I see you Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠸⣿⣯⣤⣄⢀⣀⣤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢉⣈⣷⡟⠀⠀⣿⠉⢸⡇⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⣾⣿⡿⠁⣼⠀⡟⠀⡈⠃⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠛⢉⣃⢠⡆⠀⡇⠀⣇⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠿⢿⡿⢿⣿⣿⣿⡿⣿⣟⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⢠⣤⣿⠉⢁⣤⣼⣿⠉⢁⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠘⢿⡿⠀⠘⢉⣿⡟⠀⠘⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠷⠀⠀⡇⠀⣾⡿⢿⡇⠀⣾⡿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⢀⣰⡇⠀⠀⣀⣸⡇⠀⠀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⢿⣿⣿⠟⠻⣿⣶⠿⠟⠻⣿⡾⠿⠟⠛⠀⠀⢀⣠⡀⠀⠀⠀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⢿⣰⠏⠀⠉⣿⠴⠚⠛⠛⠲⣄⠀⢀⣀⣠⣤⣶⠛⠉⠀⣇⣴⠚⠋⠁⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠘⠏⠀⠀⡼⠃⠀⢠⣶⠀⠀⠘⣶⠋⠉⠀⣿⣿⠀⠀⠀⣿⡇⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⣼⠃⠀⢀⡿⣿⠂⠀⠀⡟⠀⠀⢸⣿⣿⠀⠀⠀⣿⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠇⠀⠀⣼⡿⠀⠀⢸⢹⣿⠀⠀⠀⡇⠀⠀⢸⣯⣿⠀⠀⢠⡇⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⣰⢿⣧⠀⠀⢸⣼⡟⠀⠀⣸⣇⠀⠀⢸⣿⡿⠀⠀⣼⣷⣶⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⢀⡏⠸⣿⣆⠀⠀⠉⠀⢀⣴⢿⣿⣄⠀⠀⠛⠁⢀⣼⡟⠁⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣦⣀⣼⠀⠀⠙⢿⣿⣶⣶⣶⠟⠁⠀⠙⣿⣷⣶⣶⣶⠟⢿⣷⣤⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣄⡐⢦⣀⣲⣦⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣆⣀⡄⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠸⣿⣿⣿⣿⠏⠉⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⣿⣿⣿⣿⣟⣀⣠⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⡿⠛⠛⠻⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠉⠙⠆⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠐⢦⡀⣀⣀⠎⠀⠿⡷⠂⣲⢀⣆⠔⡄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢘⣿⠀⡟⠀⠀⢸⡁⣠⡇⣸⠏⢸⡃⣿⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠴⠃⠀⠀⠀⡜⣿⠀⠁⠀⠀⠀⠉⠁⠈⢉⡄⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣠⢶⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⣴⠞⠛⣆⣰⠞⠛⣆⠀⠀⠀⠀⠀⢻⡀⠇⠈⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣼⠇⠀⣠⣿⣏⣀⡴⠃⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⢺⡟⠋⠉⠀⣿⠀⠀⠀⠀⠀⠎⢸⠇⢀⡇⣰⠛⣥⣤⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢸⠛⣧⠀⠀⢸⣧⠀⠀⠠⠀⠀⠀⣿⠀⢸⡇⡏⠀⢠⠐⡇⠀⢸⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⠟⠀⠙⠲⠔⠁⠙⠒⠂⠁⠀⠀⠀⠙⢤⣼⡇⠓⠤⠃⠀⠑⠐⠙⠦⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣼⠀⠀⣸⠀⠀⠀⠀⣴⢳⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⣺⠛⠉⠉⠙⠛⠒⠒⠒⠒⠒⠋⠈⣢⠔⠃⠀⠀⠀⢀⡇⡼⠀⠀⣀⣸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡿⠀⠀⠀⣠⣤⢄⠀⡠⢀⢴⡄⠀⡇⠀⢀⣴⠶⡝⢺⡏⡡⣄⠉⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣼⠃⠘⢬⠴⡇⡞⣸⠃⢸⡇⠀⣾⠁⠀⡗⢸⡿⢀⣿⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣿⠀⠀⡞⢰⣿⠀⣿⠀⢸⠃⢸⡇⠀⠀⡇⣾⡇⢸⡟⠀⢸⣿⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⡼⠛⠦⠞⠁⠘⠇⠀⠙⠖⠛⢤⣞⣳⣤⣾⠄⠻⠀⠀⠻⠤⠛⠻⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣾⣿⣿⣿⣿⣷⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣧⡀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣶⣾⣗
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣍⡛⠿⠿⠰⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⡘⣯⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⢻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠈⠉⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⣿⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣞⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⢻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⠀⠘⣧⣾⣿⣿⣿⢧⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⡀⠐⣿⣿⣿⣿⢯⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣼⣿⣿⠗⢡⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠋⣠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢏⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠏

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

⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠁⠀⠀⠀⠀⣠⣤⣤⡀⢀⣤⣤⡄⠀⣠⣤⣤⡀⠀⠀⠀⢠⣄⠀⢠⣤⢀⣤⣤⡀⢠⡄⠀⣤⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠙⠷⣦⡀⣿⡷⠟⠋⢸⣿⠶⠛⠁⠀⠀⠀⠈⢻⣦⣿⠃⣿⡃⢈⣿⢸⡇⠀⣿⠀⡿
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠀⠀⠀⠀⠻⠶⠛⠁⠙⠳⠟⠛⠈⠛⠳⠛⠃⠀⠀⠀⠀⢀⣿⠃⠀⠙⠻⠟⠃⠘⠻⠟⠿⠀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠤⠤⠤⠤⠤⠄⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠉⣁⣠⣤⣤⣤⣤⣦⣤⣤⣤⣄⣀⡀⠀⠉⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣠⣴⡾⠿⠛⢻⣿⡿⠛⠛⠛⠛⠻⠟⠿⢿⣿⡿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⣋⣠⣶⠟⠋⠁⠀⠀⢰⣿⡙⠁⠀⠀⠀⡀⠀⠀⠀⠀⠈⢷⠉⠙⠙⢷⣦⣀
⠀⠀⠀⠀⠀⠀⢀⣾⡿⣫⣾⢻⡇⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⢀⣴⣷⣦⡀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠛⢷⣄⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⢿⣎⢷⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⣿⣿⣿⡟⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⢋⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠤⢉⣃⣷⣄⣤⠀⠀⠹⣧⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠡⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⠷⢤⣙⣷⣤⣀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⢛⡿⢷⣶⣶⣾⠿⠋⠀⢀⣠⡤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠉⠉

⣒⣻⣦⠤⡀⠶⠶⢶⣶⣶⣶⣶⣶⡿⠶⠶⣤⣀⣦⣴⠶⠶⠖⠶⠆⣴⣰⣶⣶⡶⠃⡀⠀⢔⠒⠶⢶⡶⢄⡐⢲⡔⢶⡀⠲⢤⣾
⣿⡿⢿⣦⣀⣂⠀⢘⣇⣼⣿⣿⡟⠀⠀⠀⠀⠀⢁⣤⣶⣦⣶⣶⣾⡿⠏⠀⣤⡶⠦⢭⠿⢿⣷⣿⣿⣟⢺⣿⣿⣁⢺⣧⠀⢸⣿
⣿⣿⣿⣏⣷⣿⠀⠈⢻⣿⣿⣿⠁⠀⣠⣤⣶⣿⠿⠛⠉⠁⠉⠀⠀⠉⠁⠀⠀⠀⠀⠂⠉⠁⠬⠵⣾⢻⣿⣿⣧⣍⡉⢿⣧⣼⣿
⠀⠀⠀⠀⢙⣿⣧⣀⣾⣿⣿⣧⣴⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⣀⡀⠃⠑⠻⠿⣿⣿⡙⢾⣿⣿⣿
⢲⣤⢦⠔⠀⢻⣿⣽⣿⣿⣿⣿⡿⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢤⠴⠟⠉⠀⠙⡇⡤⠀⠀⠙⣿⣿⣾⣿⣿⣿
⣉⠉⣹⡆⠀⢘⣿⣿⣿⣿⣿⡟⠀⠀⠀⠘⠃⠀⠀⣀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠤⢿⣷⡏⢳⢼⣿⣿⣿⣿⣿⣿
⢓⣿⣏⠊⠁⣼⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠐⠛⠼⠷⠯⠿⠷⠖⢀⠀⠈⠀⠀⠰⠀⠀⠀⡴⠀⡁⣀⣌⡸⠟⣿⡌⠻⣿⣻
⣿⡉⠛⢷⣶⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣴⠒⠊⠀⠀⠀⢀⣤⣾⣿⣃⠀⠘⣿⡇⢀⣿⣿
⣿⠅⠀⢀⣿⣻⣿⠿⠿⠟⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠖⠀⠀⠀⣴⠟⠏⢿⢯⡤⠀⢃⡜⡃⣿⣿
⣹⠤⠄⠤⣿⣿⣯⡐⣶⣦⠶⣬⣴⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⠗⠻⠯⣀⠀⠘⢺⣶⠲⢨⠇⢰⣿⣿
⣿⡀⠠⠂⢿⣿⣿⣿⢹⣿⣷⣟⣿⣿⡆⠀⠀⠀⢀⣤⣤⣤⣤⣠⣤⣠⡀⠀⠀⠀⠀⢼⣿⡄⡠⢌⢉⡛⠿⣿⣿⣿⣸⢠⣿⣿⣿
⣿⠚⠤⠶⣾⣿⣿⣿⡘⣿⣿⡿⠿⣿⣿⠐⠂⠀⠘⢹⣿⣿⣿⣿⣦⣽⠟⠀⠀⠀⠀⠀⠙⠃⢀⣴⢺⡿⠀⣷⣿⣿⣿⣿⣿⡿⣿
⣿⣦⡀⠀⣿⣿⣿⡿⠉⠉⠙⠆⠀⠈⠙⣆⠀⠀⠀⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⢀⡛⠻⣶⣿⣷⣤⣿⡀⣿⣽⣿⠿⣿⣯⣰⣿
⠿⠻⢿⣶⣿⣿⣿⡟⡤⠀⢦⢀⠀⠀⢠⣿⣷⡄⠀⠀⠈⠉⠉⠉⡿⠁⠀⠀⠀⠀⠀⢹⣿⣾⢻⣟⣻⣿⣼⣿⡾⣿⢷⣿⠨⣿⣿
⢠⠀⠈⢿⣿⣿⣿⣷⣿⣦⣾⣈⡄⠀⡬⠟⣿⣿⡀⠀⠀⠙⠛⠃⠀⠀⢃⠀⠀⠀⠀⢠⣿⣿⣯⣿⣿⣿⣿⣯⣤⢿⣾⣏⠀⣿⣿
⣸⠂⢀⠈⣿⣿⣿⣿⣿⣷⣷⠟⠀⠁⠀⠘⢿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⣧⡂⢀⣀⣼⣿⣿⣷⣿⣿⣿⣿⡿⠻⢾⣿⢇⣤⣿⣿
⣿⣅⣞⡀⣻⣿⣿⣿⣿⣿⡇⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣦⠙⢿⣿⣷⢿⣍⣹⣿⣿⢿⣿⣿⣉⣠⣿⣿⣿
⣿⣾⣿⣯⣿⣿⣿⣿⣿⣿⣤⢤⣧⣠⢀⡋⣠⣤⣄⠀⠀⠀⠀⣈⣻⣿⣿⣿⡟⠻⣿⣦⣤⣤⣾⣿⡿⠿⣾⠀⣰⢏⣿⣿⣿⣿⡏
⢚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡱⢷⣿⣍⣿⣷⠠⣽⣷⣤⣾⣿⠿⡿⣦⣹⣿⣿⡦⠄⣿⣛⣻⣿⣷⣶⣿⢿⣶⣿⣿⣿⣿⣿⣿⡟
⠙⠳⠷⢺⣿⣿⣿⣿⣿⣿⣿⣬⡽⠛⢨⡟⠟⢐⡃⠿⠻⡟⠺⠗⣮⣏⡟⠋⣁⣀⣴⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽
⢀⡀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠻⠀⢻⣤⣿⣿⠿⢹⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣉
⣬⠃⢠⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣷⣶⣿⣾⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣙
⣶⣷⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣧⣲⣶⣴⣶⣶⣴⣶⣶⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢿⠿⠿⠿⠿⢿⡿⣿⣿⣿⠿⠿⠿⣿⡿⠿⠿⣿⣿⡿⢿⣿⣿⡿⢿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠭⠉⠹

⠀⠀⠀⠀⠀⠐⠒⠀⠰⠤⠲⠤⠶⠤⡤⣤⣤⣄⣀⣠⣀⡀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠛⠛⠛⠛⠻⠿⢿⣷⣶⣶⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⢿⣿⣿⣷⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣐⠢⣤⣴⣤⣬⣦⣤⣴⣦⣤⣴⣤⣤⣤⣤⣠⣠⡀⠀⠀⠀⠀⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠽⣿⣿⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⠀⠀⠀⠀⠀⣛⣻⣿⣿⣿⣿⣿⣿⡟⠻⢿⣿⣿⢿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣧⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⠇⠀⠈⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠉⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⢀⣀⠀⠀⠟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⠀⠀⠀⠀⠀⡠⠢
⠀⠀⢀⣀⠠⠤⠤⠤⠄⢀⣀⠀⠀⠀⡜⢀⠜⢀⡀⠀⠀⡘⢀⡴⠀⠀⢰⠁
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⠋⢸⣩⣁⣀⢀⣈⡍⣹⣩⠅⢠⠀⣠⡟⣠⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠉⠉⠉⠁⠉⠉⠑⡠⠬⠮⣥⠅⢁⣉⣉⣁⠀⠀⢀⣀⣀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠤⠔⠁

⣶⣶⣶⣲⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠁⣿⡇⣴⡖⢶⢀⣶⠒⣦⠀⠀⠀⣿⡇⣠⣾⣷⢐⡒⢶⡄⣠⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⣀⣀⠀⣿⠴⠋⢸⣿⠾⠃⠀⠀⠀⠻⠷⠋⢸⡇⣿⡇⢸⣿⣿⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⠇⢿⣦⡶⠸⣿⣴⠆⠀⠀⠀⠠⣆⣀⡿⠁⢿⣇⡼⠃⢿⣾⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣿⣿⣇⡀⠀⠀⠀⠀⢾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⢀⣴⣶⠲⣦⣰⣶⡄⠀⣾⣿⡟⠉⠉⢀⣀⠀⠀⣸⠋⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⣸⣿⡇⠀⣸⣿⡟⠀⢸⣿⣿⠁⠀⣴⡿⢻⡇⢰⠇⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠸⣿⣧⢾⣿⡿⠀⢠⣿⣿⡟⠀⢸⣿⡷⠚⣱⠋⠀⣾⣿⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇⢠⡏⣿⣿⠃⢀⡞⠿⠷⠞⠁⠀⣸⣿⡇⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⡀⠻⠷⠏⠀⣿⣿⢀⡞⠀⠀⠀⠀⠀⠀⣿⣿⣇⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠙⠿⠛⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢢⢠⡷⠀⢰⡆⠀⣾⢀⡶⠻⣆⠀⣴⠛⡄⢲⡖⠒⡴⠻⣆⢰⡆⡞⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⢸⡇⠀⣼⠁⠀⣿⣸⠁⠀⡿⢸⠇⠀⡇⢸⡇⢸⠁⠀⣿⢸⣿⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⢸⡇⠀⣿⠀⠀⣿⢿⠀⠠⡄⣿⠀⠀⡇⢸⡇⢸⠀⠀⣿⢸⣯⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠒⠺⠘⣇⡤⠸⣄⠄⡿⠘⢧⡤⠃⣿⠒⠒⠇⠸⡇⠈⢧⡼⠃⠈⡿⠲⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠲⢦⡀
⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⢀⣀⡀⠀⠀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠯⣭⡁⢀⣿⠭⠁⢠⣯⡭⠀⠀⠀⠀⢧⣼⠇⠀⡏⢹⡆⠀⡇⢸⡇⠀⠀⠀⠀⠀⠘⣧
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠲⠚⠃⠀⠛⠒⠂⠈⠓⠶⠂⠀⠀⠀⠘⠃⠀⠀⠛⠞⠁⠀⠛⠞⠁⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⣀⣀⠀⠀⣀⣀⠀⠀⣠⡀⡀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠈⣧⠀⠀⠀⠀⠀⠈⠳⢦⡄⠰⡏⢸⡇⠀⡏⢈⡇⠀⡿⣇⡇⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠛⠋⠁⠀⠙⠋⠀⠀⠙⠋⠁⠀⠃⠙⠁⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠃
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠸⡇⠀⢀⡾⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢷⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⡄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⣧⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⣼⢹⣄⠀⠀⠀⣠⡶⠛⠉⠀⣿⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢾⡀⠀⠀⠀⠀⢀⣞⡥⠶⠛⠋⠉⠉⠉⠙⠛⠁⠀⠀⠀⢰⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣩⠴⣶⢦⡄⠀⠀⠀⠀⣠⠤⣤⣤⣀⠘⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⡾⠐⠿⢬⣯⡴⠟⠀⠀⠀⣾⣥⣀⣹⡥⠼⠃⡀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⡿⠀⠠⠀⣀⠀⠀⢀⣤⡀⠀⠘⠓⠂⢀⡀⠀⣤⣧⣀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⣉⣷⡀⠀⠄⠀⠀⠀⣠⣿⡀⠀⠈⠀⠄⠀⠰⣦⣞⠁⠈⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠙⠶⣶⡶⠶⠛⠁⠈⠛⠶⠶⠶⠶⠟⣟⠁⠈⠓⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⠀⠀⠀⠀⣀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣭⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣶⣿⣧⣠⡄⠀⠀⠀⠀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣦⣄⣀⣀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣦⡀
⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠈⣿⣿⢿⣿⣿⣿⠟⣿⣻⣿⣿⣿⣦⣤⣽⠿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠉⢀⣼⣿⡿⠁⠀⢹⣟⠈⢻⣿⣿⡿⠃⢀⣿⠀⠀⢹⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣽⣽⣧⣤⣀⣀⡙⢷⣥⣧⣍⣧⣴⠾⣁⣠⣴⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿⣿⣿⣾⣿⡿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠿⠿⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣀⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⢿⡿⠿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠗⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠺⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠘⢿⣶⣶⣶⣶⣶⣶⣶⣶⣿⠃⠀⠀⠀⠀⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⢀⣾⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢠⣤⠀⢀⣤⡄⠀⣤⡄⠀⠀⠀⣿⡇⠀⠀⠹⣿⡆⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⢠⣿⠟⠉⠉⠙⢿⣷⠀⠀⠀⢸⣿⠇⠀⠀⠀⠉⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⣿⡇⠀⠀⢀⣿⡇⠀⠀⠀⣾⡿⠋⠉⠉⠻⣿⡄
⠀⠀⠀⢺⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡗
⠀⠀⠀⢘⣿⣷⣤⣀⣤⣾⣿⠀⠀⠀⢸⣿⡄⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠟⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣷⣤⣀⣤⣾⣿⡃
⠀⠀⣰⣿⠟⠉⠛⠛⠋⠉⢻⣷⡄⠀⢸⣿⣿⡿⠋⠸⣿⣇⡀⢀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⢀⣿⡇⠀⢠⣾⡟⠉⠙⠛⠛⠉⠻⣿⣆
⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠘⠿⠋⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣦⣈⣿⡇⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⠀⣿⣇⡀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠻⠿⠿⠿⠿⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⢛⣛⣩⣭⣤⣶⣶⣶⣶⣭⣭⣭⣟⡛⠷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢛⣡⣾⣿⠿⠋⠁⢀⣀⣋⣉⣉⣛⣛⣻⣿⣿⣅⡀⢉⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⣴⣿⣿⣿⣦⣶⣾⣿⠿⠿⠟⠛⣛⣋⣛⣿⣿⣿⣿⣿⣷⣦⣍⣙⡛⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢏⣼⣿⣿⣿⡿⠟⠢⢄⣀⣠⣤⣴⣶⣶⣶⠶⠾⠿⣿⣿⣿⣿⣿⣿⣍⠠⣼⠏
⠀⠀⠀⠀⠀⠀⠀⣴⢟⣡⣾⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣩⣥⣤⡶⣂⡉⠀⠀⠀⠹⣿⣿⣿⣦⡹⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⣰⠏⣼⣿⠟⠁⠀⠀⠙⠻⠿⠿⣿⣿⣿⣿⠿⠷⠞⠛⠋⠁⠀⠀⠀⠀⠀⣿⣿⣹⣿⣯⢠⣤⡿⠃
⠀⠀⠀⠀⠀⢰⡟⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⣼⠇⣿⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣼⡇
⠀⠀⠀⠀⣰⡏⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢸⣿⣿⣿⣇⢸⡇
⠀⠀⠀⠀⣽⠗⢸⣿⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠿⠆⠹⣿⣿⢣⣼⡟
⠀⠀⠀⠀⢻⣆⢸⣿⣿⣿⠇⣀⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢋⣀⣠⣤⣤⣸⡿⠤⢍⠻⣆
⠀⠀⠀⠀⣾⡋⢸⣿⣿⣏⣈⣉⣉⣛⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠟⣿⣿⣿⣿⡿⣿⠣⠀⢀⢠⡿
⠀⠀⠀⠀⢸⡇⡄⠀⡿⣿⣿⣿⣿⣿⣿⠟⣛⣿⠷⢲⣤⣄⠀⣴⣾⣿⣿⡿⠁⣼⣿⣿⣿⣿⢣⣿⢀⣠⢣⡾⠃
⠀⠀⠀⠀⠸⣧⡳⣄⢤⣿⣿⣿⣿⣿⣵⣿⠟⢁⣴⣿⣿⣿⠟⠿⢿⣿⣿⠃⣼⣿⣿⣿⡿⣣⣿⣿⠚⢡⡿⠁
⠀⠀⠀⠀⠀⠈⢳⣎⢷⠹⣿⠟⣽⣿⠟⢁⣴⣿⣿⣿⣿⠇⠀⠀⠈⠻⢷⣴⣿⣿⣿⣿⠿⠛⠉⠘⣠⢻⡿
⠀⠀⠀⠀⠀⠀⠀⢿⣜⠦⠙⡿⠿⠤⣶⣿⡿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⣀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⣴⣶⠀⠀⠀⠀⠀⠀⠀⢀⡴⣱⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⡳⣄⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠃⠀⠀⠀⠀⢀⡴⣫⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⡳⢄⠀⠀⠀⠀⠀⠀⠌⠀⡆⠀⢀⡤⢂⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⡿⢗⣩⣶⠦⣄⠀⠀⠀⠀⣁⠴⣩⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⣵⣾⣿⣿⣿⠀⠈⠋⢽⡇⠀⢹⣶⣭⡻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣼⣿⣿⣿⣿⣿⣿⠀⢴⣿⡟⠀⠀⣼⣿⣿⣿⣷⣌⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣱⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⠃⠀⠀⢈⠙⣿⣿⣿⣿⡎⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢡⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠠⠀⠈⣿⣿⣿⣿⣿⡜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢣⣿⣿⣿⣿⣿⣿⣿⣿⡿⠓⠀⠀⠀⢐⠢⢀⣠⣿⣿⣿⣿⣿⣧⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡾⢣⣿⣿⣻⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣧
⠀⠀⠀⠀⠀⠀⠀⣾⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠄⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢻⡄
⠀⠀⠀⠀⠀⠀⠰⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⢮⣭⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣜⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣣⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣭⣭⣛⣛⣛⡟⠻⠿⠿⢿⣛⣛⣛⣭⣭⡶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⣿⣯⠿⠿⣿⣿⡿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣺⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣂⣤⣴⣶⣶⣿⣿⣶⣶⣦⣄⡀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⣰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⣠⡆⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠹⣿⣿⣿⡿⠟⠛⠛⣿⡿⠻⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢿⣿⣧⠀⠀⠀⠈⠉⠛⠇⠙⠋⠁⠀⠀⠀⠙⠋⠁⠀⠀⡙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠀⠀⠀⠛⠿⢿⣿⣀⡄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠟⠿⠿⠋⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢸⡿⠿⠃
⠀⠀⠀⠀⠀⢀⡀⢿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠈⣿⣾⣿⣿⣿⣿⣿⣿⣁⣤⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⣀⡆⢠⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡏⠉⠙⢯⣠⢸⣿⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡁⠀⠀⢸⡃⠈⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣆⠀⠀⣄⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣀⣀⣠⡎⠓⠀⢸⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣾⣿⠀⠀⠀⠚⠗⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠀⠁⠀⠀⣾⠿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⣠⠞⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣟⢙⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⣀⣠⡤⠶⠋⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣛⣿⣿⠟⠳⢤⣄⣀⣀⣀⣀⡠⠖⠉⠙⠲⢶⡿⠿⠋⠉⢉⡡⠤⠶⠒⠉⠁⢨⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⢨⠋⠈⠈⠀⠀⠀⠀⠀⠀⠙⢦⡀⠾⢯⣤⣄⣠⣤⣌⡀⠪⣝⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠻⣿⣭⣿⡾⠛⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣿⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⡇⠀⠇⠀⠀⠀⠀⠀⠘⢦⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠋⠀⠀⠀⠛

⣀⢀⣤⡀⣈⠉⡉⠁⠉⠉⠉⠉⢉⣭⡁⢈⡉⢉⡉⢉⢩⣩⢉⡉⣩⣍⢉⡁⣈⠉⠉⠉⠉⠉⠉⣩⣍⠉⣀⠀⣀⢀⡀⣌⣍⡉⢉⣭⡉⣉⢉⡉⠉⠀⠈
⣺⢾⠀⡿⣾⢷⡇⠀⠀⠀⠀⠀⢸⠀⡁⣼⣽⡸⡇⣸⠀⡇⢸⡿⡇⢸⢾⠷⣾⠀⠀⠀⠀⠀⠰⡇⢈⢰⣿⡆⣿⢸⡇⢸⠀⡇⣏⠀⡇⡟⣾⠃
⣈⣈⣉⣁⣉⣈⣁⣀⠀⣀⣀⣀⣈⣉⣁⣁⣀⣁⣉⣁⣀⠁⣈⣁⣉⣉⣈⣀⣉⣀⣀⣀⣀⣀⢀⣉⣁⣈⣀⣉⣈⣉⣀⣈⣀⣁⣈⢉⣀⣁⣈⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡄⠀⠈⠻⣿⣿⣦⡀⠀⠈⠻⣿⣿⣦⡀⠀⠈⠻⣿⡿⠦⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣶⡀⠀⠈⠻⣿⣿⣦⡀⠀⠈⠻⣿⣿⣦⡀⠀⡈⠻⠿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠶⣾⣿⣦⡀⠀⠈⠻⣿⣿⣦⡀⠀⠈⠻⣿⣿⣦⡀⠀⠘⠛⠋⠉
⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⠙⢿⣿⣿⣦⠀⠀⠙⢿⣿⣿⣦⠀⠀⠙⢿⣿⡿⠆⠀⠐⠉⠉
⢤⣤⣶⣆⠀⠀⠙⢿⣿⣿⣄⠀⠀⠙⢿⣿⣷⣄⠀⠀⡙⠿⠟⠛⠂
⠀⠙⢿⣿⣷⣄⠀⠀⠙⢿⣿⣷⣄⠀⠀⠙⠛⠉⠁
⣄⠀⠀⠙⣿⣿⣷⡤⠄⠀⣉⣁⣀⡀⠀⠀⣀⣀⣀⠀⠀⢀⣀⣀⣀⠀⠀⢀⣀⣀⡀⠀⠀⢀⣀⣀⡀⠀⠀⣀⣀⣀⠀⠀⠀⣀⣀⣀⠀⠀⢀⣀⣀⣀
⣿⣷⡆⠁⠈⢿⣿⣷⡀⠀⠘⢿⣿⣧⡀⠀⠙⣿⣿⣦⠀⠀⠹⣿⣿⣆⠀⠀⠻⣿⣿⡄⠀⠈⢿⣿⣷⡀⠀⠘⣿⣿⣧⠀⠀⠹⣿⣿⣆⠀⠀⠹⣿⣿⣆
⠻⣿⣿⣄⠀⠈⢻⣿⣷⡄⠀⠈⢿⣿⣷⡀⠀⠘⣿⣿⣧⠀⠀⠹⣿⣿⣆⠀⠀⢹⣿⣿⡄⠀⠈⢿⣿⣷⡀⠀⠈⢿⣿⣷⡀⠀⠘⣿⣿⣦⠀⠀⠹⣿⣿
⠀⠛⠛⠛⠂⠀⠀⠛⠛⠛⠀⠀⠈⠛⠛⠓⠀⠀⠘⠛⠛⠃⠀⠀⠙⠛⠛⠂⠀⠀⠛⠛⠛⠂⠀⠀⠛⠛⠛⠀⠀⠘⠛⠛⠓⠀⠀⠘⠛⠛⢧⣤⣤⣽⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⢶⣶⣿⣦⠀⠀⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠀⠀⠘⢿⣿⣷⣄⠀⠀⠙⢿⣿⣷⣄⠀⠀⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣤⣶⣆⠁⠀⠙⢿⣿⣷⣄⠀⠀⠙⢿⣿⣷⣄⠀⠀⢙⠿⠿⠓⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⢶⣾⣷⣄⠀⠀⠙⢿⣿⣷⣄⠀⠀⠙⢿⣿⣷⣄⠀⠀⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⠀⠀⠻⣿⣿⣷⣄⠀⠀⠻⣿⣿⣷⣄⠀⠀⠹⣿⣿⠷⠄⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⣤⣴⣶⡀⠀⠈⠻⣿⣿⣦⡀⠀⠈⠻⣿⣿⣦⡀⠀⠈⠻⠿⠟⠂
⠐⠒⠛⠃⠀⠀⠈⠛⠛⠛⠂⠀⠀⠈⠛⠛⠛⠂⠀⠀⠈⠛⠛⠋
⠀⠀⠀⠀⠀⠀⡞⠙⠀⡾⡄⢺⠀⡇⢹⠋⣾⣼⠙⣦⣿⣄⡇⠀⠀⠀⠀⠀⣰⠙⠂⣸⢧⢸⡆⢺⠘⡟⢹⣦⠏⢳⣸⣆⣷⠀⠀⠀⠀⠀⢠⡏⠓⢀⣿
⠀⠀⠀⠀⠀⠀⠳⠞⠸⠋⠳⠘⠶⠃⠸⠀⠛⠙⠶⠋⠗⠹⠇⠀⠀⠀⠀⠀⠘⠶⠃⠏⠙⠎⠷⠟⠀⠇⠸⠋⠳⠞⠸⠈⠗⠀⠀⠀⠀⠀⠈⠳⠞⠼⠉
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣠⢀⠔⡠⡪⠐⣀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⣠⣾⢂⢎⡾⢋⠔⢊⡤⢐⡀⠄⠀⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⢮⠐⢄⡢⢄⢄⡀
⠀⠀⠀⣀⣰⣿⢡⢎⣿⠕⣡⡾⠟⠋⠁⠀⠀⠀⠈⣾⠀⠀⠀⠀⠀⠠⠀⠐⡄⠀⠱⡀⠳⡀⠱⡄⠙⢄⠑⣌⢢⡀
⠀⠀⣾⠛⣿⢏⠎⡾⢃⡴⠉⠀⠀⠀⠀⠀⠀⠀⣀⡸⠁⠀⠀⠀⢰⣥⡀⠀⠙⣆⠀⢱⠀⠱⡀⠘⢄⠈⢧⠈⣆⢣
⠀⢰⡗⣀⣿⢀⠜⢁⡾⠀⠀⣀⣸⣿⠀⠀⠀⠾⡷⠦⠀⠀⠀⠀⢁⣀⡀⠀⠀⠈⢷⡀⡇⠀⠱⡀⠈⠣⡈⠆⠸⢸
⠀⢨⡯⡿⠿⠈⠈⠡⠁⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⡿⡗⠄⠀⠘⠦⡀⠘⢠⠀⡆
⠀⠈⢧⡇⠘⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⢸⠰⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⡀⢰⠀⠈⠐⣤⡤⢴⠦⡄
⠀⠀⠀⠙⠀⠈⠐⠃⠀⠀⠀⠀⠀⠀⠀⣈⣁⣠⡀⢠⠀⡇⠘⠀⣷⠀⠀⠀⠀⠀⠀⢹⣄⠈⠡⠃⠈⠒⠦⠐⠀⠀⡛⡁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣍⣒⣴⠆⠈⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢠⡀⠀⠀⠄⣧⡆⢎⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠇⢠⠀⠀⡤⠒⠀⠀⠀⠀⡇⠀⣼⠃⠀⡜⢀⣿⡇⢸⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢡⢰⡿⢀⡜⢀⢎⣾⠁⡸⠀⠇
⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⢾⡵⠋⡠⢫⠎⠈⠀⠃⡘
⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠊⠀⠜⠐⠁⠤⣀⠒⢤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠄⠡⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠤⢤⠀⢠⠄⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠐⡆⢠⣄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⢣⠐⠀⠂⡀⢠⠁⠀⠀⠀⠀⠃⢠⡆⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡌⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⠉⢱⠁⠀⠀⠀⠀⠀⠂⠀⠃⠘⡇⠀⠀⠀⠠⠐
⠲⠰⠀⠀⠀⠀⠠⢬⣀⡀⢦⡀⠈⠀⠀⣆⣠⠀⠀⢀⠀⠀⠚⠀⠀⠠⠀⠄⠀⢱⣾⠴⢤⡏⠀⢠⠎
⠀⠀⡅⠀⠠⡁⠀⠀⢠⠀⠀⠙⢮⠀⠂⡩⢉⣀⣀⠈⠀⠀⢸⠀⠀⠀⠀⠀⡀⠠⠀⣠⣄⣤⠔⠁⠀⠀⠀⠀⠄⠐⠦
⠀⠀⠀⠀⠀⠀⠂⢀⠐⠀⠀⢀⡂⠁⠀⠀⢀⡀⠩⠠⠀⠀⡛⠀⠀⢀⠤⠞⠑⠲⠶⢋⣉⠤⠐⠒⠒⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣅⠀⠀⠀⠀⠀⠀⠀⡐⠀⣀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠑⣄⡀⠀⠀⠀⢠⡧⠊⡁⢠⣤⡤⣤⣄⠀⠀⠀⠈⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⣻⢯⡻⡾⢶⠀⣸⡁⠘⠁⢺⢿⣧⣧⣽⡂
⠀⠀⠀⠀⠀⠀⠈⠿⡆⠀⠀⢥⠁⢸⠀⠀⠀⠙⢾⣼⠁⡇⠀⠀⢸⣼⡇⣟⣉⡁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠖⠁⠀⠀⠈⠀⢸⡆⠀⠀⠀⢸⠿⣤⠇⠀⠀⢸⢸⣇⣧⣦⡅⠀⠀⠀⠠⣤⠤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠸⠀⠀⠂⠀⢸⠀⢹⠆⠀⠀⠸⠜⠃⢧⠟⠇⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⢀⡄⣶⢐⢙⢸⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠐⠀⡀⠀⣀⠁⠂⢠⠐⠾⢿⠿⠾⡿⠦⣵⣏⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢹⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠈⠀⠀⠀⠀⠉⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢶⣄⠀⠀⢀⣴⡄⠀⠀⢀⣴⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡠⠍⠛⠯⠀⢿⠇⠨⠟⠋⠥⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠋⠁⣤⡦⠀⠈⠀⠠⡦⡄⠉⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠁⠀⠮⠴⠀⠀⠀⠠⠬⠆⠀⠉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠛⠛⡛⠂⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠚⡛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠁⠀⢀⠀⠈⢳⣶⣦⣶⣶⠋⠀⢀⠀⠀⠡⠒⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⢀⠀⠀⠈⢹⠉⠀⠀⢀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠦⣀⠀⢑⠐⠂⢩⢐⠂⠒⠁⢀⡠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⠄⣄⠀⢈⠀⠘⠘⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠂⠀⢸⣿⣶⣅⣴⣾⣿⠀⠀⠚⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣀⣈⠀⠸⠿⠛⠁⠙⠻⠿⠀⢈⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠙⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉

⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢐⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠒⠒⢲
⢰⠀⠀⢀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⡀⠑⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡞
⠈⢆⠀⠀⠈⢦⣉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠹⠛⠛⠁
⠀⠀⠑⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠾⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬



⠀⠀⠀⡄⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣄⣠⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣄⡀
⠀⠀⠀⠀⡠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡧⠤⡧⢤⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⠀⠀⠀⢳
⠀⠀⡀⢴⠃⠐⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣡⠤⠧⢤⣁⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⠒⠚⠦⢊
⠀⠀⠒⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤
⠀⠀⠀⠀⣠⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣾⣄⠀⠀⢀
⠀⠀⠀⠀⠉⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠛⠁
⠀⢀⠤⠤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡴⠀⠀⠣⣀⠜
⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣁⣸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠙⠄⠀⠖⠉⠱
⠰⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣠⠎⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢠⠂⢀⡔⠁⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⡴⠁⢀⠎⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⡞⠁⢀⠎⠀⠀⠀⠐⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠎⠀⢀⠎⠀⠀⣠⠊⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⡠⠃⠀⢀⡔⠁⠀⣠⠏⠀⠀⢀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢠⠞⠀⠀⢠⠊⠀⠀⢠⠋⠀⠀⠈⠉⠁⠀⠀⠀⠉⠛⠛⠃⠀⠀⠀⠀⠀⠠⡶⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⡴⠁⠀⢀⡴⠁⠀⠀⣰⠏⠀⠀⠀⠀⠀⢤⠞⠛⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠞⠀⠀⠒⠉⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠘⠡⠧⠔⠒⠂⠀⠀⠀⠀⠀⠀⠀⠬⠶⣤⠤
⠀⠀⠀⠀⠀⡀⠀⠠⡀⠀⡀⠀⠛⢀⡔⣀⡀⠀⣀⣠⠾⠃
⠀⠀⠀⠀⢀⠹⡄⠀⠻⣦⠙⢦⠀⢸⡏⠁⢹⡆
⠀⠀⠠⡀⠀⠳⣄⠄⠀⠘⣧⠈⠳⣼⡧⠀⠀⠃
⠀⠀⢀⠑⠄⠀⠙⠶⣄⡀⠙⢦⡀⠘⣷⠀⠀⠀⠀⣄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢤
⠀⠀⠈⠀⠐⣄⠀⠀⠈⠻⣄⠀⠹⠀⠘⣧⠀⠀⠀⠉⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠳⡀⠈⢧⣄⡀⠀⠙⠆⠀⠁⠀⢸⠳⣄⠀⠀⠀⣷
⠀⠀⠀⠀⠙⢆⠀⠈⠙⠆⠀⠀⠀⠀⠀⠀⢷⡀⠁⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠰⡌⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠚⠓⠆⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠁⠀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣀⣀⡌⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠙⠓⠤⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠙⠲⢤⣄⣀⠀⠀⠀⠀⢀⣠⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠁⢰⡄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢀⠤⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣤⣤⣤⣤⣤⣤⣴⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠉⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡔⠲⡌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡅⢎⡇
⠀⠀⠀⠀⠐⠒⠒⠒⠒⠂⠀⠀⠛⠀⠒⠒⠒⠒⠒⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⢊⠴⠁⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠸⠃⠀⠈⡆⡔
⠀⠠⠀⠀⠀⠀⣰⣁⡍⢀⡴⠀⢿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢨
⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠓⠒⠒⠒⠀⠀⢀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡱⠀⠀⠀⠀⠀⡠⠂⠉⠉⡇⠢⡀⠀⠀⠀⠀⠀⡇⡤
⠀⠀⠀⠙⠔⠉⠉⠉⠉⠁⡜⢣⠈⠉⠉⠉⠉⠑⠦⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⢘⠀⠀⢀⡴⢆⠀⢸⠀⠀⠀⠀⢀⠈⡀
⠀⠀⢠⠖⠉⠉⢹⠙⠢⡀⠑⠊⠀⢀⠔⠋⠉⠉⠃⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠙⢦⡀⠀⠈⢹⠈⠁⠀⠀⠸⠛⠀⠀⢀⠔⠁⠀⢈
⠀⢀⠁⠀⠀⢀⠾⡀⠀⢡⠀⠀⢀⠃⠀⠀⢀⠼⣀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠙⠦⣠⡈⢆⡀⠀⣀⠞⢸⡤⠔⠁⠀⢀⠴⠁
⠀⠀⠙⠂⠉⠁⠀⠈⢙⠈⠀⠀⠈⠛⡆⠈⠁⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⠀⠉⠈⠉⠀⠁⠀⠀⢀⠔⠁
⠀⠀⠀⢆⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⡔⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⡠⠒⠛⠂⠘⠓⠢⡀⠀⠀⠀⡤⠖⠛⠃⠘⠋⠢⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸⣁⣀⣀⣀⣀⣀⣀⣈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠁





⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠊⠙⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠈⠈⠲⢄
⠀⠀⠀⠀⠀⠀⠀⢣⠀⡴⣔⢉⢩⡀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡔⠒⢄⠀⠈⠁⠒⢄
⠀⠀⠰⠢⢄⠀⠀⡰⠛⢤⡣⠄⠃⠈⢣⠀⢀⠀⢀⡆⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠘⠢⡊⠗⢄⠀⠀⠈
⠀⠀⢐⠀⠠⠃⠀⠑⠀⠲⡀⠁⡛⠀⠘⠀⠀⠀⡜⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠙⠦⠴⠃⡀⠜
⢀⠄⣈⠀⠀⠀⠀⠀⠀⠀⠉⠊⠀⠀⠀⠀⠀⠀⠑⡲⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠲⡀⠑⠄⣀⡠⠎⢠⠒⠄
⠣⠠⠆⠀⠀⠀⠀⠀⡠⠊⠉⠉⠓⢦⠀⠀⠐⠒⠉⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⢬⣊⠀⠀⠀⠀⠀⠀⢈⡓⠅⢠⣄⡀
⠀⢠⠄⠤⡀⠀⠀⠠⣁⣀⡠⠚⠤⣀⡇⠀⠀⠀⡠⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢠⠀⠑⢄⠀⠀⠀⡰⠋⠀⠀⢠⠀⠉⢆
⠀⠸⡀⠀⠉⠢⡀⠀⢘⡀⠀⠀⠀⡇⠀⠀⡠⠊⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⣀⣀⠴⠋⠢⣀⣈⠀⠀⠀⣇⣀⣀⠤⠊⠢⢀⣈
⠀⠀⠈⠢⡀⠀⠈⠒⠤⠵⠦⠤⠾⠤⠐⠉⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⡁⠀⠀⠀⠀⢨⠈
⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⣀⣀⣠⠃⠀⠀⠀⠀⡇⠑⣄⣀⣀⣠⠊
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠉⢢⠀⠀⠀⠗⠉⠀⠀⠀⠀⠉⠲⡀
⣀⣀⣀⣀⣀⣀⣀⣘⣤⣄⣀⣤⣤⣤⣜⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣤⣤⣤⣤⣤⣤⣤⣤⣃⣀⣈⣀⣀⣀⣠⣄⣤⣤⣤⣃
⣿⣿⡟⠙⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠂⠀⠈⠐⠒⠤⣀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣊⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣘⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣪⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢡⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢀⠠⢤⡴⣂⠤⠐⠒⠒⠒⠂⠤⠀⠀⠘⠊⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⠀⢠⠞⠊⠀⠀⠀⠀⠀⣀⡀⠀⠀⠐⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡠⠈⡀⠰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠤⠈⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⡄⠈⢦⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠉⠑⢄⠀⡄⡠⠒⠒⢄⢀⣀⡀⠀⠀⠂⠀⠀⠀⠃⢸⠀⠀⣀⣈⡆⡎⠀⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠷⡀⠘⡄⢿⣧⠴⠦⠀⢧⠀⠀⠀⠀⠒⠛⠛⠛⠿⠀⠣⡎⢝⢀⡁⢨⠊⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⣹⠀⢰⠀⠀⠃⡜⠀⢀⡤⠔⠒⢠⠄⠀⠀⠀⢀⡿⠑⠂⠀⠀⣀⠼⢁⠎⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢂⡑⠋⠈⠳⣊⣀⠣⠀⢣⠀⠸⣄⠀⠘⠁⠀⠀⠠⠁⠀⠀⠀⠀⢀⡔⠁⡠⠊⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠑⠢⠤⣉⠋⠀⠀⠀⣀⠀⠀⡀⠤⠶⣭⠶⠊⠁⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠒⠊⠀⠀⠰⠁⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡑⠦⢄⣀⣀⡀⠤⠴⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠈⢏⠁⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡇⠐⢢⡄⢸⡤⡔⠒⢰⡴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⢪⣀⠉⡆⣸⡅⠀⠉⣧⠠⠼⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠉⠀⠀⠈⠳⡄⠉⠉⣽⠃⠐⣍⠗⡲⠶⠛⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬⣒⠲⠠⠤⠴⠗⣒⠬⠋⠀⠀⢟⢼⣀⠀⠀⠀⠀⡨⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠑⠢⠍⡉⣉⡉⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠴⠶⠒⠶⠚⠰⠶⠖⠒⠂⠀⠀⠀⠘⠞⠧⠅⠂⠔⠒

⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⣿⣿⣿⣷⣄⠀⠀⠀⠀⠙⣿⣿⣿⣿⣶⡄⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⡀⠀⠀⠀⠉⢿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠙⣿⣿⣿⣿⣷
⣿⣿⣿⡟⢁⢀⣀⠀⣠⣾⣿⣿⣿⣿⡛⣀⡀⣀⣀⣠⣾⣿⣿⣿⣿⢋⢀⣀⣀⢀⣴⣿⣿⣿⣿⡟⢁⣀⢀⡀⣠⣾⣿⣿⣿⣿⢟
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀
⣴⣿⣿⣿⣿⠟⠁⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠁⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋⠀⠀⠀⢀⣴⣾⣿⣿⣿⠟⠁⠀⠀⠀⢀⣴⣿⣿⣿
⣿⣿⣿⡟⠁⠀⠀⠀⣠⣶⣿⣿⣿⣿⠟⠁⠀⠀⠀⣠⣾⣿⣿⣿⣿⠋⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠃⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠈⢩⣭⡍⣍⢡⣄⣄⣄⣤⣄⣩⠍⠉⢉⡉⠁⣄⣄⣄⡄⣠⣍⣍⣍⣩⣉⣠⠄⠀⢀⡀⠈⣍⣍⢩⡍⣥⣤⣄⣤⣤⣠⡩⠍⠉
⣤⣤⣼⡙⠙⠙⠛⠙⢋⣯⣙⣽⣍⣾⠀⠈⠁⠀⠘⣙⣹⣽⣍⣭⣍⠋⠉⠙⠙⢛⣠⣬⣥⣠⣹⡙⠋⠙⠋⠛⢙⣭⣍⣽⣝⣯
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣠⣾⣿⣿⣿⡿⢃⣀⣀⢀⣀⡀⣉⠉⡉⣉⣉⣁⡀⡀⠀⢀⣴⣿⣿⣿⣿⢋⣀⡀⡀⣀⡀⣈⡉⠉⢉⣉⣉⡀⣀⠀⢀⣴⣿⣿⣿
⣿⣿⣿⡿⠋⠀⠘⠉⢋⣈⣉⣁⣉⣀⠃⠉⠑⠁⠃⣡⣾⣿⣿⣿⣿⠟⠁⠈⠋⠑⣁⣉⣉⣉⣁⡙⠈⠉⠊⠉⢈⣴⣿⣿⣿⣿⡟
⠉⠉⠉⠛⠙⠛⠋⠙⠉⠉⠉⠉⠉⠉⠛⠙⠉⠛⠙⠉⠉⠉⠉⠉⠉⠋⠛⠋⠉⠋⠉⠉⠉⠉⠉⠙⠛⠛⠉⠉⠋⠉⠉⠉⠉⠉⠛

⠛⡛⢛⠂⡀⣀⢀⣀⡈⠛⠛⠛⠛⠓⡀⡀⡀⣀⣀⢘⣛⡛⠛⠛⠃⠀⢀⠀⡀⡀⣙⣛⣛⣛⠛⠂⠀⠀⠀⡀⢈⢛⢛⣛⣛⣛⡀
⣀⣃⣋⡙⠃⠉⠙⠊⢙⣀⣀⣁⣀⣈⠒⠉⠓⠈⠈⣓⣉⣉⣀⣈⣁⠀⠙⠚⠙⠋⣈⣉⣓⣉⣃⡀⠈⠀⠀⠓⢊⣛⣃⣁⣉⣊⠙
⠛⠛⠛⠓⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠀⠈⠁⠉⠄⠀⣠⣶⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠿⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⣴⠋⠘⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡰⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⢠⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣆⠀⠰⠁⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⢸⣠⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠼⠣⠂⠀⠠⠷⢄⠀⠀⢀⣴⠒⠒⠁⠀⢠⡟⢃⠈⠟⠘⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⢠⠀⠓⠦⠘⡆⠀⢸⠿⠇⠰⠋⠹⠛⠃⠀⡆⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⣸⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠐⠁⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡁⠀⡀⢀⣎⠠⠀⠀⠀⠁⠚⠋⠁⠀⠀⠀⢀⣠⠆⠀⢹⠀⢸⠀⢀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢀⠀⣸⣿⡄⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠁⠀⢀⣬⠀⡾⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠞⣧⠾⠋⠁⠀⠀⠠⠀⠉⠉⠁⠀⠀⠀⠀⢀⣰⢿⡟⡰⠃⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣾⡄⠀⠀⠀⢠⠀⠠⢀⣀⣴⣶⡿⣿⣨⠟⠁⡠⠃⠐⠟⠀⠠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⡄⠀⠀⠈⠻⣿⣇⡀⠀⠀⠀⠒⣿⠟⢻⠛⠉⣷⠟⢁⠔⢋⢸⠀⠀⠀⠀⠀⠀⠀⠠⠴⠆⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⠀⠁⠀⡀⠀⠀⠀⠈⠫⡧⣀⣀⣠⡼⠁⠀⠀⠉⠀⠉⠀⣷⡟⠠⣿⠇⠀⢨⡆⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠈⠉⠁⢲⠀⠀⠀⠀⠀⠘⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⢱⡆⢀⡞⠀⠀⢿⣟⡄⠀⠀⠀⠀⢨⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣠⢀⠀⠀⠀⢠⠀⣀⢤⣄⡉⠃⠀⠀⠀⠀⠀⢈⡴⠚⢀⡞⠀⠀⠀⠀⠋⠙⠀⠀⠀⠀⡈⢁
⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠆⣸⠃⠈⠳⣄⠀⠈⠁⠄⢠⠃⠀⠁⠀⠀⠀⠀⣠⠏⠀⣠⠏⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠂⠀⠑⠄⠀⣰⣲⡀⠀⠀⠀⠀⠀⠀⡏⠀⡰⠃⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⣳⠀⠀⠀⠀⠀⠀⠁⠀⠁⢀⡞⠁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠉⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡆⢰⠀⠀⠀⠀⠀⠀⠀⠀⡰⠣⠤⢠⣠⣤⣀⣀⣀⣤⠀⣼⠁⠄⠁⠀⢀⡀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⣇⠈⠀⠀⠀⠀⠀⠀⠀⢠⡷⠖⠲⠦⠤⠤⠤⠤⠀⠀⠀⠛⣤⠀⠀⠀⣈⠁⠀⠀⡇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⠽⠋⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⠔⠉⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⢳⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡀⠀⠀⣇⠀⠀⠀⢠⠇⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⢀⡄⢀⠈⢦⠀⠀⢸⡀⠀⠀⠉⠀⢀⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠛⠉⠓⢌⣇⠀⢀⠷⠒⣒⣒⢲⣯⠤⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠻⣈⣋⣀⣤⠞⠑⠚⠧⣄⡀⠀⠀⠀⠀⠀⠀⠀⡎⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⢵⠀⠀⠀⠚⠁⠠⡄⠀⠀⠀⠉⠑⠢⢄⡀⠀⠀⡸⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠱⠚⠒⡀⠀⠀⠀⠀⠀⢙⢦⡀⠀⠀⠀⠀⠀⠈⠑⠢⣷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠴⠚⡇⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠑⠦⢄⣂⣀⣀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡜⠉⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⣰⠁⠀⣾⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡇⠀⠀⠀⠀⣀⠀⠀⡇⠀⠀⠀⢀⢇⡴⠚⢳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢼⡦⢧⣀⠀⠀⠈⠀⠀⠀⢻⣀⡄⠀⡌⡞⠠⣾⠟⢶⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠂⠬⣉⡒⠢⠾⡓⠲⡞⢇⢸⠈⠛⢹⡄⠀⠀⣎⣉⡉⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⡙⢷⣤⣸⡀⠟⠙⢾⣆⠀⡼⠙⠦⣀⣯⣀⡈⢙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠉⠳⣌⠓⢯⣑⢄⢩⢻⡷⠣⡀⠥⣼⣴⡒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠑⢦⡈⠙⠛⠁⢀⡔⢉⡶⢥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠁⠀⠀⠀⠀⠀⠙⠦⣤⣴⣉⠹⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡛⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠒⠶⠤⢤⣀⣀⡀⠀⠀⠀⢠⣧⣀⣀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⢀⡴⢋⠜⠉⡩⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⣀⣤⣴⣫⡔⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣜⠁⠀⠀⠀⢀⠔⢁⣁⣰⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠥⠤⠼⠿⠂⠙⠛⠉⠉⠉⠉⠛⠒⠻⠥⠬⠉⡑⠆

⠀⠀⠀⣄⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⣤⢠⢠⣀⡀⡀⡀⡀⢤⠀⠀⣠⡄⠀⣠⡄⠀⣠⣤⠀⢠⣤⠀⢀⣤⠄⠀⣤⡄⠀⣠
⠀⠀⠀⣺⣍⣉⣋⣉⣉⣉⣙⣿⠀⠀⡆⣵⣿⡿⣫⣻⣿⣿⠪⠀⠀⠃⣠⡾⠋⣠⣾⠋⢁⣴⠟⢁⣴⡿⠃⣠⡿⠋⣠⡾⠃
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡿⠋⣠⠀⢠⣤⡤⡤⣤⣤⣤⣤⢤⢤⣤⣄⠀⠋⣠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣤⣾⠟⠀⢈⡉⣁⣁⣉⣉⣉⡉⢉⣈⡉⡉⠀⣾⠟⠁
⠀⠀⠀⣠⠄⣠⢀⡤⢀⡄⣠⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣬⠀⠀⠏⢁⣴⠀⢉⣉⠉⠉⣉⡉⠉⣉⡈⠉⣉⣁⠀⢁⣴⠟
⠀⠀⠀⣠⢞⡵⢋⡴⣫⠞⡱⢚⠀⠀⣿⣇⣧⣄⣷⣦⣤⣼⡿⠀⠀⣶⠟⢁⣴⠟⠋⣤⡾⠋⣠⡾⠋⢠⣾⠟⢁⣴⠟⢁⡤
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣶⠖⢀⣶⠖⢀⣶⡶⠂⣰⡶⠂⣰⣶⠖⢀⣶⠖⢀⣶⠀⠀⣴⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠁⠴⠟⠁⠴⠿⠋⠠⠾⠋⠠⠾⠟⠁⠶⠟⠁⠴⠟⢩⠀⠀⣿⣿⡟⠛⠛⠛⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿
⠀⠀⠀⠀⠀⢀⣤⢤⡤⣤⢤⡤⣤⣠⢤⡤⣄⡤⡄⣄⡄⠀⢸⠀⠀⣿⣿⡇⢀⣷⣾⣾⢅⣧⢸⣿⣷⣾⣷⡇⣿⠂⢸⣿⣿
⠀⠀⠀⠀⠀⠸⠿⠸⠄⠿⠸⡧⠿⠿⠸⠇⠟⢧⠿⠟⠇⠀⢸⠀⠀⣿⣟⡇⢀⠤⣤⣤⢤⢠⢤⣤⣤⢤⢤⣤⣤⡀⢸⣿⢿
⠀⠀⠀⠀⣠⡤⠀⣠⣤⠀⢀⣤⠄⢀⣤⡄⠀⣠⡤⠀⣠⣤⠈⠀⠀⣿⣿⣇⣀⣁⣉⣈⣈⣀⣈⣈⣉⣉⣀⣉⣈⣀⣸⣿⣿
⠀⠀⠀⣿⠋⣠⣾⠟⢁⣴⡟⢁⣴⡿⢋⣤⣾⠋⣠⣾⠛⣡⣾⠀⠀⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿

⠀⠀⠀⡿⡿⡿⡿⡿⡿⡿⡿⡿⡿⡿⡿⡿⡻⡻⡿⡿⡿⡻⣻⠀⠀⠟⣡⠟⣡⡾⢋⡼⢋⣴⠀⠀⠆⣠⠖⣠⡶⢀⡴⢂⡤
⠀⠀⠀⠀⠀⠀⣴⣶⢲⠒⣶⢴⡒⢲⣶⣆⣦⣶⣆⡆⠀⠀⠀⠀⠀⠛⠁⠚⠋⠐⠋⠐⠋⠰⠀⠀⠟⠁⠚⠋⠐⠋⠐⠋
⠀⠀⠀⣴⣴⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⡤⣰⠀⠀⠄⣠⣀⡄⣀⣀⣄⡄⠀⠀⠀⡀⢀⣀⡀⣀⣀⣀⡀
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠃⢛⠛⡃⢛⢛⠃⡃⠀⠀⠀⠀⢘⡛⠃⣙⢛⢛⠃
⠀⠀⠀⠖⡰⢆⡶⣢⢖⡰⢂⡶⣢⢖⡴⢂⡶⣠⢖⡴⢂⡶⣢⠀⠀⡅⢯⠽⠿⣿⢿⡍⡧⠀⠀⠀⠄⠸⠸⠿⠹⢿⠸⠇
⠀⠀⠀⠉⠀⠉⠈⣥⣬⡀⣭⣬⢥⢨⣤⣍⣬⢥⣌⠀⠉⠈⢙⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣠⢄⡤⢀⡍⣩⢅⡭⢉⡄⣨⢌⡭⢉⡍⣡⢀⡤⢀⡴⠀⠀⣥⠞⣡⡾⢃⡴⢋⡴⢛⠀⠀⣤⠞⣡⡾⢃⡴⢋⡴⠓
⠀⠀⠀⠑⠛⠐⠋⠚⠑⠛⠐⠋⠚⠑⠛⠐⠋⠚⠑⠛⠐⠋⠚⠀⠀⠃⠚⠋⠐⠛⠐⠛⠐⠛⠀⠀⠃⠘⠋⠀⠋⠀⠋⠀⠘

⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⣦⣄⣀⣀⡀⢸⡇⣇⠀⠀⣇⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢻⣄⠰⠒⠒⢦
⢳⡀⠉⠁⠀⣿⠷⢹⡄⠀⢻⡄⠀⠀⠱⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢰⠉⠓⠦⡀
⠸⡇⠀⠀⠀⣿⢀⠀⢳⡄⠀⣿⠀⠀⠀⠋⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠏⠀⠀⣿⠀⠀⣠⠆
⠀⢙⠄⠀⢠⡟⠈⢧⠘⣧⠀⢹⡆⠀⠀⣿⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣸⠀⠀⢹⠀⠀⣇⠀⣀⣀
⠀⠈⣆⠀⢸⡇⠀⠸⣇⠘⢧⡀⢻⣄⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣿⠀⠀⢸⠀⠀⠈⡆⢁⣍⡽⠋⠉⠁
⠀⠀⠹⣤⡼⠃⠀⠀⣿⠀⣼⠁⢸⡏⠀⢿⡀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⡆⠀⠀⢇⣽⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣿⠀⠸⣇⠀⢸⠁⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣸⡇⡀⠘⠦⠀⠘⠄⠘⠆⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠠⠤⠿⠄⠀⢠⣇⠀⠀⠣⡀
⠀⠀⠀⠀⣀⣠⠴⣻⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠲⢸⠄
⠀⠀⠀⡖⠀⢀⣠⠏⠠⡄⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠉⠑⠒⠦⣄⣀⣀⣤⡀⠄⢺
⠀⠀⢰⡇⠀⠀⠀⠀⠀⠙⠛⠮⣟⠋⠛⠉⠁⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣀⢤⠔⠉⠁⠀⠀⠀⠈⢧⡀
⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤
⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁
⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢼⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⠿⠀⠐⠟⠛⠛⠛⠛⠛⠛⣁⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣀⣿⣿⣤⣤⣤⣤⣤⣤⣤⡾⠟⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢻⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣽⣷⠶⠞⣿⣻⡇⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⡄⠠⠐⠘⢋⣉⡄⢤⠸⡆⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣻⣭⣴⣶⡿⣿⡉⢟⣫⣿⣴⡶⠾⢿⣛⣼⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠄⠒⠛⢉⢁⠀⠰⡤⠆⠣⠘⠁⣉⣠⠠⠗⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣟⣷⣿⣭⠶⠿⣟⣛⣩⣭⣶⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠐⠈⠁⠐⣋⣁⠠⠤⠖⠒⠋⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣯⣵⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢀⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣆⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠇⠀⣀⣴⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⡀⠀⠀⠈⠙⠛⠛⠋⢁⣤⣾⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⢀⣤⣾⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠉⠛⢿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣏⠁⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣄⠀⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠉⠻⡄⠀⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣦⡀⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠈⠛⠛⠃




⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠛⠓⠶⣤⡀⠀⠀⠀⢀⣀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⣀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣸⢃⣠⠖⠶⠶⠶⠖⠛⠉⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠈⠙⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠀⠈⠙⢻⣿⠉⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⠤⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠖⠛⠁
⠀⠀⠀⠀⠀⠀⣴⠚⠛⠛⠛⠛⠛⠚⠛⠛⠛⠓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣦
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢠⡀⠀⠀⠀⠀⠀⢠⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡀⠀⠀⠀⠀⠀⢠⡄⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⡀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⣸⠇⠀⠀⣿⠀⠀⢸⡇⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣸⠃⠀⠀⣿⠀⠀⢸⣇⣀⡟
⠀⠄⠀⠀⠀⠀⠈⠉⢹⠀⠀⠀⣿⠀⠀⢸⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠀⠀⠀⣿⠀⠀⢸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⣿⣤⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⠿⠤⠴⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠉⠉⠀⠀⠀⠀⠈⠉⠑⠢⢤⡀⠀⢹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⡡⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⣧⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠎⠀⠀⠀⠀⠀⠀⣀⣀⡠⠀⢀⣀⠤⠤⠤⢄⣀⡀⠀⠀⠀⠳⠔⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⡠⠔⠊⠉⣠⣫⠔⠊⠁⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀⠄⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢀⠴⠏⣠⡤⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠠⠶⣦⠀⠀⡼⠁⠀⠀⢣⠀⠀⠀⠀⠀⠀⠦⣀⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡴⠁⡸⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢸⠇⠀⠀⢀⡼⢄⡀⠀⠀⠀⣠⢤⡈⢳⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠰⢋⡥⠤⠈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⠀⢸⠀⠀⠀⠸⡄⠀⠙⣄⣄⣠⡇⠀⢹⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠞⠙⠲⡄⠀⡇⢀⡴⠒⠲⣄⠀⢀⡀⠀⠀⠀⡔⠉⠀⠀⠈⢳⠸⡆⠀⠀⠀⠈⢢⢰⠁⠀⡝⠃⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⣄⠀⠹⣀⣱⢺⣀⣀⣀⡈⢼⠁⠉⠆⠀⠀⠤⠶⠶⠶⢶⣾⠄⠳⣄⠊⡟⠉⠛⠘⠢⠞⠁⠀⣰⠃
⠀⠀⠀⡇⢀⠔⠒⢲⠃⠀⡝⠀⠀⡇⢀⠟⠋⠁⠀⢀⣈⠁⠀⠀⠀⠀⠀⠀⢠⣤⢄⡀⠀⠀⠏⠀⢳⡖⢲⠀⠳⠀⢠⠞⠁
⠀⠰⡀⣇⢸⡀⣠⠼⣄⢀⡇⢠⠀⠀⢸⠀⠀⡴⠋⠁⠀⡨⠝⠓⠲⠤⣤⠤⠖⠁⠀⠀⠀⠀⡰⠖⠋⣠⠃⠀⠀⠀⣾
⠀⠀⠳⡈⢦⠈⠀⠀⠈⠯⣤⡤⠑⠀⠘⣄⠀⠙⢤⣀⠈⠀⢀⣀⣠⠔⠁⠀⠀⠀⠀⠀⢀⡴⠁⢀⠜⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠈⠙⠲⠤⣬⣉⣉⣉⣁⣀⣀⠠⠀⢠⡤⠖⠚⣧⡶⠖⠉⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠈⠒⠒⠉⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠺⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡓⠤⣀⠀⠀⠀⢀⣀⡤⠔⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⠦⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⢢⡤⠄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣤⣄⡀⠀⡾⢷⠀⠀⠀⣀⣀⣘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢬⣇⠀⠉⣶⠇⢸⣖⣟⠉⠉⡽⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣧⣀⡈⣳⢀⣧⡀⠀⠀⢹⡤⠔⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⠁⠀⠀⠈⠙⢦⠀⠈⠉⢩⡟⠁⠰⡋⠿⢄⣶⠶⠿⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡓⠤⢀⣀⣀⠀⣠⣇⠠⢔⡻⠃⠀⠀⣿⣠⡋⠀⠀⠀⠀⠀⠀⡳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠢⠤⠤⠔⠒⠂⠉⠀⠀⠀⠀⠘⠪⣝⡒⠤⠤⠤⠴⠚⣁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠂⠒⠀⠀⠀⠀⠰⠀⠶⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠐⠀⠒⠰⠂⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣷⣶⣄⣀⣠⣴⣶⣾⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⠋⠁⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠉⠛⠿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡿⠟⠋⠁⠀⠀⠀⠀⠀⣠⣾⠿⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⣴⡿⠛⠉⠁⠀⠀⠀⠀⢀⣀⣀⣀⣰⣿⡇⠀⠀⢸⣿⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠉⠛⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠰⠿⠿⠿⠿⠛⠛⠛⢻⣿⡇⠀⠀⢸⣿⠋⠛⠛⠛⠛⠻⠿⠿⠿⠷⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣶⣤⣄⣀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⣀⣠⣴⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣷⣾⣿⡇⠀⠀⢸⣿⣶⣶⣶⣿⣿⣿⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⣿⣧⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣿⠛⠉⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣶⣶⣶⣶⣾⣿⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⣶⣶⣶⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣶⣶⣶⣶⣾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡆⠀⢸⣿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⣿⣄⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⣹⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣶⣤⡄⠀⠀⣿⡿⠋
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢥⠀⠍⢻⣧⠀⠀⣤⣄
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠙⠋
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢰⠂
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠛⠉⠓⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢹⢀⣀⠀⠀⣀⣠⡤⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠤⠇⠀⠀⠀⠀⠈⠀⠀⠙⠊⠁⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠞⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠢⣤⠴⠋⠉⠙⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠀⠀⠀⠀⣀⠀⣀⡴⠃⢋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣆⣀⣠⡖⠚⠁⠉⠉⣠⠞⠛⠁⠀⠀⠀⠀⠀⠘⠛⠓⠆⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠚⠁⢀⡤⠶⠦⣄⡀⠀⢀⡴⠒⠒⠦⣄⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠉⠲⡏⠀⠀⠀⠀⠈⠳⡄⠀⠀⠹⣄⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢀⣹⣔⣦⠀⠀⠀⢀⡇⠀⠀⠀⠘⣶⠹⣋⡭⠭⠉⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠘⢻⣏⡁⠀⢀⣠⠞⢁⡤⠶⢤⡀⠘⣯⡽⠆⢾⣉⣌⡟
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣸⠀⠀⠀⠀⠀⠀⢀⣨⠦⠤⢤⣄⣀⠀⢠⠞⠀⠉⠉⠉⠀⢠⠏⠀⠀⢈⡇⠀⠘⣦⠔⠋⡡⠋
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠩⢍⣳⣌⣉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢧⠀⠀⠀⠀⠀⠀⠂⢠⢀⡠⠞⠀⠀⠀⢿⠀⡎
⠀⠀⠀⠀⠀⠸⡆⠀⢀⡈⢳⣦⣸⠷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡆⢹⡄
⠀⠀⠀⠀⠀⠀⠹⡄⣸⠶⠯⣄⣀⣠⡤⠴⠖⢦⡀⠀⠀⠠⠚⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣸⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡇⣗⠀⢀⣤⠤⠤⠤⠴⠦⠂⠛⠓⠤⣄⡀⠀⠀⠉⠓⠶⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⣆⡴⠋⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⢸⠀⣟⣀⣸⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣄⣀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠋⢻⡉⠁⠙⢆⠈⠳⣞⠉
⠀⠀⠀⠀⠀⠀⢸⣀⣈⣉⣉⣉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⣶⠦⠤⠤⢤⣤⣶⣧⡀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⣯⣇⣀⣠⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠦⢤⣀⠀⢀⡟⠛⠋⠉⠉⠁⠀⠀⠳⣤⣀⣀⣤⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣦⡀⠐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠈⠙⠒⠲⠦⣤⣤⣤⣄⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡜⠀⠀⠀⠀⠀⣀⣾⣰⠃⠀⢰⠃
⠀⠀⠀⠀⠀⠀⡞⠉⠀⡹⢍⢙⢓⣦⡤⣄⣉⣠⣏⣉⣁⣠⣾⢤⣤⠤⣀
⠀⠀⠀⠀⠀⠠⣷⣤⣀⣧⣠⣭⢭⣝⣶⠿⠛⠛⢶⣤⣮⡤⠤⠚⣦⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠈⠈⠉⠋⠋⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⠷⡖⠯⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⡾⣷⣄⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⣾⠏⠁⠈⠻⣷⣄⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⣾⠟⠀⠀⠀⠀⠀⠈⠻⣷⣄⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠠⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⠄⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⢀⣤⣤⡀⠀⢀⣤⣄⠀⣤⡀⢠⡄⢠⣤⣤⠀⣤⡄⠀⣠⣤⣀⠀⣠⡄⢠⣤⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⣿⡏⠹⠇⠀⣿⠉⣿⡇⣿⠀⢸⡇⠀⣿⡏⠀⣿⡇⢸⣿⠉⣿⠀⣿⣿⣼⣿⠀⠀⠙⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⣿⣇⣰⡆⠀⣿⠟⣿⡇⣿⣄⣼⡇⠀⣿⡇⠀⣿⡇⢸⣿⣀⣿⠀⣿⡟⣿⣿⠀⠀⣠⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠈⠛⠛⠁⠀⠛⠀⠙⠃⠈⠛⠋⠀⠀⠙⠃⠀⠛⠃⠀⠙⠛⠁⠀⠙⠃⠘⠋⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠐⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠂⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⣾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠙⢿⣦⡀⠀⠀⠀⠀⢀⣴⡿⠋⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠙⢿⣦⡀⢀⣴⡿⠋⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠙⢿⡿⠋⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣠⡾⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠀⠀⠀⢸⠿⠻⠛⠿⠄⠀⢀⡀⢀⡼⠈⣥⣠⢄⣦⣴⣲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣲⡞⡿⢪⡅⢻⡻⡋⡉⣿⢭⢻⣄⣾⣶⠂⢂⠀⠀⠀⠀⠀⠀⠚⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠳⠀⠀⠀⠀⠀⡀⣠⢖⣿⢉⡄⠉⣀⣤⣴⣶⣶⣾⣷⣬⣾⡗⢿⡿⡽⢦⣞⡒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣩⢱⠞⢋⣷⣿⣻⡿⣛⢿⠉⣿⢻⡙⠿⢿⣿⣤⡲⡯⠝⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡾⡿⠟⢇⣾⣿⠋⢭⡔⢡⣤⡟⡷⢲⠏⢲⡆⠻⣿⣿⡄⢠⢧⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣭⣥⣶⣃⣼⣷⠿⡴⠬⣱⠆⠿⠓⠗⢛⠞⠑⠞⠓⢿⣿⣧⢤⡴⠷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡡⣯⡯⠃⣿⣿⣤⣦⣼⣥⣴⣤⣴⣤⣦⣤⣦⣤⣧⣼⣿⣷⠀⠒⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡌⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡟⣿⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⣿⣿⣿⣿⡟⣿⣿⣽⡇⣿⢻⣿⠏⣿⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⣿⣿⣿⣿⣿⠹⣿⣻⡇⣿⠏⣿⣸⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢻⣿⣿⣿⡅⣿⢸⡇⣿⡐⣿⠻⣿⢿⣿⣿⣿⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢺⡿⣿⣿⣿⣿⣧⣿⢸⡄⣻⢂⣿⣦⣿⣿⣿⣿⣿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠦⠀⠀⢸⣿⣿⣷⣿⣿⣏⣿⡜⣷⣸⣿⣿⡯⢿⣿⣿⣿⣿⠂⠀⠀⠀⠂⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠛⠋⠉⠉⠉⠙⠋⠉⠉⠉⠛⠛⠋⠋⠉⠛⠉⠛⠃⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⡶⠀⢠⠀⣶⡆⠀⡖⠀⢤⠄⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⡏⠀⣿⠀⣿⠁⠀⡏⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⢠⢸⡇⣇⡀⢻⡀⢹⢠⡀⣇⡀⢸⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠉⠀⢉⠀⡈⠁⠈⡀⠀⢉⠀⢈⠀⡀⠀⠀⠸⢷⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⢹⠈⡿⠃⢸⠇⡇⢽⠄⢸⠀⡇⠀⠀⠀⠐⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⣄⠀⢽⠀⡿⠀⢸⠀⡇⢰⠀⢻⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠂⠸⠀⠓⠐⠘⠂⠁⠺⠒⠸⠈⠃

⠀⠀⠀⠀⠠⠐⠒⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠒⠒⠒⠒⠒⠒⠂⠀⠀⠒⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠤⢀
⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⠀⠀⡀⠊
⠀⠀⠀⠑⠂⠰⠀⠐⠒⠉⢀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠀⠀⣀⣀⣄⣀⣠⣀⠀⠐⠂⠂⠂⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠉⠀⠢⡀⠀⠀⢈⠔⠒⠃⠈
⠀⠀⠀⠀⢀⠁⠀⡰⠀⠀⠀⣠⡆⣀⣒⢀⠄⠠⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢀⠀⠀⠀⡀⠠⡐⠀⠀⠀⠈⠄⠀⠀⠤⠀⠂
⠀⠀⠀⠀⠀⠈⠀⠂⠀⢈⢠⠟⠀⠀⠉⠉⠓⠺⡄⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⡀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠔⠅⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰⠼⠀⢀⡀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⠐⠀⡼⠴⠀⠠⢬⠌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡑⠀⠉⠀⠀⠈⠉⠁⡤⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠴⠦⠆⠀⠔⠋⠁⠀⠄⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⠠⠄⠀⡆⠠⢸⡀⠀⠄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠐⢄⡀⠀⠀⡈⠀⡠⠂⢀⡀⠀⠀⠀⠀⠀⠙⢄⡁⠂⠀⣀⡄⢁⠀⡀⠳⣄⠈⠀⠁⡠⡞⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠢⢄⣀⢥⠚⡇⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠘⠀⠀⠀⠆⠙⠒⠊⣩⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣁⠀⠁⠂⣤⠐⠈⢀⡤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣤⠒⠓⢤⡀⠀⡠⠓⣥⠤⣇⠸
⠀⠀⠀⠀⠀⡄⠒⠉⠀⠀⠀⠙⠒⠚⠑⠙⠒⠉⠀⠀⠀⠉⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠁⠂⠤⠤⠅⠀⠈⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⢸⠀⠀⠀⠀⡇⠀⠒⠀⢰⠀⠀⡖⡆⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⠀⡀⡠⠂⠔⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠁⠁⠊⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠂⠁⢠⡶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠋⠀⠀⠀⠀⠀⠿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡤⠤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⢀⢀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⣠⣶⠿⢟⡃⣠⣴⡶⠀⠉⢉⡻⣷⣄
⠀⠀⠀⠀⠀⢠⡟⠀⣄⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣾⣿⣿⣿⠟⠀⠀⠀⠘⣿⢿⣿⡆
⠀⠀⠀⠀⠀⢾⠀⠀⡜⠶⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠘⢿⡟⠉⢹⣿⣿⣿⠀⠀⠀⠀⠀⠘⠋⢹⣷
⠀⠀⠀⠀⠀⢸⡄⠸⡷⠤⠀⠀⠀⠋⠙⠀⠀⣯⡄⠀⠀⠀⠀⠀⠀⣇⠀⠀⠽⣿⣿⠀⣠⣄⡀⠀⠀⢀⣼⣿
⠀⠀⠀⠀⠀⠈⡇⠀⡴⠀⠀⠀⠀⢤⡼⠇⠀⠙⡅⠀⠀⠀⠀⠀⠀⢸⣀⡀⠀⠈⠹⠂⣼⡌⠁⠲⣦⢺⣿⣿⡀
⠀⠀⠀⠀⠀⢸⡇⠀⢳⠀⠀⠀⠀⠀⠀⠀⣠⣼⠁⠀⠀⠀⠀⠀⠀⠈⢫⠷⠂⠀⠀⠀⣼⡇⠀⠀⠈⠁⠀⠘⠇⣀⡀
⠀⠀⠀⠀⢀⣾⠃⠀⢺⣶⡾⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠔⣫⣶⣿⣿⣦⣀
⠀⠀⡀⠐⠛⠁⠀⠀⠀⠈⠀⠀⠀⠙⠲⠶⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣧⡷⠄⠀⠀⠀⠀⠀⣰⣯⣾⣿⣿⣿⣯⡄⠁
⠀⣸⡇⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⢀⣀⣴⢷⡀⣰⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠷⠀⠀⠀⢀⣽⡆⠀⠀⠀⠀⠀⣼⠁⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠟⡟
⠀⠘⠂⠐⠶⠛⠋⠀⠀⠀⠀⠀⠀⠿⠈⠉⠉⠈⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠋⠀⢹⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⢀⡠⠶⣂⡞⢡⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣄⡀⡴⠂⠀⢈⡀⠀⠀⠀⢀⣶⡯⡾⡻⢹⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣟⣻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⠛⠰⢋⡀⠈⠇⢀⡴⠚⠉⠀⣀⡬⠒⠈⠀⢠⣼⣿⣿⣿⣿⣿⣟⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠐⠛⠁⢀⣠⠷⡄⠙⠛⠻⡅⣀⡀⡆⠐⣿⣿⣿⣿⣿⣿⣿⣽⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⢫⣏⠀⠀⠀⡰⢁⡿⠋⠀⡇⠀⢸⣿⣿⣿⣿⣿⡟⠃⠀⣛⣿⣿⣂⠤
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⣠⠚⠁⢀⡞⠀⠀⢀⠞⠁⣸⠀⠀⣸⠃⠀⢸⣿⣿⣵⣟⢋⡔⠁⠀⢻⣯⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⢠⡞⠁⠀⢀⡞⠀⣀⡴⠃⠀⣰⠏⠀⠀⠇⠀⢠⣿⣿⣿⣿⣇⠎⢀⡀⠀⠀⠙⣿⡿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⢠⠏⠀⠞⠁⠀⠀⣼⠇⠀⠀⠀⢀⣴⣿⣿⣿⡿⢽⠏⠀⠀⠙⢦⡀⠀⠈⢻⣞⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢀⣾⣿⣿⣟⡟⣣⠏⠀⠀⠀⠀⠈⠻⣄⠀⠀⠹⣭⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣾⣿⣿⣻⡞⢠⠏⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⣿⣿⣷⣿⡵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⣿⢀⣀⠀⠀⢀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠼⠇⠀⠀⠀⠀⠉⠉⠉⠓⠚⠉⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠔⠀⠐⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⣀⡤⠊⠀⠒⢌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡆⠀⠀⣠⠤⠞⠁⠀⠉⣠⠶⠛⠀⠀⠀⠀⠀⠀⠰⠶⠤⣄⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢋⡟⠁⠀⠀⠀⠠⠞⠁⢀⣀⣀⣀⠀⠀⠀⢀⣠⠤⢤⣀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠉⠢⣠⠊⠀⠀⠀⠀⠙⢦⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠈⢆⣠⣄⠀⠀⠀⠈⡇⠀⠀⠀⠹⣤⢔⡶⢒⣒⠒⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠸⢿⠞⠋⠀⠀⠀⡰⠁⢀⣀⡀⠀⠹⣦⢻⣍⣠⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢀⡞⠑⠂⠤⠖⠋⢀⡴⠋⠀⠙⡆⠀⠹⡏⣀⠼⠂⡰⠂
⠀⠀⠀⠀⠀⢠⠒⠀⠀⠒⠒⠲⢶⡒⠒⠶⢇⠀⠀⠀⠀⠀⠐⠚⠉⠁⠈⠈⠐⠢⢤⡚⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⣰⠃⠀⠀⢿⠁⠀⠊
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡉⠛⣟⣲⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠒⠋⠁⠀⠀⠀⢸⡄⠡
⠀⠀⠀⠀⠀⠀⠁⠀⠠⣀⣀⠉⠒⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢣
⠀⠀⠀⠀⠀⠀⠀⡇⢠⠁⠀⠀⠒⠒⠂⠉⠀⠀⠀⣄⠀⠀⠀⠀⠉⠒⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠐⠀⣨⡆
⠀⠀⠀⠀⠀⠀⢀⡇⢸⠀⢠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⢤⣀⠀⠀⠈⠉⠛⠲⠶⠤⢤⣄⣀⣀⡀⠀⡤⠤⠐⢶⠉⠂⣌⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⢸⡇⠸⠤⠼⠧⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⠳⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠘⠇⠠⠤⠤⡤⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⣏⣉⣉⣭⠭⠽⠛⠻⣄⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠧⠴⢲⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠙⠲⢤⣄⡰⠃⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠶⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠻⠄⠀⠳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠉⠉⠓⢂⡶⢶⠦⠤⢤⠾⠁
⠀⠀⠀⠀⠀⠀⠀⣀⡤⣤⣤⣼⡀⠀⠀⠀⠀⢠⣼⣥⣏⠀⠀⡞
⠀⠀⠀⠀⠀⠀⡞⠁⠀⡨⠳⣀⣉⣳⡶⠦⣤⣤⡾⢤⣬⣤⠼⠷⣶⠶⠦⣄
⠀⠀⠀⠀⠀⠀⣟⣦⣤⣷⣤⣤⣶⣒⡾⠿⠛⠉⠻⢶⣾⣗⡶⠖⠱⡄⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠷⠿⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠂

⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣾⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⠉⠉⠉⢹⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⡀⠀⠀⠸⣿⣤⣿⠇⢸⣿⠿⣶⣶⣤⣤⣀⣀
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠛⠿⠀⠀⠀⠈⠉⠁⠀⢸⣿⣦⡄⠀⠈⠉⠙⣿⡇
⠀⠀⠀⠀⣿⡇⠀⢀⠀⣴⡿⣷⡄⠙⠋⠀⠀⣀⣴⣶⣤⠀⠀⠀⠀⠀⢸⣿⣀⣀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⣿⡇⠀⣿⣇⣿⡇⠿⠃⣀⠀⢠⣾⠟⢉⣤⣿⠃⠀⠀⠀⠀⢸⣿⠛⠛⠻⠷⠀⣼⡇
⠀⠀⠀⠀⣿⡇⠀⠘⠛⠋⣠⣴⡿⠿⣷⡈⠻⠿⠟⢋⣤⣾⣷⡄⠀⠀⢸⣿⠿⢷⣶⠄⢠⣿⠁
⠀⠀⠀⠀⣿⡇⠀⠀⠀⢿⣿⠙⠟⠀⠙⠁⠀⠠⣾⠟⢻⣧⣾⠏⠀⠀⢸⣿⣶⣤⣄⠀⣼⡏
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠻⠃⢀⣴⣾⠟⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠉⠁⢠⣿⠃
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⣏⣄⠐⠟⢁⣀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⣄⣼⡟
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠙⢉⣠⣶⣿⢻⣷⠀⠀⠀⠀⠀⠀⢸⣿⠀⠉⠛⠿⣷
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠛⢋⣽⡿⠿⠋⣠⣄⠀⠀⠀⠀⢸⣿⠀⣾⡇⠀⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣠⣴⣿⡟⢻⣦⠀⠀⠀⢸⣿⠀⣿⡇⠀⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣇⠙⠃⠀⠉⣤⡀⠀⢸⣿⠀⣿⡇⠀⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⣠⣽⣿⣄⠀⠀⠀⣿⡇⠀⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠁⠈⠋⠀⠀⠀⢿⡇⠀⣿
⠀⠀⠀⠀⣻⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣿
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⡀⠀⠀⠀⡀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄⢀⠀⢀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡡⠤⠐⢂⣶⡤⠄⠈⠻⣿⢦⠀⠀⠈⠙⠳⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠠⠁⠋⠀⠀⢀⠄⣀⡈⠳⡆⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠠⠁⠀⠀⠀⢀⡖⢫⠏⢀⠀⠀⠀⠉⠉⠉⠉⠀⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣄⣈⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⣶⣷⢿⠀⠀⢠⢀⡖⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣶⠀⠂⠀⠀⠀⠀⣄⣿⡏⡎⠀⡰⢃⡞⠀⢀⠀⣤⠴⠒⠀⠀⣬⣭⠟⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢠⡇⢰⠀⠀⠀⣤⣾⣿⣿⠃⡇⡰⢡⠎⠀⠀⠔⠀⠛⠀⠀⠀⠀⠈⠁⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠂⢀⡇⠘⠀⠀⠸⢿⣿⣿⣿⠀⢱⣵⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡈⠀⠘⠃⠀⠀⠀⠀⠀⠸⢿⣧⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣻⣾⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠸
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⢠⠃⣆⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⠋⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⢿⣿⠘
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⢰⡿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠠⠁⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⣿⡇⠀⠀⠘⡇⠀⠀⡀⠀⢀⠃⠠⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡋
⠀⠀⠀⠀⠀⠘⠀⠀⠰⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠀⠳⠀⢃⠀⠀⡀⠀⠀⠀⠀⠀⢠⠀⣤⡄⢴⠃⠜
⠀⠀⠀⠀⠀⡆⢠⠁⣃⢾⣿⠇⠀⠀⠀⡆⠀⠀⠀⠀⠀⡇⠀⠀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠾⠟⣡⣎⣜⣀
⠀⠀⠀⠀⠀⣀⢆⡼⢁⣾⣿⠁⠀⡆⠀⡇⠀⠀⠀⠀⠀⢳⠀⠀⠘⣀⠀⠈⠀⠁⠀⠀⠠⠀⠀⠀⠚⠁⠸⠁
⠀⠀⠀⢀⣴⠃⠞⢠⡟⣻⠇⠀⢰⠀⠀⣇⠀⠀⡄⠀⠀⠈⠗⠀⠀⠘⣠⠎⠀⢀⣴⠁⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀
⠣⠤⠒⢉⠄⣠⠔⠊⠉⠁⠀⠀⡎⡌⠀⢸⠀⠀⡐⠀⠀⠀⢇⠀⠆⠀⡆⡆⠀⠀⠁⠀⠀⠀⠈⠀⣰⢃⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁
⠀⠀⠀⠁⡰⠃⠀⠀⠀⠀⠀⣼⡔⠀⠀⠀⠀⠀⡇⠀⡦⠀⢸⢰⡄⠀⠿⠃⠀⠀⠀⠀⢀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⢰⣿⠃⢹
⠀⠀⠀⢰⠁⠀⠀⠀⠀⣠⢞⡑⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠈⣸⠇⠀⠀⠀⠀⠤⠀⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⡣
⠀⠀⢀⠃⠀⠀⡤⠖⠋⠁⡞⠁⠀⠀⠀⠄⢀⠆⠁⢰⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠇
⠀⠀⡜⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡆⡰⠀⠌⡘⢠⡟⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠇⡘⢠⢁⣾⠁⠀⠀⢸⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡠⢁⢃⢪⠃⡔⢁⠄⡾⠃
⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢻⡗⢁⡾⢡⢃⠌⢀⠎⠜⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢃⠾⣠⠟⣠⡗⠀⣠⠋⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠐⠁⠳⢋⡴⠋⣠⠞⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣀⡩⠴⠛⠒⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠙⠓⠶⣄⡀⠀⠀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⣠⣀⣀⣀⣠⠴⠚⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⢀⡤⠖⠛⠛⠲⣼⡵⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠟⠛⠻⢿⣷⡖⠋⠉⠙⠳⣄
⠀⠀⣾⠁⠀⠀⠀⠀⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠁⠀⠀⠀⠀⢸⡆
⠀⠀⢿⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠈⠳⢤⣤⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠞⠁
⡴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⢠⠀⠀⠀⠀⠀⠀⡄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡄⠀⠀⠀⠀⠀⢠⠄
⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸
⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸
⡇⠀⢸⠀⠀⢠⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⢠⠀⠀⢸
⠳⣤⣾⠀⠀⢸⠀⠀⠀⣧⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠴⡇⠀⠀⢸⠀⠀⢸⡦⠴
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠹⣄⠀⢸⡀⢀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣼⣀⣀⡼⠁
⣶⣶⠶⠾⠿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣿⣶⣶⣶⣶

⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣠⣀⣀⡀⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣠⣤⣄⣠⣄⣤⡀⣠⣄⣀⣀⣀⣀⣀⡀⣀⣀⣄⣀⣀⣄
⠀⣽⠏⠁⠼⠂⢹⣿⣿⠀⠘⠀⠀⢠⡐⢠⡤⠄⠀⠆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⢠⣷⠀⠀⣿⣿⡀⠀⠀⠘⣿⠀⠀⣽⠟⠛⣿⣿⣿⡤⣈⣹⣿⣿⣿⣿
⠀⢸⠔⠂⠀⠀⠘⠛⠋⡈⠀⠀⠀⠀⠋⠈⠀⠀⠃⠀⠂⠀⠀⠀⠀⢰⡏⠀⣀⠀⣀⣨⣄⠀⠀⠂⠀⢿⣿⡟⠐⠚⠀⣿⠀⠀⠉⠀⠀⣿⣿⣿⡿⢿⡏⠉⠉⠙⣿
⠀⠈⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠙⠓⠻⢿⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢸⣿⣿⣃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣿⣯⡄⠉⠙⣦
⠀⢸⠀⠀⠀⠀⠀⠀⢸⠋⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⣄⣿⡚⣿⣇⢸⣷⡰⠇
⠀⢸⠀⠀⠀⠀⠀⠀⠊⠁⣽⣤⣉⠐⠀⠠⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⢀⠀⠀⢂⣻⠁⣿⣿⣿⣿⣷⣦⣶⣤⣴⣶
⠀⢸⣠⠄⠀⠀⠀⠀⢠⠀⠰⠿⠾⠃⠀⠀⠀⠀⢸⠉⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠑⠀⠀⢸⣧⡀⠀⠀⠀⠈⠁⢀⣿⣏⣿⡠⢿⣿⣿⣿⣿⣿
⠀⢸⠁⠀⠘⠂⠀⠀⠈⣠⣞⠂⠀⠀⠀⠀⠀⠀⢘⠄⠀⣤⡜⠃⠉⠀⣄⢠⣤⡾⠟⠛⠳⠻⣆⠀⠠⡇⢀⠀⣰⡍⠳⣶⠀⢀⠐⠀⠉⣻⣿⡃⣀⢸⡟⣿⡖⠻⣿
⠀⢸⡄⠀⠀⠁⠠⡄⠀⠀⠙⠁⠐⢦⠀⠀⠀⠀⠈⠀⣸⣿⠗⠀⠤⠐⣤⣎⠉⠓⠲⢦⢀⣠⣬⣦⠀⡇⠈⠀⣿⡇⠀⢿⠀⠈⡀⠀⠀⣿⣿⠁⣿⣷⣏⠛⡉⠙⣦
⠀⣾⣿⡂⢀⠀⠘⠃⠀⣿⣰⠞⣀⡠⠀⠀⠀⠀⠀⣰⣿⡟⠶⠐⠶⠄⣿⢋⡂⠀⡆⠀⠈⠀⢺⣯⡀⡇⠀⠀⢹⡿⣤⣼⣴⣿⠑⠂⠀⣿⠉⠀⣹⣧⣹⣼⣿⣷⣦
⠀⣹⠋⠉⠉⠀⢰⡇⠀⣮⡥⠀⢟⠳⠀⠀⠀⠀⣸⣿⢸⣷⡾⠆⠀⢸⣿⣆⠁⡤⠱⣦⣶⠀⠌⢉⡥⣿⠟⠃⠙⠛⠟⢻⠛⠿⣴⠆⠀⢿⠀⠀⢿⣿⣿⣿⣿⣿⣿
⠀⣿⠀⠀⠀⠀⢸⡇⠀⣤⠀⢲⣌⡴⢀⣠⣴⣾⣟⣿⣾⡟⠓⠂⠀⢿⣿⣏⣠⣦⠼⣟⣿⣀⣼⡿⢹⡷⠀⠀⠀⠀⠀⠛⠳⣀⣿⣄⡀⠘⢇⠛⠻⣉⣰⣿⣿⣿⣿
⠀⢻⠀⠀⠀⠀⢸⡇⠀⣿⣿⣿⣿⢋⣿⣫⠟⠛⢋⣠⡼⠷⠶⣶⢎⣽⣿⣿⣿⣿⣿⣿⡟⠛⢋⠤⠼⠃⠀⡀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣦⠌⠀⠰⣿⣿⣿⣿⣭⣿
⠀⢸⠀⠀⠀⠀⢸⡁⢀⡹⠿⠛⠋⣼⣿⣯⡴⠊⢁⣀⠀⠶⢞⣛⣻⣿⣿⣿⣿⣿⣿⢿⣯⣤⣤⣴⠗⠀⠀⠈⠲⣀⠀⠀⣠⠶⡄⠀⠑⢤⠀⠒⠚⢶⣿⣿⣿⣽⣿
⠀⢸⣦⡀⠔⠀⢸⣿⠟⠓⠖⢨⣾⣿⣯⣥⣶⠾⠿⣿⡿⠉⠉⠻⠿⠻⣿⣿⣿⠿⢿⣮⢹⣿⣿⣯⣆⠈⠲⡄⠀⠈⠠⠐⠟⠀⠀⠀⠀⠀⢀⣴⣶⣿⡿⢻⣟⠛⣿
⠀⢸⡧⠀⣀⣶⠸⠁⠀⠈⣰⢿⣿⣛⡿⢿⣿⠀⢰⡿⢁⣶⣤⡀⠀⠒⠛⢿⠿⠶⢆⠙⢾⣧⡀⠀⠛⠣⢀⠈⠁⠀⡀⠐⠐⠀⠂⠀⡀⠀⠙⠉⠉⠉⣿⡿⠛⢳⣏
⠀⢸⣿⡛⢋⣑⠚⠂⠀⢠⡿⢸⣿⡏⠑⠃⠻⣿⣿⡷⠛⣹⣿⣿⣿⣦⣤⣀⠀⠀⠈⠁⠈⣿⣧⡶⣠⡀⠈⢳⣄⠀⡀⠀⠠⣤⡿⠟⠀⠀⠐⣤⣤⣶⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣦⡆⢀⣿⠇⢸⣿⣷⡄⠀⠀⠀⠉⣱⠆⠙⠿⠛⠿⣿⣿⠿⣿⡶⣶⣤⣼⣿⣿⡐⠟⣷⣴⣾⠿⢷⣦⣰⣤⣠⡴⠛⠀⠀⠐⠛⠩⠌⠙⣿⣿⣿⣿
⠀⢸⣿⣿⣿⡏⠀⠇⣼⡿⠀⢸⢿⣯⣼⣿⣿⣦⣤⣀⣤⣀⣶⣦⣤⠸⣿⠃⠹⣷⣿⣿⣿⣿⣹⠇⠀⣿⠛⣧⡀⠀⠻⣿⣁⣤⣤⠿⠠⠀⠀⠀⠰⣤⣼⣿⣯⣏⣿
⠀⢸⣿⣿⣿⣇⡀⢾⣿⣷⠀⠈⣸⣿⣿⠿⡿⢆⣉⡙⠻⠿⡿⠋⠉⡀⣛⢀⠀⠉⢿⣿⣿⣷⡿⠀⠀⠉⢠⣿⣿⣶⣤⣌⠁⢠⡉⠢⣀⠀⠀⢉⢲⣶⡾⣿⣿⣿⣿
⠀⢸⣾⣿⣿⡿⢿⣿⣿⡇⠀⠀⣧⡌⠀⠐⠀⠀⣿⠋⢸⠾⠋⢈⣉⡑⣿⣿⠀⢠⣾⣿⣿⣿⠃⠀⠀⠀⢸⡟⠁⠀⣠⡏⠹⠆⠹⠆⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣷⣶⣦⡇⠸⣿⣿⡇⠀⠀⣿⣷⣦⡀⠀⢠⣿⠀⠀⠀⠀⠈⠁⠁⢿⣯⠀⢺⣿⣿⣭⣤⣤⡤⣤⡀⢸⣿⠛⠙⢨⡗⠀⠀⠀⠐⠀⠀⠈⢸⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⡇⢸⣿⣿⡟⠀⢰⣿⣿⣿⣿⡗⣿⣷⣶⠿⠃⠉⠀⠀⠘⣾⡟⠀⢹⣿⣿⣿⣿⡿⢣⣼⡈⠻⢿⣷⡄⢸⡟⠀⠀⠀⠀⠙⡀⠀⠸⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⡁⣿⣿⣿⣿⠀⢸⣿⣿⣿⠹⡇⣿⡏⢹⠂⠀⠀⠀⠀⠀⢿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⡀⢸⣻⡇⢸⡇⠀⠀⠀⠀⠀⠁⢠⠀⣿⣿⣿⣿⣿⣿⣿
⠀⢨⣿⣿⠿⡇⠘⢻⣿⣿⡆⠸⣿⣿⣿⡇⠀⣿⠇⡾⠀⠀⠀⠀⠀⠀⢸⣿⠀⣼⣿⣿⣿⣿⣿⣿⡿⣿⣿⣦⣿⡇⢸⡇⠀⠀⡜⠃⠀⠀⠘⠀⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⡟⣀⡇⠀⠀⢻⢿⣇⢀⣿⣿⣿⣿⣿⣿⣶⣶⣶⠛⠛⠛⠛⣲⣾⣿⠀⡿⣿⣿⣿⣿⣷⡈⠳⠁⠘⠏⠛⠀⠘⠋⠀⠀⠁⠀⠀⠀⡆⠀⢸⣿⠿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⠇⠀⠀⣾⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣿⣿⣿⣾⠇⢿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠿⠿⠾⠟⠋⣿
⠀⠀⣿⣿⣿⣧⣠⢷⡀⠉⠛⠟⠁⠀⠻⣿⣿⣿⣿⣿⣧⠀⠀⢠⡟⢻⣿⠉⢻⡀⠘⣿⣿⣿⣟⣿⣿⣦⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣡⡄⣷⣿
⠀⠀⣿⣿⣿⣿⢿⣿⡷⠀⠀⠠⣢⡄⠤⣀⠠⠹⣿⣿⣿⣦⡀⠘⡇⢸⡏⠀⡼⠃⠀⠻⣿⣿⣿⣿⣽⣿⣷⡀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠿⣿
⠀⠀⣿⣿⣿⣿⣾⣿⣤⣿⠶⠿⠛⠛⠒⠋⠀⠀⢹⣿⡿⣿⣧⠀⠈⣾⣿⢤⣀⠀⠀⠀⢻⣿⣿⣇⣨⠛⢿⣿⢦⡐⠀⠀⠈⡀⠀⠀⠼⠂⠀⠀⢱⠀⠠⡏⠽⠟⣿
⠀⠀⣿⣿⣿⣿⣿⡿⣿⣷⣶⠖⠀⠀⠀⠀⠀⠀⠚⣿⣷⣿⣿⠀⠀⢸⣧⠈⣉⣽⣟⠳⠌⢻⣿⢻⣯⢰⣿⠁⠀⠉⠀⣤⣄⠙⠂⠀⠀⠀⠀⠀⠘⣧⠀⢹⡄⠤⠻
⠀⠀⣿⣿⣿⣿⣛⠻⠿⠛⠁⢠⠄⢘⣀⠀⠰⠦⠀⣿⡏⢹⣿⡆⠀⠈⣿⣿⣿⣧⠿⠶⠀⢸⣿⣿⣿⢾⣿⠀⢰⠀⠀⢸⣿⣿⣦⣄⠃⠄⠀⠀⠀⡿⠀⠘⣀
⠀⠀⣿⠙⠛⡛⠡⠤⡤⢀⣠⣶⡤⠀⣁⣤⠄⠄⠀⢾⣷⠈⣿⡇⠀⢸⣿⠻⠛⣋⣰⣦⣾⣿⣿⣸⣿⣿⡏⠀⠸⡆⠀⠀⠋⠹⣿⣿⣷⠀⣀⠀⠀⠟⢸⣆⠈⣆
⠀⠀⣷⣼⡛⠿⠀⠀⠀⠈⠉⠃⣤⣾⠛⢻⠱⠶⠀⠀⣿⡀⢹⡇⠀⢸⣿⣿⣿⣯⣥⣽⣿⣷⣶⣿⣿⣿⣷⣾⣿⡷⠶⠶⡟⠚⣻⣿⣿⡀⠹⠃⠀⢀⡈⢿⣧⡈⠳
⠀⠀⣿⠽⢿⣿⣦⣴⣶⠖⠛⠻⠿⢏⣵⣿⣷⣶⣆⣴⣿⣧⣾⣇⠀⠚⣿⢿⠟⢛⣿⣿⢿⣿⣯⡿⣿⣿⣿⡟⢿⣧⣾⡾⠻⢿⣿⣿⣿⣇⠀⠛⠀⠀⢷⡀⠻⣷⣤
⠀⠀⣷⠀⠀⠿⠟⠁⠀⢠⣶⡶⢶⣿⣿⣿⣿⣿⣿⣿⣿⣧⣸⣟⠐⣛⣿⡿⣿⣿⣷⣶⣾⣿⣶⣿⣿⡿⠟⣀⣈⣥⣾⣶⣿⣶⠟⢻⣿⣿⣧⡀⠀⣦⡌⠻⣆⢉⣻
⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⣛⠉⣉⣬⣿⣿⡯⠤⢯⠀⠀⠀⠀⠀⣉⣽⣿⣛⣉⣁⣀⣠⣭⣩⠆⠈⠙⠛⠐⠒⠋⠀⠈⠉⠙⠛⠓⠤⠤⠤⠭⠿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠦⠤⠚⠀⣀⡤⠖⠀⠉⠀⠈⠉⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⣿⠂⠀⠀⠀⠀⠠⠤⠤⢤⣤⣶⡶⣀⡀⠀⠀⠠⠤⢶⠦⢤⣤⡴⠾⠥⠠⠤⠤⠤⠄⠤⠀⣠⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣄⠀⣀⣰⡄⠀⠀⣠⣤⣴
⠀⠀⣿⠠⣄⡀⠀⣀⣤⣤⣼⡿⢉⠉⠁⠈⠀⠀⠀⠀⢀⡴⠶⠎⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠈⠉⠁⣀⣀⣀⣀⡀⠀⠀⠰⠌⠙
⣤⣤⣤⣤⣤⣤⣤⣬⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠛⠻⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⡷⠀⢉⣠⣽⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠠⢀⡀⠀⠀⠀⠀⠀⣀⣤⡴⢶⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣦⣤⣔⠋⣁⣽⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣁⣒⣦⣄⣴⣿⣿⣶⡟⠩⣩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢸⡇⠴⠮⣿⣬⡹⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⢀⣤⣶⣶⠿⡟⠛⠉⠉⣯⣻⣿⣯⣽⣿⣻⡰⢻⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⠃⣿⢓⡏⠀⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⢀⣴⣿⣿⡿⠓⠀⠽⠖⢁⣴⣿⣿⣿⣿⣿⣿⣿⣧⣿⣬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣥⠀⠉⠁⠀⠀⠙⢿⣿⡟⣹⣿⣿⣦⡄
⠀⠀⣾⣿⣿⡏⠦⢠⣠⡴⢶⠿⣿⣿⣿⡏⠀⠙⠛⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣿⣷⠀⠀⠀⠀⠐⣻⣶⣴⣿⡿⠋⠙⣁⣀
⠀⣼⣿⣿⣿⢠⠀⣾⣿⣆⣐⣾⡿⢿⡻⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠁⢀⣠⣴⣴⣷⠿⢹⣿⠏⠀⣠⣾⠿⠹
⠀⠸⣿⣿⣿⡌⠀⠿⢿⣿⠋⠉⠁⢏⣩⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠋⠀⠀⠈⠈⠀⣰⣿
⠀⠀⠙⣿⣿⣵⢦⣿⣼⣿⡄⠀⠀⠈⠛⠄⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⡀⠀⠀⢀⡧⠰⣽⡃⠶⠆⡰⢿⡏
⠀⠀⠀⣟⣿⣹⣾⣿⣿⣿⣷⠀⠀⠀⠀⠀⠃⠄⠀⠑⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⣤⣾⠃⠀⠀⠀⠀⠩⠘⣦⣼⣿⣰⣿⡅⠈
⠀⠀⣠⣮⣿⡏⢿⣿⣿⣿⣿⣷⣶⣷⣶⣶⣤⣤⣈⣀⠲⣥⣤⣄⣀⡀⠀⠀⠀⠀⠠⠤⣄⣀⡀⠀⠀⠀⠂⠀⣄⣐⠇⠀⢠⣄⠀⠀⢴⠒⠛⠛⠀⠑⢟⢷⡄⢻⣦
⠀⢸⣿⣿⣿⣿⣜⠻⠿⠿⣿⣿⣿⣿⣿⣟⣭⣟⣽⡟⠛⠛⠷⠤⠯⣽⡿⠃⣀⣤⣶⣶⣿⣿⠁⠀⠀⠀⠈⠙⢿⡿⠀⢘⡏⠸⣳⡦⡞⠁⠀⠀⢀⢳⡄⢻⣿⣯⣿
⠀⠀⠐⠶⣴⠲⠿⠻⠶⣻⡿⠉⠛⠛⠛⠛⠿⠿⠿⣧⣤⣤⣄⡀⠀⠀⠀⠈⠁⠀⠀⠡⠙⣿⡀⠀⠀⠀⠀⠀⠀⢠⠿⠖⠀⣮⡿⠈⠅⠀⠀⠈⠤⠆⠙⠁⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⣤⡀⠀⠀⠀⠀⠀⠘⠻⠃⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠐⠄⠀⠀⢀⡀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉







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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢣⡹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣱⡯⢷⡹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⡰⣵⣶⣶⡽⣘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢟⣾⡇⣿⣿⣿⣟⡟⡜⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢏⣾⡟⣏⣿⣿⣿⡇⣿⠿⠎⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⢿⠛⣀⣿⢸⣿⣿⢿⡇⠀⠈⢌⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢣⣦⣤⣾⣿⢻⢸⣿⣿⢸⣷⣤⣦⣾⠏⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⡳⢻⣿⣿⣿⣿⣧⢸⣿⣿⢸⣻⣿⣿⣿⣾⣧⡹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣱⣿⣷⣿⣿⠟⣿⣿⠸⣿⡿⢸⣿⣯⣿⡟⢿⣿⢷⡹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⣼⣿⣻⣾⣿⡇⠀⢫⣿⢣⣮⣔⢟⣿⠙⣟⣶⣾⣿⣏⢻⣜⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢟⣼⣿⣿⣿⣿⣿⣧⣤⣾⠧⡸⣿⡿⢳⣿⢰⣿⣿⣿⣿⣿⣿⣿⡜⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣯⣬⣭⣭⣭⣿⣿⣭⣭⣿⣭⣥⣽⣷⣮⣿⣯⣤⣭⣭⣭⣭⣭⣭⣵⣿⣬⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⣿⡄⣼⠀⡾⣇⠀⣿⠛⣳⠄⣿⣆⢸⠀⡇⢸⣷⡀⡇⢰⠛⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁⣷⠇⣸⠷⢿⡄⣿⠉⢿⠀⡇⠹⣾⠀⡇⢸⠈⢳⡇⢹⣄⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


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

⠀⠀⠀⣀⣀⣀⣀
⠀⠀⢸⡟⠛⠛⢻⡇⠀⠀⠀⢸⠟⠛⠛⠛⠛⢻⡇⠀⠀⠀⣠⠴⣖⣷⠦⣀⠀⠀⠀⠀⠀⢤⠀⢀⣀⣀
⠀⠀⠸⡇⠀⠀⢸⡇⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⡇⠀⠀⠀⣯⢻⢫⠝⡟⢹⠀⠀⠀⠀⠀⢿⠀⢀⣿⣽⡗
⠀⠀⠀⣧⣤⣤⣼⡇⠀⠀⠀⠘⠛⢛⣿⣿⡛⠛⠃⠀⠀⠀⠈⠛⢼⡤⠗⠊⠀⠀⠀⠀⠀⠉⠀⠀⠈⠗
⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⣠⣾⣫⣟⣷⣄⠀⠀⠀⠀⠤⢒⣉⣉⡒⠤⠀⠀⠀⠀⠀⢘⣧⡴⢒⣆⠀⠀⠀⢰⠒⠒⡒⠒⣶⣶⡆
⠀⠀⡧⡧⠴⡧⢾⢾⠂⠀⠀⠀⠀⠡⢒⣒⠌⠁⠀⠀⠀⠀⢰⣫⣤⣿⣿⠏⠀⠀⠀⢸⠹⡇⡇⢦⣬⠍⡇
⠀⠀⠙⢿⣾⣷⣻⠟⠀⠀⠀⠀⠀⠀⢱⡦⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⣿⡀⠀⠀⠀⠸⠤⠤⠥⠥⠶⠦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⢀⠀⠀⡆⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⣤⠀⠀⠀⠀⠠⢲⢶⣀⣶⡲⡄⠀⠀⢠⠒⠉⠀⠉⠉⠐⡄
⠀⠀⠀⣾⣿⣿⣷⡄⠀⠀⠀⢠⢀⠵⠤⠤⢎⠙⠀⠀⠀⢰⣟⠋⣾⣿⡦⢏⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⡏⣿⣿⡿⠁⠀⠀⠀⠈⠛⠒⠒⠒⠒⠛⠀⠀⠀⠈⠹⡶⠾⠿⠷⡾⠀⠀⠀⠀⠉⠒⢦⠔⠀⠉
⠀⠀⠀⠘⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠠⡀⢀⢄
⠀⡔⠀⣲⣄⠀⠀⠀⠀⠀⠀⠀⠄⠀⡇⢸⠄⢠⠀⠀⠀⠀⣦⣼⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠑⠠⠿⢁⠔⡠⠤⣄⠀⠀⠀⡄⠀⠉⠉⠀⢸⠀⠀⠀⠀⢿⣽⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠮⠥
⠀⠀⠀⠀⠃⠚⢇⡀⠿⠀⠀⠀⡇⠀⠀⠀⢠⠞⠀⠀⠀⠀⣼⣛⣿⣟⣟⡿⠀⠀⠀⠀⠀⢰⢲⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠘⠈⠁

⠀⢄⣾⣿⣿⡟⠁⠀⣰⣿⣿⣿⠃⠈⢀⣾⣿⣿⡟⠀⠀⣰⣿⣿⣿⠃⠠⢀⣾⣿⣿⡟⠀⠀⣰⣿⣿⣿⠃⠀⢀⣾⣿⣿⠟⠀⠀⣸⣿⣿⡿⠁
⢀⣼⣿⣿⡟⠂⠀⣰⣿⣿⡿⠇⠀⢀⣾⣿⣿⡟⠀⠀⣰⣿⣿⣿⠃⠀⢀⣾⣿⣿⠟⠀⠀⣼⣿⣿⡿⠁⠠⢠⣾⣿⣿⠏⠀⠀⣼⣿⣿⡿⠁⠀⢠
⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠟⠄⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋
⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠄⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀
⣼⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿
⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⡟⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏
⠀⠂⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏
⠀⣱⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏
⣰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿
⣿⡿⠃⠀⢀⣤⣤⣤⠄⠀⠀⣤⣤⣤⡤⡀⠀⢠⣤⣤⣤⠄⠀⠀⣤⣤⣤⡤⠀⠀⢠⣤⣤⣤⠄⠀⠀⣤⣤⣤⡤⠀⠀⣠⣤⣤⣤⠀⠀⢀⣾⣿⣿
⡿⠁⠀⢠⣿⣿⣿⠏⠀⠀⣼⣿⣿⡿⠁⠀⢠⣿⣿⣿⠏⠀⠈⣼⣿⣿⡿⠡⠀⢠⣿⣿⣿⠏⠀⠀⣾⣿⣿⡟⠁⠀⣰⣿⣿⣿⠃⡀⢀⣾⣿⣿⡟

⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆
⠀⠀⠘⡏⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢋⢸⡇
⠀⠘⢿⠛⢹⠀⠉⢓⠲⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⠙⢿⠢⣄⠰⡿⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠋⢁⠔⠁⣼
⠀⠀⠈⢧⠈⣇⠀⠀⠈⠁⠒⠢⠬⢍⣛⠲⠤⣄⣠⣤⣈⣧⠈⠙⢧⠀⠉⠙⠛⠛⠛⠛⠛⠶⠶⢦⣤⡀⠀⠀⣠⡤⠶⠚⠋⠉⡀⠤⠐⠊⠀⠀⠀⣿
⠀⠀⠀⠈⣇⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢬⡙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠲⠀⠉⠐⢋⠁⠀⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠸⡆⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠹⡄⠣⡀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠙⢆⠑⢄⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢀⠃⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠑⢄⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠊⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡑⠦⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⢠⠎⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⠚⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠰⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠛⠛⠻⠿⠿⠿⠿⠿⠀⠛⠃⠀⠀⠀⠀⠀⠛⠛⠻⠿⠿⠿⠿⠿⠀⠛⠃⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⡀
⠀⠀⠀⠀⠀⢤⡴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠙⠓⠦⢤⣀⡀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠛⠿⠿⠟⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⣀⣀⣉⣳⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠴⢾⣭⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠠⠄⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠄⣀⣀⣀⠀⠀⠀⠀⠀⢀⣙⣻⣭⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠦⣄⡀⠀⠀⠐⠂⠉⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⠀⠀⠈⠉⠉⠒⢒⡿⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⡠⠔⠊⢕⠒⠢⠤⣀⣀⠀⠀⣀⡀⠤⠴⠂⠀⠉⠓⠦⣀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣔⠉⠀⠀⠀⠀⠉⠉⠉⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⣤⣀⣀⣀⣀⣠⡤⠴⠖⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢈⡁⡐⢄
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠈⠁⠘
⠠⠀⡶⠉⢫⠑⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡚⠛⠒⠒⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⢼⣉⠀⠀⢖
⡄⢻⡤⠔⠚⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡧⠴⠊
⠘⢦⣈⣓⣂⡤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠤⠤⠤⣜⣂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠠



⠀⠀⢠⠖⢦⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠄
⠀⠀⢸⣄⣬⡀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠒⡂⠀⢀⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡄⡆
⠀⠀⠸⣿⣇⠇⠀⠀⠀⠀⠀⠀⢧⣀⣈⣉⡁⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⠳⡅⠀⠜⢀
⠀⠀⢀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠘⠛⣿⠟⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡶⠄



⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠃⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡍⠉⠉⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⡘⢢⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠻⠄⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣠⢔⠂⢹
⠀⠀⠧⠤⠭⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⣒⣒⣒⣲⡤⠇⠀⠀⠀⠀⠀⠀⠀⠀⣞⠡⠨⠤⢌⣳⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⢻⡀⡀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⣦⣄⣀⣀⡀⢸⡇⣇⠀⠀⡇⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⣧⡀⠶⠒⠲⡄
⢻⡀⠉⠁⠀⣿⣷⢹⡄⠀⢻⡄⠀⠀⠳⣠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⡎⠙⠒⢦
⠘⡇⠀⠀⠀⣿⢀⠀⢻⡆⠈⣿⠀⠀⠀⡇⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⡇⠀⠀⡇⠀⢀⡼
⠀⠙⠄⠀⢠⡏⠈⣧⠘⣧⠀⢹⡆⠀⠀⣧⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⡇⠀⠀⡷⠀⢸⡄⢀⣀⡀
⠀⠀⣇⠀⢸⡇⠀⠸⣇⠘⣧⡀⢻⡄⠀⣿⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢰⡇⠀⠀⠇⠀⠀⢙⠊⣨⣭⠏⠉⠉
⠀⠀⠹⣦⡼⠃⠀⠀⣿⠀⣾⠁⢸⡇⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢸⡇⠀⠀⢸⠀⠀⠸⣨⡷⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣿⠀⠸⣇⠀⣿⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⡇⠀⠀⢸⠀⠀⢸⠏
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠘⠢⠀⠸⠀⠘⠄⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠤⠜⠧⠀⠠⢼⣀⠀⠘⡄⡀
⠀⠀⠀⠀⢀⣠⠾⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠇⡷
⠀⠀⠀⡾⠀⠀⡠⠃⠢⡄⠀⣅⠀⣀⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠉⠙⠲⠦⣤⣀⣤⣤⠤⠀⡇
⠀⠀⢰⡇⠀⠀⠀⠀⠀⠉⠛⠾⣝⠉⠉⠉⠁⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⢀⡤⡤⠊⠁⠀⠀⠀⠀⠹⣆⡀
⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠦⣤
⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠁
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⠹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⠀⠀⠀⠀⠙⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠃⠀⢰⣿⣿⡆⠀⠘⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⢸⣿⣿⡇⠀⠀⠈⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠈⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉




⣿⣿⠛⢿⠛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠋⠉⠉⠉⠉⠙⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠒⠒⠒⠒⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀
⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠛⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠄⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠤⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⠀⠀⠀⠀⠀⢀⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠔⠊⠁



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

⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠈⠂⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠙⠛⠿⠿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠠⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡀⠀⠀⠀⠀⢀⣴⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⡛⠁⠀⠀⠀⠀⠀⠛⣿⣟⠋
⠀⠀⠀⠀⠄⠁⠀⠀⠀⠐⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣷⡄⠀⠀⢀⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⠆



⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠉⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⢤⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣿⣦⡀
⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢣⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣾⣿⣿⣧
⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠀⢀⡞⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣟⣭⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢀⠎⣀⣀⣀⠤⠤⠶⣚⡉⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠿⠿⠛⠛⠉⠽⣶⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠦⠤⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡶⠦⣤⣤⣤⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠐⠾⠛⠠⣴⣶⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠁⠀⠀⠀⠀⠈⠉⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠀⠀⢀⡀⠀⠀⠀⠀⠲⠶⠄⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⣔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠸⣇⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠻⠀⣄⠀⠀⢸⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠸⡿⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠈⢦⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡊⠀⠀⠀⢾⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⡃⠀⠀⢀⡈⠀⠀⠻⣄⠀⠀⠂⠤⠀⣀⣀⣀⣀⠀⠠⠤⠘⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⠏⣤⡜⠃⠀⠀⠀⠀⠘⠳⣤⡀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⡠⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠦⣤⣤⣄⣀⣠⣄⠰⠶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠤⠤⠤⠤⠶⠄⠛⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡄⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⠑⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⠆⠀⠀⠀⠀⡆⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠉⠉⠲⠶⠆⣀⣠⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡠⠁⠀⠀⠀⠀⠀⠁⢸⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠦⠔⠃⠀⠀⠀⠀⠀⠀⡆⠀⠈⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡆⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡧⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⢀⡀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡠⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀⡆⠀⠀⢠
⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠸⡇⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⢛⡀
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠺
⠀⠀⠀⠀⠘⠛⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⡆⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⡄⠹⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠘⣀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠃⠀⠀⠀⠀⠀⠀⢻⣀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠙⢣
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠁⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠆⠼
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⣁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄
⢠⣧⣀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣌⠉⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣄⠙
⣉⣤⣤⣤⣤⣌⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⠿⠛⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣀⣀⣀⣀⣀⡀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⢀⠤⠠⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⣀⣁⣀⣤⣴⣶⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣠⣶⣿⡿⠛⠁⠀⠉⠙⠛⠛⠛⠛⠿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡴⠶⠶⢶⣤⡄⠀⠀⠀⠀⠀⣠⣴⠶⠾⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⡇⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠈⠫⠘⠿⠃⠴⠀⠆⠀⠀⠰⠆⠹⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⡄⠀⠀⠀⣀⠤⠆⠀⠀⠀⠀⠀⠀⠖⢄⡀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠞⠀⠀⠀⠀⢠⠀⠀⠠⠀⠄⠀⠘⢆⠀⠈⢂⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⢀⠀⠀⠐⠀⠀⠀⠠⡤⠀⠁⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢀⡾⠀⠤⠀⣾⣿⣿⣿⣿⣿⡦⠄⠘⣆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢁⠀⠀⠀⠈⠻⠶⠶⠾⠋⠁⠀⠀⠘⡆⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⢀⠀⠀⠀⠀⠃⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠃⠀⠀⠰⠀⠀⠠⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢰⡾⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⣶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⡀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣶⣶
⣿⣿⣇⣀⣁⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣴⠶⠶⣶⣶⣶⣶⡶⠶⠶⣶⣶⣶⣶⠶⠶⢶⣶⣶⣶⣶⠶⠶⢶⣶⣶⣶⡶⠶⠶⣶⣶⣶⣶⡶⠶⠶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⣿⠀⣰⣿⣿⣿⠟⠀⢀⣼⣿⣿⣿⠋⠀⢀⣾⣿⣿⡿⠃⠀⣠⣿⣿⣿⡟⠁⠀⣰⣿⣿⣿⠏⠀⠀⣼⣿⣿⣿⠋⣿
⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⣿⣏⣀⣀⣾⣿⣿⣿⣃⣀⣠⣿⣿⣿⣿⣁⣀⣰⣿⣿⣿⣟⣀⣀⣼⣿⣿⣿⣏⣀⣀⣾⣿⣿⣿⣃⣀⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢻⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉⣏⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣼⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣧⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⣿⠀⣠⣿⣿⣿⡟⠁⠀⣰⣿⣿⣿⠏⠀⢀⣼⣿⣿⣿⠋⠀⢀⣾⣿⣿⡿⠁⠀⢠⣿⣿⣿⡟⠁⠀⣰⣿⣿⣿⠏⣿
⠀⠀⠀⠀⠀⠀⣿⣰⣿⣿⣿⡟⠀⠀⣼⣿⣿⣿⠏⠀⢀⣾⣿⣿⣿⠃⠀⢠⣿⣿⣿⡿⠁⠀⣰⣿⣿⣿⡟⠀⠀⣼⣿⣿⣿⠏⠀⣿
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⢻⠛⢻⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡟⠛⡟⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠃

⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠰⠩⠀⠠⠑⡄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣾⣿⣾⣷⡆⠀⠀⠀⠀⠀⢠⣤⠶⡶⢶⡶⣦⣄⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠐⠂⡒⠲
⠀⠀⠀⠀⡐⡄⣦⡄⠀⢃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⡇⠀⠀⠀⠀⠀⠈⢩⣿⠩⡯⣻⣎⠉⢱⠀⠀⠀⠀⠀⠀⢸⡿⢿⣿⣿⠀⢸
⠀⠀⠀⠀⠡⠁⠈⠁⢰⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⢀⡇⠀⠀⠀⠀⠀⠀⠿⠿⠷⠾⠿⠿⠖⠃⠀⠀⠀⠀⠀⠀⡸⠅⠤⠤⠼⣀⠌
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⣠⣶⣾⣿⢶⣤⢀⠀⠀⠀⠀⠀⠀⢰⠶⡐⠂⠒⢂⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⡍⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆
⠀⠀⠀⢿⣿⣭⣿⣽⣿⠃⢱⠀⠀⠀⠀⠀⢸⠀⠈⠲⠞⠁⣠⣾⣎⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⡇⣀⣭⣯⣄
⠀⠀⠀⠀⠀⠿⠟⠫⣡⠒⠁⠀⠀⠀⠀⠀⠘⠏⠀⠠⢀⡸⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⡸⠀⠀⠀⠀⠀⠀⠀⠘⠲⠶⠆⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃



⠀⠀⠀⠀⣿⣿⠿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⠀⡄⣠⣦⡀⠀⠀⠀⠀⠀⠀⣴⠒⠒⠠⠐⠒⣦⡀⠀⠀⠀⠀⠀⠀⡖⠂⠒⣀⠂⠒⢲
⠀⠀⠀⠀⣿⣿⣀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⣘⣟⣁⠀⢃⣙⣟⣁⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⠀⠀⣽⡇⠀⠀⠀⠀⠀⢀⡥⡄⠠⠿⠄⠀⢸
⠀⠀⠀⠀⣍⣉⣉⣉⣉⣉⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠇⠺⠿⠿⠿⠂⠀⠀⠀⠀⠀⠿⠷⠶⢤⡤⠤⢼⠅⠀⠀⠀⠀⠀⠸⣿⠟⡿⢿⠿⠟⠛



⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠰⡶⠀⠼⠧⠀⢶⠆⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠿⠿⠿⠿⢿⠀⠀⠀⠀⠀⠀⠀⡛⢍⠉⠉⣉⠝⡇
⠀⠀⠀⠈⠉⠀⣈⣀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⢠⣤⣤⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘⡘⡐⢛⡘⢰⠀⠀⠀⠀⠀⠀⠀⠃⡀⠑⠚⢡⢖⡲⡄
⠀⠀⠀⠀⠀⠀⣻⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣬⣭⣭⣼⠀⠀⠀⠀⠀⠀⠀⠀⣇⠐⠀⠂⡛⠛⣰⠀⠀⠀⠀⠀⠀⠀⠛⠀⠂⠂⢣⣛⡓⠃
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢰⡆⠀⠀⠂⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡎⠀⠈⠋⠓⠉⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡀⠀⡀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠈⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠾⠟⠈⠛⠛⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠟⠀⠈⠟⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⣤⣤⣄⠀⢠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⢹⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠈⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢀⣠⡶⠁⠀⢸⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⠿⢿⠟⠛⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⡿⠿⠆⢠⣿⣿⣿⣿⣿⣷⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢳⣤⠀⡀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⡿⢿⣿⣿⣧
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠙⠛⠁⠀⠈⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠸⠇⠂⡀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀⣸⣿⣿⣿⣿⣿⠛⠿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣤⣤⣄⣀⣤⡀⡈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⢻⣿⣿⣿⣿⡇⠀⠀⠀⠀⣀⠤⠶⠂⢡⣄⠐⠶⢀⣀⡉⠘⠄⢄⡀
⠀⠀⠈⣹⠀⠀⠉⠋⠉⠂⢀⣀⠀⠘⠂⠀⠀⠀⠐⠀⠀⠀⢸⠆⠀⠀⠰⣄⣠⣽⢻⣿⣿⠃⠀⠀⣠⠞⠁⠀⠀⢤⣀⠉⠙⠶⢦⠉⠁⠀⠀⠀⠑⠄
⣶⣴⣦⣿⠁⠄⠀⠀⢠⡀⠈⢡⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠻⢿⣿⡿⠋⠀⠀⣰⠋⠀⠀⠂⠀⠈⢻⣶⣶⠀⠀⠁⠀⠀⠀⠀⠄
⣿⡏⣿⣟⣦⡴⣳⣶⡾⠁⢠⣦⣤⣼⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⡏⠀⠀⠀⠀⠀⣀⠀⠀⠘⢿⣶⠃⠀⠀⠀⠀⠀⠀⢰⡄
⣿⣧⢻⣿⣿⡷⢉⣻⡄⣀⣰⣿⡄⣼⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⡇⢰⡀⠀⠀⠘⠻⣄⡀
⠻⣿⠿⣿⣷⣿⣿⣿⣿⣿⣿⣧⣇⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠰⡇⣸⡇⠀⠀⠀⠀⠀⠱⣄⣀⠀⠀⠀⠀⠠
⢶⣴⡆⢹⣿⣿⣿⣿⣿⡟⣿⣿⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣼⣿⡏⣆⣀⢠⡀⠀⠀⠀⠉⠁
⠀⠀⠁⠀⢻⣯⣉⣉⠙⠛⠻⠿⠿⣿⣧⡀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢠⠈⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠙⠉⠉⠈⠛⢺⠗⠂⠛⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠻⠏⠛
⠀⡀⠀⠀⠀⠀⠸⣶⣶⣶⣶⣤⠶⠶⠦⠈⠉⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠉⡁⠀⣩⣄⠀⣴⡶⠀⠀⢠⣿⣶⡆⠈⠀⠀⠀⠀⠀⣀⠈⠀⠈⠀⢀⣾⣿⣿⣿⣿⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢙⡀⠙⣃⢀⣠⡤⠀⠀⣿⣿⣿⡧⠀⣷⣼⣿⠀⠀⠙⡇⠀⠠⣠⣿⡿⠋⣼⡿⣙⠓⠈⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠊⢻⠀⢨⣇⠀⢰⡟⣿⣿⡇⢸⣿⠸⣿⡇⠀⠀⢁⣴⡶⠛⠋⠀⣾⣿⣿⡧⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣸⣷⣴⠟⢰⡟⠁⢸⣿⡇⣾⣿⡇⣿⣿⡄⠠⠛⠁⠀⠀⢠⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡁⣩⠿⣆⣰⣿⠀⠀⠈⢿⣷⣿⣿⡇⢻⣿⠇⠀⠀⠀⠀⣴⠟⠘⣿⠇⡟⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣄⣤⣴⡿⠟⠃⠀⠀⠀⠈⢿⣾⣿⣫⣾⡿⠀⠀⠀⠀⡜⠁⠀⠀⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡄⣰⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⢸⡆⠀⠈⣷⡛⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⡇⠀⠀⠘⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⢉⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠱⣀⣀⠤⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠦⠀⠀⠁⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⣀⡄⠀⢀⠀⠀⠀⠀⠱⠊⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠚⣥⣄⡷⠤⢄⣁⠀⠀⠀⠀⠀⠀⠉⠉⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⡠⢺⣽⠚⠒⢤⣘⠈⢲⠻⡀⠄⠀⠀⠀⠀⠀⠸⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠉⠉⠉⠁⠒⠒⠾⠤⣮⠃⠀⠀⢀⠼⠤⠮⣀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠉⠉⠢⡀
⠀⠀⠀⠀⠀⠀⣎⠉⠐⠢⡜⠦⣀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⡨⡕⠀⠀⠀⠀⠉⢫⡤⠀⠀⠀⠀⠀⠀⠰⣤⠤⠅
⠀⠀⠀⠀⠀⠀⠀⠉⠐⠲⣜⠤⣀⢉⡶⠦⢤⣄⣀⣀⡀⠀⠀⠀⠈⠒⠠⠤⢄⡤⠔⠁⠀⠀⠀⠀⠀⠀⣀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⢀⠟⢉⡽⣯⣸⠠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠘⢤⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⢠⠃⡠⢯⡔⠋⠁⠀⠀⣀⠼⣲⠀⠀⠀⣠⡠⠦⡨⢆⠀⠀⠀⠀⠀⠀⡤⠔⠃
⠀⠀⠀⠀⠀⠀⠀⡰⠉⣇⢀⠇⡰⠓⠠⣉⠖⠒⠚⠛⣾⡾⠁⢀⠔⠋⡀⠉⡾⡁⠀⢀⢠⠦⠄⠤⠤⠊
⠀⠀⠀⠀⠀⠀⢰⠁⣼⣭⣾⢀⡇⢅⣷⠏⠀⠀⢢⣼⡿⢁⡴⠃⠀⢏⠈⠺⡄⠸⡦⢮
⠀⠀⠀⠀⠀⣀⣼⡾⢻⠁⠈⠳⠷⢿⡏⠀⠀⠀⠸⠜⡡⠊⠀⠠⠛⠒⣓⠆⢱⠀⢳⣀⡆
⠀⠀⠀⠀⡜⢝⠁⢣⠀⢣⠀⠀⠀⢸⠀⠀⠀⠀⢀⠜⠀⠀⣠⠒⠊⠁⣀⠤⠅⠒⠊
⠀⠀⠀⣤⡿⠒⠁⠈⢆⠀⢇⠠⡚⣇⣀⣀⣀⠤⣯⠶⠀⠀⠘⡄⡞⠁
⠀⠀⣞⢏⣀⣀⠀⠀⠈⢢⠀⠣⠧⢄⣀⣀⡀⢏⠀⠀⠀⠀⢴⠣⠜⡑
⠀⠀⠀⣨⡷⠋⠀⠀⠀⠀⠳⣠⠃⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⣇⠀⢡
⠀⠀⠀⠻⠢⣴⣶⠶⠀⠀⢠⡏⠀⠀⠀⠀⠀⢘⠑⠤⡈⠑⠤⣸⠀⠸
⠀⠀⠀⠀⢰⠱⣁⣠⣄⡄⠀⠉⡗⠀⠀⠀⠀⢸⡀⠀⠈⠒⢄⡁⠀⠀⡇
⠀⠀⠀⠀⠀⠁⠈⠉⡟⠥⢔⣲⡇⠀⢀⠀⠀⠀⢦⠀⠀⠀⠀⠙⠢⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠑⠵⠺⡻⡁⢀⠠⣒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠉⠉⠕⠠

⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻
⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⢀⣠⢊⡵⠋⠙⠉⣐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⠦⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⡀⠀⠀⠀⢸
⣿⠀⢀⠎⠸⡇⣾⡵⢒⠛⠉⢽⡇⠀⠀⠀⠀⠀⠀⢠⠊⠈⠈⣠⢖⠈⣧⠇⠀⣠⣴⣶⣷⣦⣰⣊⠐⠦⢄⠑⢠⠀⢸
⣿⠀⢈⠆⢸⣇⣿⠃⠀⠓⡘⢻⡇⠀⠀⠀⠀⠀⠀⠸⠀⢰⡒⠁⠇⠐⢅⠀⠸⣟⠉⢉⡈⢿⡯⠘⡁⢲⢄⠀⢄⢹⣸
⣿⠀⠻⠦⠎⠈⠛⢆⠀⠀⠅⡜⠀⠀⠀⠀⠀⠀⢤⣃⢆⣄⣥⡀⣷⠾⣿⠀⠠⡗⢂⠀⢀⠘⡇⣜⢧⣠⡎⢳⠀⢢⣹
⣿⠀⠀⠀⠀⢀⠠⣤⡟⠑⢯⢀⠀⠀⠀⠀⠀⠀⠀⡜⠛⢯⣢⣸⠇⢀⠎⠀⢀⣧⣼⡟⢉⢼⡇⡏⣺⠕⢑⡹⢲⠴⢻
⣿⠀⠀⠀⢰⠋⡄⠸⡓⠚⢻⠀⣇⠀⠀⠀⠀⠀⣸⠀⠀⣠⠟⠁⣰⡇⠀⢀⠇⡆⢀⡗⠚⠀⣭⠉⡍⠐⢌⠑⠯⣄⣼
⣿⠀⠀⢠⠃⠀⢹⠀⠑⠴⠁⢰⢻⡀⠀⠀⠀⢠⠯⠷⠴⠧⢠⠚⣧⣧⠀⡿⠒⢁⠎⢀⡤⡀⣸⢴⣁⡇⠀⠑⠤⠴⢻
⣿⠀⠀⠈⠓⢲⠇⠀⠀⢀⠀⢸⢚⠀⠀⠀⠀⡞⠀⠰⠦⡁⠛⠋⢁⡗⠀⠓⠒⡇⠀⢰⣆⣈⣠⡴⡇⡿⡄⠠⣀⡀⢸
⣿⠀⠀⢰⠆⢸⠒⢲⠀⠀⠀⢸⢸⠀⠀⠀⢠⠇⠀⠀⠀⠉⠋⢹⠀⠀⠀⠀⣰⠳⠄⠀⠈⠀⠰⡇⡧⣇⣈⡀⢀⣀⣼
⣿⠀⠀⢸⢰⠸⣀⣸⠦⠞⡦⡾⣌⠆⠀⠀⢸⡴⠦⣄⠀⠀⡇⣸⠆⠀⠀⢀⡗⠒⠀⠤⠲⠶⢤⠷⡡⡇⠀⣉⠉⠁⢸
⣿⠀⠀⠈⠙⢦⠼⢛⠀⡀⡇⢧⢏⠀⠀⠀⠀⢧⠀⠉⠙⡷⠏⢹⠄⠀⠀⡜⠀⠀⡹⠛⡆⠀⠈⡆⠙⡇⠀⠈⡇⠀⢸
⣿⠀⠀⠀⠀⡇⡄⢸⠐⡇⡇⢸⢸⠀⠀⠀⠀⣇⠀⠀⢠⡇⠀⢨⠇⠀⢠⠇⠀⢠⠃⠀⡇⠀⠀⡇⢀⡇⠀⠀⡇⠀⢸
⣿⠀⠀⠀⠠⣇⡇⡾⠀⡇⢣⠈⣏⡇⠀⠀⠀⣾⠀⠀⢘⡧⠀⣸⠀⠀⢸⡄⠀⣿⠀⠀⣗⠀⢀⡇⠈⠓⡖⢶⢱⠖⢺
⠿⡤⠤⠤⠤⠬⢽⢿⠿⠷⡿⡿⠧⠤⠤⢤⠤⠧⠤⠤⠾⠤⠤⠽⠤⠤⠼⠤⢤⠼⠤⠴⠧⡤⠼⢧⠤⠤⠧⠼⠼⠤⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣴⠀⠀⣴⠄⠀⣴⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⢧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠆⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣼⣿⣧⣼⣿⣧⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡩⠵⣤⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⡴⠭⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣇⡇⠀⠈⢷⡀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⣰⠏⠀⠀⣇⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⢯⡗⠒⠢⡀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⡠⠒⠒⣮⠇⠈⠢⣀
⠀⠀⠀⠀⠀⢀⡠⠔⢺⠖⠊⠁⠀⠀⠀⠘⡞⡄⢀⡇⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣇⠀⡼⡜⠀⠀⠀⠀⠉⠒⢾⠒⠤⣀
⠀⠀⠀⢠⢾⠁⠀⠀⠸⣼⡀⠀⠀⠀⠀⠀⠹⡝⡍⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⡝⡽⠁⠀⠀⠀⠀⠀⡸⡜⠀⠀⠀⢹⢦
⠀⠀⠀⠸⡈⡄⠀⠀⠀⠙⣷⣀⠤⠤⠒⠒⠒⠚⠮⣓⢒⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⢒⡪⠞⠒⠒⠒⠢⠤⢄⣰⡗⠁⠀⠀⠀⡞⡸
⠀⠀⠀⠀⠱⣘⣄⣀⡠⢤⢾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢦⠤⣀⣀⣜⡰⠁
⠀⠀⠀⠀⠀⠀⠈⠈⢆⠈⢣⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢣⠋⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠑⣌⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⡔⠁⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠈⠳⣌⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⡴⠋⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠘⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡜⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⡀⠤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣷⣾⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣿⣿⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣦⣠⣄⠀⠀⠀⠀⢀⣴⣿⣿⣿⡇⢿⠿⢃⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⢿⣿⣿⣦⣀⢠⣿⣿⢻⣿⣿⣇⣾⣆⣾⣿⡿⠛⠻⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⡿⠁⠈⣿⣿⣿⣿⣿⣿⣿⡁⢀⣠⣤⣽⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣅⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠘⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡏⠁⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡾⢿⣿⠄⣤⣤⣴⠿⣿⡧

⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣷⡤⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠻⠷⠀⠀⠀⠀⠀⠀⠉⠑⠂⠀⠀⠀⠀⢛⣿⡟
⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠀⠹⣿⣧⠀⠀⠀⢀⡠⠴⠊⠀⠀⡀⡀⢕⠢⡀⠀⡗⡄⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢄⠁⠀⠀⠀⠀⠹⣿⡖⠒⠊⠁⠀⠀⠀⠀⠀⠙⢄⠂⠁⡈⠢⠘⠈⢆⠀⠀⠀⠀⠀⠀⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠳⡁⠈⠃⠀⠐⠢⠀⠁⠈⠢⡀⠀⠀⠀⠠⣿⣽⠇
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢳⡀⠠⡀⠀⠹⣽⣇⠀⠀⠀⠀⠀⠀⠈⠢⣀⡈⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⢀⣾⡟⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢶⡄⠀⠀⠀⠙⢦⡀⢀⠀⠹⣾⡆⠀⠀⠀⠀⠀⠀⠀⢀⡑⣄⠂⠀⠀⠀⠀⠀⠀⠉⠐⠘⠿⢅⣀
⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠙⢦⠢⡀⢹⣿⠀⠀⠀⠀⠀⠀⠬⠤⠤⠬⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠲⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠑⠔⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠒⠠⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⣿⣷⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣍⠙⠓⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠡⡀⠠⢄⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⠀⠈⠙⠳
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠊⠀⢸⣅⠀⠠⡄⢉⠉⡐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣞⡏⠂⢳⡀
⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⢐⣻⡆⠀⠀⣀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⣾⡀⠑⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠊⠁⠀⠹⡄⠀⣀⠉⣀⠩⠤⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢳⣽⣇⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠓⡦⢄⠉⠤⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⡄⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⠤⠀⢌⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢿⡿⡀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⡠⢀⡀⢀⠰⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠇⠹⡀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⡇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢰⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⡀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⠔⠋⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠉⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢟⡻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢫⣿⣿⡝⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣣⣿⠛⠛⣿⣜⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⢀⣤⣤⡀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡿⣱⣿⣧⠀⠀⣸⣿⣎⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠸⣿⣿⠇⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⣠⡿⣽⣿⣿⣿⠀⠀⣿⣿⣿⣯⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⣿⣿⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⣰⣟⣼⣿⣿⣿⣿⡄⢠⣿⣿⣿⣿⣧⣻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⣴⢏⣾⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣷⡹⣦⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⣼⢯⣾⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣷⡽⣧⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⣾⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣷⠀⠀⠀⠀⣾⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣷
⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⠉⠀⠀⠀⠀⠉⠛⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⠿⣒⣤⣶⣶⣿⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⡟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⢛⣙⢻⣿⠟⣉⡻⣿⡿⣋⡙⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣌⣋⣼⣿⣮⣛⣵⣿⣧⣙⣣⣾⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣦⡹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⣿⣿⣿⣿⣿⣶⡍⠛⠿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⡿

⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⣀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣦⣾⣿⣿⣷⣶⣦⣄⡀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣷
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠻⣿⣿
⣿⣿⣿⢋⣥⣦⣬⣭⣭⣭⣭⣭⣭⣭⣥⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢹⣿
⢻⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⣮⣭⣍⣛⣛⣛⡛⠿⣿⣿⣿⣿⠈⣿
⠈⢿⣷⣜⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡝⢿⣿⢰⣿
⠀⠀⠹⣿⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⡎⠟⣼⣿
⠀⠀⠀⠀⠉⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣾⣿⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢙⠿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣿⣿⠿⢟⢱⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣾⠿⠸⢿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⣠⣶⣿⣿⣶⣮⣍⣛⠿⠏⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡘⠿⢛⣫⣭⣶⣿⣿⣷⣮⠉⠁
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣷
⠀⠀⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡿
⠀⠀⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⡧
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠴⡒⠉⠉⠉⠉⠒⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠒⠒⠒⠦⠤⣄⣀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⠀⠀⠀⠈⣦⡀⠀⠀⠱⡄⢀
⠀⠀⠀⠀⠀⠀⣴⡿⠿⣿⣷⣄⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⢸⠈⠓⢦⡔⠉⠁⠉⠳
⠀⠀⠀⠀⠀⢰⣿⠁⡀⠈⠛⠿⣷⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⠀⠀⢸
⠀⠀⠀⠀⠀⠘⢿⣦⣈⠛⠓⠂⠀⠙⢿⠅⠉⠙⣿⡶⠒⠤⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠘⢦⣀⡀⢀⣠
⠀⠀⠀⠀⠀⠀⢈⡿⠿⠟⠋⠀⠀⣀⣀⠀⠀⠀⠸⠇⠀⠀⢀⣀⣈⠉⠓⠶⢤⣀⠀⠀⢰⠃⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⢠⣎⠀⠀⠀⠀⣰⠾⣛⡛⠃⠀⠀⠀⠀⠀⠀⠘⢛⣛⠷⡄⠀⠀⠀⢉⣻⡁
⠀⠀⠀⠀⡰⠛⠛⠛⠛⠃⠀⠀⡜⠁⠙⡆⠀⠀⠀⠀⠀⠀⡼⠁⠈⡇⠀⠀⠘⠛⠛⠛⠛⣄
⠀⠀⢠⣞⣁⠀⠀⠀⠀⠀⠀⠀⠱⠚⠛⠋⠀⠀⣠⣄⡀⠀⠛⠛⠓⠃⠀⠀⠀⠀⠀⠀⢀⣈⣷⡀
⠀⡠⠛⠻⠿⠿⠿⠿⠟⠓⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠚⠻⠿⠿⠿⠿⠟⠛⣆
⣸⠁⠀⠀⢀⡤⠖⠊⠉⠉⠙⠒⢤⣀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⠴⠒⠉⠉⠉⠑⠲⢄⡀⠀⠀⠈⡆
⣿⠶⠆⢠⠋⠀⠀⠀⠀⠀⠀⠰⠀⠈⢿⣷⣶⣦⣤⣤⣴⣶⣾⡟⢁⡀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠶⠾⣿
⢇⠀⠀⢸⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⣈⠻⢿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⢂⡴⠋⠀⠀⠀⠀⡇⠀⠀⡞
⠘⢆⠀⠘⣆⠀⠀⠀⠀⠀⢧⠘⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠆⠀⠀⡀⠀⡾⠀⠀⠀⠀⠀⡸⠁⠀⡼⠁
⠀⠈⠳⣄⡈⢦⡀⠀⠀⠀⠈⢧⣀⠀⠘⠂⠀⣠⢾⠳⣄⠀⠀⠀⠀⣀⠜⠁⠀⠀⠀⢀⠞⢁⡤⠚
⠀⠀⠀⠀⠉⠓⠿⣦⣀⠀⠀⠀⠈⡏⠉⢩⠏⠀⣸⠀⢈⡏⠉⠉⢹⠁⠀⠀⠀⣀⡴⠿⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠲⣤⣀⣇⠀⠸⡄⠀⣿⠀⢸⠀⠀⠀⣼⣀⣤⠒⠉⢹
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢠⡇⠈⠙⢦⣀⣇⠀⠈⠀⣸⣀⣠⠴⠋⠀⢸⡀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⣰⣧⣄⠀⡾⠀⢠⣶⣦⣬⣉⠳⠤⠴⠋⣉⣤⣶⣶⡄⠈⣧⠀⣠⣼⡆
⠀⠀⠀⠀⠀⢠⠏⠉⠉⢱⠇⠀⢸⣿⣿⣿⣿⣷⣦⣴⣿⣿⣿⣿⣿⡇⠀⠸⡌⠉⠉⠹⡄
⠀⠀⠀⠀⢠⠏⠀⠀⠀⡾⠀⠀⢸⣿⣿⣿⣿⡿⠋⠙⢿⣿⣿⣿⣿⡇⠀⠀⡇⠀⠀⠀⢳⡀
⠀⠀⠀⢀⣾⣦⣄⣀⣀⡇⠀⠀⠘⠿⠿⠛⠁⠀⠀⠀⠀⠈⠛⠿⠿⠃⠀⠀⢹⣀⣀⣤⣴⣷
⠀⠀⠀⡞⠻⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠛⢧
⠀⠀⣸⠁⠀⠀⠀⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠀⠀⠀⠘⡆
⠀⢠⡏⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢻
⠀⢸⣿⣦⣤⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣀⣤⣶⣿⡇
⠀⠾⠛⠿⠿⠿⠿⠷⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠾⠿⠿⠿⠿⠛⡇
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⢡⣾⣷⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⡟⣁⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⢠⠿⣿⣿⣷⡈⢻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣰⡟⠀⣻⣧⣤⡿⠀⢻⣆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠏⣰⣿⠄⠚⠛⢰⣿⡄⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣴⠏⠀⢰⡟⣿⣧⣀⣀⠀⠻⣆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠋⣰⣿⡏⣤⠀⠹⠿⠿⣿⣄⠹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣼⠏⠀⠀⠘⠁⠘⢿⡍⠙⣧⠀⠹⣧⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃⣼⢿⣿⣧⣿⣧⡀⢶⣦⣸⣿⣦⠹⣿⣿⣿⣿
⠀⠀⢀⣾⠃⠀⠀⢀⣀⣀⣀⡈⣷⠀⠀⠀⠀⠙⣷⠀⠀⠀⢸⣿⣿⣿⢁⣶⣾⣿⡿⠿⠿⠿⣿⣸⣿⣿⣿⣿⣧⠹⣿⣿⣿
⠀⠀⠈⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠀⠀⠀⢸⣿⣿⣷⣌⣙⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣋⣠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠙⠁⠑⠀⠂
⠀⠀⢀⡼⠻⡒⠤⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣶⣶⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠃⠀⢀⢤
⠀⢆⡞⠀⢀⣱⡤⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⢆⠀⢀⡱
⢃⠨⠻⣍⡉⠈⠃⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡠⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⡛⢍⠢⠑
⠀⠓⠤⣌⣉⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠐⣉⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠬⠂



⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢉⣉⣉⡉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣄
⠀⠀⠀⣷⣶⡗⡆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⣒⣒⠒⠀⢰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢠⠒⠲⢣⠏
⠀⠀⠀⡻⣿⡷⠁⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠤⠤⣄⠀⢸⣿⣿⣟⣻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠳⡄⠀⠋⠠
⠀⠀⢀⣠⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠈⠙⣿⠟⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠿




⠀⠀⠀⢠⠀⠀⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⠤⠤⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠁⠀⠀⢀⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤
⠀⠀⠀⠀⢠⣉⡡⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠳⠄⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠦⣀⡤⢖⡉
⠀⠀⠰⠤⠬⠥⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣛⣛⣛⣉⡦⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡁⡤⠤⠤⣄⣙⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡶⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢪⠊⠁⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠙⡄⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠂⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠄⣒⡒⢴⡉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠭⢤⡴⠊⠀⠁⠀⢀⡀⠀⠀⠀⠀⠎⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣊⣙⠀⠀⠀⠀⠀⠄⠀⠀⠀⠉⠑⠢⢄⣸⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠔⡆⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⣀⣀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘⣀⠼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢐⠒⠾⡐⢞⣃⢀⠀⠁⡀⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠠⣙⠶⣷⣌⠈⣿⣄⣌⠀⠂⡔⣂⣀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⠑⢌⠑⠦⠶⠋⣈⠂⣽⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠙⠤⣠⣄⠑⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠉⠉⠒⢲⠄⡤⠤⡤⠞⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⣀⣀⣔⡡⢊⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣎⡀⠀⢀⣠⡋⣁⣔⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠂⠛⠈⠉⠉⠀⠀⠉⠀⠙⠒⠀⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⣖⠋⠉⠉⠉⠙⠓⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠷⠶⠤⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⣄⡀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣄⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⠀⢸⠓⢦⣀⢀⡼⠖⠒⠶⣄
⠀⠀⠀⠀⠀⠀⢸⣿⠋⠉⠻⣿⣷⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⢸⡄⠀⠈⡟⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢿⣇⠀⢤⣄⣈⣙⠛⢿⣷⡶⠤⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠘⢿⣷⣤⣄⣉⠀⠀⠀⠙⠇⠀⠀⠈⣿⡏⠉⠙⠓⠲⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⠦⢤⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⣠⠛⠛⠋⠁⠀⢀⣠⣤⣄⠀⠀⠀⠘⠁⠀⠀⢀⣤⣤⣀⠈⠉⠓⠦⢤⣀⠀⣠⠃
⠀⠀⠀⠀⠀⢀⣼⣁⣀⠀⠀⠀⠴⠟⣛⣙⡉⠀⠀⠀⠀⠀⠀⠀⠈⣛⣛⡛⠷⠄⠀⠀⣀⣈⣹⣅
⠀⠀⠀⠀⣠⠞⠛⠛⠛⠛⠁⠀⠀⢸⠁⠀⢱⡀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢹⠀⠀⠀⠘⠛⠛⠛⠛⢦⡀
⠀⠀⢀⣴⣃⠀⠀⠀⠀⠀⠀⠀⠀⠸⠖⠛⠛⠃⠀⢀⣀⣀⠀⠀⠙⠛⠓⠾⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⣄
⠀⣠⠞⠿⢿⣿⣿⣶⠶⠶⠆⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠶⠶⢶⣾⣿⣿⠿⠟⢦⡀
⢰⠃⠀⠀⠀⠀⣠⠴⠖⠒⠒⠲⢦⣄⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢀⡤⠶⠒⠒⠒⠶⢤⣀⠀⠀⠀⠀⢳
⣿⣤⣄⠀⡴⠋⠀⠀⠀⠀⠀⠀⢀⡀⠙⢳⣦⣤⣀⣀⣀⣀⣀⣠⣤⣶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⢀⣤⣼⡇
⡏⠉⠁⢰⠃⠀⠀⠀⠰⢤⡀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠒⠀⢀⡀⣠⠴⠀⠀⠀⠀⢻⠀⠉⠉⡇
⢹⡀⠀⠈⣇⠀⠀⠀⠀⠀⠙⡆⢠⠀⠀⠰⠂⠈⠛⠻⢿⠿⠛⠋⡀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢀⡟⠀⠀⣸⠃
⠀⠱⣄⠀⠘⣆⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⡀⠀⠀⠀⣼⡆⠀⠀⠉⠀⠀⠰⠀⣸⠁⠀⠀⠀⠀⢀⡼⠁⢀⡴⠃
⠀⠀⠈⠳⢤⣈⠣⡀⠀⠀⠀⠀⠙⠦⣄⣀⣀⣀⣤⠞⣻⠙⠦⣄⣀⣀⣀⡤⠞⠁⠀⠀⠀⠀⣠⢊⣀⠴⠋
⠀⠀⠀⠀⠀⠈⠙⠻⠶⣄⡀⠀⠀⠀⡿⠀⠀⣸⠀⠀⡿⠀⢠⡇⠀⠀⢸⡇⠀⠀⠀⣀⡴⠾⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢄⣻⠀⠀⢻⠀⠀⡇⠀⢸⠃⠀⠀⢸⣃⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⣸⡄⠀⠁⠀⣸⢀⣀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣤⣈⠙⢦⣄⡴⠋⢉⣠⣤⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣄⣤⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠟⠋⠀⠀⠀⠈⠛⠿⣿⣿⡿
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣿⣿⡷⢀⣀⣺⠆⠁⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠁⠸⣿⣿⣯⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠻⣷⣬⡟⢒⣀⣠⣄⣀⣄⡀⠀⣈⡙⠲⠦⢦⣈⡉⠉⣠⡀
⠀⠀⠀⠀⠀⢀⣴⢞⠐⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⡿⠋⣡⢟⢿⣷⡐⠂⢀⣴⠄⠀⣿⣅
⠀⠀⠀⠀⢰⡿⣁⠐⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⣾⣟⠀⡀⠘⠉⠿⠟⢻⠀⢰⣿⣿
⠀⠀⠀⠀⣾⣧⣬⣀⡀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⢁⣾⣿⣿⣿⣤⣤⣾⣆⠀⠀⢸⣀⣴⣿⡟⠀⣰⣦⣀
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⡏⠀⢰⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉⠉⠉⠀⠈⠉⢹⡗⢸⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⡆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀
⣿⣿⣫⣛⣉⣛⣻⣋⣿⣻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⢸⣿⣿⣿⠏⠉⠉⠉⠿⠿⠿⠿⠋⠉⠉⠉⠿⠿⠿⠿⠋⠉⠉⠉⠿⠿⠿⠿⠉⠉⠉⠹⠿⠿⠿⠟⠉⠉⠉⢹⣿⣿⣿⠟⠉⠉⢹⡇
⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⣰⡿⠻⠷⠀⣰⣿⣆⠀⣿⠀⢸⡇⠚⢻⡿⠛⢸⣷⢠⣾⠛⢷⡄⢺⣷⡀⣿⠀⣿⠀⢸⣿⣿⠏⠀⠀⠀⣼⡇
⠀⠀⢸⡿⠃⠀⠀⢠⣿⠀⢻⣧⣠⣤⢠⣿⣶⣿⡄⣿⣄⣼⡇⠀⢸⡇⠀⢸⣿⠸⣷⣀⣼⡇⣸⡏⢿⣿⠀⣛⠀⢸⣿⠋⠀⠀⢀⣾⣿⡇
⠀⠀⢸⡇⠀⠀⢠⣿⣿⣀⡀⠉⠉⠁⣈⣁⣀⣈⠁⠈⠉⠉⣀⣀⣈⣁⠀⠈⠁⠀⣈⣉⣁⣀⠈⠁⠀⢉⣀⣉⣀⡸⠃⠀⠀⢠⣾⣿⣿⡇
⠀⠀⠸⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⢠⣿⣿⣿⠏⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢀⣿⣿⣿⡟⠘⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⢀⣿⢻⣿⡟⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⢀⣾⠻⣿⡟⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⣼⠏⠀⢻⣆⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⣼⠏⠀⢻⣇⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⣸⡟⠀⠀⠈⢿⡄⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⣰⡟⠀⠀⠀⢿⡆⠀⠀⠀⢻⣆
⠀⠀⠀⠀⣸⡟⠀⠀⠀⢰⡿⠀⠀⠀⠀⠘⣿⡀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⢠⡿⠁⠀⠀⠀⠈⣿⡄⠀⠀⠈⢿⡄
⠀⠀⠀⢠⡿⠁⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠹⣷⣀⣀⣀⣸⣷⡀⠀⠀⠀⢠⣿⠁⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠘⣷⣀⣀⣀⣘⣿⡀
⠀⠀⢀⣿⣧⣤⣤⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣾⣧⣤⣤⣤⣾⠇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠴⠾⠉⠉⠉⠉⠉⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠉⠉⠉⠙⠓⠒⠦⠤⣀⠀⠀⠀⠀⠀⠢⡀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡀⠀⠀⢹⠦⣀⠀⢙⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠛⢿⣿⣦⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠘⡄⠈⢹⠃⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠤⣄⣈⠙⠻⣷⡶⠠⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡇⠀⢸⡀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣤⠄⠀⠀⠈⠃⠀⠀⢹⡏⠀⠉⠉⠒⠢⠤⣄⣀⠀⠀⠀⠀⡇⠀⠀⠙⠢⠤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠀⠀⣠⣴⠾⠦⠀⠀⠀⠀⠀⠀⠰⠷⣶⣄⠀⠉⠙⠒⢤⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠿⠶⠶⠀⠀⢉⡖⠒⢦⠀⠀⠀⠀⠀⠀⣰⠒⠲⡈⠁⠀⠴⠶⠾⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠈⠧⠶⠾⠀⠀⢀⣀⠀⠀⠷⠶⠶⠇⠀⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠿⠿⠷⡶⠶⠶⠂⠀⠀⠀⠀⡄⠀⠀⠈⠀⠀⠀⢠⡀⠀⠀⠀⠐⠶⠶⠶⢾⠿⠿⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⣠⠴⠒⠒⠒⠒⠦⣄⣸⠃⠀⠀⠀⠀⠀⠀⠀⢧⣠⠴⠒⠒⠒⠒⠶⢄⡀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⢀⠞⠁⠀⠀⠀⠀⠀⠠⠀⠹⣷⣶⣦⣤⣤⣤⣶⣾⡏⢀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠶⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢸⠀⠀⠀⠈⠙⢦⠀⠀⠀⢀⠙⠻⢿⣿⣿⣿⠿⠋⠀⠀⠀⠐⡠⠊⠁⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⢣⠀⠀⠀⠀⠀⢇⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠂⠀⠀⠄⢠⠃⠀⠀⠀⠀⡼⠁⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠱⣄⠀⠀⠀⠈⠢⣄⣀⣁⣀⡠⢾⠛⢤⣀⣀⣀⣀⠴⠃⠀⠀⠀⢀⠞⣁⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠷⢤⣀⠀⠀⢸⠀⠀⡏⠀⢸⠀⢀⠇⠀⠀⡇⠀⠀⣀⣤⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢼⡄⠀⢃⠀⠘⠀⢸⠀⠀⢀⡧⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠲⠾⣄⠀⠀⡼⠤⠖⢋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣬⡙⢋⣠⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠟⠋⠀⠀⠉⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⣶⣶⣶⣾⣶⡿⢶⣶⣶⡶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡿⣷⠶⠿⠶⠿⢶⣶⠶⠶⢿