Bows Emojis & Symbols

⠀⠀⢀⣠⣤⡀⠀⠀⣠⣤⣀⡀⠀⠀⢀⣤⣀⠀⣠⣤⡀⠀⠀⠀⢀⣀⠀⠀⠀⣀⡀⠀⠀⠀⣠⣤⣄⠀⣠⣤⣄
⠀⠀⢈⣿⣿⣿⢿⣿⣿⣿⣏⠀⠀⠀⢺⣿⣿⢿⣿⣿⠇⠀⠀⠀⣿⣿⣿⠶⣿⣿⡟⠀⠀⠈⣿⣀⣉⠶⣉⣀⣹⠁
⠀⠀⠈⠉⠛⠁⠀⠀⠉⠛⠉⠁⠀⠀⠀⢰⣿⡄⠟⠃⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠀⠀⠀⠀⠈⣿⡟⠀⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠀⠀⠀⠻⣧
⠀⠀⠀⣰⡶⢦⢀⣴⣶⡄⠀⠀⠀⠀⢀⣶⡄⢀⣤⡆⠀⠀⠀⠀⣰⣦⡄⠀⣠⣶⡆⠀⠀⠀⣀⣀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠙⠷⣤⣯⣤⠾⠏⠀⠀⠀⠀⠘⣿⣿⣿⣿⡟⠀⠀⠀⠀⠙⠛⣻⣶⣿⠛⠋⠀⠀⢸⣿⣿⣿⡶⢾⣿⣿⣷
⠀⠀⠀⠀⢾⡟⠀⢻⠧⠀⠀⠀⠀⠀⠀⢾⠟⠘⢿⠂⠀⠀⠀⠀⠀⣴⣿⠋⢻⣷⡀⠀⠀⠀⠛⠛⠉⠀⠈⠙⠛⠃


⠀⠀⢀⡛⠲⡄⠀⡠⠞⢋⡀⠀⠀⠐⣿⣿⣦⣤⣾⣿⡇⠀⠀⠀⣶⣶⣄⣤⣠⣶⣶⠀⠀⢰⣿⣿⣆⢀⣾⣿⣷
⠀⠀⠈⠉⠉⡟⠙⢏⠉⠉⠀⠀⠀⠀⢙⣿⠇⠀⢿⣟⠁⠀⠀⠀⣿⣿⣿⠛⣿⣿⣟⠀⠀⠸⣿⣿⣿⢿⣿⣿⡿
⠀⠀⠀⠀⢸⠃⠀⠘⣆⠀⠀⠀⠀⠀⣾⡏⠀⠀⠘⣿⡆⠀⠀⠈⢻⣿⠇⠀⠹⣿⡟⠀⠀⠀⢰⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠃⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⢠⣤⡀⠀⢠⣤⡄⠀⠀⠐⠤⣈⣱⣦⣤⣶⣉⡠⠔⠀⢀⣿⣷⣄⠀⣴⣾⣧⠀⠀⠀⣾⣿⣦⣠⣾⣿⡆
⠀⠀⠀⣿⣿⣿⣶⣿⣿⡿⠀⠀⠀⠀⠀⠞⠁⠀⠈⠂⠀⠀⠀⢸⣿⣿⡛⠿⣛⣿⣿⠀⠀⠀⢿⣿⣿⢻⣿⣿⡇
⠀⠀⠀⠈⣽⡿⠉⢿⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⣏⠀⠀⣴⣿⣿⡟⠈⢿⣿⣷⡄
⠀⠀⠀⠈⠿⠁⠀⠈⠿⠁⠀⠀⠀⠀⠈⡆⠀⠀⠀⡜⠀⠀⠀⠀⠉⠻⣿⢸⡿⠛⠁⠀⠀⠀⠹⠋⠀⠀⠈⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣴⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣦
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢀⠀⢀⡀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⢍⣴⡿⠀⠀⢿⣦⡩⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣰⠿⢿⣿⣿⣿⣛⣛⣿⣭⡷⣟⣵⣿⠟⠀⠀⠀⠀⠻⣿⣮⣻⢾⣭⣿⣛⣛⣿⣿⣿⡿⠿⣆
⠀⠀⠀⠀⠁⠿⣿⣿⡿⠿⢟⣻⣭⣷⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣾⣭⣟⡻⠿⣿⣿⣿⠿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

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



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

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



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠒⠒⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⢀⡞⠙⡆
⠀⠀⠀⠀⠀⠀⠀⡴⠛⢧⣀⠀⠀⠈⠙⠲⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣀⡠⠖⠋⠶⠂⢹
⠀⠀⠀⠀⠀⠀⡼⠀⢀⡀⠈⠛⠶⢤⣀⡀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣁⣀⣀⡤⠤⠶⠒⠋⠡⠀⠀⠁⠀⠀⢸⡆
⠀⠀⠀⠀⠀⢸⠃⠀⠫⠎⠀⠀⢀⡀⠀⠉⠙⠒⠦⢤⣬⣷⣤⡴⠚⠛⠳⡦⠴⡶⢶⡟⠉⠉⠀⠀⠂⠀⠀⠀⣞⠅⠀⠀⠰⡶⠂⠈⡇
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠣⠃⠀⢀⡀⠀⠀⣠⢄⠀⠀⠹⡄⠀⠀⠘⠀⠁⠈⡇⠀⣀⡀⠸⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⡇⠀⠮⠇⠀⢀⣀⠀⠠⠀⠓⠃⠀⠀⠙⠊⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡷⠈⠓⠉⠀⠀⠀⠀⠀⠀⢀⡠⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⢀⠀⠀⠸⠜⠀⠠⠀⠀⣀⠀⠀⢠⢒⠀⠀⣠⣷⣄⣤⡤⠶⢶⠞⠓⠦⣀⠀⠀⠚⠜⠀⠀⠀⠈⠒⠁⠀⠠⡨⠀⣼
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠐⠀⠀⠀⠀⠀⠘⠤⠀⠀⠀⠁⣠⠞⢹⠁⠀⠀⠀⢠⠋⠀⠚⠃⢸⠉⠓⠦⠤⣀⡀⠀⠀⠀⠀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⢄⠱⠀⠀⠀⠀⠀⣀⡴⡾⠃⠄⣾⠀⠀⠀⢠⠇⠀⡄⠄⠀⢸⠀⠀⠀⠀⠀⠉⠉⠛⠓⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠐⠀⠀⠄⠀⣀⡴⠞⠉⡼⢁⡀⠀⣇⠀⠀⠀⢸⠀⠁⠈⠁⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠖⠚⠉⠀⠀⣸⠃⠓⠃⠀⢿⠀⠀⠀⢸⠀⠯⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠘⣇⠀⠀⠸⡆⠀⠀⢀⡀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡤⡀⠀⠀⠀⠘⣦⠀⠀⢳⠀⠀⠧⠃⠀⢀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠒⠁⠀⢐⡱⢀⢸⠆⠀⠘⡇⠠⠀⠀⠀⠪⠃⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠈⠀⠀⣀⡴⠋⠀⠀⠀⣷⠀⢀⢢⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣠⠴⠒⠋⠁⠀⠀⠀⠀⠀⢹⠀⠈⢁⣠⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢚⣵⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⣮⡻⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠔⣩⣾⡟⣱⣿⠇⡿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠹⣿⣝⢿⡻⣷⣮⣄⠀⠤⣮⣤⣬⣀⠉⣀⣁⣁⣈⣁⣈⣆⡀
⠀⠀⠀⠀⠀⠰⣍⠙⠛⢇⠀⢻⣿⣷⡙⢦⡙⠻⠛⢀⣹⠏⠉⠉⠻⠿⠿⠿⠟⠛⢉⣠⣼⡄
⠀⠀⠀⠀⠀⠀⢣⠳⣄⠸⠀⠀⠉⠻⠈⠀⠁⣠⣶⣿⣷⣿⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⠏⡈
⠀⠀⠀⠀⠀⠀⠈⢧⠙⢆⠀⠀⠀⠀⠉⠉⡩⣟⣯⣭⡭⠴⣶⡀⠀⠀⢿⠟⣩⣿⣿⠏⢠⣧⢀
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⣀⠄⠀⢶⣾⣽⣻⣿⡷⠯⠷⠌⠛⢳⡀⠀⢀⣾⢿⣾⣿⣦⠏⡻⠐⠮⣔⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣇⢨⣿⣾⣿⣿⣝⠣⣍⠙⠂⠀⢠⣿⣿⡄⠈⠁⠋⢿⣿⡏⠀⠻⣶⣦⡠⣽⣾⣵⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢦⠹⣿⣿⢝⠻⡷⢤⡀⠀⢀⣸⣿⣿⣿⣆⠀⠀⣿⣿⠀⠀⡀⠈⠙⢿⣦⡙⢿⣿⣿⣿⣲
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⢧⡘⢿⡣⠁⠀⠀⢀⡴⠁⠛⣿⣿⣿⣿⣷⡀⢿⡏⠀⠀⣛⣿⣶⣤⣉⠛⢦⣹⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⢸⣷⡈⠁⠀⢀⡤⠋⢠⣤⡀⢸⣿⣿⣿⣿⣷⡘⠀⠀⣤⣬⣿⣿⣿⣿⣷⣦⣝⡆
⠀⠀⠀⠀⠀⢀⠞⠀⠀⣀⡤⠂⠈⣿⣿⡤⠔⠋⢀⣘⣿⣿⣿⠀⢻⣿⣿⣿⢹⠇⠀⠨⣭⣭⡽⣶⣚⡿⠶⠒⠚
⠀⠀⠀⠀⡠⢋⣠⡴⠟⢋⣀⣀⠀⠘⣿⡙⠳⢶⣿⣶⣭⣭⣉⡁⠀⠻⣝⢛⣼⣀⣬⣵⣾⠿⠛⠁
⠀⠀⠀⠀⢧⣩⣷⣞⣻⣿⣯⣭⡤⠀⠘⢷⠰⣄⠀⢉⠉⠛⠛⠛⠛⠃⠈⣹⣷⡿⢹
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠉⠉⠀⢈⡇⢿⠀⣼⢀⣧⠆⣴⣷⢂⣾⣿⠟⠁⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡆⣿⣸⣿⣼⡿⣳⣿⠿⢋⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣷⣿⣿⣿⢟⠜⠁⢄⢲⣼⡆⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⣿⠗⠋⠁⠠⡙⠷⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠑⠪⢿⠾

⠀⠀⠀⠀⠀⠀⠀⣠⡿⢻⣿⠿⠷⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⣾⡿⣿⡟⢿⣄
⠀⠀⠀⠀⠀⠀⣰⡟⢁⡾⠁⠀⠀⠀⠈⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⠁⠀⠘⣿⡜⣷⠈⢻⣆
⠀⠀⠀⠀⠀⢠⡿⠁⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠘⣷⡘⣇⠈⢿⡄
⠀⠀⠀⠀⠀⣾⠇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⣀⣀⣀⣀⣀⣀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠹⠀⠘⣷
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⣦⣤⣀⣿⠋⠉⠉⠉⠉⠙⣿⣀⣤⣴⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣿⡄
⠀⠀⠀⠀⢸⡿⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⢀⠀⠀⠀⠀⠀⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀⠀⢿⡇
⠀⠀⠀⠀⢸⣇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⠀⠀⠀⢸⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⡇
⠀⠀⠀⠀⢸⣿⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⣻⣿⣿⡇⠀⠀⣸⡇⣿⣿⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⣿⡀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣡⣴⢞⣻⡿⠿⠿⠶⠶⠿⠿⢿⣽⡳⣦⣈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⢹⣇⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⢛⣩⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣝⡳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⣸⡏
⠀⠀⠀⠀⠀⠀⢿⡆⢿⣆⠀⠀⠀⠀⠀⠐⠋⣉⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣬⣙⠲⠦⠄⠀⠀⠀⠀⣼⡏⢰⡿
⠀⠀⠀⠀⠀⠀⠈⢿⣎⢻⣧⣀⣀⣠⣴⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣦⣄⣀⣀⣼⡟⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡏⡏⠁⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠈⢮⣦
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠹⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⢳⣳⡀
⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣳⡀
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣮⡳⣄⠀⠀⠀⠀⠀⠀⠀⣰⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢣
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡕⢦⣀⡤⠤⣤⡞⣿⣿⠁⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣿⡇
⠀⠀⠀⠀⠀⡾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣼⣿⣿⣿⡞⠀⠉⠹⣿⣯⣻⣿⣦⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣿⣿
⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣿⣿⣿⣿⠀⠀⠀⢨⣿⣿⣽⣿⣅⣀⣠⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⢰⠃⡇⠀⠀⠀⠀⠀⠀⠀⢤⣠⠤⠾⠿⠿⢿⣿⣿⣻⣿⡀⠀⠀⢸⣿⣿⣿⣿⣟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠸⡀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠶⣾⣿⣿⣿⡟⠛⠃⠀⠀⣽⣿⡇⣿⣿⣿⠶⠦⠤⠄⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⡇⣷⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⣿⣿⣿⢉⣧⠀⠀⢀⣤⣿⣿⢧⡿⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⢿
⠀⠀⠀⠀⠀⣧⣿⣿⡞⡄⠀⠀⠀⠀⠀⢀⣤⣾⣿⢟⣫⣶⣿⣿⣷⣾⣛⡿⣭⣶⣿⣶⣾⣿⣿⣿⣿⣦⣀⣀⣀⣀⣤⣾⣿⣿⣿⣿⡼
⠀⠀⠀⠀⠀⢸⣼⣿⣿⣽⣦⣀⣀⣤⣶⣿⢟⣫⣶⣿⣿⣿⡿⢻⣿⣿⠇⠀⢸⡿⢿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡇
⠀⠀⠀⠀⠀⠀⢧⢿⣿⣿⣿⣿⡿⣟⡭⠚⣹⣿⣿⣿⣿⠏⢠⠏⢻⣿⠀⠀⠈⡆⠀⠀⠑⢌⢻⣿⣷⡌⠙⠿⣿⣿⣿⣿⣿⣿⠛⡜
⠀⠀⠀⠀⠀⠀⠈⣎⠙⠿⣻⠽⠊⠁⢀⣾⣿⣿⣿⣿⠏⠀⠈⠀⠸⡏⠀⠀⠀⣇⠀⠀⠀⠀⠀⠹⣿⣿⣦⠀⠀⠉⠛⠿⣟⠁⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠒⠋⠀⠀⠀⢀⢾⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⡇⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠘⣿⣯⢳⡀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣠⣿⣿⡂⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⣾⡿⣛⠽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⣻⣿⣿⣷⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢳⠀⠀⡠⢞⡵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⣿⢿⣿⢆⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢭⡳⢤⠀⠀⠸⡇⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢕⣄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⠛⠛⠛⠻⠷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠾⠿⠿⠿⠿⣶⣄
⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣠⣴⡶⠶⠶⢶⣦⣤⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠿⢶⣤⣤⣤⣤⣴⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠻⣧⣄⠀⠀⠀⠀⠀⢀⣀⣤⣴⠾⠛⠉⠀⢀⣼⠟⠉⠙⢿⣄⠀⠙⠻⢷⣦⣄⣀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠟⠿⠿⠿⠛⠛⠋⠉⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠈⠻⣷⡀⠀⠀⠈⠉⠛⠛⠿⠷⠶⠶⠿⠿⣿⣄⡀
⠀⠀⠀⠀⣀⣴⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣄⡀
⠀⢀⣴⣾⣛⣁⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣙⣻⣶⣤
⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⣻⡇⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⣿⡛⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠚⠉⠁⠐⢢⡀
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠒⠲⠤⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠚⢙⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⡋⠉⠓⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠑⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣀⠐⢟⡢⣀⡴⠒⠂⠀⠀⠒⢏⣄⠰⠀⠀⠀⠀⠀⣀⣈⣀⣀⣁⣄⣀⣀⣀⣀⣄⣀⡈⡆
⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⢀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣻⣿⣾⣧⠀⠀⠀⠀⠀⠀⢻⣷⣴⢲⠆⠉⠉⠀⠀⠀⠀⢀⠙⠈⠉⠉⠙⠒⠹⡷⠁
⠀⠀⢸⢠⡴⠖⠒⠒⠒⠒⢒⣒⣲⠶⠶⠒⠒⠉⠉⠉⠀⠎⠹⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣧⠐⠂⠀⠀⠀⠀⠀⠀⢀⠒⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠳⣅⡀⠁⣠⠴⠚⠉⠉⠁⠄⠀⠀⠀⠀⠀⠀⠀⢉⠌⣻⣿⣿⣿⢇⣀⣠⣤⣤⣤⣤⣿⣧⠭⡄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⢰⠃
⠀⠀⠀⠀⠀⡩⠋⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐⢀⣿⣿⣿⣯⣾⣿⣟⣛⣛⣻⣿⣛⠿⣿⣶⣤⣤⣀⣀⠀⠀⢀⠒⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⠀⠀⣀⣴⣾⣿⣣⣿⣿⣿⣿⣿⣿⡿⢉⡟⣿⣭⣿⣶⢭⣛⢿⡿⣿⢿⠛⠲⠶⠤⢄⣀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠐⡠⢀⣠⢶⣿⣿⣿⣻⠃⡇⠸⣿⣛⣻⣿⡇⢹⡗⠀⠈⠼⠣⠸⢧⠈⠓⢦⣛⠘⡀⠀⠀⠀⠀⠫⠉⣶⡄⡍
⠀⠀⠀⠀⠀⠀⢹⠀⠄⠀⠀⠀⢀⡤⠖⠉⠈⡎⣻⢨⣿⠃⡉⠰⣧⠟⠿⣧⣦⡇⠀⢿⠁⠀⠀⠉⠀⠀⢣⠀⠀⠈⠙⠲⠦⠤⠤⠤⠬⠦⠚⠊
⠀⠀⠀⠀⠀⠀⠘⡇⠀⢀⣤⠞⠁⠀⠀⠀⠀⣀⡵⡻⠁⠄⠈⢦⠙⠛⢂⢩⣹⠁⠀⠸⡆⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⢣⡠⡟⠀⠂⠀⢀⣀⠴⠚⠁⡼⠁⠀⠀⠂⠀⠁⠑⡌⢄⡏⠀⠀⠀⡇⠀⡄⡄⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠊⠒⠒⠋⠉⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠃⢀⠁⠀⠒⠔⠊⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢃⡀⢃⠃⠳⠚⣘⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡰⢤⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡘⢀⣨⠤⠤⠤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⠈⢟⡻⠓⢢⣛⡇⠀⠀⠀⠠⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⠭⣁⣒⡑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠃

⠀⣴⠞⠛⠛⠛⠓⠶⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠖⠛⠛⠛⠛⢶⡄
⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻
⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠉⠙⠻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⡟⠀⠀⠀⠀⣷⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣧⠀⠀⠀⠀⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠙⢻⡟⠉⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠛⠁⠀⠀⠀⣾⣇⠀⠀⠀⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠉⠳⠶⠤⠤⠤⠶⠶⠛⠋⣹⠏⠀⠀⠀⠀⠀⢠⡏⢿⠀⠀⠀⠀⠀⠀⠻⣏⠛⠳⠶⠦⠤⠤⠴⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⣼⠁⠸⣇⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄
⠀⠀⢠⣾⣏⣀⣀⣠⣤⡄⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⣤⣤⣤⣤⣤⣽⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⣦⣴⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠿⠛⠋⠉⠉⠁⠀⠀⠈⠉⠉⠙⠛⠿⣿⣿⣷⣦⣀⠀⣿⣿⡏⠀⠛⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣾⣿⣷⣶⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠁⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⢀⣼⣿⣿⠟⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠁⠀⠀⠀⠀⣠⣶⣿⣿⠿⠛⠛⠋⠉⠉⠙⠛⠻⢿⣿⣿⣦⣄⣴⣿⣿⠟⠁⢀⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠀⠀⠀⠀⣠⣿⣿⠟⠋⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠈⢙⣿⣿⣿⠟⠁⠀⠀⠀⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠏⠀⠀⠀⢀⣾⣿⡟⠁⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣴⣿⣿⠟⠁⣀⣄⠀⠀⠀⠀⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⣾⣿⡟⠀⠀⢀⣾⣿⡿⠋⠁⠀⠀⠀⠀⢉⣿⣿⣿⠟⠁⠀⠀⢿⣿⣧⠀⠀⠀⠀⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⢸⣿⡿⠀⠀⢀⣿⣿⠏⠀⠀⢀⣤⣴⣤⣴⣿⣿⠟⠁⣀⣀⠀⠀⠈⣿⣿⡆⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⡟⠀⠀⢠⣿⣿⣿⣿⣿⣿⠁⠀⠀⣿⣿⡇⠀⠀⢹⣿⡇⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⣇⠀⠀⠸⣿⣿⣿⣿⣿⣿⠁⠀⠀⣿⣿⡇⠀⠀⣸⣿⡇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⠀⢸⣿⣧⠀⠀⠘⣿⣿⣄⠀⠀⠙⠻⠿⠿⠛⠁⠀⠀⣰⣿⡿⠀⠀⠀⣿⣿⠇⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⠀⠀⠈⢿⣿⣆⠀⠀⠘⢿⣿⣦⣄⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠁⠀⠀⣼⣿⡟⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠘⢿⣿⣦⡀⠀⠀⠙⠿⣿⣿⣷⣶⣶⣿⣿⣿⠟⠋⠀⠀⢀⣾⣿⡟⠁⠀⠀⠀⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⣀⣴⣿⣿⠋⠀⠀⠀⠀⣰⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⠀⠀⠀⠀⠈⠻⢿⣿⣷⣦⣤⣀⣀⣀⣀⣀⣠⣤⣴⣿⣿⡿⠋⠁⠀⠀⠀⢀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⣿⣿⣿⣿⠿⠿⠛⠋⠁⠀⠀⠀⠀⢀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉
⣿⡿⡿⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⣿⣿⠇⢹⣿⡇
⣿⣿⢳⣺⣿⡇
⣿⣿⣷⣾⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⢦⣄⠀⣠⡤⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠘⠟⠁⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠖⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣰⡶⠛⠉⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣤⣄⣹⣷⠞⠛⠻⣯⡉⣉⣉⣉⣉⣉⣹⣧⣴⠾⠟⠛⠛⣷⡀
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠈⠉⣿⠃⢀⡀⠀⠈⢿⡛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⢀⣼⣧⣤⣄⡴⠚⠛⠻⣏⢹⡆⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⡏⠀⠀⣠⣾⡁⠀⠀⠀⣽⠋⠀⠀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯
⠀⠀⠀⠀⠀⢻⡀⠀⠿⡼⢷⣤⣤⠶⠋⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⣿⣦⣀⣠⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣧⣤⣤⣄
⠀⠀⠘⠛⠛⢻⡟⠛⠂⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢀⣠⣼⣷⡤⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⢀⡤⠤⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠐⢲⣿⠷⠦⠄⢀⣴⠶⠛⠲⣦
⠀⠀⠀⠀⠀⠀⠹⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠶⣤⣾⡁⠀⠀⣴⠏⠀⠀⠀⣀⢸⡇
⠀⠀⠀⠀⢠⡴⠞⠛⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⠈⠙⠳⠤⣟⠀⢀⣀⣴⢏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣾⣿⣋⠁⠀⠀⠀⠀⣠⡶⣫⣤⣍⣭⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣶⣤⡾⢋⡟⠉⠉⣿⠉⠉⠉⠉⠉⠁⣿⠀⠀⢹⣆⢙⣷⠶⠦⣤⡾⣫⡾⠋
⠀⠀⠀⠀⠀⢀⡾⠋⠉⠀⠈⢛⡿⠁⠀⠀⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⢹⣿⠁⠀⠀⠘⡿⠋⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣠⡴⢦⣤⠶⢦⡄⠀⠀⠀⣾⠁⠀⠀⠀⣰⠇⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢰⡏⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣿⠀⠀⠀⠹⢦⣄⣠⡴⠋⠀⠀⢀⣸⡛⣻⣇⡀
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⣾⠁⠀⠀⠀⠀⠈⠻⣄⡀⠀⣀⣼⠃⠀⠀⠀⠀⠀⠈⣷⠀⠀⢀⡶⠛⠉⠙⠉⠉⣙⣳⣦
⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⡏⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣼⣧⡴⠞⠛⠳⠾⠋⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠶⠞⠛⣻⡟⠁⠀⠘⢻⡾⠻⠛⠛⠛⠛⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⡀⠀⠀⠀⠀⣀⣽⣇⣀⣀⣀⣀⣠⣤⡾⠏⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠋⠩⠭⠩⣉⣉⣉⣉⠀⢀⣀⣀⣀⡄⠄⠤⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡟⢠⣦⠘⣧⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣴⣶⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⣠⣾⣿⡿⠿⠿⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁⠀⠀⠀⠀⠈⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠻⠿⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⣤⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢿⣿⣆⠀⠀⠀⠀⠀⢀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠛⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠻⣿⣷⣦⣤⣤⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢶⣶⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠈⠙⠻⣿⡿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠻⠿⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠶⢶⣤⡀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠉⠙⠻⣶⣄⠀⢀⣀⣀⣀⣀⣀⣀⣠⡾⠋⠀⠀⠀⠈⠻⣷⣀⣴⡾⠛⠉⠁⠀⠀⠀⠙⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠛⠛⠛⠉⠉⠉⢉⣿⠁⠀⠀⠀⢀⣀⣀⣈⣿⣯⣤⣄⡀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢰⣟⠋⣽⡟⠋⠉⠈⠉⠻⣷⣶⠶⠶⠶⠿⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠈⠻⣷⣿⠀⠀⠀⠀⠀⠀⢸⣷⣦⡀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣄⣀⣀⣠⣬⣿⣆⠀⠀⠀⠀⢀⣼⡟⠙⣿⡄⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠙⠻⠶⠶⣶⡟⠛⠛⠛⠋⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⣠⣴⢿⣇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠸⣿
⠀⣀⣀⣀⣠⣤⣼⣷⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣴⣿⣤⣤⣀⣀⣀⡀
⠘⠛⠛⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⢰⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠙⠛⠛
⠀⠀⠀⠀⠀⠀⢸⣧⣤⣴⡶⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠇⠀⠀⠀⠀⠰⣶⣤⣤⣿⠁
⠀⠀⠀⠠⣶⠾⠟⢿⡏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠛⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⣽⡟⠿⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⣠⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⢶⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⣠⣼⠿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡿⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠿⠋⠁⠀⠈⠻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋⠀⠀⠉⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢶⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠶⠶⠶⠶⠿⠿⠿⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⣀⣀⣀⣀⣀⣀⣣⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠁⣀⣀⣀⣀⣀⡈⠉⠓⠦⣄⠀⠀⢣⡷⠯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣀⠴⠚⠉⢀⣀⣀⣀⡀⠉⠑⠢⣄⠈⢱⣶⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢠⠞⠁⣠⠔⠊⠉⢀⣀⣀⠈⠉⠲⢤⣀⠽⢇⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢰⠃⢀⡞⠁⣠⠖⠉⠁⣀⠈⠉⠓⣤⡞⠱⣄⠈⢧⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠇⠀⡎⠀⡼⠁⣠⠞⠉⠀⠉⣱⢾⠁⢱⡀⠸⡆⠘⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢰⠁⠀⠇⠀⡇⠀⡞⠩⠟⡅⠈⡇⠀⡇⠀⡇⠀⣧⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⡀⠘⡆⠀⣇⠀⢧⡀⠑⠲⠚⠁⣰⠃⢀⠇⢀⡇⠀⡇⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢧⠀⠱⡀⠘⢦⡀⠙⠲⠤⠴⠚⠁⣠⠎⠀⡼⠀⣰⠃⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⢣⡀⠙⢦⡀⠉⠒⠦⠤⠤⠖⠊⠁⡠⠞⠀⡰⠃⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠙⢦⣀⠉⠓⠲⠤⠤⠤⠴⠒⠋⢀⡠⠚⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠢⣄⠈⠙⠲⠦⠤⠤⠤⠤⠖⠚⠉⢀⡤⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠙⠒⠶⠤⠤⠤⠤⠤⠴⠖⠋⠁⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣭⣩⣭⡭⠭⠭⠭⣭⣭⣭⣭⣭⣭⣍⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡷⢿⣿⣿⣷⣶⣶⣦⡿⠏⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠦⢾⣫⢻⡇⠸⡿⡿⠿⠻⡆⠈⢠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣡⣶⣶⠶⣬⣿⣿⣄⣄⠘⠛⠟⠁⢀⣼⣿⣿⣷⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⡭⠙⢻⣿⣿⣿⣷⣦⣤⣴⣾⣿⣿⣿⣿⣿⣍⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣄⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⡄⠀⣺⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣽⣿⣿⣷⡀⠀⠘⠻⣿⣿⣿⣿⣿⡟⣿⣮⣹⣶⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡟⢃⠘⠙⣿⣿⣿⣤⣤⣤⣤⡈⢉⣩⣽⣿⢸⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠝⠿⠗⠀⠠⠄⠛⢋⣿⣿⣿⣿⣿⣷⣿⣿⣿⣻⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣴⣯⡽⣯⣿⣿⣛⣵⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠃⢰⣿⣿⣿⣿⣿⡯⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⢸⠁⢀⣴⠟⠃⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⢠⠚⢿⣿⣄⣈⣙⣿⣏⠎⠀⠀⠀⠀⠀⠈⠐⠻⢷⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⡀⠀⠀⠂⠀⠐⠀⠀⠀⠀⡎⢠⣿⣿⣿⡘⠿⠿⣻⣤⣓⣶⣦⣄⡀⠀⠀⠀⠘⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢈⠃⠀⠉⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠘⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⠟⣷⡆⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢟⠷⣿⠟⡄⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⡿⠿⠟⠛⠀⠀⠀⠃⠂⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣅⡄⣹⠿⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣸⣯⠟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣰⣿⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣶⡌⠁⠀⠀⠧⢀⠀⠀⡀⠀⠀⠀⠀⢼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣁⣤⣴⠖⢢⣤⣤⠀⣀⠀⠀⢤⣅⡰⢟⣷⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠂⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣯⣅⣀⠀⠀⠀⠀⣩⡏⢁⡀⠀⢠⡟⣿⣿⣿⣷⡀⠅⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠰⣶⣷⣄⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⢷⣦⠄⠀⠈⢈⣷⣄⣽⣿⣿⣿⣿⣿⣷⡀⢦⡀⠀⠀⠀⠀⠀⢸⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣄⣀⡀⣰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢿⣎⣛⠀⠀⠀⠀⢠⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠒⠐⠛⠛⠉⠁

⢠⡶⠳⢦⡀⠀⠀⣠⠶⠳⣦⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⢠⠖⠒⣄⠀⠀⡠⠒⠲⡄
⠀⣧⠀⠀⢈⣶⣎⠁⠀⢰⡇⠀⠀⠀⠀⣸⠇⠈⠳⣀⡰⠋⠀⢿⡀⠀⠀⠀⠀⣟⠀⠀⠈⠂⣤⠐⠁⠀⠀⣻⠀⠀⠀⠀⢻⣀⣀⣈⣦⣴⡁⣀⣀⠇
⠀⠙⠛⠋⣱⠃⢻⡉⠛⠛⠁⠀⠀⠀⠀⠻⣤⡤⠖⡿⡗⠦⢤⡼⠃⠀⠀⠀⠀⠈⠁⣢⡤⠞⠉⠳⢤⣔⠈⠁⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠑⣄
⠀⠀⠀⠀⣿⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠁⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢹⡆
⠀⠀⠀⣸⡇⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⢰⡿⠟⠁⠀⠀⠙⠿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠋⠀⠀⠀⠈⠃
⠀⢠⡄⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⣰⣶⣤⡀⠀⠀⣀⣤⣶⡄⠀⠀⠀⠀⠀⢠⣶⠒⢄⠀⢀⠔⢳⣦⠀⠀⠀⠀⠀⣴⣔⡠⢄⡀⣀⡠⠄⣢⣦
⠀⢹⣿⣶⣤⣶⣶⣤⣶⣿⡇⠀⠀⠀⠀⢿⣿⣿⣽⣶⣾⣫⣿⣿⡟⠀⠀⠀⠀⠀⠸⢿⣷⣶⣶⣶⣶⣿⠿⠀⠀⠀⠀⠀⢿⣿⣿⠿⡿⢿⠻⣿⣿⣿
⠀⠸⠿⠿⢫⡿⢿⡝⠿⠿⠃⠀⠀⠀⠀⠸⠿⠛⢡⣿⣿⡉⠛⠿⠃⠀⠀⠀⠀⠀⠀⢀⣠⡾⠁⠱⣤⣀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣾⡇⠸⣷⡀⠈⠁
⠀⠀⠀⠰⠿⡇⢸⠟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠋⠀⠀⠀⠘⠻⢿⠇⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⣤⣀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⢀⣴⠲⣄⠀⢀⡴⢲⣄⠀⠀⠀⠀⠀⠀⢀⣴⠖⠻⡆⣰⠛⠲⣦
⠀⠀⣏⡀⠹⡆⢰⠋⢀⣻⠀⠀⠀⠀⢠⣿⣽⣷⣼⡆⣼⣴⣿⣽⣧⠀⠀⠀⠀⢀⣤⠿⣶⢄⢻⣇⡴⡶⠿⣤⡀⠀⠀⢰⡉⠉⠓⢄⠀⠀⡠⠒⠉⠉⡇
⠀⢰⠋⠛⠓⣷⡶⠚⠛⠙⡆⠀⠀⠀⠙⠦⣝⣻⣿⣷⣿⣿⣛⡥⠞⠀⠀⠀⠀⢿⣇⣀⣀⡁⣿⣿⣈⣀⣀⣸⡇⠀⠀⣠⠛⠛⠓⢒⠷⠶⡒⠒⠛⠛⢦
⠀⠘⠓⠒⢉⡟⢳⡑⠒⠚⠃⠀⠀⠀⠀⠀⣀⡴⠚⠉⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢀⡇⢳⠀⠉⠉⠉⠀⠀⠀⠈⠒⠒⡾⠋⠀⠀⠈⢷⡒⠒⠋
⠀⠀⠀⠠⠟⠀⠀⠷⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠙⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈
⠀⠀⠀⡠⠠⠀⡠⠢⡀⠀⠀⠀⠀⠀⠀⢀⣴⢄⠀⠀⠀⡀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⢠⣄
⠀⠀⠀⢣⠀⢀⠀⣰⠁⠀⠀⠀⠀⠀⢀⣼⣿⣆⢳⠀⡴⢡⣿⣿⡄⠀⠀⠀⠀⠀⢀⡞⠙⠓⢦⠀⠠⠚⠙⠳⡄⠀⠀⠀⠀⢸⣿⠀⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠃⣘⡐⠁⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣳⣶⣗⣻⣿⣿⣿⠀⠀⠀⠀⠀⠘⢧⡀⠀⣠⣷⣆⡀⢀⣠⠇⠀⠀⠀⢠⣾⣿⡷⡶⣶⢾⣿⣧
⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⢉⣼⡞⣷⡉⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠉⣩⠜⠙⡄⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⣿⣄
⠀⠀⠀⠀⠀⢦⠡⣄⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠀⣿⣿⠆⠀⠀⠀⣤⣄⣤⠖⠒⢛⠋⢁⣤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⢸⠿
⠀⠀⠀⠀⠀⠈⠳⠮⠉⠃⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠘⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀
⠀⠀⣾⣶⣄⣀⣤⣶⣶⠀⠀⠀⠀⣤⣤⣀⣀⣤⣤⠀⠀⠀⠀⢰⣦⣀⣀⣀⣴⡆⠀⠀⠀⠀⠐⠄⢀⣀⢀⡔⠂⠀⠀⠀⠀⢸⣿⣄⣀⣠⣾⡇
⠀⠀⠿⠿⠋⠉⠛⠿⠿⠀⠀⠀⠀⠛⠛⠋⠙⠛⠛⠀⠀⠀⠀⠸⠿⠛⠉⠛⠿⠇⠀⠀⠀⠀⠉⠔⠈⠉⠐⠦⠁⠀⠀⠀⠀⢾⠛⠉⠉⠉⠛⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠒⠒⠒⠒⠒⠚⠛⠛⠋⠉⠁⠉⠉⠉⠉⠉⠉⠙⠛⠓⠒⠒⠒⠒⠒⠒⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⡠⠄⢀⠀⠀⠱⡀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠶⠖⠒⠒⠒⠒⠒⠒⠒⠲⠶⠤⢤⣀⡀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠒⣇⡀⡤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠲⢤⢀⣸⠒⠈⠀⠀⠀⡆
⠀⠐⠂⢀⠀⠐⢯⠄⠄⢄⡂⢀⠙⠤⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠤⠋⡀⢔⡪⠤⠠⡽⠂⠀⡀⠒⠂
⠀⠀⠐⠚⠂⠀⠋⠀⠀⠀⠉⠒⠬⠂⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠐⠥⠒⠉⠀⠀⠀⠙⠀⠒⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠿⣷⣦⣄⢀⣴⣾⣿⡿⠟⠋⠙⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠙⢻⣿⡿⠛⠁⠀⠀⠀⠀⠀⢻⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣶⣾⣿⠃⠀⠀⠀⠀⣀⣀⠀⢹⣿⣄⣀⣀⠀⠀⠀⣀⣨⣿⣿⣤⣀
⠀⠀⠀⠀⠀⣰⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣶⣾⣿⣿⣿⠿⠿⠛⠛⠋⣿⡟⠀⠀⠀⢠⣾⠿⢿⣿⡿⠟⠛⠛⠿⣷⣾⡿⠛⠛⠉⠉⠙⢿⣧
⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢻⣿⡀⣾⡟⠀⠀⠀⠀⠀⠈⣿⣷⣤⣄⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠈⠻⠿⣿⡇⠀⠀⠀⠀⠀⠀⣼⡟⢙⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣀⣀⣀⣀⣤⣿⣿⣦⣀⣀⣀⣠⣾⡿⢿⡿⠟⠁⠀⢠⣿⠏
⠀⠀⠀⠀⠀⢿⣿⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠛⠛⠋⠁⠀⠉⠛⠛⠛⠛⢿⣦⣀⠀⠀⢀⣠⣾⣿⣦
⠀⠀⠀⠀⠀⠘⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⡿⠿⠟⠉⢻⣿⡆
⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣤⣶⣶⣶⣶
⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⠿⢿⣿⡏⠉⠉
⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠀⠀⠠⣶⣾⣿⣿⣿⠿⠿⠷
⠀⠀⠀⠀⠀⣿⣿⣄⣀⣀⡀⠀⠀⠀⠀⢀⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⢀⣤⣴⣾⣿⣿⣿⣿⠟⠛⠃⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣿⣿⣀⣀
⠈⠋⠉⠀⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣁⣀⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⡿⠛⠿⠿⣿⣶
⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣟
⠀⠀⠀⠀⠾⠿⠛⠉⠻⣿⣧⣀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠉⠻⠿⢿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⡏⠀⠀⠀⠀⠀⠈⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠛⠿⣿⣿⣷⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⠛⠉⠀⠘⣿⣿⣄⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠛⠛⠛⣿⡏⠁⠀⠻⣿⣦⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⢀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣿⣿⡟⠉⠁⣴⣿⠟⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠙⣿⣷⡀⠀⠀⠈⢻⣿⣧⣠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠛⠛⠙⣿⣿⣄⣾⣿⣯⣤⣤⣀⡀⠙⢿⣷⣤⣀⣀⣀⣀⣤⣾⡿⠋⠀⢀⣀⣤⣤⣼⣿⣷⣄⣀⣤⣴⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⣿⣿⣿⡿⠿⠿⠿⠿⣿⣷⣤⠉⠛⠛⠿⠿⠛⠛⠉⠀⣠⣾⣿⠿⠿⠿⠿⠿⣿⣿⣿⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣠⣾⡿⠋⠁⠀⠀⠀⠀⠀⠈⠻⣿⣷⡀⠀⠀⠀⠀⠀⢠⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣦⣤⣤⣤⣴⣾⣿⡿⠟⠛⠋⠉⠉⠉⠉⠉⠙⠛⠻⠿⣿⣷⣶⣦⣤⣴⣶⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠿⠿⠟⠛⠻⠿⠿⣿⣿⣦⣄⠀⢀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢋⣴⣿⣿⣿⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⢀⣠⣶⣾⣿⣿⣿⣷⣶⣄⡀⢀⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⠀⠀⣠⣿⣿⠟⠉⠁⠀⠈⠉⠻⣋⣵⠿⠋⠛⠛⣥⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⢰⣿⡿⠁⠀⢀⣤⡶⢄⣤⠾⠋⢵⣿⡆⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⡇⠀⠀⢾⣯⠰⣛⡥⠀⠀⢸⣿⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠸⣿⣷⡀⠀⠈⠻⠿⠟⠁⠀⢀⣾⣿⠇⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠀⠀⠹⣿⣷⣤⣀⠀⠀⠀⣀⣤⣾⣿⠏⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⣶⣦⣤⣤⣤⣴⣶⣾⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠋⠉





⣿⣿⣛⣛⣛⣛⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠖⠒⠒⠒⠒⠲⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡉⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⢉⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣁⣀⣑⡾⠟⠉⠉⠉⠉⠻⣷⣊⣀⣈⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠔⠒⠒⠤⠤⠤⠤⠤⠤⠤⠤⠔⠒⠢⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠸⡒⠒⠒⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠒⠒⢒⠗
⠀⠀⠀⠀⠀⠀⠀⣨⠗⠀⠈⠦⢴⣾⡿⠙⠒⠒⠒⠒⠒⠒⠒⠒⠋⢿⣶⡦⠴⠁⠀⠺⣁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠴⣖⠉⠉⠗⢶⣶⣦⡄⠀⠀⠀⠀⠀⠀⢠⣤⣶⡶⢾⡉⠉⣲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣃⣸⠀⠀⠉⠙⠓⠒⠦⠤⠤⠤⠒⠚⠋⠉⠀⠀⣇⢘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠠⠤⠤⠤⠤⠔⠐⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠔⠒⠒⠒⠒⠀⠁⠒⠒⠒⠒⠢⠤⢄⣀
⠀⠀⠀⠀⠀⠀⢰⠒⠲⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⠖⠒⡆
⠀⠀⠀⣤⣒⠂⢻⠀⠀⢱⠀⠀⣀⡠⠤⠔⠒⠒⠒⠒⠒⠒⠒⠒⠢⠤⢄⣀⠀⠀⡞⠀⠀⡟⠐⣒⣤
⠀⠀⠀⠀⡼⠁⠈⢷⣶⣦⣓⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣚⣴⣶⡾⠁⠈⢧
⠀⠀⠀⠘⠋⠉⠉⠉⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠉⠉⠉⠙⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠛⠻⣦⣤⣴⠶⠟⠛⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠻⠿⢶⣦⣤⣤⡴⠶⠶⣿⠃⠀⠀⣀⡈⣿⡀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢾⣉⡟⠋⠉⠛⣶⠛⠛⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⣀⣀⣽⡄⠀⠀⢀⡿⢳⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠙⠓⠶⣟⠉⠉⠀⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠛⠁⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢴⠶⣿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣤⣀⠀⠀⠀⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡷⠀⠀⠀⠀⣀⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⡅⠀⠀⠀⠀⠿⠟⠀⠀⠀⠀⣠⡤⢤⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢩⡿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠛⠻⠶⣦⣀⠀⠀⠀⠀⠀⠀⠛⠦⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⣡⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠳⢶⡇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣤⣀⣀⣤⣤⣤⣤⣤⣶⣶⣿⠿⠋⢁⣠⠴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⡌⠳⣤⣀⣤⣦⡀⠀⠀⠀⣸⡿⠉⣽⠋⠁⠀⠀⠉⠙⠁⢸⣷⡚⠉⠀⠀⢸⣿⡀⠀⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⣴⡟⠁⠉⠻⣶⠶⠟⠋⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠙⣿⠉⠉⠉⠉⣿⡏⠁⢾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢿⣦⣤⡴⢋⡀⠀⠀⠀⠀⠀⠈⠳⢦⣄⡀⠀⠀⠀⢀⣼⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⢤⣅⡀⢸⣿⣿⠆⠀⠀⠀⠀⠀⠀⠈⠉⠛⢻⡞⠋⠳⣤⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡏⠻⣦⡉⠀⠀⠀⢸⣿⣾⠆⠀⢀⣀⠀⠈⣧⠀⢸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣷⠀⠈⠙⠳⠶⢦⣄⠋⠁⠀⠀⠸⣿⡿⠃⢹⡀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡘⢷⡀⠀⠀⠀⠀⠉⠛⠲⠶⠦⣤⣅⣀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣽⣦⣀⠀⠀⠀⠀⠀⠀⢠⡿⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠶⠤⠤⠤⠶⠋

⣶⣶⠶⠾⠟⠛⠛⠛⠛⠳⣝⣿⣼⢸⠇⣿⠀⣿⠀⣿⠁⣿⠇⣿
⣦⣤⣤⣶⣶⠿⠿⠿⠻⠷⣜⡇⡇⣾⢰⡏⣸⡇⢸⡏⢰⡟⢠⡟
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠸⢷⢣⡇⣾⢁⡿⢀⣿⢀⣿⠁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⢡⡏⣸⠇⣼⠇⣸⡏⣸⡏⠀⠀⠀⢀⣀⠤⣄⠲⣮⡙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢼⢸⠇⣿⢠⡿⢠⣿⠀⡿⢥⣶⣽⡪⡕⣎⢳⡌⢷⡈⢿⡄⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⡸⡸⣇⢿⡘⣧⣸⢿⣚⠭⢿⣛⣻⣿⡼⡜⡆⢿⡘⣷⠘⣷⡈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠽⠿⠿⣟⡫⠵⣚⡭⢖⣫⡭⢖⣒⣻⠵⢷⢹⡘⣧⠹⣧⠸⣧⠘⣷⡀⠀⠀⢀⣀⣀⠤⣤⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠭⡭⠭⠵⣚⣩⠵⢚⣩⠴⢚⣩⡵⠖⠛⢻⢿⣧⡙⢧⡙⢷⣌⠳⣌⡻⢔⣪⠽⠶⠮⡝⡌⢧⢳⡜⢮⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⣒⣛⣋⣭⠴⢚⣩⡴⠞⠋⠁⠀⠀⠀⠈⣿⡼⡼⡶⢋⣗⠾⣛⠵⢚⣭⠶⠞⠛⠓⢾⢹⢸⡆⢿⠈⣧⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠭⠤⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⣇⣿⡭⠗⣋⡵⠞⣋⣤⡶⠿⠛⠛⢾⢸⡈⣇⢸⡇⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣸⡖⣋⣩⡴⠞⠋⣁⣤⣶⠾⠿⠷⣼⡇⣿⠘⣧⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣉⣥⣴⠶⠟⠋⣁⣠⣤⣶⣦⣼⡇⣿⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣤⣴⡶⠿⠛⠋⠉⠀⢨⢹⢇⢸⡇⢻⡆⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⢸⡈⣷⠘⣷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣸⣛⢌⢧⣘⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠚⣛⣩⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡿⠛⠋⢁⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⣾⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠶⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢛⠢⠀⠀⠈⠙⠻⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠙⠋⠉⠁⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢴⠦⠀⠀⠀⢠⣄⣀⢈⠩⠿⣷⢿⣿⣿⣿⣶⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠙⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠩⠉⠒⠀⠀⡁⠈⢹⣿⣿⣶⣿⠈⠉⠹⠛⣿⣿⣿⣿⣿⠟⢙⠛⠁⠀⠀⢀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣉⠐⠀⠀⠀⠤⠲⠶⠿⠻⣿⣿⣠⣤⣶⣷⣿⣟⡟⠿⣷⣿⣶⣠⣀⠁⠈⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣖⡄⠀⠀⢀⡀⢠⣤⣴⣿⣿⠿⠿⠿⠿⠿⠿⢿⣓⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⡶⠾⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢦⣤⣤⣤⣀⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠃



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣩⣝⣭⣍⣽⣭⣿⣻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠊⠀⢴⣾⣿⣿⠀⠘⠈⠙⠓⠀⠀⠒⠀⠀⠀⠐⠁⠀⠀⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠋⣼⣿⡿⠀⡀
⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠘⠀⠀⠀⠄⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⠄⠀⠀⠀⠀⠀⠀⠀⠋⡟⠁⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣾
⢀⠔⠈⠁⠈⠁⣀⠀⠀⠀⠀⢠⣖⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠲⢀⣾⣏
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠤⠀⡴⠊⠉⠉⠐⢄⠀⠀⠀⠁⢄⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⣀⡀⢀⣤⣿⣿⣟⡡
⠁⠉⠁⠉⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠄⠦⣄⡠⠲⠒⠖⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣋⣀
⡀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠈⠑⣤⢄⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⡻⣿⣿⣷
⠙⠂⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠀⠴⠊⠁⠀⠀⢀⣿⡟⠃⠀⠃⠈
⠀⠀⠀⠀⠀⠁⠀⠀⠠⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠙⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣽⣷⡀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⡀
⠀⠀⠀⠈⢱⣄⡀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣄⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣹⠁
⠀⠀⠀⠀⠈⢆⠀⠉⠁⠀⢄⠵⠶⣦⠀⠀⠀⠀⠀⢀⠀⠀⢽⣿⣿⡇⠀⣾⠇⠀⢀⠄⠀⠄⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⢀⣀⢀⡀⠉⠉⢻⣿⣿⣧⢠⣿⣷⣾⣿⡶⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠄⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠙⠻⣿⣿⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠀⠀⠀⠀⠀⠳⡀⠀⣰⠁⠀⠠⡄
⠀⡄⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⡀⠐⠃⠀⠁⠈
⠀⠐⢄⢿⡄⡀⠀⠀⠀⠀⠕⠀⠄⠀⠀⠀⠀⠀⠀⠒⢂⣩⡴⠋⡉⠀⠙⢷⡀⠈⠀⠀⠐⠒⠒⠄⠠⠄⠀⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠆
⠀⠀⠀⠀⠙⠁⠀⡎⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠋⠅⡔⠁⠀⠋⠀⠁⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠈⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣤⣴⣶⣤⣬⡄⠀⠀⠀⠀⠀⠀⠀⡀
⡠⠖⠀⠀⠀⠀⠀⠀⠁⠰⣀⠀⢀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⡀⠀⠀⠀⠈⠙⠙⠓⠚⠠⢄⢀
⠀⠀⠀⠀⠀⠆⠀⠀⠂⠀⠀⠉⠉⠒⠆⠒⠘⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⣦⢠⠀⢀⣴⣿
⠀⠀⠀⠀⠇⠑⠂⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢣⡀⣄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⢺⣧⣠⣿⣿⣿⣿⠏
⠒⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢿⣿⣄⡀⠀⠸⠁⠀⠀⢀⣤⣞⡀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣟⠋⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠁⠀⢀⣀⠀⠀⠀⠈⢻⣿⣷⣀⢠⠀⠀⣤⣿⣿⠋⢀⠀⠠⠠⠀⠠⠄⠀⡸⠉⠐⠙⠧⣐⠀⠤⠤
⠲⢀⠈⠁⠀⠁⠀⠀⠠⠀⠀⠀⢀⠂⠁⠀⠀⠀⠀⠀⠀⠙⠢⠀⠀⠈⠻⣿⣿⠛⠃⢸⣿⣿⡡⠊⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠙⠒⠒
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠄⠀⢈⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣾⣷⣶⣦⣉⡀⠃⢌⣿⣟⡡⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠎⠀⠀⠀⠐⠚⠉⠉⠉⠻⠛⠛⠛⢻⡿⢿⠁⣠⡆⢫⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠋⠁⠀⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⡄⠀⠈⠙⠷⣄⠀⠉⠠⡀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢠⠀⠀⠀⠀⠀⠠⢴⠖⠁⠀⢀⠎⠀⠀⠐⢤
⠀⠀⠀⠀⡝⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠠⣄⠑⣿⡄⢠⡧⠊⠀⠀⢀⡤⠒⠁⠀⠀⠀⠁⠀⠀⠀⠺⢺
⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢈⣙⣿⡿⠈⠉⠛⢳⣧⣴⣊⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⣀⡴⠞⠋⣿⠀⠀⠀⢸⣿⡁⠈⠹⣆⠀⠐⠲⢄⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠴⠎⠁⠈⠀⠠⠚⠉⠙⠋⠁⡀⠀⡾⠄⠀⠀⠉⠒⠤⡀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢰⠆⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠰⠶⡄⠀⠀⠀⠀⠀⠱⠋⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⢰⣼⡗⠈⠉⠉⠹⢣⠀⠀⠀⠀⠀⠀⠬⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠚⠻⣟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠸⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣛⣿

⠀⠀⠀⠀⠀⠀⠀⣾⡟⠛⠛⠻⠷⣶⣤⣄⣠⣶⠟⠋⠉⠙⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠈⣹⠟⠁⠀⠀⠀⠀⠀⠹⣷⡶⢶⣶⣶⣶⣶⣤⣤⣾⠿⠛⠋⠉⠁⠀⠙⣿⡇
⠀⠀⠀⠀⠀⢀⣼⣧⣤⣀⡀⠀⣠⣴⠶⠿⣶⣶⠟⢷⡄⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⢠⡿⠋⠉⠉⠉⠙⣿⠟⠁⠀⠀⠀⠹⣧⣼⠇⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⡞⢿⡀⠀⠀⠀⠀⢀⣿⣅⣀⣀⣀⣀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⠇
⠀⠀⠀⠀⠸⣿⠀⠀⠀⠻⣦⣼⣷⣤⣀⣀⣤⡾⠋⠉⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⣻⣧⣀⠀⠀⢀⣠⣴⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⢸⣿⠉⠛⠻⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠿⠿⠿⠿⣿⡿⠶⣶⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣿⣀⣀
⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⡆⠀⠀⠀⠀⠛⠛⣻⡿⠛⠛⠛⠛⠂
⠀⣤⣶⣶⣾⣿⠿⠆⠀⠀⠀⠀⠀⠘⠿⠟⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⢀⣀⣀⣿⠇
⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠉⠉⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⡿⠷⠶⡦
⠀⠀⠀⢀⣀⣼⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣄⣼⡟
⠀⠀⠰⠛⠋⠁⠈⠻⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠟⠋⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠷⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣴⣶⡶⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠳⣤⡀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⡀⠬⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣶⣧⠀⠀⠀⢠⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣒⣿⠋⠀⠀⠀⠈⢻⣇⡈
⠀⠀⠀⠀⠀⠀⠂⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⠀⠀⠀⠀⠀⢸⣿⣅⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢂
⠀⠀⠀⠀⠀⣠⠋⠉⠉⠉⠙⠓⠒⠀⠀⠀⢘⣿⣿⠆⠠⠀⠀⠀⢿⣿⣟⠀⠀⠀⠀⠒⠛⠉⠉⠉⠉⢻⡆
⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠠⠸⣿⡿⠫⠃⠀⠀⠀⠀⠀⠀⠯⠻⣿⡿⠢⠀⠀⠀⠀⠈⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⢨⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⡀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠓⠒⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠒⠀⠱⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠶⠶⠶⠟⠛⠛⣉⣙⠳⢜⣿⣼⢸⢃⡟⢰⡏⢸⡇⢸⡇⣸⠇
⢶⣶⣶⡶⠿⠟⠛⠛⠛⢦⡇⡇⡟⣸⠃⣾⢁⣿⢁⣿⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⠇⡿⢸⡇⣸⠇⣼⠇⣼⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢸⡇⣿⢠⡟⢠⡟⢰⡟⠀⢀⡠⣐⢤⡒⢮⡙⢦⡙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢸⠠⡇⢸⡇⣾⢇⣘⡵⣿⣶⣿⢮⢧⢹⡌⢷⡈⣷⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡮⡮⣣⣿⡬⣗⡮⢗⡪⠭⣓⣚⣷⢏⡏⣇⢷⠘⣧⠸⣧⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡯⠯⣝⣚⡭⢗⣫⠵⢚⣫⠵⣒⣫⠭⢯⣽⡜⢧⡙⣧⡙⢧⡙⢧⣀⢤⣒⣒⡺⣝⢯⢢⡐⣄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⠭⠭⠶⣚⣩⠴⢚⣭⠶⠛⠉⠀⠀⢸⢷⢧⢮⡽⠢⢽⡲⣝⡬⢚⣩⡥⠶⢭⡞⡎⣇⢻⡘⣦⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣛⣩⡭⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠻⣼⣸⣖⣺⠵⣊⡵⠞⣋⣤⡶⠶⠶⡅⡇⢿⠘⣇⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢇⡯⠖⣛⣡⡴⠟⢋⣡⣴⡶⠶⣆⣿⢸⡀⣿⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣛⣭⣥⣴⠾⠛⣉⣤⣤⣤⣤⣿⢸⡇⣿⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣤⠶⠶⠿⠛⠋⠉⠉⣿⣿⡼⣷⢸⣇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡏⣇⢻⡌⢿⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⡯⣳⣙⠬⢗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣏⣭⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢉⣁⣤⣶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠋⢁⣠

⢀⠐⢀⠘⢀⠐⠀⠀⠃⡀⠃⡀⠀⢀⠐⢀⠘⢀⠐⠀⠀⠂⡀⠃⡀⠂
⣀⠐⢀⠐⢀⠐⠀⡀⠂⡀⠂⡀⠀⢀⠐⢀⠐⢀⠐⠀⡀⠂⡀⠂⡀⠂
⢀⠐⢀⠐⢀⠐⠀⠀⠂⡀⠂⡀⠒⢀⠐⢀⠐⢀⠐⠀⠀⠂⡀⠂⡀⠂
⠀⠠⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠄
⠉⠠⠈⠠⠈⠠⠀⠀⠄⠁⠄⠁⠤⠉⠠⠈⠠⠈⠠⠀⠀⠄⠁⠄⠁⠄
⠉⠠⠈⠠⠈⠠⠈⠁⠄⠁⠄⠁⠀⠉⠠⠈⠠⠈⠠⠈⠁⠄⠁⠄⠁⡤⢤
⠀⢀⠈⢠⠈⢀⠀⠂⡄⠁⡄⠀⡀⠈⢀⠈⢀⠈⢀⠀⠀⡀⠁⡄⢁⣧⣸⡀
⠐⢀⠐⢀⠐⢀⠀⠀⡀⠂⡀⠂⡀⠐⢀⠐⢀⠐⢀⠀⠀⡀⠂⡴⢋⡤⠤⡈⢣
⠒⢀⠐⢀⠐⢀⠀⠂⡀⠂⡀⠂⡀⠐⢀⠐⢀⠐⢀⠀⠂⡀⠂⣧⠺⣄⣠⠇⣸
⠀⠀⠐⠀⠐⠀⠀⠀⠀⠂⠀⠂⠀⠐⠀⠐⠀⠐⠀⠀⠀⠀⠂⣸⠓⢤⡤⠚⢧⠀⠀⠀⠀⢀⣤⣤⡀
⠠⠈⠠⠈⠠⠀⠀⠀⠀⠄⠁⠄⠀⠠⢀⡶⠚⠲⡆⠀⠄⠁⢄⡇⠄⡟⢹⠀⠘⡄⠀⠀⠀⢸⣇⣀⡇
⠤⠈⠠⠈⠠⠈⠠⠄⠁⠄⠁⠄⠉⠤⢘⡷⠚⠠⡏⠀⠄⠁⣼⠁⢴⠃⠀⡆⠀⢧⠀⠀⠀⢸⡗⠒⡇
⠠⠈⠠⠈⠠⠈⠀⠀⠁⠄⠁⠄⠉⠠⠘⣟⠉⢠⡏⢠⣤⢁⡇⠁⡎⠐⠂⢸⡀⠘⡄⣤⡄⢸⡇⡇⡇
⣀⠀⢀⠐⢀⠐⠀⡀⠂⡀⠂⡀⠂⢀⠘⣏⡉⢠⡗⠀⣿⣿⡀⢸⠁⠈⡁⠀⣇⠀⢣⣹⠁⢸⡇⡇⡇
⣀⠐⢀⠐⢀⠐⠀⡀⠂⡀⠂⡀⠒⢀⠸⣧⣴⢀⡗⠀⣸⠇⣀⡏⡀⠂⠄⠀⠸⡀⠘⡇⠀⢸⡇⡇⡇
⢀⠐⠀⠐⢀⠐⠀⠀⠂⡀⠂⡀⠀⢀⠸⣧⠄⢀⡗⠀⡼⠀⣸⢦⣀⡶⢦⣀⡤⢇⠀⢳⠀⢸⡇⡇⡇
⠀⠠⠀⠠⠀⠠⠀⠀⠄⠀⠄⠀⠀⠀⠰⡇⠶⠈⡧⢸⠃⢀⡏⠄⠀⠓⠚⠁⠀⠸⡀⠈⡆⢸⡇⡇⡇
⠉⠠⠈⠠⠈⠠⠀⠁⠄⠁⠄⠁⠄⠉⢰⡗⠢⠈⡧⣞⠀⣼⠁⠄⠁⠄⡁⠀⠀⠀⢣⠀⢹⢸⡗⠓⡇
⠈⠠⠈⠠⠈⠠⠀⠁⠄⠁⠄⠁⠄⠈⢰⡟⠓⠈⡇⢸⠛⠅⠁⠄⠁⠠⠄⠀⠀⠀⠈⠓⡏⠈⢇⢠⠃
⠉⠀⠈⢀⠈⠀⠀⠀⡀⠃⡀⠁⠀⠈⠀⠙⢋⠛⢁⠜⠀⡀⠃⡄⠁⡀⠀⠀⠀⠀⠀⠀⠓⠀⠈⠃
⠒⢀⠐⢀⠐⢀⠀⠂⡀⠂⡀⠂⡀⠐⢀⠉⢉⠙⠻⠟⠳⠶⠲⡴⠚⠿⠛⠛⠛⠛⠋⠉⠉⠉⠉⠈⠉
⠒⢀⠐⢀⠐⢀⠀⠂⡀⠂⡀⠂⣀⠐⢀⠐⢀⠐⢀⡀⠂⡀⠂⡀⠂⣀
⠀⠀⠐⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠰⠀⠶⢎⡶⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠶⠶
⠤⠀⠠⠈⠠⠈⠀⠄⠁⠄⠁⠄⠀⠠⠈⠠⠈⠠⠈⠀⠄⠁⠄⠉⠄⠉
⠠⠈⠠⠈⠠⠈⠀⠄⠁⠄⠁⠄⠉⠠⠈⠠⠈⠠⠈⠀⠄⠁⠄⠁⠄⠁
⢤⠈⠠⠈⢠⠈⠀⠀⠁⡄⠁⡄⠀⠠⠈⠠⠈⢠⠈⠀⠀⠁⡄⠁⡄

⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣶⣄⠀⣀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⣰⡿⠉⠉⠉⠙⠻⢷⣦⣴⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠀⠈⢻⡇
⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠐⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠿⠛⢻⣇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡀⢀
⠶⠻⠟⢻⣿⠛⠒⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠐⠛⢻⣿⠛⠛⠛
⠀⠀⠀⢘⣿⣄⣀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣠⣼⣿⣀⡀
⠀⠀⠛⠛⠻⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣉⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠉⠙⠁
⠀⠀⠀⠀⣀⣽⣿⣿⡋⠀⣀⣤⣤⣤⣤⣀⠀⠀⠀⠉⠁⠀⠀⠀⣀⣤⣤⣄⣀⡀⠀⠈⣿⣿⣷⣤⡀
⠀⠀⠀⠟⠋⠁⠀⠙⣿⡿⠋⠉⠀⠀⠉⠙⢷⡄⠀⠀⠀⠀⢠⡿⠋⠉⠉⠉⠉⠻⢷⣿⠋⠀⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣿⡁⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣀⣀⣠⣤⣀⣤⣾⠏⠀⠀⠀⠀⠹⢷⣤⣤⣤⣀⡀⢀⣴⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⠉⠛⠒⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠒⠛⠉⠉⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢀⡀⠘⢶⡞⠲⠖⢦⠊⣠⠆⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣀⣀⣀⣀⡀⠀⠀⠉⠑⢲⡇⠀⠀⢸⠉⠁⠀⠀⠀⣀⣀⣀⣀⣄⢹⡄
⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠶⠶⠶⠤⣿⠸⣟⠋⠀⠀⠉⠉⠉⠀⣈⣹⠁⣼⢀⣿⣉⡁⠁⠀⠀⠀⠀⠀⠈⠹⠃⡧
⢠⣤⣶⣒⣋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢮⣥⡤⠤⠶⢖⣺⠟⢡⡟⠙⠻⡻⢄⠻⡉⠻⣗⠒⠶⣤⣤⠤⠤⠖⠃
⠀⠀⠈⠉⠉⠉⣻⠆⠀⠀⠀⠀⠀⠀⠀⠉⢉⣁⣤⠴⠞⠋⠀⡔⣼⠀⠀⠀⠘⣆⠳⣝⣄⠈⠳⢄⡀⠉⠓⠒⠶⠤⠤⢤⣤⣤⠤⢤⡖
⠀⠀⠀⠀⠀⣾⠁⣀⣠⣤⠤⠴⠶⠒⠚⠋⠉⠁⠀⠀⠀⠀⠰⠀⡇⠀⠀⠀⠀⠈⢇⠈⢮⡳⣄⠀⠉⠲⠄⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢱⢰⠇⠀⠀⠀⠀⠀⠘⣆⠀⠱⡌⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⢸⠀⠀⠀⠀⠀⠀⠀⠹⡄⢱⡙⣆⠀⠈⠙⠲⠦⣤⣄⣀⣀⣀⣀⣀⣀⣙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢷⠘⣆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠌⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠘⣇⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢹⡄⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⣧⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠘⡆⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢻⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢇⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⠂⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⣠⠞⣇⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣠⠞⠁⠀⢸⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢠⡀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣾⠛⠳⣄⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⡿⠀⠀⠈⠙⢦⣄⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡇⠀⠀⠀⠀⠀⠈⠓⢾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻

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

⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠰⣶⣿⢿⣶⠆⠀⠀⠀⠀⣶⣶⡿⢿⣴⣶⠀⠀⠀⠀⠀⠀⣠⠴⠶⢦⣄⣴⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠋
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⠶⣼⣯⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⣼⣅
⠀⠀⠠⡶⣀⡀⠀⢀⣤⠴⠖⠒⠲⠶⠶⣮⣭⣳⣀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠃⣈⠁⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣀⡀⢀⣠
⠀⠖⠔⠁⠠⠁⣰⡯⠖⠒⠒⠒⠶⢤⣄⣀⠀⠉⡿⠛⠻⠶⠶⠶⠾⠛⠋⠀⣠⣿⣧⣶⣿⠃⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⣰⣿⠛⢿⣟
⠼⠚⠀⢀⠁⠀⠏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠶⢦⣤⣤⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣥⣤⠴⠶⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠛⢻⣶⠞⠿⠃
⠀⠠⠀⠈⡀⢂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠶⠚⠋⠁
⠀⠀⢀⡀⠁⠀⠀⠀⠀⢠⠚⠉⠉⠁⠉⠙⢿⣆⠉⠉⠙⣛⠛⣿⠲⠶⠶⠶⠾⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠙⠦⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⣆⠀⣤⣿⣛⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣠⡾⠋⠉⠓⠀⠙⢋⠁⠀⠀⠈⠉

⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⣰⣆⠀⠀⠀⣠⣦⡀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢘⣿⣿⣿⢆⢀⣰⣿⣿⣿⠀⠀⢰⣿⣿⡳⣤⡴⢿⣿⣷⠀⠀⠀⢿⣿⣿⣄⢀⣤⣾⣟⡿⠁⠀⠀⣿⣿⣿⡦⣤⢰⣿⣿⣿⠁
⠀⠀⠘⠻⠛⣽⡟⠛⠻⣿⡛⠛⠀⠀⠈⢿⣿⣦⠛⠳⣿⣿⠟⠀⠀⠀⢿⣿⣿⡷⠿⣿⣿⣿⡷⠀⠀⠀⠛⣻⣿⠋⠀⠸⣯⣍⠉
⠀⠀⠀⠀⢴⣿⡇⠀⠀⣿⠷⠄⠀⠀⠀⣾⣿⡟⠀⠀⢿⣿⡶⠀⠀⠀⠀⢰⣿⠁⠀⢹⣿⡇⠀⠀⠀⠀⠰⠿⠏⠀⠀⠀⠙⠿⠃
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠁⠀⠀⠈⠿⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁

⠀⠀⠀⢀⣤⡀⠀⠀⠀⢠⣄⠀⠀⠀⠀⣠⣀⣶⣶⣶⣆⣀⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⢸⣿⣷⡀⣀⣠⣿⣿⠆⠀⢠⣾⣽⣻⣿⡽⣿⣿⣟⣿⠆⠀⠀⣰⣿⣿⣆⠀⣠⣾⣿⣿⡀⠀⠲⣶⣶⣦⣀⣤⣶⣾⣿⣿⠃
⠀⠀⢀⣼⣿⣿⡟⠟⣻⣿⣷⡄⠀⢨⣿⣿⣿⣿⣿⣽⣿⡿⣿⡇⠀⠀⠿⠿⠿⣛⠿⢟⠻⠿⠿⠇⠀⠀⠛⢉⣴⠋⠹⣧⡉⠉⠉
⠀⠀⠀⠉⠛⢻⣷⠀⣿⠛⠛⠁⠀⠘⣿⣾⣿⣿⣫⣿⣿⣿⡏⠀⠀⠀⠀⠀⣾⡃⠀⠘⣷⠀⠀⠀⠀⠀⠀⣼⣿⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠘⠛⠘⠟⠀⠀⠀⠀⠀⠈⠛⠻⠻⠿⢿⠏⠉⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢀⣴⠖⣄
⠀⠀⠀⣽⣷⣦⣀⣀⣠⣾⣿⠀⠀⠀⢠⣄⣀⠀⠀⠀⠀⣠⣤⡄⠀⢸⡏⠈⠿⣦⣀⣠⣾⣡⣤⣿⠀⠀⣴⣶⣤⡀⠀⠀⣠⣴⣶⡄
⠀⠀⠀⣻⣿⣿⢽⣿⢽⣿⣿⠀⠀⠀⢸⣿⣿⡷⣦⡶⣿⣿⣿⡆⠀⠀⠑⢲⣿⣿⠚⠻⣿⣿⣯⠁⠀⠀⣿⣿⣿⡟⠛⢟⣿⣿⣿⡃
⠀⠀⠀⠉⠉⢀⣾⢿⣦⠉⠛⠀⠀⠀⢸⣿⡿⠋⠉⠉⠛⣿⣿⠃⠀⠀⠐⠿⠞⠹⡆⢰⠛⠟⠋⠀⠀⠀⠈⣹⣿⡇⠀⠀⣿⣿⡛
⠀⠀⠀⠀⠀⢸⣿⡞⣿⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠍⢦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠋⠉
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠠⠁
⠀⠀⢀⣴⣤⡀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡾⠐⢦⡀⠀⠀⠀⣠⠴⣦
⠀⠀⠸⣿⣿⡗⣀⣤⢾⣿⣿⡇⠀⠀⠀⣾⣷⡄⠀⠀⣴⣿⣿⡆⠀⠀⢸⣿⣷⡆⢀⣀⢴⣾⣿⠀⠀⠀⣧⠀⣠⣽⠶⢖⡾⣁⠀⣿
⠀⠀⠀⣿⣿⣷⡟⢻⣼⣿⣿⠀⠀⠀⢸⣿⣿⣿⣶⢾⣿⣿⣿⠃⠀⠀⠰⣿⣿⡟⠛⠓⣿⣿⡿⠀⠀⠀⠈⠉⣴⣿⠀⢸⣿⡌⠉⠁
⠀⠀⠀⠈⣻⣿⠇⠀⢹⣿⡉⠀⠀⠀⠘⣿⡿⠋⠈⠈⠻⣿⡿⠀⠀⠀⠀⠙⢻⣷⠀⣰⣟⠋⠁⠀⠀⠀⠀⠀⣿⠙⠀⠉⣿⠃
⠀⠀⠀⠀⠈⠉⠀⠀⠈⠿⠋⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠿⠉⠁⠀⠀⠀⠀⠀⠠⡿⠀⠀⠀⣿⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠶⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠈⠛⢶⣄⣀⣤⣤⣤⣤⣤⣤⣤⣶⠟⠛⠛⣷⣤⡶⠞⠛⠛⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣴⠟⠁⢀⣀⣄⣹⣇⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⣿⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⣴⡟⢻⡿⠿⠿⢦⣄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠠⢻⣿⣿⢠⡄⢀⡀⠹⣷⣶⠶⠿⠷⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣶⣶⣿⠟⣀⠁⠻⠟⣸⡟⢻⣦⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢶⣍⣱⣿⣿⡿⣷⡾⠏⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣾⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⢿⣇⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠈⠻⣷⣶⣾⣿⡤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⠛⣿⠉⠀⠀⠀⠀⢹⡟⠀⠀⠀⠀⠀⢀⡠⢄⡀⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⣘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⠔⠃⠀⠀⠀⠀⠀⠈⡀⠀⠀⢀⢨⡿⠉⠉⠓⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠙⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠟⠳⢤⣀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣻⣷⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣞⠋⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⡶⠾⠟⠛⢁⣿⠃⠀⠈⢿⡿⠏⠉⠉⠉⠉⠉⢿⡿⠁⠀⠘⣷⡝⠻⠶⣤⣄⣀⣀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠘⢿⣤⢀⠀⣿⠇⠀⠀⠀⠈⠳⣤⡄⠀⠀⣀⣤⠞⠁⠀⠀⠀⢹⣷⠁⢀⠐⣿⡛⠙⠳⣄
⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⢛⣷⣿⣿⠏⠀⢀⡴⠶⢤⠀⠈⠉⠛⠋⠉⠀⠀⢀⣠⣤⡀⠀⢿⣤⣿⣿⡏⠁⠀⠰⣿⡀
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢘⣋⣿⠏⠀⠀⠸⣄⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⣹⠀⠘⣿⣿⠃⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⡐⠁⣸⠏⠀⣠⡴⠾⠿⠿⢧⣄⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠁⠀⠀⠸⣯⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⢷⣤⣀⣎⠁⣼⠃⢠⡞⠁⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⢀⡀⠟⠻⢻⣿⣷⣦⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⢿⣤⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⡠⠊⣿⠃⠀⠀⠀⠄⠀⠀⠀⠀⠀⢻⠀⢠⣶⡀⠀⠀⣀⢀⣼⣿⢀⣤⣶⣼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⣿⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣄⡾⠿⢻⣷⣾⠿⠿⠛⠛⠛⠻⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⣿⣇⡄⠀⢀⠁⠀⠀⠀⠀⢀⣿⡟⠃⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⢐⠎⢻⣿⣒⡞⢀⠀⢠⢀⣠⡾⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏
⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⠃⠀⠙⠻⠷⣾⣶⠾⠿⠋⠀⠀⠈⠛⠳⠶⠶⠶⠶⠶⠛⠛⠋⠁

⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠖⠒⠒⠒⠢⠤⣀
⠀⠀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠤⢤⣀⡀⢀⣀⣀⣀⣀⡀⢀⣠⠤⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠽⡇⠀⠀⠀⠀⢹⢏⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢴⠃
⠀⠀⡇⠈⠉⠉⠑⠒⠒⠒⠒⠒⠒⠊⠉⠉⠉⠁⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⠧⠤⠤⠤⠤⠾⡶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊⠁⢀⡎⠀⠀⠀⠀⠀⠀⠹⡄⠈⠑⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣇⣀⣀⣀⣀⣀⣀⣤⠴⠒⠉⠁⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠈⠉⠒⠢⢤⠤⣄⣀⡤⠤⠼
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⢀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀
⠀⠀⠲⣍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠟
⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠳⡀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡀⠀⠀⠀⠀⠀⡴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠰⢣⠀⢀⠀⢀⢀⡇⢹⣀⡠⢤⣄⡀⣃⢸⣂⣀⠤⣄⠠⡞⡀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⡼⢋⣉⣙⣷⣷⣈⣃⢸⣛⡷⣯⡋⢙⣥⠤⠻⠥⢤⡟⠉⣫⣤⣏⣀⣸⣃⣹⡶⠟⣫⠝⢷⡀
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠑⠲⢠⣮⣀⣱⡞⢟⡀⣏⣙⣯⣭⣶⢲⣤⣮⣭⣿⣉⣹⡛⣧⣼⣁⣹⣥⠚⠁⠀⠈⢷⡀
⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⣮⢤⡿⠾⣿⣀⣿⢛⣛⣿⡀⣹⠈⣅⣠⡌⢩⡝⢻⣆⣸⡷⠟⣻⣿⢳⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠐⠛⠓⠒⢢⢯⣀⣽⡞⠳⠤⠌⣽⠉⢹⡥⠞⠛⠛⣾⣅⡸⣧⠚⠀⠀⠙⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⡏⣰⠋⢤⣙⡆⠀⣼⠉⡟⣏⡑⣆⠀⢸⢀⠎⠳⣌⡱⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⠳⣴⠃⠀⠀⠀⠀⠀⠛⠁⠀⠀⠈⠁⠀⠿⠞⠀⠀⠉⠳⠀⠚⠋⠀⠀⠀⠉⠀⠀⠀⠀⠈⢷⡴⠋
⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠠⢤⣤⣤⣼⣇⣀⠀⠀⠀⠀⠀⢠⣤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣤⠀⠀⠀⠀⠀⢀⣀⣧⣤⣤⣤⠤
⠀⠀⠀⠀⠀⠸⡏⠉⠁⠀⠀⠀⠀⠀⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡇⠀⠀⠀⠀⠈⠉⢹⡏
⠀⠀⠀⣠⡤⠶⢿⡖⠒⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⡤⠤⢤⡀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠐⠒⣾⠷⢦⣄
⠀⠀⠈⠁⠀⠀⠈⢷⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⠏⠀⠀⠀⠁
⠀⠀⠀⢀⣠⡴⠶⠛⠻⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠳⠶⣤⡀
⠀⠀⠀⠋⠁⠀⠀⠀⠀⠈⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠁⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠈⠻⣿⡗⠶⠶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠶⢞⣿⡟⠋⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⠀⠸⣿⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⡟⠀⣠⣶⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠙⠿⣷⣄⠘⠾⠿⠟⠛⠻⢶⣦⣀⣤⣶⡿⠛⠛⠿⠿⠟⢀⣼⡿⠛⠉⢷
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠈⢻⣦⠤⠖⠚⠋⠉⠉⠉⠛⠉⠉⡉⠙⠙⠲⢦⢤⣿⠋⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣀⡀⠀⠀⣣⠀⠀⠀⠀⠁⢸⡇⠀⠀⠀⠀⠀⠀⠙⢷
⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⣼⣿⣨⢤⡀⠄⢡⣇⡽⣶⡾⣭⣧⠀⠀⣠⡭⣸⣿⡀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⣀⣀⣠⣾⡷⠟⠛⠻⣿⣮⣿⣿⣶⣤⣴⣾⣿⣿⣾⡟⠛⠿⢷⣿⣦⣀⣀⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣽⣿⠛⠁⠀⠀⠀⠈⠄⠁⠀⠀⠉⠉⠉⠁⠀⠀⠈⠀⠀⠀⠀⠨⠙⢻⣿⣥⣄⡀
⠀⠀⠀⠀⠀⢀⣴⠿⣶⣶⣄⠈⢦⣐⣀⣠⠤⠖⠒⠒⠒⢦⣀⠀⢀⡤⠔⠒⠒⠲⠦⢤⣄⣀⣰⠏⢠⣶⣶⣿⣶⣄
⠀⠀⠀⠀⠀⣾⣷⣿⣿⣿⣿⣦⣀⣉⣁⣤⣴⣾⣿⣭⡛⢦⡈⠙⠋⣠⢞⣩⣽⣿⣶⣤⣄⣉⣀⣠⣿⣿⣿⣿⣷⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢷⡾⠾⠛⠋⠉⠀⠀⠈⠙⢷⣽⣒⣻⣵⠟⠁⠀⠀⠈⠉⠛⠿⠿⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣾⣤⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠉⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣧⣀⣀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⣀⣀⣽⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠉⠉⠙⠒⠦⢤⣀⠀⠈⠳⡄⠀⠀⠀⠀⢀⡴⠃⠀⠀⣀⠤⠖⠚⠉⠉⠉⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⠹⣆⠀⠀⢀⡞⠀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣠⠤⠤⠄⠀⠀⠀⠀⠀⠈⠳⣽⡄⠀⣾⡴⠋⠀⠀⠀⠀⠀⠀⠠⠤⣄⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢀⢀⡞⠉⠉⠉⠉⢳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⣤⠴⠖⠒⠚⠛⠛⠛⠛⣒⣲⣶⣶⣷⢤⣤⣤⡤⣾⣧⣶⣒⠒⠂⠀⠀⠒⠒⠒⠲⠤⢄⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣷⡞⠉⠀⠀⠀⢀⣀⡤⠶⠚⠋⡽⠋⡠⢫⠃⢠⡏⢸⡇⠈⢏⢦⡉⠋⠁⠂⠀⠀⢀⣀⠀⠀⠀⠈⢑⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢶⠶⠶⠛⠉⠀⠀⠀⢀⡾⠁⡜⢁⠇⠀⢸⠇⠀⣷⠀⠈⡆⠳⡀⠠⠀⠀⠀⠀⠈⠉⠛⠛⠚⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠈⠀⠸⠀⠀⣿⠀⠀⣿⠀⠀⠾⠀⠑⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡽⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⡟⠀⠀⢹⡇⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⢸⡇⠀⠀⢸⣇⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⢀⡾⠀⠀⠀⠀⢿⡄⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡾⠁⠀⠀⠀⠀⠈⢻⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢰⡿⠉⠁⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣾⣿⣿⡃
⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠴⠛⠋⠙⣻⡄
⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠊⠁⠀⠀⠀⠀⠉⠛⡀
⠀⠀⠀⢀⣼⣿⣷⠶⠶⠤⠼⠿⢵⣤⣠⣤⣽⣆⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡄
⠀⠀⠀⠠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⡖⠒⠒⠲⢢⣴⠷⣀⣠⣤⡴⠦⠀⠀⠀⠀⣴⡿⠃
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣾⣿⣿⣿⣿⣷⣶⣾⣷⣿⣿⣏⣀⣀⣀⣀⣤⣴⣿⣿⡇
⠀⠀⠀⠀⢸⡧⡀⠀⠀⠀⠉⠉⠉⢈⣿⣿⣿⣿⠉⠼⠩⣽⣿⣿⣿⣿⣿⣭⡉⠁⠀⠲⣯⣽⡆
⠀⠀⠀⠀⢸⣿⣿⠂⠀⣀⣀⣤⡶⠋⢁⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣯⠙⠻⠿⠆⠀⠀⠈⡟⢀⡄
⠀⠀⠀⠀⠘⢹⣷⠛⠋⠝⠋⠁⠀⠀⣾⣿⡟⢹⡄⠀⠀⢸⠟⢿⣿⣿⣙⠂⠀⠀⠀⠀⠀⡇⠀⡅
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢀⡼⠋⢿⡇⠀⡇⠀⠀⡞⠀⠘⣿⣿⡍⠓⢦⡄⣀⣀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠸⠁⠀⡇⠀⠀⣧⠀⠀⠀⠙⢷⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⠀⠀⠀⠀⠀⢰⠃⠀⠀⢸⣦⡀⠀⠀⠀⢣⠀⠀⠼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠯⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⢻⡽⣆⡀⠀⠀⠀⠠⣦⠰
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠿⠁⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠈⢿⠹⣷⣦⣤⢴⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠉⠙⠟⢿⡇⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠄⠀⠀⠀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠔⢄
⠀⠀⢠⠋⡱⢤⡴⢦⠀⠀⠀⠀⠀⡃⢾⡉⣷⡸⠀⠀⣀⣄⡀⠀⣀⣠⣄⠀⠀⠀⠀⠀⣀
⠀⠀⠘⠮⠧⢴⠃⡘⠀⠀⠀⣠⣤⣌⣒⣁⣀⣀⣤⡞⠁⢀⡙⣟⡉⠀⢈⣧⠀⠀⠀⡞⢉⣹⡤⣤⠔⠲⡄
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢰⡇⠀⠀⠈⠉⠉⠀⢸⡇⠀⢯⡟⠉⠙⣿⡉⠙⣆⠀⠀⣇⠈⣺⣀⣸⠷⢀⡇
⠀⢀⣀⡀⢀⡴⠲⡄⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠈⠓⠶⠚⠳⠤⣴⠓⠃⣠⡇⠀⠀⠈⠉⠁⠈⠉⠒⠋
⠀⢸⠀⣿⠉⢹⠆⣸⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⢹⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⠦⡬⠋⠉⠉⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠈⣹⣇⡀⠀⠀⠀⠀⠰⢒⡤⢮⡈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⣟⠃⠀⠰⡿⠀⠀⠀⣠⢤⡀⠀⠀⠙⠁⠀⢀⣸⣯⣷⣶⣶⣤⣄⠰⣈⡷⠚⠒⠁
⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣷⣿⣏⣀⠀⢀⣀⣀⡀⠉⠋⠁⢀⡴⠶⠶⢦⣼⣿⠿⠛⠛⠛⠻⢿⣷⣄
⠀⠀⠀⢠⣾⣿⠟⠋⠉⠉⠉⠙⠛⠿⡶⠋⠉⠉⠙⣦⣤⣤⣿⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧
⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣤⣤⡿⠿⠛⠛⠷⢾⣷⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣦⡀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⠋⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠈⣿⣧⠀⠀⠀⠀⣾⣿⠇
⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣴⡿⠃⠀⠀⠀⢀⣿⣿
⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠉⠀⠀⠀⠀⢀⣾⣿⠇
⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣷⣄⣀⣀⣀⣀⣤⣾⣿⣧⣄⣀⣀⣀⣠⣴⣿⡿⠋
⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠙⠛⠻⠿⠿⠛⠋⠉⠉⠛⠿⠿⢿⣿⠿⠟⠋⠀⢀⣀
⠀⢀⣀⠀⠀⠙⢿⣷⣦⣄⣀⣀⣤⣶⣿⡿⠋⠀⠀⠀⢀⠎⢉⣲⣄⣠⠴⢤⠀⠀⠀⠀⠀⡔⢒⡖⢪⡄⢣
⢠⡏⣠⡵⢤⡴⠢⡙⠻⠿⠿⠿⢿⡛⠁⡠⢤⠀⠀⠀⠸⡀⠘⣏⠀⣸⠆⢠⠇⠀⠀⠀⠀⠡⣈⡳⠐⠓⠃
⠀⠳⠼⠦⢴⠋⣠⠁⠀⠀⠀⠀⠉⢸⡋⢱⢀⠃⠀⠀⠀⠉⠋⠉⠉⠑⠖⠋⢀
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠲⠋⠁⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠘⠈⠈⠉⠀⠀⠘⠈⠠⠀⠃⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠚⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠒⠒⢤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠒⠚⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠓⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⡽
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⣀⣀⣠⡇⠀⠀⠀⠀⠀⣄⠀⢀⠀⣀⡠⠤⠖⠒⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⠒⠒⠤⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⣰⠚⠉⠉⠁⠀⡇⠀⣀⣀⣤⠴⠛⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠙⠒⠦⢤⣀⡀⠀⠀⢸⠃
⠀⠀⠀⢹⡀⠀⠀⠀⠀⠑⠾⠭⠖⢺⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣸⠷⠒⠒⠲⠦⣄⣀⠉⠙⠒⢾⡀
⠀⠀⠀⢸⣇⣧⣀⣀⣀⣀⣀⣀⡠⠞⠓⠒⠢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠾⠿⠭⢭⣥⠄⠀⠉⠣⢄⣰⣆⣀⠀⠀⡏⠉⠓⠂⠋
⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠒⢒⣲⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⣀⣀⣀⣀⣀⠀⠀⠈⠉⠑⣷
⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠉⠉⠉⠉⠉⠉⠓⠲⢤⡈⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡯⠴⠒⠉⠉⠀⠀⠀⠀⠈⠉⠉⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠤⠤⠤⠶⠶⠶⠶⠶⠤⠤⠤⢤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠒⠚⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠲⠤⢄⣀
⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠢⠤⣄⣀⡀
⠀⠀⠀⢀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠤⠤⠤⠤⠤⠤⠤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠔⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⢐⡷⠤⠤⢄⣀
⠀⠀⠀⠀⠀⡧⠀⢀⣠⠤⠖⠚⠉⠁⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⣶⠤⠤⣀⢠⠇⠀⠀⠀⠈⢱⠄
⠀⠀⠀⠀⠀⣿⣋⣁⣀⣤⡴⠒⠛⠉⠉⠛⢻⣤⠤⠤⠤⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠉⠓⠚⠋⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠈⠉⠉⠁⢰⢃⣀⡤⠤⠤⠖⠚⠁⠀⠀⠲⠶⣒⠚⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠋⠉⠉⠲⠤⠤⠤⠤⠤⠶⠦⢤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣀⣀⣀⣀⡠⠤⠤⠤⠤⠤⣀⡈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠭⢍⣥⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠻⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣡⠔⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠠⠤⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠈⠱⢄⡀
⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⢀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦
⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠙⢦⡑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡫⡀⣀⣀⣤⠄⠀⠀⠀⢀⣀⣀⣧
⠀⠀⠀⠀⠀⢿⡶⢦⣄⠀⠀⠀⠀⠀⠀⠠⢿⣾⣦⡀⠀⠀⠀⢀⣀⣠⣾⣿⣿⡟⣫⠅⠀⠀⣠⡶⠞⠋⠉⣩⠟
⠀⠀⠀⠀⠀⠈⠛⢧⣬⣿⣦⣤⣀⣀⣀⣀⣀⣙⣿⣿⣷⡟⠻⠉⠉⢹⣿⣿⣿⣿⣥⣤⣴⣿⣿⣦⣦⣴⡊⠁
⠀⠀⠀⠀⠀⠀⡤⠖⠋⠋⠉⠉⠉⠉⠉⣉⣉⣉⣛⣿⣿⡇⠀⠀⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠈⠉⠓⡆
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⣩⣿⣿⣿⠀⣀⣧⣠⣿⣿⣿⣿⡶⠆⠀⠀⠀⠀⠉⠉⠉⠹⠁
⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⠉⢹⣿⣿⣿⣿⣟⠻⡶⠶⠂⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⠛⣉⡥⠖⢋⣿⣿⣿⣿⠇⠀⢸⢻⡏⣹⣟⣿⡛⠃⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⡀⠤⠒⠋⠁⠀⢠⣿⠏⡿⢹⡏⠀⠀⣾⢸⡇⠃⢿⠸⡛⠶⢄⣀⣀⣀⣀⣠⠝
⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⠀⣰⢿⠏⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠸⡆⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡟⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠧⠽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃⠀⠀⠀⠸⡇⠀⠀⡗⠀⠀⠀⠀⠀⠀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠇⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸⡀⠀⠀⠀⠀⡀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⡼⠳⠀⠘⡆⠀⠀⢣⠀⠀⠀⡼⠉⠙⠒⠾⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⢰⠃⠀⠀⠀⠺⠀⠀⠀⠳⡈⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⣀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠿⠛⠿⣷⣶⣶⠿⠛⠉⠉⠉⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠛⠛⠛⠿⢿⣷⣶⣤⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠈⢻⣧⣀⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣼⡿⠁⠀⢀⣾⡿⢿⡿⠛⠛⠛⠻⢶⣶⡶⠾⠿⠿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⢿⣤⣾⠀⠀⠀⠀⠀⠈⣿⣄⡀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠙⣿⣄⠀⠀⠀⠀⢀⡟⢹⣷⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠶⠶⠟⠛⠻⢷⣶⣶⣶⠿⠿⠿⠋⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣠⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⢹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣶⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠐⠛⠛⣿⣿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣼⣿⣤⣤⡀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⢀⣿⣿⣄⣀⣀
⠀⠀⠀⠀⠀⠚⠛⠛⠋⠙⢿⡏⠁⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠈⢻⣿⡟⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣤⡄⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣏⠉⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠟⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡿⠿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣷⣿⠃⢀⣠⣶⣿⠿⠿⠿⢶⣤⡀⠀⠀⠀⠀⢀⣤⡾⠟⠛⠛⠻⠷⣦⡀⠀⢀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⠉⠛⢿⣿⣿⣯⣥⣀⠀⠀⠀⠉⢻⣦⠀⢀⣴⠟⠉⠀⠀⠀⠀⢀⣤⡾⠿⢾⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⢠⣾⡿⠋⠉⠉⠙⢿⡄⠀⠀⠀⠙⢿⡟⠁⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⣀⠀⣀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣤⣴⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠛⠻⢿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠛⠛⠉⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⣀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠿⣷⣄⠀⢀⣴⡿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⢀⣴⠿⠃⠀⠈⠻⣷⡿⠋⠀⠀⠙⢿⣦⡀⠀⠀⠀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡤⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠿⠙⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⣾⣿⠟⠙⠉⠀⠀⡀
⠀⠀⠀⠀⢀⣼⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣄⣼⣿⠏⠁⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⣼⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⠛⣀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⡀⡀
⠀⠀⠀⠀⣿⣿⣿⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠾⠛⠻⣿⣿⠇
⠀⠀⢠⣾⣿⣿⣿⣿⣦⡂⠀⠀⡀⢀⣀⠤⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠸⠋
⠀⠀⠘⡿⢿⣿⣿⣿⡿⠿⠾⠚⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠙⣆⢤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠺⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠌⡁
⠀⠀⠀⠀⠘⡎⠇⡀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡐
⠀⠀⠀⠀⠀⠑⢄⣀⣂⣀⣠⣤⣶⣾⠋⠉⠀⠀⠀⣀⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣠⣀⠠⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢝⠿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠛⠩⣿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣿⡇⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠋⠁⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⣠⣴⣿⣿⠇⠀⠀⠀⠀⢠⣴⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠙⠿⣿⣿⣷⣄⣤⢶⣿⠟⠙⡀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠁⠀⠒⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⢈⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠉⠉⠁⠀⠈⠉⠉⠛⠲⢦⠀⠀⢀⡴⠋⠀⢙⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣠⡴⠋⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⣀⡴⠞⠋⠉⠉⠉⠉⠉⠛⠂⣠⡾⠋⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⣶⠀⠀⣠⠞⠋⠀⣀⡴⠋⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢠⡟⠀⠀⠀⢀⣤⠖⠛⠋⢠⣾⡁⠀⣠⡾⠋⢠⡀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⣾⠀⠀⠀⢠⡟⠁⠀⠀⢠⡟⣀⣹⠾⠋⠀⠀⠈⣧⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣏⠀⠀⠶⢾⡇⠀⠀⠰⣿⠛⠉⠀⢰⡶⠶⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢿⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⢀⡟⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠘⣧⠀⠀⠀⠈⠛⠶⣤⣤⡤⠶⠋⠀⠀⠀⢀⡾⠁⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠈⠓⠦⣤⣀⣀⣀⣀⣀⣤⠶⠛⠁⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⣀⣀⣀⣀⣀⣀⣠⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡠⠢⡴⠓⠶⠉⠲⠚⢦⠔⢆⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠓⢃⠀⠡⠀⢠⠀⠀⠀⡆⠀⠀⠀⡌⠀⡘⠚⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠺⡀⠈⢆⠀⢂⠀⠀⠀⠀⢐⢀⡀⢀⠁⠐⠀⠐⠀⡠⠁⢀⠗⢦
⠀⠀⠀⠀⠀⢰⠚⠣⠀⠈⠂⠀⠂⣀⠄⠚⠀⠉⠀⠀⠀⠈⠉⠑⠲⢄⡐⠀⠐⠁⡠⠞⠓⡆
⠀⠀⠀⠀⣚⠉⠑⠄⡀⠑⢀⠐⠊⠀⡠⠄⠒⠉⠁⠀⠀⠉⠐⠂⠄⡀⠈⠓⢄⠈⠀⠠⠊⠉⣓
⠀⠀⠀⣺⠁⠁⠂⠄⠈⠠⠊⠀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠲⡁⠠⠐⠈⠈⣕
⠀⠀⢸⡁⠈⠑⠂⠀⡨⠁⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠱⡄⠐⠊⠁⢈⡇
⠀⠀⣏⠈⠁⠀⠐⢠⠃⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠐⠀⠈⠁⣹
⠀⠰⡉⠈⠉⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠉⠁⢉⠆
⠀⠰⡋⠉⠉⣁⠀⢩⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡁⣉⠉⠉⢙⠆
⠀⠀⡏⠉⢀⡀⠀⠴⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⡀⠈⢹
⠀⠀⢸⠉⢀⣀⠀⠀⢳⡀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠤⣀⡀⠉⡇
⠀⠀⠀⣝⠁⣀⠤⠂⠁⠱⡀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠐⠤⣀⠈⣫
⠀⠀⠀⠀⢽⣀⡠⠐⠁⡠⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢄⠈⠂⢄⣀⡫
⠀⠀⠀⠀⢠⠿⣀⡠⠊⢀⠔⠀⠈⠑⠦⢄⡀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠋⠀⠢⡀⠑⢄⡀⡍⢣
⠀⠀⠀⡰⢃⠊⠀⠣⢤⠃⢀⠌⠀⠌⠀⠂⠀⠁⢸⠈⠁⠈⡀⠠⠀⢡⠀⠡⡀⠘⡦⠜⠀⠈⠂⠱⡀
⠀⠀⡜⢠⠂⠀⠀⠀⠈⠉⠻⡀⡜⠀⡰⠀⠸⠀⠀⠀⡇⠀⠁⠀⠇⠀⢣⢀⡜⠉⠁⠀⠀⠀⠀⠡⠘⠄
⠀⡈⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠱⠤⠣⣀⠦⣀⡤⢀⡔⠤⠎⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠘⡀
⢰⠁⠄⠀⠀⢀⣀⡤⠤⢤⠀⠀⠀⠀⠀⠀⠄⢀⠜⠁⠘⢆⠈⢀⠀⠀⠀⠀⠀⢠⠤⠤⣄⣀⠀⠀⠀⠂⠐
⣌⣠⠤⠒⠋⠁⠀⠀⠀⠈⡇⠀⠀⠀⢀⠂⢠⠋⠀⠀⠀⠀⢣⡀⢂⠀⠀⠀⠀⡟⠀⠀⠀⠀⠉⠑⠒⠤⣄
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠂⡰⠁⠀⠀⠀⠀⠀⠀⠱⡀⢂⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠌⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⠂⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⡞

⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⡼⠋⠉⠛⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠹⣆
⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⣠⣤⣤⣤⣤⣄⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⢻⡶⠶⠶⠶⣾⠛⠶⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠸⣇⠀⠀⠀⢀⣀⣤⡴⠞⣻⠟⠀⠀⠀⣾⠁⠀⠀⠀⠘⣧⠀⠀⠈⠙⢿⡓⠶⢤⣤⣀⣀⣀⡼⠃
⠀⠀⠀⠀⠘⠛⠛⠛⠉⠉⠀⠀⣴⠋⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠈⢻⣄⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⡀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣀⡀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠸⠷⠞⠛⣿⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⡿⠙⠳⣬⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡗⠺⢿⡓⠲⠦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⠀⠀⠀⠀⠀⠀⠉⠻⣝⠲⣄⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠶⠦⣤⣸⠃⠀⠈⠉⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⢄⡴⠋⠀⠀⠀⠀⠀⠈⣧⠠⣤⣤⣤⠤⠤⠴⠖⢲⠿⠒⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢰⡏⠀⠀⠀⠀⠀⠀⢲⣴⣠⡼⠧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠎⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢳⣄⡀⠀⢀⣀⣠⡴⠋⠁⠀⢀⣴⠟⠛⠳⠶⣤⣀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠈⣉⡽⠟⠋⠉⠷⣄⣀⣠⡾⣅⠀⠀⠀⠀⠀⠉⠳⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣠⠶⠛⠁⠀⠀⠀⠀⠀⠈⠉⣻⠀⠈⠑⠄⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⣠⠶⠛⠉⢹⡀
⠀⠀⠀⠀⠀⢴⠶⠿⠆⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠳⠶⠶⢚⣁⣀⣀⣀⣸⠇
⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣠⡴⠛⠉⠉⢯⠉⠉⠁
⠀⠀⣀⣠⣤⣼⢶⣿⣦⡤⠤⠀⠀⠀⠀⠀⠈⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠙⢾⣦
⠈⢉⣁⡤⠤⠴⠒⠛⢿⣉⣉⡡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠬⢷⠖
⠀⠀⠀⢀⣀⡤⠶⠒⠋⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋
⠀⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢤⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣄⣈⣹⠶⢯⡉⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡳⠶⠦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠞⠛⠳⣄⡀⠉⠳⢤⡀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⣀⠀⠀⠀⠈⠉⠉⠛⠛⠓⠲⠶⠶⠶⠶⠶⠶⠶⠖⢶⡛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠙⠳⢤⡀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⢻⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⣰⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⣰⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢀⡴⣰⠋⠏⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠈⠻⣆⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣰⠋⡿⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠲⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠈⠛⠳⢦⣄⠀⠀⠀⣀⣠⢾⣥⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⢀⣴⠿⣤⠤⠴⠖⢒⡛⠉⠉⡀⠉⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈

⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠐⠂⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⣿⡇⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⣰⠋⣷⡀⠀⢀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⣠⣤⡀⣀⣀⠀⣀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⢰⠗⠒⠻⣧⠀⣾⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⡇⠀⠁⠈⢷⣀⠏⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠠⠿⠄⠀⠰⠿⠷⠙⠿⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠿⠷⠙⠿⠶⠒⠀⢈⡟⠀⠀⢸⣿
⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⡿
⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠶⠶⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠶⠦⠤⠤⠄⠀⠀⠀⠀⠀⠀⠘⠇

⠀⣶⣷⣦⣀⣀⣠⣴⣾⡆⠀⠀⠀⢰⣶⣶⣦⣀⣤⣶⣶⡆⠀⠀⢠⣿⣷⣄⠀⠀⣴⣾⣷⡀⢠⣶⣦⣄⠀⠀⢀⣤⣶
⠀⣿⣿⣿⣿⢿⣿⣿⣿⡇⠀⠀⢀⣈⣿⣿⣿⣿⣿⣿⣿⣃⡀⠀⢼⣿⣿⣯⠿⢿⣽⣿⣿⡇⢸⣿⣿⣿⡿⠿⣿⣿⣿
⣠⣾⣿⡿⠋⠀⠻⣿⣿⣦⡐⠚⢻⣿⡿⠿⠛⠀⠛⠿⢿⣿⡿⠓⢚⣿⣿⣿⠀⠸⣿⣿⣟⠀⠈⢻⣿⡿⠁⠀⠹⣿⣿
⠀⠰⠏⠀⠀⠀⠀⠈⠻⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠛⠙⠿⠃⠀⠀⠹⠟⠙⠃⠀⠸⠟⠁⠀⠀⠀⠙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀
⢾⣿⣷⣦⠀⡀⢀⣴⣿⣿⡇⠀⠀⣿⣿⣷⣄⣀⣠⣾⣿⣿⠀⠀⢻⣿⣿⣦⠀⢠⣶⣿⣿⠇⠀⣿⣷⣦⣀⣀⣤⣾⣿
⠸⠿⠿⣻⡎⠛⣿⡛⠿⠿⠂⠀⠀⣿⣿⣿⣿⠛⣿⣿⣿⣿⠀⠀⢼⣿⣿⡿⠛⠛⣿⣿⣿⡆⠀⣿⣿⣿⣿⢻⣿⣿⣿
⠀⢀⣾⣿⠀⠀⠸⣿⣦⠀⠀⠀⠀⠉⣻⣿⡟⠀⢻⣿⣟⠋⠀⠀⠈⢩⣿⠁⠀⠀⢹⣯⠉⠀⠀⢉⣿⣿⡏⠈⣿⣿⣏
⠀⠀⠻⡇⠀⠀⠀⡿⠏⠀⠀⠀⠀⠞⠛⢿⠁⠀⠈⡿⠛⠳⠀⠀⠀⠞⠻⠀⠀⠀⠰⠟⠇⠀⠀⠸⠛⠻⠀⠀⠽⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀
⢀⡶⠛⠶⢄⣀⡴⠞⠳⣆⠀⠀⠀⢀⣄⡀⠀⠀⠀⢀⣀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣼⣿⣿⡆⠀⣾⣿⣿
⠈⢧⣤⡴⡾⠻⡶⣤⣤⠟⠀⠀⠀⣿⣿⣿⣶⣶⣶⣿⣿⣿⠀⠀⢸⣿⣿⣶⣤⣴⣾⣿⣿⠃⠸⢿⡿⢿⡋⠋⣻⠿⣿
⠀⠀⠀⢰⡇⠀⣷⠀⠀⠀⠀⠀⠀⢿⣿⣿⠟⠉⠻⣿⣿⡿⠀⠀⠸⣿⡿⠛⠀⠈⠻⢿⡿⠀⠀⠀⢠⣿⠃⠀⢿⣧
⠀⠀⠀⠛⠁⠀⠙⠳⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠸⠟

⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣷⣦⡀⣠⣤⣶⣶⣦⣄
⠀⠀⠀⠀⠀⢀⣾⠋⠁⠈⠉⠛⠻⣷⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡟⠉⠁⠀⠈⢻⣆
⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⢀⢘⣿
⠀⠀⠀⠀⠀⢿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢀⣀⣠⣤⣴⣿⣥⣤⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉⠛⣛⣿⣿⣿⣿⣿⣿⣿⠇
⠸⣿⣿⣿⣿⠏⠉⠉⠉⠉⠉⠉⠉⠛⠛⠿⣷⣶⣤⢦⣤⣤⣴⣶⡾⠿⠛⠛⠛⠛⠻⢿⣿⣿⢿⣿⣶⣶⣤
⠀⠙⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣼⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠚⠛⢻⣿⣿⠟⠓⠀⠀⠀⢀⣴⣦⠀⠀⠀⠀⣸⡟⠉⠉⠙⣿⡇⠀⠀⠀⣠⣤⠀⠀⠀⠀⠲⠶⣿⡿⣿⡿
⠀⠀⠈⢿⣿⣄⣀⠀⠀⠀⠘⢿⡿⠀⠀⠀⢠⡿⠁⠀⠀⠀⠹⣧⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⣀⣿⣧⡿
⠀⠀⠛⠛⠻⣿⣅⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⢰⡟⠛⣷⠀⠙⢷⣄⠀⠀⠁⠀⠀⠀⠀⠀⢹⣿⡿⠿⠶
⠀⠀⠀⠀⣀⣼⣿⣿⡛⠒⠖⠒⠚⠛⠉⠁⠀⠀⠈⠙⠛⠉⠀⠀⠀⠉⠛⠲⠶⠤⠤⠤⠶⣶⣿⣏
⠀⠀⠀⠾⠛⠉⠀⠙⠿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋⠙⠻⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣿⣿⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣶⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠛⢩⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⣹⣿⣿⣿⣿⡍⠻⢷⣤⡀
⠀⠀⠀⠀⠀⣠⣾⠟⠉⠻⣷⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣴⡿⠛⢿⣆
⠀⠀⠀⠀⢠⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠘⣿⡄⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠈⠻⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠛⠛⠛⠛⠛⠛⠛⢻⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣀⣀⣀⣀⣀⣀⣀⣼⣧⣀⣀⣀⣀⣀⣀⣠⣴⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠂⠀⠉⠉⠉⠉⠉⠁⠀⠐⠂⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣦⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣄⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢀⣄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⢰⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⣶⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡠⠄⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠤⡀
⠀⠀⠀⠘⣄⠀⠈⠢⢆⠀⠀⡔⠉⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⡐⠉⠁⡄⠀⢀⠦⠊⠀⢀⡘
⠠⠄⠤⣀⠀⠑⢄⡀⠀⠑⠤⡘⢄⡀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⢠⠃⠀⠀⢀⠠⠊⠀⡀⠜⡠⠔⠁⠀⣀⠔⠁⢀⡠⠄⠤
⠠⠀⠀⠀⠈⠑⠂⠌⣂⣄⠀⠈⠐⠬⠆⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⡰⠁⠀⢀⠔⠁⠀⠀⠮⠔⠊⠀⢀⣄⡪⠄⠒⠉⠀⠀⢀⡠
⡠⠀⠀⠀⢀⡤⣀⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠰⡤⠄⢀⣀⡀⠠⠄⠂⠁⣀⠔⠊⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⠠⣔⢂⠉⠤⠤
⠣⠀⣀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠁⠒⠀⠂⠐⠀⠂⠉⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠁⠀⠀⢀⣀⠤
⠀⠀⠀⡀⠌⠈⠁⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠩⠀⠀⡀
⠀⠀⠀⠂⠄⢀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠤⠀⣀⣀⠠⠔⠊⠉⠒⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠊⠉⠒⠤⢀⣀⡀⠠⠄⠐⠊
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣾⣿⣿⣿⣿⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣾⣟⡛⠛⠳⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠞⠛⢛⣻⢷
⢸⡇⠈⠛⢦⣀⠀⠀⠈⠙⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⠁⠀⠀⣀⡴⠛⠁⢸
⣾⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠉⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⢀⣠⡴⠞⠉
⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⡀⠙⢷⣤⣄⠀⠀⣀⣠⡾⠋⢀⣀⣤⠴⠞⠋⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⣿⣬⣽⣻⣭⣼⡟⠛⠉⠉
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠛⠛⠛⢻⡇
⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢛⣛⣛⣛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⢉⣉⣉⣉⢻⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠋⠁⠙⠋⣽⢿⠉⠛⠉⠙⢷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⢷⣄⣠⣤⣤⠤⠶⠞⠛⠉⠁⣼⠃⠀⠀⠀⢠⡏⠸⡇⠀⠀⠀⠀⢳⡈⠉⠛⠳⠶⠤⣤⣤⣄⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣼⠁⠀⢿⡀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢰⡏⠀⠀⠘⣇⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢻⡀⠀⠀⡀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠶⠞⠛⢷⡀⢰⡇⠀⠀⠀⠀⠘⣇⢀⡾⠋⠛⠲⢮⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠻⡿⠀⠀⠀⠀⠀⠀⢹⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣯⡻⠿⣛⡃⢘⣛⠿⢟⣽⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⢫⣮⣻⣧⣼⣟⣵⡝⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣉⣉⢈⣉⣿⣏⣹⣿⣉⣉⣉⣉⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡜⢟⣽⡟⢻⣯⡻⢣⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣟⣵⣶⣭⡅⢸⣭⣶⣮⣻⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣙⢿⣿⣿⣿⣿⣿⡻⣿⡿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣄⠀⣀⣠⣤⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣄⣀⠀⠀⢀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⠋⠉⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠉⠙⠻⠛⠛⠋⠉⢩⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣴⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⡿⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢈⣿⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠘⠋⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢠⠴⠶⢿⡷⠒⠀⠀⠀⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠂⠀⠀⠀⢶⣿⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣤⠄⠀⠀⠈⠛⠁⠀⠀⠀⠀⣴⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠙⢷⣄⣠⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⣴⡾⢿⣿⣽⣟⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣿⡀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⣤⡾⠟⣻⣿⣿⣿⡿⠷⠶⠶⠿⠿⢿⣿⣿⣿⣅⠀⠈⢷⣄⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⣁⠀⣰⣿⣿⣿⣿⣿⣦⣄⣀⣤⣴⣿⣿⣿⣿⣿⣆⠀⠀⢻⣦⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⣿⠉⠉⠉⠉⠉⠉⠉⠁⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠶⠶⠶⠿⠛⠉⠛⠻⠷⠶⠾⠿⠛⠋

⠀⣠⠖⠋⠉⠉⠓⠦⡀⠀⠀⠀⠀⢀⡤⠚⠉⠉⠉⠓⢤⡀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠙⠒⢄⠀⠀⠀⠀⠀⢀⠴⠚⠉⠉⠙⠲⣄
⡼⠁⢀⣀⠀⢀⣀⡀⠙⡄⠀⠀⠀⡎⠀⢀⡀⠀⢀⡀⠀⢱⠀⠀⠀⢀⠏⠀⣀⡀⠀⢀⡀⠀⢳⠀⠀⠀⢠⠃⠀⣤⡀⠀⣠⡄⠈
⡇⠀⠁⠈⠁⠈⠀⠁⠀⡇⠀⠀⠸⡀⠀⠉⠁⠀⠈⠁⠀⢀⡇⠀⠀⢸⠀⠀⠉⢁⣀⠈⠁⠀⢸⠀⠀⠀⢸⠀⠀⣉⣀⣀⣈⡀
⠹⡄⠈⠑⠒⠒⠋⠀⡰⠃⠀⠀⠀⢣⡀⠈⠓⠒⠚⠁⢀⡜⠀⠀⠀⠈⢧⡀⠘⠉⠉⠉⠂⢀⠞⠀⠀⠀⠘⢆⠀⠳⣄⣀⠴⠁⢠
⠀⠈⠓⠤⠤⠤⠴⠊⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠙⠢⠤⠤⠤⠖⠉⠀⠀⠀⠀⠀⠀⠑⠦⠤⠤⠤⠚⠁

⠀⠀⣠⠤⠴⠤⢤⣀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠦⠤⣄⡀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠦⠤⣄
⢠⠋⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⡴⠁⠠⡀⠀⢠⠄⠙⣆⠀⠀⠀⢀⡜⢁⠀⠀⠀⠀⠀⠑
⡏⠀⠰⠗⠀⠐⠿⠀⠀⡇⠀⠀⢰⠃⠀⠦⠖⠀⠲⠴⠂⠈⡆⠀⠀⢸⠁⣀⣉⣁⣀⣉⣁⡀⢸⡀⠀⠀⢸⠀⢈⠽⠂⠀⠺⠅
⢧⠀⠐⠒⠒⠒⠒⠀⢀⠇⠀⠀⠈⣆⠀⢠⠖⠒⠢⡄⠀⣰⠁⠀⠀⠘⡄⠳⡀⠀⠀⠀⣠⠇⡸⠀⠀⠀⠸⡀⠀⡠⠖⠒⠦⡀
⠈⠳⢄⡀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠈⠢⣄⡀⠀⢀⣠⠔⠁⠀⠀⠀⠀⠘⠦⣈⠓⠒⢋⣡⠜⠁⠀⠀⠀⠀⠙⢦⣀⠀⠀⢀⡠⠞
⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⢀⡴⠊⣡⠀⢠⣉⠳⣄⠀⠀⠀⠀⢀⠔⠋⠀⠀⠈⠉⠢⡀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠈⠙⢢⡀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠈⠑⢦
⡞⠀⢠⣦⠀⠠⣦⠀⠈⡆⠀⠀⢀⠏⠀⢢⣄⠀⣠⡔⠀⠹⡀⠀⠀⢠⠃⠐⣦⡖⠀⢢⡔⠀⢱⠀⠀⠀⢰⠁⠀⢦⡄⠀⢠⡄
⣇⠀⠀⣀⣠⣀⡃⠀⠀⡇⠀⠀⠸⡄⠀⢉⠀⠀⠀⡉⠀⢀⠇⠀⠀⢸⡀⠈⠁⠉⠀⠁⡈⠀⢸⠁⠀⠀⢸⠀⠀⣉⠀⠀⢀⡁
⠘⢄⠈⠁⠀⠀⠉⢀⡜⠁⠀⠀⠀⠳⣄⠀⠉⠉⠉⠀⣠⠞⠀⠀⠀⠀⠳⡀⠉⠉⠉⠉⠁⣠⠏⠀⠀⠀⠈⢣⡀⠈⠉⠋⠉⠀⡠
⠀⠀⠙⠒⠲⠒⠚⠉⠀⠀⠀⠀⠀⠀⠈⠑⠒⠲⠒⠊⠁⠀⠀⠀⠀⠀⠀⠈⠓⠒⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠉⠓⠒⠖⠒⠋

⠀⢀⡤⠒⠒⠒⠢⢄⠀⠀⠀⠀⠀⠀⣠⠔⠒⠛⠒⠢⣄⠀⠀⠀⠀⠀⠀⣠⠔⠒⠒⠒⠦⣀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠒⠒⢤⡀
⣰⠃⠀⠖⠀⠐⠦⠀⠱⡄⠀⠀⠀⡞⠁⠐⢒⣀⡒⠂⠈⢳⠀⠀⠀⠀⡞⠁⠰⠤⠀⠴⠂⠈⢦⠀⠀⠀⢠⠎⠀⢠⡄⠀⣤⠀⠘
⡇⠀⠳⠔⠀⠘⠴⠃⠀⡇⠀⠀⢸⠀⠀⣰⠉⠀⠉⣆⠀⠀⡇⠀⠀⢸⠀⠀⠺⠆⠀⠸⠇⠀⢸⠀⠀⠀⢸⠀⠀⠛⠀⠀⠛⠃
⢳⡀⠈⠲⠤⠤⠊⠀⣠⠃⠀⠀⠀⢧⠀⠹⣞⣉⡳⠏⠀⡼⠀⠀⠀⠈⢆⠀⠰⠊⠉⠓⠄⠀⡼⠀⠀⠀⠘⣄⠀⠉⠉⠉⠉⠁⢀
⠀⠙⠦⣄⣀⣀⡤⠖⠁⠀⠀⠀⠀⠀⠑⠦⣄⣀⣤⠴⠊⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣠⠤⠊⠀⠀⠀⠀⠀⠈⠲⢤⣀⣀⣠⠴⠋

⠀⠀⠀⠀⠀⡠⡤⠖⠒⠀⠆⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⠶⠒⠂⠆⢄
⠀⠀⠀⣠⢪⠚⢄⠀⢀⢀⣄⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⠜⢇⠀⢀⡀⣇⠀⠐⠄
⠀⠀⢰⢡⠇⠀⢀⠯⠀⠀⠀⠙⣴⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡎⠀⢀⠷⡁⠀⡇⠈⣦⠋⠄
⠀⠀⡎⡜⠀⠀⠂⠀⣱⠊⠑⡎⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢸⠀⠀⠌⠀⢱⠞⠙⢶⠁⡄
⠀⠀⡇⡧⠀⢰⠆⠠⡇⠀⠀⢸⠀⢧⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠠⠆⠀⣯⠀⠀⢸⠄⡇⠰
⠀⠀⡇⢃⠀⠀⠀⠀⡁⢄⡤⠦⠀⠈⠛⠒⠀⢀⠴⡶⠀⠀⠀⠀⢡⢰⠀⠀⠄⠀⢸⢤⡠⠿⠀⠅⠈⠀⠀⠀⣀⣈⢻
⠀⠀⠸⡘⡄⠀⠘⣔⠀⠀⡀⡠⠂⡠⠀⠀⠀⠀⠙⠋⡀⠀⠀⠀⠸⡌⡆⠀⠈⣶⠁⠀⠇⠀⠂⠀⠀⠀⠀⠐⠶⡽⠀⡖
⠀⠀⠀⠱⡑⢄⠎⠀⠁⠈⠋⠀⡀⠁⠀⠀⠀⠀⠀⢭⣺⠀⠀⠀⠀⠙⣜⢆⡔⠀⠈⠀⡏⠀⡠⠁⠀⠀⠀⠀⠳⠶⠄⠉
⠀⠀⠀⠀⠈⠚⠷⠦⠄⠦⠂⠈⠀⠀⢞⣟⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠷⠦⠤⠠⠗⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠘⠀⠀⠄⠀⠐⠒⠀⠄⠊⠘⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠃⠂⠀⠄⠁⠀⡅⠊⠈⠂⠁⠂⠄⠂⠐⠁

⠀⠀⠀⠀⡀⠀⢤⣀⠀⢀⣄⢀⠀⡀⠀⢀⣀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣀⡀⢀⣠⠀⠀⠀⣠⢀⣠⠀⢀

⠀⠀⠀⠀⠀⠀⣀⣠⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⣀⡀
⠀⠀⠀⢀⠔⡽⡋⠀⠀⠀⠄⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡫⡏⠀⠀⠀⡇⠈⠢⡀
⠀⠀⢀⠎⡎⠀⠑⡄⠂⠉⠃⠢⣀⠧⡠⠶⡂⠀⠀⠀⠀⠀⠀⠀⠀⡌⡜⠁⠈⣤⠂⠉⡏⠢⣀⠵⡀
⠀⠀⡜⢸⠀⠀⠌⠈⢢⠄⠤⡠⠠⠀⠉⠛⠁⠀⢠⣒⣆⠀⠀⠀⢸⢰⠁⠀⡐⠀⢢⡤⠧⣤⠛⠀⠁⠀⠀⠠⠤⡄
⠀⠀⡇⣇⠀⢀⡀⢀⡎⠀⠀⢡⣀⣄⡄⠀⠀⠀⠀⠉⣉⣀⠀⠀⡄⢘⠀⢀⡁⠀⣼⠀⠀⢀⣀⡀⠀⠀⢐⣛⡆⢁⡀
⠀⠀⡇⡏⠀⠈⠁⠀⢣⠀⢀⡎⢀⠀⠐⠀⠀⠀⠀⠘⠵⠯⠀⠀⢃⢨⠀⠀⡅⠀⢹⡀⢀⣌⠀⠀⠀⠀⢠⣩⣁⠸
⠀⠀⢱⢸⠀⠀⠡⡠⠂⠈⠀⢀⢦⢀⠁⢭⣭⠀⠀⠀⠀⠀⠀⠀⢸⠘⡄⠀⠐⣀⠎⠈⡏⠀⡶⠀⠄⠀⠀⠈⠁
⠀⠀⠈⢦⢣⡀⡰⠉⠂⠠⠖⠁⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡱⡀⡠⠋⠂⠀⡗⠊⢀⠍
⠀⠀⠀⠀⠑⢽⣤⣀⠀⢀⡤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠼⣧⣀⠀⠀⣇⠔⠃
⠀⠠⠀⠀⠀⠠⢀⣈⠉⠠⡠⠠⢀⠀⠀⠄⠀⢀⠀⠀⠀⠀⢠⠀⠀⡄⣀⠀⢀⠌⠉⠅⡠⠠⡀⠄⠀⠀⡀⢀⠄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠃⠀⠀⠃⠀⠈⠀⠀⠀⠁

⠀⠀⠀⠀⠉⠁⠢⠄⠀⠀⠚⠀⠀⠀⠀⠁⠚⠂⠀⠆⠀⠀⠀⠀⠀⠉⠙⠀⠜⠀⠀⠚⠂⠘⠘⠀⠀⠐⠂⠀⠆

⠀⠀⠀⣤⣤⣄⡀⠀⠀⣠⢤⣤⡄⠀⠀⠀⢀⣤⡄⣤⡄⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⣠⡄⠀⠀⠀⢠⣦⣄⡀⠀⠀⢀⣤⣦
⠀⠀⠀⠿⠶⣶⡿⠛⠻⣷⠶⠾⠃⠀⢀⣐⣿⣿⣿⣿⣿⣃⠀⠀⠀⠰⣿⣿⣦⣤⣾⣿⡷⠀⠀⠀⢸⣿⣿⣷⣤⣴⣿⣿⣿
⠀⠀⠀⠀⣴⡿⠀⠀⠀⢹⣷⡀⠀⠀⠈⣻⣿⣿⣶⣿⣿⡟⠀⠀⠀⠀⠛⣻⡿⣿⡍⠛⠃⠀⠀⠀⠈⢋⣿⡿⠋⠙⣿⣏⠋
⠀⠀⠀⠈⠛⠇⠀⠀⠀⠀⠟⠁⠀⠀⠀⠈⢻⣟⠙⣿⠋⠀⠀⠀⠀⢀⡾⡿⠀⠻⣷⣄⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠻⠇⠀⠀⠀⠀⠈⠀⡇⠀⠀⠙⠏
⠀⠀⠀⣾⣿⣶⣤⣀⣠⣶⣿⣿⠀⠀⣤⠆⢤⡀⠀⠀⣠⡄⠦⡄⠀⡠⣶⣤⣄⣀⣠⣴⣶⢄⠀⠀⣠⠴⣦⠀⠀⢀⣴⠦⣄
⠀⠀⠀⠛⢻⣿⠟⠉⠹⣿⣟⠛⠃⠀⠃⣀⣨⣿⠦⢾⣯⣀⡀⠉⠀⠿⠭⠝⡟⠉⣷⠫⠭⠇⠀⠀⣿⢀⣬⡿⠶⢯⣅⡀⣿
⠀⠀⠀⢸⡿⠋⠀⠀⠀⠘⢿⡷⠀⠀⠀⢠⡟⠁⠀⠀⠙⣧⠀⠀⠀⠀⠀⢀⡇⠀⢿⠀⠀⠀⠀⠀⠀⢉⡿⠁⠀⠈⢿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠈⠓⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠈⢷⡄
⠀⠀⠀⣀⣴⣶⡄⠀⣰⣶⣆⡀⠀⠀⣤⣤⣤⣀⣀⣠⣤⣤⣤⠀⠀⣴⣤⣄⠀⠀⠀⢀⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣿⣿⣿⣧⣤⣿⣿⣿⡟⠀⠀⢻⣿⣿⠏⠋⢻⣿⣿⠏⠀⠀⣿⣿⣿⣷⣤⣴⣿⣿⣿⡄⠀⢸⣿⣷⣶⣴⣴⣶⣾⣿
⠀⠀⠀⠉⠉⣿⡏⠉⢿⣏⠉⠁⠀⠀⢀⣿⣿⠂⠀⢸⣿⣯⡀⠀⠀⠻⢟⣿⡿⠛⠻⣿⣟⠟⠃⠀⠀⠉⢁⣴⡏⢻⣦⡈⠉
⠀⠀⠀⠀⢼⣿⠀⠀⠸⣿⠄⠀⠀⠐⠙⠿⠛⠀⠀⠀⠛⠿⠙⠀⠀⠀⠿⡟⠀⠀⠀⠹⡿⠇⠀⠀⠀⠀⠀⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⣄⡀⠀⠀⢀⣤⡀⠀⠀⠀⢸⣶⣦⣤⣤⣤⣶⣾⠆⠀⠀⠰⣾⣿⣆⠀⢀⣼⣿⡶⠀⠀⠀⢸⣯⣷⡀⢀⣾⣿⡇
⠀⠀⠀⣼⣿⣿⣤⣤⣿⣿⣧⠀⠀⠀⠀⢿⢿⣿⢿⣿⡿⡿⠀⠀⠀⣴⣶⣿⣿⠟⢾⣿⣷⣶⡄⠀⠀⠸⠿⣿⡟⢻⣿⠿⠇
⠀⠀⠀⠻⣿⡟⠉⠉⢿⣿⠟⠀⠀⠀⠀⢀⣾⣿⢸⣿⣧⡀⠀⠀⠀⠈⠉⣴⡏⠀⠈⣷⡌⠉⠀⠀⠀⠀⣴⣿⠇⠸⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠈⠿⠟⠁⠀⠀⠀⠀⣼⡟⠀⠀⠀⠘⣿⡄⠀⠀⠀⠾⣿⣿⠀⠈⣿⣿⠷
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠟⠀⠀⠀⠀⠀⠘⠟⠂⠀⠀⠀⠈⠛⠀⠀⠛⠁
⠀⠀⠀⣼⣿⣷⣦⣤⣶⣿⣿⠆⠀⠀⢰⣶⣄⠀⠀⢀⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣄⠀⠀⣠⣴⣶
⠀⠀⠀⢶⣿⣿⡿⠻⣿⣿⣷⠆⠀⠀⢨⣿⣿⡷⠷⢿⣿⣿⠀⠀⠀⢴⣶⣦⣤⣤⣴⣶⣶⠀⠀⠀⠀⣿⣿⣿⡿⢿⣿⣿⣿⠂
⠀⠀⠀⠈⠛⠋⠀⠀⠈⠛⠋⠀⠀⠀⠈⠛⠋⠀⠀⠀⠙⠟⠀⠀⠀⠘⠟⠛⠉⠉⠙⠛⠟⠀⠀⠀⠀⠉⠛⠉⠀⠀⠉⠛⠉

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

⠀⠀⢀⣤⣶⠶⠿⠛⠛⠛⠛⠿⠶⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠶⠿⠛⠛⠛⠛⠿⠶⣶⣤⡀
⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦
⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⣿⠀⠀⠀⣀⣠⣤⣤⣶⠶⠶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⠶⠶⣶⣤⣤⣄⣀
⣿⣠⣴⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⢻⡿⠿⠿⠿⠿⣿⡟⠀⠀⠀⠀⠀⣀⣴⡶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣦⣄
⣿⡋⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⡀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⢀⣴⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢙
⢹⣧⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣦⡀⢸⡇⠀⠀⠀⠀⢸⡇⣀⣴⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⣼
⠀⢻⣆⠀⠀⠀⠀⠈⠙⠻⠶⣶⣤⣤⣤⣤⣤⣶⠶⠿⠛⠉⠀⠈⠻⣿⡇⠀⠀⠀⠀⢸⣿⠟⠁⠀⠉⠛⠿⠶⣶⣤⣤⣤⣤⣤⣶⠶⠟⠋⠁⠀⠀⠀⠀⣰⡟
⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⣀⣀⣀⣀⣼⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠋⠉⠉⣻⣿⡉⠉⠙⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠙⠿⣦⣤⣀⡀⠀⠀⠀⣀⣀⣠⣤⣴⠾⠟⠋⣱⡿⠁⠀⢀⣼⠏⠘⢿⡄⠀⠈⢿⣎⠙⠻⠷⣦⣤⣄⣀⣀⠀⠀⠀⣀⣀⣤⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⢀⣴⠟⠀⠀⢠⣾⠃⠀⠀⠈⢿⡄⠀⠀⠻⣦⡀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠃⠀⠀⢠⣿⠃⠀⠀⠀⠀⠈⢿⡄⠀⠀⠘⠿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠾⠛⠁⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠈⠛⠷⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⣟⠋⠉⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠉⠙⣻⣿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣤⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣼⠏

⠀⢀⡠⠔⠒⠋⠉⠙⠓⠦⣀⠀⠀⠀⠀⠀⡠⠄⠒⠒⠒⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠒⠒⠦⣄⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠉⠙⠓⢦⣀
⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⣠⠎⠁⠀⠀⠀⠀⠀⠈⠱⣄⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⡏⠀⠠⣦⠀⠀⠀⠀⠷⠀⠀⢸⠀⢰⠃⠀⠀⠀⣄⠀⢠⡄⠀⠀⠀⠈⡆⠀⢠⠃⢀⣤⡄⠀⠀⠀⠶⠖⠀⠘⡆⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢷
⡇⠀⡄⠀⠀⠀⠀⠀⠀⡲⠀⢸⠀⢸⠀⠀⡀⠀⠉⠀⠈⠀⠀⢴⡁⠀⡇⠀⢸⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⠀⣿⠀⢸⠀⠘⡇⢠⣦⠀⠺⠗⢈⡇⠀⢸
⠹⡀⠈⠢⣄⣀⣀⣠⠔⠁⢀⡞⠀⠘⣆⠈⠣⠴⢄⡴⢄⠴⠒⠚⠁⢠⠃⠀⠘⡄⠈⢿⣭⣧⢼⡤⢿⡾⠋⢀⡏⠀⠈⢧⠀⠙⢦⣀⣀⣀⡤⠞⠀⢠⠟
⠀⠙⠦⣀⠀⠀⠀⠀⣀⡤⠊⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠘⠦⣀⠉⠛⠚⠛⠉⢀⣠⠞⠀⠀⠀⠀⠳⢤⣀⠈⠉⠁⣀⣠⠔⠋
⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⣀⡠⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⢤⣀
⠀⣠⠎⠁⠀⠀⠀⠀⠀⠉⠳⡀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⠈⠳⣄
⣰⠁⠀⠀⢀⡀⠀⣀⠀⠀⠀⠹⡄⠀⢰⠋⠀⠀⠀⠀⠀⢀⡀⠀⠀⢹⡀⠀⢀⠎⠀⠀⢀⡀⠀⠄⠀⠀⠀⠹⡄⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⡇⡀⠀⠀⠈⠁⠀⠁⠀⠀⢸⡄⡇⠀⡏⠀⠀⠟⠀⠀⠀⠈⠉⠀⠀⠀⡇⠀⣸⠀⠀⠀⠀⠀⠀⠀⣠⢤⠀⠀⢳⠀⢸⠁⠀⣠⣄⠀⠀⠀⢀⣤⠀⠀⣿
⣇⢹⡀⠀⠀⠀⠀⠀⠀⠀⣸⢡⠇⠀⣇⠀⢤⠤⠤⢒⣒⣒⣊⡟⠁⢰⠇⠀⢸⠀⠀⣠⡔⢞⠉⠉⠉⠙⠦⠀⡼⠀⠸⡄⠀⠈⠁⡠⠤⣄⠈⠉⠀⢀⡏
⠈⢦⡙⠦⣄⣀⣀⣀⣠⠔⣡⠎⠀⠀⠘⣆⠈⠻⢭⣉⣀⠤⠊⠀⣠⠏⠀⠀⠀⠣⡈⠀⠈⠉⠀⠀⠀⠀⢀⡴⠁⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠉⠒⠦⠤⠭⠤⠒⠋⠀⠀⠀⠀⠀⠈⠑⠦⢤⣤⣤⠤⠴⠊⠁⠀⠀⠀⠀⠀⠈⠒⠤⢤⣀⡠⠤⠖⠋⠀⠀⠀⠀⠀⠈⠙⠲⠤⠤⠴⠒⠋
⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⢀⡴⠚⠉⠉⠉⠉⠉⠓⢤⡀⠀⠀⠀⠀⢀⠔⠋⠉⠀⠀⠉⠳⢤⠀⠀⠀⠀⢀⡴⠊⠉⠁⠀⠉⠙⠲⣄⠀⠀⠀⠀⠀⣠⠔⠋⠉⠉⠀⠉⠙⠲⣄⡀
⢠⠏⠀⠀⠀⠀⠀⠀⡀⠀⠀⠱⡄⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢠⠎⠀⠀⠀⠀⠀⢀⠀⠀⠈⣧⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⡏⠀⠀⡹⠯⠀⠀⠿⠅⠀⠀⠀⢷⠀⢰⠇⠀⣠⠤⡄⠛⢀⠤⢄⠀⠈⡇⠀⢸⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⢸⡄⠀⢸⠃⠀⠀⠀⠀⠄⠰
⣇⠀⠀⢀⣀⣀⣀⣄⣤⡤⠄⠀⡿⠀⢸⡀⠀⠁⠀⠀⣠⡀⠀⠀⠀⢀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢸⡄⠀⠤⣤⡤⡴⠶⠒⠒⠋⠁
⠘⢆⠀⠀⠀⠀⠀⠀⠓⠁⢀⡴⠁⠀⠀⢧⠀⠀⠀⠀⠻⠃⠀⠀⢀⡞⠀⠀⠀⢣⡀⠀⠀⡴⠒⠒⠄⠀⢠⠞⠀⠀⠀⢳⡀⠀⠹⣠⠇⠀⠀⠀⠀⢀⡴
⠀⠈⠑⠦⢄⣀⣀⣀⡤⠴⠋⠀⠀⠀⠀⠀⠑⠦⣄⣀⣀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠙⠢⣄⣀⣀⣀⡤⠖⠁⠀⠀⠀⠀⠀⠉⠳⢤⣄⣀⣀⣤⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢒⠎⠀⠈⠀⠀⡀⠀⣀⠀⠐⠉⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⠀⠀⠀⠸⠀⠀⡀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠠⠄⠀⡠⠀⠀⠀⠀⠀⠐⠄⡀⠀⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

⣿⡿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠐⠒⠒⠂⠀⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢐⡨⠴⠒⠛⠋⠛⠛⠛⠛⠓⠶⢦⢤⣈⠉⠒⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣈⠴⠚⢉⣠⠤⠒⠘⠉⠉⢉⠉⠉⠙⠒⠒⠦⢤⣉⠙⠓⠦⢄⡑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣡⠞⣁⠴⠚⠉⠡⡄⡄⠀⠀⠀⠘⠈⠀⠀⠀⠀⢠⠤⠀⠉⣳⢆⠀⠙⢦⡈⢢
⠀⠀⠀⠀⠀⠀⠀⡰⢡⠎⣡⠞⠁⠀⠀⠀⠀⢀⡠⠤⠖⢒⠐⠒⠰⠒⠠⠤⣀⡀⢀⡾⠃⠀⠙⣆⠀⠻⡄⠱⡀
⠀⠀⠀⠀⠀⠀⡄⣰⠃⡴⠁⠀⠳⠒⢀⡤⠊⠁⠹⡐⢈⠀⠂⠀⠄⢈⠄⣾⢁⠞⡽⢧⡈⠙⠀⠈⢣⠀⠘⣆⠠
⠀⠀⠀⠀⠀⢀⢰⠇⡰⠃⠀⠀⠀⡰⠃⠀⡈⠀⠀⠘⣄⠴⠂⠀⠒⠤⣸⡳⢫⠞⢂⡡⠘⣆⠀⠀⠈⢧⠀⠘⡄⢆
⠀⠀⠀⠀⠀⠆⡟⢀⠇⠀⠀⠀⢠⠁⠀⠀⠐⠀⡠⠊⠀⢀⠔⠁⠀⠀⠔⡱⠳⡈⠀⠐⣉⠄⢆⠠⣦⠌⢧⠀⠰⠸⡄
⠀⠀⠀⠀⠀⠐⡇⢸⠀⠘⠙⠂⡸⠤⠀⣀⡀⢠⢁⢔⠔⠁⠀⠀⡀⢊⡜⠁⠀⠘⣆⢥⣖⡾⡛⡆⠀⠀⠈⡆⠀⡆⠇
⠀⠀⠀⠀⠀⢘⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⡀⠟⠀⠁⠀⠀⡌⣔⣲⠘⡆⠀⠀⠀⢱⠀⠁⠘⠊⢱⠀⠀⠀⢣⠀⠁⢸
⠀⠀⠀⠀⠀⠸⡇⠘⡄⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠱⢤⡠⠞⠀⠀⠀⠀⣜⠀⠀⡔⠔⣸⠰⠦⢀⡙⠀⠀⠂
⠀⠀⠀⠀⠀⢃⢣⠀⢣⠄⠈⠑⠀⢃⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⢀⢔⠅⠀⠰⣧⡁⢐⣤⣶⠏⠀⠀⠀⡇⠀⠃⡆
⠀⠀⠀⠀⠀⠀⠌⣧⠈⢇⠀⠀⠀⠀⠑⢄⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⡶⠋⠀⠀⠀⡼⠀⠠⢐
⠀⠀⠀⠀⠀⠀⠈⠜⣦⠈⢣⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣀⠀⡤⡠⠠⡠⠤⠤⠄⠀⠀⠀⠠⠤⠤⠀⡴⠁⠠⢁⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠌⢦⠀⠱⣄⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠒⠂⠒⠀⡠⠀⠀⠀⠀⠀⠀⠈⠀⠀⢁⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡑⢄⠀⠀⠀⠀⢠⠲⣢⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢰⣻⡼⠀⠀⢀⠀⠊⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣑⠢⠀⠀⠀⠁⠃⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⡀⠤⠂⢁⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢅⣐⠠⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⣂⣁⡤⠔⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠂⠤⠤⠐⠒⠒⠊⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⢀⠔⠀⡡⠂⠀⠐⠀⠀⡠⠂⢀⠀⠐⠁⠠⠊⠀⠂⠂

⠀⠀⠀⠀⠀⢀⣠⠶⠛⠛⠛⠛⠛⠛⠲⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⠒⠒⠒⠶⠦⣄⡀
⠀⠀⠀⠀⢠⣾⠁⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⢀⡴⠟⠛⠋⠉⠉⠉⠉⠉⠉⠛⠓⠲⠦⣤⣀⡀⠀⠈⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⢀⣠⡤⠶⠖⠚⠛⠛⠉⠉⠛⠛⠛⠿⢤⡀
⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⣠⠞⠉⠀⣀⡤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⠙⣷⠴⠶⠶⢿⣁⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏
⠀⠀⣿⠻⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠁⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠞⢻⠁
⠀⠀⢸⠀⠀⠈⠙⠛⠳⠶⠦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⡤⠶⠶⠛⠛⠉⠀⠀⣿
⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣇⠀⠀⠀⠀⢀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣄⠀⠀⠀⣸⣷⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠖⠛⠉⣡⠞⡿⠀⠈⠉⠉⠉⠁⢸⡆⠀⠙⣟⠛⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠘⢧⣄⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡶⠞⠛⢉⡿⠁⠀⢀⡴⠋⢰⡇⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢹⡄⠀⠀⢻⡝⠛⠷⢶⣤⣄⣀⣀⡀⠀⠀⣀⣠⡴⠋
⠀⠀⠀⠀⠉⠛⠻⠿⠿⠟⠛⠛⠉⠉⠀⠀⠀⣠⠟⠀⠀⣠⡟⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣷⠀⠀⠀⢳⡀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⣴⢿⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠈⣿⡇⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡾⠃⡿⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⡀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⠀⠀⠀⣠⡟⠁⣸⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡿⣧⠀⠀⠀⠘⣧
⠀⠀⠀⠲⣶⣶⡶⠒⠛⠋⠉⠀⠀⠀⠀⢀⣼⠏⠀⣠⠏⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠈⣧⢻⡆⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⢻⡟⠀⢀⣴⠟⠁⢀⣴⠋⢀⣀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⡌⢿⡄⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢋⣠⡴⠟⢁⣠⡴⢛⣡⡾⢛⡟⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⢷⡈⢿⡄⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⣠⣴⣶⡿⠷⠞⠛⠉⠀⠈⠛⠛⠛⠋⠁⢀⡾⢁⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢳⡌⢻⣄⠀⠀⠀⠀⠀⠈⠛⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠙⢦⡙⢧⡀⠀⠘⢶⣤⣤⣤⣤⡽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡵⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⣀⣀⣀⣹⡿⠛⢷⣄⡈⠙⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⣿⠙⠉⠁⠀⠀⠀⠉⠛⠷⢶⣤⣭⣿⣶⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠘⠷⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⡬⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠒⠲⢤⣀⣠⠚⠉⢳⡀⠀⠀⠀⠀⢀⣠⠖⠛⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣠⠼⣇⡤⡀⠀⢷⠒⠒⠒⠒⠛⠁⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⣹⡇⠀⠀⣷⡁⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢧⠀⠓⠛⡶⠚⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠑⠦⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⣼⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⣀⡀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠇⠀⠀⠒⢺⡧⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡟⠓⠒⢌⣳⡄⠀⠀⠀⠀⠀⠸⠉⠕⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⡀
⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀⠀⠀⠘⣇⠀⠀⠀⣹⣁⣀⣀⡤⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣐⣦⡞⠀⠈
⠀⠀⢰⡏⠹⠦⠤⠼⠋⢹⠀⠀⠀⠈⠳⣄⣀⣼⠈⠉⢙⡏⠉⠑⢮⡷⣶⠖⢶⡖⠚⠛⠉⠉⠀⠈
⠀⢠⠞⠁⣠⠤⠤⢤⡀⠈⢷⡀⠀⠀⠀⠀⢹⠉⠑⢤⣼⣀⣀⡴⢋⣰⠋⠙⡎⢧
⢠⠏⢀⡎⠸⠂⣚⠃⣼⢧⠈⢳⡀⠀⠀⠀⠘⢶⠖⠋⠙⣾⢟⠞⢏⡿⠤⠤⢾⠈⢧
⣿⠀⠘⣏⡃⠘⠤⠃⣤⣼⠀⠈⣷⠀⢰⡒⢦⣾⣀⡠⠞⣿⠈⠓⠺⢥⡀⠀⡼⠀⢸⣇⣀
⢷⡀⠈⠣⠷⠤⠤⠤⠔⠁⠀⣀⣿⣄⣀⣽⠟⢻⣅⠀⠀⣿⠀⠀⠀⠀⠉⠋⠁⣀⡾⠉⠉⢹⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⣇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣨⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢱⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠄⢀⡠⠤⠄⢀⡀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢀⣀⡠⠤⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡯⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠉⠈⢻⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⢀⣎⢒⣀⣶⠶⠒⠒⠒⣒⠶⠶⠒⠒⠒⠂⠐⠒⠒⠿⣿⣇⠀⠀⠀⠀⠀⠀⣸⣿⠛⠓⠒⠒⠒⠒⠒⠶⠶⠦⣖⡒⠒⠒⠶⢦⣤⣀⠱
⠀⠀⠼⠃⠀⠀⡠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠀⠘⡧
⠀⠀⠀⠈⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⠀⠀⠀⠀⠀⠀⣸⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠺⢫⠟⠋⠁⠀⠁⠀⠙⠋⠙⢷⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⣠⣴⠞⠋⠁⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢆⠀⠈⠻⠲⣦⣀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⢀⠤⢺⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠙⠓⣄⡀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠔⠀⡴⠃⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠘⣄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠂⠘⠁⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠢⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣾⠛⠛⠛⠒⠲⠶⠦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠖⠚⠛⠛⠛⣷
⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⣀⡀⠈⢻⡶⠶⠶⢶⣶⡒⠒⠶⢶⡟⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢺⡇⠀⠀⠀⠀⠉⠓⠄⢸⡟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠶⠶⠶⠶⠶⢾⡇⠀⠀⠀⠀⠀⠀⠀⢸⣷⡶⠶⠶⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⠰⣄⡀⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⢉⣽⠷⣦⣬⣿⠶⣶⣤⣤⡾⣯⡉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠉⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠘⣷⣤⣤⡤⠴⠶⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⠦⢤⣤⣤⣾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠷⢶⣤⣀⣴⣾⠿⠛⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡾⠃⠀⠀⠀⠙⢿⡏⠁⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠿⠿⢿⣷⣶⣦⣤⣶⣾⣿⠿⠿⠛⣿⠃⠀⠀⣴⡶⣶⣾⠿⠿⢶⣤⡶⠾⠛⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⣿⠀⠀⠸⣿⣠⡿⠁⠀⠀⠀⢹⣷⣦⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠈⣙⣿⣄⠀⠀⢀⣼⣯⣼⡏⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠛⠉⠙⠻⠟⠻⣿⣉⠁⠀⣠⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠈⣿⣇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⠟⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠘⢻⣿⠟⠛⠛⠃
⠀⠀⠀⠀⢀⣠⣤⣶⣿⣷⠶⠆⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣀⣾⣟⡀
⠀⠀⠀⠀⠈⠉⠁⠀⠸⣿⣄⣀⡀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⣿⣅⣬⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡿⠛⠻⠿⠦
⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⣿⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⣻⣷⣿⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⠃⠀⠀⠉⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠛⠿⣿⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣾⠿⢿⣿⡉⠁⠈⢿⣧⡀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⣠⣴⡿⠟⢛⣿⠟⠉⢻⣟⠉⠉⠉⠉⠉⠀⣸⡿⠀⠀⠹⣷⡄⠀⠈⢻⣷⡄⣀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠛⠻⣷⣤⣿⣏⣤⣀⡈⠻⣦⣄⣀⣀⣠⣴⡿⠁⢀⣀⣤⣽⣿⣄⣀⣤⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣽⡿⠛⠛⠛⠛⠿⣷⣄⠉⠉⠉⠉⠁⣠⣾⠟⠛⠉⠙⠛⢿⣿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⢀⣤⣿⣿⣶⣶⣶⣶⣶⣿⣷⣤⡀⠀⠀⠀⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⠿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠿⠛⠋

⠀⠀⠀⠀⡤⠤⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠤⡄
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠉⠓⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠒⠋⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀⠤⠤⣤⠤⠤⣄⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠉⠉⠉⠉⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠘⣇⣀⣀⣠⡤⠤⠖⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠦⠤⣤⣀⣀⣀⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠰⠖⠛⠉⠉⠉⠉⠉⠉⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠈⢷⡀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⢀⡻⣔⠛⠉⠉⠉⠉⠛⠶⣄⡀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⢠⠞⠁⠙⣦⠀⠠⣤⣄⡀⠀⠈⠳⡄⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⡏⠀⠀⣀⠨⣷⣇⠀⠈⠹⣆⠀⠀⢹⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⡏⠀⢨⡝⢹⡆⠀⢹⠀⠀⢸⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠸⣇⠀⠀⠻⣄⠀⠉⠉⠀⣠⠟⠀⠀⣸⠇⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠙⣆⠀⠀⠈⠛⠲⠶⠛⠉⠀⠀⣰⠏⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠈⠳⢦⣄⣀⠀⠀⣀⣀⡴⠞⠁⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⣀⠀⠀⠀⠀⢀⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠀⢠⡀
⠀⠀⠀⠀⠀⠒⠀⢀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠐
⠀⠀⠀⠀⠤⠂⠈⡞⢉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠾⢿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠟⠋⢂⠀⠂⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⢌⠀⠀⠀⠀⠀⠀⠈⠙⠒⠀⠀⠐⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠂⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠂⠤⠤⠤⠤⠤⠤⠤⠤⠒⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠠⣤⣄⣀⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⡉⠉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠉⠉⠐⠒⠒⠤⠤⢀⣀⠀⠀⠀⠀⣟⢠⣧⣴⣇⠼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⣴⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡫⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡥⠴⠶⠶⡟⠛⡟⠉⠉⠉⠉⠉⠁⠐⠒⠒⠒⠂⠤⠤⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠈⠉⠁⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠐⠒⠒⠒⠒⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉
⠀⠀⠀⠀⠀⠠⠀⠀⠐⠒⠒⠂⡠⠤⠤⠶⠿⠂⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠉⢁⠀⣀⣀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⢐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡠⠞⠁
⠀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠎⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠠⠤⠤⠤⠤⠄⠠⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⢠⣤⣤⡤⠗⠀⢀⣀⠀⠠⠤⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⡻⠿⠛⠻⠿⡿⢻⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿