Angery Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠖⠒⣺⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⢀⣤⠴⠚⠉⠁⠀⠀⠀⠘⠛⠛⠋⠉⠙⠛⠒⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠘⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⣠⣤⣤⣤⣄⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⣴⠟⠁⠀⠈⠳⣌⠁⠀⢀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠁⠀⠘⡆⠠⡀⠓⠤⣤⣀⠀⠈⠉⠀⣠⡾⠃⠀⠀⠀⠀⠀⠸⡆⠀⢻⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠐⢄⣢⣽⣤⠟⠲⣤⣀⣀⣀⣠⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣴⠞⠛⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢹⡀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⡀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠸⡇⠸⠏⠁⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠇⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⣾⣇⠀⡇⠀⠀⠀⠸⣧⠀⠀⣀⠀⠀⠀⢤⡀⠀⣸⡟⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⠀⠈⠙⢶⠇⠀⠀⠀⠐⣿⣿⠏⠁⠀⠀⠀⠀⠉⠿⡿⠇⠀⠀⠀⠀⣼⠀⣸⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⢿⡾⠉⢀⣀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢀⡞⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⡅⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣮⡳⢤⠄⡀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠛⠛⠛⠛⠛⢷⡄⠀⠀⠀⠀⠀⡎⢠⡇⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠦⣿⠀⠀⠀⠀⢠⣾⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⡇⠀⠀⠀⠀⡄⠀⠘⠀⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⣿⣋⣠⣤⠽⠷⠶⠚⠛⢿⡇⠀⠀⠀⠀⣿⠶⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠈⣯⠤⠶⠖⠚⠛⠛⠛⠙⠛⠻⣧⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠶⠒⠒⠀⠀⠀⠀⠈⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠳⠶⠶⠶⠖⠚⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠖⠛⠛⢳⣦⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢋⡩⠀⠀⠀⠀⠀⠁⠀⠂⠀⠘⠈⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠗⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢀⣤⡄⠀⡀⠀⢘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⡤⠊⠀⠀⠀⣠⠀⢀⣴⣿⢀⣴⣫⡴⠋⠁⢿⡀⡇⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⣀⣴⣊⣥⠶⠋⠀⠿⠟⠋⠁⠀⠀⠀⠈⢷⣧⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⡶⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⣿⠦⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡇⠀⠀⠉⠙⠳⢤⡀⠀⡄⠀⢀⣀⡴⠛⠉⠀⠀⢿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⢠⣿⣷⡦⠀⢸⠃⠀⠀⠀⣴⣶⣶⠀⠀⢸⠀⣆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢠⣙⣛⡅⢀⡟⠀⠀⠀⠀⢙⠿⣋⠄⠀⢸⣀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⠧⣤⡀⠀⠀⠀⠉⠁⠀⠀⢸⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⡄⠤⠤⠶⠶⠶⠶⠶⠀⠆⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⣀⠀⠀⠀⠀⠀⢀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⡟⠙⡟⠛⠛⠉⠉⠻⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠛⡿⠁⢸⡀⠀⠙⠀⠀⠀⠀⢠⡇⠈⣿⠓⠶⠦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠀⠀⠀⠀⢸⠇⠀⠈⢷⡀⠀⠀⠀⠀⢀⡾⠁⠀⣻⠀⠀⠀⠀⡀⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠐⡄⠀⠀⠀⠸⡇⠀⠀⠀⠱⣄⠀⠀⣠⠞⠁⠀⠀⣿⠀⠀⠀⢰⠃⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠘⡄⠀⠀⠀⠷⠖⠻⡇⠀⠈⢳⠾⠋⠀⢸⠛⠺⠃⠀⠀⠀⠌⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠦⣼⣤⡴⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠌⣀⣀⣀⣀⣀⣀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⡇⠀⠀⠸⡇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣷
⠀⠀⠀⠀⠀⠀⠀⠸⣧⣈⠙⠲⢤⣄⣰⡆⠘⣇⠀⠀⠀⠀⣰⠇⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣤⠴⠚⠉⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠛⠶⠶⣤⣽⡇⠀⠈⠙⠲⠖⠋⠁⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣤⠶⠾⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣁⣀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠒⠶⠾⢷⣤⣄⣀⣀⣤⣤⣤⣤⣀⣀⣀⡇⣿⢀⣀⣀⣤⣤⠶⠞⠛⠋⠉⠉⠀⠀⠀⠀⢻⡄
⣤⣤⣤⣤⣤⣤⣤⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⠶⣬⣭⣭⣿⠿⠿⠿⠛⠛⠻⠶⢶⣶⣲⣄⠀⠀⠀⠀⠀⠀⢸⡷⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⣻⠀⠀⠀⠀⠀⠀⠀⠀⣬⡻⣧⡀⠀⠀⢀⣠⡶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣤⣤⣄⣀⠀⠀⢀⣒⠶⣝⢮⣻⡟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠶⣯⣿⣾⡷⠟⠁

⣿⣿⣿⣿⣿⣿⡿⠟⠃⠾⠟⣿⣻⣿⡿⠿⢿⣿⣿⣧⢹⠻⠷⣏⣛⣦⣍⠐⢰⣨⡯⣯⣽⣏⣿⣷⣿⠗⠸⣗⣞⣿⣿
⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⣿⣿⣷⣾⣿⣷⣏⠻⢿⡿⠇⠘⠙⢃⠈⠻⠿⢞⡧⣾⣿⣶⡉⣿⣿⣿
⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡋⣯⣭⣿⣾⣶⣦⣿⠹⢳⣾⡸⢋⣄⠻⣷⢸⣷⣿⣶⡈⣿⡿
⠃⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣍⢻⢻⣿⣻⣿⣿⣿⣁⢘⢫⣝⣻⣍⣻⣦⠀⠀⣽⣏⣿⡌⢹
⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣹⢿⣿⣟⣿⡿⣵⣿⣸⡇⣮⣿⣿⣿⣭⠘⢶⣌⡛⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣸⣿⣟⠿⣶⣤⣿⠹⣿⣿⣿⣿⣾⡟⢸⣷⣾⣟⢮⡛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⢟⣰⣷⣿⣿⢿⣷⣿⣽⣿⣿⣿⡖⣽⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠶⣶⠶⡶⠀⢀⣼⣿⡳⢻⣿⣿⣻⡟⢿⣿⣬⣽⡿⣿⢻⣿⣥⣿⣿⣷⣽⠛⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⢸⣿⡿⢰⣾⡿⣋⣻⡿⣷⣘⠻⢿⣿⡿⣼⣿⠛⢿⣿⣇⣴⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣧⠀⠀⠀⠸⠏⣴⢺⣿⣿⣿⣿⣿⣷⣼⣃⢘⡿⢧⣻⣿⠁⣿⣾⣏⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⢿⣿⣿⢿⣿⡛⣿⢿⣧⣸⣷⣾⠀⡽⠒⢻⣿⣯⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⡟⣿⣻⣏⠙⣨⢹⣧⣦⡈⢻⣿⣷⣿⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣉⣣⣹⣷⠿⢾⡵⠂⣰⣿⡙⢿⣧⣾⣿⡛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⢀⡀⢀⣼⣿⣿⡟⢫⣶⣾⡇⣼⢿⣿⡟⣦⠙⣿⣿⣿⣦⣉⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣾⣿⡿⢿⣷⣿⡎⣿⠈⡆⣿⣿⢾⣅⡿⣿⢿⣿⣿⢿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⣫⣴⢿⣽⣿⣧⡀⢾⠰⠿⢃⣽⡽⠇⣀⡰⣿⣾⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣊⡿⣿⣷⠷⣾⠿⡛⠋⢠⠠⠀⠸⠋⣥⣾⣏⢻⣿⣽⠏⠙⣛⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡿⠁⠼⢛⣚⣉⠁⠀⠸⣀⣟⡀⠀⠀⠶⠆⠾⢨⣯⣏⢨⣭⣼⡄⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣻⣋⠃⢠⣞⠿⢻⣩⣤⣴⣶⣯⠘⠇⣀⣹⡁⡆⣀⡘⣻⡿⠘⠿⣧⡶⢡⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⢈⠹⢷⠔⢻⠟⣾⢻⣽⣿⣾⣦⠟⣿⡆⠉⠸⣣⣼⣿⣦⣷⠆⠀⣾⡿⡇⢨⣦⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠁⡜⢻⠟⢼⣟⣭⡙⣿⡾⣇⣾⣿⣷⡏⣿⣮⣛⢹⣷⡆⢞⢠⣷⣿⠇⢘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣦⣿⡆⠉⠩⣿⣿⣿⣯⡙⠣⢾⣿⡏⢇⣿⣿⠟⠀⠉⣾⠉⢸⣧⣿⣷⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⡿⣿⠰⠆⠉⣋⣥⣾⡧⣔⢻⣿⡇⡀⢄⢰⣶⣾⣤⢰⣤⣼⣯⣿⡇⣿⣟⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣙⣠⣭⣤⡝⠶⢿⢿⡽⠛⣡⣦⣼⣿⣿⡟⣡⢸⣿⣿⣿⡧⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⠉⣣⡀⠐⠨⠈⣵⡿⣿⠿⣛⣿⣿⣿⠙⡄⠻⣛⣥⡄⣾⡷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣭⣝⣷⡀⠚⢷⡟⠿⢗⣭⠹⠿⠟⡀⠃⡀⢿⣿⠃⣿⣱⣾
⣴⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣯⣽⡲⣿⣿⣿⣿⠿⠿⡗⣈⠂⠛⠀⠉⠐⣒⡛⢃⠀⣼⣌⣿⣅⣿⣾⣿
⣞⣿⣿⡏⣿⢷⣄⡀⠀⠀⠀⢀⢠⣴⣯⣛⣛⣫⣭⡤⣄⠀⠈⡁⠤⠶⣶⣷⣬⣰⣿⣷⣽⣛⣿⣤⣿⣿⣾⣹⣿⣿⣿
⠛⢛⠛⠓⢙⣛⣹⣿⣶⣶⣾⡿⢿⣿⣿⣻⣿⠶⠛⢩⣔⣿⣷⣶⣶⢦⣙⣛⠿⣿⣿⣟⢰⣿⣧⠙⣿⣿⢿⠏⢻⣿⣿
⢠⢈⡆⣠⣻⢸⡿⣿⣿⣿⣿⣼⣿⠿⣛⣭⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢳⣿⣿⣛⣿⣽⣿⡄⣷⣾⡘⢻⣿⣿⣿
⣄⣨⣀⣋⠉⣾⡾⣿⠿⠟⡁⢭⣶⣞⣿⢟⣾⣿⣿⣿⣿⣷⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⡛⣿⢳⡾⣿⣿⣿
⣈⣉⢁⣿⠀⣼⡆⣴⡶⣴⣶⣿⣿⣿⣝⣽⣽⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣆⣾⣦⣟⣿⣿
⣿⣒⣞⣿⣧⣿⣷⣜⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⡆⣽⣿⣭⣭⣽⣭⣩⣭⣿⣹⣿⣷⠰⣎⣷⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⢿⣿⣿⣿⣦⠀⠀⢀⡤⠤⠤⠖⢺⠃
⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⡔⠁⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⢀⠎⠀⠀⢀⣀⣀⣤⣤⣤⣴⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣤⣤⣤⣀⣀⠀⠀⠘⡤⠤⠖⠚⢄⡀
⠀⠀⠀⠀⠀⢸⣶⣶⣿⣿⣿⣿⣿⣿⡿⣟⣿⣿⡿⠛⣿⣿⣿⡿⢹⣿⣿⣿⣿⡿⠟⠁⠉⠛⠿⣿⣿⣷⣦⡷⢄⡀⠀⠀⠙⣲⡤
⠀⠀⠀⢀⠔⠚⣿⣿⠿⠿⠛⠛⠉⠀⠀⠉⠉⠀⠀⣰⣿⠿⠋⢀⣾⣿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡏⠲⡌⠉⠉⠉⠁
⠀⠀⠀⡎⠀⠀⣿⣿⡆⠀⠀⠀⠀⠤⠶⣦⣤⣀⠀⠉⠀⠀⠀⠈⠁⠀⠀⢀⣤⣴⡶⠦⠄⠀⠀⠀⠀⣿⣿⠁⠀⢸⡀
⠀⠀⠀⢇⠀⠀⠹⠟⠁⠀⠀⠀⠀⠚⢻⣽⠟⠛⠳⠀⠀⠀⡾⠀⠀⠀⠰⠛⠛⢯⡿⠛⠀⠀⠀⠀⠀⠻⠟⠀⠀⢸⠁
⠀⠀⠀⠈⠢⣄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠓⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠦⠤⠤⠤⠤⠤⠤⠒⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣠⣄⢀⣄⢀⠀⠀⢀⣠⠀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠹⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢹⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡆⠀⠀⠀⢻⣿⡆⠀⠀⠀⢀⣠⢤⣀⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣷⣄⠀⠀⣶⣿⣿⣿⣿⡿⠿⣅⠀⠀⠀⢸⣿⣇⠀⠀⠀⣇⠀⠀⠁⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⣸⠁⣠⠖⠚⠛⣿⡿⠏⠀⠀⠙⡿⠿⡿⠛⣦⡀⠈⣧⠀⠀⠈⢠⣮⣽⣖⣋⣀⡀⠀⠀⣀⣠
⠀⣠⡤⠖⠛⠓⣦⠀⠀⣰⠚⢭⡙⢷⠀⠀⠀⢻⠀⣇⠲⠴⣗⠞⡇⠀⠀⠀⠀⣇⠀⠻⣿⠿⠁⢀⣿⠀⠀⠀⠘⠃⣸⠇⠀⠈⠳⠶⠶⠟⠁
⢾⡁⡀⠀⠀⠀⠉⢉⡩⣽⠻⡏⣹⠀⠀⠀⠀⠈⠳⣬⣙⣫⣥⡴⠁⠀⠀⠀⠀⠙⢷⣄⣀⣠⠖⠿⠁⠀⠀⠀⣶⠚⠁
⠀⣿⠁⠀⠀⠀⠚⣯⠀⠘⠦⣥⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠛⠓⠂⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠈⠙⠻⠦⠤⠞⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠶⣒⣀⠉⠙⠛⠿⢷⣦⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠑⠄⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠒⠒⠒⠲⠶⠶⠶⠶⠖⠒⠛⠉⠉⠁

⣿⣿⣿⣿⣿⣿⡿⠟⠃⠾⠟⣿⣻⣿⡿⠿⢿⣿⣿⣧⢹⠻⠷⣏⣛⣦⣍⠐⢰⣨⡯⣯⣽⣏⣿⣷⣿⠗⠸⣗⣞⣿⣿
⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⣿⣿⣷⣾⣿⣷⣏⠻⢿⡿⠇⠘⠙⢃⠈⠻⠿⢞⡧⣾⣿⣶⡉⣿⣿⣿
⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡋⣯⣭⣿⣾⣶⣦⣿⠹⢳⣾⡸⢋⣄⠻⣷⢸⣷⣿⣶⡈⣿⡿
⠃⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣍⢻⢻⣿⣻⣿⣿⣿⣁⢘⢫⣝⣻⣍⣻⣦⠀⠀⣽⣏⣿⡌⢹
⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣹⢿⣿⣟⣿⡿⣵⣿⣸⡇⣮⣿⣿⣿⣭⠘⢶⣌⡛⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣸⣿⣟⠿⣶⣤⣿⠹⣿⣿⣿⣿⣾⡟⢸⣷⣾⣟⢮⡛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⢟⣰⣷⣿⣿⢿⣷⣿⣽⣿⣿⣿⡖⣽⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠶⣶⠶⡶⠀⢀⣼⣿⡳⢻⣿⣿⣻⡟⢿⣿⣬⣽⡿⣿⢻⣿⣥⣿⣿⣷⣽⠛⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⢸⣿⡿⢰⣾⡿⣋⣻⡿⣷⣘⠻⢿⣿⡿⣼⣿⠛⢿⣿⣇⣴⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣧⠀⠀⠀⠸⠏⣴⢺⣿⣿⣿⣿⣿⣷⣼⣃⢘⡿⢧⣻⣿⠁⣿⣾⣏⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⢿⣿⣿⢿⣿⡛⣿⢿⣧⣸⣷⣾⠀⡽⠒⢻⣿⣯⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⡟⣿⣻⣏⠙⣨⢹⣧⣦⡈⢻⣿⣷⣿⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣉⣣⣹⣷⠿⢾⡵⠂⣰⣿⡙⢿⣧⣾⣿⡛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⢀⡀⢀⣼⣿⣿⡟⢫⣶⣾⡇⣼⢿⣿⡟⣦⠙⣿⣿⣿⣦⣉⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣾⣿⡿⢿⣷⣿⡎⣿⠈⡆⣿⣿⢾⣅⡿⣿⢿⣿⣿⢿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⣫⣴⢿⣽⣿⣧⡀⢾⠰⠿⢃⣽⡽⠇⣀⡰⣿⣾⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣊⡿⣿⣷⠷⣾⠿⡛⠋⢠⠠⠀⠸⠋⣥⣾⣏⢻⣿⣽⠏⠙⣛⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡿⠁⠼⢛⣚⣉⠁⠀⠸⣀⣟⡀⠀⠀⠶⠆⠾⢨⣯⣏⢨⣭⣼⡄⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣻⣋⠃⢠⣞⠿⢻⣩⣤⣴⣶⣯⠘⠇⣀⣹⡁⡆⣀⡘⣻⡿⠘⠿⣧⡶⢡⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⢈⠹⢷⠔⢻⠟⣾⢻⣽⣿⣾⣦⠟⣿⡆⠉⠸⣣⣼⣿⣦⣷⠆⠀⣾⡿⡇⢨⣦⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠁⡜⢻⠟⢼⣟⣭⡙⣿⡾⣇⣾⣿⣷⡏⣿⣮⣛⢹⣷⡆⢞⢠⣷⣿⠇⢘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣦⣿⡆⠉⠩⣿⣿⣿⣯⡙⠣⢾⣿⡏⢇⣿⣿⠟⠀⠉⣾⠉⢸⣧⣿⣷⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⡿⣿⠰⠆⠉⣋⣥⣾⡧⣔⢻⣿⡇⡀⢄⢰⣶⣾⣤⢰⣤⣼⣯⣿⡇⣿⣟⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣙⣠⣭⣤⡝⠶⢿⢿⡽⠛⣡⣦⣼⣿⣿⡟⣡⢸⣿⣿⣿⡧⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⠉⣣⡀⠐⠨⠈⣵⡿⣿⠿⣛⣿⣿⣿⠙⡄⠻⣛⣥⡄⣾⡷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣭⣝⣷⡀⠚⢷⡟⠿⢗⣭⠹⠿⠟⡀⠃⡀⢿⣿⠃⣿⣱⣾
⣴⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣯⣽⡲⣿⣿⣿⣿⠿⠿⡗⣈⠂⠛⠀⠉⠐⣒⡛⢃⠀⣼⣌⣿⣅⣿⣾⣿
⣞⣿⣿⡏⣿⢷⣄⡀⠀⠀⠀⢀⢠⣴⣯⣛⣛⣫⣭⡤⣄⠀⠈⡁⠤⠶⣶⣷⣬⣰⣿⣷⣽⣛⣿⣤⣿⣿⣾⣹⣿⣿⣿
⠛⢛⠛⠓⢙⣛⣹⣿⣶⣶⣾⡿⢿⣿⣿⣻⣿⠶⠛⢩⣔⣿⣷⣶⣶⢦⣙⣛⠿⣿⣿⣟⢰⣿⣧⠙⣿⣿⢿⠏⢻⣿⣿
⢠⢈⡆⣠⣻⢸⡿⣿⣿⣿⣿⣼⣿⠿⣛⣭⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢳⣿⣿⣛⣿⣽⣿⡄⣷⣾⡘⢻⣿⣿⣿
⣄⣨⣀⣋⠉⣾⡾⣿⠿⠟⡁⢭⣶⣞⣿⢟⣾⣿⣿⣿⣿⣷⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⡛⣿⢳⡾⣿⣿⣿
⣈⣉⢁⣿⠀⣼⡆⣴⡶⣴⣶⣿⣿⣿⣝⣽⣽⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣆⣾⣦⣟⣿⣿
⣿⣒⣞⣿⣧⣿⣷⣜⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⡆⣽⣿⣭⣭⣽⣭⣩⣭⣿⣹⣿⣷⠰⣎⣷⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⡴⠋⠉⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠃⠀⠀⠀⢀⣠⡄⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⡏⠘⡇⠀⠀⢠⡇⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢤⡀⠀⡼⠀⠀⠱⣄⠀⠸⠴⠋⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢸⠉⠊⠁⠀⠀⠀⢹⡄⣠⡀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⡴⠋⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠼⠚⠁⣧⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠙⠆⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢀⣞⣠⠜⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⢄⣸⡃⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠀⣀⡉⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠈⡏⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⡍⣧⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⡜⠱⢶⠶⣦⡀⠀⡀⠀⣰⣟⠛⣬⠙⠃⢸⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢸⠁⠀⢧⡀⠻⣻⠞⠙⢾⡻⠟⢀⡼⠀⠀⣸⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠁⠀⠀⠀⢈⣩⠥⢤⣀⠀⠉⠉⠉⠀⠀⠀⡏⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⢰⡀⠀⠀⠈⠀⠀⠀⠙⠳⠶⢳⠀⠀⠀⢸⠃⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠠⠤⠤⠤⠤⠴⠖⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠉⠉⠉⠉⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⠠⠄⠐⠀⠐⠒⠒⠀⠀⠠⢄⡀⣠⣴⣶⣶⣦⣤⣀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣀⣀⠀⠠⠤⠤⠀⢀⣀⡈⢻⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣒⣛⣉⠁⠀⠐⠒⠂⠠⠤⣀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠉⠐⠒⠠⢄⡀⠀⠈⠑⠢⢄⡀⠀⠀⠈⠙⠻⣿⡀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⣀⠀⠈⠑⠤⡀⠀⠀⠀⠑⠢⣄⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠘⠘⣎⠓⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢀⠈⠑⠢⣀⠀⠀⠀⠉⠢⢄⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⡄⠀⠙⣆⠀⠈⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⣑⠢⠄⡀⠀⠀⠑⠤⡈⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⠃⠀⠀⠈⠳⣄⠀⠀⠀⠉⠒⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠊⠵⢦⣠⢤⢘⡘⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⡰⠉⠛⢦⣄⡀⠀⠀⠀⠈⠐⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠛⠉⡹⠘⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⣠⠊⠀⠀⠀⢀⠜⠀⠀⠀⠀⠉⠊⠕⡂⠠⠄⣀⠀⠀⠀⠈⠉⠒⠀⠀⠤⠤⠤⠄⢤⣤⡤⠤⠐⠒⣛⣱⡚⠤⠚⢁⠇⠘⢿⣿⣿⣿⣿
⣿⣿⣿⡿⠋⠀⠀⠀⣀⠔⠁⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠢⠬⢉⣐⡒⠒⠒⠒⠐⠒⠒⠂⢈⣁⡠⠤⠴⠖⢉⣉⠀⠠⠤⠔⠂⢄⠀⠀⠉⠉⠉⠉
⣿⠿⠋⠀⠀⣀⠄⠊⠀⠀⢀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⢀⡤⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠒⠠⠤⠤⠀⠀⣠⠤⡄⠒⠁
⣿⣿⣷⣶⡭⢤⠤⠔⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠸⣄⠀⠀⠀⠀⢀⣠⣾
⣿⣿⣿⡿⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢿⣿⣶⣶⣿⣿⣿⣿
⣿⣿⡿⠁⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠀⠄⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣀⣀⣁⣒⣀⣀⣒⣉⣀⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠤⣤⣠⣤⡔⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠦⡤⠞⠁⣀⣀⣠⣤⣾⣷⡆⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣠⣴⣶⣶⣶⠶⢀⣠⠌⠙⠛⠛⠛⣭⣴⣿⣿⠷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⢠⡞⣉⠈⠉⠋⢁⣴⠋⣠⣶⣦⣀⣀⣀⡈⠉⠉⠑⠶⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠁⠀⢠⡏⣾⠁⣀⣀⡀⢸⡏⣼⣿⣿⣿⣿⣿⣿⡟⠠⣄⡳⢶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠁⠀⠀⠀⢸⡆⢿⣸⣿⣿⣧⡸⠇⠿⠿⠿⠿⠛⠻⢿⣷⡄⠈⠙⠛⠛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠛⠛⢶⣄⠀⠛⠊⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⢦⣄⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢀⣼⡛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⣷⠀⠀⣿⣧⡈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠚⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠹⡆⠀⢨⡻⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠿⠀⠀⠹⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠲⣿⣿⣧⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠏⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⣉⣀⣀⣀⠀⢹⣿⣿⣿⣒⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⢠⣾⣿⡀⠀⣀⡴⠶⠒⠲⠶⢤⣤⣀⡀⠀⠀⣀⢰⡟⢁⣤⣾⣿⠛⠛⠋⠀⠀⢻⣖⢶⣽⣧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⡇⢸⣿⣿⣷⡀⠉⠀⠀⠰⣶⣶⣶⣄⣈⠙⢷⡈⠉⢿⡇⠉⠀⠉⠙⠛⠁⠀⠀⠀⢸⣿⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⣎⠻⠿⠛⠁⠀⠀⠿⠿⢙⣿⣯⠄⠉⠀⠈⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠘⢿⣧⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣭⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣀⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢠⣼⠆⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⡿⡿⠛⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠚⠛⠀⠛⠁⠀⠰⠀⠀⠀⠀⠀⣿⣿⠿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⢿⣿⢸⣿⢳⣿⣿⣽⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⢀⣀⣀⣤⣤⣀⠀⠀⠀⠀⠀⣿⣿⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣽⣻⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⠿⢛⣻⣿⡟⠉⠀⠀⠀⣸⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣧⡻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠉⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⡙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠛⠓⠶⠶⠶⠶⠶⠒⠛⠋⠁⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠾⠛⠛⠋⠉⠉⠉⠉⠙⠛⠛⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢰⠏⠀⢠⠟⢳⡶⠂⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣀⣏⠐⠋⢀⣤⠞⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠋⠀⠀⠀⣀⣤⠶⠚⠛⠛⠉⠉⠛⠛⠳⢦⡀⠀⠀⠀⠀⠀⠈⠛⣷⡀⠀⠀⠀⠀⠀⠀⢸⡇⣰⠟⠁⢀⣴⣋⣡⡴⠶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠁⠠⠾⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣠⣤⡿⠁⠀⢀⡴⠋⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢹⣷⣄
⠀⠀⠀⠀⣠⡴⠟⠉⠘⢧⣠⣴⣯⣴⠶⠚⠛⠛⠛⠛⠛⠛⠻⠶⣦⣄⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⣦⡀⢸⣇⠘⢷⡄
⠀⠀⢀⣼⠏⠀⠀⠀⣠⡾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣄⠀⠀⠀⢰⣷⡀⢀⣶⠀⢿⡿⠟⠻⣧⡈⢿⡄
⠀⠀⣾⠃⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣼⠻⣇⣸⠟⢷⣾⠃⠀⠀⠘⣷⡀⢻⡄
⠀⣼⠇⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣯⣤⡤⠞⠁⣴⠿⠃⠀⠈⠁⠀⠀⠀⠀⠘⣧⠀⢷⡀
⠀⣿⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢸⣇
⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⠶⢶⡆⠀⠀⠀⠀⠀⠀⠘⣷⠀⣿
⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⣠⡶⢦⣤⣀⠀⠀⠀⠀⠀⠀⢿⣤⣿⡄
⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⠛⠉⣠⡤⢤⣉⠛⢿⣷⣄⠀⠀⠀⠀⠀⠀⣼⡿⠃⢀⡼⠋⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠈⢹⡿⣦⡀
⠀⢹⡇⠀⢀⣼⠏⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠙⠷⣄⠈⠛⢷⣄⠀⠀⠀⣸⡟⠁⣴⢟⣀⣀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⢸⡇⢸⣧
⠀⣈⣿⣴⠟⠁⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣴⣦⡌⠻⣦⡀⠁⠀⠀⠀⠈⢠⡞⢱⣿⣿⣿⣄⡀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢸⡇⢸⡏
⣼⠏⢸⡅⠀⠀⠀⠀⠀⠀⠀⣭⠀⠀⠀⠀⣴⣿⣿⣶⣦⡀⢠⡄⠀⠀⠀⠀⠘⢧⠘⢿⣶⣿⡽⠀⣀⣠⣴⡿⠶⠛⠃⠀⠀⠀⢸⣿⠈
⣿⡀⠘⣧⠀⠀⠀⠀⠀⣀⠀⢻⡄⠀⠀⠀⠘⠿⢿⠿⠛⠀⣼⠁⠀⠀⠀⠀⠀⠀⠛⢦⡽⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠘⢷⣀⣻⡆⠀⠀⠀⠈⠉⠛⠚⠛⠛⠛⠛⠓⠒⠒⠒⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠙⠛⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠋⠉⠉⠉⠉⠉⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⡀⠀⠀⠀⢀⣤⣤⣤⣤⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⡆⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢀⣀⣀⣀⣤⣤⣤⣤⠤⠤⠠⠤⣤⣴⡀⠀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⠀⠀⠚⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠉⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⢶⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠾⣿⣏⠀⠀⠀⠀⣤⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⠉⠛⠻⣿⠶⠶⠶⠶⠶⣶⣿⡿⠿⠋⠁⠀⠀⢸⡟⣧⡀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⣀⣠⣄⢀⡀⠀⠀⠀⠘⣷⣸⣇⠀⠰⠾⠛⠉⢹⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠈⠙⢶⣤⣶⣬⣿⣿⠿⠽⠆⠀⣠⠞⠉⠉⣿⣤⣤⣤⣤⡟⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠈⠉⠛⠻⠟⠿⠿⠆⠾⠋⠀⠀⠀⢀⣀⣀⣀⣀⣿⡺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠙⠷⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠶⠶⢶⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⣈⣉⣙⣛⣿⣿⣿⣿⣿⣿⣷⠶⠶⠶⠶⠟⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠛⠛⠛⠛⠋⠉⠉⠉⠉⢉⣽⠟⠋⠙⠉⠁⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡾⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⠀⠀⠀⠀⢀⣾⠟⠛⠛⣿⠀⠀⠀⠀⠀⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠶⠶⠶⠤⣾⣇⠀⠀⠀⠘⢧⣀⠀⠀⠀⠀⡴⠟⠛⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣿⠷⢶⠶⠟⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⣶⠶⣦⣤⣤⣤⣤⣼⠇⠀⠀⠀⠀⠀⢻⣦⣤⣤⣴⡴⠶⢶⡶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠾⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠷⣤⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠈⢻⡇
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣧⣤⣤⣤⣤⣴⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣦⣤⣤⣤⣤⣼⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠶⠤⡤⣤⠖⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠖⠒⠶⠖⠃⠠⣤⣤⣴⣾⣿⢆⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢁⣤⠾⣿⣿⠟⢃⡤⠖⡈⠉⠉⠉⢶⣾⡿⢟⣓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⢠⢎⡴⠂⠀⠀⣰⠏⣴⣿⣿⣦⣴⣶⣄⠀⢀⣈⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⡇⣾⢠⣶⣶⡆⢿⢰⣿⣿⣿⣿⣿⣿⣇⠘⠲⠭⠿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣤⢤⣀⠀⠻⡜⢿⠿⠿⠟⠈⠈⠉⠁⠀⠀⠀⠙⢿⣦⠀⢀⡀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢀⣲⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠻⡆⠀⢹⣶⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢷⡀⢈⡙⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠸⠇⠀⠙⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠓⢿⣿⣷⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⢋⣥⣤⣠⡄⠸⣿⡿⠯⣇⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏⢰⣿⣿⡀⢠⠞⠛⠉⢙⠛⠒⠶⢦⣄⠀⣴⣰⡟⠠⠾⣿⣟⣭⠉⠀⠀⢹⣯⢻⣾⣾⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣾⣧⠹⣿⣿⣷⠀⠀⢠⣬⣽⣿⡟⠲⠄⠘⠧⠀⢸⡇⠀⠀⠀⠉⠁⠀⠀⠀⢸⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣾⠹⣿⠇⣈⣁⠀⠀⠀⠈⠉⠐⠛⠋⠁⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣳⠄
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠉⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣼⡿⣿⡿⠿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠰⠶⠀⠞⠋⠀⠤⠀⠀⠀⠀⣸⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⢸⣿⠑⣿⢳⣶⣿⣽⣿⣆⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⣀⣀⣠⣄⡀⠀⠀⠀⠀⣿⣿⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠻⢧⣽⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢠⡴⠿⠿⢛⣻⣿⠋⠃⠀⠀⢰⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⡜⣧⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡎⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠈⠙⠶⣤⣀⣀⡀⠀⠀⣀⣀⣠⣤⠞⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⢸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠛⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢀⣀⣤⡌⠁⠀⠀⠀⠙⠿⠟⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀⠀⠉⠉⠙⠒⢀⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⠒⢦⣤⣴⣿⣟⣿⣶⣄⣀⠒⠀⡈⠉⠰⠶⠶⠶⢶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣰⣶⠞⠛⠋⠁⠀⠀⠈⠛⠻⣶⣾⣁⣠⠤⠄⠀⠀⠾⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣦⣄⣀⡀⠀⠉⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⠙⠛⠛⢷⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⣦⡀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⣴⡿⢧⡀⠀⠀⠤⣤⣤⣤⣤⠤⠀⠀⠀⣰⣿⣧⣀⡈⢻⡦⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢰⣿⣄⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠙⢿⣷⣄⠙⢷⡀⠀⠛⠛⠛⠃⢰⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠙⠻⣷⣄⠙⣆⠀⠀⠀⣴⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣧⣤⣾⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠷⣶⡶⠾⠿⢿⣿⣷⣶⣶⣶⣿⢿⣿⣿⣿⠿⠿⠿⠿⣶⣶⣶⣶⢶⣶⣶⡦⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠙⣷⣄⠀⠀⠀⠘⠿⠿⢆⣰⣾⠋⠀⢿⣄⡀⠀⠀⠀⠻⠿⠿⢆⣸⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠛⢷⣶⣤⣤⡤⣴⣾⡟⠁⠀⠀⠀⠙⠻⣶⣶⣤⣀⣴⣶⠞⠋⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢰⡾⠛⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⡄
⠀⠀⠀⠀⠀⢰⡇⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠈⢧⣤⣀⣼⣿⠀⠀⠀⠀⠀⠀⠀⢰⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠈⠉⠉⢻⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣀⣤⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⣀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣶⣦⣤⣴⣶⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣴⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣠⡴⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠛⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⠀⠀⢀⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⠾⠿⠿⠟⠛⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠿⡋⠉⠉⠉⠉⢻⠟⠛⠛⠛⠛⢻⡿⠛⠋⢻⣿⠏⠉⠉⠉⠻⣏⡉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣿⠋⣽⡄
⠀⠀⠀⠀⠀⠀⠀⣸⠇⢰⡿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡿⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⢿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⠃⠀⠀⠀⣾⡄⠀⠀⠀⠀⠀⢀⣧⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠸⣧
⠀⠀⠀⠀⠀⢰⡎⠀⢠⡇⠀⠀⠀⠀⣿⣀⣠⣶⠆⠀⠀⠀⠛⠀⢸⣿⠀⠀⠀⠀⢴⣶⣄⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠘⠃⠀⠸⠟⠀⠀⠀⠀⣿⠛⣿⡄⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⣴⡏⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣆⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠉⣷⡄⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠠⠤⠤⠤⣾⣿⠀⠀⠀⢿⡇
⠀⣀⡄⠛⣻⡿⠛⠀⣠⣶⣾⣷⡀⠀⠀⠀⠀⠉⣷⡄⠀⠀⢸⣿⠁⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣾⣿⠀⠀⠀⢸⣇
⠀⠻⣧⡾⠋⠀⢀⣼⠟⠉⢀⣼⣿⠀⠀⠀⠀⠀⠈⢻⡆⠀⢸⣿⠀⢀⣠⣴⣿⣤⣤⣤⣤⣤⣤⣤⣴⠛⠋⠉⠉⢹⣿⠀⠀⠀⠈⣿⡗⠐⠒⠀⡀
⠀⣼⡏⠁⠀⣠⡾⠃⠀⣰⡿⠙⢿⣦⡀⣀⣄⣀⠴⠿⠿⠿⠟⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⣉⣭⡿⠿⠷⠾⠿⠷⢿⣿⠀⠀⠀⠀⣿⡷⠲⠷⠾⢧⣀
⣶⠉⠀⢀⣴⠏⠁⢀⡸⠋⠀⠀⣼⠟⠁⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⡿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⣿⡷⠰⢦⣀⣈⡛
⠙⢶⣤⣿⠇⠀⢀⣜⠃⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣴⣶⣶⣿⣿⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣿⡷⠀⠀⠉⠻⢿⡄
⠀⠀⣿⠇⠈⣴⡾⠋⠀⣠⣾⣿⣶⣶⣶⣶⣶⣶⠾⠛⠛⠛⢻⡟⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠷⠾⠾⣿⣧⠀⠑⣶⣀⣀⡟
⠀⠀⣿⣆⡀⢛⣷⣦⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠱⣦⣼⠿⠟⠁
⠀⠀⠀⠛⠻⣿⡏⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⡀⠈⢢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠈⠁
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠉⠰⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿
⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣴⣶⣿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⢿⣿⣷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣴⣿⣦⣤⣤⣴⣶⣶⣶⣾⣿⣿⣿⡿⠿⠿⢿⣿⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⡏⠉⠉⠉⢻⡟⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢸⡏⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⣆⠀⣠⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⡄⠻⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠉⠉⠋⠉⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠁⠀⠀⠀⠉⠹⠿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠿⠋⠁⠀⢀⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢠⣤⣄⠀⠀⠀⠈⠉⠙⠿⠷⣦⣄⣀⡀
⠀⠀⠀⠀⠀⣀⣀⡰⠶⠞⠉⠁⠀⠀⣀⣠⣶⠿⣧⡀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣨⡛⠛⢷⣶⣀⣀⣀⠀⠀⠀⠈⠉⠉⠛⣶⣶⣦
⠀⠠⣶⣶⣄⣉⣉⣀⣀⣀⡤⢶⣶⠾⠛⠋⠀⠀⠛⠛⠲⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠉⠛⠛⠛⠲⠶⠶⠶⠶⠶⠈⠛⠋
⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣶⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣦⣤⣤⣤⣤⣴⣶⣶
⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⠿⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠻⠿⢿⣷⡀⢀⠤⠒⠒⠊⡹
⠀⠀⠀⠀⠀⡰⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠈⢇⠀⢀⣠⡞⠁
⠀⠀⠀⠀⢰⣁⣤⣴⣶⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⠛⠻⣿⣶⣶⣤⣈⣏⠁⠀⠙⠢⣀⡀
⠀⠀⠀⡠⠒⣿⣿⣿⠿⠟⠛⠉⠘⠛⠉⠁⣸⣿⠟⠁⣼⣿⡿⠟⠋⠀⠀⠀⠀⠉⠙⣿⣿⠂⢍⠒⠒⠒⠉
⠀⠀⢰⠁⠀⣿⣿⠀⠀⠀⠀⠴⣶⣤⣄⠀⠋⠀⠀⠀⠉⠀⠀⣠⣤⣶⠦⠀⠀⠀⠀⣿⣿⠀⠈⡆
⠀⠀⠘⡄⠀⠘⠋⠀⠀⠀⠀⠉⠻⠟⠉⠑⠀⠀⢸⠁⠀⠀⠊⠉⠻⠟⠉⠀⠀⠀⠀⠙⠃⠀⢠⠃
⠀⠀⠀⠈⠒⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠉
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣄⣀⣤⣤⣤⣾⠿⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⠉⣭⣭⣭⣭⣉⣛⠋⢤⣄⡉⠀⣶⣄⡙⠻⣷⣄⣀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⠛⠉⢁⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣾⣿⣿⣶⣼⣿⣿⣦⡘⠿⠛⣋⢉⣉⠛⠛⠿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡾⠿⠛⢛⣩⣥⣶⣿⣿⠿⣛⣋⣭⣥⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣾⣿⣿⣧⣶⣶⠶⢂⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⡐⠶⣾⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣙⠿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⠟⢃⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⢯⣭⣉⣉⣛⣲⣌⣿⣿⣿⣿⡿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣷⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠒⣀⣭⣿⣿⣿⣿⣿⣿⣿⣿⣤⣼⣿⣿⣿⣿⣿⣿⣯⣉⣉⣛⣿⣿⣿⣿⣷⣬⡉⠉⠿⣷⡄
⠀⠀⠀⠀⠀⠀⢸⡿⢀⣾⣿⣿⣿⣿⣿⣿⣿⣯⣁⣤⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣁⠀⢾⡇
⠀⠀⠀⠀⠀⢀⣿⢃⡿⢿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡿⣯⠗⣀⡔⣠⡾⣿⣿⣿⣿⣿⣷⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠈⠀⣾⣿⣿⣿⢡⣾⣿⣿⣿⣿⣿⣿⣧⣾⢋⣼⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡉⢻⡇
⠀⠀⠀⠀⠀⠀⢻⡷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣼⣿⣿⣿⣿⣿⣿⣟⡉⠉⣤⣿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⠸⣿⡿⢛⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣉⣛⢿⣿⣿⣿⣿⣿⣿⣿⠛⠛⣃⣸⡿
⠀⠀⠀⠀⠀⠀⠀⢿⣆⠛⠀⣿⣿⣿⣿⣿⣿⣡⣿⣿⣿⣿⣿⣿⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⡇⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⡿⢁⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡜⡷⠛⠛⢿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⠉⠛⠿⠀⠙⢿⡟⢁⡀⠀⡸⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⡁⠀⠀⠀⢹⡏⠀⠻⡏⠉⠛⠻⠿⠿⠿⣯⡭⠭⡽⠛⠛⠛⠛⠭⠉⠛⠃⠀⢀⡠⠄⠀⠘⣧⠈⢀⡼⢡⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⢣⡀⠀⠀⠀⠃⠠⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠋⠀⠀⠀⠀⡄⢰⡟⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⢳⡄⢀⠀⡄⠀⠀⠀⢉⣻⢿⣶⣶⣶⣶⠦⠀⠀⠀⠀⠴⠦⣶⣾⣿⠟⢳⣄⠀⠀⠀⠀⠹⠈⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢱⠈⠻⡇⠀⠀⠀⠈⠉⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠠⢀⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⣀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢋⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣭⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡇⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⢳⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⠈⠓⣐⣃⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⢀⡼⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠜⣧⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⣄⣀⣀⣠⣴⣾⣿⡇⠀⠀⠀⠀⠀⢀⡾⠱⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣡⠞⠙⣦⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠿⣿⣿⡟⠀⠀⠀⠀⣠⠊⠈⠙⢦⡙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢃⠞⠁⠀⠀⠈⠢⠀⠀⠀⠀⠀⣼⡿⣿⡅⠀⠙⠀⢀⣿⡍⠁⠀⠀⣠⠞⠁⠀⠀⠀⠀⠙⣆⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢡⠎⠀⠀⠀⠀⠀⠀⠈⠢⣄⣀⠀⠹⡀⠿⣿⣷⣶⣶⣿⠿⠃⣤⡤⠞⠱⠀⠀⠀⠀⠀⠀⠀⠘⡄⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠙⠓⠒⠠⣂⠀⠠⠤⠤⠄⢘⡼⢋⠇⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⡜⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢠⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢷⠀⠠⠈⢄⠀⠙⠀⠠⠤⠄⠉⢀⠊⠈⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⢠⡁⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢄⠑⠒⠀⢀⡀⠀⠒⡁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣰⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠹⣦⣀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠀⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣂⣤⠟⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢠⡄⠉⠓⠦⠤⣀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠐⠒⠉⠉⡇⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘⡇⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⡇⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣷⢸⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⡇⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢳⠀⠀⠀⣿⠐⢌⣛⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠟⠀⠀⢉⣡⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⢀⣠⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢷⣶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⡶⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⠖⠛⠉⠉⠉⠉⠉⠛⣷⡤⠶⠖⠒⠛⠶⠶⣤⣄
⠀⠀⠀⢀⡴⠚⠉⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⢦⡀
⠀⣴⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠏
⠀⠀⠉⠛⠾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠙⠷⣤⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠶⠶⠿⠛⠛⠲⠶⠦⣤⣤⣤⣤⣤⠶⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠙⢗⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣶⣴⣆⣤⣤⡀⣀⣤⣄⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣛⣦⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡥⢤⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⣸⣿⣿⣿⣿⣿⠟⠛⠉⠉⠛⢿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⣴⣿⣷⣄⠀⠀⠀⠀⠀⢀⣴⣾⣷⣄⠀⠀⠙⣿⡇⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣹⠿⢿⣿⣷⣦⠀⢠⣾⣿⣿⣿⠿⢯⡀⠀⠀⣿⣧⡀⠀⣪⠀⠀⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣿⣿⡇⠀⠀⡜⢠⠖⠉⡙⢿⠋⠀⠀⢹⠛⢻⣩⣦⠀⣧⠀⠀⢀⡶⢿⠭⠤⣄⠀⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠛⠁⠀⠹⠤⠤⣯⡴⢦⠁⠀⠀⠰⣘⠪⡽⢛⡱⠀⠀⠀⠘⣦⡈⠉⢁⣰⠟⠀⠀⢀⡥⠞⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠂⠀⠀⠠⣆⠈⠧⣁⣨⠖⠀⠀⠀⠈⠉⠉⠉⠁⠀⠄⠒⠦⠈⠁⠚⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠤⠖⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠂⠁⠀⠉⠙⠻⠶⢄⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠁⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣛⣛⣛⣛⣒⣒⣒⣒⣒⣒⣒⣒⣋⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⠀⠀⣀⣀⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣄⠀⢠⣿⡿⣀⣤⣶⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠻⠿⠿⠟⠋⢻⣿⣿⣿⡿⠋⠁⠉⠉⠀⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡏⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠋⠀⢀⣀⣀⣀⠀⠀⣸⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⣿⣿⣿⣿⣿⠃⠀⣠⣴⣾⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣧⠀⢿⣿⣿⣿⣿⡿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢿⣿⣿⣿⣯⠀⣰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠟⠋⠙⣿⠀⠀⢻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⣿⣿⣿⡟⢠⣿⡿⠿⣿⣿⡇⠀⠀⡀⠀⠀⠀⠀⢸⣿⣿⠋⠀⠀⠀⠉⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣿⣿⣿⠁⠘⠁⢀⣠⣤⣽⣿⣦⣴⠃⠀⠀⠀⣦⣿⣿⣥⠤⠤⠤⣄⠀⠀⠀⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⡰⢋⣀⣄⠈⠛⠉⠁⠀⠤⠤⠀⠈⠉⠉⠰⣦⡴⠀⢹⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠁⠈⠛⠋⠀⠚⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠴⠟⠀⠀⠀⣸⣿⡟⠲⢦⣴⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣿⣿⣿⡀⠀⠈⠉⠉⠁⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢁⣀⣈⠻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣥⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠸⢏⡴⠚⠛⠻⠄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣯⠠⠋⠀⢈⢳⣸⡿⠀⠀⠀⠀⠀⠀⡠⠀⠀⠉⠀⠀⠀⠉⠀⠀⡀⠀⠀⠀⠀⠠⠀⣿⠀⠟⠋⠳⣆⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣴⠉⠉⠛⡇⠀⠀⠀⠀⠀⠀⠑⠦⠤⠄⠀⠀⠀⠶⠠⠒⠁⠀⠀⠀⠀⠀⢀⡏⢀⠀⠀⣠⠏⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠹⣦⣀⣤⣿⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠘⣇⠀⢉⣋⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣈⡀⣸⡄⠀⠀⠀⠀⠀⣴⠟⠉⡏⠉⢉⣏⣉⢛⡛⠛⣶⡆⠀⠀⠀⢲⡟⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠙⣷⠀⠀⠀⠀⢀⡿⡖⠚⣟⢉⣉⣏⣉⣭⣭⣿⣿⡇⠀⠀⠀⣹⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢹⣦⠀⠀⠀⠈⠷⠼⠖⠛⢉⡩⠭⠥⣄⠀⠀⠀⢀⠀⠄⢲⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠠⠀⠀⠀⠀⢰⠉⣅⡀⠀⠀⠈⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢶⣤⣀⣀⠀⠀⢸⣿⣿⣿⣧⣀⣠⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⢻⣿⣿⣿⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛

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

⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀
⠀⢀⣠⣴⣾⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⢿⡟⠛⠓⠦
⠀⠀⠀⠀⠀⡏⠉⠉⠛⠛⠿⣿⣿⣿⣿⣶⣦⣤⣀⣿⣇⣤⣴⣶⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⣧
⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠻⣁⣿⠟⠙⠛⠻⢿⡟⣿⠟⠛⣟⢙⣿⠇⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠘⣦⡀⠈⠉⠁⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⣀⡴⠛⠀⠀⠀⠈⠻⢦⣄⣀⠀⠀⢀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⣶⠒⠛⠛⠛⠛⢛⡟⠓⠲⠦⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠒
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢹⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣿⡄
⠀⠀⠀⠀⠀⠀⣠⡿⠓⠒⠚⠓⠒⠚⣿⠛⠛⠛⠛⠉⠉⠙⡏⠉⠉⠉⠉⠁⢸⡇
⠀⠀⠀⠀⠀⢠⡿⣄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢰⠀⢸⡇⠘⢧⡀⢀⣀⣤⡤⠾⠶⠒⠒⠛⠛⠛⠛⠛⠛⠛⠒⠲⠶⠿⠤⣄⡀
⠀⠀⠀⠸⡄⠀⠻⠶⠾⠛⠉⠁
⠀⠀⠀⠀⠙⠳⠦⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣄⣀⣤⣤⣤⣾⠿⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⠉⣭⣭⣭⣭⣉⣛⠋⢤⣄⡉⠀⣶⣄⡙⠻⣷⣄⣀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⠛⠉⢁⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣾⣿⣿⣶⣼⣿⣿⣦⡘⠿⠛⣋⢉⣉⠛⠛⠿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡾⠿⠛⢛⣩⣥⣶⣿⣿⠿⣛⣋⣭⣥⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣾⣿⣿⣧⣶⣶⠶⢂⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⡐⠶⣾⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣙⠿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⠟⢃⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⢯⣭⣉⣉⣛⣲⣌⣿⣿⣿⣿⡿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣷⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠒⣀⣭⣿⣿⣿⣿⣿⣿⣿⣿⣤⣼⣿⣿⣿⣿⣿⣿⣯⣉⣉⣛⣿⣿⣿⣿⣷⣬⡉⠉⠿⣷⡄
⠀⠀⠀⠀⠀⠀⢸⡿⢀⣾⣿⣿⣿⣿⣿⣿⣿⣯⣁⣤⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣁⠀⢾⡇
⠀⠀⠀⠀⠀⢀⣿⢃⡿⢿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡿⣯⠗⣀⡔⣠⡾⣿⣿⣿⣿⣿⣷⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠈⠀⣾⣿⣿⣿⢡⣾⣿⣿⣿⣿⣿⣿⣧⣾⢋⣼⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡉⢻⡇
⠀⠀⠀⠀⠀⠀⢻⡷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣼⣿⣿⣿⣿⣿⣿⣟⡉⠉⣤⣿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⠸⣿⡿⢛⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣉⣛⢿⣿⣿⣿⣿⣿⣿⣿⠛⠛⣃⣸⡿
⠀⠀⠀⠀⠀⠀⠀⢿⣆⠛⠀⣿⣿⣿⣿⣿⣿⣡⣿⣿⣿⣿⣿⣿⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⡇⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⡿⢁⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡜⡷⠛⠛⢿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⠉⠛⠿⠀⠙⢿⡟⢁⡀⠀⡸⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⡁⠀⠀⠀⢹⡏⠀⠻⡏⠉⠛⠻⠿⠿⠿⣯⡭⠭⡽⠛⠛⠛⠛⠭⠉⠛⠃⠀⢀⡠⠄⠀⠘⣧⠈⢀⡼⢡⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⢣⡀⠀⠀⠀⠃⠠⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠋⠀⠀⠀⠀⡄⢰⡟⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⢳⡄⢀⠀⡄⠀⠀⠀⢉⣻⢿⣶⣶⣶⣶⠦⠀⠀⠀⠀⠴⠦⣶⣾⣿⠟⢳⣄⠀⠀⠀⠀⠹⠈⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢱⠈⠻⡇⠀⠀⠀⠈⠉⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠠⢀⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⣀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢋⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣭⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡇⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⢳⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⠈⠓⣐⣃⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⢀⡼⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠜⣧⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⣄⣀⣀⣠⣴⣾⣿⡇⠀⠀⠀⠀⠀⢀⡾⠱⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣡⠞⠙⣦⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠿⣿⣿⡟⠀⠀⠀⠀⣠⠊⠈⠙⢦⡙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢃⠞⠁⠀⠀⠈⠢⠀⠀⠀⠀⠀⣼⡿⣿⡅⠀⠙⠀⢀⣿⡍⠁⠀⠀⣠⠞⠁⠀⠀⠀⠀⠙⣆⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢡⠎⠀⠀⠀⠀⠀⠀⠈⠢⣄⣀⠀⠹⡀⠿⣿⣷⣶⣶⣿⠿⠃⣤⡤⠞⠱⠀⠀⠀⠀⠀⠀⠀⠘⡄⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠙⠓⠒⠠⣂⠀⠠⠤⠤⠄⢘⡼⢋⠇⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⡜⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢠⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢷⠀⠠⠈⢄⠀⠙⠀⠠⠤⠄⠉⢀⠊⠈⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⢠⡁⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢄⠑⠒⠀⢀⡀⠀⠒⡁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣰⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠹⣦⣀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠀⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣂⣤⠟⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢠⡄⠉⠓⠦⠤⣀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠐⠒⠉⠉⡇⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘⡇⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⡇⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣷⢸⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⡇⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢳⠀⠀⠀⣿⠐⢌⣛⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠟⠀⠀⢉⣡⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⢀⣠⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢷⣶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⡶⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⣦⣷⣤⣦⣤⣀⣀⣤⣦⣤⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠀⠉⢻⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠟⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⠀⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣆⠀⠀⠀⢻⣿⠀⠀⠀⠀⣀⣀⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⢀⣽⣿⣿⣿⣿⣷⣄⠀⢠⣾⣿⣿⣿⣿⠿⣏⡀⠀⠀⢸⣿⣆⠀⠀⢸⡁⠀⠈⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⡎⢀⠔⠚⡛⢿⡿⠏⠀⠀⢻⠿⢿⡟⢳⡄⠀⣷⠀⠀⠉⢰⣯⣷⣚⣁⣤⠄⠀⣀⣠
⢀⡤⠶⠛⠉⢳⡄⢀⣠⠊⣩⣍⠳⠀⠀⠀⢇⠸⣔⣶⣿⠜⡅⠀⠀⠀⠸⣄⠘⠻⠿⠃⢀⣿⠀⠀⠀⠙⢁⡼⠀⠀⠘⠓⠒⠚⠁
⠻⣄⠀⠀⠀⠀⣀⠤⢶⠛⡏⠸⢀⠀⠀⠀⠈⠲⠮⠭⠶⠶⠁⠀⢀⣀⡀⠙⠧⣄⡤⠖⠙⠁⠀⠀⠀⣿⠉
⠀⢯⣀⣠⠀⠈⣹⠀⠀⠑⠲⢲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠉⠒⠂⠤⠤⠬⣽⣿⠷⢦⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠙⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠶⠶⠦⠤⠤⠤⠤⣤⣤⣤⣤⣤⠤⠤⠶⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⢻⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡟⠀⠀⠐⠶⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⠟⠋⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⢀⠔⠉⠙⠿⣿⣿⣷⣦⣄⠀⠀⢀⣤⣶⣿⣿⣿⠿⠋⠁⠀⠑⢄⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠁⠀⢠⠃⠀⠀⠀⠀⠈⠙⠿⣿⡟⠁⠀⠈⢿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠈⢂⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢿⣿⣿⡆⠀⡆⠀⠀⠀⠀⠄⠀⡀⠐⠀⠈⡄⠀⠀⢸⠁⠀⠰⠁⣠⡀⠐⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠙⠿⠛⠃⠀⢡⡀⠀⠀⠀⢄⠈⠁⠠⠀⠀⡇⠀⠀⠸⡃⠀⠠⡀⠉⢀⠌⠀⠀⠀⢸⠁⠀⣼⣿⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⢀⣠⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⠉⠀⢀⡴⠁⠀⠀⠀⠑⡀⠀⠀⠉⠀⠀⠀⠀⡠⠃⠀⠀⣿⣿⣿⡏⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢿⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠒⠂⠉⠑⢤⣀⣀⣠⠖⠈⠑⠒⠀⠤⠔⠖⠋⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢀⡤⠲⠂⠀⠒⠤⡀⠀⠀⠀⢀⠠⠔⠂⠀⠐⠒⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣠⢛⡀⠠⡤⠀⢠⡀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢄⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⠃⠈⠁⠀⡄⠀⢈⠈⠢⠄⣆⠀⠄⠂⠋⠀⠀⢹⠐⠀⣠⠀⠀⠇⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠓⠤⠒⠂⠁⠐⠲⡀⠀⠀⠃⠀⠀⣀⠤⢀⠀⠈⠀⠀⡌⠀⠀⡔⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠴⠤⠒⠉⠀⠀⠀⠈⠑⠤⣠⠀⢀⡼⠃⠀⢸⠁
⠀⠀⠀⠀⠀⠀⢀⡴⠒⠲⣄⠠⠄⠤⢄⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⡜⠀⠀⠀⠀⣠⠤⠤⣄⡀
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠐⠁⠀⠀⠀⣨⠀⠀⠉⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠀⡠⠖⠶⠾⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠈⡂⠀⠀⠀⠀⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠒⠒⠋⠉⠀⠀⠀⢣⡤⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⢉⡃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡄⠀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⢀⣀⣀⣀⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡤⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⣀⣀⣀⣀⡤⠞⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠒⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣈⣻⡷⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡏⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠿⡇⠉⠙⠃⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡸⠁⠀⠜⢉⡿⠃⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠛⠶⠶⣤⣀⠀⠀⠀⠀⠀⢸⡇⢀⣾⠞⠁⣀⡴⠋⠁⣉⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠛⠶⣄⡀⠀⠘⠀⠞⠁⣠⣼⠧⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢀⣀⡀⠀⠀⢀⣈⡙⢷⣄⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⡀⠤⠤⠠⠀⠠⠤⠄⣀⡀⠀⠀⣰⡿⠒⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷⣄
⠀⠀⠀⠀⠀⢀⡼⠃⢀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠺⠃⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⢀⡟⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠛⠛⠓⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⢠⣞⡴⠃⠀⠀⠀⠀⣀⣤⡴⠶⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⡀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣠⠞⠉⠙⠳⣄⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠙⢦⣀⠈⠳⣄⡀⠀⠀⠀⢀⡼⠁⢀⣞⣡⣤⡄⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⢻⣄⡀
⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣠⣤⡌⠳⣄⠈⠛⠀⠀⠀⠀⢁⡴⠋⣾⣿⣿⣷⣶⠀⠀⢸⡆⠀⠀⠀⠀⠀⢸⡏⢻⡄
⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⣺⠀⠀⠀⢸⣿⣿⣷⣶⠈⠳⡆⠀⠀⠀⠀⠸⡇⠀⠘⠿⠿⠿⠃⠀⠀⣼⠠⠴⠆⠀⠀⠀⢸⡇⢸⡇
⣰⠞⢺⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠈⠻⠷⠿⠃⠀⢰⡇⠀⠀⠀⠀⠀⠀⢤⣄⡤⠤⠖⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⣧⡾⠁
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠛⠓⠒⠓⠒⠒⠒⠒⠖⠶⠦⠴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠹⣆⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠈⠛⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠴⠶⠶⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠁⠀⠀⠀⠀⠀⢀⣀⣹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣤⣤⡤⠄⠀⠀⠀⠉⠉⠉⠛⠛⢷⡀⠀⠀⠀⠀⠀⠀⠼⣃
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠉⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣌⣧⠀⠀⠀⠀⣠⠟⠀⠈⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣾⡤⠤⠴⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠁⠀⠀⠀⠠⡶⠿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠉⠻⣦⡀⠀⠀⠀⠀⠀⣀⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠛⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠞⢋⡕⢁⡟⠀⠀⠈⢿⡄⠀⠀⠶⠛⠉⠉⢩⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠉⠛⢿⣿⣛⠿⠛⠛⠛⢻⡋⠉⠑⢄⡠⠋⠀⣸⠇⣠⡶⠛⠛⣧⣀⣠⣤⣤⣤⣟⠛⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣤⣁⣘⡁⠀⠀⠀⠀⠀⢀⣟⣰⠋⠀⠀⠀⠈⠉⠁⠀⢀⣀⣙⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠒⠒⠚⠛⠛⠋⠁⠀⠀⠀⠀⠀⠚⠛⠛⠛⢉⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣼⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⠀⠀⠀⢀⣀⣠⠞⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⠀⢸⣙⣿⣿⣿⣷⡶⠒⠛⠛⠛⣿⣥⣀⣽⠟⠋⠉⠉⠁⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠘⠭⠍⢻⡿⣻⣏⣀⢀⣴⠟⠉⠉⠉⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⠀⠀⠀⠀⣰⠆⠘⠀⠀⠀⠉⢹⣧⣾⠂⠀⠀⠀⠀⠀⠀⢠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡶⣤⠶⠟⢷⣄
⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣀⣀⣀⡀⡀⠀⣴⣿⣧⣤⣤⣤⣤⣼⡇⠀⠀⢀⡀⣀⣀⣀⣀⣀⣀⣀⡀⢹⣦⣀⣀⣀⣀⣉⣷⣤⣀⣀⠀⠀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢀⣀⣤⣤⣤⠖⠋⠉⠉⠁⠈⠉⠙⠷⠖⠛⠛⠛⠛⠳⠶⣤⡀
⠀⠀⢀⠔⠋⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⡀
⠀⢠⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡀
⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷
⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿
⠀⠀⠙⠲⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣤⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⣉⡛⠓⠲⠶⠶⠶⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⣩⡿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣷⣿⣶⣶⣦⣤⣶⣴⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⢀⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢛⡀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠀⣾⣿⣿⣿⣿⡿⠿⠛⠻⢿⣿⣿⣿⣿⣿⠉⠀⠀⠙⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠙⠿⠟⠋⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⣴⣶⣦⡀⠀⠀⠀⠀⣀⣴⣶⣦⡀⠀⠘⣿⡇⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢀⡽⠿⣿⣿⣷⡄⠀⣾⣿⣿⡿⠿⣅⠀⠀⡼⢧⡀⢀⠇⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣀⠀⠀⡸⠿⢧⠀⠀⠌⢰⢋⣩⣝⠏⠀⠀⢸⠉⢭⣼⠆⢸⡄⠀⠠⠗⢻⠑⠲⣄⣀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⠀⠀⠘⢒⢸⠶⡋⠆⠀⠀⠈⠮⣵⣯⡥⠃⠀⠀⠈⠳⣄⣠⠤⠟⠀⠀⢰⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣤⡀⣈⡇⠀⠑⠒⡎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠠⢄⠀⠐⠲⠶⣤⣀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣀⡀⠉⠑⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⠴⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣓⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣚⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⠛⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⢤⣤⣦⠀⠀⠀⠀⠙⠛⠋⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣶⣄⠀⠀⠀⠉⠰⡦⠀⡀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⣉⣹⣷⣾⠿⠛⠿⢷⣶⣈⣀⠤⠀⠈⠉⠉⢙⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠙⠛⢶⣾⣉⡁⠀⠈⠉⠱⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡟⠶⠶⣦⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⠦⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠷⣄⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⣸⠋⢱⡄⠀⠀⠛⠛⠒⠒⠒⠀⠀⣰⣿⣿⣦⣤⡜⠃⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠈⢿⣷⣄⠙⢦⡀⠀⢠⡤⠤⡄⢠⣴⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠙⠻⣷⣄⠹⢄⠀⠀⠀⣀⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠈⠛⢷⣾⣦⣀⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣦⣴⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣶⣶⣤⣤⡄⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠈⢻⣄⠀⠀⠀⠹⠿⠷⣈⣾⠟⠁⢿⣄⠀⠀⠀⠹⠿⠿⢆⣸⡿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠙⠷⠶⠶⠶⠶⣶⡿⠃⠀⠀⠀⠙⠳⠶⠶⣤⣴⠶⠞⠋⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⣰⠟⠉⢹⣿⠁⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠶⠶⣄⡀
⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⠀⠀⢹⡃
⠀⠀⠀⠀⠀⠘⠿⠶⢿⣿⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣤⣤⣤⣤⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣦⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⡶⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣶⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⠠⡿⠀⠀⠀⠀⠀⠀⠀⢸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⡶⠾⠿⠟⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⡰⠿⡋⠉⠉⠉⢹⡟⠛⠛⠛⠛⢿⠟⠛⠙⣿⠏⠉⠋⠙⢿⣉⠉⠉⠉⠀⠀⠀⠀⠀⣼⡏⢱⡆
⠀⠀⠀⠀⠀⠀⣸⠇⢸⡇⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⠀⠀⢀⡿⠀⠀⠀⠀⠘⢿⡀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⡇
⠀⠀⠀⠀⠀⢸⠟⠀⡸⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡆⠀⣾⠇⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⢸⡟⠀⠈⣷⡀
⠀⠀⠀⠀⠀⣷⠀⢰⡇⠀⠀⠀⢸⣧⣴⡟⠃⠀⠀⠀⠁⣀⡿⠀⠀⠀⠀⢻⣷⠾⠿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⠀⡄⠁⠀⣈⠁⠀⠀⠀⠘⠃⠹⣇⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢻⡇
⠀⠀⠠⢤⣿⣶⠞⠛⣣⣤⡄⠀⠀⠀⠀⠻⣇⡀⠀⠀⣿⡿⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠈⠉⢉⣙⣿⡇⠀⠀⢸⡇
⠀⢿⣁⡴⠋⠀⢠⡾⠛⠉⣹⡆⠀⠀⠀⠀⠙⢧⡀⠀⣿⡇⠀⠀⣀⣿⣀⣀⣀⣀⣀⣀⣀⡶⠞⠛⠛⣿⡇⠀⠀⠘⣿⡦⠠⠄
⠀⣼⠛⠀⠀⣴⠟⠀⢠⡾⠻⣷⡀⢀⣀⣀⡠⢾⣷⡶⠿⠷⠞⠛⠛⠛⠛⠛⠛⠛⣛⣿⣿⠶⣶⣶⣶⣿⡇⠀⠀⠀⣿⡦⠴⣶⣾⣀
⣶⠉⠀⣠⡾⠃⠀⣀⠏⠀⢀⣼⠟⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⡶⠟⠋⠁⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⣿⡇⠤⣄⣀⣛
⠙⢶⣼⡿⠀⠀⣬⠃⠀⢠⡞⠁⠀⠀⠀⠀⠀⣀⣀⣀⣤⣶⣶⣾⣿⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣿⡧⢀⠈⠙⠿⣆
⠀⢈⣿⠀⢰⣾⠉⠀⣴⡿⠿⠶⠶⠶⠶⠿⠟⠋⠉⠉⣻⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠻⣿⣧⠀⠰⣦⣠⡼
⠀⠈⢿⣦⣈⣿⠷⠾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⢶⠟⠋⠁
⠀⠀⠀⠉⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠰⢀⣼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢾⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⡇
⠀⠀⠀⠀⠸⣿⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣶⣶⣿⠿⠛⠛⠋
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠿⠿⠿⣿⣿⠿⣷⢿⣿⡿⠿⣿⡿⠿⠿⠿⠿⠟⠻⣿⠛⠛⠛⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣄⠀⣴⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⡄⠻⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠟⠉⠉⠉⠙⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠁⠀⠀⠀⠉⠻⢿⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠻⠋⠀⠀⣠⡄⠀⠀⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡆⠀⢰⣶⣄⠀⠀⠀⠈⠉⠛⠿⢦⣤⣤
⠀⠀⠀⠀⢠⣤⠔⠛⠃⠀⠀⣀⣤⣴⠿⠛⣷⣄⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣾⠁⠙⠛⢷⣤⣤⣄⣀⣀⡀⢀⡈⠹⣿⣿
⣀⢘⣿⣿⣦⣤⣤⣶⣶⣞⠛⠛⠉⣀⣀⣀⡈⠉⠉⠁⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⠈⠉⠉⠙⠛⠉⠁⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠋⠁⠘⠛⠉⠋⠉⠙⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣟⣻⣟⣿⣿⣿⣻⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠁⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠓⠏⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣰⠎⣿⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢰⠇⠀⢻⡀⠀⠀⠀⡟⠀⣠⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢶⣀⠀⢀⡟⠀⠀⠀⠳⣄⠀⠸⠷⠊⠁⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⡏⠑⠋⠀⠀⠀⠀⠀⢻⡀⢀⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣀⣠⠶⠋⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⣼⡤⠋⢻⡄⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⢷⡄⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⡇⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠉⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢀⡟⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⢠⣞⣠⠴⠋⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢠⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠉⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢹⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣻⣶⣞⠉⣇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⡼⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡛⠻⣿⠇⢻⠀⠐⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⡜⠁⠋⡽⠉⣻⢷⡀⢀⣄⣀⣰⡿⣧⠀⢹⡀⠀⠀⢸⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⢸⠃⠀⠀⠷⣄⣈⣛⡿⠁⠈⠙⣍⣛⠋⣀⡼⠃⠀⠀⣼⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⢸⠀⠀⠀⠀⠀⣈⣥⣤⣤⣀⠀⠈⠈⠉⠉⠀⠀⠀⠀⡟⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢸⡀⠀⠀⠀⠘⠁⠀⠀⠀⠙⢷⣤⣴⣆⠀⠀⠀⠀⢸⠇⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⣟⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠲⠶⠖⠒⠛⠋⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠉⠉⠉⠉⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⠠⠄⠐⠀⠐⠒⠒⠀⠀⠠⢄⡀⣠⣴⣶⣶⣦⣤⣀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣀⣀⠀⠠⠤⠤⠀⢀⣀⡈⢻⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣒⣛⣉⠁⠀⠐⠒⠂⠠⠤⣀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠉⠐⠒⠠⢄⡀⠀⠈⠑⠢⢄⡀⠀⠀⠈⠙⠻⣿⡀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⣀⠀⠈⠑⠤⡀⠀⠀⠀⠑⠢⣄⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠘⠘⣎⠓⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢀⠈⠑⠢⣀⠀⠀⠀⠉⠢⢄⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⡄⠀⠙⣆⠀⠈⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⣑⠢⠄⡀⠀⠀⠑⠤⡈⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⠃⠀⠀⠈⠳⣄⠀⠀⠀⠉⠒⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠊⠵⢦⣠⢤⢘⡘⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⡰⠉⠛⢦⣄⡀⠀⠀⠀⠈⠐⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠛⠉⡹⠘⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⣠⠊⠀⠀⠀⢀⠜⠀⠀⠀⠀⠉⠊⠕⡂⠠⠄⣀⠀⠀⠀⠈⠉⠒⠀⠀⠤⠤⠤⠄⢤⣤⡤⠤⠐⠒⣛⣱⡚⠤⠚⢁⠇⠘⢿⣿⣿⣿⣿
⣿⣿⣿⡿⠋⠀⠀⠀⣀⠔⠁⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠢⠬⢉⣐⡒⠒⠒⠒⠐⠒⠒⠂⢈⣁⡠⠤⠴⠖⢉⣉⠀⠠⠤⠔⠂⢄⠀⠀⠉⠉⠉⠉
⣿⠿⠋⠀⠀⣀⠄⠊⠀⠀⢀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⢀⡤⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠒⠠⠤⠤⠀⠀⣠⠤⡄⠒⠁
⣿⣿⣷⣶⡭⢤⠤⠔⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠸⣄⠀⠀⠀⠀⢀⣠⣾
⣿⣿⣿⡿⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢿⣿⣶⣶⣿⣿⣿⣿
⣿⣿⡿⠁⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠀⠄⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣀⣀⣁⣒⣀⣀⣒⣉⣀⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠖⠒⠒⠒⠒⠶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠇⡏⠀⡴⣤⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⣀⣀⡀⣤⣀⣀⠀⠀⠀⠉⠢⣄⠀⠀⠀⠀⠀⢸⢀⡼⠈⣁⠔⢋⣱⠄
⠀⠀⠀⠀⠀⠀⢠⠞⠁⢀⡠⠚⠉⠁⠀⠀⠀⠀⠈⠓⣄⠀⠀⠀⠈⠳⡄⠀⠀⠀⠈⠈⠠⠞⠛⠋⠉
⠀⠀⠀⢀⣤⠶⡏⢀⡰⣃⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠘⡆⠀⠀⢀⠀⣿⢦⡀
⠀⠀⡴⠋⠀⢀⡿⠟⠋⠉⠀⠀⠀⠀⠀⠈⠙⢶⡀⠀⠀⣷⠀⣤⢸⡷⠿⣆⠹⡄
⠀⡼⠁⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⡰⢛⣶⠛⠿⠃⠀⠈⢧⠹⡀
⢰⡇⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠀⠉⠁⠀⠀⠀⠀⠀⠘⡆⢷
⢸⡅⠀⢸⡇⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⢛⡓⠂⠀⠀⠀⠀⣷⢸⡆
⢸⡇⠀⢸⡇⠀⠀⠐⠚⣋⣩⣛⠿⣦⡀⠀⠀⠀⢀⡾⠋⡰⠋⠉⠙⢦⠀⠀⠀⠈⠛⣧⣄
⠀⣷⢀⡾⠃⠀⠀⢀⠞⠁⠀⢈⡳⣌⠙⠦⠀⠀⠾⣡⢞⣥⡄⠀⠀⠈⡇⠀⠀⠀⠀⡇⢹
⡴⢻⡋⠀⠀⠀⠀⢸⠀⠀⠀⣾⣿⣤⠁⡄⠀⠀⠀⢇⠹⣽⣿⠃⣀⣠⠧⠔⠂⠀⠀⣷⠉
⢧⠈⣇⠀⠀⠠⢤⢸⣄⣀⣀⣙⣛⣋⣰⠁⠀⠀⠀⠀⠑⠚⠋⠉⠁⠀⠀⠀⠀⠀⢸⠇
⠀⠓⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠁⠀⠉⠙⣦⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⢀⡿⠶⠒⠂⠀⠈⠉⠉⠛⢧⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⣼⣠⠤⠤⠴⠒⠒⠒⠂⠒⠲⠆⢀⣠⢾⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢾⡋⠀⠀⣸⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠉⠙⣟⠛⠛⠛⠛⠛⠋⠀⠀⢸⡿⡆⠀⣀⡤⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠈⠳⢤⣤⣿⣿⠷⠄⢀⡴⠛⢷⣤⣤⣴⢾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠈⠉⠉⠛⠃⠋⠀⠀⠠⠤⠤⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠦⣄⣀⡀⠀⠀⠀⣀⠀⠀⠠⠤⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣤⣤⠤⠬⠭⠿⣿⡿⠿⠟⠛⠓⠋⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢧⣄⣀⣀⣀⡞⠉⠉⢧⠀⠀⠀⠀⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠈⠈⡿⠀⠀⠀⣷⠦⠤⠎⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡟⢲⡖⠲⠶⠃⠀⠀⠀⠘⠖⠒⢶⠛⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠾⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠧⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⢀⣀⡈⣷⠀⠀⠀⠀⠀⠀⢰⡏⠀⢀⢈⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠛⠻⠿⢿⣿⣿⡿⠿⠿⠛⠛⠛⠻⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠲⢶⣶⣤⣀⠀⠀⠀⠀⣀⣤⣶⡾⠛⠁⠀⠘⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠔⠀⠀⠈⠛⢿⡷⠀⠘⢿⡿⠟⠉⠀⠀⠀⠂⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⠤⠸⣿⡇⠐⠀⠀⠀⠀⠤⠀⠀⡄⠀⢸⠀⠀⠰⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢅⡀⣀⠀⠀⠀⠡⡀⠀⠀⠀⠀⡠⠁⠀⠈⠀⠀⠀⠀⠀⢀⠜⠀⢸⣿⡟⠉⠁⢢
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢁⠀⠀⠀⠈⠢⠄⠐⠀⠁⡀⠀⠈⠁⠀⠀⠀⠀⢰⠦⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⢀⢞⠀⠄⢀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢏⢀⠀⠄⢀⠀⠀⠆⠀⠈⠀⠀⠁⠀⠄⠈⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠐⠤⠤⠀⠈⠀⠐⠀⣀⢀⡐⠀⠘⠉
⠀⠀⠀⠀⡴⠋⠱⠂⠀⠂⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⢀⢀⡔⠒⠲⢄
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠀⠀⠈⠑⠤⣀⣀⠀⠀⠀⠀⠀⣀⠠⠄⠐⠊⠁⠐⡥⠀⠁⠀⠀⠈⢧
⠀⠀⠀⠈⠢⣀⠀⠀⠀⠈⡅⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠔⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠒⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣤⣬⣟⠳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⠁⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣚⠋⠛⠂⠀⠉⠀⠈⢳⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣐⡁⠞⠁⣠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠉⠛⠲⢤⣀⠀⠀⠀⢼⠁⡼⠋⢀⡴⢚⣁⡤⠴⠓
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠈⠀⠴⠛⠉⠉
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢀⣀⣀⡀⢀⣀⠀⠀⠀⠀⢀⣼⠥⠒⠀⠉⠁⠀⠀⠉⠐⠚⢷⣄
⠀⠀⠀⠀⠀⣠⠞⠁⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⣰⠋⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⠛⠒⠖⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⣰⣣⠎⠀⠀⠀⢀⣀⣤⠶⠚⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠁⠀⠀⢀⣀⠈⠻⣦⡀⠀⠀⠀⠀⠀⢀⡾⠁⢀⡴⠉⠉⠓⢦⡀⠀⠀⠀⠀⠈⣷
⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠈⠳⢄⡈⠻⣤⡀⠀⠀⠀⡞⠁⡰⣫⣶⣆⠀⠀⠀⢷⠀⠀⠀⠀⠀⢸⣦⣄
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⣴⣶⡆⠙⢦⡈⠁⠀⠀⠀⢠⠞⠁⣿⣿⣿⣿⠀⠀⢸⠀⠀⠀⠀⠀⠘⡇⢹⡆
⢀⣀⡟⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⣿⣟⣿⡟⠀⢹⠂⠀⠀⠀⠈⠃⠀⠈⠙⢛⣁⡤⠄⠛⠘⠉⠀⠀⠀⠀⡇⣸⠃
⣾⠁⣷⠀⠀⠀⠀⠀⠤⣤⣹⣤⣀⣀⣈⣉⣉⣀⣀⡞⠀⠀⠀⠀⠀⠀⠙⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠃
⢻⡀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠙⠶⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠁⠀⠀⠀⠀⠀⣈⣷⡀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣤⡤⠦⠀⠀⠀⠉⠉⠉⠛⠻⡄⠀⠀⠀⠀⠀⠠⢏
⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠁⠀⠀⣀⣀⣤⡤⠤⠤⠤⠤⠷⠀⠀⠀⢀⡼⠃⠀⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠿⠖⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣏⠀⠀⠀⠀⠞⠛⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠿⣿⠀⠙⢷⡀⠀⠀⢀⣠⡴⠿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⠙⠛⠶⣦⣤⣤⣤⣤⣤⣤⣤⡶⠶⣟⠉⣡⠎⢠⡏⠀⣀⣬⣿⠀⠀⠈⠁⠀⣤⣾⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠙⠳⢮⣀⠀⠂⠸⠂⠀⠈⠛⠁⠀⣼⢁⠞⠁⠀⠈⠓⠚⠛⠛⠛⠷⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⠦⠤⠤⠴⠶⠟⠋⠀⠀⠀⠀⠠⠶⠶⠶⠲⣿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣼⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀⠀⠀⠀⣉⣡⡼⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⠀⣍⣿⣿⣿⣿⡖⠚⠛⢛⣻⣧⣤⡽⠋⠉⠉⠉⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠈⠑⠋⢹⠟⠿⣥⣀⣴⠟⡉⠁⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣴⠃⠈⠀⠀⠀⠀⢿⡾⠃⠀⠀⠀⠀⠀⠀⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣀⣀⣠⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣁⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡉⠉⠁⠈⠙⢦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⣰⣿⠁⣀⣤⣶⣶⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣬⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⠿⠻⠿⠛⠉⠛⠛⠉⠹⣿⣿⠟⠁⠀⠀⠘⢋⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣶⡄⢾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢿⣿⣿⡅⣰⣿⣿⣿⠀⠀⠀⠀⠀⠀⣼⣿⡿⠋⠹⡇⠀⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⢸⣿⡿⠡⠟⠋⠛⢿⣆⢀⡆⠀⠀⣀⣿⡟⣀⣀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⡠⢊⣉⠙⠟⠋⠀⠀⠀⠙⠛⢩⣥⠌⢻⡄⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠁⣈⡉⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠰⠋⠀⠀⣿⡟⠓⠲⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠙⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⣇⡤⠤⡌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠊⠉⣢⣹⠇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠃⠀⡀⠀⠀⠀⠀⣾⠠⠟⠲⡄⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⡏⠉⢹⡀⠀⠀⠀⠀⠑⠒⠂⠀⠀⠂⠈⠁⢀⠀⠀⠀⡿⠠⣀⡤⣣⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⡒⠛⡇⠀⠀⠀⠀⣠⠴⠶⠶⡶⠦⠤⢤⣤⠀⠀⠠⡷⠤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣳⠀⠀⠀⢰⠷⠒⣗⣉⣏⣩⣿⣾⣿⠀⠀⢰⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢧⠀⠀⠈⠓⠋⠉⠑⡲⠒⠀⠀⠀⠄⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠀⣴⣦⣷⡀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⣿⣿⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠛⠻⣶⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⣉⠀⠀⠀⠀⠀⠉⠀⡸⠁⠀⠏⠙⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣠⠔⠀⠀⠀⠀⡠⠀⢀⣴⠀⠀⣠⠎⣠⡶⠛⣆⠀⡆⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣥⠞⠁⠀⠀⢀⡤⠊⣀⣴⠟⣿⣠⣾⡵⠟⠉⠀⠀⠹⣆⡿⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢀⣤⣾⣯⠶⠛⠉⠀⠀⠛⠋⠁⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡖⡟⠛⠉⣉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠀⣿⠛⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇⠀⠀⠉⠉⠙⠳⢤⡄⠀⡄⠀⠀⣄⣤⠞⠉⠀⠀⠀⢸⡄⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⣾⣿⣷⡆⠀⢸⠃⠀⠀⠀⢠⣶⣾⣧⠀⠀⢸⡇⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠠⡘⠛⣛⡁⢀⡟⠀⠀⠀⠀⠈⠿⡿⢃⡀⠀⢸⡇⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠉⠁⠀⠸⣇⣀⡀⠀⠀⠈⠑⠚⠉⠀⠀⢸⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⢀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠘⠈⠉⠉⠉⠩⠭⠉⠉⠉⠈⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣄⣀⣀⣀⣀⣀⣠⣤⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⡴⠋⠈⡏⠉⠉⠉⠁⠈⢻⠳⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠞⠛⣻⠃⠀⢧⠀⠀⠙⠀⠀⠀⠀⠀⣸⠀⠘⣿⠛⠳⠶⢦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⡟⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⣿⠀⠀⠀⠀⢀⠈⠙⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⢣⡀⠀⠀⠀⠀⣇⠀⠀⠀⠈⢳⡀⠀⠀⢀⣴⠏⠀⠀⠀⣿⠀⠀⠀⢀⡞⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⢳⡀⠀⠀⠀⢻⣤⡤⢶⡆⠀⠙⢦⣠⠞⠁⠀⡶⠶⣼⠇⠀⠀⠀⠘⠃⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠷⣤⣀⢸⡁⢀⣀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡟⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⢣⡤⠤⠤⠤⠤⠤⢤⠀⠀⠀⠀⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢤⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣄⡉⠛⠲⢦⣄⣼⡄⠘⣇⡀⠀⠀⠀⣀⡼⠀⠀⠀⠀⣧⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⠴⠖⠋⢁⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠛⠻⠶⣦⣤⣽⡇⠀⠈⠙⠓⠶⠛⠉⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣤⣴⠶⠾⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣀⡀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠠⢤⣤⣼⣧⣀⣀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⣿⢸⠀⠀⠀⢀⣀⣠⣤⡶⠾⠛⠛⠋⠉⠁⠀⠀⠀⢸⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⣿⣯⣍⣉⣛⣛⣻⣿⣿⣾⣷⣿⣿⣿⣿⣭⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⣤⣤⣤⣤⣤
⠛⠛⠛⠛⠛⠛⠛⠛⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⡿⣿⡉⠉⠀⠀⠀⠀⠀⠈⠉⠛⢷⣿⠀⠀⠀⠀⠀⠀⣠⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⣀⡙⢶⣝⢷⣤⣤⣤⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠷⠶⠶⣶⣤⣄⣀⠰⢮⣍⡳⣝⣯⣿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠒⣲⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀⣠⠴⠚⠋⠁⠀⠀⠀⠐⠿⠒⠛⠛⠛⠓⠲⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠳⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⣀⣀⣀⣀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣴⠟⠉⠙⠻⣍⠓⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠛⠀⠀⢲⠀⣀⠐⢤⣀⠀⠀⠀⠒⠒⠉⠀⣴⠟⠁⠀⠀⠀⠀⠘⣧⠀⠰⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣄⠠⣈⣆⣨⣦⣄⡈⠉⠉⠀⠀⣀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢈⣙⣿⠿⠋⠀⠈⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡃⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣿⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢿⠀⢰⡿⠛⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠛⣷⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⠀⢸⠀⠈⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠛⠿⣦⢸⠀⠀⠀⠀⢸⣷⣤⡤⠞⠀⠀⠀⠙⠶⣤⣾⣯⠀⠀⠀⠀⢀⡟⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠦⡀⠀⠈⠛⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⢀⣀⠀⠀⠈⠛⠁⠀⠀⠀⠀⢸⡇⣴⠟⠛⠛⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠉⠿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢁⡤⠴⠆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣾⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⡞⢳⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⣛⠒⢠⡄⠀⠀⠀⠀⠀⢀⣴⠾⠛⠋⠉⢁⣀⣀⣈⣹⣧⠀⠀⠀⠀⠀⠸⠀⡼⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢻⡇⠀⠀⠀⠀⣰⣿⣿⣿⠿⠿⣿⡿⠿⢿⣿⣿⣿⠀⠀⠀⠀⢲⠀⠀⢁⣠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢿⣿⠿⠥⠴⢶⣚⣛⣛⣉⣉⣹⡏⠀⠀⠀⠀⢸⡟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⢈⠷⠖⠛⠋⠉⠉⠉⠉⠉⠉⠙⢿⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠲⠖⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⣀⣤⣶⣾⣿⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⡿⣶⣤⣀
⠀⠠⠖⠛⠛⠛⢻⡿⠿⢿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⣤⣤⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⠿⠿⠛⠋⠙⣧⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣿⣿⣴⣶⣿⣿⣿⣿⣿⡟⠛⠉⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠻⣄⣨⣿⠟⠉⠉⠛⠻⢿⡿⢻⡟⠛⠛⣿⠉⢹⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠈⢷⡄⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⣀⣀⣠⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⢦⣤⣤⣤⡴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⠶⠖⠚⠛⠛⠓⣶⠶⠶⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠋⠉⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠙⣶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⡏⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⡆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⣀⣀⣀⣿⣀⣀⣤⣤⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠓⠒⠚⠛⠛⠛⠛⠻⣿⠛⠛⠛⠋⠉⠉⠉⠉⢹⡏⠉⠁⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⢀⠀⠀⣿⠋⠻⣄⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣀⣀⣸⡏
⠀⠀⠀⠀⢸⠀⠀⢻⣆⠀⠙⣧⣠⣤⠶⠶⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠒⠦
⠀⠀⠀⠀⠸⣇⠀⠈⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠈⠳⠶⢤⡤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⣠⣶⠃⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⢰⣿⡇⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢸⣿⠀⢠⣿⠀⠀⠈⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣾⡏⠀⠀⢸⣿⣇⢸⣿⡄⠀⠀⠘⣧⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⢸⣿⡇⠀⠀⢸⣿⣿⡆⠻⣿⣦⠀⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣦⣤⣾⡿⣿⡇⠀⢹⣿⣧⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡀⠸⣿⣷⡙⠛⠛⣵⡿⠁⠀⠀⣿⣿⣤⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠈⢿⣿⢀⣾⡟⠁⠀⠀⢀⣿⣿⣿⣿⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣿⣾⣿⠀⠀⠀⠀⣼⡿⢹⣿⠏⠀⠀⢰⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣦⣴⣏⠀⠀⢀⣿⣿⣿⣤⣀⣤⠞⡉⢀⣿⡏⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠾⣿⣿⣦⣤⣾⣿⢻⣿⣿⣿⣿⠀⡇⢸⣿⡿⣶⣾⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡛⢿⣿⠀⠻⣿⣿⣿⠀⡷⠄⢀⣴⡿⠉⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠹⣧⣀⠁⠰⣆⠙⠛⠋⢰⠁⣴⣿⣿⡀⢀⣠⡿⠃⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⣠⢀⣿⣿⣷⣦⡈⠻⣶⣶⣿⠀⠻⣿⠟⣿⡿⠋⠀⠀⠀⣬⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇⢹⣿⣧⠀⢹⣿⣿⠷⠖⠀⣰⣿⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡙⠛⠋⠁⢸⣿⡏⣴⢿⡟⠁⢀⣴⣾⣿⣿⣀⠀⣀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣦⣄⠀⠘⠋⢸⣿⠘⠀⠀⣿⣿⣿⠟⠛⠛⠋⣹⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆⠘⡆⠀⣿⡀⠀⠀⠈⠉⣡⠖⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣰⣿⠇⢀⣵⠀⣽⠹⣦⣤⣤⣾⣿⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡟⠀⣾⣿⢾⣿⡀⢻⣿⠏⣿⡿⢸⣿⣧⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠄⠘⠿⠄⢉⣤⠆⠁⣰⠟⠡⠾⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠐⠀⣠⠀⠙⠀⡀⠀⠄⠀⠘⠀⡀⠈⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠐⠀⡍⠢⡉⠀⠁⢀⠈⠢⠘⠀⡀⠀⠁⢃⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⢄⠂⠐⢦⡐⢄⡰⢸⢢⡀⠆⠃⠤⣀⠡⡀⠀⡄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠔⠢⡉⠀⠒⡈⠢⠀⣌⢀⠱⡠⢮⠞⠉⠳⡘⠈⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⠒⠀⠁⣈⣲⡄⢰⡄⢹⡄⠝⢦⡹⠁⠀⠉⢠⠃⠀⠂⡠⠂⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡁⢁⣻⡈⠉⠻⡌⠋⡹⠀⠈⠇⠈⢀⣴⠁⠁⠠⠦⠶⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣬⡝⠓⠀⠀⢀⠀⠀⠀⠀⢀⠀⠁⠘⡶⠨⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⡀⠀⢰⡕⠀⠀⠀⣠⡂⠀⢰⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠉⠀⠀⠀⠉⠀⠀⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⢀⠀⠀⠀⠀⠐⠂⡀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢆⠢⠀⠤⠄⢌⡤⡁⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⠀⠀⠉⠂⠄⠀⠈⡰⠑⠀⠀⠈⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⣀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠕⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠨⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠸⢊⠄⠀⠈⠆
⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠂⠄⡰⢑⠀⢀⠀⢠⠀⠀⠀⡆⠀⠀⠀⠀⠀⠤⠀⢸⠁⡠⢀⠀⠘
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢤⢜⡉⠀⠈⠄⠀⣀⠀⠀⠤⠷⠃⣴⣄⣠⠀⠀⠀⣾⡵⣶⢗⡆⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⠒⢔⠀⡠⣿⠶⠶⠢⠂⡄⠾⠀⠀⠀⠀⠒⠘⠿⢓⡽⡂⢸⠁
⠀⠀⠀⠀⠀⢄⠀⠐⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆⠵⠚⠁⠀⠳⣀⠓⠄⠀⠀⠀⠀⠐⠘⡗⢊⠇⠘⡄
⠀⠀⠀⠀⠀⠀⠡⡀⠀⡀⠁⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠈⠀⠀⠀⠀⠀⠀⢩⡄⡠⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠄⠀⠘⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠄⡀⠀⠀⠀⠀⠈⡇⠒⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠢⠤⣀⣀⣀⠀⠀⠀⠁⠀⠀⠂⠂⠀⠀⡔⠁⢂⠄⠔⠀⠀⠀⠀⠀⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⢂⠀⡠⠀⡀⠀⡠⠈⠀⠐⠂⠀⠐⢊⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠔⠈⠔⠁⠠⠀⠠⢊⠔⠀⠔⢁⠔⠠⠁⠌⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠐⠁⠠⠀⠄⠀⠊⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠦⠀⠀⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢋⡏⡄⠀⠔⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⡰⠁⠀⠐⠀⢠⠇⠀⡡⢢⡄⢀⡴⢊⠭⢍⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⡴⢡⠀⡇⡆⡆⡿⠀⠸⣇⠀⠀⠘⠀⣇⣶⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⣇⠰⡀⢧⢸⠸⣿⡀⢆⠉⠳⣄⠐⡄⠻⣝⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠁⠈⠢⣳⡈⢆⢣⠻⣿⣌⢧⢀⠘⣷⠸⡄⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠪⡅⠀⠀⠉⡡⡄⢱⣌⢿⣾⣷⣧⠹⣿⠘⣾⣦⣿⠇⡇⣈⠀⢸⡆⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣺⢩⣛⢦⣿⣿⣿⣶⣿⢀⣿⡿⣫⠞⠁⡡⢀⣼⠃⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠠⡸⠖⠀⢸⠂⠹⣿⣿⣿⣇⣾⣯⡟⢁⣴⣿⡶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠁⢀⠀⠀⢤⠀⢿⡄⢆⠹⣯⣿⣿⣿⣿⣷⣿⣴⣆⡀⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣋⠍⢹⠒⡟⣰⠺⢿⣶⣷⣿⣿⠟⠛⠋⠉⠉⠁⠱⠉⠃⠙⠒⣶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠿⠃⣼⠀⠀⠁⠐⣠⠼⠛⠁⠁⠀⠠⠀⠀⠀⠰⢆⠀⡀⠀⣼⢡⢺⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠷⠶⠶⣧⠤⢤⣞⣭⣀⡀⠀⠀⠀⢀⡴⣋⠀⢤⠶⢦⠶⢾⣧⣷⡘⢒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡗⠀⣤⡤⡀⢀⢸⡀⢀⢮⣹⡆⠀⣨⣾⠛⢿⣃⡏⠠⠄⠀⣼⣿⡍⠹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢉⡇⢀⡛⡇⠀⣀⣸⣧⣐⣊⣿⢷⣿⡁⠠⠀⠘⣿⡷⢦⠤⠼⠋⢙⡃⢁⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠲⠶⢒⣻⠿⢛⠅⠈⢛⢽⠞⠻⢿⣴⣶⠟⠃⠐⠀⠁⠀⠃⠀⠀⠴⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠐⠈⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣶⣶⣶⣶⠤⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠟⠫⠙⠉⠉⠀⠀⠀⠈⠀⠀⠀⠈⠙⠿⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢟⠛⠳⠁⠀⠀⠀⠀⡀⠤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣧⣶⣃⣀⠀⠀⣀⣠⣎⠀⠀⠀⠀⡱⠄⠀⠤⡀⠀⠠⡀⠀⠉⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣿⠟⠈⢛⠟⠣⡴⠊⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠒⠂⡐⢀⣘⠈⠙⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣻⠀⢀⡎⠀⠀⢸⡇⠀⠀⠀⠀⢰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⠢⡮⠛⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣽⣉⣄⠘⣇⡀⠰⣾⣿⣦⣀⣳⣄⢈⣿⣆⡀⠀⠀⠀⢀⠀⠀⠀⠀⠃⠀⠀⠀⠅⠀⢈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⢶⣶⣶⣅⣄⣀⣀⡀⠘⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣮⡻⣿⡿⠻⣯⣽⣟⣻⣿⣿⣿⣿⠁⠀⠀⠀⠀⣛⣿⣿⣿⣿⣯⣭⣭⣿⡿⣛⣴⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠟⠁⠹⣿⣿⣿⣮⣹⣿⠿⢿⣿⣿⣿⡖⢤⠴⠒⠢⣼⡟⣻⣿⠿⢿⣯⣭⣿⣾⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⡿⠟⠉⠀⠀⠀⠀⠋⠙⡿⢿⣿⣿⣆⠀⠹⠉⣿⡇⠀⠀⠀⢠⠉⢠⡿⠁⣠⣾⣿⡿⣋⣵⠟⠛⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⡿⢷⣝⠳⡈⠻⣦⠀⢀⣿⡇⠀⠀⠀⢸⡀⠘⠁⣴⣿⠟⣡⠞⡛⠛⠀⠀⠈⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣻⠿⣷⣬⡑⠮⡂⠈⠷⢸⣿⡇⠀⠀⠀⠈⡇⢀⡼⠋⠠⣪⠥⠬⣡⣤⣤⣤⣤⣤⣭⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣈⠁⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⢹⡄⠀⠀⠈⠁⢀⣈⣭⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⣾⣭⣭⣁⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⢳⠀⠀⠀⠈⣉⣩⣭⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢉⠉⠉⠀⠀⡀⢰⣿⣿⣿⣿⣄⠋⠁⡤⠄⠊⠃⠄⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠐⠀⠠⣀⣀⡀⠀⠻⠟⠉⠉⠙⢿⣷⣿⣤⣄⢀⣀⣄⣡⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢾⣥⣤⣤⣤⣤⣤⡽⠿⠿⠧⣤⣤⣤⡄⠈⠉⠀⠉⡙⢻⣿⣤⣬⣭⣽⣿⣶⣶⣶⣾⠷⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣟⣹⢛⣟⢋⣿⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢿⡄⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⣿⣦⠶⠶⠛⠛⢋⠉⢁⣀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠛⠛⠉⠁⣀⣠⣤⣴⣶⠟⣿⣿⡻⣆⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⢤⡶⠞⣛⣭⠿⠟⠉⠀⣠⠞⠻⣷⡙⣧⡀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠹⣆⠀⠈⢿⡌⢻⣔⣄⣠⣴⡿⠷⠞⣛⣛⣥⡾⠗⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠹⣆⠀⠈⢻⣄⠛⣋⣩⣤⠶⠞⠛⠉⠉⠀⠀⠀⠀⣀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡄⠀⢻⣆⠀⠀⠛⠛⠉⠁⠀⠀⠀⣀⣠⣤⡴⠶⠟⠛⠻⣆⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠹⣄⠀⢻⣆⠀⠀⠀⣠⣤⠶⠞⠛⢉⠁⠀⠀⠀⠀⠀⡀⠹⣧⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠘⣆⠀⠹⣆⠀⠀⠻⣆⠀⠰⣾⡿⠿⠀⠀⠀⠀⢸⣿⡄⠘⣷⡀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠘⢦⠀⠹⣧⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠈⢧⡀⠙⣧⡀⠀⠘⣷⡀⠀⢶⣿⡿⠆⠀⠀⢀⣀⣤⡴⠾⠃⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠈⢳⡀⠘⣷⡀⠀⠈⢷⡀⠀⣀⣤⡴⠶⠟⠋⠉⢀⣀⣠⣤⢶⣄⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠳⡄⠈⢷⡄⠀⠈⠛⠛⠉⣀⣠⣤⠶⠾⠛⠋⠉⠁⠀⠀⢻⣆⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠙⣄⠈⢻⡄⠀⠀⢸⡟⠋⢉⣠⣤⣦⡀⠀⠀⠀⢠⣶⡀⠹⣦⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠈⢦⠀⢻⣆⠀⠀⠻⣆⠀⠙⠋⠉⠀⠀⠀⠀⠈⠛⠛⠀⠙⣧⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠈⢣⡀⠹⣦⠀⠀⠹⣧⠀⠀⢀⣤⣶⡄⠀⠀⠀⠀⠀⢀⣸⣷⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠳⡀⠙⣧⠀⠀⠙⣧⠀⠘⠛⠋⢁⣀⣤⣴⠶⠛⠛⠉⠀⠀⠀⠀⣹⣧
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠹⡄⠈⢷⡀⠀⠘⣷⣴⠶⠟⠛⠉⠁⠀⣀⣀⣤⣤⡶⠶⠟⠛⠉⣿
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠙⣆⠈⢻⣆⠀⠀⣀⣀⣤⣴⠶⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠘⣆⠀⢹⡟⠛⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⣠⠤⠶⠚⠋⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠈⢦⣸⡇⠀⠀⣀⣠⡤⠴⠚⠛⠉⠉⠀⠀⠀⠀⠀⢀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⢻⡖⠛⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠛⠋⠉
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣸⡇⠀⠀⢀⣀⣤⣤⠶⠞⠛⠉⠉
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡶⠶⠛⠋⠉
⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠿⢋⣭⣾⣿⣿⠟⠙⣿⣿⣿⡙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟⢁⣴⣿⣿⣿⣿⠃⠀⢀⣿⣿⣿⣷⠀⠀⠻⣿⣿⡿⠀⠙⢿⣿⣿⠏
⠀⠀⠀⠀⠻⣿⣿⠟⠁⠀⠸⣿⣿⣿⣿⠃⠀⠀⠈⢿⣿⡿⠏⠀⠀⠀⠀⠉⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣷⠄⠀⠀⠘⣿⣆
⠀⠀⠀⢀⣿⡏⠀⠀⢀⣠⣴⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⢀⣸⣿⠀⠀⢠⣿⠟⠉⠉⠉⠙⠿⣷⡄⠀⠀⠀⠀⣠⣾⠟⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⣴⣿⠿⣿⣿⠀⠀⠀⠁⠀⢠⣾⣿⣦⠀⠈⠁⠀⠀⠀⠀⠙⠋⠀⠀⢾⣿⣿⡆⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶
⢿⣧⣀⣿⣿⠀⠀⠀⠀⠀⠈⠻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣹
⠈⠛⠻⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⠛
⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⠿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠙⣿⣦⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣦⣤⣤⣀⣀⣀⣀⣠⣤⣴⣶⡿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⠀⠄⠀⠀⠘⢌⠢⠀⠀⠰⠀⠠⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠊⠀⠘⡠⠀⠀⠀⠠⠑⠀⠀⠀⠐⢄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠂⠀⠀⠀⠀⠀⣄⠀⠀⡰⠁⠀⢹⠀⠀⠀⣀⠀⠈⠀⠀⠀⠄⠁⠄⠘⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⢀⠐⢀⠁⠳⠉⢦⣠⠁⠀⠀⢸⠀⠲⢀⡶⢱⠀⢀⠀⠸⠎⠀⠀⡄⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠨⠀⡉⠀⠀⠹⡀⠀⠀⠸⠀⠀⠰⠁⠀⡇⠈⠴⠃⠐⡆⠀⠀⠀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢾⠣⣄⣠⠀⠀⠀⠀⠀⠀⠀⢇⠀⠃⠀⠀⡇⠂⠀⠀⢠⠃⠀⠀⣀⢀⡀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠈⠣⡄⠀⠀⢇⠀⠀⢠⢃⣄⣀⣤⡔⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡞⢈⠼⠊⢰⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠙⠀⠀⠀⠀⠀⢄⠀⠠⠀⠀⠀⠀⠀⢀⠔⢠⠀⠀⠱⠁⠀⠰⠁⡾⠁⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣄⣅⣀⠀⠀⠀⠀⠀⠀⠃⠄⠂⠖⠄⠀⡀⠈⠠⠄⠂⠀⠀⠠⠴⠋⠉
⠀⠀⠀⠀⠀⠀⢐⡖⣻⣉⣉⠉⣩⣽⠶⣀⠀⠀⣀⣠⠤⢤⣀⡄⡦⠀⡠⢦⠐⠠⣤⢄⠂⠠⠾⠿⢷⣿⡂⡀⢰
⠀⠀⠀⠀⠀⠀⠹⠷⠼⠷⠶⠚⢧⡇⢠⣾⣶⣛⡻⣄⣵⢠⣼⠃⣓⢲⠀⢘⣧⡂⠚⢺⣇⣀⡀⡶⡞⣮⣭⠴⣾⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠙⠛⠛⠁⠀⠙⠶⠖⠛⠉⠙⠳⠚⠋⠀⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⠶⠋⠟⢻⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠚⠛⠛⣻⣿⡛⠁⢠⣤⣤⣉⡛⠿⣷⣤⣄⠈⠉⠛⠿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢋⡵⠚⠉⠀⠀⠞⠋⠁⠐⢛⡛⠛⢿⣯⣉⠁⠠⠿⣿⣦⡀⠐⠶⣮⡙⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⢃⡼⠁⢀⡄⠤⠶⠶⠚⠴⣾⣿⣯⣁⠲⠿⠿⠛⠃⣢⡀⠻⣿⣿⣆⠀⠈⢿⠂⠈⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠈⠛⢢⣿⣾⣶⣶⠶⠖⠀⠀⠠⠾⠟⠛⠃⠀⠀⠈⢻⣷⠦⠀⠙⣿⣷⣦⡀⠀⠀⠀⠙⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠒⢦⣤⢈⣫⣿⣿⠟⣇⠀⠀⠀⠈⠉⠛⠳⢦⣀⠀⠀⠀⠀⣿⣦⣀⠚⠻⠿⠟⠃⠀⠀⠀⠀⠠⡘⣧
⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠋⠉⠁⠾⠷⠦⢍⠒⠾⠃⣠⠀⠀⠀⣀⣀⣀⣀⣙⣦⠀⠀⠀⣿⣿⣿⣧⡀⠘⣷⠞⠁⠀⠀⠀⠀⢷⠸⡆
⠀⠀⠀⠀⠀⢀⡼⣷⣋⣤⠴⠒⣁⣤⣤⣤⡗⢠⣌⣙⠛⠶⢤⣿⡞⠛⠿⠀⢨⣭⡤⢀⣴⠟⢸⣿⡏⠃⠀⢹⠀⠀⢀⡀⠈⢶⣼⡇⠃
⠀⠀⠀⠀⢀⡾⠁⠈⠉⢀⡴⠛⠉⠀⠒⠈⠁⢤⡌⠙⠿⣤⣀⡈⠉⣀⣶⡾⢛⣿⠁⣾⠃⠀⢸⢻⠁⠀⠀⣾⠀⠀⠀⢷⣀⠈⣿⠓
⠀⠀⠀⠀⢸⠻⣶⣶⡶⠛⣠⣴⣾⣿⣿⣿⡆⠈⣁⣀⣀⡀⠉⠛⣿⣿⠁⢀⣸⣿⡄⣿⡀⠀⠀⠸⠀⠀⠀⢸⢰⠀⠀⠈⢿⣷⣽⡆
⠀⠀⠀⠀⠸⣦⣨⣿⣦⣴⣶⣿⣿⣿⣿⣯⣤⣤⣬⣍⣀⣡⡴⠂⠸⣿⠀⢸⣿⣿⡇⢹⣧⠀⠀⠀⣷⢠⠀⠀⠸⣇⠀⠀⠀⠙⣿⠁
⠀⠀⠀⠀⠀⠈⠻⡖⢒⣋⣭⣴⣎⠉⢻⣿⣿⣿⡿⣿⡏⢥⡤⠀⠀⡿⠀⠘⠁⣿⡇⢾⡏⢀⠀⣀⡏⢸⠀⠀⠀⢿⡄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⢄⢫⠉⠻⣿⣆⠘⣿⡇⢻⡇⣿⣷⡄⠁⠀⣰⡇⠀⠀⠀⡟⢀⡾⠁⣼⠀⢹⠇⣾⠀⠀⢰⡜⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠧⠉⢿⠹⣧⣽⣿⠘⡇⢸⣿⣿⡀⠀⢹⠇⠀⠀⠀⢀⡾⠁⣴⠏⠀⠀⢰⡏⠀⠀⣼⠇⣿⡇⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⢱⣿⡝⠿⠀⠁⣼⣿⣿⡇⠀⠀⠀⠀⠀⣠⠞⢀⡼⠋⠀⣠⢀⣿⠁⠀⠚⡟⠀⣿⠇⡀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡙⣛⣿⣤⣤⣾⣿⣿⣿⢃⡜⠁⠀⠀⠀⠀⠀⡜⠁⠀⣠⣿⡾⠃⠀⠀⠀⠀⣼⣿⡾⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠉⠉⠀⠀⠘⠻⢫⣾⣀⡾⢸⣠⣾⡇⠀⣀⣴⣾⣿⠟⠁⠀⠀⠀⠀⣼⣿⠟⠁⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣤⣀⡀⠀⣀⣴⠏⠈⡿⠁⠸⡿⢿⣷⣾⣿⣿⠟⠁⡀⠀⠀⡀⢀⡾⠋⠁⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠛⠛⢿⡟⠀⠀⠀⠀⠀⠁⠈⠻⣿⠟⠁⣠⣾⣴⡶⠞⠁⡜⠁⠀⢠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣧⡴⠂⠀⠀⠀⣠⠀⠀⠉⠀⡼⠋⠀⠁⠀⢠⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣁⣀⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⣀⣴⡟⣠⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⣽⣿⡞⣠⡾⠀⣠⣶⣿⣾⠟⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠿⣿⣿⣿⣥⣾⢻⣿⡿⠋⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣿⡿⠁⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⢧⡤⢴⢒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣈⣞⠃⠤⠖⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⠿⠛⢏⣿⡿⣿⡄⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠉⠒⣿⡽⢿⣴⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⢿⡴⠚⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠘⠅⠽⢿⠿⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣡⢄⣿⡄⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⠿⢿⣻⡟⣿⣶⣿⣅⣞⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠈⠛⡫⢿⣿⡿⣗⣆⣠⣄⣀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣾⣧⣤⣥⣬⣭⣥⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣩⣍⣩⣬⣩⣏⣽⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠲⣄⠀⢀⣀⣀
⠀⠀⠀⠀⠀⡀⠻⣦⣀⣀⣴⠗⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⠖⠉⠀⠈⢳⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠙⢷⡀⠉⠉⠁⣴⠟⠁⠀⠀⢠⡀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⡎⠉⠱⠔⠒⠲⡀
⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠘⣧⡀⠀⠀⠲⣄⠙⠶⠴⠞⢁⡶⠂⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡆⠀⢠⠤⠧⠀⠀⠀⠀⣴⣁
⠀⠀⠀⠀⠀⠛⢁⡴⠛⠛⢦⣈⠛⠂⠀⠀⢘⡇⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠀⠀⣠⣄⠀⡗⠒⠋⠀⠀⠱⣄⣀⠀⠀⠀⠀⠀⠀⣳
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠀⣠⠟⣡⡴⢦⣄⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠊⠁⠈⢆⣧⠀⠀⠀⠀⠀⠀⣼⠔⢣⡀⢨⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⡴⠋⠀⠀⡁⠀⠙⢦⠀⠀⠀⠠⠿⠿⠿⠭⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⢈⠀⠈⠑⣄⠀⠀⠀⠸⠿⠿⠯⠭⣷⣦⡀
⠀⠀⠀⡸⠀⠀⣠⠞⠑⢄⡀⠀⢣⠀⠀⠈⣹⠋⢍⡉⢉⡠⠘⡆⠀⠀⠀⠀⠀⠀⠀⡎⠀⢀⡤⠋⠳⣄⠀⠈⡆⠀⠀⠈⡟⠩⣉⠉⢉⠠⢹⡀
⠀⠀⠀⣇⣴⠉⢱⠄⠠⣆⠉⣷⣸⠀⠀⢸⣹⠀⠺⢀⠀⠇⠀⣟⡆⠀⠀⠀⠀⠀⢸⣠⡞⠉⣲⠀⢠⡎⢙⣆⣹⠀⠀⢎⡇⠀⠟⠀⠸⠃⢀⣟⠄
⠀⠀⠀⢻⠺⣆⠈⡀⠀⠉⣠⠧⡻⠀⠀⠀⠙⣆⢠⠀⠀⠄⣠⠋⠀⠀⠀⠀⠀⠀⠀⡗⠿⡄⠉⣐⠀⠁⡼⠞⡏⠀⠀⠀⠱⡀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠈⠣⡈⢳⡶⢶⡞⢁⡴⠁⠀⠀⠀⠀⠈⣳⠯⠼⢞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⣲⠤⢴⡊⣠⠞⠀⠀⠀⠀⠀⠙⡶⠤⡶⡎
⠀⠀⠀⠀⠀⢨⠏⡸⢆⡸⡏⠀⠀⠀⠀⠀⠀⣰⠋⠜⠣⠌⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⢇⠰⣀⢿⡁⠀⠀⠀⠀⠀⠀⡞⠡⠘⠤⠙⡆
⠀⠀⠀⠀⢀⡏⢆⠀⠀⡠⠹⡄⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠓⡄⠀⠀⡌⢳⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⡼⠀⠀⢂⡰⠁⠀⢳⠀⠀⠀⠀⡼⡄⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢀⡇⡀⠘⢄⠌⠀⡈⡆⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⢠⡇⠀⠀⠀⠁⠀⠀⠸⡄⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⢸⠁⡇⠀⠈⠀⠀⡇⢹⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⡇⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡼⠀⡇⠀⠀⠀⠀⡇⢸⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⡇⠸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⢸
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠋⠀⠀⠀⠀⠀⢀⣠⣴⢶⣿⣿⣯⣷⣶⣶⣶⣾⣿⣿⠿⠿⠿⠿⠿⠿⠛⠿⢿⣷⣤⠀⠀⠙⢶⣤⡀
⠀⠀⠀⢀⣤⡾⢛⣩⡶⠛⠛⠉⣩⡽⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢤⣀⠀⠀⣿⢿⣷⣄
⢀⣤⠶⠛⣡⡴⠋⠉⠀⣀⡼⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠰⣿⢿⣢⡀
⠟⣁⣴⡞⠁⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⣄⠀⠀⠀⠘⢿⣧⣿⡛⣄
⣾⡟⠁⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⠀⣻⣿⣿⣿⣿⣷
⠏⠀⠀⢀⡴⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠉⠉⠉⠙⠳⣦⣆⠀⠘⢿⣿⣿⠟⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿
⠀⡰⣷⡟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠀⠀⠀⠀⠀⠀⠀⣌⣻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡉⣿⣿⣿⣿
⣼⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣫⣛⠁⣀⣠⡤⠀⠁⢀⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣿⣿⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢙⣷⣿⣿⣇⣠⣴⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⣿⣿⣿⣿
⣿⡿⢠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠛⠘⠲⠿⣿⣿⣿⣿⠋⠉⠁⠀⠀⠀⣠⢀⠔⣄⣠⠀⠀⢈⣿⣿⣿⣿⡟
⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢼⣮⣅⣾⣿⡟⠉⢻⡆⠀⣠⠄⢀⡼⣱⠏⠎⠀⠀⠀⣀⠈⣼⣿⣿⣿⠇
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠹⣿⠉⣣⣔⣧⣨⣧⣼⣿⣴⣿⡿⡃⠂⠀⠀⠀⠈⠛⠆⣿⣿⣿⡟⠘
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⡰⠃⣴⣿⣿⣿⣤⣿⣿⣿⣿⣿⡿⠁⡀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⣶⣿⡿⠁⢀⣿⣿⣿⠿⣿⣿⣿⣿⣿⡿⣯⠞⠁⠀⡀⠀⠀⢀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠿⡿⠋⡀⢠⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠟⢂⠀⠀⢀⡼⢃⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣷⡿⠛⡟⠀⠈⢙⣦⡙⣿⠃⠙⠻⣿⣿⣿⣿⣿⣿⣷⣄⠀⢠⡔⣴⡟⢱⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⢸⠃⠀⠀⠈⠻⣿⣿⠀⠀⠀⣿⣿⠀⠉⠛⠻⣿⣿⣷⣶⣶⠏⠘⣾⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠁⣰⡍⠨⠃⠀⠀⠀⠈⠛⠀⠀⠀⣙⠘⢄⠀⠀⠀⠀⢛⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠾⠀⠀⣸⣿⠛⣻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⣶⣾⡿⢃⣰⣿⣿⣿⡟⢻⣷⢀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣯⣴⣿⣿⠏⣹⠏⠀⢨⣿⣾⠇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠐⢀⠀⣴⣦⠄⠀⠀⢀⣈⣤⣾⣿⣿⣿⣿⣿⡿⣿⣴⠊⠁⠀⠀⣸⣿⠏⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣽⣿⣿⣿⣷⣀⣠⣴⣿⣿⣿⢿⣻⣿⣿⡿⠋⢐⡁⠀⠶⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⣼
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣭⣴⣿⣿⠟⣿⣦⡴⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢿⣯⣿⣿⣿⣿⣟⣏⣥⣾⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠐
⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿⣿⡿⢟⣩⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢾
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠝⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣩⣾
⣄⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⠟⠀⠀⠀⠀⣀⣤⢸⣿⣿
⣿⣷⣤⡀⣀⠀⡀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⣤⣶⣿⣿⡿⠛⠁⢀⣤⣶⣾⣿⠿⠃⠻⢿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣟⡿⣽⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⢻⣿⣿⣿⣸⣿⢿⣿⢣⣿⣿⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣋⣥⣶⣾⣆
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣛⠿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠈⢿⡿⠿⠇⡿⠋⠉⠙⠻⣿⣿⡿⠟⣋⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣍⣛⡿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠈⣿⣶⣿⠿⠿⢛⣛⣭⣭⣶⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠶⢦⡿⢿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠁⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣧⣄⣀⠀⠀⠀⠀⢹⣆⣀⣀⣤⣄⣀⣴⣿⡏⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣦⣴⣦⣤⣤⣽⣿⣿⣶⣤⣤⣴⣿⣿⡿⠿⠛⣿⡟⠛⢉⣠⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⣽⣿⣿⣿⣿⣿⣷⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⣿⣿⣿⣦⣤⣾⣿⢹⠉⢹⢸⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣙⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣉⣉⡁⠀⠀⠀⠀⠀⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⣛⣻⣿⣿⣿⠃⠀⠀⠀⢰⡇⣿⣿⣿⣿⣿⣿⣿⣟⣉⡉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢩⣭⣭⣭⣭⣭⡿⠀⠀⠀⠀⢸⡇⣿⣿⣯⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡆⢸⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣦⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣆⣀⣤⣄⣀⡀
⢀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⣼⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣧⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣾⣿⣿⣿⡿⠋⠀⣾⣿⣿⣿⣿⣿⣿⡆⠀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⣠⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠻⣿⡿⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢹⡿⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣷⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠻⣿⣧⡀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠹⣿⣷⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⣼⠀⠀⠀⠀⢻⣿⡇⠀⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⣿⡆⠀⠀⠀⢸⣿⠇⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⣄⣀⣠⣿⡿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢹⣿⣿⣿⣿⡿⠁⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠙⠀⠈⣿⣿⣿⣿⠃⠀⢀⣾⣿⠏⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⢹⣿⣿⣿⣷⣶⣿⣿⡟⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⢸⣿⣿⡻⣿⣿⣿⣿⡇⠀⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡀⢀⣼⣿⣿⡇⠸⣿⣿⣿⣿⡄⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠘⠛⠁⢰⣿⣿⣿⣿⣿⡀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⣿⣿⡄⠀⢰⣿⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠁⣿⣿⠃⠀⠈⠛⠀⢻⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠈⠁⠀⠀⠀⠀⠀⣸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣧⣀⠀⠀⠀⢀⣠⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⣤⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⣠⣦⣀⠀⠀⠀⠈⠉⠒⠀⠀⠀⠁⠀⠀⠸⠇⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠙⢦⣄⠀⠀⣼⠀⠀⠀⠈⠈⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢤⣠⣴⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢠⣿⣵⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢂⠀⠀⠀⠀⠀⠀⠙⢷⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⣀⠀⠀⠀⢀⠐⠢⣄⠀⠠⠈⠠⣄⠀⠀⠀⠊⠁⣾⡏⠀⠀⠀⠀⢠⡄⢀⣀⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⣈⡀⠀⠈⠙⠀⠀⠀⠙⠃⡀⠀⠀⣸⡿⠂⠀⠴⠃⠀⠹⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠤⠂⠂⠉⠉⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⣰⣿⣿⠋
⠀⠀⠀⠀⠠⠶⣎⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠉⠉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⣼⠃⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄⠀⠀⠀⢀⣠⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠉⠉⠁⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡉⠉⠁⠀⠀⠀⠀⠀⠙⢿⣅
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡶⠶⢶⣯⡴⠇⠀⠀⠤⠤⣿⣟⣠⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠒⢄⡈⠳⣌⢀⠠⣼⠓⠀⢀⣿⠃⠀⠀⠀⠘⣧⡀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠈⢳⡄⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠈⠓⠀⠀⢹⣿⡟⠛⠛⠛⠓⠒⠛⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⣿⡀⢀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣷⣿⠿⠆⠀⠀⠀⣴⣿⡷⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⣿⡟⠁⠀⠈⡄⢀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢠⣾⣿⠇⠀⠀⠀⠹⠃⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣶⣾⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉


⣿⠿⠿⠿⡿⠟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣻⣟⣻⣟⣿⣿⣿⣿⣿⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⠶⠶⠶⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⣿⣿⣆⡀⠀⠐⠀⠐⠀⠀⠀⠀⠐⠀⠂⠂⠀⠐⠀⣀⣸⣿⡇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣏⠙⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡟⢉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠳⣿⣿⣿⡏⣹⣿⣦⠀⠀⠀⠀⠀⡀⠀⠀⢀⣴⣿⡏⣻⣿⣿⡷⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣿⣿⢿⣿⣿⣛⣩⣷⣄⣠⠃⠀⢱⣄⣤⣿⣍⣻⣵⣿⠿⣿⡇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡇⠑⠒⡲⠛⠉⣠⠂⠉⠉⠉⠉⠁⢰⣄⠉⠛⢖⠒⠊⢻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣶⡞⠁⣠⣾⠻⢶⣤⡀⠀⢀⣴⠾⠟⣷⣄⠈⢷⣶⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⣴⢋⡏⠀⠀⢹⠿⢿⠿⠁⠀⠀⢸⡝⣷⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠃⢸⠃⠀⠀⠸⣷⣤⡀⠀⠀⠀⠀⣷⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣾⠀⠀⠀⠀⢉⣩⣴⠶⠶⠶⣼⣿⡆⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇⣻⣦⣤⣴⣾⣿⣥⣶⣴⣷⣾⣿⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣤⣀⡀⠀⠀⣀⣀⣤⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡟⠛⠉⠉⠉⠉⠛⢻⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠐⡇⣀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠉⠉

⠀⠀⢠⡶⠞⠛⠻⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀
⠀⠀⢸⡇⠀⠀⠀⠀⢿⡟⢧⡀⣀⣀⣀⠀⢀⡼⢿⡟⠉⠀⠈⢻⣆
⠀⠀⢸⡇⠀⠀⠀⠀⠈⣿⣬⣿⠿⠉⠉⠛⢿⣷⡞⠁⠀⠀⠀⠀⢻⡆
⠀⠀⠀⢻⡄⠀⠀⠀⢰⣿⣿⣿⡶⠄⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢻⡄⠀⢰⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⢻⡄⠈⠉⠉⠉⠉⠉⠙⠛⠛⠋⠉⠉⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⣀⣀⣀⣀⡀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣀⣀⣀⣀
⠀⣿⣿⣿⣿⣇⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣇⣿⣿⡿⣿
⠀⣿⣿⣧⣸⣿⣾⣿⣿⣿⣿⣟⣛⣛⣛⣛⠛⠛⣻⡿⢿⣿⣿⣿⡇⣿
⠀⣿⣿⣿⡛⣿⣿⣿⣿⣛⣛⣛⣛⣻⣿⣟⣛⣿⣿⣛⣻⣿⣿⣿⠃⣿
⠀⣿⡇⣿⡇⣿⡏⠙⠛⠛⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠿⢿⣿⣿⡆⣿⡆
⠀⣿⣇⣿⢧⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢘⣿⣿⡇⢻⡇
⠀⢻⣿⠹⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⡇⢱⢸⡇
⠀⢸⣿⡇⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⣿⣿⣟⣿⣿⣿⢸⡇
⠀⢸⣿⠇⡇⣿⡿⠟⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠷⠻⣿⣿⣿⢸⡇
⠀⢸⣿⡀⣧⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⢸⡇
⠀⢸⣿⡇⡏⣿⣷⣶⣶⣶⣶⣶⡶⠶⣶⣶⣿⣿⣿⣷⠶⣶⣿⣧⢻⣸⣇⠀⡀
⠀⢸⣿⡇⡇⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⢸⣿⣿⣷⣿⣿⡟⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠟⢿⣿⣿⣿⣸⣿⣿⡟⠿⡟⣿
⢰⣾⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣽⣿⣦⣤⣀
⢸⣿⣿⠛⣴⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⣿⡟⢿⣿⣿⣿⢿⣿⣿⢿⣿⣷⡀
⠘⢿⣄⠻⣿⣷⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣇⠙⡇⠀⠙⢿⣿⣿⣯⣿⡛⠿⣿⢷⡀
⠀⠀⢹⣦⡘⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣶⣶⡿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠹⣿⣿⣷⡝⠳⢿⣧⣇
⠀⠀⢸⣿⣿⣿⣿⣿⡿⡝⢟⣿⣿⡇⠈⢿⣿⣮⡳⣿⡗⠹⣿⣿⡽⣿⣿⣷⣄⠀⠀⠙⢿⣿⣽⣷⣾⡿⠿⠁
⠀⠀⢘⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⡇⢪⡻⣿⣿⣾⣿⣦⡙⠻⣿⣮⠻⣿⡻⣷⣄⠀⠈⢻⣿⡅⣿⡇
⠀⠀⠀⣿⡿⠙⣿⣿⣿⣿⣿⣿⡿⠼⢿⣆⠑⡌⠈⢿⣿⣯⠿⣆⠈⠻⣷⡌⠙⢿⡿⣧⡀⢸⣿⡇⣿⡇
⠀⠀⠘⣿⢇⢸⣿⡎⢻⣿⣿⡿⣷⣀⣈⣻⣷⣌⣢⣀⣻⣿⣧⣌⣧⣄⡙⣿⣦⣠⡝⣎⣻⣦⣿⣇⣿⡇
⠀⠀⠀⣿⡞⡀⣿⣷⣼⣿⢿⡏⣿⣿⣿⣿⠿⠛⠛⡛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⠃
⠀⠀⢸⣿⡇⣽⣿⠀⢻⣿⢺⣇⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⡄
⣶⣶⣾⣿⡇⢸⣿⣤⣼⣿⢸⣿⣹⣿⣤⣤⣀⣀⣀⣀⣀⣸⣿⣿⢠⣿⣄⣀⣀⣀⠀⠀⠀⣹⡿⣿⣿⡇
⠸⣿⣿⣿⡇⢸⣿⣿⣿⣿⠈⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡟⣿⣿⠿⡛⠛⢿⣿⣿⡇⢿⣿⣿⡶⣶⣦
⠀⢿⡿⢿⣿⣿⣿⣿⣿⣿⠀⢷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣷⣦⡀⠀⠀⢸⡇⢸⡿⣿⡇⠉⠹⣧
⠀⠸⣧⠈⠻⣿⣿⣿⣿⣿⣾⣼⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣿⡇⢸⡟⣿⡇⠀⠀⠹⣇
⠀⠀⣿⡀⠀⠀⠙⢿⣿⣿⠗⣿⣏⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣸⢨⣿⣿⡀⠀⠀⠀⢻⣆
⠀⠀⠸⣧⠀⡀⠀⠀⠙⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠆⠈⠦⠀⠀⠰⢄⠈⠙⠻⢿⢿⢿⡿⠿⣿⠇⠀⠀⡀⠀⢻⡆
⠀⠀⠀⢿⣇⠙⡄⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠤⠤⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢈⣿
⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⢿⠿⢿⡿⠛⡿⠿⣿⣿⠿⣿⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣶⣼⣷⣴⣧⣤⣿⣴⣼⣿⣴⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠃⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⡃⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠀⠀⠀⠀⠀⢀
⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠄⠀⢤⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠆⠀⠀⠐⠃⡀
⠀⢀⣀⠀⠐⣬⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠐⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⠀⣠⠏⠀⠀⠀⢀⠴⠀⠀⣁⠈⠀⠀⠀⢀⠀⠀⡴⠋
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⠀⠀⠀⠀⠀⠀⣠⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⢀⣀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠠⠆⢀⢀⡈⢉⠀⠀⠉⡠⠆⠀⠃
⠀⠀⠀⣀⠀⠀⠀⠋⠀⠈⠛⠂⠴⠓⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⠃⠀⠀⡄
⠀⠀⠂⠌⠀⠀⣀⢀⡆⠀⠀⠀⠀⠀⠀⠠⠀⠀⡒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠀⠀⠀⠀⠐⠐⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠆⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠛⡒⠤⠀⠀⢸⠀⠀⠀⠀⠂⢀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠀⠀⡀⢀⡠⠚⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠂⠀⠀⠀⠄⠄⠀⠀⠀⠀⡀⠀⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢡⠀⠀⠀⡇⠀⠀⠀⢀⠌⠀⠠⠒⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⡀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠱⠄⠀⠣⡀⠐⠁⠿⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠐⠀⣀⠀⡀⠀⠀⠀⠀⠋⠒⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⡴⠒⠀⠀⠀⠀⠀⠐⡌⠓⠀⠃⠀⠀⠀⠀⠀⠁⠂⠀⠀⠐⢢⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠐⣀⣀⡄⠐⠀⣠⠞
⠀⠀⠤⠀⠀⠓⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⣴⢟⣉⡁⢀⠀⡀⠀⠀⠀⠀⣉⡡⠤⠔⠋
⠀⣀⠀⠀⠀⠀⠀⠀⢠⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠷⠀⠀⠀⢀⡤⠀⡆⠀⠔⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠒⠋⠁⠐⢀⣡⣌⠢⣀⡤⠟
⠀⠉⠀⠀⠀⠀⠀⢠⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠚⠋⠉⠉⠈⠉⠈⡧⣈⠂
⠀⠀⠀⠀⠀⠘⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡀⣠⡄⠀⠹⡀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⡰⠃⠀⠀⠀⠄⠚⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢢⡀⠀⠈⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⠋⠀⠀⠀⠀⠀⠀⠀⡀⢠⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠢⢌⠙⢤⠀⠀⠀⠀⠀⠁⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠂⠀⠀⠀⢇⠀⠀⠀⠀⢤⣀⠀⠀⠀⣠⠂⠁⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠒⠒⠀⠀⠀⠀⠀⠐⢄⠀⠈⠀⠀⠀⠀⠀⢀⠀⡸⠁⠀⠀⠀⠀⠈⠗⢠⡎⠀⠀⠀⡀⠀⠀⠀⠀⠈⡦⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⡤⠒⢀⠀⠄⢀⠀⣀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⠤⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⢀⠤⠀⠀⠀⠀⠃⠀⠀⠠⠀⠸
⠀⠀⠀⠀⠁⠊⠀⠀⠀⠈⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠄⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣀⠀⢀⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠐⠰⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠠⠀⠄⢀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡈⢌⡀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⠿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⢠⠴⠊⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⢀⣠⣤⡤⠤⠀⠤⠄⠀⠀⠁⠀⠀⠀⡏⠀⠀⠀⠀⢀⣠⣤⣶⣞⣿⣿⠿⠗⠛⠋⠁
⠀⠀⢠⡶⣏⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⢾⣟⣿⣿⡿⠟⠛⠉
⠀⢀⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⣳⡾⠟⠛⠉
⠀⢸⠁⠀⠀⠀⢠⣤⡄⠈⠀⠁⠀⠀⠀⠀⠀⠈⡘⠁
⠀⠈⠀⠀⠀⠀⠈⣿⣿⡆
⠀⠀⠆⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠘
⠀⠀⠀⠰⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠈⠢

⣿⡟⣛⡿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢠⢹⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣿⡟⡆⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣦⢸⢸⠿⡆⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢰⢿⠇⡇⣴⠀⠀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⣿⠀⡼⠹⠘⢸⠀⣷⠀⠀⠀⠀⣶⡰⣭⡆⠀⠀⠸⠈⡇⣇⠏⢣⠀⣿⡝⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣾⣡⣾⢸⣧⣙⣡⣧⠸⣤⣿⠀⠀⠀⣰⣿⣷⣜⠃⠀⠀⣷⡆⡇⣼⣌⣋⣼⠉⣧⣜⣷⠀⣷
⠀⠀⠀⠀⠀⠀⠀⢸⠻⣼⡟⣷⢧⡄⡇⢹⣥⣨⢣⣿⣾⠀⢀⢴⣿⡏⣿⢻⡦⡀⠀⣷⣿⡜⣅⣬⠋⢸⢰⡹⡿⢿⣧⢿⡇
⠀⠀⠀⠀⢠⡄⠀⢸⣦⣹⣷⠏⣿⠀⣿⣷⢻⡏⣿⠇⡿⢀⠎⣼⢹⡇⣿⢼⢃⣵⡀⢿⠸⣳⢹⡏⣾⣿⠘⣷⠻⣾⣿⢬⡇⠀⢠⡄
⠀⠀⠀⢠⡿⠀⠀⢸⣟⣹⡏⢆⡏⢸⡏⡏⡌⠑⡎⡀⡇⢸⡟⣿⠘⣇⢹⣧⣸⣿⡇⢸⢀⣿⠸⢡⣿⢻⡆⢣⢁⢹⣏⣾⣇⠀⠀⢿⡄
⠀⠀⢠⠏⣧⠀⣠⠇⡻⣿⣱⡟⣾⢸⡇⢣⢻⣄⠁⡇⢿⡜⣧⡇⣶⢻⡀⣿⣿⣼⣣⡟⣼⠈⣰⡟⡏⢸⡇⣧⢿⡆⣿⣿⠹⣄⠀⡸⢹⡄
⠀⠀⢸⢠⡘⣿⣡⣞⢠⡿⠿⢹⢿⢸⡇⣼⣆⠛⠆⢿⢸⣙⣿⢡⢿⣾⡇⢹⡇⣿⠉⡇⡿⠸⠋⣰⢧⢸⡇⣿⠘⠃⢿⡈⣣⣎⣿⢃⡌⡇
⠀⢠⣼⢼⡇⣹⡶⡽⣾⣷⣇⢸⡿⢸⡇⡟⡿⡜⡟⡌⣼⣿⢃⡿⢸⢸⣇⣼⡇⣏⠿⣏⢃⢿⢱⢿⢸⢸⡆⢿⠀⣼⣾⣇⢯⢶⣏⣼⢱⣧⡄
⢀⡾⢻⡏⣱⠏⣀⡰⣻⢿⡟⣧⢿⣸⠃⠀⠁⡇⡷⣱⢋⣿⣏⣷⡼⢸⡿⣿⢱⢟⣄⠹⣎⢾⢸⠈⢀⠘⡇⠿⣾⢹⡟⣿⣆⣀⠻⣟⢸⡟⣿
⣼⡷⣾⡿⣱⢇⣿⠾⣇⣾⠣⢻⣄⡿⢸⣧⢠⠃⣼⠃⢸⣿⡜⠘⢁⡾⢁⡏⡿⣾⠙⣦⡹⣦⢸⣠⣼⡆⣿⢰⡟⠸⣧⣼⠧⣿⡼⣞⢿⣗⢾
⡏⢻⣿⢣⢻⡘⣯⠘⡿⢣⡶⣸⣿⠇⣦⣿⢸⡾⡏⣶⠸⢀⡖⣸⢹⣧⣾⡗⣧⢣⡆⣼⢣⣿⢿⡇⣿⣰⠸⣿⣇⢶⡘⢯⠃⣽⢣⡟⡜⣿⡞
⢷⡟⢸⡸⠞⠄⣩⢞⣡⡞⣰⠿⡏⢸⣇⢻⠘⠀⡇⠙⠘⠏⡟⢉⠘⠛⡏⢣⠉⠘⠃⠛⠈⣿⠀⠁⡇⢸⡇⢻⠻⣦⠷⣌⠳⡍⢠⠳⢷⡇⣷
⠈⢿⡈⢧⠻⣷⣯⣞⣠⡾⢃⢸⠁⢸⡘⡏⣴⡄⣿⠀⡆⠀⢰⠞⠀⠀⠀⠈⠟⣧⠐⠑⠀⣿⠀⣆⢱⢙⡇⠘⡇⡘⣷⣌⢳⣽⣦⢟⡞⢀⡿
⠀⢬⣳⡌⢳⣿⢋⡞⣿⢣⣿⢸⣸⣇⠻⢠⡏⠀⡿⡄⠀⢸⡄⢶⠀⠀⠀⠳⠀⢹⡄⠎⣸⢿⠀⢹⡌⠟⣼⣇⡇⣿⡘⣟⢿⡙⣷⡞⢠⣞⡤
⠀⢸⣿⠇⢾⡻⣟⠠⣿⢸⣻⠸⣿⠘⡆⢸⢰⢰⡇⢰⣸⡿⡇⣾⢷⡆⢾⡆⢸⠂⣷⢰⡇⠸⠀⣌⡇⢸⠁⣿⢣⣟⡇⣿⢤⣿⢿⡁⣸⣿⡇
⠀⢸⡏⢧⡾⡁⣧⠃⣿⣸⢻⡇⢡⡄⣷⢸⢃⡟⢰⢸⡿⣦⢃⡟⠸⣳⢢⣿⠸⠁⡘⣿⢳⡄⢷⡘⡇⣿⢰⡅⢸⣿⡇⣿⠘⣾⢈⢧⡽⢹⡇
⠀⢸⣷⡜⢱⢃⣿⢷⡟⣿⡟⡴⢛⠃⣿⣴⡟⣠⢿⢸⡇⡿⣸⣇⢠⣏⡎⣿⣴⢸⡇⣿⡞⢳⣌⢻⣦⣿⠸⡛⢆⢻⡇⢿⡾⣿⡸⡎⣡⣾⡇
⠀⢸⡟⣸⠀⣼⣿⢀⡁⣿⢸⢱⡏⢸⢫⢏⣾⡟⡘⣼⠉⣵⡿⡏⣾⢏⡗⢹⡅⣼⠇⡿⢧⠎⣿⣦⡹⡝⡆⢹⡞⡇⣿⢈⡀⡿⣧⠀⣇⢻⡇
⠀⠀⢧⠻⢸⠇⣿⢘⢣⠈⣤⣾⠃⡾⠃⣎⢸⢱⢣⡇⣼⠹⡇⢳⡿⢠⢗⡟⡼⠁⣸⣧⠸⡜⡜⢂⢳⠘⣷⠘⣧⣥⢁⡞⡃⣿⢸⡆⢏⡾
⠀⠀⠈⢷⣼⡆⣿⠸⡾⣇⢿⡘⣸⢃⠸⡹⠘⣿⢸⡿⣏⣆⢳⡀⣷⠺⣬⢡⣽⣼⠋⣿⢿⡇⣿⠸⢾⠀⡘⣇⢇⡟⣸⣣⢇⣿⢸⣧⡾⠁
⠀⠀⠀⠀⠉⠃⠈⢧⡹⣿⣮⢡⡏⣷⡀⣧⡄⢻⠘⠁⣿⢙⣦⠳⣌⠷⠏⢣⡿⣱⡏⣿⠘⢠⡇⢠⡼⡄⣧⢻⡈⡿⣿⢋⡾⠁⠘⠋
⠀⠀⠀⠀⠀⠀⠐⣦⡻⣍⣮⣾⡇⢿⣿⠘⠇⢸⠀⣷⠸⡎⣿⡆⠹⣷⣴⠏⢐⣯⣹⠇⣧⢸⡇⡞⠃⣿⡹⢸⣧⣽⣱⠟
⠀⠀⠀⠀⠀⠀⠀⢫⣳⡹⡏⢿⠀⡟⢸⣠⢰⣾⢰⡏⡄⣿⠘⣷⠀⢻⡏⢺⣘⠃⣿⢰⢹⠸⣇⠇⣄⠃⣷⢸⡇⢹⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣥⢻⣾⡄⡇⣾⡿⡄⢿⡘⣧⡇⣿⠃⡁⣾⢸⣷⠸⡏⢸⣿⢾⣸⢰⡟⢸⣿⢰⣶⢸⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⣏⣣⢷⠉⡇⢷⢰⣄⠻⠇⣿⢠⣿⡏⣾⠛⣧⢻⢹⣿⠸⠏⣨⡄⣷⢸⢀⡏⣜⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣎⠳⣏⢸⠸⣘⢧⠀⣿⠸⢹⣧⡟⢀⢹⣦⡟⣿⠀⠾⢹⠀⡇⣸⠟⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⡻⠄⢿⡀⡆⣿⡼⣼⢹⡇⡿⢸⣿⡕⡿⢸⣠⡟⠰⢛⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣦⣍⣃⠘⣷⡉⢾⡕⣣⡟⣩⣼⢃⣉⣭⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣾⣷⡟⢞⣵⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀⡔⣍⣁⣀⣀⣈⣹⡵⣄⣤⣤⣤⣤⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠓⢸⣿⣿⣿⡟⣳⣾⣤⣠⡈⢀⠀⠀⣀⣱⣧⣼⢻⣿⣿⣿⡆⠈⠘⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⣿⣿⢠⡟⠀⣹⢟⣣⣹⣏⣘⣿⣯⡉⢹⡄⣿⣿⣿⣧⠀⠀⠀⣿⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⢧⡀⢤⣿⣿⣿⣿⡶⣟⣽⣿⡿⠟⢛⡛⠿⢿⣿⣿⣿⣇⣿⣿⣿⣿⡄⠠⢀⣯⣉⠹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢸⣿⣿⣿⡇⢠⣿⡟⠁⠐⠀⢸⠃⠀⠂⠉⢿⣷⡌⢸⣿⣿⣿⡇⠀⠀⠀⠸⡗⢻⣿⣷⣾⣤⡀
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢸⣿⣿⣿⡇⣿⡟⠀⠛⠀⠀⢸⡀⠀⠀⠚⠈⢿⣷⢸⣿⣿⣿⡇⠀⠀⠀⢀⠟⣫⡿⠿⢿⡃⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠱⢤⠤⢼⣿⣿⣿⡿⣿⡇⠙⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⣿⣾⣿⣿⣿⡷⢤⡤⣤⣼⡼⡿⠁⠰⣟⣙⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⡇⢻⣿⡄⠿⠀⡀⠀⠀⢀⡀⠚⣰⣿⠇⣿⣿⣿⣿⠀⠀⠀⢸⣩⠼⠃⠠⣖⠻⣏⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢹⣿⣿⣿⡄⠻⣿⣶⣄⣁⠐⠂⣀⣥⣾⣿⠋⣸⣿⣿⣿⠇⠀⠀⠀⡘⠁⠀⣴⠲⣎⠛⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠋⠹⠅⠛⠻⢿⡿⠿⠿⣛⡫⣹⠙⠛⠛⠛⠛⠉⠉⢉⡉⠁⣴⠶⡜⠓⢾⡗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢦⣾⣽⢷⡞⣷⢠⡶⣦⢠⢶⡄⣿⢹⡆⠹⢦⣿⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠼⣿⡁⢸⣧⠟⠸⣠⡇⢙⣿⣇⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠚⠋⠉⠉⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢀⡠⠔⠂⠀⠈⠓⠎⠻⠁⠔⡥⠉⣁
⠀⠀⠀⢀⠔⣉⢀⣀⠄⠀⡀⣀⠤⣀⡀⠀⠀⠚⠫
⠀⠀⠰⠟⠉⣤⡊⠤⢒⠁⠀⠀⠀⠀⠠⡄⠀⠀⢴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⢢⡀⠈⠑⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠲⡆⢠⠤⡀⠀⠙⣄⢼⠀⠀⠀⣀⣀⠀⠀⡠⠉⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⡇⠀⠘⠄⠀⠈⢎⢧⢀⠊⠤⠈⠣⠘⡀⠀⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠸⡄⠀⢸⠀⠀⠈⠿⠋⠘⠁⢀⡄⠀⡇⠀⠙⠃⢐
⠀⠀⠀⠀⠀⢀⡠⠐⠀⠉⠢⠤⣴⠋⠓⠓⠀⠀⠀⢀⡀⠘⢇⠉⠀⠀⠈⠑⠤⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠀⢤
⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠈⠄⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⣀⣀⡌⠀⠀⠀⠀⠚⢢
⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡾⠁⠘⠁⡀⣀⣶⣿⡄⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡠⠀⠄⠀⠀⢤⠀⠀⠀⠐⠂⡇
⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢠⡡⠖⠂⠈⠁⠈⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣁⢎⠀⠀⠀⠀⠀⠀⠦⣀⠤⢧⡨
⢇⠀⠀⠀⠀⢀⣀⠤⠔⠂⠁⠀⠀⠈⢁⠐⠈⠉⠉⠈⠉⠉⠛⠇⠀⡆⠀⠀⡠⢄⡀⠀⣐⠩⠃⠀⠈⠢⠕⢀
⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⣀⠠⠐⠚⢦⡀⠘⠀⢰⠀⢠⠉⣳⠉⠛⢄⠈⢄⠀⠀⡴⠎⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⢰⢦⡸⠷⠤⡇⠀⠒⠈⠁⢀⢧⠀⣀⣁⡦⠑⠢⢄⣠⠀⠃⠒⠈⢱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠑⢄⣤⠆⠀⢀⣠⢊⣠⡘⢮⡁⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠒⠉⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⡨⠷⠦⠤⡤⠔⢉⡸⠋⠁⡟⠈⣳⠀⠀⠈⠢⣀⣀⣠⠴⠒⠒⠂⡏⠀⠀⠀⠀⠀⢹⡀⠀⢀⠄⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠘⢊⠝⠃⠌⠀⢸⠺⡇⢀⣀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡵⡎⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠘⠃⠐⡄⠀⠐⡄⠀⢐⠂⠒⢸⡀⢹⣍⠀⠀⠀⠀⢀⡀⠤⠐⠒⡖⠀⠀⠀⣀⣤⡞⠥⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠠⠰⡅⢀⡼⠛⠛⠛⠀⠀⡌⢣⡆⢹⡯⡑⠶⠈⠀⠀⠀⠀⣸⠀⡠⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⡄⢀⡀⢀⢒⠟⠁⠀⠀⠀⠀⠀⢈⠷⡆⠐⠀⠓⠈⠆⠡⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⢧⠀⠀⠀⠀⠀⠀⠘⠓⠢⢌⠁⠀⠀⠀⠑⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡆⠀⠀⠀⣀⣀⠠⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠉⢉⣭⠡⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣀⣀⣀⣀⣄⣀⣄⣀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠤⠤⠐⠲⠈⠙⠛⠛⠉⠃⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤⠴⠤⠐⠀⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⣀⡀⣀⡀
⠀⠀⠀⠀⡴⡋⠁⠀⠐⠋⠉⠀⠀⠀⠀⠀⣉⠁⢦⠀⠀⠀⠀⡀⠒⠊⣀⡩⢀⣁⡶⠊⠉⠉⠉⠳⠀⣍⠃⠬⣵⢦⡀
⠀⠀⢀⠞⠸⠁⠀⠀⠀⠀⠀⠀⠀⠄⣀⣦⡴⠖⣿⠀⠀⠀⢸⠀⢀⣀⠦⡄⠣⡿⠧⣀⠀⠘⣦⠀⠀⠀⢳⡀⠀⠀⠹⣆
⠀⠀⡖⢈⠁⡀⠀⠀⢀⣀⠴⠓⣻⡯⢟⠝⠀⠀⢻⠀⠀⠀⡸⠁⢀⣰⠚⠀⡘⠀⠀⠈⢷⣦⣘⢀⡀⠀⠘⢧⠀⠀⠀⡎⣇
⠀⣘⡔⠁⠀⠁⠀⡠⠁⣠⢶⣰⣿⠓⠀⠀⢹⣷⢀⠀⠀⠀⠀⠠⢴⡷⠈⠀⠱⣿⠍⠀⠀⠸⣟⣦⣽⣆⠈⠊⡀⠀⠀⠆⢸
⢠⣢⠀⠀⠀⠄⠠⠂⢾⣬⢮⣿⣿⠀⠀⠀⠀⠀⠈⠒⠒⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢫⠸⣭⡋⣦⠀⢷⡀⠀⠇⢸
⢀⢋⠐⣤⠀⠀⡞⠃⠀⢹⣿⡏⡿⠀⠀⡀⠀⢀⢣⠤⠀⠀⠀⠠⠒⠀⠑⢲⣔⡀⠀⠀⠀⠀⠀⠰⣧⡾⣿⠀⠘⢱⠀⢀⡾
⠸⠇⢀⠈⠉⠺⡇⢀⣴⢉⣿⣿⣄⡀⠀⠘⢰⠿⠟⠁⠀⠀⠀⠀⠀⠠⠀⠈⠘⢶⣌⠂⠀⠀⠀⠀⣿⡟⣿⡻⠀⠀⣸⡶⡇
⠀⢁⠘⠀⠷⣄⡀⠂⡹⡦⣽⡿⣿⣷⣷⣶⣉⡂⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠟⠀⠀⠀⠀⣴⣿⡟⣿⣷⢀⣴⣷⣷⡇
⠀⠠⠤⠂⠒⠓⢩⠴⠾⠁⠀⢹⠿⠛⠛⠛⠛⠿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡔⢘⣟⣤⣿⣿⣇⣾⣥⢄⣇⣞⣇⡾⠉
⠀⠁⠀⠀⠀⠈⠈⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠲⠀⠒⠀⠀⢀⠀⠀⠳⠿⠛⠉⠉⠙⢿⠛⣩⢿⡿⣿⣉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠻⣦⡴⣶⠂⢲⢦⣶⠾⠃⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⠋⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠛⠋⠈⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃

⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢄⡀⠈⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠉⠙⠳⡤⠶⠠⠀⡀⠀⠀⢀⠀⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠠⣼⣿⣓⠦⣄⡀⠀⢀⣀⡀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⠳⣄⠉⠉⠀⠀⠉⠻⠋⠉⠩⠋⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⠴⣚⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣾⠷⠀⠀⠀⠀⠐⢦⣀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢋⣥⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⣷⣦⡀⠀⠀⢸⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⠳⢦⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠁⠀⠀⠀⠀⠁⢐⡟⠀⠀⢀⣀⣠⡴⠖⠋⠁⠀⠀⠀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃⠀⠀⠀⠈⠛⠻⢯⣍⠀⠀⠀⠀⠀⠀⠀⠘⠓⠚⣻⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢀⣀⣠⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠷⠶⠤⠄⠀⠀⠀⠀⠀⠀⢾⡛⠉⠋⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠃⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣤⢰⣴⣮⡀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡼⠃⢿⠹⣿⡏⠇⠙⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣋⣠⢤⡖⠒⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠛⠉⠁⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣆⠀⡤⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⢀⠀⠀⠈⠉⠁⠀⠀⠀⠙⠲⣝⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢣⡏⢸⡄⠀⠀⠀⠀⢀⣄⠀⠀⠀⢘⡆⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡼⠀⠀⠙⠶⣤⣤⠶⠋⠹⣄⣀⣠⡾⠁⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⠀⠸⠇⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⢻⠀⠀⠀⠀⡇⣸⠃⢀⠀⢠⡈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠁⠀⠀⣰⠃⠀⠀⡞⠀⢈⡇⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠸⠁⠀⠁⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠚⢸⡆⠀⢰⡆⠀⠀⣠⠀⢀⡀⠀⠈⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣞⠀⠀⠀⢻⠀⠀⢸⠆⠈⠁⠀⠀⣇⡔⢰⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⠀⠀⠋⢠⠀⡄⠀⠀⠀⣀⠀⠀⠀⣼⠀⢉⠀⢸⣧⡇⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠇⣼⠃⠀⡠⠀⢀⡀⠆⠀⢀⢉⠀⠀⣧⠁⠀⠈⠀⠏⠏⠀⢰⠀⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⣇⠐⡇⠀⠀⠁⠀⢸⣷⠀⠃⣾⠀⠀⠂⠈⠀⠀⠁⠆⠀⠀⠀⢸⢃⡁⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠁⣸⡆⣦⢠⠀⠀⢠⠀⠁⠀⠀⠛⠀⢀⢰⠀⡄⠀⣆⠀⠁⠀⠀⠀⠸⠃⢸⠆⠋⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠈⡀⣠⠠⠈⡃⠞⠀⠀⠁⡀⠀⠀⢀⠀⢀⠸⠆⣆⠁⡞⠑⠀⡀⣤⠀⠀⣠⠀⠊⠀⣠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⢳⣉⠀⠁⠑⠰⠓⣠⣷⠇⠀⠰⡌⠂⠰⠀⡄⠉⠀⠇⠐⠾⠁⣸⠀⣶⠀⣶⠄⠘⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠋⡤⠛⠐⣷⡀⠜⠿⣿⡏⣻⣧⡐⠁⢀⠀⢸⡇⠀⠀⠀⠀⢠⣘⡧⣶⠛⣷⣤⠄⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⡀⠘⠄⢀⡀⠀⣄⣀⠈⡄⠙⠉⣿⡄⠉⠀⡠⢁⡯⠀⢠⠀⣸⡟⠁⠇⠞⠟⢁⣀⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⡄⠙⣻⡆⢨⢉⣠⣿⣾⣏⢀⣔⢃⠈⣁⠀⡀⠇⠋⠛⠀⠀⠀⠋⠅⢡⣴⡀⢸⡇⢦⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡾⠓⠉⠙⠛⢾⡌⠙⣿⠇⠁⠈⠛⠙⢷⢈⣤⣅⠀⣄⢂⣼⣷⡀⢰⣾⠆⡿⠃⣾⣿⠆⠘⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠀⠀⠀⠀⢠⣿⡿⠂⠉⠁⠒⢶⡄⠀⣤⣼⡿⠋⠁⠀⠈⠙⢿⣿⡆⠀⠀⠆⠉⠘⠳⣦⣤⡠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣗⡒⣶⠞⣻⣟⡀⢀⠀⣀⣼⣮⠀⠀⢀⣼⡁⠀⠀⠀⠀⠀⢀⣿⣄⠀⠀⣸⣆⠀⠀⢀⣟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠯⢉⣛⣛⡽⣿⣦⢠⠞⣿⣿⠦⡀⢀⡀⣰⡿⠿⣿⣿⣯⣾⣽⣷⣀⣫⣾⢣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠓⠒⠦⠤⠔⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠿⠿⠿⠿⠀⣀⣀⣀⣀⡀⠀⢀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⡀⢠⣴⡶⣦⡄
⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣸⣿⣿⣿⣷⣶⣶⣶⢂⣼⣿⠟⣿⣿⡇⢸⣿⣿⣯⣭⣍⠁⠛⣿⣿⡟⠛⠁⣼⣿⢿⣿⣤⡄
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠣⠿⠿⠿⠁⠀⠀⠀⠠⠿⠿⠛⠛⠿⠿⠇⠶⣾⢿⣿⡿⠟⠀⠰⠿⠿⠀⠀⠀⠙⠛⠛⠙⠛⠃
⠀⠀⣸⣿⣿⣿⣿⣤⣤⣤⣤⣤⡄⣾⣿⠇⢰⣿⡿⢠⣿⣿⠿⢿⣿⡷⢀⣾⣿⢂⣴⣾⡿⠿⣿⣷⡄⣸⣿⡗⢠⣿⣿⢂⣶⣿⣿⠿⡿⠟
⠀⢠⣿⣿⣿⣿⠟⠛⠛⠛⠛⠛⢸⣿⣿⣠⣿⣿⠃⣾⣿⠿⣿⣿⣍⠁⣼⣿⡏⢸⣿⣿⣀⣴⣿⡿⢠⣿⣿⣁⣼⣿⠏⣨⣛⣛⣻⣿⣿⠆
⠀⠘⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠉⠉⠀⠈⠉⠉⠀⠉⠉⠀⠀⠈⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉










⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢨⡀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠄⠀⠊⠀⠀⠀⠀⠀⠀⠙⠙⠂⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠒⠋⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⡀⠈⠀⠐⣂⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⡖⠀⠀⠀⠀⠀⣼⡆⠀⠀⠰⡀⠁⠋⠀⢹⡵⠋⠀⠰⢀⠀⢀⡁⢀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡭⠁⠀⠃⠀⠈⠓⠀⠀⠉⠠⠄⠀⠀⠀⠀⠀⠊⠰⠄⠀⠀⠀⠀⠊⣭⢤⡼⢃⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣼⡇⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠠⠄⠀⠀⠀⠸⣦⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠛⠒⠒⢿⡣⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢴⡾⠛⠓⠀⢀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⡄⢸⣷⠀⣿⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠀⠀⣀⡏⠉⠁⠐⠘⠀⠀⠀⣤⠠⠞⠃⠰⠆⢠⣤⣌⡶⡚⠀⡆⠀⢩⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⣼⠃⠁⠀⢰⠆⡴⠃⠰⠦⠐⠤⠄⠀⠃⠀⣀⣀⣤⢅⡆⠇⢴⡀⢈⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⣾⠗⠀⢠⡖⠐⠺⢦⡀⡦⠤⢿⣷⣶⣶⣦⣭⣥⣴⣾⣿⣷⣆⣀⠘⣿⣯⡴⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⠻⣦⣒⠲⣽⣮⣻⣯⣙⣭⢙⠉⢻⣭⣿⣤⣯⣛⣛⡛⠨⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠚⠛⠉⠀⣯⣽⠙⣈⣁⢉⡞⠂⠨⣻⣿⡃⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⢊⠉⠹⠻⡇⠳⢷⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠉⡛⣻⡆⢀⠤⡆⢄⢟⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⣿⢸⠀⣘⡇⠸⡀⣴⣿⣨⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⣼⣿⡏⠗⢀⣀⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⠃⠿⠿⠁⠶⠌⠁⠀⠙⠋⢹⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠰⠀⣀⠀⠀⠉⠙⠃⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣐⣲⣿⣠⣄⢲⣴⣤⣤⣦⣬⣿⣟⡾⣠⣤⣀⣰⣽⣿⣵⣆⣀⠀⠠⠤⠶⡤⣦⣤⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠠⠤⠤⠬⠿⠿⠶⠌⠐⠂⠒⠚⠛⠛⠛⠛⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠈⠉⠉⠁⠀⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡀⠠⠄⠀⠀⠀⠄⢀⡀⡀⠀⠀⠤⢀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⣀⣀⡀⠀⠀⠠⠄⠀⠈⣂⠀⠀⠀⠀⠑⡀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠛⠉⠁⠀⠤⠀⡀⠁⠒⠀⠄⡀⠀⠁⠂⠄⡀⢁⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢦⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⠀⠁⠢⢀⠀⠀⠙⠢⢀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠈⢫⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⡈⠂⢄⠀⠀⠁⠠⠀⠀⠄
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠆⠀⠳⡀⠀⠁⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠨⠐⠠⣀⠀⠑⠌
⠀⠀⠀⠀⠠⠁⠀⠀⠀⡈⠀⠀⠀⡘⠲⣄⡀⠀⠈⠐⠂⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠿⠒⠔⡀
⠀⠀⠀⠀⠃⠀⠀⢀⠌⠀⠀⢀⠔⠀⠀⠀⠙⠢⠀⠠⠀⠀⠀⠀⠁⠂⠀⠀⠤⠀⢀⠤⠤⠀⢒⣫⠖⠆⠚⡀⠐⢄
⠀⢀⠔⠁⠀⢀⠔⠁⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠌⠡⠄⠀⠀⠨⠀⠠⠔⡂⠛⠀⠀⠐⠂⠈⡰
⠐⠁⣤⡐⣈⠀⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⡉⢃⠀⠀⠀⠀⢀⠔
⠀⠀⠠⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⡁⠐⠂⠈
⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠂
⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠁⡄
⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠂⠁

⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢲⣤⡀⠀⠀⠀⠀⠀⠀⠀⣄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⢀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠄⣱⣿⢻⣧⠀⠀⠀⠀⠀⠀⢀⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣿
⠀⠀⠀⠀⢸⣿⠘⣿⡀⠀⠀⠀⠀⠀⠐⢠⠘⣏⢻⢿⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⢁⣾⡏⠀⠀⠀⠀⠀⠰⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⣦
⠀⠀⠀⢰⣼⠀⣤⠛⡇⠀⠀⠀⠀⠀⢰⢿⣿⢃⡀⢸⣶⠂⠀⠀⠀⠀⠀⢸⡇⠋⢀⢸⣯⣔⡇⠀⠀⠀⠀⣾⠉⡀⢸⡇⠀⠀⠀⠠⢠⣠⡾⣿⢠⡌⠹⣷⡀
⠀⠀⠀⢈⣏⠃⡝⣆⣷⠀⠀⠀⠀⠸⣿⠆⢁⡞⣇⠀⢛⡟⠀⠀⠀⠀⠀⠘⢿⣄⢼⣦⣿⡟⠀⠀⠀⠀⠀⣾⠄⣧⣈⡇⠀⠀⢀⣴⠛⢛⣄⣁⣼⣧⡆⡹⠁
⠀⠀⠀⠀⠉⠛⠓⠋⠁⠀⠀⠀⠀⠀⠈⠑⠂⠉⠉⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠚⠃⠀⠀⠀⠀⠀⠈⠑⠛⠉⠀⠀⠀⠘⠛⠀⠛⠈⠉⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠰⢃⣴⡎⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠴⠀⠠⠀⠀⠀⠀⣶⣄⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠠⠄⢀
⠀⠀⠀⠀⢾⡟⢿⣷⡈⠀⠀⠀⠀⠀⠀⠀⢠⠀⣰⠀⣠⣴⣶⡿⠀⠀⠀⠀⠀⠀⢿⣿⠿⣷⣆⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣴⠁⠀⢀
⠀⠀⠀⠀⣸⡇⠀⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣆⡿⠋⣜⣥⣠⡌⠀⠀⠀⠀⠀⠻⣆⠀⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⢃⡴⣿⠠⣶⡿⢻⣿⢀⡴⣯
⠀⠀⠀⠀⠿⣟⢸⡄⡿⠀⠀⠀⠀⢀⣠⣰⣶⠏⣸⠟⢡⠜⠙⠻⢛⡵⠆⠀⠀⠀⠰⡶⣿⢠⡄⢸⣦⠀⠀⠀⣠⣴⣤⡴⠋⣸⡿⢚⡉⣠⢿⡿⠋⣴⠏⡀
⠀⠀⠀⠀⠀⠈⢿⠟⠁⠀⠀⠀⠠⣿⢋⣛⣡⢦⣠⠶⠻⠴⣦⣤⣀⡶⣧⡀⠀⠀⠀⢹⡦⣾⡁⡸⠃⠀⠀⠞⡋⡉⣩⣞⣀⡀⢴⣋⣈⣥⣠⡄⠚⠟⣻⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉
⠀⠀⠀⢀⣅⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠘⣟⣻⣦⢀⠀⠀⠀⠀⠀⣀⠂⢀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀
⠀⠀⠀⠃⠘⣿⡏⢻⣆⠀⠀⠀⢸⣿⣷⣄⡠⢀⣁⣀⣒⣠⣤⣌⡁⠀⠀⠀⠀⠀⠀⣿⡿⣦⣤⣇⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⣧⠈⡀
⠀⠀⠀⣠⣦⡿⡁⠘⠉⡇⠀⠀⠀⠻⢿⣟⠛⠟⠛⠻⠿⣿⣿⠛⣿⣤⡀⠀⠀⠀⠀⠹⣧⠈⠙⠻⡿⣦⣷⣦⠀⣂⠀⠀⠀⠀⠀⠀⠀⣿⠻⣷⣿⣦⣤
⠀⠀⢼⡏⠋⣼⣧⠄⠸⣿⠀⠀⠀⠀⠐⠉⢻⣷⣶⡄⢲⣤⣈⡆⠙⢿⠇⠀⠀⠀⢀⢳⣾⢷⡄⠰⣤⣤⣍⣉⠻⡟⢷⠀⠀⠀⠀⢀⣗⠾⣆⢉⡛⠈⠻⠿⣆
⠀⠀⠈⠻⠦⠿⠸⠧⠖⠁⠀⠀⠀⠀⠀⠀⠀⠸⠿⠧⠼⠷⠽⠷⠦⠟⠀⠀⠀⠰⠾⠿⠋⠁⠠⠶⠾⠧⠉⠉⠳⠤⠌⠦⠀⠀⠀⠸⠿⠆⠠⠾⠟⠓⠛⠲⠼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣋⠵⢊⠓⣧⠀⠰⠄⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⣩⠔⢊⡤⠚⠉⡆⢿⠀⠀⠀⠀⢀⣤⣀⣀⡀
⠀⠀⠀⠀⠀⣀⡴⠚⡡⠔⠋⢠⠊⣁⠀⢺⡇⡇⢸⠀⣸⣿⣿⣿⢸⡏⡍⠭⢝⢦⣤⣀⠀⠀⠀⠀⣀⣠⠄
⠀⠀⢀⡴⢞⡡⠖⢉⡤⡆⠀⠘⡀⣿⠀⠈⠀⡇⢸⠀⠛⠿⠿⢿⢸⡇⠀⢀⡨⣸⣿⠿⢿⣟⣶⣯⠿⠚⠃
⠀⢸⢯⣖⢉⡤⠒⢁⣤⢰⠀⠀⡇⠀⣠⣄⠀⡇⢸⠀⠀⠛⠛⢻⣼⠷⠿⠿⠞⠃⠀⣠⢟⡽⠛⢮⢷⡀
⠀⢸⢸⠀⡇⢴⡆⠀⠿⢸⠀⠀⡇⠀⢛⡡⠖⣩⡾⠀⢀⠀⠀⠠⡀⠀⠀⠀⠀⠀⢠⣯⡏⠀⠀⠈⣿⡇
⠀⢸⢸⠀⢣⠘⠃⠀⠀⢸⠀⠀⠧⠚⢁⡴⠞⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⣇⠀⠀⣠⣿⡇
⠀⢸⡌⡄⢸⠀⢰⡶⢂⡸⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢬⣿⣿⠽⠛
⠀⢸⡇⡇⢸⢀⡠⠖⠉⣠⠴⠋
⠀⠈⡇⡇⡈⠁⣠⠴⠋⠁
⠀⠀⢷⣤⠷⠋⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⡿⠛⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⠿⠿⠿⣿⡿⠋⠀⠀⠉⠛⠋⠁⠀⡀⠀⠀⠙⠛⠋⠉⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢰⠀⠀⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡏⠉⠋⠉⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠘⠀⠀⠀⢠⠀⠈⠛⠉⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠆⠀⠀⢿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⠀⠀⠀⣀⠤⡀⠀⠀⠀⠃⠀⡀⣀⠀⠀⠘⠀⠀⠀⠸⣿⣿⣿⣿
⣿⣿⣿⣿⠁⠀⠀⣀⠔⠒⠒⠀⠠⠀⠤⠄⠀⠔⠁⠀⠈⠑⠂⠁⠀⠀⠈⠑⠠⠐⠊⠁⠀⠀⠉⠐⠠⣀⠀⠀⠀⢿⣿⣿⣿
⣿⣿⣿⣿⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠸⣿⣿⣿
⣿⣿⠋⣁⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⣆⠙⣿
⣿⡟⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡷⠀⠰⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢹
⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈
⡇⠀⡇
⣿⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠐⠂⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿
⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⣀⣀⣑⣂⢀⣒⣀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⢀⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢗⠴⠀⢛⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠻⡏⣽⣛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⠠⠔⠂⠈⠉⢡⠼⠉⠈⠙⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⡄⣀⢀⡀⢈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂
⣶⣿⣿⣿⣷⣮⣿⣦⣶⣭⣶⣶⣶⣖
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡃⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⢀⠀⡀⣀⡀⠀⣀⠀⢀
⠀⠀⢀⡗⠀⠀⠀⠀⢀⣴⣿⣾⡆⠀⣼⣿⣷⠀⠀⠀⢀⡀⡀⠀⠀⠐⣾⣿⣷⣿⣿⣿⣾⣿⣿⡿⢀⣉⡇⣨⡽⢃⡀
⠀⢠⣾⣿⣴⡄⠀⠀⣸⣿⣿⣿⣿⡀⠁⠹⠏⠀⡐⡜⣨⡵⣿⣷⣤⡠⠙⠛⠛⡛⠛⣛⣛⣛⣋⠀⡸⣿⣜⡿⢳⡞⣖
⢀⣿⡟⠉⢿⣷⠀⠀⠻⣿⠿⢿⣿⠇⠀⠀⣠⣾⣷⣿⣿⣿⣿⣯⣿⣿⣦⣄⢼⣿⣙⢺⣟⡋⠛⠁⠘⠿⠻⣷⣾⠟⠋
⠀⠛⠄⠀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩⣼⣿⣿⠋⠀⠁⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣠⢰⣠⢼⡰⢠⢠⣀⠀⠀⠁⠊⠘⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠛⠀⠢⠀⠀⠰⡄⣠
⠀⠀⠐⡄⠀⢰⣺⣽⣿⣿⣾⣽⣼⣿⣿⠀⠀⠀⠀⠀⣠⣦⣤⣄⠀⠀⠀⠀⠐⠀⠈⠼⠷⠁⣨⢷⣿⢕⠗⠀⢀⣘⡆
⠀⡄⣾⢁⠀⠈⠙⠛⡉⠉⠉⠙⠛⠛⠋⠀⠀⠀⢠⣾⣿⣿⣿⣿⣧⠀⠀⣠⢞⢢⡇⠀⠀⢀⣈⣿⣿⣏⣀⠰⣞⣮⣷
⠈⢷⣙⣾⣦⠀⠀⣴⣻⣶⡄⠀⢀⡺⣿⣎⠠⣄⠈⠻⣿⣿⣿⣿⡿⢠⣶⣿⡏⣿⣷⡇⠀⠉⠉⠉⠉⠉⠉⠀⣤⣿⠇
⢀⣾⣼⣿⡏⠀⣼⡏⠹⢿⣿⠀⢢⣹⣯⣟⣵⡝⡄⠠⣿⡿⣿⣿⠀⠈⣿⣿⣿⣿⣿⣗⠀⠀⢀⣠⡄⠀⠀⠈⠻⣧⣄
⠀⠛⠿⠟⠃⠀⠈⠓⠀⠘⠃⠀⠘⠛⠛⠛⠛⠛⠋⠀⠘⠇⠸⠋⠀⠰⠿⠿⠿⠿⠿⠟⠀⠶⠿⠿⠿⠷⠶⠀⠶⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⡶⠿⠿⠿⠿⠿⠿⠿⠿⠶⢶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⢀⣴⠟⠁⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠻⣷⡀
⠀⠀⠀⢠⣾⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠚⠠⠻⢾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢫⠈⠀⠰⠄⢀⠀⡀⠀⣄⣀⡀⢠⠚⠆⠀⠀⠀⠀⡘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠋⠂⠱⠎⢿⡟⠉⠃⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⢸⡇⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠸⡇⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣀⣴⣧⣀⣀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠀⠹⡅⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠁⠀⠀⢀⣼⠃⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢀⣼⠃⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣉⣉⣉⣉⣉⣉⣉⣉⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠈⢿⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣄⣀⠀⠀⠀⠀⠀⣴⠞⠛⠉⠉⠉⠛⠓⠦⠀⠀⠀⠀⢀⣀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠉⠑⠲⠶⠦⢤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠶⠒⠋⠉⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣀⠀⠲⠶⠤⠄⠀⠀⣀⡀⠀⠀⠀⠄⠰⠖⠀⢀⣀⣤⠶⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠙⠛⠓⠶⠶⠶⠶⠤⠤⠴⠶⠶⠖⠛⠛⠛⠉⠁⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣀⣙⡓⠒⠂⠠⠤⠤⠤⠤⠤⠤⠀⣀⣀⣀⣀⣤⡤⠴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠈⣉⡉⠛⠛⠓⠒⠒⠒⠒⠒⠒⠋⠉⠉⠉⠉⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠶⢤⣬⣉⣉⣐⡒⠒⠒⠒⠒⠒⠒⢂⣀⣀⣀⣠⣤⡤⠶⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠈⡉⠉⠉⠉⠙⠛⠛⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠲⠶⠤⣬⣉⣉⣉⣉⣀⠀⠀⠈⢉⣈⣉⣉⣠⡴⠶⠞⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣦⣤⣤⣤⣄⣀⣀⣤⣤⣤⣤⣤⣶⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠈⠉⠉⠉⠉⠉⠉⠉⢉⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣤⣤⣤⣤⣤⣤⡾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⠀⡀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⣾⣿⣿⣷⣿⣾⣦⣾⣿⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡛⠉⠉⠙⢿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠈⠻⣿⣿⣿⠟⠉⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠁⠀⠀⢀⣤⡀⠀⠀⠀⠀⠁⠀⠀⢀⣀⣀⠀⠀⠀⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠐⣿⣿⣿⣦⣀⠀⠀⠀⣠⣶⣿⣿⣿⡧⠀⠀⠈⣿⡇⠀⠀⣠⠴⠲⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⡰⠛⣛⣿⣿⣿⡷⠀⠸⣿⣿⣿⢟⡋⠳⡄⠀⠠⠟⡳⣄⡠⠗⠀⠀⠀⠚
⠀⣀⣠⠴⠦⡀⠀⢀⠜⣛⠻⡀⠀⠀⡇⢾⢠⣴⣮⢏⠀⠀⠀⢻⠀⢷⣾⠇⠀⣿⠀⠀⠸⠟⣹⠊⠉⠯⣤⣴⠟
⢾⣁⠀⠀⠀⠉⣀⢽⠲⡋⡇⠀⠀⠀⠙⢬⣿⣯⣥⠎⠀⠀⠀⠈⢷⣄⣠⠤⠺⠃⠀⠀⢰⠚⠁
⠀⢧⣀⡀⠀⣹⠆⠈⠓⠶⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠤⣤⣤⣀⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠠⠒⠁⠀⠁⠒⠒⠂⠮⠿⠟⠢⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠖⠚⠛⠛⠋⠉⠉⠛⠛⠛⠓⠲⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠶⠄⣀⠀⠸⠀⠀⠒⢦⡀⠀⠀⠀⠀⠀⠀⢀⡟⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠹⡇⠀⠀⠀⣠⡴⠯⠚⠉⠀⠉⠙⠒⠒⠶⠦⠤⠤⠤⠤⠤⠤⠤⠴⠶⠶⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡤⢤
⠀⠀⠀⣀⡀⡀⢀⣀⣰⠲⢾⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣷⣰⠶⢤⠴⠒⠲⢤⣀
⢀⣴⣭⣍⡑⠀⠉⠀⢡⠘⣾⠃⠀⠀⠀⢠⣀⣀⡼⢤⣤⠚⠀⠀⠀⠘⣿⠀⠈⠁⢀⠞⠉⠉⠙⡀
⠎⡠⠤⠤⣙⠀⠀⡄⠀⢁⣿⠀⠀⠀⠀⠀⡤⠈⠀⠀⠁⠐⠄⠀⢀⣠⠏⠀⠀⠀⠀⠀⠀⠄⠠⢵
⠀⡆⠀⠀⠀⣁⣇⡇⠀⠀⠻⣤⣀⡀⠀⢀⡑⠚⠤⠤⠜⢁⡠⠀⢂⣿⠀⠀⠀⢀⣌⠐⠂⠉⠙⡞
⠀⢿⠀⠀⠀⡏⠁⠉⠀⠀⠀⠘⣇⠠⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡞⡿⠀⢠⠔⠉⠈⣧⠀⠀⢠⡇
⠀⢸⡄⠀⠀⣇⠀⠀⠀⠰⡀⠀⠹⣆⠫⠀⡀⠀⠀⣀⠨⠁⠀⠀⣹⠃⠀⣼⠀⠀⢀⡏⠀⠀⢸⠃
⠀⢸⠇⠀⠀⠈⠦⣀⠀⠀⠹⣄⠀⠘⢧⡉⠀⠀⠀⠀⠀⠀⠀⣶⠏⠀⢀⡇⢀⡤⠚⠀⠀⠀⢻⡀
⢠⠇⠀⠀⠀⠀⢀⠈⢣⠀⠀⠙⣦⠀⠀⠙⠚⢷⣤⣤⣤⡴⠞⠁⠀⢀⣾⠁⢀⠀⢄⠀⠀⠀⠀⢱
⠇⠂⠀⠀⢉⡀⠙⣤⠂⠀⠀⠀⠈⢷⣄⠀⠀⠈⠀⠀⠀⠈⠂⠀⢀⡾⠃⠀⠈⡧⠋⠀⠀⠀⠀⠀⡵
⠈⠲⠦⠑⠦⠜⠒⠋⠀⠀⠀⠀⠀⠀⣹⠷⢤⣤⣀⠀⠀⣤⡤⠴⣏⠁⠀⠀⠀⠈⠒⠅⠠⠤⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⢀⡤⠴⠀⠀⢠⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠁⠀⠀⠀⠀⠁⢄⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⢠⠀⠀⠀⠀⠀⠈⣧⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠈⠉⠀⠀⠀⠀⠀⡀⠃⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣯⣽⣵⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣶⣶⣷⣶⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⣳⣤⠄⢀⣠⣴⡾⠛⠚⠛⢶⣶⣤⡀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⢕⢶⣿⡿⠟⢻⠿⠒⠣⠀⠀⠀⢉⠙⢿⣤⡄⠲⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢾⣷⠋⠀⠃⠁⠀⠀⣀⣄⡀⠀⠀⠽⣿⠿⣾⣷⠀⠀⠀⣀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡿⠖⠀⠀⠀⠔⠉⠀⠀⠈⠁⠀⠀⠈⠀⠈⢻⣶⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⣾⡿⠻⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠸⢿⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠼⡇⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣇⠀⠀⠀⠙⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣼⡢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⣫⣷⡁⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⢾⡿⠉⠛⠦⣌⠐
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢫⡻⢿⣿⣷⡷⣄⠙⠚⠳⢤⣤⠒⠅⠀⣀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠉⠛⠚⠛⠛⠋⠙⠿⠿⠿⠿⠛⠓⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠓⠒⠒⠊⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⣰⣿⣿⡇⠀⠀⠀⠀⣼⡟⠁⠀⠀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠁⠀⠀⢀⣼⠀⠀⠀⠀⢀⣼⣿⣿⡟⠀⠀⠀⠀⣼⣿⠁⣸⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠁⠀⠀⣠⣾⠇⠀⠀⠀⣠⣿⣿⣿⡟⠀⠀⢀⠃⢰⣿⣿⢀⡇⢸⠏
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠃⠀⢀⣼⣿⡏⠀⠀⣠⣾⣿⣿⣿⡟⠀⠀⠀⣿⠀⢸⣿⡟⠸⡇⠋
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡏⠀⣰⣿⣿⡏⠀⣠⣾⣿⣿⣿⣿⠋⢠⠂⠀⢰⡿⠀⢸⡿⢁⡆⠁
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⣴⣿⡿⢿⣿⣿⡇⣸⢿⣿⡟⢀⣾⣿⣿⣿⣿⠟⠁⡰⢃⠆⢠⣿⠇⠀⡞⢁⡾⢱⠆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣸⣿⠟⠁⢸⢻⡿⢰⡏⣸⣿⣰⣿⡿⢋⣾⡿⠋⠄⡼⣡⠟⣴⡿⠃⠀⠀⣰⣟⣷⠏
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⣿⠏⠀⠀⣼⠈⡇⣿⠀⣿⣿⣿⡟⢀⣾⣿⢃⡏⣼⣷⣯⣾⡿⠁⢀⣴⣾⣿⠟⠁⠀⠀⣀
⠀⠀⠀⣶⠀⣼⢿⡀⠀⠀⢸⣿⠀⢠⠀⡟⠀⠃⠟⢸⣿⣿⠛⠀⣾⣿⠇⣾⢡⣿⣿⣿⡟⠀⣰⣿⣿⠟⠁⠀⣠⣴⣿⠇
⠀⠀⠀⣿⠀⠸⣟⣧⠀⠀⢸⡇⠀⣾⣴⡇⠀⠀⠀⢸⣿⠇⠀⢸⣿⡿⢸⣿⢸⣿⣿⣿⡇⣼⣿⣿⠃⢀⣴⣾⣿⡿⠃⢀⠆⢀⠄
⠀⠀⠀⣿⡄⠠⢹⣿⡆⠀⢸⠃⢠⣿⣿⠃⠀⠀⠀⢸⡏⠀⠀⣿⡿⡇⣿⣿⢸⣿⣿⣿⣷⣿⡿⠁⢀⠼⡫⠓⢉⣠⡶⣋⠔⠋
⠀⠀⠀⣿⣧⠀⣧⣻⣷⠀⢸⠀⣼⣿⡿⠀⠀⠀⠀⢸⠃⠀⢠⡿⢰⢱⣿⣿⠸⣿⣿⢹⣿⣿⡇⢀⡾⠋⣠⣶⣿⡿⠛⠁
⠀⠀⠀⢻⣿⣧⢸⣿⣿⠀⡀⢠⣿⣿⠃⠠⠀⠀⠀⣎⠀⠀⢸⢇⡟⣼⣿⣿⢸⣿⣿⠸⣿⣿⡇⢸⠇⣰⣿⣿⡩
⠀⠀⠀⠈⢿⣿⣿⣿⣿⠀⢃⣾⣿⠃⠀⢰⠂⠀⠀⠋⠀⢸⣿⡼⢡⣿⣿⢇⣾⣿⡟⣸⣿⣿⡇⠀⢰⣿⣿⠋
⠀⠀⠀⠀⠀⢻⣻⣿⡟⠀⣾⡿⠃⡀⠀⣸⠃⠀⠀⠀⢠⣿⣿⢃⠟⣸⢏⣾⣿⣟⣵⣿⣿⡟⠀⣴⣿⠟⠁⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠈⢿⣿⠃⣼⡿⠁⣼⠃⣰⣿⠀⠀⠀⠀⣾⣿⢃⢂⢠⣫⡾⣻⣿⣿⡿⠟⠋⡠⠞⠋⠀⠀⠀⠀⣀⣤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⡟⣸⡟⠀⣼⠃⣰⣿⡏⠀⠀⢀⣼⡿⠁⢂⣾⣰⣟⣽⣿⠿⠋⠀⢀⣠⠄⠀⢀⣠⣴⠾⠟⢉⣴⠋
⠀⠀⠀⠀⠀⠀⢀⢱⡟⠀⣸⢃⣼⣿⡿⠁⠀⢀⣾⠟⠁⠀⣼⡇⣏⣿⡟⠁⠀⣠⡾⠋⠀⣠⡶⠟⠉⠀⣴⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⡟⠀⢰⣿⣿⣿⡟⠁⠀⢀⣾⠋⠀⠀⣼⣿⣇⢻⡟⠀⢠⣾⡿⠀⢀⠾⠋⠀⢀⣴⣾⢿⣿⡏⢀⡿
⠀⠀⠀⠀⠀⠀⢸⠁⠀⣾⣿⣿⠋⠀⠀⠀⡾⠁⠀⠀⣼⣿⡟⢻⢸⡇⣰⣿⣿⠃⠀⠁⠀⠀⣰⣿⣿⣳⣿⡟⠀⡜⣯
⠀⠀⠀⠀⠀⠀⡄⠀⠀⣿⣿⠃⠀⠀⠀⠀⠁⠀⢀⡾⢋⣿⡧⣀⣼⣠⣿⣿⠏⠀⠀⠀⢀⣼⣿⡿⣷⣿⠟⠀⡰⠁⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⣰⠏⣴⣿⣿⡇⣿⣿⣿⣿⠋⠀⠀⠀⢀⣾⣿⣿⣿⡿⠋⠀⡰⠁⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⡰⢡⣾⣿⣿⣿⣼⣿⠿⠋⠁⠀⠀⠀⣰⣿⣻⣿⡿⠋⠀⢀⡔⠁⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢠⡄⠀⠀⢀⣴⢟⣽⣿⡿⠛⠉⠀⠀⠀⠀⠀⣠⣾⣿⡷⠛⠁⠀⣠⣴⣿⣶⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⠀⣼⠁⠀⢀⠞⣱⡿⠋⠁⠀⠀⣀⣤⣔⣀⡴⠞⠛⠉⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⠁⣿⣳⠏⠀⠀⢊⡞⠁⠀⢀⣤⡾⠿⠋⠁⠉⣁⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⡘⣼⣿⠏⠀⠀⠀⢋⠀⢀⡴⠟⢁⡀⠀⠀⠔⢊⣩⣴⣶⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⢀⡴⢃⠤⢊⣠⠞⠁⠀⠀⣠⣾⡿⠿⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⣼⡿⠋⠀⣠⣴⣾⠟⠀⢀⣴⡿⠁⠀⠀⣠⠾⠛⠁
⠀⠀⠀⠀⣼⠋⢀⣴⣿⡿⠟⠁⣠⡴⠿⠋⠀⠀⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠐⠀⠴⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠚⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣤⣀⣀⣀⣀⡠⠄⠒⠒⠛⠓⠢⠤⠄⣀⠀⢀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠙⠳⠶⣶⣶⠊⢀⣠⣶⠆⢀⣠⣀⡱⡄⠀⡀⡀⠀⡀⣀⠀⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡵⡄⠀⢀⡄⠀⠚⠁⠀⡄⠉⢸⡅⢿⠗⠘⠶⠇⠟⠲⣣⣍⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⡸⠆⠀⣀⡼⠀⠸⣇⠱⣌⠁⠀⣉⣉⠤⠴⠖⣙⢆⣩⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⣸⠁⠀⠀⣨⠗⠀⢸⠃⠠⠃⠀⠜⡽⠉⠁⠀⠀⠉⢿⣿⣿⠟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⡇⠀⠠⠟⢻⠃⠀⢰⣅⠀⠈⠆⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⢻⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⡇⠰⢲⠀⠈⠳⣄⠀⠀⣠⠏⣰⣿⣷⣶⡖⣶⣶⣶⡾⣤⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⡎⢠⡄⠀⠀⠀⠀⢿⣄⠈⣆⡄⠀⠘⠄⠀⣵⢒⢻⣿⢻⢻⣷⣿⡿⠁⣸⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⣤⡼⠧⠈⠃⠀⣠⠀⠀⠀⠈⠃⠹⠅⠀⠀⡄⢠⡏⠈⣾⣿⠇⢸⣿⣿⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠴⠶⢶⡄⠀⢀⣿⣀⡀⠀⢀⠀⢀⠀⠀⣼⡁⠸⣧⣦⢹⣿⠸⢿⡉⣿⠇
⠀⠀⠀⠀⠀⠀⣠⠖⠠⠥⠤⠤⠖⠒⠳⡄⠻⢦⣌⠙⠻⣿⠀⠸⡆⠸⣠⡆⠀⢻⡀⠻⡇⢼⣽⣷⣿⣿⣥⣆
⠀⠀⠀⠀⠀⠸⣡⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠳⡄⢹⡇⠀⢿⣄⠁⠸⣄⡄⠁⢀⠉⢸⣷⣤⠤⠤⠤⡔⡴⠒⠐⠒⠲⠖⠀⠶⠚⣉⣑⣢⡴⠶⠶⣄⡀
⠀⠀⠀⠀⠀⣰⠇⣼⠃⢠⡇⢠⣤⠀⠀⢹⣆⡀⠀⠀⡀⠈⡇⠀⣾⠋⣀⣀⡙⠃⠀⠘⠂⠀⠻⠷⠤⠤⠔⠁⠀⠀⠀⠀⡄⢄⠀⠀⠬⣄⠀⠀⠈⢳⣤⣀⡬
⠀⣠⠶⠦⣾⣿⢀⣿⠀⣾⡀⠸⣿⡆⠀⢿⠉⠉⠙⠲⣣⠀⠃⠀⠁⡴⠋⠉⠉⠁⢀⣤⠖⢀⠼⠀⠀⣀⠀⢴⠀⢀⣷⢠⡗⢸⣷⠷⠶⠙⠿⢶⣄⢠⠦⣹⠟
⡀⡁⣠⠞⠛⢃⣼⣿⢰⣿⡇⠀⡏⡄⠀⠈⡇⠀⠀⠀⠸⣄⣀⣀⣀⣁⠀⣀⡤⠶⠿⠏⠀⠀⠒⠒⠚⠉⠁⠈⢶⣾⣷⢂⢄⠸⠟⣤⡀⢠⡖⢤⠿⠿⣦⠟
⠷⣷⢃⡀⡔⠉⠁⢰⠟⠋⠀⠀⠈⢮⠚⣨⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣸⣥⠾⠛⠿⠿⣇⠎⠀⠘⠉
⠀⠸⠺⢶⠇⣄⠀⢸⣀⠀⠀⣠⣼⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⠀⠀⠀⠘⠃
⠀⠀⠀⠀⣶⠿⣶⣾⣏⢿⣻⣛⡵⠋
⠀⠀⠀⠀⠈⠀⠀⠀⠈⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣶⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⣤⣤⣶
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠈⠁⠀⠀⠀⠀⠂⢈⡏⠁⣠⣆⣠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠢⠒⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣿⠁⠃⣨⠙⢉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤
⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠶⠀⠸⠇⢀⡾⢡⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⣱⠄⠀⠀⠀⠀⠈⠠⡄
⠀⠀⠀⠀⠀⡱⣦⠀⢀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠧⠀⠀⠋⠠⠋⠔⠁⠀⠀⠀⠀⠀⠀⢀⡀⡠⠀⠃⠀⠀⠀⢠⣄⣂⣀⡈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠰⡄⠇⠀⠀⠁⠀⠀⠀⠉⠙⠛⠻⠷⠿⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣴⣀⠀⠀⠀⡄
⠀⠀⠀⠀⠈⠒⠶⠲⢀⣤⠄⣠⠴⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⠟⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠰⠀⠐⠂⠘⠢⣀⣀⠀⠀⠀⠀⠀⠀⠀⠂⠰⠂⠙⢻⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⣀⡠⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠆⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠞⠢⠄⠀⠤⠊⠉⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⢄⡀⣀⡀⠠⢀⠐⠃
⠀⠀⠀⠀⠀⠀⡬⢀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣷⡾⠷⣿⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⣈⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠰⣤⡀⠀⠀⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠠⢛⠀⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣀⠀⠀⡄⠐⠃⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⢤⡀⢤⠈⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟
⠀⣠⠀⠀⠀⠀⠈⠀⣀⠒⣀⠀⠂⡐⢂⣤⡄⠀⠀⠀⠀⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⡀⣀⠀⡀⣀⡠⠆
⠘⢋⡙⠉⠉⠁⡀⠀⠻⠿⣃⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠁⠀⠀⠁⠂⠀⢀⠀⠀⠀⠀⠀⠀⠚⠃⠰⠂⠉⠀⠀⠀⠀⠀⠀⠁⠒
⠀⠼⢇⡨⠍⠀⠀⢀⠄⠀⠛⠀⠀⠠⠀⠐⢢⡤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡃⠐⢀⠀⠀⡁⢠⡤⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⠛⠂⠀⠀⠘⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⡼⠃⠀⠚⠃⠀⠀⠀⠁⠚⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⢀⡀⢠⣤
⡄⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠋⠁
⠁⠀⠀⢠⡀⢃⡰⠐⠶⠀⠑⠲⢄⣠⡄⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡰⠚⠎⠀⣠⣀⠀⠀⠀⠀⣀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠘⡀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀
⠀⠀⠐⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠉⠀⠀⡰⠀⠀⠀⠀⠠⠤⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⣀
⣀⣀⣀⣀⣈⣉⣉⣁⣀⣘⣢⣄⣀⣀⣐⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣈⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡶⣻⣽⣏⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣶⣤⣄⣀⣀⣠⣤⣶⡶⠶⠦
⠀⠀⠘⣠⣾⣿⣿⠏⠀⠈⠉⠛⠛⠛⠛⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⣼⣿⣿⣿⣿⢶⡆⢐⣒⠀⣲⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣾⣶⣿⣿⣿⣷⣶⣤⡀
⠀⠀⢸⣿⣿⣿⣿⡜⠁⠈⠉⠤⠉⠉⡽⠛⠋⠛⠛⠙⠙⠉⠋⠋⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⠿⠟⠛⠛⠋⠁
⠀⠀⠘⣿⣿⣿⣿⠃⠀⠀⣀⣀⣀⡀⠙⢤⣀⡀
⠀⠀⠀⠘⠻⠿⣥⠤⠖⠊⠁⠀⠀⠈⠉⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡴⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢦⠀⠀⠀⠀⣆⠀⢃⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⠀⡏⠳⢄⠐⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡈⠀⠀⠀⠀⠹⣀⠄⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠌⡇⢀⡄⠀⠀⠀⠐⢤⠀⠀⠙⢆⠀⠀⠀⠀⣠⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⡏⢸⠱⡀⣀⠀⠀⠀⠑⣄⠀⠈⣇⠀⠀⠌⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠤⠐⡇⠀⠑⠹⡌⣏⢢⡀⠀⢹⢀⠈⢸⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠲⡀⠻⡄⠀⠀⢀⡇⠐⡧⢠⠏⠸⠀⢸⠀⢠⡾⡀
⠀⠀⠀⠀⠸⡗⢄⡀⠀⠀⢳⠀⠡⠈⠢⡈⢢⣀⠞⢠⠀⠿⠁⡐⢀⠀⡜⠀⢸⠁⢇
⠀⠀⠀⠀⠀⠑⢄⠈⢦⠀⠀⠱⣄⠡⠀⠈⢄⠛⡐⠁⠃⡠⠊⠀⠌⡸⠁⣠⠏⠀⢹⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠐⠲⡤⣍⠀⠀⠀⠘⡆⠑⠀⠈⣤⠁⠀⠞⠀⠀⠌⣰⠵⠚⢁⡄⠀⣼⠀⠀⡠⢊⡾
⠀⠀⠀⠀⠀⠀⠀⠈⢂⠉⠢⡀⠀⢸⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⡁⠔⠂⡑⠀⣰⠃⠀⠨⠐⣉⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠐⢆⠹⡄⡞⠠⠀⠀⠀⠀⠀⠀⠀⠰⠎⠀⠀⡎⢀⡔⠁⣠⢀⡴⢋⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠂⠛⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡞⣰⢾⢃⡎⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠀⠘⡀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⠀⡿⢨⢸⣼⠰⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢁⠈⡄⠁⠁⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠒⠀⢃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠅⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢔⡵⠋
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠄⠀⠀⠀⣀⠀⠀⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠨⠀⠄⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠠⠊⠁⠀⠈⠐⠂⠂

⠀⠀⠀⠀⢠⠁⠀⠀⢀⣀
⠀⠀⠀⣤⢰⣦⣀⠀⣀⡭⡹⠶⡄
⠀⠀⢀⣭⣿⣿⣏⠠⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆
⡤⠾⠉⠿⣿⡇⣅⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇
⠀⠀⠀⢆⢸⣇⣿⠬⠭⠛⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⣢⣴⡿⣿⡯⡶⠈⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣧⡄⠰⠖⠛⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠟⠉⠁⣧⣸⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢾⣿⡟⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣒⡀⢀⣬⣀
⠀⠀⠈⠁⣿⣯⣿⠛⡛⠲⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠰⣸⣯⠧⠭⠓⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠽⣿⣭⣄
⣠⡤⢺⣿⣿⣇⣛⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⢻⡗⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡤⠶⡦⣴⣿⡡⠈⠉
⠉⠂⢠⠀⣿⡏⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⢸⣷⡶⠦⣤⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠈⠀⠀⠀⠲⣿⢿⠶⣀
⠀⠀⠠⠁⣼⡇⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢾⣿⣏⡃⠀⠀⠑⠀⠀⢀⠒⠯⠄⣴⡦⠀⠀⠀⠀⢀⠤⠶⣶⣿⠛⠀⠈
⠀⠀⣆⠀⢿⡗⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢹⡏⠁⠀⠀⠀⠀⠀⠀⢀⣀⡀⣹⡟⠳⠤⡀⠀⠀⠀⠀⣭⣿⠿⠒⠤
⠀⢠⠊⠠⣿⡇⢂⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢻⡇⢣⠀⠀⠀⠀⠀⠀⠊⠈⠉⢽⣿⣤⡀⠀⠀⠀⠀⠀⡁⢿⡘
⠀⠐⡀⠀⣯⡇⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣼⡇⠠⠀⠀⠀⠀⠀⠀⠀⠤⠀⢞⡟⠀⠈⠁⠀⠀⠀⠀⠀⣿⠀⠂
⠀⠃⠀⠢⣯⡏⠂⡰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢀⢹⡇⠀⡃⠀⠀⠀⠀⠀⠈⠀⠀⢿⡟⠃⠒⠀⠀⠀⠀⠘⠁⣼⠁⢠
⢰⢣⡀⣉⣧⣷⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣛⣇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⠁⠀⠀⠀⠀⠀⡐⠀⣿⠆⠈⡀
⢼⠛⠉⢶⣟⡿⠘⠓⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠣⣿⡯⠛⢺⠀⠀⠀⠀⠀⠀⠀⠐⢈⡇⠀⠀⠀⠀⠀⢀⡶⠾⣿⡷⠛⡆
⣿⣷⠶⠶⠿⢷⣷⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣿⣿⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣿⣶⣶⣶⣶⣾⣷⣾⣿⣷⣾⣷
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒⢲⣤⣬⣭⣍⡉⠉⠉⠉⠉⣉⣭⣭⣤⣤⠒⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠼⣇⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠈⠒⠄⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣀⣀⠔⢤⠲⡚⠉⠹⢻⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁⠀⠀⠘⢧⡀⠀⣤⣀⠀⠀⢸⡆⠙⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⢸⠻⠉⠙⡖⢢⠔⢄⣀⡀
⡔⠃⠀⠈⠆⠈⡧⠀⠞⠂⠸⡇⠀⣠⡶⠋⠁⠀⠀⠀⠀⢀⣀⠀⠀⠙⠶⠏⠉⠛⠲⠾⢃⣀⠀⠀⠈⠙⠓⠶⠦⣤⠤⠀⡿⠀⠚⠆⠀⡋⠀⠎
⠧⠤⠄⠤⠈⠀⠀⠀⠀⠀⠀⢻⠀⢸⡇⠀⠀⠀⠀⠀⣀⣀⣉⠻⠂⠀⠀⠀⠀⠀⠀⠺⢋⣁⣀⡀⠀⠀⠀⠀⠀⣿⠀⢸⠃⠀⠀⠀⠀⠀⠈⠠⠀⢤
⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠈⢧⣸⡇⠀⠀⠀⠀⠘⠋⠙⠋⠙⠀⠀⠀⠀⠀⠀⠀⠘⠉⠛⠉⠙⠀⠀⠀⠀⠀⣿⣀⡏⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⢹⠀⠉⠓⠲⠤⠤⢤⢀⠀⡞⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⡆⢀⡀⠤⠤⠴⠒⠛⠁⢸⠂
⠀⠀⠈⡷⢄⠀⠀⠀⠀⠀⠈⠑⡙⠦⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠞⡑⠁⠀⠀⠀⠀⠀⠀⠴⡏
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠰⡀⠈⣧⠀⠀⠀⠀⠀⠀⡤⠰⢤⣤⡤⠤⠤⣤⣤⠴⠢⡄⠀⠀⠀⠀⠀⢠⡏⠀⡴⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠙⢄⠘⢧⠀⠀⠀⠀⠀⠹⣠⠎⠀⠀⠀⠀⠀⠈⢦⡸⠁⠀⠀⠀⠀⢠⠟⢀⠜⠀⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠲⣆⣀⣀⣀⡀⢀⣙⡾⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣞⡁⠀⣀⣀⣀⣀⡴⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢀⠝⠋⠀⠀⠀⡾⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣶⠋⡀⠀⠀⠈⠛⢅⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠈⠉⠀⢤⠒⠘⢺⠃⠀⠀⠈⠉⠙⠛⠒⠒⠒⠒⠒⠚⠛⠉⠁⠀⠘⣦⣗⠐⠋⢢⠄⠈⠁⠘⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⢂⠟⠂⢠⠜⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡄⠀⠘⠄⠀⠚⠑⢢⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠰⠁⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⣤⣄⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡉⠉⠉⠛⠉⠙⠋⠉⠉⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠢⠇⠀⠀⠂⠀⠀⠇⠀⠀⠦⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣁⣠⡀⣀⣀⣀⡄⠀⠀⣅⣀⣀⣀⣠⣀⣉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⣀⣀⣀⣀⣀⡽⠀⠘⣅⣀⣀⣀⣀⣀⣀⡁
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⢀⣤⡀⠀⠀⣿⠉⢳⡀
⠀⠀⠀⠀⢺⡉⢹⡄⠀⢸⡇⠈⣧
⠀⠀⠀⠀⠸⡇⠀⢷⠀⠈⣷⠀⢹⣆⣀⡀
⠀⠀⠀⠀⠀⢿⡀⠸⣧⡤⠞⣧⣀⣿⣽⢷⣄
⠀⠀⠀⠀⠀⣨⢷⣠⣿⠀⠀⢈⣭⠟⠋⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠓⠶⣄⡀
⠀⠀⠀⠀⠐⣅⣀⣀⣀⣤⠶⠋⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⣀⣀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠈⠫⡉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⡿⢠⡟⠉⠉⠙⣦⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⡀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣷⠘⣇⠀⠀⠀⠈⣇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠲⣄⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠘⢧⣄⠀⠀⡟⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣞⠃⠙⢦⡀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠳⣾⠇⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠙⠶⣤⣀⡀⠀⠀⠀⠀⠙⣶⡴⠋⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠉⠉⠛⠛⠛⠛⠋⠉⠀⠀⢀⣼⡏⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠶⠤⢤⣤⣤⣤⠴⠶⠋⠀⠙⣦⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⢦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡉⠲⢽⡶⠶⢤⡤⠴⠞⠋⠁⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣥⣀⣀⣀⣀⣠⡴⠞⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣤⣀⣀⠀⠤⣴⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀
⢰⠆⣠⠾⠁⠉⠀⠈⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⡿⢶⣄⡀⠀⠰⣦⡀⠀⠀⠀⠀⢀
⠙⠚⢿⡀⡀⠀⡀⠀⡄⠀⢀⡆⢀⣆⡀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠳⣿⣿⡿⣷⣄⢉⣿⣽⡲⠀⣀⠉⢀⣤⣤
⠀⠀⠘⠓⠙⢾⡿⢶⠿⣿⣿⣯⣭⣿⣿⣒⠀⠹⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⠀⠀⠀⠀⠀⠙⢿⣟⣿⣿⣶⡟⣵⠋⢸⡇
⠀⠀⠀⠀⠀⠈⣿⡄⣿⣫⣵⣾⣿⣿⣿⣯⣥⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢷⣤⣤⣀⣀⣤⣾⣿⣿⣏⣹⣿⡟⣠⡾⠇
⠀⠀⠀⠀⠀⠀⣿⣧⢻⣿⡿⠿⠛⣶⡿⠛⠉⠛⢪⣷⠑⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡄⠀⠈⠛⠉⢉⡷⢟⣿⣭⣿⣿⡿⢿⡋
⠀⠀⠀⠀⠀⠀⠙⣿⣸⣿⣏⠀⠚⠁⠀⠀⢀⡄⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⡜⠋⢀⣠⡤⠶⠶⣤⣾⣶⠽⠷⠵⣯⡹⠀⠁⠰
⠀⠀⠀⠀⠀⠀⢰⣿⢱⣿⣟⡳⣶⣶⠶⠖⠉⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠃⠀⠀⠀⠈⠉⠻⠿⠷⢶⣿⣭⠶⠒⠂
⠀⠀⠀⠀⠀⠀⠈⡟⣾⡽⢯⢁⣿⠃⠀⠀⠀⣀⢠⡔⢿⡀⠀⠀⠀⠀⠀⠀⡰⠏⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⢏⡔
⠀⠀⠀⠀⢠⡤⣼⠿⢷⣿⡯⢿⡉⠤⣤⠔⠃⣌⠑⠇⡿⠃⠀⠀⠀⠀⠀⠀⠀⣸⣧⣀⠀⠀⠀⠴⠀⠐⠂⠐⠆⢰⡏⣾
⠀⠀⠀⠀⢹⡇⠈⠀⠀⠻⠀⠸⠀⠀⢸⠀⠀⠈⢀⣸⠃⠀⠀⠀⠀⠀⠀⠀⣰⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢳⠋⠰
⠀⠀⢠⣾⠛⠓⠚⠒⠠⠤⣄⣀⣀⠀⠈⣄⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⢸⠉⢸⣿⠀⢀⣤⣠⡦⠶⠶⠾⢶⠾⠷⠾⠷⠶⣤
⠀⠀⠀⠛⠶⣤⡀⠀⠀⠀⠀⠈⠙⠒⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠟⠛⠋⠁⠀⠀⠀⠀⠒⠶⣾⠷⠷⠒
⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄
⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⢀⣴⣋⣷⡀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣿⢣⡀⠀⢠⡆⠀⠀⠀⡴⠋⠀⢀⡴⣿⣿⣿⡏⠃⠀⠀⠀⠀⠀⠀⠀⣿⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣖
⠀⠀⠸⢿⣆⠹⣦⡀⠁⠀⠀⣀⠀⢀⠴⠋⠀⠀⠉⣹⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⣧⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠋⠁⣠⠋⠂
⠀⠀⠀⠀⣉⡷⣾⣿⡗⠚⠺⣷⡖⠁⠀⠀⢀⣠⣞⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡙⢦⣀⣹⣿⣦⡷⠀⠀⣼⣷⣾⣿⠁⠀⣀⠀⠔⠁
⠀⠀⠀⢀⡈⠁⠠⠏⠀⠀⠀⠻⣿⢗⣲⠒⠛⠁⣋⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡿⠿⠛⠃⠀⠀⠈⡿⡝⠛⠒⠒⠂⠀⠢⠂
⠀⠀⠀⢈⣦⡀⠀⠀⠀⠀⠀⠀⠉⠂⠀⣠⡀⠈⠙⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠘⣿⠳⠶⠦⡤⠤⡦⠖⠞⠉⠙⢻⡟⢦⡀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢡⣿⡁⠹⠛⢦⣄⡀⣀⣀⣠⣤⢶⡶⠛⠛⣷⠸⠆
⠀⠀⠀⠀⢈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣎⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⢷⡄⠀⠀⠀⠉⠉⠁⠉⠀⠀⠁⢀⡾⠀⠰
⠀⠀⠀⠀⣼⡟⠀⠀⠉⠁⠛⠂⠒⠠⠶⠀⠀⠼⣿⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢻⠄⠤⠀⢀⠀⢀⡀⠀⠀⣠⡎⢸
⠀⠀⢀⣄⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢣⡏⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⢷⡈⢆
⠀⠀⠺⢯⡉⠉⠙⠓⠶⢤⡄⠀⠀⠀⠀⣀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣿⣠⣆⣠⣦⣤⣴⣦⣦⣀⣄⣀⠈⢣
⠀⠀⠀⢨⡟⠀⣰⣦⠄⡀⠈⠐⠛⠐⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠋⠉⡁⣀⡀⣠⣠⡀⠀⠉⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉

⠀⠀⠀⣠⣿⠀⠀⠀⠀⣸⣧⠀⠀⠀⢀⣰⣷⡄⠀⠀⢀⢰⣧⠀⠀⠀⠀⣾⣧⠀⠀⠀⢀⣼⣧⡄⠀⠀⠀⣾⣄
⠀⠀⢾⠿⡹⣿⠀⠀⢰⣿⣿⣿⠀⠀⢾⠟⠙⣿⠀⠀⢸⣾⣯⡤⠀⠀⢰⣿⣿⡆⠀⠀⣽⡟⠻⣷⠀⠀⣿⠏⢻⣶
⠀⠀⠈⠘⠃⠉⠀⠀⠈⠛⠛⠁⠀⠀⠈⠃⠐⠁⠀⠀⠈⠛⠛⠁⠀⠀⠈⠛⠛⠁⠀⠀⠈⠐⠃⠉⠀⠀⠈⠀⠀⠁
⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⣄
⠀⠀⣰⣾⣿⣧⠀⠀⠀⣼⣿⠀⠀⠀⢀⣿⣷⡀⠀⠀⢀⣿⢷⣄⠀⠀⢠⣾⢿⡆⠀⠀⣠⣿⣿⡀⠀⠀⢠⣶⢿⡄
⠀⠀⠻⢿⡽⠟⠀⠀⠰⣿⣿⠟⠀⠀⠸⡏⠘⠟⠀⠀⠸⣧⣆⠟⠀⠀⢸⡇⠸⡇⠀⠀⢻⣿⣿⡟⠀⠀⢸⣣⡌⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⣰⣷⡄⠀⠀⠀⢰⣦⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⢀⣴⡄⠀⠀⠀⢰⣶⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⣾⣆
⠀⠀⢴⠟⡈⣷⠀⠀⢨⡟⢹⣶⠀⠀⢰⡿⠹⣧⠀⠀⢀⣿⢿⣀⠀⠀⣰⣿⣿⡇⠀⠀⣤⣿⢿⣧⠀⠀⣤⡟⢹⣧
⠀⠀⠘⠚⠃⠋⠀⠀⠘⠃⠀⠋⠀⠀⠘⠃⠀⠃⠀⠀⠈⠃⠈⠃⠀⠀⠘⠻⠿⠋⠀⠀⠙⠋⠈⠋⠀⠀⠙⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡀⠀⠀⠀⠀⠄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠀⡄⠁⠀⠀⠀⠀⠢⡅⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣅⡌⠀⠀⠀⢐⠋⠀⠐⢰⠐⢦⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠂⠀⠴⠋⠑⠒⠂⠈⠑⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠘⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠢
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠘⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⣄⢠⠈⠂⠀⢀⠄⠐⢀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⣸⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢀⢈⢁⣼⡇⠈⡀⠈⢀⠀⠈⠁⠀⠀⠐
⠀⠀⠀⠀⠐⠁⢀⠐⢁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠉⠪⢙⡿⠀⠀⡀⠀⠀⠁⠀⠀⠈⡢⠀⠈
⠀⠀⠀⠜⠀⠐⠋⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠂⠀⠀⠀⡠⠂⠀⠀⠑
⠀⡀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⢀⢂⡀⡉⠲⡢⠤⡤⠄⠀⢀⠜⢀⡀⡘⠀⠀⠀⠀⠀⠀⡆⠀⠀⠙⢀⠀⡤
⡈⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠸⡀⠙⠄⢠⠋⠳⡅⠀⠔⠁⡀⠄
⠁⠀⠀⠀⠀⡰⠂⠰⠀⠀⠀⠀⠀⠀⠁⠢⢕⣄⠴⠋⠔⠈⠁⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠒⠄⠤⠠⠀⠀⠀⠀⠃⠂⠀⠀⠀⠀⠠⡄⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⢏⣁⢀⡀⣀⡀⠀⠀⠀⠂⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠈⠁⢀⣧⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣹⣦⣀⣴⠀⠀⠈⠀⠀⢰⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⢙⡛⠛⠴⣶⣶⣵⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠁