I love your Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⢰⣭⠑⠀⣠⡄⢹⠀⡆⢠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡾⠁⠀⢸⡟⠘⠁⣿⠇⠈⠲⠐⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠣⢀⠀⠈⠃⠁⠀⢻⠀⡐⠁⠀⠀⠁⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢳⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⣰⡆⢀⡤⠂⠄⢀⣤⠀⣠⡄⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⡀⢹⡇⢸⣾⡇⡤⢼⡏⢰⣿⠃⢀⢃⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⣾⡇⠀⣿⡠⠁⠸⠇⠁⡟⢀⠂⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⢄⡀⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣀⣼⠋⠀⠀⠀⠀⠀⠀⠀⢀⡀⣸⡟⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠁⢘⣷⡀⠠⣴⡦⠀⠀⢰⡟⢡⣿⠃⠀⢰⡟⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢸⡿⠀⠀⣿⠃⠀⠁⠈⠃⢸⡿⠀⠀⠸⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⣠⠟⠁⠀⠀⠈⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⡀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣰⠟⣿⠁⠀⠀⠀⠀⠀⠀⢸⡟⢻⣴⠟⠙⣿⠆⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡀⢹⡄⠀⠀⠀⢠⣟⡼⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠋⢀⡴⠋⠀⠀⢸⡇⠀⠀⢾⡿⢧⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡙⠋⠁⠀⢀⣠⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣠⠞⠁⠀⠀⠀⠀⢿⣄⡀⠀⢀⣼⠀⠀⠀⢰⡟⢀⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠷⠶⠛⠋⠁⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⣀⣼⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠛⠁⠀⠀⠀⠈⢧⡈⠃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⣠⠎⣁⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⣿⣄⣙⡇⣿⠀⠀⠀⠀⡴⢰⡟⣷⢰⡆⠀⢠⣦⣴⠞⠛⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠉⠉⠀⢻⡆⠀⠀⢸⠇⠀⢨⡟⢹⡇⣰⠏⢨⡟⠲⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠲⣦⠀⠀⢻⡟⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⣤⡿⣧⡴⠟⠀⢸⡿⠃⠀⠘⢷⣄⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⠃⣿⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣌⠉⠁⢀⡠⠋⠀⠀⠀⣤⡄⠀⠀⢠⡄⠀⣀⡤⠤⣄⠀⢠⡄⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⢠⡏⠀⠀⢸⡇⢸⡇⠀⠀⣿⠀⠀⠀⠀⢀⡠⠔⠒⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠒⠦⣄⠈⠻⣦⡤⢾⡇⠀⠳⣄⣠⡾⠃⠈⢷⣄⡠⠿⢦⣀⡠⠔⠉⠀⣶⡟⠛⢦⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣾⡻⠷⢸⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⣀⣫⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⡈⠙⠚⠁⠀⠀⠀⠀⣼⠃⠀⠀⢀⡴⠖⠛⠛⠲⣦⠀⠀⠀⣀⣤⣤⣀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⠤⠤⠐⠊⠁⠀⠀⠀⣿⠁⠀⠀⠀⠀⠈⢧⡴⠋⠁⠀⠀⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠀⣀⣄⣀⠀⠀⠀⠉⠉⠒⠦⣀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠿⠗⠈⠃⠀⠀⠀⠀⠀⢀⣀⡑⢄⠀⠈⠻⣶⡀⠀⠀⣀⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣄⣀⣠⠄⠀⠀⠀⠀⠀⠰⣇⠘⠁⣸⠀⠀⠀⠈⢿⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⣶⠶⠶⣶⣄⢻⣷⠀⠀⢰⡿⢰⣶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢠⡿⠁⠀⠀⢸⣿⠘⣿⡆⢠⣿⠃⢸⣧⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⢸⣇⠀⠀⢀⣼⡟⠀⢻⣿⣿⡏⠀⢸⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠇⠻⠷⠶⠿⠋⠀⠀⠈⠿⠟⠀⠀⠸⠷⠶⠾⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⢠⣾⡷⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣤⣿⡟⣠⣶⠶⢶⣦⡀⣿⡇⠀⠀⢸⣿⠀⣠⣴⠶⢶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⣸⡟⠁⠀⠀⢹⣇⢸⡇⠀⠀⣸⣿⢠⣿⣀⣤⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⣿⡇⠀⠀⢀⣿⠇⢸⣷⠀⢠⣿⣿⢸⣿⠿⣿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠘⠿⠶⠶⠿⠋⠀⠈⠿⠿⠿⠋⠿⠎⠿⠀⠈⠻⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠆⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣇⠀⠀⠀⣤⣶⣶⣶⢸⣿⠀⠀⠀⢀⣤⣤⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⠀⣿⣀⣠⣄⢸⣿⠀⠀⠀⢸⣿⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⣸⠃⣿⠛⠉⠉⢸⣿⠀⣀⡀⢸⣿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡾⠟⠀⣿⣤⡶⠶⠸⣿⡿⠿⡿⢸⣿



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⢀⠤⢄⠀⠀⠀⢠⡇⠀⡆⠀⣠⡴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⣷⡶⢈⡀⢴⣶⠀⣇⢰⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⢾⠇⡘⠀⢸⡇⠀⢈⠋⠁⢸⡇⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠠⣀⠀⠈⠉⠀⠀⢸⡇⠠⠂⠀⠀⠀⠉⣤⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠻⡀⢨
⠀⠀⠀⠀⠀⠠⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⣶⡆⠀⡠⠤⣄⠀⢀⣀⠀⣠⡄⠀⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⡐⣿⡇⢸⣵⡆⢈⣀⣾⠇⢰⣿⠇⠀⡜⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠘⢀⣿⡇⠈⣿⠉⡙⠀⣿⠠⢹⡿⠀⡘⢠⣿⠃⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠙⠊⠀⠀⠉⠀⠸⣇⠔⠀⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⣼⠏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣠⣴⠖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⠁⣀⠀⠀⠀⠀⠀⠀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣾⡇⡠⠁⢀⠀⢀⡀⠀⠠⠄⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⡀⠀⠀⠀⣿⠃⠁⠀⣿⡆⠀⢠⣴⣦⠀⠀⠀⣾⠏⣹⣿⠃⠀⢠⣿⠀⢾⡇⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⣿⡏⠀⠀⣿⠃⠀⠂⠈⠻⠀⢸⡏⠀⠀⠸⡇⠀⢸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣼⠏⠀⠀⠀⠻⠄⠈⠀⠀⠀⠀⢸⡇⢀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⡀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠶⠶⠶⠶⠶⣶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢀⣾⠋⠀⠀⣀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⣴⠶⠛⣛⣿⠆⠀⠀⠘⣷⡾⠁⠀⠀⠈⠻⠶⠶⢦⣤⣉⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣾⣯⣤⠶⠛⠉⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⡾⢻⣇⡀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠛⠷⠶⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⠏⠀⠀⠈⣩⡟⠀⠀⠀⠀⠀⠀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠋⢉⣉⣡⣴⠶⠛⠶⢦⣼⠇⠀⠀⢀⣠⣴⠞⠋⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠠⣶⡶⠶⠶⠶⠶⠶⠶⠞⠛⠛⠉⣥⣄⠀⠀⣀⣀⣠⣤⠶⠞⠛⠉⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⢴⡟⠋⠈⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⡀⠀⠀⠀⠀⠀⠉⠻⣦⣀⣀⣀⣀⣘⡷⣶⣾⣛⣛⣛⣻⣿⣿⣡⣤⣤⣤⣤⣤⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⣤⣄⠀⣠⠀⢠⡄⣠⣤⡀⠀⠀⢠⡀⢰⡀⣠⣤⣀⡄⣠⠶⣤⠀⣤⣤⡀⢰⡄⠀⣰⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡷⢿⠟⢹⣇⣿⠀⣿⣼⠇⠀⠀⠸⣧⣿⡇⣹⠞⣿⠀⠙⢦⣄⢸⣧⡴⠃⢸⠀⠶⢿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⠌⠳⠟⠀⠈⢿⠇⠀⠻⠶⠄⠀⠀⠰⣤⠾⠁⠻⡶⠟⠀⠷⣤⠟⠈⠳⠶⠂⠸⠷⠂⠸⠂


⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⡠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡟⢀⣠⣴⣾⠉⢹⡆⢀⣤⡀⣸⣾⠋⢹⡇⠀⠀⠀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⣿⣿⡟⣰⣿⣿⣿⣇⠀⣸⡇⣸⣿⣿⠹⣿⡄⣸⠇⠀⣠⣾⣿⡿⠟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢯⣿⣿⡿⣽⢻⣿⡿⠁⠹⣦⢻⣡⣿⣿⣿⠀⠙⢿⣿⣀⣞⣽⣿⡟⠁⣠⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣾⣿⣿⠃⣿⣿⣿⡇⠀⢠⣿⠋⢹⢹⣿⡏⠀⢠⡿⠉⣿⣼⣿⣟⣠⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡏⠀⠻⣿⣿⣇⣴⡿⠃⠀⣾⣿⣿⡇⣠⡿⠁⠸⣿⣿⣿⠛⠛⣹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⢸⣿⣿⣴⡿⠁⠀⠀⢻⣿⣿⣤⣾⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⣗⣒⣶⣿⣿⣿⣶⣶⣄⠀⠀⠀⠙⠻⢻⣴⣶⣄⠀⠀⠙⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣿⠂⠀⠀⠀⠀⠈⠻⢶⣬⣷⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡉⠀⠀⠀⢀⠀⠀⠀⢀⣴⣤⣄⠀⢠⣾⣿⠀⢰⣿⡆⠈⣿⡏⢹⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⡄⢀⣾⣿⡆⣰⣾⣿⣀⣿⣷⣏⣿⡏⢀⡿⣿⡇⢰⣿⣱⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⢡⣿⣿⡇⣼⣿⣿⣷⣿⡇⠈⣹⣿⢻⣿⣿⢡⣾⣇⣿⢠⣿⣿⣳⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⣼⣿⣿⣰⣿⣿⣿⢿⣼⣇⣴⣿⠏⠘⢿⣿⣿⠟⢿⣿⣿⢿⡇⣿⡟⠀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠀⣿⣿⣿⣿⠃⣿⢻⣄⣙⡛⠛⣡⣤⣄⠀⠉⠁⠀⠀⠉⠁⢸⣿⣿⠁⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⢃⣴⣿⣼⠛⠋⣱⣾⡿⠋⣹⡧⠀⠀⠀⠀⠀⠀⠸⣿⣿⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢿⣿⡟⢀⡾⣿⠏⠀⣴⣻⠃⠀⠀⠀⠀⠀⠀⠀⣬⣟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣻⠃⣾⣿⠇⡞⣽⠏⣠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣏⣰⣿⠏⣼⣼⣿⣾⡿⠋⣰⡿⠛⢻⣆⠀⣠⣴⣶⡿⣿⡇⢀⣾⣷⠄⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠋⢰⣿⣿⠟⠋⠀⣼⣿⣟⠀⣸⣿⣼⣿⡟⣻⣷⣿⠁⣼⣿⣿⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣏⣤⣀⣸⢻⡟⠉⢻⣿⣿⣽⡏⢠⣿⢸⣿⠰⣧⣿⡏⣠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣿⣿⡏⢛⣿⣿⣺⡇⣠⣿⠏⣿⣿⢀⣾⣿⣾⡇⠀⣿⠿⣿⠿⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⣿⣿⡇⣼⡿⠘⢿⣿⡿⠋⠀⠹⣿⣿⠟⢹⣿⡇⣸⡟⠀⠀⠀⣼⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠘⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣴⣶⠾⢛⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠁⠀⣼⣿⡿⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡀⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠿⠛⠛⠛⠛⠶⢦⣀⠀⠀⣀⣤⣶⡶⠶⠶⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⣀⠀⠀⠀⠀⠀⠙⣦⣾⠟⠉⠀⠀⠀⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⢠⣾⢫⠇⠀⠀⠀⠀⠀⠈⠁⠀⣀⠀⠀⠀⣠⣶⣤⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⢀⣿⠟⠁⣠⣴⡶⠶⢶⣄⠀⠀⠀⣿⡄⠀⣸⡟⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡷⠀⠉⣹⡟⠀⢰⡿⠋⠀⠀⠀⣿⡆⠀⠀⣿⣇⢰⣿⠀⣠⠟⠁⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⢠⡿⠁⠀⢻⡇⠀⠀⠀⢠⡿⢁⣴⡆⣿⠉⢺⡷⠚⠁⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⢀⣤⣤⣿⡁⠀⠀⠈⠣⣄⣀⣠⡾⠗⠋⣿⢁⣿⠀⢸⡇⠀⣸⠇⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⢿⣵⠟⠁⠙⢦⣀⣀⣤⡴⠟⢉⠀⣤⢠⣿⣸⠇⠀⠈⠳⠶⠋⠀⢿⣄⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢤⣀⠀⠀⠀⠀⠀⡆⠀⣤⣠⣞⣶⣟⡼⡏⠈⠿⠏⠀⢀⡾⡧⣾⣷⠀⠈⣿⡆⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⣀⣀⠧⠎⣽⠹⠿⠙⠟⠀⠙⠄⠀⣶⡀⣼⡷⠃⢨⡟⣶⡄⢸⣷⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⢠⠶⢾⡟⠉⠁⠀⣾⡟⡶⡦⠀⢀⣴⡄⣶⠀⢹⢶⡟⠀⢀⡞⣰⡿⠀⢸⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⣿⠃⠀⠀⣤⣌⠀⣻⣇⠀⣾⡽⢣⣿⠀⡿⠸⡇⢀⡞⢱⡿⠁⠀⣸⡏⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⢀⣿⠛⠋⣰⠧⣿⠞⢡⡟⢰⣟⣠⠏⣿⣼⠃⠀⠙⠋⠀⣿⠃⠀⢠⣿⠁⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⣿⢠⡏⢀⡿⠁⣾⠈⠁⠀⠘⠃⠀⣀⣤⡀⢸⣿⠀⢠⡾⠃⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠸⣿⠀⠀⠻⠞⠁⣼⠃⣰⠃⠀⣴⡶⢤⡴⠾⠛⣿⠁⠘⠿⠴⠟⠁⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠛⠀⢀⣤⡀⠀⢿⡴⠃⠀⣰⡿⠀⠀⠀⠀⠀⠻⣇⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⣀⣀⣠⣤⡶⠟⠹⣧⡀⠀⢀⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠙⠛⠛⠉

⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⡗⠀⡆⠀⢸⠀⠀⠀⢀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣠⠀⠀⡄
⠀⠀⠀⡇⢸⡆⠀⠏⢸⡀⡄⡼⢸⡇⠁⠀⠀⠀⡇⠀⡇⠀⡀⠀⢸⡆⢸⠁⠀⠀⠀⠀⠀⣺⠀⡰⢢⢰⣢⡇⡟⠁⠀⠀⠘⠁⣿⠀⢸⠇⢀⣤⡀⢀⣀⣄
⠀⠀⢰⡇⢸⣇⣀⠆⠰⠀⢻⠇⢸⡀⠀⠀⠀⠀⠃⢀⠃⠀⢋⡀⢈⠀⠘⠂⠀⠀⢸⡾⠀⠹⡼⠣⣏⠀⠛⣀⠈⠁⠀⠀⠒⠴⠋⠀⠛⠑⠛⠚⠙⠋⠻⠗
⠀⠀⢀⣀⠠⠀⠀⠠⠒⠀⡄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠳⠄⡼⠀⢘⠀⠀⠀⠀⠀⠀⠀⠣⣴⣆⢮⣩⠟⢥⠿⡄⠀⠀⠀⠀⠀⠈⢦⣿⣠⢄⣠⢠⡄
⠀⠀⠈⠛⠀⠘⠀⢀⠀⠀⠃⢀⠄⠙⠃⠀⠀⠀⠀⠀⠀⠀⢁⡀⠠⡄⠀⠀⠀⠀⠀⠀⡤⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠋⠓⠞⠙⠚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣤⣤⠄⡤⢤⡀⡤⠀⠄⢠⠤⠀⠀⣤⠀⠀⢠⢦⡠⡄⢀⠄⡤⠤⠀⠀⢀⡟⠀⠀⢰⠏⢹⠀⣿⢀⡿⢀⣏⡉⠁
⠀⢠⣠⢰⠊⢹⠀⢀⡆⠀⠸⣿⠀⠀⠀⠀⢸⡇⠀⡟⢯⠀⣷⢰⡇⣼⠒⠀⠀⣿⠀⠀⣯⢨⡇⢻⡾⠀⡷⠂⠀⠀⢸⣇⣀⡀⢾⣀⡟⠀⣿⡾⠁⣸⣉⡁
⠀⠀⠋⡀⠀⠈⡆⢸⠀⡄⢠⡆⢤⡖⠀⠀⠈⠁⠀⠃⠘⠂⠘⠋⠀⠘⠒⠀⠀⠛⠛⠃⠘⠛⠀⠈⠁⠀⠛⠓⠀⠀⠈⠉⠉⠁⠈⠉⠀⠀⠈⠁⠀⠈⠉⠉
⠀⠀⢎⠈⠀⣰⠁⢸⡼⠧⠈⠧⠂⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⢑⡚⠁⡆⢀⣠⠂⡀⢠⠄⢲⠃⠀⢀⠠⣗⠀⠀⢀⡾⢷⡄⢿⡀⢀⡿⠀⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠁⠀⠷⢻⠹⠎⠳⠋⠂⠀⠀⠀⠀⢰⡇⠀⠀⢸⡇⢸⡇⠘⣷⣸⠇⠀⣿⠶⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡆⢀⠆⠀⠈⠛⠀⠀⠀⠀⠀⠀⠸⣷⣤⣤⠘⢧⡾⠃⡄⢹⡿⠀⠀⣿⣤⡌
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⡴⠦⠀⠀⠀⠀⠀⢰⣿⠀⠀⢠⣾⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⡆⢰⡄⢿⣍⠀⠻⢶⣄⠀⠀⠀⠀⢸⣿⡀⠀⢸⡇⢀⣿
⠀⠀⠀⠀⠀⢸⡇⣰⠋⠑⠞⠙⣦⠹⣆⣸⢇⡞⠉⢷⢀⡇⠀⢿⡀⠀⠀⠀⠀⣿⣿⡀⢸⡇⣄⢈⣿⢤⡶⠟⠀⠀⠀⠀⠘⠟⠟⠟⠋⠻⠿⠋
⠀⠀⠀⠀⠀⢸⡇⠸⣄⠀⠀⣰⠃⠀⢹⡟⢸⡇⠀⢸⠿⣧⠀⣸⠃⠀⠀⠀⠀⣿⠆⠑⠈⠁⢈⡉⢥⠰⣖⣿⠀⠀⠀⠀⢸⣧⢀⣿⠃⣾⡟⠓
⠀⠀⠀⠀⠀⠸⠇⠀⠈⠓⠾⠁⠀⠀⠿⠀⠈⢷⣤⠟⠀⠻⣴⠏⠀⠀⠀⠀⠀⢰⣴⢹⠀⡟⣿⣿⣏⠄⠙⠃⠀⠀⠀⠀⠈⣿⣾⠇⠀⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⠛⠛⠓
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠄⢶⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡇⠀⠀⠁⠀⢠⠋⠙⡄⣆⠀⡔⢘⡁⠁⠀⠈⠢⡰⠁⠆⠀⢃⢰⠀⢸⠀⠀⠀⠀⠀⡀⠀⡿⠀⢠⠇⢀⡘⣳⣆⡼⣞⣡⠄⠀⢧⡦⣏⡿⣧⣾
⠀⠀⠠⠄⠀⠠⡄⣀⠘⣀⡠⠁⠘⡜⠀⢀⣀⡀⠀⠀⠀⠃⠀⠘⠤⠂⠀⠑⠃⠀⠀⠀⠀⠀⠉⠚⠁⠀⠉⠐⠋⠉⠁⠈⠀⠈⠁⠀⣀⡼⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠰⠀⠀⠀⠈⣿⡏⠀⠀⠀⢠⣿⢿⣧⠀⠹⣷⠀⣸⡏⠀⢸⠟⠉⠛
⠀⠀⠀⠀⠀⢸⠀⠀⢰⠀⠀⠀⠄⡇⠀⠱⣼⠀⠰⢀⠁⠀⠀⢁⠁⠀⡆⢸⠀⠠⠀⠀⠀⠀⠰⣿⡇⠀⠀⠀⢸⡇⢸⣿⠀⠀⢿⣄⣿⠀⠀⢸⡷⠺⠇
⠀⠀⠀⠀⠀⠼⠀⠀⠘⠀⠄⠀⣠⠇⠀⣠⠇⠀⠐⢀⡀⠀⠀⠀⠀⠐⠤⠊⠀⠀⠈⠀⠀⠀⠀⣿⣧⣀⣠⡆⠘⣿⣿⡏⠀⠀⠘⣿⠃⠀⠀⢼⣧⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠈⠉⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠈⠁⠲⣄
⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣾⣿⡿⠿⠟⠁⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⣿⣿⠀⠀⠀⠀⠀⠀⢠⣿⢻⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⢠⣿⠃⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠⣴⣤⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⣾⡇⢠⡿⠀⠀⣠⣴⣤⣤⠾⢻⣿⠆⢰⣿⢇⣴⡟⠁⣼⡇⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢠⣿⢃⣾⠃⠀⣴⡟⠛⢹⣿⠀⣼⡿⠀⣸⣿⣾⣿⠀⣰⡟⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡾⠃⠀⢸⣿⠀⠀⣾⡏⢠⣿⠁⢠⣿⠋⣿⡇⠸⠋⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⠁⠀⢠⣾⣿⠀⣸⡿⠁⢸⣿⣰⣿⠃⠀⠸⣷⣠⣤⠾⠋⠀⠀⠀⠀⠀⢠⡶⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⠀⣰⡿⠈⣿⣶⡿⠁⠀⠘⠿⠟⠃⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⣴⣦⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣧⣶⣾⠟⠀⠈⠻⠿⠛⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠻⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⣄⣀⣀⣠⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣼⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠴⢾⣿⠀⣾⢇⡤⢹⣿⣀⡤⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠆
⠀⠀⠀⠀⣀⣤⡀⢠⡄⢠⣬⠁⣿⡇⣾⡏⢠⣿⣼⠃⣾⣿⠋⠀⡾⠀⠀⠀⠀⠀⢲⣷⡦⠀⠀⣰⣿⠃⡏
⠀⠀⠰⠞⠉⣽⡇⢸⡇⢸⣿⢠⣿⠁⣿⣧⡟⣿⡏⠀⠻⠃⠀⢰⠃⠀⠀⠀⠀⠀⣿⡟⠁⠀⢀⣿⡇⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⣿⡇⣾⡇⠟⠻⡿⠏⠀⠙⠋⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣸⣿⠀⠀⠀⢀⣾⡿⠁⣿⡇⠀⢠
⠀⠀⠀⠀⠀⠻⠟⣿⠇⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⣤⡀⠀⣴⣾⣦⠀⢀⣿⣿⡇⠀⢀⣿⡇⠀⠀⠀⣾⣿⠁⢠⣿⠁⠀⡞
⠀⠀⠀⠀⠀⢀⣠⣿⠶⠖⠂⠀⣴⣿⣿⢃⣠⣴⣶⣤⣿⠋⣿⣾⡟⢩⣿⠀⢸⣿⣿⠃⠀⣸⣿⠃⠀⠀⢸⣿⣿⣠⡿⠃⢀⣼⠃
⠀⠀⠀⣠⠶⠋⢩⣿⠀⠀⠀⠀⣿⣿⠁⠈⠉⠀⣿⣿⠋⠀⣿⡟⠀⢸⣿⠀⡸⢻⣿⠀⢰⢿⣿⡇⠀⣴⡿⣿⣿⠋⠀⣠⡿⠃
⠀⣠⡞⠁⠀⠀⢸⡏⠀⠀⠀⠀⠘⣿⣦⠀⠀⢠⣿⡟⠀⢰⣿⡇⠀⣾⡟⣰⡇⢸⡏⢠⡟⠈⣿⣧⣾⠟⠁⠙⠛⠷⠞⠋
⣸⠏⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠸⣿⣇⠀⢸⣿⠇⠀⢸⣿⠃⠀⢿⣷⡟⠀⠸⣿⠿⠁⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⣿⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⢸⣿⠀⠀⢼⡿⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣶⣶⠾⠿⠿⠛⠛⠉
⠈⠓⠶⠒⠛⠁⠀⠀⠀⠀⠀⠀⣦⡀⢸⣿⠇⠸⠿⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⡶⠿⠟⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠃⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⠾⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠟⠛⠉⠁
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣀⣀⣴⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠋⠛⢿⠿⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣃⠀⠀⠡⠂⠀⠉⢹⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡅⠂⠰⠄⣠⠀⠀⣾⣿⣿⣿⠀⢸⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣄⠐⠛⠋⠀⢀⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⠛⠻⢷⡖⠐⠂⠈⠹⡀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⢰⡧⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠂⠀⢨⠁⢠⢷⡀⠀⠣⡀⠀⠀⠀⢀⠇⠀⠀⡤⠊⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡒⠬⠄⢳⠀⠀⠉⠂⠤⠐⠂⠀⠀⡎⢀⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠑⢄⠀⠀⣶⠂⠀⠀⢠⡵⢵⢟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡓⠒⠢⢄⣀⠀⠉⠢⢄⠀⠀⣀⠜⠁⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠄⠓⠐⠒⠒⠋⠉⠀⣀⡠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⡠⠤⡒⠈⠁⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠣⠤⠔⠂⠉⠀⠀⠀⠈⠑⠢⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢀⠂⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⢀⡿⠐⠋⠈⠀⠀⠀⢀⠂⠀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠰
⠀⠀⠀⠀⠀⠀⢰⠋⡾⠁⢸⡇⠀⠀⠀⡀⢀⠀⡀⢀⡀⠀⠠⢲⡆⠀⡀⢐⡀⠐⣀⡖⢠⠝⠆⣄⡄⠀⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣿⡇⠄⣰⠆⢠⣿⠠⢠⡟⠀⠀⠰⢸⡇⢸⡇⣸⠃⠘⣿⢁⣿⠀⡀⣼⠃⠀⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢀⣿⢀⠃⠸⡆⢻⠁⠀⠀⠀⠸⣇⢺⡇⠘⠀⠀⠘⠀⠙⠂⠀⢿⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠊⠈⠀⠀⠀⠀⠈⢁⡤⠀⠄⠀⢀⠊⣷⠠⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠎⢀⡇⡆⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡿⠀⠀⠀⠀⠀⠁⠈⠀⢰⡿⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⢸⡇⠀⠀⠀⡀⢀⠀⢀⡤⢼⡏⠀⣠⠁⣄⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠊⣧⢘⣶⠈⡆⣾⠃⢺⠃⢰⡇⢰⣿⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⣿⠈⣇⠠⠀⠹⡠⠻⣄⠜⠓⠱⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠠⠏⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⢀⠀⠠⠂⠰⠦⣷⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣤⢄⣦⠄⠀⠀⠀⠀⠀⢠⣿⢿⣰⡀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠆⠀⣶⠀⠀⠀⠀⣀⣤⣸⣿⣶⡟⣿⣯⠳⠀⠀⠀⣸⣿⣯⠀⠈⠉⠀⠀⠀⠀⢀⡀⠸⢿⣿⠿⠁⡴⠒⠰⠆⣴⣰⡄
⠀⠀⠀⠒⠀⠀⠊⠀⠀⣤⡈⣿⣥⣼⣿⣼⣶⣶⡿⠀⠀⠀⣬⣿⣿⣿⣆⣀⠀⢀⡀⠶⣿⡟⠛⠀⠀⠈⠀⠀⠀⠀⠀⠀⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⡛⠿⣛⠉⠀⠀⠀⠀⠀⣰⡆⠀⠼⠿⡿⠻⡟⢄⠀⢁⠾⢿⣿⣷⣦⠀⠀⠀⠠⠄⠤⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣆⣈⣅⢀⣤⡞⣰⠶⠋⠀⠀⠀⠀⠑⠀⠀⢻⣿⡟⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢺⣿⣷⡄⠈⠀⠹⠯⡀⣀⡬⠀⠁⠀⢠⠀⢠⢦⡀⠀⢤⢤⠢⡉⠀⢠⠀⠀⠴⣄⣀⠀⡄
⠀⠀⠀⠀⠀⠐⠈⠻⣿⠁⢀⡀⣀⣤⣖⢴⣶⠄⠀⠀⢸⠀⢰⠀⠁⡂⠀⢸⠄⠀⠀⠸⣀⡇⡀⢈⢸⠀⡇⠀⠀⠀⠀⠀⠀⢠⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢡⣾⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⢸⠀⢸⠀⠀⡀⢀⢸⠀⡇⠀⢠⠀⡇⡅⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠰⠿⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠿⡿⠁⡇⠉⠉⠀⠀⠀⠈⠉⠀⠉⠀⠈⠁⠀⠉⠀⠀⣤⣍⣀⠈⠁⠀⠉⡑⢾⣦⣼⣷⣤⣄⠀⣀⣤⣤⣄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀⠱⠛⣃⠄⣠⠄⠈⠁⢾⣿⢿⣏⠻⠿⠀⠸⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠉⠢⠿⠀⠀⠀⠀⠀⠈⠀⢀⠀⠁⠀⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠲⠠⠄⠀⠀⢠⣿⡀⠳⠦⠀⠶⠶⠟⠀⢰⣿⣿⡍⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠉⠀⠀⠀⠀⢠⣴⡆⠀⠙⠻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀⡀⢀⡠⠔⠒⢒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠠⠄⠀⠀⣀⠀⠀⠺⡯⠍⠀⠀⠀⠀⠀⠀⠉⡩⠛⠉⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⣾⠃⠀⠀⠀⠀⠀⢀⡾⠋⠀⢠⠤⣄⠀⠀⠀⠀⠀⠀⢠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⣠⡟⠁⠀⢀⣼⠗⡞⠈⢉⣿⠋⢀⡔⠁⢀⣴⢖⠆⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⣰⠏⠀⠀⡠⣾⢏⡔⠁⢀⡾⠁⣠⠎⠀⢠⡿⠑⠉⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⢰⡏⠀⣠⠊⠀⠙⠋⠀⠀⣾⣡⡾⠁⡀⠀⢿⡀⣀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣾⠇⠒⠒⠒⠂⠀⠹⠤⠞⠁⠀⡠⠀⡀⠀⠀⠈⠁⠀⠘⢁⡀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⢀⣄⠀⣠⣤⠀⣠⣧⣠⡇⣴⡆⢀⣴⠆⢀⠌⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⡡⢊⣿⠃⠨⣿⡵⠃⢰⣯⢴⡿⠁⡠⠊⣴⠟⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠟⡊⢰⡿⠁⠀⠀⠈⠀⠀⠀⠀⠘⠓⠊⠀⣼⠃⠀⠀⢀⡜⠁⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⢀⡴⢊⣴⠟⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⢻⣀⣠⡴⠋⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠉⠀⠀⠸⠴⠟⠁⠀⢀⠔⢹⡆⢀⡴⠚⢁⣾⠃⠀⢀⠀⠀⠈⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⣰⡆⠀⠐⠁⢠⣿⡵⠋⠀⢀⣾⠃⠀⠀⡀⠀⠀⠀⢠⡾⠀⠀⢀⣀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡃⠀⠀⠀⠀⠀⠀⠤⢺⡿⢁⠌⠀⠀⣾⠟⠀⠀⠀⣾⠃⠀⠀⣾⠇⠀⠀⢤⡦⠀⠀⣴⡿⡿⠃⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢦⡄⠊⠀⠀⣾⠇⡄⠀⠀⠈⠛⠀⠀⠀⣰⠇⠀⠀⢸⣏⡤⠒⠀⢾⠀⡀⢺⡏⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢿⡼⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠉⠀⠀⠀⠈⠋⠁⠈⠳⠤⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠⡟⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⣀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢁⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠒⠒⠲⠦⠤⣀⡀⠀⠀⠀⠀⠈⠶⠋⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠉⠁⠀⠀⠒⠀⢄⠙⢶⣤⣤⣀⡀⠀⠀⠀⢏⡼⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⢀⣀⣤⣤⡤⠴⠼⣷⠢⠀⠀⠈⢉⠳⣄⠀⠀⠁⠀⠨⢈⡽
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⢊⣤⠞⠋⠁⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⢀⣠⣯⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠗⠛⠺⠿⣝⣧⣿⣀⣀
⠀⠀⠀⣀⡴⢾⠟⣾⢿⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢸⠀⡎⣙⡶⠚⠉⠉⠝⠉⣛⠷⢦⡤⠤⠤⢤⣤⡀
⠀⣠⠞⠁⠐⠁⢰⠻⣜⣦⠀⠀⠀⢤⣀⠀⠨⣁⠐⠆⠀⠀⠸⣄⠟⠉⠚⠂⠀⠘⠋⠀⣤⠀⣠⢻⡍⢉⠗⣸⠃
⠸⣇⠀⠀⠀⠀⢆⢰⡏⠙⢣⡀⠀⠘⡦⢭⣒⠤⣁⡀⠀⠀⢰⠏⣀⡀⠀⡠⠤⠼⣁⡓⠀⠀⠁⠀⢻⣡⠞⠁
⠀⠈⠳⣄⡀⠀⢸⢸⡇⠀⠀⠉⠳⢤⣈⠳⠤⠭⠶⠞⠓⠀⣿⠒⠉⠀⠀⠳⡀⠀⠀⠈⡷⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠉⠛⠛⠚⠁⠀⠀⠀⢰⠟⠛⠛⠳⣶⣶⠤⠤⠴⠻⢦⣀⠀⠀⠀⠈⡖⠒⠉⠀⠀⠹⡐⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⢻⠦⢄⡀⢀⣭⢿⡶⠶⢤⣤⣤⣤⣄⣤⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠃⠀⠀⠀⠀⢀⡞⠀⠀⣹⠏⠻⠀⢉⣠⣄⣀⣠⣀⣠⠟⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠰⣄⠀⣠⠊⠀⠀⠀⠻⣇⡀⣠⠟⠻⡄⠈⠃⠈⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⡹⠦⣄⡀⠀⠀⠈⠉⠁⠀⢠⡇⠀⠀⠀⢀⡾⠃⠀⠀⣀⡤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠱⢤⣀⣉⣛⡶⠤⣄⣀⣤⣞⡛⣿⢶⣿⢛⣧⣀⣠⡾⢇⣾⣀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⠋⢳⠀⡜⢧⡠⡏⠳⡄⢀⣼⠋⠤⣾⠈⢿⡳⢄⣽⣅⠴⠛⠙⣏⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠈⠓⠋⠀⠓⠞⠁⠙⠚⣰⡏⠀⠀⢀⣾⣤⠾⠃⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠛⠒⠒⠶⠶⠶⠤⠤⠴⠶⠶⠛⠷⠶⠶⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⠛⠲⠶⠤⢤⠤⣤⣤⣤⣤⣤⡤⠤⠶⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠀⢀⣀⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢀⣄⣀⠀⠀⠘⣧⠀⢻⠿⣋⣭⡉⠀⣈⣙⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣠⠴⠖⣚⣻⣟⡲⢦⣄⣿⣴⠃⠘⢵⡿⠛⠉⠁⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⣀⣒⣒⡒⠆⠀⠈⢿⡇⢀⡴⠃⠀⠀⠀⠀⠀⠀⣀⡼⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠒⠉⠉⠁⠀⠀⠈⠉⠉⠒⠶⣿⡞⠀⠀⠀⠀⠀⣀⡴⢞⣩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣶⡶⣶⣒⣻⡿⠷⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠯⠭⠭⠭⠭⠭⠽⠟⠋⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⣰⣏⣠⣿⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣿⣷⡾⢽⠿⠷⠾⠿⠷⣴⣮⣭⡏⢍⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣽⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠘⠏⠛⠦⣝⡳⢤⡉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⣧⣬⠲⣄⠑⣄
⠀⠀⠀⠀⠀⠀⠀⡠⣺⣿⡋⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⢇⠐⡕⢌⢧⡀
⠀⠀⠀⠀⠀⢀⡼⣵⠟⠁⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡌⢆⠑⢷
⠀⠀⠀⠀⢀⢾⣼⣓⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⡟⠐⢿⣠⠃⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⠄⠈⢧
⠀⠀⠀⠀⠂⢰⣿⣯⠁⠀⠀⠀⠀⠀⢠⠄⡀⠸⠇⠀⠉⠀⠘⠃⠀⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢹⠸⠀⠘⡆
⠀⠀⠀⠀⠀⢸⣟⣏⡄⠀⠀⠀⠀⠀⢸⡄⢁⣴⠀⢀⠀⢀⢠⡆⠀⢸⡇⠀⢺⠂⠀⠀⠀⠀⠀⠀⠘⣠⡷⡄⣷
⠀⠀⠀⠆⠀⢸⣏⣏⠀⠀⠀⠀⠀⠀⠀⣿⡁⠻⡤⢿⣀⡌⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⠀⣿
⠀⠀⠀⡀⠀⠸⣇⢻⠀⠀⠀⠀⠀⠀⠘⠘⠃⢀⠀⠀⠉⠀⠀⠛⠀⠈⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⠈⠟⠃⣿
⠀⠀⠀⠁⠀⠀⣧⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡶⣄⣸⡄⡔⣰⣏⠇⠀⠀⠀⠀⠀⠀⠀⢘⡿⡎⠀⡘⡼
⠀⠀⠀⠀⠀⠠⣶⣫⣇⠂⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⠧⠄⢸⣷⠁⠹⠧⠤⠀⠀⠀⠀⠀⠀⠐⣿⣷⠀⡰⢡⠇
⠀⠀⠀⠀⠃⠀⠘⢇⠘⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡣⠁⡔⢁⡞
⠀⠀⠀⠀⠀⠂⠀⠀⠢⡀⠙⢷⣆⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣺⡎⡠⠊⣠⠏
⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⠢⡀⠟⢫⡓⠦⣼⠦⢀⣤⢰⣄⣄⠀⠀⠀⠀⢀⣀⣤⠖⡫⢞⡧⠂⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠈⠑⡂⠌⣋⣁⡀⠿⠃⠞⢥⠾⠭⠍⢙⣛⠉⠅⣂⡡⢐⣉⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠒⠈⠙⠛⣋⣙⣿⢹⡯⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠛⠋⢸⠇⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠁⠀⠀⠀⠀⣿⠀⠀⠀⣠⡶⠶⣄⠀⣾⠁⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⠃⠀⣴⠏⢰⡇⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣿⠀⠀⠀⠀⣾⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣸⡇⠀⠀⠘⠷⣤⣀⣰⣧⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠋⠉⠉⠉⢹⡟⢶⣄⠀⠀⠀⠉⣽⠃⠀⠀⠀⢀⣴⠶⣤⠀⠀⣠⡀⠀⢀⣤⡀⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⣿⠀⠀⠹⠃⠀⠀⠀⣿⠀⠀⠀⢠⡿⣿⢦⣼⣷⠞⢻⡇⢀⡾⢻⣇⡾⢁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⡿⠀⣿⠀⣸⡇⠀⢸⣇⣾⠃⠀⢹⡟⠋⠁⣠⡶⠂
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⢠⡿⠁⡀⠀⠀⠀⠀⠀⠀⢻⡄⠀⡾⠁⠀⠘⠿⠋⠀⠀⠈⣿⠇⠀⠀⢸⡇⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⠾⠋⠀⠀⣿⠀⠀⣶⠀⠀⠀⠈⣧⣼⠃⠀⣦⣤⣤⡀⠀⠀⣀⣀⠀⠀⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠙⢦⣼⠇⣠⣤⡀⢠⡌⠁⡄⠀⣿⣤⣾⠃⠀⣼⠋⢹⡆⠀⠀⣠⡴⠞⠛⠙⠻⣆
⠀⠀⠀⠀⠀⢠⡾⠉⠉⠉⢻⡄⠀⠀⠀⠀⢸⡇⠀⠻⣼⠇⠀⠻⠾⠃⠀⠟⠀⠻⠀⣸⠇⠀⣼⣣⡴⠛⠁⠀⠀⣴⠖⠀⣿
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⠃⠀⠀⠀⣠⡴⠖⠳⣦⠀⠀⠗⠀⢀⣿⠀⣰⡿⠋⠀⠀⣠⣤⡄⠙⢦⡴⠏
⠀⠀⠀⠀⠀⢹⡆⠀⠀⢀⡾⠃⠀⢀⣤⣶⠀⢀⡴⣦⣰⠏⠀⠀⣰⠇⠀⣴⡄⠀⢸⡇⠀⠉⠀⠀⢠⣾⣉⣼⠏⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠹⣦⡀⠈⠀⣠⡾⠋⠁⢿⣠⡟⠁⣿⡏⠀⠀⣰⠏⠀⣸⣿⠇⢀⣾⡇⠀⠀⠀⢠⣿⠋⠉⠁⢀⣠⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡞⠉⠀⠀⠀⢸⣿⠁⠀⠿⠀⠀⢰⡏⠀⢰⡏⢸⣇⣼⢻⡇⠀⠀⢀⣾⢻⡆⣀⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠙⢦⡀⠀⠀⠈⠋⠀⠀⠀⠀⠀⣾⠁⢠⡟⠀⠀⠉⠁⠸⣧⠀⠀⣾⠃⠀⠙⠉
⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡟⠀⠀⠀⠀⠀⠀⢿⣄⣼⠃
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⡔⠀⣸⠁⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠠⡼⠀⡸⠁⢀⡏⠀⢠⠁⠀⠄⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢰⠁⠰⡃⠀⡸⠀⡰⢣⠆⠸⠃⠀⠀⠈⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⡏⠀⠷⠀⢁⡴⠊⢤⠎⡠⠁⠀⢀⠀⡎⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡄⠘⠀⣀⠤⠚⠋⠀⡴⠋⠀⢠⠂⢀⠆⣼⠇⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢁⠀⠊⠁⣠⡤⠐⠁⠀⣀⠔⠁⢀⠀⡔⡙⣰⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢐⠀⡀⠊⠁⠀⣀⠴⠋⠁⠀⠀⡪⠊⢄⡔⠁⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡊⠉⠁⠤⠀⠴⠦⣤⣀⡰⠚⠉⠔⠂⠠⠢⠴⢚⣠⠾⠋⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠓⠶⠖⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡄⢀⡠⠖⠀⢉⠝⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⡠⠤⠓⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠸⡷⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⣤⡀⢀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢀⣴⡆⠁⠠⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠒⠒⠂⠉⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠈⠉⣁⠥⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠰⡷⢿⣿⣿⣿⣿⣷⣿⣿⡿⠀⠀⠀⠀⠀⠛⠃⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢴⠤⠀⢀⣀⠀⠀⡈⠉⠉⠙⠁⣤⣤⢸⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡀⠀⠈⠀⢄⠑⢂⣀⠀⠉⠀⠀⢿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠠⡄⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠁⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠋⠀⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠄⠠⠄⠀⠀⢠⠂⠉⠉⠙⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⡀⡐⠒⢦⠀⠀⠀⠈⠻⣗⠒⠊⣡⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣿⠁⠀⠀⣇⣰⡏⠀⠀⢹⠀⣾⠁⠀⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⣿⠓⠒⢰⠃⢸⡇⠀⠀⡜⢸⣿⠂⠀⠀⠀⠰
⠀⠀⠀⢀⣀⣀⣀⡀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠻⣧⠴⠃⠀⢸⣇⣀⡔⠁⠀⠻⣦⡀⣠⠐⢁⡤⠤⠴⠦⢤⣄⡀
⠀⢠⡞⠉⠉⠉⠉⣩⡿⠿⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢀⡭⠒⠉⠀⠀⠀⠀⠀⠀⠈⠙⣦
⠀⠘⠷⠶⠶⠖⠋⠁⠀⠀⠀⠈⠛⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠶⠶⢤⣌⠙⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠈⡆⠀⠀⠉⠛⠳⠶⣤⣀⣠⡒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠋⠀⣀⣀
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠉⠉⠛⠛⠒⠒⠒⣲⣒⠒⠒⠚⠉⠉⠀⠀⠀⠀⠁⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⣀⠀⠀⠀⣀⣀⡠⠤⠒⠉⠀⠙⣷⠀⠀⠀⢰⡏⢠⠊⠀⠙⣧⠀⢠⡟⠀⠀⣿⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⡠⢻⠇⢹⣏⣀⠀⣹⡄⣾⡇⠀⢐⣿⡀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⠔⠁⣿⠀⢸⡏⠉⠉⣹⠃⢿⣇⡠⠊⠙⠻⠶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠚⠁⠀⣸⠇⠀⠸⣷⣄⣠⠏⠀⢀⣁⠤⠄⠒⠒⠒⠒⠒⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠒⠒⠒⠲⠶⠤⣤⣀⡀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠈⢉⣡⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⣰⠏⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠓⠦⣤⣊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠋⠁⠀⠀⢀⠞⠀⠈⠉⠓⠲⠤⠤⢤⣄⣀⣀⣀⣀⡠⠤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠒⠶⠖⠚⠛⠋⠁⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠔⠛⠛⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⡀⠀⠀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⣠⠞⠃⠀⡀⠀⠀⠀⢀⠔⢡⡤⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⢰⣆⢀⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⡇⠀⠠⠀⠻⠥⠤⢍⠀⢸⡏⠀⠀⠀⠀⠀⠉⣬⣶⠖
⠀⠀⠀⠀⠀⠀⠳⠤⠷⠀⢹⣷⢀⠀⠈⠁⠀⠀⠀⠀⣾⡇⠀⠀⠂⠀⠈⡀⠀⠀⠀⠀⠀⠀⠁⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠠⢶⠆⠒⠸⣿⡎⢀⢤⡄⠀⣾⠋⠀⢿⡇⠀⢸⡇⢠⣿⠩⠀⣶⡀⠀⣒⠀⣀⡀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠂⠹⣷⠂⢸⣿⠴⣿⣀⠠⠺⣧⠀⠘⣇⣈⢿⡀⠀⢹⣇⠀⠙⣾⡟⠉⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⡿⠀⠀⠀⠀⠀⡈⠉⢀⠀⠀⠀⠀⠀⠀⠿⠆⠀⠻⠄⠀⠀⠀⠀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠤⠼⠦⠀⠀⠀⠰⠟⠁⠀⠀⠀⠤⠐⠇⠟⠻⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠐⢠⡉⠴⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡯⠈⢀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⢀⢄⠀⠀⠀⠀⣾⠀⢀⣀⠀⠀⠻⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⣷⠀⡘⣿⡀⣸⣅⡀⣀⠀⣦⠀⠋⣲⠀⣷⡀⠀⣠⡶⠸⣿⠀⣾⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⠀⢻⡇⣿⠁⠁⢿⡘⣿⠈⠀⣿⠐⠈⠿⡆⢿⡀⢀⣿⠀⣿⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⡀⠀⠠⠀⠀⣼⠇⠈⠁⠀⠈⠀⠈⠁⠀⠷⠀⠐⠄⠁⠈⠉⠁⠹⠴⣿⠂⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠴⠖⢠⡄⠁⢠⠀⠀⣠⡿⠀⠐⢂⣤⡀⠀⣠⠆⣕⡄⢀⠀⠀⢀⢀⣄⢑⡀⠀⠻⡄⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠈⠲⠞⠋⠀⠀⡿⣬⠉⠀⠀⢱⡏⠈⠛⠀⠀⠀⠀⠚⣵⠘⠿⠀⠄⠄⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⠀⡀⠀⠀⠁⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠛⠛⠛⠛⠷⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⣠⣴⢶⡄⠀⠀⠀⢰⣿⠀⠀⣠⣾⠛⠛⠁⢿⣿⣄⣤⡶⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣾⠛⠛⢶⣤⠀⠀⠀⠀⢠⣿⣧⡾⠋⠀⢸⡇⠀⠀⠀⣾⣿⠀⣴⣿⠁⠀⠀⠀⠀⣿⠉⣿⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⠙⠻⠶⠟⠋⠀⠀⠀⣰⣿⣿⠋⠀⠀⢀⣿⠁⠀⠀⣼⠇⣿⣾⢿⡇⠀⠀⠀⠀⢰⡟⢸⡏⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⠀⣾⠃⠀⢀⣼⠏⢀⣿⠃⢸⣇⠀⠀⠀⢀⣾⠃⣿⠁⠀⠀⠀⣿⡇⠀⢀⡄
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢫⡿⠁⠀⠀⠀⢰⡟⠀⣠⡾⠃⢠⣿⣿⠀⠀⢿⣆⣀⣠⡾⠃⢠⣿⠀⠀⠀⢸⣿⠀⢀⡾⠃
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠙⠀⠀⠀⠀⠀⠈⠻⠞⠋⠀⢠⡿⢹⡏⠀⠀⠀⠉⠛⠉⠀⠀⠀⣿⡀⠀⢠⡿⣿⣶⠟⠁
⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⣴⠆⠀⠀⠀⠀⢀⣤⡄⢠⡿⠀⣼⠇⠀⢀⣠⣤⣄⠀⠀⠀⠀⠘⠻⢾⣿⠿⣿⠁
⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⣀⣴⠟⠁⢀⣀⣀⣼⠃⠀⢀⣤⠆⢀⣾⣿⠇⢸⡇⣰⡟⢀⣴⠟⠉⠀⢹⡇⠀⠀⠀⢠⡾⠋⠁⢸⡿⣰⠟⣛⠛⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠟⠛⠋⢡⡶⠟⠛⠉⣻⠟⠛⠛⠋⠁⠀⣾⡿⠃⠀⠘⠛⠋⢀⣾⠃⠀⢀⣠⡾⠃⠀⠀⣰⡟⠀⠀⠀⢸⡇⠻⣦⣹⡇⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⣿⠀⠀⠀⠀⣠⣾⣿⡾⢻⡷⠀⢠⣿⣾⣷⡶⠾⠛⢉⣠⡶⠂⢀⡿⠀⠀⠀⢀⣿⠁⠀⠈⠉⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⢉⣼⠋⢻⣦⣤⣴⠟⢋⣿⠋⠀⣾⠃⣠⡿⠁⠈⠻⢶⣴⠶⠟⠉⠀⢀⣸⣇⠀⠀⣠⡾⠃⣠⡶⠟⣿⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣡⡶⠟⠁⠀⢀⣈⠁⠀⠀⠾⠃⢀⣤⠻⠟⠋⠀⠀⢀⣾⠂⠀⠀⠀⢀⣼⢻⡿⠙⠛⠟⠋⢀⣼⠏⢀⣼⠏⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠿⠛⠉⠀⠀⠀⠀⣨⠉⠀⢠⡶⠟⣻⣿⠿⢶⣤⣤⣴⣶⡿⠿⠛⠛⠋⢠⣿⣷⠟⠀⠀⣠⡀⢀⣾⣃⣴⠟⠁⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠸⠛⢹⡟⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⣠⣾⣿⠃⠀⠀⠀⣠⡾⣿⠁⠀⠀⣀⣤⡿⠋⠁⠀⢀⣴⠟⠛⠻⣿⣉⣁⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⣴⣿⠃⠀⣀⣤⡾⠋⢹⣇⣀⣠⣴⠟⠉⠘⣿⣠⣴⠾⠛⢹⡇⠀⣠⣴⠟⠁⠀⠀⠀⢈⣭⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣴⠾⠋⠀⠙⠻⠛⢋⡿⠀⠀⠀⠙⢋⣿⢿⣷⠀⠀⠈⠉⠀⠀⠀⠈⠛⠛⠋⠁⠀⣠⡶⢀⣦⠘⣿⠿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣿⣥⣤⡶⠆⢠⡿⣡⡾⠁⠀⠀⠀⠛⠀⠀⣶⣠⡾⢻⡷⠀⣴⡟⠀⣼⣇⣾⠋⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⠀⠀⠀⢀⣿⠉⠉⠀⠀⠀⣾⣿⣿⡄⠀⠀⢀⣾⠃⠀⣼⣿⠏⢀⣾⠃⣴⠟⢷⣾⡿⢛⣅⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣼⡿⠃⢀⣿⠀⢀⣾⡇⢀⣼⢿⡏⢠⡾⠃⣼⠏⣴⠟⣿⠃⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣀⣀⣴⢿⡟⠀⠀⣾⠇⣠⡾⠛⠷⠟⠁⠿⠁⣿⣁⣾⠋⣼⡏⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁⡿⠁⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠈⠁⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡄⠰⡄⠃⡇⠀⠀⠸⠈⢠⠆⢠⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢘⢇⠠⣠⠀⠇⠐⠄⠰⠀⡾⠀⡘⡃⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⡜⡌⢸⠂⠈⠃⠣⠄⠠⠜⠘⠃⠀⡇⢡⢣⠁⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡑⠀⠅⠔⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⠤⠁⠪⢈⠀⢂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠃⠑⣣⠴⠒⠒⠒⠢⢄⡀⢀⣠⠴⠖⠒⠲⠤⣈⠊⢘⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢖⠞⠀⠀⠀⠀⠀⠀⠀⠙⠏⠀⠀⠀⠀⠀⠀⠈⢳⡳⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣧⠠
⠀⠀⠀⠀⠀⠀⢦⡈⠣⡀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠀⢀⠜⢁⡴
⠀⠀⠀⠀⠀⠀⠰⡈⠃⠈⠢⡈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⠔⠁⠘⢁⠆
⠀⠀⠀⠀⠀⠀⠀⠑⢦⠀⣀⡈⢢⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡔⢁⣀⠀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠛⣡⣯⠛⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠙⣽⣬⡙⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣤⡾⠋⠀⣡⠄⡀⠀⠉⠂⠄⡀⠀⠀⣀⠤⠒⠉⠀⠀⣠⣘⣷⣌⣻⣦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢀⣼⡋⢠⡾⢷⣆⣆⢠⣤⢱⣊⡤⠄⣴⡂⣾⠸⣏⠙⣷⡉⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⢼⣧⡾⠋⣿⣼⠃⣿⡯⠀⠀⠈⢻⡏⢀⡩⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠛⠣⠤⠽⠿⠥⠄⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢀⣠⣤⡀⠀⣠⣦⡀⢠⠏⣠⣴⡄⠀⠀⢀⡀⠀⠀⠀⠀⣠⢦⠀⣠⡶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠸⣷⡄⣿⠂⢸⣿⣇⡟⣼⣿⠾⠃⣠⠾⡋⣹⡆⠀⠀⢰⡇⠘⢰⣿⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⢰⡄⡠⠂⢸⣿⠀⢰⣿⡄⣿⡄⢸⣿⣿⠁⣿⣇⡤⠞⠁⣀⡉⠉⠀⠀⠀⣾⡣⠁⣼⣏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⠉⠀⠀⢸⣿⠀⡜⢿⣷⡿⠀⠀⠛⠛⠀⠙⠋⠀⢠⣿⢉⡇⠀⢀⣀⠀⣿⠃⠀⣿⡇⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⣦⣄⠀⢠⡔⢦⠀⠀⣿⣠⠃⠀⠀⠀⠀⠀⢴⣶⡀⣠⣄⠀⣿⣿⠎⠀⠀⣾⡿⢀⣿⠀⠀⣿⡧⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣸⣿⣠⣿⠇⠁⣄⠀⠙⠋⠀⢀⣀⠀⢠⡄⠀⢹⠃⣿⣿⠀⢿⣿⡀⠀⠰⣿⣀⡎⣿⡇⢠⣿⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠛⠁⢸⣿⠀⣸⣿⢀⣷⡈⡦⢸⣿⠀⣿⣧⠀⣸⢰⣿⣿⠀⠀⠉⠻⣦⡀⠈⠉⠀⠸⣇⡈⢸⣧⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡤⢻⣿⢸⣿⡅⢰⠀⠛⠃⢻⣿⢠⠇⢸⣿⡇⠀⣠⡖⣦⠈⢻⡄⠀⠀⠀⠈⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣼⡟⠘⣿⣇⡞⠀⠀⠀⠈⠙⠃⠀⠈⠻⠟⠀⢿⣆⣸⠀⣸⠇⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠝⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⠀⠀⠀⠀⠀⣠⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⣿⠀⠀⠀⠀⠀⣼⠇⣼⠀⠀⠀⢀⣠⠤⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⡴⠃⠀⠀⠀⠀⣸⡟⢰⠏⠀⠀⠠⣿⠀⠀⠀⣠⣍⠙⠳⢶⣤⣔⠊⠁⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡧⠊⠀⠀⠀⠀⠀⢠⣿⣡⠏⢀⡤⠀⠀⢙⣀⡀⢰⣿⡜⠀⠀⡔⠉⢙⡛⠓⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⢀⣴⢶⠄⣾⡿⠃⠀⢠⡅⠀⢠⡿⢁⠇⢈⣿⡄⠀⡜⠀⣴⡿⠋⠀⠀⠄⠀⠀⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣄⠀⠀⠀⠀⢠⣿⣇⡀⠀⢀⣾⡿⠋⢠⣿⡇⠀⠠⣿⡇⠀⣿⡟⠉⣠⠋⠘⣿⡸⠀⢼⡟⢀⡤⠊⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠐⡄⢀⡿⠀⠀⠀⠀⢸⣿⠙⢿⡞⠉⣿⣠⠔⠀⣿⠀⡴⠁⣿⡠⠊⢿⣧⠜⠁⠀⠀⠈⠁⠀⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡀⢸⣿⠀⣸⠁⠀⠈⠁⠀⠀⠙⠋⠁⠀⠈⠁⠀⠀⠀⠀⠀⢰⢴⣴⠂⠀⠀⢀⡀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠙⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠘⠙⠙⠀⢀⣴⡿⠁⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣶⣀⢀⣀⠀⢰⣿⡭⠀⠀⢀⣤⡀⣾⡿⠁⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢠⡄⠀⢾⣀⡈⣷⣠⡄⢀⡀⠀⠉⠀⣿⡄⢻⣿⡀⠀⢠⣿⠃⢸⣿⠁⠀⣼⣯⠠⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⢘⣷⠀⢸⣿⠁⡏⣿⠇⣼⣷⠀⡀⢸⣿⠁⠘⢿⣧⠀⠸⣏⡠⢻⡟⢀⢘⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠖⠂⢀⣿⠁⠘⠿⠞⠁⠈⠃⢿⣿⢀⠀⠘⢿⠖⠀⠈⣿⡄⠀⠈⠀⢸⡇⠆⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⣿⠄⠀⠀⠀⠀⠀⠀⠈⠻⠌⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠘⠛⠀⣿⣇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠐⠋⠀⠀⠀⠀⠀⠀⠀⣿⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⢀⡤⠖⠒⠀⠀⠀⠀⠐⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⠄
⠀⠀⠀⠀⠀⠊⣠⣴⠶⠛⢛⠓⠒⠒⣲⣿⡟⠁⠀⠀⠀⠀⠀⠀⠉⣠⡶⠒⣄⠀⠄⠀⠀⠀⠀⢀⡤⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠠⡏⠀⠀⢀⣼⡇⠀⢠⣿⡟⠀⠀⠀⠀⠺⠇⠀⢀⣾⠟⠀⣰⠃⠀⠀⠀⠀⠀⠀⡟⢀⡄⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠉⠓⢒⣛⠉⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⢠⣿⠏⢀⠜⠁⠀⠀⠀⠀⠈⠀⠀⠁⡆⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠆⢀⠂⠙⣿⣧⠀⢀⣿⡟⠀⠀⡀⠀⠀⢄⣠⣿⡟⠀⠁⢀⣤⣶⠂⠀⠀⢀⣶⡆⠀⣇⠁⠀⣠⣶⠖⣦⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠃⠀⣸⣿⠃⠀⠈⠁⠀⠀⠀⣼⣿⠁⠀⠀⣾⣿⠃⠈⠃⢀⣾⣿⠃⢀⡿⣄⣰⡿⠁⡠⠏⠐⠁
⠀⠀⠀⠀⠀⣇⠐⠀⠀⠀⠀⣰⡿⠋⡀⠀⣀⠲⣄⠀⠀⣿⡏⠀⢀⢺⣿⠇⠀⠀⠀⣾⣿⠃⢀⠎⠀⢸⣿⠃⠈⠀⠀⠄
⠀⠀⠀⠀⠀⠙⢦⣄⠀⠒⠛⠉⡴⠋⠀⢀⣿⡇⢹⠀⠀⠘⠳⠐⠁⠀⠛⠀⠀⠀⠀⢿⣇⡴⠋⠀⠀⠘⠿⡄⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⣼⠁⠀⢠⣾⡿⠁⠀⣠⣶⡿⠃⢠⠄⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀⣀⠀⠀⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠟⠀⢠⣿⡿⠁⢀⣼⣿⠏⢀⣶⣿⠀⠀⠀⣠⣿⠆⠀⣠⣦⠀⠈⠋⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠠⣶⠄⠀⠀⠀⢸⣿⢁⠀⣼⣿⠏⢠⣿⡟⠈⡛⠀⢩⣿⠏⠀⣰⣿⡟⠀⠀⠀⠐⣶⡄⠀⢶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⣿⠏⠀⠸⣿⡀⠌⠀⠀⣿⡏⠀⢸⣿⡟⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡴⢋⡡⠀⠐⠂⠀⠀⣰⣿⠯⢀⡀⠀⠀⠀⠀⡡⠀⠈⠁⠀⠈⠛⠀⠀⠀⢀⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠺⠁⡞⠀⠀⠀⠀⢀⣴⠟⠁⢀⣤⠈⠙⠲⢦⣤⣄⣀⣀⣀⣀⣀⣀⣠⣴⣾⣿⣿⡿⠋⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⠤⠤⠶⠛⣡⠄⠀⠈⠉⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⢿⣿⡿⠿⠟⠛⠉⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠞⠛⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⣀⣀⠀⠲⣄⠀⠠⣴⣶⠀⠀⠀⢿⠇⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⢠⣾⡿⠋⠀⠀⣿⠀⠀⣿⡟⠀⠀⢰⠊⢀⣴⣿⠟⠉⡗
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠁⠀⣾⣿⠁⠀⠀⢀⣏⡤⢺⡿⠁⠀⣰⣇⣀⣼⣿⣁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠻⢿⣄⡤⠖⠋⠁⠀⢸⣧⡴⠊⠀⠀⢸⡏⠁⠀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠐⢦⣯⣤⣤⣤⣤⠴⠂⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠘⠳⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣉⣤⠄⠀⢀⣿⡗⠀⠠⣎⣁⣀⠙⣦⠀⠀⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⣼⡿⠀⠀⣴⣿⠛⠒⠂⣿⡖⢺⣿⡍⠀⠀⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢠⣿⢁⣠⢼⣿⠃⠀⠀⢠⡿⠀⣾⠋⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⠤⣾⠃⠉⠀⠘⠿⣦⣤⠴⠋⠀⢰⡿⠀⢀⢼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠁⠘⢇⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃



⠀⢀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⣠⡾⠋⠀⠀⠀⠀⠀⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⢈⣀⣀⡴⠲⣦⠠⣤⡼⢟⣀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⢰⣿⠏⢹⠀⢸⣿⢠⠇⣼⣿⣻⡗⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣟⡀⠀⠈⢿⣄⠜⠀⠈⠓⠋⠀⠙⢯⣉⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠐⠏⠩⠽⠛⠙⠛⠿⢷⣦⣄⣀⠀⢀⣰⠀⢠⣤⠀⣀⠀⢀⣤⣤⡤⢦⡀⣠⡄⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠁⠀⠈⠛⠚⢉⡧⢾⣟⣡⠇⣿⡥⢿⠁⣀⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢿⣦⠀⠀⠀⠀⢀⣴⣶⡄⢀⠴⠒⠚⣩⠏⠀⠀⣉⣁⡀⠀⠀⠀⣉⣉
⠀⠀⠀⠀⡀⠀⣀⣼⠟⠁⠀⣾⣿⠀⠀⠀⡰⢻⣿⣿⠀⠈⠓⠒⠊⠁⢀⡴⠛⠉⠉⣿⡆⠀⡴⢋⣿⡆⠀⠀⠘⣿⣷⣆
⠀⠀⠀⠘⠷⠾⠟⠁⠀⠀⠀⣿⡏⣠⣤⡰⠁⣾⣿⠇⠀⠀⣴⡟⢳⣠⠏⠀⠀⢀⣼⣿⠃⡸⠁⣼⣿⣀⣀⠀⠀⠸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡱⠁⠿⠁⠀⣿⣿⠀⠀⠀⣿⡄⠐⠇⠀⠀⣴⣿⢟⡁⢠⠃⠀⣿⡟⢻⡟⠀⠀⣸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢀⡴⠉⢷⣄⠀⢀⣾⡟⠁⢸⣟⡎⠀⠀⠉⠀⠸⢇⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠈⠀⢸⡏⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⢸⡟⢀⠀⠀⣠⣤⢰⢑⢀⣶⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢀⣿⠁⢸⡟⢸⠁⣿⢀⠏⣹⡏⢀⠀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠀⣀⣾⣇⡀⠘⠓⠁⠀⠙⠊⠀⠘⠗⠁⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡠⠊⣸⠏⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣴⠋⠔⢻⡆⠀⠙⠳⢤⣀⣀⣀⠤⠚⠁⣰⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣸⠀⢸⡇⠀⣶⠀⣀⠀⢀⡀⣸⡇⠀⣿⡇⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⡇⢀⣿⠀⢣⣶⠀⢻⠸⣧⣰⠛⢧⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⢚⣿⠄⠹⣿⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢺⡿⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉



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

⠀⠀⠀⠀⢀⠀⢀⠀⡀⣠⡀⠀⠀⠀⡀⠀⡀⡀⠀⢠⡀⠀⠀⠀⠀⠀⠠⠤⠤⠡⠤⠀⠀⠀⠀⠀⠀⢀⣥⠲⡆⠄
⠀⠀⠀⠀⠿⠔⣹⣽⢵⣤⣹⣦⠀⠀⢡⢤⠷⡛⠷⠇⣟⡤⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⢰⣶⣶⣶⣶⣾⣾⡇
⠀⠀⠀⠀⢠⠞⡟⠨⠤⠄⠉⠁⠀⠀⡬⢤⣤⡄⣀⡔⣤⠄⠀⠀⠀⡝⢉⣅⠙⣣⢤⠤⡀⠀⠀⠘⠛⠻⠛⢛⡛⢛⠁
⠀⠀⠀⠀⠿⠼⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠑⠁⠑⠌⠁⠚⠂⠀⠀⠀⠀⠚⠁⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡰
⠀⠀⠀⠀⠀⠀⠂⣂⣤⡄⠀⠀⠀⠀⠀⣏⡠⠜⠱⣤⡶⢀⠀⠀⠀⠐⡁⣀⣀⣇⣀⣠⠀⠀⠀⠀⠏⢃⣈⣡⣦⣀⣀
⠀⠀⠀⠀⢀⣀⣁⣀⣄⣘⣁⣀⡀⠀⠀⣀⢄⢀⢀⣀⠑⠯⠀⠀⠀⠀⠘⠀⠄⢁⠀⡤⠀⠀⠀⢀⡈⠈⠁⢁⢀⡀⡁
⠀⠀⠀⠀⣿⣿⣯⣿⣿⣽⣿⣿⡇⠀⠘⢶⡟⢻⡡⠉⠒⠊⠀⠀⠀⢨⠀⠉⠋⡉⢉⠉⠀⠀⠀⠀⠐⡻⠻⠉⠈⠹⠁
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠴⠒⢄⠀⢀⠀⠀⠀⠀⠀⢠⠔⠘⠄⠀⠀⠀⠀⠀⠀⠠⠏⢜⣨⣁⣀
⠀⠀⠀⠀⠀⢰⠋⠀⢈⡆⠀⠀⠀⠀⠠⡃⢀⡠⣮⡀⢠⡀⠀⠀⠀⠀⠈⠢⡨⠷⠒⡆⠠⠀⠀⠀⠔⠘⠞⣉⠉⠙⠡⡀
⠀⠀⠀⠀⠀⠘⢤⣀⣭⣴⣶⠐⠀⠀⠀⠑⢌⢴⢃⡴⢿⣡⠆⠀⠀⠐⠐⡆⣡⢆⠈⡆⡀⠀⠀⠀⠸⣀⡐⢆⠴⠶⡆
⠀⠀⠀⠀⠀⠐⢠⢳⠐⠒⠉⠁⠀⠀⠀⠀⠉⠀⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠀⠀⠀⠀⠀⠙⠁⠔⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢄⠀⠀⣤⢦⠀⢸⠉⢙⡆⢀⠈⠉⠉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢰⡏⣠⢤⡈⣷⠀⢡⠀⢃⣸⠀⡼⠀⣸⠠⣏⠉⠋⠀⣰⠋⢳⠀⠀⣀⣤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡆⠀⠀⠀⢸⡆⠹⣼⠇⣿⠀⠸⡇⠘⠏⣸⠁⠀⡏⠰⣞⣁⠀⢀⡏⠀⡾⢀⣴⠋⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠸⡄⠀⠀⠀⠳⣄⣀⣰⠏⠀⠀⠳⣤⡤⠃⠀⠀⠳⠒⠒⠛⠀⢸⠇⣼⠁⣸⡇⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠹⡄⠀⠚⣦⠈⠉⠁⣰⢶⡀⠀⠀⠀⣤⠤⠀⠀⢠⡴⡄⠀⣸⡾⠁⠀⣿⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣠⠤⠖⠋⠁⠀⠀⠀⠘⢦⡧⢶⡄⠀⢳⡀⠀⠀⣟⡰⠃⣠⡟⠀⢶⣾⣿⡿⠒⠓⠲
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⣴⠖⢦⢀⣶⠀⢸⠇⢰⠇⣸⠃⢠⠖⠛⣏⠉⢻⣤⠞⢹⡇⠀⡜⢨⣿⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣿⠀⢰⠀⠳⣤⣼⡏⢹⣀⡾⣇⡟⠀⢻⣠⢿⡀⠀⢸⡇⠀⠀⠀⢸⠇⡼⠁⠀⣿⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⣸⣇⡜⢇⡼⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠑⢤⡾⠃⠀⠀⠀⠘⠾⠁⠀⠀⠈⠁
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢩⣏⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⢤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⣿⡤⠤⠖⠒⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠲⠶⢤⡄
⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⢾⠋⢩⡿
⠀⠀⠀⠀⠀⠒⠉⠁⠀⠀⠘⠶⠟⠁

⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⢀⣴⣶⣶⡄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⡇⠀⣄⣿⣿⣿⣿⣿⣿⢸⣿⣧⠀⢸⣿⢀⣴⣿⣿⣿⣮⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡇⡅⢹⢸⣿⡏⢸⣿⣿⠀⣿⣿⠀⣿⢇⣼⣿⣧⣶⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢺⣿⣁⠒⠂⢸⣿⡇⢸⣿⡿⢰⣿⣿⣼⡏⢸⢻⣿⣄⣀⣀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢸⣿⣿⣿⡇⡼⣿⣿⣿⣿⠃⠜⢹⣟⡿⠱⢈⠈⢻⣿⣿⣟⠒⠐⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢌⣴⣶⠀⣴⡆⠑⠙⠿⠟⠣⡶⣴⣦⣤⣤⣥⡄⢸⣿⠿⠿⠿⣿⡄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣾⣿⠃⣩⣾⣿⣷⡀⠆⣿⣿⣿⣿⣿⢃⣼⣿⣶⣶⣾⣿⢧⠇
⠀⠀⠀⠀⠀⠀⠀⢠⡴⠛⡆⣿⣿⠁⢸⣿⣿⣿⣿⡇⡄⣿⣿⣿⣿⡿⢸⣿⣾⣿⣿⡋⠑⢀⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠉⡛⠁⣿⣿⠀⢸⢻⣿⣿⣿⠁⢃⢻⣿⣿⣿⠃⢸⣿⣿⡿⣿⣷⣞⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢼⠀⣿⡿⠀⢀⡈⠻⠿⠋⠁⠀⠄⠛⠗⢋⣠⣌⣾⣿⡇⠈⠻⣿⢃⣖⡆
⠀⠀⠀⠀⠀⠀⠀⠐⠯⠝⣡⣴⣶⣿⣶⡄⢰⣿⣿⣶⡶⠀⢸⣿⠪⠟⠯⢹⣿⠇⠄⠃⠀⠠⣗⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⠉⢙⣅⢰⣿⣇⣀⠀⢰⢸⣿⡀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⠿⣿⣶⣄⠉⢸⣿⡟⠛⠰⢾⢸⣿⠂⠀⠀⢸⣿⣿⣛⣛⠛⠣⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣤⣶⠊⢿⣿⣆⢸⣿⣧⣤⠀⠘⢸⣿⣬⣭⡁⢸⣿⣿⠿⠇⢄⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢨⣭⣤⣾⣿⡟⣈⣙⡉⠉⠁⠀⠘⠛⠛⠛⡁⢸⣿⡇⠀⣠⣀⣉⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠿⠿⠿⠛⠀⠄⢤⠀⡬⢩⡍⢩⡍⢥⠀⡤⢸⣿⣧⠀⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠸⠴⠶⠶⠒⢒⣒⣒⠒⠲⠶⠦⠘⡛⠁⠀⠀⠁



⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢀⣶⡿⠛⠙⣿⠀⠘⣿⣄⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⢠⣿⡏⢀⣠⣼⠏⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⠿⠛⠋⣁⣤⣤⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⢀⣀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡟⠁⠀⠀⢸⣿⡇⠀⢠⣾⢿⣇⣸⡇⣠⣾⡟⠀⢸⣿⠀⣠⣾⣿⡿⠃⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⣿⣿⠁⢠⣿⡃⠀⣹⡿⠋⣿⡟⠀⣠⠟⠉⢹⣿⠋⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣯⡀⠀⠀⢀⣿⡟⠀⠀⠛⠿⠛⠋⠀⠀⠙⠓⠛⠁⠀⠀⠈⠻⠷⠶⠶⠞⠋⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣼⣿⣅⣀⡀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠶⠶⠿⠿⠶⣶⣤⣀⠸⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠋⠉⣰⣿⠏⠉⠉⠛⠻⠶⣤⡀⠶⠟⠋⠁⠀⣧⠔⡴⢲⢰⡆⡶⠉⠻⣧⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣀⣠⣾⠟⠁⠀⠀⠀⠀⠀⢀⣶⠟⠻⢦⣤⣀⠀⠏⠀⠧⠞⠸⠴⠃⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠈⠙⠛⠷⠶⣶⣦⣤⣴⣶⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⠃⠀⠀⠀⠀⣿⣿⣿⠟⠰⣿⣤⣶⣶⣶⣄⠀⠶⣶⣶⡶⠀⠀⠀⠀⢠⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⣴⣿⣿⡟⠀⣠⣿⡿⠋⠁⠈⣿⡇⠀⣿⣿⣧⠀⠀⠀⢀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠻⣿⣷⠀⢀⣼⠏⣾⣿⠁⢰⣿⡟⠙⠷⠦⢀⣿⡇⢰⡿⣿⣿⡀⠀⢀⣾⢋⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣿⣿⣄⣾⠃⢸⣿⡟⠀⣿⣿⠀⠀⠀⠀⣼⣿⠁⣾⠇⢹⣿⡇⢠⣾⠃⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⢸⣿⡿⠁⢀⣿⣿⠁⠀⢿⣿⠀⠀⣠⣾⡿⠁⢰⡿⠀⠘⣿⣿⡿⠁⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠰⠾⠿⠿⠂⠀⠈⠻⠿⠿⠟⠋⠀⠰⠾⠷⠀⠀⠙⠛⠁⣠⣾⣿⣧

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⢙⣋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠚⠛⠛⠛⠲⠶⣤⡀⠀⠀⠀⠀⣀⡤⠔⠒⠒⠒⠒⠶⢦⣄⡀
⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡄
⠀⠀⠀⠀⢠⠎⠀⡤⣰⡄⠀⠀⠀⢀⣤⣄⠀⠀⠀⠠⣿⡇⠀⠀⢀⡴⢲⡄⠀⠀⠀⠴⣦⠀⠀⠹⣦
⠀⠀⠀⢠⡏⠀⠈⠳⢿⣾⠏⠛⢀⡿⢸⣿⠀⠀⠀⠀⢀⠀⣴⡆⢾⣇⡼⠁⣶⠄⠰⠈⡋⠙⠆⠀⠹⣇
⠀⠀⠀⣼⠀⠀⢠⠁⢸⣿⠀⠀⣸⡇⢸⡇⠀⢀⡀⠰⣿⠀⢹⡗⠘⠛⠓⠋⢀⡤⠀⢸⡇⢂⣀⠀⠀⣿⡀
⠀⠀⠀⣿⠀⠀⢧⠀⢸⣿⠀⠀⣿⡇⡾⠀⣼⠋⢻⡆⡹⢷⠾⠁⠀⣠⠴⣤⠸⣿⠀⢸⣧⣛⡸⠂⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠈⠳⣾⣿⠀⠀⢿⡿⠁⠀⢙⠷⣸⠟⠀⠀⢠⣶⠀⣿⡀⢻⡧⠛⠓⣊⡀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠸⡇⠀⠀⢀⡬⣻⡆⢀⡼⣿⡀⢀⡟⠛⠁⢰⡟⠀⢸⣿⠀⣮⣭⠟⠀⠀⢿⡠⡌⢷⠀⠀⢀⡿
⠀⠀⠀⠀⢻⣄⠀⠸⣼⠵⠁⡊⣀⠙⠛⠋⠀⠀⠀⡸⣧⣀⡼⣿⡰⠃⡀⣀⠀⡀⠀⠀⣠⠟⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠈⠳⢭⠈⠝⠀⢀⡤⠊⠀⣈⠭⠔⣿⡀⠘⠧⣌⣧⠽⠋⠉⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠛⠒⠛⠉⠀⣠⡞⡵⣊⠁⣿⠉⠑⠒⠂⠁⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⠀⠀⠀⠀⠀⢿⣇⠳⠽⣰⠏⠀⠀⠀⠀⠀⢀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣀⠀⠀⠙⠛⠚⠁⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⡀⢀⡠⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠈⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⣀⡤⠤⠒⠒⠒⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠘⣄⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠘⡜⠀⣰⢲⠤⡀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠒⣿⠈⠀⠁⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⢀⡴⢦⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠏⠀⡸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠐⢠⠃⠀⠀⠀⢺⡖⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⡤⣀⠀⠀⠀⢸⠁⠀⠁⠀⠀⢀⣀⠉⡀⠀⣠⡀⢄⠀⠀⠠⡄⠀⠀⢀⣀⡀⣀⠀⡠⠊
⠀⠀⠀⠀⠀⠢⠀⠃⠀⠀⢼⠀⠀⢰⠋⠙⠃⠈⡷⣿⠀⠆⠁⡸⡎⢀⠼⠁⠀⠀⢮⠠⠇⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠘⢦⡀⠀⠠⠃⠘⢦⡀⠈⠈⢯⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠂⠊⠉⠁⠀⢀⢀⡀⠀⢀⡀⣉⣅⣂⣀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⡏⠘⠱⣾⠻⡃⠀⠃⠀⠷⡏⠀⢸⡇⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⣿⠀⠑⠠⠤⠀⠈⠳⠤⠜⠣⣀⡘⢈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⢠⢁⠁⠈⠈⠗⡴⡆⡠⠒⢴⢦⢶⠖⢲⡔⠇⠉⠀⠁⢠⢈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠋⢀⠀⡀⡍⠓⢮⣇⣛⣧⣛⡼⣆⡼⣁⣸⢯⠖⠋⡁⡀⢀⠈⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠒⠣⠠⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠮⠠⠓⡊⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢈⠣⣀⠆⠀⠘⣿⠀⠀⠁⣰⠋⠐⣆⠀⢹⡇⠀⠀⠈⠀⠀⣆⡠⢃⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠀⡄⠛⠀⠀⠀⠘⣧⠀⢰⡏⠀⠀⢹⡄⢸⡇⠀⠀⠀⠀⠀⠘⠃⡄⠰⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠦⠱⠀⣀⠀⠀⠀⠀⣿⠀⠸⡇⠀⠀⢸⠇⢸⡁⠀⠀⠀⠀⠀⢀⡀⠰⠑⢆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⠆⠁⠀⠀⠀⠀⣿⠀⠀⢷⡀⢀⡟⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⢆⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠉⠀⢠⠀⠀⠀⠈⠉⠁⠀⠀⠉⢉⡀⠀⠀⠈⠁⠀⠀⠀⠀⢠⠀⠈⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠲⢃⠀⠀⠀⠀⠀⠘⠷⠒⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠳⢆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠒⠦⠀⠀⠀⠠⣔⠁⢠⠮⢗⡀⠀⢒⣀⠀⠀⠠⠒⠂⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡄⠀⢄⡀⠀⠀⠀⠀⠃⢔⠊⠁⠘⠉⠉⠀⣀⠄⠀⣤⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠞⠉⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠈⠙⠆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⢀⣦⡀⢘⡁⠲⠀⠀⠀⠠⠂⣙⠀⣠⣆⡠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢻⡳⠙⠅⢘⡀⠀⠀⣸⠀⠝⠱⣻⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢳⠀⠈⢀⡁⠀⠀⣁⠌⠀⢰⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⠰⠀⢰⠁⠀⠀⢱⠐⠰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠉⠀⠀⠀⠈⠁⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣤⣴⣶⢶⢶⣶⣶⣶⣶⣧⣤⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣴⣾⣿⣿⡟⣚⣷⡛⢓⡛⢻⣿⢻⣿⡿⣻⣿⣷⣦⣔⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣊⠀⠀⢹⣿⣷⣶⢰⣿⠟⣱⣿⢱⢟⡁⡾⣋⣴⣿⠛⣭⡶⢏⣿⣾⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⡄⠀⢻⣿⣿⣷⠷⠿⠟⠛⠛⠻⠿⠿⠿⣮⣶⡸⠷⣚⣿⣿⠟⣵⣌⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡌⣿⣿⣿⣿⣷⠀⠈⢉⣵⣾⣿⣿⣷⣤⠀⢾⣿⣶⣄⠀⠈⠙⠻⢿⠟⠡⠞⢛⣋⣁⡁
⠀⠀⠀⠀⠀⠀⠀⣠⣄⣿⣿⣿⣿⣿⣿⣿⠀⠀⣺⣿⣿⡟⠁⠹⣿⣷⠘⣿⣿⣿⡀⢠⣾⣷⡀⢀⣴⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣹⣿⣿⣷⠀⠀⣿⣿⡇⢿⣿⣿⠁⢼⣿⡿⢡⣾⣿⣿⡿⠋⢹⡿
⠀⠀⠀⠀⠀⠀⡸⢀⣀⡉⠛⠿⣿⣿⣿⣿⠀⣹⣾⣿⣿⣿⣧⠀⣿⣿⡇⢸⣿⣿⠀⢸⣿⢣⣿⣿⣿⠏⠀⣠⡾⠃⣿⡆
⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣦⣄⣸⣿⣿⡿⠀⣿⣿⣾⢿⠿⠋⢸⣿⡿⠀⣼⣿⢇⣴⠟⠻⣿⣿⣿⡏⣠⣴⠟⠁⢰⣿⣇
⠀⠀⠀⠀⣴⣿⣿⣿⠟⠋⠉⠛⢿⣿⣿⡇⠀⠹⣿⣿⣆⠀⣠⣿⡿⠃⠀⣿⢿⣛⡁⠀⠀⢿⣿⣿⡟⠋⢄⠠⠀⣾⣿⡏
⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⣾⣿⣿⣿⣦⣀⠙⢿⣿⣿⣿⠟⠁⢀⣠⣴⣿⣿⡇⠐⠀⠘⣿⣿⣇⠀⠈⠌⣼⡿⠋
⠀⠀⠀⠸⣿⣿⣿⣧⡀⠀⠀⣰⣿⡿⠉⠻⣿⣿⣿⣶⣮⣥⣤⣶⣾⣿⣿⣿⣿⠟⢀⣤⡀⠀⠈⠻⠿⠷⠶⠞⠋⢡⣄⣾⣷
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⣿⡿⠟⢀⡀⠤⣈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⣴⣿⣿⣷⠀⣠⣶⣿⣿⣿⣿⣶⣌⢿⣿⡟
⠀⠀⠀⠀⢀⣤⣭⣙⠛⠉⢁⠔⡪⢕⣈⣥⣴⢶⡲⣎⠻⠿⣿⣟⣯⣭⠀⣠⡽⣿⣿⣿⣧⢠⣿⣿⣿⣿⣿⣿⠟⠁⠀⠙⠃
⠀⠀⠀⠀⠈⢛⣿⣿⣿⡶⢑⣥⣾⢿⣿⡿⣇⣿⡇⣿⡸⠷⣸⣿⣿⣿⡌⠻⣧⠙⠻⣿⣿⣆⢿⣿⣿⣿⣿⣷⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠺⡿⠟⣡⣾⢿⢣⣶⢡⡻⡗⡏⣯⣵⢿⣱⣮⣛⣼⠿⢛⣣⣤⣦⠀⠀⢹⣿⣿⣆⠉⠛⢛⣯⣉⠙⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠐⣽⡟⢿⡎⡞⣿⠞⣷⡁⢿⣌⣋⣾⠿⣋⣭⣶⡸⣿⣿⣿⣿⡄⢀⣴⣿⣿⣿⣆⢴⣿⣿⣿⣶⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢠⡻⡔⡐⣾⡻⣷⣼⠿⠋⠀⢺⣿⣿⣿⡧⠿⣿⣿⣯⠀⣾⣿⣿⢿⣿⣿⣎⢿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⢳⣄⠹⣮⣵⢟⣵⣶⣶⣶⣼⢿⣿⣿⣷⡀⢸⣿⣿⡘⣿⣿⡇⠀⢹⣿⣿⣧⣴⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠻⣶⠟⢡⣿⣿⣿⣿⣿⣿⡎⣿⣿⣿⣿⣄⣿⣿⣷⣿⣿⣧⠀⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡷⢋⡀⣿⣿⣿⢡⣾⣿⣿⠁⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣸⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠘⣿⣿⡏⠻⣿⣿⣿⣧⠻⣿⣿⣿⣿⣿⠏⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣧⣴⣾⡇⠀⣻⣿⣿⣤⡙⣿⣿⣿⡆⠙⠿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣇⠈⢻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠁⠈⢿⣿⣿⣿⣿⠄⠀⠙⠉



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣏⠄⢾⣋⣙⣛⣛⣉⣛⣙⣋⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢠⣿⡗⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⢲⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠑⣶⡄⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢁⣿⣿⠁⣼⠇⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠊⠀⠀⠀⠀⠀⠀⢀⠎⠀⢀⠄⠀⣼⡗⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣸⣿⡟⠚⠉⠙⢦⡀⠚⠋⠉⠉⢸⣿⠁⠰⠆⠀⢀⣀⠀⣀⠀⢀⠎⣠⡤⡜⠲⠞⠛⢉⡃⠐⣟⠁⢀⡀
⠀⠀⠀⠀⠀⢀⠎⠁⣹⡆⢠⢠⣿⣿⠁⠀⠀⠀⣼⡇⣰⡞⠉⢀⣿⡇⢰⣿⠁⣴⠏⠀⢸⣿⠀⠎⣼⣿⠀⠁⠀⠀⢠⡿⠁⢠⣿⢁⣾⠇
⠀⠀⠀⠀⠀⢸⠀⠸⠿⠁⢸⣾⡿⢃⡀⠀⠀⣰⡿⢻⣿⠇⠀⢼⡿⢀⣿⡇⢸⣿⠀⠀⠸⣧⠜⠘⢿⣇⠀⠀⠀⠀⠻⠧⠰⡿⠃⠺⠏
⠀⠀⠀⠀⠀⠈⠦⣀⣀⣠⠾⠛⠧⢄⣹⣧⠾⠛⢁⡈⠉⠁⠁⠈⠁⠀⠈⠁⠀⠉⠉⠀⠀⢀⡠⢀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣼⡏⠀⠀⣾⠇⠈⠀⠀⠀⠀⠀⠀⠀⣴⡎⠲⠀⠀⣾⠃⡸⠀⠐⣆⠀⢀⣾⠇⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢠⣿⠇⠀⣸⡿⠀⣀⣠⠀⠀⢀⡀⠀⠀⠻⠧⣤⣀⠀⣿⣧⠁⠀⣀⡬⡁⢸⡿⢠⡀⢰⣿⠄⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⣿⢇⣾⡟⠙⢲⢃⣾⠇⣸⡏⠀⢀⢢⣾⠇⢸⣿⡀⣼⣿⠀⠀⣿⡇⠀⢩⣿⣿⠀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠉⠉⠲⣤⣸⡿⠈⠿⠧⠴⠃⠸⣟⠀⠿⣇⠴⠃⢿⡏⠀⠀⣿⡇⠻⠧⠄⠊⠻⠤⠂⠀⣿⡏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⢨⣿⣧⡀⠀⠠⡖⠒⠄⠀⠀⠀⢀⡀⠀⢀⣀⣀⠀⣿⡿⠀⢀⣤⠖⠠⢄⠀⢀⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣠⡴⠟⠁⠉⠻⠦⣄⡠⠴⠁⠀⠀⡞⠁⠀⠀⠀⠀⠉⢙⣿⢷⣤⣈⣻⠂⢀⣸⠃⠸⣿⣇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⢀⣠⡾⠋⠀⠀⠉⠉⠉⠉⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣄
⠀⠀⠀⠀⣿⠀⠀⠰⠆⠀⢀⣀⡀⠀⢰⣌⠿⢀⠀⠀⠀⠀⠀⢴⡄⣤⣠⡄⣠⠀⣤⠀⡴⢦⡀⠀⠀⣶⣤⡆⠀⣶⣶⣿⣿⣿⣿⣿⣿⣿⡟⠛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣶⣦⢸⡇⠀⢿⣭⣿⡀⠀⠉⠂⠀⣀⠀⠀⠀⠀⢸⠟⣿⣻⡀⣿⣀⣿⣀⣷⣼⠇⠀⠚⢿⡿⠷⠀⣿⡟⣹⣿⠀⠋⢸⡯⣿⡇⠀⣿⣿⠉⣿⡏
⠀⠀⠀⠀⠻⠿⠋⠘⠷⠇⠺⣦⡿⠀⠀⠀⠀⠀⠨⣀⣷⠀⠀⠘⠀⠉⠈⠁⠉⠉⠉⠉⠈⠁⠀⠀⠀⠈⠀⠀⠀⣿⣟⢊⣿⠀⣿⠈⣇⢛⣧⠀⠉⢻⣘⣿⡇
⠀⠀⣿⠀⠀⣾⠆⣰⡋⢳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢲⣆⡀⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⡇⠉⠁⠘⠿⠟⠃⠀⣶⣴⣖⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣶⣄⡀⠈⢿⣦⡀⠀⠀⠀⠀⠠⣶⣞⡁⠀⠀⠀⠀⣠⣄⡀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⠟⠃⠀⣼⣿⠿⡿⢿⠿⡟⠻⣿⠟⠻⣿⡄⠁⠉⠛⠷⣶⣶⠾⠀⣶⣴⠀⠀⠀⣀⠈⠻⠿⠀⠀⠀⠀⣀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⣿⣧⠘⣧⠘⣠⣧⡘⢻⡀⠔⣿⣿⠀⠀⠀⠀⠈⠀⠀⠀⠤⠾⠆⠀⠀⣿⠀⢐⠠⡷⠐⣟⠀⣿⠀⢰
⠀⠸⡄⠐⠃⠂⠀⠆⠀⠛⠲⠔⠖⠂⠀⠀⠘⣿⣟⠛⠛⠛⡟⢟⢹⡟⠙⢙⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠙⠛⠈⠁⠙⠃⠉⠃⠙⠛⠈
⠀⠀⣴⠀⣤⣤⢠⣷⠀⣴⠶⠂⣠⠄⠀⠐⠀⠘⢿⣿⣿⣿⣷⣾⣶⣷⣾⣿⠟⠀⠐⠀⣀⠐⠃⠀⠀⠀⠤⠿⠂⠀⠀⡿⢿⣮⣵⣿⡗⣧⢹⠹⣇⣷⠂
⠀⠀⠮⠇⢻⠉⢸⡟⡇⢻⣦⡄⢹⠄⠀⠀⡀⠀⢀⡈⠙⠛⠛⠛⠛⠛⠉⠀⠀⣴⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠁⠀⢀⣄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣾⠁⠀⠈⠁⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠛⠁
⠀⠀⠀⠀⠿⠗⠀⢰⣾⠀⡎⠉⠂⠈⠁⠀⢸⡏⠙⣶⠄⠀⣿⠉⢱⡆⢰⠀⣿⣿⢰⣧⣷⠘⣇⣼⠀⠒⠀⠀⠀⠀⠀⠸⡃⢰⣷⢠⣿⡀⣾⠛⠁⣶
⠀⠀⠀⡄⠀⠀⠀⠀⠉⠀⠼⠾⠥⠼⠵⣤⠈⠉⠉⠁⠀⠀⠇⠀⠈⠳⠎⠀⠏⠻⠘⠋⠋⠀⠒⠃⢤⡖⣴⢰⡆⢶⡄⠙⠋⠘⠀⠘⠃⠃⠙⠟⠂⠓⠂
⠀⠀⠀⣷⣀⡀⢀⡤⢤⡀⢀⡴⠒⣤⠀⠈⠦⢠⣤⢀⣤⢀⣠⡀⢀⣀⢀⡀⣀⡀⣤⣤⡄⣤⡄⣀⣀⠀⣤⣄⠉⠀⠀⣶⠂⣶⠀⠂⣠⣄⢠⣀⢾⠂
⠀⠀⠀⣿⣿⠇⠘⠷⠾⠃⠈⢿⣶⠿⠀⠀⠀⢸⡿⠿⣿⣾⡁⣹⣾⣿⣾⡇⠯⣅⠈⣿⠀⣿⠀⣿⡼⢋⡓⣦⠀⠀⠀⠁⠀⠛⠂⠑⠸⡽⠘⠉⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣿⠘⠿⠟⠘⠃⠛⠊⠳⠛⠀⠋⠀⠉⠁⠿⠙⠆⠙⠋⠀⠀⣀⣀⣤⣤⣤⣤⣤⣀⡀⠀⠀⢰⡦
⠀⠀⠀⠀⠀⠛⠃⠀⠀⠁⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢠⡀⠀⠀⡀⠀⠀⠀⣠⡶⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣦⣄
⠀⠀⣤⠀⣤⣄⠀⣶⠀⣠⣴⡆⢀⣀⠀⢠⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⠀⠀⠶⠀⢀⣓⣙⠀⢀⣾⣿⡇⢸⣿⡯⣟⠉⣿⡁⣸⣿⠀⣿⣿⢹⣧
⠀⠀⣵⡆⣿⠛⢰⡟⡇⢻⣇⣀⢸⣇⠀⠸⣿⡟⢿⢻⡇⣿⢫⣿⠹⢻⠛⢻⣻⡇⠀⠛⠆⢠⣈⣛⡀⢸⣿⣿⣧⣈⣹⣧⣽⣆⣋⣷⣌⣼⣄⣉⣿⣼⣿⡇
⠀⠀⠀⠀⠈⠀⣬⢁⡁⡈⢉⠉⢀⡁⠀⠀⣿⣿⣿⣿⣷⣾⣷⣾⣷⣿⣾⣾⣿⡇⠀⠰⠆⢐⣊⣉⠁⠸⣿⣿⡟⣿⠛⣿⠛⡇⣿⠹⠏⣿⢉⡏⡏⢹⣿⠇
⠀⠸⠀⠿⠴⠋⣿⣜⠷⠏⠻⠸⠽⠇⠀⠀⣿⣯⢿⣛⣿⣇⣟⣬⢠⣇⢺⣏⣿⡇⠀⠨⠀⠸⠏⠹⠃⠀⠙⢿⣧⣿⣴⣷⣾⣧⣨⣷⣾⣿⣶⣷⣷⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠐⠆⠐⠉⠓⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⡿⠿⠛⠉⢀⣤
⠀⢠⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉
⠀⢸⣿⢛⣿⣿⠟⡟⣿⣟⢿⡻⣻⣿⡆⠀⠀⠀⠀⠀⢴⢀⣀⠀⢰⢀⣤⠀⢸⢾⢠⡖⣦⢠⡒⣤⢴⡶⠀⠀⠀⠀⣠⡶⠆⣶⠀⣶⠰⣿⠖⠀⣴⣟⣷
⠀⣼⣿⣶⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠁⠀⠀⠈⠈⠉⠛⠋⠈⠛⡁⡈⠁⠀⢠⡄⠀⠸⣧⡤⠹⣦⠟⠀⢹⣦⡶⠻⣯⡵
⠀⣿⣏⢽⡏⢉⡿⠀⣿⠁⣤⣿⠉⣿⡇⠀⣤⣤⡄⠀⠀⠀⢸⡆⣶⡆⣴⡇⠀⢠⡀⣼⢤⡞⠂⣿⠁⠀⠀⠀⠀⠀⢠⣤⣠⣄⣀⢠⡄⣰⠄⢠⣤⣴⡤
⠀⠿⢦⣾⣧⣿⣧⣼⣽⣦⣬⣽⣬⡿⠇⠀⠻⠿⠋⠀⠀⠀⠉⠁⠀⠁⠀⠀⠀⠘⠁⠉⠈⠙⠃⠛⠛⠀⢠⡄⠀⠀⠀⠘⠑⠋⠉⠲⠟⠘⠶⠛⠿⠚⠓
⠀⠀⠀⠀⠈⠉⠙⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢢⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣳⠄⠀⠃⠀⠀⠀⢠⣴⣤⣀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⢀⣴⠀⠀⠀⠀⢰⣗⢶⣷⣀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⣦⣤⡥⣤⠀⠀⠀⠀⠀⢸⣿⡿⣿⣠⣧⣴⠀⠘⢛⢾⠀⠀⠀⠀⠀⠀⠀⢀⢻⣿⢸⣿⣦⣶⣶⣴⣾⡏⣻⡟⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⡏⣴⢻⡄⠀⠀⠀⠀⠈⢩⣦⣿⣿⠇⣷⣿⣇⣆⡊⠀⠀⠀⠀⠀⠀⠀⢸⠸⠟⠘⣟⡛⣿⣿⣿⣿⣧⣿⡇⣸⣿⣦⣶⣊⠄
⠀⠀⠀⠀⠀⠀⠀⡸⠃⣽⠻⠇⠀⠈⢉⠀⣤⣶⣿⣿⣷⣴⣿⠟⣻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣿⣿⣶⣷⣾⣿⢻⣿⡆⣿⣿⣿⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈⠁⠀⠀⣈⠁⣿⣿⣿⣿⣿⣿⣿⣦⣽⣿⣿⣿⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⣿⣽⣿⣾⣿⡟⠚⠛⠛⠛⠛⣿⣿⣿⣿⣯
⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠀⣿⣿⣟⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢻⣛⠃⠀⠀⡾⣛⣿⠇⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣿⣿⣿⡏⠁
⢸⣧⡶⣆⠀⡄⠀⠀⠀⠀⠀⠀⠀⡐⣈⠀⣿⣿⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⠂⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣠⠹⠉⣿
⠘⣿⣇⣷⣲⡆⠀⠀⠀⠀⠀⠀⡀⠂⠐⠂⣿⣿⣫⣻⣭⣿⡇⠀⠀⠀⠀⠀⣶⣶⣿⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢦⣄⡀⠀⢹⣄⡇⠰
⢸⣿⣿⣿⣍⣇⣤⣤⣤⣤⣤⡞⠻⣀⡤⣀⠉⠉⠉⣽⣿⣿⠁⠈⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣬⣭⣤⣬⣤⡀⠀⡨⢎⡙⠿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡘⠡⠶⢿⡇⠀⠀⣼⣿⣿⡄⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⡆⠉⠀⠸⣾
⢀⡆⠨⢽⡄⣿⣿⢿⣿⣿⣿⣿⣿⣶⣤⣤⡄⠀⠀⣼⣿⣿⡇⢀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠈⠀⠉⠙
⠀⣀⣤⡑⠆⣿⣿⣾⣿⣿⣿⠄⣄⡄⣀⠈⠀⠀⠀⢼⣿⣿⡏⢽⣷⣶⡖⠘⣿⣿⣿⡿⠛⢹⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠘⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠐⠀⢠⡄⠀⠀⠀⠀⠘⠉⠉⠓⢻⣿⡿⠷⠄⠈⠙⠛⠀⣰⡆⣠⣴⣶⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⢀⡉⢉⣛⣛⡃⠀⠀⠀⢠⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⡖⠀⡀⠘⠿⡿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⣿⣿⣿⣭⣩⡍⠁
⢸⡿⠋⠙⠻⣿⡇⢘⡀⢈⠁⠑⣂⠀⠀⠈⠋⠀⠀⠀⠀⠀⢀⣿⣿⣿⢿⣿⠀⢰⣷⣶⣤⡀⠀⠀⠘⣿⠂⣴⣿⣿⣿⣶⣿⣿⣿⣿⣟⣧⣤⣤⣤⣤⡄
⢸⣄⣄⠀⠀⢸⡇⢨⠀⠀⠀⠰⣿⣷⣾⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣴⣿⠠⡾⢿⠿⠿⠇⠀⢀⣼⣿⠀⠻⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢸⣿⣿⣷⣶⣾⡇⠀⠐⠀⠀⢰⣿⣿⣿⣿⣷⣀⣀⡀⠀⢀⢀⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣤⣤⣼⣿⣿⢶⣶⣾⣿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢸⣿⣿⣿⣿⣿⡆⠓⢀⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⢿⣿⡄⠻⣿⣷⡆⠘⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⡇
⠈⠉⠉⠉⠉⢛⣩⣽⣿⢿⣿⣿⣿⣿⣿⣿⠟⠋⣠⣨⣻⣉⡁⠀⠀⠀⠀⠀⠀⠀⠈⠸⢸⣿⣿⣿⣿⣿⣧⠔⠋⢟⡔⠀⠀⢀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⠟⢀⣼⣇⠈⠈⠁⢁⣈⠀⢸⣿⠛⠻⣿⡇⠀⠆⠄⠀⠀⣀⡀⠀⠀⢸⣿⣿⣿⣿⣿⡗⠀⢀⠀⣹⣆⠀⠀⡀⠂
⠀⠀⠀⠀⠀⣿⣷⣶⣾⣿⡇⢀⡀⠀⠘⠟⠀⢣⠀⠀⢰⣿⡇⠐⠀⠂⠂⠀⠉⠇⢢⠐⢾⠛⠛⠛⠛⠟⢃⠄⢤⡌⠀⠛⠀⠠⠀⠂
⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠃⠘⣧⠀⠘⠃⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠀⣢⣦⣾⣶⣯⠸⠃⠀⠀⠀⢀⣴⡄⢿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡄⡾⡀⣀⣓⡲⠾⣿⣂⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠰⣀⡄⠀⠀⠛⠛⠀⠘⠊⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠛⠛⠛⢿⣿⣟⣻⣶⣭⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣚⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠻⢿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠐⠰⠇

⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⣿⣟⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⡇⠀⠀⠀⠀⠀⠀⢀⣠⣾⢿⣿⡿⣿⡟⠛⠛⠷⠶⣤⣄⣀
⣿⡇⠀⠀⠀⠀⣠⡴⠛⣩⣴⠾⣿⠗⠻⠁⠈⠁⠀⠀⠀⠉⠻⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣄⣀
⣿⡇⠀⠀⢀⣼⠋⢀⡾⠟⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣦⡀⠀⢀⣠⡤⠖⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⢻⢷⣤⡀
⣿⡇⠀⠀⣼⠃⢰⡟⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠾⠋⣡⠔⠀⠈⠀⠀⠀⠀⢀⠀⠀⣀⣀⣹⠞⠉⠻⣦⡀
⣿⣿⠀⢸⡏⠀⢾⠀⠀⠀⠀⢀⡄⠀⠀⠀⣰⡗⠀⠀⣠⣶⡆⠀⠀⠀⠀⠀⠉⠁⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆
⣿⣿⠀⢸⡇⠀⠀⠀⠀⠠⠀⢸⡇⠀⠀⣰⣿⠃⠀⣼⠏⠘⣧⢰⡆⠀⣠⠀⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⣿⣿⠀⠸⣇⠀⠸⡄⠀⠀⠀⢸⡇⢀⣀⡻⠿⠿⣾⡟⠀⠀⣿⡜⣿⢠⡟⢠⣿⡉⠛⠟⠀⠀⢀⡀⠀⠀⡀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⣿⣿⠀⠀⢿⣆⠀⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠻⣿⣶⣾⣿⠇⣿⣿⠃⣼⣟⣁⣠⣤⡀⠀⠈⢿⡄⣼⣿⡟⣿⣴⡄⣶⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇
⣿⣿⠀⠀⠀⠻⣷⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠿⠟⠛⠉⠉⠀⣀⠀⢘⣿⢣⡟⠀⣿⣿⠀⢸⣧⠀⠒⠀⢀⣴⣿⣿⣿⠃
⣿⣿⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠛⠷⠶⢿⣿⣶⣾⣿⠀⢀⣴⣾⣿⣿⡿⠃
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣤⣀⠀⠀⠀⡠⠞⠃⠉⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⡟⠁
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣞⠀⠀⠀⠀⠀⠘⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⠿⠉
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣤⡀⠀⠀⠠⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⣿⣿⣿⣿⣦⠀⠀⠀⣿⡀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁
⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡏⣻⠋⠻⣿⣦⡀⣸⠆⣠⣶⣿⣿⣿⣿⠿⠟⠋⠁
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠠⣿⠇⢩⣿⣀⣸⣿⣿⣿⣾⣿⣿⡿⠟⠉
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠁⠀⢠⣿⡏⠀⠺⠛⢿⣿⡿⠁
⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣶⣿⣿⣿⣦⣴⣶⣾⣿⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⡄⢨⠒⠆⠄⢠⡈⡶⠁⢀⡔⠀⠀⠀⠀⠀⡼⠀⠠⡀⡈⡀⢀⠀⠀⠀⠀⢀⠀⠀⠃⠇⠀⠆⠇⠸⠀⠰⠀⠀⠀⠀⠀⠀⠀⡏⣦⠠⠐⠢⠭⣼⡀
⠀⠀⠀⠙⣌⠧⠀⠋⠸⠑⡇⠞⡟⠀⡻⠀⠀⠀⠴⠇⠠⠆⠙⠇⠘⠇⠀⠀⠀⠀⡨⣧⢠⢠⡀⠁⡾⢰⡇⢀⠀⡀⠀⢀⠀⠀⠀⠀⡇⠉⡷⡏⢺⡄⡏
⠀⠀⠀⠀⢀⠈⠚⠀⢀⡆⡴⠀⠃⣤⠀⠀⠀⢀⠀⡀⠀⠀⠀⢂⡀⠀⡀⠀⠀⠀⠀⠇⠼⢸⡧⠇⠣⢸⢻⠺⠀⠸⠁⠘⡆⠀⢀⢠⣀⣀⣅⢀⢈⡁⣈⣄
⠀⠀⠀⠀⢸⢱⢾⢸⠺⡇⢿⠀⠏⢸⠁⠀⠀⠈⢰⢃⡏⡇⣼⠡⠇⠏⡆⠀⠀⠀⠀⠀⠀⠈⡁⡄⠀⡄⢀⠀⡀⠀⠀⠀⠀⠀⠈⢸⢡⡇⡏⠲⠸⠇⠇⡾
⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠃⠓⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠀⠀⠆⠘⠘⠀⠃⠂⠐⠀⠀⠀⠀⠀⠠⠠⠄⠠⠀⠀⠆⠂⣄⠀⠀⠀⠀⠀⠀⠀⠦⠀⠇⠟⠸⠃⠳⠀⠀⠀⠀⠀⠠⠀⡟⠂⣿⠀⠣⡀⡇⠠⠄
⠀⠀⠀⣸⢰⢸⢂⢠⠏⣾⣤⠄⣧⠠⡀⠀⠀⣄⢤⡰⡤⣀⢰⢤⠄⣠⡄⡀⠀⠀⠀⠀⡄⡄⡄⢀⢰⢢⣆⢀⠀⡀⢤⠀⠀⠀⢠⠠⡠⡀⠈⢠⠄⣴
⠀⠀⠀⠀⠉⠺⡬⠈⠊⠙⠈⠀⢃⡈⠁⠀⠄⠘⡎⠃⠧⠉⠀⠈⡃⢀⠜⠰⠣⠀⠀⠀⠁⠃⠇⠃⠘⠈⠹⠉⠀⠁⠈⠁⠀⠀⠸⠸⠁⡃⠆⢘⠀⠓⠀⠀⠸
⠀⠀⠀⠀⠀⢸⡃⡾⠪⡄⢶⠆⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡈⠠⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⢀⣱⠢⠰⣤⠀⠁⠀⠀⠀⠀⢠⡤⣤⠰⡄⢠⡄⢠⡀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠈⠁⠓⠀⠓⠘⠃⠘⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⡆⢶⡆⢠⢰⢰⣰⢰⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠐⠆⢸⠗⠀⠦⠀⠀⠀⠀⢀⠠⠔⠒⠂⠂⠠⠂⠠⡀⠀⠀⠀⠀⣸⢡⢏⢇⡝⢡⢏⡏⡍⡀
⠀⠀⠀⢀⡀⢁⣨⡀⡌⠀⠀⠀⠀⠀⠀⠀⠀⡆⣶⢲⢀⢀⡇⣾⡀⡄⢦⠀⡆⠀⠀⠀⠀⢤⡠⣄⢶⢰⢀⣀⣀⠈⠀⠀⠀⠀⠀⣀⡀⡈⢉⢁⢀⠀⢀⣉
⠀⠀⠀⠀⠑⢸⠁⡇⡏⣶⢸⢿⠖⠀⠀⠀⠀⠁⠁⡸⢈⣀⠃⡁⠇⢁⠈⡀⠁⠀⠀⠀⠆⠌⡇⠟⢸⠘⠏⠞⡜⡜⠀⠀⠀⠀⠀⠁⡁⠁⡈⠈⡈⠀⡀⠉
⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠈⠛⠀⠋⠀⠘⠀⠁⠈⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠘⠃⠀⠀⠀⠀⠘⠩⡗⠀⡇⠀⢰⠄⡇⠘⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣭⣛⣍⣍⣉⣯⣍⣿⣻⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢀⡤⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠉⠉⠉⠉⠉⠳⡄⠀⣿⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠶⢖⣒⣛⣓⣒⠒⠲⢦⣄⠀⠀⢸⠀⠀⠀⠀⢀⣴⠃⠀⢹⡆⠀⠐⠞⠋⠀⠀⣠⢔⣫⢖⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢛⣽⣿⠀⠀⢸⣀⣀⣤⠶⠋⠁⠀⠀⠀⢻⡄⠀⠀⠀⠀⠺⣿⠯⣭⠤⠴⠞⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⡡⠖⠁⣿⠏⠀⠀⢸⠋⠡⠀⠀⠀⣰⡞⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠑⠲⠿⠀⠀⡆⠀⣸⢄
⠀⠀⠀⠀⠀⠰⡀⡠⡄⠉⡉⠀⠀⠀⠁⠀⠀⠀⢸⡄⠀⠀⠀⠀⠹⣧⣤⣤⡴⠏⠀⣀⠀⠀⠀⣠⠀⠀⠀⢄⠤⠛⠋⠁⢀
⠀⠀⠀⠀⠀⠀⠉⠙⠤⠼⡏⠀⠀⢀⠀⠓⠀⠀⠘⠠⠀⠘⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⣀⣤⣤⠄⠀⠈⠁⠀⢀⣀⣀⣠⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣷⠆⠠
⠀⠀⠀⠀⠀⠀⡏⠀⢀⠀⣴⣿⣿⣿⡿⣿⣿⣿⡿⠛⣉⣤⣤⣤⣍⠻⣿⣿⠹⣿⣿⣿⣿⢁⣿⣿⡿⠋⣠⣴⡆⢹⣿⣿⣿⣿⣿⠆⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢈⣽⣿⣿⠀⣿⣿⡟⢠⣾⣿⡿⢛⡻⢿⣷⡸⣿⠀⣿⣿⣿⡇⣼⣿⡟⠀⣴⣿⣿⠇⣸⣿⣿⣿⣿⣧⡄
⠀⠀⠀⠈⠀⠀⡇⠀⠀⠰⣿⣿⣿⡟⠀⣿⣿⢠⣿⣿⢋⣾⡿⣿⡎⣿⡇⣿⡀⣿⣿⡿⢠⣿⡿⠀⣼⣿⠟⢃⣼⣿⣿⣿⣿⣿⡿⢃⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠠⠀⢈⣽⣿⡇⢸⣿⣇⢸⣿⣿⢸⣿⡀⠟⣱⡿⢱⣿⡇⢻⣿⠇⣾⣿⡇⢠⣥⣴⣾⣿⡿⢁⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠿⣿⣿⠁⣼⣿⣿⣼⣿⣿⣦⡛⠿⠿⣛⣵⣿⣿⣧⢸⡟⢰⣿⣿⡇⠸⣿⣿⣿⡿⢁⣾⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠰⠶⠿⠖⠀⠀⣠⣾⣟⠀⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⢡⣿⣿⣿⣿⣄⠙⠛⣉⣴⣿⣿⣿⣿⣿⣅⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟
⠀⠀⠀⠀⠀⣤⠈⡷⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⡀⠀⠈⢉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⢹⠛⠁⠀⡀⠈⠁⢀⠀⠀⠃⠀⣼⠃⠘⠀⣠⣾⣿⣿⣿⣿⣷⡄⠀⠀⣤⠀⢀⠀⠀⢰⠄⠙⠀⠀⠳⢾⠉⢠⡆
⠀⠀⠀⠀⠈⠁⠀⠀⠀⢹⣆⠀⠉⠀⠀⠀⣰⠏⠀⠀⣰⣿⣿⣿⣿⠟⢻⣿⣿⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠉⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢰⠏⠀⠀⢰⣿⡿⠀⠘⠃⠀⣼⣿⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠹⣦⠀⢠⡏⠀⠀⠀⢸⣿⣿⡀⠀⢀⣾⣿⣿⡏⠀⠀⣿⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡏⠀⠀⠀⠀⢸⣿⣿⣷⣤⣾⣿⣿⡿⠁⠀⠀⢻⡇⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠈⢷⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⣴⠊⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⣾⠀⠀⢀⣿⠀⠀⡀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠣⠀⠀⠀⢿⡇⠀⢸⣿⠀⠀⠈⢀⠆⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡏⢀⡶⢠⡄⠀⠸⣧⠀⢸⡇⠀⠀⠀⠈⠀⣸⡇⠐⣦⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⡇⠀⠀⢿⡄⠸⣇⠀⠀⠀⠀⠀⣿⠇⢠⡿⢸⡏⠀⠀⠀⣠⠊⠰⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠈⠁⠀⠁⠀⡀⢸⡇⠀⠻⠌⠀⠀⠀⠀⠿⠊⠼⠃⠘⠃⠀⠀⣼⠃⠀⠁⡴⠃⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠠⠀⠀⣿⢸⠇⣴⠉⠂⠀⠀⠀⢠⡖⠀⠀⡤⠂⠀⠀⢰⡟⣠⢂⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣿⠁⠀⠀⠐⠀⠀⡠⠏⠀⣿⠀⠀⠀⠀⠀⢸⣇⠀⠸⠀⠀⠀⠀⣿⢸⠁⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⠀⠈⣿⡀⠀⢀⡠⠀⢰⡏⠀⢰⡿⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⢤⡀⢰⡆⢣⡄⠀⡄⣾⠀⣶⠀⠉⢳⡆⠀⠀⠻⣧⠠⣾⠁⠀⢸⡇⠀⣼⠇⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣇⢸⡇⠀⠸⣏⢸⡇⠀⠀⣿⠀⠄⠌⠀⣿⠀⣿⠀⡀⢻⡇⢀⣿⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠈⣿⡈⠓⠈⠀⠉⠀⠙⠈⠘⡏⠀⠿⡀⢀⡟⠀⠙⠂⠀⠈⠓⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡇⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⠉⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇⠀⢀⣠⡴⢶⣦⣄⠠⢤⣴⣶⡖⠀⠠⢤⠄⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇⢠⣿⣿⡇⠀⣿⣿⣧⠘⣿⣿⣷⠀⠀⡟⢠⣿⣿⡟⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠘⣿⣿⡇⢸⣿⣿⡇⠀⢸⣿⣿⠀⢹⣿⣿⡆⣰⠃⢸⣿⣿⡟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣿⣿⡇⠘⢿⣿⣧⠀⢸⣿⠏⠀⠀⢻⣿⣿⠃⠀⠈⢿⣿⣧⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠒⠀⠀⠀⠚⠛⠛⠓⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠈⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⡆⠀⠰⢤⣄⢀⣴⠶⢶⣦⣄⠀⠠⣤⣤⣄⠠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⡀⠀⡇⢰⣿⣿⠀⠈⣿⣿⡆⢰⣿⣿⡏⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣠⠀⢺⣿⣿⠀⠀⢹⣿⡇⢸⣿⣿⡇⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡏⠀⠘⢿⣿⣇⠀⣼⡿⠁⠸⣿⣿⡇⣠⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣦⠀⢀⠟⠀⠀⠀⠀⠉⠉⠋⠉⠀⠀⠀⠈⠙⠋⠁⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠔⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣡⠀⠀⠀⠀⠈⠀⠸⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⠀⠀⡀⠀⠀⣠⢦⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠐⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⢠⡏⠈⣆⠀⣷⣸⢰⡉⠙⢧⢠⣯⠀⢠⠏⡇⠀⠀⠀⠀⣈⠀⠀⠀⠠⢀⠄⡀
⠀⠀⠀⠀⠄⠀⠆⠀⠀⠀⠀⠓⠐⠀⠀⠀⠀⠘⢇⠲⠘⠦⢿⡇⣨⡷⠶⢺⣿⣧⠀⢸⠠⠃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⡄⠀⠀⠀⠀⣀⠠⠀⡀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠦⢼⢷⡇⠁⠀⢸⠋⣿⡀⠘⡆⠘⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡄⡀⡄⠈
⠀⠀⠇⡄⠀⠀⢀⠁⠀⡀⡠⢀⠠⢰⡄⠀⠀⠈⠃⠀⠀⠴⠋⠀⠙⠦⠔⡃⢀⠈⣛⠋⠉⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠞
⠀⠘⠐⠂⠂⠀⢎⢸⠃⠾⠇⠘⠑⠞⠃⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀⠁⢘⠀⠇⠈⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠈⠐⠀⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈⡡⠀⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⡄⠂⠀⡈⡆⠜⠀⠀⠀⣀⠀⠀⢀⡤⠄⠀⢀⡀⠀⠀⠀⠀⡎⡠⡇⢰⡁⠀⠀⠀⠀⠀⠀⠀⢠⡹⢦⣰⠀⢴⡀⠀⠃⠠⠀⠄⠂⠀⠑⠰⠀⠂
⠀⠱⡄⠀⠀⠉⢀⠀⢄⠀⠛⠁⠀⠘⢦⠀⠀⣩⠄⠄⡠⠤⣄⠀⠀⠀⠀⣠⠄⢠⠄⠀⡄⢠⡠⡄⢦⣤⠙⡄⠀⠀⠀⡄⠀⢠⠀⠀⡀
⠀⠁⠀⠀⣄⠐⠀⠀⠀⡀⠄⠀⡀⠀⠀⠠⣿⣡⠔⣾⠉⢋⢙⠥⢤⠄⣾⢸⠁⢿⢀⡄⢳⣈⠧⠹⣄⠘⠦⠃⠀⠀⢐⢀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⢁⠀⠀⡔⠉⠀⡀⠀⢃⠪⠀⠀⠀⠀⠀⠈⠉⢡⣏⡠⠺⣤⣿⢸⡆⢸⣄⠹⠀⠉⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈⠀⠀⠀⠁⠀⠀⠤⡀
⠀⠀⠀⠠⠜⠀⠠⠀⠀⠀⠣⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⠴⠋⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆⠠⠐⠀⠆⠐⠄⡀⠇⠂
⣴⣴⣤⣦⣦⣴⣶⣴⣴⣶⣦⣶⣾⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣶⣶⣦⣴⣤⣶⣦⣶⣶⣶⣶⣶⣶⣶⣦⣤⣶⣶⣴⣴⣦⣴⣦⣴⣶⣦⣴
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣽⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠖⠠⠈⢿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⡠⢄⠀⢄⠀⠀⣠⠘⠿⠻⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠠⠎⠀⠀⠉⢉⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠒⠒⠀⠀⠙⠃⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢀⣾⡛⠿⣿⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠖⠄⣤⣾⣿⡇⣿⢏⢸⡇⢟⡽⣭⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢞⣫⣭⣻⣳⣏⣬⢸⢡⢻⣇⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣏⣾⣿⣿⡷⣛⡛⢿⠹⠿⢳⣦⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⣻⡸⣿⣿⠟⣰⡿⢡⢏⡞⣴⢫⡶⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠹⣿⡿⣿⣷⢭⣵⣾⣦⣶⣭⡞⣼⡿⢨⠾⣫⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⣡⣾⡟⣰⣶⠘⣿⢛⣿⣿⠿⡶⠾⣷⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢁⣿⣿⠁⣿⡿⢰⡏⣼⣿⡟⢸⡇⣶⡦⢹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⢿⣧⣭⣴⣿⣇⠻⠿⢁⣿⢠⡄⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡸⣷⡟⣴⣾⡏⣿⣿⠞⠿⣷⣾⣧⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡻⠷⣹⢇⣶⣿⢸⣿⡟⢸⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣬⣽⣏⡻⢿⣇⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠘⢷⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⢣⣆⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⣠⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠸⣦⡏⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⢘⣧⡀⠀⠀⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢾⡿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣷⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠬⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣶⣷⡄⠀⠀⡠⢺⣶⠀⠀⡏⢺⣿⠛⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⠀⢠⡂⠀⠈⠓⠂⠊⠀⢻⠟⢀⠴⠃⠀⡽⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⠟⠳⠤⠤⠤⠤⠔⢉⡴⠁⠀⣰⣿⠁⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⡄⡷⣄⡀⠀⣀⣤⣴⡋⢁⣠⡾⠋⠛⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠿⠃⠀⠉⠉⠉⠻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢀⡇⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⡠⣀⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡾⠃⠄⠇⠔⠓⢠⠓⠀⠀⡀⠓⠃⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢄⡤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠄⢤⡤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠠⠒⠁⠚⢦⡰⠁⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠈⠓⢾⠃⠀⠀⠉⠲⠄
⠀⠀⠀⠀⠈⠂⠀⠀⠀⡠⢄⢇⠀⡐⠓⡄⠀⠀⠀⠱⡄⠀⠀⠈⢼⠁⠀⠀⠀⡆⠰⢈⢦⡐
⠀⠀⠀⠂⠀⠀⠀⠀⠸⠠⡯⠉⠣⠁⡶⡇⠀⡈⠀⠀⠐⢠⠄⠈⠆⠀⠀⠀⠀⢓⠐⢫⡺⠛⠀⠀⢀⣼⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠉⠁⠂⡤⠰⡆⢙⡈⢱⢀⠠⢄⠀⡊⠠⠀⢠⠃⡾⠏⠉⠁⡂⢀⡤⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢙⠀⠦⠚⠀⠦⡇⢨⠀⢹⠃⣀⡀⢡⢡⠀⠄⡄⢀⡡⠂⠈⡄⠠⠊⠀⠀⠀⣀⢄⡃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠼⠀⠥⢀⠐⠁⡇⠸⠤⢬⡀⠢⠄⢸⢿⡄⠘⠀⡼⠸⠃⢈⣁⣢⠔⠚⠉⠙⢇⠰⠃
⠀⠀⠀⠀⢀⡀⠀⠈⡀⠺⠄⠀⡤⠞⠁⠰⠧⠀⠠⣤⠵⣤⠴⠯⡈⠰⠤⡰⠃⠀⡦⠤⣠⠧⢤⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠒⣲⠋⠁⠁⠠⡐⠁⠀⠀⣼⡒⠀⠀⠸⡠⠂⠀⠠⠟⢅⠀⠐⠀⠀⣡⠖⠁⠀⢀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠳⠄⠀⠀⠀⠈⠂⠤⠘⡍⠹⠎⢁⢆⠞⠁⠢⣰⠉⡇⠰⢰⠄⠛⠀⠀⢐⢛⠋
⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠈⡆⢠⡁⢈⠰⡚⠀⠹⠀⢧⠄⠈⠀⠀⢀⣠⡃⢘⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⡇⢸⠗⠲⣄⢀⡰⣷⢆⠀⢀⢦⣤⠔⠁⢸⠁⠈⡧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⣐⡒⠒⢲⡌⠁⠉⠀⡰⠋⠉⠈⢰⠀⠙⠁⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡣⠀⠀⠤⠂⠀⠀⠍⠆⠀⠀⠀⢂⠀⠒⢢⠇⠀⢀⢘⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠖⠈⠑⠠⠀⠘⠀⠀⠐⢘⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡖⢙⡆⠀⠀⠀⠙⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨⡆⠀⠀⠀⠀⠀⡠⠊⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠕⠁

⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⡀⠀⠀⠀⠀⡠⠀⢘⣦⢃⡀⢀⡀⣀⡀⠀⠀⠀⠀⢈⠇⣇⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠐⡿⠀⣠⠀⣤⢀⣰⠢⠀⠀⠀⠀⠰⠏⠃⠫⠀⠁⠀⠀⠀⠀⠣⠀⢸⠁⠊⢁⠈⠁⡽⠛⠀⠀⠀⠀⠈⢱⠋⣁⣉⢃⡀
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠈⠀⠘⠂⠈⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠹⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠉⡀⠀⠈⠀⠈⠉⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡸⠆⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠁⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸⡖⡜⡠⣠⡄⣇⡠⠖⡘⠈⠁⠀⠀⠀⠀⠀⠀⠀⢰⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠐⣇⣤⣤⢁⢰⢣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⢄⢃⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠖⢰⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠈⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠙⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢰⢰⣀
⠀⠀⠀⠀⠀⠀⠀⠠⠞⠉⠘⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⠟⠃⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢢⠀⡀⠰⠀⢀⢀⡀⠀⠀⠀⠀⠀⠘⡆⢠⣠⢸⢠⢠⢄⡆⣄⡤⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠰⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡜⠃⠱⠇⠃⠁⠋⠇⠀⠀⠀⠀⠀⠀⠣⠀⠈⠈⠀⠀⠈⠁⠁⠘⠁⠨⠂

⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠈⠉⠉⠉⠉⠉⠙⠛⠻⠀⠀⠀⠀⣰⡿⠋⠉⠀⠀⠈⠲⣄⣴⣾⠟⠋⠉⠛⢲⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣶⣶⣴⠀⠀⠀⣶⣶⣶⡾⠁⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠈⠿⠁⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿⣷⠀⠀⠀⡏⠈⠈⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡯⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⡇⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⢰⣿⣏⣀⣀⠀⠀⠀⠈⠉⠉⠑⡆⠀⠀⠀⠀⠀⠉⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⡷⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊
⠀⠀⠀⠀⠀⠀⠀⣶⣟⠓⠢⠀⠀⠀⢀⣶⡖⠒⠒⠂⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠱⡀⠀⣾⡟⠀⣠⢃⣴⣾⠟⠋⠉⠉⠑⢄⠀⣴⣿⠛⠁⡆⠀⠀⢀⣴⡖⠒⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠘⣼⡟⠀⣠⠃⣾⣿⠁⠀⡄⠺⣷⡄⠈⠂⣿⡿⠀⢀⠀⠀⠀⢘⣿⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠈⠀⡠⠀⠀⣿⡟⠀⠀⠀⠀⣿⡇⠀⠃⣿⡗⠀⠸⠀⠀⠀⢸⣿⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆⣀⠀⠀⢸⣿⠁⠀⠀⠀⠀⣿⠃⠀⢶⣿⡇⠀⡄⠀⠀⠀⢸⣿⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠰⠀⠀⢸⣿⠀⠀⠀⡀⣸⠟⠀⢰⢸⣿⠃⠀⠇⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡆⠀⠀⠉⠁⠀⢀⡏⢸⣿⠀⠀⡇⠀⠀⣰⣿⡇⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⡆⠀⠀⠘⢿⣿⣦⣀⣀⣀⣤⠞⠁⢸⣯⠀⠀⠙⠲⠾⠛⠋⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠈⠿⣷⣤⣀⡀⠀⢀⣀⡠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⣿⡟⢀⣴⠿⢡⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠠⢝⡠⠀⡀⠀⠀⢶⡜⠁⠀⢰⣿⢿⣍⠁⠀⣮⡁⠀⣦⠀⣠⣤⠀⢀⣴⣤⣸⣿⠀⠀⠘⣧⠖⠀⠀⢀⠀⣄⡫⠄
⠀⠀⠀⠀⠀⠀⠀⠺⣶⣧⢰⣿⠀⠀⠀⠀⠀⠀⣾⡏⠀⣿⡇⢀⣿⢁⣼⣿⠟⢰⡿⠀⣾⡏⢈⣿⡇⢀⡀⠀⠀⠀⠀⢰⣿⡄⣬⣶⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠘⠿⠀⠆⢸⣷⠀⠀⢻⡇⠀⢿⣇⡀⠙⠉⠘⠋⠀⠘⠧⠔⠉⠛⠉⠿⣤⠞⠁⢀⣾⡇⡐⠀⠿⠣⡄
⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠴⢿⠞⠋⠒⣿⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠚⠙⠻⡶⠦⠀⠀⠀⠀⡖
⠀⠀⠀⠀⠀⠘⠓⠇⡀⢀⡀⠀⠀⠀⣡⣴⡾⠂⠀⢠⣀⠀⠀⠀⠈⠱⡀⠀⠀⡠⠈⠉⠀⠀⣀⣄⠀⠀⠐⢷⣦⣍⠀⠀⠀⠀⡀⠀⡰⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠑⠶⠓⠀⠀⠉⠀⠀⠀⠀⠈⠙⢦⡀⠀⣴⣿⣿⣿⣷⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠉⠀⠠⠞⠶⠚⠉⠛
⠀⠀⠀⠀⠀⠀⣀⣀⠤⢤⠤⠤⠤⠄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠳⣄⣹⣿⣿⣿⣏⣠⠞⠁⠀⠀⠀⢀⣀⣀⣀⣀⠠⠤⠤⠤⡤⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠣⡄⠀⠠⠕⠂⠀⠀⠀⠑⠶⠄⠠⠄⠀⠀⣙⢛⣿⣿⣿⣿⣿⣿⣿⣋⣉⠀⠀⠤⠄⠠⠖⠀⠀⠀⠀⠐⠣⠄⠀⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢆⠈⠀⠲⠖⢂⣀⣀⡬⠔⠒⠪⢉⡠⠊⠹⣿⣿⣿⣿⣿⠏⠳⢄⡉⠕⠒⠨⢧⣀⣀⠀⠲⠂⠀⢁⡰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡒⠋⠁⡠⠄⠴⢁⠴⠋⢁⡴⢒⣿⣥⣶⣶⣶⣦⣿⡒⢦⡈⠛⠦⡈⠤⠠⣄⠉⠑⢺⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠥⠤⠒⠋⠁⠀⣠⢟⡴⢻⣡⣤⣤⣤⣤⣄⣈⡙⢦⡹⣆⠀⠈⠙⠒⠤⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢸⡇⡞⢋⣉⣉⣉⣉⡙⠛⡇⠸⣇⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⣼⠀⢹⠿⠛⠛⠛⠛⠿⡿⠁⠀⢿⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠱⢿⣿⣿⣷⠖⠁⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⣴⠟⠐⠁⢀⣠⢤⠀⠀⢀⣠⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣼⠏⠀⠀⡠⠋⢀⠌⠀⡴⠋⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⡠⠔⠒⠒⢦⡀⢸⡁⠀⣼⠃⠀⠀⡼⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⣰⠋⠀⠀⠀⠀⣸⠇⠈⢳⣾⠃⠀⠀⣼⠃⠀⡠⢀⡄⢀⡄⢀⠀⡠⠶⠀⠀⣴⠋⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⣟⠀⠀⠀⠀⠚⠁⠀⣠⠟⠁⠀⠀⠀⠛⠂⠈⠓⠊⠉⠘⠓⠈⠉⠓⠂⠀⢼⡃⠀⢀⣤⡟⣴⢀⣟⠠⣖⡀⣴⡂⢀⡀⠁
⠀⠀⠙⠦⢀⣀⣀⠠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣉⣉⣱⣟⣀⣉⣁⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠋⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⢠⡿⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⣴⠋⢹⡿⠁⣀⢀⣀⠀⠀⠀⠀⠀⠂⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠃⠀⣾⢣⡋⠀⠀⠀⠈⠑⠢⣤⣀⣀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡟⠙⠀⠀⠀⠀⠀⡀⠰⢀⣀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⡇⠀⠀⣴⠀⡠⣾⠡⣱⠏⠀⠀⠀⠟⠀⠁⢀⡴⠋⠄⠀⢀⡴⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢸⡇⠀⢸⡏⢀⠁⢿⡆⣿⠀⠀⡠⠊⠀⠀⠀⣾⠇⠀⠀⣴⠏⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⡠⠠⡄⠀⠱⡀⠀⠓⠐⠁⠁⢀⠀⠈⠀⠉⠂⠀⠀⠀⣴⠂⠈⢿⠀⠀⣸⡏⠀⢊⡴⢉⠂
⠀⠀⠀⠀⠀⠀⠰⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠃⠀⠀⠀⠀⠀⣯⠀⠀⠀⠁⢰⣿⠁⠀⣾⠕⠁
⠀⠀⠀⠀⠀⠄⣆⠀⠀⣿⠀⠀⢀⢠⢂⡀⡄⠀⠙⠲⠤⢰⠄⣤⠀⡀⣄⠸⣇⢠⡞⠉⢺⡇⠀⢰⡏
⠀⠀⠀⠀⢠⠀⠻⠀⠀⣿⡀⠀⣿⠈⣾⠀⠖⡾⠀⡾⠀⠈⢸⡇⢠⠀⣸⠆⡿⣿⠃⠀⣿⠃⠀⢸⡇⢀⠄
⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⡇⠀⢿⡆⢻⣀⠀⢧⠼⣇⠀⠄⢸⠁⠘⠄⡩⠜⠁⠘⠂⠂⢿⠀⢀⢿⣇⠊
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠓⠘⠜⣷⠀⠀⠀⠀⠀⠈⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢃⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⣿⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢹⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀
⠀⠀⠀⡠⠂⠀⠂⠂⠐⠂⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⠀⠁⠀⠀⢀⠄⠀⠀⠀⠙⢆⠀⢂⠀⠆⠀⡜⠉⠁⠁⠘⡀⠀⠀⠀⠀⠀⠔⠀⠀⠒⠀⠐
⠀⠀⠀⠅⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡔⠈⠐⡀⠀⠀⠀⠚⠀⢠⠀⠐⠏⠈⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠤⢄⠑⠤⠔⠁⠀⠰⠀⠀⠀⠸⠀⠀⣆⣒⣊⡀⠀⠀⠀⠀⠀⠑⢄⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠸⠄⠀⠀⠤⠈⠀⠐⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠁⠊
⠀⠀⠀⠀⡀⠤⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⡀⡀
⠀⠀⡎⣀⠵⠀⢎⠉⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠑⠀⠀⠀⠀⡀⢴⠙⣈⣁⠋⠼⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢐⠉⡩⠄⣜⠒
⠀⠈⣀⡅⢎⢈⠀⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⣀⡞⠁⠀⢱⠌⢡⠂⠀⠀⠀⠘⠐⢄⣀⣀⠄⠚⠀⠀⠀⠀⢀⠁⡉⢀⠁⡄⠒⠑
⠀⠀⠁⣈⠆⠤⡊⢁⠃⠀⠀⣀⠠⠐⠀⠉⠉⠉⠉⠀⠂⠠⢀⠀⠀⢣⠆⡱⢤⡠⠚⣐⢒⠀⠀⠀⡀⠴⠄⠐⠂⠘⠀⠰⠄⠀⠀⠀⠀⠏⠉⡄⢨⠔⠑⡊
⠀⠀⠀⠀⠑⠊⠁⠀⠀⡠⠊⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠑⢄⠀⠸⠴⠂⢆⡑⠊⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠁⠄⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠡⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠃⠀⠀⠀⠀⠱⡀
⠀⠀⠔⠂⡀⠀⠀⡈⠀⠨⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠑⠀⢡⠀⠀⠀⠀⠰⠀⠄⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠑⠀⠐⡀
⠀⠀⠂⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠁⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠘⠍⠈⠁⢀⠽⡁⠀⠈⠀⡨⠛⡀⠁⠉⢀⠜⠙⠀⠀⢈⠇⠀⠀⠐⡃⠀⠐⠀⠶⡀⠂⠒⠐⠰⢆⠐⠐⠂⢐⠒⢒⠂⠒⡳
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡠⠁⠀⠈⢂⠀⠔⠀⠀⠘⢄⡠⠃⠀⠀⠈⡤⠁⠀⠀⠀⠀⠈⢄⡠⠂⠀⠑⠄⠀⠔⠀⠀⠠⡀⠐⠁⠀⠀⢢⠔
⠀⠀⠀⠀⠗⠀⠀⠀⠀⠑⡄⠀⠀⠠⠛⡀⠀⠀⢀⠜⠣⠀⠀⠀⠔⠀⠀⠀⠘⠁⠀⠀⠀⠡⡀⠀⠀⡠⠃⡀⠀⠀⢀⠛⠄⠀⠀⢀⠊⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⡐⠁⠀⠈⢂⡠⠃⠀⠀⠁⣀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠁⠀⠐⢄⠐⠁⠀⠈⢆⡰⠁
⠀⠀⠀⠀⠀⣄⠀⢀⢄⠀⠀⠀⠑⡄⠀⢀⠌⠑⡀⠀⠀⡰⠁⢀⠄⠐⠀⠐⠠⠀⠀⠀⠐⢄⠀⠀⠢⡀⠀⢀⠘⢢⠀⠀⢀⠌⠀⠀⠀⡠⠀⠀⣠
⠀⠀⢀⠀⡀⡂⠠⠂⡘⢀⣀⠀⠀⠈⢢⠀⠀⠀⠈⢆⠜⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠐⢤⠁⠀⠀⠑⣄⠂⠀⠀⡀⢀⢡⠘⠄⣠⢀⣀⡀
⠀⠀⠀⢁⠠⠀⠂⠂⢡⡀⠂⠀⠀⠀⠀⠑⡀⠀⠠⠊⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠣⡀⠀⡐⠁⠀⠀⠀⠈⡠⠄⡄⢰⠈⣄⠂
⠀⠀⠐⠀⠐⠆⣈⠀⡂⠬⠆⠀⠀⠀⠀⠀⠈⠶⠁⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⣀⡀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠂⠄⡲⢈⡁⠔⠠⠱⠄
⠀⠀⠀⠀⠸⠔⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⢀⠔⠁⠀⠀⠀⠐⡀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠐⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⡀
⠀⠀⡀⠀⠀⡠⢄⠀⠀⠀⠀⠆⠁⠀⠀⢨⠁⠀⠀⠀⡄⠂⢂⠀⠠⠂⢠⠄⠂⠀⠄⣠⠠⢄⠀⠄⠐⡄⠀⠀⠀⠈⠇⠀⠀⠈⡰⠀⠀⠀⠀⠀⠀⢀⠠⡀
⠀⠐⡀⠀⠉⠀⢀⠃⠀⠀⠀⠀⢠⠠⠤⠼⠀⠀⠀⠀⠐⠀⠀⠖⠀⠠⠁⣀⠤⢄⠀⠈⠀⠘⠀⡆⠀⠁⠀⠀⠀⠀⠦⠤⠄⡘⠀⠀⠀⠀⠨⠀⠀⠃⠀⠘
⠀⠀⠐⠄⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⡄⠀⢰⠀⠀⢇⠀⢀⠃⠀⠀⢀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⡠⠁
⠀⠀⠀⠈⠤⠂⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠇⠇⠀⠀⠀⣇⠀⡀⠐⢄⠀⠈⠀⢀⠴⢄⠀⠀⠀⡠⠁⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠔
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣀⣐⣠⡤⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡒⡛⣂⠀⠀⠀⡆⢰⡀⠀⠀⠀⠤⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠣⠩⠤⠤⠤⠀⠀⠀⠈⡐⠂⠀⠀⠀⢃⣉⢉⡍⡙⠀⠀⠀⠻⢌⡁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡀⢀
⠀⠐⡄⢥⡤⢶⠀⠀⠀⠒⠐⠒⠂⠀⠀⠙⢉⣨⣉⠕⠀⠀⢐⢂⠀⠀⠀⠀⠀⠀⠠⠆⠄⠀⠀⢀⣀⢤⠛⡛⠃⠃⡯⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢡⡆⠠⣤⢫⡏⠂
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠴⠴⠤⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠄⣠⣤⣄⠀⠀⠀⡞⠠⢞⠀⠀⠀⠞⠒⠧⣀⣉⡄⠁⠁
⠀⠀⠀⠀⠠⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠥⠖⠒⠀⡀⠸⡙⣋⣉⣠⣀⡀⠀⠀⠁⣡⠼⡓⠀⠀⠀⠀⠄⢀⠀⠄⢀⡀⡀
⠀⠀⢀⡀⣀⡀⠀⠀⠘⠘⠒⢚⠖⠁⠀⠑⣂⠂⣉⢞⠁⠀⠀⠤⠼⠤⠥⡆⠀⠉⠘⠉⡊⡙⠀⠀⠀⠈
⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠉⠀⠉⠈⠀⠀⠐⠲⢦⠞⠁⠀⠀⠀⠓⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⣰⣾⣿⣶⣤⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠿⠿⠿⠿⣿⣿⣦⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⠈⠻⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⢀⣤⣄⣠⣶⣦⡀⠀⢀⣿⣿⡟⠀⠀⠀⣴⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠙⠋⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⢸⣿⣿⠁⠀⠀⠐⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡟⠁⠀⣾⣿⡿⠀⠀⠀⠀⠻⣷⣤⣀⣀⣠⣿⠏⠀⠀⢀⣶⣦⣴⣄⠀⠈⢿⡿⠟⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠈⠙⠛⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠉⠛⠿⠿⠛⠁⠀⠀⠀⠀⠻⣿⠿⠃
⠀⠀⠠⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⢀⣤⡶⠶⢶⣤⠀⢠⣤⣄⠀⠀⢀⡾⢹⣧⠀⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠙⠿⣿⠋⠀⢠⣶⣶⣶⠀⠀⣿⡟⠀⠀⠀⠀⠀⢰⣿⣧⣴⡖⠀⢹⡇⢸⣿⣿⠀⠀⠸⣧⣸⡏⠀⠀⣴⣿⡿⠋⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⠟⠀⢸⣿⡇⠀⠀⠀⠀⠀⠘⣿⣿⣿⣠⣤⣴⡿⢻⣿⣿⠀⠀⠀⢹⣿⠧⠶⢺⣿⣿⠧⠤⠞⠁⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣼⣿⡀⠀⠀⠀⠀⠀⢰⣿⣿⡏⠉⠁⣼⠃⠘⣿⣿⡄⠀⠀⣾⠃⠀⠀⢸⣿⣿⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠘⣿⣿⣇⣀⡼⠃⠀⠀⢻⣿⣷⣤⣾⠋⠀⠀⠀⠈⢿⣿⣆⡀⣀⡴⠋⢀⣴⣶⣄⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠉⣩⣿⠟⠛⠿⠿⣿⣿⣿⣿⣶⣬⡙⠛⠉⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠀⡀⠉⠉⠉⠉⠀⠀⢾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠻⣦⣴⣾⠿⠋⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠿⠿⢷⣦⡀⠀⠀⠈⢻⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣦⣄⠀⠀⢀⣴⣿⠿⠋⠁⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠛⠋⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣄⠀⠀⣠⣄⠀⠀⣀⠀⢀⡄⠀⠉⠻⢿⣿⣿⣶⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣿⢀⣾⠃⡞⣆⣰⣟⠀⣾⠁⠀⡖⠀⠀⠈⣻⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣴⣾⡟⠁⠀⠀⣶⣶⣶⡄⠀⠙⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢞⣿⠚⣿⡀⣱⠟⠉⣿⡼⣿⣄⡼⠁⠀⠀⠀⣿⡏⠀⠈⠉⠛⠛⠿⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠻⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢺⣿⠀⠈⠛⠁⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣤⢀⣄⠀⠀⠀⠀⣀⣤⠀⠀⣤⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⣿⠇⠀⠀⠀⠀⣠⣶⣦⣀⣤⣄⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⢠⠀⠀⣿⠇⣿⡎⣿⠀⢰⡏⣜⣃⣴⣷⠇⣿⡎⣿⠀⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⣷⡀⠀⠀⢸⡇⢰⣿⠀⠿⠀⣿⣤⠿⣧⣨⠋⢸⣿⠀⠿⠀⣿⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠈⠛⠶⠾⠟⠀⠀⠋⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣦⠀⠀⢻⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠿⠿⢁⡉⢑⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⢮⠿⢿⡙⢳⡐⠇⢤⡌⡴⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡗⠀⠀⣀⠀⣹⡷⣦⣼⣿⣿⡀⠈⢂⠚⠁⡰⢆⠀⢀⠀⠀⢸⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠋⣠⣾⡿⠿⠛⠿⣿⡝⢹⣿⡿⣿⢷⣾⡛⣦⣿⣧⣄⠀⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠰⢿⣶⡆⣹⣿⣮⣧⣠⣼⡿⠿⠿⣿⡿⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡂⠀⣰⣿⣿⣿⣿⣧⣸⣉⣷⣿⣟⠻⣿⣿⡯⢴⣧⠤⣾⣣⣾⣿⣿⣿⡄⠀⢰⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠃⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣷⣴⣥⣾⣿⣿⣿⣿⣿⣿⣿⡄⠘⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢑⣤⣞⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣤⣬⣿⣿⣁⣈⣰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠷⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢛⡛⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡬⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠍⡿⠀⠄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⠸⠱⠀⠂⠀⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠴⠚⠁⠀⠰⠀⠈⡠⠃
⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⡇⠀⠀⢀⡆⠀⡞⠄⠀⠀⠸⣷⠀⠀⠠⣄⣀⠜⠀⠀⢠⠏⠀⠀⠀⢠⠃⠚⠁⠀⢠⠖⠢⢀⠀⡖⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠱⢀⡀⠟⡠⢿⠀⠀⡌⢡⣤⠀⠈⠄⠀⠀⠹⣧⠀⠀⠈⠋⠀⠀⢠⡟⠀⠀⠀⠠⠁⠀⠰⣀⠀⠉⠀⠁⠠⡀⠛⠐⠃⢁⠄
⠀⠀⠀⠀⠀⠀⠀⢈⡁⠀⠀⠀⠑⠊⠀⠈⠀⠄⠀⠈⢆⠀⠀⢻⣧⠀⠀⠀⠀⠠⡟⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠸⠏⢦⡀⠀⠀⠀⠘⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠱⣄⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠀⠤⠊⠑⠲⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠞⠉⠉⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⣀⡤⠤⠤⠤⣀⡞⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⣼⡟⠀⠀⠀⢀⡾⠉⠓⠦⣀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠇⠀⠀⠀⢻⣿⡄⠀⢠⡞⠀⠀⠀⠀⠈⢙⡿⣤⣀⡀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠙⢿⣶⣏⡀⠀⠀⠀⠀⢀⡾⠁⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⢀⣴⡿⢁⣴⣾⣷⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⢀⡾⠉⠉⠉⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠘⣿⢀⣿⣿⠟⠁⠀⠀⣿⣿⡏⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣀⡀⠀⢰⡏⠀⠀⣀⣤⣤⣄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇⠀⠀⠀⢸⣿⣿⠃⠀⠀⢀⡴⣋⠤⠄⢤⡀⠀⣸⣿⡇⠀⡾⠀⢠⣾⡿⠁⢀⡟⠀⢠⣾⢋⡟⠀⢀⡴
⠀⠀⣀⡀⠀⠀⠀⠀⠹⣇⠀⠀⠀⣼⣿⣿⣀⠤⠚⣡⠞⢡⣴⡆⠀⢹⠀⣿⣿⠀⢰⠇⢠⣿⣿⣃⡴⠋⠀⠀⠘⣿⣾⢁⣠⠞
⠀⢀⣭⡁⠀⠀⠀⠀⠀⠈⠛⠐⠀⣿⣿⡏⠀⠀⢰⠇⢠⣿⣿⠁⠀⢸⡗⣿⣿⡄⣾⠀⢸⣿⡿⠁⠀⠀⠀⠀⢀⡞⠉⠉
⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠸⣄⣸⣿⣟⣠⠔⣻⠀⢹⣿⣧⡇⠀⢸⣿⡇⠀⠀⠀⠀⡴⠋
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣸⠿⣿⡏⠀⢠⠏⠀⠀⢿⣿⠃⠀⠈⢿⣧⠀⢀⡴⠊
⢸⣿⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⠀⣰⠃⠀⠙⠷⠴⠋⠀⠀⠀⠀⠛⠀⠀⠀⠀⠉⠋⠉
⢸⣿⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣤⡾⠃
⠘⣿⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀
⠀⠈⠛⠁⠀⠀⠀⠀⠀⣠⠴⠚⠳⣦⡄⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠈⠙⣆⠀⠀⠀⣠⣴⠄⠀⢀⣾⡟⠀⠀⠀⢀⣶⢶⡄⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⡟⠀⠀⠀⣀⡀⠀⠘⡆⠀⣼⣿⡏⠀⠀⣾⣿⠁⠀⠀⠀⠘⢿⣾⣡⠞
⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⣧⠀⢀⣾⣿⠃⠀⢀⣷⠘⣿⣿⠁⠀⠀⣿⡟⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⣠⠊⠉⣿⣆⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢠⣶⡆⠈⠑⣾⣿⡯⠄⠒⠉⡇⠸⣿⡟⠀⢀⡼⣿⡇⠀⠀⠀⠀⡴⠃
⠀⠀⡼⠁⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⠇⠀⠀⠀⡇⠀⢿⣧⢀⡾⠁⠘⢷⡀⠀⣠⠞⠁
⠀⢸⠃⠀⠀⠀⢿⣿⠇⠀⠀⠀⢰⣿⣿⡇⠀⠀⣿⣿⣇⠀⣰⢿⣿⠀⠀⠀⣸⠃⠀⠀⠉⠉⠀⠀⠀⠀⠉
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣴⠃⠈⢿⣆⠀⣰⠃
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣸⠉⣿⣿⡇⠀⠀⠀⠉⠉⠁
⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⢠⡏⠀⣻⣿⣿
⠀⠀⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⡾⠀⣰⠛⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢠⠇⠀⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣀⡴⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣽⣿⣷⣞⣩⣿⣹⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠐⠀⠄⡀⠀⡀⠀⡻⠿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⡟⢠⠄⢸⠀⣦⡶⠺⠋⠀⢀⠘⢍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⢃⠀⡀⠑⠤⠅⠊⡀⠄⠀⠰⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠄⠊⠢⠈⠀⠈⠲⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⠄⠀⠀⠐⡀⢀⢔⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⡐⠁⢈⠀⠈
⠀⠀⠀⠀⠀⡠⢊⠄⠤⡈⢂⠀⡀⠀⠀⠈⡆⣸⣤⣤⠀⠈⠀⠀⠀⡄⠀⢀⠀⠠⠆⠒⠀⠃
⠀⠀⠀⠀⠨⠀⠇⠀⠀⡀⢀⠆⠘⠄⠀⠀⠐⢍⣀⡘⣀⠈⠀⠀⠠⠀⠐⠓⠀⢰⠒⠀⠀⠅
⠀⠀⠀⠀⠀⠑⢄⠁⡈⠀⠄⠀⠀⠈⡄⠀⠀⠂⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠑⠄⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⡐⠗⡀
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⣷⠀⠀⢆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢘⠐⠀⠤⠊⠀⠃
⠀⠀⠀⠀⠀⠀⠘⠄⢀⠀⠽⠃⠀⠀⠈⠉⠀⠂⠀⠀⠐⠐⠂⠸⠁⠀⠀⠢⠤⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠋⢀⠜⠋⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠈⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⡀⠀⠀⠀⠀⠀⡀⠄⠊⠈⠛⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠠⠀⠠⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢿⣟⣿⢿⣿⣿⣦⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⠦⠀⢰⣿⡿⣈⡛⠇⣿⡹⠋⣿⣿⠀⣘⣣⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣛⣯⠤⠈⢿⣷⣿⣧⣶⣽⣿⣽⣿⣯⡀⠤⠬⣥⣤⣀⣀
⠀⠀⠀⠀⠀⠀⢠⡤⠖⠚⠋⠉⠉⠀⣀⣀⢀⣈⣉⣻⡛⢛⣭⣥⣄⠀⣁⠀⠀⠀⠀⠉⠉⠙⠓⠲⢤⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠁⢰⣦⠀⠀⠘⢿⣟⠈⠙⣿⡏⠁⠈⢹⣿⠉⢀⣿⠁⠀⢀⣿⠿⠷⠆⠀⢘⣿
⠀⠀⠀⠀⠀⣠⠔⢻⡅⠈⣿⡆⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⢸⣿⠀⢸⣿⠀⠀⢸⣿⣤⡀⠀⠠⣲⡟⠢⣄
⠀⠀⠀⠀⠛⢷⡟⠄⣿⠀⢸⣷⣠⣤⢠⣿⡦⠀⠹⠇⠀⠀⠸⠿⠀⠸⣿⣶⣦⣼⣯⣍⠀⠀⠨⣿⠀⣨⡿⠛
⠀⠀⠀⠀⠀⠀⠳⡶⢿⡄⣀⣻⣭⠍⠀⢀⣴⢶⣄⢠⣤⢀⣤⡀⢠⣶⢦⣄⠀⠈⠩⣭⣄⣀⢽⡏⢤⠟
⠀⠀⠀⠀⠀⠀⠐⡗⠨⣿⠿⣿⣿⠆⠀⢸⡇⠀⣿⢸⣿⡏⢹⣷⢸⣿⠸⣿⡄⠀⠰⣿⣿⢿⡛⠁⣺⠂
⠀⠀⠀⠀⠀⠀⠀⣇⡤⠖⠋⢉⡀⠀⠀⣾⡇⠀⣿⢸⣿⠃⢸⣿⢸⣿⢰⣿⠁⠀⠠⣀⡉⠙⠲⢤⣻
⠀⠀⠀⠀⠀⠁⠀⠁⠤⠶⠛⢋⣥⠆⠀⣿⡟⠛⣿⢸⣿⠀⢸⣿⢸⣯⡼⠋⠀⠀⠰⠌⠙⠓⠀⠀⠈⠀⠀⠄
⠀⠀⠀⠀⠀⢰⣤⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⣭⣈⠋⢀⣤⣅⡈⠉⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⣄⡄⠀⠀⢸⣏⠀⠹⢃⣿⡏⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⡀⣀⣤⣤⡀⠀⠀⠉⠓⠀⠀⠈⠻⣧⡀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⢈⣾⡿⠛⢻⣿⡆⠀⠀⠀⠠⠀⠀⡞⠈⣿⡆⢿⡆⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢸⣿⡃⠀⢸⣿⡇⣴⣶⠀⠀⠀⠀⠙⠛⠋⠀⠈⠙⣉⣤⣀⠀⠀⠀⢀⣠⣄⡀⣶⣶⣿⣷⣶⡄
⠀⠀⠀⠀⠘⢿⣷⣄⠈⠋⢰⣿⡇⠀⢀⣀⠀⠀⣿⣧⠀⣰⣿⡿⢻⣿⡇⢀⣾⡿⠟⣿⣿⠀⢸⣿⠋⠉
⠀⠀⠀⠀⠀⣈⡙⢿⣷⡄⢸⣿⡇⠀⣾⣿⠀⢀⣿⡿⢰⣿⣿⣶⣿⠟⠁⣾⣿⣷⣾⡿⠋⠀⢸⣿
⠀⠀⠀⠀⢸⣿⣇⣨⣿⡿⠈⢿⣷⣼⣿⣿⣷⣾⡿⠃⠈⢿⣿⣤⣴⣶⠀⠹⣿⣧⣤⣶⡆⠀⢸⣿⣇⢰⣿⡇
⠀⠀⠀⠀⠈⠛⠿⠟⠋⠀⠀⠪⠙⠋⠁⠈⠉⠉⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠈⠛⠛⠋⠀⠀⠀⠻⢿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡆⠸⣿⣿⠇
⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⣁⠘⢁⢰⣿⠿⠉⠗⢠⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠉⠁
⠀⠀⠀⠰⣿⣿⣿⡀⠀⣴⣶⣶⡍⠡⠍⢹⣤⡀⠀⡀⠈⠁⠀⠀⣠⣿⣿⡆⠘⢿⣿⣵⣶⣿⣿⣿
⠀⠀⠀⠀⠈⠙⢛⠁⠀⢻⣿⣿⠇⣴⣿⡟⠛⠁⠈⠁⠀⠀⣀⡈⣿⣿⣿⡇⠀⠀⠉⠛⠛⠿⠿⠋
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡾⢻⣯⡀⠿⡟⠀⠀⠀⠀⠀⠀⠀⠙⠁⠈⢿⣿⣧⣤⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠛⠋⣩⡴⠻⢿⣇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⣻⢶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠽⠿⠥⠿⠧⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⢌⠁⠀⠀⠀⠀⠀⠀⠀⠤⠤⠆
⠀⠀⠀⠀⠀⠀⠈⠈⠉⠁⠂⠀⠀⠀⠉⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠂⠀⠀⡔⠀⠀⠀⠀⠀⠂⠃⠄⢀⠠⠀⢀⠁⢀⠀⠀⡀⠈⠀⡐⠄
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠨⡓⢀⠀⠈⡄⠒⠂⢀⠀⠀⣀⠁⠂⠐⠈⢂⢲⠀⠀⠐
⠀⠀⠐⠀⠐⠂⠀⡠⠂⠂⠁⠀⢂⡄⠁⠀⠀⠀⠓⠰⣁⠀⠀⠀⠀⠀⠄⠀⠂⢀⠀⠀⡡⠀⠀⢀
⠀⠀⠀⠀⠀⢠⠀⠈⡔⠂⠀⢨⠔⠂⠀⠀⠀⠀⠀⣅⣀⠄⠀⠀⠀⠄⡁⠀⠀⠐⠀⠉⢡⠠⠞⠫⢈⡂
⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⡄⠠⡆⢀⣈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠀⢁⠀⠀⠀⣁⠀⠀⠀⠘⠃
⠀⢠⣟⠀⢐⠀⠄⠀⢘⠀⢀⠀⠀⠰⠀⠆⠀⠀⠀⠀⠀⣀⠀⠃⠀⡆⠀⠒⠐⠆⠁⠈⠀⠉⠱⠀⠀⠐
⠀⠀⠛⠁⠀⠀⠀⠐⡄⠀⠈⠀⠀⠀⡠⠤⡜⠂⠀⠀⡊⡙⢀⢄⣴⠆⠀⠋⠁⠡⠀⢀⠀⠀⡞
⠀⠡⠀⠀⠲⠊⠀⠀⠈⠀⠅⠀⠈⠰⠁⡰⠃⠀⠘⢷⠀⠠⡑⠁⢡⠀⠰⠰⡐⠺⠂⠘⠊⣱⢀
⠀⠀⠀⠁⠘⠓⠀⠀⠀⠀⢂⠐⢂⠀⠀⠀⠁⠀⠀⠀⠀⠄⡘⠀⠘⠅⠐⠀⠁⠲⢔⠁⠀⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁⠂⠀⣰⡀⠈⠉⠀⠀⡀⠌⠀⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠈⠀⠀⠀⠀⢂⠀⢆⠀⠀⠐⠈⠀⠘⠁⠀⠈⠹⡀⠔⠤⠘⠁⠠⠄⡄⢠⢐⠐⠑⠂⡂⠀⠛
⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠄⠀⠘⣠⠔⠁⢀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠐⠠⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⢈⠀⠉⠀⠀⣀⠀⠀⠀⠀⠐⠀⠠⠂⠀⠀⠐⠡⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⢃⣉⠁⠄⠀⠈⠀⠠⠀⠀⡀⠉⠀⠀⢂⠄⢠⠀⠱⠀⠠⠤⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠀⠁⠀⠀⠒⠂⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⢠⠇⠀⠀⢀⠀⠀⢀⣠⠤⠄⠀⠀⡏⠋⢰⡇⠀⠒⠒⠲⠤⣤⣀⣀⣠⡴
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢠⣿⣿⠃⢀⡞⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢰⠟⠁⠀⠀⠀⣸⣿⣿⡠⠋⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢄⠀⠀⠀⠀⣀⣤⡶⠶⢤
⠀⠀⠀⠀⠈⠦⣀⠀⠀⣀⣿⣿⡏⠀⠀⢀⠔⠋⠉⠀⠀⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣇⠀⠀⣴⣿⠏⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣿⣿⡇⠀⢠⠃⠀⠀⠀⢀⣀⡀⠀⠀⢳⠀⠀⢰⣾⣶⡆⠀⠀⠱⣄⠀⡏⠀⢰⣿⡿⠀⠀⡰⠃⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠸⣄⠀⠀⢠⣿⣿⣿⠃⠀⣸⡠⠆⢿⣿⣿⠇⠀⠀⠀⠈⡳⠧⠤⣿⣿⡧⠖⠋⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠈⠙⠒⣾⣿⡿⠧⠒⣺⠃⠀⠀⢸⣿⣿⠀⠀⠀⠀⡴⠁⠀⠀⢻⣿⡇⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⣿⡟⠀⢀⡴⠁⠀⠀⠀⠘⣿⣯⠀⠀⢀⠞⠀⠀⠀⠀⠀⠙⠿⠦⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠘⠓⠒⠉⠀⠀⠀⠀⠀⠀⠙⠿⡤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣀⠤⢾⣿⡿⠁⠛⠒⠒⠢⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠋⠀⣸
⠀⠀⠀⠀⠀⢀⡎⠀⣠⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠢⠤⣄⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⢰⣿⣿⠁⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠑⠚⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠚⠋⠉⠀⠀⣴⣿⠟⠉⡇⠀⠀⢀⣿⣿⠃⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠘⢿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⢠⠇⠀⠀⣼⣿⣏⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⢀⣤⣦⠀⠀⠘⠇⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⢀⠎⠀⠀⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⢸⢻⣿⣧⠀⠀⠀⠀⠀⣴⣿⠟⠉⠃⣀⡀⠀⠀⢰⣿⡟⠀⢠⠎⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⣠⠃⠀⢿⣿⡆⠀⠀⠀⣸⣿⡏⠀⠀⣰⣿⡿⠀⠀⢸⣿⢇⡴⠁⠀⠀⠀⣰⣿⣿⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⣀⠔⠁⠀⠀⠈⣿⣷⠀⠀⠀⣿⣿⠁⠀⡴⣿⣿⠃⠀⠀⢿⣿⠋⠀⠀⠀⠀⡰⢹⣿⡇⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠞⢢⠞⣷⠀⠀⠈⠁⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠘⠿⠦⠊⠀⢻⡏⠀⣀⠴⢻⣿⡄⠀⠀⠀⡰⠁⢸⣿⠁⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠘⠄⠀⡟⠀⠀⣀⠴⠒⠊⠁⠒⠢⢼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠁⠀⠈⢿⣇⠀⢀⡜⠁⠀⢸⣿⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⣼⠋⠓⠦⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠈⠛⠒⠋⠀⠀⠀⠀⠻⣧⣀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠈⠙⠢⣤⡖⠉⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⡸⠳⠚⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⡴⠁⠉⠑⠒⠤⠤⠒⠋⠀⠀⠀⠀⠀⠘⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢴⡧⠤⠤⠤⣼⣤⣄⣀⣀⣀⣀⣀⣀⣞⠁⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢠⠿⢷⣦⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⡭⣍⠉⠙⠛⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠘⢄⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠉⣹⠇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⠖⠂⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢀⡴⠒⠲⢄⡤⠶⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⡇⠀⠀⡄⠀⢀⡤⢄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠑⠶⣾⠁⢠⡇⠀⢘⡇⢰⠃⠀⠀⡟⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠘⠲⠖⠃⠀⠀⠧⣤⠞⠁⠸⣆⣀⡜⠁⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠠⣾⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠒⠒⠶⠦⠤⠤⠤⢤⣤⣤⣤⣤⣤⣤⠤⠴⠚⠉

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

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣀⡀⢀⡀
⠀⠀⠀⠀⣴⢚⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠱⣲⢤⠆⠠⣦⡔⣤⠀⠀⠀⠀⠀⠀⠈⡍⠀⠸⡋⠙⠟⣹⠆
⠀⠀⠀⠀⠀⢠⡇⢠⢤⣄⡀⢄⢰⠀⠀⠀⠀⠀⠀⠀⢠⢠⣠⠆⢠⠆⡄⠀⠈⠀⠀⠀⠀⠀⠀⣀⣧⡀⠀⠙⢦⡾⠋
⠀⠀⠀⠀⠉⠉⠀⠈⠀⠈⠙⠉⠙⠁⠀⠀⠀⠀⠀⢰⣈⣈⣀⣀⣈⡀⢡⠄⡀
⠀⠀⠀⢸⡂⠢⡼⠡⠎⢁⣴⠀⡆⡘⠀⠀⠀⠀⠀⢨⡆⡁⢀⠈⠁⠉⠉⠉⠁⠀⠀⠀⠀⢰⡆⠠⣴⢠⠖⠂⡆⢠⠀⡄
⠀⠀⠀⠐⠂⠀⠁⠀⠘⠋⠈⠛⠁⠭⠀⠀⠀⠀⠀⠘⠀⠧⠸⠄⠤⠈⠒⠩⠦⠀⠀⠀⠀⠘⠀⠀⠃⠈⠶⠃⠑⠋⠀⠇


⠀⠀⠀⠀⢀⠀⢠⠄⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠴⢚⡤⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠰⠤⠀⠔⠐⠃⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠑⠓⠀⠒⠰⠆⠀⠓
⠀⠀⠀⠀⠀⠴⠠⠤⠄⡤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠓⠣⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⡶⠲⠒⠂⠐⠒⠓⠶
⠀⠀⠀⢴⠠⢶⠰⠀⠐⠂⠄⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠸⠞⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⣟⢰⡆⢰⠆⢰⠂⣒⡐⠆⢰⡀
⠀⠀⠀⠈⠛⠛⠛⠋⠉⠉⠒⠒⠚⠂⠀⠀⠀⠀⠨⠅⠋⣱⡖⣰⡿⠃⢘⠁⠀⠀⠀⠀⠁⠀⠀⠈⠀⠈⠁⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠁⠙⠃⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠈⠁⢀⡄⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢋⡵⣠⠆⠀⠀⠀⠀⠀⠀⠀⡄⠀⢟⣀⡦⢣⡾⢹⡷
⠀⠀⠀⠀⢠⣠⣌⣥⣆⡠⣄⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠀⢨⠁⣈⢉⣁
⠀⠀⠀⠀⢠⢀⣀⢀⠀⣀⢀⣀⡚⠀⠀⠀⠀⠀⠀⡂⢰⣆⢰⠂⠒⠐⠆⠠⠀⠀⠀⠀⠀⠘⠃⠾⠢⠵⠀⠸⠐⠇⠫⡴⠂
⠀⠀⠀⠀⠘⠢⠭⢼⣦⠋⠈⠘⠂⠀⠀⠀⠀⠀⠀⠀⢐⠚⠙⠂⠴⠀⠀⠘⠀⠀⠀⠀⠀⢰⠀⢰⢠⣄⣤⠀⡄⠶⠶⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠅⣁⢘⠁⠈⠁⠀⠀⠀⠀⠀⠙⠛⠈⠈⠈⠀⠀⠁⠀⠀⠐⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⡇⢠⠟⣷⢷⣰⢳⣏⠁⠸⣴⢣⠞⣦⣶⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇⠘⠷⠚⠓⠃⠘⠃⠘⠓⠂⠀⠃⠘⠓⠋⠓⠞
⠀⠀⠀⠀⠀⣴⠶⢦⡤⠒⢦⠀⢠⣶⠀⣠⣤⠀⣠⣴⡄⠀⣠⣤⡀⠀⢀⣾⠃⣠⣶⡄⣠⢶⡆
⠀⠀⠀⠀⠸⡇⠀⠀⠀⢀⡾⢀⣿⢇⡾⢻⣿⣾⢋⣾⢡⣾⢿⣌⣿⣤⣾⣏⡾⢻⣿⡿⢃⣾⠃
⠀⠀⠀⠀⠀⠙⢦⣀⡴⠋⠀⢸⣿⡿⠁⣿⡿⠃⣾⠇⢺⣇⣠⠞⠁⠀⣿⡟⠁⠻⠟⠀⢸⡏⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣀⣈⣄⣀⣀⡈⠋⠀⠀⠈⠀⠀⣿⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠛⠁
⠀⠀⠀⣠⠞⠋⠉⠀⠀⠀⠈⠉⠓⢦⣀⠀⠀⠀⣿⠀⠀⣀⣀⣤⣤⣤⣄⡀⠀⠀⢀⡤⢤⡠⠤⡄
⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⣠⣽⣷⠛⠉⠀⠀⠀⠀⠀⠉⢷⡀⢺⠀⠀⠀⣰⠇
⠀⠀⠀⠙⠳⢤⣀⣀⣀⣀⣀⣤⡤⠴⠖⠛⣿⠁⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠓⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠉⠳⠦⣤⣤⡤⠞⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡟⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠖⠛⠛⠛⠀⠀⠀⠘⠛⠉⠛⠻⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⢤⣀⣠⡤⠀⣧⠀⠠⣤⣤⣤⠄⠈⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡄⠀⠉⠉⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⡀⠈⠀⢀⡀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠙⠛⠛⠋⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠓⠦⣤⣤⣤⡤⠴⠚⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⢿⣏⠛⢧⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢦⡈⠳⣄⠙⢦⠙⣆⠀⠀⠀⠀⠀⠀⠀⣠⠖⣦⠴⠋⣩⡿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡷⣤⡈⠳⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢀⡼⠁⠚⠁⠴⠋⣁⠴⠻⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠷⣤⡙⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⢠⡞⠀⠀⠀⠀⠀⠋⣡⡴⢿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡇⠀⠙⠷⣄⠀⠀⠀⠀⠀⠈⢧⣠⠏⠀⠀⠀⠀⠀⠀⢀⣡⠴⠋⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⣠⠖⠋⠁⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠙⢦⡀⢀⡾⠁⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠈⠛⠞⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⢀⣽⢧⣄⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣤⣤⠶⠛⠁⠀⠉⠛⠶⠶⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠄⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⢻⡀⢀⡼⠃⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠈⢷⡏⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣆⠀⠀⠀⠀⠀⠀⠀⣷⣄⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⢧⣀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⠀⢠⣴⣄⠀⠀⢀⢠⣤⠀⠀⡏⣶⣿⠟⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⢠⠀⠉⠻⠂⠰⠁⢺⡿⠀⣠⠃⠈⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⢹⠓⠦⠤⢤⡤⠤⢊⣤⠊⠀⣰⡞⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠠⢸⠦⣀⠀⣀⣠⣴⡊⢁⣠⡾⠋⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠟⠀⠈⠉⠁⠙⠿⣿⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⣠⡁⢀⠀⢀⠀⣨⠀⠀⠀⣀⡀⢔⡤⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠐⠀⠉⠀⠏⠀⠀⡄⠈⡁⠀⠀⠀⠁⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠀⠀⠀⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠂⣉⣠⡤⠴⠶⠒⠒⠲⠶⠶⢶⣆⣤⡤⣌⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⢠⠶⢛⡉⠄⡐⠒⠛⠉⠉⠉⠁⠉⠓⠉⢉⠛⢶⣍⡓⢄⠈⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠃⠀⠀⠀⢀⣠⣦⠞⠀⠀⡊⠁⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢌⠳⣅⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⣾⣟⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠳⣌⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⣾⣧⠀⢸⣿⣿⣿⣾⣷⠀⠀⢸⡇⠀⠀⢠⣿⣧⠈⡿⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⢻⣆⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡧⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⠙⠢⠤⣾⠏⢰⡶⢰⡶⠀⣜⣵⣶⠀⠀⠀⠀⠀⠀⠀⠹⣧⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠙⠛⠛⠛⠁⠀⢰⣿⣿⢶⣾⣏⡀⣿⢁⣿⢁⢎⣿⡿⠊⢀⠆⠀⠀⠀⠀⠀⠀⠹⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣉⣶⠋⠉⠛⠿⢧⣭⣁⣈⠻⣄⣔⡡⠀⢠⣿⠃⠀⠀⠀⠀⢻⣏⡧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢾⣿⠏⠀⠀⠀⠀⣴⡄⠀⠂⢠⣿⡟⠀⡝⢀⣀⠀⠀⢀⡀⠀⢉⠙⠛⠋⠉⣀⠀⣾⡟⢀⡀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢏⣿⡏⠀⠀⠀⠀⠀⠻⣇⠀⢠⣿⣿⠡⢊⣴⡟⢉⡃⠀⣾⠇⢠⣿⠁⣰⣾⠋⡟⢸⣿⡅⣸⡇⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠈⢱⣿⡿⠁⠀⣼⣿⠁⣾⡟⢸⡿⠀⣾⡇⢰⣿⠇⢰⢃⣿⠏⢰⡿⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⣴⠶⢶⣤⣿⣟⡀⠀⠀⠙⠧⠈⠻⠇⠚⠧⠀⠻⠇⣼⠿⢐⣁⣸⠟⠀⣿⠃⠠⠀⠀⠀⠀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⠒⠶⠛⠉⠉⠛⠻⠶⣦⣾⣷⣄⠀⠀⠀⠀⣼⠀⣙⡚⠋⣠⣿⠆⣀⣿⠴⠃⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⣶⠆⠀⠀⢠⣿⠃⣽⠛⠛⠷⠶⠿⢣⣏⣁⣴⡿⠟⠉⠀⠙⠁⠀⠀⠀⠀⠀⠀⣼⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠘⢿⡀⠀⠀⣾⣧⠞⢁⣄⡀⢀⡀⠀⡠⠽⢯⣁⠀⠀⢶⡶⠀⢿⣿⡿⠀⠀⠀⠀⢃⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠋⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⡿⠀⢠⣿⠉⢳⣿⢃⠎⣼⡿⢛⠘⠆⠀⢀⣤⣤⣄⠉⠀⠀⠀⠀⡼⠸⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⣿⡇⠀⠀⠀⠀⠀⠀⡖⠲⣶⣿⣅⣀⠸⡇⠀⢸⡇⠈⢰⣿⠗⠃⡀⠀⢠⡌⢿⡿⠟⠀⠀⠀⠀⠸⢁⠱⠁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣻⣿⣄⠐⠀⠀⠀⠀⠙⠚⠋⠀⠉⠙⠻⠷⣶⣤⣤⣄⣀⣛⡖⢊⣀⣀⣼⠃⠀⠀⣀⣤⣤⣀⠀⠀⠀⢀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡹⠿⣷⣄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠋⠁⠀⠀⣾⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢝⠿⣗⠠⣈⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡏⠀⠸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢅⡪⢿⣿⣝⡳⠶⢷⣦⣤⣤⡄⠀⠀⠀⠀⠄⠀⠤⠄⠀⣀⣴⢶⣂⣀⡀⠀⣸⣿⠇⠀⠀⠀⠀⢈⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⠝⠻⣿⣶⡶⣤⣬⣭⣭⣍⣙⣿⠿⠿⠟⠛⢛⢋⣉⡭⠟⠁⣠⢾⠿⠋⠀⠀⠀⢀⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁