Cunny Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢦⡀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠙⠢⠤⠚⠁⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠠⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠮⠥⠄⠀⡀⠀⠀⢠⠀⠀⠳⣄⣀⣀⣤⠄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡰⢻⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⠀⠀⠀⠓⣜⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣠⠿⡄⠀⢀⠜⠀⠀⠓⠒⢒⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠈⢄⠀⠀⠀⠄⡞⠀⠀⠀⠀⠀⡰⢃⠃⠙⠒⠁⠀⣘⠠⠐⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⡊⠀⠀⠀⠀⢈⡠⢤⡀⢠⠋⠉⠉⠑⢄⡴⣁⣌⠀⠀⡀⠄⠀⠁⠀⠀⢌⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠦⢄⣎⠀⠀⢸⢾⡀⠀⠀⠀⡸⡏⠀⠈⢳⠈⠀⠀⢀⠔⠚⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠑⠚⠁⠀⠙⠒⠒⠚⠁⠙⠒⠐⠋⠑⢤⣀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⣀⠴⠒⠒⢦⠀⠀⠀⢀⡴⠒⠙⠢⡀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠤⠤⠖⠊⠁⠀⠀⠀⠀⠑⠶⠖⠋⠀⠀⠀⠀⠑⢄⡠⠊⠉⠲⢄⡀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢠⣤⠀⠀⠀⠀⠀⠚⠀⠀⢀⡀⠀⠀⠠⣶⠀⠀⠀⠀⠀⠀⠀⣀⠈⠉⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠉⠀⠀⢀⡀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣠⡀⠀⠀⠀⠀⣤⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢴⠆⠀⠀⠀⠀⣤⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠸
⠀⠀⠀⠀⠀⠀⢏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠁⠒⠒⠛⠛⠛⠛⠛⠛⠓⠒⠚⠓⢲
⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⣄⢀⡇⡰⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣜⣾⣷⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⢚⣿⣿⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⣾⣽⣻⣹⡏⣧⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⡽⢧⣯⣯⣽⣏⣽⡴⠟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣼⣛⣿⠒⡧⠴⣧⣠⣷⠶⢻⡆⠀⢀⡴⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⡼⢦⣿⣉⡟⢺⣷⠞⣷⠶⢛⡶⠚⣧⣴⠟⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢳⡄⠀⢠⡗⢾⠧⢼⡿⣭⣏⢉⣿⡛⠛⣟⠉⢹⡇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⢦⣺⣙⣿⠦⢾⡇⠘⣯⢉⣿⡙⢛⣏⠛⣹⣇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⣧⣉⡟⠲⢺⡇⠀⢸⣏⣹⣝⢋⣿⡉⣁⣿⢸⢇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⣦⣹⡄⢠⡷⢬⣟⣛⣻⠃⠀⠀⣯⣽⣏⠉⢹⣏⣩⣿⣿⠏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⠻⣧⢸⡓⢺⢧⣠⡟⠀⠀⠀⣿⢼⣯⣙⣹⣧⣤⢾⡿⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠹⣾⣙⢛⣧⣼⠁⠀⠀⠀⡷⢾⡷⣤⣤⡿⢤⣼⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⣬⣩⣿⡇⠀⠀⠀⢸⡟⠛⣷⠶⠶⣿⢶⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢷⡴⣿⠀⠀⠀⠀⢸⣋⣋⣟⠒⠒⣿⠏⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⢧⣿⠀⠀⠀⠀⢸⣍⣉⣿⣲⡾⠁⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠈⣿⠀⠀⠀⠀⢸⡧⣥⡿⠋⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢹⡄⠀⠀⠀⠀⣷⠟⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠈⢷⡀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣄⡀⠈⢳⣄⠀⠀⠀⣷⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣍⣷⣤⣴⡷⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠴⠁⢃⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠢⣀⠇⠀⠀⠀⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡝⠢⠔⢢⢀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠢⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠉⠈⢆⣸⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⠤⣀⠖⠦⠔⢊⠝
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡢⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⣀⣆⣠⠀⠑⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⠀⠀⢠⠋⠻⠟⡆⠀⢰⠁⠛⠋⠆⠀⠀⠀⠀⠀⠀⢎
⠀⠀⠀⠀⠀⠠⡤⠤⠔⠢⠼⠀⠀⠀⠀⠓⠤⠐⠁⠀⠀⠑⠒⠊⠀⠀⠀⠀⠀⠀⢀⡤⠃
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠉⢱⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⣐⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⣠⣀⠔⠊⢻⠐⠢⠤⠤⣀⡠⢀⣘⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠊⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⡆⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠸⡄⠀⠀⠀⣰⡯⠽⠷⠖⠒⠚⠓⠓⠒⠶⠦⢴⣍⡙⣧⠀⠀⠀⡰⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⣳⠴⠒⠉⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡄⠀⠉⠻⢦⣄⡰⣣⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⡎⠉⠓⠀⢀⠴⠶⣄⠀⠀⠀⠁⠀⠋⠀⠀⢀⡀⠈⠻⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⢖⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⡼⠀⠀⠀⠀⠀⠀⠀⠞⠁⣹⠀⡠⠊⠀⣸⠃⣀⣀⡀
⠀⣠⠖⣤⠀⠀⠀⡸⠋⠀⡔⠋⠙⣦⠀⠀⠀⠀⠀⠀⠀⠋⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⠋⠉⢧⡀⣠⠃
⠀⢿⣀⣘⢦⣀⡜⠁⠀⠀⠛⠲⢦⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⢀⡈⢿⠀⠀⠀⠀⣀⣀
⠀⠀⠁⠉⠙⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⢸⠀⠀⠀⠀⠀⢀⡤⠞⠉⢻⠈⣇⢀⡠⠞⠁⠉⡗
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠲⠶⠞⠀⠸⡷⠊⠉⠉⠛⠁
⠀⠀⠀⢠⣾⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⢸⣿⡇⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠷⠲⡆
⠀⣀⠀⠈⠻⠀⠀⠘⡍⠙⠒⢦⣤⣤⣤⣀⡀⠀⠐⠙⠳⠤⣄⣀⣈⣉⠤⠞⣁⠀⠀⠀⠀⠀⠀⠀⢀⠤⢤⡀⠀⠙⢳⠚⡁
⣎⠉⢉⡉⢴⠀⠀⠀⠙⣆⡀⠀⣛⣏⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⠁⠀⠀⢾⡀⢹⡆
⠈⠉⠁⣠⢸⡄⠀⠀⠢⣄⣉⣉⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠒⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡀
⠀⠀⠀⠹⠦⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡠⠤⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⢹⡀⣾⡁⠀⠀⠀⠀⠀⠀⠰⣇⠈⣧⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠃⠀⠀⣄⢈⡷⠦⣄⠀⠀⠻⡄
⠀⠀⠀⠀⢠⠊⢳⡈⠉⠁⠀⠀⣞⠉⢧⣠⠽⠊⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠚⣼⠁⠀⢸⡀⠀⠀⢧
⠀⠀⠀⠀⠈⠓⠚⢳⡄⠀⠀⠀⢘⠶⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⢻⠀⠀⠙⠲⣄⠸⣄⡀
⠀⠀⠀⠀⠀⠀⡆⠓⠙⣦⡀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡇⠀⠀⠀⣠⠊⠹⡔⠋⠀⠀⠀⢠⠇⠀⠀⠙⢦
⠀⠀⠀⠀⠀⢀⡧⠴⠊⡥⠛⢦⡀⠀⠀⠀⣇⢀⡇⠀⠀⠀⠀⡀⠠⡄⠀⠀⠀⠀⠈⠛⠋⠀⢀⡠⠾⡅⢧⣄⢳⠀⠀⠀⠀⢸⢠⠞⣄⢸⣜⡇
⠀⠀⠀⠀⠀⢯⣀⣀⡾⠀⠀⠀⡟⠳⢦⣄⣉⠁⠀⠀⠀⠀⠀⠳⠤⠃⠀⠀⠀⢀⣀⣠⡔⢾⡉⠀⠀⢻⣀⠼⠘⡆⠀⠀⠀⠸⠏⠀⠘⣶⠙⠇
⠀⠀⠀⠀⠀⠀⡍⢉⠆⠀⠀⠀⣧⠀⢀⡼⢩⠛⠛⢲⠖⡶⠲⡶⠒⠒⣖⡛⠉⠉⡇⢧⡉⠀⠹⣤⡀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⢰⠀⢚⣤⡀⠀⠀⢹⠀⣏⣀⡷⠀⢠⠏⠈⠀⢠⠁⡇⠀⠀⠀⠀⠀⢦⣤⠽⠂⠀⠈⠙⢦⣀⡀⠀⠈⢳
⠀⠀⠀⠀⠀⡎⢀⠈⣷⢹⠀⠀⠘⣇⠈⠉⠀⣠⠋⠀⠀⠀⠈⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠆⠀⠀⠸⡀
⠀⠀⠀⢀⡀⢧⣈⡳⣉⣿⠀⠀⠀⢹⠄⠀⠘⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠁⠀⠈⠉⠉⠀⠀⠀⠀⠘⢻⡆⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠒⡩⠷⠲⠤⠶⠬⠇⣠⠷⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠈⢠⣆⡐⡤⠤⠤⣄⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣵⡿⣡⠴⣆⣀⡴⠲⠦⢤⠴⠓⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡮⠅⠒⠦⣄⣠⠴⠒⢠⡀⠀⣀⣸⣦⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⠟⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⢦⡏
⣶⣶⡶⠾⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠉⠲⡀
⡴⠋⠉⢈⣉⡳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣮⣍⠙⣦⠀⡷
⣇⠀⡼⠁⠀⣉⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠸⣿⣿⠇⣼⣶⠃
⠻⣄⢳⡀⢸⣿⡿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠈⠛⠛⠉⠁
⠀⠉⠓⠿⠶⠟⠑⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢻⠀⠀⠀⠀⢸⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠃⠀⠀⠀⢸⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⠤⠤⠤⠤⠬⡇⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⢛⠽⠒⠀⠀⠀⠀⠀⠈⠳⣌⠳⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡆⠠⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠡⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠙⢱⠒⠒⡖⡖⣾⣻⠒⠦⢤⡀⠀⠀⠱⡌⢣⡱⣄
⠀⠀⠀⠀⠀⠀⠀⠂⠥⠴⠞⠚⠛⠉⠉⠉⠉⠙⠛⠼⠀⠀⠀⠀⢸⡆⠣⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⠒⠲⠶⠶⠖⠒⠁⠀⠀⠀⢿⠀⢥⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠢⢤⣤⣤⡤⠤⠖⠀⡸⢀⡾⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢔⠟⢀⡀⠀⠀⠀⢀⠠⠔⣊⡴⠛⠁⢸⡧⠤⠶⠛⠛⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢚⠁⠀⠀⠉⠉⣰⢋⠤⠀⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠠⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡟⠁⠀⠈⠓⠦⠀⢠⢞⣟⣁⣀⡀⣀⣠⣄⣀⡀⠘⢷⣦⣤⠴⣚⣁⣠⡤⢤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠄⠈⠓⠶⠀⠀⣠⣕⣻⠼⠐⠚⠒⠓⠾⣭⣏⣛⢳⠶⢤⢶⣻⣿⣿⠟⠛⠛⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣶⣤⣤⣴⣶⣿⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣛⣿⣿⣛⣛⣉⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠔⡖⢲⠒⢒⠖⡶⠠⡤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⢛⠁⢸⠘⡀⢀⣇⣬⣀⣼⣰⣁⡰⢁⠎⠀⢩⠖⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢾⢹⡀⢠⣸⡤⠞⠋⠉⠉⠀⠀⠀⠀⠀⠈⠉⠛⠲⢤⣃⡞⠀⠈⣳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡏⢆⠘⣤⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡴⢡⠊⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⠀⠱⣬⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⢀⡴⢉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢦⣑⡖⠁⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡴⠃⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⡀⢀⡟⠀⠀⠀⠀⠀⠏⣌⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⡴⢋⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠢⣝⡞⠀⠀⠀⠀⠀⢠⢉⠀⠣⡳⡀⠀⠀⣠⢔⠚⣆⣀⠠⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠹⡴⠃⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢄⠀⢸⠁⠀⠀⠀⠀⠀⡸⢸⠀⠀⠑⢌⣀⡩⠒⣡⠛⣉⠤⠒⣒⣒⡒⠠⢍⠲⣄⠀⠀⠀⠀⠀⢧⠀⣠⢇
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠢⣙⡫⡞⢩⠭⠭⠭⠍⠒⠁⣀⠤⠤⠤⣀⠀⡸⠁⡜⡵⠊⠉⠀⠀⠈⠉⢢⡱⡌⢇⣀⣠⣠⣄⣸⠞⡡⢺
⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⠈⡇⠘⢮⡲⢄⠀⠀⢠⠎⡡⣒⠒⣐⢬⠱⣧⠴⠚⠣⢤⡀⠀⢀⡤⠠⣄⡃⣳⠞⠁⠀⡀⠀⠉⢻⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡷⢬⣙⡇⢀⠤⠘⠊⠋⠘⡇⢸⣴⠁⠀⠈⠣⡑⠒⡊⠍⡒⠄⠙⡆⠸⡀⠀⠀⠙⠁⠀⠀⠠⡀⠐⠀⠀⡧⢺
⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠀⢻⣡⠔⡂⠭⠥⠼⡆⢀⣤⠀⠀⠀⠀⠀⣀⡀⠀⠸⠸⠀⢸⠀⢙⣄⠀⣀⣀⠀⢀⢤⠀⠀⠀⣴⠖⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⣑⢚⣥⠊⠀⠀⠀⠀⢣⠸⡸⡄⠀⠀⠀⡎⠀⠈⠂⠾⣴⣀⣮⣬⠵⠊⠉⠀⠀⠘⡌⠊⠀⠀⠀⠈⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣨⠍⠷⢀⣀⡀⠀⣀⣈⣦⢱⡼⠦⠤⠤⠷⠤⠤⡄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢀⡄⠤⢤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠈⠀⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠋⠉⠉⠉⠉⠉⠉⠒⣞
⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⢣⣀⣠⠒⠉⡁⠀⡀⠀⠀⠀⠈⠑⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠒⠢⠤⣀⣀⢀⡀⣀⣀⣀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠀⠀⠈⠈⠁⠀⠀⡀⢀⡀⠀⠀⠉⠒⠤⢤⣀⣠⠤⠒⠉⠀⠀⠀⠀⠀⢀⠀⢀⠀⠉⠉⠀⠀⠀⡛⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢄⣠⠄⠀⠉⠉⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠘⠒⠋⠀⠀⠀⠈⠓⠋⠀⠢⣀⡤⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠁⠀⠰⣄⣠⠄⠀⠀⠀⠀⢦⣀⡤⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠈⠉⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⡀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣜⠒⠂⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠁⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠑⠒⠃⠀⠀⠀⢄⡠⠀⠀⠀⠀⠈⠉⠁⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠠⠤⠤⠤⠤⠤⠤⠐⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢄⡀⠀⡜⠀⠀⣠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⡼⠁⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠢⢄⠀⠀⠇
⠀⠀⢠⣤⡶⠚⠛⠛⠺⣧⠀⢂⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⣀⠀⠀⠱⡄⡇
⠀⠀⠀⠙⠷⣀⠀⠀⣠⡟⢷⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠜⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠁⡇⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⠤⠤⠒⠒⠒⠀⠒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠉⠁⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣼⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠒⠒⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢺⣿⣿⣿⣷⡀⠀⠀⠀⠀⣰⠏⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢿⣿⣿⣿⡇⠀⠀⠀⣰⠋⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⣈⣩⠝⠁⠀⣠⠞⠑⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⡀⠀⠀⠀⠀⠀⠘⠡⠊⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⠟⠀⠀⠀⡀⠔⠁⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠁⠀⠀⠀⠀⠀⠀⣠⠔⠀⠀⠀⠀⢀⡖⠀⢀⠀⠀⠀⠀⠁⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⡜⠁⠀⠀⠀⣀⣴⠟⠁⠁⠀⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢤⣝⣓⠖⠛⠛⠋⠀⠀⠀⠀⠀⣀⣀⠤⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠋⠁⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⠆⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⣼⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⣠⣤⠀⠀⢸⣿⡳⣄⡀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣽⡟⠀⠀⠀⠉⠻⠿⠇⠀⠀⠀⣰⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⢀⣠⣤⡦⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢸⣿⣂⣶⣄⣤⡀⣠⡄⣠⣤⠐⠋⠈⠻⡅⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⣹⠀⣿⠤⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⢸⡿⢛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣿⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⠀⠈⠻⢿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⣄⡀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⣿⣿⣿⣿⣿⣷⠀⠀⢀⣾⣿⣿⣿⣿⠿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⠆⣼⣿⣿⣿⠟⠉




⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⢦⠀⠀⡆⠀⠀⢰⡇⣿⢀⡏⢀⡇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⢿⡷⣄⢀⠈⢷⡀⣇⠀⢢⣾⠀⢸⣿⠀⢸⡇⢠⡷⢾⡇⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⣦⡀⠀⢦⡈⠳⣬⣧⠙⢿⡆⠈⢿⣿⠀⠸⡏⠀⠘⠇⠀⣿⠃⠈⠀⣿⡴⠃⡼⢁⣀⡤⠆⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⢤⡈⠻⠮⠻⣦⡀⠙⣆⠈⠙⠂⠀⠙⠘⣎⠙⠀⣇⠁⠀⡆⡀⠀⠈⠀⣴⢰⡿⠁⠈⠁⡼⠋⠐⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⠙⢦⠀⠙⠳⣦⣄⠨⣙⠂⠢⣁⠀⠑⠾⡄⠀⠙⠀⠀⡿⠀⡄⠙⠁⣠⠀⢠⠟⠀⠀⣠⠆⠀⠀⠀⣼⣯⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣦⣼⣿⡛⠛⠃⠙⢷⠀⠘⠂⠀⣀⠀⠀⠱⠇⠀⠀⠀⠃⠀⠀⠛⠀⠀⣠⠆⠀⠉⢀⡔⠁⠈⡽⠛⢁⡠⠖⠊
⠀⠀⠀⠀⠒⢤⣀⠀⠀⠈⠐⠲⢮⣿⣆⠠⣉⠁⠀⢦⡀⠀⠀⠐⢦⠀⠈⠀⠀⠀⡄⠀⠀⠀⠀⠘⠀⢀⡀⠀⡀⠀⢠⠀⠀⠀⠀⠠⠴⠾⠯⠶⢺⡟
⠀⠀⠀⠀⠀⠀⠀⠉⠓⢶⣤⣀⡒⠦⣍⣀⠈⠳⠀⠈⠓⠀⠀⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⣷⡀⠀⠀⣾⢀⡔⠀⠀⠀⠀⠀⠀⢐⣛⣓⣀⡀
⠀⠀⠀⠀⠀⠀⠠⠤⣤⣤⣀⣉⡁⠀⠀⠉⠛⠂⠀⠀⣦⡀⠀⠙⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⣿⡟⠀⠀⠀⠀⠀⠀⢀⣤⠴⠛⠋
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠉⠉⠀⠀⢳⣄⠀⢀⣀⡀⠀⠀⠀⠄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡈⠳⣦⣿⠀⠑⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⠁
⠀⠀⠀⢠⣤⣤⣤⣽⣿⣷⣦⣤⡌⠉⠉⠉⠉⠁⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣷⡄⠀⠙⠀⠀⠀⠀⣠⠀⠀⠀⠠⣄⡀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠐⠒⣲⡶⠖⠀⠀⣀⠀⠀⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠞⢋⡀⠀⠀⠀⠬⡝⢸⡿
⠀⠀⠀⠀⠀⠐⢛⣛⣉⠉⠀⠀⠊⠁⣠⠄⠀⠈⠁⣀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⢹⡄⠀⢠⣴⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠉⣉⣿⡿⠷⢂⣠⣤⣶⠄⢀⠟⠁⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠄⠀⠀⠀⠀⠀⠀⠀⢨⡉⢸⠇⠀⠸⣤⡞⣼⣁⣀⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠈⠉⠁⢀⡄⠀⠁⠀⠀⡴⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⢄⣀⣀⣠⠀⠀⠉⠉⠀⡠⠔⠚⠋⠉⠉⠛⠛⢷⡟⢛⣿⡆
⠀⠀⠀⠀⠀⠒⠒⠚⠋⣠⣴⡇⠀⠔⢻⠀⠀⠴⠁⠈⡀⠀⣤⠀⠀⠀⠰⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢹⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠛⠁
⠀⠀⠀⠀⠀⠀⣠⡴⠟⣻⡟⢀⣤⠀⠀⠀⠀⠀⠀⠸⠃⢀⡀⠀⠂⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠈⠁⠀⣰⣿⠾⢻⡏⣠⣶⣷⡴⠻⢦⡴⠂⠈⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⡀⠉⠛⠷⠶⠶⣤⣤⣤⣤⣤⣴⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⣼⠟⠋⢀⡅⠀⠀⠀⠀⠀⠀⢳⢠⣀⣀⣀⣀⣰⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⣷⣠⡿⠷⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣼⣄⣤⣤⡿⢶⡟⠀⠀⠉⠉⠉⠀⠛⠷⢶⣶⣶⣾⣦⣀⠀⣈⣦⣰⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠏⠉⠈
⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶
⣿⣿⣇⣀⣈⣈⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⡀⢀⣀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣃⢸⣯⠅⠘⢿⣄⢹⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟⠹⠿⠗⠺⠿⠋⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⣠⣤⣄⠀⣶⠀⣀⣀⡀⣀⡀⣀⡀⣠⣀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⡅⣿⣿⠟⢸⣿⡂⣿⡯⠁⣿⡿⣿⡇⣿⣿⣷⠘⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠻⠿⠆⠘⠿⠃⠻⠿⠂⠿⠃⠻⠃⠛⠛⠁⠛⠿⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠄⠀⠀⢄⠄
⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⡂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀⣀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠘⠟⠀⡀⠀⠀⠀⣀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠁⠀⠀⢀⠀⠀⠀⠀⢠⣶⣿⣦⠀⣴⣿⣶⡄
⠀⠀⠀⠀⠁⠀⠘⠿⣤⣤⡄⠛⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⣦⣀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠈⠻⠿⠛⠚⠛⠿⠟⠁
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⣿⠀⢿⣻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⠀⣴⣶⣶⠄⠀⠘⣿⣿⠟⠁⠀⠀⠙⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⣈⡀⠁
⠀⠀⠀⣠⡶⠿⢷⣦⡀⠀⢉⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢿⠁⠀⠰⣶⡆⠀⠀⣹⡇⠀⠀⠀⠠⠊⠠⠞⡁⠀⠐⠦⠈⠿⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢸⣿⣷⣤⣴⣿⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡿⠇⠀⠀⠐⠃⠀⠀⢶⣶⣶⠆
⠀⣀⣘⣿⠿⠉⠿⠿⠏⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠂⢀⠄⢠⡆⠀⠄⠈⠀⠀⠉⠁⠀⠀⠀⠀⠀⡀
⣾⣿⣿⣿⠀⠀⠀⣶⣦⣶⣷⣀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠠⠀⠀⠀⠀⠀⡀⠀⠐⠋⣸⣧⠹⠂⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⣿⣿⣿⠏⠀⠀⠀⠙⠛⢿⠉⠛⠷⡄⠀⠀⣀⣬⣶⡆⠀⡀⠀⠀⠀⠀⠀⠃⠀⢀⣾⡿⠿⣷⣤⠀⠀⠁
⠟⠋⠁⠀⢶⣷⠀⠀⠀⠸⠀⠀⠀⠈⠀⣼⣿⣿⣿⣿⣠⠗⠀⠀⠀⠀⠠⣄⠀⣾⣿⣷⣶⣿⣿⡆⠀⢄⠀⠀⠀⠀⠀⠀⠀⣀⢴⠇
⠅⠀⠀⠀⠀⠀⠀⠀⣠⣴⠀⠀⢖⠀⠀⢿⣿⢭⣿⣿⣿⠀⣭⠄⠀⣬⠄⣿⣿⣿⢿⠿⠿⡿⢿⠃⠀⠀⠐⠂⠀⠀⠀⠐⠂⠁⠸⠀⠁⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠀⠀⠈⠁⣤⣬⠙⠚⢻⣿⣿⠀⠀⣤⣤⡄⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⢀⠀⠠⠆⠀⣀⡀⠰⠆⠀⠀⠈⢀⣀⣀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⠏⠀⠀⠀⠹⣿⡀⠀⠀⠁⠀⠀⣻⡟⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⣼⣿⣍⣀⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⠂⠀⠀⠀⠀⠀⢠⡶⠀⢴⣟⠛⠛⢻⡟⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠻⢿⣿⣿⣿⡿
⣶⡄⠀⠀⢀⡄⠀⠀⢿⣧⡄⠿⠆⠀⣴⠆⠈⢀⠀⡀⠉⠐⠂⠀⢁⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⢰⡶⠦⠤⢴⣶
⣿⠇⠀⢰⣿⡿⠀⠀⢸⡇⠀⠠⢠⠀⣤⡀⠸⠏⠈⣿⡆⠀⠒⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘⠛⠈
⠟⠀⠀⠈⠙⠁⠀⠀⠈⠁⣠⠀⠀⠀⠛⠃⠀⠀⢀⡀⡐⠒⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⠙⠃⠸⣦⠀⢠⣶⠀⢈⠀⠁⣀⠀⠁⠀⠀⠈⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠘⠇⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⡀⠀⠀⠀⠋⠐⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⣷⠶⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠈⠉⠐⠾⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣯⣤⣄⠀⣀⣀⠀⢈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⣿⠁⠉⠉⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣾⣿⣤⣤⠀⠀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣯⣤⠀⢀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣦⣶⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠃⠀⠀⢠⣿⣿⣿⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⢻⡟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⠘⠃⠀⣴⣿⣿⣉⠋⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⠘⢃⣴⣿⣿⣿⣿⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣀⣀⣀⣰⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣷⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠙⢿⣟⢿⣿⢟⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢫⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠙⢷⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣠⣴⡾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣼⣿⣿⣿⣿⣿⣿⣯⣹⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣠⣴⣿⡧⠀⠀⠀⢼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⡧⠀⠦⢤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡿⠶⢾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⣸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣩⣿⣿⣿⣿⣿⣿⣷⣶⣾⠟
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠉⠙⠛⠛⠿⠿⠛⠁
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣰⣟⠉⠉⠻⣿⣿⠏⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢰⣿⡿⢿⣦⣤⣿⠏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣰⡿⠃⠀⠀⢻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⡿⠿⠿⠿⠿⠿⠿⢿⡿⠿⢿⡿⠿⠿⠿⠿⠿⠿⣿⠁
⣿⡇⣤⢸⣦⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢸⡀⠀⠀⠀⠀⠀⢠⣿
⢸⡟⢻⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⢸⡇⠀⠀⠀⠀⠀⣾⡿
⠀⢹⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢰⡇⠀⠀⠀⠀⠀⠀⢠⡿⠈⣿⠀⢸⡇⠀⠀⠀⠀⠀⢿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⣼⠃⠀⠀⠀⠀⠀⢀⣿⠃⢰⣿⠀⣿⠁⠀⠀⠀⠀⠀⣾⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⡟⠀⠀⠀⠀⠀⢀⣾⠇⠀⢸⣿⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⣾⠁⠀⠀⠀⠀⢠⣾⡿⠀⠀⢸⡇⢸⡿⠀⠀⠀⠀⠀⠀⣿⠇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⢰⡏⠀⠀⠀⠀⠀⢈⣿⠃⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⢰⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⡸⠃⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⢸⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⡇⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⢸⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⠁⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⣼⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠘⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⢸⣿⠀⠇⠀⠀⠀⠀⠀⣿⠇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⢠⣿
⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⡘⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⣼⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢀⠇⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⠀⠀⠀⠀⢠⡿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⣼⡇
⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢠⡏⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⠇⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣸⡟
⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⢱⡏⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠁⠀⠀⣠⣿⠁
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⣇⣀⣠⡴⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠷⠖⠚⣋⡙
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣧⣤⠴⣶⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⡿⠛⠛⣹⡇
⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣾⢏⣙⠻⠁⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠤⠈⢀⣼⠟⠁
⠀⢿⠇⠀⠀⠀⠀⠀⢀⣾⠁⠀⢉⣴⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣃⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣃⣴⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⠉⠉⠉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣷⡶⠀⠀⠈⢻⣿⡿⣿⣿⡿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠉⠀⢻⡆⣹⣿⣿⣿⠇⢀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⠀⠀⠀⠀⣴⠃⠀⠀⠀⢉⡡⠋⢹⣿⠏⠀⠀⠀⠀⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⡀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣾⠏⠛⠀⠀⠀⣀⡤⠂⣀⣤⠆⠔⠓⢚⣧⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠘⠿⠶⠗⠒⠺⣿⡯⠐⠋⠁⠀⣀⣤⣾⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⡠⠁⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠐⢀⠄⠀⠀⠀⠀⣦⣀⣩⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠄⠀⠄⠀⠀⠈⢿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠠⠂⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⢿⡿⠋⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣾⣦⣾⣷⢹⡆⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢖⣫⠿⠛⠭⠴⠶⢾⣛⣛⣵⣫⣿⣼⣷⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢟⣵⣶⣿⠿⠭⣽⣭⣝⣛⢿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣡⡿⠙⠂⠀⢀⣢⣤⣤⣾⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⣴⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣦⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⢿⡇⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠷⣀⠈⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠒⠿⣿⣿⣿⣿⣿⠋⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣄⠀⠀⠀⠭⡛⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠈⢳⣦⣀⣀⣈⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣆⢻⣿⣿⡿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⢿⣿⣿⡟⣴⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣆⠋⠥⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠬⠙⣰⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣶⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⣟⣛⣛⣛⡿⢯⡿⣷⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣭⣽⣷⣶⣶⣶⣶⣾⣷⣯⣭⣍⡙⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠊⣬⡍⠻⣿⣿⠀⠀⠀⠀⠠⠖⠿⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⣿⣿⣿⣿⣿⡏⢰⠈⣼⡇⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡆⠀⢧⡘⠟⠷⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢤⡴⠟⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⠂⠀⠀⠀⠀⠘⢿⣿⣿⠁⠀⠀⠙⠳⠂⠀⠀⠠⡀⠀⠀⠀⠀⢀⣴⣿⠷⠀⠀⠀⠀⠀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⠀⠀⠀⠀⠀⠰⣄⠀⠀⠈⠒⠀⠀⠀⠀⠀⢴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⣀⣀⠀⠀⠀⠀⠈⢷⣦⣄⣀⠀⠀⠀⠀⢀⣼⣷⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠓⠀⠀⠀⠀⠀⠀⢲⣤⣀⠀⠈⠙⠓⠲⠦⣤⣀⡙⣿⣿⣿⠃⠙⠛⠛⠛⠋⠀⠀⠀⠀⠀⠚⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⢁⣤⣉⠛⠃⠀⢀⣾⠀⠈⠻⢿⣶⣤⣄⣀⠀⠀⠀⠉⠛⢿⣧⣶⣇⠀⠀⠀⠀⠀⠀⠸⢟⣉⣤⡈⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⠃⣾⣿⣿⣿⣷⣦⣼⣿⡀⠀⠀⠀⠉⠛⠿⣿⣿⣶⣤⣠⣦⡀⢻⡇⢈⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⡸⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣌⣙⣿⣿⣿⣿⣿⣮⣷⣿⣾⣛⣛⣻⣿⣟⣛⣻⣛⣛⣛⣃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡿⣢⣶⣾⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠟⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⣻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⣿⣿⣧⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⡇⠀⠶⠶⢤⡀⠀⢀⡤⠶⠖⠀⢀⠹⣿⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠇⠸⣶⣤⡤⠀⠀⠀⢤⣤⣶⠇⢸⣷⣿⣼⣿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣿⢏⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⠜⣡⣫⣵⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠟⠃⠀⣠⣴⣾⣿⣿⡟⣡⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠄⢀⠀⠀⣸⣿⣿⣿⢟⣵⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢿⡿⣫⣷⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣨⣾⣿⢟⣽⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢰⣿⣿⣿⣿⣿⣯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠚⠁

⠀⠀⠀⢰⡀⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠆⠀⠀⠀⠀⠄⢠⠀⡀⠀⡄⢸⢀⢏⣾⢇⢻⣼⡎⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷
⠀⠀⢠⣼⣧⠸⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠠⢂⠀⠘⠸⢹⠇⡀⠚⠀⠈⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢰
⣄⠀⠸⣿⣯⢆⠙⠹⣄⠀⢀⠀⢀⠀⠀⠀⠀⢰⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠱⠆⠀⠀⢀⠀⠏⠃⠁⠀⠀⠀⠁⠀⠀⢰⣿⠿⠶⠄⠴⠖⣶⠀⠀⢀⠃⠀⠀⣴
⣿⠐⠆⡹⣿⣾⣇⢣⠸⣂⣄⡿⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⢿⣦⠀⠀⠀⠀⠀⢤⣶⣦⣄⣧⠀⣀⡀⣀⠞⠁⠀⠀⠈⢧⡠⣼⣿
⣿⡆⠣⡄⠙⣿⣿⡞⣧⡘⣌⠴⢖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢨⠀⠀⠀⠀⠀⠀⣿⠄⠀⣹⡿⠋⠉⠁⠀⠀⠀⠀⠀⠀⢳⣸⣿
⣿⣿⣦⡈⢦⡈⢿⣿⣦⡄⣿⣏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⣏⠀⢀⣱⣿⡴⢀⡴⣊⠀⠀⠀⡌⠀⢀⡿⠀⠀⠀⠀⣄⠀⠀⢀⡆⠀⠀⠙⣿
⣿⣿⣿⣷⣎⢿⣷⣿⡷⢾⣿⣻⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠁⠆⠈⠁⠃⠀⠀⠀⢸⣥⠞⠋⠀⠀⠈⠁⠶⠟⠉⠀⠘⠛⠏⠁⢀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⢻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣤⠀⠀⠀⠠⠦⠀⠀⠀⠀⠀⠀⠐⠒⠆⠀⣤⡶⠤⢼⣆⠖⠀⠾⡍
⢷⣿⣿⣿⣿⣿⣿⣿⣿⢿⣫⡅⣾⣣⠂⣀⢠⣾⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠷⠒⠻⢆⠠⠙⠛
⠸⣯⡻⣿⣿⣿⣿⡻⢿⣿⣟⡁⢳⡝⠂⠽⢿⡇⠀⠀⠀⠀⠀⠀⠀⣴⣶⣤⣦⣤⣤⡀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠛⠀⢀⠀⠀⠀⠀⢀⡀
⠜⠘⢿⣮⣻⣿⣽⡿⣴⠊⢿⣷⢿⡏⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠹⡿⠉⣿⠟⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀
⠀⠀⠀⢙⣿⣿⣿⣁⡈⠆⠘⡿⢘⣷⡀⠀⢀⣴⠋⠁⠀⠀⠀⠀⠀⡰⢇⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠁⠠⠿
⠂⠀⠐⠉⠀⠘⢿⣧⡙⣿⢿⢆⣼⣿⠇⢠⡟⠁⠀⠀⠀⠀⠠⠄⢸⣟⣿⡋⠒⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⣷⠀⠀⠠
⠀⠀⠀⠴⣣⠀⠀⠙⢻⣦⢳⡚⣶⣿⣿⣁⡴⢞⠉⠀⠀⠀⠀⢠⢋⡉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣯⠻⣿⣿⣯⠞⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⣀⠶⡀⡀⣄⣀⠀⠙⢷⡝⣿⠃⠀⠄⠀⠀⠀⠀⠀⠀⡴⠀⠀⣿⣦⡀⠀⠀⠠⠖⠀⠀⠀⠀⠄⠀⠀⡄⠀⠀⠀⠀⠀⠀⣿⣄⡠⠄
⠀⠀⠀⠈⣿⣷⡄⠺⠋⠉⠀⠀⠀⠻⠉⠀⠈⠇⣴⣄⡆⣠⣶⠞⠀⠀⢸⠿⠟⢁⠠⠆⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⡏⠀⣀⠀⢠
⠀⠀⠀⠀⢻⣷⡘⢿⣆⠀⠀⠈⢀⣀⠀⢠⠴⢗⣿⣾⡿⢡⣿⣤⡀⢀⣼⡤⠀⠀⠀⢀⡐⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⢨⣿⣡⡠⢟⣀⡤⠤⠤⠆
⠀⢠⣤⠀⠘⠿⠆⠀⠘⠳⠀⠰⣯⣿⣿⣿⣿⣯⣷⡜⠃⣀⣄⣼⡟⠚⠁⣡⠤⣄⡙⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠜⠘⢿⣿⠯⡽⡆⠀⡆
⢀⡘⠙⠀⠀⠀⠀⠀⣀⣴⣦⣼⣿⣿⣿⡏⣹⡏⢸⣧⣸⡿⣷⣤⡽⠶⢾⡱⠷⣄⡁⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⢀⠀⢈⣶⣤⣷⡄
⢺⣿⠗⠀⠀⠀⡀⢀⡹⣿⣿⢿⣾⣿⣧⠠⣾⣷⣯⣽⣿⣳⣿⣿⣧⣾⣿⣷⡶⡟⠇⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠆⠸⣆⡞⠛⠏⠰⣷
⠀⠈⠀⠀⠀⠀⠉⢰⡾⠛⠳⠿⣏⣿⣥⡔⠛⣿⣦⣾⡛⣷⠂⣿⣿⣿⣿⠯⣷⡀⢠⡃⠀⠸⣶⠴⠂⠀⠰⠷⠐⠃⠁⠐⠃⠆⠀⠀⠁⠀⠀⢣⡅⠀⠆⠀⠀⠀⠈
⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⣘⢿⣯⠳⣾⣍⣭⣤⣿⣿⣷⢬⣙⠟⢻⣽⣿⣁⠀⡿⠤⠀⠈⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣰⠀⣴⣿
⠀⢀⠀⠄⠀⠀⠰⠤⣤⣀⣀⠀⠉⠀⠻⣄⢉⣹⣿⡟⣻⣛⣋⣤⣙⣦⡀⠙⠋⠈⠀⠀⠀⠀⠀⠀⣠⣴⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣯⡀⠙⠋⠁
⠀⠀⠀⠀⠀⠄⠀⢀⡀⠉⠍⠻⡓⠈⣠⠼⣟⣿⡿⣿⣿⣿⣿⣿⣿⣻⣿⠏⠀⠀⠀⢀⣤⣤⣦⠘⠙⠻⡾⠋⠀⠀⠀⠀⠘⠶⣦⣶⣾⣶⣤⡀⣿⠀⠀⣂⡖
⡠⡇⠀⠠⢀⡄⠀⠀⠀⢀⣰⡿⠋⠀⣡⠞⠉⠀⡴⢿⠟⠉⠉⠉⠉⠉⠈⠂⠀⢀⡐⣿⣿⣿⣿⣇⠀⠀⠀⣦⣤⡤⠄⠀⠀⠀⠈⠻⣿⣿⣿⣷⣿⡀⠀⡇
⠀⠀⠀⠀⠚⠐⠀⠀⣠⣿⠟⠀⠀⠐⠁⠀⠀⠞⠁⠉⠀⣔⠀⠀⠀⠀⠀⠠⠀⠀⣁⣸⣿⡟⣿⡟⠀⠀⣼⣿⣭⣤⣄⣀⡀⠀⠀⠀⠈⠻⣿⣿⣿⣷⢦⠁
⠀⠨⠀⠂⠀⠀⢀⣰⠟⣱⡇⠀⠀⠀⠀⡀⠀⢄⣀⡀⡰⠏⠀⠀⣁⠀⠀⠀⠀⠀⠉⠻⠁⢸⡏⠁⠀⡿⡀⠀⠉⠉⠉⠉⠉⠉⠒⠀⠀⠀⠘⢏⠁⠱⣼
⠀⠀⠀⠀⠀⠀⠀⠟⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⢍⡹⠁⠀⠰⠛⠁⠀⠀⠀⠀⠀⠀⠀⣠⡌⠀⠀⠀⢷⣌⡱⠀⠂⠀⠀⡁⠀⠀⠀⠀⠀⠀⠈⢂⠀⢹⣄
⠀⠀⠀⠀⠀⣤⠄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠟⠀⢠⣾⡔⢠⣾⢻⡌⢷⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠆⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⢀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⡠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣀⠠⢲⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣯⣷⡄⢠⡩⠻⠃⠀⠀⠀⣠⠔⠚⠉⠀⠀⠙⠬⠱⠊
⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠿⢥⣆⡀⠀⠀⢀⢴⣞⠉⢠⣐⡀⠀⣿⠉⢺⣤⠏⣀⠀⢀⡠⠋
⠀⠀⠀⠀⠈⡳⠉⠀⣢⡄⠀⠀⠸⡀⠀⡈⠀⡇⢷⠳⡄⣰⠿⠟⠅⢀⣄⡙⠙⣶⣿⡠⣾⠏⠘⠁⠀⠀⠁⠀⠤⠀⠲⠲⡽⠈
⠀⠀⠀⠀⠀⠙⠁⠀⡉⢉⢢⠀⠀⢣⠀⣧⠀⡇⢸⠀⢸⣿⡀⣀⠤⢴⡛⢿⣷⣿⣿⣿⣿⠴⢃⣠⠊⠀⠀⠀⠀⢰⠢⠦⡶⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⡄⠀⠈⠉⣹⡃⠀⡈⣧⣼⣀⠧⣾⠄⣯⡟⠓⠋⠹⠭⣭⣭⣨⣿⣿⣿⣇⡀⠈⠀⠀⠀⠀⠀⢈⠼⠷⣿⣻⣿⡿⠿⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠙⡄⠀⢀⣨⣍⢶⣬⣾⣯⠟⠲⢿⡏⠋⠀⡄⠐⣀⠀⠲⣶⢾⠿⣿⣿⢻⣓⣋⠩⠂⠀⠀⠘⠋⢍⡿⠋⠉⠀⠀⠈⠂⣶⣿⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠙⣽⣽⠋⠙⢳⡀⢸⣷⢸⡆⢣⠀⠘⠁⠐⢹⡯⢶⡟⠋⠙⢉⠉⣠⠀⠀⠀⠀⡆⡸⡇⠁⢀⣤⢊⠠⠎⡍⡽⢀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣘⠿⠶⡒⠻⣿⣿⣽⣿⡔⢻⣦⡀⣀⠀⠀⠱⣸⢷⣄⣶⡏⠐⣁⠀⠃⠌⢀⣼⣰⣿⣤⠞⠛⣍⡠⠞⣼⠃⠈⢾⡇
⠀⠀⠀⠀⠀⠀⡶⡦⢀⣠⣄⣾⡃⢀⠶⠶⣿⣿⣿⣿⣿⡿⠛⠀⠀⠌⣭⠎⣹⣼⢿⣷⣿⣎⢚⡡⠂⠀⠀⠀⠀⠙⢯⣠⣤⣼⣯⢶⡿⠃⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢘⣤⠴⠻⠛⠻⠉⠀⠀⡱⢒⣫⠿⣿⣿⢧⣀⡀⠀⠠⠦⣄⣠⣿⣿⣿⣿⢿⡟⠁⠀⠀⠀⢀⣠⣾⠿⣡⠭⠄⠚⠟⠀⠀⢀⣀⠀⠀⠀⣌⠃
⠀⠀⠀⠀⠀⢀⡟⠱⠂⠀⠀⠰⢤⣰⡛⠙⠋⠁⠚⠫⢨⠜⠁⠀⠀⠀⠀⠒⠭⣙⡿⣻⡿⠿⠃⠤⣤⣶⣶⢻⢿⡆⠀⠠⠖⠀⠀⠀⠂⠀⠸⠋⠀⠀⠀⠊
⠀⠀⠀⠀⠀⢸⡞⠀⠀⠀⠐⠀⢀⡅⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠁⠀⠁⣀⡘⠁⠁⠙⠀⠀⠀⠀⠙⢷⠀⢤⢳⠄⠀⠀⠀⠀⠀⠀⡔⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⣁⠀⠦⣀⣀⠀⠦⠀⠀⡀⠀⠀⠀⠀⡔⢩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠄⠈⠀⠐⠀⠀⠂⠀⠀⠦⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠉⠡⠀⣀⠨⢧⣜⣿⣄⣀⠀⠀⠀⠤⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠚⠲⠄⠘⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢉⠓⢂⣈⣉⣼⣿⣿⣿⣿⢵⡤⠠⣀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡠⣑⠦⣀⠀⠀⠀⠘⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⢄⠿⣏⠈⠙⢛⡿⣿⣿⡿⣿⣆⡅⠈⠁⠀⠀⠀⠐⠀⠀⠂⢤⣄⣤⢴⣾⣧⣤⡴⠾⠥⠈⣆⣀⠑⠢⠀⡀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡱⠀⠀⠀⠈⠰⣾⣷⡯⠏⣝⢀⡇⠁⠀⠀⠀⠀⠀⠀⠀⠘⠰⠈⠉⠺⡿⣿⡿⠙⠋⠁⠀⠀⠙⣌⠑⠲⠤⣬⠭⢄⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢿⠌⠀⢈⡍⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠙⠚⠒⠒⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢀⡀⠈⣷⣀⡀⠀⡈⠐⢲⠖⠒⢲⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡿⡀⡼⠛⢏⣿⣾⡇⠈⡿⠈⠀⡾⢡⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡮⢞⠵⠋⠀⠀⠈⢻⣿⡇⢠⠇⠀⡴⠛⢀⠠⡀⠀⠀⠀⠀⢀⠔⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠁⠁⠀⠀⠀⠀⠀⠀⢹⠀⣾⣀⡼⣣⣤⡤⠰⠆⠀⠂⠠⠐⠉⠀⠦⢶⠑⠒
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⡅⣫⠞⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⠈
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠉⠁
⠀⠁

⠀⠀⡄⠀⠰⠂⠀⠀⠂⠀⢀⣀⠀⠀⠀⠄⠲⠀⠀⠀⠀⡀⠀⣠⠆⠀⠀⠆⠀⢀⠀⠀⢀⠀⠀⠀⠰⠀⠀⡠⠀⠀⡀
⠀⠀⠀⠀⠀⣀⠀⡀⠁⠀⠀⠁⠓⠀⠀⠁⡄⢀⠆⡰⢡⣷⣊⡄⠀⠔⠀⢀⠀⠀⠀⠄⠈⠀⠐⢤⣄⠀⠀⠀⠐⠈⠁⠀⢀
⠀⠀⠈⣄⡠⠁⠀⠀⠉⠀⠃⠀⡄⠀⡇⢰⠃⠈⢠⣶⡪⠥⠂⠀⠃⠀⠠⠀⠐⠀⠀⠔⠠⠠⠄⠀⠀⠉⠓⠤⡄⠀⠀⠀⠉
⠀⠈⠁⠀⠁⠀⠠⠤⠀⡀⢰⠀⠙⠸⠀⡀⢠⣶⠋⠄⠄⠒⠂⠐⢁⠀⠀⠀⠠⠆⠀⠀⠀⠠⣠⠎⢀⠀⠀⠀⠀⠺⢆⠀⠈
⠈⠀⣄⠀⠀⠀⠀⢰⠀⡇⠈⠀⡆⢸⡰⠷⠇⠀⠀⠀⠒⠒⠀⡀⠀⠀⢀⢀⡂⠀⠀⠀⠀⠀⠀⠀⠘⠀⡤⠀⢠⠂⠘⠃
⠀⠀⠀⢄⠀⡆⠀⢀⡀⠀⢰⠀⣧⣈⣉⠓⠂⠀⠈⠒⠒⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⠀⠠⠀⢀
⠀⡀⠰⠸⠀⢇⠀⠀⡇⢻⣘⠁⠀⠀⠀⠁⠀⠉⢉⡁⠀⣀⠀⠀⠄⠀⠀⠀⠀⠟⠂⠀⠀⠐⠀⠀⠀⠈⠂⠀⠀⠀⠂⠐⣄
⠀⠓⠀⠀⠀⢸⠀⠀⠣⠤⣬⡉⢀⠁⠒⠦⠄⠀⠞⠙⢦⡈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁
⠀⠀⠀⠀⣇⠘⣤⣘⣓⠒⠂⠠⠤⣈⠛⢂⡤⢨⢥⠛⡀⣁⠐⠆⠀⠀⠀⢀⠆⢀⠀⠀⠊⠀⠀⠐⠒⠀⠀⠿⠄⠀⠀⠀⢁
⠀⠀⠀⠀⠐⠓⠒⠀⠠⢄⣈⡑⠂⠀⠀⣉⠝⣇⠈⠆⠁⣨⠀⠘⠃⠈⡡⠄⢀⠀⠀⠈⠀⣀⠄⠀⠀⠀⠛⠀⠑⡄⠀⠀⠈
⠀⡠⠂⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠈⠁⠀⢠⠀⠺⠀⠀⠀⠀⠀⠼⠃⠈⠊⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠐
⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠘⠀⠘⠀⠀⠀⡠⠚⠀⠀⠀⠀⢀⠠⠋⢀
⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⣢⡄⠉⠀⠀⠀⠀⠄⠀⠐⠂⠁⠀⠀⠀⠀⠀⠀⡀⠔⠉⠀⠀⠀⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⢆⠀⠀⠀⠀⣰⠦⠤⠄⠀⠀⠀⠤⠀⡀⠄⡀⠀⠀⠊⠀⠀⠉⠀⠀⠀⠈⠁⣀⠜⢏
⠀⠀⠀⢀⣀⠄⠈⠀⠀⠀⠀⠀⢠⠀⠀⠀⣠⡜⠓⠀⠂⠀⢀⣀⠐⠀⠐⠃⢀⡠⠀⢰⠀⠁⠀⢀⡠⠀⠀⠰⠀⣠⠒⠀⠘
⠠⠤⠒⠊⠉⠀⠀⠈⢀⠀⠈⠀⠐⠄⠀⣴⣋⡁⠀⠀⠀⠀⢠⠅⠀⠀⠀⠀⠀⠀⠀⠈⠃⣠⠔⠋⠀⠂⠀⠔⠀⠀⠤⠀⠈
⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠂⠀⠀⠰⠯⠀⠀⠀⡀⠀⡋⠀⠀⠂⠀⠀⠱⠀⠀⢠⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⡲⠄⠀⠰
⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢣⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠦⡀⠀⠊⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡀⠜⠊⠀⠀⡀⠀⠀⢀⠔⣴⣾⣿⣿⡿⣿⣿⣿⣿⣷⣦
⠀⠀⠁⡀⠀⣐⠆⠂⠄⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠡⠐⠀⠊⠀⠀⢆⠀⠀⠈⡠⠀⠁⢸⣿⣿⡿⢿⡷⣿⣿⡿⣿⢿⣿
⠀⠀⠃⠀⠀⠀⠿⠰⠙⠀⠒⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠱⠀⠀⠀⠀⠀⢸⣿⢸⡇⣻⡇⢹⠃⡆⣿⡸⣿
⠀⠀⠀⢀⠠⠊⠀⠄⠀⠀⠀⠀⠀⢠⠀⣤⢀⣀⣐⠀⠀⠀⠀⠀⠀⠈⠁⡠⢐⠊⠔⠃⠀⠘⣿⣼⣧⣻⣴⣾⣴⣇⣿⣣⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠠⠀⠌⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢄⣠⣤⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠋⠚⠛⣻⣿⠿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⡁⢰⣿⣶⣭⣛⠿⣿⣳⣾⣟⢿⣿⣿⣷⡡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢣⡾⠃⣀⠻⣿⣿⣿⣽⣲⣭⡳⢯⡻⣮⣻⣿⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣴⠟⠀⠀⠸⣿⣮⣙⠿⣿⣿⣟⣷⠦⣝⣮⣳⣝⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠙⠻⣽⣛⡶⢍⠙⢻⣿⣶⣮⣝⡻⢾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠉⣛⡲⠀⠀⠀⠴⣛⡛⠛⠳⢦⣝⠻⣿⣿⣿⣷⣌⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡷⠛⢿⠆⠀⠀⠐⠛⠿⠻⠖⠀⠉⢷⡈⢻⣟⢿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡙⡄⠙⣯⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧⠀⠀⠷⠦⠀⠀⠀⠀⠀⠀⠀⠀⣄⢱⠰⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣆⠀⣤⣤⣤⠀⠀⠀⠀⠀⠀⣴⣿⣇⠇⠁⣧⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣦⠐⠒⠀⠀⠀⠀⣀⡴⢺⣿⣿⢇⡶⠀⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣤⣤⣤⡔⠊⠁⠀⠸⠟⢁⡾⠁⢰⣿⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⣾⣿⣿⣿⣿⣿⡇⠀⠀⢀⡴⣻⠟⢡⠋⢛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢟⣾⣿⣿⣿⣿⡿⠟⣀⡤⢞⣽⢞⣵⢯⣾⣌⣟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢏⣾⣿⣿⢿⣫⠭⣖⣫⣵⣾⣿⣿⢟⣵⢟⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⣫⣵⣿⣿⣿⣿⣿⢟⡵⢟⠵⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⣿⢰⣿⣿⣿⠟⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠛⠿⠃⠀⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⣡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⢴⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⡟⠙⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣼⠖⠑⠀⡀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠀⠉⠀⠟⠀⠃⠀⠀⠉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠿⠋⣤⠄⡆⠀⠀⢀⠀⠓⠠⠀⠛⢱⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣉⠀⣳⠂⠃⠀⣄⠐⠁⣀⠙⢦⢻⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡆⠻⠟⡀⠛⠀⠀⠀⠉⠀⠀⢉⡶⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠴⢾⡷⠃⠀⡀⡀⠀⠃⣴⠐⠀⠁⠀⠛⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡷⠁⠀⡤⠀⢠⡀⠁⠈⠀⠄⠁⠠⡦⣾⢧⡆⡘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡴⣧⣄⢠⣶⠄⠀⣿⢀⠀⠀⢀⠀⢤⡄⠀⠀⢾⠇⠠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣤⡈⠉⡌⠁⠀⠀⢀⠈⢞⠗⢁⣀⠀⡀⢀⡄⠀⠀⠀⠀⢆⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠛⠋⠁⠀⠀⠆⠠⠶⡜⢀⡀⢰⠎⠛⠂⠉⠀⡀⠀⢀⠈⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠰⢸⡆⠁⠀⠀⡠⠞⠒⠘⠳⠶⠾⣀⡀⠘⠈⠀⠠⡌⠓⠢⡀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢀⠀⠀⠂⢀⠊⣠⣾⠏⠶⢸⣦⠐⣮⣀⠈⣾⠆⡈⠠⠴⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⠀⠂⠠⠄⠲⣿⠰⠈⣥⣰⣀⡌⠁⠖⠉⠁⠀⠀⠻⠃⡐⢃⡄⡀⠂⡀⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⠡⣶⠂⢫⡶⠀⠙⣆⢀⠀⠟⡡⣴⡶⠁⣼⠇⣀⡻⠟⠱⠿⣯⣂⡐⠓⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣿⠆⠴⢫⡍⡑⠂⢈⡉⢈⡑⢤⣎⠴⠚⠓⣿⡐⠓⠻⠤⢴⠐⠃⢬⠍⠁⡳⠂⣠⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣌⠁⠀⠀⠀⢰⣄⠀⠀⠘⣃⡘⠿⠇⠉⠀⠙⡃⣠⣦⠸⠇⠀⣦⡈⢳⠂⢐⠋⢢⣄⠈⣠⣭⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠀⠘⣷⠀⠈⡉⠁⢀⣘⣿⠇⠐⠚⡛⠃⠛⠂⠘⠤⠄⠀⠆⠛⣵⡎⢡⡾⢠⡍⣀⡀⠹⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡦⠀⠀⠀⣠⡌⠈⠡⠦⠀⣩⣄⠙⣠⣤⠀⠠⣦⣄⢢⣛⡤⠀⢀⣈⠀⠀⠀⢀⠈⠟⠣⠤⠄⠐⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⠘⠇⣛⠋⠈⣿⠀⠀⠈⠉⠁⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⢿⡛⠃⢐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠄⠈⣡⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠰⠞⠱⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠀⠲⠀⠀⠀⠂⠂⠂⠂⠀⠀⠂⠀⠂⠐⠐⠐⠀⠀⠀⠀⠀⠀⠈⠀⠁⠃⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡎⠀⠰⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠇⢠⠁⠀⠀⠀⠀⠀⠘⠿⣷⣦⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⣸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⣻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠒⠀⠀⠀⡄⠀⠀⠀⡠⡋⠀⢠⣁⡙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⣠⠋⠀⠀⠀⡀⠀⠀⠀⢠⣮⡄⠛⣀⠀⢠⣷⠞⠁⠈⠪⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡶⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⢿⣿⣦⣄⣈⣤⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⢀⣀⠀⣀⠠⠚⠟⠁⠀⠋⠉⠛⠉⠛⠿⠦⠌⠉⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⡐⠁⢸⡇⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠒⡠⣥⣄⡀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣼⠀⠀⣼⡇⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠈⢿⣻⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠇⠀⠀⢰⡏⠀⢠⡇⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠘⣿⡇⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⡀⠀⢸⠃⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⣿⠃⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⠇⠀⣿⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⣿⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠁⠈⣦⠀⡏⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢸⡇⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣴⠃⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠙⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢘⡁⠠⣸⠃⠀⠀⢀⢀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠈⡄⠀⣿⢦⠀⠀⠀⠀⠀⠀⠀⠈⢇⠜⠁⠀⠀⠀⢀⡌⢀⡀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⢹⡜⠳⣀⡀⢀⡠⠀⠀⠘⠁⠀⠀⠀⠀⠀⢿⡿⠫⢔⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⢀⡴⡸⣼⠃⠀⣇⣴⣉⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡵⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢫⢆⠗⣎⢀⣄⣿⡉⡁⠈⠧⠜⣀⠀⠀⢀⣤⣠⡶⣺⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⢮⠔⢹⣟⣟⢁⠈⠈⢋⣑⠟⣁⠽⠭⠩⠤⠿⠾⠻⠙⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢇⡄⠞⢻⠀⠃⠀⠀⠈⠁⠈⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠐⠁⠛⠃

⠀⠀⠀⠂⠀⠀⠀⠀⢠⣄⣠⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠠⣤⣤⠀⠀⠄
⠀⠀⠀⠀⠀⠀⣴⣾⠋⠇⠳⢠⠏⢭⢿⣄⠄⠒⣀⠀⠦⠀⠀⢀⠀⠀⡀⠀⢠⠄⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣤⠜⠛⠻⢒⡖⠲⠻⢷⣀⠒
⠀⠀⠀⢠⡐⢄⣶⠉⠙⠷⣼⣜⡅⠀⠈⠙⡦⣌⡠⠖⢫⡡⠀⠁⠔⠀⣀⠀⢸⢴⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣯⠛⠛⠶⣼⣜⠄⠀⠀⢠⣉⠅
⠀⠀⠀⡀⢹⣷⣈⢆⠀⠀⠈⣙⣣⣀⠄⢘⡱⠉⠀⢰⣾⣁⠠⡀⠁⡀⣥⡀⢨⢰⣿⠁⠠⠀⠀⠁⠄⠀⡘⢷⣞⡙⣄⠀⠀⠀⣩⣶⣤⣒⢨⢎⡄
⠀⠀⠀⠠⠀⢿⣿⣿⣓⡁⣞⠭⠼⢛⣳⠎⠁⠙⠲⣜⣝⣦⣶⢷⠀⠰⣿⡇⠈⣼⣿⠀⠀⠀⠀⠀⣀⡤⢓⠈⣿⣷⣮⢓⣃⣾⠫⠔⢛⣽⠃
⠀⠀⠀⢢⡄⠘⡼⠔⠋⢻⣷⣄⡠⣪⠃⠀⠀⠀⠀⡘⠩⢒⣲⣼⣧⠀⢻⠀⠀⣿⠋⣰⣶⣶⠂⠘⡁⠐⢤⡂⠸⡉⠀⠝⢻⣷⣤⡐⠹⠁
⠀⠀⠀⢴⣿⣄⠸⢌⣠⣔⡽⣿⣿⣷⣄⡀⠀⠀⢀⣁⢉⣀⠉⣼⣿⡆⠀⠀⠀⠁⠐⢋⡩⠽⠗⠿⠿⣤⣬⣿⣄⡡⣔⣀⣄⡙⣿⣿⡷⣤⡄
⠀⠀⠀⢠⡟⢷⣀⡨⠍⢁⡈⠬⢿⢀⠈⠃⠀⡍⢈⢉⣯⣭⣛⣍⠻⣷⠀⠈⠁⠹⢆⡞⠛⢦⣄⠀⠀⠀⢈⡿⢷⣀⣈⠏⢀⠀⠬⢿⢅⢀⣂
⠀⠀⠀⠘⠀⠂⠁⠀⠈⠀⢀⠀⢰⠀⠁⡀⠀⠃⠘⠙⣻⣿⣿⣿⡷⠈⠁⠀⢢⠠⢀⣬⣀⣀⣬⠟⣠⢀⠼⠁⠘⠉⠀⠈⠀⢀⠀⠠⠀⠁⡀⠀⡇
⠀⠀⠀⢀⠀⡐⢀⠀⠀⠀⠁⠀⠀⠀⠀⢀⢠⠀⠟⠛⠛⠛⠻⠛⠁⠀⠀⠓⢈⡴⡯⠼⡏⠉⠀⢁⡈⣵⡄⠂⠀⠂⠀⠀⠀⠁⠀⠀⠃⠀⢠⠠
⠀⠀⠀⠘⠀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠤⠤⠄⠀⠠⠀⡀⠀⠔⠧⠄⠠⠀⢈⣦⣊⣥⠞⢋⠞⠑⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠘⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠰⢰⠀⠁⡠⠁⠀⡠⠡⠊⣀⠀⠂⠀⠐⢦⡰⠿⠿⢿⡧⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⢈⠀⢀⠠⠀⢀⠔⣤⢇⣔⠠⣄⣀⢠⢬⢏⠒⠦⣄⡊⠀⠈⢃⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠂⠀⠀⢻
⠀⠀⠀⠐⣄⡤⢀⠀⠀⠐⢀⠀⠀⠈⢆⢻⠸⠈⠰⠄⢁⠄⡈⠸⣿⡆⠀⠙⡆⠀⠀⢀⡀⠀⠁⠀⠀⠀⠑⢄⡤⠀⠀⠀⠀⢀⠀⠀⠈⢄⢸⠸
⠀⠀⠀⢸⡟⠘⢦⡀⠀⠐⠁⠀⠐⠀⢘⢻⡇⠀⣶⢀⠂⢰⣶⣦⣝⡧⠤⠚⠋⠲⣴⡶⢶⣤⣀⠐⠀⣠⣶⡿⠰⢧⡀⠔⠀⠀⠀⠀⠄⠸⢻⣷
⠀⠀⠀⢘⠁⠀⠀⡈⠒⠄⣀⠀⠐⠀⢸⡌⣇⠀⣿⠇⢀⣶⣮⠍⠉⢁⠀⠀⠀⠀⠘⡏⠀⠙⣤⣷⠞⣿⢟⠃⠀⠀⣈⠒⠄⣀⠀⠐⠀⢰⡄⣿
⠀⠀⠀⠀⣠⣠⡞⠀⢻⣶⣏⠩⠙⠛⠠⢄⣿⠀⠏⠀⣼⣧⣈⠋⠁⠚⠀⠀⠀⠀⢀⣸⣴⣾⠟⠈⠁⢀⠃⣠⣠⡞⠁⢺⣶⣞⠋⠉⠛⠀⢤⣿
⠀⠀⠀⢰⣿⣿⣿⡟⢻⣒⣫⣧⡢⢎⢠⠖⠈⠀⠀⢰⣿⣏⣛⣭⡭⠉⠃⢃⠀⠈⠉⠉⡿⠑⠲⠶⡤⠃⢰⣿⣿⣿⡿⢻⣄⣫⣧⡂⢞⣀⠞
⠀⠀⠀⣾⡿⠛⠀⢷⢈⢂⠀⠐⠚⣠⡑⢄⣠⠀⠓⣾⠏⣴⣿⣟⣛⠛⠓⠀⠀⠀⠀⠁⠙⠒⠀⠠⠒⠦⣾⣿⠿⠁⢻⠰⣂⠀⠒⠚⣢⡁⢤⠠⠄
⠀⠀⠀⠟⠀⠅⠀⡈⢈⢽⡋⠓⡭⣉⣹⣦⡑⠴⠃⠀⠘⠻⠿⠿⠷⠶⢾⠈⡄⠐⠁⢀⠀⠀⠄⠀⠀⣼⠟⠁⠄⠀⠈⢃⠽⡏⠓⡭⢍⣍⡦⡙⠔
⠀⠀⠀⢀⠀⠐⠀⣜⠷⣔⠺⣄⠀⢀⢀⣀⣩⣄⠐⡀⠀⠄⠈⢀⣉⡀⢀⠀⠗⠃⠀⠀⢀⠀⠀⠀⡼⠋⠀⠀⠀⢀⣔⢿⣗⡻⢦⠀⠀⣀⣠⣬⠦
⠀⠀⠀⠁⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠈⠀⠀⠀⠀⠀⠀⠁⠁⠀⠁⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠈⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⣽⣿⣶⣶⣿⣯⣲⣄
⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣤⣤⣀⡀⠀⢠⣾⣿⣿⣿⢿⣿⣿⡿⣿⣿⣿⣷⡄⠀⢀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣓⣒⣦⣬⣭⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣭⣥⣴⣒⣚⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠯⠭⣯⣿⣻⣿⣿⣿⣿⠯⠶⠶⠶⠶⠽⣿⣿⣿⣿⣟⣿⣭⠭⠭⢿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣛⣽⣶⣿⣿⣿⣿⣿⣿⣶⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠈⠉⢙⡋⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠏⢀⣀⠀⠐⣏⣹⠆⠀⣀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢠⣄⣈⡉⠀⠈⠁⠀⢉⣉⣀⡄⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠈⠛⠛⠋⠀⠀⠀⠀⠛⠛⠛⠁⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠹⣿⣿⣿⣿⣿⡿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⣿⣿⣿⣿⠏⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣺⡄⠀⠀⠀⣀⣀⡀⠀⠀⢠⣗⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡏⣿⣿⣦⣀⠈⠛⠛⠁⣀⣴⣿⣿⢹⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣻⣿⣿⠈⠑⠢⠄⠊⠁⣿⣿⣿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢴⡞⠛⢿⣧⠇⠀⠀⠀⠀⠀⠀⠸⣼⡿⠋⢱⣦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣿⣾⣿⣿⣦⡀⠹⣧⡀⠀⠀⠀⠀⢀⣾⠏⢀⣴⣿⣷⣷⣿⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠯⠶⠦⢾⣭⣛⢿⣷⡀⠘⣿⣦⣄⣠⣴⣿⠃⢀⣾⡿⣛⣭⡷⠤⠶⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⣄⣠⣶⣝⣇⢿⣷⠀⠘⣏⢿⡿⣹⠃⠀⣾⡿⣸⣯⣶⣄⣠⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠛⡟⢸⡿⠇⠀⠹⢎⡱⠏⠀⠘⢿⡇⢻⠛⠟⠉⠉

⠀⠀⠀⡾⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁
⠀⠀⠸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠼⢿⣦
⠀⠀⠀⠉⠳⢤⣆⣰⠐⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣀⣠⡄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠃⠓⠀⣸⠏
⠀⠀⠀⠀⠀⢀⠽⠋⡱⠛⣻⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢰⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⠿⣿⠛⠈
⠀⠀⠀⠀⠀⠀⠤⠀⢰⣲⣶⡦⠀⠀⠀⠀⠀⣀⠀⠀⠠⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⢚⡭⣞⠿⠁⠀⣇
⠀⠀⠀⠀⠀⠀⡠⠻⢋⡵⣿⢦⡀⠀⠀⠀⠠⠈⠰⠀⡠⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⢠⣿⢋⡞⢡⢤⡀⠀⠀⠀⠀⢰⠚⠳⡄
⠀⠀⠀⠀⠀⢺⣏⠙⣁⠤⠋⣲⣇⠀⠀⠀⠀⠈⡁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡼⡆⠀⣀⡇⠀⠀⠀⠀⢈⡀⣠⠁
⠀⠀⠀⠐⡻⢓⢊⡏⡉⠎⠛⠏⠁⠀⠀⠀⠘⡜⠁⠀⢢⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠶⢯⢯⣰⡿⣦⣶⣲⡥⠚⡇
⠀⠀⢀⣰⠧⠱⠪⢆⡀⠄⣍⢶⡅⠀⠀⠀⠀⠇⠀⠒⠀⠀⠀⠐⢀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⢰⢮⣋⢦⡏⢐⢢⣧⡐⠁
⠀⠀⡊⠃⢚⡐⠳⠈⢱⠤⠣⢭⣁⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⠈⠁⠈⠀⢄⠠⠀⠒⠀⠀⠈⠙⠤⠜⠛⡹⣝⡫⣿⢟⡷
⠀⠀⢫⠫⠇⡀⣋⠘⢈⠃⠇⡘⢁⠀⠀⠀⠀⠘⡀⠘⡖⠀⠄⡱⠀⠐⠋⠁⠀⢀⡄⠀⠀⠀⢀⣠⣄⡀⠘⠋⣿⣻⣛⣿⡆
⠀⠀⡞⠒⢂⠉⡅⢤⢈⠧⢸⠀⠃⠁⠀⠀⠀⠀⠑⢀⠓⢦⢐⡀⣢⠀⠀⠀⠉⠉⠀⠀⠁⠀⡟⠀⠀⣷⠖⠲⠟⣹⡳⠋⠁⡀
⠀⠐⠙⡧⠬⠂⡑⣄⡼⠐⠞⢀⠁⠀⠀⠀⠀⠀⠀⠠⠀⢈⠁⠉⢂⠀⠀⠀⠁⠁⢈⠂⠀⠀⢳⣀⠀⠘⠦⡤⢔⡵⣟⠁⡰
⠀⠀⠀⠇⢠⣠⣁⠎⡙⠃⡇⠃⠀⠀⠀⠀⠀⠀⠀⢁⠠⠈⠠⢄⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠚⠋⠀⠀⠉
⠀⠀⠀⠀⠈⠀⠁⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢀⠀⡀⢀⠀⠀⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⢀
⣿⣿⢛⠿⢛⠛⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⠟⢛⣉⣭⠥⣤⣶⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⡿⠟⣡⠴⢚⣩⠴⠒⠉⣁⢤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⢟⣵⢞⣡⠖⠋⠀⢀⠔⣨⠶⣛⣭⡽⢿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣵⣟⣵⣟⡅⠀⡠⠂⢠⢞⡵⡿⠋⣁⢤⣭⣭⣿⣿⣿⣿⡍⠈⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣩⡴⠊⠀⣴⣿⢟⣠⡶⠋⢀⣴⣿⣿⣿⣿⣿⠟⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣥⢖⣡⣾⣿⣿⣿⡟⣡⣾⣿⣿⣿⡿⠿⠟⠁⠰⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡼⣻⣿⡿⠋⠁⠀⠀⠀⠀⠀⢰⣮⣭⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠙⣽⢏⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣧⣻⣿⢟⣯⣿⣿⣿⣿⣿⣷⣝⣿⡿⣻⣯⡤⣶⣦⣤⣄⡀⠀⠀⠀⣀⣼⣿⡏⠠⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⢟⣵⣿⢿⣿⣿⣿⣿⣿⣿⣿⣫⠞⢽⣷⣶⣿⣶⣯⠛⠃⠀⢠⣾⣿⣿⣿⣇⢀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠅⠀⡄⠐⠿⡯⠤⠁⠀⠀⠀⣿⣿⣽⣿⣿⣿⣬⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⡟⠿⣿⣿⣿⠛⢾⣿⣧⠆⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⣇⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠈⠿⣿⣿⣿⣿⡟⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⢄⣠⣿⣿⣿⣏⢷⣬⡿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡟⠛⣻⣿⣿⣛⣛⣛⣽⣿⣿⣧⠀⢀⡾⠁⠀⠀⠀⠀⢀⣀⣈⣿⣿⣿⣿⡌⢀⣐⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣶⣌⣛⣿⣿⣿⣿⣵⣾⣿⣿⣷⣏⡀⠀⠀⠀⠀⠛⣻⣿⣿⣿⣿⣿⡿⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣱⣎⣉⣡⢟⣿⣿⠟⠉⠉⠻⣿⣿⣶⣄⡀⠀⠀⠉⢙⣿⣥⣥⣭⣥⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣵⣾⡿⠃⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣷⣧⣸⣿⣾⣶⣼⣷⣾⣶⣷⣾⣼⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷⣯⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣥⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣷⣄⠻⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣭⣿⣿⣷⣽⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉⠁

⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡴⢶⣶⡄⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠾⠿⠷⡄⠀⠀⠀⠠⠀⠀⡀⠀⠀⢰⣷⣶
⠀⡄⢀⡄⠀⠀⠀⠀⠀⠠⣶⣎⡉⡇⡓⢒⠷⣄⠐⢀⢤⣆⠐⠱⠀⡀⠀⣠⢀⡴⠀⠀⠀⠀⠀⠀⠰⣿⣭⡄⣫⡍⠻⠻⢦⡀⡡⢴⣣⠀⢀⠂⣠⢸⣿⣿
⢈⢱⡏⠀⠀⠀⠀⠀⣄⡠⠇⠨⠛⠿⠃⠀⢁⢄⠙⠁⣼⠃⢀⠐⠈⣅⠀⢣⣿⠀⠀⠀⠀⠀⠰⣤⠼⡁⠀⠙⠛⢃⢀⠀⡢⠋⠀⣼⠃⢠⠀⠀⣬⢸⣿⣿
⠈⣾⡇⠀⠀⠀⠀⡰⠘⣷⣌⡤⠀⡴⢚⡷⡤⠃⠲⢄⣫⣢⣼⡄⢰⣿⠀⢸⣧⠀⠀⠀⠀⣠⠇⢸⣷⣴⠤⢠⠞⠫⠗⠜⠁⠒⠤⣽⣶⡾⣇⠀⣿⢸⣿⣿
⠀⡿⢁⣦⡆⠐⠉⢄⡀⠩⠛⠱⣷⣅⠀⡼⠃⠀⠀⠌⢉⠨⡱⣷⠀⠿⠀⣿⠃⣴⣶⠒⠉⠠⣄⠀⢝⠭⠺⣷⣅⡀⡞⠁⠀⠀⢠⡍⢒⣞⣿⡄⠹⠸⣿⣿
⠀⡠⠚⡻⠷⠶⢧⣤⣷⡀⢠⠀⡈⢿⣷⣦⡀⠀⠀⢘⡁⣋⢢⣿⡇⠀⠀⡀⠘⡉⠽⠲⠿⢦⣬⣧⠀⠄⣀⡜⢿⣿⣦⡀⠀⢀⢀⣉⣩⡀⣿⣧⠀⢰⣿⣿
⠀⠣⣘⠛⢦⡀⠆⠈⡿⢧⣀⠛⢁⠀⢿⠅⠉⠀⠄⢈⣙⣙⣭⡙⢷⠀⠀⠘⢦⠋⠳⣄⢈⠀⢨⠿⣄⡈⠋⠀⠈⢿⠀⠁⡀⠅⣈⣽⣯⣍⣍⠻⡀⢸⣿⣿
⢀⠀⣰⣀⣤⡞⢑⠤⠁⠈⠁⠀⠀⠀⠀⠀⢀⠠⠀⢉⣉⣿⣿⣿⠈⢁⠀⠔⡠⣇⣤⠴⠃⡢⠘⠀⠉⠀⠈⢀⠀⠀⠈⢀⠀⠁⠏⣉⣽⣿⣿⠂⢉⣸⣿⣿
⠀⠚⠓⢇⠀⠀⢈⣦⡀⠀⠀⠀⠀⠁⠀⠀⠀⡤⠀⠉⠉⠉⠉⠁⠀⠈⠁⠼⠳⠺⡀⢀⡤⣼⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⣰⠀⠋⠉⠉⠉⠁⠀⠀⢹⣿⣿
⠐⠂⠀⣠⣦⡴⠋⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⡠⠛⠒⠀⢀⣶⣤⠞⣡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢺⣿⣿
⠀⢨⣞⡛⠛⠗⢺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠁⠀⠄⠃⠀⠀⠀⠀⣶⡛⠛⠛⠒⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠅⢸⣿⣿
⠐⠚⠀⠉⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠀⡀⠠⠀⠚⢾⡉⠲⡤⠒⢁⠉⠒⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⢠⠂⠞⢶⡅⠒⣼⣿⣿
⠤⡄⣀⣂⠀⠀⠀⢀⣴⢪⠀⠀⠀⢂⠀⠀⢦⣇⠀⡃⠐⢀⣀⡺⡇⢀⡴⢤⢀⡂⠀⠀⠀⠀⣀⡶⢁⠀⠀⠀⠀⠀⠀⢤⣇⠀⡋⠠⢀⢀⠘⢇⠀⣸⣿⣿
⠀⠘⡷⠉⠟⣤⣤⣿⠏⠀⠑⠄⠀⠀⠀⠀⡀⣿⢸⣷⠁⣸⠻⠛⠎⠁⠀⠈⢿⠌⢻⢂⣄⣴⣿⠁⠈⠂⢁⠀⠀⠀⠀⡈⣿⢸⣷⠁⣘⣛⠿⠶⠉⢸⣿⣿
⠀⠀⢨⣠⣾⠋⠒⠱⢀⢀⡞⢠⣤⡲⠶⠦⠉⢸⠸⠏⢰⣏⠿⠆⠤⠀⠀⠀⠈⣄⣴⡿⠣⠊⠇⠀⣠⠖⢤⣐⢦⠤⢄⠃⣹⠘⡏⢠⡟⠷⠤⠠⠀⢸⣿⣿
⠀⠘⠋⣿⠣⢤⡴⠀⣼⣿⣷⠾⣛⣳⡂⡄⢠⠋⠀⠀⣾⡟⢂⠰⠲⠀⠀⠠⠞⢻⡟⠠⣄⠈⢠⣿⣿⣶⢿⢛⣦⢀⠄⣨⠛⠀⠀⣼⡿⢶⡴⠶⠂⢸⣿⣿
⠀⠀⠋⠐⠀⠀⠐⢴⣿⠿⠹⡐⡉⠀⠥⣈⠡⡀⠀⢰⡟⣹⣿⣷⠶⠆⠀⠀⠐⠉⠂⠀⠈⢤⣼⣿⡏⢷⠘⠉⠨⠥⠈⢋⠀⠀⢤⡿⢫⣿⣶⡶⠦⢸⣿⣿
⡀⠂⠀⠂⠀⠀⢀⡾⠁⠀⠀⢁⠽⠘⠪⠉⡵⡈⠰⠂⠈⠻⠿⠿⠶⠆⠀⠀⠀⠀⠀⠄⠀⢰⡟⠁⠀⠈⢊⢗⠒⢙⣛⣦⠁⠴⠊⠀⠿⡿⠷⠶⡄⢸⣿⣿
⢉⠁⠀⠀⠀⢀⠞⠀⠀⠀⢰⡻⢤⠳⠀⠀⠠⠮⢄⠀⠀⠀⠀⢬⠄⠀⢈⠅⠀⠀⠀⠀⣠⠏⠀⠀⠀⢔⢷⡂⢦⠀⠉⣊⣅⡄⠂⠀⠀⠀⢀⡁⠐⢘⣿⣿
⠘⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠻⠌⠳⠾⠆⠔⠂⠀⠄⠀⠐⠀⠀⠀⠀⠈⠄⠀⠀⠀⠰⠃⠀⠀⠀⠀⠈⢧⠙⢦⣶⡄⠤⠄⠌⠄⠀⠰⠀⠀⢀⠀⠸⣿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣦⡀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠈⠻⣦⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢀⣠⣾⣋⡀⠀⠀⠀⢠⣾⠏⠈⢿⣧
⠀⣶⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⣀⡀⠀⠀⠀⠀⣀⣿⣾⡿⠿⠛⠛⠻⠿⢿⣶⣤⣿⠃⠀⠀⠈⢻⣧
⠀⢻⠀⣠⠞⠁⠀⠀⠀⣀⠀⠀⠉⠙⠛⠶⣦⣾⡿⠋⠁⢀⣾⣿⣷⡀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⢻⣷⡀
⠀⠸⣷⣏⣤⡴⠶⠂⠈⠙⠻⢶⣄⠀⠀⣰⣿⠋⠀⠀⠀⠀⠻⠿⠟⠀⠀⠀⠀⠀⠈⢻⣿⡄⠀⠀⠀⠹⣷⡀
⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣴⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⢉⣿⣿⠟⠉⠻⣿⣿⠋⠉⢻⣿⠀⠀⠀⠀⠈⣿⣷⢶⣶⣶⣶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣿⣿⣴⣿⠟⠁⠀⠀⠀⠈⠻⣷⣴⣾⣿⢀⣠⣄⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢈⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣾⣿⣿⣿⣼⣿⠃⠐⢶⣄⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣠⡾⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠘⠻⠿⣻⣿⠏⠀⠀⠈⢿⣆⠀⠈⢃
⠀⠀⠀⢠⣤⣤⣤⣤⣼⡷⠟⠋⠀⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⣠⣾⡿⣿⣿⠀⣀⣴⡿⡏⠀⠀⠀⠀⢸⣿⠀⠀⣩
⠀⠀⠀⠀⠀⠈⢉⡿⠁⢀⣠⣤⠀⣿⣏⠀⠀⣿⣷⣄⠀⣠⣾⣿⠋⠀⢸⣿⣾⠿⠛⠲⠿⠀⠀⠀⠀⣾⠇⠀⠀⠤
⠀⠀⠀⠀⠀⢠⡟⠁⠘⠿⣿⡇⠀⠿⠿⠷⠿⠿⣿⠿⠿⠿⠿⠿⠿⢷⣶⡿⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠐
⠀⠀⠀⠀⣰⣯⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣄⣠⣹⣦⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⣼⡟⠀⠀⣘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠙⢿⣧⡀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⣸⡟⠀⠀⣠⠁
⠀⠀⠀⠀⠀⢀⣠⣴⡶⠿⠛⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠠⠆
⠀⠀⠀⠀⠀⠹⣿⡿⢶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠉⠙⠛⠿⢶⣦⣤⣀⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣄⢌⠀⠰⡘⢃⠀⠀⠋⣰⠟⣱⡿⣻⣿⡟⠀⠀⣰⣿⣿⣿⣄⠀⠘⢀⢀⡄⠐⣶⣌⣿⣿⣿⠿⠟⠋⠁⣿
⡮⣑⣇⠀⠀⠈⠀⠀⠘⠃⠀⠋⣈⣼⣿⣀⣀⣀⣿⣿⣿⡻⡿⢷⣦⠀⠳⣿⣶⣾⣝⣿⣿⡇⠀⠀⠀⠀⠻
⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠈⠋⠿⣿⣿⣿⣿⡿⣿⡝⢿⡄⠈⡯⠳⠀⠀⠉⠛⠿⢿⣿⡏
⠀⠉⠢⠄⣈⢦⡀⠀⠀⠀⠠⠒⠁⠈⣦⣍⠙⠻⣷⠈⢿⡄⠀⠀⠀⠀⠤⠀⠀⣠⣄⠀⠈⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠙⠀⠀⠒⢦⣄⡀⠀⠘⢯⣝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉⢙⣧⣤⣄⣠⣼⣿⡿⢷
⠀⠀⠐⠂⠀⠀⠀⠀⠀⢠⣐⡽⠃⠀⠀⠀⠙⠻⠇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠲⢿⣿⣿⡷⢞⡫⠅
⠂⠀⠀⠀⢐⣤⠀⢀⢨⣦⣿⡆⠀⣠⣶⣬⡄⣀⣀⠀⣀⠠⠔⠒⡫⠉⠁⠀⠀⠀⠀⠈⣉⠭⣭⡶⠋⢴⣿
⣴⠋⠀⡄⢹⣿⣦⣇⢿⣿⠟⠀⠀⠒⠂⢀⡬⢔⣈⠝⠀⢀⠀⠎⠀⠀⠀⡀⠀⠀⠸⣿⣿⡿⠋⠀⠀⣰⠟
⠇⣀⣴⣃⢀⡉⠙⠛⠈⠁⠀⠀⠀⠀⠐⠻⠶⠿⠟⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡀⠀⠦⡄⠀⠀⣼⡇⣿
⣾⣿⣿⡿⡼⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣿⢠⢀⢠⣀⡄⠀⢧⡀⠀⣿⠀⠀⠘⠛⠛
⠟⠸⠁⠃⠀⣿⣿⠀⠞⠙⠙⢣⣶⣶⣤⣀⠀⠀⠀⠀⠀⢿⢏⡿⣿⢸⡘⣿⠷⣦⢲⣽⣷⣾⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠍⠁⠀⠤⠤⠞⠫⠪⠫⣾⠇⠀⠀⠀⠀⡀⣤⣾⣷⢿⠜⠗⠀⠀⠘⢷⣗⣽⠇⠀⠀⠀⠈⠻
⠀⢀⠀⠀⠀⢀⡀⠀⠀⠤⣤⣾⠛⠉⣟⢿⡗⣄⡀⣠⡹⡌⣇⣿⣾⣦⠀⠀⠀⠀⠀⠀⣈⣠⠤⠀⠐⠂
⠀⢸⡇⡀⠀⠀⠀⠀⠀⠐⣛⣛⣡⣴⣷⣿⣛⠌⠻⢿⣷⠸⠸⢹⡿⠏⠀⠀⠀⢀⣤⣾⣿⠋⠙
⠀⠀⡇⠈⢢⠀⠐⠶⣬⣴⣒⡶⣶⣖⢰⡀⠉⠳⡄⠈⠁⠀⠀⠈⠃⠀⠀⠀⠸⡿⢏⡝⠁⣀⡍⣀⣠
⠀⠀⢡⠀⠈⠂⠀⠈⢺⣿⣿⠝⠛⠉⠈⠳⠦⠀⠀⠀⠀⠠⠀⠀⠀⡔⠁⠀⣘⢤⠩⠂⡴⢶⣞⣉⡀
⠄⠀⠞⡀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣦⠀⠀⠀⠀⠀⠘⣀⠴⠊⠀⠀⠠⢀⣀⣠⡿⠋⣠⢀⣼
⠄⣶⠀⢷⢹⡿⠀⣀⣴⣿⣦⡀⠀⠀⢠⡿⠭⢉⠀⠀⠀⠀⠀⠴⠋⠁⠀⠀⠀⠀⠀⠛⠻⢿⣵⣾⠷⠟⠁
⢸⡟⡄⠀⠻⢁⣾⡟⢸⣿⣿⣧⣀⣠⣤⣿⣿⣿⡦⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠛⠿⣦⡀
⠎⣿⠡⠄⠠⡿⠘⠁⠼⠿⢻⡿⢿⢿⢿⣿⠛⠻⠿⢿⡕⣠⠞⠀⠀⠀⠀⠀⠀⠻⣦⡈⠿⡿⣷⣶⣶
⠀⢀⠄

⠀⠀⠀⠀⠀⠀⢀⠀⢀⣱⣴⣴⢛⣋⠐⠒⠢
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⣿⣿⠿⣭⡉⠉
⠀⠀⠀⠀⠀⠀⠀⡟⠈⣾⡗⣌⣷⡈⠙⠈⠀⠀⢀⣰⣿⠋⣓⠀⣠⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠶⢻⡇⠈⢻⠙⠠⠀⠀⠀⡾⠁⢻⣷⡌⠸⢳⠏⣦⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢄⠑⢤⡀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠑⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠎⠉⠠⢄⠀⠀⠀⠀⠉⠒⠦⢄⡀⠀⠈⠑⠦⡀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠓⠺⡞⠛⠛⠒⠂⠀⠀⠀⠀⠀⠀⠈⠑⠠⣀⠀⠈⠓⢄⡀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⢦⣀⠀⠙⢶⠀⠾⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⢐⠤⡹⣧⣠⣄⡈⠶⣬⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⡤⠖⠋⠉⣀⠀⠚⢑⠻⣮⡾⠛⠓⠾⣧⡘⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠟⣹⢀⣴⠡⠜⡉⣤⣦⣾⣷⣿⣿⣿⣶⣄⠺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠖⠟⠳⠿⠿⠾⠷⠷⠛⠁⠁⠀⠀⠐⠀⠐⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡖⠂⠈⢉⡀⣾⣤⣶⣿⡶⠦⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⣴⣿⠯⠍⠩⢤⢐⣮⡟⣉⣛⣿⣿⣿⣷⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⢀⡠⢿⣦⣌⠉⢻⡿⣦⠙⠣⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⡀⡀⠀⢀⡠⠞⠉⣴⠏⠙⣿⣧⠈⠉⠉⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣼⡜⣁⠔⠋⣀⡴⠞⠁⠀⠀⢹⣿⡆⢧⠀⢠⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡾⠗⠚⠉⠁⠀⠀⠀⠀⠀⢈⣿⡇⡘⠀⠀⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠸⣟⠿⣦⠀⠀⢀⣤⣤⣄⣀⣸⠻⣧⣷⡄⠀⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⡏⠀⠉⠁⠀⣸⡟⠙⠿⢓⠄⢀⣿⣿⣿⡆⢿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⣿⡇⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢿⣿⣻⡟⡇⢸⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⡙⣧⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⣸⣿⣽⣷⠣⢸⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣴⡘⢷⡈⢆⠒⠍⠍⢀⠀⠀⠀⠒⠉⠁⠘⣼⣿⠀⣸⢠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣆⠻⡎⠆⠀⠣⠔⠊⠉⠀⠀⠀⣠⡼⣿⢏⡇⣿⣿⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣭⡻⠆⠘⠀⠀⠀⠀⠀⢀⣶⣿⣵⢞⣁⣾⣷⡏⣿⢼⡟⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣦⣭⡛⠃⠀⠀⠸⣿⠀⠀⠸⣿⣿⡿⢫⡾⣿⣿⢣⠿⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⣦⡍⠀⠀⢀⡀⠀⠀⠀⠀⠀⢻⣷⣾⢟⣼⣿⠃⠈⠸⢸⠃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣠⡶⠞⠉⠀⠀⠀⠀⠀⠀⢸⣯⣥⡾⠛⠁⠀⠀⢀⢸⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢹⡄⠀⠀⡀⠀⠀⠀⠀⣸⣿⣏⠁⠀⠀⡰⡠⡘⣸⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣘⠿⣷⣶⣶⣶⣿⣿⣧⢀⣷⢇⣇⡇⣷⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠉⠙⠛⠛⢸⡿⠸⣿⡇⠹⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠐⠳⠤⣄⠀⠀⠀⠀⣀⠀⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⢄⠘⡦⠰⢤⣴⡥⠀⠠⠀⢑⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠳⠔⠒⡿⠒⢧⣀⡟⠓⠴⠲⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠑⠒⠊⠀⠀⠀⠀⠀⢈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢀⡀⠀⢀⡴⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇⢀⠏⠱⠀⣸⠀⠐⠧⠋⡇⢰⠛⠀⡎⢣⠀⠀⡀⠀⠀⣶⡸⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠈⠀⣸⣇⡞⠀⡆⠀⡇⠀⠀⠀⠀⠃⠸⠀⠉⠀⠰⡀⢸⢹⡀⠀⡇⣇⢧⣈⡉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⡴⠊⠁⢸⠀⢠⠁⢰⠃⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠳⠞⠈⡇⠀⡇⢻⠋⠀⠉⠲⣄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⡜⠀⠀⡴⠋⠀⠀⠀⢸⠀⢸⣀⡞⠀⠀⠀⠀⠀⢧⡼⠀⠀⠀⠀⠀⠀⠀⢻⣀⡇⢸⡄⠀⠀⠀⠀⠳⡄⠈⢣
⠀⠀⠀⠀⠀⡘⠀⠀⡜⠀⠀⠀⠀⠀⣿⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⡀⠀⢇
⠀⠀⠀⠀⠀⢇⠀⠀⢇⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠾⠁⠀⠀⠀⠀⠀⢠⠃⠀⢸
⠀⠀⠀⠀⠀⠘⡜⠆⠈⢆⠀⠀⠀⠀⠀⠈⠙⠲⠤⣤⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣠⣤⠴⠖⠋⠁⠀⠀⠀⡀⠀⠀⣠⠎⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⢁⡠⠞⠁⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠈⠑⠢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠒⠉⠀⢀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣀⡀⠀⠉⠉⠒⠒⠲⠶⠤⠤⠤⠤⠤⠶⠶⠒⠒⠚⠉⠉⠀⠀⢀⣠⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠦⠤⢤⣄⣀⣀⣀⣀⣀⣀⣀⣠⠤⠤⠴⠖⠒⠉⠁




⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣗⣒⠭⠉⡝⢿⠿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠏⠀⠀⠀⠠⠀⠁⠀⢠⣽⣼⡇⠀⣠⣽⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠏⠁⠀⠀⡀⢠⣾⠽⠾⠙⠪⠛⠹⠧⠶⣄⡘⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠀⢀⣠⠖⠋⠁⠩⣴⠞⠁⠀⠀⠀⠀⠀⠀⠈⢳⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠀⢸⡇⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢀⣿⣿⣿⣿⣿⢘⢆⠈⣃⣀⣴⠆⠀⠀⠀⠤⠤⣤⣄⣀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣆⢕⡅⣈⣭⡽⠁⠀⠀⠀⢤⢶⡿⠿⠉⠀⢰⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦⡑⠂⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠈⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡏⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡂⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡏⢢⠀⠀⠀⠀⠀⠀⠀⣀⣈⣋⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠱⣄⡀⠀⠀⠀⠈⠙⠿⢦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠻⣶⣤⣀⣀⣀⣠⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠈⠻⢿⣿⣿⢿⡿⠿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠠⢋⣪⣤⣤⣾⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠿⠛⠇⠀⠀⠀⠠⢐⣺⣽⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠙⢷⣄⠀⠀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠘⢷⣾⡿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢀⣠⣾⣋⣀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠈⢿⣷⡀
⠀⢠⡄⠀⠀⠀⠀⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣴⣾⣿⣿⠿⠿⠿⢿⣿⣷⣶⣤⣠⣿⠟⠀⠀⠀⠈⢻⣿⡄
⠀⢸⡇⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠚⠷⢶⣤⣤⣀⣀⣴⣿⡿⠛⠉⢀⣤⣶⣦⡀⠀⠀⠉⠙⠿⣿⣿⣄⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠘⣧⢠⡾⠃⠀⠀⢀⠀⠀⣴⣦⣄⡀⠀⠀⠀⠀⠈⣹⣿⡿⠋⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠈⠻⣿⣷⡀⠀⠀⠀⠀⠻⣿⣆
⠀⠀⢹⣿⣥⣶⠾⠛⠛⠁⠀⠀⠈⠙⢿⣦⡀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠄⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⠀⠀⠹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡈⠉⠙⢛⣿⣿⣿⠟⠻⣿⣿⣿⡟⠉⠙⢿⣿⡀⠀⠀⠀⠀⠀⢹⣿⣷⣶⣶⣶⣶⣶⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢸⣿⣇⢀⣴⣿⣿⠟⠁⠀⠀⠈⠻⣿⣧⣀⣀⣼⣿⡁⠀⠀⠀⠀⠀⢸⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⢄⣴⣶⣶⣦⡀⣸⣿⡇⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⣽⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣾⣿⣿⣿⣿⣧⣿⣿⠀⠀⠹⣿⣆⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⣠⣾⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡀⠛⠿⠿⢛⣿⣿⠃⠀⠀⠀⠘⣿⣆⠀⠀⠙⠃
⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣠⣿⣵⡾⠟⠁⢸⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠀⠀⠀⣠⣿⣿⠃⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠛
⠀⠀⠀⠀⠈⠛⠛⠻⠛⣿⡟⠋⠉⠀⠀⠀⢸⣿⡟⠋⠙⢻⣷⣄⠀⠀⠀⠀⣠⣾⣿⠟⠉⢿⣿⣇⣴⣿⣿⣏⢹⡄⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⣀⣤⣶⣿⠀⢸⣿⣧⣀⣀⣼⣿⣿⣷⣄⣠⣾⣿⣿⣅⣀⠀⠸⣿⡿⠟⠋⠀⠉⠙⠃⠀⠀⠀⠀⣰⣿⠃⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠈⠛⢿⣿⡏⠀⠸⠿⠿⠿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠛
⠀⠀⠀⠀⠀⢀⣾⣃⣀⣀⣀⣀⣀⣈⣃⣀⣀⣀⣀⣀⣀⣀⣀⣻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢿⣿⣍⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⣼⡿⠋⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⣀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣀⠀⢠⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡃⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠈⢻⣿⡻⠿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄⠀⠈⠉⠛⠻⠿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠈⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢶
⠀⠀⠀⠀⠀⡀⠛⣍⠈⠁⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠄⠀⠀⠀⠀⠀⠀⢢⡀⠘⣿⠁⠀⠔⠁
⠀⠀⠀⠀⡀⠀⠀⠸⣄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠻⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⣿⠁⠁⠀⠊
⠀⠀⠀⠀⠙⠀⠂⠠⢸⡂⠠⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⣿⡤
⠀⠀⠀⠀⠀⠂⣀⠀⠌⣇⢀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡄
⠀⠀⠀⠀⠀⠀⠁⢀⠀⢿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠚⠲⠒⠓⠚⠀⠛⠒⠓⠒⠒⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠰⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠐⠠⡀⢠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⡄⠀⠀⣸⠅⠀⠀⠀⢤⠠⠀⠙⠀⠀⠁⠉⠃⠀⡀
⠀⠀⠂⠀⠀⠀⢲⡆⡄⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⡼⠃⠀⠀⠉⠀⠀⠀⠳⢤⡄⠀⠀⠀⠁⠀⠀⢀
⠀⠀⠠⠤⠤⢴⣾⡿⠃⠈⠉⠂⠀⠁⠀⠀⠀⢠⡼⠄⠀⠀⠀⠀⠀⠈⡓⠀⠈⠈⣙⡦⣄⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⣟⠂⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠀⡀⠐⠉⠹⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠈⠀⠙⠨⠀⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢦⣤⣄⡄⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠻⠍⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣰⢂⡇⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠛⢇⠈⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣈⣆⠀⢠⠀⠁⠀⠠⡆⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡑⢀⣫⡽⢿⠶⠂⠀⠀⢀⡇⠐⠂⠚⠃⠂⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣛⡥⣀⡀⠈⠀⣤⠀⠀⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠀⢧⡀⠀⠙⠒⢶⠊⠐⠄⡀⠀⠂⠠⠤⣀⠀⡀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⢠⠀⠀⠙⢦⡄⠀⠀⠑⢦⠀⠁⠀⠀⠀⠀⠸⠇⠀⠀⠀⢠⠋⠀⠀⠴⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠆⠀⠀⠀⠀⠀⠀⠁⢄⠀⠀⠉⡄⠀⢀⠂⣀⡀⠀⠁⠀⠀⢀⡣⡌⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠛⠔⠁⠀⠀⠀⠐⢒⠐⠀⠈⣄⠀⠀⠀⠀⠈⠀⠃⣴⠀⡀⢀⠀⠓⠒⠀⠀⢀⠀⠁⠀⢲⣀
⠀⠀⠀⠀⠀⠀⠄⠀⣀⠄⠀⠐⠀⢀⠀⠄⠀⠀⠀⠀⢹⡂⠀⠁⠀⠀⠀⠇⠈⠛⡇⠈⠀⠀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢴⣤⡀⠀⣚⣚⢉⣠⡜⠂⠀⠀⠀⠀⢁⠀⠀⠀⠄⠀⠁⠀⠀⠘⠑⢀⣘⡃⣄⣠⠀⠁⠈⠉⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⠀⣉⡡⡬⠐⠙⡀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠈⢈⡿⢄
⠀⠀⠀⠀⠀⠀⠈⡄⠐⡗⠉⡉⠉⠀⣁⡤⠔⠒⠒⠒⠛⢣⡀⠀⠀⠀⠐⡖⣦⣆⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡤⠚⠉⠀⠀⠀⣤⠀⠀⠀⠉⠉⢀⠤⠀⠘⠃⠿⠋⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⣀⡀⢀⡀⠁⠀⠀⠀⢀⠄⢰⠁⠋⢳⡀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠐⢶⠆⠛⣶⠑⠂⠀⠀⠀⢘⠁
⠀⠀⠀⠀⠀⠋⡀⠹⡟⠀⠀⠀⠀⠀⠀⠀⠘⠧⠀⠀⠇⠐⠑⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠰⠓⠾⡏⠐⠂⠀⠀⠠⠀⠀⠈
⠀⠀⠀⠀⠠⡀⠒⠹⠀⠀⠀⠀⠀⠀⠸⢀⠀⠀⠀⠢⡜⠁⠀⠀⠀⠀⠱⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠠⠀⢄⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠠⠠⠦⠤⠶⢄⣀⠀⠀⠀⠀⠀⡠⢰⠈⠓⠀⠀⠀⠀⠀⠀⠀⠈⠀⠗⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣾⠀⠀⠀⠀⠀⣤⡞⠙⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⡀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⢸⡇⠀⡏⠀⠀⠀⠀⠀⠀⠓⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠏⠙⢉⡗⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⠀⠀⢸⠁⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣤⠆⠓⠀⠀⠀⠀⢸⣿⠻⣿⣶⣄⠀⠀⢸⠀⠸⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣼⣿⠀⠈⠻⣿⣿⣶⣼⣤⣤⣴⣿⣿⣧⣄⣀⣠⣤⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡏⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠀⠈⠛⠛⠛⠛⠋⣽⣿⠃⠀⢠⡤⠤⠶⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣶⣶⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣿⡀⠀⠘⣧⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⣠⠞⠋⠁⠀⠀⠈⠉⠓⣶⣿⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠃⠀⠀⣿⡇⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⣼⠀⠈⣿⣟⠄⣠⣴⣶⣶⣦⣄⣼⣯⣼⡇⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⡾⠟⠛⠛⠲⠦⣄⣿⡵⡛⢦⡀⠀⢀⡏⠀⠀⠘⣿⢲⡟⠁⣶⠚⠉⠉⠁⠀⣿⣿⠀⠀⢻⣤⣄
⠀⠀⢀⡀⢀⡄⠀⠀⠀⠀⠈⠙⠻⣿⡟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⡎⠈⠻⣄⢸⠇⠀⠀⠀⠹⣟⣧⡀⢻⣄⣤⡄⠀⠀⣿⣿⠶⠒⠛⣇
⠀⠀⠈⡯⠁⢙⠀⠀⠀⠀⠀⠀⢠⣿⣷⡅⣴⡾⠟⠻⢷⣆⠀⠀⠸⣧⠔⠀⠈⠿⠀⠀⣤⠀⠀⢹⣎⠛⠛⠛⠉⠀⠀⣼⠇⣸⣇⣀⣀⣿⠿⠷⠶⠶
⠀⠀⠈⠉⠑⠞⠀⠀⠀⠀⠀⢀⣼⣿⣿⢾⡏⠀⠀⠀⠀⠹⡆⠀⠀⣿⣸⣦⡀⠀⠀⠀⣿⣆⠀⠀⢻⣤⣄⣄⣤⣴⠿⢣⡾⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⣠⠇⠀⠀⣿⣯⣿⢿⣄⠀⢸⡇⢻⣦⣴⣾⡟⠛⠉⠉⠀⠀⢿⡀⠀⠀⠀⠀⠀⢠⢀⣀
⠀⠀⠀⠀⠀⠤⠴⠾⠿⠿⠿⠿⢿⣿⣿⣿⣯⠛⠶⠶⠞⠁⠀⠀⣰⣿⣿⣿⠀⠹⠿⠟⠀⠀⠉⠀⠀⠀⢀⣘⣦⡤⢤⡬⣷⣄⠀⠀⠀⠀⠎⡀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢈⠿⣿⣿⣇⡀⠀⠀⣀⣠⣾⣿⡛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠳⣄⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⢀⡴⠋⠀⠀⢹⡿⠻⠿⠿⠛⠋⠉⠙⢿⣆⠀⠀⠀⠀⣠⡴⠲⠒⠲⢤⣸⠄⠀⠘⢦⡈⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⢠⡞⠁⠀⣀⣤⣄⡀⠘⠃⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠆⠀⠀⠀⠀⠀⠀⢘⡇⢠⡏⠀⣠⠟⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠀⢈⡺⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠶⢦⣼⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⢈⡆⠀⡤⠴⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡤⠚⠤⣞⣦⣜⣱⢲⣋⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠓⠚⠲⠶⠻⣿⠗⠛⠱⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠀⡤⠒⢲⡀⠀⡴⠁⠀⠀⠙⡆⢰⠃⠀⠙⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡞⠀⠀⠈⠧⠴⠃⠀⠀⠀⠀⠑⠜⠁⠀⠀⠘⢞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⠒⢚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⠑⠲⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡃⠀⠀⠑⠒⠤⢤⣒⣀⣀⣀⣀⣀⣀⣀⣀⣠⠤⠤⠒⠁⡀⠠⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠦⠤⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠤⠤⠀⠊⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⢠⠖⠚⠒⢀⠀⠀⠀⠀⢀⣀⣠⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠘⠇⠀⠶⠀⠀⢰⡞⡂⠀⠀⠉⠉⠁⣀⣨⠉⣯⠒⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠘⢦⠄⠊⠁⢀⡴⢋⣱⠿⠛⠚⠒⣒⣾⣿⣿⣶⡏⠙⠢⠔⠺⠟⠑⠒⢦⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⣟⡰⠀⢀⣴⡿⣡⣾⠁⠀⣠⣶⣉⣼⣿⡋⠃⠀⠉⠳⣤⡇⣀⢀⡀⠀⣊⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣦⣴⠋⢀⣴⣿⣿⢰⣿⠿⠗⠉⠀⠀⠀⢉⣄⠁⠀⠀⠀⠀⢹⣿⣿⣶⠀⢀⡾⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠀⣿⣿⡏⠃⢸⡁⠀⠀⠀⠀⠀⢠⣴⣿⣗⣴⣦⡄⠀⣾⠙⠿⠛⢁⠈⠓⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣿⡏⠰⠀⠘⣿⠏⠀⠒⠰⠎⣧⠈⠻⣿⢰⣿⣿⣶⣄⠁⢀⣷⣦⣤⠾⠟⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⣿⡄⠀⢿⡁⣶⠈⣰⣿⡦⠀⠀⠀⠀⣿⡗⣸⢃⣼⣿⣿⣿⣿⣧⢼⣿⡅⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⠋⠉⠉⠀⠘⢿⢿⣆⠻⠟⠁⢀⣀⣠⣤⣿⠗⣱⣾⣿⣿⣿⣿⡿⢉⣈⣉⠁⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢶⠀⠈⠃⢻⡀⠀⢀⣌⡁⠀⠀⠙⠛⠛⠛⠟⠿⢿⣿⣧⣿⣿⣿⣧⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⣇⠀⠀⠀⠀⢀⣽⠖⢲⡌⠙⠢⠉⠻⢿⣷⡀⠀⠀⠀⠀⠀⠂⣈⣥⣴⣶⣾⣿⡟⠁⠀⠀⠀⠀⢙⠶
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⣾⣇⡀⠀⠠⠄⣿⠁⠀⠀⢿⣖⣄⣀⠀⣠⣭⠙⢀⣀⣰⣤⣶⣿⣿⣿⡿⠟⠿⣿⣿⠀⠶⠀⠀⢰⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⡶⠀⠐⢻⡆⠀⠀⠀⢻⣯⣉⣉⣀⣀⡑⠾⣿⠿⠿⡟⠛⠛⠋⠀⠀⠀⢹⣿⠀⢢⣴⣶⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠈⠓⠀⠀⠀⠀⠉⠉⠹⣿⡁⠀⠀⢀⠈⠀⠀⣂⣁⣤⣤⣀⠠⠾⠃⢀⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠠⢦⡾⢶⡄⠀⠀⠀⠀⠀⠈⠑⠀⠀⢹⣿⣿⣯⡉⠀⠁⠉⠀⠀⠀⢀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠉⠛⢡⣿⣷⣶⣤⣤⣄⠀⠻⣿⣿⣿⣧⡈⡀⠀⢀⣀⡠⢾⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠉⠹⠛⠉⠁⣛⠆⠸⣧⣹⣿⠻⣇⠉⠋⠙⠉⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⡀⠀⠤⠓⠀⠰⣿⠋⠱⠍⣷⣤⠠⠾⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠜⠃⠉⠙⠀⢄⡃⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠖⠒⠋⠙⠒⠲⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⢀⣀⣠⣤⣤⣤⠶⠶⠶⠶⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣤⣖⣚⠋⠉⣉⣈⣀⣠⣤⣤⣤⣤⣤⠾⠧⠤⠤⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠾⠿⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣬⣤⣷⣶⣶⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠛⠉⠉⠀⠀⠀⠀⠀⠀⢀⡀⠤⢐⣀⣤⡴⠶⠛⢻⡿⣿⣿⣿⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⣊⡭⠶⢟⣋⠁⠀⠀⠀⠀⠀⢡⡿⠟⠋⠁
⠀⠀⠀⠀⠀⣠⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣀⡤⣒⡭⠖⠋⠁⠲⠶⣶⠪⠉⠀⢀⣰⣄
⠀⠀⣠⡖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⠷⣪⣵⣞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠙⢿⣦⡀⠀⠀⢀⣀⣀⣀⡤⠔⣡⣾⣏⣴⣿⡋⢦⣉
⠀⠀⠀⠀⠉⠻⢿⣶⣦⣭⣭⣷⣾⣿⣿⢿⠟⢻⣿⢷⣤⣀⣼⠆⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠸⢦⠀⠹⣧⡀⣍⣁⣰⠀⠀⠀⠀⠀⠈⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣆⠈⠻⢶⣭⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣮⡉⢀⡀⠨⣿⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣝⠋⠀⠀⠘⣿⣮⣧⣀⠀⠐⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⡿⢹⡿⠈⣷⡄⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣸⢿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢚⣫⠸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⡇



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣯⣿⣭⣭⣯⣬⣭⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿

⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⢀⠀⠀⠈
⡈⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠈
⠀⠀⢀⠀⠀⠀⠀⢠⡡⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠈⠻⣧⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣆⡀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⢮⣛⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣌⠳⣻⡵⡃⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠐
⠀⠦⡆⠘⢄⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠈⠯⢳⡵⠛⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢰⡄⠙⠠⠂⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⢰⠀⠀⠁⠘⢃⠉⣀⡀⠛⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈
⣨⠃⠀⢲⣁⡀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠁⡴⠂⠈⠈⣌⠁⢋⣠⠆⠀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⢐⠀⠱⠀⠈⢑⠀⠈⠉⠀⠀⠠⠖⡄⠀⠞⠀⠀⠀⠀⠈⣄⠩⠖⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠾
⠠⠀⡀⠀⠀⣨⠆⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠊

⠀⠀⠀⠀⠀⣠⣀⣠⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⡠⠞⢉⣤⣴⣶⣾⣿⣶⣬⡓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⢀⡀
⠀⣼⣯⣾⣯⣹⡿⡿⣿⣿⣿⣿⣿⣝⡆⠀⠀⠀⠀⠀⠀⠀⠠⠂⠐⠟⠸⡣⠔⠀⠀⠀⠀⠤⠉⢷⡀
⠀⢸⣿⣿⣿⣾⣶⣿⣾⣾⣷⣿⣿⣿⣷⠀⠀⠀⠀⢀⣠⢴⡿⠷⠶⠀⠘⠀⠀⠀⠀⠀⣰⠀⡀⣴⣧
⠀⢸⣿⣿⣿⡿⢿⣿⣿⢫⢾⣿⣿⣿⡟⠀⠀⢀⡴⢩⡞⠋⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡆⠀⠀⣸⣿⣦⡀
⠀⠀⠹⣯⡻⢿⣿⣿⣿⣾⣿⡿⠟⠿⠃⠀⣴⠃⠈⣸⠧⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢰⠶⠖⠀⠉⠀⠉⢇⡀⠔
⠀⠀⠀⠈⠛⠖⢒⠛⣻⣭⠥⠐⠁⠀⢀⠚⢁⣷⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣅⣀⣴⠃⠀⠀⢀⣀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣾⠿⠏⠛⠛⠋⠀⡉⠉⢩⣀⡀⠀⢀⡀⠀⠀⠀⠚⠑⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠋⠁⠀⠰⠂⠀⠒⠀⠙⠓⠀⠉⠉⠀⠠⡔⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⢰⡏
⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠤⠶⠶⣤⣀⠀⠀⠀⠀⠀⠀⢶⣿⡇
⠠⣠⡄⢀⠀⠸⠟⠁⠀⠀⠀⠀⠀⢀⣠⣦⣾⠋⠀⠀⠀⠀⠀⠀⠙⢢⠉⠀⠀⠈⠳⣄⠀⠀⠀⠀⠸⡿⣥
⠀⡵⣕⠘⠄⠀⠀⠀⠀⠀⠀⢀⠸⣞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠙⣷⣤⣀⠀⠀⠃⣿⡆
⠀⠉⡄⠁⠀⠀⢀⠀⠀⠀⠐⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⣿⡛⠿⣧⠀⠰⢻⢸
⠀⠢⣑⠀⠀⠀⠀⠀⠀⠀⡇⣠⠜⠂⠀⠀⣀⣀⠀⠀⠠⠐⠀⠀⠀⠀⠀⣧⡀⠀⠀⡠⠋⢷⡀⠈⠇⠀⠨⠀⠄
⠀⠐⡄⠀⢀⡀⠀⠀⠁⠀⣷⠇⠠⠁⠀⠀⡏⡯⠀⠁⠀⠀⠀⠀⠀⠀⠀⢣⠉⠀⠀⠀⠀⠈⠻⡄⢀⠀⠀⠣⣸⡄
⠀⠀⠉⣖⣒⡀⠁⠀⠀⠴⠁⠀⢀⠈⣄⠀⠐⠆⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢹⣼
⠀⠀⠀⠀⠙⠋⢁⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠀⠠⠀⠠⢄⡈⣷⣤⣀⣀⣀⣿⡏
⠀⠀⠀⠒⢒⡿⠁⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠨⣿⡟⠛⠛⣿⣛⣋⡀⠂⡀
⠀⠀⠀⠀⠀⠃⠤⠄⠀⠶⠒⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⠿⠻⠛⠉⠃⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⡀⠀⠀⠘⢳⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠘⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠓⠒⠒⠚⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣶⡆⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⢾⣷⣹⣿⣷⡇⡿⢸⣿⡇⣿⠀⠀⠀⠀⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⡠⡏⣰⡗⠀⣀⠀⠀⠈⠛⠳⠿⠅⢓⣃⣬⡿⠧⠟⠃⠀⠀⠠⠆⢸⣾⣿⡅
⠀⠀⠀⠀⠀⠀⠄⣰⡿⣿⠇⠐⡿⠓⡄⠀⠀⢄⣀⠀⠀⠀⣀⡀⠀⠀⠤⠖⢰⣿⣦⢸⣿⠈⠻⡄
⠀⠀⠀⠀⢀⠀⣴⠟⢠⡏⣰⠀⢱⠀⢠⡿⡆⠀⣏⠀⣤⣦⡈⠇⢠⣾⣦⡀⢸⣿⡀⠂⢻⣧⠀⢿⣆⠀⡀
⠀⠀⢀⣴⣽⣿⠏⢠⣾⡠⠃⠀⣹⡆⢸⠁⢹⠀⡏⣸⣿⢻⣷⠀⣸⡿⣿⣇⠘⣿⣧⣀⢠⢿⣿⡄⠹⣷⡴⣄
⠀⠀⠘⣿⣿⣯⣠⣾⣿⠃⠀⠀⣿⣷⠇⠀⢸⡇⠁⢹⠉⣹⣿⠀⣾⣧⠈⢹⡀⢿⣿⡧⠈⢾⣿⣷⡄⠙⣿⡿⠃
⠀⠀⠀⠈⣿⣥⣿⣿⡇⠀⠀⢠⣿⣿⠀⢀⢻⡇⠀⢸⠀⠀⢿⡇⣿⡿⠀⠀⢇⠸⣿⣧⡀⠸⣿⣿⣷⣷⡞
⠀⠀⠀⠀⠈⠻⣿⣿⡀⠄⠀⣴⣿⡏⠀⠨⣿⣧⠀⢸⠄⣈⣼⡇⣿⣿⠀⠀⢸⡄⣿⣿⣷⣄⣽⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠈⢿⣼⣮⣱⣿⣿⣧⠀⣹⣿⡏⠀⣾⠀⣐⣾⡇⣿⣿⠇⠀⠸⣷⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⣧⣤⣿⣿⡇⠀⡟⢂⣹⣿⣧⣿⣿⣯⣤⣴⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠓⠈⠙⠛⠛⠛⠛⠋⠙⠋⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡿⠿⢟⣛⣛⣓⣒⣲⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢢⣾⣿⠟⣫⣵⣾⣿⣿⣿⡿⢿⣿⣿⣿⣿⡿⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢣⣿⠏⣴⣾⣿⡿⣛⣯⠷⣞⣻⣿⣿⣿⣿⠋⠀⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡿⢣⣾⣿⢟⡵⢟⣩⣶⡿⢟⣻⣿⣿⡿⠃⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⢃⣿⠏⣱⢋⣴⡿⢛⣵⣾⣿⣿⡿⠛⠀⠀⠀⠒⠒⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⣼⡟⠐⣱⡿⢋⣴⣿⣿⡟⢉⠁⠀⠀⠀⠀⠰⢿⢧⣿⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘⣿⠃⢰⡿⣱⣿⢟⣿⠟⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⣿⠀⣿⣱⡟⣱⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⣿⣷⠹⠀⡏⡟⣼⡟⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⣰⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⢿⣧⠀⠃⡇⣿⡁⠀⠀⠀⠀⣠⣾⣿⣶⣤⣀⣠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠌⣿⣧⡀⠁⢻⣧⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡝⢄⠀⠻⣧⡀⠀⠸⣿⣿⣿⣿⡿⠟⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣦⡁⠀⠙⢷⣄⠀⠀⠉⠉⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⣦⡀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠈⠣

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢶⣄⢀⣤⢦⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠱⠆⡀⠈⣧⡀⠀⢀⣄⢀⠀⠄⠻⡿⣶⣇⠇⢭⠇⣳⡾⡽⠃⢠⠀⠀⣇⠀⠈⢠⡎⠀⣀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠄⣠⠘⠓⠨⡀⠀⠈⢃⠤⣍⠁⠀⣴⣷⡿⠀⣲⠠⣷⣶⡄⠀⢉⡅⠠⠈⠀⠀⣈⠓⠚⠀⣄⠄⠈⠉
⠀⠀⠀⠀⠀⡔⠓⠚⠂⠈⠉⠀⢹⣶⣾⣇⠀⠀⠘⠲⠀⢇⠀⣀⣋⠂⠶⡵⠆⢊⣓⣀⢀⡇⠰⠚⠁⠀⠀⣽⣶⣿⠁⡈⠉⠁⠘⠒⠓⣂
⠀⠀⠀⠠⠂⠀⢀⠀⠀⠀⢀⡄⠉⣀⣁⣉⣁⣙⣲⣦⣁⣈⣳⣉⣽⣷⣾⣹⣷⣼⣏⣹⣋⣀⣡⣴⣖⣁⣈⣁⣉⡈⡉⣀⡀⠀⠀⠀⠀⠀⠠
⡤⠠⠀⠀⠀⠠⠀⠄⠀⠀⠀⠤⠀⠠⠀⠀⠄⠠⠀⠀⠀⠄⠀⠠⠠⠠⠄⠀⠀⠀⠠⠠⠄⠄⠠⠀⠀⠀⠀⠀⠠⠄⠀⠀⠠⠄⠤⠀⠀⠀⠄⠀⠄⠀⠠
⡇
⡇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⣈⡄⢢⢠⢢⢇⢀⢠⢎⢀⡀⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠈⠀⠀⠈⠀⠀⠏
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢠⡟⣠⢀⡄⣠⢀⡄⣠⢀⠄⡄⣠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠘⠐⠃⠘⣰⠃⠊⡼⠁⠊⠘⢱⠇
⡇⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠰⠃⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⡇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⡇⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⡇
⠇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢠⠀⠀⠐⠀⠀⠀⠒⠖⠒⠒⠒⠲⠲⢒⣶⠶⠒⠒⣶⠒⣶⢶⣶⢲⡶⣶⠖⢲⡒⠒⠲⣶⡒⠶⠲⠒⠒⠒⠂⠖⠀⠀⠀⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠑⡂⣀⣀⠀⠀⠀⠊⣹⣾⣷⠟⠈⠁⢀⢠⠁⡾⠙⢛⠻⠁⣝⢯⣄⠹⡛⠋⠹⡄⢁⢀⠈⠁⠙⣷⣿⣮⠑⠀⠄⠀⣀⣀⡀⡆
⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠘⠜⠦⢉⣁⡘⠃⠀⠀⠙⠠⠞⠁⠀⣬⣿⡦⢰⡢⢠⣾⣥⠆⠈⠑⠠⠜⠀⠀⠀⠻⢀⡉⠱⠏⠈⠀⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⣀⠰⠊⠀⡿⠀⠀⠘⡆⠀⠹⠁⣠⢦⣾⠃⡼⠄⡈⣷⣧⣦⡀⠉⠂⠐⠟⠀⠀⠺⡆⠉⠂⣠⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆⠈⠁⠐⠀⠀⠁⠀⠀⠀⠛⠉⠸⣵⠗⠰⢷⡅⠈⠙⠃⠀⠀⠀⠀⠀⠐⠀⠉⠐⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⣀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⣄⢠⠏⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣰⢿⡄⢰⠇⠀⠹⠏⠀⠀⣀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠋⠉⠉⠙⠛⢳⣤⡏⠀⣧⣸⠀⠀⠀⠀⢠⠾⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠶⢶⣾⠿⡍⠀⢹⡇⠀⠸⠟⠀⠀⠀⠀⠀⣀⠞⠀⠀⠀⠀⠀⢀⠀⢀⣠⠴⠒⢂⠼⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⣶⣖⡚⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡴⠖⠛⠋⠀⠀⠀⠀⣀⠔⠋⠟⠉⢀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠈⢹⡷⣀⣀⣇⢀⢸⡇⠀⢀⠀⠠⠾⢷⣤⡤⠶⢚⡗⣠⠞⠁⠀⠀⢠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡳⣄⣠⣯⣬⣉⣉⣻⠈⢾⠃⠀⣯⡴⢁⡴⠛⠁⠀⠀⠘⠾⠃⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⣽⡟⠉⠀⠀⠀⠉⠁⠀⠀⠀⠀⠋⡴⠏⠀⠐⠀⠀⠀⠀⣠⣤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠉⣠⢆⣤⠀⣀⡄⠀⢀⣀⣀⣀⠀⠀⢠⣄⣘⣛⣀⣀⣩⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⢀⡴⠋⡀⠀⠀⠐⠟⠉⠟⢋⣥⡶⠿⠿⠿⠿⠿⠇⠀⠀⠈⢉⢹⠉⠉⢥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠐⠋⡠⠞⠁⣠⠴⠃⠀⠀⠰⠋⠀⢀⡴⢾⠟⠒⢦⣄⠀⠀⣶⣿⠀⠀⠀⠀⢙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠞⠀⠠⠚⠁⠀⠀⠀⠀⠀⠀⣴⡯⡻⠃⠁⠀⠀⠙⢧⠀⢈⣹⡷⣦⣄⣀⣘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⡡⠒⠁⠀⠀⠀⠀⠘⣷⡿⡫⠃⠀⠙⣧⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠔⠃⠀⠀⠀⢸⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣟⣩⠁⠀⠀⢀⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣤⠀⠀⠀⠀⠀⡜⢡⡏⠀⠀⠀⠀⡄⣿⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠘⠋⡟⠀⠀⠀⠈⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⣄⢿⡄⠟⠀⠀⠴⠃⢊⣰⡏⣰⡆⠀⠀⠀⢠⡏⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢄⡀⠞⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣤⣤⣤⡶⠟⠛⢿⣯⣁⣀⣀⣤⠟⢠⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣾⠀⣠⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣁⣤⠴⠶⠶⠶⠿⠿⠿⣿⣟⠂⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠉⠀⠒⠮⠙⠓⡀⠠⠤⠄⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠄⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠀⡷⠀⡀⠀⠈⠢⣌⠳⣄⠰⣄⠀⠀⢠⡟⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢫⢿⡤⣶⠀⠀⠁⠀⠀⠈⠁⠀⢸⣿⣯⢻⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⡼⠳⣤⣽⢶⣤⣀⣸⡟⠛⠻⣶⣘⣛⠛⣻⣶⣶⣶⠶⠾⠛⢳⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⣤⠀⠀⢀⡀⣹⣧⠀⠀⠀⠉⠉⠛⠛⠋⠉⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢠⣿⠀⠠⠤⠄⠈⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢀⣾⡟⠂⠤⠤⠀⣰⡇⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢀⣠⣾⣟⠉⠒⠀⠤⣤⣿⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠖⠋⠁⠈⢻⣮⣁⣀⣴⠛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡿⠉⠉⣙⣧⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⠶⠶⠟⢛⣿⣿⣿⠷⠄⠈⠉⠛⠛⠿⢿⣿⣟⣛⠛⠛⠛⠻⠷⠶⠶⣦⣤⣤⣤⣀⡀
⠀⠀⢀⣀⣤⣴⠶⠟⠛⠋⠉⠁⠀⠀⣠⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣤
⣠⣶⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⣴⠞⠉
⣭⣤⣤⣤⣤⣤⣤⣤⣤⣴⣦⣾⣧⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣍⣹⣭⣭⣥⣭⣽⣏⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣭⣿⣯⣽⣿⣯⣭⣽⣯⣿⣯⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠟⠙⠛⠛⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠉⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠳⠶⠖⠋⠻⠦⣤⣤⠀⢀⣤⠞⢃⣤⠀⠀⣀⣀⢠⡞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢻⣰⣏⣤⠤⠞⣉⠉⠛⠂⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠦⣄⠀⠐⠛⠛⠲⢿⣶⠋⠙⣧⣴⣖⡛⠉⠓⠲⠖⠚⠉⠹⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠳⣄⣀⣤⣴⡏⠉⡿⠛⢻⡯⣭⡉⠛⠒⠀⠀⠀⠀⠀⢀⡽⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠐⠋⠁⠀⠀⠛⠚⠳⠤⠞⠁⠀⠙⢦⡀⣠⣤⢤⣤⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⢷⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⣠⣄⠀⠀⢠⡞⣷⠀⠀⣴⢦⠀⠀⡞⢦⠀⠀⠀⣾⢧⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⡼⠁⡇⠀⠀⡾⠀⢸⣄⣰⠇⠘⣇⠀⣿⠘⣇⠀⠀⣿⠸⣇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣠⠟⠄⢸⡇⠀⢠⠇⠀⠀⠈⠀⢀⠀⠘⢦⡟⠀⢹⡄⠀⣿⠀⠙⢶⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠸⡇⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⠏⠀⠐⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⡀⠀⠀⠛⠋⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠓⠒⠒⠶⠶⠶⠶⠷⠒⠒⠛⠛⠉⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠁⠀⠀⠤⠀⢀⣄⠸⠀⠀⠀⣤⣶⡄⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡘⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠻⣿⠗⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢰⡖⢰⣶⢄
⠀⠀⡼⠁⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡽⠟⠛⠘⡅⢆⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠤
⢀⡌⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠐⡀⠀⠀⠀⠀⠐⠆⠀⡀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠇⠀⠀⠀⠀⠀⠦⣄⠀⠄
⣿⠇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⢤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠐⠤⠄
⠋⠀⠘⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⢀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⢄⠀⠠⠀⡀⠀⠑⠩⣢⡀⠀⢀⠀⠀⠀⠀⠀⠀⢠⡔⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⡠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀
⠈⠀⠀⣦⡈⡢⠄⡀⠀⠀⠀⠀⠀⢉⣉⣙⡛⠙⠋⠀⠀⠀⠀⠐⠠⢀⠀⡠⡀⠰⡃⠀⠠⠎⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⠂⠀⠀⠀⠀⠀⢀⣀
⠀⠀⢶⢶⣦⣃⡀⠀⣤⣀⣉⣩⣭⣍⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠲⣆⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹
⠀⠀⠀⡃⠸⣿⢿⣿⣿⣿⣿⠿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠉⠁⠘⣄⠀⠄⠀⢀⡀⠀⠀⠀⢀⠀⠂⠁⠠⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⢀⠀⠀⠘⠛⢿⣏⣛⠙⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢈⡁⠀⢀⡀⠈⠧⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⢤⢀⠀⠀⠀⠀⠀⠈⠙⠳⠀⠀⠀⠀⠐⠀⠀⠀⠀⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢐⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁
⠺⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠰⠂
⣄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⣦⡀⣿⣄⡄
⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠘⣣⠈⠓⠘⡦⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⢰⠀⠀⠀⠀⠢⠀⠐⢄⠐⠀⠀⠀⠘⠋⠁⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠄⠘⡀⣄⢀⢃⠀⠀⠀⠀⠁⡞⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠂⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⢠⠼⠏⡐⡶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠈⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⡈⠀⣧⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠰⣟⣿⣿⣿⡿⠟⡻⠶⠖⠲⢆⣀⣀⠀⠀⠀⢃⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠠⠔⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⣠⣾⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠃
⠀⠀⠀⠀⠀⠇⠀⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣶⣄⠀⠀⠀⠀⠀⠃⢠
⠀⠀⠂⢄⣀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣄⠘⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉
⠀⠀⠀⠀⠀⠙⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠂⠀⠀⢀⠔⣩⠖⠈⠉⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⢠⣶⠠⠀⠀⠀⠀⠀⠁⠠⢁⠎⠀⢀⠔⡀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠌⠀⢀⢋⠞⡴⠁⢀
⠉⠀⠀⠀⠀⣘⠿⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡌⠐⠀⢠⠃⡠⢀
⠀⠁⠀⠀⠀⠉⠲⠠⠄⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⠂⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰⢁⠂
⠀⠀⠀⠀⠀⢀⡄⣧⠀⠀⠁⠄⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂
⠀⠀⠀⠀⠀⢸⠇⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡄⡀⠠⠖⣂⡄⢴⡦⢀⡐⠲⠄⢀⢰⠤⢤⡄
⠀⠀⠀⠀⠀⠀⣠⡞⣫⠭⢥⣬⣤⣶⠶⠷⣥⡼⣻⣉⢷⡞⣉⣟⢧⣬⠾⠶⣶⣤⣥⡬⠭⣝⢳⣆
⠀⠀⠀⠀⠀⠀⢻⡸⣽⣧⡾⠫⢬⣽⡿⠿⢯⣥⡭⢞⣼⣧⡳⢯⣬⡽⠷⢿⣯⡥⠝⢷⣼⣦⢇⡟
⠀⠀⠀⠀⠀⠀⢠⡦⢠⡟⢨⠖⠀⠀⠂⠁⠀⠉⠛⠋⠁⠈⠙⠛⠉⠀⠈⠐⠀⠀⠲⡅⢻⡄⢴⡄
⠀⠀⠀⠀⠀⠀⡾⠀⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⢷
⠀⠀⠀⠀⠀⠀⠇⡇⣿⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣿⢸⠸
⠀⠀⠀⠀⠀⠀⠀⠥⢻⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡟⠬
⠀⠀⠀⠀⠀⠀⠀⣙⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢩⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⡍
⠀⠀⠀⠀⠀⠀⠀⡒⣼⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣧⢒
⠀⠀⠀⠀⠀⠀⡆⡃⣿⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⢸⢰
⠀⠀⠀⠀⠀⠀⢳⠀⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⡞
⠀⠀⠀⠀⠀⠀⠘⠗⠘⣧⢘⠦⠀⠀⡄⡀⠀⣀⣤⣄⡀⢀⣠⣤⣀⠀⢀⠠⠀⠀⠴⡃⣼⠃⠺⠃
⠀⠀⠀⠀⠀⠀⣼⢱⠻⡟⢷⣔⢚⣻⣷⢶⣞⠛⣓⢮⢻⡟⡵⣚⢛⣳⡶⣾⣟⡓⣢⡾⢻⣟⡎⣧
⠀⠀⠀⠀⠀⠀⠹⢧⣝⣒⡚⢛⠛⠿⠶⠶⡛⢳⣽⣉⡾⢧⣉⣯⡞⢛⠶⠶⠿⠛⠛⢓⣒⣫⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠇⠁⠐⠦⠌⠁⠺⠗⠊⠡⠴⠂⠈⠸⠖⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⣛⣟⣜⢳⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣠⡞⠋⣩⠉⠙⢿⣧⣽⣻⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠻⠻⣦⡀⠠⠀⠀⠈⠡⢘⡁⠄⠙⠻⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣷⣦⡀⢀⡔⠒⠰⠂⠀⡠⢾⣿⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⠟⠿⣿⣿⡋⠀⡀⢴⣼⠿⠛⠋⠉⠹⣿⠿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⡆⢈⠟⠈⠀⢀⠻⣤⠃⠀⢀⠀⣄⢘⠁⠆⡜⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠈⠁⡺⠀⢸⠛⣄⠸⣏⢸⣎⢺⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠠⡄⢿⠑⢿⡕⠬⣿⣽⡿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⠀⠀⠀⠀⠀⠈⢀⡘⠳⡄⣻⣄⣻⣮⣽⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠴⡶⠹⣦⡀⠻⣿⣏⣺⣿⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢶⣤⣄⣠⣴⣷⣄⣿⢻⣶⣿⠙⣿⣻⢟⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡍⣿⡿⠿⣥⡾⠹⣷⣴⣿⢿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣲⣷⠟⠛⢾⣼⠷⣎⠻⢽⣿⣾⣼⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⣼⢟⣄⣀⣾⢻⠛⠿⢿⣵⣶⣭⣻⣿⣟⠻⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⢹⡟⠛⠠⢄⠀⠉⠛⢿⣿⣷⣬⣵⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠻⣿⡻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⡿⣿⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢻⣯⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣴⣶⣂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡼⠛⣿⣛⡛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢫⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⢢⣧⢸⣿⣇⣿⣏⣿⢻⣿⡗⠯⡂⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠡⢹⡿⠋⠉⠁⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⠐⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠈⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠺⠿⠶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣶⣼⣢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣠⣴⣶⠿⢶⣤⣄⣤⣶⠟⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣥⣴⣿⣿⣿⣿⣿⣶⣶⣮⡁⠀⠀⠈⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣾⠿⢻⣿⣿⣿⣿⣯⡟⠛⠋⠁⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⣻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠈⢿⠋⣤⣀⣀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣿⡻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣿⡿⠿⠼⠻⣿⣾⣷⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠈⣟⠋⣁⢉⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⢁⣡⣤⣀⣀⣸⣿⣿⣿⣿⣿⣦⣀⡄⠀⠀⠀⠀⣰⣿⣷⣤⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢿⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⡇⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣶⣿⣿⣿⠟⠀⠈⢹⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣾⡟⠉⠁⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⣴⣶⣦⣴⣶⣾⣿⣿⣿⣿⣿⣿⡿⠿⠛⠙⠂⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⠟⠛⠉⣴⣿⠟⢁⠀⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⢠⣿⣿⣶⡿⠋⠀⠀⢁⡴⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⣠⣿⡿⠛⠃⠀⠀⢀⡀⠉⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⣾⡿⠋⠀⠀⠂⡀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠋⢀⠀⠀⠀⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠀⢙⣆⣐⡛⢏⣿⣦⣴⣾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠉⢳⣶⠿⢟⣿⣿⡈⢿⣭⡭⣟⣻⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣷⣤⣤⣾⡿⣱⣾⣿⣿⣿⣷⣶⣶⢌⣿⡟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠿⠿⣿⣿⣿⡗⣽⣾⣿⣿⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠿⢮⣿⣳⡄⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⣀⣀⣀⣴⠆⠀⠀⠀⣴⠀⠈⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⣀⠹⡄
⠀⠀⠀⠀⠀⠀⡴⠁⠀⢀⣴⣿⣿⣿⡟⠀⠀⠀⢠⣿⣆⣀⣀⣴⣷⣤⣴⣾⣧⡀⠀⠀⠈⢷⡇
⠀⠀⠀⠀⠀⠀⠳⣴⣾⣿⣿⣟⣿⣿⡁⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣽⣿⣿⣿⣿⣷⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣭⣿⣿⣿⣟⣿⣿⣷⣾⣧
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠉⠁



⣿⣿⠟⠿⠛⠿⡿⢻⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣀⡤⠄⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⠀⠀⢀⣤⡞⢁⣠⣾⣟⠋⠡⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⡠⠀⢀⣴⠟⢡⣴⡿⠛⠉⠀⠀⠂⢩⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡠⠊⢀⣴⣿⣿⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣰⣿⣿⠟⠉⠀⢄⣀⣀⡀⠠⠔⠒⠠⢄⣀⠀⠀⠈⢀
⠀⠀⠀⠀⢠⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⣠⣾⡿⠟⠁⠀⠀⡀⠄⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣾⣿⠟
⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠰⢄⣀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⢀⣤⡾⠋⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡷⢶⣶⣶⣶⡶⠖⠶⡟⠋⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠐⢀⠀⠀⠀⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣇⠀⠀⠘⢆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀⠀⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠒⠶⡀⠀⠀⠀⠰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⡄⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠑⢤⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠤⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢧⣠⡿⠁⣠⣽⠛⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡁⣸⣿⣿⣵⠀⣀⡀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠛⠃⠰⠋⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣺⡟⢋⣠⠼⢿⣿⣾⣿⢗⠝⠋⠁⠀⠀⠀⢀⠴⠿⠿⢛⠁⠀⠠⠖⠟⠉⣧⠐⠎⠓⠀⠀⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠋⡄⡶⣷⣿⣿⣿⣿⢧⡀⠂⠄⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⢰⠁⠀⡘⢧⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣥⠟⢹⡿⢹⠉⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠐⠃⠈⠆⠀⠀⡄⠀⠀⣧⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⢼⣧⠌⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣡⠀⠀⢹⡀⠀⡀⠁⢰⠃⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠉⠀⠀⠀⠉⠁⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⣧⠀⡀⠀⡞⠀⠠
⠀⠀⠀⠀⠀⠀⢀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠁⢠⠃
⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸
⠀⠀⠀⣰⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢼⠦⣄⣀⣠⢿⠀⣿
⠀⠠⣴⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠜⠓⠀⠾⢊⡴⢂⣉⣿⡄⣿⡀
⢀⣈⣽⣿⣷⣷⣤⠄⡀⠀⣠⡀⠀⠀⠀⢀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡰⠚⢙⠣⣃⠁⠚⣉⡐⢂⣔⣿⣿⣿⣷⠦⠄
⠀⢴⣿⣾⡿⠽⣿⡧⢹⣿⣿⣿⣿⡤⠀⡼⠙⠁⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣶⣿⣤⣴⣶⣮⣭⣥⣿⣿⣿⢿⣿⣿⢿⡟⠋
⠀⢸⣿⡟⠀⣀⡀⣸⣿⣿⠿⣯⠝⠂⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠻⠯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣭⢿⣄⠙⠆
⠀⣞⠏⠀⠉⠙⣿⠛⢉⡃⠈⠈⣀⠀⠀⡅⡀⢨⠀⠀⠀⠀⠀⠀⠀⢈⣉⣰⣾⡿⣽⣛⣉⡻⠦⠚⠛⠉⠉⠉⠉⠀⠹⣄
⠚⠁⠀⠀⠀⠀⠠⠀⠉⠀⠄⠠⢅⠀⠀⣧⡇⡈⠀⠀⠀⠐⠛⠙⠷⠿⠯⠶⠶⣿⣇⡱⠯⠻⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣉⡭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠄⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠈⠈⠁⠐⣄⠀⢰⠀⠀⢀⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠂⢀⠀⣀⣀⣴⣀⠀⠀⠀⠈⠁⠀⠉⢴⣿⣇⣸⡿⠲⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⣵⣿⣿⣿⣿⣷⡶⠒⠂⠉⢷⡀⠀⠉⠛⠉⠛⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠠⡄⢙⠤⠤⠀⢼⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢀⡅⠈⠂
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⢀⡀⠀⢃⡄⢠⣿⡄⠀⠀⠈⣻⢿⣿⣿⡿⠟⠓⠦⣄⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⣤⣀⠔⠁⠀⠈⠈⢇⠀⠀⢀⢘⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡄⠀⠀⠀⡌⢦⣴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠉⠁



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣽⣯⣮⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⠀⠀⠀⠠⢤⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠁⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⡇⠀⠀⠇⠸⠀⠀⢘⢸⠇⠀⠀⢰⠁⡹⠀⠀⡌⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⣇⠀⠀⠈⣀⣧⠤⠼⠾⠦⠤⠤⣬⣴⣁⠀⡐⡼⠀⠀⢀⠔⢙⠇⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠤⣄⠀⠀⠙⣾⠆⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠧⣄⡠⢊⠔⠁⠀⠀⡠⠂⠉⢹⠘
⠀⠀⠀⠀⠀⡇⠈⠳⣄⠀⠈⠢⣌⠳⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⣀⠀⠀⠈⠙⢦⡀⢀⠠⠊⣀⠤⠒⠁
⠀⠀⠀⠀⠀⠘⠦⣀⠈⠣⣀⠀⡸⠃⠀⠀⠀⠀⠀⢀⠤⠖⠂⠀⠀⠀⢀⠤⠶⠦⢤⡈⠑⠀⠀⠀⠀⠙⢧⣠⠞⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢬⠏⠀⠀⠀⠀⠀⠀⠀⠁⢀⡤⠤⣄⠀⡴⠁⢀⣀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠈⢧⡠⣐⢈⡡⠃
⠀⠀⠀⠀⠀⠀⠀⡄⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢠⡎⡇⡇⠀⢿⡿⠃⠀⠀⡼⠀⠀⠀⠀⠀⠀⠈⣯⠉⠀⠀⠀⢀⠠⠤⠐⢦
⠀⠀⣀⠤⣀⠀⠀⠘⠲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⣀⡠⠃⠙⢦⣀⡀⠀⣀⠴⠁⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⠶⠭⠤⠤⠤⠔⠊
⠇⢸⣀⠀⠀⠑⠦⠤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠤⠤⠄⠀⠀⠐⠪⠭⠭⠓⠂⠀⠀⢀⣀⣀⡤⠂⠀⠀⣇
⠀⠀⠈⠉⠉⠉⠙⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡤⠤⡤⡴⢒⠖⢞⠉⡏⠹⣤⠃⠀⠀⠀⣯⣍⣉⡉⣳
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠙⣽⠙⡏⡟⠙⡏⡝⢹⠁⡏⢹⢀⠏⢣⣸⠀⢹⣼⠀⠘⣶⠁⠀⠁⠀⠀⠀⢠⡇⠀⠀⠈
⠀⠀⠀⠀⠀⢠⠊⠉⠽⡄⠀⠀⠀⠈⠀⠹⠁⠀⢳⠃⠀⠿⠀⠀⠟⠀⠈⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢭⣒⠤⠄⠀⢀⣄⣀
⠀⠀⠀⠀⠀⠀⠁⠁⠀⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⠀⠈⠙⠲⣄⡀⠀⠀⣱
⠀⠀⠀⢀⡠⠤⠄⠒⢈⡠⠟⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠮⣍⡙⠒⢤⠀⠀⠈⠉⠉
⠀⠀⣔⠀⠀⠀⡠⠚⠁⠀⡠⠄⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠉⠓⠚⠁
⠀⠙⠜⠲⠴⠊⠀⠀⠀⠀⠧⠤⠊⠀⢙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠓⢝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⡡⠋⠀⠙⢓⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠳⡀⠀⠀⠳⣍⠓⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⡼⠁⠀⡔⠁⡎⠀⠀⡌⢹⠋⠛⣒⠒⡖⠒⢺⠹⡅⠀⠀⢸⡄⢹⠀⠀⠀⠈⠲⣄⣈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠤⠊⠀⠀⠀⠳⠊⠀⠀⠠⠁⡇⠀⠀⢻⡠⠃⠀⠘⡆⢣⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⢀⠇⠀⠀⠀⠀⠀⠀⠀⣇⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡸⢀⠆
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡀⠀⢹⡄⠀⠀⠀⢀⡀⡂⠀⠀⠀⣠⡄⠊⠀⠀⠀⡀⠈⢿⣇⠀⠃⢀⣀⠀⠛⠻⣐⠿⢤⠔⠒⠤⠄⣿⡾⠟⠀⢊⣿⣿⣿⠏⠈⣯⡐⢀⠀⠀⠀⠈⠒⠤⣤
⠁⠀⠀⠙⡄⠀⠀⣤⡀⠡⠶⢶⣼⣯⠕⠀⠀⢂⠀⠀⡀⣸⡿⠀⠘⠀⢩⣄⠀⠀⠎⣤⢤⡴⠶⠟⠘⠈⠀⠀⢀⣌⢶⡟⡻⣄⠀⠋⠐⢦⣣⣄⡀⠀⠀⢀⣀
⡀⠀⢀⠀⠙⠂⠒⢿⡈⠳⢤⡈⠛⠻⠀⡀⠠⡤⠀⠀⠀⡿⠃⠐⠀⠀⢾⠋⡥⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠊⠁⠈⠋⠑⠫⡀⠀⠀⠈⠉⠻⠻⠆⠉⠉
⡇⠀⠸⣆⠀⠀⡀⢈⣷⡀⠀⠙⠣⢀⠈⠈⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⣇⢀⠀⠘⡆⠀⣇⠀⢿⠛⠢⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠟⡌⠀⠀⠋⠂⠄⠀⢠⡀⠀⠈⠓⢆⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀⠠
⠀⡀⠀⠀⠠⠀⠀⠀⠀⠈⠒⠄⡀⠀⠉⠓⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋
⣘⠀⠀⠀⢀⠀⢂⡐
⠟⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠠⡀
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢻⡄⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣿⣦⣤⣀⠉⢷
⠀⠀⠀⠀⠃⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠈⣉⣙⠛⠶⢬
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⢠⡤
⢆⠀⠀⠀⣶⣠⣥⡀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣬⡀
⡛⣷⠀⠀⣮⣽⣷⠞⠁⡀⠀⣠⣤⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠤⠄⠀⠀⠀⢀
⡻⣾⡇⢸⣿⣯⣿⠀⣸⠗⠊⠖⠉⠉⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠘⠓⠲
⣿⠿⣿⣼⡇⣸⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⢹⢶⣽⠙⣷⣿⣥⣄⣤⡴⣆⠀⢰⠴⣦⠠⢄⡀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠠⡀⢀⣦⠀⠀⠀⠠⠀⣴⣟⠉
⠀⣿⣽⣷⡌⠻⠟⠛⠉⠁⠀⠀⡀⢅⣨⣤⡀⠀⠀⠀⠀⢀⠃⠀⠀⠂⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⣿⣃⠀⠀⠀⠀⢀⠀⠈⠻
⠂⢿⡏⠙⠟⠧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⡄⠀⠀⡜⠀⡠⠀⣼⠏⠀⠀⢀⡤⠤⠎⠀⠀⠀⠀⠀⠁⠀⢀⡅⠀⠀⠀⠀⠀⠀⢰⡖⠒⠁⠊⢄⠈⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠞⣀⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣧⢀⣀⢠⣤⣤⣴⣶⣶⣤⣄⠀⢨⡿⢯⡽⠉⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢠⣀⢰⣤⣄⣎⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣿⣿⣷⣿⣿⣿⡻⠆⠀⠀⠢
⠀⠀⠀⠀⠀⠀⢀⣯⣿⣿⣧⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣧⣖⢳⣶⢢⡄
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⡯⣯⠻⢿⣿⣿⡿⣶⣿⣿⣿⣿⣿⣿⣿⣿⡗⣿⣆⠈⢻⣿⣧
⠀⠀⠀⠀⠀⠠⣹⣿⣿⣿⣿⣿⣿⠿⣿⣇⠜⣐⣲⣦⠐⢿⣿⣿⣿⣿⣿⣶⢟⣻⣿⣿⣿⣿⣿⣿⣤⡙⢿⡧
⠀⠀⠀⠀⠀⠐⠹⣿⣿⣿⣿⣿⣧⣾⣭⣿⣟⠝⠇⢱⠇⣿⠿⣆⡿⢫⣶⠉⢛⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⠗
⠀⠀⠀⠀⠀⠘⣴⢿⣿⣿⣿⣿⣿⣟⣿⡋⣩⣶⣿⡄⡘⠻⡄⣹⣗⠉⠁⠀⠀⠉⠙⠛⣿⣿⣏⣻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠘⢿⡟⢿⣿⣿⣿⣶⣡⣾⣷⣦⡀⣽⡶⠂⠠⠀⠀⠀⠀⠀⠀⠀⠺⢿⣿⡏⠛⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠘⣉⢙⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡆⣈⣻⣿⣃
⠀⠀⠀⠀⠀⠀⠀⢐⠀⡀⠈⣿⣿⣿⣿⡿⣿⣏⣛⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡿⣿⣿⣽⣿⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢽⠉⠦⠙⠻⢿⡿⢉⣝⡟⢿⡿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠀⢫⡄⡹⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠈⠀⠉⠒⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠋⡑⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠁⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀




⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡞⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠁⠀⡆⠈⠀⠘⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣲⡏⠩⠁⣿⡀⠀⢇⠀⠀⣰⣖⠀⠉⢉⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣋⣡⠤⠖⡉⠁⢀⠗⠈⠁⠻⡅⠀⠀⠐⠤⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⢠⠁⢀⠾⠁⢉⡿⢠⡏⠈⢳⡅⢀⠄⠴⠀⢀⠸⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢠⡆⢠⠋⠁⠀⢸⠇⠘⠁⠀⠀⢻⠘⡧⡀⠀⠸⡄⢽⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠸⠀⣼⠃⠀⠀⣸⠀⠀⠀⠀⠀⣸⠀⢹⠁⠀⠀⠁⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢨⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠘⣇⠎⠀⠀⠀⠀⡄⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠁⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠈⣇⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⠀⡼⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠛⠒⠢⡄⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⢀⣀⣤⠼⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⢤⡀⠀⠙⠛⠛⠲⠦⠴⢥⣀⣀⣀⣀⣀⣠⡤⠤⠶⡶⠶⠖⢉⣁⣠⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡠⠤⠒⠢⣤⠀⠀⠈⠁⠛⡉⠉⠀⠀⠀⠀⠀⠀⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠚⠉⠁⠀⠉⠒⠤⠄⠐⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡜⢀⡿⠀⠀⠀⠀⠀⣾⠀⠀⢰⡇⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣁⡾⠃⣿⠀⢀⡞⢈⡟⢰⡇⣸⡇⣿⢸⡄⢺⢆⡾⠁⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠋⠀⠀⢿⡠⠘⣧⠼⡇⢹⣤⢿⡇⣿⢸⡇⢺⠘⣧⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⠀⠀⠀⢀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⢀⠀⠀⡀⢀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠙⢦⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣟⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⢠⢃⡴⣆⠀⣹⢦⡚⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣟⣯⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠐⡗⠲⣾⣾⣤⣞⣏⠁⠀⠙⢆⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣽⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⣱⣴⠟⠁⠀⠀⠉⢿⠒⠢⠤⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡛⠛⠛⠛⠛⠛⠛⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⢴⡋⣩⡏⠀⠀⠀⠀⠀⠀⢳⡀⠀⣿⠀⢙⣿⡂⠀⠀⠀⠀⠀⠀⣰⣿⣼⣷⡄⠀⠀⠀⠀⣰⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡜⠛⠀⠀⣤⡜⠃⠀⠀⠀⡹⢻⢿⣀⠀⠀⠀⠀⢀⣀⢷⡤⢿⠖⠋⠉⠀⠀⠀⠀⠀⠀⣰⣿⣧⣿⣿⣿⡄⠀⠀⣰⣿⣿⣿⢳⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⡼⢁⣏⣠⣯⠿⠲⣎⡉⠁⠀⢀⠿⣯⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣼⣿⣿⣿⣿⣆⣼⡿⢿⣛⡏⣼⣷⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⢀⣜⣁⣀⣁⣀⣸⣀⣀⣀⣉⣳⣶⣃⣀⣽⣄⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠛⠛⠛⠛⠋⠉⠙⠛⠛⠛⠉⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⣤⡟⠃⠀⢀⣤⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢀⠜⠉⠀⠈⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡖⠋⠀⠀⠀⠀⠀⠀⢄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠱⡀⠃⠀⠂⢀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠨⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡈⠂⠌⠈⣠⠀⠀⠐⠁⡀⠄⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠛⣧⡄⠀⠈⠛⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣻⣿⣿⣗⡀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡠⠀⠙⠻⠫⠂⢈⠐⠀⠄⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢃⠑⢄⠁⠠⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠢⠀⠀⠀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⣶⡏⠁⠀⢠⣶⠉⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠐⠂⠀⠀⠀⠘⡄⠀⠉⠉⠁⠉⠉⠀⠂
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⠈⡇⢦⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠑⢦⣤⣀⡀⠅⠀⠀⢰⡀
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠸⣇⢻⣌⣓⢧⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠇
⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⢻⣆⡉⣡⢎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⡀⠔⠋⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⡙⠛⣡⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇

⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠍⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢯⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣤⠀⠖⠡⠙⠛⠳⢽⣶⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠴⠣⠀⢃⠘⠂⠀⠈⣿⠁⠀⠀⠘⣿⣠⠀⠁⢯⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡰⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠠⣸⣿⣷⡀⠀⠀⡼⠡⠇⠀⠀⠇⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⣏⢷⣠⠛⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠘⢠⡏⣿⣻⣷⠀⠀⠇⠀⠀⠁⢠⢀⡀⠀⠀⠀⠀⠛⠀⠦⠌⣉⠻⢼⣏⣴⣣⡴⢦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡿⠀⠹⡟⠃⠀⠈⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⣩⡵⢈⣿⣶⣤⡙⣦⣤⣁⠀⠈⣿⠂⢀⠠⢒⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢴⠶⢾⡏⠠⠞⠈⢿⡈⢻⣸⣇⣙⣧⣴⣼⣿⡏⠶⢭⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣼⡀⠁⠀⠀⠘⣷⠀⠉⠉⠁⡰⢋⣴⣿⣿⣧⠘⡇⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠹⠆⠀⠀⣟⡤⠋⣸⠛⠟⠘⠷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠁⠀⢠⡏⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢉⣉⠁⠀⠀⡁⢀⡀⠀⠀⠀⠉⠉⠉⠉⢙⠃⠀⠀⠠⠤⠄⠀⠀⠀⠉⠉⠁⠐⠒⠠⠀⠀⠀⠀⠹⡇⠀⢾⠆⣾⢰⠚⠈⠉⡉⠈⠠⠤⠸⠋
⠀⠀⠀⠠⣤⡀⠘⠛⠻⠶⡶⣡⢤⣴⣷⣶⠀⠀⠀⠀⠀⣤⣤⣤⣤⠀⠀⠤⠈⠙⠁⢀⠀⠠⢀⣀⡀⠐⠒⠠⠄⠈⣷⡀⠐⢀⡟⠀⡛⠂⠀⠀⠀⠠⠿⠛⠃
⠀⠀⠀⠀⢴⣟⣩⣤⣶⣆⢀⡤⠀⠀⠈⠉⠁⢀⠀⠀⠀⠀⠀⠀⠛⠀⢤⣶⣆⣥⣄⣀⢀⢀⠀⠀⠀⠁⠀⠀⠙⠃⢙⠈⠂⢸⠃⠀⠶⠶⠶⠒⠂
⠀⠀⠀⠀⢠⣶⣤⡀⠀⠈⠸⣇⠀⠀⢴⣶⣶⣯⣤⣿⢠⢣⣶⡤⠄⡄⠀⠀⠛⠋⠉⠉⠀⠀⠤⠶⠶⣧⣶⣤⣠⣀⢈⠀⢄⣉⠀⢀⠀⣀⡀⢀⣤
⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠴⡦⡴⠴⠆⢦⡄⢀⠀⣤⢀⣀⡈⠀⠈⠀⠀⠃⢠⠟⠛⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠈⠁⠉⠉⠁⠀⠀⣦⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠔⣧⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠁⠶⠶⠶⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣷⡾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⣠⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⠀⠀⠉⠀⠀⠀⠀⠀⠉⠉⣹⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠰⠆


⣿⣿⡿⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⠁⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣹⣁⣚⣃⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢀⡙⡛⣿⠿⢻⣿⣺⡟⢫⣬⡝⣿⣬⣟⠿⣈⢻⠳⡴⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣈⠅⣟⠃⣾⢏⣥⣺⣿⡞⠋⡴⢧⡘⢡⡰⡶⣽⣭⣌⠛⢤⠖⢫⠈⣒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢿⣿⣾⣥⣍⣠⣭⡥⣭⣃⣼⣯⣥⣴⣤⣤⣷⣶⣷⣶⣶⣶⣶⣿⣿⣿⠙⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣽⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⢏⣽⣿⣿⡟⣡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⢿⣿⣿⣙⢿⣇⡣⣿⣿⣿⣿⣿⣬⡝⣿⣿⣿⠉⣥⡸⣥⣾⣿⡿⣗⣩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣼⣿⣿⡻⣿⢠⢽⡇⣿⣼⣿⡏⣭⣿⣿⣿⣟⠺⣧⡼⡗⣿⣿⣟⣿⢁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣷⣾⣾⣿⠈⢧⠔⢰⣾⣿⣿⣿⠷⠬⢻⣿⣿⣧⡀⡌⣖⢋⢹⣿⣿⢞⡊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣻⣿⣿⣿⣿⡤⠴⢀⢻⣿⣿⣿⣶⣟⢿⣿⣿⣿⡇⢠⣬⠅⢺⣿⣿⡿⠑⡇
⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⣿⣿⣟⣿⣿⣿⣾⣙⢶⣸⣻⣿⣿⣿⣟⣿⣿⣿⣿⣿⣷⣯⢘⣿⣾⣿⣇⣿⢟⡇⢀
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣻⣿⣿⣿⣿⣿⣸⣶⣶⣾⣿⣿⣿⣿⣷⣟⣿⣿⣿⣿⣿⣿⣿⣸⣾⣿⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⢀⠰⡌⠀⢴⡶⢶⣿⣻⣿⣿⣿⣿⣯⣿⣭⣯⣿⣿⣿⣿⡷⢘⣹⣿⣿⣿⡯⣽⢿⡿⣿⣿⣿⢿⡿⠥⠋⠀⠀⠈⢂⡞⠇
⠀⠀⠀⠀⠙⠒⠂⠠⠍⣘⠉⠙⠛⠛⠻⠿⣷⣯⣝⣈⣿⣿⣿⣿⣓⣙⣛⣿⣿⣟⣛⣏⣭⣓⣋⣋⣋⣁⣀⠀⣀⠒⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⢄⠐⡂⣬⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠤⢤⣬⣤⠤⢤⡶⢶⣶⣄⠴⣶⣶⣶⣶⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠖⠆⠤⠤⠜⠯⠍⠉⠛⠋⠉⠉⠉⠁⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⣠⢠⣾⢀⡄
⠀⠀⠀⠀⠀⢠⡦⠀⡀⠀⡜⠛⠀⠀⠀⢰⡇⢿⣇⡞⢀
⠀⠀⠀⠀⠀⠋⢤⠀⠛⣾⡟⠆⠀⠀⠀⢸⣃⣇⢸⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠠⠀⠀⠀⠀⢠⣿⡧⠀⠀⠀⠀⠘⡋⠉⠻⣣⣾⣀⡀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢘⡇⠀⠀⠀⠀⠀⠁⠀⢠⠈⠹⠿⠃⠀⠀⠀⠀⡠⠊⠁⣀⠴⠂
⠀⠀⠀⡤⢀⡀⣿⠀⣀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⠀⢀⣼⠋
⠀⠀⠀⠀⣀⠙⡷⣴⢱⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⣄⣤⡐⠀⠀⢠⣼⠉
⠀⠀⠀⢱⠋⣼⣿⢁⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄⠝⠛⠙⠀⠀⠈⠊⢅⣀⣀⣠⠤⠐⠀⠀⠂
⠀⠀⠀⠘⣰⡟⢻⡏⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣄⡵⠂⠀⠀⠀⢤⣤⣤⣬⣿⣿⣧⣀⠀⠀⠀⠈
⠀⠀⠀⢰⡏⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⣏⣹⠛⢿⠷⠤⠄⠀⠀⢀⠠
⠀⠀⠀⢸⢷⣾⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠖⡠⠐⠂⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⡇⠀⡜⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠢⣀
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢁⠈⡃⠀⡃⡾⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠈⠳⢦⣙⣽⠾⠧⠇⠀⠀⠀⠀⠀⠀⣶⣥⡒⠂⡐⢾⣶⢄⣀
⠀⠀⠀⠀⢸⡼⠀⠀⠀⠀⠀⠀⠀⠒⠒⠲⠛⠛⣁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡄⠀⢸⠀⠀⠛⡀
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠁⠉⣴⣯⡆⠀⠀⠀⠀⠀⠰⣀⠿⣿⣦⡟⡂⢴⠈⠖
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⠂⠋⠈⠁⠀⠄⠱⠈⡄
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠰⠅⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠤⡀⠈⠁⠀⠀⠀⣠⠱⢦
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠀⠁⠀⠀⠀⠀⠀⠸⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣶⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠙⠿⢶⣇⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⣸⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣦⠀⠀⢿⣤⣀⠀⠀⠀⠀⣼⠿⠛⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠉⢻⡷⠟⢷⣴⣿⣀⣀⣠⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣦⣄⢸⣧⣀⣠⣿⠉⠉⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠿⠛⠛⠙⠳⣼⡟⠉⢻⣶⣶⠟⠛⠻⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠘⢷⣤⣼⠏⠀⠀⠀⠀⠀⠈⠙⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⣤⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢃⡾⠋⢸⡇⠀⠀⣠⣴⠶⠿⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢡⡿⠁⠀⢸⡇⢀⣾⠋⢀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣾⠃⠀⢀⡿⠀⣾⠇⠀⠸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣾⠏⠀⠀⣾⠁⢸⡟⠀⠀⠀⠀⠀⠀⠻⣷⣤⣤⡄⠀⠀⠀⣾⠛⠻⣧⠀⠀⠀⣴⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠁⠀⠀⣸⠃⠀⣼⡇⠀⠀⠀⠀⠀⠆⠀⠈⠉⢻⣇⠀⠀⠀⣿⠀⠀⢻⡄⠀⠀⣿⠀⠹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⡀⠀⢀⡏⠀⠀⣿⠃⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⣿⡇⠀⢸⡇⠀⠀⣿⠀⠀⢿⣧
⠀⠀⠀⠀⢀⣤⣶⢿⣿⡇⢠⡇⠀⣸⠁⠀⢰⡿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡗⠀⣿⡇⠀⠀⢿⡄⠀⣿⡄⠷⢸⣿⣶⣤⡀
⠀⠀⣠⣾⠟⠉⠀⢸⣿⠁⠀⠀⠀⢿⣦⣤⡾⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣿⠁⠀⣿⠁⠀⡀⠈⣿⠀⢹⣧⠀⢸⣿⠂⠈⠻⣶⡄
⠀⢰⣿⡏⠀⠀⠀⢸⣿⣆⠀⠀⡆⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣧⠀⢿⡆⠀⣧⠀⠸⣇⠀⢻⣷⣿⡿⠀⠀⠀⠘⣿⡆
⠀⢹⣿⡆⠀⠀⠀⠈⠻⣿⣷⣦⣤⣀⣀⠀⠀⣰⠀⠀⠘⠀⠀⠀⠉⠀⠀⠀⢠⡀⠀⣿⡀⠈⣿⡀⠀⣀⣀⣿⣾⣿⡿⠛⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣿⣿⣾⣿⡿⠿⠿⢻⡿⠛⠀⠀⠀⠀⠀⢀⣴⣿⠟
⠀⠀⠀⠈⠛⢿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠿⣿⣦⣄⠀⠀⠀⠀⢿⣧⣀⡀⢀⣠⣤⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣶⣶⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣃⣀⣀⣤⣤⣿⣿⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣚⡛⠛⠛⢛⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⢀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⢣⡀⣆⠀⣴⡿⠛⢣⠴⢶⡄⢀⣠⠂⡔⠻⠳⡀⢠⡀⠀⣠⡶⢠⠛⠻⣶⡄⠀⡆⢰
⢄⠀⠀⠀⠀⠈⠋⠙⠧⣽⣆⡘⠦⣴⣾⡄⠀⣿⢯⣥⡾⢅⡀⢈⣹⣦⡹⣷⡇⠀⣼⣦⣄⠌⡀⣈⣠⠿⠋⠙⠁⠀⠀⠀⢀
⠘⠧⠰⣄⠀⠀⠀⠀⠀⠈⢻⣥⣀⠙⠟⠁⢀⠀⠀⣰⠆⠈⢩⠉⠀⢴⠀⠀⠀⠀⠻⠟⢃⣀⣿⠋⠀⠀⠀⠀⠀⢀⣴⠂⠞
⠀⠀⠀⢘⣷⡄⠠⢴⣟⡀⠀⠈⠉⢉⣜⣛⣛⣿⠟⠁⠀⠀⠿⠇⠀⠈⠻⣿⣟⠛⢳⣌⠉⠉⠁⠀⢘⣵⠤⠀⣰⣿⡁
⠀⠀⠀⠈⠁⢀⣠⠜⠻⣟⠻⢦⡔⣱⠏⣹⣋⣄⣈⡙⠭⠃⠂⠐⠭⠝⣃⣀⣍⣿⡉⣯⠱⣠⡶⢛⣿⠛⠦⣄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠾⠁⠯⠴⠿⠭⠭⠭⠭⠤⠤⠯⠥⠤⠭⠭⠭⠭⠽⠷⠼⠇⠽⠶⠟⠁











⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⣖⢲⡖⢶⣖⣒⣒⣒⣒⠒⢲⢲⠒⠒⣒⣒⣒⣒⣶⠖⣲⢒⡶⣤⡀
⠀⠀⠀⢀⡀⠀⠙⠲⣤⣿⣵⠾⠛⢄⣻⣈⣿⣙⠉⢩⡔⢒⠄⠡⢠⣢⣌⠉⠙⣩⣏⣰⢏⠜⠷⣮⣽⣦⡔⠋⠀⢀⡀
⠀⠀⠀⢨⣿⠏⠀⠒⢟⡅⠀⢀⣀⣀⡙⢥⣭⣽⣷⣦⠀⠀⢰⡶⠀⠀⢀⣴⣿⣭⣭⡝⢁⣀⡀⠀⠈⡻⠓⠂⠙⢿⡍
⢀⡴⠡⠟⠁⠀⠀⠀⠀⠀⣠⣟⠉⢡⣴⣆⠀⡀⠀⠀⠗⠀⣈⣃⣀⠰⠏⠀⠉⠁⢀⣴⣍⠉⢹⣆⡀⠀⠀⠀⠀⠀⠙⠆⢲
⠘⠁⠀⠀⠀⠀⣄⣠⣶⢻⡝⠈⡵⠛⠿⡟⠀⢸⢿⣜⠻⣏⠁⠈⢑⡾⢓⣲⣟⠀⠘⡿⠟⠲⡈⠙⡝⢲⣄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⢐⠃⠸⠀⠘⢿⣦⣤⠃⠶⠋⠀⠈⠃⠈⢦⣆⠌⠠⠋⠁⠘⠶⠖⢣⣠⣾⠟⠀⠹⠈⢃⠈
⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠞
⣿⣿⠟⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠙⡄⣠⠞⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢀⣿⠃⠀⠀⠀⠆
⠀⠀⠀⣠⠄⠠⣀⣶⠀⠀⣾⠁⠀⠀⠀⣾⡇⠀⠀⠀⡸
⠀⢀⡾⠁⠀⠀⢹⡇⠀⢠⡟⠀⠀⢀⠞⣿⠀⠀⠀⡰⠁
⢀⣾⠁⠀⠀⠀⣾⠁⠀⢸⡇⠀⢠⠊⢰⡏⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⣸⡇⠀⠀⠀⡸⣿⠀⠀⣼⣇⠔⠁⠀⢸⣇⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡆
⣿⠁⠀⠀⠰⢸⣿⠀⠀⢸⡇⠀⠀⠀⢸⡏⠀⠀⠀⢀⣿⠀⠀⢠⡄⠀⠀⠹⠷⢤⣤⠀⠀⠀⣰⡆⠀⠀⠀⣰⡆⢀⡴⢲⡄⠀⠀⠀⣾⠀⡸⣇
⣿⠀⠀⢠⠃⢸⣿⠀⠀⠘⣧⡀⠀⡴⠋⢿⡀⠀⣠⠋⣿⠀⢀⢿⣇⣀⠔⠁⠀⢸⡏⠀⠀⡰⠙⠷⠤⠤⠊⢹⣇⠎⠀⢸⡇⠀⠀⢰⠙⠒⠁⣿
⠻⣀⡠⠃⠀⢸⣿⠀⠀⠀⠈⠙⠉⠀⠀⠀⠙⠋⠁⠀⠈⠛⠉⠀⠈⠁⠀⠀⠀⠈⢧⡤⠞⠁⠀⠀⠀⠀⠀⠘⡿⠀⠀⠈⢧⣀⡴⠁⠀⣠⠔⣿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠎⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⢀⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⢀⣀⣠⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡿⠟⣋⣤⣭⣭⣭⣭⣭⣙⡛⠿⣿⣿⣶⣄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣿⣿⣿⣿⡿⢋⣴⣿⡿⠿⠛⠛⠛⠛⠛⠿⢿⣷⣄⠙⠻⣿⣿⣦⡙⢦⡠⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣾⣿⣿⣿⡟⢀⣾⠟⠉⣀⣤⣶⣶⣶⣶⣦⣄⡀⠈⠛⢧⡀⠈⠙⢿⣿⣎⢻⣦⡙⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡻⣿⣿⣿⡇⢸⠏⢀⣾⣿⠿⢿⣿⣿⣿⣿⣿⣿⣦⡤⣤⣑⠈⣷⣤⡙⢿⣿⣿⣷⡸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⢿⣿⣷⣌⠛⠿⠷⠼⠀⠛⠁⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⡌⣿⣷⣼⣿⣿⣌⢻⣿⣿⡗⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦⣍⣛⣛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⡎⣿⣿⢧⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⡗⣿⠏⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⠓⠁⣰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏⠀⠀⠠⠤⠶⠶⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢸⣿⣿⣿⠏⠀⣴⣿⠟⣹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠻⡇⠀⠀⢀⣀⣀⣀⠀⠀⠉⠛⠿⣿⣦⡄⠀⠀⠀⢸⣿⣿⣿⣿⡇⣼⣿⣿⠃⠴⠛⣫⠔⢠⣿⠟⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠙⢿⣿⣿⣷⣶⣤⡀⠈⠙⠃⠀⠀⠀⣸⣿⣿⣿⡟⢀⣿⣿⢁⡞⣠⣾⠋⣰⠟⠹⣿⣷⣾⢃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⠀⠀⠀⠺⢾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁⢸⣿⡇⢸⣷⣿⣧⡀⠠⣤⣤⡾⠟⣡⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠷⠶⠶⠒⠀⠀⠀⠀⠀⣿⣿⣿⢁⡆⠸⣿⣇⠸⣿⣿⡇⠻⣷⣶⣶⣶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣄⠈⠻⠆⣿⣿⡇⠀⠀⠉⢹⣏⠁⢀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡁⠸⣿⣿⣿⣦⣄⠸⣿⣿⡄⠀⠀⠘⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⠀⠙⢿⣿⣿⣿⣷⡹⣿⣿⣶⣤⣴⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠀⠀⠙⠻⣷⣤⣀⠙⢿⣿⣿⣷⣹⣿⣿⡿⠿⠿⠛⠁⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⣀⣤⣤⣄⡐⢮⣍⡁⠀⠹⣿⣿⣿⣿⣿⢧⣤⣀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢾⣿⣿⣦⠀⠘⣿⣿⣿⡿⠀⣿⣿⣿⣆⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠀⠀⠙⠻⠿⣿⡿⣿⠿⠋⠀⣸⣿⣿⣿⡇⠀⣿⣿⡿⠁⢰⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠘⣷⣶⣿⣿⣿⣿⣿⠃⢀⣿⡿⠁⣠⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣶⣶⣶⡶⠃⠈⢉⣭⣿⡿⠟⠁⣠⠾⠋⣠⣴⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠿⠋⠁⠀⠀⢠⡿⠋⠁⠀⠀⠉⠀⠔⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠈⢿⣿⣿⠃⠀⠀⠀⠀⠀⠘⠷⠚
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣴⣶⣶⣶⣶⣶⡾⠟⠋⠀⠀⠀⠻⣿
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆
⠀⠀⢠⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠹
⠀⠀⢸⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠀⠃
⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠇⠀⠁
⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⢰⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠉⠙⢶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⣴⣀⠀⠀⢠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⢓⠳⣤⠀⡀⠀⠈⠛⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢻⣿⣆⣿⣊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡘⠷⣧⣴⡂⣀⠻⡇⠀⠀⠀⠀⠘⠠⡀⠀⠘⠶⢦⣴⣿⡧⠔⠁⠀⠀⢳⣸⣿⣿⣥⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠒⠛⠻⣥⣈⣻⣿⣿⡾⠀⢠⠤⠀⠀⠀⠀⠀⠀⠂⣾⣿⣿⣿⣗⠀⠀⠀⠀⢹⣿⣃⡠⠉⠙⠋
⠀⠀⠀⠀⠀⢠⢁⡤⠴⠆⠶⠶⠿⣿⠿⣿⣧⡠⣷⠀⢀⠀⠀⠁⠀⣠⣤⠶⠋⠀⠀⠛⢆⡀⠀⣤⣾⣿⢻⣍⢛⠠⢄
⠀⠀⠀⠀⠀⠈⠈⠒⠤⠀⢀⣤⠤⠾⣄⠀⠉⠧⢿⣄⡽⠒⠀⠂⠀⠑⢆⠀⠀⠀⠀⠀⠈⠁⠀⠘⠿⠿⠏⠉⠙⠷⠬⢵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⣿⣟⢠⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⡀⢀⠀⡀⢀⠄⡄⡄
⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡝⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢣⠈⣄⠀⠀⠀⡞⠀⡄
⠀⠀⠀⠀⠁⢀⠀⢀⠀⢀⠄⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠈⠀⣼⡀
⠀⠀⣀⠀⠀⠈⠁⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⢄⠀⠀⠀⢨⣀⣴⣿⠷⣠⠀⣠⠔
⠀⠈⠀⠀⣀⠀⠀⢧⠀⠀⠀⠀⠐⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠄⢀⣄⠤⣙⣿⣿⣿⣷⣶⡈⠀⠀⠂
⠀⢀⡠⠜⠁⠀⠀⠼⡖⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⡇⣆⡏⠀⡀⠀⠀⠀⠐⠂⠀⠉⠿⡿⠟⠿⠿⣿⡂
⠀⠸⠀⠀⣄⠍⠀⠀⠉⠀⡀⠀⠀⠲⠄⠀⠀⠀⢀⡃⢀⠹⣿⣠⣨⣾⡶⡄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠁⠀⠀⢀⠀⢁⠉⠚⠉⠀⠀⠀⠀⠀⠰⢄⡪⣤⣰⢳⣿⣿⠃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠠⣄⠈⣑⢿⡻⡘⣿⣗⡼⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠰⡇⠀⢸⠀⠀⠁
⠀⠀⠀⠀⢀⠀⠀⠀⡰⠀⠀⠈⠀⠀⠀⠀⠀⠀⢄⡀⠘⣿⠇⢇⠘⢻⣃⠌⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢠⠃⡀
⠀⠀⠀⠀⣤⡀⠆⠸⠀⢐⡤⠂⢐⠀⠀⠀⠀⠀⠠⠄⠉⠙⢳⠄⢧⡘⡵⢊⠤⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠠⣸⣿⣰⠁⡠
⠀⠀⠂⠀⠨⣍⠳⡟⢤⣠⣦⠄⠚⠚⠀⠘⠁⠀⠀⠀⠄⠚⣉⡥⠄⢣⣘⣕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡟⣿⣇⠎
⠀⠀⠀⠀⢢⣝⣦⣶⠈⣿⣡⣴⣷⡶⠂⢀⠀⡀⠀⠀⠀⠈⠐⠚⠿⠶⣛⠅⠀⠀⣀⠀⠀⠀⠀⠀⣀⡀⢈⣻⣷⣿⠃
⠠⠀⠀⠀⠀⢉⡽⠿⠿⣿⡿⠑⠀⠀⠀⣀⡠⠊⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠁⢀⡀⠀⠀⠀⠬⢝⣿⡏
⠀⠀⠀⠀⠀⠨⣀⠀⠠⡿⠛⣦⣤⣴⣶⣍⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠐⠶⠦⠤⠴⠾⢿⡦⠂
⠀⠀⠀⠀⠀⠀⣦⠀⠀⣏⠉⣡⠋⠯⠙⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⣤⣤⣤⣤⣤⣤⣬⣤⣤⣥⣤⣤⣤⣤⣤⣦⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤
⣿⣿⣁⣍⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢶⣶⣿⣿⣿⣿⣷⡶⠆⠀⠀⠀⠀⠀⣀⣠⣤⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⢀⣼⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠒⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃⠈⠁⠚⠛⠛⠓
⠀⠀⠀⠀⠀⠘⠉⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀
⠀⠀⠀⠀⠀⠴⣆⠀⠀⠀⠁⣠⣶⡀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⢀⣶⣄⢀⣠⣦⡀
⠀⠀⠀⠀⠀⠀⠹⣿⣶⣤⣾⣿⠋⠀⠀⠀⠀⠀⠀⠸⣿⣷⣶⣶⣿⡟⠁⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⡿⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣤⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢐⡅⠀⢀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠀⠀⠀⠀⠀⠘⣿⡟⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢒⣶⣿⣶⡂⠀⠀⢤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢹⠏⢻⠉⠉⠷⠈⡿⠋⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠀⠈⠐⡀⠀⠈⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣾⢿⠓⠀⠀⠀⠀⢠⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⢠⡀⠀⠀⠀⠀⣆⡀⣠⣷⣴⣦⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣴⣶⣽⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠑⠒⠓⠒⠉⠏⠉⠍⠉⢉⡙⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠒⠂⠂⣉⣁⣀⣤⡴⠒⠚⠀⠀⢠⢤⠖⠂⠁
⠀⠀⠀⠀⠀⢀⣀⣀⣢⣤⠤⠶⠶⠶⠾⠿⠟⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠈⠑⣶⣤⢤⠈⠁
⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠾⠿⠛⠛⠉⢀⡸⠀⠀⠀⠀⠀⣀⠔
⠀⠀⣀⠀⠤⠤⢄⡠⠤⠶⠞⣛⣩⣭⣤⠶⠶⡶⡶⠶⠶⢶⢦⣤⣤⣤⣤⣤⠤⢤⡶⢞⣉⣦⠀⣀⠠⠔⠈⠠
⠠⠠⠀⠒⠒⠒⠒⠚⠛⠉⠉⠉⠀⣀⣀⣀⡤⠤⠒⠒⠴⠒⠉⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠉⠀⣦⣤⣤⣶⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣡⠤⠂⠀⠀⣀⡠⠤⠐⠀⠘⠇⠀⠀⠀⠀⠀⡠⢎⣤⡷⢖⣸⡟⢈⣩⡁
⠀⠀⠀⠀⠀⠉⠉⠉⠁⠚⠛⠉⠉⠀⠠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠈⣑⡫⠔⠋⠀⠉⠁⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⡿⠻⣧⣄⣀⡀⢀⣄⣠⣾⠟⢿⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⠀⠀⣀⡈⠉⢙⣿⣿⡉⠉⢁⣀⠀⠈⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣀⡀⠙⠻⣶⡟⠋⠙⢻⣶⠟⠋⢀⣀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣹⣿⣆⣀⣿⣆⠀⠀⣸⣿⣀⣴⣿⣏⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠛⠉⠉⠙⠉⠉⠛⠿⠿⠛⠉⠙⠋⠉⠉⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⢀⣴⡶⠶⣶⣄⠀⠀⠀⠀⠀⠀⣤⣶⠶⢶⣤⡀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⢀⣠⡿⠋⠀⠀⠀⠙⣷⣄⠀⠀⣠⣾⠋⠀⠀⠀⠙⢿⣄⡀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠛⠛⠋⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠺⠂⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣈⣿⣦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣴⣿⣁⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢰⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡶⢀⣬⣿⣶⣄⣀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⡿⣧⣿⣿⣿⣭⡉⠉⠀⠀⠀⢠⣄⣠⣤⣤⣤⣤⣤⣤⡄⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣰⡀⠈⠉⠀⠉⠻⣿⠻⣿⣿⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣟⣧⣸⣿⣿⣿⣿⣿⠇⣶⣦⡀⠰⣾⠛⢣⡄⢻⣷⣶⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⡈⢙⣿⣿⣿⣿⣷⣾⡿⠧⠀⠘⠛⠛⠀⠹⠇⠘⠿⣾⣿⣿⣿⡏⠙⠻⡟
⠀⠀⠀⠀⠀⠀⠸⣷⣦⣀⠀⣿⣿⣿⠿⢿⣿⣿⣿⣿⣄⡀⠀⠒⢤⡤⠀⠀⠀⠀⠀⠈⠛⢿⣿⣉⡄
⠀⠀⠀⠀⠀⠀⢀⣈⣿⣿⣷⣿⣿⣿⣦⡈⣉⣩⣇⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡟⠉⠀⠙⢁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢉⣿⣷⣶⣦⣀⠿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣹⣿⣤⣤⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⠁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠆⠈⠻⣿⡟⢿⣿⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠉⠁⣄⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠛⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⣀⡛⠉⠀⢠⣶⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⣀⣠⠶⣶⡶⢿⠿⠂⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡀⠹⣿⣀⠀⠀⠸⠋⣹⡿⢿⣿⣿⣿⡆⠀⠟⠋⠛⠒⣾⣷⣿⡷⠶⠆⠀⠀⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠈⡉⠀⠀⠀⣼⡿⠇⠀⣽⣿⠛⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣷⣼⣷⠀⢀⣀⠸⣿⡀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⠋⣻⣟⣀⣈⣡⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣹⣿⣿⣿⣤⣽⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣿⣿⣿⠋⣿⠿⠃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠁⠈⠉⠁⠈⠉⠁⠀⠀⠀⠀⠉



⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣿⣿⣷⣶⣶⣄⠀⠀⠀⠀⠀⣴⡄⣠⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣶⡀⡶⣤⣄⠀⠀⢀
⣿⣿⣿⣿⠁⠀⣸⣿⠿⠻⣿⣿⠿⠻⣿⣿⣷⠆⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣶⣿⣿⣧⡀⢀⣀
⣿⣿⠿⣿⣶⣀⠈⠁⠀⢤⣄⣡⣄⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣆
⣿⣿⣶⣿⠟⠻⠋⠐⣶⡾⠋⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡏⣉⠀⠂⢠⡄⠻⢷⡄⢀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡏
⣿⣿⣠⡏⢀⣴⡆⠀⢀⣤⣄⣘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⣿⣿⣿⣿⣿⣿⣷⡸⠿⠉⠛⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣯⡀⠻⠏⣸⠁
⣶⣟⣿⣿⣿⣿⠙⣿⣧⣀⡀⠀⣀⣀⣠⣤⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⣿⣿⣿⣿⠟⠀⠈⢻⡄
⡌⠛⢿⣿⣿⣷⣼⣿⣿⣿⣿⠿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠀⠀⠈⠻⣿⣿⣿⡴⠃⢠⡾
⣀⠀⠀⠙⢿⣿⣿⠋⢿⣿⣷⣶⣿⣿⣿⡿⠡⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣇⢀⣾⠇
⠙⠣⡄⠀⠀⢙⣿⣿⠟⠿⠿⠻⢿⠿⠿⢇⣀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⡇
⡀⠀⠀⠂⠀⠈⠹⠟⠀⣠⣤⠀⠀⠀⠀⠀⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠿⠛
⣹⣶⣀⠀⠀⢠⡄⠀⠘⣭⣉⠿⠟⠀⢀⠀⠀⢀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣤⣾⣗⣀⡀
⣿⣹⣿⣧⡀⠈⠋⢀⣈⡉⠟⠀⣴⠀⠀⣠⣶⣾⡿⠿⢿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⡿⠻⢿⣿⣿⣿⣍⣿⡿⠷⠄
⣿⣿⣿⣿⣁⡀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠘⠋⢀⣴⣿⣿⣿⣿⣟⠿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⢻⣿⣦⣾⣶⣶⣶⣿⣏⣀⣿⣿⣁⣤⣤
⣿⣿⣿⣿⣿⣿⠆⠀⠘⣿⣷⣄⠀⣰⠟⠀⠀⣼⠃⠀⠀⣤⣌⢻⣶⣿⣿⣿⣿⣿⠿⠛⠀⢘⣛⠧⠈⠉⠉⠀⠀⠀⣁⣸⣿⣋⢼⣿⣿⣷⠶
⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠹⠿⠂⠁⠀⠀⡆⠀⢰⣾⣿⣿⡟⢀⠉⠀⠹⣿⣟⣿⣶⠿⠇⠀⠀⠀⠀⠀⢠⣤⣶⣾⣿⡉⣿⣿⣼⠟⣯⣄⣤
⣿⣿⣿⣿⣿⠿⠟⢰⡆⠀⢀⣠⠀⠀⠀⠀⣀⡀⢾⣿⣿⠟⢁⡼⠀⢸⣿⣿⣿⠋⠉⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⠿⣿⣿⣿⣿⣷⢿⣿⡟
⣿⣿⣿⣿⣿⡖⠒⣿⣷⠀⢸⣿⣦⠀⠀⠀⠀⠀⠀⣽⣧⣴⣾⠀⠀⢠⣿⣿⣤⣶⣾⣷⣶⣿⣿⣿⣿⣭⣿⣿⣿⣿⣶⣿⣿⣿⣿⡟⢛⣉⡀
⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣾⣿⣿⣄⠀⠀⣸⣷⡀⠘⣿⣿⣁⣀⡂⢠⣿⡿⠻⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⣿⣿
⠛⠋⠛⠛⠛⠋⠛⠛⠛⠛⠛⠛⠛⠋⠀⠁⠙⠛⠁⠀⠛⠛⠛⠛⠁⠀⠉⠉⠁⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠙⠛⠛⠛⠛⠛⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠾⢦⣀⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⢧⣶⣿⣶⣤⣄⡀⠉⠀⠻⠿⠃⠀⣀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣯⣿⡟⠋⠹⢷⣄⣰⣶⣶⣾⣿⣿⣷⢆⣀⠸⠷⣨⣧⡶
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⠇⣀⣽⣿⣿⣯⡝⠛⣛⡻⠿⢿⣧⠸⢿⡏⠛
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠛⣿⡛⢿⣁⣠⡿⣿⣿⣿⣿⣿⣿⣿⣁⠀⠀⢁⡀⠀⠀⠀⠀⠀⠻⢿⣿⣦⣶⣲⣶⣾⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠻⢿⠿⠿⣿⣿⣿⣌⣿⣿⣾⣿⣿⣷⣿⣿⣿⠛⢿⣋⣉⣛⣤⣤⣌⠁⠀⢀⡀⣀⠀⠀⣠⣶⡛⠛⠛⠉⠹⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣟⢡⣾⣿⣿⣿⣿⣿⣿⣿⣆⠻⣿⣿⣿⣿⣿⣷⡄⣈⣿⠋⠀⠀⠉⠙⠳⠀⠀⠀⠀⠀⠹⣿⡟⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢿⣿⡟⣿⣿⠛⠛⣏⣉⣷⣏⣿⡿⢿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣠⡿⢿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣤⠞⢻⣿⠟⠁⡾⠟⢩⣿⣿⣿⣿⣾⡿⠛⠿⣿⣿⡿⠿⠉⣿⣿⣿⣿⡗⠀⠀⠀⣿⣿⠀⠀⠉⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇⠀⠉⢰⣿⡿⣷⣿⣗⣶⣮⣼⡿⣿⣁⠁⣀⣴⣾⣿⣿⡀⠀⠀⢿⡿⢿⣿⠟⠀⠀⠛⢻⣿⡇⠚⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣦⣤⣀⣀⣵⣿⣿⣿⣿⣿⣟⣿⠿⢿⣿⣿⣛⣿⣿⠿⠿⠿⢀⠀⣠⠀⠀⠸⠗⠀⠀⠈⠙⣷⠀⠀⠙
⠀⠀⠀⢠⣦⣄⡀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣿⣿⡿⠋⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈
⠀⠀⠀⠈⢿⣿⣿⣷⣶⣄⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠙⠛⠒⢾⣿⡆⢠⣦
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣶⣄⡸⣿⣿⣿⣿⣿⡀⠀⠀⣿⡿⠛⠛⢿⣿⠋⠹⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⣦⣄⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣷⣤⣀⣀⡀⠀⢠⣤⣤⣄⡀⠀⠉⠉
⠀⠀⠀⠀⠙⢿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⠟⠉⠉⠹⣿⠟⠋⠉⠙⠁
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠏⣀⡀⠘⢢⡴⣿⣦⡀⠀⢠⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢀⡔⠉⠙⡆
⠀⠀⠀⠀⠀⠠⠠⢄⠠⠀⠀⠀⡞⠀⠀⠈⠉⢑⠀⠀⠀⠒⠤⡤⠂⢱⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠢⡀⠀⠁⠀⢄⣼⡠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣨⠔⠁⠀⢰
⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⢐⣿⡭⠏⠉⠉⠁⡆⢈⠁⠢⡀⠀⠀⢠⠃⠀⠀⠀⠈⠢⢤
⠀⠀⠀⠀⠀⠙⠂⠐⠀⡠⠊⠀⢹⡄⠀⢠⠒⢁⠜⠀⠀⠈⠲⣤⣦⣀⣤⡤⠐⠒⠒⠈
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠈⡛⣆⢀⡱⠘⠤⣤⢄⣤⠶⠊⢻⡁⠀⠈⠉⠑⢢
⠀⠀⠀⠀⠀⠀⠀⠑⠚⠉⠳⣀⡰⠀⠈⢦⠀⠀⠀⡴⠋⠀⠀⠀⢸⢱⠀⠀⢀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠑⢄⠀⠑⠒⠢⡀⠀⡼⠀⢇⠀⡯⠑⡄⠈⢦⡀
⠀⠀⠀⠀⠀⡰⠋⠉⠆⠀⡠⠄⠠⡀⠀⠀⠈⢄⠙⢦⡀⠀⠙⠚⠁⠀⠀⡉⠁⠀⠘⢆⠀⢋⠢⡀
⠀⠀⠀⠁⠉⠃⠀⠀⠘⢸⠁⠀⠀⠑⠤⠄⡀⠄⠁⢀⠈⠓⢤⡀⠀⠀⠀⠑⢀⠀⠀⠈⢧⠈⢆⠘⢆
⠀⡀⢀⡱⡀⠀⠀⠀⢰⣼⠀⠀⠀⠀⠀⢀⡔⠀⠀⡀⠀⠂⠄⡉⠳⢤⡀⠀⢀⠀⠢⡀⠈⢧⠘⡀⠀⢣
⠘⡀⠀⠀⠙⠢⡀⠀⠈⡇⠀⢀⡠⠔⠊⠁⠀⠀⠀⠈⠢⡀⠀⠀⠑⠢⢝⠦⣀⢢⠀⠈⢄⠘⡆⢡⠀⠀⠃
⠀⢱⡄⠀⠀⠀⠈⠑⣶⢿⣖⠛⠀⠐⠒⠒⠤⡀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠈⠪⡳⢧⡀⠀⠡⣽⠸⠀⠀⠘
⠀⡎⠀⠀⠀⣀⡤⠖⠉⠉⠧⠀⠀⠀⠀⠀⣠⠞⠐⠢⠤⣀⡀⠀⠑⢄⡀⠀⠀⠈⠢⡑⡄⠀⢹⠀⠀⠀⡆
⠀⠉⠉⠉⡩⠊⠀⠀⠀⠀⠠⠀⠀⠀⠀⢻⠀⢄⡀⡀⠀⠀⠉⠉⠒⠲⠬⢦⣄⡀⠀⠘⢌⠂⠀⢆⠀⠰
⠀⠀⠀⠀⠀⠀⡠⠶⡄⠀⠀⠆⠀⠀⠠⠜⠀⠀⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠶⢤⡀⠑⡅⠸⡆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⠀⠀⠐⠀⠀⠀⠠⠄⠈⣙⣾⣄⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠉⠉⠛⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡎⠀⠀⠀⠀⡤⢀⣀⡀
⠀⠈⣈⣤⣤⣄⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣨⡝⢆⢀⡀⠀⠀⢠⡆⠀⠀⢈⣄⡀
⠀⠀⠉⠀⠰⠷⠄⠀⠀⠀⠈⠙⡏⢀⡅⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠒⣀⡈⠩⢓⡏⡈⠀⠀⠀⢠⡀⠀⠕
⠀⠀⠀⢈⣵⡆⠀⠀⠀⠀⠀⠀⠘⣾⡇⠀⠀⠀⠀⠀⠡⠀⡀⠀⠈⠐⢩⢠⣶⠀⠉⠁⠀⠀⠀⠘⠃⠂⠒
⠀⠀⢈⡧⠄⣠⡔⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⡎⡔⠀⠀⣁⣤⠄⢅⠉⠁⠀⠀⠀⠀⠰⢢⠀⡀⠙⠷
⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⡀⠀⡿⠁⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾
⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠈⣽⣶⠃⠀⠀⠀⠀⣾⡗⠀⠀⠀⠀⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨
⠀⠀⢀⠱⢟⣛⠀⠀⠀⠀⠈⣱⡀⠀⠀⠀⠀⠄⠀⢀⡄⠀⠀⠀⠀⠀⠀⢈⣻⡩⢊⠁⠄⠀⠀⠀⠠⡀⠠⠒
⠀⠀⠈⣇⠀⠉⡤⠀⢀⣶⣿⡿⠅⠀⠀⠉⠉⢀⠀⠰⠀⠆⡂⠀⠀⡰⠀⠾⢿⡿⠀⠀⠀⠀⠀⠉⠂⠀⢀⠈
⠀⠀⠀⢹⣶⡟⠁⠀⠘⢛⣿⣿⡆⠀⠀⠀⠀⠃⠀⠀⠀⠀⢳⠀⠀⠀⢰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣌⡤⢀
⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⡀⠀⠘⠀⠀⡄⠀⠀⠀⠀⠀⢱⡁⠀⠀⠀⠀⠀⠀⠀⠉⡃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠛⠻⢶⣤⠀⠀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣷⣄⡀⠀⠀⢹⣧⣼⠟⠉⠀⣩⣿⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠿⠛⠛⠋⠉⠛⠿⠶⠶⠿⠿⠶⠾⠟⠋⠉⠛⠛⠻⠿⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠉⠙⠛⠷⢶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⠶⠾⠟⠛⠉⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⡿⠁⢀⣤⣶⣦⡀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣻⣿⠶⣶⣦⣤⣀⣀
⠀⠀⢀⣠⣴⠾⠟⠋⠉⠀⣰⡿⠿⠶⠟⠋⠀⠹⣷⡀⣠⡾⠟⠛⠻⣦⠀⠀⠀⠀⣠⡾⠛⠙⠻⢷⣄⠀⠀⠀⣀⣴⠟⣿⡄⠀⠀⠉⠉⠛⠿⢶⣦⣄⡀
⢀⣴⠟⠋⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠙⢷⣤⣤⡾⠋⠀⠀⠀⠀⠀⠙⠻⠿⠟⠛⠁⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄
⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿
⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠛⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠷⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣶⣶⡶⠶⠿⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠋
⠀⠀⠀⠉⠛⠷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠷⢶⣶⣤⣤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣴⣶⠶⠾⠟⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠋⣉⣷⣖⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠘⠟⠀⢬⣾⡋⠁⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⠔⣰⠇⣰⠒⠛⠀⠀⠽⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⡖⠀⠀⠀⠈⠉⠠⡄⠀⣀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠎⡻⠁⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⠻⣾⣳⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢰⣷⣖⠃⠀⠰⡿⣏⣊⠀⠀⠌⢻⡧⠨⢻⣷⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣭⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⣴⠎⢻⣧⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢽⣿⢰⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠶⠓⡦⠀⠀⠀⠀⠀⠀⠀⠉⣺⡙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢢⢰⡇⠀⠀⠉⠀⠀⠀⠀⢠⡖⣦⢔⣿⡟⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢸⣼⣇⣤⠀⠀⠀⡀⠀⠀⠀⡠⠎⡸⠀⠀⠈⢑⡄⠀⠄⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⣿⣿⣽⣿⡆⠀⠰⠟⢛⠂⠉⢀⡔⠁⠀⠤⠚⠉⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⢛⣻⡟⠁⠀⠀⠠⠇⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠏⡻⠁⠀⠀⠀⠼⠀⠀⠀⠀⠀⢀⡆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣿⣶⢏⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣵⡾⠟⠀⠀⠐⠛⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⣿⣿⡘⠀⠀⠀⢰⡋⠀⠀⠀⠀⠀⠸⠟⠉⠀⠀⠀⠀⣀⡄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⢸⠃⠀⠀⠠⡀⠰⡀⠀⠀⢓⣶⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠠⢄⣀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⣄⠀⠀⠀⠁⠀⠀⠀⠂⢈⣙⠻⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣀⣠⡜⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠊⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢠⠟⢠⠀⠀⠀⠉⠙⢯⣀⢦⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠄⡀⢁⣄⠲⣽⣷⠦⣿⢳⣔⠏
⠀⠀⠀⠀⠉⠉⠙⠛⠋⠙⠋⠉⠉⠋⠐⠀⠀⣀⣇⠀⠀⠀⠀⠀⡙⣦⠙⢷⠀⠹⡷⢻⠆⠐⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡀⠀⠀⠀⠀⠁⣈⣠⣬⣇⣀⠀⠀⣧⢦⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠹⣷⡿⣟⣿⣿⣄⠰⣻⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠹⣯⠙⢿⣿⣿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣴⠀⢰⠂⢰⠀⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢀⠇⠄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠀⠀⠀⠀⠀⠀⢰

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣄⠐⣶⣤⡄⠀⠀⣀⢰⡶⣾⣶⠀⠀⠀⢠⣤⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠀⢀⣆⡀
⠀⠀⠀⠀⠀⠀⣼⡞⠀⠀⠀⠀⡀⢿⣼⢟⣿⣛⡆⠀⣤⢹⣼⣿⣿⡩⠄⣠⣴⣿⣿⣿⣿⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⣿⣿⡿⡃⠀⠀⠐⢿⣷⡄
⠀⠀⠀⠀⠀⠲⣿⠂⠀⠀⠀⠀⠻⣾⣫⣾⠿⢦⠀⢰⣸⣿⣿⣽⣿⡗⠀⣿⣿⣿⣿⣿⣷⣿⣿⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣾⣯⣴⡶⠂⠀⠈⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡟⢯⠷⠀⠀⠀⠉⠙⠿⠟⠃⠀⠀⠻⢿⣿⣧⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠙⣿⡓⠀⠀⠀⠠⣾⣿⡇
⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⡡⠄⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠹⢿
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⢠⣿⠂⠀⠀⠀⠀⢀⡴⡛⠈⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⡀⣄⣀⠀⠀⢠⣄⠀⠀⠀⠀⣀
⠀⠀⠸⢿⣿⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢷⣦⣿⡟⢁⣴⡿⠀⠀⢀⠎⡀⠁⠀⡀⠄⠀⢸⣿⡆⠀⠉⠀⠠⣉⣻⡥⢆⡀⠘⣿⣷⠀⣠⣾⣿⠀⠀⠀⢀
⠀⠀⢻⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⢬⣷⡿⣋⣤⡤⠀⠀⠘⠁⠀⠀⠀⠀⠀⠒⠓⠒⠀⠀⢐⠲⠶⣽⠋⠩⠀⠀⠈⠛⠃⣿⣿⠟
⠀⠀⠻⣿⣿⣿⡗⠀⣤⣤⠀⣼⣷⠀⣠⣤⠸⠿⡶⠦⣿⡴⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠻⠀⠀⠠⠭⢭⣯⠴⠞⣋⠀⠀⠀⠀⠈⠁⠀⢀⠀⠀⠁
⠀⠀⠈⢿⣿⠆⠀⠀⡻⣿⣷⣿⣿⣾⠿⠋⠀⠀⠀⠀⠀⡆⠀⠀⠀⠁⠘⠀⠀⡀⠀⠀⢠⠀⠐⣻⡭⠀⠠⣤⣀⡒⣶⣚⣯⠗
⠀⠀⢠⣀⠀⠀⠀⠘⠿⠿⣿⣿⣿⡷⠿⠷⠀⠀⠀⠀⠀⠃⢀⠀⠀⠀⠐⢢⡤⠙⠀⠄⠀⠀⠀⢻⡯⠀⠲⠦⣤⣄⣋⣡⠴⠚
⠀⠀⠸⣿⣿⣦⠀⠀⠀⠰⠿⠛⠻⠿⠀⣴⣄⢠⣿⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠁⠂⠐⠠⠀⠀⠀⠁⠀⠑⠶⢦⣄⣉⣴⠾⠋⠀⡀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠻⣿⣿⡇⠀⠀⠀⠀⢰⠀⠀⠀⣬⣽⠿⡇⠀⠀⠀⠀⠀⠀⣾⡆⠾⠡⡤⠆⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠁⠀⠀⠀⠛⠏
⠀⠀⠀⠀⠈⠉⠁⠀⢠⡀⣴⠀⢁⣀⣠⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠹⣦⡘⠳⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⡀⠀⢰⣴⡀⠀⣄⠀⠀⣶⡧
⠀⠀⠀⠀⠀⣀⠸⣆⢸⠇⣯⣴⣿⡉⠁⠀⠀⢀⡄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠿⠛⠀⢼⣯⠄⠈⠉⠀⠀⠿⢀⡀⠈⠁⠀⢠⡐⣴⢃
⠀⠀⠀⠀⢦⡙⢦⠻⣼⣿⠯⠄⠀⢀⢠⣶⠶⠘⣿⣖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⢀⣀⡙⠶⣋⣤⠄⠀⠀⠲⣝⣗⣫⣔
⠀⠀⠀⠳⣮⡻⣦⣍⣛⡛⠁⠀⠠⢜⠛⠁⣀⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⣿⡇⠀⢠⣛⠯⢈⠥⠶⡂⠀⣈⡻⢮⡭⢟⡋
⠀⠀⠐⠲⣞⣿⡓⢦⡄⠀⠀⠀⠀⠀⣶⡿⣿⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢩⡿⠇⠀⠀⢭⣍⡀⢚⡋⠁⠀⠠⣭⣹⣛⡭
⠀⠀⠀⠉⠁⠀⠙⠀⠀⠀⠀⠀⠀⠀⠱⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⠑⠋⠁⠀⠀⠀⠀⠈

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣯⣤⣭⣥⣦⣭⣷⣥⣿⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢻⣄⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡿⢷⣄⠀⠀⠀⢀⡴⠋⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠹⣦⣀⠀⣀⡼⢻⡆⠀⠀⠀⠀⠀⢀⣼⠃⠀⠈⠛⠛⠛⠋⠀⠀⠀⠈⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠈⠙⠛⠋⠀⠈⠳⣄⣠⣄⣀⣤⣾⡁⠀⠀⣀⣠⣤⡴⠶⠶⠛⠂⠀⠀⠀⢉⣿⠆
⠀⠀⠀⠀⠀⠀⣤⣶⠟⠛⠁⠀⠀⠀⢠⣤⡶⠶⠶⠶⠶⢤⣤⣀⣿⠁⠉⣿⠁⢀⣿⣾⡛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠿⢻⡟⠉⢻⡏⠁⠈⠹⣦⣠⡶⠛⠓⠲⢦⣄⠀⠀⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⢀⣴⠾⠛⠛⠛⢦⡀⢀⡾⠁⠀⠀⠙⠒⠛⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠈⠻⣾⡇⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠻⣦⡾⠁⠀⠀⠀⠀⠀⢻⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆
⠀⠀⠀⣿⠻⣦⣀⣀⣀⣤⡾⢻⣆⠀⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠻⣦⡀⠀⠀⠀⠀⣠⡿⢹⡇
⠀⠀⠀⣿⠀⠀⠉⠉⠉⠀⠀⠀⠙⢦⣀⣀⣀⣀⣤⠞⠁⠙⣦⡀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠙⣦⣀⠀⠀⠀⢀⣀⡴⠟⠁⠀⠈⠙⠓⠶⠖⠛⠉⠀⢸⡇
⠀⠀⠀⣿⠀⠀⢀⡴⠶⣦⡀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠈⠙⠷⠦⣤⡤⠶⠛⠁⠀⠀⠀⠀⠉⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢸⡇
⠀⠀⠀⢿⡆⠀⠸⣇⡀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⢦⡀⠀⠀⠀⠀⠀⠀⠻⡽⠇⠀⣿⠁
⠀⠀⠀⠸⣇⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⢻⡄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠁⠀⠀⠀⣠⣤⣀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⢿⡀⢾⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣿⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⣴⠶⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠘⣧⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠈⠳⢶⡶⠏⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠿⠶⠶⠶⠶⠒⠛⠛⠛⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠦⠤⣤⣀⡀⠀⠀⠀⠀⣤⣄⡀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⡀⠀⠀⠈⠙⣿⣿⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠉⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⢻⡇⠀⠀⠀⠀⠀⠀⠉⠀⣀⣠⣤⣴⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⢸⠇⠀⢀⣀⣤⣴⣶⡾⠿⠟⠛⠉⠉⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⡏⠉⠉⠉⠉⠙⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⢢⡀⠀⡾⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣳⣦⣀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠳⡸⠃⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⠛⠋⠙⠳⣦⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠂⠀⢸⠇
⠀⠀⠀⠀⠀⠀⢀⡹⡄⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⢀⡟
⠀⠀⠀⠀⠀⠀⣸⠁⠙⠄⠀⣿⡇⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣷⣶⣤⣄⣀⠀⠀⠀⠀⢀⣴⡿⠋⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠶⠶⠶⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠻⠿⣶⣦⣤⣾⠟⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢀⣿⡏⠀⢠⣤⠌⠙⠋⠀⣰⠏
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣦⣾⡟⠀⠀⢨⣿⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠈⢿⣯⠀⠀⠀⢸⣿⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⢤⣤⣤⣄⣀⣀⣀⣠⣬⣽⡿⣷⣶⣤⣤⣤⣤⣴⣿⠋⠀⠀⠈⢻⣷⡀⠀⢸⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣹⣿⠃⠀⠀⠀⠀⠀⢻⣷⡀⠸⣿⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣿⣿⣿⣿⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠈⠓⠶⠤⠤⠄⠀⠀⠀⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣯⣥⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢛⡛⠻⢟⣯⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠢⠜⢀⢌⠛⠁⣤⠄⠀⠈⠛⠻⢫⣶⡍⢋⣥⡀⠀⠃⠐⠀⠠
⠀⠀⠀⠀⠀⠀⠠⠡⣎⣹⣧⠀⠀⠉⣷⡄⠀⠀⢠⡦⠀⠀⠀⣡⣄⣠⠈⣀⠀⣿⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⢋⡴⣦⢸⣿⣿⣀⣤⣶⠖⠤⠀⠀⠹⣿⣿⣄⣬⡀⠻⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣀⡙⠃⠘⠟⠛⣿⣿⠟⢴⡶⠠⠐⠀⠘⢷⠘⠛⠁⣩⣤⠀⡀⣠⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣶⣾⣷⣽⡥⠀⠀⣀⡀⠈⠁⠀⠀⠀⠐⠠⠀⠁⢀⣀⡀⢠⣬⣯⣶⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠃⠀⠀⠀⠈⠛⠂⠀⠀⠀⠀⠀⠒⠋⠁⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠠⢶⡿⣿⣷⢄⠀⠀⠀⠀⠀⠀⠠⣾⡿⢿⡶⠼⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢷⠀⠀⠀⠁⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠛⠣⡀⠀⠀⠀⠀⢀⣤⣤⡤⢤⡀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⢻⣄⠀⠀⠀⠙⠏⡴⠂⠿⠃⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⢆⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⣆⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⢸⡇⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⠘⣷⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠰⡟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⢿⣿⣿⠟⠋⠉⠙⠛⠿⢿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠛⠿⣧⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⣰⡦⢴⢴⢄⠴⣴⡦⣶⡄⠀⠀⠀⠀⠀⠀⢀⠀⣀⡠⡤⢀⠀⣤⢄⡀⣀⣀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⢀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⣽⣿⣗⠀⢸⠀⢄⡮⣛⡇⠀⠀⠀⠀⠀⢸⣾⢿⣿⢧⣳⣽⢠⣟⡞⣷⡏⡿⡇⠀⠀⠀⠀⣼⡻⠤⠽⠷⢾⣿⠴⠿⠥⠴⣟⡀
⠀⠀⠀⠀⠀⢹⣖⠍⠀⢸⠁⢐⣭⣶⡂⠀⠀⠀⠀⠀⢸⡜⡜⣟⢯⣻⣿⣿⣯⡞⡽⣱⣱⠆⠀⠀⠀⠀⣿⠅⠀⠉⠍⢹⣿⠉⠩⠥⠬⢽⡇
⠀⠀⠀⠀⠀⢿⣛⡫⠁⢸⡄⠉⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⡝⣌⢳⡙⣿⣿⣫⢞⡵⢣⢿⠃⠀⠀⠀⠀⣿⡛⣛⣛⣒⣻⣿⣛⣛⣛⡓⣻⡇
⠀⠀⠀⠀⠀⣿⣿⣓⠂⢸⡇⣐⣿⣿⡃⠀⠀⠀⠀⠀⢨⢳⡹⣌⢳⣽⣿⣿⣣⢞⣵⢋⢎⠂⠀⠀⠀⠀⣿⠄⠀⠠⠤⢾⣿⠤⠤⠤⠄⢽⡇
⠀⠀⠀⠀⠀⣲⣿⠿⠗⢸⡇⠾⣿⣿⠇⠀⠀⠀⠀⠀⢸⣧⣳⣌⢷⣽⣿⣿⣫⠟⣡⢟⣾⠇⠀⠀⠀⠀⣿⡍⠉⠩⠭⢽⣿⢭⣭⠍⢩⣽⡇
⠀⠀⠀⠀⠀⣸⡷⢏⠁⢸⡇⠨⣿⣿⡇⠀⠀⠀⠀⠀⢸⣧⠳⣽⣦⡙⢿⡿⣡⣾⣿⢋⢾⠇⠀⠀⠀⠀⣿⣄⣒⣒⣒⣚⣿⣒⣒⣒⣒⣺⡇
⠀⠀⠀⠀⠀⣻⠽⡣⠂⢸⠀⢔⣽⣿⠃⠀⠀⠀⠀⠀⠀⢧⠳⣬⡳⣽⣿⣿⡵⣛⣵⢏⡞⡀⠀⠀⠀⠀⣿⣒⠲⠶⠶⢺⣿⠒⠒⠒⢒⣺⡇
⠀⠀⠀⠀⠀⣼⣭⡶⠀⢸⠂⢨⣒⣿⡇⠀⠀⠀⠀⠀⢸⡘⠷⢮⣽⣿⣿⣿⣞⣯⠵⠟⣡⡇⠀⠀⠀⠀⣿⠧⠤⠭⠭⢭⣿⠭⠭⠭⠴⢿⡃
⠀⠀⠀⠀⠀⣾⣿⣡⣀⡸⣠⣤⣝⣿⡅⠀⠀⠀⠀⠀⠈⠛⠛⠒⠒⠻⢿⡿⠖⠒⠚⠻⠟⠁⠀⠀⠀⠀⠛⠁⠀⠀⠈⠸⠿⠘⠛⠁⠈⠙⠃
⠀⠀⠀⠀⠀⠈⠁⠈⠈⠀⠈⠈⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⣀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠈⢀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡤⣁⠂⠄⠐⠃⣐⢀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⠈⠉⠀⠃⠀⠊⠙⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠠⠃⠀⠀⠳⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⡀⠀⠂⢆⠀⠀⠀⠀⠠⠠⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠶⠒⠀⠉⠀⠁⠀⠀⠀⠲⠖⠶⠊⠀⢶⠁⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠘⠃⢠⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠠⣄⠀⠀⣤⠊⠀⠂⢤⡐⠂⢳⡄⠀⠀⠤⠀⢡⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠨⠶⠀⠙⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠎⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⣀⠀⢀⠁⠀⠡
⠀⠀⠀⠀⠀⠀⠀⡷⠊⠁⠀⠀⠘⡄⠀⠀⠀⠃⠀⠀⠐⠀⠀⡠⠘⢅⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢰⠄⣄⠀⣠⠐⠈⠀⠀⠀⠀⠀⠀⣴⠆⠀⠉⠀⠀⠈⠄⠀⠀⠀⠀⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠸⣆⡠⠀⢸⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠡⡄⠀⠀⠀⠀⠀⠀⠀⠘⡀⡶⠁⠄⠀⠀⣐⡠⠔⠊⠀⠀⠀⠰⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠄⡀⠀⣠⠄⠈⠀⠀⠃⠀⠀⣀⠌⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠐⠒⠂⠄⠀⠀⠰⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⡄⠀⠣⠀⠀⢠⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⡄⠀⠀⠀⢠⠀⢣⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠆⠀⠀⠀⢨⡀⠀⠁⠂⠐⠊⠁⠀⠀⠀⢀⠆⢰⡏
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠠⠈⠀⠉⠐⠂⠤⢀⡀⠠⠀⠖⠁⠀⠌⠂
⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠉⠓⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠓⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⢐⠒⠀⠀⠀⠀⠀⠀⠀⠑⠂⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⢸⠁⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡿⣋⢙⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣯⣼⣿⣿⣶⣾⣷⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣶⣿⣿⣧⣭⣿⣿⣿⣿⣿⣿⣀⠂⣛⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⢉⡛⣿⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢀⣀⣈⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠿⢿⣿⣿⣿⢉⣾⣿⢻⣿⡷⠶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⠀⠀⠀⠀⢠⣾⣿⣭⣥⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠀⠉⠑⠿⠟⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⡾⣿⣿⠿⡟⡻⢲⣶⣬⣝⣿⣿⣟⣉⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠛⠓⠮⣝⣛⡋⠙⠈⣷⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⠝⡛
⠿⠁⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣁⣀⡀⠀⠀⠤⣬⣭⣙⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠀⠁⠀⠐⠉
⠀⠰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⣀⣙⣓⠛⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣋⣙⠛⠛⠛⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⡀⢀⢀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣁⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⢛⣛⢛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣨
⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⣶⡟⠉⡿⣿⡿⣻⣿⣿⣿⣏⠉⠙⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠛⢿⣿⣯⠉⠉
⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡇⠈⣰⡥⠚⣳⡟⠉⠀⠈⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠸⣟⢷⣄
⠀⠀⠀⡠⠂⠀⠀⠠⡶⠶⢶⣦⡞⢿⣿⣎⠛⠿⢿⣿⣿⣿⣿⣿⣸⣿⣤⣤⣿⣿⡃⠀⠀⢀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣾⣟⡙⠟⢿⣿⣿⣶⣬⠁⠛⠲
⠀⣠⠊⢀⣠⣤⣤⣄⡀⠀⣀⣹⣇⣤⡿⣿⣿⣿⣦⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡿⠈⠀⠀⠠⢶⡟⠛⢿⣿⣿⡟⢉⡙⠛⠿⠿⠿⠿⠛⠛⠛⠛
⡜⠁⣰⡿⢩⣿⢝⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣋⢠⣀⣴⣾⣿⣻⡛⠳⠀⠙⠻⠿⣶⠿
⠀⠀⠻⠿⢟⣣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣽⣿⣿⡿⠋⣭⣭⠶⠿⠟⠣⣄
⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿