Sitting Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠂⠲⠉⠑⠊⠑⠶⠒⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⣀⠶⠒⠉⠒⠊⠉⠑⠉⠳⠤⠤⢤⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢿⣀⣴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠖⢢⡄⠀⢀⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡀⢹
⠀⠀⠀⠀⠀⠀⠀⢠⠏⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢧
⠀⠀⠀⠀⠀⠀⠀⠘⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡼
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢸⠉⢦⣠⠞⣹⡶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡼⠀⠀⠀⠘⠋⣠⠟⢛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣣⠞⠁⠀⠀⠀⠀⠀⠀⠴⢛⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⣡⠊⠁⠀⠀⠀⠀⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⣀⣸⠉⠙⠒⠒⠒⠒⠒⠒⠉⢉⣇⠴⠋⠈⠳⣄⠀⠀⠀⠀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⣸⠋⠑⠒⠒⠒⠒⠒⠒⠊⠙⣇⠀⠀⠀⠀⠈⠓⢤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠗⠲⠤⠤⣀⣀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠁
⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⣸⠁⣀⡤⠤⠤⠤⣀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠒⠢⢄⣷
⠀⠀⣴⠋⣀⠀⠀⠀⠀⠀⠀⢻⡜⠁⠀⠀⠀⠀⠈⠳⡄⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⢻⣆⣀⣀⣀⡀
⠀⠀⠈⢹⣇⣴⠂⢰⠀⡴⢦⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⢤⡇⠀⠀⠀⠀⠀⠀⠀⠀⣯⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢙⡇
⠀⢀⡔⠃⠀⠈⠓⠋⠛⠁⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⡼⠤⠤⢷⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⢰⠋⣿⠊⢰⠗⠒⠒⠒⠒⢺⠇⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⡶⢺⡄⠀⠀⠀⠀⠀⠀⠀⠘⡧⠤⠤⠤⠤⠤⠤⢤⡤⠤⣤⠤⠤⣾⣿⡄
⡇⠀⡏⠀⣾⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢸⠀⢀⠃⠀⢠⣿⣿⣷
⡇⠀⣇⠀⢻⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⢨⡆⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⢸⣿⣿⡏
⠸⣄⣿⣀⣨⣆⣀⣀⣀⣸⡧⠀⠀⠀⠀⠀⠀⠀⢠⣟⣄⠀⣇⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⢸⡀⠀⡇⠀⠘⣿⣿⡇
⢸⠁⣸⠁⢠⡇⠀⠀⣠⠷⠤⠤⠤⣀⣀⣀⠀⠀⠻⠏⠉⠉⠉⠉⠉⢿⣁⣀⣠⠤⠤⠴⠒⠒⠊⢻⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⣇
⢸⠀⣿⠀⢸⡇⠀⢠⣇⡀⠀⠀⠀⠀⠀⠉⠉⠉⢻⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⣀⣀⣀⣠⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⢸⡀⢹⠀⠀⡇⠀⢠⡟⠙⣿⣿⣟⣒⡶⢶⣤⠤⡼⠀⠀⠀⠀⠀⠀⠹⡶⢲⣾⣛⣯⣭⣿⣿⣧⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏
⢀⠷⢸⡗⠒⡿⠂⢾⢧⡘⠛⠛⠭⠭⠿⣻⣭⢀⡷⠤⠤⠤⠤⠤⠤⠴⣇⠨⠿⠯⠓⠒⠊⠉⢁⡴⣹⠤⠤⠤⠤⢤⣤⣤⣤⣀⣿⣿⡇
⢸⠀⢸⠀⠀⡇⠀⠀⠳⢽⣲⣤⣄⣀⣀⡠⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠙⢷⣖⣤⣤⣤⣴⡾⠽⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷
⢸⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠉⠋⠉⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿
⠈⣦⢸⣆⢀⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣤⣶⣶⣦⣴⣿⣷⣤⣾⣿⣧⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠉⠛⠛⠉⠈⠙⠛⠉⠛⠛⠉⠻⠛⢿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠐⠚⠻⠃⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠸⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⠟⠿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣿⣿⠂⠀⠀⠀⠀⠀⢰⣶⡄⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⣤⡄⠻⠃⠀⠀⠀⢨⡇⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⢀⣿⠇⠰⠂⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠰⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠉⠀⠰⠀⠀⣀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠛⠛⠦⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠠⠶⠀⠀⠀⠄⠀⠀⠤⡄⠀⠀⠄⠀⠠⠴⠄⠰⠆⠀⠀⠀⠀⠀⠄⠀⠀⠘
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠰
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠉⠀⠐⠐⠂⠀⠒⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠘⠃⠈⠉⠀⠀⠀⠈⠉⠉⠘⠁⠀⠀⠀⠀⠟⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠈⠋⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⡈⠀⠀⠦⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣸⣧⣀⣠⣿⣿⡷⣶⣄⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⠿⣿⣿⡿⠿⡎⠀⠀⠀⠛⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⢀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠀⢀⡀⠐⠀⠀⠀⠀⠤⠀⠀⠀⠒⠒⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠤⠠
⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡟⠛⠉⠛⠛⠛⠙⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣇⣠⣀⣠⣤⣤⣤⡀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⡿⠟⠋⠀⢸⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡟⠽⠏⠉⠠⠤⠀⠀⠀⢻⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⡄⠀⠀⣤⠀⠀⠀⠈⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⢨⣿⠏⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⡠⣤⠀⠀⠀⠀⠀⢫⡄⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠦⠤⢤⡤⠦⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠾⣃⢀⣀⣀⠠⠖⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣶⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠁⢆
⠀⣀⠀⠀⠀⠀⠀⢶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡐⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⢸
⡏⠉⠒⢤⠀⠀⢀⡿⢀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⣻⠀⢠⡞⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⡘
⠀⡆⠀⠀⠀⠉⠉⠀⠈⣄⠀⠀⠀⠀⠇⠀⠙⠒⠀⢠⣤⠏⢠⠾⢃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁
⠀⠈⠒⠂⣀⣀⡀⠀⠀⠈⠉⠐⠆⢀⡘⣅⡀⠀⠀⠀⠀⠀⠀⢦⣼⡄⠈⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠛⣡⠀⠀⠀⠀⠀⠀⢀⡔⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⠆⣀⠀⠀⠀⠉⠛⠳⠶⠆⠀⠀⣀⠀⠀⠈⠉⠳⠆⣀⡀⠀⠀⠀⡠⠊⠀⠀⠈⠆⠀⠀⠀⠀⡠⠋⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠦⣀⠀⠀⠀⠀⠀⠀⠀⡸⠇⠤⣀⠀⠀⠀⠀⠉⠁⠊⠀⠀⠀⠀⣀⠞⠢⡀⠴⠊⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠤⠄⡀⠤⠋⠀⠀⠀⠀⢹⡷⠤⣀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠛⠢⠄⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠐⠒⠓⠛⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠒⠀⠀⠀⠀⠈⡹⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠴⠆⢉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠒⠒⠦⠤⠤⠤⠤⠤⠦⢻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠻⠀⠀⠀⠀⠀⠀⠀⠸⠗⠲⠒⠒⠒⠒⠒⠒⡖⠒⠒⠒⠒⠒⠒⠖⠒⠒⠒⠒⠒⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⡆⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠇⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⢠⣤⠇⠀⠀⢇⡀⠀⠀⠀⠀⠀⢀⠿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣇⠀⣦
⠀⠀⠀⠀⠀⠀⠀⡤⠊⠀⠈⠉⠉⠁⠀⣀⠀⠀⣤⠊⠁⠉⠁⠐⠂⠘⠋⠀⢀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣧⠀⡟
⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⢠⠶⠒⢁⠔⠉⠀⠀⠀⠀⠀⢀⡄⠰⠂⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠦⣀⣀⣀⣀⠰⠊⠁⠀⠀⢸⡀⠀⠀⠀⠀⣀⠴⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡿⠋⠙⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣴⣦⣴⣶⣶⣆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣋⠤⠤⠄⠤⠤⢌⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢩⠀⠤⠄⠀⠀⠤⠄⠸⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⠀⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠵⢤⡀⠉⠀⡠⠾⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠐⢣⠉⢡⠓⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠒⠠⣱⡡⠐⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⢠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⢀⠧⢄⡀⠀⠀⣠⣋⡼⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠸⣌⣣⡀⠀⠀⣀⠤⢇
⠀⠀⠀⠀⠀⣠⠏⠤⡀⠘⡄⠔⠉⠀⠀⠀⠀⠉⠒⠂⠄⠀⠒⠈⠁⠀⠀⠀⠈⠑⠤⡜⠀⡠⠌⠧⡀
⠀⠀⠀⠀⡸⡁⡀⢀⠈⡎⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡎⢀⠀⡀⡹⡀
⠀⠀⠀⠀⢧⠁⠃⠾⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡅⢀⣀⣀⠄⣀⠀⠀⠀⠀⠀⠰⡼⠆⠃⢡⠇
⠀⠀⠀⠀⠀⢳⡚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁⠀⠀⠀⠑⠢⢄⡀⠀⠀⠀⠘⢒⡇
⠀⠀⠀⠀⠀⠀⠙⠒⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠤⠤⠤⠤⠤⠤⠤⠬⠧⠤⠤⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣦⠊⠀⠀⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⣤⠀⠻⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⣾⠀⠸⣷⣤⣤⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠘⡀⢸⣦⣀⣼⣋⣉⣉⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠙⡴⠿⠟⠛⠉⠉⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⣄⢸⠿⠷⠶⠾⠟⠛⠋⠉⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠲⠦⠤⠤⠤⠴⠖⠛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠤⠤⣤⡤⠤⠖⢛⣆
⠀⠀⣠⣤⣄⡀⠀⢠⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡲⠲⠶⠒⢒⣉⣉⣇
⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⢿⣿⣿⣿⣿⣿⣿⣷⠾⠛⠛⢋⣡⣼
⠀⠀⠀⢈⣿⣿⣿⡿⣿⣿⣿⣿⣿⡇⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠸⣿⣿⣿⣷⣿⣿⣿⣿⣿⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠈⠉⠛⠛⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⠬⡧⠈⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⢸⠀⠀⠁⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠠⠀⠁⠃⠁⠀⠃⣐⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⠀⠀⠀⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡠⢄⡲⠥⠈⣂⡉⠤⠀⠀⠀⠄⠀⠠⠀⠀⠠⠄⠀⠀⠄⠀⡄⠠⠌⣁⡊⠩⠕⣂⠤⡀
⠀⠀⠀⠀⡔⡡⢊⠅⠂⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠁⠂⢍⠢⡑⡄
⠀⠀⠀⠀⡄⢇⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠇⡄
⠀⠀⠀⠀⠈⠢⢕⡪⠅⡒⠠⠤⠄⢀⣀⡀⠀⠀⠀⠀⠀⡀⢀⠀⢀⡀⡀⠀⣀⣀⠀⠤⠤⠐⡂⠭⣒⠥⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠋⠤⠀⠒⣂⡠⠤⠉⠩⠭⠍⠉⠍⠩⠍⠍⠥⠤⣀⡐⠢⠠⠌⠓⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠟⢉⣃
⠀⠀⠀⠀⠀⠀⠄⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠨⢳⡄⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⢠⣦⣺⣟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⢀⣸⠃⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠺⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠚⠒
⠀⠀⠀⠀⢨⣧⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡀⡧⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⠀⠀⠀⢂⡆
⠀⠀⠀⠀⠈⠉⢁⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⡆⠀⠀⠠⠘⣆⠀⠀⠀⠀⠀⠀⢀⢠⠞⠀⠀⠀⠂⠁⡄⠀⠀⢀⡀⠐
⠀⠀⠀⠀⢀⠀⡁⢦⠀⠰⠼⡔⠢⡀⠀⠀⠀⠀⠀⢠⢰⡇⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠛⠂⠀⣤⣐⣀⣀⣠⡤⠈⠉
⠀⠀⠒⣾⠏⠔⠁⠺⠀⠀⠐⡄⠐⠘⠄⠀⠀⠀⠀⢀⢾⡇⠒⠀⠀⠀⠔⠉⣆⣠⠤⠂⠂⢀⠀⢒⣀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠐
⠀⠀⠀⡹⠀⣸⠀⠀⠂⠀⢀⢀⠄⠐⠊⠆⠀⠐⠠⠄⣠⣈⡀⡀⠀⠀⣧⣀⣉⣀⣔⡩⠔⠋⠉⠉⡀⠠⠀⠊⢀⡀
⠀⠀⢰⡁⢂⠙⠓⠀⡀⠀⢸⠀⠆⠀⠠⠠⢴⣤⠀⢃⠛⠓⠀⠈⠁⠀⠀⠀⢭⡉⡁⠀⠀⢀⠁⠀⠁⠀⠀⠀⠈
⠀⠀⠁⣈⠉⠙⠀⠀⠀⠀⠈⡀⠂⠀⠘⠒⢠⣄⣴⡊⠷⠆⠀⠀⠀⠘⠀⠀⠀⣀⠄⠈⡢⠀⢰⠀⢀⠀⠀⣀⡀
⠀⠀⠀⠀⠉⠆⠒⢠⣠⣀⠀⠉⠀⠒⠶⠓⠋⠉⠁⢀⡀⢠⣖⠒⠖⠦⣄⡀⠠⠇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡰⡉⠉⠉⠉⠱⠒⢪⠿⠀⠶⠾⠋⠀⠀⠀⠈⡇⠀⠀⢠
⠀⠀⠀⠀⠀⠠⢿⣄⣀⡀⠀⡆⠀⠀⠠⠀⠀⠀⢰⡀⠀⠀⠈⠀⠂⠤⢾
⠀⠀⠀⠀⠀⠐⠀⠐⠨⠍⣙⡄⡀⠠⠷⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠢⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠐⠓⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣂⣤⣀⠀⠀⢀⣼⠟⠻⣥⡀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠛⠧⡀⠘⠋⠀⠀⠈⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠄⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠙⡀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⢧⠀⠀⠀⠀⠰⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢠⡞⠳⢄⡀⠨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠑⢦⣀⢸⡗⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⡆⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⣤⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢺⣀⣤⠊⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢛⣿⠀⠀⠀⠀⠛⠒⠒⠢⠠⠀⠀⠀⠰⠟⠋⠉⣭⣽⣿⢿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⡀⠀⣴⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠠⠤⠴⡾⠟⣻⣷⣶⣶⣶⣶⡶⠶⣾⣿⣿⠛⠉⠉⠉⠀⠈⠁⢺⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠁⣠⣶⣶⣶⣾⣿⣿⣯⣭⣭⣁⣤⣶⣾⣭⣵⣶⣧⠀⠀⠀⠀⠀⠀⣸⡴
⠀⠀⠀⠀⢀⣠⢔⡢⣴⠦⣀⣤⠀⠀⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠠⠄⠀⣠⣾⣅⣀⡀⢀
⠀⠀⠀⠀⠋⠑⠉⠊⠵⡾⢫⢶⠿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⡏⢴⣶⣿⠟⠛⠋⠉⠁⠒⠒⠒⠚⠛⢳⡶⢾⡷⠦⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⡿⠿⠟⠋⠉⠛⠛⠶⣤⣜⠶⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢀⡄⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣀⡠⠄⠖⠚⠁⠘⢄⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣠⡎⢈⣉⡁⠀⠀⢉⣉⠀⡓⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠑⢇⠀⠀⠀⡇⠀⠀⠀⢠⠗⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠈⣆⠀⠀⠤⠄⠀⣠⡊⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡰⠋⠡⢽⠒⠤⠔⢺⠥⠈⠳⡀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⠓⠀⢀⡀⠱⡄⡰⠁⡀⠀⠐⠳⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠈⠑⢿⠗⠉⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⡼⡟⠀⣠⠤⠤⠄⣼⠀⠤⠤⢤⠀⠘⡿⡄⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⢀⡼⢱⠣⠚⠁⠀⠀⠀⢸⠀⠀⠀⠀⠑⡆⢳⠹⣄⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠒⢄⠀⠀⢠⡾⠴⠞⠓⠢⢄⡀⠀⠀⠸⠀⠀⠀⣀⠤⠓⠚⠦⠼⣦⠀⠀⢀⠔⠊⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠒⠢⣄⢀⡧⠔⠁⠀⠀⠀⠀⠀⠀⠈⠉⠐⠚⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠑⠤⣄⢀⡤⠐⠚⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⠄⡀⢀⠈⠋⠐⠦⡀⣀⡀⠀⠀⠀⢀⣀⣀⠀⠀⢀⣀⡀⠀⠄⠀⣀⣀⣀⣀⣀⠔⠈⢋⢀⠀⡄⢌⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⢰⠀⣏⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡅⠀⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⣈⡇⢸⠈⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⡜⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡏⠉⠀⠀⠈⠢⢄⡀⠀⠀⠀⠀⠁⠘⠼⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⢀⣀⣇⣀⣀⣀⣀⣀⣀⣈⣙⣢⣀⣀⣀⡤⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠈


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠶⠶⠖⠛⠛⠛⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⣀⣤⣤⣤⣤⠤⣤⣄⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⠀⠀⢻⡟⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣿⠀⠀⣤⣄⠀⠀⠀⠤⠤⠀⠀⢻⡄⠀⠀⢸⡇⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢷⣤⠶⢾⡁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⢿⠀⠀⠀⠛⠀⠀⠀⠈⠉⠀⠀⠀⠀⢠⠖⢈⡧⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢸⡄⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣼⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣇⠀⠀⠀⠠⣄⡠⠄⠀⠀⠀⣰⠏⣿⠉⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⢀⣠⡞⠁⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣉⣻⡶⠶⠶⠞⠋⠹⣇⣀⣀⣽⣦⣤⣀⣀⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠋⠉⠉⠉⠉⠹⣧⡀⠀⢀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠉⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣶⣤⣀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠉⠙⠳⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⣴⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢀⡾⠃⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠹⣇⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠹⣦⡀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⣠⡴⠞⠋⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢈⣻⡄⠀⠀⠀⠀⠈⣧
⠀⣠⠟⠁⠀⠀⠀⠀⢠⡞⠉⠙⠓⡶⣤⣀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡷⠞⠋⠉⣿⠀⠀⠀⠀⠀⢸⡆
⣸⠏⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠒⠭⣛⢶⣄⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢻⡄
⣿⢠⠀⠄⡸⠀⣿⠀⡿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠪⣟⠦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⢸⠃⢸⡆⢀⠀⠀⠀⠀⢿⡀
⣿⡘⠀⠀⠇⠀⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣉⠛⠲⠶⢶⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣿⡇⢸⠃⢠⠀⡇⢸⡇
⠈⠛⠶⢦⡼⢴⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡇⢸⠀⡿⠀⣇⡼⠃
⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣠⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠫⢼⡟⠳⠶⠳⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⠉⠉⠉⠉⠛⠛⠛⠛⠋⠉⠁⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠄⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣉⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⢀⣀⣤⠶⠟⠛⠷⠶⠶⠶⠶⠶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣖⡀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠶⠛⠉⠙⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠠⠤⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⣀⠀⠀⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢡⠀⠁⠒⠤⣀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⢠⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠢⠔⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠒⠒⠒⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⢸⡆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⠀⡀⡀⠐⡇
⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠁⡀⡁⡀⢨⡇
⠀⠀⠘⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠒⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⢀⡀⡀⠀⢸⠇
⠀⠀⠀⠃⠀⡆⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⢆⠀⠀⢀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣼
⠀⠀⠀⢐⠀⢡⢰⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠢⡴⠁⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠘⡀⠐⠀⡆⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠃⠀⡆⢰⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠸⠀⢠⠈⡄⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠐⠐⠀⠀⢋⣧
⠀⠀⠀⠀⠀⠇⠈⡄⠆⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠉⠉⠙⡟
⠀⠀⠀⠀⠀⠰⠀⢡⢠⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⡾⢦⠔⠈⠀⠘⢄⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠸⣀⣀⣀⣀⣀⣀⣠⠁⢃
⠀⠀⠀⠀⠀⠀⠀⠈⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠘⢷⡀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⡈⠁⠒⣤⡄⠀⠀⠀⢀⡄⠀⠀⠄⢃⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠃⠀⠅⢡⠀⠀⠀⠀⠀⡀⠀⠀⠬⠀⢀⠀⠙⠶⠴⡶⠶⠾⠿⠷⠤⠴⠾⠿⠶⠿⡤⠤⠤⠤⠀⠸⠷⠤⠤⠦⠼⠄⠀⠵⠀⡄
⠀⠀⠀⠀⠀⠀⠸⠀⢠⡨⠅⠛⡋⠭⠒⠒⠒⠛⠏⢉⣐⡕⣄⠀⠀⠀⢹⠋⠀⠀⠀⠀⠈⠙⠛⠉⠉⠉⠉⠉⢻⣟⠉⠉⠉⠉⠉⠹⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠇⠈⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡍⠀⠈⠉⠉⠑⢄⠀⠀⠀⠀⣿⠂⠀⠀⠀⠀⠀⠘⢛⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠀⢁⠠⣀⣀⣀⣀⣀⣀⣀⣀⣀⡐⣸⣀⣀⣀⠀⠀⠀⡜⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠒⠒⠒⠒⠒⡶⢲⠒⠲⡖⠚⢲⠒⠋⠀⠀⠀⡘⠀⠀⠀⠀⣜⠃⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠼⠀⢠⠀⠀⡎⠀⠀⠀⠀⢐⠀⠀⠀⠀⠠⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢠⠁⠀⠀⠀⢀⠆⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⢘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢶⠀⡇⠀⡌⠀⠀⠀⠀⠌⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡆⠀⠀⠀⠀⠀⠈⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠤⢃⡘⠴⠍⢉⡗⠐⠒⡲⠚⠀⠀⠀⠀⠀⣜⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢱⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⡔⡂⠁⠀⢏⣹⢘⠛⣺⠀⠀⠀⠉⠁⠒⢄⡀⠀⢀⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣁⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠞⠉⠉⠙⠉⠑⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⠴⠒⠒⠒⠒⠒⠲⢤⡴⠒⠒⠒⢤⡀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⣴⣶⠶⠶⠶⣖⣒⠒⠷⣤⣄⡀⠀⢳⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠋⠉⠉⠉⠉⠐⠒⠢⢤⣉⠑⠲⢤⣈⠉⠓⠾⣆⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠻⣷⡦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣉⡓⠦⣀⠈⠙⠦⣄⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣶⠈⢷⡌⠙⠲⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠯⠿⣦⣄⣈⠙⠦⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢠⠏⠀⠀⣹⢷⣄⡀⠈⠛⠶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣻⠿⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⣴⠋⠀⢀⡾⠃⠀⠈⠛⠷⣶⣦⣀⣀⡈⠛⠓⠶⠶⢦⣤⣤⠶⠶⣨⣿⡿⠶⣻⡟⠳⣄⣀⣤⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⢠⡴⠞⠁⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⠷⠟⠻⠷⠶⠶⠶⢿⣝⠣⠌⣀⣋⣩⠷⠀⠀⠀⢀⡵
⠀⠀⠀⠀⠀⠈⠙⢶⣿⣥⠶⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢩⡄⠀⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣏⣧⣄⡀⢠⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⡼⢣⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡼⣇⠉⠁
⠀⠀⠀⠀⠀⠀⠘⣇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠉⠙⣄⣰⠋⣹⡤⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢠⡇⠀⠈⠁⠘⠋⢀⡾⠛⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⣰⠋⠀⠀⠀⠀⠀⠀⠉⢠⠾⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢞⡿⠋⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠦⢤⣤⣤⣦⣤⣥⠤⠴⠶⠞⠋⣀⡴⢿⡀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠶⠶⢾⣄⡀⢀⠀⠀⠀⠀⠀⢀⠀⣠⣵⠞⠁⠀⠀⠙⢦⣀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⣼⠃⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⢷⠀⠀⠀⠀⠀⠀⠉⢳⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠤⠤⢄⣀⣀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠈⠉⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⣰⠎⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢹⡄
⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠃⣠⠶⠊⠉⠉⠑⠶⣄⠀⠀⠀⠀⠀⣴⠞⠉⠀⠀⠉⠑⢾⣥
⠀⠀⢰⣏⣀⡀⠀⠀⠀⠀⠀⠀⠨⡾⠁⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢈⣿⣠⣿⠀⣶⠀⣰⠢⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠤⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠁
⠀⢀⡴⠋⠀⠀⠘⠒⠋⠛⠃⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠤⠴⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷
⢰⠟⢳⡖⠛⣶⠒⠒⠒⠒⠒⠲⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠦⢤⡤⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠤⠤⠤⠤⠤⠤⠤⣀⣀⢀⣄⣀⣀⣠⣿⣿
⡇⠀⡸⠀⠀⡟⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⣬⠁⠀⡟⠀⠀⣹⣿⣿⣦
⡏⠀⢛⠀⠀⡇⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⡏⠀⢸⡇⠀⠀⣿⣿⣿⡟
⢡⡄⢸⡄⠀⣿⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠸⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⣷⠀⠘⡇⠀⠀⢿⣿⣿
⢈⡷⠾⡷⠶⠾⠶⠶⠶⠶⣾⠛⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠷⠦⠴⠷⠤⠴⣧⡄⠀⠀⠀⠀⠀⠀⠀⢀⣹⣧⣤⣤⣤⣤⣤⣼⣤⣤⣽⣤⣤⣼⣿⣿⡆
⢸⠃⢸⡇⠀⣾⠀⠀⠀⡼⠋⠁⠒⠶⠶⠦⠤⣤⣤⣀⣘⡃⠀⠀⠀⠀⠀⠀⢻⣤⠤⠶⠶⠖⠋⠉⠉⠉⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⢠⠀⢸⡇⠀⣿⠀⠀⠰⣥⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⢴⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠸⡇⠸⡅⠀⢻⡀⠀⢠⡟⢈⣿⣯⣭⣝⣛⣲⣶⣶⡢⣼⠃⠀⠀⠀⠀⠀⠀⠘⡶⢲⣿⢿⣭⣭⣷⣶⣾⣧⡄⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡃
⢀⡿⠒⡿⠒⢺⡗⠰⢾⢧⡈⠛⠛⠫⠭⠭⢿⣿⣯⠁⣴⠤⠤⠤⠤⠤⠤⠤⠤⣧⠀⠿⠿⠝⠓⠒⠊⠉⢉⡠⢻⡤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣈⣿⣿⡇
⢸⡇⢰⡇⠀⣼⠀⠀⠈⠳⣝⣦⣄⣀⡀⠀⣀⣈⣡⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⢤⣀⣀⣠⣤⣔⡾⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣧
⢸⡄⢸⡇⠀⢿⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠸⣇⠘⣇⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃
⠀⠙⠶⠻⠶⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢋⡀⠀⠉⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⢀⠀⠉⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⢈⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡋⠘⣗⠲⠖⣯⡟⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠙⠢⣈⣷⣞⣽⣴⡞⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢈⡋⣿⢰⢹⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣇⣃⣀⡀⠀⠠⡇⡇⡆⢸⠀⠀⠀⠘⣟⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⠤⢄⠀⠉⢻⢸⠃⡃⢸⡆⠀⠀⠀⢻⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠲⣇⣀⡜⠓⠂⢾⣎⣠⣆⣸⣧⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣴⣧⡤⠭⠶⢶⣾⣿⠁⠀⠀⠀⠀⠀⠀⢀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠈⠑⠒⠢⠤⢄⣀⣀⢼⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡴⢁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠒⠒⠊⠉⠉⠁⢀⡠⠔⠚⠑⠒⠠⠤⠤⠤⢤⠤⠄⠺⢏⣉⣏⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⠦⠤⠤⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⡴⢃⣀⣀⣀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⡛⢹⠀⠀⠀⠀⣠⠤⡤⠐⠒⠒⠛⠛⠒⠒⠒⠚⣿⠉⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⡇⢸⠀⠀⠀⠀⣟⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⡇⠀⡆⠀⠀⠀⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣠⠃⠀⢇⠀⠀⢀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⢤⢼⡀⠀⠘⣤⠄⢼⣇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⡠⠔⠒⠋⠉⠀⠀⢀⠇⠀⣀⠷⠀⢾⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠈⠑⠒⠀⠐⠒⢒⡶⠛⠊⠁⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠓⠈⠁⠙⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠉⠀⠀⠀⠀⠉⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠉⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣠⠔⠒⠊⠡⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡄⠚⣁⠀⠀⣘⠀⢃⣰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡃⠀⠀⢸⠀⠀⠀⢀⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠻⡜⣣⡟⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠈⠀⠀⠀⠇⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⣤⠀⠀⠀⠀⠶⠀⣬⠀⢷⠆⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠉⠉⢸⠀⢠⠀⠀⠀⢃⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣄⣠⣀⣼⡋⠀⠀⠀⠀⢀⠴⠶⢠⣄⠀⢸⠀⠀⠀⠀⠈⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠁⠚⠉⠑⠶⠖⠋⠀⠀⠀⠀⡈⠑⠺⣄⢀⡀⢀⠀⡸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡤⠔⠊⠀⠀⠀⠀⢰⠚⠉⠉⠲⠄⢈⣉⣿⣉⠁⣡⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⢀⡀⠀⠀⠀⠈⠠⠄⠠⣤⠄⣀⡈⠀⠀⠀⢸⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⠒⣀⣀⣀⠒⠉⠀⠀⠀⠀⠙⠉⢛⠉⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⠡⠉⡍⢩⠌⡗⠶⢲⡟⣼⢧⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⡆⠷⠴⠦⠣⡾⢶⠃⠶⠸⠧⠶⠰⠟⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⢰⡆⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠘⠀⠀⠀⠸⠂⠀⠀⠀⠀⠑⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠎⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡰⠾⠦⠤⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⣄⢀⣀⣀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠉⠀⠀⠀⠈⢡⣼⡧⠐⠒⠒⠒⢄⢀⡜⠒⠒⠒⠒⠒⠒⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⡏⠀⠁⠀⠀⠉⠁⠈⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠈⠀⢰⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⢀⣀⣀⡄⠀⠘⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⣀⠀⠀⠀⠉⣶⠉⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠙⠀⠀⠀⢠⠛⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠎⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢀⠘⠀⢀⡈⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠄⠒⠉⠀⠀⠠⠀⠠⠔⠊


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠁⢀⠀⣿⠻⡉⠛⢹⠟⠛⡉⡟⢫⠉⠙⠋⢙⠋⡿⠛⠋⠙⢉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣛⣛⣛⣛⣛⣛⣻⣟⣛⣛⣛⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣴⢾⣍⣉⣉⣉⢐⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⢉⣤⠴⢿⣄⠑⢌⠳⢤⡉⠙⠳⢤⡉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠎⡠⣚⠥⠒⠊⠙⢻⢦⠀⠙⢤⡉⠳⣄⠀⠉⠢⡈⢢⡀
⠀⠀⠀⠀⠀⢀⠔⣡⢊⠞⠁⠀⠀⠀⡴⠋⢻⡳⣄⠀⠙⠢⣄⠉⠒⠤⢝⣦⢑⡄
⠀⠀⠀⠀⢠⢫⢞⡵⠋⠀⠀⢀⡤⠊⠀⠀⠀⠙⢮⡳⣄⡀⠀⠉⠛⠒⠶⠤⢭⣍⣶
⠀⠀⠀⠀⣹⣧⣊⣀⡠⢤⡖⠋⠀⠀⠀⠀⣃⠀⡐⠉⠲⠭⣓⡒⠠⠤⠤⢤⣤⡶⣏⢦
⠀⠀⠀⣰⡏⠀⡜⠁⠀⢸⠇⠀⠀⠀⠀⠀⠩⠠⠁⠀⠀⠀⠀⠀⠉⠉⡟⠉⠈⢇⠈⣮⢧
⠀⠀⢠⢿⡆⢠⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢘⡄⢸⡜⡂
⠀⠀⣼⢸⠇⢸⠀⠀⠀⡻⠀⠀⠀⠀⠀⠀⠓⠖⠋⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⡇⢸⡇⢣
⠀⠀⡇⡾⠀⣸⠀⠀⢐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠈⣷⢸
⠀⠀⣷⢃⡴⠹⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⣧⡀⢻⣄⡂
⠀⠀⢹⡏⠀⠀⣇⠀⡞⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⡇⠀⢰⠁⠙⣮⡏⠁
⠀⠀⠸⣇⠀⢰⠉⣦⡇⠀⠀⠈⠑⡶⠠⠤⠤⠴⠶⠤⠄⢲⠊⠉⠀⠀⠀⠇⢠⠋⡄⠀⢻
⠀⠀⣼⣸⠀⡌⢠⠁⣠⠖⠉⠉⠉⡟⠒⠒⠠⠤⠤⠐⠒⢺⠉⠈⠉⠳⣄⠘⠻⠈⡇⠀⡛⡀
⠀⠀⠀⢯⡼⢡⠇⡰⠃⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠘⢆⠀⢇⢹⡀⣷⠇
⠀⠀⠀⠀⠉⠋⡺⠁⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠈⡇⠙⠢⠟⠋
⠀⠀⠀⠀⠀⢰⠉⠉⠉⠒⠒⠿⣄⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢹⡖⠈⠉⠉⠱⡀
⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠉⠙⠒⠚⢩⠟⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⣉⣳⣀⡠⠤⠔⠖⠚⣆⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣏⣽⠓⠢⢄⠀⠀⠠⢤⡽⠁⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠋⣤⣾⡀⠀⠀⠑⢄⣹⠚⠙⣆⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⡠⠊⠁⠹⣿⠀⢾⣿⣻⡄⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⣀⣰⣃⠴⡄⣤⠐⣦⡿⠋⠐⠒⠒⠒⠒⠂⡶
⠀⣠⠊⠀⠀⠀⠀⢿⡄⠰⣿⣿⡟⣄⠀⠀⠀⠀⠀⠀⠈⣳⡉⠉⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣸⡇
⡘⢩⡏⢩⡏⠉⠉⠙⢮⣦⣉⣤⢧⡬⢆⠀⠀⠀⠀⠀⢀⡠⠿⢖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡖⠒⡶⠒⢰⣿⣇
⡇⠠⠀⢸⡇⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⢷⠂⣀⡴⠚⠉⠀⣠⡼⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢸⣿⣿
⢣⠐⡆⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠁⢀⣠⣖⣯⢿⡿⢎⡷⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢸⣿⡇
⢸⠉⡟⠉⡟⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠊⠓⣶⠛⣿⠯⠒⠉⣠⣾⠟⠒⠒⠒⠒⠒⠒⠒⠒⠓⠒⠛⠒⢺⣿⣏
⢠⠰⡇⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡶⡶⢶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠸⡀⣃⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⢰⠉⡛⠉⡿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣿⣿
⢸⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠘⡄⣧⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣀⡤⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⣠⠇⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠋⢀⣄⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⡡⣍⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⡜⢸⠀⠈⣧⡇
⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡿⠀⢀⠇⠈⢦⡤⢼⠇
⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⢠⣿⣿⡿⠋⠁⠀⠀⠀⠀⢸⠀⠀⠈⢆⠈⣇
⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⣀⣀⣼⣿⣿⣀⣀⠀⠀⠀⠀⢠⠧⣀⠀⠀⢈⠷⢌⡳⢄⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠹⣿⣿⣿⣿⣿⠋⠀⠀⠀⣰⠃⠀⠈⠙⠒⡞⠀⠀⠉⢢⠈⢷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⣿⣿⣿⠃⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⢱⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠛⠛⠛⠉⠉⠉⠉⣿⡇⠀⠀⠀⠀⡞⠀⠀⠀⡰⡻
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢰⠁⠀⠀⢰⢃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢀⠇⠀⠀⢀⠏⡸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⠀⠀⡜⠀⠀⠀⣜⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠁⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⡷⠦⠤⣔⣁⣉⣒⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠓⠲⠤⠤⠽⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣿⣿⣿⣿⣿⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠋⠀⠈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠟⠉⠉⠉⠉⠒⢤⠀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠃⠀⠀⠀⠀⠠⠀⠀⢦⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡆⠀⠀⠸⠀⠀⠀⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡄⠙⠦⠶⠞⠁⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡦⠤⠤⠄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠐⠲⣃⠀⠀⠀⣘⠶⠂⣠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠉⠀⠀⠀⠀⠀⠚⠛⠓⠀⠀⢠⠈⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣀⣀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⠈⠛⠛⠛⠣⠤⠄⣠⡤⠐⠉⠀⠀⠀⠀⠙⠠⢄⡀⠀⠘⡀⠀⢀⡀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⢀⠤⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠤⢸⡃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⡀⠀⣀⣀⠠⠼⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠦⢀⡀⠀⠀⠀⠀⡤⠛⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠄⣀⠀⠢⣄⠀⠔⠈⠁⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠓⠐⠂⠀⠀⠀⠄⢤⠤⠀⠀⠐⠂⠀⠈⠉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⠸⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣨⠃⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢠⡍⠉⢉⣀⠀⣉⡈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠲⠖⠒⠂⢤⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠈⡇⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⠄⠀⠀⢀⡍⠢⠾⠆⢉⠉⠉⠉⠁⣶⠋⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡋⠀⠀⠀⠉⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⠠⡌⠀⠀⠀⠀⠀⢀⠦⠤⡤⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⢠⡆⠀⠀⠀⡐⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢀⣀⠤⠛⠁⠀⠀⡄⠀⠀⢀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋

⣤⣤⣤⣤⣤⣤⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣾⣃⣹⣯⣹⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣛⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠈⠉⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡐⠿⠀⠈⠀⢀⣷⣆⠀⠀⠀⠀⠈⢀⣿⣷⠀⣙⡥⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠉⠓⠀⠀⠀⣿⣿⠟⠀⠀⠀⠀⠀⣿⣿⠟⠀⡏⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⠀⠀⠀⣀⠠⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⣉⣉⡅⠈⠉⠁⢩⣍⣉⠙⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠒⠋⠈⠁⠪⡅⠒⠂⡵⠋⠄⢸⠒⠃⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠈⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡀⠀⠀⠉⠒⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠃⠀⠀⢀⡄⢻⠀⠁⠀⠀⠀⠀⠀⠀⡇⠈⠃⡀⠀⠀⢠⠉⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡞⠤⢾⣙⢀⠾⠰⡄⠀⠀⠀⠀⠀⠀⡷⠲⢆⢹⡇⢤⣮⠦⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠤⠀⠀⢀⡈⢩⡀⢀⡇⠀⠀⠀⠀⠀⠀⣅⢀⡨⢤⣄⠠⠤⠤⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⡜⠉⠏⠀⠀⠀⠀⠀⠀⠀⠈⡍⠡⡀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠠⠄⢀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠤⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⠀⠀⠀⠀⢀⠎⠀⢁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡖⠀⠤⢤⡤⠉⠀⠀⠈⠤⢤⡤⠤⠄⢲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠁⣩⠇⠀⠀⠀⠀⠀⠀⠀⢸⡍⢨⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⡉⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⣉⡀⠇

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠛⠿⠟⠿⠻⡿⢫⡙⠛⢿⠿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣛⣛⣛⣻⣟⣙⣿⣿⣛⣛⣛⣛⣛⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡈⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⢻⣽⣿⣿⣿⠿⠟⠛⠉⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠀⠀⢀⣠⣴⣶⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣷⣦⣄⠀⢸⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡏⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠺⠿⣿⣿⠃⠀⢀⣴⣚⡛⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠚⠛⠒⠦⡀⠀⢻⣿⣿⣏⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢉⣦⡈⡇⠀⠀⣿⣿⣿⣁⣀⠀⠈⠣⠀⠀⠀⠀⠀⠀⢀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠘⣿⠟⣡⠖⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠙⣧⠀⠀⢿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⠀⠀⠀⠀⠀⢰⣇⠞⠉⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⢹⠀⠀⠈⠻⢿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⠀⠀⠀⠀⠀⢸⠉⠆⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⣤⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢠⠤⠤⣄⣀⣀⣠⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠈⠢⣄⣀⡀⢀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⣄⣀⠀⠀⠀⣀⣠⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡾⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢚⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠧⡄⠀⢀⣀⣀⣀⣀⡀⣀⣀⣸⠀⢇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⠁⠒⠒⠒⠂⠤⠤⠤⠤⠬⣿⢠⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢸⠼⠧⠬⢉⣉⣁⣒⠒⠒⠒⢴⡇⡄⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢺⠀⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⣺⠀⠐⠒⠠⠤⠄⣀⣀⠀⣸⢠⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠂⠂⠐⠐⠦⠤⣤⣀⣀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠇⡀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣉⡁⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠄⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠁⠒⠒⠬⢄⣒⠂⠤⢀⡀⠀⠀⠀⣀⠤⠐⣀⠥⠒⠉⣀⡀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢈⢹⠒⠢⠤⣀⡈⠉⠐⠲⠤⣍⣉⣩⡤⠒⠉⠀⠀⠐⠺⢤⡇⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡤⢆⣀⡤⠶⠋⠁⣒⠤⢄⡄⠀⠀⡧⠔⠊⠉⠐⠂⠤⢀⡇⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠆⠀⠀⠁⠂⠶⡯⣁⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⣀⡤⠖⠉⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡟⠀⠂⠤⣀⠀⠀⠉⠓⢿⠀⠀⡇⢀⣠⠴⠊⠁⢀⡠⠆⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⡇⠀⠀⠀⠀⠈⠐⠢⢄⣸⠀⠀⠿⠋⠁⣀⠤⠚⠁⠀⢧⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁



⣿⠿⠿⡿⢿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣿⣿⣻⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠈⢀⡒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠈⡷⠤⢎⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⢰⢀⡀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⡀⠀⠀⢰⢸⠀⠀⠀⣀⠀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣇⠀⠀⡸⠖⢲⠾⠓⠯⠀⠀⠘⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⠦⠤⢄⡠⡌⡹⠊⢁⣀⠤⠔⠒⠂⠉⢃⠀⢱⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣈⣷⠊⠁⠶⣺⠀⠀⠀⢀⣀⣀⣀⡗⣿⠣⣄⠀⠀⠀⠀⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⡄⠈⠹⣧⣄⣀⣆⠀⠀⠀⠸⣦⠶⠺⢶⣅⠀⠞⠂⡐⠲⡀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢖⡏⠢⣈⠁⠀⠀⠉⣀⡈⠉⠈⣛⠀⠀⣈⣏⣐⣐⣂⣉⣀⣈⣢⡀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡬⠗⠈⠉⠐⠒⡖⠓⠛⠒⠒⠉⠹⣧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⠃⠀⠀⠀⢸⠓⡇⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⡖⠒⠒⠒⣶⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠐⠋⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣌⣳⣔⣦⢀⣠⣼⣄⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⠀⠉⠁⠀⠀⠈⠉⠒⠭⣒⠦⡀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠣⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢄⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⠟⠂⠀⠀⠤⠤⠤⠤⠤⠭⠭⠍⠁⡖⠲⠒⠤⡄⣀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣇⣀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡜⠉⢺⡄⠀⢸⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⢄⣀⢼⡇⠄⣞⠣⢄⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⡊⠁⠀⠉⠀⠉⡷⠊⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠢⡀⠀⠀⠀⢰⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⢆⡀⠀⠀⠑⡄⠉⠉⠉⠁⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄⠀⠀⠀⠀⠀⠉⠒⠒⠊⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⣀⣀⡀⠤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣾⣿⣶⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠉⠛⠻⠟⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⢠⣤⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠁⠀⠀⠁⠀⠀⠃⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⢠⣀⡀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡴⠃⠀⠐⠤⣀⣀⡀⠴⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠂⣠⣄⡀⠀⠀⣄⠀⠈⠉⢠⡄⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⢠⣄⠀⠀⠙⠛⠳⣆⢨⣵⣦⣴⣿⢉⡶⠚⠛⢀⣤⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⡎⠁⠀⠀⠀⠀⠀⠹⢯⡷⣏⢭⡍⢠⡇⠀⠀⠀⠈⠁⠂⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠰⣯⢰⡇⢸⠀⠀⠀⠀⠀⠀⠀⣸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⣿⣧⡀⠀⠀⠀⡇⢈⡻⢈⣿⣼⠀⠀⠀⠀⠀⠀⢰⡏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢸⡏⠈⢿⡀⠀⠀⢰⡶⣿⠀⣾⣿⡄⠀⠀⠀⠀⡀⣾⠁⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⢸⠁⠀⠘⡇⠀⠀⢸⡍⣿⡄⣿⡏⠀⠀⠀⠀⠀⢱⠏⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢁⠀⠀⢸⡗⣻⢶⣿⡦⡆⠀⠀⠀⠀⡘⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣠⡟⠀⠀⠀⠸⠀⠀⢸⡟⣭⣻⡇⠀⠀⠀⠀⠐⢰⠇⠀⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠙⠻⢿⡇⠀⠀⠀⠀⡄⠀⢸⣷⣬⢻⡄⠈⠙⠀⠀⠀⣾⣤⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡟⠀⠀⣟⠀⢸⣿⡽⢺⡇⠀⢸⡇⠀⢀⣿⠖⠒⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⣿⠃⠀⠀⠀⠐⠻⡄⢸⣿⣾⣻⡷⠀⢸⡇⠀⢸⡇⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⢨⡅⠀⢸⣷⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⢺⡀⠀⠀⢠⠄⠀⠀⢸⡷⣽⣻⣇⠈⠈⠁⠀⢿⡏⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⣉⣀⣀⣹⡇⠀⠀⠀⠀⠀⠀⣿⣟⣺⣿⣿⣠⣤⡖⠀⠠⣿⠀⠀⠀⠐⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠋⠉⠁⠉⢳⡀⠀⠀⠀⠀⠀⢻⣭⣗⣯⣽⣿⡽⢧⡄⠀⣿⡤⣤⣤⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⠀⠀⠤⠤⠶⢺⣧⣿⣿⣿⣶⣶⣿⣧⣼⠃⣀⠀⣀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢦⡀⠘⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⢠⡟⠀⢄⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠉⠰⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠘⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠃⠀⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣟⡛⠻⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣤⣆⡀⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣷⣤⢔⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠉⠛⠿⢿⠿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⡚⣿⢧⠀⠀⠀⠀⠀⠀⡔⡩⣚⡟⠙⠋⣻⣔⢕⡄⠀⠀⠀⠀⠀⠀⡼⣿⢓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣉⣉⡜⠀⠀⠀⠀⠀⢠⣱⡟⢁⡀⠀⠀⢀⣈⢻⣿⡀⠀⠀⠀⠀⠀⢠⣉⣉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⡆⠀⠀⠀⠀⠀⣸⠹⡇⠭⠍⠀⠀⠩⠭⢸⡏⣇⠀⠀⠀⠀⠀⢸⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠰⠀⠀⠀⠀⠀⡏⠒⡀⠀⠀⠀⡄⠀⠀⢀⠒⢹⠀⠀⠀⠀⠀⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢃⡀⠀⠀⠀⠇⠀⠱⡀⠻⣉⣉⠟⢀⠎⠀⡘⠀⠀⠀⢀⡸⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠉⠉⠁⠘⠒⠒⠚⠷⡂⢒⠾⠓⠒⠒⠃⠈⠉⠉⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠒⠒⠒⠒⠒⠒⠲⠯⡉⠈⠉⠉⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠑⠤⣀⢀⡀⣸⠀⠀⠀⠀⣄⡇⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠒⢂⠀⠀⠀⢸⠀⠀⠀⢸⠒⠂⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠈⡶⡾⠁⠀⠁⢹⠀⠀⠀⢸⢶⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡈⢰⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠃⢸⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⣀⣀⣀⣀⣀⣀⣀⠏⠉⢏⣀⡰⠉⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠄⠑⠍⠂⠤⠌⠱⠄

⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢐⣾⡇⠀⢄⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀⠀⠀⠀⣀⣀⣀⡠⠤⢔⣲⠁⡰⡀⠀⠀⠀⣠⣶⠶⢦⠀⠀⠀⡔⡄
⠀⠀⠀⠀⠀⣯⣶⣀⠔⠃⠈⠢⣀⣿⣳⠀⠀⠀⢰⣫⠤⠂⠤⠬⢦⠀⠊⢰⢱⠃⢀⠤⠀⠈⢢⠘⠇⠀⠀⡞⣁⠀⠀⠄⢰⡄⠀⡽⠃
⠀⠀⠀⠀⠀⠀⠈⡄⠉⠁⢈⠁⢠⠏⠀⠀⠀⠀⢼⠁⠂⠀⠐⠀⢸⠀⠈⠻⠹⡚⠃⠀⣒⠈⣹⠀⠀⠀⠀⠡⣀⢒⣲⢂⣼⡠⠎
⠀⠀⠀⠃⠗⠀⠠⠼⠆⠈⠁⢲⠅⠀⠀⠀⠀⣠⢼⠦⣈⣩⣛⡠⣧⣀⠀⠀⠀⢈⡶⠛⠓⢺⠁⠀⠀⡀⣠⠖⡫⠀⠀⠁⢱
⠀⠀⠀⠀⠀⠀⠀⡀⢀⡶⠄⡄⢃⠀⠘⢏⠟⠀⢸⠀⠀⠀⠀⠀⢸⠀⠑⡶⢶⢁⡇⢀⠀⡀⠑⡀⠹⣡⠇⠀⡇⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠘⠀⠳⣦⠬⠇⠀⡆⠀⠀⠀⠀⠸⡆⠀⠀⠀⢀⣸⠀⠀⠉⠉⢸⠀⠫⣋⡽⠀⠘⡄⠀⠀⠀⡧⢀⣀⢀⢠⠇
⠀⠀⠀⠀⠀⠀⠣⠤⢤⠤⠤⠖⠚⠁⠀⠀⠀⠀⢀⡏⠉⢉⡉⠁⣸⠀⠀⠀⢠⠇⠀⠀⠉⠀⠀⠀⠘⡄⠀⠀⡇⠀⣀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢩⠏⠉⠈⠏⠀⠀⠀⠀⠘⠒⠂⡖⠒⢖⠂⠀⠉⠀⠀⠀⢣⠔⠛⠲⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢰⡤⠀⢄⠀⠀⠀⣀⢸⠀⠀⠀⢸⣀⣀⠀⠀⠀⠀⣸⠁⠀⠈⢣⡀⠀⠀⠀⠀⣸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠪⠀⠜⠃⠀⠀⠀⠁⠀⠁⠀⠀⠸⠤⠝⠀⠀⠀⠘⠃⠘⠀⠀⠯⠭⠃⠀⠀⠀⠀⠓⠤⠓⢰⣂⠇⠀⠀⠀⠰⡂⡦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⣀⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣦⠀⣿⣿⢇⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣴⣶⣶⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⠏⠉⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠷⢦⡀⠀⠀⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠿⠿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⠔⢢⡀⠈⢻⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⡔⠁⡝⡆⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠀⠀⠀⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠁⠁⠀⢧⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⢀⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠤⠤⠀⠤⠤⠤⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠲⠤⣀⠀⠀⠀⠀⢀⡠⠤⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠞


⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⠶⢶⡶⠶⣶⣶⠶⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠟⠛⠛⠋⠛⠛⠛⠻⠟⠳⠶⠶⠾⣿⡿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠈⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣅⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠛⠉⠺⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠶⠛⠉⠶⠶⠾⠿⠛⠻⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠋⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠙⢷⣤⡀
⠀⠀⠀⠀⠀⢻⣷⣄⣠⣤⣴⡶⠾⠟⠉⠀⠀⠀⣠⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢦⣤⠀⠀⠀⠉⠙⠿⣶⣤⣤⣄⣀⣠⣾⠏
⠀⣀⣤⣶⡶⠞⠋⠈⢹⡇⠀⠀⠀⣀⣀⣤⡴⠞⠋⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠉⠹⠷⢦⣤⣀⠀⠀⠀⠈⣿⠉⠁⠛⢶⣦⣤⣤⡀
⢶⣏⣁⢀⣠⣠⡄⠂⠘⠿⠶⠾⠚⠋⠉⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠛⠚⠛⠀⠀⠀⠀⠀⠀⢙⣻
⠀⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠙⠃⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⣿⠟⢿⡆⠀⠀⠀⠀⢸⡟⠉⠛⠛⠛⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣿⣤⣼⡇⠀⠀⠀⠀⢸⣷⢶⣶⣶⣆⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⣿⣿⠀⠀⠀⠀⠀⣿⠋⢻⡇⠀⠀⠀⠀⢸⣿⣿⠛⢻⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠆⠀⠀⠀⠀⣿⠀⢸⡷⠀⠀⠀⠀⢸⣿⣿⡄⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣧⠐⣿⣿⡆⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⢸⣿⣿⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠠⣿⣿⡏⠀⠀⠀⠀⣿⠀⠀⣶⠀⠀⠀⠀⢸⣿⣿⠃⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⡇⠀⠀⠀⢈⣿⠀⠀⢻⠀⠀⠀⠀⣸⣿⣿⠀⠰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡇⠀⢻⣿⡇⠀⠀⠀⢸⡟⠀⠀⣿⠀⠀⠀⠀⣿⠿⣿⠀⠐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⢸⣿⡇⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⣸⣿⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠸⣿⣟⠀⠀⠀⣸⡏⠀⠀⢹⡃⠀⠀⠀⣿⠁⣿⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣿⣽⠀⠀⠀⢿⡇⠀⠀⢸⡇⠀⠀⠀⣿⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣿⠻⠷⠷⠶⢿⣇⠀⠀⠘⢿⠶⠶⠾⣿⠀⣿⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣿⠀⠀⠀⠀⢈⣿⠀⢀⠀⠀⠀⠀⠀⣹⡇⣿⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⠛⠚⠛⠛⠛⠛⠙⠃⠈⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⢸⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠙⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢂⢠⣰⣾⠋⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣾⣿⡿⠟⣼⡏⡀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠉⠉⠉⠉⠀⠀⠙⢳⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⢀⣤⠖⡂⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠲⠞⠿⠦⠀⠘⠁⠉⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡂⠀⠀⢀⡼⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠘⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠃⠞⠁⠤⡀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣰⠾⠐⢦⠖⠀⠀⠐⢉⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⢠⣿⡟⠁⠀⣤⠾⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣾⣿⣷⣿⣿⣧⠀⠋⠀⠀⠀⠀⢀⣾⣿⣿⢯⣿⣿⣿⣿⣿⡟⢽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⢸⣼⣿⣿⣿⣿⣿⣿⠀⠀⠴⠺⠀⢀⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠸⣿⣿⡿⣽⣿⣿⣿⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⢿⣿⡟⣷⢸⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⢻⡿⣽⣿⣿⣿⣿⡇⠀⠀⢰⣾⣿⣿⣿⣿⡿⠟⣋⣥⣴⣿⡛⡄⢸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⣿⣿⣿⣿⣷⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠃⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠉⠈⢿⣀⣾⣿⣿⣿⣿⡿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡼⠛⠛⠿⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⡶⠶⠲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⡇⠀⠀⠀⠀⣀⣀⣹⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢹⣿⠛⠛⠉⠉⣯⠉⡇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠸⣿⡀⠀⠀⠀⢸⠀⢳⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⡿⡇⠀⠀⠀⠸⡇⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢸⢛⠀⠀⠀⠀⡇⢸⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠸⡼⡀⠀⠀⠀⣿⣀⡇⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⡷⠷⠶⠛⠛⠉⠉⠇⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢶⣀⣀⣠⣤⣤⠖⢻⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢸⣹⡉⠀⠀⠘⡇⢸⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠘⣏⣇⠀⠀⠀⡇⠈⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣿⢸⣀⣤⣤⣿⣠⣦⣤⣷⣤⣤⣤⣤⣤⣤⣤⣤⡴⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⠋⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠶⠤⠤⠤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠤⢶⠖⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⣤⣤⣤⣤⣤⣤⣤⣄⣀⣈⡇⠀⢸⢉⣀⣠⣤⣤⡾⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⡄⣧⠀⠀⠀⢸⡇⠀⢹⢻⡇⠀⢸⡏⠉⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⡇⣿⠀⠀⠀⠀⡇⠀⢸⢸⡇⠀⢸⡇⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⣿⠀⠀⠀⠀⡇⠀⢸⢸⡇⠀⢸⡇⠀⠀⠀⠀⣇⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⣿⠀⠀⠀⠀⡇⠀⢸⢸⡇⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⡇⣿⠀⠀⠀⠀⡇⠀⢸⢸⡇⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠃⡟⠀⠀⠀⠀⡇⠀⢸⣸⠃⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢰⣀⡇⠀⠀⠀⠘⠷⠶⠞⣿⠀⠀⢸⡇⠀⠀⠀⠀⣿⣀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠛⠶⠶⠾⠁⠀⠀⠀⠀⠈⠉⠉

⣿⣟⢹⣟⠻⣿⢟⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠢⢍⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠢⣝⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⡤⢝⡻⣷⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⡀⠀⠀⠘⢝⡦⣈⡎⠓⢯⣒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⢲⣄⡀⠀⠈⠉⠁⠀⠀⠈⢹⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⡄⠉⢳⢤⡀⠀⠀⠀⠀⢸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡇⠀⢿⠀⡟⠦⣀⠀⠀⢸
⠀⠀⣠⣤⠀⣀⡠⣔⢥⠝⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⢸⠀⡇⠀⢸⠓⢦⣸⡀
⠀⠀⣧⠾⠛⠋⠉⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢻⢸⡇⠀⢸⠀⡇⠀⢸⠀⢸⠉⠉
⢈⡇⠀⠂⠀⠀⢀⡠⡔⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠴⠶⠖⠒⠒⠒⠋⠉⠉⠉⠉⠉⠑⠾⣀⡇⠀⢸⠀⡇⠀⢸⠀⢸
⠈⡇⢀⣠⠔⠊⠁⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡉⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣸⠀⡇⠀⢸⠀⢸
⠈⠉⠁⠀⠀⠀⠀⠀⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⢤⣙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠧⣄⣸⠀⡟
⠀⠀⢸⠀⠀⠀⠀⠀⡇⢀⡀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⡇⠀⢸⠓⢬⡓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣧
⠀⠀⢸⠀⡄⠀⢀⣠⠗⢊⠔⠒⣦⠒⢢⠀⠀⠀⠀⡠⢚⠇⡇⠀⢸⠆⠀⠙⠲⢍⡳⣄⣀⣀⣀⡤⠤⠤⠀⠒⠒⢒⣒⣋⣉⣉⣽
⠀⠀⢸⢀⡧⠖⠉⠀⠀⢈⠗⢛⣞⠛⣫⠀⣀⣴⠊⠆⠁⠀⡇⠀⢸⠀⠀⠀⠀⠀⠙⢾⡤⠤⢤⢴⠖⠒⣲⠋⠉⠉⡏⠀⠀⣿
⠀⢠⠾⢥⣤⣤⣤⣤⣤⣤⣭⣥⣤⢭⡕⠟⣹⣶⠆⡂⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⣿⡇⠀⢸⠀⠀⠀⢹⠀⠀⣽
⠀⠈⢹⠁⣩⡀⡀⢈⡁⣨⣀⡍⠀⢹⡴⠟⢻⠁⠀⡇⠀⠀⡇⠀⢸⡄⠀⠀⠀⠀⠀⠸⡇⠀⠀⡇⡇⠀⢸⠀⠀⠀⢸⠀⠀⣿
⠀⠀⢸⠈⢲⠒⠚⢛⠛⢳⠒⡇⠀⢸⠀⠀⢸⠀⠀⡇⠀⢀⡇⠀⠈⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠙⠒⠚⠀⠀⠀⢸⡄⠀⢸
⠀⠀⢸⠀⢸⠀⠀⠸⣆⡼⠀⡇⠀⢸⠀⠀⢸⠀⠀⡇⠀⠈⠧⠤⠼⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸
⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⡅⠀⢸⠀⠀⠘⠦⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢿⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸
⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⣇⣀⣼
⠀⠀⠘⠒⠃⠀⠀⠀⠀⠀⠀⠙⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠤⠞⠁⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠛⠛⠛⠛⠛⠁⠠⢤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠶⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⡀⠤⠤⠤⠤⠤⠤⠄⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣠⠶⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠉⠉⠶⣆⠀⠀⢾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢸⡇⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢘⡅⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⢀⠀⣀⡀⠃⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢨⡇⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢸⡇⢴⠋⣻⣿⣿⡟⢣⠀⠀⠀⠘⡟⣸⣿⣿⠋⣷⠀⢸⡇⢸⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⣧⣼⡇⠀⠻⢿⣿⣿⡿⠃⠀⠀⠀⠀⠻⣿⣿⣿⠿⠉⠀⢸⡇⢸⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢀⡏⠀⠀⠀⠀⠀⠈⠁⠀⢠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠛⢿⠃⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢦⣄⡀⠀⠀⠀⠀⣀⣀⠰⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡎⠉⠉⠀⠀⠀⠀⢈⡟⠓⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠃⠀⠸⣆⠀⠀⠀⠀⠀⡸⠃⠀⠀⠈⠛⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠁⠀⠀⠀⠀⠀⠈⠳⠶⠶⠶⠎⠁⠀⠀⠀⠀⠀⠀⠈⠉⠶⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠇⠀⠀⠀⣷⠀⠀⢀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⡾⠁⠀⠀⠸⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⢀⡷⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠒⣾⡂⠀⠀⠀⠀⠘⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠁⠀⠀⠀⠀⣴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠛⣦⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⣾⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⣧
⠀⠀⠀⠀⢸⠇⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠸⡇
⠀⠀⢀⣰⠇⠀⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠰⣆
⠀⠀⠸⠇⠀⠀⠀⠈⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣄⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠰⠶⠚⢋⡀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⠸⠇
⠀⣠⣇⠀⠀⠀⠀⣠⠶⠀⠽⠃⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠀⠘⢯⡀⠶⣀⠀⠀⠀⠀⣸⡆
⣄⢋⣾⣇⡀⣄⡀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢀⣠⢄⣸⣧⡜
⠉⢹⣧⣸⡇⣿⠇⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⣾⠸⣿⢸⣇⣹⡇
⠀⠀⠉⠙⠿⠁⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠈⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠄⠀⠠⣤⣤⣤⣤⣤⣤⣦⠀⠈⢳⣤⣤⣤⢤⣤⣤⣤⢤⣤⣤⣤⣴⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⠀⠀⠀⠹⡁⠀⠀⠀⠀⠀⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠁⠈⠉⠉⠛⠒⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠔⠛⠁⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠢⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠖⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠺⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠰⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠒⠲⠶⠶⠶⠶⠶⠶⠶⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣿⣾⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⠿⠿⢿⣿⡿⠿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡗⠀⠀⠂⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠿⠋⠁⠀⠙⢿⡿⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠁⠤⢤⠀⢚⣋⣁⠀⠀⣠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠹⠋⡇⠈⠙⠁⠀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⡀⠁⡤⠄⠀⢠⠎⢨⠟⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠋⠁⠙⠶⢤⡤⠖⢋⡴⠋⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠓⣦⣀⣀⡄⠀⠀⠉⠋⠉⠀⠀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢠⣞⣥⠤⠤⣽⣞⠉⠉⠙⠳⣄⢠⠧⢤⡀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠉⠁⠀⠀⠀⣀⡬⠿⠦⣤⣤⠼⠋⠀⠀⢙⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣀⣀⣀⣤⠶⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠁⠀⢀⡇⠉⠛⠲⠦⠤⣤⠤⠴⠺⡅⠀⠀⣇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⡟⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⢳⠀⠀⣿⠀⠉⠙⠓⠲⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡤⠖⠛⠉⠀⠀⠀⢠⡇⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠂⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣿⠙⠓⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣶⣤⣶⣿⣄⣀⣀⣀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⢿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢤⣴⣦⡤⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡸⠀⠈⠉⠉⠓⠚⠿⠿⢷⣖⣂⡤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡖⢫⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⡎⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡟⡶⠦⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣸⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢣⠇⠀⠀⠀⠈⠉⠓⠒⠦⢤⣀⣀⡞⢠⢇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡜⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠁⡞⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠒⠛⠉⠉⠒⠢⠤⣄⣀⡀⠀⠀⠀⢀⠇⢰⢡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠂⡞⠀⡏⡜
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢶⠞⡇
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢠⡇
⠀⠀⠀⠀⠀⢸⠉⠑⠒⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡔⢡⠇
⠀⠀⠀⠀⠀⢸⠒⠢⡤⣀⡀⠀⠉⠉⠓⠲⠤⢤⣀⡀⠀⠀⠀⠀⢀⠔⠁⢀⠴⡃⠀⠸⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⡆⠉⠉⠓⠲⠤⣤⣀⣀⠀⠈⠉⠓⠒⡶⠃⢀⠔⠁⢨⡇⢠⢀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⡇⡇⠀⠀⠀⠀⠀⡇⢸⠈⡏⠑⠒⡦⠤⣥⠜⡇⠀⠀⢸⡇⢸⢸⡃
⠀⠀⠀⠀⠀⢸⠀⠀⡇⡇⠀⠀⠀⠀⠀⡇⢸⠀⡇⠀⠀⡇⠀⢰⠀⡇⠀⠀⢸⡇⢸⠘⡇
⠀⠀⠀⠀⠀⢸⠀⠀⡇⡇⠀⠀⠀⠀⠀⠳⠼⠞⠁⠀⠀⡇⠀⢸⠀⡇⠀⠀⢸⡇⢸⢀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⡇⠀⠀⠘⠧⢼⠞⠁
⠀⠀⠀⠀⠀⢸⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠘⣇⠀⣇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣌⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⢀⣀⡀⢠⣿⣿⣷⡄
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⢀⣸⣿⣿⣿⣿⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡀
⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣇
⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡇⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⡇⠀⠀⠀⢻⣿⣿⠃⠀⠀⠀⠀⣠⣶⣶⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢴⣿⣇⣿⣿⣿⣿⣿⣿⣿⣷⠈⣿⣇⠀⣰⣿⣿⣿⣿⣷⣄⠀⠀⠀⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣇⣰⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠙⠛⠛⠋⣿⣿⣿⣿⣿⣿⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⢸⣿⣿⣿⡏⠸⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⣠⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⢸⣿⣿⣿⠀⠀⢻⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⠟⣻⣿⣿⣿⣿⠇⢹⣿⣿⣿⣿⣿⡏
⠀⠀⢸⣿⣿⠇⠀⠀⠈⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⢻⣿⣿⣿⡿⠀⢸⣿⣿⡿⣿⣿⡇
⠀⠀⣼⣿⣿⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠘⣿⣿⣿⣿⡟⠀⠀⠀⢸⣿⣿⣿⡇⠀⢸⣿⣿⣇⣿⣿⡏
⠀⠀⣿⣿⣿⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⢹⣿⡇⠀⠈⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⡆⠀⠀⠀⣼⣿⣿⡀⠀⠀⠀⢻⣿⣿⣿⠃⠀⠀⠀⣼⣿⣼⣿⠆⠀⠀⠸⣿⣟⣿⣿
⠀⠀⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠙⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡀
⠀⠀⢨⣿⣿⡁⠀⠀⠀⠀⢻⣿⣷⠀⠀⠀⠈⠛⠛⠋⠁⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠈⠛⠋⠈⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣤⣤⠀⡠⡀⠀⠀⠀⢀⣤⣶⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⢀
⠀⠀⠀⠀⢀⣤⣶⣶⣦⣀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠘⣧⡈⣷⠀⠀⠀⣾⣿⠟⠉⠉⠀⠙⠃⠀⠀⠀⠀⣴⣿⠿⠿⠷⢾⣧
⠀⠀⠀⢰⡿⠟⠛⠛⠛⠻⡆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠛⠿⠀⠀⠀⠻⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⢻⠇
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠓⠚⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⣄⡤⠀⠀⠀⠀⠀⠀⠀⣤⣾⣯⣥⡄⣀
⠀⠀⠀⠀⠀⢀⠂⠄⠢⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢈⠁⠀⠀⠈⠹⣿⡆
⠀⠀⠀⠀⠀⣠⡌⣨⣄⢄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠔⠁⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠠⢉⣤⡈⢡⣬⣇⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡆⢰⡀
⠀⠀⠀⠀⠀⠤⠛⢤⠝⠓⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⣠⠃⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠄⠘⠱⠆⠺⢂
⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢨⣤⣠⣄⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⣴⣆⠀⠀⠀⠀⠀⢀⡿⠄⠀⠀⡯⠆⠀⠀⠀⠀⠀⠰⢖⠀⠀⠻⠄⠀⠀⠀⠀⠀⢀⣄⠀⣥⡀
⠀⠀⠀⠀⠀⠀⠐⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠰⠒⠒⠒⠒⠲⠶⢶⡠⠄⣀⠀⠹⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣇⢰
⠀⠀⠀⠀⠀⠀⠀⡰⠃⣀⡤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠰⠢⠤⣄⢀⠀⠘⠆⡇
⠀⠀⠀⠀⠀⠀⣰⡵⠚⠁⠀⠀⠀⠀⠀⠀⠀⣀⡼⠦⢐⣖⣒⡀⠀⠈⠙⢛⡺⢾⡀
⠀⠀⠀⡔⠉⣩⠏⠉⠀⠀⠀⠀⠀⢀⠤⣒⠭⠒⠊⠉⠀⠁⠀⠀⠀⠀⠈⠀⠈⡗⢌⢦
⠀⠀⠀⠧⠊⡈⠀⠀⠀⠀⠀⢀⡔⡡⠊⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘⠈
⠀⠀⠀⠀⠀⠇⠀⠀⠀⢀⣴⠛⢘⠁⠀⠀⠀⠀⠀⠀⠀⣠⢔⡒⠠⢤⣈⠁⠀⢀⡟⣸
⠀⠀⠀⠀⠀⢰⠀⢀⠔⣃⣨⡿⡘⣆⠀⠀⠀⠀⠐⢁⢞⡵⠋⠀⠠⠓⠮⠉⠅⠭⢺⡉⡆
⠀⠀⠀⠀⠀⠀⢶⠋⡎⠀⠀⠈⠻⠮⡓⠤⠤⠤⢒⡥⠊⢀⠀⠀⠀⠲⠀⠀⠀⠀⡸⢃⢰
⠀⠀⠀⠀⠀⠀⠈⢮⠑⢄⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠨⡀⠀⠀⠀⠀⠀⠀⡠⠁⢸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠠⠈⠀⣀⣔⣁⣤⢊⡜⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣕⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠊⠑⢧⣠⣔⣋⠤⠤⠞⠙⢶
⠀⠀⠀⠀⠀⠀⠀⠠⣾⠁⠈⠒⠒⠂⠬⠭⠥⢀⣀⠉⠿⠉⠈⠡⠤⠄⠒⠒⠈⠁⠀⠀⢀⡠⠴⠴⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠂⠤⣄⠀⠀⠀⠀⠀⠀⠀⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡅⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⣀⡀⣃⣀⡀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠘⣿⠉⡉⢉⠉⠉⠉⠛⢑⠒⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣠⢼⡀⠐⠉⢉⠀⠀⠀⢈⠃⡀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠢⠇⠀⢀⢆⣼⠉⣀⣐⣃⣀⡀⠀⠉⠀⠀⠂⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⡎⠀⠁⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠂⠈⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⠀⠀⠀⠀⠀⠸⠃⡤⠒⠬⣿⠒⠒⠚⡖⠈⠁⠀⠀⠀⠀⠀⠀⠀⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠰⠁⡠⢤⣀⡀⢀⡴⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠘⠀⠀⢂⠈⠀⠀⠀⠈⠀⠉⠀⠀⠀⡀⠄⠀⢀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣐⣀⣀⣀⣣⠀⠀⠈⠢⣤⠀⠀⠀⡤⠄⢠⣤⡤⠤⢤⠤⠐⠁⠀⡠⠴⠤⠤⠀⢀⡀
⠀⠀⠀⠀⠈⠃⠠⠤⠀⠀⠀⠀⠭⠀⣀⠀⠀⠀⠀⠈⠁⠀⠀⠰⠊⠀⠀⠜⠂⠀⠀⠀⠐⠦⡄⠔⠂⠲⠘⠛⠠
⠀⠀⠀⠀⠀⠀⠐⠠⠀⠠⠤⠀⠀⠂⠐⠒⠀⠈⠉⠉⠉⡈⠉⠒⠓⠛⠒⠚⠛⠛⠉⢩⡉⠀⠀⠀⠀⠀⠘⠁⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⢿⡿⠟⠛⠛⠉⡙⠩⢙⠛⠿⡟⠛⠛⠛⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢠⣴⠖⠈⢷⠂⠄⡀⠀⠀⠑⠤⡈⠂⠄⡐⠶⣄⠀⢻⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⠏⠀⠌⠀⠗⢤⣀⠁⠐⠀⠀⠀⣉⣐⡠⠶⢻⠀⢸⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣠⣋⠀⣈⠄⠀⠀⠀⠀⠐⠤⠀⠠⠄⠠⠍⡘⢛⣚⠄⢸⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢀⣿⠁⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠘⣿⡀⠘⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⢠⣿⣇⠀⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣉⣀⣼⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣦⣀⠙
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿
⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠁⠸⠟⠍⠻⢿⣿⣿⣿⣿⣿
⣟⡂⠀⠀⠀⠀⠀⠀⠀⢰⣒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢀⠀⠀⡀⠀⠘⠀⢀⣿⣿⣿⣿⣿
⣯⣭⠀⣀⣀⣀⡀⠀⠀⢉⣉⣿⡿⠟⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⠁⠀⠀⠘⡆⠀⠀⢸⣿⣿⣿⣿⣿
⣿⡖⠀⠀⠀⠛⠠⠤⠤⠤⠿⠾⠓⠀⠀⠀⠀⠀⠲⠶⠖⠞⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⡏⠀⠀⢈⠀⠀⠀⣀⡇⠀⡄⢸⣿⣿⣿⣿⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⠹⠁⣀⠀⠜⠀⠀⠀⠈⠀⠀⠁⢸⣿⣿⣿⣿⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡎⠀⠀⠀⠀⣀⡀⠀⠀⠁⠀⠀⠀⠘⣿⣿⣿⣿⣿
⣿⣿⣤⣤⡄⠀⠀⠀⢠⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠤⠀⠀⠀⠠⠄⠘⠛⠄⠐⠀⠀⠀⠀⠀⠀⠀⢐⠯⣴⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⢠⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠬⠠⠤⠥⠬⠤⠤⠀⠀⠀⠀⠀⢒⣨⣶⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠸⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡶⠀⠀⠀⠀⠀⣷⣶⣶⣶⣶⣶⣶⣶⣶⡆⠂⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⢸⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⢸⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⡏⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣇⣀⣀⣀⣸⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⣀⣀⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣦⣠⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⣔⢀⢀⠀⣠⠄⠀⢆⠀⠀⣦⠀⡆⡆⣀⢀⢀⣀⣀⡀⠀⢀⣀⡀⠀⢀⣀⣀⠀⡆⠀⠀⠀⡀⠂⢀⣀⠀⠀⠀⠀⠀⢀
⠀⠀⡄⠀⠠⠆⢠⠀⠀⠀⠱⠊⠑⠁⠂⠀⠚⠀⠑⠋⠀⠀⠀⠃⠀⠈⠃⠀⠈⠒⠙⠀⠘⠀⠘⠋⠀⠑⠂⠀⠂⠐⠀⠈⠊⠀⠀⠀⠀⠁⣀
⠀⠀⠙⠒⠦⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠁⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠒⠀⠠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠄⠠⠂⠀⡀⠀⡀⠘⢤⠀⠂⠀⠤⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠠⠀⠁⠌⠁⠀⠀⠁⠉⠀⠁⠀⠄⠀⠀⠀⠉⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⢀⠨⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⡱⠂⣀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠂⠀⠀⣀⡰⠂⣀⠀⠀⠀⣀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⡀⠀⢀⠀⠀⡄⠀⠀⠀⠀⠀⠁⢠⡄⠁⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠁⢤⡌⠁⠀⠀⠀⠀⠀⠨


⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠿⠿⠿⠿⠿⠿⠿⣿⣷⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⠉⣹⠉⠙⠛⠓⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⡏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡎⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣄⡀⠀⠀⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢢⣄⣃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠖⠚⠧⡤⠧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⢸⠃⣼
⠀⠀⠀⢀⣤⠴⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣀⣸⠀⣿
⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣦⡏
⢰⡃⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⡉⢴⡿⠁
⢈⡟⠲⣤⡀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⣋⡥⢖⣩⠞⢯⡄
⢸⡇⢸⠇⠉⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢚⣭⠶⢋⣥⠖⠋⠉⢳⡄⢻⡀
⢸⡇⢸⠀⠀⠀⠀⠉⠓⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣡⠞⢋⣤⠞⠋⠀⠀⠀⠀⠀⣷⠸⡇
⢸⠇⢸⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀⠀⠀⠀⢠⠞⣠⢞⣥⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⣿
⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣤⡏⡰⣳⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢹
⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠸⡇
⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣤⣇
⣼⠒⠺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⢘⠄
⢿⣀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⣤⣼⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠢⠤⠉⠃⢀⠊⠉⠙⠓⠢⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⡀⢁⡀⠒⠌⠀⢾⠀⠀⠀⠤⠤⠀⡤⡖⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠐⣷⠁⠀⠉⠀⠸⠶⠴⠟⠀⠀⠄⢱⡉⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠣⠀⢩⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠈⠁⠃⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣥⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠃⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠃⠤⠂⡄⠀⠀⣄⠶⠀⠐⠒⣧⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠫⠿⠿⠀⠀⠀⠸⠿⠯⠃⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⢠⠄⠀⠀⡠⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⡆⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⡄⠀⠀⠀⢀⡝⠁
⠀⠀⠀⠀⠀⡀⠄⠂⠀⠀⠀⠢⡀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡠⠃⠀⠀⠀⠀⠄⡀
⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢴⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡯⡂⠤⠀⠀⠀⠠⠄⡪⣿⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠉⠀⠀⠀⠈⠁⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⡘⠁
⡄⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠁⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠐⠀⠄⠤⠤⠀⠀⠀⠀⠀⠀⣀⡀⢀⣈⣆⣎⣀⠀⣀⡀⠀⠀⠀⠀⠀⠠⠤⠄⠄⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢆⠀⠀⠀⢈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣨⢀⣼⠶⠀⠲⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢴⠋⢹⠀⣀⢀⡤⢾⢣⠉⢉
⠀⠀⠀⠀⡠⢄⠀⠐⠀⠀⠈⢄⡰⣠⠾⡏⠁⢰⠀⢸⢸⠀⠀⠀⠀⢀⠀⠀⡠⠶⠠⠄⠀⠀⡠⠠⡀
⠀⠀⠀⡈⠀⢈⢀⣀⠀⠐⠞⢻⢁⡇⠀⠁⠀⠀⠁⠸⠸⡀⠀⠀⠀⠈⠀⢬⣲⣶⠀⠀⠐⢳⢶⢈⠇
⠀⠀⠀⡅⢀⠾⠛⠀⠀⠀⠀⢸⠐⡄⠈⠀⠀⠀⠀⠘⡆⣇⢀⣀⠀⠀⠠⢴⣖⡓
⠀⠀⠀⡇⠰⠀⠀⠀⠀⠀⠀⠸⠄⡇⠀⠀⠀⠀⠀⠀⢁⣞⣁⣀⡀⠀⠀⠀⣊⡀⣵⠀⠀⠀⣢
⠀⠀⠀⢉⠀⡇⠰⠀⠀⠀⠀⠀⣇⢸⠀⠀⠀⠀⠀⠀⠈⣆⢻⣿⠤⢤⣀⣑⣀⠡⠜⢀⠀⡤⠟⡅
⠀⠀⠀⠸⡀⠅⠀⠀⠀⠀⠀⠀⢸⡈⡦⠺⠂⠀⠀⠠⠌⣙⠀⠉⠉⠈⠐⠛⠓⠭⢓⣰⣾⣥⣾⠃
⠀⠀⠀⠀⢇⠀⡀⠀⠀⠀⠀⠀⠀⢿⡲⢦⣤⠄⢀⠀⠀⠁⠠⣴⠀⠀⠀⠀⣥⣰⢿⡆⠈⠉
⠀⠀⠀⠀⠘⡀⢱⢀⣤⣄⠀⠀⠈⣄⣷⣼⠿⢖⣂⠴⠶⠦⢤⣋⡀⠀⢀⣴⣿⣽⣾
⠀⠀⠀⠀⠀⠹⡄⡏⣴⣀⣀⠀⠀⠀⠁⢬⡆⠀⠉⠉⠉⠩⠑⢦⢱⡴⠿⠿⣽⠏⠁⠁
⠀⠀⠀⠀⠀⠀⠈⠆⠀⡿⡿⠥⢍⣢⣀⡠⠃⠀⡀⠀⠀⠀⢸⣾
⠀⠀⠀⠀⠀⠀⠀⠰⠢⠑⠉⠉⠙⠒⠒⠬⢌⠳⠄⠀⣢⣞⡩⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡌⣶⣦⢤⣠⣀⣀⡀⣾⣃⣶⣟⣩⡟⢃⡟⠁
⠀⠀⠀⠀⠀⠀⠔⡾⣱⠋⠁⠉⠀⠉⠛⠛⢶⢷⠀⠀⠉⠉⠡
⠀⠀⠀⠀⠀⠀⠁⠐⠂⠤⠝⡂⠤⣀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠢⠤⠕⠃⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠛⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣄⠐⠒⠂⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⠉⠁⠀⠛⠿⣷⠦⢾⡿⠃⠉⠛⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠈⡀⠀⡄⠀⠀⢸⡇⣸⡇⠀⠀⡄⠀⡘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡇⠀⢡⠀⠀⢸⡧⢯⣷⠀⠀⡇⢸⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⢸⡁⠀⠘⣧⠀⢸⢇⣾⣿⡀⠀⠁⠈⣧⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢈⠁⠀⠀⢿⠀⣸⣏⣭⢿⡀⠘⠀⢰⡇⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⡼⢂⡶⢄⠈⠀⣿⣽⢻⢾⡇⠀⢠⣾⡇⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣿⠏⠀⠈⠣⠀⣿⢿⡎⢸⡇⠐⠀⢿⡆⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠈⠿⠀⠀⠀⠀⢀⣿⣿⠅⣀⡏⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡀⠀⡏⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⢶⠀⠀⠴⠾⢹⣿⣿⣿⣿⡏⠛⠚⡷⠀⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⠓⠺⣿⣿⣷⣶⣶⣼⣿⣿⣿⣿⣿⣿⠇⠈⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠐⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠦⣽⣿⣿⣿⣿⡟⠙⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠁⠀⠸⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠹⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⠀⠀⠀⠀⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣴⣾⠇⠀⠀⠀⠀⠀⠀⠀⠸⠻⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣮⣵⣾⣭⣵⣦⣶⣄⣠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠀⠀⠀⠀⠀⠀⠈⠻⠿⡿⠿⡿⢿⣿⠿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠑⠀⠀⠀⠀⠠⠄⣀⠀⠀⢸⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠙⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠲⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠔⠀⠐⢀⠀⠀⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠈⠀⠀⠀⢠⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠐⠀⣺⡦⠂⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⢄⡀⠀⢀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢰⣆⠈⠠⡀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠸⣿⣿⣦⣌⡁⠒⣠⡤⠄⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠙⢙⠛⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠁⠂⠠⠄⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠒⠂⢸⡇⠉⠛⠀⠀⠠⣤⣤⣤⣶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠈⠛⠂⠤⠤⢀⣀⣀⡀⠶⠄⠀⠛⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠤⢤⣤⡤⠤⠐⠂⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⡆
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠹⣄
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⣿⢀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢻⠆
⠀⠀⠀⠀⠀⠙⡿⠿⠿⠿⢷⣦⣤⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣤⣶⡶⠉⠁
⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠈⢻⠿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⡇
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢇
⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⢀⣱⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡄
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀⠀⠀⠀⠀⠀⠀⠨⡆⠀⠠⠁⣄⠀⠀⠀⠀⠀⠀⠠⡯⠀⠀⣷
⠀⠀⠀⠹⢤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢤⣀⡀⠀⠀⡠⠴⠄⠀⠀⠈⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠀⠀⠨⡄⣀⣀⡸⡄⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⢦⡤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠂⠀⠀⢀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠉⠉⠋⢻⢉⣁⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠴⡄⠀⠀⠘⢓⡶⢉⠛⠿⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠈⠐⠒⠋⠉⠀⠀⠃⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣓⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣯⠁⡤⠀⠀⠀⡿⣅⠺⣅⠐⣇⢰⡛⢔⣫⡤⠟⠛⠃⠀⢀⣉⣷⡖⠶⠤
⠀⠀⠀⠀⣀⢴⠷⠀⠘⠧⠭⣒⡆⠳⡈⢪⡳⣮⠓⢀⠁⣚⠩⠵⢶⣶⣖⣛⣛⣛⠫⠭⠴⠏
⠀⠀⠀⠐⠁⠈⣧⠴⢖⣀⣤⣤⣤⣤⣈⠲⣝⠊⠡⣲⡿⢶⣒⣩⠭⠭⣶⠶⠦⢤⣤⣤⡤⠶⣫
⠀⠀⠀⠀⠀⣼⣛⣩⣭⣤⣶⠶⠶⠶⣭⡛⢌⠠⣪⢷⣿⡭⠴⢶⣛⣛⣻⣿⣿⣿⣷⣒⣂⣭⣭⠆
⠀⠀⠀⠀⠀⡺⠿⠶⠶⠶⠶⠿⠿⠒⠂⠙⠮⠀⠕⡋⠤⠖⣛⣩⣭⣶⣶⣶⣟⣻⡭⠭⠷⠶⢾⠗
⠀⠀⠀⠀⠀⢍⣛⣛⣛⠛⠒⠒⢛⣛⣭⠭⢀⡤⢀⠒⠩⢟⣫⠵⣺⡿⠟⢟⣛⣉⣙⣛⡛⠓⠚⣋
⠀⠀⠀⠀⢀⣢⡤⠭⠭⠭⣭⣛⡋⣉⠶⡻⢼⡰⠟⣚⠩⠁⣚⠭⠄⣐⣪⡭⠤⠤⠤⡤⠭⠭⢭⡧
⠀⠀⠀⠀⢀⣍⣉⣉⣙⣛⣒⡺⠭⣙⠻⣿⡳⡙⣧⠐⣦⠹⡌⣷⢩⡅⣶⢆⣴⠟⣋⣉⣉⣙⣛⣫
⠀⠀⠀⠀⠈⣯⠭⢴⣶⡾⠭⠟⠷⢮⣵⣌⢷⡁⠘⣰⡿⣹⣷⣿⢿⠾⠋⠈⠑⠛⢛⣭⠭⣴⣾⡷
⠀⠀⠀⠀⢰⣶⣤⣤⡶⠖⣛⣭⡭⠤⣌⠙⠏⠇⣰⠿⠞⠁⣫⣵⠶⠾⠿⠿⣿⣷⣶⣤⣤⣴⠶⠟
⠀⠀⠀⠀⠀⣉⣩⡥⠞⣫⣽⠶⠋⢩⡉⢍⢁⡀⢃⠰⠶⠟⣋⣥⣤⣶⣶⣦⣄⣀⣉⣉⣭⠴⣚⣛
⠀⠀⠀⠀⠀⣭⠵⠞⠛⠋⣁⠀⠀⣀⢻⣈⡳⣝⠦⣙⠶⣟⣋⣥⡴⠶⠶⠶⠬⣭⣿⡭⠶⠟⠛⠋
⠀⠀⠀⠀⠀⢩⣭⣭⣝⡲⣬⡳⡄⢻⣿⣿⡻⣮⣝⠺⡭⢿⣭⣀⣴⣦⡤⢶⣺⣿⣭⣭⣭⣭⣟
⠀⠀⠀⠀⠀⠈⢉⣉⣉⣙⠪⡙⢹⡸⣿⣎⠳⣄⠙⣦⢼⡿⢕⣪⡭⠶⠛⠛⠉⣉⣉⣉⣉⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠱⠶⠮⢭⣺⡆⠣⡙⢮⡻⣮⠻⢐⠁⣚⡫⠵⢶⣶⣿⣛⣛⣛⣛⡍⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠢⡝⠎⢡⡶⣻⣷⠶⠊⢉⡭⠥⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠿⠛⠛⠛⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢐⡀⠈⣅⠈⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡶⠛⠁⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢶⣶⣤⣤⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣯⣽⣿⣿⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⣿⣿⣽⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣾⣷⣶⣷⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠏⠀⢻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣸⣿⣿⡏⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⢿⣿⡅
⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠸⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠁⠀⠀⠀⠸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠩⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠤⠀⠀⠀⠀⠻⠗⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣦⢀⡀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣇⣠⠀⠀⠀⠀⢀⣶⣶⣤⠄⠀⠀⠀⠀⠀⠀⢛⣿⡟⠋⠀⠀⠀⠀⠀⢰⣾⣿⢧⠔
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠟⠋⠀⠀⠀⠀⠀⡨⠟⠛⠁⠀⠀⠀⠀⠀⢀⣄⠉⡁⠀⠀⠀⠀⠀⠀⢀⣴⠚⠿⠁
⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⢀⠀⠀⣼⣷⠀⠒⠀⠀⠀⠀⠀⠀⣺⣿⡀⢐⢲⠀⠈⡆⠀⢠⣿⡏⠃⢰⡀⢰⢢⣄
⠀⠀⠀⢀⣀⣀⣸⣿⣧⣤⣤⣄⣸⣇⣀⣿⣿⣁⣁⡄⠀⣷⡄⣀⣀⣻⣿⣿⣬⣤⣤⣴⣿⣆⣼⣿⣿⣀⡿⠀⡈⣼⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⠀⠀⠀⡏⠀⢸⣿⣿⡿⢅⠀⠀⠀⠀⠀⠀⣿⣿⣿⢫⠀⢸⠀⠀⠀⢸⣿⣿⣾⠓
⠀⠀⠀⠀⠠⠤⠤⣼⣿⣿⣶⣾⡦⠤⡷⠤⣾⣿⣷⠷⣿⣇⢳⣦⡤⠤⢤⣿⣿⣿⣿⡇⢼⠤⠤⢤⣿⣿⣿⣤⣏⠀⠤⠤⠄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⢸⣿⡇⠀⡇⢰⣿⣿⡇⠀⠀⢈⣽⣿⡇⠀⢰⣿⣿⠈⣿⡄⢨⠀⠀⣼⣿⠿⠋⢻⣿⠃
⠀⠀⠀⠀⠒⠒⣾⣿⡟⠒⢸⣿⡏⠒⡗⠚⣿⣿⡗⠒⢲⣾⡿⣿⣿⠒⢺⣿⡿⠒⣿⣷⣿⠒⣾⣿⣿⡗⠐⠚⣿
⠀⠀⠀⠀⣀⢠⣿⣿⣿⢀⢠⣿⣏⠀⢁⣀⣿⣿⣷⣀⣿⣿⣀⣛⣁⠀⢸⣟⣇⠀⢸⣿⣿⣰⣿⣿⣿⠀⠀⢰⣟
⠀⠀⠀⠀⠈⣽⣿⣿⡏⠈⢿⣿⣿⠀⡏⠉⣿⣿⣟⡉⠉⠉⠉⠀⠁⠈⣹⣿⠋⠉⢸⣿⣿⢉⣿⣿⡏⠉⠉⠉⢹
⠀⠀⠀⠠⠴⠿⢿⠟⠧⠤⠼⠿⡿⠧⡧⠤⠯⠿⡿⠧⠤⠤⠤⠤⠤⠤⠿⡿⠠⠤⠼⠿⢿⠶⢿⡿⠇⠀⠠⠤⠼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠈⠉⠉⠉⠈⠉⠈⠐⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠁⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠁⠀⠉⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠑⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠈⠀⠑⢄⠀⠀⠀⠀⠀⠀⡀⠤⠉⠒⠁⠀⠀⠱⠔⢢⡀⠀⠀⠀⠀⠀⠂⢠⠀⡠⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠰⣀⠊⠁⠀⠀⠀⡀⡐⠉⠀⠀⠀⠀⠲⡀⣀⠄⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠤⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠄⠈⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠀⠀⠀⠀⠈⡖⠦⠤⠖⠊⠀⠀⢂⠀⠙⡆⠀⠀⠀⣀⣤⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠠⡌⠁⢉⠀⠀⠀⠀⠀⠈⠩⡀⡆⠀⢃⠀⠀⠀⠰⡴⣝⠀⣄⠀⠊⠡⢀
⠀⠀⠀⠀⠀⡄⠊⠀⠑⡲⠂⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢱⠐⠫⠀⢄⠀⢀⠀⢶⣶⢰⢻⡄⡔⠀⡰⠎⡀⠀⠀⠀⠈⠂⠀⠀⠖
⠀⠀⠀⠀⠀⠀⢁⡀⠄⣃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠐⣿⡇⠂⠈⠂⢚⠁⠁⠀⡖⢠⠊⠀⠀⡜⠀⠀⠀⠈⠉⠀⠑⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⢀⠄⣄⠃⠀⠀⠈⠀⣄⣈⠄⠀⠀⢠⡖⠁⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠌⢤⠙⠒⣄⡀⠀⠀⠀⠀⠀⡀⡴⠋⠀⠀⢀⠎
⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⢄⣬⣿⢶⣒⣸⠇⢴⠁⠀⠀⢠⠃
⠀⣀⠔⠊⠁⠀⠀⠀⠀⠀⠉⠐⠤⣀⠀⠀⠀⠀⠀⣀⢄⠮⠂⠁⠘⠓⠓⠛⡅⠀⠀⢣⠀⡐⠁
⡎⠀⠀⠀⢀⠄⠀⠒⠒⠀⠄⡀⠀⠀⠉⠐⠒⠒⢉⠔⠁⠀⢀⣠⡔⠀⠲⠚⠀⠙⠋⠀⠞
⠰⡀⢀⠔⠁⠀⣀⣀⣤⣤⣄⣈⠁⠒⠒⠂⠀⠀⠚⠀⠀⠐⡡⠂⡇⠀⠀⠴⠈⠉⠂⠀⢰
⠀⠈⠃⠀⠱⠈⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⠤⠤⠤⠷⡀⠀⢱⣼⠃⠤⠤⠤⠤⢤⠄⠐⢚⡄
⠀⠀⠀⠀⠀⢠⠰⠒⠒⠛⠛⠲⠶⢤⣄⣀⣀⡀⣀⣀⣬⢮⠁⠘⡂⠒⠀⠀⠀⠈⠀⠐⠙⣇
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⢠⡤⣬⠟⠒⢦⡀⠀⢀⡤⠀⠀⢥
⠀⠀⠀⠀⠀⢠⠤⠖⠒⠒⠒⠲⠶⠦⢤⣤⣤⣤⡤⠤⠖⠋⠀⠄⠀⠀⠀⡘⠂⢊⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠀⠀⠀⠀⠁⠀⠈⡆⠀⠀⠀⠃
⠀⠀⠀⠀⠴⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠃⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢣⠀⠀⠸⠀⠀⠀⠀⠀⠐⠀⠀⡦⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢬⠀⠀⡄⠀⠀⠀⠀⠀⢠⡄⠀⢩⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠯⠏⠜⠀⠁⠀⠀⠀⠀⠀⠀⠈⢡⠘⠞⠽⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⣤⣤⣮⠶⠀⠀⠀⠀⠀⠀⠀⠲⠶⣦⣤⣤⣤⠶

⠀⠀⣠⣶⣶⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣶⣦⡀
⢠⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣿⣿⣿⣿⣿⣿⠿⣟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⢿⠿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠋⠉⠙⢿⢿⣿⣿⣿⣿
⢻⣿⣿⣿⣿⣿⣿⣣⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⣽⣃⣿⣿⣿⣿⣿⣿⠃⠀⠀⣿⣿⣷⣀⢿⣿⣿⣿⣿
⠀⠻⣿⣿⣿⣿⣿⣿⣿⡿⠃⢀⣀⣤⠄⠀⣤⣆⠈⢿⣿⣿⣿⣿⣿⣿⣿⡿⠃⢠⣶⠀⠘⢿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠈⣩⣯⡿⠏⠉⠀⠀⢨⠟⠁⠀⠀⠈⢏⠀⠀⠉⠙⠿⢯⢫⡉⠀⠀⠀⠀⢻⡀⠀⠀⡉⠹⢧⣴⣶⣤⣄
⠀⠀⠀⣼⣷⣿⣯⡀⠰⢦⣤⠏⠀⠀⠀⠀⠀⠘⣦⣠⡴⠀⣸⣼⣿⣷⡄⠀⠀⠀⠀⢳⣤⡾⡁⢀⣿⣿⣿⣿⣿
⠀⠀⢰⠋⣽⣿⣯⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢈⣙⣿⣿⠉⢣⠀⠀⠀⠀⠀⠁⠀⢉⣩⣩⣿⣿⣿⣿
⠀⣠⠁⢀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢃⡀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿
⠸⠋⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠈⠻⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡍
⠀⠀⠀⠈⢻⠿⠉⠿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠏⠻⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠙⠟⡟⠁
⠀⠀⠀⢠⡷⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⢀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡀⠀⢀
⠀⠀⠀⣿⠃⠀⠘⠻⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠠⠿⠟⠃⠀⠚⣷
⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⣀⣤⣤⣤⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⢹⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⣿⡟⣽⢙⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣾⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⠿⠛⠋⠀⠀⣿⠆⠀⠀⠀⠀⢺⡇⠀⠈⠙⠛⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡀⠀⢄⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠙⢦⣀⣤⠄⠀⣻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⢁⣀⣀⣽⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣅⣀⣁⢹⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣾⠻⣿⣿⣿⣿⠟⠁⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠙⣿⣿⣿⣿⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠿⠁⠈⠻⢿⣿⣇⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠀⢠⣿⣿⠿⠋⠈⢻
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣀⣌⣿⣿⣧⣤⣀⣀⣀⣀⣬⣝⣭⣉⣄⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡇
⠀⠀⠀⡄⠀⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠐⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⢀
⠀⠀⠀⠣⡤⢤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠤⠤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠤⢤⠔
⠀⠀⠀⠀⢱⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⠤⠎⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢢⠂
⠀⠀⠀⠀⠀⠢⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠤⠃⠣⢤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠊
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⡠⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⢄⠀⠀⠀⢀⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠧
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⡤⠤⠴⠶⠶⡶⠲⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠉⢉⣁⣀⣀⣠⡤⠤⣤⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢩⠁⠉⢠⠁⠇⠀⢸⠀⢀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢸⠀⢰⢸⠀⡀⠀⢸⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⠀⢸⢸⠀⡇⠀⢸⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⠀⠸⢸⠀⡇⠀⢨⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⢸⠀⠀⢸⠀⡇⠀⢸⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⠀⡄⢸⠀⡇⠀⢸⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⢸⠀⡇⢸⡄⡇⠀⢸⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠸⠀⣇⠘⡇⡇⠀⢸⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⣿⠀⡇⠁⠀⠸⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⢸⠀⢇⡆⠀⢀⡇⢸⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡆⢸⡄⢸⡇⠀⠈⡇⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣘⣀⣇⢸⡇⢸⡇⠀⠀⡇⠸⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠶⠒⠛⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⢉⡛⢾⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠛⠛⠛⠩⠥⢀⣀⣀⡀⠀⠀⠀⢀⣀⡠⠤⠄⠒⠒⠋⠉⠀⠀⣸⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⠤⣄⣀⣀⡀⠀⠀⠈⠉⠉⡟⠉⢀⣀⣠⣤⠤⠶⠒⠒⠙⡏⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢉⣈⡉⠉⠓⡖⠢⠤⠷⡾⡏⠀⢀⡿⠛⠓⠒⠒⠶⣇⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢀⡞⠉⠉⠛⢲⡇⠀⠀⢀⡏⢳⡄⢸⠀⠀⠀⠀⠀⠀⢸⡆⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⡀⠀⠀⠀⠀⠙⡆⠀⡟⠀⢸⡇⢸⡄⠀⠀⠀⠀⠀⠀⡇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠃⠀⠀⠀⠀⠀⡇⠀⣷⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡀⠀⠀⠀⠀⠀⣧⠀⢻⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡇⠀⠀⠀⠀⠀⣿⠀⢸⠀⠘⣇⣸⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⠀⠀⠀⠀⠀⢹⠀⢸⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢿⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⢸⡀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠚
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⣿⣿⠿⠻⡿⠛⠿⢿⡿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣼⣿⣶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠉⠛⠛⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠶⠀⠠⡤⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠃⡀⠈⠒⠂⠠⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡙⠁⠀⠀⠈⠁⠒⠈⡁⠈⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠲⠉⡉⠉⠉⠛⠦⣄⠘⣁⣤⣠⣼⠃⣤⠤⠄⣉⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠘⢷⢉⣟⢹⡟⢸⡁⠀⠀⠈⠉⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠠⣮⠐⣇⠘⠁⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⣿⢦⡀⠀⠀⠀⢐⣾⢃⣿⡇⠀⠀⠀⠀⠀⢀⡿⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢰⠇⠈⣷⢠⠀⠀⣧⢾⡅⣾⡇⠄⠀⠀⠀⣤⣸⠁⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⡄⠀⢸⣇⡹⠆⣿⠁⠀⠀⠀⠀⠀⡇⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⠀⠀⠀⠃⠀⢸⣿⡶⣏⡿⠋⠃⠀⠀⡀⣤⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠨⢴⣾⡇⠀⠀⠀⢰⠀⠀⣿⡛⢷⡃⠀⠦⠀⠀⢠⡏⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢹⡇⠠⡦⠀⢸⠀⠀⣿⣿⣿⡇⠀⣠⠀⠀⢸⣟⠦⠄⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣼⠇⠀⠀⠀⠺⠇⠀⣿⣿⣾⣧⠀⣀⠀⠈⣿⠁⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠠⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠀⢩⡆⠀⣿⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⡀⢰⡄⠀⠀⠠⠄⠀⢠⣷⢸⣻⣇⠈⠙⠁⢸⢹⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⣈⣤⣤⣬⡇⠀⠀⠀⠀⠀⢸⡟⣓⠿⠻⡴⣶⠆⠀⢸⡇⠀⠀⠀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠘⡀⠀⠀⠀⠀⢈⡇⢅⣲⠈⡟⢒⣶⡀⣺⠧⠴⠶⢤⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣤⣌⣀⣀⣈⣩⣷⣾⣿⣿⣿⣿⣿⣷⠏⢰⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠤⠏⣀⠴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⢹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠿⣻⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢰⣿⣇⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣧⣄⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⣦⣤⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⢿⣿⣿⣦⢔⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠛⠛⠛⠛⠛
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠛⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠿⠿⠿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣁⡀⠀⠶⠀⠘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣋⢿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠘⠃⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣟⣛⣻⠇⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠋⠁⠀⠀⠀⠀⠀⠀⠺⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣶⣾⣷⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⠛⣺⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣧⣿⣿⣿⡿⣏⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣴⠶⠶⠶⠾⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣿⠀⠀⠀⠀⣀⣀⡀⣠⣤⣤⡀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⠀⠀⠀⡾⠉⠈⠙⠏⠀⠘⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⢷⡀⠀⠀⠀⢠⡾⠃⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠉⠛⠳⣶⠏⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡶⠾⠷⠶⠶⣦⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⣀⣠⣤⣴⠶⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⢷⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⡿⠿⣿⣭⣅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠾⣛⣩⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡿⠲⠶⣬⣍⣙⡛⠻⠶⣦⣤⣄⣀⠀⢀⣀⣤⣴⠶⠛⣋⣭⡴⠞⠛⣉⡁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⡇⠀⠀⢀⣈⠉⠙⠛⠲⠶⣤⣍⣉⠛⢛⣩⣥⠶⠞⠋⣉⣤⣴⢾⣿⢻⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⣿⠟⠛⠷⢶⣤⣤⣀⡈⠉⢻⠋⣉⣤⣴⠾⠛⠋⠁⠀⠀⣿⢸⡇⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠈⢹⣷⠀⢸⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⢸⣷⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣿⠁⠀⢠⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢱⡟⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣿⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⡇⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣀⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢸⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⣿⣿⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⢠⡄⣘⢣⡞⢢⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦⠀⠀⡔⢣⡜⣃⣠
⠰⣄⠻⡄⠿⠀⡆⣠⢤⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠻⠟⢿⣿⣿⣿⣿⠋⠁⡤⣄⢸⠀⠿⢠⠟⣠
⣖⠺⠆⠀⠀⠀⠈⢁⡜⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠀⠀⢳⡈⠁⠀⠀⠀⠰⠗
⠈⠑⠢⣄⡀⠀⠀⡞⠁⠀⠀⢸⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣏⠀⠀⠈⢳⠀⠀⢀⡠⠔⠊
⠀⠀⠀⠀⠈⢣⠀⠘⢆⢠⠊⡨⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢅⠑⡄⡰⠋⠀⡜⠃
⠀⠀⠀⠀⠀⠀⢣⠀⠈⢿⠀⠉⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠉⠀⡼⠁⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠳⡄⠈⠢⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⠔⠁⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠈⢣⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡜⠁⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⡰⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢯⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⢣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⡟⠲⢦⣤⣤⣤⣀⢠⣤⣄⣤⣤⠶⢾⠁⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⡀⠀⠀⠀⢀⠀⠀⠀⠈⡏⠀⠀⠀⠀⢹⠁⠀⠀⠈⡀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢀⣼⠀⠀⠀⠀⠳⣄⡀⢀⣠⠞⠀⠀⠀⠀⡃⢀⡠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⠌⡸⠁⠉⠉⠉⠉⠉⠉⠉⠁⠈⢃⠣⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⡞⠁⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠈⢳⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⡆⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠻⠤⢀⡀⠠⠖⢻⡟⠢⠄⢀⠀⠰⠟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠤⡄⡀⠀⠀⠀⣸⣇⠀⠀⠀⠀⣀⠤⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⣦⡉⣶⣶⣿⡇⢸⣿⣷⣶⢎⣤⣶⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡿⢻⣞⠻⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠛⠛⠿⣿⠿⠛⢉⠴⠃⠈⠳⣈⠛⠻⢿⠿⠟⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣤⣀⣠⣤⠒⠉⠀⢀⡀⠀⠈⠑⢤⣄⣀⣠⡔⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⡀⠄⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡄⠀⠀⠀⠈⡂⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡣⠀⠀⠀⢜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠘⢁⠈⠀⠠⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠀⠈⢀⠉⠀⠱⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠈⠁⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠉⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⢠⠊⠀⠀⡜⠀⠀⠀⠀⠀⠀⡆⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢠⠇⠀⠀⠀⠀⠀⠐⡆⠀⠈⢢⡀
⠀⠀⠀⠀⡰⠁⠀⢀⣄⡇⠀⠀⠀⠀⠀⠀⣍⣀⠀⠀⠈⡄⠀⠀⠀⠀⠎⠀⠀⣠⡫⠀⠀⠀⠀⠀⠀⠀⠝⣄⠀⠀⠹⡀
⠀⠀⠀⡜⠀⢀⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⡄⠘⢄⠀⢠⠊⠀⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢢⠀⠙⢄
⠀⠀⠸⣀⠰⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⢀⠀⠀⠀⠰⠆⣸⠀⢸⡀⠲⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⢀⠀⠀⠀⠼⣀⡼
⠀⠀⠀⠈⠢⠄⠀⠀⢀⣀⣀⣀⣨⣀⣅⢀⠀⠀⠀⠄⠠⠤⠃⠀⠀⠑⠤⠄⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢁⠀⠄⠠⠔⠁