Princess Peach Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡄⠀⣀⡀⠀⢀⣠⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⠇⢀⣿⣗⠀⣼⣝⣿⡀⢰⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣽⣿⣟⣧⣿⣿⣿⣧⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⣿⣯⣠⣬⣽⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠉⠒⠚⢿⣿⣿⣿⠿⠿⠟⠻⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠀⠀⠁⠀⠀⠀⠉⠛⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢤⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠸⡿⠞⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣀⠀⠀⠀⣤⣴⣿⣿⣿⡿⠀⠀⠀⢰⡀⢀⢠⣷⣿⣿⣿⣿⣿⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣴⣦⣄⣘⣛⣢⣼⣿⣿⣿⣿⠟⠋⠀⢒⣆⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⡿⢿⣧⡀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠿⣿⡿⣿⠹⣿⡟⠁⠙⠛⠛⠋⠀⠿⠟⠛⢿⣿⣿⢛⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣷⡇⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⡇⢠⠸⡏⠀⠀⠀⠀⠀⠀⠀⠳⢿⣿⠿⣆⠀⣿⣿⠃⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠘⣄⠀⠀⠀⠀⠀⠓⠦⠐⠒⠲⠶⣧⣿⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢻⣷⣷⡄⠀⠀⠀⠀⠙⠿⠿⡿⢋⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣿⣿⣇⠈⠛⠀⢀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢀⣴⣿⡄
⠀⠀⠀⢰⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⢸⣿⣿⣿⣿⡀⠀⠀⠈⠻⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣤
⠀⠀⠀⢸⣿⣿⣀⡀⠀⠶⠈⣉⠀⣤⣥⣿⣿⣿⡇⢻⠖⢦⡀⢀⣠⢬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⡿⠏⠁⠈⠀⠀⢱⠋⠀⠀⠈⠛⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠰⠿⠿⠿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⡢⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣴⠞⢛⣳⣦⣄⠀⠀⠉⠙⠛⠻⠿⠿⢯⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠘⠂⠀⠀⠀⠀⠀⠀⠀⢘⠀⢰⣿⣿⣿⣿⢆⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⣿⣿⣿⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⢳⡀⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠳⡘⢿⣿⣿⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⣶⣄⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⡖⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠙⠦⠤⠦⠀⠀⠀⠈⠁⠚⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⢹⡧⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⣁⢀⠄⠀⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⣋⣀⣤⣈⡍

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⢄⠀⠀⠀⠀⢀⡴⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠢⠄⠔⠋⡠⣤⡈⠣⣀⠤⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢡⠄⠀⠀⠀⠀⠀⡅⠈⢳⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠘⠂⠀⠀⠀⠠⢀⠑⣀⠟⠀⠀⡀⠾⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠂⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠄⠀⠀⠀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠈⢡
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢤⣃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠙⢆⡀⠀⠀⠀⠀⠀⠀⣠⠞⢧⠀⠀⠀⠀⠀⠀⠈⠉⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⡁⠐⡈⠀⠀⠙⠢⣄⠀⢀⡤⠚⠁⠠⠈⢣⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠒⠀⠀⠀⠀⡇⠀⠈⡇⠀⢄⡽⠋⠉⣳⡄⠀⠀⠀⠉⠁⢀⡤⢤⡀⠀⡞⠁⠀⣀⡠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠸⠀⠀⡇⠀⠀⡇⠀⡼⠀⠀⢰⣿⣿⠀⠀⠀⠀⢀⠋⣰⣿⣟⢱⠀⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠣⠀⡇⠀⠀⢠⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⡆⠀⠀⠹⠿⣟⡇
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠰⠀⠀⢠⠒⠉⢡⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠀⠀⠀⠀⢹⠁⠀⠀⠀⠆
⠀⠀⠀⠀⠀⡌⠀⠱⢄⡀⠀⡠⠀⠀⡎⠀⠀⠈⡄⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡆⠀⠀⢠⠎
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠙⠦⣀⢀⣸⡄⠰⢦⣌⡄⠀⠀⠀⢄⠀⠀⠀⣀⠀⠀⠀⡠⡁⠀⢠⠋⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠙⠦⡀⠀⠀⠀⢌⣀⡨⠀⢀⡠⠶⠁⠀⠀⠘⠀⠀⠀⠀⣀⢤⠈⡑⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢸⠀⠀⠀⠈⠑⠢⠄⣀⡀⠤⠒⠉⠀⠀⠀⠀⠀⠈⠐⠂⠒⠚⡀⠺⡄⠐⠈⡀⠋⡄
⠀⠀⠀⠀⠀⠀⡐⠁⠀⢀⠤⠒⠊⠉⠉⠁⠐⠤⣀⠙⢼⡔⠂⠠⢀⠄⠠⠀⡼⠂⠰⠀⠀⠂⠀⠐⠒⠀⠀⡀⠀⠀⠀⢱⡀⠡⠀⠀⠡⠀⡇
⠀⠀⠀⠀⢀⠞⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⠀⠁⠀⠀⠓⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⢠⠌⣁⠀⠁⠀⠀⠀⢱
⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠉⠁⠐⠒⠒⠒⠂⠉⠁⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠈⢢⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠘⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⡜⠛⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⡇⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⣄⠀⠀⠀⡘
⠀⠀⠀⡇⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠣⠀⠀⠀⣳⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⠐⣄⠀⠀⠊⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠄⠀⠁⠀⠀⠀⢃⠀⢀⠌⠑⣆⠔⠁⠀⢸
⠀⠀⠀⠀⢄⠀⠉⠉⠃⠀⠀⠁⠂⢰⠆⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠄⠁⠀⠀⠘⣆⢀⡀⠇⠀⠀⠀⡄
⠀⠀⠀⠀⡜⠦⣀⠠⠀⠀⠀⠀⠀⡞⠀⠀⠐⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠜⠀⡿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢱⠀⠈⠱⡄⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣣⠼⠖⠒⠒⠒⠊⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠇⠀⠀⠈⠂⠀⠉⠉⠉⠁⠁⠀⢀⠀⠬⠗⠒⠒⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠈⠓⠦⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⡰⠃⠈⠙⠢⡀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡄⢀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣄⡀⣠⠟⠁⠈⢷⡄⢀⡴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⠿⠿⠿⠿⢿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡷⡈⠙⠁⢜⠭⢝⡄⠙⠋⣠⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣠⡤⢤⠤⢠⣠⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣧⣿⣉⣉⣟⠤⢜⣟⣉⣩⣯⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⣤⣤⣤⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠉⠀⠀⠀⠉⠙⠿⠟⠉⠁⠀⠀⠈⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠶⠟⠂⠀⠀⠀⠀⠀⣼⢢⣄⠀⠀⠀⠀⠀⢀⣠⠾⡄⠀⠀⠀⠀⠀⠚⠓⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⢀⡞⢁⣀⡈⠓⠦⣤⣤⠶⠋⣀⣀⠙⢦⡀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣀⠀⠀⠐⣏⢳⡏⢉⡽⣆⠀⠀⠀⠀⡾⣍⠉⢷⡋⡿⠀⠀⠀⣠⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢹⡇⠀⣀⠀⢻⡸⠇⠸⡿⣿⠀⠀⠀⢸⡿⣾⠀⠸⢰⠃⢀⡀⠀⣿⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢷⠰⡟⣧⠀⠳⡄⠀⠉⠁⠀⠀⠀⠀⠈⠁⠀⣰⠃⢀⡞⡿⢰⠏⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠘⡇⣹⠞⠳⡤⣽⢦⣀⠀⣀⠈⢀⡀⠀⣠⢞⡥⡴⠛⢾⡁⡟⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣤⣤⡇⢻⣀⣠⢷⢇⢻⢹⣀⠈⠭⠉⠀⣸⠹⠇⢧⢧⣀⣰⠇⢧⣤⣤⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢸⡄⣠⠞⠋⠉⠙⠦⣄⠀⠀⠉⠙⢶⠞⠋⠁⠀⢀⡤⠞⠉⠉⠛⢦⡀⣼⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢹⡏⠀⠀⠀⠀⠀⠈⠙⢶⡄⠀⣸⡄⠀⣤⠞⠉⠀⠀⠀⠀⠀⠈⢿⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢸⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⢀⣟⣇⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⢸⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⠀⢀⡿⠶⢶⣚⡾⠁⠀⣸⡧⢿⡀⠀⠹⣞⣳⠶⠾⣇⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⣠⠷⠶⠋⡽⠁⠀⢀⣿⠀⢸⣇⠀⠀⠹⡏⠳⠶⢧⣄⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠥⢤⠀⠸⠁⠀⠀⣼⡛⣤⡼⣹⡄⠀⠀⠹⠀⢠⡤⠽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⣰⣇⣙⣒⣚⣁⣿⡀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡼⠋⠈⠉⠉⠉⠉⠀⠻⣦⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⣈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠆⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠞⢷⣄⠀⠀⠀⠀⣀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⣀⣴⢋⠤⢄⠈⠳⣄⣴⠞⢉⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⡀⠙⢃⣧⠊⢳⢡⠀⠈⣁⣠⣏⡊⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⣏⣉⡉⠻⡢⢞⣼⠾⠛⠋⠉⠉⠛⠛⠷⢦⣄⡀
⠀⠀⠀⠀⠀⢀⣴⠟⠛⠉⠉⠉⠙⠛⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⣤⣠⣤⣤⣤
⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣽⠏⠀⢀⡀
⠀⣴⣄⣀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠛⣿⡇
⣄⢹⡍⠙⠃⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢀⣴⠋⠨⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⣿⠛⠿⠦⠀⠀⠀⠀⠀⢀⡾⠽⢶⣄⡀⢀⣴⠟⠁⠀⢀⣠⣌⡙⡶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢻⣦
⢹⣆⠀⠀⠀⠀⠀⠀⢀⡾⢁⣤⣀⠉⠛⠋⠀⠀⠀⣰⠋⢁⡈⢹⡷⠂⣿⠀⠀⢀⠀⠀⠀⠴⣿⠋⠀⠀⠹⣧
⠀⢻⣦⣀⠀⠀⠀⠠⢿⣝⡿⠀⢈⢷⡀⠀⠀⠀⠀⠇⢰⢡⡌⡀⠏⢀⡏⠀⠀⣾⠙⣧⠀⠀⠘⢷⡀⠀⠀⢹⣧
⠀⣼⠏⠉⢻⣟⠁⠀⠈⣷⠿⠀⠃⣦⠳⠀⡄⠀⠀⠀⠀⠮⠵⠀⠀⣸⠃⠀⢠⡏⢠⡏⠀⠀⠀⠘⣷⠀⠀⠀⢿
⢠⣿⠀⠀⠀⢿⡄⠀⠀⠘⣇⠀⠘⠒⠁⠀⡁⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣼⠷⢿⡀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⢸⡟⠀⠀⣶⡦⠿⠦⠀⠀⢹⡄⠀⠀⠀⠀⠉⣀⠀⠀⠀⠀⠀⢠⣿⣤⠀⣼⠃⠀⠈⣷⠀⠀⠀⠀⣿⠀⠀⠀⠈
⢸⡇⠀⠀⠈⠳⣦⡄⢀⡾⠋⢻⣄⠀⠀⠀⠀⣄⣲⠆⠀⠀⠀⠀⣴⣯⠾⣇⣀⣀⣴⠏⠀⠀⠀⠀⠹⣦⣤⣾⣧
⠸⣿⠀⠀⠀⠀⣿⠀⠸⣇⠀⠀⠙⢳⣤⣀⠀⠀⠤⠂⠀⣀⣴⢿⡏⠁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡄⠀⠀⠀⣿⠀⠀⠙⠳⠶⠶⠋⠀⠉⠛⠶⣶⣶⣞⠋⣠⣼⣧⡴⢶⣦⡶⠟⠛⠛⠛⠷⣦⣄⢤⣤⣴⠟⣽
⠀⠸⣷⠀⠀⠀⢿⡄⠀⠀⣠⣤⣤⣖⣻⠶⢾⣏⡻⠏⠉⠻⠁⠀⠉⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠙⣷⡿
⠀⠀⢹⣧⠀⠀⠈⢿⣶⠟⠋⢀⡾⣽⣟⣛⠒⠈⢻⣤⣤⣤⣤⣤⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣠⡿⠁
⠀⠀⠀⠹⣧⡀⠀⣾⠏⠀⠀⣿⠞⠓⠂⠀⠀⠀⠀⣿⠦⢤⡀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠁
⠀⠀⠀⠀⠘⢿⣴⣿⠀⠀⠘⡷⢚⣳⡆⠀⠀⠀⠀⢹⠛⢦⢳⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠙⢿⣄⡀⠀⣿⠀⠙⣧⠀⠀⠀⠀⢸⡇⢸⢸⠀⢿⣆⣰⡄⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣼⣇⠀⠹⣄⠀⠀⠀⢸⡧⢃⡞⣠⣤⣽⡿⠛⠶⣤⣤⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣶⣽⣷⣀⣀⡈⢳⣏⠀⣸⣇⣈⣿⣴⡶⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠖⢤⡀⠀⣠⢴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠲⠃⠀⠀⠉⠋⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣷⣀⡀⢀⡰⠄⠒⠛⠘⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⣼⠴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠋⠀⠀⢀⠀⠀⠀⠀⢀⠐⠀⠢⡀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠌⢀⡀⠐⠊⠀⡴⠒⠦⠈⠀⠀⡀⠀⠀⢔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡄⢠⡄⠁⠐⠀⠀⠈⠀⢀⠀⢁⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠐⠐⡀⠀⠀⠀⠀⠀⠀⢠⠎⠀⡜⠒⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢤⡼⠂⢀⠛⠤⠀⠀⠀⠀⢠⢿⣦⡞⠀⡀⠂⠀⠈⠲⠴⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢀⡜⠂⠀⠹⣇⠀⠘⠃⠉⣠⠟⠈⠀⠑⢦⡀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡎⠀⠀⡠⢊⠴⠳⡀⠀⠈⠃⠀⠀⠀⠀⠀⠱⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠃⣄⢰⠀⡏⠀⠀⠸⠀⢀⡀⠀⠀⠀⠀⡀⡰⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢗⠓⠼⡄⠑⠀⠀⠁⠀⠀⠑⢤⢴⠒⢂⠹⢤⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⣷⡀⠀⠀⠀⠀⢀⡠⢆⡊⢄⠉⠀⠆⠀⠘⢧⠴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⠼⠁⠀⠀⠀⠀⠉⠀⠀⠀⠈⢆⡀⠀⠘⡀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⢡⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠁⠀⠀⠀⣀⡤⠊⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠐⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⡁⡘⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠈⠳⢦⣄⣀⠀⠀⠀⢀⡇⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⡞⣁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⡄⠦⡀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⡀⠓⢄
⠀⠀⠀⠀⣞⠀⠀⠈⠀⠐⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠁⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠘⠲⢤⣀⣀⢀⡐⠀⠀⠁⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠴⠋⠉⠛⠒⠒⠒⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡴⠳⣤⡀⠀⠀⠀⣠⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢦⣠⢏⡤⠤⡈⠻⢦⣴⠟⢁⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡜⣇⢀⣉⣼⡎⠙⢼⠤⢤⣤⣤⣧⣋⣻
⠀⠀⠀⠀⠀⠀⠀⣀⣧⣿⣤⣀⣘⠗⣺⡼⠟⠋⠉⠉⠉⠉⠙⠛⠶⣤⣀⡀
⠀⠀⠀⠀⢀⣴⠟⠋⠉⠀⠀⠈⠉⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⣦⣤⣶⣶⠆
⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣏⣀⣠⡄
⠀⣿⠶⣼⠇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣹⡇
⣶⣽⣆⠈⠀⠀⠀⠀⠀⢈⣷⣄⠀⠀⠀⠀⣠⠞⠁⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡀
⣿⠈⠉⠁⠀⠀⠀⠀⢀⡼⠋⠉⠳⢦⣴⠞⠁⠀⢀⡴⠶⠷⣮⢛⠓⡦⠀⠀⠀⠀⠀⠀⣠⣴⠟⠉⢿⣄
⠘⣧⡀⠀⠀⠀⠀⢠⡾⢥⡞⠓⢦⡀⠀⠀⠀⠀⡞⢀⠔⠢⢸⡯⢠⡇⠀⠀⡷⣤⠀⠈⠻⣦⡀⠀⠈⢿⡆
⠀⣼⡟⠶⣤⡤⠀⠈⢻⣺⡄⢠⢋⢳⠀⠀⠀⠀⠁⠘⡘⢃⠈⠀⢸⠃⠀⢸⡇⢸⠇⠀⠀⠘⣷⠀⠀⠈⣿
⢀⣿⠀⠀⠸⣧⠀⠀⠈⣧⠁⠈⢬⠅⠀⠘⠀⠀⠀⠀⠉⠁⠀⢠⡏⠀⠀⣾⣤⡟⠀⠀⠀⠀⢻⡀⠀⠀⢸
⢸⡟⠀⠀⣤⣙⣧⡀⠀⠸⣆⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⢀⣾⡀⠀⣼⠃⠈⢻⡄⠀⠀⠀⢸⡇⠀⠀⠘
⢸⡇⠀⠀⠻⣯⣉⡁⢀⡴⠻⣆⠀⠀⠀⠀⣈⠠⡄⠀⠀⠀⠘⣹⣇⣾⡃⠀⢀⣼⠃⠀⠀⠀⠸⣇⣀⣠⣄
⢸⣧⠀⠀⠀⢀⡟⠀⢾⡀⠀⠙⢷⣄⡀⠀⠈⣉⠄⠀⠀⣠⣾⡛⠋⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠈⠉⢩⣿
⠈⣿⡀⠀⠀⢸⡇⠀⠈⠳⢤⣤⠾⠋⠙⠳⠦⣤⣤⠶⠛⣁⣸⣧⡤⣤⣤⡴⠶⠟⠳⠶⣦⣀⢀⣀⣠⡾⣻
⠀⢹⣧⠀⠀⠘⣷⠀⠀⢀⣠⣤⣤⢖⡶⢴⣟⠻⠟⠙⠾⠉⠈⠋⣴⠟⠁⠀⠀⠀⠀⠀⠀⠙⢷⡌⠙⣷⣿
⠀⠀⢻⣆⠀⠀⠙⣷⣴⠟⠉⣰⣯⣟⣓⠲⠈⠻⣄⣀⣀⣠⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣿⠃
⠀⠀⠀⠻⣦⠀⢠⡿⠁⠀⢸⡷⠛⠒⠀⠀⠀⠀⣿⠤⢤⡀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃
⠀⠀⠀⠀⠙⢷⣼⡇⠀⠀⢻⠖⣛⣷⠀⠀⠀⠀⢸⡛⢦⢳⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠈⠻⢷⣄⠀⢸⡆⠈⢻⡀⠀⠀⠀⢸⡇⣸⢸⠈⢿⣄⣲⡀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣷⡀⠘⢧⡀⠀⠀⢸⡷⢣⠏⣴⠦⣽⡟⠙⠲⣶⣶⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢾⣿⣦⣤⣀⣹⣧⣀⣼⣦⣬⣿⠾⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡀⣴⣶⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣜⠛⢱⣮⠋⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠻⠄⣈⠋⠀⠋⠙⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⡐⢮⡙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⠀⠙⢿⣷⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠻⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠞⠁⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄⠀⠤⣬⣭⣭⣾
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡦⠂⠀⢀⣴⢣⢀⣟⣄⠀⠀⠀⠀⠀⠀⡰⢋⣙⣷⣝⣷⣶⣶⡶⠚⠁
⠀⠀⠀⠀⠀⠀⠓⠋⠅⣤⠤⢶⣿⣡⣾⣋⣤⣬⡑⠄⠀⠀⠀⣸⣿⣟⣿⣿⡿⣻⣿⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣴⡿⡿⠋⠘⣿⢹⣧⡽⠆⠀⠀⠠⠛⢿⢛⣏⣈⠇⣾⣿⣞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⡿⣤⡀⠰⡀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⣸⢰⡟⠻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⣷⠟⢦⡄⠀⠀⠀⠀⠀⢴⠒⠀⠀⠠⠃⡌⠀⢠⡟⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡀⠀⠀⠀⢴⣟⣻⣿⠀⢠⣾⣿⣿⣧⣠⡀⠙⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢐⡿⣷⣦⣄⡀⠀⠂⢀⣴⣿⣿⣿⣿⡿⢿⣿⣷⣭⣙⢇⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡡⠀⠀⠀⣰⠃⢠⠃⢻⣿⡿⠻⣿⣿⣯⢻⣿⢸⡿⣹⢿⣦⣛⢿⣿⣿⣏⠉
⠀⠀⠀⠀⠀⠀⣀⣥⣾⠟⠀⠀⢀⣼⠇⠀⠀⠀⣼⣿⠂⠀⠈⠙⠁⠌⠃⠀⢠⡟⠋⠉⠉⠻⣾⡿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⢀⡴⠫⠟⠀⠈⠀⠀⠀⠀⠀⠀⡀⢠⡔⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⣷⣯⡀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⠈⣧
⠀⠀⠀⠀⠀⠀⠀⡠⠋⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠑⣿⡄⠀⠀⣀⣿⣿⡧⠻⡆⢸
⠀⠀⠀⠀⠀⠀⡐⠀⣰⣿⡇⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠳⣦⣄⣴⢂⣵⡄⠀⠀⠘⢿⡟⢛⠋⢭⣭⣿⡶⢿⢸
⠀⠀⠀⠀⠀⢠⠁⠰⠁⣿⣷⣀⣀⣀⣤⣧⡴⠶⢶⣶⣤⣤⣹⣿⢷⣿⠛⠛⡄⠀⠀⠀⠙⢾⣿⠶⣿⣿⠂⢸⣿⡀
⠀⠀⠀⠀⠀⠹⠀⠀⠀⢿⢟⣯⣟⣩⡭⠶⠚⠛⠉⠉⠉⠉⣿⠃⠂⣽⡄⠀⠀⡀⠀⠀⠀⠀⠙⢿⣻⣿⡇⠈⡇⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠃⢶⣵⣿⣿⠁⢾⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⡴⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢻⣿⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢈⣙⣿⣯⣄⣀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠸⣧⢸
⠀⠀⠀⠀⠀⠀⠠⠀⣿⠁⢸⡿⣇⠀⢳⣄⠀⠀⠀⠀⠀⠠⠿⠿⠿⠿⠿⠿⣷⣶⣶⣷⣄⠀⠀⠀⠀⢀⡿⣿
⠀⠀⠀⠀⠀⠀⣠⠞⠁⣧⣼⠇⣿⡖⠉⢻⣿⣿⣶⣿⠛⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣷⣦⣀⡀⣸⣶⣬⡁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡯⠞⢼⠁⠀⠘⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣟⣟⢻⣿⡿⠿⠦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠖
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⢼⡧⣾⣾⣵⣦⣇⡴⠚⠇⣀⠘⢂
⠀⠀⠀⠀⠀⠀⠀⢠⣛⠟⠁⢈⢠⣟⡋⢸⣏⣥⣤⢐⣿⡽⠃⠰⢀⠔⡆
⠀⠀⠀⣦⠀⠀⣴⣿⣇⣼⣦⣿⣷⠟⣿⠾⢫⡿⠳⠟⠉⠀⣴⣿⠊⠀⣷⠀⢀⢀
⠀⠀⠀⣆⠑⠤⡏⡁⢹⣿⠋⠘⣁⠀⣥⢀⡞⠀⠀⠀⣠⣞⣿⠃⠀⠀⠻⡶⠗⠚⠦⡄⢀⡠⠊⣸
⠀⠀⠀⠸⣦⡼⣿⣷⣷⢃⣤⠀⣿⣸⣿⣿⣿⣄⠐⠻⢛⡟⠻⣷⣄⣀⣶⠃⠀⢄⠀⣼⠉⠀⢴⡇
⠀⠀⠀⠀⠘⣇⢈⠻⣿⣸⣧⣼⢿⣿⣿⣿⣽⡿⠿⠛⠻⠧⡀⠈⠛⢿⠃⠆⠀⡾⠕⠁⠀⠀⠀⣃
⠀⠀⠀⠀⠀⢸⣿⣷⠍⠳⢏⣁⣿⣿⡿⣹⡏⠀⠀⠀⠀⠀⡘⣆⢀⡎⠀⠀⠘⣴⠇⠀⠀⠀⢦⠋⣆⠀⡄
⠀⠀⠀⠀⠀⢸⢻⡿⠃⠀⠈⢻⡿⠶⠊⠛⠃⠀⢀⠀⠀⡠⠛⠛⠉⠀⠀⠀⠀⣧⠖⠂⣀⢠⠆⠀⣸⢿
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠀⢀⡇⢀⣀⣀⠀⠀⣀⣴⠋⢠⣾⡿⠉⠀⣼⡿⠒⠗⠂⠠⠄⣀
⠀⠀⠀⠀⠀⠘⡀⠀⢰⣔⣪⠄⠀⠀⠀⣠⣸⡏⠁⠀⣸⣧⡉⠁⢤⡀⠓⠿⠟⠻⢿⠀⢉⣵⣢⡿⢸⢂⣆⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠉⢿⠱⡟⣿⠶⠶⠰⡿⠈⠉⠁⣰⡞⠛⣿⠿⣧⣴⣿⡿⢿⣿⣿⠶⣶⣿⣿⣿⡗⣣⣿⣼⠋
⠀⠀⠀⠀⠀⢠⠒⢿⠁⠀⠀⣀⡀⠀⠀⢀⣠⡰⠟⠛⠀⠋⢱⣘⠯⠙⣅⠾⢿⣿⣶⣽⣿⡿⣫⣾⣿⡿⣿⢀
⠀⠀⠀⠀⢠⠣⠛⠀⡷⠶⠿⣿⠋⠛⠒⠒⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣨⣿⡿⠙⡏⣹⣿⣿⣇⡷⠒⠁
⠀⠀⠀⢠⠃⠀⠀⠈⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠸⢻⣿⠀⠀⠀⣿⡿⠀⡈⢧
⠀⠀⠀⠈⢂⣲⡶⠀⢀⡴⠂⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⣼⡆⠀⠀⣿⣇⡘⠁⠈⠑⠄⡀
⠀⠀⠀⠀⠀⠈⠃⠓⢤⣄⣠⢾⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⣷⣾⣿⣧⠀⠀⣿⣿⣷⡀⢦⣤⢤⡬⠦
⠀⠀⠀⠀⠀⠀⢰⣿⡷⣽⣿⣿⣼⣆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠾⣿⡄⣰⣿⣿⣿⠖⢋⡀⠒
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢹⣿⣿⣦⠀⠄⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⠹⣿⡏⣹⡟⢄⠈
⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠄⢸⣿⣿⣿⣶⡄⢀⠀⠀⠀⢀⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠏⡚⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠂⠀⠈⠛⣿⣿⣿⣦⡀⠀⠠⠴⣿⠘⠀⠀⠀⠀⠀⠀⠀⢀⡦⣿⣿⣇⠘⠃⠠⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢀⡒⠀⢁⣠⣼⣿⣿⣿⢻⡟⢿⣿⣿⣿⢀⠀⠀⠀⠀⠀⠀⠠⢿⣷⣿⣿⣿⣷⣦⣤⠄
⠀⠀⠀⠀⠀⠀⢀⡠⠘⢁⡁⠈⠅⠉⠿⢿⣿⡞⣷⠢⠙⠻⢿⣸⣂⠀⠀⠀⠀⠀⢔⣦⣿⣿⣻⡏⠯⠍⠒
⠀⠀⠀⠀⠀⢀⣈⡀⠠⢤⡀⡀⢀⣀⣀⣤⣼⡷⣿⠠⡦⠀⢸⠏⠛⠻⣦⣾⣿⣶⣾⣿⣻⡁⠀⣠⣠⣌⢀
⠀⠀⠀⠀⠀⠈⢈⢣⣴⣾⣿⠻⠿⠞⠛⠉⠙⠂⠁⠀⠀⠀⡠⠊⠉⠀⣤⠬⠩⠩⠆⠾⠛⣇⠀⠏⠻⠏⠩
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣈⠀⠀⢀⠀⣀⢠⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠀⣾⡏⡀⠀⢿⢻⣇⡀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠘⠉⠙⠛⠦⠞⠋⠉⠙⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⢠⣾⣿⡿⠿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠟⣩⡽⠿⢷⣦⣿⣿⢃⣾⣤⡆⢹⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⢸⣿⣾⣿⣿⣿⣿⣿⣾⣭⣉⡇⢸⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠰⠶⠞⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣀⢀⡀⠀⠉⢹⣿⣷⡄⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣯⣉⣴⠾⠛⠛⠛⢿⡁⠀⠀⣾⡏⠉⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡏⠀⠀⠀⠀⠀⠁⠀⠀⡿⠃⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⠀⢾⣿⣿⣿⠋⠉⠈⠙⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣶⣤⣤⣶⣿⣿⣿⣿⣿⡿⠀⠀⠛⠛⠁⠀⢀⡴⠂⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⢀⡀⠀⠘⣄⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡙⢿⣿⣿⠃⠀⠀⢀⣴⣿⣿⣿⣿⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⣄⠀⠈⣻⣿⣿⣷⣿⣥⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣷⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⣻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⢿⣿⣿⣿⣿⣿⣿⢁⣿⣿⣿⡟⠁⠈⢻⣿⣿⣿⣿⣿⣿⣿⡥⠀⠀⠀⠀⠈⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣠⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣰⡄⠀⢤⡀⠹⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣞⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣅⣀⣸⣷⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠛⠛⠋⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠈⣿⣿⣿⣿⣿⣿⣿⣧⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠛⠛⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠶⠶⠖⠋⠁⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⢶⣶⣼⣆⠀⠀⠀⢘⣧⣤⡶⠶⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⣠⠶⠛⠉⣩⡽⠛⠋⠉⠉⠛⠻⣭⡉⠙⠳⢦⣈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠟⠁⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠙⢷⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⡼⠃⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠹⣆⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⡼⠁⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠹⣆⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣸⠃⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢹⡄⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢀⡏⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⣧⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢸⠇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣼⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢸⡇⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⡆⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⣸⠀⠀⠀⡏
⠀⠀⠀⠀⠀⢀⡤⠶⠿⢦⡦⡼⠗⢦⡀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠋⠛⣲⣦⣀⣟⠀⣠⢼⠷⠲⣄
⠀⠀⠀⠀⢰⣋⣀⣀⣾⠻⣣⠖⠻⣼⣁⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣸⡝⢦⣰⠃⠧⠟⠹⠿⢻⡀⠀⠀⢿⡀
⠀⠀⠀⣾⠉⠁⣸⡽⠿⣤⡇⠀⠀⠸⣏⣠⣴⡒⠻⢦⣀⣀⣀⣀⣀⠀⢀⠀⢰⡓⠟⠀⠉⠀⠘⣿⣀⣀⣀⡀⠀⣠⣙⣦⢴⡞⠓⠲⢤
⠀⠀⣀⣼⣦⣞⣡⡤⠤⠤⣧⠀⡶⢶⠇⠉⠀⢹⠆⡴⠃⠀⢰⡏⠀⠙⢹⣷⣏⠀⣀⣀⢸⡳⣤⢻⣞⠁⠈⠉⠳⣿⠈⠀⠀⣿⠀⠀⣀
⢤⣬⡿⠉⠸⡅⠀⠀⠀⠀⢾⠳⠟⠀⠀⠀⢀⣼⡿⣥⣀⣀⣾⣅⡀⠀⠀⠁⣿⡶⠿⠈⠉⠁⠘⣿⡿⣄⡀⠀⠀⢸⠀⠀⣠⡾⠷⠯⣅
⠀⠈⢉⣙⣿⡳⣄⡀⠀⠀⠈⣳⣄⠀⢀⡤⠬⠗⠀⣸⠋⠉⠉⠛⢦⡀⠀⢠⡟⠀⠀⠀⠀⢀⣀⢸⡀⠀⠉⠿⣖⠚⠶⡏⠁⠀⠀⠀⢈
⠀⠀⠈⠙⠻⠶⢶⠭⠭⠿⠿⠏⠈⠓⢮⣽⣄⣠⠾⣅⠀⠀⣀⡤⣾⣥⠴⠾⠳⠶⣄⡴⣆⡞⠙⠞⠓⠒⠚⠏⠻⠛⠲⠿⠶⠤⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠛⠛⠻⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠖⠶⠶⣏⡔⠉⠀⠈⠑⡄⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢻⣄⠀⠀⠀⣀⡴⢾⡇⠀⠀⠀⠠⡈⠙⢦⣀⠀⠀⢸⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⢀⣴⠋⢻⡄⠀⠀⠹⣆⣤⠞⠉⠀⠸⡇⠀⢀⣀⡀⠈⢦⡀⠙⢧⣠⠎⠀⠀⠀⠸⣧
⠀⠀⠀⠀⣼⠋⢹⣤⠶⠛⠒⠒⠀⠹⣇⠀⠀⠀⠀⠻⣦⡟⢛⠻⠉⠉⣻⣶⣶⣽⢦⠀⠀⠀⠀⢻⣷⡄
⠀⠀⠀⣠⠿⣆⠸⣇⠀⠀⠀⠀⠀⠀⢹⡄⠀⣀⣤⣤⣾⣜⣉⡇⠀⣠⠖⠺⣯⡙⠈⢷⡀⠀⠀⢸⡇⠹⣆
⠀⠀⠀⢻⣄⠙⠓⣾⠋⠹⠀⠀⠀⠀⠘⢻⡾⠁⠀⠀⠀⠙⢿⡅⢠⠇⢠⣲⡇⠉⠓⠚⢧⠀⠀⠀⣷⣄⢹⡆
⠀⠀⠀⠀⠉⠻⣞⠋⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠈⠧⡽⠟⡀⢀⣠⡤⠾⡇⠀⠀⠇⠙⣧⣿
⠀⠀⠀⠀⠀⠀⠈⠛⢳⡖⠁⠀⣀⡴⠟⠉⣿⣦⣀⣀⢀⣀⣤⣾⣷⣶⣾⣿⣿⡾⠀⠀⣀⣿⣤⡀⠀⠀⣸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢿⣍⠀⠀⠀⠸⣏⠿⠿⢿⣿⣿⠿⣿⡿⠋⠉⠘⠧⠴⠊⣡⠄⠀⢹⡶⠞⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠙⢷⣄⠀⠀⠹⣦⡀⠙⣯⡭⠭⢽⡇⠀⠀⠀⠀⠀⠾⠋⠀⠀⣸⠇⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠙⢷⣀⣠⡴⢿⣄⠘⠢⠴⠊⠀⣀⣀⣀⣠⡤⣤⣤⣴⠞⠁⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢀⣠⣴⠛⠁⣴⣋⣉⣳⣶⣶⠞⠋⡽⠛⠉⠙⠳⣦⡶⠋⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⠁⣹⢰⡯⠉⠉⠀⠠⣜⠋⠓⢾⠛⢶⣤⣄⡀⠀⠙⣦⡀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣱⡏⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢷⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣼⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣤⣤⣤⣴⣇⣠⡿⢤⣤⣤⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠋⠀⠨⠙⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣧⣀⣀⣰⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠤⣄⠀⠀⠀⠘⠏⢥⣼⣶⣿⣷⣿⣿⣿⣿⣿⠿⠇⠀⢀⣠⣤⣶⣶⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⣿⢿⣯⠁⠁⣴⡿⣿⣟⠻⣦⣶⣿⣟⣿⣏⣹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣴⣤⠀⠀⣥⣦⡀⠉⣻⣿⠛⢷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠛⠉⠀⠀⠉⠛⠁⠀⠈⠉⣠⡈⠻⣿⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣷⣄⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣦⣬⡄⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣷⣿⣿⣿⣿⣿⡀⢀⡀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⡿⠿⣿⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡇⠀⠠⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⣴⡿⣷⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣌⣥⣈⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠛⠁⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠁
⠀⠀⠀⠀⠀⠀⡀⠙⠁⢀⣀⡄⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣀⣠⠋⠀⢀⠠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠙⣿⣿⣿⣿⣿⡿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠈⠑⠠⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⢀⣈⣿⣿⣿⣿⣿⡇⠀⠀⠀⢻⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠙⠿⠟⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠉⠀⠉⢈⣉⣉⣩⣿⡷⠀⠀⠀⠊⠉⠉⠀⠀⠉⠉⠙⢛⡻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡾⠿⠛⠛⠛⠛⠛⠿⢿⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠻⢿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣤⢵⣷⣼⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣂⣀⣀⣷⣀⣀⣀⣂⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣛⠻⣿⣛⣻⣟⣿⣿⣫⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠉⠁⠉⠉

⠀⠀⠀⠀⠄⠀⠀⠀⠰⡀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠑⢤⡯⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠄⠀⠀⠀⢸⣣⠚⢀
⠀⠀⠀⠀⠀⠀⠀⠡⣼⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠆
⠀⠀⡴⠀⠀⠀⠀⠠⢻⠐
⠀⠀⠂⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠠⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⢶⠶⠶⡤⢤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠟⢣⢀⣠⣤⣄⢀⣀⣊⡉⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢌⣰⣾⣿⡿⣿⣿⣿⣛⣳⣄⣠⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⠈⢯⣻⣿⡇⠀⣿⣿⣟⡟⠛⠉⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈⠀⡛⣹⡁⢰⣿⢿⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⣤⣶⡶⠟⣻⡶⣄⢀⡀⢠⣔⣮⣷⣿⣿⣷⣇
⠀⠀⠀⠀⠀⢀⣴⣿⣷⣷⣾⣿⣿⣿⣿⡿⠿⠋⣁⠈⠠⢀⣤⣿⣛⣻⣿⣻⣿⡆⣈⠁
⠀⠀⠀⠀⠀⠊⠁⠚⣻⣿⣿⣿⣿⣟⠋⠉⠁⠠⠀⠀⢠⡾⣿⣟⣿⣿⣦⣉⠙⢛⢿⣿⣦⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢟⣿⡇⣿⠿⠀⠀⠀⠀⣠⡀⠈⠑⣻⣿⣿⢻⣿⣿⠋⠀⢸⣿⣟⣓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡄⠀⠀⠀⣤⣴⣾⣿⣿⣄⠈⠉⠉⡟⢸⠋⠀⢀⣴⣿⣿⣛⠡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠨⡑⠿⣿⣿⠿⠟⠁⣶⡆⠐⠁⠰⢾⣿⣿⠟⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠘⠏⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⣾⡿⠀⠀⡰⠀⠀⠀⠘⠻⢿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⡆⢀⡄⣰⢀⣿⣿⢁⣼⠁⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣆⢐⣷⣘⣿⡿⢸⣿⣯⣼⣏⡠⠀⠀⠀⠀⢳⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⣳⣾⡿⣾⣿⣿⠀⢀⣠⣤⣶⣾⣿⣿⣦⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⢶⣿⡿⢿⠟⢿⣿⣿⣿⣿⣿⣿⡷⢞⡿⠁⠉⠛⡇⠘⣏⠙⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡀⠶⠋⠉⡧⣼⠀⠰⢇⠈⢿⠈⣿⠙⠀⢺⠃⠀⡼⠶⠁⠈⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⠙⠀⠀⠀⠤⠃⠁⠰⠀⠎⠀⠸⢠⡋⢃⠀⡄⠀⡴⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠁⠈⢆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢾⣿⡄⠀⠀⠀⠀⢀⣀⣠⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠚⠁⠀⠀⢀⡤⡺⠛⠀⠨⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠢⠁⢂⣼⡍⡞⠀⠀⢀⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠉⠄⠄⠀⣾⣿⢀⣿⣆⣀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠤⣴⣹⣿⡈⢿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣦⠬⣅⠽⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⣛⣋⠔⠋⢬⣔⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢨⢁⢌⣁⣴⠮⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠀⠀⠀⠀⠀⠀⢸⣏⠃⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠑⠀⠀⠀⠀⠀⠀⡠⡿⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠡⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⠀⠀⠀⢀⣠⣾⠛⠓⢀⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠀⠀⠀⢀⣿⡋⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠀⡠⠀⠀⠀⠸⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠺⠖⠂⠂⠀⠻⠦⠴⠤⠆⠀⠀⠀⠀⠀⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣻⡟⡛⢻⡛⣻⢿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣠⣀⣄⣄⢀⡀⠀⠀⣲⡀⠀⠀⠀⠲⡤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠈⠉⠀⠀⢠⣄⠀⠙⠂⢀⠈⠀⠁⣉⠋⣷⡆⠀⠀⣬⠖⠠⣇⠀⠈⡲⠄⡉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⣄⠀⠀⣀⠀⠀⠄⣆⢬⡼⠓⠀⢠⣰⡧⣀⣀⣻⣇⡀⠀⠀⠀⢀⠉⢀⠙⣄⠈⠉⡒⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡻⠀⣧⡄⡈⣥⢾⣜⡓⢉⣁⡀⠀⡀⠈⠁⠀⠀⠀⠈⢛⣁⠐⠤⠤⢀⡉⠁⠳⢽⡄⠀⡇⠀⠀⣷⣼
⠀⠀⠀⠀⠀⠀⠀⢰⡈⢳⣶⣿⣦⣿⠿⢋⠁⡠⢈⣨⠍⣁⠐⠦⠀⠀⠀⠀⠀⠈⢥⡆⢀⠀⠨⡇⠢⣀⠀⣿⠢⣅⠀⠀⠹⣿⠀⢀⡄
⠀⠀⠀⢀⠀⠀⠀⠎⠀⣠⣿⡏⢱⢋⣴⠥⠐⠀⠀⠀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢀⡆⣷⢀⠈⠀⢹⠆⢺⣷⡀⢣⢹⣦⠞⢹⡆
⠀⠀⠀⢸⡄⠀⡄⠀⣸⠿⣸⠷⣿⠁⣀⢂⡂⣔⣬⣰⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⢿⠓⠾⢬⣤⡈⢧⡑⣼⢠⣿⣦⣿⠁
⠀⠀⠈⡀⢧⣠⠓⣤⣿⢟⡟⣸⡿⠬⠋⠣⠑⠈⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣳⠌⢳⣄⠸⡋⠙⢫⣷⣼⠁⢸⣿⠁⢰
⠀⢢⠀⢧⠸⣧⠑⣾⢧⡾⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠙⣿⡗⢾⣿⣊⣋⠄
⠀⠈⣧⢨⠃⠸⣿⣣⡾⢛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠶⣄⣿⣧⠸⣿⠈⣯⠆
⠀⠀⠘⣷⣀⠀⢻⣿⡡⣺⣻⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣌⢻⣿⠀⣿⣇⡳⠂
⢦⡀⢰⡛⣿⣦⣾⣯⣠⣿⢫⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣍⠹⣿⠳⣿⣧⣟⠁
⠈⢷⡌⡾⠰⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⣱⣿⣧⣿⣿⠃
⠀⠈⢷⡸⡇⢸⣿⣿⠁⣀⡀⠀⣤⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⢹⣿⣿⣿⠇
⠀⠀⠘⣿⠷⣸⣿⣿⠘⡿⠛⡸⡯⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣤⠀⠈⠣⣼⣿⣿⣿⣆
⠠⣼⡇⣹⣯⢸⣿⣿⡏⠀⠐⣮⣿⢅⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⢷⣫⣴⣶⣼⣿⣿⡇⢸
⠀⢨⣟⡵⢻⣿⣿⣿⣇⠀⢐⡫⢷⢝⡂⠀⢄⣧⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢭⣷⡋⠡⣼⣿⣿⡇⢸
⠀⠀⢻⣫⠌⢻⣿⣿⣿⡆⠀⢩⠽⣝⣃⣨⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣗⡂⢀⣿⣿⣿⡃⣾⠀⢸⠁
⠀⠀⠈⠧⣞⢄⢹⣿⣿⣿⣄⠈⠬⣯⢲⣵⡿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠬⢽⣿⠒⣾⣿⣿⣿⣶⠃⠀⠅
⠀⠀⠀⠀⠻⢎⢔⣿⣿⣿⣿⣦⠨⣧⣾⣿⢓⣂⣤⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣯⣄⠀⡄⠀⢀⣦⣄⢺⣿⣃⣼⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠈⢏⣿⣿⣿⣿⣿⣷⣾⣿⠻⠛⢻⣟⣿⣿⣤⣤⠀⠀⠀⢀⠀⠀⠐⡩⣻⣽⠇⡇⣇⡼⣻⠏⡽⢷⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠿⠿⣿⣿⣷⣬⣧⣼⣽⣽⣿⣿⣀⠀⣈⣄⣄⣀⣈⡴⠖⠓⣚⡛⠛⣿⣾⣾⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⠷⣤⣀⡀⢹⣿⣟⠁⠀⠀⠀⡀⢀⠌⠋⠙⠻⣧⡊⠁⢀⣅⠚⢻⣛⣠⣿⣿⣿⣿⣿⠋⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣽⠿⢿⣯⣭⣭⠾⢿⣤⣀⢀⠠⠊⢀⣱⡤⢶⣿⠖⡛⠛⠋⠛⠛⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⣿⣿⣧⣼⣾⣾⠾⠷⣦⠾⡉⠐⢣⡏⠀⠀⠀⣈⡥⠖⠚⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣙⣇⣀⣜⣁⣀⣈⣠⣎⣈⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⠭⣟⡛⣛⣛⡛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀⠀⢹⠆⠀⠀⢀⣤⣠⡏⠉⠃⠀⠈⡗⠒⠀⠐
⠀⠀⠀⠒⠶⡶⢼⡶⣶⣦⡀⠀⠀⠀⠂⠀⠀⠀⠂⢤⠤⣘⣤⡟⠲⡁⠠⠒⠱⠋⡇⠀⠀⠀⠀⠀⠀⠀⢀⡼⢅
⠀⠀⠀⠀⠀⠀⠀⡇⠈⡌⢿⠄⠈⠀⠀⠀⠀⠀⠀⡸⢐⠛⠀⠀⢀⣄⠀⠀⣤⠐⠉⠁⠀⢠⣾⢻⣻⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣣⠀⡇⢸⠞⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠈⠀⠀⠀⠏⠈⠲⠒⠊⠉⠙⠋⠁⠀⠀⠀⠈
⠈⠺⡄⠀⠀⠀⠜⡿⠾⡟⠦⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄⠀⠀⠄⠀⠀⠉
⠀⠀⠈⢩⠖⠀⠀⡅⠀⢀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠈⠀⠀⠀⠋⠀⠀⣠
⠀⠀⠀⠈⠀⠀⠤⢰⡤⠼⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠒⠶⠀⠃⠆⠀⠇⠶⠘⠰⠆⠖⠒⠓⠀⠀⠀⠀⠀⠀⠀⠤
⠀⢀⢠⡴⠶⣶⠉⠙⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡀⠀⠄
⢀⠔⠚⠂⠀⠈⠀⠀⠜⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣄⣤⣾⡛⠛⠯⣃⢄
⢸⠄⠀⠀⠀⢀⣤⣤⡄⠀⠐⠋⠃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⣀⣴⡜⠛⠦⠀⠙⠦⡈⢳⣃
⢀⠀⠀⢀⣨⣿⣭⣷⠁⠀⠀⠠⢄⠀⠀⠂⠐⢀⡀⠉⠀⠀⠀⠀⠆⠠⢤⣿⣤⣔⠉⠀⠇⠀⠀⠑⠄⠀⠱⠄⣿
⠈⠟⠉⠁⠀⠈⠉⠀⠀⠀⢀⣀⠈⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⢡⠈⢇⠀⠀⠀⠀⠀⠈⠆⠀⣧⡟
⠀⠀⡀⢀⠄⠀⠀⢠⠀⠀⠀⠁⣠⠎⠀⠀⠀⠠⠀⠀⠀⠀⠢⣀⡀⢈⠀⠀⡰⢀⣿⠄⠀⠐⠀⠀⠀⢸⡴⠋
⣤⣤⣥⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣥⣤⣽⣶⣶⣿⣬⣥⣤⣵⣤⣤⣵⣦⣥⣤⣤⣤
⣿⣯⣭⣭⣭⣽⣭⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠶⠿⠿⠿⠿⠿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠋⢁⣠⡤⣦⣴⢤⣄⡀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⣴⠛⠁⠀⢀⠀⠀⠈⠙⢷⡀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⢀⣠⣤⡿⠶⠟⠛⠛⠛⠿⠿⢶⣶⣄⣽⠃⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⣠⡶⠟⣩⣤⣄⣀⣀⣀⣀⣀⣀⡀⠀⢀⣉⣙⠻⣦⣄⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢀⣾⠏⠀⣾⣿⠿⢿⣿⡉⠉⠉⠉⠉⢙⣿⣿⣿⣿⣷⡌⠙⣷⡀⠀⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣇⣠⡾⠋⢠⡴⠶⣌⠃⠀⠀⠀⠀⢛⣩⣥⣈⠙⢿⣿⣄⠈⢿⡆⠀⠀⠘⡈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠈⣛⣿⠁⢰⠏⠠⠔⣘⣧⠀⠀⠀⣴⠋⠀⠄⠙⣷⠀⠛⠻⣦⣸⣿⡀⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣧⡿⣿⠀⣿⠀⣆⣼⣯⣿⣤⣀⣰⡇⠰⢀⣶⣷⣿⠄⠀⠀⠈⠛⠻⣧⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⣈⣵⡟⠀⣻⡶⠋⠁⠀⠀⠀⠈⠙⢷⣆⢸⣷⣼⣿⠀⠀⠀⢀⣠⡾⢻⡄⢀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣄⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣎⣻⡿⠃⠀⣀⡀⣿⡁⠀⣸⠟⠛⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢸⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣁⣠⣴⣾⣿⣿⢼⣇⣸⢋⣠⠤⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡟⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⠏⠀⠛⢁⣾⡧⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡈⠛⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠁⠀⠀⠀⠸⡇⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⣰⡾⠛⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠉⠙⣿⠿⠼⠿⠿⠿⢿⡖⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢁⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⠻⣦⣄⣠⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠸⣿⠟⠛⠛⠛⣻⠁⠀⠀⠀⠀⠀⣠⡾⠟⢿⡿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⢽⠟⠁⢠⣬⣿⣤⣶⠶⠶⠾⠿⠟⢻⣿⠿⢻⣷⡀⠀⠙⢦⣄⣠⠞⠁⠀⠀⢀⣠⣴⣾⣿⣷⠶⠟⠁
⠀⠀⠀⠀⣠⡶⠟⢻⡇⠀⠀⠀⠀⠈⠀⢠⡟⠉⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⢀⣾⠃⠙⢶⣤⣀⣈⣀⣀⣠⣴⡾⠟⠉⠀⣿⠉⠙⠻⢶⣤⣀
⠀⢀⣴⠟⠋⣀⡀⠀⠃⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⢀⣾⠃⠀⠀⠀⠈⠉⣙⡛⣽⠟⠁⠀⠀⠀⢸⡏⠀⠀⠀⠀⠈⠛⢷⣤⡀
⢠⣾⣫⡶⠟⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⣀⣀⣤⣾⠷⠶⣄⠀⣸⣇⣠⣤⣤⣀⣀⣀⠀⣸⠏⠀⣀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⣸⡿⠃⣠⡴⠚⢿⡀⠀⠀⠀⠀⠀⠀⣤⠀⠈⠛⣻⡟⢉⡿⠁⢸⡆⣿⢀⡿⠉⠉⠉⠉⠉⠉⠉⣻⣏⡴⠛⠉⠙⢦⡀⣿⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⢹
⣿⣧⡾⠋⢀⣤⠞⠃⠀⠀⠀⠀⢀⣠⡿⠿⠶⠾⠟⠁⢸⣇⠀⠜⣡⠏⣼⠇⠀⠀⠀⠀⠀⠀⠀⣿⢸⠁⠀⠀⣴⠘⡇⣿⡀⠙⢷⣄⠀⠀⠀⠀⠀⡘⢸
⠈⠻⣷⣴⣿⠉⠀⠀⠀⣀⣤⡶⠟⠉⠀⠀⠀⠀⠀⠀⣼⣿⣳⢚⣫⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⢿⡘⣦⡀⠐⣁⡞⣹⡟⢿⣤⣾⡟⠛⠀⠀⠀⠴⣱⡿
⠀⠀⠀⠈⠛⠷⠾⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣉⣉⣡⡶⠋⠀⠈⢿⠛⣷⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⢰⠀⣿⣦⡿⢿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣿⢄⣴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣸⡇⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣰⣿⠷⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠠⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠈⠉⠛⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠜⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠛⠛⠛⠻⢶⣤⣀⠀⠀⠀⠀⣸⡟⠀⠀⣀⣤⣴⠶⠾⠛⠛⢻⡿⠿⢷⣬⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣶⣶⣿⣧⡴⠟⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣼⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⠉⠉⠛⠛⠛⠛⠛⠉⠉⣉⣩⣉⣀⣸⣿⠛⠳⠶⠦⣤⣤⣤⣤⣤⣤⣶⣶⡶⠾⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⠿⠿⠟⠛⠛⠉⠉⠉⠉⠉⠛⠿⠷⣦⣄⣀⣀⣀⠀⠀⠀⣀⣀⣀⣠⣴⡾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⣺⠧⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠇⠀⠀⠀⠀⣀⣀⣠⢤⠒⠉⣡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⡀⢀⠴⠚⠅⠀⠀⢈⣖⣋⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⣰⠀⠀⠠⡀⡴⠀⠐⠺⢉⠁⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢮⠀⠢⣀⡀⠀⠀⠀⠀⠈⢀⣁⠤⠞⢁⣀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠉⠑⠒⠢
⠀⠀⠀⠀⠀⠀⢀⠤⠶⠒⠒⠚⠋⠒⠒⠒⢢⡇⣠⣀⣧⡠⠤⠤⠤⠴⠟⢳⠋⠛⠉⠀⠀⠘⠓⠢⡄⠉⠓⠪⢂⣠⡮⠁⠀⠀⠀⠀⠀⠘⠒⡄
⠀⠀⠀⠀⠀⠀⣜⡀⠄⠀⠠⠤⠀⢀⣀⣠⠞⠁⢀⠔⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡀⢲⡞⠀⠀⠀⠀⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠻⡤⢂⣤⡄⢀⡼⢋⡭⠟⠀⠀⠘⠀⠀⢀⡠⡤⠒⠚⠛⠛⠳⠷⠤⣤⠀⣦⣹⠋⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⢀⡴⠛⠛⣜⠤⠏⣴⠋⠀⠀⠀⠀⠘⠤⣤⢴⠃⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠀⠀⣀⡴
⠀⠀⠀⢀⠴⠟⠀⡠⠊⠈⠳⣤⠯⠀⠀⠀⠀⠀⠀⠀⠀⠘⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠤⠔⠒⠉
⠀⠀⡰⠃⠀⠀⠰⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⣀
⠀⠀⢻⣀⠀⢀⣼⣤⡀⠀⠀⠁⠀⠀⢀⣠⢴⣀⠰⠞⠁⠀⠙⢷
⠀⠀⠀⠉⠛⠋⠁⠀⠈⠳⢦⠤⠶⠚⠉⠀⠀⠈⠳⡄⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠁⠀⣂⡀⣀⢀⢀⠀⣀⡀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢀⡼⠛⠋⠻⣆⠀⠀⠀⠀⠐⠆⠚⠞⠓⢃⣾⠄⢸⠀⣀⢠⣼⠐⢽⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡉⠀⠉⠁⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠚⠳⠖⠺⠛⠻⠻⠟⢻⡟⠛⠟⠛⠿⠛⠿⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠛⠛⠛⠉⠀⠀⠀⠀⠠⢸⠀⠀⠐⢀⢈⠸⠀⠸⢸⡇⠀⠂⠁⡇⡇⠰⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣶⣶⣶⣶⣾⣶⣾⣶⣾⡷⣶⣷⣶⣶⣶⣷⢶⣷⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⡂⠀⠀⠀⠀⠀⠀⠀⢠⡀⠑⠈⠁⠈⠀⣀⣀⠀⠀⠀⠀⠀⠦⠰⠢⠠⠀⠀⠀⠀⠀⠠⠈⠁⠢⠄⠀⡀
⠀⠠⠀⠈⠀⠘⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⠄⠀⠀⠂⠀⠐⠤⠀⠄
⠀⠀⢀⠀⠀⠀⠠⣀⡠⠀⠀⠀⡰⠀⠀⠀⠲⠇⠀⠀⠾⠀⠀⠀⠂⠀⠀⠱⠔⠀⠀⠘⠂⠀⠀⢨⠄⠀⢤⠇⠀⢀⣨⠁
⠀⠀⠈⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠐⠀⠀⠀⠀⠀⣀⠀⠀⠈⠓⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⡀⠀⠢⠌⠀⠂⢤⠀⡼⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⠀⠀⠀⠀⠀⢀⡀⢠⠠⠀⠶⡒⠇
⠀⠀⠐⠁⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠙⠀⠀⢢⣄⠀⠦⠀⠀⠀⡈⠀⠀⢁⠀⠀⠀⢠⠇
⠀⠀⠀⠈⠀⠀⠀⠑⠦⠤⠤⠞⠀⠀⠀⠀⠈⠃⠀⠠⠴⠀⠀⠀⠀⠀⠠⣸⢿⠀⠀⠀⠀⠀⠐⠄⠠⠜⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⡐⠠⠄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠃
⠀⠀⠀⠈⠐⠂⠀⠀⡀⠀⠀⠀⠐⠀⠀⠈⠃⠀⠀⠄⠄⣤⠄⠄⠀⠀⠀⠀⡠⠀⠀⡀⠀⠀⠀⠀⠀⠂⠂⢸⡇⠒⡄⠆
⠀⠀⠀⠀⠀⡀⡀⠀⠁⠰⠀⠂⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠜⢹⠓⠚⠩⣤⡆⠀⠀⠀⡀⠀⣺⣁⠔⢀⠀⠄
⠀⠀⠀⠈⠀⠙⠃⠉⠀⠀⠀⠀⠄⠀⠠⡴⠃⠀⠀⠀⠀⠙⠊⠀⠀⠘⠂⠀⠘⠶⠚⠐⠚⠀⠀⠀⠉⢀⣤⣽⠇⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠶⢖⠛⠛⠛⠒⠶⠦⣼⣍⡉⠉⠉⠉⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠙⠳⢶⣄⡑⠄⠀⠀⠀⠀⠀⠉⠛⣧⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣍⡁⡠⠀⠀⠀⠈⠹⡗⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠤⠄⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠾⠛⠛⠛⠛⠓⠲⢦⣧⠀⠀⠀⢈⡉⣀⠤⠀⠚⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠿⡛⠒⢦⠀⠀⠀⢀⡤⠖⠿⠃⠀⡾⠛⠙⠙⠛⠛⠛⠳⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠉⠉⠃⠀⠀⣼⣷⠞⠛⠋⠉⠛⢛⣷⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠸⠉⠛⠷⣤⣸⣆⠀⢹⡛⣶⡄⠀⠀⡤⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣴⣛⡫⠭⠤⠭⠶⢽⣷⣴⣇⣀⣀⣀⡀⠀⠀⠀⠀⠒⠠⠤⠠⠬⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡬⠒⠉⠀⠀⠀⠀⠀⢀⢷⡿⠛⠉⠉⠉⠉⠙⠳⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠘⠟⠉⠈⠁⠀⠀⠀⠀⠀⢰⢡⡿⠤⡀⠀⠀⠀⠀⢀⢀⠤⠽⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⢘⡆⠀⠀⠀⡼⠧⣄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠤⠤⡈⠀⠀⠀⠰⠁⢀⣬⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⠀⠠⠂⠂

⠀⢀⠀⠀⢀⡀⠀⠀⠀⠠⡀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⢀⠂⣄⠀⠀⠰⠀⠀⠀⡄⢀⠀⢠⡀⠀⠀⠀⠄⢠⠄⠀⢀
⠀⡮⠷⠀⣠⠈⠀⠀⢀⡀⠀⠙⢆⠀⠈⠁⠀⡛⢌⡀⠀⠤⠀⠀⢻⡁⢲⠀⠈⠱⠀⢁⡁⠀⠺⠛⠁⢀⠖⠀⠘⠀⠀⠫⢠⣫⠂
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠈⠺⠀⠠⠀⠀⣴⠼⠧⢹⠀⣁⠁⡀⠐⠆⠀⠂⠀⠀⠉⠐⠀⠀⠀⡀⡂
⠀⡐⢖⠩⢀⠀⠀⢀⠴⠒⠈⠉⠉⠓⠂⠀⠆⠀⣄⣀⣠⡄⠀⣈⠉⠉⠁⠀⣤⠀⠃⡀⠀⠀⠀⠀⠀⠀⠊⠀⠠⣠⣀⠈⢉⡆
⠀⣚⣅⠤⠑⠀⣀⣀⡀⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢾⣿⠄⠀⠀⠄⠀⠀⠘⠀⠀⠇⠸⠀⡀⠉⠀⠄⠀⠠⡀⢉⠁⠀⠁
⠀⠠⡈⠁⡐⠀⡉⠀⠉⠉⠁⢻⠂⠉⠉⢁⣍⡁⠘⠒⠊⠀⢀⠁⠀⠤⠄⠀⡐⠂⣂⠀⠂⠁⠀⠀⠈⡀⡋⠁⠈⠏⣈⣁⠄⢶⢀
⠀⠠⣈⡁⠀⠀⢹⠃⠉⠀⠀⢸⠀⢤⠀⠨⢱⠀⠀⢰⢈⣄⠊⠀⠀⠀⠀⠀⠂⠀⠉⣉⠰⢢⡄⠀⢀⠁⠁⠀⠈⠱⠛⠓⠀⢠
⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⢸⠀⠀⠀⠀⢪⡇⠀⠀⠀⠀⢈⠀⠀⢀⠀⢀⠄⣴⡀⠀⡀⡀⠀⡀⠘⠛⠃⠀⠀⡀⠠⣀⠂⠄
⠀⠠⠁⠀⡄⠂⠁⠀⢀⡀⢴⡸⠀⣀⠀⠠⠈⡇⠀⠁⠀⠀⠘⠀⠤⠚⠃⢀⣸⠁⡱⠀⢠⣴⠀⠹⢄⠀⠀⢀⠀⠀⠠⠐⠐⡀
⠀⢐⡂⠀⠈⠈⠀⠀⡉⠀⠀⡀⠘⠀⠄⠀⠠⠐⠄⠄⡀⠀⠀⠀⠒⠀⠀⠀⠉⠀⠀⢀⣤⣌⡄⠀⠒⠋⠀⢀⣄⠀⠀⠀⠀⠇
⠀⠈⠃⠀⠁⠐⠀⠺⠗⢀⢀⠁⠀⢀⠀⠠⠂⠀⢈⠂⠡⠀⠀⠀⠀⢠⠄⠀⢶⣦⣤⡄⠀⠀⠁⠀⢀⣶⡤⠐⢀⣉⢓⠄⠀⢀⠄
⠀⠤⠀⠀⠐⠀⠂⠀⠀⠀⠀⢀⠀⠈⠂⠀⠁⠐⠀⠤⠐⠂⠀⠠⠀⠢⠄⠀⡈⠀⠀⠉⠀⢰⡇⠀⣄⢻⣺⣟⣃⣞⣿⣯⡄⢀⡀
⠀⠀⠀⠀⢂⡁⡀⠀⢦⠀⠀⠢⠘⠀⠀⡅⠀⠀⠂⠀⢈⠁⠀⢀⠢⠀⢐⡀⠂⠀⠀⠀⠀⠈⠁⠘⠉⠃⠈⡏⡍⠋⠈⠁⠑⠀⠁
⠀⢌⠤⠀⠈⣄⠁⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢜⠀⠈⠀⠀⠀⠑⠀⡀⠂⠀⠀⡀⠀⣤⣄⡄⠀⠀⠃⠀⠀⠠⠀⠀⠦
⠀⠠⠎⡄⠀⠘⠀⠀⠒⠋⢀⠀⠃⠉⠘⠀⠃⠉⠐⠁⡀⠈⠁⠀⠀⠀⠀⡀⠈⢴⡥⠀⠀⠃⠀⠁⢛⠃⠀⠀⠀⠀⠂⡀⢠
⠀⢁⢚⡠⢡⠀⠀⠀⢀⠀⠉⢀⡁⠀⠁⠉⠁⢈⡁⠈⠁⣠⠘⠠⣀⣀⠶⠃⡀⣈⠈⠁⠀⠀⠀⡀⠘⠶⡄⠓⠃⠠⠀⠉⠠⠠⠆
⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⢀⠃⢀⠀⠀⠀⠀⠀⠀⢀⠈⢠⣀⠀⠀⠀⢠⠅⡊⠁⠀⠀⠀⠀⢉⡁⢁⠁⠀⠂⠀⠄⠠⣌⡀⠂
⠀⡎⠀⠀⠀⠆⠀⠰⣶⡴⣧⡄⠩⠤⠀⠀⠐⠋⢱⠀⠀⠈⢐⠃⠀⣀⠀⠐⠒⠁⠀⠒⠂⢠⠀⠀⢂⡊⠒⠀⠀⠀⡐⠀⠉⣀⡀
⠀⠀⠘⠈⠣⠀⠀⠀⠋⡏⠉⠇⠀⠀⠁⠄⠀⠀⢊⢄⠀⠁⠁⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⡐⠀⠀⠇⠐⡁⠀⢠⠀⠀⠀⠩⠐
⠀⠀⠦⠀⠄⠀⠲⣒⠚⢁⣐⡇⠀⠀⠤⠀⠀⠄⠀⠽⠇⠀⠠⠄⠁⢀⠀⢀⡂⠈⡆⠙⠂⠁⢰⠘⠀⢩⢨⠅⠀⠒⠀⠀⠀⠀⠅
⠀⠀⣀⢠⠀⠀⢠⡌⠉⠉⠉⠁⠀⠄⣷⠲⠀⣠⢶⣖⠤⣄⠀⠲⠆⠀⠀⠶⢄⠀⠀⠀⠀⢤⠖⣆⠀⠒⠰⠂⠀⠀⠀⠐⠔⠁
⠀⠀⢘⠬⠆⠀⢈⠉⠢⠀⢀⠈⠀⠀⠀⠀⢨⣶⠋⠉⠙⢶⠆⠀⠄⢀⠘⠈⠁⠀⠀⠀⠀⢊⡗⢤⠀⠀⠂⠀⠤⣥⢀⢤⠀⠀⢄
⠀⠀⠈⠓⡯⠂⠈⠠⢀⠀⠀⠉⠀⠗⠀⠀⠈⢭⣷⣠⣶⡿⠂⠀⠂⢁⠀⢀⠀⠘⠉⠀⠀⠼⠧⣐⠀⠂⠀⢤⠀⠁⠀⠀⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣈⣁⣈⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠹⣆⠀⠀⠀⠀⠀⣰⡏⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⢻⡄⠀⠀⠀⢀⡿⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⣷⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠘⠛⠛⠛⠃⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢀⠤⣤⠀⠀⠀⠀⠀⢀⢤⣄⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣿⡿⠀⢀⠀⢀⠀⠸⣿⠟⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⠀⢀⡄⠀⢙⣿⣏⠁⢀⣄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣿⠀⠀⠀⠀⠙⠛⣟⣹⣙⡟⠛⠁⠀⠀⠀⠸⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠋⠉⠉



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

⠀⠀⢠⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢰⠀⠀⠀⡀⠀⠀⣿⠀⠀⠀⠀⠆⠘⠃⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢀⣇⠀⠀⣿⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠂⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠈⠁⠈⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⣀⣠⣦⡀⠀⢀⠀⠀⠀⠈⠉⠀⡀⠀⠀⠀⠄⡀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠈⠙⢉⡟⠁⠀⠀⠀⠀⢤⠀⠀⠀⣷⠀⠀⠀⠀⠁⠀⠀⠀⠂
⠀⢰⣤⣠⠀⠀⠀⠄⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⢀⣿⡄⠀⢰⣿⡇⠀⠀⢠⠀⠀⠀⣀⣂
⠀⠀⠁⠀⠀⠀⠀⠀⠐⠿⠂⠀⠀⠀⢰⠛⡚⣅⠀⠀⠀⠀⢸⣿⡯⠀⠀⡞⠃⠀⠀⣿⣧⠀⠀⠈⠉
⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣤⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠻⠉
⠀⠀⠀⠀⢀⠀⠀⠀⠁⢶⣀⠀⠀⠀⠀⠹⡿⡛⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠄⠈⢆⡀⠀⠀⠐⢍⣣⡀⠀⠻⠿⠿⢷⣿⣽⠄⠀⠘⠀⠶⠀⠀⠀⠄⡠⠀⠀⠢⣄⣠⡆⠉
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠉⠁⠀⠀⣀
⠀⠀⠀⠐⠀⠀⠀⠀⠖⠛⠇⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠁⠀⢈⣴⠛⣹⣤⠀⠀⠀⣠⠖⢀⠠⠀⠀⠀⠀⠀⠀⠀⢠⡩⠘⠁
⠀⠀⢠⠀⠀⠀⠀⠀⢼⣦⡄⠀⠀⠀⠀⠈⢹⡿⣿⡇⠀⠀⢀⣒⣀⡀⠏⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠛⠀⠀⢰⠶⠶⠆⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠠⠄⠀⠀⠀⠀⠀⠶⠦⠄⠀⠀⠀⣅
⠀⠀⠀⠀⠰⣛⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠛⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠾⠷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⡠⢤⡀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣘⣶⠳⢿⣆⡀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⣚⠻⠿⠿⣿⣿⢿⣶⣤⣄⣀⠀⠙⠟⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠴⢪⣷⣦⣀⠀⠘⣿⡏⡤⢸⠏⠙⠺⢟⡢⣄⡀⠀⠀⠈⣳⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠻⣆⠀⢸⡇⡜⠡⣀⠀⠘⢹⠀⠀⣿⡆⠀⠀⠈⠑⣭⡲⣤⡴⠋⠀⠈⢷
⠀⠀⠀⠀⠀⠀⢀⡴⠚⢅⡀⠀⠈⠳⣄⠹⣇⣀⣽⠷⠀⠘⡀⠀⠻⣿⠀⣀⢀⣼⣿⣿⣿⠃⠀⢾⠃⢸
⠀⠀⠀⠀⠀⣰⠟⣀⣀⡀⠙⣶⠴⠲⢬⡷⠋⠁⠀⠀⠀⠀⠑⢄⣡⠞⣠⣿⡾⠿⣿⣿⡟⠀⠀⠘⢀⣾⠶⠖⠛⠛⠛⠛⠓⠶⣄
⠀⠀⠀⠀⢸⡇⠀⠈⠉⠙⡟⠁⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⣀⡤⠼⠷⢤⣀
⠀⠀⠀⠀⢸⡇⠀⠴⠒⢺⠀⠀⠀⣦⣿⡄⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⢹⣆⡀
⠀⠀⠀⠀⠸⡇⠀⠀⠀⢸⡀⠀⠀⣸⠿⣿⣦⣤⣤⣤⣾⣿⣿⡿⠛⡹⠀⡿⢆⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⢀⠴⠖⠋⠀⠀⢈⠙⢦⡀
⠀⠀⠀⠀⠀⠙⢦⠀⡀⠀⠙⣳⠛⢁⡜⠙⣿⢿⠉⠛⠛⠋⠀⢠⣞⣡⠞⠀⣼⢰⣧⠞⣇⠀⠀⠀⠀⠀⣠⡞⠉⣿⡀⠀⠀⠀⠀⢠⡄⢷⠈⠁⠈⣧
⠀⠀⠀⠀⠀⠀⠸⣤⣙⣶⠞⠁⠴⠋⢀⡼⠃⠀⠳⢄⣀⣀⣲⣄⠈⠛⠦⠤⢋⣾⠁⠀⢸⡀⠀⠀⠀⠺⠃⢿⠀⡿⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣤⣶⠋⠀⠀⠀⠀⢠⡏⠉⢹⡏⠛⠶⠶⠶⠋⢸⡆⠀⠀⢳⠀⠀⠀⢀⠀⢸⡇⢷⣼⡆⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⢸⡇⠀⠀⢧⠀⠀⠀⢀⡀⣼⡇⢀⠤⡈⢧⣀⣀⡜⠀⢸⡇⢠⡟⠁⠀⣤⠀⠀⠀⣤⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣸⣇⢀⠤⢌⠳⡂⠀⠀⠀⢸⢡⡏⠀⣸⡼⡆⠀⠀⠀⢸⣇⣈⣧⣀⣴⡛⠶⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡜⡀⢠⢱⠹⡆⠀⠀⢸⡆⢧⣤⡿⢁⡇⠀⠀⠀⢸⣏⣁⣀⢀⠈⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠤⢿⣷⠚⠚⠳⠿⣄⡀⠀⠳⣤⣤⣤⠾⠁⠀⠀⢀⣾⣅⣀⠀⠀⠈⠓⢤⡀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣋⣉⡉⠛⠓⠶⢤⣈⠙⠂⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠉⢳⡄⠀⠀⠀⠙⣆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡀⠀⠈⣉⠒⠤⢀⠈⠓⢦⡀⠠⡀⠀⢀⡇⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⢰⠇⠀⠀⠀⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠈⠐⠀⢙⠢⣄⠙⣆⣿⠀⡼⠁⠀⣀⣠⣴⠿⠋⠀⠀⠀⢀⡤⠿⠤⠤⠴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⣀⠀⠀⠀⣠⠑⢌⠣⡈⣿⣾⣷⣟⣛⣛⣛⣛⡒⠲⠶⠒⠛⠉⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠛⠷⠶⣇⡀⠀⢣⢱⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠀⠉⠷⠾⠾⠧⠤⠄⠀⠊⠉⠀⠀⠀⠀⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠈⢀⢄⡄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⠔⢡⠋⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡎⠀⡆⠀⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⠀⠀⠁⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠠⡀⠀⠆⠀⠠⠃
⠀⠀⠀⠀⢀⠔⠋⠁⡠⠴⠄⠀⠠⡀⠀⡄
⠀⠀⠀⢠⠎⠀⠀⠀⠸⣭⠄⠈⠀⠐⠆⠀⠄⠀⠀⠀⠀⡀⠀⠤⠄⠀⣄⠀⣀
⠀⠀⣠⠟⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠅⠀⠨⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢤⡀
⠀⠀⡗⠒⠁⠀⠀⠐⡀⠀⠀⠀⠀⠈⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀
⠀⠀⠘⠔⠒⠤⣀⡼⠀⠄⠀⠈⠁⡠⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠉⠀⠀⠈⠀⠉⠁⢆⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠢⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⡁⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠕⣠⠤⣀⠀⠀⠀⠀⣆⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⣂⠀⠀⠀⢀⠌⠌⠀⠀⢀⠙⢆⣀⠊⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠄⠀⢀⣁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡯⠥⠬⠤⠄⠀⠀⠀⠀⠀⠀⠸⠴⠂⠈⠐⠤⠤⠎⢀⣤⠄⠠⠤⠴⠒⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢼⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠠⠁⣀⣠⣂⠀⡤⠃⠀⠀⠀⠤⠀⠀⠀⠤⠀⠀⠀⢀⠈⢸⡆⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡰⠃⠄⠀⠁⠀⠁⠏⠀⠀⡤⠂⣠⠾⠅⠀⠀⠊⠭⠋⠀⣈⠉⠛⢤⡀⣧⡀⠀⠘⣿⢃
⠀⠀⠀⠀⠀⠀⠀⢠⢦⡟⠀⢁⠞⠁⠀⣢⠎⠀⣂⣴⡤⠊⠁⠀⠈⠈⠁⠀⠀⠀⣸⣿⣿⡧⠀⠙⢿⡄⠑⢀⣿⡈⣞
⠀⠀⠀⠀⠀⠀⡰⠃⡿⠀⢸⠃⣠⡶⢋⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣷⡄⠘⠀⠉⣷⣾⣿⣧⡿⡄
⠀⠀⠀⠀⠀⣰⠁⢰⡇⠀⣿⣟⠁⣰⣿⠡⠄⠀⠀⠀⠀⠀⠀⠒⠠⠤⣭⡽⠦⠤⡿⣿⣿⣿⣧⠀⠀⠀⣌⣿⣿⣿⡇⠘⡄⠀⡀
⠀⠀⠀⠀⢠⣿⠀⢸⣧⣰⡿⠈⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⢦⡀⠀⠀⣿⣿⣿⡇⠀⢸⣠⠃
⠀⠀⠀⠀⠘⣿⠀⠀⣿⣿⣿⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣾⣿⣿⣿⡇⢀⡾⠁
⠀⠀⠀⠀⠀⢻⡄⠀⣿⣿⣿⡏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣷⣼⠁⠀⢸⡆
⠀⠀⠀⠀⠀⠘⣧⠀⣿⣿⣿⡇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣇⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠘⣦⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣿⣿⣿⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢠⣿⣿⣿⣿⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢇⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⡀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡄⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⢠⠃⠀⡊⣉⣷⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣧⣧⠀⢀⠀⠀⠀⠀⠀⢈⣽⡇⠀⣿⠀⠀⣄⡟⣠⣶⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠟⠛⠻⠿⡿⠶⡷⠶⠶⣤⣬⣾⣿⣸⣿⣶⡶⠿⡿⣿⠟⠋⠉⢙⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠿⣧⣤⣤⣴⣷⣤⣧⣦⣀⠠⠬⣿⣿⠛⡲⣗⡂⢠⢠⣿⠀⠀⠒⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣨⣿⣾⣿⠁⠀⠀⠴⠿⠿⢭⣭⡭⠀⠼⠿⠷⠦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠼⠶⠒⠾⠖⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠄⠀⢠⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢴⠶⣚⠛⠂⢤⣶⠖⣿⡵⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣀⠀⣀⡤⠄⠒⠂⠀⠐⠁⠀⠀⠉⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠝⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠑⢦⣄⢀⡀
⠀⠀⠀⠀⠃⠀⢀⣰⠆⢐⡢⠠⡤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠉⠁⠀⠁
⠀⠀⠀⢀⡠⠔⠋⠱⠚⣉⣀⣞⠁⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⣀⡤⠄
⠀⠀⠀⠉⠀⠀⢀⡀⠀⢉⢀⢈⠄⡀⣀⠀⢔⠀⣴⠃⢀⣀⣀⣀⣐⣦⡀⢠⣤⣄⣀⣈⠣⡤⣾⠟⠃
⠀⠀⠀⠀⠀⠀⠛⠁⢠⣇⣀⣶⣿⣷⣿⣠⣾⣦⡟⠰⣿⣿⣿⣿⣿⣿⣿⣏⠙⢿⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⣿⣿⠟⠁⠈⢿⣿⣿⠃⠀⣿⣿⣿⣿⡿⠟⠋⢈⣾⣿⣭⣿⢻⡁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠈⠻⣭⡀⠀⠈⠛⠛⠉⠀⠀⠀⣾⣿⣿⠀⠀⠀⢳⠀⠠
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⢿⣿⣶⢰⠶⣤⣶⣾⡤⠀⠀⠉⣁⣀⠀⠲⠄⠉⠉⠉⢰⣇⣀⡍⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⠟⠀⠀⣼⠿⣿⣿⢿⣶⣿⣿⠄⠙⠒⠂⠀⠀⠀⠀⢹⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣀⣀⣶⣤⡄⢠⣿⣿⣿⡟⣾⣿⡆⢻⣿⡞⣿⣿⠄⠀⠀⣾⡇⠨⠨⠠⠤⢹⡿⠛⠿⠆⠐⠦⠤⠤⡄⠶⠒⠈⠁⠁
⠀⠀⠀⠀⠹⡏⠀⠈⠉⠁⠀⠈⢻⣿⡜⣿⣷⡄⢹⣿⡈⠻⡀⡀⠀⠁⠐⠀⠀⢀⠀⣾⢁⣿⣤⠀⠀⠀⠀⢤⣄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣿⣿⣷⣿⣼⠷⠀⢢⡁⠀⠀⠀⠀⠀⠀⠀⢧⠸⣿⣏⣼⣤⣴⠂⣀⣤⣶⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣾⣿⡻⣿⣿⣿⡿⠋⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⣀⣼⣶⡿⡟⠛⠛⠁⠾⠟⠿⠿⠋
⠀⠀⠀⠀⠀⠀⢰⡶⣾⣿⣷⣦⣍⠁⠉⠉⢋⡿⢶⣦⢄⣤⣤⣤⡾⣷⣾⠾⠿⠻⠛⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠌⠉⠉⠁⠀⠀⠀⠀⠀⠈⠛⠀⠈⠁⠉

⠀⠀⠀⠀⠀⠀⠖⠈⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠈⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⢑⣄⡤⢠⡀⠀⠀⠀⠀⠆⠈⢰⣄⠑⠀⠀⢣⠀⠀⠀⠀⠀⠀⠐⠃⢀⠐⠀⠀⠀⠂⡀⠠
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠈⠀⢠⠞⢀⡄⠀⠀⠀⠀⢀⠀⠀⠉⠀⠀⠀⠈⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠
⠀⠀⠀⡤⠀⠀⢀⠈⠀⠀⢈⠀⠀⠀⠠⠁⠀⠀⢀⠁⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠉⠓⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠂⠤
⠀⠀⠀⠉⠀⠀⠀⠉⠈⠀⠀⡄⠀⠀⠀⢀⠤⢄⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠞⠈⣀⡀⠄⠀⠀⠀⠀⢀⠄⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⠊⠉⠆⠀⠀⠀⠉⡎⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠄⠀⡀⠀⠀⠀⠀⠢⡀⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠆⠁
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠈⠀⠀⠈⠠⡀⠊⢲⣀⣀⠴⠧⠤⠃⠀⠀⠀⠀⠀⠀⠈⡆⠠⠔⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠑⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠙⠂
⠀⠀⠀⢀⠤⠐⠒⠒⠀⢄⠀⠀⠀⠀⠀⠀⢀⠀⠰⠖⠒⠠⢄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠠⠄⢀
⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠤⠤⠊⠀⠈⠁
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠘⠄
⠀⠀⠃⠀⠀⢠⠀⢠⠀⠀⠀⠃⠀⠀⠀⠁⢀⠀⠀⡀⢀⠀⢀⠀⠁⠀⠀⠀⠀⠐⠀⠠
⠀⠀⠀⠈⠀⠈⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠰⠂⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠦⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⡓⠀⢉⠀⠃⣀⢈⣄⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠉⠈⠲⡀⠀⠀⠀⠀⠠⠚⠂⠐⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠋⢿⠋⡙⡟⠉⡉⠻⠋⠻⠛⠃⠀⠀⠀⠀⠐⠀⠇⠀⠀⠘⠂⢀⠉⠀⠀⠀⠀⢀⣀⠀⠀⠈⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⣤⠄⠀⢀⡁⠈⠀⡁⢸⠀⣤⠀⠛⢀⡆⠀⠀⠀⢀⢀⡀⢤⠖⠀⠻⠑⠀⠀⠀⠀⢀⠀⡀⣤⡶⠐⠻
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠃⠀⠚⠛⠋⠘⠛⠛⠙⠛⠛⠉⠙⠛⠉⠀⠀⠀⠀⠀⠀⠐⠄⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠰⠀⠐⠚⠁
⠀⠀⠀⠀⠐
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠠⠄⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢘⠀⠀⠛⠉⠉⠉⠉⠙⠀⠀⢉⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠀⣀⠄⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠁⠀⠀⡃⢘⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢘⠀⠀⣁⠀⠀⠂⠀⣀⠀⠀⢈⠀⠀⢘⠀⠀⠀⠀⠁⡐⢓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠈⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠁⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠛
⠀⠀⣀⣠⣈⣡⣴⣆⣀⣀⣤⣀⠀⠀⠘⠄⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⢰⡎⠀⠀⠀⠀⠀⢀⠔⠒⠒⢦⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⢀⡀⠀⠄
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠔⠉⠁⢀⡀⢘⠀⠐⢄⠀⠀⠀⠑⠄⠀⠛⠘⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠈⠁⠈⠀⠀⠈⠄⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠀⢰⠁⠀⠁
⠀⠀⢐⣇⣀⣀⣀⣀⣀⣀⣘⡇⠀⠀⠣⣀⠀⠀⠀⠀⣀⣀⡠⠞⠀⠀⠀⠀⠘⠈⠀⠀⠉⠚⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠑⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢠⣧⠀⠀⢠⣀⠠⣿⣿⠀⠀⢨⣶⣦⣀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⢶⡿⠀⠀⢘⢿⣷⣿⣯⣶⣶⠀⠛⢿⣯⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⢠
⠀⠀⠀⠘⣾⡷⠂⠀⠛⣻⡿⣿⣿⣯⡀⠰⢶⣿⠿⣇⣴⣤⡦⠹⣿⠇⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⣤⠚
⠀⠀⠀⡀⠀⠀⣠⣾⠀⠈⠀⠘⠃⢩⣤⣤⣴⡄⠀⠉⣁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠃⠀⢢⠇⠀⢀⠄⠀⢀
⠀⠀⠀⣿⣧⣰⣿⠃⠀⠀⠀⣤⣾⣿⣿⣿⠟⢀⣴⣾⣯⡀⢰⣿⣧⠀⠀⠀⠠⡀⢰⠁⠈⠣⣀⡼⠒⠋⠁⡠⠏⠁
⠀⠠⣤⣼⣿⣿⣷⣶⡷⠂⠀⠛⠛⠟⠛⠉⢠⣿⣿⣿⣿⣿⠀⠻⠿⠀⠀⠀⠀⠀⠘⠢⣀⡀⢸⡇⣀⠤⠯⣤⠤⠤⠄
⠀⠀⠀⠀⠉⠻⣯⡀⠀⠀⢼⣿⣦⣶⣿⠀⠀⠉⠙⠻⠿⠛⣀⣧⣀⠄⠀⠀⠀⠀⠉⠙⠉⠙⣾⡿⠁
⠀⠀⠀⠀⣠⣤⣄⡁⠀⢀⣼⣿⣿⣿⣿⣿⠆⠀⠀⢀⣴⠀⣺⣷⣻⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠸⣿⣿⣿⣿⣆⠈⠛⣿⣿⠟⠉⠀⠀⠀⠀⢿⡿⠀⢺⣿⣺⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣠⡉⠉⠉⠉⠀⠀⠙⠁⢰⣤⣄⠀⠀⣤⣬⣶⣆⠀⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠾⠟⠣⣦⡿⢿⣠⣤⡤⠀⣾⣿⣿⡇⢸⣿⠇⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠀⠀⠈⠛⠋⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢰⣀⣶⡀⢸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠏⠈⣀⣉⣼⣿⣿⣿⣷⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⠾⠛⠛⠷⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣷⡀⠀⠀⣿⠿⢯⡀⢿⣿⠿⣿⠋⠀⠀⠀⠀⠀⠀⣌⣳⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⣡⠋⠁⠉⢣⠀⢀⣏⣀⠀⢹⣼⣷⣴⣿⣤⠀⠀⠀⠀⠀⠀⠈⠻⣧
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⡇⠀⣰⣿⣿⡄⠀⣿⣿⣧⢠⡇⣿⣷⣶⣤⣄⠀⠀⠀⠀⢀⡀⢀⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⣇⠀⢻⣿⣿⠂⠀⢻⣿⠿⠚⠛⠛⠛⠿⣿⣿⣧⠀⢠⠀⣸⣿⡟⠁
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠿⠟⠻⢿⣿⣿⣿⣿⣷⡄⠀⣀⡙⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣠⡾⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠘⢿⣿⠃⠤⣄⡀⠈⢿⣿⣿⣿⠿⠃⠀⣿⣿⣶⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⠴⠿⣦⠀⠙⠛⠛⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣷⣦⣄⣀⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⣿⣿⢿⣿⠿⣿⡿⠛⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⢿⣿⣿⣷⣦⣄⣀⣀⣀⣀⣀⠀⠸⣿⣿⣿⣿⣃⣤⣴⠾⠛⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠿⠛⠉⠀⠀⠀⢀⡀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠠⢤⠀⠀⠀⠈⠙⠻⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣠⢶⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⠈⠓⠂⠀⠀⠀⡀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⣠⣴⣫⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠘⣷
⠀⠀⣠⢞⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⡀⠀⠰⣆⠀⠀⠈⢧⡀⠀⣿
⠀⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣸⣿⣿⣶⣶⠀⠀⢿⠀⠀⠘⣧⡀⠀⢘⡿⠛⠁
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣿⣿⣿⣿⣿⣄⣀⣾⣷⣤⣤⡿⠛⠛⠛⠁
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣶⣦⣄
⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠟⠛⠿⣿⣷⡀
⠀⠀⠀⠈⠉⠛⠛⠛⠻⢿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦⣤⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⢶⣭⣍⣛⣛⠛⠛⠛⢛⣛⣻⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠤⡤⠄⠀⠀⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠐
⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⠁⠀⠀⠀⠀⢰⠀⠀⠠⣠⡶⢶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠉⠉⠀⠀⠠⠤⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠉⠉⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠠⠠⠀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠇⢨⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢎⣁⡀⢢⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠂⠀⠀⣂⡤⠀⠀⠢⠈⠇⠆
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠉⠃⠀⠈⠉⠉⠁⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠠⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⡔⡀⠀⡠⡀⠀⠀⠀⠽⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠰⡁⠀⠀⠀⠁⠀⠀⠉⡀⠘⠁⠱⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠂⠠⢄
⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠤⠀⡨⠀⠀⠠⠀⢀⡀⠀⠀⠄⠂⠀⠀⠀⢸⠀⢐⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠈⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠂⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠀⠒⠒⠊⠉⠀⠀⠀⠂⣀⣀⣀⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠀⠀⠂⢠
⠀⠀⠀⠀⠤⠦⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠢⠔⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠛⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠨⣳⠄

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠸⠀⢀⡄⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢮⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠉⢣⡄⠀⠀⠀⣤⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡀
⠀⢀⠀⠈⠁⠠⠂⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⢀⣀⡴⡼⠁⠀⡠⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠀⠀⠀⠀⢀⠊⢹⣷⠻⡆⠸⣴⠏⠠⠔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄
⠀⠀⠀⠀⡄⠰⠏⠀⠀⡀⠀⠀⠀⢀⠀⠀⣿⢷⣿⣿⣱⢿⣾⢷⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁
⠀⠀⠆⠀⠀⠀⠂⠀⠀⠁⠀⠀⢀⠈⠀⠀⡹⣾⣿⡿⠁⣾⢃⠌⣰⠀⠀⠀⢠⡜⠁⠀⠀⠀⠀⢀⡀
⠀⢸⣀⡆⠀⡀⠀⠀⠀⢀⠤⡀⠀⠑⠀⠀⠁⢿⣿⡇⢠⠛⡏⣴⠿⠒⠀⣴⠁⠀⠄⢠⠈⠀⠀⠈⠙
⠀⠐⠀⠀⠈⠇⠀⢀⠔⣋⡀⠄⠁⠷⠛⢷⡉⠷⣼⣷⡶⢻⣠⡿⣢⠴⠆⠀⠀⠀⢶⠂⠀⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠖⠈⣩⠖⠈⠀⠀⠀⠀⠀⠀⠚⠳⣦⣴⡬⠛⠛⠿⣵⠄⠀⠀⠛⠓⠚⣁⡴⠆⠰⢤⠤⠄
⠀⠰⡄⢰⠀⡔⠁⠀⠐⣦⡠⡄⠀⠀⠀⠀⠐⠉⣙⡕⠊⠉⠑⣖⠓⣞⡤⡶⠐⣊⠁
⠀⠀⠄⡌⡀⠀⠀⠀⠀⠈⠻⣶⣃⡀⠀⠀⠀⠀⡿⠀⠴⠀⠛⠹⡆⣉⢷⡄⢀⢈⠙⠢⣀⠀⠀⠀⣠⠄⠶⠀⡄⠀⠀⠀⠄
⠀⠰⡎⢀⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠇⠀⠀⢀⡝⠳⡄⠰⠟⠀⢿⣸⠎⠷⠆⢘⣋⡄⢹⠗⠂⠉⠐⠖⠒⠈⠀⢠⠾⠂
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠰⠀⠉⠒⣰⣷⡶⠟⠊⣠⣔⣻⣧⢈⣁⣙⣧⠳⣄⢦⣄⠠⠀⠉⣸⢇⣀⡀
⠀⢰⠆⡄⠷⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣆⣼⠆⣠⣤⡀⢠⠞⢢⣴⢇⣼⠿⠿⡄⣉⡻⣹⡆⠈⠴⣉⣦⡰⠞⠁⠀⠈⢉⠆
⠀⢨⡂⡇⠘⢇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⡧⠴⠇⠀⠳⢶⣢⣶⣶⣿⣷⠄⣀⠿⢿⠙⢦⣛⡶⡘⠗⣤⠸⣄⠀⠰⠆⠈
⠀⢠⠍⠀⠀⠈⠷⢀⠀⠀⠀⠀⠀⠈⢉⠅⡲⣓⢛⠶⠹⠃⠌⢹⡄⣀⠙⠶⡫⠄⠰⢇⠠⡏⢳⠚⣤⠀⠙⢿⡓⠳⣄
⠀⢠⠀⣆⠐⠀⣄⠀⠙⠙⠢⠄⣠⣖⣁⡴⢉⠋⡽⢛⢄⣼⢳⣷⡾⠙⠥⠀⡿⣤⡚⠁⠀⠿⠀⠃⢻⡉⠩⣈⠣⢤⣁
⠀⠈⠀⠉⠠⡀⠈⠀⠀⢀⠀⠉⠉⣠⠞⣴⠋⣼⠳⠟⣘⡣⠙⠁⢀⡐⠂⠤⠭⢙⣡⡴⠂⠀⠀⠙⢮⣹⠶⠺⢿⠂⣉⠓⡀
⠀⠀⢀⠀⠰⠁⠀⠀⠴⣿⠀⠀⠀⠀⠘⠀⠢⠙⠒⠎⠁⠴⠣⠾⣦⡀⠀⠀⠀⠈⠳⠶⢞⡉⣤⡄⡀⠈⢷⡒⠦⠀⢉⡜
⠀⠀⠈⠂⠀⠀⠀⠀⢠⣄⠙⠆⣰⡆⠀⠂⠀⠀⠈⠀⠀⠀⢰⡞⢻⣿⡧⠀⠀⠀⠀⠀⠈⡡⠟⠐⣻⣿⡀⠘⢦⠁⠈⠓
⠀⠈⠀⢀⠄⢰⡆⠀⠰⣿⠀⠘⠉⠇⠀⠀⠀⠀⡄⠀⠀⠀⢠⡿⢇⣈⣿⣀⠀⠀⠀⠀⠀⢀⠀⠀⣿⠏⠿⣄⠈⠼
⠀⠀⠀⠈⢀⣈⠹⠀⡀⠁⣾⠀⠀⠃⠐⠀⠀⠀⠃⠀⠀⠀⢷⣄⣤⠙⣟⠻⡶⣦⣤⣄⣰⡁⠀⢸⣿⣿⠛⡇⢠⠀⠀⠀⡀
⠀⠀⡄⠀⠈⠟⠀⢠⡁⠢⠘⠄⠀⠀⠐⣦⠠⡀⠀⢀⠀⠀⠀⠀⢠⡁⣘⣇⠠⡏⡹⢏⡿⢻⢦⡏⠀⠀⠀⡀
⠀⠀⠁⢀⡈⠀⣀⣾⠑⠊⠀⠀⠀⠀⠈⠈⡠⡏⠀⠘⠀⠀⠀⠀⠀⣉⣹⠟⢻⡷⠇⠈⠱⡟⢬⣿⣶⠟⣿⣀
⠀⠀⠀⠸⠓⠁⠙⢉⠀⠀⠠⢦⠀⢠⡀⠄⠡⣁⠀⠀⠀⠀⠀⠀⠈⠉⠀⣤⣼⢁⢦⡀⠆⢻⡆⣿⠻⣿⣽⣯⣶⡖⠀⠠
⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠈⣇⠄⠻⢠⠀⢈⡇⠀⠐⠀⠀⠀⠀⠀⠀⠤⠜⣡⡜⠀⠈⠀⡌⢛⣦⡁⠈⠉⢻⡎⣠⠀⠇
⠀⠀⠀⠀⢳⡤⠤⣄⡈⠀⠀⠈⠀⡀⣦⠈⢀⣊⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⠁⠻⡄⠀⠀⠸⡇⠙⠻⣦⣡⣤⢹⡗⠀⡀
⠀⠠⠀⠀⠈⠃⠀⠈⠁⠀⠀⠀⠞⠉⢈⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠙⠄⠀⠀⣷⢀⡀⠟⣀⠀⠀⠀⠀⠁
⠀⢰⣤⠐⠒⣋⠟⣚⣡⡾⠲⣟⣵⠞⣍⣓⣶⣤⣼⡵⠦⠠⠀⢠⠀⠠⠄⠠⠤⠆⠤⠤⠔⠉⣉⣠⣾⠷⢦⠟⠷⣦⡄
⠀⢸⡟⠛⠛⠟⠛⠉⠏⠑⠛⠋⠉⠀⡁⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠙⠋⠴⠁
⠀⠸⡇⣴⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⣀⣀⡀⢘⡀⢀⠐⠀⡀
⠀⢠⡀⡀⢀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠋⠁⠁⠀⠈⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡆⠰⣤⠀⠀⠀⣿⠀⠄⢸⠉⠲⠤⠤⠶⢊⡴⠋⠁⠀⣀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠈⠙⠳⢶⡚⠁
⠀⠀⠀⠀⠧⠀⢤⢤⠤⢤⣥⡀⢀⠸⠀⠀⠀⠀⣰⠋⠀⠀⠀⣰⠋⠳⣄⣠⠞⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⢤⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠶⢤⡾⠁⠀⠀⠀⢠⠇⣠⣄⠈⠁⣠⣆⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠈⠉⠉⠉⠙⠲⠤⣤⣤⣤⠤⠖⠋⠀⢀⡞⠀⠀⠀⠀⠀⣟⣠⠏⠈⠳⠾⠁⠸⠦⠼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⠳⠦⣤⣠⣤⠤⠶⢤⣤⣤⡤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⢀⣀⣬⠥⠶⠒⠚⠛⠛⠛⠛⠒⠲⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡴⠚⠉⠁⣀⣤⠤⠴⠶⠶⠶⠶⠶⠤⢤⣄⣀⡀⠉⠙⠶⣄⡀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⠋⢀⡠⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡈⠉⠛⠶⣤⣀⠙⢦⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⡴⣿⢀⣴⣶⣶⣶⣄⠀⠀⠀⠀⢀⣴⣿⣿⠿⣿⣷⡄⠀⢼⡉⠛⠾⣇⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⡤⠤⡖⠒⠲⣄⠀⠀⠀⢈⡟⢰⡏⣾⡟⢉⠝⢿⣿⡇⠀⠀⠀⡿⠿⢋⠔⠀⠢⡙⢿⠀⠀⠻⣄⠀⠈⠙⢦⣀⣸⣇⣤⣀⠀⡀
⠀⢀⡴⠛⠛⢋⠀⠀⠈⢂⠀⠘⡆⠀⢀⡾⠛⣿⠀⠋⢠⠁⢰⣶⡀⠀⠀⠀⠀⠀⠀⡌⢀⣾⣦⠰⠈⠀⠀⠀⢹⡆⢀⡼⠋⠉⡽⠁⠀⡨⠋⠉⠳⣤⣤⡀
⣰⠛⠢⡀⠀⠀⠑⡀⠀⣴⠛⠋⠉⠛⠻⣄⠀⡟⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡇⠃⠀⠀⠀⠀⣇⣾⣥⠶⠚⠓⠶⢴⡁⠀⠀⡐⠀⠀⠙⢦
⡟⠀⠀⠐⠄⠀⠀⠐⠀⢸⡆⠀⠀⠀⠀⢸⣂⡇⠀⠀⠠⢸⣿⣯⠷⠶⠶⠶⢤⡀⠀⡄⣿⣿⣿⣷⣇⠀⠀⠀⠀⠿⣿⠁⠀⠀⠀⠀⢨⠇⠀⡐⠀⠀⠀⠔⠉
⠻⣄⠀⠀⠈⣀⣠⠴⠓⠁⠙⠶⠖⠁⣠⠏⡿⠀⢠⣦⢘⡾⠋⠀⠀⠀⠀⠀⠀⠙⢦⢐⣹⡿⠟⠁⣿⠀⠀⠀⠀⠀⢹⡄⠀⢦⣀⡴⠛⠦⣴⡀⠀⠀⠊⠀⢀
⠀⢹⡉⠉⠉⠀⠀⠀⠀⠀⠈⢀⣠⠞⠁⢰⣇⠀⣼⠝⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⠀⠀⢘⣿⠀⣀⠀⠀⠀⠀⢹⣦⡀⠀⠐⠄⠀⠀⠈⠑⠒⠶⢲⡏
⠀⠀⠳⣤⡀⠀⠀⠀⠀⠐⠈⢹⡇⠀⠀⠈⣏⠙⢾⡇⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠰⣿⠟⠋⠁⠀⠀⠀⢀⡟⣀⣽⡗⠤⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⣇⠀⠀⢀⣀⣠⠤⠶⠺⢧⣀⡀⠀⠹⡄⠀⠻⣆⠻⢦⣀⡀⠀⢀⣀⣠⠞⠃⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢀⡞⠛⣁⡼⢤⣄⣀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠙⠻⡏⠉⠁⠀⠀⠀⠀⠀⠈⠉⠛⠓⠻⣄⠀⠈⠛⠶⡾⠿⠟⠛⠛⠳⠶⠶⠚⠋⡗⠀⠀⠀⠀⠀⠀⣠⠞⠛⠉⠁⠀⠀⠀⠉⠙⠓⠲⣶⠶⠃
⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠹⠒⠲⠶⠶⠶⠒⠚⠙⠚⠁⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠙⣷⢤⡀⠀⠀⠠⣄⣀⡤⠀⠀⠀⢠⠄⣀⣤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋
⠀⢦⢄⡀⠀⠀⣰⠀⠈⠓⠦⣤⣀⡀⠀⠀⠀⠀⠀⡿⠀⣿⠀⠙⣦⣀⠀⠀⠀⠀⠀⣀⡴⠿⡟⠁⢰⡇⠀⠀⠀⢠⡦⠤⠶⠒⠛⠉⠁
⠀⠘⢆⠉⠓⠞⢁⣖⣒⣒⠒⠒⠉⣩⠛⠛⠓⠒⠲⡇⠀⢿⠀⠀⡇⠉⢙⣛⣛⣛⣋⣉⣀⣘⡇⠀⢸⡇⠀⣠⠴⠞⠓⠒⠲⢤⣄
⢀⡠⠚⠀⠀⠀⢿⡷⠞⠉⠀⠀⠀⠑⢤⡀⠀⠀⠀⡇⠀⢸⠀⣠⠿⠚⠋⠉⠉⠁⠀⠀⠉⠉⠛⢒⣼⣤⠞⠁⢀⠤⠤⣀⡀⠀⠈⢳⡀
⠯⠀⢰⠞⠓⢄⣘⣳⠶⢾⠀⣴⠗⢶⡆⠽⠂⠀⠀⣧⡤⢾⠟⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢹⠀⡔⠁⠂⠄⡀⠙⢦⠀⠀⢳
⠀⢠⣋⣴⠞⠟⠁⣰⡒⠚⢟⠷⢤⡀⢹⡀⠀⠀⠀⣿⠀⠸⣄⣀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣴⠇⡼⠁⠒⠤⢀⠀⠑⠺⡆⠀⣸
⢀⡼⠁⢠⠟⠒⠚⠁⣅⠀⠀⡸⠂⠛⢮⡀⠀⠀⠀⢹⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠒⠠⢀⡀⠈⠐⢰⠃⢠⡟⠉⠉⠉⡏⠉⠉⠉⡋
⠟⠒⠒⣡⠤⢄⠀⢀⣀⣉⠉⠀⠀⠀⡞⠙⣆⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡏⠑⠢⢄⣀⠈⠑⠢⣞⡴⠋⠉⠉⠋⠉⠉⠉⠋⠉⠉
⢱⡄⢸⠁⠀⠈⣷⠋⠀⠈⢣⢴⠂⠐⠷⣂⣼⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⢸⡃⠂⢄⡀⠈⠉⠒⣼⠋⠀⠀⠐⠒⠓⠒⠖⠒⠓⠒⠖
⠻⢤⣌⣦⣀⣴⣋⣦⣄⣤⣋⣦⣠⣤⠴⠞⠁⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠳⣄⡀⠈⠁⢂⡴⠏⠠⡄⠀⠆⠀⡤⠀⠒⠀⠤⠀⠂
⠀⠀⠀⠀⣨⡏⠀⠙⢦⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡬⠉⠛⠛⠉⠛⠛⠉⠀⠀⠀⣇⣀⣀⣀⣇⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠁⢘⣀⡠⠤⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠊⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⠊⠀⢀⢤⡤⠆⢁⠠⢄⣰⡭⡄⣤⡤⣌⢑⡄⠀⢠⠀⠀⢠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⣸⠉⠈⣖⠁⢹⢊⠇⢸⢪⡀⠀⣇⠀⢰⡏⠀⠘⠁⠘⠆⠉⡶⠂⢻⠤⢖⢃⠠
⠀⠀⠀⠀⠀⠀⠘⡤⠶⠀⢿⣧⠀⣽⠀⣜⢸⠀⡏⠁⠇⢠⠇⡇⢸⢹⠀⡴⠀⢀⡆⠀⡟⡄⢠⡄⠸⠂
⠀⠀⠀⠀⠀⠉⢨⠸⠸⠀⢸⡏⠀⣿⠀⣇⣾⠀⢧⠀⡇⢸⠀⡇⢸⢸⠀⣧⠃⠸⡟⠀⢷⠃⢸⠃⢈⠄
⠀⠀⠀⠀⠀⠀⠀⠑⠊⠓⠤⠣⡀⢀⡀⠈⣿⠀⠈⠲⡇⠀⠉⠀⠈⡎⠀⢹⡀⢀⡷⠤⠺⢤⠎⢉⠉⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⢂⠉⠙⠈⠢⠤⠔⢉⠒⠊⠒⠊⠑⠒⠁⠍⠉⠂⠀⠀⠀⡐
⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⠀⠀⠰⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢂⠀⠂⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠚⠋⠉⠉⠉⠉⢛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠰⡀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠐⠉⠀⠀⠀⠀⠀⢀⣀⡀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡴⢋⠔⠁⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠈⢢⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⣰⠋⠉⠀⠀⠀⠀⠎⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⢠⠀⠀⠀⠀⠱⣄
⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⢆⡀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠒⠁⠀⠀⠀⠀⠀⠘⢦
⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⣀⠤⠐⠒⠢⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠒⠄⠈⢧
⠀⠀⡼⠁⡀⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠁⠈⢧
⠀⣸⢣⠊⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⣀⣀⠴⠁⠀⠘⡇
⢀⡏⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⢸⠁⠀⠳⢄⡀⣀⡠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠤⠠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠆
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠇
⢹⡀⠀⠀⠀⠀⢀⠀⠀⠀⠘⢦⡀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⣠⠏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠙⢦⣀⣀⣀⣀⣱⣦⠤⠤⠴⠟⠒⠒⡶⠛⠛⠛⠛⠉⠉⠉⠉⠉⠙⠛⠛⢻⠓⠒⠒⠦⠤⠾⢤⣠⣴⣂⣀⣀⡤⠊
⠀⠀⠀⠈⠉⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⢧⡀⠀⠀⠀⠀⠀⠀⢀⣞⣀⣀⣠⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⡵⠀⠀⠀⠀⢀⣼⠋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢃⠞⠀⠀⠀⠀⠀⣾⡏⢸⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁⠀⠀⠀⠀⠀⢸⣿⢠⠃⣰⡟⡸⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⣤⣶⠆
⠀⠀⠀⠀⠀⠀⢠⠜⣿⡇⡠⣾⡧⠀⠀⠀⢸⣿⠏⢠⣿⣳⠃⠀⠀⡤⣤⡠⠊⠀⣠⠔⠊⢉⡴⠋⡾
⠰⡿⡢⠖⣄⡀⠀⠀⢿⠟⣰⣿⢃⣴⢦⠀⢸⣿⠀⢸⣿⠁⠀⢸⠃⢩⣿⠃⠀⢰⠃⢀⡴⠋⠀⢐⡇
⠰⣉⡘⠢⣁⢾⠖⢦⠀⠀⣿⡏⢸⣯⠜⢠⠏⠘⠛⢹⣿⠀⣠⠋⠑⠚⠋⠀⢀⡤⠛⢯⣀⣀⡤⠎
⠀⠺⠄⡤⠈⠷⣄⣚⠆⠀⠻⠧⠜⠷⠔⠁⠀⠀⠀⠘⠿⠚⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣠⠴⠂⢤
⠀⠀⠘⠦⡤⠛⠃⠉⠲⠄⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⣀⠠⠀⢀⣀⠤⣤⡤⢲⣦⠀⠘⢄⣀⡤⣿⡆
⠀⠀⣠⡤⡀⠀⠀⠀⡠⠒⠒⠛⠒⠶⢶⣿⡷⠴⠴⠶⠚⠋⠁⠀⠀⣿⡏⠀⣿⠃⣼⡟
⢀⣾⠋⣠⣿⠀⠀⣀⡀⠀⢀⣀⠀⠀⢸⣿⠀⠀⣤⡄⠀⠀⠀⠀⠀⣿⠁⠀⠛⠀⣿⡇⠀⣸⣷⠴⠲⣦
⢸⣯⠔⣿⡏⠀⣰⣿⠃⢀⣾⣯⠀⢀⣾⣿⠀⣰⣿⠀⢠⣿⡇⠀⡸⠁⠀⠀⠀⠀⣿⡇⠀⣿⡏⠀⢀⣿
⠀⠀⠀⣿⣇⡴⢻⣏⡠⢛⣿⡇⢀⠞⠀⠙⠋⠘⣿⣀⢼⣿⠀⣰⠃⠀⠀⠀⠀⠀⢽⡇⢠⣿⠇⠀⣾⡟
⠀⠀⠀⠈⠉⠀⠀⠉⠀⠘⣿⣧⠏⠀⠀⠀⠀⠀⠈⠁⠀⠻⠞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⢋⣬⠹⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⣋⣴⣿⣿⣿⣿⣿⢟⣿⣯⣿⡿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⢇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣗⣿⣽⣿⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣁⣌⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣿⠿⣿⣿⣿⣿⣿⣿⣿⣇⠐⢿⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣒⣸⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⣿⣿⣿⣟⠋⡽⠋⠉⢻⣿⡿⠿⣿⡿⠛⣿⣿⣿⣿⣿⡟⣭⢹⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⣓⣽⣿⣿⣿⣿⣇⠀⠀⣼⡿⠁⠀⢈⡇⢀⣼⡷⠿⢿⣿⣿⣶⣾⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⠋⠈⠙⣿⣷⣾⣿⣷⣿⣿⠿⠿⠿⠋⢀⡴⢊⣽⣿⣿⡿⠿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⡿⣠⠴⠚⠙⠻⣿⣿⣿⣿⣿⣷⣶⡶⠋⠉⠉⠁⢿⣿⣿⣆⣣⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣧⣿⣿⣷⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢾⣾⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⠿⣷⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠂⢐⡁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠎⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠸⡇⢀⠀⠀⠀⠀⠠⢲⠀⡄⢠⣿
⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠍⠀⠀⠀⡁⠈⣄⠀⠀⠀⢰⡿⠰⡁⣹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠘⣇⢡⠀⠐⡆⠀⣿⠀⠀⣥⣾⠀⢸⡞⣡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡿⣾⠁⣠⠿⠃⠸⠿⠟⣄⠘⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠏⡌⠁⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⣻⣿⡄⢀⣤⠀⠄⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠄⡴⠛⠁⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⢀⡔⠁⠀⠀⡸⠀⠈⠛⠀⠀⣰⡿⢷⣀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠁⠈⠀⠀⠀⢰⡅⠀⢠⠇⠀⡠⠂⠀⠀⠛⡄⢰⡜⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⢹⡃⢀⡄⠀⠀⠀⠘⣷⣆⡈⠄⠀⠀⠘⢶⠷⠂⡠⢀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠿⣿⣄⡀⠀⠀⠸⠀⠀⢀⡾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢵⣠⣤⣶⢤⡴⠎⠁⠀⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠤⠤⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⠀⠀⠀⠀⢇⠀⠀⠀⠀⢰⣴⣤⣾⡆⠀⠀⢠⣿⡅⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣧⣤⣀⣄⣼⠀⠀⠀⠠⣿⠋⢻⣿⡿⠀⠀⠘⠿⠛⠓⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠷⠄⠀⠠⣄⣸⠀⠈⠉⠙⠺⠷⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠙⠳⣦⡟⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡷⠶⠶⠾⠃⢰⠛⠳⣄⠀⣸⢷⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡤⣄⡀⣤⣸⣧⣤⣏⡤⠞⠉⠉⠀⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⠛⠉⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠸⣧⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⢟⣳⡶⢄⣀⣀⣰⠞⠉⡉⠳⣦⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣯⣀⠀⠀⠀⠀⠀⠀⠀⡸⠀⣁⣤⣄⠀⠈⠀⠀⠿⠉⣤⣻⣦⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡉⠁⠀⠀⠀⠀⠀⡀⠺⠇⠾⢋⠁⠌⠓⠀⠀⠀⠻⣀⠀⣿⣙⠀⣿⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⡀⠀⠀⡼⢷⠀⠸⣇⠉⢀⠸⠀⠀⠀⠘⠀⠀⢀⣿⣏⠀⠻⠷⠛⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠹⣎⣷⡀⠘⠦⣀⠀⠀⠀⢀⡔⡄⠀⡼⠃⢘⡇⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⢸⠋⠙⢦⣀⢠⡿⢆⣀⡀⢁⣠⠞⠛⠚⠋⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⢀⣸⠦⠦⠴⣿⣟⠻⠧⣌⠽⣯⡀⣀⡀⠤⠤⣄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣠⣾⣧⠋⠁⠀⠀⠀⠀⠹⢶⠤⠤⠄⢈⣴⠿⣁⠀⠀⠀⠱⣆⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⢸⡇⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣠⠟⠛⠳⣌⠳⡄⠀⠀⠸⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡸⢇⣀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢿⠀⠀⠀⣻⠀⠻⣀⣤⣸⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡈⢼⡓⠤⢤⣄⣀⡴⠋⠀⠀⠈⠷⣤⣠⠟⠀⣾⣩⠟⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣱⠀⠙⠒⠚⣯⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⡉⠁⠠⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠀⠀⠠⡎⣠⠛⠛⠶⢤⣤⣀⠀⠀⣀⡴⢿⣧⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⣴⠿⠀⠀⠀⠀⠀⠀⠉⠳⣶⠃⠀⠀⠙⠦⣄⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢤⡀⠀⠂⠠⠤⠀⠈⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠅⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠈⠙⠷⣤⣄⣀⡀⢀⣨⠟⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠀⠀⠀⠀⣷⣀⣀⣀⣀⣀⣠⣤⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢦⡴⠬⡙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡝⣂⣴⠆⠘⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢴⣢⠙⢷⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠓⠉⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣸⡇⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⢠⡾⠋⠱⣄⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡜⠛⢷⣄
⠀⠀⣿⠀⠀⠀⠀⠉⡉⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠠⠠⠴⠒⠉⠀⠀⠀⣿
⠀⠀⠙⠷⣤⣄⣀⣰⠁⠀⠀⠈⠁⠒⠒⠢⠀⠀⠀⠀⠀⠒⠲⠛⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠈⢦⡀⠀⠀⠀⠀⣀⣴⠟
⠀⠀⠀⠀⠀⠈⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠈⠉⠉⠁⠀⠀⠒⠒⠒⠂⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⣷⠶⠶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⢶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠋⠉⠁⠈⠛⠷⢶⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⡶⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣴⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡟⣻⣿⣿⣿⣿⣯⣁⣹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠟⠉⢉⡉⣀⣪⣱⡈⣄⡀⣀⠵⠩⠉⠙⠛⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣀⣌⡀⣐⠀⠘⢿⡟⢩⢉⠙⣿⣦⣶⣿⣶⣤⣲⣦⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⢻⡟⠂⠀⠀⢻⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤⠶⡴⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠄⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡂⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠄⡞⠀⠀⢠⠀⠀⢸⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⢸⡼⢁⡄⢰⡿⠀⢠⣾⡴⠋⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠹⣾⣇⢸⣧⡇⣇⣦⣮⣼⡷⣫⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣧⢻⠻⠸⠀⠟⠿⠉⠋⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⣀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⢠⡧⢠⡀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⢀⠀⠀⠀⣇⢠⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢝⠁⠀⠀⠀⣴⠈⠀⠀⣄⢀⡄⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⡸⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣖⠓⠀⠀⠂⣴⠈⠀⠈⠃⠠⡃⠀⠀⢧⡀⠀⠀⠀⠈⠁⠀⠁⢀⡤⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠄⠑⠀⠋⠁⠀⠠⣅⣠⡄⣀⣙⠶⠔⠢⡀⡀⠀⠠⡎⠀⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠜⢀⠀⠀⠰⠀⠀⠀⣦⠀⡤⠐⠈⠸⣅⢀⡀⠤⠄⠀⡃⠀⠀⠴⣠⡭⠃⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠓⠀⡣⠀⠀⢰⠘⣠⠀⠀⠀⠀⠈⠀⢹⠐⠂⡘⠀⢀⠀⡸⠁⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⢈⠄⠙⠀⠀⠈⠧⣈⢢⡀⠀⠀⠀⠀⠊⠃⢼⠀⢀⣰⠂⠄⠀⣨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠂⢐⡂⠀⠈⠉⠙⠒⠲⡄⠀⠁⠀⠀⠀⠀⠂⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢸⡄⠀⢠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠥⢳⡷⢄⡀⡤⡀⠀⠀⠀⠀⠀⠀⣆⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠁⢩⠔⠉⠀⠀⠌⣦⣀⢀⡀⢀⠴⠋⠀⠀⠀⠀⣠⠔⠀⠀⠋⠀⠅⠈⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠃⠀⠀⠀⣠⣼⣍⣤⠠⠔⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⣠⠊⠙⠁⢠⠘⠖⠈⠑⡄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠂⢺⣿⣤⠆⡈⠉⢀⣀⡀⠀⠃⠀⣄⣰⠖⠂⠀⠀⠂⡀⠀⠀⢀⣀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠘⠀⣉⠉⢸⣿⣶⠿⠉⠉⠢⠀⠀⠗⠚⠗⠲⣤⠄⣖⡿⢿⣿⣧⠖⠚⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠊⠈⢣⡀⠨⠋⠀⠀⠐⢲⡀⠜⠀⠀⢀⠀⣾⢧⡞⢹⣿⣄⣴⠀⠀⠀⠺⢚⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠈⠙⠳⣧⠀⠂⢉⠘⠀⠘⠂⢪⡔⠅⠌⢙⡛⠦⠤⠄⢉⡁⠀⢸⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⡸⢧⣤⡠⢒⡏⠀⠀⠀⠀⠀⡀⠉⢁⠀⠂⠂⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠙⠒⠢⢤⡤⠖⠚⠳⢄⠁⡄⠀⠀⠈⠀⠀⠐⠀⠁⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢀⠠⢄⠉⠉⡇⢸⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠈⠣⡰⠀⠠⢄⠁⠀⢀⠳⠄⢄⠀⠀⠀⠀⣀
⠀⠀⠀⡀⠐⠲⢂⣊⡴⠊⠀⢺⣆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠗⠀⠠⣰⠗⠁⢈⣠⠀⣰⠓⣀⡀⠀⢆⠀⡦
⠀⠀⠐⠣⠀⢐⠒⡏⢸⣀⠂⠁⡈⠂⡤⢳⣄⡴⠐⣥⡤⠤⠚⠁⠀⠀⠀⢧⡐⠉⠁⠠⣞⠀⠘⢫⡅⠦⡀⢠⠁
⠀⠀⠀⠀⢱⠉⠃⡇⢸⡰⠀⠀⡧⠊⠀⢸⡴⢁⡆⠡⠣⠄⠒⡐⠂⢶⠛⠃⢇⠀⠃⠄⢨⠄⡀⠋⡀⠀⠐⠆
⠀⠀⠀⠀⠘⠀⠂⠁⠘⠃⠀⠀⠃⠀⠀⠘⠁⠘⠁⠀⠋⠀⠛⠘⠁⠘⠃⠀⠘⠀⠀⠘⠘⠘⠁⠀⠐⠃⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡔⠓⢄⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢇⠉⣐⠀⠈⠘⡄⡴⠊⠗⡙⢉⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣏⠂⢄⠉⠰⠃⡰⢧⢇⡁⢸⠆⢁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⡕⠒⠒⠪⢕⠤⡀⠀⢿⡟⡇⠀⠀⠀⠈⠫⣍⠉⡴⠊⢨⠄⠀⡵⠂⠸⠥⢲⡒⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⡎⠙⠻⣶⡀⠀⠉⠪⡳⢌⢯⣿⠀⠀⣀⡤⠴⠞⡗⢳⡒⠛⠚⠳⠠⠆⣀⣤⣊⠀⠠⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⡇⡁⠀⠀⠘⣵⠀⠀⠀⠈⠳⣽⣝⣷⠋⠁⠀⡠⠊⠁⠘⡀⠁⣬⣀⠃⠠⠀⡶⢭⣀⣐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠱⡳⣄⣀⢴⡝⠀⠀⠀⠀⠀⠈⣯⠞⣆⠀⠀⣅⠄⠈⠐⢙⠀⢧⢸⡀⢒⠀⠈⡆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⣸⣝⣛⡟⣆⠀⠈⢆⣰⡀⠚⢢⣈⠎⢦⠮⣡⣹⠁
⡰⠋⠑⠂⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠫⡏⣀⠹⠊⢧⡀⠀⠁⠙⠢⠇
⣇⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⣠⣤⡶⠶⠶⣤⣴⠾⣾⣷⢦⣤⠿⠛⠷⠻⠿⣦⣤⣤⣤⣤⣀
⠘⢦⣀⣀⡤⠞⠁⠀⠀⠀⢀⡴⣛⡉⢢⡀⠄⢡⢬⡸⣌⣛⡿⠀⠁⠔⡠⣒⡪⡔⠘⡆⢉⢰⠋⢨⢿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⡏⣿⣷⠄⡇⣰⠿⢀⠅⣂⢀⢀⡈⢿⡜⣇⠳⠟⢣⣴⠣⠘⣾⡀⠻⡿⢃⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⢄⠑⠬⠵⠚⣰⠯⡡⢀⡈⠄⠀⠃⠀⢸⡇⡈⠙⠋⠉⡲⠓⠓⢾⡝⠲⠒⠋⠂⠙⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣧⡈⠂⡬⠀⡀⣾⡟⢢⠐⠀⡠⠚⠉⢉⢉⢺⡇⢀⢄⠰⠶⢇⠿⣿⠀⣷⡠⠖⢛⠓⣅⠾⣇
⠀⠀⠀⠀⠀⠀⠀⣼⠋⢅⣁⡤⢴⡀⣿⠿⠠⠀⠀⡇⢴⣫⢯⢣⢸⡇⠁⠣⡠⠄⡀⢝⠩⡀⣿⠡⢸⣟⡿⢸⢆⢻
⠀⠀⠀⠀⠀⠀⠀⢻⠇⡔⠁⣴⠲⡌⣿⠦⠴⣥⠤⡱⣤⣉⣩⢼⢸⣇⠔⠋⡉⠑⣔⣘⠀⢈⣿⠀⠘⠲⡖⠁⠜⢻⠇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢧⠀⡳⢭⠖⣿⣤⣦⡄⢱⠈⠂⢠⣴⡅⣿⡇⠀⡎⣟⡇⣸⣀⣪⣦⣿⡢⠂⡻⠋⢋⡄⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢙⠲⠤⠤⢿⣜⣚⣃⠜⢀⠄⢀⢠⠡⣺⠇⡁⡑⣢⢒⡏⣡⣤⣨⡏⣴⣕⠐⢈⠚⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣅⠖⡡⢌⡘⡇⢩⠝⡋⠉⠑⢦⡌⡄⣿⢀⠂⠐⠙⠉⢣⣙⣛⣼⠏⣤⣌⠑⣬⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣴⣶⣁⣹⢏⡀⡔⡭⣷⢈⡗⡈⠃⡸⣋⡙⢦⡕⣠⠀⡸⠁⡧⣝⣝⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⢤⣧⣉⣡⣞⣀⣤⡾⢧⣿⣿⣀⣏⣥⡶⠶⠶⠶⠟⠋

⠀⠀⠀⠀⡠⣀⡀⠀⠀⠀⣀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⣀⠀⠀⣠⡄
⠀⠀⠀⠀⠱⠕⠃⠀⠀⠀⢆⡼⠁⠃⠀⠀⠀⠠⠡⠁⠐⢀⣀⠤⡠⠀⠀⢬⡳⣀⢇⠇⠀⠀⣖⢆⣸⣿
⠀⠀⠠⠤⡄⠠⠒⠆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠙⠟⠂⠈⠀⡈⠜⠀⠈⠙⠄⠀⢀⠜⠛⠓⠇
⠀⠀⠈⠲⣈⡶⡤⡁⠀⠀⡀⢸⢓⠀⠀⠀⠀⠀⠔⠀⠠⡄⠀⠀⠀⠀⢀⠆⠀⠀⠠⡂⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠤⠀⢸⣄⠀⠀⠂⠶⠄⠁⠀⠀⠀⠀⠸⢄⣀⠀⠘⠀⠀⠘⡀⠀⠀⠎
⠀⠀⠀⠈⠀⠀⠖⠀⡀⠀⠀⠀⠘⣧⠀⣟⠀⠀⠀⡀⢤⢀⢠⡀⣀⡄⢀⠀⢀⠀⠀⠀⠀⠀⠢⠤⠦⠞
⠀⠀⠀⠈⠂⠀⠰⠀⠈⠀⠀⠀⠀⣿⠀⠀⣷⠿⠷⠁⡸⠈⠸⠏⠡⡇⠘⠚⠿⠃⠀⡀⠀⠀⢠⣄⠀⡔⡢
⠀⠐⣄⡀⡤⠀⡀⠀⠀⠀⠀⠀⠀⣿⣄⣰⠟⣼⠃⢸⠁⠀⢠⡐⠘⠁⠀⣼⠀⠀⠀⠌⠔⠀⠹⡨⠷⠤⡅
⠀⢠⢎⣹⠀⠀⠐⠂⠀⠀⠀⢤⡄⠀⠈⡁⠀⢿⠀⢿⡘⠗⡗⠃⠿⢆⡆⣇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢄⠀⠀⠀⠈⠀⠈⣴⣂⣴⢛⡞⠀⠈⠁⠀⠀⠀⠁⠄⠀⠀⠁⠈⠀⢦⠲⣼⣩⠀⠐⢤⡶⠄⢲⡀
⠀⠀⠀⠀⠀⠀⠒⠈⠀⠀⠰⠁⠁⠈⠫⠠⢶⠀⠀⢀⠀⠀⢀⡔⢅⠀⠀⠀⠀⠀⠕⠋⠋⢂⠀⠘⠀⠂⠀⠁
⠀⠀⠀⡎⠲⣲⠀⠀⠀⠀⡥⠀⠀⠀⢠⠀⠀⠀⠀⠛⠿⠂⠙⠰⠁⡀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⡠⡄
⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⢠⠀⠀⠀⣰⠁⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⡄⠀⠀⠀⠁⠸⠴⠀⠀⠀⢔⣁⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠂⠁⠀⠀⠀⠀⠀⠀⠁⠓⠀⠀⠈⠀⠀⠀⠀⠈⠈⠀⠁

⠀⠀⠀⣀⠤⣤⡀⠀⠀⠀⣀⣀⠀⠀⢀⡤⠠⣄⠀⠀⠀⢀⠠⡄⠀⣀⡀⠀⢠⡤⣤⡀⠀⠀⠀⠀⣤⡀
⠀⢀⣼⣇⣀⣀⣿⡄⠀⡜⠈⣿⣿⣧⠈⠗⢺⡏⠀⠀⢀⠈⢉⠁⢴⣥⣷⠠⣿⣀⣸⠏⠀⣤⡀⠘⠿⠇
⠀⣼⣥⣽⣿⡿⠛⢋⢼⣻⣷⣭⠋⠁⠀⠑⠚⣡⠔⠈⠙⢷⣄⠀⠀⢸⠀⢤⡈⠉⠧⠀⢀⣁⡿⣆
⠀⠁⠟⡟⢹⢧⠀⠈⠛⠾⠃⢹⢒⣄⢀⢄⠘⢿⣤⣄⣀⡸⠿⢶⡀⠀⠒⠾⠋⠀⡋⢀⣲⡄⠠⠿⠆⢠⣄
⠀⠀⠸⡇⠉⡇⠀⠀⠈⠀⠀⢘⢿⠋⠛⢺⠆⠀⠘⠋⠟⠋⠁⠈⠁⠀⠀⢸⠀⠐⠃⠈⠉⠀⠀⠀⢴⣛⣿
⠀⠀⠀⡇⠀⡇⠀⠀⠀⢳⣦⡜⡌⠀⠀⢽⠀⠀⢄⡾⢀⠀⠀⢤⣄⠀⠰⡜⠀⠀⠀⠀⠀⠀⢰⡆⠈⠙⠋
⠀⠀⡀⠷⣶⠇⡄⣠⠀⠀⠀⠀⠀⠀⠀⠈⢀⣠⠞⠙⠩⣷⣦⣼⠶⢓⣤⣔⠒⠒⢦⡀⠀⠀⢀⡆⢀⢨⡄
⠀⡞⢿⡀⠀⠀⠈⢉⡀⠀⠀⡀⠠⡰⣤⠉⠉⠀⢀⣀⠀⠀⢰⠀⠀⠻⣟⠛⠻⢶⣶⣿⣄⠚⠊⢀⣈⣧
⢀⡇⢸⣧⠀⠐⡲⠈⠁⠀⡸⡧⡑⣽⠻⡎⠀⢠⣿⣻⣷⡀⠘⡄⠀⠀⠀⠙⣺⠛⠛⠉⠁⣴⣶⡄⠉
⠘⠷⣾⠟⠃⠀⠀⠀⣤⠀⠈⢺⣿⢿⠨⠆⠀⢾⣾⣽⡿⠋⣤⣊⠉⣷⣦⠠⣽⠀⠀⠈⠻⢿⡶⠷
⠀⠀⠾⠀⠀⠀⠀⠀⠀⣴⠀⢴⡄⠘⡄⠀⠀⠀⠀⠈⡇⠀⠛⠻⣻⠟⠋⠀⣿⠀⢶⡷⠀⠈
⠀⠀⡄⠀⠀⠀⠈⠉⠀⠻⠿⠗⠃⠀⠀⠀⠀⠀⠀⢀⡷⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠁⠀⣆⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⡄⢲⡾⣧⡠⢀⣀⠐⣀
⠀⠀⠀⠀⠐⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠢⢤⣁⡀⢂⣴⣂⣄⣇⢦⣯⣷⣿⡗⣣⡍⠘⣤
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⢤⡄⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣵⢿⣿⣿⣿⣿⣿⣷⣿⣷⣿⣿⣿⣶⣿⣾⣶⡿⣶⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠡⠀⠀⠀⠀⠀⠀⠂⠀⡒⣴⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣯⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣐⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⠃⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣮⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣷⡦⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⠿⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⡄⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⡇
⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠄⣸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⠟⠀⠀⠀⢠⡒⠶⣮⢺⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠡
⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣧⣄⡀⠀⠀⠈⠀⠀⡾⠻⠝⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⢰⡟⠉⢩⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣤⣶⣤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⢾⡯⣿⣿⣿⣿⣿⢿⠿⣿⠋⣽⣾⣿⣿⢿⣿⣿⣿⣿⠻⡞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠉⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠷⢿⠿⣟⡿⠿⣿⣿⣿⣿⣿⣍⡬⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠁⠀⠀⠄⠱⠏⠉⠀⠂⠀⣛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡀⠀⠀⡠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠃⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⣀⡤⢤⣴⣶⣷⣿⣿⣶⣶⣿⣶⣖⣶⣶⣶⣾⣶⣶⣶⣤⣦⣤⣦⣄⣀⡀⠀⠀⠀⡀⣿⣿⡇
⠀⠀⠀⠀⢤⣠⣦⣴⣶⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣟⣋⣇
⢀⡤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣄⣀
⠘⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⠿⢃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠶⠰⠖⠢⠳⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⡌⢨⡏⣳⠤⠶⢄⠀⠀⠀⠀⠀⠀⠀⠄⠣⠤⠦⠤⠼⠴⠬⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠖⠋⠛⠁⠀⠀⠈⠤⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⢀⠔⣀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠅⠀⠀⠀⠘⡠⠀⠀⢠⠙⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⢂⠓⠀⠀⠀⠀⣿⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠑⣤⡑⣤⡀⠀⣠⡁⠡⠧⣀⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠇⡠⠒⠙⠻⡃⠊⠉⡁⠠⢀⢉⣑⢀⠀⢀⡀⠄⠒⠂⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣄⣇⠀⠀⠀⡁⠀⠀⠔⠲⠀⠏⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠛⠸⢤⠷⠤⠴⢇⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠒⠁⢀⣠⠖⠈⠁⠀⠀⡨⠧⠠⠀⢀⠸⢄
⠀⠀⠀⠀⠸⣥⠀⠀⠀⡀⠄⢐⣨⠏⠀⠀⠀⡠⠴⠋⠀⠀⠀⠀⠈⠢⡀⠑⣄
⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠈⡱⠀⠀⣠⠚⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠢⣀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠺⣳⣇⣀⣀⠤⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡏⠉⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠳⡄⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠓⠢⠤⣀⣀⣀⣀⣀⣀⣠⠤⠖⠋

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⢀⢤⠀⠀⡀
⠀⠀⠀⡅⠧⠀⠤⠤⠤⠭⠩⠶⠹⠏⠀⠀⠀⠀⠀⠀⠀⡌⠓⠖⠁⠀⠳⢚⠁
⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠒⠒⠦⢀⢠⠼⠧⢤⣄
⠀⠀⠀⠀⠰⠀⠉⠉⠉⠀⠈⠉⠀⠀⠀⡤⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠛⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠢⢄⡠⠐⠀⠀⠀⠉⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡦⠀⠀⠀⠀⠁⡚⠉⠁⠀⠀⠀⠐⠂⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⠘⠀⠸⢀⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢄⠀⡸⠀⠀⠠⠛⡄⠀⡀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠁⠀⠀⠀⠀⠀⠈⠘⠃⠀⠀⠀⣀⡠⠗⠚⠐⣄⢸⡐⠦⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡀⠀⠠⢹⠀⠀⡀⠀⠈⠀⠱⠀⠀⢀⣀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⢠⠖⠈⠉⠉⠢⡄⠠⠔⠀⠀⠀⠀⠁⢢⡘⠱⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠠⢦⡜⠢⡀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡠⠜⠀⠢⣄⠚⠂⠴⠞⠙⢦⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⡠⠄⠛⠀⠀⠀⠀⣀⠱⠀⠀⠀⠀⢠⡄⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⣀⠞⠁⠀⠀⡀⠀⠀⠀⠉⠉⠀⠀⠀⠢⡀
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢊⡏⠀⠀⠀⠀⠀⡰⠊⠁⠀⠐⠀⢁⠄⠈⠀⠀⠀⠀⠀⠢⢀⡀⠈⡄
⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠲⢉⡀⠀⠀⠀⢀⠼⠁⠀⠀⠀⠐⠈⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉⢃
⠀⠀⠀⠠⠀⠂⠁⠀⠀⠀⠀⠀⢠⠁⢀⡀⢠⠏⠀⠀⠀⠀⠀⣀⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠞⠙⠋⣹⣄⠀⠤⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⣴⠋⠉⠁⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⢻⡀⠀⠀⠐⢄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠳⣄⠀⠀⡀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠈⠙⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣤⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠶⠶⠖⠒⠚⠓⠤⢄⣀⣀⢀⠀⠀⢀⣀⣀⣀⡠⠤⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠠⠀⣴⣶⣶⣶⡄⠉⠀⠀⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⣰⡀⠻⠿⠿⠟⠃⠸⠂⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠘⢿⠟⠀⠀⠀⠰⠆⠀⠀⠘⠁⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣤⣾⣿⠟⠀⠃⠀⠈⠀⣠⣶⣶⣦⣄⠀⠀⠀⠀⠀⠻⣿⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠉⢀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣄⠀⢰⠄⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠆⠀⠈⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠠⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠺⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠛⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠿⠟⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⡀⠀⠈⠙⠛⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⡟⠀⠀⠀⠀⣠⡀⠘⠋⠀⣤⠀⠀⠀⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠈⠀⠀⠀⠺⣿⠗⠀⠀⢹⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣽⣿⣧⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠾⠿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣭⣥⣤⣶⠾⠃⠀⠀⠀⠈⠻⠿⠋⠛⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣛⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣠⠖⢩⠭⡉⠲⡄⠀⠀⠀⠀⠀⠰⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⢀⣥⠁⠀⠀⠀⠀⢰⡥⣚⣩⣭⣍⡒⣼⡆⠀⠀⠀⣴⡇⠐⡳⡄⠀⠀⠀⠀⠀⠀⡐⠊⠅⠀⠀⠀⢀⠉⣠⠐⡂⠠
⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⡼⢿⠉⣍⣀⣩⠉⡿⡇⠀⠀⢆⣀⣡⣿⠄⠈⠀⠌⠀⠀⠀⡀⠙⠒⠃⠀⠀⠀⠈⠀⠈⢳⣤⠧⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⢶⣧⣀⣼⡖⡧⠃⠀⠀⠈⠉⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠲⣠⠄⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠮⠭⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⢉⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠘⠀⠀⠻⣄⣨⠄⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⡀⠈⠐⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠑⠂⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡒⠀⢐⡄⠀⠀⠀⠀⠀⡀⠀⢀⡴⠠⠄⠀⠀⠀⠀⠀⠑⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠩⠶⠎⠈⢄⠀⠀⠀⠀⢀⡍⢩⠅⡉⡰
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠰⢧⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠄⠴⠀⠀⠀⠀⠀⠙⠚⠅⠀⢈⡅⢄⠀⠀⠀⠀⠀⠉⠃⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠊⠉⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡀⠀⠐⠄⠀⠀⠀⢀⠴⡋⠀⠴⣽⢀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⠁⠀⠂⠀⠀⠀⠀⠀⡍⠤⠔⢀⠀⡀⡞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⡄⠀⠀⠀⠀⠀⠀⠈⠃⠸⠀⠈⠡⠈⠀⠈⠡⡀⠀⠈⠁⠓⠁⢠
⠀⠀⢀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⠄⠀⠀⠀⠀⠀⠐⢀⣀⠠⠀⠀⠀⠈⠀⠈⠠⠘⠂⠀⠊⠁⠄
⠀⠀⣠⠈⢧⠟⠁⠈⠢⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⢀⠂⠀⠀⠀⠀⠀⢀⠆⠀⡠⠀⠂⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠄
⠀⠘⢆⠀⠀⠐⠀⠒⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣙⡤⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠄⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠷⠶⢤⣠⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠑⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠏⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠉⠙⢳⣄⡐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⡤⠀⠉⠉⠀⠀⠀⠀⠀⠈⠉⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣁⣤⠶⠓⠀⠑⠀⠀⠀⡇⠀⠀⠀⠀⢀⡤⢶⣞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⢠⣥⡀⠀⢠⣶⣵⠞⠛⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠤⠤⠶⠶⠶⣶⣿⠇⠀⢻⣶⣿⣟⠩⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣠⠜⠛⠻⠟⠇⠀⢸⡿⠵⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⠈⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⠀⠀⠀⠂⠉⢇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⢀⡶⠿⠳⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠞⠁⠀⠀⢆⠀⠀⠀⠀⠀⠀⢀⠀⢠⡞⠁⠀⠀⠀⠡⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢿⢲⡏⠳⣈⣰⣤⣠⡄⠀⠀⣰⣟⣀⠤⠀⠀⠠⠠⣶⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠟⠉⠀⠈⠁⠀⠀⢟⡫⣀⣠⡴⠖⣤⣀⠼⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⠃⠀⠸⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡇⠀⠀⠀⢀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⢱⠀⠀⠀⢸⡇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠺⠧⠄⠤⠼⠁

⣿⣿⢛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡀
⠀⠀⠀⠐⠆⠀⠰⠀⠀⠄⢀⠀⠀⠀⠆⠰⡀⠐⢆⠀⠀⠀⠠⠠⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠈⠐⠂⠃⠀⠀⠂⠀⠀⠀⠈⠃⠈⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⡉⠙⠻⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣠⡶⠞⠛⠛⠖⠤⣄⡈⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠛⠛⡛⠿⣦⡀⠀⠀⠀⠀⠀⠀⢸⠋⡠⠐⠀⠂⠒⠂⢄⠉⠓⢄⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠠⠂⠁⠀⠁⠨⠻⡄⠀⠀⠀⠀⠀⠸⣆⡀⠀⣠⠀⠒⠲⠆⢙⡦⡀⠣⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢀⡠⠴⠒⠛⠛⠛⠓⠲⠦⣷⠀⠀⠀⠀⠀⠀⠀⠉⠲⡏⠠⠀⡀⠀⠊⠁⠹⣆⢡⠘⠙⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⡇⣠⠞⣁⡤⢤⡞⠏⡁⡀⠀⠂⢄⠈⠻⣆⠀⠀⠀⠀⠀⢀⠼⣇⣼⣀⡀⣠⡆⠀⠀⠈⠚⠓⢤⠏
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⡰⠡⢈⠋⠀⠀⠐⠄⠀⠀⠘⢨⡄⠑⠀⢹⡆⠀⠀⠀⠀⠘⡞⠁⠀⠀⠉⠉⠀⠀⠀⠀⡴⠁⡸
⠀⠀⠀⠀⡴⠒⣆⢸⠋⡈⠙⠀⢸⠀⢸⣷⣰⡌⠀⣠⠌⡂⢱⣤⡶⠟⠀⠀⠀⠀⠀⠠⠛⠦⢄⡀⠄⠊⠈⡿⢀⣤⡤⠞⠁
⣰⠞⠲⣤⡇⠀⢸⢻⡀⠳⠠⠴⢚⡀⠈⠟⠉⠁⠉⠻⡴⠃⣸⢧⡀⠀⠀⠀⠀⠀⠀⠀⠑⠒⣖⢛⠲⠖⠊⢠⢋⣈⡇
⢻⡀⠀⠘⣇⠀⢸⠤⣙⡖⠂⠀⠀⡄⠘⢄⠀⠀⠀⢠⡄⢴⠇⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣦⡤⢀⣤⡿⡻⠟⣦⣄⢀⣀
⠀⠙⣦⠤⠼⡷⠁⠀⠸⣏⣪⡤⠀⠐⠤⠤⡉⠁⢉⠁⠜⣱⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡞⠀⠨⠁⢧⡁⡄⠀⡘⠉⣩⡇
⠀⠀⡿⠒⠠⣱⡤⠴⢡⠋⡏⠉⠛⠒⣄⠑⠭⢟⣡⢶⣍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢁⣴⠉⡶⠀⢀⡾⣄⠉⠀⠁⢿⠁
⠀⠀⠹⣤⡢⠾⠀⠀⣀⡼⠁⠀⠀⠀⠸⡀⠈⢢⠠⢆⠱⡝⢦⡀⠀⢀⣠⣤⣄⣀⡼⠃⣷⣚⠏⠈⠤⠤⠿⠰⡜⠦⠤⠖⢹⡆
⠀⠀⠀⠀⠉⠉⠙⠛⠉⠉⠀⣹⠀⠀⡰⣇⡔⠘⠄⠀⢳⢈⡀⡳⡎⠑⢌⠀⠈⠏⢇⢸⠃⠀⠀⠀⠀⠀⠀⠛⣿⢦⣄⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠢⠞⠁⠸⣑⣰⠂⠀⠀⠲⡟⠀⠘⠲⢤⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢨⡇⠀⡴⢻⡄⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⣀⠀⡼⠁⠀⠀⠀⠀⠈⠳⠴⠃⠀⠻⡆⠠⣀⠀⠀⠀⠀⢾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⡴⠊⠁⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⣇⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⡼⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⣼⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠈⡇⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⢛⣁⠀⠀⠀⢰⣿⠀⠀⠀⠀⣘⣧⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠥⣀⠀⠀⡇⠀⠀⠀⢀⣘⣷⣄⡀
⠀⠀⠀⠀⠀⠀⢠⠖⠉⠁⠀⠈⠙⠲⢤⣿⣀⡤⠒⠉⠀⠀⠀⠹⡄⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠙⠺⡦⠤⠴⠚⠉⠀⠀⠈⠙⣆
⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⣠⣿⡄⠀⠀⠀⠀⠀⢀⣠⡇⠀⠀⠀⠀⢻⡄⣀⠀⠀⢀⣀⣀⡧⢀⡀⠀⠀⠀⠀⠀⣀⣽
⠀⠀⠀⠀⠀⠀⠈⠻⠥⣤⣤⣬⠬⠽⠚⠁⠙⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠈⠛⠲⠶⠾⠛⠛⠉⠙⠲⠮⣭⣭⣤⠽⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⡌⠀⡀⠀⡥⠒⠉⠀⢀⡤⢲⠎⢀⠴⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠙⣟⠀⠀⠙⣆⠠⡇⡇⢀⡔⠁
⠀⠀⠀⠀⡀⠀⣸⠃⢀⠀⢰⠁⢀⣀⣰⣋⣠⣤⡴⠧⠒⠀⠀⢀⡠⠤⠒⣀⠀⠀⠐⠂⢀⡂⠄⠀⡀⠀⠀⣿⠀⠀⠀⠘⣿⣷⢷⣞⡀⠀⢀
⠀⢘⡐⢰⠃⡰⠃⢀⠈⡠⠃⠀⠘⠁⡝⠁⠀⢀⣄⠤⠀⢀⡴⣋⠀⠀⠀⠰⠋⣩⠟⠉⠉⠉⠉⠛⠧⣄⠀⢸⡇⠀⠀⠀⠸⣿⡾⡁⠀⣠⠋
⠀⢸⡇⣿⠞⠁⠶⢉⡼⠀⠀⣈⠀⡼⠁⣄⠐⣠⠃⣀⡶⠫⠤⠅⡦⠤⠖⢊⡉⠀⠀⠀⣰⣷⣶⣶⡊⠈⠳⣌⣇⠓⠠⣄⠈⣿⡇⢧⡴⠁
⢠⡞⣸⡏⠀⣀⡰⠋⠀⠀⣀⡵⠊⠁⣀⣭⣴⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⣿⣿⠃⠀⠀⡈⠻⣦⣀⠉⢠⣿⣧⠘⡇
⠏⢰⡿⠀⠀⣽⠁⢀⣴⠟⢉⣴⡶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡄⠈⠉⠐⠀⣹⣷⣿⣿⣿⣆⡿⣄
⠀⢸⡇⠀⢰⣿⡾⠋⠀⣰⣿⢋⣤⡁⠀⠀⠀⠀⠀⠀⠀⠐⠢⢀⣉⣉⣉⣙⣳⣤⣀⣘⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⡈⢹⣿⣿⣿⣿⠇⠙⣆
⠀⢸⡇⠀⣼⣿⠱⠆⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢲⣤⠤⠤⠤⠞⠋⠉⠁⠈⠉⠉⠛⠻⣿⣿⣿⡀⠀⠀⠀⠉⠹⣿⣿⣿⣿⠀⠀⠸
⠀⢸⣿⣰⣿⡏⢀⣶⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠋⠙⢶⣄⠀⠀⢠⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣿⣿⣿⣿⣿⡆⠀⣸
⠀⢸⣿⣿⣿⣿⡏⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣇⣰⡟
⠀⠈⣿⣿⣿⣿⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⡄⠀⣿⣿⣿⣿⣇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡀
⢷⣴⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇
⠈⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇
⠀⢹⣿⣿⣿⣿⣿⣷⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠄⠀⣿⣿⣿⣿⣿⡇
⠀⠘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⠍⢸⣿⣿⣿⣿⣿⣷
⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⡿⢣⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣃⡤⠾⡟⣻⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⢀⠀⠀⠀⠀⢀⡞⠀⠀⡇⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡼⣿⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢛⣻⣧⠀⢰⣯⠄⠀⠀⡞⣼⠃⣠⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣧⣤⣦⣤⣀⣀⠀⢀⢒⣻⣷⡒⣶⣻⣧⣤⣤⣿⣿⣾⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣴⣿⣿⠟⠉⢹⣿⡏⣿⣿⡃⠀⠈⠉⠉⡿⡉⠉⡏⢹⠛⠛⠻⠷⠭⣽⣯⣷⣿⠿⣿⠟⠛⠛⡛⢻⡿⠁⠀⠀⠀⣠⣿⣿⠿⠛⠛⠛⠻⠿⠿
⠛⠉⠀⠀⠀⠀⢿⣷⣜⢿⣷⣦⣤⣤⣴⣷⣷⣶⣷⣾⣤⣀⠀⠶⠶⣿⣿⡛⢛⡛⣟⣋⡁⢰⡇⣼⣇⠀⠐⠒⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠉⠀⠀⣿⣷⣾⣿⡏⠀⠈⠉⣉⣽⣿⣿⣝⣋⣛⣏⡁⢀⣼⣷⢿⣿⣤⣤⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠒⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠁⠀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⢀⡠⠄⡀⠀⠀⢰⠀⢀⡀⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⡔⠁⠀⠀⢰⠀⢀⠜⠊⠁⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢀⠎⠀⠀⠀⠀⡸⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢀⠎⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡄⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢰⠁⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⡄⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠒⠀⠈⠉⠉⠉⠉⠐⠒⠇⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡉
⠀⠀⠀⠀⣃⣀⡀⠀⠀⠀⠀⠀⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⢢⡀
⠀⠀⠀⠰⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠈⠁⠒⠂⠤⢀⣀
⠀⠀⠀⠀⢫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠲⠀⠀⠀⠒⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠊⠁⢀⡨⠃⠀⠀⠀⠀⠀⡌⠀⠀⠀⣀⠀⢡⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⡠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⡐⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠈⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⠤⠀⠤⠔⠊⠀⠀⠀⠀⠈⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠐⠒⠈
⣿⣿⡿⠿⣿⡿⢿⢿⡿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣶⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣱⡭⣄⢆⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠉⡒⠚⠮⣀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⣤⡔⠒⠿⠻⢟⡑⣄⠀⠳⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣧⣰⠀⠀⣼⡆⢉⠝⢣⣀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⣀⣐⣾⡤⠨⡤⢨⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⢶⡾⣿⠿⡏⠀⠀⠀⣈⣤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠷⢤⣈⠑⢀⣠⠴⡾⢤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡁⠀⠀⡠⠚⢿⠫⣵⠟⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠑⢶⠸⣁⠀⠀⠉⠀⠀⡠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠶⠞⠉⠉⢳⣦⣤⠴⠚⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠉⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⢳⡀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣧⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠼⠧⢄⣀⣸⠆⠀⣀⡤⠤⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⢹⠊⠛⠁⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⡤⠤⣤⣄⣺⣤⣀⣀⣀⣀⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠙⠚⠒⠒⠒⠒⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣦⣶⡇⠀⣀⠀⢀⢀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠟⠛⠁⠺⠁⠃⠀⠛⠻⠈⡌⠉⠛⣄⠀⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡼⠛⠁⠀⠀⠀⠀⠀⠀⠀⣥⠀⠀⠄⠀⠀⢀⡈⠀⢹⡅⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⡆⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢸⣿⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠲⠀⠘⡇⠡⠀⠘⠀⠀⠀⠀⠀⢀⡠⢠⡀⠀⢀⠀⠀⠀⠀⠀⠀⠁⡦⠀⠨⠀⠘⠓
⠀⠀⠀⠀⠀⠀⠀⢀⡆⠸⠁⣀⠀⠀⠀⠀⠀⡀⠒⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠰⡆⣤
⠀⠀⠀⠀⠀⢶⡴⠘⠀⡄⠀⠛⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠁⠀⠀⠀⠀⠀⠁⢸⣇
⠀⠀⠀⠀⢀⣾⡷⠀⠀⢣⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠈⣠⣤⣬⣯⠤⠄⢀⡀⢠⠁⠸⣿⣷
⠀⠀⠀⠀⣾⣿⠂⡀⠀⠘⠀⠀⠀⠀⣠⣤⠤⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣿⠯⣟⠯⠁⠀⠀⡰⠁⢘⡄⢸⣿⣿
⠀⠀⠀⠀⣿⣶⠀⠁⠀⠀⠀⠀⠀⣠⡏⠛⣀⡴⣿⣿⣍⣫⣹⣾⣾⣿⣿⠅⢚⡭⠀⠀⠀⢀⠀⠀⠀⠀⣄⠘⣿⣿⠃
⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⣠⣾⣿⣶⡯⣩⣵⠾⢿⡿⣿⡿⠿⠟⠋⣘⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠘⢱⣿⣿
⠀⠀⠀⠀⢸⣿⡇⢤⠀⠀⠰⣿⣿⣿⣯⣄⠀⠀⠈⣡⣾⣵⣶⣷⠶⡾⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟
⠀⠀⠀⠀⠈⣿⣷⡀⠀⠀⠀⠉⠙⠛⠛⠛⠲⠀⡾⠛⠛⠛⠛⢿⡀⠀⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠸⣿⣷⣀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⡇⠀⠀⠀⣀⡕⠀⠂⠀⠀⠀⠀⢀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠙⣿⣷⡈⡓⢀⠀⠢⠀⢄⣀⠀⠀⠉⠀⠀⡌⠀⣇⠀⢠⣾⣇⠀⠀⠀⠀⠀⢰⠖⢹⡟⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣇⠀⠿⠄⣀⢀⣿⣆⠀⣀⡀⠀⠇⠉⢿⣆⠈⢿⣿⣧⣄⡀⠀⠀⠀⠀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣏⣫⣤⣿⡞⣿⣿⡿⢻⣿⣢⠇⠀⢸⠇⠀⡀⢁⡾⠻⣯⠋⢉⣵⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠿⣿⣿⣏⠙⠿⢅⣐⣼⣧⣄⣡⠖⠁⢀⣼⡇⢸⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠚⠆⠸⣿⣿⣿⣿⣿⠿⠼⠟⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠛⠛⠳⣤⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣰⠟⠁⢀⠖⠉⠈⠑⢌⣷⡶⠚⠋⠉⠙⡆
⠀⠀⠀⠀⠀⣰⠏⠀⠀⢸⠀⠀⢀⡴⠛⠁⣀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⢰⠏⠀⠀⠀⠈⢣⡴⠋⢀⡴⠊⢀⡀⠀⣠⡟⠙⠛⠶⣤⣀⠀⠀⠀⠀⢀⡴⠛⢹⡆
⠀⠀⠀⠀⡿⠀⠀⠀⠀⣰⢏⣀⣠⠾⠶⢲⠟⠻⢶⠏⠀⠀⠀⠀⣠⣭⣷⣦⡀⣴⠏⠀⠀⣼⡥⠖⠚⢶⡄
⠀⠀⠀⢰⡇⠀⠀⠀⡼⠃⡿⠋⣉⣁⠀⠀⡔⠒⢜⣇⠀⠀⠀⠀⣯⠀⠀⠙⣿⡃⠀⠐⠾⡋⢀⡤⠖⠛⢻⡄
⠀⠀⣴⠛⠀⠀⠀⣼⠛⠛⠃⢸⠁⠈⢣⠀⢻⣦⡼⠟⠒⠶⣄⠀⠈⢳⡄⠀⠀⠉⠣⠀⠀⠈⢯⣀⡴⠖⠛⣷
⠀⠀⣿⠀⠀⠀⢰⠿⡄⠀⠀⠸⣯⣧⠈⡇⠚⠃⠀⠀⠀⠀⠸⡇⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⡴⠋
⠀⠀⢸⣧⣄⣤⣿⡄⠉⠓⣆⠀⡈⠓⠚⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⣟⠙⠆⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉
⠀⠀⢸⡇⡟⢀⡤⣜⠦⣤⠏⢰⣿⣷⣦⣤⣶⣦⣤⣤⣤⣾⣿⠀⣴⠻⣦⡀⠀⠀⠀⣀⣀⣤⠾⠋
⠀⠀⢸⡄⢷⡀⠀⠚⠳⠀⠀⠸⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⢏⡞⠁⠀⠈⠛⢦⣤⣀⣼⣿
⠀⠀⢸⡇⠀⠛⢦⣄⣀⣤⢤⠀⠈⠛⢿⣟⣛⣓⣲⠎⢩⣷⠋⠀⠀⠀⠀⠀⣰⠏⠁⢀⡟
⠀⠀⠀⢷⠀⠀⠀⠀⢀⡗⠈⢳⡀⠀⠈⢇⠉⢉⡟⢀⡾⠁⠀⠀⠀⠀⢠⡾⠁⠀⠀⣸⠃
⠀⠀⠀⠘⣧⠀⠀⠀⠸⣤⣴⡤⢷⠀⠀⠈⠉⣉⣠⠾⣅⠀⠀⠀⠀⣴⠏⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠈⢷⡀⠀⠀⢀⣴⡞⠛⢷⡤⠤⠤⠥⠞⣆⠈⢷⡀⢀⡾⠁⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠙⢾⡋⠙⢧⣽⡄⠀⠳⡄⠀⠀⠀⠙⡄⠀⠹⣿⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⣸⠂⢻⡙⢷⠛⢲⣀⣀⣠⣼⣄⡶⢻⣦⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢧⣤⣼⠀⣸⠃⠀⣿⠭⠴⠶⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠓⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠀⠀⠀⣐⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⡀⠜⠀⠀⢙
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⢀⡀⠀⠀⠀⠀⣀⡶⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⡀⢀⣀⣀⣀⣀⡀⠀⠉⢀⡠⠐⠛⠊⢀⡀⠀⠀⢀
⠀⠀⠀⠀⠂⠤⠤⠤⠬⠤⠄⠀⠀⠒⠠⠤⠭⠴⠀⠀⠀⢠⡄⠀⠀⠀⣀⠤⠤⡼⠿⠊⢃⠁⠀⣥⣿⡅⢀⡤⠞⠉⠀⠀⠀⠀⠀⡌⠀⠀⠉
⠀⠀⠀⠀⠁⠐⠂⠀⠀⠂⠀⠁⢒⠛⠛⠒⠓⠀⠀⠀⠀⡆⠉⠒⠀⢉⡴⠚⠛⢣⠧⠤⣈⣁⣁⣠⣟⠃⠉⠀⠐⠢⣄⠀⠀⠀⠀⣸⠀⠁⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡎⠀⠀⠀⠁⠀⠀⠈⢣⡀⠀⠈⢢⠀⠀⠀⠀⠈⠑⠂⠠⠖⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢪⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡃⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⢀⠄⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠓⣶⢧⠐⠀⠀⠀⠀⣀⠜⠁⠀⠀⠀⠀⠀⠉⠲⠦⢤⡄⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠞⠁⠀⢰⣿⢻⡃⠄⡶⠶⠾⣛⣦⡂⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⡀⢰⠺⡟⢩⠏⠘⠁⠈⠀⠀⢰⠿⠉⣙⣶⢿⣾⡿⠤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⠶⠞⠄⢀⡸⠂⠁⠰⡇⠸⣆⠀⠀⠀⢀⣠⠀⠀⣰⠽⣯⡿⠏⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠉⠁⠈⣦⣿⡾⠉⠀⠀⠀⠀⣷⠒⣿⣦⣀⣀⣈⣁⣀⣬⣴⡶⢋⠐⠂⠰⠀⠂⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⠈⢀⣹⠖⠋⡉⠀⠀⠀⣠⠔⠚⡵⠋⠀⠀⠈⠹⣤⣉⠁⡖⢻⣷⡀⣷⠀⡄⡀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⢀⡴⠊⠀⣠⡶⠛⠁⠶⠋⠁⢢⡀⢰⠁⢀⡜⠀⠀⠀⠀⢀⣠⠃⡴⠋⠰⠛⣌⢹⡇⠘⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⢀⡠⠂⠁⢀⡤⠚⠁⠰⡄⠀⠀⠠⠀⠀⣹⠞⠛⠫⠤⠤⢤⣶⣶⠟⢁⡴⠃⣠⠆⠀⣼⢀⡇⠀⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁
⢤⠀⠁⠀⣠⠜⠉⠀⠀⠀⠀⠐⢄⠄⠀⡴⠑⠋⠀⠀⠀⠀⢀⣠⡞⠁⠐⠃⠒⢉⠄⢀⠞⠛⠉⠙⠄⡀⠁⠀⢀⣠⠀⠀⠀⠀⠀⠘⠁
⢎⢳⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠣⣸⡇⠀⠀⠀⢀⣠⣴⠟⠉⠀⠀⠀⠀⠀⠚⡠⠋⠀⠀⠀⢀⡀⠀⠉⠁⠉⠀⠀⠀⡔⠀⠀⠘⢄
⠀⠛⠁⢦⠶⠖⠒⠒⠢⠦⠤⣤⣀⡀⠈⠉⠒⠒⢺⠏⠈⠈⠋⠁⠀⠂⢤⡄⣤⠎⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⢄⠀⡰⠃⠀⠰⠀⠀⠀⠀⠀⠀⢞⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠇
⠀⠀⠀⡀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠂⠀⠀⠀⠘⣆⠀⠐⠁
⠀⠀⢀⡡⠛⠚⠚⠛⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠒⠒⠉⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢹⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢆⠀⠀⠀⠀⠀⠀⠘⡈⡇⠙⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢁⣿⡀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠈⠈⣇⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢠⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠟⢧⢎⣱⠐⠂⠏⠰
⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡇
⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣿⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠛⣛⡛⠛⣻⠿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠖⢋⡀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠋⠉⠛⠻⠷⢶⣄⡀
⠀⠀⠀⠀⢀⣠⡞⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⣠⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⣶⣦⣶⣤⡄
⢀⡾⠟⠁⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⣿⣧⠀⠀⢻⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣾⣿⣿⣿⡆⠘⠈⣿⣿⠏⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⠿⢿⡷⠀⠀⢉⣉⣐⣿⣿⠂⠀⠀⠀⠀⡴⢶⣾⣿⣿⠏⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠛⠂⠀⠀⢄⣠⠾⠟⣻⣿⣟⠋⠁⠀⠀⣠⡺⣿⠦⠌⠋⠻⠷⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠙⣿⣷⠀⠀⠀⠐⣙⡛⠻⠄⠀⣀⠀⠐⣜⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⢾⣿⡇⠀⠀⠉⠙⠡⠶⠂⠀⠙⠷⠸⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠉⠀⣰⣶⣤⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⡇⠀⠀⠀⠀⠠⢄⣄⠀⠉⢀⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣹⣿⣧⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠙⠛⠿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢽⣿⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣿⣿⣧⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢨⣿⣿⡄⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⣼⣿⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣺⢙⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠰⠟⢽⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣃⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣾⢛⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡼⣣⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣮⣵⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠈⠢⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣤⢄
⠀⠀⠀⠀⠈⠢⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢀⠀⢹⠀⡀⣀⠀⠀⠀⠀⠆⣊⠑⣢⢉⢛⠐⢺
⠀⠀⠀⠀⠀⠀⠈⢳⣿⡒⠂⠀⠁⠀⠀⠀⠀⠀⣄⣀⡈⠢⢄⣿⣷⠲⠋⠀⠀⠀⠈⠸⠬⣱⣇⠸⢸⣂⠉
⠀⠀⠀⠀⠀⠀⠠⠊⠃⠈⠑⠤⠀⠀⠀⠀⠀⠀⠀⠙⠒⢷⣀⣡⡦⠤⣀⣄⡄⠀⠀⠘⠒⠋⠘⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠠⢼⠖⠀⣄⣬⣄⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠘⡳⠤⣤⡀⠪⠑⠂⠦⠄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠢⣿⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠠⣨⠆⠁⠄
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⡀⠈⠑⣃⠴⢁⠂⠐⣢
⠀⠀⠀⠀⣐⠀⠀⠑⢀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡛⣫⣕⣶⡆⠀⠀⠀⠀⠀⠀⠈⠲⢤⣞⣀⣣⡊⠑
⠀⠀⠀⢐⣀⡀⠠⠑⠖⢨⠁⠀⠀⠀⠀⠀⢀⣀⢉⡚⢬⣭⡼⢿⣜⣇⠀⠀⠀⠀⡠⡀⠓⣤⢵⣿⡿⢇
⠀⠀⠀⠠⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠐⢴⣿⠟⡁⡏⢩⡟⠈⠀⠀⠀⠀⠀⠜⠐⢋⣱⠗⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠩⠀⠖⡀⠚⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣭⣭⣬⣭⣯⣿⣽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠂⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠛⠃⢹⣿⣷⣀⢲⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⢠⣿⡄⠀⠀⠀⣀⠂⠀⠀⢀⣀⠀⠀⠀⠀⠛⠘⠃⠉⠉⠀⠉⣍⠉⠳⠦⣄
⠀⠀⠀⠀⠀⠀⣰⠟⣀⣤⣤⣤⡞⠛⢯⡻⣶⡾⠻⡿⡶⢾⡷⣿⣿⢷⠶⣾⠷⣿⡾⣧⣠⣶⡶⠀⡀⠀⠀⠤⠘⠓⢄
⠀⠀⠀⠀⠀⣸⣷⠶⠫⠌⣀⡐⠀⠀⡂⢙⢮⡃⠁⠈⠞⣤⠃⢷⢸⡘⣇⠻⠀⣦⢡⢃⡜⢉⠟⣻⢿⣶⡤⠀⣶⠀⡀⠳⡀
⠀⠀⠀⠀⠀⣟⣁⠀⠐⠀⠤⢈⡑⠂⢤⡑⠢⠸⣶⣌⠄⢱⣿⢸⡆⣷⣌⣣⣾⣧⡡⣊⡔⢉⡾⠓⡉⢀⡽⢺⣟⣦⡀⠀⢱⡀
⠀⠀⠀⠀⠀⢷⠀⣒⠒⠠⠤⢄⣒⣡⠔⣨⡑⠦⣄⡹⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣯⣼⣛⣡⣔⣪⣴⣍⣂⣥⣚⡿⣇⠘⠋⢃
⠀⠀⠀⠀⠀⠈⢷⡍⢁⣄⣄⣠⣤⣤⣉⡙⠿⣿⣿⣿⡏⠀⠀⠀⢹⣿⢿⣿⣿⣉⣉⠉⠉⠙⠋⢙⠻⠟⠽⠚⠉⣷⡼⡗⠛⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠣⣴⢒⣂⠭⠍⢛⣛⣛⣷⣿⣿⡇⠀⠀⠀⢸⠃⢸⣿⠿⢿⠿⢿⠿⣿⣯⡭⣭⢤⣀⣒⣛⠭⠄⣺⣿⣱⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⠶⣿⢛⣉⣭⢶⣿⣿⠀⠀⠀⠀⢸⠀⢈⣿⣶⣌⠘⠄⠈⠢⢝⠻⡲⢌⠊⣝⠋⠛⣥⣿⢈⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢳⣾⣿⠀⠀⠀⠀⠀⣦⠘⣿⣿⣿⡕⢄⠀⠀⠠⠑⠌⢆⠑⣌⣣⣼⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⠀⠀⠀⢀⣇⢰⡿⣿⣩⣯⣬⣶⠶⠞⠛⠛⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢀⣀⣼⣿⣸⡇⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠁⠀⢙⠿⠟⠙⢿⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⡀⢰⡏⢰⣷⣄⡌⠻⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣴⣵⣿⣿⣿⣿⢿⣿⣷⣥⣮⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡋⠀⠀⣠⠀⢀⣾⠿⠟⠋⠁⣾⢸⣼⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠉⠉⠛⣿⠁⠀⠀⠀⠀⡈⢈⠇⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⣠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣶⢰⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠠⢸⡆⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢠⣾⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡦⠤⠀⣙⣵⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⠟⠁⢻⣿⠃⣀⣢⡀⣶⣈⠉⡒⣋⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣦⣀⣤⣄⣼⣀⣈⣲⣿⠏⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⢀⠠⠀⠀⢀⡠⠀⠈⠉⠉⠙⠒⠦⢄⡙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠟⠙⠁⠐⠁⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴
⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⢺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠘⠋⢀⣿
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢀⡀⠀⠀⠸⢿
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⣿⡌⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⣺⡿⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢀⠀⢻⣧⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⠚⠋⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡐⠛⣉⣩⣠⢓⡵⠀⠑⢄⠀⠀⠀⠀⠀⠘⠲⠝⠦⠤⣤⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠠⣾⢿⣿⣿⣿⣞⠢⡀⠀⠀⠀⢀⢀⡀⠀⠀⠀⣴⡾⠃⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⣌⠘⠀⠀⠀⠀⠀⠀⢀⡟⣰⣿⠀⠀⠀⡾⠁⢸⠻⠟⠛⠋⠀⠈⢳⠦⣴⡤⢤⡤⠖⠒⠋⠁⠀⠀⠀⠀⠀⠴⢸
⠀⠀⠀⠀⠀⠰⣂⠀⠀⠀⠀⠀⠀⢸⣷⡏⣿⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠐⡸⠀⡾⠓⢠⡇⠀⣆⠀⠀⠀⠀⠀⠀⠶⠶⣿⣦
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⢸⡿⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠇⠘⠁⣰⠋⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⢐⠀⢸⡇⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠌⣰⡶⢺⡏⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡇⠀⢨⡀⢸⡇⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣰⢁⣼⢿⡀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣴⣧⢸⡇⠀⠀⣉⠁⠀⠀⠀⠀⠀⠀⣰⡵⡿⢻⣸⡇⢀⣠⠿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⢷⢀⠀⠉⠀⠀⠀⠀⣠⡴⠞⠉⠀⡷⢚⣁⣄⠛⠁⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢎⠀⣱⠄⣤⣤⣶⠋⠁⠀⠀⠀⠀⢧⢨⠘⣿⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣳⣶⢿⣾⡀⠀⠀⠀⠀⠀⠸⢧⡀⢯⡀⠀⠀⠀⠀⠙⠲⠤⠴⠚⠀⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⢳⡀⠣⠀⠀⠀⠀⠀⠠⠔⠊⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣆⠑⣤⡀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠈⠷⣶⠶⠒⠂⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠈⠁⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣁⡀⠻⠀⣀⡤⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⠀⢀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⢉⣷⢿⡿⢗⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⣀⠉⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢡⣄⡼⠙⠛⠉⢨⡗⢦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠚⠱⢨⠄⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣰⠻⠇⠘⢄⣀⣀⡼⠁⢠⣂⠀⠉⠉⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠈⢆⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠿⠷⣶⡈⠉⡅⣠⣿⠿⠿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠈⢿⡄⢰⠟⠉⠀⠀⠈⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡁⠀⠀⠀⠀⠀⠀⠈⠲⠦⠴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠈⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣤⣄⣀⣠⣤⡤⠖⠋⠀⠉⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠄⠀⠀⠐⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢶⣳⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠫⠔⠋⣙⠒⠝⡄⠀⠀⠀⠀⠀⠀⠀⠰⣄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣈⣘⣆⣀⣀⣀⣀⣀⣀⣀⣀⣱⣄⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠀⠄⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠲⠗⠚⡗⠛⠖⠳⠞⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣛⣲⡉⠈⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠠⣤⠶⢤⡤⢤⣥⣤⣧⡤⢌⣡⠄⠀⠀⠀⠀⠀⠀⢠⣤⠟⠙⠛⠉⠀⠀⠀⠉⠉⠒⢬⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠉⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⠀⣈⠁⠀⠓⠒⠒⠈⠇⠀⠀⠀⠀⠀⣽⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡟⠑⠀⠀⠀⢤⣄⡈⡤⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠴⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣵⠀⠀⠀⠀⠀⠀⠈⡰⠀⣠⡀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⣀⣰⠌⡄⠀⠁⠁⠀⢀⠔⢀⣆⡔⠁⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠉⠠⣄⣀⣈⠦⣤⣀⣠⡽⠴⠉⠼⠀⢠⠃⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⢁⡍⠛⣧⠸⠉⢩⡷⠂⠉⠑⢦⠈⠓⠊⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀⠀⠀⡈⠠⣤⠀⠀⠀⡜⠀⠀⠀⠀⠀⡇⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢺⠏⠀⡄⠀⠜⢹⡀⠀⠀⢧⣀⡀⠀⠀⢀⡇⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⢰⢇⠀⠀⠀⠑⢄⠈⠒⠿⣢⠝⢻⡯⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡠⠊⢈⡄⠀⠀⢀⣀⣡⡄⠀⠀⠰⠇⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⡟⠁⠀⠀⠈⠙⠷⡂⠋⠀⠀⢸⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⡠⠂⠰⡀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⢀⠤⠊⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠒⠺⢦⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠤⠚⠁⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠈⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⣀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠁⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠘⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⣀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣠⠞⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⠤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⠞⠓⠲⠲⠶⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⠤⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠙⢱⠒⡌⠊⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠙⠢⠼⡾⠆⠒⠋⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⡐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢐⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡈⠁⠀⠀⠀⣦⡀⠀⠀⠀⣀⣤⠀⠀⠀⠈⢩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠄⠀⠠⢊⡤⢬⠒⠒⢊⡤⣤⡳⡄⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠰⢀⠘⠟⠧⠂⣀⠸⢟⠌⢁⢢⠆⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢥⠀⢶⣑⡀⠀⣠⢤⡀⠀⣠⣣⠦⠀⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠣⠼⠀⠢⠘⠧⣄⠙⠚⢀⡤⠛⠇⠀⠀⠧⠾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⠤⠤⠄⣴⡤⠗⠖⠻⢔⡆⠤⠤⠤⡀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢠⠋⠉⡀⠖⡋⠈⠙⡀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢠⡅⠀⣴⠀⠀⢁⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢳⠐⠺⣇⡀⠀⠆⡀⠌⠀⢀⣼⠖⠒⡛⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢰⠂⠎⢹⡀⠀⠀⠀⠀⠀⢨⠎⠒⢨⠛⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢑⠀⠀⢷⠠⡀⠀⡠⠔⡟⠀⢀⠄⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡓⢚⡤⠎⢣⠀⠘⣄⠰⠛⢄⡸⠀⠀⡜⠰⢌⠓⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠎⠀⠀⠀⢡⡀⠹⡄⠀⢠⠃⢀⡌⠀⠀⠠⠷⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣀⠄⠂⠁⠑⣄⢘⡤⠊⢠⠎⠉⠒⢤⣀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠉⠁⠀⠀⠀⠀⠀⣸⠇⠀⠀⣤⡀⠀⠀⠀⠀⠉⠋⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣴⣽⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠁⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠠⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⡰⠁⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⡄⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢠⠁⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠰⡀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⡆⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⢣⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡘⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠈⡄⠀⡇
⠀⠀⠀⠀⠀⠀⢀⡇⠀⡆⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢃⠀⢰
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠈⡆
⠀⠀⠀⠀⡴⠂⠙⠦⣸⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⣄⠔⠉⠐⣆
⠀⠀⠀⠀⠹⣄⢀⡔⠙⠦⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡠⠞⠙⢂⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠉⢩⠛⠒⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠛⡈⠁⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⡀⠠⢧⡀⠀⠀⠀⠈⠉⠛⠓⠒⠒⠂⠀⠉⠛⠉⠉⠉⠉⠁⠀⠀⣣⢄⠠⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠔⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢶⡶⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣾⣿⣷⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⣼⣿⣿⠿⠿⠛⠻⠿⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⣿⡟⠁⠐⢑⠀⢠⡉⠂⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⢻⣯⠉⠩⣿⡛⠿⢿⣿⡆⠀⢀⠿⣿⡄⠀⡼⠓⠺⢇⠀⢸⡿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⢰⠁⣾⣀⣠⣧⢈⠀⢸⣿⡇⠀⠸⡜⢾⣿⣶⣄⣀⣀⣴⣾⣿⣟⡠⠀⠠⠒⡶⣲⣆
⠀⠀⢀⡀⠀⡾⠛⣦⠀⠀⣰⢹⣿⣷⣦⣰⠋⠀⠀⢹⣤⣴⣼⣯⢍⡆⠀⠈⠛⢿⣿⡿⠟⠻⠿⢿⡿⡯⣄⠀⡇⠸⠐⣁⣼
⠀⣴⠟⠛⢿⡇⠀⣿⠀⠀⠘⢾⣿⠘⢿⣿⣿⣶⣾⣿⣿⠟⢠⡿⠋⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢰⠉⢃⠀⠑⠃⠀⠰⠃⣿
⠀⠹⣦⣀⠈⣷⣠⠟⠛⣦⠀⠀⠈⠳⢤⣀⠉⠉⠉⣁⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠈⣧⣄⣀⣠⣇⠀⢸⡀⠀⠀⠀⠀⣰⠃
⠀⠀⠈⣿⣉⡹⣇⣀⡴⣹⣷⠀⢀⡴⣿⣿⠉⠉⠉⢻⣿⡟⢦⠀⠀⠀⠀⠀⠀⢠⣾⣿⠉⠉⠉⣿⣷⡜⣿⡶⠿⠶⢚⡟
⠀⠀⠸⣧⣠⣽⡏⠉⠀⣡⠿⣤⠎⡜⠉⣿⣿⣿⣿⣿⡇⠙⡄⢣⠀⠀⠀⠀⡰⡻⠛⣿⣶⣶⣶⣿⠻⣟⠟⠉⠙⢛⠋
⠀⠀⠀⠈⠛⠛⠶⢶⣞⡉⠀⠀⣸⣄⣴⣿⣿⣿⣿⣿⣷⣤⣷⣀⡇⠀⠀⢀⣡⣇⣠⣿⣿⣿⣿⣿⣄⣸⣄⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⣀⡞⠩⠳⣄⣠⠴⠚⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣬⣼⣋⣇⣌⠡⠄⡞⡇⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠛⠉⠉⠉⠉⠉⠉⠙⠻⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⡄⠀⠀⠀⠀⠀⠀⣼⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣠⢞⡡⠤⣽⠦⣀⠀⠀⠀⠀⢀⣼⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⠀⠀⠀⠀⢀⡴⠃⠊⠀⠴⠟⢀⡇⠀⣄⠀⢹⠉⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢧⠀⠀⠀⠀⢀⡞⠁⠀⠀⠈⠉⠀⣼⢁⡾⢹⠇⡞⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠈⢳⣄⠀⠀⣼⠁⠀⠀⠀⠀⠀⡰⣷⣿⠒⠉⣾⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠒⣯⢻⠶⠟⣶⢤⣀⣀⡤⠒⠀⣿⠘⡲⠚⣧⣀⣠⡄⡾
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⢧⡀⠗⢹⣛⣿⣁⡤⠤⠼⠀⠀⠺⢷⣶⡃⠀⠀⠀⠘⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡼⠁⠙⡶⢶⡶⢍⢧⣀⣀⣀⣈⣉⣷⠲⣤⠞⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣟⣟⢪⣀⡸⣤⠾⣅⡀⣀⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠛⠚⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠋⠀⠂⠀⢀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠤⠄⠀⠈⠀⠒⠁⠐⠀⠀⡌⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠎⠈⢁⡨⠬⠔⢲⡖⢲⢒⢖⢲⠂⠄⠬⣀⡉⠡⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠴⡘⢪⠑⠠⣀⣠⣼⡗⢷⣦⣾⣿⣦⣄⡤⠀⠀⠁⠛⠐⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠁⠐⠖⠀⢀⡠⠼⠿⢿⡇⠀⠈⣿⣿⣿⣟⡛⠛⠀⢀⡠⠖⠚⡓⠲⠶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⢤⣀⢀⣀⣀⠀⠀⠒⠀⠇⠀⠀⢸⠟⠘⠈⠊⣀⡴⠛⠀⠀⠀⠁⠀⠀⠂⠝⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠚⠺⢤⢍⣉⠉⠍⢉⠉⠐⠁⡆⠀⠀⣼⠁⠉⠉⠁⡼⠗⠠⢂⣀⣠⣤⣬⣤⣤⣤⣬⣭⡑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢡⡉⠱⠀⠀⠀⠄⢀⠝⠉⠒⢧⡀⠀⠢⠄⠀⠀⡟⣧⠀⢀⣺⠴⠒⠛⢭⢿⡏⠉⢻⣿⣿⢷⣥⣌⡿⠇
⠀⠀⠀⠀⠀⢀⡴⠊⠀⢀⣀⡀⠀⠈⠂⠈⠀⠁⠀⠀⢄⣁⠽⠳⡞⠀⠀⢸⠯⣏⢣⡘⠧⠮⢤⡦⠖⠒⣃⠀⢠⠉⠉⠡⡄
⠀⠀⠀⠀⠰⠉⠆⠀⠀⢀⡀⠈⠀⠀⠠⣀⡸⠀⠀⠀⠘⠀⠀⠠⣵⠀⠀⠸⣀⣨⡽⡥⠴⢲⣿⠁⣀⡵⠃⢀⣞⠓⠀⠄⢾⡀
⠀⠀⠀⠀⡏⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⡒⠂⠐⡼⠂⠂⢡⠻⠁⢁⢄⣀⣸⣟⣼⠯⠤⣄⡼⢹⡏⢄⣠⠞⠁
⠀⠀⠀⠀⠰⠉⠀⣀⡤⠶⠶⣶⣶⡟⠛⠛⠛⠙⣽⡉⢈⠠⠄⠊⠁⠀⠀⡬⠠⠼⣿⣿⠋⡽⠋⠁⢀⠼⠁⠀⠔⠂⠉
⠀⠀⠀⠀⠀⠈⠃⠀⠠⠤⠤⠤⠀⠆⠀⠀⠀⠀⠀⠠⠀⠀⡆⠂⠀⠀⢠⠃⠲⠈⠀⣯⠜⠀⢀⡴⠋⢀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⢀⣑⠀⠠⡐⠀⠀⣘⠖⡁⢹⡴⠋⠀⢠⠎⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡀⠁⠀⠀⠀⠀⢠⣨⡇⠀⠀⢿⣐⣠⠖⠁⠀⣠⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢅⠳⡄⠀⠀⠀⠀⠀⢻⣧⡆⢀⠠⠿⡇⠀⢀⣴⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠨⣑⠻⡄⠀⠀⠀⢀⠀⠉⠙⣍⢁⢐⣁⣰⣾⠃⣜⡅⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣢⣄⣱⠟⡖⠒⠦⠁⠀⠒⢹⡄⡀⡿⠿⠋⠙⠉⢳⠊⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠉⢲⠇⠀⠓⠰⣖⣁⠀⢀⡸⠗⠻⣁⢤⡴⠒⠋⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⢏⠀⠀⠀⠀⣃⣨⡽⠯⠥⠤⠤⠤⠼⠷⠖⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⠛⠉

⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠁⢀⡤⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⣰⠀⢰⠀⠀⠀⢀⠀⠶⢀⡀⠀⠀⠛⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢈⠅⠀⠀⢰⡀⠀⠀⢠⡀⠀⢠
⠀⠀⡅⠉⡄⠈⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢸⠃⢠⠞⠉⠂⣀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⢉⠀⠠⡐⠋⠘⠃⠈⠀⠏⠀⠈⠒⢀⠀⢨
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠾⠒⠲⣄⠈⠳⡀⠀⣠⡄⠀⠀⠀⠀⠀⣄⠀⠀⠉⣤⣀⡂⠐⢤⣄⠀⠀⠘⠋⠁⠘
⠀⠀⡄⠀⠀⠀⠀⠀⢰⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⠀⠀⠀⠈⡾⠃⠀⣤⢀⠁⠀⠀⢿⡆⠘⠀⠀⠀⠀⠠⠒⠉⠀⢠⠻⠋⣉⣹⣶⠀⠙⣷⠀⠀⠋⢱⣾
⠀⠀⡇⠀⠀⠀⠀⠐⣏⠀⠀⠀⢱⠀⠀⠀⠀⠈⢡⠃⠀⠀⠰⠃⣠⠜⠀⠀⠚⠀⠷⠀⢷⡀⠀⢠⠀⠰⠛⢐⠤⠀⠀⠀⠀⠉⠀⢸⣧⠀⢸⡇⠀⠖⢱⣾
⠀⠀⡇⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠃⡤⠂⡲⣘⠛⣗⣤⠘⠣⠀⠙⠀⢀⣄⠙⠃⠀⠠⡀⠀⣠⡄⣠⡏⠀⣸⠃⠀⢤⡄⣩
⠀⠀⡇⢠⠆⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣀⠀⠘⣠⠂⣠⠟⠁⠈⢁⣈⠁⠐⢀⡆⠀⠀⠛⢁⠑⡶⠀⠀⣿⡛⠿⠿⠋⢀⣰⠏⠀⠀⣀⡸⢿
⠀⠀⡇⠉⡆⠀⠀⠀⠰⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣥⠀⠐⠃⢰⣯⠀⠀⠀⠀⠀⢀⣶⡄⣥⠀⠀⠀⠈⠣⡀⣀⠀⠀⠑⠒⠲⠚⠛⠁⠀⠤⡀⠸⣀⣈
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠄⠦⠀⡄⠀⠀⠀⢀⠀⣀⣾⠇⠀⠛⠇⠀⠐⠿⣿⡇⢸⣇⠀⣿⢆⣠⠀⠉⣀⠴⡂⢀⠂⡄⠀⣷⠀⡜⠋⠛⠉⡸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢰⡖⠁⠀⠀⠀⠀⣰⣧⡟⢰⠄⠀⠀⡀⠄⠠⣿⢠⡇⣿⠀⠑⠈⠉⠂⡀⠋⣤⠈⣸⡇⣧⡜⠀⠐⠶⣠⣶⣾⢷
⠀⠀⡇⠀⠀⣰⠃⠀⠀⠀⠀⢀⡀⠀⠀⢀⡶⠀⠘⠀⠀⢠⡟⠁⢃⣼⡀⠀⠀⠀⠀⣸⣷⡌⢃⠘⡆⢻⣏⣀⣀⡀⠚⠛⣦⠀⣤⡈⠁⣦⣽⣦⠟⢻⡇⢘
⠀⠀⡅⠀⠀⠈⠃⠀⠀⠀⠘⠄⠀⣠⣶⡏⠀⠀⠀⠰⡄⠁⣰⣶⠚⢅⠱⢦⣀⣤⣴⡉⠏⣛⠉⠴⣣⠀⢉⠉⠀⣄⢹⠀⠘⢦⠛⡧⠘⠃⠙⠃⢴⣬⠀⢸
⠀⠀⣇⠀⠀⠀⠀⠀⠀⢀⡶⠀⢻⣿⡌⠛⠂⠀⡀⢾⠀⣀⠀⠘⠋⢼⣿⠀⣉⣤⢷⡆⢸⡍⠛⣀⠙⢆⡀⠁⠀⢿⣥⡄⠀⡀⠀⠇⠘⠀⠀⢀⠾⢋⡀⢸
⠀⠀⣯⠀⠀⠀⠀⢈⡷⠀⡴⠀⠀⣀⣤⠐⠒⠋⢤⡈⠀⡀⠉⠀⠀⠀⢻⡇⠀⣨⡼⣷⣾⠕⠊⡙⠰⠚⠉⢂⡀⠀⠀⠰⣶⡅⡀⠀⠀⠀⠀⠈⢰⡾⠃⢸
⠀⠀⠉⠀⠀⠀⠀⠈⠁⠸⢧⠄⠒⠂⢠⠀⠳⢄⠀⠹⣆⠀⠀⠀⠈⠂⢸⣇⠀⢈⠀⢻⡃⠀⣹⡍⠂⢀⡤⠚⠡⣄⠀⠘⣶⣴⡍⠆⠀⠀⠀⠀⢩⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡼⠖⠀⠀⠀⢋⣶⡄⠓⡀⠉⠳⢦⡁⢀⠀⢸⡟⠀⠈⠀⣸⠁⡂⢹⣇⡀⠂⣠⡈⣰⠛⣀⠀⠿⣤⠀⢄⠀⠀⠀⣠⠞⠁⡄⣠
⠀⠀⡄⠀⠀⠀⠀⡾⢠⣾⠃⢲⣶⣟⠀⠈⠁⣿⠀⠀⣠⣤⣄⠙⢦⡀⢨⡗⠂⠔⠸⢷⣈⣱⡞⢋⠀⣤⠼⠈⢉⡴⢇⠀⠂⠀⠀⢸⠁⠀⣐⠁⠀⠈⠓⠹
⠀⠀⡧⠤⠔⠀⠤⢇⣼⠟⠀⣿⢿⣿⠚⣤⠀⠙⣧⡀⢀⠋⠀⠛⠲⠷⣌⡛⠛⠛⠛⠂⡉⢡⡀⣹⣶⠏⠠⢠⡾⡇⠀⢣⡾⠦⣀⠈⣦⠀⡉⠠⠤⠤⠄⣸
⠀⠀⠃⠀⠀⠀⣴⣾⠇⠀⠀⣹⢿⣿⠐⢸⡄⠀⠈⢷⡀⠻⣷⣄⡀⠀⠈⠙⠛⠧⠼⠶⠷⠾⠿⠿⠟⠓⣐⠿⡄⠈⠒⡋⠀⠞⠛⠛⡤⠟⠀⠀⠀⠀⠀⢿
⠀⠀⡆⠀⠀⠈⢻⠋⠀⣤⠀⠇⢸⣿⠀⠈⠀⠀⠀⠈⠳⣄⠈⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠚⠟⣇⠈⡖⠀⠀⣾⠀⠘⣾⠤⠀⠦⠀⠀⠀⠀⣸
⠀⠀⡇⢠⠀⠀⠇⢀⡴⠉⠀⠃⣼⠇⠀⠀⢸⡀⠀⡀⠀⠻⣆⣀⠈⠛⢿⣶⣄⡀⠀⠀⠤⠄⢀⣀⣀⡀⠀⡈⠓⠀⡀⢸⣿⣧⣀⡀⠀⠋⠀⠀⠀⢀⠀⢨
⠀⠀⡁⠘⠆⠀⣀⡌⠁⠀⠁⠀⣿⠀⠀⠄⠀⣷⠀⠹⡀⠀⠙⢾⡦⠀⠈⠻⣿⣿⣷⣤⣤⣤⣴⡾⠋⣡⡞⠃⣠⡾⠃⢸⣿⣿⣿⠀⠀⠀⠀⠆⠀⠀⠀⢼
⠀⠀⠟⠃⠘⠃⢸⠀⢀⣿⣠⡄⢩⣷⡄⠀⠀⠘⣧⠀⠀⡀⠀⠈⠷⡼⢆⡀⠈⠙⠷⠾⠟⠉⠁⢀⣼⡟⠀⣠⠁⠘⠃⠸⣿⣿⠏⠠⠖⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠎⠀⠈⠁⠀⠶⠾⢻⣿⢀⡄⢻⣷⠀⠀⠀⢹⣄⠀⠘⠀⠀⠀⠈⠺⢏⠂⡀⠀⠀⠀⡀⠀⢠⡿⠀⡄⢙⡂⠀⠀⠁⠈⢡⣤⡀⢀⠀⠀⠀⠞⠁⡆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣹⣿⣆⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠈⠀⠉⠀⠈⠀⣈⣡⠏⠱⠀⠀⠀⣠⣌⠟⠻⢮⣧⣆⣰⠇⠀⢀⣿
⠀⠀⠇⠀⠀⠀⠠⠀⠀⣰⠊⠙⢿⣿⣿⡟⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢿⣿⡏⠀⢠⠀⢠⡎⠁⢈⣅⢠⣼⠋⠙⢿⡎⢉⠉⢻
⠀⠀⠄⠀⠀⢠⠆⣠⠎⡉⠁⠀⠀⠙⢿⡇⠀⢆⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠈⠒⠄⠀⠠⠒⠋⣠⣼⠟⠀⢠⡞⠂⠺⣁⠀⠸⢧⣸⣧⡀⢀⠸⢅⣸⣇⣤
⠀⠀⠆⠀⢸⡅⠀⠹⡄⠁⠤⠀⠘⠇⣠⠀⠀⠘⢄⠀⠀⠀⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠠⡾⢂⠀⠛⣷⣤⣤⣠⡄⠀⠀⠀⢤⡄⠀⠆⢸
⠀⠀⠂⠀⠃⠰⣦⡄⠈⢑⣆⣠⣤⡄⢸⠀⠀⠀⠀⢂⠀⠀⠀⠄⡀⠀⠀⠀⠀⠈⠉⠿⠿⠛⠉⠀⠀⣠⠀⣷⠈⡆⠀⠸⠁⢘⡁⢀⠀⠄⠀⠀⠉⡀⠀⢘
⠀⠀⠁⠐⠂⠄⠘⢷⠟⠛⠋⠉⠀⠀⢸⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠰⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⢱⡄⠀⣦⠎⠁⠐⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⢾⣧⣀⡀⣀⣄⠀⠀⠸⢂⡀⠀⠀⢮⠀⣤⠀⠀⠀⠀⠀⠀⠈⠓⠄⠀⠀⠀⢀⡜⠀⠀⢸⠖⢲⠆⠀⠐⠛⠁⠀⠀⠀⣦⡀⠀⡤⠀⠀⢠
⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠉⠁⠀⢠⡆⠈⠷⠀⠀⠆⠀⡏⠑⣄⣀⢀⠀⠀⠀⠢⠀⠀⠀⣠⡞⠀⠀⠀⡞⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠁⠀⠂⢀⡄⠀⠀⠸⠁⠀⠀⢆⠀⠐⠀⡇⠸⠟⠉⠉⠻⠷⣤⣀⠀⠀⠈⠀⠀⠀⢀⡰⢿⠄⠀⡅⠀⠁⠀⠀⠀⠉⠀⠀⠀⠠⠀⠀⠀⠐
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⡄⠀⠀⠀⠐⡀⠀⠀⣀⣴⡄⠀⡇⠀⣿⡟⠳⠦⣤⣤⡾⠛⠙⠀⢸⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀
⠀⠀⣁⣀⣀⠀⠠⠤⠤⠤⠀⠀⠀⢰⣿⣟⠀⢱⡄⣠⠀⠀⢹⣿⠀⢀⠀⠀⠿⣿⠀⠀⠘⠁⡇⠀⢸⡆⠘⣷⠀⠀⡀⠤⠛⠀⠓⠀⠀⠤⠤⠤⢀⡀⠀⢨
⠀⠀⣙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠈⠁⠈⠁⠀⠦⠀⠀⠛⠀⠸⠀⠀⠂⣿⠀⠀⠀⠀⢱⠀⢸⡇⠀⢻⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠐⠀⠒⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠄⠀⠀⠀⢳⣄⠀⠀⠄⡇⠀⠀⢹⡀⠀⠀⠀⢸⡀⠈⣷⠀⠀⢻⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠒⣰⣤⡀⠀⠀⢸⣿⠟⠉⠀⣷⠀⠰⣿⣧⠀⠀⠀⢸⠃⠀⣹⣦⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠶⠞⠛⠁⠙⠁⠶⠚⠛⠦⠤⠴⠿⠀⠀⠿⠗⠀⠀⠐⠚⠓⠘⠛⠹⠶⢶⡴⠆⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠻⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣦⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⢲⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿
⠀⢀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⣿⣿⣿⠿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣄⣴⣿⣿⣷⣾⣿⣿⣇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⣼⠟⢒⡆⠀⠀⠀⠀⣀⣀⣄⣀⣤⣾⡷⢿⣿⣿⡿⠟⠉⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣄
⡟⠀⣿⣵⠀⠀⠀⠈⠙⠛⠻⠿⠿⢿⣄⣨⣿⣿⣷⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠿⢿⡿⣷⡀
⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣟⠛⠛⠛⢙⡃⠀⠀⢁⣿⣿⣿⣶⣶⠶⠆
⠀⢠⡿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⡿⠁⠀⠀⠀⠀⠈⢿⣿⣿⣷⣾⣿⣿⣷⣿⣿⣿⣿⣿⣟⠁
⣴⡾⡗⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡀
⣽⠟⢿⣶⡆⠀⠀⠀⠀⠀⠀⢸⣟⣹⣿⣟⠁⠛⣿⡏⠉⠻⣿⣿⣿⣧⣀⡴⢾⣿⣷⣴⣾⢿⡿⣿⣿⣿⡿⠛⠹⢿⣟⢾⣿⣿⡇⠀⠀⠀⠀⠀⢦⡻⠄
⡟⡀⠀⢈⠀⠀⠀⠀⠀⠀⠀⢸⡿⠻⣾⣽⣶⣤⣭⣳⣠⣼⠺⣿⡟⣹⣿⡦⣴⣿⣴⠿⠇⣯⣿⣿⣫⣯⣤⠠⣤⣶⣛⡮⣿⡻⠀⠀⠀⠀⠀⠀⢾⣷⡀
⠤⠀⠀⣞⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⢠⣽⣿⣿⡿⣏⣾⡟⢿⣧⣿⣻⣾⡀⢈⣛⢿⣿⣿⣿⣿⠿⢫⣙⢿⣿⡿⢛⣿⣷⣼⡇⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄
⡿⠂⠰⢿⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣟⣺⡿⢧⣌⢁⡀⠀⢿⢿⣿⣿⣶⣾⣿⣿⣟⡍⣤⣴⣿⣿⢛⣅⣀⣿⣧⣆⠾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦
⠓⠀⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡽⠧⢥⠡⣟⢢⣙⣿⣿⣿⣿⡿⣿⣿⣿⠿⠯⢽⣿⣿⣿⣿⣿⣈⣮⣿⠵⠼⣿⠵⠶⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣌⣸⣶⣠⣿⡶⠿⣿⠿⠛⢛⣉⡉⢰⣿⣿⣿⣿⣿⡷⠀⠀⠙⠻⠿⠷⣷⣿⣾⣮⣷⣾⣁⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⣿⣿⡿⢀⣴⣇⠀⠀⠀⠀⢙⣿⣿⣋⣉⣁⣸⣟⡏⠀⠘⣛⡏⠀⢀⣉⣭⣿⣿⣿⣿⣿⣶⣄⢠⣀⣈⣿⣿⣻⣿⡿⣿⣾⣿⣿⠓
⢸⣿⣿⠇⣾⣿⡿⠀⠀⠀⠀⠀⠉⠛⢿⣟⠻⣿⣷⣶⡶⣶⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⡾⠿⠋⠀⣭⡉⠉⠉⢁⣘⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⣀
⣾⣿⣿⣸⣿⣿⠇⠀⠀⠀⠀⠀⢀⡀⠀⠈⠙⢿⣿⣷⣗⣷⣾⣿⣿⣿⣿⣿⡿⠏⠀⠀⢀⣤⣤⣤⣤⣤⡿⣿⣿⠋⠟⣡⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣇
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠰⣿⡷⠀⠀⠀⠀⣿⢿⣯⢹⣿⣿⣿⣿⣿⣿⣶⣴⣤⣴⣾⡿⠿⣿⡟⣹⣶⣿⠟⠁⠀⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿
⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠀⠰⠆⠀⠀⣿⠀⡙⢾⣿⠟⣿⣿⠁⡘⣿⢿⠋⣼⣯⣴⣷⡟⠩⣿⣿⣿⠀⠐⠋⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢶⡦⠀⠀⣠⠀⣿⠀⣀⣼⣿⣿⣿⣿⣾⡇⢹⠘⡀⣸⣽⣿⣿⣿⡿⠋⢹⣿⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⠤⠀⠻⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣱⣿⣿⣿⣿⣿⠄⠀⢸⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⣶⣿⡄⠈⢹⣿⣿⣿⡿⣿⣿⣿⠀⢿⣿⣿⠟⠉⠃⠀⠀⣸⣿⠈⠛⠉⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⣿
⣿⣿⣿⣿⡇⢀⣴⣦⡀⠀⠀⠀⠀⠀⠃⠀⠀⠻⡿⠃⠀⠘⣿⣿⣿⡇⠘⣿⣿⣶⣾⡿⠁⠀⠀⢀⠀⠀⣻⡿⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣟
⣿⣿⣿⣿⣧⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⣿⣿⣿⡇⠀⣿⢿⣿⠏⣹⣷⣷⡞⠁⠀⠀⠿⠁⠠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣿⣿⣿⡇⠀⣿⡇⠁⠀⠹⠟⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠄⠀⠉⠉⠃⠀⠘⠃⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⢀⠔⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣄⢀⠀⠀⢀⠂⢠⡎⣰⣾⣿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠻⣷⣲⡦⣤⣿⢉⡻⣿⠏⣁⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⠤⣥⣿⠑⢶⡿⣸⠷⡋⡉⢐⣤⣆⠁
⠀⠀⠀⠀⢀⣀⢠⠾⣍⣄⣿⣺⣠⡳⣿⣧⣿⡾⠿⠓⠈⣀⣀⣀
⠀⠀⠀⢴⣎⢿⣯⣠⡀⢿⣿⢟⢷⣗⣽⣛⢿⣣⢖⣶⠟⠁⠀⠈
⠀⢀⣴⣷⣿⣿⣿⣿⠖⠂⢹⡗⢌⢻⣿⣿⢛⠻⡅⣃
⠀⠈⠁⣻⣿⣿⣿⢉⡈⠙⠃⠳⣾⣷⢼⡿⣿⡴⢛⠳⠔⠂
⠀⠀⠀⢭⣯⣝⣿⣵⠄⠀⠀⢲⢱⣹⣿⣷⢲⣾⣿⢀⡀
⠀⠀⠐⣾⣼⣿⣿⢟⡁⠀⠀⠙⣬⣿⣏⠹⣿⣿⣿⡉⠏
⠀⢀⣼⣿⣿⣿⣿⣿⣇⠄⠴⣏⣾⡻⣿⡽⣿⢟⣻⡟
⠀⠛⠉⠛⠛⠛⠟⠛⠙⣁⢠⣠⢿⣿⣿⣧⠻⢩⢰⣷⢖⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⣽⢻⣿⢻⣍⠟⡆⢹⣆⣿⡈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⣿⢾⢏⢸⠿⣰⣧⣘⣿⣿⣷⣷⣇⡄
⠀⠀⠀⠀⠀⠀⠀⠘⢁⣻⣟⣿⡿⣷⡐⣿⣿⣿⣿⣿⣿⣇⣿⣧
⠀⠀⠰⣵⠞⢈⠭⡻⢿⣿⢿⢿⣷⢿⣿⡞⣿⣿⣿⣿⣾⣿⣿⣯⡤⢤⢤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢸⣷⠃⡎⠀⢠⡏⣆⡆⣿⠙⢿⣷⣾⣯⣺⣿⣿⣿⡿⢁⠄⠠⠈⠉⡟⡍⠛⣆⠀⠀⠀⢀⠀⢠⠊⠁
⠀⠀⠐⠀⡙⣽⡅⣴⢿⣛⠋⠛⠑⢶⣾⣬⣿⡿⣿⣿⣿⡿⣷⣸⡤⡇⣀⣼⣡⣇⣷⣿⣰⣷⠶⣗⡘⠾⠂
⠀⠀⠀⠰⣟⡿⠋⡙⠋⠀⢲⡈⣦⣿⣍⣛⡉⡑⢘⠿⢫⣵⠚⠉⠦⣿⠉⣴⣿⡿⣿⣧⣤⣌⠑⠾⠃
⠀⠀⠀⠀⠀⢁⢡⣹⡘⢦⡤⢿⣙⣷⣤⣤⠾⠵⠿⠛⠚⠉⠀⠀⠀⠈⠉⠁⠩⢭⡍