Pathology Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⢐⣒⣀⣐⠒⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢈⡵⢤⠾⠛⠻⢏⣿⣿⣿⣷⣄⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⣈⣥⡤⠒⠉⠀⠀⠈⣤⠄⠠⣼⣿⣿⣟⣹⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⡀⣤⣤⣤⢤⣤⣤⣤⣤⣤⣶⣶⠟⠋⠀⠀⠘⠁⠀⢠⠞⠁⠀⠀⣿⣿⠉⠉⠁⢹⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡼⠞⠉⠀⠀⢉⡈⢾⣉⢱⣿⣟⠻⣿⣿⡀⠀⠀⣤⡀⠀⠀⠁⠀⠀⠀⢘⣿⠁⠀⢤⣤⣹⣿⣿⣿⣧⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣲⣯⡄⠠⠀⠀⠀⠈⠠⠀⣸⣿⣧⣿⡿⣏⣿⡆⠀⠀⠸⠿⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠩⣿⢻⣿⣿⡆⡇
⠀⠀⠀⠀⠀⠀⠀⢆⡏⡗⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣇⣬⣿⣪⣽⣧⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠈⡙⣿⣿⣿⣇⣧⣀⡀
⠀⠀⠀⠀⠀⣴⣾⢹⡄⣙⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠈⠟⣻⡿⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣖⣿⣿⣿⡿⢸⣿⣿⠉⠳⣄
⠀⠀⠀⠀⠀⣼⣿⠘⣯⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠋⠁⢠⣿⡁⠀⠀⣳⣤⢀⣀⢀⣀⣀⣤⣴⣤⣿⣿⣿⣣⠿⢋⡽⠃⣾⣿⣿⣧⠀⠈⡄
⠀⠀⠀⠀⢸⣿⣿⣇⠹⣿⡀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⣿⠃⠀⠈⠛⠋⣽⣿⠿⠟⠛⠛⣻⠃⢠⣾⠃⣼⡟⠉⠀⢸⠀⠀⠃
⠀⠀⠀⠀⠈⢿⣿⣿⣆⠉⠷⣴⠱⣇⠀⠀⠀⠀⠀⠀⢒⡻⠄⠀⢿⣿⡇⠀⠀⢻⠀⠀⡀⠀⣠⣿⡏⠀⠀⠀⡾⠁⠀⣼⢃⣾⣿⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⠐⠀⠀⠋⠿⡜⣷⣤⠁⠂⢭⡄⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣿⠃⠀⠠⠋⠀⠀⠁⠀⠀⣿⣿⣆⣴⠞⠁⠰⣾⢃⣼⣿⠁⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⣿⣧⣄⠳⣆⠀⠀⢀⣀⡀⣤⣴⡿⠋⠉⠉⠈⠁⠀⠀⠀⠀⠀⣾⣿⣿⠟⠁⠀⣠⣾⢣⣾⡟⠃⠀⠀⠀⢹⠗⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣷⡄⠻⢧⡈⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠁⠀⠀⣶⡾⢳⣿⠏⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣈⡀⠀⠀⠈⠉⠋⢦⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣀⣾⡿⠶⠟⠋⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄⠀⠀⠀⠀⠀⠙⠲⣽⣧⠀⠀⠀⠀⠀⠀⢀⡖⠀⠀⠀⠀⡈⠁⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦⡀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠀⠀⠀⠀⢀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣶⣼⣾⣷⣿⠿⠿⣷⡀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⢺⣿⣗⣄⢀⣴⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣼⡇⠀⢰⣿⡇⠀⠀⠀⣦⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⢸⣿⣿⣷⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣿⣿⣿⡁⠀⣸⣿⡷⠀⠀⠀⣅⠀⠀⠀⠀⠀⢨⣿⣧⡀⠀⢸⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⡿⠀⠀⠀⠀⢀⢸⣿⣿⣿⣷⣾⣧⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠿⢻⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⢸⡆⠀⠀⢘⠁⠀⠀⠀⠀⠀⢸⡄⠀⠉⠻⠿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠇⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠁⣧⠈⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠋⠉⡉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠀⠀⠉⠛⢿⣦⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⢠⣿⠛⠉⠙⠛⠿⣶⣶⣶⡾⢿⣿⠁⠀⠀⠀⠀⠀⠀⠈⣿⡿⢷⣶⣶⣶⠿⠛⠛⠉⠛⣿⡄
⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣤⣄⡀⠀⠸⣿⡀⢀⡀⠀⠀⠀⠀⢀⣿⠇⠀⢀⣠⣤⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠸⣿⡄⠀⠀⠀⣿⣏⣹⡇⠀⠀⢹⣿⠟⠻⣷⡄⠀⠀⣾⡏⠀⠀⢸⣏⣹⣿⠀⠀⠀⣠⣿⠇
⠀⠀⠀⠀⠈⠻⢷⣶⣶⡿⠿⠛⠁⠀⠀⢸⣿⣄⣀⣾⡇⠀⠀⠛⠁⠀⠀⠈⠛⠿⢿⣶⣶⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠙⠛⠉⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢀⣴⡶⢶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⢸⣏⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠈⠛⠿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠛⣿⡄⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⣤⣿⠃⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⠁⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⡲⢖⠶⠖⠒⣉⡩⠿⠟⢳⠢⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⢀⠀⣴⠃⠀⠀⠀⠹⡄⠘⢲⡎⠈⣿⣝⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⢀⠄⠐⠁⠀⠀⣀⡄⢚⠁⠼⡁⠀⠈⠀⡠⠐⠁⠀⢠⣧⠉⢁⣿⡄⠛⠉⢻⣿⣶⣄⡀
⠀⠀⠀⢀⠰⠋⠀⠀⢠⡄⠀⠈⠱⠊⡀⠀⡼⠂⠀⢰⠃⠀⠀⢰⠃⢸⠃⠉⠘⣷⣀⣷⣿⣿⠉⠙⢿⣦⡀
⠀⢠⡞⠁⠀⡀⠀⡀⠀⠈⠢⡀⡀⠈⠀⡌⠀⠀⠒⠛⢄⠀⠀⡄⠀⣿⠀⠀⢂⣼⣿⡏⣈⢿⡶⡄⡈⢿⣷⡀
⢀⣾⠀⠀⢈⠁⢀⡇⠀⡀⢀⡟⠀⠀⢠⣇⣀⣀⣤⡤⠤⠤⡴⠇⠀⣻⠐⢇⠈⢹⡿⣇⠅⣼⣿⡷⠌⣾⣿⣷⡀
⣿⡻⢀⣴⡞⠀⠈⠀⠀⡈⣿⣧⠤⠔⠋⠉⣅⠀⠀⠀⡀⠀⠠⠴⠚⠻⣦⣤⣤⣼⠧⠛⣙⣿⡍⠀⣴⣿⣧⣿⣷
⣿⡇⠀⠈⠉⠓⠄⠀⠀⠀⠟⠁⡆⠀⢀⡶⠛⠀⠀⠈⠀⠠⢤⡤⠀⠀⠀⠈⣇⣠⣶⡿⢟⠻⣄⣠⣿⣿⣿⣿⣿
⠹⠷⢤⠀⠘⢤⡄⠊⢁⣼⠀⠀⡛⠲⢋⠀⠀⠀⠰⣄⠀⠤⣂⡼⣳⣀⣐⣀⠘⠋⢉⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢢⠀⣠⣤⣬⣤⣴⣿⣷⠀⠀⡇⠐⢁⣴⡀⠁⡠⣋⣴⣾⣿⣾⡿⠿⠿⠛⠛⠿⠿⠿⣿⣿⣿⣿⣿⡟⠿⣿⠏
⠀⠀⠑⠻⢿⣿⣿⠿⠛⠉⢄⣠⣉⣉⣭⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠉⠁⠒⠶⣤⣀⣀⣲⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢽⣫⣴⣾⠟⠀⠙⠿⢿⣿⡿⣿⣿⣦⣀⠰⣶⣜⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠙⠦⣍⡙⠿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠿⠟⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⢀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⡶⠢⢄⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠘⡇⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣴⣴⣶⣦⠴⡟⠻⢗⣸⣯⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⢿⣼⡿⡿⠦⣼⡧⠀⢨⠿⣿⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢿⣿⣾⢿⣶⠤⣼⡧⠔⡿⣯⣵⣿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣟⣯⢽⡷⣤⢺⢃⡼⣳⢷⣢⡿⣯⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣽⢷⣻⣽⡭⢿⣳⡛⣟⣞⡾⡼⡗⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢿⣞⣯⣶⣻⣿⣿⣯⣽⢷⡿⣌⣿⣵⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣼⣷⣿⡿⠋⡗⠊⠐⢚⡟⢾⣯⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⣷⠀⠀⢀⡇⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣇⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠛⠛⠛⠛⠛⠛⠛⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠦⠴⣦⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣀⣀⡀⣀⡀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠉⠉⢉⣉⡉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢾⠀⠀⠀⠤⠟⠘⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⡄⠀⠀⠀⢘⡷⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡿
⠀⡤⠶⠞⠿⣿⣟⠲⢦⣽⡆⠀⠻⣆⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠔⠁⠀⡼⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀
⣼⡁⣠⡶⠒⠶⠬⢿⣷⣄⣙⣦⡀⠈⠳⢤⣤⠞⠛⠋⠉⠉⠉⠙⠛⠁⠀⠉⠉⠉⠀⠀⠁⠀⠈⠉⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⠉⠉⠉⠉⠙
⠈⢹⣯⣤⣀⠀⠀⠀⠈⠉⠛⠦⠽⢦⣄⡀⢹⣦⣤⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠙⠢⢥⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠻⠷⣶⡶⠦⠤⢤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠒⠤⣤⣤⣤⣤⡤⢤⣤⣄⣠⣤⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣀⡀
⣿⣿⡟⠛⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⣦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠁⠀⢠⡄⠀⠈⠉⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠰⢿⡿⠗⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡾⢛⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣀⣈⣁⣀⡀⠀⠀⢸⡿⠀⠀⠀⠀⠀⢀⣴⠟⠉⠻⣿⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⡇⠀⠀⠀⣀⣼⠟⠁⠀⠀⠀⢈⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⣤⣤⣾⠟⠁⠀⠀⠀⢠⡾⠛⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣿⣅⠙⢿⣦⡀⠀⠀⡀⠸⣷⣤⣴⡿⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠻⢷⣤⣀⡀⢀⣀⣠⣾⠟⠀⢈⣿⣷⣄⠙⢿⣦⣾⠟⠀⠀⠉⠁⣤⡈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢿⣿⠿⢷⣄⡉⠙⠛⠛⠛⠉⠀⠀⢰⣟⢁⣼⡿⣷⣄⣹⣷⠀⠀⠀⠀⠀⠹⣧⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢁⣾⢋⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠋⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⣿⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠘⠿⣿⠇⠈⢿⡄⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠘⠻⣶⣄⠈⣿⡄⠀⢸⡟⠉⠉⠉⠉⠉⠉⠉⠉⣿⡆⠀⠀⠀⠀⠀⠀⠀⣼⡏⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠙⠿⣾⣷⡀⠈⠛⠿⠿⣿⣿⠿⣿⣿⠻⠛⠀⠀⠀⠀⠀⢀⣠⡾⠋⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠙⠃⠀⠀⠀⠀⠈⠻⣷⣄⠙⠻⠂⣰⡾⠻⢷⡄⠙⠋⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡟⠛⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣙⢻⣶⡀⢻⣧⣀⣼⠏⢠⡟⣋⢁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⣇⣀⣠⣿⠇⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠓⠀⠉⠉⠁⠀⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠤⠀⠀⢀⠀⠀⠀⢀
⠀⠈⠀⠀⢀⣴⠾⠗⣣⡄⠀⠀⠊⣠⡞⠛⠛⠀⠀⡀⠠⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷
⠈⡁⠀⣴⠃⠀⠀⠀⠉⠁⡦⠀⠘⠿⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡿⢿
⠀⠀⣼⡇⠀⠀⠀⠀⣠⠾⠃⠀⠀⠀⠤⠀⠀⠈⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢾⣻⡞
⠀⣸⣿⣠⣤⣤⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⢿⢿⣸⠟⠉⠀⠀⢴⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⣀⠀⢀⣄⠂⠀⠀⠑⠀⠀⠀⠀⠈⠓⠆⡤
⠀⠀⠀⠀⠀⠘⠟⣤⠌⠀⠀⠀⠘⠋⠀⠙⠒⠉⠁⠁⠀⠀⢀⣿⠁⠁⠀⣠⡾⠃⣀⡀⠀⠀⠀⠀⡀⢠⡟
⠀⠀⡀⢀⣄⠀⠀⠀⠐⠂⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠘⠿⢣⣾⠟⢡⣀⠀⠀⠀⡠⠈⡁
⠀⠘⠏⠈⠟⠀⠀⠀⠀⠀⠐⡏⠈⢳⠀⢀⢀⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠲⣧⠔⠛⠅⠀⢻
⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⣇⣠⠜⠀⣼⠬⡇⡀⠀⠀⢸
⠀⣰⣿⣷⡀⠀⠀⠀⠀⠀⠀⡇⣴⣳⣆⢸⠀⡏⢹⢴⢹⢼⣴⢛⡦⡞⣳⡘⣼⠃⠈⠀⢀⡀⠀⣀⢀⡃
⠀⣿⡿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠁⠀⠈⠉⠈⠀⠉⠀⡽⣻⠀⣸⠀⠀⠀⠈⢙⣛⡉⠀⠂
⠸⣛⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠉⠁⠤⠇⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⣠⠀⠀⠀⢀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠸⠷⠶⢲⠖⠾⠟⢋⣀⣀⡤⢀⠤⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠠⠆⠁⠀⣄⡐⠀⠀⢿⡀⣴⣶⣶⠆⠀⡜⠡⠤⢤⡀⣸⢸⢀⡴⢂
⠀⠀⠀⠀⠀⠀⠀⠒⠃⠢⡄⠃⠀⠀⣄⠀⠀⠀⠈⠛⠀⠀⠘⠃⠠⠶⣶⣶⣶⠌⠒⠦⠤⠭⠕⠃⠈⠱⠎
⠀⠀⣀⠰⠈⠈⠀⠀⠀⣠⠋⠃⡀⣨⡏⠀⠀⠀⢠⢤⡆⠀⠀⢀⢠⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠲⠐⢣⠄⣀⠀⢀⡼⡄⠁⠀⠀⠈⠀⠀⢀⡴⠂⠁⠀⠀⠀⣿⠆⢸⡇⢸⡇⢰⣿⠀⣿⡆⣿⠀⠀⣾⡇
⠀⠀⠈⠙⠐⠁⠭⠤⠓⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠉⠀⠈⠁⠈⠀⠈⠁⠀⠈⠀⠀⠀⠀⠉
⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠁⠀⠀⢀⣄⠰⣆⣤⣀⣠⣤⣄⣠⣤⣦⡶⠶⠾⠓⠒⠈⠟⠉⡠⠃⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢠⣶⣤⡶⠶⠿⢻⡟⠋⠏⠀⠁⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⣤⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠸⠏⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⠀⣤⣄⣬⣿⣤⣀⡀⠀⠙⣿⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⣃⣤⣶⢿⣿⣿⣿⣿⣿⣦⣀⣈⣠⠀⣽⣿⣿⣿⣿⣿⣿⣶⣄⠀⠐⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⢿⣷⣼⣿⣿⣿⣿⣿⣿⠋⣁⡀⠙⠒⣿⣿⣿⣿⣿⣿⣿⢣⣿⠆⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⡏⣸⣿⣥⣤⠀⢿⣿⣿⣿⣿⡟⣯⣿⣿⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏⡀⠀⠠⢽⣿⠿⠿⠟⠉⠀⣿⣿⣿⣿⣅⠀⠐⣍⣹⡿⢃⠛⠑⠛⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⡇⣀⣠⡶⣤⡀⣀⣤⡖⠀⢸⣿⣿⣿⣿⣿⠇⠰⣾⣧⣴⣬⣀⣄⣀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⠥⣄⡙⢿⣿⣿⠇⠀⣿⣿⣿⣿⣿⣿⡏⠀⠘⠧⠼⢟⣻⣿⣿⣡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣿⡌⢻⡿⠀⠀⠛⠿⠛⡇⠿⠟⠑⠀⠀⡁⠂⣾⣿⠏⣿⣄⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢐⡇⠀⠂⠀⠀⠀⠙⠀⣿⡿⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⣿⡷⠖⠖⡖⡲⢆⣴⣮⣷⣷⡤⢴⡶⣶⡶⠘⣿⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣧⠈⢿⣇⢤⡤⡄⡀⢀⣿⣿⣿⣿⣷⠼⠷⢚⠇⠀⡿⠁⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⡆⠀⠀⠀⢤⣿⣧⣸⣿⣿⣿⣈⡇⢀⠀⠘⠃⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⢀⠀⠸⡏⣿⣿⣿⣿⣿⣗⠮⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣀⠀⠀⠀⠀⠙⠹⠟⠿⠀⠋⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠼⡄⠀⠀⠀⢀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣶⣦⣴⣿⣆⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠈⠙⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠒⠚⠻⠿⠟⠳⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠠⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠿⠦⣕⡀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣠⣊⠁⠀⠀⠀⠙⠐⠀⠀⠀⠀⠀⠀⠀⠾⠋⠁⠀⠀⠀⠃⠀⠈⢮⡢⠘⠳⣄⣤⡴⢶⠒⢒⣿⠿⠯⠽⠗⠲⢄
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⠀⠒⣶⡈⣷⡅⡀⠘⢿⣟⠂⠥⠚⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⣿⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⢀⣷⣾⡷⣾⣿⡔⠀⠸⣧⣀⡆⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠻⣿⣧⢤⡩⣝⣧⠁⠀⠀⠀⡀⠀⠀⠀⣠⠞⠁⠀⠀⢀⡾⢀⣍⠐⢉⡙⢿⣆⠀⢻⣿⢿⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢿⠮⡀⠁⣻⠘⡠⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⢠⠞⠴⠋⠉⠉⠅⠁⠀⢹⣟⢸⡏⢰⠀⠀⠀⠀⠀⠀⠀⠃⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠘⠆⠛⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠘⠁⠀⠀⠀⠐⠀⡿⣿⣷⠛⠀⠀⠀⠀⠀⠀⠀⡠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠹⡷⣀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠁⣿⠏⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣀⡀⠀⠀⠽⣦⡀⡀⠀⠈⠀⠀⠀⠉⠀⠉⠀⠀⠀⡤⠄⣠⢀⡼⠋⠀⠀⠀⠀⠀⢸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠛⢆⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠭⠟⠊⠀⠀⠀⠀⠀⠀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣀⣤⠴⠟⠳⠀⠀⠀⠀⠀⠀⠀⣠⢯⡤⢤⣀⡀⠀⠀⠀⢀⡠⢚⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡄⠛⠻⠃⠀⠀⠀⠃⠀⠀⠀⠀⠀⢠⡗⠸⡀⠀⠀⠀⡗⡴⡶⣾⣆⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⢺⣧⡀⢥⣄⠀⠀⣃⣀⣤⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⡞⠙⢿⡒⣙⢿⣿⠍⠁⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⠀⠈⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠙⠻⣤⣿⣷⣃⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣉⣡⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣆⠘⢿⣿⣿⠟⢀⣾⣿⣿⣿⣶⣤⡀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣧⡈⠿⠋⣠⣿⣿⣿⣿⣿⣿⣿⡏⢰⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣿⣿⡿⢿⠟⢉⣿⣏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⣹⡇⠀⠀⠀⣿⣿⣷⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢁⣀⡀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣿⠀⣿⣿⡏⠛⠋⣠⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⣁⣀⣈⠙⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣉⣉⣿⠆⠸⢿⠟⢉⣿⣯⣀⣀⣀⣀⣀⣀⣀⣄⣀⡀⢿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠉⢻⡏⢠⣷⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⢸⣿⣿⣿⠿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⢸⣧⡉⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⢸⣇⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⢀⣀⣽⣷⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣐⣀⣠⡖⠂⠠⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠘⢿⣿⣿⣿⣿⡋⣇⡀⠀⢀⠀⠀⠀⠀⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠶⠖⠂⠀⣀⣥⡄⠀⠀⠀⠘⠿⠀⠙⣿⠇⠉⠀⢀⣄
⠀⠀⠀⠀⠀⣠⣾⢟⠒⠀⠀⠀⠀⣻⡏⠀⠀⠀⠀⠀⠀⠀⣀⠈⢣⡄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡃⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣼⣿⡟⠈⠁⠀⠀⠀⠀⣿⣅⠀⠐⠀⠀⠀⠀⠀⠘⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢠⡅⠀⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⢠⣿⣿⣅⡄⠀⠀⠀⠀⢸⣿⠄⠀⠀⠀⠙⠦⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⣇
⠀⠀⠀⢸⣿⣿⣾⣿⣧⣀⣀⣴⣿⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠆
⠀⠀⠀⢻⣧⠾⣾⣿⣿⢿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣾⡄⢀⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠈
⠀⠀⠀⠀⠙⠢⠌⠙⢛⣼⣿⣿⠿⠛⠓⠒⠒⠀⢉⣉⣁⣰⡿⠡⢶⣿⣷⠬⢷⡦⡤⠾⠛⠋⠉⠀⠒⠖⣿⡷⠶⢶⣶⣦⣄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡙⣫⣤⣿⣿⠛⠞⢋⣁⣬⣥⣾⢹⣿⣤⣾⠿⠟⠷⠷⢶⣶⣶⣤⣀⡄⠀⢀⠐⢠⣀⠀⢀⠉⠋⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣾⣟⣛⡹⢆⣼⣿⣿⡿⠁⠛⠈⠉⠁⠁⠀⠀⣀⢀⣠⡄⢹⠿⢿⣧⡀⠀⠀⠐⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣷⡿⣿⢁⣿⣿⣿⡟⠀⠀⠀⢠⣿⣄⠤⠒⠽⠋⢠⡾⠁⠀⡶⠈⠛⢻⣷⡀⠀⠀⢘⡷⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⣿⣾⠋⣿⣿⣿⣏⣀⣀⡠⠀⠙⠟⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⣿⣧⡄⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⢻⣿⢻⣿⣿⣿⣿⣿⣥⡃⠀⡀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠠⡚⣿⢿⡇⢰⡆⠑⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣿⢼⣿⣿⣿⢁⡝⢩⣿⣟⠳⣷⣴⣾⣵⣶⣶⣦⣤⣴⢷⣦⣤⣀⣹⣿⣶⣤⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣷⣿⡿⢷⣿⣅⣈⠀⠟⠸⠙⣿⣛⣿⢿⡿⣿⡋⠛⢟⠉⣯⢣⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠷⣌⡛⠻⣶⣶⣦⣄⣈⣀⣀⠂⢀⠉⢠⣄⣼⠁⣦⣹⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣯⣘⠳⠤⣌⡉⠝⠛⠛⣛⠛⠛⠛⠟⢯⠘⢇⡈⠻⡟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⣿⣿⣿⣿⡉⠃⣀⡈⠁⠀⠰⠃⢉⠙⢳⡶⠿⠞⢣⢈⡛⢶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢸⣿⣿⣿⣿⣿⡍⢀⡺⠤⠖⠲⠶⠆⢹⡄⠘⣦⠶⢶⣤⠘⣁⣼⣿⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣿⣿⣿⣿⡇⠳⣌⣁⠲⠶⣶⡤⠞⠀⠳⣄⠘⢀⣀⢸⣇⢻⣿⣽⠇⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⣿⣿⣆⣠⠟⣉⠛⣦⠈⣷⡆⠀⠀⠉⡄⣸⠿⣿⣿⡿⣻⣿⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢹⣿⣧⣿⣿⣿⣷⣋⣤⠈⢉⣉⣉⣀⣤⠾⣷⣇⣰⣿⣯⣽⡿⠁⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⡀⢿⣿⣿⣿⣿⣿⣿⣿⣃⣏⠉⣩⡿⢀⣤⣿⣿⣷⣯⣿⡿⣡⡾⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡄⡙⢿⣿⣿⣿⣿⣽⣿⣯⢵⣿⣷⣿⣟⣷⣿⣿⡟⢡⣾⠋⠀⠀⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣮⡙⢿⣿⣿⣿⣿⡗⠆⠉⢻⣿⣿⣿⠟⠉⣠⠟⡀⠀⠀⠀⢉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣦⣈⡙⠛⠛⠳⠆⠀⠘⠛⠉⢠⣤⠞⠃⠀⡁⠀⠀⠀⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠛⠃⢽⣿⣷⣶⣦⣤⣶⣴⣿⣿⡿⡁⠀⡀⠎⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣇⠀⠀⠀⠀⠈⢿⣿⣍⠙⠏⠈⣿⣿⡀⠉⢄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⣤⣄⠀⠀⠈⢛⣿⣿⣷⡖⣿⣿⣿⡀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠷⠾⠿⠿⠿⠖⢠⡈⣿⣿⣿⡇⢹⣿⣿⣧⣰⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠀⠹⣿⣿⣿⣤⡀⠀⢧⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⢰⡄⢁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣾⡖⠠⣀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡞⠳⠿⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣂⣸⠤⠤⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣁⣀⣀⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢈⡅⠈⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢀⠔⠉⠁⠀⠀⣴⠆⠀⢱⣶⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⢠⠀⠠⢤⡏⠀⢰⣿⣿⣿⣟⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠆⠀⠀⠀⡐⠃⠀⣠⠟⠀⠀⣿⣿⣿⣷⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠀⢂⣵⣿⣤⡀⠀⠈⠀⠀⢀⡟⠀⠀⡼⠉⠁⠀⠀⠁⠈⠙⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣂⣩⣤⣶⣞⠉⠁⠀⠈⠻⣧⡀⠀⠀⣾⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠈⠅⠉⠻⢿⣷⣄⠀⠀⠀⠈⢻⣄⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣄
⠀⠀⠀⠀⠀⠐⣿⡏⣠⠀⣆⠀⠁⠀⠀⢀⡀⠨⡀⣿⡆⠀⠀⠀⠀⢿⣿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⣿⣷⡄
⠀⠀⠀⠀⢠⢾⣿⠀⣟⠈⡷⠄⡀⠀⠀⡀⢀⡼⠁⢹⡇⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡄⢻
⠀⠀⠀⠀⣆⢀⢻⡠⣏⣨⠀⠶⠆⠀⢀⠶⠉⠀⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⣾⣿⣿⡇
⠀⠀⠀⠀⢿⣿⣿⣧⢹⡭⡁⠂⠐⠓⣶⢶⠆⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⣴⣿⣿⣿⡟⠀⢠
⠀⠀⠀⢄⠘⣿⣿⣿⣧⡙⠣⣄⠀⠀⠈⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⣡⡾⢟⠉⠉⢿⡧⠔⠃
⠀⠀⠀⠀⠈⠉⠙⢻⣿⣿⣦⣄⣉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣊⣴⡿⠛⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢈⠛⣿⣿⣿⣯⣟⠣⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢞⣡⣾⠟⠋⠀⠀⠀⠀⢠⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢂⣉⠉⠁⠀⠀⠀⠀⣸⣷⡄⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⢡⣿⣿⠆⠀⠀⠀⠀⠀⠀⣠⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⠄⠀⠀⠀⠓⢼⡧⠀⠀⠀⠀⠀⢰⡟⣿⣿⢁⢿⡿⣟⠀⠀⠀⠀⠀⠀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⠉⢤⣤⣶⣾⠋⠛⣧⠀⠀⠀⠀⣾⠃⢹⣿⠀⠀⠀⠀⠀⠀⢀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣻⢿⣇⠀⠐⠿⡄⠀⠀⢸⣿⡗⠛⣿⠈⢠⡀⠀⢀⣿⣇⣰⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⣿⡂⣿⣿⡆⠀⠇⠀⠀⠀⣿⡃⠹⣿⡌⣿⣽⣦⣬⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⢃⠀⡅⠀⠀⠀⣿⠀⠀⣿⡞⠻⢿⣿⣿⣼⣷⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣟⡠⠀⠁⠀⠀⠀⣿⠀⠀⣹⡇⠃⠀⠻⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢠⡿⠀⠈⣿⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠼⠻⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⢂⣀⡐⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢾⣧⣌⠈⠙⠿⠣⠑⠶⣬⡑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⣤⠞⠡⠴⠌⣠⣧⣉⢠⣆⢠⡄⠀⠘⠛⣦⡱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣼⠃⠐⢄⠀⠸⣛⣿⣫⠁⠈⠽⣉⡀⢳⣦⣌⢳⡙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣸⡏⠂⠀⠸⡇⠈⠃⠈⣄⢁⠄⣀⣿⣿⣞⣧⣟⣯⣷⡹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣿⠁⠰⠃⡀⠓⠘⠊⠀⢃⡈⣶⣹⣽⣟⣿⣯⣿⣿⣿⣗⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⡄⠘⠐⢤⣜⡀⠀⠀⢲⣼⡏⣴⢸⣿⣟⣿⣿⣿⡟⢻⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⣿⡇⠀⢠⡄⢿⠀⠀⠀⠈⢋⣽⣿⣾⣿⣿⣿⣿⣿⣇⠈⡆⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⣿⡇⠀⢀⠀⠂⠀⠀⠀⠀⠿⢿⡉⠉⠉⠉⠉⠛⢿⣿⣦⢳⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⠗⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠙⢿⣾⡆⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⡀⣀⣀⠤⣴⣒⡒⠀⣐⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣟⣠⡭⢳⡆⠀⠀⠀⠀⠀⠈⣀
⠀⠀⠀⠀⢀⣠⠔⠛⠛⠛⠉⠀⠑⠠⠃⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣯⢿⣽⣛⡇⠀⠀⠀⠀⠀⡄⠙⡀
⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⠟⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣿⢿⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⣿⠟⡏⠀⠀⠀⠀⠀⠀⡀⠀⠀⠃
⠀⠀⠀⢹⡇⢯⠻⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢸⡇⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠁⠀⣷⣌⣧⠘⢆⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⡆⠀⠀⠀⠀⡠⠃⠀⢠⡟
⠀⠀⠀⠙⠋⠉⣻⢷⣟⠠⠀⠀⠀⠀⠘⢦⠄⠠⡀⠀⣠⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠲⣿⣿⡀⠀⠀⠀⠀⠃⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡅⠀⠀⠀⠀⢻⡼⡄⢹⠋⠉⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢧⣧⢿⡄⠀⠀⠀⠰⠄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠈⠳⣄⣼⡀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢯⣿⡼⠁⠀⠀⠀⠀⠀⣁⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⡄⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⣠⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠰⠄⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠇⠀⠀⢀⣼⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⣀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣄⣠⣴⠿⣋⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢄⣠⣴⢫⢃⡀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣿⣿⡿⣿⣷⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣄⣠⠟⠘⠀⣾⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠋⠀⠙⡿⢣⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⢻⡄⠀⠀⠀⠂⠈⣰⠦⠀⠀⠀⠠⠶⣾⣿⣽⡷⠤⢎⠘⢿⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⢿⣾⣦⢠⣴⣾⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣉⣀⣤⡴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠛⠉⠁⠛⢯⣿⠀⠀⠈⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠠⡀⣨⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⡿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡁⠀⠐⠄⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠈⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⢶⣶⣿⣿⡇⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠁⠈⡗⠀⠀⠀⠀⣰⣿⣾⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⡇⢀⠀⠀⠘⠀⠍⠛⠿⣷⢀
⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⢷⣾⠂⠀⢸⣷⡀⠐⠀⣽⢹⠘⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⣸⣿⣿⣥⣆⢯⣿⣷⣦⢶⠟⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⢰⣿⡏⣿⢿⡟⠛⠚⠓⠿⣸⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⢻⡿⣷⠒⣀⠀⠀⣿⠁⠀⠃⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⣿⠞⣿⣟⣷⠿⢷⡇⣿⢀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠿⣿⣿⣕⠪⡼⣿⣿⡁⠂⠠⣿⣿⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢷⣤⡀⠀⠀⠀⠀⠀⠀⠁⠀⡠⠴⠛⠛⠛⠻⠿⣾⡽⣿⣧⣶⣶⢻⣍⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⠂⠀⡀⠀⠀⣀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠚⠛⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⣙⢟⡀⢴⡥⠄⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⣿⣿⣥⣴⣼⡿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⢰⡿⣿⡏⠐⢩⠊⣵⡾⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠾⣷⣿⣸⢿⣹⠆
⠀⠀⠀⠀⢀⠔⠈⠀⠀⡀⠀⠀⢸⣵⣋⠁⠷⠓⠀⢨⡤⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⣴⣧⡿⢈⠁⠃
⠀⠀⠀⢀⠂⠀⠀⠀⠠⠀⠀⠀⢺⡕⠁⠣⢌⡐⠓⠶⠁⠺⢗⡖⡀⠀⠀⠀⠈⠑⢿⣶⣿⣿⣿⠅⠓⠂
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⠾⡁⠀⠀⠂⠀⢀⡉⠵⣴⠁⠀⠀⠀⠀⠀⠈⣿⣿⡿⢫⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠠⠔⣶⡞⢻⡏⠑⢟⣿⣷⣎⣴⣶⣄⣼⢿⠛⠃
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢿⣾⣄⣜⣀⣐⢟⣛⣫⣽⡿⠟⠉⠉
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⡛⠋⠉
⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠠⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠖⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠙⠛⠛⠛⠲⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣯⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣽⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣿⢸⠈⢹⠉⠉⣻⠛⠛⠛⡟⠛⠛⢿⠛⠛⢻⠛⠛⢻⠛⠛⢹⡏⠉⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⢾⣦⣼⣄⣀⣽⣀⣀⣠⣇⣀⣀⣼⣀⣀⣸⣀⣀⣸⣄⣀⣼⣄⣤⣧⠾⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠧⠤⢤⣉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣁⡠⠿⣆
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⣼⠷⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣼⠇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠁⠀⠀⠀⠀⠘⠛⠛⣶⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣷⠀⠀⠀⠀⣴⠟⠛⠃⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠶⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠘⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡏⠀⣼⠋⠉⠉⠉⡏⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣄⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣴⠟⠀⠀⢿⣤⣤⣤⣤⣿⣤⣤⣤⠾⠁
⠀⠀⠀⠀⢀⣤⠶⠶⢶⣤⣤⣭⣍⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⡴⠶⢶⣤⡀
⠀⠀⠀⠀⣟⠀⠀⠀⡼⠀⠀⠀⠈⠙⣦⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠛⠁⠈⢹⣧⠀⢀⡼⠛⢶⣄
⠀⠀⠀⠀⠙⢦⣄⣀⣳⡀⠀⠀⢀⣰⠟⢀⣀⠀⠀⣿⠁⠀⠀⠙⣳⣦⣀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠙⢦⣠⡾⠋⠳⣮⡀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⢛⣻⣷⠞⠋⠉⢻⡄⠹⣦⡀⢠⠞⠁⠀⠙⢷⣄⠀⠀⠀⢿⡀⠀⠀⣠⡾⠋⠀⠀⠀⠈⠻⢦⣄⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠈⢧⠀⠀⣠⠟⠀⠈⠻⣧⡀⠀⠀⠀⠀⢹⡇⠀⠀⠈⠛⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣀⣼⠷⠛⠁⠀⠀⠀⠀⠈⠻⢦⣀⣀⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠤⠶⢶⣾⣿⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠛⢻⣿⠁⠀⠀⠀⠀⠸⠿⣯⠏⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁⠀⠀⠀⣠⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⣀⣿⣿⣟⣿⣿⣯⣤⣤⡙⡀⠀⠀⢠⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣤⡶⠶⠿⣿⢿⣿⢟⣿⣿⣿⡿⣿⣿⣟⠀⠄⢀⣼⢷⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⠇⠉⣹⢀⡀⣸⢿⣿⣿⣿⣿⣿⣧⠒⢛⣿⣝⣶⣿⠸⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣧⣿⠀⠀⢹⡆⣠⣿⣿⢻⡾⣿⣿⣟⠃⠐⢿⣿⣿⣿⡟⠠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠈⢻⡅⢹⣧⣿⣿⣿⣿⣿⠙⠻⣿⣤⣴⣿⣷⣿⣿⣿⣷⠈⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢺⣿⣦⣿⣿⣿⣿⣿⣿⣾⡷⣉⠛⠿⣿⣿⣿⣿⣿⣿⠀⠆⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣧⣭⣍⠛⢿⣿⣶⣦⠘⣏⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠏⢼⡿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⣉⣻⣆⡿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠉⠀⠛⠛⠛⠋⢹⡿⠛⣿⣿⠉⣿⣿⣿⡇⣸⡏⢸⡇⣿⣽⣿⣷⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢀⠀⠀⠀⠀⠀⠁⣸⣿⣿⣆⢹⣿⣿⢣⣿⠃⢸⣧⡧⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡆⠈⢧⠀⠀⠀⠀⢾⣿⣿⣿⢹⣮⣿⣿⣿⣿⣇⠈⣿⣣⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣅⣷⡤⣄⢨⡛⠂⠀⣼⣿⣿⡇⢸⣿⣿⣿⣿⣿⣧⣸⣿⣿⣿⣿⣿⣿⣿⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣼⣀⣷⣤⣴⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⡏⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠛⠷⣦⡙⣏⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⠇⠘⠛⠻⠙⠋⠕⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⡼⠁⠹⢷⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⣇⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⣿⣿⡆
⠀⠀⠀⠀⢀⡾⠋⠛⠳⠀⠀⠈⠛⠛⠶⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢸⡇⠹⣿⡇
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣿⣿⣿⣿⣿⡇⠀⠀⢻⣿⣿⣿⣾⣇⣼⠟
⠀⠀⠀⠀⠀⠘⠷⣶⢀⣠⠂⠀⠀⠀⠈⣩⣿⣿⣿⣿⡿⢷⠀⠀⠀⠉⠻⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⢨⣰⡏⣄⠸⠿⠃⡴⠋⠉⣿⣿⣿⡁⠈⠀⡀⠀⠀⠀⠙⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⠻⠀⠀⡜⠁⠀⢠⣿⣿⣿⣧⣴⠀⠁⠀⢠⡄⠀⠘⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢀⣠⣿⡿⠻⣿⣿⣿⠀⢠⠀⣸⣿⣇⣀⠀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠁⠀⠀⠀⢉⠧⢶⣿⢀⡇⠈⠛⠻⢿⣦⣤⣨⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣻⣟⣛⣻⣯⣙⣛⣟⣛⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣯⣭⣿⣿⣯⣭⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⢠⣀⠀⠀⢠⡀⠀⠀⠀⣀⣶⠤⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⣤⠘⢦⡴⠊⠁⠠⠟⠉⢺⡁⠀⣀⡶⠟⠉⢀⡠⠒⠒⠤⡀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠘⠐⢓⠚⣓⠐⣟⠀⠀⠀⠀⢀⡤⠞⢿⣾⠋⠀⠀⢠⠃⠐⢠⠀⠀⠈⢳⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠁⠄⠻⠉⣆⠀⠀⠂⠈⢀⡤⠾⡿⠀⠀⠀⠁⠀⠀⣰⣀⠀⢀⣰⢷⡀⠀⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣧⠀⠀⢀⣀⠀⠁⠚⣲⠿⢷⣄⠀⠀⠀⠀⣀⣴⣀⡴⠾⠤⠤⢾⣿⣿⣿⢿⣿⡄⠑⡀⡷⠀⡬⡀
⠀⠀⠀⠀⠀⠀⡘⠀⠀⠂⢹⣷⣀⣀⣽⢶⠒⣨⣶⣈⣤⢿⣷⣄⠀⠁⠀⣿⣏⠉⢈⣀⣴⣄⣿⡉⢁⠀⠈⠙⣦⣼⡧⠤⠴⠃⠠⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠟⣀⢽⣿⡄⣼⣿⣿⣿⣿⣿⣀⣀⠽⢧⣙⣰⠛⢻⣷⣼⣿⣻⣅⣠⣈⠟⣠⡄⠀⠹⢧⡴⠀⡀⠒⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡑⢰⣧⣤⣈⠻⠛⠙⠛⠉⠉⠉⡀⠠⠛⢿⡟⠛⢿⠛⠛⠻⣏⣭⡉⠃⠈⠁⠀⠀⠀⢠⡌⠁⠰⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠡⣀⣠⠀⠐⠛⢙⡆⠀⠀⠀⠀⠘⡀⠀⠀⠉⠓⠲⢾⣷⢄⢲⡄⠀⠀⠘⢻⡄⠀⠀⠖⠀⠀⠀⠀⡀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⡁⢀⣆⠀⠁⠀⠀⠀⠀⢈⣴⣶⣶⣶⣦⣄⠈⢿⡄⠀⡃⠰⠀⠂⡼⡇⠀⠀⠀⠀⠀⠀⠈⠁⡈⠀⠈
⠀⠀⠀⠀⠀⡐⠂⠀⠰⠤⠀⠀⠁⢠⣾⣿⣿⡆⢆⠈⢿⣿⣿⣿⣿⣿⠂⠀⣿⣤⣙⠘⠃⣷⣾⠇⡀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡄
⠀⠀⠀⠀⠀⠉⠨⢋⠴⠂⠀⠀⢰⡌⠛⣻⠿⢳⣶⡄⠈⠳⢿⠿⠃⡼⠀⠂⢠⢈⡅⠒⢠⣾⠃⠀⡆⠀⠀⠀⢀⡃⢀⠁⠂⠁⣰
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣤⣼⣇⠰⡁⢀⡘⢻⡿⣦⡴⠏⡠⠙⠑⠦⠀⣼⠉⠀⣷⠉⡏⠀⣼⣶⡦⠀⠀⣌⣀⠀⠁⠀⡰⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⠉⠀⠘⣿⣇⡊⠀⠋⠙⠛⠁⠂⠄⡄⠇⣸⣿⡹⠀⠘⣿⡎⠀⢠⣿⠟⠀⢀⣼⡿⠋⠀⢠⣾⡀⠁
⠀⠀⠀⠀⢸⠀⠀⣀⢰⣿⣆⠀⠀⠀⠀⠀⢸⣤⣘⣄⠁⢦⠐⠖⠞⢋⠐⡿⠁⠀⠀⠎⠀⢀⣿⠃⠀⢠⡿⠋⠀⠀⣴⣿⠟⠃⠀⣠⠎
⠀⠀⠀⠀⠸⡄⠀⡉⠙⣾⠉⢄⢠⣈⣰⢲⣾⡀⠈⠈⠀⢀⡦⠾⠊⣡⠌⠀⠠⠄⠈⠀⠀⠘⠁⠀⠀⠊⠀⠀⢀⣾⠟⠁
⠀⠀⠀⠀⠀⠱⡔⠻⢃⠙⢴⠃⠀⠀⢮⢀⣼⠿⣞⠋⣀⣉⡃⠤⠂⠁⠀⣰⠃⢀⠄⢀⡀⠀⠀⠀⠀⠀⠀⠰⠋⠀⢀⡤⠎
⠀⠀⠀⠀⠀⠀⣴⠖⠚⠓⠛⠀⠀⠀⢀⣁⠀⠑⡄⠀⠀⠀⠀⠀⠄⠀⣿⠏⣀⠴⠲⠒⠒⠒⢠⡀⠀⠀⠀⠀⠀⡤⠊
⠀⠀⠀⠀⠀⡀⣇⣀⠀⠄⠀⣠⡀⠀⠈⣡⣴⠀⠣⣓⢲⣶⡆⢠⠀⣷⠋⠔⠁⣤⣴⡶⢶⣶⣾⡷⠃⠀⠀⢀⣼⣅
⠀⠀⠀⠀⠀⠀⠁⢀⢃⣀⠤⠤⡁⠀⠘⠛⠉⠀⠀⢃⣀⣉⡑⠈⠷⠖⢋⣠⡿⠋⠀⢀⣠⡉⠉⠰⠔⠂⣠⠊⠈⢹⣧⠀⠰⠁
⠀⠀⠀⠀⠀⢀⠘⠱⠈⠁⣴⣾⠇⠀⠀⠀⠀⠀⠈⠉⠱⣾⡝⢷⣶⠚⠛⠉⠀⠀⠀⠀⠀⠐⠀⠀⠀⠰⠿⠧⢤⡴⢿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢡⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢠⣀⠀⠈⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠠⠄⠀⠀⠀⢀⠐⠘⠀⠀⣿⣯⡌⣤⣤⠾⢿⣆⠀⣠⣤⣤⣀⣀⣀⣠⣤⣴⣶⡟⠉⢻⡷⣦⡘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣾⣦⡀⠀⠁⠀⠀⢀⣴⣿⣿⣡⣿⣉⠞⢿⡿⡻⢿⣿⠉⢿⡏⠀⠀⢿⣿⠟⠁⢀⡾⠤⠀⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⠛⢿⣿⣷⡾⠿⢿⡟⡉⣿⣷⣾⣟⣿⢻⣷⠀⢻⡅⢺⡇⠀⠀⢸⠁⢀⣤⠞⢀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠔⠉⠀⠀⣀⣀⣘⣿⣧⣤⢿⡟⣿⣟⣟⣛⣿⣤⣦⣍⠁⠂⢀⡾⠖⠋⠁⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠃⢀⡀⠀⡀⠸⣿⣋⡼⢷⣟⡏⣹⣿⠿⡛⠛⠁⠀⢀⡞⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠄⠀⠈⠛⣿⣿⣬⠿⡏⠁⠀⠁⠀⢓⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⢷⡀⠈⠳⡖⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⡀
⠀⠀⠀⠀⠀⠀⢠⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣶⣶⣶⣶⡶⠶⣶⣶⣶⣶⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⡿⠿⢰⣇⠸⠿⣿⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣇⣛⢻⡿⢛⣃⣿⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣿⣿⣘⣃⣸⣿⣿⣿⡇⠀⠀⣿⡴⠖⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣡⣴⣾⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣧⡴⠶⠶⠶⠶⣿⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠘⠷⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠹⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠙⠳⠶⠶⠦⠴⢿⣿⣿⣿⡿⠟



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠪⢒⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠊⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠈⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠁⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⠀⠀⠐⠒⠶⣶⣄⡀⠀⠀⠀⠀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⠀⠀⠀⠉⠻⢷⡂⠤⠒⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⢼⡇
⢀⡄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣃⢸⣦⡀⠀⠀⠀⠀⢀⡠⠛⠲⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⡈
⡟⠁⠀⠀⠘⡄⠀⠠⠄⠀⠀⠀⠀⣻⠀⠈⠳⡀⠀⢠⠔⠁⠀⠀⠀⠀⠙⢄⢀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠁
⢹⡄⠁⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⢳⣤⣶⣷⣦⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠁
⠀⢹⡄⢰⣄⢤⣼⡄⠀⠀⠀⠀⠀⣸⣽⡄⠀⣀⢀⠆⡶⣄⡀⢐⡤⠀⣀⣀⣨⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠃⠀⠀⠀⠀⠀⣰⠄
⠀⠀⢻⣿⣷⡈⢿⣿⡆⠀⠀⠀⣠⡿⠻⠃⠚⠉⠉⠈⠐⠋⠻⡿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⠟⢋⣡⣤⣶⠄⠀⢀⣠⡴⠞⠋
⠀⠀⠀⢿⣿⣷⡈⢻⣿⡄⠈⠈⠍⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠉⠉⠣⠛⡄⠀⠀⠀⠀⠀⠘⣧⣴⣾⣿⣿⣿⡿⠷⠛⠉
⠀⠀⠀⠈⢿⣿⣷⣼⠛⡇⢀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢼⠇⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⢸⡉⠀⠀⠀⠠⡙⣶⣤⡀⠀⠀⠀⣀⣀⡀⣠⡴⠔⠋⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡀⢢⣼⠿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⣿⠈⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣠⡀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡱⠂⠈⠛⢿⣿⣤⣌⠻⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠀⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠚⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢆⠀⠙⠻⣿⣧⣄⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⣸⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠀⢀⢂⢀
⠀⠀⠀⠀⢺⢆⡀⠀⠀⠀⠀⣀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⠀⠈
⠀⠀⠀⠀⠀⠢⣍⣿⣷⣏⣍⣇⡴⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⢻⡷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡛⠛⠋⠉⠉⠁⠀⢐⣤⢴⣤⣀⠀⠛⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⢀⣯⣀⡴⣤⡈⠃⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⣿⣤⡍⢡⢛⣋⡇⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⡇⢿⡼⢤⣌⣿⡇⡀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢼⡇⢀⣸⣇⣸⣿⣦⣿⣿⢁⠈⡤⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣷⡏⡿⢯⡻⣿⣿⠇⢇⠎⠞⣵⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⢀⡙⠾⣿⣼⣯⣾⣀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢻⡎⣿⣿⣦⡀⠙⢿⡿⣯⣲⣶⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢸⣷⢸⣿⣞⢿⣆⠀⡀⢈⠉⠻⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣹⣿⣿⠀⣿⡄⣿⣿⣆⠹⣷⣼⣶⣷⣄⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡅⣿⣿⢹⠀⠘⣧⢹⣿⣿⣦⡘⢧⠉⡛⠛⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢣⣇⡟⣿⠈⢠⠀⠹⡆⠻⣿⣿⡳⣄⠱⣌⠃⢸⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢇⣾⣿⡇⢿⠀⣿⢰⠀⢻⣆⠹⣿⣿⣌⢓⢬⡛⢸⡇⠘⡖⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⣾⣿⢹⡇⠀⠀⣿⠸⣧⠀⢻⣷⡙⢏⢻⣿⣝⢦⢻⣧⠀⢹⡘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⢰⡟⣿⢸⡇⠀⢸⣿⠀⣿⣆⠀⢻⣟⠦⠀⢿⣿⣮⢻⣿⠀⣼⡇⡈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⣿⠘⠁⠀⢸⣿⡄⠘⣿⡄⢠⠙⣦⠀⠈⣿⠈⠋⠃⣴⡟⢇⢻⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣿⠀⠀⠀⣿⣿⣇⠀⢹⣿⡀⢳⡄⠀⠀⣿⠀⢠⣾⢿⠁⠀⠈⣧⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢱⠀⠀⠀⣿⡏⠹⡆⠀⢹⣧⠀⢻⣆⠀⠁⢀⣿⣷⡈⠀⠀⠀⠸⣇⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠙⣆⠀⢻⡄⠀⠹⣷⣄⠘⢻⡿⣷⡄⠀⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠘⠂⠀⢧⠙⣿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⣿⣏⢽⣟⠻⣿⢟⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠀⠈⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⣀⡤⠖⢒⣶⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠘⠁⢀⠤⠚⣛⡛⢟⣟⣻⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⡄⠀⠀⠀⠁⡰⠟⢛⣿⣮⣭⣧⣤⣤⣝⡲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢰⢱⣤⣆⡴⢊⠔⢺⣿⣏⡙⣿⣿⣿⣿⣿⣿⢦⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠛⢿⡴⠁⠀⠀⢻⠀⠀⢹⣿⣿⣿⣿⣿⡇⡟⢶
⠀⠀⠀⠀⢀⣠⡤⠶⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡷⠺⠤⠈⠷⡦⠾⠆⠀⣠⣿⣿⣿⣿⣟⡏⢱⣾
⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⣤⣤⠄⠀⠄⠀⠘⠻⣿⣿⣿⣿⡟⠀⣾⡟
⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⣉⢿⣿⣀⠀⢀⠀⠀⡀⠀⠈⠙⣿⣿⣷⡦⠉
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠆⠀⠀⠀⠀⠀⠀⠻⢆⣿⡿⠋⠉⠁⠉⠁⢀⠀⠀⠘⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢈⠳⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠈⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈
⠈⠳⣦⣌⡉⠙⠛⠛⠛⠛⠛⢉⡉⠤⠀⡾⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⡀⠀⠀⠀⠐⡆⠀⠀⠀⢹⡤⡤⠐⠁⠀⠀⠀⣆⡀⠀⠀⠀⢸⠤
⠀⠀⠈⠙⠒⠷⠠⠀⠠⠈⠄⠄⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠄⠀⠀⠀⢀⣧⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⢴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣦⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠐⢶⠶⠆⠀⠠⠀⠀⠀⠀⠙⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠂⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠦⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠠⠤⠄⡤⠤⣶⣶⣶⣶⣄⣀⣤⣄⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⣶⣲⡆⠈⠙⣿⣿⣿⣿⠛⣍⣩⣽⣉⣝⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⡽⠃⠀⠀⠁⠙⣿⠿⢷⣿⣿⣿⣿⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠁⡜⠃⠀⠀⠀⠀⠀⣰⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣍⠐⠁⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⢠⡇⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠸⣿⠀⠉⠛⠿⠀⡀⠀⠀⠀⠀⠰⠿⠿⠷⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⡄⢀⡀⠀⠀⠀⢀⣴⢣⢼⡇⠀⠠⠐⠟⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⢀⣀⣤⠀⣽⣿⠟
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⣠⠃⠈⢻⡌⣡⠺⣿⡿⠗⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠐⠉⠀⡰⠧⡄⣦
⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣴⣿⠀⣢⣾⣏⡛⠁⠠⠁⠰⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⢘⠀⠑⢾⠆⠐⣑⣿
⠀⠀⠀⠀⢠⠃⠀⠀⢠⡞⠁⡟⣰⣿⣏⠛⠁⠶⠤⠄⠠⣴⡂⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢤⣼⣶⡶⠚⡏⠸⡟⠉
⠀⠀⠀⠀⠸⠀⠀⡰⠋⠀⠀⡃⣿⣦⡍⠀⠚⠒⠒⠂⠀⠄⠈⠈⠉⠀⠀⠀⠈⠉⠉⠉⠉⠛⠃⠂⠼⠀⠀⠀⠀⡼⠏⠀⢀⡆
⠀⠀⠀⠀⣠⠤⠚⡁⠀⠀⢸⢠⡿⠶⠤⡀⣛⣗⠀⡇⠀⡑⠦⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠤⡆⠀⢸⠁
⠀⠀⠀⠸⢆⠄⠠⠀⠀⠀⡸⢸⣿⡷⣡⠭⠈⡉⠀⠀⢆⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⣀⡾
⠀⠀⠀⠰⠼⠀⡅⠀⠀⠀⡇⠰⣶⢤⣙⣤⡠⣌⠲⣀⠀⡅⢀⠃⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⣼⡼⠀⢹⡇
⠀⠀⠀⠀⣧⠄⣀⠈⠀⢰⠇⠰⣿⡙⠷⠈⠋⠈⠑⠚⠁⠋⢁⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢇⡀⠀⠄
⠀⠀⠀⠀⢹⠀⡇⠀⠀⢸⠀⠐⣿⡁⢜⠠⠀⠀⠀⠤⣐⠺⢿⠛⠉⠀⠚⠴⠦⠀⠀⡌⠀⠀⢀⣠⠄⠋⠀⠰⣿⡈⠁
⠀⠀⠉⠁⢸⡀⡇⠀⠀⡄⢸⣦⣌⠳⠠⠀⠁⣿⣿⡦⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠈⠀⣁⣴⡖⢀⡰⣄⠀⠗
⠀⠀⠀⠀⠘⡇⢀⠀⠀⠇⠘⣿⣿⣷⡀⠀⠀⣼⣿⣟⠀⠐⠀⠀⠀⠀⠀⠀⣁⣤⡀⠶⠚⠛⠿⠁⠈⠉⣤⠀⠀⢰⠀⡀
⠀⠀⠀⠀⠀⠹⠀⡀⢸⠀⠀⢸⡿⠓⢄⡈⠀⠘⡷⠛⠀⣠⣦⡤⠴⠒⠋⠉⠁⠀⠀⢀⣠⣤⣶⣶⣦⣼⣿⠁⣜⣷⣾⠃⠀⠃
⠀⠀⢀⣤⠶⣷⠂⠙⣿⢠⣾⣿⡷⠀⢸⣧⣠⣆⡐⠆⠘⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⢸⡇⠀⠀⠈⠉
⠀⣄⠀⠀⠀⠈⠆⠐⠋⠈⠛⢿⣧⡀⣠⣈⠉⠉⠉⠲⠲⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⣿⣷⡀⢤⣀⠀⠀⡄⠀⠀⠀⠀⠈⠉⠙⠻⣿⡶⢤⣄⡀⠀⠈⠹⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡀
⠀⣿⣻⣧⠻⠿⠘⣿⢆⡄⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⠈⠉⠙⠒⠦⢤⣄⡀⠀⠀⠀⢀⠀⠀⠀⠀⣾⢠⠃⠁
⠀⣿⣟⢿⠉⠁⢸⡇⢸⣄⣦⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠈⢇⠀⣀⠑⠎⠀⠀⠀⠀⡇⠘⠀⠀⠀⠀⠀⢀⡆
⠀⢽⣾⣧⡀⠀⢸⠀⠘⠿⠿⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⡀⠃⡈⠉⠒⠃⠀⠀⠀⢸⠁⡄⠀⠀⠀⠀⠀⢸⣷⣄
⠀⠈⠉⢿⠇⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢀⡠⢖⣩⠔⠉⣤⠶⠶⠀⠀⠀⢠⠏⠀⠁⠀⠀⠀⠀⠀⠒⢹⠁
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣸⠀⢀⠴⣊⡬⠞⡋⠄⠒⠉⠹⣆⠠⠀⡀⠀⡷⡄⠴⡀⠀⢀⣦⡀
⠀⠻⣿⣿⣇⢠⠇⢀⣷⠀⠀⠀⠸⠧⠀⢀⡀⢀⣠⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⡠⠺⢿⣿⣆⠈⡄⠉⠛⠻⠷⠶⢦⠤
⠀⠀⠀⠈⠉⢸⠀⣿⣄⣀⡀⠠⠀⢀⡚⢁⣄⠉⡀⢀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠶⠦⢭⣝⣧⣈⠠⠄
⠀⠐⢤⡀⠀⡇⠀⠸⣿⣿⡏⠘⠛⠿⣇⠀⠈⡏⠀⢈⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠘⠔⠂⠠
⠀⠀⠀⠙⠲⠀⠀⠀⠀⠙⠻⢶⣤⣤⣠⣦⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠷⠶⣧⠤⠦

⠀⠤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⡐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣼⣧⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠲⢇⣀⣀⣠⡀⠉⠻⣦⣀⣨⠀⠀⠀⢴⣀⣉⠋⠐⠒⠤⣀⠀⠊⡀⢠⡾⠗⠈⠢⡄
⠀⠀⠀⠀⠀⠀⡠⠤⠤⢼⣄⣀⣤⡄⠀⠈⢻⣿⣿⣶⡾⠋⠉⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⠁⠀⣾⡿⠋⢀⣦⣾⡷⠾⠖⠒⠢⡀
⠀⠀⠀⠀⢠⠞⢀⣴⣄⡀⠀⢈⣿⣿⣦⣄⣀⣿⢋⠹⠀⠀⠀⠀⠀⠀⠀⢠⠁⣹⡇⢡⠀⠇⠀⣸⣿⠁⠀⣿⣿⠏⠀⣠⣶⣾⣶⠡
⠀⠀⠀⠀⠁⣰⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⡋⢡⠋⣶⠀⠀⠀⠀⠀⠀⠀⣟⣛⡛⠛⣻⢲⣀⣼⠋⡇⠀⣸⣿⡏⢀⣴⣿⣿⣿⣿⡆⢃
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⡀⣘⠸⠛⠉⠉⣶⠀⠀⠀⠀⡇⠀⠄⠀⢼⡂⠉⠀⠀⢧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣧⠘⡄
⠀⠀⠀⢐⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣎⣠⡏⠀⢀⣴⠟⠁⠀⠀⠀⠀⣇⡀⠀⢀⣀⡄⠀⠀⠀⠀⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢳
⠀⠀⠀⢰⠀⠀⢀⣡⣾⣿⣿⣿⣿⢫⣿⣿⣿⣿⣾⣿⠃⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣧⣶⣶⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠈⡆
⠀⠀⠀⠸⠰⣾⣿⣿⣿⣿⢿⡿⢱⣿⣿⣿⣿⣿⠋⡜⠀⠀⠀⠀⠀⠀⠀⠻⠯⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻
⠀⠀⠀⠀⡄⢹⣿⣽⣿⣟⢸⡇⢻⣿⣿⡿⠿⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠋⠋⢙⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⢸⡄
⠀⠂⠀⠀⢵⣦⣼⣿⣿⣿⣌⣷⣈⠻⠉⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣬⣻⡿⣿⣿⣿⣿⣿⣶⣶⡄⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣜⠱⠟⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⣿⣿⣷⢿⡄⠀⢸⣿⣯⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⣿⣿⣿⣿⣿⢹⡆⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣹⣿⣿⣿⣿⡏⣾⠁⠀⣰⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣸⣿⣻⡟⣿⠇⣯⢀⣼⣿⣿⠟⠻⣿⣷⡝⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢸⣿⡏⣿⣿⣿⡿⠀⢻⡟⠻⣿⠁⠀⠀⠈⢳⣿⣦⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠘⣿⢀⢿⣿⣿⡇⡷⢸⡇⠇⠀⠀⠀⠀⠀⠀⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⡇⡂⣿⢸⢸⣿⡿⢸⡇⣾⢿⠀⡆⠀⠀⠀⠀⠀⠀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠃⢸⡟⢸⣿⣧⣿⠃⣿⢸⢸⡇⠀⡄⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣧⠀⠈⡗⠘⣿⢹⡿⢰⡿⢸⢸⡇⠀⡄⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⢰⠀⡇⡆⣿⣸⠇⣾⡇⡆⢸⡇⠀⡿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠰⣿⢸⠀⡇⡇⣿⠛⣸⣿⠇⡇⢸⡇⠀⣷⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⣿⢸⠀⡇⡇⣿⡄⣿⣿⢠⡇⢾⣧⠘⡟⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⣿⠘⣿⢸⠆⠃⡇⣿⠀⣿⡟⢸⡇⣼⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠙⠃⠀⠀⠀⠀⠀⢠⣿⠀⣿⢸⡃⠂⡇⣿⢰⣿⡇⢸⡇⣿⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⠹⠆
⠀⠀⠀⠀⠀⠀⠒⠲⠔⠐⠒⠒⠲⠛⢢⡆⠀⠀⢠⣿⠛⠀⣿⢸⡇⠁⡇⣿⢸⣿⠁⣾⣿⣿⣇⡄⣿⣄⠀⠀⠀⠀⢲⠛⠔⠂⠘⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠖⠐⠶⠀⠀⠂⠀⣨⢏⣾⢰⡇⢸⡇⡄⡇⣿⢸⣿⠀⣿⣿⣿⣏⣽⣻⠏⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣐⣒
⠀⠀⠀⠀⠀⠓⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠈⠁⣸⣧⢸⡇⡇⡇⢻⢸⣿⠀⣿⡿⣻⣿⢸⠷⠀⠀⠀⠀⠀⠀⠀⠛⡤⠀⠀⠀⠀⠀⣹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠐⠛⠂⠀⣼⠏⠀⠀⠀⠛⣿⢸⣧⡇⡇⣸⠸⣿⠀⣿⡇⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⢲⡆⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠛⠛⠈⢙⣿⢇⠀⠀⠀⠀⠀⡹⣸⣏⡇⡇⢹⠀⣿⠀⣿⡇⣿⣿⠀⠴⠆⠀⢀⠀⢠⡀⠀⠀⠀⠉⠀⢘⡁
⠀⠀⠀⢀⠀⠛⠒⠒⠒⠋⠁⠀⠀⣴⢻⠏⠀⠀⣀⠀⠀⢠⡀⣿⡀⡇⡇⢸⠀⣿⠀⢹⣷⢸⡿⡇⠀⣠⠈⠛⢀⣤⠉⢁⠄⠀⠒⠞⢋⣡⡖⢀
⠀⠀⠀⡄⠀⠊⠠⠴⢶⠶⣤⡤⢾⣷⣾⠀⠐⠚⠛⢁⡀⢸⡇⣿⢰⠇⡇⣼⠀⣿⡇⢸⣿⡘⣧⣿⠀⠀⠒⠒⣿⣀⣀⣀⠐⠲⠶⢶⡞⠉⢛⣉
⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⣠⠀⠸⢿⡿⠀⠀⠀⠀⠀⢀⣾⢹⣿⢸⠀⡇⣿⠀⣿⡇⠈⣿⡇⢻⣾⣷⡀⠴⠊⠉⠙⠳⣿⡿⠷⠶⠚⠛⢱⠿⠛
⠀⠀⠀⠀⠤⠄⠁⠤⣤⠤⠤⠆⠀⣸⣇⠀⠀⠀⠀⠀⣼⡟⣸⡇⢸⢠⡇⣿⠀⣿⣷⠀⢿⣿⠈⣷⣹⣷⡀⠀⠀⠀⠀⠛⠓⠀⠀⢀⣠⣧⣄⡀
⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣴⣶⣜⣣⡏⠀⢀⣀⣠⣾⡿⢁⣿⠁⡏⢸⠁⣿⠀⣿⣿⡀⢸⣿⣇⠸⣷⡹⣷⣔⠀⡀⠀⠀⠀⠈⠻⠟⠁⠀⠈
⠀⠀⠀⠀⠀⠈⠉⠉⠀⢺⣿⡏⠈⠁⠀⠸⢿⣿⣿⣟⣡⣾⡟⢸⠃⣾⢀⣿⠀⣿⣿⣇⠀⢿⣿⣆⢹⣷⣽⣿⣷⡤⠆⠀⠉⠐⠦⣾⣶⡤⠐⠊
⠀⠀⠀⠀⠀⠄⠀⠀⠄⢸⣿⠃⠀⠀⠀⠀⠀⢹⣿⣿⣿⠟⣠⠇⢰⡇⢸⣿⡄⢿⣿⣿⡄⠘⣿⣿⣾⣿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠀⠀⠀⠀⠀⢠⣿⠏⠠⣿⣶⣏⣴⡿⢀⣾⣿⡇⣸⣿⣿⣿⣤⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠈⠀⠀⢠⣿⠃⢠⣿⠏⠉⠿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⡷⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣴⡿⠁⣰⣿⠃⠀⠀⠰⠟⣿⣿⠛⠛⢻⣿⣿⣿⡁⠈⢿⣦⠀⠙⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣴⢤
⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⡠⣯⠟⢀⣼⡿⠃⠀⢀⣶⣤⣤⣿⣿⣦⣾⣿⣿⣿⣿⣷⣄⠀⠹⣿⡄⠀⠈⢻⣓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡿⠃⢐⣏⡹⠀⠀⠀⢸⣿⣿⣿⢻⡏⠀⣤⣈⡿⣿⣿⣿⣿⠀⠀⠘⢏⣵⠀⠀⠈⢹⣛⣦
⠀⠀⢀⢀⡤⠀⠀⠀⣔⠛⠿⠉⠀⣰⣛⡏⠁⣀⠀⠂⠈⢁⣽⡿⠼⠷⠠⣿⢻⡀⠀⠨⠀⣀⣤⡄⠀⠀⠹⡛⣆⠀⠠⠈⠩⡟⢢⡀⠀⠉⠛⠀⠐
⠀⠀⠈⡜⠀⢀⠴⣾⠟⠋⠀⢀⠶⣿⠉⠀⣾⣿⣷⣦⣰⣾⣿⣄⣠⣤⣴⣾⣿⠟⢲⣾⣿⡿⠋⢀⣶⣦⡀⠈⠿⠛⣆⠀⠀⠀⠉⢻⠟⠀⡀
⠀⠀⢀⣤⣤⡿⠖⠋⠀⠀⣠⣾⠟⠂⠀⣠⠟⠻⠟⠋⣿⣿⣿⣿⣿⡦⣀⣸⣿⣀⣀⣼⣿⣿⣿⣿⡿⠟⠓⠀⣄⠚⢿⣤⡀⠈⠁⠈⠀⠶⢿⣤⡄
⠀⠀⣾⡀⠋⠀⢐⠂⠀⢾⣤⠋⠀⠀⣴⣯⣤⣄⣠⣴⡿⠟⣿⣿⣍⣀⠹⣿⣏⢉⡛⢻⣿⣿⣿⣧⣤⣤⢀⣀⣤⣷⣞⢀⣼⡀⠀⠐⠀⠀⠈⠳⣀⡀
⠀⠀⠈⠀⠀⠉⠉⠀⠀⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠀⠀⠉⠁⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣛⣛⣛⣲⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠞⠻⢦⣤⣀⣉⣉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣁⣬⠟⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⠶⣶⣤⣬⣭⣭⣉⣉⣉⣉⣉⣉⣉⣉⣩⣥⣤⣶⡾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⢸⡇⢸⡏⢹⡏⢹⡏⢹⡏⢹⡏⠉⡏⠉⡇⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣄⣸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⠀⡇⢀⣇⣠⣧⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡟⠙⠛⠛⠻⠿⠷⠾⠷⠾⠷⠾⠿⠿⠛⠛⠛⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⢷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⡾⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣴⣦⣤⣤⣀⣀⣀⣀⣤⣤⣤⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⣤⣤⣤⣀⣀⣤⣤⣄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣉⣉⣉⣛⣛⡋⠉⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡇⠀⠀⠀⠀⠀⠀⢀⣴⠶⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠛⠿⢿⣶⣶⣤⣤⣤⣤⣤⣴⣶⣶⠿⠟⠛⢻⡇⠀⠀⠀⠀⢀⣴⣟⡱⠋⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠛⠻⡿⠷⠶⣦⡀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⢰⡿⠃⠈⠹⣦⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⡀⣠⡇⠉⠛⢸⡇⠀⠀⢻⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⡇⠀⠸⣧⣀⣠⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⢿⣿⣷⣾⠟⠁⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠈⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢟⣛⡛⣟⠛⠙⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠉⣀⣿⣀⣀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣔⣶⣒⣒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⢉⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣻⣿⣽⣶⣶⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⠋⣿⣏⡉⡺⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⠛⣿⠀⠈⣹⡏⠀⠀⢀⣀⣤⣴⡶⠿⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⣧⣴⣿⣿⡄⠘⠻⠶⠶⠾⣋⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠉⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣦⣀⣀⣠⣴⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢛⢻⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠸⢿⠈⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⣿⣛⠻⠿⢿⣭⣽⣭⣿⠣⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢻⡀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠟⠻⡉⠑⡀⣤⣤⣤⣤⣷⠀⠀⠀⠠⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠈⣇⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣛⡁⠀⢀⣿⣶⣿⣿⣿⣿⣿⣿⢰⣷⣦⡀⠀⠈⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⣆⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣿⣿⣾⣷⣸⡟⠻⣦⡀⠀⠀⠀⠀⠀⡜⢠⡀⣽⣦⢻⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⠻⣆⠀⠀⠀⣼⢁⠸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣴⡄⠈⣻⠓⠾⠁⣠⣷⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣟⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣷⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡇⠀⡀⠀⢀⢀⠄⣀⣄⢀⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠠⣙⠶⣿⠃⢾⡿⠫⢈⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⡇⠶⣄⠈⢁⠀⠚⢣⣿⣿⣿⣿⠉⠛⠿⢿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠻⢢⣼⣿⣯⡐⠒⣸⠏⢝⣿⣿⡇⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣖⣴⣂⣱⣾⣾⣿⠀⠀⠤⠀⠙⠃⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⣤⣶⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠋⡇⠹⢿⡇⠹⠈⡏⠏⠿⠿⠙⠿⠿⠿⠙⠏⠁⠈⠁⢠⠈⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠹⣿⣦
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷
⠀⠀⠀⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⠀⠀⠀⣿⣿
⠀⠀⢀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣾⣿⠇⠀⠀⢻⣿⣇
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⡄⠀⠀⠀⢿⣿⡄
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠈⠛⠁⠀⠀⠀⠈⣿⣿⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇
⠀⠀⠀⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⠉⠉⠉⠉⠁
⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣷⣦⡀
⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣤⣀⠀⠙⢿⣿⣦⡀
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣀⣀⣀⣀⡀⠀⠀⣀⣀⠀⠈⠙⠻⢿⣷⣄⠀⠙⢿⣷⡄
⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⣿⣿⠀⠀⠿⠿⣿⣷⣦⣄⠀⠙⣿⣷⡀⠈⢿⣿⡄
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⠀⠀⠀⠀⠀⠈⠻⣿⣧⠀⠘⣿⣷⠀⠘⣿⣷
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⢹⣿⡆⠀⣿⣿
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⠀⢸⣿
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠘⠛⠃⠀⢸⣿
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢠⣤⡄⠀⢸⣿
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⠀⢸⣿⠂
⠀⠀⠀⠀⠸⠿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠸⠿⠇⠀⠸⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⡿⣻⣿⣷⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⠿⠟⠛⠉⠁⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠙⢿⣿⣿⣦
⠀⠀⠀⠀⠀⢶⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠙⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⣀⣤⣶⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⣀⣤⣴⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⠿⠟⠛⠉
⠀⠀⠀⢸⢇⡼⣿⠏⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣹⣿⣧
⠀⠀⠀⠈⢸⡎⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣹⣽⠇
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠊⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⠶⢒⣊⣉⣩⣭⣭⣥⣤⣤⣤⣤⣤⣭⣭⣍⣉⡁⠒⠶⠦⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⣉⣭⣤⡶⠶⠟⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⠉⠓⠲⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣊⣩⣴⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⡤⠤⠤⠤⠴⠶⠒⠒⠶⠤⠤⠤⠤⢤⣤⣤⣄⣀⣀⣈⠉⠓⠢⣤⣀
⠀⠀⠀⠀⣤⣤⣴⡞⣩⣴⠾⠛⠉⠀⠀⠀⠀⢀⣀⡤⠤⠖⠒⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠤⠭⢷⣦⡄
⠀⠀⠀⢺⣏⢿⣿⣷⣉⠀⠀⠀⠀⣀⡠⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⣿⣦⡀
⠀⠀⠀⠀⠻⡾⢿⣿⠉⢷⣠⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⡼⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠊⣿⣷⠀⠀⠀⠀⠀⠀⠀⣸⣟⣾⣿⣿⠟⢋⣤⣶⣶⣶⣦⣌⠙⠿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡸⣿⣇⠀⠀⠀⠀⠀⢰⣿⣿⡿⢛⣡⣾⠟⠉⠀⠀⠀⠈⠙⢿⣦⣌⠻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⣿⡆⠀⠀⠀⢀⣿⣿⢏⣴⣿⠋⢁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡈⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⣿⡀⠀⠀⢸⣿⢡⣾⣿⡟⢰⣿⣇⣰⣤⡄⠀⣤⣆⣸⣿⡧⢻⣷⡌⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣿⣷⠀⠀⢸⠉⣾⣿⣿⣧⠈⠻⠿⠿⠟⠀⢀⠛⠿⠿⠿⠁⣼⣿⣿⡈⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⣿⣧⠀⢸⠀⣿⡟⣿⣿⣷⣄⡀⠀⢠⣿⣿⡆⠀⢀⣀⣼⣿⣿⣿⡇⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⣿⡆⠸⡇⢸⣿⢹⣿⣿⣿⢸⡀⠀⠁⠈⠁⣰⣯⣸⣿⣿⣿⣿⡿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢿⣿⡄⢱⠈⣿⡌⣿⣿⣿⠛⢿⣧⣤⣄⣥⡿⠋⣿⣿⣿⣿⣿⠇⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣨⢿⣷⣈⣧⠘⣧⠸⣿⣿⣦⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⣋⣰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡜⣿⣿⣿⣆⠹⣧⠹⣿⣽⣿⣶⣾⣷⣶⣾⣿⡿⢛⣡⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⠹⣿⣿⣿⣦⠙⣇⠹⣿⣮⡻⣿⣿⣿⡟⣡⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⣷⡈⢧⡘⢿⣿⣿⣿⡿⣴⣿⣿⣿⣿⠟⣩⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣦⣳⣌⠻⣿⣿⣷⣿⣿⣿⡟⢡⣾⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⣩⣭⣭⣽⣿⣿⣿⡘⣿⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣿⣿⣿⡀⠸⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠿⠟⢛⣛⣛⣧⡼⣿⣿⠿⣿⣿⣿⣿⣷⣄⡹⣿⣿⣿⣧⠀⢹⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢁⣴⣿⣿⣿⣿⣿⣧⢻⣿⣷⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⢠⣿⣿⣿⡿⠿⠛⠟⣩⣌⠿⠛⢠⡆⢻⣿⣿⣿⣿⣿⣿⣿⡿⠃⢠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏⢠⣶⡿⠀⢉⣠⡴⣾⠿⢁⢸⣿⣿⣿⣿⣿⣿⣏⣴⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣿⣏⠀⠈⢋⠉⢠⠴⣶⠟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⢿⣿⣿⣤⣄⣀⠀⢒⢸⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠸⣿⣿⣿⣿⣿⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⢿⣿⣿⣿⣿⣿⣿⣿⠜⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠸⣿⣿⣿⣿⣿⡿⣱⣷⠹⣿⣿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣿⣿⣿⣿⣿⡇⣿⣿⣧⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⣿⣿⡿⠂⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡟⣸⣿⠁⠘⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢇⣿⠃⠀⠀⠁⡼⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠟⡼⠃⠀⠀⠀⠀⠠⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠿⠿⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⢿⡿⢿⠿⢿⠿⠋⠿⠿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡿⢻⠟⠛⠻⣿⣿⣿⠿⠿⠿⠿⠿⠿⢿
⣿⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣆⠀⠀⠀⠀⠀⠀⠀⢸⡴⠋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠳⡇⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢠⠞⣧⠀⠀⢀⣀⡤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠉⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⠀⠀⣠⠋⠀⠘⠚⠉⠉⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠈⠙⠲⢤⠞⠉⠉⠙⠣⡀⠀⠉⠳⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⡞
⣀⣀⣀⣀⣀⣀⣀⣀⣀⡈⣧⠀⠀⠀⠀⡰⣋⣞⠉⣱⣄⡀⢸⡄⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⡼⠁
⠙⣄⠀⠈⠉⠉⠉⠀⠈⠉⠉⠀⠀⢠⡾⣽⠣⣼⠃⢌⣇⡜⡸⠀⢀⡤⠲⡈⠳⣄⠀⠀⠀⠀⠘⠧⠤⠤⢤⣤⣀⣀⣀
⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠒⠲⢧⡦⢴⠛⣬⣶⠁⡰⠛⢢⠰⢇⣠⠟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠈⢾⣠⣿⠏⠻⣅⣀⠼⣀⡠⠃⢱⠛⡆⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⣨⠆⠀⠀⢠⡇⠀⠀⠀⠀⠠⠁⠀⣼⠟⠁⠀⠀⠘⠷⣤⢼⠒⠒⡇⢀⠇⠀⠀⠀⠀⠀⠐⠛⠲⠤⣄⣀
⠀⢀⡠⠖⠋⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠈⠛⠳⠿⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡨⠽⠗
⠶⣉⡀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣀⣀⣤⠞⠉⠀⠀⠀⠀⠀⠀⣀⣤⣤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉
⠀⠀⠉⠙⠒⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢶⡛⢏⣙⡦⢵⣈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀
⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠉⢫⠈⣇⣸⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⡏⣧⣞⡁⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⣀⣀⣀⣈⣶⡄
⠀⠀⠀⣴⣁⣤⠤⠤⠶⠒⣶⠂⠀⠀⠀⡠⢿⡀⠀⠀⠀⢀⣠⠴⠋⠁⠀⠉⠙⠲⢤⣀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣠⠔⠋⠀⠀⠙⠓⠒⠚⠋⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠑⠢⢌⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⡠⠞⠁⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡽⠃⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⠀⠀⢀⡴⠋⠸⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⣁⣀⣀⣀⣀⡀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⣩⡤⢤⠔⠦⣤⡉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⣿⡇⠀⠀⠀⣱⠀⠈⡇⠀⠀⠀⠀⠀⠀⢠⠏⠰⢥⣠⣋⣀⣀⣀⣙⡆⢹
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠀⠸⡯⠩⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠒⠚⠀⠸⡄⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠐⠒⠃⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠈⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡏⠉⠓⠋
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠀⠀⠀⠀⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠀⠀⠀⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡩⠤⣴⢶⣤⣬⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢋⡁⠴⢶⡄⣀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⣰⢡⣎⣿⣿⣿⣿⣟⣻⡇⠈⡇⠀⠀⠀⠀⠀⠀⢠⠏⠀⠋⡟⠀⠟⠀⣽⠇⠀⢳
⠀⠀⠀⠀⠀⠀⠀⡇⠔⢯⡿⡧⠟⠉⠂⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠀⠁⠀⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢳⠈⠁⠀⠀⠀⠀⠀⠒⠒⠀⠸⡄⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠐⠒⠂⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠃⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠈⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠉⠓⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡏⠙⠒⠊
⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⠀⠀⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⡝⠿⣿⣿⣿⣿⣿⣿⠟⣫⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⢟⣱⣿⣶⣭⣭⣍⡱⣶⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⢰⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇⣫⣭⣭⣿⠿⠿⣿⣭⣭⣝⠸⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢇⡶⣼⢧⡝⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢻⡿⣸⢧⡟⣼⢣⡝⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢇⣿⣼⣣⡟⣼⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⣿⣿⣿⣿⡏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡼⣿⣿⣿⣿⣿⣿⢸⢟⣵⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡷⣹⣿⣿⣿⡿⣣⣴⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⠇⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠚⠀⠀⠀⠙⠿⠃⢠⣶⣆⠘⢿⣶⣎⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣠⣾⠂⠀⠀⢠⣴⡆⠀⠀⡰⠋⠉⢰⡧⠄⠀⠙⠻⢿⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠛⢷⡀⠀⠀⠀⢉⣴⠶⡜⠀⠀⠦⠞⠛⢠⠀⠀⠀⢠⣿⡗⠻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠈⠁⠀⣿⣄⠀⠀⢄⣀⣾⣧⡀⠀⠉⠀⠀⠐⣾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣁⣀⠀⢘⣿⠿⢳⣴⠀⠀⠀⠰⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠠⠀⠾⠄⣼⣿⣿⠿⡿⢿⠁⠀⠀⠁⠈⠙⠛⢀⠞⠀⢀⡿⠙⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣶⣼⣆⣀⡀⠐⡋⠀⢀⣿⡉⠛⠂⠀⢸⠁⠀⠀⠀⢀⣤⡴⠃⠀⣠⣿⠁⢠⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠭⠿⣿⠅⠀⠈⠿⢋⣽⡿⠶⠒⠈⠉⠻⠉⢠⣉⡽⠷⡀⢀⣴⠛⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣄⣥⣤⡤⠤⠶⠦⠼⠿⣷⣿⣯⡅⣲⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣷⣶⣼⣿⣷⣾⣷⣶⣆⢀⣀⣀⣼⣿⣏⢀⣹⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠙⢿⣿⣿⣯⠻⠛⠁⣶⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣷⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡻⢁⡴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⣷⠋⣼⣡⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣾⠁⡼⢫⠟
⠀⠀⠀⢀⠀⠀⠀⠀⣸⠃⠀⠁⢈⡴⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣰⠋⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢀⡇⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⢋⡤⠖⠒⠦⣄⠀⠀⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣏⣀⡀⠘⠤⢄⡀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⡾⠁⠀⠀⠀⠙⣶⠻⠅⠀⠈⢦
⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁⠀⠀⠐⠳⠤⣀⠀⠀⠀⠙⢦⡀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣄⣀⣀⡠⠋
⠀⠀⠀⠀⠀⢸⡇⠀⣠⢾⠁⣴⠀⠀⣠⣤⣟⣦⡀⠀⠀⠉⠳⢾⡇⠀⠀⠀⠀⠀⢰⡏⠉
⠀⠀⠀⠀⠀⠈⢷⣼⠃⣾⠀⡟⣆⡼⠁⠀⠉⠻⣿⣦⠀⠀⠀⠀⠻⣄⠀⠀⠶⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠈⠀⠀⡴⠒⠛⠛⠿⢧⠀⠀⠀⠀⠈⠙⢛⡿⠋⠛⠶⢤⣀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠴⠖⢲⢤⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠲⣄⠀⠉⠙⠓⠶⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠉⠉⠒⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡗⢤⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠉⠓⠦⣌⣀⡉⠂⠀⢂⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠛⣦⣄⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠄⠐⣯⠙⣧⠈⠳⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡰⠀⠀⠀⠀⠀⠀⠈⣇⠈⠙⠾⣦⡀⠹⣮⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣤⠀⠀⣹⣦⣙⠀⠀⠀⠈⠉⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠒⠒⠒⠒⠛⠦⣄⡀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢺⣁⣀⡽⠟⠢⠤⣄⣀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣲⠶⢤⣀⣈⡏⠁⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠈⡷⣄⠀⠀⠀⠀⠳⣄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣉⠀⠀⠈⠉⠉⠒⠒⠤⠤⢤⣀⣀⠀⠀⠀⠀⠀⣷⢤⡿⠆⠀⢀⣀⣘⣦⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⣄⡀⠈⠀⠂⠀⠀⠀⠀⠈⠉⢛⠶⣄⢠⠇⠀⠙⠚⠛⠉⠉⣿⣿⠱⡄⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠛⠁⠀⢉⡙⠢⣤⣀⠀⠀⠀⠀⣰⠏⠀⠘⠻⢦⡀⠀⠈⢦⡀⠀⣿⣿⣿⠛⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡉⠀⠀⠀⢀⡴⠋⠁⠀⠀⠉⠛⠲⢦⣴⠏⠀⠀⠠⢴⣾⣷⠀⠀⠀⠙⢶⣿⡛⠛⠒⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⣦⣿⡄⠀⢠⠞⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠷⠛⡄⠀⠀⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⣳⠞⠉⢳⡄⠀⣠⣿⣿⢿⣠⠏⠰⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⣸⠁⢀⡖⠈⡇⣰⢻⡟⠉⠻⠏⠀⣠⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠖⠒⡙⠶⠞⠀⠀⣿⠃⡾⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⢶⠚⠉⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣰⠻⣄⣐⣦⠀⠀⠐⢇⡀⠀⠀⠀⠉⢻⠖⠲⠲⠶⠶⠶⠒⠒⠒⠛⠛⠉⠉⠁⠀⢸⠁⡼⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠚⠁⠀⠈⠉⢈⡷⠒⠲⡬⠻⡄⠀⠀⢀⡟⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢀⡇⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⣀⣀⡴⠇⠀⠀⠀⢙⡷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⣼⣀⣤⡤⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠃⠄⠀⠀⠀⠀⠀⠀⠐⠚⠛⣛⠾⢻⠿⠍⡁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣑⣀⢤⡐⠄⠀⠀⠀⠐⠂⠉⠀⠀⢀⡏⠀⠀⠀⠀⠀⣸⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠉⣻⣿⠿⠛⠛⠲⠤⠴⠦⠤⠶⠶⠒⠚⠛⠓⠒⠒⠶⠞⣽⣩⠏⠉⠙⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⠾⠿⠿⠿⠿⠿⠿⠿⢷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠛⠉⣁⣀⣀⣀⣠⣶⠶⠶⣦⣄⠀⠀⠈⠉⠛⠻⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⣀⣴⠟⠛⠉⠉⠛⠋⠀⠀⠀⠀⠙⠟⠛⠛⠻⢶⡄⠀⠈⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⢀⣶⠟⠋⠉⠀⠀⣠⣤⣤⣄⠀⠀⢠⡶⠟⠛⠻⣦⣀⡈⠿⠶⣦⣄⠀⠙⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⣿⡁⠀⠀⣠⡶⠾⠟⠁⠀⠁⢀⣠⣤⣤⣄⠀⠀⠈⠉⢿⡄⠀⠀⠙⣷⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⢠⣾⠛⠁⠀⠀⣿⡀⠀⣠⣴⣦⣴⡟⠉⠀⠀⠙⠿⠀⠀⠀⣸⡇⠀⠀⠀⠻⢷⡄⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠸⣧⠀⠀⠀⣾⠏⠁⣾⠋⠀⠀⠈⠁⠀⠀⣀⠀⠀⠀⣼⠟⠋⠀⠀⣤⠀⠀⢘⣷⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠙⣿⡆⠀⢿⣄⣠⡿⠂⠀⢠⣄⡀⢀⣴⡟⠛⠛⠛⠋⢀⣀⣀⣴⠟⠀⢀⣼⠇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣧⣄⣀⣽⡏⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠈⢉⣿⠁⠀⠀⢀⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠘⠋⠀⣽⠍⠙⣷⣀⡀⠀⣀⠀⠀⠀⠀⢠⣶⣤⣀⣀⣤⡾⠛⣶⡶⠶⠟⠁⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡗⠀⢠⡶⠀⠠⡿⠀⠀⠛⠙⢻⡿⠛⢷⣶⣶⣶⣿⠇⠈⠉⢩⣿⠀⢰⡟⠀⢰⡖⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠈⠁⠀⣤⠀⢀⣿⠀⠀⣀⠀⠀⣼⣿⣿⣿⠏⠀⠀⠀⠸⠃⠀⣈⠁⠀⠿⠁⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠈⠋⠀⣈⠁⠀⢰⡟⠀⢠⣿⣿⣿⣿⣶⣶⠂⢰⡆⠀⢸⡏⠀⢠⡀⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠈⠁⠀⠘⠛⠛⣿⣿⡿⠁⠀⠿⠃⢀⣌⠀⠀⠛⠁⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⢿⣦⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣸⣿⠟⠀⠀⣰⡆⠀⠘⠃⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣄⣀⣀⣀⣀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠋⠉⠉⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⠲⣿⣿⣿⡗⢲⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠛⢛⣛⣛⠛⢻⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣶⣼⣿⣏⣴⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢫⣿⣿⣿⣾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣼⣿⣿⣿⣭⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣾⣿⣛⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣻⡿⣭⣹⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⠟⠁⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠟⠛⠀⠀⠈⠻⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟





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

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠴⠦⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠶⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠋⣤⠤⠤⠤⢤⠈⠱⡄⠀⠀⠀⠀⠀⠀⢠⠞⡵⠒⡲⠊⠲⢢⡉⢦
⠀⠀⠀⠀⠀⠀⢸⠁⠀⢿⣁⣀⣀⡴⠃⠀⢱⠀⠀⠀⠀⠀⠀⡏⠈⢓⠚⢦⠤⠤⠤⠟⠘⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣠⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠂⠀⠈⠁⠀⢀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠒⡇⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠐⢺
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡖⠒⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠒⠒⠒⠃
⠀⠀⠀⠀⠀⠀⣠⠤⠊⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠁⠀⠀⠀⠳⢄⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡰⢋⣐⣢⣞⣷⣶⣬⠳⡀⠀⠀⠀⠀⠀⠀⢀⠞⣁⣀⠚⣳⢀⣀⠙⢆
⠀⠀⠀⠀⠀⠀⢰⠁⣷⢿⡿⡿⡟⠿⠷⠀⢡⠀⠀⠀⠀⠀⠀⡎⠀⠀⢏⠈⡃⢰⠋⠀⠘⡆
⠀⠀⠀⠀⠀⠀⢸⠈⠓⠛⠛⠑⠁⢀⣀⡀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠶⡏⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠠⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡔⠒⠤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠒⠢⠤⠃
⠀⠀⠀⠀⠀⠀⣀⡤⠞⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠃⠀⠀⠈⠣⣄
⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠉⠉⠙⠛⠛⠻⠿⢿⣿⣿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠉⠻⣿⣦⣄⣠⣤⣤⣴⣶⣶⣶⣾⡿⠿⠿⠿⠟⠛⠻⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠙⠻⣿⣯⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣽⣿⣷⣄⡀⠀⠀⠀⠙⠻⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣟⡛⠛⠋⠉⠉⠉⠙⠻⣿⣶⣤⣀⠀⠀⠀⠉⠛⠿⣿⣷⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⣻⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠉⠀⠈⠙⠻⣿⣷⣤⣀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣄⣀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣷⣶⣶⣶⣶⣿⣿⠁
⠀⠀⠀⠀⠀⣴⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣄⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢉⣿⠃
⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣶⣶⣤⣤⣤⣤⣤⣤⡾⠋
⠀⠀⠀⠀⣼⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠉⠉⠉⠉⠁
⠀⠀⠀⠀⣿⣿⡿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣷⣦⣄⣀⣠⣾⠟⣿⣿
⠀⠀⠀⠀⠘⣿⣧⡈⠙⠻⣿⣿⣶⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⣿⣿⣿⠟⠋⢠⣿⣿
⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠈⠉⠙⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠀⠀⣠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⡀⢀⣤⣦⣶⣷⣾⣿⣶⣿⣿⠟⡠⠖⠈⠀⠀⡀⠀⢰⣤⡤⢀⠜⠉⢀
⠀⠀⣠⠔⠒⠀⣀⣽⣿⣏⣴⣿⠿⠛⣋⡶⣊⠤⠒⠐⢚⡛⠉⠉⠉⠩⠍⢛⡀⠂⠠⠤⢀
⠀⠀⠀⠀⠚⣉⣿⡿⠟⣋⣡⡴⠚⠋⢁⡄⣀⣀⣤⣀⣀⣤⣀⡀⠀⣀⡄⠈⠀⠀⠉⠀⠐⢆⢁⣒⢦⡀
⠀⠀⠀⠀⢀⢁⣭⣶⠟⣋⣠⡴⠊⠉⣁⣘⣋⣵⣿⡿⡿⠻⠏⡻⠋⢁⠉⠑⠒⠚⠧⣄⡀⢠⣾⢩⣯
⠀⠀⠀⢀⢰⣿⣿⣣⣶⡿⠗⠀⠠⣴⣟⣿⣿⣶⣶⣾⣿⣿⣵⡃⠀⠈⠈⠑⠂⠀⠀⠀⠁⢈⡿⠓⣿
⠀⠀⠀⣼⣿⣿⣿⢿⣿⠃⠀⣀⠶⣊⣿⣫⣿⣿⣯⠭⠘⠉⣂⡀⠛⢗⡢⣄⡀⠂⠀⠈⠀⠀⡟⢐⡃
⠀⢀⡼⢀⣿⣿⣧⣾⡇⡄⠀⠔⢪⣿⣿⢟⡭⢀⡤⠖⣋⣉⠉⢍⣛⠲⢌⢪⡙⢦⠀⠀⣀⣠⢷⡾⢃⡀
⠀⠋⣠⠞⡿⣿⣿⣿⢃⠁⣠⠾⣿⠿⠣⠏⡐⢁⡴⣛⠣⠩⢉⣹⡇⡌⢲⡄⡌⢈⡆⢐⠚⢹⡞⠋⣸⡇
⠀⠊⠠⠚⣱⣿⣿⣿⠛⠀⠀⠀⣀⣞⠀⡟⣰⢟⡃⢻⢤⣷⡆⡁⢿⠘⡆⢡⡸⣸⡄⠙⠁⢸⠈⢰⠍⠃
⠀⠀⢀⠴⣿⡿⢋⣭⣜⡀⠀⠚⡟⡞⣰⢡⣿⣶⣶⡌⠈⠉⠀⠀⠈⢣⡀⠺⡅⢸⠁⠀⠀⣿⠘⡟⠁
⠀⠀⠀⡖⣾⣿⡿⢿⣿⢰⡀⠞⡐⡇⢸⡟⠿⣿⠿⢃⠈⣤⣶⣤⣰⢎⡗⠂⣯⡆⣤⠐⢁⠏⢸⠃⡄
⠀⢠⠎⢈⣿⣿⣷⣾⣿⣷⡞⡵⢳⣇⣾⠀⡦⣤⣾⣿⢇⠻⣿⣿⣿⣿⠙⣸⣽⠠⠀⣀⣼⢄⡎⠀⠈
⠀⠀⣴⣫⣿⣿⡿⢹⣿⡟⡜⢡⠟⢹⣿⣇⠀⡛⠻⡿⢠⠦⠈⠩⢡⠇⠀⢿⣏⠴⠦⢤⠏⢸⠃
⠀⠀⣠⠴⢾⣿⣷⣿⢏⡞⣡⡏⠐⢺⡿⣏⠛⢦⣤⣤⣶⠓⣢⡝⡝⢠⢃⡼⠘⠖⢠⠏⢠⡿⠁⠲⢦
⠀⠀⣴⢿⣿⣿⡟⣡⠛⣴⡿⠁⠂⠀⡖⠈⣍⣓⣖⣿⣠⡿⢟⣠⣶⠇⡉⢓⠶⢲⡟⢀⡞⠀⠀⡀⠼
⠀⠀⠁⢘⠿⢁⡞⠡⣼⢿⣧⣤⣾⠛⣧⣄⣀⢙⣉⣤⠋⢐⡻⢟⠔⠁⠘⢣⡴⠋⢠⢟⡼⠃⠸⠒⠁
⠀⠀⠰⢇⣤⠏⢠⡾⠟⡿⢉⡶⢿⡔⢛⣛⡻⠥⢄⣐⣋⡭⠶⣧⣦⢀⣴⠾⢁⡔⠁⠸⣶⠠⡄
⠀⠀⢠⠞⠀⢀⡞⠁⡼⠃⣾⣧⣿⠒⣯⡴⠟⡯⠉⣽⣇⠿⣛⣭⣿⡶⠁⣠⡿⣦⣀⠶⠬⣁⠈⠓
⠀⠀⠘⢤⣤⡿⠁⣾⠇⢸⢃⣼⣿⠈⣧⣿⠾⠗⠟⣋⣭⢾⣿⣾⠿⣁⣰⣟⠓⣤⡙⡄⠀⠸⡟⢉⡇
⠀⠀⠀⡰⠛⣀⢄⡟⠀⡿⢸⡏⢻⣶⣿⣽⡶⢛⠿⢉⡱⣻⣏⣱⣟⠳⢮⣿⠀⣿⠇⡼⢀⡀⣈⣁⠠
⠀⠀⢰⠃⠈⢵⣾⠁⢰⡗⢸⢿⡇⢹⣏⣤⣿⣾⡾⢋⣼⣏⡶⠳⣍⣽⡻⣿⡄⠻⣾⡁⠈⠉⠀⣥⠆⣀⡀
⠀⠀⠘⣿⣦⣾⠃⠀⢸⠇⢸⡾⣿⠈⣿⣿⡇⠀⠉⠙⠛⣿⡇⢀⣿⣿⡇⢻⣿⢦⣀⡋⢉⡐⠒⠤⣄⡀
⠀⠀⠰⠋⢰⣇⣀⡀⣿⠆⢸⠃⢻⠀⢻⣧⠉⡻⣶⣶⣴⡿⠀⢸⣿⣷⣱⣿⣿⢣⢹⠿⣿⡙⠯⣤⣾⠟
⠀⠀⡗⠀⢺⢿⣿⠏⡟⠐⠸⣆⣿⠀⠈⢻⣎⠯⠻⠻⣿⠀⠀⣾⣿⣿⠷⠞⢋⡴⡿⢸⠹⣿⣀⠛⠻⡷
⠀⠁⠟⣄⡄⣸⢿⡟⢠⡦⣤⡏⢿⣦⠀⢼⣿⣀⠀⠂⢸⠀⢰⢻⣿⣥⡴⠶⠟⢁⡥⠾⢅⠛⠛⣷⡄
⠀⠀⠀⢨⣿⣿⡜⠁⠉⠠⢿⣯⣲⠟⠀⠈⠻⢥⠔⠆⠈⠓⠊⠚⠻⠯⠤⠰⠖⠀⠀⠀⠀⠛⠾⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⠸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⡈⠻⣦⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⢷⣌⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡈⣿⠈⠻⣦⣈⣻⣶⡆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⠀⣷⣄⠙⢷⠌⠉⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠻⣯⡙⠃⣿⡏⠻⠿⠛⠛⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢠⣄⠰⣶⣾⡄⢻⣤⣤⣤⣴⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡋⠻⣦⡈⠻⣦⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢶⣄⠙⢷⣌⠻⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠻⣦⡈⠻⣦⣨⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣦⡙⢷⣄⠙⠷⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⣤⣤⣶⣤⣸⣿⢠⣌⣻⣦⣾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠛⠛⠛⣧⠘⡿⠿⠆⠙⠃⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⠿⣯⡉⠻⣦⡀⣿⡈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣀⡐⢷⣄⠙⢿⠀⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠻⣦⡈⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⡙⢷⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠋⠉⠙⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠛⠛⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡾⠥⠤⠤⣀⡀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠏⠀⠀⠀⠀⠀⠉⠓⣄⠀⢀⡼⠁
⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡞
⢀⡴⢏⠉⠋⠉⠙⠓⠦⣄⠀⣷⣀⣀⣀⣀⣀⣤⡤⠤⠤⠶⠒⠚⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⣾⡇⠀⠑⢦⡀⠢⡀⠀⠈⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⣯⠱⡀⠀⠀⠙⢦⡈⠢⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠹⡄⠱⣄⠀⠄⣀⠙⠢⡘⢦⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠰⠐⠀⢀⡾⠁
⠀⠙⣆⠈⠣⣄⠀⠀⠀⠈⠢⡙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡴⠏
⠀⠀⠈⠛⢦⣀⠉⠒⠦⠤⠤⢾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠶⠴⠶⠚⠉⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠛⠉⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢀⣠⠴⠚⠋⠉⠁⢀⡠⠞⠋⣻⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⢀⣴⠋⠁⠀⠈⠁⡠⠖⠉⣠⠔⠋⠁⢀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠈⠛⠲⠶⠶⠒⠋⠁⠀⠀⠀⠀⣼⠃⠀⠀⣠⠔⠁⢀⡴⠊⠡⠀⠀⢀⢼
⠀⠀⠀⠀⣠⡞⠻⢭⣀⠀⠀⠀⠀⠉⠛⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠦⣴⠋⠀⣀⠔⠁⠀⢀⣀⠤⠞⢁⡾
⠀⠀⢀⡾⠉⠙⠲⢤⣈⠉⠒⠤⣀⡀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠘⢧⣀⡉⠉⠉⠋⠉⣉⣉⣀⡤⠶⠋
⠀⠀⢸⣧⣀⠀⠀⠀⠈⠙⠲⢤⣀⠉⠒⠤⣄⡀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⢿⡈⠓⠦⢄⡀⠀⠀⠀⠈⠙⠲⢤⣀⣉⣳⠖⢻⡆
⠀⠀⠀⠈⠳⣄⠀⠀⠈⠉⠑⠒⠒⠒⠒⠚⠋⠉⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣿⣶⣶⣶⣶⣾⣯⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣤⣶⠀⠀⠿⠂⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣛⢉⣿⣿⣀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣫⣿⣟⡽⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⠞⠁⣨⣮⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡔⣯⣾⢟⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣫⠞⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠻⣯⡺⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⡙⢅⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢦⡴⠋⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡛⢋⣴⣦⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠛⠉⠉⠉⠉⠛⠻⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣦⣀⡀⠀⡀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⢠⣶⣤⣀⣠⣤⣶⡿⠟⠈⠙⠻⠿⠿⣿⡆⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⢸⣿⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠸⣿⡆⠀⠀⣀⣀⠀⠀⣀⣀⠀⠀⢰⣿⠇⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⢻⣷⣾⠿⠿⠿⢿⣿⠿⠿⠿⣷⣾⡟⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣾⡟⠁⠀⠀⠀⢸⣿⠃⠀⠀⠈⢻⣷⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⣠⣼⣿⡁⠀⠀⠀⢀⣾⣿⡀⠀⠀⠀⢈⣿⣧⣄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣴⣿⠋⢿⣿⣷⣦⣴⣿⠟⠻⣿⣦⣴⣾⣿⡿⠙⣿⣦⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠁⠀⠀⠉⣻⣿⢿⣷⣄⣠⣾⡿⣿⣟⠉⠀⠀⠈⠻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⢀⣾⡿⠋⠀⠈⠛⠛⠁⠀⠘⢿⣷⡀⠀⠀⠀⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠀⠀⠀⠀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⠀⠀⠀⠀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁⠀⠀⠀⢠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄⠀⠀⠀⠈⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⠀⢀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⣠⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣄⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣀⣠⣼⡿⠻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠙⢿⣧⣄⣀⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠙⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠛⠛⠛⠉⠉⠉⠉⠙⠛⠛⠻⠷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⣀⣤⡴⠶⠛⠛⠛⠛⠷⠶⢦⣤⣀⠉⠙⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⣠⣾⣋⠀⠀⣠⠟⠋⠛⣷⣦⡶⠶⣦⡉⠛⢶⣄⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⣠⠾⢻⡟⠉⠉⠛⢻⡿⢦⣴⡟⠉⠈⢻⣦⣸⣇⣀⠀⠙⣧⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⣼⠃⠀⢸⡇⠀⠀⠀⣾⠁⣸⠏⠙⣷⣾⠋⢩⣿⡍⠉⠛⢦⣸⡇⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⣿⠀⠀⣨⣷⡟⠛⠛⣿⢶⣿⣤⣼⣏⡙⣷⡟⠀⢻⡀⠀⢀⣿⣇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠸⣧⢠⡟⠈⠻⢦⣴⣇⣼⠃⢨⡏⣉⡿⢿⡷⠶⢾⡷⠶⠛⠁⢻⡄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠈⢻⣧⣀⡀⠀⣸⣏⣿⣻⣿⡿⠿⠶⢾⣧⣤⠀⢻⣄⠀⢀⣾⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠈⣷⡉⠙⢻⡟⠋⣿⠁⢹⣇⠀⢀⣼⠃⢸⡇⠀⠉⠛⠋⠁⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠈⠛⠾⠋⠀⠀⠹⣦⡀⠙⠛⠛⢁⣠⡾⠁⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⢿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠈⢹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣶⣶⣶⣶⠶⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣫⠄⠀⠀⠀⠀⠀⠄⠀⠀⢄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠁⢛⡀⠀⠀⠀⢀⠀⣑⠠⠵⠕⠃⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠁⠁⠀⠀⠀⠀⠘⠘⢀⡀⠈⠁⠀⡄⠀⠈⠈⠀⠀⢸⠀⠀⠀⢠⢫
⠀⠀⠀⠀⠀⠀⠀⣰⣻⠀⡀⠀⡀⠀⠂⠀⠀⠀⠀⠈⠿⠢⠀⠀⡀⠀⠀⠀⣀⢼⣆⠉⠁⢸⡑
⠀⠀⠀⠀⠀⠀⣰⠿⢹⠉⣃⡁⠅⢀⡀⠈⠠⢠⣤⠄⠈⠈⠀⡴⠇⠀⠰⠊⠀⢸⠀⠀⢒⡞
⠀⠀⠀⠀⠀⣰⡝⠃⠀⠀⠀⠁⠀⠀⠉⢣⠀⠀⠀⠈⠂⠀⠀⢀⠀⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣰⡴⠿⣶⣷⣴⣤⣤⡄⡆
⠀⠀⠀⣀⣀⡀⢀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠂⣖⠀⠐⠞⠁⠆⠀⠀⠀⠰⠀⠈⠃⠣⡀
⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠈⠁⠉⠀⠀⠘⠁⠉⠉⠉⢻⠆
⠀⠀⠠⠤⠜⠗⢠⣄⢠⣐⣛⠻⠅⠀⠀⢠⠀⠀⠀⠀⠠⠷⠆
⠀⠀⠀⠀⠈⠉⠛⠙⠻⠉⣹⣿⣯⢭⣿⠩⢥⡦⠄⢦⠽⠿⠶⠶⠮⠷⠖⠒⠺⠥⠟⠿⠷⠮⠥⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠀⠂⡆⢰⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣘⣀⣁⣃⣘⣘⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⣀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢸⠉⢉⣉⣉⣉⣈⡀⠀⠀⠀⡴⠊⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⢨⡇⠀⢀⠎⠈⠢⡀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠘⣀⠤⡄⠒⠒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠤⠤⠤⠤⠤⠤⡇⢠⠊⠀⠀⢱⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⡇⠣⢀⣀⣀⣀⠠⠔⠊





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢤⠄⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠀⠐⠂⠀⠉⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠀⠸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠂⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠊⠀⠀⣠⠀⢀⠴⢀⣴⣤⣀⡉⠓⠀⠀⠀⠘⢷⢶⠖⠐⣊⡉⠴⠒⠉
⠀⠀⠀⠀⠀⠀⣴⠁⠸⣿⣿⣦⡑⠀⢀⡀⠈⢹⡿⠯⣆⠉⠛⠛⠫⠁⠀⠀⠀⠀⠀⠴⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠙⠀⠀⣿⣿⣿⠇⢰⢿⣽⣶⣾⣿⣷⡇⠠⣁⠤⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢿⠾⣿⣿⠅⡀⠈⣸⣿⣿⣿⣿⡿⠁⠀⠀⠓⠀⠐⢢⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⣘⠂⠀⠀⡜⠁⠀⠁⠀⣿⡆⠋⠄⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⢀⡃
⠀⠀⠀⠀⠀⠀⣤⣄⠀⢠⢖⣹⠀⡀⠀⠄⠀⠀⣄⠀⠁⠀⢀⣐⠓⠄⠘⠂⠀⠀⠀⣀⠁
⠀⠀⠀⠀⠀⠈⢹⡟⡀⠀⠸⠋⣠⠽⡖⢀⢊⡄⣴⣹⣺⠃⣿⢿⣷⠀⠀⡀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⣾⣷⡁⠀⣀⣾⣧⠀⠀⠀⠘⠈⢻⣿⣿⣠⣿⣦⠹⣧⡇⠘⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠙⠿⣷⠀⢿⠃⠀⠁⠀⠘⠀⠀⢸⣿⣿⣿⠻⣟⠀⣿⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⢄⡀⢰⣄⠀⠀⠀⣿⣿⡇⠙⠟⠙⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟⠉⠀⠀⣸⠏⡋⠀⠀⢠⠸⣿⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢸⠀⠀⣿⠇⠈⠀⠀⢈⠢⡟⠄⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢺⣀⣼⠏⠀⢀⡀⠀⣸⠃⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣏⠀⠀⠈⣠⣞⡰⣇⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣿⠋⠀⠀⠙⠿⠆⢹⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠣⠽⠯⢿⣛⡯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠉⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣦⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣏⣼⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⡿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⡀⠑⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⠿⠛⠛⠛⠋⠉⠉⠉⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠋⣁⡤⠶⠖⠛⠛⠛⠛⠛⠛⠛⠓⠲⠶⣤⣄⡀⠉⠙⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢟⣡⠞⠋⠁⠀⠀⠀⢀⣴⣾⣿⣿⣷⣤⡀⠀⠀⠀⠈⠉⠛⠦⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⡵⠋⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⣼⠉⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠉⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⡟⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠟⠋⠉⠉⠙⠻⢿⣿⣿⣿⣿⣦⠀⢰⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡇⢸⠇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢠⣤⣤⣤⣤⠀⠀⣴⣶⣤⣤⡄⠈⣿⣿⡇⣼⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⣿⡿⠀⠀⢿⣿⣿⣿⠇⢸⣿⣿⠁⣿⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣀⠉⠛⠋⠀⡴⢦⠈⠙⠋⠁⣠⣿⣿⡏⢀⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡀⣠⡄⢠⡄⣰⣄⣠⣿⣿⣿⡟⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡏⠙⠋⠙⠋⣹⣿⣿⣿⠏⠀⠀⢸⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⣾⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣿⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣇⣠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣯⣡⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⠧⠤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡏⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⡀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⣦⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠈⠻⣿⣦⡀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠈⣿⣿⣆⡀⠀⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠻⣿⣦⣠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠙⢿⣦⡀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠈⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣦⡀⠀⠀⠀⠙⢿⣿⡋⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠙⢿⣦⡀⠀⠀⠀⠙⢿⣦⡀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠙⢿⣦⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠋⠀⠀⠘⢻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠻⣿⣦⡀⠀⠀⠀⠉⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣦⡀⠀⠀⠀⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠙⢿⣦⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⢿⣦⡀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠃⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⣠⣾⡟⠋⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⢠⣾⠿⠁⠀⠀⠀⠀⠀⠀⣹⣿⣷⣄⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⣠⣾⠟⠁⠙⢿⣷⣤⣾⠟⠁
⠀⠀⠀⠀⠀⣠⣿⣿⣦⡀⣠⣾⠟⠁⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⣠⣾⠟⠁⠀⠙⠿⠟⠁
⠀⣠⣾⠟⠁
⣾⡟⠁
⣿⣿⡿⠻⠟⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢦⠀⠀⠀⠑⢦⣁⠚⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⢀⠀⠘⡐⠓⡀⠀⠀⠀⢻⠻⣄⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⠂⠀⠂⠀⠋⠀⠀⠀⠀⠀⢿⠏⠳⡄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡔⠈⠂⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠊⠀⠀⠠⠀⠀⠀⠁⠆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢰⣄⠀⠀⠁⠂⠀⡀
⠀⠀⠀⠀⠀⣱⡀⠜⠀⠀⠀⠀⠀⠈⠤⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠓⢀⠀⠀⠈⠉⠰⠆
⠀⠀⠀⠀⠀⠀⢉⠀⢄⠀⠀⠀⠀⠀⠀⠉⢀⡀⠄⠀⠘
⠀⠀⠀⠀⠀⠀⢸⡀⢈⡆⠀⠀⠀⠀⠠⡄⠀⢛⣆⠀⠸⣅⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠰⠇⠈⠁⠀⠀⠀⠀⠈⢠⠄⠘⢀⠱⡀⠀⢰⠄⠀⠀⠀⠀⢐⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠙⣷⠀⠀⠐⠁⠀⠁⠈⠀⠀⠄⠀⠀⠲⣄⠀⠀⢐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠳⣮⣾⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⣿⣦⡀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠙⣆⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣴⠏⠳⣄⠀⠀⠀⠀⠀⠀⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣂⣀⣀⣀⣈⣁⣀⣀⣀⣀⣠⣚⣁⣀⣀⣌⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡀⠘⣿⡹⣧⠹⣿⡻⣿⣷⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣾⣷⠹⣧⠘⢷⡙⢷⡝⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⢹⡆⠈⣷⡈⢿⡄⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣿⡀⠸⣧⠈⢿⡄⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⡇⠀⣿⠀⠸⣧⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠈⠀⠀⠁⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⣠⡉⠉⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⢀⠰⠀⠒⠄⠀⠀⠀⠀⠀⠠⠰⠀⠀⠄⠆⢀
⠀⠀⠀⠀⠀⠀⠸⠤⠈⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠐⠶⠂⠀⠀⠀⠀⠀⢃⢀⠞⡍⠌⣂⠄⡀⠀⠀⠀⠀⠐⢯⢨⠭⠅⠭⠝⠹⡆
⠀⠀⠀⠀⠀⠓⠖⠀⢀⡤⠃⠀⠀⠀⠀⠀⠀⠀⡤⢁⡀⣤⠀⠀⡖⠒⠀⠀⠀⠀⠀⢀⠉⠲⡠⠔⣇⡄⠴⠀⠀⠀⠀⠀⠇⡀⠀⢆⠂⠀⠠⠇
⠀⠀⠀⠀⠀⠰⠉⠈⠉⠡⠀⠀⠀⠀⠀⠀⠀⠨⠀⠨⠤⠄⠅⠀⠑⠀⠀⠀⠀⠀⠀⠈⠚⠉⠁⠠⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠈⠀⠀⠀⠱

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠀⠀⠀⠈⠀⠠⢀⠄⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡉⡈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠛⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠠⢥⠄⠀⡤⠒⠀⠀⠀⠀⠀⠀⠀⠀⣈⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠖⠊⠀⠀⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠰⣤⠚⡖⣰⠒⠀⠀⠀⠀⠀⠀⢠⣾⠒⢠⡘⢳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⠀⡀⠆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠈⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠁
⠀⠀⠀⠀⠀⠀⡐⠁⠠⡢⠀⣠⠀⡔⠄⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡆⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠐⠀⠀⠈⢬⡱⡀⠰⠹⠀⠀⠀⠀⡔⠁⠀⠈⣤⡀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠰⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠉⠉⢸
⠀⠀⠀⠀⠀⠀⠀⣀⠄⣈⠉⠀⡀⠡⠂⠀⠀⠀⠀⡄⠀⣠⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡠⢀⠁⠐⢄⠀⡠⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⢦⠀⢤⠂⠀⠀⠀⠀⠀⠀⢀⡔⠓⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⢰⣲⣀⠠⠜⠣⢄⣀⣖⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣀⢤⠐⣓⠠⢄⡰⠆
⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⢰⣡⢘⡡⡯⢘⣣⣵⡀
⠀⣏⠶⡁⠀⠀⠀⠀⠀⠀⢈⠶⡱⠀⠐⢷⠔⢺⣏⠉⠁⡢⣾⠂⠀⠰⠒⢇⣬⣨⡑⣖⣬⠱⠆⡗⠺
⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢠⠃⣴⡤⠀⠀⠈⠋⠈⡄⠀⠀⠀⠘⣤⠾⢠⡝⣤⣛⡜
⠀⠀⠸⠽⠉⠒⢢⡔⠒⠉⠯⠇⠀⠘⢻⣤⡂⠁⠐⠽⠀⢢⣄⡟⠛⠀⠀⠛⠀⠉⠚⣖⠊⠁⠘⠃
⠀⠀⠀⠀⠀⠀⠓⢋⠀⠀⠀⠀⠀⠀⠀⣣⡀⠀⣦⠆⠀⢈⣝⠀⠀⠀⠀⠀⣠⢤⠀⠉⢸⡽⠂
⠀⠀⠀⠀⢦⠀⢀⣸⣀⠀⢀⠆⠀⠀⠈⠋⠈⠑⢻⡖⠊⠁⠙⠁⠀⠀⠀⠀⠈⡽⠉⢝⣱⠈⠑⠤⡮⠧
⠀⠀⡀⠀⡠⠋⡧⢰⠆⣭⢣⡀⠀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣔⣢⠜⠀⠀⠈⠈⠀⢀⠀⠇
⠀⠀⠉⢹⠙⠃⠶⣙⡁⠶⣛⢻⠉⠁⠀⠀⢀⡀⠀⣀⡠⠻⢄⣀⠀⢀⠈⠀⢇⠀⢀⣄⠀⠀⠺⣋⡯⢞⡇
⠀⠀⡤⠼⡟⢑⡗⠨⠜⢣⡍⡸⠤⡄⠀⠀⠘⡏⠉⠀⠀⠀⠀⠀⠙⢹⠁⢔⡾⠤⣑⠛⠀⠀⣀⡜⠁
⠀⠀⠀⠀⢈⠮⢅⣗⣘⠧⢎⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠈⠁⠀⢀⠵⡏⠉⠀⠯⠃
⠀⠀⠀⠀⠃⠀⠀⢸⠀⠀⢀⡃⠀⠀⠘⠟⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠻⠂⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⡔⡆⢀⣀⠧⡏⠀⠀⡀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⢠⠄⢤⠀⡄
⠀⠀⠀⠀⠀⠀⡨⠋⠱⢭⡆⠈⠑⢺⠬⠂⠰⠕⠢⢄⣀⠀⣀⡠⠔⢾⠆⠀⣄⡹⢚⡉⢭⠍⣓⢎⣠⡀
⠀⠀⠀⠠⣋⢖⠁⠀⠀⠀⠈⠀⡀⠈⡄⢀⠀⠀⠀⠀⠈⠾⠀⠀⠀⠀⠀⠖⡞⣤⢈⡁⠶⢀⣉⠴⢧⠴
⠀⠀⠀⠀⠀⠈⡄⢀⣴⡄⠀⠈⠺⢽⠘⠭⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡧⡤⣨⡁⢟⢠⡍⠶⢸⠒
⠀⠀⠀⠀⠰⣊⠗⠪⣒⠁⢀⣀⣠⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡱⣤⡾⢠⡝⢡⡘⣣⢎⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣻⠀⠀⠘⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢱⠚⡥⢺⡋⠷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢴⣶⣶⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢹⣿⣿⣿⣿⣏⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⡀⠀⠀⢀⢀⡷⢠⣼⣿⣿⣿⣿⡟⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣾⣿⣿⣿⣿⣿⣶⣶⣾⣶⣦⣄⣒⣒⣀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⣿⣿⣿⢿⣟⣿⣿⣿⠿⢿⡏⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠊⠙⣿⠛⠛⠛⢫⣿⣿⡇⠀⣼⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠣⠄⢀⣈⣓⠈⠳⣠⣴⣾⣿⣿⣯⣿⣷⡀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠚⠒⢀⣈⣭⣛⠷⡹⣿⣯⡙⠿⠭⠿⢿⣿⣦⣄⡙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠊⠀⢀⣈⠄⠂⢀⣡⡶⢟⣫⡥⠀⢀⢀⠻⣯⠈⠄⠀⠀⠀⠀⠈⠙⠻⣦⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢛⠃⠀⣠⠞⠋⣀⣴⢞⣫⡵⠞⣫⡿⣧⣿⣾⣿⡆⢹⡆⠙⢆⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢻⢶⡶⠔⠁⡠⠞⠋⠊⠈⠁⣠⣾⣫⣾⣿⣿⣿⡟⢿⡄⢳⡀⢆⡱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠈⠀⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠺⢷⡔⠈⠀⠀⢀⣤⣴⣾⣿⡿⢫⣾⠇⣿⣿⣿⠘⣿⡀⣷⠀⠙⢮⡓⢤⡀⠀⠀⠀⠀⠀⠀⠠⠭⠃⡄⠀⠀⠀⢠⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠴⠿⣿⣇⡙⢿⣿⠿⠟⠋⣭⣴⣿⡟⢸⣿⣿⢹⣇⢹⣧⠘⡄⢢⡀⣹⠞⠃⠵⠤⡀⠀⠀⢢⣄⣴⣖⣷⣤⣀⡁⢀⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡇⠐⡲⠞⢿⣟⣿⠦⠤⣤⣿⣿⣿⣿⡇⣼⣿⣿⡿⣿⠈⣿⠀⠘⡄⠻⠃⠀⢀⡀⠀⠤⠚⢛⣀⠀⠀⠀⠀⠉⠤⠶⢛⣋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢠⡆⠙⢿⣾⣟⣧⣀⣠⣾⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣠⣾⣷⣾⣿⠉⠁⠘⠂⢡⣠⣤⣴⠾⠛⠉⠓⠒⠠⠟⠃⠬⠩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣷⡴⠸⣿⣿⣿⣿⣿⡏⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⡿⣧⣤⣴⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢯⢼⡇⣿⣿⣿⣿⣧⣟⢻⣿⣿⡿⠿⣿⣭⣽⡿⢴⠟⠔⢷⣹⡇⠻⢿⣿⠇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⣽⡿⣶⣿⣿⣿⣿⡟⡌⢸⡟⣽⣿⣿⡟⣿⡿⣿⣴⡦⠄⠶⠋⠑⠈⠁
⠀⠀⠀⢀⠀⠀⡐⠀⠀⡼⠀⣿⣧⣿⣿⣿⣿⣿⡅⠀⢸⢃⣿⣿⣿⢱⡜⢷⠘⢿⣿⣆⠀⠀⠢
⠀⠀⠀⠀⢀⠀⢈⡦⠊⢀⣠⣿⠃⢻⣿⣿⣿⣿⠁⡇⣿⢸⣿⣿⣿⠈⣿⣦⡀⠀⢻⣿⣷⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⣄⠀⠠⡀
⠀⠀⠀⠈⠂⣄⠉⣠⣾⣿⡟⠁⠀⣼⣿⣿⣿⡇⢀⣇⠀⢸⣿⣿⣿⣇⢻⣿⢷⡼⣆⠻⣿⣦⡙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠀⠀⠀⣀
⠀⠀⠀⣐⡤⠜⣹⣩⡟⣫⠀⠀⠀⠹⣿⣿⣿⣧⣼⣯⠀⠰⣿⣿⡞⣿⣦⡙⠧⢻⣜⠆⠹⡻⣿⣶⣄⡙⠃⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠋⠀⠠
⠀⠀⠀⠈⠉⠛⠋⠉⠁⡉⠀⠀⣠⣮⣙⣿⡏⢻⣿⣿⣧⠀⢻⣿⣧⠹⣿⣿⣦⢤⣛⣦⡀⠱⡈⢻⠛⣿⡷⠦⣌⡁⠀⠀⠀⠀⠀⠀⠚⠁⢀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⢉⢻⣿⣟⢷⣸⣿⣿⣿⣷⡌⢿⣿⣶⣿⣿⣷⣶⣮⡙⠻⣦⠰⡸⠂⡿⢻⣆⠀⠠⣀⡀⠀⠀⠀⠀⠠⠖⠁⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣮⡻⣿⣄⢘⣿⣿⣿⣿⣿⡜⣿⣏⣿⣿⠟⣥⣝⢿⣷⡈⢧⠹⣶⣄⠀⣹⡶⡀⠀⠉⠓⠦⠀⠀⠀⠀⢒⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠙⡙⠃⡽⠋⣴⢿⡿⢛⣭⣿⣿⣿⣿⢻⣧⣼⣷⡜⢎⢿⣷⡈⣇⠹⠻⣮⢿⣧⡿⠄⠈⠑⢦⣀⠀⢤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠛⠀⠸⣦⣉⣤⣤⣾⣿⣿⣿⣿⣿⣿⡙⢻⣿⣿⡌⢮⣿⣷⣹⣦⢳⣬⣿⣿⣇⠸⣆⢢⡀⠉⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠡⡄⠘⠛⠿⠿⣿⣻⣿⣿⣿⣯⣾⠃⠀⠈⠙⠛⠦⣜⣈⣛⣛⣳⡛⠛⡧⣻⣆⣿⣦⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠿⡿⠄⠀⠓⠒⠊⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⣟⣻⡿⣿⠿⠿⠿⣿⡿⢿⣟⣻⢿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠶⢮⣭⣭⣭⣭⣭⣭⣭⣭⡵⠶⠛⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⡶⠛⠛⢶⡞⠉⠀⠈⠉⢉⣭⣭⣿⣿⡛⠉⠉⠉⣴⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣼⣷⡀⠀⠀⠹⢦⣤⣤⣤⣴⣿⣿⣿⣿⣇⣀⣀⣤⣿⡿⣿⠟⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠁⠀⠻⢶⣄⣠⣶⣽⣶⡉⠁⢻⣿⣿⣿⡟⠉⠉⠀⣨⣽⣿⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢰⣀⠀⠀⠀⠀⣩⣛⣿⣿⣇⣠⣴⠿⠿⢿⣻⣿⣶⣤⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⣙⠻⠶⣦⣤⣿⣿⣿⣿⡟⠉⠀⠀⠀⢻⣿⣿⣿⠇⠀⠉⣉⣽⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣤⣌⣉⡙⠛⠋⣄⣀⣠⣤⠶⠟⢛⣉⣀⣤⣶⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠉

⠀⠀⠀⢀⣤⣴⣤⣄⡀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣦⣾⡿⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠞⠿⠿⠟⢿⡿⣦⣠⣶⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣷⡄
⠀⠀⠸⣋⠈⢉⠀⣿⣿⡏⢀⠀⠀⣻⢿⣿⣿⠀⠀⠀⠀⠀⠀⢖⠂⠐⠦⢸⣿⣿⣿⢿⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⣴⣶⣶⣬⡋⠁⠘⣿⢿⣿⡧
⠀⠀⠀⠎⠂⠘⠄⣠⠜⠁⠸⠃⠐⡁⢼⡿⠃⠀⠀⠀⠀⠀⠀⡏⠓⠈⠃⠘⣩⠏⠀⠠⡀⠙⣹⣿⣿⡇⠀⠀⠀⢰⡿⣿⢿⣿⣿⣿⡶⠠⠉⠸⡿⠃
⠀⠀⠀⠐⠀⡴⠛⢅⠀⠀⣀⣠⡰⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠎⠁⠀⢈⡀⢀⣂⣽⠛⠋⠀⠀⠀⠀⠈⢃⡉⢀⡁⢻⢿⡄⢀⢬⠀⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠈⣑⠂⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢱⠒⠉⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠈⣱⠐⠈⢀⣀⠀⠀⠀⠀⠁
⠀⠀⠀⢈⠃⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠻⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣳⣧⡚⠁⠀⠁
⠀⠀⢀⡤⠆⠠⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⡎⠀⠈⠀⡀
⠀⠀⠀⠜⠁⡴⡈⠧⠀⠀⠀⠀⢠⠢⡀⢀⡀⠁⠄⠀⠀⠀⢸⣿⣿⣿⡋⠣⠀⠀⠐⠈⡤⢤⠤⣀⠈⠠⠀⠀⠀⠀⠋⠇⣴⠚⠏⠀⠀⠀⢠⠢⡄⢀⡀⠁
⠀⣀⣔⣦⢠⠤⠄⠀⠀⠀⠀⣀⣸⠶⣆⣠⣀⠀⠀⠀⠀⣀⣼⣿⣿⠧⠤⠀⠀⠀⠀⢀⣌⠔⣄⣀⡀⠀⠀⠀⠀⠀⣼⡷⢧⠄⡀⠀⠀⢀⣈⠶⢆⣸⣀
⠀⠉⠙⠂⠉⠀⠀⠁⠀⠀⠂⠀⠉⠘⠃⠙⠋⠛⠀⠀⠀⠉⠙⠋⠉⠁⠀⠈⠀⠀⠀⠀⠉⠙⠉⠛⠁⠃⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠉⠈⠛⠘⠋⠙⠃
⠀⠀⠀⢀⣴⣶⣶⣶⣤⡀⢀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣶⣦⣄⢀⣤⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀
⠀⠀⢠⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣻⣝⠛⠋⢻⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣆
⠀⠀⣿⢠⠈⢹⡛⣿⣿⣿⠙⠋⠻⡿⣿⣿⣿⠀⠀⠀⠀⠘⠇⡀⠀⣉⢈⣿⣿⡿⣦⠄⡸⠋⣿⣿⠃⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣿⠻⠟⢿⡿⣿⣿⣿
⠀⠀⠙⡈⠁⠈⣴⣿⣿⠟⠅⠲⢀⢰⣿⣿⠟⠀⠀⠀⠀⠀⠐⠓⠀⠩⠈⠛⠋⠀⠉⠀⡰⠈⢟⠁⠀⠀⠀⠀⠀⢸⡏⠸⡏⠈⢻⣿⣆⠀⠄⠠⣾⠿⠟
⠀⠀⠀⠀⡰⠎⠙⣢⣀⢀⣶⣤⡖⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣵⣤⣀⣠⣶⡤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠆⠰⢆⠈⡹⠃⠀⠸⠄⠛⢀
⠀⠀⠀⠀⡇⠀⠀⣤⣭⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣯⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣄⡄⠺⠿⠃⡀
⠀⠀⠀⢀⡁⠀⠀⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡛⠛⠂
⠀⠀⠀⠁⠀⠀⠠⡁⠀⠀⠀⠄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⠩⠋⡉⢹⡃⠀⠀⠀⠂⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠴⣿⣿⣿⣧⠀⠀⠀⠄⠀⠀⠀⠀⠄
⠀⠀⠀⢤⡠⢳⠀⠁⠀⠀⠀⠘⠉⡋⢹⠂⠀⠀⠀⠀⠀⠀⠀⣸⢸⢒⠀⠁⠀⠀⠀⠀⡍⢀⢰⠀⠀⠀⠀⠀⠀⠀⣰⡦⢹⠀⠁⠀⠀⠀⠘⠉⡋⢱⠂
⠀⠀⠰⡜⠾⠛⠀⠄⠀⠀⠀⠲⢌⡽⢶⡶⢤⠀⠀⠀⠀⠀⠾⡿⠽⠚⠀⠄⠀⢀⠐⠶⢯⡭⢾⡦⣤⠀⠀⠀⠀⠘⠻⠟⠙⠀⠄⠀⠀⠀⠲⣮⣽⢾⡶⢤

⠻⣦⡀
⠀⠈⠻⣦⡀
⠀⠀⠀⠈⠻⣦⣤⡀
⠀⠀⠀⠀⠀⠻⣄⠙⠷⣄⡀⣠⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠘⢧⡀⣨⠟⢁⣽⠟⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣨⠟⠁⣴⠟⠁⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠙⣷⠟⠁⠀⠀⢀⡀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣟⢁⡴⠀⢀⡴⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⡴⠋⣀⠀⠀⣀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣾⠋⣠⠞⠁⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣅⣠⠖⢀⣴⠗⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣴⠟⢁⡄⠀⢀⡄⠀⠀⠀⠀⠙⢦⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣏⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠙⢦⡴⠋⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣤⡾⠃⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⢀⡴⠋⣠⠞⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡴⠋⣠⠞⠁⠀⠀⠈⠻⣦⢀⣤⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞⠻⣦⡀⠀⠀⠀⢀⣼⠟⣡⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠞⠁⠀⠀⠈⠻⣦⡀⣴⠟⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⣡⡾⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣷⣞⣋⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠶⠖⠒⠒⠒⠒⠒⠲⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⣉⣡⣤⣄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡿⠁⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣩⠔⠒⠛⠛⠓⠒⠤⠌⠉⠛⠛⢗⠒⠒⠲⠶⠦⣤⣀⡀⣼⠁⢀⢀⣴⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠠⠊⠀⡠⠁⣼⠿⠋⠑⢦⣀⠀⠀⠠⡀⡄⠀⠑⣄⠀⠀⠀⠀⠈⠙⠣⠶⣾⣿⠇⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠐⢀⠔⠋⠁⣸⡣⠀⠀⠀⠀⠉⠟⠦⢀⠈⠻⡀⠀⠈⢦⡀⠀⠀⠀⠀⠀⢀⡿⠑⢦⡀⠀⠀⣠⡿⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣟⡇⠀⠀⢀⡠⢔⣦⣀⠼⢷⡄⠈⠀⠀⠀⢳⡄⠀⠀⠀⠀⣼⠁⠀⠀⠙⠶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠟⢯⡅⠀⠐⢿⣿⡯⠃⢰⣿⣆⠀⠀⠀⠈⣿⡄⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⢻⠟⢱⣦⠈⣉⠁⣠⡯⢻⣿⡄⢠⡆⠀⢹⠏⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣷⣶⣆⠀⠀⠀⢈⣽⣿⣷⠹⣿⡇⢨⡇⣰⠘⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡸⣿⣿⣧⣶⡞⠀⢠⠟⠉⠀⣿⢣⠟⢁⡿⠀⢸⣿⣾⣷⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡻⣞⢿⣿⣿⠡⠄⠂⠀⣰⡟⠁⠀⠀⠘⠃⢠⡿⣿⡟⠻⡄⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣕⣈⠻⣇⣠⣀⣼⠏⡔⠀⡄⠀⠀⢠⣿⡟⢻⡇⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⣿⣿⣽⣮⣝⠿⡟⣸⡇⠀⣿⠀⠀⣿⠋⠀⢸⠃⠀⠀⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⠀⠀⠇⠁⣿⡇⢹⣶⡇⣿⡇⠀⡔⠀⣼⠇⠀⠀⠘⠀⡆⠀⠁⠀⢀⡰
⠀⠀⠀⠀⠀⠀⠀⢰⠟⢠⠸⡆⠀⠀⠀⡫⢷⡞⡓⠀⢛⣃⡰⢁⣼⢿⠃⢠⠀⠀⢠⡇⠀⡀⡄⠈⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸⠀⢗⣠⠎⠀⠀⠊⡞⠁⢀⢠⣾⡑⢿⣿⣿⣤⣿⣄⣺⣾⡇⠘⣸⣧⣼⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢀⢀⡭⠖⠉⠇⠀⢀⠀⢀⡱⡀⣰⡟⢦⣙⡶⡹⣿⡿⢿⣿⠿⣿⣿⣾⡿⢿⠏⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠀⡀⢀⠀⠀⠀⢸⠀⢸⠹⠿⠧⢙⢳⣞⠓⠁⡌⠃⠀⢀⡄⠀⠉⠁⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠚⣧⣾⣗⠄⣠⡿⣦⡟⣦⠀⠐⠀⠙⠀⠀⠰⡇⠀⠀⠈⠁⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠁⠉⠛⠉⠀⠀⠀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠤⠤⠤⠤⢤⢤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⠀⠀⠀⣀⣸⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠉⠉⠉⠉⠈⠹⢿⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠠⠤⠤⠤⠤⠶⠾⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠏⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣴⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡰⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⣀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⣇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠛⡌⠐⢲⡶⠲⠂⠒⠒⢒⣾⢻⠀⠀⠀⠀⠀⠀⠀⣧⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣄⡔⠺⣾⡟⢀⣦⣀⡸⠟⡀⣸⠀⠀⠀⠀⠀⠀⠀⣥⣿⡆⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⢈⣿⡋⢁⣼⡁⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⣽⣿⡇⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠘⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠓⠲⣶⠶⣾⠟⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠃⠀⠀⢤⣧⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢙⡆⠀⠀⣞⠉⣷⠀⠀⣚⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢀⣀⣀⣠⣤⣬⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠈⠋⠁⣀⡀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⢸⡇⣼⣿⣿⣿⣿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡄⢀⠀⡤⠠⡄⠀⠀⣠⠒⢠⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠹⠉⠉⠉⠉⠉⠉⢸⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠼⠃⠀⠀⠢⠴⠃⠀⠀⠙⠶⠞⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣋⣸⡇⠀⠀⢨⣀⣵⠀⠀⠺⣤⡿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢾⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⠐⠀⠀⢠⣤⣤⣤⣤⣤⣶⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢰⡄⠀⠀⣔⠀⣆⠀⠂⢠⡁⣰⠄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⢹⠙⠛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠙⠛⠃⠀⠀⠈⠛⠋⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⣀⣈⣀⣀⣼⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉⠁

⣷⠛⠛⣶⢶⡶⣶⠶⠶⣾⠶⢶⠶⠶⣾⠷⠶⢿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⢶⡶⡶⣾⢿⣿⣿⠾⠿⠷⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠛⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠊⠉⡆⠀⡠⠚⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠵⡒⠁⠀⢇⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢦⣀⡼⠀⠀⠸⡯⠀⠁⠀⠀⢀⠀⠀⢠⣿⣄⠀⠀⠀⠀⠓⠀⠀⠠⠄⠀⠈⠛⢇⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⣼⠁⠀⠁⠀⠀⢸⠀⠀⠘⡏⡇⠀⢀⣤⡈⠀⠑⠀⠀⠀⡀⠀⠀⠀⢱⠀⣸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⠘⠻⢤⡀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢸⠀⠀⢸⠇⠀⢀⠊⠈⠁⠀⠀⠃⠀⠀⠙⠿⠿⠿⠋⠀⠛⠉
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⡖⠀⠀⡀⠀⠹⡆⠀⠀⡇⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠠⠇⠀⠀⠀⠀⡄⠀⠀⠀⢠
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢣⠀⠀⠅⠀⣠⠇⠀⠀⡇⠀⠀⠀⠀⠀⣸⠘⣦⠀⠀⠀⠀⡄⠀⠈⢩⠉⠉⡇⠀⠀⢠⡮⡀⠀⠀⢘
⠀⠀⠀⠀⡆⠀⢠⠀⠀⠀⠈⠟⠛⠟⠛⠏⠀⠀⠀⡇⠀⠀⠀⠀⠉⢻⠀⠉⠻⠉⠉⠋⠁⠀⠀⢸⠀⣗⡇⠀⡖⠉⠀⠀⢀
⠀⠀⣠⣤⣿⣀⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣷⣶⣶⣶⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣤⣷⣧⣤⣷⣤⣤⣤⣼⣤⣬⣤⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢶⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠈⢇⠈⢧⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⣧⠈⢇⠈⢆⠈⢇⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠠⣦⠘⢇⠘⣇⠘⣇⠙⢧⠈⢆⠈⢆⠈⢆⠈⣦⠈⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⢿⣈⣿⣿⣿⣿⣿⣿⡙⡇⠘⣄⠈⣆⠘⣆⠈⣆⠸⡧⠸⠇⠸⠇⠸⠇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠛⠛⠋⠈⠛⠛⠿⣿⣿⣿⣿⣧⠘⡄⠈⡄⠘⡄⠈⢀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⢋⣬⠐⡄⠘⣄⠸⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣿⣿⡷⠞⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡠⠞⠋⠁⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣤⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⣀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣧⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡆⠀⢶⣦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠈⣿⣿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⢿⣿⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠟⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⣿⣷⣄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣴⢋⡴⣫⡼⠛⠋⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠙⢿⣯⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣀⡿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢞⡥⠤⣤⠄⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⢰⡏⣠⠞⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣭⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣟⣷⣝⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠶⠶⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡴⠞⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣭⡅⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⢸⡇⣒⡂⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠶⠶⠶⣦⠀⠀⠀⠀⣸⠀⣧⠀⠀⠀⠀⢸⡇⠭⠥⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣼⠀⠀⠀⠀⡟⠀⢹⡀⠀⠀⠀⢸⡇⣛⡃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠉⠉⠳⡄⠀⠀⠐⣇⠀⢸⠇⠀⠀⠀⢸⡇⠶⠆⠀⢸⡇
⠀⠀⠀⠀⠀⢠⡏⠈⣧⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣤⣤⣤⣤⡄⣿⠀⠀⣴⠋⠀⠈⢷⠀⠀⠀⢸⡇⠉⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠈⠳⠴⠃⡞⢹⠄⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⡇⣿⠀⠀⣿⠀⣾⠛⢻⠀⠀⣠⡼⣧⣤⣤⣤⣼⣧⣤
⠀⠀⠀⠀⣰⠻⡄⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⡇⣿⠀⠀⣿⠀⣿⠀⢸⠀⠀⠙⠛⠻⡟⣿⢻⡟⠛⠋
⠀⠀⠀⠀⠳⡤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⡇⣿⠀⠀⣿⠀⢿⠀⢸⠀⠀⠀⠀⢰⡇⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣤⣤⣤⣤⠇⣿⠀⠀⣿⠀⠘⠒⢾⠀⠀⠀⠀⣸⣇⣿⣸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠞⠃⠀⠀⠙⠶⠶⠶⠋⠀⠀⠀⠘⠷⠿⠿⠿⠾⠇

⠀⠀⠀⠀⠀⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠢⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢰⡇⠀⠀⠀⠀⠉⢳⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⡐⡇⠀⠀⠀⠀⠀⠞⠁
⠀⠀⠻⣤⡀⠈⠳⣶⣤⣈⠻⢿⣶⣤⡀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡥⠀⠀⢀
⠀⠀⠀⠈⠙⠷⣾⣿⣿⣿⣷⣦⣉⣻⣟⠂⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣼⣷⣔⡁⠀⠀⠀⢀⡸
⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣟⣋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣟⣿⠡⣓⣤⡻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣴⠇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢜⠁⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠
⣀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠒⠉⠁⢀⣤⠏⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢷⠀⢀⠀⠀⢀⠀⣀⣴⣿
⣿⣶⠦⠄⠀⠂⠀⠈⠀⠀⣀⡠⠴⠚⠋⠁⠀⠀⠀⠀⢠⣄⣀⣀⣀⣤⣆⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⢉⣻⣿⣿⡿⠃
⠙⠻⠶⣤⣤⠤⠤⠖⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠂⠄
⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠒⠐⠚⡆⠀⠀⠀⠀⠀⠀⢠⣾⣿⠞⠓⠂⠐
⠀⠀⠀⠉⢁⢤⠁⠀⠀⠀⠀⠀⠀⠘⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣟⡽⠁
⠀⠀⣾⡤⠴⠐⢦⠀⠀⠀⠀⠀⠀⠀⢻⡧⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠀⡀⢸⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⣿⢳⡴⠀⠚⠇⠰⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠸⡇⢹⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⢹⣿⢀⣄⣄⣰⡶⠀⠀⠀⠀⠀⠀⢹⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⢸⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠻⣏⡭⠵⢿⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⣭⣿⡇⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⣀⡠⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠙⠶⠦⠴⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⡀⠀⠀⠠⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⡀
⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡁⢀⠀⣁⣠⢀⡅⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠤⠀⠤⠤⠤⠤⠜⡀⡀
⠀⠌⠀⡄⢠⠆⢀⡀⢠⠀⠀⢹⠁⠀⠉⠈⠀⠀⠀⠀⠈⡇⠀⠆⠰⠀⠠⠤⠄⠂⠁⡐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠁
⠀⠉⠀⠁⠈⡏⢈⠀⠠⡀⠀⣸⠀⣀⠀⠀⡀⠀⠀⠤⠀⠇⠀⠀⠀⢀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⢀⡀⠠⣈⠃⠀⠀⡐⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⠀⠀⢀⠀⠀⠠⠀⠂⣀⠐
⠀⠀⠀⠀⠠⠆⠀⠀⠀⢀⡀⠀⢁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠒⠂⡑⡄⡅⠀⠀⠀⠐⠠⡤⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⣁⡒⠀⠠⠀⠒⠈⣀⠀⠐⠀⠀⠀⠀⠈⠀⠐⠄⠀⠀⠠⠄⠀⠆⡄⢀⠀⡆⠀⠑⠁⠀⠀⠀⢎⠀⠀⠀⣀⠘⠀⢀⡂⠶⡅
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠐⡱⠦⠤⠔⠊⠀⠀⠀⣇⢀⠙⣠⠀⢀⠠⠦⠀⢰⠀⠭⠃⢠⠠⡄⠀⠁⠀⠀⠀⢀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠔⠁⠀⠀⠐⠀⠀⠀⠀⠘⢦⡀⠈⠁⠉⠁⢂⡠⠬⠀⠀⠉⠀⠋⠀⠠⡀⠀⠀⠐⠈⠐⠉⠐
⠀⠂⠀⠀⠀⠀⠃⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰⠂⠰⠀⠀⠀⠀⠀⠀⠉⡁⠒⠒⠊⠉⠀⢠⠀⠀⢰⠀⠑⠀⠠⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢂⠀⢀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠔⠀⠈⠂⠀⠀⠀⠀⠀⡀
⠀⠀⠁⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⡊⠐⠀⠄⠀⠘⠀⠑⠁⠦⠀⠀⢀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠂⠂⠀⠀⠀⠀⠀⣀⠀⠀⢀⣠⣤⣄⡀⢀⣀⣀⡀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⢀⠈⠉⠉⠁⠁⠀⠀⠀⠈⠁⠀⠀⠀⡀⢸⣿⠀⠀⢸⠟⠉⢻⡇⣿⡇⢹⡇⠀⠀⠰⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠍⠀⠁
⠀⠀⠀⠀⢨⠈⡅⠀⠀⠀⠀⠀⠀⠀⠃⠐⠀⠀⠃⢸⣿⠀⠀⢸⣆⣀⣸⡇⢸⣧⣾⠇⠀⠀⢀⣀⣀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢉⠆⠺⠀⠆⡀⠠⡄⠀⠀⠀⠀⠱⡀⢀⠔⠀⢸⣿⠀⠀⢸⡇⠀⢸⡇⢸⡏⣹⡇⠀⠀⢸⠉⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠘⠃⠀⠘⠁⠘⠛⠛⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⢠⠀⠀⠀⠀⠂
⢀⠀⠐⠉⠀⠀⠀⡆⠀⠂⠀⠀⠀⢠⢀⠀⠀⣀⠀⠀⢸⡛⡟⣻⠻⢿⠛⢟⣿⢻⡇⠀⠀⠀⡄⠀⠆⠐⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠈
⠀⠉⠀⠀⠒⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⣷⣷⢾⡾⣾⡷⣷⡿⡾⡇⠀⠀⠀⠁⠈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠑⠀⠀⢸⠀⠠⠀⠀⠀⣀⠀⠀⠠⠀⠀⠐⢀⠀⠌⠐⢆⡃⠂⠀⠡⠠⢀⡐⠀⠀⠀⠀⠀⠠⡀⠠⠂
⠀⢀⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠆⠀⠀⠀⠰⠀⢀⣀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠵⠀⠀⠀⠀⠊⢔⠀⠀⠀⠀⠀⠙⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠂⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⠀⠀⠁⠀⠂⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠀⠀⠀⠂
⠀⠀⢰⡆⢰⡆⠀⠀⢠⡾⢷⡄⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⠐⠁⠊⠁⡐⠀⠀⠀⠉
⠀⠀⢸⣧⣼⡇⡶⣦⠸⣧⢸⡇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠕⠈⠀⠀⠀⠀⠀⠈
⠀⠀⢸⣿⢸⡇⢀⡿⠀⣿⣸⡇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⡄⣡⢠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠
⠀⠀⠈⠛⠘⠁⠙⠛⠀⠙⠋⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠿⠛⠉⠉⠩⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣀⣀⡀⠀⠀⣾⣿⣿⣦⠘⡏⣿⡀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣱⣿⣿⣷⠀⠀⠻⣿⣿⡿⠀⠃⣹⡇⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠻⠿⠃⢰⣤⡄⢨⣭⢀⡎⢐⣿⣷⠀⠀⠈⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣾⡇⠘⠛⠃⠀⢡⣿⣿⡿⢻⣿⣷⡀⠀⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠀⠀⠀⢰⠘⣿⣿⠇⣼⣿⣿⠟⠀⠸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡀⠉⠛⠀⠐⠉⢉⣾⠛⢃⠀⠀⠀⠀⠃⠉⠉⠑⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢿⣿⣿⣇⠀⢀⣀⣀⣴⣿⠃⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢀⣄⠉⠁⠙⠏⠻⢿⣿⣿⣷⣿⠃⡰⠈⠀⠀⢀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⠈⠁⠀⠀⠐⣄⠀⠉⠛⣿⣿⠀⠀⠀⠀⢠⠾⠄⠄⠠⠤⠄⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢀⠀⠐⠚⢿⡄⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⢸⠤⠤⠦⠤⢿⣄⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⢸⡇⠈⣠⣾⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠄⡠⣠⣤⣤⣘⣦⡀⠀⠀⠈⠀⠀⠀⠀⠀⠈⡓⣷⣿⠋⠻⣿⣿⣿⣿⣧⠀⠀⠀⠀⠈⠏
⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠘⠀⠀⠈⠀⢀⣝⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠁⣤⣷⣿⣆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢹⡎⠉⠉⠉⣠⣾⡛⠉⠂⠀⠀⠀⢄⠠⠀⠃⠁⠀⢠⠆⢰⠀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢳⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣀⢈⡇⠀⡐⠀⢠⡏⢀⡇⠀⣿⠈⠻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⢿⣿⣿⣿⠟⠉⠀⠀⠀⠀⡟⠿⠇⣶⠀⠀⠈⠀⠸⠀⠀⢿⣄⣀⣈⠹⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣀⠀⠀⠈⠛⠛⠧⠀⢀⠀⠀⠀⠀⠁⠀⠀⠸⡇⢸⣷⠀⣶⠂⣠⣾⣿⣱⣿⡆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⣀⡀⠀⠀⠀⢀⠠⠈⠀⠀⠀⠀⠀⠁⠈⠽⠀⡟⢰⠏⠀⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣦⣤⣄⠀⢸⠃⢀⡤⢐⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠛⠛⣛⣿⣿⣿⣷⣄⣰⣯⣾⣿⣛⣉⣙⡆⠀⠒⢤⡀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢒⡩⠒⠂⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠿⣦⣄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣴⣿⣿⣿⣿⡿⠟⣏⢈⠀⠙⢿⣿⣿⣿⣿⠻⠿⣿⣿⣶⣄⠙⢷
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠊⢩⣿⢛⣽⣿⡿⢻⣠⠞⡏⢸⠀⠀⠘⣿⣿⠹⣿⡄⠘⠾⡙⢿⣿⣷⣤⣄⣀
⠀⢇⠈⠉⠐⠒⠈⠁⠀⠀⠀⠀⠀⡟⢁⣾⣿⡿⢁⢸⠫⠔⠁⣀⣤⠀⠁⢸⠏⠄⡿⢘⣀⠘⢹⠀⢻⣿⣿⣆⠈⠙⠒⠤⣀⡀
⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⣿⣿⡇⠀⠃⠀⣴⡿⢟⣋⣁⡀⠀⠀⠀⠀⠹⣿⣷⣤⠀⠈⢻⣿⣿⣆⠀⠀⠀⠀⠉⠓⠤⣀⠀⠀⠀⠄
⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢠⡞⠍⠎⠂⠍⠀⠀⠀⠀⠀⠸⡹⠏⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢀⠠⡤⠀⣿⣿⣿⠇⠀⠀⠀⣾⠁⠀⠈⣈⣠⠄⠀⠠⣦⠀⠀⡇⠀⠀⠀⣿⣿⣿⠚⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⣼⠇⠀⠀⠀⠀⠠⣃⠎⡠⢶⡿⠿⣟⠀⠀⠀⠀⠸⣆⠀⠁⠀⠁⠀⣀⠀⠀⢀⡴⠁⠀⠀⣸⣿⣿⡟⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠁⡾⠀⢀⣃⠀⠀⠀⠀⠀⠀⠐⠬⣭⠶⠋⠀⠀⠈⣳⠾⠯⠔⠁⠀⠀⢻⡙⡛⡙⢳⠀⠷⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⣇
⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⣿⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⣿⡗⢹⢸⡇⠒⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣿
⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⢀⣼⠃⠀⢉⡼⠁⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣴⡿⣄⠀⠀⠀⠀⣀⣴⡶⠖⠲⠴⠚⠁⠀⠀⣀⣤⠶⣿⣥⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⡄⠠⠀⢀⣀⠤⢜⣫⠭⢑⣰⠋⢀⣀⡀⠀⢀⣤⣶⣾⣿⣿⠀⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢻⠃⠀⠀⠀⣼⠃
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠁⢀⡠⢴⣮⡥⠖⢊⡡⡴⠾⠛⠛⠛⠻⡟⠀⠀⡾⠁⠀⠉⠻⣿⠀⠀⠀⠈⠑⠲⢄⡀⠀⠀⠀⠀⢠⢣⠇⠀⠀⠀⢀⡿
⠀⢸⡇⠀⠀⠀⠀⠀⢀⠆⠱⠶⠞⣡⠴⠊⠉⠀⣳⠟⠈⣹⣷⣄⠃⢀⣎⣴⣿⠉⠓⠚⢃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣞⡞⠀⠀⠀⠀⣼⠇
⠀⠸⣷⠀⠀⠀⠀⠀⠀⠑⠒⠒⠉⠀⠀⠀⢠⠾⣫⠊⣴⣿⡟⢙⣧⣾⠁⢿⣿⡆⠀⣄⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⣸⡏
⠀⠀⣿⡄⠀⠀⠀⢀⣀⣀⡠⠒⠒⢄⠀⡔⠁⡼⢁⣼⣿⡿⠁⢸⣿⣿⠀⠸⣿⣿⡀⠘⣆⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⣰⡟
⠀⠀⠸⣧⠀⠀⢰⠁⠀⣼⠁⠀⢀⠀⠀⢌⣸⠁⣼⣿⣿⠇⠀⣾⣿⣿⠀⠀⢻⣿⣷⡀⢘⣇⠀⠀⢷⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣼⡟
⠀⠀⢀⣿⡦⠶⢾⠀⠀⢸⡄⠀⢸⠀⠀⢸⠉⠉⠉⠉⠙⠒⠦⢿⣿⡿⠶⠚⠋⠉⠉⠉⠉⠉⠙⠓⠚⠷⢤⣀⠀⠀⠀⠀⠃⠀⢀⣾⠏
⠠⠖⠉⠀⠀⠀⠀⠳⣄⡠⠷⢄⡼⠓⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠟⠁
⣿⠿⠿⠿⣿⢿⠿⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣻⣟⣿⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⣀⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴
⠀⠀⠀⠀⠘⠋⣡⡀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢈⣤⡀⠈⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⡼⠟⠿⣿⡀⠈⠑
⠀⠀⠀⠀⠀⠀⡛⠁⠀⠀⢠⣤⣤⡌⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠁⠈⠀⠢⡀⠀⠀⠀⠀⠀⠀⢧⠀⠀⣈⣥⡦
⠀⠀⠀⠀⢀⡀⡁⠀⠀⠠⠿⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⣄⠋⠁⠀⠀⠻
⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠀⠚⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡂⡂⠀⠀⠀⠀⠀⠀⠃⠻⠀⠀⠀⠀⠀⢰⣶



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣩⣙⣋⣍⣹⣟⣽⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣤⣶⣶⣿⣶⣶⣄⠀⠀⠀⠀⠀⢀⣤⢶⣶⣿⢶⣶⣤⡀⠀⠀⠀⠀⠀⣠⣴⣶⡾⠓⢶⣤⡀
⠀⠀⠀⣰⣿⡏⣻⣿⣿⡮⣿⣿⣧⠀⠀⠀⢰⣿⣿⣟⡿⣿⣿⣯⣿⣷⠀⠀⠀⢠⣾⡟⠋⢌⢤⣷⣾⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠿⠛⡃⠀⠀⠀⣿⣿⣿⣿⣧⣿⠿⠿⠟⠻⠀⠀⠀⣼⣿⡶⣀⣷⣿⠿⠿⠟⢻
⠀⠀⠀⢿⣿⡟⠛⠿⠁⠀⢠⠄⢡⠀⠀⠀⢻⣿⣿⠛⠻⠃⠀⢠⠄⠰⡀⠀⠀⢸⣿⣿⠛⠻⠏⠀⠀⡄⠘⡄
⠀⠀⠀⠘⢿⣿⡄⠀⠀⠀⠀⢀⡞⠀⠀⠀⠈⢿⣿⡆⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠻⣿⣦⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⡏⠀⠀⡤⠤⠠⠜⠀⠀⠀⠀⠀⠀⡉⠁⠀⢠⠤⠤⠴⠁⠀⠀⠀⠀⠀⢹⠁⠀⢠⠤⠄⠤⠃
⠀⠀⠀⣀⣴⣇⣀⣀⣷⣄⡀⠀⠀⠀⠀⠀⣀⣤⣇⣀⣀⣸⣄⡀⠀⠀⠀⠀⠀⢀⣠⣮⣀⣀⣸⣄⡀
⠀⠀⠚⠿⠿⠿⣿⠿⠿⠿⠛⠀⠀⠀⠀⠘⠿⠿⠿⣿⡿⠿⠿⠟⠂⠀⠀⠀⠐⠻⠿⠿⢿⡿⠿⠿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣰⣆⠀⢠⡀⠀⢀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠀⠀⠀⠀⢠⠀⠀⢠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⣸⣿⣿⣷⣼⣿⣸⣺⣾⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⢀⣼⣧⠀⣸⠀⠀⣾⣿⣿⣷⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣽⡿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡺⣿⡇⠀⡏⠀⢰⣿⣿⣿⡟⣇⣇⡀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⡇⠀⠼⣿⣿⣿⣿⣿⣿⣿⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⢆⣿⠅⢀⠀⠀⣸⣿⣷⣿⡇⣿⣿⠇
⠀⠀⠀⠀⠀⣸⡇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣼⣿⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡇⡀
⠀⠀⠀⠀⢠⡿⠲⡄⣀⣀⠀⢀⣤⣀⣀⡀⠀⣀⠤⠓⠒⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⡀⠀⣼⣿⣿⣿⣿⣿⣿⣿⢹⡇
⠀⠀⠀⠀⣽⣧⣤⡇⣧⣀⣾⣿⣿⣿⣿⣿⠀⡇⠀⠀⠀⠀⡀⣀⠀⠁⠀⠠⠄⣀⠀⠀⢘⣿⣿⣿⡇⠀⠁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣼⡇
⢀⣀⣤⡶⢿⣿⣿⣷⣿⣿⣿⡿⠛⠿⠿⣿⣧⣷⣤⡀⠀⠀⠈⠻⣷⣤⠀⠀⠀⠀⠉⠒⢸⣿⣿⣿⠇⠆⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠈⢱⡤⢤⣥⣭⣭⣽⡿⠉⢻⣷⣟⣦⣴⣹⣿⣿⣮⣳⣶⣶⡀⢁⠈⠙⢿⡧⡀⠀⠀⠀⠈⢿⣿⣿⣿⣾⠛⠋⠉⠁⠈⠈⢀⠈⠛⠛⢿⣿⡇
⠀⣼⡇⠀⡖⠒⢺⣿⣧⣯⡿⠛⠿⢿⣿⣿⣿⣿⣿⣿⡋⠛⣛⣛⣙⣷⣾⣿⣇⠀⠀⠀⠀⢸⣿⣿⣿⣟⠀⣠⣦⠼⠂⠙⢶⡷⢶⣒⣞⣉⣇
⢀⣿⡇⠀⣇⣀⣀⣉⣛⣉⣀⣀⣀⡀⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣄⠀⠀⠀⠈⣿⣿⣿⣿⣾⠟⠁⠀⠀⢠⣿⣿⣿⡿⡿⠿⣿
⢘⣿⡇⠀⠆⠀⠀⠐⠆⠀⠰⠀⠀⠰⣷⠖⠲⠞⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⠀⡄⣾⣿⣿⣳⣿⣿⣿⣦
⠀⣿⡇⠀⠀⠀⠰⣆⠂⠀⠀⠀⠀⠀⣯⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢠⣿⣿⣿⣿⢿⠂⠀⡇⠀⣿⣿⣷⣿⣿⣿⣿⡿
⣠⣿⡇⠀⡀⣀⢀⣀⣀⠀⠀⠀⠀⣘⣟⡀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⡄⠀⠀⢸⣿⣿⣿⣟⣸⡷⣾⡅⠃⣿⡟⣿⣿⣿⣿⣿⡇
⠟⣿⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⡇⠀⠀⢀⣿⣿⣿⣿⣹⡷⣏⣷⣴⣿⣽⣿⣿⣿⣿⣿⡇
⠈⣿⠃⠀⠃⠀⢀⡤⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⡇⡷⠀⠀⠘⣿⣿⢻⣿⣿⣃⣯⣿⣿⣿⣿⣿⣿⣿⣿⡿⡅
⠰⣿⡆⠀⡄⠀⠀⠉⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠀⠀⢰⣿⣿⣿⣷⣶⣶⣶⣦⣭⣽⢛⣛⣉⣽⣿⣷⡇
⠘⣿⡇⠀⠓⠐⠒⠒⠒⠒⠒⠂⠐⢲⣿⠒⠒⠆⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢉⣀⣠⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⡁
⢠⣿⡇⠀⣴⣶⣶⣦⣴⣤⣶⣶⣷⣿⣽⣷⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣼⣿⣿⡄
⠈⢿⣧⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⠉⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠿⢿⠻⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣇⣿⣾⣿⡿⣿⣿
⠀⠀⠈⠛⠉⠫⠿⢟⠻⠿⠿⡿⠿⠦⣤⣤⣾⣿⣿⣿⣿⣿⣟⣋⣱⣦⣴⡿⢿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⠿⢿⠿⢿⣶⣷⡀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣉⣈⣁⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣈⣉⣀⣀⣀⣀⣉⣉⣛⣛⣇⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠐⠲⠤⠤⠤⠤⢤⠤⠤⠴⣶⢶⢾⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠸⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡽⠦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣾⣟⣲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠁⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠉⠉⠉⠛⣛⣿⢿⣿⠑⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣙⡎⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠠⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⠿⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠹⠀⡆⡆⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣱⢳⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢼⣟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡆⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠦⠴⠶⠞⠳⠻⠛⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡖⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠤⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡇⢰⡆⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡼⠇⠸⣤⣄⣄⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣤⣤⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣮⠉⢧⠀⠀⡎⠘⢿⣿⠉⠉⢏⠉⠉⠉⠉⢩⠉⣴⣦⡶⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢀⣡⡀⣷⠀⣸⣿⡖⠀⢈⢄⠀⠀⠀⣸⢴⡿⠈⢅⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣿⡀⠉⠻⣿⡻⣿⡇⣠⢞⣺⣧⣤⡰⠣⠋⠁⡐⠕⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣄⠀⠈⠻⣾⣿⡿⠋⠁⠀⣨⡧⠁⠀⡰⠋⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣟⣷⠤⣠⣿⣿⣀⡠⣤⣾⢟⣷⣀⣠⣦⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠿⠿⠶⠾⠿⠛⠛⠛⠉


⡿⠿⠿⠿⡿⡿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣛⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⣿⣷⣶⣄⠀⠀⠀⠀⢀⡾⠋⠁⣤⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣏⡻⣿⣿⣿⣿⣿⡆⠀⣠⠴⠋⠀⠀⠀⠓⢤⡀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠑⢻⣿⣿⣿⣿⢿⠾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠐⢆⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠿⣿⣷⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⢀⣹⣻⣿⣷⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣟⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠈⠃⠉⠙⢻⣿⣿⣿⣟⢿⣿⡿⣿⣾⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠰⢦⣄⡀⠀⣠⡤⠂⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⢺⣾⡇⡀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠰⠇⠁⠑⠺⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣽⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣽⣿⢿⣿⣿⣿⣻⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠹⣿⢿⣿⣿⣹⣿⣿⣿⣿⣿⣿⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠻⠟⠀⠉⠨⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣿⠟⠋
⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢢⣴⣾⣿⣶⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠁⠀⢀⡆⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⡜⡡⠂⢀⠀⠀⣿⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⠘⠃⢠⠆⢀⠰⠈⠇⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠈⢧⡄⠀⢀⠘⠀⠀⠀⢄⢀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠀⢸⡇⠐⠀⠀⠀⠧⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⣧⠀⡄⠀⠐⠀⠣⡀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢃⠀⠂⠀⠀⠀⣠⣾⣤⢴⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠈⠓⠤⣾⠶⣿⠿⠉⢿⡄⠙⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠘⢦⡀⠀⢁⠀⠈⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠾⠷⠤⣦⣤⡴⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣶⡟
⣓⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣁⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠶⠶⠶⠒⠛⠛⠓⠲⠶⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⢛⣩⣤⣶⣶⣾⠛⠋⠉⠀⠀⠉⠙⢛⣿⣶⣶⣤⣍⣙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣋⣴⣾⣿⣿⡿⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠿⣿⣿⣿⣦⣍⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣡⣾⣿⣿⣿⣿⣿⣟⡋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣛⣿⣿⣿⣿⣿⣷⣦⡹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⣾⣿⣿⣿⣿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⣿⣿⣿⣿⣿⣌⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⣿⣿⣿⣿⣿⡿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⠄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣣⢰⡸⣿⣿⣿⣿⡯⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢽⣿⣿⣿⣿⡟⡄⣠⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣿⡆⣧⢿⣿⣗⠪⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠅⣺⣿⣿⢱⠃⣿⡜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿⣇⢸⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⡟⡟⢸⣿⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⣿⠀⣿⣿⣿⣿⣷⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣿⢿⣿⢿⡇⢸⡏⡄⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⣿⠀⣿⣞⠻⠿⠶⠯⣥⣤⣀⠀⢰⡏⣆⠄⠀⠀⡀⠀⠀⡄⣾⣆⠀⢀⣠⣤⡭⠷⠾⠿⣫⣿⠀⣿⡇⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⣿⠀⣿⡿⠟⠒⠀⠀⠀⠀⠠⣄⣼⡁⠽⢰⢸⣠⡇⣿⣦⠐⠇⣿⣠⡔⠀⠀⠀⠐⠒⠚⠿⣿⡀⣿⡏⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣰⣿⠠⢋⣶⠀⠀⣀⣀⠀⠉⠉⠙⠻⠷⣦⣾⣿⣿⣿⣿⣿⣶⠾⠟⠛⠉⠉⠁⣀⣀⠀⠀⣰⣌⠇⢻⣇⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⢰⣿⣧⣾⣿⣿⣿⣿⣿⣶⣄⡈⡑⢄⡹⣿⣿⣿⣿⡟⣡⢄⡁⣠⣴⣾⣿⣿⣿⣿⣿⣮⣻⣦⠘⣿⠃
⠀⢀⠀⡀⠀⠀⢀⠀⣀⠀⢠⣏⣀⢨⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣝⣷⣽⣿⣽⣿⣽⣯⣵⣫⣽⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣏⣀⡙⣦⣀⢀⠀⠀⠀⡀⢀⡀
⠀⠈⠈⠁⠈⠁⠈⠉⠉⠀⢹⣙⣿⡏⡛⢙⣿⣿⣿⣀⣘⣻⣿⣿⣿⣿⣿⣿⣋⠈⠉⢛⣿⣿⣿⣿⣿⣿⡛⠄⣘⣿⣿⣿⡟⠙⠋⣿⢇⣿⠁⠈⠈⠀⠀⠁⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣽⠇⣤⣰⡌⣿⣙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢰⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⡄⠛⠈⠲⠶⠿⢿⣛⣛⣿⣿⣟⣛⣣⣿⠏⣸⣿⣿⣷⢘⣿⣮⣛⣛⡿⣿⣟⣛⡻⠿⠷⠖⡃⠘⢁⡀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⣿⣿⠀⠀⢽⣿⣭⡁⠀⠉⠉⢻⣿⣿⣿⣿⢃⣿⣿⣿⣿⡎⢿⣿⣿⣿⣿⠋⠉⠁⠀⣭⣿⣿⠇⠀⢸⣿⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠈⠁⠀⠀⠀⠈⠉⠉⠉⣿⢇⣾⣿⣿⣿⣿⣿⡌⣿⡍⠉⠉⠁⠀⠀⠀⠈⠉⠀⠀⠀⣹⣾⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⡷⣦⣦⣤⣶⣲⠦⢆⣀⡀⠀⠀⢸⡜⣿⣿⣿⢿⣿⣿⢇⡟⠀⠀⠀⣀⡐⡤⣶⡶⣤⣤⣶⢾⣻⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣽⣻⡿⣿⣿⣛⣥⣾⣿⣿⣿⣷⣆⠈⠙⠘⠿⢻⡟⠿⠏⠚⠁⢠⣾⣿⣿⣿⣿⣮⣛⣻⣿⡿⣿⣻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⢿⠉⠉⠁⣿⢿⣿⢽⡇⠀⠀⢀⠘⠇⡀⠀⠀⢠⣿⣿⡿⣿⡇⠛⢻⣿⢿⣿⠿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⢀⠀⠀⠁⢹⡿⣷⣼⣷⣆⣼⣬⡀⠀⣧⣹⣤⣼⣿⣴⣿⣿⠇⠀⠈⠈⠀⠀⠀⣤⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠸⣿⣿⡏⢿⢻⠙⢿⢳⡶⢻⠋⡟⢹⠙⣿⣿⣿⠀⠀⠀⢸⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣾⣼⣆⡘⠀⡇⢸⣀⣧⣼⡾⠻⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⣿⣿⣾⣿⣿⠟⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡆⠀⠉⠋⠘⠉⠀⠀⢀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠛⠛⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠦⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢀⡟⠛⠶⣤⣄⣀⣀⣀⣠⣤⠴⠞⠉⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢀⣠⡴⠋⠀⠀⠀⠀⠀⠈⠉⠉⠳⣤⣀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠉⢻⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠋⠙⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⣸⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢸⠄⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⣄⣼⠖⠋⠉⠉⢉⣀⢀⡤⣄⠀⠈⠉⠉⠉⠓⢾⣄⣤⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⢀⡟⠉⡿⠀⣼⠶⣄⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣸⠃⢸⠇⢠⡏⠀⣟⡀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠷⣦⠀⢀⡟⠀⡿⠀⣼⠁⢸⠏⣹⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⡆⣸⠁⢸⠇⢠⡇⠀⡞⠀⣿⡴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⣧⡏⠀⠀⠀⠈⠁⠸⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠳⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠖⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣠⠟⠉⠉⠛⠓⠶⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢈⡗⠀⠀⠀⠀⠀⠀⣀⡼⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧



⣇⢸⠆⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠂⠒⠒⠒⠒⠒⠒⠒⠒⠂⠂⠀⠐⠒⠒⠒⠒⠒⠒⠐⠂⢾⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢨⡅⠀⡇⠈⡆⡇⠈⠁⢠⢻⡀⡧⢼⢸⠀⠀⡇⠀⠀⠪⣄⠀⡇⠀⣅⡕⣸⠀⢱⢈⣖⠁⠬
⣤⣤⣤⣤⣤⣤⣤⣤⣶⣦⣷⣶⣥⣧⣤⣤⣦⣤⣥⣷⣮⣼⣶⣤⣷⣦⣤⣴⣮⣤⣥⣤⣤⣼⣬⣶⣮⣤⣤⣥⣴⣦⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⡷⠀⠀⠀⠙⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡄⠀⣠⣿⣿⣿⣿⣿⣥⠀⠀⠀⠀⠈⢿⣿⣿⣿⣆
⠀⠀⠀⠀⢸⡿⠿⠿⠿⠿⠿⣿⠀⣿⠿⠿⠿⠿⠿⣿⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣤
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠄⢸⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⡅
⠀⠀⠀⠀⢸⣷⣶⣶⣶⣶⣶⣿⠀⣿⣶⣶⣶⣶⣶⣾⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⢻⣿⣆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⣀
⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⡿⠟⠀⠻⢿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡿⢤⢤⣀⣀⣀⣀⣀⣀⣀⣠⡽⡇
⠀⠀⠀⠀⠀⠀⣀⠴⠚⠙⢧⡀⠻⣤⣀⣀⠀⠀⠀⠁⢈⣀⣀⡴⠃
⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠙⣆⡧⢀⣈⡉⠉⠉⠉⠉⠁⠀⣶⣛⣉⢉⣙⣦
⠀⠀⠀⢠⣶⠋⠙⢫⡳⡀⠀⠀⠘⡇⠀⠀⢩⢭⠉⢹⠀⠀⢸⡷⠤⠤⠤⠤⢾⣀
⠀⠀⠀⠇⡇⠀⣠⠤⣷⢹⠀⠀⠀⣷⠀⠀⠘⡫⠆⢸⠀⠀⢸⠀⠀⠀⠀⠀⠈⡇⠉⠓⠢⢄⡀
⠀⠀⠀⡇⢧⡠⠗⣶⣹⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⣾⠑⠒⠲⠶⠒⠊⡇⠀⠀⢀⠔⠉⡆
⠀⠀⠀⠘⢾⣤⣞⣡⡧⠏⠀⠀⠀⢸⠉⠒⠦⠤⠴⠚⠀⠀⢻⠀⠀⠀⠀⠀⠀⡇⡌⡀⡜⢰⠁⢩⣦
⠀⠀⠀⠀⠀⠈⢉⣈⣱⡦⠤⠤⠴⠚⠓⠤⣄⣀⣀⣀⡤⠔⢻⡄⣀⣀⢀⣀⣠⠏⠙⠓⠷⠞⠛⠛⠉⠑⠢⢄⡀
⠀⠀⡠⠤⢤⠴⢅⣀⡤⠟⠀⠀⠀⠀⠀⡴⠶⠶⣆⠀⠀⠀⠀⠉⠛⠛⠚⠛⠁⠀⢠⠤⠖⠢⠤⢤⡀⠀⠀⠀⠀⠁
⠀⠀⠑⠲⠦⠤⠤⠃⠀⠀⠀⡆⢉⠉⣹⡯⠭⢭⡿⠶⡉⣹⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠄⣤⡤⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠭⠭⠟⠁⠀⣎⠀⣀⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⠟⣋⣤⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⢁⣶⣶⣦⣡⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠟⢃⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣦⡙⠻⢿⣿⣿⣿⣿⣿⣿⡿⢡⣶⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⠈⠉⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⢋⣩⣤⣤⣤⣍⣁⠀⠀⠀⠀⠀⠀⠈⢿⣟⠻⣿⣿⣦⡈⢻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠈⢿⣧⣙⣿⡿⢿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢰⣿⣿⣿⡿⢋⣥⣤⣌⠻⣿⣿⣧⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣆⠹⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⣾⣿⣿⣿⠆⣿⣿⣿⡆⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⣼⣿⣿⣿⡟⣰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⣸⣿⣿⣿⡟⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢰⣿⣿⣿⣿⠁⢿⣿⣿⣿⣿⠃⠀⠀⣀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⣿⣿⣿⣿⣿⣿⣶⣤⣉⡛⢿⠀⠀⠘⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣌⡛⠻⢿⣿⣿⣿⣿⣿⣿⠄⠀⠀⣴⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⣤⣉⡛⠿⠿⠋⡀⠀⢀⡉⠛⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡇⠀⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠙⠿⠃

⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⣀⣀⣀⣀⣀⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡶⢶⣶⡶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠛⠛⠛⠛⠛⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣶⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠛⠛⠛⠛⠛⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⠛⠛⠛⠛⠛⠛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠛⠛⠛⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠿⠿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣬⣤⣤⣬⣥⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣉⣉⣉⣩⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣉⣉⣉⣩⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣉⣉⣉⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣶⣶⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⣀⠋⠛⠛⠋⣛⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣤⣤⣤⣼⣿⣿⡀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⣻⣿⡿
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣤⣤⣤⣤⣤⣤⣿⣿⡇⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣤⣤⣤⣤⣿⣿⡇⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⣿⣿⡅
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣤⣤⣤⣤⣽⣿⡇⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠉⠉⠉⠉⠉⣽⣿⡇⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⣄⠀⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠐⣧⠀⠀⠀⠀⢀⣠⣶⣿⠿⠛⠛⠉⠉⠉⠉⠉⠉⠛⠻⠿⣿⣶⣄⡀⠀⠀⠇⢰⡌⢧⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠉⡠⠀⣠⣶⣿⠟⠁⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⡄⠀⠀⠀⠙⠻⣿⣦⣄⠱⠴⠢⠌⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⡜⠁⠈⠉⣻⣿⣿⣦⡀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⢰⣷⣤⣼⡿⢛⣿⣿⣿⣿⣦⣄⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⢀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣟⣧⣾⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠘⢿⣷⡀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠿⢋⣻⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠘⣿⣇⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠸⣿⡇⠸⢿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣦⠘⠃⢻⡄⢠⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿⡆⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⢸⣿⠋⠀⠀⠀⠀⠀⠄⠰⠀⠈⠷⠁⠀⠀⠈⠁⣽⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢙⣿⡇⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⢐⣿⡇⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠐⡀⠘⠇⠈⢀⠀⠀⣀⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠨⣿⡇⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⢠⣀⣀⢀⠀⠑⢄⠉⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢐⣿⣧⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⡾⠞⣧⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡟⠀⣠⠝⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⢀⡟⠀⠀⢈⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠐⣿⣿⠀⠈⣿⣿⣿
⣿⣿⣿⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⣾⣷⣄⠀⠀⠂⠀⠀⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⠃⠀⠀⠀⠨⣿⣿⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⢸⣿⠄⠀⠀⠀⢸⣿⣿⣿⣦⣀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣿⣿⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⢸⣿⠅⠀⠀⠀⠘⣿⣿⣿⡿⣿⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠠⣿⣿⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⢸⣿⡂⠀⠀⠀⠀⠈⣿⣿⣿⣿⠱⡀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡿⠀⠀⠀⠀⠈⣿⣿⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡿⠁⠀⠀⠀⠀⢐⣿⣿⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⢸⣿⣷⡶⣶⣦⠀⠀⣹⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⠀⠀⠀⢰⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⢸⣿⣿⣧⣾⡟⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⣦⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠴⠖⠒⠒⠲⠦⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⣀⣤⠤⠀⠀⠀⢀⡄⠀⢀⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⢠⠖⠃⠀⠀⢸⠀⠀⠀⠀⠀⢳⠴⠃⡔⠉⢢⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⣠⠤⣄⠸⡖⠁⡀⠀⠸⠀⠀⠀⠀⠀⠘⠀⠸⣄⣠⢎⡞⠀⢀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⢀⡞⠁⢰⠁⢀⡤⡄⠳⠊⠀⣀⡤⠴⠒⠒⠒⠒⠦⢤⣀⠀⠀⠸⣤⠴⢋⡴⢦⠈⢳⡀
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠸⣄⣀⡴⠃⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠰⢛⠗⠉⠀⠀⢳⡀
⠀⠀⠀⠀⠀⣼⠀⠀⢀⠄⠀⠁⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠸⡀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⢀⡇⠀⠀⡼⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡶⠋⢶⡄⠀⢱⡀⠀⠀⠘⡆⠀⠀⢸⡀
⠀⠀⠀⠀⢸⠁⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⠴⢏⣀⠜⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⢻⠀⠀⠈⡇
⠀⠀⠀⠀⢸⡀⠀⠀⢳⠀⠀⠀⢸⠀⠀⠀⠀⠀⣴⠚⢁⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⢀⡇
⠀⠀⠀⠀⠈⡇⠀⠀⠘⡆⠀⠀⠈⢇⠀⠐⣶⣾⠛⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⡞⠀⠀⢸⠁
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠘⠆⠀⠀⠘⢦⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⢀⡄⠐⠁⠀⠀⡞
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢀⡀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⢀⡐⠛⢦⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠉⢨⠷⠄⠀⠉⠑⠲⠤⠤⠤⠤⠖⠊⠉⠀⢀⢰⡃⠙⡆⠈⠃⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⡰⠓⠂⣠⠏⠀⢠⠀⠀⡀⠀⢀⡤⠆⠀⣆⡸⣆⠳⠤⠃⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠰⣅⡄⠀⡇⠀⡸⠁⠀⠘⠒⠒⣦⠘⠄⠈⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠓⠚⠁⠀⠀⠐⠒⠋⠁⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠦⠤⠤⠴⠖⠒⠛⠉⠁