Occupied Emojis & Symbols

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⣴⣿⣿⡿⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⡀
⠀⠀⢰⡏⠉⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠛⠋⠁⠹⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁⠉⠙⠋⠙⢿
⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣷⣄⠀⠀⣠⣼⣿⡇
⠀⠀⠀⡼⢿⡄⢀⣿⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢟⠋⠁⠀⠉⢛⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠾
⠀⠀⠀⢧⠈⠛⠙⠣⠞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⡤⠴⠋
⠀⠀⠀⠈⠷⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈
⠀⠀⠼⠀⠀⠀⠁⠀⠀⢰⠀⠄⠀⠀⠀⠀⠀⠀⡌⠄⠀⠀⠀⠀⠀⠌⡄⠀⠀⠀⠀⠀⠀⠄⠠⣧⣤⣤⣤⣤⣤⣴⠄
⠀⠀⠏⠀⠀⠀⠂⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⢠⠂⣀⠀⠀⠀⠀⠀⠁⢂⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⠀⣿⣿⣿
⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⡀⣿⣿⣿
⠀⠀⢀⣿⣿⣿⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿
⠀⠀⣠⣿⣿⣿⢸⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⢠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⣿⣿⣿⣀
⠀⠘⠛⠛⠛⠛⠘⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠘⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠃⠛⠛⠛⠛

⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠈⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⣀
⠀⠰⣤⣤⣤⣤⣤⣤⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠛⠛⠉⠁⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠿⠿⠛⠹⣿⣧
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠀⠀⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄⠀⠀⠀⢀⣾⠟
⠀⠀⠀⣤⣄⠀⣰⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡄⠀⡴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⠀⢀⣄⡀
⠀⠀⢀⣿⣈⣈⣁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⣶⡟⠓
⠀⠀⢸⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁
⠀⠀⣼⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⢤
⡀⠀⣿⣽⣿⣿⣿⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⡰⠀⡀⠀⠀⠀⠀⠀⡀⠀⠇⠀⠀⠀⠀⠀⠘
⠁⡈⣿⣿⣿⠛⣿⣿⣿⠁⠈⠀⠀⠀⠀⠀⠀⠉⢹⠀⠀⠀⠂⠀⠀⠡⠉⠁⠀⠀⠀⠀⠀⠉⠹⠀⠀⠀⠀⠀⠀⠀⠏
⠈⠀⣿⣿⣿⠀⣿⣿⣿⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⢀
⠀⠀⣿⣿⣿⢠⣿⣿⣿
⠀⠀⣿⣿⣿⢼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⡿⢉⣭⣭⣴⣶⣤⣭⢹⡇⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡆⠀⠀⠀⠀⣗⢸⣿⣿⠿⢿⣿⣿⠘⡇⠀⠀⣸⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡆
⠀⠀⠀⠀⣿⠿⠿⠿⠿⢛⣩⣴⣌⠻⢿⡇⠀⠀⠀⠀⣿⣬⣭⣴⣶⣶⣬⣭⣴⡇⠀⠀⣿⠿⠿⠿⠿⢛⣩⣴⣜⠻⢿⡇
⠀⠀⠀⣠⡇⢶⣶⣿⣿⣿⣿⣿⣿⣿⡆⣹⣄⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢀⡇⢶⣶⣿⣿⣿⣿⣿⣿⣿⡆⣱⡀
⠀⣴⣾⣿⣿⡆⢻⣿⣿⣿⣿⣿⣿⡟⣰⣿⣿⣷⣄⠀⠀⠀⢉⣙⣛⣛⣉⠁⠀⠀⢠⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⡏⣴⣿⣷
⠘⣿⣿⣿⣿⣿⡌⢿⣿⣿⣿⣿⠟⣡⣿⣿⣿⣿⣿⠀⣠⡄⢾⣿⣿⣿⣿⡆⣠⣀⠀⠙⠿⣿⡜⢿⣿⣿⣿⣿⠟⣱⣿⠿⠋
⠀⠹⣿⣿⣿⣿⣿⠆⣬⣭⣭⣅⢸⣿⣿⣿⣿⣿⢃⡜⢿⣿⡈⢿⣿⣿⡟⣰⣿⠟⣰⣶⣤⡀⠉⠀⣬⣭⣭⣄⠈⠉
⠀⠀⠙⠟⣩⡀⣶⡘⢿⣿⣿⡿⣠⣶⢨⡙⠿⢃⣿⣿⣶⢆⣵⡌⣿⡟⣰⣇⢲⣾⣿⠟⣩⡀⣶⡘⢿⣿⣿⡟⣠⣶⢠⡀
⠀⠀⠀⢸⣿⣧⣙⠻⣎⠻⠏⣴⠛⣋⣼⣿⡆⣿⡿⣿⣿⣌⠻⣿⡈⣰⡿⢋⣼⣿⡟⢸⣿⣧⣙⢻⣎⠻⢏⣴⠛⣃⣼⣿⡄
⠀⠀⠀⣾⣿⡏⣩⡘⢿⣷⣾⠟⣃⣏⢻⣿⡇⢹⡇⣿⣿⣿⣷⣌⠻⢋⣴⣿⣿⣿⡇⣿⣿⡏⣭⡘⢿⣷⣾⠟⣠⡍⢿⣿⡇
⠀⠀⠀⣿⣿⡀⣿⣿⣷⡌⣡⣾⣿⡿⢸⣿⣿⢸⡇⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⠃⣿⣿⡐⣿⣿⣷⡌⣡⣾⣿⡿⢸⣿⣷
⠀⠀⠀⣿⣿⡗⢨⣿⣿⡇⣿⣿⣯⡄⣿⣿⣿⠸⠇⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢰⣿⣿⡇⢨⣿⣿⡇⣿⣿⣯⡄⣿⣿⣿
⠀⠀⢸⣿⣿⡇⣾⣿⣿⡇⣿⣿⣿⡇⢻⣿⣿⡀⠀⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢸⣿⣿⡇⣾⣿⣿⡇⣿⣿⣿⡇⣿⣿⣿
⠀⠀⠘⠛⠛⠃⠛⠛⠛⠃⠛⠛⠛⠓⠘⠛⠛⠃⠀⠛⠛⠛⠛⠛⠃⠛⠛⠛⠛⠋⠘⠛⠛⠃⠛⠛⠛⠃⠛⠛⠛⠃⠘⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢹⡦⡏⠉⣾⠉⢰⡇⢸⢸⣯⠇⣾⣧⢹⡏⢸⣷⡏⢹⡦⡿⣼
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣤⣿⣷⣼⣿⣦⣿⣿⣼⣧⣼⣧⣼⣾⣧⣼⣧⣿⣿⣴⣧⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠈⢤⡤⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⢀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠘⠃⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠀⠂⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠍⢁⡀⠀⢀⣀⠩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠉⠉⠉⠉⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠰⡂⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⣀⡀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠀⠀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⢀⠀⠰⢄⡀⡀⠴⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢦⡀⢀⡴⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠇⡀⣀⠸⠂⠄
⠀⠀⠀⠀⠀⠔⠁⠀⠘⠈⠁⠁⢒⠀⠀⠑⡀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠳⠞⢠⠤⠤⠈⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠂⠙⠋⠐⢀⣀⠐⠄
⠀⠀⠀⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠃⠀⠀⠀⠀⠀⠃⠄⡀⠀⠀⠀⠀⠈⠀⢀⠠⠘⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠘⠃⠀⠘⢉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠉⣂⢀⡉
⠀⠀⠀⠀⠀⠀⠀⠠⢈⠈⠁⠈⢆⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⠈⠁⠈⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⣖⠔⠀⠀⠀⠐⢐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠀⠀⠄⠀⠀⠀⡊⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⢀⠝⠀⠐⠈⠁⠐⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⢈⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠊⠀⠉⠀⡇
⠀⠀⠀⠀⠀⠀⢇⢠⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⡀⠘⠃⢀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠢⠄⠀⠤⡰⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⣀⣀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⡀⢀⠴⠁
⠀⠀⠀⠀⠀⠀⡠⠀⢚⢂⡤⣄⢚⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢒⠃⡠⣤⠘⡒⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⡗⢠⡄⢺⠒⠢⡀
⠀⠀⠀⠀⠀⠌⠀⠀⠈⠀⠓⠂⠉⠀⠀⠐⡀⠀⠀⠀⠀⠀⢀⠁⠀⠈⠅⠐⠂⠨⠁⠀⠈⡀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠉⠡⠌⠈⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠁⠀⠤⢀⣀⡀⣀⡀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠈⠐⠠⠀⣐⡄⢰⣀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠈⠂⠤⢀⣀⣐⣂⣀⠀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⣀⡄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠁⢤⡤⠀⠁⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠐⡠⠂⠉⠈⠁⠢⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠢⠤⠤⠤⠤⠐⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⢀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⢉⡉⠉⠁⠫⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠁⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠠⠀⠠⠐⠁⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠂⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⢙⢞⠠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⢀⣀⣀⠔
⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⠪⠀⠚⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡐⠧⣀⣀⠸⠲⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠉⠙⠋⠉⠂⠠⡀
⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⡐⠁⠀⠕⠈⡕⢊⠁⠪⣀⣈⠂⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⢴⡦⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠑⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠐⠀⠀⠀⠀⠀⠑⠀⢀⠈⡄⠃⠸⢀⢃⡀⠄⠊⠀⠀⠀⠀⠀⠀⠑⠠⢀⠀⠀⠘⠃⠀⢀⡀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠉

⠀⠀⠀⠀⠀⠀⠀⢠⣴⢶⢤⡀⡠⢘⣶⡃⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⢠⣤⣶⣴⣦⣴⣦⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠑⠊⠂⠃⣿⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣻⣾⣿⣯⣇⠀⠀⢸⣿⣿⣿⣿⣷⣯⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇⠀⣀⣀⡀⣿⠀⠱⠇⢈⠆⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡏⠀⠀⢸⣿⣿⣿⣷⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⡍⠀⢏⣀⣀⡀⣿⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡏⠀⠀⠀⢸⣿⣾⣿⣿⣭⣽⡇
⠀⠀⠀⠀⠀⣠⣴⣾⡿⡿⣿⣶⣤⡀⠀⠀⢻⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡻⡿⣷⣦⣄⡀⢠⣿⣿⣷⢿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣴⣶⣾⣿⣷⣶⣦⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣭⣿⡇⢰⣿⣿⣟⣿⣿⣽⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣽⣿⣿⣿⣿⡇⣿⣯⣭⣭⣾⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣻⣿⣿⡇⠈⠀⠉⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠃⠙⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠻⠛⠟⠛⠾⠻⠟⠻⠃⠀⠀⠀⠈⠻⠻⠛⠁


⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢺⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣾⣭⣭⣽⡆⠀⠀⢸⣟⢽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡆⠀⠀⠀⣶⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢸⣿⣵⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⣿⣿⣯⡇⠀⠀⠀⢿⣯⣻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⠿⠀⠀⠠⣄⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣿⡇⠀⠀⣤⣤⣽⣿⣯⣤⣄
⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣧⣀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣧⡀⠀⠀⠀⣿⣬⢿⣿⡿⣿⣿
⠀⠀⠀⠀⠀⣾⣿⣿⣧⣧⣿⣿⣿⡆⢰⡇⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣭⣿⣿⣷⡆⠀⣿⣷⣿⡶⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⢶⣶⢾⡿⣿⣷⣤⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣯⣻⣽⣟⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣽⣿⡿⣿⡇⢿⣿⠏⠁⢿⣿⠿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣾⣿⣿⣿⣤⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠛⠳⢦⠶⠛⠛⢷⣦
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠤⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠉⢿⡀
⠀⠀⠀⠀⠀⠀⠻⣦⣀⣀⣤⡆⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⣷⡀⠀⠀⠀⠀⠀⠈⢷⣀⣤⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⣀⣴⠟⡉⠉⠙⣏⠉⠙⠓⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠶⠶⠛⠉⢹⡜⠟⠀⢀⡿⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⢀⣹⡶⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣠⣤⡈⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠳⠄⠐⣿⣀⣀⣴⣿⡿⠿⣿⣷⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠁⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⣀⡀⠀⠈⠙⠛⢩⡏⠷⣄⣈⣉⠉⠉⣿⣿⠛⣛⠒⠛⠛⠛⠉⣠⠊⠁⠀⠀⠈⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋⠀⠀⠀⠀⠀⠘⣷⣄⣀⣹⠀⠀⠀⡟⠻⣤⡾⠀⠀⠀⠀⠀⢷⢫⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢻⡿⠿⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⡜⢸⠉⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⠷⠖⠚⠀⠀⠀⣻⡀⠀⠀⠀⠀⠀⠀⢶⡇⡇
⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⣶⡶⠶⠶⠶⠤⣤⣶⠶⠶⢒⣛⡿⠛⣿⠶⣤⣄⣀⣀⣀⣼⣷⣿⣷⠆
⠶⣤⣬⣷⣶⣄⣤⣤⡤⠶⠟⠋⣿⠀⠀⠉⠛⠒⣶⢾⡉⣉⣷⡛⠉⠀⠀⠀⢸⡆⠀⠈⠉⠉⠉⢻⢻⠿⠇
⠲⠶⣚⡿⠛⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⣁⣰⠏⢻⣄⣻⡀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠸⣸
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠈⠉⠀⠤⢸⠈⠉⠀⠀⠀⠀⠀⢻⠀⠀⣏⢄⠤⡣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⣤⣤⣤⣤⣤⣄⣤⣤⣼⣤⣤⣤⣤⣤⡤⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⠀⠀⠀⠀⢸⣇⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡋⣥⣉⡉⢙⡦⠀⢰⣋⣥⣄⣀⣀⣹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣷⣆⠀⠀⠀⠀⢀⣀⣤⣶⣶⡷
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣷⣿⠿⠿⣿⣷⡶⢶⣦⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⣰⡿⠋⣀⡀⠀⠀⠙⣿⡄⠀⠀⠈⠉⠙⠛⠻⠷⣶⣤⣄⡀
⠀⣀⣀⠀⠀⢀⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣿⡇⢸⣭⣿⡆⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣶⣤⣀
⣾⣿⠋⠀⢀⣼⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠹⣧⡀⠙⠉⠀⠀⣠⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣦⣄
⢻⣿⣦⣴⣾⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣶⣶⡾⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠙⠛⠛⠛⠛⢁⣿⡇⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⣠⣾⣧⣀⠀⠀⠀⠀⠀⣼⡟⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⢠⣾⡿⠋⠉⠛⠷⠀⠀⠀⢰⣿⠀⣸⠻⣿⣄⠀⠀⠀⠀⠀⠀⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⣰⠇⠀⢈⠿⢷⣄⡀⠀⠀⠀⠈⠉⠙⠛⠛⠻⣿⣿⢶⣶⣶⣶⣦⣤⣤⣴⣶⣶⣶⠾⠟⠋
⠀⠀⠀⠘⢿⣷⣤⣠⣴⣶⠀⠀⠀⣿⣧⠏⠀⢀⡞⠀⠀⣹⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠙⣿⡿⠛⠁⠀⠀⠀⣿⡟⠀⢀⡞⠀⠀⣰⠏⠀⢠⡟⠙⠛⠃⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠸⣿⣀⡾⠁⠀⢠⡏⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⣄⠀⠀⠙⣿⣇⡀⢠⡟⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡄⢸⡇⢀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢻⣦⡀⠀⠈⠙⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⣠⣜⠛⠛⢹⣷⣟⣱⡿⠁
⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠙⠿⣦⣀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⡟⡀⣠⣿⣿⣯⣭⡵⠄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⣀⣠⠖⠉⢳⣼⣿⣤⡀⠀⠀⠀⠀⠀⢀⣼⡿⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡏⠀⠉⢳⡏⠀⠙⣧⣠⠄⢸⡇⠸⣿⠻⣷⣤⣀⠀⣴⡿⠋⠀⠀⠀⠀⠀⢠⡞⠉⠉⠉⠉⠙⣦
⠀⠀⠀⠀⠀⢀⣤⣾⡏⠀⠉⠉⠉⠉⠉⠉⢹⠋⠉⠉⡏⢱⠀⠉⢉⡷⢤⣴⠳⣄⠀⢸⡇⠀⣿⡆⠀⠉⠻⢿⣋⣀⣀⣀⣀⠀⠀⠀⣿⠉⠛⠒⠒⠚⠛⠙⡇
⠀⠀⠀⢀⣴⡿⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣧⢈⣧⡴⠋⠀⠀⢸⡀⠈⠳⢾⡇⠀⢻⣿⠀⠀⢰⣏⣁⡀⠀⢀⣉⣿⡀⢰⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢀⣴⡿⠋⣀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠟⠁⠀⠀⠰⠖⢸⡇⠀⠀⠀⣷⠀⢺⢹⡇⠀⡟⠀⠉⠉⠛⠉⠉⠈⣇⠈⣷⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠸⡿⡻⠋⠉⠉⠁⣢⣄⣀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢻⡄⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢿⠀⢸⠘⣷⠀⡇⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⠳⣦⣤⣤⣤⠾⠋
⠀⣰⢁⡇⠀⠀⠀⢠⡏⠉⠙⣿⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠶⠀⣿⠀⠀⠀⢸⡇⢸⠀⣿⡄⢳⡀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠙⠈⠁
⠀⡇⢸⠁⠀⠀⠀⢸⣇⣠⣴⣿⠀⠀⠀⠀⠀⠸⣇⠀⠀⣷⣀⣀⣀⣤⣤⣤⣿⣤⣤⣤⣼⣧⣾⣦⣿⣧⣤⣽⣷⣦⣴⣶⣏⣁
⢸⠀⣼⠀⠀⠀⠀⢸⠨⣿⠿⠋⢀⣀⣤⡤⠶⠶⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠉⠉⠉⠙⠛⠛⠲⠶⢴⡄⣄⡀
⢸⡀⣿⠀⠀⠀⠀⢸⣆⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡥⣯⣉⡛⢶⣄⡀
⠀⠉⠳⠦⠤⢤⣤⢼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠰⢮⣁⣈⣉⣁⣀⣤⠟⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢴⣗⡶⠄⠤⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣌⠛⠿⢷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⣤⣤⣤⡤⠶⠋⠀⠀⠀⣀⣀⣤⣴⣶⠿⠟⢁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣌⡉⠛⠛⠿⠿⠿⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⡿⠿⠿⠛⠛⢉⣁⣠⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠷⣶⣦⣤⣤⣄⣀⣈⣉⡉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⣉⣉⣉⣁⣀⣠⣤⣤⣴⣶⠶⠿⠟⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠉⢉⣓⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠒⠤⢄⣰⠁⠀⣠⠞⠁⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡈⠲⢄⡀⠙⢦⡾⠁⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠇⢴⣤⣍⣉⠙⠉⠓⠦⣝⡲⣄⡀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠷⠶⠚⠛⣿⠟⠛⠛⢿⡟⠻⠶⣯⣙⠛⠷⣶⡤⠞
⠀⠀⠀⠀⢀⣠⡴⠞⠛⠉⠁⠀⠀⠀⠀⢸⡇⠀⠰⣿⡆⣿⠀⠀⠀⠙⢷⡄⠀⠘⠉⢆
⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⣀⣴⣿⠤⠄⠀⠀⠈⢿⡄⠀⠈⡃
⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠁⠀⠀⠀⠀⠀⠀⠘⣧⠀⡲⠁
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢶⣦⣤⡀⠀⠀⠀⣿⠉⠁
⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢀⡾⠹⡆⠀⠀⠠⠾⠛⢿⣦
⠀⠉⠛⠲⠶⠶⠶⠶⠶⠶⠾⣿⠛⠋⠉⠁⠀⠀⣀⡴⠋⠀⢀⣧⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⣠⠶⡆⠀⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠛⢻⡋⠁⠀⠀⣠⣾⡇⠀⠀⠈⠻⣶⡟⠃⠀⠀⠀⠀⢸⠇⠀⡇⠀⢸⠇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢷⡶⠶⠛⠉⡼⠁⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⢸⣄⠀⠙⠛⠋⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⠲⠶⠛⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⢳⡆⢠⡶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠲⣦⣤⣴⠖⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢀⣠⢶⡿⠉⠙⣿⠛⢶⠶⣶⡶⠶⣶⡶⢶⠶⡶⠶⠟⢻⣇⠀⠀⠀⠀⠀⠀⠸⣶⡿⠀⡏
⠀⠀⠀⠀⣠⡴⠞⠋⠀⣼⠃⠀⢀⠏⠀⠘⡆⢠⠞⡟⠣⣄⡟⠀⢳⡀⠀⠈⣿⡶⣤⣄⣀⠀⠀⢠⣾⣿⣾⣿⡄
⠀⠀⠀⠀⠈⠙⠷⣤⣰⡟⠒⠚⠁⠀⠀⠀⠙⣇⡀⡇⣀⠞⠀⠀⠀⠳⢄⣀⣸⣇⠀⠉⠉⠛⠛⠻⢿⣿⣿⡏
⠀⠀⠀⠀⠀⣠⣀⣼⣿⡇⠀⠀⠀⠀⠀⠀⣀⣀⣉⣉⣁⣀⠀⠀⠀⠀⠀⠈⠉⣿⠀⠀⠀⠀⠀⣀⣼⠁⣽
⠀⠀⠀⠀⠀⠛⠛⠁⣿⠉⠉⠉⠉⠉⠁⠀⡏⠔⠒⠒⢢⢹⠒⠒⠒⠒⠒⠒⠒⢻⡆⠀⠀⠀⣼⢋⡴⠒⢮⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⡀⠀⠀⠀⠀⣇⢧⠤⠤⠼⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿⡘⢦⣤⠾⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠛⠛⠛⠛⠛⠛⢻⠛⠛⠛⠛⠛⡿⠛⠛⠛⠛⠛⢻⡟⠃⠀⠀⠀⠀⠙⠲⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⢲⣶⠒⠚⠀⠀⠀⠀⠀⠛⠛⡿⣞⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠛⠛⠓⠚⠉⣇⠀⠀⠀⡴⠚⠛⠛⠛⠁⠸⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣛⣶⣶⣶⣖⣻⣦⣞⣀⣀⣀⣳⣶⣶⣖⣛⣹⣶⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠔⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡆
⠀⠂⠀⡆⠀⠐⠀⠰⠂⠀⠀⠀⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠃⠤⢄⡑⠀⠀⠄
⠀⡆⠀⡇⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠐⠂⠠⠤⠄⠀⠀⠀⢰⡁⠠⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠁⠀⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣸⢴⠦⠤⠐⠒⠂⢂⠀⠀⠀⠀⠀⠀⠈
⠀⠁⢰⠃⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⡀⠀⠈⠁⠀⠘⠒⠊
⡀⠀⢸⠀⠀⠀⠸⠀⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸
⡇⠢⠜⠐⢄⠔⠀⠴⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠸
⢇⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡀
⠀⠉⠐⠆⠀⠒⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⣀⣀⡀⠀⢣⠠⣄⢀⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡇⠀⣿⠉⢆⣤⡴⣻⠀⣇⠀⠘⢦⡀
⠀⠀⠀⢧⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡇⠀⡏⠀⠁⠌⠁⠿⠀⢠⠠⡄⠈⠉⠒⢄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢀⡷⡇⢠⠇⠀⠀⢐⠀⠘⡀⢸⠄⡛⢠⡀⠀⠀⡙⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠆⠀⠐⠰⡊⣨⠃⢸⡃⠀⠀⣨⢀⠀⡇⢸⠂⡇⠀⠙⠖⠁⠀⣠
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡠⠔⠊⠉⠀⠀⣡⡀⠐⠁⣿⠀⠸⠂⠀⠀⠸⠐⠀⡇⢸⠀⣃⠀⠀⠢⣠⡜⠁
⠀⠀⠀⣀⡀⠚⡀⠡⠐⠊⠀⠀⠀⣀⠠⠚⠁⠀⠀⠰⠙⠀⠀⠀⠀⠀⠨⠈⠀⡇⢘⠤⢸⠀⠀⢠⠟⠒⢀
⠀⠀⡀⠑⠀⠶⠁⠀⠆⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⡤⡄⠀⡀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠈⡇⠠⠇⠀⢀⠂
⠀⠀⣍⠀⠀⣒⢫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠧⠀⢀⢠⡅⢀⢘⡃⠀⣇⢀⡧⠀⢀⠔⠁
⠀⠀⠢⠈⣉⣭⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁
⠀⠀⠀⠐⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢳⠄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⠘⡀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢈⠀⢸⠀⠀⠀⠀⠀⠘⡁
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠈⡀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠘⡃
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢡⠀⢸⣀⣀⣀⣀⣀⢠⠅
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡸⠁⠀⠀⠀⠈⠃⠀⠀⠀⠀⠡⠐⠛⠒⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠁⠀⠂⠀⠀⠀⢀⡇⠀⠀⡤⠄⠀⠀⠀⠀⠀⠀⣡
⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠶⣠⡄⠀⠀⢤⣀⡀⠴⢤⡀⠉⢈⣡⠀⠀⠁⠘⠷⣀⠄⠀⠄⢐⠡
⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠁⠋⠉⣉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠛⠻⠿⣿⣿⣿⣿⣶⣶⣬⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣦⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠈⠉⠀⠄⠀⠀⠀⠀⠀⠈⠉⠙⠻⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⡀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠁⠀⠉⠉⠛⠃⠀⠀⠀⠘⠋⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡶⡆⠀⠰⠀⠀⠀⠀⣶⡶⣦⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⠀⠸⢿⣿⠏⠀⠀⠀⠀⠀⠁⠿⣿⠿⠀⠐



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⢀⣴⠾⠛⠉⠙⣷⡀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⣰⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⢀⣴⡄⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⢀⣴⣿⣿⣿⣆⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⣰⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠙⣿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠴⢁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠘⣷⣄⡀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡀⠋⠁
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠃⠀⠀⠀⠈⣿⣿⣦⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⣡⣾⣷⡀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠀⠀⡀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠁⣀⣤⣿⣿⣿⣿⣇⣤
⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⠏⠀⠀⠀⠘⠛⠛⠿⣿⡿⠟⠀⣨⠁
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣶⡀
⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⡆⢹⣿⣷⡀
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠘⣿⣿⡇
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⢠⣿⣿⠇
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣶⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⣿⣷⣶⡦⠄

⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⠒⠛⠉⠉⠉⠉⠛⠳⡆
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠒⠒⠒⠒⠒⠋⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣤⣀⣰⡬⠥⠶⠚⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠈⢉⡳⠒⠒⢒⣛⣻⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠈⢀⣴⠞⠻⣝⠿⠋⠉⡿⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢠⡾⠽⠇⣠⠃⠉⠐⠉⠀⠀⠈⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⠈⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠙⠀⠖⡖⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡶⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⢷⡀⠀⢀⠀⠀⠀⠙⠛⠻⣿⠿⠿⠿⠿⠛⠛⠁⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⠈⠛⠞⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⢡⠇⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣸⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠖⠒⠒⡶⠒⢶⢲⢶⡖⢲⠒⠒⠒⣿⡄⠀⠀⠀⠀⢻⡛⠒⠒⠀⡏⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⢀⣴⠟⣿⠀⠀⠀⣡⡀⢸⡏⢲⡇⣸⠀⠀⢠⢏⡷⣄⠀⠀⠀⠀⢻⡄⠀⠀⢧⣀⠀⠀⠸
⠀⠀⠀⠀⠀⢠⡾⠁⠀⣿⠀⠀⠈⢣⡀⠀⣇⡼⠀⢠⠏⡴⢋⣾⠀⠈⠳⢦⣄⡀⠸⣿⡿⠇⢠⠋⠓⢦⣄
⠀⠀⠀⠀⢠⡿⠁⠀⠀⣿⠀⠀⠀⠀⠙⣄⢹⠁⣠⡧⠚⡠⠋⣿⠀⠀⠀⠀⠉⠙⠛⢹⡇⢀⠏⠀⢀⣤⠾
⠀⠀⢀⣤⣾⣇⠀⠀⠀⣿⠀⠀⠀⠀⠀⢨⣿⠖⢋⡴⠊⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⣷⡦⠶⠛⠉
⢀⡆⢺⣿⣿⡟⠳⠿⠋⢻⡄⣀⣀⠤⠒⣉⣤⡞⡁⢀⣀⠀⠤⣿
⣼⠀⠸⠟⢿⠿⡆⠀⠀⣸⣿⣉⡤⠖⠛⠛⢲⠖⠓⠒⠂⠈⠉⣿
⡏⠑⠒⠒⢺⣀⡗⠒⠉⠁⢸⠷⠾⢿⡿⠿⠾⠷⠶⠶⡶⠶⠶⠿
⢧⣄⣀⣀⣀⣉⣀⣀⣠⣤⠼⠀⢀⢸⡇⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠈⠉⠉⠉⠁⢀⡖⠋⠙⡉⠉⣙⢷⠀⠀⣰⣛⠉⠉⠙⠛⠲⣤
⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⡀⣤⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⠀⠀⠀⠀⠒⠂⠀⠈⠀⠈⠉⠁⠀⠀⠀⠀⠀⠒⠂⠀⠐⠶⠂
⠀⠀⠀⠀⢀⣾⣿⣿⣼⣿⣷⣾⣿⣶⣶⣦⣤⣦⣤⣀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣶⣦⣶⣤⣀⣀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⠀⡀
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀
⢀⣀⡀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠤⠤
⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⠆⠠⠄
⠀⠀⣸⢹⣿⣛⣿⣿⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠄
⠀⢰⠇⠘⠛⢿⣿⣿⣿⣿⡍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⡌⠀⠀⠀⣸⣿⣽⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⢘⠀⠀⠀⢀⡿⠹⣟⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠗
⡞⠀⡠⠶⣾⠷⠶⣿⣿⣿⣿⣿⣿⠿⣿⣿⣏⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⠙⠃
⠁⣠⣄⣸⣇⣁⣨⠽⠿⠾⠻⢿⠿⠛⠿⠿⠿⠿⠿⡿⠿⠿⠿⠽⠿⠿⠻⠿⠟⠿⢿⡿⠿⠛⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠻⠿⠟⠂⠈⠁
⠊⠉⠀⠀⠠⠤⠤⠤⠤⠖⠒⠺⠤⣀⣀⣀⡀⣀⣀⣀⣀⣀⣤⣀⣀⣀⣀⣀⣀⣀⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠄⠠⠤⠤⠶⠄⠀⡀⠀⠀⠀⠀⢈⣉⠍⠉⡉⠙⠓⠲⠶⠶⠶⠶⠦⠤⣄⢀⠤⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⢄⡀⠂⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⣤⣤⣤⣤⣤⣀⡀

⠀⠀⠀⠀⠀⠀⠀⣀⡀⣤⣀⣀⣀⡀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣽⣧⣽⣿⣿⣿⣧⣼⣿⠀⣠⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡄⠀⢠⣿⡿⣟⣧
⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⡟⣿⡿⣿⣿⡿⠸⣿⣿⣿⣿⣯⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣿⣿⣿⣹⡇⠀⠘⣧⣭⣯⠟⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿⣽⣿⢿⣿⠶⣿⣿⢿⣯⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠱⣿⣿⠀⢿⣿⡄⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠃⠀⢰⡟⣭⡝⣷⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠁⠀⠈⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡽⣿⢏⣀⣀⠀⠘⢮⣉⣣⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⢿⢻⣿⣷⠀⢶⣮⣼⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣿⣿⣿⣿⣿⣷⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣻⣿⡗⠀⢺⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣷⣿⣳⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠑⠛⠛⠀⠀⠀⠘⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠋⠘⠧⣿⡯⠟






⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠰⢤⡀⠀⠀⠀⣤⣴⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⡆⠀⢹⡄⢀⣾⣻⣿⣿⣼⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠛⠀⠠⠆⠀⠀⠸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠁⠀⡸⠁⠙⢿⣿⣿⣿⣿⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢀⡀⣤⣴⣴⣆⠀⠖⠀⠀⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢙⠒⠀⠋⠀⢀⠀⠈⠻⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⢿⠿⢿⡆⠀⠀⠀⠈⠏⠀⢐⣿⡆
⠀⠀⠀⠀⠀⠀⠠⣶⣘⣶⣶⣶⣖⣸⡶⢴⣶⣽⣿⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⡟⠀⠀⠀⠠⣶⡆⢨⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠋⠀⠠⣶⡆⠀⣿⡇⠘⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⢾⣿⡷⣿⣿⢾⣿⣿⣿⣻⣽⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⡇⠀⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⣿⣿⣾⣿⣾⣽⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⣿⡇⠀⣿⡇⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠿⠃⠘⠻⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⡀⠀⠈⠉⢀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣩⣉⡽⣷⣾⡿⣿⣾⣼⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⣀⣴⣾⡟⣴⢳⣿⢸⡜⣮⢻⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣷⣿⣼⣿⣸⣧⣽⣶⡇⠈⠉⠉⠉⣉⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠋⠁⠀⠈⠹⣿⣿⣿⡟⣤⡀⠀⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠠⠴⠖⠒⠶⣿⠟⠙⠁⠛⢸⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢨⠀⠀⠀⠻⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣶⣄⣀⣰⣤⣠⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣉⣉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠲⣄⣠⡤⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠚⠁⠳⡄⣀⣀⣤⣤⠤⠴⠶⠶⠖⠒⠒⠛⠛⠛⠛⠒⠒⠲⠶⠶⠦⠤⣤⣤⣀⣀
⠀⠀⠀⢀⡴⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⢦⡀
⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⣾⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⣷
⠀⢰⡇⠀⠀⣠⣾⣿⣿⣿⣿⣟⡳⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⣞⣿⣿⣿⣿⣿⣷⣄⠀⠀⢸⡆
⠀⢸⠁⠀⣼⠿⠿⠿⠿⢿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡿⠿⠿⠿⠿⣧⠀⠈⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠾⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⣀⡀⠙⢿⡀⠀⠀⠀⠀⠀⠀⢠⡿⠋⢀⣀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⢿
⠀⡇⢀⢀⠀⠀⣠⣞⣭⣶⣶⣶⣿⣿⣷⣦⠃⢀⠀⠀⠀⠀⠀⠊⣴⣾⣿⣿⣶⣶⣦⣭⣷⣄⠀⠀⡀⡀⢸
⠀⡇⠛⠣⠷⠿⠿⣬⣝⣛⣛⣛⣛⣿⡿⠟⠀⣼⠀⠀⠀⠀⡇⠀⠻⢿⣟⣛⣛⣛⣛⣫⣵⠿⠿⠿⠜⠛⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⢀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠀⢸
⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢹⡟⠀⠀⠀⠀⣿⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢻⠀⢰⣄⡀⠀⠀⢀⣀⣤⡶⠟⢳⠃⠀⢸⡇⠀⠀⠀⠀⢸⠆⠀⠸⠜⠻⢦⣤⣀⡀⠀⠀⣀⣠⠆⠀⡟
⠀⢸⡄⠀⢻⣏⢿⣿⡉⠉⠀⠀⠀⠀⠀⠘⠆⠛⠦⠤⠤⠶⠋⠲⠂⠀⠀⠀⠀⠀⠉⢩⣿⡟⣽⡟⠀⢠⡇
⠀⠈⣇⠀⠈⢿⣧⢻⣿⡄⠀⠀⠀⠀⠀⢀⣴⣿⣿⠷⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢠⣿⡟⣼⡿⠀⠀⣸⠁
⠀⠀⢻⡀⠀⠈⢿⣦⠻⣿⣶⣶⣶⣶⣾⣿⣿⡿⠋⠀⠀⠙⢿⣿⣿⣷⣶⣶⣶⣾⣿⢏⣼⡟⠁⠀⢀⡟
⠀⠀⠈⣧⠀⠀⠀⠻⣷⠦⡉⠉⠉⠉⠉⠹⠿⠶⠶⠶⠶⠶⠶⠿⠏⠉⠉⠉⠉⢉⠴⣿⠟⠀⠀⠀⣼⠁
⠀⠀⠀⠈⢧⡀⠀⠀⠙⢧⡈⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠈⢁⡼⠃⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠈⢻⣿⣿⡟⠁⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⠀⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠸⣿⣿⠃⠀⠀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠤⠽⠧⠤⠶⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠋⠈⠋⠉⠛⢹⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠛⠛⠃⠀⢸⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⠀⠀⠒⠒⠒⠀⢸⣄⣀⣴⠃⠀⠀⠀⠀⠀⢀⣠⠔⠳⣄⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣸⣄⣀⣀⣀⣀⣀⣸⣄⢰⠇⣰⠒⠒⠒⠚⠋⠉⠀⠀⠀⠈⠓⠲⡄⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠀⠹⣄⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⣠⠃⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣀⣀⠀⣀⣀⡀⠀⣀⣸⠀⠀⠀⠀⠳⣄⠀⠀⠤⠄⠀⠀⣠⠏⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⣤⡽⠀⢿⣤⡇⠨⣧⣼⠀⠀⠀⠀⠀⠈⡷⠦⠤⠤⠴⣞⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠖⢶⠀⣶⠒⡆⢠⡖⢯⣀⣀⣠⣤⠤⢾⠁⠀⠀⠀⠀⠘⡦⠤⣤⣀⣀⣠⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢘⣓⡛⠀⣻⣶⣧⣬⣷⡚⢡⠏⠀⢹⠀⠈⢧⠀⠀⠀⢀⡞⠀⢀⡇⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣉⣽⠀⣿⣲⡤⣄⣠⠼⢿⠀⠀⠈⠓⢲⠈⢻⡙⢹⠏⠠⡖⠋⠁⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣸⠇⠀⠈⣇⠀⢠⠇⠻⣄⠀⢳⡏⢀⣤⠇⢰⡄⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⡀⢀⣀⣀⣀⣀⣸⠀⠀⠀⢹⠀⣿⠀⠀⠀⠙⢾⡇⠈⠀⠀⠀⡿⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠰⠶⠶⠶⠀⠸⡇⠀⠀⢸⡆⢸⣦⠀⠀⠀⢸⡇⠀⠀⠠⣴⡆⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡇⢸⣿⠀⠀⠀⢸⠀⠀⠀⠀⡏⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⠃⡾⣻⠀⠀⠀⢸⠀⠀⠀⠀⡇⢷⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠽⠦⠤⠾⠥⠿⠤⠤⠤⠼⠦⠤⠤⠤⠧⠼⠤⠤⠤⠿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠋⠋⠙⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠛⠛⠛⠛⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠁⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠐⠠⢀⡔⠁⠀⠀⢙⠾⡀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⡜⠁⠢⣄⢠⠋⠀⠈⠢⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠘⠀⢄⡸⠀⠀⠀⢈⠗⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⢤⡀⠀⠀⡉⠢⠀⠀⡘⠀⠀⠁⠀⠀⠀⢀⣀⣤⠄⠀⠀⠈⠛⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠒⢄⠀⠀⠀⠀⠁⣀⣠⡤⠶⠖⠋⠉⠁⠀⢀⣀⣤⡤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⣀⣠⠤⠶⠚⠋⠉⠀⢀⣀⣤⠴⠖⠺⣿⠛⠻⠷⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠉⠁⠀⢀⣀⣤⠶⠖⠛⠙⣷⠰⣾⠏⠉⣹⠂⠀⠀⠀⠀⠉⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⣠⡤⠶⠒⣿⣏⢫⠀⠀⠀⡀⠀⠙⠶⣤⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣠⣼⣯⣸⡌⣇⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠈⢻⡇⢸⣟⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣴⢼⡇⡟⠈⠙⠲⢬⣳⣤⡀⠠⣦⣤⣄⣀⣀⠀⠀⠀⠀⣀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢸⠀⡇⠀⠘⠶⢤⠈⠙⠻⣦⠀⠀⣿⠋⠙⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠒⠉⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠰⠩⠽⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢀⡀⢀⣀⠀⠀⠀⠀⠀⠀⢀⡷⠶⠶⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇
⠀⠀⠀⢀⣤⣤⣿⡉⠀⠀⠀⠀⣀⣤⣾⠓⠤⣤⣀⡟⣿⠳⢦⣤⣤⣤⠶⠞⣻⠟⢷⣤⣀
⡀⠀⠀⠘⢿⣿⣿⡷⠶⠶⠞⠛⠉⢀⡿⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⢀⣠⠞⠁⠀⠘⣧⠉⠛⠶⣄⡀
⡏⠙⠒⠒⠚⠋⠹⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⢳⣄⣴⠟⠁⠀⠀⠀⠀⢻⣆⣤⠞⠋⠁
⠧⣤⡤⠤⠶⠂⠀⡇⠀⠀⠀⠀⠀⣿⠈⠉⠉⠑⠒⠒⠒⣿⠉⠉⠉⡏⡗⠒⠒⠒⠈⠙⣿⣇⣀⡄
⠀⠀⠀⠀⡔⡆⡂⢠⠀⠀⠀⠀⠀⣟⠓⠲⠶⠦⠤⠤⣤⣼⣤⣤⣤⣧⣧⡤⠤⠤⠶⠶⣿⠈⠛⠋
⠀⠀⠀⠀⣢⠟⠁⠠⢄⡀⠀⠀⠀⠛⠛⠳⢶⡶⠶⠦⢤⣤⣤⣤⣤⣤⣤⣤⠶⢶⠶⠶⠿
⠀⠀⢠⠞⠁⣀⡴⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠓⢦⠤⡤⠜⠁⠀⠈⠢⡤⢤⠤⠞
⠀⠀⠈⠢⠞⠉⣇⠀⠀⠈⣇⠀⠀⢠⠔⠲⠤⠤⠼⠆⣇⠀⠀⠀⠀⢀⡗⠼⠀⠠⠴⠶⢤
⠀⠀⠀⠀⠀⠀⢹⠀⣀⣀⡼⠀⠀⠸⣄⣀⣀⣠⠤⣄⣈⡇⠀⠀⠀⣟⠀⣤⣤⣀⣀⣀⣼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠠⢄⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡏⠀⢠⠀⠄⢀⠔⠁⠀⠢⢔⠀⢼⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⢞⠠⡂⢆⠂⡈⠀⠀⠀⠀⣀⡀⠑⠠⠓⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢡⠀⠤⠶⠄⠀⠀⠀⠀⣶⠤⠀⠀⠀⠀⠌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣸⠀⠀⢐⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣠⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠸⡀⠌⣰⠀⠀⡈⠉⠁⠀⠉⢉⠂⠀⠀⢪⡄⣦⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⡀⠀⠈⢢⠐⠒⠲⠁⠀⢀⣀⠯⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣀⠈⠚⠓⠂⠉⠉⡩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢸⡂⠈⠑⠖⡀⠤⠒⠹⢸⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣌⠀⠀⠳⡀⠀⠈⠈⠀⠀⢈⠜⠀⠀⡗⠀⢤⣀⣤⡤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣴⡞⡟⣟⢴⣀⡏⠈⠈⡀⠀⠀⠈⠢⣀⠀⢀⠠⠀⠀⠀⢰⠁⠀⠀⠻⣞⣡⣞⣿⣻⠱
⠀⠀⠀⠀⠀⠀⠄⠀⠙⠓⠿⠛⠉⠛⠉⠀⠀⠈⢄⠀⠀⠀⠀⠄⢦⡀⠀⠀⢀⡎⠀⠀⠀⠉⠈⠛⠛⠛⠻⢋⡄⢸⡄
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣠⣞⢊⡀⠳⡝⣡⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡜⠀⠀⢷
⠀⠀⠀⠀⠀⠂⠀⠀⢸⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡪⠙⢺⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢘⠂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⠁⠈⡆⠀⠀⢶⣒⠨⡁⠁⣶⡆⠀⡏⡄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠐⠀⠀⠀⣟⢇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠢⠐⡇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡅⠁⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⡀⠀⠗⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⢂⡰⠀⠀⠀⠀⢸⣇⠀⠀⣩⢞⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠁⠀⠀⠘⣾⡿⠀⠀⠙⠋
⠀⠈⠁⠂⠀⠀⠰⣼⣿⠀⠀⢴⣯⠇⠀⠀⠀⡀⠀⠀⠀⠘⠛⠀⠀⠀⠾⣷⠄⠀⠀⠀⠙⡇⠀⠀⢀⣿
⠀⡴⣶⡄⠀⠀⠀⠻⡇⠀⠀⠀⣀⠀⠀⠀⠀⡁⠀⠀⠀⠀⣾⡇⠀⠀⢊⡴⡃⠀⠀⠀⠀⠃⠀⠀⢠⣿
⠀⢞⣵⠀⠀⠀⠀⠀⠃⠀⠀⠀⣿⠀⠀⠀⡠⠁⡀⠀⠀⠀⢸⣇⠀⠀⣩⢞⡅⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣫⢾⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠈⠨⠮⡈⠀⠀⢠⣸⣿⠀⠀⢴⣯⠇⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⣵⣻⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣀⠀⠀⠀⠀⢿⡟⠀⠀⢀⣀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣾⡆
⠀⠈⠉⠀⠀⠀⠀⠆⠀⠀⠀⠈⣉⡀⠀⠀⢋⠿⡃⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⡐⠁⠀⠀⠀⠀⢹⡇
⠀⢸⣿⠀⠀⠀⢠⠃⠀⠀⠀⠀⢿⡇⠀⠀⡡⢚⡅⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠊⠩⢎⡁⠀⠀⢠⣸⣧
⠀⢸⣿⠀⠀⡐⢓⡜⠂⠀⠀⠀⢸⣷⠀⠀⠔⣫⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⡀⠀⠀⠀⠈⢿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠃⠀⠀⠸⣼⣿⠀⠀⠺⠷⠃⠀⠀⠀⡄⠀⠀⠀⠈⣉⡀⠀⠀⢚⡿⠂⠀⠀⠀⠈⡇
⠀⠈⠋⠀⠀⠀⠴⣷⠄⠀⠀⠀⠹⡟⠀⠀⠀⣤⠀⠀⠀⢀⠆⠀⠀⠀⠀⣿⡇⠀⠀⣩⢞⡁
⢀⠀⣿⡇⠀⠀⢚⡵⠂⠀⠀⠀⠀⠃⠀⠀⠀⣿⠀⠀⠀⠬⡠⠄⠀⠀⠀⢸⣇⠀⠀⠴⣫⡄
⠀⡀⢿⡇⠀⠀⣩⢞⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠑⠄⠀⠀⠸⣼⣿⠀⠀⠺⡷⠃⠀⠀⠀⡄
⠀⢠⣸⡇⠀⠀⢴⣯⠇⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⣠⣦⡀⠀⠀⠀⠻⡇⠀⠀⠀⣤⠀⠀⠀⠀⠆
⠀⠈⢿⡇⠀⠀⠀⣁⠀⠀⠀⠐⠂⠀⠀⠀⠀⣤⡄⠀⠀⡴⣫⡄⠀⠀⠀⠀⠃⠀⠀⠀⣿⠀⠀⠀⢌⠠⠄
⠀⠀⠈⠇⠀⠀⠀⣿⠀⠀⠀⠘⠁⠀⠀⠐⠀⢻⣇⠀⠀⢞⡵⠆⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠐⠑⠄
⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠔⢣⡜⠂⠀⠀⢀⢸⣿⠀⠀⣫⣞⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠇⠀⠀⠈⣾⣿⠀⠀⠈⠋⠀⠀⠀⠀⠆⠀⠀⠀⠀⣤⡄⠀⠀⡵⣫⡅
⠀⠀⠄⠀⠀⠀⠈⣉⡀⠀⠀⠾⣿⠆⠀⠀⠀⠘⡇⠀⠀⠀⣿⠀⠀⠀⢠⠃⠀⠀⠀⠀⢻⡇⠀⠀⢞⡵⠆
⠀⠰⠂⠀⠀⠀⠀⣿⡇⠀⠀⢚⡵⡇⠀⠀⠀⠀⠁⠀⠀⢠⣿⠀⠀⠔⠓⡜⠂⠀⠀⠂⢸⣧⠀⠀⣫⣞⡃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠛⠉⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⢁⣀⣀⡀⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⠞⠂⠈⢁⡤⠀⠘⣿⠛⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠚⣰⠀⢀⠀⢀⠠⡉⠐⠋⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢓⡶⢽⡲⡶⠑⠘⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣴⣶⣶⣤⣤⡀⠀⠘⢦⡓⠀⠚⢁⠀⠀⠀⠀⠀⢀⡷⠦⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠋⠉⠀⠀⠉⠙⠛⠻⣷⣦⡀⠙⢦⡁⠁⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠏⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⠴⠆⠀⠀⠤⣄⠀⠀⢿⣿⣿⢀⡤⠟⠲⢤⣤⠴⠚⠁⠀⠀⠀⠀⡄⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⢴⢀⠀⠀⣄⠈⠀⠀⠸⣿⣿⢯⡤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣓⠄⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠾⣡⢤⠀⠁⠀⠀⠀⠀⠙⠙⣷⢍⣻⠅⡉⠉⠒⠲⠤⣄⣀⣀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣀⠘⠿⣖⣩⠿⠂⠀⠀⠀⣼⡛⠛⠁⠉⠓⠶⠔⠲⣞⠒⠶⢴⡇⠈⠉⠉⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠟⢋⠲⣦⣤⡴⠒⠒⠻⢤⣉⡽⠃⠀⠀⠀⢠⠔⠚⣦⡀⠈⠉⠛⠒⠲⠦⠤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⡀⠙⠒⠀⠒⠈⣀⠄⠀⠪⠭⣓⢦⡄⠀⠉⠀⠈⠀⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠙⠧⣄⠀⠀⠀⠀⠀⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⡀⠀⠀⠀⠀⠉⠒⠢⠤⠤
⠀⠀⠀⠀⠀⣠⠞⠋⠀⠱⡐⠒⠢⠤⠤⠤⠤⠶⠒⠋⠁⠀⠀⢀⣠⡤⠶⠖⠚⠛⠉⠉⠉
⠀⠀⠀⠀⠺⢧⣄⡀⠀⠀⠈⠑⠦⢤⣀⡀⠀⠀⠀⠀⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⡀⠀⠀⠀⠈⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⣠⣤⡀⠀⢤⣤⣤⠄⠠⣤⣤⣄⠀⢀⣀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠘⣿⣛⣷⠀⢸⣯⣽⣧⠀⢸⡷⠆⠀⢸⣿⣶⡟⠀⢿⣭⡉⠀⣿⣃⣀⣿⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢰⡶⢿⣦⠀⠀⣼⣿⣧⠀⠀⣿⠉⣿⡀⢸⣷⣶⠿⠁⠸⠷⠶⠇⠸⠇⠸⠷⠀⢶⣼⠟⢀⣿⠙⢻⡇⢰⡟⠁⢹⡇⢀⣿⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣛⣻⡇⠀⠿⠉⠙⠃⠀⠉⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡁⠀⠙⠁⠈⠻⠶⠟⠁⣸⡟⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠁⠀⣀⣠⠤⠴⠖⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠓⠶⠦⢤⣀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⢀⡀⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣷⣿⣿⡟⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⢀⡀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠈⠁⠀⠀⠑⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠊⠀⠀⠈⠁⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠈⢘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠈⣻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠁⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠫⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⠝⠁⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠑⢄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠨⡃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠈⠢⠄⠈⢠⠀⠀⡠⠗⠁⠀⠀⠀⢐⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠕⠀⠠⠪⠂⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⣀⡀⣀⡠⠬⠭⣥⡄⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠠⠼⠓⡉⢠⡆⠀⠀⠀⠙⣴⠟⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⣀⣄⣦⠸⡧⠈⣻⠂⠀⠀⢀⣇⢐⠆⠈⠀⢠⣴⠘⢧⡄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡶⠾⠟⠁⢀⡠⡄⠀⣿⣝⢿⠀⠛⠁⣠⡶⠶⠶⠟⠋⠀⣀⢰⡸⡇⠹⡄⠘⠁⠀⣠⣴⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠠⣿⡘⣗⢷⠀⠋⢉⡤⠐⣫⡾⠋⠀⠀⠀⠀⠀⠐⣇⡈⠳⠛⠀⢀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠸⠃⠉⢀⣠⠔⣉⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠠⢄⠉⠢⣑⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡀⢴⣾⠟⢋⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡙⠳⣿⡷⢉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠉⣈⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⢶⣄⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⢤⡀⠚⠡⠶⠀⠀⠹⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣄⠠⠖⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⢠⠎⠈⠃⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡎⠣⠄⠤⠀⠇⠀⠀⠀⠈⠃⢸⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠐⢀⢀⡀⠀⡀⢀⡀⢀⠀⠂⠀⠀⠀⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⡉⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⡄⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠃⠀⠀⠀⠲⡀⠀⢠
⠀⠀⠀⠀⠀⠀⣤⢄⡀⠀⡀⣠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡐⠁⠀⠀⠀⠀⠀⠀⠙
⠀⠐⠀⠠⣴⡶⠂⣄⠀⠀⠀⠀⠀⠀⠈⠡⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠢⢀⠀⠀⣄⠠⢩⡟⠀⣼⣦⠀⠀⠀⠀⠀⠠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠈⠐⠀⠀⣍⢀⡛⠛⡋⠆⠉⠀⠀⠀⣀⣀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⠂⠈⠉⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⣀⡴⠁
⣀⣀⢀⣀⡀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⡀⠀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣾⠉⠉⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠿⠆⠀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⣀⠤⠀⠸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠃⠀⠀⠀⠀⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⡁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠛⢤⡤⢊⠀⠀⠀⠀⠀⠀⠀⣸⠛⠉⠀⣀⡤⠰⣆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡂⠁⣓⠀⠀⠀⠀⠀⠀⠈⠳⡆⠀⠉⠀⠀⠈⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠉⢿⡆⠈⢆⠀⠀⠀⠺⢷⡀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠈⠳⡀⠀⠑⠈⠉⠀⠀⠀⢀⠀⠀⠀⠐⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠘⢆⠄⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠈⠳⣄⠀⣠⡎⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⣀⣀⠀⣀⣀⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠙⣧⠁⠀⠀⠀⠀⠀⣴⣺⠃
⠀⠀⠀⠀⢀⣠⣴⣶⡂⢹⣿⢻⣄⠉⢙⣿⡿⠿⠛⢉⣩⣍⣀⠁⠀⠀⡼⣷⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⡾⢡⡇
⠀⠀⠀⠀⠘⠛⢿⣿⣷⣾⣯⣬⡭⠁⡄⢰⣶⠺⠇⠈⠳⣼⣫⣄⠁⠀⡇⠻⢿⣦⠀⠀⠀⠀⠀⠀⢀⣤⣾⠋⠁⣾
⠀⠀⠀⠀⠀⠐⢌⢻⣿⣿⣿⣿⡦⢄⠁⢸⡿⠀⠀⠀⠀⠈⣿⡞⢀⣸⠃⠀⠸⣿⣷⣤⠀⢀⣤⣴⣿⡿⠁⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠺⠛⠛⠛⠋⠛⠀⠀⠀⠀⠀⠀⠀⠀⣿⢡⣿⣿⠀⠀⠀⠈⠻⢿⣿⣿⡿⠛⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣝⡿⣿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⢀⠀⠀⠀⠀⢀⢀⠀⢀⠀⠀⠀⢀⣀

⠀⠀⠀⠁⠀⠈⠃⢀⠀⠀⠀⠀⣠⡀⠠⣤⣄⣠⡀⠀⠀⠀⠀⡂⣠⠀⠀⠀⠀⠀⠀⠀⡀⠠⠨⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠞⠁⠀⠀⠀⠀⠿⠦⣤⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠠⠳⠀⠀⠀⠀⠀⢧⣷⡀⢰⣦⠆
⠀⠀⠀⠂⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠻⠁


⠀⠀⠀⠀⡴⠛⢦⠀⠀⠀⠀⠀⠀⡠⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡏⠉
⠀⠀⠀⢀⡷⠀⢾⡀⠀⠀⠀⠀⠐⠀⣭⠰⠇⡈⠀⠀⠀⠀⠀⣦⣀⢀⣴⡄⠀⠀⠀⠀⡇⠀⠀⢰
⠀⠀⠀⠿⠶⠤⠶⠷⠀⠀⠀⠀⠀⠑⠢⠤⠀⠁⠀⠀⠀⠀⠀⠿⠿⠿⠿⠃⠀⠀⠀⠀⣧⠄⠠⣼


⠀⠀⠀⠠⠊⠀⠑⠄⠀⠀⠀⠀⣾⡿⣿⣦⣤⣤⡀⠀⠀⠀⠠⢚⡴⢦⡀⠀⠀⠀⠀⠀⣤⡀⡶⠦
⠀⠀⢰⡇⠀⠀⠀⢼⡄⠀⠀⠸⣿⣧⣼⣿⣇⣹⣷⠀⠀⠀⠀⡏⠀⠀⢸⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠈⠁⠀⠴⠄⠈⠁⠀⠀⠀⠀⠙⠛⠈⠻⠿⠿⠀⠀⠀⠀⠈⠒⠒⠁⠀⠀⠀⠀⠀⠀⠙⠶⠄


⠀⠀⠠⠀⠀⠀⢠⡄⡀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⡔⠂⠢⠀⠀⠀⠀⠀⠀⠀⣠⡐⢠⡀
⠀⠀⠀⠀⢠⣤⡾⠛⠁⠀⠀⠀⠀⠘⠃⠀⠛⠀⠀⠀⠀⠀⠀⠑⠤⠜⡀⠀⠀⠀⠀⠀⡜⠋⠀⠈⠛⡆
⠀⠀⠀⠀⡴⠛⢷⡀⠀⠀⠀⠀⠀⣴⣦⣦⣶⡄⠀⠀⠀⠀⠐⠮⠽⠟⢻⣆⠀⠀⠀⠀⠘⢷⡄⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠠⢀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠐⠒⡲⠒⠊⠁⠀⠀⠀⠀⠘⣤⠆
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⣀⠤⠠⠤⠨⠄
⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⢀⠀⠁⠀⠀⣀⠄⣊⣁⠄⠀⠀⣭⣔⢂⡂
⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⠋⠀⠀⢠⠞⢁⠞⣁⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⢣⡀⠀⠀⠁⡀⠀⡰⠁⠸⠀⠈⢠⡆⠐⠀⠀⡡⣷⠠⡇
⠀⠀⠀⠀⠀⠀⠉⠒⠂⣉⠀⠈⠁⢳⠐⡄⠈⠘⠁⠀⠀⠀⠉⢀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⣠⠀⠀⠀⠀⠀⢀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⡆⠀⠀⠁⠙⠯⡀⢈⠩⡈⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠠⣀⠀⠈⠒⠭⠤⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢶⠀⠀⠑⠦⢀⡀⢤⡤⣊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠉⡇⠉⠀⠑⠠⢀⣄⠀⠄⢪⢠⡂⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠃⠀⠁⠀⠀⡰⠀⠈⣀⡖⠀⡠⣇⣀⣀⢂⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠤⠀⢴⡠⠾⡌⠈⠀⠀⢄⣀
⠀⠀⠀⠀⣀⣂⡁⠀⠀⠀⠀⠀⢠⠃⠀⠀⢰⠀⠀⠇⡍⠉⡇⠸⠀⠀⠀⠀⡥⠈⡡⠔⠂⠙⠿⠓⠀⠀⠀⡢⠀⠀⠉⠉⠁⠀⠐⠀⠒⠞⠀⠀⠀⠄⠁⠠⡀
⠀⡠⢀⠠⠀⠂⠠⢁⡢⠀⠀⠀⢸⡔⡒⠚⠦⣧⠀⠀⢸⠀⠡⡀⠃⠈⢠⠆⠀⠠⡱⢺⡄⢀⠤⢄⠀⠉⢉⠡⠀⠠⠤⠤⣆⠀⠀⠀⡈⠔⠒⣆⣈⣽⠆⠀⢹
⡜⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡿⠂⠀⡸⠀⠀⠉⠲⠠⠾⢧⡀⠀⠙⢅⣀⠓⠤⠔⠀⢄⠤⠀⢀⠂⠶⡀⠀⠢⠤⠤⢞⣀⠨⠒⠋⠁⢀⡤⣳
⣇⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠂⠱⠶⠊⠀⠀⠀⠀⠀⠀⠀⢱⡝⡲⢤⣀⡀⠈⠉⠓⠒⠒⠂⠲⠀⠛⠚⠑⠒⠒⠋⠉⠁⢀⣀⣠⠤⣖⡭⠚⠁
⢸⡈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢴⣇⣀⣀⣀⣀⣀⣀⣀⢀⣀⡌⠈⠢⣧⣰⣯⣭⣷⣒⣲⡶⣶⣶⣤⣤⣶⣶⣶⣖⣒⣺⠯⠽⠒⠚⠉
⠀⠑⠢⠭⠐⡒⠠⠦⡀⠀⠀⠀⠀⠀⡠⠤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠋⠀⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣄⣀⣠⠔⠊⠁
⣿⣿⡟⠛⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣴⣄⣿⣿⣾⣷⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠎⠻⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⣴⣿⡿⠋⠁⠀⠀⢈⣽⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⣀⣴⣿⡿⠛⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷⣤⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⢿⣿⣿⣿⣿⣿⡏⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣿⢿⣿⡏⠛⠃⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉







⡟⠻⠿⠻⠟⡛⠟⠻⠛⠻⣿⣟⣛⣟⣻⣿⣿⣛⣿⣿⣿⣛⣿⣻⣿⣿⣿⣿⣛⣿⣿⣿⣻⣟⣿⣿⣿⣛⣿⣛⣿⣿⣟⣻⣿⣿⣟⣿⣿⣿⣛⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⢟⣥⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣓⣧⡐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⡀⠀⣠⣤⠀⠸⣿⠁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡁⠭⠽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡏⠋⠀⠓⠤⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡍⣈⡶⠁⠀⠀⠀⠀⠀⠀⡠⠊⢁⡫⢹⣿⣧⣢⠤⠀⠀⠀⠀⣸⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠓⠀⡇⠀⠀⠀⠀⣠⠔⠊⠀⠀⠀⣴⠂⢾⣿⣯⠁⠈⠀⣀⡾⢁⠏⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠠⢀⡀⠀⠋⠀⠀⠀⠀⠀⠀⣻⠀⠉⢻⠿⣷⡶⣞⡋⠀⢀⠀⠁⠈⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢀⣴⡶⢿⠿⣿⣿⣶⡀⢀⣴⠃⢀⡀⠈⠀⠈⠟⠹⡇⠰⠃⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⡟⠻⠿⢿⠿⠿⠿⠛⣿⡋⠁⠀⠈⣹⠀⠀⠀⠀⠀⠛⠒⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠻⠃⢐⡆⠄⠀⢴⡀⠂⠈⣏⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⣨⣄⢀⡀⠤⠀⠀⠀⠀⠀⠀⢸⡾⣇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢀⠀⠠⣤⠔⠀⠀⢸⡝⢩⣟⠂⠁⠀⠀⠀⠀⠀⠀⠈⢷⢙⡀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡌⠀⢶⠂⠒⠚⢀⣿⣿⣎⣻⡇⠀⠀⣇⠀⠀⠀⠀⠀⢈⣟⡇⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣆⣀⣀⣀⣴⠋⣿⣿⣿⣽⢿⣤⣀⠀⠉⠉⠁⠐⠒⠚⠫⠴⠶⠶⢦⣄⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣿⡻⠿⠟⠋⠁⣠⣿⢿⢻⣿⡸⣿⣿⡿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣇⠹⣯⠁⣀⠠⢥⡟⡟⠈⢸⡟⠻⢿⣿⡧⢸⢛⢻⣷⣦⣤⣀⠘⠚⠶⢆⣀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢶⡟⠹⡇⢻⡄⠹⡆⢠⡴⠋⡜⠁⠀⢸⢿⠀⠀⠛⠿⣾⣿⣿⡞⢻⢿⠋⠙⠓⠲⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠃⠆⢸⠇⠀⣧⠈⢷⠀⠙⠀⠀⠈⢠⠆⢠⠇⢸⠀⠀⠀⠀⠀⠙⢿⣗⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢸⡄⠀⠘⠀⠈⠀⠀⢀⣠⠶⠁⢠⠎⠀⠸⠀⠂⠀⠀⠀⠀⠀⠙⣿⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠈⣧⠀⠠⣄⡀⠀⠺⠛⢁⠠⠊⠁⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠉⠀⠀⠋⠁⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠈⠈⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠐⠊⠁⠠⠞⠁⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡸⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⠄⠤⠀⠀⠤⠀⢖⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠐⠁⣤⠶⠦⠀⠀⠀⠀⠀⣤⡀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠔⢀⠂⢹⡠⠀⠀⠂⠐⠄⠀⠀⠀⠠⠄⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⡀⢰⡇⠙⠁⠀⢠⣾⣿⡆⠀⢠⢠⣴⣶⣼⠀⠀⠀⠀⢠⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠌⠀⡄⠀⠀⠀⠀⠉⠁⠀⠀⡈⢳⠉⠉⡁⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣠⡼⠁⠄⠀⠀⠀⠀⢠⡀⠀⡉⠉⠀⡰⠀⠀⠀⠀⢠⣬⠍⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠀⠀⠘⠀⠀⠈⠢⡀⠀⠀⠀⠀⠙⠁⢀⠜⠀⠀⠀⠀⠀⢩⠖⣒⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠸⡄⠀⢠⠀⠀⠀⠻⠐⡤⠤⠤⠤⠒⠁⠀⠀⠀⠀⠀⠀⠈⠯⣥⣴⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡣⠖⠀⠲⡀⢁⠀⢀⣑⡒⣲⣡⣄⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡱⠁⠀⠀⠀⢱⠀⠞⠁⠀⠉⠁⠀⡏⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠀⡈⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢁⠃⠀⠀⠀⠀⠸⠀⠀⢁⠀⠀⠀⠀⠇⠀⠀⠀⠉⢢⠄⣤⠤⠀⠀⣔⡡⠿⠠⠧⠼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⢆⠀⠀⡀⠀⠀⠀⠀⠠⠂⣰⠃⠰⣀⡙⢤⣡⠷⢛⣡⡌⡁⢘⠛⠳⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠁⠈⠀⠳⢄⠀⠐⢁⣴⠇⠀⠀⣢⢴⡶⠋⢔⠉⡍⣿⣿⠊⢗⢄⠑⠻⣷⢄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢃⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⢀⡩⠖⠉⠀⠀⣀⣐⠖⣿⢡⠜⣡⠎⠀⣿⡝⠀⠈⠳⣄⠢⢨⠇⢂
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡀⠀⢀⠔⠀⢀⠤⢇⠀⣀⡠⠔⠋⠁⠀⠀⠀⡠⠊⠀⠀⢸⣷⡷⠚⠥⠆⠀⣿⠇⡇⠀⠀⠾⠹⠂
⠀⠀⠀⠀⠀⠀⣀⣸⠀⠀⠀⠐⡄⠀⡠⠊⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠠⣇⣇⠀⣴⣿⠆⠆⠀⡇⣼⣿⠗⠀⠀⣶⡴⡆
⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⢈⠞⠑⢦⡀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠈⣈⣿⠀⠉⠉⠄⢀⠄⠀⠈⠉⠀⠀⠀⡟⡄⠃
⠀⠀⠀⠀⠀⠀⢺⣿⠀⠀⢀⠔⠁⠀⠀⠀⠙⠲⠤⡤⠴⠚⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⣆⠀⠀⠀⠈⣀⠄⠀⠀⠀⢀⡜⠻⣜⡀
⠀⠀⠀⠀⠀⠀⠘⠿⠆⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡟⡏⠑⣠⣀⣀⠀⠀⣀⡀⠰⣁⠀⠀⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠈⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⡴⣿⠋⣼⠆⠙⠛⠻⢤⣤⡼⠁⠀⠹⠓⣀⠌⡇
⠀⠀⠀⠀⠀⠀⠀⠣⡌⢂⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⣴⣶⠀⠀⠀⠙⠉⠀⣠⣤⠀⠈⢦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⡀⠃⠈⠄⠀⠀⠀⠀⠀⠀⠔⢠⣀⠈⠈⠀⢼⣶⠀⠀⠀⠙⠛⠀⢀⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣡⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣁⣀⣀⣀⣈⡀⠀⠀⠀⠀⠜⠀⠹⠛⠀⠀⠀⠈⠀⠀⢤⣶⠆⠀⠀⠈⠛⠃⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡀⡄⢂⢠⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⠃⠀⠌⠋⠁⠻⡂⠀⠀⠛⠀⠀⠀⢠⣀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠖⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠙⠙⣤⡀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢠⠭⢿⣀⠱
⠀⠀⠀⠀⠀⠀⠈⠛⠃⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢱⣿⠣
⠀⠀⠀⠀⠀⠀⠀⢠⡄⣾⠄⠴⡛⠶⠶⢄⠀⠀⢠⣾⢿⣿⣻⡷⣄⠀⠀⢰⣿⠸⣿⣦⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠐⠒⠂⠂⠀⠀⠘⠁⠀⠉⠀⠋⠀⠀⠀⠈⢡⡆⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠞⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠦⠆⠀⠀⠀⠀⠀⠀⠀⠤⢀⣾⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⣴⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢸⣿⣿⠚⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣼⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⠃⣸⠇⢦⢠⡀⠀⠀⠀⠀⠀⢀⣴⡀⠊⠀⠀⡼⢋⣼⣿⣷⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⢀⣿⠂⢸⡆⠉⠲⢶⣾⠟⠛⠋⠡⠀⠀⠀⢜⠁⣸⣿⣿⡿⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣿⠀⡨⢳⡀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣰⣾⣿⣿⡿⣱⣿⠋⠈
⠀⠀⠀⠀⠀⠀⠂⠀⣿⠃⠀⢻⡆⢠⢜⣧⠀⠀⠀⠈⠁⠀⠀⠀⢰⣾⣿⣿⣿⠏⢀⡟⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣻⠗⠢⠬⣿⣯⣕⠘⢇⠀⠀⠀⠀⠠⠄⣪⣿⣿⣿⣿⣻⣿⣿⡿⠁⠀⠀⠀⠠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠙⢿⣿⣶⣌⢢⠠⠆⠷⣁⣰⣿⣿⣿⢯⣴⣿⣿⡛⠁⠂⠀⠀⠈⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠼⢀⠀⠀⠀⠉⠻⢿⣿⣷⣶⣾⣿⣿⣿⠿⠭⢅⣼⣿⣿⡟⠅⠛⣶⡆⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⢀⣠⣴⣿⡿⠻⠛⠿⠿⣏⠠⣵⣶⣿⢻⣿⠟⡑⢷⡟⢤⣿⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⠾⠋⢉⣀⡀⠀⢀⡐⢑⣏⣴⣻⠿⣿⡟⢛⠀⡙⠛⡀⠈⢆⡁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠒⠐⠀⠤⢽⣛⣧⢿⡻⣿⣿⣾⡟⠁⠨⠄⠛⢂⢨⠀⠀⠀⢠⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⢀⡠⣴⡧⣿⣷⣶⠾⠿⠏⠷⠀⠐⠀⠁⠀⠂⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣈⣙⢍⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠤⢀⣀⠉⠓⠄⠙⢆⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⠩⠭⡠⠤⠄⣀⡁⠱⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢹⡄⠀⠀⢈⠚⠋⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠛⠓⠦⣴⠉⠱⡄⠀⢸⠁⠀⣔⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⢸⠀⠀⠀⠀⢠⢸⣠⠀⠘⡄⢸⣳⢂⣁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠉⠉⠁⠚⠓⠂⠀⠄⠐⠊⢧⡀⢀⠃⣾⢋⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠈⠀⠀⠀⢀⣈⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠊⠁⠐⠀⠀⢀⡀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⢸⢁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⡆⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⡜⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠚⠁⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣧⠀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡊⠇⠀⠀⠀⡧⠒⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠝⣇⡠⠔⣮⠀⢠⠚⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣱⣀⠀⠣⢔⣒⣉⣀⡓⠁⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣴⠃⠀⢀⠔⣩⡀⠤⡠⣌⢃⠀⢡⡀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⠁⠒⠀⢾⠰⠥⢇⡶⡧⢂⠦⠄⠀⢟⡉⡨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠁⠉⠀⠉⠢⢤⣤⣤⡴⠧⠤⡤⠤⠤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⢸⠀⠐⠆⠀⠀⠈⠂⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢸⡎⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠒⠛⡗⠓⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣚⣉⣉⣛⣛⡓⢩⣀⣂⣤⣼⣄⣀⣀⣠⠤⠤⠘⣒⣄⣠⡤⠤⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠂⠚⠶⠀⠠⠤⠶⠞⠁

⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⢠⣤⣄⠀⠀⢀⢀⡀⢀⡤⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠄
⠀⠀⠀⢰⠟⢋⣩⠀⠀⡇⠀⡱⢮⠀⢨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢋⣡
⠀⠀⠀⢰⡾⠟⢛⠀⠀⠀⢀⠀⠀⠡⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⢰⡾⠟⢋
⠀⠀⠀⢨⣴⡾⢿⠀⠀⠀⠀⠂⢠⡀⠀⠀⠀⠢⠀⠀⠀⠌⠀⠀⠀⢠⠁⠈⠀⠀⠀⣚⣿⠀⠀⢸⣴⡾⠟
⠀⠀⠀⢘⣡⣴⠿⠀⠀⠀⠀⢰⠈⠀⠀⠀⠀⠀⠊⠉⠱⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⣒⣾⠀⠀⢘⣡⣴⠿
⠀⠀⠀⠸⣋⣥⣾⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠤⣾⠀⠀⠸⢋⣥⣶
⠀⠀⠀⠸⠛⣉⣴⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⢰⠁⠀⠀⠀⠀⠠⠭⣽⠀⠀⠸⠟⣉⣴
⠀⠀⠀⠁⠈⠉⠀⠁⠀⠀⠀⢠⣾⠀⠀⠀⠀⠀⣀⣀⣀⣀⣐⣂⣩⠭⠀⠀⠀⠀⠀⠉⠭⠀⠀⠁⠈⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⢀⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠤⢰⡾⠀⠀⠀⠀⠀⢷⠀⠄⠀⠀⠀⠀⠀⠀⢀⡁
⠀⠀⠀⠀⠀⠀⢠⠀⣦⠀⠀⠀⠀⠀⠀⠀⣾⡘⠀⠀⠀⠀⠀⢢⡞⠀⠀⠀⠀⠀⢀⠔⢋⣉⡙⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠆⠀⠀⠈⢼⣷⠄⠀⠀⠀⠄⣿⠇⠁⠀⠀⠀⠀⣠⠀⣾⣉⡷⠀⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢶⣄⠀⣤⠎⠡⠀⠀⠀⠀⠶⠁⠘⠦⡈⡉⢡⠴⠁⠨⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣠⣄⢀⡀⢀⠄⣤⡀⣀⡀⣀⢀⠀⣤⣀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠚⠑⠊⠚⠛⠛⠐⠃⠙⠒⠀⠛⠋⠒⠉⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠚⠉⠉⠳⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠴⠒⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢴⠄⠀⠀⢄⣀⣠⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠣⢀⡞⠙⢆⡀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣎⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢅⠀⠘⠧⠒⢦⠉⢙⠎⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⡀⠉⠐⠒⠒⠒⠉⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠑⠢⠤⣤⣬⠶⠻⡄
⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⡟⣄⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⣠⠾⠁⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢰⠤⢄⡀
⠀⠀⠀⠀⠀⠀⢸⣠⡠⠀⠀⡇⠈⠑⠒⢤⡘⠄⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⣀⠴⠖⠁⢸⠀⠀⠠⢀⠂⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠹⡙⡆⢀⠁⠀⠀⠀⠀⠈⠉⠉⢒⠒⠋⠀⠀⠀⠀⠀⠀⠀⢐⠅⢒⡊⠉⠉⠀⠀⠂⠀⠈⠦⠤⠖⠾⣀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢟⣌⠊⠀⠀⠐⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠈⡄⠀⠀⠀⢀⠀⠀⠀⠀⣠⠊⠀⠞⠳⠴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠩⢵⣆⢄⣀⣈⢒⣀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠷⠄⣀⠐⠃⠀⡀⠄⣖⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡁⣈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⡍⠒⠊⠁⠀⢹⠄
⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠄⠚⠀⠀⠈⡗⢄⡀⠀⠴⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⣚⠠⠁⠀⠀⠀⠐⠻⠀⠀⠀⠀⡼⠠⣰⠇
⠀⠀⠀⠀⠀⠀⢀⠴⣶⠀⣤⠀⠘⠀⠀⠁⠀⠁⠀⠀⠤⠀⠀⣧⡍⠲⣤⣤⣤⠄⢡⣀⣈⣀⠀⠀⠀⡤⠤⠁⠀⠀⠀⠤⠚⣆⢀
⠀⢀⠀⠀⠈⠀⠘⢦⣼⠖⠁⠀⠒⠒⠀⢰⠀⠀⣶⣈⠁⠀⡬⠃⠐⠃⠀⢀⡁⠀⠃⢈⡿⠉⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⢧
⡀⢠⠀⠀⠀⠀⠐⠠⡉⠀⠀⠀⣠⠀⠀⠀⡄⠀⢱⡈⠉⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠃⠄⣤⠂⠀⢠⣀⠀⠀⠀⢀⣄⣤⣤⣺⣀⣀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⡀⠉⠉⠍⠠⠄⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⣀⠉⠉⠉⠉⣗⠀⠤⠤⠤⠀⣂
⣇⠀⠀⠀⠀⠀⠐⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢢⣄⣀⡀⠀⢀⣀⣀⡤⢶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠋
⡟⠶⠦⠤⣤⣄⣀⡀⠀⡐⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⡤⠛⠛⠲⠶⠭⠭⠭⠵⠖⠒⠋⠀⠀⠀⠀⠀⠀⠰⠦⠤⠤⠶⠩⡀⠀⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠉⠓⠚⠛⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠛⠁
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⡇
⡇
⠃
⡿⠿⢿⠿⠿⡻⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠿⣿

⠀⠀⠀⠀⠀⠀⣠⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⣤⡀⠐
⠀⠀⠀⠀⠀⣈⣩⣭⣅⣚⠃⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⢠⣿⣶⣄⣉⠛⠒⠆⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⢠⡏⢁⠀⠀⢈⢹⡇⠀⠀⡀⠀⡏⢉⠁⠈⢉⢹⠃⠀⠀⠀⢸⠁⡀⠈⠉⡉⡿⠀⠀⠀⠬⡏⡉⠀⠈⡉⢹⠄
⠀⠀⠀⠀⢸⣧⣄⠀⠆⣀⣼⡇⠀⠀⠽⠆⠐⠀⠀⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⠀⠀⠈
⠀⠀⠀⠀⠈⠛⡟⢰⡆⢻⠟⠃⠀⠀⠀⢀⠀⣠⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⢐⠀⢰⠀⠀⠀⠀⠈⢀⠈⣴⣦⣴⣄
⠀⠀⠀⠀⠀⠰⠇⠸⠇⠸⠇⠀⠀⠀⠀⠐⠐⠷⠼⠷⠼⠃⠀⠀⠀⠀⠀⠀⠼⠀⠨⠀⠀⠀⠀⠀⠸⠸⠿⠿⠿⠿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⣠⡀⠀⠀⠀⠀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⢟⣉⣉⡻⢿⡧⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠿⠀⢀⣤⣾⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠰⡏⡙⠛⠛⡋⢳⠀⠀⠀⠀⢸⢋⠉⠉⠁⠈⡇⠀⠀⢸⣿⡟⢋⠉⠉⢀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡆⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⡄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢷⠀⠀⠀⠀⠀⠀⢀⡿⠸⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠸⠿⠿⠛⠋⠻⠀⠀⠀⠀⢠⡞⠛⠋⠛⠛⢶⠀⠀⠀⠀⣾⠿⠿⠛⠋⠹⡇⠀⠀⠀⠀⡖⠛⠋⠙⠛⢲⡀
⠀⠀⠀⠀⠰⣧⠘⠀⡀⠘⢀⣿⠀⠀⠀⠀⠃⠀⢀⡀⠀⠘⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠃⠘⠀⡀⠘⠘
⠀⠀⠀⠀⠀⠀⠀⠦⠠⢆⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡶⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣾⣧⢀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⢺⣿⣭⣿⠂⠀⠀⠀⢠⢶⢀⡆⠀⠁⢰⡄
⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠀⠀⠀⠀⠈⠉⠈⠁⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣤⣀⠀⠀⠀⠀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠈⠑⠲⢄⠀⠀⠀⣀⠴⠚⠉⠉⠙⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠈⠙⢦⡀⢀⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⣀⠀⠀⠀⠀⠀⠀⠙⠺⣁
⠀⠀⠀⠀⠀⠀⣠⠤⠴⠒⠉⠀⠀⠠⠤⣀⣀⣠⠤⠂⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠈⠙⠒⠒⠒⠒⠂⣀⢀⠈⠙⠒⠦⡄
⠀⠀⠀⢀⣀⣀⡬⠶⠖⠒⠉⠈⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠉⠙⠒⠮⢤⣄⣀
⠀⠀⠐⢧⣤⠀⠀⠀⠀⠀⠂⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⡰⠃⠀⢀⠀⠁⠐⠀⠀⠠⢤⣤⠝
⠀⠀⠸⢥⣀⡀⠤⠤⠒⠂⠀⢀⠀⠀⠈⣆⠀⠀⢀⣀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢤⣀⣰⠃⠀⠀⠄⣀⡉⠐⠒⠢⠤⣄⣀⠽
⠀⠀⠀⠀⠘⠲⠶⢆⣀⣈⣩⠤⠤⠶⠒⢺⡒⠉⠉⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠈⠉⠓⠒⠦⠤⣌⡴⠖⠲⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⢰⠋⠀⠈⠙⠢⡴⠚⠉⠀⠈⢣⠀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢤⣀⡀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣠⠤⠴⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠢⣄⠀⠀⠀⠀⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢆⢠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠛⠱⢾⡋⠉⠻⠶⠶⠶⠤⠴⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠠⣄⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠹⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠏⠀⠀⠀⣟⠀⢀⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⢦⣄⠀⠀⢀⠀⠀⣇⠀⢸⡇⢰⡆⠀⠀⢀⣼⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⡀⠀⠀⠀⠙⠷⠀⠀⢀⣠⣿⣦⣼⣷⣤⣥⣠⣇⣾⠃⠀⢘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠻⣦⡀⠀⢀⣤⠶⠛⠋⠉⠀⠀⠀⠀⠀⠉⠙⠛⢷⡀⠀⡼⠇
⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠴⠶⣮⣹⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇⠶
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠛⠛⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⣀⠀⠀⠀⠀⣾⡇⠀⠀⠀⣀⣤⡶⠶⠾⠛⠛⠛⠶⠶⣦⣀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠶⣤⣿⡇⢀⣴⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠟⣡⣤⣶⣦⣤⠀⠀⠀⠀⠠⡴⠶⠿⠷⣦⠈⢻⣇⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣿⡏⠀⢉⣴⣶⣤⡉⠀⠀⠀⠀⠀⢠⣶⠿⢷⡄⠀⠀⣿⡉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡐⣿⡇⠀⠉⠉⠐⠉⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣿⠀⠀⠀⠀⠀⠀⠀⠰⣄⡠⠾⠃⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠈⢻⣄⠀⢘⣿⠁⠀⠀⠀⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⡀⠀⠀⠀⠛⠀⠀⠀⠀⢀⣀⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠿⣟⡛⠷⣦⣤⣤⣤⡶⠞⠛⣉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤⠺⣷⡀⠈⠙⠳⣄⠀⠀⢀⣤⡶⠟⠋⠁⠻⢷⣤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠛⠛⠉⠉⠀⠀⠀⠈⣁⠀⠀⠀⢘⡄⠀⠘⣏⠀⣀⣀⣀⣀⠀⠀⠀⠉⠉⠛⠛⠦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⢦⣀
⠀⠀⠀⠀⠀⢀⡾⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦
⠀⠀⠀⠀⢠⡟⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⢿⣬⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⡟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠛⢿⣧
⠀⠀⠀⠀⣾⠃⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢻⣇
⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣤⣿⡄⠀⠀⠀⠀⣴⠟⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣽⠇⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣼⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⠾⠛⠙⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⠶⣶⠞⠁⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⣾⠇⠀⠀⠀⠀⠺⠿⠿⠛⠛⠳⢦⡀⣿⡀⠀⠀⣴⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⣦⢰⡿⠁⣸⠇⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣸⡟⠀⠀⠀⠀⠀⠀⠐⠒⠛⢛⣃⣀⣀⣸⡇⠀⠀⠙⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣰⠏⠀⣰⣏⣀⠀⠀⠀⠀⢀⣼⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⣿
⢰⡟⠀⠀⠀⠀⠀⢀⡤⠶⠶⠛⠉⠉⠉⠉⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠒⠒⠒⠶⠾⠿⠾⠛⠋⠉⠉⠉⠉⠙⠛⠿⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣾⠃⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣀⠀⣀⣠⣤⡴⠶⠿⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣆⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣏⠁⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸
⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣦⡀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠈⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠀⠉⠛⠛⠿⠿⠿⠿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⠟⠁
⠀⠀⠀⠀⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣄⣀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣴⣶⠾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣦⣤⣀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣽⠿⠛⠛⠙⠛⠛⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⣶⣶⣤⣤⣤⣤⣤⣶⣶⠶⠞⠛⠋⠁
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⣀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠛⠋⠉⠉⠉⠓⠦⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⣀⡴⠏⠀⠀⠈⣧⠀⠀⠀⠀⢀⣤⠶⠛⠛⢳⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⡴⠒⠈⠉⠁⠀⠀⠀⠀⠀⠘⣧⣠⡤⠔⠊⠁⠀⠀⠀⠀⠀⢷⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢘⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣿⠀⠀⠀⠀⡉⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠈⣇⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢩⠀⠀⠀⠜⠋⠁⠴⠉⠓⠄⠄⠀⠀⠀⠔⠚⠉⡭⠉⠒⠄⠀⠀⣿⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢸⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⡟⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡠⠔⠀⠀⠀⠀⢀⣾⠇⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣠⠏⡜⢠⠇⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⡈⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⢁⡜⢁⡞⠀⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢃⠙⢦⣈⠙⠿⣶⣤⣤⣀⣠⣤⣴⠞⢋⡴⠋⣰⠋⠀⠀⠀⠀⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⢡⠃⠀⢦⡀⠈⠳⣄⠀⠈⠉⠉⠉⠁⣀⡴⠋⢀⡼⠁⠀⠀⠀⠀⠀⠀⠁⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠋⠁⠀⠀⢠⠏⠀⠀⠈⠀⠀⠀⠀⠑⣶⡤⠴⠶⡟⠁⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⡀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠂⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⠛⢠⠇⠀⠀⠀⠀⠀⠀⣠⡶⠟⠉⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⠇⢀⠌⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠰⡆⢸⠖⠛⠓⠲⣤⠊⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⣷⡎⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡧⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⣶⣾⣿⣷⣿⡷⢾⣾⣿⣿⣿⣿⣿⣿⣿⣾⣶⣿⣿⣶⣿⣾⣿⣿⣿⣷⣷⣿⣶⣾⣷⣷⣾⣷⣿⣷⣷⣷⣿⣿⣿⣷⣷⡾⢷⣾⠷⢿⠶⢶⣿⠷⠶⠿⢶⠿⠶⠷⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⡴⠋⠀⡠⠀⠀⠀⠀⠀⢀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢀⡴⠋⠀⣠⠞⠁⣠⠄⠀⢀⡴⠋⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠋⠀⣠⠞⠁⣀⠞⠁⢀⡴⡏⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣠⢞⡥⢦⡄⣡⠤⣄⢋⣀⠙⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡵⠋⠀⣠⠟⠁⢀⡼⠋⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⠞⠁⢀⠴⠋⢀⡴⠛⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢠⠞⠁⢠⠔⠉⢀⡴⠋⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠞⠁⢀⡴⠋⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⡤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤
⠓⠐⠃⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠁⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⢠⠤⢀⠀⡀⢤⠤⠀⡄⠀⡤⢄⠀⡀⠀⡤⠄⠀⢀⠤⠀⢤⠄⢠⠤⡀⣀⠤⡀⢠⢀⡄⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⢸⠀⢨⠀⡇⢸⠀⡼⠸⠄⡗⢺⠀⡀⠀⡗⠀⠀⠈⠳⠄⠸⠀⢸⠲⡅⡇⠀⡸⢸⠻⡀⡗
⣶⣶⣶⢶⣶⣶⣶⣶⣿⣿⣾⣿⣷⣶⣷⣾⣶⣷⣶⣶⣿⣿⣶⣿⣷⣿⣿⣶⣾⣿⣶⣾⣶⣾⣶⣷⣾⣿⣷⣾⣶⣷⣿⣿⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡽⠿⠿⠿⠿⡓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡀⠀⠸⣆⡀⠀⠀⠀⠀⠀⠀⠀⢣⠘⣆
⣀⣀⣀⣀⣀⡔⠎⢘⠇⢸⠉⣣⣔⣆⣀⣀⣀⣀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠉⠛⠛⠋⢷⠀⠀⡀⠀⢸⡇⢻
⢹⡏⠉⠉⠉⠣⠜⠼⠧⠺⠄⠉⠉⠉⠉⠉⢉⡟⠀⠀⠀⠀⠀⠄⣀⣀⠀⠀⠀⠀⠀⠘⢷⣀⠘⠲⠞⣣⡟
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣒⠂⠀⠈⠙⠲⣦⣀⠀⠀⠀⠙⢻⠖⠛⠉
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣀⡉⠁⠀⠀⠀⠀⢀⠙⠿⣦⣀⣠⠏
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣾⠋⠀⠀⠀⠀⠀⠀⠸⠃⠃⠀⠈⠻⠏
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⡄
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠷⣦⡤⣖⠁⠀⠀⣠⠆⠀⠀⠀⢨⠴⠃
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠠⠽⠛⠋⠁
⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠁⢆⣈⡧⠤⠤⣴⣶⣶⣚⡛⠉⠁⠀⠈⠙⢦⣀
⠀⠀⠀⠀⠉⠳⠒⢤⣿⡤⠒⠞⠉⠀⠀⠀⢸⡇⠀⢸⢆⠡⣄⡀⠀⠉⠉⠉⠉⠙⠛⠛⠷⠶⣶⣤⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣸⡇⢀⡎⠘⡆⢹⡙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⢉⡤⠋⡄⠀⣿⡀⢧⠈⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣠⠊⢀⠀⠈⠳⣬⣁⣈⣙⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠖⠁⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠂⠒⠶⠶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⡿⢹⣿⣿⣦⢀⣀⣠⣤⣶⣦⣤⣶⡏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠁⠀⢸⣿⣿⢽⡿⢀⣈⡻⣿⣿⣿⣿⢿⡛⠻⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠋⠿⠀⢸⣿⣧⣀⣙⣻⣛⢳⡎⠙⣿⡏⠀⢠⣴⣾⣿⣿⣿⣿⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠧⠀⣿⠇⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠋⠙⣴⣿⠃⠒⠋⠨⣽⣻⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠹⠿⣿⣿⣶⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢻⡿⠋⠠⠀⢲⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠘⠛⠀⡠⠔⠁⣴⣷⣶⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣿⡿⠀⠀⠀⡠⠊⠁⠀⠀⣿⣿⣿⣿⣿⡿⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⢀⠔⠁⠀⠀⣀⣠⣶⣿⣿⣿⣿⣷⡄⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠⠊⠀⢀⣴⣿⣿⢿⣿⣿⣿⣿⡜⠿⣿⣦⡀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⣴⠃⢀⣴⠿⠋⠡⣾⠘⣧⣿⣿⣿⣿⡄⣈⠙⢳⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣾⠉⢿⡿⠁⠀⠖⣴⣿⣦⣝⣿⢻⣿⣏⣿⡏⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠀⣸⣷⣄⣠⣾⣿⣿⣿⣿⣿⣷⣽⣿⣿⠀⣿⣿⠛⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠸⠀⢀⣧⣼⣿⠀⡟⠙⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣠⡆⣼⠀⠀⠀⣰⠞⣿⣿⠛⠟⠃⠸⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣿⠀⢀⠀⠀⠸⠋⡿⠀⢀⣷⢶⣶⠄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣿⣿⣤⢸⡇⠀⠀⠀⣠⣶⣿⠇⢾⡀⣔⣻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣼⣧⠀⠀⠘⣿⠟⠀⠙⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⢿⠿⠿⠿⠿⠿⠿⠸⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣴⣦⣴⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⡿⠟⠁⠙⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠐⠀⢲⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠛⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⡠⡤⠊⠀⠀⠀⠓⣄⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠁⣷⠀⠘⠆⠀⠇⢸⡏⠂⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠂⡀⠁⡎⢀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠀⠀⣧⠀⠀⢀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⠒⡁⠀⠀⠈⠈⠶⠁⠀⠈⠂⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠁⠀⠀⠀⢹⣂⠄⠀⡀⢰⡟⠛⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠀⠆⢀⠀⠀⠀⠀⢸
⠀⠀⠀⣇⣀⣀⣀⣈⠀⠀⢀⡀⢸⣉⣦⠀⢀⢸⣀⠀⣹⢘⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⠀⠀⠀⠂⠀⠀⠃⠀⡀⠀⣀⣀⣀⣀⣸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠀⡼⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢠⠂⠀⠀⠀⠄⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⣠⣀⠁⠀⣶⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⢄⠀⠀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠔⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠑⡎⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⡀⠠⠀⠀⠎⠉⠀⠀⠀⠑⠤⠀⡀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠠⠂⠈⢸⡇⠀⠀⠀⠀⠈⣷⢤⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠈⢁⠀⠂⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⠳⠀⠀⢘⠀⠀⠀⠀⠀⠠⠁⠀⠀⠈⡇⢀⠀⠀⠀⡀⢸⠀⠄⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⣇⢈⠀⠀⠀⢠⡾⢷⠀⡄⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⠀⠀⠀⠀⠂⠀⠀⡐⠀⢠⡝⠺⡄⠀⠀⡿⠀⠈⡇⠁⠀⠀⢸
⠀⠀⠀⠓⠒⠚⠂⠐⠐⠓⠒⠐⠐⠂⠚⠒⠚⠀⠂⠂⠐⠓⠚⠒⠒⠚⠓⠒⠒⠓⠐⠀⠐⠛⠐⠐⠛⠙⠃⠂⠐⠛⠓⠚⠃⠓⠒⠒⠚

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠈⠉⠲⠶⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠓⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⢁⡀⢻⣦⣀⢰⣦⣀⣀⣀⡀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠍⠉⠉⠉⠉⠉⠉⠉⢸⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠓⢒⡖⠀⢲⣒⠛⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣿⠀⠙⢻⠀⡟⠋⠁⣻⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠀⣤⣬⣤⣥⣤⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠢⡀⠀⠤⠀⢀⠔⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠚⢤⣀⢀⡗⠀⢺⡅⣀⡤⠓⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡞⠁⠀⠀⠀⠀⡀⠉⢃⡶⡘⠉⠳⠤⡀⠀⠀⠉⢳⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠡⠀⠀⠀⠀⠐⠖⢒⣉⠀⠈⠒⠲⠂⡼⠀⠀⠀⠆⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡆⠀⠀⠐⡺⠁⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣀⣀⣀⠤⠔⠁⠀⠈⡙⣆⣱⠁⠀⠀⣠⠄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⣠⠔⠊⠉⠀⠀⠀⠀⣀⣀⠀⠀⡾⠧⣤⣴⠚⠃⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠘⡖⠀⡞⠁⠀⠀⠀⠀⠀⢀⡔⠉⠉⠓⠊⠀⠀⠀⠈⠳⣄⠀⠀⠀⢠⡾
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠘⡇⠀⠀⠀⠀⢀⡠⢏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢘⣦⣤⡶⠓⠉⠀⠀⠀⠉⠓⠦⠤⣄⣀⣀⣰⠦⠤⠤⠷⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢀⡀⠖⠊⠑⠀⠀⠀⠁
⠀⠀⠀⠀⢀⠠⠂⠀⠀⢀⠊⠀⠀⠀⠀⠀⢂⠀⢀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠴⠒⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⠘⠉⠀⢀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡋⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⢀⣠⠶⢲⢶⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⢰⣤⠔⡩⢰⠟⢢⡾⢡⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡴⠋⠽⣦⠼⠚⠴⠏⠴⠃⠀⢀⡠⠔⢀
⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡰⣫⢰⡼⢘⣥⣾⢛⠂⠁⠀⢠⣼⡉⢀⠺⢝⡛⣳⠄
⠀⠀⠀⠀⠀⠀⠐⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⣉⣭⠾⢛⣡⣶⠟⠀⠀⢠⣿⣿⠇⠀⢠⣰
⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢥⣶⣿⣿⣧⣦⣦⣾⣿⡿⠋⠀⠀⠘⣿⡒⢀⣀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠛⠿⣿⣿⡿⠟⠉⠀⠀⠀⣤⣴⣿⣦⡄
⠀⠀⠀⠀⠀⠀⣿⣀⣲⡀⠀⠀⣶⣦⣿⣻⠭⠡⢤⣄⣤⠖⠒⠦⣶⣶⣶⠄⠀⠀⠀⠀⢻⣿⣿⣿⠇
⠀⠀⠀⠀⠰⣶⣿⠿⠿⠷⠾⠿⠿⠿⠿⠿⠿⠿⠿⠷⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠾⠿⠿⠿⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒⠉⠉⠉⠉⠙⠒⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣀⣀⣀⡤⠤⢤⣀⣀⡀⠀⣠⠤⢄⣀⡀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠁⣀⣀⣀⣀⣀⣀⣀⣉⢻⠀⠀⠀⠀⠉⢦⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠷⠾⠯⢍⣉⣉⠓⠒⠦⠤⣀⠉⠉⠒⠦⢄⡀⠈⡆⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠓⠢⢄⡀⠉⠒⠦⣀⠀⠈⠑⠿⣀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠉⡿⡗⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢤⣈⠑⠢⣄⠀⠀⠙⠢⣄⠀⠘⡄
⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⢸⠁⠹⣄⠈⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢝⣲⢄⡀⠀⠑⢦⡹⡄
⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⡜⠀⠀⠈⣳⣄⠀⠀⠉⠒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⣤⣽⣷⡀
⠀⠀⠀⠀⣰⠁⠀⠀⢀⡼⠁⠀⠀⡰⠃⠈⠻⢶⣤⣀⡀⠀⠉⠑⠒⠤⢄⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⣅⣠⢧⠳⡀
⠀⠀⢀⡴⠁⠀⠀⡠⠊⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⠙⠻⠭⣒⣒⠤⠤⠤⠤⠤⠭⠭⠟⢛⣋⣩⣤⣒⣚⣋⣉⠤⣎⡀⠈⠲⠴⠒
⢀⡴⠋⠀⣠⠔⠋⠀⣀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢄⡉⠉⠉⠉⠉⠉⠉⠉⠉⠓⠲⠤⠬⠭⢤⠤⡤⠚⠁
⠀⠉⠛⡿⢲⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢸⣤⣀⣀⣠⠴⠊
⠀⠀⡰⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠤⢤⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢧
⠀⠀⡇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢨⠇
⠀⠀⠘⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢾⡁⠀⠀⠀⠀⠀⠀⣠⠟⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⣀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⣄⣀⡀⠀⠤⠤⠤⠤⠔⠀⣀⣀⡤⠴⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠠⠊⠁
⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢠⣀⡀
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠙⠳⡄
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⢠⠞⠃⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⡴⠁
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣴⠦⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢿⡆⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠛⠛⠲⢤⣀⣀⡀
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⡁⠀⠀⠀⠀⠀⠀⠀⠸⣷⣴⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡿⠶⣶⡿⠃
⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣅⣀⠀⠀⠀⠀⠀⠀⢀⣤⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣲⡟⠁
⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣏⠁⠠⣄⠀⠀⠀⠀⠈⠙⠻⠂⠀⠀⠀⠀⠀⠀⣤⣤⣤⠬⠿⠶⡦⠤⢤⣤⣶⣶⣶⣶⡶⠾⠿⢷⣶⣤⡦
⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠈⠳⢦⣤⣄⣀⣀⣤⡷⠀⠀⠀⠀⠀⠀⠀⠉⠛⣳⠶⢶⣶⠾⠟⢛⣫⣭⣴⠾⠶⢖⣺⠾⠋⠁
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠹⣧⠀⠀⣀⣀⣀⢸⣿⠉⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⢠⡟⠁⠀⠀⠀⠀⣀⣐⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⠉⠉⠉⠉⠉⢻⣧⣾⣿⣷⣤⣦⠀⠀⠀⠀⠀⣼⡏⢠⡿⠀⠀⢀⡶⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣷⣦⣀⠒⠿⠿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣰⡿⠀⣾⠇⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠈⠉⠛⠿⣿⣦⣄⣰⡿⠁⣼⡏⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠙⠻⣿⡅⣠⡿⠀⠀⠀⣿⡇
⠀⠀⠠⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠁⠀⠀⢸⣿⠁
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢿⣷⡀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⢀⣿⡏
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠋⠀⠀⢀⣽⠿⣷⣄⡀⠀⣴⡿⠁⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣀⠀⠀⣰⡿⠁⠀⠀⠙⠿⣾⠟⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⢿⣿⣿⣾⣿⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⣿⡟
⠀⢀⠐⠀⠀⠀⠀⠀⣼⠿⢷⣶⣤⣀⡀⠈⢿⣿⡄⠙⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⣠⣿⡟
⠀⠠⠀⠀⠀⠀⠁⣼⡇⠀⠀⠈⠙⠛⠿⣿⣶⣿⣿⡀⠀⠀⠈⠙⠻⠿⣶⣤⣄⣴⣿⠟
⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣧⠀⠀⠀⠀⠀⠀⣼⣿⠏⠛⠁
⠀⠀⡀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⣀⣀⣤⣤⣽⡿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠠⠀⠀⢸⣿⡄⢀⣠⣴⣾⣛⣭⡵⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠛⢿⠿⢿⡟⠛⠉⠁⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢿⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⣄⢖⡶⠆⡰⢀⡄⠠⢰⠀⠇⠶⠀⠆⠀⣴⠆⢴⠦⢴⡄⢰⡢⠀⢄⣴⠀⡦⢠⠶⠀⠠⠀⠠⠆⠰

⠀⠀⠀⠀⠀⠚⠂⢀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⡀⠂⠀⠀⠀⡀⠁⠀⠀⢀⠊⠀⠀⠈⠋⠑⠄⠀⢀⠐⠁⠀⠀⠑⠄⠀⠀⠄⠀⠁⠈⠑⣄⠀⠀⠀⠀⠊⠀⠠⠀⠀⠑⠀⠀⢀⠰⡂⠀⠀⡱⠠⡀
⠀⠔⠃⠔⢄⠀⠀⠈⠚⡀⢐⠀⢀⣀⠁⠂⠀⢆⡀⠀⣆⠄⠤⠆⠀⠠⣠⡀⠀⡤⠀⡀⠠⠈⠊⠄⠀⠀⡎⠤⠀⠘⠃⠠⣀⢂⠀⠀⠀⢀⡧⠰
⠀⠐⠒⠀⠀⠀⠉⠉⠒⠃⠀⠐⡁⠁⠀⠀⠀⢸⠀⢠⢠⠀⠀⠀⠀⠀⡃⡅⠀⡁⠀⠀⠀⠉⢈⠁⠀⠘⢈⠁⠀⠀⠀⠀⡇⠌⠀⢌⣹⠀⠀⠀⠀⠃⠁
⠀⠀⠉⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⡄⠀⠀⢀⠄⠀⠀⠀⠠⡀⠀⠀⡐⠀⠀⠀⠡⡴⠀⠀⠄⠁⠀⠀⠘⡀⠠⢀⠀⢀⠔⠀⠊⠀⠀⠝⠀⠀⠀⢀⠊⠁
⠀⠀⠀⠈⠒⠀⢊⡀⠀⠀⠀⠀⣠⠀⠈⢍⡀⠀⠀⠸⠀⠀⡨⠁⠉⢄⠀⠀⠀⠀⠈⠁⡁⠀⠰⡀⠀⠀⠇⢀⠜⠀⠱⢄⠐⠁⠀⠀⠀⢈⠒⠐⣅
⠀⠀⠔⠉⢀⡠⠂⠈⠢⠀⠠⠊⠀⠈⠀⠀⠈⢂⠀⠀⡱⠉⠀⠀⠁⠀⠉⠀⠀⠀⡀⠈⠈⠀⠀⠈⠂⠀⠜⠁⠈⠉⠀⠀⠁⠀⠀⢀⠂⠁⠈⠀⠀⠐
⠀⠘⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⡀⠄⢊⠁⢢⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠉⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⢠⠖⠉⡉⠐⠌⡘⢒⠀⠀⠂⠀⠀⠀⠀⠐⡄⠀⡴⢅⣐⡀⠉⠉⣐⣄⠒⠀⠀⠀⠀⢂⡞⠻⠻⠛⠛⢻⠛⡆⠀⠀⠀⠀⣴⠰⡁⠀⠀⠘⠉⢦
⠀⠀⠀⠀⢾⠇⠀⠀⠈⠂⢀⠀⠚⠲⠄⣀⣀⠲⡆⠀⠄⡊⢀⢀⠔⠄⣀⡀⡆⠀⠀⠀⠀⢸⣇⢀⣀⣀⣀⣘⣀⣟⠀⠀⠀⠐⡄⡤⠃⠀⠀⠐⢤⢰
⠀⠀⠐⢄⡀⠀⠀⠀⠀⠀⠨⣸⠀⠀⠀⠀⠈⡁⠇⠀⠉⢅⡉⠀⠀⠀⠀⢁⡁⠀⠀⠀⠊⠀⠙⠛⠛⠛⠛⠛⠋⠁⠈⢂⠀⠀⠁⢃⠀⠀⠀⠀⠰⠈
⠀⠀⠀⢀⠃⢄⣀⡀⠀⠀⠀⠀⢀⠀⠀⢀⠔⠀⠀⠀⠀⠀⣀⠄⠀⠀⡠⠀⠅⠀⠀⠀⠀⡳⡷⢸⡏⡍⠅⡁⡟⢹⣆⢨⠀⠀⠀⠀⠁⡤⠠⠔
⠀⠀⡠⠈⠀⠀⢇⠀⠀⠀⠀⠀⣀⠋⠡⠡⢄⠀⠀⠀⠀⠀⠀⣠⢍⡩⠤⠀⠀⠀⠀⠀⡀⠀⠃⠐⠁⠂⠐⠂⠃⠋⠈⢸⠀⠀⢀⠔⠚⠡⠤⠘⠒⢀
⠀⢰⣝⣲⠤⣤⠤⣶⠀⠀⠀⠊⠀⠀⠀⠀⠀⠑⡀⠀⠀⢀⠊⠀⠀⠀⠀⠈⢂⠀⠀⠀⠐⠀⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⡆⠴⠀⠀⠀⠀⠰⠄⠇
⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢐⡈⢐⡄⠀⠀⠀⠀⣀⠔⠀⠈⣀⠜⠀⠀⠀⢀⠄⢂⡐⢄⠀⠀⠀⠀⠠⠀⠊⠑⠀⢄⠀⠀⠀⣠⣶⣿⣷⣶⡄⠀⠀⠀⠀⠄⢀⡀⠀⠠⡀
⠀⠀⣔⠁⠀⠀⠈⠦⠀⠀⢈⢀⡄⠀⡠⠈⠀⡦⠀⢀⠁⠀⣐⠂⡀⠑⡄⠀⠀⠀⠀⠀⠂⠀⠀⡆⠀⢸⡿⢛⡻⠿⣿⣿⣦⠀⠀⣄⠀⠀⠀⠀⡠⢅
⠀⢈⠉⣀⡦⠔⢄⡀⠐⠀⠀⠚⡄⣀⠴⡈⠕⢠⠀⢰⣿⠷⠆⠴⡶⣿⡆⠀⢐⢂⠦⠀⠀⠤⢡⣱⠀⠘⠃⠁⠊⠈⠛⠻⠋⠀⠀⠀⢈⠠⠦⠢⣀⢀
⠀⠈⢉⠀⠀⠀⠀⢈⠁⠀⠀⠰⠀⠀⠀⠀⠀⠸⠀⠀⢈⠀⠀⠀⠀⢀⠠⠀⠠⢈⠀⠀⠀⠀⢀⡠⠀⠀⠆⠀⠀⠀⢠⡎⠀⠀⠀⢉⡃⠀⠀⠀⠀⢉
⠀⠀⠈⢀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠁⡀⠀⡀⠂⠀⠀⠀⠀⡀⠀⡀⠂⠀⠀⠀⠈⠀⡀⠀⡀⠄⠀⠀⠀⠀⠂⠀⠄⠈⢀⠀⠀⠠⠴⠠⡀⠀⠀⡠⠝⠖
⠀⠀⠀⠀⡙⠀⠣⡀⠀⠀⠀⡀⠂⣀⣨⠁⠀⠀⠀⠀⢀⠄⠎⠀⠓⠠⠀⠀⠀⢀⠄⠎⠀⠱⠠⣀⠀⠀⠀⢠⠈⡀⠀⠌⠃⠀⠀⠀⠀⢨⡁⢘⠄
⠀⠀⡐⠉⠀⠀⠀⡀⠢⠀⡈⠀⠀⠀⠀⠈⠀⠀⠀⠠⠁⠀⠊⠀⠐⠀⠈⠀⡐⠀⠐⠊⠀⠐⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠊⠀⠀⠁⠀⡀⠄
⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⡈⢐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢐⠂⢈⡠⣌⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢅⠘⠿⠿⣦⣕⡆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⣬⡐⠤⣀⠈⠻⢶⢿⡖⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⢸⣿⣄⠀⠁⠐⠠⠠⠔⠹⣟⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠺⣿⡄⠀⠀⠀⠀⠀⠀⠀⠑⠲⣽⢷⣦⡖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⢀⡀⠤⢐⣢⣍⣤⣶⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣄⣀⣠⡤⠵⠚⠛⠛⠠⣭⣤⣶⣾⣿⣿⠿⠛⠋⠉⠉⠋⠙⠖⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠉⠉⠀⠀⢀⣠⣉⣠⣤⣼⣿⣿⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢦⣤⡾⠶⠶⠶⠿⠛⠛⠛⢫⡀⠀⣺⣿⣿⠆⠀⠀⣀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡼⠉⠁⠀⠀⣼⠏⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⢉⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⠀⣸⣿⣠⠊⠀⠀⠀⣠⣴⡿⢳⡄⣼⡿⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⣼⢁⠀⠀⢀⣤⡤⢼⣿⠀⣴⣿⢁⣿⡏⠀⣿⣿⠟⢉⣿⠀⢰⣿⠟⣠⠞⠀⡿⠷⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⠞⠋⢻⣇⢠⣿⡏⠀⣿⣿⠀⣸⠿⢻⣿⠁⢀⣿⡿⣦⣼⡏⢀⣿⣿⠋⠁⣠⠞⠀⢰⣿⠇⡠
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣠⡿⠁⢸⣿⡇⢠⣿⡏⡠⠁⠀⣾⣿⣠⠚⣿⣇⡼⠋⠀⠈⠿⣿⠤⠞⠁⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠞⣸⠃⠀⠸⣿⠷⠋⠿⠋⠀⠀⠀⠙⠋⠁⠀⠉⠁⢀⣀⠀⠀⠀⢀⣠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠉⠁⠀⣴⣿⠃⢠⠇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⠀⠀⠀⣸⣿⣯⠔⠁⠀⠀⣴⣾⠷⣆⠀⢠⣿⡖⠒⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⣀⠴⣿⣿⠅⣴⡦⠀⣸⣿⡇⠀⣿⠀⣼⣿⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡅⢸⣿⡟⢸⣿⠁⣠⣿⣟⢷⣠⡧⠈⣿⡟⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢸⣿⡇⣼⣿⠇⣾⣿⠜⠁⠻⢿⡶⠟⠀⢸⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⣼⡿⠃⠻⠟⠀⠀⠁⠀⠀⠀⠀⠀⠐⠊⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣤⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⢿⠿⠿⢿⣟⡙⠿⡿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣛⣻⣿⣿⣛⣛⣛⣿

⠀⢠⣶⣦⠀⠀⠀⠀⠀⢀⣶⣶⠀⠀⠀⠀⠀⠀⢰⣾⣿⡦⠀⠀⠀⠀⠀⠀⣿⡗⠀⠀⠀⠀⠀⢀⣶⡄⣀
⠀⠈⢿⡏⠀⠀⠀⠀⠀⣘⣿⣿⡁⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⣀⣠⣼⢧⡄⠀⠀⠀⢀⣬⣿⣵⣶
⣴⣿⡶⣾⢿⡄⠀⠀⠀⣿⣿⠿⠿⠶⠀⠀⠀⢠⣾⣿⣿⣷⡄⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⠀⢸⣿⣿⣿⣯

⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠠⣿⡇⠀⠀⠀⠀⠀⠀⢾⣿⠇⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠰⣿⠇⠀⠀⠀⠀⠀⠀⠸⡿
⢠⣤⣿⣧⣄⠀⠀⠀⠀⣤⣶⣿⣦⣤⠀⠀⠀⢠⣶⣿⣿⣶⡆⠀⠀⠀⣶⣿⣿⣿⣶⡄⠀⠀⠀⣴⣾⣿⣶
⠘⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⢺⣿⡆⠀⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢹⣿⠄
⢀⣀⣿⣇⣀⠀⠀⠀⠀⣀⣨⣿⣄⡀⠀⠀⠀⠀⣀⣼⣧⣀⡀⠀⠀⠀⢀⣀⣿⣄⣀⠀⠀⠀⢀⣤⣾⣿⣦
⠞⠿⠿⠿⠟⠇⠀⠀⠐⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠛⠛⠃⠀⠀⠀⠛⠛⠛⠛⠛⠀⠀⠀⠈⠛⠛⠛⠛

⠀⠠⣴⣄⠀⠀⠀⠀⠀⠀⢰⣦⡄⠀⠀⠀⠀⠀⠀⡶⣶⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⣶⣶
⠀⠾⣿⠿⠀⠀⠀⠀⠀⠀⠸⣿⠃⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠀⢻⡿⠁
⢰⣶⣿⣷⣟⠀⠀⠀⢀⣶⣿⣿⣷⣦⠀⠀⠀⢰⢶⣛⣟⡷⣆⠀⠀⠀⣾⣿⣿⡟⡧⠀⠀⠀⢠⣾⣿⣿⣿

⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠐⣿⡇⠀⠀⠀⠀⠀⠐⢿⣿⠗⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢹⣿
⣴⣶⣿⣷⡶⠆⠀⠀⢀⣤⣾⣿⣦⣄⠀⠀⠀⢰⣾⣿⣿⣷⡆⠀⠀⠀⢰⣶⣿⣷⣶⠀⠀⠀⠰⣿⣿⣿⣿
⣿⡿⠻⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢀⡠⠒
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠏⠉⢛⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⠿⠿⢿⠿⠿⣿⣿⣿⡿⠿⣿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⢦⣡⣌⣛⣈⣛⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣭⣯⣯⣭⣭⣽⣯⣽⣽⣭⣿⣭⣭⣽⣯⣭⣭⣭⣿⣭⣭⣭⣭⣭⣿⣿⣿⣿⣯⣯⣽⣯⣭

⠀⠀⠀⠀⣴⢾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⣾⣿⣿⡖⠂
⠀⠀⠀⢠⡖⢛⡛⡓⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣗⠠⠄
⠀⠀⠀⠀⣻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿
⠀⠀⠀⠀⢈⣿⣿⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣟⠁
⠀⡀⢠⣰⢾⣿⣿⣿⢶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⢯⣛⣫⢷⣶⣤⢐⡒⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣟⣭⣿⣶⣶⣤⣄
⢸⡏⢚⣿⠰⠖⠖⠶⢺⣟⠊⣿⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡾⠫⣵⡆⠩⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠃⠉⠙⠈⠛⠛⠋⠘⠋⠁⠛⠃⠀⠀⠀⠀⠀⠀⠀⠛⠋⠛⠛⠛⠋⠛⠁⠃⠁⠁⠚⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⣾⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣅⣿⡇
⠀⠀⠀⢀⣺⣷⣾⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡗
⠀⠀⠀⠀⢸⡏⡭⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡝⠿⡿⢋⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣟
⠀⠀⢀⣤⡷⣿⣿⣾⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⣵⣻⣟⣷⡉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣾⣿⠿⣿⣷⣶⣶⣤
⢰⣿⣿⣿⣗⣯⣽⣞⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣯⣶⣿⣼⣷⣏⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢾⣿⣷⣿⣿⡻⣿
⣿⢿⡼⣯⠽⡿⡿⠧⢭⡿⢾⢿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠿⡿⠿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠨⣿⡿⠯⣿⠼⣿⣿⣿⢯⡿⢽



⠀⠀⠀⠀⣀⡰⡶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣦⣄
⠀⠀⠀⢰⣿⣿⣿⣏⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣿⣽⣿⣿⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⣣⡟
⠀⠀⠰⣾⣾⣾⣿⣶⣗⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⡗
⠀⠀⠀⠈⠙⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⢿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣟⠁
⠀⠀⠀⢀⡸⣾⣿⡾⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⣴⣾⣷⣶⣷⣦⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣿⣿⣿⣆⣶⡶⣾
⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡛⢟⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣿⣯⢳⣿⣿⣿⣿⣗⣿⣏⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⠢⠤⠂⠢⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⡀⠀⢠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣘⡍⠀⠀⠀⠀⠀⠀⠀⡦⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⠤⠤⠄⠀⠤⠄⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣛⠀⢐⠦⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠘⠁⠑⡄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡠⠶⣄⡠⠏⣀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⠃⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⣀⠴⢉⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠸⠀⠀⠘⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠄⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠈⠒⡤⠀⢀⡉⠄⠂⠛⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢁⠀⠀⡆⠀⣀⡠⡚⠉⡉⠑⢆⡀⠀⢀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠊⠀⠀⠀⡸⠀⢃⠀⠀⠀⠀⣆⢸⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠀⠀⠁⠀⠙⡆⠣⠤⠔⠁⢉⠋⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⠳⢅⠀⠈⠁⠒⠒⠂⠁⠀⣷⠀⠀⠈⡆⠰⠲⠶⠶⠶⢌⠉⠉⠉⠉⠀⡸⠀⠀⢠⡀⠀⠀⠀⡘⠀⠀⢸⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⡀⠀⠀⠱⠤⡀⠀⠀⣀⡀⢤⠞⣧⡀⠀⠸⠀⠀⠀⣠⡶⠚⠛⠛⠛⠋⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠈⠉⠁⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⡀⠀⢦⣧⢔⡩⢄⠖⠃⣠⠟⠁⠀⢰⠀⠀⢘⡯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢹⢋⣼⠿⠓⠁⢀⡠⠋⠀⠀⡴⠃⠀⣭⠾⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠓⠤⣄⡀⠿⠁⠀⢀⡤⠊⠀⠀⠀⠀⠀⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⠀⠀⠀⠀⠈⠀⠐⠂⠉⠀⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⢁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡆⠇⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⢀⠏⠈⠈⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠀⠀⠀⠈⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠁⠀⠀⠀⡇⠀⠀⠀⠀⢳⠀⠀⠀⠈⠑⠢⣄⠀⠀⠀⠃⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣎⣀⡀⠀⠀⢀⣀⣠⠇⠀⠀⠀⠀⠸⠤⣤⣀⣀⠤⠤⠜⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠐⠒⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⣤⣦⣽⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠃⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⢀⠁⣸⣿⣿⠇⠀⢀⣶⣿⡛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠨⣿⠿⠃⢀⣰⣿⡻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣀⣀⣀⠀⠐⠄⣠⣶⡟⡭⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⣣⠾⢿⣯⣎⢰⡏⠉⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠉⠛⣄⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢴⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣦⡀⠀⠀⠀⠀⠀⢀⡄⠚⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠉⠉⣿⣿⣷⡄⠀⠀⠀⠀⠀⢀⡀⣠⣟⡿⠶⠶⣤⣀⣀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠐⢦⡤⠀⠰⡿⣿⣿⠆⠀⠀⠀⠀⠀⣾⣴⣿⣿⣿⣓⣀⣨⣍⣛⢿⣗⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣤⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢰⣦⡀⠸⢿⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠘⢿⡋⠙⣿⣿⣆⢻⣿⣿⣶⡀⠀⠀⠀⠀⠀⢰⠀⢨⣿⢿⡏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⢛⣿⣿⣿⣿⣿⡻⡄⠀⠀⠀⠀⠈⠀⠘⠻⣪⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⢸⣿⣿⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢾⡇⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣸⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⢠⣜⣻⣿⣿⠀⠀⠀⠀⠉⠐⣥⣀⡀⠆⠀⣹⣯⠙⣿⡿⢋⣿⣟⣽⣧⣿⡿⠀⠀⠀⠀⣃⠀⢠⣿⣿⡟⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⢸⣿⣷⣆⠘⠀⠀⠀⠀⠀⠠⠜⠁⠁⠀⠀⠘⢿⣦⠈⢿⢸⠌⣿⣼⢃⣿⠀⠀⠀⠀⠀⣷⡀⢸⣿⣿⣇⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠀⠀⠀⠀⢀⠒⣀⠀⠀⠀⠀⠀⠘⣳⠀⠀⠀⡼⢿⣿⣾⠟⠀⠀⠀⠀⠀⣿⡇⠈⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠄⢻⣿⣿⠀⠀⠀⠀⢺⣗⠚⢧⣄⠀⢀⣠⣼⡉⠀⠘⣿⣧⣞⢋⠁⠀⠀⠀⠀⠀⠀⣿⠃⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢀⣶⣿⡟⢻⠀⠀⠀⠀⢘⡻⣿⠿⠟⠀⢾⣟⠟⠃⠀⢸⣿⣿⣷⡿⠀⠀⠀⠀⠀⠀⠀⡧⠀⢠⣿⣿⡟⢹
⠀⠀⠀⠀⠀⠀⠀⢈⠈⠁⠀⢻⣿⣿⣿⡇⠀⠀⠀⢀⡛⠁⠀⠀⣾⡿⠗⠀⠀⢀⣼⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠉⢹⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⢿⣿⣿⡿⠁⠀⠀⠀⠈⢧⣶⠀⣤⣭⢴⣦⣶⣿⠿⢿⣿⣿⡿⠄⠀⠀⠀⠀⠀⠀⠘⠂⠒⠲⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡃⠀⢰⣿⣿⣷⣤⣤⣶⣤⣀⠀⠀⠀⠀⠀⠀⠼⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠁⠀⠸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣶⣤⣀⡀⢀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣦⡀⠀⠀⠀⠀⣀⣤⣴⣆⠈⠀⠠⣤⣶⣤⣤⡿⠟⣩⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⣦⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣜⢿⣿⣼⡇⢠⣴⣶⣿⣿⣿⣿⣿⣷⣮⣤⣈⣹⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢹⣾⢟⣾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠉⠀⠙⢿⠗⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠹⢟⣵⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠛⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⣿⣆
⠀⠀⠀⠀⠀⠘⣿⡧⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠈⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠓⠄⠀⠀⠀⠀⠀⡾⠋⣴⣶⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢴⣶⡘⢷⣄⠀⠀⠀⠀⢀⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠰⡽⠇⣼⡟⢠⣿⡇⠀⢠⣶⠄⣤⠀⣤⡄⢤⣤⠀⠀⣾⣧⠸⡦⡈⢿⡷⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠾⠿⠀⠀⢸⡇⢰⣟⠀⢽⡇⠀⣷⠀⠀⢹⡿⠆⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠀⠉⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠛⠛⠛⠛⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣄⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣾⣶⣶⣶⣶⣶⣶⣷⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⣠⣶⡄⠀⢿⣯⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣦⣰⣿⣿⣿⡘⣟⣿⣛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡏⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣷⣿⣿⣷⣿⣾⣿⣽⣽⣿⡆⣰⣾⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣧⣟⣿⣿⣿⣿⣿⣿⡿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⢿⣿⣾⠩⣿⢿⣿⣽⣿⣿⠿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠉⠙⠟⠋⠹⠟




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣟⣹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢸⣿⣿⣿⣿⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣿⣷⠀⠉⠛⠛⠁⢠⣿⡿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⡿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠟⠀⢀⣤⣀⡀⠘⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣿⣿⣿⣯⣿⣲⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠀⠀⢰⣿⣿⣽⣷⠀⠀⠈⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⣿⣿⣿⣿⣻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀⢸⣾⣿⣿⣷⠀⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣿⣿⣿⣿⣿⣿⣿⣤⣤⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣾⣷⣾⢶⣿⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠿⣽⣿⣿⣷⣿⠻⠿⠿⠿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣶⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡿⠁⠈⡇⠀⢹⡄⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣧⠀⠀⢻⣄⠸⠇⢀⡾⠁⠀⠀⢠⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠟⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠘⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣾⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣄⣴⣿⡇⠀⢀⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⣿⣿⣿⣿⣿⡎⣿⣿⡿⠋⢀⣴⣿⡿⠋⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢹⣿⣿⣿⣿⣿⣷⠘⠛⠀⢠⣿⡿⠋⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣭⣥⣬⣭⣭⣭⣭⣭⣭⣤⣤⣤⣤⣭⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⣿⣿⣿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠛⠿⠃




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠻⠥⠵⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣠⣘⢤⣬⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠘⠛⠁⠀⠹⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠁⢂⣠⡔⠀⠀⠻⣀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⡿⢷⡤⢀⣀⡴⠂⠀⠉⠓⠠⢄⣀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⢵⣶⣿⠥⠤⠀⠀⢄⠀⠀⠈⠉⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣽⠉⠀⠀⠺⠂⠆⡄⠈⠀⢁⠀⠀⠀⠐⣀⢀⠐⠁⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⢉⠐⠀⠀⠀⠀⢤⠀⠁⠀⠀⠀⠈⠢⠄⠀⠀⠙⢄⠀⠀⣴⢳⣀
⠀⠀⠀⠀⠀⢠⠖⡙⠀⠤⢀⠀⠐⡳⠀⢧⠀⠀⠀⠀⢀⡀⠈⠢⡀⠀⠉⠲⡤⡄⢩⣆
⠀⢀⣀⠀⠀⠁⠀⠀⠐⠂⠀⠀⠉⠛⢦⠘⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⡀⠀⢃⠘⠄⢻⡆
⠀⠈⡿⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠖⣮⠐⣤⠼⡧
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠚⢻⡟⠁⠀⠁⡼⠁
⠀⢸⠀⡆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⡄⠀⠀⠀⢠⠟⠀⡀⡠⠂
⠀⣾⠀⡥⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠻⢄⡀⠀⢰⠃⣠⠖⡹⣧
⢀⣿⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠱⠤⠁⠀⡀⢘⠃⢹⣆⡀
⢸⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡤⠄⠁⠀⠀⢛⡀⠘⠳⡈⠁⠀⠈⡁⠈⠉⠙⢦
⣼⡇⠰⠀⡆⣠⠤⠤⠤⠤⠄⠀⠀⡀⠤⠴⠖⢩⠀⠀⣀⣀⠴⠛⢧⣀⡰⠚⠀⠉⠉⠀⠀⠀⠀⠀⠳⣦⡀
⠀⢅⣀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡿⠉⠁⠀⠀⠀⠈⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡀
⠀⠀⠸⠋⢲⡷⢿⣿⣿⣿⣿⣿⣿⣿⠟⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣆⡀
⠀⠀⠀⢟⡰⢰⣻⣿⣿⣿⣿⣿⠏⣴⣿⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠹⣟⣿⣧⣿⣿⣿⣟⣾⣿⣿⣿⣿⣿⡟⢲⢠⣤⣄⡀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣏⠸⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢿⣿⣿⣿⣶⣶⣯⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢸⣿⡏⠛⠛⠁⠛⠛⠛⠛⠛⠛⠛⠛⢳⡆⠛⠛⠛⠛⠋⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⢸⣿⡇⢀⡾⢧⣄⠀⠀⠀⠀⠀⠀⠀⠈⠃⡆⠀⠀⠀⠀⠀⣀⣀⣀⣿⡏⢻⣿⣿⣿⣿⡀⠀⠀⠀⠀⠸⣿⣿⣿⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⢸⣿⡇⠸⠧⠒⠛⠓⠒⣶⣶⣶⣶⣶⣶⣄⣻⣲⠖⠿⢯⣿⣟⣿⣛⣿⠀⠀⠙⢿⣿⣿⣷⣀⡴⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⢩⢿⡇⣷⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⡿⠆
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢸⡿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠛⠉⢀⣠⠖
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢑⠂⠉⠉⠉⠹⣃⣹⣿⣿⣿⣿⣿⣿⣿⣏⣿⠀⠀⠀⠀⠈⣿⠿⠛⠉⣱⣤⣴⣾⠟⠁
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢿⣿⡿⣿⣿⠿⠟⠛⣉⡠⢿⣶⣶⣠⣤⠚⠁⡄⢀⣼⣿⠿⠏
⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠘⣿⣇⡠⡄⠒⠈⠋⠘⠄⠒⠈⠀⠍⢀⠰⡟⠉⠉⠁⡌⡀
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣠⣶⣞⣿⠀⠀⠀⠀⠀⢐⢸⢇⡐⠈⠆⠂⠈⢀⡠⠀⢀⣀⣤⣾⡷⢇⣒⣂⠠⠚
⠀⠀⠀⠀⠀⠉⠉⠀⠀⣀⡠⠽⠟⠛⠟⠀⠀⠀⠀⠀⠀⢂⣁⣌⣹⡟⣓⣦⡀⠀⢀⠠⠄⠀⠀⠤⠃
⠀⠀⠀⠀⠀⠀⠀⣰⡏⠁⠀⠀⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⠁⠈⠉⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣤⣄⣈⡉⠃⠀⠀⠀⠀⠐⠂⠀⠀⠈⠉⢀⣀⣤⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⢿⣿⣷⣶⣾⣿⣿⣿⡿⠿⠿⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣦⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠟⠛⠉⠉⠉⠛⠛⢿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠁⠀⣀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠈⠙⠟⠀⠀⠠⠀⡀⠀⠀⠀⠂⡁⠘⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠴⣼⡿⠀⠀⡀⣤⣿⠾⠟⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠂⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠰⠀⠀⠀⡀⠀⠀⢀⡶⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠛⠋⠈⠋⠁⠙⠉⣽⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠶⣄⡀⢀⣤⡀⠁⡀⢾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠁⠀⠀⠀⠀⠀⠂⠐⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⡈⠻⣿⣿⣿⠃⢵⠈⡖⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⣦⡀⡀⠀⣬⣷⠀⠃⠈⡉⠀⠁
⠀⠀⠀⠀⡠⠀⠈⠁⠀⡁⠀⠱⠀⠈⠛⠁⠀⠤⠄⣀⠀⠀⠑⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠙⢁⠃⠀⠀⠉⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠳⡀⠀⠀⠀⠀⠄⠄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠠⠀⠀⠂⠀⡠⣾⣿⣷⣄
⠀⠀⠀⠄⠀⠀⠀⠀⠀⢿⠀⠔⠀⠰⡀⠀⡡⠀⠀⢘⠐⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⣦⣀⣀⡄⠀⠈⠀⣿⣿⣿⣟⠿⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠈⠐⢄⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⣹⣿⠿⠛⠁⠀⠀⠀⠈⠛⠻⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⢤⡱⠆⠡⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠏⠀⠀⠀⠄⣀⡀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⠀⡀⠀⠆⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠐⠰⡄⠜⠀⠈⢄⣀⡆⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠂⡀⠂⠒⠀⠀⠀⠀⠠⠀⠀⣠⢆⠀⠀⠂⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠰⡆⠀⢡⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠂⢆⠀⠈⠁⠠⡆
⠀⠀⠀⠄⠀⢀⠁⠀⠀⠑⠒⠖⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠄⠀⠀⠆⠀⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⢱⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡄⠀⠁⠀⠀⠈⡅⠀⢠⠈
⠀⠀⠀⠀⠀⠙⠦⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠓⠄⠀⠀⢀⠀⠀⠐⠀⠂⠊⠀⠀⠠⠤⠠⠤⠀⠂
⠀⠀⠀⠀⠀⢠⡤⢤⠤⠄⠄⠤⠠⠄⠀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣀⣀⠀⠀⠀⠀⠁



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

⠀⠀⠀⠀⠀⠀⠐⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢸⡏⡉⠉⠉⡉⢻⡆⠀⠀⠀⢿⠀⠀⠀⢈⠙⡇⠀⠀⠀⠀⢾⢁⠉⠉⠛⠛⡇⠀⠀⠀⢸⠋⡅⠀⠀⠉⢹
⠀⠀⠀⠀⠀⣼⣅⠀⠠⠄⢀⣬⡇⠀⠀⠀⠈⠀⠀⠤⠀⠀⠁⠀⠀⠀⠀⠈⠈⣿⣿⣿⠀⠁⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠹⠿⠿⠂⠘⢿⠿⠃⠀⠀⠀⠀⠀⠠⣀⠄⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⢸⢘⡛


⠀⠀⠀⠀⠀⢀⣴⣿⠟⢿⣿⣦⡀⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀
⠀⠀⠀⠀⠀⠿⠛⣋⣤⣌⣙⠻⠿⠀⠀⠀⢰⣿⣿⣿⡿⢿⡄⠀⠀⠀⠀⢀⣠⠤⠤⠤⢄⣀⠀⠀⠀⢰⡿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠸⡇⡄⠀⠀⡀⢸⠀⠀⠀⠀⠚⢉⠉⠀⢀⠀⠇⠀⠀⠀⠀⢸⠀⡀⠀⠀⡀⡿⠀⠀⠀⢸⠀⡀⠈⠉⡍⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⢠⣶⣄⡀⠂⠀⣤⣶⠀⠀⠀⠀⢀⡀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠐⢠⠀⠄⠀⠀⠀⠀⠀⠙⢿⠋⠀⢑⡾⠋⠀⠀⠀⠀⠀⢁⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠸⣷⣶⣾⠇⠀⠀⠀⠀⠀⠸⠀⣦⣤⣴⡇

⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⣀⣤⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣦⣤⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⢿⡆⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢠⣿⣿⣿⣿⢿⣷
⠀⠀⠀⠀⠀⠘⡉⡉⠉⠉⡉⢹⠁⠀⠀⠀⢿⢉⠉⠁⢀⡀⡗⠀⠀⠀⠒⢲⠀⠀⠉⠀⡈⡿⠒⠀⠀⠸⠋⡉⠉⠀⡀⢹
⠀⠀⠀⠀⠀⠀⠡⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⣠⣸⣧⣄⠀⠀⠀⠀⠀⠀⠀⠤⠀⡆⠀⠀⠀⠀⠀⠐⠂⠀⣄⠀⣠⠀⠀⠀⠀⠀⠀⠀⠘⠃⠉
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣤⣿⡆⠀⠀⠀⠀⠀⠈⠇⣶⠻⠛⠟⢺⠀⠀⠀⠀⠀⢸⡀⠐⠀⢸⡄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⣫⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢏⣾⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣄⠻⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣷⣌⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣷⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠘⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣹⣿⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⣾⠋⠙⠛⠋⠙⢿⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠃⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⡟⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⢻⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣾⠟⠛⠛⠛⠛⠛⠛⢿⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡟⢻⣿⣿⣿⡏⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠘⢿⣿⣿⠇⠀⠀⠀⠀⢻⣿⣿⡿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠇⠘⣿⣿⣿⠇⠀⢿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣦⡆
⠀⠀⠀⠀⠀⠀⢄⣀⠠⠀⠠⢄⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⣈⣁⡀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⡀⠀⠀⠀⠀⢠⡅
⠀⠀⠀⠀⠀⠀⠐⠀⠤⠀⠠⠤⠐⡂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠉⠀⠀⠁⠸⡇
⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡂⠐⠂⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣑⠀⠀⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠂⠈⡁⠆⠀⢩⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⢆⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⣦⠆⠉
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠁⠸⠉⠁⢀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡥⠀⠀⠶⠆
⠀⠀⠀⠀⠀⠈⠀⠀⠠⠄⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠀⠀⠀⠀⠠⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠐⢘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⡀⠀⠀⢐⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠀⠀⢸
⠀⠀⠀⠀⠀⠰⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠐⠂⣀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠈⠑⠤⠀⠠⠠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡃⡤⢤⢚⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠄⢲⡃⡤⢤⢚⡖⠢⡀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠈⡃⣠⡀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠃⡈⢁⠘⠀⠀⢀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡇⠁⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠆⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠤⠅⠤⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠠⠤⠀⡀⠮⠅⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠛⠛⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡴⠡⠤⠤⠤⠄⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢉⡉⠁⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⡄⠂⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠈⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⡤⠀⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⠀⠀⢹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠶⠖⠣
⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠎⠀⢰⠐⡫⢝⠂⠖⠀⠱⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⣤⡄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘⠀⠀⠀⠄⠃⠸⢀⠁⠉⠁⠃⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠘⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢌⠆⠣⢋⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣼⡿⠂⠀⠀⠀⢀⣀⡀⠀⠀⠀⣤⣤⣤⣤⣀⣠⣄
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡠⢸⣇⠠⡘⠀⠀⣀⣤⣄⣤⠀⠀⠀⢸⡿⠿⠿⢿⠃⠀⢀⣾⠿⢿⠋⠁⠀⠀⠀⣩⢿⠿⠿⢿⣯⡈⠓⠂
⠀⠀⠀⠀⢛⣿⣛⣠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⠿⠓⠀⣴⣿⣿⠿⣿⣿⣦⡀⠀⠀⠁⢀⡀⠈⠀⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠁⠋⠀⠀⠋⠈⠁
⠀⠀⠀⢠⠟⠉⠁⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠇⠈⠁⣿⣿⠶⣿⣿⡇⠁⠀⢀⣤⣼⠀⣤⣰⠀⣄⡤
⠀⠀⠀⠀⢰⡤⡆⠀⠀⣻⡃⠀⠀⠀⠀⠀⠀⢸⣀⣀⣸⠀⠀⣿⣇⠶⢉⣿⡇⠀⠀⠀⠈⠁⠀⠈⠁⠀⠉⣁⣀⣠⣴⣄⠀⠀⠀⠰⣤⡄⠰⣤⡶⣷⣴⣀
⠀⠀⣀⣀⣘⣃⣗⣀⡀⠀⠀⠶⠂⢒⣒⣩⣥⣥⣤⣶⣶⣶⣶⣶⣷⣶⣾⣶⣶⣶⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⠂⠀⠀⠀⣀⣀⣀⣠⡄⠀⠋⠋
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⡆⣿⣿⣿⠿⠻⠿⠿⣿⠏⠀⠀⠀⠀⠀⢩⢟⠛⡻⠯
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠁⢻⢻⡄⠀⠀⠀⠀⢿

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠐⠺⡿⠒⠰⣾⡷⠂⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣶⣿⣷⡀⠀⠀⠀⠀⣴⠄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⣱⡟⠛⡇⢠⠟⠛⠼⠇⠀⢀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃⠈⠁⠀⠀⢠⣿⣷⣯⣥⡀⠀⠀⠀⠈⠟⠟⢿⣿⡇⠀⠉⠂⠈⠀⠀⠀⠀⠀⢀⣴⣄
⠀⠀⡁⣠⣤⠉⠈⢠⡄⠀⠀⠀⠚⢾⣿⡟⠛⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⣷⣽⣿⣷⠀⣀⠀⠀⠀⠀⠀⢄⣴⣶⣶⡿⠃⠀⢀⡴⠛⣿⣿⣿⣿⣟
⠀⠀⠀⠺⠿⠀⠀⠸⠇⠀⠀⠀⠀⠘⠏⠇⠺⠃⠻⠼⠾⠀⠀⠀⠀⠠⠄⠰⠃⠻⠙⠿⠋⠁⠀⠀⠀⠀⠀⠀⠏⠈⠉⠂⠀⠀⠀⠀⠸⠙⠧⠀⠘⠎⠓

⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠒⢒⡒⢦⡄
⠀⠀⣠⣽⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⣀⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣾⣷⣾⣿⣠⣽⣁⣄⣽⣄⣀⣀⣀⡀
⠀⠀⢾⡷⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠇⠀⠀⠀⠀⠀⢠⣤⣤⣤⣴⣶⡆⠀⠀⠀⣾⣿⠿⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⠻⣿⣷
⠀⠀⢸⡟⣆⠀⠀⠀⢀⠀⡀⢀⠡⠀⠀⠀⠀⠀⢸⡇⠈⢷⡀⠀⠀⠀⢀⡼⠿⠿⢿⣿⡇⠀⠀⠀⠀⠛⠃⣾⣉⣷⠘⠛⠛⠛⠛⠛⠛⠛⠋⣾⣋⣿⠘⠛
⠀⠀⠉⠀⠉⠀⠀⠀⠉⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣤⣤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣧⣂⣯⣬⣭⣭⣭⣭⣬⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡀⣀⣀⣀⠀⠀⣀⣀⣀⠀⢀⠀⣀⣀⢀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠤⣄
⠀⠀⠀⠀⠀⠓⠓⠋⠐⠀⠛⠓⠛⠘⠃⠀⠛⠁⠊⠓⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠊⠀⠀⠀⠀⠈⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡗⠄⠀⠀⠠⣀⣀⣀⡤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣞⠀⠀⠀⠀⠀⠀⠉⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠑⠢⠤⣄⣀⣠⠤⠜⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⢴⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠖⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠑⡖⠒⠒⠦⠤⠒⠚⠓⠉⠀⡏⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⣀⢠⣤⣤⠄⠀⠝⡂⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠄⠀⠀⡀⢀⢂⡀⠘
⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡤⢯⠀⠀⠀⠀⠀⠀⠀⠪⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠳⠋⠱⡳⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠢⣀⠈⠲⠒⠁⠀⠀⠀⣀⠤⣺⠁
⠀⠐⠀⠀⠀⠀⠀⠈⠆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⣰⡢⢄⠀⠀⢀⡽⡶⠶⠶⣶⠊⠙⢊⠉
⡇⢹⠁⠀⠀⠀⠀⠈⡇⠀⠀⢨⣀⠀⠀⠀⠀⠀⣀⠀⠨⠁⠀⠀⠱⣻⡆⠙⠀⠟⠬⠈⠁⠐⡇⠀⠀⠀⠁
⡇⠸⡀⠀⠀⠀⠀⢀⠇⠀⢀⡘⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡤⠶⡊⠑⠀⠂⠀⠑⠒⠠⢀⡑⠠⢤⠜
⡇⠀⠙⠒⠒⠒⠒⠋⠂⠀⠀⠆⠀⠀⠀⢈⡀⠈⠁⠲⠄⠀⠈⢆⠀⠀⠀⠀⠀⡀⠀⠤⠒⠋⠁⡠⢷⠞⢣
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠸⠀⠳⠆⠈⠀⣄⠀⠀⠠⣛⠄⠀⡠⠔⠊⠀⢸⡫⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠉⠁⠀⠀⠀⠀⠀⣧⡅
⡇⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠉⢱⠒⠒⠀⠐⠀⡆
⡇⠀⠀⠀⢀⠈⠉⠁⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡀⠀⠀⢸⠀⠀⠀⠀⠀⢹
⠓⠒⠒⠒⠂⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠀⠈⠉⠒⣏⠀⠀⠀⠀⠀⠀⡗⠒⠒⠒
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠠⠈⠂⠀⠀⠰⡉⠉⠒⠂⠦⠤⢤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠐⠐⠒⠁
⣿⣿⣿⣿⣛⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣟⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⠟⣹⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡯⢹⣿⣿⣿⡿⠁⣴⡿⠟⣡⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⡇⠘⣿⣿⠟⠀⠾⠟⢀⣼⡿⠛⣩⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⡿⢟⣿⣿⣿⣿⠃⠀⠉⠀⠀⠀⠀⠀⠛⣁⣴⣿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⢋⣥⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⣡⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⣀⣴⣿⣿⣿⣿⢟⣯⣿⣿⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⠟⣫⣶⣿⣿⡿⠃⠀⠀⠀⣠⣴⣾⣿⣿⠿⠛⠙⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⢋⣡⣾⣿⣿⣿⠟⠀⠀⠀⢀⣾⣿⣿⣿⠟⠁⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⣴⣿⣿⣿⣿⠟⠁⠀⠀⠀⢠⣾⣿⣿⣿⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⢠⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⣠⡴⠶⠖⠲⠶⢦⣄⠀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠉⣻⣯⣄⠀⠀⠀⠀⠀⠹⠟⠉⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠈⠙⢿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠿⠶⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢴⣞⣷⣶⡶⠷⠶⠶⠶⠶⠶⠶⠶⠦⣤⣤⣀⣀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⢸⡟⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⢻⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⡇⠀⣿⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠀⠀⢸⡇⠀⣼⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⡿⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⣟⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⠀⢹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣬⡇⠘⠛⠛⢉⠉⠃⠀⠀⠀⠀⠘⠉⣉⠛⠛⠃⢸⣤⡤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⡠⠄⠈⠇⠀⠀⠀⠿⠇⠀⢀⡤⢤⡀⠀⠐⠿⠇⠀⠀⠸⠁⠠⢤⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠘⠇⢀⣾⡁⠀⠀⢀⣠⣶⣶⣿⡀⢀⣿⣶⣶⣄⡀⠀⠀⠀⣷⡀⠸⠃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣸⣿⣷⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣾⣿⣇⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠿⠋⠙⠿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢯⡋⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⢙⡹⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡛⠛⠒⠲⢶⡶⠶⣶⠖⠒⠛⠻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠉⠙⠒⠒⠚⣁⡤⠟⠁⠁⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⣾⠉⠉⠉⠁⠀⡀⠀⣟⣛⣛⣳⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢀⠀⠀⠀⣿⠀⠀⢀⠀⠀⡀⠀⡏⠉⠉⢽⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠼⠤⠤⠤⠿⠤⠤⠤⠤⠤⠤⠤⠿⠶⠶⠿⠤⠤⠤⠼⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠒⠦⠀⠀⠀⠀⠀⠀⠀⠒⠂⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠒⠂⢀⠀⠀⠀⠀⠀⠀⡤⢄⠀⠀⠀⠀⠀⠀⠀⠈⠉⡀⢄
⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠁⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⡀⠀⠀⠀⠀⠠⠀⠖⠂⠐⠀⠶⠀⠀⠀⣀⠀⢤⠄⡀⠀⠀⠀⠀⠓⠘⠀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠔⠐⠄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠘⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂
⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠘⠂⡄⠀⣠⠀⠃⠀⠀⠘⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⠀⣨⠍⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠆⠀⠀⠀⠰⠁⠀⠀⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠈⠈⠁⠉⠁⠀⠀⠀⠀⠀⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠐⡄
⠀⠀⠀⠀⠈⠏⠒⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠲⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⡄⠀⢀⠄⢠⠀⠀⠀⠀⠀⠁⠀⠀⠃⠐
⠀⠰⠀⠐⠐⠆⠁⠐⠃⠀⠀⠀⢀⠒⠒⠀⠄⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠂⠁⠀⠃⠐⠂⠀⠁⠐⠐⠆⠃⠀⠀⠀⠀⠀⠀⠐⠂⠃⠀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠂⠁⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠁⠠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⠤⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀
⠀⠀⠀⠀⠊⠁⠂⠀⠀⠀⠀⠀⡟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢒⠀⠀⢀⠀⠀⠀⠀⡄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠓⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⡄⠀⢈⠀⠀⠀⠀⠀⡄⢀⠰⠀⣀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠇⠀⠐⠀⠀⠀⠀⠀⠀⢀⠉⢘⠇⣘⠠⡀⠀⠀⠀⠀⠐
⠀⠀⠦⢀⠐⠈⠀⠁⠀⠀⠀⠀⠓⠀⠀⡈⠀⠛⠀⠀⠀⠀⠈⠀⡈⠄⠀⠀⠀⠉⠀⠁⢘⡉⠀⠲⠀⠀⠀⠀⠈⠻⢯⠄⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⡗
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠋⠉⠉⠉⢩⠉⠻⣿⣿⣶⣶⣶⣦⣤⣤⣤⣀⣀
⠀⠀⠀⠀⣿⣿⣿⠿⣟⣛⡻⢿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⡿⠁⡄⡀⡀⣠⣞⡓⠒⠣⣀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⢟⠕⠉⠀⠀⡁⣽⠉⠉⠉⠉⠉⠑⠒⠒⠒⢲⠁⣤⣿⣿⠿⣇⡘⠚⠧⣖⢌⡆⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⠫⣤⣤⣀⣀⠙⠓⠦⠤⣾⣄⣀⣀⠀⠀⠀⠑⢮⡹⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠟⠑⠚⠫⢽⣓⠿⣶⣤⣀⠀⠙⠻⣧⣤⠀⠀⢀⡄⣧⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⠀⠀⠀⠀⠀⠀⠀⡤⣬⠟⢶⣄⠀⠀⠀⠈⠑⠶⣬⣙⣷⣦⡀⠈⠻⣿⣍⣉⡀⣇⢸⣿⣿⡇
⠀⠀⠀⠀⢐⣒⣲⣶⣶⣶⢦⢺⢤⣄⠀⠀⠀⠀⣸⡱⠃⢤⣄⢙⡄⠀⠀⢀⣠⣤⣈⡙⠿⣿⣿⣶⣄⢹⣿⣇⠀⢹⣸⣿⣿⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢳⢹⣿⣷⣳⠀⠀⣰⠏⠁⠀⠙⠿⠋⠀⠀⠠⣀⣈⡉⠉⠙⠦⠀⣤⣿⡁⢘⣿⣷⡄⠀⡁⣿⣿⠇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⢸⣿⣿⡇⡇⢠⠋⣀⡠⢶⣆⠀⠀⠀⢀⠈⠛⠿⠻⠀⠀⠀⠀⣿⣿⣧⣼⣿⣋⢵⣮⠸⣿⣿⡃
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⢸⣿⣿⡇⣧⡏⠀⠁⢲⠤⣀⠈⠙⠀⢸⡄⠀⠀⠀⠀⠀⠀⢠⡿⠿⢥⠀⠀⢠⢸⣿⡆⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⢸⣿⣿⡇⣿⡅⢰⠀⢮⠢⢈⠉⠓⣢⠤⠙⠦⠀⠀⠀⠀⢀⢴⡾⣇⣼⣴⣦⣸⢸⣿⡇⣿⣿⡁
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⢸⣿⣿⡇⣹⡇⢸⠀⠀⠑⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⣰⣫⣬⡡⣫⣿⣿⣿⣿⡜⣿⣇⢻⣿⡁
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⢸⣺⣛⣣⡧⣇⠘⣦⣀⠀⠀⠀⣀⣠⠔⠀⢀⣠⡴⠿⣶⣞⣿⣞⠝⢉⣵⣿⣯⡇⣿⣿⢸⣿⡄
⠀⠀⠀⠀⠈⠭⣭⡽⠿⠛⢻⢸⠉⠁⣰⣶⣿⠀⠀⠉⠛⢛⣋⡥⠶⣞⣛⠯⠕⢒⣽⠯⠋⠄⢉⡑⠾⡛⠛⠋⡇⣿⣿⢸⣿⡃
⠀⠀⠀⠀⠰⣮⣁⠀⠀⠀⢸⢸⡴⢋⣿⣿⣿⣧⠤⣀⣤⣭⡖⠛⠉⠁⠀⣠⢔⡽⠓⠐⠒⢂⠀⠀⡆⢠⠀⠀⢡⢹⣿⡌⣿⡇
⠀⠀⠀⠀⠀⠈⢹⡗⠀⣠⣾⢸⣴⣃⣨⣽⠿⠛⡾⣿⣿⡿⠁⠀⢀⡤⠞⠁⡜⠀⠉⠁⠀⠘⠁⡜⠀⣆⠀⠀⢸⢸⣿⡇⣿⡇
⠀⠀⠀⠀⠀⢀⡾⢡⡾⠋⢸⢸⠀⠁⠀⠀⣹⠋⢹⡈⠛⠃⣀⡴⠋⠀⠀⠀⣇⣳⠀⠀⠀⠆⠀⠃⡼⠀⢧⡀⠘⡸⣿⡇⣿⠁
⠀⠀⠀⠀⠀⠚⠐⠟⠁⠀⠸⠸⠀⠠⢶⠞⠁⠀⠘⠧⠴⣺⢋⡴⠿⠿⢷⣤⣬⡇⠀⠀⠀⠀⠀⢸⠃⠠⢤⡷⡄⠁⣿⣿⢸⡇
⠀⠀⠀⠀⢨⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣿⣿⣶⣿⣿⣾⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⣠⣴⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣷⣿⣶⣦⣄
⠀⠀⠀⣸⡿⠿⠟⠛⠉⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⠟⢻⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠿⢿⣿⣿⢿⣿⡄
⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠛⠁⠁⠀⠀⠀⠋⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣀⠀⠀⢀⣴⣿⣿⠄
⠀⠀⠠⢐⡾⣿⡀⠂⣸⡟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢿⠟⠛⠁⠀⠘⠛⢿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠘⢧⣀⣼⠓⠒
⠀⠀⠁⢸⡇⠹⢷⠰⠟⣡⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠳⣄⡀⠀⣀⣴⠏⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢳⣄⠀⠀⡐⠁⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠈⠉⠀⠄⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠆⠀⡀⠀⠀⠀⡀⠀⠀⠀⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⢠
⠀⠂⢸⠃⠀⠀⠀⠀⠀⠀⠀⡷⠀⠆⠀⠀⠀⠀⠀⠀⡆⢠⠱⠀⠀⠀⠀⠀⠀⠰⢡⠄⠄⠀⠀⠀⠀⠀⠀⠲⠀⢿⣶⣶⣶⣶⣶⣶⣶⡾⡄
⠀⠄⠼⠃⠀⠀⠀⠆⠀⠀⠀⠻⡄⠂⠀⠀⠀⠀⠀⠀⢀⡰⢘⠀⠀⠀⠀⠀⠀⢸⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢺⣿⣿⣿⠀⣿⣿⣿⡟⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢘⠀⠁⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣿⣿⣿⡇
⠀⠀⠀⢻⣿⣿⣿⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣿⣿⣿⡇
⠀⠀⠀⢾⣿⣿⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣿⣿⣿⡇
⠀⢀⣴⣿⣿⣿⣿⢸⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⢰⣦⣬⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡅⣿⣿⣿⣷⣤
⠀⠈⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉


⠀⠀⠀⠀⠻⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⠿⠟⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣿⣿⣿⣿⢿⣾⣷⡀
⠀⠈⢻⡞⠛⠛⠛⠛⠛⠷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠟⠙⠋⠁⠀⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣧⡀⠀⠀⠀⢀⣼⡿⠋
⠀⠀⠀⢀⣤⡀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⢤⡄⠀⡤⢤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⠂⠀⠀⣁⣀
⠀⠀⠀⢸⣇⠙⠐⠋⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠘⢳⣶⠂⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣷⣤⣾⠿⠄⠀⢀
⠀⠃⠀⣿⣻⣿⣿⣿⣏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇⠀⠀⠀⠈
⢀⠀⠀⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠁⠁⠁⠀⠘⣷
⡇⢀⢿⣵⣿⣿⣿⣿⣿⣬⡇⠀⣀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠇⠀⠀⠀⠀⠀⠀⢀⢇⣀⡄⠀⠀⠀⠀⠀⠀⣂⣀⠘⠀⠀⠀⠀⠀⠀⠀⠘⣀
⠅⢨⢸⣿⣿⣿⠛⣿⣿⣿⡇⡅⠉⠀⠀⠀⠀⠀⠀⠀⠉⢹⡆⠀⠀⠀⠀⠀⠀⠀⢬⠉⠁⠀⠀⠀⠀⠀⠀⠈⢉⡅⠀⠀⠀⠀⠀⠀⠀⢨⡍
⠈⠃⢸⣿⣿⣿⠀⣿⣿⣿⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢸⣿⣿⣿⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠈⠉⠉⠉⠈⠁
⠀⠀⢸⣿⣿⣿⢀⣿⣿⣿⡇
⠀⠀⠘⠛⠛⠛⡜⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠄⠠⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣂⣠⠀⣤⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠖⠚⠛⠉⠁⠉⠉⠉⠑⠒⠠⠄⣀⣀⣀⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠤⠶⠶⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣇⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⢹⣿⣄⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣄⡈⣿⣇⠐⣄⠀⡀⠀⠀⠈⠀⢀⠀⢀⠀⠀⢰⠃⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣝⠿⣿⣿⣦⡈⠳⡌⢄⠀⠀⠀⠀⡇⠘⡀⠀⢧⠤⠤⠿⠦⠄⠀⠀⠀⠀⢀⡀⠃⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣾⣿⣿⣿⣆⠸⠈⢠⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⠀⠀⠀⢀⣀⡤⠴⠒⠒⠉⠉⠉⠉⠉⠉⠉⠉⠙⠲⢆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⡴⠞⢋⠁⠀⠤⡄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⡔⢠⠿⠿⣷⣦⡀⠀⡀⠀⠀⠀⣠⣴⣶⡶⣦⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⣿⠐⠋⠀⠀⠀⠈⠙⠛⣤⠓⠀⠁⠾⠟⠁⣀⠀⠈⠀⠀⣿⠛⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⡙⢿⡿⢿⠃⠀⠀⢀⣄⠀⠀⣼⡆⠀⠘⠀⡄⣡⡈⢳⣄⠂⠀⠈⢴⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢁⣸⣧⡄⣩⠄⠀⢌⣉⠅⠈⣿⡁⠀⠀⠀⠤⠉⠉⠄⠈⠁⠀⢤⡀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⠀⠀⠀⠀⠀⠀⠀⠸⣧⢋⢻⣿⣿⣤⠰⠀⠀⣀⢀⡞⣿⠋⠀⠁⠀⠀⣀⣠⠀⠀⠀⠀⠻⠀⡸
⠀⠀⠀⠀⠀⠀⠀⡴⠳⠄⠞⠉⠁⣼⠆⢀⣀⡀⠀⠀⠀⠀⢻⠈⢸⣿⢿⣿⠖⢾⡿⢯⣌⣿⣿⠷⠆⠘⠈⢹⣿⣿⡇⠀⣆⠀⠈⠟⠁⠲⣄
⠀⠀⠀⠀⠀⡠⠊⠀⣸⠀⠀⡺⠉⢀⣀⣈⡨⠃⠀⠀⠀⠀⣸⣦⣼⡇⠈⡟⠀⠈⠧⠠⣬⣉⠻⠶⢰⣾⣛⣧⠄⠋⠁⠀⢸⠀⠀⢰⠀⠀⠘⢷⣤⣄
⠀⠀⠀⠀⢰⠁⠀⠞⠁⣠⡞⠀⢀⡘⠉⠀⠀⡔⠳⣀⣠⣾⣿⡿⢿⣇⠀⢿⠀⠀⠀⠀⠈⠻⣶⣤⣤⣴⠞⠁⠀⠀⠀⠀⡈⠀⠀⢸⠀⠀⠀⠀⠀⡟⢦
⣠⣄⡀⠀⡌⠀⢀⡄⠀⠁⢸⠀⣩⣅⣠⣤⣶⡀⠀⠙⡏⢸⣿⠁⠸⣿⣆⠘⢧⡀⠀⠀⠀⠳⣤⣉⣁⡅⠀⠀⠀⠀⠀⡰⠁⠀⢠⠎⠀⠀⠀⠀⠘
⢣⡀⠘⢾⡇⠉⠉⢱⣄⡠⠀⠉⠁⠸⢿⣿⡿⡿⠦⠀⣧⠸⣿⡆⠀⠻⣿⣄⠀⠙⢦⣀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣠⠞⠀⢀⡰⠋⠀⠀⠀⠀⠀⠀⠀⢰
⠀⢫⡀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⢄⡀⠟⠀⠻⢧⠀⠀⠈⠻⣶⣦⡀⠉⠳⢦⣀⠀⠀⠀⣀⠔⠋⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠈⢳⡄⠀⢙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠉⡆⢠⡀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠦⣤⣙⢛⡛⢛⣁⣠⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⣠⠃⢀⠎
⠀⠀⠀⢳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠾⠿⠷⠶⠦⡤⢀⠀⠈⢫⡉⣉⠅⠈⠀⠀⠀⠀⠀⢀⡀⠀⢀⠔⠁⡴⠃
⠀⠀⠀⠀⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠸⠦⣌⣀⣀⣉⢀⣀⣀⣤⣤⣴⠶⠞⢋⣠⠖⢃⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠈⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠈⠛⠿⠶⢶⣶⣤⣤⠶⠶⠖⣿⡟⠙⠻⠿⣯⣭⣭⣤⣴⡶⣛⣩⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⢠⣤⠀⠀⠀⠀⠀⠀⢼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⡰⢉⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣬⡘⠇⢷⡆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⢹⣞⢳⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠆⣎⣱⠟⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠸⣾⠟⠀⡆⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡍⡉⠀⠀⣼⡰⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⢰⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠙⢿⣁⠀⣀⠠⠛⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⢠⠇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠙⠫⡀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⡴⠃⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢠⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⣠⠞⡄⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠞⠁⠀⢰⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠈⣧⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⢀⣤⠶⣄⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠉⣠⣶⣿⣿⡿⠃⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠀⠀⢀⣴⣿⣿⣿⣿⠟⠀⠀⠀⡐⠄⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⣾⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢰⠈⢂⠀⠀⣠⣾⡟⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⣠⣾⣧⠈⠻⣿⣿⠟⠀⠀⠀⠀⠀⢰⡃⠀⠀⢣⣼⣿⡟⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⢀⣴⣿⣿⣿⣿⣧⡀⠘⠃⠀⠀⠀⠀⠀⠀⠀⣇⢀⣴⣿⣿⢏⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠗⠄⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⠟⠁⠈⠑⢄⡀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠈⠠⣤⣀⠀⠀⠀⣾⣿⣿⡿⣇⠀⠀⠀⠀⠀⠙⠲⣤⣀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠙⢿⣿⣷⣦⣜⢿⠟⠀⠈⠳⣄⠀⠀⠀⠀⠀⠈⠉⠛⢳⠀⠀⠀⠈⠳⣄
⠀⠀⢠⡞⠁⠀⠀⠀⠀⢸⣿⣀⣨⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣀⣴⣿⠿⣥⣿⣷⣦⣀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠻⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⣀⠴⠚⠋⠉⠙⠃⠈⠻⠿⠿⠿⠷⣦⣄⡀⠀⠈⠙⠒⠲⠤⠼⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⣠⠞⠋⠉⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠉⠻⣦⡀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⡾⠁⠀⠀⢀⣼⡇⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣷⣶⣦⣀⠀⠀⠀⢿⣦⡀⠀⠀⠈⣷⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣯⡀⣀⣴⣿⠟⠀⡤⢴⣾⣿⣿⣿⡿⡿⢿⢻⡿⣿⣿⣿⣷⣶⢦⡄⠙⠿⣶⣤⣴⣿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⠛⠋⠀⠀⠀⡇⢸⠈⡇⢿⢸⠀⡇⢸⠘⡇⡇⢸⠀⡇⢻⢸⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠃⠘⠀⠇⠠⠸⠀⠇⢸⠀⠇⡇⠸⠀⠇⠐⠘⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⣻⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣃⣤⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⣿⡿⢿⡿⠋⠛⠿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣁⠀⠀⠀⠀⠈⡱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠰⠖⠀⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣀⣠⣤⠴⣋⠜⣷⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⣻⡿⠋⢹⣁⣰⣊⣁⣾⡿⠁⠀⠉⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢿⡀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠻⡄⠀⠀⣾⠏⢙⣿⣿⣶⣶⣶⣿⣿⡇⠀⠀⠀⣄⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⢠⣇⢀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢠⠴⣿⠴⠞⣿⠀⢐⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠹⣦⣤⠖⠚⠛⠛⣾
⣾⠀⠀⠀⠀⠸⣏⠻⣆⠀⠀⠀⠀⣸⠃⠀⠀⣰⠋⠀⠀⠀⠀⣿⠀⠹⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢰⡟⢳⠀⠀⠀⠀⢸
⣿⡇⠀⢀⠀⡴⠿⠀⠸⣿⠀⠀⢰⠃⠀⢀⡜⠁⠀⠀⠀⠀⠀⣿⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣀⡼⠓⠉⠀⠀⠀⠀⢸
⢿⣿⡄⢸⣆⢃⠀⠀⢀⡏⢀⡴⠃⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠚⠁⠀⠀⠀⠀⠀⣠⡴⠋
⠸⡄⢻⣞⣸⣦⠑⢒⠋⠀⡏⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠲⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠹⣿⠙⠿⠹⣆⣾⠀⣀⢧⢠⣶⣠⠏⠀⠀⠀⠀⠀⠀⠀⣴⠃⣰⣾⣿⡿⠋⠁⠀⠀⠀⢀⣠⣴⣶⣏⠁
⢀⠀⢹⣆⠀⠀⢻⡟⣇⣿⣮⡏⠋⠁⣤⠀⠀⠀⠀⠀⠀⡼⠁⣸⣿⣿⡿⣶⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡀
⢸⣷⣨⣿⡄⠀⠈⠃⠘⠛⠘⠧⣤⡾⠋⢷⡀⠀⠀⢀⡞⣁⡿⠟⣋⣥⣴⣃⣜⣴⣤⣼⣿⣿⣿⣿⣿⣿⣇
⠀⣇⠙⠛⠃⠀⠀⣌⣧⣀⢠⡀⠀⠀⠀⢸⣿⣿⣷⣾⣛⣡⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢹⡄⠀⠀⠀⢸⣾⣿⠟⠿⣿⣆⡀⠀⣸⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⢹⣦⡀⠀⠘⢿⣧⣀⣀⣀⡿⢧⣐⣯⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠈⢿⠿⢻⣶⣬⣭⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠳⢤⡈⠻⢿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⣶⣶⡶⣶⣶⣶⣾⣷⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢤⠔⠋⠉⠉⠑⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠉⠑⢬⡗⠤⢄⣀⣤⡀⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠈⢹⢅⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠑⠮⢌⣐⡐⠀⠤⡜⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⢀⢖⠝⡒⢐⡺⢝⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠈⢔⢐⠈⠈⡞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀⠠⠁⠀⡰⡹⠀⠀⠀⠘⣄⡠⠧⠤⠔⠦⠤⠦⣀⣜⠀⠀⠀⠀⠠⡄⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⠷⢶⢯⡞⠁⠀⢘⠩⠉⠀⠶⠆⠀⠀⠀⠀⠶⠆⠈⠩⢱⠀⠀⠀⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣧⠀⠀⠀⠉⢦⡀⠀⠀⠀⠄⠄⠀⠀⠀⣀⠎⠁⠀⢀⡐⣩⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣗⣢⠑⡤⠤⠤⠤⠽⡖⠲⠦⠠⠤⠖⠒⠺⡓⠒⠊⠉⢉⣆⣹⡻⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠻⠿⠴⠧⠤⢀⡀⡼⠀⢀⢋⡁⠀⠀⠀⠀⢧⠔⠒⠊⠁⠀⢀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣆⠀⠀⠀⠀⢠⣇⠀⠘⠀⠀⠀⠀⠀⠀⣈⡄⠀⠀⠀⠀⡸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⠤⡎⡗⠻⠛⢫⠭⠩⠟⠛⡟⠺⢱⠒⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⣀⣀⣀⠀⠀⢀⣀⣀⣀⡠⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⢲⠟⠒⢲⠶⠖⠒⢲⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡖⠒⢺⢸⠒⠒⣺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣤⣬⣶⡾⠸⣷⣮⣤⣬⠇

⠀⠀⠀⠀⠀⠀⢀⠤⠔⠢⢤⣀⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡤⣤⢤⠀⡠⢠⣄⢀⣄⢀⣀⠀⢠⣤⡀⣠⠄⡤
⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠀⠁⠉⠉⠉⠉⠈⠁⠁⠈⠉⠀⠉⠉⠁
⠀⠀⢀⡴⠊⠁⠀⠀⢀⡤⣄⠀⠀⠀⠀⡼
⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠙⠒⠒⠊⢳
⠀⠀⠈⠓⠒⢷⠒⠂⠸⠤⠖⠉⠛⣄⢀⡮⣀⡀
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢀⡨⠋⠀⠀⠉⠓⢤⡀
⠀⠀⠀⢀⠎⠈⠙⠒⠒⠒⠒⠋⠁⢀⠀⠀⠀⠀⠀⠀⠘⢉⡆
⠀⠀⠀⡜⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠑⠄⠀⠐⠠⠀⣀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠃⠀⠀⠀⠀⠀⢳⠤⣀⣈⡠⢄⣀⠈⣀⣤⣀⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠈⡄⢀⠀⠀⠀⠀⠀⢧⠀⠤⠀⠀⠈⠉⠁⠐⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢐⠀⠀⠀⠀⡀⠠⠤⣌⠢⡀
⠀⠀⠀⡇⠠⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⠀⠠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠉⠉⠉⢨⠀⡠⡖⠁⠀⠀⠀⠀⠀⠈⡖⢄
⠀⠀⠀⡇⢠⠀⠀⡆⠀⠀⡇⠀⠀⢀⡀⢀⡠⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠳⡌⠉⠭⠭⠭⠭⠍⠋⠥⠎
⠀⠀⢴⠃⠠⠀⠀⠀⠀⢀⣇⣀⣀⠀⠈⣉⣠⣴⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⡀⠀⠀⠉⠳⣄⣠⣄⡀⢸⠉⠙⢏⠙⠛⢡⡟⢦⡀⠀⣀⠄⢢⢧⠄⠀⠀⠀⡀⠀⠇⠀⠀⠀⠀⣤⠁⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢘⡀
⡇⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠘⡏⠉⠀⠰⠚⠂⠀⠀⢴⠋⠂⠄⡀⠀⠀⠐⠂⠤⠔⠒⣁⠀⣇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠀⠶⣼⣄⣀⠤⢞⡥⠒⡂⠉⠁⠙⠁⠀⠊⠓⠲⠦⣅⠀⠀⠁⠀⠀⠀⠈⠀⠈⠗⠀⣤⠀⠀⣸⡀⠀⠀⠀⠀⠀⢸⠗
⠇⠀⠀⠀⠀⠀⠈⠂⡤⠏⠡⡀⠈⠑⠨⠑⣒⠀⠀⠀⠀⠒⠒⠂⢁⠅⠀⠀⢠⣀⣀⣀⣠⠀⠰⡀⠀⠇⠀⡀⠩⣷⣦⣤⣤⣤⣶⡧
⡆⠀⠀⠀⠀⠀⠀⡔⢸⠁⠀⠘⠢⠄⡀⠀⠀⠰⢁⠀⠀⠀⢀⡠⠂
⠁⠀⠀⠀⠀⠀⠀⣗⠃⣀⠀⠀⠀⠀⠈⠐⠒⠚⡳⣶⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⡬⠤⠭⢭⡭⠄⡾⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠙⢄⠀⢀⡾⠄⡀⢥
⠁⠀⠀⠀⠀⢀⠞⠁⠉⢦⠤⠉⣽⠂⠠⠐⠚⡦⡀
⠁⠀⠀⠀⠀⠘⡀⠀⠀⠘⢄⣀⠼⡄⠀⠉⠈⠀⠈⠆
⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⡳⠄⣀⢀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠤⠚⠁
⣶⡶⢶⢶⡶⠶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⣶⢶⣶⣶⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄
⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠲⡿⠿⠿⡷⢢⠀⡆⡆⠀⠀⠀⡹⠿⠿⠏⠀⠀⠀⠀⠀⠀⠈⡿⠿⠳⡆⠀⠀⠀⠀⠀⠀⠰⠟⠿⠿⠇⠀⠀⠀⠀⠀⠀⢲⠉⠈⡖⠂
⠀⠀⠀⠀⠀⢂⠀⡀⠀⠘⢄⣇⠇⠀⠀⠀⠑⡀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠡⠠⢄⠌
⠀⠀⠀⠀⢲⣈⠃⠈⣰⠂⠀⡇⠀⠀⢠⣤⣼⡟⢻⡧⣤⡀⠀⠀⠀⠠⠒⠹⡏⡟⢐⠢⠀⠀⠀⠀⡀⠸⡏⠛⢸⡗⠀⠀⠀⠀⢀⠀⠀⠉⠉⠀⠀⡀
⠀⠀⠀⠀⠘⠛⠛⠛⠛⠐⠀⠃⠀⠀⠘⠛⠛⠃⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠚⠛⠛⠛⠓⠀⠀⠀⠀⠐⠀⠀⠐

⠀⠀⠀⠀⠀⢠⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⣤⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠰⠆
⠀⣀⣠⣤⠀⡛⠛⠛⡃⠀⠀⠀⠀⠀⠀⠀⠘⣛⣛⡃⠀⠀⠀⠀⠀⠈⠹⠟⠛⠿⢩⠁⠀⠀⠀⠀⠀⢀⠏⠛⠻⡀⠀⠀⠀⠀⠀⢰⡶⠒⠒⢲⣆
⠘⡿⠟⢻⡐⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠐⡈⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠈⢁⣀⡘⠀⠀⠀⠀⠀⠆⠸⣷⡀⢀⣼⡏⠀⠒
⠀⠀⠀⢠⢳⡐⡆⣲⢰⢳⠀⠀⠀⠀⢠⣾⣿⣷⣿⣿⣷⡀⠀⠀⠀⣴⣦⡑⣷⣠⣶⣆⠀⠀⠀⠀⡀⣾⡅⣾⢀⣷⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠈⠈⠁⠁⠉⠈⠉⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠃⠀⠀⠘⠛⠛⠛⠛⠛⠋⠛⠀⠀⠀⠀⠓⠛⠛⠛⠛⠛⠒⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⢰⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⡱⣿⣷⡁⠀⠀⠀⠀⠀⠀⠀⢸⠛⠻⡇⠀⠀⠀⠀⠀⠀⠙⡿⣧⡿⡟⠀⠀⠀⠀⠀⠀⠈⣿⣷⣿⠃⠀⠀⠀⠀⠀⠀⢸⠂⠀⡇
⠀⠀⠀⠀⠠⣆⠀⢀⡦⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⢠⡦⠀⠀⠀⠀⠀⠀⠑⡀⢀⠋
⠀⠀⠀⢠⣴⣾⣴⣾⣶⣄⠀⠀⠀⠀⢀⠔⠐⢵⡮⠂⠀⠀⠀⠀⠀⠀⠴⠚⢵⠝⠲⠄⠀⠀⠀⠀⢠⣀⣌⣴⢡⣄⣄⠀⠀⠀⠀⠀⠘⠷⠾⠂⠢
⠀⠀⠀⠸⠿⠿⠻⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠸⠂⠈⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠶⠿⠀⠀⠀⠘⠘⠀⠀⠀⠀⠒⠃

⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⡾⠿⠿⡆⠀⠀⠀⠀⠀⠀⠀⣼⡿⠻⣦⠀⠀⠀⠀⠀⠀⠀⡯⠉⠹⡇⠀⠀⠀⠀⠀⠀⠀⡞⠿⠻⡄⠀⠀⠀⠀⠀⠀⢸⢿⣾⡇
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢶⡀⢀⣾
⠀⠀⠀⠀⠂⠈⠀⠉⠀⠀⠀⠀⠀⠀⢀⣴⡊⢵⡯⢰⣦⡀⠀⠀⠀⠀⢰⡎⢱⠉⣶⠄⠀⠀⠀⠀⣠⣶⣆⣾⢰⣶⣄⠀⠀⠀⠀⠀⠀⠁⠊⢀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘⠿⠿⠾⠷⠿⠿⠃⠀⠀⠀⠀⠸⠷⠾⠶⠿⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠟⠀⠀⠀⠐⠀⠀⠀⠆⠉⠀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣿⣏⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣭⣽⣭⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠄
⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⢀⣀⠀⠁⠢⢄⠀⠀⠀⠀⡀⠤⠤⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠄⠂⠁⠀⠀⠀⠂⡄⢀⠄⠒⠒⠒⠤⡀
⠀⠀⡠⠊⠀⠀⣤⡄⠀⠀⠘⢉⠁⠀⠀⠀⠑⡀⠀⠀⡇⠀⠀⠀⠀⠀⡴⠆⠀⠐⠛⠃⠀⠀⠀⠀⠰⠀⠀⠀⠀⠊⠀⠀⠀⠀⢠⢶⡄⠈⠁⢀⠀⠀⠀⠀⠀⠢
⠀⠰⠁⠀⠀⠀⠁⠄⠀⠀⠀⢁⡀⠀⠀⠀⠀⠈⡄⠀⢠⠀⠀⠀⠀⠀⡠⣤⠀⠀⣸⣯⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠝⠃⠀⠀⠀⠀⠀⡄
⠀⡆⠀⠀⠀⠀⠀⢀⣶⣄⠀⠹⠛⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠿⠿⠀⢛⠋⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⣦⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠇
⠀⡄⠀⠀⠀⠀⠀⠈⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠱⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠉⠃⠀⠀⠀⠀⠀⡠
⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠼⠀⠀⠀⠀⠀⠈⠣⢄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠒⢄⠀⠀⠀⠀⠀⠀⠆⠂⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⢀⣤⡀⠀⠀⠤⡄
⠀⠀⢀⠀⠀⢀⣀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠑⠀⡀⠀⠀⠀⠸⡀⠀⠀⠈⠁⠐⢀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⡄⠀⠀⡠⠄⠀⠀⠀⠀⠐⠄
⠀⠰⠉⠀⠀⠘⠀⠀⠀⠀⠈⢁⡀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⢸⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠸⠶⠀⠀⠳⠂⠀⠀⠀⠀⠀⠌
⠀⢰⠀⠀⠀⠀⠀⢀⣤⡀⠀⠻⠹⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢘⠄⠀⠀⢀⢤⠀⠀⠀⠐⠉⠀⠑⠤⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠠
⠀⢸⠀⠀⠀⠀⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⠄⠀⠀⠀⠀⠈⠒⠄⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠊
⠀⠀⠀⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠁⢀⣀
⠀⠀⠀⠀⠀⠀⠑⠤⠤⠀⠀⠀⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⢈⠉⠀⠀⠚⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⡠⠀⠈⠀⠀⣀⡀⠀⠀⠀⣀⠀⠀⠐⠤⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⣯⠀⢀⣲⡂⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⢉⡀⠀⠀⠈⡉⠁⠀⠀⠀⠈⠑⠀⠀⠨⡀⠀⠀⠀⠀⠀⠈⠉⠀⠌⠛
⠀⠀⠀⠀⠀⣠⠀⠀⠀⠐⠄⠀⠀⠀⠘⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣧⠀⠀⣬⡷⠀⠀⠀⠀⠀⠀⠀⡀⠀⠑⡀⠀⠀⠀⢀⡀⠀⠀⠀⠓⠀⠀⠀⠀⠀⢡
⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠂⠀⠀⠒⠆⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⡄⠀⠀⠀⠰⠦⣤⠄⠀⠀⠀⠀⢀⠔
⠀⠀⡰⠁⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠈⡄⠘⡄⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁
⠀⠎⠀⣀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠘⡀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⢀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠤⠤⢀⣀⡀⠀⢀⠀⠀⠈⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠒⠒⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠄⠠⠀⠠⠀⠀⠄⠄⠂⠄⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠠⠆⠂⠀⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠿⠟⠛⢻⣿⣆⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣥⣤⣶⣶⠾⠿⠛⣿⣟⣉⣉⣠⣬⠿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⠶⠿⠟⢻⣿⣏⣉⣡⣤⡤⠶⠖⠚⠛⠋⣉⣁⣠⣤⣴⣶⡿⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠉⣠⣤⠴⠶⠖⠛⠛⢋⣉⣀⣤⣤⣶⣶⠾⠿⠛⠛⠋⠉⠉⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⢠⡞⠁⢀⣤⣶⡶⠿⠟⠛⠛⠉⠉⠁⣀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣾⠀⢠⡿⠋⠀⠀⠀⠀⠀⠀⢀⣴⠞⢉⣠⣤⡀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⣾⡇⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⢸⠁⢸⡇⢀⣀⠈⣷⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⢀⡟⢠⣤⠴⠾⠀⠘⠛⠉⢹⠀⣸⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⢸⣇⣤⣤⠀⢰⡖⠛⠋⢀⡟⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠘⣧⠈⠁⠀⢸⠀⣸⡇⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠘⠉⣉⣠⠾⠋⠀⠀⠀⠀⠀⠀⢀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣿⠀⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉⠀⢀⣀⣀⣤⣤⣶⣶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣿⣇⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⡶⠿⠟⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⡶⠾⠿⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠛⠛⠛⠲⠶⣤⣄⣀⡀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⣀⢄⠀⠀⠀⠑⠤⣄⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠃⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠈⠛⠶⠤⣤⡤⠴⠶⠤⢄⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢦⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠉⠉⠉⠉⠉⠉⠛⠓⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⡀⠀⣠⠖⠁⣀⠘⢛⣉⡤⠇⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⢠⠀⡠⠂⣀⡴⠟
⠀⠀⢀⠀⢹⣰⠃⢀⡖⢁⣠⠞⠁⠀⣀⠀⠀⠀⠀⠀⠈⣡⣴⣶⡤⠀⠀⠈⠛⠛⠋⠙⠻⣆⠀⠀⣿⣷⣄⠀⠀⢠⡿⠭⠶⠚⠁
⠀⠀⢸⠀⢸⡇⠀⢺⠀⢸⣷⠉⠉⠉⠉⠙⠶⡄⠀⠀⢰⡏⣉⣡⣄⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⢻⣿⣿⣧⣀⡞
⠀⠀⢼⠀⠈⢧⠀⠘⠀⠛⢁⡀⢐⠏⢡⠟⠁⣸⠂⢀⠜⠃⠀⠄⢈⠆⠀⠀⠀⠀⢛⠛⣇⠀⠀⠀⠈⣿⡟⣫⠟⢧
⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡟⠀⣸⠁⠀⡟⠀⠀⠀⡔⠁⠀⠀⣀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠛⢈⡿⢳⣼
⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠘⢾⡇⠀⣏⠀⠀⠇⠀⣀⡠⠳⣤⡤⢤⡽⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠺⣶⡿⠋
⠀⠀⠀⠀⣧⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢀⠠⢿⣷⣴⡶⠠⣶⣦⣀⣯⡹⡇⠀⣀⠔⠀⠀⢰⠾⡏
⠀⠀⠀⠀⡿⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⣰⠏⠀⠀⠸⡄⠀⠀⠉⠒⠶⠾⠟⠛⠋⠀⠀⠁⡇⠀⠀⠀⡼⢰⣇
⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⡧⠀⠀⠀⢰⠇⠀⠀⠀⠀⠙⡆⠀⠀⠉⠙⠒⠀⠀⠀⠀⠀⠈⠀⠀⠀⠜⠁⢸⣿⡆
⠀⢠⡖⢲⡏⠁⠀⠀⠀⠀⠀⠀⣀⣴⠏⠀⠀⠀⠀⠀⠀⣿⡸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡼⠀⣧
⠀⣸⠀⠙⠶⢤⣤⣀⠀⠀⢠⢾⣭⠀⠀⠀⠀⠀⠀⠀⢸⡏⢻⣌⠛⠶⠶⠶⠶⠂⠀⠀⠀⠀⣀⡼⠋⠀⠀⠀⣸⠇
⢠⡏⠀⠀⠀⠀⠀⠉⠙⠛⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⣇⡆⠙⣷⣄⠀⠀⠀⠀⠀⣀⣤⠞⠋⠀⠀⠀⠀⣴⠏
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠿⠈⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⡽⠟⠛⣩⠟⠛⢂⠶⠾⢯⣥⡤⠴⠶⠀⠀⢀⡾⠃
⠈⠻⢦⣄⠀⠀⠀⠀⠀⢀⣀⣼⠇⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⢠⡏⠀⠀⢸⣶⠆⠀⠈⠛⢦⡀⣀⡴⠋
⠀⠀⠀⠈⠉⠙⠛⠛⠛⠉⠉⠀⠀⠀⠀⢀⡾⠋⢀⡤⠖⠛⠟⢛⡇⣀⣀⣾⢦⡀⠀⠀⠀⠀⢻⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣰⣥⠴⠶⠚⠋⠉⠀⠀⠁⠉⠳⣝⠦⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠙⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⣴⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣶⣶⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠈⠐⠐⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸⡀⠀⢀⣷⠀⠀⠀⠀⠀⠀⣊⠛⠛⠉⠂⠀⠀⠀⠀⠀⠐⠁⠂⠂⠑⠀⠀⠀⠀⠀⠀⢀⠋⠛⠛⠆
⠀⠀⠀⢀⢁⣀⠴⣦⡷⢄⡀⠀⠀⠀⠀⠀⡀⢼⣶⣿⢇⡀⠀⠀⠀⠔⣱⡦⣶⣴⡦⣄⠀⠀⠀⠀⠀⠀⠐⣲⡆⠀⠄⠀⠀⠀⠀⠀⠀⠀⡄⡄
⠀⠀⠀⢸⢸⣿⠀⢸⠛⢻⡇⠀⠀⠀⠀⢀⣧⠐⡿⠋⣼⡃⠀⠀⠀⠈⠉⢟⠻⡿⠛⣿⠀⠀⠀⠀⠀⠴⠟⣻⡟⠳⠤⠀⠀⠀⠀⠀⠁⠀⢿⠁
⠀⠀⠀⠈⠉⠀⡆⣸⠀⣼⡇⠀⠀⠀⠀⠭⢭⠽⠀⠅⣿⡇⠀⠀⠀⠀⠀⢸⣀⣿⢈⣿⠀⠀⠀⠀⠀⣶⣼⣿⣿⣴⡆⠀⠀⠀⠀⠀⠈⠀⠈⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠶⠋⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠁

⠀⠀⢠⣶⡄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠻⡃⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠓⠒⠚⠀⠀⠀⠀⠀⠀⠀⡴⠟⠻⢧⠀⠀⠀⠀⠀⠀⠠⡾⠿⠟⠛⠄⠀⠀⠀⠀⠀⠀⡼⠟⠛⠿
⠀⠀⠀⠀⠃⠀⠀⣤⡤⠀⠀⠀⠀⠀⠀⠀⢀⣦⣠⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣤⡀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠁⠁⠀⢸⡅⠀⠀⠀⠀⠀⠀⣼⣿⡉⣿⣿⣿⡆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢹⣀⣀⡏
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠘⠟⢻⣿⠟⠛⢉⡃⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠎⠀⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠉⠈⠃⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁⠂


⠀⠀⠀⠀⠀⠀⠀⡴⠞⠶⡄⠀⠀⠀⠀⠀⠀⠶⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠐⢒⣒⡄⠀⠀⠀⠀⠀⠀⣄⠠⠄⢤⠀⠀⠀⠀⠀⠀⠀⡰⠶⠲⢄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣀⣠⡆⠀⠀⠀⠀⠀⠀⠘⣄⡈⣀⡄⠀⠀⠀⠀⠀⢠⢱⢤⡠⢬⠃⠀⠀⠀⠀⠘⠀⠀⡀⢀⠈
⠀⠀⠀⢀⠀⠀⠀⠒⣷⠇⠐⠀⠀⠀⠀⠀⣸⡇⠉⠀⢽⠄⠀⠀⠀⠀⠀⠠⡚⠴⠌⡒⠀⠀⠀⠀⠀⠙⠁⠐⢁⠈⠉⠀⠀⠀⠀⠄⢈⠀⢲⡖
⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠐⡬⢹⢀⣀⠾⡺⠀⠀⠀⠀⠀⢰⠑⣦⣀⡡⠀⠀⠀⠀⠀⢠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠃
⠀⠀⠀⠘⠖⠒⠒⠒⠳⠒⠀⠀⠀⠀⠠⠾⠆⠼⠶⠧⠰⠷⠄⠀⠀⠀⠐⠁⠴⠉⠥⠤⠀⠀⠀⠀⠶⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣴⣤⣶⢶⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠗⠓⠛⠿⠿⠟⠋⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠈⠃⢠⣄⡀
⠀⠀⠀⠀⠀⠠⠞⠁⠀⢀⢅⠀⣀⣀⣀⣙⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠓⣠⣴⡶⠞⠛⠃⠄
⠀⠀⠀⠀⠀⠀⣾⠻⠻⣿⠛⠛⢻⡾⣿⠏⢣⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠒⠒⠛⠫⢹⠷⠀⡀
⠀⠀⠀⠀⠀⠀⠹⣎⢁⢸⡆⠀⠀⠉⠡⣤⡜⠀⠀⠀⠀⠀⠀⠀⢀⣶⢸⠂⠀⠀⡀⢀⢀⡀⣾⡀⠀⣀⡤⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⣤⣤⣤⡶⢷⣟⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⠆⡾⢹⡇⠿⣿⢰⡿⡼⢰⡟⠊⠘⣿⡁
⠀⠀⠀⠀⠀⠀⢀⠂⠀⠾⢿⣯⣫⡽⢳⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾⡇⠍⠀⠓⠙⣁⣀⠋⢈⠍⠀⠈⠁⠀⠀⠈
⠀⠀⠀⠀⠀⢐⡏⠀⠀⠀⠘⣿⣿⣿⣿⣿⡋⠀⡀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⢷⡀⢰⡿⡈⢠⡶⡄⢰⠆⡄
⠀⠀⢚⢀⣴⣿⣇⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⣽⢿⡇⢾⣧⠇⣿⠔⠁
⠀⠀⢸⣿⣿⣿⣿⠀⠂⠠⠀⣼⣿⣿⣿⣿⣿⣿⡷⠀⠀⣤⠁⠀⠀⠀⠀⠀⠠⣋⡴⠏⠛⠈⠀⠀⠀⢸⡇
⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠲⠷⠿⣿⣿⣿⣿⣿⠇⠀⠸⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁