Beauty parlour Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠟⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣤⣤⣀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡍⠙⠲⠄
⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⠉⠉⠁⠠⠄⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⡟⠋⠃⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠙⠛⠿⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣟⣹⣿⣼⣿⣙⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠴⠀⠀⠀⠀⠀⣰⣿⣤⠟⠋⠉⠉⠉⠛⢧⡼⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⣶⣆⠀⠀⠀⣿⣶⡏⠀⠀⠀⠀⠀⠀⠀⢷⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣽⠁⠁⠀⠀⠀⣷⣾⡇⠀⠀⠀⠀⠀⡀⠀⢸⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡿⠋⠉⠀⠀⠀⠀⠀⣷⣼⠃⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣇⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣸⣥⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡏⠀⠀⠀⣀⣀⣀⠀⣯⣽⣯⣼⣿⣬⣿⣭⣿⣭⣿⣥⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣴⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣶⠄⢠⣤⣤⡖⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠁⠀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠈⠙⠻⣿⣿⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸⠿⠿⣶⢠⣴⣶⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣆⠀⠀⢸⣿⣿⣇⠀⡀⠘⣿⣇⠀⢸⣿⠀⢀⣴⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡹⣯⣿⣆⠀⢿⣿⣿⣿⣠⡇⠀⠹⢿⣆⠈⣿⠀⠘⢿⣿⣿⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠿⢷⣹⣿⠿⠂⠸⣿⣿⣿⣿⡿⠀⠀⠻⢿⡄⠈⠀⡶⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⣠⣀⠀⠀⡩⠤⣀⠀⠀⠻⣿⣿⣿⠃⠀⠀⠀⠀⠹⡀⢸⠁⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣼⣿⡟⠀⠘⠄⠤⠤⠝⠀⠀⠸⣿⠿⠇⠀⠈⠀⠀⠀⠱⠀⠀⡄⠀⠀⢀⣶⣿⣿⠿⠿⠟⢋⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⣠⣾⣿⠋⠀⢀⣠⡴⠖⠋⠉⠀⠀⣠⣴⣦⡄⠀⠘⢦⡀⠀⠀⠡⢀⡇⠀⣠⣿⣿⣿⠏⠀⠠⠖⢛⣋⣉⣀
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣴⣿⡿⠁⠐⠿⡻⠋⠀⢀⣠⣴⠷⠀⢿⣶⠼⢷⡄⠀⠀⢻⣦⠀⠀⢸⡇⢀⣿⣿⡿⠃⠀⣠⣶⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⣠⠟⠋⠀⠀⠀⠀⣀⣤⡾⠟⠉⠁⠀⠀⠈⢿⣷⣻⣿⡂⠀⠀⠋⠁⠀⢸⠇⢸⣿⠟⠁⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢠⣿⠁⢀⡼⠋⠀⠀⠀⠀⠰⣿⡿⠛⠀⠀⠀⢀⣀⡀⠀⠀⠿⣿⣾⡧⠀⠀⠀⠀⠀⣿⡇⠘⠁⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⣠⠂⣀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⣠⣾⣿⣮⠀⠀⠀⠉⠛⣧⠀⠀⠀⠀⢹⣿⡇⠀⠀⢰⡄⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠠⡃⢉⡤⠛⠄⠀⢠⣿⠇⢀⣠⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⠅⠀⠀⢀⡀⠹⣷⡀⠀⠈⢸⣿⣷⡀⠀⢀⣧⠀⠛⠓⠶⠤⠉⠙⠛⠋⠁⣠⡄
⠀⠀⠀⠀⠀⠐⠄⠀⢠⣿⠇⣰⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⢡⣄⠈⠀⠐⣿⣿⣧⠀⠘⣷⡀⠸⢻⣿⣿⣿⠂⠠⠿⠀⠀⢀⡤⠀⣤⡄⠀⢀⣾⡿⢁⣴⡷
⠀⠀⠀⠀⠀⠀⠁⢠⣿⠏⠀⢻⣿⣿⣿⣿⣿⠿⠋⠀⠀⢀⡿⠋⠀⠀⠀⠻⣶⡞⠀⠀⠈⡠⠤⢀⠀⠈⠁⠀⠀⣀⣤⠞⠋⠀⠀⣿⣿⣠⣿⣿⣷⣿⣿⣴⡶
⠀⠀⠀⠀⢀⡔⢻⣿⡟⠄⠀⠀⠙⠛⠻⣿⣿⣦⠀⠀⠀⡜⠁⠀⠀⣀⡀⠀⠘⣷⠀⠀⠀⠀⡒⠘⢁⣠⣤⣶⠟⠋⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤
⠀⠀⠀⠀⠘⠀⣿⡿⠁⠀⠀⡠⠀⠄⠀⢻⣿⣿⣏⠀⠐⠀⠀⠐⣾⣿⡟⠂⠀⠸⡄⠀⢐⠃⢈⠟⢁⡠⠊⠁⠀⣄⢾⣇⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⣰⣾⡿⠁⠀⣠⠎⠀⠀⠈⠄⠀⢻⣿⣿⡄⠀⠀⢀⣾⣿⠟⠀⣀⡀⠀⢣⠀⠀⠉⡝⠀⠻⠀⠀⣄⠀⠘⢾⣿⡀⠾⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⢠⣿⡿⠁⢀⣼⠋⠀⠀⣀⣤⠀⠀⠄⠹⣿⡇⠀⠀⣾⣿⡏⠀⣰⣿⡿⠀⠈⠀⠀⠔⠃⠈⡁⠀⠈⣿⣧⠀⠈⢿⡇⠀⠈⠛⢿⣿⠏⢁⣼⣿⡶⡄
⠀⠀⠀⠈⠉⠁⣠⣿⠇⠀⣠⣾⣿⣧⡀⠀⠀⠄⠙⠁⠀⣰⠟⠁⠀⢠⣿⠟⢀⣄⠀⠀⠀⢰⣦⡈⠫⢧⡀⠘⣿⡆⠀⠈⣿⠀⠀⢀⠀⠀⢀⣾⣿⣣⠏
⠀⠀⠀⠀⠀⣰⡿⡻⠀⠾⣯⣽⡛⠙⠛⠦⠀⣠⣾⠇⣴⠋⠀⠀⠀⠿⠃⠀⠘⢋⠑⡂⠀⠈⠻⣿⣆⠀⠉⠀⢹⣿⡀⠀⢹⠀⢐⠉⣳⢀⣾⣿⡿⡡
⠀⠀⠀⠀⠀⠉⠁⠁⠀⠀⢸⣻⣿⣀⠀⠀⠾⠋⠁⡼⠃⠀⠛⠷⠦⣀⠀⠀⠱⡀⢀⢐⠀⠀⠀⠈⢻⣷⣄⠀⠀⢿⣧⠀⠀⠀⢈⣀⠆⣼⣿⢏⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣦⡀⠀⠀⠁⢴⣄⠀⠀⠀⠀⠀⠀⠀⢡⠈⡄⢧⠀⠀⠀⠀⠙⢿⣦⡀⠘⣿⣇⠀⠀⠀⠁⣼⡿⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⢿⣶⣌⢿⣷⡄⠀⣾⣆⠀⠀⠀⠃⣤⣶⡄⠀⠀⠀⠐⠈⠻⣿⡄⠈⢿⡄⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⠿⠟⠁⠻⣶⣽⣿⣿⣿⣿⣦⣿⣿⠀⠀⠀⠈⢿⣿⣿⠀⠈⠲⣤⡀⠀⠈⢿⣦⠘⣷⡀⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣮⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⡈⣿⣿⣧⠀⠀⠈⠻⣶⣄⠀⠙⢧⠾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣦⠀⣰⡏⠸⣿⣿⣆⠀⠑⠀⠘⣿⣷⡄⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⠟⠁⢰⡿⡇⠀⢹⣿⣿⡄⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠻⠟⠀⠀⠀⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣷⣶⣶⣶⠖⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣠⡴⣫⣷⣿⣿⣿⣿⡿⠶⠾⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢖⣵⢞⣽⣾⣻⣿⣿⡿⠋⠁⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⣫⣷⣿⣿⣿⣿⣿⠋⠀⠐⠈⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⣫⣾⡿⠁⠀⠀⠀⠾⢿⣿⡄⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⣳⣿⣿⣿⡿⢋⣴⣿⢏⠔⠀⠀⠀⠀⠀⠀⠁⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⣼⣿⣿⡿⠋⣠⣿⣿⡵⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣸⣿⣿⠏⢀⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⠃⣰⢟⣿⠏⠀⠀⠀⠀⠐⢦⣀⠀⠀⠀⠀⠀⠙⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣿⣿⢠⣏⣾⠇⠀⡴⠀⢀⡄⠀⢸⣿⣿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢸⢸⣿⠀⣸⡇⢠⣿⠃⠀⢸⡿⠋⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⢸⣧⠀⣿⡇⣿⣿⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡘⣿⡀⢿⣧⣿⣿⡀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣜⢷⡈⠻⢿⣿⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡻⣆⠘⣿⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡌⠣⢻⡇⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠈⠀⠀⣰⡿⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⢦⣀⣀⣀⡼⠋⣴⣿⣿⣿⣿⣿⢟⣿⡿⠁⠙⢿⣟⢦⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠐⠬⣉⣠⠴⣾⣿⣿⣿⠿⣋⣵⣿⠟⠁⠀⠀⠈⣿⣧⢻⡌⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣤⣤⣤⣿⣟⣫⣭⡶⠿⠻⠋⠀⠀⠠⠔⠂⠀⢸⣿⡆⡷⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣿⣿⣿⣿⡏⠲⠶⠶⠀⠀⠀⠀⠶⠶⠶⣿⣿⢣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⣵⣿⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⣿⣿⡇⣿⣿⡄⠀⠀⢀⣀⣀⠀⠀⢀⣿⣱⢃⡴⠲⣦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣿⠸⣿⡇⣿⣿⣿⣆⠀⠉⠭⠍⠁⢀⢼⣿⣿⡘⢷⣤⡿⠁⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⢿⡇⣿⣿⣿⣿⣷⡦⠤⠤⠖⠁⠀⢿⣿⣿⣦⣤⣠⣤⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠈⢷⡹⣿⣿⡻⣿⣷⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣻⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡹⣿⣦⡀⠙⢿⣿⣿⣮⡙⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠙⠿⣿⣷⣦⣍⡛⢿⣿⣦⡄⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠈⠙⠛⠿⣿⣗⢽⣿⣿⣦⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡤⠖⠒⠀⠙⣧⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢸⡾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣖⠤⠤⠴⣛⣵⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⠶⢶⣶⣄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣛⣯⣿⡿⢿⢽⣿⣶⣢⣩⠳⡄⣹⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣩⣴⣪⣭⣭⣭⣉⣲⣮⣯⣻⣿⣿⡇⣽⣯⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⡀⣱⢫⣾⡿⠟⣛⣳⣦⣍⡉⢽⣿⠿⠟⠛⢿⠿⢿⣿⡵⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⡀⣷⣿⠿⣳⣿⡿⠟⠛⢻⡗⢀⠔⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠝⡻⢋⣵⡾⣛⣭⡶⠿⣿⣿⡋⠁⣀⢤⣄⣀⣀⠀⠀⠀⣸⣟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⡴⠞⣫⡵⠋⠁⢀⣤⣿⣿⡿⠀⠀⣀⣠⣤⣌⠉⠀⢠⢫⣤⣿⠇
⠀⠀⠀⠀⠀⢠⠆⢀⠀⠀⢀⡾⢫⣿⣿⣿⣶⣞⣛⡿⠿⣷⣿⣁⠀⠀⠉⠻⠿⠈⠀⠀⢸⠻⡿⠋⢙⡆
⠀⠀⠀⠀⠀⡞⠀⠘⢤⡤⠌⠀⢘⣯⣥⣦⣴⣿⣿⣿⡀⢥⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⡣⢹
⠀⠀⠀⠀⠀⠃⢀⣠⡴⠶⢶⡤⢋⡾⣿⡿⠿⠷⠚⣽⣿⢠⣤⡀⠀⠀⠀⠀⠀⠀⣀⣤⣬⣀⡏
⠀⠀⠀⠀⡰⢪⣏⡵⠚⣩⡭⣡⡌⠘⣡⣴⣶⣷⣾⣯⣾⠸⣴⠏⢦⣀⠀⠀⠀⠀⠉⠩⢿⡞
⠀⠀⠀⠰⡇⢸⡏⣠⣾⢋⢸⡉⣇⡾⢟⣫⣭⣭⠷⣿⣿⡆⠀⠀⡀⠘⢿⣶⣤⣄⣀⣀⡼⠁
⠀⠀⠀⠀⠑⣿⣽⣵⢣⢸⡈⣷⠾⣴⣋⣡⣤⣤⣴⣿⢿⡇⠀⠀⢣⠀⠘⣿⠁
⠀⠀⢸⠀⠀⣿⣿⢸⡼⣧⣳⣽⣿⣿⣿⣿⣿⣿⣟⡽⠛⠀⠀⠀⠈⢧⡀⡏
⠀⠀⠈⢇⠀⠙⠺⠿⠿⠦⢬⢭⣉⣉⠉⠭⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠷⣀
⠀⠀⠀⠀⠑⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠒⠒⠾⠷⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⠠⡀⠂⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⢀⡺⠭⠛⠲⡹⡆⣄⠀⠀⠀⠀⠀⠀⡠⠖⠂⡁⠀⠀⠀⠢⡀
⠀⢀⡤⠬⠶⠊⣠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢫⠖⠀⠂⠀⠉⢡⠋⠀⠀⠀⠀⢀⡞⠁⣠⠊⣀⡤⠞⠀⠱⡈⣆
⠀⢸⠶⠀⢀⣀⢦⠀⢤⠀⠀⡄⠰⠜⡄⠀⠀⡏⠏⡀⠠⠄⠀⠀⠨⢉⠀⠀⠀⠀⢸⠁⣰⠃⠈⠁⠀⠒⠀⠆⢧⢸⡀
⠀⠈⠆⠂⢙⡦⠟⠀⠉⠀⠈⠀⢠⢊⠀⠀⠀⠱⡀⢀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⢸⠀⡇⠈⠃⠀⠀⠀⠠⠀⢸⠀⡇
⠀⠀⠀⠀⠇⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡙⠦⡀⠀⠀⠀⠀⠀⡼⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐⣁⠎

⠀⠀⠀⢀⣠⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠆⡠⢠⠤⡀
⠀⠀⢰⢣⡒⠏⢀⣉⣩⠽⠂⠀⠀⠀⠀⠀⠀⢀⣀⣠⠶⠋⢀⠔⣩⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠐⢀
⠀⠀⠘⢄⠀⢤⡭⠟⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠁⠀⢀⡴⢃⡜⣁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠐
⠀⠀⠀⠀⠀⠐⡄⠀⠀⠈⠁⠃⠀⠀⠀⠀⣠⠴⠒⠚⠉⠀⡼⠁⠉⠀⠘⠀⠘⣆⠀⠀⠀⠠⠀⠤⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⢀⡠⠴⠚⠁⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠠⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠃⠰⠈⠰⢀⡀
⠀⠀⠀⠀⠀⡀⢠⠀⡄⠀⠃⠈⠃⢠⠀⠀⠀⠀⠀⠘⠀⠀⠀⡄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣀⡱⠀⠀⢀
⠀⠀⠀⠀⢠⠁⡌⢰⠀⠢⠀⠠⢴⡎⠀⠀⠀⠀⠀⠈⢠⠀⠁⡀⠀⢃⠇⠀⠀⠀⠀⠀⠀⠢⠀⣧⠄⠀⠴⠀⠢⡅⠀⢈
⠀⠀⠀⠀⠄⠀⡇⢸⠀⠀⢀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠚⠀⠈⠀⠀⠀⠀⠀⠀⠀⡞⠀⡄⠀⠀⠸⠅⠀⠀⠂
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠐⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠗⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⡶⠆⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⢻⣿⠙⣿⣏⣻⣿⣿⣿⣍⣛⣳⣤⣠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣌⠛⠃⠈⠉⠛⢷⣶⣿⣿⣿⣿⣿⣿⠏⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣛⠿⠿⠿⠿⣷⣶⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣟⠁⣠⣾⣷⣤
⠀⠀⠀⠀⠀⠀⢸⣿⣟⠻⣧⣄⣀⣀⡀⣀⣉⣉⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠛⣹⡿⢀⣠⡄
⠀⠀⠀⠀⠀⣠⡀⠻⢿⣷⣤⣭⣿⣿⣿⣿⣿⣿⡿⠿⠟⣿⠿⠿⠿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣡⣤⣀⣤⣄⣀⡀⠚⠂⠀⠀⠙⠛⠻⠿⠟⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠙⠛⠁⣻⣿⢿⡿⠛⠊⠁⢈⣹⣾⣋⣡⣈⣁⣀⣉⣉⣉⣽⣿⣿⣿⣿⣷⣷⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠎⣠⣤⣶⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣟⣛⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣴⣿⣿⣿⡿⠋⠛⣉⣴⣾⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣤⣤⣠⣍⣙⣿⣾⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠼⢛⣿⣿⣿⡟⠱⠛⢀⣴⣾⣿⣿⣿⠿⠿⠷⠤⢤⣈⣱⣶⣶⣾⠿⠿⠛⠛⠛⠛⠙⠋⠀⠀⠙⠻⠿⢿⣧⡀
⠀⠀⠀⠀⠀⢰⠀⠈⢻⣿⠏⠁⢀⣼⣿⣿⣿⣿⣥⣤⠄⣀⣠⣴⣤⣬⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠈⢀⠀⢸⣿⣄⣾⣿⣿⣿⣿⡷⠆⠀⣰⣾⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⢉⣻⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⠟⠛⠛⢋⠉⠳⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⣿⣿⣿⣿⣿⣿⣿⡶⠶⠀⣠⣾⡟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣿⣿⣿⣿⣏⣉⣩⣤⣴⣿⣿⣿⡞⠀⠀⠀⠀⠀⢠⣤⣤⣴⣶⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠀⠀⠀⠀⠀⠀⠺⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣶⣤⣤⣾⣿⣿⣧⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠰⡆⠈⣿⣿⣿⣿⣿⣿⠋⣉⠉⢣⡘⢧⡀⠀⠀⠀⠀⠘⠿⢿⣏⢹⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⢇⡹⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠸⣿⣿⣿⣿⡇⠀⠙⠇⠀⠹⢾⣷⣄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣧⠀⠀⠀⣀⠀⠀⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣆⠀⠀⢻⣀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⡀⠘⠿⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠈⠁⠀⠀⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣸⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣾⣧⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⢠⣾⣿⣿⠁⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⡿⠿⠃⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⠟⠁⠀⠀⠀⠀⠘⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣷⣶⣤⣤⣀⣀⣀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡟⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠈⠋⠉⠛⠛⠛⠿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣶⣦⣶⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣴⣶⣶⣴⣶⣦⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⡟⣁⣀⣹⡿⠟⠿⠿⠿⠿⠿⠟⠿⠿⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣏⣀⣹⣷⣶⣶⣶⣶⣶⣶⣶⣦⣴⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡇⠀⣸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣾⣿⣿⠟⠀⣰⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣿⣿⠿⠋⢁⣤⠞⣿⣿⡿⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⡿⠿⠟⠛⣋⣡⣤⣶⠾⢛⣥⣾⣿⡟⢁⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣯⡵⠶⠾⠟⠛⠛⠉⢁⣠⣶⣿⣿⣿⣫⣶⡿⠃⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⠟⣴⠞⠋⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣶⠞⣋⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⣤⣤⣴⣯⡤
⠀⠀⠀⠀⠀⠀⢀⣾⣟⠵⣫⣿⣿⣿⣿⡿⠟⣩⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠈⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠸⣿⠋⣾⣿⣿⣿⣿⠟⢀⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣸⣿⣿⣿⠃⣿⠀⢸⣿⣿⣿⡟⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠀⣿⡀⠘⣿⣿⣿⡇⡄⠀⠀⠘⠿⠿⠿⠃⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⠘⢷⣄⠘⢿⣿⣿⡇⠀⠐⢄⡀⠀⠀⡠⠎⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡈⠻⣷⣄⠙⠿⣷⡀⠀⠀⠈⠉⠁⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠉⠀⠀⠙⠷⣦⣈⠙⠂⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡑⠦⣀⠀⠀⠈⠙⠳⢦⣄⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣬⣙⡲⠤⣄⡑⢦⣬⣛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠛⠿⣿⣿⣶⣿⣦⡙⢿⣷⣮⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⠤⣍⠻⣿⣿⣿⢸⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡿⠟⣡⣾⣿⠟⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⣶⣿⣿⠿⠗⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡏

⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣴⡆⣴⣿⣟⡳⣄
⠀⠀⠀⠀⢀⣄⢿⣿⡿⢫⠾⣿⣶⣝⡢⠀⠀⠀⠀⠀⠀⣴⣶⣄⠙⢿⡽⣇⠏⠥⣤⡙⠺⡆
⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠘⢿⡗⠀⠀⠀⠀⠀⢘⣭⡯⠭⠙⠀⠉⠈⠀⠀⠈⣿⢰⡀
⠀⠀⠀⠘⣋⡀⣀⣀⣀⣀⣀⠀⣀⡈⠛⠀⠀⠀⠀⠀⣫⢶⠾⠃⠀⠀⠀⠀⠀⠀⠀⣿⣾⢹
⠀⠀⠀⠀⣇⠀⠀⢠⠁⢹⡀⠀⢀⠇⠀⠀⠀⠀⢀⡾⣣⣆⣾⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣿⢦⡀
⠀⠀⠀⠀⢇⠉⠉⠁⡀⡀⠉⠉⠁⡎⠀⠀⠀⠐⠋⠵⡋⣼⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⢺⡘⠳⠍
⠀⠀⠀⠀⢸⡣⠚⠛⠁⠙⠛⠢⣶⡀⠀⠀⠀⠀⠀⠸⣷⢟⣤⡄⠀⠀⣤⣤⣤⠀⠀⠻⣦⣝⣊⡛⠚⠁
⠀⠀⠀⠀⢿⣿⣧⣄⣺⣆⣠⣿⣿⠃⠀⠀⠀⠀⠚⣫⡵⢾⣼⣧⠀⠀⠉⠉⠁⠀⠀⢠⢷⣔⠆
⠀⠀⠀⠀⠀⠙⣿⣾⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⣿⢰⠀⠀⠀⠀⠀⠀⡼⣿⡻
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣾⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⣿⠀⠀⠀⠀⠀⢠⢯⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠘⠘
⠀⠀⠀⠀⠀⢺⡆⢲⠆⠀⠀⠀⠀⠀⠀⢸⡆⣦⠀⠀⢀⣤⠤⣄⠀⠀⡠⢤
⠀⠀⠀⠀⠀⣸⠉⣷⢠⡖⣰⣫⢫⡇⡞⣸⠃⡟⣸⠀⢿⡇⠀⢈⡀⢸⡇⢜⡀⠀⣄⡀⣤⣠⡀
⠀⠀⠀⠀⠀⠛⢧⡿⠈⠉⠈⠙⠊⠉⠓⠛⠊⢩⡟⠀⢨⢿⣴⣏⡞⢸⡇⢺⢉⡿⡏⣹⣿⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣽⠃⠀⠳⠼⠃⠈⠉⠀⠓⠊⠉⠀⠉⠁⠉


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

⠀⢀⣠⣴⣶⡾⠟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄
⢰⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣿⢿⣷
⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣦⣀⡀⠀⣼⡟⠀⣿⡀
⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢻⣿⣿⣿⣿⣴⡟⠀⠀⢻⣿⡆
⠈⠻⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣴⠀⠀⢰⣿⡅
⠀⠀⠈⠙⠻⠿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣯⢿⣧⢆⣤⠾⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠦⣝⠋⣴⣟⠛⠳⣄
⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠑⠜⠳⠤⠴⢼⣧⠀⠀⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠘⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣤⣀⠀⠀⠀⠀⠀⠉⠛⠶⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⢈⣠⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄
⠀⠀⠀⠀⠘⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠈⢹⣋⡀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⢠⣿⡿⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣾⡿⠁⠀⠀⠀⠀⣀⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠀⠀⣼⡿⠁⠀⠀⢀⣠⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠿⠋⠁⠀⢀⣾⡟⠁⠀⢀⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⣠⣿⠋⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⢀⣼⡟⠁⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⣴⣿⠋⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⡄⠀⠀⠀⠀⢠⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣀⠀⠀⠀⠀⢀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⠀⠀⠀⣾⣿⣿⡟⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣿⡶⠶⠶⠾⠿⣯⡟⣤⠀⢠⣿⣿⣿⣧⠀⠀⠀⣴⠛⠛⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⣈⠿⣿⠀⠈⣿⣿⣿⣿⣆⠀⠀⠻⣦⡀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⡿⣳⠛⢧⠀⠀⠙⢿⣿⣿⣿⣷⣤⣀⣀⣀⣤⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⠎⠇⠀⠀⠀⠉⠛⠻⠿⠿⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠗⠶⢦⡙⢇⣠⣤⣤⠄⡀⢀⣴⢞⣫⣭⣬⣥⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠁⠀⠀⠐⢠⣶⡿⣽⠋⣴⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣦⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠋⣠⣾⣿⣿⣷⠹⣿⣇⣯⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⣼⣿⣿⣿⣿⣿⣷⣌⠻⣾⣦⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠇⢰⣿⣿⣿⣿⣿⣿⡿⣿⣿⣾⣽⣿⣷⣾⣿⣿⣿⣿⣿⣿⡿⠟⣫⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣏⡄⢸⣿⣿⡿⣿⣽⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠉⠙⠛⠶⠾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢸⡇⠘⣿⣿⣿⣮⣻⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢸⣷⡀⠹⣿⣿⠋⠀⠹⠟⠀⠀⠀⠒⠚⠛⠻⢶⣄⡀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡀⣿⣧⠀⢻⣿⣇⠀⠀⠀⠀⢲⣿⣶⣶⣶⣤⡄⠉⠁⠀⠠⠖⢒⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⢻⣿⡄⢠⢻⣿⣄⡀⠀⠀⠀⠙⠳⡮⣭⠟⠃⠀⢀⣶⣶⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣼⣿⢿⣿⣿⡇⣼⡞⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣻⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣟⣿⣿⣣⣿⣇⣿⣿⣿⠙⢦⡀⠀⠀⠀⢤⣀⡀⠀⠁⢠⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⣠⢞⣛⣲⠹⡿⢋⣥⡉⢻⣿⡿⣿⣿⠀⠀⠙⣦⡀⠀⠀⠍⠉⣠⣶⣯⣿⣿⣿⡏
⠀⠀⠀⠀⠀⣼⣏⢺⣿⡿⢠⠃⣿⣿⣿⢸⣿⣧⣿⣿⡄⠀⠀⠈⢻⣶⣤⣴⣿⣿⣿⣿⢟⣥⣶⣶⣄
⠀⠀⠀⠀⠀⣿⣿⣦⣭⡤⣋⣾⣿⣿⠏⣼⣿⣿⣿⠋⢧⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡸⠿⣿⡿⣹⠃
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡿⠟⠁⠀⠈⠃⠀⠀⠀⠙⠻⠿⣿⣿⣿⡿⠿⠿⣋⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢠⣤⣀⣤⣀⣤⢠⣤⣤⣤⣤⣄⡄⠀⣤⡀⣄⣄⣤⣄⠀⣤⣤⢀⣄⢠⣠⣤⡀⢀⣤⡀⣤⢠⡄⢀⣤⣀⣄⡄
⠀⠀⠸⠿⠿⠿⠼⠿⠾⢿⠇⠿⠁⠿⠁⠰⠿⠧⠿⠧⠿⠿⠀⠿⠿⠼⠿⠼⠼⠿⠇⠸⠿⠾⠿⠿⠷⠾⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣮⡛⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⠉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣝⠀⠀⠀⠀⠲⠟⣛⣉⡉⠉⠉⠀⢸⣿⣧⠀⠀⠈⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠻⠗⠀⠀⠀⠀⠾⠋⣿⡿⢻⡶⠶⠊⣿⣿⡆⠀⠀⠈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⢀⣿⡇⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⢀⣾⠟⣰⠇⣻⡶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠀⠀⣠⡿⠋⠔⠁⣰⣿⡿⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⠀⣠⡾⢋⠄⠀⣠⣾⠿⠋⠀⠀⠀⣨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣶⣤⣀⡀⠀⠀⣾⡿⠁⢀⣼⠋⡴⠋⣤⡾⠋⠁⠀⣀⣤⣶⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠛⠉⠀⠀⢰⣿⠃⠀⣼⢁⠎⣠⡾⠋⢀⣠⣶⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⣇⠎⣴⡟⢀⣴⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢸⣿⢸⡟⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠈⣿⣿⣧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠄


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣽⣏⣭⣩⣭⣩⣟⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⠶⠶⠶⠒⠶⠶⠖⠒⠒⠲⠖⠶⠲⠶⠶⠶⠒⠲⠶⠶⠶⠖⠒⠲⠲⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠏⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢀⡞⠀⢀⣠⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢀⣾⠃⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⢠⠃⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣸⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠐⣇⠀⠀⢀⣿⣿⠃⠐⠂⢀⣠⡴⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢹⡀⠀⢸⣿⡿⢷⣦⣄⠾⣿⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠿⣄⠘⣿⣧⠘⣿⣿⣷⣦⡀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠢⡙⣿⣧⡸⣟⡌⣿⣿⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢎⢻⣷⡜⡆⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢹⣧⢻⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠉⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣤⣿
⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡻⣿⣿⣿⡃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣟⣻⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣯⣿⣌⣩⣩⠉⠀⠀⠀⠀⠀⣤⡤⢰⣶⢶⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣛⣰⣿⣷⣾⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣤⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣆⡛⢿⣿⣟⢻⣿
⣿⠿⠿⠿⠿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⣛⣩⣭⣴⣶⣶⣶⣶⣦⣭⣍⡙⠻⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢋⣥⣶⣿⣿⣿⡿⢿⣿⡿⠛⢿⣿⣿⣿⣿⣷⣄⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⣡⣴⣿⣿⣿⣿⡇⢹⠁⠀⠉⠀⠀⠈⠉⣿⣿⣿⣿⣿⣷⡄⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢃⣼⣿⣿⣿⣿⣿⣿⣿⡘⣤⣄⢈⣟⡩⢶⡄⠈⢹⣿⣿⣿⣿⣿⡄⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢠⣿⣿⣿⣿⣿⣿⣟⠉⠉⠁⠙⠈⠙⢲⣳⣋⡄⠀⠀⢉⣿⣿⣿⣿⣷⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠠⢤⠞⠀⠈⢇⠀⠀⠐⣾⣿⣿⣿⣿⣿⢻⡄⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣀⣬⠀⠀⢨⡛⠤⠤⣤⣼⣿⣿⣿⣿⡟⢨⡇⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⢟⣵⣿⣿⣿⣆⢠⣾⣿⣷⣦⣾⣿⣿⣿⡿⠋⠀⢸⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣯⣿⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠸⣿⣿⣿⣿⣿⣿⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢻⣿⣿⣿⣿⡇⣿⠁⠀⠀⠀⠀⣀⣘⣛⣷⣦⣸⣿⣷⣀⣤⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⢿⣿⣿⣿⣧⠃⠀⠀⠀⠀⠀⠙⣿⣿⣿⣵⣾⣿⠟⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⢰⠀⠀⠀⠀⣶⣦⣆⠀⠀⠀⠀⠀⠀⠀⢻⡌⢿⣿⣿⣿⠀⠀⠀⠀⠀⣴⣾⣿⣏⣭⣥⠻⣿⣾⣿⣿⣿⣦⠄⠀⠀⠀⠀⠀⢀⡾⠁
⠀⢸⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡙⣿⣿⣇⠀⠀⠀⠀⠈⣽⣿⡏⢴⣦⡟⣶⣝⣈⣏⠉⠁⠀⠀⠀⠀⠀⣰⠟⠁
⢀⡇⠀⠀⠀⠀⡿⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡙⢿⣆⠀⠀⠀⠀⠈⢹⣿⣿⣿⣶⣿⣿⣿⣿⡆⠀⠀⠀⢀⣠⠞⠁
⣾⣶⡄⠠⡄⠀⣷⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣬⣀⠀⠀⠀⠀⠀⠉⠹⡿⠋⠈⠻⠇⠉⠁⣀⣤⠶⠛⠁
⣿⣿⡂⠀⣧⢠⣿⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⢤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠛⠉⠁
⣿⣿⡇⠀⡿⢸⣧⡄⢠
⣿⣿⣷⣾⡇⢸⣿⣯⢠⡇
⣻⣿⣿⣿⡁⣸⣿⣿⣸⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⡅⠀⠿⠿⠿⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡥⠀⣸⡇⠀⠀⠀⠀⢨

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠿⣷⣦⣄⡀⠀⠀⠀⣠⣤⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣽⢻⣶⣿⣷⣦⡍⠙⠻⠿⠿⠟⢛⣛⣛⣃⡒⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠃⢿⠉⠙⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠆⠀⠀⠀⠀⠙⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠈⢛⠶⠤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣦⣤⣤⡶⠶⠛⠿⣿⣿⣷⠀⠈⠙⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠄⠀⠀⢰⣌⠉⠉⠀⢠⠀⠀⠀⠘⣿⣿⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⡤⠀⠀⠀⣾⡟⣧⡀⠀⠸⣧⡀⠀⠀⠸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠹⣿⣦⣄⡻⣿⣦⣀⠀⢁⡀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠈⠻⣿⡿⣾⣝⠿⣿⣖⢿⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⣾⣿⣿⣦⠈⢳⡄⠀⣿⣿⠈⡿⠀⠀⠙⡇⣿⣧⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⡀⠈⠉⠉⠁⢠⠘⣿⣦⣿⠋⠀⠁⣠⣴⡾⠁⣿⠏⢀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠐⠒⠊⠀⠀⣿⣿⠃⡄⣴⣿⣿⡟⢁⡾⠃⣠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⢿⡿⣿⣿⠀⢸⢁⣾⡿⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠌⣰⣿⠏⠻⣿⣿⣌⠻⣆⠈⢾⡏⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⢀⠔⢱⣿⠀⠀⠈⢻⣿⡷⣌⣷⡈⠛⢦⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡰⠀⠀⠀⠀⣴⣯⠀⣿⣿⡀⠀⠀⢀⣿⠇⠨⣿⣧⠀⠘⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢿⣿⣆⠈⠙⠻⢦⣠⠞⠁⠀⣴⣿⣯⠎⠀⠀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⠀⢀⣼⡿⢇⡴⠛⠉⣾⣿⣦⡀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⣼⡏⢠⡏⠀⠀⠀⠹⢿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡏⠈⠻⢾⡇⠀⠀⢀⠄⠎⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⢿⡀⠀⠸⣷⡀⠀⠘⢆⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠈⠻⡀⢀⣿⡿⠀⠀⠀⠉⢶⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠀⠀⠀⠀⠀⡴⠞⠋⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉

⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⢰⡿⠛⠻⣿⣿⡛⠓⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⡛⢛⣿⣿⠟⠛⢷⡆
⠀⠀⠘⠷⠿⠀⢸⣿⣷⣵⣄⢀⣈⣝⠷⣤⡀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⢀⣤⠞⣫⣀⡀⣠⣯⣾⣿⡇⠀⠿⠾⠃
⠀⠀⠀⠀⠀⣤⣾⣿⣧⣘⠙⡷⡦⠬⢵⡝⣝⢷⣄⠀⠀⠑⠀⠀⠀⠀⠀⠀⠋⠀⠀⣠⣾⡻⢣⡮⠥⣴⢾⢻⣁⣼⣿⣷⣤⠄
⠀⠀⠀⠀⠀⠙⢿⣿⣿⡈⠑⣧⠯⣦⡀⢡⠘⡇⢛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡛⢸⠁⡎⢀⣼⠽⣼⠋⢁⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⣀⣿⣿⡟⢲⡁⡴⠶⠽⣜⠦⣸⡈⣸⠙⣧⡀⠀⣴⣆⠀⢀⣼⠋⢣⢀⣇⠴⣡⠯⠖⢦⢈⡖⣻⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣶⠷⣷⠄⣀⡈⢷⣄⠑⢾⡀⠘⣿⣔⣿⣿⣢⣿⠁⢠⡷⠊⣠⡾⢁⣀⢀⣿⠾⣶⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣦⣤⣽⡶⣦⣀⣌⣑⡢⣝⣲⣾⣿⣿⣿⣿⣷⣖⣫⣔⣋⣥⣀⡴⢶⣯⣤⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣷⣶⣶⡶⠾⣿⠟⠛⠟⠋⣼⣿⣿⣧⠙⠻⣟⡿⡿⠷⢶⣶⣶⣾⡿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢛⡫⠁⡠⠞⠡⢠⠀⢀⣼⣿⡻⢿⣿⣧⡀⠀⡝⡌⠳⠄⠈⢝⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⠋⡰⣋⣠⠤⣧⣾⠶⠋⣸⢿⣿⣿⡿⣇⠛⢶⣷⡜⢤⣄⡙⢆⠙⢾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⡾⢋⠎⠀⡰⢁⠃⣀⡴⢻⠟⣿⣿⠛⡟⢦⡀⠙⡘⢆⠀⠱⡙⢷⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣞⠀⡰⠀⣸⠔⢁⠁⣸⠀⠻⠟⠀⣇⠈⡌⢢⣇⠈⣆⠀⣷⣾⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⣿⣿⣴⡇⢠⡟⠠⢹⢀⡟⠀⠀⠀⠀⢻⡀⡇⠄⢻⡄⢸⣦⣿⣿⡿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣧⣶⣿⠞⠀⠀⠀⠀⠀⠈⢳⣿⣶⣼⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⡆⠀⢸⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⡇⠀⢰⡟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢤⡶⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⢶⡦⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡤⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣟⣡⠶⢚⣓⠒⠬⣙⠢⣀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡘⢿⣟⡯⠶⠿⢿⣿⣿⣿⣾⣷⣌⡳⢄⡀⠀⠠⠤⠖⠋⡀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⢠⣈⣉⠁⠁⠈⠉⠻⣶⣌⠻⣿⣿⣿⣿⣿⣶⣭⣙⣒⣒⣒⣩⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣫⡅⠀⣀⠀⠀⠀⣈⢿⣷⡌⢿⣿⣿⣿⣿⣝⠻⠿⠿⠟⣻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⣶⡾⠁⠀⠨⣷⣶⠝⢿⣦⡙⢿⣿⣿⣿⣿⣷⣾⡿⣫⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⣛⠂⠈⣉⣛⣛⣩⣵⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣀⣤⣀⠀⢀⣾⣿⣦⣌⡛⠻⠿⠿⣿⣿⣿⢟⡟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⣄⠈⠛⢁⡠⠋⠠⢬⣉⣛⠻⠿⠿⠿⣛⣫⡷⠋⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠈⠁⠉⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⢋⣁⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⢀⡤⣄⣠⡤⢠⡄⣤⢠⣤⠤⡤⢠⠤⡄⡄⢀⡤⣄⣠⠤⠄⣄⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠘⠦⠞⠸⠟⠺⠛⠻⠸⠿⠀⠇⠙⠶⠋⠷⠜⠷⠞⠙⠾⠃⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠶⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢸⡇⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⣠⠤⠴⢻⣇⣀⣀⣀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⣰⠀⣿⠈⠈⢸⡇⠉⠉⣹⠀⣧⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠛⠉⠁⢿⠀⠘⣆⠡⢸⡇⠀⣠⠏⠀⡛⠈⠉⠛⠲⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⣸⠶⠖⠚⣷⢾⡷⣾⠛⠲⢤⣇⠀⠀⠀⠀⠀⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⢻⡏⠁⠀⠀⠀⠈⣧⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⢁⣤⣀⣀⠀⠀⠈⠀⠀⠀⠀⠀⣀⣬⣤⣄⣀⢠⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣣⠴⠒⠛⠶⡇⢸⡇⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⡧⠀⠛⠒⠒⠒⠒⠛⢸⡇⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠶⠖⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠙⣆⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠸⠇⠀⠀⠀⠀⣠⢤⡟⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣘⣾⣭⣭⣭⣤⣤⡐⠀⠀⠀⠀⠀⢃⠀⠀⣀⣾⡷⠊⠀⠀⢀⣹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣓⣒⣒⣒⣺⣇⢻⡀⠀⠀⠀⠀⢈⢧⠀⠈⡃⠁⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡏⠀⠘⢦⣷⡤⠤⠠⠶⠛⣸⠀⠀⠙⠲⠤⡤⢤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠚⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠘⢩⠶⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡀⠀⠀⡐⠀⠣⠀⠀⠀⠰⠘⢃⡀⠀⠀⢉⠖⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷
⠀⠀⢴⠌⠁⠀⢀⠁⠀⡃⠀⠀⠀⠀⠇⡀⢡⡀⠀⠀⠀⠁⠞⡀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⠘⡏⢀⣴⠆
⠰⣙⠂⠀⠀⠀⠠⠀⡐⠅⠀⠀⠀⠀⠈⡥⠨⡅⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⢄⡇⠘⠋⣀
⠓⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠴⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⠀⠀⡇⠀⣾⠟
⠀⢀⡄⠶⠃⡛⢐⡋⢐⠃⠰⠆⠀⠀⠀⠒⠰⠆⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠢⢷⠊⠁⣤
⠚⡀⢤⡘⠂⠙⠀⠁⠈⠀⠺⠉⠀⠀⠚⡷⠲⠀⠀⠀⠀⢸⣿⡆⠀⣴⡇⠀⠀⠀⠀⠀⣤⡀⠀⠘⡄⣸⠃
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣷⣶⡙⠓⢸⡿⠃⠀⠀⠀⠀⠀⠈⠉⠁⠒⢺⠁⢠⣾
⠄⣀⠀⠀⠀⠀⠀⠈⠳⣾⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠊⠀⢀⡀⠀⠀⠀⠀⠀⠰⣶⣄⣀⣣⡜⠁
⣀⠀⠈⠜⠑⢂⡄⠀⠀⠀⠉⠀⠀⠀⢀⣴⠆⠀⠀⠀⠀⠰⣦⣄⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠹⣄⡼
⠀⠀⠀⠐⠐⠂⠀⠀⠀⠀⠀⢰⡄⢀⣿⠟⠀⠀⠀⠀⠀⠀⠉⠛⢃⡼⠋⠀⠀⠀⢘⣷⠶⠃⠀⠙⠛⠋⠈⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢿⣷⣯⡄⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⣴⡿⠀⠀⠛⠁
⠀⠀⠀⠀⠀⢀⣄⠀⢸⣿⣔⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡀⠀⠛⠁
⠀⠀⠀⢠⡀⠘⣿⣠⡊⠛⠛⠋⠀⠀⠀⣦⠗⠀⠀⠀⠀⣶⣄⠀⠘⠁⢀⣤⠄
⢀⡄⠀⢸⣇⣔⠛⠻⠟⠁⠀⠀⠀⠀⠛⠈⡶⠁⠀⠀⠀⠘⠻⢧⠀⢰⡿⠟⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠠
⢸⣇⡤⠊⠙⠛⠛⠀⠀⠀⠀⣀⢀⠀⠀⠳⡆⠀⠀⠀⢲⣦⡀⠀⠛⢁⣤⡦⠀⠀⠀⠀⠀⠀⠈⠠⢠⠄⠄⡀⠅
⠚⠻⠿⣷⠆⠀⠀⠀⠀⠀⢠⡹⠟⣀⠀⠀⠀⠀⣀⠀⠈⠻⡗⠀⢀⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡈⠁⠀⠀⠀⠀⢹⣷⡄⠀⠐⠔⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⣀⠀⠉⠃⠀⢀⣴⣾⠗⠀⠀⠠⣄⠆⢀⡀⢀⠠⡄⠤⠈⠃⢠⡆
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢿⣦⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠰⠂⠤⠃⡬⠀⡤⠰⠆⠘
⠀⠀⣠⡿⡀⠻⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣀⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢤⢀⠀⠀⠀⠀⠀⠀⠀⣬
⠀⠀⠹⠁⠐⣤⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠈⡀⠀⠀⠀⠀⠤⡝⠂
⠀⠀⠀⢀⣴⡀⠳⢟⠛⠓⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⡁⠀⠀⢠⠜⠃
⠀⠀⠀⠸⠟⠁⠀⠀⠑⠦⣔⠶⠿⠖⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠆⡀⠀⠀⠀⠀⠀⢬⠁⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡈⠑⠒⢤⣤⣄⠀⠀⠀⠀⠀⠀⠠⡅⠀⠄⠀⠀⠀⠀⡀⠦⠠⠆⠀⠀⠀⠀⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠂⠠⢂⠀⠀⠀⢠⠙⠀⠖⠀⠀⠀⡄⢦⠉

⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⡀
⠀⣿⠀⢹⣿⣿⢿⣷⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠁⣰⣿⣿⣿⣿⣷⣦⣌⣿⣻⣻⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⡾⠻⠛⣩⡿⢿⣿⣿⣿⣯⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⡇⠀⢀⡟⡰⠶⢾⣿⣿⣿⣿⣿⣿⣿⠃⢰⣿⣿⣿⣿⡖⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠳⣔⠞⠁⠈⢀⣼⣿⣿⣿⣿⣿⣿⣿⠀⣼⣿⣿⣿⣿⣿⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠁⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⣾⠟⠛⠛⠻⢿⣿⣿⣟⠁⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⢻⣦⣄⠀⠀⠀⠙⣏⢿⣷⣤⡀⣌⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠉⠃⠀⠀⢀⡿⢀⣿⣿⣿⣿⣿⣷⣬⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠞⠗⠛⢻⣿⣿⡿⠟⠛⠿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠘⢆⡶⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠿⢷⠀⠘⣿⣿⣿⣿⣿⡿⣿⠟⠛⠛⠉⠁⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠈⡷⡀⠀⠘⣄⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢻⠻⣿⠀⠘⠦⠀⢀⡀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⢀⠀⠈⢦⢀⠈⠑⠒⢀⣤⣤⣤⣤⣤⣤⣀⣀⡀⠘⢦⡣⠀⠀⠀⠀⠀⠀⠈⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠬⠀⠀⠀⠉⠀⠀⠀⠸⠿⠿⢿⢿⢿⣿⣿⣿⣿⠇⠀⠈⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢶⣶⢶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⣤⣶⣶⠶⠶⣦⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠠⣤⣽⣿⣷⣄⡰⣿⣡⣤⣶⠾⡿⢏⡁⠈⠒⣶⣽⣿⣬⣼⢿⣖⣦⡀
⠀⠀⣠⡾⠃⣀⠀⠀⠙⣯⣷⣶⣔⣿⡶⠿⣿⣿⣿⡿⢟⣿⣧⡤⠖⢋⣷⣶⣭⣷⣾⣿⠟⠛⢿⣿⣿⣧⡀
⠀⠀⢿⡐⣦⠀⠚⢯⠐⣄⢀⣀⢽⣯⡾⣿⣶⣾⣿⣿⣿⣿⣹⣶⣶⣿⣿⣿⣿⡟⠉⠀⠀⡀⠀⣿⣿⣿⠃
⠀⠀⠈⠉⠁⢤⣲⣦⣷⣽⣄⣹⣦⣿⣧⣯⣽⣿⣭⣤⣾⣿⣿⣿⣿⡏⣹⢻⡏⠀⠀⠲⢾⠟⢀⡜⣯⠛⠳⣄
⠀⠀⠀⠁⠀⠀⣩⣿⣿⣿⣯⣯⣙⡲⠿⣿⣝⣛⣛⡿⠟⣻⣿⣿⣿⣿⣅⡈⠛⠀⠀⠀⠀⠀⠘⠃⠹⣦⠀⠈⠁
⠀⠀⢀⣴⣤⠘⠍⠊⠁⠋⢸⣷⣯⣝⣿⣿⡿⢟⣫⣴⣿⡿⣋⣭⣿⡿⡿⣇⠀⠀⠀⠀⠀⢤⡏⠀⠐⣾⡀⠀⡠
⠀⠀⠀⢄⣸⡇⠀⠠⣯⠤⣤⣽⡿⢿⣿⠷⠿⢟⣿⡿⢋⣼⣿⣿⡾⠁⡇⠙⣷⣦⣄⣀⠀⠛⠀⠀⠈⠀⠉
⠀⠀⠀⢮⣩⢶⣿⠃⣲⣷⣾⣤⣴⣶⣶⣾⣿⠟⣡⣾⣿⣿⣿⡿⠁⠀⡇⠀⢸⣿⠉⠉⠉⠁⠀⠒⢤⡀⠤⠄⢀
⠀⠀⠀⠠⣿⣿⣿⣟⡛⠿⣿⣯⣾⣿⡟⣽⣧⣾⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠈⣿⣀⠈⠛⠶⡰⣴⣌⣿⢠
⠀⠀⠀⠀⠿⠟⠉⠛⣙⣿⣿⣿⣿⣋⣴⣿⣿⡿⠛⠉⠉⠁⣤⠀⠀⠶⠀⢀⣼⠻⣿⣷⣄⠀⠁⠘⣿⠏⣼
⠀⠀⠠⣶⢤⡄⣤⠀⠨⢽⠿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠃⠀⠀⠉⠛⣖⡤⢚⣡⡾⣣⠂⢀⠄
⠀⠀⠀⠁⠀⠀⠻⣤⡴⢀⡴⣹⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⢀⡠⠔⣊⣥⡶⣛⣥⣚⡥⠞⠁
⠀⠀⠀⠀⠀⠀⢀⠀⡎⠀⡻⣧⣏⢹⣿⣿⣧⣤⣤⣤⣤⢾⣻⠀⠀⠀⠠⢐⣨⣵⠶⣛⣫⡵⡾⣛⠯⠅⠂⠈
⠀⠀⠀⠀⠀⠀⠈⠳⢦⠀⠛⠛⠛⠛⠛⠛⠛⠋⠉⠐⠋⣁⣀⣤⠴⠶⣛⣯⠵⠖⣛⡭⠖⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠈⠤⠤⠀⢒⣚⡛⠯⠭⠭⠗⠒⠋⠉⠁⠄⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣄⣴⡆⡀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢀⢰⣦⢠⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⡿⢻⠃⠿⢌⣥⢺⡇⠀⠀⠀⠀⠀⠀⢹⡗⣬⡡⠿⠘⡿⢿⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⠉⠁⠹⡄⠀⠀⣳⣯⣇⣤⣶⡚⢓⣦⣤⣸⣽⣎⠀⠀⢠⠇⠈⠉⣁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡭⠄⢀⣲⣽⣻⡎⠀⠀⡡⠤⠤⠶⠶⠤⠥⣌⠀⠀⢱⣞⣯⣆⡀⠜⢭⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣤⣾⡈⠈⡌⠛⠙⠒⠤⡄⡴⡆⢇⡰⢑⢆⢠⠤⠒⠋⠛⢡⠉⢑⣶⣤⡈
⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⢠⢫⣟⢀⡠⠀⠀⣻⠤⣅⠐⠁⠂⠐⠈⠂⣨⢤⣟⠀⠀⢆⠐⣻⡽⡈⠉⠓⠦⣄⣀
⠀⢀⣠⡴⡺⠋⠁⠀⢀⠆⣠⠿⠛⠁⠚⠘⠦⠘⠱⣧⠈⠀⠀⠀⠀⠀⠀⠁⣾⠎⠃⠔⠁⠓⠈⠛⢷⡄⢠⠀⠀⠉⠻⡶⢤⣄⡀
⠀⣼⢻⣇⠰⠆⢐⣾⠋⡼⠃⠀⠀⠀⡠⢄⠄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⡠⠄⠀⠀⠀⠹⣧⠓⣲⡆⣠⠄⣽⡿⡇
⠀⣿⠀⠙⠧⠀⠀⠀⡼⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠉⠀⠀⡰⠟⠀⡇
⠀⣿⣠⣦⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⣤⣄⡇
⠴⣿⣩⠈⢦⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⡠⠋⣝⡿
⠀⣿⣅⡆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠁⢠⣹⡇
⠀⣾⠉⢧⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡸⡙⡇
⠀⣿⠘⠰⠈⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠁⡄⠆⡇
⠀⣿⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⡇
⠀⠿⣀⠀⠀⠀⠀⠀⠀⢹⢦⡀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠀⠀⢀⣠⡇
⠀⠀⠀⠉⠓⠦⣤⣀⠂⢻⣼⣟⣷⠀⠀⠀⢘⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡃⠀⠀⠐⣮⣻⣧⠗⠐⣀⡤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣌⡋⠳⢤⡄⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⡴⠋⢘⣥⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡈⠀⠀⠉⠻⠶⢭⣛⠶⢤⣀⣀⣀⠀⠀⠀⢀⣀⣀⣠⡤⢖⣋⡥⢶⠟⠉⠀⠀⢁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣬⣶⠀⢠⠖⠙⠈⠙⠛⠖⠦⣍⣉⠉⠉⠉⠉⣉⣭⠴⠺⠛⠉⡡⠋⠢⡄⠀⣾⣵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠛⠛⠇⠀⠀⠀⣀⠀⠀⠀⠀⣉⠹⠶⠶⢋⣁⠀⠈⠀⠀⣈⠀⠀⠀⢹⠛⠛⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠤⠀⠁⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡤⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⢀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡀⠐
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣍⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠙⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⡀⠀⠀⠀⠀⣀⣀⣀⣤⣀⣀⡈⢿⣷⡀⠀⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢲⣅⠀⠀⠀⠈⢉⣭⣤⣤⣀⡀⠀⢸⣿⣷⠀⠀⠈⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠘⠁⠻⠟⠹⠛⠋⢀⣿⣿⠀⠀⠀⢘⣿⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⣸⡿⢸⢿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⢀⣴⠟⢡⠏⢠⣿⡟⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⣠⣿⡟⠀⠀⣠⠞⢋⠔⢁⣴⣿⠟⠁⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣄⡀⠀⠀⠀⠀⢰⣿⠏⠀⢠⡞⢁⠔⢁⣴⡿⠋⠁⠀⢀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠷⠂⠀⠀⣿⡏⠀⣰⠏⡰⢁⣴⠟⠁⢀⣠⣴⡾⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⡟⡜⣠⡿⠁⣠⣶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣷⢠⡿⢁⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠘⣿⣾⣇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣌⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣔⣿⣿⢿⣯⠀⠀⠀⣀⣠⢄⡉⣍⣛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣪⣵⠞⠁⠘⡍⢸⡏⠀⠀⠀⠘⠾⣿⣿⣿⣿⣿⣦⣝⢿⣦⡀
⠀⠀⠀⣠⣶⣿⠿⣿⣿⣿⣿⡏⠀⡆⡆⠘⢦⡁⢄⠀⠀⠀⠀⠈⠻⣿⡌⢻⣆⠙⢿⣿⣿⣆
⠀⠀⠰⢫⠏⣠⡿⠻⠛⣫⣿⣧⠀⠈⢹⣄⣠⠈⠙⠷⣤⡀⢄⢠⡀⠙⣿⡆⠙⡆⠀⠹⣿⣿⣧⡀
⠀⠀⠀⢸⠀⣿⢀⣠⠞⠉⢿⣿⣦⣦⠀⠙⠻⣿⣧⠰⣦⡙⢾⣧⢱⡄⠘⣿⡀⠈⠀⠀⠹⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢿⡿⠁⠀⠀⠈⠻⣏⠻⣷⣦⣀⠀⠉⠻⢮⣿⣦⡹⣧⢻⡄⢹⣧⠰⡄⠘⣆⢻⣿⣿⢷
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⢀⣉⣹⣿⠷⣶⣤⡈⠙⢷⣿⣾⣷⠈⣿⡀⣿⡄⢻⡌⣿⣿⣎⠃
⠀⠀⠀⠀⠀⠀⠐⣛⡲⠀⠀⠀⠐⠛⣩⣥⣤⣤⣄⣤⠉⠻⣦⡀⢻⣿⣿⠀⢿⡇⣿⣇⢸⣷⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠘⠉⠹⠷⠀⠀⠀⠀⠃⠘⠛⠘⠋⠀⠀⠀⠸⣷⠀⣿⣿⠀⣿⣿⣿⣿⣼⢻⡿⣿⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡄⣿⣿⠀⣿⢸⡇⢻⡏⠈⠇⣿⣿⡇⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣾⣰⣿⡇⢰⡏⠈⠀⠸⢡⡄⣸⣿⣿⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⡾⢡⢾⡿⣿⡿⢀⣿⠁⠀⢀⠀⣼⣰⢯⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⣀⠀⠀⠀⣸⠁⣿⠸⣹⣿⠃⣼⡏⣸⠀⣾⣰⣿⢋⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟⠋⠀⠀⢠⡏⠀⡇⢠⣿⡏⣸⡿⣰⢇⣼⣿⠟⣡⡿⣹⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⣿⣰⡿⣱⣿⣾⠟⢃⣼⠋⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣼⣿⣿⣷⣿⠟⠁⠔⠋⠀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⣿⣿⠿⠋⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣹⣿⠇⢀⡀⠀⢀⠔⣡⢟⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉⠀⠀⠀⠀⣸⣧⢱⣿⣶⣿⣔⡴⢋⡼⢋⡞⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⡇⠘⣷⣿⣿⡿⢋⣴⠟⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⠎⠻⣿⣷⡟⢁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣾⣶⡜⢻⣷⠏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢣⢸⣿⣿⣧⣿⡿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣿⣿⣿⢿⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠘⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠤⣤⡤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠒⠀⠉⠁⠀⠀⠀⠀⠀⠙⠁⠓⣤⠄⠂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠸⡶⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡁⢠⣄⠀⠀⠀⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠰⣾⣿⣶⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡔⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⡘⠁⢶⣄⡀⠂⢀⡀⠀⣀⣠⡄⠀⠀⠀⠠⠄⣀⠀⢀⣀⣠⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡟⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠃⠐⣤⠉⠻⠓⣤⡤⣤⣤⣤⠇⠀⠀⠀⠀⠀⠀⠀⠻⢿⣏⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠾⠁⠤⢺⡿
⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠉⠀⡄⠀⣄⠀⠰⠆⠠⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⣀⣰⠦⠘⠛⠃⣼⣧⢀⣄⠂⠀⣼⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣶⣦⣄⣀⣀⠀⠀⠀⠁⠀⢠⡚⠉⠀⠄⠀⠀⠀⠀⠀⠀⢸⢿⣶⠉⣉⠉⠁⠸⠻⢆
⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠉⠉⠉⠙⠲⠶⠦⣤⣼⣅⣂⣴⣾⣿⣦⠒⠀⣠⣿⡯⠛⠁⠾⠛⢠⠆⠰⠀⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠐⣶⣥⡄⠋⠹⠿⢿⣿⣿⣥⠞⣋⣁⡀⠀⠀⠀⠀⢁⣴⣦⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⣿⠷⠄⠀⠀⢉⠳⣶⣿⣿⣿⣧⠙⠁⠀⠀⣘⠋⢀⣴⠞
⠀⠀⠀⠀⠀⠀⠃⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠁⣴⡟⠻⣷⣌⠀⠈⠛⣿⣿⣿⢀⣤⣾⣷⣾⠟⠙⣉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣄⠙⠃⣠⣿⢋⣴⡀⠀⠜⠻⣿⣉⣤⡀⠀⠀⠀⠘⢻⡇⠈⣡⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣀⠃⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠉⠉⣉⡙⠛⡛⠷⢾⣿⣿⡄⠈⠄⠹⣿⣿⣿⡄⠃⠀⠀⠀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠁⡄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠈⠻⣿⣁⠟⢢⣌⠙⠻⠂⠀⠂⡹⡿⠋⠀⠀⠐⠀⠾⠟⢢⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠀⠀⠀⠀⠘⠏⣀⡄⠀⠑⢦⡀⠀⠀⠠⢿⣀⣀⣀⣀⠀⠠⣬⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢁⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡀⢤⣀⡀⠀⠀⠀⠲⠄⠙⠶⣶⣿⣦⣸⣦⡀⠀⠸⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣀⠀⠀⢠⣴⣿⢿⡟⢉⡙⢷⡈⢻⣭⡔⠒⠒⢤⣤⣀⠀⠀⠀⠈⠛⠿⣷⡄⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⣀⠀⢈⠙⠿⡆⢿⣅⣼⡿⢀⠘⣅⠚⢧⣀⡐⠛⣧⡀⠀⠀⠀⠀⠈⢻⡄⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⠦⠄⢀⡀⠈⠛⠛⠰⢿⣷⣄⡉⠶⠋⣡⣶⢄⣙⣤⢀⠀⠀⠀⠀⠑⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⠇⠀⠀⠀⢻⣷⣤⣄⣀⣀⣌⠀⠀⠀⠀⠈⠙⠋⠙⠟⠳⠦⠀⢀⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣤⡄⠀⠀⠀⠀⣧⣿⣿⠿⣿⡿⠷⠚⠛⠻⣯⡄⢠⡶⠂⠴⠤⠄⠀⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⠀⠀⠸⢸⣿⠀⣿⢁⡀⠀⠀⠀⠈⣄⣴⡾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣴⠃⠀⠀⢀⠀⠠⢶⡚⢿⣷⣿⠎⢡⠀⠀⠀⠀⠈⠃⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠃⠀⠀⠀⡀⠂⣈⣿⣸⣿⣿⢠⡆⣤⠂⠀⠀⠀⠈⡴⠷⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣶⠋⠀⠀⠈⠠⠈⠓⠀⠉⠹⣿⣿⡀⠀⠁⠞⢉⠀⠰⠀⠰⢆⡀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠃⣿⣟⠁⠀⠀⠀⠉⠀⠂⠀⠀⠈⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣦⠀⠀⠀⠀⠀⠒⢯⡄⢀⠀⣿⣿⠀⠀⠀⢇⡈⠀⠀⠀⠀⠀⣥⡄⢠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⣠⣤⣿⣷⡀⠀⠀⠀⠀⠁⣰⠋⢶⣿⣿⡸⠇⠳⠀⠡⠄⠀⠀⠀⢸⣿⣿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠉⢡⣿⣡⣿⣿⡦⠀⠀⠀⠠⠀⠀⠴⣿⣿⡄⠞⠀⠀⠀⠀⠀⠀⣰⣿⣏⣹⣿⢠⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢷⣸⣿⠿⠋⠉⠀⠈⠁⠀⠀⠀⠁⠀⣿⣿⠇⡀⠈⠀⠀⠀⠀⠚⠛⠛⠿⢿⣿⣏⡛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⠀⠸⠃⢀⠄⣀⣀⣤⣤⣼⣿⣿⣆⣁⣤⣀⡀⠛⠀⠀⠀⢤⡀⠀⠈⠛⢣⣄⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⡇⠀⠀⠐⡄⠀⠀⢠⣶⡿⠛⠉⠉⠉⣿⡿⣿⠋⠉⠙⠛⠷⣷⣀⡀⠀⠹⠃⡄⠀⠀⢨⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⢊⠀⠀⠀⡄⠀⠀⢰⣿⠃⠀⠈⢀⠀⠰⣿⠀⢻⣆⠀⡀⠁⠦⠈⢿⣄⠠⠀⠀⠐⢄⠀⠀⡀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠋⠀⠀⠀⠁⠀⠠⣿⠇⠀⠀⠠⠈⠀⠀⡏⠀⠘⡇⠀⠳⢆⡀⠀⠀⣿⣄⠀⠀⡇⠘⠀⠀⠻⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡆⠀⠃⠀⠀⠈⢠⣿⠀⠀⠀⠠⠄⠀⣦⡀⠀⠀⣤⡀⠀⣈⠀⠀⠀⢿⣯⠀⠀⡇⠀⠀⠀⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⡀⠀⡀⠀⠀⣿⡄⠈⠀⢀⠈⠀⠀⡾⠀⢸⣇⠀⠀⠉⠠⠀⠀⣿⡏⠀⠀⡆⠀⠀⠀⣿⠾⣷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠀⠀⡀⠳⡀⠁⠸⢿⣆⡀⠀⠀⠀⠀⢳⣄⡰⠏⠀⠀⠀⠀⢀⣼⡿⠄⠀⠀⠃⠾⠀⠀⣈⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠩⠉⢀⠀⠁⠀⢣⡀⠀⠈⢻⣿⣦⣀⠀⠀⠈⠀⠀⠀⠀⢀⣴⣾⣿⠿⠁⠀⢀⠈⢠⠂⠀⢠⣼⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣭⡄⠀⠀⠀⠀⡄⠀⠀⠙⢻⡟⣷⣄⠀⠀⠀⠀⣴⣿⣤⡿⠇⠀⠀⡰⠆⠀⠈⠀⠠⡌⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢀⠁⢄⡀⠀⡀⠀⢆⡀⠀⠀⢻⣿⠙⣧⡄⠀⣿⣏⣸⡟⠛⠀⢀⠐⠀⣠⠂⠀⢀⢷⣦⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠳⠄⠘⠠⠼⠃⠀⠀⠈⢽⣿⠟⠷⢴⡍⢻⣿⠁⠀⠀⠠⠦⠄⠁⢀⡰⠄⠀⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡤⠀⣥⡄⡀⠀⠀⠀⠉⣼⣿⠀⠀⠀⡀⢸⣿⣵⠀⠀⠀⠀⢠⠀⠾⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠏⠈⠁⠴⣉⣁⣾⠟⠙⣷⣦⣤⣴⣿⠛⢿⠇⣀⠒⢰⣧⡄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⣿⡏⠁⠀⢰⡿⠋⠉⠉⣩⡟⠓⠒⢾⣧⠠⠄⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⢽⡟⠀⠁⠂⣿⠇⠐⠀⠈⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠷⠍⠉⠒⢄⡠⠒⠉⠩⠾⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠄⣀⡇⠀⢀⡶⣠⢠⣨⣅⡄⣄⣶⡀⠀⢹⣀⠠⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠏⠀⠀⢀⣳⡼⡲⣼⠾⣇⣺⣗⣸⠷⣭⢖⢧⣞⡀⠀⠀⠩⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⣻⢚⣾⣑⣵⠶⠿⠶⠟⣁⣈⠻⠶⠿⠶⣯⣊⣷⡓⣟⠀⠀⠡
⠀⠀⠀⠀⠀⢀⡤⠼⢤⣰⢞⢻⣅⣽⠃⡴⣖⣦⣼⣭⣽⣧⣴⣲⢦⠘⣯⣨⡟⡳⣆⡤⠧⠤⡀
⠀⠀⠀⢠⢴⡇⠀⢀⢺⡫⢱⡟⢉⡤⣤⡿⣛⣫⣲⣾⣷⣶⣝⣟⢿⣤⢤⡉⢻⡟⢝⡖⡀⠀⢸⡦⡄
⠀⠀⠀⠈⡇⠁⠘⣫⣲⡟⢻⡇⠸⣽⡿⣰⣴⣿⡿⣿⣿⢿⣿⣦⣗⢿⣯⠇⢸⡟⢻⣔⣝⠃⠈⢸⠁
⠀⠀⠀⠀⠘⢄⡀⣲⣩⣹⠟⢡⠖⣿⣨⣾⣿⣯⣿⡿⢿⣿⣽⣿⣷⣅⣿⣲⡌⠻⣏⣍⣖⢀⡠⠃
⠀⠀⠀⠀⢠⠊⠁⠽⣙⣹⣦⡘⠧⣿⢹⢿⣿⣟⣿⣷⣾⣿⣿⣿⡿⡋⣿⠽⢃⣴⣏⣋⠯⠈⠑⡄
⠀⠀⠀⢀⡇⡀⢠⣝⡝⣧⣼⡇⢰⣻⣷⢽⠻⣿⣷⣿⣾⣾⣿⠟⡯⣾⣟⡆⢸⣧⣼⢫⣫⡄⢠⢸⡀
⠀⠀⠀⠘⠺⡇⠀⠈⢼⣕⡼⣧⣈⠓⠛⣷⣽⣝⠝⢿⡿⠫⣫⣯⣾⠛⠚⣁⣼⢧⣪⠯⠁⠀⢸⠗⠃
⠀⠀⠀⠀⠀⠈⠒⢲⠚⠹⢮⣼⡋⣻⡄⠳⠯⠟⢻⣛⣟⡟⠻⠽⠞⢠⣟⢙⣧⡵⠏⠓⡖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⣽⢬⢿⡩⣻⠶⣶⠶⣦⡉⢉⣴⠶⣶⠶⣟⢍⡿⡥⣯⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⢘⣧⣂⠀⠀⠈⡽⢳⠵⣻⢶⡏⢽⡯⢹⡶⡟⠮⡞⢯⠁⠀⠀⣐⣼⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠂⠉⡇⠀⠈⠿⠙⠘⢙⡋⠃⠋⠾⠁⠀⣸⠉⠐⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡶⣂⣀⠤⠊⠑⠤⣀⣐⢶⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣦⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣴⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠊⠉⠁⠀⠀⠀⠀⠈⠉⠉⠙⠛⠙⢳⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠞⠋⠛⠋⠉⠉⠁⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠲⣄⣀⣀⣀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⣀⣠⠖
⠀⠀⠀⠀⠀⠀⠿⣿⣿⡿⠟⡛⢿⣿⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⢿⡿⢛⠻⢿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠾⠶⠴⠾⠐⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠒⠷⠦⠶⠶⠚⠋⠁

⠀⣴⡿⠻⢷⠆⣰⡾⠟⢿⣦⠀⣾⡟⠻⠆⢸⣿⣷⠀⣼⣿⡇⢾⡿⠛⠟⠘⠿⣿⠟⠃⣴⡿⠻⢷⣦⠰⣿⠀⠀⢠⣶⠟⠻⣶⡄⣠⣾⠟⠿⠦⠙⣷⣀⣼⠟
⠀⣿⡄⠀⣀⡀⣿⣇⠀⢀⣿⠇⣙⠻⢿⣶⢸⣿⢻⣿⡟⣿⡇⣿⡟⠛⠛⠀⠀⣿⠀⠘⣿⡀⠀⣸⣿⢈⣿⠀⠀⢸⣧⠀⠀⣼⡇⢿⣇⠰⢿⣷⠀⠘⣿⡏
⠀⠈⠛⠛⠛⠁⠈⠛⠛⠛⠋⠀⠛⠛⠛⠋⠘⠛⠀⠛⠀⠛⠃⠛⠛⠛⠛⠀⠀⠛⠀⠀⠈⠛⠛⠛⠁⠘⠛⠛⠛⠀⠙⠛⠛⠋⠀⠈⠛⠛⠛⠋⠀⠀⠛⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡶⠶⠶⠶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠁⠀⠀⣰⡄⠀⠀⠀⠉⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⢀⣠⣴⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⢀⣿⠿⠿⠛⠋⠁⠀⣿⣿⣶⣤⣀⣀⣀⣀⣤⡄⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢸⣿⠀⠺⠆⠰⠶⠀⣿⡇⠀⠉⠛⠛⠛⠋⢹⡇⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⢸⣿⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢿⡇⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣤⣿⠷⠿⠛⠃⢸⡏⠀⠀⠀⢻⣇⠀⠶⠆⠀⠀⣿⡇⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⢸⡇⠘⠻⠿⢷⣦⣾⣇⡀⢀⡀
⠀⠀⠀⠀⢀⣾⣶⠿⠛⠉⠁⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠈⢿⣦⠀⠰⡶⠀⣿⡇⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠉⠙⠛⢿⣧⣄⣀
⠀⠀⠰⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⢀⣻⣷⣄⡀⠀⣿⡇⠀⠀⣠⣴⣟⠁⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡄
⠀⢀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣷⣶⠾⠿⠛⠛⠙⣿⠿⠷⣿⣷⠾⢿⣿⠉⠛⠻⠿⢶⣶⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡷
⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠉⠀⠀⠀⠀⠀⠀⢿⣄⣀⣸⣇⣀⣼⠟⠀⠀⠀⠀⠀⠈⠙⠻⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀
⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠆
⠀⠘⠛⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁
⠀⠀⠀⣼⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠿⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠿⠿⠿⠿⢶⣶⣤⣀⠀⣾⠟⠛
⠀⠀⠀⠀⠀⢹⣿⣷⡄⠀⣀⣤⣤⣶⣿⣥⣄⣰⡿⠁⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠈⠁⢀⣴⡿⠛⠉⠁⠀⠈⠉⠛⠿⣧⡀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄
⣶⡷⢶⣦⠀⠀⠀⣾⣯⣤⣤⣄⣤⣤⣤⣤⣤⣤⣸⣿⡀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠈⢿⡆
⣿⠆⠀⣿⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠀⠸⣿
⣿⣶⣶⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣿⠿⠿⠆⠀⣿⡇
⠘⣿⡄⠈⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⡟⠁⠀⠀⠀⠀⣿⡆
⠀⠘⣿⡄⠀⠹⣷⡀⣠⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⣿⠇⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠘⢿⣄⠀⢻⣷⣿⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⡀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠰⣾⣿⠛⣿⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠈⢿⣆⣼⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣿⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⢰⣦⣾⠿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠰⣿⡾⠿
⠀⠀⠀⠀⠈⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡄⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠙⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠈⢿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠾⠻⣿⡀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢹⣧⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠹⣷⡀⠘⢿⡄⠀⠀⠀⠀⠀⢀⣀⣴⡾⢿⡇⢸⣿⠀⠀⠀⠀⠹⢷⣦⣤⣀⣀⡀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡿⠀⠀⠀⠀⠀⠙⠻⠶⠿⠿⠿⠿⠛⠛⠀⠛⠻⠏⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢻⡿⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⢠⣤⣤⣾⣷⣤⣤⣾⣷⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇
⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣀⣐⣀⣀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡟⠒⠢⢠⣤⣄⣐⣒⣒⣂⣠⣤⠠⠚⠚⣧⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣽⠷⠚⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠙⠒⢫⡉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠺⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢢⣧⡀
⠀⠀⠀⠀⠀⠰⠴⠦⣤⡴⠁⠿⡀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣸⠀⠱⣤⡤⢶⠶⠂
⠀⠀⠀⢀⣰⡶⠖⣸⡽⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠙⣽⡔⠶⣶⣀
⠀⠀⠀⠀⠀⢀⣼⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⢆
⠀⠀⠒⠶⢤⣼⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣶⠞⠻⠃
⣠⣶⠶⠂⣼⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠄⠙⠾⢦
⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⣠⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⡀
⠀⠰⡁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡱
⠀⠀⠱⣼⠀⠀⠀⠀⠀⠀⢀⠶⠂⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⢸⡴⠁
⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⢹⣄⠀⠐⣆⢀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢠⠄⠀⢀⣼⠁
⠀⠀⠀⠀⠀⢎⢆⢀⡌⠃⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠩⢁⠀⢀⢎⠆
⠀⠀⠀⠀⠀⠈⢦⢫⠉⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠘⢡⢋⠎
⠀⠀⠀⠀⠀⠀⠀⠣⣳⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣰⡣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⡍⢀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡿⣏⠏⢂⠀⠀⡠⢄⡀⠡⠀⠀⠎⠁⠀⣀⢹⡤⠀⠄⠀⡐⡺⢸⡷⡟⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⣟⠛⣻⢤⠊⢰⠇⡎⣲⢉⠟⣳⣄⣸⡑⠎⢰⠀⠈⣠⣿⠛⢟⡸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⡞⠋⠀⡷⠒⠀⣄⠀⠀⢫⢄⠤⠊⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣁⠘⠛⡚⠀⠐⠎⠁
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣤⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡀⠠⠀⢀⣀⣃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣙⣧⡙⡇⢴⣿⠿⠟⠃⠀⠘⠖⠤⣐⠤⢀⡀⠀⠉⡠
⠀⠀⠀⠀⢈⣭⣇⣛⢰⣶⣄⠀⠀⠚⠀⠀⠀⠀⠉⠀⠀⠩⠿⠃
⠀⠀⠀⠀⡾⠟⠅⠟⡠⠙⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠠⠄⠶⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⣤⡾⠁
⠀⠀⠀⠀⢰⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣤
⠀⠀⠀⣰⣧⠁⢠⢠⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠠
⠀⠀⠀⠈⢁⠀⠀⠳⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠿⠤
⠀⠀⠀⠀⠈⣄⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠘⡟⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠠⠀⠄⢀⣀⡀⠀⠀⢀⠀⣰⣷⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⢁⡀⠀⠀⠀⠀⠀⠠⠽⣿⣷⣦⣼⣓⣿⣧⡊⠀⡼
⠀⠀⠀⠀⠀⠤⠀⠀⠉⠓⠶⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⣀⠀⠓⠀⠀⠀⠁⠀⠀⠀⠈⠉⣩⢜⡛⣙⡻⡥⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠁⠂⠀⠀⠀⢔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡷⠋⠻⠈⠛⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠛⠛⠛⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠘⠛⠶⣄⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⢹⡆⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠾⠷⠦⢤⣤⣄⣀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣶⡶⠿⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠻⣦⠀⠈⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢹⡆⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⣸⡇⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⣀⣀⣀⣀⣠⣴⠞⠁⣠⣿⡄⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠶⠛⠉⣉⣀⣀⣉⡉⣉⣉⣉⣁⣀⣤⠾⠋⠀⢻⡄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⡴⠞⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣠⡞⠁⠀⠀⠀⠐⠶⠶⠶⢤⡄⠀⠀⠀⠀⠀⠀⣤⡶⠶⠶⠖⠀⠀⠀⠙⢷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⣗⠀⠀⠀⠀⠰⠟⢻⣏⡿⠳⣄⠀⠀⠀⠀⠀⢠⡞⢿⣿⡟⠛⠆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠳⣤⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⣤⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠉⢙⣳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠈⢻⡄⣸⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠃⠀⠀⠀⠀⠈⠉⠉⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⡾⠻⠶⠛⣰⠶⣬⣿⣏
⠀⠀⠀⠀⠀⠀⣾⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠟⢷⡆⠀⠀⠀⠀⣤⣀⠀⠀⠀⣀⣤⠀⢿⡀⠀⡶⢿⡆⠹⣇⠹⣆
⠀⠀⠀⠀⠀⠀⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠰⠶⠶⢾⡇⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠈⢳⣬⣷⠀⣿⠀⢿⡀⢿⡀
⠀⠀⠀⠀⠀⠀⣿⠀⢠⣤⣤⣤⡄⠠⣤⡤⠀⣤⣄⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢹⡆⢸⡇⠸⡇⠸⣇
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢸⡏⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⣷⠸⣇⠈⣷⠀⣿⠀⢿⡟⣷
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⢸⡇⠀⣿⠙⠳⢦⣄⣀⣀⣠⣤⡶⠛⠉⠀⣿⠀⣿⠀⢻⠀⠻⠀⠸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠘⠛⠛⠛⠒⠛⠛⠛⠛⠛⠋⠀⢸⡇⢠⡟⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⣿⠀⠉⠀⠘⠁⠀⠀⢠⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠟⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠸⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙


⡏⠉⡧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡏
⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡂⠀⡖⠐⢢⢰⠀⢲⠂⢀⠶⠀⢰⢒⡆⣶⠀⢰⡆⠂⠀⢰⡂⠀⢰⡆⢐⡆⣲⢠⠂⠒⡄⢰⡰⠂⡖
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⡀⣇⣀⡜⢸⡀⢸⠀⡼⠒⢇⢸⣉⡗⣻⣀⣸⣇⣀⠀⢀⣈⠆⢸⡇⢈⡏⢳⠘⣄⣠⠆⣘⠑⣄⣏⣁
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣶⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠛⠛⠛⠛⠶⣤⡀⠀⠀⠀⠀⣀⣤⠶⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⣠⣤⠶⠶⠶⠦⣬⡻⣦⠀⣠⠞⣩⣤⣤⣄⡀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠟⠁⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠙⢿⣷⣿⠞⠉⠀⠀⠈⠻⣦⡀⠀⠛⢶⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⢉⣠⡴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣄⣀⣉⠉⠙⠛⢶⡄
⠀⠀⠀⠀⠀⠀⣼⠃⢠⡾⠋⠁⠀⠀⠀⠀⣠⡀⠀⠀⠀⣀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠉⠉⠛⠷⣄⠀⢻⡄
⠀⠀⠀⠀⠀⠀⣿⢠⡟⠀⠀⠀⠀⠀⠀⠘⠿⠟⠀⢠⡞⢻⣷⠀⠀⠀⣿⠀⠀⣴⡿⠳⣆⠀⠻⠟⠁⠀⠀⠀⠀⠀⠀⠸⣧⣸⠇
⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠘⣧⠈⠉⠀⠀⠀⣿⠀⠀⠙⠃⠀⣿⠀⠀⣀⣤⣤⡄⠀⠀⠀⠀⢀⣿⡟
⠀⢀⡴⠞⠛⣛⣛⣿⣷⡀⠀⠀⠀⠐⣿⣄⡀⠙⢷⣄⢹⣆⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡿⠀⣴⠏⠠⣼⡟⠀⠀⠀⠀⣼⣟⣀⣀⣀
⣰⠏⣠⡶⠛⠉⠉⠉⠙⠛⠳⠶⣤⣀⠈⠉⠀⠀⠀⠻⣦⢻⡄⠀⠀⠀⣿⠀⠀⠀⠀⣼⢇⡾⠃⠀⠀⠀⠀⠀⢀⣠⠾⠛⠛⠛⠿⣭⡛⢶⡄
⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⡀⠀⠀⠙⣿⣷⣤⣄⠀⣿⠀⣠⣤⣴⣿⡟⠁⠀⠀⠀⣀⣤⠶⠛⠁⠀⠀⠀⠀⠀⠈⣷⡀⢻
⢻⡀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⢰⡏⠀⠀⠉⢷⣾⡿⠁⠀⠀⢹⡆⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸
⠸⣧⠘⣧⠀⠀⠀⠀⠀⣾⡿⡷⠀⠀⠀⠀⠀⠀⠀⠙⣾⣧⡀⠀⠀⠈⣿⡇⠀⠀⠀⣼⠟⠋⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⢠⡾⢡⡟
⠀⣿⠀⣿⠀⠀⠀⠀⠀⠻⠷⣤⣤⣀⣀⡀⠀⠀⣠⡴⠾⠿⣿⣦⣤⣶⣿⣧⣀⣴⡾⠷⠶⢦⣄⠀⠀⠀⢀⣀⣠⣤⠟⠁⠀⠀⢠⡿⠀⣿
⠀⡿⠀⢹⠀⠀⣾⣿⠆⠀⠀⣀⣈⣩⣭⣽⣿⣿⡏⠀⠀⠀⠀⠹⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢹⣶⣟⣛⣋⣉⠁⠀⠀⠠⣾⣷⠘⣧⠀⣷
⢸⡇⠀⢸⡆⠀⠈⠁⢀⡴⠛⠉⠉⠉⠀⠀⠀⠘⢷⣄⣀⣠⣴⡾⠟⠿⣿⠛⠻⢶⣤⣤⣠⡴⠟⠉⠉⠉⠉⠉⠙⠻⣦⠀⠉⠁⠀⢻⡀⢻⡆
⢸⡇⠀⢸⡇⠀⠀⠀⠘⢷⣴⡶⠀⠀⠀⠀⠀⠀⣀⣤⠿⣿⠋⠀⠀⠀⣿⠀⠀⠀⠈⢿⡟⠳⠶⣤⣄⣀⠀⠀⣴⣤⣼⠇⠀⠀⠀⢸⡇⠈⣷
⠈⣷⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠋⠁⠀⣧⠀⠀⠀⣸⣿⣧⡀⠀⠀⣸⠇⠀⠀⠀⠈⠉⠛⠶⣌⠉⠁⠀⠀⠀⣀⣼⠃⢠⡟
⠀⠘⢷⣄⠀⠙⠷⢦⣤⣤⣤⠶⠞⠛⠉⠀⠀⠀⠀⠀⣠⣿⠷⠶⠾⠋⢸⡏⠙⠛⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣶⣶⣿⣯⣤⡶⠟
⠀⠀⠀⠙⠛⠶⠶⢶⣿⡏⠀⠀⠀⠀⠀⣀⡀⠀⣠⡾⣻⠏⠀⠀⠀⠀⣸⠃⠀⠀⠀⢿⡝⣧⡀⠰⢶⡄⠀⠀⠀⠀⢻⡎⠙⣧
⠀⠀⠀⠀⠀⠀⠀⣿⢹⣇⠀⠀⠀⠀⠸⣯⣡⡾⠋⣰⠏⠀⠀⠀⠀⠀⡿⠀⠀⠀⢀⡈⣷⠈⠛⠶⠞⠁⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢿⡄⠙⢶⣄⣀⠀⠀⠀⢀⡀⠀⣿⣀⣶⡄⠀⠀⢸⡇⠀⠀⠀⠻⣷⡿⠀⢴⣶⠀⠀⠀⠀⠀⢀⡾⠁⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠈⠉⠛⠷⣦⡿⠿⠀⠘⠻⠛⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣡⡴⠶⠶⠶⠶⠟⠁⣀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⢈⣷⡀⠀⠀⠀⠀⠀⣰⠟⠁⢀⣀⣤⡴⠶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠈⢷⡄⠀⠀⠀⠀⠀⣠⣾⣿⡿⣦⣀⣀⣤⡾⠃⣠⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠙⠷⢦⡴⠶⠛⣡⠟⠀⠹⣦⣉⣉⣁⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣤⣤⡴⠞⠋⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣕⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡈⠙⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡏⠉⠙⠛⠛⠿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣦⣈⠙⢦⡀⠀⠀⢀⣤⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠘⣿⣿⣿⣷⣦⡙⢄⠀⠀⠑⡺⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢹⣿⣿⣯⡻⣿⣷⣄⠀⠀⠨⡺⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢸⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣌⢻⣿⣷⣄⠀⠈⢙⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢀⣤⡶⠖⠛⠒⢸⣿⣿⣿⣿⣧⡙⣿⣿⣆⠀⠈⢑⣽⡿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣾⣿⣿⡿⠛⠁⠠⠴⠶⣶⠞⣿⣿⣿⣿⣿⣷⡈⢿⣿⡆⠀⠀⠀⠀⠈⢻⣷⣄
⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠠⣤⣤⣴⣯⠀⣿⣿⣿⣿⣿⣿⣧⠘⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⣧
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠉⠉⠀⢸⣿⣿⣿⣿⣿⣿⣿⡆⠟⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇
⠀⠀⠀⠀⢸⣿⣿⡿⠛⢉⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠘⣿⠏⠀⣴⣿⣿⣿⣿⠟⣩⣾⣿⣿⣿⣿⠿⠋⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⣿⠃
⠀⠀⠀⠀⠀⠘⠀⢰⣿⣿⣿⠟⢡⣾⣿⣿⣿⣿⡿⠁⠀⠀⠀⠲⣿⣿⣿⡷⠂⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡏⢠⣿⣿⣿⣿⣿⣿⠁⠀⢳⣄⠀⠀⠈⠉⠉⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⢸⣿⣿⣿⣿⣿⣿⠀⠀⠈⢿⣷⣦⣄⡀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠸⣿⡏⣿⣿⣿⣿⠀⠀⠀⠈⢿⡟⠉⠀⣰⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢻⣿⣿⣿⣧⠀⠀⠀⠘⠇⠀⣰⣿⣿⣿⠋⢸⣿⣿⡇⠰⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠻⣿⣿⣿⣷⡀⠀⠀⠀⢀⣿⣿⣿⠃⠀⠀⠙⠻⠷⠄⠈⠻⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠙⢿⣿⣿⣷⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⢶⣶⣤⣀⡀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠇⠀⠀⠀⠈⠻⢷⣄⠀⠀⠀⠀⠀⠉⠛⠻⢿⣶⣤⡉⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠶⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣀⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣾⡏⠁
⠀⠀⠀⠀⠀⠀⠀⡔⠉⡿⠋⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣴⣄⠀⠀⠀⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡉⢿⣴⣶⣤⢀⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠪⣿⠋⠁⠀⠀⢀⣴⣿⣦⣄⡀⠀⠀⠀⠀⠘⠁⣠⣸⣯⡛⠻⠟
⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢀⡀⠀⠀⠠⣤⣿⣿⣿⣿⣷⣶⣶⠏⠀⣀⡀⢉⣹⣃
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⣿⣦⠞⠁⠀⣦⣾⢻⣯⡻⢿⣿⣿⣿⣿⠀⠘⠿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⢀⠀⢹
⣿⣶⣦⠀⠀⠀⠀⠀⢀⣺⣿⣿⣿⣿⠃⠀⠀⠀⠈⠻⣿⢿⡻⡄⢻⣿⣿⣿⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣷⣤⡀⣶⣾⡆⣾
⣿⣿⣿⣧⠀⠀⠀⠀⠙⠿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⢸⣆⠓⣿⡈⣿⣿⣿⡀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣽
⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⠾⡇⠀⣠⣤⣤⣈⣿⣶⢸⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣦⡀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣀⠈⠿⠞⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠿⠟⠻⢿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠨⣿⣷⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣞⣛⣻⣿⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠿⢿⣷⣶⣶⣾⣿⣷⣤⣄⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠛⠛⣛⣫⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠤⠀⠀⠀⠀⠀⣔⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠉⠀⠀⠐⠆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⠀⢀⣤⠀⣠⠞⠛⠑⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢱⠞⠁⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠛⢦⡀⠀⠀⠀⠀⣀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠻⣷⣄⣀⠘⠃⠀⠀⠀⠀⠂⠄⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⣾⣿⡿⠧⣼⡀⠀⠀⠀⠀⠐⠂⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⡀⠘⠋⠉⠃⠤⢤⠄⠤⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⢦⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢠⢤⡄
⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡀⠀⣷⠀⠀⢠⠞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠂⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢹⣿⣧⣀⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠆⠀⠀⠀⢘
⠀⠀⠀⠀⡀⠀⠀⠀⠛⠀⠀⠈⠘⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁⠄⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠁
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠁⠀⠀⠀⠀⠀⠀⠰⡤⠂⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⠖⠋⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠊⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣇⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⢀⣠⠀⠀⠀⠀⠖⠛⠁
⡏⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⢨⣿⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⠀⠀⠀⠀⠀⠈⠁
⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠋⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⠉⠁⠀⠀⠀⠰⠒⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡇
⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡆⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠐⠂⠀⠀⠀⠀⠀⢀⢄⠀⠈
⢀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠊⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠐⠒⡀⠀⠠⠴⢣⡄⡀⠀⠀⠀⠀⠀⠀⣠⠶⠊⠈⠁⠈⠀⠀⠀⠀⣠⠴⠋
⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠔⠂⠉⠀⠁⠀⢠⠊⠙⡎⠀⠀⠉⠚⢰⠋⢩⠄⠀⢠⣀⣀⣀⣀⣀⡀⠀⠄⠀⢄⣀⣀⣀⣀⢀
⣀⣀⣀⣀⣈⣒⣒⣒⣂⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⣒⣂⣙⣋⣛⣁⣀⣀⣀⣀⣈⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣈⣁⣀⣀⣀
⣿⣿⣿⠿⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣥⣤⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠆⠻⠂⣤⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢀⣠⣤⡀⠀⠴⠂⠀⠀⠀⢀⡜⡿⢀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠖⠀⠀⢀⣀⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴
⠶⣶⣿⣕⣽⣿⠀⠒⠋⠀⠀⢠⢏⡾⢡⢫⡟⡴⡦⠀⠀⠀⠀⠀⢀⣴⡿⠛⣿⠟⢀⣴⠾⣿⣿⣿⠇⠀⠀⠀⣀⣠⣤⣴⢶⣶⠾⠟⢿⣿⣿⣿⣿⣿⡿
⠀⠀⠉⣿⣿⡟⠀⠀⠀⠀⡰⣣⡟⡱⣱⢏⢞⡽⢡⢶⠀⠀⠀⣠⠞⠋⠓⢚⣩⠾⢿⣁⣀⡼⠟⣋⣤⣴⣾⣿⣿⡿⠋⠀⢸⣏⠀⠀⣠⣿⣿⣿⡿⠋
⠀⠀⠀⣿⣿⠁⠀⠀⢀⡜⣵⣟⢜⡼⣣⣫⡞⡱⢣⠟⣰⣶⠰⠋⠀⣠⡴⠋⠀⠀⣀⣬⣴⣾⣿⠿⠿⠿⠿⠿⠿⠷⠼⣀⣀⣙⣛⣛⣛⣉⣉⣀⣠⣤⣴
⠀⠀⢰⣿⡇⠀⠀⢀⢞⣾⢯⣪⢟⣵⣽⢋⢜⡵⣫⣪⣾⠋⣀⡠⠞⠁⢀⣠⡶⠟⠛⠉⠁⠀⢀⣀⣤⣤⣶⣶⣿⣿⠟⠉⢉⣿⠋⠉⠉⣿⣿⣿⣿⡿⠋
⠀⠀⣾⣿⠀⠀⢠⣿⣿⣳⡿⢫⣮⡿⣵⣵⣫⢞⣿⢟⡵⣫⡿⠃⠠⠞⠉⠁⠀⠀⣀⣤⣶⣿⡿⠿⠿⠿⠿⢿⣿⣧⣐⡆⠈⠿⣷⣶⣾⣿⠿⠟⠋
⠀⢠⣿⠇⠀⣰⡏⢙⡻⢿⣷⣿⣯⣫⣿⣿⡿⣻⢕⣥⠾⣋⣤⣤⠀⠀⢀⣤⠶⠛⠋⠉⠀⠀⠀⠀⢀⣀⣀⣀⣤⣭⣭⣤⣤⣤⣤⢤⣤⣤⣶⣶⣶⣶⡶
⠀⣼⡿⠀⢰⣿⣿⣿⣷⣶⡙⢿⣿⣿⢿⢯⣪⡶⣋⢵⣺⡿⣟⣡⣄⠚⠉⠁⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⡟⣁⠀⢾⣧⣀⢀⣽⣿⣿⣿⡿⠋
⠀⣿⡇⠀⢸⣿⣿⣿⣿⣿⣯⣤⠹⣿⣷⣯⣷⡯⣟⡯⢗⣫⡵⠞⠁⢀⣤⠶⠛⠋⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⣙⣛⣋⣀⣀⣙⣛⣛⣛⣛⣋⣉⣀⣀⣀
⢰⣿⣇⠀⢸⣿⣿⢋⣍⢻⣿⣷⡆⢿⣟⣯⣷⣯⣷⠾⢯⣷⣾⡿⠂⠈⠀⢀⣀⣠⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⢛⠉⣹⣿⠛⠙⠻⣿⣿⣿⣿⣿⣿⡟
⣸⣿⣿⣄⠈⢻⣿⣦⡼⢸⣿⣿⠆⣸⣿⣯⣭⣴⡖⠛⠯⣭⣤⡄⠠⠖⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⢈⠉⠙⠛⠻⠼⠃⠛⠿⣦⣤⣶⣿⣿⣿⣿⠿⠋
⢿⣿⣿⣿⣷⣤⣬⣭⣴⣿⣿⣿⠄⣿⣿⣷⣶⠶⠿⠿⠟⣍⡉⢀⣠⡤⠴⠶⠶⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢙⣿⣷⣶⣶⣶⣬⣥⣉⣀⡀
⢸⣿⣿⣿⣿⣿⡏⡛⠿⠟⠿⢀⣾⣿⣿⣛⣛⣛⠿⢿⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣇⢲⡄⢸⣿⠉⠈⠉⢻⣿⣿⣿⣿⣿⣿⣷⣶
⠀⠻⣿⣿⣿⣿⣇⢹⣶⣶⣿⣿⣿⡿⣿⣭⣛⣩⣭⠙⠉⠈⠉⠉⠙⠛⠛⠛⠻⠿⣿⣶⣶⣶⣶⣦⣤⣤⣍⣉⠀⠈⠻⢷⣶⣴⣾⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠈⠛⠻⠿⠿⣿⣿⡿⣿⣿⢯⣟⠶⠿⠆⢩⣥⠤⠤⠶⠶⣶⣶⣶⣤⣤⣤⣄⣀⡈⠉⠛⠿⣿⡟⣍⠉⣽⣿⠿⠷⣶⣤⣭⣉⡉⠉⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣶⣦⣬⣙⠛⠀⢿⣧⡀⠀⠀⢹⣿⣿⣿⣿⣶⣶⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⠩⡈⢿⣿⣶⣤⡙⠿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡇
⠀⣀⠤⠤⣄⠀⢀⡤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡇
⡼⠁⠀⠀⠈⣦⠋⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⣿⠛⣿⡇⠀⠀⠀⠀⢀⣀⣀
⡇⠀⠀⡴⠒⢿⠊⠑⡆⠀⢸⠀⠀⠀⠁⡀⠀⠀⢀⡤⢤⠤⠤⡀⠀⠀⢀⣀⣼⡇⢠⣿⡀⠀⠀⠀⡰⠋⠀⠘⣆⠖⠛
⢣⠀⠀⡇⠀⠀⠀⢀⠇⠀⡎⠀⢰⣊⣉⡥⢽⠀⠘⢧⡩⢉⣩⠇⠀⣴⣉⣰⣿⣿⡿⢿⡧⡀⠀⢠⠃⠀⢀⣀⣏⡀
⠘⡆⠀⠹⣄⠀⢀⡞⠀⢠⠃⠀⠘⠦⢤⡴⠊⠀⣀⣀⣩⣏⣀⣰⣋⡇⠀⠀⠀⠀⠀⢸⡴⠁⠀⢸⠀⢰⠃⠈⠁⡇
⠀⢣⠀⠀⠈⢦⡞⠀⠀⡌⠀⣀⡤⠴⢚⣏⡉⠉⠉⢀⠼⠧⡀⠀⢹⡇⠀⠀⠀⠀⠀⡟⠧⣄⠀⢸⠀⠘⡆⠀⡰⠁⢠
⠀⢸⡀⠀⠀⢸⠀⠀⠀⣧⡎⠁⠀⠀⠯⠥⠟⠀⠀⠈⠓⠚⠁⠀⠀⡇⠀⠀⠀⠀⢸⡁⠀⠀⠙⣾⠀⠀⢸⣰⠃⠀⡞
⠀⠀⡇⠀⠀⢸⡄⠀⢀⡇⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠧⠴⠒⠒⠊⠉⠁⠀⣠⠔⢹⡄⠀⢀⡇⠀⢰⠁
⠀⠀⣇⣤⣤⣧⣷⣤⡜⠀⠀⠈⠙⠒⠦⠤⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠤⠔⠒⠉⠀⠀⢸⢳⠀⢸⠁⠀⡏
⠀⠀⡟⠻⠿⡿⠿⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⣈⣇⣾⡀⢸⠁
⠀⠀⡇⠀⠀⡇⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⢠⡇⠀⠀⡇⠘⠒⠋⢱⠦⣀⠀⠀⠀⠀⣀⡤⠤⢤⣀⠀⠀⠀⠀⠀⢀⠠⠴⡷⠿⢿⣟⣛⡛⢻⣟⣛⡿⠋⡇
⠀⢸⠀⠀⠀⡇⠀⠀⠀⡘⠀⠈⠙⡖⠚⠉⠁⠀⠀⠀⠈⠙⣶⠤⠤⠒⠋⠀⠀⠂⠀⠀⡏⠉⠉⠉⠏⠁⠀⠀⡇
⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⢀⡤⣄⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠃⠀⠀⡇⠀⠀⠀⢹⠀⠀⠀⣇
⠸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣟⠀⠸⠀⠀⢠⣿⠀⠀⠀⠀⠀⢸⠀⠀⢠⡇⠀⠀⠀⠸⡄⠀⠀⢻
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⣀⣠⡴⠋⣿⠀⠀⠀⣠⠖⢺⣦⠀⠸⠁⠀⠀⠀⠀⡇⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠓⣄⠈⠀⠀⣸⠋⢧⡀⠀⠻⠄⡾⢸⠆⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣀⡏⠀⠀⢠⠏⠀⠀⠙⠦⢤⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠁⠀⢀⣴⠋

⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣦⠀⢠⣦⣄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠑⠀⠀⠀⠀⠑⠀⠀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡠⢄⡀⠀⠀⠆⠀⠀⠀⠑⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠁⠀⠄⠀⢀⠉⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡆⡆⡆⢰⠶⠀⡒⠠⡆⡆⡔⣶⡀⠀⠀⠀⢖⢀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠛⢷⠳⡸⢳⠀⣘⠞⠣⣇⢣⠋⠇⢠⠆⠀⠀⠃⠀⢰⡿⠁
⠀⠐⠁⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⡀⠀⠀⠑⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠤⠖⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠁
⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠞⠱⠀⠠⡀⡆⠀⠀⢀⣠⠏
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣰⢁⡀⠀⠀⠀⠹⡧⣀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠈⠄⠀⠀⠀⠄⠀⠀⠀⢄⠀⠀⠀⠰⠗⠀⠀⠀⣞⡀⠈⠉⠄⠀⠀⠀⠀⠀⢀⠀⠀⡃⢀⡤⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠢⠄⠤⠀⠀⠀⠀⠀⠀⠀⠐⣃⠀⠀⠀⠀⢈⠁⠀⠁⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠛⠛⣃⠀⢟⠀⠸⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠰⠀⠒⠃⠀⠦⠰⡄⠠

⢰⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣤⡀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀
⢸⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢦⣷⣿⣿⣿⣟⣿⡿⣶⣆⢀⡀⣄⠀⣠
⠸⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠐⣿⣿⠉⠻⣿⣿⣿⠟⠉⠻⣿⣿⠟⢹⣿⣿⠀⠀⠈⠻⣿⣿⣿⣯⣹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠀⠀⣿⣿⣧⡀⠘⠿⠁⠀⠀⠉⣻⠏⣠⣿⣿⣿⠀⡄⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⡟⠛⠛⠛⢦⡀⠠⣼⣠⣠⢟⣴⠟⠛⢿⣿⠀⣷⣦⡀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠹⠆⢈⡁⣫⣾⣿⡂⣶⣿⣿⠀⡿⣿⣿⣷⣦⣄⡀⠀⠀⠉⠛⢿⣿⠃
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⣴⣎⣽⣿⠿⣿⡇⠿⠛⢿⢸⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⣙⡯⣷⣾⡟⠀⣤⠀⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣷⠀⠀⠀⡇⠀⠀⠀⠀⣼⣞⣿⣿⣿⣿⣾⢯⣿⣷⣿⢸⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣶⣤
⢀⠀⢀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⠀⠀⠀⣷⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⢸⣿⣿⣿⠿⢛⣥⡄⠀⣸⣿⣿⣿⣿⣿
⣽⣸⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠀⢰⣿⠀⠀⠀⣿⣿⣶⣀⢰⣿⣻⣯⣿⣿⣿⣿⣿⣿⣿⣿⢘⣿⡿⢁⣠⣾⡿⠇⣰⣿⣿⣿⣿⣿⣿
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣼⣿⣀⣤⠀⠿⠿⠿⠿⠿⠷⠾⠿⠿⠟⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣊⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⢃⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡿⡹⠂⠀⠀⠀⠀⠀⠀⢀⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⡠⠀⠀⠀⠀⣀⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢛⡡⢀⠀⠀⠀⠀⡅⠑⠄⠈⡿⢋⡠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣟⣂⡁⠀⠀⣦⣄⠱⡀⢨⡔⢠⡁⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⡿⢐⡀⠀⠀⣿⣿⡆⠈⢹⠗⢁⣀⡀⠀⠀⢀⠠⠴⠀⠀⡀⠌⢒⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠧⣀⠀⠰⣦⠀⠀⠀⠀⢐⣿⠄⡃⠀⠀⢈⠙⢇⠀⡎⣴⣿⣿⠀⢀⠒⡁⠀⣠⠔⢁⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣑⢄⠀⠆⠀⢠⣄⠘⢽⡆⡈⠀⠀⢀⠉⢢⢱⣸⣿⠟⠁⢀⠂⠀⠀⣰⠕⠊⠁⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠳⣜⠀⢘⠋⠀⡒⣶⡡⠀⠀⠀⢱⠞⠚⠓⠠⠄⣀⣌⠀⠀⡞⠁⣀⣀⡔⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢄⡀⠀⠈⢃⠠⠀⢰⣶⠻⣌⠅⢀⠔⠁⠀⠀⠀⠀⠀⠀⠈⠑⢞⠁⠀⡠⠊⠀⠠⠊⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡈⠁⠐⠚⢳⡀⠀⣆⠬⠛⠞⠁⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠈⠇⠛⠤⡀⠀⣃⣠⢊⣀⡔⠈⠉⢁⠝
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠙⠡⢄⡀⠀⠀⢣⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠈⠂⣀⠄⠀⠀⠀⠙⠻⢤⠀⠠⣁⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡉⠐⠄⡈⠉⡰⠋⠀⠀⠀⡠⠒⠉⠉⠉⠀⠒⠇⢀⠤⠤⡀⠀⠈⠂⠀⠀⠀⠀⠀⠃⠱⢄⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⣀⢘⡏⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢀⡠⠴⢁⠀⠀⠈⠆⠀⢸⠃⠢⢄⠀⠀⠘⠐⢖⠒⢤
⠀⠀⠀⠀⠀⣀⠄⠐⠐⢒⠀⠒⠨⢟⡁⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡎⠀⠀⠀⢡⠀⠀⠸⢀⠾⠀⠀⠀⠑⣀⢏⠑⠌⠑⠀⠃
⠀⠀⣠⠖⠁⠈⠢⣀⠀⠀⠱⣀⠀⠀⠈⠑⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⡀⠀⠙⠁⠀⠠⠋⠁⠀⡇⠀⠀⠀⠘⠀⠑⠼
⠀⠜⡡⠔⠒⠈⢉⣉⡭⠭⠒⠒⠳⢤⣤⡀⠵⠀⠀⠀⠀⢂⠀⠀⠀⢀⠔⠚⢄⠀⡠⠊⠀⠀⠀⢠⠁⠀⠀⠀⠀⡗⠀⠀⣀
⡼⠬⠄⠀⠠⠖⣉⠀⠀⠀⢀⠤⠂⠀⠀⠀⣀⠕⣄⠀⠀⣀⠟⠐⠂⠉⠣⡀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠘⠀⣤⠴⠏
⠀⠀⠀⠀⠀⠀⠀⠉⠐⠐⠃⠀⠀⠀⠐⠈⣔⣪⡗⠟⣍⠀⠀⠀⠀⠀⠀⠈⠒⠠⠤⠄⠤⠊⠁⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣲⠾⢭⠆⠈⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠆⠀⠒⠒⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡨⠟⡈⠄⠀⠀⡠⠒⠉⠑⢄⠀⠀⠀⢀⠠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢾⠑⠐⠁⠀⢠⠊⢀⡀⠤⣲⠋⠀⠈⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣂⠿⢠⠁⠀⠀⢰⠋⠁⠀⢀⠜⠁⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⡗⠄⠉⠀⠀⠀⡌⠀⡀⠰⠃⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⢔⣸⡗⡄⠀⠀⠀⠀⠀⡏⠁⠀⠏⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⢠⣫⡟⠉⠁⠀⠀⠀⠀⠀⠃⠀⠰⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠠⣈⠌⠁⠀⠀⠀⠀⠀⠀⠈⠀⡆⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠀⠠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠠⣀⠔⠁⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠑⢄⡴⢄⡀⠀⢀⠜⠁⠀⠀⠀⠈⡄⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠠⠙⠋⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠒⠚⠒⠒⠒⠒⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⠒⠒⠒⣚⣓⣒⣒⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡉⠀⢀⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⠉⠉⠉⠉⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠎
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠉⠲⢄⢀⠔⠉⠀⠀⠈⡆
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⣀⡀⠤⠒⠊⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⡀⣀⣀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠢⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠉⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠒⠒⠒⠀⠐⠚⠃⢢⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣁⣛⣛⣛⣛⣛⣛⣛⣛⣓⣒⣒⣒⣂⣀⣀⣀⣀⣀⣀⣀⢀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠈⠉⠉⠉⠀⠒⠖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠁⠈⠉⠛⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣆⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⠁⠈⠉⠻⣿⣿⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣠⠖⠋⠀⣀⣤⣴⣶⣶⣶⣄⠈⢿⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣶⡾⠋⠁⣠⣴⣿⣿⠿⠛⠉⠁⠀⠈⠀⠀⠈⠠⢤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⣫⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠄⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠀⡀⠀⠈⠙⠳⠄⠀⠀⠀⠚⠉⠀⠀⣠⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⢼⣦⣤⣀⣀⠈⠀⠀⠀⠠⣤⣴⣾⠿⠂⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⡆⠀⢡⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⢀⠆⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⢣⡀⠀⠡⡀⠀⠠⠶⠶⠶⠄⠀⢀⡞⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠈⠀⠀⠑⢄⠀⠓⠒⠊⠀⡴⡟⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠑⠠⠤⠔⠉⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⡶⢦⡀
⠀⠀⠀⠀⢠⠞⡿⣷⡿⢫⣶⢰⡏⡾⠁⠀⠀⠀⠀⠉⠙⠛⢷⣝⣆
⠀⢠⠞⢠⠏⡜⣱⡿⢁⣿⠇⣿⠀⡇⡆⠀⠀⠀⠀⢀⣠⣤⠀⠈⢺⡆
⠀⠃⠀⡞⠀⠀⣿⠁⢸⡟⢹⡇⠀⣇⢱⡀⠀⢀⠞⠋⠁⠀⠀⠀⠀⠙
⠀⠀⢰⠇⡀⢸⡇⠀⣿⡇⢸⡇⠀⢿⡄⢳⡀⠊⠀⠀⠀⣀⣤⣿⡾⠄
⠀⠀⣾⠀⠀⢸⠇⠀⣿⡇⠀⡇⠀⢸⡷⡄⠹⣄⠀⠈⠛⠛⠉⠉⠀⠀⠀⠀⠀⠁
⠀⠀⣿⠀⠀⢸⠀⠀⢻⣧⡀⢱⠀⠀⢷⡹⣆⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⡀
⠀⠀⣿⠀⠀⢸⠀⠀⢸⣿⣇⠀⠄⠀⠈⢧⠘⣧⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠹⠋
⠀⠀⣿⠀⠀⠘⡇⠆⠈⣿⣿⡄⠈⢆⠀⠈⢧⡘⣧⠈⢇
⠀⢠⣿⢰⢀⠀⣇⠰⠀⢻⡿⣿⣄⠈⢂⠀⠈⠳⡜⣆⠈⠄⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠿⠃⣼⢸⡀⢻⠀⡆⠘⣷⢣⠻⣦⣄⠑⢄⠀⠙⢿⡄⢠⣀⡀⠀⢀⣠⡴⠚⠁
⣶⠶⠿⠃⢸⡇⠈⡀⠸⡀⢹⣧⡳⡈⠻⣷⣦⣑⣤⡀⠙⢾⣿⣿⡿⠛⠁
⠈⠲⣤⣴⡿⠃⠀⢳⡀⢱⠈⣿⡷⣜⢦⡈⠻⢿⣿⣿⣶⣄⢻⡏
⢤⡀⠀⠀⠀⠀⢠⣿⢳⡀⢇⠸⣿⣌⠳⣌⠳⢤⣙⠾⣝⠿⣷⣇
⣶⣝⠳⠶⠶⢞⣻⣿⡎⢧⠈⡄⢻⢿⣧⡈⠻⢦⣉⠓⢮⡳⡌⢿⡄
⠻⢿⣿⣷⣾⣿⣿⢿⠇⠀⠀⠰⠸⣯⢻⣿⣦⡀⠉⠳⣄⠱⡙⡌⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣀⣀
⠀⠀⠈⠉⠉⠉⣁⡞⡼⠀⡀⠀⢠⣿⢦⠹⣿⣿⣦⡀⠘⡆⠠⠁⢸⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠋⣉⣉⣩⣉⣙⡛⢷⣦⡀
⣿⣶⣶⣶⣶⣞⣯⠞⠁⣠⡇⠀⢸⣿⡈⢧⡘⢿⣿⣿⡆⠀⠀⠀⠈⢂⠱⢄⠀⠀⠀⠀⠀⢀⣀⣤⠾⠋⠁⣀⣤⣶⠿⢛⣛⣿⣿⣿⣿⣷⣌⢷⡀
⠈⠻⢮⣭⣉⣉⣠⣤⣾⡟⠀⡌⣾⣿⣇⠀⠻⣄⠙⣿⣿⠀⠀⠀⠀⠀⢧⡀⠉⠒⠒⠒⠛⠋⠉⠀⣠⣴⠾⢛⣩⠴⠛⣋⣉⣁⣀⡀⠉⠙⢿⡆⢣
⠳⣦⣀⠉⠛⠛⠛⠛⠉⣠⢞⣼⣿⣿⣿⣆⠀⠘⢆⠘⣿⠀⠄⠀⢸⡇⠈⠻⢦⣤⣄⣠⣤⡴⠶⠛⠉⣠⠴⠋⣡⡶⣛⣩⣭⣭⣭⣙⠳⣄⠀⣿
⠀⠀⠉⠛⠶⠶⠶⣶⣿⣵⡿⠿⠛⠛⠛⠿⣦⡀⠈⠆⢹⠀⠁⠀⢸⣹⡀⠈⠲⢤⣀⣀⣀⣀⣤⠴⠊⢁⣴⣾⠗⢋⣁⣀⠀⠀⠀⠈⠑⢌⡆⣿
⠙⠛⠓⠶⠖⢛⣛⡿⠟⠁⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⡎⠀⠀⠀⢸⣇⠳⣄⠀⠀⠀⠈⠁⠀⣀⣤⡾⢟⡫⠖⠊⠉⠉⠀⠉⠐⢄⠀⠀⠈⢧⢸
⣾⣶⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠃⠀⠀⠀⠘⣿⣆⠈⠛⠲⠶⠶⠾⠛⢋⣩⠖⢋⡠⢖⡢⠬⠤⢀⠀⠀⠀⢣⠀⠀⠘⢸⣇⠀⡴⣶⣶⣄
⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡾⠀⠀⢠⠀⠀⠹⣿⣷⣤⣀⣀⣠⣤⣾⠟⣡⠴⡫⠚⠁⠀⠀⠀⠀⢳⡀⠀⠀⢦⠀⠀⠀⣿⣄⡀⣨⣿⣿
⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡼⡅⠀⢀⠈⢧⠀⠀⠈⠛⠻⠿⠿⠛⠉⣠⠾⣫⠞⠀⣠⣶⣿⣟⠀⠀⣸⡇⠀⠀⠘⣷⡀⢳⣌⠛⠿⠿⢛⡽
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⢀⣇⠀⠈⢧⠈⠳⣦⣄⣀⣀⣀⣠⡴⠞⣡⠞⠁⢀⡾⢹⢻⠻⣿⣿⣿⡿⠃⣰⠀⣸⣿⠟⢦⣭⣻⣶⣾⠟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠡⠀⢸⣿⡄⠀⠈⠳⣤⣀⡉⠉⢉⣉⣡⡴⠟⠁⠀⣠⣿⡇⠸⡌⠳⣤⣉⣉⣤⠞⠁⣴⡿⠋⣠⣾⣿⣿⣥⣤⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠙⢰⠋⣳⠰⢯⡃⣿⣦⣿⢸⡯⠅⢹⡏⢰⠋⢳⢸⡇⠀⡞⢙⡆⡞⢩⡃⢳⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠈⠛⠋⠘⠛⠁⠉⠀⠉⠘⠛⠃⠈⠁⠈⠛⠋⠈⠛⠃⠙⠛⠁⠙⠛⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠚⠉⠙⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⣠⡄⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣤⠤⠤⠶⠒⠊⠁⠙⢦⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣇⡇⢠⢖⢦⡀⢠⢖⣦⡀⣯⣻⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⡇⡇⠘⠮⠞⡤⠘⠪⠞⠁⡏⡹⠈⠙⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⠁⠀⠀⠀⠀⢻⢻⡘⠟⠀⠙⠂⠘⠟⢠⢟⠇⠀⠀⠀⠀⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢢⡀⠀⠀⠀⠀⠀⠑⠷⣌⠀⠉⠁⢈⡱⠟⠋⠀⠀⠀⠀⣀⣠⣬⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠚⠉⠀⠀⣠⡤⣤⡴⢊⣹⠛⣗⣮⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠿⡄⠀⠀⠀⢶⡶⠶⠶⠶⣷⠀⠀⠀⠀⢀⢞⡧⣉⠓⠶⡭⠚⢣⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⡇⠀⠀⢀⠞⠈⠉⠉⠉⠙⣆⠀⠀⢠⣿⣋⠴⣫⣷⡀⠀⠀⠈⢿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⠀⡇⠀⠀⢸⠛⠛⣻⢿⡛⠛⢻⠀⠲⣞⣩⠝⠿⣝⡇⠱⢤⠀⠀⢸⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠶⡇⠀⠀⢸⠀⠀⢧⣤⠇⠀⢸⠀⠀⠈⣠⠎⠀⠀⠈⠳⠤⢤⡀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⠷⠀⠀⢸⠀⠐⣖⠲⡖⠀⢸⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⢸⣉⣉⣉⣉⣉⣉⣻⠲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠷⡄

⠀⠀⠀⠀⠰⠁⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠈⠀⠠⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⡀⠠⠀⠀⢀⠆⠀⠀⠀⠀⠄⢀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠐⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠁
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡑⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠂⢠
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠁⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⣸⡅⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣄⣯⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣿⠉⠛⠛⣄⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⡏⠙⠛⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣾⣿⣷⣷⠄⠀⠀⠀⠀⣠⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠏⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⣼⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠋⠀⠀⠀⠀⣴⣿⣿⣻⣿⣿⡿⠁⠀⠀⠀⠀⢀⣼⢿⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠈⠉⠿⣿⣿⡿⠁⠀⠀⠀⠀⢀⣾⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠁⢀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣥⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣻⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠏
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠠⠂⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⠣⠒⠤
⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡇⠀⠀⠀⠀⡠⢂⠀⠀⠀⠀⠀⠀⠀⠈⠒⣄⡀⠀⠄⠐⣦⡀
⠀⠀⠀⢉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠰⢸⠁⠀⠀⢀⡜⡡⠃⠀⠀⠀⠀⠀⠀⠀⠖⠑⠀⠀⡀⠂⢀
⠀⠀⠀⠀⠁⠒⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠏⠀⠀⢀⠎⡐⠁⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⢸⣶⣶⣾⡇
⠀⠀⠀⠀⠀⢠⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣸⠁⠀⡰⢃⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠉⠉⣇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡌⢀⡜⡡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠌⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢣⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣽⠂
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡠⠂⠀⢀⡤⢖⣿⢁⣴⣾⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣵⠋⡠⣪⠞⠉⣰⣿⢇⡿⣫⣽⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣫⡟⣡⢾⠞⢁⣠⣾⣿⠏⣼⣿⡏⠈⠉⣿⣷⣦⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣳⡟⣴⡷⢃⡴⣻⣿⢿⣷⣾⣿⣟⣀⣀⠀⢘⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡿⣰⠟⣱⣯⣾⣿⣷⣿⣿⠿⠟⠉⠉⠙⠃⣾⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⡇⢃⣾⢋⣼⣿⣿⠿⣫⠗⠁⠀⠛⠛⠃⠀⠙⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣾⢁⣾⣿⡟⢁⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⡇⠃⣾⣿⣿⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⢿⡀⣿⣿⡝⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⡇⣿⣿⣷⠹⣿⡆⠈⢶⣤⣄⣀⡀⠀⠀⣸⣿⣿⣿⣿⡿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⡏⣿⣿⠀⣿⡷⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⢸⠟⢰⡿⠃⠀⣠⣿⣿⣿⣿⣿⣿⣿⣟⡽⣫⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢁⠏⣰⠟⠁⣠⣾⣿⠿⠟⠛⠛⣻⠽⣺⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢃⣏⡾⠁⠀⠀⠋⠁⠀⠀⢀⣰⠽⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣀⣀⣀⣀⣀⡀
⠀⠀⠀⣠⣴⣿⣿⠿⠿⠛⠛⠪⡉⠉⠛⠛⢿⣿⣿⣷⣶⣤⣄
⠀⠀⠸⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠛⠿⣿⣿⣿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⡿⣾⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣶⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⣤⣤⣶⣶⣧⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⠿⠿⠟⠋⠁⠀⠉⠙⠄
⠀⠀⠀⠀⠀⢤⣤⣾⣿⣿⣿⡿⣿⡿⠿⠿⠿⢿⡿⢶⣽⠔⠢⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠛⣍⠁⠀⢿⡄⠰⠾⠃⣸⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⣒⣶⣶⣶⣆⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⡈⠻⠓⠶⠞⠋⠀⠀⠀⠀⠰⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣉⣽⠙⣿⠈⠙⠻⢿⣿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠃⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡘⢧⣍⣡⡼⠃⠀⡀⠄⢠⡝⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠌⠤⠤⢴⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠉⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⡀⠔⡰⠃⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠉⠀⡴⠁⠈⠉⠳⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢂⠖⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠄⠒⠀⠐⠣⡰⠀⢀⡔⡀⠁⣰⢃⡔⢠⠂⢀⡔⣠⠊⠀⠀⠲⣄⡔⣀⠃⣼⠀⣠⢆⠁⢠⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠉⠀⠁⠈⠁⠈⠁⠀⡥⠧⠀⠀⠈⠀⠀⠁⠉⠀⠉⠁⠈⠁⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠠⢲⣆⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡄⠀⢈⣿⣿⣇⢀⡀⠀⣿⡆⠀⣽⡆⢀⣴⣿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⠹⣿⣷⡀⢸⣿⣿⣿⣼⡇⠀⠈⣿⡄⠸⡇⠈⠛⢿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠛⠷⠿⠋⠁⠀⢿⣿⣿⣿⡇⠀⠀⠉⠳⡀⠀⡼⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠉⠁⢀⣶⣦⠀⡔⠂⠒⠤⡀⠈⢻⣿⣿⡀⠀⠄⠀⠀⢣⠀⠁⠀⠀⠀⠀⣙⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⣀⣩⠥⠐⠀⠀⢛⡉⠁⠀⢄⠀⠀⠀⠣⠀⡄⠀⠀⣴⣿⣿⡟⠉⢉⣠⡴⠾⠃
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢀⣾⣿⠏⢀⣴⣶⠟⠉⠀⢀⣤⠀⣾⡯⢿⣦⠀⠀⠳⣄⠀⠀⠀⡇⠀⣾⣿⣿⠏⠀⢀⣨⣴⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣿⠿⠁⠀⠀⠐⠁⣠⣴⠾⠛⠉⠀⠘⣷⡞⣿⣆⠀⠀⠹⠃⠀⢀⡇⢰⣿⡿⠋⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢰⡏⠀⢠⡞⠁⠀⠀⠀⢠⣶⣿⠿⠀⠀⠀⠀⠀⠀⠘⣿⣫⣿⡅⠀⠀⠀⠀⢸⡇⠸⠛⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⡠⠉⡇⣴⠋⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢀⣰⣿⣵⠀⠀⠀⠛⢻⡅⠀⠀⠀⠐⣾⡇⠀⠀⠀⡀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⠁
⠀⠀⢠⠊⢃⡐⠃⠀⢀⣾⠇⠀⣀⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⡯⠀⠀⠀⠀⢻⣆⠀⠀⢀⣿⣿⡀⠀⢈⡇⠰⠶⠶⠤⠙⠛⠛⠛⠋⣠⡄
⠀⠀⠀⠀⠀⢢⠀⢀⣾⡟⢰⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⣤⠉⠀⠐⣿⣿⣅⠀⠻⣆⠀⠸⣿⣿⣿⠂⠸⠟⠀⠀⣀⡤⢠⣄⠀⠀⣼⡿⢃⣴⡖
⠀⠀⠀⠀⠀⠈⣀⣼⡟⠀⠸⣿⣿⣿⣿⣿⠟⠁⠀⠀⣼⠋⠀⠀⠀⠻⣶⡏⠀⠀⠡⠀⠄⡀⠈⠁⠀⢀⣠⡴⠚⠁⠀⣸⣿⣆⣾⣿⣷⣿⣯⣶⡗
⠀⠀⠀⠀⣠⢚⣿⣿⠂⠀⠀⠈⠙⠻⣿⣷⣆⠀⠀⠰⠁⠀⠀⣤⣀⠀⠘⣧⠀⠀⠀⣀⡀⣀⡠⢴⡾⠛⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣭⣦
⠀⠀⠀⠀⠁⢨⣿⠃⠀⢀⡤⠀⠀⠀⢻⣿⣿⡅⠀⠀⠀⢀⣼⣿⠏⠀⠀⢹⡀⠀⢊⣀⡝⣁⡤⠋⠀⠀⣄⢿⡆⠀⣼⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⣼⣿⠃⠀⣠⠏⠀⠀⠀⠀⠀⢻⣿⣷⠀⠀⢀⣾⣿⠋⠀⣤⣄⠀⠇⠀⠀⢸⠁⡸⠀⠀⣦⡀⠈⢿⣿⠈⠻⣿⣿⣿⣿⡿⢋⣁⡀
⠀⠀⠀⢺⣿⠃⢀⣾⠃⠀⢀⣤⡶⠀⠀⠀⠹⣿⠀⠀⣹⡿⠧⠀⣼⣻⠋⠀⠀⠀⠐⡁⠀⣤⡀⠀⢻⣷⠀⠀⢿⡆⠀⠀⠙⠿⠋⢠⣾⣿⠟⠆
⠀⠀⠀⠀⠀⣠⣿⡇⢀⣴⢿⣿⣷⣄⠀⠀⠀⣀⠀⣰⠏⠀⠀⣸⡿⠁⢰⠦⡀⠀⠀⢿⣧⡀⠙⠦⠈⢿⣇⠀⠘⡇⠀⠠⢄⠀⢠⣿⣿⡃⠂
⠀⠀⠀⠀⠐⠟⠹⠀⠈⢹⣿⢦⠀⠈⠁⣠⡾⠊⡴⠃⢠⣄⡀⠉⠀⠀⢂⠐⠘⡄⠀⠀⠙⣿⣦⠀⠀⠘⣿⡄⠀⠃⠀⡅⢰⢠⣿⡿⡕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣶⣦⡈⠁⠀⠘⢁⣀⠀⠈⠉⠑⠀⠀⠈⠆⠂⠐⠀⠀⠀⠈⠻⣷⣄⠀⢹⣿⡀⠀⠀⠓⢁⣼⠿⠯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡄⠠⣦⣈⢿⣧⡀⠀⣤⡄⠀⠀⠀⢈⣠⡄⠀⠀⠈⠠⠘⢿⣦⠀⠻⣧⠀⠀⢠⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠿⠀⣶⣝⣿⣷⣿⣿⣄⣿⣷⠀⠀⠀⢸⣿⣿⡆⠀⠢⣄⠀⠀⠙⣷⡀⢻⡆⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⣬⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢹⣿⣷⠀⠀⠈⠳⣦⡀⠈⠻⣮⣟⡀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣼⠇⢿⣿⣧⠀⠀⠀⠘⡿⣦⡄⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⠟⠁⢰⢻⠀⠈⣿⣿⣇⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠛⠃⠀⠀⠘⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠛⠓⠶⡜⣃⣤⣤⠤⢄⣀⣠⡞⣫⣥⣶⣦⣄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⢀⡈⠀⢰⣾⡟⡿⢁⣾⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⣠⣾⣿⣿⣿⡘⢿⣧⢧⠸⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⣰⣿⣿⣿⣟⢿⣿⣦⣙⢿⣷⣝⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⣿⣿⣿⣿⢿⣷⣝⣿⣿⣿⣾⣿⣿⣿⠿⠿⣿⣿⣿⣟⣋⣵⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⣆⠀⣿⣿⣿⣻⣷⣽⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠈⠉⠉⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⣿⡀⢹⣿⣿⣿⠿⢿⣿⠟⠉⠉⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⢿⣧⠀⢻⣿⡆⠀⠈⠁⠀⣀⣄⣀⣀⣀⠉⠻⠶⠀⠀⢀⣀⣰⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⢸⣿⡆⠈⢿⣿⡀⠀⠀⠀⠘⢿⣟⡻⠿⣿⠀⠀⢀⣀⣉⣠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣿⣇⣿⣿⣧⣿⡇⢸⡎⣿⣿⣆⠀⠀⠀⠀⠈⠋⠋⠀⠀⠀⠞⣻⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣯⣿⣿⡇⣼⣧⣿⣿⣿⢧⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢰⢿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢀⡤⠤⡙⣿⡿⠛⠓⢿⣿⣿⢹⣿⠀⠙⢦⣄⠀⠀⠙⠷⠒⢀⣴⣯⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣴⡏⣾⣷⡇⡸⢰⣿⣿⡌⣿⡿⣾⣿⡀⠀⠀⠙⢷⣄⣀⣡⣶⣿⣿⣿⢟⣩⣭⣀
⠀⠀⠀⠀⢰⣿⣷⣙⣛⡡⢣⣿⣿⡿⢠⣿⣿⣿⡟⣧⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣾⣿⣿⣟⣵⣿⣿⡿⠋⠀⠈⠆⠀⠀⠈⠻⢿⣿⣿⣿⡿⣿⣶⡾⣋⡾⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠁
⠀⠀⢀⣀⣀⣀⢀⡀⣀⣀⣀⣀⣀⣀⠀⢀⡀⣀⣀⣀⣀⠀⣀⣀⢀⡀⣀⣀⣀⠀⣀⡀⢀⡀⣀⠀⣀⣀⣀⡀
⠀⠀⢸⡿⢼⣯⡼⢿⢿⡿⠹⡏⠹⡏⠀⡿⡧⡿⡿⢿⡿⠀⣿⡿⡾⠷⣿⢿⢿⠀⢽⡷⡿⣧⣿⡶⣿⠿⡿⡇

⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀⢀⣀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠘⣇⠀⣸⠇⠀⠀⠀⠀⠀⡇⢸⠘⠀⠀⠀⠀⠀⠐⣧⣤⣿⠀⠀⠀⠀⠀⡟⠹⣿⣿⣷
⠀⠀⠀⠀⠀⠙⢀⡟⠀⠀⠀⠀⠀⠀⠃⢸⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⢣⣄⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠸⠭⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠿⠟⠋

⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⣩⠶⡀
⠀⠀⠀⠀⢀⣾⣷⣄⠀⠀⠀⠀⠀⠀⠐⡴⠁⠀⠀⠀⠀⠀⠀⠀⠹⠛⣿⠀⠀⠀⠀⠘⢾⣁⠔⠁
⠀⠀⠀⠀⢿⣅⣀⣿⠂⠇⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠠⣿⣷⣶⣶⣶⡄⠀⠀⠀⡎⠻⣿⣟⣦⠀⠀⠀⠀⢰⣋⣹⠴⠀⠀⠀⠀⠀⣾⣇⡆⣤⡤
⠀⠀⠀⠀⠀⠀⠙⣿⠟⠃⠀⠀⠀⢠⣄⣿⣿⡟⠀⠀⠀⠀⢰⡏⢻⢸⠀⠀⠀⠀⠀⠘⣿⣟⣿⡷
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠟⠛⠛⠙⠧⠀⠀⠀⠀⠸⣧⡼⠸⠀⠀⠀⠀⠀⠀⠛⠿⣿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⡄⠀⠀⠀⠀⠚⣥⠀⢠⡙⠀⠀⠀⠀⠀⢽⢵⠇
⠀⠀⠀⠀⢠⣤⡴⠂⠀⠀⠀⠀⠀⢠⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣝⡅
⠀⠀⠀⠀⠈⠟⠁⠀⠀⠀⠀⠀⠀⠘⠋⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁

⠀⠀⠀⠰⠀⠀⡀⠀⡀
⠀⠀⠀⠀⣄⣼⡾⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠂⠀⠀⠄⠠⢀⡀⠐⢀⠄⢀⡀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠪⠓⠄⠀⠀⢲⠃⠔⠃⣄⣀⠀⢡⣰⣿⠆
⠀⠰⠀⠀⠀⣜⡀⠀⢤⡿⣋⣵⠟⠋⢙⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢
⠁⡲⠃⠠⣰⣧⡖⢣⢿⡟⠃⢀⠔⠂⢦⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⢰⠁⠀⢡⠇⠀⢠⢇⡞⠀⠀⢨⠀⢾⠚⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⢰
⡏⠀⢠⠇⠀⢠⡿⠸⡇⠀⠀⠈⠢⠀⢀⣠⣤⡀⠀⠀⢰⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⡄⠐⠈⠉⠲⡆
⠇⠀⠀⡎⠀⠐⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⣠⠤⠄⠀⠘⠡⡦⠼⠅⢀⠀⠀⠺⡷⠀⠄
⣆⠪⢠⠁⠀⢠⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⢀⣭⢇⡀⡠⠀⣦⠊⠀⠀⠀⠀⠑⢂⠀⠉⠀⠰
⡇⠀⠃⠀⠀⠸⡃⣄⠀⠠⠘⠢⢀⡀⠀⠀⠀⠀⢀⣀⡤⠐⠋⠉⠀⠀⠘⢆⢀⣾⠀⠀⠀⠀⠘⠐⠋
⡇⢀⠀⠀⠀⠀⠿⡀⠀⠀⠀⠀⠈⠉⠉⠀⠒⠺⠉⠀⠀⠀⠀⢀⠠⠀⠀⡈⠇
⠹⠁⠀⠀⠀⠀⠀⠹⢮⡁⠀⢀⡄⠀⠀⠂⠤⡀⠀⠀⠀⡀⠀⠁⠀⠒⢢⡉⡆⠀⠁⢀⠀⠀⠀⠀⢀⡤⠤⠀⡀
⠀⠁⠀⠄⠀⠀⠀⠀⠃⠙⠲⣦⣁⣁⣈⠐⠉⢀⡀⠄⠊⠊⠀⠀⠀⠀⠈⢳⣧⠀⠠⠀⠒⠄⠀⠁⠀⠂⠀⠀⠈
⠀⠀⣧⠀⠀⠀⠀⠀⠈⠂⠀⠀⠉⠉⠉⠙⢡⠋⠤⠀⠀⠀⠀⠀⠀⠁⠀⠠⢿⡄⠀⠀⠀⠀⢠⡄⠉⠽⠤⢥⡀
⠀⠀⠘⢷⡐⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠈⠀⠀⠀⠀⢀⡀⠀⠈⠀⠀⠀⠀⠠⣀⠀⠈⠁⠀⠀⠀⠀⠀⠈
⣀⠀⠀⠀⠹⣄⡀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣝⠂⠒⠒⠂⠈⠀⢢
⠀⠀⠀⠀⢀⠀⠙⠦⣈⣦⣀⠀⠠⢀⣒⢤⡤⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢨⡤⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣪⣭⣿⣽⣿⣿⣷⣠⣾⣿⣿⣿⣿⣭⣙⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠿⠛⣿⣻⣿⣿⣶⣤⣌⠉⠻⣿⣧⡘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣻⣿⣿⠋⣠⣴⠶⡟⠋⠀⠈⠹⣿⣿⣇⢠⠹⣿⡇⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⡾⠋⠀⠀⠁⠀⠀⠀⣠⣿⣼⡟⢸⢠⣿⣧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠀⠀⠀⠤⡀⠀⠲⠾⡿⠿⢛⡱⢃⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⢀⣀⠀⠈⠀⠀⠈⠀⣩⣵⡾⠿⠟⠋⠁⠲⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠁⠀⠀⠀⠀⠀⣾⣿⣏⣤⢤⣀⠀⢭⢻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⣿⣿⢧⣼⠇⣿⣧⢸⣧⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠀⠀⠀⠀⠀⠘⢿⣷⣶⣾⣿⢏⣾⠇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⠉⠉⢙⣡⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠻⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⡹⣧⡸⣷⢰⣄⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠞⣣⣿⠇⢀⣾⣿⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠻⢯⣥⣴⢿⣿⡟⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣀⠒⠚⢛⣡⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡌⠻⠿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠄⠠
⠀⠢⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⣴⣶⡄⠀⠹⣧⠀⢶⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣿⣧⠀⣿⣿⣧⣿⠀⢻⣆⢸⠀⠻⢿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠞⠋⠙⠚⠋⠁⠸⣿⣿⡿⠀⠈⠙⡄⢠⠃⠀⠙⢿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢀⠀⠉⠀⢠⣾⠗⠐⠡⠈⠐⠀⠹⠿⠇⠀⠀⠀⠐⡀⠀⠀⢀⣴⣿⡿⠿⠛⣠⡄
⠀⠀⠀⠀⠀⡞⠀⠀⣴⡿⠋⣠⡴⠚⠉⢀⡀⣴⣺⣦⠀⠑⣄⠀⠀⢰⠀⣾⣿⡟⠀⣀⣭⣥⣤⣤⣀
⠀⠀⠀⠀⢸⠃⠀⡸⠟⠁⠀⠈⣠⡴⠞⠋⠁⠘⣷⣻⣧⠀⠘⠃⠀⣿⢰⣿⠋⠀⣼⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠠⠿⢀⡞⠁⠀⠀⠐⠿⠛⠁⠀⢠⣄⠀⠘⢻⣿⠂⠀⠀⣀⣿⠈⠁⠀⠀⠘⢿⣿⣿⣿⣿⡿⠋
⠀⠀⠐⣃⠶⠉⢠⡦⠀⣀⣀⣤⣤⣴⣶⣿⣿⡆⠀⠀⠘⣧⠀⠀⣸⣿⡀⠀⣧⠠⠤⠌⠛⠿⠿⠛⣁
⠀⠀⠀⠀⠄⢠⡿⢡⣾⣿⣿⣿⣿⠿⠋⢡⡌⠁⢸⡿⣧⠈⢧⠀⠻⠿⡷⠀⠛⠀⣀⠤⢠⣄⢀⣼⣟⣴⠆
⠀⠀⠀⢀⢤⣾⠁⠈⠻⠿⣿⣟⠁⠀⢠⠏⠀⠀⠀⠙⡇⠀⠀⠀⠀⠀⢀⣠⠴⠊⠁⠀⣼⣿⣾⣿⣿⣿⣿⡁
⠀⠀⠀⠃⣿⠏⠀⢀⠀⠀⠹⣿⣷⠀⠀⠀⣰⣿⠇⠀⢻⠀⠐⢀⠖⣈⠜⠁⢀⣶⡀⢠⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⢀⣿⠏⢀⡴⠁⠀⠀⠀⠹⣿⡆⠀⣰⣿⠏⢠⣤⠈⠀⠀⠨⠉⠃⠀⣦⠀⠻⣇⠙⢿⣿⣿⠟⣫⣄
⠀⠀⠘⠋⣠⡞⠀⣠⣾⣇⠀⠀⠙⠃⢀⡟⠉⢠⡿⢃⡀⠀⠀⣤⡐⢢⡀⢻⣇⠀⢻⠀⠀⠉⠁⣴⣿⠏⠁
⠀⠀⠀⠰⢿⠇⠚⣿⣏⠉⠓⢀⡾⢣⠏⣀⡀⠚⠁⢼⠉⢢⠀⠈⢿⣆⠈⠀⢿⡄⠘⠀⠌⠱⣸⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣹⣿⣶⣄⠁⠀⠃⣄⠈⠉⠑⠀⠈⣆⠀⢇⠀⠀⠙⢷⣄⠘⣷⠀⠀⠐⣠⡿⠳
⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⠇⡸⣶⣽⣷⡀⣶⡆⠀⠘⣴⣾⡀⢀⠀⠈⠻⣆⠘⣧⠀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⢀⣻⣿⣿⣿⣿⣿⣷⠀⠀⠘⣿⣧⠀⠙⢦⣀⠘⢷⣽⡆⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣧⠀⡾⠹⣿⣆⠀⠀⠹⡷⠄⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡿⠋⠁⣼⡇⠀⢻⡿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢠⠟⠛⠀⣿⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠄⡀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⡼⢻⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠙⠿⠟⠉⠿⠇⠀⠀⠀⠀⠀⠀⢠⠞⠃⡞⠻⣿⠇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⣘⠋⠉⠉⢀⠀⡀⠀⠀⠀⠀⠀⠰⢾⣿⣿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠁⣰⡿⢛⠁⢶⠀⠀⠀⠀⠀⠀⠻⣿⣶⣾⣿⠇⡀⠀⠀⠀⠀⠀⠀⢀⣻⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢰⣶⡜⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣷⠀⠀⠀⠀⠀⠀⢸⣇⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢳⡀⢰
⠀⠀⠀⠀⠀⠀⠀⢀⡹⣿⣻⣿⠏⠀⠀⠀⠀⠀⠀⠈⢹⡏⣴⡔⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠃



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⢁⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠐⠀⢀⠀⢞⠰⠋⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠌⡀⠀⠁⠀⠀⠀⠉⣂⣠⣤⠶⣟⣁⣀⠄⠀⡩⠋⠉⡁⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣢⠞⠀⠀⠀⠉⣙⡿⠟⠛⠛⠋⠈⢁⠀⠶⠉⠉⠔⠀⠀⠀⠀⠂⠀⠀⠀⠻⣿⡾⠁
⠀⠀⠀⠒⠻⠷⠤⠤⠖⠒⠋⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠉⠚⡇
⠀⠀⠀⠀⣠⠄⠠⣤⠤⠒⠐⠐⠒⠎⠀⠀⠠⠄⢀⠀⠀⠂
⠀⠀⠀⠘⠀⡶⣋⠁⠀⠂⠉⠀⠀⠀⠐⠈⠄⠀⢀⠀⠄⢁
⠀⠀⠀⠀⠈⠈⠀⠀⢀⠠⠄⠀⠀⠂⠀⠀⡁⠒⠦⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠘⠠⠊⠐⠉⠀⢐⡀⠀⠀⢠⣀⠀⠀⠀⠀⣸⠄
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢺⣦⣀⣠⠛⠩⠀⠆⠐⠀⠀⠀⠀⠛⠉⠀⢀⠀⣴⠃⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣷⣿⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⡈⠐⠀⠠⠈⢻⡀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣌⣿⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠅⠈⠚⠿⡏⠀⣀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠏⠀⠀⣸⢔⣱⣄⡀
⠀⠀⠀⠀⠀⠀⠒⡐⠁⠀⠠⣠⣠⠈⢉⠂⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⢤⡀⢳⣀⢰⠃⠔⠋⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⣿⣷⠀⠐⠀⣀⠀⠀⠁⠃⠀⠀⠀⠀⠀⣹⡄⣿⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⠀⠀⠀⣠⠂⠀⢂⠀⠀⠀⠀⠀⠀⠈⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉