Alphabetic Emojis & Symbols

⠀⠀⠀⠀⠀⠀⢰⣿⠿⠛⠛⠛⢛⣿⡿
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⣠⣿⡟
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠠⡀⠀⠠⡧⢀⡤⢄⠠⣄⢤
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⣼⠁⠀⠀⠀⠀⢸⠄⢌⠛⠆⠀⠀⡇⠸⡄⠸⠀⡏
⠀⠀⠀⠀⠀⠀⣠⣿⣟⣀⣀⣀⣠⣼⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢄⠸⡏⣿⣿⣿⣷⡄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡕⠻⠋⠉⠛⣋⣴⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣠⠀⠄⠈⢢⡈⢻⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣋⠁⠀⠁⠂⢀⡆⢀⠎⡀⠘⠉⠙⠑⡖⠲⠆⠤⢤⠤⢤⣤⡤⢤⠤⢤⠴⣏⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠁⠐⣀⡡⠄⠈⢾⠀⠡⠀⠀⠀⠰⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠒⢀⣀⣸⠖⠁⠀⠈⢇⠃⠀⠀⢠⠃⠀⠀⠺⠀⡄⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠀⠀⠀⠀⢀⣸⡀⢀⠀⡆⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠄⠀⡞⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢒⣫⠵⠚⡇⢈⣤⣇⣀⣀⣂⣀⣁⣀⣬⣠⡤⠞⢻⣄⠙⢳⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡜⠁⠀⡜⠱⠃⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠰⢐⡇⠈⠦⣈⡼⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣰⠀⠀⡼⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠇⣀⠞⠀⠀⠘⣅⣀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠀⢠⠇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢡⡶⠋⠀⠀⠀⠰⢂⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⢠⠟⡚⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡶⠋⠀⠀⠀⠀⣴⣦⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⣰⠇⠀⣀⠀⢸⡆⡀⠀⡀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣀⢼⠀⠛⢸⡇⢸⠀⣟⠈⢈⠅⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠈⠁⠀⠈⠁⠁⠀⠉⠀⠈⠀⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⢸⣿⣷⣿⣿⣿⣿⠀⠀⠀⠹⢿⣿⣾⣿⠗⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⡿⠁⠀⠠⣾⣿⠿⠛⠿⠀⠻⠟⠀⠀⠘⠛⠃⠀⣼⣿⣿⣿⣿⡟⠀⣠⡤⣦
⣤⡀⠀⠀⢻⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠛⣿⣿⡿⠁⠀⠀⠀⠘⣿⡷⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠿⠿⠿⠿⠛⠀⢾⣿⣤⣼
⠉⣿⡆⠀⠀⠈⠉⠁⠀⠈⠉⠀⠀⠀⠀⠀⣶⡆⠀⢠⣄⡀⠙⠛⠁⠀⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⡟⠉⠉⣿⣄⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿
⣾⠟⠀⠀⠀⠀⠀⢀⣠⡶⠻⣦⡀⠀⠀⠀⢻⣷⣠⣿⠏⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣼⣿⡆⠀⢀⣀⠀⠀⣿⣿⣿⣷⣶⣶⣿⣿⠀⠀⠀⠀⣀⡀⠀⠀⠹⣿
⠁⠀⣠⣄⡀⠀⠀⠹⣿⣷⣶⣿⣷⠀⠀⠀⢸⣿⡿⠁⠀⠀⠀⣴⣿⣿⣿⣦⡀⠀⣼⣿⣿⣷⣠⣾⣿⡇⠀⠹⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠘⣿⣷⡄⠀⠀⠙
⠀⣸⣿⣿⣷⠀⠀⠀⠘⢿⣿⡟⠁⠀⠀⠀⢸⣿⠇⠀⢀⣠⡄⠘⣿⡀⠈⠋⠁⣰⣿⡿⣿⣿⡿⣿⣿⠀⠀⠀⣿⡏⠉⠁⠀⠀⠀⢀⣀⠀⠀⠈⢿⣿⣆
⢰⣿⣿⣾⣿⣇⠀⠀⠀⠈⢻⣿⡦⠀⠀⠀⠀⠉⠀⠀⠘⢿⣿⣶⣿⡗⠀⠀⠀⠛⠟⠰⢿⠏⢸⣿⡟⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⠿⠃⠀⢀⣈⣿⣿
⣾⣿⠛⢻⣿⣿⡄⠀⠀⠀⠀⠉⠀⠀⢲⣿⣆⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠇⠀⠀⠀⠀⠀⠀⠀⠿⠋⢻⣿⣆⠀⠀⠘⠿⠟⠃
⠛⠋⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⢀⣴⣄⡀⠀⣠⣤⣄⠀⢰⣶⡆⠀⠀⠀⠀⢰⣤⡀⠀⠀⠀⠀⢹⣿⣆⠀⠀⠀⠀⠀⡀
⣄⠀⠀⠀⠀⠀⠀⠀⢀⣼⣶⠄⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⣰⣿⣿⠋⠀⠀⢸⣿⣿⣧⡈⣿⣧⠀⠀⠀⢀⣿⣿⠀⢀⣴⣄⠀⠀⠻⠟⠂⠀⠀⢠⣾⣿⠄
⢿⣷⣄⠀⠀⠀⠀⣴⣿⣿⡁⣠⣶⣄⠀⠀⠈⠿⠗⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⢸⣿⣯⠻⣷⣿⣿⠀⠀⠀⣼⣿⣃⣴⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠠
⣼⣿⠏⠀⠀⣴⣿⡿⠛⢻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⠀⠀⠀⠀⠀⣿⣿⠀⠘⣿⣿⡄⠀⢰⣿⣿⣿⠟⠁⠀⠀⢠⣴⡾⠛⢷⣄⠀⢻⣿⣷⣤
⠿⠋⠀⠀⠀⠈⠋⢀⣴⣿⠟⠁⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠛⠋⠁⠀⠈⠉⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠘⣿⣿⣶⣾⣿⠀⠀⠙⠻⠿
⠀⠀⢀⣀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⢠⣿⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠸⣿⣿⠟⠁
⣄⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠷⠂⠀⠀⢀⣶⠞⠛⢻⣶⡄⠀⠀⢸⣿⡿⠉⠀⠀⣄⡀⠀⠀⢿⣿⠿⣿⣶⣄⡀⠀⠀⢹⣿⣦⠀⠀⠀⠀⠠⣴
⣿⣷⣜⣿⡇⠀⠀⢴⣿⣄⠀⠀⠀⣾⣿⣧⣄⠀⠀⢠⣿⣿⣦⣤⣾⣿⡟⠀⠀⠘⣿⣿⣄⠀⢠⣿⡿⠀⠀⠘⣿⣷⣿⡿⣿⣿⡷⠀⠀⠉⠀⠀⢰⣶⡆⠀⢿
⣧⠙⢿⣿⣿⠀⠀⠈⠻⣿⣦⠀⠀⠀⠉⠛⠋⠀⠀⢹⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠈⠻⢿⣿⣿⠿⠁⠀⠀⠀⠘⣿⣿⡀⠈⠋⠀⠀⠀⠰⣿⣦⡀⣿⡇⠀⢸
⣿⡄⠀⠻⠿⠃⠀⠀⠀⢹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠹⠟⠁⠀⢠⣶⣶⢦⣄⠈⠻⢿⣿⣧
⠁⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⠏⠀⠀⠀⣴⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⣴⣦⠀⢀⡀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⣸⣿⣧⣤⣿⠀⠀⠀⠹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⢿⣿⣿⡖⠀⠀⢠⣿⣟⣴⣿⠟⠀⠀⢰⣿⣿⠋⢀⡀⠹⠿⠃⠀⠀⠀⣿⣿⠿⠿⣧⠀⠀⠀⠀⠈⠀⢀
⠃⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣶⡄⠉⠀⠀⠀⣼⣿⣿⣿⡁⠀⠀⠀⠘⢿⣿⣤⣼⣿⡦⠀⠀⠀⠀⣼⣿⣿⣤⣴⡟⠀⠀⠀⢀⣤⣶⣿
⡀⠀⠀⣾⣿⡄⠀⠈⠻⣿⣆⠀⠀⠈⠙⠿⣿⣿⣭⡙⠁⠀⠀⠀⢀⣿⡿⠘⣿⣧⠀⠀⠀⠀⠀⠙⠻⠛⠁⠀⢠⣶⣤⣀⠀⠈⠉⠉⠁⠀⢀⠀⠈⢻⡿⠛⣿
⣿⣄⠀⠈⢿⣿⣦⣀⣀⣿⣿⠀⠀⠀⠀⠀⠈⠙⠻⠋⠀⠀⢀⡀⠀⠉⠁⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡋⢹⡇⠀⠀⢀⡀⠸⣿⣷⡀⠀⠀⠀⠈
⠻⠛⠀⠀⠀⠙⠻⠿⠿⠟⠁⠀⣠⣿⣦⣄⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⡖⠀⢀⣾⡿⣿⡿⠿⠟⠁⠀⠐⢿⣷⣄⣼⣿⣷⣄
⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡋⠉⠿⠃⠀⠀⠀⣰⣿⣶⣿⣿⣷⣶⣤⣤⠀⠈⠉⠉⣼⣿⠃⠀⠙⠟⠀⣿⡇⠀⠀⠀⠀⠀⠈⠻⣿⣿⠁⠹⣿⣧
⡿⠁⠀⣀⠀⠀⠀⠀⢀⣴⣿⠟⠙⠟⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⣷⡉⠙⠃⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠈⠛⠃⣀⣀⠀⠀⠀⠀⠀⠙⣿⣷⠄⠈⠁⠀⢀
⣁⣴⣿⡿⠃⠀⠀⠀⠀⠉⠋⠀⠀⠀⢀⣴⣶⣶⣦⠀⠀⠀⠀⠀⠀⠻⡿⠀⠀⠀⢠⣾⣿⣷⣶⣶⡂⠀⠀⠀⠀⠀⢿⣿⡆⠀⠀⠀⢀⡀⠈⠁⠀⠀⠀⢀⣾
⡿⠟⠉⠀⣠⡶⣶⣷⣦⡀⠀⠀⠀⣰⣿⡿⠋⠛⠋⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⢀⡀⠀⠀⠸⣿⣧⠀⠀⠀⣾⣿⠛⢶⡄⠀⠀⣾⣿
⠀⠀⠀⢸⣏⠀⢈⣿⣿⣧⠀⠀⠀⣿⣿⡁⠀⢀⡀⠀⠀⠀⢸⣿⡇⣠⣶⡆⢀⣤⡄⠀⠀⠀⠀⣠⣴⣾⣿⣷⡀⠀⠀⢻⣿⡄⠀⢰⣿⣿⣀⣠⣿⠀⠀⠈⠛

⠀⠀⠐⢄⠀⠀⠀⠀⡠⠐⠀⠀⠉⡀⠀⢤⠐⠤⡀
⠀⠀⠒⠚⠠⡀⡠⠀⠀⠀⠀⣠⣿⣿⣿⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⠐⠒⠒⠂⢤⢀
⠀⠀⠀⠀⡠⠃⢓⠀⠀⠀⠀⠛⠻⠛⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣠⠀⠒⠀⡀⠂⠀⣄⡀⠀⠀⠀⠀⢀⠔⠨⡤⠈⡄⡄⢠⡆⠇⡀⣈⠢⡀
⠀⠀⠀⡎⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣤⠘⠀⠀⠀⡃⠀⡔⠁⠈⢆⠀⢀⡇⠀⡔⠁⠈⢂⠀⠀⢀⠲⢄⠀⠃⠀⠰⠘⠚⡙⠀⠳⢁⡤⠓⡀
⠀⠀⠀⢳⡀⠀⠑⢄⠀⠀⠀⠀⡠⣿⣿⣿⠏⠀⠀⠀⠀⠃⠌⠀⣶⢴⠈⢆⠀⠀⡔⠀⢸⠀⠀⠂⠀⠂⠀⣤⠛⠢⣀⢀⠧⠰⢇⣠⠔⡅⣠⢀⢠
⠀⠀⠀⠀⠉⠓⠶⠤⠤⠾⢿⠟⠁⠀⠀⠀⠀⠀⠀⡠⠊⠰⠢⠤⢀⡈⠀⠈⡦⠼⠀⠈⣈⡀⠤⠼⠀⣀⠀⠠⠟⠀⢨⠁⠀⠀⠀⢁⣀⣘⣋⣛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠢⠤⣀⣀⠀⠠⠊⠀⠀⡆⠀⡖⣀⠈⠉⠚⠀⠘⠉⢁⣀⠀⠀⢸⠀⠀⠀⡟⡫⠉⠙⠀⠀⠀⠀⠀⠀⠐⣤⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⠀⠉⠁⠀⠀⠀⡥⠀⣉⣊⣀⡀⡅⠀⡄⠀⠘⠑⠀⠀⡄⠀⢂⠀⣋⣱⠤⠔⠇⠀⠀⠠⠲⠤⠬⢈⣀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠄⠁⣈⡀⡐⠀⠀⠀⠀⠀⠀⢰⡒⠀⠀⠁⢠⠉⠉⠉⠉⠉⠉⠁⠀⠘⠁⢰⠉⣳⠀⢠⠀⢀⠁⠀⢦⠂⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡄⠀⠁⠀⠀⠀⢀⠆⠤⣘⣀⠀⠀⠀⢸⠀⠀⠰⢀⣀⠴⠀⠀⠀⡦⠤⠍⠭⠄⢸⠀⠖⠀⠤⠬⠄⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠥⠔⠊⠢⡀⠠⡆⠀⡤⠄⠈⠉⢱⠀⠈⠖⢲⡍⣡⠀⢀⡵⢠⠘⢄⠀⣯⠵⠀⠈⠀⠀⠀⡰⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠉⠀⡴⠒⠀⢠⠁⣀⡘⢄⠫⠁⠀⢀⠂⠀⠀⢁⠘⠙⠃⡣⣊⠀⠸⠀⡠⣕⢧⡀⢀⠂⠀⡀⢀⡁⠕⣚⣄⣀⡶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠑⠂⠀⡎⠀⣇⠝⠈⢢⠀⡠⠁⠀⠀⠀⠀⠣⣀⠞⠀⠏⢲⠀⡀⠤⠜⠀⡸⠃⠀⠀⠘⣁⠴⢊⠜⡙⡉⢋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠴⠅⠀⠀⠀⠠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠠⠇⠠⠐⠈⠀⠀⠀⠀⠀⠈⠐⠡⠴⠥⠒

⠀⠀⠀⠀⣠⠆⢀⠀⠀⠀⠀⣶⠛⠉⢑⡀⠀⠀⠀⣠⠖⠢⠆⠀⠀⠰⡖⠂⠀⠀⠀⠀⠀⢠⠖⠒⠒
⠀⠀⠀⠠⠇⠀⠐⡄⠀⠀⠀⢓⣀⣄⠞⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⢘⣠⡄
⠀⠀⠀⡴⠉⠉⠉⣸⠀⠀⠀⣛⡀⠀⣠⠇⠀⠀⠨⣄⣀⣠⠀⠀⠀⠀⠇⢀⣠⠟⠀⠀⠀⢘⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠉⠀⡀⠀⠀⠀⢀⡈⠉⠉
⠀⠀⠀⢠⠖⠋⠁⠀⠀⢠⡞⠉⠐⠃⠀⠀⢰⠇⠀⠀⢁⠀⠀⠀⢸⠁⠀⠀⠀⣿⠀⠀⠀⢰⡇⠀⡸⠂
⠀⠀⠀⢈⡴⠄⠀⠀⠀⠞⠀⠠⢠⠀⠀⠀⢨⠖⠒⠒⣾⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠈⡵⠚⠁
⠀⠀⠀⠨⠆⠀⠀⠀⠀⠀⠤⠤⠞⠀⠀⠀⠀⠷⠀⠀⠟⠀⠀⠀⠼⠀⠀⠐⠦⠏⠀⠀⠀⠀⠧⠀⠸⠇
⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⠤⠤⡀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡼⠀⢠⠞⢰⠀⠀⠀⢸⠋⠂⠀⡆⠀⠀⠸⡇⠀⠀⢰⠀⠀⠀⢼⠀⢀⡆
⠀⠀⠀⠀⠇⠀⢀⡀⠀⠀⢣⠀⠉⠀⢰⠃⠀⠀⠸⠂⠀⢡⡇⠀⠀⠀⠃⠀⢀⡟⠀⠀⠀⠀⠞⠉
⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠃
⠀⠀⠀⢀⡖⠒⠢⡀⠀⠀⠀⡴⠃⠒⡄⠀⠀⠀⡞⠉⠁⠃⠀⠀⠲⠒⠲⡋⠒⠀⠀⠀⣲⠀⠀⢰
⠀⠀⠀⠾⠀⠀⠀⢱⠀⠀⠀⠇⠀⣀⠗⠀⠀⠀⠘⠤⠄⢀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢃⠀⠀⢘⡄
⠀⠀⠀⠈⢀⣀⡥⣫⠀⠀⠀⡟⠁⠁⣤⠀⠀⠀⢀⣀⣀⡼⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠘⣀⣀⡾⠁
⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢠⠀⠀⠀⢀⣀⡀⢀
⠀⠀⠀⢸⠁⠀⠀⡟⠀⠀⠀⡿⠀⠀⠀⣾⠀⠀⠀⠱⢠⠖⠋⠀⠀⢱⠀⢀⡼⠀⠀⠈⠋⠁⠉⡼
⠀⠀⠀⠈⢢⠀⣰⠇⠀⠀⠀⢳⡀⣠⣀⡾⠀⠀⠀⣰⠋⠄⠀⠀⠀⠈⢀⡞⠁⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠳⠃⠀⠀⠀⠀⠀⠞⠁⠚⠀⠀⠀⠘⠁⠀⠀⠃⠀⠀⠀⠸⠇⠀⠀⠀⠀⠢⠿⠶⠚⠃
⠀⠀⠀⠀⣠⠤⢄⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⣠⠄⡄⡀⠀⠀⠀⠴⠒⠄⡀⠀⠀⠀⠀⡴⠂
⠀⠀⠀⢰⠃⠀⠈⡄⠀⠀⠀⠚⠁⡇⠀⠀⠀⠈⠁⠀⣠⠇⠀⠀⠀⠀⣀⣀⡟⠀⠀⠀⢸⣁⠰⣀
⠀⠀⠀⠀⠂⣀⣰⠇⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣀⣼⣁⡄⠀⠀⠀⣀⣀⣀⡇⠀⠀⠀⠈⢈⡾⠉
⠀⠀⠀⠀⠀⢉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⢸⠉⠉⠃⠀⠀⠀⠀⡖⠒⠃⠀⠀⠀⠑⠒⣢⠇⠀⠀⢠⡟⠉⢰⠀⠀⠀⠀⡞⠉⠁⣀
⠀⠀⠀⠀⠘⠁⢠⡀⠀⠀⢠⡞⠉⠁⣢⠀⠀⠀⠀⢰⠏⠀⠀⠀⢠⠗⠒⠈⡀⠀⠀⠀⠃⣀⣀⡏
⠀⠀⠀⠀⠦⠶⠞⠀⠀⠀⠀⠙⠖⠶⠃⠀⠀⠀⠀⠚⠀⠀⠀⠀⠘⠤⠤⠾⠁⠀⠀⠀⠠⠤⠞

⠀⢀⠤⣦⡄⠀⠀⠀⠀⠀⢀⡄⠢⡄⠀⠀⢀⡤⠒⠒⢠⡤⠀⠀⢠⡄⠐⠦⣄⠀⠀⢠⡄⠀⠀⠀⠀⠀⣀⠤⠶⣄
⠀⢶⡀⠈⢿⡆⠀⠀⠀⠀⢸⡟⣦⡃⠀⠀⣿⠀⠀⠀⠈⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⣧⠄⢠⠀⠀⠀⢿
⠀⠘⡽⠁⡼⣿⡄⠀⠀⠀⢸⡟⠁⣿⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣼⠃⠀⠀⡇⠀⠀⠀⠀⠀⣿⠉⠑
⠀⠀⠀⢠⠏⢹⣷⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠉⠓⠒⠉⠈⠀⠀⠈⠉⠀⠉⠁⠀⠀⠀⠁⠀⠀⠁⠀⠀⠉
⠀⠀⠀⣟⣀⣀⣻⣆⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠀⢠⡄⠀⠀⢠⠀⠀⠀⠀⠀⡄⠀⠠⠄⠀⠀⠀⠀⠰⣶
⠀⠀⢰⠉⠁⠀⠈⣿⡄⠀⠀⣇⠀⠀⢤⠀⠀⣽⢀⡤⢸⠇⠀⠀⢸⠀⠀⠀⠀⠘⡇⠀⢸⡇⠠⡀⠀⠀⠀⡃
⠀⢠⡏⠀⠀⠀⠀⢸⠿⡄⠀⠈⣰⡄⠘⠀⠀⣿⡀⠀⠀⠇⠀⠀⢈⡄⠀⠠⡦⠘⠀⠀⢸⡇⠀⠱⠄⠀⠀⠇⠀⠸
⠀⠛⠋⠀⠀⠀⠀⠈⠋⠙⠂
⠀⠀⡀⠀⠀⠠⠰⡄⠀⠀⠀⠀⢰⣖⠀⠖⣶⠀⠀⠀⢀⠄⠀⠀⠒⢦⡀⠀⠀⢺⡟⠈⠳⡄⠀⢠⡖⠁⠈⠉⠲⣄
⠀⠚⠁⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⡿⣆⠀⣿⠀⠀⠀⡎⠀⠀⠀⠀⠀⣿⠀⠀⢸⣷⠦⠴⠃⠀⣿⠀⠀⠀⠀⠀⣹⠆
⠀⠀⢠⠀⠀⡀⠀⠈⠧⠀⠀⠀⠀⣧⠘⢧⣿⠀⠀⠀⠱⡀⠀⠀⠀⢠⠏⠀⠀⢸⡇⠀⢠⡄⠀⠙⠆⠀⠀⢀⣠⠟
⠀⠐⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⢰⣶⠒⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣤⠤⠄⠀⠀⠀⠀⠒⡤⠀⠀⠐⣶⠀⠀⢲⣆⠀⠀⠀⠰⠂
⠀⠀⣿⠀⢀⣿⠇⠀⠀⠀⠀⠀⢸⣄⡀⠀⠀⠀⠀⣰⡞⠁⣠⡄⠀⠀⠀⢀⡜⠀⠀⠀⠀⣯⠀⠀⠀⢿⡄⠀⢠⠃
⠀⢀⣿⠘⠫⣄⠀⠀⠀⠀⠀⠀⠀⠉⢺⡆⠀⠀⠀⣿⡀⠀⢟⠇⠂⠀⠀⢸⡇⠀⠀⠀⠠⣿⠀⠀⠀⠈⣷⡀⠆
⠀⢨⣿⠀⠀⠐⠷⣄⡿⠀⠀⠰⢤⣀⠰⠃⠀⠀⠀⠙⠷⠤⣀⡠⠊⠀⠀⠘⠻⠦⣤⠄⠀⠿⠄⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠰⠟⢓⡙⠛⠙⢒⠛⠀⠈⣇⡏⠀⠀⠻⣿⡄⠀⠀⠛⠉⠀⠀⠀⡏⢹⣧⠀⠀⠀⣽⡏⠀⠈⠁⡀⠉⠋⢉⣽⠎
⠀⠀⢀⣾⠀⠀⢀⡟⠀⠀⠀⣿⡇⠀⠀⠀⠙⢿⢆⡞⠁⠀⠀⠀⠘⠁⠈⣿⡀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⢸⣯⠀⠀⢼⡇⠀⠀⠀⣾⡇⠀⠀⠀⠀⢀⢿⣷⡄⠀⠀⠀⠀⠀⠀⢿⣇⠀⢀⣿⡇⠀⠀⠀⠀⢀⣤⠊
⠀⠀⠈⢿⣦⣀⢘⣿⣄⣀⣠⣽⣇⠀⠠⡤⢀⣋⠀⠙⣿⣦⠀⠀⢀⣄⠀⠀⠉⠉⠁⢸⡇⠀⠀⠀⢀⡾⠃⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠉⠁⠀⠈⠈⠀⠁⠀⠀⠀⠀⠛⠉⠀⠀⠘⠉⠉⠀⠈⠉⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠓⠛⠉⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠤⠤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢠⢿⡄⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠋⠉⠉⠉⠉⠓⢦⡞⠛⠛⠛⠛⠒⢶⡄
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⢀⡟⠀⢿⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣸⠁⠀⠘⡇⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠻⠤⠤⠴⠇⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣠⠴⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⣺
⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⠛⠻⡆⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠳⠦⠤⠤⠤⠤⠤⠦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⠴⠤⠤⠤⠤⠴⠟⠀⠀⠀⠀⠀⠈⠛⠶⠤⢤⡤⠤⠖⠋⠙⠶⠤⠴⠤⠤⠶⠋
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠶⠶⠒⠶⠶⠶⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠶⠶⠒⠲⠒⠶⠶⢤⣄⡀
⠀⠀⠀⠀⣠⠞⠉⠉⠉⠉⠉⠙⡆⠀⠀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢸⡀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣆⠀⣀⡤⠖⠛⠉⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠙⠛⢋⡉⠉⠈⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠸⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢠⡞⠉⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡁
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠚⠛⠛⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⡤⠴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠋⠋⠉⠉⠉⠉⠻⡆⠀⢠⠞⠛⠓⠶⣤⣤⣤⣤⠾⠃
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢨⡇⠹⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁
⠀⠀⠀⠘⠲⠶⠶⠶⠶⠶⠶⠖⠃⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠋⠀⠀⠀⠈⠉⠛⠛⠒⠲⠶⠶⠶⠶⠒⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠁
⠀⠀⠀⠀⢀⣀⡀⠀⠀⣴⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⣸⡇⠀⠀⣀⣀⠀⢤⣼⣧⡤⠀⢀⣀⣀⡀
⠀⠀⠀⢰⡋⢸⡏⠀⠀⣿⠚⢲⡄⠀⢰⡏⠈⠁⠀⡏⠁⣹⡇⠀⢰⣧⠼⠀⠀⢸⡇⠀⢠⡏⠀⣿
⠀⠀⠀⠈⠛⠋⠃⠀⠀⠙⠳⠚⠁⠀⠈⠓⠶⠂⠀⠙⠚⠙⠃⠀⠈⠳⠚⠀⠀⠘⠃⠀⠀⠙⠚⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠋
⠀⠀⠀⢰⡆⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⣶⠀⡀⠀⢠⡆
⠀⠀⠀⠘⣧⣤⡀⠀⠀⢠⡅⠀⠀⢘⠃⠀⠀⣯⡞⠁⠀⢸⡇⠀⠀⠐⣿⠛⣶⠖⣦⠀⠀⢸⠞⢳⡄
⠀⠀⠀⠀⣯⠀⣿⠀⠀⢸⣇⠀⠀⢸⡇⠀⠠⡏⢷⠀⠀⠸⣧⡄⠀⠀⢿⠀⠿⠀⠿⠀⠀⢸⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⣀⣄⡀⠀⠀⣀⣀⠀⠀⠀⣠⣄⡀⠀⠀⣤⢀⠀⠀⠀⣠⣄⠀⠀⢀⣿⡀⠀⠀⡀⢀⡀
⠀⠀⠀⢸⡏⠈⣷⠀⠀⣿⠈⣷⠀⢸⡇⢸⡇⠀⠀⣿⠉⠓⠀⠀⠳⣌⠀⠀⠉⡏⠁⠀⢸⠁⢸⠃
⠀⠀⠀⠈⠛⠞⠃⠀⠀⣿⠋⠁⠀⠈⠛⢻⡇⡀⠀⠛⠀⠀⠀⠀⠲⠞⠁⠀⠀⠛⠆⠀⠘⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢸⠟⠁

⠀⠀⠀⠀⣷⠀⣸⠀⠀⢸⠀⣠⡀⢸⠇⠀⠀⠙⣶⠋⠀⠀⣼⠁⣸⠀⠀⠈⣩⠟
⠀⠀⠀⠀⠘⣦⠏⠀⠀⠘⠶⠃⠳⠏⠀⠀⠠⠞⠙⠆⠀⠀⠙⠚⣿⠀⠀⠚⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠄⢄⠁⠀⠀⠀⣿⡏⢻⣧⠀⠀⠀⠀⠀⢀⠀⠀⢸⣿⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠃⠀⢰⠯⠚⣬⡆
⠀⠀⠀⠀⢠⢠⣀⠈⠀⠀⠀⣿⣧⣾⣇⠀⠀⠀⠃⠀⠀⠀⠀⢸⡿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣧⣴⡄⠀⣮⠃⡤⠬⡤
⠀⠀⠀⠀⡄⡀⠀⠂⠃⠀⠀⣿⣇⣨⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣠⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠀⠀⢫⢦⣀⢿⠁
⠀⠀⠀⢀⣒⠁⢀⡐⠈⠀⠀⠛⠛⠋⠁⠀⠀⡀⠀⠀⣀⡀⠀⣘⠛⠛⢉⢀⠀⠀⠀⠀⡀⡀⠀⠛⠃⠀⠀⠀⠈⠁⠃⠁
⠀⠀⠀⢸⣿⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡃⢜⡞⠁⠀⠀⢠⠀⠀⠀⠀⣿⣾⠀⢸⣿⣿
⠀⠀⠀⢸⣿⣶⣾⡇⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⣿⣯⡟⠀⠀⠀⠠⢠⠀⠀⠀⠀⣿⣿⣇⣼⣿⣿
⠀⠀⠀⢸⣿⠀⢹⡇⠀⠀⠀⠀⠀⠀⢂⠂⠐⣐⠀⠀⣿⠗⣻⡄⠀⠀⠨⠠⠀⠀⠀⠀⣿⡈⣿⣿⢹⣿
⠀⠀⠀⠈⠛⠀⠘⠃⠀⠀⠀⠀⣀⣀⡀⠂⠐⠀⠀⠀⣙⡀⠘⠙⠀⠀⠈⠀⠀⠀⠀⠀⠛⢁⣉⠉⠈⠛⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢨⠀⠸⠁⠀⠀⠀⣿⡟⢿⣷⠀⠀⢀⣾⠿⢿⣦⠀⠀⠀⠀⡔⠀⠀⠀⠀⢸⣟⢻⣷⠀⠀⠀⠛⢻⡿⠛
⠀⠀⠀⠀⠀⢰⠃⠀⠀⡂⠀⠀⣿⣇⣼⡿⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⣀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠴⠂⢠⠀⠀⠀⠀⣿⡟⠉⠀⠀⠀⠘⣿⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣧⢈⢿⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠈⠛⠉⠛⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠛⠛⠋⢀⡀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⢽⢷⠀⢸⡿⠀⠀⠇⠅⠐⠀⠂⠨⠨⠀⠀⢫⡧⡀⣽⠀⠀⢻⣷⢀⣿⠇⠀⠀⠃⠀⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⣿⠇⠀⠀⠅⢀⠂⠤⠠⠆⡂⠀⠀⠈⣇⣗⡃⠀⠀⠀⢻⣿⡏⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⢻⣿⡿⠀⠀⠀⠰⠀⠰⠀⠄⠀⠀⠀⠀⣐⡗⠓⣶⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠸⠟⠃⠀⠀⠀⠈⠂⠂⠀⠓⠐⠀⠀⠀⠁⠃⠀⠿⠀⠀⠀⠘⠿⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⡀⠀⡄⠀⠀⢀⠀⢀⠀⠀⠀⠀⡀⢀⠀⠀⠀⢠⠀⣄⡀⠀⠀⡄⠀⢀⢀
⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠁⠈⠁⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠈⠀⠀⠁⠀⠀⠈⠁⠈

⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣤⣀⡀⠀⠀⠀⣀⣀⠀⠀⠀⣠⣄⡀⠀⠀⢀⣤⣤⣄⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⢸⣇⣈⡷⠀⠀⣾⠁⠙⠃⠀⠀⣿⠀⢹⡆⠀⢸⣇⣀⠀⠀⠀⢸⣁⡀
⠀⠀⠀⠀⠀⣼⠿⣧⠀⠀⢸⡏⠙⣧⠀⠀⢿⠀⢀⡀⠀⠀⣿⠀⢸⠇⠀⢸⡏⠉⠀⠀⠀⢸⠋⠉
⠀⠀⠀⠀⠀⠛⠀⠛⠀⠀⠘⠛⠛⠁⠀⠀⠈⠛⠛⠁⠀⠀⠛⠛⠋⠀⠀⠘⠛⠛⠛⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡴⢶⡄⠀⠀⣴⠀⢰⡆⠀⠀⢶⡶⠄⠀⠀⠀⠰⣶⠶⠀⠀⢴⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣤⣅⠀⠀⣿⣤⣼⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠸⣇⣠⡟⠀⠀⣿⠀⢸⡇⠀⠀⣸⣇⡀⠀⠀⣦⣀⡿⠀⠀⠀⣾⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠀⠈⠁⠀⠀⠉⠉⠁⠀⠀⠈⠉⠀⠀⠀⠀⠈⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣿⡄⢀⣿⠀⠀⢸⣧⠀⣿⠀⠀⣴⠟⢳⡄⠀⠀⣿⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣧⣸⣿⠀⠀⢸⡟⣧⣿⠀⠀⣿⠀⢸⡇⠀⠀⣿⠶⠛
⠀⠀⠀⠀⠀⠀⠀⢿⣦⣤⡄⠀⠀⡿⠹⡏⢿⠀⠀⢸⠇⠸⡿⠀⠀⠹⢦⡼⠃⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⣀⣀⠀⠀⠀⣀⣀⣀⡀⠀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⣧⠀⠀⢸⣇⢈⣿⠀⠀⢸⣏⠙⠃⠀⠀⠀⣽⠁⠀⠀⠸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡿⠀⠀⢸⡏⠙⣷⠀⠀⢀⡉⢹⡆⠀⠀⠀⣿⠀⠀⠀⠀⣇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠀⠀⠘⠃⠀⠛⠀⠀⠈⠛⠛⠁⠀⠀⠀⠙⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⢰⡆⢠⡆⠀⠀⢶⠀⢰⡆⢀⡆⠀⠀⢰⡄⣰⠆⠀⠀⢶⡀⣴⠀⠀⠀⠶⠶⣶
⠀⠀⠀⠀⠀⠀⠈⣧⣼⠃⠀⠀⢸⡇⡿⣷⣸⠇⠀⠀⠀⢿⡏⠀⠀⠀⠈⣿⠃⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⢠⡟⣷⡀⠀⠀⠀⣿⠀⠀⠀⠀⣾⣃⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣶⡀⠀⠀⣴⣶⣶⣤⠀⠀⢠⣴⣶⣶⡄⠀⣶⣶⣶⣄⠀⠀⣤⣤⣤⣤⠀⠀⣤⣤⣤⣤
⠀⠀⠀⢀⣾⡿⣷⠀⠀⣿⣿⣿⣿⡀⠀⣿⠋⠉⠀⠁⠀⣿⠉⠙⣿⡆⠀⣿⣭⣭⡁⠀⠀⣿⣭⣭⠁
⠀⠀⠀⣼⣿⣿⣿⣧⠀⢿⣯⣭⣿⡇⠀⢻⣷⣤⣴⡆⠀⣿⣤⣴⡿⠁⠀⣿⣯⣭⣄⠀⠀⣿⠋⠉
⠀⠀⠀⠙⠁⠀⠀⠛⠀⠈⠛⠛⠋⠀⠀⠀⠙⠛⠋⠁⠀⠛⠛⠋⠀⠀⠀⠛⠛⠛⠋⠀⠀⠛

⠀⠀⠀⣠⣾⡿⣿⡦⠀⢰⣿⠀⠀⢸⡇⠀⢸⡇⠀⢸⣇⣠⣾⠃⠀⢸⣷⠀⠀⠀⠀⢸⣧⠀⠀⣰⣿
⠀⠀⠀⣿⡇⢰⣶⣦⠀⢸⣿⣿⣾⣿⡇⠀⢸⡇⠀⢸⣿⣿⡁⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⣧⣰⣿⣿
⠀⠀⠀⠹⣿⣶⣾⠟⠀⢨⡿⠀⠀⢸⡇⠀⢸⠇⠀⢸⡏⠹⣿⠆⠀⢸⣿⣶⣶⠆⠀⢸⡏⠻⡿⢹⡿
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⣀⡀⠀⢀⡀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⣿⣷⣄⢸⡇⠀⠀⣾⡿⠛⠻⣿⡆⠀⠀⣿⡟⢻⣿⠀⠀⢀⣾⡿⠛⠻⣿⡄⠀⠀⣿⡿⢿⡇
⠀⠀⠀⣿⡟⢿⣾⡇⠀⠀⣿⣄⠀⢀⣼⡇⠀⠀⣿⣿⣿⠟⠀⠀⠸⣿⣄⢰⣶⣿⠇⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠿⠁⠈⠻⠃⠀⠀⠙⠻⠿⠿⠋⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠀⠀⠀⣿⠁⠹⡟

⠀⠀⠀⢠⣴⣶⣤⠀⢠⣶⣶⣶⣶⡄⢠⣄⠀⠀⣠⡄⢠⣦⡀⢀⣤⠀⢠⡄⠀⢀⣤⠀⢠⣤⣤⣤⣤
⠀⠀⠀⢿⣿⣭⣿⠀⠀⠉⢹⡟⠉⠀⠘⣿⡄⣰⡿⠀⠈⢻⣷⣿⠋⠀⠹⣿⣦⣾⠏⠀⠘⠛⣻⡿⠃
⠀⠀⠀⣶⣭⣽⣿⠃⠀⠀⢸⡇⠀⠀⠀⠸⣿⣿⠁⠀⢠⣾⠟⣿⣦⠀⠀⠘⣿⡏⠀⠀⢠⣾⣟⣁⣀
⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠘⠃⠀⠀⠀⠀⠙⠁⠀⠀⠘⠋⠀⠈⠛⠀⠀⠀⠻⠁⠀⠀⠺⠿⠿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠉⠓⢄⠀⠀⢰⠋⠉⣭⡉⠙⣆⠀⠀⣠⠞⠉⠉⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⡶⠀⢸⡆⠀⢸⠀⠀⣉⡁⠀⡇⠀⠀⡇⠀⠸⣭⠶⢮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⡴⢲⣀⣸⠃⠀⠸⣄⣀⣛⣁⣠⠏⠀⠀⠙⢦⣀⣀⣀⠼⠁
⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⢀⣉⣀⣀⣈⣁⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⠤⠤⣄⠀⠀⠀⠀⣀⡀⠀⢀⣀
⠀⠀⢸⠁⠀⠀⠈⠑⡄⠀⠀⡏⠀⢠⠶⠶⠞⠁⠀⢸⠁⠀⡶⠶⠶⠋⠀⠀⡜⠁⢀⣀⣀⡇⠀⠀⢸⠁⠹⣆⡏⠈⣧
⠀⠀⢸⠀⠐⠿⠂⠀⡽⠀⠀⡇⠀⢨⠭⠭⠽⠂⠀⢸⠀⠀⡭⠭⠭⠗⠀⠀⢧⠀⠘⠊⠉⢹⡆⠀⢸⠀⢀⡭⡀⠀⣿
⠀⠀⠘⢤⣀⣀⡤⠞⠁⠀⠀⠣⣄⣌⣉⣉⣹⠀⠀⠘⢤⡴⠃⠀⠀⠀⠀⠀⠈⠑⠲⢦⡀⣸⠇⠀⠸⢤⡼⠃⢧⣤⠏
⠀⠀⢠⣒⡒⠒⠒⣒⡦⠀⠀⣔⣒⣒⣒⠒⢦⠀⠀⢠⠖⠲⣀⠔⠒⣄⠀⠀⡴⠒⣆⠀⠉⠁⠀⠀⡴⠒⢦⣀⡤⠒⢦
⠀⠀⠀⠀⣿⠀⢸⠁⠀⠀⠀⢀⣀⡀⣹⠀⢸⡇⠀⢸⠀⠀⠁⠀⣰⠋⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠉⠀⠀⢸
⠀⠀⢀⡤⠟⠀⠸⠤⣄⠀⠀⢏⠀⠉⠁⢀⡼⠀⠀⢸⠀⢀⣦⡀⠈⡧⠀⠀⡇⠀⠸⠤⠤⢤⡀⠀⡇⠀⣰⢤⣦⠀⢸
⠀⠀⠈⢛⡛⠛⠛⡛⠁⠀⠀⠀⠉⢛⣋⠉⠀⠀⠀⠀⢙⣉⣀⣉⠋⠁⠀⠀⠙⢛⣛⣛⠛⠋⠀⠀⠈⢋⣁⣀⣈⠙⠉
⠀⠀⢰⠉⠉⠲⠎⠉⢣⠀⠀⡰⠊⠉⠉⠙⢦⠀⠀⢰⠉⠉⣭⡍⠉⣆⠀⠀⡔⠉⠀⠀⠙⢦⠀⠀⢸⠉⢉⣭⡉⠙⣆
⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠸⠷⠀⢸⠇⠀⢸⠀⠀⡭⠤⠴⠃⠀⠀⡇⠀⠻⠇⠀⢸⠀⠀⢸⠀⠀⠉⠀⢰⡃
⠀⠀⠸⣄⣠⠟⢦⣀⠞⠀⠀⠙⠦⣀⣀⡤⠞⠀⠀⠸⣄⡰⠃⠀⠀⠀⠀⠀⠙⠦⣤⣤⣀⣨⠇⠀⠸⣄⡰⠓⢄⣠⠇
⠀⠀⠀⡠⠤⠤⠤⢤⡀⠀⠀⡤⠤⠤⠤⠤⢤⡀⠀⢀⡤⢤⡀⡠⠤⡄⠀⠀⣠⠤⣄⢀⠤⢤⠀⡤⠤⡀⣠⠤⣄⢀⡤⢤⡀
⠀⠀⢸⡁⠀⠯⠷⢖⠋⠀⠀⠉⢹⡆⠀⡏⠉⠀⠀⢸⠀⠀⡇⡇⠀⣿⠀⠀⢧⠀⠘⠋⠀⡸⠉⣇⠀⠙⠃⠀⠈⠋⠀⣰⠃
⠀⠀⢠⡽⠶⣶⠆⠀⣳⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠸⡄⠀⠙⠁⢀⡏⠀⠀⠈⢧⠀⠀⡰⠃⠀⠘⣆⠀⠀⣤⠀⠀⣰⠃
⠀⠀⠀⠑⠒⠒⠒⠚⠁⠀⠀⠀⠀⠓⠒⠁⠀⠀⠀⠀⠈⠓⠒⠒⠋⠀⠀⠀⠀⠈⠓⠚⠁⠀⠀⠀⠈⠓⠚⠀⠓⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠳⠖⠉⢹⠄⠀⢰⠋⠙⠶⠊⠉⡦⠀⠀⠯⠭⣭⠍⠉⢱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠀⠀⠰⣏⠀⠀⠀⠳⡄⠀⢠⠞⠁⠀⠀⢠⠞⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⡴⠦⣀⣸⠂⠀⠀⠀⢿⣀⡸⠀⠀⠀⠀⢧⣀⣀⣛⣒⣲⠄

⡏⠉⠉⠉⢉⡉⡉⢩⠡⠍⠉⠉⠉⠩⡍⠭⠉⠉⠉⠉⠉⠉⢩⢹⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡅⠀⡀⢀⡈⠁⠇⠀⡄⠀⣀⡀⠀⠀⠀⢀⠀⠀⠀⢀⣀⡀⢀⡈⠀⠀⠀⠀⣀⠀⢀⢂⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠴⡴
⡇⠀⠻⢎⠦⠀⡇⢀⠇⠈⠁⡿⠟⠀⠀⡿⠁⠀⠀⢸⡷⠄⢰⣿⡿⠀⡄⣹⣍⡰⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠿⠀⣶⠤⡄⢠⣶⢤⢴⣶⣴⡟⠓
⠇⠾⡶⠏⠀⢰⣆⣸⣷⡄⢠⣧⡤⠈⢀⣏⠀⣾⡤⢨⣿⠀⠼⢅⡀⠄⠁⢀⣿⡆⠀⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣀⠆⣻⢠⣇⠨⣿⠃⣸⣍⠀⣷⡦
⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⡄⠀⠀⠈⠉⠁⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠀⠉⠀⠀⠀⠈
⡇⠀⠀⠀⠀⢠⣄⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠃⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⡇⠀⠀⠀⠠
⡇⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⠀⢰⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⢰⠆⠠⡀⠀⠀⣀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠴
⡇⠀⠀⡼⠁⢣⠀⢦⣨⡆⠠⠀⠀⣏⢸⠆⢸⣁⡆⠀⠀⠀⢄⣀⡀⠰⣄⡀⠀⠀⢸⡅⣠⠇⠀⣀⣭⠀⠀⠀⣏⢀⠀⢸⣁⠂⠀⠀⠀⠃⠀⠀⠀⡇
⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡤⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⡄⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⡄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢀⣀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠄⠀⠀⢰⠆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠠⠄⠀⠀⠁
⡇⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⡃⠀⠀⠀⠀⡀
⡇⠀⠀⠀⢆⠀⠀⡄⠀⠔⢢⠀⡀⠀⠀⢺⠀⢺⠀⠸⠂⡄⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠇⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⣶⠈⠀⢀⡤
⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠃⠀⠀⠈⠀⠀⣀⠃⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠙
⡇⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠀⠃⠀⠀⠀⠀⣀⡀⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⡇⠀⠀⠀⠀⡆⠀⠀⠠⠀⠀⠀⠀⠀⢠⡄⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⢠⠀⠀⠀⠀⠀⡀⠀⠀⡔⠂⠢⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠹⠀⣀⢀
⡇⠀⠀⠀⠀⡇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠰⡄⢹⠀⠐⠙⠃⠄⠀⠀⠀⠀⠀⢾⠀⠞⠱⠄⠀⡂⠀⠀⢄⠀⡠⠁⠆⢸⠄⠀⠀⠀⠀⠘⠃⠀⠀⡧⠘
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⡀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⢀⠄⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀
⡇⠀⠀⠀⢨⠀⢀⡄⠀⠒⣲⠀⠀⠀⠀⠀⠷⠞⠀⡴⠀⠀⠀⠀⠀⠀⠈⠂⡄⠀⣄⠀⠀⠀⠇⠀⠀⠀⠀⡇⠀⢲⠀⠀⠀⠀⠀⠀⢠⠀⠀⡇⢠⠀⡄
⡇⠀⠀⠀⠀⠈⠁⠁⠀⠉⢹⠀⡀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠊⠀⠈⠘⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡶⠀⠀⠈⠁
⡇⠀⠀⠀⠀⠀⠀⢀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠰⠦⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾
⡇⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠀⠈⠁
⡇⠀⠀⠀⠰⠀⡄⠀⠀⠀⠀⠀⠀⠀⡄⢀⠶⠀⡄⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⡁⠀⠀⢢⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⢀
⡇⠀⠀⠀⠀⠻⠀⠀⠀⠆⠀⠀⠀⠀⠐⠏⠀⠻⠀⠀⠺⠷⠀⠀⠀⠀⠰⠉⠄⠀⠜⠂⠀⠀⡅⠀⠀⠸⠇⠀⠈⡎⠀⠀⠀⠀⠀⠀⠮⠀⠀⠠⠃
⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣶⣶⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣽⣿⣿⣿⣿⣿⣿⣻⣿⣿⣟⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣼⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣛⣿⣻⣿⣟⢯⠿⢻⢛⢙⣽⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

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

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




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠉⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⢀⣀⣀⡀⠀⠊⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢜⠇⠀⠀⠀⠀⠉⠳⣬⣵⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠈⠀⠀⠀⣀⠀⠀⠀⠈⢻⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢿⠃⠀⠀⠀⣤⣨⠷⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣶⠞⡟⠀⠈⡀⠀⠀⠀⠀⠁⣤⣠⠽⠾⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠋⠀⠀⠀⡤⠀⠃⠀⠀⠀⠀⢸⡢⠐⡂⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⢀⠊⢀⣤⣘⡄⢀⣼⡓⠚⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢁⠘⠋⠉⠀⠈⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠧⠂⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠜⠁⠀⠈⠠⣠⣠⡤⠤⠄⠀⠀⢠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⣀⣀⣤⣤⣿⣿⡆⠀⠀⠀⠀⠀⢀⠄⠉⠃⠐⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠃⠀⠒⠈⠀⠀⠠⠈⠛⢷⣄⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣹⣑⠈⠉⠉⠙⠛⠒⠒⠶⠒⠒⠋⠁
⠀⠀⠀⠀⣀⠀⠈⠛⠛⠒⠂⠠⠤⠀⠀⠠⠤⠄⣈⡿⠛⠉⠉⠉⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢤⡢⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⢤⣀⣀⣄⣀⡀⠀⠀⠀⠀⢀⣰⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⣿⣶⡶⠖⠋⠁


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

⠀⠰⠶⠂⠀⣤⢦⠀⠀⠐⠒⠀⠀⢀⡤⢄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡖⠆⠀⠀⠀⠀⠀⠀⣠⢦⠀⠀⠐⠂⠀⢠⠲
⠀⢠⣦⠀⠀⠁⠈⠀⠀⡀⠀⠀⠀⠈⠁⠘⠀⠀⠈⠀⠀⠀⣠⠄⠀⣄⣤⠀⠘⠁⠀⠀⢹⠃⠀⠀⠠⠀⠀⠀⠻⡤⠀⠀⠃⠀⠀⠘⠿⠂⠀⣶⡆⠀⢰⡆
⠀⠈⠇⠀⠀⠀⠀⠀⢀⣛⣘⡀⠀⠀⠀⢀⠀⠀⠀⢠⣄⠀⠘⠂⠀⠙⠁⠀⠀⣁⠀⠀⠈⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡆⠀⣼⣧
⠀⠀⠀⠀⠀⠴⡶⠄⠸⡿⢹⡇⠀⠀⠀⠄⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠋⠙⠁⢸⡇⣿⠀⠀⠁⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⣐⠀⠆⠀⠀⣀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠃⠙⠀⣀⠀⠂⠐⣀⡀⠀⠀⠀⠘⠃⠙⠀⢀⠀⠠⠀⠘⠛⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠
⠸⠋⠀⠀⠀⡈⠉⠁⠀⠀⠐⠄⠀⠀⠸⠙⠂⠀⠀⠀⠀⠀⠉⠁⠀⠘⢏⡹⠀⠀⣴⠋⢹⠀⠀⠉⣷⠀⠸⠓⠀⠀⢛⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠾⠛⠗
⠀⣀⠀⠀⣿⣿⣿⣷⣀⠀⠀⠀⠀⠰⠿⠂⠀⠀⣘⣁⠀⠠⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠺⣿⠇⠀⢸⡇⠀⣾⣶⡂⠀⢀⣀⠀⠀⠀⠠
⠀⢀⡿⠀⢻⠟⠛⠛⠋⠀⣾⣶⣖⠀⠀⠀⠀⠀⣿⣿⠀⠈⠁⢰⠄⠀⣾⣿⠀⠀⣤⣤⣤⠀⠀⠒⠂⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠰⢂
⠀⠀⠀⠀⠑⠋⠀⢀⠀⠀⠉⠁⠁⠀⠀⠀⠀⠀⠸⡧⠀⠀⠀⠈⠀⠀⠟⠸⠀⠀⠙⠋⠛

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



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

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

⠀⠀⡀⠀⠀⢰⣿⡧⠀⠀⠀⠀⠀⠀⢾⣿⡆⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⠀⠰⣿⣷⠀⠀⠀⠀⠀⠀⢰⣿⡇
⣠⣤⣶⣤⣠⣴⣿⣶⣄⣠⣴⣤⣄⣠⣶⣿⣦⣀⣤⣶⣤⣄⣤⣾⣷⣦⣀⣤⣶⣤⣀⣴⣾⣷⣄⣠⣴⣦⣤⣠⣴⣿⣶⣄⣤⣶⣦
⠉⣿⣿⡍⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⠉⠉⠉⢹⣿⣯⠉⠉⠉⠉⠉⠉⢹⣿⡏⠉⠉⠉⠉⠉⠉⣿⣿⡍⠉⠉⠉⠉⠉⠩⣿⣿
⣠⣼⣿⣤⠀⣠⣤⣤⡀⣠⣽⣯⣄⢀⣤⣥⣄⢀⣤⣿⣥⡀⢀⣤⣤⣀⢀⣬⣿⣤⡀⣠⣤⣤⡀⣠⣼⣯⣄⢀⣠⣤⣄⡀⣠⣽⣧
⠉⠛⠛⠋⠉⢹⣿⡏⠉⠉⠛⠋⠉⠉⣹⣿⡍⠉⠉⠛⠋⠉⠉⣽⣿⠉⠉⠙⠛⠉⠉⢉⣿⣯⠉⠉⠙⠛⠉⠉⢹⣿⡏⠉⠉⠛⠛
⢀⣀⣄⡀⠀⣨⣿⣅⡀⢀⣈⣀⡀⢀⣨⣿⣅⠀⣀⣁⣀⠀⢀⣽⣯⣀⠀⣀⣈⣀⠀⣠⣽⣯⡀⢀⣈⣈⡀⢀⣨⣿⣅⡀⢀⣀⣁
⠛⣻⣿⡛⠛⠛⠛⠛⠛⠛⣿⣿⠛⠛⠛⠛⠛⠛⢛⣿⣟⠓⠛⠛⠛⠛⠛⢻⣿⡟⠛⠛⠛⠛⠛⠛⣻⣿⠛⠛⠛⠛⠛⠛⠛⣿⣿
⠀⣙⣟⡁⠀⢀⣐⡀⠀⢀⣻⣟⡀⠀⢀⣂⠀⠀⣈⣻⣋⠀⠀⣀⣁⠀⠀⣘⣿⣃⠀⢀⣐⣀⠀⢀⣹⣟⡁⠀⢀⣀⡀⠀⢀⣻⣟
⠿⠿⠿⠿⠖⢻⣿⡟⠓⠿⠿⠿⠿⠚⢻⣿⡛⠺⠿⠿⠿⠷⠚⣿⣿⠓⠺⠿⠿⠿⠗⠛⣿⡟⠓⠿⠿⠿⠿⠖⢻⣿⡟⠲⠿⠿⠿
⠀⠀⠀⠀⠀⠘⣿⡃⠀⠀⠐⠀⠀⠀⢹⣿⠃⠀⠀⠀⠂⠀⠀⣻⣟⠀⠀⠀⠀⠀⠀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠘⣿⡃
⠶⢿⣿⠷⠴⠿⠿⠿⠧⠾⣿⣿⠷⠾⠿⠿⠿⠦⠿⣿⡿⠦⠾⠿⠿⠷⠴⢿⣿⠿⠶⠿⠿⠿⠷⠾⢿⣿⠷⠴⠿⠿⠿⠦⠾⣿⡿
⠀⠻⡿⠃⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠘⢿⠗⠀⠀⠠⠄⠀⠀⠸⣿⠇⠀⠀⠀⠄⠀⠀⢿⡿⠂⠀⠀⠠⠀⠀⠐⢿⡿
⣴⣿⣿⣷⣤⣶⣿⣶⣤⣾⣿⣿⣶⣴⢾⣿⣶⣤⣾⣿⣿⣦⣴⣾⣿⣦⣴⣿⣿⣷⣦⣶⣿⣷⣦⣴⣿⣿⣷⣤⣶⣿⣶⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⢾⣿⠆⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠰⣿⡷⠀⠀⠀⠀⠀⠀⠸⣿⡇
⣠⣶⣶⣦⣠⣶⣿⣶⣄⣴⣶⣶⣤⣰⣶⣿⣶⣄⣴⣶⣶⣄⣴⣾⣷⣦⣠⣶⣶⣦⣠⣶⣾⣷⣦⣠⣶⣶⣦⣠⣶⣿⣶⣄⣴⣶⣶
⠀⣾⣿⡆⠀⠀⣁⠀⠀⠀⣿⣿⠀⠀⠀⡈⠀⠀⢰⣿⡧⠀⠀⠈⠁⠀⠀⢼⣿⡆⠀⠀⢁⡀⠀⠀⣾⣿⠄⠀⠀⠀⠀⠀⠠⣿⣿
⣠⣾⣿⣦⣀⣤⣶⣤⣀⣤⣾⣷⣄⣠⣤⣦⣤⣀⣴⣿⣶⣄⣠⣴⣦⣄⣠⣴⣿⣦⣀⣤⣴⣤⣄⣠⣾⣷⣦⣀⣤⣴⣤⣀⣴⣾⣷
⡿⠛⢻⠿⡿⠿⠿⢿⠿⠿⢿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⡿⣿⡿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡾⠿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠛⠉⠁⠀⠀⠀⠈⠙⠻⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⢀⣤⣶⣶⣶⣶⣤⣀⠀⣀⣀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠠⠮⡭⢗⣚⡛⠿⠿⠋⠘⣿⣿⣿⣷⣄⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡟⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⠿⠏⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣼⣿⣧⠀⠀⠀⠀⠀⣤⣤⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡆⠀⠀⠀⡠⠈⠘⠛⠉⠉⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⠻⣟⠛⠷⠶⣦⣤⡀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⣷⡀⠀⠈⠙⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⢿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⣀⣀⣠⣤⣶⣿⣿⡄⠀⠻⣿⣿⣿⣿⡿⢿⣀⡲⠶⠾⠋⣼⡆⢠⣤⣀⣀⣀
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠻⢟⣿⣷⣿⡿⠿⠶⠄⠀⢈⠃⡀⢻⣿⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣧⠀⠀⠰⢯⠈⠀⠈⠁⠁⠀⠀⠀⢀⣴⡾⣧⠀⢻⣿⣿⣧⣼⣇
⠀⠀⠀⠀⠠⡝⣿⡻⢿⣿⣷⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⠟⠻⠋⡉⡍
⠀⠀⠀⠀⠀⣹⣯⡫⠬⣻⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣟⡏⠈
⠀⠀⠀⠀⠀⠈⠘⢟⠳⢿⣯⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣯⡋⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠈⠃⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠀⠉⠛⢦⡀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣲⣶⠶⠂⠀⠀⠀⠀⠀⢹⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⣰⠋⠠⠙⠃⠀⠀⠀⠀⠀⠛⣅⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠛⢹⠐⠀⠀⠀⠀⠀⢠⡙⠲⢀⡼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⢻⡁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⡴⠉
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠚⢻⡄⠀⠀⢳⡀⠀⠀⢠⠀⠀⠀⠀⢀⠀⢀⡀⠇
⠀⠀⠀⠀⠀⠀⠘⠁⢠⣀⠀⠀⠀⣷⠀⠀⠀⢳⡀⠀⡈⢦⡀⠀⠀⠼⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⢸⡀⠀⠀⠈⣧⠞⢃⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⡟⠙⢣⡈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⣾⣀⣠⠚⠇⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢠⣿⠂⠀⢤⡄⠀⢸⡆⠀⠀⠀⠀⢠⠀⠀⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⣿⠀⠀⡠⣣⠊⠁⠀⠀⢀⣀⣸⣇⠀⠀⣀⠤⠬⠷⢔⣄⡀⠀⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠐⠚⠒⠛⠛⠛⠛⠛⠚⠋⠉⠉⢉⠀⢻⠁⠀⠀⠀⠈⠉⢛⠛⢦⡀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⠁⢸⡇⠀⠀⠀⠀⠀⠤⣍⣲⢽⣆⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣯⣤⣤⠞⠓⠲⠶⠶⢤⡭⠶⠋⠁⠀⠨⠇⢁⣠⠔⠂
⠀⠀⠀⠀⠀⠀⠘⠶⠶⠶⠶⠶⠶⠶⠶⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣄⢠⢰⣎⡄⡆⠄⡠⡀⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⡪⠑⣺⠇⡇⡿⠀⢘⢹⣷⡇⡖⣧⡟⣦⡦⡠⠀⠀⠀⠀⠀⠀⠀⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣴⣷⢄⣂⡄⠱⠂⡁⠒⠀⢸⠂⠘⠁⠼⠃⠥⠧⠷⠶⠀⠀⠀⠀⠰⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠻⢯⢥⡽⠧⡀⠀⠗⡆⣠⢸⡇⠹⡆⢠⡀⢷⠀⢒⣛⠀⠉⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⠰⠂⠌⣁⠬⠉⠀⠀⠂⠁⠐⠀⢱⠀⢻⡄⠁⠘⢆⣨⡥⠢⠄⠀⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣣⠄⠒⠀⠧⢈⠁⠒⡐⠌⠁⠐⠃⠀⠡⠜⠁⠀⠀⠤⠄⠰⠂⠀⠀⠹⢸⠇⠀⠀⠈⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠃⡨⠀⠁⠀⠀⠀⠌⠉⠀⠀⠅⠀⠀⢈⠀⠐⠀⠀⣀⠘⠂⢣⢀⠗⢁⣍⣲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠰⢻⣼⠊⠀⠉⠀⠀⢠⠀⣤⢀⢠⡌⡤⠱⠀⠆⡈⠰⢈⠐⠀⡞⣭⠤⡀⣀⣠⡿⡿⡄⠀⠀⠀⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠢⢆⠐⣀⢀⡀⠀⠢⠉⠀⠨⠐⠀⠂⢁⠤⠤⣸⠓⢿⠀⠧⣇⣽⣽⣧⢘⡙⡷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢪⠓⠀⠡⢀⣀⠤⡤⡐⣶⡞⡏⣳⢿⠸⣿⠆⠻⠋⠠⠀⣶⠋⠼⣹⣎⠡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⢾⡖⢣⢸⢸⡇⣟⠀⢾⣻⣳⣝⠺⣓⡶⠳⣆⠀⠛⡭⠌⠶⣔⠈⡥⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⢻⡌⢿⡴⠞⣜⣹⣾⠬⢕⣺⣭⣭⢵⡖⢋⣏⠀⣿⡀⡴⣧⣧⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄⠀⠈⠁⣴⣺⣟⡭⠳⠂⣻⠋⡍⢃⢈⣀⣧⣉⡽⠶⠶⠛⠛⠓⠀⠀⠠⢖⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠊⠀⠀⠀⠈⠁⢁⣈⣷⣣⣐⢋⡶⢰⡞⢻⣣⢭⣶⠮⠎⣠⡀⠐⠀⠀⠀⠀⠀⠉⠊⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⣥⢆⢀⣙⣀⣀⡁⣗⣭⣥⡖⣾⡿⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡝⢋⡭⣭⡻⠩⡊⢧⣸⣲⣵⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠺⣝⣓⣝⡃⣀⣁⡁⡤⣵⢛⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣮⣝⣲⣥⣬⠿⠓⡲⠏⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣀⣀⣠⣤⠴⠟⠂⠉⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢉⣀⣀⣠⣤⠶⠞⠋⢁⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⡉⠉⣁⣀⣠⡤⠶⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⠉⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠁

⣿⡟⠉⠈⢻⣷⡆⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⢀⣤⣤⠀⠀⠀⠀⣸⣿⠋⠁⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⢠⣤⣄
⣿⡄⠀⠀⢈⣿⡏⠀⠀⠀⠀⠀⠘⠿⣿⣿⡶⣾⣿⣿⣷⣶⠶⣤⣿⣿⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠸⢿⣿⣿⢶⣾⣿⣿⣶⣶⢶⣼
⣿⣷⠀⠀⢿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠈⠟⠙⠀⠀⠀⠹⢿⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠸⠏⠁⠀⠀⠂⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢉⠈⠉⢱

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤
⠀⢀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠛⠛⣷⠀⠀⠀⠀⠀⠀⠀⢀⡄⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⡛⠛⠻⣧
⣾⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⠀⠘⣳⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⣿⠀⠀⠀⠀⠘
⡏⠂⠀⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⠘⠟⠁⠀⠀⠀⠘⢿⠇⠀⠐⠹⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠂⠀⠚⠛⠀⠀⠀⠀⠘
⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠈

⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⣾⡿⠟⠛⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢰⣾⡿⠟⠻⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀
⣿⠃⠀⠀⠸⣿⡷⠀⠀⠀⠀⠀⠠⣾⣿⣦⣤⣼⣿⣿⣦⣤⣤⣀⣾⣿⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⢰⣿⣿⣦⣤⣼⣿⣿⣤⣤⣄⣰
⣿⣧⠀⠀⣴⣿⡁⠀⠀⠀⠀⠀⠀⠉⡿⠛⠉⠹⡿⠛⠉⠉⣉⢻⣿⣿⣄⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⢹⡟⠋⠉⢻⡟⠏⡉⠉⡉⣿
⠙⠋⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠛⠟⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⠟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⠿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠿⣿⣆
⣴⣾⣿⣶⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡆⠀⠀⠀⠠⢤⣴⣾⣷⣾⣇⡀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠁⠀⠁⣿⡀⠀⠀⠀⠤
⡟⡋⠛⢻⣿⠉⠀⠀⠀⠀⠀⠀⠀⠸⣷⡄⠀⠠⣿⠇⠀⠀⠀⠠⣿⡟⠉⠛⢻⡿⠉⠀⠀⠀⠀⠀⠀⠀⠹⣧⠄⠀⢴⡿⠀⠀⠀⠀⢰
⠇⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠈⠁


⣤⣶⣶⣶⣶⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣤⣶⣾⣶⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄⣀⣀⣀⣀⣀⣀
⣟⣻⣛⣛⣟⣙⣻⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⣆⡤⠖⠚⠛⢳⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠾⠛⠁⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⠋⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠀⢠⣤⣶⣶⣾⡋⠉⠉⢉⡽⣿⣶⣄⠀⠀⠙⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠟⠃⠀⢀⣼⡵⠋⠹⠿⢿⠀⢠⡏⠀⠛⢛⡟⣇⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡾⢹⠁⠀⠀⣠⠞⠀⠸⠷⣤⣠⡞⠀⢹⡄⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣥⡶⠀⠀⢀⡞⠁⠘⠷⠶⠚⠁⠀⠀⠀⠀⠀⠈⢳⡄⠀⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣤⣶⠟⠀⠀⣇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡀⠀⣾⠀⠀⠀⣠⠄⠀⠀⠀⠈⠉⠉⠉⠁⣴⣤⡀⣿⠟⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠛⢲⡄⠀⠈⢻⣄⣿⠀⠀⠚⠉⠳⢦⣀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢡⠖⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⣭⣽⣇⠀⢀⡏⡙⠻⠀⠀⠀⠀⠀⠀⠉⠛⠒⢀⣀⠜⠁⣀⣤⣤⣩⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣠⣿⠀⠀⣧⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣋⣁⣀⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⡤⣶⣿⠀⠀⠘⢷⣄⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⠽⠛⠛⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⣿⢸⡇⢀⣤⠤⠄⠀⠿⠿⠧⣄⡀⠀⠀⠀⠀⠀⠀⢀⣿⡶⠚⠓⠀⣿⠙⠛⠛⠛⠛⠒⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⢸⡇⣸⠀⠀⠀⠀⠀⠀⣀⠠⠿⣦⡄⠀⠀⢀⣶⡛⣯⣤⡶⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇⢻⢸⡇⣿⠀⠀⠀⠀⣠⠞⠁⠀⠸⣿⡁⠀⠀⢸⣿⠛⣾⠁⢀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣸⡇⣿⣠⡴⠖⠒⡯⠀⠀⠀⠀⠈⠛⠒⠚⠛⠋⠀⠹⣶⠟⠁⣸⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣇⢸⣿⡷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠶⠞⠋⠈⢷⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⢻⣿⢸⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠻⣾⡟⠈⣿⠀⢰⠃⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⢳⣄⠀⠀⣿⡇⢀⡿⠀⣿⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣴⡞⢹⡇⠀⠙⣶⣾⢿⣿⡏⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢸⠁
⠐⠛⠛⠓⠲⠶⣾⡏⠸⡇⠀⢷⡀⠀⣸⣿⠷⣿⡿⠶⠿⢤⣤⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣦⣤⣤⣶⡿⠃⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠙⠷⠞⠻⢦⣼⠿⣶⡿⠛⣿⣿⠛⠲⠶⠤⣤⣄⣀⣉⣉⣉⣉⡉⠉⠉⠛⠛⠒⠒⠶⠶⠶⠤⠴⠿⠿⠧⠶⠤⠤⠤⢤⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠁⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⠏
⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠉⠛⠛⠛⠓⠲⠶⠦⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣽⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣟⣵⣿⡿⠯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⣫⠟⣵⣌⣛⡟⠁⠀⠀⠉⠓⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢫⡀⠀⣠⠞⣡⣾⣿⣿⣿⢠⡞⣠⠆⠀⠀⠀⠀⠀⠉⠓⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⣀⣧⡞⣡⣾⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣀
⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⢀⠞⢃⣴⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⡀
⠀⠀⠀⠀⠀⢠⠏⢠⣄⡴⢃⠴⠋⠀⠘⣜⣆⠀⠀⠀⠀⠀⢀⢄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⠃⣾⣷⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⡞⠀⢀⡏⠐⠣⡀⠀⠀⠀⢈⣨⣤⣶⣶⣶⣿⣿⡼⢟⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⡸⢁⡴⣯⣀⣠⣐⣁⣤⣶⣾⣿⣿⣿⡿⣿⡏⢹⣿⣿⣷⣄⡀⠙⠲⢦⣤⣀⠀⠀⠀⠀⣠⡆⠐⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱
⠀⠀⠀⠀⢣⣞⣵⠟⢹⣿⠛⠛⠻⣿⢿⣿⣫⣿⣾⣿⣷⢘⣿⣿⣿⣿⣿⣷⣶⣦⣤⣭⣶⣦⣴⡾⢻⣿⠇⠀⢸⣿⣿⣿⡁⠟⣿⢹⣿⣿⠸⡸
⠀⠀⠀⣰⣿⠟⠁⠀⠈⠻⢶⡀⠢⡘⣷⠾⢿⣿⡿⠣⠬⣉⠛⢦⣌⠁⡹⣿⣿⣿⡿⣿⡿⠿⣟⢹⣿⡟⠀⠀⣿⣿⣿⣿⣷⣤⡇⣾⣿⡏⠀⡃
⠀⠀⠠⠛⠁⠀⠀⠀⠀⠀⠀⠙⠦⠴⣯⠀⠀⠙⢯⣀⠀⢀⡵⠀⠉⠛⠶⠶⣭⣭⡭⠿⠟⠲⠶⠤⢽⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢄⡀⣸⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⣿⣿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠋⠉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣰⣶⣦⠀⠀⠀⠀⠀⢀⣸⣿⡇⢠⠀⠀⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡄⠀⠙⠿⢿⣷⣶⣴⣶⣿⣿⣿⣿⣧⣀⣠⣾⣭⣿⣤⣤⣀⡀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠻⣿⣿⡏⠀⠤⢾⣿⣿⣿⣿⣿⣿⣿⣿⡏⠻⠿⣿⡿⠿⣿⣿⣿⣿⡆⢰⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣶⣿⡟⠁⠀⠀⠀⣾⡟⠹⣿⣿⣿⣿⣿⣿⡀⠀⢋⣴⣤⠀⣉⡉⢹⡟⣿⣿⣿⡟⣉⠉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⣿⠿⠿⠿⢷⡄⠀⠉⠀⠀⠻⣿⣿⣿⣿⣿⣧⠀⣌⣛⣋⡘⠿⠿⢘⠁⢿⡿⠙⣧⣉⣤⡉
⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⣿⣿⣿⣶⣋⣠⡄⠠⣶⣶⣷⣾⣿⣿⡿⠂⠈⠀⠀⠉⠉⠙⠀⢹⣿⣭⣿⣶⣶⣿⣿⣿⣷⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⣀⣢⣽⣿⡛⠛⢁⣤⡀⢀⠸⠿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢸⣿⡿⠉⠛⠿⢿⣇⣤⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣶⣿⣿⡿⠛⢹⠀⢶⣿⣶⣿⠏⠀⠀⠀⠀⠀⠀⠤⠀⠀⣤⠀⣀⣸⣶⣤⣤⣴⣶⡅⠘⠃⠀⢲⣼⣿⠟⠛⠿⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢥⠾⠿⣣⣾⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣼⣿⡿⣀⠀⠀⡀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣿⣣⣿⡿⠟⠉⠹⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠀⠈⣿⣿⢀⣤⣤⣤⣀⣿⣷⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⢀⣀⣀⠀⠀⠀⢣⡀⠀⠒⠀⢼⢿⡿⢿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠹⢿⣿⠏⠀⠀⣀⣠⣿⣿⣧⠀⠀⣴⣿⡿⠿⢷⣀⣀⠀⠀⠀⡀⠀⠈⣼⢠⡀⠀⣀⢸⡟⣧⣀⣀⣠⡦
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⠁⠘⢿⠀⢠⣿⣿⣿⡿⠛⠉⠶⣤⣿⠁⠀⠀⠀⢹⣿⣧⣾⣿⣿⣿⣷⣾⣀⡤⣤⢉⡼⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄⠀⠀⠀⠠⢶⣿⣿⣿⠇⠐⠀⠓⠶⠆⠒⠀⠀⠒⠀⣿⡏⠉⠉⣻⣿⣿⣿⡿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢻⣿⣿⡆⠀⠀⠛⠛⠛⣶⣟⣷⣶⡾⣿⣇⠀⠀⠋⣿⣿⠟⠃⠀⠀⠹⠀⠀⠀⠀⠀⠶⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠀⠀⢘⠛⠛⠛⣛⠛⠛⠋⣾⣿⣿⣷⣤⠄⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⣶⣿⣏⠉⠉⠀⠐⣶⠂⠈⠉⣉⣿⣿⣦⣤⣾⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⢠⣿⡄⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠸⣿⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣸⡆
⠀⠀⠀⠀⣾⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⣦⣤⡄⠀⠀⠀⠀⢀⢀⢤⠇⢀⣄⡈⢀⡀
⠀⠀⠀⣰⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣏⠙⠀⠐⢦⣾⣠⣾⣿⠠⣾⣤⡤⠂⡟⠷⣆⡀
⠀⠀⠀⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⡀⠙⠿⣿⣿⣷⣼⣿⣡⡟⣡⠤⠉⢻⡴⣆
⠀⠀⢰⣿⣿⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⠉⢌⠙⠛⠷⣥⠰⢚⣒⡀⢿⣿⡃
⠀⠀⡼⣿⣿⣧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠿⢿⠁⠀⠀⢿⡿⣿⣿⣆⠀⠱⣄⠈⢱⢆⣼⠟⠛⠪⣯⠈⢀
⠀⠀⣶⣿⡏⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠷⠾⠿⠇⠸⠶⢾⣿⣮⣿⣿⣧⡄⠈⠃⠈⠨⠟⠁⢈⢂⣿⠀⠆
⠀⠀⣿⣿⣟⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⣿⣿⠀⠀⠀⣀⢐⡀⠀⣹⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠎⠉⢹⣿⣂
⠀⠀⣴⣿⣿⢀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⣿⣿⣿⣶⣷⣾⢻⣿⠷⣶⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⡔⠀⠠⠋⢺
⠀⠀⣿⣿⡟⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣏⠻⣿⣿⣷⣉⣉⣴⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠐⠀⣼⣀
⠀⠀⢸⣿⣿⣦⡈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣯⠉⠁⠉⠩⢁⡌⠻⢿⣿⠟⢻⡟⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠛⠋
⠀⠀⠘⣍⣿⣿⡅⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠛⠿⠿⣿⣿⣷⣤⣀⡀⠘⠛⠟⠀⠀⠁⢺⣇⠘⣿⣿
⠀⠀⠀⣿⣿⣾⣧⡄⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠿⣉⣷⢯⡄⠙⡿⣿⣻⣄⣀⣀⡀⠀⠀⠀⢿⣏⠄⠈⠋⠐⣀⡀
⠀⠀⠀⢹⣿⣿⣿⡇⢘⢷⡀⠀⠀⠀⠀⠀⠀⣀⣼⣥⣤⣶⣿⡷⠀⣸⣴⣾⡏⠻⣯⠁⠉⠳⣤⣤⡴⠿⣧⢤⡄⠀⣀⣼⡴⣦
⠀⠀⠀⠀⢻⣿⣿⣽⣇⠀⡆⠀⢀⠀⠀⠀⣰⣿⣿⣿⣿⢿⣿⡇⠸⣿⢉⣿⣟⠀⠘⢿⡁⠀⣈⠿⣽⡇⠀⠛⠛⠚⢏⣬⡆⣦⡀⢀
⠀⠀⠀⠀⠀⠙⢳⣾⣿⡀⣠⣖⠁⠀⠀⣸⣿⡿⢿⣿⡣⢾⣿⠃⢠⡟⣸⣿⣟⠂⠀⠀⢹⣿⡿⠛⠚⣶⠀⠀⠀⠀⠀⢻⡷⠛⢷⡀⢇
⠀⠀⠀⠀⠀⠀⠚⢛⢘⣿⡾⠿⢷⠎⡛⠥⣠⡄⠘⢿⣷⡾⠛⠂⣸⣷⣿⣿⣾⠲⢤⠀⠀⢩⣷⣾⣾⡿⢧⣀⠀⠀⠀⢸⠇⠀⢸⢿⣦⣑⡀
⠀⠀⠀⠀⠀⠀⢀⣦⣤⢻⣿⣿⣿⢾⣄⠀⣿⠛⠂⢶⣿⣑⣴⣿⠟⣹⣿⣿⣇⡰⣄⠀⠀⠘⠙⠙⣏⠂⠀⢻⣦⠀⢰⣿⡖⠀⡼⠦⣼⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠸⣮⣿⠄⣿⣿⣷⣮⡛⠀⣿⣿⣀⢨⣿⠟⢋⣿⣿⣿⣿⣟⣾⡿⠀⠀⠀⠀⠀⠀⠨⣷⢶⣡⠼⣧⣞⣿⣟⡏⠀⠀⠙⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠻⣽⣦⠈⣟⣿⣇⣴⣧⣿⣿⣯⡙⢛⣿⣿⣿⣿⣿⣿⣿⣿⣧⣰⡆⡠⠃⠀⠀⠀⣿⣿⣅⣤⡿⣏⣿⡿⢁⣤⢤⣶⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⡿⠾⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣼⣿⣴⣿⣯⣿⣯⡅⠀⠤⢀⡀⣉⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣭⠉⠉⠁⠀⢸⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⡖⠀⠀⠀⢨⣷⡝⣌⠿⢀⣀⠈⠙⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦⠄⠀⠉⢿⣷⢻⣄⣿⡟⠃⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⢀⡇⠈⢿⠀⠀⠀⠈⠉⠉⠙⡟⠉⢿⣿⣿⣿⣧⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢯⠃⠀⢸⡄⠀⠀⠀⠀⠀⠀⣿⠀⠈⢿⡟⣿⠿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣞⠂⢀⡀⢣⠀⠀⠀⠀⠀⠀⣿⠂⠀⠈⢿⡘⣇⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⢸⣷⠈⠃⠀⠀⠀⠀⠀⢹⠃⠐⠀⠈⣿⢻⣧⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⡴⠶⢶⣶⢶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡾⣿⣿⣿⣿⣶⣾⣿⣿⠿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠛⠁⢀⣠⣬⣤⣤⣿⡿⠉⠁⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣟⠀⢀⣫⣥⣤⣶⣶⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠟⠋⣁⣠⠤⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣯⠴⠒⢋⣡⣤⣶⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣶⡶⠟⢛⣯⣷⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣟⣋⣥⣶⣿⠿⠟⢋⣽⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡄⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡝⠟⣿⣿⡿⠿⠿⠇⠀⠀⠀⠀⠀⣠⠶⢛⣽⣽⣿⠇⢄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡜⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠋⠉⠉⠇⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡹⠟⣻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣷⣾⠿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⣴⣦⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣧⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣷⣟⡿⠋⠁⠘⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠲⢶⣄⠀⠀⢴⣏⠭⠛⢶⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⢟⢍⠢⡈⠀⠈⠻⣦⠀⠀⠀⠀⠈⠻⣶⣄⢀⠄⡠⡘⢶⣄⣀⣈⣿⣷⡄⠹⣧⣀⣤⡿⠀⠀⠸⠟⠶⣆
⠀⠀⠀⠀⠀⣴⣿⣿⢿⡙⠨⠂⠑⠁⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠈⠙⢷⣦⣈⣫⣲⣿⣿⣿⠿⠿⣿⣆⣈⠛⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣼⡿⣏⠢⡕⠌⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⢘⡯⣻⣿⣿⣿⠁⠀⠀⠀⣿⣿⣃⡄⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⣰⡿⣕⠮⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣄⠀⠀⠀⢫⣾⣿⡯⠽⣿⠀⠀⠀⠀⣿⣿⡧⠄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢠⣿⠁⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠿⢿⣷⣦⣴⣿⣿⣧⢖⣫⡿⠀⠀⠀⠀⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠄⠉⠛⢿⣿⣿⣿⠿⣇⠀⠀⠀⢀⣠⣽⣿⡿⠀⠀⠀⠀⠀⠀⣼⠃
⠠⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⡿⢋⣡⠴⢿⠀⠀⢠⣾⣿⣿⣿⣿⣷⣤⣄⣀⣀⡼⠃
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⠋⠀⠀⠘⣇⢠⣾⡿⠛⠁⠀⠀⠀⠀⠈⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃

⣀⢀⣠⡀⠀⠀⣀⡀⠐⠂⠀⠄⠀⠚⠀⣀⡀⠀⠀⡈⠛⠙⠉⠃⠀⠀⠀⠀⠀⠁⠀⠀⠢⡀⠁⣠⣀⠀⠁⠘⢀⡀⢀⠑⠈⠉⠀⠀⠀⠀⠁⠀⠘⡟⠄
⠈⠉⠀⢀⣾⡁⣀⠀⠱⠀⠀⠀⡴⠋⠩⣽⣿⣿⣯⡽⣆⠀⠀⠀⢨⠿⠟⠷⢤⠀⠀⠀⠀⠀⢰⠃⠬⠽⣆⠀⠀⠘⠀⠀⡀⠁⠈⠲⠖⠀⠈⠀⠀⣀⠉⠀⡈⣷
⠀⠀⠀⠸⡄⠀⠈⠀⢸⠀⠀⠐⠃⢀⠀⠆⠀⠘⠉⠄⡋⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⡞⠀⠀⠀⣼⠀⠀⢠⠀⠀⣧⠴⠄⠴⠤⣖⠒⠒⠒⢣⠂⠀⠀⠀⢷
⠀⠀⠀⠀⠁⠀⠥⠀⠀⠀⠀⢨⠀⠈⣀⠀⠦⠀⠀⢀⠁⠀⠀⠀⠁⠀⠤⠄⠀⠁⠀⠀⠀⠀⠈⠀⠤⠀⠁⠀⡄⣤⠄⠀⢻⡀⢠⠄⠐⣉⣉⣉⣉⣉⡀⠠⠤⠀⠊
⠀⠀⠀⠀⠀⣅⣀⣠⠀⠀⠀⢈⠀⠀⢨⠂⠀⢺⠀⠀⠀⡆⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀
⠀⠀⡠⢄⠀⠙⠛⠋⠀⠀⠀⠈⠢⠇⡄⠀⠀⠈⣇⣤⠜⠁⢀⠀⠀⠒⠦⠒⠂⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⡤⣀⣀⣢⣄⠀⣀⡀⠐
⠀⠀⡯⠀⠀⠀⠀⠀⠀⠀⡙⠯⠯⡲⡄⠀⠀⠀⠀⠠⡀⣀⡀⠈⠀⠀⠀⠀⠀⠀⠚⠝⢀⣶⠀⠀⠈⠀⠀⢰⠋⣡⠉⡆⠠⠉⠂⠋⢀⠶⠝⠧⡠⣵
⠀⠀⠐⢃⡘⠀⠀⠀⠀⣦⠠⡤⠤⢴⡅⠀⠀⠀⠸⡀⠀⠀⠀⣀⢀⠀⠀⠀⠀⡌⠀⠀⠀⠀⡅⠀⠀⠀⠀⡟⠀⠀⠀⢻⡆⠀⠀⠀⠘⡆⠀⠀⠀⠸⠀⠀⠀⠀⠐
⠀⠠⠀⠈⠀⠠⠀⠀⠠⠐⠄⠴⠄⠐⠀⠀⠀⠀⠐⠳⠀⢂⠀⠊⠀⠀⠀⠀⠀⠁⠠⠔⠤⠀⠃⠀⠀⠀⠀⡇⡀⠀⠀⣸
⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⡐⠀⠀⡀⠀⠀⠀⠀⠀⠀⡖⠀⠐⠀⠀⠀⠀⠀⠀⠀⡄⡀⢀⡄⠀⠀⠀⠀⠀⣇⡗⠀⠘⡯⠀⠀⠀⠀⠀⢰⠁⠀⠈⡀
⡀⠀⣀⣤⣀⣤⠀⣀⡤⠈⠂⠠⠀⠄⠀⢀⠀⠀⠀⠐⠀⠄⠀⠀⢀⠀⠀⡀⠀⠀⠒⠠⠄⠓⢀⡄⠀⢠⡀⠋⠀⢀⠠⠉⠂⠀⠀⠀⠄⠀⠠⠤⠄⠀⠠
⠉⡛⠉⠉⠇⠉⠁⠀⢰⠂⠀⠀⠀⠀⠀⢺⠁⠀⠀⡀⠀⠀⠀⠀⠨⠏⠭⠟⢦⠀⠀⠀⠀⠀⠿⠤⣀⠀⢙⡄⢀⠀⣀⠀⠠⠁⡏⠁⠀⠀⠀⠀⠀⢀⡛⠣
⠀⠁⠀⠀⠏⠉⠀⠋⠸⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠃⢀⠀⠠⠀⠀⠀⠀⢻⠀⠀⠀⠀⠐⡂⠀⡀⠀⠸⠁⠢⡐⠂⠀⠀⠄⠁⠠⡁⡞⢀⠀⠀⣾⠀⠀⠀⠀⡬
⠀⠀⠀⠀⠀⡀⠋⢠⠀⠀⠀⠀⠀⠁⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠘⣀⠸⡄⠀⠄⠀⡸⠈⠤⠀⠀⠈⠤⢾⠆⠠⠁
⠀⠀⠀⠀⢈⠀⠀⠀⠅⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⢰
⡤⣠⠀⠀⠀⠄⠒⠀⠀⠀⠢⢄⣀⠀⠀⠀⠀⠁⠀⣠⠀⠀⢄⠀⠁⠂⠀⠂⠁⠀⡄⡀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠘⠘⣫⡄⠀⠀⠀⢀⡀⢾⡀⠀⠈⠀⠀⠀⠀⠰⠂⠓⠀⠀⠀⡐⠀⠀⠀⠀⢀⣀⠀⢸⡗⠀⠀⠀⠀⠀⢠⡶⠪⠳⣶⡉⠀⠀⠀⠀⢈⠃⠈⠉⣷⠀⠰⠋⠐⢴⡳
⠀⠀⢰⡆⠀⠀⠀⢠⡇⠀⠈⠀⢲⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡄⠀⠀⠀⠀⣺⡿⡄⢤⣿⣄⠀⠀⠀⠀⢿⢔⠀⠐⢿⡇⠀⠆⠆⠆⢰⠂⠀⠈⠀⣿⠀⠀⠀⠘⡇
⠁⠀⠘⠀⠀⠀⠀⠀⢣⠀⠤⠀⣈⠂⠀⠀⠀⠀⢲⣀⠢⠄⣰⠃⠀⠀⠀⠀⡟⡀⠰⠀⢨⠅⠀⠀⠀⠀⣏⠁⠉⠉⠉⢃⠀⠀⠀⠀⠘⠆⢀⣢⠀⠟⠀⠀⠀⠀⠁
⠀⠐⠀⠀⠀⠀⠀⠀⠈⢗⣀⣺⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠇⡘⠀⠀⣧⠃⠀⠀⠀⠀⢻⠆⠘⠁⠀⠻
⠀⠀⠆⠀⠀⠀⠀⠀⠀⠙⠓⠛⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⢀⡀⢀⠠⠤⠤⠤⠼⠠⠀⢀⡄⣀⠓⠃⠀⠀⢀⠋⠀⠀⠀⠀⠀⠚⠢⠔⠚⠀⢀⣀⣀⡤⡄
⡄⠀⠀⠀⠀⠄⢉⡁⠦⠀⠀⠀⠀⢠⠔⠇⠶⢁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⡝⠊⠉⠚⡆⢀⡠⠀⠀⠠⠔⠋⠢⣤⡀⠀⠀⣀⡀⢎⡁⢩⡀
⠉⠀⠀⠀⢪⠀⠀⠀⠀⣤⠀⠀⢲⣎⡉⠀⠀⠁⢁⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡔⠐⠒⢂⠂⠂⡇⠀⠀⠀⡇⠈⡇⠀⠀⢰⣶⠀⠈⣧⡐⠒⠒⠺⣗⣅⠀⠈⠉⠉
⠀⠀⠀⠀⠸⡄⢀⡀⢠⡧⠀⠀⠀⠈⠆⢀⣦⡄⠠⠁⠀⠀⠀⠀⠈⠣⠼⠆⠜⠁⠀⠀⠀⠀⠀⢰⠀⠄⢰⠀⠀⠀⠀⠀⢿⠋⢃⠙⢙⠇⠀⠀⠀⠘⠻⡄⠨⠀⠸
⠀⠀⠀⠀⠠⠸⠆⠠⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠆⠀⢦⠀⠀⠀⠀⠀⠀⢈⠁⠀⢀⠀⠀⠀⠀⠀⡇⡄⠀⠀⡼⠐⠂⠒⠒⠒⠂⠈⠀⠈
⠀⠀⠀⠀⠁⡀⠠⡀⠁⡈⠠⠄⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠉⠒⠒⠈⠀⣠⠀⠀⣀⠐⠀⢀⣤⡀⠐⠀⣀⣀⠀⠉⠋⠀⠀⢛⠉⠀⠀⠁⠈⠉
⠀⠀⡌⠓⣄⠈⠒⣄⡐⠁⠀⠈⠑⣀⡄⠀⠀⠀⠀⠈⠀⠐⣶⣤⡀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠁⠀⠁⡐⠉⠄⠹⢻⠀⠘⠡⠈⠀⠀⢴⠀⠀⠀⠀⡀⣀
⠀⠀⠑⠄⢠⠈⠉⠉⠚⣆⠁⠀⠀⢀⠁⠀⠀⠀⢘⠀⠠⠀⠀⢹⡇⠀⠀⠀⠀⡆⠁⠀⠀⠀⡔⠀⠀⠀⠰⢖⠀⠀⠀⢘⡇⠀⠀⠀⠀⣠⠈⠀⠄⣠⠰⠀⠀⠀⡀
⠀⠀⠀⠰⠈⠀⠀⠀⠁⠛⠄⠲⠂⠋⠈⠈⠀⠀⠈⣤⠀⣤⠀⠸⠁⠀⠀⠀⠀⢠⡀⠂⠂⣐⠇⠀⠀⠀⠐⠘⠀⠠⠀⠠⠡⠀⠀⠀⠀⠇⠀⠄⠀⠘⠈⠈⠉⠀⠉
⠀⠚⡠⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠉⠀⠀⠀⠀⠀⠀⡄⠀⠀⡍⠀⢂⠀⠀⠀⠘⡄⠉⢰⡎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣮⣵⣾⣯⣵⣶⣦⣄⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠤⣄⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠄⠀⠄⠀⠀⢀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠁⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠛⠀⠀⠂⢐⡶⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣀⡀⢀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣧⡀⠀⢀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⢿⣿⣷⣦⡄⠀⠴⠄⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠈⠣⡉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠉⠘⠂⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⡷⠶⣤⡀⠀⠀⠀⠀⠀⣀⣤⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠈⠁⠀⠐⠧⠄⣀⡀⠀⠈⠛⠛⣛⠋⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠈⣀⠋⠈⣩⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⣇
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣆
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⢧⡀
⠀⠀⠀⠀⢸⣇⣀⣀⣀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡹⠆
⠀⠀⠀⠀⠀⣹⠟⠛⠛⠿⢷⣦⣀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⠶⡾⠉
⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⡹⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣿
⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⡀
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠁⣄⠐⠚⠆⠀⠀⠀⠀⠀⠀⠀⢸⠃⠘⣇
⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⠀⠈⠱⠦⣀⠀⠀⠀⠀⠀⢛⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙
⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⠎⠀⠁⠀⠀⠀⢢⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠐⠊⠉⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠈⠁⠉⢹⠀⠀⠀⠀⠀⠒⠒⠒⢶⠚⠃⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠓⠤⢄⠀⠀⠐⠒⡶⢋⠻⠇⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠛⠉⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠂

⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣯⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⠙⣿⠉⡟⠽
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⠋⠙⠢⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⢀⡀
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⡁⠹⡿⠛⠛⠀⠀⠀⠀⠀⠀⠀⡇⣏⣫⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⠗⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰


⠀⠀⠀⠀⡀
⠀⠀⠋⠀⠡

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣷⣀⣁⣈⣜⣀⣀⣡⣀⣔⣤⣀⣳⣶⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿
⣿⣿⣁⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⡠⢄⠰⢦⣰⠀⣎⣇⠈⡏⢠⢂⡠⢄⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢎⡀⠱⡘⠆⠂⠘⠀⠁⠈⠀⠀⠈⠸⡀⡰⢀⠶⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡐⠙⢦⡈⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⠊⡀⣀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣤⠨⠶⠄⠀⠀⠀⠀⠀⠀⠀⣰⣲⣄⣰⣦⣄⡀⠀⠀⠀⠀⠉⡱⢃⡠⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣀⠀⠀⢰⣿⡇⠐⠦⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣘⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠐⠉⠀⠀⢸⣿⡆⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢿⣿⣦⢸⣿⣷⡀⠀⠀⣠⣶⣄⢀⡀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣟⣡⣴⣷⠀⣠⣶⣄⠀⠀⢀⣾⣿⡇⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣦⣴⣿⣿⡏⢸⣷⡄⢰⣄⣼⣿⣿⣿⣿⣿⣿⣿⡿⠷⡀⢹⣿⣿⣦⣴⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⣿⣿⣷⣿⡉⠉⠻⣿⣿⣿⣿⡏⠀⠀⡀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⠛⠁⠀⠈⢿⣿⣿⣿⣿⡆⠀⢠⣿⣿⣿⣷⡄⠐⠀⠀⠈⠛⢿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⢿⡟⠉⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡇⢀⣴⣿⣿⣿⣏⠔⠁⠀⡀⠀⠀⠀⠀⠉⢻⡿⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠤⢞⢉⡆⠀⠀⠈⠙⠻⠿⠾⠿⠿⠟⠋⠁⠀⠀⢀⣘⣆⡀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⣩⣞⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡲⣤⡀⠉⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠈⠣⠈⢀⡔⡿⠀⡠⠤⢀⡄⠄⠀⢖⠉⠑⡄⠻⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠁⠈⠇⢸⠀⠀⠀⡖⠂⠀⠈⢦⠴⠃⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠓⠂⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣤⣀⣀⣤⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉


⡿⢿⠿⠿⡿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⢁⣤⣄⣀⠠⠶⠶⡄⠀⠀⠄⢰⠂⠈⠀⠐⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢤⣾⠋⠉⠓⣄⡀⠀⠀⠀⠑⢄⡀⢰⡀⡀⠈⢠⠒⠒⣤⣄⠰⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢄⠉⠠⠀⠈⣷⣄⠀⠀⠺⠂⠉⠉⠈⠹⢸⠏⠖⠒⠀⣾⠐⠮⠗⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⡉⠀⠀⠄⠀⠙⠷⣦⣤⡄⠀⠀⠰⠊⠉⠉⠙⠤⠒⢳⡈⠀⠈⠷
⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⡀⠀⠙⠻⢄⠀⣄⠀⢀⠠⡤⢬⠤⠶⢾⡦⠤⡲⡄
⠀⠀⠀⠀⠀⠀⠀⣎⡤⠴⢤⡤⠀⠀⠀⠀⠁⠀⠀⡔⠀⠀⠂⢀⣄⠀⠈⢣⣶⣽⣉⠀⢀⡄⣱⠗⠸⢁⡀⢻
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠈⠀⠀⠜⠈⠋⡓⢯⡵⠋⠙⠂⢠⣀⠀⢺
⠀⠀⠀⠀⠀⢠⠞⠛⠳⣶⣤⡄⠀⠀⠀⠀⠄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢨⢁⡞⠧⡈⠓⣄⣤⠄⢉⢠⢯⡆
⠀⠀⠀⣠⠐⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠠⠀⠘⠀⢀⡀⢐⣦⣀⣀⠀⢀⠀⠰⣈⡂⠑⡌⠲⣄⡉⢠⠄⠀⠈⠉
⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠖⠋⠀⢹⣤⢈⡆⠢⢼⠳⣄⣸⠞⠈⠃⠀⢰⠂⡀
⠀⠀⠀⠻⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⣾⡏⣹⠶⠀⠫⣰⡆⠉⡘⢷⡄⢴⠴⠂⠀⠀⠀⠁
⠀⠀⠀⣠⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⡟⠀⠈⠀⠐⠏⠀⡀⠀⢸⠄⠈⢰⡃⠀⠀⠀⠀⠁
⠀⠀⠀⠈⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡀⣰⠋⡾⠀⠀⠀⠀⡀⢀⠶⣄⠀⠀⠀⠀⠛⠲
⠀⠀⠀⠀⠘⣟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢫⢃⡜⠁⠀⠀⠀⢠⠉⠛⠀⠀⠉
⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠱⠤⣄⣀⣀⣀⣀⣀⣠⣄⣀⠀⠀⠀⠀⡰⠀⠐⠀⠀⠀⠀⠔⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠈⣃⡆⢤⣀⡀⠀⠇⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠴⠿⠶⠆⠲⠶⠶⠶⠷⠶⠄⠆

⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠰⠀⠂⠀⠀⠀⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠀⠈⠀⠀⠁⠀⠐⠃⠈⠀⠠⠀⠀⠌
⠀⠄⡀⠀⠀⠀⠩⡀⠄⠀⢀⠌⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⢀⢠⠀⠀⠀⠀⡀⠀⠐⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠄⠂⠃⢀⡀⠘⠒⠠⠀⠀⠀⠠
⠀⠠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠐⠄⠐⠁⢀⠀⠀⠀⠀⠀⠀⡃⠈⢢⢠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠐⠈⠀⠀⠀⠀⠁⠄
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⢀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢀⠈⠀⠀⠀⠐⠠⠐⠂⠄⠀⠀⠀⠀⠀⡀
⠀⢡⠀⠀⠀⠀⠠⠂⠀⠀⠀⢀⠀⠀⠀⢀⡐⠀⠀⢊⡀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⡀⢌⠀⠀⡱⢀⠀⠀⠀⠁⠀⠁⠉⠀⠂⠀⠀⠀⠀⡈
⠀⢁⠌⡄⠃⠀⠐⡀⠀⠀⠀⠄⠀⠐⠀⠀⠐⠠⠀⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⠁⠌⡑⠊⠁⠀⢀⠀⠀⠀⠀⡀⠀⠀⠘⠠⠁⡈
⠀⠈⠀⠀⠀⠐⠀⠐⠂⠐⠆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠀⠀⢊⠀⠁⠀⠀⠀⠀⠁⠊⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠑⠀⠀⡀⠀⠀⠃⠀⠀⠘⠁⠀⠀⠀⠀⠠⠒⠐⠂⠄⠀⠀⠀⠃⠀⡰⢆⡀⠀⠐⠀⠀⠀⠐⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠢⠀⢀⠐
⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⡀⠖⠂⡀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠑⠄⡀⠀⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡐
⠀⠀⡁⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢈
⢀⡀⠁⠀⠀⠀⠊⠀⢄⠀⠀⠐⠀⠑⠄⡀⢀⠀⢀⡀⢀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠢⠡⠀⠀⡄⠰⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠡⣀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠠⢀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⣀⠀⠐⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠤⠀⠀⠀⠀⠂⠈⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠄⠑⡀⠀⠀⠀⠀⠠⠀⠠⡀⠀⠈⠁⠈⢀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠰
⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⠁⠈⢉⡁
⠈⢠⠀⠀⠔⠀⠀⠀⠀⠀⠁⠀⡀⠠⠀⠠⠈⠉⡀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠈⠈⠀⠀⠀⠊⠈⠁⠀⠂⠀⠄⠨⡀⠂⠀⠀⠰⢀⠁⠠⠀⠀⡄
⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠈⠢⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⡀⠀⠀⠀⠀⠀⢅⠀⠀⠀⠀⡈⠀⠀⠀⠈⢀
⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⡀⠐⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⠀⡄⠀⠀⠀⠃⠁
⠀⡠⠁⠀⠀⠀⠀⠠⠀⠀⠄⣀⠀⠀⠄⠀⠡⠀⠠⠄⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠄⠀⠀⠀⠡⠀⠀⠈⠀⠀⠀⠀⢁⠀⠀⠀⠀⠈⠀⠀⠀⠈⢄
⠀⠐⠂⠀⡀⠀⠁⠀⠀⠀⠀⠨⠀⠀⠀⠤⠊⠀⠀⠑⠤⠀⠀⠀⠃⠀⠀⠀⠀⠘⠀⠀⠠⠀⠆⠀⠀⠀⠀⢀⡀⠰⢀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠆
⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢄⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣰⠉⣄⠀⠀⡠⠉⢆
⠀⡀⠀⠀⠀⠀⢤⠆⠀⠀⠰⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⢆⠀⠀⠰⢀⠀⡀⠀⠀⠀⢀⡀⡀⠀⠀⠀⡀⠤⠃⠀⠀⠘⠄⢈⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠜⠣⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠰⠨⡄⠀⠁⢠⠄⠂⠀⠀⠀⢂⠀⠀⣁⢐⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⢀
⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠐⠚⠀⠀⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⡄⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠡
⠀⢊⠀⠀⠀⠀⣄⠀⢀⡀⢀⣠⠀⠀⠀⢀⡕⠀⠀⢊⠀⠀⠀⢠⠀⠔⠑⠀⠀⠀⡀⠀⠀⠈⡅⠀⠀⣨⠀⠀⠀⠀⡂⠈⠀⠠⠂⢤⠀⠀⠀⠀⡑
⠀⠐⠁⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠐⠣⠜⠀⡐⠣⠜⠂⠀⠄⠠⠀⠀⠀⠀⡀⠀⠠⠐⠈⡁⢉⡅⠈⠁⠒⠄⠀⠂⠀⠀⠀⠀⠄⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣝⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣻⣥⣝⢆⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⣹⣇⠀⡉⠛⠻⢷⣤⡄⠈⠣⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠄⠐⠉⢴⠿⠛⠻⢿⠀⡀⠀⠀⠈⠉⠳⢤⡙
⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⣀⣠⣾⣄⠀⠀⠀⠀⠛⢷⣆⡆⣀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠃⠀⠀⠀⠀⠀⠙⢿⡓⠦⣄⡀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣾⣟⠀⠀⠀⠱⣦⣄⠀⠀⠀⠈⠓⠺⢿⣦⣀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢈⡙⡛⠿⠿⣆⣠⡁⠙⢻⠶⣷⣂⡀⠀⢨⠟⢻⣟⠁
⠀⠀⠀⢀⣠⣤⣴⣶⣶⣽⣄⡻⣿⣧⣀⠀⠘⠿⣿⣦⣦⣔⣾⣿⣿⣿⣿⡶⠶⠛⢷⣄
⠀⠀⣸⣿⣿⣿⣿⣟⣿⣿⣿⣿⣷⣿⣿⣿⣴⣤⣌⣏⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠈⠑⠚⠓⠚⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣀⣤⣤⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣠⣀
⣿⣯⣤⣥⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠤⠁⠈⠁⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠈⡇⢠⠀⠀⣀⠀⡇⢈⠀⢈⠀⢁⠐⡂⠀⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠸⡄⢸⠀⠃⡇⡇⢻⡇⡇⡇⡇⢸⢸⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠈⠓⠀⠃⠑⠈⠓⠁⠃⠃⠘⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠠⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠂⠀⠀⠀⠀⠛⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠂⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠘⠆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠠⠊⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⠀⣀⡀
⠀⠀⠀⡤⠀⠀⠀⠀⠀⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⣀⠄⠓⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⣀⠀⠀⠲⠆⣈⠁⠀⠁⡠⠁
⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠊⣀⠤⠀⢂⣀⠩⠓⡄⠀⠀⠼⣀⢤⡀⢇
⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠢⢰⠂⠀⠀⠀⠀⣄⠀⣀⠀⠀⡬⠉⠔⠈⠉⠁⠀⢀⠀⣠⠀⠀⠉⠁⠀⠈⠉
⠀⠀⠀⠀⢀⡔⠀⠉⠀⢆⠀⠀⠀⠀⠀⠠⠀⠀⠀⠋⢲⠔⠘⠂⠀⠀⠀⣴⡀⢁⣏⠀⠀⢣⡠⠀⠤⡰⠍⠀⠑⢒⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠙⠤⡈⣀⣠⣤⠔⠃

⠀⠀⢀⠤⠐⠒⠂⠠⢀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠔⠁⠀⠠⠒⠈⠉⠈⠁⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⢠⢻⠀⠀⠀⠀⠀⣀⡤⡒⠒⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣉⣔⢤⢄⠡⡀
⠀⡀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡆⡘⠀⠀⠀⢀⠎⢯⠤⠞⠀⠘⠿⠑⡄⠀⠀⠀⠀⠀⡠⠂⣨⠃⢠⠢⠗⡢⠁
⠀⠡⡀⠈⢄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⡜⠀⡇⠀⠀⠀⠸⠀⠀⠀⢠⣒⠄⢸⣲⢄⠀⠀⠀⠀⢺⠀⠀⠰⠟⣠⠀⠆⡠⠁
⠀⠀⠈⠂⠀⢢⡀⠀⠋⠀⠀⠰⠂⠀⠀⠀⢠⠀⠀⡰⠀⠸⠀⢀⠀⠀⠸⡼⣂⠀⠀⠀⣀⠀⢁⡎⠀⠀⠀⡠⠁⠀⠀⠀⢀⢀⣄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣼⣡⠃⢠⠇⣴⠏⠀⠀⠀⠙⠢⢄⣐⢣⣴⠷⠋⠀⠀⢀⠌⠀⡀⡤⣀⠔⣿⠂⠈
⠀⢠⢄⡀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠸⠀⠀⠃⠀⠸⢫⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡃⣬⡨⠊
⠀⣠⠀⢀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢰⠏⠃⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢛⡠⠋
⠀⠉⠈⠃⠀⠀⢠⣄⡀⣀⠀⠀⠀⢨⠀⢀⢀⣀⠀⠐⢳⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠠⢜⡀⣁⣏⠀⠀⠀⢸⠋⠀⠀⠀⢣⠀⠎⠀⠀⠀⠀⠀⠀⠀⠰⢀⢠⡀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠤⣄⣴⣖⡉⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⢘⣄⠀⠀⠀⡜⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠈⢳⡤⣣⠞
⠀⠀⠀⠀⠀⠠⠀⡄⠀⠀⠀⠀⠀⠈⠪⢁⠀⡩⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠀⠀⠀⠀⣠⢴⣫⡞⠁
⠀⠀⠀⠀⠀⠴⢠⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡤⠀⠀⠄⡀⠀⠀⠀⠀⠀⢠⣾⠋⠙⣆⣠⢔⡮⠗⠉⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡎⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⢇⠑⢒⡺⠝⠛⠣⠤⠤⠚⠁⠀⠀⡘⠠⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠰⠀⠀⠀⠀⠀⠀⠃⠀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⢀⡀⠤
⠀⠀⠀⠀⠀⠆⠈⢀⠀⠀⠀⠠⢄⡀⠘⠀⠀⠀⠀⠀⢄⠀⢃⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠁⠀⡘
⠀⠀⠀⠀⠀⠃⠰⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠢⢄⡑⢂⣤⡤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠔⠂⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠦⠶⠦⠳⠶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡌⠛⠁⣰⢶⣾⠊⢈⢈⣤⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣧⡀⢛⣉⠉⢨⡁⠛⠚⢺⡏⠉⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣿⠉⠉⠛⠻⢧⠆⣄⠀⡅⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡖⠀⠀⠀⠀⢀⠷⢧⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⢈⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⣠⠞⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⢫⠀⠀⠀⣠⠟⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠉⠛⠲⢤⣀⠀⠀⠀⠀⣰⠏⠀⠀⢂⠈⠓⢲⠎⠀⠀⠀⠈⠑⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⢀⣀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⡾⢫⠀⠀⠄⠈⢀⠀⠠⠁⢀⠀⢈⡀⠉⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠀⣠⣜⣹⠟⡉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠎⠀⠀⠀⠀⠀⠐⢀⠀⠀⡀⠈⠀⠀⠀⢀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⠀⣀⠤⠒⠈⣩⠿⠃⠢⡀⣠⡏⠝⠉⠛⠛⠓⠒⠀⠀⣤⡟⢀⡇⠀⠀⢀⠁⠀⠀⠀⠀⢀⠀⠰⣄⠀⠀⠀⠀⠀⠘⢣⡀
⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⢞⠱⣅⠀⠀⠀⠀⠀⠀⠈⠙⠶⠦⣀⡀⠀⠁⠀⠀⠂⠀⠀⡶⠀⠀⡁⠀⠀⠀⢀⠀⠀⠀⠐⢀⠟⠳⣄⠀⠄⠀⠠⠀⢳
⣀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠹⣆⠉⠓⠶⣀⠄⠈⣀⣼⠃⠀⠀⠀⠄⠂⠀⠀⠄⠀⠈⠀⡜⠀⠀⠈⠳⣄⠀⠀⠄
⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠊⠉⠣⡈⠢⡀⠀⠀⠀⠀⡀⠰⡅⠀⠀⠀⢈⣱⡿⢛⡿⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐
⠀⢸⡟⠻⣦⣄⠀⠀⠀⠀⢀⠀⠀⠀⠳⡀⠀⠙⣆⢹⣦⣠⠖⣉⠤⠚⠁⢀⣤⡶⠛⠁⣠⡞⠁⠉⠀⠀⠈⠐⠀⠈⠀⠀⠤⣴⠀⠀⠀⠀⠀⢀⡟
⠀⢸⡇⠀⠈⠛⢷⣄⡀⠀⢸⠘⡄⠀⠀⠹⣄⠀⠀⣡⠤⠒⠋⠁⣀⣴⠾⠛⠛⠒⠛⠉⠉⠳⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠞⠀⠀⠀⠐⡼
⠀⢸⡇⠀⠀⠀⠀⠉⠻⣦⡀⠁⢱⡄⠀⠀⠈⠲⠋⠀⢀⣠⣴⠟⠋⠁⠀⠀⠀⠀⢀⣤⠖⠀⠀⠙⠲⢦⣄⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣴⠋⠀⠀⠁⣠⠞⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠐⢰⡾⢧⡔⢃⣀⠈⠉⣧⠰⠞⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣠⣴⠾⢻⡇⠀⠀⠀⠀⣈⣢⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠀⠀⢻⠖⠁⠀⡼⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠀⠀⢸⡇⠀⠀⠀⠀⠈⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠒⠋⠁⠀⠀⢀⣀⣴⠛⠦⠤⠚⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣀⠢⣤⡤⠖⢺⣿⠋⠉⠛⠛⠛⠛⣿⠋⠉⣿⡆
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⢻⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡄⠀⣿⣇
⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠸⣧⠀⠀⠀⠀⢀⡿⣆⠀⠀⠀⠀⠀⠈⢧⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡏
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⠁⠹⣆⠀⠀⠀⠀⠀⠘⡇⢸⡿⠀⠀⠀⠀⠀⠀⣿⠃⠀⢸⣷
⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⠀⠀⠹⣆⠀⠀⠀⠀⠀⢻⣼⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣿
⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣿⣀⡀⣀⣀⣼⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⣿⣀⣤⡶⠟⠋⠉⠉⠉⣿⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⣿⠉⠀⠀⢀⣀⣀⣀⣠⡿⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⣿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⢀⣠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠋⠉⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⢀⣀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⢰⡟⠋⠁
⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⣶⡾⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣶⣦⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⢀⠔⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠲⣄⠀⠀⢠⠖⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠲⡄
⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⡞⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣏⠀⠀⠀⠀⠀⠀⢰⠉⠉⠉⠉⠉⠉⠉⠉⠑⢦⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⡜⠀⠈⡄⠀⢀⣀⠀⠀⢀⡀⡀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⡆⠀⢸⠉⠉⠉⠑⡄⠀⠀⡇⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⡰⠁⢀⡀⢀⡀⢠⠮⠐⣻⠀⢧⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣀⣀⣀⡠⠃⠀⢀⠇⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⢰⠁⠀⠎⠇⠁⠀⢀⠠⠀⡄⢀⠀⠈⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⣀⣀⠀⠀⠀⠘⢆⠀⠀⣿
⠀⠀⣿⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠛⠰⠊⠁⠈⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⢀⡀⠀⡇⠰⠞⠀⠀⠀⠉⢆⠀⠀⠈⣇⠀⣿
⠀⠀⣿⠀⠀⢀⠎⠀⡐⠉⠉⠉⢳⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠂⠁⠠⢴⠀⠀⠀⢀⡜⠀⠀⠀⡟⠀⣿
⠀⠀⣿⠀⢸⠁⠀⠀⢱⠀⠀⠀⡔⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠐⡰⢂⡀⠀⣀⣈⠁⠀⠉⠉⠀⢀⡠⠊⠀⠀⣿
⠀⠀⣿⠀⠀⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡄⠠⡃⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠄⢅⡀⠅⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢄⡀⠄⠠⢀⠜⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠐⣀⡨⠀⠈⠀⣈⣀⠀⠀⢀⣈⠀⠁⠀⠘⣀⠁⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⣅⡈⠐⣀⡉⢨⣀⠁⠀⡄⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠸⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠞⠀⠀⠹⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠏
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠸⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢄⢀⣤⣴⣯⣶⣿⣿⣿⣧⡄⠂⣀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠙⣷⣿⣾⢿⣿⣿⣿⣯⣿⣿⣿⣿⣵⣶⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣦⠶⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⣙⢵⣶⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⠿⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⣁⣾⣭⢿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠛⣧⣾⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⡿⣶⠒⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠁⢠⣬⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠟⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣏⡁⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠓⠛⢷⣞⡩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⣿⠏⠓⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⡀⡼⠧⠏⢻⣿⣿⣿⣿⣿⣿⣿⣷⠴⣿⣿⡉⠿⠃⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠛⠚⠋⠿⣿⡯⣉⠗⠙⠾⣁⠙⠖⠂⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠂⢈⠈⠑⠃⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⡠⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣾⣶⣦⣴⡁⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣾⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⣻⣟⣏⠿⣿⣿⣿⡆⠀⠀⠀⠀⢰⣿⣿⣿⣿⣟⠻⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣄⠠⢀⣨⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⢿⠉⣀⡀⢛⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣯⣷⣦⣽⣿⣿⣿⣿⣿⡀⠀⠀⠀⢸⣿⣿⣿⣷⣴⣿⣷⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠈⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠖⠒⠛⠉⠉⠉⠉⠉⠙⠓⠒⠶⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⢀⣠⠤⠴⠒⠚⠋⠉⠉⠉⠉⠛⠒⠲⠦⢤⣀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⣠⠴⠛⠉⠀⠀⠀⠀⠀⠀⢠⠟⣧⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠈⢦⡀
⠀⠀⠀⢀⡼⠁⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣶⡀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠹⣄
⠀⠀⢀⡞⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠈⢧⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠘⣆
⠀⢀⡞⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠈⢧⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠘⣆
⠀⡼⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠘⣆⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠸⡄
⢰⠇⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣁⣀⣀⣀⣀⣀⣸⡆⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢷
⣼⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠉⠉⠉⠉⠉⠉⠉⠹⡄⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣇⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⢋⣀⣙⢦⣤⡄⠀⠀⠀⠀⠀⠀⣤⣤⡞⣁⣀⣀⡈⢳⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀
⢻⠀⠀⠀⢹⠀⠀⠀⢀⣀⣤⡤⠤⢬⢭⢭⣍⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⣩⢭⠭⠤⠤⢤⣤⣀⠀⠀⠀⢸⠁⠀⠀⢸
⠸⡇⠀⠀⠈⣇⢀⡴⠋⠀⢸⠀⠀⠀⠀⠀⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠈⠙⢦⢀⡏⠀⠀⠀⡿
⠀⢻⡀⠀⠀⠸⡞⠀⠀⠀⢸⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠈⣿⠀⠀⠀⣸⠃
⠀⠀⢧⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⢠⠇
⠀⠀⠈⢧⡀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⣠⠏
⠀⠀⠀⠀⠳⣄⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⡴⠃
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠾⠶⠶⠶⠶⠶⠞⠀⠀⠀⠀⠀⠀⠀⡴⠖⠚⠓⣆⠀⠀⠀⠀⠳⠶⠶⠶⠶⢶⣷⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡤⠴⠖⠺⡆⠀⠀⠀⠀⠀⠀⢀⣴⣯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢀⣧⠀⠀⣀⣠⡴⣾⠽⠋
⡀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⡘⠒⠺⣯⣭⣭⣍⣭⡽⢒⡋⣁⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⡀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣀⣤⣤⣿⣿⣿⠇⠀⡀⣫⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⡿⠈⢿⣿⣿⠁
⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⣏⣀⣜⣳⣿⣿⣿⣇⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣿⣿⣿⣇⣀⣈⣻⠏
⠀⠀⠀⠀⠘⠿⣿⠿⠟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣽⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⣤⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤
⣿⣿⣥⣬⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣠⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⡀
⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⢸⡇⠀⠀⢀⣠⣤⣤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⢀⣁⣸⣇⣀⡀⣿⠛⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠈⠉⣽⣏⡉⠁⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⢀⣾⢻⡏⠻⠆⣿⠀⠀⢸⡇⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣠⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⢸⡇⠀⠀⠈⠁⢸⡇⠀⠀⣿⠦⠤⣾⣷⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣆
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠃⠀⠀⠉⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⢻⣷⡀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠈⣻⡟⢀⣴⡿⠛⠛⠛⠛⠛⠛⠛⣿⡇⠀⠀⠀⠀⠀⠀⣰⠃⠀⢿⣧⠀⠀⠀⢰⣿⠀⠈⣿⡆⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣼⡿⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢠⠿⠶⠶⠾⣿⣆⠀⠀⢀⣠⡤⠖⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠸⣿⡄⠀⣾⣏⠀⢀⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠘⠛⠛⠛⠀⠀⠐⠛⠛⠛⠃⠈⠛⠛⠋⠈⠛⠃⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣆⠀⢲⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁




⠀⡁⣈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠄⠀⠤⠤⢀⡀⠀⠀⠀⠀⡀⠄⠂⠀⠉⠉⠉⠁⠀⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⢰⠊⠉⠉⠀⠀⠈⠤⡀⠀⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢠⠁⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠃⠀⡠⠀⠀⠠⢤⠔⠒⠂⡄
⠀⠀⠀⠀⠀⠋⢦⡀⠀⠀⠀⠀⢀⠄⢸⠀⠀⠀⣁⣀⣀⣀⣀⣀⣀⣀⣀⠀⠈⡄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡆⡊⠀⠀⠀⠀⢠⡀⠀⠀⡘
⠀⠀⠀⠀⢸⠀⢀⠀⢨⠃⠩⠌⢘⠀⠘⢄⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⡼⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡓⢄⣠⡀⠀⢸⠉⠲⡼⠆
⠀⠀⠀⠀⠀⡄⢅⠁⠘⡀⠀⢁⣰⠀⠀⡜⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠀⠀⠉⠒⠄⠠⢀⣀⡀⠠⠤⠤⢄⡀⢘⡄⠀⠀⠀⠀⠑⢾⡱⢠
⠀⠀⠀⠀⠀⠈⠺⡀⠀⠈⠹⢙⠃⠀⠘⠒⠒⠊⠁⠉⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⡢⠀⠀⣸⠕⠃
⠀⠀⠀⠀⠀⠀⢀⡏⠐⠂⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⡖⠈⢨⣀
⠀⠀⠀⠀⢀⣔⡁⡈⠒⠒⠊⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠓⠊⠁⠀⠱⡀
⠀⠀⠀⢠⠏⠁⠀⠈⡀⠀⠀⠀⠈⢛⢦⠀⠀⠀⠀⠀⢀⡠⢾⢷⡴⠏⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⡐⠁⠀⢡⠰
⠀⠀⠠⠃⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⠀⠱⡀⠀⢀⠔⠁⠈⢢⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡐⠀⠀⠀⠘⠀⡇
⠀⢠⠃⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡄⠀⠐⠔⠁⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠽⣓⠄⡀⠀⠀⠀⠀⠀⢰⠀⢠⠁⠀⠀⠀⠘⠀⡇
⠀⡌⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠌⡀⠈⠢⣀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡌⠀⡇
⠀⠁⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠓⢄⣀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠈⠀⠐⠰⢤⠁⠀⠀⠀⠰
⢸⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⡀⠀⢁⠀⠀⠠⠁⠀⠀⢰
⠈⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⠆⠔⠁
⣧⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣴⠈⠀⠀⠀⠀⠈⠙⣿⣿⣿⡏⠐⠀⢤⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⡇⠀⠀⠘⠃⢀⣠⣴⣿⣿⡿⠃⣀⠀⠀⠀⠈⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⠉⢻⡿⠛⠉⠛⠿⠿⠿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⣳⣶
⠁⠀⠀⠀⠀⣴⣿⣿⣿⠟⠁⠀⡀⠀⠀⠀⠀⣾⣿⠟⣹⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠸⢠⣶⣶⣄⠀⢀⣀⡀⠈⣿⡟⣿⣿⣿⣿⣿⡅⣤⠀⠀⢤⡎⠁
⠀⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣧⠀⠀⠀⠀⠙⠁⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⡆⠀⢀⠘⠿⠿⠛⠐⣿⣿⣿⠀⡟⠀⣻⣿⣿⣉⣹⣇⣀⣀⣄⡀
⣴⣴⣦⠀⣾⡿⠋⠉⠁⠀⠈⢉⢻⣧⠀⠀⢀⣤⣤⣤⣤⣄⠈⠛⠛⠿⠿⠿⠿⣿⠀⠸⣷⣶⡶⢶⣄⡈⠙⢉⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠞⠁⢀⣀
⣿⣿⣿⣷⣏⣠⣤⣄⠀⢰⣿⣿⡿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠻⣿⣷⣶⣿⣿⡿⠿⢿⣿⣿⣿⠛⠻⢿⣿⣿⣿⣧⣶⣿⣿⣿
⢐⣍⣻⣿⡟⠿⠟⠛⣀⣀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠸⣿⡿⢃⣀⠀⢀⣀⠈⠁⢻⣿⣿⠫⢄
⣿⣿⣷⠿⡇⢀⣀⣴⣟⣛⣷⣶⡞⠁⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠁⠀⠸⡏⣢⣥⣥
⣟⡙⠑⢺⠇⠈⣻⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠐⠶⠀⣠⣤⣶⣿⣿⣿⣶⣾⣿⣿⣿⣷⣦⣄⠀⠀⠀⠤⠀⠀⠀⠀⠀⢻⣿⣿⠟
⣿⠏⣂⣴⣿⣿⣿⣿⣿⣿⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠿⠷⠀⠀⣀⣼⣿⣷⣦⣄⣀⣣⡃⠞
⡤⣵⣿⣿⣿⡿⠿⠿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁⠀⢸⡿⠛⠛⠛⠋⠉⠉⢿⣿⣿⡏⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠲
⣾⣿⡿⠛⠁⠀⠀⠀⠹⠿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠸⣿⣿⢡⣶⣿⣿⣿⣶⣾⣿⠁⠀⠈⠉⠀⠈⣿⣿⣿⣿
⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣆⠀⠀⠓⣠⡀⠀⠐⠛⠀⠀⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣧⠀⠀⠤⣴⣾⣿⣿⣿⣿
⣿⣿⣿⡿⠀⠀⣀⠀⠀⠀⢻⣿⠟⠃⠆⠀⠀⢰⣿⣿⠟⠛⠛⠛⠻⣶⣶⣄⠉⠀⠀⠂⠀⠀⠀⠛⢸⠇⠀⠀⠉⠉⠉⠀⣿⡇⠀⠀⠀⠸⣿⠟⠋⠉⠙
⢹⣿⣿⠀⠀⠀⣁⣀⣤⣴⣿⣿⣿⣶⠀⠀⠀⠸⡿⠁⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠸⢺⠐⠖⠀⠀⠠⠦⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠙⣿⠀⠀⣴⣿⣿⣿⣿⣿⠿⠛⠋⠿⠶⠶⠶⠇⣤⡄⠀⠀⣤⡄⠀⣿⣿⣤⣤⣤⣤⣄⣀⣤⣴⣿⣆⣠⡴⢶⣄⢀⣼⠂⠀⠀⠀⠀⠀⢸⡔⠀⠈⠁
⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⡇⠀⠀⢀⣴⣦⣤⣤⣆⣀⣀⣤⣄⡀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠈⠧⠀⢠
⠀⣠⣴⣤⣭⣿⣿⣿⣿⡟⠃⠙⠁⠀⢠⣤⣤⣤⣼⣿⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠛⠛⠛⠇
⠐⣿⣿⣿⠿⠿⠿⣿⣿⣿⠀⠀⠀⢢⣾⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠐
⠀⠻⠋⠀⠀⠀⠀⠀⠹⣿⣿⡆⠀⣿⣿⠿⠿⠿⢿⣿⣿⣄⣀⣀⠐⠚⠿⠿⢿⡀⠀⠀⠉⣿⣿⣿⣿⣿⠁⠀⡀⠀⠀⠀⠀⠁
⠀⠀⠴⠀⠀⠀⣀⠀⠀⣿⡏⠀⠀⣿⣥⣥⣲⣴⣶⣽⣿⣩⣿⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢻⣿⣿⡿⠀⣠⣴⣶⣶⣶⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⡀
⣀⠀⣀⢀⣀⡀⠈⠀⠐⢻⣿⣿⣿⣿⡿⠿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠃⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣀⣤⣤⣤
⣿⣧⢻⣿⣯⣽⣶⣤⣶⣿⣿⣿⣿⣿⣷⡰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠀⠀⠀⠀⠛⢁⠀⠀⣿⠟⠁⠀⠀⠀⠈⠹⡟⠀⣠⣾⣿⣿⣿⣿
⠈⢿⡇⠙⠛⠿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠉⠉⠉⠉⡟⣠⣶⣿⣿⣿⣷⣄⣀⣀⣴⣄⡀⠃⢠⡤⠀⠀⠠⠦⠀⢰⣴⣿⣿⡟⠁⠉⠉
⣠⠈⠇⠀⠀⠀⠀⠘⠛⠿⢿⣿⣿⣿⣿⣿⣿⡟⠛⠀⠐⠆⠀⠠⡤⢠⣿⣿⣿⠿⠛⠉⠉⠉⢿⣿⣿⣿⣿⣷⣄⣀⣠⣤⣤⣀⣴⣦⣿⣿⣿⡇
⠉⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⢨⣿⣿⣿⣿⣿⣿⡀⠀⠀⢀⡀⠀⣠⣿⣿⡿⠁⠀⠀⠀⠀⠖⠈⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⡿⢶⣿⣿⠟⠁⠛
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣆⠀⣾⡟⢿⣿⣿⣿⣿⣧⣤⣤⣤⣾⣿⣿⣿⣿⣿⡄⠶⠂⠀⢀⣠⣤⣼⣿⣿⣿⣿⣿⣿⠙⠛⠋⠁⠀⠀⢻⣿⣷
⠀⢀⠄⠀⠈⢿⣿⣿⣿⣿⣦⣙⡀⠀⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⣴⣟⣡⣿⣿⣿⣿⣟⡋⠉⠉⠠⣤⡆⣠⣴⣶⣾⣿⣿⣇⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⣶⣶⣿⣿⣿⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⣻⣿⣿⣿⣿⣿⠿⠿⣿⢿⣿⡿⢿⡿⠿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣘⠛⢿⢀⠛⠁⢸⣀⡛⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣛⣣⣾⣸⣤⣇⣸⣛⣋⣼⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠟⠋⢹⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠋⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣭⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣾⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⢀⣿⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⣀⠀⠀⢀⠀⠀⢀⠀⣠⣤⣠⠀⠀⢠⠤⣀⠀⢀⣀⢤⠀⠀⣀⣀⠀⣀⣀⣀⡀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⢹⠀⠉⢢⠀⠀⠀⠀⠀⢸⠀⢠⠋⠀⣇⣀⡀⠀⠀⣧⠤⠚⠀⠺⣄⡀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠘⡦⠊
⠀⠀⠀⠀⢸⠀⠀⡜⠀⠀⠀⠀⠀⠈⣤⠃⠀⢠⠃⠀⠀⠀⢰⠏⠢⡀⠀⠀⢀⡽⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠸⠔⠊⠀⠀⠒⠒⠂⠀⠀⠉⠀⠀⠈⠀⠉⠁⠀⠈⠀⠀⠈⠀⠈⠁⠀⠀⠉⠉⠁⠀⠀⠀⠁⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣄
⠀⠀⢸⣶⣄⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶
⠀⠰⣾⣿⡿⠀⠀⣶⡿⣷⡄⠀⢹⣿⣿⡏⠀⠀⣶⣾⣶⡄⠀⢹⣅⣤⣇⠀⠀⠀⡀⠀⠀⠀⠀⢹⡉⢙⡖⠀⠀⣄⣦⣶⡆
⠀⠀⡝⡩⠛⡄⠀⠈⣷⣞⣳⠀⡰⠿⠛⢲⠀⠀⢻⣿⡏⠀⠀⠸⠒⠒⠚⡄⢻⣿⣿⣇⡀⠀⠀⡼⠿⠯⠽⢄⠀⢹⣿⣿⣧⡆
⠀⢸⡜⢁⠴⡇⠀⠀⡇⠀⠀⠃⠧⠀⠀⠸⡄⠀⢺⢿⡇⠀⠀⢠⠀⠀⠀⡇⠀⢻⣿⠋⠁⠀⢸⠀⠀⠀⠀⠸⠀⠀⣿⣿⣿⣤
⠀⢸⡴⠁⢠⠀⠀⠀⡇⠀⢸⠀⢸⠀⠀⠀⡇⠀⢸⣿⡇⠀⠀⢸⠀⠀⠀⡇⠀⢸⣇⠀⠀⠀⢀⠶⠤⠄⠲⡁⠀⠀⣿⠉⠀⢸
⠀⠸⢀⠔⢹⠀⠀⠀⡇⠀⠘⠀⢸⠀⠀⠀⡵⠀⣼⣿⡇⠀⠀⢸⠀⠀⠀⡇⠀⢸⡿⠀⠀⠀⢸⠀⠀⠀⠀⢱⠀⠀⣿⢰⢸⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠤⣀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠂⠒⠂⡆⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠃⠀⠀⢃⠐⠀⠐⢀⠐⡁⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢢⣀⠗⠒⠒⣥⣅⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⠤⣶⢢⠸⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠌⡆⡀⠀⠄⢨⣒⠘⣀⠔⠀⠒⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠀⠀⢖⠄⠀⠀⠀⠀⠀⠈⠘⢪⣄⡡⠚⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢀⡀⢠⠈⡇⠀⠀⠀⠀⠀⡤⢶⠄⠀⠀⠼⠵⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢀⠀⠃⢀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⢠⢀⣀⡄⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣚⣀⣀⣠⣀⣳⣤⡀⠀⠀⠀⠀⠀⡀⠡⠤⠦⢵⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠂⠀⠒⠸⡅⠀⠀⠀⠀⢀⠄⠀⠀⠤
⠀⠀⠀⠀⠀⠀⡄⠀⠄⠀⠤⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠑⠀⠰⠀⠀⠀⠁⠠⠄⠁⠐⠁
⠀⠀⠀⠀⠀⠈⠀⠀⢄⢀⠄⠸⠂⢀⠀⠂⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠰⠢⠠⠴⠆
⠀⠀⠀⠀⠀⠀⣀⡆⢀⣀⠴⢄⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠤⠤⠬⠤⠤⠤⠬⠦⠠
⠀⠀⠀⠀⠠⠦⠘⠶⠅⠶⠴⠎⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂⠐

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

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

⠀⠀⠀⠀⢀⣠⣶⣶⣄⣀⡀⢠⣄⠰⠀⢠⡠⠿⣻⣿⣶⣶⣶⠄
⠀⠀⠀⡾⠿⢿⣿⣿⣿⣿⣿⣮⣿⣵⡭⢻⡆⠀⣸⣿⣿⡿⠋⠀⠀⠀⠁⠀⠀⠠⠂⡀⣠⣾⣷⣤⣤⣤⣴⣤⣤⣤⣶⣦⡤
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡟⠉⣡⡔⠋⠁⢸⠟⠋⠉⠤⠀⠀⢀⣴⣿⣶⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡿⠃
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣗⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⡄⠀⠻⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠹
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠿⠄⠛⠀⠀⠀⠀⠀⠀⠈⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡀
⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠋⠉⠫⠿⣧⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⢧⢀⡆
⠀⠀⠀⠘⣿⡿⠉⠉⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣤⣤⣤⣼⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠻⢧⣴⠀⠑⠤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣞⢿⣿⣾⠍⠙⢿⣿⡿⠛⣿⣿⡟⠋⢀
⠀⠀⠀⠀⠀⠀⠙⢇⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣛⠁⠀⠀⠀⢿⠁⠀⠈⡟⠷⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠈⠋⠉⠙⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠁⠀⠀⢶⠀⣴⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠌⠈⠀⠀⠙⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠃⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣇⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠃⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⢻⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠤⠦⠴⠶⠾⠿⠷⠦⠶⠾⠿⠿⠶⠶⠶⠦
⠀⡁⣈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣖⢠⠖⠒⢦⢹⡇⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⠤⠘⠦⡤⠞⢸⡇⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠘⠈⠐⠁⠀⠒⠂⠉⣭⠈⠛⠛⠛⠨⠇⠄⠠⠠⠤⠄⠤⠀⠴⠄⠀⠀⠀⠀⠀⠀⢠⠶⠶⢦⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠉⠁⠀⢬⢸⣉⣉⠉⢠⢼⡇⣇⡤⠤⣿⢬⠀⣿⡇⣿⠸⠆⠀⠀⠀⠑⢶⣶⠖⢐⣒⠂⣀⢠
⠀⠀⠐⢲⠂⣖⢳⠄⡗⢲⡠⣖⣲⠀⠀⢉⣉⡙⠈⣛⡃⠋⠙⠃⠛⢙⣄⣿⠇⠙⠀⠀⢈⠀⠀⢠⣶⣶⡶⣞⢹⡇⢸⠼
⠀⠀⢰⠙⠀⠑⡒⢀⠓⠚⠐⠃⠙⢹⣰⡏⠀⣹⠀⢤⡤⠴⠀⣀⠀⡀⠁⠁⠠⠀⡄⠄⡄⡄⠀⢠⡶⠶⡶⣈⢛⢛⣋⠀⠐
⠀⠀⠀⠐⠀⠂⠥⢸⢠⢠⠰⡄⢠⠰⠉⣉⣉⡁⠶⠶⠶⠶⠀⣒⣒⠂⢀⣀⣬⠀⠁⠁⠁⢁⠀⢰⣶⣶⣶⣿⢤⠤⠤
⠀⠀⠀⠀⡂⠀⡀⠘⠈⣈⠀⠁⠈⠀⠸⣁⢀⣹⢀⠀⠀⠨⠤⠭⠭⠠⠾⠷⠷⠨⠅⠀⠀⠡⠠⢄⣤⣤⣶⠉⢸⢨⡅⡄⡄⣿⢠⣠⢠
⠀⠀⠀⠨⠧⠠⡂⠀⢾⣿⢻⣽⢼⣿⠠⠬⠯⠤⠀⠀⠂⠀⢀⡙⠛⠁⠸⣧⡿⠠⠀⠀⠀⠀⠰⠲⠤⢦⡶⠀⠘⠈⠃⠁⠀⠋⠈⠈
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⣀⣀⢠⠶⠶⠦⠈⠀⠃⠀⠈⠉⠀⡒⠒⠾⠷⢈⡃⠉⠉⠉⠁⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠲⢤⠀⠈⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠐⠂⠘⠂⠀⠀⣰⣿⣿⡿⠋⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠏⡀⠀⠀⢹⣿⣀⣤⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⢛⡁⣨⣽⠃⠀⠀⠀⡀⠀⠀⢀⣠⣶⣷⣿⣾⣶⣼⣿⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⠋⠉⠉⣠⣴⣿⣿⣿⣿⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠈⣿⣿⣿⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠂⠈⠁⠙⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⠋⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠆⠀⠀⣠⣿⣾⡷⠶⡎⠉⠁⢹⣿⣿⣿⡟⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⣿⣿⠃⠀⠀⠈⠄⠀⠀⠀⠀⠹⠁⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⡟⠁⠀⠈⠹⡆
⠀⠀⠀⠀⠀⠀⣀⣤⣴⣤⣼⣿⣿⡀⢀⠀⠈⠀⠀⠀⡀⠀⣤⢸⣿⡿⠿⠿⠿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠤⠀⢠⡄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠁⠀⠀⣼⣿⢷⣶⠄⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀
⠀⠀⠀⠀⢸⣿⡟⠛⠿⠿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢻⣿⣦⣤⣤⣤⡌⠀⠀⠀⣀⣀⣀⡀⠀⣼⣿⣿⣧
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⣿⣿⣿⣿⡿⠀⠀⠀⢀⣠⣴⣶⣦⣿⣿⣿⣾⡿⠀⠀⢠⣾⣿⣿⣿⣿⣶⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠘⢿⣦⡄⠀⠀⠙⠉⠀⠀⠀⠀⠀⢠⣿⣿⡿⠛⠋⠻⣿⡉⠉⠀⠀⠀⣿⣿⣿⣿⠛⠻⠿⡟⠋⠉⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠙⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠠⣿⠇⠀⠀⠀⠀⢿⣿⠉⠁⡄⠀⣀⠀⠀⠀⠀⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡇⢀⡀⠀⠆⠀⠀⠀⣿⣿⡏⠙⠀⠐⠂⡿⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⣠⣿⠀⠀⠀⢠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠈⠁⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠲⠀⠀⠖⠀⠀⠀⠀⠀⣿⣿⡿⠀⢠⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠀⠀⠀⠀⠀⠀⠀⠙⠻⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿

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



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



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

⠀⠀⠀⡀⠔⠂⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠒⠤⣀
⠀⠀⠊⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⠀⠀⠀⢀⡀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⡼⢫⠭⠽⠿⠿⠿⠿⠿⠿⠯⠭⠵⠶⠿⠿⠿⠿⠿⠿⠽⠿⠿⠶⠶⢶⣿⣷⣄⠀⠀⠀⠳
⠀⠀⠀⠀⠀⣧⣘⣒⣒⡒⣒⡒⠒⢒⡒⠒⢒⢒⡒⣒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠀⠀⠀⠰
⠀⠐⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠉⠉⠡⡀⠀⠀⠀⠀⠀⢠⣾⣿⠯⠀⠀⠀⡆
⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢁⠆⠀⠀⢀⣴⡿⣿⣅⣃⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠁⠈⠁⠀⠀⠀⡠⠋⡰⠃⠀⢀⣴⡿⠁⠀⠉⠁⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⡠⠊⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠞⠁⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⣠⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⡾⠉
⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢡⣀⣤⣄⣤⣀⣀⣠⣠⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠠⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠵


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠆⠀⠀⠀⣀⠀⠀⠠⣀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣍⣘⠲⢶⣁⣟⢃⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⣇⡄⠀⡀⠸⢀⣞⢉⣀⠀⠐⠀⢨⣦⠉⢿⣦⣿⣀⣻⠀⢿⡄⢠⠴⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣸⣿⡆⠀⠄⠸⣾⣼⣜⣿⣧⠀⠀⠈⢻⠳⠬⢿⣭⠟⠷⠛⠁⡰⠞⣲⡛⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣽⣿⣿⡄⠈⣦⢽⣟⢿⣿⣃⠀⠀⠀⢸⣷⡛⣇⠀⠓⢰⢭⡄⠀⠀⣛⢲⠟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⡘⠁⠺⠓⣽⣿⣦⡈⠓⠈⠙⠛⠛⠀⣀⠴⠛⠛⠛⢯⡛⠷⣶⢨⡀⢀⠀⢛⡤⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠞⣡⠿⠿⠉⢲⠦⠤⠘⠂⠉⠁⠀⠀⠀⠀⠀⠘⠛⠻⣌⠳⡘⡵⢴⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢚⡷⢶⠞⠱⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠳⡇⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠄⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣦⠀⠀⠀⡄⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠰⣶⠀⠀⣠⠎⠀⠀⠀⡤⣴⡒⢲⣄⠷⠀⠀⠉⠂⠀⢠⡖⣻⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠑⠤⣉⡉⠙⠀⠀⠀⠀⠀⠀⠈⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠦⣶⣶⢶⣶⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⠀⠀⠃⠀⠀⠀⡼⠛⡠⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⡃⠠⠇⢈⢷⡀⠀⠀⠀⠀⠀⠀⢢⣄⡀⠀⠀⠀⠀⠀⣀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢤⡔⣻⡿⣝⣶⠄⠘⠀⠈⠠⣤⣀⠀⠀⠀⠀⠻⠿⠟⠛⠛⠛⠛⠛⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⡁⢨⠌⣯⡙⠒⠠⢶⡄⠀⠈⢻⡁⠰⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠿⠛⣧⠙⠻⣥⠴⠞⣡⠂⣀⣀⠀⠀⠁⠉⠉⠛⠻⠦⣤⡀⣀⣀⣤⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠃⠀⢀⠁⠖⠒⠎⠠⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⡀⢣⠆⣤⡀⠿⠀⠢⠄⠈⠱⡄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢲⠆⠁⢠⡏⣰⣶⠦⡴⢤⠦⠀⣤⠀⠀⠃⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣤⡄⢸⡇⠠⠄⠈⠙⠄⣀⣈⠟⢀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢀⣹⠃⢠⡄⠰⠚⢤⠿⣭⡴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠠⠤⣀⣀⡀⠤⠄⠀⢀⣀⡀⠀⣀⡀⠀⣀⠀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⣀⡀⠀⠌⠀⠀⠀⡇⠸⠀⠆⡀⡒⢀⣀⡠⡄⠦⣄⠀⣆⠀⡤⢂⢀⣀⢠⣠⠀⡶⢠⡤
⠀⠀⢀⡐⡒⢸⠅⣸⡯⠭⠍⢂⣥⠭⠍⠁⡇⣿⣸⢿⠀⡇⢸⡟⢻⣿⣿⣧⣼⣿⡇⢺⣭⠀⣋⠐⠁⠀⠠⠤
⠀⠀⡀⢰⠀⢸⡇⢿⣿⡇⢸⣿⣻⠀⡆⠀⢿⢿⡏⣼⠁⡇⢸⣇⣸⣿⣿⣿⣿⣿⡇⢼⣿⢀⠛⡀⡄⣒⣐⠒⡀⠂⠒
⠀⠀⠇⠘⠀⠐⠡⣝⣙⡏⢰⣿⣿⠀⢁⠀⠘⠈⢃⣙⡂⠓⠪⠛⡍⠛⢙⠛⡽⡝⠓⠼⠿⠘⠏⠇⠇⠿⠛⠀⡇⢨
⠀⠀⠀⠂⠒⠂⠀⢻⣿⡇⢰⣿⣛⠐⢾⣷⣾⢷⣿⣾⡿⢾⣿⡾⣷⡇⠸⠿⠻⠿⠿⠿⠿⠟⠘⠛⠓⠛⠓⣀⢁⣀
⠀⠀⣉⣉⣉⢩⢩⠤⠤⠤⢐⡿⢿⢪⠈⢋⣙⣉⡉⢉⡙⡉⠉⣙⢉⣃⡀⠒⠲⠀⠶⠲⠆⠰⠶⠆⠶⠔⠖⠐⢂
⠀⠀⠀⠀⠀⠀⠤⠤⠤⠰⢿⣿⣿⠠⠄⠤⠄⠠⠤⠠⠁⢰⠀⠤⠄⠀⠀⢘⠉⠈⠀⢾⠀⠀⡇⠈⠩⠡⠅⠁
⠀⠀⠀⠀⠀⠀⠀⠌⠤⢘⣶⣷⣦⠀⠈⠉⠉⠀⠂⣀⣀⠸⠀⠤⠉⠈⠁⠬⠩⠉⠁⢈⠠⠀⠃⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠐⠀⠸⠠⠈⠀⠁

⣶⣶⡶⣶⠶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⠀⠂⠀⠄⢀⠀⠀⠀⠀⠠⠄⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢺⣿⠆⠀⠀⣀⠐⠓⠋⠀⢀⣀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⣤⣌⠁⠂⠄⢀⠀⠀⠀⠁⣴⠏⣷⠀⣴⡄⢠⠀⢠⡲⢶⡿⢹⡇⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⡿⠃⠀⠀⢀⠃⠀⡄⠰⣿⣴⠇⣴⣿⡅⠀⣥⠈⢶⣁⣀⣸⠃⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠆⠀⠀⠀⠒⣤⣄⣈⠛⠾⠀⠀⠁⠆⠀⠉⠛⠁⠀⠀⠀⣡
⠀⠀⠀⠀⠀⠀⠒⠤⠀⠀⠀⢾⠆⠀⣼⣤⢄⡀⢀⡜⠿⠁⠀⠈⠑⠚⠀⠀⠘⡀⢀⠠⣤⣴⠈⠉⠀⠀⠀⠀⡀⠤⠄⠐⠒⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠓⠀⠤⠴⠀⠉⠋⢥⣬⠞⠒⠂⡀⠀⠠⣄⠀⠀⠀⠁⠀⠀⠘⠹⠀⠀⠀⡆⠀⠀⠀⠠⣤⠒⠲⡄⢡
⠀⠀⣀⡀⣀⠀⠀⠠⠤⡀⠀⠐⢐⠢⡀⠀⠀⠙⣍⠠⢀⡀⠀⠈⠠⡈⠆⡀⠀⠀⠠⠀⠔⠁⠀⠀⠀⢰⠀⠄⠰⠆⣀⣰⠒⠻⢧⢄⣀
⠀⠀⠁⠀⣠⠀⣄⠀⠀⠁⠘⡀⢠⣄⡄⠀⠀⠀⠈⠑⠀⣌⢢⠀⠀⠈⠢⠈⢠⠂⣴⠁⠀⠀⠀⠀⠀⣠⡆⠂⠐⠒⠉⠙⠀⡀⠐⠚⢤⢉⠉⠑⠆
⠀⠀⠀⠀⢻⣿⣿⣻⡷⢸⣿⠹⠦⢌⡃⢄⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠃⡞⠛⠡⢀⠂⠤⣂⠅⠛⠙⣄⣀⡀⠤⡆⠀⡇⠰⠶⠄⣗⠀⠘
⠀⠀⠀⠀⠘⠿⠿⠿⠇⢸⠀⠀⠀⠀⠈⣂⠷⠠⡄⣠⠜⢡⡀⠀⠀⠀⠀⢀⡄⠃⣀⠀⠀⠉⠴⡷⣄⠀⠀⠀⠀⠀⠰⠀⠈⢀⣷⣾⠋
⠀⠀⠀⠦⠄⠀⠀⠀⠐⠋⠀⠀⠀⢀⠔⣡⡶⠆⠀⡌⢠⣟⠋⡼⣣⣴⣤⣬⡙⢎⡠⠤⣙⡞⣀⢰⠃⣄⡀⠀⠀⠀⠦⠀⣀⠈⠙⠃⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣽⠀⠋⢁⠔⡙⠇⢸⠛⣦⣌⠃⣬⡗⠡⣼⣦⣢⡼⢉⠇⡀⡾⠀⠛⠫⠐⠤⣀⠠⠐⡻⠏⠉⠐⠒⠒
⠀⠀⠀⠀⣀⠴⠲⡄⠀⠀⢀⠰⠷⠀⣤⣡⣤⡄⢇⣈⣤⡙⠛⠋⣣⠘⠋⠳⣦⣤⡤⢶⣧⡤⢋⣴⣧⢀⡀⠈⠑⠶⠆⠊⠀⢀⢀⣀⡀⠀⠠⠶⠴⡆
⠀⡠⠐⠉⠠⠀⠀⠀⢦⣾⠄⠂⠁⠒⢄⠙⢮⣻⡊⢹⠋⢉⠟⡟⢴⠀⠀⢠⢏⠳⣚⡺⠷⣝⢯⡹⠁⢆⡆⠂⠀⠀⠀⠀⡏⠀⢀⠒⢄⠀⢀⡀⠀⡇
⠻⡀⠀⢠⠊⠈⢂⠠⠈⢃⠀⠀⠀⠀⠀⠉⠐⢶⣶⣡⢞⣉⠁⡾⢃⣜⠳⢭⣒⡷⢈⣄⢀⣍⣷⠑⣶⣯⣴⣷⠐⠒⠤⠿⠇⠀⠙⠂⠋⡠⢒⢉⠴⡇
⠀⠡⡀⠀⡂⠐⠀⠐⠀⢀⠃⠀⠀⠀⣀⠀⢨⣽⣯⣇⣉⣻⠟⣰⣿⣿⡷⣦⠙⣿⣾⡟⢿⣿⣾⢀⣿⣿⣭⠉⢋⠁⠀⠀⢠⠀⠀⠀⠸⡐⠉⠋⠀⠇
⠀⠀⠁⡀⠀⣁⠤⠒⠉⠀⠀⠀⠀⢠⠁⠉⠒⠩⠙⠛⠛⠿⠾⢧⣼⣏⣴⣿⣰⣿⣵⣿⣤⡿⠧⠾⠟⠛⠩⠐⠉⠀⠀⠀⠸⡀⠀⠰⠶⠧⠤⠄⠀⠚
⠀⠀⠀⠙⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⡁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⣀⠣⠤⠤⠤⣈⢢⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠂⠉⣑⠂⢄⠠⡂⢤⡠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠑⢾⠀⠀⠾⠁⠀⠀⠀⢠⡀⠙⠃⠀⠀⠀⠀⠀⠀⠸⠉⠀⠀⠀⠀⠉⠺⠦⠀⣿⢀⣖⠁⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠘⣦⠠⠀⠀⢠⡾⠛⡢⣼⡗⠻⠳⡀⠀⠀⠀⠀⠀⠀⣰⠾⡖⣶⣶⡦⣶⡄⠀⠅⡻⢬⣶⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠋⠀⠀⠀⠀⠑⠢⠤⠐⠊⠠⠐⠁⠀⣄⠀⠀⣦⠀⠁⠦⡀⢋⣛⣂⠤⠃⠀⠀⢵⠠⠃⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠁⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣇⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠴⡎⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⡅⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡿⠀⢦⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣀⠀⠀⠀⠀⢀⣰⠧⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠙⠶⠶⠟⠛⠐⠃⠀⠀⠀⠀⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠊⣈⣐⡠⠈⠁⠐⠢⡄⠀⠀⠀⠀⠀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠾⡯⣭⣄⣤⣀⣀⣈⣉⣉⠐⠚⢷⡄⠀⠀⠀⡀⠂⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠰⣀⠙⠳⢬⣭⣡⣽⡽⠟⠛⠛⠋⠀⠀⠀⠀⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⣀⡄⠈⠀⠀⠀⠀⢐⠑⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢆⣀⡀⠀⠀⠀⠀⠀⠀⠉⠘⠙⠋⠉⠀⠀⠀⠀⠀⠀⣀⢾⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡤⠚⠃⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠜⠣⢄⣀⠀⠀⠀⠀⠀⣠⡀⠄⢠⡟⠀⠄⠀⠀⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠊⠌⡙⠛⠓⠰⠴⠴⠤⠞⠋⠀⠌⠀⠀⢀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠘⠀⠐⠀⠃⠚⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠠⠀⠐⠖⠀⠞⠑⠞⠳⠶⠳⠞⠰⠗⠿⠞⠷⠋⠞⡑⠟⠺⠒⠷⡺⢞⠶⢃⠾⡚⢟⠳⠛⠚⠛⡛⠟⠻⠟⠛⠀⠆⠀⠂⠀⠀⠀⠠⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⢸⠀⠀⠰⢿⠿⠿⠿⣿⣧
⠀⠀⠀⠀⠠⠔⠂⢠⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍
⠀⠀⢀⠀⠀⠄⡀⠀⠆⠀⠀⢳⠀⠀⠀⢀⣤⠄⣽⣿⡒⠒⢲⡄⠀⠀⠃⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠤⣀⡀⠀⢄
⠀⠀⠀⠀⠀⠂⠀⠀⠇⠀⠀⢸⠀⠀⢀⡼⠋⠀⠛⠓⠶⣿⣾⣇⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⣙⢦⠈⠆
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠈⣻⡴⠆⠀⠘⠃⠙⢻⣦⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢨⣿⣿⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢤⣿⡀⠀⠤⠄⠀⢈⡼⣿⣿⡀⠈⠈⠀⠀⠀⠀⠀⢄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠛⠟⠁⠈⢂
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢄⠀⠀⠀⠂⣳⣹⣆⠀⠀⠈⠀⠑⢿⣮⡿⠀⠀⠀⠀⢀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠦⠈⠃⠀⠀⠀⡿⣿⠏⠀⠀⠀⠀⢠⣿⣯⠁⠀⠱⠄⠀⠋⠁⠀⠀⠀⢱⣆⠀⠀⠀⠀⠀⣀⣀⠀⠈⠀⠀⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠊⠀⠂⠂⠀⠂⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠙⠧⠀⠀⢀⣄⢀⠀⠀⠀⣀⣠⠎⠟⠀⢀⣀⣔⣛⣛⡛⡳
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡤⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠸⣿⣶⣤⣾⡿⠿⠛⠀⡠⠀⡿⠟⠿⠿⡿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠓⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠈⠉⠛⠋⠀⠀⠀⠀⠀⠀⠴⠀⠐⠓⠂⠈⢛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⢀⠀⠀⠀⣠⣤⣄⣶⣶⣄⠑⠀⠀⠄⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡄⠀⠈⠀⠀⠀⠀⠱⠀⠀⠀⠈⡆⠀⣼⣿⡿⠟⠻⣿⣿⣄⠁⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣻⣻⣿⣟⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠁⢠⡟⠁⠀⠀⠀⠈⠻⣿⣧⠀⠀⠀⠀⠀⠀⢱⠒⠀⠠⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣷⣼⣇⠀⠀⠀⡬⠄⠆⠀⠀⠀⡀⠀⢸⡇⠐⠀⠀⠀⠁⢀⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠉⠋⠉⠉⠉⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠠⠄⠀⣼⣟⢿⣿⡆⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⠀⡀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠩⣿⣇⠀⠀⠘⣯⣿⡂⡿⠟⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀
⠀⠀⠀⢰⡶⣦⣤⣤⣴⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣭⣽⣤⣷⣶⣶⣾⣿⣿⣷⡿⠄⡀
⠀⠀⠀⠈⣿⢠⣿⣻⣦⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠟⠛⠿⡿⣿⣿⣿⣿⣿⣿⣣⡞⡁
⠀⠀⠀⠀⢸⣾⣿⢀⣾⡿⠿⣿⣿⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣴⣿⣿⣿⠿⣿⡿⠋
⠀⠀⠀⠀⠘⣿⢻⣼⣟⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⢃⣴⢋⠔⠁
⠀⠀⠀⠀⠀⣿⣸⡿⣻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠋⣠⢟⠕⠁
⠀⠀⠀⠀⠀⣿⡿⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⣡⠞⡡
⠀⠀⠀⠀⠐⡟⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣳⠟⢁⡼⡣⠊
⠀⠀⠀⠀⠀⡇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⡿⠏⡴⢋
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⣡⠞⡑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢏⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡳⣫⠞⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⢟⣽⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⢻⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣡⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣏⣀⣀⣀⣠⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣯⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣠⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣟⣵⣾⢻⣿⡆
⠀⠀⠀⠀⠀⠰⠿⠿⠷⣿⣿⣿⠿⢿⣿⠿⠿⠿⠟⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠻⠟⠉⠻⢿
⠀⠀⠀⠀⠀⠀⠀⣶⣟⠿⠷⠖⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠑⣾⣿⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠚⢌⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠒⠈⠀⠁⠀⠀⠄⠀⢀⠔⡠⠀⡴⢿⡟⠁
⠀⠀⠀⠀⠀⠀⡄⡒⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢁⠈⣰⠎⡠⠋
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⡀⠄⠤⠠⠤⠀⠒⠊⠁⠀⠀⠀⠠⠈⠀⢁⡴⢃⠜⠁
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠈⠋⠈⠉⠁⢀⣀⣀⠠⠄⠀⠐⠂⠀⠀⠀⠔⠁⢊⡠⠃⣠⠋
⠀⠀⠀⠀⠀⠀⠙⠀⠐⠀⠀⠒⠂⠁⠁⠀⢀⣀⣀⡀⡀⠤⠒⡰⠀⠔⣠⢟⢀⠞⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠒⠂⠊⠁⠀⠀⠀⠀⠀⢀⠀⠠⢲⡴⣏⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠤⠤⠤⠀⠂⠉⠁⠀⢂⠀⢠⡎⣠⠯⠤⠄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠤⠤⠄⠐⠂⠸⠣⣀⡴⣧⠞⢁⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠠⠤⠀⠐⡆⠀⡠⠗⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢆⠀⠀⠤⠤⠴⠟⠡⠀⠀⠀⠀⢀⢀⣀⣀⠀⠀⢰⠖⠒⠚⠻⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⡀⠀⠀⠀⣀⣀⠤⠔⠒⢛⣋⣉⣩⣥⣥⣤⣶⠾⡿⠽⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⣝⣒⣪⡭⠤⠖⢒⠚⠋⠭⢍⣀⣀⣠⣤⣤⣔⣶⣶⡾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⠛⢉⣐⣉⠠⠎⠁⠚⠋⠏⠋⠀⠂⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠞⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠄⠄⠤⠄⠀⠀⠀⠀⠀⠄⠔⠔⠐⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂

⠀⢠⠤⣤⣤⣤⣤⣤⣤⣠⣤⢴⣶⢲⣖⣰⣴⡞⠀⠀⠀⠀⠀⠀⢀⡤⢤⣤⣤⣤⣤⣤⣀⣤⡤⣶⣖⣖⣲⣦⣶⠃
⠀⢏⠻⣾⣿⣿⣿⣯⣿⣿⣻⣿⣿⣿⣿⣿⣽⠇⠀⠀⠀⠀⠀⠀⢸⡸⢿⣿⣟⢯⣯⣿⣿⣟⣿⣽⣿⣿⣿⣭⡟
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩⣿⣿⣮⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣽⣿⣷⣱⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣧⡴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣻⡬⢾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣽⣟⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣯⣿⡻⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣻⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⡟⣯⠟⠁
⠀⠀⠀⠀⠀⢀⣴⣟⣿⣿⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣻⣿⡯⠋
⠀⠀⠀⠀⣀⣿⡿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠾⠿⠿⠿⠷⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⢾⣿⣿⣿⣿⠃
⠀⢀⢼⡬⢿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⣲⣖⢶⣶⣶⠂⠀⠀⠐⠒⠀⠀⠐⠒⠒⠒⠒⠒
⣠⣶⣮⣷⢿⣻⣴⢶⠶⢾⠒⢶⣶⣶⣦⣴⣴⡆⠀⠀⠀⠀⠀⢀⣾⣶⣿⡾⣿⣧⡄⡴⠰⡖⢲⣶⡦⣤⣤⣤⣤
⢿⢿⢻⣷⢿⠥⣾⢛⣶⣿⣷⣧⣩⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⢿⡟⣿⣾⡯⠾⡛⣧⣾⣷⣾⡉⣿⣟⣿⣿⣿
⣶⣷⠿⠿⠷⣷⣿⣿⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣾⣶⣶⣶⣶⣶

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⢠⠃⠀⡀⠀⠀⠀⠀⠀⣩⠀⠰⠴⠀⡀⠀⠀⠀⡜⠀⢀⠐⢠⠄⠀⠀⢀⠂⠀⠀⠐⢀⠀⠀⠀⠀⣰⠀⠠⠭⠬⠎⠀⠀⠀⠀⠀⠀⠶⠶⠉
⠀⡼⠀⠐⠃⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⠁⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⡾⠀⠀⠀⠀⡄⠀⠀⠀⢠⠀⠀⠀⠀⡇⠀⠀⡄⡤
⢰⠇⠀⢠⠐⡄⠀⠀⠀⠀⡏⠀⠀⠹⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⢀⠄⠀⢘⠀⠀⠀⠀⠁⢀⠀⠀⠈⠇⠀⠀⠁⠁⠀⠀⠀⠀⡇
⠾⠳⠖⠀⠘⠄⠐⠂⠀⠀⠖⠀⠀⠠⠖⠃⠀⠀⠀⠀⠒⠤⠀⠴⠊⠀⠀⠠⠀⠀⠀⠤⠖⠁⠀⠀⠀⠦⠱⠖⠲⠒⠒⠀⠀⠀⠕⠤⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⢀⠎⠀⢠⡔⠠⠀⠀⠀⠨⡃⠀⠀⠸⠄⠀⠀⠀⠀⡼⣀⡀⠀⢀⡀⠀⠀⠠⠦⠀⠀⣀⠀⠀⠀⠀⠸⠀⠀⡴⠃
⣼⠀⠀⣠⠀⠀⠀⠀⠀⠘⡇⠀⢀⡄⡄⠀⠀⠀⠀⠉⢑⡇⠀⠊⠀⠀⠀⠀⠁⠉⠀⣿⠀⠀⠀⠀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢡
⠏⠄⠀⠻⠴⠖⠀⡀⠀⢸⠇⠀⠈⢰⡇⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠏⠀⠀⡆⠀⠀⠀⠀⠀⠀⣿
⠈⠂⡀⠀⡀⠀⠔⠁⠀⢘⡴⢠⠄⢸⠣⡠⠀⠀⠀⡼⣀⡀⣀⢀⡀⠀⠀⠀⣼⣀⣀⣠⠔⠀⠀⠀⠍⡀⡀⠑⠔⠂⠀⠀⠀⠀⡿⠤⠄⠀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠁⠀⠈⠉⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠈⠀⠀⠀⠀⠉⠉⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠾⠀⠀⠀⠠⠏⠀⠀⠀⠀⢈⡇⠀⠀⠠⣤⠀⠀⠀⠀⡴⠁⢀⡤⠀⡀⠀⠀⠀⢈⠃⠀⠰⢦⡆⠀⠀⠀⢀⠔⠁⢀⢀⡀⠀⠀⠀⠀⠚⠀⠀⢄⣤
⡤⠀⠀⢠⡀⢀⡄⠀⠀⠀⠘⡄⠀⢆⠀⠱⠀⠀⠀⠐⠂⠀⠀⠀⢫⡀⠀⠀⠀⢠⡆⠀⠀⠀⢀⡤⠀⠀⠎⠀⠀⠀⠱⣰⠀⠀⠀⠀⡇⡀⠀⡀⠀⢀⡄
⠂⡇⠀⠸⠀⠛⡇⠀⠀⠀⢰⡆⠀⠠⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⢳⠀⠘⠀⠁⠀⠀⠀⡄⠀⠀⠀⠂⠀⠀⠀⠀⠀⢁⡇⠀⢬⢆
⢰⣦⣠⠀⠀⠀⡇⠠⠀⠀⢸⣧⠠⠀⠡⡱⠄⠀⠀⠀⠡⢁⠀⠀⣀⡠⠂⠀⠀⢘⡫⣤⡄⠀⠀⠀⠀⠀⠈⠄⠀⠠⢠⣤⣤⢠⡄⠀⢸⣧⠤⠀⢫⡢⢤
⠈⠉⠁⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠈⠁⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠰⠃⠀⢤⣄⢀⠀⠀⠋⣀⡀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠠⡆⠀⠀⠀⡜⠀⠀⠀⢰⠃⠀⠀⠀⡖⠀⠀⢀⡇⠀⠀⠀⢀⡇
⠐⢄⡀⠀⠉⠂⠀⠀⠉⢰⡆⠀⠈⠀⠀⠀⠀⡖⠀⠀⠀⡅⠀⠀⠀⢸⠰⠀⠠⡎⠀⠀⠀⠀⢫⡆⠀⢸⠀⠀⡄⠀⢸
⣠⠀⠀⠲⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⡟⠀⠀⠠⠇⠀⠀⠀⠀⠆⡄⠀⠁⠀⠀⠀⠀⢈⣀⠀⠀⠀⠸⣁
⠡⡀⠀⠀⣠⠄⠀⠀⠀⢰⣤⢀⠄⠀⠀⠀⠀⢁⠀⠀⠀⣠⠔⠀⠀⠀⠸⡔⢤⡤⠀⠀⠀⠀⠀⣽⠆⠀⠀⠀⢉⣂⣠⡄
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠁
⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠠⡠⠀⠀⠎⠀⠀⠀⠀⠠⠑⡀⠀⠴⠁⠀⠀⠀⢸⡤⠀⠤⡤
⠀⠡⠁⠀⠀⠀⠀⠀⠀⠀⠐⠌⢄⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⣠⠃⠀⠠⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⢀⠎
⣠⠇⠤⠀⠡⠁⠀⠄⠀⠀⠀⠀⣴⠄⠄⠀⠀⠀⠠⠂⠀⠀⠀⠀⢀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⣀⣀⣀⣦⣀⡠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⣒⡒⠋⠋⣉⣉⣙⣚⠙⣓⣚⠍⣉⣉⣡⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⣿⣿⣿⣿⣿⡗⢸⠟⡛⣟⢘⣛⣫⡵⠾⣿⠟⣛⣿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣤⣿⡇⢼⢸⡇⣿⢸⣿⣿⡄⠭⢾⣇⡿⣿⣦⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢝⣿⣿⣃⣾⣿⣷⣿⠸⡿⢢⣿⣿⣿⣿⡇⣿⣿⣿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢺⣿⣿⡟⡻⣿⠆⢿⠳⣾⣿⣄⠻⢿⠼⡇⣿⣿⠛⢾⣿⡏⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠙⣿⣿⣯⣧⣿⣤⣾⣔⣻⣼⣷⣼⣴⣶⢇⣿⣿⠷⠞⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣝⣻⣿⡿⠿⠿⠿⠛⠛⠻⠿⠿⢿⣿⣿⣿⣟⠷⢾⣿⡗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⠋⠉⠀⠀⠒⠀⠀⠀⠀⠀⡀⠀⠀⢀⠈⠙⠛⢷⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⠀⠀⠀⠀⠀⠀⡅⠀⠃⠀⠓⠃⠀⠀⡄⠀⠀⣀⣼⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠠⠈⠀⣀⣀⣀⣀⣀⣀⠀⠀⠐⠀⠀⣾⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣅⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⣿⣿⡟⢿⣿⣿⣿⠏⣄⢻⣿⣿⣿⣿⣿⣭⣟⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⠉⣾⡿⣛⣿⡉⡇⢘⡧⢻⢹⣿⡿⣿⣿⡇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣾⣿⢸⣿⣿⠟⣰⣜⢃⣌⡘⢛⡠⣿⠟⣱⡟⣛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣮⣭⣵⣾⣿⣿⡟⣭⣭⣭⢡⣷⣿⣏⣠⣀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠛⠻⠿⠿⠿⠿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⡿⣿⠿⡿⣿⡿⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣧⣠⣇⣹⣥⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⣙⡟⢻⠋⢹⡟⠉⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⠛⠉⠛⠉⠉⠀⠀⣀⣤⣶⣶⣾⣿⣷⣶⣶⣤⣀⠘⢿⣦⣾⣷⣾⣶⣶⣷⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⡟⢻⠟⢻⡟⣿⡟⢻⡿⠛⢷⡀⠀⠉⠉⠉⠙⢻⣿⠿⠋
⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣶⣤⣀⠀⠀⠀⠘⢿⣇⣄⣾⣠⣸⣇⣹⣇⣘⣧⣀⣼⠇⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⢀⣾⠟⢿⠟⣿⠿⣿⢿⡿⠛⠛⣧⠀⠀⠀⠀⠉⠛⠿⠿⠿⣿⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠸⣟⠂⣿⣀⣸⠘⢏⠘⢸⡇⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⢰⡏⠿⠙⢋⠹⠏⠋⠛⠟⡋⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⣾⣿⣿⣾⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠋⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⡄⠀⠀⠀⢀⣤⣶⣶⣤⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠰⠀⠀⠀⠀⢸⡿⠋⡉⢍⢝⢿⡇⠀⠀⣴⣿⣿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢿⠇⠀⠐⣿⣿⠁⣀⠀⠀⠋
⠀⠀⢀⣶⣷⣾⣿⣷⣴⣆⠀⠀⠀⠀⠀⠀⢳⡑⣾⡷⠀⡜⣠⣴⣶⣿⣿⣄⠉⢀⣴⠖⣿⡆
⠀⣰⣿⣿⣿⣿⡿⠛⠛⣿⣶⣿⣶⣿⣷⣆⠀⢹⠶⠀⡎⢰⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⣀
⢼⣿⣿⣿⣿⡏⣤⡠⢀⣽⣿⠋⠉⠈⠉⠙⣶⣾⣔⣐⣹⡾⣿⣿⣿⣿⣿⣟⠉⠙⠛⠋⠁⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣆
⣼⣿⣿⣿⣿⣄⠀⣠⡀⣿⡇⠤⠀⠀⣀⠀⢸⡟⠛⠛⠋⠐⡟⣩⣁⣠⣭⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠉⠀⢘⣿⡿
⠈⣿⣿⣿⣿⡟⠀⣠⣾⣿⣷⣄⣠⣄⠀⠀⠘⠁⠀⠀⠀⠀⠈⠛⠋⠉⠋⣴⡀⠀⠀⠀⠀⠀⠈⠁⠀⢹⣿⡇⠂⠀⠐⠂⠈⢹⠇
⠀⠹⣿⡿⠟⠃⠀⠛⠋⠉⠉⠻⣷⣾⣿⡿⠁⠀⠀⠀⠀⠘⠒⠂⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣄
⠀⠀⢠⣿⣷⣶⣴⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣻⣿⣷⠀⢀⣤⣄⣠⣤⣾⣿⣿⣿⠿⠿
⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠈⠛⠋⠁⠀⢸⣿⣿⣿⣿⣿⡟⠁
⠀⢸⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿

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



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

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


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



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



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠖⠚⠛⠛⠛⠓⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣶⣿⡿⠁⠀⠀⢹⣿⣶⡄⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠉⣿⣿⣄⣀⣠⣾⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠻⠟⢻⣿⡿⠛⠿⢁⣄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⣶⡟⠛⢿⡖⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣦⣠⣼⢧⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⢀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠶⠶⠖⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠛⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠉⣉⣉⡉⠛⠓⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢉⣠⠴⠂⠀⠀⠀⠀⠀⢀⣴⠛⣧⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢁⡴⠋⠁⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠘⣆⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⡴⠋⠀⠀⠀⠀⠀⣠⡴⠞⠋⠀⢀⣀⠀⠀⠘⣦⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⣀⣠⡴⠖⠋⠁⠀⠀⠀⠀⠀⠉⠉⠛⠲⠎⢳⣄⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠴⠶⠚⠋⠉⡙⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⠷⣄⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡏⠀⠀⠀⣀⡻⣤⢷⣶⣤⣀⠀⠀⠀⠀⠀⢀⣤⣶⡾⣤⢟⣀⠀⠀⠈⠙⢿⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⢸⠃⠀⠀⠀⠈⢻⠃⠸⢿⡿⠹⡆⠀⠀⠀⢠⡏⠻⣿⠇⠈⣿⠀⠀⠀⠀⠀⠈⡇⠀⠈⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡴⢻⡟⢿⠀⠀⠀⠀⠀⠸⣆⠀⠀⢀⣸⡇⠀⠀⠀⠈⢧⡀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⡿⢻⡟⢦⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠈⠷⣼⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⢹⡇⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⣯⠼⠁⠀⣸⠇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⡶⣶⠿⣶⠶⠶⣶⡶⢶⡶⠶⣶

⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⢀⡠⠒⢄⠀⠀⠀⢄⡤⠒⠒⢤⣠⠄⠀⠀⣠⠀⠐⢦⣄⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢀⠤⠤⢄⡀
⠀⢸⠀⠈⢻⣧⠀⠀⠀⠀⠀⢸⡟⣦⡜⠀⠀⢰⡟⠀⠀⠀⠈⠉⠀⠀⠀⣿⠀⠀⠀⢹⡆⠀⠀⣿⣀⡀⣄⠀⠀⠀⣿⠀⠀⠈
⠀⠘⣿⠗⢠⣿⣧⠀⠀⠀⠀⢸⡧⠋⢹⡆⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⡇⠀⠀⣿⠀⠀⠁⠀⠀⠀⣾⠒⠲⠄
⠀⠀⠈⠀⣼⠘⣿⡆⠀⠀⠀⠘⠇⠀⠘⠁⠀⠀⠛⠳⠤⠤⠔⠀⠂⠀⠀⠿⠆⠤⠞⠋⠀⠀⠀⠟⠀⠠⠴⠀⠀⠀⠿⠄
⠀⠀⠀⢰⠇⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀
⠀⠀⠀⡾⠶⠒⠒⢻⣧⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⡆⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⢀⡇⠀⠀⣏⠀⠀⠀⠀⠀⠀⡿
⠀⠀⣸⠃⠀⠀⠀⠈⣿⣆⠀⠀⠻⣄⠀⢈⡇⠀⠀⣟⠰⠛⠚⡇⠀⠀⠀⡇⠀⠀⠀⠀⢀⠇⠀⠀⣿⠀⠑⡄⠀⠀⠀⣄⠀⠈⡀
⠀⣰⣯⠀⠀⠀⠀⠀⢸⣹⣆⡀⠀⠰⠛⠀⠁⠀⠠⠿⠀⠀⠀⠳⠀⠀⠀⠴⠀⠀⠘⠇⠁⠀⠀⠀⠿⠀⠀⠙⠀⠀⠀⠃⠐⠒⠁
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⡄⠀⠀⠀⠒⢲⡀⠀⠀⠀⠀⠀⣿⡁⠐⢺⣟⠀⠀⠀⢀⠔⠁⠀⠈⠑⢦⡁⠀⠀⢹⡿⠀⠉⣷⠀⠀⣴⠋⠀⠀⠀⠙⢦⡀
⠀⠘⠁⠀⠀⠀⠀⠘⣾⡀⠀⠀⠀⠀⣿⢷⡀⢸⡇⠀⠀⠀⢺⠀⠀⠀⠀⠀⠘⡷⠀⠀⢸⣷⠲⠤⠛⠀⢸⣯⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⡄⠀⢀⠀⠀⠘⠧⠀⠀⠀⠀⣿⠀⠻⣼⣷⠀⠀⠀⠹⡄⠀⠀⠀⠀⣸⠃⠀⠀⢸⣇⠀⠀⡠⠀⠀⠻⢄⠀⠀⢀⣀⡾⠁
⠀⠐⠒⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠈⠉⠀⠀⠀⠀⠀⠑⠒⠀⠈⠀⠀⠀⠀⠉⠉⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠈⠉
⠀⢠⣤⡤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⠀⠀⠀⠀⠠⢤⣄⠀⠀⠀⢤⡄⠀⠀⢤⡄⠀⠀⠀⢠⡤
⠀⠀⣿⡇⠀⢸⣿⠆⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠈⢉⣴⠋⠁⡀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⢸⡃⠀⠀⠘⣷⡀⠀⠀⡜
⠀⠀⣿⡧⢤⡞⠋⠀⠀⠀⠀⠀⠀⠀⠳⢗⣆⠀⠀⠀⠀⣿⡃⠀⣞⠿⠤⡀⠀⠀⣸⡇⠀⠀⠀⠀⣸⡇⠀⠀⠀⠹⣧⠀⡰
⠀⠀⣿⡇⠀⠙⣂⡄⠀⣤⠀⠀⢀⡀⠀⠀⣹⠀⠀⠀⠀⢻⣇⠀⠈⠉⢀⡄⠀⠀⢹⣧⡀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢿⡦⠁
⠀⠐⠓⠃⠀⠀⠈⠙⠒⠛⠀⠀⠘⠙⠒⠒⠋⠀⠀⠀⠀⠀⠉⠛⠒⠒⠋⠀⠀⠀⠀⠉⠛⠛⠋⠀⠛⠋⠀⠀⠀⠀⠀⠁
⠀⠀⣠⣀⣀⣀⣠⣄⢀⣀⠀⢠⡄⢠⠀⠀⢤⡤⡄⠀⠀⠀⣠⣤⠀⠀⠀⠀⠤⣄⠀⠀⠀⠀⢤⢀⠀⠀⣄⠀⠠⣤⣀⣀⣀⣤⡄
⠀⠈⠃⢉⡤⠀⠁⠀⡽⠁⠀⠀⣧⣝⠀⠀⠀⠹⣷⣄⠀⢀⡈⠀⠀⠀⠀⢸⠇⢸⣷⠀⠀⠀⢸⣿⠀⠀⠀⠀⠄⠀⠀⢀⣾⠟
⠀⠀⠀⣾⠃⠀⠀⣴⡇⠀⠀⠀⢸⣯⠀⠀⠀⠀⠙⢿⢢⡏⠀⠀⠀⠀⠀⠁⠀⢸⣻⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣞⠏
⠀⠀⢸⣿⡄⠀⠀⣿⡇⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⣠⢫⣵⣄⠀⠀⠀⠀⠀⠀⠈⢿⣇⡀⢀⣼⣿⠀⠀⠀⠀⠀⢀⣭⠂
⠀⠀⠀⢻⣧⣄⠀⢹⣷⣄⣀⣀⣼⣿⠀⠀⢠⡤⢰⣃⠀⠙⣿⣦⡀⠀⠀⣀⡀⠀⠀⠉⠉⠉⢸⢿⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠉⠉⠉⠀⠉⠉⠁⠉⠉⠁⠀⠀⠘⠛⠃⠀⠀⠘⠒⠛⠀⠀⠛⠃⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠐⠖⠓⠛⠛⠲⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠆⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⣿⠛⠙⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⢹⣽⡏⡉
⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⢰⣶⠶⠶⢶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢀⠄
⠀⢸⣿⡶⠀⣼⣿⡇⢠⢠⡀⢠⣄⡤⣄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠋⠙⠂
⠀⢸⣿⣀⣈⣹⣿⡇⠈⠈⠁⠈⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⡄⠀⠘⠀⠀⠀⠀⠀⠈⠢
⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡂⠀⠀⠀⠀⠶⠶⠶⠴⠆⠀⠀⠐⠖⠒⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠒⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠿⠇
⠀⠠⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣽⠄⢷⣿⣏⡁
⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠁⣆⠀⠀⢸⣛⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⠀⠀⠀⠀⠀⠁⢀⣨⣤⣤⡈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠛⠛⠛⢳⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡸
⠀⠀⠀⠀⢀⠀⡀⢀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠠⠀⠅⠨⠀⠅⠠⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⡠⠐⠒⠀⠀⠘⠓⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠤⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠈⠢⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠛⠚⠀⠀⠀⠀⠀⠀⠘⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢆⣰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣵⣟⡓⠒⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠐⠂⢁⠜⣡⡾⠍⠉⠉⠑⠀⠀⠉⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⠛⠛⣿⠟⢿⣿⠿⡿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢉⡠⠠⠐⠂⢉⠴⣡⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣀⠀⠀⠀⠀⠀⣀⠀⠀⣿⣿⣄⢀⢠⠀⠀⢩⠀⢘⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣠⠞⠁⠀⠀⢀⡴⣢⡾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⣤⣼⣿⠉⠁⠀⠈⢹⡞⠀⠀⠠⠀⡆⣞⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⡼⠁⠀⠀⣤⡴⢋⣴⣯⣷⣇⠂⠐⠀⢀⡄⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢧⣀⠀⢃⡄⣿⣼⣿⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣞⡵⠋⠀⠀⢀⡼⢋⣴⣿⣿⣿⣿⣿⠿⡿⣶⣿⣿⣴⣏⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠇⠀⠎⡇⣏⢹⣿⡄
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⢠⣤⠞⣵⣿⡿⠿⣏⣧⣾⣯⣴⣵⣩⠥⣿⣿⣻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⣤⡷⣿⢸⣿⢧
⠀⠀⠀⠀⠀⢠⡏⠀⢀⣠⣾⢞⣵⠞⣛⠛⠿⠛⡛⢿⡯⢍⣙⡛⠿⢾⣿⣿⣿⣇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⣧⠀⣯⢱⡟⣿⢿⢸
⠀⠀⠀⠀⢠⣏⠀⠀⢀⡿⢫⣿⠁⠀⠀⣹⠁⢿⠋⠁⠈⠂⠉⠉⠉⠉⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡠⢤⠀⠀⢸⣿⣦⣽⡈⣧⣿⣾⡇
⠀⠀⠀⣠⠟⡘⠀⢀⣾⣵⠏⠸⣦⣖⡒⡛⠀⢟⣆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠣⠜⠀⠀⠸⣸⡿⣿⣇⣿⣿⡿⠅
⠀⠀⠀⢧⣊⣤⣶⡿⣿⣿⠁⠀⠙⢿⡙⠛⣿⣿⣿⣾⡅⠀⠒⠓⠒⠲⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡴⢤⣦⣼⠀⠀⠀⠀⢠⣟⣿⣿⣾⣿⣷⡿⣳
⠀⠀⠀⣠⠟⣩⣎⣴⡿⠃⠀⠀⠀⠀⠉⠳⣬⣉⣿⣿⣷⣶⣶⣶⣴⣶⣶⣦⣽⣿⣿⣿⣮⣶⣶⣶⣾⣿⣿⣦⠤⡄⣄⠉⣉⣙⠛⣛⠿⢾⣾⣿⣽⣿⣯⣿⠾⠋
⠀⢀⣴⣷⠞⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⠿⣿⢿⡿⠧⠬⠉⠭⠿⠽⠏⠛⠛⣿⠛⠛⠛⠿⠉⠉
⢀⣾⠟⠁

⣿⣿⣿⡿⠛⠋⠀⠙⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⡟⡠⠄⠀⠀⠤⡌⢿⣿⣿⣿⡀⠰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⣾⣿⣦⣄⡀⠀⠀⠀⣰⣶⣾⣿⣶⡆⠀⠀⠀⠀⠀⠀⠀⣠
⣿⣿⡇⠀⠀⠀⠀⠈⠀⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⣿⣿⣿⣷⣶⣤⣶⣼⣿⣿⣿⣿⣷⣴⣶⣶⣴⣄⡀⢠⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⣿⣿⣿⡿⠿⢿⣿⣿⡟⣻⣿⢿⢿⡿⣛⠿⠛⣿⣿⣿
⣿⣿⣿⣦⠀⠀⠀⠀⣴⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⠋⠀⠀⠀⠀⠹⣿⡏⢹⢁⣤⡄⠀⠀⣤⣄⠹⣿⣿
⢿⣿⣿⡿⠀⠀⠀⠀⢹⣿⣿⡿⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠐⠀⠀⠀⠐⠀⠀⠙⠿
⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠉⠙⠛⠋⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠇⣀⣀⠀⠀⡠⠤⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣠⣶⢗⣀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠉⠐⠂⠀⠀⠒⠊⢹⣿⡄
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠁⠀⠀⠈⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢶⡶
⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⠿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⣿⢋⣀⡀⠈⣩⣍⢻⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣶⠀⠀⠀⠀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿
⡟⠀⠀⠀⠀⠀⠀⠘⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⣿⠀⠀⡀⠀⠀⠀⠀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐



⠀⠀⠀⣠⣤⣤⣤⣄⣀

⠀⠀⠀⠀⠀⠀⢀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⡆⣤⣠⡼⠀⠀⠠
⠀⠀⠀⠀⠠⠀⡀⠀⠁⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠉⠀⠃
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠜⠀⣀⢤⠀⠀⢰⣀⠀⣤⢠⢦⠷⠀⠱⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠾⡾⡄⠘⠏⠛⠛⠋⠉⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⣀⡀⠀⢠⠐
⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⣼⣠⡶⢯⣧⣿⠞⠃
⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠋⠉⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠐⡀⠀⠀⠀⠀⠀⠄⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⣄⢠⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⣀⠀⢀⠀⠀⠀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣿⢾⢨⡜⡇
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣣⢴⠿⣴⡧⠶⠶⠃⠀⠀⠀⠀⠀⠀⠠⠀⠀⠙⠘⠉⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠂⣀⠀⡄
⣤⢸⡇⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⢸⠉⣧⣿
⠞⠆⢥⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠏⠻
⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠠⠐⠂⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡄⢀⣠⡀⢀⢠
⠀⠀⠀⠀⢠⠖⠀⠀⣀⠀⣀⢰⡆⠀⠀⠀⠀⠀⢨⠀⠐⠛⠛⠛⠙⠛⠛⠋⠀⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢺⣠⢰⣶⣿⡴⣿⠏⡃⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡆
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣆⢠⣤⣤⣦⢱⣰⣳⣧⣯⢻⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠾⠛⠚⠙⠈⢸⠏⠁⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡀⠀⢠⠀⢀⢸⡄⣇
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⣁⣾⡞⡾⡼⠌⠧⠈⠀⠄
⠀⣆⡆⣀⣀⡆⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠈⠈
⠀⠏⠷⠯⠻⣷⣿⠘⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⢎⡴⣶⠸⠼⠛⠛

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⢸⡏⢡⢿⣇⣿⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⣀⠀⢁⣿⢳⢼⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠀⠈⠀⢰⡏⢸⢸⠣⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡶⠈⠉⣷⣧⠞⢠⡿⣄⠀⠀⠀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠰⠶⠶⠶⠶⣶⣶⠆⠀⠀⠈⠉⠉⠀⠀⠐⢋⣤⠶⠛⣰⡇⣸⠉⣇⠹⣇⠘⣏⢰⣄⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠠⠚⠙⠦⢻⡄⢻⡄⡟⢠⡇⠸⢉⡄⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢀⡀⣠⣤⣤⣄⣸⠇⣸⠇⠀⠋⠀⠖⠛⡇⠀⠀⠀⢸
⣿⣿⠀⠀⠀⢠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣏⣁⣌⡙⠟⠾⠋⢚⣶⠶⠶⠾⠇⡃⠀⠀⠀⢸
⣿⣿⠀⠀⢰⣿⣯⣤⣤⣤⣤⣤⣤⣤⡄⠀⣴⣿⣥⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⠋⢩⠬⠁⠀⠀⠀⠙⢋⣡⣤⡶⠢⣷⡄⠀⠀⢸
⣿⣿⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⡄⠀⠀⠀⠀⠀⠐⠷⠃⠰⠶⢿⡇⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠰⠦⠀⠀⠀⠀⢀⡐⠟⠀⠰⠶⠈⠁⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⠀⣶⣤⠀⠀⠀⠀⠿⠟⠃⢴⣷⡆⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣌⡠⡄⠀⠟⢰⠖⠂⣤⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⡀⢷⠀⠒⠊⠤⣀⠃⠀⢀⡽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡔⠀⠀⠀⠀⠀⠀⠀⠐⢀⠈⢡⠞⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⣧⠀⠀⠐⠀⠀⠀⠐⣄⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠠⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠔⢠⠉⠀⠀⠀⡄⠀⠊⢡⠀⠀⠀⠀⠀⠀⢣⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠄⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠙⠓⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⡀⢀⠀⡀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⠈⠁⠀⠀⠁⠀⠀⠁⠀⠀⠁⡄⠀⠀⠀⠐⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠟⠁⠐⠀⠀⠚⢀⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢱⠂⠀⠐⠀⠀⠒⠁⡿⢻⠃⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠂⢆⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⡀⡄⠀⠀⠀⠀⠀⠀⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢂⠀⠀⠀⠀⡐⠂⠒⠈⠉⢡⡀⠐⢄⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠊⠰⡀⠀⡰⠀⠀⠀⠀⠀⠀⠐⢦⠈⠑⢄⠀⠀⠸⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠓⢀⠀⢘⠀⠀⠗⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⠀⠀⢠⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠀⢸⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⠀⡄⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⠀⠀⢰⠀⢠⠀⠘⠁⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡀⢀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⢀⠜⠃⢤⡀⠀⢀⠌⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡉⠀⠰⠃⠐⠆⠀⠀⠀⠀⠀⠘⠀⠀⠊⠀⠀⠈⠈⢁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡏⠉⣠⣌⠈⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣅⠸⢿⡿⠀⣼⠿⠿⣿⠟⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠻⠴⣄⣰⡶⠿⠀⢀⣀⣀⠀⠛⠛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⢠⣶⡄⠙⣿⣄⠀⢰⣿⣿⣿⣷⡀⢰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⠈⡛⢁⣨⣿⡏⠀⠘⢿⣿⣿⠟⠀⠀⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡆⠀⢀⣀⠀⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠟⠛⠛⠛⠋⠉

⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿