Enjoyment Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣲⣶⣦⣤⣶⣦⣠⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠄⠀⠀⣸⣿⣿⣿⠟⠿⠿⢿⣿⣿⣿⠿⢷⠀⠀⠘⢻⣿⣿⣿⣿⡿⣿⡿⠿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠀⠀⠈⠙⠛⠋⠈⠛⣿⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢸⡏⠛⠁⠋⠀⠉⠀⠀⠈⠙⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢀⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⠃
⠀⠀⠀⠀⠀⢰⠷⡞⠛⠀⠀⠀⡀⠀⠀⠀⢀⡄⠀⠂⠀⠀⠀⢇⡟⠀⠀⠀⠀⠉⠀⢀⠀⡀⠈⠀⠈⡄⠀⠀⠀⠴⠀⠀⠀⠀⠴⠀⠀⠀⠈⠸⠛⠃
⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⢀⡁⡀⠁⠀⠁⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠛⠿⠍⠀⠀⠀⠇⠀⡁⠀⠀⢰⡂⢐⠆⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠂⢉⠅⠛⠅⡘⠦⡤⢀⣀⠀⠀⢀⠀⣀⠤⠺⡐⠤⣁⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⢢⠀⠠⠂⠰⠊⠀⢀⠔⠁⠀⠀⠀⠀⠉⠋⠂⠀⠃⠀⢃⠀⠀⠀⠀⠀⠑⠄⡉⡗⠄⡓⡄⢠⡊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢎⠐⠩⣤⠂⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠈⠂⠤⡄⠀⠀⢏⠑⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠑⢄⡔⢄⢻⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⠀⠀⠀⠘⡧⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠘⡷⢈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠑⢌⡢⢠⡄⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡀⠀⠀⠀⠀⢡⢂⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡔⢮⠚⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠛⠶⡇⠀⠀⠀⠀⠀⢀⠌⠀⢀⡔⠀⠀⠀⠀⠀⢸⣇⣢⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠀⠀⡂⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⠠⠄⢀⠀⢀⠠⡇⠀⠓⠄⣀⣀⡠⠐⠁⡀⠔⠈⠁⠀⠀⣀⠀⠈⡇⠀⠁⠀⠀⠀⠀⠀⠒⠂⠀⠀⠒⠂⠀⠱⠐⠛⣱⢑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⢀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠒⠠⠄⢀⣀⠠⠊⠈⠐⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠔⠁⠀⠀⠢⡀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡆⠀⡄⢀⡃⠀⠀⠀⠀⠀⠀⡆⠀⠔
⠀⠀⠀⠀⠀⠀⠰⡙⢶⡇⠀⠀⠀⠀⠀⠱⣂⣱⣂⠶⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠇⠀⠧⢠⣇⣀⠀⠀⠰⡉⠻⢤⡜
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠀⠀⠀⠀⠀⠀⠛⠀⠂⠁⠀⠀⠀⠀⠀⠀⠐⠍⠐⠋⠀⠀⠓⠒⠒⠀⠃⠀⠀⠀⠁⠂⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⡂⢀⡴
⠀⠀⠀⢠⠈⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠉⠉⠃⢈⠐
⠀⠀⠀⠀⠣⠀⢀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠚⠤⠈⠉⠒⠒⠄
⠀⠀⠢⡈⠢⣁⡄⠛⠁⢠⠀⠁⠐⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠇
⠀⠀⡀⠈⠓⠀⠀⠖⠉⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠲⠀⠠
⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠡
⠀⠀⠀⠀⠀⠀⠈⠉⠐⢄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⣆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠸⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠂⠁⠘⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⡀⠀⠀⠀⠀⢀⡔⠉⠲⠂⠉⠉⠙⠛⠶⣷⡀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠠⠀⢀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⡱⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣇⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⢠⠁⢠⡤⠄⠠⠤⠤⠒⠒⠉⢀⣠⣾⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢸⠀⠀⠰⣤⣀⣀⣀⣤⣴⣶⣿⣿⣿⣿⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠗⠠⠄⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⢸⠀⠀⠀⠙⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡌⠁
⠈⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⠻⣿⠁⠀⠀⠀⠘⣩⠏⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠈⡓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠉⠁⠒⠒⢊⡁⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣠⢾⣅⣀⡀⢀⣀⡠⠔⠋
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣤⣤⣠⣤⡤⠶⠛⣁⠰⠌⢣⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢹⢄⠀⣤⠀⠈⠐⠃⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⠀⣀⣀⠀⠠⠖⠋⠀⠀⠀⠘⣴⠁⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠈⠹⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢋⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⢀⡰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠚⠉⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡄⠀⠀⢸⠈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢋⣤⠞⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣌⠓⠒⢁⡖⠀⢀⠇⠀⠀⠀⠀⠀⠀⣀⠩⠴⠚⠉⠈⠳⡄⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢶⠛⡀⠰⠿⠤⠤⠤⠐⠐⠛⡏⠀⠀⠀⠀⠀⠀⠀⠹⣄⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠑⢄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠓⢤⣀⣀⡴⠃⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣤⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠴⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠩⣿⡿⢿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣁⣸⠀⠀⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣼⢇⢼⣷⣇⢸⣀⣀⡀⢥⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠬⠿⠿⠿⠿⠏⢹⣿⣿⠈⡡⣿⣿⠿⠿⠷⠾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣄⡀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡋⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣍⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⠧⠔⠊⣩⠿⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢖⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠈⠁⣷⡀⠀⢰⡒⠀⠀⡴⠶⠖⡀⠒⠐⠤⠀⠀⠨⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣥⠬⠁⠀⢘⠁⠀⠉⡰⠿⣿⣷⠤⠤⠴⠆⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡭⣶⣤⡼⠀⠀⠀⠀⠀⣀⡀⠀⢀⠀⠀⠀⠐⠘⡀⠀⠈⣤⠞⠉⠀⢀⡆⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⣧⡀⢰⡀⠀⢀⣾⡷⠋⠀⢋⣄⣀⣰⣆⡠⡯⠀⠀⢿⣥⡴⠞⠋⣷⢀⡟⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢎⡛⠳⣀⢹⣉⡝⣉⣤⣨⣿⣞⡏⢀⠜⡡⠂⠈⡿⠉⢖⡘⣿⡃⠀⢻⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡟⠀⠀⠀⠻⠈⢻⣏⣷⢨⣾⣿⣧⡤⠚⠐⠉⡰⠁⠀⠈⠈⣴⠐⣄⠚⠐⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠿⠀⠒⠒⠚⠒⣲⣿⣿⣿⣿⠟⠁⣤⣰⣦⠔⢲⡤⡤⠀⠀⢋⠁⠉⠉⠓⢦⠀⠶⠄
⠀⠀⠀⠀⠀⠀⢀⡠⠖⠂⢀⣠⢀⡔⠁⠴⠶⠖⠲⠦⠤⠇⠀⠀⣥⠂⢀⣞⣟⢓⣺⣏⣾⣷⢤⠆⡀⠀⠐⠠⠄⠸⢑⣤⠆⠦
⠀⠀⠀⠀⠀⠴⠉⠁⠀⠀⠀⠈⠀⠢⠤⠤⠤⠤⠤⠀⠀⠤⠤⠠⠤⠤⠤⠄⠀⠉⠉⠉⠀⠀⠒⠊⠀⠀⠀⠐⠒⠛⠉⠉⠁⠰⠠⠤⠤⠔⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠈⢳⠀⣀⣀⡀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⢀⡴⠖⠋⠀⠙⠚⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢀⡼⠀⠀⠀⠀⢠⠞⠋⠀⠈⠉⠙⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠈⠙⣛⠉⠀⠳⣄⠀⠀⠸⣄⣀⣀⣀⣀⣀⣹⠆⠀⠀⠙⠒⠒⠒⠒⠒⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠈⠁⠀⠀⠀⠉⠉⠉⠉⢉⡁⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⢀⡤⠤⣄⠀⠀⣀⠀⠀⠀⠀⢿⠉⢦⢰⡏⠀⠈⡇⢠⠎⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠹⡄⢺⠀⠀⢸⢆⡞⢁⡇⠀⠀⠀⠈⢧⡸⣆⠓⠦⠖⢁⠯⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡤⠿⣄⣙⣛⣁⡼⢤⡼⠀⠀⠀⠀⠀⠘⣆⠈⠉⠉⠉⠉⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠉⠉⠉⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⡾⡄⠀⠀⠣⠴⠖⠒⡏⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢠⣦⠀⠀⠘⡆⠀⠀⠀⠀⠀⡇⠀⠀⣞⠀⠹⣄⡀⢀⣀⣠⠤⠿⣀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⣼⠈⢷⡀⠀⠹⡄⠀⠀⠀⠀⠙⣄⠀⠈⠳⡀⠀⠉⠉⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢀⡇⠀⠀⢳⡀⠀⢹⡀⠀⠀⠀⠀⠈⢳⡀⣠⠟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣸⠀⠀⠀⠀⠳⣀⡤⠷⣄⠀⠀⠀⠀⠀⣟⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣉⣑⡇⠀⠀⠀⠀⠀⢯⡴⠚⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠐⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠃⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⠀⡄⡀⠠⡄⢀⡄⠀⡄⡄⡀⠀⡤⠀⠀⠠⠀⠠⡄⢠⠀⡄⡄⠠⡀⡄⡄⠠
⠀⠀⠀⠀⠀⠀⠀⣋⡀⣀⣀⠆⡂⢀⡁⡞⠰⢀⣉⡇⣃⡀⣏⣀⠀⣈⡳⠀⡃⢸⠘⡌⢄⣀⠃⡏⢆⣈⣀
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠠⢊⡆⠀⣤⠀⠤⢤⡀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡶⠚⠁⠀⣴⠠⠁⢸⠀⠀⣿⠀⠀⣼⠃⣼⡁⢹⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢀⠴⠂⣷⠂⠀⢸⡄⢀⡿⠀⠈⠃⠀⢋⡤⣾⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠠⢔⠁⠀⠀⠀⠀⠀⠀⡳⢢⠇⠀⠀⠀⡰⠋⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⡀⡟⠀⠀⠀⠀⠀⠀⡌⠀⡞⠀⠀⠀⢰⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡟⠐⢰⡧⢻⢠⡟⡐⢀⡸⣀⠞⠁⠀⣀⠀⣨⠋⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣀⣘⠃⠸⠯⠁⠀⣾⠁⠀⠊⠉⢩⡏⠙⡟⠀⡟⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⡿⠿⠿⢿⠆⠀⠀⢰⡇⢀⠔⢸⠂⣼⠀⢰⡇⢸⡇⣼⠏⢠⠁
⠀⠀⠀⠀⠀⠠⣾⣿⠿⠛⠉⢁⣿⠁⠀⢠⡖⠀⠀⠀⠀⢸⡧⠂⠀⠸⠀⠿⠀⢸⡇⠘⠇⠻⠔⠁
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢸⡏⠀⠀⡿⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⠸⡇⠀⠀⠀⠀⠀⣀⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⢸⠇⡠⠀⣷⠀⠀⠔⣽⠂⢠⣇⠎⢠⡟⠀⢀⣠⡾⣷⠄⠈⠯⣀⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠰⠁⠀⣿⢀⠎⠀⣿⠀⢸⠇⠀⣾⢁⡴⠁⢘⢓⡟⠀⢠⠂⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⠀⠈⠁⠀⠀⠙⠊⠀⠀⠉⠀⠀⠀⠀⠻⠋⠀⠠⠁⣼⠃⠀⠧⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡼⠋

⡅⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠁⠈⠂⠑⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡠⠀⠀⠀⠠⠀⠀⠂⠔⠁⠀⠁⠈
⡇⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⢀⠁⠀⠀⢀⠀⠈⠢⡂⠀⠘⠀⠀⠢⠀⠀⠀⠀⠀⢁⠀⠐⠀⠀⠀⠀⠀⠊⠀⠀⠀⡠⠊⠀⠠⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠄⠀⠑⠤⡀⠀⠂⠀⠀⠈⠢⠀⢁⠀⠀⠐⠤⠃⣀⠀⠼⠘⠂⠀⠠⢄⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⡇⠀⠀⡈⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⢀⠄⠈⠁⠀⠈⠁⠐⠉⠀⠀⠉⠀⠀⠀⠀⠀⠈⠐⠠⣃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⡇⢀⠁⠀⠀⠀⠀⢀⠅⠀⠀⠈⠢⡀⠀⠀⠀⠀⡨⠂⠀⢀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀⢀⣤⠖⠛⠛⠛⠲⢄⠀⠀⠀⠀⢀⠂⠀⠀⢀
⡇⠀⠉⠁⠈⠉⠁⠉⠀⠀⠀⠀⠀⠈⠀⠀⢀⠄⠀⠀⠐⠁⠀⠚⢉⣀⠀⠙⠀⠀⠀⢠⠎⠀⣠⣍⡑⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠈⠀⠀⢀⠔⠁⠀⠀⢀⠠
⡇⠀⠀⠀⠘⠢⠀⠀⠂⠀⠤⠀⠀⠀⠀⣑⠂⠠⡠⠀⠀⠀⠀⢰⠋⠉⢷⠀⠀⠄⠀⡜⠀⠸⠁⠈⣷⠀⠀⠀⠀⠀⡴⠄⠐⠀⠀⠀⠀⠐⠀⣀⠠⠒⠈⣀⠠
⡇⠀⠀⠐⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠞⢄⠀⠀⠀⠀⠀⠘⢁⠀⠀⠀⠀⡆⠀⠁⠀⠀⠀⢠⡈⠀⠀⠀⠀⠀⢀⠒⠃⠀⡀⠂⠀⠐⠉⠀⠀⠈⠁
⡇⠀⠀⠀⠀⠠⠄⢈⣁⠒⠂⠁⠀⠄⡌⠀⠴⠈⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠀⠙⠁⠀⢀⣠⠔⠋⠈⠀⠀⠀⠀⠀⠀⠁⡖⠀⠑⠀⠀⠒⠥⠁⠀⠀⠀⠀⠴⠂
⡇⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢷⠀⢃⣄⠀⡀⡤⢠⠄⠾⠇⠶⠽⠻⠟⠻⠟⠛⠐⠃⠖⠏⠻⠚⠇⠆⣠⣀⠈⢴⡗⣠⠀⠀⠀⠀⠀⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠱⠐⠃⠔⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠉⠸⠓⡷⢠⢄⢴⡰⠀⡀
⡇⠀⠀⠀⢤⠠⠷⠒⠎⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢀⠴⠀⠀⠀⠉⠁⠀⠀⠐⠁⠀⠀⠀⠀⠄⠆⠄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠈⠑⣰⣦⠰⣄
⣇⡔⣖⠨⠆⠀⠀⠀⠀⠀⠀⠀⢀⡸⠃⠀⠀⢠⠠⠠⠀⠀⢐⣄⢀⣈⠀⠀⠀⠀⠀⠀⠲⢦⠀⢀⠥⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠄⠁⡒⡀⠀⠀⠀⠀⠀⠀⠈⠑
⡧⠉⠀⠀⠀⠠⣂⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠊⠈⠈⠀⠀⠀⠒⠬⢨⠀⠉⢀⠀⢀⣁⠀⣂⠐⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⡇⠂⠸⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠈⠋⣀⡉⠁⠀⠄⠂⠀⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⣂⣤⡆⡠
⡞⣣⠄⠈⠡⠀⠀⠀⠀⠈⠄⠁⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠢⠀⠀⠩⢄⣀⣀⠀⢠⣀⣠⢖⢀⡀⠀⠀⠀⠀⢼⡦⢀⣀⡀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⢈⡀⡷⢆
⢇⡀⡬⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⢠⢰⠆⠀⠀⠀⠀⠘⠷⠅⡼⣉⠛⠋⠉⣷⠕⠊⠀⠀⠀⠀⠀⠝⠳⢺⠗⠂⠀⠀⠔⠂⠪⠐⠀⠀⠀⠀⠠⠘⠂⢀
⡇⠀⠀⠀⠀⠀⠐⠂⠄⡀⠀⠀⠦⠄⠀⠀⡃⠈⠰⠆⠀⠀⠀⠀⠀⠂⢀⠇⡴⠶⠶⠶⠲⡄⠐⠊⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠰⠄⠀⠀⢄
⡇⠀⠀⠘⠀⠀⠀⠘⠀⠙⠀⠠⣤⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠐⠀⠺⠃⠀⠀⠃⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡈⠦⠂⠀⠀⠀⢄⠀⠉⠀⠈⠒⢄
⡳⣄⠀⠀⠀⠀⠀⢘⠐⢃⠁⢀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠂⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⢦⡀⠃⠶
⡇⠘⣎⠀⢀⠄⠀⠈⠀⠈⣤⠀⠒⠀⠀⠀⠀⠂⠠⠀⠀⠄⠀⠳⠙⢇⣴⣄⠀⠀⠀⠀⣠⡆⡹⢋⠞⢀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠎⠨⠀⠀⠠⠀⢛⠳⡈
⠅⠐⠺⣆⠞⠀⣠⠀⣠⠀⠀⠘⠀⠈⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠦⠓⠀⣀⠐⠉⠠⡄⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠑⡄⠀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠐⠀⠈⡆⢡⠀⠁⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠜⠀⠚⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠌⠀⠀⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠠⠤⡅⠀⠤⢤⠁⢀⣀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡁⣄⠈⠐⡁⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢠⠔⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠑⡄⢀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⢀⣀⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢀⣀⣠⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⠀⣰⠋⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢡⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠤⠤⢄⣀⠀⢀⠎⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠆⠀⠠⠀⠀⣀⣀⣀⣀⣀⡀⠀⠈⢆⠀⠀⠈⠙⢮⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⢄⡴⠚⠉⠀⠀⡀⠀⠀⠈⠉⠓⠬⡆⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠋⠀⠉⠁⣀⠀⠀⠑⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠐⣲⢤⡀⠈⢦⡀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢄⣾⡏⠀⠁⠀⠀⠙⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡇⠀⠀⠨⣭⣁⣀⣈⣉⣉⣤⣶⣿⣿⠁⠀⠀⠀⢰⢤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠹⣿⣿⡿⠟⠟⠉⢙⡿⠁⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀⠀⠈⠛⠧⠤⠠⠔⠋⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣄⣀⣀⣁⣀⣀⣀⣠⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠈⠉⠉⡹⠁
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣰⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠆⡠⠤⡀⠶⢀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠘⣄⠀⠀⠌⠢⡀⠀⠀⠘⠀⠀⠈⠃⠀⠑⢄⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠈⢢⡀⢄⠀⠈⠲⢤⣠⢄⠀⠀⢀⠀⠀⠀⠀⠻⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠙⢦⡑⠠⣀⡐⠁⠀⡇⠀⠈⠑⠢⢄⡀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠤⢀⡀
⠀⠀⠀⠀⠀⠈⠒⠬⣀⣀⣀⡆⠀⠀⠀⠀⠀⠈⠁⠒⠠⠄⣙⠲⢤⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⡄⢠⠞⠉⠀⠂⠀⢁⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡈⠙⠓⠲⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠦⣄⠀⠀⠀⠈⠢⣐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠈⠂⠀⡀⠀⠀⠀⠀⠀⡀⠀⠇⠈⠳⡀⠀⠀⠀⡙⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⢀⡟⠂⠀⠀⠠⠐⢀⠜⠀⠀⠀⠙⢆⠀⡰⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠌⠀⠈⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠉

⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⣦⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⣿⣿⣿⣿⠂⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾
⣿⣿⣿⣿⣿⣷⣶⣶⣄⠀⢀⣀⡀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣀⡀⣤⣴⣾⣿⣿
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠾⠿⣿⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠠⠾⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠉⢠⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢰⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠋⠹⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠐⠛⠀⠸⠷⠶⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣯⡿⠋⣀⣭⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⣾⠋⢀⣾⣿⣿⣿⣿⣿⢉⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠃⢀⣾⣿⡿⠁⠀⠀⠈⢻⣷⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⡿⣷⣶⢪⣿⠏⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣾⣟⡽⠃⣾⡟⠲⠤⣄⠀⠀⠀⠀⣿⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⠟⠀⣼⠟⠈⠙⠒⠄⠀⣴⣚⣹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡍⣶⣾⠉⠀⠀⠀⠀⠀⢀⠇⢠⣿⡿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⣿⣿⣇⠀⢀⣀⣐⣊⠊⣴⣿⡀⢿⢸⡸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣒⣡⣾⡟⢹⡏⢢⠀⢉⣙⣛⡿⠻⡇⣳⣜⣇⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣀⡉⠉⠉⠀⢸⠁⢈⣛⣉⡍⢀⠠⠤⠤⠤⣤⡛⠒⠒⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀⠈⠀⠋⠁⠀⠀⣃⠂⠀⠀⢰⠟⣀
⠀⠀⠀⠀⠀⠀⠀⡔⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡤⠊⣉⣩⣭⣱⢤⡀⠀⠈⡑⢾⠀⠀⠀⠀⡆⢠
⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⢀⠴⠉⢠⠎⣉⣩⣅⣀⠀⠉⠲⣟⠂⠘⡄⠀⠀⡰⠁⠀⡆
⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠈⢆⠀⠀⠀⡠⠋⣀⠔⢁⠎⠉⠀⠉⠉⠀⠀⠀⡇⠀⠜⠀⠀⢰⠁⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠔⠁⠀⠁⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢄⡀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡺⢦⣤⣼⢛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⢲⠀⠠⢀⠀⠀⠀⢀⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣾⠀⠀⢲⡆⢸⡄⢸⡇⣾⠀⠐⢸⠃⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⠀⣀⠀⢿⠀⠀⠸⠁⢸⡇⢸⡇⠻⠀⠀⠈⠃⠈⣿⠈⠆
⠀⠀⠀⠀⠀⠀⢠⠖⠀⢸⠇⠀⣼⠁⠈⠀⠀⠀⠀⠀⠈⠓⢈⣇⠀⠀⠀⠀⢠⠀⣿
⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⢀⡧⠀⠀⠀⢀⡠⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢠⡄⠀⣾⠀⠀⠀⠀⠠⠊⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣷⠀⣷⠀⣿⠀⠀⠀⠀⢀⣀⣿⠀⣀⡀⠀⢀⡤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣿⡆⢻⣀⠙⠦⠃⠀⢰⡏⢸⡇⢰⡇⢰⡇⣸⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠈⠃⠀⠁⠐⢤⡀⠀⢸⡄⠸⣇⠸⠇⢾⡀⢻⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⢱⠀⠀⠉⠀⠉⠀⠀⠀⢥⡀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣿⣶⣶⣿⣶⣴⣶⣤⡀
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄
⠀⠀⠀⣿⣿⠟⠁⠀⠉⠙⠛⠛⠙⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡿⣿⠟⠿⠟⠙⠿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠤⢢⠛⠁⠀⠀⡄⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠛⡭⡏
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢀⠀⠀⠀⠀⠈⠁⠀⠀⡀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠚⠁⠤⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⡈⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢁⡀⢠

⠀⠀⠀⠠⢂⡀⠀⠀⠀⠀⠀⢀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀
⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢢⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠐⠀⠰⠀⠆⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠐⠁⠉⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠀⠀⣐⠀⠠⠀⢒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⢲⠀⠐⠒⠓⡀⠤⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡐⢆⡀⠡⣂⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠘⢀⠂⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡤⠒⠈⠐⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡕⡴⠒⠋⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⡜⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⡜⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠆⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠈⠉⠑⠦⡜⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠘⢄⣈⠦⠒⠉⠉⠉⠉⠉⠒⠢⢌⡂⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠜⠁⠂⠂⠀⠊⠉⠂⠀⠀⠀⠀⠉⠢⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠚⠉⠀⠀⠀⠀⠀⠐⢲⢤⡀⠘⢆⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣼⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⣰⡿⠀⠁⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢾⠀⠀⠘⣥⣍⣉⣉⣉⣥⣴⣿⣿⠃⠀⠀⠀⠸⠦⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠘⢿⣿⠟⠛⠋⢉⡹⠃⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠉⠓⠒⠚⠉⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠗⠒⠤⠤⠤⠤⣤⠶⠚⠁
⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⣰⠃
⡄⠁⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⣀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢠⠇
⢣⠀⠀⠀⠴⡀⠀⠀⠇⠊⠉⠲⡇⠐⢄⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠳⡄⠆⠀⠘⠢⣀⡆⡀⠀⠀⠇⠀⠀⠈⠢⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠈⠢⡑⠄⡀⢀⠋⠉⡆⠀⠘⠢⢄⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀
⠀⠀⠀⠀⠈⠓⠬⣅⠀⠀⡠⠀⠀⠀⠀⠈⠑⠂⠠⢄⡑⠢⣄⡀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠘⡄⢀⠜⠁⠀⠈⠄⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠉⠓⠲⠾⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠰⠦⡀⠀⠀⠈⠢⣌⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠑⠢⢀⠀⠀⠀⠀⠀⡃⢠⠀⠘⢦⠀⠀⠀⢸⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⡬⢑⡂⠀⠀⢊⡠⠎⠀⠀⠀⠱⣄⡠⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⡸⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⠀⠀⠐⠀⠂⠀⠀⠀⠂⠂⠀⠀⠀⠐⠀⠘⠀⠀⠘⠀⠀⠀⠂

⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡅⠀⢸⣿⣿⣄⠀⠀⣷⣾⠀⠀⠀⢸⣶⡇⠀⠀⣴⣾⡿⠿⢿⣷⣦⠀⠰⣶⣧⠀⠀⣼⣷⠆
⠀⠀⠀⠀⠀⣿⣿⣀⣀⣀⠀⠀⢸⣿⢻⣿⣆⠀⣿⣿⠀⠀⠀⢸⣿⡇⠀⣼⣿⡏⠀⠀⠀⢻⣿⣧⠀⠹⣿⣧⣸⣿⠏
⠀⠀⠀⠀⠀⣿⣿⠿⠿⠧⠀⠀⢸⣿⠀⠹⣿⣆⣿⣿⠀⠀⠀⢸⣿⡇⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠘⣿⣿⠃
⠀⠀⠀⠀⠀⣿⣿⣀⣀⣀⡀⠀⢸⣿⠀⠀⠹⣿⣿⣿⠀⢀⣀⣼⣿⡇⠀⠹⣿⣷⣀⣀⣠⣾⣿⠇⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠇⠀⠙⠙⠀⠀⠀⠙⠿⠿⠀⠸⠿⠿⠋⠀⠀⠀⠈⠛⠿⠿⠿⠛⠁⠀⠀⠀⠀⠋⠙
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣀⡀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⣤⡾⠟⠀⢟⣋⡙⢷⣴⡿⣋⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢦⡀
⠀⠀⠀⠀⠀⠀⣹⡿⢃⡴⠋⢸⡇⢸⢏⣼⠇⡼⣡⣶⠾⢶⣠⣶⢶⣶⠶⣦⣤⣀⣤⡶⠶⢶⡶⣶⡶⢶⣴⠿⠁⢾⡆⣿⣶
⠀⠀⠀⠀⣠⡾⢋⠄⡼⠁⠀⣼⡇⢠⢾⡟⢰⣷⠟⢡⣾⢀⡿⠁⢚⡄⢀⠁⣼⡿⠋⢐⣣⡞⠀⣩⢀⣾⠟⢠⣿⢿⡇
⠀⠀⠀⠠⣿⠱⢋⡾⠁⠀⠀⠻⠴⠋⠘⢧⣈⡥⢦⣙⡥⢦⣆⡴⠛⠖⢿⣄⣭⢦⣜⣋⣥⣠⠾⢧⣈⡥⣦⣨⡵⠋
⠀⠀⠀⠀⠉⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠶⠶⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡴⠚⠋⠉⠀⠤⠶⠶⣿⣿⡶⣦⣄⢠⣸⡇
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠛⣀⠀⠠⠒⢛⣛⣛⣛⠲⠄⠉⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⣴⠟⠁⠀⢀⡴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸⣧⢖⣀⠀⠘⠙⠶⡤⢴⣺⢷⠀⠀⠀⣠⡶⠛⠳⠀⣷
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠸⣱⠋⣁⠀⠀⢀⣤⣖⡿⣟⠉⠀⠀⠀⠋⠀⠀⣄⠀⣿
⠀⠀⠀⠀⣷⠀⠀⠀⠀⣠⡄⣩⡾⠁⠀⠀⢚⡭⢿⡿⠁⠀⠀⠀⠀⠠⣤⣤⣽⣄⠘⢷⣄⠀⠀⢀
⠀⠀⢀⣠⠿⠀⠀⣤⡾⠁⢸⢻⡇⠀⠀⠐⠉⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣶⠟⠛⢶⣤⡴⣦⣀⡀
⠀⢰⡏⢱⠟⣰⢾⠃⠇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠏⠀⠀⠀⠈⠓⠻⣬⡿⢶⠀⢀⡴⡲
⠀⠸⣇⠙⠀⠇⠘⠀⠀⠘⢦⣄⣀⠀⠙⠶⣤⡤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⡆⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣯⣞⠁
⠀⠀⠙⢷⠀⠀⠀⠀⠤⣄⡀⠈⣉⡿⠟⠛⠙⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢳⠄
⠀⠀⠀⠸⡇⠀⠀⠀⠙⢮⣭⠭⢩⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢿⣛⡻⠯⢭⣍⣉⠉⠉⠙⠛⢛⡟⢠⡏
⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⢀⡾⠉⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⢧⡀⠀⠻⠈⢉⡷⠶⢤⣜⣠⡞
⠀⠀⠀⠀⠀⠙⠛⠶⠶⠶⠞⠋⠀⢠⣼⣤⣤⣤⡀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠸⡟⠀⠀⢠⡟⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠈⡁⠀⠀⠹⠟⠻⣶⣿⣅⠉⠉⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠉⠛⠳⠶⣤⣀⠈⠉⢩⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣁⣤⠤⠀⠀⠀⠀⠈⠛⢦⡀⣉⣿⡄⠀⠀⣀⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠞⠉⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⢭⣏⢛⡶⢛⣡⡏⠀⠀⠀⠀⣿⠁⠀⠀⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⡞⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠈⢻⣾⠁⣈⣹⡟⠓⠶⠦⢴⡟⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢻⣄⣽⠋⠀⠀⠀⠀⠀⠁⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⢿⣭⣭⠑⠲⣦⣀⠀⠀⠀⢀⣤⡶⠂⠼⠃⠀⠀⠀⠈⢿⡷⠦⢤⣤⣄⣀⣀⣀⣀⣀⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠦⢤⡄⠙⠳⡄⣠⠞⠻⣆⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠈⠳⢤⣴⣄⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⢠⠞⡇⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⢠⡇⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⠷⠀⠀⠀⠈⠷⠏⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠘⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠿⠛⠁⠀⢠⣶⡄⣀⣤⡀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣠⣶⡄⠀⠀⢠⣦⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⢋⣡⣤⣶⣤⡀⠀⣾⣿⣿⣿⣿⡿⠀⠀⠀⢸⣿⡟⠈⢁⣾⣿⢻⣥⡀⢠⣿⡿⠁⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠿⠟⠃⢠⣿⣿⢡⣿⣿⠃⠀⢀⠀⣿⣿⠁⠀⢸⣿⠃⣸⣿⠇⣿⣿⢡⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⠉⠀⠀⠀⠀⠸⣿⠃⢸⣿⡏⠀⣴⠋⢰⣿⠃⠀⠀⢸⣿⣰⣿⠏⠀⣿⣷⣿⡿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠏⠀⠀⠀⠀⢀⣤⠶⡄⠀⠀⠘⣿⡃⣸⡇⠀⣿⠇⠀⠀⠀⠀⠉⠉⢁⣀⠀⠈⠉⣾⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠻⠏⠀⠀⠀⢀⡴⢛⣡⠞⠁⠀⠀⠀⠀⠀⣿⠁⢰⡏⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣠⣤⣤⣶⣯⡶⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿⠀⡾⠁⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣿⠞⠁⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⣠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⠀⣴⣦⠀⣶⡟⠁⣠⣶⡄⢠⣦⢀⣠⣄⢀⣾⡗⢀⣼⡷⢸⣿⣀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠋⣼⣿⠋⣸⡿⢀⣾⣿⠟⢁⣿⣿⠋⠛⠋⣾⡟⡴⣻⣿⠁⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⢸⣿⠃⣴⡿⠁⣾⡟⠀⠀⣼⣿⠃⠀⠀⢸⣿⡟⢱⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⢻⣿⡾⠋⠀⢸⣿⠁⠀⠀⠿⠇⣠⡶⠂⠈⠁⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠿⠄⠀⠀⠀⣼⡿⠁⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⣠⣄⢀⣤⣤⠀⠀⣀⣤⡄⠀⢠⣄⢀⣠⡀⢀⣀⠀⠀⢀⣀⠀⢀⡀⠀⢀⣤⣤⣾⣿⠿⠿⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡟⣠⣾⠿⢋⣥⢀⣿⣿⣿⣿⣷⣿⡿⢃⣴⣿⡿⢀⣿⣟⣠⣌⠋⣹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⣼⣿⢋⣿⡏⢰⣿⠋⣠⡿⠁⣾⡿⢱⣿⡟⣽⠿⢁⣾⡿⠋⢀⣾⣿⡿⢻⡟⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠻⠃⢸⣿⠀⠸⢿⠾⠋⠀⠸⠿⠁⠿⠏⠀⡹⠀⢸⣿⠀⠀⣸⣿⠇⠀⠘⠀⠀⢿⣧⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠻⠤⠀⠙⠋⠀⢄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⢠⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠏⣤⠘⠀⠀⢀⡄⠀⠀⠀⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢠⠀⣠⠞⠀⠀⠂⣰⣄⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡌⢶⣿⡄⡀⠀⠀⢿⡀⠀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⣮⠁⠀⠀⢀⡿⠀⠀⠀⢀⡈⠗⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡄⠀⣇⠷⣤⠀⠀⠨⠀⠀⠀⠀⠸⠁⣀⣁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣧⣾⣟⣏⢲⣿⠐⠄⠰⠀⠀⠀⠀⢠⠠⡤⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢠⡟⢟⠙⠛⣅⡿⠀⠀⠀⢦⠀⠀⣰⡟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⢸⡇⠀⠀⠀⢨⡄⣤⡿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⡄⢸⡇⠀⠀⠀⣈⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢁⣤⠬⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⢰⡶⠈⡄⠪⠛⡃⣙⠀⢠⡀⠀⠀⠀⠄⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡠⡁⣶⠑⡏⠡⡘⢋⠿⢐⠙⠞⠳⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴⠌⠀⢴⡮⠺⠇⠚⠁⠋⠠⠋⢁⡰⠦⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⡀⡀⣴⠤⡄⠀⠁⣩⡿⠳⣦⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠲⠊⠀⠢⠀⠀⡀⠙⠛⠳⢶⡀⠀⠸⠟⣅⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⡄⠁⢸⡷⢴⡶⣤⠙⣶⡛⠀⢀⡄⠈⠃⠀⠀⢠⠄⣴⡆⠽⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠆⠀⠰⢟⠉⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣠⣿⣿⡀⠈⠀⢠⠀⠀⣰⢿⡜⢿⣀⡀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠈⠀⠀⠀⡀⠈⠃⣀⠀⠀⠠⢠⠀⠀⠀⠀⡘⠁⠿⠛⠉⠉⠓⠴⣠⢞⣥⣾⣿⣾⣿⣷⠀⠀⠓
⠀⠀⠀⠀⢀⣔⣡⠄⠀⠀⠐⣄⠀⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠘⠳⣷⢶⡾⡰⠆⠀⠀⠀⠀⠀⠀⣤⣛⡟⠀⠉⠈⠉⠉⠀⣤⠊
⠀⠀⠀⠀⠻⠿⠛⠀⠀⠰⣶⡌⠀⠰⠊⠀⠀⠼⠃⠐⠁⠀⣠⠋⠀⢀⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢸⠛⠀⠀⠀⠀⠀⢀⠸⠆
⠀⠀⠀⠀⣄⠀⠀⣀⢠⠶⠋⠀⠀⢴⠄⠀⠀⠀⠀⠀⢡⡶⠀⣠⠶⠉⠁⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⡆⠀⢀⡀⠀⠀⡿⠃
⠀⠀⠀⠀⢸⣉⣋⡹⠏⠀⠀⠀⠀⠘⢧⣀⣁⠂⠀⢠⡘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠀⠀⡀⢡⠟⠀⣀⡠⠟⠁
⠀⠀⠀⠀⠀⠙⠈⠋⠀⠀⠀⠀⠀⠀⠀⢨⠹⢳⣦⣨⢁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡤⠚⢩⣵⣦⢰
⠀⠀⠀⠰⠀⠚⠉⠉⠉⢠⣄⠀⠀⢀⠰⠏⢙⣁⡉⢘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠃⢀⡅⢨⣿⠼⢦⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣈⠶⢤⣉⡷⠾⣯⣼⡷⣤⡜⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠶⠆⠛⣋⣙⠳⠻⠷⠄⠡⠂⠈⡙⠂
⠀⠀⠀⠀⠀⠀⢀⡖⠉⠁⠛⠲⠚⠋⡁⣐⠂⠨⠷⢴⡋⠙⠂⠙⠙⡶⠄⠀⠀⠀⠀⠀⠀⠀⣼⡆⠁⠀⠁⠉⠉⠀⠁⠀⠀⠀⠃⠀⠁⠘
⠀⠀⠀⣀⣤⠄⣎⡀⠉⢙⡂⡴⠚⣂⠐⠛⢲⠾⡀⠀⠀⢂⠤⣤⣐⣀⣼⠀⠀⠀⠀⠀⢀⡴⠂⠉⠠⠆⠀⠀⠠⠀⠠⠄⠀⣀⡀⢠⠁
⠀⠠⠂⠉⠀⡰⠀⣑⠟⢈⠁⠆⠘⠁⢶⡎⠀⠃⢁⡀⡤⠉⣧⢘⣀⡅⠸⠀⠀⠀⠀⢀⠢⠄⣸⠀⠠⠤⠤⠤⠤⠤⠤⠤
⠰⠀⡀⠀⣰⠃⣔⠉⠖⠉⠀⠀⠛⠂⠒⠠⠠⣆⠀⣄⠁⠀⢀⡜⢿⠌⢀⠦⠀⠀⠀⡜⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⢀⠀⡙⠲⠶⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠷⠚⠂⢿⠤⠶⠌⠀⠈⠂⠰⠁
⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠡⠤⠤⠤⢀⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣀⣀⣀⣀⣀⣀⣀⣐⣒⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣶⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣌⣤⣀⡤⢾⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣶⣤⣿⣿⣿⡿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣷⣄⣘⣿⣿⣿⣿⡟⠛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⡀⢿⣿⣿⣿⣿⣿⡿⠛⠛⢻⣿⣿⣧⣶⣾⡀
⠀⠀⠀⠀⠀⠀⠠⣆⠀⢹⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣿⣿⣿⣿⣿⣿⢀⣀⣤⣤⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠶⠟⢻⡄⠉⠛⠻⠿⠿⠿⠿⢟⣿⣿⡿⠋⢻⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⣶⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⢀⣠⣴⣿⣿⠟⠁⣀⡜⠀⠀⠻⣿⣿⣿⡿⢁⣠⣤⣶⡿⠀⠈⣍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣶⣿⣿⣿⠿⠋⢁⡶⢲⣿⣄⠀⠀⠀⠈⠻⢿⣷⣿⣿⣿⣿⠃⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠻⣿⠿⠇⠀⠀⠀⠀⠩⠋⢹⡯⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠙⣄⠀⣠⠄⠒⠂⠀⠀⠐⠒⠒⠈⠉⠀⠀⠀⢡⠀⢀⠖⢉⠤⠄⣉⡑⡄
⠀⠀⠀⠀⠀⢀⠤⠊⠀⢀⣤⠶⠒⠂⠘⡖⢂⠀⠀⠀⣀⣠⣀⠀⢀⡤⠶⠤⣄⠀⢸⠀⢸⠀⠇⠀⡠⠚⣉⢝⠠⢄
⠀⠀⡠⠔⠊⠁⠀⣀⠀⠸⠀⢀⣤⠤⠄⠘⡌⡄⠀⠚⠁⠀⠈⠁⠈⠀⣀⣀⡀⠀⢨⠔⠁⡸⠀⠀⢧⡞⣊⠥⠒⠉
⠀⢎⠀⠀⠀⣰⠟⠉⠁⠀⠀⠸⠁⠀⢀⡀⠘⣷⠀⢠⠶⠛⠓⠄⠀⠘⠉⠉⠉⠀⡖⠒⠊
⠀⠈⠢⡀⠀⠃⠀⣴⠞⠃⠀⠀⡀⣠⠎⢱⠀⢹⢃⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠇
⠀⠀⠀⠈⡶⢄⠀⠁⠀⠀⣀⡭⠞⠁⠀⠸⠀⢸⠈⢆⠀⣠⢶⣒⠛⠿⠟⠻⠄⠸
⠀⠀⠀⢠⠀⠎⠑⠤⡀⠙⠤⣀⣀⣀⠔⠁⠀⡌⠀⠈⢦⡉⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠈⣼⡀⠀⠀⠈⠒⠤⠤⢀⣀⣀⡠⠔⠁⠀⠀⠀⠉⠢⣀⠀⠀⣀⠜⠁
⠀⠀⠀⠀⠈⠫⣁⠒⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⣈⠆
⠀⠀⠀⠀⢐⣉⡁⠬⠄⠒⠊
⠀⠀⠀⠀⠈

⠀⠀⢘⣆⢠⣾⠆⠛⠋⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠠⣾⣆⣀⣀⠀⠀⠀⠀⠸⠿⠃
⠀⠀⠻⠛⠀⠀⠀⠀⠀⠀⠀⠀⡞⣷⠀⠀⠀⢀⡤⡞⠿⠗⠀⠀⢀⡾⠿⠂⡼⠿⠆⠀⠀⣾⠁⠀⠀⠈⠙⠿⠟⠁⠀⠀⠀⠀⣀
⠀⠀⠀⢀⡴⣦⠀⠰⣞⡇⠀⠀⣵⠏⠀⠀⠀⠈⠙⠁⣤⠀⠀⠀⡜⠁⠀⡼⠁⠀⠀⠀⠠⡞⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣹
⠀⠸⣿⠞⠚⠁⠀⠈⠃⠁⢀⡞⣽⠶⢶⡄⠀⢀⣤⣴⠃⠀⠀⠀⠉⠙⠚⠁⣀⠀⠀⠀⠀⠀⠀⡤⢄⣰⣩⠆⠀⠀⢠⣶⣦⠀⢰⣿
⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠸⣆⠧⢁⣸⠇⠀⠈⠙⠃⠀⡠⠤⡀⠀⠀⢀⣴⣏⠀⠀⢀⡤⣄⠀⠙⠊⠉⠀⠀⠀⣠⠏⠈⠁⠀⠈⠁
⠀⠀⠀⠯⠵⠫⠖⠃⠀⠀⠀⠈⣙⠛⠁⠀⢀⠀⠀⠀⠀⢻⠉⠀⠀⠀⠻⣴⠋⠀⠀⣸⣼⠃⠀⢀⣶⠀⠀⠀⣀⠀⢀⣴⣄⠀⢠⣖⡦
⡀⠀⠀⣀⠀⣀⡴⠛⢦⡀⠀⠀⠉⠋⠀⢠⠏⠉⢹⠃⠀⠸⠀⠀⢀⣀⠞⠁⠀⣰⢞⣽⣄⡀⠀⢘⣧⠀⠀⣼⠻⠃⡼⠉⠁⠀⠸⡇
⠃⠀⠀⠻⠟⢡⣤⣠⠜⠻⣄⠀⠀⠀⣀⡼⢀⣄⡟⠀⠀⠀⣠⣄⠻⡟⠀⠀⠰⡇⠏⠀⢨⡇⠀⠘⠀⠀⠰⢿⣗⣺⠃⠀⣼⠃⠀⠃⠠
⠃⠀⠀⠀⠀⠈⠛⡁⠀⣠⠞⠁⠀⠘⠿⠃⠈⠛⠁⠀⣴⡆⠻⣅⠀⣹⣆⠀⠀⢙⠲⠶⠛⠁⠀⡤⣤⠀⠀⢠⣄⠀⠀⣸⣧⠀⠀⠀⢰
⠀⣠⣖⠁⠀⠀⠘⠿⠋⢁⡀⢠⢞⡆⠀⠀⠀⠀⠀⠀⠹⣄⢀⡼⢾⡥⠞⠁⠀⠙⠓⠁⠀⡔⢠⡟⠁⠀⠀⢸⡸⠃⠀⠈⠉
⠈⠁⠋⣀⣀⠀⠀⠀⠀⠸⠿⠋⠉⠀⠀⠀⣠⢾⡅⠀⠀⠈⠯⠖⠋⠀⠀⠀⠀⢀⠀⠀⠀⠈⣉⠀⠀⠀⠰⣉⠇⠀⢠⠏⠙⢲⠆
⢀⡀⠈⠓⢮⣇⠀⠀⠀⠀⢀⣠⣴⣦⠀⠀⠀⠉⠀⢠⡤⠤⣤⠀⠀⠀⣀⣤⡞⠉⢳⡀⠀⠸⡯⠭⢽⠀⠀⢀⣤⣴⠋⣀⣠⠏
⣻⠏⠀⠀⠸⡿⠃⠀⠀⠐⠋⠀⠈⠉⢠⣶⡄⠀⠀⢸⠋⠉⣿⠀⠀⢺⡁⠀⣳⡀⢀⣷⡄⠀⣷⣦⢸⡇⠀⠀⠉⠁⠈⠛⠋⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⠀⢠⣶⡆⢸⠉⠀⢀⣤⡿⠰⣿⠇⠀⠀⠀⢳⡐⠿⠃⠈⠛⠁⠀⢀⠀⠀⠿⠗⠀⠀⠀⠀⡞⣇⠀⠀⠉⠋
⠀⠀⡀⠀⠀⠀⠀⠘⡷⢄⠀⢸⡥⠤⣾⠀⠀⠀⠉⠁⠀⠀⣤⣆⠄⠀⠿⠗⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢀⣤⡀⢈⠽⡄
⠀⣷⣿⠄⠀⠀⠀⠀⠛⠃⠀⠈⠛⠉⠉⠀⠀⠀⢀⣀⠀⠘⠱⠋⠀⠀⣤⡄⠀⠺⣝⣦⠀⢠⣶⡇⠀⢠⣶⡄⢸⠋⠁⠈⠊⠁⠀⣄
⠀⠉⠈⠀⠀⠀⢠⡒⡄⠀⠀⠀⢠⠖⣛⣛⡶⠾⡥⠾⠃⠀⠀⢠⣦⠀⡟⠁⠀⠀⠀⣿⡆⠀⠉⠀⠀⢸⣉⣤⣿⠀⠀⠀⠀⠀⣿⠋
⠀⠀⢀⡞⠀⠀⢸⡇⠀⠀⢀⣀⢿⠈⠁⢛⣷⠀⠀⠀⡄⠀⠀⢸⡁⠀⢻⠀⠀⠀⠀⠈⠀⠀⣤⣄⠀⠘⠒⠒⠋⠀⠀⠀⠀⠟⠙
⠐⣿⡟⠀⠀⠀⠘⠃⠀⠀⠈⠫⠘⠛⠲⠛⠁⠀⠀⣼⣥⡀⠀⠘⣧⠴⠞⠀⠀⣀⡀⠀⠀⠀⣿⣉⠀⠀⠀⢀⡄⠀⠀⢠⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠛⠒⡆⠀⠀⠀⠀⠀⠀⠀⠸⢧⡞⠀⢠⣦⡀⠀⠀⠀⡴⠛⠛⠀⠀⠀⠸⠜⠀⠀⣰⡟⠁⠀⠀⣾⠀⠀⡇
⢠⣶⡄⠀⠀⠀⢰⠇⣀⣸⠃⠀⠀⢀⣀⠸⣿⠃⠀⠞⠀⠀⣀⡯⠇⠀⠀⠘⠁⠀⠀⠰⣖⢦⠀⠀⠀⠸⠉⠃⠀⠀⠀⢿⣶⠀⣇
⠋⠉⠁⠀⠀⠿⠟⠀⠻⠛⠀⣠⣤⠸⣟⠀⠘⣦⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⢀
⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠙⣏⠀⢈⣶⣚⡭⠇⠀⢠⠤⣄⡴⣻⠆⠀⢰⠯⣟⣷⠀⠈⠛⡁⠀⠀⢸⡭⠃⠀⠀⢀⡞⠓⢦⣤⣀⠈
⠀⢀⡤⠖⢿⠉⠈⢧⠀⠀⠀⠀⠈⢿⣭⠞⠉⠀⣀⠀⠀⠉⠁⠉⠁⢀⣀⡏⣠⣤⡏⠀⠀⣿⣿⠀⠀⠸⡇⠀⢠⣄⡾⢀⣄⡾⠀⢉⡟
⣀⣈⣧⣀⣈⣧⣀⣨⣇⣀⣀⣀⣀⣀⣀⣀⣀⣼⣻⣇⣀⣀⣀⣀⣀⣻⣟⣀⣈⣉⣀⣀⣀⣘⣀⣀⣀⣀⣁⣀⣈⣛⣁⣘⣛⣁⣀⣼⣁
⣿⣿⠋⡛⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢐⣛⣉⣉⣉⣛⠲⣶⣤
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢰⣿⣿⡿⠿⣿⣿⣿⣿⣶⣿⣇
⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠉⠉⠉⠋⠙⠛⠛⣿⠀⠀⠀⠀⢻⡏⠁⠀⠀⠀⠀⠀⠹⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠙⢫⢵
⠀⠀⠀⠀⠀⠰⠲⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣠⠀⠀⠀⠈⠠⣤⡄⠈⠀⠀⠀⢨⡆⢠⠴⣦
⠀⠀⢠⡟⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠤⠒⢶⠄⠀⠀⠀⠀⠀⠀⣀⣤⡾⢛⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⡙⠻⠷⠀⢾⠟⢋⣡⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠄⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⢰⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢻⣿⠻⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠂
⠀⠀⠀⠀⠀⠀⠢⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠉⠈
⠀⠀⠀⠀⠈⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠊

⣾⡷⠻⠗⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣾⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠀⠀⣠⡶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠤⠤⠤⠤⠴⠒⠊⠉⠀⢀⠀⠀⠻⣦⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢻⡆
⠀⠀⠀⠀⠀⠀⠐⠀⠈⠛⠛⠁⠀⠂⠀⠀⠀⠀⣴⠓⣦⠀⠀⠀⠠⠀⡊⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠆⡀⠀⠀⠀⠀⣰⡿⠛⠁⠀⠀⢀⠀⠠⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠐⠀⠂⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠃⠀⠀⠀⢸⡏⣛⢛⡛⡧⠀⠀⢰⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⢸⡇⠉⠈⢰⡇⠀⠀⠘⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⣦⡀⠀⢰⡄⠀⠀⠘⠛⠛⠛⠛⠃⠀⠀⠂⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡾⠃⠀⢠⠆⡆⣀⣀⣀⠀⢤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⠀⣠⣶⣇⡀⠀⠀⠀⠀⠀⢀⣠⣴⢺⣿⡏⣿⣿⢻⣦⡀
⠀⠀⠀⣿⣿⣿⣿⣦⣀⣥⢴⣿⣿⣿⣿⣸⠿⠷⠿⠿⢼⣿⡏
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠉⠛⠻⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡏⠀⠀⠀⠀⠐⢎⣀⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢦⣤⡄⠀⠀⠀⠈⣿⣄⠀⠀⠀⠀⠀⢸⠇⢰⠇⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢏⠀⡐⡅⠀⠀⠀⠀⣿⣿⣷⣄⠀⠀⠀⢸⠀⢸⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⣿⡇⠙⢿⣷⣦⣄⣘⠀⣂⣀⣴⣷⡀⠀⠀⠀⣀⣤⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠉⠛⠿⠿⠿⠿⠛⠉⠻⣿⣿⣿⠿⢿⣿⡟⠀⠀⣀⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⣀⣠⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠸⡏⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠙⢿⣏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣠⠖⠛⠉⠉⠉⠓⠶⣤⣿⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠇⠀⢸⣿⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣾⠁⠹⣾⡣⠀⣠⣴⣶⣤⣀⣰⣟⣹⡆⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣀⣀⣤⡾⠟⠓⠒⠶⣄⣹⣯⠝⢦⡀⠀⢀⡇⠀⠀⢻⡗⣾⠋⢰⡖⠛⠉⠉⠉⣿⣷⠀⠀⣧⣀⡀
⠀⠀⢀⣀⣠⠀⠀⠀⠀⠀⠉⠻⣿⡟⠁⠀⠀⣀⠀⠀⠀⠙⢿⡯⡈⠻⣄⢸⠃⠀⠀⠀⢻⣻⣄⠸⣇⣤⡄⠀⠀⡿⣿⠖⠒⢻⡀
⠀⠀⢈⡏⠀⡣⠀⠀⠀⠀⠀⢠⣿⣷⢅⣴⠿⠛⠻⢷⡄⠀⠈⣿⣔⠀⠈⠿⠀⠀⣆⠀⠈⢷⡉⠛⠛⠉⠀⢀⣼⢃⣸⣄⣀⣼⡿⠛⠓⠒⠂
⠀⠀⠈⠁⠙⠃⠀⠀⠀⠀⢀⣼⣿⣿⣺⡇⠀⠀⠀⠈⢿⠀⠀⢸⣿⣷⡄⠀⠀⢰⣿⡆⠀⠘⣷⣤⣤⣴⡶⠟⣱⠟⠉⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣧⡀⠀⠀⣠⡞⠀⠀⣼⣷⣿⠻⣦⣀⣸⠀⢻⡶⠾⠿⠉⢉⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⡆⠠⡄
⠀⠀⠀⠀⠐⠚⠛⠛⠛⠉⠉⣹⠿⣿⣿⣾⠉⠛⠛⠁⠀⠀⣰⣿⣿⣿⠄⠘⠛⠋⠀⠀⠀⠀⠀⠐⣤⠷⣖⠒⠒⠛⠦⠀⠀⠀⠈⢠⡤⠓
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⡰⠛⠻⣷⣧⣤⣤⣤⣴⡾⢿⣯⡁⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⡎⠈⢳⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣠⠞⠁⠀⠀⠜⠋⠉⠉⠉⠀⠀⠀⠙⣷⡀⠀⠀⣠⠞⠋⠉⠉⠉⠲⣇⠀⠀⠙⠆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡼⠃⢀⡴⠞⠛⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠒⠈⠦⡄⠀⠀⠀⠀⠀⣿⡼⠁⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣠⣀⡎⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠃⠀⠀⠀⠀⠀⠀⠏

⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣦
⢸⣿⡿⠻⠿⠿⠿⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠉⢣⠀⠀⠀⠀⠀⠀⢀⠎⠉⠉⠙⢦⠀⠀⢸⣿
⢸⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣀⣀⣞⢲⣶⣶⣶⣶⣶⣾⠟⠢⣀⠀⢸⣶⣶⣾⣿
⢸⣿⣿⡿⠋⠉⠙⢿⣿⠛⠛⠻⢿⣿⣿⠿⠛⠿⣿⣿⣿⠟⠉⠉⣹⣿⣿⠉⠉⠉⠉⣿⡟⠉⠀⠙⢻⣿⣷⣄⡈⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣧⠤⠤⠤⠼⡇⠀⠀⠀⢸⠿⠧⠤⠤⠤⠼⣿⣏⠀⠀⠘⡕⢨⡧⠤⠤⠤⠼⢿⡀⠀⠀⠀⢀⣿⣿⣛⡻⠿⠿⣿⣿⣿⣿
⣸⣿⣅⡠⠤⠤⠴⠂⠱⢤⢤⡤⠊⡔⠉⠉⠉⠉⣲⣨⣿⣿⣿⣛⣷⡊⡰⠒⠒⠒⢒⣤⣕⣀⣀⣀⣺⣩⠒⠒⠒⠒⠒⣆⣹⣿⣉
⡧⠤⣁⠉⡆⠀⢰⠊⣉⣉⣉⣙⣒⠒⠢⡀⠀⡎⢡⠤⠤⠤⠤⠤⠤⣍⠙⠆⠀⡎⠁⣀⡀⠤⠤⠤⢄⠈⢱⠀⠀⡰⠉⣠⠤⠤⡼
⡇⠀⢸⠀⡇⠀⡇⢰⠁⠀⠀⠀⠀⢹⠀⡇⠀⣇⢰⠀⠀⠀⠀⠀⠀⡜⢸⠀⠀⠇⢸⠁⠀⠀⠀⠀⠀⡇⡸⠀⠀⢇⢸⡁⠀⠀⢣
⠑⢄⣸⣀⣯⣷⣷⣈⡆⠀⠀⠀⠀⡎⢸⠀⠀⠸⡀⢇⠀⠀⠀⠀⢠⠃⣾⡆⠐⣾⣈⣆⡀⠀⠀⠀⡰⢡⠃⠀⠀⠘⡄⢣⠀⣀⠼
⣶⣶⡶⣷⣶⣶⣾⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⣿⣿⣤⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠇⠀⠀⣼⣇⠀⠰⠀⠖⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⢑⠚⠀⠀⠀⠀⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡀⠈⠀⠀⢀⣠⣠⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠘⠃⢀⠀⠀⠰⣴⣶⣼⣷⣷⣿⡻⣿⣯⠀⠂⠶⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠠⠄⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦⠀⠀⠀⡀⣴⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠻⠆⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⠟⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠰⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣦
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⢏⣥⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣵⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⠛⠛⠛⠿⣿⣞⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠟⠉⠉⠉⠻⣿⣿⣿⣿⣿⢀⣤⣶⣶⣶⣿⣿⣎⢿⣿⡿⠛⠋⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣏⣴⣶⣾⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣯⣴⣶⣶⣦⣄⣹⣿⣿⣿⡿⠛⠉⠙⠻⣿⣷
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣄⣹⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣾⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣣⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣍⠛⠻⠿⠿⠿⠛⠋⠀⢰⣿⣿⣼⣿⣿⠟⠋⠁⠀⠀⠀⠙⠻⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣾⣿⡟⣿⣿⠃⢀⣀⣤⣀⡀⠀⠀⠀⢹⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣷⣄⠀⠀⠀⣰⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣷⡀⠀⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣶⣾⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡟⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⠿⠟⠋⠀⠀⠀⠙⠻⠿⣿⣿⠿⠟⠉




⡿⠿⠿⠿⡿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⠿⣿⡟⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠉⠉⠙⠛⠻⠿⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠲⠦⣤⣤⣸⣷⣤⣀⣀⣀⣠⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠿⠛⢛⢿⠏⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠈⠆⣸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠐⠀⠺⢽⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⡀⠀⣀⠀⠀⠀⠀⣄⡶⡀⢠⣀⠙⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⣠⣶⡶⠛⠋⢉⡙⠿⣿⢿⣿⣿⣿⡀⣤⠀⣰⣿⣿⣷⣿⣿⠿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢘⢿⠁⣤⠴⠒⢋⣉⣩⣝⡳⣝⣿⢿⢷⡜⠢⣿⣿⡷⢛⣁⣘⠢⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠉⠃⣨⣤⠶⣿⣿⣿⣿⣿⣿⣿⠿⢷⠀⠉⠈⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢀⣠⠤⢤⡀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠉⠀⡿⠿⣿⣿⠿⠟⡁⠒⠀⠀⠀⠀⠈⢻⣿⡿⠟⠛⣻⡏
⠀⠀⠀⠀⠀⣴⠟⠟⠓⠄⠙⡀⠀⠈⡆⠀⠀⡀⠄⠊⠀⣠⠎⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡛⠿⣿⣧
⠀⠀⠀⠀⠀⢿⣇⠀⠀⢠⠀⠃⠀⢀⠧⠀⠀⠀⡀⠴⡊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡵⡤⡸⣿⡄
⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⢀⣠⠔⢁⡠⠔⠂⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⢀⣀⣀⠀⠀⢸⣷⡸⣿⣿⡇
⠀⠀⠀⠀⣼⣉⠙⠧⡀⢸⡿⢁⡔⠉⡗⡀⠀⠀⠀⠀⠀⠂⠀⠀⢠⠀⠀⠀⠀⠀⠛⠿⠿⠿⠿⣿⣿⣦⣼⣿⣿⣻⣿
⠀⠀⠀⣼⡟⡟⠓⠄⠘⢌⡇⣼⣀⣠⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⢿⣿⡟⣿⣿⡏
⠀⠀⠀⢰⣵⣇⠀⠀⠐⠪⡿⣛⣿⠿⢷⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⣴⣋⣍⠳⡶⣻⣯⣠⡉⢷⣽⠟
⠀⠀⠀⠘⣿⣿⠀⠀⠀⢰⠳⣾⣷⣄⠈⢆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠻⠿⠛⠛⠉⠛⢿⣾⠛⠿⣿⢟⡿⠁
⠀⠀⠀⠀⢹⣻⣦⠀⠀⢸⠀⠘⡿⣿⣷⠈⢢⡀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⢤⣀⣀⣀⣘⣿⣤⣾⣿⣼⠃
⠀⠀⠀⠀⠘⣿⡽⣄⠀⡀⠃⢁⡇⢿⣧⢠⡀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠿⠿⣿⡿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠙⢾⣷⣤⣿⡤⠞⠀⠘⣿⢦⣕⢤⡀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⢷⣿⠉⠁⠀⠀⠈⠛⠷⣦⣤⣄⣀⠀⠀⠀⠀⡀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠆⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠒

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠄⠈⠀⠈⠠⠀⠄⠠⠀⠀⠀⠀⢀⣤⡀⡀⠠⠀⠀⠀⠀⡤⡀
⠀⠀⠀⠀⠀⠀⢢⡀⠈⠁⡀⠀⠈⠀⠠⡌⡙⢿⡛⢿⡀⢙⣷⠀⢰⣦⡼⢷⣀
⠀⠀⠐⡆⠀⠀⢨⠁⠀⠀⠀⣦⣦⣄⡀⠀⠀⠂⢥⠰⠌⡛⣁⠀⠼⠏⣾⡿⠛⠉⠀⠞
⠀⠀⠀⢁⠀⠀⠀⠀⠀⢀⠰⠫⢻⢻⠛⢦⡣⠀⠀⠀⢡⠁⠀⠀⠀⢆⠋⠀⡀⠔⢐⣶⣤⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠡⡄⠎⠁⠀⡀⠸⠦⠀⠀⡄⠈⢃⠀⠂⢁⢣⠂⠁⠂⠀⣀
⠀⠀⠀⠂⣈⠀⠂⠀⠀⠀⠀⠀⠀⠢⠀⠘⠀⡀⠄⠋⠢⢠⠸⡦⠜⠀⠀⠀⠐⠃⠄⠀⠀⠈⣡⠆⢔⠀⠀⠂
⠀⠀⠀⡾⠁⠀⣴⡏⠐⠂⢤⣤⣀⣂⡙⠀⠴⣤⡈⠠⢖⠚⡠⠅⢂⠄⠀⠂⠨⠄⠄⣠⠔⢾⡇⢴⠂⠐
⠀⠀⠀⠀⢶⣤⣿⡇⠀⠀⠈⠛⢻⠋⠀⠄⠀⡀⠈⣂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠒⢺⠇
⠀⠀⠀⢀⣰⣿⡏⠀⡓⠊⠐⢀⣈⠀⠠⠎⠀⠀⠨⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠐⠀⠄⣁⣴⡿
⠀⠀⠀⠈⡿⠋⠀⠁⠀⠀⠀⢀⠉⢠⡄⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣀⠀⠀⠀⠀⠀⠙⠂⠰⢿⣽⠇
⠀⠀⠀⢷⠔⠀⢀⣤⣶⡎⣤⠜⠁⠀⢀⠄⣀⡂⣀⠀⠀⠒⠀⠀⠀⡀⠀⢀⠉⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠨⠄⠀⢸⣏⣌⣁⠙⢶⠀⠀⢸⠀⢇⠀⠀⠀⣠⠧⠀⠀⠐⢃⠀⠘⠠⣁⡀
⠀⠀⠐⠁⠀⠀⡀⠉⣱⠏⠀⠰⠀⠀⠈⡇⠈⠃⠌⠁⣀⠀⠀⠀⠐⣠⢦⡤⠚⠻⢟⡀⠖⡤⠀⠀⡠
⠀⠀⠀⠀⠐⢾⣶⣴⣿⠁⠀⠀⠀⣤⣴⣷⣄⡀⠀⠀⠅⠀⢂⣠⠶⢹⣥⣂⢐⡘⠃⠁⠾⠷⢂⣠⡀
⠀⠀⠀⠶⠀⠀⠻⣿⣃⣀⣀⣀⠈⢨⠟⠉⠉⠍⠈⠀⠸⢂⠯⠀⠩⠀⡟⠟⠅⠈⢢⠲⢟⠛⠋⠉⠀⣠⠀⡀
⠀⠀⠐⠀⢰⠀⠀⠈⡿⠏⠛⠉⠀⢀⡀⠀⡘⢆⠛⠠⠠⢹⣥⢤⠴⣎⠁⠈⢃⡔⠁⠁⡀⣘⡂⣴⡞⠛
⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⣰⣾⣿⣗⠀⠄⠈⠦⠤⠀⣸⢕⠥⡊⠀⠑⡤⠋⣀⣤⠅⣬⣭⠃⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡈⡻⠷⠠⣤⣤⣤⣾⡃⠁⢀⠄⠠⡀⡏⠀⠀⠋⠀⠈⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠛⠛⠉⠢⠁⡡⠠⢴⣶⣶⡞
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠶⠿⢻⣿⠿⠿⠿⣿⡛⠛⠷⠶⢦⣤⣀⡀
⠀⠀⠀⠀⣠⣴⠾⠛⠉⡉⠀⠰⢶⠆⠙⠶⣦⣤⣤⡴⠟⠀⣳⠆⣠⠄⠉⠛⠷⣤⡀
⠀⠀⠀⣼⠏⠀⣀⣠⣤⣄⣉⡑⢒⣤⣶⢶⣿⡏⣿⣿⣶⣤⡚⠉⣠⣔⠶⣶⡲⢬⡹⣆
⠀⠀⠀⢿⣄⠻⣯⣭⣛⣉⣭⡿⠸⢿⣿⣼⣿⡇⣸⣸⣿⡿⢃⡀⠻⣤⣬⣥⣤⡾⢃⡿
⠀⠀⠀⣼⠛⢷⣤⡀⠉⠍⢡⠖⣠⠂⣩⠿⣿⣿⠿⣏⠉⡘⠪⣉⠛⠢⠀⢀⣠⣴⡿⣧
⠀⠀⠀⢿⡀⠸⣌⣙⢳⣶⣦⣤⣄⣈⣿⣷⣯⣷⣾⣿⣣⣬⣤⣬⣤⠶⣾⣿⡭⠾⢃⡿
⠀⠀⠀⠈⠻⣦⣀⣼⠟⢙⣿⣤⣿⠟⣫⡿⠿⠿⢿⣍⡉⠉⠁⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣤⣾⣿⣥⣀⣈⠳⣤⣀⣀⣤⣼⣇⣠⣠⣤⣤⡶⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⢸⡿⣿⣶⣤⣬⣍⣉⣛⣛⡛⠛⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠉⠉⢻⣿⠿⠿⠿⢿⣿⣿⣷⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠈⢷⣼⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠉⠉⠙⠛⠻⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⣼⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⣶⣶⣶⣿⣤⣤⣴⣿⡿⠛⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠿⠛⠩⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠟⠉⠀⠀⠀⠀⣿⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠿⠿⣿⠁⠀⠀⠀⠀⠀⠀⣿⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠋⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣰⣿⡿⠋
⠀⠀⠀⠀⢀⣴⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣠⣴⣿⠿⠋
⠀⠀⢀⣴⣿⠿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣅⣴⣿⡿⠛⠁
⠀⣰⣿⠋⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠟⠁
⣸⡿⠁⠀⠀⠀⢸⡇⠀⠀⠀⢀⣠⣾⡿⠟⠋
⣿⡇⠀⠀⠀⠀⢸⣿⠀⢀⣴⣿⠟⠋
⣿⣷⡀⠀⠀⠀⣸⣧⣼⡿⠋⠁
⣿⣿⣿⣦⡀⣠⣿⠿⠋
⣿⠉⠻⣿⣿⣿⣇⡀
⣟⠀⠀⠈⠙⠿⣿⣿⣿⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤
⣯⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⣶⣿⣿⣿⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠉⠉⠉⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠙⠿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣷⣶⣤⣤⣤⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣤⣤⣴⣶⣶⣶⣿⣿⣿⡿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣤⣶⣶⣾⣿⣷⡴⠛⠉⠁⠀⠀⠀⠈⣳
⠀⠀⠀⠀⠀⠀⠀⣴⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢿⣿⠻⠿⣿⡀⣴⣶⣤⣤⣀⣠⣔⣃⡀
⠀⠀⣠⠖⠒⢂⣾⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠉⣿⣯⣤⣾⣿⣿⣾⣿⣷⣿⡿⠿⣿⣿⣿⣿⣿⡇
⠀⠀⢧⡀⢠⣿⣿⣿⣿⣿⣿⣿⣷⣶⡞⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠦⢿⣿⣿⠟⠉⠉⠻⣿⣿⣿⣶⣶⣿⣿⣤⣬⣿⡤⠖⠒⢲
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠿⣤⣾⣿⣶⠏⢻⣿⡿⢿⣿⣿⣿⣿⣿⣇⡤⠖⠋
⠀⠀⠐⢸⣿⣿⣿⣿⣿⣿⣿⡿⠁⠙⠛⣫⣿⣤⣵⡀⠀⠀⠀⠀⠀⡎⠀⠀⠀⣈⣻⣟⠋⠀⠀⢻⣧⡀⠀⢀⣴⡿⣻⠏
⠀⠀⠀⠈⣿⣿⣿⣿⡿⢟⠏⠀⠀⠀⣾⣿⣿⣿⣿⣿⣦⣄⠤⠤⡀⠈⠉⠉⢉⣉⣀⣼⠇⠀⠀⠀⢻⣿⣿⣿⠏⠀⠻⠤⠤
⠀⠀⠀⠀⢘⠛⠿⢋⡀⠘⣶⣉⣉⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠳⢤⣿⡉⠑⣬⠁⣾⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣤⣶⣿⣿⣷⣶⣤⣤⣄⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⢀⠤⣌⠉⠀⢹⣿⣿⣿⣿⣿⣿⣿⠏⠀⡇⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠿⠿⠿⠿⠟⠋⠁⢤⠴⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣩⣽⣿⣯⣶⣾⣿⣟⣍⣻⣿⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢀⡤⠤⣄⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣟⠻⠿⠋⣿⣿⠟⠛⠻⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⢀⣠⠴⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣼⣿⣷⣿⣿⣦⠟⠁
⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⠋⠁⠀⣾⣿⣶⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠛⠛⠁⠙⠻⠛⠁
⠀⠀⠀⣰⣿⣿⣿⣿⣿⡟⣻⣄⣴⠉⢸⣿⣿⣿⣿⣿⣿⣿⠛⠛⣦⠀⣤⣄⣀⣀⣤⣤⣶⣿⡇
⠀⣴⠞⠉⠉⢻⣛⣿⣿⣿⣿⣿⣯⣩⣿⣶⣽⣿⣿⠿⣿⣁⣤⠾⠋⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⣠⣤⣤⣤⣤⣄⣀
⠘⣧⣤⡤⠖⠒⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⣠⠤⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⡿⢿⡏⣠⣴⣾⣿⣷⠈⠛⠛⣻⡟⠁⠉⠉⠁⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣞⣥⠖⢿⠿⣿⠿⠉⣿⣷⣶⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠔⠋⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⡿⠗⢿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠎⢀⡴⠀⣠⠏⡀⠀⢄⣀⠒⠦⠤⣄⣀⣀⣀⡤⠤⠀⠀⢀⣀⡠⠤⠀⠙⣆⠀⢢⡀⠑⢄⡀
⠀⠀⠀⠀⣀⠴⠋⣀⡴⠋⣠⡞⠁⠀⠉⠲⢤⣀⠉⠉⠒⠒⠒⠒⠒⠒⠒⠚⠉⠉⢀⣠⠴⠊⠁⠈⠳⣄⠙⠦⣀⠙⠦⣄
⠀⠀⠀⠀⠀⠠⢚⣡⡴⢾⠁⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⢤⣤⣤⣤⡤⠴⠖⠚⠉⠀⠀⠀⠀⠀⠀⠈⡗⢦⣌⣓⠆
⠀⠀⠖⠒⠚⠋⠉⠁⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠉⠙⠛⠒⠒
⠀⠀⠀⠀⠀⠀⢰⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⡆
⠀⠀⠀⠀⠀⠀⡟⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢹⡀⠀⡆
⠀⠀⠀⢰⠀⡸⣡⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢧⠀⡇
⠀⠀⠀⡘⡰⠵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⢧⢡
⠀⢀⣶⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣶⡀
⢀⢛⠩⠁⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠈⡍⡙
⢸⢸⠀⡇⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢰⠀⡇
⢸⢸⠀⡇⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢸⠀⡇
⢸⠀⠀⢁⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠸⠀⠁
⠈⠀⠀⢸⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡆
⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠊⠀⢀⡀⠀⠀⠀⠀⠀⣰⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⡀⠀⠀⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣄⣹⠇⠀⠀⠀⠀⠀⠀⠀⢦⣤⣾⢃⣠⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠸⣿⠟⠋⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠛⠛⢋⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⢱⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣤⣤⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⢀⡸⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠚⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⢠⣿⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⣤⣴⣾⣿⡄
⠀⢀⣼⠏⡀⠁⡈⠁⡀⢀⡀⢀⡈⢓⡦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷
⠀⣸⠋⡄⠁⡄⠁⡄⠁⡄⠁⡄⠁⡄⢁⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇
⢀⠃⠄⠀⠄⠀⠄⠠⠄⠠⠄⠠⠄⠠⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿
⢸⠂⠀⠂⠀⠂⠀⠂⠐⠂⠐⠂⠐⠂⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇
⠚⣷⡃⡀⠃⣀⣧⣄⠃⡐⠃⡀⠃⡐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣷⡀
⠀⠈⠙⠷⣶⠟⠁⠉⠿⣅⡀⢁⡀⢁⣈⣥⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠹⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠛⠛⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠟⠁⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠇⠈⠿⠟⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡾⢷⠀⠀⠀⠀⠀⠀⠘⠛⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠋⠉⢹⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣮⡻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠋⠀⠀⣠⣾⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠻⣿⣿⡌⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠁⢀⣀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣯⣠⣶⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡁⢈⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠛⣿⣿⣿⠟⠁
⠀⠀⠀⢀⣠⣤⣤⣤⣤⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⣿⣷⣴⣿⡿⠋
⠀⢀⣾⣿⡿⠿⠟⠿⢿⣿⣿⣿⠀⠀⠀⠀⣀⣠⣤⣄⡀⠀⣠⣾⣿⣿⣧⣀⣼⣿⡿⠋
⠀⠸⣿⣿⣄⡀⠀⠀⣀⣼⣿⡟⠀⢀⣴⣿⡿⠿⠿⠿⣿⣿⣿⡟⠁⢙⣿⣿⡿⠋
⠀⠀⠙⠿⣿⣿⣿⣿⣿⡿⠟⠁⢠⣿⣿⠋⠀⠀⣠⣾⣿⠟⢿⣿⣶⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣶⣾⣿⡆
⠀⠀⠀⠀⠀⠈⠉⠁⣀⣠⣤⣴⣿⡿⠃⢀⣠⣾⣿⡟⠁⢀⣴⣿⣿⣿⡀⠀⠀⣴⣶⣾⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⣿⣿⡇
⠀⠀⠀⠀⢀⣤⣶⣿⡿⠿⠟⠛⠉⠀⣴⣿⡿⠛⢿⣿⣶⣿⡿⠋⢸⣿⣷⠀⠀⣿⣿⡏⠉⢉⣀⣀⣤⣤⣤⣶⣶⣾⣿⣿⡇
⠀⠀⢀⣴⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⣠⣿⡿⠋⠀⠀⣸⣿⡏⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠉⣿⣿⡇
⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠟⠁⣀⣤⣾⣿⠟⠀⠀⠀⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⢸⣿⡏⠀⠀⠀⠀⠰⣿⣶⣀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠛⠁⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠘⣿⣷⠀⠀⠀⠀⠀⠈⠻⣿⣶⡄⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣿⣿⡇
⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠈⠻⠟⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⠀⣀⣀⣀⡀⣿⣿⡇⠀⠀⠀⢀⣾⣿⡿⠿⠿⢿⣿⣿⡇
⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠸⣿⣿⣄⣀⣀⣠⣿⣿⠇
⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⠀⠀⢺⣿⣯⡁⠀⠀⣉⣿⣿⠇⠀⠀⠀⠀⠙⠻⢿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣶⣾⣿⡿⠟⠁⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠶⠶⠤⠤⠶⠶⠚⠛⠛⠲⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⢀⣀⡄⠀⠀⠀⢀⣀⣀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⢀⠞⠁⠀⣀⡤⠭⣭⣝⡛⠫⠛⠂⠰⣾⡀
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⣠⠖⠂⠀⣠⠖⠚⠛⠀⠉⠀⠀⠀⠀⠀⠻⡆
⠀⠀⠀⢀⡏⠀⠀⠀⠀⢸⣧⢄⡀⠸⠳⣤⣀⡤⣶⠄⠀⠀⢀⡴⠶⠄⢻
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠾⡵⢋⠀⠀⢀⣠⣖⣿⡛⠁⠀⠀⠋⠀⢠⡀⢸⡆
⠀⠀⠀⢸⡄⠀⠀⢀⣤⣄⣵⠏⠀⠀⢚⡽⣻⠏⠀⠀⠀⠀⠠⣤⣬⠧⠈⠳⣄⠀⠀⣀
⠀⠀⣠⠼⡃⢀⡤⡞⠁⡞⠿⠀⠀⠀⠁⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣾⠋⠙⢶⡖⢶⣤⣄
⠀⣼⠁⡞⢰⢻⡇⠃⣀⠀⠀⠀⢀⡀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⠃⠀⠀⠀⠉⠙⠶⣯⣷⣠⢔⡽
⠀⠘⢦⡀⠈⠀⠀⠀⠈⠛⠶⢤⣤⣭⢿⣛⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢤⡀
⠀⠀⠈⣧⠀⠀⠀⢬⣓⢶⡶⡚⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣦⠤⣤⣀⡀⠤⠤⠤⠤⠤⢤⠖⣰⠃
⠀⠀⠀⠸⣆⠀⠀⠀⠉⠉⢠⡟⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠉⠻⡎⠉⠹⠛⠒⣦⣤⣀⡞⣰⠃
⠀⠀⠀⠀⠘⠳⠶⠤⠴⠖⠋⠀⣼⣆⣠⣀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢹⡆⠀⠀⡞⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⣉⠀⠈⠳⠚⢷⣾⣏⠉⠉⠙⠛⠛⠛⠁⠀⠀⠀⢸⠇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠈⠉⠛⠲⢦⣀⠉⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣡⠶⠒⠀⠀⣀⠀⠈⠑⠆⠤⢿⣆⢀⣠⣤⠀⠀⠀⠀⣽⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢘⡅⠀⠀⠀⠀⠈⢷⠀⠀⠀⠸⣷⣽⠉⢠⣭⣀⣀⣀⣠⡏⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⣧⠐⡶⠛⠀⠉⠉⠙⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣴⣒⡚⠣⣄⡀⠀⠀⠀⠀⣀⡤⢀⡞⠀⠀⠀⠸⣿⣥⣀⣀⣀⠀⠀⠀⠀⠀⣀⡟
⠀⠀⠀⠀⠀⠀⠈⢹⡧⣤⣤⠙⠳⡀⣠⡾⢿⡀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠙⠦⣤⣄⠹⡄⠀⠁⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⢀⡜⢣⠈⠙⠁⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢾⢀⠟⠀⠀⣀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠋⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁

⠀⠈⠳⡄⠐⢦⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠄
⢀⣀⣠⣇⠀⢸⠇⠀⠀⠀⢰⡆⣋⢉⡀⡀⣀⠀⠀⠀⠀⢀⣀⣤⠖⠀⠀⠀⠀⠀⣤⣶⣋⣻⣟⠒⣦⡀⠀⢀⣀
⠀⠈⠉⠛⢳⣿⣦⠤⡀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠰⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⣯⣤⡿⠷⣿⣉⣹⣇⣴⠿⠿
⠀⠠⢤⣄⣸⠟⢿⣀⠃⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠛⠿⡀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢰⣶⡶⣶⠠
⠀⠀⢀⣴⣿⣦⡀⠀⠀⠀⠀⢠⢠⡄⡄⣤⠀⠀⠀⠀⠀⣿⣷⣿⣷⣿⣇⠀⠀⠀⣤⣤⣤⣶⣿⣶⣄⣀
⢀⣴⢏⢞⣹⣿⣷⣦⠀⠀⠀⢸⢸⡇⡇⡟⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣻⣿⠁
⠈⠻⠿⢿⣿⣿⣿⡿⠀⠀⠀⠘⡏⡇⡇⡇⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠘⠿⠶⠟⢻⣝⣻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣦⣤⣄⡀
⡞⠲⠶⠤⢤⣤⣤⡤⠤⠶⣶⠒⠛⠉⠉⢻⠀⠀⠀⠀⢸⠀⠀⠀⠉⠁⠀⠀⠀⣠⣾⣿⣿⡟⠛⣿⣿⣿⣦
⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⡿⠀⠀⠀⢀⣼⣀⣠⣤⣤⣼⣤⣄⣀⣀⠀⠀⠀⢠⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⡇
⠘⠲⠦⠤⣤⣧⣤⡤⠤⠴⠿⠒⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠸⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⡇
⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⢠⣴⡤⠀⠀⠻⣿⣿⣿⣧⣼⣿⣿⣿⡟⠁
⣠⡏⠉⡟⠉⢻⣆⠀⠀⣲⡀⠀⠀⠀⠀⢀⣰⠀⠀⠀⢹⣿⡃⠀⠈⣿⠇⠀⠀⠀⢸⣿⣿⠿⠿⢿⣿⣏
⣿⡇⠀⠀⠀⣉⣿⠀⠀⣿⠈⠙⠒⠒⠊⠁⢸⠀⠀⠀⢸⣿⠁⠀⠀⢻⠀⠀⠀⠀⠜⢹⠃⠀⠀⠈⠏⠘
⠙⢿⣷⣶⣿⡿⠋⠀⢀⣿⡀⠀⠀⠀⠀⠀⣸⡀⠀⠀⣼⣿⡆⠀⠐⣿⡆⠀⠀⠰⣦⠄⠀⠀⠀⠀⠀⢶⡆

⠀⠀⠀⠀⠀⠀⢠⣴⣠⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣀⣤⠤⠤⠄
⠀⠀⠀⠀⢠⣶⣿⣏⣿⡜⠿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢡⡞⢁⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠈⠿⢋⣫⣬⣷⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⣾⠟⢋⣉⣉⡛⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠛⠿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣴⣤⡀⠀⠸⣿⣿⣏⡜⣁⠤⠄⡉⠳⣌⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⣀⣀⣀⣀⣈⣙⣿⣷⣦⣹⣿⣿⣕⡡⠒⠒⢌⢦⠈⢦⡙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣶⠿⠿⠛⠛⠛⠛⠙⢙⣿⠟⠛⠛⠛⢿⣿⠿⠿⣶⣾⣦⡱⡀⠑⢄⠙⢿⣦⡀
⠀⠀⠀⠀⣠⣶⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢰⡶⠆⢻⡆⠀⠀⠈⠙⢿⣿⣢⣌⠳⡌⢻⣷
⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠈⠙⢁⣿⠃⠀⠀⠀⠀⠀⠙⢿⣮⢆⢹⢸⣿
⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠶⣿⣟⡁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡞⢸⡿⣇⣀⡄
⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡸⣷⡈⠉
⠀⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⣶⣶⣤⡀⠀⠀⠀⢀⣿⣿⣿⣾⣿⠖⡀
⠀⠀⠹⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⢿⡄⠁⠀⠀⢰⠟⠛⠻⣿⣿⠟⠋
⠀⠀⠀⠀⠉⠛⠻⠿⠷⢶⣶⣶⣶⣶⣶⣶⣶⠶⠿⠗⠀⠀⠀⠀⢀⣼⠋⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⣀⣀⣀⣀⣤⠶⠛⠁⠀⠀⣴⣿⠀⠀⠀⠸⣷⣤⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠉⠉⢿⡇⠀⠀⠀⢀⣤⣾⣿⡏⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⢀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⢿⣶⣶⣾⠟⠋⢁⣾⠃⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣤⣤⣀⣠⣿⣡⡶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠛⢿⣥⣤⣴⠟⠁⣠⡆⠀⠀⠀⢸⣿⠀⠀⠀⠀⠘⠛⠛⢿⣿⣿⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⢀⣴⡾⠛⠿⠛⠁⠙⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠶⠤⢤⣤⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⣿⠃⠀⠀⣀⣤⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠓⠲⠶⠶⠤⠤⣤⣭⣷⣤⣟⣉⣉⣉⣉⣉⣉⣁⣸⣿⣴⡶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢻⣏
⠀⢀⣴⠟⢳⡄⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⣸⠇⠀⢸⡇⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⡼⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⣴⠿⠇⣰⣃⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠿⠶⠞⠛⠛⠻⠿⣶⣶⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⠿⠷⠶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣍⣉⠉⠉⠉⣿⡍⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣈⠉⢿⡿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣴⠇
⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣀⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠉⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⢸⣿⣷⣄⡀⣀⣤⣤⣼⣿⠀⠂⢠⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣤⠄⢈⣹⣿⣿⣿⣿⣿⣿⣿⡿⣶⣿⣿⣿⣿⣿⣶⡄⢀⣴⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣼⣿⠇⠀⠈⠈⣿⡟⣿⣿⣿⣿⣿⡇⠀⠙⣩⣈⠙⠛⠻⣿⣾⣿⣿⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣿⠿⠿⢷⡄⠀⠉⠀⠘⢿⣿⣿⣿⣿⡀⢌⡛⣋⠸⣿⠆⡃⢻⡿⢻⣌⣡⡉
⠀⠀⠀⠀⠀⠀⣤⣶⣾⣿⣿⣷⣋⣤⡀⢴⡶⢿⣾⣿⣿⠓⠈⠀⠀⠉⠉⠃⠸⢿⣯⣷⣶⣾⣿⣿⣷⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⢿⣋⣠⣤⣾⣿⠉⢉⡴⢦⣠⠜⠛⠻⠇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠻⠟⣀⠉⡛⢿⣴⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠈⠿⢟⣿⡍⢸⣠⣿⣿⣿⠃⠀⠀⠀⠀⠀⠘⠀⠀⢂⣴⣶⣿⣷⣾⣿⣷⠀⠀⠀⢹⣿⡏⠁⠉⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣻⡭⣵⣿⡿⠿⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠛⠛⠙⣿⡿⠙⢀⡀⠀⣿⡏⠑⠂⠀⠒⠁
⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣍⠀⠀⠈⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⠠⠄⣹⣷⣿⣿⣿⣿⡿⠗
⠀⠀⠀⠀⠀⠘⠟⠉⠙⢿⣿⡟⠀⢀⣀⣼⣿⣧⠀⠀⣾⣿⠿⢷⣀⡀⠉⠀⠀⠀⠈⡟⣀⠈⢉⢸⡿⣦⣀⣀⡤
⠀⠀⠀⠀⠀⠀⢠⠀⠉⠀⠻⠀⢰⣿⣿⣿⠋⠉⡲⠼⡏⠀⠀⠀⢹⣿⣶⣿⣿⣿⣷⣧⣠⢦⣈⡼⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠘⠀⠂⠀⠀⠐⢾⣿⣿⡟⠐⠀⣙⣛⣉⣁⣀⠉⣈⣿⠀⠀⢹⣿⣿⣿⠟⠿⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢹⣿⡏⠀⢠⣭⣭⣽⣿⣿⣿⣻⣿⣧⡀⠀⢻⡟⠉⠀⠀⠈⠁⠀⠀⠀⠉⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣤⣤⣴⡶⠆⠒⠂⣤⠄⠛⠛⠿⣿⣇⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⠀⣽⡀⢠⣶⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠐⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠛⠲⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⢀⡤⠤⠤⣄⡀⠀⠀⠀⠀⠀⣀⡤⠤⠤⣄⡈⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⠞⠁⠀⢀⣠⣤⣽⣦⠀⠀⢠⣾⣥⣤⣀⠀⠀⠙⢦⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⡞⢀⣠⣾⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣦⣄⠈⡇⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⢳⣾⣿⣿⣿⣿⣿⣿⣿⠇⠀⢿⣿⣿⣿⣿⣿⣿⣿⣷⠇⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⠋⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣠⠴⣶⣶⣶⣦⣀⠉⠉⠉⣡⠔⡪⣉⠉⡩⢖⢤⡉⠉⠉⢁⣠⣴⣶⣶⡶⢤⡀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⡜⠁⢀⣿⣿⣿⣿⣿⣦⠀⣰⠃⠈⠠⠘⠀⠣⠜⠀⠱⡀⢠⣾⣿⣿⣿⣿⣧⠀⠙⡆⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡃⠀⢸⣿⣿⣿⣿⣿⣿⠀⡇⢎⢉⠆⢰⣶⠰⢰⠉⡂⠁⢸⣿⣿⣿⣿⣿⣿⠀⠀⣻⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⢧⠀⠘⣿⣿⣿⣿⣿⡿⠀⢳⡀⢀⠔⢌⢉⠕⢤⠉⠀⠀⠘⣿⣿⣿⣿⣿⡿⠀⢀⠏⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠑⠦⣽⣿⣿⠿⠋⠀⢀⡀⠙⠬⡒⠈⠀⠑⠊⠀⠀⣀⠀⠈⠻⠿⣿⣿⡧⠔⠋⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣶⡄⠀⠀⠀⣤⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⣇⠈⠻⢿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⡿⠋⢀⡇⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⡀⣀⢠⡀⡄⣤⢠⡄⣀⣀⠈⢢⡀⠀⠘⢆⠀⠀⠉⠛⢻⣿⢿⣳⠆⢾⣻⣿⣟⠋⠁⠀⢀⠜⠀⠀⣠⢾⡤⡤⢤⢤⣄
⠀⠀⠀⠀⣴⡞⡏⠁⠀⠀⠀⠀⢸⠀⠀⠁⠈⠉⠃⡿⠲⢤⡄⣙⠒⠶⠖⣿⢏⡀⣇⠀⠀⠀⠈⡟⠛⠸⢞⡧⣤⣞⠛⠉⠀⠀⠀⢸⡇⣹⠂
⠀⠀⠀⢸⠛⠀⢻⠀⣀⣀⡤⣶⠾⠹⠏⠛⠰⣶⠶⣧⡀⠀⠀⠉⣾⠱⠶⡧⣨⣗⡙⠘⠋⠛⡴⠳⣤⣀⠀⢨⡏⠘⠓⠛⢶⡲⠆⠾⠽⡟⡁
⠀⠀⠀⠘⢆⡠⡼⠟⠋⢹⡇⣿⠀⡀⠀⢠⠆⡟⠀⠀⢹⣦⡄⣀⡟⠀⠀⠀⠈⢸⠋⠛⠶⠰⠇⢦⢿⡟⠺⠾⣧⣠⣀⣀⢸⠀⠀⠀⢀⣧⡇
⠀⠀⠀⠀⠸⡷⣆⣤⣀⢸⡀⠈⢸⡃⠉⣹⣴⡇⣄⣶⡟⠀⠈⠁⠛⠱⢆⣤⣀⣸⠀⠀⠀⠀⠀⢸⣨⡇⠀⠀⠀⠈⠉⠑⠛⠙⠋⠛⠚⠉
⠀⠀⠀⠀⠀⢧⢸⡆⠉⠙⠓⢻⠟⠛⠛⠛⠚⡟⢁⡼⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠓⠓⠰⠆⠖⠚⠋
⠀⠀⠀⠀⠀⠀⠛⠵⣆⣤⣄⣸⡀⣀⢀⡀⣼⠳⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠁⠀⠁⠐⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠀⠂⠐⠈⠁



⠀⠀⡀
⠀⠀⠀⠨⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠡


⠀⠀⠈⠀⠀⠂⢀
⠀⠀⠀⠀⣀⣀⡸⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠣⣀⣀⡀
⠀⠀⠘⠋⠃⠀⠀⠀⠀⠲⠠⠖⠐⠀⠄⠄⠒⠺⢛⣿⠟⠛⠛⢿⡛⠛⠒⠠⠄⠖⠒⠐⠂⠐⠀⠀⡀⠠⠀⠋⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰⡄⡀⠈⠉⠀⠀⠀⣄⣾⠁⠀⠀⠀⠈⢱⣄⠆⠀⠀⠈⠉⠀⡀⡎⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⡆⣠⡍⠀⠈⠉⠀⠈⠉⠀⠘⣤⢀⡇⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠟⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠛⠒⠒⠶⠶⠦⠤⠴⠶⠶⠒⠓⠚⠒⠤
⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⣄⣶⡐⠨⠁⠈⠑⠄⠁⡤⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⡒⠃⠀⠀⠀⠀⠀⢂⠀⡀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢠⢈⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⢀⡀⠀⠈⠂⠀⠐⠀⠀⡠⠄⠀⠈⡆⠀⠀⢀⠈⠈⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⣀⠠⠪⠀⠀⠀⣠⡁⠀⠀⠈⢑⠀⡀⠠⡇⢀⠔⠁⠱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣐⡄⢰⣦⠀⠀⠘⣿⠀⠀⠀⢀⠢⠀⠆⠘⠁⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠈⠿⠃⠀⠀⠀⠀⠀⠀⡈⠀⡀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠃⠐⠀⢠⠊⢀⠰⠁⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⢠⠀⢉⠠⣥⠒⢁⠌⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠃⢠⠀⠀⢀⣀⠀⠤⠄⠒⠂⠉⠄⣦⣤⡆⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⡕⠂⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠉⠀⠀⠀⠈⠉⠉⠉⡙⠀⠀⠘⠋⠉⢹⠤⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠄⠀⢸⠷⠿⠿⢿⠀⡀⢀⠄⠂⠀⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⢠⣷⢶⣤⣤⣼⡆⡀⠌⠳⠤⢔⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢲⠙⠿⢿⠠⢍⣯⣽⣌⢁⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⠈⠋⠉⠉⠈⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠉⠁⠚⠊⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠒⠀⠁⢢⣀⡀⠀⠀⢡⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡔⠁⠑⣼⠄⠀⠈⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⡶⠃⢀⡇⢀⠥⠀⠊⣡⠤⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⢀⣠⠶⠋⠀⠰⠀⡀⠊⠁⠀⢀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⠉⠀⠀⠀⠀⠀⠁⠈⠈⠀⠉⠄⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⠀⠀⠙⢷⠀⠀⠀⠀⠀⠀⠀⠀⢷⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⣀⣤⣶⠾⠿⣷⣶⣾⣇⣀⣤⣴⡶⠶⠶⠶⠶⠶⢶⣤⣄⣀
⠀⠀⠀⢠⣾⠟⢉⣠⣴⣶⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣶⣄
⠀⠀⠀⣿⣥⣶⣿⣿⣿⣿⣿⣿⡿⠁⣀⣤⣶⣶⡿⠿⠿⠿⢿⣶⣶⣤⣀⠀⠀⠙⢷⣦
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠿⠋⣠⣾⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣷⣄⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠉⣹⡟⠉⠀⠀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀⠀⠈⢿⡄
⢀⣀⡏⠂⠀⢰⡟⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠈⣿⡄
⠿⠿⠃⠀⠀⣾⠃⠀⠀⢀⣿⡇⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠸⣿⡆⠀⠀⢸⣧⣴⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣧⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣦⣤⣿⡇⠀⠀⢸⣿⡿⣿
⠀⠀⠀⠀⠀⣿⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠀⠀⣿⣶⣤⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡄⠀⣾⠏⠀⣿⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⢻⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⢹⢹⣿⣷⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⢸⡟⠀⢠⠇⠀⠀⢀⢜⣥
⠀⠀⠀⠀⠀⠘⣿⡀⠸⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣾⡾⠟⠋⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⢸⣇⣴⠋⠀⠀⠀⠘⢌⣡
⠀⠀⠀⠀⠀⠀⠘⣷⡀⠙⠻⢿⣿⣿⣿⣿⠀⠀⠙⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⢋⣴⠿⣿⠁⠀⠀⠀⠀⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⣰⡟⠁⠀⣹⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠀⢀⣾⡿⣿⠛⠻⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⣀⡀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠾⠋⢹⣇⠀⠘⣿⠀⢹⡀⠀⠀⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠁⢠⡄⠀⠀⢿⣄⠀⠈⠁⠸⣇⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠀⠀⠀⠀⠙⠿⢶⣦⣤⣿⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⢾⢳⡄⠀⠀⠀⠀⣠⣤⣄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠋⠀⠀⠀⠀⢰⣿⠿⠟⣸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⣾⣿⣿⣿⣷⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠛⠿⠟⠛⠿⠿⠿⠿⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠳⣦⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠈⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⢙⢟⡿⣿⣿⠃
⠀⠀⠀⠀⣄⣄⢀⡀⠂⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⡀⣀⠀⠀⠘⡿⠀⠀⠶⠀⠠⠠⠀⠀⠶⠀⠀⠀⠀⠀⢿⢈⣿⣿
⠀⠀⠀⢹⣟⠉⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢾⡿⠛⢿⡂⠀⠀⡁⠀⠀⠀⠠⠄⠠⠤⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠢⡑⠢⢀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢃⠤⢂⠝⠛⠒⠚⡑⠠⣀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⠁⠂⢬⢀⠀⠢⠀⠀⡔⠒⢂⡀⠕⠈⡀⠐⠁⠀⠀⠀⠀⠈⠐⢄⠁⠒⢆⣠⢀⣀⡀⠀⢀⣠⣤⣴⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡅⠀⠁⢊⠐⠒⠉⠊⠀⢢⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢤⠁⠑⠽⠳⠀⢸⠿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠒⠈⠀⠀⠀⠱⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠎⠀⠀⠈⠢⡀⢉⠿⠿⠟⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡻⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢔⠔⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠒⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⡠⠢⢤⡀⠀⠧⠄⣀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀⠠⠰⠂⢀⣤⢖⠢
⠀⠀⠀⠀⠀⠀⡬⠀⣤⡅⠈⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⢦⡤⢱
⠀⠀⠀⠀⠀⠀⠑⠒⠁⠈⠂⢄⠀⢀⡀⠄⠐⠈⢂⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠊⠁⠂⠠⠄⣀⡠⠐⠁⠀⠑⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠔⠈⢣⡴⠢⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⢲⡎⠁⡺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠑⢈⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠋⢉⣉⣉⠉⠙⠻⢿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⡠⠀⣀⣤⣤⣴⣤⣤⣤⣄⣠⡀⠀⠀⠀⠀⠀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣶⣤⣌⠻⣿⣧
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⣸⣿⣿⣿⡿⠟⠙⠿⣿⣿⣿⠿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣿⣿⣿⠛⠉⠀⠀⠀⠀⠉⠉⠛⠛⠿⣿⡿⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠟
⠀⠀⠀⠀⣸⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⢰⠁⠒
⠀⠀⠀⠀⠧⠒⡄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣠⡀⠀⠀⠀⠀⠀⠀⢠⣶⠁⢤⡠⢦⢂
⠀⠀⠀⠀⠀⣤⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⡀⠀⠀⠀⠀⠀⠐⠂⠈⠀⠀⠀⠀⠀⠀⢀⣾⣿⡄⠀⠀⠈⠓
⠀⠀⠀⠀⢰⣿⣆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⡅⠀⠄⢠⣀⣀⢀⠀⠀⠀⢀⣀⣤⣤⣾⠿⠛⣁⡄
⠀⠀⠀⠀⢸⣿⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠂⠀⠁⠠⠈⠙⠟⢿⡿⠀⠸⢿⡿⠛⢋⢀⣤⣾⣿⠁
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣾⣷⣿⣿⣿⠀⠀⠀⠀⠀⢻⣄⣀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⡿
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠘⢿⣿⣿⣿⡇⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡃
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣝⠻⢷⠄
⠀⠀⠀⠙⢿⣿⣿⣿⣿⡏⠘⠁⠀⠀⠀⠀⠀⠉⢿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠐⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠒⠁⡀
⠀⠀⠀⠀⠀⠀⠈⠑⠉⡘⠀⠂⠀⠀⠀⠀⠐⠀⠋⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠰⠄⣀⠀⠀⠀⠀⠒⢆⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠠⠤⡄⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⡉⠀⠀⣁⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⡅⠀⠀⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠋⡹⠂⠀⠀⠤⠬⠅⡂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣶⠶⠾⣿⣟⠛⠉⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⠾⠻⢿⣟⠋⠛⢿⣄⠈⠙⢷⣄⠈⠙⢷⣄⣀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠉⠙⢷⣄⠀⠙⢷⣄⠀⠙⣷⣄⣠⣽⣷⣴⠶⠿⠿⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣀⣠⣤⣽⣷⡶⠶⠟⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⢻⡿⠛⢛⣿⠟⠛⣻⡿⠛⢛⣿⠛⠛⣻⡟⠛⢻⡿⠛⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣴⠟⠀⣠⡾⠃⢀⣼⠏⠀⣴⡟⠁⢀⣾⠋⢀⣴⡟⠁⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠀⠀⠀⠀⠀⢸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡾⠇



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

⠀⠀⢢⠰⠀⡆⠦⠰⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡆⢰⢠⠆⡄⠴
⠀⠀⠈⠀⠀⠆⡆⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠨⠘⠀⠅⡇
⠀⠀⠠⠀⠐⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠁⠁⠀⠃
⢀⣠⠂⠄⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠘⢄
⠀⣿⣿⠻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡟⡻⣻
⠀⠘⡠⢂⢋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠐⡙⢾
⠀⡀⡀⠈⡌⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠃⠃⢀⣸
⠀⠉⠘⡂⢄⢁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡠⢐⠆⠘
⣈⣉⠉⠦⠮⣀⡀⠲⠖⠩⡉⡱⠶⠀⠠⡀⡶⠊⠉⢉⣲⡆⠀⠀⢰⣖⠈⠉⠑⢦⣀⠀⠀⠶⢆⢉⠉⠲⠤⢄⢀⠱⠶⠨⣉
⠈⠁⢲⡔⠀⠉⠁⠀⢢⡄⠉⠁⠀⠐⣤⠉⠁⠀⠀⠲⡟⠀⠀⠀⠀⢻⡄⠀⠀⠈⠙⣤⠂⠀⠈⠉⠲⡔⠀⠈⠉⠀⢢⡄⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⣦⡤⣀⣄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠋⠀⢀⠀⠀⢤⡤⣀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠃⠉⠳⣤⣀⠈⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡃⢽⡶⣄⠈⠑⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⡀⠀⠀⠀⠀⠰⠂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠀⠀⠀⠀⠀⢯⡛⠶⣿⣯⣌⣲⣤⣄⠲⣴⡆⣀⡀⢤⣤⠤⠀⣀⡀⠀⡠⠀⠀⠈⣊⣤⠎⠳⠠⠄⠀⠙⠡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠓⠊⢤⣴⡒⠦⣛⣴⣤⣤⣄⡀⠀⠈⠓⣤⣤⣀⠞⠀⠀⡀⠚⢩⡼⠂⠀⠁⠀⣀⠴⢚⠤⣴⠶⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠁⠀⠀⡐⠦⣤⣀⡀⠀⠈⠛⠷⠾⠇⣻⢿⣤⣿⣿⣟⡷⣮⣞⠉⠉⠒⠊⠀⠀⣀⣀⠠⠀⠂⠩⠐⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠆⠠⠄⣸⠍⠙⠻⠻⢏⣹⠷⢋⣙⠛⠛⠁⠀⠀⠠⠖⠋⠉⠀⠀⠀⠒⠀⠁⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⠤⡀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠈⠀⠈⠉⠉⠴⠶⠾⠀⠀⠉⠀⠀⡀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠙⠛⠓⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠊⠳⡈⠳⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠤⠤⠤⠀⠀⠀⠒⠂⠈⠉⠉⠉⠉⠐⣀⠤⠷⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢿⡀⠸⡄⠀⡀⠐⠠⣄⣀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⢫⣖⠴⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠁⠀⣧⣄⣈⣓⠒⠀⠉⠉⠒⠠⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠐⠂⠀⠀⠀⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠃⠀⠤⠄⠉⠉⠉⠈⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠼⠴⢦
⠀⠀⠀⠀⠀⠀⠀⡴⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⣴⢉⡁⠁⣦⣄⠶⡄
⠀⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡂⠀⠀⠠⣀⠋
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠁⠀⠀⠀⠈


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⣝⡛⣋⡙⢻⡟⣫⣿⢻⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠿⢿⣿⠿⠟⠛⠛⢿⣿⣧⣠⣾⣿⣿⣿⣿⣿⣶⣅⠄
⠀⠀⠀⠀⠀⠀⢸⣿⡿⠻⠛⠛⠛⠉⠋⠙⠉⢻⠆⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⡿⠿⠻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⡾⣍⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⢀⠀⠀⠸⠿⡟⠉⠁⠀⠀⠀⠀⠀⠙⣿⣿
⠀⠀⠀⣤⣄⡀⢀⡇⠀⠀⠀⠀⠀⣤⣠⡄⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠂⠀⠁⠀⠂⠀⠈⠀⠀⠀⢀⠀⢠⠀⠀⠀⢀⠀⠀⠀⢸⠿⠄
⠀⠀⠀⠛⠤⠅⢼⣿⣄⡀⠀⠀⠀⠉⠉⠀⠀⢀⡄⡀⠘⠉⠛⠉⡀⠐⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠠⠀⡠⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠑⠈⡙⠿⣿⠶⣦⠀⡶⣶⡾⢛⠋⠉⠀⠀⠀⠀⠈⠂⠀⠈⠲⠐⢀⠀⡐⠀⠈⠀⠈⢠⣀⠀⠀⠀⠀⣀⣠⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢀⣈⠍⠀⢉⢀⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠏⢀⠝⢂⠀⠀⠀⣼⣿⣿⡟⠀⢻⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⢘⠔⠱⠐⠉⠂⠀⠰⣿⣿⡟⠀⠀⠠⡈⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠀⢂⠀⠈⠄⠀⠀⠀⠀⠀⠀⣶⢢⡄⠁⠀⠀⠀⠈⠄⠀⠙⠟⠀⠀⠀⠀⠈⡸⠟⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠢⡀⠀⠀⠀⢀⡍⠁⠀⠀⠀⠀⠀⠀⣠⠀⡀⡎⠀⠀⠀⠀⠀⢙⡀⡀
⠀⠀⠀⠀⠀⣄⣈⡋⠈⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠐⠒⠂⠁⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠉⠋⠀⠀⠐⠀⠀⠩⠉⠁
⠀⠀⠀⠀⠀⠐⠁⠀⠀⢲⢒⣋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠊⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢤⠂⠴⠁⠀⠀⠀⠈⢢⠼⠂⠀⠸⠬⠀⠡⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⠙⠃⠀⠀⠀⠀⠀⠡⠂⠀⢤⣾⡗⠀⢸⡿⢄

⣶⣶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⣀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢆⣼⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣧⡘⠀⠄⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣻⡿⢿⣿⣿⣿⠉⣴⣿⣿⡿⠗⠉⢋⠉⠙⠁⠀⠉⣡⣤⣄⢀⡀⠈⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠐⠿⢇⣬⣥⣭⡹⣄⠹⠿⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⢿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⣿⣿⡿⠃⣹⡷⠀⣿⢾⣿⣿⣿⣟⣷⣶⡄⠀⠀⡈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⣸⠁⡀⠀⣩⣷⡏⣰⠘⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣴⣶⣦⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⢸⡟⢾⣿⣏⣠⣥⣼⣿⣛⣟⣛⣿⡟⠀⠀⠀⠀⠹⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⠇⡀⢹⣿⠋⠁⠀⠂⠙⠀⠉⡛⢿⣷⣦⡄⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣧⠐⣥⡾⠃⠄⠈⠛⠀⠀⠀⠀⠐⠰⡦⡉⠁⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣙⢤⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡐⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡿⣿⠀⢰⣦⡀⠀⠈⠀⣀⡀⠁⠀⠀⠀⣀⠀⠆⢹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⠀⢸⣿⣃⠀⠀⠀⠈⠙⠲⠖⠂⢹⣿⢷⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠁⢸⣿⣋⡓⣶⡶⠦⣀⡀⢀⣸⡿⣿⣆⠀⢊⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡿⣿⡿⣿⣿⣿⣿⣗⣀⣛⢋⡿⣿⣿⣿⢿⠀⣘⠧⢒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠛⠻⠷⠛⠋⣿⡻⣿⡋⠉⢉⣿⣿⣿⣿⢿⡀⠑⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢻⣷⡷⢠⣇⢨⡦⠂⣠⣄⡀⠀⠉⠉⠘⢣⡱⢤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧⣀⡄⠀⠁⠀⠀⠀⠀⠀⠀⣟⡀⢀⣀⣀⡴⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⢿⡿⠾⠿⠶⠀⠀⠀⠸⢦⣿⠿⢿⢟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⢒⠒⠒⡂⠠⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊⠔⢠⠄⠦⠘⠷⠀⠀⠀⠓⠉⣒⣄
⠀⠀⠀⠀⠀⠀⡠⠨⢄⠀⠀⠉⠀⠀⠀⠐⠀⠒⢣⡀⠠⠶⢠⣑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡰⠵⠄⠀⠹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢫⣅⠉⠹⡀⠀⠠⢀⡔⠉⠀⠂⡤⠐⡄⠂⢐⡠
⠀⠀⠀⠀⢀⢁⡌⢉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢬⣥⠄⠁⣘⡀⠀⠦⢀⣈⠁⠈⢁⠸⠠⠁⠀⠠⡀
⠀⠀⠀⠀⢸⢋⡀⡜⠁⠀⠀⠀⠈⠂⠀⠀⠀⢠⣤⣤⣤⣤⣀⠀⢻⠚⢗⡤⠐⠊⠀⠀⠀⠀⠀⠀⠐⠠⡤⠀⠑⣌⠄
⠀⠀⠀⠀⠀⠪⣥⠃⠈⢀⣠⣤⣴⢞⢠⠖⠒⣀⠉⣋⠭⠭⢽⠉⠘⡧⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠢⠌⠁
⠀⠀⠀⠀⠀⠀⠸⡄⠴⠫⣋⠭⠽⢤⡙⡄⠀⠻⠘⣤⣔⣒⡿⠀⠘⡇⠈⠀⣀⣀⡠⢂⠄⠀⠀⠀⠀⣄⠀⠈⢯⠋⠳⡆
⠀⠀⠀⠀⠀⠀⠀⢇⠀⠘⠥⣄⣒⠾⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢿⣚⠛⣛⡿⠛⠡⠄⣀⠰⢦⣀⡀⠐⢇⢾⣑⢂⠃
⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒⢤⣶⡄⢰⣼⢇⠤⠄⣹⠂⡇⠀⣰⢄⣘⣛⡻⢶⣄⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⡀⢀⣄⡜⠁⠶⣦⣤⡼⠋⠀⠀⡘⣇⣾⡗⠀⠉⠉⠁⠀⠀⠀⡄⠩⣦⠄⠈⢢⠉⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠱⡌⢿⠠⠀⠀⠀⠐⠛⠀⠀⠀⢄⣼⣿⣀⡀⡠⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠚⢤⠢⡀⠀⠀⠀⠀⡠⠊⠈⢁⣷⠘⠀⠁⠈⠻⣤⣀⣴⠌⠳⢤⡀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡈⠁⠀⠩⣉⣡⡤⠔⠁⠀⢠⡟⠀⠔⠀⠒⠀⠼⠧⢀⡀⠀⣌⡏⣠⡧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⢀⣀⢀⢀⠜⡃⠀⢂⠠⠄⢀⡀⠀⠀⠈⣂⣸⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠴⠶⠾⠿⠛⠁⠀⢱⠀⠀⠀⠲⢶⠲⠭⠒⠜⣻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⡀⠀⣀⠀⠀⠀⢰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠷⠖⠒⠁

⠀⠀⠀⣶⣶⣶⣴⣄⣀⣴⣴⣤⣤⣤⠤⢤⣤⣴⣴⣶⣶⣤⢄⡶⢶⡤⡶⠶⠄⠒⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂
⠀⠀⠀⣿⣿⣿⣿⣿⠿⡇⣹⡟⢋⣽⡶⠾⢿⣻⣿⣽⣿⡛⠒⠒⠿⣶⣿⡟⠀⠀⠅
⠀⠀⠀⣿⣿⣿⣿⣷⣦⣿⣿⡇⢻⣷⣄⣠⣼⣿⡿⢿⣿⣷⣤⣤⣤⣸⣿⣿⠙⠆⢠⠌
⠀⠀⠀⣟⣉⣽⣉⣼⣿⣿⣿⣻⡟⠿⣿⠿⣿⢉⠀⡺⠟⠻⠿⠿⢿⣿⣿⣿⠿⣄⠣⠀⠀⢀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⣾⣿⠿⣿⣻⣻⢿⡟⣹⣿⣾⣭⣦⣿⣿⣿⡛⠓⣶⣶⢶⣦⣄⡉⢿⣿⠊⠠⠄
⠀⠀⠀⣼⣿⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣧⣿⣿⡇⠀⠀⣀⡈⢻⣽⣷⣿⣿⢛⡿⣦⠄
⠀⠀⠀⣼⣿⣿⣿⣿⣟⣫⠛⢻⣿⣺⣿⣿⠟⠛⠉⠀⠓⠦⡽⠲⠟⢿⣿⣿⣟⡈⠉⡚⠃
⠀⠀⠀⢿⣿⣿⣿⢻⣿⣾⣷⣿⣿⡟⠛⠁⣤⠞⠀⠀⠈⠀⠀⣀⡤⠀⣻⣟⠻⠶⠋⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⢼⣿⣿⣿⡧⢹⣿⣯⣽⣿⣿⣷⣤⣤⡄⠀⠀⠀⢀⡞⠁⠀⣄⣼⣿⡧⠀⢦⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠠⡆
⠀⠀⠀⣿⣿⠋⢿⣵⣦⣾⡟⢻⣿⣿⣿⣿⣿⠇⠀⠀⠀⢸⣧⣦⣶⣿⣿⡿⠃⢀⣿⡘⢛⡻⣿⡷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⢸⡇
⠀⠀⠀⣼⣿⣿⣿⣿⣌⣭⣥⣼⣏⣀⣹⠁⠈⠀⠀⠀⠀⠈⣿⣿⠿⠟⠋⣀⣴⢿⣿⠟⠙⠛⠛⢛⣫⣾⠆⠴⠂⢰⡦⠀⢀⠋⠀⠀⠀⢀⠀⣦⠸⡇
⠀⠀⠀⣿⡙⢋⣿⣿⢟⣽⣍⠻⣦⣀⡿⠀⢀⠀⠀⠀⠀⠀⠘⠵⠦⢦⢚⠻⠙⣶⣿⡇⢠⣶⣿⣿⣿⣾⡆⠐⠀⠀⠑⠀⠀⠂⠀⠘⠀⠘⠀⢈⠀⡇
⠀⠀⠀⣿⣻⣟⢻⣿⣿⣿⣿⣶⠞⣻⠃⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⣿⣿⣿⣧⣿⣿⣿⣿⡿⢟⠘⠀⠀⠀⠀⢀⡘⡀⠀⠀⠀⠀⠀⠈⠀⡁
⠀⠀⠀⣿⣟⣿⣿⣿⣿⣧⣤⣏⣹⡟⠀⣴⠇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣏⣁⣤⣽⠛⢿⣿⣿⡷⠏⠀⠀⠀⡠⠀⠈⡀⢣
⠀⠀⠀⣿⣿⣿⠿⠟⣿⠟⣟⠫⡿⠃⣸⣟⡅⠀⠀⠀⠀⠀⠀⠀⠤⣄⠀⣛⣛⠛⢻⣿⣷⣤⣤⣀⣀⣀⣀⣀⠼⠛⢀⡴⠃⣼
⠀⠀⠀⣿⣿⠯⠛⠛⢡⡀⠉⢀⠣⣠⣿⣿⠀⠀⢠⡇⠀⠠⠀⠯⢠⣋⣛⡸⠃⠀⣼⠏⢈⡛⣷⣬⣿⣬⣉⠑⠲⠒⢋⣴⣾⣿
⠀⠀⠀⡿⢿⡖⠛⠡⠀⠈⢻⠇⣴⢢⣽⠟⢠⡴⠋⠀⡀⢀⣀⢀⠉⠹⠟⢀⣤⣞⡥⣶⡾⣥⣤⣿⣿⣍⣉⠋⢲⣶⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣿⣈⢠⣈⡠⣬⡎⢤⡶⠇⢘⣿⡟⢌⠠⡄⠀⠐⣊⠿⡿⠒⢴⠚⣵⣿⡿⣿⣾⣻⣿⣷⣿⣷⣿⣯⣽⣿⣿⣿⣷⣿⣷⠖⣯⣄⡀⠀⠀⠀⠃
⠀⠀⠀⣿⣿⣿⠁⠀⡘⢳⠞⣧⣰⣿⣿⣩⡊⣧⡌⢙⣈⣛⡦⣼⠳⣿⠹⡴⣛⣻⠿⣿⣟⣫⡿⠟⡉⣟⣉⣴⣴⢿⣿⣿⣏⣻⡛⢳⢤⣍⠀⠾⠘⡇
⠀⠀⠀⣿⣿⠾⠿⣿⣯⣾⣾⣿⣿⡿⢿⣇⣀⣶⣯⡘⡟⢿⣦⡄⠀⠉⠀⠀⠤⢼⡧⡶⣶⣶⠾⣿⡟⠻⡘⢀⡀⠠⠷⠼⠂⠛⠃⠄⠜⢃⡈⠀⠀⡇
⠀⠀⠀⣟⣻⣓⢜⠻⠻⣏⢿⣿⡿⣿⣿⣿⣿⣿⢿⠟⡿⢰⣿⣿⣥⣶⣶⣄⣀⠀⠛⠣⠀⠩⠶⡮⢃⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠃⠀⠀⠀⠀⠀⠇
⠀⠀⠀⡏⠉⢁⠈⠓⢀⣤⣄⠉⡽⢿⣿⣿⣿⣿⠶⠿⠖⣻⠿⡿⢫⡟⠿⠥⢉⣻⣤⣀⣤⢄⠀⠀⡁⠀⠠⠀⣀⠀⠀⢀⠀⠀⠀⠀⠐⠂⠂⠀⡀⡆
⠀⠀⠀⡇⠈⠀⠈⣓⡄⢁⠀⠌⠲⣿⣻⣿⣿⣿⣤⣤⢙⠛⠀⠠⠐⠀⠆⡐⠳⠈⡈⡉⢿⠀⣀⠀⠁⠀⠀⠠⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⡇
⠀⠀⠀⡛⣆⠀⠤⠈⠀⠁⠐⠀⢈⣻⡏⡼⠻⢻⣾⡃⣺⠟⠛⠁⠈⠂⠀⠈⠀⠀⠀⠀⠀⠀⢄⠔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⣇⠈⠰⠱⠆⠀⠀⠐⠀⢀⠃⠤⠉⠀⠦⠁⢉⠋⢰⡷⠤⠄⠀⢠⣀⣌⠀⠄⠀⠀⠀⡀⡀⠑⠄⡀⠀⠈⠁⠐⠚⠂⢀⢈⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠡⢓⠂⠀⠀⠁⠢⠀⠀⠸⠓⠀⠀⠠⠜⢠⣤⣀⠡⠾⣁⡄⠆⢠⠀⠆⢀⣤⠀⠀⠀⠀⠃⠀⢀⠀⠀⠰⠀⠁⠀⠁⠉⠉⠀⠀⠄⠀⠀⠀⡀
⠀⠀⠀⡄⠃⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠀⠀⠈⠀⠃⠼⠋⠀⠀⠐⠆⠈⠀⠉⠘⠏⠀⠀⠀⠆⠐⠤⠎⠀⠀⠐⠀⠀⢀⠘⠂⡠⠤⣄⠀⠀⠀⢠
⠀⠀⠀⠓⠀⠄⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠂⢀⣀⠀⠈⠀⠀⠀⠀⠀⢐⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣆⡙⠻⣿⣶⡄⠄⠀⠀⡠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡐⠺⢿⣷⣤⡙⠿⠟⠂⠉⠀⠀⠀⠀⠀⠀⠁⠠⠼⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢶⣶⣤⡉⠻⡿⠷⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠒⢢⡙⣋⣉⡀⣀⣀⡀⢀⣀⣀⠀⣀⣀⡀⠀⣀⣀⡀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠃⣳⣿⡏⢠⣿⡟⢁⣼⡿⠁⣰⣿⡟⢀⣼⣿⠏⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢹⠶⠂⠦⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⠀⠒⠷⣦⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⣿⠖⣤⡜⠛⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠈⣟⢦⣽⣎⣒⢠⣈⡉⢄⣀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠫⠆⡠⢄⣀⠀⠈⠶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠓⠪⠋⣙⡯⣭⡹⣠⣉⣀⡉⠉⠉⠣⣀⣁⠀⠔⠁⠄⠠⠞⣡⠅⠀⠀⠀⠀⡮⢄⣠⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠑⠀⠀⠐⢤⣄⡀⠀⠈⠛⠷⠼⡏⣻⣯⣽⣿⣷⣶⣼⡭⠙⠂⠒⠀⠀⠀⢁⡀⠀⠐⠃⠅⠂⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠳⣄⠀⠀⠀⠀⠀⡀⠈⣙⠳⢂⣦⣜⣎⡙⠛⠻⣛⣿⢾⣍⢛⡛⠁⠀⠀⠠⠚⠉⠁⠀⠀⠂⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣷⢤⣀⠀⠀⠀⠀⠀⠙⠒⠀⠈⠀⠀⢈⠐⡓⠺⠛⠃⠙⠁⠠⠤⠀⣀⣤⣅⡀⠀⠈⠀⡘⠋⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⢧⠈⢷⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠓⠒⠒⠂⣀⠈⠉⠉⠀⠀⠈⠀⠀⣤⡐⠛⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠙⢧⠀⡇⠀⢠⣈⠒⢦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠸⡗⠃⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠟⠓⠲⠤⠭⠀⠀⠀⠈⠶⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⢄⡀⣀⡈⠁⠉
⠀⠀⠀⠀⠀⠀⡉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⠍⠉⠀⡈⠝⠀⠁
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠈⠃⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⠤⣌⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⢰⠁⣘⠄⣀⠛⠄⡀⠂⠌⠀⢀⣀⣀⣀⠀⠤⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠌⡘⢀⡏⠔⠂⠚⠂⠉⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠌⡰⠤⠌⠴⠃⣘⠴⡄⠀⡤⣤⠒⣶⣒⢆⣜⣡⣌⡹⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠐⢠⠁⡊⠀⠀⠀⠀⠀⣯⣱⡇⠀⠁⢉⣉⣉⢉⡁⢤⢄⠀⢤⠐⡄
⠀⠀⠀⠀⠀⠀⠀⣀⠂⠌⣑⠦⠄⠀⠒⠚⢶⢯⡇⠀⣭⡿⣼⣛⠾⣽⢃⠎⣐⠢⠌⡁⠀⠉
⠀⠀⠀⠀⠀⠈⡐⠠⡎⠁⠀⣀⡤⣴⡖⣶⣸⢿⡇⠀⣿⣽⣳⣯⣿⢣⣏⡲⢠⠋⠴⠅⠀⡦⢀
⠀⠀⠀⠀⠀⠀⢀⠂⠴⠀⠀⠯⠓⣋⣩⣭⢯⣿⡇⠀⣿⣿⣟⡾⣭⣛⡾⣱⠃⡌⣑⠇⠀⡵
⠀⠀⠀⠀⠀⡀⠂⠌⡰⢇⠀⢰⢏⡟⣼⣟⡿⣽⡇⠀⣿⣿⣿⣽⢧⣟⡳⢧⣝⢰⣁⠄⠀⣇⠂
⠀⠀⠀⠀⠀⢀⠀⠠⢁⠺⡀⠀⢿⡽⠗⠛⠛⠋⠁⠀⣿⣟⡿⣯⣟⡶⠋⠁⠀⠀⠀⠀⠀⡏⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠂⡘⢃⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⡾⣹⠿⣼⠀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠠⠀⠐⠠⢙⡀⠀⡆⠀⠀⠀⠀⠀⢀⡟⡳⢯⡟⠰⢦⡀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣱⠀⠸⡤⢤⣀⡠⢴⠛⠤⢁⠒⡬⢁⠆⡌⠃⠎⡐⠈⠉⠐
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠉⠀⠀⠀⢇⠂⠬⣑⠂⡌⠑⠎⡜⠠⢌⡐⠠⠉⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⢸⡌⠠⢀⡘⠠⢁⠒⠠⠁⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⡜⠀⠁⠀⠀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⡆⢠⡆⠀⠀⠀⠀⡠⠷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⢸⡇⢸⠃⠀⠀⠀⠀⠱⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⢝⠀⠀⠀⠀⣿⣷⣄⠀⠀⠀⢸⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠞⠈⠀⠀⠀⠀⣿⠙⢿⣷⣤⣀⠸⠀⠄⣀⣼⡄⠀⠀⠀⢀⣠⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠈⠛⠿⢿⠿⠿⠛⠻⢿⣶⣶⠿⢿⣿⠏⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⣤⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⠀⠀⢿⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠻⣯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⣠⠞⠋⠉⠉⠉⠙⢲⣾⡆⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠞⠀⢀⣿⣆⣀⠀⠀⣤⢶⡀⠀⠀⠀⡇⠀⢻⡧⢀⣴⠾⢿⣶⣤⠿⢤⣧⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠲⢦⣴⣶⡶⠛⠉⠉⠉⠙⠲⣿⣞⠙⢦⡀⢸⠃⠀⠀⢻⣾⡁⢸⡏⠀⠀⠀⢸⣿⣄⣀⣗⠒
⠀⠀⢲⠒⠣⠀⠀⠀⠀⠀⢹⣿⡄⢀⣤⣶⣦⣄⠀⠈⢿⡃⠈⠳⣾⠀⢀⠀⠈⢿⠳⠤⠿⠟⠁⠀⡼⢹⡏⠀⢸⣦⣤⣤⡀
⠀⠀⠚⠒⠜⠀⠀⠀⠀⠀⣼⣿⣯⡿⠁⠀⠀⠙⡆⠀⠘⡗⣦⡀⠀⠀⢸⣧⠀⠘⣧⣀⣀⣀⣤⠞⣵⠞⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⢷⡀⠀⠀⢠⠇⠀⢰⣿⣿⢿⣦⣄⣿⠸⣦⣶⣾⠛⣛⠉⠁⠸⣇⠀⠀⠀⠀⢠⣄⣤
⠀⠀⠀⠀⠐⠚⠛⠛⠛⠛⡻⢿⣿⡿⡙⠓⠚⠉⠀⢠⣆⣄⡩⠅⠙⠙⠉⠀⠉⠀⠀⢱⣬⢷⣶⠶⠿⠶⠄⠀⠀⠛⣦⠼⠆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⠜⠙⠿⣾⣤⣤⣤⡶⢿⣯⡍⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠙⢃⠙⠃⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠴⠃⠀⠀⠘⠉⠈⠁⠀⠀⠙⠊⢻⡆⠀⢀⡼⠛⠟⣛⡊⠛⢿⠀⠀⠛⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⣷⢠⡞⠀⡴⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠍⠀⡑⠂⠀⠀⠀⠀⢹⡞⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠚⠲⠇⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⢈⢤⠞⠫⠒
⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄⠀⢃⠀⠀⢸⣟⣳⣯⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣤⠆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠈⠉⠉⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠉
⠀⠀⠀⠀⠂⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠐⢄⠀⠀⠀⠀⣴⣿⣿⣿⣷⣄⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠙⠋⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢿⠋⠉⠉⢹⡟⠐⠀⠀⠂⠀⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠈⠂⠀⠀⠀⢸⣇⠐⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠠⠀⠀⠀⠚⠋⠀⠈⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣈⣀⣄⣉⢡⡀⠀⠊⠂⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⣤⠀⠀⢠⠀⠀⠊
⠀⠀⠁⠀⢀⣀⣤⣀⡀⠀⠀⠁⠓⠀⠈⠠⠃⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⠀⠀⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠸⠋⠙⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣀⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠉⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡛⡟⢟⣹⢛⣛⡛⣿⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣿⠛⠋⠉⢳⡄⠀⠀⡰⠊⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢀⠂⠀⢇⠀⠀⠀⠀⡇⠀⡼⠁⠀⠀⠀⢸⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡜⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠘⡇⠀⣀⠀⢨⠆⠀⣤⡤⠒⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣼⣿⠟⠛⠻⣿⣦⠐⠁⠈⡞⠉⣴⠶⣷⣯⣿⠶⢦⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⣀⠀⣠⣿⣿⠾⠿⠿⠛⠷⠧⠤⢾⠭⠟⠤⠤⠿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠈⠻⡿⡿⠿⠛⠁⠂⠀⠐⠀⠀⠂⠀⠀⠐⠀⠀⠂⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣠⢶⡛⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣮⠉⣳⠦⣀
⠀⠀⠀⠀⣴⣯⣾⣿⡇⢸⣆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⣿⣿⣮⡷⡄
⠀⠀⠀⠀⣋⡿⣿⣿⡇⠸⠿⠗⢄⡀⠀⠀⠄⠀⠀⣀⡤⢞⣻⣜⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⠿⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢺⣿⣯⣽⡳⠿⣏⣉⣩⣩⡤⡦⢶⣺⡉⣴⡿⠟⠋⢁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠈⠑⠳⢮⣙⡿⣿⡇
⠀⠀⠀⠀⣾⣿⣿⣿⡇⢰⣶⣶⣶⡆⢰⣿⣿⣿⡗⠁⠀⠀⠚⠃⠛⠀⠐⠢⠘⡀⢀⠂⠔⡃⠀⠂⠀⠃⠀⠀⠈⠓⢯⡅
⠀⠀⠀⠀⠑⢟⡻⢿⡇⢸⣿⣿⣿⡇⢸⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠀⠁⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠉⠲⢥⣜⣛⡛⢛⢃⣜⣋⡭⠵⠚⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠃⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠂




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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿
⣻⣿⣿⣿⣽⣟⠾⣭⣭⡿⢟⣿⣟⠶⣿⣽⡷⣿⣿⣿⣻⣿⣿⣿⣽⣟⠾⣭⣭⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠯⣭⣯⢿⣿⣟⢮⣭⣭⣿⢟⣿⣿⢿⣿⣿⣿
⢾⣿⣿⠿⡋⢙⠿⣳⣶⣻⢿⡋⣛⢟⣿⣾⡻⠟⠉⠻⢷⣿⣿⠿⠋⠙⠿⣳⣶⣺⠿⠋⠻⢟⣾⣿⡿⠟⠉⠻⢿⣿⣿⠿⡋⢙⡿⣶⣶⡾⢟⡉⣻⢟⣶⣾⣻
⣷⢰⣿⣿⡇⣾⣿⢰⣿⢰⣿⢠⡿⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣿⠘⣿⡐⣿⡐⣿⡇⢹⣿⣷⠀⣿
⡏⢸⣿⣿⢁⣿⡇⣾⡇⣾⠇⣼⢣⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠸⣇⢻⣇⢻⣇⢻⣿⠸⣿⣿⡄⣿
⡇⣸⣿⡟⣸⣿⢰⡿⣰⡟⣸⠏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢻⡌⣿⡜⣿⡘⣿⡆⣿⣿⡇⢻
⠃⣿⣿⢁⣿⢇⣿⢣⡿⢡⡟⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⢿⡜⣷⡹⣧⢹⣷⠸⣿⣧⢸
⢸⣿⡏⣼⠏⣾⢣⡿⣱⠟⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⢻⡜⣷⡹⣧⢻⣇⢻⣿⡘
⣾⡟⣼⢏⡾⣣⡟⣵⠯⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡻⣌⢷⡽⣧⢻⣎⢿⣇
⣟⣼⣫⣞⡵⣫⣾⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⡻⣾⢷⡻⣎⢿
⣾⣿⡿⣫⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣮⣷⢿⣾⣿
⣛⡹⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣝⣛
⣭⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⣽
⣿⣏⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿
⣹⣽⡼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣹⢿⢿
⣇⣧⣿⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣏⡟⡿⣾
⣿⢸⡸⡞⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣸⢱⢇⣿
⣿⡎⡇⣷⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⡟⣼⢸⡿
⢿⡇⣿⢹⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢹⡇⡏⣾⡇
⢸⣿⢸⢸⡟⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣾⢸⡇⣿⡇
⢸⣿⢸⣇⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣸⢠⣿⢰
⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣴⣶⡆⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⣿⣿⣧⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢠⣿⣿⣿⣶⠀⠀⠀⠈⢻⣿⠿⣿⣿⡟⣱⣷⣶⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣾⡿⣿⣿⣿⡄⠀⠀⠀⠀⢁⣤⣻⠁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⢠⣶⣿⡇⠋⠀⢸⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠀⣨⣍⡛⢧⡹⣿⣿⣿⣿⣇⡇⠀⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠐⠻⢿⣿⢿⡏⢸⣿⣿⣿⣷⣄⣸⣿⣿⣿⢹⡇⠀⣸⣦⣽⣿⣿⣿⣿⡟⠿⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⢘⣯⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⡿⢻⣿⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣷⣿⣿⡿⠟⠁⠀⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣯⣥⣷⣾⣧⣀⠀⣠⢽⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣶⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢻⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠘⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣙⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣫⡽⠖⠀⠀⠘⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣰⣿⣍⠳⣜⠿⣿⠿⣟⣛⣛⡿⣿⣿⡿⣫⡵⢋⣥⣶⣿⡧⠀⠀⠀⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣷⣍⡛⢛⣋⣭⣭⣭⣙⠲⠶⠚⣫⣴⣿⡿⠋⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡀
⠀⠀⣠⣤⣴⣿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⡄
⠀⢺⣿⣿⠟⠁⠀⠀⠀⣰⣿⣿⣿⣿⣿⠏⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⢠⣾⡿⠋⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⡿⢰⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⣿⣿⣿
⠘⣿⠇⠀⠀⢀⣴⣶⣾⣿⣿⣿⣿⣿⡇⠈⠟⠛⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⡿⠟⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⠳⣄⠀⠀⠀⠀⠀⠀⠈⢦⣀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠘⣷⡄⠀⠀⠐⠶⠀⠈⢏⠛⢦⡀⢸⣧⣀⠀⠀⠀⢰⡀⣦⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⢹⣿⡆⢀⣀⠀⠤⠤⠼⠿⠶⣇⢸⣎⠻⣆⠀⠀⢸⢷⡸⡿⢦⡀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣿⣿⠀⡀⠤⠄⠒⠒⠘⡖⠚⠺⡿⢿⣿⠀⡀⢸⠀⢻⢃⠀⠃⢸⢿⣄⢀⡀⢀⡀
⠶⠦⣶⣶⠞⠛⠛⠓⠢⢀⡴⠊⢸⣿⣿⡉⢀⠤⠾⠂⠉⣁⣩⡅⠈⣇⠀⢻⢽⣿⣿⠀⠚⠈⣏⠉⢹⣀⢹⡆⠀⠛⢸⡆
⠀⠀⠀⠈⠉⠓⠢⠤⣠⣋⠀⡠⢺⣿⣿⠛⢧⣴⣶⣾⣥⣈⠿⠁⠒⡧⠠⠉⡐⠄⣺⡐⢨⣾⡷⠀⢺⠂⠘⣥⣐⠛⣻⠴⣆
⠀⠀⠀⠀⢶⠆⠀⢰⢣⣿⣿⣇⠸⣿⣇⢀⣿⡟⢧⡞⠙⢻⡆⠸⠿⠷⠀⠀⠀⢁⣼⠀⠉⡩⢐⣬⣽⢴⣶⣍⣿⠛⠁⠀⠈⢷⡀
⠀⠀⠀⠀⠀⢀⣤⣇⠀⢻⠿⠟⡼⠙⢿⣧⡙⡿⠥⠈⠲⣼⠃⠐⢾⡿⠀⠀⠘⠛⠋⠐⠍⡀⠿⠿⠋⠀⠉⠁⢠⡄⠠⣦⣤⠖⠁⢴
⠀⠀⠀⠀⠀⢸⣿⣿⠏⣻⠶⢤⣇⣀⢸⠉⠻⠷⠖⠒⠋⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠜⢵⣶⠎⠈⢀
⠀⠀⠀⠀⠀⣼⣿⣿⠀⣿⠀⠀⡇⠉⢙⠓⠶⣤⣀⠀⣶⣷⣀⠟⠀⠀⠀⢀⡀⣀⢀⡀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⡞⢉⡟
⠀⠀⠀⠀⠀⣿⣿⡏⠀⢸⢀⡴⣷⣶⣼⡀⠰⣿⣿⡆⠀⠉⠀⠀⠀⠀⠀⢸⡇⣿⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣧⠉⡄
⠀⠀⠀⠀⢠⣿⣿⣧⠀⠈⢿⣴⣶⡀⠙⣷⠀⠙⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣸⣿⢿⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢀⡀
⠀⠀⠀⠀⢸⣿⣿⠘⡆⠀⠘⡄⠀⢱⡀⠘⢆⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⣶⣶⣞⢲⡖⠀⠀⠀⠀⠀⢠⡆⢀⣀⡿⣶⡆⣾⡏
⠀⠀⠀⠀⠈⢉⡉⠙⠛⣖⠒⠻⡦⠤⠷⣤⣌⣢⠀⠸⣤⡀⣀⠀⠀⠀⠀⢸⡇⢿⡿⣿⢩⠿⣧⠀⠀⠀⠀⠀⠈⠁⡘⠻⢿⣤⠁⠉
⠀⠀⠀⠀⠀⠛⠃⢀⣀⡈⢶⣦⠘⣆⠀⢿⣿⡷⠑⣔⠿⠳⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢆⠌⠲⡄⠠⡤⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠨⠿⠶⠈⠻⣃⡠⠷⡌⢉⡱⣶⣾⡧⡀⠈⣢⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠖⠙⣦⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡟⠢⣀⡼⠫⡀⢈⣻⣁⠈⣿⢋⣉⠫⡉⠀⠀⠀⣴⠇⠀⢰⠗⣼⠇⠜⣿⡠⠀⣨⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⣿⣧⠖⠉⢢⡠⠚⠻⣿⠿⠓⢷⠾⠛⢄⠀⠑⢄⠀⠀⡄⠀⠀⢢⠜⣄⡠⠊⢣⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⣼⠟⢁⣤⡊⠁⠈⠢⡀⢈⡷⣶⣾⣷⢤⡀⠑⠢⣠⣷⢆⡘⡤⢊⠔⠱⡈⢣⣾⡗⠙⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⢺⣧⠹⣷⡄⣀⠴⠊⠓⢄⠈⠙⠯⡀⠈⠢⣀⠹⠫⡀⢉⢘⣅⣠⡾⠋⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠼⠋⠀⠀⠀⠀⢻⣧⡜⣿⣄⠀⠀⠀⠀⣙⡦⡖⣿⣿⣷⡌⠑⢧⣈⢔⣴⡿⠋⠀⠀⠰⣤⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡘⢿⣦⣠⠔⠊⢁⣀⠈⠛⢍⣩⠗⢆⡘⣷⡿⠋⠀⠀⠘⠃⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡷⠚⠉⠀⠀⣤⣈⠙⠋⢠⣾⠛⠢⡀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⡇⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⠼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠚⠛⠻⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠙⢣⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠀⢀⠂⠉⠛⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠉⠀⠐⠒⢠⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢐⡡⠴⠛⠀⠈⠁⠀⠀⣽⡊⠀⠰
⠀⠀⠀⠀⠀⠀⣀⢄⠈⠁⠀⠀⠀⠀⠀⢀⣰⠄⠂⠙⠤⣀⠀⠀⢤⣀⢀⠀⡤⠰⠆⠉⠀⠀⠈⠀⠀⠫⡅⢃⠀⠤⣈⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⢀⡴⠋⠉⠉⢣⠀⠈⠑⠀⠀⠈⠀⠉⠁⠀⠀⠠⠀⡀⠀⠀⡀⠀⠈⠑⠀⠀⠘⠁⠀⠀⠀⣠⡰⡀
⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⡄⠾⠁⠠⢠⣄⣘⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠠⠊⠀⠀⠀⠀⠠⠂⠀⠃⠀⠒⡛⠀⠀⠈
⠀⠀⢀⢀⡀⠠⠉⠁⠀⠐⠈⢀⡤⠰⠂⠀⠉⠐⠡⠀⠀⠀⠀⠀⠀⠡⠀⠀⡀⠁⠀⠒⠀⠈⠈⠀⡂⠀⠀⠀⠀⠶⠁
⠀⠀⠈⠉⠀⠁⠀⠀⠠⠈⡘⠉⠀⠀⠾⠃⠰⠟⠃⠀⠀⠀⠀⠀⠀⡀⠂⠁⠀⠀⠀⠤⡀⠀⠀⠀⠀⡀
⠀⠀⠀⡂⠀⠀⠘⠀⠀⠀⠀⠒⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠂⠐⠁⡀⠀⡀⡀⡀⠀⠐
⠀⠀⠠⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠲⠧⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠁⠂
⠀⠈⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣀⠄⠀⠀⠘⠒⠒⠦⢤⠀⠀⠀⠀⠀⠀⠠⢀⡀⠀⣀⠄⠴⠁⠀⠀⠀⠀⠀⠀⠘⠠⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠆⠀⠩⠡⠀⠀⠀⠀⠀⠀⠀⠈⠱⢶⣖⠣⡀⠀⡀⠈⠱⠆⠀⠀⠀⢂⠀⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣘⠠⠀⠀⠀⠂⠀⠄⠁⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⢀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡁⠀⠀⠀⣞⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⡄⠀⠀⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡜⢙⢁⣤⠤⣶⣾⣀⡴⢴⠦⣒⡨
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣆⡴⡶⣧⣦⠼⠃⠀⢈⡤⠐⡵⠃⠀⢙⣥⣦⠀⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠒⠁⠀⠚⠚⠛⠛⠛⠛⠐⠁⠘⠛⠑⠊⠀⠐⢼⡇⠉⠙⠳⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠢⠀⠀⢀⣴⣶⣦⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⣶⣿⡏⣿⣿⡟⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣸⣿⠀⠀⠀⠤⠀⠤⠀⠠⠀⢀⣤⣴⣾⣿⣿⡇⣿⣿⡇⣿⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠘⠃⣸⣤⢶⣾⣿⢹⣿⣿⣿⣿⣿⠷⠛⠋⠉⠉⠉⠙⠻⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⣴⣿⣿⢸⣿⣿⢸⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⠿⠾⠟⠛⠉

⠀⠀⠀⢀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⢸⣿⣇⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠛⠛⠛⢷⣄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡀
⠀⠀⠀⠘⣿⣿⣦⡀⠀⠀⠀⠀⣠⣤⡀⣰⣿⡟⠀⠀⠀⠀⠀⠀⣿⠀⢀⣴⠞⠲⣄⠀⠀⠀⢴⠀⢀⣴⣿⣿⡿⠟⠋⠉⠉⠉
⠀⠀⠀⠀⠈⠻⣿⣿⣶⣄⡀⠀⠻⠿⠃⣿⣿⠀⢠⠀⠐⢄⣀⡤⠏⠀⣿⣇⠀⠀⠋⠀⠀⠀⠀⢀⣿⣿⣿⠏
⠀⠀⠀⠀⠐⠂⠈⣻⣿⣿⣿⣷⣶⣶⣾⣿⣿⣦⣤⣄⡀⢀⣴⡄⠀⠀⢹⣿⣦⣤⣴⣶⣶⣶⣶⣾⣿⣿⣏
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡉⠁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠸⠷⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⢿⣿⣿⣿⣟⡻⢿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣶⣦⣽⣿⣿⣿⣋⣭⣭⣿⣿⣿⡏⢸⣿⣿⣿⢟⣵⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣤⣶⣿⠿⠿⠿⣶⣄
⠀⠀⠀⠀⠀⠀⣿⣿⣥⣤⣬⣿⣿⣿⣿⣿⡿⢛⠛⠻⣿⡿⠀⠈⣿⣿⣿⠟⢛⣻⣿⣿⣿⣿⣿⣭⣤⣬⣽⣿⡿⠋⠀⠀⠀⠀⠀⢹⣧
⠀⣰⡟⠉⢳⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠃⢸⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢠⡟
⠀⢿⣧⡀⠀⣾⣿⣿⣿⣿⣿⣟⡿⣿⠿⣟⣿⣿⣿⣿⣿⣿⣤⣴⣾⣿⣿⣿⣿⣿⡿⠿⢿⡿⣛⣿⣿⣿⣿⣿⣿⡇⠀⠀⠑⠲⠖⠋
⠀⠀⠛⠿⢿⢿⣿⣿⣿⡿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⡉⠀⠸⣿⣿⣿⣿⣿⣿⣿⡷⠾⠿⢿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠉⠙⠛⠛⠛⠉⢩⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠈⢻⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣆⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⣀⠉⢉⡛⠿⣿⣿⣿⣿⡏⠀⠀⢀⣀⣀⣀⠀⢻⡿⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣤⣤⣤⣴⣿⣿⡟⠉⠛⢿⣷⡈⠁⣠⡤⠿⣿⣿⣥⣴⣾⣿⣿⣿⣿⣿⣿⠁⠀⠻⣿⠟
⠀⠀⠀⠀⠀⠀⠰⠆⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢹⣷⠀⣿⡀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠰
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠻⢿⣿⣿⣿⠟⠁⠀⡅⠀⠀⠀⢸⡟⠀⠻⣿⣶⣾⣿⣿⡿⢿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣠⣀⠀⠀⠀⠀⠙⠦⠤⠶⠛⠀⠀⠀⠀⢹⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠘⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛


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

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⣴⣶⣾⣿⣿⣿⣿⣿⣿⠏⢿⠀⠉⠙⢿⡆
⠀⠀⠀⠸⣿⡇⣿⣿⡟⠘⠋⠋⠅⠀⠀⠀⠉⠀⠈⡇⠀⠀⢀
⠀⡀⠀⢀⣿⣧⣸⣿⡇⠀⠀⠀⡆⠀⠀⠀⠂⠁⠀⡇⠀⠀⡄⠂⡄
⢆⠀⣠⣾⣿⣿⠏⢻⡇⠀⠀⠀⢀⠀⠀⠀⠀⡠⠀⢧
⠈⠛⠛⠛⠛⠁⠀⠀⢹⣄⢀⠀⠀⢸⣶⣶⡇⠀⠀⡸⠀⢀⠁⠇
⠀⠀⠀⠨⠐⠄⠀⠀⠘⠮⣅⡀⠀⠈⠁⠁⠀⣀⡜⠓⠆⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠄
⠀⠀⠀⠐⠂⠨⡢⠀⠀⡀⡔⠉⠑⠒⠒⠒⠋⠁⠘⡆⠁⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⠀⠀⠛⠀⠠⠖
⠀⠀⠀⠀⠀⠀⠈⠂⠄⠂⠉⢲⠁⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠶⠎⠀⠀⠁⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣀⣤⠤⠤⠶⢲⠾⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠒⠒⠲⣶⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠀⠀⢀⣀⣴⣏⣀⣾⠄⡼⠀⠀⠀⠀⢸⣿⣭⣭⣍⣉⣹⣿⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢑⢦⠞⠉⠉⠁⠀⠀⣠⢣⠎⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠱⡀⠀⠀⠀⠀⠀⠘⠛⠻⠷⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⣵
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠉⠑⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠒⢦⡀⠈⢁⡔⢁⢍⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢨⣀⡠⠶⠖⠚⣛⣛⣉⣉⣉⢭⡽⠃
⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⣇⠐⡩⢦⡘⠒⣡⡈⠢⡀⡰⣡⠔⣒⣒⢠⠃⠀⠀⠀⠀⠀⠰⠯⣥⠶⡖⠉⠙⢉⠁⠀⠈⢸⡄⠀⡇
⠀⠀⠀⠀⣶⡋⢻⡆⠙⠲⠤⢤⠔⡡⣊⡴⣛⣳⣄⠉⢶⠄⠈⠳⡇⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠠⠈⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠘⢎⡴⣫⣾⢟⣵⣿⣦⡀⠀⠀⡤⠜⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠠⠄⣿⠀⡇⠀⠀⢸⠃⠀⠀⠀⠃⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣏⢔⣹⠿⣋⣴⡷⡄⠀⠘⢆⠀⠻⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢹⡀⢸⣀⣀⣰⣦⠦⠤⠚⠛⠉⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠈⠻⡥⣪⢟⣭⡾⢟⣦⠀⠀⢳⣀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠰⣟⣉⣉⣉⠥⠤⠤⠒⠒⠚⠋⠻⣧⣤⣤⣤
⠀⠀⠀⡠⠂⠀⠀⠐⠄⠄⠀⠾⠿⠀⠀⠀⠘⢷⠟⡫⣴⡿⡫⣣⡀⠀⠉⠀⢱⣀⡀⠀⠤⠠⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⢀⢀⠠⠇⠀⠀⠀⠀⠀⣀⠤⠤⠒⠒⠈⠉⡂⠀⠻⣊⢕⡪⣞⣭⣷⣂⠀⠀⠀⢷⢓⠲⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⣿⣿
⡎⠀⠀⠀⠀⠀⠀⠀⠸⣴⡀⠀⠀⠀⠀⢠⠇⠤⠤⢽⣯⡪⢟⣫⣶⣷⠂⠠⣶⢸⠀⠀⠀⢔⣢⠒⣼⡂⣀⠀⠀⢀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⡇
⠈⠀⠀⢒⢀⡀⠀⠀⠀⠈⠙⢶⣤⡀⠤⡼⢦⠤⣤⣃⠘⣾⣟⠟⣋⢼⡧⠤⠒⠉⠀⢰⠒⣤⠿⣋⡡⠄⠈⢷⣾⣷⣄⠈⠉⠏⠀⠙⢯⡁
⠀⠀⠀⠈⠁⠙⠦⣄⣀⠀⣰⡪⠛⢧⢠⠇⠈⠲⠃⠈⠉⢙⣶⣿⣿⡉⠀⠀⢀⡤⠒⠉⠱⡜⢧⠀⠐⠂⢀⡼⠈⢿⣿⣆⠎⠀⠀⠀⠀⠘⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡹⠒⠒⠒⠢⣤⠀⣾⣿⣿⠋⣡⠴⠊⠁⠀⠀⠀⢀⣵⢈⡧⢴⠊⡿⣦⠀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠘⠂⠤⠼⠁⠀⠀⠀⠀⢹⠀⢻⡿⠗⠉⠀⠀⠀⣀⠴⠒⠉⠉⠈⢋⠀⠛⠃⠈⠁⠀⠀⡟⣠⡆⠀⠀⠀⠀⠀⡟⠫⠯⠃
⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠒⠉⠀⠀⢀⡤⠖⠛⠁⠀⠀⠀⣠⣴⣶⣦⣄⠀⠀⠀⠀⠰⣟⠄⠃⠀⠀⠀⠀⠀⠃⠀⢀⣤
⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⢰⠃⠀⠀⣠⠴⠚⠉⠓⠦⡄⡀⠀⠀⢰⣿⢋⠽⢻⣾⣧⡀⠀⣀⠀⠄⠀⠈⠀⠀⠀⠀⢸⣶⣤⡀⠉
⠀⠀⠀⠀⡄⠀⣀⡀⠀⣬⠀⠀⠹⡆⠀⠀⠀⠈⠳⣍⣉⠀⠀⠀⠀⠀⢄⠈⠳⠀⠀⠸⣷⣝⠊⣠⣿⡴⠋⠓⢤⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡷
⠀⠀⠀⠀⣷⠈⣿⣇⠀⠙⠧⣄⠼⠀⠀⠀⠀⠀⠀⠈⠙⣿⣷⣶⣶⣬⣉⠀⠀⠃⠀⠀⠈⠓⠶⠿⠿⠅⠀⠰⠆⣉⠢⣄⠀⠀⠀⠀⢠⡿⠿⠟⠁
⠀⠀⠀⠀⢸⡆⢹⣜⡄⠀⠀⠀⠀⠀⠀⢀⠔⠉⢱⣶⣿⣿⣿⣿⣿⠟⠉⠛⢷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠈⠳⢄⠀⠀⡜
⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⢀⣿⠿⠿⠻⣿⣿⣄⣀⣀⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠄⠀⠒⠒⠾⢷⣈⠀⢀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠒⠉⣀⡤⠔⠊⠁⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣄⣠⣤⡶⠖⠉⠉⠀⠁⠢⣀⠀⠀⠠⡖⠒⠢⣝⢆⠀⠀⢱⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡀⠀⠀⠰⠋⠈⠀⠙⢦⠀⠘⣿⣿⣿⣿⣿⣿⣿⠟⣉⣀⣀⠤⠤⠀⢠⣤⣈⡑⢦⡔⠋⠀⠀⠀⠀⢣⠀⠀⠀⠠⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡄⠀⢣⠀⠀⠀⠀⡀⡇⠀⠘⢿⣿⣿⠿⠋⠀⠀⠹⡄⠀⡰⠁⠀⢸⡿⠋⠁⠀⡍⠓⠢⠤⠤⠴⠋⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠑⠤⠤⠤⠜⠁⠀⠀⠈⢟⠁⣀⠤⠤⠤⣄⠘⣴⠃⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠠⡄⠀⠀⠀⢀⠏⠒⠲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⡠⠤⡄⠀⠀⠈⢎⠃⠒⠒⠒⠊⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠓⠤⠂⠀⣀⣠⣼⠀⠾⠭⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣤⣤⣴⣾⡿⠿⠿⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣹⣛⣩⣙⣙⣟⣩⣟⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⣀⣴⣶⣄⣴⣾⣯⣴⣶⣶⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠋⠛⠛⠛⠛⠛⠛⠛⠻⠿⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⣼⣿⣿⣿⣿⡿⠋⠀⠙⠻⠏⠻⣿⠻⣿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠷⢳⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⢟⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢄⣦⢀⡄⠀⢌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠘⠋⢧⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⢴
⠀⠀⠀⠩⠟⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠒⠀⠀⠀⠀⠀⠂⠑⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠾⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠠⠀⡩⠄⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⡀⠔⠠⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠐⢠⠄⠀⠐⡀⠀⢀⠒⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠠⠀⡀⠀⠀⠀⣀⠀⠀⠄⠠⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠚⠁⠀⠉⠓⠀⢐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⣀⠐⠄⠀⠘⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⡠⠀⠀⠀⢠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⢀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠄⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡀⠒⢤⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠠⣀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠤⠀⣤⡅⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡌⢀⣠⠀⠁⠊⠀⠀⠀⠁⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈⠀⢀⠀⠀⡀⠠⠐⠈⠢⠀⠀⠀⠀⠀⠀⠐⠊⠀⠐⠀⠀⠀⢀⡀⠄⠂⠢⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢁⣄⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠱⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⡁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢁⠈⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠇⠀⠀⠀⠀⣠⠀⠐⠇⠀⣀⠀⢀⡀⣲⠈⡿⠀⢠⡎⡝⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢨⠘⠇⠀⠈⠁⠀⢸⠀⠀⠀⠀⢀⠀⣙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠀⠐⡷⢀⠀⠘⠀⢐⡆⠀⢠⠀⠸⡄⠀⠀⠀⠈⠘⠃⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⢀⣀⡀⠀⣀⣀⡀⢀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⢸⣿⠃⠀⢹⣿⢻⣿⠁⠙⣿⡏⠀⣿⡏⠁⠙⣿⠛⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⠾⣿⠀⠀⣸⣧⣤⣿⡀⠀⢹⣧⢰⡿⠀⠀⠀⣿⡗⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⢠⣿⡀⣠⣿⡁⢀⣹⣧⡀⠈⣿⣿⠃⠀⠀⢠⣿⣤⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠉⠀⠀⠈⠁⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠧⠉⣬⠬⢍⣹⡇⠨⠉⢩⡟⠸⠁⢨⡟⠀⠘⡤⠸⣟⠘⠇⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠿⢰⡆⠀⠀⠀⡀⢸⡇⠀⡄⢸⡇⠀⠀⡗⠀⠹⣆⡃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⠀⠁⠀⠀⠀⠁⠸⣇⠠⠀⣼⠁⠀⠀⡇⢸⠆⡙⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣖⣀⣻⠆⠀⠀⠀⠀⠐⢀⠈⢀⠴⠃⠀⠀⢠⠃⣸⣦⣗⣀⣘⡧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⢀⡐⠀⣄⠀⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡄⢉⠻⠈⠓⠀⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠣⠸⢣⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠈⢁⠔⠂⠀⠸⠆⡇⡀⠀⠀⠈⠂⡠⠃⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⢀⠔⠚⠁⠀⠤⠺⣷⡇⠇⠅⠀⠀⠀⠘⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡌⠀⠀⠀⠀⠀⢀⠈⢠⢑⠀⠀⠀⠀⠀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠀⠀⠀⠀⠀⡘⠀⠀⠂⡈⠀⡄⢀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢁⣀⠄⠀⠀⠃⠀⠙⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡠⠀⡠⠀⠀⠀⠠⠠⠈⣀⣀⠘⠀⠀⠀⠀⠐⠢⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠄⣀⡀⠤⠂⠀⠀⢀⡤⠒⠁⠊⡀⠀⠀⠀⠀⠀⡬⠈⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠄⢀⠠⠂⠀⠈⠀⠀⠀⠀⠀⠍⡰⠞⠁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⢀⠆⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⣌⡴⢋⠀⢰⠀⠣⡀
⠀⠀⠀⠂⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠉⠒⢄⣀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠠⠏⡰⠉⡄⠀⡈⠢⣈⠢⢀⠂⠀⢀⠤⠀⠀⠀⡀
⠈⠀⠁⠐⠠⠤⠤⠐⠒⠠⠄⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠈⢄⠀⠀⢈⠁⢄⠈⠁⠂⡚⢀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠤⠤⠠⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢂⠀⠀⠀⠀⠁⠐⡆⢀⠈⢂⡴⠋⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⡤⠠⠤⠐⠋⠉⠉⠉⠁⠀⠀⠑⠀⠀⠀⠀⠈⢁⠌⠈⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠄⢂⣡⡶⣺⣿⣟⢿⠟⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⡴⠉⣡⠞⢝⡵⢡⢎⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠤⠐⠊⠀⠈⠀⠀⠁⢀⡞⡰⢡⣾⡿⢛
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⢠⠟⡼⣡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⢀⠤⠊⠀⠀⠔⢡⣪⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣽⣖⣒⣂⣁⣀⠤⠤⠄⠒⠉
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠈⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⣀⠴⢖⡲⠦⣄⠀⠀⠀⢀⡴⠚⣉⠉⠓⠢
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠤⢄⠀⠀⠀⠀⡎⢀⠔⢦⡀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡈⠃⠀⠀⠀⠸⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⠀⠀⠈⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⢠⠈⠀⠀⠀⠀⠀⠳⣄⠀⠀⠃⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠁⡄⠀⠤⢠⠀⠶⠠⠭⠛⠗⠺⠛⠑⠂⠒⠛⠂⠃⠆⡄⣀⠠⡆⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠰⠄⠑⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠙⠠⠀⠢⠀⡀
⠀⢀⠀⢀⠀⠉⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠌⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠑⠨⢆
⣖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠐⠀⠉⠀⠊
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠁⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⢠
⠘⡤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣠⠀⣄⡀⠔⢠⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢄
⠡⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠐⠀⠀⠀⠀⠉⠂⠾⠉⠉⠁⠫⠂⠁⠀⠀⠀⠀⠈⠃⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⡄⠀⠀⠀⠰⠀⠀⠁⠀⠀⠠⠄⠈⠀⠀⠀⠀⠀⢀⡦⠀⢀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠅⠀⣀
⢆⠀⠀⠀⠀⠀⠄⠤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠄⡄
⡀⠱⡀⠀⡄⠀⠃⠈⢡⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠺⢠⣄⠀⠀⠀⢀⡄⠘⠢⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⢑⢦⡒
⡁⠀⣣⡜⢀⡄⠠⣀⡀⠘⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠒⠙⡀⡀⠊⠠⠄⠱

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⠛⠻⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⢹⣿⣧⣀⣠⣿⡿⠿⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣶⣿⣿⠟⠛⢿⣿⣅⠀⣸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⠈⣿⣿⡇⠀⠀⠀⣿⣿⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣿⣿⣷⣤⣤⣴⣿⣿⡀⢀⣼⣿⣶⡶⢶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠐⣿⣿⣿⡏⠀⠈⣿⣿⠋⠉⢻⣿⣿⣿⣇⣠⣿⣦⣼⡟⠻⣦
⠀⠀⠀⠀⢀⠀⢰⣶⣷⡄⠈⢿⣷⡄⠈⢿⠿⣿⣶⣾⣿⣿⣦⣤⣾⣿⣿⣏⢉⣿⡇⠀⢸⡿⠛⣿⠄
⠀⠀⠀⠀⠘⣧⢀⣿⣿⣿⡄⠈⠛⠉⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠉⠘⢿⣿⠛⣿⡟⠻⣿⣶⡟
⠀⠀⠀⠀⠀⢹⡿⣿⣿⡇⠀⣿⣿⠀⢸⣿⡇⠀⣿⣿⠀⢸⣿⠀⢸⣿⠀⠈⠛⠿⣿⣷⡾⠿⠋
⠀⠀⠀⠀⠀⢸⡟⢃⣿⡇⠀⣿⣿⠀⢸⣿⡇⠀⣿⣿⠀⣸⣿⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣴⠂⢠⣤⣤⣄⣀⠀⠀⣰⣧⣬⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣽⣿⣶⣿⣿⣿⣿⣿⣆⡀⣿⣯⣄⢠⣀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠹⠇⠹⢿⣿⣿⣿⣷⠸⡅⣩⠀⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡉⠀⣘⠃⠸⣿⣿⣿⣿⠤⣧⡙⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣦⣁⡴⠀⠻⣿⣿⣿⣶⠟⣿⠟⢀⣾⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⠻⢻⠯⣤⡟⠀⠀⣀⣈⣋⠀⠀⠀⠘⣇⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⠄⠀⠀⠀⠀⠀⠀⠀⠄⣌⢠⢠⠀⢈⠄⠀⠐⠙⠿⠈⠳⡆⠀⠀⢻⣿⣿⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⡾⠃⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⢇⠀⠀⢸⣿⣿⡆⠀⢌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣰⣤⣞⠰⣧⠚⣷⣄⣄⠀⠀⠃⠀⢀⣲⡏⢳⣿⠀⢸⢸⣿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠈⢿⡟⠁⢠⣿⣿⡿⠀⢰⠄⠀⣨⢷⡅⢈⣻⠀⢸⣾⣿⣿⡇⢠⣟⠃⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⣠⠟⠈⢻⡀⣰⠃⠀⣴⠋⣾⣯⣨⣿⠀⢘⣿⣿⣿⡇⠀⠈⢁⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠻⠶⢀⠈⠃⠋⠐⠈⣧⠚⠉⠛⢿⡿⢀⣾⠛⠉⢹⡇⠐⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠓⠧⣀⣠⡀⡇⠀⢀⠄⠈⡠⠊⠃⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡛⣡⣤⣴⡾⠁⠀⠀⢀⣴⡟⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⢠⣤⣴⠋⠀⠀⠀⠀⠠⠽⠋⠙⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⢀⠀⣿⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⣰⡏⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠟⠀⠀⠀⢠⠀⠀⠀⠀⠱⡀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⣀⢀⡀⠉⠀⣀⠴⠦⠀⠀⠀⠀⠀⠀⠕⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠠⣻⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠻⠇⠀⡟⠃⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣭⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀⠀⣗⡆⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣤⣿⣧⡀⢫⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣷⡛⢃⣺⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⡟⠉⠀⢠⡤⢠⡄⠀⠀⠀⠀⣾⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠛⠻⠿

⣷⣷⣿⡟⠁⣠⡶⣿⣿⣿⣿⣮⣷⡍⠻⣿⣶⠶⣖⣻⣻⣛⣰⡆⠦⣬⢈⡛⢲⢤⡙⠑⠆
⣿⣿⠏⢀⠞⡙⠂⠘⢹⣿⣿⣿⣿⣿⣿⣮⣻⣿⠁⣇⠉⠈⠹⠙⢋⠗⡶⣍⡑⠦⠛
⣿⡟⠀⡞⠰⠀⠀⠖⢙⠟⠛⢯⡿⣿⣿⣿⣿⣿⠿⠛⠿⠛⠙⠟⡶⢦⣤⣀⠛⠓⠲⢄⠀⠀⠀⠀⠀⢀⣤
⢭⠇⠀⣧⠁⠀⢻⠎⠁⠀⠀⣾⡿⢿⣾⣿⢿⣿⣯⡙⠿⢦⡘⠷⣷⣿⣭⣇⠀⠀⠀⠀⢣⠀⢀⣠⢾⣿⣽
⣧⣾⡀⢲⡀⡐⠀⠀⢰⣿⡆⢀⣴⣾⣿⣕⢌⣿⠟⣮⡢⡈⠟⢶⣽⠿⣿⣿⣿⣤⠀⠀⢠⠐⢿⣿⣿⣿⣿
⣷⣿⣇⠀⠛⠀⣠⡤⡀⠉⣰⠟⣵⣯⣙⣿⢿⣟⠻⣿⣺⡯⡢⣈⠻⣷⣜⠿⣿⣿⣷⣤⠜⠃⣀⢿⣿⣿⣿
⣿⣿⣿⣷⠀⠀⠻⠷⠇⣼⡇⣽⣿⣿⣻⣼⡻⣿⣷⣝⣻⡮⡪⡻⣅⣍⠙⢳⣞⠻⣿⣿⡄⠀⠉⣈⣿⣿⣿
⣿⣿⣿⡇⣠⡖⢦⠀⣸⠿⠗⣿⡿⠛⠛⣿⡿⠿⢿⣟⣛⣿⣛⠊⠪⡳⠕⢌⠹⣷⣮⡉⠙⢄⠀⢀⣸⣿⣿
⣿⣿⣿⠀⠻⣿⠿⠃⣿⠃⠘⣿⣁⣴⣾⡟⢂⣴⡌⣿⠿⠿⢿⣯⣦⡁⠀⠈⠳⣌⠿⡿⠀⠀⠀⠀⢼⣿⣿
⠿⣿⣿⠀⠀⡠⢄⠀⢿⡀⠀⢻⣿⣿⣿⣷⣿⣿⣷⢸⠀⠀⢀⣿⣿⣼⡢⡙⣆⠈⠀⠀⠀⠀⠀⠀⡀⣿⣿
⣷⡿⢿⡄⠸⣷⡾⠂⠸⣧⠀⠈⣿⣿⣦⢿⣿⣿⣿⣤⣷⣤⣾⣿⣿⣿⣿⣮⡺⣷⠄⠀⠀⠀⠀⠀⠑⢎⡛
⡟⣷⣾⣇⠀⠀⣠⠤⠀⢻⣇⠊⠈⢿⣿⣷⣿⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢮⠀⠀⠀⠀⠀⠀⣇⣀⠑
⣟⡿⣿⣿⡄⠐⣿⣷⣶⠀⠁⠀⢀⡀⠀⠙⠿⣿⣾⣿⣿⣿⣄⣀⣽⣿⣿⣷⣤⣼⡇⠀⠀⠀⠀⣨⡃⡝⢴
⣿⣿⣿⣿⣿⣄⠈⠉⠁⠀⢀⣾⣯⣀⣡⠀⠀⠀⢀⣩⡿⠿⢿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⢀⡪⣻⣮⡓

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡀
⠀⠀⠀⢤⣴⣧⠄⠀⡀⠀⠀⠀⠀⠀⢀⣠⡴⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣀⣳⣾⡅
⠀⠀⠀⠘⠉⢃⣴⣿⣿⣿⣦⣤⡶⠞⠋⠁⠀⠀⢀⠀⠆⠀⠂⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⣔⣾⡯⢤⡿⡝⠛
⠀⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠘⠄⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⣷⠿⣿⢾⡇⠙⠁⠀⢰
⠀⢿⣿⣿⣿⣿⡿⠛⠿⠿⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠀⠀⠀⣀⣀⠔⠈⠈⠐⡌⠧⠉⠁⠀⠀⠀⡠⠃
⠀⠀⠛⠿⠿⠋⠀⠀⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⣣⣤⡀⠸⠟⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⣰
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣰⠾⣶⡶⢶⠿⣶⢶⡶⠶⢾⡿⢼⡿⠛⢷⡄⠀⠀⣠⡎⠀⠀⠀⠀⠀⢈⠠⠠⠃
⠀⠀⠀⠀⠁⠀⠀⠀⡄⠀⠀⠀⠀⣿⢀⡈⡀⣸⢠⣧⠈⠱⣯⣼⡇⢸⢁⡿⠛⠃⠀⠛⠋⠀⢀⡀⠀⠀⠀⠠⠒⠒⢢
⠀⣀⣠⡄⠀⠀⡀⠀⠀⠀⠐⠂⠀⣿⢸⣿⡇⢹⠈⠋⢸⣷⣄⠙⡇⢸⡀⢿⢶⡄⠀⠀⠀⠀⠈⠡⠀⠘⠁⠠⠟⠒⢲
⠀⠘⠿⢁⡄⢦⠙⠀⠀⠀⠀⠀⠀⢿⡼⠛⠷⠾⢷⣴⠟⢷⣤⠾⢷⡾⠷⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡭⣅
⠀⢀⠔⢤⠙⠀⡀⠀⠀⠀⢰⡆⣾⠀⠀⢰⡆⣶⢠⣶⠀⠀⣶⠀⡷⠀⠀⣶⠀⣶⢰⡇⠀⠀⠀⠠⠀⠀⠀⠀⠔⠠⢬
⠀⠈⠙⠀⠀⠀⠀⡠⠀⠀⠸⠇⠻⠀⠀⠸⠇⠿⠘⠟⠀⠀⠻⠐⠇⠀⠀⠛⠀⠛⠘⠃⠀⠀⠀⠀⣠⣄⠀⠠⠖⠒⢦
⢰⠆⠀⠀⠐⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠠⠧⠤⠤
⢀⡚⠀⠖⠄⠀⡄⠀⠀⠀⢀⣠⣄⣠⣾⣶⠀⢠⠀⠀⢀⠀⠰⠶⠶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣄⣈⣤
⠀⠉⠉⠲⠒⡑⣉⠤⠖⠦⢺⣿⣿⣿⣟⡟⢀⣾⣿⡖⠈⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢡⣙⢻⢒⡀⠀⠀⠙⠿⣿⠏⠀⠀⠈⠹⠀⠀⠀⠀⠛⠀⡿⠀⠀⠀⢀⢀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠒⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⣿⣷⣾⣿⣯⣤⡀⠀⠀⠀⠀⢀⣀⣤⣴⣦⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠶⠶⣤⣤⣀⡀
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⢠⣮⣴⣶⣶⣿⣶⣶⣦⣌⡻⣿⣆
⠀⠀⠀⢸⣿⣿⣿⡿⠋⠙⠻⠟⢿⡿⣿⢿⡇⠀⢠⣿⣿⣿⡟⠉⠛⠉⠛⠛⠛⠛⢿⡇⠀⠀⠀⢠⣿⣿⠿⠟⠛⠻⠿⠿⠿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⢿⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿
⠀⠀⠀⢨⣯⡝⠃⠀⠀⢀⠀⠀⠀⠀⣀⠀⠃⠀⡾⣯⠋⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠃⠀⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⡀⠀⠀⠙⢫⡿⡆
⠀⠀⠀⠈⠲⠇⠀⠀⠀⠈⠀⡀⠀⡀⠉⠀⠈⣤⠑⢾⢴⣤⡀⠀⠀⠀⠀⠀⢀⠀⠀⠈⣤⣀⣄⣴⣇⠀⠁⢠⣄⣤⠀⠁⠀⠀⠀⢨⣖⠠⡤⣦⣆
⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠑⠐⠀⠀⠀⡠⣼⠄⣢⠤⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢣⡬⡃⣬⠿⠀⠀⠈⠉⠉⠀⠀⠀⠀⣠⣾⡿⠒⠄⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⡄⠀⡤⢀⢴⠚⢈⠔⠈⠢⡁⠊⢐⡂⠠⡀⢀⠤⠀⣀⠎⢂⠡⠊⠈⠢⢁⠻⢶⣶⣦⠀⢠⣶⡾⠟⢋⣡⣾
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠋⠉⠹⣁⠠⠋⠀⠀⠀⠀⠀⠑⠆⡀⠑⠁⠈⠹⠀⡡⠂⠁⠀⠀⠀⠀⠀⠁⣮⣀⠍⠁⠁⣉⣀⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢔⠩⡖⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⠀⠈⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⠀⠀⠀⢹⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠱⠠⠴⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠿⠓⠉⠛
⠀⠀⠀⠀⣀⠀⠀⡁⠀⠈⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢈
⠀⠀⢠⣠⣭⠙⠐⠁⠁⠂⠀⠀⠀⠺⠀⠀⠀⡠⡈⠲⠄⡘⠂⠠⠄⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠒⠀⠒⠀⠁⠀⠂⠐⢊⠀⡀
⠀⠀⠈⠚⠀⠑⠄⠀⠤⠀⠂⠢⡀⠀⠃⠀⠀⠭⠟⠑⢀⠀⠀⠀⡀⠤⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠚⠓⠤⠤⠀⠂⠙⠣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⡨⣀⣀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠈⢄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠉⢇⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⢛⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⣈⡷⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠒⠉

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣠⣶⣤⣶⣦⣶⣖⣀⠀⠀⠀⠀⠀⠀⠀⣒⣛⣉⣉⣛⠲⢶⣤⡀⠀⠀⢀⣠⣤⣶⣶⣤⣄
⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⣼⣿⣿⠿⢿⣿⣿⣿⣷⣿⣧⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⢸⣿⣿⠟⠋⠈⠛⠙⠟⠻⢻⠇⠀⠀⠀⠀⢿⠃⠀⠀⠀⠀⠀⠈⢿⣿⡟⠀⠀⠈⠉⠀⠀⠀⠀⢿⣿⡇
⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡄⠀⠈⢫⠾⡄⠀⠀⠀⠀⠀⠀⠀⠀⡿⠇
⠀⠀⠀⢉⠃⠀⠀⠐⠀⠀⠀⠐⠀⠀⢰⣶⣤⡀⡀⠀⠀⠲⠶⠀⠀⠀⠀⢰⣇⠀⠀⠐⠶⠆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠛⠀⢠⣄⣀⠀⢀⣠⣤⡶⢛⣩⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⡂⠠⠀⠀⠀⠀⠀⠀⠀⠠⣍⠙⠁⠘⠋⣡⣶⣿⡏⠀⠀⢀⠌⠀⠑
⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠹⠇⠀⠀⠈⢿⣿⣿⡁⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠀⢠⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠂⢀⡀⠘⠚⠂⠀⠀⠀⠐⠈⠚
⠀⠀⠄⡌⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣀⠀⠀⠲⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⠀⠀⠀⠀⠀⠀⠐⠀⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣈⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠜⠀⠘⠏⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⣖⡶⠶⠶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣀⣴⡆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠸⣿⣿⣿⣿⡿⢿⣷⣿⣷⣦⣤⣀⣀⣠⠀⠀⠀⣿⣿⣿⡀⠀⢀⠀⢠⣀⢰⠁⠈⠙⠛⠿⠿⠿⢿⣿
⠀⠀⠀⢻⣷⣾⣿⣿⣿⢸⣿⣮⣿⣟⣛⣿⡏⠀⠀⠀⣿⣿⣿⣧⠀⠉⠀⠀⠀⢸⠀⠙⠂⠀⠤⠄⢀⣿⣿
⠀⠀⠀⠀⠻⣿⣭⣬⣽⢸⣿⠿⣿⣿⣿⠟⠀⠀⠀⠀⣿⣿⣿⣿⣧⡀⠴⠛⠓⣸⡀⢄⣀⣀⠀⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠻⣷⣶⠿⠋⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⡷⢶⣶⢶⣿⣷⣈⣉⣉⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⣿⠉⠉⢹⠉⠉⢹⡏⠉⢹⡏⠉⠉⡏⠉⠉⡏⠉⠉⣿⠉⠉⣿⠉⠉⢹⠉⠉⢹⡋⠉⢹⡏⠉⢹⡏⠉⠉⡏⠉⠉⣿
⠀⠀⣿⠀⠀⢸⠀⠀⢸⡇⠀⢸⡇⠀⠀⡇⠀⠀⡇⠀⠀⣿⠀⠀⣿⠀⠀⢸⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⡇⠀⠀⣿
⠀⠀⣿⠀⠀⢸⣀⣀⣸⡇⠀⢸⣇⣀⣀⡇⠀⠀⣇⣀⣀⣿⠀⠀⣿⣀⣀⣸⠀⠀⢸⣃⣀⣸⡇⠀⢸⣇⣀⣀⡇⠀⠀⣿
⠀⠀⢻⣄⣀⣼⠋⠉⠹⣇⣀⣸⠏⠉⠙⣧⣀⣠⡟⠉⠉⢿⣀⣀⡿⠉⠉⢻⣄⣀⣼⠋⠉⠹⣇⣀⣸⠏⠉⠙⣧⣀⣠⡟
⠀⠀⣾⠈⠉⣦⠀⠀⠀⠈⢉⡅⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢨⡉⠁⠀⠀⠀⣴⠉⠁⣷
⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⣿⠀⠀⡇⠀⡖⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢲⠀⢸⠀⠀⣿
⠀⠀⣿⠀⢸⡇⢠⡇⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢸⡄⢸⡇⠀⣿
⠀⠀⣿⠀⣼⠁⣼⠁⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⣧⠈⣧⠀⣿
⠀⠀⣿⢀⡟⢰⠏⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠹⡆⢻⡀⣿
⠀⠀⣿⣸⢣⣏⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣹⣜⣇⣿
⠀⠀⣿⣉⣉⣉⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣉⣉⣉⣿
⠀⠀⣿⠀⣧⢿⡻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⡿⣼⠀⣿
⠀⠀⣿⠀⢹⡎⢷⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⡾⢱⡏⠀⣿
⠀⠀⣿⠀⠈⡇⢸⡇⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢸⡇⢸⠁⠀⣿
⠀⠀⣿⠀⠀⡇⠈⠁⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠈⠁⢸⠀⠀⣿
⠀⠀⣿⣀⣀⣃⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣘⣀⣀⣿
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁