Soccer Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⡶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⢀⡠⠔⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠒⠦⢤⡀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⣠⠞⠁⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠈⠳⣄
⠀⠀⠀⢀⡴⠁⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣄⣈⢦⡀
⠀⠀⢀⣾⣶⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡄
⠀⢀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡀
⠀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷
⢸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⣿⣿⣿⣿
⣾⣿⣿⣿⣿⣿⠟⠉⠓⠲⠤⣤⣀⡀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⡤⠴⠚⠉⠁⠀⠈⢻⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿
⡏⠻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠋
⣷⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⡀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢀
⠀⢧⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠿⠿⠿⠿⠿⠿⠿⠿⠿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⡼
⠀⠈⣆⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣰⠃
⠀⠀⠘⢆⠀⠘⣦⣀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⣀⣤⣶⣿⠀⡰⠃
⠀⠀⠀⠈⢳⡀⣿⣿⣿⣶⣤⣤⣤⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣶⣾⣿⣿⣿⣿⣿⡞⠁
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⡶⠤⣤⣤⣤⣤⣤⣤⠤⠤⠾⢿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠧⢤⣤⣀⣀⣀⣀⣀⣀⣤⠤⠤⠒⠊⠉
⣿⣿⠿⢿⢻⠿⡿⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⠒⠒⠂⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠿⠿⠿⠇⡀⠀⠀⠀⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⢹⠟⠉⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⢈⠷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣏⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢀⣴⣿⣷⣦⡀⠀⠀⠀⠀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡗⠤⣴⣿⣿⣿⣿⣿⣿⣶⠒⠊⠉⠀⠀⠀⠉⠻⣿⣿⣿⣿⡿⠛⠓⢄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠟⠁⠀⠀⠀⠀⢳⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢸⠟⠛⠛⠛⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢸⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⡀⢠⣾⡀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠰⡶⡶⠶⣷⢾⡿⡿⢦⡀⠠⠖⣫⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣾⣿⠗⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⣒⡒⠒⠙⠛⠿⠿⠿⣿⣟⠥⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢛⣛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣴⣟⣭⣤⡴⠚⠛⠛⠿⣿⢦⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢋⠱⠾⠟⣫⣭⣶⣤⣀⢈⡙⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣰⠏⢰⣿⣿⣿⣧⣀⠀⠀⠀⠘⡆⠻⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⣴⡏⠀⠀⠀⠘⢿⣿⣿⣿⣧⢻⣎⢻⣿⣿⣿⣿
⠀⠀⠀⢠⣏⡴⠉⠛⣿⠋⠀⠈⠙⠒⢶⣾⣿⣦⣻⡀⠀⠀⠀⢸⣿⣿⣿⡟⢸⢫⣶⣤⢀⣴⣿⣷⣶⣭⡛⠁⠀⠻⠄⣿⣿⣿⣿
⠀⠀⠀⢸⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣿⣿⡿⢹⡇⠀⠀⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⡇⠀⠀⢠⡆⣹⣿⣿⣿
⠀⠀⠀⠘⣿⣇⠀⠀⢸⣦⣀⠀⣠⠖⠉⠉⢹⡇⣼⠁⠀⠀⠀⢸⣿⣿⣿⣧⠀⢻⣿⣿⡇⠻⢿⣿⢟⣩⣾⣷⣆⣾⢃⣿⣿⣿⣿
⠀⠀⠀⠀⠙⣮⠑⢶⣿⣿⣿⡿⠁⠀⠀⢀⣼⣴⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⠡⣬⡍⠀⠀⠀⢰⣿⣿⣿⣿⠇⢏⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠳⢦⣹⡿⠛⠓⠶⢤⣶⡿⠟⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄⡛⠆⣠⣤⣬⣟⡛⠋⢁⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⣁⡈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠐⠿⠇⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡟⠁⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣋⣤⣴⣿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆⠀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠉⠉⠉⢹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓
⠀⠀⠀⠀⠀⠀⢀⣾⠟
⠀⠀⠀⠀⣠⠾⠿⠋
⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠀⠀⠀⠀⠀⣠⣴⡿⠋⠁⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⣠⣾⡿⠁⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠇⠀⠀⢀⣴⣿⠇⠀⠀⠀⠀⢀⣼⣿⡟⠀⢀⣤⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⠏⠀⢀⣴⣿⣿⣿⣦⡀⠀⣀⣴⣿⣿⡿⠀⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠛⠋⠉⠙⢿⣿⣿⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠛⠋⠉⠁⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⡤⠶⠶⢿⡟⠉⠒⠒⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠟⠋⠀⣀⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠟⠉⢀⣠⣶⠟⠉
⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠀⠀⠀⢀⣷⣤⣤⣀⡀⢀⠔⠛⣆⠀⠀⠀⢀⣼⣿⡿⠋⠀⠀⢠⣿⣿⡏
⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡇⠀⠀⠈⢆⠀⢀⣾⣿⡏⠀⠀⠀⠀⠘⠛⠋
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠓⠒⠒⠦⠴⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠘⡄⠘⣿⣿⣷⣤⣤⣾⡆
⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠿⠳⢤⡀⠀⢀⠃⠀⠈⠛⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⡍⢧⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠁⠀⠀⠀⠀⠙⢦⡾⠀⠀⠀⠀⣸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢳⡈⢆⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣠⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠑⢌⣶⣤⣤⣤⣤⣾⡀⠀⠀⠀⠀⠀⠀⢀⡼⢁⣠⠴⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣦⡀⠀⣀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠶⠶⠶⢶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠉⠁⠀⠀⠀⠲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠂⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠟⠋⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⣀⣀⣀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣷⣦⣄⡀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⣷⡎⠀⠀⠀⠈⠻⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⠿⠛⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣀⣾⣿⣷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢀⣀⡴⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠿⠿⠶⠒⠚⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠤⠰⠀⣰⠆⢀⡴⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠃⢤⣠⣤⣄⣀⣀⣤⣈⣴⣿⢆⣯⣵⣴⣦⡄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠁⠈⠥⢀⣂⣴⣇⣾⡀⠸⢿⣿⣿⣿⣿⣷⣠⣄⣩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠟⠋⠁⣤⣶⣶⣶⣶⣤⣭⡛⠿⣿⣿⣿⠤⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⡆⠀⠀⠀⢨⣿⣿⣿⡟⠋⠀⠀⠀⣠⣽⣙⣛⡛⠿⠟⠛⠛⠂⢮⡙⢿⢷⣀⠾⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢦⣽⣿⢫⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠙⢆⠡⣻⣄⣤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠀⢸⣾⣿⠇⣼⣿⣟⢺⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢀⣤⢻⣿⡇⢶⣄⡄⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣸⠿⢿⡟⣰⣿⣿⣿⡈⣿⣿⣿⣿⣿⣿⡿⢋⣥⣤⣀⡀⠀⢠⣿⣿⡆⣯⢽⣾⣿⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⣿⣿⣿⣿⣿⡿⠃⠛⠿⠿⣿⣿⢟⣴⣿⣿⣿⣿⣿⣿⢹⣿⣿⡇⢹⣮⡿⠅⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⣦⡄⠁⠞⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⡇⣾⣯⡄⢀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢈⡁⠀⠘⣿⣯⢴⣴⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠃⠛⠿⢰⣿⣿⢏⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣼⠟⠉⠀⠒⢩⣿⣧⠻⣿⣦⠀⠀⠀⠀⠀⣀⣝⡻⠿⠿⢿⠿⠉⠀⠀⠀⠠⡟⢉⡘⠎⡉⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢻⠷⡝⠿⣿⡆⣤⣶⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⢀⣴⣧⣉⣀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⢁⡀⠀⠀⠨⣰⣾⣷⣬⡑⠸⠿⣿⣿⣿⡿⠟⠩⠵⢂⣠⣶⣿⡯⠈⠚⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠈⠉⠀⠘⠀⠈⠼⣿⣿⣿⣿⣶⣶⣦⣤⣴⣶⣶⣾⣿⠟⠿⢿⠀⢃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣀⠀⠀⠀⠘⠃⠠⠿⢿⣿⣿⢿⡿⣿⠿⣿⣽⣷⡁⠀⠿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠁⠀⠈⠛⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠁⠀⠁⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠡⠅⣀⠀⠀⠀⠀⢀⡀⠡⠃⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠈⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠀⠀⠀⠀⠀⣤⡄⣤⠖⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⢀⣀⣠⣤⣤⠄⠀⡐⠄⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄⣀⠔⠈⠲⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠈⠺⠓⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠉⠁⣼⠏⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⢀⠤⢚⠶⠶⢯⣤⡒⣦⠶⠶⠤⠤⠤⡤⠤⠀⠐⠖⠒⠉⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⣠⠀⢠⠁⠀⢰⢡⠁⠝⠁⠁⠀⠀⠀⢸⣅⠀⠀⣀⡀⠤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠰⠋⠀⣿⣟⠃⠆⡃⠀⠤⡄⢀⣀⣀⣀⠈⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠇⠀⠀⠀⢸⣷⢈⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⣀⡀⠀⠀⠂
⠀⠀⠀⠀⡇⠀⡟⠷⠟⢹⢸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠲⠶⣄
⠀⠀⠀⠀⠁⠀⠃⠀⠀⢸⢈⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢰⠋⠳⠶⡞⠘⣆⣳
⠀⠀⠀⠀⠠⠸⠐⠀⠀⠈⣸⠀⠈⣀⠀⠀⡀⠠⠀⠀⠦⠄⠒⠋⠁⠀⠀⠙⡀⠀⠀⠀⣾⢹⠾⡎⢉⡟⣏⢙⡇
⠀⠀⠀⠀⢾⣤⡤⣤⣤⣤⣿⣴⣶⡿⠛⠇⠀⠀⠀⠀⠀⠈⠆⠀⢀⡀⠀⠀⢁⠀⠀⠀⠘⢾⡃⢸⠋⠓⣧⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠐⠨⡇⠀⠀⢸⡆⠀⠀⠀⠀⠉⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡨⠷⠂⠀⠀⠱⢤⣴⢻⠘⢦⣀⡀⣠⣎⠙⢴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡀⢀⡀⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⡄⠀⢛⣿⠗⠈⣴⠟⠀⠀⣀⢤⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⡀
⠀⠀⠀⠀⠀⠈⠀⢤⡄⠀⠀⠀⠀⠀⠉⣨⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡗⠟⠁⣠⠞⠁⡀⠐⠉⠀⠀⠉⠄⣀⣀⠀⢀⣀⠁⠀⠊⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠡⣤⡀⠀⠀⣠⠞⠁⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠊⠁⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠴⢏⠁⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⢀⡤⠖⠲⢦⡀⠀⠰⠂⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠇⠰⢯⠔⠛⠛⡆⣱⠀⠀⠀⠀⠂⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⣠⡼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢡⠃⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡅⠀⢻⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⣊⣥⢖⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣤⢞⠽⣦⣀⣀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸⠦⠬⠭⠭⠭⠭⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠈⠙⠁⢀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣀⢤⡤⠖⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢤⠤⠤⠤⠠⡄⠐⠶⢶⡲⠶⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠐⠂⠀⠀⠒⠀⢤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠃⠀⠤⢀⣀⣤⣤⣦⣀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠁⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣧⡀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠁⠁⠲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠄⠊⠀⠀⠉⠙⠛⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠰⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠘⢿⣿⣿⡿⠟⠂⠄⠀⠀⢀⣴⣿⣿⣿⡿⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠹⠁⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢤⣤⣀⠀⠀⠀⠀⢀⠸⠛⢛⠩⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠖⠈⠀⠒⠈




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⠶⠿⠿⠿⠿⠿⠿⠿⠿⢶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⠟⣻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡙⠛⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠈⠛⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠉⠀⢀⣠⣴⡾⠿⠿⠷⣶⣶⣤⣄⣀⠀⠀⠀⠀⢀⣠⣤⣶⡶⠿⠟⠛⠻⠿⣶⣤⣀⠀⠈⠙⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⢀⣤⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⢷⣶⡾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣦⣄⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠈⢻⣧⡀
⠀⠀⠀⠀⢀⣼⡟⢁⣀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣶⣶⣽⣿⣄
⠀⠀⠀⢀⣾⣿⡿⠟⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠉⠛⢿⣦
⠀⠀⢠⣿⠟⠁⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⢻⣧
⠀⢀⣾⠏⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⢿⣇
⠀⣼⡟⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⣿⡄
⢠⣿⠁⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠋⠀⠀⠀⠀⠙⠻⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣇⠀⠀⠀⠀⠀⠀⢹⣧
⢸⡟⠀⠀⠀⠀⠀⠀⣴⡿⠿⢶⣶⣤⣄⣀⠀⠀⠀⠀⠀⣠⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣤⡀⠀⠀⣀⣤⣴⣶⠿⠛⠋⠉⠻⣷⣄⠀⠀⠀⠀⠈⣿
⣿⡇⠀⠀⠀⠀⢠⣾⠟⠀⠀⠀⠀⠉⠙⠛⠿⢷⣶⣤⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⢀⣿
⣿⣿⡀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⢀⣼⣿
⣿⡿⣷⣄⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠋⣿
⢿⡇⠈⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⣿
⢸⣷⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⣿
⠈⣿⡄⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣼⡏
⠀⢹⣷⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⢠⣿⠁
⠀⠀⢿⣇⠀⠈⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠉⠁⠉⠀⠀⠀⠀⠉⠁⠀⠀⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⢀⣿⠇
⠀⠀⠈⢿⣆⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠁⢀⣾⠏
⠀⠀⠀⠈⢻⣧⠀⠈⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⢀⣀⣤⣶⠿⠛⢹⣿⢠⣿⠋
⠀⠀⠀⠀⠀⠹⣷⣔⣿⠀⠉⠛⠿⢷⣶⣤⣤⣤⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠾⠿⠛⠋⠉⠀⠀⠀⢸⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆⠀⠀⠀⠀⠀⠈⠉⠉⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⡀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⣀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣤⣀⡀⠀⠀⠀⣙⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣿⣯⣄⣀⣀⣠⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣟⣋⡁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⣀⣀⣤⣽⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠟⠛⠿⠿⠿⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠈⣉⠛⠁⠀⠀⠀⠀⠄⠀⠐⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠈⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠐⠠⠀⠀⡀⣀⡤⠊⠀⠀⠀⠀⠀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠐⠀⠀⠂
⠀⠀⠀⠀⠀⠁⢲⡆⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁⠀⠁⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢨⣥⠀⠀⠀⠀⠀⠀⠀⢠⣵⣄⠀⠀⢴⣿⣿⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⣶⠀⠀⠤⠤⠖⠊⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸⠛⠋⠀⠀⠀⠉⠁⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢿⣄⣀⣀⣤⣤⠶⠶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⣤⣶⡀⠀⠠⡿⠁
⠀⠀⠀⠀⡠⠀⠀⠀⠐⠀⠣⣀⠉⠉⠁⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠍⠹⠤⠄⠊
⠀⠀⠀⠔⠀⠐⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠁⠀⠐⠀⠀⠀⡁⠤⠀⠉⢉⠝
⠀⠀⠜⠀⢀⠔⢀⠀⠀⠀⠐⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠁
⠀⣌⣀⠔⠀⠀⠀⢠⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠊⠁⠀⠀⠀⠀⠠⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣟⣂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣦
⠀⠀⢠⣶⣤⣀⣀⣤⣴⣿⣮⣿⣛⣿⣷⣾⣶⣶⣤⣤⣶⣤⣤⡀⠀⠀⠀⠀⣴⣿⣿⣷⣯⣯
⠀⠀⣸⢯⣾⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠹⠄⠀⠀⠀⠀⡙⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⣫⠉⠱⠆⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⠈⢿⣿⣿⣿⣿⣿⣦⡶⡦⠀⢀⠴⠛⠻⠟⠛⠙⠨⡑⠢⡀
⠀⠀⠀⠙⠛⢋⡍⠉⠰⡆⠴⠶⠋⠉⠉⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⢿⣿⣿⣿⣿⢾⣿⠀⡴⠁⠀⠀⡂⠀⠀⠀⠀⢸⣷⣾⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⣠⣦⣤⣀⠠⠐⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣾⣿⣿⢰⠓⠠⣴⣿⣿⣶⡤⠤⠐⠛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠄⠠⢾⣿⣿⣿⡏⠀⠀⠀⠈⡉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣆⠀⠀⢹⣿⣿⣿⠃⠀⠀⠀⢈⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢻⠿⠛⠓⠤⡀⠀⣠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠸⣿⣿⣿⡟⢄⡀⠜⠉⠉⠉⢄⠀⠀⢀⡐⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠆⠀⠀⠀⠀⢸⣿⣿⣧⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⢻⣿⣷⡌⠻⣦⡀⠀⠀⢀⣿⣿⣿⡷⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠊⣶⣿⣧⣄⡀⢀⠠⠾⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠙⢿⣧⠀⠈⠉⠓⠂⠬⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⠁⠀⠀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄
⠀⠀⠀⣀⠄⠂⠒⠈⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⡸⢆⡙⠙⠒⠤⠤⠀⢀⣀⣀⡀⢀⣀
⠀⠀⣼⠁⠀⠀⠈⢙⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢠⠇⠀⣈⠁⠀⠀⠁⢈⠋⠉⠉⠉⠁⠈⠉⠀⠀⢉⣸⠄
⠀⠀⡸⢧⠀⠀⠀⠈⡀⠣⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⢀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⣀⣀⣄⡠⡶⢿⠛⠜
⠀⠀⠈⠉⣧⡀⠀⠀⢣⠀⠈⠗⣌⠰⠀⡠⠞⠉⠙⠋⠉⠉⠒⢌⠒⠄⠀⠀⠀⠀⠘⣇⠀⠀⠀⣀⣀⣤⡤⠤⠖⠒⠋⢻⠹⠉⠓⠁
⠀⠀⠀⠈⠒⢳⡄⠀⠘⣀⠈⡀⢸⠁⡜⠁⠀⢀⣄⠀⠀⠀⠀⢸⣿⣾⡆⠀⠀⠀⠀⠈⠛⠙⠭⠉⠉⢒⠕⠚⡋⠁⠀⠉⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠁⠡⢘⠰⠋⠒⣶⣿⣿⣷⡦⠒⠀⠙⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⣠⣶⣶⣤⡄⠂⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⢇⢏⠀⠀⢹⣿⣿⣿⠁⠀⠀⠀⠀⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠒⣿⣿⣿⣿⠃⠀⠀⠀⢸⢡
⠀⠀⠀⠀⠀⠀⠀⠑⠚⣦⠀⠀⠀⣾⢢⣀⠎⠀⠀⠈⠢⡀⢀⣀⡌⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⠟⠛⠉⠑⠄⢀⣠⣼⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⢷⡄⠀⠈⡇⠙⢷⣄⠀⠀⢀⣾⣿⣿⡿⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡣⡀⠀⣰⠀⠀⠀⠀⠀⣿⣿⣿⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⣧⡀⢻⠀⠀⠈⠉⠑⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢼⣿⣿⣦⣄⡀⠤⠘⠻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠧⠀⠤⠐⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠻⢿⣿⣿⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⡶⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡼⠿⢿⣿⠿⠿⠿⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⠖⠀⠀⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⣠⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⠂
⠀⠀⠸⠏⠙⠋⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⢀⠀⣀⡀⠀⠀⠀⠀⠋⠀⠀⠀⠄⠤⢄⠀⠀⠀⠘⣀⡀
⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢀⣶⣶⣶⣶⡄⠁⠀⠁⠀⠐⣰⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⢄⠀⣴⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⢶⣶⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠇⠀⣀⠀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⡹⠟⠐⠁⠀⠀⠤⠤
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠀⠀⠈⠙⠿⠶⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠉⣀⣀⣄⡤⠶⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠶⣯⢥⣤⣴⡾⠿⣿⢿⣿⣶⡄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡿⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣲⣦⢈⣶⣟⠉⠅⠙⣶⠢⢤⣼⣿⣿⣇⠀⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⠺⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠫⣠⣿⣿⣿⢷⡤⣷⣄⣸⣷⣾⣿⣿⣿⡄⣿⣎⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⡸⣰⣿⣿⣿⣿⣿⣿⣾⣿⣾⣿⣿⣿⣿⣿⣧⠘⠚⠛⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣴⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠤⢀⡀⠀⠉⠙⠳⢶⣿⣦⣀⡴⠖⢋⣈⣭⣤⣾⣮⣃⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⡀⠂⠀⠀⠀⠀⠀⠉⠻⣶⢆⣀⡀⠀⣩⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠱⠆
⠀⠀⠀⠀⠀⠀⡠⢀⠴⠛⢳⣯⣝⡻⢿⣿⣿⡿⠟⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠹⡆⢻⣿⣿⣿⣿⣿⣷⢿⣷⣿⣿⣿⣿⣿⣦⡀⠀⠈⣦
⠀⠠⠀⢀⡴⢊⡴⠉⠀⠀⠀⠀⠀⠁⠙⠢⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⢣⣬⢣⡀
⠀⢀⣤⡾⡁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣿⣿⣿⣿⣿⣽⣟⣛⡿⣿⣿⣿⣿⣿⡿⠿⣷⣦⣶⣡
⣰⣯⣛⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⡿⢉⠳⢦⣍⡛⢼
⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿⠃⠉⠀⠠⠙⣻⠊
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⠁⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⣿⣿⣿⡄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣤⣀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⣦⣠⣴⠟⠉⠀⠠⠾⠂⠡⠤⠀⠀⠉⠙⠻⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠦
⢿⣿⣿⣇⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣶⣦⣄⣁⠲⠴⢶⣶⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠕⢂⠀⠀⣀
⠸⣷⣿⣿⡸⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣭⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡈⠆⠀⣹
⠀⣿⣻⣿⡗⠢⠒⠤⣀⡀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡆⠀⠸⡀
⠀⠸⣿⡿⠀⠀⠀⠀⠈⠙⠛⠁⣰⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢀⢿⣦⣄⣀⡀⠀⠠
⠀⠀⢻⡇⠃⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣴⡇⠀⢀⠟⠽⡿⣻⣿⡿⡶⠁
⠀⠀⠀⢡⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⣻⢇⣶⣾⣷⣀⠀⢀⠈⡷⠁
⠀⠀⠀⠀⢲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡜⡹⢶⠷⢚⣴⣿⣿⣿⣿⣿⣿⣾⡿⠂
⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠿⠿⠟⠛⠋⢉⠤⠦⢀⠈⠙⠻⢶⣴⡾⡮⠚⣣⡻⢟⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⡝⠁
⠀⠀⠀⠀⠀⠀⠈⢷⡄⡀⠀⠀⠀⠀⠀⠀⠀⡰⣸⢠⣤⣤⠤⠶⠶⠊⠐⠀⠀⠀⠀⠀⠀⠰⠆⣠⠝⣋⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⠀⠀⠀⢀⠠⠇⡇⢸⠛⠇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣝⣛⠶⠶⠿⢼⡃⣠⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣶⣾⣧⣈⠓⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⣿⣿⠿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣶⣭⣿⡻⣿⣦⣵⣦⣤⣤⡤⡤⠖⠚⢃⣁⣤⠤⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⠋⠁⠀⠀⠨⠛⢿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠈⠙⠢⣄
⠀⠀⠀⠀⠀⠀⢀⡴⡪⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠻⠟⡋⠁⠂⠁⠀⠀⠀⠀⠀⠂⢑⢦⡀
⠀⠀⠀⠀⠀⣠⡎⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠱⣄
⠀⠀⠀⠀⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦
⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣇
⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆
⠀⠀⣾⣿⣿⣿⡿⠃⠀⠠⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠄⠀⠂⠻⣿⣿⣿⣷
⠀⠀⣿⣿⠟⡁⠊⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠠⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿
⠀⠀⣟⠡⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⠿⠿⠿⠿⠿⠿⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠳⡁⠀⠀⠀⠀⠀⠀⠀⡀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠙⢦⣤⣤⣤⣤⣤⡎⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣴⣶⣶⣶⣶⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠤⢤⣄⣀⣀⣀⣀⣀⡤⠤⠟⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠚⠋⠉⣉⣉⣉⣉⡽⠛⠓⠲⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠾⠯⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠤⣤⣀⡀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠈⠉⠳⢦⣀⣴⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⣀⡴⠋⠁⠈⠉⠙⠛⠿⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⡿⣼⣶⣶⣦⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠛⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣦⣄⣀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠈⣿⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡞⠋⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠸⡟⢿⣿⣿⣿⠋⠈⠉⠓⠲⠦⣤⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⢹⡌⠻⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠆⠀⠀⠀⠀⠀⠀⣰⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⠿⠟⠛⠦⣄⡀⠀⣀⣴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣳⣄⣀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⢉⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣶⣶⣤⣤⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣷⣤⣀⣀⣀⣀⣠⡤⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⣿⣿⠿⠟⠛⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢀⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠿⠿⠿⠿⠿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢛⠽⠟⠿⠿⢿⣿⣿⡿⠿⠿⠻⠯⡛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⡡⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢌⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡧⣀⡀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣦⣀⠀⠀⠀⣀⡀⢼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⠀⠀⠀⠉⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠉⠀⠀⠀⠙⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢿⠁⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⡆⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢰⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠸⡀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠛⠛⠛⠛⢧⠀⠀⠀⠀⠀⠀⢀⠇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⢣⡀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⢀⡜⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣸⣿⣿⣷⣶⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣾⣿⣿⣧⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⡷⠤⠤⠤⠤⠤⠤⢾⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠶⠶⠶⠶⠶⠶⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⠶⢶⣶⣦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠿⠛⠛⠛⠛⠛⠛⠒⠲⢶⣶⣶⣾⣿⣿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⡿⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⠏⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⢳⣤⣤⣴⣶⣶⣶⣶⣶⣶⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣶⣄⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣀⣤⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠓⠶⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⡀⠀⠀⣰⣿⡇
⠀⠀⠀⠀⢸⣿⣧⡀⣠⠴⠛⠉⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⢤⣀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠞⠁⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⣿⣿⣿⠿⠿⠿⡏⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡝⠲⣄⠀⠀⠀⠀⠀⠀⢀⣤⣾⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡈⠙⠶⣄⣠⣴⣾⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⡀⣀⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣌⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠦⣤⣄⣤⣠⣤⡶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉


⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢀⣴⣿⠿⠛⠤⢄⠀⠀⠀⡹⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⢠⠏⢠⠁⠀⠀⠀⠀⢸⣿⣿⣿⣆⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⡆⡟⠀⢸⠀⠀⠀⠀⣀⢾⣿⣿⣿⠟⠢⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣠⢴⡇⣇⣰⣿⣿⣶⠒⠉⠀⠀⠈⠙⠁⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⣠⠞⢿⣨⣇⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣠⠞⢁⡴⠺⡏⠹⣆⠢⣹⠉⠉⠲⢤⡀⢀⣠⣾⣦⣄⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⠞⠁⠀⠁⠀⣀⣹⣶⡾⠷⣌⡣⣄⣀⠀⠀⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⡸⠁⢇⠀⣀⠤⠖⠋⠉⠀⠀⢈⣁⣀⣩⡤⠬⠍⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠠⠁⠀⠈⠋⠀⣀⣠⠤⠶⠚⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠋⠁

⠀⠀⠀⢀⡄⠀⠀⡀
⠀⠀⠀⣻⡇⠀⢀⣥⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⢄⠀⢄
⠀⠀⣸⣽⣷⠊⠁⡸⠤⠀⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠑⠦⠀⠶⣀⣇⡀
⠀⠀⠩⣼⣿⣷⣄⠀⡡⠒⢐⡩⣄⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⢹⣀⣴⣆⠹⢎⡇
⠀⠀⠀⠙⣟⣦⢿⢮⣀⣞⠻⠆⣛⠉⠸⠏⣹⡶⠶⠄⠀⠠⠀⣤⠄⣤⣰⣣⢤⣤⡴⣿⠿⡃⠙⡦⢾⡅
⠀⠀⠀⠀⠘⣿⣏⣹⣏⡝⠀⠀⠀⠑⢆⠀⠀⠀⠀⠀⠀⠠⠀⠙⠃⠉⡀⠿⠾⢷⡀⠈⢦⡸⠚⠁⢈⣙
⠀⠀⠀⠀⠀⠀⠸⠳⢄⡀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠈⠤⣤⠞⠁⠀⠀⡌⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹

⣶⣶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀
⠀⢀⣴⣶⠞⠭⠰⡖⠨⠝⢶⣶⣄⠀⠀⠀⠀⢰⠶⠽⣯⡀⠶⣀⣿⡧⢶⡄⠀⠀⠀⠀⠀⢀⣾⠟⢫⣽⣿⣍⠻⣷⡀
⠀⠈⠋⢸⠟⠲⢦⣤⠶⢺⢯⠉⠋⠀⠀⠀⠀⠈⠋⢁⣼⡼⣶⠇⣧⣉⠙⠀⠀⠀⠀⠀⣴⡾⠟⢰⣯⣻⣿⣽⡆⠹⢷⣦⡀
⠀⢀⢀⡾⠇⠀⠀⠀⠀⠐⠿⣆⢀⠀⠀⠀⠀⣠⢊⠉⣁⣼⣟⣦⡈⢈⠳⡄⠀⠀⠀⢼⣏⣠⣼⣆⣸⣿⣿⣇⣠⣧⣄⣹⣧
⠀⠈⣮⡄⠀⡀⠀⠀⠀⡀⠀⣼⡞⠀⠀⠀⠀⠀⡋⢰⠏⣠⣿⡀⠻⡆⢹⢀⠀⠀⠀⠸⣯⣾⣿⠋⡋⣿⣿⢙⠉⣿⣷⢿⠇
⠀⠠⢹⣵⣿⣿⣧⢰⣿⣿⣶⣽⡁⠀⠀⠀⠀⠛⠓⢾⡾⠉⡀⠉⢶⠷⠛⠛⠀⠀⠀⠀⠈⠛⠛⠻⢿⣎⣱⡿⠟⠛⠋⠁
⠀⠀⠀⠀⠟⣯⠈⠽⢿⡟⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣶⢿⣶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠙⠃
⠀⠀⠀⠀⠀⠸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡀
⠀⠀⠀⠀⠀⠈⢳⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣽⠇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢸
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢸
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⢼⣿⢤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⡀⣷⢾
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣈⣉⣅⣰⡶⡦⠀⠀⠀⠀⠀⢐⣓⣿⣧⣿⣾
⠀⠀⠀⠀⠀⠀⣼⣿⡼⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠨⢍⣩⡹⠋⠻⠶⠖⠀⠀⠀⠀⠀⠸⠇⣿⡿⢿⣿
⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠀⠸⠿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠁⠀⠉⠲⣄
⠀⠀⠀⠀⢀⠔⠁⠀⠀⣀⣀⠀⠀⠀⠘⡆
⠀⠀⠀⢠⠋⠀⢀⠀⠀⠀⠀⠙⡄⠀⠀⡇
⠀⠀⢠⠇⠀⢠⠆⠀⠀⠀⠀⢀⡇⠀⢀⡇
⠀⠀⣼⠀⠀⢸⠀⠀⠀⠀⢀⡼⠁⠀⣼⣁
⠀⠀⣿⠀⠀⠘⢧⣄⣠⡴⠛⠀⢀⣼⡁⠈⠙⠢⣄⡀
⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⢀⣤⠟⠀⠙⠲⢤⣀⠀⠉⠓⠤⣀
⠀⠀⠀⠈⠳⠦⣤⡤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠈⠙⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠈⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠁⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡀⡀⠈⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠌⡜⠀⠀⠀⠀⠀⢹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢌⠞⠠⢂⢄⠌⡱⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠎⡠⢁⢊⠎⡰⡡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⢼⡱⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠿⠿⢿⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠂⠒⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⡀⢀⡄⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⣀⢀⣀⠀⢀⣈⣀⣀⣿⣄⣟⣅⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣟⠉⢉⠉⣉⣨⣍⣾⡈⢹⣾⣿⣿⣶⣶⣄⣄⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠁⠀⣼⣿⣿⠟⠋⠉⣤⣶⣶⣶⣦⣭⣙⠻⣿⣿⣷⠄⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⠀⢀⠀⢰⣿⣿⡿⠋⠀⠀⠀⣠⣿⣭⣍⣛⠟⠛⠙⠃⠦⡙⢿⢧⡐⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢲⣿⡿⢅⣀⢀⣴⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠘⢤⠫⣷⡴⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢄⣸⣿⡿⢠⣿⣿⡘⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⣴⣆⢻⢿⣸⣷⠆⠀⠀⠀⢠⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢋⣽⣧⣾⣿⣿⣧⣹⣿⣿⣿⣿⡿⣫⣶⣷⣶⣤⣄⣼⣿⣿⠸⣟⣿⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⡀⠺⣩⣿⣿⣿⣿⡟⠁⠀⠉⠉⠛⢻⣴⣿⣿⣿⣿⣿⣿⡸⣿⣿⢁⣿⡋⠁⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⠀⠠⣿⡟⣙⣛⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⢿⡿⣸⣿⣿⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⡾⠛⠀⠤⢹⣿⡜⣿⣧⡀⠀⠀⠀⠀⢀⡿⠿⢿⣿⣿⠟⠁⠀⠀⢠⠟⠋⢦⢛⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠈⠑⢿⢎⠻⢿⡆⢠⣤⣾⣿⣿⣿⣶⣶⠀⠀⠀⢀⣴⣎⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡢⠄⡀⠀⠀⢀⣶⣷⣦⣝⡘⠟⢿⣿⣿⠿⠋⠽⢂⣠⣶⣿⢇⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠘⠀⠀⢻⣿⠿⣿⣷⣶⣶⣶⣶⣶⣿⣿⡛⠻⠿⣈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣤⠀⠀⠀⠉⠀⠛⠻⠿⠿⠻⢻⠟⠻⣶⣿⡆⠀⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⠀⠀⠄⠀⠀⠀⠘⠂⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⣿⣷⣦⣀⣀⣀⣀⣀⣀⣀⣀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣶⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣷⣶⣄⣀⣀⣀⣀⣀⣀⣀⣀⣴⣾⣿⣿
⠉⠉⢉⣽⣿⣿⣿⠿⠿⠿⠿⠿⣿⣽⣿⣯⠉⠉⠉⠉⠁⢩⣿⣿⣿⡿⠿⠿⠿⠿⢿⣯⣿⣿⡍⠉⠉⠉⠉⠉⣽⣿⣿⣿⠿⠿⠿⠿⠿⢿⣭⣻⣿⡉⠉
⠀⠀⢸⣿⡿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⠂⠀⠀⠀⠀⢿⣿⢿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⠐⣿⣿⣿⣿⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇
⣀⣀⡠⢛⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⣀⣀⣀⣀⠨⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣀⣀⣀⣀⣀⢝⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀
⠶⠶⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠶⠶⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠷⠶⠾⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠶⠒⠒⠒⠒⠒⠒⠶⠶⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠉⠉⠉⠙⠓⠦⣄⡀⠀⠀⠀⣀⣤⠴⠖⠛⠋⠉⠛⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⠶⠶⣟⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⡾⠋⠙⠛⠛⠋⠛⢷⠶⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⣀⣤⠴⠞⠋⠉⠁⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠈⠉⠛⠲⠦⣄⣀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⡀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⣧
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡄
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠒⠶⠶⠶⠶⠶⠖⠒⠛⠋⠉


⣀⣀⣀⠀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⡤⠴⠒⠒⠒⠒⠒⠲⠤⣄
⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⣰⠒⢲⡀⢀⣀
⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⢸⠁⠀⣧⡇⠀⠀⣷⡏⠈⡧
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢳⡀⠀⠀⠀⠀⢸⠀⠀⢹⡇⠀⠀⣿⠀⠀⣇⣠⢤⡀
⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢄⠀⠀⢹⡄⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⣿⠀⠀⣿⠀⠀⡷
⠀⠀⠀⡏⠀⢠⠀⠀⢀⠀⠀⠀⠀⠀⢸⡇⠀⠑⠤⠞⠁⠀⠀⠀⢸⡇⠀⠸⡇⠀⠀⣿⠀⠀⡟⠀⢠⡇
⠀⠀⢸⠃⠀⣼⠀⠀⣿⠀⠀⢸⡆⠀⢸⡇⠀⠀⠀⢀⡴⠒⢤⠀⢸⡇⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⣸
⠀⠀⢾⠀⠀⣿⠀⠀⣿⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠘⣇⠀⠀⠑⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠈⠓⠋⢹⠀⠀⣿⠀⠀⢸⣇⠀⠀⡇⠀⠀⠀⠀⠈⢧⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⢸⡀⢸⢿⠀⠀⢸⢻⠀⢀⡇⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠉⠁⠈⠧⠤⠏⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⠤⠤⠤⠤⠤⠖⠚⠁

⠀⠀⠀⢹⣄⢲⡲⢡⡇⠀⢠⠘⣴⠢⢼⣠⠂⠀⢧⡙⣏⡙⢣⣸⠀⠀⡇⢸⢨⡇⢸⠀⠀⣣⢰⡌⡴⠀⠀⢂⣱⠞⠲⣀⡇
⠀⠀⠀⡄⡟⢦⣽⡎⠃⠀⠘⢳⣌⣵⣎⠹⡀⠀⡏⢻⣿⣽⡏⢱⡀⠀⡃⠸⣼⠃⢰⠀⠀⢹⠦⠵⠈⠀⠀⡤⣻⣤⣼⡏
⠀⠀⠀⢷⡘⣤⣭⢃⡆⠀⢸⣴⠿⢍⠙⢶⠁⠀⡷⣬⡰⣇⡠⡞⠀⠀⣳⠄⠐⢀⣿⠀⢸⣈⣤⡬⢠⠃⠀⡀⢏⡉⣍⠁
⠀⠀⠀⡎⣷⡌⢢⠙⡇⠀⢃⠈⣷⠺⢷⢈⠂⠀⢁⠹⡟⡟⠇⢸⠀⠀⢱⠀⢸⣿⡏⠀⠀⡋⢸⣴⡏⠀⠀⢱⢾⡀⢠⢖⠁
⠀⠀⠀⡇⢣⡙⠚⡇⡀⠀⠘⢣⣌⣱⣄⠹⠀⠀⡞⢶⣩⣷⡞⠣⠀⠀⣼⣆⢰⡌⠣⠀⠀⢇⢰⠬⡼⠀⠀⠇⡟⠿⠟⠃⡄
⠀⠀⠀⡸⣦⡹⣙⢰⠇⠀⡆⢸⠀⠀⢸⢀⠀⠀⢢⡈⢻⣅⠳⡼⠀⠀⠁⢿⣿⡁⠶⠀⠰⢹⡶⢷⠑⡀⠀⢦⣀⠊⢑⣠⠇
⠀⠀⠀⡁⣟⢳⣾⡆⡇⠀⣧⢸⠀⠀⠸⠉⠀⠀⡄⣹⠷⠾⡗⢛⠀⠠⡀⠈⢸⠃⠀⠀⢰⡘⡤⠤⢠⠂⠀⠐⣽⣶⣾⡏
⠀⠀⠀⢻⡈⠦⠤⢠⡇⠀⡇⢸⠀⠀⢠⠀⠀⠀⡼⣦⡑⢢⡬⢻⠀⠀⡇⠀⢸⠀⠈⠀⠀⠉⠸⢶⣯⠀⠀⡆⠡⠔⠣⠁⡀
⠀⠀⠀⡊⠿⣧⣵⡌⡇⠀⢇⠸⣄⠀⢼⣠⠀⠀⢧⡘⣏⡉⢳⣸⠀⠀⡇⣰⢸⠀⢀⠀⠀⣠⠰⡞⠇⠀⠀⠈⢓⣀⣠⡿⡁
⠀⠀⠀⢃⡹⣌⣙⢁⡇⠀⡜⢳⣬⣹⣄⢹⠀⠀⡏⢻⣷⣼⡏⢹⡄⠀⡇⠙⣼⠀⣿⠀⢠⢹⠇⠺⠈⠀⠀⡄⠏⣻⡋⠇⡄
⠀⠀⠀⡜⣷⡝⢮⡙⡇⠀⢰⡈⠳⢤⠙⣾⠀⠀⡵⣬⡰⣧⡠⢞⠀⠀⠷⠀⡰⢠⡿⠀⠘⣌⠆⡤⢀⠆⠀⠑⣬⠁⢨⡤⠃
⠀⠀⠀⡇⢣⠹⠿⡇⡗⠀⢆⠘⣗⠚⢷⣨⠀⠀⢄⠹⣟⡟⠇⣸⠀⠀⢳⠀⢿⣿⠃⠀⠀⡍⢸⢷⣯⠀⠀⠀⡼⢿⠾⡇⡄
⠀⠀⠀⢸⣆⢳⡒⢱⡇⠀⠜⠳⣌⣱⣄⠹⡀⠀⡞⢷⣽⣷⡞⠃⠀⠀⣞⡄⢠⠈⢆⠀⠀⣡⢰⡆⡴⠀⠀⢂⣰⠊⠳⣀
⠀⠀⠀⡄⣟⢦⣼⡎⠃⠀⠀⠀⠀⠀⠈⠀⠁⠀⢳⡀⢫⣅⠱⡴⠀⢸⠈⣿⢾⠃⣸⠀⠀⢹⠆⠿⠀⠀⠀⡤⣹⣤⣴⡟
⠀⠀⠀⢳⡈⣤⢬⢁⡇⠀⢀⠀⠀⠀⢸⠈⠀⠀⣄⣽⠷⠾⡗⢻⠀⢸⠀⢾⢸⠀⢼⠀⢸⣌⠄⠸⢠⠃⠀⡀⢋⡝⢍
⠀⠀⠀⡎⣷⡌⢲⠙⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⡜⣦⣀⣢⡼⢻⠀⢸⡀⢸⢸⠀⢸⠀⠀⡉⢸⣶⡏⠀⠀⢙⢾⡀⣰⣟⠁
⠀⠀⠀⣇⢣⡙⠛⡇⡀⠀⢠⡈⢶⠂⠤⣀⠄⠀⢧⡘⣏⡉⣓⣸⠀⢸⠁⢸⢸⠀⣌⠀⠀⢧⠸⠈⡬⠀⠀⠇⠟⠻⠛⠃⡆
⠀⠀⠀⡼⣦⡹⣝⢰⡇⠀⠰⢓⣤⣹⣦⠸⡄⠀⡏⢻⣷⣼⡇⢹⡄⢸⡇⢸⣸⠀⣹⠀⠀⢹⡶⢿⠁⠀⠀⢦⣄⠈⢙⣴⠃
⠀⠀⠀⡁⢏⢳⣾⡆⡇⠀⢘⣌⠙⢤⡙⢶⠁⠀⡽⣬⡐⣧⡨⢟⠀⠸⣷⠂⡤⢠⡟⠀⠸⡈⠄⠤⢰⠆⠀⠐⣽⣶⣾⡏
⠀⠀⠀⢹⡈⠢⠴⢠⡇⠀⢨⠈⣿⠒⢿⣀⠆⠀⢇⠹⣟⡟⠇⣸⠀⠀⢱⠀⢿⣿⡇⠀⠀⡉⠸⢷⣯⠀⠀⣇⠡⠜⠡⢀⡄
⠀⠀⠀⡆⠻⢧⣵⡌⡇⠀⠘⠱⣤⣙⣆⠙⡀⠀⡞⢷⣭⣷⡞⠣⠀⠀⣾⣄⠸⠄⢳⠀⠀⣠⠰⡞⢏⠀⠀⠈⢻⣠⣰⡋⡀
⠀⠀⠀⢇⡹⣌⣉⢁⡇⠀⠀⠀⠀⠀⢀⣀⡁⠀⢳⣈⢺⢄⠱⡾⠀⠀⡏⢿⣿⠀⣴⠀⢠⢹⡗⠛⠈⠀⠀⡀⠋⣹⣋⠇⡄
⠀⠀⠀⡞⣷⣝⢶⡙⡇⠀⢀⣴⠀⠀⢸⠉⠀⠀⣄⣽⠿⠾⡗⣙⠀⠀⡆⢸⠇⠀⢫⠀⠘⣌⠆⡴⢀⠆⠀⠐⣮⡁⠨⡴⠃
⠀⠀⢀⣇⣃⠙⢿⡇⣇⠀⡈⢸⡀⠀⢀⢀⡀⢀⡞⣶⣁⣲⣜⣫⡀⠀⣇⣸⣰⣀⣸⡀⠀⣍⣸⣷⣟⠀⠀⣀⣾⣿⣿⣇⣄
⣿⣿⠛⢿⠛⡟⢿⠟⣿⢟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣛⣛⣛⣻⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠙⠛⠻⢿⣿⣿⣿⣿⣿⠟⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣟⠁⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠈⢧⠀⢀⣀⣀⣀⣀⣀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣷⣦⡀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⡄⠀⢀⣠⠴⠚⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠈⠉⠓⠲⠤⣄⣀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣷⣄⣠⠞⣿
⠀⠀⠀⠀⣿⣿⡶⠋⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣿
⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢠⡟
⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣼⠃
⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠃⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠛⠛⠛⠉⣇⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠙⣯⡳⣄⠀⠀⠀⠀⠀⠀⢀⣸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣿⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠙⠦⣄⣀⣤⣶⣿⣿⣿⣿⣷⣦⣤⣀⣀⠀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣿⣿⣯⣭⣭⣿⣶⠶⠶⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⣉⣉⣭⣍⣁⡒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣡⡴⠛⠉⠁⠀⠈⠉⠙⠷⣝⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣼⠀⣀⡠⠤⢄⡀⠀⠀⠀⠀⠀⠀⢸⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣿⣏⣁⣤⣤⣀⠙⠶⣾⠛⠛⠛⠛⠛⠛⠛⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⣤⠿⣿⢯⡀⠀⢙⣧⡀⠈⠳⣶⣲⠖⢲⡖⢦⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠸⣧⣀⣘⣷⣝⡲⠿⣿⠇⠀⠀⣼⣁⡴⣋⡤⠞⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠈⠉⠉⠉⠉⠉⠋⠁⠀⠀⢠⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢠⠎⠁⣰⠳⡄⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⡆⠀⢸⡀⠀⢀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡾⠀⢠⡇⠀⠀⠀⣰⠏⠹⣄⠀⠀⢀⡀⠀⢳⡀⠀⢻⡀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢸⠃⠀⡾⠀⠀⣾⠛⠁⠀⠀⠈⠓⠚⠉⠉⠛⢦⢳⡀⠀⢻⡄⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⡏⠀⢰⠇⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⢳⡀⠀⢻⡄⠻⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣸⠁⢀⡟⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⢳⡀⠀⢻⡄⠿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⡏⠀⣼⢁⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢱⡄⠀⢳⡄⣼⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣼⠀⢠⡏⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⢻⡄⠀⢻⡄⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢰⠇⠀⡾⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢻⡄⠀⢻⡼⢧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡾⠀⣸⠇⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢻⡄⠀⢻⡄⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢸⠃⢀⡟⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠹⣆⠀⢻⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡟⠀⣼⢁⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⣆⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⣋⡙⠋⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⡶⠟⢻⣇⣀⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣼⠋⠀⠀⠀⢹⡟⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠔⠛⢦⡀⢀⡴⠋⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣀⡼⠛⣉⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⢲⠖⠁⠀⣠⡾⠛⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⣵⠾⣿⡁⠀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣴⠏⠁⠀⠘⢻⡟⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢐⠧⢤⡀⢀⣴⠟⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠁⠀⣠⠗⠋⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠶⣤⠀⠀⠀⣠⠾⢷⠶⠟⠋
⠀⠀⠀⠀⠀⠀⣠⠖⣩⣶⣶⡎⠁⠀⣠⠞⠁
⠀⠀⠀⠀⢀⡴⢡⡞⠉⠀⠘⢷⣶⠞⣿
⠀⠀⠀⠀⣼⠁⡿⠤⣄⠀⣠⡾⢃⣼⠏
⠀⠀⠀⠀⠟⠃⠀⢀⡼⠞⢋⣴⠟⠁
⠀⢀⠔⠁⠀⣀⣴⣿⣶⠾⠛⠁
⣴⠁⠀⣠⡾⠋
⠹⣷⠞⠉
⠀⢀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠒⠒⠒⢉⣩⣭⣵⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠒⠉⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⡰⠋⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⡜⠀⠀⠀⢀⣴⣶⣾⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⣸⠁⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⢸⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⢸⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠈⢻⡇
⠀⠀⢸⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠃
⠀⠀⠈⣶⡇⠀⠀⠀⠀⠈⠻⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠰
⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⡄
⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⡐
⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠿⠛⠁⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⣠⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⢴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⠿⠿⠀⣀⣀⠀⠀⠠⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⠛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠁⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⡀⠀⠀⠊⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣀⠃⠀⠀⠐⠀⠀⠑⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠐⠀⠀⠀⠂⠀⠀⠙⠒⠢⣄⡀⢀⡀⠀⠀⢀⡠⠐⠶⠶⠖⠶⠶⠶⠂⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠂⠒⠒⠶⠦⣤⣤⡄⣀⠀⢀⠀⠙⣿⣦⡀⠆⠀⠜⠁⠀⠀⠀⠀⠈⠁⠉⡉⠙⢻⡅⠀⠀⣀⢀⣀⣀⣀⣀⣠⣈⢳
⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⠀⠀⠀⢘⠃⠘⠉⠋⠉⠉⠉⠉⠉⠁⠈⣷
⠀⠀⠀⠀⠀⣬⡁⠀⠀⠀⡀⠀⠀⠀⠋⠀⠀⠀⠈⠀⠀⠐⠂⠀⠀⠀⠀⡀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⢸⣤⣄⡀⣀⣀⣀⣀⣤⣤⡤⠘⠣⡀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢴⡃
⠀⠀⠀⠀⠀⢨⡀⠀⠀⠀⠖⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠰⠀⠀⠀⠀⠁⠀⠀⠀⠀⡆⠀⠀⠀⢻⡤⣄⣀⣤⣠⣤⠀⠠⠶⠶⠶⠀⢹
⠀⠀⠀⠀⠀⠈⠻⡄⠀⠀⠀⡀⠀⠀⢠⠀⠀⠀⠉⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠃⠀⠀⠀⢸⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⣯
⠀⠀⠀⠀⠀⠀⠱⣏⡆⠀⠈⠥⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⡄⠀⠀⠀⢠⠀⠀⠀⠈⣷⠰⢄⣤⣠⣄⣠⣄⣤⣤⣶⡀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⢀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣸⡁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣣⡀⡀⢠⠀⠀⠀⢢⡀⠀⠀⢠⠀⡀⢀⠘⠇⣀⢠⣀⡘⠆⢀⢠⢤⣰⣴⣤⣿⡁⠈⠀⠚⠛⠛⠓⠛⠳⠖⠛⠐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠶⣾⠿⠶⢷⣿⣿⣶⣶⣶⣿⡾⢿⣤⠾⠄⠌⠷⠂⠴⠿⠂⠛⠛⠛⠙⠻⣷⣶⣤⣀⣀⣀⣠⣤⣠⡶⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⣀⠀⠀⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠀⠀⠀⢀⡀⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠞⠁⠀⠀⠈⡀⠀⠀⢣⡼⠋
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠛⠛⠛⠛⠛⠛⠋⠐⣄⣀⣀⣀⢤⣤⠐⠚⠉⠁⠀⡀⠀⠃⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡼⢠⣴⣤⣤⣤⣤⣄⣤⣠⢀⣠⠟⠉⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠰⠾⠋⠀⠄⠀⠤⠰⠶⠶⠚⠛⠉⠉⠀⠙⠀⢄⡀
⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢡⠀⠀⠀⢠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠈⠃⠀⠀⢹⠆
⠀⠀⠀⠀⢠⡞⠁⠒⠿⠶⠦⠤⡤⣦⠤⠶⠤⡟⠀⠀⠀⢀⠀⠀⠀⠀⠳⠀⠀⠀⠀⡄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⡄⠀⠀⠀⢾⡀
⠀⠀⠀⠀⢀⡟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⢻⡀⠙⠛⠛⠊⠀⠶⠶⠶⠦⠴⠲⣞⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⡄
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⠀⢀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⠀⠀⣀⢾⠁
⠀⠀⠀⠀⡿⠐⠾⠶⠾⠶⠶⠦⠶⠴⠒⠂⠺⡅⠀⠀⠀⡐⠀⠀⠀⠐⠆⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠶⠀⠀⠀⡀⠀⠀⠐⠂⠀⣸⢂⠛
⠀⠀⠀⠀⠳⣤⣤⡀⠀⣀⡀⣀⣠⣀⠀⡄⠀⡟⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠃⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⣤⡀⢸⠾⠃
⠀⠀⠀⠀⢰⣏⠉⡉⠉⠉⠉⠉⠉⠉⠁⠀⢀⣹⣦⣃⣾⣤⣆⡰⠄⢉⡤⠄⣠⠐⢁⠄⣀⡀⠰⠀⢀⠀⠘⠁⢀⡀⠼⢀⠔⡄⢦⠝⠃
⠀⠀⠀⠀⠀⠉⠛⠿⠶⣶⣶⠶⣴⠾⠶⠿⠛⠁⠈⠉⠉⠉⠉⠑⠈⠙⠂⠙⠚⠲⠗⠚⠧⠶⠾⠷⠧⠞⠚⠃⠛⠛⠚⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠉⣉⣉⠉⠑⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⡴⠚⠋⠉⠉⠉⠉⠙⠲⢄⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢸⡀⠀⠀⠀⠀⠀⠀⠀⡠⠒⣉⣉⣉⣉⠙⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢧⠘⣷⣨⣍⣁⣒⠻⢭⠩⠭⢛⠲⣶⣶⣄⠻⣆⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⡴⣪⡶⠛⣧⡈⣷⡆⠀⠉⠻⡆⢻⣶⢄⠑⢌⢿⣾⠇⠙⡆⠀⠨⣳
⠀⠀⠀⠀⠀⢂⠼⠉⠀⠀⠈⠉⠡⠤⠀⠐⠀⡄⣼⡯⡀⠀⠀⠈⢫⠀⠈⠁⠀⠠⢸⡄
⠀⠀⠀⠀⢸⠸⣅⠀⠀⠀⣀⣈⣐⣈⠏⢀⠜⣰⣟⠥⡂⠄⠀⡆⣿⡴⠃⠀⠐⠀⠀⢱⣄
⠀⠀⠀⠀⠈⡆⠀⠉⢁⣲⡬⢵⣂⡠⢐⡡⣾⠞⡣⠑⠂⢀⡰⣱⠁⠐⠀⠀⠄⡠⣠⡳⢦⣍⡒⠤⡀
⠀⠀⠀⢠⡊⢀⣾⢣⢶⢌⣤⣴⣶⣾⢷⣿⣫⠵⠬⠒⠊⢩⡔⣇⠂⣁⣁⣁⣪⣜⠉⠛⠶⡬⣛⠷⣮⣑⠦⣀
⠀⠀⠀⢈⢏⢾⣮⢧⢣⣿⠃⠀⠈⠻⠿⣶⣖⣦⢶⡜⡿⡔⣱⡸⠛⠛⠋⠉⠉⠙⠳⢶⢗⡮⣙⣷⣦⣍⡛⢶⣭⣒⠤⡀
⠀⠀⠀⢃⣾⣿⢣⢣⣿⠃⠀⠀⠀⠀⠀⠀⠀⠹⡎⢷⠐⡹⡽⣷⢃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⣗⠉⣛⢷⣮⣙⠳⣮⣕⡢⣄
⠀⠀⢰⣿⣾⢣⢣⡻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⠱⢱⡹⣟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠛⠉⠿⡻⣦⣿⣿⣗⣷
⠀⣔⣽⢯⢣⣳⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣷⣧⢻⡸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠛⠋
⡴⣿⢯⡷⣱⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣯⢿⡆⢷⡱⡀
⢻⣳⡿⣽⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣎⣿⡘⣧⣧
⠈⠛⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⡷⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠁
⠀⣀⡀⠀⠀⡀⢀⠀⡀⢀⠄⡀⠀⢀⠠⠀⢄⠀⡢⠐⠖⠠⠀⠔⠠⠂⢄⠠⢀⢄⠠⡀⢄⠄⡠⡀⠔⡠⢀⢄⡠⣀⠄⡀⡀⢀⠀⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣿⡇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣧⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢰⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⠤⠐⠷⠼⣀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣀⡀⠀⠈⠑⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠿⠿⢷⣶⣴⣶⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠙⠿⣆⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⢸⣼⠖⠖⠒⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⠀⠀⢸⢟⣷⢄⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣧⣄⣦⣤⣠⢠⣿⣿⠃⠀⣠⡟⠹⠃⠘⠒⠊⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣷⠠⡆⠀⠀⢠⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣶⣿⣿⢀⣃⣀⣠⡏⢀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⢿⣯⡽⠟⣿⣿⣷⣶⣿⣿⣿⣿⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠛⠛⠛⠛⠻⣿⠿⣿⡄⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣴⣤⣴⣤⣴⣶⣿⣦⠐⣴⣿⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣽⠯⠛⠁


⣿⠿⠿⠿⠿⠻⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⢤⡔⠲⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠙⣧⠀⠘⡆⠀⠙⣄⡑⠢⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠙⢿⡄⠀⢿⡄⠀⢿⡄⢠⣿⠿⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⢶⣄⠀⠣⠄⠀⣀⣤⣾⣩⠿⠿⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣿⣧⣤⣾⣷⣾⠟⠉⠉⠀⠀⠈⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⢹⠿⢿⡟⠇⠀⠐⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠼⣶⣄⣀⣀⣴⣦⣄⣴⢶⣿⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⣷⡀⢰⣦⠉⠌⠓⣈⠙⠛⠃⢘⠐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣙⣦⣿⣧⣤⡀⢉⠲⠄⢀⡞⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡔⣻⣼⢷⣦⣙⣻⢿⣷⣤⣾⡇⢸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⣿⡇⢙⠣⠉⠻⣿⣉⠡⡙⢻⣿⣇⠸⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠦⣹⣶⣤⣉⣻⣿⣿⣤⡏⢹⣷⣿⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠀⡿⠸⣿⡋⠛⠿⠿⣭⣼⡷⣾⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣄⢂⣤⣉⡙⢲⣴⣿⠟⠋⡙⣯⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠸⢷⣧⣠⣽⠋⠉⠉⠳⠁⡠⢺⣋⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣁⣛⣽⣿⣿⡿⡷⣺⣶⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⣛⡀⠤⠿⠿⠟⣹⣿⠙⠙⢿⣆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣕⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠐⠒⠂⢈⣉⣈⣁⣀⣒⣂⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠉⠁⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡉⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠑⠠⢀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠑⠠⡈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⣠⣤⣤⣤⣤⣤⣤⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⣀⠔⠉⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡠⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠉⠀⠂⠀⠀⠀⡀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠈⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠀⠀⠈⠻⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠉⠉⠉⠉⠉⠁⠀⠀⠱⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⣡⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⣀⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⢀⣴⣿⣿⣿⣷⣶⣤⣤⣀⣀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠛⠉⣉⣈⠉⠀⠤⠜⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⢷⣶⣶⣶⣶⣿⣿⣶⣶⣾⣾⣿⣷⣶⣶⣶⣶⣦⣶⣦⣤⣤⣀⣀⡀⣀⣀⣀⣄⣀⡀⢀⣀⡀
⠀⠀⠀⠀⠠⣤⣶⣿⣷⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⢀⣀⣤⣄⣀⡀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⡉⠂⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣥⡀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⠧
⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⡿⣿⠃⠀⠸⠋⠰⠟⠿⠿⢿⠿⠾⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣆⣀
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠂⣶⣶⡶⣶⣶⣶⣦⣤⣌⣉⠟⠻⢿⣿⣀⠀⠀⠀⠀⠤⠈⠛⢿⣿⣿⡯
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⣰⠀⠀⠀⠀⠀⡀⠀⣿⣿⣿⣿⣿⣿⣻⣿⣾⣿⣿⣦⣄⡉⠀⠀⠀⠀⠃⠀⠋⢀⣤⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠉⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⡸⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⢤⡀⠙⢿⣿⣿⣿⣿⣿⡟⢿⣿⡿⠋⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣯⡟⠁
⠀⠀⠀⠀⠀⠠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡄⠀⠙⠛⠿⠿⠿⠿⠛⠉⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠙⠿⠟⠻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠖⠈⣧⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠂⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠤⠤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⠁⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⠿⠋⠀⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⡼⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⢰⠃⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⡎⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⡀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢿⣿
⠀⠀⢸⡇⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢻
⠀⠀⠀⡇⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢸
⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠈⠻⢿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⡆
⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡿⠿⠿⠛⠋⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⢀⣴⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⡡⠖⠋⠉⠉⠛⠛⠻⢿⣿⡿⠟⠛⠉⠉⠐⠠⣈⠓⢄
⠀⠀⠀⠀⠀⠀⢠⠞⠁⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣙⢆
⠀⠀⠀⠀⠀⣴⣣⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀
⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡄
⠀⠀⠀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡄
⠀⠀⢠⣿⣿⣿⣿⡿⠋⠒⠒⠦⠤⢄⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⡤⠤⠖⠒⠙⢿⣿⣿⣿⣧
⠀⠀⢸⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿
⠀⠀⣿⠘⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠋
⠀⠀⢻⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠸⡄⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠛⠛⠛⠛⠛⠿⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⢧⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠈⢧⠀⢹⣤⣄⣀⡀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢀⣀⣤⣶⡁⢀⠆
⠀⠀⠀⠀⠈⢧⣸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣇⠊
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⢀⣀⣠⣾⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⡟⠋⠀⠉⠉⠉⠉⠉⠀⠀⠙⣛⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠒⠒⠒⠒⠒⠋⠉
⣶⠖⡶⠶⠶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⣀⣀⣀
⠀⠀⠀⡇⠀⢸⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲
⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⣀⣀⠀⠀⠀⠉⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⢀⡤⠖⠋⠉⠁⠀⠉⠉⠓⠦⣄⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣄⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠀⠀⠀⠀⠀⡾⠤⠄⠔⠤⠤⠤⢄⣀⣀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⢀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⣸⠛⠉⠉⠉⠉⠉⠓⠒⠶⠬⣍⡓⠦⣄⡀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⢸⠀⠀⠀⠺⢍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣍⠲⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠂⠀⠀⢸
⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠉⠲⢤⡀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢾⠁⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠉⠒⠦⠤⣀⣀⣀⠤⠔⠚⠁⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⣀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣇⣀⣸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠦⠼
⠀⠀⠀⣽⣉⡇
⠀⠀⠀⣾⠈⡇
⠀⠀⠀⣿⠀⡇
⠀⠀⠀⣿⠀⡇
⠀⠀⠀⣿⠀⡇
⠀⠀⠀⢼⠀⡇
⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣠⣴⣶⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⢀⠄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠒⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣷⣄⠀⠀⠠⡀
⠀⠀⠀⠀⠀⡀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⠂
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⢿⣿⣿⣿⡟
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⡀
⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾
⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⢀
⠀⠀⠸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣶⣤⠄
⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟
⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠑⠀⠀⠀⡀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⡤⠄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠂⠉⠐⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢊⣀⣤⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡁⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣨⣴⣾⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣄⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠊⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠠⠄⣀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣷⣄⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⡐⠁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠈⠢
⠀⠀⠀⠀⠀⠌⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠑⡀
⠀⠀⠀⠀⠊⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠐
⠀⠀⠀⡀⠀⠀⡠⠈⠉⠉⠉⠉⠙⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠑⢲
⠀⠀⠰⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢁
⠀⠀⠃⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠈⡄
⠀⠸⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠉⠉⠛⠻⠿⢿⣿⣿⡟
⠀⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⡔⠁
⠀⠀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⢸⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡘
⠀⠀⠀⢿⣿⡷⠄⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣶
⠀⠀⠀⠈⢿⣷⠀⠀⠈⠀⠀⠒⠒⠒⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠈⠁⠐⠒⠒⠒⠒⣾⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠲⠤⢤⣴⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⣥⠶⠚⢛⣡⠶⠞⣛⣉⣉⣛⡛⠶⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣴⠋⠁⠀⣴⠟⣡⠾⠛⠉⢉⣩⣭⣝⣷⣯⡙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣼⠃⠀⠀⣼⢃⡾⠁⠀⠀⠀⣾⣁⣴⠟⣷⠈⢻⡌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⣿⠀⠀⢸⡇⣼⠁⠀⠀⠀⠀⠻⣥⣤⣴⠏⠀⠀⣿⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⡀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣌⢷⣄⠘⣧⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⣴⢟⣦⣙⠳⠘⣧⡙⢧⣀⠀⠀⠀⠀⠀⠀⢀⣴⠏⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⣴⢟⣵⠟⢁⣽⠗⠂⣼⠻⣦⣍⡛⠶⠶⠶⠶⠞⣋⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⣴⢟⣵⠟⠁⠀⡟⠁⠀⣼⠃⢠⡟⣽⠛⢻⡖⠒⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢁⣴⢟⣴⠟⠁⢰⡟⠛⠃⠀⣰⠏⢠⡟⣼⠃⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⢋⣵⠟⡁⠀⣾⠛⠃⠀⠀⣰⠏⢠⡟⣸⠃⣀⣤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠟⠁⠀⣿⠙⠉⠀⠀⠀⣰⠏⢠⡟⣸⠇⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⢰⡟⢀⡟⣰⠏⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡾⣱⠏⠐⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣏⣤⡴⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⠛⠉⠉⠉⠈⠉⠉⠛⢿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣦⣄
⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣷⣦⣄⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣷⣤⣀⣀⡀⢀⣀⣀⣤⣾⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⡿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠒⠒⠚⠛⠉⢉⣻⠶⠶⠲⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠛⠉⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠙⣄⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠘⣆⣠⣤⡦⢴⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⣴⡿⠋⠉⢙⡎⣇⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣀⣠⡤⠖⢺⣿⠀⠀⣠⠞⡼⠃⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠒⠒⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⣞⣠⠔⠋⠉⠀⠀⠀⠘⣮⣛⣩⠵⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⣄⡀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀⠀⠈⣧
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡄⠘⡆
⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣷
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⢸⡀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⢧⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠈⢧⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠶⠦⠤⠤⠤⠤⠤⠤⠴⠖⠚⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠈⠀⠀⣀⣠⣤⣾⣦⣄⡀⠈⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⠉⠉⠓⢲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣠⡀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣛⡒⠒⠸⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠿⢿⣿⣿⣿⣿⣿⠟⠁⠊⢿⣿⡿⣿⣞⢥⠄⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠄⠀⠀⠀⠨⢽⣿⣿⣷⣖⡄⠀⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⢩⣿⣿⣿⣿⣕⢔⠈⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⢀⣠⣴⣶⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢀⢕⣿⣿⣿⣿⣿⢕⣴⠇⠉⢣
⠀⠀⠀⠀⠀⠀⠁⠈⣦⣶⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣶⣿⣽⣿⣿⣿⣿⣿⡥⢋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣯⡊⠀⡃
⠀⠀⠀⠀⠀⠈⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡫⠀⠀⠅
⠀⠀⠀⠀⠀⠀⡤⠀⠁⠉⠛⠻⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣕⡁
⠀⠀⠀⠀⠀⠀⠀⢂⠀⠨⡻⣮⡀⢄⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣴⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⢏⡄⠉⢢⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⢿⢿⣿⣶⣕⣷⡶⡡⣢⣆⢁⢄⣴⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅⡊⠀⢀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡤⠼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡻⡯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠌⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠈⠂⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⡀⠠⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢀⠀⡠⠒⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⣾⠀⠀⠀⠀⣬⠄⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠠⠁⠀⠀⠀⡖⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠃⠀⠀⠀⠈⠀⠀⠀⠂⡇
⠀⠀⢀⠤⠤⠤⡀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⡜⠀⠀⠀⠠⠁⠀⠀⠀⠐⡠⠒⠀⠒⠄
⠀⠀⡃⠀⠀⠀⠈⢢⡀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⢠⠏⠀⠀⠀⠀⡆
⠀⠀⠐⡄⠀⠀⠀⠀⠑⢄⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⢀⠔⠒⠳⡀⠀⠀⠀⡸⠁⠀⡠⠒⠉⠉⠳
⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⢑⠄⠚⠃⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⡇⠀⠀⡰⠁⡠⠚⠀⠀⠀⠀⠰⣠⠔⠒⠢⡄
⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⡼⠀⢀⢞⡤⠊⠀⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⢀⣇⡶⠟⠁⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⡠⠜⠓⢄
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⣀⡀⠜⠋⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⡠⠘⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⡔⠁⠀⠁⠐⠀⡀⡀⠀⠐⣂⣴⣃⠀⢠⡖⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⣀⣔⡩⠤⠤⡀
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠉⠀⠀⠈⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠚⠉⠀⠀⠀⠀⢸
⠀⠀⠀⢀⠇⠀⠀⠀⣀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⡜⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠚⠁
⠀⠀⢸⣀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠒⠉
⠀⠀⠈⠻⢿⡶⢤⣽⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠄⠒⠉
⠀⠀⠀⠀⠀⠈⠑⠛⠻⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠊⠁

⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣄
⠀⠀⠀⠀⠀⣰⠏⠀⡀⠀⠈⢳
⠀⠀⠀⠀⠀⡟⠀⢨⡟⢃⠀⠀⣇
⠀⠀⠀⠀⠸⣧⠀⢸⡷⢾⣾⡀⢸
⠀⠀⠀⠀⠀⣿⡄⠈⢷⣄⡈⠻⣾
⠀⠀⠀⠀⠀⡟⢻⣄⠀⠀⣹⣦⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢣⠀⠸⣟⢉⡭⢹⡇⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠨
⠀⠀⠀⠀⠀⠈⢦⡀⣸⠃⠀⣺⠃⠀⣿⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠑⡀⠀⠀⠀⠀⠀⢰⠏⡐
⠀⠀⠀⠀⠀⠀⠈⠻⣿⠀⠸⡟⢀⣰⣿⡄⠑⢄⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⢠⣟⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠹⣷⣁⣬⠿⠶⠟⠷⠶⢦⣤⡀⠀⠀⠀⠀⠈⠧⢠⡆⠀⡾⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠈⠛⠶⠖⣾⠞⠶⠶⣤⡀⠙⢶⡶⠤⠤⢄⡀⠀⠁⠀⡀⠀⠀⠴⠞⠛⠋⠉⠁⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⣀⣠⣎⡔⠋⢻⡄⠀⢱⠄⣀⡎⢹⠆⠀⠘⢋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣥⣉⠨⣇⠀⠘⢯⡀⠊⠀⠈⠙⠋⠀⠀⢸⠋⡿⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢧⡀⠈⠛⠦⣄⡀⢀⡀⠀⠆⠈⠛⠧⣄⡀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣾⣁⡵⠀⠀⣀⣀⠀⠀⢠⠊⢳⡄⠀⢹⠨⠝⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣀⠠⡇⢸⠋⠒⢇⠀⠙⣇⣰⡸⢹⠛⠛⠙⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣒⣶⡶⠶⠛⠃⠀⠛⠁⠀⠁⠀⠉⠓⠲⠶⠾⢧⣀⠈⠻⢶⣦⣴⣮⡶⢦⣄⡀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⣤⡀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣤⣤⣤⣶⠞⢋⣿⣆⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⢿⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠸⣷⡏⠀⣹⡆⠀⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⡀⠀⠀⠀⢸⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⠀⢠⡿⠅⢀⡟⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠐⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢸⣧⠐⣋⣹⡄⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠻⣏⠉⠉⠙⣦⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣦⡉⠛⢷⡀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡿⣶⣾⡇⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢡⣸⡇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⠉⠁⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠴⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⣤⡶⠒⠒⠲⠦⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⢗⣶⣾⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢩⠎⣠⣶⣿⣿⣿⣿⣿⣿⠿⠒⠤⣀⡀⠀⠀⠀⠀⠀⠈⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢠⣿⣾⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠈⠉⠒⠦⣤⣤⣤⣤⣷⡝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⡰⠋⠉⠉⠙⠛⠻⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣆⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢀⠎⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣧⡹⡄
⠀⠀⠀⠀⠀⠀⠀⢰⢁⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⢷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠻⢿⣿⣿⣿⣿⡏⠈⣇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠒⠦⣤⣀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠈⠛⠿⣿⠁⠀⢿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣠⠏⣀⣤⣤⣤⣤⣬⣉⣓⣶⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠤⠤⣀⣀⠀⢀⣾⣷⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡇⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣇⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣴⣄⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣦⣤⣤⣤⣴⣾⡿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣶⣆⣀⢀⣀⣴⣶⣹⠛⠋⠉⠁⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣼⣿⣿⢟⣟⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣦⣤⣤⣶⣾⣿⣿⣷⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢛⣻⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠓⠚⠋⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠖⠒⠒⠒⣤⣤⠤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠁⠀⢀⡤⠴⠒⠉⠉⠉⠀⠈⠙⠲⠤⣉⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁⠀⠀⢀⣠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣬⣻⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⣀⣤⣴⡾⠟⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢳⣴⣶⠿⠟⠋⠁⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⢀⡞⠁⢠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⢠⠏⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣀⣀⠤⠤⠤⠖⠒⠒⠢⠤⠤⢤⣰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⢠⠏⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⢀⡏⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⡄⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⣸⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⣆⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⡧⠖⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⢠⠃⠀⠀⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠦⠤⠤⠤⠤⠔⠻⡛⡄
⠀⠀⢸⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠙⡇
⠀⠀⠘⡄⠀⠀⠀⠘⣾⣄⣠⡤⠤⠖⠒⠉⠁⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢹⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠈⣇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠘⣆⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠘⣆⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢦⡀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠈⢦⣠⣾⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⢀⣠⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢅⠀⠈⠑⠦⣄⡀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡏⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠈⠉⠓⠒⠦⣴⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠈⠻⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠛⠷⠤⠤⠴⠒⠒⠒⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣏⠛⠃⠀⠀⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣷⣿⣽⣦⣰⣿⣦⡐⣦⡀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣧⡀⠀⠀⠀⠠⣤⣤⡽⣿⠻⣿⣿⣿⣟⠧⠿⣷⡀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣦⡀⠀⠀⠸⣿⣿⣦⣇⢿⠿⣧⣹⠷⣰⣍⠃⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⣿⣿⡎⢹⡀⢠⠴⠍⠰⢟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠟⠂⠀⢄⣠⠙⣿⣿⣶⣿⡄⠀⠠⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⠀⠀⠀⠀⠉⠛⠺⣶⣷⣾⣿⡿⣆⣈⠛⠛⠛⠙⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⣴⣾⣿⣿⣯⣭⠐⠉⠉⠁⣠⣤⣾⢷⣦⣸
⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⠀⠀⢀⣴⣿⣿⡟⣿⡍⢈⠙⠃⠀⢀⡈⠂⢉⣵⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠙⠀⣀⠀⠀⣴⣿⣿⣿⣿⣿⣮⠙⠋⠀⣰⣾⣋⣀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠃⠀⣴⣿⣿⠟⠋⣠⣶⠆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⡦⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣯⣄⣰⣿⡟⢁⣠⣼⣿⠁⠈⠉⢈⡁
⠀⠀⠀⠀⠀⠀⠀⣰⣦⡓⠤⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⡾⠁⠘⣭⡍⠁⠀⠀⠁⠸
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠈⠉⠛⠁⣀⡀⣼⠁
⠀⠀⠀⠀⢀⠀⣴⣏⠚⢿⣿⣿⣿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠐⠒⢂⣀⡀⠉⠑
⠀⠀⠀⠀⡄⣼⣿⡇⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠙⠻⠓⠀⢸⡟⡩⠟⠠
⠀⠀⠀⠀⠇⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣄⡠⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠐⠹⣿⣧⠀⠀⠀⠀⠀⠀⢰⡿⠋
⠀⠀⠀⠀⠀⡆⣿⣿⠃⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⢠⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⢸⠘⣿⡄⠀⠀⠀⠀⢀⠏⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠸⠀⢹⣧⠀⠀⠀⠀⢸⠀⠀⠀⠙⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⡇⠸⣿⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣄⣀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⢾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⣶⣶⡶⠶⠶⢶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣀⣀⣉⣀⣀⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡤⠖⣋⠭⢥⢄⡀⠀⠀⠉⠉⠙⠶⣄⠀⠠⡀
⠀⠀⣠⢈⣀⡴⠉⢀⡯⠶⠒⠾⣶⢾⣆⠀⠀⠀⠠⠄⠈⠳⣄⠈⢄
⠀⣬⠇⠀⡟⠀⠀⠈⢄⠀⠀⠀⠀⡩⠋⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠄
⢨⣾⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰
⠀⣷⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢠⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠃⢀⠛
⠀⠸⣏⠦⡀⢁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠁⠀⠀⠀⢆⠂
⠀⠀⠜⢦⣙⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣄⡀⠀⢶⢄⠀⠑⠄
⠀⠀⠀⠈⠛⠛⢷⡦⣭⣩⣒⣒⣒⣒⣦⠀⠻⣷⡀⠑⢿⢢⡈⠳⡕⢄⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⢋⢿⣳⣤⣿⣷⣄⠀⠑⢌⢢⡈⠣⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠍⠩⠻⣾⣷⣄⠂⠙⢽⣦⣌⢮⣧⣄⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⣶⣶⠁⠤⠤⡄⠿⠷⠦⠵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣄⠀⠙⢟⡷⡘⣾⡢⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣓⣴⣶⣝⢜⣮⡙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡆⠑⢝⢦⡑⢟⢄⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧⡀⠀⠳⣽⣌⠳⣽⢦⡀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣶⣤⣼⣿⣷⡌⠷⢽⣿

⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠚⠀⠀⠀⠀⠀⢠⠂⠀⠠⢤⣤⣄⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⣠⣠⠄⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣻⡟⠸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢠⣿⢧⠇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⣠⠆⠀⠀⠀⡴⢛⡇⠀⠀⠀⣾⠀⡦⠂⠑⣾⢏⡞
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠉⠁⠀⠀⠀⠊⢁⣼⠃⠀⠀⠀⣿⣶⠀⢠⣾⣿⣾⡅
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣩⣇⠀⠀⠀⠐⠀⣴⡟⠁⠀⠀⠀⠀⠙⠿⣇⠘⣿⠿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⢉⣿⣿⠄⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠡⡀⠹⣿⣳⠀⠀⠀⠀⠀⣠⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣴⣿⣿⣴⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠱⣴⡉⠉⠓⠀⠀⠴⢯⣠⣤⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⣠⣾⣿⠿⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⢘⣿⣷⣶⣦⣤⣤⣀⣀⣨⡍⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣤⣾⣦⣾⡟⢻⣧⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠋⠀⠈⢻⣿⣿⣏⠀⠀⠈⢿⣄⣆
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠙⠻⠃⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣬⠓⣄⠀⡴⠻⣿⣧⡀
⠀⠀⠀⠀⢸⠀⣾⠟⢤⡀⠀⠀⠀⣡⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣬⢻⣿⡀⠹⣿⡇
⠀⠀⠀⠀⠀⢿⠋⠀⣼⣷⣠⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣇⠀⠛
⠀⠀⠀⠀⣀⠀⠀⣼⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣆⠘⣯⡈⠢⡀
⠀⠀⠀⢸⣟⠀⣰⣿⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡿⣿⣦⠈
⠀⠀⠀⡿⢿⣿⡿⢿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⢯⠉⠀⠈⢻⣷

⠀⠀⠀⠀⠀⠀⣴⣴⠦⠤⠀⢰⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣶⡀⢀⣀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡤⢬⣿⣿⣿⣿⣿⣿⣿⣿⣾⣉
⠀⠀⠀⠀⠠⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣷⣥⡖⠀⡐⢀
⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⢀⣠⣤⣤⣤⣤⣀⡀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣾⣷⣾⣷
⠠⠀⣴⣤⣾⣿⣿⣿⣿⣿⠟⠋⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠠⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠐⣛⣛⣛⡟⠛⠻⠿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠈⠙⠿⣿⣿⣿⣻⡿⠟
⠀⠀⢠⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣦⣬⣙⠻⢿⣇⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⠁
⠀⣴⣿⣯⣷⣤⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡁⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿
⠀⠀⢘⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⣠⣴⣿⣿⡿⡿⠿⠇⠐
⠀⠀⠀⣿⡿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣠⣴⣿⣿⣿⣿⣿⣷⡇
⠀⠀⠀⠀⠃⠃⢸⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠈⠉⠉⠉⠉⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡛⠻
⠀⠀⠀⠀⠀⠀⠸⡿⠛⣻⡿⣿⣿⣿⣿⣿⣿⣦⣄⡀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠃⠀⠀⠀⠀⠀⠈⠲⠾⠿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⣷⠀⠀⠀⠀⠈⠛⢿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒⠛⠒⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠤⠠⠀⠀⠠⡄⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⢀⢠⢰⡼⠚⠉⠉⠙⠢⡀⠐⢥⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⢠⢐⡨⠞⠁⠀⠀⠀⠀⠀⠀⠹⡀⠠⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢀⢐⣨⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠘⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⢲⠎⠀⡜⣵⣯⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢔⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢁⢀⠄⡘⡀⠀⢸⣯⣦⠀⠙⡆⠀⠀⠀⠀⢀⡤⠞⢁⢠⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠄⢎⡥⠞⠉⢷⠲⣀⠈⢻⡮⠁⠀⢸⠀⢀⡠⠖⠁⢤⣰⠜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠌⣨⠞⠁⠀⠀⠀⠈⢣⣙⢷⣼⠇⠀⠀⣿⢏⢡⠠⣐⠼⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠴⠞⠚⠓⠤⢄⡀⠀⠀⠀⠀⢈⠽⠁⠀⡀⣼⣼⡼⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⣰⢰⣶⡶⣦⣙⣦⣀⡤⠞⠁⢀⠠⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⡀⠄⣪⡷⡻⡳⠤⠬⠿⠛⠋⢁⢀⡀⡜⣊⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⠄⢒⡥⠞⠙⠧⣇⣡⢱⢠⢰⠰⠐⣊⡬⠵⠊⠁
⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⢠⠰⣨⠔⠉⠀⠀⠀⠀⠀⠉⠙⠓⡶⣾⡛⡏
⠀⠀⠀⠀⠀⡴⠁⠀⢀⡜⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠻⡲⠃
⠀⠀⠀⠀⢰⠁⠀⢀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⡐⣣⠞⠁
⠀⠀⠀⠀⡇⠀⠀⠼⠐⣇⠀⠀⠀⠀⠀⠀⣠⠖⠉⡀⢆⡴⠊⠁
⠀⠀⠀⠀⢳⠒⣒⡂⠀⠈⠢⢤⣀⡤⠖⠊⠁⡠⢘⡴⠋
⠀⠀⠀⠀⠀⠳⣄⠒⡤⢀⡀⡀⠀⠀⢠⠐⣢⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠥⡮⢝⣁⠥⠖⠋⡁⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⢀⡀⢀⡀⢀⡀⢀⡀⢀⡤⢀⠀⡀⠀⠀⣀⠀⠀⠀⢀
⠀⠀⠀⠂⠵⢊⡴⢊⠴⢊⡴⢊⠔⣣⠔⣉⠔⢋⠴⣊⠕⣫⠔⣩⠖⣩⠔⣩⠖⣩⠞⣡⠖⡡⠞⣡⢖⡥⢊⠔⣫⠞⡵⣫⠞⡵⢋⠞⡵⣣⠞⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠁⠀⠁⠀⠁⠀⠁⠀⠁⠀⠀⠀⠁⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠛⠻⢟⠉⠉⠙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣇⠀⠀⠀⠀⠀⢠⣾⣿⣷⣶⣤⠚⠁⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣄⣀⣀⣀⣰⣿⣿⣿⣿⣿⣿⡀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠏⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣇⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢇⠀⠀⠀⠀⠀⠀⠀⡹⠛⠋⠉⠀⠀⠙⢦⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡘⡄⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⣦⣤⣤⣤⣴⣇⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣧⣄⢀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠖⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠓⠲⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⢉⡁⠀⣠⡀⠀⣤⠀⠀⡠⠄⠀⢠⠄⠀⢀⢤⠀⢀⢄⠀⢀⡉⠛⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⡏⠁⢀⣠⠒⡏⢐⢚⠃⣟⣹⠀⣋⣉⡇⣨⣉⡇⢸⣉⣟⢸⣋⡋⢸⣒⡇⠐⢲⣄⡀⠈⠏⠓⢦⣄
⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠀⠀⠷⠚⠉⠉⠉⢧⠏⠉⠦⠁⠈⠖⠁⠀⠓⠂⠀⠰⠆⠀⠐⠞⠀⠰⠌⠀⠧⠏⠉⠀⠉⠙⠃⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⢀⣴⢿⠁⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣙⢷⣄
⠀⠀⠀⣠⣿⠗⢻⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠑⢽⣷⡀
⠀⢀⣾⠟⠁⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠙⢿⣆
⢀⡾⠃⠀⠀⠀⠘⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠹⣇
⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢹
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸
⢹⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣼
⠀⢻⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣼⠃
⠀⠀⠹⣦⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡇⠀⠀⢠⡾⠁
⠀⠀⠀⠈⠳⣄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠇⣠⡴⠋
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠦⢤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⣤⡤⠴⠶⠖⠛⠋⠉
⣿⣿⣿⢿⣿⠿⢿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣷⣦⣶⣶⣴⣧⣾⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠉⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠑⣿⠀⢀⡤⠴⠶⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠈⠄⠙⢿⡄⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⣠⠷⠊⠁⠀⢀⡀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠈⠂⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣽⠀⢀⡜⠁⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⡿⠠⠚⠛⠷⣦⣀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠉⠢⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⢀⠈⠙⢷⣏⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠁⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠈⠢⡀⠙⣷⡄⠀⠀⠀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠈⢻⣦⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⣡⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠹⣷⡀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠠⡀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠑⢄⠀⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠘⣷⡀⠀⠀⢀⠤⡲⠋⠁⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠂⢌⣆⠘⣿⣶⡨⠕⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠘⣿⠀⠀⠀⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢻⡀⠀⣀⢤⠞⠋
⠀⠀⠀⠀⠀⠀⡠⠃⠰⡀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠖⣩⠔⠁
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠣⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠊
⠀⠀⠀⠀⢰⣯⠀⠀⠀⠀⠱⡀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⢀⣾⡇⠆⠀⠀⠀⠀⠈⠢⡀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣾⡘⣿⡘⢄⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠙⢿⣦⣀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠸⣧⠘⣷⡈⠢⡀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⣿⠛⠷⣮⣄⡀⠀⠀⠀⠀⣀⠔⠈
⠀⠀⠀⠀⠹⣷⡈⢿⣆⠉⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠉⠛⣻⠖⠚⠉
⠀⠀⠀⠀⠀⠘⢿⣄⠙⢷⣄⡀⠈⠁⠀⠀⠀⠀⠀⡰⠃⠈⠒⠤⢠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡙⠻⢶⣤⣄⣀⣠⠤⠞⣁⠀⠀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣄⡈⠙⠻⠶⣤⣄⡠⠜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⠔⠋

⠀⠀⠀⢀⣤⠶⠚⠛⠛⠛⠳⢶⣤⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⢠⡟⠁⢀⣠⣒⡒⠢⡄⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⠳⣦⣀
⠀⢀⣿⠄⠀⣾⠛⠉⠙⣧⢸⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠙⠷⣄
⠀⢸⣿⢠⠀⢻⣄⠀⢀⣿⠟⠀⠀⢠⡟⠉⠀⠀⠀⠉⠓⢄⠀⠀⠀⠀⠙⢷⡀
⠀⠸⣿⣤⠃⠀⠉⠙⠉⠁⠀⢀⡴⠏⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⢸⣷
⠀⠀⣿⡟⠀⠀⠀⢀⣀⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⡄⣿
⠀⢠⡿⠀⠀⢠⡞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠘⣰⡟
⢀⣾⠁⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⢀⣰⡿
⣼⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠁⠀⠀⠀⠀⢸⣿⠁
⣿⡀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⡁⣮⣷⠄⠀⠀⠀⠸⣿⣆
⢻⣧⠀⠀⠀⠀⠙⠤⢄⣀⡀⠀⠀⠐⠈⠀⠀⠀⠐⠘⣏⠢⡱⡄⠀⠀⠈⠛⠶⠶⣦⡀
⠈⢷⡑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⢣⡑⢌⢢⠀⠀⠀⠀⠀⢾⣷
⠀⠈⠛⠷⣵⣦⣤⡤⠀⢀⣀⣈⣡⣌⣉⡲⣄⠀⠰⡉⠀⠑⡙⢄⠢⠑⠀⠀⠀⣇⣿⡿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠛⠿⣿⣧⠀⠙⢄⠀⠘⢎⠳⣄⢌⠀⠀⠈⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠈⠢⡀⠀⠣⡙⢦⡡⡑⠄⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣦⡑⢄⠀⠀⢄⠳⣌⢈⠢⡀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⣎⣀⠤⠤⡑⣜⢦⡑⠑⢄⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣷⠈⠣⡱⣌⠨⠢⡀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠤⢀⠈⢮⠳⡄⢄⢄⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣦⡑⡄⠑⢝⢦⡣⡠⡀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠠⡱⣌⢎⠠⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣍⣄⡀⣀⠈⢌⢣⡑⡑⢄⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣥⣬⣅⠀⠱⡝⢮⡢⠣⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣬⣉⡋⢪⡳⣔⢄⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣇⣀⣱⣌⢷⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⣶⣶⣾⡿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣶⣿⡿⠀⠀⠀⠀⠀⠀⢀⣴⡂
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⢀⣤⣿⣿⣿⣧⣀⣤⣶⣶⣾⣶⣾⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣤⣀⣀⣀
⠀⠀⠀⠀⢀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⣴⣾⣿⣿⣿⣶⣄⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⣤⣴⣦⣭⣭⣛⠻⢿⡆⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠉⠻⠟⣿⣿⠿⠋⠁⢘⣿⣿⣿⣿⣶⣄⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣶⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠛⠛⠿⢿⣿⣿⣿⣷⣤⡀⠙⢿⣿⣿⣿⠿⠋⣀⣴⣾⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣤⡀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⢛⡽⠟⠛⠿⠿⣿⣿⣿⣿⠿⠿⠛⠻⢯⡛⠿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣡⠖⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠲⣌⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣵⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣦⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠟⠒⠦⢄⣀⠀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣶⣄⣀⠀⣀⡠⠴⠒⠻⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠿⠿⠿⠿⠿⠿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⢇⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⡸⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠘⣦⣤⣀⣀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⣀⢠⣴⡇⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣵⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⠦⠤⠤⠤⠤⠤⠤⠴⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠷⢶⣶⣤⣤⣤⣤⣴⡶⠾⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⡿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡼
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⣿⣿⡏⠻⠿⠟⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣹
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻
⠀⠀⠀⠀⠀⠀⣿⣿⣇⣴⣶⣦⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⡿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢷
⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣸
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⡿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻
⠀⠀⠀⠀⠀⠀⣿⣿⡏⠻⠿⠟⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢷
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡾
⠀⠀⠀⠀⠀⠀⣿⣿⣇⣴⣶⣦⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⡿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣼
⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⡿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢷
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡾
⠀⠀⠀⠀⠀⠀⣿⣿⡏⠻⠿⠟⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣼
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⣿⣿⣇⣴⣶⣦⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⡿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣹
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⢼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠉⠋⣿⣿⣿⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⢀⣀⣀⣀⠀⠀⣀⡀⠀⢀⣀⣀⡀⢀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣼⡇⢸⣧⣤⡿⠀⢰⡟⣷⠀⠸⣧⣬⡁⢸⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣼⠟⢸⡇⠘⢷⣠⡟⠛⠻⣧⠲⣦⣬⠟⢸⡇⢿⣧⣤⡄

⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⢀⣀⠀⠀⣀⣀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⢀⠴⠊⠁⠀⠀⠀⠀⠀⠉⠢⣄⠈⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⢠⢋⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠈⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⢠⠇⠘⠉⠚⠘⠁⠀⠐⠖⠂⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠰⢏⡁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⢎⣄⠀⡰⠀⠀⠀⠀⠀⠀⠀⢈⡱⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡇⠀⠀⠀⠀⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠈⠓⢤⣀⠀⠀⠀⠀⢀⣠⠔⠁⢀⡤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠈⠉⠉⠉⠉⠁⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣀⣀⣀⠀⠀⢀⡀⠀⢀⣀⣀⡀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣍⣩⡗⢸⣏⣉⣽⠆⢠⡟⢷⡀⠈⢩⡿⠃⢸⠅⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣥⣬⡿⢸⡏⠙⢷⣄⣾⠛⠛⣧⣴⣿⣤⣄⢸⠆⣿⣤⣤⡀

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠂⠀⠀⠀⠀⣀⣴⡾⠛⠉⠀⢀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠁⠀⠀⢀⣴⣿⠃⠀⠀⠀⠀⢀⣴⣿⡟⠁⠀⣠⣤⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⠋⠀⢀⣴⣿⣿⣿⣄⠀⠀⢀⣠⣾⣿⣿⠀⠀⠈⠻⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠿⠛⠛⠛⢿⣿⣿⣧⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠿⠿⠛⠛⠉⠉⠉⠉⢻⣿⣿⣿⣿⣿⣷⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⣁⣠⣤⣶⣶⠖⠲⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠟⠁⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡵⠊⠉⠀⠀⠀⠱⡄⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠛⠁⢀⣤⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣰⣟⠀⠀⠀⠀⠀⠀⠀⢹⣄⣀⡀⠀⠀⢀⡼⢦⡀⠀⠀⠀⠀⣴⣿⣿⠟⠁⠀⢀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣶⠉⠀⠈⢳⡀⠀⢀⣾⣿⡟⠁⠀⠀⠀⠸⠿⠿⠁
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣧⢀⣀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢳⠀⢸⣿⣿⣦⣀⣀⣴⡀
⠀⠀⠀⠀⠀⢸⣿⣿⠟⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⢸⠀⠀⠻⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⡟⢏⠀⠀⠀⠀⠀⠀⠀⠀⢹⠿⠛⠋⠉⠀⠀⠙⢦⣀⡜⠀⠀⠀⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢣⠘⡄⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⣠⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⢧⡘⢄⣀⣀⠀⢀⣀⡜⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⣀⣴⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⣠⠞⠁⠐⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⡷⠤⠔⠒⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣲⣯⣭⣩⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⡀⠀⠙⠻⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣄⠀⠈⠻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣟⠙⢿⣿⣿⣦⡀⠈⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠙⠿⠃⣀⡉⢻⣿⣿⣆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠛⠿⠃⢀⡈⢻⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠛⠻⠇⢀⡀⢻⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡛⠻⠗⢀⡀⠻⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠘⢿⣿⣷⣦⡀⠻⣿⣾⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⠙⢿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣄⠀⠈⠻⢿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣦⣀⠀⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⠶⠶⠶⠶⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣶⠦⠤⠴⢂⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠉⠀⠀⠀⢀⣠⣴⣾⣿⣿⣧⣤⣤⣴⠶⠖⠒⠒⠂
⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠖⢻⣟⣻⣶⣶⣦⣿⣿⣷⣶⣶⡶⠛⠉⢉⣙⡛⠛⠛⠋⠉⠀⠒⢻⣿⡟⠛⠛⠉⣉⣁⣀⡀⠀⢀⣀
⠀⠀⠀⢀⣤⠞⡫⠏⢱⡤⠛⠭⣁⡣⠬⠥⠪⠦⢝⡿⣿⡏⠠⠚⠛⠿⣿⣿⠟⠛⠒⠶⠤⠶⠾⠿⣟⣓⣁⣀⣉⣿⣉⣉⡭⠉
⠀⠀⣠⠟⢡⢎⣅⡵⡫⢈⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣤⣤⣾⣿⡷⠾⠛⣋⣠⡴⠾⣛⣉⠉⠉⠈⠉⡉⠉⠁
⠀⣼⠋⡐⠇⢆⠞⠖⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠁⠀⠀⣀⣤⣶⡿⠛⣩⡶⠛⢁⣴⣶⣿⣛⣩⠴⠒⠂
⢰⠇⢰⠇⠆⡞⠊⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⠿⣿⣤⣾⠏⣠⡾⠿⠿⠿⣛⣩⡤⠤
⣿⠀⢼⠘⢸⠙⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⢀⣀⣤⠿⠋⣡⣾⣿⣷⣶⠶⠛⠉
⣿⠀⡃⢘⣾⢈⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣀⣀⣀⣤⣾⣿⣿⣏⣁⣀⡤⠒⠁
⢿⡀⢯⢈⢹⢈⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠿⠿⠿⠿⠿⠛⠋⠁
⠘⣧⠸⡀⡙⡎⣄⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⠀⠀⠀⣠⡴⠖⠒
⠀⠘⣧⣓⡀⡹⡄⢌⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⣀⠀⠀⣠⣾⡋⣀⣤⠶⠛⠉⠁
⠀⠀⠈⠻⣷⣀⠜⢦⡢⢙⠦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠉⠛⠿⠿⠿⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠈⠛⠶⣷⣙⡪⢅⣦⢛⡲⢢⣄⣀⣤⠶⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⣋⠬⠭⠬⣍⡛⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣏⠕⠉⠀⠀⠀⠀⠀⠙⢷⣬⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢧⣤⣤⣤⣄⣀⡠⠤⢒⣚⣻⡶⢯⣅⡐⠊⠩⣒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣒⠍⠚⡇⢸⣥⠈⣿⠉⠛⢝⡻⣯⣐⠲⢦⣌⡐⠬⡓⢤⡀⠁⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠔⡁⢄⠳⡹⣮⣿⣤⣿⡤⢄⡀⠉⣪⢲⣙⢳⣬⡙⢦⡈⢪⢳⣄⠀⢣⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠤⣈⠳⠁⠀⠘⢿⣽⣿⣿⣦⠱⠀⠤⠧⢹⡧⡹⣍⢌⢳⡄⠑⢻⡄⠀⡌⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⢐⠒⠠⠀⠀⠀⠠⡸⣇⣿⣿⣿⢀⢄⡒⠂⡀⡇⠠⢹⡄⢆⢿⠀⡇⣷⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠄⠀⠀⠀⠀⠀⠑⠬⠽⠟⠕⠁⡲⢌⠀⡇⡿⠘⢸⡇⠈⢸⡇⠀⣿⢀⢳⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡞⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢬⡑⠀⢡⡇⠀⢸⡇⢠⢹⣥⢃⣟⡋⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠀⠀⢈⣢⡀⠀⠀⠀⠀⢄⠣⡱⠌⠂⢠⢃⡞⢀⡌⡾⠀⣌⡾⢣⡞⠣⡙⠸⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡞⠃⣠⢤⡖⢁⡜⠈⠢⡀⢠⢌⠢⡣⠑⢀⠔⣡⠊⡠⢊⡾⢁⢜⡞⣱⣿⡄⠀⣽⡄⠸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡴⠡⠊⣰⣿⠂⠀⠀⠧⢤⠤⣄⣐⣪⠔⣋⠔⣊⣴⡋⢔⣥⡞⣹⣧⡹⣧⡀⠘⢿⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢊⠜⣡⠟⡉⣻⣷⣾⢛⠟⠫⣿⠋⢡⣈⣥⠲⣿⠸⠠⣾⣿⣿⠇⠸⡈⢿⣌⢳⣄⠀⠙⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⡵⢃⡾⠃⢐⡼⢃⣿⠇⠋⠀⠸⠇⠂⣸⣿⣟⢸⡿⠸⠀⣿⣿⠿⠀⠀⢇⠀⠹⣿⡮⣧⡀⠘⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡡⢊⡴⠋⢀⡼⠋⠀⣌⡗⡘⠁⢰⣟⢘⣶⣾⢿⠀⢼⡏⢰⠚⢿⡱⠆⡆⠀⠘⡄⠀⠈⢿⣕⢅⠀⠀⠻⠶⣄
⠀⠀⠀⠀⠀⠀⣀⠼⣡⠞⣠⠏⠀⣰⠏⠀⠀⣸⡿⢡⠁⠀⡾⠋⠁⣼⡏⣽⠀⣸⡏⡃⠀⠀⣩⢀⠰⠀⢀⢱⠀⠀⠀⠹⣾⣺⣦⠀⠑⣜⣇
⠀⠀⠀⠀⠀⠸⢃⡜⣡⡞⠁⢀⠞⠁⠀⠀⢠⣟⠇⠄⠀⣸⣷⠀⠀⣿⡇⡟⠀⢿⡷⠁⠀⠀⠻⣟⠀⠆⠀⣺⡆⠀⠀⠀⠘⢿⣷⡱⡄⠈⢆⢻
⠀⠀⢀⣴⠖⡡⢀⢴⠉⢀⡴⠁⠀⠀⠀⠀⣾⠏⡐⠀⢰⡟⠁⠀⠀⣿⡇⡇⠀⣿⠁⠀⠀⠀⠀⣿⡸⠸⠀⠉⢥⠀⠀⠀⠀⠈⠻⣶⠿⣆⠀⢻⡶⢤⡀
⠀⠀⣷⢣⠊⡠⡿⠀⡠⠋⠀⠀⠀⠀⠀⢸⡯⠀⠀⠀⣿⡇⠀⠀⠀⣿⠃⡇⠀⣻⡇⠀⠀⠀⠀⠲⡆⠇⠃⠀⠪⡆⠀⠀⠀⠀⠀⠙⢷⡝⣧⡈⠿⣈⢻⡆
⠀⠀⢸⠁⣴⣗⣱⠎⠀⠀⠀⠀⠀⠀⢀⣿⢃⠁⠀⣼⣟⠀⠀⠀⠀⣿⠇⡇⠀⡿⠀⠀⠀⠀⠀⠰⣿⡀⠸⠀⠁⢷⠀⠀⠀⠀⠀⠀⠀⠻⣎⠟⢦⡙⣶⠁
⠀⠀⠘⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣮⡇⠀⠀⣰⠟⠃⠀⠀⠀⢀⣿⠢⠇⡞⠀⠀⠀⠀⠀⠀⠀⠈⡧⠀⠃⢹⡘⡇⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡜⠠⠀⢠⣿⡀⠀⠀⠀⠀⠘⣿⣴⠀⢱⡄⠀⠀⠀⠀⠀⠀⢸⡁⠀⠘⠘⡏⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⣾⣣⠃⠀⠀⠀⠀⠀⠈⢮⢀⡼⠃⠀⠀⠀⠀⠀⠀⠈⢷⣧⣀⠀⢱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠙⢦
⠀⠀⠀⠀⢠⡞⠙⢦⣰⢳⡅⠀⠀⣧⠞⠉⠻⡄
⠀⠀⠀⠀⢰⡅⠀⢸⡏⠯⠀⠀⠒⡏⣐⠀⢰⡇
⢀⠶⠒⢦⡈⠇⠀⢸⣥⠦⠤⠤⣤⡇⠿⠀⠘⡇
⡎⣰⠀⢸⠇⠒⠒⠺⡇⠀⠀⠀⣠⣄⣀⣀⣀⡇
⡇⠀⣀⣈⡂⠀⠀⢸⡇⠀⠀⠀⣼⠀⠀⠀⠀⡧
⡟⠁⠀⠀⠁⠀⠀⠸⡇⠀⠀⠀⠈⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⡠⢄⡀
⡇⠄⠀⠀⡇⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡤⡲⡑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠈⢣⡰⢋⠄⠀⢻
⢳⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⡇⢠⠋⡘⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡜⡼⠀⠀⣸⠣⠜⠀⠀⣾⠖⠒⢆
⢸⠀⠀⠀⣿⠀⠀⠀⠀⢀⣠⡀⠬⠡⠀⠀⠀⣾⡀⠀⠁⠀⣄⡞⠀⠀⠀⠀⠀⠀⠀⠀⡾⣈⡀⠀⢤⠇⣀⠀⠀⢸⣏⠆⠀⣸
⠸⣀⠀⠀⢿⠀⠀⠀⠀⠈⢪⡀⠂⡠⠂⠀⠀⠁⠘⢦⡀⣠⠎⠀⢀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠉⠓⣎⠁⠀⠉⠓⠯⠘⠀⢰⡇
⠀⣷⠀⠀⠈⠑⣆⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⣠⠟⠁⢀⠎⠉⠙⢦⠀⠀⠀⡼⠡⠀⠀⠀⠰⠌⠀⠀⠀⡌⠓⠤⣀⣸⢗⠒
⠀⠧⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⢸⠀⡆⠀⢈⣧⠀⡐⢦⠁⠀⠀⢀⠗⠀⠀⠀⠻⠀⠀⠀⣸⢃⠏
⠀⢸⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢸⢠⠃⠀⠀⠸⣢⡑⠂⠀⠀⠀⡜⠀⠀⠀⠀⠁⠀⠀⣠⠍⠑⠦⣘
⠀⠀⡆⠀⠀⢀⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⡇⠂⣠⡴⠶⠻⠈⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣀⡟
⠀⠀⢣⠀⠀⠀⢸⡇⡀⣀⣀⣀⣉⣀⣀⣁⣀⣿⠀⠀⠀⠀⠀⠘⡎⠁⠀⠄⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⢸⡗⠒⠚⠚⠛⠛⠛⠯⠭⠭⠍⠉⠉⢉⣿⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡽
⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣈⡿⠀⠀⠀⠀⠀⠀⡘⠆⠀⠀⠀⠀⠀⠀⣸⠁⠈⠂⢀⢨⠐⠆⠀⠀⠀⡾⠁
⠀⠀⠘⡏⠉⠉⠉⡛⠉⠛⠙⠙⠉⠉⠉⠙⠁⠀⠀⠀⠀⠀⠀⢀⠷⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠈⠀⠁⠀⠀⠀⢤⡾⠁
⠀⠀⢘⠁⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⣿⢰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣇⠀⠀⡼⠛⠒⠦⣄⣀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⣆⠀⠀⠀⠀⣿⢈⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣼⣁⡀⠀⠀⠀⠉⠛⠺⢧⣄⡀⠀⠀⠀⠀⠰⡱⠋
⠀⠀⣿⠀⠀⠀⠀⣿⢠⠔⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠉⠉⠛⠿⣤⣒⡤⢠⡀⠀⠀⠉⠛⠶⢤⣀⡞⠁
⠀⢠⢹⠀⠀⠀⠀⣿⠸⢀⡃⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠠⡐⠀⠟⠙⠳⣦⣤⣀⠀⠀⠀⠀⣌⠂
⠀⢸⡇⠀⠀⠀⠀⣿⢸⠘⠇⠀⠀⠀⠀⠀⡄⣰⠏⠀⠀⠀⠀⡰⠡⠗⠀⠈⠀⠀⠉⡙⢛⠶⢦⡤⠎
⠀⡇⠀⠀⠀⠀⠀⣿⠘⠀⠀⠀⠀⠀⠀⢈⣼⡧⠀⠀⠀⠀⡰⠁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠱⡼
⢰⠘⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⡼⢣⡻⠵⠀⠀⠀⠀⠀⠀⠀⠀⣶⠃
⣆⣀⣀⡀⠀⠀⣀⣿⣀⣀⣀⣀⣀⡴⠁⠀⠀⠀⠀⠀⡼⢁⠳⡙⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⠟⠀⠀⠀⠀⠀⢈⡼⠡⠢⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠄⠀⠀⠀⠀⢢⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣄⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠦⣤⣀⣀⡼
⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣠⣄⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⣚⡓⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢡⠖⠉⠁⠉⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢐⡁⠀⠀⢀⡜⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⡽⠒⢚⣁⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⣢⠊⠁⣠⠞⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡴⢫⠔⠉⠈⠳⢶⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢃⠀⠀⠀⠀⠀⠀⣇⠮⡀⣀⡠⠔⢁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣀⠀⠈⠳⡀⠘⠀⡠⠔⠒⡲⠊⢁⣴⣉⣥⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠀⠀⠀⢮⣀⢠⡞⢀⡴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢙⡤⠈⠀⠀⡙⠋⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠔⠁⠀⡠⢤⡈⠱⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⢋⡷⠒⠊⢀⡠⠋⠀⢄⠀⠙⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⡤⠊⢀⡴⠋⠉⠒⠉⠁⠀⢰⠀⠈⢧⠀⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡡⠂⣏⣠⡔⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣨⡀⠀⢀⡴⡱⠀⠀⠀⠀⠀⠀⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⣠⣓⣒⡩⠜⠁
⠀⠀⠀⠀⠀⠀⡠⢔⣩⠎⢀⢴⠁
⠀⠀⠀⠀⠀⡜⡴⠁⠈⢊⠇⡼
⠀⠀⠀⠀⠀⡄⠓⠤⠒⣁⠜⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣴⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣿⣟⣉⣉⣉⣉⣙⡛⠒⠲⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠉⠉⠉⠉⠉⠛⠛⠳⠶⢦⣭⣙⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟⣵⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⡴⣿⣿⣿⣿⣿⣿⣿⣷⠶⠶⠾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣠⠏⢀⣽⠟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⡼⠃⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⣛⡿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⢀⡾⠁⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⡾⠁⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⡄
⠀⣼⠃⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⢿⡀
⢠⡏⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡜⣧
⣸⣡⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢻
⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸
⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸
⣿⣿⣿⣿⣯⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣾⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸
⢿⣿⣿⣿⣿⡏⣿⣷⣤⡤⠴⠶⠒⠛⠉⠉⠀⠹⣿⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⢸⣿⣿⣿⣿⣱⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿
⠈⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠈⠙⠶⣤⡀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡟
⠀⢹⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣠⣾⣿⣿⣿⣿⣿⠁
⠀⠀⢻⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠈⢿⡟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠳⡌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠙⣆⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡉⠳⣦⣀⣀⣀⣠⣤⣤⣤⣶⣶⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⣡⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠋⣡⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠞⠛⣉⣡⡴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠞⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⠋⢳⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣉⣉⠳⣿⡷⠛⣉⣙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⢠⠋⠁⢈⡇⢸⠀⢮⠁⠈⢳⠈⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⡴⢇⠘⣄⠀⠻⠴⠋⠳⠼⠃⠀⡼⢀⡷⣌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⡞⠀⠈⢦⡘⠦⣄⡀⠀⠀⢀⣤⠞⣡⠞⠀⠈⣇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣧⠀⠀⠀⠙⠢⢌⡉⠒⠚⣉⡤⠞⠁⠀⠀⢀⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⠣⣄⣀⢀⣀⡤⠛⡹⣏⠣⣄⣀⠀⣀⣠⠞⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣭⣭⣭⡴⢋⣠⣈⠓⣦⣭⣍⡩⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡉⢩⠤⢤⡉⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠻⣀⣠⠇⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⠤⠴⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠛⠛⠛⠛⠛⠛⢟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡭⠗⠀⢠⡖⠒⢟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢲⣆⠀⢸⡇⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠁⠀⠸⣧⣀⣽⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⣀⣀⣀⣀⣀⣯⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠶⠶⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⠶⠶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠤⠤⠞


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡟⣿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⠉⠙⠛⠛⠋⠛⠁⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⡿⠿⠝⠛⠟⠒⠶⠶⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣦⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠈⠉⠉⠁⠈⠉⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣾⣿⡿⠛⠋⠉⠉⠉⠉⠛⠻⢻⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠻⣿⡟⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⣿⣿⣽⣿
⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⣀⣴⢰⣿⣿⣵⣾⣿⣴⣶⣶⣦⣤⣤⣤⣀⡀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠛⠛⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠉⠄⠀⡀⠀⠈⠉⠉⠛⠛⠿⣶⣶⣀⠀⠸⢻⣿⣿⣿⣿⣏⣿⣷⣤⡀⠀⠀⢩⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠈⠻⢿⢹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠠⠄⠀⠁⠀⠀⠉⠛⢿⣾⣿⣿⣿⣿⣴⣿⣷⣿⠟⠋⠀⠀⠀⠛⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣷⣾⣿⡆⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⣽⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⡿⣿⣆⠀⠀⠀⠄⠀⠈⠀⢀⡀⠀⠀⣀⣾⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠰⢀⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢴⣿⣷⣦⣀⡀⠀⠀⠀⠈⢁⣠⣴⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣶⠾⣗⣰⣂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣂⣀⣀⣀⣀⣀⣀⠀⠀⣀⡀⠀⣀⠀⠀⠀⢦⣬⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠚⠋⠉⢉⣭⠟⠋⠉⠉⠉⠙⠛⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⢀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣇⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣠⠤⠤⠄⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠛⠿⢶⣦⣤⣤⣤⣾⣯⣤⣤⠶⠶⠟⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣦⣹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠉⠀⣽⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣿⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣀⣤⣴⣶⠿⠿⠟⠛⢛⣿⠟⠛⠛⠛⠛⠛⠿⠿⠿⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣷⠿⠛⠉⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠄⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣠⣶⡿⠿⠿⣿⡿⠿⠿⠿⢷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⢀⣶⠟⠉⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠉⠛⢿⣥⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢿⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⢄⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣄⡀⣴⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣦⡀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣈⣝⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣟⣦⣄⣀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⣤⣶⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣿⣷⣐⣿⣶⣦⣥⣿⣶⣿⣷⣿⣿⣿⣿⣿⣿⣿⣥⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠩⠽⡿⢟⣿⠿⣿⣿⣿⣿⣿⣿⡿⠿⢿⠿⠿⡿⠻⢿⠛⠛⠋⠉⠈

⠀⠀⠀⠀⠀⠀⠀⡐⠉⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠣
⠀⠀⢀⠒⠀⠢⡀⡀⠀⠀⠘⠀⢀⠔⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠢⠀⢀⠁⠀⠀⠰⢀⠔⠀⠐⠄
⠀⠀⠀⡀⠀⠀⢠⠁⠀⠀⠀⡆⠌⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⢸⠀⠀⠀⠐⡌⠀⠀⠀⡆
⡀⢀⠀⢡⠀⠀⠀⡈⡀⠀⠀⠇⡆⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢘⠀⠀⢀⠃⠀⠀⠀⡐⠀⡀⠄
⡇⠀⠑⡄⢂⠀⠀⠃⠡⠀⠀⢀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠸⠘⠀⠀⡌⠸⠀⠀⠠⢁⠎
⢃⠀⠀⠘⡌⡀⠀⠘⠔⠀⠀⠉⠀⠀⠀⠀⡔⠉⠑⠀⠀⠰⠁⠉⢆⠐⠀⠀⠀⠉⠀⠀⠓⠂⠀⠀⢃⠎⠀⠀⢠
⠈⡄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⢀⠀⠀⢡⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠆
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⢠⠃⠀⠀⠀⠡⠀⠀⠈⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⡀⢤⢪⠀⠀⠀⠀⠀⠀⠀⠀⢀⢓⠤⢀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊
⠀⠀⠀⠀⠀⠏⠑⠂⠂⢒⠀⠈⠅⠀⠀⢣⠱⠀⠀⠀⠀⠀⠀⠠⢁⠊⠀⠀⠨⠉⠁⠒⠀⠐⠊⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⢡⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠠⣜⠀⠀⠀⠂⠀⠀⠅⠀⠀⡅⠀⡄
⠀⠀⠀⠀⠀⠢⢀⣄⣀⣀⣄⠀⠬⠄⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠰⠤⠀⢠⣀⡀⢠⠄⠄

⠀⠀⠀⠀⢀⡤⠒⠒⠒⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠒⢲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⠤⢤⡀
⠀⠀⠀⡴⢡⠔⠊⠉⠓⢄⠀⠈⠳⡄⠀⠀⠀⠀⢀⠾⣇⠀⠀⠀⠀⣼⠀⠈⢣⡀⠀⠀⠀⠀⠀⢀⡤⠚⠁⣠⣴⣤⠐⠂⠀⢹
⠀⠀⢸⢠⠃⠀⠀⠀⠀⢸⡀⠀⠀⢱⠀⠀⠀⠀⡞⠀⠈⢷⡀⠀⠀⣿⠀⠀⠀⣇⠀⠀⠀⠀⢠⠋⠀⣴⣿⠟⠛⠈⠀⠀⠀⣸
⠀⠀⢸⢸⠀⠀⠀⠀⠀⠸⡇⠀⠀⣸⠀⠀⠀⠀⣇⠀⠀⠈⡇⠀⠀⠹⣇⠀⠀⡟⠀⠀⠀⢠⠇⢀⠔⠉⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠈⢯⣇⠀⠀⠀⠀⠀⠳⣄⣠⠃⠀⠀⠀⠀⠘⣆⠀⢀⡇⠀⠀⠀⠈⢲⡴⠁⠀⠀⠀⢸⢠⠊⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠙⠳⢄⣀⣀⣠⠤⠚⠁⠀⠀⠀⠀⠀⠀⠈⠓⠾⣅⣀⣀⣠⠴⠊⠀⠀⠀⠀⠀⠘⠧⣄⣀⣀⣀⡤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⢀⡴⠿⠥⠤⠤⣽⡉⡶⣄⠀⠀⠀⠀⠀⠀⢠⠖⢹⠿⠻⡏⠉⠙⢢⡀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠀⠌⠁⠍⠳⢦⡀
⠀⠀⢠⠏⠀⠀⣀⠤⠖⠁⡸⢣⡜⣆⠀⠀⠀⠀⢰⣁⣴⠁⠀⢀⣴⣤⣤⠊⠛⡄⠀⠀⠀⠀⡼⠁⠀⠀⠠⠀⠀⠀⠀⠀⢱⡀
⠀⠀⢸⡠⠚⠉⠀⠀⠀⡴⠁⢸⠙⣼⠀⠀⠀⠀⣿⣿⣿⠗⠂⠹⣿⣿⣿⠀⠀⣷⠀⠀⠀⠀⡇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠸⡀⠀⠀⠀⣠⠎⠀⠀⡄⠀⡟⠀⠀⠀⠀⢳⠉⠫⡀⠀⢀⠎⠀⠀⢱⣾⠇⠀⠀⠀⠀⢇⠀⠀⠀⠀⠠⠀⢀⠀⠀⢨⠃
⠀⠀⠀⠙⢄⡠⠚⠁⠀⠀⢠⣣⠞⠁⠀⠀⠀⠀⠀⠳⣄⣸⣿⣿⣦⠠⠄⡺⠏⠀⠀⠀⠀⠀⠈⠣⣄⠀⠂⠀⠄⠀⢀⡵⠋
⠀⠀⠀⠀⠀⠉⠒⠒⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠛⠉

⠀⠀⡠⠐⠐⠠⠀⠀⠀⠀⡀⠐⠂⠤⡀⠀⠀⠀⢀⠠⠀⠀⢠⠀⠀⢀⠤⠐⢲⠤⠀⠀⠀⠀⠠⠶⠆⢀⠀⠀⠀⢀⠤⠒⠢⢄
⠀⠈⠀⠀⠀⠀⠀⠀⠀⡌⠑⡀⠘⡀⠀⠀⠀⡀⠡⠶⠋⠀⢀⠀⠀⡀⠀⠀⠀⠑⢡⠀⠀⣶⡆⠰⣷⠎⠁⠀⠀⠃⠀⠀⠀⠀⠂
⠀⠘⢄⠀⠀⢡⠔⠀⠀⠀⢀⡇⠀⢑⠄⠀⠀⡄⠀⠀⢀⠀⠂⠀⠀⠡⡀⠀⠀⡀⠈⠀⠀⠉⣠⣤⡀⡸⠃⠀⠀⠢⡀⠀⠀⡠⠁
⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢯⣍⣉⣙⡒⠦⣄⡀⣀⡤⠶⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠉⢉⣉⣷⣤⢿⡁⣀⡴⢋⡙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠰⣏⣁⣠⠼⠖⠋⠁⠀⠀⠙⡌⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠠⢤⣀⠀⠀⢹⣇⠰⡂⠀⠀⠀⠀⠀⠀⣸⢌⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢴⣶⠾⠛⠉⠀⠀⠀⣸⢳⢖⣡⣿⡆⢳⡀⠀⠀⠀⠀⣀⣴⠞⠙⢦
⠀⠀⠀⠀⣠⠶⣟⠉⢀⡴⠋⠀⠀⠀⠀⠀⠀⢠⣃⣈⡥⢿⠺⢿⡄⢉⣠⠴⠖⠛⠉⠀⠀⠀⠈⠳⣄
⠀⠀⢀⣾⠥⢄⠀⠀⠉⠀⠀⠀⢠⠇⠀⠀⢀⠞⠛⠒⣾⣏⣃⠀⡿⢿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⢀⡠⠋⡤⠤⢿⠹⡈⠉⡼⢁⠞⠀⠙⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠈⢷⣀⡶⠂⠀⠀⢀⡴⠛⠒⠚⠋⠀⠈⠋⠉⠿⠷⠇⣸⠟⠁⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠈⠻⢧⣤⣤⠶⣿⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⢤⣌⣉⣓⡲⠤⠤⣄⣀⡀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⢸⣆⠹⢦⣀⠀⠈⠉⠙⠒⠦⢤⣉⠉⠛⡇⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣴⢀⠩⠟⣳⠶⣤⣤⣀⣀⣈⣉⣹⢿⡀⠀⠀⠀⠀⠀⡏⣠⣾⠛⢧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⠀⠀⠃⠀⠁⠀⠀⠙⠉⠁⠈⣼⢷⣄⡀⠀⢀⣰⣷⡏⠈⠀⢸⠇⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠈⠉⠉⠉⠁⣿⠹⢶⠶⡏⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣈⠀⠀⠀⠀⢀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠛⠉⠉⠉⠉⠙⠓⣲⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣧⣄⣀⣀⣤⣤⣤⣤⣄⣴⣾⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠏⠉⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠘⣦
⠀⠀⠀⠀⠀⠀⠀⢀⡟⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠉⠉⠉⠻⣄⠘⣇
⠀⠀⠀⠀⠀⠀⠀⣼⢀⣿⠛⠉⠁⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠹⣦⢹⡄
⠀⠀⠀⠀⠀⠀⠀⡏⣸⠇⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠸⣾⡇
⠀⠀⠀⠀⠀⠀⠀⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣯⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣄⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⣤⣴⠶⠞⠛⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⠟⠋⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣄⣀⣀⣠⣴⣿⣤⣤⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠘⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠀⠀⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠀⠀⣀⣀⣀⣀⡀⠀⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠀⣀⠔⠊⠁⠀⠀⠀⠀⠈⠑⠢⣀⠀⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⡞⡈⣀⣀⠀⠐⠒⠒⠠⠤⣀⡀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠉⠲⢄⡀
⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⢠⠁⠈⠀⠀⠀⠈⠉⠉⠒⠒⠤⢈⡑⠢⢄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠑⠢⣄⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⣹⠁⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠉⠒⠤⠄⣀⣀⠠⠤⠒⠉⠀⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⢀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣟⣿⣻⣛⣻⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣻⣿⣟⣻⣿⣻⣿⣟

⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣉⣙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣌⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⡉⠛⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⣴⠛⠛⢶⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠻⣆⠀⠀⣙⣦⣀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣀⠈⠳⣾⡉⣉⣉⡛⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠈⠛⠛⠋⣻⡆⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣻⡶⠶⠞⢹⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣿⠀⠀⠀⣾⠁⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠗⠶⠶⣿⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠻⣤⣤⣄⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠳⣦⣤⣤⣽⣇⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢻⡄⠀⢹⡆⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠈⣷⣀⣄⣿⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠟⠁⣀⣀⣽⠇⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢠⡞⠉⠋⢁⣠⢾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢰⡏⢠⡟⠙⠛⠛⣿⠁⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠙⠻⢾⣤⣾⠁⠀⠀⣸⠏⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢁⡴⠚⠚⢶⣤⡈⠙⠶⣄⢠⡟⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⢻⣟⠛⢿⣅⠹⣷⣤⣼⠟⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⢻⣦⡉⠻⣦⢹⣧⣤⣀⣀⣠⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡍⠙⠶⣿⡌⣿⣤⣽⠟⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⡇⠀⠀⠈⣷⢸⣦⣤⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣴⠟⠛⠛⠶⣾⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣀⡴⢶⣶⣤⠾⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠞⠻⢦⣤⣤⣤⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣠⣾⣥⡶⣤⣴⡋⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠹⣯⣤⠶⣤⡈⠻⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⣀⣠⡾⢿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⡏⠀⠀⢺⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣧⣤⣤⣾⡀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⣯⣀⡀⠈⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡈⠛⠛⣷⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈⡟⠛⠛⢻⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣇⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠹⢦⡴⠟⠀⣸⠇
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⣀⣀⣴⠾⠋
⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⢤⠴⠖⠒⡻⣋⢛⣒⡶⠙⢢⠀⠀⠁⢔⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⠁⠀⠀⠀⣀⣤⠴⠚⠋⢉⡀⣠⠀⠲⠛⢉⠋⠁⡀⢨⣿⣛⣛⣉⠃⠃⠉⠙⠛⠛⠛⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⢠⡀⠀⣀⣠⡴⠞⠉⠀⣦⠀⠘⠁⠈⢀⠀⡀⠂⠀⠂⠁⠁⢀⠄⢀⠭⣹⠇⠰⡀⠀⠀⢳⡄⠀⢝
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠁⠀⢀⣠⠴⠞⣫⠣⡤⠓⡈⠁⠀⠁⠀⡄⠀⠀⠆⢀⠐⠀⣈⣡⠤⠶⠿⣷⠚⠡⡀⣠⢳⠀⠀⠀⢣⢆⣼
⠀⠀⠀⠀⠀⠀⠀⠔⠁⠂⢀⡶⡏⠰⠈⠛⠉⠀⠀⠀⠠⠀⠀⠀⠣⠀⠀⢐⢈⣄⡴⠚⠉⡅⠀⠄⠀⢳⡄⢸⢴⡿⡣⢇⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠔⠀⠀⠀⠙⣿⣷⡿⢇⡀⠀⠐⠈⢀⠐⢀⣩⣰⣤⣦⡵⡾⠛⢋⢁⣦⢦⠐⠅⢰⣠⢌⠀⢩⡀⠛⣪⢄⢬⡆⠀⠀⣿⡟
⠀⠀⠀⢠⠎⠀⠀⠀⣸⡾⠏⠁⣠⢞⡁⡀⠀⣀⣴⡟⠛⠋⠁⠁⠀⢀⠐⠘⠁⠁⠹⠁⠂⠈⢢⢏⠆⠀⡀⠿⢑⣵⣿⣿⣿⠀⠀⡾⠁
⠀⠀⡰⠃⣰⣀⡤⠖⠁⢹⠀⠸⠧⡮⣽⡿⠟⡉⠀⢀⢸⣀⣀⠐⠀⢣⠀⡀⣴⡀⠶⢪⣀⡰⣦⠴⣦⢈⣋⣤⣿⣿⣿⣿⣿⠀⡰⠁
⠀⢰⠃⣤⡿⡇⠀⠀⠀⠈⣆⡿⠧⠡⡚⢘⣤⢂⡶⡜⡆⠁⢈⡤⠶⢆⠊⠈⠐⠅⠀⢍⣀⣾⣾⣥⣴⣿⣿⣿⢿⣿⣿⣿⣿⠞⠁
⢠⣧⠞⠁⠀⠸⡀⠀⠀⠀⠸⡘⢰⣇⢓⠡⢱⣄⣘⡱⠜⡀⢧⢁⡔⣼⣡⣥⠠⢶⣴⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⡿⠃⠀⠀⡀⠀⢧⠀⠀⠀⠀⢡⡙⢹⣯⣯⠌⣿⣮⣴⢴⡄⠨⣮⢹⣌⣿⣿⣾⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣧⣠⣠⣦⣡⣂⠈⣆⠀⠀⠀⠀⢻⣃⣨⣦⣠⣾⣿⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣿⣟⣿⠟⠋
⠈⠿⣿⣿⣿⣿⣷⣿⣦⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣷⣿⣿⡿⠛⠁
⠀⠀⠈⠛⠿⣿⣿⣿⣿⣆⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠷⠄⣀⣀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⠁