Diaphragm Emojis & Symbols

⠀⠀⠀⠀⠂⠀⠶⠂⠶⠀⠐
⠀⠀⠀⠀⠀⠈⠉⠉⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⣖⡶⢶⡖⠆⣀⣀⢀⡐⠒⠀⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⢴⡆⠉⠉⠉⠉⠉⠉⣹⣿⢉⡿⠁⠀⠀⠀⠁⠂⠠⢄⢈⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠈⠉⠓⠒⠀⠀⠀⠈⠁⠋⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⣀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠑⠄
⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠂⡼⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⠀⠀⢀⠤⠂⠄⠀⢀⡀⠀⠀⠀⠀⠀⠄⠤⡤⠤⢤⡤⠤⠀⠀⠀⠀⠀⠀⠈⢣⡀⠄
⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⢀⡶⠞⠉⠠⠘⠃⠀⠀⠀⠒⠒⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠄
⠀⠀⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡋⠀⠁⠐⠁⠀⠀⢀⡶⠈⡄⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠀⡆⠀⡏⠀⢠⣧⡇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢸⣶⡇⠻⢇⣠⣿⣿⣧⡆
⠠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣧⡇⠀⠈⢻⢱⠛⠨⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡄⣿⣶⠀⠀⠈⡞⠀⢸⠁⠀⠀⠀⢀⣀⡠⠤⠤⠤⣀⡀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣿⠀⣠⣶⡀⠀⡸⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⡌⢠
⠀⠀⠀⠀⠀⠀⢰⠏⢆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢻⡄⢿⡐⣏⣾⡇⠀⣟⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢀⡿⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠷⠀⣽⡁⠀⠁⢠⡇⣀⣀⡀⠳⡄⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠙⠉⠙⠀⠀⣿⠀⠀⠀⢘⡯⠁⠉⠀⠈⠁⠒⠈⠀⠁⠀⠀⠂⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⢀⣀⣀⣀⠀⠀⡄⠀⢻⠂⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⢃⠒⢒⠐⠒⡀⠂⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠨⠀⠀⠙⢸⡃⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡔⠀⠀⠀⠂⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠑⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⣀⠀⠀⡀⠀⠀⣤⡤⠀⠀⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣾⣶⣿⣿⣿⣿⡏⠛⠉⡿⣻⣿⣵⣢⡔⣒⣁⣈⣑⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⣿⣿⣿⣿⡿⠋⠿⠋⠀⠀⠀⠘⠋⠙⠛⣿⣿⣿⣾⣿⣧⡈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣲⢿⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣛⣗⣳⣤⣬⣵⣷⣤⡁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣯⠻⠛⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣯⣤⣀⡈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣻⣿⣿⠿⣿⣿⠟⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⣿⣿⣿⣿⣿⢙⣿⢿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠗⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠺⣿⣿⣽⣿⣶⣄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠆⣾⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣼⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠠⠷⣾⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠀⠀⠀⠀⢀⣤⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡤⢢⣀⢀⡾⢋⢧⡹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣷⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡿⣿⣿⣿⣇⣉⣸⡀⢀⣽⣷⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡆⡀⢀
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⣿⣷⣿⣿⣿⣿⡿⢿⣷⣿⣿⣿⣿⢃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣽⣿⣿⣧⠸
⠀⠀⠀⠀⠇⣸⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣷⣾⣿⣿⣿⣿⠃⣠⣾⣟⣻⣿⠛⢇⣾⣧⡄⠀⠀⠀⠀⠀⠀⠀⢉⡻⣷⣾⣿⣿⣽
⠀⠀⠀⠀⠃⢹⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⣾⣿⠿⣣⢾⣿⡆⣸⣿⣿⡇⣦⡀⠀⠀⠀⠀⠀⢈⣙⢶⢶⣿⣿⠉⡇
⠀⠀⠀⠀⢀⣾⡿⢟⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⣾⣟⣿⣿⣿⣯⣿⣿⠏⣀⣴⣏⠸⣿⣷⣿⣜⠿⢾⣿⣇⠀⠀⡀⠀⣴⣾⣿⣿⢻⣿⣿⡇⠃
⠀⠀⠀⠀⢸⣿⣾⢿⣿⠟⠀⡀⠀⠀⠀⠀⢀⠀⣾⣼⣿⣏⣿⣿⣿⣿⣿⡿⠃⣴⣿⣿⣿⠇⣿⣿⣿⣯⡔⢰⡿⡿⢠⣰⣿⣷⣿⣿⠿⢻⣿⣿⣿⠇
⠀⠀⠀⠀⣿⣿⣿⣾⢿⣷⡿⡅⠀⠀⢀⣶⣿⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⡷⣿⣿⣿⡏⣴⣿⡷⠁⣸⣿⣿⣿⣿⡻⣷⣿⣿⣿⠛⠀⡇
⠀⠀⠀⠚⣿⣿⣿⣿⣿⣫⣼⣿⣿⣿⣿⢻⣿⣿⣹⣷⡿⠛⠛⠛⢿⣿⣿⣰⣯⣿⣿⣿⣿⡷⣿⡏⢻⣼⡿⠩⢅⣴⠟⢿⣿⣿⣿⣿⣿⢿⡿⠇⢠⣿⠃
⠀⠀⠀⠘⠉⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣾⣿⣿⣿⡿⢁⣾⢿⣿⣾⣿⣿⠟⠃⠀⠸⠛⠟⢢⡿⠀⣾⠟⠠⠐⠛⠄⠀⠸⣿⣿⣿⣿⣿⠈⠇⠀⢂⡟
⠀⠀⠀⠀⠒⠉⢺⠿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠛⠁⣼⣿⣾⣿⣿⣿⠿⣿⡧⠀⢼⠒⠊⢺⡇⢸⠁⠂⠀⠀⠀⠀⠀⠀⠈⠙⢿⡿⠉⠀⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⠟⢃⣤⣾⣿⣿⣿⣷⣿⡿⠛⣿⣿⣷⣠⣨⣷⠀⣾⠂⠀⣿⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⣺⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢁⣴⡿⢟⢻⡛⠊⢻⣿⣿⢀⡀⡏⢹⣿⠀⡉⣿⡀⢸⡂⢀⣤⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⢢⡁⠈⢚⣿⣿⣿⣼⣅⡄⣼⣿⠀⠛⣻⠉⢹⡇⢈⡏⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠂⠀⠈⠀⢶⣿⣿⡟⢧⡝⢡⣿⣿⠂⡿⣿⡟⣿⠇⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠁⡈⢡⣼⣿⣿⡇⣁⢸⡇⡏⠀⢰⣿⡿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡷⠛⠻⠟⠓⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣦⢾⡷⢾⠂⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⣿⣿⠏⠁⠀⠀⡦⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⠟⠀⠀⠀⢀⣿⣾⠛⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠉⠀⠉⠉⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠒⢦⡄⠀⠀⡗⢺⠀⠀⣠⠶⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢹⡀⠀⡇⢻⠀⡞⠁⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡇⡇⢸⠃⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢠⡇⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠃⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣋⣤⢷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠈⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⡀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣸⠟⢿⡛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⢳⡹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠳⣌⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠘⢦⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣯⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⠤⠤⢤⠴⠃⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧
⠀⠀⢸⡇⠀⠀⠀⣠⢴⣫⣵⠶⠶⠛⠛⠛⠲⠶⠶⣶⣤⣿⣷⣶⣦⡤⣤⣤⣤⡴⠟⠹⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠸⣇⢀⣠⣾⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠙⠛⠛⠛⠛⠛⠛⠿⡿⢿⣿⣶⣤⣄⠀⠀⠀⡇
⠀⠀⠀⠉⣩⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢸⡟⠿⣿⠲⠞⠁
⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠈⡇⠀⠘⢿⣄
⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣆
⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣿⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⡿⢹⡆
⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⣶⡚⠻⠷⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣀⡼⠃⢸⡇
⠀⢸⣇⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠖⠚⠉⠁⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠛⠒⠋⠁⠀⠀⢸⡅
⠀⢸⣟⠋⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠒⠒⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠈⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢢⠀⠀⠀⡇⠀⢸⠀⠀⠀⢀⠃⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⡀⠀⠀⡇⠀⠘⡀⠀⠀⢸⠀⠀⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠀⠀⡠⠀⡇⠀⠀⡇⠀⠀⢇⠀⠀⢸⠀⠘⣇⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠠⢼⠀⠀⣇⠀⠃⠀⡰⠀⠀⠀⠈⠢⡀⠸⠀⠀⢏⠀⣆⡄⠀⠑⡄
⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠘⢆⢠⠁⢀⢀⡔⠁⢀⡠⠤⠤⣀⠈⢢⡄⠲⢜⣀⠟⠀⡀⡀⠈⠢⡀
⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠢⣀⠀⠙⢦⡸⢋⠤⠚⠁⠀⠀⠀⠈⠑⢄⡑⢄⠀⡇⠀⠀⢸⢇⠄⠀⠱⡀
⠀⠀⠀⠀⠰⠀⠀⠀⡄⠀⠈⠉⢉⣽⡭⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠪⣮⣀⣰⡠⠞⠣⢤⡤⠀⢣
⠀⠀⠀⠀⡇⠀⠀⠀⠼⡄⠀⣰⡏⢠⢧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡘⢗⢢⡤⠤⠄⠀⠉⠀⢸
⠀⠀⠀⠀⡇⠀⣀⠈⠒⠓⡾⠱⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⣆⠙⢦⡤⠤⡀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠝⢒⢿⡇⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⣏⠫⡈⠉⢶⠀⠀⢸
⠀⠀⠀⠀⡇⠀⡤⠤⡆⢸⠑⣄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠞⡮⡃⠈⢢⡀⠁⠀⢸
⠀⠀⠀⠀⡇⠈⠀⠠⡇⠀⠀⠈⠁⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠃⠁⠀⠸⠱⡤⠀⠸
⠀⠀⠀⠀⢰⠀⠀⠃⠀⠀⣀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢄⣀⠀⠁⢀⠇
⠀⠀⠀⠀⠀⠓⠒⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠾⠿⠛⠛⠿⠿⠶⠶⣶⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠾⠿⠿⣿⣭⣤⣤⣤⡤⠤⠴⠦⠤⣤⣄⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠛⢿⡍⠛⠻⣦⣄⡀
⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠦⢤⣄⣠⣤⣤⡾⠟⠛⠛⠛⠛⠲⠶⣤⣤⣤⣤⣤⠴⠿⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠾⠿⠿⠿⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⢰⠁⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣥⣤⣤⣤⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠉⠉⠉⠉⠛⠳⠶⢾⣿⣷⣤⣀⡀⠀⠀⠀⢀⣿⠄
⠀⠀⠘⣧⣄⣀⣀⣠⣤⣶⠾⠿⠟⠛⠉⠀⠀⠀⠉⠙⠻⣦⠙⢷⡀⢀⣀⣀⣀⣀⣀⠀⣰⡟⣀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣷⣶⡿⠟
⠀⠀⠀⠈⠙⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠘⣧⣀⡀⠀⢀⣀⣀⣰⡟⣾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠈⠉⠉⠉⠁⠀⣽⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⡷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢠⣶⣶⠶⠶⠖⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠋⠉⠒⠦⡀⢸⡇⢸⡇⢀⠴⠒⠉⠙⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠘⣶⠃⠘⣶⠃⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠒⠺⡆⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⢰⠗⠒⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠙⠋⠉⠱⣄⣀⡤⠴⡇⢸⠶⢤⣀⣠⠎⠉⠙⠋⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⢰⠃⠀⠀⠀⠀⠤⢶⠖⢲⠖⠶⢲⡏⠀⠀⢸⠃⠘⡇⠀⠀⢹⡖⠶⠲⡖⠲⡶⠤⠀⠀⠀⠀⠘⡆
⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢰⡁⠁⠀⠀⢸⠀⠀⡇⠀⠀⠈⢈⡆⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠙⠄⠀⠀⢸⠀⠀⡇⠀⠀⠠⠋⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⣾⠀⠀⣧⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠈⣆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⠀⠀⠀⠀⠈⠣⣄
⠘⠦⣄⣀⣀⣀⣀⣀⣀⣀⣀⠤⠤⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⠤⣀⣀⣀⣀⣀⣀⣀⣀⣠⠴
⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⢀⣀⣀⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣀⣀⡀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⢀⡠⠤⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠤⢄⡀
⠀⠀⠀⠀⢼⠁⠀⣀⣀⠤⠤⠖⠒⠒⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⠒⠒⠲⠤⠤⣀⣀⠀⠈⡧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⠀⠙⠐⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⡀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⢴⣤⣤⣤⠴
⠀⠀⠀⠀⡠⠊⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠄⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠈⢂⡀
⠀⠀⡄⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠐⡄⠈⢄⠑⢲⡀
⠀⢀⠀⠀⠸⠀⠐⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⢰⠀⢀⠀⠀⠀⠀⠀⢀⠀⢁⠀⠘⡄⠈⠄⠰⣷⡀
⠀⡀⠀⠀⠇⠀⠀⠈⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢨⣄⡜⠀⠀⠀⠀⠀⠈⠄⠘⡄⠀⠐⠀⠈⠀⠙⣇
⢠⠁⠀⢸⠃⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢧⢸⡂⢻⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄⠀⠂⠀⢹
⠀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⣶⠘⡇⢻⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡞⠩⣿⡆⣿⡆⡁⠈⡇⢨⢗⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⡀⢻⣧⠁⠹⠁⠀⠠⠄⠠⠀⠃⠀⢹⢧⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⢘⡀⠃⡸⣷⡀⠀⠀⠀⠀⢢⠃⢈⠇⡶⠁⠄⠘
⢀⠀⠀⠀⠀⠀⠀⢀⡴⢒⠾⢂⠴⣤⡞⠀⠘⣷⡘⣦⠀⠀⡈⠀⠊⠈⡀⢀⡠⣀⣱⢤⡀
⠀⢆⠀⠀⠀⠀⠀⠾⠋⠉⠉⠁⠀⠀⠀⠀⡀⢹⣧⠀⢀⣾⠁⠀⠀⠀⠀⠁⠀⠈⠋⠉⡁⠀⠀⠀⠀⠀⣰
⠀⠘⢄⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠻⠀⢸⡉⢠⡀⡠⠘⠀⠀⠀⠀⠀⠀⢇⠀⣀⠀⠀⢠⡟
⠀⠀⠈⠢⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠁⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⠁⡧⢢⡄⠀⡤⠥⢄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⢺⠿⠀⡷⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢅⡀⠀⣮⡀⠀⠀⣄⣀⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⡇⠀⠰⠀⢱⠠⠠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⡿⠀⡐⠀⢀⣸⡆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣈⣿⠠⠀⠉⠙⠒⡆⢀⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠯⠤⠤⢤⣀⡥⠎⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⢶⣦⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣤⣶⠶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠁⠀⠀⠀⠙⣷⡄⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⣰⠟⠁⠀⠀⠀⠉⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⢠⣿⠀⠀⠀⢻⡇⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢀⣾⠃⠀⠀⠀⠈⢿⣆⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠁⠀⠀⠀⠀⠀⠀⠹⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣠⣶⠟⠛⠛⢷⣄⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠟⠀⠀⠀⠀⠀⠙⢷⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀
⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⣾⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⣿⡅
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣿⣇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠈⣿⡄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄
⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠉⠀⠀⠀⠀⣠⣤⣤⣴⡶⠟⠋⠉⠙⠻⢶⣤⣤⣤⣄⡀⠀⠀⠀⠈⠙⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠘⣷⡀⠀⠀⠀⠀⠀⣀⣴⠿⠋⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣦⡀⠀⠀⠀⠀⠈⠻⢷⣄⡀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠈⠻⢶⣤⣤⣴⠾⠋⠁⠀⠀⠀⢀⣠⣴⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⣄⣀⠀⠀⠀⠀⠉⠻⢶⣤⣤⣴⡾⠋
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣴⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣠⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠛⣦⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢶⣤⣴⠟
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣦⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠚⠉⠉⠉⠙⠳⢄⠀⢸⠀⠀⡇⠀⡠⠞⠋⠉⠉⠉⠓⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡼⠀⠀⢧⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⢤⣀⣦⠀⠀⠀⠀⠀⢸⡞⢳⡇⠀⠀⠀⠀⢀⣔⣠⡤⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⣤⣀⣀⣽⡇⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⢸⣯⣀⣀⣤⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⣦⡤⠶⠚⢹⠁⠸⡏⠓⠶⢤⡴⠃⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⢀⡞⠀⠀⠀⠀⠀⠠⠴⢲⠟⢙⡗⠒⠒⣫⡏⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⢹⣝⠒⠒⢺⡛⠳⡖⠦⠄⠀⠀⠀⠀⠀⢳⡀
⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⣼⠁⠙⠀⠀⠀⣾⠀⠀⣧⠀⠀⠀⠋⠈⣧⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢳⡀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢠⠞⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠓⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡾⠀⢀⡖⢲⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⢠⠖⣦
⡇⠀⠀⠉⠉⠀⠀⠀⠀⠻⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠴⠃⠀⠀⠀⠈⠙⠁
⣇⠀⠀⠀⠲⣀⡀⢀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⣀⣠⠆
⠙⣆⡀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢀⣀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⡀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⢀⡴
⠀⠀⠉⠙⠒⠒⠒⠒⠒⠒⠚⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠒⠒⠒⠒⠒⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠶⠒⠒⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠒⠒⠦⠤⣤⣀⡀
⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⢮⡀⢀⣀⣠⠤⠴⠖⠒⠚⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠲⠦⢤⣄⣀⡀⢀⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣀⣸⣄⡀⢖⣂⡠⣀⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠉⠁⠄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣀⣤⣴⣶⣾⡟⡷⣆⡀⣿⣖⣀⣀⣒⣾⣵⡤⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⣠⣾⠟⠻⠇⠙⠉⠉⠙⠎⢁⡿⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⣾⡟⠁⠀⠀⠀⠀⢀⣀⡀⠀⢸⣆⣀⣀⣀⣠⣴⡦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣠⡿⠉⢀⣀⣤⣾⣿⡿⠟⠻⢿⣷⣬⣷⣤⣤⣤⣴⣾⡷⠖⠚⠉⠂⠢⢤⡀⠀⣾⣿⣿⡄⢰⡀
⠀⠀⠀⠀⠀⠀⡜⢠⡟⠁⣠⡾⠛⠁⠀⠉⠀⠀⠀⠀⠈⢿⡋⠈⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⡀⠈⠓⢥⠑⢯⣷⠀⢷
⠀⠀⠀⠀⠀⠰⢠⡟⢀⡾⠏⠀⢀⣠⢴⡚⠋⠉⠉⠻⠷⢾⣛⡒⣶⣤⢒⣺⣿⠗⠾⠓⠒⠦⣈⠑⠠⡀⠘⣄⠙⢇⠘⣧
⠀⠀⠀⠀⠀⠃⡟⢀⣾⠀⢀⣴⣿⡉⠞⠃⡀⠀⠀⠀⠀⢀⠉⢩⣶⠞⢻⠉⠃⠀⠀⠀⠀⠀⠀⠱⢶⡀⠳⡌⠣⡈⠆⢹⡄
⠀⠀⠀⠀⠀⢰⢠⣾⠃⣰⣿⡿⠁⠀⣀⣠⣰⣖⣒⢠⣀⠘⣒⣺⣟⠀⠼⢾⣉⠴⠂⠀⠒⠦⣄⠓⢄⡙⢦⠀⠀⠱⣄⡈⡇
⠀⠀⠀⠀⠀⠀⣼⢃⣼⣿⠃⣠⡴⠛⠉⠉⡉⠉⠉⠙⠛⠻⢭⣽⣿⣷⠀⢨⣇⣀⣠⠤⠤⠤⠤⠷⣄⣁⠀⠐⣄⠀⠹⡇⣿
⠀⠀⠀⠀⠀⢸⠇⣼⡟⣡⡾⠁⠀⠀⠀⠀⠁⠀⠀⠘⠛⠶⣤⠈⠙⢧⠀⠚⠋⠁⠀⠀⠀⠀⠸⠀⠀⠉⠳⣄⠈⣆⠀⡇⢸
⠀⠀⠀⠀⠀⢈⣸⣿⡾⡯⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⣶⡶⠶⠞⠶⢤⡶⣿⣷⡘⡆⠀⢸
⠀⠀⠀⠀⠀⢸⣿⡿⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠌⠀⠈⠀⠀⠀⠀⠀⠈⠛⠻⣿⡄⢸
⠀⠀⠀⠀⠀⢸⡟⠀⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣽⡇⢘⡀
⠀⠀⠀⠀⠀⠈⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⣽⡇⢻⡇
⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣴⡇⠂⢸⡀⠘⢲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡿⠁⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⣬⠅⡀⢶⡟⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣸⠏⣀⢸⡇
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣄⠌⠀⣼⡥⠆⠁⣼⣀⣸⣷⡀⠀⠡⢤⣀⡠⡀⠀⠀⠀⣦⣽⡷⠀⢠⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⢴⠂⡨⠒⠋⠁⣠⠆⠸⣇⠀⠀⢃⡉⣧⢙⠁⠀⠀⠀⠉⠳⣿⠀⢀⢐⣿⠟⠀⢀⡇⢨
⠀⠀⠀⠀⠀⠀⠘⠣⡀⠀⠿⢦⡿⠋⠀⠀⠀⠀⡼⠃⠀⢰⣾⠄⢠⡏⣱⡇⢹⠆⠈⠠⠀⠀⠀⠈⠣⠿⠚⠁⠀⢀⡼⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆⠀⠀⠀⠀⠀⡠⠂⡀⠁⠀⠀⠰⣯⡀⢸⠓⣾⠀⢸⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⣀⢀⣽⠃⣼⡇
⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠧⠄⡔⠁⠀⠸⠀⠇⠀⠈⡆⠀⣀⠀⠀⠀⠀⠈⠛⠷⠿⠛⢁⡴⢣
⠀⠀⠀⠀⠀⠀⠈⠙⠆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢻⢷⠆⠰⠖⠶⠴⢶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠜⣁⠌
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣇⠀⠀⠀⠀⠀⠀⠀⣑⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠟⠂⠶⠶⠶⠂⠛⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣤⣤⣈⣁⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣤⣤⣼⣿⣿⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣴⣿⣷⣦⣄⠈⣿⣿⣿⣿⣽⣶⣦⣄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣻⠀⢸⣿⣿⣿⣿⣿⣇⣸⣿⣿⣿⣿⣿⣿⣿⣷⣈⠳⢂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣦⡈⠛⠿⠿⠋⢁⣿⣿⠟⢁⡉⢿⣿⣿⡿⣧⣿⣧⡄⢰⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣴⡏⢰⣿⣿⣿⣿⣿⣿⣷⣤⣤⣄⡘⠿⠇⠀⢸⣿⣾⣷⣤⣿⣿⣿⣿⣿⣇⡼⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣾⣿⣷⣾⢻⣾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣾⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠏⠈⢻⡋⣿⣿⣿⡇⠀⠀⠀⠀⠉⠙⣿⣿⣏⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⡶⠿⠟⠛⣯⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⣿⣿⣶⣶⣤⣴⣤⣄⣸⣥⣿⣿⣍⣿⡆
⠀⠀⠀⠀⠀⠀⢰⣿⣼⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣷⣾⣿⣾⣿⣿⣿⣿⠟⠉⠉⠉⠉⠙⢻⣿⣿⣿⣿⠛⣿⡄
⠀⠀⠀⠀⠀⠀⣸⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⡛⠛⣿⣿⡿⠇⢠⣤⣤⣤⣤⣤⣀⣹⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠞⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⣿⡟⢿⡇
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣈⣉⣽⣿⣿⣿⣿⣿⢿⣄⠛⣿⣿⣿⣿⡖⠒⠠⠶⠄⠀⣀⣀⣀⣤⣿⣿⣿⣧⣸⣿
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⠉⠉⠉⠉⠙⠻⠿⢿⣤⣿⣴⣿⡿⠿⠏⠀⣀⠀⠀⢀⣴⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⡠⢴⢄⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿
⠀⠀⠀⢰⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣷⣶⣶⡌⠋⠁⠀⠀⠀⠙⠳⣄⠐⢄⡀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⡀
⠀⠀⠀⢸⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⢀⣀⠀⠀⠀⣀⣀⠉⠳⣎⣷⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣧
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⢠⣾⣿⣿⠃⠀⣼⣿⣿⣷⡀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⣿⡀
⠀⠀⠀⣿⠁⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁⡀⠀⢠⣿⣿⣿⡟⢣⣴⣿⣿⣿⣿⣿⡆⢀⠈⠀⢤⣀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣿⡇
⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡶⠋⠀⣴⣿⣿⣿⣿⣇⣾⣿⣿⣿⣿⣿⣿⣧⠈⠳⣄⠈⠻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠚⠋⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⡄⠀⠈⠳⢦⣀⣈⠉⢙⣿⣿⣿⣿⣿⢿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⣀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣏⣨⣿⣿⣿⣿⣿⣿⣆⠐⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿⣟⣦⣿⡇
⠀⠀⠀⠘⣿⡎⢿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⢿⣿⢿⣿⠛⠛⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠙⠿⠿⡿⠟⠁⣼⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣄⠙⠛⠿⠟⠛⠁⠀⠀⢀⣶⡿⠟⠛⠉⠉⠻⡇⠀⢋⣜⣿⡆⠀⣿⠋⠀⠀⠈⠛⣷⣄⠀⠙⠶⣤⣄⣀⣴⡾⣿⣿⡟⠁
⠀⠀⠀⠀⠸⣿⣿⣿⣦⣤⣤⡶⠞⠀⢀⡾⠋⠀⠀⠀⠀⢀⠀⠈⠇⠘⢿⣿⠇⠀⢡⣤⣄⠀⠀⠀⠈⢻⣦⠁⠀⢀⣀⣩⣤⣤⣿⣿⠇
⠀⠀⠀⠀⠀⠹⣿⣿⣤⠤⠤⠀⢂⣠⡞⠁⠀⠀⠀⠀⠀⠈⠀⠀⡄⢠⣿⠛⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠘⢷⣤⣤⣉⣿⣏⣨⣿⠏
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣶⡤⢺⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣷⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠺⠂⠦⠀⠀⢀⣼⠁⠀⠾⠆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⣄⠀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣤⣤⣀⠀⠀⠀⠀⠀⠉⠀⠘⣷⠀⠀⠀⠤⠴⠄⠀⠀⠀⠀⠀⠀⣠⡶⠛⠛⠛⠛⠓⠒⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⣠⡖⠂⠀⠀⠀⠉⠛⢦⣄⠀⠠⣶⡀⢰⡏⠠⠄⠀⠀⣄⣰⣄⣼⡿⠀⢠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⢠⠟⠘⡏⣠⣶⣤⡀⠀⢀⣰⣂⡈⢣⡄⠀⠓⢾⡟⠂⠀⢘⣿⠻⣿⣿⠟⠁⢠⣿⠇⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⢿⠀⢸⣿⣿⣿⣿⣿⡆⢹⣿⣿⡟⢧⣿⡄⢀⡜⠛⠀⠤⠀⠈⠛⠋⣰⣦⣤⣾⣏⠀⠀⢀⠀⠀⢀⣿⣿⣤⡀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⢿⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡟⠁⢈⣷⣶⡄⣤⣤⣴⣾⣿⣿⣿⡿⠋⢿⡇⠀⢸⠀⣶⢿⣿⣿⡿⠋⠀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠻⣄⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣵⣄⠀⢘⣿⣿⣿⣿⣿⣿⣤⠀⠀⣃⠀⠀⠀⣿⣼⣿⡏⠀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⣦⠉⠻⢶⣌⠻⣿⣿⣿⣿⣿⣿⡿⣶⡄⢸⣿⣿⣷⣾⣿⣿⣿⣿⠟⠛⢧⠀⠤⠛⢠⣾⣾⣿⣿⡿⠀⣸⡿⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡼⣿⣿⠛⠛⠫⠻⣦⠿⣿⣇⣈⠩⣿⣿⣿⣿⣿⣿⣷⣼⣷⣀⠀⠈⠁⠼⠛⠛⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢹⣏⠀⠀⠀⠸⣿⣍⣳⣿⣿⣶⣾⣿⣿⣿⣿⣿⢿⡟⢋⡝⠁⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠛⠳⣄⣀⠀⠀⠀⠲⠾⠛⠻⠿⠿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠁⠀⠀⠀⠀⢀⣴⠛⠿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣄⠀⠀⢹⣦⣄⡰⢶⣶⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠴⠄⠀⠀⠀⢠⡟⠀⣀⣤⣶⣿⡏
⠀⠀⠀⠀⠀⠀⢠⣤⣶⠞⠉⠙⣿⣷⣄⠈⣿⡟⢻⣿⠃⠀⠙⢿⠀⠐⠃⠀⠀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⡆⣰⣿⣿⣿⣿⣷⣦⡄⣀
⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⢀⣿⣿⣿⠿⣿⣿⠁⠈⣽⡆⠀⢠⣴⠃⠀⣀⣤⣀⣿⡿⢿⣆⠀⠀⢠⡘⠂⠀⣠⡿⠛⢿⣿⡟⠀⠀⠀⣰⣿⡆
⠀⠀⠀⠀⢀⣇⠀⠀⠀⣴⣿⡟⠉⠀⠀⠘⢿⡶⠃⢸⣿⡄⠬⠉⢀⡴⠟⠁⠺⣯⡄⠀⠙⠷⠄⠴⠀⠀⣴⡋⠀⠀⠀⢙⣷⠄⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠘⣿⠀⢀⣴⣿⣿⠁⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠈⠻⣦⣄⣀⣀⣀⣠⡾⠋⠀⠀⠀⠀⠈⣯⣰⣦⣿⣿⣿⡇
⠀⠀⠀⠀⢸⡇⠀⣸⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠘⣿⠉⠙⣿⣿⠁
⠀⠀⠀⠀⣴⣤⣤⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⢹⣿
⠀⠀⠀⠀⠈⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡿⠋⠉⣹⣿⠿⣿⣿⣿⣿⣿⠿⣿⣿⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⡟⠉⢻⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠐⡀⠐⠒⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠒⠈⠁⠀⠀⠀⠉⠒⠦⣄⣨⡀⡀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠉⠀⠀⠈⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⢀⠜⠁⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀
⠀⠀⠀⠠⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢍⢢
⠀⠀⢀⠂⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡄⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⡄⠈⢣⠈⢂⢷⡀
⠀⠀⠎⠀⠀⣸⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⢰⠀⠀⠀⠈⠀⠀⠀⠀⠰⠀⠘⡀⠀⠸⡄⠀⢣⠀⢪⣷
⠀⠸⠀⠀⠀⡇⠀⡀⠀⠂⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠃⠀⢱⠀⠀⢑⠀⠀⠆⠀⢾⣧
⠀⡆⠀⠀⢸⠁⢠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠦⢸⡓⢺⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⠇⠀⠀⠆⠀⠘⡀⠀⢹⡄
⢠⠇⠀⠀⢸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⠇⢸⡄⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠃⠈⠈⣇
⢰⢠⢰⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠘⣷⣵⠈⡇⢙⠟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⣿
⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⣝⣷⠸⣿⡄⠃⠀⢫⡀⣩⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠘⢿⣆⠂⠸⡕⠄⠉⡀⡅⢀⠀⠠⠁⠉⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⠖⠀⠈⢻⣦⠀⠀⠀⠀⠈⠀⡀⢡⠂⢠⠂⣸⠜⠁⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⡈⠀⠀⠀⣠⠊⢻⣧⠀⣀⠀⠀⠀⢀⠇⠀⠌⢠⠃⠀⠀⠀⢨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡀⠀⠀⠀⠀⠀⠀⠀⡴⢋⣤⣛⣡⠊⠳⠶⠃⠀⠀⢻⣧⠱⠇⠀⢀⡌⠀⠈⢠⠢⠄⣀⠴⠲⣖⣽⣲⡄⠀⠀⠀⠀⠀⠀⠀⣾
⠀⢣⠀⠀⠀⠀⠀⡀⠼⠛⠁⠀⠀⠁⠀⠀⠀⠀⠀⡀⠨⣿⡄⠀⢠⣞⠁⠀⢀⠂⠀⠀⠀⠀⠀⠉⠉⠁⡁⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠈⢣⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠺⣧⠀⣶⠈⠀⡆⠐⡠⠃⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⡀⠀⠀⢀⣾⠁
⠀⠀⠀⠳⣄⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠀⡏⢣⠀⣇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⡆⡆⠀⠀⠁⠑⠐⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠐⢆⡈⠃⣺⣶⠀⣴⠈⠉⠓⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡃⠁⠻⠙⠀⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣐⠀⢰⣦⡀⢀⡀⡤⣀⡰⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣽⡅⠀⠰⠆⠉⠦⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⣼⡎⠀⠠⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄⠐⢿⣆⢐⠒⠒⠶⡫⢩⡀⠀⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣾⠇⠉⠀⠀⠀⢠⠰⠤⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⡀⠀⠀⣿⣿⠀⠀⢀⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⠀⣠⣿⣿⡄⠀⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡰⠟⠉⠙⠻⣆⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣴⣶⣷⣦⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣼⣿⣿⣿⣿⣧⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⢟⣩⣾⣿⣿⠿⠛⠻⢿⣿⣿⣷⣝⡻⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠿⣫⣴⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣦⣝⠿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀⠀⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣠⠆⣄⣌⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣽⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⢿⣿⢿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⢻⣻⣿⣿⡿⡧⣿⣿⣭⣙⡛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣫⣼⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣶⡙⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢏⣾⣷⣿⣿⣿⣿⣻⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣸⣏⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣿⠿⣿⠟⠿⠿⣿⢿⣿⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢷⣿⣼⣿⣿⢏⠙⠆⠀⠀⢸⣿⣿⡜⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⣾⣿⡿⠋⠈⠋⠀⠀⠀⠀⢸⣿⣿⣿⠱⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣾⣷⣟⡷⠁⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣄⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡸⣿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣿⣿⣿⣿⣿⡿⠈⠀⡀⠀⠁⠀⢠⠀⢸⣿⣿⣏⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠚⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢿⣿⣿⠟⠓⠀⢡⠀⠄⠆⠀⠀⠀⣼⣿⣿⣿⠋⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣾⡟⠁⠀⠀⠀⠈⠐⠀⠁⠀⠀⠀⣿⣿⣿⣷⠾⣿⣿⣽⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⢸⣧⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⣿⠻⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⢶⣿⣿⣿⣿⣿⣟⢻⣿⣯⣾⣿⣿⣿⣿⣿⣿⣧⣼⡝⣷
⠀⠀⠀⠀⠀⣼⡷⢿⣋⣗⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣥⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣿⡄
⠀⠀⠀⠀⣾⣿⣧⣾⡿⠌⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣿⣅⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣙⡟⣷
⠀⠀⠀⢰⣿⢸⠟⡋⢙⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⢸⢿⡟⣯⢻⣝⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⢀⡿⣿⢾⣿⡀⢀⡑⢀⣤⡤⠔⠒⠈⠉⠉⠁⠒⠶⠤⡄⢈⡶⢛⠁⣋⠉⢘⣿⣿⣿⡿⠿⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣟⢿⣦
⠀⠀⣸⣧⣸⣷⣿⣅⣤⡿⠋⠁⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⠀⢀⠈⠉⠁⠈⠉⠁⠀⠀⡀⡆⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⠊⣿
⠀⢠⡟⢻⣿⣧⣿⣿⡫⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠛⠛⠛⠛⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣼⢻
⠀⣸⡇⢸⣾⣿⣿⣿⠁⠃⠀⠀⠀⠀⢀⠀⣀⣀⣴⣦⣡⣦⠖⠁⠀⠀⠀⠀⠀⠀⠈⢣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⢸
⢠⣿⣿⣿⣿⣿⡿⠁⠀⠀⣠⣾⣿⣷⣿⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⡷⠀⢠⣶⣷⡆⠀⠙⢶⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣇⣼
⢸⡏⣹⣿⣿⣿⡷⢧⣨⣿⣿⣿⣿⣿⠿⠿⠿⠛⠋⠁⢀⢰⣿⣿⡟⠁⠀⢸⣿⣿⣿⡀⠀⠀⠙⠂⢠⣄⣿⢀⡀⠀⠀⠀⠀⠀⠀⢿⣿⣽⢸
⣿⡇⣿⣿⣿⣿⡷⣸⣿⣿⣿⣿⣏⣀⣀⠀⢠⠄⠀⢠⢾⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠈⢉⣡⠀⠀⡇⠿⠀⠀⣿⣿⣏⣼
⣿⡇⢹⣿⣿⣿⡖⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢀⢀⢩⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠰⣿⣻⡆⠰⠀⠀⠀⠀⢿⡾⣧⠛
⣿⣧⢹⣿⣿⣿⣿⣏⣿⣿⡿⠛⠋⠁⠀⠴⠂⠀⢀⣾⣿⣿⣿⣿⣿⡟⣹⡟⢹⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠉⠐⠄⠀⠀⠀⠀⠈⣿⣿⠄
⢿⣿⡀⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠁⢸⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣤⣼⣿⡟
⢸⣏⣧⡈⠻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣾⠃⣀⣾⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡟⠁⢰
⠸⣿⣧⡙⠀⠈⠙⠛⠛⠛⠋⠀⠀⠀⢠⣶⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠿⠿⣆⠀⠀⠀⠀⠻⠿⣷⣿⡿⠿⠋⠀⠀⢸
⠀⢿⣿⣇⠀⠀⠐⠂⠀⠀⠀⠀⠀⠰⣼⠏⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾
⠀⠘⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣄⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠸⣿⣿⣿⣧⣤⣤⡀⢠⣬⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠀⠀⠀⠀⠀⠈⢻⣷⣷⡀⠀⠀⠀⠀⢤⣸⡿
⠀⠀⠀⠹⣿⣷⣹⣿⣉⣿⡾⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣟⠲⠤⠞⣼⢟⣿⠃
⠀⠀⠀⠀⠈⠛⠿⠲⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⣿⣿⢿⣿⣿⣿⣿⡿⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣠⣞⣥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣤⣜⣿⣿⣯⣽⣿⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⡿⣽⠏⣿⠋⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣯⣥⣤⣥⣿⣿⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠉⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡄⢀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣠⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⡀⠀⠀⠀⠀⠀⣨⠅
⠀⠀⠆⠀⢠⣿⣦⣴⣾⣿⣿⣿⣿⣴⣿⡷⢁⣴⣦⠂⣴⢆⡄⠀⠀⠀⠈⠐⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡄⠀⠠⡄
⠀⠜⠀⢀⣿⣿⣿⣿⣿⣿⣿⢫⣿⣿⡿⣱⣿⣿⣯⣾⣧⣿⡎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣄⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣯⠀⠀⢧
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⢁⣿⣟⣿⢡⣿⣿⡿⣿⣿⣿⡿⢰⣧⣦⠀⢀⡀⠀⣀⡀⢀⣤⣀⣿⣿⣿⣿⣆⠀⠀⡀⠠⠀⢠⠀⣠⠸⣿⣿⣿⡆
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⣼⣿⢸⣇⣿⣿⣿⣷⣿⣿⣿⣧⣿⣿⡿⠋⠉⠉⠉⠉⠻⣧⣿⣷⣿⣿⣿⣿⣿⣷⣷⣿⢰⡹⡌⣷⣜⣷⣿⣿⣿⣧⠀⠀⡄
⢸⡀⠀⠈⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⡀⠀⢀⣄⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣏⣇⢿⣾⣿⣿⣿⣿⣿⣟⠀⠀⢷
⠈⢣⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⣠⣾⣇⠀⢸⡿⠀⣀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⡇
⠁⠀⠑⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣷⣿⣧⣾⣿⣷⣦⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⢸⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣇⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠁⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠉⠛⠛⠛⢻⣿⣿⣿⣿⣿⡏
⠀⡏⠀⢀⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣯⢹⣿⣇⣿⢻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣷⠄
⠀⠠⠀⠈⢿⣿⣿⣿⣿⣿⣷⠄⠀⠀⣠⣾⣿⣿⣿⣿⣿⠿⠿⠿⣿⠘⣿⣿⣧⣾⣿⣿⣟⣛⣙⠹⠿⢿⣿⣿⣿⡆⠀⠀⠀⢿⣿⣿⣿⣿⣿⠟⠀⡀
⠀⠀⠷⣄⠀⠙⠿⠿⠿⠟⠋⠀⠀⢠⣿⣿⣿⡿⢛⣉⣴⠶⠿⠷⢶⡄⢸⣿⠋⢹⣿⡿⠟⠛⠛⠛⠲⢮⣙⠿⣿⣿⡄⠀⠀⠀⠈⠉⠉⠉⠀⠀⠆⠃
⠀⠀⠰⡈⠃⢄⠀⠀⠀⠀⠀⠀⣰⣿⣟⣯⡿⠟⠛⠉⠀⠀⠀⠀⢸⡇⠸⢻⠀⠉⢼⡅⠀⠀⠀⠀⠀⠀⠉⠛⠦⣍⣻⣆⠀⠀⠀⠀⠀⠀⠈⢠⡾
⠀⠀⠀⠹⠢⡄⠀⠀⠀⢀⡠⣾⡿⠟⠉⠀⠀⠀⠀⠀⢰⣿⣷⣶⣦⠄⠀⢨⡀⠀⢨⣴⣟⣭⣫⡷⠀⠀⠀⠀⠀⠈⠉⠛⠳⠀⠀⠀⠀⠀⠀⢷⡇
⠀⠀⠀⠀⠆⠀⠁⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠃⢀⠈⠇⠀⠸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⡀⣤⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣿⢿⣿⡇⠀⠀⡀⠁⠀⠘⢿⠿⠶⠞⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀







⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠄
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠈⠀⠀⠀⣀⣤⣤⣿⣿⣷⣄⠀⢁⠄⡀⠁⢤
⠀⣿⠀⠀⠀⠀⢤⠀⠀⠠⢺⣿⣴⣿⣷⡄⠀⠠⠀⡠⠄⠀⠠⣇⠀⠀⠀⢀⣾⣩⣷⡇⠆⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⣴⡤⣀⢱⡀
⠀⢹⣏⡶⣀⣰⠁⠂⠄⣴⣿⣿⣿⣿⣿⣿⣦⠀⢤⢖⡤⢴⣺⡇⠀⠀⠀⣿⣾⣿⡟⢁⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣌⠹⣿⣶⣗
⠀⢸⣿⣿⣼⡿⠁⠀⠀⣻⣿⣿⣿⣿⡛⠉⣿⣧⡈⢿⣭⣽⣿⡇⠀⠀⠀⠿⠟⠁⠒⠋⠉⠉⠉⠛⠻⢿⣿⣿⣿⡋⠀⠘⠉⠀⠀⠀⠁⠈⠛⠿
⠀⠈⣿⣿⡿⢡⠾⠊⠁⠀⠈⣿⢻⣇⡀⠀⠀⠉⠓⠌⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣯⡀
⠀⠀⠻⠋⠀⠁⠀⠀⠀⠀⠀⠉⠎⠉⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠴⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠒⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠔⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⢠⢀⡀⢀⣀⣀⡀⠀⡄⢀⢀⣀⡠⠀⢀⣀⠠

⠀⠀⢘⣆⢠⣾⠆⠛⠋⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠠⣾⣆⣀⣀⠀⠀⠀⠀⠸⠿⠃
⠀⠀⠻⠛⠀⠀⠀⠀⠀⠀⠀⠀⡞⣷⠀⠀⠀⢀⡤⡞⠿⠗⠀⠀⢀⡾⠿⠂⡼⠿⠆⠀⠀⣾⠁⠀⠀⠈⠙⠿⠟⠁⠀⠀⠀⠀⣀
⠀⠀⠀⢀⡴⣦⠀⠰⣞⡇⠀⠀⣵⠏⠀⠀⠀⠈⠙⠁⣤⠀⠀⠀⡜⠁⠀⡼⠁⠀⠀⠀⠠⡞⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣹
⠀⠸⣿⠞⠚⠁⠀⠈⠃⠁⢀⡞⣽⠶⢶⡄⠀⢀⣤⣴⠃⠀⠀⠀⠉⠙⠚⠁⣀⠀⠀⠀⠀⠀⠀⡤⢄⣰⣩⠆⠀⠀⢠⣶⣦⠀⢰⣿
⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠸⣆⠧⢁⣸⠇⠀⠈⠙⠃⠀⡠⠤⡀⠀⠀⢀⣴⣏⠀⠀⢀⡤⣄⠀⠙⠊⠉⠀⠀⠀⣠⠏⠈⠁⠀⠈⠁
⠀⠀⠀⠯⠵⠫⠖⠃⠀⠀⠀⠈⣙⠛⠁⠀⢀⠀⠀⠀⠀⢻⠉⠀⠀⠀⠻⣴⠋⠀⠀⣸⣼⠃⠀⢀⣶⠀⠀⠀⣀⠀⢀⣴⣄⠀⢠⣖⡦
⡀⠀⠀⣀⠀⣀⡴⠛⢦⡀⠀⠀⠉⠋⠀⢠⠏⠉⢹⠃⠀⠸⠀⠀⢀⣀⠞⠁⠀⣰⢞⣽⣄⡀⠀⢘⣧⠀⠀⣼⠻⠃⡼⠉⠁⠀⠸⡇
⠃⠀⠀⠻⠟⢡⣤⣠⠜⠻⣄⠀⠀⠀⣀⡼⢀⣄⡟⠀⠀⠀⣠⣄⠻⡟⠀⠀⠰⡇⠏⠀⢨⡇⠀⠘⠀⠀⠰⢿⣗⣺⠃⠀⣼⠃⠀⠃⠠
⠃⠀⠀⠀⠀⠈⠛⡁⠀⣠⠞⠁⠀⠘⠿⠃⠈⠛⠁⠀⣴⡆⠻⣅⠀⣹⣆⠀⠀⢙⠲⠶⠛⠁⠀⡤⣤⠀⠀⢠⣄⠀⠀⣸⣧⠀⠀⠀⢰
⠀⣠⣖⠁⠀⠀⠘⠿⠋⢁⡀⢠⢞⡆⠀⠀⠀⠀⠀⠀⠹⣄⢀⡼⢾⡥⠞⠁⠀⠙⠓⠁⠀⡔⢠⡟⠁⠀⠀⢸⡸⠃⠀⠈⠉
⠈⠁⠋⣀⣀⠀⠀⠀⠀⠸⠿⠋⠉⠀⠀⠀⣠⢾⡅⠀⠀⠈⠯⠖⠋⠀⠀⠀⠀⢀⠀⠀⠀⠈⣉⠀⠀⠀⠰⣉⠇⠀⢠⠏⠙⢲⠆
⢀⡀⠈⠓⢮⣇⠀⠀⠀⠀⢀⣠⣴⣦⠀⠀⠀⠉⠀⢠⡤⠤⣤⠀⠀⠀⣀⣤⡞⠉⢳⡀⠀⠸⡯⠭⢽⠀⠀⢀⣤⣴⠋⣀⣠⠏
⣻⠏⠀⠀⠸⡿⠃⠀⠀⠐⠋⠀⠈⠉⢠⣶⡄⠀⠀⢸⠋⠉⣿⠀⠀⢺⡁⠀⣳⡀⢀⣷⡄⠀⣷⣦⢸⡇⠀⠀⠉⠁⠈⠛⠋⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⠀⢠⣶⡆⢸⠉⠀⢀⣤⡿⠰⣿⠇⠀⠀⠀⢳⡐⠿⠃⠈⠛⠁⠀⢀⠀⠀⠿⠗⠀⠀⠀⠀⡞⣇⠀⠀⠉⠋
⠀⠀⡀⠀⠀⠀⠀⠘⡷⢄⠀⢸⡥⠤⣾⠀⠀⠀⠉⠁⠀⠀⣤⣆⠄⠀⠿⠗⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢀⣤⡀⢈⠽⡄
⠀⣷⣿⠄⠀⠀⠀⠀⠛⠃⠀⠈⠛⠉⠉⠀⠀⠀⢀⣀⠀⠘⠱⠋⠀⠀⣤⡄⠀⠺⣝⣦⠀⢠⣶⡇⠀⢠⣶⡄⢸⠋⠁⠈⠊⠁⠀⣄
⠀⠉⠈⠀⠀⠀⢠⡒⡄⠀⠀⠀⢠⠖⣛⣛⡶⠾⡥⠾⠃⠀⠀⢠⣦⠀⡟⠁⠀⠀⠀⣿⡆⠀⠉⠀⠀⢸⣉⣤⣿⠀⠀⠀⠀⠀⣿⠋
⠀⠀⢀⡞⠀⠀⢸⡇⠀⠀⢀⣀⢿⠈⠁⢛⣷⠀⠀⠀⡄⠀⠀⢸⡁⠀⢻⠀⠀⠀⠀⠈⠀⠀⣤⣄⠀⠘⠒⠒⠋⠀⠀⠀⠀⠟⠙
⠐⣿⡟⠀⠀⠀⠘⠃⠀⠀⠈⠫⠘⠛⠲⠛⠁⠀⠀⣼⣥⡀⠀⠘⣧⠴⠞⠀⠀⣀⡀⠀⠀⠀⣿⣉⠀⠀⠀⢀⡄⠀⠀⢠⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠛⠒⡆⠀⠀⠀⠀⠀⠀⠀⠸⢧⡞⠀⢠⣦⡀⠀⠀⠀⡴⠛⠛⠀⠀⠀⠸⠜⠀⠀⣰⡟⠁⠀⠀⣾⠀⠀⡇
⢠⣶⡄⠀⠀⠀⢰⠇⣀⣸⠃⠀⠀⢀⣀⠸⣿⠃⠀⠞⠀⠀⣀⡯⠇⠀⠀⠘⠁⠀⠀⠰⣖⢦⠀⠀⠀⠸⠉⠃⠀⠀⠀⢿⣶⠀⣇
⠋⠉⠁⠀⠀⠿⠟⠀⠻⠛⠀⣠⣤⠸⣟⠀⠘⣦⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⢀
⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠙⣏⠀⢈⣶⣚⡭⠇⠀⢠⠤⣄⡴⣻⠆⠀⢰⠯⣟⣷⠀⠈⠛⡁⠀⠀⢸⡭⠃⠀⠀⢀⡞⠓⢦⣤⣀⠈
⠀⢀⡤⠖⢿⠉⠈⢧⠀⠀⠀⠀⠈⢿⣭⠞⠉⠀⣀⠀⠀⠉⠁⠉⠁⢀⣀⡏⣠⣤⡏⠀⠀⣿⣿⠀⠀⠸⡇⠀⢠⣄⡾⢀⣄⡾⠀⢉⡟
⣀⣈⣧⣀⣈⣧⣀⣨⣇⣀⣀⣀⣀⣀⣀⣀⣀⣼⣻⣇⣀⣀⣀⣀⣀⣻⣟⣀⣈⣉⣀⣀⣀⣘⣀⣀⣀⣀⣁⣀⣈⣛⣁⣘⣛⣁⣀⣼⣁
⣿⣿⠋⡛⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⡄⠀⠀⠀⠀⠀⢀⣴⡶⢶⣄⠀⠀⠀⠀⠀⣠⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠸⠫⠀⠌⠛⠀⠀⠀⠀⠀⣞⠡⠀⠄⢻⡆⠀⠀⠀⢰⡟⠩⠉⠙⢻⠀⠀⠀⠀⠀⠀⠟⡛⠉⠿
⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀⣴⡄⠀⠀⠀⠀⠀⠘⠂⠀⠀⠋⠀⠀⠀⠀⠀⢛⣦⣠⣴⡋⠀⠀⠀⠀⠀⠀⢠⡀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣹⡏⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠐⠞⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠠⣤⠀⠀⠀⠀⠀⠀⠈⢋⡥⠀⠀⠀⠀⠀⠀⠀⢠⡄⠿⠀⠀⠀⠀⠀⠀⠋⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠿⢷⡄⠀⠀⠀⠀⠀⣴⣿⠷⣶⡀⠀⠀⠀⠀⠀⣾⣿⡿⣿⡄⠀⠀⠀⠀⢠⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠈⠁⠀⠀⠀⠀⠀⡍⠀⠐⢈⡅⠀⠀⠀⠀⠀⠉⠀⠐⠈⠁⠀⠀⠀⠀⣟⠡⠀⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠸⣷⣴⣦⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⢰⣶⣴⣦⠀⠀⠀⠀⠀⠙⠂⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠐⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⣴⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠙⠿⠖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠈⠁⠀⠀⠀⠀⠀⠠⠖⠰⠆⠀⠀⠀⠀⠀⠀⠈⠃⠶⠀⠀⠀⠀⠀⠲⠖⠁⠀⠀⠀⠺⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠛⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠈⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⣀⡈⡸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⡛⣵⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⢟⠀⢘⠧⠤⠀⠀⠀⠀⠀⠀⠀⢀⠄⠖⡇⢹⠀⢈⠑⠒⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠠⢰⠈⠀⠀⠀⠀⠀⠀⠄⠀⢲⡧⠬⠈⠁⠀⢰⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡀⠀⠀⠀⠀⣵⠀⠀⠀⠀⠀⠀⡈⡀⠐⢉⠧⠰⠀⠀⠀⠈⠧⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⢠⠁⠀⡠⢃⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠅⠀⣀⡀⠠⡆⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡰⡀⢠⠀⡀⣘⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢈⠿⣁⠘⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢠⠀⠀⠀⣰⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⠜⠒⠒⠴⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢰⣿⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⡀⠀⠀⠀⢠⠟⣽⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠐⠀⠀⢧⠂⠀⣿⠁⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠓⠀⠀⠀⠐⠇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡁⠀⠐⠉⠓⠉⠈⡈⠉⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠰⠀⢀⠀⠀⠀⠀⠀⠘⠀⠠⠀⠀⠀⠀⠀⢀⠀⢸⡥⠤⠥⡴⣀⡀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠃⠁⠃⠀⠀⠀⠀⠀⠸⣇⠀⠆⠀⠀⠆⡉⠉⠉⠉⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⢸⠀⠀⠈⠉⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠘⢩⠁⠀⠀⠀⠀⠀⢸⡾⠃⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⠀⠀⢈⣷⠋⠁⠀⠀⠀⠑⠄⠀⠀⠀⡍⢶⡇
⠀⠀⠀⠀⠀⠔⠋⢡⠀⠀⡼⢹⠑⠠⠆⡱⠀⣠⣛⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⣆⢀⡜⠛⢀⡀⠀⠀⠀⠀⠈⠢⠀⠀⡁⠘⠙⣄
⠀⠀⢀⠔⠁⠀⠁⠁⠀⠀⠈⠈⠀⠈⠐⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠈⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠑⠀⠉⠁⠀⠉
⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠇⠀⠀⣼⣇⠀⠰⠀⠖⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⢑⠚⠀⠀⠀⠀⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡀⠈⠀⠀⢀⣠⣠⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠘⠃⢀⠀⠀⠰⣴⣶⣼⣷⣷⣿⡻⣿⣯⠀⠂⠶⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠠⠄⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦⠀⠀⠀⡀⣴⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠻⠆⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⠟⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⡀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⢀⣤⣤⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⡔⢆
⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⣿⣿⡇⣿⠇⠐⠒⠒⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⢀⡀⠀⠾⢿⣿⣿⣿⣿⣿⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢻⣿⣿⣿⣿⣰⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⠿⠛⠈⠻⣷⣔⣄⠀⠀⠀⠀⠀⠀⣠⣴⣶⣄⠀⠀⣤⠋⢀⣼⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠦⣄⠀⠀⠀⠀⣿⣿⣿⣿⡇⠈⠁⠐⠋⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣌⡂⢰⣤⣠⣿⣿⣿⣿⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣹⣿⣿⣿⣧⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣒⠈⠙⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⡿⣋⣽⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⠛⠛⠛⠛⠛⠋⢉⣼⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣴⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠛⠻⠿⠿⠿⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠊⠀⢀⡀⠀⠀⠀⠀⠀⣰⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⡀⠀⠀⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣄⣹⠇⠀⠀⠀⠀⠀⠀⠀⢦⣤⣾⢃⣠⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠸⣿⠟⠋⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠛⠛⢋⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⢱⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣤⣤⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⢀⡸⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠚⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⢠⣿⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⣤⣴⣾⣿⡄
⠀⢀⣼⠏⡀⠁⡈⠁⡀⢀⡀⢀⡈⢓⡦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷
⠀⣸⠋⡄⠁⡄⠁⡄⠁⡄⠁⡄⠁⡄⢁⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇
⢀⠃⠄⠀⠄⠀⠄⠠⠄⠠⠄⠠⠄⠠⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿
⢸⠂⠀⠂⠀⠂⠀⠂⠐⠂⠐⠂⠐⠂⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇
⠚⣷⡃⡀⠃⣀⣧⣄⠃⡐⠃⡀⠃⡐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣷⡀
⠀⠈⠙⠷⣶⠟⠁⠉⠿⣅⡀⢁⡀⢁⣈⣥⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠹⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠛⠛⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠟⠁⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠇⠈⠿⠟⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡾⢷⠀⠀⠀⠀⠀⠀⠘⠛⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠄

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⢶⣄
⠀⠀⠀⠀⠀⠀⣾⣻⣭⣥⠴⠶⣦⣹⣦⣄
⠀⠀⠀⠀⠀⢰⡿⢃⡀⠠⠤⠶⠾⣿⠇⣹⡀
⠀⠀⠀⠀⠀⣸⠃⢏⣀⠀⠤⠤⣤⣾⣽⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠶⠶⠒⠒⠒⠒⠀⠀⣀⡤
⠀⠀⠀⠀⠀⣿⠀⠈⠁⠐⣦⣤⡤⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋⠁⠀⠀⠐⠦⣤⣤⣤⠶⠟⠉
⠀⠀⠀⠀⠀⡿⠀⠀⠀⠈⠙⣧⣴⢟⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⣷⠀⣶⠀⠀⠀⢸⣧⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⣿⠀⠛⠶⠶⠶⢋⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⡴⢒⡦⠖⣻⠓⠆
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠙⠻⡻⣌⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣴⣿⣤⣘⣵⣋⡉⠉
⠀⠀⠀⠀⠀⠀⢻⡀⢢⡀⠀⠀⠙⣦⠙⣧⡀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠈⠉⠛⢿⣟⢷⣄⠐⢶⣄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢷⡀⠀⠀⠈⠃⠈⢷⣄⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠹⣆⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢷⠀⠀⠀⠀⠀⠀⠙⢷⣄⣾⠃⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⣧⣄⠈⠻⣦⣄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣸⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠉⠛⠦⣌⠙⠻⠷⢶⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⢸⡇⡀⠀⢠⡿⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⢸⠀⢠⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⣷⠸⣄⠀⣿⣿⢧⣀⣀⡀⠀⠀⠀⠀⠀⣿⡀⣿⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⢸⡇⣸⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢹⡎⠒⠀⢹⡄⠑⠚⠒⠊⠀⠀⠀⠀⠀⣿⣿⢯⠐⠦⣤⣤⣤⣴⠾⠋⠀⡠⣞⠄⠀⢸⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠸⣧⡀⠀⠀⠀⠀⠀⢠⣤⠾⠃⢹⡄⠀⠀⠀⠀⠀⠀⢄⠀⡼⡼⠁⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠈⠛⠷⠶⢶⣤⡤⠤⠄⠀⠀⠸⠃⢰⡄⠀⠀⠀⠉⠢⣭⣥⣷⠄⠀⢀⣾⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⣾⠋⠀⠙⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⠙⠦⣤⣤⠤⠀⠈⠁⠀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⠶⠟⠋⠀⠀⠀⠀⠀⠉⠛⠛⠿⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄⡀⣀⣠⣴⠾⠛⠛⠋⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠠⠈⠐⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠠⠐⡠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠔⠄
⠀⠀⠀⠀⠈⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢣⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠊⠀⠀⠈⠵⠿⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠴⠢⠤⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⣠⠄⠀⠀⠄⢁⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠁⡀⠈⠙⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡁⠀⠀⠁⠀⠀⡀⠀⠀⠙⠔⢫⠂
⠀⠀⠀⠀⠀⠀⢠⠗⠀⠀⠆⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⡸⢀⠁⠀⠐⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠎⠀⢠⣀⠀⢀⡀⠄⠊⠀⠸⠀⠀⠀⠁⠀⠀⠂⠀⠀⢸⢠
⠀⠀⠀⠀⠀⠀⡈⠁⠠⡀⠀⠀⠁⠀⠀⡠⠊⠀⠘⠅⢀⡀⠚⠀⠠⠂⠀⠀⢄⠀⠀⠈⠀⠀⠀⠀⢀⡬⠃
⠀⠀⠀⠀⠈⠃⠄⠀⠀⡀⡁⠰⣀⠀⠉⠀⠀⠀⠀⢀⠁⠚⠀⠀⡀⠀⠀⠀⠀⠁⡦⢀⣀⡀⠠⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠔⠈⠈⠀⠑⠆⠀⠀⠀⠀⠸⠈⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠤⣀⢺⣀⡀⠀⠀⠀⠀⡄⠀⡐⣂⠀⡠⠀⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠄⠀⠀⠀⠠⠀⠀⠀⠀⠈⠈⠀⠨⠉⠀⠀⠀⠀⠀⠀⠠⣄⠀⣀⠌⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠂⠘⠂⠈⠀⠀⠀⠀⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⡠⠀⠀⠘⠀⠰⠀⠀⠀⠀⠀⠘⠀⠂⠀⠂


⣿⣿⣿⣿⣟⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣟⣛⣛⣛

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠄⠈⠀⠈⠠⠀⠄⠠⠀⠀⠀⠀⢀⣤⡀⡀⠠⠀⠀⠀⠀⡤⡀
⠀⠀⠀⠀⠀⠀⢢⡀⠈⠁⡀⠀⠈⠀⠠⡌⡙⢿⡛⢿⡀⢙⣷⠀⢰⣦⡼⢷⣀
⠀⠀⠐⡆⠀⠀⢨⠁⠀⠀⠀⣦⣦⣄⡀⠀⠀⠂⢥⠰⠌⡛⣁⠀⠼⠏⣾⡿⠛⠉⠀⠞
⠀⠀⠀⢁⠀⠀⠀⠀⠀⢀⠰⠫⢻⢻⠛⢦⡣⠀⠀⠀⢡⠁⠀⠀⠀⢆⠋⠀⡀⠔⢐⣶⣤⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠡⡄⠎⠁⠀⡀⠸⠦⠀⠀⡄⠈⢃⠀⠂⢁⢣⠂⠁⠂⠀⣀
⠀⠀⠀⠂⣈⠀⠂⠀⠀⠀⠀⠀⠀⠢⠀⠘⠀⡀⠄⠋⠢⢠⠸⡦⠜⠀⠀⠀⠐⠃⠄⠀⠀⠈⣡⠆⢔⠀⠀⠂
⠀⠀⠀⡾⠁⠀⣴⡏⠐⠂⢤⣤⣀⣂⡙⠀⠴⣤⡈⠠⢖⠚⡠⠅⢂⠄⠀⠂⠨⠄⠄⣠⠔⢾⡇⢴⠂⠐
⠀⠀⠀⠀⢶⣤⣿⡇⠀⠀⠈⠛⢻⠋⠀⠄⠀⡀⠈⣂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠒⢺⠇
⠀⠀⠀⢀⣰⣿⡏⠀⡓⠊⠐⢀⣈⠀⠠⠎⠀⠀⠨⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠐⠀⠄⣁⣴⡿
⠀⠀⠀⠈⡿⠋⠀⠁⠀⠀⠀⢀⠉⢠⡄⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣀⠀⠀⠀⠀⠀⠙⠂⠰⢿⣽⠇
⠀⠀⠀⢷⠔⠀⢀⣤⣶⡎⣤⠜⠁⠀⢀⠄⣀⡂⣀⠀⠀⠒⠀⠀⠀⡀⠀⢀⠉⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠨⠄⠀⢸⣏⣌⣁⠙⢶⠀⠀⢸⠀⢇⠀⠀⠀⣠⠧⠀⠀⠐⢃⠀⠘⠠⣁⡀
⠀⠀⠐⠁⠀⠀⡀⠉⣱⠏⠀⠰⠀⠀⠈⡇⠈⠃⠌⠁⣀⠀⠀⠀⠐⣠⢦⡤⠚⠻⢟⡀⠖⡤⠀⠀⡠
⠀⠀⠀⠀⠐⢾⣶⣴⣿⠁⠀⠀⠀⣤⣴⣷⣄⡀⠀⠀⠅⠀⢂⣠⠶⢹⣥⣂⢐⡘⠃⠁⠾⠷⢂⣠⡀
⠀⠀⠀⠶⠀⠀⠻⣿⣃⣀⣀⣀⠈⢨⠟⠉⠉⠍⠈⠀⠸⢂⠯⠀⠩⠀⡟⠟⠅⠈⢢⠲⢟⠛⠋⠉⠀⣠⠀⡀
⠀⠀⠐⠀⢰⠀⠀⠈⡿⠏⠛⠉⠀⢀⡀⠀⡘⢆⠛⠠⠠⢹⣥⢤⠴⣎⠁⠈⢃⡔⠁⠁⡀⣘⡂⣴⡞⠛
⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⣰⣾⣿⣗⠀⠄⠈⠦⠤⠀⣸⢕⠥⡊⠀⠑⡤⠋⣀⣤⠅⣬⣭⠃⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡈⡻⠷⠠⣤⣤⣤⣾⡃⠁⢀⠄⠠⡀⡏⠀⠀⠋⠀⠈⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠛⠛⠉⠢⠁⡡⠠⢴⣶⣶⡞
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠋⠉⢹⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣮⡻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠋⠀⠀⣠⣾⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠻⣿⣿⡌⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠁⢀⣀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣯⣠⣶⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡁⢈⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠛⣿⣿⣿⠟⠁
⠀⠀⠀⢀⣠⣤⣤⣤⣤⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⣿⣷⣴⣿⡿⠋
⠀⢀⣾⣿⡿⠿⠟⠿⢿⣿⣿⣿⠀⠀⠀⠀⣀⣠⣤⣄⡀⠀⣠⣾⣿⣿⣧⣀⣼⣿⡿⠋
⠀⠸⣿⣿⣄⡀⠀⠀⣀⣼⣿⡟⠀⢀⣴⣿⡿⠿⠿⠿⣿⣿⣿⡟⠁⢙⣿⣿⡿⠋
⠀⠀⠙⠿⣿⣿⣿⣿⣿⡿⠟⠁⢠⣿⣿⠋⠀⠀⣠⣾⣿⠟⢿⣿⣶⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣶⣾⣿⡆
⠀⠀⠀⠀⠀⠈⠉⠁⣀⣠⣤⣴⣿⡿⠃⢀⣠⣾⣿⡟⠁⢀⣴⣿⣿⣿⡀⠀⠀⣴⣶⣾⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⣿⣿⡇
⠀⠀⠀⠀⢀⣤⣶⣿⡿⠿⠟⠛⠉⠀⣴⣿⡿⠛⢿⣿⣶⣿⡿⠋⢸⣿⣷⠀⠀⣿⣿⡏⠉⢉⣀⣀⣤⣤⣤⣶⣶⣾⣿⣿⡇
⠀⠀⢀⣴⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⣠⣿⡿⠋⠀⠀⣸⣿⡏⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠉⣿⣿⡇
⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠟⠁⣀⣤⣾⣿⠟⠀⠀⠀⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⢸⣿⡏⠀⠀⠀⠀⠰⣿⣶⣀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠛⠁⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠘⣿⣷⠀⠀⠀⠀⠀⠈⠻⣿⣶⡄⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣿⣿⡇
⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠈⠻⠟⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⠀⣀⣀⣀⡀⣿⣿⡇⠀⠀⠀⢀⣾⣿⡿⠿⠿⢿⣿⣿⡇
⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠸⣿⣿⣄⣀⣀⣠⣿⣿⠇
⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⠀⠀⢺⣿⣯⡁⠀⠀⣉⣿⣿⠇⠀⠀⠀⠀⠙⠻⢿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣶⣾⣿⡿⠟⠁⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠉⠰⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⢀⣤⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⣠⡟⠉⠳⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠈⠉⠉⠀⠀⣇
⣴⣶⣦⡿⠀⠀⠀⠀⠀⠉⠑⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣠⣯
⠉⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⢀⣴⣶⣴⣾⣿⣿⣿⣿⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡄⠀⠀⠘⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣶⠟⠀⠀⠀⠀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⡿⠃⠀⠀⠙⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡆⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣈⠄⠀⠀⠀⠀⠘⠛⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠚⠲⠄⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⠆⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⢀⠤⣀⠴⠳⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⡿⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⢸⠟⠉⠦⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⡄⠀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡼⠁⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣾⠁⠀⠀⠀⠀⣼⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠋⠉⠉⠁⠀⠻⠇⠀⢠⡿⠀⠀⠀⢀⡼⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⣠⣾⡥⢤⡴⠞⠁⢀⡴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⣿⡀⠀⠈⢷⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠙⠛⠶⣦⣤⣤⣤⣤⣴⣶⠶⠞⠛⠁⠀⠘⣷⡀⣨⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢀⣵⣤⣤⣴⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⡴⠋⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠃⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢰⠏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⢹⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣠⡀⢸⡀⠀⠀⠀⠸⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣶⠶⠿⢿⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣥⡼⠏⠉⠛⠀⠀⠀⠀⠀⠈⣿⡉⠉⠛⠛⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡆⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣤⣤⣤⣭⣬⣯⣬⣥⣤⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠀⠈⡏⠀⠀⢹⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⡇⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠦⢾⡇⠀⠀⢸⡷⠴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣶⣾⡇⠀⠀⢸⣷⣶⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣇⠀⠀⣼⣿⣿⣿⣿⣿⣷⡄
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⢀⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⣄⣶⡐⠨⠁⠈⠑⠄⠁⡤⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⡒⠃⠀⠀⠀⠀⠀⢂⠀⡀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢠⢈⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⢀⡀⠀⠈⠂⠀⠐⠀⠀⡠⠄⠀⠈⡆⠀⠀⢀⠈⠈⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⣀⠠⠪⠀⠀⠀⣠⡁⠀⠀⠈⢑⠀⡀⠠⡇⢀⠔⠁⠱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣐⡄⢰⣦⠀⠀⠘⣿⠀⠀⠀⢀⠢⠀⠆⠘⠁⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠈⠿⠃⠀⠀⠀⠀⠀⠀⡈⠀⡀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠃⠐⠀⢠⠊⢀⠰⠁⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⢠⠀⢉⠠⣥⠒⢁⠌⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠃⢠⠀⠀⢀⣀⠀⠤⠄⠒⠂⠉⠄⣦⣤⡆⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⡕⠂⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠉⠀⠀⠀⠈⠉⠉⠉⡙⠀⠀⠘⠋⠉⢹⠤⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠄⠀⢸⠷⠿⠿⢿⠀⡀⢀⠄⠂⠀⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⢠⣷⢶⣤⣤⣼⡆⡀⠌⠳⠤⢔⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢲⠙⠿⢿⠠⢍⣯⣽⣌⢁⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⠈⠋⠉⠉⠈⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠉⠁⠚⠊⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠒⠀⠁⢢⣀⡀⠀⠀⢡⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡔⠁⠑⣼⠄⠀⠈⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⡶⠃⢀⡇⢀⠥⠀⠊⣡⠤⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⢀⣠⠶⠋⠀⠰⠀⡀⠊⠁⠀⢀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⠉⠀⠀⠀⠀⠀⠁⠈⠈⠀⠉⠄⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⡀⠀⢀⡀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢤⣼⣧⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⣾⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡃
⠀⠀⠀⠀⠀⠀⠘⣽⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠛⡵⠃
⠀⠀⠀⠀⠀⠀⢀⡟⢿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣋⡉⠉⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠱⡏⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡆⠀⢠⡟⠿⠂⠀⠀⠀⠀⠀⠀⢸⠃⢻⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠀⢧⠘⣆⣀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠚⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⡦⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢇⡀
⠀⠀⠀⠀⠀⠠⣤⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣇⢹⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⢿⣿⡀
⠀⠀⠀⠀⠀⠀⢹⣾⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢷⡀⠈⣿⣷
⠀⠀⠀⠀⠀⢀⣼⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⡀⠀⠀⠀⠀⠀⠀⢰⡋⠀⣰⡋⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠙⣦⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠑⢾⣷⠄⠀⠀⠀⠀⠀⠀⠀⠘⠻⡄
⠀⠀⠀⠀⠀⠀⣀⣼⣿⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣿⠉⠁⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣿⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠋⢛⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠛⠂⠀⠀⠀⠀⠀⠀⠘⠟⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠙⠿⠟⠃⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠄

⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣆
⠀⠀⠀⠀⠙⢣⣁⣸⣇⣹⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠾⢿⡿⠷⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠉⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⠏⡇
⠀⠀⠀⠀⠀⠀⠀⡏⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⢷⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⠀⠀⠙⢷⠀⠀⠀⠀⠀⠀⠀⠀⢷⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⣀⣤⣶⠾⠿⣷⣶⣾⣇⣀⣤⣴⡶⠶⠶⠶⠶⠶⢶⣤⣄⣀
⠀⠀⠀⢠⣾⠟⢉⣠⣴⣶⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣶⣄
⠀⠀⠀⣿⣥⣶⣿⣿⣿⣿⣿⣿⡿⠁⣀⣤⣶⣶⡿⠿⠿⠿⢿⣶⣶⣤⣀⠀⠀⠙⢷⣦
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠿⠋⣠⣾⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣷⣄⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠉⣹⡟⠉⠀⠀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀⠀⠈⢿⡄
⢀⣀⡏⠂⠀⢰⡟⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠈⣿⡄
⠿⠿⠃⠀⠀⣾⠃⠀⠀⢀⣿⡇⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠸⣿⡆⠀⠀⢸⣧⣴⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣧⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣦⣤⣿⡇⠀⠀⢸⣿⡿⣿
⠀⠀⠀⠀⠀⣿⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠀⠀⣿⣶⣤⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡄⠀⣾⠏⠀⣿⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⢻⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⢹⢹⣿⣷⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⢸⡟⠀⢠⠇⠀⠀⢀⢜⣥
⠀⠀⠀⠀⠀⠘⣿⡀⠸⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣾⡾⠟⠋⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⢸⣇⣴⠋⠀⠀⠀⠘⢌⣡
⠀⠀⠀⠀⠀⠀⠘⣷⡀⠙⠻⢿⣿⣿⣿⣿⠀⠀⠙⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⢋⣴⠿⣿⠁⠀⠀⠀⠀⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⣰⡟⠁⠀⣹⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠀⢀⣾⡿⣿⠛⠻⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⣀⡀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠾⠋⢹⣇⠀⠘⣿⠀⢹⡀⠀⠀⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠁⢠⡄⠀⠀⢿⣄⠀⠈⠁⠸⣇⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠀⠀⠀⠀⠙⠿⢶⣦⣤⣿⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⢾⢳⡄⠀⠀⠀⠀⣠⣤⣄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠋⠀⠀⠀⠀⢰⣿⠿⠟⣸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⣦⡤⣀⣄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠋⠀⢀⠀⠀⢤⡤⣀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠃⠉⠳⣤⣀⠈⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡃⢽⡶⣄⠈⠑⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⡀⠀⠀⠀⠀⠰⠂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠀⠀⠀⠀⠀⢯⡛⠶⣿⣯⣌⣲⣤⣄⠲⣴⡆⣀⡀⢤⣤⠤⠀⣀⡀⠀⡠⠀⠀⠈⣊⣤⠎⠳⠠⠄⠀⠙⠡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠓⠊⢤⣴⡒⠦⣛⣴⣤⣤⣄⡀⠀⠈⠓⣤⣤⣀⠞⠀⠀⡀⠚⢩⡼⠂⠀⠁⠀⣀⠴⢚⠤⣴⠶⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠁⠀⠀⡐⠦⣤⣀⡀⠀⠈⠛⠷⠾⠇⣻⢿⣤⣿⣿⣟⡷⣮⣞⠉⠉⠒⠊⠀⠀⣀⣀⠠⠀⠂⠩⠐⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠆⠠⠄⣸⠍⠙⠻⠻⢏⣹⠷⢋⣙⠛⠛⠁⠀⠀⠠⠖⠋⠉⠀⠀⠀⠒⠀⠁⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⠤⡀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠈⠀⠈⠉⠉⠴⠶⠾⠀⠀⠉⠀⠀⡀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠙⠛⠓⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠊⠳⡈⠳⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠤⠤⠤⠀⠀⠀⠒⠂⠈⠉⠉⠉⠉⠐⣀⠤⠷⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢿⡀⠸⡄⠀⡀⠐⠠⣄⣀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⢫⣖⠴⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠁⠀⣧⣄⣈⣓⠒⠀⠉⠉⠒⠠⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠐⠂⠀⠀⠀⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠃⠀⠤⠄⠉⠉⠉⠈⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠼⠴⢦
⠀⠀⠀⠀⠀⠀⠀⡴⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⣴⢉⡁⠁⣦⣄⠶⡄
⠀⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡂⠀⠀⠠⣀⠋
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠁⠀⠀⠀⠈


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⣝⡛⣋⡙⢻⡟⣫⣿⢻⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢰⠋⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢱
⠀⢠⠒⢺⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡗⠒
⠀⢼⠀⠸⡇⠀⢸⠀⠀⠀⠀⠀⢀⣠⢤⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡤⢤⢄⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⡶
⣤⢸⡀⠀⡇⠀⠸⣒⡒⢒⣒⣒⣇⢠⡠⠀⢗⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠀⠂⡤⢠⡁⠀⠀⠀⢈⡇⠀⢸⡇⠀⡇
⠛⠀⡇⠀⡇⠀⠀⠁⠀⠀⠀⠀⠙⣆⣀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⣴⠋⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⡇
⠀⠒⡇⠀⡇⠀⠀⡆⠀⠀⠀⠀⠀⢸⠀⠀⣀⡤⢤⣤⡄⠀⢀⢤⡖⠲⠤⣀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡇⠀⠘⡇⠀⠉
⠀⠀⠣⠤⡇⠀⠀⡇⠀⠀⠀⠀⠀⢸⡴⠋⠀⠀⠘⠉⠀⠈⢑⠈⠉⠀⠀⠈⠳⣼⠃⠀⠀⠀⠀⠀⠀⡇⠀⠈⠓⠒
⠀⠀⠀⠀⠣⠄⠤⠃⠀⠀⠀⠀⠀⡸⣇⠀⠀⠀⠀⠀⣀⠀⠎⣤⠀⠀⠀⠀⠀⢉⡀⠀⠀⠀⠀⠀⠀⠓⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠈⠀⠀⠀⡄⠘⠛⠀⡆⠉⠁⢀⠃⠀⠀⠈⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠂⠀⠀⠈⠂⠀⠀⠀⠀⠀⠁⠀⡠⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⡀⠀⠀⠀⠑⠠⣀⡀⠀⠀⠀⡀⡀⠐⠀⠀⠀⠀⠀⡀⠆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠂⠀⠀⠀⠀⠀⠀⠖⠂⠀⠃⠠⠃⠀⠔⠀⠀⠀⠀⠀⠀⠼⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠀⠂⠀⠀⠀⠀⠠⠀⣀⡉⠦⠀⠀⠀⠐⠤⠀⠀⠀⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⣀⢀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢀⡤⡀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠑⠒⠒⠒⠉⣿⠀⢹⡏⠉⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣿⠀⢸⣇⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⢂⠀⢀⡹⠂⢎⡀⠈⡐⠉⠈⠲⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣦⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠉⠁⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⠀⣼⣿⣿⣿⣿⡿⠛⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠒⢸⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⣿⣿⢻⡿⠋⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠘⠊⠀⠈⠙⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢶⣶⣿⣿⠁⠀⠀⠀⠹⡟⠂⠀⠀⠀⣀⡸⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⡇⠀⠀⠀⠐⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⡙⠁⠀⠀⠀⠀⠀⢀⣴⠅⠂⠐⠶⣄⠀⠀⠋⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠉⠀⠈⣿⡦⢀⠀⡃⠀⠀⠀⠀⢀⡠⠐⠈⠀⠀⠀⠀⢀⠆⠰⠀⠙⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠈⠀⠀⠀⠀⠀⢠⡛⢱⠀⠈⢱⡶⠢⡀⠀⢁⠀⢀⠤⡄⠀⢡⠄⠊⠀⠀⠑⠤⠄⠈⠈⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡤⠀⠂⠈⣥⣴⠗⠀⠀⠓⢀⡸⠃⠁⠠⠀⠀⠀⠀⠆⠐⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠰⠀⠀⠀⠀⢰⠊⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢃⠀⠀⠀⠏⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⠀⠡⡀⠀⢠⠊⠀⠑⠄⠀⠀⠀⠀⠀⠀⡰⠀⠃⠀⠀⢀⠜⢢⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠸⠀⠀⠀⢰⢱⠀⠀⠀⠀⠀⠀⠀⢀⠆⠘⠀⠀⠱⣀⡬⢠⣤⠊⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⣀⠔⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⡆⠀⠀⣼⡎⠀⠀⠀⠀⠀⠀⠔⡁⠀⠀⠃⢀⡴⠋⢠⡎⠈⠁⠒⠀⠄⠀⡤⠁⠀⠁⠀⠀⠁⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⠰⠙⠂⠤⠀⣀⡀⢀⠌⠀⠐⡀⠀⠈⠀⢀⠴⢤⣿⣀⡀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢘⡂⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠈⠢⣀⠔⠁⠰⠋⠀⠀⠈⠈⠓⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠟⡒⠂⠠⡤⠄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡌⠀⠀⠁⠀⠀⠈⠒⠣⠀⢀⠀⠀⠀⠀⠀⠀⢀⠂⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠐⡀⠀⢅⡂⠤⢀⠀⠀⠀⠈⠁⠁⠂⠄⣀⢀⠎⠀⠀⠀⢀⠎⠈⠉⠀⠒⠒⠒⠒⠂⠄⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡰⡙⠃⠂⢌⣀⡀⠀⠁⠀⢧⠀⠀⠀⠀⠀⠉⠂⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⠕⠀⠀⡠⠃⠀⠀⠀⠉⠉⠉⠉⠁⡐⠀⠀⠀⡆⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠛⠤⣀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠂
⠀⠀⠀⠀⠀⢀⠊⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⢀⠔⠃⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠷⠺⡀
⠀⠀⠀⠀⢀⠂⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠹⠶⠁⢠⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⣀⠀⠈⢄
⠀⠀⠀⢀⠂⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡠⡀⠀⣠⢾⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠭⠂
⠀⢀⣔⣇⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢦⡍⠀⠀⠈⢄
⠴⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠫⢐⡤⣀⠑⢄
⠈⠃⢑⡄⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋
⣀⣀⣀⣙⣲⣌⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣿⣿⡿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠗⣁⠀⢹⣧⣿⣿⣿⣿⣿⣸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠼⣿⢷⣿⣸⡟⣿⣧⣻⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠰⣿⣾⣿⣿⣧⢹⣿⣿⣿⣿⣿⣏⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢹⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢠⣀⢲⣶⣶⡓⡄⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣎⢿⣆⠿⠛⠁⠁⣤⣿⣿⣿⣽⢿⣿⣿⣿⣿⣿⣏⢻⣿⣿⣷⡄
⠀⠀⢀⡀⠀⠄⠀⠀⠀⠠⠷⠿⠛⠛⠋⠉⠟⠛⠛⢦⠀⠀⠀⠀⠀⢻⣿⣻⢹⣿⣾⣿⣿⣻⢿⣿⣿⣆⣿⣿⣿⠇
⠀⠀⠀⠀⠈⠀⢲⠀⠙⠢⡀⠀⠀⠀⠀⠀⠈⢣⠀⢻⣥⡄⠀⠀⠀⣼⣿⣿⣾⠿⠛⠉⠁⠙⡏⢢⣿⡻⣿⣼⣿⡀
⠀⠀⠀⠀⠀⢠⡊⠀⢀⡀⠑⠆⠀⠀⠀⠀⠀⠐⡗⠀⢸⡇⠀⠀⣨⣵⣿⣿⠋⠀⠀⠀⠀⠀⠃⠀⠙⣿⣷⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠛⠻⠄⠀⠀⠀⠀⠀⠀⠀⠙⠀⢸⡇⢠⣿⣿⢟⡿⠉⠀⢠⡤⠴⠛⠀⠀⠀⠀⢻⣿⣏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⣏⡵⠋⠀⢠⣀⠀⠀⠀⠀⢀⣜⣥⠠⣯⣿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠈⣷⠘⠋⠀⠀⠀⠀⠙⣿⣶⣤⡀⣼⢿⠁⣼⠸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠷⢤⣾⣿⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⠏⠙⠈⡡⢂⣘⣿⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣀⠀⠀⠀⣀⣠⣶⡛⣿⠿⠛⡛⠋⠀⣀⡴⢡⣾⣿⣿⣿⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠷⠛⠉⠙⣿⣿⣿⣷⣷⣻⣆⣀⠙⣶⣿⣟⣹⣿⠁⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣈⣿⣿⣿⡙⠛⡋⠁⠛⠿⢾⣿⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣿⠋⠉⠉⢱⣿⣿⣳⣶⣬⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠾⠿⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢠⣠⣤⣠⡄⠘⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣠⠀⠠⢄⣀⣀⣨⣿⣿⡏⠉⠉⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣽⣶⡛⠋⠙⢛⢿⡛⠛⠓⠒⠒⠒⠒⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣿⣿⣾⣿⣶⣻⡶⠀⠀⠀⢠⣤⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢉⣉⣭⡌⣇⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠛⠉⠀⣀⣾⠂⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆⠀⠀⠀⠈⠓⠀⠀⠀⠀⠲⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠃
⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠠


⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⣹⠳⡀
⠀⠀⠀⠀⠀⢀⡞⠁⡟⠀⢠⠃⠀⡇
⠀⠀⠀⠀⠀⡼⠀⢸⠃⠀⡎⠀⠀⠃
⠀⠀⠀⠀⢰⠇⠀⡟⠀⢰⠃⢀⢰
⠀⠀⠀⠀⣸⠀⢰⡇⠀⡾⠀⣾⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢦⠤⣀
⠀⠘⠀⠠⡿⠀⣸⠀⢀⡇⢠⡿⠧⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡸⠀⢸
⠀⠀⠀⠀⡇⠀⡿⠀⢸⠁⢸⣿⠃⠐⠂⠀⠨⢍⠋⣰⠀⠠⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢠⡇⠀⣸⠑⣆
⠀⠀⠀⠀⡇⠀⡟⠀⢸⠀⢸⢋⠀⠀⠀⠀⠀⢐⠀⠀⠀⠉⠐⠚⠀⠬⢐⡂⠀⠄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠜⡤⠀⡼⠀⢠⡇⠀⡏
⠀⠀⠀⠀⡇⠀⡇⠀⣾⠀⠀⠆⠀⠀⠀⠀⠀⠸⡄⠀⠀⠤⣤⡉⣿⠆⠀⠀⠀⠁⠒⠇⡀⡐⠀⠄⠀⠀⡜⣼⠇⢰⠃⠀⡾⠀⢰⠃
⠀⠀⠀⠀⢃⠀⡇⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⢄⡉⠈⠛⠑⠀⠈⠑⡄⠀⠀⢀⡴⠀⠀⠈⠑⠂⠨⣙⡟⢀⠏⠀⣸⠃⢀⡟
⠀⠀⠀⠀⠈⠉⢧⠀⢧⢠⠃⠀⠀⠀⠀⠀⠠⡀⠀⠀⠙⢄⠀⠀⠀⠀⠠⣼⡤⠔⠋⠀⠀⠀⠀⠀⢀⢱⡿⢀⡞⠀⢰⠏⠀⡼⢀
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠁⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠙⢶⠢⢲⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠞⠁⡜⠀⢠⠏⠀⡼⠁⠂⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠖⠉⠉⣽⣦⣄⣀⣠⣴⣄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡼⠁⢠⠏⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⠻⠿⣿⣿⠿⡇⠑⣄⠀⠀⡀⠈⠀⡎⠀⡜⠀⣠⠋⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡿⣿⣦⣤⣀⣀⣴⡇⠀⠈⡠⠊⠀⠀⠀⣇⠎⢀⡴⢇⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡈⠙⠻⠿⢻⠏⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠙⠷⠶⠞⠁⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠁⠀⠀⠀⠀⠀⠁⠀⠐⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⡴⠻⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣀⣠⠞⠓⠤⣉⢒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡋⠀⢠⠾⠄⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠱⠤⠞⢁⡤⢄⣀⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⡀⠑⠴⠂⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠟⠉⠉⠉⠉⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣁⡹⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠀⠀⠠⠄⠀⠀⠀⠀⠀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠜
⠀⠰⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⡰⠁⡜⠉⢢⡀
⠑⢮⠀⠀⣎⢉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣤⣴⣶⣶⣤⡀⠀⠄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⣱⢦⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠙⢿⣿⣿⣻⠗⠚⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢳⣤⠖⠛⢤⡙⢦
⠀⠀⠣⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠂⠀⠀⠀⠀⠁⠸⣄⣠⣶⣇⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢠⠤⡄⢀⣹⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠁⠁⠀⠀⠀⠀⠀⠒⠀⠀⠈⢻⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢴⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣹⠤⠖⠙⠀⠀⠀⠀⠄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠹⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⡄⠀⠀⠀⠼⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⢠⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠈⠂⠄⣠⠁⠀⠀⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⣿⡏⠛⢧⡀⠀⠀⠀⠀⠀⠀⢹⡇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⢀⡀⣀⠀⢸⣷⣄⠀⠈⠀⠀⠀⠀⠀⠀⢾⠁⠀⠀⠠⢀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠛⠷⠄⣀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⣶⣶⣶⣶⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⢻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⠀⠈⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⠀⠀⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠉⠛⠛⠀⠀⠀⠀⣿⠿⠟⠋⠉⠂⠈⠓⡀

⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⠀⢀⣀⡀
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⡀
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠿⠿⠿⠿⠿⠟⠁⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣤⣀⣬⣶⣿⣷⣄
⣸⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠇⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⢻⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠈⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠸⣿⣿⣿⣿⣿⣿⣦⣄⠀⠸⠀⠀⢴⡦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠀⣀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠈⠉⠻⣿⡏⠉⠁⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡏⠘⣿⣿⣿⠿⠃⠐⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠘⠋⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠋⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣀⣀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠀⣹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣦⣤⣾⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣠⣴⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡒⢢⣤⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢹⠶⠂⠦⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⠀⠒⠷⣦⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⣿⠖⣤⡜⠛⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠈⣟⢦⣽⣎⣒⢠⣈⡉⢄⣀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠫⠆⡠⢄⣀⠀⠈⠶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠓⠪⠋⣙⡯⣭⡹⣠⣉⣀⡉⠉⠉⠣⣀⣁⠀⠔⠁⠄⠠⠞⣡⠅⠀⠀⠀⠀⡮⢄⣠⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠑⠀⠀⠐⢤⣄⡀⠀⠈⠛⠷⠼⡏⣻⣯⣽⣿⣷⣶⣼⡭⠙⠂⠒⠀⠀⠀⢁⡀⠀⠐⠃⠅⠂⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠳⣄⠀⠀⠀⠀⠀⡀⠈⣙⠳⢂⣦⣜⣎⡙⠛⠻⣛⣿⢾⣍⢛⡛⠁⠀⠀⠠⠚⠉⠁⠀⠀⠂⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣷⢤⣀⠀⠀⠀⠀⠀⠙⠒⠀⠈⠀⠀⢈⠐⡓⠺⠛⠃⠙⠁⠠⠤⠀⣀⣤⣅⡀⠀⠈⠀⡘⠋⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⢧⠈⢷⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠓⠒⠒⠂⣀⠈⠉⠉⠀⠀⠈⠀⠀⣤⡐⠛⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠙⢧⠀⡇⠀⢠⣈⠒⢦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠸⡗⠃⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠟⠓⠲⠤⠭⠀⠀⠀⠈⠶⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⢄⡀⣀⡈⠁⠉
⠀⠀⠀⠀⠀⠀⡉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⠍⠉⠀⡈⠝⠀⠁
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠈⠃⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⠤⣌⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⢰⠁⣘⠄⣀⠛⠄⡀⠂⠌⠀⢀⣀⣀⣀⠀⠤⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠌⡘⢀⡏⠔⠂⠚⠂⠉⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠌⡰⠤⠌⠴⠃⣘⠴⡄⠀⡤⣤⠒⣶⣒⢆⣜⣡⣌⡹⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠐⢠⠁⡊⠀⠀⠀⠀⠀⣯⣱⡇⠀⠁⢉⣉⣉⢉⡁⢤⢄⠀⢤⠐⡄
⠀⠀⠀⠀⠀⠀⠀⣀⠂⠌⣑⠦⠄⠀⠒⠚⢶⢯⡇⠀⣭⡿⣼⣛⠾⣽⢃⠎⣐⠢⠌⡁⠀⠉
⠀⠀⠀⠀⠀⠈⡐⠠⡎⠁⠀⣀⡤⣴⡖⣶⣸⢿⡇⠀⣿⣽⣳⣯⣿⢣⣏⡲⢠⠋⠴⠅⠀⡦⢀
⠀⠀⠀⠀⠀⠀⢀⠂⠴⠀⠀⠯⠓⣋⣩⣭⢯⣿⡇⠀⣿⣿⣟⡾⣭⣛⡾⣱⠃⡌⣑⠇⠀⡵
⠀⠀⠀⠀⠀⡀⠂⠌⡰⢇⠀⢰⢏⡟⣼⣟⡿⣽⡇⠀⣿⣿⣿⣽⢧⣟⡳⢧⣝⢰⣁⠄⠀⣇⠂
⠀⠀⠀⠀⠀⢀⠀⠠⢁⠺⡀⠀⢿⡽⠗⠛⠛⠋⠁⠀⣿⣟⡿⣯⣟⡶⠋⠁⠀⠀⠀⠀⠀⡏⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠂⡘⢃⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⡾⣹⠿⣼⠀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠠⠀⠐⠠⢙⡀⠀⡆⠀⠀⠀⠀⠀⢀⡟⡳⢯⡟⠰⢦⡀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣱⠀⠸⡤⢤⣀⡠⢴⠛⠤⢁⠒⡬⢁⠆⡌⠃⠎⡐⠈⠉⠐
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠉⠀⠀⠀⢇⠂⠬⣑⠂⡌⠑⠎⡜⠠⢌⡐⠠⠉⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⢸⡌⠠⢀⡘⠠⢁⠒⠠⠁⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⡜⠀⠁⠀⠀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠉

⠀⠀⠀⠀⠀⠀⠀⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⠳⣄⠀⠀⠀⠀⠀⠀⠈⢦⣀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠘⣷⡄⠀⠀⠐⠶⠀⠈⢏⠛⢦⡀⢸⣧⣀⠀⠀⠀⢰⡀⣦⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⢹⣿⡆⢀⣀⠀⠤⠤⠼⠿⠶⣇⢸⣎⠻⣆⠀⠀⢸⢷⡸⡿⢦⡀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣿⣿⠀⡀⠤⠄⠒⠒⠘⡖⠚⠺⡿⢿⣿⠀⡀⢸⠀⢻⢃⠀⠃⢸⢿⣄⢀⡀⢀⡀
⠶⠦⣶⣶⠞⠛⠛⠓⠢⢀⡴⠊⢸⣿⣿⡉⢀⠤⠾⠂⠉⣁⣩⡅⠈⣇⠀⢻⢽⣿⣿⠀⠚⠈⣏⠉⢹⣀⢹⡆⠀⠛⢸⡆
⠀⠀⠀⠈⠉⠓⠢⠤⣠⣋⠀⡠⢺⣿⣿⠛⢧⣴⣶⣾⣥⣈⠿⠁⠒⡧⠠⠉⡐⠄⣺⡐⢨⣾⡷⠀⢺⠂⠘⣥⣐⠛⣻⠴⣆
⠀⠀⠀⠀⢶⠆⠀⢰⢣⣿⣿⣇⠸⣿⣇⢀⣿⡟⢧⡞⠙⢻⡆⠸⠿⠷⠀⠀⠀⢁⣼⠀⠉⡩⢐⣬⣽⢴⣶⣍⣿⠛⠁⠀⠈⢷⡀
⠀⠀⠀⠀⠀⢀⣤⣇⠀⢻⠿⠟⡼⠙⢿⣧⡙⡿⠥⠈⠲⣼⠃⠐⢾⡿⠀⠀⠘⠛⠋⠐⠍⡀⠿⠿⠋⠀⠉⠁⢠⡄⠠⣦⣤⠖⠁⢴
⠀⠀⠀⠀⠀⢸⣿⣿⠏⣻⠶⢤⣇⣀⢸⠉⠻⠷⠖⠒⠋⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠜⢵⣶⠎⠈⢀
⠀⠀⠀⠀⠀⣼⣿⣿⠀⣿⠀⠀⡇⠉⢙⠓⠶⣤⣀⠀⣶⣷⣀⠟⠀⠀⠀⢀⡀⣀⢀⡀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⡞⢉⡟
⠀⠀⠀⠀⠀⣿⣿⡏⠀⢸⢀⡴⣷⣶⣼⡀⠰⣿⣿⡆⠀⠉⠀⠀⠀⠀⠀⢸⡇⣿⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣧⠉⡄
⠀⠀⠀⠀⢠⣿⣿⣧⠀⠈⢿⣴⣶⡀⠙⣷⠀⠙⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣸⣿⢿⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢀⡀
⠀⠀⠀⠀⢸⣿⣿⠘⡆⠀⠘⡄⠀⢱⡀⠘⢆⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⣶⣶⣞⢲⡖⠀⠀⠀⠀⠀⢠⡆⢀⣀⡿⣶⡆⣾⡏
⠀⠀⠀⠀⠈⢉⡉⠙⠛⣖⠒⠻⡦⠤⠷⣤⣌⣢⠀⠸⣤⡀⣀⠀⠀⠀⠀⢸⡇⢿⡿⣿⢩⠿⣧⠀⠀⠀⠀⠀⠈⠁⡘⠻⢿⣤⠁⠉
⠀⠀⠀⠀⠀⠛⠃⢀⣀⡈⢶⣦⠘⣆⠀⢿⣿⡷⠑⣔⠿⠳⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢆⠌⠲⡄⠠⡤⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠨⠿⠶⠈⠻⣃⡠⠷⡌⢉⡱⣶⣾⡧⡀⠈⣢⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠖⠙⣦⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡟⠢⣀⡼⠫⡀⢈⣻⣁⠈⣿⢋⣉⠫⡉⠀⠀⠀⣴⠇⠀⢰⠗⣼⠇⠜⣿⡠⠀⣨⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⣿⣧⠖⠉⢢⡠⠚⠻⣿⠿⠓⢷⠾⠛⢄⠀⠑⢄⠀⠀⡄⠀⠀⢢⠜⣄⡠⠊⢣⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⣼⠟⢁⣤⡊⠁⠈⠢⡀⢈⡷⣶⣾⣷⢤⡀⠑⠢⣠⣷⢆⡘⡤⢊⠔⠱⡈⢣⣾⡗⠙⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⢺⣧⠹⣷⡄⣀⠴⠊⠓⢄⠈⠙⠯⡀⠈⠢⣀⠹⠫⡀⢉⢘⣅⣠⡾⠋⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠼⠋⠀⠀⠀⠀⢻⣧⡜⣿⣄⠀⠀⠀⠀⣙⡦⡖⣿⣿⣷⡌⠑⢧⣈⢔⣴⡿⠋⠀⠀⠰⣤⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡘⢿⣦⣠⠔⠊⢁⣀⠈⠛⢍⣩⠗⢆⡘⣷⡿⠋⠀⠀⠘⠃⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡷⠚⠉⠀⠀⣤⣈⠙⠋⢠⣾⠛⠢⡀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⡇⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⠼⠁

⠀⠀⠀⠀⣠⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⢦⡀⠀⠀⠀⠀⠀⠀⠈⡋⠀⠀⠀⠀⠀⠀⣴⣴⠀⠀⠀⠀⢠⣤⣄
⠀⠀⣠⣿⣿⣄⣄⣼⣿⣧⡄⠀⠀⠀⣰⠛⢻⣷⣶⣾⡇⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⠀⡏⠁
⠀⠀⠀⠿⣿⠁⠀⢸⡿⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣾⣿⣷⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠉⠉⠀⠀⠀⠀⠀⠀⠈⠛⠿⠋⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠚⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣶⣄⣀⣀⣰⣶⠀⠀⠀⠀⡷⠶⠶⠶⠶⠶⡇⠀⠀⠀⠀⠀⠀⢠⣥⡄⠀⠀⠀⠀⠀⢰⣿⣿⡆
⠀⠀⣤⡄⠿⠏⢉⣯⡹⠿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⢀⣹⣏⣀⣀⣀⣀⣀⣀
⠀⠀⣿⡟⠛⠛⠻⣿⠅⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠈⢹⡏⠉⠉⠉⠉⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠘⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠘⢷⠂⠀⠀⢠⡾⠏⠀⠀⠀⠀⣀⠀⣴⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⣀⠄⢀⠤⣀⢰⣄
⠀⠀⣤⣾⣶⣶⣶⠾⢧⡀⠀⠀⠀⠈⣿⠀⠉⠉⠍⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⣈⡀⠀⠀⠀⠀⠀⢈⡁
⠀⠀⢿⣯⣼⣿⣿⣄⣠⠇⠀⠀⠀⠸⠿⠸⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⡇⠀⠀⠙⠀⠀⠀⠀⠀⠙⠉⠉⠉⠉⠉⠉⠙⠋
⠀⠀⠀⠉⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⢠⣄⢀⣤⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⣠⣤⣤⣤⣤⣿⣇⠀⠀⠀⠀⣿⣿⣤⣄⣾⣿⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣄⠀⠀⠀⠀⠙⢻⠉⢹⣿⡍⠉⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣄⣅⣸⣿⠀⠀⠀⠀⠀⢸⠀⢸⣿⡇⠀⡇
⠀⠀⢶⣶⣶⡶⡶⣶⣿⡇⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣭⣭⣭⣭⣍⠀⠀⠀⠀⠀⠼⠄⠨⠿⠄⠤⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣴⣶⡆⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⣿⣿⣧⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢠⣿⣿⣿⣶⠀⠀⠀⠈⢻⣿⠿⣿⣿⡟⣱⣷⣶⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣾⡿⣿⣿⣿⡄⠀⠀⠀⠀⢁⣤⣻⠁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⢠⣶⣿⡇⠋⠀⢸⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠀⣨⣍⡛⢧⡹⣿⣿⣿⣿⣇⡇⠀⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠐⠻⢿⣿⢿⡏⢸⣿⣿⣿⣷⣄⣸⣿⣿⣿⢹⡇⠀⣸⣦⣽⣿⣿⣿⣿⡟⠿⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⢘⣯⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⡿⢻⣿⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣷⣿⣿⡿⠟⠁⠀⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣯⣥⣷⣾⣧⣀⠀⣠⢽⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣶⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢻⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠘⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣙⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣫⡽⠖⠀⠀⠘⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣰⣿⣍⠳⣜⠿⣿⠿⣟⣛⣛⡿⣿⣿⡿⣫⡵⢋⣥⣶⣿⡧⠀⠀⠀⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣷⣍⡛⢛⣋⣭⣭⣭⣙⠲⠶⠚⣫⣴⣿⡿⠋⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡀
⠀⠀⣠⣤⣴⣿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⡄
⠀⢺⣿⣿⠟⠁⠀⠀⠀⣰⣿⣿⣿⣿⣿⠏⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⢠⣾⡿⠋⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⡿⢰⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⣿⣿⣿
⠘⣿⠇⠀⠀⢀⣴⣶⣾⣿⣿⣿⣿⣿⡇⠈⠟⠛⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⡿⠟⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⣴⣶⣾⣿⣿⣿⣿⣿⣿⠏⢿⠀⠉⠙⢿⡆
⠀⠀⠀⠸⣿⡇⣿⣿⡟⠘⠋⠋⠅⠀⠀⠀⠉⠀⠈⡇⠀⠀⢀
⠀⡀⠀⢀⣿⣧⣸⣿⡇⠀⠀⠀⡆⠀⠀⠀⠂⠁⠀⡇⠀⠀⡄⠂⡄
⢆⠀⣠⣾⣿⣿⠏⢻⡇⠀⠀⠀⢀⠀⠀⠀⠀⡠⠀⢧
⠈⠛⠛⠛⠛⠁⠀⠀⢹⣄⢀⠀⠀⢸⣶⣶⡇⠀⠀⡸⠀⢀⠁⠇
⠀⠀⠀⠨⠐⠄⠀⠀⠘⠮⣅⡀⠀⠈⠁⠁⠀⣀⡜⠓⠆⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠄
⠀⠀⠀⠐⠂⠨⡢⠀⠀⡀⡔⠉⠑⠒⠒⠒⠋⠁⠘⡆⠁⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⠀⠀⠛⠀⠠⠖
⠀⠀⠀⠀⠀⠀⠈⠂⠄⠂⠉⢲⠁⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠶⠎⠀⠀⠁⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣀⣤⠤⠤⠶⢲⠾⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠒⠒⠲⣶⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠀⠀⢀⣀⣴⣏⣀⣾⠄⡼⠀⠀⠀⠀⢸⣿⣭⣭⣍⣉⣹⣿⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢑⢦⠞⠉⠉⠁⠀⠀⣠⢣⠎⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠱⡀⠀⠀⠀⠀⠀⠘⠛⠻⠷⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⣵
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠉⠑⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠒⢦⡀⠈⢁⡔⢁⢍⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢨⣀⡠⠶⠖⠚⣛⣛⣉⣉⣉⢭⡽⠃
⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⣇⠐⡩⢦⡘⠒⣡⡈⠢⡀⡰⣡⠔⣒⣒⢠⠃⠀⠀⠀⠀⠀⠰⠯⣥⠶⡖⠉⠙⢉⠁⠀⠈⢸⡄⠀⡇
⠀⠀⠀⠀⣶⡋⢻⡆⠙⠲⠤⢤⠔⡡⣊⡴⣛⣳⣄⠉⢶⠄⠈⠳⡇⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠠⠈⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠘⢎⡴⣫⣾⢟⣵⣿⣦⡀⠀⠀⡤⠜⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠠⠄⣿⠀⡇⠀⠀⢸⠃⠀⠀⠀⠃⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣏⢔⣹⠿⣋⣴⡷⡄⠀⠘⢆⠀⠻⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢹⡀⢸⣀⣀⣰⣦⠦⠤⠚⠛⠉⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠈⠻⡥⣪⢟⣭⡾⢟⣦⠀⠀⢳⣀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠰⣟⣉⣉⣉⠥⠤⠤⠒⠒⠚⠋⠻⣧⣤⣤⣤
⠀⠀⠀⡠⠂⠀⠀⠐⠄⠄⠀⠾⠿⠀⠀⠀⠘⢷⠟⡫⣴⡿⡫⣣⡀⠀⠉⠀⢱⣀⡀⠀⠤⠠⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⢀⢀⠠⠇⠀⠀⠀⠀⠀⣀⠤⠤⠒⠒⠈⠉⡂⠀⠻⣊⢕⡪⣞⣭⣷⣂⠀⠀⠀⢷⢓⠲⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⣿⣿
⡎⠀⠀⠀⠀⠀⠀⠀⠸⣴⡀⠀⠀⠀⠀⢠⠇⠤⠤⢽⣯⡪⢟⣫⣶⣷⠂⠠⣶⢸⠀⠀⠀⢔⣢⠒⣼⡂⣀⠀⠀⢀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⡇
⠈⠀⠀⢒⢀⡀⠀⠀⠀⠈⠙⢶⣤⡀⠤⡼⢦⠤⣤⣃⠘⣾⣟⠟⣋⢼⡧⠤⠒⠉⠀⢰⠒⣤⠿⣋⡡⠄⠈⢷⣾⣷⣄⠈⠉⠏⠀⠙⢯⡁
⠀⠀⠀⠈⠁⠙⠦⣄⣀⠀⣰⡪⠛⢧⢠⠇⠈⠲⠃⠈⠉⢙⣶⣿⣿⡉⠀⠀⢀⡤⠒⠉⠱⡜⢧⠀⠐⠂⢀⡼⠈⢿⣿⣆⠎⠀⠀⠀⠀⠘⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡹⠒⠒⠒⠢⣤⠀⣾⣿⣿⠋⣡⠴⠊⠁⠀⠀⠀⢀⣵⢈⡧⢴⠊⡿⣦⠀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠘⠂⠤⠼⠁⠀⠀⠀⠀⢹⠀⢻⡿⠗⠉⠀⠀⠀⣀⠴⠒⠉⠉⠈⢋⠀⠛⠃⠈⠁⠀⠀⡟⣠⡆⠀⠀⠀⠀⠀⡟⠫⠯⠃
⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠒⠉⠀⠀⢀⡤⠖⠛⠁⠀⠀⠀⣠⣴⣶⣦⣄⠀⠀⠀⠀⠰⣟⠄⠃⠀⠀⠀⠀⠀⠃⠀⢀⣤
⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⢰⠃⠀⠀⣠⠴⠚⠉⠓⠦⡄⡀⠀⠀⢰⣿⢋⠽⢻⣾⣧⡀⠀⣀⠀⠄⠀⠈⠀⠀⠀⠀⢸⣶⣤⡀⠉
⠀⠀⠀⠀⡄⠀⣀⡀⠀⣬⠀⠀⠹⡆⠀⠀⠀⠈⠳⣍⣉⠀⠀⠀⠀⠀⢄⠈⠳⠀⠀⠸⣷⣝⠊⣠⣿⡴⠋⠓⢤⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡷
⠀⠀⠀⠀⣷⠈⣿⣇⠀⠙⠧⣄⠼⠀⠀⠀⠀⠀⠀⠈⠙⣿⣷⣶⣶⣬⣉⠀⠀⠃⠀⠀⠈⠓⠶⠿⠿⠅⠀⠰⠆⣉⠢⣄⠀⠀⠀⠀⢠⡿⠿⠟⠁
⠀⠀⠀⠀⢸⡆⢹⣜⡄⠀⠀⠀⠀⠀⠀⢀⠔⠉⢱⣶⣿⣿⣿⣿⣿⠟⠉⠛⢷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠈⠳⢄⠀⠀⡜
⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⢀⣿⠿⠿⠻⣿⣿⣄⣀⣀⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠄⠀⠒⠒⠾⢷⣈⠀⢀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠒⠉⣀⡤⠔⠊⠁⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣄⣠⣤⡶⠖⠉⠉⠀⠁⠢⣀⠀⠀⠠⡖⠒⠢⣝⢆⠀⠀⢱⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡀⠀⠀⠰⠋⠈⠀⠙⢦⠀⠘⣿⣿⣿⣿⣿⣿⣿⠟⣉⣀⣀⠤⠤⠀⢠⣤⣈⡑⢦⡔⠋⠀⠀⠀⠀⢣⠀⠀⠀⠠⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡄⠀⢣⠀⠀⠀⠀⡀⡇⠀⠘⢿⣿⣿⠿⠋⠀⠀⠹⡄⠀⡰⠁⠀⢸⡿⠋⠁⠀⡍⠓⠢⠤⠤⠴⠋⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠑⠤⠤⠤⠜⠁⠀⠀⠈⢟⠁⣀⠤⠤⠤⣄⠘⣴⠃⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠠⡄⠀⠀⠀⢀⠏⠒⠲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⡠⠤⡄⠀⠀⠈⢎⠃⠒⠒⠒⠊⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠓⠤⠂⠀⣀⣠⣼⠀⠾⠭⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣤⣤⣴⣾⡿⠿⠿⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣹⣛⣩⣙⣙⣟⣩⣟⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣦⣀⣀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠿⠛⣻⣿⠿⠟⠛⠋⠉⠉⠀⠀⠻⠿⠟⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿
⣟⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠉⠙⠛⠻
⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣰⣏⣽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⡿⠁⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⢁⣦⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⣀⣀⠀⠀⠐⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⣷⣶⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣉⣉⣉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣖⣯⣿⣿⣿⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⣠⢀⣆⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⠀⠀⠀⠀⠀⠀⠀⣰⣀⣀⣀⣀⣀⣶⡀
⠀⠀⠀⠀⠀⠙⠶⣾⣷⡶⠞⠁⠀⠀⠀⠀⠀⢰⣿⠻⠦⠄⣴⠂⠀⠀⠀⠀⠀⠠⠞⣿⢦⣠⢀⡿⠀⠀⠀⠀⠘⠐⣆⡶⣠⠆⠛
⠀⠀⠀⠀⠀⠀⣀⣼⣿⣄⡀⠀⠀⠀⠀⠀⠀⢶⣭⣷⡀⢠⠃⠀⠀⠀⠀⠀⠀⠀⣸⠟⢶⡀⣸⠇⠀⠀⠀⠀⠀⠀⢘⣿⡏
⠀⠀⠀⠀⠀⠀⢻⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢹⠶⠿⠻⣗⠀⠀⠀⠀⠀⠀⢠⣭⣥⣴⣾⣶⡿⠀⠀⠀⠀⠀⠀⠀⣼⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢉⡳⢄⠀⠀⠀⠀⠀⠀⠀⠐⢲⡂⢶⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣸⣃⣀⡽⠂⠀⠀⠀⠀⠀⠀⠀⠹⣶⣦⡄⠀⠀⠀⠀⠀⠀⢈⣭⣀⣠⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣥⡀
⠀⠀⠀⠀⠀⠀⢹⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠟⠂⠀⠀⠀⠀⠀⠀⠻⣿⣥⣄⠈⣻⡇⠀⠀⠀⠀⠀⣠⠿⣿⣿⢷⣄
⠀⠀⠀⠀⠀⣤⣬⣿⣏⣿⣤⣄⠀⠀⠀⠀⠀⠀⣾⠁⢻⡀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣷⡟⣷⣿⡇⠀⠀⠀⠀⠀⠘⣿⠿⢿⣿⠃
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠂⠀⠀⠀⡀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⢠⡴⣿⣿⠦⡄⠀⠀⠀⠀⠀⠀⠀⢠⢾⣷⣆⠀⠀⠀⠀⠀⠀⠠⡾⣿⢧⡀⢀⡾⠃⠀⠀⠀⠀⢶⠎⣷⣦⣤⣄
⠀⠀⠀⠀⠀⢰⠀⣿⣿⠀⡰⠀⠀⠀⠀⠀⠀⠀⠘⣼⣿⣮⡅⠀⠀⠀⠀⠀⠀⢹⡟⢳⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⣶⣮⡙⣷
⠀⠀⠀⠀⠀⠈⠢⡌⣿⠴⠁⠀⠀⠀⠀⠀⠠⠴⠚⠉⠀⠸⠁⠀⠀⠀⠀⠀⠀⢸⠃⠼⢤⣾⡇⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠃⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠹⣄⡀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣪⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⣰⠛⣿⡇⠀⠀⠀⠀⠀⠀⠰⡶⣻⠄⢀⣄⡀⠀⠀⠀⠀⠀⠀⠈⢉⣹⣷⡘⠟
⠀⠀⠀⠀⠀⠀⠸⢿⣟⢻⠆⠀⠀⠀⠀⠀⠀⠀⠸⡶⢛⡇⠀⠀⠀⠀⠀⠀⠀⠐⠿⣷⣾⡿⢻⡄⠀⠀⠀⠀⠠⣔⣛⡯⠽⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠒⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁

⣷⣷⣿⡟⠁⣠⡶⣿⣿⣿⣿⣮⣷⡍⠻⣿⣶⠶⣖⣻⣻⣛⣰⡆⠦⣬⢈⡛⢲⢤⡙⠑⠆
⣿⣿⠏⢀⠞⡙⠂⠘⢹⣿⣿⣿⣿⣿⣿⣮⣻⣿⠁⣇⠉⠈⠹⠙⢋⠗⡶⣍⡑⠦⠛
⣿⡟⠀⡞⠰⠀⠀⠖⢙⠟⠛⢯⡿⣿⣿⣿⣿⣿⠿⠛⠿⠛⠙⠟⡶⢦⣤⣀⠛⠓⠲⢄⠀⠀⠀⠀⠀⢀⣤
⢭⠇⠀⣧⠁⠀⢻⠎⠁⠀⠀⣾⡿⢿⣾⣿⢿⣿⣯⡙⠿⢦⡘⠷⣷⣿⣭⣇⠀⠀⠀⠀⢣⠀⢀⣠⢾⣿⣽
⣧⣾⡀⢲⡀⡐⠀⠀⢰⣿⡆⢀⣴⣾⣿⣕⢌⣿⠟⣮⡢⡈⠟⢶⣽⠿⣿⣿⣿⣤⠀⠀⢠⠐⢿⣿⣿⣿⣿
⣷⣿⣇⠀⠛⠀⣠⡤⡀⠉⣰⠟⣵⣯⣙⣿⢿⣟⠻⣿⣺⡯⡢⣈⠻⣷⣜⠿⣿⣿⣷⣤⠜⠃⣀⢿⣿⣿⣿
⣿⣿⣿⣷⠀⠀⠻⠷⠇⣼⡇⣽⣿⣿⣻⣼⡻⣿⣷⣝⣻⡮⡪⡻⣅⣍⠙⢳⣞⠻⣿⣿⡄⠀⠉⣈⣿⣿⣿
⣿⣿⣿⡇⣠⡖⢦⠀⣸⠿⠗⣿⡿⠛⠛⣿⡿⠿⢿⣟⣛⣿⣛⠊⠪⡳⠕⢌⠹⣷⣮⡉⠙⢄⠀⢀⣸⣿⣿
⣿⣿⣿⠀⠻⣿⠿⠃⣿⠃⠘⣿⣁⣴⣾⡟⢂⣴⡌⣿⠿⠿⢿⣯⣦⡁⠀⠈⠳⣌⠿⡿⠀⠀⠀⠀⢼⣿⣿
⠿⣿⣿⠀⠀⡠⢄⠀⢿⡀⠀⢻⣿⣿⣿⣷⣿⣿⣷⢸⠀⠀⢀⣿⣿⣼⡢⡙⣆⠈⠀⠀⠀⠀⠀⠀⡀⣿⣿
⣷⡿⢿⡄⠸⣷⡾⠂⠸⣧⠀⠈⣿⣿⣦⢿⣿⣿⣿⣤⣷⣤⣾⣿⣿⣿⣿⣮⡺⣷⠄⠀⠀⠀⠀⠀⠑⢎⡛
⡟⣷⣾⣇⠀⠀⣠⠤⠀⢻⣇⠊⠈⢿⣿⣷⣿⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢮⠀⠀⠀⠀⠀⠀⣇⣀⠑
⣟⡿⣿⣿⡄⠐⣿⣷⣶⠀⠁⠀⢀⡀⠀⠙⠿⣿⣾⣿⣿⣿⣄⣀⣽⣿⣿⣷⣤⣼⡇⠀⠀⠀⠀⣨⡃⡝⢴
⣿⣿⣿⣿⣿⣄⠈⠉⠁⠀⢀⣾⣯⣀⣡⠀⠀⠀⢀⣩⡿⠿⢿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⢀⡪⣻⣮⡓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠀⠀⣠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⡔⠂⠄⣀
⢿⡷⠁⠀⠀⠀⠉⣲
⠀⠀⠀⠀⠀⣶⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠶⠒⠒⠛⠋⠉⢿⣠⣄⠀⠀⠀⠀⠀⠀⣀⠠⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⢉⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠿⠇⠀⠈
⠀⠀⠀⠀⢀⡶⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⣠⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠛⠛⠓⢦⣄⣧
⠀⠀⠸⢧⡄⠀⠀⠀⠀⠀⢀⣠⠤⠖⠚⠛⠉⠉⠛⠒⠦⣄⣰⠏⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⣾⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⡏⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠀⠀⠀⠈⡇
⠀⠀⠀⣇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠸⣇⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⠀⢸
⠀⠀⠀⠀⢻⣄⣤⣄⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⣾⠁⠄⢀⠈⠁⠀⠀⠀⠀⠀⠐⠒⠋⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡆⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠓⠶⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣼⠀⠀⡠⠖⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠿⠟⢿⡆⠀⠀⠀⡼⠃⠰⣯⠛⠄⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⢀⡽⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⢾⡯⢴⢶⣤⠶⠋⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠟⠉⠉⠛⠶⠶⠶⠖⠛⠛⠛⠉⠉⠀⠀⠈⢷⠀⠸⡇⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢧⡐⢄⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠶⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢹⡶⢛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠻⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⣦⡼⠚⢷⡶⠶⠶⠶⠶⠶⠒⠚⠛⠛⠋⠉⢉⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠷⠃⠀⠀⠈⠳⣦⠀⠀⠀⣀⣀⣀⡀⠀⠀⢴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢀⡀⢀⣈⣹⡅⣀⡀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⡖⠒⠒⢻⠗⠒⠒⣶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢸⡅⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠁⠉⢉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣴⠂⢠⣤⣤⣄⣀⠀⠀⣰⣧⣬⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣽⣿⣶⣿⣿⣿⣿⣿⣆⡀⣿⣯⣄⢠⣀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠹⠇⠹⢿⣿⣿⣿⣷⠸⡅⣩⠀⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡉⠀⣘⠃⠸⣿⣿⣿⣿⠤⣧⡙⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣦⣁⡴⠀⠻⣿⣿⣿⣶⠟⣿⠟⢀⣾⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⠻⢻⠯⣤⡟⠀⠀⣀⣈⣋⠀⠀⠀⠘⣇⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⠄⠀⠀⠀⠀⠀⠀⠀⠄⣌⢠⢠⠀⢈⠄⠀⠐⠙⠿⠈⠳⡆⠀⠀⢻⣿⣿⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⡾⠃⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⢇⠀⠀⢸⣿⣿⡆⠀⢌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣰⣤⣞⠰⣧⠚⣷⣄⣄⠀⠀⠃⠀⢀⣲⡏⢳⣿⠀⢸⢸⣿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠈⢿⡟⠁⢠⣿⣿⡿⠀⢰⠄⠀⣨⢷⡅⢈⣻⠀⢸⣾⣿⣿⡇⢠⣟⠃⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⣠⠟⠈⢻⡀⣰⠃⠀⣴⠋⣾⣯⣨⣿⠀⢘⣿⣿⣿⡇⠀⠈⢁⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠻⠶⢀⠈⠃⠋⠐⠈⣧⠚⠉⠛⢿⡿⢀⣾⠛⠉⢹⡇⠐⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠓⠧⣀⣠⡀⡇⠀⢀⠄⠈⡠⠊⠃⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡛⣡⣤⣴⡾⠁⠀⠀⢀⣴⡟⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⢠⣤⣴⠋⠀⠀⠀⠀⠠⠽⠋⠙⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⢀⠀⣿⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⣰⡏⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠟⠀⠀⠀⢠⠀⠀⠀⠀⠱⡀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⣀⢀⡀⠉⠀⣀⠴⠦⠀⠀⠀⠀⠀⠀⠕⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠠⣻⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠻⠇⠀⡟⠃⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣭⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀⠀⣗⡆⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣤⣿⣧⡀⢫⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣷⡛⢃⣺⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⡟⠉⠀⢠⡤⢠⡄⠀⠀⠀⠀⣾⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠛⠻⠿

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡀
⠀⠀⠀⢤⣴⣧⠄⠀⡀⠀⠀⠀⠀⠀⢀⣠⡴⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣀⣳⣾⡅
⠀⠀⠀⠘⠉⢃⣴⣿⣿⣿⣦⣤⡶⠞⠋⠁⠀⠀⢀⠀⠆⠀⠂⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⣔⣾⡯⢤⡿⡝⠛
⠀⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠘⠄⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⣷⠿⣿⢾⡇⠙⠁⠀⢰
⠀⢿⣿⣿⣿⣿⡿⠛⠿⠿⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠀⠀⠀⣀⣀⠔⠈⠈⠐⡌⠧⠉⠁⠀⠀⠀⡠⠃
⠀⠀⠛⠿⠿⠋⠀⠀⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⣣⣤⡀⠸⠟⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⣰
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣰⠾⣶⡶⢶⠿⣶⢶⡶⠶⢾⡿⢼⡿⠛⢷⡄⠀⠀⣠⡎⠀⠀⠀⠀⠀⢈⠠⠠⠃
⠀⠀⠀⠀⠁⠀⠀⠀⡄⠀⠀⠀⠀⣿⢀⡈⡀⣸⢠⣧⠈⠱⣯⣼⡇⢸⢁⡿⠛⠃⠀⠛⠋⠀⢀⡀⠀⠀⠀⠠⠒⠒⢢
⠀⣀⣠⡄⠀⠀⡀⠀⠀⠀⠐⠂⠀⣿⢸⣿⡇⢹⠈⠋⢸⣷⣄⠙⡇⢸⡀⢿⢶⡄⠀⠀⠀⠀⠈⠡⠀⠘⠁⠠⠟⠒⢲
⠀⠘⠿⢁⡄⢦⠙⠀⠀⠀⠀⠀⠀⢿⡼⠛⠷⠾⢷⣴⠟⢷⣤⠾⢷⡾⠷⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡭⣅
⠀⢀⠔⢤⠙⠀⡀⠀⠀⠀⢰⡆⣾⠀⠀⢰⡆⣶⢠⣶⠀⠀⣶⠀⡷⠀⠀⣶⠀⣶⢰⡇⠀⠀⠀⠠⠀⠀⠀⠀⠔⠠⢬
⠀⠈⠙⠀⠀⠀⠀⡠⠀⠀⠸⠇⠻⠀⠀⠸⠇⠿⠘⠟⠀⠀⠻⠐⠇⠀⠀⠛⠀⠛⠘⠃⠀⠀⠀⠀⣠⣄⠀⠠⠖⠒⢦
⢰⠆⠀⠀⠐⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠠⠧⠤⠤
⢀⡚⠀⠖⠄⠀⡄⠀⠀⠀⢀⣠⣄⣠⣾⣶⠀⢠⠀⠀⢀⠀⠰⠶⠶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣄⣈⣤
⠀⠉⠉⠲⠒⡑⣉⠤⠖⠦⢺⣿⣿⣿⣟⡟⢀⣾⣿⡖⠈⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢡⣙⢻⢒⡀⠀⠀⠙⠿⣿⠏⠀⠀⠈⠹⠀⠀⠀⠀⠛⠀⡿⠀⠀⠀⢀⢀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠒⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠖⠲⠦⣄⠀⢀⣿⣿⡀⠀⣠⠴⠖⠲⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⢀⢀⡀⠀⢸⣶⣿⡿⢿⣿⣶⡇⠀⢀⡀⡀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠠⣴⠈⣉⣽⣿⠟⡏⠁⠀⠀⠈⢻⠻⣿⣯⣉⠁⣦⠄⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠐⣤⡼⢖⡿⠋⣱⠟⡏⠀⡇⠀⠀⠀⠀⢸⠀⢹⠻⣎⠙⢿⡲⢧⣤⠂⠙⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠘⠁⠠⠎⠇⣰⣏⠘⣿⠄⡇⠀⠀⠀⠀⢸⠠⣿⠃⣹⣆⠸⠱⠄⠈⠁⠀⠘⣆
⠀⠀⠀⠀⠀⠀⣰⠃⢰⣆⣀⣷⠤⠤⢾⠃⣸⣆⡙⠀⡇⠀⠀⠀⠀⢸⠀⢋⣰⣇⠘⡷⠤⠤⣾⣀⣠⡆⠘⣆
⠀⠀⠀⠀⠀⢠⠃⢀⠴⠋⢰⠿⠀⣠⣇⠠⠃⢹⡌⠀⡇⠀⠀⠀⠀⢸⠀⢡⡏⠘⠄⣸⣆⠀⠾⡆⠙⠦⡀⠘⡄
⠀⠀⠀⠀⠀⡏⠀⠀⠀⢤⠀⢀⡴⣇⠸⣦⣀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⣀⣴⠇⢸⢦⡀⠀⡤⠀⡀⠀⢹
⠀⠀⠀⠀⢸⠁⠀⢈⣩⠟⢉⡟⢀⣿⠀⢸⣎⠀⠀⢰⠃⠀⠀⠀⠀⠘⡆⠀⠀⣱⡇⠀⣾⡀⢻⡉⠻⣍⡁⠀⠈⡇
⠀⠀⠀⠀⢾⠀⠀⠀⣤⣴⠿⡆⠚⡿⠃⠘⠈⠃⣠⠎⠀⠀⠀⠀⠀⠀⠱⣄⠘⠁⠃⠘⢿⠓⢰⠿⣦⣤⠀⠀⠀⡷
⠀⠀⠀⠀⠸⡄⠀⠀⠰⠃⠀⣁⣀⡥⠴⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠦⢬⣀⣈⠀⠘⠆⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠈⠒⠒⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠒⠒⠁



⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⢀⠀⡠⠤⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠈⠀⡀⠀⡇⠀⠀⠀⢀⠠⠠⠀⠀⠀⠀⠀⠀⡤⠀⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⢸⠀⢢⢄
⠀⡄⠁⠀⡇⠀⡇⠐⠂⠰⡃⠀⢀⡘⢂⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠘⠒⠀⠀⠀⠰⠁⣀⠀⠀⠡⠄⢸⠀⢸⠸
⠀⠀⠁⠀⡇⠀⡇⢸⠀⠀⠹⡁⠀⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠎⠀⠧⠤⠡⠆⠈⠀⠸⠀⣩
⠀⠀⠘⠀⠃⠀⡇⢸⠀⠀⠀⡁⠀⡜⠀⠀⠀⠀⠀⣀⠀⠀⠢⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⠀⠀⡇⠀⠀⠀⡆⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠐⢤⠀⠐⠀⠄⠈⠁⠀⡄⠄⠀⠀⠁⠀⠀⠀⠁⠀⢠⣠⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠱⡀⠀⠁⠀⠀⠀⢆⠀⠀⠐⠀⠀⡰⠀⠀⠤⠀⠀⠠⠁⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠉⠐⠂⠀⢠⠤⡕⠢⠀⠠⡊⠀⡤⡀⡀⠄⠂⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠄⣀⢀⠎⠀⡇⠀⠀⠀⠂⠈⡅⢩⠀⠀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠰⢄⡀⠀⡀⠎⢠⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠀⠀⡆⠀⠀⠈⡀⠠⠀⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣤⣶⣶⣷⣆⠀⠀⠃⠀⣃⣉⢡⠀⢃⢀⣠⣼⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠈⢻⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⠋⠁⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠖⢪⠉⠻⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⠇⠀⡠⠀⢄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠑⢄⣹⡿⠿⠟⠛⠛⠛⠛⠿⠿⠿⣿⡿⠦⠊⠀⠀⢠⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡤⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠠⠤⡌
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡯⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⢶⡀
⠀⠀⠀⠀⠀⢀⠊⠁⠈⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣀⠈⠈⠉⠒⠄
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠂⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⣼⢃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⣶⡄⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⣀⣼⠟
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⢱⣾⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⢠⣤⣶⠟⠋⠁⣿⣿⣿⠟
⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠈⢿⠀⠀⣠⣾⣿⣿⠏
⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⢸⣇⣼⣿⣿⣿⡏
⠀⠀⠀⠀⠀⣿⣿⠛⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⠟⠋⠁⣸⣿
⠀⠀⠀⠀⠀⢿⣏⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠈⠙⢶⣤⠀⠀⠀⠀⠀⠀⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢴⡟⠿⠿⣷⡄⢀⣤⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠲⡄⠀⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⢀⣷⡤⠜⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⢠⡇⠀⠀⠀⠃⠀⠀⠄
⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣀⠤⠶⠒⠒⢤⣄⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠠⣤⣀⠀⠀⠀⣰⠏⠀⠀⢀⠊⠀⠀⠀⠀⠀⡴⠀⠈⠢⡀
⠀⠀⠀⠀⠀⢠⣀⣀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⠛⠢⠚⠋⠀⠀⠀⠀⠂⠁⠀⠀⠀⣰⣃⣄⣠⠖⠛⢇
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠂⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⠿⠿⣆⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠤⠤⠄⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣥⣤⣤⣽⡧⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⣿⡏⠉⠉⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡹⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣏⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣟⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠒⠶⠶⠾⠿⠛⠻⠿⠿⠿⠿⠷⠶⠶⠶⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣶⣦⣤⣤⣀⣀⣠⣤⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⠿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠐⠒⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠁⢀⠙⠿⠋⠀⠀⠀⠀⠀⣤⣦⠀⡀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠡⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⡐⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠐⠠⠄⠄⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣆⠹⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣌⡛⢿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⣿⣿⠁⢠⣤⣀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣬⣙⠻⣿⣿⣿⣿⣿⣧
⠀⠀⣿⠃⠀⠘⣿⣿⣿⣶⣶⣶⣶⣦⣾⣿⣿⣿⣿⣿⣿⣿⠿⣷⣦⡙⢿⣿⣿⣿⣇
⠀⢸⣿⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⣿⣿⡄⢿⣿⣿⣿
⠀⢸⡿⠀⠀⢄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸⣿⣷⠘⣿⣿⣿⡆
⠀⢸⡇⠀⠈⠊⠉⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠈⣿⣿⡆⢹⣿⣿⣷
⢂⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠘⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⣿⣿⣿⡿⠟⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠷⠘⠋⠁⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⡶⠟⠛⠛⠋⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⢠⣶⡀
⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⡷⠃⣿⣿⣿⠻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⣀⣤⠀⠀⠀⢸⣼⡇⠀⠀⢰⣄
⣿⣿⣿⣿⣿⣦⣶⣶⣿⣿⠋⣀⣾⣿⣿⠟⢹⣿⣿⡟⢙⣿⡇⠀⠀⠀⠀⠀⠀⡟⠉⢹⡀⠀⣴⣿⡋⠀⠀⢠⣼⠈⠀⢰⣶
⣿⣿⣿⣿⠟⢻⣏⣁⣿⡇⢸⣥⣦⠘⣿⣷⣿⣿⣿⠗⣿⣿⡇⠀⠀⠀⠀⢀⣤⡇⠰⡿⠇⢸⣯⡞⡟⢻⡄⠈⠉⠀⠀⣾⡇
⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡮⢿⣹⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢙⡦⢷⠞⠁⠀⠀⠀⠀⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣘⣹⣿⣿⠉⣹⣏⣀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠻⣿⡾⠀⠀⣾⡷⢰⠿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢽⠿⠟⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣇⣀⣸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠐⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠃⠀⠀⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠂⠕⠄⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⡆⠈⠀⠀⠀⠀⠀⠀⢀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠲⠀⠀⠀⠁⠀⠀⠀⠀⠇⠀⠀⡀⠂⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆⠊⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠈⠀⠠⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠰⠄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⡀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢡
⠀⠀⠉⠐⣆⠄⠀⠀⠀⠀⠀⠂⠂⠂⠀⠈⠐⠄⠂⠀⠀⠀⠀⠀⠀⠀⠂⠐⢃
⠀⠀⠀⢠⠇⠤⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⡐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⠀⠖
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⢀
⠀⠀⠀⠀⢰⣶⣶⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠓⠼⣀⡄
⠀⠀⠀⠀⢘⡉⠉⣁⣸⠀⠀⠀⠀⠀⠀⠀⢠⢊⡔⠛⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠸⣿⠇⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠚⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗⠀⣠⠿⢄⠀⠀⠀⣀⣠⣔⣀⡀⠀⠀⠀⢠⠒⠩⠿⠔⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠙⢻⣱⠊⠀⡄⠀⠈⠈⠋⠀⠀⠀⠀⠈⠉⠉⠉⠁⣀⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣤⡎⠁⣀⡼⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠉⠉⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢀⠷⡄⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢀⡎⠀⠀⠀⣄⣀⣀⣀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⡠⠃⠀⠀⠈⠢⣀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⢸⠀⠀⡘⠁⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠙⢄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠖⠋⠀⠀⣧⠄⣀⠠⠤⠤⠒⠒⠂⠉⠉⠹⡀⠀⠈⣇⠀⠀⠀⠀⠉⠉⠑⠒⠂⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⠤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⠎

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠤⠤⠀⠀⠀⡀⠄⠀⠀⠀⠀⠄⢀⠀⠄⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⠐⠂⠐⠒⠀⢀⠐⠀⠀⠈⠐⠠⡀⠀⢀⣀⣤⣤⠀⢈⠀⠀⠀⠐⠀⠀⠀⠀⠀⣠⠞⠂⠐⠠
⠀⠀⠀⠀⠀⠒⡄⠄⡰⡋⠄⠂⠀⠀⡀⠀⠀⢠⡾⣶⣄⡀⠀⠘⡈⠛⠻⣿⣿⣷⣶⡀⠀⠀⡀⠀⠀⠀⠀⢐⠁⠀⠈⠈⠀⠀⡀
⠀⠀⠠⠀⠀⡈⠀⠀⢁⠀⠄⠀⢡⠀⠡⠄⠠⣈⠀⠀⠀⡄⠀⢁⠑⠀⠀⠈⠉⠁⢻⣿⡀⠊⠀⠂⢀⠀⠀⡜⢂⠀⠀⢀⡂⠀⢄
⠀⠀⠀⠀⠈⠀⠀⠀⢰⠀⠀⠀⠀⠄⠀⠒⠀⢎⠙⣧⣀⠀⢀⠌⢣⠀⠀⠀⠀⢀⣸⣿⠇⠀⠀⠀⠂⠈⠞⠀⠄⠠⠀⠒⠉⠀⠄⠢
⠀⠀⠀⠀⠈⢃⠁⠀⠀⢡⠀⢠⠀⠐⠀⠈⡈⠀⢠⡈⠻⣿⠀⡀⢀⠋⠀⠡⡈⠛⠉⠉⠀⠈⠀⣘⣁⠜⢀⠀⠀⠀⡇⢄⠐⠀⡀
⠀⠀⠀⠀⠂⢈⢂⠱⣿⣿⣆⠀⠄⠀⠁⡀⠀⠠⠀⠕⣄⠙⡆⠁⡀⠇⢀⠀⠈⠀⠀⡠⠋⠙⣠⢠⡀⠀⠄⢀⠠⢀⣇⠀⠀⠀⠄
⠀⠀⠀⠀⠀⣠⠈⣷⡙⠿⠿⡦⠐⠀⠀⠀⠀⠀⢀⣼⠟⣠⣿⠂⠀⠀⡌⠐⠠⠀⢀⡂⠀⠀⢄⠶⢀⣀⡴⠁⠀⠀⠀⡀⠀⢀⠐⠰
⠀⠀⢀⠀⢀⠀⢤⡜⢹⣄⠲⣤⠠⣄⣀⣠⡥⠶⠟⣡⡔⠙⢿⣤⣤⡌⠀⠑⠠⠀⠐⡆⠀⠀⡘⠐⣼⠁⢺⡈⠀⠀⠴⠁
⠀⠀⠂⠀⣴⠰⣿⣇⠻⢿⣧⣘⢂⣿⣟⣿⠟⠒⠓⠈⡙⢢⡰⠟⢅⣠⢶⡷⢠⠀⣰⠁⢁⢀⠃⠼⠳⠞⠉⣿⡶⡃⠀⡀⢠⡠⠀⠀⡀
⠀⠀⠐⠀⠘⣧⡙⠿⣿⣿⠿⢫⡜⣿⠟⠁⠀⠀⠀⠀⠀⠂⠔⠁⣠⠁⡸⠃⡢⠔⠁⠀⠀⠒⠌⠀⠀⠀⣠⡿⠁⢊⠕⠄⠅⠀⠠⠁
⠀⡄⠀⠐⠄⠹⢿⣶⣶⡶⠾⢿⡷⠀⠀⠀⠀⠀⠠⠊⠄⠀⠀⠀⠌⠉⠀⠊⠐⠀⠀⢀⠀⠈⠀⠀⡠⣾⡟⠁⠀⠈⠐⢂⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⣴⡆⣧⡈⠃⠀⠀⠀⠀⠀⠀⠈⠁⢂⠀⠱⠀⠀⠂⠀⠀⢀⠁⠀⠰⢰⣷⡌⡄⠀⡔⡀⠞⠁⠀⠘⣷⡀⡀
⠀⠀⠀⠀⠀⠀⣿⡿⠓⠄⠙⠛⠋⣡⣄⠀⠀⠀⠀⠨⠀⠀⠀⠐⠀⣰⡀⠀⠀⢄⠀⠀⠁⠀⠀⠈⣭⡇⢸⡰⢧⣴⣉⢂⠀⣰⡿⠃⠁
⠀⠀⠀⠀⢀⠀⡛⠁⠀⠖⢢⢻⡿⠛⠋⠁⢄⠈⠂⢀⠒⠄⠀⠀⠦⠐⠈⢤⡀⠀⠀⠀⠀⠁⠀⣸⡿⢁⠇⠘⡦⡙⣇⣠⢸⠉
⠀⠀⠀⠀⠀⡐⢤⣴⡊⠀⢀⡌⠀⠀⢀⡀⠀⠐⢢⡤⠓⠀⠁⡅⡈⠉⠂⢔⠙⠒⠦⡄⠀⠦⢮⠩⠔⢍⠃⠠⣌⣷⢨⠄⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠸⢋⡵⠋⡰⠀⠀⠄⡛⠋⠀⠀⢁⠀⠁⠀⠀⠛⠀⠀⡷⠄⠰⡀⢀⡁⠀⠀⢀⢀⡀⠐⢄⡔⠛⢶⢸⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡴⠁⠆⠈⠣⣀⠀⠀⠀⢀⠁⣶⡞⠀⠀⡀⠀⠀⠀⠠⠄⠂⡱⠁⠀⣘⠼⠀⠙⣄⠔⠃⠂⠁⡜⡀⢀⠀⠂⡄
⠀⠀⠀⠆⠀⡉⢐⠂⡆⠀⠠⢁⣾⣿⣶⡾⣿⠸⣿⠁⠀⠀⠘⠿⢀⠀⠀⠀⢨⡀⠀⠠⢉⠇⠀⠙⠏⣤⣤⡤⢊⣼⠃⠀⠀⠀⠄
⠀⠀⠀⠀⠂⡑⠘⡃⡇⠀⢀⠸⢻⡿⠛⠛⣿⣆⢻⣷⠄⠀⠈⠠⠀⠀⣤⣄⠀⣼⣦⣄⠁⠀⠠⠒⠦⢩⣭⣶⡿⠁⠠⠁⠀⠀⠈
⠀⠀⠈⠈⠀⢀⠜⠡⠈⡀⠀⠂⠹⠵⠾⠃⣾⣿⣦⡀⠨⣁⣶⣶⣶⡿⠟⢁⣼⣿⣿⣿⣷⣦⡈⠀⠀⠊⠈⢉⠀⢠
⠀⠀⠈⠂⠠⠄⠀⠁⠀⠐⠦⣄⢀⠀⠀⠄⠙⠋⣹⣿⡶⠤⠍⠩⠄⠀⠀⠀⠈⡣⠊⠉⠻⢿⣿⣿⠇⠰
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠈⠀⠒⠀⠠⠀⠀⠀⠈⠀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡄⠀⢠⢶⡀⠀⢠⡆⢀⡄⠀⢠⡄⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⢀⣦⠀⠀⣴⡆⡰⢿⣠⠊⢧⡴⠃⠈⢷⡴⠃⠸⠏⢻⡴⠛⡇⠀⠀⠘⣧⠴⠛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢰⠏⠘⣤⠞⠀⠛⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠠⡤⡤⠀⠀⠀⣧⠀⢀⡞⢁⡞⠓⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏⠀⠀⠰⠖⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⠿⠁⣾⠖⢾⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⠀⠸⠏⠀⠀⠁⠀⢿⡀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠀⢀⣀⡀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠙⠓⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠉⠉⠉⠁⠀⠀⠠⡴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣁⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣀⣤⢤⣄⡀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡞⢁⡀⠀⠀⠹⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣞⠀⡀⢀⣤⣿⠀⠀⠀⠀⠀⠀⢀⣤⡎⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⠀⠀⠀⠀⣾⠋⢉⡟⠓⠿⠁⠸⠆⠀⠀⠀⠀⣠⠟⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⣀⠀⢀⠜⠁⠀⠞⠀⠰⠃⠀⠀⠀⣀⡤⠶⠋⠁⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠙⡻⠛⠶⠶⠾⠤⢤⡶⠶⠶⠞⣯⠝⠛⠳⣄⠀⢠⠃⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡠⠊⠀⠀⢀⠊⠀⠀⠸⡇⡰⠚⢾⡇⠀⠀⠀⠈⠷⠃⠀⠀⠀⠀⠀⠀⢿⣀⡤⠶⢾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⢲⡤⢀⣠⣁⡀⡇⠀⠆⠿⠁⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠀⠀⠀⣁⠌⠀⠀⠀⢁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⠀⠀⠀⢸⠷⣤⡀⠁⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠙⠳⢤⣀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⣠⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢦⣀⡎⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠉⠀⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠉⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⣀⣹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⣀⣀⡤⠶⠞⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠖⠛⠉⠁

⠀⠀⠀⠀⢀⢰⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣆⣀⡆⠃⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠐⢻⡟⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⣦⡀⠀⠀⠀⠀⠀⠸⣿⠁⠀⠀⠀⠀⠀⠀⠀⢾⡷⡴⠂
⠀⠀⠀⠀⠀⣞⡷⠀⠀⠀⠀⠀⢀⣯⣍⠁⠀⣻⣇⠀⠀⠀⠀⠀⢰⢿⠀⠀⠀⠀⠀⠀⢀⣀⡿⣇⢣
⠀⠀⠀⠀⡀⠘⠃⢀⠀⠀⠀⠀⠘⢿⡿⠃⠀⢏⠉⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⢠⡄⢠⣼⣼⣦⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⣻⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣷⡟⠀⠀⠀⠀⠀⠀⡞⣿⡦⠓⠀⠀⠀⠀⠀⠀⣠⡾⣷⡄
⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⣰⠿⣬⠁⠀⠀⠀⠀⠀⠀⠓⣿⡄⠀⠀⠀⠀⠀⢀⡼⢻⡄⠸⣿⡀
⠀⠀⠀⠀⠰⠁⢸⠀⠀⠀⠀⠀⠀⠞⠁⠀⠈⠳⡄⠀⠀⠀⠀⠀⠰⠁⡇⠀⠀⠀⠀⠀⠏⠀⠞⠀⠀⠻⠇

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣶⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢈⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠐⣿⠂⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⣤⣾⣿⡇⠀⠀⠀⠀⠀⣀⣤⠟⠻⢿⠀⠀⠀⠀⠀⠀⠼⣿⠄⠀⠀⠀⠀⠀⣶⣾⣷⣶⣶⣦
⠀⠀⠀⠀⠘⠿⠿⠁⠀⠀⠀⠀⠾⠉⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢶⠠⢠⡄⡄⡦⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠙⢾⡗⠋⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⠀⠀⠁
⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⢀⣠⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡄⠀⠀⠀⠀⠀⠀⠀⡎⠋⡇
⠀⠀⠀⠀⠀⡃⠃⠀⠀⠀⠀⠀⠀⣸⠀⠀⠓⠂⡄⠀⠀⠀⠀⠀⡞⠀⣇⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀

⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠛⢡⠜
⣿⣿⣿⣿⣿⣿⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⣆⣸
⢻⣿⣿⣿⣿⣿⡿⠀⢀⣤⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠘⠟⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠉⠀⣠⠿⠛
⠈⣿⣿⣿⣿⠏⠀⠐⠉⢀⣀⡀⠀⠀⠀⠀⢶⣯⠁⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⢀⠎
⠀⠸⣟⠙⠿⠆⠀⠀⠈⠻⠷⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⡴⠁
⠀⠀⢉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⡠⠋
⠀⠀⢨⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⡄⠀⢠⠔⠂⠀⠀⠀⠀⢀⠔⠊
⠀⠀⣼⣿⣿⣿⣷⡄⠀⠀⠀⠀⠙⢲⣶⣶⣾⣁⣠⣠⣴⠗⠈⠀⠀⠀⠀⠀⢀⠖⠁
⠀⣴⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⢀⡶⢁⠀⠀⢷⠀⠀⠀⠀⠀⠀⡰⠋
⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⡔⢀⣐⣦⣾⠀⠀⠀⠀⢠⠊
⢀⣼⣿⣿⣿⣿⣿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠻⢻⣿⣿⠄⠀⠀⡰⠂
⠈⠻⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣦
⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠶
⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠀⡄
⠀⠀⠈⠉⢻⣿⡀⠀⠀⠀⠀⠰⠀⠀⠀⠠⠀⠀⠀⠀⠈⠻⠿⠀⠃
⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠈⠻⠿⢧⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡄⠀⠀⠀⢈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⣴⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⡏⠙⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢠⡇⠀⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⢸⠇⠀⠈⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣤⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠻⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⣿⡿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⣅⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠉⣄⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⡁
⠀⠀⠀⠀⠀⢀⣤⣥⣀⣠⣠⡆⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣄⣠⣥⣄
⠀⠀⠀⢀⣴⣟⣋⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡛⢻⣷⡄
⠀⠀⢠⣿⣿⣿⣿⣿⣿⠿⣛⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠄⡻⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠈⣿⠛⠛⠛⠉⠅⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠉⠛⠛⠛⢻⠇
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣡⣤⠐⠒⠒⠒⠒⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⢀⣀⣀⠠⠦⠶⠖⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇

⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠖⠲⠾⠤⠄⣀⡀⠀⠀⠀⠀⠀⠸⣦
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⢠⡤
⠀⠀⠀⠀⠀⠀⡴⠂⠀⠀⠀⠀⠀⠈⠀⠀⣘⠁⠀⠀⠀⠀⠄⡀⢀⠀⠀⠀⠀⡀⠀⠱⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⢃⣀⠠⠴⠒⠄⠀⡁⠸⠆⠀⠀⠀⢠⡀⠀⡇⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠉⠀⠀⢀⢿⠟⠁⠀⡀⠀⠀⠀⠀⠈⠂⠀⠁⢉⠓⠃⢀⠇⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠴⡋⡨⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡜⠀⠠⠤⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⡐⠠⠀⠀⠀⠘⠁⠀⠀⢠⠜⠀⢠⠁⠀⠘⡁⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠜⠛⠒⠓⠀⠤⡀⠀⢀⠠⠀⠔⠒⣁⠉⠀⢄⣑⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠀⠀⡀⠀⢀⢐⠎⠤⠉⠀⠀⠀⢱⠟⠲⠞⢋⡵⣲⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⡦⠀⠀⠀⠀⠀⠀⢀⣦⡀⠈⠓⠒⠒⠋⠀⠁⣸⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠠⠀⠤⠠⢤⣀⣘⠀⠑⠄⡄⠤⠄⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠈⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⢴⠀⠁⠀⠀⠐⢢⡠⠶⠦⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠈⡶⡄⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠄⢣⠈⡦⢀⡟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⣉⣀⣈⣨⠤⠃⢹⠀⠄⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⡾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢃⠑⡀
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢄⠀⠀⠀⢠⣾⣿⣿⣿⠾⡄⠰
⠀⠀⠀⠘⢄⣼⡂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡞⠀⠀⠀⣿⣿⠟⠉⠁⠀⣷⠒⠓⡄
⠀⠀⠀⣠⣶⣿⠿⠢⢄⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⡄⠀⠀⠹⣿⣄⡀⠀⢀⡍⠀⠀⢰
⠀⠀⠀⣿⣿⣇⠀⠀⠀⠱⡄⠉⠒⣄⠀⠀⠀⠀⠀⠀⡆⠰⡀⣀⣀⠈⢿⣿⠏⠀⣉⠏⠒⣼
⠀⠀⠀⠈⠻⣷⣄⣀⡀⠤⠦⠔⠒⠈⠧⣀⠀⠀⠀⠀⠰⡰⠉⠀⠀⠀⢀⠎⠉⠉⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⠠⠃⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⠣⠤⠄⠒⢩⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⢀⠜⠁⠡⡀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠣⠀⠀⡠⠄⠐⢲
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠱⢮⠀⠤⠤⠤⢄
⠀⠀⠀⠀⠀⠀⢹⠤⡤⠔⠓⣄⠀⠀⠀⠀⠀⠀⣀⠤⠊⠀⠡⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠸⠀⠃⠀⠀⠀⠑⠄⣀⠄⠊⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⡠⢦⠀⠀⠀⢰⠀⠀⢠⠃⠀⠀⢀⠔⠲⢄⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠈⠄⡇⠀⠀⠀⠀⡜⠀⠀⡠⠊⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠑⢄⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠊⢉⠂⠀⠀⡘⠀⢠⠞⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠃⡔⠁⠀⠀⠀⠀⠀⠀⠀⠱⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡰⢁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠩⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠲


⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⢉⠇⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⠞⠁⠀⠀⠚⠀⠴⠛
⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⣤⣄
⠀⣴⠟⠉⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠻⣦
⣼⠃⠀⠀⢀⡆⠀⠙⢦⣤⣀⣀⣤⠞⠁⠀⠀⢀⡾⠀⠘⣇⠀⠀⠀⠙⠦⣄⣀⣀⣠⡴⠋⠀⢳⠀⠀⠀⠘
⣿⠀⠀⠀⣼⣅⠀⠀⠀⢀⣠⡿⠃⠀⠀⢠⣴⣿⡁⠀⠀⣙⣷⣦⠀⠀⠀⠈⠻⣍⡀⠀⠀⠀⣸⣧
⠙⠷⠶⠾⠋⠙⠛⠒⠛⣫⠟⠀⠀⠀⠀⠀⠻⣷⠖⠀⠘⢿⡶⠂⠀⠀⠀⠀⠀⠈⢿⡓⠒⠛⠋⠙⠻⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⣀⣤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠶⠤⢤⣄⡀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠳⣤⡀⠈⢱⠀⠀⠀⠀⣰⠂⠀⠀⣠⡟⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⠀⠀⠀⠀⠀⠙⠶⣾⣀⠀⠀⠀⢿⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡿⠒⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⣄⡀⠀⣀⣤⠖⣲⠟⠛⢛⡿⢶⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠙⣿⠘⣷⠃⡼⠁⠀⠀⣾⣡⡼⠁⣀⣀⣠⣤⡴⠶⠛⠉⢷⡄
⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠙⢷⡏⢰⠃⠀⠀⢰⡏⣁⣤⡾⠋⠉⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⢸⡇⠘⠀⠀⠀⢸⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣴⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣸⡇⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⡇
⠀⠀⠀⠀⣿⠇⠀⠀⢿⣄⣀⣀⣀⣀⣀⣤⡤⠴⢶⡄⠀⠀⠀⣠⠶⠶⠶⠶⠤⣤⡤⠞⠁⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⡽⠋⠉⠁⠀⠀⠀⠀⢷⠀⠀⣼⠃⢠⠀⠀⡀⠀⢸⡇⠀⢀⡟⢸⡇
⠀⠀⠀⠀⢿⣤⣄⣐⠦⠤⣤⡇⠀⣧⠀⠀⣴⠀⠀⢸⡇⢰⡇⠀⣾⠀⠀⣿⠀⢸⡷⠶⠋⢀⣸⡇
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠶⢿⣿⣿⣦⣤⣿⣤⣤⣶⣧⣘⣛⣻⣧⣤⣤⣿⣶⣾⣷⠶⠿⠛⠋⠁

⡏⠉⠉⠉⠉⠉⠉⠉⠉⢩⠉⠉⠉⠉⠉⠉⠉⡏⠉⠉⠉⠉⠉⠉⡏⠙⢻⠋⠉⡏⠉⠉⠉⠉⠉⠉⡏⠉⠉⠉⠉⠉⠉⢩⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⡇⡀⠠⠤⠤⠔⠚⡣⠖⠉⠑⠶⡓⠒⠀⠠⠤⠤⠤⣅⠀⠀⠃
⡇⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠐⠒⠉⠀⠀⠀⠀⠀⠈⠑⠢⠤⢤⠄⠀⢸
⡇⠀⠀⠸⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢄⠀⣠⡴⠒⠦⣄⡀⠀⠀⡇⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠸
⡇⠀⠀⠀⡄⠀⠀⠀⠀⠃⠀⠀⢀⠃⠀⢸⠁⠀⠀⠀⠀⠀⢀⣀⡤⣜⠀⢀⠙⠁⠀⣏⠀⢀⠛⡄⠀⢹⠀⠀⢃⠀⠀⠀⠃⠀⠀⠀⠀⡄
⡇⠀⠀⠀⠇⠀⠀⠀⢠⠀⠀⠀⡘⠀⠀⡎⠀⠀⠀⢀⡴⠊⠉⠀⠀⠈⠉⠁⠙⠢⠴⠫⣑⡆⠀⣻⠀⠈⡄⠀⠘⡀⠀⠀⢠⠀⠀⠀⠀⠇
⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⢰⠁⠀⢰⠒⠋⡀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⡟⠀⠀⢱⠀⠀⠁⠀⠀⠘
⡇⠀⠀⠀⢸⠀⠀⠀⠇⠀⠀⠀⠀⢀⡇⠀⠀⢮⡀⣀⠠⠐⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢻⠀⡤⡀⠀⡇⠀⠀⠀⠀⠀⠇⠀⠀⢸
⡇⠀⠀⠀⢸⠀⠀⢠⠀⠀⠀⠆⠀⡸⠀⠀⠀⠀⢳⠤⠤⠤⣄⣀⠤⠴⠦⠖⢚⡇⠀⠀⠀⠀⢸⡐⠁⢲⠀⠸⠀⠀⠆⠀⠀⢀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⢠⠃⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢀⠀⣼⡁⠀⡈⠀⠀⢣⠀⠸⠀⠀⠈⠀⠀⠈
⡇⠀⠀⠀⢀⠀⠀⠀⠀⢀⠁⢀⠆⠀⠀⠀⠀⠀⠸⡀⠁⠀⠀⠀⠀⠘⠀⠀⢸⡀⡴⠚⢣⣾⠕⠀⡐⠀⠀⠀⠀⢆⠀⢁⠀⠀⠀⠀⢀
⡇⠀⠀⠀⢸⠀⠀⠆⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠠⠄⠀⠀⠀⠀⠀⢉⣠⡶⠻⠋⠀⡰⠁⠀⠀⠀⠀⠘⡄⠀⡀⠀⠆⠀⢸
⡇⠀⠀⠀⢸⠀⢰⠀⢀⠁⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⢤⣤⠤⢲⣶⡈⠭⠡⠐⠁⠀⡰⠁⠀⠀⠀⠀⠀⠀⠱⡀⢡⠀⢰⠀⢸
⡇⠀⠀⠀⠀⠀⡄⠀⠂⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠀⡏⢰⠀⡸⠐⠀⠀⢀⡀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠂⠀⡄
⡇⠀⠀⠀⡆⢠⠀⡈⡰⠁⣀⠀⣒⢀⢔⠈⠀⠀⠀⠀⠀⡀⠠⡇⠈⠉⠀⠘⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⡀⢠⠀⡆
⡇⠀⠀⠀⠁⠄⠰⡰⠁⠀⠈⠀⠈⢻⠛⡎⠁⠈⠛⠀⠀⠀⠀⠇⠀⠀⠀⠀⢰⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡰⠈⠄⠁
⡇⠀⠀⠸⢀⢀⡞⠁⠀⠀⠐⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠒⠤⠤⠒⠁⠀⠈⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⢀⠸
⡇⠀⢀⢂⣼⠯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⢿⣄⣀
⡷⡏⠉⢹⣳⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠟⠒⠢⠤⢤⣤⢤⠦⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⢠⣻⠉⠈⡷
⡇⠑⠲⢾⠁⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠁⠀⣠⠔⠚⢹⠉⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⢹⠶⠒⠁
⡇⠀⠸⢸⠀⣇⠱⠀⠀⠀⠀⠀⠀⣀⡴⢚⣧⢀⣠⢒⣉⡀⡀⢀⢸⡄⠀⣇⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⢡⡇⢸⢸
⡇⠀⡆⢸⠀⢹⣀⡇⠀⠀⠀⢀⠚⣃⣀⠤⠐⢛⠿⠂⠀⠀⠀⠀⠀⠀⠀⢣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡉⠀⠀⠀⠀⠀⠀⣇⣸⠀⢸⠀⡆
⡇⢸⠀⢸⣤⣸⠉⢣⠀⠀⠀⢠⣀⡠⠔⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡀⠴⠀⠀⠀⢠⠋⢹⣠⣼⠀⢰
⡇⡇⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⡴⢁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⡇
⣿⠁⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠰⠶⠀⠈⠀⠀⠀⠀⠀⣟⣠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢹
⡇⠀⢠⠀⠀⢰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⠀⠀⢠
⣗⢒⣂⣤⣒⣒⣐⣒⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣈⣑⣒⣐⣒⣠⠤⣒⣒
⡿⢿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡋⣽⡿⠿⠏⠤⠿⣷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠋⠉⠀⠐⠀⠀⠀⠀⠀⠈⠻⡦
⠀⠀⠀⠀⠀⠀⢠⡿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⣼⡖⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⣿⠁⠀⣿⣶⣤⣀⡀⠀⠀⠀⢀⣀⠠⠀⠀⠀⡇⠀⠀⠾⠛⠃⠀⠀⠰⣦
⠀⠀⠀⠀⠄⡀⠋⠀⠀⢋⠉⢹⣿⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠇⠀⡀⢀⡀⠰⠆⣀⡀⠘⠷
⠀⠀⠀⠀⠂⠀⠀⢄⠀⠘⢿⣿⡏⠑⠀⠐⠁⠀⠀⠀⠀⠀⠐⠂⠐⠀⠙⠁⠀⠀⠉⠁⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠙⠧⢀⡀⠀⠀⠀⠀⠀⡠⠂⣴⡿⢁⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠑⠂⠤⠀⣀⣀⣀⣤⣄⠀⠀⠀⢿⠀⣾⡿⠁⠐⢀⠀⠀⠀⠀⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠀⠀⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁⢲⠠⢀⡄⠀⠀⣂⢡⣀⠠⠄⠀⠑⠀⠀⠀⠓⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠉⠁⠀⠉⠀⠐⢄⠀⠀⠀⠀⠀⠠⠄⠀⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠂⠀⢀⠀⡀⠐⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠑⢄⠀⠀⡐⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠁⢄⠀⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠈⠃⠊⠀⠀⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠄⠀⠐⠤⢀⣀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠶⠶⠀⠀⠀⠀⠳⣶⣶⡄


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

⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡛⢶⣄
⠀⠀⠀⢠⡾⣿⣿⣿⣿⣿⣿⣆⢻⣧⡀
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣄
⢶⣆⣚⠛⠛⠻⠉⠙⢻⣿⣿⣿⡇⢸⣿⡦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⣿⣿⣿⣷⡌⣷⡄
⠀⠀⢸⣿⣿⣿⣤⣾⣼⣿⣿⣿⡇⢸⣿⠀⢠⠋⠻⠿⠿⠿⠶⢶⣶⣶⣶⣤⣤⣤⣤⣀⣤⠚⢿⣿⣿⣿⣿⣿⣷⠸⣿
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣿⣧⡀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠻⡁⠀⢯⣀⣈⣿⣿⣿⢀⣿⣦⣤
⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡿⣡⣿⠏⠀⡇⠀⢳⡀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⠀⠀⠀⠀⢷⠀⠘⣿⣿⣿⣿⡿⣸⡟⠉⠁
⠀⠀⠀⠀⠈⠛⠿⣿⣯⣵⣾⠟⠁⠀⠀⢸⠀⠀⢳⡄⠀⠀⠀⣾⣿⡟⠁⡤⠙⡄⠀⠀⠀⠸⠁⠀⣿⣿⣿⢟⣵⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠻⣄⡀⠀⣿⠋⠁⠀⠁⠈⣧⠀⠀⢀⣇⡀⠀⡝⠻⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠱⣄⣘⡇⠀⣀⡀⢈⣧⣠⡔⠁⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⣀⡀⠉⠑⠲⣤⡙⠛⢷⣄⠁⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠛⠛⠆⠀⠀⠸⣷⡀⠀⠙⣦⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢠⡀⠀⣿⣿⣿⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠈⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⢀⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣿⣿⣿⣿⣿⣿⣿⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡿⠟⠉⠀⠀⠀⠙⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠉⠀⠀⠀⠀⠀⢰⠂⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⣠⣾⣿⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡦⠰⣤⣀⣀⡀⠀⢠⠛⠙⠒⠉⠉⠀⢸⡀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢧⡸⣇⠀⠀⠉⠗⠁⠀⠀⠀⠀⢀⡴⠻⣓⢀⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⡏⠉⠉⠳⢤⡀⠀⠀⠀⠀⣠⠖⠉⠀⠀⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠻⠿⠿⠟⠉⠉⠉⠉⠉⠓⠶⠶⠊⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠋⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠂⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣤⠄⠊
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣾⣿⣶⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⣨⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠇⡀⢠⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⡀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡇⠀⠁⢠⠀⠀⠀⠀⠀⠀⢀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢉⠂⠠⢬⠀⠀⢠⡾⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣿⠀⠀⠀⠘⠣⡀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⢠⢀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠄⢠⠀⠂⢳⣄⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡔⠐⣿⣷⣤⣿⣏⣤⣤⣌⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠙⢿⣿⣿⣿⣿⣿⣿⠈⠀⠈⠳⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠙⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣄⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣯⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠁⠀⠈⠉⠛⠛⠛⠛⠛⠻⠿⠿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠴⠢⣄⡀⠈⠀⠀⠀⢀⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠿⠿⣿⣿⣿⣷⣶⣀⠢⠄⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⠒⠂⠸⠂

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠤⠾⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⣃⡸⠆⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣤⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡿⠟⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠁⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⡌⡁⠀⠀⠀⢀⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠃⠀⠀⡂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢻⣿⣇⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⣽⣿⣿⣿⣶⣶⣦⣶⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⣿⣿⣿⣏⠁⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⣴⠟⣿⠟⣿⣿⣿⣿⠇⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⡀⡾⠁⣼⠃⢸⣿⣿⣿⡈⠀⢀⣼⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⡀⣸⠃⠀⣿⣿⣿⣿⡇⠀⣼⣿⡿⣿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠑⠃⠀⠀⠁⣼⠏⢸⡇⢠⣿⡿⠃⢻⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠘⣷⢸⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠛⣼⡏⢸⣿⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⡀⠀⠀⠀⠀⠀⢿⠀⠸⡏⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣄⡀⠀⠸⣇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣶⣿⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣏⠛⠻⠿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠻⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠛⠛⠛⠳⢦⡄
⠀⠀⠀⢀⣠⡶⠞⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣿⡆
⣼⠃⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⠛⠛⠋⠉⠉⠀⠸⣇
⣿⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢿⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣤⠀⠀⠀⢸⡆⢸⡆
⢸⡇⠀⠀⠀⣀⣤⣤⠿⠀⠀⠀⠀⠹⠂⠀⠀⠈⠁⠘⣧
⠀⢷⡀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢿⡀⠀⠀⠀⢀⣠⡴⠶⠞⠛⠶⠶⣤⡀
⠀⠘⢷⡀⠈⠷⣤⣄⠀⠀⠀⠀⢀⣤⡴⠞⠛⢉⣿⠀⢸⡇⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀
⠀⠀⠀⠙⠢⣤⣤⣿⡀⠀⠀⠀⠀⠹⢦⣤⣤⠞⠃⠀⣿⠀⠀⣼⠃⠀⠀⢀⣤⠶⠶⢦⣄⠀⠀⠈⢿⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⢀⡟⠀⠀⠀⣾⠁⠀⠀⠀⢻⡆⠀⠀⢸⡇
⠀⠀⣠⠀⠀⠀⠀⠀⣠⠿⠳⢦⣤⣤⣤⣤⡴⠾⢯⡀⠀⠀⠀⣿⠀⠀⠀⢻⣄⠀⠀⣀⡾⠁⠀⠀⢸⡇
⠀⣼⠋⠀⠀⠀⢠⡾⠁⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠙⣦⡀⠀⠹⣆⠀⠀⠀⠉⠛⠛⠋⠀⠀⠀⣠⡟⠀⠀⠀⣸
⢰⡏⠀⠀⠀⢠⡟⠀⠀⠀⣠⡾⠛⠉⠉⠉⠙⠷⣤⡀⠘⣧⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⢀⣠⡴⠋⠀⠀⠀⠀⠉
⣼⠁⠀⠀⠀⢸⡇⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣿⡀⠀⠀⠀⠈⢹⡟⠛⠛⠛⢻⡍
⣿⠀⠀⠀⠀⢸⡇⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⣨⣿⣄⣀⡀⠀⢸⡇⠀⠀⠀⠸⣇
⠉⠀⠀⠀⠀⢸⡇⠀⠀⠘⣧⡀⠀⠀⠀⢀⣴⠏⠀⠀⣴⠏⠁⠈⠉⠙⢷⣼⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠻⢦⣄⣠⡾⠁⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠻⣦⣀⣰⠿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠈⠙⢷⣶⣤⣀⠀⠀⠀⢰⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⡇⠈⠉⠛⠳⠶⣼⣧⣀⣤⠾⠃
⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠶⠶⠶⢾⡃
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣤⣤⣤⣤⣤⣼⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠉⠉⠁⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⣀⣱⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣀⠔⠉⠀⢸⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⣀⠸⠎⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠐⣶⠏⠒⠀⠀⠀⠀⣄⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⡡⠔⠒⠀⢲⠲⡀⠀⠀⠀⠸⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠊⠁⠀⠀⠀⠀⢈⠀⠱⡦⣀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⡜⠀⠀⠲⡀⠀⠀⡍⠙⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢀⠤⠴⠒⠈⠁⡧⠤⠊⠀⠀⠀⠀⠑⠤⠤⠁⠀⢡⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠘⡀⠈⢆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢰⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠇⠀⠀⠑⠢⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣂⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⢰⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠘⡀⠀⠀⠀⠀⣠⠎⢀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡉⠀⠀⠀⠃⠀⢀⡠⠊⠁⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠓⢄⠀⢰⠤⠄⢀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠱⡀⠀⠀⠸⠖⠁⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢈⡅⡆⠀⠀⠈⠁⠐⠒⠂⠀⡴⠋⠀⠀⠀⠱⡀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⢀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠈⠢⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠋⠀⠈⠛⠓⠀⠀⠠⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠦⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠠⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠉⠒⢠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⢠⠎⠉⠁⠀⠒⠒⠒⠂⠀⠀⠀⠤⠤⡄⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⠉⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢀⠀⠀⠀⢀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠈⠳⣀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠋
⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡌
⠀⠀⠀⠀⠀⡰⠁⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠀⡸
⠀⠀⠀⢀⡰⠁⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠘⠛⠹⡀
⠀⠀⣴⢉⣧⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⡀⠀⠀⠀⠘⢆
⡴⣏⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠖⢋⡀⢂⣀⠀⠑⢦
⠑⠪⡓⢄⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠛⠳⠶⠋
⠀⠀⠈⠲⣷⣄⠈⢦
⣀⣀⣀⣀⣀⣉⣳⣦⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠻⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⣤⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⣼⣷⢄⡀⠤⠐⠂⠉⠉⠀⠀⠀⠀⠀⠉⠀⠐⠒⠠⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⢠⠶⣾⣿⢡⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠘⠋⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⣤⠀⠒⡄⠠⡖
⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢤⣀⢸⡆⣠⢣⡄
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀⠞⠋⠁⠀⠉⠂⠤⠀⣀⡀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠈⠉⠀⠀⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠠⠒⠀⠂⠀⠀⠀⠀⠀⠐⡳⠦⡀⠀⠘⠄⠀⠀⣠⠞⢳⡄⠀⢀⣉⣀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠠⣾⡇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⠁⠶⠚⣠⠈⠁⠀⠀⣇⠠⠊⠀⠀⠀⠀⢶
⠀⣷⣶⠀⠀⠀⠀⡆⠀⠀⠀⢇⡠⠔⠠⢤⠜⠀⠀⡒⣠⠀⠀⠉⠀⠀⠀⠀⡀⠀⠀⠸⠇⠀⠀⠀⣡⠞⠁⠀⢀⡴⢣⠀⠀⠀⠀⠀⠀⢔⡃
⣤⠆⢀⠀⠀⠀⠀⠃⠀⠀⠀⠸⡇⠀⠀⠛⠀⠀⠀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠠⠊⠀⠁⢴⠄
⠀⠀⠟⠀⠀⠀⠀⠸⠀⠀⠀⠀⠘⣄⢳⡀⠀⠀⠀⢄⠀⠿⡒⠤⢀⡀⠀⠀⠀⠀⢀⠀⠐⠧⠀⠆⠀⠀⡰⠁⠀⠀⠠⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⠹⣿⣄⠀⠀⠈⠲⡄⢀⠀⠀⠀⠀⣁⣾⣟⣻⣿⣆⢀⠊⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠂⠬⢤⣀⡀⣸⣻⡶⢟⣻⣿⡁⠀⠀⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠈⠛⠷⠶⣤⢀⣀⣀⡀⢼⢹⡿⠋⠉⢉⡉⢃⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⢻⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠛⣄⠀⠀⠀⣼⠃⠀⠀⠀⠀⡀⢻⠁⠀⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⢠⡌⠁
⠀⠀⠀⠀⠀⠀⠢⠒⠤⠊⠀⠀⠀⠀⠀⠀⡠⠂⠰⠀⠀⢠⠀⠈⢠⣿⡴⠋⠀⠀⠀⠀⢀⣈⠷⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠈⢤⠆
⠀⠀⠀⠀⠀⠀⠀⠉⠂⠠⠀⠀⠀⠤⠀⠊⠁⠀⠀⣀⠀⠙⠄⠒⢻⣧⠀⡀⣠⠄⠤⢴⣷⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠄⠀⠀⠀⢸⣿⣶⣛⡟⠀⠀⠀⠻⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⣡⣄⠈⠀⠀⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⢻⣿⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠱⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠢⡉⠢⡀⠀⠀⠀⠀⠀⠀⠁⠚⠳⠶⣤⣤⣀⣀⠀⢀⣤⣇⠦⠑⠁⠀⠀⠉⠀⠒⠶⠒⠶⠀⠀⠈⠆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣨⠄⢑⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠈⠉⡱⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠎⠀⠇⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⢠⡄⠘⡀⠀⠀⠀⢀⡐⠁⠀⠀⠀⠀⠀⠒⠉⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⡘⢀⠮⠤
⠀⠀⠀⠀⠀⠀⠐⠘⠀⠀⠀⠀⠐⢀⠈⠒⠂⡐⠮⡑⣠⡦⠐⠂⠀⠀⠀⠀⠀⠀⠀⡠⣔⢂⠁⠀⠓⢀⠔⠇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⡼⠋⢸⠀⠀⠀⠀⠠⠔⠂⣤⡴⠓⠁⠀⠀⣸⠒⠛⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠉⠀⠀⠐⠂⠤⠭⡌⠤⠅⠒⠒⠊⠁⠀⠀⠀⣠⠚⢋⡄⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡆⠀⠀⣠⠾⡈⢀⠈⢌⡰⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠒⠐⢇⠖⢲⠉⠢⡐⢠⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⡀⠀⢻⠀⠗⠐⢶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡡⠒⠀⠺⠀⠐⣌⡙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⡠⠤⠇⠸⡇⠈⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢀⠰⢺⡀⠈⠀⠁⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⢁⠤⢷⡀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠂⡠⠿⢶⡄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢆⡘⠀⣇⠐⡛⠦⡤⠤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠁⠈⠣⠴⠁⠔⠁⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠙⣉⡙⣏⢀⣿⢉⣹⣙⢉⣿⠟⣭⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿