡌⡆⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠈⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃

⡟⠉⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⠿⡿⣿⣿⠿⣿⡿⡿⢿⢿⠿⠿⢷⣿⣿⣿⣿⣏⣿⣹⣻⣿⣿⣿⣿⣯⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⠀⠁⠂⠘⠰⠀⠂⠂⠰⠐⠀⠛⠇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⢋⡀⠀⠀⠀⢀⠀⠄⠢⠀⢀
⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢁⠂⠠⠠⠀⠀⠀⠀⠀⠢⠔⠔⡒⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠅⠪⠚⠀⠀⠀⠀⠀⢀⠀⠈⠒⠄⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠖⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠠⠀⡀⠀⠀⢀⠀⠀⠊⠀⢀⠁⠀⠀⠐⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠚⢀⠀⠀⠐⠒⠀⠀⠀⠂⠠⠀⠴⢁⡡⠀⡄⠘⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠐⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠠⠀⠀⠄
⠀⠀⠀⠀⢀⠀⠀⠀⠘⠀⠀⠀⠈⢀⠀⠈⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣀⡐⠂⠀⠀⣢
⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⡄⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠂⠀⠀⠉⠀⠀⠀⠀⠀⣆⣈⠠⠘
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠦⠤⠄⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠉⠀⠀⠠
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣤⠖⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡆⠀⠀⣤
⠀⠀⠀⠀⢠⣿⡇⠐⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠿⡿⠃⠀⠀⠀⠀⢀⢠⡀⡄⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠙⠶⠞⠃
⠀⠀⠀⠀⣾⣿⠓⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠾⠁
⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣷
⠀⠀⠀⠀⠛⠛⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠩⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⢀⠀⠀⠀⠀⠀⢈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣸⢀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠅
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠻⣦⣤⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠀⠀⠀⠄⠀⠾⠉⠉⠓⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠂⠀⣀⠰⣆⢀⠓⠂⠈⠀⠀⠠⠄⠀⠈⠀⠠⠐⠁⠀⠀⡀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⡞⠁⠀⠀⠀⡀⠀⠀⠀⠘⠀⡀⠀⠸⠄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⢂⣀⣤⣧⢹⢀⡀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠃⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠃⣤⠴⠖⠛⠉⠁⢠⡄⠀⠀⠈⠶⠏⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⣰⣷⡀⠀⠀⠀⠤⠀⠀⠀⢸⠷⠃
⠀⠀⠀⠀⠀⠀⠠⠚⠁⢠⡄⠀⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡄⢀⠀⢠⡄⠀⠀⠀⠻⠏⠠⠀⠀⠀⢶⠁⠀⠸⠛⠂
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠛⠋⠉⠀⠀⢠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠃⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟
⠀⠀⠀⠀⠀⣠⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢀⠞⠀⠀⠀⠀⠀⢀⡀⠀⠐⠂⠀⠀⠀⢶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣿⠟⠟⣶⣄⠀⠜⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠀⠀⠐
⠀⠀⠀⠰⢾⣯⣬⣾⡿⡉⢳⡀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢾⡀⠀⢠⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⣴⣦⠀⢀⠀⢠⣾⣿⣿⡄
⠀⠀⠀⠀⢇⠀⣙⢿⣆⣤⣿⣷⠴⣿⡀⠀⠀⠀⠀⢀⣠⣼⣿⠦⣄⢸⡇⠀⠋⠀⠀⠀⠀⣀⣀⡀⠀⠀⢰⣿⡏⠀⠛⡑⠴⣟⠉⢻⡇
⠀⠀⠀⠀⠘⡄⠀⠏⢿⡿⠟⣿⡿⣿⣿⣶⣶⣶⣿⠿⣿⠿⠃⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢁⠶⣶⡄⣿⡄⢸⡆⡭⡄
⠀⠀⠀⠀⠀⠈⢦⡀⠈⢷⣤⣾⠷⡉⠛⠛⠛⠉⠀⠀⠀⠂⠀⣰⡖⣭⠆⠀⠈⡄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠘⠃⡿⠗⠈⠉⠁⠈⠙⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠑⣄⠈⠻⡿⠛⠛⠦⡤⠀⠉⠙⠁⠀⠀⢠⠃⠇⠸⣄⠀⠠⠜⠂⠀⠀⠀⢀⠀⠉⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⠈⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡴⠁⠀⠀⠛⠹⢶⡆⠀⠀⠀⠘⠚⠀⠘⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠒⠀⡀⠀⠔⠲⡆⠀⠀⠀⠁⠀⠀⠐⠀⢘⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⣠⠀⠀⠀⠙⠀⠶⠀⠀⠀⠛⠀⠀⠀⠀⡶⠀⠘⡄⠀⠀⠀⠀⠂⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢉⡄⠀⠘⠋⠀⡀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢦⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠁⠀⠀⠂⠈⠁⡨⠄⠀⠀⠀⢀⠀⡄⠀⠀⠀⢰⡀⢀⣹⣦⣀⠀⠀⠀⠈⠀⠀⠀⠀⠖⢀⣴⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣦⣀⠀⠀⠠⠊⠀⠐⠢⠀⠁⣀⡄⠁⠀⠀⢠⠿⠿⠿⠋⠀⣼⣷⠦⠄⠀⠀⠀⢀⣡⣶⣿⢋⣝⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠻⣿⣷⣦⡇⠀⠀⠘⠟⠀⠀⠀⠈⠠⠀⢀⠂⠀⠀⣀⣤⣾⡿⠋⠀⠀⣹⡿⠛⠋⠉⠉⢈⡥⠍⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢏⠉⣶⠄⠙⠋⠁⠰⠀⠀⢀⣈⣥⣤⣄⠀⠀⢹⣶⣶⣿⡿⠟⠋⠀⣀⣴⡾⠋⠀⠀⠰⡦⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⣜⣋⡒⢃⠀⠀⣴⣶⠖⠚⠛⠉⢉⠀⠙⠋⠀⠀⡸⠉⠩⢤⣤⣴⣶⣿⠟⠁⠀⠀⠏⢣⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠠⢦⣤⣀⣎⠀⠀⠀⠀⠈⠀⢀⡶⢘⠒⠀⠀⢀⡀⠰⠊⠀⠀⠀⠀⡉⢤⠚⣥⣤⣤⣤⠀⢠⠀⡀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠉⠁⠀⠀⢪⠀⢀⠀⠁⠀⠀⠂⠀⠔⠀⠈⠀⠀⠀⠀⠈⠀⡇⢠⣤⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡆⠀⢀⣠⠤⠀⠲⠉⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⢀⣠⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋⠈⠁⠃⠰⠞⢀⡴⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⠉⢀⣀⠀⠈⠑⠂⠁⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⠾⠿⠾⠿⠟⠛⠁

⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢿⣿
⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⠿⠿⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢾⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠉⠉⠉


⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⡿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⡿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣯⣽⣿⣿⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠉⠉⠉⠉


⣷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶
⣿⣷⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⣿⣿⣿⣿⣶⡷⢿⣾⣿⡿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣷⡾⠿⠿⠶⡿⠷⣶⡿⠷⢾⠿⠿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⣡⢻⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⡰⠃⡏⢸⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠑⠒⠂⣴⠁⢀⠇⠈⡆⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠑⡎⢩⣒⠤⣄⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⠃⠳⣸⣠⠴⡷⠤⣘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⡱⣄⡈⠁⠀⠉⣷⠀⠀⠀⠀⡞⠀⠀⠀⡎⠀⢀⡏⠀⢸⠀⠀⡴⠍
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⣰⠁⠀⠉⠓⠦⡼⠀⡆⠀⠀⢀⡇⠀⠀⢸⠁⠀⡸⠀⠀⡇⠀⡰⡹
⠀⠀⠀⠀⠀⠀⠀⢸⡑⠦⣤⠃⠀⠀⠀⠀⠀⣿⠀⣇⠀⠀⢸⠀⠀⢀⠇⠀⢠⠇⠀⣸⠁⠀⢱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠸⡄⠀⠀⠀⠀⠀⢻⠀⢸⠀⠀⡏⠀⠀⡼⠀⠀⡼⠀⢀⡇⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢧⠀⠀⠀⠀⠀⢸⠀⠸⡀⢠⠃⠀⢠⠇⠀⢰⠃⠀⢸⠀⠈⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠸⡆⠀⠀⠀⠀⢸⠀⠀⡇⣼⠀⠀⡼⠀⠀⡞⠀⠀⡇⠀⡸⠀⠀⠀⣀⣤⣤⣤⣤⣄
⠀⠀⠀⠀⣀⣀⣀⣀⠀⠸⡄⠀⢳⠀⠀⠀⠀⢸⠀⠀⢷⡇⡎⢀⠇⠀⢰⠃⠀⢸⠁⢠⡧⠖⠚⠉⣀⢾⠒⠋⡹⠙⡆
⠀⠀⠀⢠⠿⡙⠮⣝⠉⢩⢷⠀⠘⡇⠀⠀⠀⢸⠀⠀⢸⣱⡀⢸⠀⣀⣏⠀⠀⡏⠀⡾⣀⡤⠖⠋⠀⠀⠑⡶⠥⢤⠟
⠀⠀⠀⢸⠀⢣⠀⢀⡹⢧⠸⡆⠀⢻⠀⠀⠀⢸⡆⠀⢸⣯⣃⣿⣍⠀⠀⢙⣶⡧⠚⢯⡁⠀⠀⠀⠀⢀⡞⢡⡾⠋
⠀⠀⠀⣾⣠⠤⢷⠋⠀⠈⢇⢻⠀⠘⡇⠀⠀⣸⡇⠀⠀⡟⠁⠸⡄⣹⠚⠉⠘⡇⠀⢀⣵⡀⠀⠀⢠⠏⣠⠟
⠀⠀⠀⠈⠢⡀⠀⠳⡀⠀⠈⢿⣇⠀⢹⣀⣀⡿⠧⠴⠧⣷⣠⠤⠟⠳⠤⢤⣀⡧⠖⠉⠀⢣⠀⣰⠋⡴⠃
⠀⠀⠀⠀⠀⠈⠳⣄⠙⢆⣀⠤⠟⠚⠉⠁⠀⢀⣀⡤⠖⠊⢳⡀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠘⣴⢧⡞
⠀⠀⠀⠀⠀⠀⠀⢈⡟⠯⣄⡀⣀⡤⠴⠒⠋⠉⠀⠀⢀⣠⠴⡇⠀⠀⠀⠀⠀⠘⡆⠀⢀⡞⠀⠎⡇
⠀⠀⠀⠀⠀⠀⢠⡿⢀⡠⠖⠻⡀⠀⠀⠀⣀⡤⠴⠚⠉⡀⢸⠁⠀⠠⠄⠀⠀⠀⢹⣀⠞⠀⢀⣀⡇
⠀⠀⠀⠀⠀⠐⢷⡾⠥⠤⠄⡀⣧⠴⠒⠋⠁⠀⣀⣠⠴⠒⠙⣏⠉⠛⠓⠒⠒⠒⣲⠟⡊⣉⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠢⣔⠀⡰⢃⣀⣤⠶⠞⠉⠉⠒⠤⣀⠀⠘⣆⠀⠀⣀⠴⠋⣐⠽⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠀⠈⠉⠓⠲⠤⣄⣀⡀⠙⠒⠾⡦⢊⣡⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠐⢒⠺⠓⠢⠒⠘⠏⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⣀⡀⠀⠀⡰⣴⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠰⡤⠤⠀⠤⡴⠀⠀⢠⠂⠁⠀⢀⠔⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡛⠚⠒⠐⠢⠐⠚⠁⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠔⠛⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠠⠇⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠛⠀⠀⠷⠀⠀⠀⢀⣠⠖⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠃⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠂⢠⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢳⣶⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠠⡴⠿⠁⠙⠉⠁⠀⠀⠀⠩⠆⠀⠀⠀⠀⠒⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠙⠉⠃
⠀⠀⠀⠀⠀⠀⢐⠀⢓⠀⠀⠀⠀⠀⠘⠘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠜⡁⡐⠲⠂⠀⠀⠀⠈⠇⢀⣄⠀⠀⠀⠀⠀⠀⢰⡶⣶
⠀⠀⠀⠀⠀⠀⠈⢤⠀⠀⠀⠀⠀⠀⡀⣈⠙⠀⠀⢐⣖⣶⡞⠆⠀⠀⠀⠀⢤⠚⣁⡚⣠⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⢀⠌⣄⡀⠀⠈⠉⠈⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠯⣣⠅⠀⠀⠀⠀⠀⠀⠀⣩⠃
⠀⠀⠀⠀⠀⠈⠛⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⡴⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠰⠆⠀⠀⠀⠀⠉⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠄⢺⠠⠄⠀⠀⠀⠁⢀⣦⢤⣤⡤⣤⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠜⠀⠀⣠⠰⠿⠒⠛⠋⡉⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⢤⡘⡛⡀⠜⠁
⠀⠀⠀⠀⠤⠴⠦⣐⡀⠁⣠⠦⠤⢀⣀⡸⠂⠘⠀⠙⠤⠖⠀⠀⠀⠛⠋⠙⠁⠀⠀⠀⢀⡀⠀⣴⠀⠀⠀⠀⢀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢐⡀⠀⠀⠆⣄⠈⠀⣠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⡀⠀⠀⠀⠀⠙⣃⠅⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠐⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠈⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠑⡐⠋⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠈⠄⠀⠀⠀⠀⣤⢘⠃⠀⡀⠀⠀⠀⠀⠈⠁⠀⠂⠀⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡄⠀⠀⠀⠀⠠⣶⢈⠀⣠⢫⠄⠀⠀⠀⠀⠒⠤⢄⣀⠠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣷⡌⠰⠀⠀⠀⠀⠀⠲⢿⡆⣀⡌⠀⠀⠀⠀⠀⠀⠐⠘⠛⡀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠌⣁⢴⡂⠀⠀⠀⠀⠀⢭⣿⠋⠁⠀⠀⠀⠀⠀⢀⡤⠤⣤⠤⡀⠀⠀⠀⠀⠀⠀⠠⢴⣶⠄⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠋⠀⠁⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⣘⠉⠀⣚⠀⠛⣄⠀⠀⠀⠀⠀⠀⢨⡏⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⢸⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⢸⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⢸⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⢸⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⡇⣿⣿⢸⢸⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⠿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠙⡧⣿⣿⢸⢸⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⢿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣿⢸⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡴⣻⢿⡟⣽⣱⢧⡏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢸⡏⣯⢳⡹⣝⣿⢦⡀
⠀⠀⠀⠀⠀⠀⢠⡞⣵⢃⡿⢸⡇⣿⡸⣧⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣸⡇⣿⢼⣷⢹⡎⣷⡹⣆
⠀⠀⠀⠀⠀⣰⡿⣸⡟⣸⡇⢻⣇⢻⣧⡹⣷⣍⡻⢶⣤⣤⣀⣀⣀⣠⣤⡶⠟⣫⣾⠟⣰⡿⢸⣿⢸⣿⢸⣷⢹⣧
⠀⠀⠀⠀⢠⣿⠃⣿⡇⢻⣷⠘⣿⣆⠻⣿⣮⣙⠿⢷⣶⣭⣭⣭⣭⣭⣴⣶⠿⢛⣥⣾⠟⣡⣿⠏⣼⣿⢸⣿⡆⣿⡆
⠀⠀⠀⠀⣼⣿⠀⣿⣧⠘⣿⣧⡘⢿⣷⣬⡙⠻⢿⣶⣦⣬⣭⣭⣭⣥⣴⣶⣾⠿⢛⣡⣾⡿⠋⣰⣿⡏⢸⣿⡇⣿⣿
⠀⠀⠀⠀⣿⣿⠀⢿⣿⣆⠙⣿⣷⣄⡙⠻⣿⣷⣦⣤⣍⣉⣉⣉⣉⣉⣉⣥⣴⣾⣿⠿⠋⣠⣾⣿⠟⢠⣿⣿⠁⣿⣿⡀
⠀⠀⠀⠀⣿⣿⣇⠘⣿⣿⣦⡈⠻⣿⣿⣶⣤⣉⠙⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⣉⣠⣴⣿⣿⠿⠃⣰⣿⣿⠏⢠⣿⣿
⠀⠀⠀⠀⢻⣿⣿⡄⠘⢿⣿⣿⣦⡈⠙⠻⢿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⠿⠛⢁⣠⣾⣿⣿⠋⢀⣾⣿⣿
⠀⠀⠀⠀⠈⣿⣿⣿⣆⠈⠻⣿⣿⣿⣷⣦⣄⡈⠉⠛⠛⠛⠻⠿⠿⠛⠛⠛⠉⢁⣀⣤⣶⣿⣿⣿⠟⠁⣠⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣄⠀⠙⠿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⠿⠛⠁⣀⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣦⣀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠁⢀⣤⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣾⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⢀⢠⠒⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡖⢄
⠀⠀⠀⠀⠋⠉⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠐⢠⠄⠂⠁⠂⠤⠐⠆⠀⠀⠀⠀⠀⢀⠀⡀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐⢣⠊⠈⢠⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⢗⠒⠗⠒⠒⠗⢒⠗⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠦⣤⠀⠰⡤⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⠂⠄⢂⠁⠔⠃⠀⠀⠀⠀⠀⠀⠁⢈⠀⢐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⢎⠢⡠⢊⠘
⠀⠀⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠓⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠠⠻⠄⠀⠀⢼⠃⠀⠀⠀⠀⠀⠀⠐⠃⢀⠜⠙⢄⠀⠓⠀⠀⠀⠀⠀⠀⢀⠆⡨⠁⠹⡐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠳⡀
⠀⠀⠀⠁⠄⠀⠀⠀⠈⢄⠡⠀⠀⠀⠀⠮⠀⠴⠡⠤⠤⠤⠰⠆⢨⠆⠀⠀⠀⢠⠵⠂⠀⠀⠀⠈⠢⢡⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄
⠀⠀⢉⠁⠄⠀⠀⠀⠀⠔⠩⠁⠀⠀⠀⠀⠐⢄⠢⠀⠀⠠⢂⠔⠁⠀⠀⠀⠀⠰⠠⡀⠀⠀⠀⠀⡠⠰⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠍
⠀⠀⠀⠡⣠⠄⠀⠀⢼⡐⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠑⢄⠲⠀⢰⢂⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢲⣠⠃
⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠔⠩⠀⠀⠀⠤⠑⠄⠀⠀⠀⠀⠀⣠⢼⡅⢈⣏⠀⢽⣤⡀⠀⠀⠀⠀⣠⠞⠉⠀⠰⠈⠁⠙⢦⡀⠀⠀⠀⠀⠀⠴⣈⡀⡇⣌⡰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⡌⠀⠀⠈⣄⣠⠇⠀⠀⠀⠘⢁⠒⠘⡃⠒⠂⡛⠐⢒⠝⠀⠀⠀⠀⣌⣐⣊⢢⣧⢊⣓⣈⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣀⡡⢀⣀⠠⣁⣈⡇⠀⠀⠀⠀⠀⠑⢀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠡⠀⣢⡊⡍⣢⡖⠠⠁
⠀⠀⠀⠑⢬⡁⠀⠀⣉⠜⠁⠀⠀⠀⠀⠀⠈⠹⣅⣠⢧⣀⡽⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⢦⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢍⣇⠡⠘⠃


⠀⠀⠀⠀⣠⡄⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠤⠄⠠⡀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠄⠀⢠
⠀⠀⠀⠄⣁⠠⠒⠢⢀⡑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡀⠀⠀⠀⠀⠀⢠⠚⠀⡴⠀⠐⡄⠐⢢⠀⠀⠀⠀⡠⠊⢀⠂⠀⠀⠀⢂⠈⠢⡀
⠀⠀⣖⠀⠁⠀⠀⠀⠀⠁⢸⡄⠀⠀⠀⠀⡤⠤⠔⠁⠀⠑⠤⠬⡄⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠑⢀⠈⠅⠀⠀⠀⠍⢀⠀⠁
⠀⠀⠀⠄⠓⠀⣀⡠⠐⠃⠌⠀⠀⠀⠀⠀⠈⠢⡀⠁⡄⠁⡠⠊⠀⠀⠀⠀⠀⠰⡀⠈⢆⠀⢀⠎⠀⡰⠀⠀⠀⠀⠀⠀⠑⢌⠀⠀⠌⠔⠁
⠀⠀⠀⠀⠷⠂⠉⠑⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠇⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠁

⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡐⠁
⠀⠀⠀⣀⠀⠀⢤⢈⠲⡀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡰⢊⠀⠄⠀⢀⡀
⠀⠘⠀⠈⠀⢀⡀⠀⠒⠸⣼⠀⠐⠁⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠑⠀⢸⡼⠐⠂⠀⣀⠀⠈⠀⠘
⠐⠂⠠⠽⣄⡣⠀⠪⠰⠄⣹⡀⡐⠀⠀⡁⣠⠰⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⢠⡀⡀⠀⠐⡁⣸⡀⠴⠈⠂⠀⢃⡼⠥⠀⠒
⠴⠀⠀⢂⣈⡉⠙⠮⠥⣤⠌⢳⡁⠀⡐⢀⢅⡀⢰⢀⠅⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠅⢰⠀⣀⢅⠐⡀⠀⣱⠋⢤⡤⠭⠞⠉⣉⡀⠂⠀⠰
⠀⠀⠐⠁⠈⠀⠈⢄⢂⠲⠙⢮⠷⣔⠀⠈⢀⡉⠚⣇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⡟⠊⣁⠈⠀⢐⡴⢯⠞⠱⠂⢂⠌⠀⠈⠀⠑
⠀⠀⠀⠀⠀⠫⠅⢠⣈⠁⢠⣤⡥⠮⠽⣶⢦⣍⣉⡙⣦⠁⠀⠀⠀⠀⠀⠀⠀⠀⢡⡞⣉⣉⣤⢶⡾⠭⠦⣤⣤⠀⢉⣠⠀⠬⠃
⠀⠀⠀⠀⠐⠖⠠⠤⠬⢛⠏⢠⠁⠠⠞⠥⠠⢄⣨⠽⠛⠛⠶⠶⠶⠶⠶⠶⠶⠖⠛⠻⢭⣀⠤⠠⠝⠦⠀⢠⠈⢟⠫⠤⠤⠰⠖
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠁⣐⠀⠀⡇⡦⠠⡤⠞⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠦⡤⠤⡆⡇⠀⢜⡀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠆⠀⡇⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⡇⠀⠦⠀⡀

⠀⠀⠀⠀⠀⣤⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⢠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⠀⠀⠰⡆⠀⠄
⠀⠀⠀⠁⠀⠀⠋⠀⠀⠀⠀⢠⠄⠀⠂⠀⡠⠀⢀⠙⠛⠋⠀⠀⠀⠀⠀⠃⠜⠀⠐⠃⠀⠀⠀⠈⠁⠀⠻⠔⠀⠀⠀⠀⠸⠷⠆⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠡⣄⡀
⠀⠀⡀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠒⠁⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠈⠃⠀⠀⠀⠈⠁⠀⠀⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⣀⣀
⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠐⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⡍⠉⠐⣇⠀⡉⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠀⠠⠀⠈⠓⢠⠜⠃⠀⡱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠧⠘⠂⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠈⠁⠉⠁⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠢⡀⠀⠀⠒⠄⠀⢁⠈⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠙⠀⠇⠀⠘⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠐⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⡄⠀⠀⠀⢰⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⡀
⠀⠀⠿⠈⠁⠀⠀⠤⠁⠀⠀⠀⠀⠀⠃⠀⠀⠂⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡐⠀⠀⠠⡀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠊⠀⢹
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠐⠀⠀⠀⠑⠢⣀⠾
⠀⠀⠀⠈⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠐⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉
⠀⠀⠈⠂⠒⠀⠚⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⢃
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠭⡀⠋⠀⠀⠀⠀⠀⠀⠐⢆⠐⠀⡴⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡀⠀⡠⢈⡶⠀⠀⠀⠒⠠⠤⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢦⣀⡤⠋⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠈⠀⠰⠆⠀⠀⠀⠲⡶⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠠⠒⠆⠠⠀⠷
⠀⠀⠉⠉⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠀⠠⠄⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁⡀⠀⣌⢣⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⣀⡉⡑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡀⠀⠀⠀⣀⡀⠀⡼⢀⠀⠀⠁⠘⠃⠀⢠⠀⠀⠀⠀⠀⠀⡄⠀⠸⢇⠀⢀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠄⠀⠀⢀⠄⠁⠀⠀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⢡⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⠣⠤⠀⠐⠓⠉⠀⢀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠏⠀⠀⠀⣄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠞⢁⠔⠂⠒⣄
⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡈⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⡏⠀⠀⠀⠈⡃
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠉⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢣⢀⡀⢀⡶
⠀⠀⠀⠀⠀⠀⢀⠀⠒⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠂⠐⠀⢡⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠠⠶⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣰⢫⠄⠀⠤⠹⢦⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⡀⠈⠁⠀⠁⠀⠀⠀⢠⠂⠃⢀⠀⢀⡀⠈⠀⠀⠉⢀⡀⢸⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠰⠀⠀⠀⠁⢡⡄⠁⠠⠄⠋⠀⠀⠀⠀⠀⠀⠀⠚⡍⠉⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠴⠀⠀⠠⠀⠀⠀⣤⣤⠇⠀⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠴⠒⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠁⠒⠉⠄⠀⠀⠀⠀⠀⠠
⠀⢠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣀⠀⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⣈⠠⠈⠀⠁⠄⢨⡄⠀⠐
⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡆⢀⠀⠀⠀⠈⠀⢀⠀⠀⠀⢠⣶⣶⣶⣄⠀⠂⠀⠀⠂⠁⠠⠜⢀⠐⡀⠘⢁⠐⡀⠄
⠀⠀⠀⠀⡀⠤⢀⡀⠀⠀⠀⠀⠀⠴⠄⠀⠀⠀⠀⣀⡃⠀⠈⠈⣁⡀⠀⠀⠀⠀⠀⠀⠀⣷⠁⠀⠈⣿⠀⠀⠠⠐⠀⠀⠀⠀⠉⠈⡁⠒⣈⠁⡍
⠀⠀⠆⠘⢁⠀⠀⠙⠋⠀⣄⠀⠀⢀⡀⠀⠀⢄⠀⠉⡄⠀⠀⢀⡉⠁⠀⠀⡀⠀⠀⠀⠀⠹⣷⣤⠾⠟⠀⠀⠀⠀⠤⠀⠐⡀⠓⠀⠁⣀⠈
⠀⠀⡔⠀⠀⠀⡀⠀⠁⠀⣨⠀⠀⠀⠈⠠⢀⣀⠶⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⣰⠀⠐⠠⠀⠀⢀
⠀⠀⠀⠒⠀⠀⠀⠀⠄⠀⠀⠀⢠⣔⠳⣦⠀⠀⡠⠆⠌⠑⠠⠀⠀⠀⢆⠀⠀⠀⠀⣤⠂⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠁⠀⡏⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⢀⡀⢠⡎⠀⣤⢆⠸⠇⡀⠁⠀⠂⠀⠀⠀⠀⠘⠇⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠀⢀⡀⠀⠀⠀⠀⠠⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠚⠛⠀⠀⠀⠈⠱⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉⠀⠉⠁

⠀⠀⠀⠀⡀⠐⢲⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀
⠀⠀⢠⡆⢘⡧⠌⡃⠘⠆⠀⠀⠀⠴⠊⠱⣀⡈⠶⠁⠢⡄⠀⠀⠀⠀⠀⠀⠀⣎⣩⢢⠀⠀⠀⠀⠀⠀⡠⢹⠆⡙⠆
⠀⠀⢿⡤⠎⠀⠀⠱⣤⡌⠀⢀⣦⣀⡠⠂⠁⠀⠀⢢⠄⡟⠂⠀⠀⠀⠀⠔⢠⠂⠈⣌⠳⡀⠀⠀⠀⠠⡧⠂⠀⠘⣮⡀
⠀⠀⡄⢰⠀⠀⠀⠀⣇⢘⠂⠸⣁⠠⠀⠀⠀⠀⠀⠀⠄⢀⡶⠀⠀⢀⠆⢠⠂⠀⠀⠘⣆⠀⠀⠀⠀⢰⠻⠀⠀⠀⡇⡳
⠀⠀⢴⠒⢇⠀⠀⡠⠸⣆⠀⠀⢤⡇⠀⢄⡀⠀⢀⠴⠒⠃⠀⠀⠀⡞⠀⠁⠀⠀⠀⠀⠘⢇⠳⠀⠀⠨⡼⡄⠀⢠⢾
⠀⠀⠘⠄⢈⡔⢺⡁⣠⠇⠀⠀⠈⠣⠀⠰⡍⠁⠶⡤⠐⠃⠀⠀⣼⣓⠶⠒⠒⠀⠀⠒⢖⣸⡙⣧⠀⠀⢣⣼⢦⣀⡌
⠀⠀⠀⠀⠰⢥⡀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠑⠄⠃
⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⡀⠀⠀⠀⠀⢀⡀⢀⠀⣀
⠀⠀⠀⠀⠐⡘⢣⡠⠀⠀⠀⠀⡠⠀⡒⠖⢤⡀⠐⠀⠧⠤⡀⠀⠀⡐⠀⠲⢄⡀⠀⠁⠠⠀⠘⡀⠀⠀⢸⡇⠉⠳⣧
⠀⠀⠀⡜⠋⠌⠠⡇⠱⡀⠀⠀⢓⡀⠟⠉⠑⠂⠉⠑⠄⣀⠀⠀⠀⠈⢳⠖⠆⠈⠉⠓⣆⠀⠃⡀⠀⠀⠠⠤⠶⠚⠤
⠀⠀⢠⣤⠊⠀⠀⠘⣄⠁⠀⠀⠛⢃⠃⠀⠀⠀⠀⠀⢎⠵⠃⠀⢰⡀⠀⣂⠀⠀⠀⠀⢸⠀⠀⠸
⠀⠀⣀⢀⠀⠀⠀⠀⡏⢲⠀⠀⠀⠐⢜⠦⠀⣀⡤⠃⡁⠀⠀⠀⠠⠑⡤⠉⠁⠀⣀⡤⠦⢤⡀⠀⠀⠀⢀⡈⠂⢌⠠
⠀⠀⠉⡛⢣⣀⡀⡰⢾⠊⠀⠀⠀⠀⠈⠙⢦⣈⠓⠋⠀⠀⠀⠀⠀⢡⣀⠴⣀⠀⠀⢀⠰⣤⣤⠁⠀⠀⢤⣧⡀⡠⠤
⠀⠀⠀⠈⠁⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠁⠀⠈⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠈⠍⠐⠒⠷⡉⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣠⡀⠉⠀⣼⠋⠑⠀⠀⠀⢈⣈⠓⠀⠀⠀⠀⡰⣆⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠛⣀⣀⡀⢂⡒⠠⣀⣀⠱⡆
⠀⠀⡠⠤⠃⠈⠉⠁⠈⠁⣇⠰⡀⠀⠋⢙⣷⠂⠲⣌⡀⠐⢀⡠⠀⢸⣄⠀⠀⠀⠀⠀⠛⡀⠂⠀⠀⠀⠀⠀⠀⢘⠢
⠀⠀⣁⣤⡇⠀⠀⠀⠀⠀⠈⢢⡁⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠀⠂⡄⠄⠀⡀⢀⣄⣀⡀⣀⠀⠀⣀⢼⡀⣦
⠀⠀⢇⡈⡣⠀⠀⠀⠀⠀⣔⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⣇⠀⠀⠀⠀⢀⠂⠀⠀⠙⠡⠤⠤⠤⠶⠶⠦⠤⠤⠄
⠀⠀⠀⠁⢉⡉⠓⢤⡈⠁⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡤⠄⠀⢠⡀
⠀⠀⠀⠀⠀⠈⠀⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠒⠂⠀⠘

⣿⣿⣿⣿⠋⠀⠀⠀⣴⣿⣿⡿⠛⠉⣀⣴⣶⠿⢛⣩⡴⠟⣱⠇
⣿⣿⣿⠃⠀⠀⠀⣼⣿⣿⡟⠁⠀⣼⣿⠏⢀⣴⡟⢉⡴⠛⠁
⣿⣿⣿⡆⠀⠀⠀⢿⣿⣿⡇⠀⠸⣿⣿⠀⢸⣿⠀⣿
⣿⣿⣿⣿⣄⠀⠀⠈⢿⣿⣿⣄⠀⠹⣿⣆⠈⢿⣆⠻⢦⣀⠀⠀⠀⢀⣀⣴⣮⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿
⠻⣿⣿⣿⣿⣷⣄⠀⠀⠙⠿⣿⣷⣄⡈⠛⠷⣦⣍⣛⠲⠯⢭⣽⣿⣿⣿⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠈⠙⠻⠿⣿⣿⣿⣶⣤⣀⣀⠉⠛⠻⠷⠶⠦⣬⠭⠭⠭⣟⣿⣿⡿⠋
⣤⣄⣀⡀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠶⠶⠶⠶⢒⣚⣫⢽⡲⠟⠉
⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⠶⠶⠖⢛⣛⣭⠵⠚⠉
⠀⠀⠈⠈⠈⠉⠉⢉⣀⣀⣠⣤⣤⡶⠶⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠙⠛⠁⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⡤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣿⢸⡃⠈⡷⠴⡏⠀⢹⠀⡟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢿⠀⠙⠛⠁⠀⠉⠛⠉⢀⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣦⡀⠀⠀⠀⠀⠀⢀⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠈⡟⠒⠶⠶⠶⠚⡏⠈⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⣇⠀⠀⠀⠀⢀⡇⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⣻⠇⠀⢻⠉⣻⠿⡏⢹⠃⠀⢿⡳⠤⣄⣀⡤⠖⠚⠛⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢴⡃⠀⠀⠈⣇⢳⣶⢋⡏⠀⠀⠀⣹⠄⣰⠋⠳⡄⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠙⢦⣀⠀⢹⡞⠘⣿⠀⠀⣠⠞⠁⢠⡇⣾⠁⢿⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⢷⢰⢇⡴⠋⠁⠀⠀⠘⡇⠉⠀⢸⡄⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠋⠀⠀⠀⠀⠀⠀⠹⣄⠀⠈⠇⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣤⣤⠴⠞⠙⡶⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣿⣾⢹⡟⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣿⢹⣿⣿⡟⣷⡹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⢸⣿⡹⣷⣝⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡘⣿⣧⢻⣿⣜⢿⣦⣙⢿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⢻⣿⣦⠻⣿⣦⡙⢿⣷⣍⡻⢿⣷⣦⣤⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠿⢫⣧⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⣠⡇⠐⠙⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣥⣴⣾⠋⠀⠰⠷⠈⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢾⣋⣭⣵⣾⣯⡐⠷⠀⠀⡀⠀⣿⣿⣿⣿⣦⡀⠀⠘⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠛⠛⠉⣿⣦⣄⠈⠁⠀⠀⠉⠛⠿⣿⣿⣶⣶⢟⣿⠇⠀⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠅⠈⠀⠃⡸⠋⡣⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣏⣀⣠⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⣫⣿⠃⢀⠀⠐⠜⠂⠆⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣽⣿⡿⢛⡛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠁⠀⠠⠀⠤⠜⠀⢀⡠⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⢹⡿⣷⣌⢻⣿⣿⣿⠿⠿⢿⣦⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⡀⠀⠀⠐⠀⠀⠠⠄⠅⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⣾⣿⡟⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠵⡀⢄⣴⣶⣿⣿⠥⢀⠀⠄⢀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣿⣿⡿⣏⣿⡇⠀⠀⣀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⣴⢬⣿⣿⣝⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠀⠀⢰⣿⣿⠀⠻⣾⣿⣦⡀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⠋⠈⢿⣝⡻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣼⣿⣟⢇⠀⠀⠉⠉⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠉⠛⣿⣷⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠰⠿⠟⢻⣮⡓⠦⡤⣶⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠙⠓⠀⠀⠀⠀⠀⠀⠄⠀⠀⣠⣴⡿⠟⠿⣶⡀⠙⠻⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣾⣿⠃⣴⢿⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠹⠦⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⢀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⠀⢠⣾⡿⠙⣷⠀⣀⣤⣤⡴⠶⢦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁⠀⣴⣿⠟⠀⢀⣿⣿⣿⠟⠁⢀⣤⣾⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣝⡲⠶⢶⣶⣶⠿⠟⠋⠀⢀⣠⣾⡿⠋⠀⣠⣾⣿⠟⠁⣠⣶⣿⣿⡿⠛⠁⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⠳⣝⡶⣤⣤⣤⣤⣤⣴⣶⡿⠟⠋⣠⣴⣿⠿⠋⢁⣴⣾⣿⣿⠟⠋⠀⢀⣴⠶⠶⢤⡈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢮⣛⠶⣯⣭⣍⣉⣩⣥⣤⡶⠿⠛⠉⢁⣤⣾⣿⡿⠟⠉⠀⠀⠀⠀⣾⡇⢰⣆⣼⡇⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣍⣉⣉⣁⣀⣤⣤⣶⣾⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⣉⣉⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⠀⠀⠉⠉⠛⠛⠛⠛⠃⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣡⠴⡾⠛⠁⢀⡄⠀⠀⢀⠀⠀⠀⠀⢄⡀⠀⠠⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡷⢊⡵⠂⠀⠀⣴⠋⢀⡴⠀⣾⠀⠀⡇⢠⡀⠹⣦⠘⢦⡹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⣴⠋⠀⠀⣠⡞⠁⢠⡾⠁⡸⠇⣰⠀⣿⠀⢷⡀⠈⢷⡀⠹⣌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⢠⡾⢁⡔⢁⣼⠏⠀⠀⠈⡀⠠⠁⢠⡟⠀⠈⢠⡌⢷⠰⡌⢿⡄⠹⣞⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣾⢡⡿⢁⡞⢀⡞⠁⠀⠀⢀⡼⢡⠇⢠⠿⠃⡤⠀⡼⣿⡘⡆⢿⡸⢷⠀⠻⡆⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⣿⡇⠃⣾⠁⡞⢀⡼⢀⡴⠋⣰⠋⠀⠁⣨⣞⢁⠘⠀⠙⢧⠃⠈⠃⡞⡄⠀⢱⢸⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⣿⠀⠀⠁⠘⣰⣾⠴⠚⠛⠛⠙⠛⠛⠛⠉⠉⠛⠛⠋⠙⠻⡰⠾⣶⢁⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⡀⢀⠂⢰⣯⠴⠷⠶⣤⣄⣀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⡹⢦⢹⣯⠀⢰⠀⢸⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢃⢸⡄⣾⠁⢀⣠⣄⡀⠉⠙⠣⠀⠀⠀⠀⠘⠋⠁⢀⣰⣇⡀⠈⣟⠀⡜⠐⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞⣿⣇⣿⠀⢻⡟⠋⣿⠷⢦⠀⠀⠀⠀⠀⢀⠴⢻⣿⣿⢻⠟⢠⡇⣰⢃⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣇⠈⠳⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠵⠛⠀⣼⣿⣧⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣎⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢟⣥⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⡻⣿⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠚⠛⠁⣿⢿⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣜⠿⡄⠀⠀⠀⣀⣀⡀⠀⠀⠀⠈⣲⣄⡀⠀⠀⠀⢠⣿⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠳⣿⣄⠀⠀⠀⠘⢥⣀⣀⣀⣀⡜⠁⠀⠀⠀⣠⠟⢿⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⡀⣻⣦⡀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢀⡴⠫⣷⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠶⢛⣿⠉⠓⠦⣄⣀⣀⣀⣀⠠⠶⠞⠹⡗⢶⣾⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣛⣿⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠠⣶⡸⢿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣭⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢣⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠜⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡈⠎⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡱⢀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡄⢿⡌⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢠⡟⢠⡄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⣧⠀⢣⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡜⠀⣼⢷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢿⠀⠀⠆⢹⡄⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⡿⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠀⡀⣶⣼⠁⠀⠀⠀⠀⠀⠀⠈⣇⣧⢠⢀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⣷⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣾⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣹⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣏⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁

⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣄
⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⡄
⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣤⡄
⣤⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠁
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇
⠈⣻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⣿⣿⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⢻⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⣿
⠀⠀⠉⣽⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇
⠀⠀⠀⠙⢿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠟⠁
⠀⠀⠀⠀⠀⠻⢿⣿⣿⣆⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣀⠀⠀⠀⠀⠀⣴⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⡀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⢿⣿⣶⣄⣼⠿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡄⠀⠀⣾⣿⠋⠁⠀⠀⠀⠀⠈⠙⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⡶⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⡤⠀⠉⠀⠄⢀⠀⠀⠀⠀⠁⠄⠀⣰⡟⠀⠀⠴⠆⣹⠷⢶⣶⣄⡘⠷⠦⠈⠀⠺⣆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠲⣿⣄⣜⣿⠋⠒⢀⣀⠀⡀⠀⠀⠀⠙⢤⢴⢶⣿⣿⣿⣶⣿⣠⣤⣝⠶⠄⠈⣿⣿⣷⡶⠀⡀⠀⠀⡁⠢⡀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⢀⡜⣻⠙⣼⣡⣤⣶⡉⠀⠀⠀⠀⠀⠒⡇⠘⢻⣰⣿⣿⢿⣿⣿⣿⣝⡋⠐⠀⠀⠀⠉⣿⣷⡤⠀⠠⣳⡌⡀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⠀⠐⠿⣷⣿⡏⢩⠿⢋⣩⠤⢀⣀⢄⡁⢀⣴⣿⡿⣿⠉⣼⣿⣿⣿⠛⠋⠀⠂⠈⠙⠢⠜⢿⣧⡠⠊⢉⠓⠏⠰⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⠓⠹⣶⣿⣿⣾⡁⣀⡉⣀⠐⣎⠉⢹⣷⣿⣿⡟⠷⢃⣴⣿⣿⠿⠁⢶⡀⣀⣉⣀⠀⠌⠀⠀⠹⣷⣾⣥⠀⠀⠰⠆⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢉⣭⠑⢦⣹⣿⡛⣿⡗⠡⠈⠋⠲⡾⠛⢿⣿⣿⣿⡿⠀⣼⣿⢛⣤⣤⣾⣿⣯⣿⣽⣿⣿⣶⣆⡈⢻⣿⣿⣭⡇⠀⠈⠀⣈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣎⠻⣹⣿⣿⡀⠀⠀⠀⠃⢈⣿⣿⣿⣿⣿⠄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⣾⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢶⣽⣦⣿⣿⣧⠀⠀⠄⠁⢠⣿⣿⣿⣿⣿⣡⣼⣿⣭⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠈⢹⣻⣷⣬⡀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣬⣷⣾⣿⣿⣿⣿⡟⠀⠀⢄⣠⣼⣿⣿⣏⣽⣿⣿⣿⣟⠿⣹⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣏⣿⢃⣶⣿⣿⣿⣷⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣬⡿⢿⣿⢣⡄⠀⠀⣼⣿⣿⡿⠉⣹⣿⣿⣽⣿⣷⣾⣿⣥⣼⣿⣿⣿⣿⣿⣿⣿⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣦⡜⠿⣀⣹⣷⠐⣻⣾⣿⡟⣛⢡⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣟⣩⣥⣶⢾⠛⠛⡿⣟⠓⢼⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⣿⢿⣿⣿⠷⣊⣀⣿⢿⡻⠃⠘⢿⣿⣿⣿⣿⣿⠛⠂⢤⣄⣤⡙⠛⠿⠿⡿⠿⠿⡟⠀⠀⠃⢿⣠⢾⣯⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢼⡿⣿⣿⣿⠟⠿⣿⣞⠀⢐⠀⢸⣿⣿⣯⣿⣿⣥⢤⡀⠙⠋⡄⠤⠀⠤⠀⣀⠀⠀⠀⠰⢄⣿⢀⣼⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⣂⠀⢛⣥⡶⠊⠁⢸⣿⣿⣿⣿⣿⣿⡂⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡀⣿⣿⣟⢹⡗
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠈⣿⣿⣿⡿⣇⣰⠞⢃⠀⠠⢳⣾⣿⣿⣿⣿⣿⡛⠿⡦⠄⠀⠀⠀⠀⠀⠀⠀⠀⢴⠛⢠⣼⣿⣿⢋⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠛⣿⣣⠆⡡⠔⣁⠄⠾⢿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠋⣠⢾⣿⡡⢷⠂⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢠⠛⣻⠋⠀⠀⠀⣻⣧⠴⢊⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⢀⠀⠃⠀⠀⣠⣇⣼⣳⣾⠵⢁⠘⢀⠗
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠀⠀⢘⡷⣟⣡⠀⠀⣄⠉⠈⠶⣼⣷⣿⣿⣿⣯⣷⠀⠀⠀⠀⠀⠀⢠⣀⣴⣶⣟⣽⣿⡿⣫⠐⡁⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⢀⡀⠀⠐⣰⣥⡜⢀⣯⠄⠀⣶⡼⣿⣿⣿⣷⣿⣿⡀⠀⠀⠀⠠⠰⣞⠶⠜⠛⢿⣿⢟⡼⢋⠀⠀⠂⠀⠀⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠈⠹⠁⠠⡆⠀⠀⠀⠈⣿⣾⣿⣿⣿⣿⣞⢶⡀⢀⡀⠀⠲⠀⠀⢷⠈⠿⢣⠈⠀⠀⠆⠀⢀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⢀⠀⡀⠀⠀⢉⠠⢠⠄⢀⣽⣿⣿⣿⣿⣿⣿⣿⣦⣷⣸⣧⢀⣾⣿⡃⠀⠀⢰⡄⠀⠠⠀⠋⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠈⠀⢀⡵⠀⠀⠤⠤⠈⠀⡘⢲⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠂⢀⠈⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠐⠚⠀⠀⠀⢀⠀⣤⡤⠶⠒⠀⠰⡦⠉⠀⠁⢀⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⡇⠐⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣀⠀⠄⠒⠐⡎⠂⠀⢀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁⠀⠀⠀⠀⢠⠀⠙⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠾⠂⠉⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⣿⣿⣿⣿⣿⡟⣿⡟⠉⠟⢿⣿⣿⣿⠀⠀⠀⠄⠀⠀⠀⢀⣤⡀⠀⠀⣀
⠀⠀⠀⠀⣀⠀⢂⠀⡄⠀⠀⠀⠐⠀⠀⢀⠀⢰⣰⣦⣵⣿⣿⣿⡿⣻⣥⣿⣿⣷⠀⠘⠡⠆⣈⣿⣿⣿⡄⠊⠀⠀⠀⠀⣼⡟⠉⠁⠀⠀⠁
⠀⠀⠀⢀⣒⣿⣶⣤⢁⢀⠀⠀⠀⣀⣀⣤⣴⣿⠿⢋⢛⣉⣿⣿⣿⣿⣿⡟⠋⢭⠤⠚⠁⠀⠈⠠⢿⣿⣀⡄⠀⢀⣦⡾⢋⠞⠀⠀⠄
⠀⠀⠀⠈⣻⣿⣿⣿⣯⣿⣷⣶⣆⣤⣯⣤⣾⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠒⠲⠀⠀⠀⢴⣾⠿⠿⠄⣰⣾⠋⡅⠀⠀⠀⡀
⠀⠀⠀⢺⡿⠛⠛⢿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣻⠁⠘⣻⣥⣿⣿⡿⠀⠀⠈⠉⠀⠀⠀⠀⠀⢠⣾⣷⢶⣾⣿⠏⠈⠀⠀⠀⡘⠁
⠀⠀⢀⣼⣿⣶⣿⣶⣦⣀⢀⣁⡀⢸⣿⠟⣠⣼⣣⣴⣿⣿⣿⣿⣿⡿⠒⠀⠄⠀⠀⠀⢀⡄⠀⣸⣯⣇⣛⣷⠃⠂⠴⠀⠀⠆
⠀⠀⠀⠀⠟⠛⢉⣿⣿⣿⣶⣉⣋⣠⣷⣌⣩⣾⣿⣿⣿⣿⣿⢿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣸⣟⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠐⠿⠛⠹⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠛⠧⠀⠀⠈⠓⠂⠀⠀⠀⠀⠀⢀⢀⣽⠛⠛⢠⡞⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⣿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠿⠏⠀⠀⠀⠀⠈⠀⠀⠁⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⣿⣿⣿⡏⠁⠀⠀⠀⠀⠲⣤⡀⠀⠀⠀⠀⠀⡠⠀⠈⠛⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣆⠀⠀⠀⠠⢀⠀⠀⠀⠃⠐⠒⠆⠸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠐⠁⠄⠀⠀⠀⠀⠘⠏⠘⠀⠁⠀⠀⠀⠀⠀⠈⠒⠘⣦⡀⠀⠂⠀⠉⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡆⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⢀⣠⠤⢖⣲⠶⣶⣶⣤⣄⡀
⠀⠀⢀⣤⠾⢋⡵⢞⣭⢖⡯⢗⣻⠿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤
⢀⣴⠟⢁⣾⠏⣴⠟⣱⢏⡴⢫⢞⡭⠟⠻⢿⣿⣿⣿⣾⣿⣶⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⡿⠃⣴⡿⢁⣾⠏⣼⠇⣾⢡⡏⡾⠁
⢀⣼⡿⢁⣾⡏⢸⡟⢸⡏⢸⠁⡇
⣾⡿⠁⣼⣿⠀⣿⡇⢸⡇⢸⡆⣷
⣿⠃⢠⣿⡇⢠⣿⡇⢸⣇⢸⣧⠸⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣿⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣿⡛⠻⣿⣿⣿⣿⡿⠀⠹⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠏⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣏⡀⠀⠀⢤⡤⠀⠀⠐⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⡟⠉⢀⣀⠀⠀⠀⢠⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣽⣿⣶⣄⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⡀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣠⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣄⡀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢷⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣶⠀⠀⠀⠈⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⣀⣾⢿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡤⠟⢿⣶⣿⣿⡏⣿⢺⣿⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣼⣿⣿⣿⡛⣿⢸⣿⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣼⣿⣿⣿⠁⣿⠼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠠⠀⣿⣿⣿⣿⠀⡿⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⠿⣿⡿⠿⠿⠛⠛⠻⠿⠟⢿⠀⠀⠉⠉⠉⠙⠀⠀⠈⠁

⠀⠀⢀⣠⣤⠴⠶⠶⠖⠒⠒⠒⠒⠶⠶⠶⢤⣤⣀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠶⠖⠒⠒⠒⠒⠲⠶⠶⠤⣤⣀
⠀⠀⠻⣷⣶⣶⣶⡦⢤⣤⣤⣄⣀⣀⣀⣤⣤⣴⡿⠀⠀⠀⠀⠸⣿⣶⣶⣦⡦⢤⣤⣤⣤⣀⣀⣀⣠⣤⣤⣿⠇
⠀⠀⠀⠀⠉⠉⠉⠛⠚⠛⠛⠛⠛⠛⠋⠉⠉⡅⡯⠤⢤⡄⠀⠀⠀⠈⠉⠉⠛⠒⠓⠚⠛⠛⠛⠋⠉⠉⣥⡨⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡇⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣸⣿⣇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠰⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣲⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠄⠠⠀⠀⠀⠀⡠⠐⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠤⠀⠀⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠤⣀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠤⣀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠙⠊⠀⠀⠀⠒⠲⣄⠀⠀⠀⠀⠀⠀⡀⠀⠓⠠⠀⠨⠄⠂
⠀⠀⠀⠀⠑⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠠
⠀⠀⠀⠀⡠⠒⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⡀⠤⠀⠀⠈⠐⠀⠀⠈⠉⠉⠀⣀⡀
⠀⠀⠀⠀⠂⠀⠁⠉⠉⠀⡀⠠⠔⣈⡀⠀⡀⠀⠠⠄⢂⢠⠀⠄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠨⡠⠤⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⢾⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠄⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠨⠤⢄⣤⣬⣥⣤⠄⠀⠀⠀⠀⢀⡀⠀⠀⠠⢔⣽⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠄⣀⠀⠉⠐⠒⠀⠄⠠⠤⠀⠀⠀⠀⠤⠄⠐⠷⢤⣤⣀⣀⣀⣀⣐⣐⣨⡧⢀⠠⣊⠔⠒⠉⠉⠉⠙⢌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠒⠒⠠⠄⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢁⠀⡀⠀⡴⠃⢀⠠⠀⠀⠄⠠⢌⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠄⡀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢰⢧⡔⡥⠚⠉⠉⠉⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣈⠀⠀⠀⠀⠠⠀⠐⡀⠔⡯⠥⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣟⠒⠤⠤⣤⠭⢤⣴⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣌⡀⠉⠉⠙⠛⠛⠛⠛⠛⢑⣂⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠲⠶⠖⠒⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠀⠀⠀⠄⠠⠈⠠⠀⠀⠠⠂⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢀⠀⣀⣀⢀⠀⡀⠀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠰⡄⠇⡰⢄⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⡿⠁⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⠊⠀⠀⠀⠀⠄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠂⠀⠀⠀⠀⠂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠠⡀⡆⢠⠄⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠈⠀⠘⡶⠁⠀⠀⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠇⡠⠀⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠰⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢠⡀⡇⢠⡀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠠⠈⢰⠂⠐⠂⠁⠬⠐⠀⠁⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠈⠂⠀⠱⣶⠃⠈⠂⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠂⠀⠀⡨⠠⠞⠀⠆⠇⠀⢀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠁⠦⠀⠅⠀⠔⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠠⠈⡌⡔⠀⡤⠤⠤⢤⣀⠀⠀⠈⠀⢳⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣃⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡫⠭⠭⣙⣒⠀⠸⣿⣇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡪⠭⢝⣒⣂⠀⠰⠿⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣊⣩⣍⣙⣓⣒⠦⠥⣄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣒⣛⣛⣒⣒⣒⠭⠭⣓⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣒⣒⣒⣒⣒⡦⠭⢭⣒⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠈⠉⠙⠒⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣁⣉⣀⣉⣉⣉⣉⣓⣒⣤⣼⣤⣤⣐⣀⣀⣀⣀⣀⣀⣀⣈⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⡀⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢊⠭⠙⠋⠛⠚⠳⠶⢦⣤⣄⣀⣀⠀⠀⢙⣿⣶⡴⣚⠩⠐⠂⠈⠉⠀⠉⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⡡⠊⠁⠀⠀⢀⠤⠀⠠⢄⠀⠀⠉⠉⠙⠛⠛⠃⢚⡙⢷⣿⣷⣄⠀⠀⠀⠀⢀⣀⡀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡰⠋⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠛⢿⡿⣷⣀⡴⢛⣩⣭⣿⣿⣦⣄⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⣰⢃⠎⠀⢀⠔⠂⠒⠒⢄⠑⢤⣀⣠⡴⠏⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⢮⢻⣽⣿⢁⣾⣿⣿⣿⣿⣿⣿⠂⠀⠙⡄
⠀⠀⠀⠀⠀⠀⢰⢁⠃⠀⠀⢊⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣿⠘⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⠃⠄⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠆⡈⠀⠀⠀⠈⠳⠤⣤⠴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣘⣦⢠⡘⣜⢧⡀⢀⡜⠉⠻⠟⣛⣛⡉⠕⠊⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⢠⢀⠀⠀⠀⠀⢀⠤⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⣭⣽⣿⣿⣿⡖⠛⡏⠁⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠠⠃⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠠⣏⢰⣿⣿⣿⣿⣿⣿⡟⡀⠘⡀⠀⠘⢄⠀⢀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠱⢄⣀⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⡠⠁⠀⠱⠀⠀⠈⣮⡴⠖⠒⠀⠈⠳⣄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠛⠛⠭⠚⠁⠀⠀⠀⢳⣴⣾⣿⣆⠀⠀⠀⠀⠀⢹⡆⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢀⠄⣒⣲⡒⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⠀⠀⠀⢠⢎⠃⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠰⢡⡟⠉⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⢀⣀⣠⣀⡀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⣔⠪⠕⠁⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⢃⠘⠀⠀⠀⠀⠀⠀⢀⠽⠀⠀⠀⢠⡞⢉⣴⣶⣿⣿⣷⣄⠀⠀⠀⠈⢿⣿⣿⣿⣷⡄⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠓⣄⣀⠀⣀⢀⠴⡡⠃⠀⠀⠀⣿⣠⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠈⣿⣿⣿⣿⣿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⢡⠃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠯⠭⠅⠒⠁⠀⠀⠀⢀⡠⠖⠙⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠔⠊⠁⠀⠀⠀⠘⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠽⠥⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⡶⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣶⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠘⠛⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⢶⠄⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠋⢙⣿⣿⣿⣯⣬⣿⣿⣿⡿⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣶⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⣶⠀⠀⣶⢀⡴⠖⢶⡄⣶⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣼⣿⠀⣿⠓⠒⣿⢸⡇⠀⢀⡀⣿⠀⣿⣧⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⢿⣿⠀⠛⠀⠀⠛⠀⠛⠶⠞⠁⠛⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣯⣬⣿⣿⣿⡏⠙⣿⣿⣿⣶⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣠⡤⠤⠤⣤⡀⠀⠀⠀⠀⠀⣀⢴⢄⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⠀⢸
⠀⠀⠀⡠⠊⢇⡜⠉⢢⡜⠑⢄⠀⠀⠀⡾⢤⠗⠒⠒⣥⠼⡆⠀⠀⣰⣊⢁⠼⢄⢉⣢⠀⠀⠀⢀⡞⢅⡠⠷⢄⡨⢳⡀⠀⠀⢸
⠀⠀⠈⠫⡉⠹⡉⠉⢩⠏⢉⠝⠁⠀⠀⡇⢸⠀⠀⠀⢸⠀⡇⠀⠀⡇⠀⡇⠀⠀⡇⢀⠀⠀⠀⣼⢠⡏⠀⠀⠀⢱⡄⣧⠀⠀⢸
⠀⠀⠀⠀⠙⢦⣣⠀⣎⡴⠁⠀⠀⠀⠀⣇⣸⡀⠀⠀⣼⣀⡇⠀⠀⢧⠤⠣⣀⡠⠣⢤⠀⠀⠀⠸⡀⡱⢄⣀⡠⢎⢠⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠙⠿⠋⠀⠀⠀⠀⠀⠀⠙⢤⣏⣉⣉⣆⠜⠁⠀⠀⠀⠑⠦⣸⡠⠖⠁⠀⠀⠀⠀⠙⠲⠤⣯⠤⠔⠋⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠛⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠙⠋⠲⡄
⠀⠀⠀⠀⢀⡠⠊⠓⢄⠀⠀⠀⠀⠀⠀⠀⠞⢁⠀⠊⠀⢀⠙⠆⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⢰⠉⢀⠀⠀⡀⠘⡆
⠀⠀⠀⠀⡜⣀⠐⠃⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⡁⡆⠑⡈⣥⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡀⢸
⠀⠀⠀⠀⠳⣀⣘⣃⠰⠃⠀⠀⠀⠀⠀⠸⠀⡠⠀⠀⠀⠤⡀⠇⠀⠀⠀⠀⠀⠀⠈⢦⡁⣊⡠⠃⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠐⠀⣀⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⠐⠂⠑⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠔⠁
⠀⠀⠀⠀⢀⡀⠀⠤⣤⡤⠤⢤⡀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠤⣄
⠀⠀⠀⢠⠎⠐⣀⠔⠀⠐⣀⠀⠘⣄⠀⠀⠀⠀⡰⠁⠱⡀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈⠠⠈⠀⢳⠀⠀⠀⠀⠀⠀⡴⢁⠐⠄⠀⠠⢈⢷
⠀⠀⠀⠈⢆⠀⠘⠀⠀⠀⠁⠀⡠⠃⠀⠀⠀⢰⠁⢀⠀⢱⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⠁⢸
⠀⠀⠀⠀⠈⢣⠀⠀⠀⠐⠀⡔⠁⠀⠀⠀⢠⠋⠖⠈⠐⠁⢣⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰⡀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠱⡀⠀⢠⠎⠀⠀⠀⠀⠀⢸⠀⢂⡀⡀⠂⢸⠀⠀⠀⠀⠀⠀⠙⠢⡈⠀⠌⡠⠋⠀⠀⠀⠀⠀⠀⣧⠀⠃⠄⠀⠄⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠘⠾⠁⠀⠀⠀⠀⠀⠀⠀⠑⢌⣀⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡄⠤⠤⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢀⠶⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠀⠈⠑⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⡈⣧⠀⠀⠀⠀⠀⢠⠖⠒⠉⠁⠉⠇
⠀⠀⠀⠀⡬⢠⠰⠌⡇⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠠⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠁⡆⠀⠀⠀⠀⠀⡌⠀⠖
⠀⠀⠀⠀⢡⠀⠀⠀⡼⢚⢍⢻⠀⠀⠀⡇⠀⠄⠆⠀⠀⠈⠁⠀⣃⠀⠀⠀⠀⠀⢀⠎⠀⠁⢰⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠉⣠⠖
⠀⠀⠀⠀⠀⡇⢀⢼⢁⡤⢊⠎⠀⠀⠀⠣⡀⠀⠀⠆⠀⠘⠀⣠⠋⠀⠀⠀⠀⠀⠘⡓⠂⢠⠞⠀⠀⠀⠀⠀⢸⠈⢀⡠⠔⠉
⠀⠀⠀⠀⠀⠈⠚⠚⠚⠒⠋⠀⠀⠀⠀⠀⠘⢤⣀⣀⣀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠳⠚⠁⠀⠀⠀⠀⠀⠀⠿⠚⠁

⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢰⡄⠀⢀⠀⣾⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣵⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡏⠈⣅⠀⠧⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⢿⣿
⠀⠀⠀⠀⠀⠸⠁⠈⠋⣠⠀⠀⠛⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣷⣻⣏⠀⣿
⠀⠀⠀⠀⠀⢠⡀⠀⣶⣿⣶⡀⠀⢡⡟⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⣿⢫⡟⣼⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⠀⣸⣿⡟⢳⡆⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣭⡝⣟⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⠰⠟⣿⡇⠘⣿⡿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣽⣷⣿⣿⣿⣯⣿⣇⣿⣿⢺⣿⣿
⠀⠀⠀⠀⠀⢸⢁⣼⣿⣿⠅⣦⡿⢿⡿⣹⣿⡏⣠⣿⣾⣿⣿⠟⢋⣿⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣟⢹⡿⢲⣽⠃⣸⣿⠁⢻⣿⣯⣿⣷⣟⣿⣃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠸⢿⠛⢼⣷⣺⣧⣴⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠸⢸⡠⡿⣟⣿⣿⢿⡟⣽⣽⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠠⠾⠿⠿⠿⠿⠿⠾⠷⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣻⣿⣿⣿⣿⣽⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣏⣿⣟⣻⣿⠿⠛⠋⠀⠀⠀⢀⣴⠶⢤⣤⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣠⣤⣴⣦⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⡿⣿⣿⣿⣽⣿⣦⣄⣀⣸⣯⣾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⣿⣌⡚⠉⠻⣿⣿⣿⣿⣯⣿⡇⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣟⣿⣶⣿⣿⣿⣿⣿⣿⣿⣃⠀⠀⠀⠉⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣿⣿⣿⡋⠙⢻⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣼⣿⣿⣿⣷⠀⠀⠁⢻⣿⣿⡇⠠⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⣿⣿⡿⠁⠀⠀⣀⣤⣿⣿⡇⠀⠈⠛⠉⣿⠀⠠⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣧⠀⠀⢸⣿⣿⣿⠿⠓⠀⠀⠀⠀⠀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣄⠀⠀⠈⠁⠀⠀⣀⣡⣄⠀⠀⣤⣴⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣷⣶⣾⣿⣿⣷⣦⣤⠖⠀⠀⠀⠉⠋⠀⢀⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡿⠿⣿⡿⠿⣿⣿⣿⣿⣄⢢⣛⠛⠓⠀⣀⣀⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠻⠿⠛⠉⠋⠙⠓⢓⣾⠿⣿⡿⠿⠛⠋⠛

⠑⢤⣀⠀⠤⠐⠀⠀⠀⠃⠀⠀⡠⠰⠂⠀⠂⠀⠀⠀⠀⢠⣟⣷⡿⣃⠀⠂⠀⡀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠂⡀⠀⠀⡇
⣿⠏⣉⡓⠤⣀⡐⢠⠬⠆⡈⠣⡀⠀⢋⠀⠤⠄⡀⠀⣴⣿⣯⡫⠚⠁⠀⠂⠆⠀⢁⠀⠀⢈⡡⠆⡘⠣⣀⣐⠀⠠⠀⠐⠀⡇
⣿⠀⢹⣿⣷⣤⣉⠢⣄⣂⡬⣤⣟⠣⠀⡀⢒⢀⢄⣞⣿⡿⠋⠀⠀⣠⣔⣩⣶⣊⠡⣤⣞⣓⢦⡰⢜⣂⢉⣣⣄⢃⡀⠀⠂⡇
⡼⠄⣀⣿⣐⠸⡿⣿⣷⣭⣕⣪⢾⣾⠿⢷⣄⣵⣿⣿⣿⢭⣉⣳⣶⣶⣭⣽⢘⣻⣾⣶⣯⣻⢘⣿⡿⣿⣿⣿⣿⣮⣶⢘⠂⡇
⡇⠀⠈⡟⠉⠉⣿⠛⡞⡿⡿⣿⣿⣿⣿⢺⣿⣷⣿⣿⢷⡾⢻⣿⡟⢸⢻⡇⢸⣿⣿⣿⠿⣿⠈⠈⡏⡿⣧⢇⣸⠀⠏
⡇⠀⠀⡏⠀⠀⠹⠀⠁⠀⡇⣗⣸⢘⣿⠀⣉⡋⣋⣿⣸⣀⣈⡉⠃⢸⢸⡇⢸⣿⣇⣟⣀⣹⠀⠀⡇⡇⣿⡿⣸⠀⣀⢉⡏⠁
⡇⠀⠀⠧⠀⠀⠐⠂⠓⠒⠋⠛⠛⠻⠟⠉⠉⠉⠉⠉⠍⠩⠍⠉⠙⠛⠉⠉⠩⠉⠉⠉⠉⠻⠶⠐⠓⠛⠛⠛⠋⠉⠛⠛⠛⠃
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒

⣹⣦
⠈⢳⡀⠉⠲⣄⡀
⠀⢸⡇⣄⡀⠼⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠶⠦⢤⣄⣀
⢀⡏⠸⠀⠁⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠋⠉⠀⠀⠀⠀⣀⣠⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀
⢸⠁⠀⠀⠀⠀⣸⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⢀⣀⣀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠉⡧⠀⠀⠀⠀⢀⡤⠤⢤⡄⠀⠀⠙
⠈⣆⠀⠀⠀⡼⠁⠀⠹⣄⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⡴⠛⠉⠉⠉⢻⡆⠀⠀⠀⠱⢤⣤⣤⠤⠞⠁⠀⠀⠀⡔⠁⠀⠀⠀⡿
⠀⠈⠳⢤⣴⠃⠀⢀⡴⠟⣦⡶⠛⠁⠀⠀⠀⠀⠀⠸⢅⣀⣀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⣀⣀⡤⠞⠁⠀⠀⢠
⠀⠀⠀⠀⠀⢇⢠⠏⠀⠀⠈⢧⣠⠴⠶⠶⣦⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠈⢷⡀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⢠⡿⠳⡀⠀⠀⠀⠀⢳⡔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠲⡆⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠈⣇⠀⠙⣆⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⢀⡴⠃⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠹⢦⣀⠈⢳⡀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠶⠋⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡙⣆⠀⠀⠀⠈⢧⣠⠤⠔⠒⠒⠒⠲⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠳⡀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⡽⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⢀⡟⢦⡀⠀⠘⣧⡀⠀⢀⣠⡴⠋⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠘⢦⣀⣉⣹⣦⣼⡷⠞⠋⠁⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣄⣀⣀⣀⣠⣤⣤⣤⡴⠶⣾⠛⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠘⢷⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⡀⠀⣠⣤⣤⣄⡀⠙⠏⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⣼⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⢿⣿⣿⣿⣿⣿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⢀⣸⣧⡀⠙⠛⠛⠛⠁⠈⠋⠀⣀⠀⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠈⢻⡟⠉⠀⠀⠀⠰⠶⠀⠀⠀⠙⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⠇⠀⠃⠀⠈⠁⢀⣤⣶⣷⣦⣄⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⣠⡀⠀⠀⠉⠉⠉⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⡀⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠈⠋⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠈⠁⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠟⠉⠁⠉⠀⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠉⠉⢻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⣤⠀⠈⠋⠀⢰⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠰⣿⣿⠆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣟⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣯⡿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣦⡀⣀⣤⣤⣄⣤⠤⠀⠀⠀⣀⠄⠘⠛⠉⠹⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⡾⠟⠋⠀⠀⠀⠀⠀⠈⠛⠟⠀⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣛⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠻⢿⠛⡿⠿⣿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢤⡀⠋⠶⠴⠴⣄⡄⠀⠀⠧⢀⠀⡀⠑⢖⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠒⠂⢠⢀⣤⠀⠀⣰⠀⠘⣧⡄⠐⠛⠁⠛⠄⠀⣅⠂⠬⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠠⡔⡖⢻⢀⠠⠒⣄⢼⠝⠏⠋⠀⠀⠶⠋⠑⠒⠚⠷⠂⢀⣀⠀⠂⠰⣌⣆⠀⡀⠈⢹⣄⡄
⠀⠀⠀⠀⠀⠀⢠⠹⣦⣼⣧⣶⡿⠟⠉⠀⢓⡳⠂⠡⣄⠀⠀⠀⠀⠘⣧⠀⠀⢉⡧⢀⠀⢹⡦⣀⠀⠈⢿⡇⠀⡀
⠀⠀⠀⡀⢸⠀⠎⢀⣸⣿⢳⣏⣤⠤⠚⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠲⢇⣆⣷⢀⠁⠘⡧⢸⣦⡀⠎⢷⡴⢻⡆
⠀⠀⠀⣧⠀⡆⠀⡸⢋⡿⣾⣉⣆⢠⢠⡠⣴⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⢿⠳⠷⣧⡄⣣⡑⡄⢼⣷⣾⠁
⠀⠀⢀⠸⡔⡉⢶⡟⣹⣱⡟⠓⠁⠁⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠵⠂⠳⣄⣏⠘⠻⢷⣿⣀⣿⡃⣨
⠀⢳⠈⡦⢻⣦⡟⣿⣧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⢸⣿⡝⣿⡷⡴⠂
⠀⠈⢷⡄⠈⢿⣾⢃⣾⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣌⠳⣿⡇⢻⣇⢽⠄
⣄⠀⣸⢿⣆⣼⣟⣱⣿⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡑⢻⡷⣼⣿⣷
⠘⣧⡽⣎⢿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢟⣸⣿⣼⣿⠏
⠀⠘⣧⣿⢸⣿⣿⢀⣄⠀⢠⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⢹⣿⣿⡏
⠀⢀⢹⣾⢾⣿⣿⣨⠋⠡⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣀⠀⡓⢾⣿⣿⢳
⠐⢾⡤⣿⣤⣿⣿⡄⠀⢸⣿⣫⡀⠀⢠⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣟⡻⡟⣾⣿⣿⢸⠇
⠀⠸⣿⢏⢿⣿⣿⣧⠀⢒⢸⣟⡆⣠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣿⡇⢈⣿⣿⡇⢸⠀⠠
⠀⠀⢹⡥⡂⢻⣿⣿⣧⠀⢖⣿⢩⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣛⣿⠥⣼⣿⣿⣇⡏⠀⠏
⠀⠀⠀⠘⢷⡈⣿⣿⣿⣷⡨⣯⣾⣿⢘⣠⣀⡆⠀⠀⠀⠀⠀⠀⠀⢶⣷⣆⡀⠀⠈⢀⣤⡸⣿⣏⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣾⣿⣛⠛⣿⣻⣿⣶⣄⢀⠀⢀⠀⠀⠘⢝⢯⣇⢆⣇⡞⣿⠩⢿⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠛⠻⣿⣷⣾⠼⠿⠿⠿⠯⣄⣠⣤⡄⢠⠴⠓⢉⣩⣍⠙⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠓⠦⠤⣼⣿⣥⠤⠄⣨⡤⠐⠀⠀⢈⠗⣅⠀⣲⣊⣹⣥⣾⣿⣿⣿⣿⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡏⠉⢻⣿⣟⣩⣹⣷⣶⣊⣠⠶⢻⣯⠟⠉⠀⠀⢤⣤⠼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠙⠿⠏⠀⠊⠁⠐⢤⠞⠂⠀⠐⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⡤⣿⢻⡏⠻⠃⠰
⠀⠀⠀⠀⠀⢀⡀⣲⡄⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⡿⣶
⠀⠀⢀⡀⠀⣾⣇⣿⠙⣶⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠤⡒⠸⡝⣿⣥⣷⢾⠉⠉
⠀⠀⠀⣙⡗⠿⣿⣿⣶⣿⣄⠀⢀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⡄⠒⢹⢹⠇⠀⠈⠀⠀⠀⠀⣿⠣⡇⠰⡆
⠀⠀⠀⠙⣟⣿⣿⣿⣿⣭⡓⠠⠈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⠀⣖⡲⠀⠽⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠴⡷⠀⡇
⠀⠀⢠⣼⣷⣿⣿⣿⠅⠈⠁⠀⠀⠉⠁⣀⣠⡄⢠⡄⡆⠀⠿⠿⠤⠓⠋⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⡷⢠⠇
⠀⠀⠀⠈⠋⢽⣿⣿⣂⡀⢠⣤⠠⡎⠃⠘⠘⠇⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠈⣿⠉⡆
⠛⠋⠀⠀⠁⣹⣿⣿⡿⠓⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠄⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣀⣿⠀⡟
⠀⠀⠀⣤⣤⣿⣿⣿⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠈⣿⠚⢶
⠀⠐⠨⠀⠋⣿⣿⣿⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢀⣿⠀⢸
⠀⠀⠀⢸⣠⣿⡏⡿⠀⠀⠀⠀⠀⠀⠠⡿⠀⠀⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠈⣿⠱⣿
⠀⠀⠀⠘⢻⢿⣿⡇⠀⠲⠶⠀⠠⠀⠀⠁⠀⠀⡟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢀⣿⣀⣼⡤⠆
⠀⠀⠀⠀⣼⣿⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⠀⣯⠿⡿⢉⠿⡟
⠀⠀⠀⣶⣿⣿⣭⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢰⣇⠀⣾⠷⠶⡷
⠀⠀⠀⠀⢸⣿⣉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠿⠖⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠺⠿⠛⠁⠠⡟⢠⣾⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢠⣴⠒⠀⢀⣶⢀⣤⡀
⠀⣀⡀⣀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠂⠀⠈⠀⠀⠶⠁⠰⠀⠀⠤⢺⡁⠀⠁⠈
⠀⠀⠹⠿⡺⢭⡉⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⢰⠀⢰⠀⠁⠈⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀⣰⣤⠔⣴⣶⣧⣾⡷⠲⠄
⠀⠀⠀⠀⠛⠳⠌⢉⡷⠦⣀⣀⠀⠀⠀⠀⠀⠀⠸⠀⢸⡀⠠⠛⠻⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠋⠁⠈⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠈⠀⠀⠀⢄⣀⣀⠀⠼⠂⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣶⡄
⠀⠀⠀⠀⠀⡀⠀⠱⡆⠀⠀⢀⡀⠐⢦⡀⠉⠀⠄⢀⣄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠁
⠀⠀⠠⠄⠀⠀⠀⡁⠁⠀⠀⠁⠤⡄⠀⠈⠀⠀⠑⠀⠁⠀⠒⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠙⠀⠀⠀⢲⣶⠆⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸⠟⠀⠀⠀⢰⠀⡌
⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠠⡴⢶⠯⡟⠀⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠆⠀⠀⢀⠀⠈⠀⠀⠀⢀⣿⡴
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠋⠀⠈⠁⠀⢸⠑⠇⣤⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠈⠀⠀⠀⢀⠀⢸⠉⡇⠀⢀
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠀⠀⠀⠀⠃⠐⠀⡇⠖⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠋⠰⠀⠀⠀⠀⢠⠄⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⢀⠴⠀⠠⡇⠀⠐⠀⠀⠒⠖⠐⣷⠚⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠒⠁
⠀⠀⠀⠀⠀⠀⣸⡎⠁⠀⠀⣀⡀⠀⠀⠀⣀⠀⠂⠀⠻⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⢦⣤⣤⡤⠶⠞⠛⠋
⠀⠀⠀⠀⠀⠂⠉⡀⠀⠀⠀⠈⡅⠀⠀⢠⡄⡈⠀⠀⣾⠡⠐⠛⠙⠈⠒⠄
⠀⢀⠀⠈⠠⢠⠆⠰⠆⠀⠐⢰⠇⠀⡘⠼⠘⠛⠉⠙⠚

⠀⠀⢠⠀⠄⣀⣤⣄⣴⣦⣄⠐⠀⠀⠀⠀⡤⠲⢄⣠⣤⣲⣶⣤⡀⠀⠀⠀⠀⢠⠄⠦⣀⣤⣴⣶⣦⣄⠐
⠀⠀⠘⡄⣀⠞⢋⣿⣿⣿⣿⣦⠠⡀⠀⠀⢥⢀⡴⢙⣹⣿⣿⣿⣷⡄⣄⠀⠀⠈⡅⢀⠞⣋⣽⣿⣿⣿⣧⡀⡀
⠀⠀⠀⠹⣠⡟⣹⣿⡟⣿⣿⣿⣿⣧⠀⠀⠈⢇⣾⢋⣿⣿⢻⣿⣿⣿⣿⡄⠀⠀⠹⣤⡟⣹⣿⡟⢿⣿⣿⣿⣯
⠀⠀⠀⣀⣽⣧⣿⣾⣧⣼⣿⣟⣹⡃⠀⠀⢀⣨⣿⣼⡿⣿⣤⣿⣿⣋⣟⠀⠀⠀⣀⣬⣧⣿⢿⣧⣼⣿⣿⣹⡇
⣠⣷⣾⠁⣿⣿⣿⣟⣙⣷⣟⣩⣃⢀⣼⣶⡏⣼⣿⣽⣿⣋⣻⣾⣋⣝⡄⢠⣽⣶⠁⣿⣿⣿⣟⣙⣷⣟⣩⣫
⢿⣽⣿⡀⢿⢾⣿⣿⣿⣿⣿⠟⠃⠸⣯⣿⣇⠻⡷⣿⣿⣿⣿⣿⡿⠛⠀⢺⣽⣿⡂⢫⢿⣿⣿⣿⣿⣿⠗⠃
⠸⣿⣿⣷⣞⣈⣻⣹⣿⣿⠋⠀⠀⠀⢿⣾⣿⣶⣃⣙⣩⣿⣿⡟⠁⠀⠀⠘⣿⣻⣷⣾⣈⣻⣹⣿⣿⠏
⠀⠈⠻⢽⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠈⠻⢿⣿⣻⣯⠿⠟⠁
⠀⠀⠀⠀⠀⠀⣤⣶⣿⣯⣿⣷⣦⡀⠀⠀⠀⠀⠀⢀⡴⣾⣿⣽⣿⣶⣄⡀⠀⠀⠀⠀⠀⣠⢶⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⣠⣿⣿⣟⣯⡉⡽⢿⣿⢿⡄⠀⠀⠀⣴⣿⣿⣋⣏⣉⡿⣿⡿⣷⡀⠀⠀⢠⣾⣿⣟⣹⣍⡹⢿⣿⢿
⠀⠀⢠⣴⣿⣿⣿⣿⣿⡷⣷⠈⣿⣟⣷⠀⣤⣼⣻⣿⣿⣿⣿⣿⣣⢹⣿⣿⡇⢠⣤⣿⣿⣿⣿⣿⡿⣿⡌⣿⣟
⠀⠀⢫⣋⣽⢿⣍⣽⣿⣻⣿⢀⡿⢿⠋⠀⣍⣩⡿⣯⣍⣽⣟⣿⣿⣸⠿⡟⠁⣨⣋⣼⢿⣭⣩⣿⣿⣿⣇⡿⢿
⠀⢨⣏⣽⣿⡟⢻⡿⣿⢻⣟⠉⠀⠀⠀⢹⣉⣿⣿⡛⢿⣿⡿⢿⡛⠉⠀⠀⢈⣏⣽⣿⣟⠻⣷⣾⠿⣟⠋⠁
⠀⢻⣿⣿⣿⣿⣼⣿⣏⣼⠋⣆⠀⠀⠈⣿⣿⣿⣿⣧⣼⣿⣁⡾⠹⡄⠀⠀⢹⣿⣿⣿⣿⣼⣿⣏⡰⠏⢧
⠀⠈⠂⠻⣿⣿⣿⣿⣩⡰⠋⠘⡀⠀⠀⠙⠈⢿⣿⣿⣿⣏⣡⠞⠁⢱⠀⠀⠈⠃⠹⣿⣿⣿⣿⣩⡔⠋⠈⡂
⠀⠀⠀⠀⠙⠻⠟⠙⠛⠉⠲⠄⠃⠀⠀⠀⠀⠀⠙⠿⠋⠛⠛⠉⠲⠘⠀⠀⠀⠀⠀⠈⠻⠟⠝⠛⠉⠑⠂⠋

⠀⠀⡟⡉⠉⢉⠉⢉⢉⣉⠉⣉⡉⢉⣉⠉⢉⡉⢉⠉⡉⡉⠉⢉⡉⢉⣉⡉⡉⢉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡆
⠀⠀⡇⡷⢾⡏⢳⠋⣿⣉⣿⡷⣟⣿⣈⡷⣏⣉⣹⠶⣿⣇⣰⣏⣹⢾⠷⣟⡷⣏⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠁⠈⠀⠈⠀⠈⠉⠈⠀⠈⠈⡉⠀⣈⡁⡈⣀⡁⠉⠁⠈⠁⠈⠀⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢷⣸⡉⣉⣿⣏⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⣧⣀⣠⣀⣀⣀⣀⣀⣠⣀⣀⣁⣀⣁⣉⣁⣁⣉⣁⣠⣠⣀⣀⣄⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠠⠒⠐⠦⠀⠀⣀⣴⡆⢴⣤⣤⣴⡄⣤⣶⣦⡄⢀⣴⢶⣤⠀⠀⡇⠀⠀⠀⣠⣤⣤⣬⣭⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠛⣿⡇⠀⠀⣸⡟⠀⢻⣦⣾⠇⢸⣇⣀⣿⡇⠀⡇⠀⠀⢾⣿⢽⡝⢽⡝⢹⢹⢹⢙⢹⡝⣍⠭⢩⠻⡍⣹⣿⣷⠄⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢰⣿⠁⠰⣿⡉⣹⡷⢠⣍⣋⣿⠇⠀⡇⠀⠀⠀⠙⢶⣾⣶⣾⣾⣷⣿⣿⣷⣾⣷⣾⣾⣷⣿⣶⠟⠁⠀⠀⠀⡇
⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣙⣃⣀⣛⣁⣀⣀⣉⣛⣋⣁⣀⣙⣛⣁⣀⣀⡇⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠒⠆⠔⠐⠒⠶⠆⠀⠀⠀⠀⣠⣤⣤⡀⣤⣤⣤⣤⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⢿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⣼⠇⣿⣇⣀⣽⠇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣇⣜⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣁⡀⣿⡏⠉⣿⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣁⣉⣁⣀⣈⣁⣀⣀⣀⣀⣇⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⡏⠀⣤⣀⢠⡄⣄⣤⡄⣤⣄⠤⣠⢠⣀⡠⠤⣀⠀⡤⣄⣠⠀⠀⠈⡇⠤⠦⠤⠠⠬⡵⠶⠆⠶⠶⠴⠿⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⡇⠀⠉⠉⠈⠃⠉⠉⠁⠉⠉⠘⠉⠈⠈⠉⠂⠀⠀⠁⠉⠉⠙⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⡇⣴⢲⡔⢢⡖⡆⢰⣖⣦⡖⠳⣴⠀⢰⣴⠓⢶⣖⡂⢀⠤⣠⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠙⠊⠙⠋⠑⠃⠎⠃⠀⠑⠚⠙⠒⠚⠙⠚⠉⠓⠂⠈⠚⠙⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⢰⣒⢲⣶⣲⢶⣒⠂⠀⣖⣦⣖⣲⢴⡴⢒⡂⣴⡄⡖⢲⣴⣒⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠘⠀⠘⠋⠘⠚⠒⠂⠀⠓⠋⠃⠘⠛⠙⠚⠛⠉⠙⠛⠚⠙⠒⠂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠃

⠀⠀⠠⠂⠀⠉⠀⠂⠀⠀⠀⠠⠐⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐
⠀⠠⠁⡀⠁⠈⢢⠀⠂⠀⠀⢀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⠈⠀⠀⠊⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠜⠁⠀⠀⠀⠀⠀⢀⠁⠀⠀⡆
⠀⠀⠂⠀⠐⠒⠊⠀⠂⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠤⢀⠀⠀⠀⠀⢀⠈⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠠⣀⠄
⠀⠀⠀⠈⠀⡀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣀⠀⠀⠀⠀⢀⣀⡀⠁
⠀⠀⠀⠀⠀⠃⠈⠂⠀⠀⠀⠀⠀⠀⠀⡠⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢂⠉⡁⠈⠙⡀⠀⠀⡰⠁⠀⠀⠹⡄
⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠂⠀⠀⠉⠠⣀⠀⠃⠀⠀⢇⠀⠉⠀⠀⢶⠈⠈⠂
⠀⢀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⠁⠁⠲⠈⠉⠃⠀⠀⠀⠀⠀⠀⢡⠀⠠⠇⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⢠⠃⠀⢀⠈⢤⠀⠀⠀⠀⠀⠀⢰
⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠸⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⡄⠀⠀⠀⡀⢀⠤⡅
⠀⠀⠀⠀⠠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⡠⠊⠀⠀⠀⠀⠉⠀⠄⠀⠄⠈⠟⠀⠠⠂
⠀⠀⠀⠀⣀⣠⠄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠠⢀⠀⠀⠀⠀⠀⠀⢠
⣀⠔⠒⠉⠠⠁⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⡀⠃⠀⠆⠀⡀⠀⠀⢀⠎⠀⠄⠀⠈⠱⢦
⢰⠙⠢⣀⠃⠀⡐⠀⠀⢂⠀⠀⠀⠀⠀⠀⡀⠊⠀⠂⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠆⠄⠀⠀⣾⠀⠀⠘⠒⠒⠆⠀⣄⡀⠀⠀⠀⠀⠀⠂⡀⠀⠀⢀⡀
⠀⡆⠀⠀⠑⠲⠂⠒⢒⠴⠂⠀⠀⠀⠠⠌⠉⢢⠀⢀⠀⠀⠀⠀⠂⠐⠀⠀⠀⣀⠀⠄⠀⠀⠘⡄⠈⠢⠀⠀⠠⠰⠃⡠⠀⠀⠀⠀⠀⠀⠈⠲⠄⠀⠡
⠀⢃⡘⠀⠀⣀⠠⠂⠁⠀⠀⠀⠀⠀⠘⠆⠀⠀⡁⠀⠀⠀⠀⠀⠑⠀⠃⠐⠂⠁⠀⠀⠀⠀⠀⠘⠀⠀⠐⠀⠀⣰⡔⠁⠀⠀⠀⠀⠀⠁⠤⢀⠀⠀⣐
⠀⠀⠓⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠐⠒⠛⠂⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⢀⠴⠂⠒⠀⠐⠒⢠⡀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠁⠀⠐⢠⠀⠀⠀⠀⢀⡴⠁⢀
⢠⠃⠀⠘⠁⠀⠀⠓⠁⠐⠀⠀⠈⠁⠂⢤⠊⠀⡄⠉⠀⠀⠀⠀⠇⠀⠈⠴⠄⡰⢀⢀⡄⠀⠀⠀⡠⠊⠀⠀⢀⠀⠀⠘⠀⠀⠀⣠⠂⠀⠀⠀⠑⡀
⠅⠘⠀⠀⠀⠀⠀⢀⡄⠒⠁⢀⡀⠠⠀⢀⡀⠀⠁⠀⣀⡀⠀⡀⢤⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠔⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡿⠀⠀⠀⢀⠌⠁
⢃⣀⢡⠀⢀⠀⡠⠂⠀⡙⠀⠀⠀⠈⠛⠀⠀⠂⠀⠀⢀⠂⠀⢧⠊⠀⠀⠀⠀⠀⠀⠲⡀⠀⠘⡄⠂⠁⢀⠀⠀⢀⠀⠋⠀⠀⠀⡄⠀⠋⠀⠁⠀⡀
⠀⠑⠤⢀⢀⠈⢡⡠⠊⠀⠀⠀⠀⠀⠀⠀⢀⠆⠉⠁⠂⠀⠀⠀⠂⢀⢀⣀⠀⡁⢀⢀⠀⠀⣟⡀⠠⠀⠁⠠⠊⠁⠀⠀⠀⠀⠀⠈⢄⡀⠀⢀⠘⠁
⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⣀⠈⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠁⠀⡉
⠀⠀⠀⡠⠊⠈⠂⠀⠀⠀⠀⢸⠀⠀⠀⠰⠄⣠⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠁⠀⢩⠀⠀⠀⣴⠀⠉⠀⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⡄⠀⠀⠁⢸⠁⠀⠀⢈⡆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠑⡀⠂⠀⠔⠀⡌⠀⠀⠀⠇⠁⠄⣀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠂⠀⡠⡘⡅⠠⡀⠘⠃⠀⠀⠀⠁⠂⠀⠀⠀⠁⠀⠀⠀⠀⠜⠀⠃⠈⠢⠄⠀⢠⠁⠀⠀⢸⠀⠀⠀⠌⠈⠉⢀⠃
⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠄⠁⠈⢷⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠈⣁⡠⠝⠋⠀⠀⠀⠘⡀⠀⡐⠀⠀⠀⡐
⠀⠀⠀⠀⠠⠄⠁⠀⠀⠀⠀⠀⠈⠂⠀⠀⠤⠀⠠⠠⠼⠀⠀⠀⠀⠀⠢⠀⠠⠀⠀⠀⠀⠀⠷⠆⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠃⠈⠀⠶⠁
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⣀⢀⡖⢉⢉⡁⠀⠀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⣠⠤⣤⣌⠙⢳⡄
⠀⠀⠀⠀⢺⣤⣍⡼⠋⢋⣉⣛⢻⣶⣈⣲⣶⡆⠀⠀⠀⠀⠀⠀⣾⠀⠀⢰⠃⢠⠀⢈⡇⠀⢹⡀
⠀⠀⠀⠀⠘⢣⠇⢠⠀⠀⠙⠻⢿⣿⣿⡿⣿⠀⠀⠀⠀⠀⠀⢸⠇⣤⠀⠸⣴⢻⠑⣬⠃⢠⡄⣷
⠀⠀⠀⠀⠀⡇⠀⠏⠀⠀⠐⣄⠀⠈⠉⠁⠸⠀⠀⠀⠀⠀⠀⢸⠇⠁⢠⠀⢌⠉⢋⠁⢤⠈⠀⣯
⠀⠀⠀⠀⣠⣏⠈⠂⠀⠀⠀⠃⣀⠀⠀⠀⠶⠆⡘⠀⠀⠀⠀⢸⡀⠀⠘⠀⣶⣶⣶⡄⠘⠀⠀⣿
⠀⠀⠀⠸⣇⣉⠱⠦⣾⣷⣦⣶⣶⡦⠘⢷⠶⠚⠀⠀⠀⠀⠀⠘⣧⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⠇
⠀⠀⠀⠀⠀⠙⠢⢤⣀⣀⣀⣀⣀⡴⠳⠞⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣧⣼⣼⣧⣼⣧⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠀⠠⢔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⢸
⠀⠀⠀⠀⠀⡎⠁⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠒⠇⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠗⠀⡞
⠀⠀⠀⠀⠀⢿⡃⠴⠇⠀⠀⠀⠀⠀⠀⠀⠠⣥⣶⣶⣾⣾⣶⣇⠀⠀⠀⠀⠀⠀⢰⣿⣿⣾⣶⣶⣤⢄
⠀⠀⠀⠀⢀⡼⠓⡊⠉⠑⠲⢄⠀⠀⠀⠀⢸⡏⠐⢼⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⢿⣿⣿⡏
⠀⠀⠀⢰⢁⡀⠀⠀⠀⠀⢰⣾⡆⠀⠀⠀⠘⡟⢓⣋⣿⣭⣭⢼⡇⠀⠀⠀⠀⢀⣿⡿⣗⠀⣸⣿⣿⠃
⠀⠀⠀⠸⣉⣵⢆⠶⢊⢠⣼⣾⡇⠀⠀⠀⠀⢿⢩⡿⡮⢈⠠⠹⣷⠀⠀⠀⠀⣸⣿⣯⣴⣬⣳⣿⡟
⠀⠀⠀⠀⢹⣶⢧⢤⢪⣾⣤⣿⡇⠀⠀⠀⠀⢸⡶⡷⢶⡟⣛⣋⣿⡄⠀⠀⠀⡿⢿⣿⣿⣼⣏⣿⠇
⠀⠀⠀⠀⠀⢻⠦⠶⠻⢁⣠⣽⠃⠀⠀⠀⠀⠀⢟⣳⣟⣻⠯⠽⠚⠀⠀⠀⠀⠈⠛⠲⠿⠿⠭⠟
⠀⠀⠀⠀⠀⠀⠳⢼⣾⣷⡼⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⡤⣤⣤⠄⢠⢤⣤⠀⣤⡤⣤⡤⣤⠀⢤⡤⢠⣤⡤⠤⢤⠠⣤⠠⣤⠄⣤
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣌⡂⢠⡄
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⣉⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡠⠀⠒⠐⠐⠂⢠⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠂⠐⠀⠀⠀⠀⠐⠺⣀⣀⡀⡀⢀⣠⠔⠃⠀⠀⠀⠀⢣⠠⠐⡆⠐⠒⠒⢂⢀⠐⡒⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠄⠀⣀⡄⠀⠀⠀⠀⠀⠀⠆⢠⣼⣧⡼⠇⠉⠁⠀⠀⠀⠆⠀⠀⢋⠦⠀⠇⠀⠀⠀⠀⠀⠠⡄⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠡⠄⠎⢙⣉⡀⠀⠀⠁⠀⣄⣠⠈⠀⠀⠀⠀⠀⠃⠀⠀⠴⡄⠀⢤⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⡙⠉⠁⠀⠀⠀⠀⠠⠞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⡀⠀⠁⠀⠦⢀⠀⠀⠀⠔⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠄⠀⢠⠀⣂⡀⠦⠀⢀⠠⠤⡄⠀⠀⠀⠀⠀⡀⠦⠘⠛⠂⠁⣀⠀⠀⡀⢇⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⡀
⠀⠀⠀⠈⡀⠀⠂⠉⠀⠉⠀⠀⠀⠑⡀⠀⠐⠀⢠⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠰⠀⡃⠸⠀⠀⠀⠀⠂⡬⢁⡤⠂⠘⠓
⠀⠀⠀⠀⠀⠀⡄⠀⠀⢱⡾⠃⠀⠀⡇⠀⠉⠀⠀⠰⠀⠀⠀⠀⠊⠀⠀⠀⡀⠌⢀⣉⡤⠂⠀⠀⠈⢚⠁⢸⡄⣀⠀⠀⠀⡤⢠
⠀⠀⠀⠀⠲⠄⠈⠀⠀⠀⠀⠀⠀⢀⠀⢠⠜⠂⠀⠀⠀⠀⠐⠀⠐⠂⠠⠊⠁⠀⠊⠁⠀⠀⠀⠀⠀⠛⠀⠈⠀⠀⢀⢀⡴⠇⠺⠢
⠀⠀⠀⠀⠀⠤⣄⠀⠁⠒⠐⠒⠀⠀⡠⠐⠠⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡁⠈⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠘⠒⠈⠁⠂⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠰⠶⠲⠆⠐⠗⠶⠂⠲⠖⠲⠖⠶⠀⠶⠶⠰⠶⠲⠚⠲⠀⠖⠒⠆⠳⠆
⠀⠀⠀⢀⠔⠛⠃⠀⠧⠔⢠⠁⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⢢⠀⣴⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡁⠀⠀⠀⠀⠀⠀⠈⠤⢤⡄⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠉⣥⠀⠀⡄⠄
⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠱⣀⡀⠀⠀⢠⠄⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡞
⠀⠀⠀⠙⡇⠲⢲⠒⠆⠀⠒⣠⠛⠄⠀⠀⠀⠀⠀⠀⠉⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠁
⠀⠀⠀⠀⠀⠈⠋⠠⠛⠃⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠆⢆⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢀⠀⡀⠀⣀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣈⠀⠀⢀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠂⢈⠉⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠘⠁⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃⠐⢢⠘⠓
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠀⠈⣈⠀⡜⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠒⠈⠘⠁
⠀⠀⠀⠀⠀⠠⠄⠀⢀⠂⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠂⠀⠉⢈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠑⠀⠉
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣌⠁⠀⠀⠀⢀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠂⠀⠀⠀⠀⠀⠀⡀⠚⠢⡄⠀⠀⠴⠜⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣠⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⣤⣤⣤⠂⠀⠀⠀⠀⠙⠶⣄⣀⣀⣀⣠⠤⠤⠶⠞⠀⣀
⠀⠀⠀⠀⠂⠀⠀⠤⠄⣄⣀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠇⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠤⢀⠄⣀⢀⣏⣍⣶⠆⠘⣦⠠⣄⡀⠀⢠⠾⠅⠜⠻⣤⠠⠄⠶⣦⠀⠀⠀⢠⣇
⠀⠀⠀⠀⢸⣨⡝⠙⠂⠄⠀⠚⠉⠁⠱⠿⢷⣾⠾⠶⣂⠀⢃⠀⠀⠀⠊⢹⣿⠦⢴
⠀⠀⠀⠀⠈⣤⣴⡀⠀⠤⠀⠁⠀⠈⠁⠂⠀⢠⣤⡂⠼⠋⠘⠛⢃⡄⠀⠈⠋
⠀⠀⠀⠀⠘⠁⠉⠀⠀⢀⠐⠂⠀⠀⠀⢡⠀⠉⠀⠀⠃⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣤⣄⠀⠀⠀⠀⠀⣤⣤⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠈⠡⠀⠛⡰⠈⢀⡤⠤⣤⡈⠰⣘⠓⠀⠇⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡫⡍⡄⣠⣶⠶⢮⣄⠨⣴⢱⡸⢀⣤⠶⢶⢶⡄⢘⠭⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⢸⠏⣀⣤⠖⠈⢳⡜⣿⣰⠋⠑⢢⣄⣀⢻⡄⠀⣤⣄
⠀⠀⠀⠀⠀⠀⢠⣶⠦⢠⡟⢴⠲⡅⢿⠈⣀⣿⣷⣻⣿⣿⣿⣿⣷⣳⣾⣧⡈⢸⡇⡝⢒⠎⣷⠀⢶⢦⡀
⠀⠀⠀⠀⢀⡀⢸⠎⠭⣀⣈⣁⣙⡻⣾⣿⣿⣾⣯⣿⣿⣯⣿⣿⣿⣯⣽⣾⣿⣿⣽⣛⣉⣉⣀⡠⠍⠺⠁⢀⡀
⠀⠀⠀⠀⠙⠑⠂⣰⢾⠋⢩⠀⣨⣿⣿⡿⣯⣿⣿⣟⡿⠉⠉⠹⢟⣿⢿⣿⡿⣿⣿⣿⡀⠸⡉⢙⢷⡄⠐⠉⠃
⠀⠀⠀⠀⣴⢒⢠⢸⡇⢠⠟⣾⣿⡿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⡿⢷⣿⡞⠳⠀⣿⢣⢜⡳⡄
⠀⠀⠀⠀⠙⠊⣁⣀⡹⣦⣀⣿⣾⣿⣿⣿⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣇⣀⡼⢃⢀⡉⠘⠃
⠀⠀⠀⠀⠀⠀⡏⣔⠓⣦⣭⣿⣿⣯⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡯⣿⣿⣿⣭⣖⠓⡎⣿
⠀⠀⠀⠀⠀⡀⠳⠪⢊⡴⠚⢿⣿⣷⣿⣷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡿⠲⣌⠻⠰⠏⢀
⠀⠀⠀⠀⣿⣐⣢⢠⡟⢡⣀⣿⣾⣏⣟⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣯⣹⣿⣇⣀⠈⢷⠠⣐⣊⡇
⠀⠀⠀⠀⢈⠉⠀⢾⣧⠀⢻⠉⠿⣿⣿⣯⡿⣶⣤⣴⣄⠀⠀⢀⣴⣴⣤⣾⣿⣿⣿⣳⠟⢹⠇⢀⣹⡇⠘⢉
⠀⠀⠀⠀⠺⠏⢁⡀⢙⠳⠶⠶⠶⢟⣿⣿⣿⣿⡿⣿⣿⡷⣶⣾⣿⣿⣿⣿⣿⣿⣿⡳⠶⠶⠶⢛⡉⣀⠈⠱⠃
⠀⠀⠀⠀⠀⠀⢸⣕⣃⠠⡔⣷⢋⠟⣸⠛⠿⣽⣷⣿⣿⣿⣿⣿⣿⣷⣾⡽⠟⢻⡎⢏⠱⡛⣤⠀⣓⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⢮⣥⠃⢻⡐⠦⢾⡉⠙⢻⡟⣿⢿⡋⠉⣹⡷⠔⢸⡇⠒⣭⡴⠋⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⡄⠌⢷⣄⣄⣬⠶⢋⣼⢻⢊⠻⢮⣄⣄⣲⠟⠤⣠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠂⢀⠉⢉⢖⠀⠴⣬⣭⡴⠆⢀⢊⠉⢉⡀⠘⠶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠥⠀⠻⢮⠴⠀⠀⠀⠀⠘⢮⠼⠃⠀⠵⠂


⠀⠀⡀

⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡴⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠃
⠀⠀⣾⣷⣴⣷⣴⣿⡆⠀⠀⠀⣤⣶⣿⣿⣿⣿⣶⣄⠀⠀⠀⣠⣿⣷⣦⠛⣹⡇⠀⠀⣷⣤⣤⣄⣠⣾⣿⡟
⠀⠀⠻⣿⠛⣿⠛⣿⠇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠹⣬⣿⣧⡾⢷⠃⠀⠀⣾⣿⣿⣾⣿⣿⠟⠃
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠸⣷⡗⠐⢪⢸⠀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⡿⠟⠋⠁⠀⠀⠀⠀⠙⢷⣬⣭⠾⠀⠀⠀⠀⠀⠉⠛⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣄⣀⣀⣄⡀
⠀⠀⠀⣰⣦⣕⣢⣄⡀⠀⠀⠀⠀⠀⣰⣿⣿⣷⡀⠀⠀⠀⠀⣰⢚⣿⣿⣿⣿⢳⣄⠀⠀⢸⡿⣟⣿⣿⣿⣿
⠀⣾⠉⣹⣿⣿⣯⣛⡉⡒⠀⠀⣀⡀⣿⣿⣿⠿⣧⣤⠄⠀⢠⡟⣷⣿⣿⣿⣿⣿⢻⡄⠀⠘⣿⣿⣿⣿⣿⡟
⠀⠈⠙⣿⣿⣿⣿⣿⣿⣧⠀⠀⣨⣽⣻⣰⣤⣯⠛⠿⠄⠀⠀⢷⡟⣿⣿⣿⣿⠛⣿⠁⠀⠀⠹⣿⡿⣿⡿⠁
⠀⠀⠀⠙⢿⠿⠿⠿⠿⠟⠀⠀⠀⠁⠀⣿⣿⠇⠀⠀⠀⠀⠀⠈⠳⢿⣧⣠⡿⠟⠁⠀⠀⠀⠀⠹⣿⡿⠁
⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⣶⠀⢸⣿⣿⡇⠀⢰⡄⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⣾⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣀⣄⣤⣤⡄
⠀⠀⠙⣶⣾⣿⣿⣿⣶⡞⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡿⠿⣿⠿⢿⠇⠀⠀⣀⣤⣾⣟⣿⣟⣯⣠⡉
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠸⢧⣴⠛⣤⣴⡆⠀⠀⣿⣿⣿⣿⣟⢿⣿⣹⣿⡇
⠀⠀⠀⠘⣿⣿⣿⣿⡿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢲⣝⣛⣫⠆⠀⠀⠀⠂⠊⠒⠐⠚⠿⠏⠙⠊
⠀⠀⠀⣼⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢿⠛⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠀⠀⣀⣤⣶⣶⣶⣦⣤⣄
⠀⢀⣶⣶⣶⣶⣶⣶⣶⣶⣆⡀⠀⠀⣶⣠⠀⠠⢲⡄⠀⠀⣀⣀⣀⠀⣦⣰⣿⣿⡋⠀⠀⢿⣿⣿⣿⣿⣿⣿⡏
⠀⢸⣯⣭⣥⣭⣭⣭⣭⣭⣽⡇⠀⠀⣿⢫⡻⣛⢻⠁⠀⠙⢿⣿⣿⣿⣿⣿⠻⠋⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿
⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⣿⣺⣿⣿⣼⡇⠀⠀⠀⠈⠉⠉⠀⠻⠀⠀⠀⠀⠀⠀⢻⣿⠛⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣤⣶⠞⠂⠲⣄⡠⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣠⣾⡁⠀⣹⠏⡰⢻⣿⠃⠀⢀⡴⠋⢀⣀⣀⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡷⣄⠙⣶⠁⣸⠁⡃⣿⡼⢀⣿⣶⣿⣿⡿⣿⣿⣶⣤⣽⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⠿⠛⠂⠀⢨⣧⣿⣆⣿⣿⡇⣸⣿⣿⣏⣀⣠⠤⠾⠿⡿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣁⣀⠀⠀⠀⣈⣿⣿⢿⣿⣿⣿⡿⠿⠛⠛⢿⣟⠒⠐⠀⠈⠺⣿⣯⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠋⢈⣩⣿⣿⣿⣿⠋⠀⠀⠈⠻⠟⠀⠀⠀⠀⠈⠛⢿⣗⡂⠀⠀⠙⢿⣇⣈⣢
⠀⠀⠀⠀⠀⠀⢠⠞⢁⣤⠖⠋⣁⣠⣾⣿⣿⣿⠇⠀⠀⠀⢠⡄⢀⡀⠀⠀⠀⠀⢻⡙⢿⣿⡟⠛⠛⠢⡁⠀⢣
⠀⠀⠀⠀⠀⢠⠃⡤⠋⠀⢠⣾⣿⣿⣷⣿⠟⠉⣀⡀⠀⠀⠀⠙⡟⠀⠀⣀⣠⣄⡀⣧⠘⣿⠙⠷⣦⣄⠈⠢⡈⡃
⠀⠀⠀⠀⠀⡌⠀⠀⠈⣉⣵⣾⣿⢃⣼⢻⠑⢿⣿⠿⣞⡧⠀⠀⠀⢀⠐⣿⣿⠟⠁⠹⣆⣿⠀⠀⠀⠙⠦⡀⠈⣇
⠀⠀⠀⠀⠀⠁⠀⠔⠀⠀⣩⣿⣿⣿⡏⠀⠀⠀⠀⣦⢹⡷⠂⠀⠀⠊⠻⣯⣷⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⢉⠆⢼⡆
⠀⠀⠀⠀⠀⢰⠁⠀⢠⡞⠉⠁⢻⣿⣿⡄⠀⠀⠀⢫⣟⣵⠃⠀⠀⠀⢲⡌⢿⠂⠀⣰⢻⣿⣆⠀⠀⠀⠀⠈⢂⢸⣷
⠀⠀⠀⠀⠀⣿⠊⣰⣾⠁⠀⠆⠘⡟⣧⣑⠀⠀⡶⠀⠻⣿⡟⢛⠛⣹⣿⠗⠈⢱⡄⠁⢀⠟⠙⢆⠀⠀⢦⡀⠈⢧⣻⡄
⠀⠀⠀⠀⠠⣿⡸⢱⡇⠀⠘⠀⠀⢃⠘⢿⣶⣾⣇⡀⣠⡬⠿⢿⡟⢡⣤⣤⣤⡬⣷⣾⡏⢳⡀⠀⠣⠀⠈⢷⡀⠈⢿⠛⠦⠄
⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⣸⡤⠖⣿⣿⡥⠖⠚⠫⠀⣸⣧⣸⣿⣿⣚⣳⣾⡿⠤⢌⣿⠀⠐⠂⠀⢸⣧⠀⢸⠂
⠀⠀⠀⠀⠀⠹⠀⠘⣆⢀⡀⠐⢿⠁⣀⠼⠟⣹⣦⢂⣡⠴⢛⣿⣿⡿⠳⢦⣭⣿⣿⡧⠀⠀⢸⡆⠀⢸⠀⠀⣿⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⠀⠰⡼⡏⠀⣠⣿⣿⣿⡏⡏⢀⡈⠛⠈⠀⠀⣄⢨⡏⠁⠻⣦⡀⠘⠇⢀⣞⠀⠀⣿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢩⣿⠞⢽⣿⡿⠈⡇⣇⡎⠻⣄⠀⢀⡜⢹⠏⠁⠀⠀⠈⢷⣼⣷⡿⠃⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⡾⣿⡇⠀⠀⢸⣄⠀⠈⠀⢀⠀⠀⡆⢸⣆⠀⠀⢸⣿⡿⠁⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠃⢙⠁⠀⣦⡟⠹⡆⠀⠀⡿⡇⠀⠀⠸⠻⣇⠀⢠⣿⠇⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢀⣿⡇⠀⠹⡄⢠⠇⢿⠀⠀⠇⠀⢸⣄⡾⣿⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣴⡀⢸⠀⠃⠀⠀⠹⣾⠀⢸⠀⡜⠀⠀⢸⡟⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣸⡀⠀⠀⠀⠀⠙⡀⢸⡟⠀⠀⠀⠎⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠀⠀⠀⠀⠀⠁⠘

⣿⣏⢹⣟⠻⣿⢛⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣶⣄⠀⠀⠀⠀⠀⠀⠁⠄⠄⠀⢀⣴⣿⣿⣷⣄⠀⠀⠀⠤⠀⠐⠂⠀⠤⠀⠀⣴⣶⣶⣶⣶⣦
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠌⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⡁⠀⠀⠄⠁⠀⠀⠀⠠⡄⠀⣿⣿⣿⣿⣿⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠃⠀⠀⠈⠀⠀⠀⠀⠈⠀⠙⢿⣿⣿⣿⠟⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠄

⠀⠀⠀⠄⠠⠀⠀⠄⠄⠀⠀⢀⣤⣶⣶⣶⣄⡀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⢀⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠐⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠐⠀⠠⠀⠀⠀⠀⢀⣾⣿⣿⣷⡀⠀⢀⣄⠈⠁⠈⠀⢤⠄⠀⣴⣾⣿⣿⣿⣷⣦
⠀⠀⠀⠄⠠⠀⠀⠀⠄⠀⠀⠛⢿⣿⣿⣿⠿⠃⠀⣀⠀⠀⡀⡀⠀⢀⠀⠀⣿⣿⣿⣿⣿⡷⠀⠀⠀⠘⠂⠐⠁⠀⠀⠀⠈⠙⠛⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠢⠂⠂
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⡀⠄⠠⣀⠀⠀⠀⠀⣠⣶⣦⡀⠀⠀⠀⠀⠀⢀⠠⡀⡀⠀⠀⠀⠀⣀⣀⣠⣀
⠀⠀⠂⠁⠀⠀⠀⠀⠱⠀⠀⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⡀⠀⠀⢀⠀⠀⠀⣼⣿⣿⣿⣷⡀⠀⠐⣬⠀⠁⠀⠀⠂⠀⢀⡴⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠃⠀⠀⠀⠀⠘⡀⢰⣿⣿⣿⣿⣿⣧⠀⠈⠐⠂⠢⠀⠄⠅⠀⠀⠈⠻⠿⣻⣿⠿⠋
⠀⠀⠁⠀⠀⠀⠀⠀⠋⠀⠀⠛⠿⠿⠿⠿⠿⠛⠁⠀⠁⠁⠀⠀⠈⠁⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠀⠀⠠⠒⠀⠀⣤⣾⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠨⠘⠁⠀⣽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⢂⠀⠰⠂⠀⠀⠈⢻⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠐⠂

⢦⣼⠀⡌⣿⣿⡆⢻⡰⣴⣃⡶⠋⠹⢲⡼⠀⢠⡸⠀⠀⢸⡄⢴⠥⣰⣷⡞⢻⡈⢸⣿⣿⣧⢸⣿⣀⢼⢿
⣛⣿⣇⡿⣿⣿⣷⠋⣷⠋⢸⠀⣥⠟⡼⠓⠂⢩⡤⣰⠀⠈⢷⡘⠮⢼⡞⣷⠀⢻⣾⣿⣿⣿⣾⣿⣸⣽⣿
⠘⢧⣸⣧⣿⣿⣿⣼⠉⣧⡯⣂⡴⠚⠁⠀⢐⠘⠆⣬⡳⢦⡈⢷⡤⣉⣿⡜⣄⡄⠹⣿⣿⣿⣿⣿⢉⣿⣿
⡦⣹⡟⠈⣿⣿⣿⣇⣾⣿⡞⠉⠀⢬⠌⣠⠀⠤⠄⠰⠭⠵⢿⣿⣷⣌⣐⠿⣎⠻⣄⣿⣿⣿⣿⣿⣼⣿⡷
⣷⣼⣷⣞⣿⣿⣿⠞⢡⡿⢀⣤⣉⣤⣞⠚⠯⠀⠀⠉⠊⠀⠀⠀⠉⠻⢾⣝⠾⣿⡟⣿⣿⣿⣿⣿⣿⣯⣏
⠸⣏⣿⣿⣿⣿⣿⢸⡿⣷⣩⣠⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣷⣿⣿⣿⣿⣿⣿⣦⣾
⠀⢹⣏⣸⣿⣿⣿⣿⣋⡽⠯⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⢹
⢀⡾⠙⣿⣿⣿⣿⣏⠿⣽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡇⢸
⣼⠁⢰⣿⣿⣿⣿⠋⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣿⣿⣿⣿⡇⢸
⣏⢀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡜⣿⣿⣿⣿⣿⣧⢠
⣿⡟⢰⣿⣿⣿⣿⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠤⢻⣿⣿⣿⣿⣿⣿
⣿⠀⢸⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣻⣿⣿⣿⣿⣿⡏
⣿⣇⢸⣿⣿⣿⣿⣧⡀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⡇⠉⣿⣿⣿⣿⣿⣿⡇
⠘⣿⣾⣿⣿⣿⣿⣿⡘⣌⠂⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡇⣴⣿⣿⣿⣿⣿⣿⣇
⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⣰⡿⣝⡞⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢰⣿⣿⣿⣳⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡃⢸⡷⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢹⣸⣿⣿⣿⣿⢿⣿⣿⡏⣿⣿
⢋⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⡕⠀⠀⡀⢀⣀⣀⣠⣤⣤⡤⣶⣷⣿⣿⡟⢻⣿⣿⣆⣿⣿⣧⢻⣿
⣾⠃⣸⣾⢻⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⣦⣴⡷⠛⢋⣭⢿⣉⣴⢿⣿⣿⣿⡿⣠⣾⣧⢻⣇⠟⣿⣞⣷⣻
⡟⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⣷⡟⢣⣾⣱⣷⣟⣿⣿⣿⢇⣿⣿⣿⣾⣿⣷⣻⣿⡟⣿
⣷⠀⣿⡟⢸⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠻⣿⣷⣿⣿⣿⣿⡏⣽⣿⣿⠸⣿⣿⣿⢣⣿⢹⣿⣿⣿⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠢⢈⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠛⢄⠀⠀⠈⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠄⢴⡁⠀⠀⠀⠈⢺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠑⠢⡀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⠤⠄⠠⠄⡀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠰⠁⡠⠐⠢⡀⠙⡄⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢃⠀⢠⡀⣀⠄⠀⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠈⠢⢀⣈⣁⡠⠊⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠈⠐⠠⣤⠀⠀⠤⠔⠚⠣⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡃⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣙⠳⣤⡀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠐⠤⠠⠄⠤⠄⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣭⠆⠀⠈⢩⠋⠝⣋⡛⢍⢫⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⢁⢠⠸⡘⢅⠸⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠀⢀⠔⠁⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⡜⢠⣤⡄⠊⠀⡈⢆⠢⢍⣒⣒⡡⠊⠳⡀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⢄⠁
⠀⠀⠀⠀⠀⠀⡜⠠⠁⠀⠀⢀⠀⠘⡌⢆⠀⠀⠉⠢⡀⠀⠈⠢⡀⠀⢀⠊⠀⠀⠀⢠⠊⠉⠑⡄⠀⠀⠈⢢
⠀⠀⠀⠀⠀⡼⠠⠁⠀⠀⠀⠑⠃⠀⠘⡌⢆⠀⠀⠀⠈⠦⡀⠀⠈⢦⡌⠀⠀⠀⠀⠐⢄⣀⡠⠃⠀⠀⠀⠀⡆
⠀⠀⠀⠀⡰⢡⠃⠀⢠⡆⠀⠀⠀⠀⠀⠘⡜⡆⠀⠀⠀⠀⠈⠢⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⡰⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⣦⠀⠀⠀⠀⠀⠈⢢⡇⠀⠀⠀⠀⠀⣀⡀⠀⢤⣄⣀⡠⠤⠿⠐⠒⠺⠄
⠀⠀⡰⢁⠂⠀⠀⠀⠀⠀⠀⡔⠉⡆⠀⠀⠀⠱⡘⣄⠀⣀⣀⡀⠤⢤⣗⠒⠦⠭⠙⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⢰⢃⠂⠀⢠⢢⠀⠀⠀⠀⠈⠛⠁⠀⡠⠠⡀⠱⡘⡏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⢠⢃⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠴⠁⠀⠡⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⣣⣤⣭⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣤⣍⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⠂⡸⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠻⡝⠻⢧⣄⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⡀⡏⢠⡎⠹⡆⠀⠀⠋⠒⢤⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠡⡘⢷⣠⡇⡀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣼⠸⣿⡀⠀⠈⡢⣝⠛⠧⢀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⠙⠷⠾⠮⣅⣉⡒⣤⣀⠁⠓⠺⢅⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣤⠞⠟⠓⣲⣦⣤⣄⡀⠈⠙⠺⣟⡲⣦⡂⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠁⠀⢀⣽⡿⠛⠉⠈⠹⣆⠀⠀⠀⠹⡄⠙⢦⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⣸⣟⠂⠀⠀⠀⢠⡿⠀⠀⠀⠀⡇⠈⠈⢷⣁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢠⡋⠉⠙⢻⠒⢤⣾⡟⠁⠀⠀⣠⠞⠀⢀⠀⢸⣿⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠙⢄⡈⢿⡻⠶⠃⠀⠃⢈⢿⠿⢷⣿⣅⡀⠀⠈⢀⣾⠇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠓⢦⢹⡿⢽⣶⣄⣀⠀⠈⠁⠀⠀⠀⠈⠉⠓⠶⢮⣈⡴⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠆⠙⣿⠁⠄⠈⠹⣿⣭⡒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢻⣦⡀⠀⠀⠻⣷⡙⢦⡉⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢤⣀⠀⠀⠀⠀⠀⢈⡄⠀⠈⡆⠉⢳⠢⠄⡘⢷⡼⠟⠋⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢤⣝⢦⠈⠳⢤⣀⠀⠀⠀⢇⡀⠀⢱⣀⣨⠦⠖⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⡀⠀⠀⠈⠓⢄⣀⠼⠖⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡹⠶⠶⠚⠋⠉
⠀⠀⠀⠀⠀⢀⠴⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠄⠀⠀⠀⠀⠀⠀⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⡗⣊⣉⣦⢠⠤⠤⠤⠀⠛⠚⠉⠙⠛⠓⠒⠠⠤⠤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣭⣭⡵⠤⠤⠤⠤⠤⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⠐⠒⠒⠒⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣀⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠟⠋⠉⠉⠉⠉⠉⠉⠉⠙⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠿⠛⠋⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢠⣄⡻⢿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣛⣦⡀⠹⣿⣦⣿⣯⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠧⠀⠰⠶⣦⣤⣤⣤⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠎⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣋⠀⢀⣼⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⠋⠁⠈⠉⣉⣼⣿⣿⣿⢟⣿⡙⣿⣿⣿⠛⢋⠻⣿⣿⠿⣿⣿⣦⡄⠀⠀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢋⠀⣠⣴⣾⣿⣿⣿⡟⠃⢾⣿⡇⢹⣿⠋⢰⣿⣷⡌⣿⣷⡘⣿⣿⣿⣤⣘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣿⣿⣿⣯⡍⠉⠁⢸⣿⣍⣁⠘⠿⠀⣼⣿⣿⣿⡸⣿⡇⣿⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣾⣧⣼⣿⠏⣀⣄⠘⣿⣿⣏⣿⣷⢻⡇⣿⣾⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⣿⡟⢿⣋⣴⣿⣿⣷⣌⢿⣿⡿⣿⡧⠀⣿⣿⣿⣿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣋⣁⣀⣉⡁⠀⢿⡿⠿⣿⣿⠟⠈⣉⣠⣤⣤⣤⣉⣭⣽⡇⠀⠻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡏⣴⣿⣿⠿⣿⣿⣿⣿⣦⣤⡄⢈⣡⣴⣿⣿⡿⣿⠿⣿⣿⣿⣿⣷⡈⢿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣏⢿⣿⣿⣿⣿⠃⢀⣄⡀⠈⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⠀⢀⣠⠀⢹⣿⣿⣿⣿⣷⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡶⣿⣿⣿⣿⣄⠸⠿⠀⢀⣿⣿⣿⣿⢃⡘⣿⣿⣿⣿⡀⠸⠿⠂⣸⣿⣿⣿⣿⣿⡿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡇⠻⣿⣷⣶⣶⣾⡿⣿⣿⣷⣿⣿⣯⢉⡻⢿⣿⣶⣶⣾⣿⢋⣽⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣾⣧⡈⣭⣥⣶⣶⣿⣷⣿⣿⣿⣿⣯⣿⣶⣦⣮⡉⣩⣤⣼⣿⣧⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⡿⢸⣿⣿⣯⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⢶⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⡿⢿⣿⣶⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣾⣿⣿⣿⠿⠥⠒⠛⠛⣿⣿⠛⣛⣓⡀⢻⣿⣿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣳⣶⣶⣿⣿⣿⣿⡶⠿⣿⣿⣿⣿⡏⢨⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣟⠛⠛⠙⠙⠂⠹⠈⠒⠐⠑⠛⢛⣻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣷⣶⣶⣾⣾⣿⠿⠿⡿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⣤⣬⡀⠉⢉⡁⢀⠀⠀⢐⣣⣠⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣯⢿⣿⣿⢿⣿⠿⠟⠿⢩⠗⠺⠛⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡆⢋⡯⠜⠉⠀⠀⠀⠀⠀⠀⠀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡈⠁⡀⠀⠀⠤⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠾⠒⠒⠚⠓⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⢿⢿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⢫⢎⡎⡎⡞⢾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣱⢣⡏⣼⢰⡇⢿⠘⣆
⠀⠀⠀⠀⠀⠀⠀⣠⠏⣰⠇⣼⠁⣿⠀⣷⠘⣧⠹⣦
⠀⠀⠀⠀⠀⠀⣰⡟⢠⡿⠀⣿⠀⣿⡄⢻⣆⠹⣧⠘⢷⡄
⠀⠀⠀⠀⠀⣰⡿⠀⣾⡇⢸⣿⠀⢿⡇⠈⣿⡄⠹⣷⡈⢻⣦⡀
⠀⠀⠀⠀⢠⣿⠁⢸⣿⠀⢸⣿⠀⢸⣿⠀⢹⣷⡀⠹⣷⡄⠙⣷⣄
⠀⠀⠀⢠⣿⡇⠀⣾⡿⠀⢸⣿⠀⠘⣿⡇⠀⢿⣷⡀⠹⣿⣆⠈⢻⣷⡀

⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⢀⣀⢀⠘⠛⠀⠀⠀⢀⣀⡀⡈⠛⠀⠀⠀⠀⢀⣀⠀⠛⠁⠀⠀⠀⢀⣀⠀⠙⠃⠀⠀⠀⠀⢀⣀⠘⠋
⠀⠀⠀⠸⠺⠸⠹⠀⠀⠀⠀⠸⠿⠧⠿⠀⠀⠀⠀⠀⠧⠤⠺⠀⠀⠀⠀⠀⠀⠇⠾⠇⠀⠀⠀⠀⠀⠧⠼

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⣀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⣀⠀⠘⠟
⠀⠀⠀⠀⠽⢯⠀⠀⠀⠀⠀⢀⠿⡆⣆⡆⠀⠀⠀⠀⡏⢰⢲⢲⠀⠀⠀⠀⡟⣾⢿⡲⠀⠀⠀⠀⠀⡏⢸⢲


⠀⠀⠀⠀⠀⠀⠸⠷⠀⠀⠀⠀⠀⠀⠰⠿⠂⠀⠀⠀⠀⠀⠀⠿⠆⠀⠀⠀⠀⠀⠀⠾⠇⠀⠀⠀⠀⠀⠀⠸⠷
⠀⠀⠀⢀⣾⡄⡆⠀⠀⠀⠀⠀⢰⠀⡆⠀⠀⠀⠀⠀⢸⠅⡴⠀⠀⠀⠀⠀⢈⡟⢰⠀⠀⠀⠀⠀⢸⢽⢂⣦
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⠠⣶⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⢴⡆⠀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠐⡖⢰⢤⠀⠀⠀⠀⢠⠒⣆⡤⡄⠀⠀⠀⠀⣦⡆⡤⡆⠀⠀⠀⠀⢰⣔⢠⠄⠀⠀⠀⠀⠰⣖⢰⢤
⠀⠀⠀⠀⠁⠈⠉⠀⠀⠀⠀⠈⠙⠁⠿⠃⠀⠀⠀⠀⠁⠁⠉⠁⠀⠀⠀⠀⠈⠈⠈⠀⠀⠀⠀⠀⠈⠋⠈⠉

⣀⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⣀
⢚⣻⣉⠥⠀⣀⣤⣤⠄⠀⠚⠛⠃⠀⠛⠛⣫⣤⣤⠀⢀⣤⣤⡄⠛⠛⠃⠀⠛⠛⠃⢠⣤⣤⡀⠀⢤⣌⣟⠛⠀⠈⠛⠓⠂
⠟⢻⣿⣿⣿⠟⠛⠓⠒⠂⠀⠀⠒⠒⢲⣾⣿⣿⠟⠓⣿⣿⣿⡟⠒⠒⠂⠀⠀⠐⠒⠒⠛⣿⣿⣷⡟⠛⠿⣿⣶⣖⡒⠒⠀⠀⠐
⠉⢉⣩⣿⣶⣶⠶⠂⢠⣴⣶⣶⣶⠀⠉⠉⠉⠉⠀⠈⠉⠉⠉⢱⣶⣶⣶⣶⠄⠠⢶⣶⣶⣾⡍⠉⠉⠉⠀⠉⠉⠉⠉⠁⠐⢶⣶
⣶⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣥⣤⣤⡄⠀⠀⠀⣠⣤⣤⣄⣼⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣦⣄⣤⣤⣤⣀⠀⠀⠀⣤⣤⣬
⣿⠿⠛⠛⣻⣿⣿⣿⣿⡿⠋⠁⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠁⢹⣿⣿⣿⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⣅⠉⠉⠉⠀⠀⠀⠀⠉⠉

⡆
⡇
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⡇⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣀⣀⣤⣤⣄
⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⡟⡆⢰⣰⠀⣶⣖⣢⢰⡖⠆⢰⡄⢰⢰⠀⡖⡆⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠌⠀⡴⠆⠀⣤⠀⡦⡤⡄⣤⠆⢸⠂⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁⠁⠈⠉⠈⠁⠁⠁⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⢸⠁⢰⡄⣆⠰⠀⡤⠒⣄⢰⡆⣤⠠⡆⡰⠀⣴⢠⡄⣠⠔⢤⡀⡤⠢⣄⢰⡄⠴⠀⢸
⡇⠀⠀⠀⠀⠀⠀⢸⡀⠀⠷⠹⠎⠈⢧⢀⠟⢸⡟⣧⠀⡟⢧⡀⣿⢉⡇⠻⡀⡸⠛⠇⢀⠟⢸⡟⣦⠀⢸
⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⠀⡠⠠⠀⠀⠄⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⣀⣀⠀⡀⡀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠁⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠐⠒⠒⠒⠒⠀⠀⠀⠐⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛
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⢸

⠰⢶⣶⣦⡀⣶⡄⣤⡀⠀⠀⢀⡤⣠⡶⣠⣶⣦⠶⢦⣆⢠⡀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⢀⣄⣠
⠀⠀⣛⠋⢽⠟⢫⡿⢓⠶⢶⠛⢿⡝⠿⡯⠛⣓⠀⠠⣿⡟⣽⡾⣱⣯⢣⢰⣄⠀⢠⠆⡴⣹⣞⡟⣯⢯⣷
⣤⣾⣌⢃⡀⣀⠀⣤⡀⠀⠀⢠⡄⢀⡀⢀⢘⣼⣧⣤⣠⣾⢿⢷⡿⡻⣿⢽⣼⠀⣧⣯⣷⢷⣻⡿⡿⣗⣤
⠀⢙⣿⣽⣧⣹⣷⣼⣗⡶⢶⢿⣧⣿⣇⣾⣹⣷⡋⠀⠉⠀⠟⢫⡮⠞⡵⡺⠛⣶⠛⢿⣪⡳⢽⣍⠺⠂⠈
⠀⣈⠁⠸⠃⠘⠋⠉⠘⠀⠀⠋⠉⠙⠀⠹⠆⠈⣁⠀⠀⠀⠀⠀⠀⠈⠉⠀⡜⠀⢳⡀⠉⠃
⠀⣻⣴⡆⣤⢀⡀⠀⠀⠀⠀⢀⠀⢀⠀⣦⢰⣦⣟⢰⢲⣀⣄⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⢠⣄⡔
⠀⠉⣩⣷⣿⣼⣇⣸⡇⠀⠀⣼⣇⣼⣧⣿⣮⣉⠁⠈⠙⣻⣿⢾⣶⣷⣴⡄⢄⠀⣠⢤⢶⣞⣷⣿⣿⣟⠛
⠀⠀⠀⠀⠰⠟⢡⠿⠋⡶⢖⠙⢿⡍⠻⠄⠀⠀⠀⠀⠺⠛⣡⣿⢗⣿⢾⣿⠚⠋⠓⣿⡷⣿⣞⢿⣬⠓⠓
⢰⠓⡄⡠⡄⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⢠⣄⡰⢻⠄⠀⠀⠉⠀⠛⠁⠛⠉⠀⠀⠀⠈⠛⠈⠛⠀⠉
⠀⠛⢯⣧⡷⡟⡆⡴⢄⠀⠀⡴⣆⣰⢹⢷⣾⡿⠋⠀⠀⠐⢖⡄⠀⣶⡄⡴⡄⠀⢀⣦⢀⣶⠀⢠⣶⠂
⠀⢮⠽⠋⣓⣿⡧⣵⣾⠀⠀⣷⡯⢿⣿⡊⠻⢯⡧⠰⣿⢿⣾⡿⣲⣿⢧⣳⣷⠀⣼⣜⣸⣿⣶⠻⡷⡿⢿
⠀⠀⠀⢰⠥⠛⣴⣻⠏⡽⢯⠹⣟⣎⠻⠼⡀⠀⠀⠀⠀⠺⠊⠾⠟⡵⣊⠝⢛⡶⡛⠻⣕⢦⡳⠷⠙⠿
⠺⣷⣦⣿⣆⣤⠉⠀⠈⠀⠈⠀⠀⢁⣤⣸⣷⣴⣾⠇⠀⠀⠀⠀⠀⠈⠁⢠⠋⠀⠘⣆⠀⠁
⢠⣺⣿⣿⣻⣽⣷⣦⡀⠀⠀⢀⣶⣾⣯⣿⣿⣿⣳⡄⡶⡀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡶
⠈⣉⣭⢤⣽⣿⣉⣿⡇⠀⠀⢾⣿⣸⣿⣧⡤⣍⡉⠀⠳⣿⣿⡔⡶⣜⠿⣢⣤⠛⣧⣴⡿⣣⢴⢆⣾⣿⠞
⠀⠙⠛⠛⠉⣤⣾⡿⡹⠓⠺⣝⢿⣶⣄⠉⠛⠛⠋⠀⠀⠈⢛⣿⡬⠾⣷⡚⢏⣷⣹⠓⣶⠷⢥⣾⡟⠁
⢀⣀⣐⣷⣈⣯⢁⡀⠀⠀⠀⠀⢀⣈⣽⣠⣿⣀⣀⡀⠈⠓⠿⠚⠁⣴⡷⠃⠘⠁⠓⠀⢯⣧⠀⠑⠾⠞⠃
⢉⣿⠟⠛⢻⡿⣿⣧⡄⠀⠀⣠⣼⣿⢿⠟⠛⠻⣿⢁⢤⣿⣷⣼⣷⣾⣦⢶⡄⠀⢠⣶⣰⣶⣾⣿⣼⣿⡤
⠀⠉⠀⠀⠀⠙⠿⢿⣿⡷⢿⣿⡿⠿⠉⠀⠀⠀⠉⠈⠉⠾⠋⠾⠿⡿⢫⡿⢳⠶⢞⢿⡝⢿⡟⠵⠙⠷⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠓⠺⠟⠋

⢄⠀⠀⢷⠇⠀⠀⠀⠀⠁⠃⠁⠀⠀⠙⢄⠀⠀⢀⠼⠀⢠⡁⠀⠈⠀⠀⠠⠀⠀⢀⠀⠀⡀⠈⠈⠀⢀⠠⠊⠀⠀⠀⠁⠀⠀⠁⠈⡄⠀⠀⣠⠐⠈⡀
⠀⠇⠀⠀⠀⢀⠄⠤⢄⡀⠀⡞⢂⠦⡀⠀⠙⠋⢁⡠⣀⡀⠑⡀⠀⠀⠆⠀⠀⡠⠋⠀⠀⠉⠐⠂⠛⠁⠀⠀⣀⠀⠀⢆⠁⠀⠀⡠⠃⠀⢀
⠀⠃⠀⠀⠠⠃⠀⠄⠀⠹⡄⠙⡀⠀⠘⢄⠀⢀⡋⠄⠠⠉⡄⠈⠒⠀⠀⠐⠊⠀⢀⠄⠒⠒⠐⠢⡄⠀⢠⠊⢠⠑⢆⠀⠉⠀⠈⠀⠀⠀⣌⠀⠄
⡤⠀⠀⠀⣆⢀⠀⠀⠈⠀⣰⠀⠐⡄⠀⠀⢵⠀⠢⣁⡂⠔⠀⠀⠀⠀⠀⠀⢀⠔⠁⠄⠀⠀⠀⠀⢈⠀⢨⠐⠀⠄⡠⠀⠀⡀⠀⠶⡄⠀⠑⢄⠀⠂
⠠⢤⡀⠀⠇⠈⠀⠀⠀⠀⠸⠀⠀⠈⠒⠥⣼⠀⠀⠀⠀⣠⠐⠚⠛⢄⠀⠀⡜⠀⠂⠀⠀⣀⠈⢀⠃⠀⠀⠑⠚⠈⠀⠀⢰⠀⠠⠀⠈⠢⡀⠀⠑⠒⠈
⡄⠒⠘⡂⠘⡄⠀⠀⠀⣠⠃⠀⡤⠄⠠⢄⠀⠀⠀⠀⠀⠁⠈⠠⠀⠀⠧⠀⢣⡀⠀⠐⠀⡠⠔⠁⣠⠤⠄⠀⢤⠀⠀⢀⠋⠀⠀⠀⠀⠀⠐⢄⠀⠀⠠
⢀⣈⠠⠇⠀⠈⠐⠐⠊⠁⠀⠎⠠⠉⠈⢀⠱⡄⠀⠀⠰⠀⠀⠀⠀⢀⠃⠀⠀⠈⠀⠉⣁⠀⠀⠎⢀⠀⠀⠄⠀⢱⠀⢨⠀⠘⠀⠀⠀⠀⡀⠼⠗⠀⠐
⣄⠀⠀⢀⡄⠀⠒⠒⠢⡀⠀⠡⡄⠀⠀⠘⠀⠘⠄⠀⢧⠠⠐⣀⠀⠌⠀⠀⠀⠀⠔⡉⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⠀⢂⡈⠁⠀⠀⠀⢀⠤
⠈⠄⠀⠈⠀⠂⠈⠁⠀⠈⠢⡄⠀⠁⠒⠂⠀⠀⠼⠄⠈⢤⡤⠔⠋⠀⠠⠖⡠⣜⠀⠀⠁⠃⠀⡀⠄⠀⠀⢀⠀⡆⠀⠀⡴⠒⠉⠀⠉⠐⠠⡀⠠⡅⠈
⠚⠀⠀⢦⠀⠂⠀⠀⠀⠈⠀⡀⠀⠀⠴⣄⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠘⠍⢠⡈⣆⠀⡼⠀⠀⠣⣈⣀⣈⣀⡰⠃⠀⡘⠀⠀⠄⠀⠃⠀⠀⢠⠀⠑⠠
⢠⣦⣀⠀⠑⠢⡀⠐⠀⣀⠠⠆⠀⢸⠀⠂⠣⠀⠀⡠⠚⠀⠀⡀⠈⠑⢄⠈⠒⠛⠓⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠐⠀⢀⠀⠀⠀⢨
⠎⠀⠀⠑⢠⠀⠀⠉⠁⠀⠀⠀⠀⠘⡀⠀⠀⠀⠈⠁⠀⠠⠀⠀⠀⠄⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⡖⠨⡍⠒⠄⠀⠀⠀⠱⣄⡀⠈⠀⣀⠔⠁
⠀⡄⠀⠂⠀⠀⠀⣠⠐⠒⠂⠀⢤⠀⠰⡀⡄⡅⠀⣀⠀⠀⠠⠀⠀⠀⠀⠂⢀⠚⠁⠐⢢⠀⠀⠀⠆⠰⠀⢃⠍⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣤⡀
⠢⡀⠀⠀⢰⠀⠀⠁⠀⠊⠀⠐⡀⠵⠀⠑⠤⠅⠀⠈⠂⢄⣀⢀⡀⠤⠎⡠⠃⠀⠘⠂⠀⢦⠀⠀⠈⠆⠂⠁⠀⠀⠀⠀⠀⢸⠋⠩⠅⠢⡜⠆⠚
⠀⠈⠢⡀⡄⠀⡘⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀⠀⣀⠀⠈⠀⠀⠀⡰⠁⠀⠀⠀⠀⣄⠀⢣⠀⠀⣀⠤⠐⠂⠐⠒⠢⡀⠀⠢⡁⠀⠀⢠⡠⣦⠇
⠤⡀⠀⠙⠃⠀⡇⠀⠀⠀⡀⠄⢠⠀⠀⠀⠀⠐⠡⠒⢠⢱⠀⠀⠀⠘⠇⠀⠤⠀⣀⡀⠀⣀⠌⢀⠌⢀⠀⠂⠀⠁⠀⠀⠸⠀⠀⠀⠙⠢⠀⠞⠁⢰⠦
⠀⠈⠒⢄⠀⠀⠐⠂⠀⠠⠤⠔⠃⢀⠶⢄⠀⠈⠤⣀⡠⠞⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠉⠀⠄⠀⠠⢀⡀⠀⠀⠁⡰⠁⠀⠀⠀⣀⠠⠤⢀⠀⠀⢦
⠂⢄⡰⠄⣡⠀⠀⠀⠀⠀⠀⠀⠐⠣⠄⠄⡃⠀⠀⠀⠀⠀⢀⠔⠁⠀⢄⡊⠀⠐⡖⡏⠀⠂⡄⠈⠢⠤⠄⠀⠤⠐⠈⢀⣤⠤⠂⠉⠀⠀⠂⠀⠑⣄
⠀⠀⠈⠉⠉⢠⠔⡉⣉⠙⡇⠀⢠⠆⠀⢠⠷⡆⢶⠀⠀⠔⡁⠀⢈⠤⠊⠀⠀⢀⠸⠦⠄⣽⢻⠀⠀⠀⠀⡀⠤⠄⡀⠀⠈⠑⠠⢀⠀⠰⠀⠀⡀⠞⠁
⠉⠉⠢⡀⠀⠸⣄⠠⠀⢀⠇⠀⠘⢅⠘⣃⠄⢈⠗⠀⠸⠂⠰⠊⠁⠀⠀⠀⢐⠡⢀⠀⢀⣠⠜⠀⠀⡤⢊⠀⠂⠀⠉⡆⠀⠀⠀⠀⠈⠐⠒⠈⠀⠀⡠
⠈⠈⠀⡈⢦⠀⠀⠁⠚⠁⡀⠤⢀⠀⠉⠉⠛⠁⠀⠀⣀⣠⠀⠰⠢⡀⠀⠀⠛⠂⠰⠐⠉⠀⠀⡠⠊⠀⠀⢀⠀⠄⢁⠃⠀⢠⠄⠶⢄⠀⠀⢀⡔⠊
⠀⠀⠐⢠⠎⠀⠀⢀⠖⠁⠀⠀⠀⠈⠲⠀⠀⢀⠊⠁⡀⠀⠀⠆⠀⠸⠀⠀⢀⠄⠤⢀⠀⠀⠘⠓⠀⠀⠀⠒⠀⠚⠁⠀⠐⡧⠆⢈⠈⡁⠀⣌⠀⠐
⠀⡠⠚⠁⠀⠀⠀⡈⠀⢀⠀⠁⠀⠆⠀⠆⠀⠰⣀⠈⠁⠀⠀⠀⠀⠁⠀⡐⠀⠀⠀⡈⠑⢄⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠈⠣⠠⠤⠄⠆⠀⠈⠢⡀
⠋⠀⢀⣄⠀⠀⠀⢓⠀⠀⠀⠀⠠⠀⢀⠇⠀⠀⠈⠑⢄⠀⠀⠀⡘⠀⠀⠃⠀⠂⠀⠀⠰⠈⢂⠀⠸⣁⠐⠀⠀⠀⠀⠙⢠⡀⠀⢀⡀⠀⠀⠀⠀⠈⠑
⠀⡠⠊⠈⠑⠄⠀⠀⠑⣀⠀⠀⠀⣠⠊⠀⢀⡠⢆⠀⠀⠑⠦⣴⠁⠀⠀⠈⠄⠐⠀⠀⠀⠂⢈⠀⠀⠀⠉⠐⠂⠀⠰⠇⠋⠁⠀⠋⠁⡕⢆⠀⢀⣄
⢰⠁⡀⠀⢠⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠔⠡⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠁⢀⠌⠀⢀⠤⠠⢄⣀⠀⠀⢀⡀⠀⠀⢃⠁⠀⠀⠏⠓⠝⠆
⠀⢆⠀⠀⠀⡘⠀⠀⡰⢊⢍⠰⡀⠀⠀⠀⠀⠈⠀⠀⠇⠀⠀⣀⠤⠀⠲⠠⣀⠀⠀⠉⠈⠁⢀⠔⠁⠀⡂⡀⠩⠀⠀⡈⠉⠢⡀⠀⠳⡀⡸⣤⢇⡜
⠀⠀⢂⠀⣀⠁⠀⠀⢇⠢⡠⢀⠇⠀⠘⠀⠀⡀⠀⠀⠀⠀⡘⠁⠀⠀⢀⠀⠀⠙⡆⠀⠀⠴⠁⠀⠀⠊⠀⠀⡰⠀⠀⠃⠀⡀⠐⠄⠀⠉⠉⠉⠁⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠤⠤⠤⠶⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠈⠻⢷⣶⣤⣴⡶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⣤⣦⣠⣤⣤⣤⣬⣭⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠉⠑⠂⠀⠀⣸
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠒⠒⠲⠶⣤⣤⣀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠿⢻⣿⣿⣿⣿⡟⠩⠐⠆⢻⣿⣿⠇⠀⠀⠀⠐⠛⠻⢷⣤⣄⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠁⣆⣴⠀⠸⣿⣿⠀⠀⠀⠀⠀⠲⠆⠀⢸⠛⠛⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⢸⣿⣿⣿⣿⡆⠸⠇⠀⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠠⠋⠀⠈⠙⠻⢿⣿⣿⣿⡿⠠⡄⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⠆⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⣀⣠⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡟⠲⠄⠀⠀⠀⠀⠀⠀⣈⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣄⠀⠉⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠂⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠓

⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣆⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠻⣿⣷⣾⣿⣿⣿⣿⠟⠀⠀⠀⠀⠈⢲⢟⢰⠲⣶⢦⣷⠂⡞⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠙⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠈⠈⠪⣿⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢀⠀⠀⠀⡈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⡸⠀⠀⠀⡇⡆⡌⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠙⠀⠀⠀⡇⡱⠁⠒⠠⠬⣿⡙⣳⠖
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣀⣀⣀⣈⣠
⠀⠀⠀⠀⠀⠀⠀⢰⡔⠁⠀⠀⠀⠀⡠⠊⢀⠝⠛⠋⠁⠀⠭⣀
⠀⠀⠀⠀⠀⠀⣰⠋⡇⠀⠀⠀⠀⡐⠁⡰⠁⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⡼⠁⠁⣿⡄⠀⠀⡐⠀⡰⠁⢠⠂⠒⠠⢄⡀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠳⠲⢦⣋⠇⠀⢠⠁⢠⠁⠀⡇⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⢸⡀⠰⠂⠀⠀⠨⠀⡄⠀⠀⠉⠈⠀⠒⠠⠄⡀⠈⠢⠢⡀⠀⠘⠄
⠀⠀⠀⠀⣜⡧⡵⣀⡷⢋⡃⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⡐⣀⠠⠜⠀⠀⠈⠢⣀
⠀⠀⠀⠀⠁⠂⠁⠘⠓⠉⠀⡆⢸⠀⠀⠐⠦⢤⠼⠵⢠⡦⡂⡧⣤⣤⢄⡠⠀⣠⡀⡀⡌⣽⢆⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠓⠀⠀⠀⠡⠄⠤⠄⠀⠀⠤⠤⠠⠤⠤⠤⠍⢁⡙⢳⡏⢺⣿⡀

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⢰⠆⠳⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⠃⠀⠀⠀⠀⠀⢤⢰⠀⠀⠀⠀⠀⠀⠀⠂⡘
⠀⢀⡎⠠⠀⢸⡁⠀⠀⠀⠀⠀⣄⠀⠺⢢⠚⠃⢀⡄⠀⠀⠀⠲⠀⣀⣀⠐⠃⠀⠀⠀⠠⠴⠀⣀⠄⠀⠀⠤⠀⠒⠂⠀⠀⠀⠈⡟⢻⠀⢐
⠀⠀⢶⣶⠟⡇⡏⠀⠀⠀⠀⠀⠘⡗⣬⣿⡹⡄⡯⠁⠀⢀⣐⠀⠃⢴⣄⠘⡁⢂⡀⠀⣰⣦⠀⣤⠟⠀⠚⡴⠂⠈⠀⠀⠀⠑⢘⣓⣘⡣⠊
⠀⠀⠨⣐⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠿⠗⠉⠀⠀⠀⠀⠀⠠⠧⣙⣁⠴⠑⠂⠀⠈⠘⠁⠀⠀⠉⣀⡘⠁⠀⠀⠀⢠⠀⠀⠀⠏⠹⠄⠀⠀⢀⡀
⠀⠀⠀⢿⡵⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢤⡤⠀⠀⠀⠀⠀⠀⠶⠁⠀⠀⠘⠖⠀⠀⠀⡀⠀⠀⠀⠀⢭⡥⠀⠀⠀⠀⣡⢣⠀⠀⢀⣀⠀⢠⠌⠙⠆
⠀⠀⠀⣘⢠⠀⠀⠀⢀⡷⢯⡀⠀⠀⠀⢀⢀⠀⠀⣀⠤⠄⡄⡀⠀⠸⠗⠀⠀⡀⠠⠀⡁⠠⠀⠀⠀⢰⡆⠀⢀⠀⡘⠔⢥⡏⠀⠨⠍⠀⢰⢚⠀⠈⠀⠀⡀
⠀⠀⠀⢸⡖⠀⠠⣁⠁⣷⣯⡇⠀⠤⠀⠘⢉⠀⠮⢰⡶⠳⣶⠹⡄⠀⡂⠀⠀⠀⠛⠀⡅⠘⠃⠀⠀⠈⣷⠒⠋⠀⠀⣄⣢⠀⠀⢀⣆⠀⠀⠙⠶⣀⡈⠟⠁
⠀⠀⠀⢰⢺⠀⢄⡅⢰⢾⠿⡇⠤⠄⠀⠸⠚⠀⣇⢹⣦⣤⣟⢁⠃⠈⠋⠀⠠⠀⣀⠈⠂⢀⣀⠈⠀⠀⢸⠀⠀⠀⠀⠰⠖⠀⠀⠘⡞⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⢬⡷⣾⠔⠡⠀⠀⢰⣠⠀⠀⠃⠠⡅⠄⠋⠀⠀⠀⠀⠀⠐⠁⠀⡀⠸⠁⠂⠀⠀⡟⠀⠀⠀⠀⠈⡏⠀⠀⢀⢦⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⣴⠄⠀⢐⣸⠀⠐⠀⠀⢠⠀⡀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠰⠇⠀⠀⠀⡏⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⢀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠅⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢐⠀⠀⠀⠘⠿⠀⠀⠀⠀⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠁⠀⠀⠀⠘⠦⠆⠀⠀⠀⠆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠠⡄⡄⠀⠀⠀⡁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠰⡆⠀⠀⢠⠉⠄⠀⠀⠀⠘
⠀⠀⠀⢸⢉⠀⠀⠀⠖⠐⢠⠁⠀⠀⠀⠀⠇⠁⠀⠀⠀⠆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠷⡆⠀⠀⠀⠀⡇⠀⠀⠀⠉⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢦⡆⠀⠀⠀⠂⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢐⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⠀⠀⠀⠀⠈⠀⠀⡀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠴⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⡞⠰⠀⢠⢠⠄⢄⠄⠀⠀⠀⠀⠀⠀⢠⢠⠄⠀⢀⠄
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠄⠰⠒⠚⠀⠀⠀⠀⠀⠞⠀⠐⠀⠰⠠⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣽⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠈⠉⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣧⣴⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠿⠋⠀⠸⠀⠀⠉⠉⠙⠻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠁⠀⠀⣈⡀⠀⠀⠘⣇⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣷⣶⣶⣶⣿⣿⣶⣦⣤⣟⠃⢠⣷⣤⣄
⠀⠀⠀⠀⠀⣠⣤⣤⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣸⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⡀⠀⣠⣤⣀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠛⠛⠛⠋⢙⡋⠉⠁⠀⠉⠉⠉⠉⠉⡋⠉⠉⠉⠙⠛⣛⣛⣿⣻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠸⣿⠟⣻⣿⣿⣿⣿⣿⠟⠻⢦⣼⠛⠋⠙⠛⠈⠉⠁⡴⡆⠀⠉⠉⠛⠉⠈⠙⢷⡾⠉⠙⠟⠛⢿⣿⣿⣏⢹⣿
⠀⠀⠀⠀⣿⠀⢿⣿⣿⣿⣿⣿⡇⠀⢈⡁⠀⢰⣿⡶⠀⠀⠀⢀⡄⠀⠀⣰⣦⣄⠀⠀⣠⡀⠀⢀⣴⡀⣸⣿⣿⡟⢸⣿
⠀⠀⠀⠀⣿⡇⠈⢿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠘⠋⢰⠇⠀⠀⢸⡇⠀⠈⠛⠿⠋⠀⠀⣽⠃⠀⠘⣿⡇⢸⣿⣿⠁⢸⡟
⠀⠀⠀⠀⢸⣧⠀⠸⣿⣿⣿⣿⣿⣷⠀⡇⠀⡆⠐⠀⠀⠀⠀⢸⠃⢠⣶⠆⢀⠀⡃⠀⢹⠀⠰⢶⡿⠇⣼⣿⣇⡀⣾⠇
⠀⠀⠀⠀⠀⣿⡘⣶⣿⣿⣿⣿⣿⣿⣤⠃⢰⣷⣾⣠⣶⣠⡀⠀⠀⡀⢀⣴⣿⣿⣇⣀⠀⠀⣀⡉⣿⣄⢹⣿⡿⢡⡿
⠀⠀⠀⠀⠀⠸⣇⣿⣿⣿⣿⣿⣿⡋⠹⠃⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠎⣿⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠹⣾⡟⢿⣿⣿⣿⣷⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢹⣶⡏
⠀⠀⠀⠀⠀⠀⠀⢻⣷⢈⣿⣿⣿⣿⣿⣹⣷⡀⠀⠀⠀⠀⣀⣼⣧⣄⠀⠀⠀⠀⠀⢀⡍⠀⠀⠀⢀⣾⣷⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣼⣷⣄⣿⣿⣿⣿⣤⣼⣿⣤⣴⣾⣿⣦⣾⣷⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⣹⣿⣿⣿⣿⣿⣿⣛⣛⣉⣛⣹⣿⣿⣿⣿⣿⣯⣬⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣯⣭⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠿⡟⠙⡟⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⡀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠋⠀⠃⠀⠁⠀⠁⠀⣀⡀⠀⡀⣠⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣄⣾⣿⣿⣿⣿⣷⣶⣶⣴⣶⣶⣿⣦⣸⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⡀
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠨⠛⢃⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠿⠿⠿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⠀⢸⡿⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡏⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣾⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⢸⠁⢠⣠⢄⡤⡤⣤⣤⣠⢤⡠⠤⣄⣤⠀⡤⢄⣤⣤⡠⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⢸⠀⠈⠘⠈⠀⠓⠙⠉⠙⠊⠙⠊⠃⠛⠒⠑⠋⠉⠛⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⢯⠭⠿⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠖⠒⠒⠒⠒⠒⢒⠒⠒⠒⠒⠒⣒⠒⠒⢒⠒⠒⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠒⣿⠈⠉⢹⠇⢾⣉⡷⢸⡋⢻⡄⣿⠀⢀⣀⣀⣘⣛⣚⣛⣃⣀⣀⣀⣀⣀⣀⣀⣠⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⢠⡟⠀⢾⣉⣿⢨⣝⣽⠃⣿⠀⠈⠻⣿⣿⣽⣯⣭⣭⣽⣭⣩⣿⣏⣿⠟⠁⠀⢸
⠀⢸⠆⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⣿⠀⠀⠀⠈⠋⣽⣿⣯⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⣿
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠟⢹⡇⣿⣉⣹⡆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠻⣿⡿⢻⣿⠟⠃⠀⠀⠀⠀⠀⢻
⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⡀⣿⠉⢻⡄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⠞⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠸⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠾

⠀⠀⠀⢀⣤⣤⣤⣤⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠐⠄⠀⠠⠀⡄⠈⠩⢻⡇⠀⠀⠀⠀⣀⣀⣀⣀⢀⣀⣀⣀⣀⡀⠀⠀⠖⡶⠲⠒⠒⠲⢒⠒⢚⣷
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠈⠀⢸⡇⠀⠀⢀⣯⣩⣟⣿⣿⡶⣷⣾⣯⡟⡇⠀⠀⢀⡂⢩⠖⣄⢀⠀⠘⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡛⢌⠙⢯⠿⣿⣟⣭⣽⣟⡇⠀⠀⢀⣠⠇⣰⣋⢈⠀⠉⠁⢸⡇
⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢠⡌⡉⣣⠜⣇⢁⣟⣤⠽⡇⠀⠀⠀⠀⠀⢅⣀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⡇⠀⠀⠀⠂⠈⠂⠾⠓⢹⣎⠝⣷⡄⡇⠀⠀⠀⠀⠀⢀⢀⣘⢦⣀⡀⢸⡇
⠀⠀⠀⠀⠠⠀⠀⠀⢷⣾⡇⠀⢸⠇⠀⠀⠀⠀⠀⠊⠀⠀⢸⡟⢹⡿⠁⡇⠀⠀⢀⠠⠒⢡⠂⠀⠉⠑⠛⢺⡇
⠀⠀⠀⠠⠀⢡⠀⠀⠈⣿⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡇⣰⠇⠀⡇⠀⠀⠂⠀⠀⠘⢆⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠐⠀⠂⠀⠀⠹⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⣦⠎⣀⠿⠃⠀⠀⡇⠀⠀⣧⠀⠀⠀⠀⢣⠀⠀⠀⢸⡇
⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⢹⠎⠁⠀⠀⠀⢸⡇⠀⢀⣿⡇⠀⠀⠀⠀⢣⠀⠀⢸⡇
⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠐⠄⣸⠀⠀⠀⠀⠀⡐⡡⠁⠀⠀⠀⠀⠀⢸⡇⠀⠘⣿⠄⠀⠀⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠉⠀⠀⠀⠀⢀⣾⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡇⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡋⢙⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣤⣀⠀⠀⠀⠀⠀⣾⣿⣿⣏⣹⣿⣿⣿⣿⣯⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣧⣽⣿⣷⣄⠀⠀⠀⠘⠛⠛⠛⠛⢿⠟⠛⢻⡟⠉⠉⠉⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⢿⣿⣿⣿⣿⢿⣿⣦⠀⠀⠀⠀⣀⣀⡀⣼⠀⠀⢸⡇⢀⣴⣿⡿⠟⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿⣿⣷⣾⣿⣿⠀⢀⣴⣿⠟⣿⣿⣿⡀⠀⢸⣧⣾⣿⣿⣿⣦⣿⣿⣻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢉⡍⠙⡏⠉⠉⠀⣴⣿⣿⣷⣦⣾⣿⣿⣿⡄⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢦⣄⠀⠈⡇⠀⡇⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡸⣿⣷⣼⣿⣿⣿⣶⣿⣿⣿⣾⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣾⣿⣦⠀⣷⠀⢻⠀⣼⣿⣿⣿⣿⣯⡸⣿⣿⣿⣿⣿⡇⡿⠛⠛⣻⠛⠛⠛⡟⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣙⣿⣟⣡⣿⣿⣷⡰⡇⠘⠀⠘⠛⠛⠛⠻⡿⠿⠿⣿⠿⠿⠟⢡⡇⠀⠀⡏⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠹⠉⠉⠀⣃⣄⣀⣾⠂⠀⠀⠀⠀⡇⠀⠀⡟⠀⠀⠠⢸⠇⠀⢸⠱⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡆⠀⠀⠁⢣⠀⠙⡆⠀⠀⠀⢰⡇⠀⢠⡇⠀⢀⡀⣾⠀⢀⡏⠁⢀⣴⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢰⣴⣶⣦⡘⡆⠀⢸⡀⠀⠀⢸⠀⠀⠘⣡⡾⣿⣿⡆⠀⡼⠀⢠⣾⣟⣿⣅⣽⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⣿⣿⣿⣷⠀⠈⡇⠀⠀⡞⠀⢰⣾⣿⣿⡿⣿⣿⡌⠁⠀⠘⣻⠛⠿⠿⡿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠿⣿⠿⡟⠋⡇⠀⢸⣶⣶⡇⠀⢸⡟⣻⠛⢻⠿⢿⠇⠀⠀⢠⠇⡌⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢹⡀⢹⡄⢹⢀⣿⣧⣿⣿⠆⢸⠁⠇⠀⣾⡴⠃⠀⠀⢠⠏⣸⠁⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣾⣄⢧⠘⢷⡘⡆⠀⡏⢸⠁⠀⣿⡜⠀⣴⠋⠀⠀⡠⣠⢳⣌⠁⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣼⣟⠈⢆⠀⢳⣇⠀⡇⢸⠀⢠⡿⠁⢰⡏⠀⣠⢞⡴⠇⡁⠈⢳⢤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣆⠈⢦⠘⣆⠀⠙⣄⡷⢸⠀⡼⠁⢀⡞⠀⡪⣐⢩⠴⠊⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣄⠀⠀⠈⠣⡈⠧⠈⢧⠀⠈⠃⠘⠋⠀⠀⠚⠋⠉⠈⠀⠀⣠⠔⠋⠀⠀⢠⡤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡀⠀⠀⠀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠆⠀⢸⣿⣷⡄⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠰⠀⠈⠛⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡇⠀⡀⠀⣼⣿⣿⣿⣿⣷⠀⢀⠀⠸⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠋⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠙⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⢸⠀⡇⠤⢀⠀⠀⠀⠀⣈⠂⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⢀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂
⠃⢹⠀⡗⠢⠠⢨⣀⣀⣀⡀⠀⠠⠀⠀⠀⢠⠀⠀⠀⠑⢀⠀⠀⠀⠀⠀⠑⠦⣀⠀⠀⠀⠀⡆⠱⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇
⠀⢸⠀⡇⠀⠀⠸⠈⠇⢹⠘⠑⢲⠦⣤⣴⣤⣷⣶⣦⣤⣾⠗⠒⠲⢦⢤⢤⣐⣬⣷⠖⣸⢻⢹⠉⠏⡏⠛⠒⠲⠒⠖⠒⠆⠇
⠀⢸⠀⡇⠀⠀⠠⠀⡀⢸⠀⠀⢸⠀⠁⢸⠪⠁⡃⣿⢨⠠⠀⠀⠀⢸⢸⠀⢸⣿⠛⣛⠉⠸⢸⠀⡄⡇
⠀⢸⠀⡇⠀⠀⠀⢀⡇⢸⠀⠀⢸⠀⣆⠀⢀⡀⣇⣟⣸⣀⠀⠀⠀⢸⢸⠀⢠⣸⣀⣏⣀⠀⢸⠀⠁⡇
⠀⢸⠀⣇⠀⠠⠰⠐⠓⠛⠈⠘⠋⠋⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠐⠚⠀⠃⠛⠋⠘⠉⠉⠉⠉⠃⠃
⠁⢸⠀⡇
⠀⠸⠠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠋
⠀⠀⠀⠀⠀⠀⣸⣟⣟⢟⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠹⡜⢮⣓⠭⣿⣛⣛⣛⡿⣖⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠳⣬⣙⣓⣒⡒⠲⢭⡒⢝⠪⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠲⣝⢦⡱⣌⢮⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠻⣌⢳⡙⢮⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡙⣦⡙⣦⠙⣮⡓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡈⣳⣌⢳⣌⢻⣦⡙⣶⣄⣀
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠖⠛⠛⠛⠛⠳⢦⣄⡀⠀⠀⠀⣀⣤⠶⠛⠛⠛⠛⠓⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡴⠛⠉⠉⠉⠙⠛⠶⢤⣤⣤⣤⣤⠶⠞⠋⠁⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢠⡟⠀⠀⠀⠀⠀⠀⣴⠟⠉⠙⠛⠛⠛⠋⠉⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣼⠁⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⣷⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣿⡄⠀⠀⠀⠀⠀⠀⠀⣷⠀⢀⣠⣤⢤⣄⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣤⣤⣤⣀⡀
⠀⣀⣠⡤⠶⠛⠋⠁⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⣿⡴⠋⠀⠀⠀⠀⠙⢶⣿⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠉⠙⠓⠶⣤⣄
⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡾⠀⠀⠀⣠⣤⢤⣤⡀⠀⠀⢻⡄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⢀⡾⠋⠀⠀⠈⠻⣆⠀⠈⣷⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⣿⠀⠀⣸⠃⠀⠀⠀⠀⠀⢻⡀⠀⢹⡄⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⡿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢠⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⡇⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢸⡇⠀⠀⡿⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⡇⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⡇⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣇⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣸⣇⣠⠟
⣶⣶⠶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣠⣤⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⡄⠘⡄⢠⢃⡨⠛⠧⣽⡤⠃⠠⠿⠄⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⣧⡟⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠛⠯⠤⠤⠔⠊⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠰⢿⠀⠀⠀⣄⠫⠑⠀⡀⠀⠀⠀⠠⠀⠀⠈⠇⠒⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟
⠀⠀⠀⠀⠀⢀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠀⠀⠀⢿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠐⠐⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⠀⠉⠉
⠀⠀⠄⠀⠀⠀⠀⠾⠆⢿⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠠⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠄⢀⣴⣿⠟⠁
⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠁⠤⢘⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠇⠀⠀⠀⢻⣿⣿⡀
⠀⠀⠘⠀⡒⠈⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⡀⠀⠀⠀⣾⡇⠀⠀⢠⡶⣆⠀⠀⠀⠛⠛⠛
⠀⠐⠂⠀⠀⠀⠀⠳⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠀⠀⠀⠈⠀⠀⠀⢸⣽⣿⡇⠀⢀⢀
⠀⠀⠀⠀⠀⣀⡀⠳⠆⠀⠀⠀⠈⠀⠁⠀⠀⠀⡀⠀⠀⣠⣶⡧⠀⠀⠀⠀⠀⠻⠓⠀⠙⠛
⠠⠀⠀⠀⠀⢙⡃⠀⠀⠀⠐⠂⠀⢠⣤⡄⠀⠻⡟⠀⠀⠘⠁⠀⠀⡀⠀⢀⡴⡶⢢⠀⢀⣦⡀⠀⠀⠀⠀⢴⡷
⠀⠐⠢⠀⢰⣏⣷⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⣤⣤⡄⠀⣾⠿⠀⠘⢷⣏⡿⠁⠈⠰
⠀⢈⠡⠀⠹⣾⡿⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠑⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠾⠮⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠛⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠆⠀⢰⠀⠀⠆⠀⠀⠈⠀⢀⡤⠒⡻⣏⠑⠢⣄⠀⠀⠀⠀⢒⠀⠗⠀⣀
⣐⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⣏⣤⣤⡧⠼⣦⠤⢬⣇⠀⠀⠠⠈⠀⠀⠀⠐⠀⠀⢀⠠
⠀⠅⠀⠂⠀⢲⡶⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢈⠉⠲⣾⣀⠀⣹⡖⠉⢸⠀⠀⠙⠀⠀⠀⠁⣰⣰⡀⠘

⠠⠀⠤⣄⠀⠠⡄⠠⡖⡄⣄⠀⠀⠀⠀⠀⡄⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⡄
⢰⣤⣄⡈⠙⠢⠈⢆⠁⠁⡯⢢⠀⠀⠀⠀⣑⣀⢠⠆⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠄⠒⠀⠉⠠⠋⠀⠀⠀⠀⡇
⢨⣀⣀⣈⣙⡆⣀⡝⠀⠊⠁⢀⣴⡀⠀⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⠀⠔⠀⠁⣀⣤⡶⠶⠶⠄⠃⠀⠀⠀⠀⡇
⢘⠉⠁⣠⡤⠶⠀⠀⠀⠀⠀⠘⠠⣤⡤⠀⠀⠀⠀⢀⡀⠀⣀⠀⢠⣮⣤⣤⡄⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⠟⠛⢉⡲⠆⠀⠀⠀⠀⠀⠀⠚⣄⡀⠠⠀⠀⠀⠀⠡⠖⠃⠀⠉⠉⠉⠉⠁⠄⠀⠀⠀⠀⠐⠓⠫⣷⠄⡐⡇
⢀⢀⣴⠿⠛⡷⣀⡀⠀⠀⠀⠀⢎⠉⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⡇
⠸⠿⠁⢀⣼⠟⠉⣷⠋⢳⣿⠓⢬⣣⡀⠈⠀⠤⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠘⢠⠀⢀⠩⠥⠀⠛⠂⠀⠿⠄⠀⢹⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇
⢘⡀⠐⠀⠰⠈⢠⠈⢀⠰⠆⠠⠆⠐⠄⢀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⣀⣀⣀⣀⣀⡀⠀⠖⠁⠀⠂⡇
⢀⠀⠦⠀⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⡈⠁⠈⠈⠐⠐⠂⠄⠌⡀⣈⠙⠋⠉⢉⠉⠙⠢⠤⠄⠀⠠⡇
⢈⠁⠄⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⢀⠠⠒⠅⠀⠀⢀⠀⠈⠁⠘⢂⣀⠤⠭⠉⠑⢒⠺⠇
⠘⢀⠐⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⡛⠷⠀⠡⠤⠀⠀⠐⠒⠺⢋⠉⠃
⠘⠀⠐⠈⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠋⠛⠋⠃
⠐⠀⠄⠈⣠⠐⢀⠠⠐⠤⠀⢀⠀⢀⡤⠀⠤⠤⠎⠀⠀⡀⠠⠀⢦⡀⠀⠒⠒⣶⣶⣖⣶⠆⠀⢿⢽⣛⣻⠷⡇
⠘⠁⡀⠓⡀⠂⠀⠤⠈⣀⡈⠥⠤⠒⠒⠘⠈⣉⣉⢭⢓⢄⡀⠀⠀⠁⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠿⠇
⢠⠀⢀⡀⣥⣐⠦⠻⣽⠒⢙⡆⠀⠀⠀⠀⠀⡱⡀⢠⡗⠁⡊⢆⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠰⠦⢽⡍⠀⣿⡄⠀⢿⡀⠈⡇⠀⠀⠤⢀⡀⠙⠙⠓⢀⣄⡀⠅⣺⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢘⠒⠒⡇⠀⡯⡇⠀⣿⡇⠀⡇⠀⠹⣤⣼⡅⠀⠀⠰⠟⡣⠈⠰⡒
⢨⡭⠭⡇⠀⡭⣇⣀⠿⢧⡴⠷⠀⠀⠁⠀⠄⠄⠀⠀⠀⡀⠀⠀⠀⠀⣂⡀⢀
⠰⠖⢒⡓⠠⣭⠉⠐⠾⣁⡨⠆⠀⡀⠀⠀⠶⠞⠀⠀⠀⠀⠂⠀⢠⡐⠠⠯⢭⣆⠆
⠘⠛⠙⠂⠐⠛⠑⠛⠐⠒⠒⠒⠂⠠⠒⠒⠒⠒⠒⠒⠒⠒⠒⠙⠒⠈⠛⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣠⣤⣤⢤⣤⡤⠤⣤⡤⠤⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⢉⡿⡋⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⠴⠤⣤⠤⣤⣤⣄⡀
⠀⠀⠀⣼⢯⡉⠒⠀⠀⣀⣀⡀⠀⠐⠊⢁⡽⡆⠀⠀⠀⠀⠀⠀⣠⡿⠋⠁⠛⠁⠛⠂⠛⢂⠀⠀⠀⠀⠀⠀⠀⣜⡏⠈⠋⠁⠀⠀⠀⠀⠀⠐⠋⢹⣻⡄
⠀⠀⠀⣿⠀⠁⠈⠀⠀⠈⠉⠀⠀⠊⠀⠘⢦⣷⠀⠀⠀⠀⠀⣰⡯⣀⡄⠀⠀⠆⠀⠀⢤⡐⢂⠀⠀⠀⠀⠀⠀⣿⠟⠦⣄⠀⡀⠀⠀⠀⠀⣀⠤⢻⣽⡇
⠀⠀⠀⡿⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⢠⣿⠜⠉⠁⠀⠀⠁⣀⡀⢜⠓⢸⡆⠀⠀⠀⠀⢰⡟⠀⠀⠈⠳⢤⠄⢠⡔⠊⠁⠀⠀⢿⡇
⠀⠀⠀⣿⣆⢠⡃⠀⠀⠀⠀⠀⠀⠀⢁⠀⣸⣿⠀⠀⠀⠀⢸⣧⣄⠀⢀⡿⠀⠀⠸⢅⢸⣇⣸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸⣤⡴⠇⠀⠀⠀⠀⣸⠇
⠀⠀⠀⣿⣻⣿⡶⠃⠀⣀⡀⠀⠀⢠⣼⣷⠏⡟⠀⠀⠀⠀⢸⣿⣿⣿⣧⡀⢶⣠⡇⣌⣻⣟⣿⠇⠀⠀⠀⠀⢸⣿⣄⣤⠖⣠⠎⠀⠀⠐⢄⠀⢄⣰⣿
⠀⠀⠀⢿⣿⣴⣿⣞⠋⠀⠀⢙⣲⣾⣿⣻⣷⠇⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣾⣿⡶⣟⣿⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⣤⣔⣁⠀⠀⠀⠀⣈⣳⣤⣿⣿
⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⠿⠽⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠷⠶⠿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⣀⡀⠀⠀⠜⠑⠖⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⠄⠀⠰⠄⢠⣀⣀
⠀⠀⠀⠀⡐⠁⠀⠀⢀⠤⠀⠐⠂⠀⠀⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢏⡀⠀⠀⠀⠀⠄⠀⠈⠋⠉⠉⢄
⠀⠀⠀⢀⡠⠐⣀⠋⠀⠀⠀⠀⠙⢆⠐⠃⠀⠀⠀⠀⠀⠀⢀⣤⣦⠄⣴⡅⢠⣄⠀⢠⡄⠀⢄⠀⠀⠀⠀⠀⢠⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⢸⡀
⠀⠀⠀⢧⣀⠀⢇⠀⠀⠀⠀⠀⠀⠸⣆⣀⣤⠗⠀⠀⠀⠀⣸⢷⠀⢠⠇⠀⡠⣅⠀⡀⢣⠀⠸⡆⠀⠀⠀⠀⢸⡷⣄⠌⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⣸⡇
⠀⠀⠀⠈⢿⣿⠛⢳⡀⣀⣀⠀⣶⠉⢻⣯⠏⠀⠀⠀⠀⠀⡏⠘⣄⡞⠸⣄⠇⢰⣠⠃⢸⢀⠇⢡⠀⠀⠀⠀⠘⣿⣿⠃⢶⠀⢀⡀⣀⡀⢠⣦⣼⣾⣿
⠀⠀⠀⠀⠈⢿⣤⣶⣿⡍⢀⣽⣛⣶⡿⠟⠀⠀⠀⠀⠀⠘⠓⠶⠟⠳⠒⠛⠲⠚⠻⠶⠾⠿⠶⠶⠂⠀⠀⠀⠀⠘⠿⣿⣿⣿⣯⣷⣽⣷⣿⣿⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠒⣲⢶⡒⢢⡄
⠀⠀⠀⣴⠫⠀⠀⢤⡄⠀⠀⠉⠀⣀⣴⡳⢦⡀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠀⠘⠀⠀⢀⡂⢄⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠃⠀⡄⠀⠀⢡⡈⠀⠍⣀
⠀⠀⠀⡇⠓⣦⠀⠠⠀⠀⠠⠄⠠⢉⠁⢰⣿⡇⠀⠀⠀⠀⠀⠀⡙⠢⡀⠑⣠⡄⠈⢀⡵⠛⠀⠀⠀⠀⠀⠀⠀⡼⢁⣇⢀⣠⠴⠚⠛⠢⣄⠀⢠⡏
⠀⠀⠀⡧⠈⠁⢸⠀⠀⠀⠀⠀⠀⢸⣷⣿⣯⣡⠀⠀⠀⠀⣔⠴⠀⠠⠞⢧⣸⠁⡹⠛⠀⠔⠄⠃⠀⠀⠀⠀⢰⣁⣤⠜⣿⠃⠀⠀⠀⠀⠘⣿⠛⢴⣦⡀
⠀⠀⠀⡷⡀⢀⢾⠀⠀⠀⠀⠀⠀⢸⡿⠙⣟⣿⠀⠀⠀⠐⢓⡒⢒⣿⠓⢶⣿⣿⠦⠼⣿⣶⠶⣾⠆⠀⠀⠀⢸⣿⣯⢀⣿⠀⠀⠀⠀⠀⢀⣿⠀⣀⣽⠇
⠀⠀⠀⣇⣵⠾⢿⣦⣀⣀⣀⣀⣠⣾⣿⣦⠋⡿⠀⠀⠀⠀⢀⡞⢻⣾⣮⣱⠏⢿⣛⣾⡧⠼⣿⡿⠀⠀⠀⠀⠈⣿⣷⣿⠿⣧⣀⠀⠀⣀⣼⠿⣿⣽⠻
⠀⠀⠀⢿⣏⣀⣸⣿⠟⠁⠈⠙⢿⣧⣈⢻⣷⡇⠀⠀⠀⠀⣴⣞⣉⡿⠗⡿⣟⡿⣿⢾⣿⣧⣿⠇⠀⠀⠀⠀⠀⠹⣿⣯⣤⣼⡿⠛⠻⣿⣃⣤⣹⡷⠃
⠀⠀⠀⠀⠙⠿⣿⣿⣷⣤⣤⣴⣯⣽⣿⠿⠋⠀⠀⠀⠀⠀⠀⠙⠻⣷⣸⣤⣿⣧⣹⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠙⠿⣏⣿⣷⣄⣾⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠟⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠆
⠀⠀⠀⠀⠀⡟⢲⡶⣦⡭⣭⡭⣥⡶⣶⡞⡇
⠀⠀⠀⠀⠀⣇⣸⡇⣿⡇⣿⡇⣿⡇⣿⣇⡇
⠀⠀⠀⠀⠀⠈⢙⣛⣛⣓⣛⣓⣛⣛⣛⠉⠁⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠻⣿⡄⠀⠀⠀⣿⠉⡍⢩⠉⡍⢩⠉⡍⢩⠉⡍⢩⠉⡿
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣷⡜⢿⣆⠀⢠⡿⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣧
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⠻⣷⡟⠀⠙⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇⠀⠀⠀⠀⠈⠙⠓⠶⠤⣤⣤⠤⠖⠋⠈⣷
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢿⡇⠀⠀⢀⣠⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⡀⢹⡄
⠀⠀⠀⣿⣿⣿⣿⣿⠉⠙⠛⠛⠋⠉⠉⠁⠀⢸⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⠰⠶⠶⠶⠆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣏⣙⣛⣛⣛⣛⣉⣭⣭⣵⣶⣾⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣀⢀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⢸⣠⣧⡏⠉⢸⢸⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠛⠛⠉⠉⣿⣿⡆
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠘⠀⠃⠙⠋⠈⢸⣿⣿⣿⣿⣿⣿⠀⠿⠿⠿⠟⢛⣂⣀⣤⡀⢻⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⠠⣶⣶⣶⣶⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣤⣘⣛⣛⣩⣭⣭⣭⣶⣶⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⠀⣶⣶⣶⣶⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣶⣶⣦⣤⣴⣶⣶⣾⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠛⠃
⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠁

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠀⠲⣴⠄⠀⠀⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⠀⠀⠀⠀⣠⣤⣀⡀
⠀⠶⠐⢲⠆⠀⠀⡴⠔⢢⠆⠀⠀⢰⣄⢰⣄⠲⣀⡀⠀⠀⠸⣿⣷⡄⠀⢐⣆⠀⠀⢐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣿⡀⠀⣠⣿⣿⣿⣿⣷
⠀⠳⠘⠏⠲⡄⠀⢻⠚⡎⠲⡄⠀⠈⠿⠀⠟⠀⠈⠻⠄⠀⠀⠸⠏⢿⣦⠈⢻⢷⡀⠘⣿⣦⡀⣶⣾⣶⠀⢠⣶⣶⡆⠀⢰⣿⣽⣿⣿⠂⠀⠠⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠊⠃⠐⠲⠞⠃⢲⠶⠘⠁⠙⠟⠛⠁⠈⠻⠟⠋⠀⠈⠀⠉⠉⠉⠀⠀⠀⠉⠙⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⡿⡿⢿⣆⠀⠀⢰⠀⠀⠀⠀⡆⠀⠀⢠⠁⠀⠀⣡
⠀⠠⠤⣤⡀⠀⠠⠤⣄⠀⠀⣀⣀⠀⠀⢺⣿⡿⣿⡗⢀⣠⣷⣄⠀⣠⣶⣄⢸⣿⡉⠙⠳⠮⣿⣠⣞⣿⣿⣆⣰⣾⣿⢷⡄⢀⠈⠉⠁⣀⠀⣀⠂⠀⢠⣚⡀
⠀⠳⢤⣴⣷⡌⠲⣤⣼⣷⡜⢦⣬⣷⡄⣼⣯⣭⣿⡧⢀⣿⣻⣏⢀⣿⣻⣏⠐⠿⣿⣿⣿⡿⠿⠸⣷⣾⣿⡿⢿⣶⣾⣿⣇⣼⣶⣶⣶⣶⣀⣆⣠⡀⣰⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠉⣁⡀⠉⠩⡁⢀⣀⣤⣬⣄⣀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠈⠉⠉⠉⠁⣈⡉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣤⣤⣤⣤⡶⠛⠉⠀⠀⢀⣴⣿⣿⣤⣴⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⡀
⠀⣤⣤⣤⣤⡴⠚⢁⣤⣤⣤⣤⡖⠋⠀⠉⠻⣿⣿⣿⠟⠋⠀⠀⠀⠀⣸⡿⠛⠻⠏⣁⡀⠀⠀⠀⢀⣤⣖⣤⡄⠀⢀⣤⢖⣤⡄⠶⣶⡟⣿⡿⠁⠀⣰⣦
⠀⠀⠙⢻⠋⠀⠀⠀⠈⠛⣿⠉⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠙⢍⠀⠀⢀⡟⠛⠋⠍⠀⢀⡟⠛⠋⠍⠀⠀⠉⠋⠙⠁⠐⠶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⣄⠘⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠘⠀⠉⠀⠀⠂
⠀⠀⠀⠀⠀⣴⣿⣟⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⡄⠂⣀⠀⠀⢀⡀
⠀⠀⢀⣄⠀⣿⣿⣿⣿⣿⣿⡀⢀⣶⣶⣶⡄⠀⣴⣾⣶⣆⠀⢹⠿⣟⠆⢸⣷⣔⠈⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⡀
⠠⣤⣻⡿⠀⣿⣿⣿⣿⣿⣿⠆⢸⣿⣿⣿⡿⠈⣿⣿⣿⣿⠀⠀⠠⠹⡀⠀⠋⢏⠀⠘⡿⡁⢀⡄⣀⠀⠀⡄⢀⠀⠀⣾⣷⣶⠛⠁⠀⢀⣾⣀⡦
⠀⠀⠉⠀⠒⠛⠿⠿⠿⠿⠿⠋⠼⠿⠿⢿⡿⠠⠿⠿⠿⡿⠀⠀⠀⠀⠡⠀⠀⠈⠀⠀⠀⠁⠸⣿⣧⠀⠸⢿⣯⠀⠰⠿⢿⣿⡶⠶⠂⠸⢿⣿⣤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠢⡀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢠⠜⢀⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠑⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⠭⠭⠿⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣴⢦⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡶⣦⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠀⣠⣀⠀⠀⠀⠉⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠻⠟⠀⠀⣀⡀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⠟⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠟⠛⠉⠉⠫⠹⠿⣿⡿⠟⠛⠉⢏⡹⠉⠉⠉⠛⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣀⣸⡇⣴⠖⠒⡛⠛⢛⠛⠛⣛⡛⣓⠒⠳⣦⠀⠀⠀⣼⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣠⣼⢠⡏⠁⠁⣿⠀⠀⣿⣶⡦⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠿⠀⢸⠈⠷⣤⡤⠿⠀⠀⣿⣿⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡟⢻⠏⠛⠷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣯⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⣀⣤⣿⣿⣿⣿⣿⣤⣿⣿⡋⠀⠀⠀⣨⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⠿⠋

⠀⠀⠀⠀⠈⠉⠙⠻⢿⣿⣿⡿⢽⣾⡿⣿⣿⣿⢫⡙⢧⡍⢿⡇⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠊⠀⢰⣿⣿⠁⣾⣷⠈⢃⣾⢹⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠯⢿⠃⠰⣿⣿⡀⠈⣿⣾⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢻⣏⡄⠂⢸⢹⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⣌⣿⡌⠸⢻⠘⡏⢹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡡⣄⣆⣿⣵⠸⣾⢰⠃⠹⣷⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡤⠤⠄⠈⣷⠛⠉⠁⠈⠈⠙⠸⠆⣶⠀⡉⠃⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⠘⡀⠈⢿⣩⣿⣿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢷⣶⣸⣯⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⠿⠿⢿⡄⠀⠈⠄⠀⠀⠠⠀⠐⡇⢀⠀⢹⡼⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢰⡀⠀⢰⣿⠀⣹⠇⠂⠐⠸⣷⣌⠁⠀⠆⢠⠀⡸⢰⡇⢸⠂⢸⡇⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢰⡖⡁⠀⣀⣽⠟⠉⣠⠀⠁⢠⣽⣻⡽⢿⠿⢿⠀⣷⢠⡇⢸⢇⢸⣧⣿⣿⣿⣿⣿⣿⣿
⠀⢀⢠⣾⣿⣦⣾⣟⢀⣴⣾⣿⠀⠀⢸⣿⣿⣿⣆⠹⠜⠀⣿⢸⡇⢸⡀⣽⣿⣿⣿⣿⣿⣿⣿⠃
⠀⡀⢻⣿⣿⣿⣿⣿⣾⣿⣿⡇⠀⠀⣸⣟⣿⣿⡏⠩⣷⠀⠀⠈⠃⠸⠷⠹⠿⡿⠿⣟⣿⡟⠁
⠀⢻⣾⣇⣭⣽⣷⠏⣿⣿⣿⡇⠀⠀⣾⣿⣿⣿⣧⠀⣿⠀⣿⢰⡇⢰⣶⢰⣶⣿⣿⡿⠋
⠀⠶⣿⡿⣿⢿⠁⠀⢸⣿⣿⣷⠀⢀⣿⣿⣿⡿⠛⢰⣿⠀⣿⢈⡇⢸⣿⣾⣿⣿⠋
⠀⠀⢀⢹⣾⠗⠀⠀⢸⣿⣿⣿⣄⣼⣿⣿⠟⠀⢀⣾⣿⡀⣿⢸⡿⢺⣿⣿⣿⠃
⣴⣦⣶⣶⣾⠆⠀⠀⢸⣿⣿⣿⣿⣿⣿⠋⠀⠀⢸⣿⣿⡀⣿⣰⡇⢼⣿⣿⠃
⠛⠻⠿⠻⠋⠀⠀⠀⠀⣿⣿⣿⣿⡿⠁⠀⠀⠀⢸⣿⣿⠏⣿⣿⣧⢸⣿⡏⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠸⣿⣿⣿⣿⠻⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⢿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡻⣟⢿⡻⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢹⣧⢻⣎⢷⡝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠈⣿⡎⣿⣎⢿⣎⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⢸⣿⡇⢸⣿⡌⣿⣦⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⢠⣿⣿⡇⢸⣿⡇⢸⣿⣆⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡿⠋⣠⣿⣿⡟⠀⣿⣿⣷⠘⣿⣿⡀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⠿⠋⣠⣾⣿⣿⡿⠁⣸⣿⣿⡇⠀⣿⣿⣇⠘⣿⣿⡄
⣶⠶⣶⢶⢶⢶⣶⠶⢶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣷⣾⣿⣿⣿⣷⣾⣿⣿⣿⣾⣿⡿⣿⠶⣶⣶⢶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠆⠉⠙⠉⠉⠠⠦⠄⢀⣀⣀⣀⣀⣀⣀⣀⢀⣀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠈⠉⠙⠉⠛⡀
⠀⠀⠀⠀⠀⠀⢀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠰⠧⠤⠄⣀
⠀⠀⢀⣠⡄⠒⠛⠓⠒⠒⠒⠒⠒⠂⠀⠀⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠒⠒⠒⠒⠒⠒⠂⠀⠈⠉⠉⠀⠀⠀⠀⠀⠉⠑⢢
⣀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠛⠦⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣌⠳⠆⠈⠉⠉⠉⠉⠠⠾⢋⡴⢆⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠴⠖⠉⠁
⠀⠀⠀⠀⣿⠛⡟⠀⠠⠤⠤⠤⠤⢀⡴⣛⡾⠁⣀⡀⣀⠀⢀⠀⠀⠀⠀⠀⠠⠴⠶⢌⡷⡄⠚⠛⠋⠁⠈⠻⢄⣘⡷
⠀⠀⠀⡌⢿⠏⠀⠀⠀⠀⠀⠀⢀⢮⡙⠁⠀⠀⠀⠀⠀⢰⠄⠀⢴⠄⠀⠀⠀⠀⠀⠈⠳⣡⡄⠀⠀⠀⠀⠀⠀⠹⠇⣴
⠀⠀⣴⠃⠘⠀⠀⠀⠀⠀⠀⢠⡙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠋⡸⠂⠀⠀⠀⠀⠀⠀⢸⠉⡆
⠀⠀⣅⢠⡆⠀⠀⠀⠀⠀⠀⡈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⣴⠀⠀⠀⠀⠀⠀⢸⠀⡘
⠀⠀⠈⠲⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠟⠁
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠻⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⢰⡟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢻⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⢴⣣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡝⠃⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠙⢾⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢾⠓⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣀⠀⠀⠀⠀⠙⠿⣦⣤⣀⣀⠀⠀⠀⠀⣀⣀⣠⣴⠷⠋⠁⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢄⡀⠀⠀⠀⠈⠛⠛⢷⡾⠷⠾⠟⠛⠉⠀⠀⠀⠀⢀⣠⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⠂⢤⣤⡄⠀⢸⡂⠀⠀⣿⠂⢠⣤⡤⠰⡖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⢹⠃⠀⠀⣟⠀⠀⠀⠀⢀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣼⡁⠀⠀⠉⠉⠉⢹⡇⠀⠀⣿⣉⡉⠉⠉⠀⠙⣧⡀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⢸⠉⠳⣄⠀⠀⠀⠘⣇⠀⠀⠟⠈⢡⠀⢀⡴⠛⠋⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⢀⡌⠀⠀⠈⢳⠄⢀⡜⠛⠳⠤⠀⠀⠈⠾⠯⠤⢀⣸⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⣼⣧⣤⡤⠰⠛⠲⠞⠙⠲⠀⠀⠀⠀⠸⠀⠀⠀⠈⠉⠑⠒⢠⣄⠀⠀⠘⣆
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⠷⠶⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠶⠆⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡦⠤⠤⠀⣾⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠈⢤⠤⠀⠀⠀⠀⢀⣀⡀⣀⣀⣀⡀⠀⡀⠀⠀⣇⣀⣀⣀⡈⠉⠿⢂⠒⠒⠀⢤⣤⣄⣀⡀⡀⢀⡀⢀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠉⣟⠉⠉⢻⠋⠹⠟⠛⢩⣽⠛⠏⣭⡅⠴⡉⢹⠖⠒⠐⡟⠈⢿⡁⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠒⠋⢉⠉⠹⢇⠿⠲⠶⢶⡄⠀⠀⣰⠴⠶⢧⣸⠇⠀⠈⠙⠆⠛⢃⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⣰⣄⣠⣾⡦⢾⣿⣦⡀⠀⠀⠀⣠⡟⢹⡟⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠠⣿⡟⠛⠿⠀⠙⠁⣠⡛⠀⠱⠀⠀⠀⡛⠁⡾⢷⡄⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢿⣥⠄⠒⠒⠈⢀⣍⠉⠀⢿⠀⠀⣠⠃⢸⠀⠀⠳⡀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⢸⣷⠲⠂⠈⠉⣹⠏⠉⢹⣿⠀⢠⡋⠀⠿⢢⡄⠀⢣⣀⡬⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠶⠛⠉⠠⠷⠤⠰⠾⠛⠳⡄⠙⠾⠆⠾⠇⠀⠈⠁⠖⠁⠶⠶⠄⠀⠀⠀⠀⠠⠶⣷
⠀⠀⠀⠀⠀⠀⢰⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠
⠀⠀⠀⠀⠀⢀⡆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡄
⠀⠀⠀⠀⠀⡌⠁⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠘⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢡⡀
⠀⠀⠀⠀⠀⠏⠀⠀⠱⢆⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠸⡇⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠈⢃
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠸
⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠧⠤⡤⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣤⡀⠐⠀⣠⣄⣠⣀⣀⣀⣀⣀⡀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠤⢞⣛⡿⢿⠿⣛⣛⣫⣿⣬⡁⣿⡏⣨⣿⣯⣛⣛⣛⠿⠿⢿⣛⠢⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⡷⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢥⠶⢂⣀⣀⣀⣀⡰⠶⠬⠓⠉⠘⠫⡴⠖⣀⣀⣀⣀⣀⠶⢦⠝⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠁⠀⠀⠂⠀⠀⠀⣀⡀⢀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠀⠀⠤⠤⠤⠀⠀⢀⠀⠀⢹⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠩⠉⢉⣁⣀⠀⠠⢼⡯⡀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠉⠐⢄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣒⣒⡒⠒⠆⠀⠀⡜⠧⡀⠰⠀⠀⠀⠀⠀⠌⠀⠀⠁⠀⠀⠄⠀⠠⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠤⠀⠤⠭⠭⢀⣀⣀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠁⠀⠀⠐⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠉⠉⠉⢁⣐⡒⠢⠠⠭⣁⣺⢀⣀⠀⠀⠀⡂⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣒⣒⡒⠒⠢⠬⠍⣉⣒⠒⠼⣿⠉⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⡠⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⠤⠤⠤⠭⠍⣁⣒⠂⠠⠬⢹⠿⠀⠀⠀⠀⠀⠂⠁⠂⠀⠀⠀⠐⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠉⠉⠃⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠉⠉⠉⠙⠒⠂⠤⢤⡨⣄⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⡿⠋⣴⣾⣿⠟⣩⣴⣿⠟⣩⣴⠿⠛⠉
⣿⠏⢠⣾⣿⡿⢁⣾⣿⠟⣵⣿⠟⠁
⡟⠀⣿⣿⣿⠁⣾⣿⠇⣼⣿⠁
⡇⠀⣿⣿⣯⠘⣿⣿⠀⣿⣇
⣷⡀⢻⣿⣿⡄⢻⣿⣧⡙⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣄⠻⣿⣿⣦⣙⠿⣿⣦⣝⣛⠿⠶⠶⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢿⣿⣿⣷⣤⡙⠻⢿⣷⣶⣭⣝⣛⣛⣛⣻⣯⡽⣿⣿⣿⠟⠋⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣦⣌⡙⠻⠿⣿⣿⣶⣶⣭⣭⣭⣭⣭⣭⠷⣞⣿⠿⠋⠁
⢿⣿⣿⣿⣶⣶⣦⣬⣭⣭⣭⣭⣷⠶⠞⠛⠉
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣊⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠉⣉⠉⠉⣉⠉⢉⣉⣉⠉⣉⠉⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⣿⣀⣀⣿⢰⡟⠉⠙⠇⣿⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⣿⠉⠉⣿⠸⣇⡀⣠⡦⣿⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣀⣉⣀⣀⣉⣀⣈⣉⣉⣀⣉⣀⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣤⣤⣤⣀⡀⠀⠀⠚⠏⠀⠀⠀⣄⣀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⣀⣤⡶⠿⠛⠉⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠉⠛⠿⣶⣄⠀⠀⠀⠚⠻⠁⠀⠀⠠⠴⢿⡟
⠀⠀⣠⡾⠛⠉⠀⠀⢀⣠⡤⠶⠚⠛⣋⣉⣩⣭⣭⣭⣛⣦⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠁
⢠⣾⠋⠀⠀⢀⡴⠞⠉⢀⡤⢖⣪⠭⠒⠒⢋⣩⡭⠿⠛⠉⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣼⣿⠏
⣿⠃⡎⣖⡆⡞⠀⢠⠞⡡⠚⠁⢀⣠⠶⠛⠉⠀⠀⠀⠀⢀⣀⣠⣤⣶⠟⠁⠀⠀⢀⡾⣻⠏
⣿⡄⢣⠳⠃⠳⢤⣏⣞⣀⡤⠞⠋⠀⠀⠀⠀⣀⣤⣶⠟⢛⣿⠉⠁⠀⠀⠀⠀⢀⡞⣰⠃
⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⠁⣠⢎⣿⡄⠀⠀⠀⠀⢠⡞⣴⠃⢀⣸⣦⠄
⠀⠈⠻⢷⣤⣀⣀⠀⠀⢀⣀⣠⣴⡾⠛⠉⢀⣠⠴⢊⡵⢋⠘⣧⠀⠀⠀⢠⠟⡼⠃⠀⠀⠀⠙
⠀⠀⠀⠀⠈⠉⠛⣿⣛⠿⠿⠯⠗⢒⣒⣉⠥⠔⠊⡥⢖⡡⠂⠹⣆⠀⢠⠏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠐⡩⠔⣡⠀⢻⣴⢏⡾⠁
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⡴⣢⣴⢃⡞
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠚⣵⢣⡞
⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⢡⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⣀⣀⣀⣠⣤⡴⠶⠛⡿⢡⠏⠀⠀⣰⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⢀⣾⣿⠏⠀⠀⠈⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠃

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢠⢷⡄⢀⠀⠀⠀⠀⢀⣠⢉⡀⠰⡄
⠀⠀⠀⢀⡀⠀⠀⣾⠀⢠⡆⢀⡖⠀⠀⠀⠃⢸⡇⠈⢺⠀⠀⠀⠈⡇⠈⠀⢸⡇
⠀⠀⠀⠀⠸⣶⡄⢿⠀⢠⣧⢫⠇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠤⠈⢿⠗⠀⣘⠷⡁⠀⠀⠀⠀⠀⢸⡇⠀⡆⠀⠀⠀⢀⠀⣀⣘⣼⡇⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠐⠒⠚⠱⠾⠻⠟⠁⠀⠀⠀⠀⠀⠘⠷⠘⠁⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⢐⡄
⠀⠀⠀⠀⠆⡀⠀⠐⠃⠀⠀⠰⠀⠀⠀⠀⡁⠀⠀⠈⠕⠄⠀⠀⠐⠀⠀⢀⡈⡀
⠀⠀⠀⠀⠀⢹⠀⠀⢀⣾⠁⢲⠀⠀⠀⠀⠉⠦⠀⡀⢀⠀⠀⠀⢰⠀⠀⠈⠀⠃⠀⠀⠀⠀⢰
⠀⠀⠀⠀⣿⠀⠀⠀⢸⡏⠀⠋⠀⠀⠀⠀⠀⠀⠀⣷⡌⠀⠀⠀⢸⠀⠀⠸⠸⠀⠀⠀⠀⠀⠛⠀⠀⡀
⠀⠀⠀⠀⢸⡦⣀⠔⢺⣷⡎⢠⡾⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠸⡀⠀⣬⡼⠀⠀⠀⠀⠀⠟⠀⠸
⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠀⠉⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⣷⠀⠀⠀⢀⠀⠀⠁⣠⡀
⠀⠀⠀⠀⠁⠀⡴⠀⠀⠀⢹⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⢻⠇⠀⠀⠀⠀⠀⠘⣿⠇
⠀⠀⠀⠀⠀⠚⠥⠐⠊⠁⠉⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⣠⠄⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⣀⠀⠀⣤⣤⣤⡄⢠⣤⣤⣤⣤⣤⣤⣤⡤⠀⠂⠠⡄
⠀⠀⠀⣤⣤⣤⣤⡤⠥⠤⢤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡶⠤⠤⠤⠤⠤⠾⠿⠦⠴⠿⠿⠿⠷⠿⠿⠿⢿⣿⣿⣿⣿⣷⠶⠶⡖
⠀⠀⠀⠀⡀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣻⣿⣟⣟⠉⠀⠀⠁
⠀⠀⠀⠀⠰⣄⣀⣀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⡬⢉⣽⣋⣩⠙⠁⡜⠁
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢠⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠠⠙⠛⠀⠘⠿⠿⠭⠍⠹⠻⠿⣿⣿⡿⣿⠗⠠⡴⠁
⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⡿⠻⡟⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣷⠂⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣁⠂⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣯⡿⢫⠃⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣾⣿⣳⣏⣀⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠻⠟⠛⠛⠻⣷⣾⣿⠿⠓⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠇⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⠀⠀⠀⠀⠀⣲⣶⣶⣛⣻⣏⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠞⠁⠈⠁⢈⡈⠩⠉⣣⠈⠁⠀⠉⠈⠁⣀⣫⡟⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠁⠉⢣⡉⠈⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠔⢉⣴⠀⠒⠢⠀⠀⠀⠀⠀⠁⢊⢀⡄⠒⣠⣀⣸⠷⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⡀⠀⠈⠓⣤
⠀⠀⠀⠀⠀⢀⣴⠋⢀⡉⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⢉⠉⣀⠉⠁⠀⡀⣿⠀⡀⠀⠀⠀⠀⠀⠀⢶⣶⣶⠾⠿⣾⠏⢿⣿⣄⢀⣉⣉⡱⣤⠀⡀
⠀⠀⠀⡠⠒⠀⠁⠀⠈⣿⠀⠀⢀⡐⠘⠁⠀⠀⠀⠀⠀⠏⠐⢩⠌⣧⠚⠁⠋⠀⣿⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⡁⠀⠀⠀⠀⠙⣦
⠀⠀⢰⠁⠀⢴⡶⢢⣴⣧⣠⡄⠤⢁⠁⠀⢰⠀⠀⠀⠀⢰⠀⠸⠀⠇⠀⡄⠀⠀⣿⣴⡆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⡄⠀⢿⣿⣷⣀⣀⠶⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠉⡿⠁⠠⠀⡤⠀⠀⢤⠀⣀⣠⠀⠀⠀⣴⢠⠀⠀⠀⡇⠐⠷⠟⣧⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢄⠀⠘⣿⣿⡟⣿⣦⡆⠀⡸⣄
⠀⡄⠋⡀⠈⠀⠀⠀⣼⠁⠀⠀⠠⠁⠀⠐⠂⠘⠁⠀⠸⠀⠀⠀⠸⠇⠀⠀⡀⠀⡀⠀⣿⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⢰⣿⣿⣿⣾⣾⠃⠀⡃⠘⣆
⠰⠁⢸⠃⠀⠸⣆⡼⠃⠀⠀⢀⡄⠠⠀⠰⡄⠀⠀⠰⠀⡀⠀⠀⠀⠀⠀⠜⠃⠀⣷⢠⡇⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⢸⣥⢿⣿⡿⡇⠀⠈⠱⡀⠘
⡆⠀⣇⠀⠀⣰⠟⡀⠀⠀⡀⠴⠁⠠⠂⡠⠁⠄⠀⣴⠰⠇⠀⢠⠀⠀⣁⠄⡀⢠⣞⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⣷⣻⣿⣿⡿⣤⡀⢠⡇
⡇⠀⠀⢶⠞⠁⣜⠀⠀⠈⣠⠀⠀⠶⠂⠔⢀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠉⠂⡹⢋⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢀⠁⠀⠚⠛⢻⣿⣿⣿⣿⡀⠻
⢇⣠⡄⣁⣀⡀⠠⠤⠄⠉⠤⠬⠴⠉⠄⠀⠁⠠⠤⠤⠤⠤⠀⠀⠈⠉⠉⠀⣽⣯⣀⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣭⣤⣤⣾⣿⣿⣿⣿⣿⣿⠃⠀⢠⡖
⠈⠻⢟⠛⠛⠛⠒⠒⣂⡒⠂⠂⠀⠀⠀⠒⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⣉⣉⣉⣉⣳⣾⣿⣿⣾⣿⣿⣾⣿⣿⣿⣯⣽⣭⣿⣿⣿⣿⣿⣿⣃⠜⠃
⠀⠀⠀⠑⠂⢠⣄⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣾⣵⣮⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠉⠛⠯⢍⣉⠈⠀⠃⠤⠀⠀⠈⠉⠉⠉⠉⠉⢉⣉⣯⣩⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣯⠿⢋⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣶⣿⣶⣶⣾⣿⣛⣛⣛⣿⣯⣭⣤⣧⣴⣮⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⡛⠛⣻⠿⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡁⠀⣖⣰⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠠⠀⠀⠀⠀⠈⠅⠰⠙⣿⣿⣿⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⠀⠀⠐⠒⠂⢲⣾⣷⣶⣶⣾⣿⣿⡟⠰⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠸⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⣿⣿⣧⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠂⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠻⣿⣿⡄⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠹⢿⣷⡀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠉⣹⣆⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠴⠏⠡⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠸⡇⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⡟⠙⠓⠦⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡶⠞⠓⠉⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠤⠰⡦⠤⠄⠠⠤⠤⠤⠤⠤⠀⠤⠄⠀⠘⣳⣀⣤⣤⣴⣿⣷⡆
⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣥⣤⣤⣥⣤⣤⣤⣬⣥⣤⣤⣤⣤⣬⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠟⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⢸⣿⢸⣇⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⢸⣿⢸⡇⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⢸⣿⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⢸⣿⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⢸⣿⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⢸⣿⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⢸⣿⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⠸⣿⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢸⣿⢸⣿⢸⡇⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⢡⣿⢃⣾⡇⣾⡇⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⢠⣿⡏⣼⣿⢰⣿⢡⣿⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢀⣴⣾⡷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⢠⣿⡟⢰⣿⠇⣾⣿⠸⡿⢤⣴⣶⣾⣿⣿⣿⣿⣿⣿⡿⠿⠋⠉⣴⣾⣿⣿⡟⠀⣠⣶⣿⣿⡟⢱⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⢠⣿⣿⠁⢿⣿⣦⣝⣛⣛⣿⣿⣿⣻⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⣼⣿⣿⣿⣟⣴⣾⣿⣿⣿⠏⠀⠈⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣅⠘⢿⣿⣷⣶⣬⣭⣥⠿⢶⣚⣿⠽⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡿⣽⣿⣿⢿⣟⡿⠁⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣷⣶⣶⣦⣴⡶⠶⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣟⠾⠿⢷⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿

⡀⡀⡀⢀⠀⠀⢀⠀⡀⢀⢀⡀⡀⠀⣀⣀⣀⣀⣀⣀⡀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⡀⡀⡀⢀⡀⡀⢀⡀
⣿⣿⠿⡻⠿⢛⠿⢿⡿⣻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿