Winking Emojis & Symbols

⠀⠀⠀⠀⢀⣴⣶⣶⣶⣦⡀⠀⠀⠀⣠⣶⣾⣶⣦⡀⠀⠀⠀⣠⣴⣾⣶⣦⣄⠀⠀⠀⢀⡴⣶⣷⡶⣄⠀⠀⠀⢀⣴⢶⣶⡶⣤
⠀⠀⠀⠀⣼⣿⣽⣿⣯⣿⣧⠀⠀⢸⣿⣭⣿⣿⣽⣿⠀⠀⢠⣿⣯⣿⣿⣽⣿⡄⠀⠀⣿⣧⣾⣿⣤⣿⡇⠀⠀⣼⣷⣽⣿⣧⣾⡧
⠀⠀⠀⠀⠸⣿⣟⣿⣻⣿⠇⠀⠀⠘⢿⣫⣷⣮⣿⠟⠀⠀⠀⢿⣜⡿⢿⣣⡿⠀⠀⠀⠻⣿⣾⣿⣾⡿⠃⠀⠀⠸⣧⣻⠿⣟⣼⠇
⠀⠀⠀⠀⠀⠈⠉⠛⠉⠁⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠈⠉⠛⠉⠁
⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⢠⣞⢿⣿⡿⣳⡄⠀⠀⢀⣾⢿⣿⣿⢿⣆⠀⠀⠀⣴⡿⣿⣿⢿⣦⠀⠀⠀⣰⡿⣿⣿⠿⣷⡀⠀⠀⢠⡾⢿⣿⡿⢷⡄
⠀⠀⠀⠀⢺⣿⣿⣛⡻⣿⡿⠀⠀⢸⣿⡟⠉⠙⣿⣿⠀⠀⠸⣿⣯⢿⡿⣿⣿⠂⠀⠀⣿⣿⢿⣿⢿⣿⡇⠀⠀⢿⣿⡾⣿⢷⣿⡗
⠀⠀⠀⠀⠈⠻⣾⣿⣷⠟⠁⠀⠀⠀⠻⢧⣄⣤⠿⠃⠀⠀⠀⠙⢿⣿⣿⡿⠋⠀⠀⠀⠘⠿⣧⣢⡿⠟⠀⠀⠀⠈⠻⣿⣿⣿⠟⠁

⠀⠀⠀⠀⠀⣠⣤⣶⣤⣄⠀⠀⠀⠀⢀⣤⣴⣦⣄⠀⠀⠀⠀⢀⣤⣴⣦⣤⡀⠀⠀⠀⠀⣠⣴⣶⣤⡀⠀⠀⠀⠀⣠⣤⣶⣤⣄
⠀⠀⠀⠀⣸⣏⣉⣭⣉⣹⣇⠀⠀⢰⣏⠿⣿⡯⢏⣷⠀⠀⢀⣿⣫⣿⣿⣝⣷⡀⠀⠀⣾⣿⣿⣿⣿⣿⡆⠀⠀⣰⡹⠦⣶⠰⢏⡆
⠀⠀⠀⠀⠹⣟⢿⣿⡿⣻⡏⠀⠀⠸⣯⢿⣿⡿⢯⡿⠀⠀⠈⣿⡻⣿⣿⢟⣿⠁⠀⠀⢿⣿⡟⠻⣿⣿⠇⠀⠀⢹⣿⡿⠿⢿⣿⡏
⠀⠀⠀⠀⠀⠙⠛⠷⠛⠋⠀⠀⠀⠀⠈⠛⠷⠟⠋⠀⠀⠀⠀⠈⠛⠳⠞⠛⠁⠀⠀⠀⠀⠙⠻⠿⠛⠁⠀⠀⠀⠀⠙⠛⠿⠛⠋

⠀⠀⠀⠀⢀⣴⣿⣿⣿⣦⡀⠀⠀⠀⣴⣾⣿⣿⣶⡄⠀⠀⠀⣠⡾⣿⣿⢷⣄⠀⠀⠀⢠⣶⣿⣿⣷⣦⠀⠀⠀⢀⣴⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢼⣿⣾⣿⣷⣿⣷⠀⠀⢸⣿⣶⣿⣷⣾⣿⠀⠀⢰⣿⣷⣿⣿⣾⣿⠆⠀⠀⣿⣷⣿⣿⣶⣿⡇⠀⠀⢾⣿⣾⣿⣷⣿⡧
⠀⠀⠀⠀⠘⢿⣯⣿⣭⡿⠃⠀⠀⠈⢿⣿⣶⣿⣿⠏⠀⠀⠀⠻⣶⣲⣞⣶⠟⠀⠀⠀⠹⣿⣀⣀⣸⡿⠁⠀⠀⠘⢿⣿⣿⣯⡿⠃
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⠀⠀⣀⣤⣀⡀⠀⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀
⠀⠀⠀⠀⢰⣿⢻⣿⡟⣿⡆⠀⠀⢠⣾⠻⣿⣿⢻⣦⠀⠀⠀⣾⡛⣿⣿⢛⣷⠀⠀⠀⣴⡟⣻⣿⡛⣷⡄⠀⠀⢰⣿⢻⣿⡟⣿⡆
⠀⠀⠀⠀⢻⣿⡟⠉⢻⣿⡟⠀⠀⢸⣿⢟⣿⣟⢿⣿⠀⠀⠘⣿⣿⣿⣿⢿⣿⠃⠀⠀⣿⣿⣭⣭⣽⣿⡇⠀⠀⢻⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠈⠻⠷⣤⠾⠟⠁⠀⠀⠀⠙⠶⢶⠶⠞⠁⠀⠀⠀⠙⠿⢿⡿⠿⠋⠀⠀⠀⠈⠳⠶⡶⠶⠋⠀⠀⠀⠈⠻⠿⣿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠟⠛⠛⠻⠿⠿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⡇⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠤⠤⢤⣀⡀⠀⠹⣿⡇⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⣤⣄⣀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡘⠃⠀⢀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⠴⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠇⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠦⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠘⠣⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⢤⣀⠀⠀⣸⠇⠀⠘⠲⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠍⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠃⠀⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁
⠀⢰⡿⠟⢿⣏⠛⠛⠳⢤⠀⠀⠀⠀⠉⠓⠶⠤⠤⠤⠤⠶⠚⠋⠁⠀⠀⠀⠀⠀⠉⠓⠶⠦⠤⠤⠶⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠛⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⣸⠟⠛⠷⠶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⠶⠝⠋⢻⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢠⡏⠀⠀⠀⠀⠀⠀⠈⡯⠉⠉⠉⠉⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⡤⠤⠴⠾⠷⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⢻⡀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣾⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣧⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⣀⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⠀⠀⢀⣀⣤⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠾⠷⠶⠶⠤⠤⠶⠶⠾⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠒⠚⠛⠛⠛⠛⠓⠒⠶⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⢠⠴⠒⠚⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠖⠲⠶⢤⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣴⣿⣿⣷⡄⢷⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⣿⣿⣿⣿⣿⣸⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠉⠉⠓⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠘⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠙⢿⣿⠟⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⣿⣿⡟⠒⠲⠤⢤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠤⠶⣾⣿⡗⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠘⣿⣿⣷⣶⣤⣤⣀⣀⣉⣉⣉⣉⣉⣉⣡⣤⣤⣶⣶⣿⣿⡿⠁⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⠤⢤⣤⣤⣤⣤⡤⠤⠶⠒⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠐⠀⢉⣉⠐⠒⠒⠀⠀⢀⣤⡭⠭⠙⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢠⣼⣛⠛⠿⠀⠀⠀⢀⣉⠁⠀⠐⣚⣹⠒⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠖⠒⠒⠒⠠⠤⠤⣄⠈⠛⠂⠀⠀⠩⡭⢁⠀⣴⣧⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢟⡄⠈⠘⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⣤⣀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⢴⡟⠁⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣦⢤⣅⠀⠀⠀⠘⠉⢉⣀⣀⠀⠀⠀⠀⢀⡴⠃⢸⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠉⠁⠀⠀⠀⠀⠂⠹⠏⠝⠃⠀⠀⠀⢨⠟⣀⠸⣼⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⣡⠀⠿⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⡀⠃⡀⠀⠀⠑⠌⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢃⠠⣶⡶⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⣿⢏⣀⣀⡀⡄⡍⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡥⡎⣇⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⣠⠔⠙⢀⡄⢠⠾⠋⠋⠀⠻⠧⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠶⠼⡀⠀⠀⠀⢀⣀⣤⣤⣶⠟⠁⠀⢸⠀⢀⡀⠀⠖⣠⡠⠀⣤⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣶⡎⠙⢿⣿⣿⣿⠿⠋⠁⠀⠀⠀⡜⠀⠛⠃⠐⠦⠈⠀⣸⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣶⢯⠉⠁⠀⠀⠀⠀⠀⠀⠀⣇⡐⣚⡄⠩⢶⣢⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠴⠒⡋⠑⠨⠐⠀⠄⠀⠀⣀⣀⣀⣀⠈⠩⣅⣀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠑⠒⠀⠒⠂⠨⠩⠍⠉⠉⠁⠀⠀⠀⠈⠙⠓⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢠⡤⠤⠤⠤⠌⠙⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢴⡒⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠟⠛⠛⠛⠛⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠟⠋⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣸⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠉⠻⣦⠀⠀⠀⠀⠀⠈⠳⢦⣴⣿⠟⠁⣀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⡀⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠿⢿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠁⠀⠀⠀⠀⠀⠉⠙⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣷⣦⡀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⣀⣤⣤⣤⣄
⠀⣾⣿⡿⠁⠀⠈⠁⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠈⠁⠀⠈⣿⣿⣷
⢸⣿⣿⣷⣄⡀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⣠⣿⣿⣿
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠉⠛⠻⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠟⠛⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠆⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⠉⠉⠉⠛⠉⠀⠈⠉⠉⠑⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢀⡄⠀⡴⢦⣀⠜⠳⣤⠔⠁⠘⢦⠞⠉⠙⢦⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢠⠏⢘⣊⡤⠀⠉⠀⠀⠀⢀⠴⠂⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢀⡏⠘⠞⠁⠀⠀⠀⠀⠀⠰⡣⢂⡠⠖⠀⠐⠒⢦⣌⣇⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣸⠀⠀⠀⠀⣰⣶⡄⠀⠀⠸⣷⣭⠤⠄⠀⠀⠀⠀⠹⢹⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠖⣾⠀⠀⣿⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢉⣿⠶⡄⠀⠸⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⠇⠀⡟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⡝⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣸⠀⢀⣷⡀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣻⠞⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⢸⠹⡇⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⣀⣠⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⢀⣠⠤⠤⠿⢤⣸⠀⣇⠀⠀⢸⡛⢦⡀⠒⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡇
⠀⢠⡋⠀⠀⠀⠀⣸⠻⡆⣿⠀⠀⢸⡇⠀⠙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⢹⡆⠀⠀⠀⢸⠇
⠀⠀⣟⠀⠉⠁⠀⣼⠁⣿⣿⠀⠀⢸⠁⠀⠀⠀⠀⠉⠛⠓⠶⠶⠦⠤⠶⠶⠒⠋⠁⠀⠀⠘⡇⠀⠀⠀⢸
⠀⠀⢸⣁⠈⠁⠀⢸⡤⠃⡏⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢸
⠀⠀⠀⠻⠤⠤⠔⠊⠀⢰⠇⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢀⢎⡀⠀⢀⣄⠀⠀⢀⣤⡀⠀⠀⠀⡠⣄⠀⠀⠀⣀⢤⠀⠀⠀⢷⡀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⡠⠋⠀⠳⠔⠁⠘⠦⠤⠋⠀⠙⠲⠤⠊⠀⠈⠓⠦⠞⠁⠈⠳⣄⣠⠎⠳⣄⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠙⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣠⠤⠶⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⢹⡀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿⣿⣿⣿⣷⣦⡀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠀⠀⣿⠀⠀⠀⠀⠀⣠⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⠀⣿⠀⠀⠀⠀⢰⡏⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠹⡿⠀⠀⢹⡇⠀⠀⠀⢸⣷⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠉⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠈⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⣾⣿⠛⠛⠛⠛⠛⠻⠷⠶⣦⣤⣈⠳⣄⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⢀⣀⣤⣤⣤⣤⣍⡻⢶⣄⠀⠀⠀⠀⠀⠀⠹⢦⣀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠞⠋⠁⠀⠀⠀⠀⠈⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⢤⣤⡴⠶⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠛⠛⣷⠚⠛⠛⠛⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⡰⠋⠈⠀⠂⠀⠠⢄⡀⡀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠈⢤⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢗⣩⡞⠠⠶⢦⣤⡄⠀⠀⠀⢀⣀⡈⣧⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠊⣹⠇⡧⠀⣠⣤⡀⠀⠀⠀⠛⠋⠀⠀⢹⡄⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⡾⠃⢀⣿⠀⠀⠙⠀⠀⠀⠀⠀⠀⠈⠀⢸⣧⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠁⢀⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡅⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⣆⡿⠛⠌⡆⠀⠀⢀⣤⣤⣤⠤⠀⠀⢀⣾⣷⠇⠌⢀⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡿⠇⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⡵⠃⡠⠋⢈⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡇⢀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠄⠊⣻⢯⠋⢀⢊⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣰⣄⠀⡰⠀⠀⠀⠀⠀⠀⠀⡍⢀⠀⣸⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⣀⣭⠁⠀⠀⠀⠀⠀⠀⠀⠷⢫⣍⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⣿⣿⣿⣿⣿⣷⣦⣤⣀⣠⣀⣠⣤⣴⣿⣿⣿⣿⣿⣷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⢿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠟⠛⠛⠛⠛⠛⠛⠿⣷⠄⠀⠀⠀⠀⠀⠀⢠⣶⡾⠿⠿⠛⠛⠛⠻⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣆⠀⣿⡇⠀⣾⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣆⡈⣿⣦⣾⣷⣶⣿⣁⣼⡟⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣌⣿⡿⣿⡏⠿⠿⣿⡟⢻⣿⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣄⡘⠿⢷⡾⠟⢀⣠⣽⡿⠀⠀⠀⠀⠀⢀⣾⣦⣀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠷⠶⠿⠿⠛⠉⠀⢀⣀⠀⠀⠀⠿⠋⢉⣿⠿⢿⣿⠿⢿⣿⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠿⠋⠀⢸⡿⠀⠘⠿⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⠀⣠⣴⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠛⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣦⣶⣶⣾⣭⣭⣭⣭⣽⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢩⠞⣿⣿⣿⣿⣿⣿⡿⠟⠛⢋⣯⣽⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠘⣧⣿⣿⣿⣿⣿⡽⢁⣴⣶⣿⡿⠛⠉⠙⠉⡉⠙⢳⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢿⠿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣶⣶⣶⣼⣾⣿⣿⣶⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⣀⣤⣄⡀⠀⠸⣝⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣝⡆⠀⠀⠘⠉⠁⢀⠀⠀⠀⠀⢻⣝⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠘⠻⠀⠀⠀⠀⠉⠙⠃⠀⠀⠀⠀⠉⠉⠈⢹⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠿⡿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣶⣦⣴⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⣟⣛⣉⣭⡟⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠘⠯⣭⠿⠋⠀⠀⠀⠀⢀⠐⠰⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⠛⠛⠛⠛⢿⠋⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣦⣦⣴⣆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠲⡒⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡐⠄⠨⠁⠑⠒⠢
⠀⠀⠀⠀⠀⠀⢀⠎⠡⠀⠀⠼⠁⠀⠡⢀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠁⠀⠀⡠⠃⡀⢀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡰⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠨⠑⠐⠂⠂⠉⠑⠤⠀⠀⢀⠀⢀⠠⠔⠉⠈⠀⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠐⠈⠈⠀⠀⠀⠂⠀⡀

⠀⠀⠀⠀⣠⣴⣶⣶⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣄
⠀⠀⠀⡼⠋⠉⠉⠉⠛⠛⠿⢿⣿⣷⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⠿⠿⠛⠋⠉⠉⠙⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣖⣶⣶⣤⡀
⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠘⠿⣿⣿⣭⣄⣹⣿⣶⣾⣇⣤⠌⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠋⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⠿⠿⣿⠿⠿⢿⢿⡿⢿⠿⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣾⣿⣿⣿⡿⣛⣴⣶⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⣾⣿⣿⣿⡿⣩⣾⣿⣿⣿⣿⣿⢟⣻⣽⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⢸⣿⣿⣿⠟⣼⣿⣿⣿⣿⡿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⣫⣥⣬⣤⣤⣤⣤⣭⣭⣭⣉⣋⡰⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣽⣿⣿⣿⣿⣿⡿⢟⣋⣭⣭⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣴⣿⣿⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠏⣼⣿⣿⣿⣿⣿⣿⡟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣙⣛⣡⣾⣿⣿⢏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡌⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⡃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢣⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡜⢿⣿⣿⣿⣿⣧⠹⣿⣿⣿⣧⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠿⠃⠈⠻⢿⣿⣿⣿⣷⡘⢿⣿⣿⣿⣮⣝⣛⣿⣿⣿⣿⡿⢟⣡⣶⣿⠟⣫⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠐⠛⠓⢆⠘⣆⠀⠈⠙⠛⠻⠿⠷⠝⠻⢿⣿⣿⣿⣿⠿⠟⣩⣶⣟⣛⣭⣶⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⣠⡴⣶⣶⣦⡈⠂⣿⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠟⠻⣿⠿⢛⣡⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣏⠀⢿⡟⣿⣷⠀⣿⠀⠀⠀⠀⢰⡏⠀⠀⣀⣠⣤⠞⠁⠀⢀⣿⠏⠀⠀⠀⢹⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠈⠓⠦⠤⠔⠋⢀⡏⠀⠀⠀⠲⠿⠿⠿⠿⢿⣉⠀⠀⠀⠠⡿⠋⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⢿⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠉⠉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⢀⣴⣷⣾⣿⣿⣶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠈⠉⣿⣿⣯⣽⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣀⣀⠀⠀⢀⣀⣠⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠚⠛⠿⢿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⡒⢛⣿⣶⣦⣀⠈⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠟⠻⣿⣿⣿⣿⣷⡐⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠈⠻⢿⣿⣿⣿⠛⣿⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠀⠀⠠⠖⠀⠀⠉⠛⢇⠀⣸⣶⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠓⠂⠀⠈⠉⠀⠀⠀⠀⢠⡘⠛⢿⣏⡻
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠈⢿⣷⣾⡿⡅
⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢤⡤⠤⠤⠀⠙⣆⣠⠂⠀⢀⠀⠀⠈⢁⣿⣷⠇
⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠋⠀⠂⠀⠀⠀⠠⠤⢬⠤⠤⠔⠁⠀⠀⠀⠀⠉⠁⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠒⢐⠉⠉⠡⡆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠂⠨⠥⠄⠈⠘⡄⠚⠢⢤⡀⠀⠀⡄⠄⠀⠠⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢸⡇⠁⠀⢱⠀⠀⠀⠀⠈⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠒⠤⠀⠀⡇⢀⠿⡇⠀⠀⠈⣇⢀⣠⣤⡤⢯⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢱⠃⠀⢠⠀⠀⠀⠸⡍⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⢳⡀⠀⠀⢠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⣠⡾⠛⠻⢿⣿⠿⠛⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠻⢿⣿⠿⠛⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⢰⡏⣾⣥⣶⣿⣧⣾⣶⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢰⣏⣾⣵⣶⣿⣧⣾⣾⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢈⠁⠻⠿⠿⠛⠋⠉⠛⠛⢻⣿⣭⢿⣿⡄⠀⠀⠀⠀⢈⠁⠿⠿⠿⠛⠋⠉⠛⠛⢻⣿⣭⢿⣿⡄
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠀⢻⣷⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠀⣻⡷
⠀⠀⠀⠀⠸⡇⠛⠛⠂⠀⠰⠾⠛⠛⠂⠀⠉⣿⣶⣿⠇⠀⠀⠀⠸⡇⠛⠛⠂⠀⢠⡴⠶⠶⠄⠀⠉⣿⣶⣿⠇
⠀⠀⠀⠀⠀⠃⠚⣿⠀⠀⠀⠘⣿⠗⠐⠀⠐⢛⠭⢹⡄⠀⠀⠀⠀⠃⠘⡿⠀⠀⠀⠴⠒⠄⠄⠀⠐⢛⠭⢹
⠀⠀⠀⠀⠀⡆⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠨⢇⡜⠀⠀⠀⠀⠀⡆⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠨⢂⡜
⠀⠀⠀⠀⠀⢧⠀⠀⠑⠀⠁⠀⣠⡄⠀⠀⢠⠴⠊⠀⠀⠀⠀⠀⠀⢧⠀⢀⣑⣀⣁⣀⣤⠒⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠈⢧⠀⠉⠭⠍⠁⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠙⠶⠶⠋⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⢀⣀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠉⠁⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠋⠉⠀⠀⠀⠀⠀⠉⠙⠻⣦⣀
⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠙⠿⢿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⢀⣴⣿⣿⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⢰⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣤⣀⡀⠀⠀⠀⠀⠀⣀⣠⣴⠿⠋
⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠛⠛⠛⠉⠁








⠀⣀⣴⢶⣿⠟⠛⠛
⠘⠋⠀⢸⣿⣦
⠀⠀⠀⢸⡇⠙⢷⣤⡀
⠀⠀⠀⣾⠃⠀⠀⠉⠻⢶⣄⣀
⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠉⠛⠳⢶⣤⣀⡀
⠀⠀⠈⠛⠿⣶⣤⣀⣀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠶⠶⢶⣶⣶⣤⣤⣤⣶⣶⡶⠶⠶⠾⠟⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠛⠻⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠛⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠀⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢀⣀⣣⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠴⢊⠀⢨⠀⠢⠄⣀⠔⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡆⣺⡀⠈⠢⣀⣀⣨⣦⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢡⡄⣏⠀⠀⠀⠀⠀⠁⠄⠑⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠠⣀⠰⠆⠈⠉⠋⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠈⠒⢄
⠀⠀⠀⠀⠀⡀⠠⢁⢉⡖⢱⠪⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠤⠊
⠀⠀⡠⠖⡋⠹⠡⠀⠀⢈⡒⢧⡑⢤⣂⣈⡁⡅⡈⠈⠉⠀⡜
⢠⠞⠀⠀⠠⠌⣀⠀⡤⠁⠀⠀⠹⡄⠁⠀⠀⠀⠀⠀⣠⠎
⡎⠀⠐⠤⣛⠰⠄⡐⠀⠀⠀⠀⠀⡇⠠⠀⢈⡢⣤⢴⡵
⡇⠀⠀⠀⠀⠉⠰⣁⠀⠐⡀⠀⠸⡅⠀⠈⠁⠐⠗⡾⢇
⢧⠀⠁⠈⠐⠤⢼⠄⢀⠉⠉⠂⠀⠈⠀⢀⡠⢠⠰⢅⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡀
⠘⣆⠀⠀⠀⠀⠐⡀⠀⠀⢉⡡⠞⠓⠠⡎⠀⠸⡐⠀⢡⣄⡓⠀⡀⠀⠀⠀⠀⠀⠀⠐⠍⡠⠼
⠀⠈⠳⠤⣤⣤⠤⠼⠶⠊⠁⠀⠀⠐⠒⠞⠀⠈⠱⡀⠈⠬⡑⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠰⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢝⠐⢀⠀⢠⠖⢁⣰⣖⣉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢚⡣⠂⠨⠥⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙

⠀⠀⠀⠀⠀⢀⣠⠤⠤⡀⣠⠄⠴⡖
⠀⠀⠀⠀⠀⣠⣅⠈⣠⣷⣧⣈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠟⠿⠋⡅⠀⠀⠀⢤⠤⣤⠤⣴
⠀⠀⠀⠀⣴⣿⣿⣯⣯⣯⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠃⠀⠀⠀⢠⣂⣤⣒⣬⠀⢀⡀
⠀⠀⠀⣼⣿⣿⣿⡿⣿⣿⡷⡷⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⣿⣿⠿⢿⣿⠀⢸⣇⡾⢀
⠀⠀⢰⣿⣿⣿⢷⠻⣿⠿⠃⠀⠀⠊⠉⠹⢦⡠⢚⠙⢃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⢀⠀⢹⡇⠘⠛⣷⠟
⠀⠀⢸⡿⠋⠙⡏⢹⠁⠀⠀⠀⢀⠄⠀⠀⠈⠁⠀⠀⣨⠅⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠤⠄⠂⡇⢠⠇⢠⡆⢀
⠀⠀⠈⠀⠀⠀⠭⠖⠂⠀⠀⠀⢨⢋⡎⠀⠀⠀⠈⣿⣟⡂⠀⠀⠀⠀⠀⠀⠀⠸⡧⠐⠀⠂⠀⣇⣌⠀⠉⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣽⠁⠀⠀⠀⠐⢘⢷⠁⠀⠀⠀⠀⠀⢰⣄⣀⣧⣂⣀⡠⠼⠃⠀⠀⠙⠷⠄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠩⠀⡈⢀⠁⠀⠘⠇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⡆⠠⠐⢰
⡎⡤⣤⣀⣮⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⡄⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠸⣄⢀⡏
⠄⠀⠈⠛⠁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠞⠁⠀⠀⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠖⠚⠛⠋⠁⠡⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠁⠀⠀⠀⠀⠀⠀⠈⢿⡇
⠀⠀⠀⠀⠀⡎⠀⠀⠀⡓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠶⠖⠖⢊⠂⠀⠀⠀⠀⣤⠀⠈⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⣤⣻⣿⣈⣘⣈⢤⠼⠀⠀⠀⠀⡟⣉⠐⣶⠞⡋⢹
⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠈⠆⠀⠀⠀⠀⠀⣘⢮⣽⣋⠀⠈⠀⠈⠀⠀⠀⠀⢀⣧⣈⣰⣿⣄⣁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠢⣀⠠⡀⠀⠀⠀⢠⣍⣛⣾⡋⠙⠛⠛⠟⠁⠀⠀⠀⢸⠘⠋⠀⠀⠈⢿⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠈⠑⠃⠀⠀⠀⠼⠛⠋⠙⢿⡶⡶⠀⠀⠀⠀⠀⠀⢸⣅⠀⠀⠀⠀⣸⡼
⠀⠀⠀⠀⡀⢠⠄⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠔⢠⠀⠀⠀⠀⠀⠀⠈⣿⠀⠉⠩⠅⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⣀⢀⣀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⣀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⢈⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡀⠉⠒⢌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⡠⠋⠀⡠⠐⠀⠀⠀⠀⠀⠀⠐⠠⢀⠀⠀⠉⠒⢄⠑⠌⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰⠁⢀⠎⠀⡀⢀⠄⠀⡐⠢⢀⠁⠢⢄⡈⠑⠂⠀⠠⠈⠢⠈⠄
⠀⠀⠀⠰⠢⡞⢂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢈⠔⠁⡠⠊⠀⠀⠀⠑⠢⠄⡀⠁⠒⠠⠤⠤
⠀⠀⠘⣅⠳⠉⠀⠈⠈⠀⠀⠀⠀⠀⠀⢠⠀⠈⠠⠔⢊⡡⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠂⠠⠄⠀⠂
⠀⠀⠘⢌⡀⠀⠀⢀⠔⠲⠄⠀⠀⠀⠀⡘⡆⡰⠂⠉⠀⠀⠀⠘⠃⠀⠀⢀⡀⠀⠀⠘⠃⠀⠀⠀⠀⢀⣈⠅⠐
⠀⠀⠀⠀⠈⠑⡊⠁⠀⠀⠈⠢⡀⠀⠀⢡⠈⠂⢧⠀⠀⡠⠄⠀⠀⠀⠠⣀⣠⠄⠀⠀⡠⠠⡀⠀⠀⡌⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠒⢤⠏⠀⢀⢀⠧⡀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⢀⠜⢠⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠉⠒⠮⣀⠀⠈⡦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠁⠀⠀⠲⡖⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢇⠀⠈⠁⠒⠀⠀⠀⠀⠒⠈⠁⢀⠑⢦⠀⠀⠀⢡⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣬⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠄⠀⠤⠤⠤⠀⠀⠠⠔⠁⠀⠀⠑⠄⠀⠀⢱⣊⣥⣦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣡⠋⠿⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠰⢻⠸⡈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⢣
⠀⠀⠀⠀⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⣴⢤⣤⣤⡀⠀⠀⠀⠀⣀⣀⣀⠤⠄⠀⠀⠠⠃⠀⠀⠀⠀⢃
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⠀⡠⠒⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡕⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⢈⡆⠀⠀⠀⠀⠀⡌⠀⠀⠈⠄⠀⠀⠀⠀⠀⢀⠔⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠈⠑⠄⡀⠀⡘⠀⠀⠀⠀⠐⡀⠀⠀⡠⠔⠊⠢⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⠲⠁⠀⠀⠀⠀⠀⢡⠔⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠐⠱⠀⠀⠂⠀⢀⠀⠀⢠⠁⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠯⣈⠙⠢⢄⡀⠸⠀⠈⠄⠄⠀⠀⠀⠠⡸⠀⠀⠀⢐⢁⠅⠀⣈⡤⠖⠉⣁⢽
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠉⠲⢄⠈⠓⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⢁⡤⠚⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠑⣞⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⡏⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢠⠜⠉⠁⠉⠀⢶⠒⣲⣦⠀⡤⢤⠀⣤⠄⠀⠐⠐⢢⡄⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣀⡼⠁⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠉⣇⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠒⠒⠲⡁⠀⠀⠀⠀⢾⠋⣾⣿⡎⢱⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣠⠀⠀⠀⠀⢸⠒⠒
⠀⠀⠀⠀⠀⠀⠉⠉⣙⡇⠀⠀⠀⠀⠈⠓⠼⠿⠕⠋⠀⠀⣀⡀⠀⠀⠈⠙⠖⠻⠈⠀⠀⠀⠀⣸⠉⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢻⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠶⠤⠤⠤⠤⠤⠴⠒⠒⠉⠁

⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣻⣟⣻⣿⣟⣻⣻⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣽⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻
⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠤⠴⠶⠒⠚⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠒⠒⠶⠦⠤⣤⣤⣀⣀⣘⡇
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣀⣭⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠛⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠋⠉⠉⠙⢦⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢙⡆⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢾⠉⠛⠛⠚⠛⠉⠉
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠈⢿⣿⡿⠿⠛⠛⠛⠛⠛⠻⠿⣿⣿⠏⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠻⢦⣀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣾⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⠁⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⢠⡇⠀⠀⢀⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣄⡀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣡⠴⠚⢉⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎⠙⠳⢤⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⠟⠛⠉⠀⠀⠀⠀⠀⠀⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⠋⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⣼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣶⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⣿⣿⣿⣿⠿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏
⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠉⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⣿⣿⣿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣷⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡠⠒⠉⠀⠈⠑⠴⠋⠁⠉⠑⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡔⠁⠀⠀⠀⢁⠄⡐⡠⣠⢀⡄⠠⢂⠆⡄⢀⠀⠈⢂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⡠⢀⠔⡡⡡⢪⢊⠔⡡⢊⠔⡱⢁⠎⡴⡡⢣⢂⠄⡈⠱
⠀⠀⠀⠀⠀⠀⠀⢀⠦⠀⢐⠔⡱⢡⢊⢜⠔⡡⢊⠔⡡⢃⠌⡰⢁⠎⡔⡑⡡⢊⢜⠄⡈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⢎⠀⠀⣪⢊⢔⡕⣡⣋⠞⡡⢋⢞⡰⢃⠎⡰⡡⢊⢎⠔⡱⣡⢊⢎⠔⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠎⠀⠔⡡⡣⣻⠚⠫⠙⠺⠻⠉⠈⠙⠛⠻⣜⣑⣁⡬⢦⠐⠡⡣⢊⢊⠄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢆⠀⢊⢔⣎⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⡦⠰⢕⡡⠳⠎
⠀⠀⠀⠀⠀⠀⠀⠀⣅⣐⡹⠿⠁⡠⠒⢀⠐⠒⠀⠀⠀⠀⠀⠔⢒⠐⠒⠤⠀⠑⡤⠩⡊⠎
⠀⠀⠀⠀⠀⠀⠀⡏⢦⡈⢿⡢⡤⢲⠽⠏⠡⢕⢦⠀⣀⠀⡴⣪⠽⠫⠝⣦⣀⢄⡷⢊⡝⢧
⠀⠀⠀⠀⠀⠀⠀⠳⣴⠃⠈⠘⢣⠇⢶⡰⡶⠂⡇⡿⠶⡞⡜⠑⢖⢲⡖⠜⣿⠋⢠⢿⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠃⠀⠀⠢⡓⠤⠍⠤⣊⡼⠁⠀⠰⡙⠦⣈⣉⡠⣊⠏⠀⠰⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠑⡶⡇⠀⠀⠈⢙⠉⠉⠁⠀⠊⠈⠁⠈⠑⠒⢒⠚⠁⠀⠀⢸⢶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠁⢰⠐⠒⠂⠂⠐⠖⠒⠄⠤⡄⠑⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⢸⠝⡝⡹⡻⡻⢫⢃⢂⢞⠃⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠈⢯⣎⠌⡔⡱⣱⣡⣣⠎⠀⠀⢠⣦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠉⠚⠚⠚⠚⠃⠀⠀⢀⣤⣢⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠤⠀⡀⠀⠀⣤⡤⠤⣨⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣗⠶⠤⡤⡤⡤⣶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⢎⢜⠔⡑⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡤⠃⠀⠀⢪⢊⢔⠵⢤⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⡁⠙⢄⠑⣄⠀⠀⠀⠱⡿⢁⠞⢀⣎⢍⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⡊⠕⡡⢀⡱⣤⡑⠦⣀⣀⠔⢁⣴⢟⠕⡡⢊⢔⡄⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠋⠴⠋⠔⠡⠊⠕⠩⢳⠦⡥⠚⠕⡱⠛⠊⠁⠙⡧⡪⢲⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣻⣿⡿⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⢋⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⡃⣽⣯⣵⣶⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠶⣏⡁⠀⡙⢿⣿⣿⣿⣿⠟⣡⣤⣤⡉⠛⣿⣹⣿⣿⣿⡟⣩⣟⣛⣷⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣗⣦⣹⡆⠀⠓⠈⡿⠛⠃⠀⣼⣿⠹⠸⡇⠀⢙⣿⣿⡿⠋⣰⠟⣿⣶⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠻⢿⣿⣤⡾⢆⠙⣦⡀⠘⢻⣿⠀⢈⠁⣠⣾⣿⠛⢁⣴⣷⣼⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣉⡥⡝⢿⡳⢦⣪⣝⣿⣶⣌⣀⠀⠸⢿⣿⡟⢡⣶⣿⣿⡟⡻⢿⠟⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡾⣛⣫⠟⠻⣿⣿⡟⠙⣭⠉⠳⣷⣞⠻⣿⣼⣿⣿⠏⠀⠀⠀⠐⢫⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⠘⠋⣴⣿⡞⣡⣿⡇⢸⣿⣱⣦⡿⠟⣷⣿⣿⣵⡦⠀⠀⠀⠀⠀⠀⠀⠚⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⢿⣵⣿⠑⣶⣿⣿⣥⣤⣄⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠹⠏⠀⣏⣽⡟⠁⠉⠨⠙⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠠⠀⠀⠄⠀⢋⣿⠁⠀⢰⡀⢰⣿⣿⠀⠀⡀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⡀⠀⢸⣟⠀⣀⣸⣧⢸⡿⢻⡗⠀⠈⠸⠿⣿⣤⣂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠀⠀⠀⠀⠀⠀⠀⠹⠆⠀⣿⡞⢹⣿⣿⣿⠇⢸⣧⠀⠀⠀⠀⠛⠊⢿⣧⣦⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣿⣷⠠⣿⣿⣿⡃⡘⣿⠀⠀⠀⠀⠀⠀⠘⠻⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠛⠊⠻⣿⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠛⠛⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡆⠀⣿⣦⡙⠇⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢁⢬⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⣿⣿⣿⣦⠀⢼⠀⠀⣀⠀⣀⣤⠀⠀⠀⠀⠙⢿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣁⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⢿⣿⣿⣿⠀⠀⣄⠲⣿⣦⠷⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠁⠁⠀⠀⠀⠀⠀⠤⢦⣹⣷⣴⣾⣿⠿⠉⢀⣠⣿⠀⢸⡏⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⡰⠀⠉⠀⠀⠀⠈⠛⡋⣽⣿⣷⣤⣤⣾⠙⣿⣤⠀⣿⡄⠀⠈⢷⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⣠⣴⣯⡗⣏⣿⣷⠡⢙⣿⣶⣾⣿⡶⣿⣿⣆⡀⠈⣷⣞⡁⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠛⠒⠲⣤⣤⣶⡻⣾⣏⣽⡿⢿⣿⣿⣾⣿⣿⣤⡙⣷⣿⣷⣿⣤⡌⢿⣿⣿⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⠉⠙⢻⢾⣿⣿⣿⣧⣜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠒⠂⠘⠛⠛⠛⠛⠛⠛⠓⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣡⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢟⣴⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠟⠉⠈⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣇⠈⠹⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⣿⣿⣿⣿⠈⠉⠉⠛⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢸⡟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠹⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡀⠀⢀⣀⣘⡁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠙⠻⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢹⣿⣿⣿⣿⣿⠀⠘⠛⠛⢁⣴⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣇⠀⠲⢶⣿⣿⡿⠛⣉⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣧⡀⠈⠁⠉⣠⠿⠟⠻⠟⠛⣿⣿⣿⣿⣿⣿⣿⡏⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⢿⣿⣿⡿⠀⠀⠀⠠⢴⠂⢘⣿⣿⣿⣿⣿⣿⣿⣿⡿⡄
⠀⠀⠀⠀⠀⠀⠀⣀⡠⢀⣀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⣀⡀⠀⠈⢙⣻⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⢀⣤⣿⣿⣿⣿⣿⣿⣿⡟⠇
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⡔⣿⣧⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠀⠀⠀⠀⢰⡇⠙⡿⠇⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⢀⣿⣆⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢘⣿⣿⣧⢀⠀⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⢿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⡀⠄⠂⠉⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⢠⠀⡄⣠⢠⢀⡆⣰⠀⡔
⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⠀⢤⡐⢄⠳⠙⠈⢁⡀⠀⢀⠀⢀⠈⠈⠀⠃⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⡀⠐⢤⡙⠦⠉⢀⣤⠒⠀⣐⠂⢲⡐⡐⢈⠉⡑⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠉⠀⡴⡉⠇⠈⠢⢉⠘⢀⡀⠄⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⣠⠒⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠘⠠⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣴⡟⢱⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⣼⣦⣶⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⠦⠤⠤⢤⡤⠤⣤⡴⡾⠀⣰⣩⡵⣾⡿⠄
⠀⠀⠀⠀⠀⣀⡤⠴⢒⣿⣿⣿⢿⠁⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣡⣋⣿⣛⣩⣣⣼⣐⣎⡵⣧⡿⠵⠓⡀⣠⢚
⠀⠀⠀⠀⠀⢏⠛⣠⠟⠉⠉⢳⣾⠱⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠯⠉⠛⠛⠛⠻⠵⠷⣾⣁⡀⢲⣾⡽⢺⠄
⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⢉⠀⠾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡠⡆⢏⢛⠡⣔⠓⠀⠌⠀⣚⠩⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠓⠒⠒⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠀⠢⢰⠀⠀⠀⢰⢀⠠⢠⠀⠀⡄⠀⠀⠀⢀⡄⡄⠀⠀⠀⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡑⠄⡗⠆⠀⠀⢠⡧⠰⠃⠠⢊⢆⡀⡴⠈⣼⠊⠀⢀⠠⢀⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⠘⣇⢰⡄⠀⠸⣣⡶⠊⢐⠍⢉⠌⢀⡼⠁⣐⠈⢤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠝⠂⠘⣈⡟⣲⠞⣋⣶⠿⠗⠚⠃⠰⢋⡀⠄⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⣿⠛⣡⡾⠡⠁⣀⠠⠴⡒⢠⠋⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⣤⠞⡇⣰⡟⢀⣠⠊⠀⠀⠀⢈⠃⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⢀⣧⡟⠀⡾⠁⣀⣀⣀⣠⠾⣆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣠⢞⠟⠁⡸⣳⢋⠉⠉⠉⠀⠈⢹⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⢞⡵⠋⢀⡤⠗⡵⠁⠀⠀⢀⡴⠁⢈⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞⠀⣠⠎⠀⢀⠀⠀⢀⠄⡘⠁⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠌⡈⠀⠀⡆⠀⣀⡡⠊⠀⠠⠊⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡾⠀⠀⠀⣦⡞⣡⠴⠊⠁⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡀⢼⣿⠏⠁⠀⠀⠀⢀⠂⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠈⠀⣸⣿⠀⠀⠀⣠⣀⠅⠂⡐⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⡀⠐⠈⡄⡿⠀⡠⠊⠍⠀⢀⡔⢡⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠁⠀⡠⡾⠀⣧⠊⢀⠌⢀⡴⠋⠠⠈⡀⠂
⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⣠⠎⠀⡇⢠⠇⠀⣈⡔⡱⠁⠠⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠃⠀⠈⢠⠀⡄⡴⡋⢰⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣤⠁⠀⠀⠀⡀⠸⠇⢡⡗⢁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢀⠀⠀⢠⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⢻⣿⣿⣿⡿⠟⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⣿⣿⣿⣿⠿⢻⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣾⢫⣶⣁⣤⣾⣿⣣⣶⣴⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢰⡟⣵⣎⣠⣴⣾⣟⣴⣦⣾⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠯⢙⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⠿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠸⠏⣻⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⡿⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢰⠂⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⣿⠛⠛⢿⣿⡀⠀⠀⠀⠀⠀⡶⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⠻⣿⣇
⠀⠀⠀⠀⠀⠀⡟⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡆⢈⣿⡇⠀⠀⠀⠀⢸⠃⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⠀⣹⣿
⠀⠀⠀⠀⠀⠸⡇⠘⠛⠛⠂⠀⠀⠾⠟⠛⠛⠓⠀⠀⠈⣿⣶⣿⣿⠁⠀⠀⠀⠀⢿⠀⠛⠛⠗⠀⠀⢠⣴⡶⠶⠶⠄⠀⠀⢹⣿⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠳⠠⢺⣷⠀⠀⠀⠀⢾⣿⡶⠠⠄⠀⠸⠟⣋⠙⣧⠀⠀⠀⠀⠀⠘⠆⠴⣿⡆⠀⠀⠀⣠⡤⢤⠀⡀⠀⠀⠿⢛⡉⢻⡅
⠀⠀⠀⠀⠀⠀⡄⠀⠈⠁⢠⠆⠀⠀⠈⠛⠁⠀⠀⠀⠀⠸⡅⢀⡟⠀⠀⠀⠀⠀⢠⠀⠀⠉⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⣰⠃
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠰⣏⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠞⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢾⡁⣀⡀⠀⠀⢀⡀⠀⠀⠀⠀⢈⡴⠃
⠀⠀⠀⠀⠀⠀⠸⡄⠀⢀⣀⣀⣀⣀⣠⠾⠆⠀⠀⢀⡔⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⢀⡁⠠⠀⢤⣶⠟⠋⠀⠀⣰⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠤⠤⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠈⠛⠶⠟⠋⠁⠀⠀⣀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⢀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠈⠛⠉⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠒⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢹⡇
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠁⠖⢶⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣼⠃
⠀⠀⠀⢀⣤⡖⠉⠀⠀⠀⠀⠀⢐⠊⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠄⢠⣿
⠀⠀⠀⠈⣿⣿⢂⠀⢀⠀⠀⠀⠋⣀⠰⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠘⠄⣼⠇
⠀⠀⠀⠀⠘⣯⣿⡆⠈⠀⠂⠀⠈⠁⠀⢀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⡐⠀⣰⡟
⠀⠀⠀⠀⠀⠘⣿⣿⣄⠀⠐⠀⠀⠀⠀⠘⠆⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢤⣶⠀⢀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⠀⠀⠄⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⢠⡞⠁⠰⣤⠉⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⡀⠂⠠⡄⠀⠀⠀⠂⠄⠘⢦⠀⠀⢀⡴⢋⠀⢀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⠀⠀⠙⠀⠀⠀⠀⠀⢀⡬⠷⣴⠟⠰⠊⠀⠈⠁⠀⠀⣸⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄⠀⠈⠀⢀⡤⣾⡁⠰⡆⠙⢷⣤⠀⠀⠀⠀⢠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡶⠞⠁⠀⢀⣿⣗⡀⠀⠀⠹⣷⣦⣲⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣰⠋⣀⠹⣷⣄⠲⠀⠈⢧⡙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⣡⡼⠋⠻⣧⣄⢀⡀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠰⠃⣼⠏⠀⠀⠀⠙⣿⠈⠉⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠾⠋⠀⠀⠀⠀⠀⠹⠶⠶⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠁⠀⠀⠈⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢀⡴⠞⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠈⠀⢀⣤⡀⠀⠀⠀⠀⠐⠋⠉⠙⠂⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢠⡏⠀⢹⡄⠀⠀⠀⣠⡤⠶⠂⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠈⢧⣀⡼⠀⠀⠀⠀⠈⠙⠳⠆⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⢻⡙⠲⠶⠶⠖⢋⡟⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠙⠲⠶⠶⠖⠋⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⣀⣀⣀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠃⠒⠢⠀⠀⠀⠀⠀⣠⣴⠚⠉⠉⠙⠳⠞⠛⠻⣿⠙⣻
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⣾⣿⣿⠀⣠⣤⡶⠶⠶⢶⣶⣿⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⢠⣾⣿⣿⣷⣼⣿⡟⠀⢀⣀⣀⣀⡉⡄
⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⢸⣿⣿⣿⣿⡿⣿⣄⠀⠈⢵⢻⣯⡵⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣥⣞⠹⣯⣻⣄⠀⠀⠈⠓⠰
⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣶⣿⣿⣧⣤⣴⣭⡄⠈⠻⡛⠳⣤⡀⠂⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⣽⣿⣧⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠛⣿⣿⣷⡖⠠⣤⣄⣠⣛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀⠉⠐⠒⢿⡁⠉
⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠋⣉⡉⠙⠻⣿⣿⣿⣿⣿⣆⠀⠀⢸⣿⣦⣴⡄
⠀⠀⠀⠀⢀⣾⣿⣾⣧⡤⠀⢀⠤⠤⠤⠶⣽⣿⣿⣿⣿⣿⡆⠀⠛⣿⣻⠁⣀⣀
⠀⠀⠀⠀⢈⣿⣿⣿⠏⠀⣴⠁⣀⣤⣶⣄⣈⡉⠍⠁⠙⠛⠤⣄⡀⠈⣻⣿⣟⢷⡄⠉⠀⠠
⠀⠀⠀⢀⣾⡿⡟⠁⢀⡼⣣⣾⠿⣿⢟⡿⠛⠛⠛⠳⣦⡀⠀⠈⢳⣶⡇⢋⢿⡷⠀⠀⠀⠠
⠀⠀⠀⣾⣿⣮⣠⣴⣾⣰⣿⣧⣼⡁⠈⠀⠀⠀⠀⠀⢹⣿⣶⣦⠘⢿⡇⠘⠘⣷⣤
⠀⠀⠈⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⡇⠐⡄⢹⣿⡇
⠀⠀⠀⠘⣿⣿⣟⣿⣿⢿⣿⣿⣿⣿⡇⢀⡀⢀⡀⢀⣼⢿⣿⠀⠀⠀⡧⠀⠈⢺⣿⠁
⠀⠀⠀⠀⢸⣿⣿⣿⡷⠈⣿⣿⠿⠀⠀⠈⠀⠿⣿⣿⣿⡌⣿⠐⠀⠀⣷⡀⠀⠀⠉⢆
⠀⠀⠀⠀⢸⣿⣿⣽⣿⡆⣿⣿⡟⠀⠀⠀⠀⠰⣿⣿⣿⣿⡏⠡⡀⠀⣿⣇⠀⠀⠀⠈⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠀⠀⢨⣿⣿⣆⢀⠀⠀⢳⠲⡀
⠀⠀⠀⠀⢸⢹⣿⣿⣿⣿⣿⡟⠀⠀⢀⡀⢀⣾⣿⣿⣟⠻⣿⡄⢀⣼⣿⣿⣿⣿⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⢸⡆⣿⣿⣿⣿⣿⢁⣿⣿⠼⣿⣿⣷⣾⣿⡗⠚⠫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⡇⣠⣿⣿⡁⢸⣿⣇⠀⢠⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⢸⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣸⣿⣧
⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢻⣿⣿⠻⢇⡙⠛⠻⡀
⠀⠀⠀⠀⢸⡇⣸⣿⣿⡿⠋⣼⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⠀⠀⠀⣿⡏⠣⠀⠙⠦
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃⣼⣿⣿⣿⣿⣿⣿⡟⢻⣯⣷⣿⢿⢿⡁⠀⠀⠀⡹⢧⠀⠁
⠀⠀⣴⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠟⠛⣿⣿⣿⢿⣿⣿⣿⣆⢠⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠈⠀⣿⣿⣿⠈⣿⣿⣿⡿⢸⣿⣿⡿⢿⣿⣿⣿⢸⣿⡇⣿⣿⣿⡟
⠀⢀⣈⣻⣿⣿⣿⡇⢸⢻⣿⣿⣿⣿⣿⣿⣿⢹⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⢀⣉⣿⣿⢸⣿⣿⣿⢸⣿⡇⣿⣿⠁
⠀⠸⣿⣿⣿⣿⣿⣧⣦⣼⣿⣿⣿⣿⣿⣿⣿⣽⣿⣇⣤⣼⣿⣟⣿⣿⣿⣿⣿⣷⣸⣿⣿⣿⣸⣿⣿⣿⣸⣿⡇⣿⣿
⠀⠀⠛⠿⠋⠈⠛⠟⠛⠻⠿⠿⠿⠛⠿⠿⠟⠻⠿⠿⠿⠿⠿⠟⠿⠟⠛⠛⠛⠛⠛⠻⠿⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠠⠔⠉⠀⠑⠊⠈⠐⠠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠂⢉⠀⡀⠀⢀⢤⡄⢴⠀⠂⠀⠀⠀⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠄⠘⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡂⠀⠐⢀
⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠰⡁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠘⠤⠄⠄⠀⢨⡀⠀⠐⠄
⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡡⠀⠘⠤⡄⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐
⠀⠀⠀⠀⠀⡂⠈⡂⠀⠀⠠⡠⣀⠄⠀⣀⡠⠂⠄⢀⠈⠙⠂⠀⠀⠀⠀⠠⣀⡀⡄⠆⠀⠀⠀⡀⢀⢩⠎⣁⢤
⠀⠀⠀⠀⠀⠁⠀⠈⠒⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⠀⠑⠠⡀⠸⣀⢄⢀⡄⠀⠀⠐⠁⠀⠀⠀⠈⠂⠈⠀⠡⠦
⠀⠀⠀⠀⠀⠀⠆⠐⠖⠊⠁⠈⠁⢀⡀⠀⠀⠀⠀⠀⠀⣀⠈⠂⠀⠀⡀⡠⠜⢹⠁⠀⠀⠀⡠⠄⣀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠑⠠⠠⠀⠀⠀⠀⠈⠃⠀⠀⢀⡀⠀⠀⠉⠀⠀⠈⠈⠀⡄⠄⠂⠀⠀⡠⠊⠀⠀⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠐⣉⠆⠀⠀⠀⡀⡠⠀⠀⠀⢎⡁⠀⠀⡜⠀⠀⢀⠄⠊⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡤⣀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⡠⢪⠠⠔⠊⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠓⠄⣈⠐⠂⠤⠀⠀⠀⠠⠄⠂⡁⠔⠁⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠈⠁⠀⠀⠒⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁
⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠂⠁
⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠙⢷⢄⣀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⡀⠐⠤⠋⠉⢉⢲⡀⠀⠀⠀⠀⣠⠤⠔⠒⠒⠒⠒⠒⠦⠤⣄
⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⢈⠀⠀⠠⠄⠀⠀⠀⠀⠠⠌⠐⢄⡰⠋⠀⠄⠀⠀⠐⠀⠀⠤⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⡿⢿⠿⠿⠙⠁⢀⣼⡿⢛⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠛⠿⣿⣿⣍⣻⣿⣿⣿⠃⠐⢛⡿⣿
⡷⢸⡄⠢⠀⣰⡿⠁⣡⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⠄⣠⣆⢼⣷⡼⠁⠀⠀⠀⠀⠀⠈⠙⢿⣷⣄⠿⢿⣷⡆⠠⠻⣶
⠠⡞⣄⢀⡼⠋⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⡏⠴⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣌⢿⣷⡔⢀
⠶⠀⢠⣿⠃⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⠷⢠⢶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⢻⣿⣷⢟
⡀⢀⣼⠇⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠝⢷⡄⢀⣹⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⢻⣿⡍
⠈⢹⡧⣰⠣⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣍⠄⢀⢈⣉⣉⠩⢛⢿⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡎⣿⣿
⣠⡿⠀⡥⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣴⣾⣿⣿⣿⣿⣌⣿⣾⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢹⣿
⣼⡇⠠⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢙⣿⣿⢧⣤⣿⣎⠻⣦⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣿
⢼⡇⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⠖⠁⣸⣿⡿⠟⢿⣿⣿⣆⢹⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⣺⡇⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢀⣾⢻⠉⣓⣄⣂⣻⣿⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢻
⣿⡿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⣸⣿⣷⣦⣽⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢰⣿⢿
⢿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡀⠀⣿⣿⣉⣼⣿⣿⢿⣟⢳⠇⣿⡿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢀⣿⢸
⣻⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⣿⠁⠠⠸⠿⢿⣿⣿⣇⢻⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⢸
⣻⡇⢸⡇⠀⠀⠀⠀⢀⣔⡀⢀⠀⠀⡄⢰⣿⡇⢸⣿⣿⣿⣿⣧⣶⠀⠀⠈⣿⣿⣿⢸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⢸
⣿⡇⢸⡇⠀⠀⠀⠛⠛⠛⣧⠀⠀⠀⠃⠀⣿⠁⣼⣿⣿⣿⣿⡉⢿⠀⠀⠀⣿⣿⣿⣀⣿⣿⣿⣿⣧⠀⠀⡀⠀⠀⠘⣿⢸
⣿⠁⢸⡇⠀⠀⠀⠀⡆⠀⣿⡀⡂⠀⡆⠐⣿⠀⣿⣿⣿⣿⣿⠙⣾⡀⠀⠀⠙⠁⣦⣿⡿⣿⣿⣿⣿⡀⠀⠀⠁⠀⠀⣿⢸
⣿⣆⢸⡇⠀⠘⠦⢌⣁⡘⠿⣡⡇⠁⠃⣻⡿⢠⣿⣿⡿⣿⣿⠂⣏⣰⣤⣤⣦⡞⠡⠛⠻⠛⠛⠛⠻⣷⣤⡀⠀⠀⠀⣿⣸
⣿⣿⠺⣇⡀⡀⢀⡀⢸⣏⣤⣦⣽⣷⡇⣿⡏⢸⣿⣿⣿⣿⣿⣰⣿⣿⣧⣤⣴⣿⣶⠀⠐⡢⠴⣉⣙⣊⠿⡿⠦⢀⠀⣿⣿
⣯⣿⣇⠀⣿⡧⢻⠀⣾⠟⠉⣙⣿⡭⠅⣿⣧⢺⣿⣿⣿⣿⣿⣬⣿⣿⡧⣽⣿⣿⣧⢐⠏⣀⡀⠀⢈⢛⠿⣶⣶⣶⡶⢿⣿
⡏⣿⣿⡆⡉⣴⡇⢠⣿⠀⠀⡀⠛⡀⠺⣿⡇⢸⣿⣼⣿⣿⣿⢸⣿⣿⡇⠘⣟⣿⣋⡠⠔⣡⠈⡀⠁⠐⢐⠎⢿⣿⣷⢊⣉
⢠⣿⡿⣿⣟⠉⡅⣸⣿⠂⣀⠙⢤⣣⡤⣿⣿⡿⠛⣿⣿⣿⣿⠼⣿⣿⠁⢨⣟⣿⠀⠀⢈⠁⣠⣿⢾⢂⠄⢶⡆⣿⢻⣯⣽
⡿⠷⣿⡉⠉⠉⠀⣩⣿⣿⣛⢲⣾⡿⠓⠀⢲⡦⣰⣿⡏⠈⢉⢾⡉⠉⡖⢻⡿⠋⠈⠐⠂⢭⠼⡁⠄⠨⢀⠨⡧⠀⣸⣡⠉
⣿⣿⢬⣭⣽⣿⣿⣿⣟⣻⡏⢹⣿⣿⣧⣶⣾⡟⢹⣿⣿⡆⢸⢸⣓⣀⣹⣿⡀⣠⠀⠀⠀⢀⣀⡠⠀⣷⣾⣽⣷⣦⣿⢃⣼
⣰⣯⢯⢽⣿⣿⠏⠉⢻⡿⢤⣤⡿⢿⣿⠿⣿⣿⡿⣿⢟⣣⣤⣼⣤⣿⣿⣿⣿⣿⣧⡀⠆⣘⠉⢰⠦⣼⢳⣿⣿⣿⣿⣾⣿
⣿⣿⣛⡤⢛⡁⢰⣤⣿⣿⣿⣿⣷⣿⣇⡀⠘⣻⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⣙⣿⣿⠁⠫⣏⣻⣸⣛⣿⣿⢿⣿⣿⣿⣿⡿
⣿⣭⣥⣤⣾⣟⣀⣿⣿⣿⠿⢻⣿⣿⣿⣛⠋⠉⠉⢉⠉⠈⠉⠉⠉⠛⠛⠛⠙⠛⠛⢻⡛⠛⠛⠛⢻⣿⠛⠚⠛⠛⠛⣋⣉
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⠯⣄⡈⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⢁⣠⢽⡀
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠉⠓⠦⣀⠉⠳⢤⡀⢀⣀⡤⠤⡤⡶⠶⢶⠤⡤⢤⣀⡀⢀⡤⠚⠉⣀⠴⠊⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠈⠳⣄⣀⡉⠋⢻⠀⠀⣷⡇⠀⢸⢸⡇⠀⢸⠟⣁⣀⣠⠞⠁⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢀⣰⠛⠃⠁⠀⢸⡀⠀⣿⡇⠀⢸⣼⠀⠀⣿⠀⠉⠘⠓⣆⡀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⡖⠚⠀⠀⠀⠀⠀⠈⠳⠶⠋⠑⠶⠚⠈⠳⠖⠃⠀⠀⠀⠀⠀⠓⢢⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠒⠒⠒⡇⠀⠀⠀⠀⠀⢾⠋⢸⣿⣷⠈⣧⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢸⠒⠒
⠀⠀⠀⠀⠐⠚⠛⠛⣧⠀⠀⠀⠀⠀⠈⠳⠬⣿⠯⠔⠁⠀⠀⠀⠀⠀⠀⠀⠋⢱⠖⠻⠀⠁⠀⠀⠀⠀⡞⠛⠓⠂
⠀⠀⠀⠀⠀⠒⠋⠉⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⢻⠃⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠙⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠚⠢⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠉⠉⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣠⢖⣜⣩⣥⣤⡼⣮⡟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠳⠞⠁⠀⠀⠀⠘⠋⢠⠾⠁⣾⣧⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠻⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣄⠀⢀⣮⣿⣷⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⢤⣀⠀⢰⣿⣿⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣀⣼⣿⣿⣿⣿⣾⠀⢀⣀⠤⠐⣂⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⣌⠱⣦⠹⣿⣆⣼⠇⠀⠀⠲⡄⠀⢠⣀⠀⢀⣠⡖⣩⢽⣻⣿⢿⣿⣿⣃⡴⢋⣴⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡝⢿⣧⠈⢻⣿⡿⠃⠀⠀⠀⠀⠙⢦⡐⠬⠭⠍⣀⣾⡃⠀⠉⠁⢀⣙⢿⡃⣰⣿⠟⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣇⢰⣿⣦⣼⣿⣿⣷⣦⣤⣀⡂⠀⠈⠻⣵⣒⣪⠟⠁⠀⠇⣠⣾⣿⣿⣿⣿⡿⣿⣧⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⠻⣏⣿⣿⡿⣿⣿⣿⣿⣿⣧⡀⠀⠉⠀⣤⡀⠀⠀⣀⣼⣿⣿⣿⠿⠋⣿⣷⢜⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣎⠛⣿⡇⠀⠝⠻⢿⣿⣯⣿⣶⣶⠞⠛⠳⣶⣾⣿⣿⣿⠟⠉⠀⣰⣿⣉⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠻⣿⢻⣦⣄⣀⣀⠼⣋⣽⢁⡘⠣⣀⠜⣃⣈⣿⣯⡱⢲⢶⣾⣿⣿⣍⠁⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡂⠀⣞⣫⡞⠛⠟⠋⣠⣾⡟⠉⠛⠿⣷⣄⣴⣿⠟⠉⢙⣿⣮⠳⣝⡧⣻⣏⣁⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⣿⡿⠸⡞⢰⣿⣿⢿⠲⣶⣦⢬⣿⢥⡴⡶⠛⠻⠿⠻⡇⣿⣧⣿⣟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡘⣗⢻⣇⣼⢿⣦⣶⣴⠦⣶⠐⣂⣳⡾⢟⣄⣸⡇⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣹⣎⢿⡁⣰⣏⣉⣉⣁⣈⣀⣈⣉⣄⠈⡿⠋⣰⣿⣿⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢿⣷⣧⣯⢽⡟⠋⠉⠉⠉⠉⠻⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠙⣷⡀⠈⢁⠀⠀⠀⠀⠀⠰⠋⣼⡿⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣿⡿⣿⣿⣷⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡿⢷⡜⣿⣿⣿⡿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠌⠛⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠛⠷⢬⣙⣻⣿⡿⠿⣿⣿⣷⣶⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠟⠂⠀⣉⠉⢤⣤⣤⣼⣿⣿⣿⣿⣿⡟⠀⢀⠀⣀⣹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣽⣿⣿⠤⢠⣐⣾⠏⠀⠈⠉⠿⣿⣿⣿⣿⣿⣅⡀⣼⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠾⣿⣿⡷⠒⣾⣿⣿⣤⠀⠀⠐⠂⣬⣿⣿⡿⠛⢿⣿⠛⠹⠟⠉⣹⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠚⢛⡯⢤⠀⢿⣿⣿⣿⣦⣤⣀⡀⣿⡿⠿⣿⡆⢀⡇⣠⣶⣤⣈⠙⠛⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇⠰⣻⣾⣤⣤⠜⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠈⠈⠿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⣿⣿⣿⣿⣿⣿⣮⡟⠛⠿⠿⠿⠿⠿⠿⢿⡤⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠴⣿⠋⠀⠙⠛⣻⣿⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡟⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡛⠳⢤⣀⣈⣼⣿⣿⣿⡿⢋⠇⠀⢀⣠⣴⣶⣤⣤⣀⠀⠀⠀⠀⢀⣴⠿⠿⣿⣷⣿
⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣶⣤⣉⡙⠻⢿⣿⣻⠂⠁⠀⠘⠋⠉⠀⠀⠈⠉⠛⠃⠀⠀⢀⡌⣁⣤⣤⡿⢟⡇
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⡛⢻⣶⣾⣿⣿⣿⣿⠀⠀⠀⠀⣠⣴⣶⠿⠦⣬⠏⠀⠀⠀⣼⢿⣯⣉⠻⠗⠊
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠉⠉⠀⠀⠐⠒⠀⠀⠀⠀⠸⠀⠉⠀⠀⣄
⠀⠀⠀⢠⣾⣶⣿⣿⣿⣿⣿⡿⠛⢿⡛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⣻
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡄⠺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣤⢀⣠⠇⠀⠀⡿
⠀⠀⠀⠀⠈⠻⣿⣟⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠁⠀⠀⠀⠀⢈⣁⣀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣧⣼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠲⣖⣛⠛⠛⠛⢻⡿⠁⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠛⠉⣹⠁⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⡿⠃⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣤⣀⣀⠀⠀⠀⠀⢀⣼⠀⠀⣤⣶⡤⠶⠶⢤⣀⠀⢀⡆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠙⠛⠛⠛⠁⠀⠀⢘⠓⢲⡶⢶⣄⠈⠛⢿⡅⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠸⠿⠷⠀⢀⠠⠰⠒⠒⠚⠓⠶⠿⢦⡈⢻⡀⢰⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⣀⣴⣿⣿⣿⡆⠈⠓⣶⡶⠶⠖⠶⠶⠶⣄⠙⠿⡀⠈⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠙⠻⠤⠒⠒⢤⣶⡀⠈⠳⠄⠈⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⣿⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⢸⣇⡿⣦⡀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⣸⡏⣰⠉⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣷⡄⠀⠀⠀⠀⢠⡟⡰⠃⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡟⠁⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⡟⠁⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠟⠋⠀⠀⠈⣿⣿⣿⣿⣿⡇⠀⠀⢰⡟⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡄⠀⠀⠀⠈⠃⠀⠀⠀⠀⠈⠣

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢘⠃⠀⢀⣀⠀⠀⠉⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣀⣴⣿⣿⣾⡆⣀⠉⠉⠛⠻⣷⡞⠻⡆
⠀⠀⢀⡴⣞⣿⣿⣿⣿⣿⠉⣀⣀⡰⣀⡀⣿⣿⠀⠗
⢀⡀⢫⣴⣋⡵⠶⠋⢁⣠⣤⣿⣿⠧⠈⣙⣿⣿⡇
⢊⣤⣪⡿⠋⠀⢀⣼⡿⣿⣿⠿⠁⠀⠲⠘⠿⠿⣧⡀⠀⠃
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⣄
⣸⣿⠏⠉⠉⠁⠐⠒⠙⣿⢷⣄⠀⠀⠀⠀⠀⠀⢶⡀⠀⠙⠿⣿⣿⣤⡀
⣛⣿⣄⢠⣤⣤⣴⠶⢶⣿⣮⣿⡂⠀⠀⠠⠐⠇⠘⢿⣶⣄⣀⡀⠈⠛⠿⢧⣤⣀
⣿⣿⡿⣨⣿⠀⢀⠀⠀⠀⠙⠿⣷⡀⠀⠀⠀⠀⠀⠀⣼⣹⡏⣿⡗⢶⣶⡈⣿⣿⣷⣦⣤⣀
⠀⠈⠹⣿⣿⡇⠉⠈⠉⠐⢄⠀⠈⢻⣄⠀⠀⠀⠀⠀⣿⣿⠀⢹⠀⠺⣿⣷⣮⠉⠉⠈⠒⠽⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠐⢠⣀⣿⡳⣤⣤⣤⣤⣿⣿⣘⣸⣿⡒⠚⠛⠛⣺⠟⠁⠐⠀⢈⠛⢿⣿⣯⣁⡀⠀⢠⣤⣤⣤⣤⢤⣤⣴⣾⣿⣧⡀
⠀⠀⠀⠘⡞⣿⡇⠀⠀⠀⠀⠀⠀⠙⠛⠿⢶⣴⣗⣉⣿⣿⣿⠿⣏⠻⠻⠟⢻⣏⡛⠛⠛⠛⠋⠁⠀⢉⡉⠉⠉⠛⠻⡍⢉⢻⡍⠀⠁⠀⠉⠉⠙⠻⣧⡀
⠀⠀⠀⠀⣷⠘⣿⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠒⠛⠟⠒⠒⠲⠦⠬⠿⠶⠶⣶⠶⠶⠚⠛⠛⠛⠉⠛⠛⠚⠺⠷⠀⠀⡀⠀⠀⢀⣀⡈⠻⣦⠴⢠
⠀⠀⠀⠀⠘⠇⠈⢳⣄⣈⣀⢀⠀⢀⢆⡀⢀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣄⠀⠀⠀⠀⢀⣾⠨⢯⢤⣄⠤⠀⠩⠄⣹⣿⡇
⠀⠀⠀⠀⠀⠸⠟⠉⠉⠊⠛⠛⠷⢶⡂⠷⠛⠿⡛⠒⠿⠶⢿⣶⡾⠶⣾⣷⣿⣿⣶⣶⣶⣶⣖⣒⠒⠒⠶⣒⣆⠻⠾⢿⣶⡶⠾⠗⢒⣀⠰⠠⢠⣤⣤⣤⠙⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠉⠀⠀⠘⠋⠀⠀⠀⠛⠃⠀⠀⠘⠛⠀⠀⠀⠉⠉⠒⠛⠒⠒⢲⣿⣿⡃⣀⡈⡛⠛⠀⠀⠘⠿⠃⠀⠀⠻⠿⠁⠀⠈⠉⠛⠛⠁⣿⡁
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣛⡛⠛⠓⠛⠻⠛⠛⠛⠛⣛⡛⠛⠉⠙⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⢿⠿⠟⠛⠛⠛⠛⠛⠿⠿⠿⠿⠷⢾⣷⣶⣶⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣼⣿⣿⡄⠈⠛⢿⣿⣿⣿⣿⡿⠛⠉⢠⣿⣿⣧⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢰⣿⣍⠙⢪⣿⣿⡿⠀⠀⠶⠄⠙⢿⡿⠋⢀⠀⠀⠀⢿⣿⣿⡕⠋⣠⣿⡆
⠀⠀⠀⠀⠀⠀⠈⣟⢿⣧⣰⣿⣿⣠⡀⠈⠂⠀⠢⣄⣀⠔⠁⠀⠁⢀⣤⣿⣿⣆⣼⡿⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣹⣿⣿⢿⣿⣿⣦⣀⡀⢀⣙⣫⡀⠀⣚⣴⣿⣿⡿⣿⣿⣟⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢹⣿⡇⠀⠙⠛⠷⢟⡿⠛⠙⠋⠛⣿⡛⠺⠛⠋⠀⢸⣿⡏⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠀⣿⣷⣶⣾⡟⠋⠻⣿⣶⣶⣿⠟⢛⢻⣷⣶⣾⣿⠀⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢟⣿⣿⡏⢷⣤⡈⠛⠛⢡⣼⡏⢿⣹⣿⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⠋⠈⣏⣼⢧⣄⣧⠯⣽⠯⠽⣽⣁⡼⣧⣼⠀⠙⣹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣴⣿⡿⣦⣤⣤⠶⠾⠶⠶⣤⣭⡼⢿⣿⣦⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢷⣴⣯⠴⠟⠛⠛⠻⠶⣶⣦⡼⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀⣀⣀⠲⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⢀⡴⠖⠛⠛⠛⠿⢿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⠿⠛⠛⠛⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣀⡤⡤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⠾⣿⡿⣿⣿⣾⣤⣀
⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⢿⡛⢻⡟⠈⠙⠛⠓
⠀⠀⠀⠀⠀⠈⠻⢿⣷⣶⣦⣤⣴⣶⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠘⠏⢸⣇⢻⣿⠟⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⢀⣾⣿⡀⠀⠀⢀⣶⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⢠⣤⣤⣼⣿⣿⡇⠀⢸⣿⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠸⣿⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠹⣿⣿⣿⣿⡇⠀⠀⣿⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⣿⡀⠀⠀⠀⠹⣿⣿⢻⣿⠀⠀⢸⣿⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⠙⢿⡄⠀⠀⠀⠹⣿⣆⣿⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠙⣏⠛⠀⠀⠀⠈⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠠⠀⠀⠐⠇⠀⣘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠰⡿⣤⡄⢠⣤⢻⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⢿⡆⠀⢠
⠀⠀⠀⠀⠀⠀⢁⣉⠡⠄⣩⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⢀⡀⠀⠀⠘⣇⠀⠀⣆⡀⠀⠘⣿⣇⢸
⠀⠀⠀⠀⠀⠀⠈⠣⣴⡶⠞⠁⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠙⢦⠀⠀⠹⠀⣼⣿⣿⡄⠀⠘⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⢾⣿⣿⣧⠀⠀⠏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠛⠉⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⢷⣄⠀⠀⠀⠘⢿⣿⡄⢷⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⡇⠀⠀⠀⢀⣼⣏⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠻⣆⠀⠀⠀⠈⣿⡿⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡇⠀⠀⠀⠛⠩⣿⠻⠿⠿⠿⠿⠻⠿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⡽⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣿⠷⠀⠀⠀⠀⠀⢻⣦⣬⠋⠐⠀⢠⠄⠈⠛⢿⣿⣿⠟⡁⠀⠀⠁⠀⣶⣤⠃⣠⣴⣿⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⠁⠀⠀⠀⠀⠀⠻⠟⠛⠋⠀⠀⠀⣠⠴⠒⠀⠺⢿⣇⠈⠛⠀⠀⠀⠀⠀⠉⠀⠛⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⡀⠀⠀⢀⠄⠀⠀⠀⠀⢀⠞⣠⡖⠂⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⡠⠀⠀⢀⣀⣴⡾
⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⠟⠀⠀⠀⠀⠀⠂⠁⡀⠀⢀⡤⠔⢁⠀⠛⣿⣶⣦⣀⠀⠠⠀⠀⢤⠀⢀⣀⣀⠤⠶⠾⠟⠿⠟⠁
⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⡀⠀⠀⣤⣄⠀⠀⣴⠏⠁⠰⢻⣿⣾⣿⣧⣿⡤⢸⣿⠀⠀⠘⠆⠀⠉⠁⢀⡁
⣿⣿⣿⣿⣶⣶⣦⠀⠀⠀⢀⠀⣀⡴⠾⠿⠚⠃⣰⡖⣰⣿⣿⣠⡞⠠⠀⠀⠰⠿⠛⠛⢹⠛⣾⠀⣼⣿⣆⠀⠀⠀⠀⠀⠐⠋⠁⣀⡤⢊⣠⣶
⣿⣿⣿⣿⣿⣿⣿⣿⡀⢰⣿⣿⡇⢠⣀⠀⠀⠙⣿⣿⣇⠉⠛⠛⠀⠀⠀⠀⠀⡀⢀⣤⣼⣨⡋⢀⣿⣿⣿⡄⠀⣠⣿⣶⣶⣤⣤⣤⣶⣿⠟⠉
⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣷⣸⡇⠀⣀⡀⠙⠻⣿⣦⡄⠀⠀⠀⠀⠀⠋⠁⠀⠈⣻⣿⠃⣼⣿⣿⣿⡇⠀⠀⣽⣿⣿⣿⣿⠟⢻⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠆⢸⣿⣿⣿⡇⠀⠀⠈⠱⠶⣤⡀⠙⢷⣄⣀⡀⠀⠃⠀⠀⠀⣠⣿⡏⢠⠈⠈⠙⠋⠀⠀⣴⠟⠻⠿⠏⣿⠀⢸⣿⠀⢠
⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣇⠀⣤⣀⠀⢠⣿⡟⠛⠐⠛⠛⠛⠷⣜⠃⠀⣾⣿⣿⣧⠈⠀⠀⢶⡀⠤⠾⠟⠀⠀⠀⠀⡏⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣶⠋⠀⣹⡇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⡿⠛⠃⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⣿
⣿⣿⣿⣿⣿⣿⣿⠿⠂⠘⠛⠉⠀⠉⠀⠀⠀⠚⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣃⠀⠀⠀⣷⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡁⠀⢠⠰
⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⢸⣷⠀⢸⡄⠀⠀⢠⣤⠈⠀⣁⠀⠈⠉⠁
⣿⣿⠋⠀⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡥⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⣸⠿⢧⠀⠁⠀⠀⠾⠟⠀⠠⠿
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⣶⠀⠀⣾⠀⠀⠀⠀⠀⢤⣿⠀⠀⠀⠀⠍⠀⠈⠁⠀⠀⠀⠐⠀⠄
⠀⠀⠀⠀⠀⢀⡦⠀⠀⠀⠸⠏⠀⠀⠀⠀⢀⡀⠊⠀⠁⠉⠉⠀⠀⠀⠀⠀⣰⠧⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⢠⡾⣤⣤⡀⠀⠀⠀⠀⣤⠤⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠐⠄
⠀⠀⠀⠀⠀⠀⢸⣿⠃⡴⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠉⠰⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠉⠀⠀⠁

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

⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠘⠀⣹⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣋⣥⣶⣿⠿⠛⣿⡟⢿⣿⡇
⠀⠀⠈⠀⠉⠛⣆⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⠿⢛⣉⢩⡴⣯⣴⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠰⢶⢦⣬⡓⠈⠀⢘⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⢉⣁⠈⠉⠉⡿⢻⡟⣱⣿⣿⣿⡿⣿⡿⢩⡶⠾⢿⣧⣭⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢠⠤⠬⢥⡿⠍⠠⢺⡄⠘⣿⣿⣿⡿⠟⠋⢁⣴⣿⠿⠻⢷⡆⠁⠸⣿⢾⣿⣽⣶⣿⠟⢡⡿⣶⠿⠶⣯⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢨⠙⠻⠥⢽⣆⠀⠈⠀⠀⢻⡉⠙⠃⠀⢀⣼⣿⣯⠘⠁⣻⠁⠄⠀⣨⣿⣿⣿⡟⠁⢀⡿⠓⢻⣾⣛⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢨⣿⣿⡿⣿⣿⠠⢵⣌⣃⠈⢷⣄⠀⠀⠙⢿⣿⡏⠀⢀⡙⠀⢀⣼⣿⣿⠿⠋⠀⣰⡿⠤⢯⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢰⣶⠶⠶⠋⢿⣿⣿⢦⡉⠤⠘⢻⣷⣤⣀⡘⠏⠁⠀⠈⣧⣴⣾⣿⡿⡏⢀⣤⣾⣟⣻⣿⣾⣿⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣷⡶⠞⣻⣐⣹⣶⣏⣜⣶⣹⣶⣿⡿⣿⣿⣶⣄⡀⢀⡉⣽⣿⣿⡄⢹⣿⣿⣷⣿⡛⠀⠊⠙⠋⠐⣻⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⠛⣙⣋⡽⠖⢋⢿⣷⣿⡟⠁⢩⣷⡀⠈⠻⣷⣼⡨⠻⣿⣧⣼⣿⡿⣴⡏⠀⠀⠀⠀⠀⠘⠁⡿⢻⣿⣿⣿⣿⣿⡇
⠀⠀⢀⠈⠃⠈⠿⠏⢡⣴⢾⣥⠟⢀⣿⡧⠀⣸⣿⠟⣤⣀⣹⣿⡿⣦⣾⣿⣿⡉⣠⣬⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣼⠿⣿⣿⣿⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠛⢫⡿⣥⡾⢟⣿⣷⡀⢙⣿⣷⡽⠟⠋⠁⠀⣿⣿⣿⣝⣿⣟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠙⠿⣿⡇
⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠰⢿⣷⣿⣿⠀⢘⣿⢿⢿⣿⣶⣶⣷⣦⣽⣾⣿⣿⡿⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡇
⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠉⢁⡾⠇⠀⢸⠋⣸⣿⠟⠀⠙⠀⢟⠙⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⠟⣿⡟⠀⠀⠈⡆⠸⢀⣿⣿⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠃
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⣶⣿⡇⠀⠀⠠⣷⡀⢸⣿⣿⣿⣀⠀⢀⠂⠴⣶⣄⡀⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⡀⡄⣿⣇⢸⣿⡟⣿⣿⠃⠀⠀⠀⠹⡛⣿⢧⡴⢃⠆
⠀⠀⢠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣤⠀⠘⣿⣀⡶⢹⣿⣿⣿⢸⡿⠁⢻⣿⠀⠀⠀⠀⠀⠀⠹⠁⡼⢻⣾⡡⢊⡤
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡀⢰⣿⣿⡁⠘⣿⣿⣿⣿⣇⠀⢾⣿⠆⠀⠀⠀⠀⠀⠀⠈⠀⢉⣿⢿⣿⣷
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣋⡗⠈⣿⣿⣿⣿⣏⢠⠂⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢈⣿⢻⣥⡀
⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠛⠞⠢⠹⣿⣿⣿⡇⡞⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠼⡃⢛⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢸⣇⠀⠀⣰⣄⠙⢿⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠂⡸⠗
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢸⣿⠀⠀⢹⣿⣿⣆⡉⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⢔⣿
⠀⠀⠀⠀⢸⣿⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠾⠠⢘⡿⠀⠀⣿⣿⣿⣿⣷⡀⠀⠨⡇⠀⠀⢀⡀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠉⢴⡂
⠀⠀⠀⠀⣼⣿⠟⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⣇⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⢉⠠⣦⣼⣿⡄⣯⣤⣿⡄
⠀⠀⠀⢸⣿⠿⠶⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⢦⣿⡄⠀⢸⣿⣿⣸⡿⣿⠀⠀⢨⡗⠈⠻⣿⢧⡏⠑⢯⣿
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢼⠸⣿⣿⡿⣿⡿⣿⠿⠃⠀⢀⣠⣾⣷⠀⠀⢯⡚⠀⠀⠈⢻⣧
⠀⠀⠀⠀⢿⡀⠀⠀⠀⡀⠀⠚⠀⠀⠀⠀⠀⠈⠣⠟⢀⣻⣿⣿⣧⣀⡄⢠⣿⠃⢿⣿⢀⡀⢸⣿⡀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⢸⣧⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢊⣭⡾⣿⣯⡙⠛⠹⣿⣯⣀⡘⣿⣿⣇⣸⣦⣽⣤⡀⠀⠀⢻⣧⣀⡴⠃
⠀⠀⠀⠀⠸⢿⣷⣄⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⣷⣯⢑⣿⣼⢿⣷⣎⠑⣸⣿⣿⡟⢻⣿⣿⣛⣿⡿⠿⣷⠄⠀⣠⣿⣯⣤⣤⣴⡇
⠀⠀⠀⠀⠀⠀⠙⠛⠓⠲⠶⢖⣒⣶⣿⡌⣷⣿⡏⣤⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢉⣸⣿⣿⣿⣿⣿⣶⣦⠘⣿⣿⣿⣿⣍⠁
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⠏⠙⠻⠿⣎⣹⣿⣿⣿⣷⣦⣈⠹⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡇
⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⡄⢀⠀⢸⢏⣹⣿⣿⣿⣿⣿⣿⣷⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣴⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⡌⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣏⣩⣥⠈⣉⣹⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠾⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣏⣀⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠅⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣏⣫⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⣹⣾⡿⠿⠿⠿⠛
⠀⠀⠀⠀⠀⣀⣀⣤⣦⣤⣤⡄⢉⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⢹⠃⠀⠀⠐⣰⠟
⠀⠀⠀⠀⠀⠉⠉⠀⠉⠀⠀⠀⠈⠻⣦⡀⠠⣀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽⣿⣿⠟⢻⡅⠀⢀⠀⠔⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢸⣧⣤⡿⢢⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣧⠀⠀⣿⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠟⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠈⠠⠀⣀⠞⢯⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠲⠒⠒⠒⠘⢙⣁⡀⠀⠙⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⣖⡉⣹⠀⠀⡔⠁⠀⠈⣆⠀⠀⡧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⢞⣓⡇⠀⢠⡏⠿⠭⠝⡟⠀⠀⡿⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠘⠁⢀⡇⠙⠒⠐⠚⠁⣤⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠓⠚⠀⠀⠀⠀⠀⠚⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠑⠢⠤⠤⣤⢀⣀⡀⣀⡀⠠⠤⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⡖⠒⠠⠤⠤⢤⡋⢦⠤⠤⠖⠒⣞⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡴⡇⠀⠀⠀⠀⠀⡟⡏⠀⠀⠀⠀⡏⠳⡌⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡞⠀⡇⠀⠀⠀⠀⠀⡇⢱⠀⠀⠀⠀⡇⠀⠱⡀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⡇⠸⠀⠀⠀⠀⣇⠀⠀⡇⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡌⠀⢠⠁⠀⠀⠀⠀⠀⠁⠈⡇⠀⠀⠀⣿⠀⠀⡇⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡖⠳⣤⠐⠀⠀⠀⠀⠀⠀⢣⣰⢃⠀⠀⠀⠁⢠⡞⠉⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣸⠊⢦⡀⠀⠀⠀⠀⠀⡹⢠⠀⠀⠀⡤⠘⠤⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠛⢵⣒⣂⡤⠤⠧⢄⣐⣲⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠈⠉⠉⠉⠉⠹⡄⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠸⡀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢇⠀⠀⠀⠀⠀⠀⡜⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⠀⠀⢿⡧

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣚⣭⢽⣿⣄⠀⠀⢀⣴⣾⣿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠬⣤⣀⠙⣿⠀⠀⠚⠋⠁⠀⠀⠀⠐⠒⠶⠶⣶⣶⣤⣄⣉⣿⠿⣿⣿⣦⡀⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠮⣽⣇⠈⢤⡈⠙⠶⣤⣀⡀⠀⠀⠀⠁⠀⠀⠈⢛⣿⠟⠻⢦⣄⠙⢿⣿⣆⡠⠀⣿⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠀⠀⠀⠀⠻⣶⣄⡀⠉⠙⠻⠷⣶⣦⣤⣤⡶⠟⠁⠀⠀⠀⠉⠳⣦⣹⠟⠁⣰⠏⢰⡏
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⡀⠀⡀⠀⠀⠀⠀⠈⣿⠻⢿⣶⣤⣄⣀⣀⣀⣠⡤⠀⠀⠀⢀⣠⡤⠖⠋⠉⠀⣀⣁⣠⡾⣱⠇
⠀⠀⠀⠀⠀⠀⢰⣿⠀⠠⣈⠑⠢⠄⠀⠀⠀⠀⣿⠀⠀⠀⠉⠙⠛⠻⠿⠿⣿⣦⣄⣀⠀⠀⣀⣠⣤⣶⣿⣋⣉⣡⣾⣿⡞
⠀⠀⠀⠀⠀⠀⣿⣿⢶⣀⡈⠛⠶⢤⡤⠖⠒⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⡿⠿⠛⠛⠉⠘⡛⠛⢹⣿⡟⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣧⣀⠀⠀⣀⣠⣴⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡑⢸⠋⠁
⠀⠀⠀⠀⠀⠀⠈⣿⣇⠙⡛⠟⠛⠋⠛⣉⢠⠞⠁⠀⠀⠀⠀⠀⠶⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⡌⣷⢸
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⣍⡛⠶⠖⢋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⣦⣤⠀⠀⠀⠀⠿⠋⠉⠙⠇⠁⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢉⣧⣬⣙⡛⠛⠋⡀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⢀⠄⢠⣤⣤⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣩⣭⣙⠃⠈⠁⠀⠀⠈⠓⠒⠈⠑⠶⠖⠛⠛⠦⡄⠄⠀⠀⢰⠇⢰⠏⡀⠈⡇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠘⢁⣄⡙⣆⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠈⣉⠀⠀⢀⠀⠀⣿⠀⠈⢰⡿⠀⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⣯⠈⠙⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠀⣀⠈⠀⠄⢀⡀⠀⠸⠃⠐⠁⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡹⢄⡀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⡿⠶⠖⠀⠀⠀⠀⢀⡀⠀⢈⣷⠀⠃⠰⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡉⠋⠀⠀⠀⠀⠀⠈⠁⠀⠈⠳⢦⣀⠀⠀⠀⠀⠈⠁⠠⡾⠟⠀⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⢶⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠶⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠤⠬⠑⠲⠤⠒⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡜⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⢀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣹⣧⠀⠀⠉⠻⢶⣄⣀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠑⢬⣿⡄⠀⠀⠀⠀⠈⠉⠛⠳⢶⠶⠶⠛⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠈⠙⠓⢶⠦⡀⠀⠀⠀⠀⢸⡷⠰⠶⠞⣩⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⢀⡏⠀⠹⡄⠀⠀⠀⢸⣿⡀⠀⠀⣧⡀⠈⠓⠶⣶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡿⢦⡀⠀⠀⠀⠀⣼⠁⠀⠀⠙⢦⡀⠀⠀⢿⣷⡀⠀⡏⠹⣶⡴⠞⠁
⠀⠀⠀⠀⣀⣀⣠⣤⣤⠞⠋⠻⢧⣄⡀⠹⣄⠀⠀⣴⠃⠀⠀⠀⠀⠀⠙⢤⡀⠀⠻⣷⢀⡿⠦⣤⣄⣀⣀
⢀⣴⣾⣿⣿⠿⠛⠻⣿⣦⣄⣀⠀⠈⠙⠛⠛⣦⠞⠷⠤⢤⣤⣤⣤⣤⣀⣀⣙⣦⣄⣨⣿⣡⡴⠟⢁⣩⣿⣿⠿⠷⣶⣄⡀
⣿⠿⢿⣿⣿⣶⣤⣀⠀⠉⠛⠿⢿⣿⣶⣶⣦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣤⣴⣶⡿⠿⠋⠁⣠⣴⣿⣿⠿⣦⡀
⠀⠀⠀⠙⠻⢿⣿⣿⣿⣶⣤⣄⡀⠀⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⢁⣀⣤⣶⣿⡿⠟⠋⠁⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣿⣿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠋⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⢶⣦⣄⣀⣴⠏⠀⠀⠀⠀⠀⠀⢿⣦⣤⣤⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠈⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⢀⡄⠀⠀⢀⡄⠀⠀⠀⣤⠀⠀⠀⢿⣏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣸⡇⠀⠀⢸⣿⠀⠀⠀⣿⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠙⠀⠀⠀⠈⠃⠀⠀⠀⠘⠁⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠰⣿⠆⠀⠠⠔⠒⠁⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⢀⣤⣶⠃⠀⠀⠀⠀⢸⣶⣤⡀⠀⣸⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⣰⣿⣿⣿⣷⣄⣀⣀⣤⣿⣿⣿⣿⣶⠃⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⣼⣿⣿⣿⡿⠿⠟⠛⠙⠻⠿⢿⣿⣿⣿⡇⠀⠀⣼⡿⠁
⠀⢀⣠⣤⣤⣴⣶⣶⣾⣿⣿⣄⠀⠀⠀⠀⠦⣀⠀⠀⠀⠀⣀⡤⠊⠀⠀⠀⣠⣾⣿⣷⣦⣤⣤⣤⣀⣀
⠀⢸⣿⠉⠉⠉⠁⠀⠀⠀⠈⠻⢿⣦⣄⡀⠀⠈⠉⠙⠋⠉⠁⠀⢀⣀⣴⣾⠟⠉⠀⠀⠉⠉⠉⠉⢻⣿⠃
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣶⣶⣶⣶⣶⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⠋⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⢸⣿
⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠋⠀⠘⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢸⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠿⠿⠛⠁⠀⠀⠀⠀⠙⠿⠿⢿⠆⠀⠀⠀⠀⣶⡆⠀⠀⠀⢸⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣼⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⣿⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿
⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠉⠉⠉⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⣀⣠⣤⣶⠶⠾⠟⠛⠛⠛⠋⠁⠀⠀⠀⣠⣾⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣶⣶⣤⣀⡀
⠀⢀⣴⣾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠋⢻⡆⠀⠀⠀⣠⡶⠀⠀⣀⣠⣤⣄⠀⠀⠉⠙⠛⠿⢿⣶⣤⣀
⠠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣦⣾⡇⠀⠀⡴⢋⣠⡴⠛⠋⠁⠐⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣶⣄
⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⣠⠶⣋⣥⡤⢤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡄
⠀⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⢶⣄⠀⠀⠀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠙⠆⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠃⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠁⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠐⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⠿⠛⠉⠀⠀⠉⠛⢿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⢿⣷⣤⣤⣤⣤⣴⣶⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣶⣤⣤⣀⣀⣀⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣚⡋⠉⢉⣙⣓⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠋⣀⠤⠴⠿⠋⠉⠉⠉⠉⠉⠉⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⡠⠴⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠊⠀⠀⠀⠀⠀⠀⠠⢎⠔⣀⡴⠋⠀⠐⠒⠲⢦⡀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⣰⣶⣄⠀⠀⠀⢿⣮⣥⠤⠄⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⢀⡴⢦⠀⠀⢸⡇⠀⠀⠀⠀⢹⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢚⣷⡶⡄
⠀⠀⠀⠀⠀⢀⡎⠀⣼⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣹
⠀⠀⠀⠀⠀⣸⠀⠀⡟⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⡼⠁
⠀⠀⠀⠀⠀⣿⠀⢠⡇⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⣿⠀⢸⠃⠀⠀⠀⠀⠹⣦⡀⠀⣤⣤⣤⠤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⢠⠔⠊⠉⠉⠉⢹⣾⡄⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⢧⡄⢀⣀⡀⠠⢾⡄⢷⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠋
⠸⣄⠀⢠⠄⠀⢾⡄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠋⠉
⠀⢯⣀⠀⠀⠀⢚⡷⠃
⣀⣌⣷⣶⣶⣞⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄⣀⣀⣀
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⣀
⠀⠀⠀⢀⡴⠛⠉⠉⠉⠉⠛⠛⠿⠿⣿⣶⣤⡄⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⡿⠿⠟⠛⠋⠙⠛⠷⢦
⠀⠀⠀⠀⠀⠀⡀⢀⣠⡀⡄⠠⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠛⠉⠁
⠀⠀⠀⠀⠲⣴⣿⣿⡿⣿⢻⣿⢿⠷⣵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤
⠀⠀⠀⠀⠀⠀⠾⢿⣦⣙⠨⠭⠊⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⢶⣶⣶⣶⣶⣿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠃⠛⠓⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠿⠛⠉⠉⠉⠉⠉⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠁⠀⠀⠀⠀⠀⠀⢀⠰⣬⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⢀⠀⠀⠀⢀⣾⡿⠃⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣴⠆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣯⢀⠀⠀⠀⠀⡠⠴⠿⠟⠉⢻⣿⣿⢵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡏⠉⠃⠀⠚⠁⠀⠀⠀⡄⠛⣿⡏⠀⠉⢛⣛⣋⡩⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣴⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣀⠀⢙⡟⠀⠀⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣷⣦⣤⣿⡿⠛⠁⠀⠀⣀⣀⠀⠀⠀⠀⢀⣀⣠⣾⣿⣿⣿⡿⠀⠀⠀⠀⠔⠋⡩
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠛⠋⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠐⢿⣿⣿⠿⠛⣻⠿⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠛⠀⠀⠀⠁⠀⠀⠀⠀⣐⣀⣠⠤⠖
⠀⠀⠀⠀⠀⠀⠐⢾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠂⢀⠀⠀⠀⠂⡄⠀⠀⢀⣽⠃⣠⣴⡶
⠀⠀⠀⠀⠀⠀⠀⣀⣻⣿⡇⠀⠀⠀⠀⠐⢾⠀⠀⠀⠀⠀⠤⠞⠁⢱⠀⠀⠀⡀⠀⠀⠀⠀⡰⠛⠁⣰⣿⣿⣀
⠀⠀⠀⠀⠀⢀⣤⣼⣿⡿⠁⠀⠀⠀⠀⠀⠀⠐⠁⢠⠊⠀⠀⠀⠀⠘⠁⢀⠀⠀⣢⣀⣰⠾⡁⠒⠀⠈⠛⢿⣿
⠀⠀⠀⠈⠻⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⠖⢁⠀⠀⣠⣾⡟⠉⠁⢀⣸⣦⣀⠂⢀⡶⣭
⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⢀⠄⠀⢀⠄⢀⣤⣀⣀⡴⠃⡀⠀⢠⣾⡋⠉⠉⠉⢋⣽⠟⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣾⣿⡇⢀⣀⣤⠖⠁⠀⣀⠈⡄⣼⣿⣿⡆⠀⡀⠀⣴⠛⠋⠃⠀⠀⠀⠀⣠⣼⣇⣾⣟⣼⣿⡿
⠀⠀⢠⣴⣿⡿⠿⠿⠷⠈⠋⠁⠀⣾⣿⡏⠸⣰⣿⣿⣿⣷⡀⢀⡾⠃⠀⡦⠠⡀⠀⠀⢠⠿⣿⡏⠼⠛⡿⠃⣴
⣤⣶⠟⠁⠀⠀⠀⠀⠀⠠⣄⡄⣾⡿⢟⣣⡴⠛⠿⣿⣿⣿⣿⣿⠃⠀⠚⠁⠀⠉⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠰⣄⡀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣷⡀⠀⠈⠙⠛⠛⠣⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣠⠀⠀⢀⣴⡆⣿
⣿⣿⠀⢸⣿⡉⠓⠒⠂⠀⠀⠀⠸⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠇⠀⠘⠛⠿⡷⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⠒⠯⣅⡀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢆⣴⠀⢙⣧⢸⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣷⠟⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⡀⠀⠀⠛⠛⣡⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣛⣿⣆⠼⠉⠛⠓⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⠀⣠⣴⣶⠀⠀⠀⠀⠀⠀⣧⣻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⡾⠿⣿⠏⠀⠀⠀⠀⢤⣤⣽⡅⠙⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣬⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⡟⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠉⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣷⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠐⢹⣿⡿⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⡎⠀⠁⠀⠀⠀⠀⠙⢷⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⣷⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⢼⣿⣿⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⣿⣿⣿⣊⣽⣿⣿⡿⠦⣤⡄
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⠀⠀⠀⠀⢀⠄⠀⠀⣧⠿⣯⣿⡿⢷⢿⣿⣿⣯⠽⢿⡅
⠀⠀⠀⣠⠴⢟⣫⣴⡮⠀⣈⡁⠠⠿⠀⠉⢀⡄⠈⢿⢍⣰⣾⠆⠀⠞⠃⢬⣶
⠀⠀⠀⠐⠛⠋⠉⠉⠉⠀⠉⠁⠀⠀⠀⠘⠻⠡⠀⣼⠿⣴⣶⣶⡀⣾⡦⣰⣿⣿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣻⣥⣬⣭⣶⣀⣽⣧⣼⠋⠀⠀⢢⣤⠀⠠⢤⣄⣀⡀⠀⠀⡠⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣔⣋⠩⣞⣿⣿⣋⣩⣽⣿⣿⣯⢿⡄⡈⠈⣿⡇⠀⠀⠀⠀⠉⠉⡻⠃⠀⠤⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠛⠃⠚⠒⠋⠙⠛⠃⠙⠷⠛⠛⠛⠿⠛⠛⠓⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣶⣶⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠟⠛⠛⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠛⠛⠿⢿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⠿⠇⠀⢰⣿⠿⢿⣷⡄⢰⣿⠿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀
⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⣿⡇⣾⡏⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣴⡦⠸⣿⣄⣴⣿⠃⢻⣿⣤⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠈⠛⠋⠁⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣴⣿⣶⣄⠀⣿⣿⠀⣠⣾⣷⣦⡀⣿⣇⠀⣸⣷⠀⣴⣿⣿⣦⠀⣶⣿⢿⣷⡄⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⣼⣿⠀⠉⠉⠀⢿⣿⠀⣿⣧⣼⣿⠃⢹⣿⠀⣿⡏⢸⣿⣧⣾⡟⠀⣿⣿⣴⣿⠇⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⢻⣿⡀⠀⣀⠀⢸⣿⠀⢻⣿⣋⣁⠀⠘⣿⣾⣿⠁⠸⣿⣏⣁⣤⡀⣿⣿⢿⣷⡀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠈⠻⢿⡿⠿⠃⠸⠿⠀⠈⠻⠿⠟⠁⠀⠙⠿⠋⠀⠀⠙⠿⠿⠛⠁⠻⠟⠀⠻⠟⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⣾⡿⠛⠃⠀⣠⣤⣄⡀⠀⢠⣴⣶⣤⡀⠀⠀⠀⣶⡆⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⢀⣀⣿⣧⣤⡄⢸⣿⠉⠻⣿⡄⢸⣿⢉⣿⡷⠀⠀⠀⣿⡇⠀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠘⠛⢻⣿⠛⠁⢸⣿⠀⠀⣿⡇⢸⣿⣿⡟⠁⠀⠀⠀⣿⡇⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⢸⣿⡄⠀⠘⣿⣶⣾⡿⠁⢸⣿⠛⣿⡦⠀⠀⠀⢻⣿⣿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡿⠿⠿⠿⠿⠿⠿⣿⣿⣷⣶⣶⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠂

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⢿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣄⣦⣠⣠⣀⣈⠙⠻⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠿⠛⠛⠛
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⢿⣷⣦⣌⠛⠻⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⢿⣧⣼⣟⣾⠟⠀⠙⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠛⠛⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⣤⣤⣤⣶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⠿⠿⠗⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠛⠛⠲⡄⠀⠀⢀⣠⠞⢛⣛⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡘⠄⣠⢎⡵⠿⠋⠉⠛⢦⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠑⣦⡉⠞⣁⣤⣀⠀⠀⠀⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⢠⠂⣀⠀⠀⠀⠀⠠⣏⣉⢉⠉⠙⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠃⢰⠁⠐⠩⠵⣶⠶⡯⠄⠤⡤⣼⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣽⣦⡘⢼⣆⠀⠀⠈⠁⠊⠁⠀⠀⠛⠻⡝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⠋⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠐⢀⢀⣤⣠⠄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⡇⠈⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⣀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⠙⠃⠀⠀⠀⠠⣦⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠖⠀⠀⠀⠀⠀⠀⠀⢀⠛⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠄⢶⠿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⢾⣀⠀⢀⣨⠉⠑⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠱⢤⣋⣛⣉⡕⠋⠀⠀⠀⢙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⢵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢸⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣇⣀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣟⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⢶⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣀⡀
⢀⣠⣤⣤⣶⣶⣶⡶⠾⠿⠿⠟⠛⠙⠋⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠒⠲⠶⠤⢤⣤⣤⣄⡀
⣞⠕⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠈⠙⠆⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻
⠙⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏
⠀⠈⠙⢿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠙⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠒⠛⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⣴⠏
⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⣈⡥⠶⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠻⣾⠿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠓⠒⠀⠀⠀⠀⠀⣰⡏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠘⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡎⠀⠀⠀⠀⠀⠙⠒⠒⠒⠶⠶⠶⠖⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡇⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠹⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⠿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⠃⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠟⠋⠉⠀⠀⠀⠈⠉⠙⠻⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡞⠀⠀⠀⢀⣀⣺⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣷⣄⠀⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣦⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣂⣴⠏
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⠂⢀⢈⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢁⡀⠀⠉⠸⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⡄⠉⠀⣠⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⠠⣄⡈⣩⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠀⢠⠞⣷⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⡇⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠈⠙⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠈⠑⠠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣠⡤⠶⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏⣠⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡟⣰⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⠿⠿⠛⠛⠛⠿⠶⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⣿⣷⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⣀⣀⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉⠁⠀⠀⠈⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⢀⣠⣤⣖⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⢀⡤⠚⠛⠚⠛⠉⠉⠛⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⡷⣀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢟⣿⣽⣿⣷⣄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⡕⣧⣾⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢵⣿⣿⣿⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⡎⠉⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠀⠀⠈⣿⡆⠀⠀⠀⠀⢀⣆⣀⣸⣻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣷⣤⡀⠀⠀⠀⠀⠘⠇⠀⠀⠀⢹⣧⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⢀⣀⣀⠀⣀⡀⠀⠀⠈⠙⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠙⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟
⠀⠀⠀⠀⠀⠀⢰⣯⣾⠛⠋⠈⠛⠛⠛⠛⠿⠿⠷⠶⣤⣄⣈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠗⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠈⠙⠿⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠞⠋⠉⠉⠉⠉⠉⠙⠂⠈⠙⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣦⣄⣀⠀⠀⠀⢀⣠⡴⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣤⠤⠼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠾⠿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣤⠶⠶⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠒⠶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣴⠶⠒⠾⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣉⣛⠳⣾⣿⠀⠀⠀⠀⢀⣤⣾⡟
⠀⠀⠀⠀⣤⣄⠀⠀⠀⢹⣿⣁⣤⠶⠖⠚⣿⣧⠀⢀⣀⣤⣴⣶⣶⣦⣤⣄⣀⠀⠀⠀⢀⣾⡿⣥⣈⠉⢳⣾⡿⠀⠀⠀⣠⣾⢿⣿
⠀⠀⠀⢠⣿⣿⡆⠀⠀⠈⣿⣿⣥⡶⠟⢛⣉⣿⣿⡿⠟⠛⠉⠉⠉⣉⣉⣽⣿⣷⣀⣴⣿⣟⠓⢦⣌⠻⣾⣿⠇⠀⣠⣾⡿⠃⢸⣿
⠀⠀⠀⣼⡇⠹⣿⡄⠀⠀⠸⣿⣿⣶⠿⠛⠉⠉⠉⠀⠀⠀⠀⠐⠛⠉⠉⠉⠉⢛⣿⡿⠉⠛⠷⣦⡙⣿⣿⣿⣴⣾⡿⠋⠀⠀⢸⣿
⠀⠀⢰⣿⠃⠀⢻⣿⡄⣠⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣦⣄⠀⠈⢻⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⣿⡆
⠀⢠⣿⠏⠀⣶⡀⢻⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠀⠀⣀⣩⣽⣿⣦⡀⠀⠈⢻⣿⣆⠀⠀⠀⠀⠀⢻⣧
⢀⣿⡟⠀⠀⣸⣷⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠉⠉⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⢸⣿
⣸⣿⠁⠀⠀⣿⣿⡇⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⣿
⣿⣿⠀⠀⢸⣿⣿⣧⠀⠀⠻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠾⢶⣿⣷⣄⠀⠀⢠⣿
⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⠈⢻⣿⣷⣀⣾⣿
⢿⣿⠀⠀⣼⣿⣿⣿⡄⠀⠀⠿⠃⠀⠀⠀⠀⢀⣴⠟⠋⠉⠉⠙⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣠⣤⣦⣤⡀⠀⠀⢿⣿⣿⡿⠃
⠸⣿⡇⢸⣿⣿⠿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⡾⠃⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠁⠀⠹⣿⠀⠀⢸⣿⡏
⠀⢻⣿⡄⠻⠃⠀⠘⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⡀⠀⢀⡄⠀⠀⠀⢠⣟⣠⣤⡄⠀⣿⡇⠀⢸⣿⠃
⠀⠀⠻⣷⣄⠀⠀⠀⠀⠈⠻⠆⢠⡀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣦⣤⣈⠛⠛⠟⠛⠁⢠⠀⠀⢸⣿⣿⣿⣿⠀⣿⠃⠀⣾⡿
⠀⠀⠀⢹⣿⣷⣦⣄⣀⣀⡀⠀⠀⠙⠛⠃⠀⣀⣴⣿⡿⠟⠛⠛⠛⢛⣿⣷⣄⠀⠀⠀⣿⠀⠀⠸⣿⣿⣿⡿⢰⣿⠀⢀⣿⠇
⠀⠀⠀⢸⣿⡏⠉⠉⠉⠉⠁⠀⠀⠀⢙⣻⡿⠛⣉⣠⣤⣤⣶⡶⠿⠛⠛⠛⠛⠂⠀⠀⢻⣇⠀⡴⠟⠛⠛⠻⢿⠏⢀⣼⠟
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠉⣱⠿⠋⠁⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣷⡀
⠀⠀⠀⠘⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠰⢿⢁⣤⡄⠀⢀⣀⢸⣷⡄⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄
⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠿⠇⠀⠀⠀⠀⢴⣦⠉⠻⣿⢴⣿⠏⣸⣿⠇⠀⢀⣀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠛⠻⢶⣤⣄⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⣷⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⣤⣀⣀⣀⣀⣀⣠⣤⣶⠟⣿⡟⠀⠘⠀⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣽⣏⣻⡟⢻⡏⢹⣇⣽⣿⡟⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣦⣤⣤⣤⣀⣀⣀⣀⣷⡀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢠⣷⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⣿⣷⣄⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣤⣄⣀⡀⠀⠀⢀⣀⣴⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡏⠙⠻⠿⠿⣿⣿⡿⠿⠿⠛⠁⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⠀⠀⠀⣧⠀⠀⢀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣄⣻⣧⣠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣬⣉⠛⠦⠀⠀⠀⠀⠀⠤⠚⠉⠡⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣟⢻⣿⣟⢿⡄⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠻⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⢦⡀⠀⠀⠀⢠⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣾⢿⣿⣶⡄⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⡶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣈⣉⣙⠛⠦⠀⠀⠀⠀⠠⠶⠛⠋⣉⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡛⢿⣻⡷⠀⠀⠀⠀⠀⠀⠀⠞⢻⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣤⣄⡀⠀⠀⠀⠀⢠⣶⠿⠛⠉⡀⠀⠀⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠈⠉⠁⠀⠀⠀⠀⠀⢠⣶⣿⣶⡾⣶⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⠟⠿⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⡟
⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠋⠉⠉⠉⠉⠙⠛⠷⣤⡀
⠀⠀⠀⠈⠻⠿⠿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣤⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠹⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⢠⣾⣿⣿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⣰⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣄⣀⣀⠀⠀⢀⣀⣠⡴⠟⠁
⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠋⠉⠁







⢀⣤⠶⢿⡟⠛⠛⠁
⠈⠁⠀⣸⡿⣆
⠀⠀⠀⣿⠁⠉⠳⣦⡀
⠀⠀⢸⡏⠀⠀⠀⠈⠙⠷⣦⣄⣀
⠀⠀⠻⢦⣄⣀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠈⠉⠛⠳⠶⢶⣤⣤⣤⣤⣄⣀⣀⣤⣬⣭⣭⣭⡽⠿⠿⠿⠿⠛⠛⠛⠛⠛⠻⠿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣆
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣾⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠐⠒⠤⢄⣀⣀⣀⣀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡺⠻⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⡜⠑⠀⠀⠀⣰⣶⣦⣤⡴⠤⢤⣽⣦⣤⣄⣀⣀⣀⣴⣶⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠁⠈⠀⠈⠀⢀⡀⠀⣼⣿⣿⣿⣿⣷⣄⠀⠸⣏⠉⠏⠉⠉⠉⠁⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⣿⣿⣿⡆⠀⠘⢦⠀⠀⠀⠀⠀⢰⣴⡆⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⣿⡆⢹⣿⡿⠟⠛⠛⣹⡇⠀⠀⠈⠳⡀⠀⠀⠀⠀⡿⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⡀⣿⡟⢩⡄⠀⠻⣆⣀⣠⣴⠿⠁⠀⠀⠀⠀⠈⣆⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠹⠾⠃⠀⠀⠀⠉⠁⠀⠀⠀⢀⣀⡀⣀⡴⠋⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢰⣦⣀⠀⠀⠀⠀⠀⢀⠀⣾⠉⠉⠉⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠜⠋⠙⠃⠀⠀⠀⢀⣼⠃⣿⡀⢠⡀⢀⡈⠛⢊⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠔⣿⠏⠀⢸⡷⡀⡉⠈⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣈⣈⣩⣷⣾⡟⠀⠀⢸⡇⢣⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠿⠋⠀⠀⠀⣾⡇⠀⠂⡠⠀⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⢀⠜⢻⠇⠀⢠⠁⠀⠀⠀⣸⡿⣳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡔⠠⢀⣀⠎⠀⠀⠑⢄⠂⠀⠀⢁⣾⡏⣾⣿⣣⣼⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠈⠙⠂⠀⠀⠀⠨⠀⠀⣠⣟⣼⣧⣿⣿⣿⣿⡟⢯⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⣤⠁⠀⠀⠀⠠⠁⠀⣼⢏⢾⣷⣿⣿⣿⣿⣿⣷⣻⣷⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⢀⣾⢑⣢⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⣦⢀⡾⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡡⠄⢸⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡾⣵⣿⣿⣿⣿⣏⡷⣻⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣉⣾⡿⠿⠟⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡐⠰⡧⠋⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣀⠀⣠⣴⣞⣛⢿⣿⣿⣶⣴⡀⢀⣤⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣱⣾⣿⣾⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣎⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣽⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣽⣿⠟⢹⠻⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⡈⠙⣿⣯⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣠⡏⠀⠀⠉⠊⢻⣿⠿⠋⠀⠀⠀⠀⢉⡆⢻⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢤⣸⣿⣿⢡⠞⠛⠲⠀⠀⠀⠘⠁⠀⠀⣀⠀⢀⠔⠛⠳⡌⣿⣿⣇⡤⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣏⡿⢷⡙⢿⣷⢯⣴⣦⣀⠀⠓⢄⠀⠒⠒⠈⡠⠖⠁⢀⣠⣤⡽⣾⡿⢋⡾⢿⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣰⣷⣼⠇⢠⣿⣿⣿⣷⣄⠀⠃⠘⡇⠘⠁⣀⣴⣿⣿⣿⡄⠙⣧⣾⣦⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⣿⡏⠀⠈⠹⢏⡓⢭⣽⣿⣿⠟⠳⣿⣿⣿⡉⢒⣿⠏⠁⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⣈⣿⣶⣆⠀⠈⢉⣭⣝⣁⡀⠀⠀⢁⣌⣳⣭⡉⠁⠀⣰⣶⣿⢁⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢻⣿⠿⣷⢞⠟⠋⠈⠛⢻⣦⣤⡿⠛⠋⠉⠻⡳⣾⠿⣿⡏⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢸⡏⠀⣏⢖⣾⢶⣧⣤⣄⣉⣉⣠⣼⣿⡶⣶⡶⣹⡀⢸⡇⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⢸⣅⢻⣛⣀⣿⣨⡄⠽⠯⠤⣅⣽⠀⣙⣿⣈⡇⠀⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡘⣷⣯⣙⠳⣖⣩⣤⣴⣶⣶⣦⣤⣍⣓⡞⢋⣽⣾⣇⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⡻⠛⣹⡿⠋⠀⠀⠀⠘⡿⠛⢟⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⣀⠀⢀⠄⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣷⣶⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠴⠒⠋⠉⠉⠁⠀⠐⠐⠒⠠⠤⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠔⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⣈⣉⠐⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠒⠂⠠⠄⣉⠁⠐⠢⠀⠀⣁⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⢬⣐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣⠀⢸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠞⢻⡉⡹⠒⠒⠠⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⡏⢸⠀⠸⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠈⢧⠇⠀⠀⠀⠀⠀⠀⠈⠉⠑⠂⠀⠒⠊⠁⠀⠀⠀⣱⢾⠀⠀⢆⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠈⠲⢤⣈⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠚⢁⡃⠀⠀⠀⡙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢀⠀⢀⣀⣀⡀⠈⢭⣽⣟⣗⣒⣒⣛⣋⣉⣉⡠⠤⠔⠒⠉⢹⠀⠀⢸⠀⢹⠀⣀⣀⣤⣤⣄
⠀⢀⠴⠒⠒⠶⢤⣀⠀⢠⡇⠀⠀⣀⠸⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⠣⡸⠋⠁⠀⠀⠀⠀⢳
⢰⠃⠀⠀⠀⠀⠀⠈⠑⠾⠀⠀⠀⡸⠀⠀⠀⡠⠚⠉⠉⠁⠲⣄⠀⠀⠀⠀⢀⡴⠒⠀⠀⠒⢦⡀⠀⣾⢾⠠⡐⢮⠆⠀⠀⠀⠀⣠⠞
⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢸⣁⣀⣀⣀⣀⣀⣨⡆⠀⠀⠀⣌⣀⣀⣀⣀⣀⣀⣱⠰⠿⠿⠣⡍⠛⠀⠀⠀⠀⠛⠓⢤
⢀⡼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢱⠤⠼⠤⠤⡼⣤⣤⠇⠀⠀⠀⣧⣄⣧⣀⣀⣿⣀⣸⠁⠀⠀⡂
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⣀⣀⣠⠴⢫⠖⠚⠒⢢⠳⣄⣀⠀⣀⣠⠔⠁⠀⠀⠀⡄⠀⠀⠄⠀⠀⠀⠀⠶⣶
⠙⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⢀⠇⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⡏⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠳⣄⣀⣀⣠⡴⠞⠋⣿⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢠⡦⣤⣄⣀⣤⠴⠞
⠀⠀⠈⠉⠉⠀⠀⠀⠀⣿⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠱⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠈⠐⠠⠤⠄⣀⣀⣀⠠⠤⠔⠊⠁⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⠗⠂⠤⣄⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠴⠒⠉⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⣀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⢩⡇⠀⡀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⢯⣤⠿⣄⡼⠃⠀⠀⠀⠀⠀⠀⠘⣧⡴⠳⢤⠼⠓⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢾⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠠⠤⠴⠶⣶⣶⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠉⠀⠀⠀⠉⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⢀⣀⡀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⣠⣾⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⣷
⠀⠀⠈⠙⠛⠛⠲⠶⢶⣤⣄⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢷⣦⣷⡀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠿⣿⣧⠀⠀⠀⠀⠀⢸⠃⠀⠀⣿⣿⡀⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⡸
⢀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢀⣀⣀⣈⣛⣿⣿⡟⠋⠀⠀⠀⠀⠀⣰⠃
⠈⠉⣿⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠓⠲⠤⣀⡰⠃
⠀⠀⡞⣷⠸⡄
⠀⠀⣇⢹⡄⠹⡄
⠀⠀⢿⠀⢧⠀⠙⣆
⠀⠀⢸⡆⠈⢧⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⣦⡀
⠀⠀⠀⢿⡀⠈⠣⡀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⠋⠈⠛
⠀⠀⠀⠘⣷⡀⠀⠑⠄⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⢁⡞⠁
⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠦⠤⣤⣤⣤⠤⠴⠖⠚⠉⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣶⣦⣤⣀⣀⣀⣀⣀⣤⣤⣶⠿⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣿⣿⣿⣿⣿⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣥⣤⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠄⢂⡒⠒⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⡹⡀⠀⠀⠐⢻⣷⣤⠄⠤⣾⡀⠺⠗⠀⣿⠛⠻⢶⣄⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣃⣷⠀⠀⢰⣾⣿⣿⢦⣤⢰⡷⠒⠀⠀⣿⡀⢀⣰⠏⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⣀⢹⡏⠀⣠⠀⠀⢈⣽⣿⢷⣿⣿⣿⣄⣙⢯⢿⡟⠋⠁⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠠⣿⣼⠁⠊⠉⢻⠀⢸⣿⠁⢸⣿⡻⣿⣿⣷⣖⣺⠴⢤⡀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢠⣿⣿⣰⠀⠠⠞⣄⢾⡇⠀⣾⠴⢿⣋⢻⣿⣿⣷⡋⠀⢹⡄⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⠀⣿⢸⣿⣾⠟⠚⣦⡿⣼⠿⣿⣿⣷⠀⠸⣧⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⣿⡟⡇⠀⠀⠀⢿⣿⣿⣇⠀⠀⢅⣉⠀⢿⡟⣿⣿⢧⢀⣿⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢢⣼⣥⠩⠀⠀⢀⣄⣽⣿⠉⢴⡀⡾⣿⡷⢸⡟⣿⣿⡎⢸⡿⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣸⣿⡟⢲⠶⠀⠸⣿⠟⢻⣆⠈⢷⡇⠉⠁⢸⡟⣛⡷⠇⣾⡇⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢹⡿⣿⠙⣶⣶⡶⣦⡀⢸⡿⢀⣘⠁⣼⣶⣾⣟⡿⠤⣼⡿⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢸⣿⢹⣾⠿⠿⠿⢛⣶⣿⡟⡛⣟⣦⣿⣿⣿⣿⣿⠆⣿⠂⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⢏⣿⠟⣶⣾⡟⣿⡟⡻⢿⣷⣾⣿⣯⢿⣻⠟⢷⣠⠏⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⣬⣿⡆⠿⠉⣧⠸⣄⠁⠀⣿⣿⣿⣷⠁⣿⣿⣿⠁⠀⠇⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠙⣧⣤⣼⣿⣧⢻⣄⣴⣟⣸⣿⣯⡆⣻⠿⠿⠀⠀⠃⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠢⣍⡿⠃⣿⠈⢿⡜⠻⠿⠀⣀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠁⠀⠀⠁⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠒⠉⠉⠉⠒⠦⢀⠀⣀⡤⠔⠂⠉⠉⠙⠛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠰⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠠⠾⠿⠿⠋⠀⠀⠀⠀⠀⠸⠿⢷⡄⠀⠀⠀⠀⠀⠄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢀⣠⣏⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⣁⠀⠀⠀⠀⠀⢈⡀⠀⢠⡾⠦⣄
⠀⠀⠀⢀⣴⠋⠀⠈⠓⣤⠄⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠈⢆⣠⠊⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠻⡄
⢀⡴⠛⠉⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⢰⠁⠀⠀⠀⣀⣀⣀⣨⣇⣀⣀⡀⠀⠀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣾⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠸⡋⠉⠉⠛⠿⠇⠀⠀⢩⠈⠛⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⡟⠀⠀⠀⠀⣀⠄⠀⠀⠐⠺⠓⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⢀⡾⡀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳
⠸⡆⠀⠀⣌⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠳⣄⠀⠀⠀⡠⠊⠀⠉⠂⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠓⢦
⠀⠹⣄⠀⠟⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈
⠀⣠⠞⠳⠄⠀⠀⠀⠀⠀⠀⠀⠈⠗⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠸⡏⠒⠒⢠⡤⠤⠤⠤⣤⠐⠒⠂⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⠀⠀⢀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠈⢾⡀⠀⠀⢠⠇⠀⠀⠀⢸⠀⠀⠀⠀⡆⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⣶⠚
⢿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠢⡀⣨⠀⠀⠀⠀⢸⠀⠀⠀⢀⡿⠺⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⠤⢀⣀⣸⡀⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠘⢿⣤⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡴⠋
⠀⠀⠀⠈⠉⠻⣏⣁⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣹⠆
⠀⠀⠀⠀⠀⢠⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡄
⠀⠀⠀⠀⠀⢀⡟⠛⠒⠛⠛⠛⠛⠛⠚⠛⠒⠚⠛⠛⠛⠒⠒⠒⠛⠛⠛⠒⠛⠛⠓⠒⠛⠛⠛⠛⠛⠛⠛⢻⡀
⠀⠀⠀⠀⠀⠈⠳⢶⡶⠦⠤⠤⠤⠤⢦⡤⠶⠤⠴⠤⠤⠤⠤⠤⠤⠴⠤⢴⡤⠴⠦⠤⠤⠶⠶⠶⠶⠶⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣠⡼⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡃⣀⠴⠞⠋⠁⠀⠀⠈⠳⢦⡀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠙⠳⣄⠀⠀⠀⠀⢀⡰⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡴⠚⢉⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡟⢿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣴⣿⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠿⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣠⡤⠶⣶⣾⣿⣿⣿⣷⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠉⠀⠀⠀⠀⠀⠉⠛⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⢀⣤⣶⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠿⢷⣶⣤⣀⡀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⢶⣽⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⠁⠈⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⣿⣿⣶⣾⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⢀⣤⣤⣶⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣀⣤⣤⣤⣤⣭⣿⣿⣟⣋⡁⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠉⢹⣿⡿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠦⢤⣀⣠⠟
⠀⠀⠀⠀⢸⣿⣧⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⢸⡇⣿⡀⢻⡄
⠀⠀⠀⠀⢸⡇⠸⣇⠀⠻⣄
⠀⠀⠀⠀⠸⣷⠀⢻⡆⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣿⡄⠀⢻⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢿⣶⣄
⠀⠀⠀⠀⠀⠸⣷⠀⠀⠻⡄⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⠟⠉⠻⠇
⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠙⢦⡀⠀⠀⠈⠙⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⢁⣴⠋
⠀⠀⠀⠀⠀⠀⠀⢿⣧⡀⠀⠀⠑⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠟⠋⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠳⠶⠶⠶⠶⠶⠶⠶⠛⠛⠉⠁⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⣀⣠⣤⣤⣠⣄⣄⣤⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣤⣀⣤⣤⣤⣄⣀⣀⣀⣀⣠⣤⣤
⣿⣛⣛⢻⣛⣻⣋⣛⡛⡟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣛⣛⣿⣟⣻⣿⣿⣟⣛⣛⣿

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

⢶⣦⣀⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⣤⣄
⠀⠛⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣢⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣟⡿⠓
⠀⠀⠀⠈⠛⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠉
⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣿⣶⣦⣀⠤⣀⠀⢻⣿⣿⣯⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⢴⣤⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣶⣬⣭⡍⠻⣿⣿⣿⣷⠒⠒⢂⣀⣽⣤⣶⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣄⠀⠙⢿⣿⡇⠀⢀⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣾⣿⣷⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣙⣿⣿⣿⣿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠟⣡⣾⣿⣿⠏⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣫⣷⣾⣿⣷⣿⡉⢀⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡻⠿⠿⣿⣿⣷⣶⣶⣶⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⠄⠀⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣫⠀⠀⠀⢸⣿⢿⣟⣵⣶⣉⣻⣿⣿⣿⣿⣷⣄⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣧⣠⣤⣄⣸⠟⠼⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠤⠄⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⡠⠔⠂⠁⠀⠀⠀⠀⠀⠀⠐⠠⢁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢠⠊⠀⡀⠀⠀⠀⠀⠀⠠⢀⠈⠀⠂⠄⡑⢌⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠃⠠⠊⠀⢀⠄⢀⠄⢄⠐⠠⢈⠐⠂⠤⠈⠀⠀⠂
⠀⠀⣀⡖⢜⢂⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠄⣊⠔⢁⠔⠁⠀⠀⠁⠂⠄⣈⠐⠂⠠⠄
⠀⠀⡱⠝⠈⠀⠁⣀⠀⠀⠀⠀⠀⡀⠀⢐⠈⠄⠊⢀⡀⠀⠀⠀⠀⠀⣠⠀⠀⠉⠁⢐⣐⢃
⠀⠀⠈⠂⢄⡠⠂⠀⠑⢄⠀⠀⢐⠩⠚⡀⠀⠀⠀⠈⠀⠀⠀⠒⠀⠀⠈⢀⠀⠀⠀⠔⠊⣸
⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠑⢄⡠⠂⠀⠑⡄⠐⠤⠃⠀⠀⠉⠊⠀⠀⠀⠀⠃⠀⠴⡀⠈⠣
⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠁⠂⠜⡁⠀⢠⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠊⠀⠐⠖⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠈⠱⡀⠀⠁⠐⠀⠀⠐⠀⠉⢀⠍⠢⡀⠀⠐⡄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢒⠶⣷⣄⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠐⠐⠂⠐⠒⠁⠀⠀⠐⢄⠀⠘⣵⡾⢗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠁⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢂⠈⠀⠘
⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⢀⣀⢀⠆⠀⠀⠀⢂
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠠⠂⠉⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣖⣠⣩⣭⣽⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠛⠿⠻⠙⠻⠿⠿⣿⣿⣿⣿⣿⠟⠋⠁⠀⣤⣠⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣀⣀⣀⣷⣶⣶⣶⣶⣿⣿⣿⣿⣁⣀⣤⣵⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠛⢿⣿⣿⠿⠟⠛⠉⠉⠉⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠿⠿⣿⣿⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⠀⣰⣿⣿⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⡀⠀⠀⠀⠸⣧⣤⡿⠛⠻⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠷⢦⡀⠀⠀⠐⠟⠛⠉⠉⠉⠙⠓⠀⠀⠀⣿⠟⡔⠁⠀⢰⠀⠀⠀⠀⠀⠀⢀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠋⠹⠗⢱⠀⠀⠀⠀⠀⠒⠒⠶⠄⠀⠀⠀⢸⡿⢀⠀⢠⠀⡌⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢯⠀⠀⠀⠀⠠⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠀⠀⠀⠀⠀⠈⠌⠀⣀⣁⡀⠀⣀⣀⣀⠠⣤⠆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⢧⣌⣉⣁⣤⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢃⠀⠀⠉⠛⠋⠉⠁⠀⠀⠀⠀⠀⢀⡴⠁⠀⠈⣦⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠉⠀⠀⠀⠀⠀⡀⠔⠉⠀⠀⡀⠀⠀⠱⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⠦⣄⣀⣀⣀⣀⡠⠔⠊⠀⠀⣀⡤⠊⠀⠀⠀⠀⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⣿⣤⡈⠓⠀⠀⠀⠀⠀⢀⣊⠉⠁⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣧⣴⣶⣶⡶⣧⣿⣿⣿⡶⢀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠶⠷⡶⢶⣿⣿⣿⣿⡏⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣧⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⡿⠿⢻⣿⠟⣿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠠⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣟⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠴⠒⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⢤⣀⡀
⢀⣀⣠⣤⠴⠶⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⢤⣄⣀
⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠼⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠦⢤⣀⡀
⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⢤
⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠀⣈⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠚⠛⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠲⠒⠋⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⠉⠛⠒⠒⠶⠶⠶⠶⠶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⠦⠤⢄⣀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠈⠳⣄⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠁⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡇
⣶⣶⣶⢶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣿⣷⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⠟⢁⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣫⡾⠟⠋⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣀⣀⣀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡏⠀⠀⢠⣾⣿⣿⡿⠛⢦⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣿⣿⣿⣿⣧⣤⣼⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠘⣿⣿⣿⣿⣿⠟⣴⣿⣿⣿⣿⣿⣭⡙⠛⠛⠛⠛⠋⠉⣀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣉⣉⣭⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⡉⠉⠛⠛⠛⠛⠋⢉⣁⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⣿⡿⠿⠿⠿⠟⠛⠉⠁

⠀⠀⢀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣷⣶⣦⣄
⢀⡴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠙⠿⣆
⠊⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⢄⣀⠲⣶⣋⣻⣯⣽⣭⣍⣉⣉⣉⡉⠛⠳⢤⡀
⠀⢒⠶⣬⣿⣿⣿⣿⠿⠿⡟⠛⢿⣿⡏⢻⡷⢦⣄⠉
⠐⣲⣿⣿⣿⣿⣿⣷⣄⡀⢣⠀⠸⠿⠃⣸⠃⣀⣬⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢶⣤⣤⣀
⠀⠈⠉⠉⠛⠉⠉⠈⠙⠛⠷⠿⠶⠶⠾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⣶⣤⣀⣀⡀⠀⠀⢀⣀⣠⣴⣾⣿⣛⠻⠶⣄⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⢹⠛⣿⢻⡟⢿⣟⢿⣍⠻⣦⡉⠛⠦⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠻⠀⠙⠄⠙⠆⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⣴⣦⣤⣤⣤⣤⣽⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠈⠻⣶⣶⣿⣿⣿⣿⣟⠻⢿⣿⣿⡿⣾⣧⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣄⣠⣿⣿⣿⣿⣿⡿⣿⠛⠿⢿⣿⣷⡀⠁⠈⢻⡄⠹⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣝⡻⣿⣿⣮⡳⣄⢀⠹⣿⡆⠀⠀⠀⢿⡀⠈⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣧⣤⣿⣧⣾⣧⣇⣼⠃⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠟⠛⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⢟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠟⠉⠀⠈⢿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡈⠹⣿⣿⣿⣿⣿⣿⣁⠀⢀⣀⣤⣀⠀⠀⣀⠀⠀⠀⠁⠀⠀⠀⠀⠸⣿⠞⠁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣍⢥⣄⡙⠲⠿⣿⣿⣿⣿⣿⡇⢰⡏⠀⠀⠀⠑⢄⠈⠍⢓⣒⣒⣒⠀⡠⠂⠀⢿⣤⠞⣉⣭⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡌⣿⣿⣷⣦⣄⠀⢿⣿⣏⠰⣿⣷⣷⣤⡀⠀⠈⠋⢕⠒⠒⢀⡠⠎⠀⠀⠀⡀⢳⣾⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠁⣿⡟⣿⣿⣿⣾⣿⡟⠂⢻⣿⣿⣿⠿⣆⡀⠀⠀⢀⣾⠀⠀⢀⣠⣴⣾⣿⣿⠏⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡜⣧⣼⣿⣁⣿⣿⠀⠀⠈⠉⠙⠿⢷⠾⢿⠿⠿⠋⠙⣿⣿⣿⣥⡿⠏⠀⣿⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠿⡟⣿⢿⣿⣷⣤⣀⠀⡠⣠⣴⣾⣿⣶⣦⡀⠀⢠⣾⡄⠀⢀⣀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⢧⡟⣸⣿⠿⣿⣿⡿⣼⡟⢀⠀⢀⠀⠈⠙⠛⠋⠉⢹⣆⣿⣿⣯⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣶⣿⣧⠀⠘⣿⣽⡟⣷⠈⣾⡧⠴⠖⠒⠒⠤⠾⢽⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣻⣧⢰⣿⠁⠧⠼⠋⠁⣠⡴⠶⠖⠛⠶⢶⣤⡙⢣⡟⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣾⣿⣧⣄⣀⣤⣾⣿⣤⠤⠀⠀⠀⠰⣿⣷⣄⢉⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡛⢿⣿⣿⣿⣟⠋⠉⠀⠀⢀⣠⡀⠀⠈⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣶⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⡿⠟⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡇⠀⣇⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣷⢠⣧⣴⣶⣶⣶⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠉⡇⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿⠀⣿⡝⠁⠀⠀⠀⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢷⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠋⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⡤⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⡶⠾⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⣀⣤⣴⣶⣦⡀⠀⠀⠀⠀⣀⣴⡾⠿⠛⠻⣷⣆⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣼⠏⠁⠀⠀⠈⣿⡀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⢸⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣿⠇⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⢸⢸⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⢸⣸⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⣀⣼⣿⠃⠀⠀⠀⠸⣿⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠛⠿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⢻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣼⡆⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣤⣶⣶⣶⣶⣶⣶⣶⣾⠿⠿⠿⠿⠟⠛⠛⣛⣋⣉⣉⣭⣭⠭⠭⠭⠽⠶⠶⠶⠿⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠤⢤⡶⠖⠒⠒⢚⣛⣛⣛⣛⣋⣉⣉⣉⣉⣁⣀⣤⣤⣤⠤⠴⠶⠶⠶⠖⠒⠒⠲⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠒⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣬⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⠀⢠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⡀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣀⢀⠰⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣥⠺⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⣤⣤⣶⣂⣄⠐⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣠⣾⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡀⠀⢻⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠛⠀⢤⣀⣤⡼⢿⣷⣎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠰⠚⠿⣯⠀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠰⠶⠀⠀⠀⠰⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣀⡀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⢿⣃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠰⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣾⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠘⣿⣿⣿⣿⣿⣿⡿⠋⠁⠸⠋⠻⠚⠛⢉⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⠀⣿⣿⡿⠿⠙⠛⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣈⢻⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠛⠈⠛⠻⢿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡶⠂⢀⡀⠙⠛⠻⠛⠟⠻⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣤⣤⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠿⠛⣤⠀⢤⠄⠀⠀⠙⠶⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⣤⡀⠀⠀⠀⠀⠀⠀⢀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣀⣼⡄⠀⢀⠀⠀⠀⠀⠀⠈⠁⠰⣶⣄⣀⣄⠀⢠⣄⠀⠀⣀⡀⠁⠻⢦⣄⠀⡀⠀⢠⠀⠀⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠒⠛⠉⠉⢀⣦⠘⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠹⠶⠤⠀⠀⠻⠦⠈⠉⠠⠻⡾⢯⣼⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣶⣤⡆⠘⠻⠋⠛⢣⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠙⢄⠤⡄⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢀⡤⠖⠉⠀⠀⠀⠀⡿⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠁⠀⠄⠐⣷⠀⠀⢸⠀⡠⠚⠉⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⡇⢸⡴⠁⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠀⠀⠀⣠⡉⠀⢀⠞⠈⠀⠀⠀⠀⠀⢀⡠⠐⠁⠀⠀⠀⠄⠂⠼⣃⠼⢄
⠀⢀⠀⠏⠀⠀⠈⢀⡐⠄⠠⢌⡡⡀⠣⣀⣀⠥⠿⠈⠁⠄⠀⠀⠀⠀⣖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⣤⠎
⠰⡁⠁⠠⠄⠠⠀⢌⠃⠈⢁⡀⠨⠿⠆⠈⠠⢶⡂⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠈⠄⠒⠠⠀⡄⡎⠀⢠⡃⠶⠴⢋⡀⠀⠀⢹⡱⠄⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢠⠎⡀
⠀⠀⠁⠀⠀⠁⢀⠁⠀⠀⠑⠒⠊⠁⢱⠀⠀⠈⣧⠀⠁⠠⠀⠀⠀⠀⠀⠈⡆⠀⢀⢀⣠⠋⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠠⡀⢀⣤⡀⠀⡆⠀⠀⠈⢄⠀⠀⠠⠛⠐⠂⠀⠀⢱⢀⠇⠀⠈⠘⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢌⡗⠁⠑⡄⠸⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢀⠀⠀⠀⠀⠉⠒⠂⠤⣀⣀⡄⠀⠈⠚⠃⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠤⠀⠤⠄⠀⠤⠀⠤⠤⠀⠀⠲⠂⢀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢠⠃⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣥⡆⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣤⣶⠚⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠒⠒⠒⠋⠁⠐⠛⠀⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣦⣭⣿⣷⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⢀⣠⣤⣤⣥⣼⣿⣿⡟⢁⣀⣤⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠟⠛⠿⠿⠿⠛⠋⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠋⢻⣿⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⣿⣠⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢛⡳⢄⠀⠀⠘⠛⠋⠉⠙⠒⠀⠀⣿⠟⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠃⠀⠀⠀⠀⠈⠉⠃⠀⠀⠀⠟⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠤⠤⢀⠖⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠳⠶⠶⠟⠁⠀⠀⠀⠀⡠⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠑⡄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡕⠠⡄⠀⠀⠄⠂⠁⠀⠠⠂⠁⠀⠀⣸⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣾⣿⣿⣶⣬⣦⣠⣤⣤⣴⡤⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠁⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⠦⣤⡀
⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⢀⡴⠞⠉⠡⠚⠀⠈⣉⡛⠦⣤⠖⣛⢶⣄
⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⡾⠁⠀⠀⠀⠀⠐⠞⣉⡉⠉⣻⣆⣌⢩⣟⢳⡄
⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠀⠀⠀⢿⠃⢰⡄⣆⢀⣀⠀⣰⣿⡼⢿⣟⣿⣿⣻⣿⡠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⢦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠶⠖⠛⠉⠀⠀⠀⠀⠨⣑⡶⠙⣿⢯⣛⠀⢿⡆⠹⣯⢿⣾⡯⢻⢈⢷⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢿⡋⠀⠀⠀⠀⢿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠉⠉⠙⠛⢫⠙⠓⠋⡠⠚⢋⠉⠛⠿⢧⣿⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠂⠀⣧⠄⣼⠁⠀⠀⠁⠀⠀⠀⢙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠶⠿⠷⠶⠤⠄⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠃⣠⠟⠸⢿⠀⠐⠃⠀⠀⠀⠀⣸⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡤⣤⡴⠾⠃⢠⣀⡌⠛⢦⣀⣀⣀⣤⠶⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⣶⣤⣄⡀⠀⠉⠋⠛⠀⣼⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⡘⢙⣛⣏⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠙⠶⣄⣀⠀⣀⣀⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⢀⡿⢿⠟⠭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠞⢲⢄⣿⡇⠙⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠋⠃⠚⢪⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⢷⠶

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣼⣽⣭⣿⣼⣯⣯⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣶⣶⣿⣿⣷⣷⣾⣶⣿⣷⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⣠⣶⡄⠀⠀⠀⠀⣠⣤⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠟⠉⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⢀⠔⠀⠁⠀⠢⡀⠀⡠⠀⠀⠀⢄⠀⠘⠶⠤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠴⠖⠚⠛⠉⠁⠀⠀⡎⠀⠀⠀⠀⠀⠱⡎⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠒⠒⠶⠦⠤
⡠⠶⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⣞⣦⠀⠀⡇⣜⣧⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢣⠀⠀⠻⠟⠀⢀⣇⠘⠃⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠠⠎⢻⠀⠀⠀⠀⠀⡀⠳⢄⡀⠀⡠⠊⠀⠑⠀⠀⠐⢁⡀⠁⢠⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠘⡉⠳⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠱⣘⠀⠀⠉⠉⡟⠛⠒⢻⡋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠹⣄⣀⠤⠄⠚⠒⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⡴⠲⠇⠲⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣠⡴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣦⣴⣤⣤⣤⣶⣦⣴⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⢁⡙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠿⠖⣀⣨⣿⣿⣿⣿⣿⣿⣿⡆⡄⠈⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⡆⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣠⣴⡾⣿⠛⣿⣿⣿⣿⣿⣿⣿⡇⠘⠛⠂⣠⣿⣿⣿⣿⣿⣿⣿⣇⣼⣧⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠠⣿⣿⡿⣻⡟⠀⣛⠀⢹⣿⣿⣿⣿⣿⣿⡿⠀⢀⣀⡛⠛⠛⠛⣿⡿⣿⣿⣿⣿⣇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣦⡀⠀⠀⠀⣼⣷⣴⡟⠀⠘⣿⣿⣿⣿⠛⠁⠀⠛⠛⠋⠁⠀⠀⠀⠈⠀⣠⣽⣿⣿⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣟⠀⣀⣶⣾⣿⣿⡿⠀⠀⢹⣿⡏⠁⢐⠀⠀⠀⡐⠀⠀⠀⠀⠀⢠⠜⠁⠸⣿⣿⡟⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠛⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠈⠉⠀⠘⠛⠀⠀⣀⡃⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⢻⣷⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⢁⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⣠⣴⣶⡾⠛⠁⠀⠀⠀⠀⢀⣠⣴⣶⣶⣤⡀⠛⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⡾⢁⣾⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠘⠻⣿⠿⣿⡿⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠰⣿⡿⠀⠀⣴⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠼⠟⠛⠛⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⢀⣾⢿⣷⣄⣶⣿⣿⣿⣿⣀⣿⣿⣟⣽⣿⣿⣿⣿⡿⠏⠉⢻⡆⠀⠀⣀⣀⡤⠀⠀⢀⠀⠀⢠⣾⣧⠀⠈⠟⠻⠀⣿⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⢘⡁⠘⢿⣿⣿⣿⣿⣻⣿⣿⠿⣻⣾⣿⣿⣿⣿⣿⡷⠀⠀⣾⡇⠀⠘⣿⣿⣴⠄⣀⠈⠀⠀⡀⠀⡉⠁⠠⡴⣶⣶⠉⡉⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠘⣿⣶⣸⣿⣿⣿⣿⣿⣻⣯⣾⣿⣿⣿⣿⣿⡿⠋⠀⠀⠈⢻⣿⣷⣾⣿⣿⣿⣿⡟⢀⣠⣶⣿⣷⡝⠁⣀⣠⣾⡿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢠⡙⣿⣿⣿⣿⣿⣿⣿⠋⠉⣿⣿⣿⣿⣿⡟⠀⠀⠀⣤⣾⣿⣿⣿⣯⣽⣿⠿⠛⠓⠙⠋⠉⠻⣿⣷⣤⣿⡟⠻⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣨⣷⣿⣿⣿⣿⣿⣿⠉⣤⣀⣿⣿⣿⣿⣿⡅⠀⠄⠐⢿⢽⣿⣿⠿⠋⠁⠀⢀⡤⠀⠀⠀⠈⠀⢈⠉⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⣷⣿⣿⣿⣿⣿⣯⣤⣿⣿⣿⣿⣿⣿⣿⣿⣶⠄⠀⠄⢴⡿⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣟⡛⠋⠉⠙⣿⣿⣿⣿⣿⣠⡄⠀⢸⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣿
⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠈⠛⢿⣿⣿⣿⡿⠉⠀⠶⠿⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡏⠁⣸⣿⣿⣿⡄⠀⢸⣿⣿⡿⠁⠀⢢⡀⠀⠀⡸⠋⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⠀⣾⣿⣿⣿⣿⣿⠀⠘⣿⠋⠀⠈⠀⢺⡇⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠁⢠⣿⣿⢿⣿⣿⡟⠀⠀⢸⠀⠀⠀⣈⠀⣿⠀⣀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⢻⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠘⢿⣿⣟⣿⡿⢀⣿⡿⠃⠘⠛⠋⠀⠠⠀⠀⠀⠀⠀⠀⠀⠻⣷⠟⣼⣿⣆⠸⠆⣰⣲⡄⠀⠀⠀⠀⠀⢀⠀⠀⢸⡧⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠈⢻⣿⣿⡇⣾⣿⠀⠀⠀⠀⠀⠀⠀⣤⣾⡄⠀⠀⠀⠀⠀⠀⢶⣾⣿⢿⣆⠀⠈⢻⣶⡆⠀⠀⠀⠀⠈⠀⠀⢸⡇⠺⠂⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⣼⡏⠀⠀⠀⠀⠀⠀⠀⠍⢉⣀⣶⠆⠀⠀⠀⠀⢻⣿⠃⢰⣿⣷⣄⠀⠛⠁⠀⠀⠀⠀⠀⢷⠀⠀⠃⠀⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣈⢁⣵⣿⡂⠀⣀⠀⠀⠉⠠⠈⠡⣌⡉⠀⠀⡠⣴⠋⠀⠀⠀⠀⠀⠀⢸⠀⢳⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⢠⡾⠿⡽⢿⣿⣿⣿⣿⣿⠙⠀⠀⠁⡀⠂⠉⠛⠦⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠁⠰⣶⠘⣿⣿⣿⣿⣷⡶⠊⢀⡀⠟⠀⠀⠙⢠⡄⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⡄⠀⣆⣴⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⣽⡿⠀⠀⠀⣤⡀⠀⠀⠀⠼⠿⠿⢿⣋⠉⡹⢧⡀⠀⣥⠸⣇⡾⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡇⠀⣸⣧⠀⠿⡟⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠆⢸⣿⡆⠀⠀⠀⠀⠀⠈⣻⣟⠀⣠⣶⣾⠟⣀⡈⢳⠀⠀⠀⠐⡶⠀⠀⠀⠀⠀⠀⢸⡟⠿⠀⠘⢿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡿⣿⡁⠀⠀⠀⠀⠀⠀⣸⠁⢰⡟⢿⣿⣦⠘⣿⣆⡀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⢼⠇⣼⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠁⠀⢀⣾⡇⣾⣿⣦⣼⣦⣤⣦⢼⣿⠀⣾⠿⣿⡟⠋⢀⠈⢿⣿⣦⣀⠘⠁⠀⠀⠀⠀⠂⠀⣾⠀⢻⠀⢀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⢸⣿⠃⣿⣿⣿⣿⣿⡟⠿⡶⣿⣄⡀⠠⢿⣷⠀⠞⢀⠀⣾⠻⣿⣷⣄⠀⠀⠀⢰⠀⢨⣿⠀⣿⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣘⡷⢘⣟⠀⣿⣿⣿⣿⣿⣿⡄⠀⠀⠉⠀⣤⣴⣿⡆⠀⠸⣧⡈⠂⠈⠻⣿⣷⣦⣀⠁⠀⣿⡇⠀⢿⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⡞⢫⡅⣾⡿⢨⣿⣿⣿⣿⣿⣿⣷⣄⠀⣀⣤⡾⢿⣿⠁⠀⠀⢻⣷⣄⠀⠀⠛⢸⡿⠿⣷⣾⣿⠃⠀⠀⡄⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⡄⢨⣑⣿⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⠉⠁⢀⡰⡇⠀⠀⠀⠀⠻⣹⣷⡆⢠⣤⣤⣤⣶⣿⡟⠀⠀⠀⡇⢸⡇⢠⣿
⠀⠀⠀⠀⠀⠀⣾⣿⠿⣿⣦⣾⣿⣿⡀⠈⠉⠉⠉⣿⣦⣉⣿⣿⡇⠀⠀⢸⣇⡀⠀⠀⠀⠀⠀⠀⠈⢱⣾⠿⠟⠛⠻⠛⠁⠀⠀⠀⠀⢸⣿⠀⣿
⠀⠀⠀⠀⠀⣰⣷⡀⢀⣷⢼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠻⠻⣿⣤⠀⠀⣸⣿⡇⠀⢀⡄⠀⠈⠀⢀⠀⢻⣷⣦⣸⠧⠀⠀⠀⠰⠆⠀⠈⢻⠀⣿
⠀⠀⠀⠀⣾⣿⡟⢷⣿⣇⢀⣿⡟⠁⠀⠀⢀⣠⣶⣶⣄⡀⠀⠀⣿⣧⣤⣿⣿⠗⠀⠈⣷⡌⢀⠀⠉⢱⡀⠙⣿⡟⠠⠄⠀⠀⢀⡀⠀⠀⣿⠀⣿
⠀⠀⠀⣰⣿⣿⣿⣿⡋⢻⣿⠋⠀⠃⠀⠀⢸⣟⠻⣧⣼⣿⡆⢀⣤⣿⢿⣿⣿⣇⠀⠀⠘⢧⠈⢠⣄⠀⠻⡀⠈⢷⡀⠀⠀⠀⠈⢱⡾⣡⡿⢀⣿
⠀⠀⠀⣨⡿⠿⣼⣛⣷⡟⢃⣾⣇⠀⡀⠀⣾⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⡿⠉⣧⠀⠀⠀⠀⡀⠙⢧⡀⠻⣶⣀⠻⣦⡀⠀⢰⣾⠶⢩⣷⠀⣿
⠀⠀⠀⠿⢣⣿⣭⣿⣏⣸⣯⡘⣿⣇⠁⠊⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡉⠀⠀⠀⠀⠹⣷⣌⠠⢶⡈⠛⢿⣯⡿⢷⣶⣤⣤⣼⡇⠀⣿
⠀⠀⠀⠀⢀⣿⣿⢃⣿⣿⣿⣧⡼⣿⣄⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡳⠦⠀⠀⠀⠀⠀⠃⠈⠛⣷⣌⡹⠖⣤⡈⠙⠦⠈⠻⢟⣻⠀⠀⣿
⠀⠀⠀⢠⣾⠿⣡⡿⣿⣿⠏⣹⣷⣉⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⣄⠀⠀⠀⣴⠂⠘⠂⣿⣿⣦⣬⡟⠠⡦⠘⠓⣸⡇⠀⠀⣿
⠀⠀⠀⢾⣷⣦⠉⣴⡿⢃⣼⡿⢛⣿⣶⣹⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢤⣶⡟⠀⢡⣶⣄⠹⠿⠃⢰⢶⣦⣬⡟⣻⣿⡷⢴⣦⣀⠈⡁⠀⠀⣿
⠀⠀⠀⣼⡿⠁⣸⡟⣡⡿⣫⡴⢿⣿⣿⣿⣿⢯⣽⣿⣿⣿⣿⡿⠛⠉⠙⢢⡙⢿⣶⣿⣿⣿⣿⣷⣶⣶⡆⠉⠉⠀⠉⠁⠀⠀⣿⣿⣿⣿⡗⠀⣿
⠀⠀⢰⠛⠁⡸⠏⢀⣿⡿⠃⢀⣾⣿⣿⡏⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⣨⠇⠀⠈⠀⠉⠙⠛⠿⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠉⠀⠀⣿
⠀⠀⠀⠠⠀⠀⣴⠞⠁⠀⠀⣾⣿⣿⡏⢡⣶⣤⡿⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣤⣴⠀⣿
⠀⠀⠀⣀⡴⠊⠁⠀⠈⢳⣄⣛⣋⡈⠛⠛⢻⡏⠀⠀⠀⠀⠀⠘⢻⣷⣿⣶⡤⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠻⣿⡧⠀⣿
⠀⣤⠞⠋⠀⠀⠀⠀⠀⣼⡿⠋⠉⠓⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣤⣤⣤⣿⠇⠀⣿
⠀⠁⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⠀⠂⠀⠀⠈⣿⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⠉⠛⠉⠛⠛⠘⠋⠉⠒⠒⠀⠀⠀⠀⠀⠀⠠⠄⠤⠤⠤⠄⠤⠤⠤⠤⠤⠤⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣦⣶⣶⣾⣭⣭⣭⣭⣽⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⢩⠞⣿⣿⣿⣿⣿⣿⡿⠟⠛⢋⣯⣽⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠘⣧⣿⣿⣿⣿⣿⡽⢁⣴⣶⣿⡿⠋⠉⠉⠉⡉⠙⢳⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢿⠿⣿⣿⠿⠿⠿⠿⠿⢿⣟⣿⣶⣶⣶⣽⣾⣿⣿⣶⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠸⣝⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣝⣆⠀⠀⠘⠉⠁⢀⠀⠀⠀⠀⢳⣽⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠘⠻⠀⠀⠀⠀⠉⠙⠃⠀⠀⠀⠀⠉⠙⠈⢹⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠿⠿⡿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣷⣶⣦⣴⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⢹⣟⣛⣋⣭⡟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠜⠯⡭⠿⠋⠀⠀⠀⠀⢀⠐⠸⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⣿⠻⠛⠛⠛⢿⠋⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠀⠀⠀⠀⠀⠀⠈⢭⣽⣭⣵⣧⣶⣾⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠲⡒⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡐⠄⠨⠀⠑⠐⠢
⠀⠀⠀⠀⠀⠀⢀⠎⠡⠀⢀⠼⠀⠀⠡⢀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠁⠀⠀⡰⠠⡀⢀⠀⠠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡰⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠑⠐⠐⠂⠉⠑⠤⠀⠀⠀⠀⢀⠠⠔⠁⠈⠀⠀⠅⠈⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⡐⠈⠈⠀⠐⠀⠈⠀⡀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⢀⠀⠀⡀⠀⠀⣀⡀⠀⠀⠀⣀⠀⠀⢀⣀⣀⡀⠀⣠⣤⡀⠀⣠⠶⣦
⠀⠀⠀⠀⠀⠀⠀⢿⣤⣙⠀⢸⡆⠘⡇⠀⣾⠋⠻⠆⢠⡟⠉⠳⠀⢸⣏⣉⡁⠘⣧⣤⡁⠀⠻⠶⣦
⠀⠀⠀⠀⠀⠀⠀⢶⣌⣽⠀⢸⣇⣀⡇⠀⢿⡀⣠⡄⠘⣇⠀⣠⠀⢸⣏⣉⡀⠰⣦⣨⡟⠀⠷⣤⠾
⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠁⠀⠀⠉⠉⠀⠀⠈⠉⠁⠀⠈⠉⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡍⠉⠀⠒⠂⠉⠉⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⣀⣀⣀⣠⣤⣀⣀⣀⣀⣠⣤⣠⣤⣠⣤⣤⣄⣠⣤⣤⣤⣤⣠⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣤⣤⣤⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿