Fly Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⢀⡤⢶⣶⣾⣶⣶⡠⣄⠀⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠰⣿⣶⣿⣯⣽⣽⣿⣿⣾⡷⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣾⣿⡿⠋⠉⠛⢛⣿⣿⣦⠃
⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇⣀⠀⠀⣨⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⡄⠤⠄⠀⠀⠈⠉⠀⠀⠀⠀⣠⣿⣿⠿⠿⠛⠛⠿⣿⣿⣧⡀⠀⠀⠀⠈⠉⠁⠀⠀⠤⠤⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢿⣿⡿⠗⠀⠀⠀⠠⢴⣿⣿⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⣿⣿⣶⣦⣤⣤⣴⣿⣿⡟⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⢐⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠁⠁⠘⠁⠘⠀⠀⠀⣄⢸⣿⡛⠁⠉⠻⣿⣆⠀⠀⠀⠘⠀⠁⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⡼⠀⠀⣰⠀⠀⠀⠀⠀⠀⢸⣾⣿⡃⠀⠀⢠⣽⣿⣄⠀⠀⠀⠀⠀⢰⡀⠀⠹⡄
⠀⠀⠀⠀⠀⡘⠁⠀⠀⠃⠀⠀⠀⢠⡆⣄⣼⣿⣿⠛⠛⠛⢿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠃⠀⠀⠱⡀
⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⣿⣿⣿⣿⣀⣀⣀⣨⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠉⠁⠀⠉⠹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠐⠀⠀⠀⢠⠁⠀⠀⠀⡠⢹⣿⣿⣿⣿⡿⠿⠷⠶⠶⢾⣿⣿⣿⣿⠠⡀⠀⠀⠀⢣⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⡀⠀⠀⢻⣿⣿⣷⣧⣄⣀⣀⣤⣤⣿⣿⣿⠃⠀⠀⡀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠻⣿⣿⣉⣻⣟⠉⣿⣹⣿⡿⠁⠀⠀⠀⠉⢦
⠀⠀⠀⠀⢂⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⡀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠝
⠀⠀⠀⠀⠀⠈⠻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁⠀⠀⠀⠀⣰⣿⣿⣷⡀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⡀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢀⣴⣾⡿⢿⣿⣿⡟⠉⠳⢦⣀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢰⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣷⡀⠀⠀⠀⠀⠰⣿⣇⠀⠀⣾⣿⣿⣦⠸⣿⣿⣧⣴⣦⣴⣿⡆⠀⢠⣿⠀⠀⠀⠀⠀⣀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡀⠀⠀⠀⠀⠈⢻⣆⠀⠻⠿⠿⢿⣶⣿⣿⣿⣾⣿⣿⠿⠃⣰⡿⠁⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠻⣦⣼⣶⣾⣿⣿⣿⣿⣿⡋⣿⣖⣶⡾⠋⠀⠀⠀⠀⠀⠀⣼⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠧⣿⠃⣿⡇⠀⠀⠀⠀⠀⠀⣰⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣿⣾⣿⡅⠀⠀⠀⠀⠀⢠⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⣶⣤⣄⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⠁⣿⣀⣀⣀⣤⣶⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠟⠉⣿⣿⣿⣿⣿⣿⢿⣿⣯⣿⣤⣿⡟⡛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣴⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⠇⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨⠇⠀⠀⢻⣿⡿⣿⡿⢿⣿⠿⣿⡟⢸⡆⣷⣽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡌⠙⠀⠺⢃⠀⠹⣷⣿⣷⣾⡷⣶⠋⢁⠸⠷⠹⠟⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣈⡟⢁⠃⠀⠀⣽⣄⣈⣽⣿⣿⣿⢿⣇⠀⢀⡀⠀⠀⢸⡀⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢋⢹⠗⠈⠀⠀⣼⣟⣀⣌⠹⠘⠀⣼⢠⣍⡀⡄⠁⠀⠀⠈⡶⠦⢈⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⠑⠀⠂⠾⡶⠀⠀⢿⣉⠿⢿⢆⣠⣦⠙⣠⡏⢰⡗⠀⠆⠀⠀⠧⠱⠈⢀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠈⠀⠋⡀⠀⠁⠀⠀⢸⣜⣀⣤⣿⡇⠋⠸⠙⢀⠂⠐⢸⠁⠀⠀⠘⠀⡆⠈⡄⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⡐⢸⠆⠀⠁⠀⠀⠀⠀⣀⣿⡷⠹⣿⠁⡦⢰⠀⠌⠀⢰⡄⠀⠀⠀⠀⠃⠘⠀⠉⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⢁⡁⢠⡄⠀⠋⠀⠀⠠⣿⣽⣶⡶⣿⣼⣿⠁⢠⠀⢀⠀⡁⠀⠀⠀⠀⠀⠀⠰⡄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠂⠘⢁⠛⡀⠀⠀⠀⠀⢠⢻⣿⣿⢻⣿⢟⣿⣿⡾⡠⠚⣤⠆⠀⠀⠀⠀⠀⠀⠀⢣⠀⡆⠘⢿⣶⡀
⠀⠀⠀⠀⠀⣠⣶⠛⠋⠀⢠⡠⠀⠄⢀⠀⠐⠦⠀⢀⣸⠴⢿⣧⡞⣹⣾⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠃⢃⠀⠀⠈⠻⣷⣆⣀
⣀⣤⣤⡾⠟⠁⠁⠀⠀⠀⢸⠃⠀⠀⠀⢂⠀⠴⠚⠁⠀⡇⠈⣿⣾⣿⣿⣿⣿⣿⣿⣶⡆⠀⠈⠉⠐⠀⠀⠀⠘⠀⣬⢠⠀⠀⠀⠀⠀⠈⢯⠶⣤⡄⣀
⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡠⠚⢿⣿⣿⣿⣿⣯⣭⣽⡿⠁⠀⠀⠀⠀⣢⠀⠀⠀⢰⠀⣻⡆⠀⠀⠀⠀⠀⠀⠀⠈⠛⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣿⠀⠀⠀⠀⠃⠀⢀⠞⠁⠀⠈⠻⣿⣿⣿⣿⣿⠿⠀⠀⠀⢀⡀⠀⠈⠀⠠⠀⠈⠂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢘⠀⠀⠊⠉⢰⡀⠊⠀⠀⠀⠀⠀⠘⢿⣿⠟⠉⠀⠀⠀⠀⠀⠣⡀⠐⠦⠀⠀⠀⠘⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡀⠰⡖⠈⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⢶⣰⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⠁⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠐⠀⠀⢀⠀⠂⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢀⠀⠈⠀⠀⠀⢀⠘⣄
⠀⠀⠀⠀⠀⠀⣀⣀⣤⢤⠤⠄⠤⡤⠄⢀⣀⣀⣀⣘⡦⠴⠖⠒⢀⠖⠑⠐⠆⠘⣄⠀⠀⠀⠀⣴⡆
⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠌⠉⠄⠀⠀⠀⢈⠀⠀⠈⠀⣠⡀⠑⣤⣠⡶⠋
⠀⠀⠀⠀⠹⣄⠀⠤⠀⠀⠀⠀⠠⠄⠀⠈⠀⠤⠀⠀⠀⠀⠂⠀⠀⠀⠀⢴⠛⠁⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠈⠣⣀⠀⢠⡀⠀⠀⠀⠀⠀⠐⢆⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠗⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣄⠀⣀⡀⠐⠀⠤⠄⠐⠤⣀⣀⠀⠀⠀⠀⠰⢆⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡸⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠄⣀⣀⣀⣀⣀⣀⣀⡀⠬⠿⠿⢿⡾⠛⣧⡀⠀⠖⠀⢄⠀⠈⠙⡍⠲⣌⡶⢤⣀⣀⣀⣠⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠒⠒⠒⡖⠩⠁⠐⠊⢡⡇⢠⡀⠀⠀⡤⢠⡞⠁⠀⠀⠉⠁⠈⠉⠒⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠋⠁⠈⠁⠈⢆⠢⡀⠀⠘⠦⡜⠅⠉⠉⢳⡉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠱⠈⠄⠀⠀⠘⡈⠀⠀⠈⠓⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢴⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⠀⠘⡄⢄⠀⠾⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡜⠀⠀⠀⠀⠀⠀⠀⠙⠻⠀⠀⠀⠀⠳⠔⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⢢⡱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠇⠀⠀⠀⠀⠀⠀⠀⢌⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠖



⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⢀⣀⡄⣄⣀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠸⠼⠃⠻⠼⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢲⣤⠀⠀⠀⢠⣴⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠠⡋⢠⠀⢠⠈⡣⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡃⠀⢠⠀⠀⡇⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⡇⠀⢸⠄⠀⡇⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢰⠀⠳⣀⠼⢄⡸⠃⢰⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⠀⠊⠙⠤⠘⠉⠂⢠⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣿⣿⣿⣿⣯⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠛⠉⠀⠀⣿⣿⣿⠁⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣴⣾⠳⣶⣄⢿⣶⣤⣄⣶⣿⣿⣿⣿⣿⣤⣤⣀⣼⠇⢀⣤⡴⣶⣤⣄⣰⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠻⣷⣌⠉⣽⡿⠿⠛⠛⠿⢿⣿⣿⣛⢀⣴⣿⠟⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣷⣄⠀⣀⠀⠀⠀⣙⡯⢻⠉
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠿⠟⠛⠛⠻⠿⣿⣧⣼⡀
⠀⠀⠀⠀⠾⠛⠋⠉⠉⠻⣶⣄⡀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢘⣿⡻⣿⣷⣤⣀⠀⠀⠀⠀⢀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⡴⠿⠛⢩⣾⠋⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣠⡇⣇⠨⠹⣿⣿⢷⣦⣤⠶⠛⠉⠀⠉⠑⠶⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣇⣤⣽⣿⣿⣿⣧⣤⣀⢀⣀⣠⣝⢰⣝⣆⣅⠡⠘⣧⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⢿⡏⢉⠋⣿⣿⣿⣿⣿⣿⣿⣿⣻⣯⣫⣾⡛⣿⠛⢿⣷⡌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢡⠞⠀⡌⢰⣿⣿⣿⣿⣿⡏⠉⠉⠙⣿⣿⣿⣧⢸⡆⡄⠹⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢋⣾⡵⠁⠀⢠⠁⣾⢿⣿⣿⣿⣿⣻⣦⣶⣿⣿⠿⠃⢛⣰⡇⠐⣄⠀⠙⢿⣿⣄
⠀⠀⡀⠀⠀⣀⣀⣄⡠⠜⠛⢁⡾⡹⠀⠸⠀⡌⠀⣿⣿⣿⣦⣭⣽⣿⣿⣿⣿⣷⣶⣾⣿⣿⠁⠀⠘⢦⠀⠀⡏⢻⡆
⠀⠈⠛⠒⠛⠛⠛⠉⠀⠀⠀⡾⠡⡆⠀⠆⢰⠃⠀⣿⣿⣿⣿⣿⣿⣿⡟⡟⢻⢿⡟⠛⣿⢿⠀⠡⡀⠀⠳⣄⣇⠈⢿⡻⢶⣶⣴⣤⣄⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⡜⠀⡌⠀⠾⠀⠀⠸⣿⣿⣿⣿⡟⠹⠇⠁⠀⠈⡇⠃⣼⠎⠀⠀⠡⠀⠀⠈⠻⡄⠘⣧⠀⠀⠉⠉⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⠊⠀⡚⠀⠀⠀⠀⠀⠀⢿⣮⣿⡟⠃⠀⠀⠀⠀⠀⢹⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⡐⠀⠀⠀⢰⠀⠀⠀⠈⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠻⣿⡿⡽⠏⠙⡿⠋⡿⠁⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠈⠓⠦⣀⣀⡤⠊⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⢀⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠆⠠⠞



⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢼⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⣼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⡾⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡇⡇⠀⠀⠀⠀⢀⣠⣴⣶⣶⡏⢻⣻⠇⢨⡷⢶⣄⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⠁⠀⠀⠀⣰⡿⠝⠃⢈⠻⣷⡟⠅⣰⡏⢰⣀⡙⢷⡄⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⠀⣿⣿⣴⠿⠿⣶⢾⠃⠘⣙⣿⣿⢿⣿⣸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⣿⣤⡀⠀⠀⢸⣿⡯⣿⣿⣿⣿⡿⢛⣿⣶⣷⣶⣿⣻⣷⠀⠀⣠⣴⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣿⣷⣤⣽⣿⢇⣿⢏⠟⣿⡇⠙⢻⣿⣏⢿⣿⣽⣿⣤⣾⢋⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠛⣿⣿⣿⢫⣿⡏⢈⣿⠀⠀⠀⣿⣟⠆⣿⡶⣿⣿⣷⢯⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣾⣧⠸⣿⡥⠚⣿⠀⠀⠀⣿⣯⠴⣿⡷⣾⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣷⣠⣿⡎⠀⣿⠀⠀⠀⣿⡗⠀⣸⡿⣽⣇⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢴⡾⣿⡟⣹⢤⣿⡖⢠⣿⠄⠀⢶⣿⡇⠘⣿⣧⣿⣿⣽⡟⠿⣶⡶⢆⣀⡀
⣤⣀⣀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⣺⣭⠖⢋⣴⠿⢀⡟⢉⣿⣿⢉⣿⡇⠤⢰⣿⡇⢨⣿⣿⡟⣠⡿⢮⣁⠈⠙⠓⢾⣅⣚⣧⣤⣀⣠⣤⣤⣤⣤⣶⡶⠾
⠻⠋⠉⠟⠛⠻⠷⠶⠿⠲⠶⠾⡿⢷⠖⠋⠉⠁⣠⠏⢹⣶⠟⠀⡾⠻⣿⣧⣿⣷⣶⣿⢿⣶⣾⡿⠋⣠⣀⠹⡀⠙⢦⡄⠀⠀⠀⠉⠻⣍⠉⠉⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⣤⢋⠏⢀⣼⡇⣷⡘⢿⡟⡍⠛⣭⠛⣿⠋⢠⡌⣹⡟⡆⠳⣄⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠱⡟⣡⢋⠔⡛⣏⣹⠎⡡⠀⢙⣿⣶⣷⣿⠁⠇⠰⣿⢉⣿⠸⡄⠈⠓⡦⣌⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡡⠊⡰⢃⠆⠸⠀⣯⣿⣆⠙⣤⣾⣿⢿⠿⣿⣧⣀⣴⣸⡌⣿⡇⠹⣗⢄⠳⠦⢍⡙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠊⢀⢾⠡⡆⢀⠇⣠⣿⡭⣷⡋⢾⣿⣿⡖⠫⢽⡏⠈⠀⢹⡿⢻⣿⡋⠙⢆⠻⡔⠦⡍⠺⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢀⡴⠃⡌⠠⢀⣼⣶⣝⣿⣇⢾⢷⡈⣿⣿⣇⣠⣼⡇⢈⢉⣹⡁⣾⣟⣿⣦⡈⣆⠈⢦⡀⠰⣌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⡠⠖⠉⠀⣠⣥⡗⣩⣿⠿⠋⠹⣷⡞⠎⠑⣾⣿⢿⠱⣿⣧⠈⢨⠇⢹⡟⠈⢸⣧⡉⢸⡄⠀⠱⣄⠈⢧⡙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣣⠊⠀⠀⠀⣴⣟⠿⠟⠙⡏⠀⠀⠐⣿⠛⠶⢀⣿⣍⣂⠀⢊⣻⣆⠘⢶⣏⣿⠀⠀⢹⠛⠿⣷⣄⠀⠈⢧⠀⠑⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠁⠀⠀⢀⠎⠘⣿⡀⠀⢠⡇⠀⠀⠀⢹⡏⢀⣾⣿⣿⠿⡿⢿⣿⣿⣶⡊⢻⡇⠀⠀⠈⡇⠀⠈⣿⡅⠀⠀⢳⡀⠈⠊⢷⡀
⠀⠀⠀⠀⠀⠀⠀⡾⠃⠀⠀⠀⡀⠁⠀⢠⠻⣷⠀⠸⠀⠀⠀⠀⠀⢠⣾⣟⣿⣟⣛⠀⠀⠚⣋⣉⣿⣏⠀⠀⠀⠀⡇⢀⣾⠟⢧⠀⠀⠀⠹⣄⠀⠈⢷
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⡄⠀⠀⢀⠇⠀⠘⢻⡇⠀⠀⠀⢀⡴⠋⠿⣿⣿⡿⣿⠿⢻⡿⣿⣿⣿⠙⠳⢦⡀⠀⢰⡾⠃⠀⠈⢧⡀⠀⠀⠈⠣⣀⠈⣇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⠈⠀⠀⣀⠈⠀⠀⠀⠨⡗⠀⣠⠴⠋⠀⠀⠀⠈⠹⣿⣷⣦⡬⣤⡼⠿⠈⠀⠀⠀⠙⢷⣿⠁⠀⠀⠀⠈⠳⣄⠀⠀⠀⠈⠉⢻
⠀⠀⠀⠀⠀⠀⡟⠀⠐⠁⡀⠀⠈⠀⠀⠀⠀⢀⡴⢿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⡻⢻⢉⠃⠁⠀⠀⠀⠀⢰⣧⠋⠓⢤⣀⠀⠀⠀⠈⠳⣄⣀⠀⠀⢸
⠀⠀⠀⠀⠀⢰⣇⠀⠠⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⢸⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⠀⠀⠉⠳⢦⣄⣀⠀⠀⠈⠉⢓⣾
⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⣀⣤⠶⠋⠁⠀⢀⣄⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠈⠉⠛⠷⠶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠛⠉⠀⠀⠀⠀⣀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠢⡁⠀⡀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠢⣤⡀⠈⠲⣄⡀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠘⢦⣌⠳⠬⣙⣶⣄⣀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣘⠀⠲⠿⠞⠳⢶⣍⣛⣊⣝⠐⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⢞⣾⣿⡿⢿⣿⣿⣿⣯⠂⣈⣿⣿⠟⠛⠻⣿⣦⣴⠖⠒⢲⣄⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣿⣿⠃⣾⣿⣿⣢⣾⣿⣿⣿⣧⣼⣋⢀⡉⠀⠀⠀⠈⣿⢹⡀⠀⢈⣿⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠛⠈⠛⠋⠀⠙⠋⠀⠙⡁⢻⣿⣿⠈⣿⣿⠈⡉⠀⠀⠒⠸⣿⣷⣿⣷⣿⣿⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣤⣤⢀⣤⡖⠀⣴⣶⡄⣰⣿⣿⣿⣿⣧⡘⢿⣦⣷⣦⣀⢀⣴⣿⡟⠀⠀⢹⣿⣷⡏⢵⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣜⣿⣿⣧⣿⣿⣿⣿⡷⠿⠿⠿⠿⢟⣛⣉⣹⢟⡛⣿⣿⣿⣷⡀⠀⣸⡿⠙⣷⡄⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡿⢿⢛⡭⢽⠾⡷⣿⠿⢞⣛⠿⣋⢝⣥⣾⣯⡍⢹⣃⠘⠿⠿⠟⠁⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠁⢘⠊⠐⣀⣤⢴⠞⡩⠴⡊⠀⠀⠀⠉⠁⠀⣼⡿⠿⢷⣶⣦⣤⡀⡀⠀⠀⠣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠠⠒⠔⢊⡴⣉⡴⠃⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⠀⠀⠀⠉⠀⠘⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠕⡂⢀⡔⠩⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢊⠴⠋⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⡰⠁⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⣁⠍⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⡠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠊⠉⣉⣉⡉⠉⠉⠐⠛⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⢀⠔⠊⠉⠀⠀⠈⠉⠲⡄⢀⠤⠚⠉⠒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠒⣉⣉⣀⣀⣀⡀⠀⠀⠉⠉⠑⠒⡴⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣠⠔⣋⠵⠒⠋⠉⠉⠁⠀⠀⠈⠉⠉⠙⠒⠒⠒⡶⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⣿⠟⠀⠀⠀⠀⠀⠀⢀⣶⣦⠀⠀⠀⠀⡇
⠀⠀⠀⡰⣫⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⢠⢇
⠀⠀⡸⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⣀⠴⠣⣀⠀⠀⠀⠀⢀⠴⠁⠘
⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠉⡉⠉⠉⠁⠀⠀⠈⠉⠒⠒⠊⠁
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⢦⣀
⠙⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⡜
⠀⠀⠈⠙⠒⠤⢄⣀⣀⣀⣀⣠⠤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠋⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⣄⣀⣀⣀⣀⣀⣠⠤⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⢀⡴⠋⢀⡠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⣀⡤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⢀⣯⠤⢶⠉⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠛⠣⠤⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠒⠻⡏⠁⠘⣆⠈⣇⠀⠈⠙⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠉⣩⠟⠉⠉⠉⠀⠀⠀⠀⠀⠀⡝⠀⠀⡞⠀⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠆⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣚⡫⡽⠕⢫⡙⢿⣗
⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⣤⣾⣉⠀⣉⠁⡀⠠⠀⢰⠲⠀⢹⠀⢀⠤⠂
⠀⠀⠀⠀⣴⢻⠋⣼⠟⠉⠻⠿⣿⣾⠙⠈⠁⠐⠀⠁⠀⠀⢀⣶⠁
⠀⠀⠀⠀⣧⣼⣰⣸⡦⠀⠀⠀⠸⣿⠀⠀⠀⠀⡀⠀⠀⣰⡾⠿⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠈⠁⡮⡇⠹⣦⠀⠀⢴⣿⠆⢰⣾⣷⡻⣖⡛⠋⢠⡆⠂⢀⡐⠀⢱⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠙⣝⠓⣦⣽⣿⣿⣿⣷⢸⠀⢿⣿⣿⡟⠃⢠⣿⠘⠰⠾⠶⠶⠈⡖⠚⠉
⠀⠀⠀⠀⠀⠰⠨⠀⠁⠀⠀⠛⢿⠟⠸⢀⣸⠁⠙⢧⡴⠛⠿⠃⠁⢃⠀⠀⠠⢇⡀
⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⢸⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠀⠀⠀⠉⠢⣤⣀⡀
⠀⠀⠀⢀⡰⠄⣠⣤⠀⠀⠀⠀⠀⢠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠓⠢⢤⣄⣀⡈⠉⠉⠛⠓
⠀⢶⠶⠋⠀⠀⠀⠀⠉⠀⠀⠀⢀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠤⣴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⡀⠀⠀⢀⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠼⢿⣟⠷⣞⣻⡿⠧⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⣿⡞⣱⣿⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⣰⣿⣧⢿⣿⣆⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⠀⠻⣦⣀⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⣀⣼⠏⠀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⣸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁⠀⣠⣿⣿⠃
⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠙⢿⣿⣷⣶⣦⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣴⣶⣾⣿⡿⠃
⠀⠀⠀⠈⠉⠛⠛⠛⠛⠻⠿⠿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⣠⣴⣿⣿⠿⠿⠟⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⣷⣦⣄⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⢀⣠⣴⣿⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣶⡞⠁⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠈⢷⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡞⠁⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠛⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⠻⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢱⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡜⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠹⣿⣧
⠀⠀⠀⠀⠀⢀⣾⣿⠏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠻⢿⣿⣿⣿⣿⡿⠋⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢻⣿⣧
⠀⠀⠀⠀⠀⠺⠟⠋⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠙⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⣀⣤⡶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢦⣤⣀⣀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣠⢟⣡⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⣠⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⣀⣿⣀⣠⣤⣾⠟⠋⠀⠀⣀⣀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⢿⡿⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣋⡀
⠀⠀⠀⢴⣾⣿⠿⠋⢻⣿⢾⣿⣿⣿⣿⣿⢶⣏⣙⠟⢳⡿⢧⣿⣿⣿⣿⣿⣿⣛⠛⠟⠛⠛⠻⢿⣶⣶⣦⣤⣤⣤⡄
⠀⠀⠀⠀⠀⣠⣴⣾⡿⠿⠷⠾⣿⣿⣿⣿⣷⣼⣯⣿⣿⠾⠋⠀⠉⠙⠛⠛⠻⣿⡆⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁
⠀⠀⠀⠀⣠⣿⡏⠀⠀⠀⣠⣼⣿⠏⠀⠉⢉⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆
⠀⠀⣠⣾⣿⡿⠀⢀⣠⣾⣿⡿⠁⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣄⡀
⠐⠻⠟⠛⠋⠀⠀⢻⣿⣿⡟⠀⠀⠀⠀⠀⣸⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⣰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋


⣿⣿⣿⣿⣻⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣿⣿⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠀⠀⠀⠀⠀⠀⢀⣠⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⡀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⢀⣤⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠈⣷⡀⢀⠀⣴⣋⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⠖⠛⡙⢯⣿⣯⠋⢩⡉⢦⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠿⠀⣿⡇⣀⠀⠟⣸⠀⠀⠀⠀⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⣀⠀⠀⠈⣷⣬⣷⢰⣧⣾⢴⣮⣽⡃⠀⣀⣠⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣿⢣⡏⢹⣇⣿⣽⣿⣿⡷⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢸⡇⣿⡇⣿⠘⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣿⡂⣿⡏⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢸⣇⣿⡇⣿⡇⣿⣸⣿⣸⣿⠀⠀⢀⣀
⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⠿⠛⠙⡿⣋⠙⢯⣿⡇⣿⡇⣿⣥⣿⣽⠏⣕⢾⠟⠙⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠙⢿⣦⣀⣠⣴⠾⠻⠛⠉⠀⠀⢀⡤⠞⠉⠀⢱⡈⢹⣇⣿⡯⣿⣋⡟⢁⠄⠣⠙⠳⣄⠀⠉⠋⠻⣶⣤⣀⠀⠀⠀⠀⠀⣠⣴⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⢀⠀⠀⡙⠈⠈⢻⣄⡟⠉⠑⠉⠂⠑⠌⠂⠈⠣⡀⠀⠀⠀⠉⠙⠻⠶⠶⠾⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⡀⠁⠊⢀⠀⠀⠀⣿⣿⣷⡀⠀⠄⠀⠀⠀⢂⠡⡁⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣀⠁⢀⠌⡰⠓⠒⡇⠀⠀⠀⣻⣿⣿⡇⠀⠘⡀⠐⠄⠄⠁⠐⢄⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠠⠋⢀⠂⢰⠀⠀⠘⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⢁⠀⠀⣌⣄⠐⣄⠳⣌⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡰⠁⡠⠁⠠⢃⠀⢤⠁⠀⠀⠀⢀⣿⣿⣿⡃⠀⠀⠠⣤⣀⠀⡈⢠⡀⠙⢮⣧⡻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⢞⡰⠊⠀⢠⠋⠀⠀⢰⠀⠀⠀⠄⣾⣿⣿⣿⡟⠀⠀⠀⠲⠀⠀⢶⠀⠈⠒⠦⣄⠰⠄⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡻⠊⠀⠀⠀⡜⠀⠀⠀⡆⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄⠀⠀⠈⡄⠀⠀⠇⠀⠀⠀⠀⠁⠢⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠇⠀⠀⡐⣀⡤⠞⣧⠻⣻⣿⣿⡿⣿⣿⠷⠤⣄⣨⣄⡀⠈⢄⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⡀⠔⢁⣤⠞⢿⣿⠀⠀⠙⢶⣏⡘⡏⢀⣵⠏⠀⠀⣸⡟⠁⠉⠳⠦⣅⣀⣤⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠛⠒⠛⠉⠀⠀⠈⣿⠀⠀⠀⠀⠙⠻⠿⠛⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢿⣿⣿⣿⡗⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠉⢦⠹⣿⣿⣷⡀⠈⢧⠀⣀⠀⠀⠀⣠⣶⠿⠿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢘⣴⠛⡋⠉⠉⠻⠟⢻⢿⣦⣀⢔⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⣴⠟⠂⠅⡐⠀⠀⠀⠀⠀⣤⠽⣧⣞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⡿⣉⡄⡼⠠⡈⠀⢀⣀⠤⠴⠶⠿⢿⣮⣕⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣃⣸⡑⣴⣴⠛⠉⠀⠀⢀⠁⡰⣿⣯⣙⠻⣮⣓⢤⣠⣤⣤⣤⣀⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣯⠭⢽⡟⢸⠉⡿⣿⣣⡌⠀⠂⠁⠀⢀⠈⠇⢶⣟⠛⢷⢤⡉⠳⣬⡻⣍⠈⠙⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢀⣾⠇⣼⠀⡇⠈⡷⢤⣀⣰⠘⠀⣰⣰⣴⠾⣌⠳⣄⠳⣿⣶⣄⡙⠲⣵⣦⡀
⠀⣀⣠⣤⣴⠾⠋⠀⠀⠀⠀⠀⠀⡼⡟⠀⣿⢸⣿⠀⢸⠀⢿⡍⡍⠉⠉⠈⠀⠀⠑⠑⣌⠳⡌⢿⢷⣽⡲⣄⠙⢿⣷⣄
⠿⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⣿⢸⣿⡆⠈⣇⠘⡇⠁⠦⠀⠀⠦⠀⣠⣁⣤⣱⣌⠢⡱⣝⢿⣮⡑⢤⡈⠻⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡾⢏⡁⠀⣿⠘⡟⣧⠀⠸⡄⢓⢀⢀⡾⠖⠚⠛⠉⠀⠀⠀⠈⠳⣌⡪⣢⡙⢿⣦⡉⠢⡀⠓⢝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢾⠇⡇⠃⠀⣟⠀⣧⠸⡄⠀⢷⣌⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠛⠾⣦⡙⢟⢦⡈⠢⣀⡿
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⢸⠀⢳⠀⠀⣿⠀⣿⠀⢷⠀⢸⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠉⠉⠛⠋⠛⢿⣦⡀
⠀⠀⠀⠀⠀⣠⣾⠏⠁⠀⠀⠘⢰⡘⠀⠀⣿⠀⢸⠀⠘⣧⡟⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢿⣦⣄
⠀⠀⢀⣤⣾⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⠀⠸⡆⢠⠋⠹⢶⢀⠀⠀⠀⣀⡤⠔⠒⠉⠉⠉⠉⠉⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⢸⣸⡆⠀⠀⢹⠀⠀⡗⠁⠀⠀⠈⠙⢦⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡤⠊⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣤⣄⠴⠖⠚⠙⢋⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⢤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠖⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠂⠀⠙⠲⠤⣄⣀⠀⠀⠀⣀⣀⣤⡀⣀⡤⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠀⠀⠀⠀⠒⠿⢿⣶⢌⡉⢀⣀⣀⠎⠀⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡷⠔⣰⣿⣿⣄⣴⣾⡗⡻
⠀⠀⠀⠀⠀⠀⠀⠄⠀⢁⣀⠀⠀⠀⠀⠀⢀⣽⣿⡿⠿⠛⣳⣾⣿⣿⣿⠿⣿⣿⣹⢇⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢚⡟⠉⣉⣷⣔⣾⣮⣹⣿⡿⣿⣯⣴⠶⠻⣏⠀⠉⠓⠒⠒⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡇⢀⣿⣿⣧⣿⣿⣿⠛⠛⡎⠉⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣌⣿⣶⣿⣿⣿⣿⣯⠾⠀⠀⠛⠐⠂⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠿⢿⡟⠭⢭⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢀⡿⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠴⠋⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁


⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠂⠀⠀⠀⠀⠀⠒⠂⠀⠒⡂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠂⠀⣀⠀⠀⠀⠀⠒⠀⠀⠀⠘⠐⠛⢶⣦⡴⣲⣶⣿⣶⣶⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣹⣿⣿⡆⢀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣧⣤⡤⣤⠽⠶⢶⣿⣿⣿⣿⠿⠋⢻⣿⣿⣿⡿⢿⣏⣿⡏⠱⡀⢣⠀⠀⡆⠀⠀⠀⠀⠀⢀⡠⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⣿⣿⣶⣶⣶⡶⢣⣹⠗⢻⣾⣿⣷⣶⣿⣿⠿⣿⣷⣬⣾⣇⠀⠁⠀⠀⢀⠁⠀⠀⢀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠀⢻⡼⡟⢛⣩⣽⣿⣿⣿⣬⣷⣵⣽⣿⣿⣿⢿⡇⣀⣬⡉⠛⠛⠛⣷⠀⠀⠀⢸⡆⠀⢠⠞
⠀⠀⠀⠀⠀⠀⢸⣿⣆⠈⢓⣾⣿⣿⣯⠉⢻⣿⣿⣿⢻⣿⡿⠛⠻⠿⢻⣿⣽⣧⢺⡤⣠⣎⣥⣴⣠⡟⠀⠠
⠀⠀⠀⠀⠀⢠⠀⠻⡿⣷⣦⣉⢝⣿⣛⣴⢟⣩⡟⣱⢻⡟⠀⠀⠀⠀⢸⡟⢠⣼⣾⣻⡿⠿⣿⣿⡟⠀⡐⠁
⠀⠀⠀⠀⠀⠸⢶⣤⣽⣮⣟⣛⣿⣿⣟⣵⠟⠉⠀⢹⣼⣄⠀⠠⠤⠤⣼⣄⣰⣿⠟⠁⠀⠀⢨⣿⡶⠋⢀⣰⣖⣉
⠀⠀⠀⠀⠀⣄⠀⣈⣙⣻⣿⣷⡷⠛⢭⣬⣷⣤⣄⠀⠹⣿⣷⣄⠀⠀⢇⣽⡿⠻⠀⠀⠀⢀⣽⣿⡷⠒⠛⠛⠉⠀⠠⢄
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠞⠀⠀⠀⠈⠉⠉⢿⣿⣦⡉⠁⠸⠧⢾⣿⣿⣿⠀⢀⣀⣽⣿⡯⠿⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣠⠝⠛⡝⡄⣀⣠⠴⢿⣿⣿⡷⠟⠋⡞⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢥⠤⠒⠙⠉⠁⢀⣠⣼⢟⡟⠀⠀⠀⠀⠀⠹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁⣀⠠⠤⢴⠾⣿⡟⠁⢈⡧⠀⠀⠀⠀⠀⠀⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠔⠉⠀⠀⠐⠁⡰⠛⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠈⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡀⠀⠄⢀⣀⣀⢀⡈⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠋⠉⢀⡠⠄⠠⡀⠀⠀⠀⠈⠃⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⢀⠴⠉⠀⠀⠀⠈⢢⠀⠀⢠⠃⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⣁⠈⠂⠊⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⡆⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠈⠣⡀⠀⠀⡠⠂⠐⠒⠀⠐⠁⠀⠀⠀⠃⠀⠀⠤⡀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠘⡄⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⣀⠤⠤⣀⠀⣀⢀⠀⠀⠀⡇⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⢀⡤⠒⠀⠣⡀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠉⠀⠀⠑⡀⠀⠈⢆⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⣎⢄⡀⠀⠀⠣⠠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⣀⣰⠀⢀⡠⠜⠒⠒⠈⠁
⠀⠀⠀⠀⢀⠔⠁⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠉⠁⠀⠀⠀⡃⠈⡆⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⢰⠂⠐⠒⠚⠢⠔⠊⠁⢱⠀⠀⡎⠀⠈⢆⠀⠀⠀⡠⠠⠔⠁⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⢃⠀⠀⠀⠑⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠴⠁
⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠘⣄⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠠⠄⠒⠀⠉⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠂⠒⠂⠐⠒⠒⠒⠒⠒⠂⠒⠒⠒⠀⠒⠒⠀⠒⠂⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠐⢄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠠⠤⠄⠠⠄⠀⠒⠒⠒⠀⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⢀⠂⠀⣀⠀⠀⠀⢠⣢⡘⠋⢈⠷⣣⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⢃⡸
⠀⠀⠀⠀⠀⢤⣈⠂⠀⠀⠎⠁⣏⠹⢗⣋⢁⠙⠧⢔⡄⠀⠀⠠⠺⠇⢀⠀⠀⠀⠀⠈⠁⣠⣞⠱⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠙⠁⠛⢣⣄⣓⢬⠄⣬⡒⠇⠀⠠⠀⠀⡀⠀⢉⠖⠀⠀⠾⢿⡟⠉⠁⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⢶⢄⣤⣼⢀⠁⠾⠧⠕⡀⡄⠀⠀⡀⣠⢴⠛⢋⠠⢂⠀⠤⣀⡄⠀⠀⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠡⠐⠂⢌⠉⠰⠆⠁⠀⠈⠉⢿⡁⠊⠁⠀⠀⠀⠀⠈⠉⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠂⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠓⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⢃⡄⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠶⠈⠀⠀⠀⠀⠀⠈⠑⠂⠤⡀⠀⠀⠁⠈⠼⠀⠀⠀⠈⠙⠷⠤⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠢⣄⠉⠒⠄⣐⠂⠀⣆⡀⠀⢀⠀⣀⠂⠀⠀⠀⠀⠀⠈⠓⠶⢤⣀⠀⠄⡠⠀⠀⠤⠀⠀⠀⠠⢠⠐
⠀⠀⠀⠀⠉⠀⣠⣿⡟⠓⠚⢷⣾⡚⠛⠋⠙⠲⠀⠀⠀⠠⡀⣠⡆⠀⠈⢹⣦⡇⣀⠀⠂⠠⠠
⠀⠀⠠⠊⢀⠾⠛⣿⠷⠒⠂⠀⠀⠘⠯⠶⠲⢾⠆⠀⠐⣎⠂⠀⠀⠀⠀⠈⠨⠛⡧⢶⣠⣇⣰
⠀⠀⠀⠈⠁⠜⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠋⠁⢀
⠀⠀⠀⠀⠘⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⠖⠦⢀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠁⠀⠀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⢡⣶⡀⢰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠆
⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠰⣦⠀⠉⠀⣐⠆⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠉⠈⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⢀⣜⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣠⠔⣠⣤⠊⣠⠀⠀⣨⡕⢶⣤⠤⣤⠣⠁⢨⣄⡀⢀⣀⣀⣀⣀⡄⢀⣀
⠀⠀⠀⠀⠈⢢⡉⠠⠈⠻⣄⠀⠙⢆⠀⠐⡄⠈⠉⠓⠀⠀⠀⠈⢰⢧⠀⠈⠀⠐⠈⠒⠀⡀⠀⠈⠁⠉⢸
⠀⠀⠀⠀⠀⠂⢀⣀⠈⠉⠀⠀⠁⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢘⠀⠁
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠸⢶⡦
⠀⠀⠀⠀⠀⠀⢸⠐⠀⢀⡄⢃⠀⢐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠁⠀⢀⡀⠀⠰⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠬⠥⠴⠤⠤⠴⠧⠤⠴⠾⠄⠀⣶⣾⡄⠀⠀⠤⠔⠒⡲⢀⠾⣆⠀⠀⠙⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠅⠀⠀⠀⠔⠉⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠉⠛⠿⢿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⣾⣿⡷⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣾⣿⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣴⣶⣶⣿⣿⣷⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⣴⣶⣾⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣀⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⡍
⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⣿⡇
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⢠⣤⣶⣶⣷⣿⣿⣶⣄
⠀⠀⠀⠈⠿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠙⠲⢄⠀⠀⢀⡠⠖⠋⠀⢀⡹
⠀⠀⠠⠤⠠⠀⠀⡀⠀⠀⠀⠀⠙⠲⡄⠀⠀⠀⠀⠀⢈⣳⡄⡎⠀⠀⠀⠀⠀⠀⢠⡴⠊
⠀⢀⠀⠀⠐⠀⠢⠈⠙⠲⢤⣀⠀⠀⠈⠛⠦⢄⡀⢰⣏⣨⣥⣅⣼⡀⠀⣀⠴⠊⠁⠀⠀⢀⡠⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡙⠒⠤⣀⠀⠘⢷⢀⣽⣿⣿⡿⢯⣀⡼⠁⠀⣀⡤⠔⠞⠁⠂⠀⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⡀⠀⠆⠀⠀⠀⠈⠁⠀⠈⢙⣶⡜⢿⣿⣿⣿⡁⠀⡿⠧⣒⠉⠀⠀⠨⠀⠀⠀⠈⠂⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡁⢀⠀⠀⠉⠁⠀⠠⠬⠿⠓⣞⣿⣿⣿⣿⣚⣻⠶⠥⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠡⠂⣀⠤⣤⣾⣿⣿⡿⢿⣿⣿⣤⣤⢦⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⢀⣩⣿⣿⡇⣨⣾⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⠀⠀⣠⠋⢽⣿⣿⣿⣿⣷⠄⠙⢆⠀⠁⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢰⡇⠀⣤⣤⣼⣿⡇⣤⣴⡀⢹⠀⠀⠈⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠋⠀⠀⠀⢸⡇⠀⣉⣙⣛⣛⣛⣋⣩⡁⣸⡇⠀⠀⠀⠙⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣭⣭⣭⣭⣝⣉⣥⠄⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⢉⣉⣉⣭⣙⣉⣥⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠈⣉⣛⣛⣛⣋⠁⠀⠀⠀⠈⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠰⣄⡀⠀⠀⠀⠀⠀⠀⢀⡴⠆⠀⠀⠀⠀⠀⢀⣴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢷⡄⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢀⣴⠏⠁⠀⠀⠀⠀⢠⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⣀⣽⣦⣤⣤⣾⠥⣄⠀⠀⠀⠀⠀⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⠎⠁⢢⠙⣿⡟⠀⢰⡄⢣⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠘⢆⡠⣌⠀⡟⣣⢀⡨⣥⠃⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⡀⣸⡟⣿⢹⡏⣿⣿⣿⣿⣤⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⣿⢿⣿⣼⡏⣿⢹⡟⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣇⣻⡇⣿⢺⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⣿⢸⡏⣿⡇⣿⣸⣿⣸⡇
⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠾⠋⢻⡷⡚⢿⣿⡏⣿⡇⣿⣹⣿⡿⢛⡲⣾⠛⢷⣦⣄
⠀⠀⠀⠀⠈⠻⣷⣄⣀⣤⡶⠻⠛⠉⠁⠀⢀⡴⠋⠉⠈⡄⠙⣗⣿⡯⣿⣸⠟⢡⠜⠈⠙⣦⡀⠉⠙⠻⣶⣤⣀⠀⠀⠀⠀⢀⣤⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠈⠂⡀⢹⣤⡎⠉⠀⠀⠀⠢⠀⠀⠑⣄⠀⠀⠀⠉⠙⠿⠶⠶⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⢀⠈⠀⠀⠀⠀⡆⣾⣿⣧⡀⠀⣀⠀⠀⠀⠄⢀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡴⠁⠠⢁⠎⠉⡙⠀⠀⡇⢸⣿⣿⡅⠀⠨⠁⠑⠐⠈⠄⠱⣄⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡌⠅⡔⢀⠎⠀⢠⠁⠀⠀⠁⣿⣿⣿⡇⠀⠀⡄⠀⠀⡀⡀⠂⣈⠢⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢋⠞⡡⠊⠀⡌⠄⠒⡂⠀⠀⠠⢠⣿⣿⣿⡆⠀⠀⠐⠂⠤⣁⠈⠢⣀⠑⠰⣜⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣻⡣⠊⠀⠀⡜⠀⠀⢀⠃⠀⠀⢀⣾⣿⣿⣿⣇⠀⠀⠀⢆⠀⠈⠆⠀⠀⠑⠢⢌⠣⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠊⠀⠀⠀⠠⠇⠀⠀⡸⠀⢀⣼⢿⣿⣿⣿⣿⣿⣷⣄⠀⠘⠄⠀⠘⡄⠀⠀⠀⠀⠁⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⡰⠀⢀⣴⣶⡎⠁⠹⣎⡸⠻⡿⠋⣫⡟⠁⠉⣷⡿⠲⢤⣘⠠⠄⡀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⢤⣴⣬⠴⠚⠉⠀⢻⡇⠀⠀⠈⠻⣷⣷⠾⠋⠀⠀⢰⡏⠀⠀⠀⠈⠉⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠁

⠀⠀⠀⠀⠀⠠⠄⠀⠀⠈⠛⠛⠋⠀⠀⣀⠀⠀⠀⠘⠉⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠈⠛⠛⠉⠀⠀⣀⠀⠀⠀⠘⠉⠛
⠀⡀⡄⣠⠀⠀⠈⠀⠀⠀⠀⠀⠤⠤⠈⣹⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⡄⡄⣄⠀⠀⠈⠀⠀⠀⠀⠀⠤⠤⠈⣹⠀⠀⠀⠀⣶
⠀⠐⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠃⠉⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠃⠩⣆⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢠⡄⠈⠛⠁⠀⠀⠈⠗⠣⠶⠂⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠈⠛⠁⠀⠀⠈⠟⠣⠶⠂⠀⠀⠀⠀⠤
⣤⠀⡀⠀⠀⠀⠙⠾⠤⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠋⡘⠒⠒⠀⠀⠁⠀⠀⣤⠀⡀⠀⠀⠀⠙⠺⠤⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠋⡐⠒⠖
⠀⠀⠁⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠁⠀⠀⠀⠀⡄⠐⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠁⠀⠀⠀⠀⡄⠐
⠀⠀⠀⠀⠀⠁⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠠
⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⢠⠖⠒⠒⠦⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡅⣀⡄⠀⢠⠖⠒⠒⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡆⠠⠄⡀⠀⡁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡄⠠⢌⡀⠀⡃
⠆⠀⠀⢀⠀⣀⣀⠀⠀⠀⠀⠀⠠⣖⡶⠂⠀⠀⠀⠘⠂⠠⠠⠔⠁⣀⡀⠀⡂⠈⠀⢀⠀⢀⣀⠀⠀⠀⠀⠀⢠⣖⡶⠂⠀⠀⠀⠐⠂⠠⠠⠔⠁⣀⡀
⠃⠀⠀⢸⠰⠋⠘⠃⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡍⠙⢠⠆⠀⠀⢸⠐⠋⠘⠃⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡍⠉
⠀⠀⠀⠈⠳⠤⡤⠞⠁⠀⢀⣀⠀⠀⠀⣾⣷⠀⠀⠀⠈⡁⠈⠀⠀⠀⠀⢸⠀⠀⠀⠈⠳⠤⡤⠞⠁⠀⢀⡀⠀⠀⠀⣸⣷⠀⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠹⡿
⠀⠀⠀⠤⠀⠀⢀⠀⠀⢶⣶⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⣀⠄⠀⢲⣶⣶⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠠⢄⠀⠀⠈⠛⠛⠉⠀⠀⡀⠀⠀⠀⠘⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠈⠛⠛⠉⢀⠀⡀⠀⠀⠀⠘⠉⠛
⠀⡀⡄⢠⠀⠀⠈⠀⠀⠀⠀⠀⠤⠤⠈⣻⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⣠⠀⠀⠈⠀⠀⠀⠀⠀⠤⠤⠈⣛⠀⠀⠀⠀⢶
⠀⠐⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⠃⠉⣦⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⠃⠩⣦⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢠⡄⠈⠛⠁⠀⠀⠈⠕⠃⠶⠂⠀⢀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠈⠙⠁⠀⠀⠈⠟⠣⠶⠂⠀⢀⠀⠀⠤
⣤⠀⡀⠀⠀⠀⠙⠺⠤⠄⠀⠀⠀⠀⢀⠀⠀⠰⠀⠋⡐⠖⠐⠀⠀⠁⠀⠀⣤⠀⡀⠀⠀⠀⠙⠲⠤⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⢋⡐⠖⠚
⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠁⠀⠀⠀⠀⡤⠐⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠁⠀⠀⠀⠀⡄⠐
⠀⠀⠀⠀⠀⠁⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⢠
⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢠⠖⠒⠒⠦⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⢠⠖⠒⠒⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣆⠠⠌⠁⠀⡃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡆⠠⠌⡉⠀⡃
⡆⠈⠀⢀⠀⣀⣀⠀⠀⠀⠀⠀⢠⣖⡶⠂⠀⠀⠀⠐⠂⠠⠠⠔⠃⣀⡀⠀⠂⠀⠀⢀⡀⢀⣀⠀⠀⠀⠀⠀⢠⣖⡶⠂⠀⠀⠀⠐⠂⠠⠠⠔⠁⣀⡀
⠃⠀⠀⢸⠐⠋⠘⠃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠝⢰⠆⠀⠀⢘⠐⠋⠘⠃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠰⡝⠙
⠀⠀⠀⠈⠷⠤⠤⠞⠁⠀⢀⣀⠀⠀⠀⣾⣷⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⢸⠀⠀⠀⠘⠳⢦⡴⠞⠁⠀⢀⣀⠀⠀⠀⣸⣷⠀⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠹⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠹⡿
⠀⠀⠀⠤⠀⠀⢀⡀⠀⢲⣶⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⡀⠀⢲⣶⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠸⣦⠀⠀⠀⣀⣤⣽⢿⣯⣄⡀⠀⠀⢀⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠲⣆⠀⠀⠀⣀⣬⣿⣿⣥⣄⡀⠀⠀⢠⡾
⠀⠀⠀⠀⠀⠀⠸⣇⠀⢰⡏⠈⣻⣾⡏⠈⣿⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢸⣿⣿⣿⣿⣿⣿⣷⠀⢀⡿
⠀⠀⢀⣀⠀⠀⠀⠛⠶⣤⣙⡿⠋⠈⠻⣿⣫⣴⠞⠋⠀⠀⢀⣀⠀⠀⠀⢀⣀⠀⠀⠈⠛⠶⣬⣻⣿⣿⣿⣿⣟⣩⡴⠞⠃⠀⠀⢀⣀
⠀⠀⠈⠙⠛⠻⣆⠀⢀⣾⣿⠶⠶⠶⠶⢾⣿⣦⠀⢀⣶⠛⠛⠉⠀⠀⠀⠈⠛⠛⢻⡆⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣆⠀⢀⡾⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠹⣶⠟⢸⡇⠀⠀⠀⠀⠀⣿⠈⢿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡾⠁
⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⣿⡶⠶⠶⠶⢶⡏⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠹⣦⡀⠀⢀⡼⠃⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⣠⣶⡏⠀⠀⠀⠀⣠⡟⠀⠀⠘⢷⡀⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠉⠘⣧⣀⣀⣠⡼⢿⣄⡀⠀⣀⣼⠿⣦⣀⣀⣠⡿⠈⠁⠀⠀⠀⠀⠉⠸⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⡟⠈⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠉⠙⠛⠉⠁⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠉⠙⠛⠉⠁⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⢿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⠁⣼⠏⠀⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠋⠀⣠⣴⠾⠋⠀⠀⠀⢀⣿⠛⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠛⠉⣀⣤⡶⠟⠋⠁⠀⠀⠀⠀⢀⣼⠏⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢁⣠⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⣹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⣠⡶⠟⠉⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⠀⠀⠀⢀⣴⡿⠉⠻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠀⠀⠀⠀⠀⢀⣠⣶⠟⠋⠁⠀⠀⠀⠀⢀⣤⡶⠟⠉⠀⠀⠀⣸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⢀⣠⡶⠟⠉⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁⠀⠀⠀⠀⢀⣴⠟⠉⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣷⡀⠀⣰⡿⠁⠀⠀⢀⣀⣀⡀⠀⣴⠟⠀⠀⠀⠀⠀⣀⣴⡾⠛⠁⠀⠀⣀⣤⣾⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡅⠈⢻⣶⣿⣤⣴⠶⠟⠛⢻⡏⠻⢿⣯⣀⣀⣤⣴⠾⠛⢻⣿⠛⠷⢶⣶⡟⠋⠁⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⢻⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⡟⠉⠁⠀⠀⠀⢸⡏⠀⠀⠀⣿⠁⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠈⢿⡄⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⡿⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠈⢿⡄⠀⠀⠀⠘⣷⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⣸⡇⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠘⣷⡀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢀⣿⠁⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⢹⣧⠀⠀⠀⣿⠀⠀⠸⣿⠀⠀⠀⠀⠀⣼⡇⠀⢸⡏⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⡀⢿⡆⠀⠀⣿⡄⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⣿⠁⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⢶⣦⣾⣧⣤⣤⣿⣄⣀⣀⣀⣴⣿⣤⣴⣿⣶⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⡀⠈⡂⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠂⠀⠀⠂⠖⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢤⠄⠛⠈⠐⢂⣀⣼⠯⣼⡀⠀⠌⠁⠀⣸⠆⠐⠴⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⢧⠄⣤⠀⠐⠎⠀⠀⠳⠀⠈⡁⠀⢀⡴⠔⠀⠈⠐⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⣍⠀⠀⠛⠀⠀⢰⠂⢂⠩⡀⠀⠠⠀⠀⠰⠉⠂⢀⢓⠀⠀⠀⠀⠀⡁⠀⠀⠐⠘⠳⠦⠄⢀⠔⡀⠀⣠⠀⠀⢀⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄⠰⠄⢊⠀⠂⠀⡇⠂⠁⣶⣀⠀⠠⠀⠀⠀⢠⠀⡀⠀⠐⠒⢦⣀⠀⠀⢀⡀⠀⢀⣀⡀⠼⢇⣀⡀⣈⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡏⣠⣄⠈⠀⠀⣰⣾⠉⠀⠐⠠⡄⠦⠀⠀⠀⡠⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⢸⡆⠈⡁⠁⠈⡠⠊⠁⠈⠑⠀⠀⢀⢁⡀⠋⠈⠩⠁⠀⠀⠥⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠈⠂⠀⠀⠀⠅⡆⠀⠀⠀⠥⠀⠀⠄⠄⠀⠀⢤⡀⠠⠀⠀⠀⠀⠁⠀⠆⠀⠁
⠀⠀⠀⠀⠀⠀⠐⠒⠶⠦⣠⢀⠀⣡⡈⠀⠀⠀⢀⠀⣀⠀⠃⢀⠀⠀⣰⣀⠀⢀⡀⠙⠆⠀⠀⠀⠀⠧⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠆⠆⠈⠋⣃⠉⠀⡀⢰⡟⠀⠁⠀⢀⠍⡀⠊⠁⠀⠀⠚⠄⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠠⣗⠀⠀⠀⠀⠀⢭⡀⠰⠛⠸⠇⠀⠀⠀⠉⠀⠁⢠
⠀⠀⠀⠀⠀⠀⠀⢠⢠⡖⠡⣄⠠⠀⠙⡠⠀⠀⠈⠪⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀
⠀⠀⠀⠀⠀⡄⠐⢶⠀⠱⠆⢸⠂⠀⠀⠉⢈⡁⠂⣴⠆⠀⠀⠐⠲⡈⠀⠐⡊⠄⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⠀⠀⠤⣁⣀⣀⣀⠀⠀⠀⠀⢧⡀⠈⠫⠁⠀⠀⠀⠀⠀⠀⠕⠀⠀⠀⠀⠀⠀⠀⢀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠂⠀⠉⠙⠛⡦⠴⠖⠒⠋⠀⠀⠀⢠⠀⠀⠠⠐⠀⠀⠂⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠠⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠂⡀⢉⠴⠡⠄⠀⡠⠀⠑⠄⠀⡀⠀⠈⠀⠀⠤⠂⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠻⠄⠀⠀⠀⡤⠀⢀⡀⠀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⣤⠿⠀⠄⠀⡁⢀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠁⠀⠈⡄⠁⠀⠄⢦⡰⠀⠙⠂⠉⠀⠴⢠⡀⣀⠀⠚⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣈⠀⢢⡀⠀⠀⡀⣘⡄⠀⠀⠀⠀⠀⢆⡇⠀⡀⠀⡀⠸⠋⠶⠆⠀⠈⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠒⡐⠋⡉⠉⠐⠉⠑⢳⣦⣴⣤⣚⡉⠀⠀⢈⡁⢆⠀⠀⠀⡅⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠈⠋⠔⡲⠦⠄⢸⣇⢧⠙⡇⠐⠀⠉⠀⡈⡗⢀⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠂⠀⠀⠀⠀⠇⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣼⣀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⣄⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡢⠄⣲⣂⠤⣖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣀⣿⣿⣿⣇⡀⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠤⠃⡀⠀⢀⡿⢋⣾⣿⣿⣯⠻⣇⠀⠀⡀⠣⠄⠠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡧⣾⣿⣿⣿⣿⡦⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢷⣿⡿⣿⡿⣿⣷⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢾⣳⣶⣿⣿⣶⣻⠇⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡼⢁⠄⠨⠿⡛⠛⠛⡻⠯⠀⢄⠹⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠈⠀⢀⣤⣼⣷⣿⣥⣄⠀⠈⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⢠⡏⠁⢉⣿⣯⠁⠉⣷⡀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⡔⠁⠊⠹⠛⠷⢾⣿⣿⠶⠟⠻⡉⠂⠜⡄⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⠚⠀⠀⠀⠂⠸⠗⠸⣿⡿⠀⠿⠂⢃⠀⠀⠘⠢⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠐⠀⠀⠀⠀⠘⠛⠶⠶⢶⣶⣴⠶⠶⠞⠛⠀⠀⠀⠀⠐⠈⠄
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠇⠀⠀⠀⢀⡀⠘⠃⠈⢿⠟⠀⠛⠀⣀⠀⠀⠀⠀⠇⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠉⠛⢛⠓⣶⡖⢚⡛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠶⠤⠤⣬⡥⠤⠴⠶⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⠤⣬⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠗

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⣀⣀⣴⣿⣿⣿⣷⣄⠀⢠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠁⠐⠤⠀⣀⣠⣴⣾⣿⡟⣿⡛⢿⡛⣿⣯⣿⣿⣿⣿⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⠛⠯⣉⣚⢿⢾⣿⣿⣿⣟⣯⣻⣿⣷⠦⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠂⡀⠀⠀⠀⠀⣠⣴⣿⢻⡄⠁⠀⠀⠄⢀⠜⣫⠞⢋⣭⣿⣯⣿⣿⣿⣄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣫⣬⡅⠉⢿⣷⠀⠀⢠⠏⠰⠁⡄⠠⢻⢫⣟⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⠀⡀⠂⠀⠀⣀⣀⣀⣠⡴⠖⠀⠀⢻⣿⣷⢇⡀⠀⢿⠀⢡⡇⠀⡇⣾⣟⣶⣴⣿⣿⣽⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⡸⠟⠛⠛⠛⠋⠁⢠⣽⣿⣗⠀⠀⠀⠈⠻⣧⡾⣷⡤⣈⣷⣤⣦⣴⣶⣿⢿⣻⣽⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠁⠻⠀⢠⠀⠀⡀⠀⠁⠀⡀⡹⣏⣵⣆⠀⠀⠀⠀⢈⡷⢿⡤⠟⠻⢿⡷⠾⣷⣿⣫⡿⠾⠿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢈⡇⠀⠚⣇⠀⠄⠈⢀⣀⠤⠀⢀⠀⢿⣭⡶⢣⡀⠀⠀⠴⠛⡡⠂⠁⠀⢸⣛⣿⠿⠛⠉⠀⠀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡄⠈⣇⠁⠸⢿⡄⠀⠀⠉⠀⣀⠔⠀⠀⡀⠴⢶⠟⠋⠁⠤⠐⠀⠀⠀⠀⢸⣯⡁⠀⠀⠀⠀⠀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡜⣿⣯⢳⠀⠹⡀⠀⠀⠻⡀⠁⠀⠉⠀⠀⠀⠐⣧⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠹⣿⣷⠀⠀⠀⢀⡞⠋
⠀⠀⠀⠀⠀⠀⠀⣾⡖⣿⡿⣅⢇⠀⠙⡀⠀⠀⠙⠦⠀⠐⢀⣀⠀⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡸⠠⠾⣿⣮⢱⣄⠀⢄⠈⠏⠀⠙⠦⣤⠛⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⡎⠍⠡⠹⣄⠾⢛⣳⣜⢦⡀⠢⡅⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⢀⣿⣆⠀⠀⢌⠢⡈⢳⣾⣿⣔⡦⣀⡙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠸⣿⣿⣶⣆⣠⣵⣼⣣⣉⣻⣿⡿⠿⠿⠾
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡠⠤⡧⡧⠤⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢌⣄⡀⠾⠆⣀⣌⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠂⠂⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠄⠀⠀⠀⠀⣼⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢫⠞⠕⠂⠠⠀⠒⡝⢮⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢋⠸⠀⠀⠀⠀⠀⠸⢈⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢸⡇⠀⠠⠀⠠⠄⠀⣾⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⢸⢣⡀⠀⠀⠀⠀⣠⢻⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⡠⠂⠀⠢⡀⠀⠀⡠⠂⠀⠢⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠈⠈⠉⠀⠀⠀⠀⠈⠂

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢠⡶⠟⠛⢟⡛⠲⠦⣤⣀
⠀⠀⠀⠀⠀⣿⠓⠒⠠⢀⠈⠢⡀⠀⠹⡟⠶⢤⣀⡀
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠑⢤⡘⠄⠀⠸⡀⠀⠈⠙⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠻⣧⡁⠐⠤⣀⠑⢬⡀⠀⠹⣕⠂⠤⠀⡀⠸⡛⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⡀⠑⠢⣉⠢⠄⣀⠙⠲⢤⡀⠑⠹⣦⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⣉⠒⠢⠵⣦⡀⠈⣂⣀⣈⣧⣀⡀⠙⢦⡀⠀⠀⠀⠳⢧⣬⣽⣷⡞⢛⣿⠃
⠀⠀⠀⣀⣤⣤⣶⠶⠶⠶⠶⠶⠶⠶⢾⠿⠿⢷⢶⣶⠟⣛⣛⡛⠒⠦⠬⢍⣙⠒⠷⣄⣠⠶⣛⣛⡛⠛⠶⣤⣬⣟⡛⠲⢦⣀
⠀⣴⠟⠻⢤⡀⠈⠙⠒⠢⠤⣀⡀⠀⠀⢣⠀⢀⢺⣯⡍⠀⠀⠉⠉⠉⠒⠢⢤⣉⢢⡾⠡⢚⠭⠥⣄⡀⠈⣩⠛⠓⢤⠀⠀⠙⢷⡀
⣼⠏⢀⣀⠤⠬⠓⠢⢄⣀⡀⠀⠈⠉⠑⠒⠒⠞⠓⣿⡷⢤⣀⠀⠀⠀⠀⠀⠀⠈⢹⡇⠀⡏⡤⣄⠀⢳⠀⣿⠾⠇⢸⠀⠀⣠⠞⠁
⣿⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠒⠒⠉⠉⠍⠁⠀⠈⠛⠓⢶⣤⣄⣀⠀⢸⡇⠀⠹⣍⣁⣠⠎⠀⠈⠓⣾⡥⠴⠛⠁
⠈⠻⣦⡀⠀⠀⢀⣀⡤⠔⠒⠉⠉⠉⠉⠉⠙⠓⠒⠒⠒⠒⠈⣉⡤⠞⠁⠀⠉⠉⠛⠻⣦⡀⠀⢰⡀⠀⠀⣀⣴⠟⠀⠀⠀⠀⠀⣴⠓
⠀⠀⠀⠉⠛⠻⠥⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠴⠖⠋⠉⠀⠀⠀⠀⣀⣀⣤⠴⢾⡛⠳⠶⠷⠶⢻⠸⣅⠀⠀⠀⠀⠀⠀⡸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⡛⠉⠉⠉⠁⠀⢈⣭⣽⠿⠛⠛⠛⠉⢉⡽⠉⠙⢷⣄⡀⠙⣦⡀⠀⠀⢰⡇⢈⣽⠇⠀⢠⣄⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣄⡀⠀⠀⠈⠙⠳⢶⣤⣀⠀⠐⠻⠷⠶⠦⣄⣭⡿⠆⠈⠙⠷⢦⣌⣉⣉⡁⠀⣀⣠⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠾⠿⠟⠛⠛⠀⠀⠀⠀⠀⣀⣤⣼⣿⡷⠀⠀⣠⣴⣾⣟⡁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⢛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠟⠛⠉⠁⠀⠀⠀⠈⠉⠛⠛⠿⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡤⣄⠀⠀⠀⡠⢤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢂⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠃⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣤⠀⣀⠈⢧⠀⠐⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄⣞⠀⠀⠀⠀⠀⠱⠀⠀⡀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡆⠀⠀⠀⠀⠀⢸⠍⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣠⡍⠠⠇⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠤⠀⠀⠐⠉⠀⠀⢀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠒⠀⠀⢘⠂⡢⠀⠀⠀⣇⣸⠁⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠄⠀⠀⠔⠀⠀⠀⠀⠀⠈⠱⠀⠀⠄⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠐⠀⠀⢠⣀⠀⠁⠀⡀⠀⠀⠀⢀⣀⠀⠀⠄⠁⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣅⠀⠀⠀⠚⠋⠀⠀⠄⠁⠀⠀⠀⡀⠀⠀⠠⠀⢠⠀⠀⠈⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠈⠀⠀⠎⢀⡀⢀⠀⠆⠀⣀⣀⠇⠀⠀⠄⠂⠀⠒⠐⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠀⠀⢀⠔⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠒⠀⠠⠤⠤⠤⠤⠤⠤⠀⠒⠒⠶⠒⠒⠚⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡇⠀⠀⠀⠀⠀⠀⠀⣤⡀⢀⣤⠀⠀⠀⠀⠀⠀⠀⢸⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⢀⡤⢼⣿⣿⡧⢤⡀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢰⣡⣾⣿⣿⣿⣿⣷⣌⡆⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠹⠶⠶⢶⣌⢿⣿⣿⣿⣿⣿⣿⡿⣡⡶⠶⠶⠏⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠛⠷⣤⡀⠀⠀⠀⠀⢀⣀⡙⣿⣿⣿⣿⣿⣿⣿⣿⢏⣀⡀⠀⠀⠀⠀⢀⣤⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⣀⡴⠟⠙⡿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠋⠻⢦⣀⣤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⡞⣰⢿⣿⣿⣿⣿⣿⣿⡿⣆⢳⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢴⢻⠾⠘⢿⣿⣿⡿⠃⡷⡟⡦⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⡎⡄⡄⣠⠈⣿⣿⠁⣄⢀⢠⢱⠈⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⣹⠁⣼⣰⠃⡿⢻⠀⣿⣿⠀⡟⢿⠘⣆⣧⠈⣏⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢠⣇⠞⡿⢁⠀⡇⠘⣾⣿⣿⣷⠃⢸⠀⡈⢿⠳⣸⡄⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣸⠏⢠⠃⠀⡏⢱⠀⣿⣿⣿⣿⠀⡎⢹⠀⠘⡄⠹⣇⠸⡆
⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⣿⠀⡞⠀⠀⡇⠀⣿⢿⣿⣿⡿⣿⠀⢸⠀⠀⢳⠀⣿⠀⠙⢦⣄
⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⢿⢰⠀⡀⡴⠻⣤⠏⠀⠀⠁⠀⠹⣤⠟⢦⠀⠀⡆⡿⠀⠀⠀⠙⠿⣦⡀
⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠘⣿⠀⢿⠁⢠⠏⠀⠀⠀⠀⠀⠀⠹⡄⠈⡟⠀⣿⠃⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣼⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣧⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⢰⣦⠀⡟⢦⣠⢄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠑⣤⡭⠧⣧⡚⠁⠀⢹⠁⠹⢄
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣦⡀⠀⠀⠀⠲⣦⣄⡀⢇⠀⢰⠍⠀⠀⠀⠙⢆⠀⠀⠀⠀⢨⡦⡴⢤⣀⣀⢀⣀⠤⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡍⠓⠦⣀⠀⠈⢇⠈⢻⣄⣸⣆⠀⠀⠀⠀⠈⠀⣄⣆⣠⠋⠀⠀⠀⠋⠘⢿⣇⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⣾⣧⡴⠚⠿⣿⣿⡿⠿⣄⣖⠙⠢⣄⠀⠉⠛⢿⣦⡀⣠⠖⠛⠛⠻⣿⠓⠒⠲⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠋⠁⠀⠀⡿⠁⠀⣠⠿⣯⠄⠀⠘⠫⠿⢶⢤⣿⢿⣇⠀⣠⣄⣠⣾⣥⠴⠚⠋⠉⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠛⢷⡀⠀⠀⠇⠀⠞⠁⠀⡼⠷⢖⣰⣋⠙⢻⡀⠁⠀⠈⣾⠃⠀⠈⠛⠖⣤⣴⡊⠉⣻⠟⠉
⠀⠀⠀⠀⠀⢀⣠⣤⠒⣾⠀⠀⠀⠀⠀⠁⠐⢓⡀⠀⣀⡤⠾⢧⠀⣜⠉⠉⠉⢢⠱⣄⠀⢀⡎⠀⠠⠄⠀⠈⠉⠛⣇⠴⠁
⠀⠀⠀⠀⠀⠈⠉⢉⣲⣿⣄⣀⡠⠲⡤⣥⣚⣛⣉⠵⠿⠀⠀⣰⠋⣹⣤⠶⣦⣸⣶⠿⣶⣿⣦⠀⠀⣀⣠⣤⣤⣴⣧⠤⣀⡀
⠀⠀⠀⢀⣤⣴⣞⣁⠬⠿⠛⠛⢣⡄⡠⠋⠁⠀⢀⠀⠀⢀⡾⠁⠀⠙⢦⣴⠄⠌⠃⠲⣶⣾⠻⡏⠉⠁⠀⠀⠹⣿⣏⣀⣀⠈⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠦⠔⠋⠉⠓⠋⠀⠀⠀⠀⠀⠙⠶⢦⣖⡆⠄⣅⣸⡧⠀⢀⣀⠀⠀⠀⣽⣿⣿⣋⠉⠉⠉⠷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠾⠿⢤⡀⠀⢘⡇⠀⠀⣿⣿⣯⠛⠻⣕⠒⠊⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠧⠴⣾⣿⣿⣷⡀⠀⠈⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠌⢿⡌⠙⠒⠒⠾⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻

⠀⠀⠀⠀⠀⢀⠠⠄⠀⢠⢄⡀⡚⠑⠀⠂⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠠⣀⡀
⠀⣂⠳⣠⡴⢯⡁⠀⣈⠆⠀⢙⡤⠞⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⠃⠴⠙⡌⠄⢉⠁⠉⢳⠆⠀⡆
⡠⠊⠀⠀⢀⠝⠻⠀⠀⠠⣀⠇⣷⢴⠈⡡⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠊⢀⡈⣣⠀⢸⠀⢸⡄⣈⡿⢗⡔⠊
⠣⠀⠖⠣⡈⢑⣔⡌⡈⢂⢐⡶⠙⠢⠤⠃⠀⠀⣠⠄⠰⠒⣒⡲⠂⠤⢄⠀⠉⣟⡴⠈⠁⠁⠋⠁⠉⠉⠉
⠀⠀⠀⠀⠈⠁⠀⠉⠒⠊⠁⠀⠀⠀⠀⠀⠀⠘⠂⠬⠄⣋⣉⣉⣑⡍⠴⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀
⠀⠀⠀⠀⢀⡠⠊⠁⠀⠀⠉⠢⡀⠀⠀⠀⠀⠘⠀⣐⠀⠀⣀⠀⢀⣇⠜⠁⠀⠀⠀⠀⠀⠀⠠⠎⠀⠀⢠⠒⡋⢂
⠀⠀⠀⢀⡀⠣⡀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠔⠀⠀⠆⠈⠀⢎⡀
⠀⠀⠀⠀⡓⢄⡈⠢⠄⣀⢀⣀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠂⠠⠄⣀⠀⠀⠀⠀⠀⢆⠠⠀⠀⠀⠤⠜⡀⠁
⠀⠀⠀⠀⠘⢄⡈⠀⠂⠠⠀⡴⠊⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⡀⠀⠠⠖⠉⠀⠀⠉⠰⠆⠀⠀⠙⠠⠠⣠⠶⠊⣑⡢
⠀⠀⠀⠀⠀⡸⠈⠗⠒⠒⠉⠀⠀⠀⠀⠀⣀⣈⠓⢆⠀⠀⠀⠀⠁⠀⠲⠄⠀⠂⠀⠀⡸⠀⠀⠀⠀⠐⠂⠑⠂⠐⡄
⠀⠀⠀⠀⢠⠃⡌⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⡀⠛⠶⠄⣀⣀⠤⣄⠀⢸⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⢠⠃⡘⠀⠀⠀⠀⠀⠀⠣⢄⡂⣄⣀⣀⠬⠄⣀⡀⣀⠐⠉⠂⢈⡄⠈⠁⠀⠀⢀⡠⠴⠲⠶⢤⡀⠀⠀⠀⠂
⠀⠀⠀⠁⠴⠁⠀⠀⠀⠀⠀⠀⠑⠤⣈⡀⢀⡅⠀⠀⣇⡀⢀⡁⢀⠜⠚⠀⠀⠀⢀⠞⠁⠀⢀⣀⡀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⡀⠉⠀⠀⠀⠀⠀⠀⡟⢀⣀⠔⢁⣄⢹⠃⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣦⠀⠢⡀⠀⠀⠀⠀⠀⢧⡀⢸⡀⡨⠶⠚⠠⣀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠐⠂⠛⢍⢂⠀⢱⡄⠀⡃⠀⠀⠀⠀⠀⠀⠑⠯⣙⣦⡤⠴⠞⢁
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣒⠈⠁⠀⠀⠑⡀⠀⢀⣦⠴⠋⠉⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠱⡀
⠀⠀⠀⠀⠑⡒⠠⠤⠀⠑⠄⠑⠠⣀⡠⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠘⢆
⠀⠀⠀⡏⠀⣈⣒⡖⠊⠤⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠁

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⢀⡀⡠⠀⠀⠀⠀⠀⠀⠀⢀⠰⠦⢀
⠀⠀⠀⣰⣇⣘⢆⡆⡀⠀⠀⠀⠀⠈⢢⠶⠘⠀⠀⠀⠀⠀⠠⢿⣥⣜⣹⢢⡀⠀⠀⠀⠀⠈⡹⢧⠉
⠀⠀⠀⠀⠛⣽⣿⢞⣱⠀⠀⠀⠀⠀⣞⣚⣧⠀⠀⠀⠀⠀⠀⠈⢡⣻⠷⣫⠞⠀⠀⠀⠀⢠⣟⣻⡇
⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠈⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂

⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⢀⡢⠔⣦⣀⣀⠀⠀⠀⠀⠀⠀⣠⣶⠀⠀⠀⠀⠀⠀⣠⡳⢸⣦⣤⡄
⠀⠀⠀⠀⠀⡟⠯⠀⠠⠀⠀⠀⠀⠈⢻⢙⠙⢻⠯⠆⠀⠀⠀⠀⠰⠛⠒⠘⠢⠀⠀⠀⠀⠈⡍⢹⠉⠟⠛

⠀⠀⠀⠀⠀⢘⡔⢁⠀⠀⡠⠃⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢰⣶⣾⣛⠀⠈⠢
⠀⠀⠀⠀⢀⡈⠀⡜⢀⠴⠡⣚⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠒⠂⠄⠐⠖⠋⠀⠘⣿⠋
⠀⠀⠀⢠⣼⣧⡄⠀⠀⠀⠉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠀⠰⠋⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢰⠏⠉⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏
⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⡗
⣶⣤⣾⠂⠀⠒⡄
⠉⢻⡿⠃
⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⡀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠑⠦⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠑⠤⣉⠊⡆
⣀⣴⣄⣠⠔⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⣁
⠀⣽⣿⡏⠁⠐⠄⠀⠀⡠⠊⠠⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⢀⣷⣶⡾⠆
⠴⠋⠀⠀⠀⢀⡁⢀⣌⠀⢐⠀⠀⠠⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⣿⠟
⠀⠀⠀⠀⡠⠜⢻⣿⡇⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠤⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠶⠶⣤⠠⠤⠴⠄⠀⠀⠀⠀⠀⠰⠄⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣷⣶⡆⠁⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣠⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣺⣏⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⡿⠄⠠⠀⠀⠀⠀⠀⣠⠖⢡⠛⠀⠀⢸⣿⣾⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠟⠛⠻⣿⣧⣤⡀⠰⣬⢀⡴⠋⠀⠀⣏⡀⠀⢀⣰⣿⣿⡿⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣷⠿⠉⠐⠁⠀⣼⡻⣯⣆⣾⢿⡟⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠛⡇⠀⠀⠀⠀⣼⡿⠃⢻⡟⠉⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠀⢀⣠⣤⣤⣄⣠⠴⠋⠀⠠⠃⢠⠅⢀⡴⠛⠁⠀⠘⣇⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠙⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⢿⣆⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠈⢻⡏⠀⠀⠘⠀⠀⢀⣿⠇⢸⡇⠀⠀⠀⠀⠀⠿⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠈⣿⢻⡅⢠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣠⣾⠇⠀⢸⣿⠀⠀⠀⠀⡰⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣿⣾⣶⣶⣴⣼⣷⣶⣶⣾⣿⣿⡏⠀⠀⣸⣿⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠁⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⢙⠃
⠀⠀⠀⠀⠀⠀⠀⠸⡃⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣝⣿⡏⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠾⢦⣲⢢⣶⣶⣶⣾⣿⣿⡿⠁⠀⠀⠀⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠛⠛⠻⡿⣿⡿⠋⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣠⣤⣄⣴⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣻⣟⡛⢻⣻⣻⢿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣾⣟⣿⣿⣿⡶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠙⢲⡈⢻⣿⣿⣿⡄⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⢹⠈⣿⣿⠿⠷⢶⣤⣼⣴⣴⣀⠀⠀⣠⢊⡿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡼⠛⠅⠂⠀⠀⠀⠉⠃⠨⠛⢻⣤⠞⣱⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⣠⣿⠛⢁⠂⢄⡁⠀⠀⠀⠀⠀⠲⣖⣻⡾⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⢇⡄⡔⡾⠁⠄⠈⣀⣀⠤⢴⠒⠒⠚⢿⢿⣖⢍⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣎⢣⢼⠙⣌⣦⠞⡍⠈⠀⠀⠀⠀⡀⢾⣿⣯⡙⠻⣮⣝⠦⣄⣠⣤⣤⣤⣄⣀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣖⣒⣚⣫⣭⣿⠃⡟⠻⣯⣿⢻⣌⠃⠰⠈⠀⠀⠀⠁⠻⠰⣬⡟⠻⣷⣄⡉⠳⣦⣙⢿⡋⠉⠙⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁⠉⠉⠁⠀⣸⡏⠀⡇⢰⡇⠙⢯⣁⡇⠀⣄⢰⠰⣠⡁⣤⣬⣟⠙⢦⡈⠺⣿⣦⣄⠙⠳⣌⡳⣄
⠀⠀⠀⠀⠀⣀⣴⡿⠋⠀⠀⠀⠀⠀⠀⢰⢿⠇⢰⡇⢸⣇⠀⢸⠈⢻⡶⠺⠦⠖⠚⠛⠋⠁⠸⢗⢆⠙⢆⠙⢿⣿⣝⠦⣄⠙⠿⣷⣦⣀
⣤⣴⣶⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⠀⢸⡇⢸⣿⠀⠈⡇⠈⣿⠀⠂⠠⠀⠀⠀⠀⠀⠀⠄⠛⢦⡑⢌⠣⡙⢷⣮⡓⢤⡈⠛⢿⣷⣄⡀
⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⠉⠀⢸⡇⢸⡿⡇⠀⢹⡀⢸⡄⡀⠉⠀⠁⠀⣀⣠⠴⠶⠒⠒⠳⣄⠑⣝⢆⠙⢿⣦⡙⠢⣀⠙⠫⡻⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢶⡟⣁⡄⠀⢸⡇⢸⡇⢷⠀⠀⢧⠀⣇⠀⢀⡶⠛⠉⠀⠁⠀⠀⠀⠀⠀⠈⠳⣬⣳⡵⣄⠙⢿⣦⡈⠱⢄⠈⠢⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⢿⠃⡇⠇⠀⢸⡇⠸⣇⠸⡆⠀⠸⣧⡧⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢽⠛⠮⢷⣄⠑⣝⢦⠀⠳⣄⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⡘⠀⡧⠀⠀⢸⡇⠀⣿⠀⢻⡀⠀⡿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⠉⠙⠛⠛⠛⢿⣣⣄
⠀⠀⠀⠀⠀⠀⣀⣼⣿⠋⠀⠀⠀⡇⢠⢸⠀⠀⢸⡇⠀⣿⠀⠈⣧⣰⠇⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⣄⡀
⠀⠀⠀⠀⣠⣾⠿⠉⠀⠀⠀⠀⠀⡇⠘⠸⠀⠀⢸⡇⠀⢸⡄⠀⡸⠛⣤⡇⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠖⠒⠒⠛⠒⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠋⠉
⠀⠀⠐⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⡇⣼⡀⠀⠀⠸⡇⠀⠘⣇⠜⠁⠀⠈⠳⣾⡀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⢘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣼⡇⠀⠀⠀⡇⠀⡠⠋⠀⠀⠀⠀⠀⠈⠻⢯⣄⢠⠀⠀⠀⠀⢀⣀⣀⣀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢀⡷⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⢶⣚⡍⢁⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠴⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠐⠀⠀⠀⠰⠀⠆⠀⠀⠐⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠀⠂⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⡎⠀⠉⠁⠉
⠀⠀⠀⠀⢀⠀⠄⠒⠒⠢⠤⢤⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠱⡄⢠⣷⠦⡀
⠀⠀⢀⡼⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠟⠿⠋⠛⣛⡾⣶⣿⣿⣶⣶⣶⣦⣼⣶⡿⢿⣿⡷⡟
⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⢿⡛⠓⠶⠤⣄⡀⠀⠀⣀⣀⡀
⠀⠀⠀⡀⠴⠚⠛⠋⠉⠉⣉⣧⣷⣿⣟⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⠿⢭⠟⢀⠏⠀⠸⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠉
⠀⢠⠈⠀⠀⠀⠀⠀⢠⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠋⣠⠏⠀⠀⠈
⠀⠈⠀⠀⠀⢀⡴⣾⣟⣻⣟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢡⠞⠁⠘⠃⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠋⠙⠙⠋⠛⠧⣶⣼⣿⣿⡿⠿⠟⢛⣿⠟⠑⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠎⣤⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⡀⣠⠞⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⡱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢄⡤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⡀⠀⠀⠀⠈⠠⠀⠠⠈⠀⠀⠀⠀⣀⡤⠄
⠀⠀⠀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠈⠙⣦⡀⢰⣽⣶⣶⣶⣾⣷⠀⣠⡞⠉⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⡀
⠀⢔⠁⠀⠑⢒⠖⢒⣀⡩⣒⢂⠤⢀⡀⠀⠘⠿⣆⡽⠉⠉⢛⣿⣅⡾⠟⠀⠀⣀⠠⢤⢒⡪⣁⣐⠒⢖⠒⠁⠀⢑⡄
⠀⠈⠁⢄⠈⠈⢁⠄⠤⠄⣔⠂⠓⠂⠠⠽⣷⣶⣾⡁⠀⠀⢰⣿⣿⣶⣶⡿⠥⠀⠒⠃⠐⡎⠠⠤⠜⠉⠈⢀⠄⠉
⠀⠀⠀⠀⠉⠩⢁⡠⠄⢂⢠⣎⣁⡀⢤⡾⢟⣿⣿⣷⣀⠀⣸⣿⣿⣿⣟⠷⣦⠄⣀⣉⣢⢀⠂⠤⣀⠩⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠙⠤⢤⣊⣠⣔⠉⢄⣒⣭⣭⣿⣿⣾⣿⣿⣯⣭⣕⣂⠌⢑⡠⣈⣢⠤⠜⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠋⣩⣿⣿⣿⣏⣉⣻⣿⣿⣟⣯⡉⠻⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⣠⣿⠟⡟⠙⠛⠛⣿⣿⣯⡟⢿⣧⡀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⢰⣧⣤⣀⣀⣼⣿⣿⢰⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⠘⣧⣉⣉⣉⣹⣿⣿⡘⠀⠘⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢯⣉⠛⠛⢻⣿⣏⠇⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⢯⣭⣽⣿⣿⠞⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢿⣿⣿⡿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⣌⣡⣾⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣛⠿⣿⣿⠿⣛⣛⠿⣿⣿⠿⣛⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣮⣵⣿⣿⣿⣿⣦⣵⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣝⡛⢿⡿⣛⣫⣝⣛⢿⡿⢛⣫⣝⣛⢿⡿⢛⣋⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣭⣝⠻⢟⣭⣭⣭⣍⡻⢟⣩⣭⣭⣍⡻⠟⣫⣭⣭⣭⡻⠟⣫⣭⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣤⣤
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠈⠙⢏⠛⠃⢤⠀⠀⠉⠉⠳⣄⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣶⡟
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⠳⠀⢄⣠⣄⡈⢦⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⢺⠝⣧
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢦⣀⣃⠀⠉⠋⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⣿⠞⠋⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⣀⠀⣨⠟⠒⢀⣀⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⠘⠁⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠿⠗⠒⠛⠉⠀⠈⠲⢝⢢⡀⠀⡀⠀⠀⣀⠀⡼⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠳⡑⡾⠁⠀⠀⠈⢏⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠒⠂⠛⠐⠒⠢⢄⣀⠀⠀⠙⣜⡄⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠘⣞⡄⠐⡜⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⡏
⠀⠀⢀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠤⠤⣄⡀⠀⢀⠀⠀⠀⢳⡀⠘⢂⡔⠓⠂⠀⠀⠀⠀⠠⠶⠊⠁⠀⣷
⠀⠀⠈⠑⠤⣄⣤⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⢤⡀⢳⠀⣄⢹⠀⠀⠀⠀⡠⠞⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠉⠓⠦⣌⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡇⢰⠸⠀⠀⣠⠚⠁⠀⡀⡠⠊⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠏⢲⣄⣀⠠⣄⡰⠖⠛⠁⠈⠛⢦⡀⡆⢰⠃⠀⢀⢀⠟⠁⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⡤⠧⢤⣿⠀⠀⠈⠿⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠈⠑⠚⠛⠒⠦⣄⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠛⠓⢦⡦⣾⠤⠖⠛⠛⠛⠲⣄⠀⣀⣀⣀⡈⢳⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠚⠋⠉⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⡠⠤⣤⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⠃⠀⠀⢀⡀⠀⠀⠀⣼⢰⡖⣪⡅⠪⣽⢃⠀⢀⡤⠀⠀⠀⣤⣟
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⣆⠀⣀⡇⠉⠓⣤⡾⢻⣌⠳⢭⣠⠞⠉⢧⠞⠁⠀⠀⠀⠀⠈⠉⠑⡄
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣨⠟⠋⠀⠀⣼⠋⠀⠀⠉⠳⣄⠟⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠤⠐⠒⠠⡏⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⡏⠀⠀⣠⠞⠒⣒⢦⡄⠀⣠⡶⠶⠋
⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠹⣦⡀⠀⠀⣀⣉⠤⠞⠁⣠⠎⠹⣄⠀⢠⠏⠉⠳⣄
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠘⠳⢤⣄⣀⡠⠞⢋⠟⠉⠀⠀⠀⣀⡾⠁⠀⠀⠀⠉⠉⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⢤⣄⠀⠀⠀⠀⠰⢟⣁⢀⣠⢞⠟⠒⠒⠉⢩⠑⢀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠳⣤⣀⣀⣀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠉⣉⡩⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠴⠶⠚⠉⠂⠈⠙⠓⠛⠛⠲⠶⠴⠖⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡊⢉⣩⠥⠀⠒⠋⠉⠉⠉⠉⠉⡉⠉⠁⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⡠⠗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡴⠊⠀⢀⠤⠐⠀⠀⠀⠀⠀⠀⠐⠈⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠕⠁⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⡠⠴⣇⣄⣀⣀⣀⣀⡀⠀⠄⠀⠠⠤⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣦⣀⣴⡿⢻⠏⢠⣴⠏⠈⣻⠟⠛⠻⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⡏⠟⠛⣿⢿⣻⣿⣯⠌⢸⣏⠀⣰⣿⣆⣤⣾⢏⢀⣽⣯⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣄⣰⡟⠀⠉⢳⣾⡶⠒⠛⠛⢻⡟⠛⠿⣧⣷⣿⣏⠈⣿⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⡠⢫⠞⠀⠀⠀⠀⢸⢹⡀⠀⠘⢽⢿⠯⣼⣟⣿⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡃⠀⠀⠀⠀⠀⡆⠀⠃⠀⢐⠁⢀⠟⠒⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠲⡄⢀⡠⠊⠁⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡶⠋⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⢠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢀⡴⠛⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⣠⠴⠒⢲⡄⠀⣀⣠⣀⠀⣸⠇⣴⡏⡤⠂⠀⠁
⠀⠀⠀⠀⠴⢋⡠⠒⠶⣄⣻⣾⣿⠟⣿⡆⠋⢸⢻⠗⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⣰⠋⠀⠀⠀⠈⣿⣗⣿⣇⣿⣅⢄⠀⣀⣤⣾⣧⣯⡻⡄⠙⣷⡄⠈⣳⣦⡀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢿⣿⣧⣼⣿⣿⣼⣿⣿⣿⣿⣿⣿⣷⣀⢴⣿⡇⠀⣺⣿⡗⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠟⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣀⣾⣿⡿⢃⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠇⠀⠀⢻⣿⣿⣻⣿⡿⣿⢿⣿⣿⣤⣿⣿⡿⣻⢆⣿⣿⣟⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⡿⠛⠀⢸⢾⣿⣿⡽⠛⣠⣾⣯⣾⡿⣫⣾⡿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣛⣥⣾⠟⢿⣿⣾⣿⣿⣯⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⣿⣿⡖⠄⠈⠉⠹⢿⣯⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠘⢧⣀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠗⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⢀⣴⠞⠛⠿⡛⠓⠶⢤⣄⡀
⠀⠀⠀⠀⠀⣼⠗⠒⠠⢄⠈⠑⢄⠀⠉⢿⠳⢦⣤⣀
⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠉⠢⡀⠣⠀⠀⢣⠀⠀⠉⠛⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠻⣯⡀⠀⠤⣀⠈⠢⣁⠀⠈⢷⡂⠤⠀⡀⠀⢹⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠑⠦⣈⠲⢄⡀⠉⠲⠤⣀⠀⠂⠷⣄⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣤⣈⡙⠒⠬⣕⣢⣄⠈⠑⠄⠀⠙⣆⠀⠀⠙⢄⠀⠀⠀⠀⠲⣼⣉⣙⣿⣠⠶⢶⡶
⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣬⣭⣿⣷⣶⣦⣤⣭⡷⠶⠿⠿⠭⣍⣙⡓⠒⠦⢷⡀⢀⣠⡴⠶⠿⠿⢯⣍⣀⣸⠷⠶⣤⣀
⠀⢀⣴⣿⡋⠉⠙⠲⠤⣄⣀⠀⠀⠀⠀⠱⡀⠈⠉⣿⡟⠒⠒⠓⠒⠒⠢⠤⣍⡙⠒⠄⣹⠟⡡⢞⣛⣛⠀⠀⠚⣻⠿⢟⡁⠀⠉⠳⣄
⣰⡟⠁⠀⠈⣑⣢⣄⡀⠀⠀⠉⠙⠒⠠⠤⠬⣄⣀⣈⣳⣄⡀⠀⠀⠀⠀⠀⠀⠈⠙⢲⡟⠀⣰⠋⣀⠈⠳⡄⢸⣧⣶⡄⢹⠀⠀⢀⡼⠂
⣿⣁⠔⠊⠉⠀⠀⠀⠈⠉⠒⠒⠤⠤⠄⢀⣀⣀⣀⣀⣀⣀⣉⣓⣦⣤⣀⡀⠀⠀⠀⢸⡇⠀⠹⣼⠾⠃⢠⠇⠈⠳⢭⡤⢊⣠⡴⠋
⠹⣯⡀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠶⠦⠤⠤⠤⢤⣄⣀⣀⡤⠤⠤⠐⣨⠟⠛⠲⠶⢤⣿⣄⠀⠈⠓⡚⠉⠀⠀⠀⣸⡟⠋⠁⠀⠀⠀⢀⢤
⠀⠈⠛⠶⢤⣤⡔⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⣙⡷⣤⣈⣓⣂⣀⡆⡾⠛⠀⠀⠀⠀⠀⠀⢻⡤
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠲⠶⣶⡶⠶⠶⠶⠶⠚⠿⠿⢭⣤⣤⣤⣤⡤⠤⢤⡴⠶⣞⡉⠉⢣⡀⠉⠉⠉⠁⢹⠘⠦⣤⡀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⠀⠀⠀⠀⠀⠰⣿⣟⡉⠀⠀⠀⢀⣴⣏⣀⡀⠈⠻⣦⡀⠙⢦⣄⡀⠀⢿⣤⠶⠛⠀⠀⠘⣷⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣬⣿⣿⣴⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⠀⠉⠉⠉⣉⣷⡶⠟⠋⠀⠀⠈⠛⠛⠲⠦⣤⣤⣤⣴⢶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⢀⣤⣶⣾⡿⠿⠛⠁⠀⠰⣾⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⣶⣶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢄
⠀⠀⠀⠀⢀⡠⠔⠂⠉⠁⠄⠀⠀⡀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⢠⠃⠀⠂⠐⠈⠠⠀⠈⠀⠀⡀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊
⠀⠀⠀⢠⠀⠂⠀⠐⠀⠂⠀⠈⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠊⢥⡿⠳
⠀⠀⠀⠀⢣⠀⠁⠀⠀⠈⠀⡰⠐⢆⠀⢩⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠩⠀⢘⣄⢀⠀⡰⢅
⠀⠀⠀⠀⠈⡄⠁⢸⠉⠑⠒⠁⠀⠈⠂⠔⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⡀⠀⠀⠉⠈⠁⠠⠱⡀
⠀⠀⠀⠀⠀⠸⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢀⣤⡦⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢱
⠀⠀⠀⠀⠀⠀⠘⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢀⢤⡞⡻⠁⣿⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⡡⠊⡜⠀⠀⢇⠁⠚⠤⡔⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢰⠁⢐⠁⠀⠀⠀⠳⡄⠀⠀⠻⡝⢧⡇⣐⢲⣄⣀⡤⡀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣀⣀⡀⡤⠒⢺⡅⠀⢠⠀⠐⡆⠀⠀⠀⠀⢸⠀⠀⠀⣟⠻⠀⢏⠁⠈⢢⣠⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢰⡀⡜⠑⠤⠌⢦⡀⠘⣦⣀⢀⡀⠜⠀⠔⠊⠇⠓⢖⠈⠙⢍⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠕⠒⢘⡛⠊⠍⣢⡠⠊⠢⡠⠚⠳⣠⠞⠒⠤⠒⢆⡤⠲⢖⡚⡳⣆⣞⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡹⠁⢙⠒⠶⠀⢵⠀⠀⢆⠀⠀⡆⠀⠀⠇⠀⣘⡠⠚⡉⠁⠄⠀⠀⠀⠠⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠘⡀⠘⡀⠀⡆⠀⢸⠀⠀⡇⠀⢸⠀⡠⠋⠀⢂⠀⠀⡤⠤⠊⠢⢄⡀⠁⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢱⣀⡵⠤⣰⠀⠘⡂⠀⠀⠀⡌⢠⠁⡀⠀⠀⢪⠝⠉⠈⠉⠈⢰⠇⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⡔⠉⠈⠹⠆⠀⠺⡁⠈⠛⠀⠀⠈⡇⠀⠇⠀⠀⠀⠇⢰⣀⡀⢀⡁⠀⡑⠲⠄⠒⠒⠅⠀⠀⡘⠑⢲⠃
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠈⠑⡌⠁⠘⣄⡠⠤⡀⠀⡤⠤⠰⠋⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⣀⠠⠊⠑⠠⠄⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⣀⠀⠀⠈⠉⠀⠀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠉⠉⠉⠉⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠈⠢⡀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⡠⠴⠒⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠙⣄⢣⠀⢸⠀⡏⠀⠀⡰⣡⠔⠁⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢌⢇⠘⠀⠀⢀⣾⠞⠁⣀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡈⠓⠄⢀⠼⠚⠓⠒⠲⡟⠁⠠⠊⢀⣀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡏⠀⢛⢀⠘⠃⢸⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⢒⣒⠭⠛⠧⡀⠘⠚⠀⢀⡼⢍⠑⠒⠤⣄⣀⣀⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠉⠁⢀⠤⠀⠀⢨⣷⠒⠒⠁⠀⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠴⠁⡰⠁⢀⣿⣿⠀⢰⠀⠳⡀⠀⠀⠀⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⠻⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢄⡀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⡿⠟⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⡸⢰⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⣀⠤⠒⠉⠉⠀⢹
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠈⠳⢄⡀⢀⡰⠁⠸⡇⢸⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⡠⠊⢁⡠⠖⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡇⠀⠣⡀⠀⢀⡠⠴⠊⠁⠀⣠⠊⢀⠴⠋⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠑⢄⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠁⠀⠀⠀⢀⠜⢁⡔⠁⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⠣⡀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⡠⠃⠴⠋⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠈⠢⣄⠀⠳⡄⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⢀⠜⠁⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡈⠁⠀⠈⠢⡀⠀⠀⠀⠘⡌⡄⠀⠀⠀⢠⠎⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠀⠈⢦⡀⠀⠀⢃⢱⠀⠀⢠⠃⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⠈⠢⣀⠸⡈⡆⢠⠇⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢌⡓⢧⠱⣎⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⣤⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠱⢃⡴⠖⠶⢤⣀⠀⠀⠀⠀⠀⢠⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⡀⠀⠀⢌⠳⢄⡀⠀⢀⣼⣜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢰⠏⠉⠉⣩⠍⣓⠳⢤⣵⡒⣮⠻⠟⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣆⣀⣈⡴⠲⢤⣵⢤⠤⠃⣼⡷⠖⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣴⡙⠷⣤⡗⣿⠁⠈⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⡄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⣛⡷⡟⢸⠀⠀⠀⠀⠀⠀⣴⣋⠭⠖⠒⠂⠈⠁⠀⠉⠀⡨⢿⠿⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⠟⠀⠛⠀⠀⠀⠀⠀⠀⢸⡷⠛⠛⠓⠶⠶⠶⠶⠶⠶⠶⢦⡀⠀⢀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣥⣤⣄⣁⣀⣀⣀⣀⣀⣀⣸⣇⠀⠠⣺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⡉⠙⠯⠍⠉⠙⠛⠛⠛⠉⠉⠩⠽⣶⣬⢹⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠑⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠀⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡓⠆⠀⠀⠀⠀⣀⠤⠤⠶⠤⣄⣀⡀⠀⠸⣧⠿⢽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣦⣤⣀⣀⣠⠞⠁⢀⣠⣤⡀⠀⠙⣦⠀⠀⢰⢾⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⠀⣴⣿⣿⣿⣿⣷⡄⠘⣿⣿⣿⣛⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⡿⢭⣿⣧⠀⢿⣿⣿⣿⣷⣿⠁⢠⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣒⡢⠔⠛⣦⣬⠉⠛⠿⠋⠡⣪⠟⢩⠤⢌⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣗⡀⠀⠀⠙⠳⠶⠶⠖⠛⠁⠀⠠⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠐⠩⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣤⣄⣀⣀⣤⣤⣄⣀⣀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠷⠶⠶⣤⣌⡒⠦⣄⡀⠀⠀⠀⠀⠀⠸⣯⣀⣉⣛⠙⠛⠛⠛⠉⢙⣻⣿⠇⠐⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠤⠬⠙⠓⠶⣭⣓⠦⢄⡀⠀⠀⠀⠈⠉⠙⢛⣛⠛⢛⣛⣋⣉⣁⠀⠀⣀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢙⡻⢶⣮⣀⠤⢤⣤⣄⣀⡀⠀⠀⠀⠀⠠⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠢⡈⠙⠽⠳⢶⣄⡀⠀⠀⠈⠑⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡈⢦⣄⣐⣐⠉⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣄⣭⣤⡁⢂⣡⡼⣁⣉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣻⣿⠛⠋⠥⠤⠤⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠳⢄⣀⡤⣤⣄⣠⠊⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠾⡤⠤⠤⠤⡼⠇⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⣦⣤⣄⣀⣤⣤⣶⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢤⣄⣀⣀⣀⣀⣠⠤⠖⠚⡻⣿⣯⡟⠻⣿⠿⢻⢿⡿⠟⣿⣿⢿⠓⠒⠤⣄⣀⣀⣀⣀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⣴⣿⣿⢿⠻⡿⣿⣿⣾⣿⡿⠿⡹⣿⣷⣧⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡩⢱⢿⠈⣷⣄⢧⣼⣿⣤⢇⣴⠃⡟⡏⢏⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢠⡁⡎⡆⢇⠘⣿⡆⠻⠹⠅⣿⠇⠰⠠⢸⡈⡆⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋⣼⡠⠃⡟⠁⢁⠘⢆⢳⣶⣶⣶⣶⡞⣠⠃⠰⠀⢻⠘⢆⣻⡙⢷⣤⣀
⠀⠀⣠⡤⠶⠚⠋⠉⠀⠀⡜⡋⠀⣰⣧⢼⠸⡀⠀⠙⣦⣤⣤⣤⠟⠁⠀⡆⡄⢸⣧⠀⢩⢣⠀⠀⠉⠙⠓⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣸⠁⡔⢹⢻⢸⠀⠱⡀⠀⣿⠛⠛⠿⠀⢀⡜⠀⠇⡌⣏⢢⠀⣆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⣿⠜⠀⣼⠘⠈⡇⠀⠙⢦⣿⣿⣿⣿⡴⠋⠀⢰⠃⡇⡹⠀⠱⣿⡄⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸⡉⠀⠰⢸⠀⡇⠘⡄⠀⡰⢸⣭⣭⡇⢣⠀⢀⠇⢠⠁⡇⢇⠀⢹⢇⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠆⠀⡇⠈⡄⠸⠀⠘⣴⠁⢸⡟⢻⡇⠀⣧⠋⠀⡞⠀⠇⠘⡀⢸⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢀⢠⠀⠀⢣⠀⠣⡴⠉⢢⢠⣿⣿⡇⡰⠉⢧⡜⠀⡸⠀⠀⡇⠸⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠅⢸⠸⠀⠀⠈⣦⠾⡀⠀⠀⢹⣶⣶⣏⠀⠀⢀⠷⣤⠃⠀⠀⡇⡆⠀⡆⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠙⢦⠤⠒⢇⠀⠘⣄⢠⠛⠿⠿⠟⡄⢠⠊⠀⣨⠓⠠⠴⠋⡅⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⢸⠀⠸⡄⠀⠈⠣⡀⣨⠃⠀⠀⠀⠀⠈⢇⢀⠔⠁⠀⢀⠇⠀⡇⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡀⠀⠱⡀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⢀⠎⠀⠀⡃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⢣⠀⠀⠘⢤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣤⠊⠀⠀⡸⠁⢰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠰⠀⠈⢢⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⡰⠁⠀⡎⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣃⣀⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⣸⣠⠎


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣭⣥⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣽⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⢯⣉⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣯⣽⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠰⣧⡤⢤⡤⣤⡷⠀⠀⠀⠀⠀⠀⠀⢠⠟⠲⠤⠬⡇⠀⠀⠀⠀⠀⠀⣠⠼⠁⠀⠀⠈⠦⣄⠀⠀⠀⠀⠀⠀⡟⠲⠊⠙⠊⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡗⢦⣙⡞⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠒⠒⠒⢻⠀⠀⠀⠀⠀⠀⣇⢀⣀⠀⢀⣀⠀⣸⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡦⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⠛⠛⢛⡏⠀⠀⠀⠀⠀⠀⠈⢻⡀⡍⢩⠄⡏⠁⠀⠀⠀⠀⠀⠀⠈⠒⠶⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⠤⠼⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠋⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣇⡖⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠒⠢⣄⠀⠀⠀⠀⠀⠀⠀⢀⡤⠾⣄⣠⢧⡀⠀⠀⠀⠀⠀⠀⡴⠉⠉⠉⠱⡖⠁⠀⠀⠀⠀⠀⠀⢀⡤⠧⠾⢤
⠀⠀⠀⠀⠀⢠⠴⠋⠀⠀⠀⠀⠀⠀⠓⢤⡀⠀⠀⠀⡞⠛⠒⠛⠛⠛⠋⣻⠀⠀⠀⠀⠀⠰⣧⣠⢄⣤⣤⡷⠀⠀⠀⠀⠀⠀⠀⣸⠖⠦⠔⠺⡇
⠀⠀⠀⠀⠀⠘⠲⣍⣙⣚⣉⣙⣒⣋⡹⠚⠁⠀⠀⠀⡿⠿⠿⠿⠿⠝⠛⢿⠀⠀⠀⠀⠀⠀⠘⢦⣶⣤⡞⠀⠀⠀⠀⠀⠀⠐⢾⣛⣒⠒⠒⣓⣛⡶
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠈⠉⠁⠉⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠙⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⡎⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡗⢯⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠷⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡖⠒⠲⢄⠀⠀⠀⠀⠀⠀⢀⡔⣩⠥⠬⣝⢦⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⣿⣿⣽⣯⣿⣿
⠀⠀⠀⠀⠀⠀⢠⠎⣡⠤⣍⠓⣄⠀⢳⠀⠀⠀⠀⠀⢸⢸⠁⡏⠵⣸⢀⡇⠀⠀⠀⠀⠀⢸⡁⢀⡀⢀⡀⢀⣧⠀⠀⠀⠀⠀⠀⣿⣿⣼⣧⣿⣿
⠀⠀⠀⠀⠀⠀⢸⢸⡁⣄⠈⡇⢸⡆⣸⠀⠀⠀⠀⠀⠈⢮⡳⢬⣯⡵⠾⠃⠀⠀⠀⠀⠀⣿⠙⣯⢹⡏⢹⡇⣹⠂⠀⠀⠀⠀⠀⣿⣿⣯⣍⣉⣽⡇
⠀⠀⠀⠀⠀⠀⠈⢦⡳⣌⣙⣁⡼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠸⡆⢿⢸⡇⣼⢠⡏⠀⠀⠀⠀⠀⠐⡏⠉⠉⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠚⠓⠛⠛⠀⠀⠀⠀⠀⠀⠈⠷⠶⠶⠶⠶⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠲⢄
⠀⠀⠀⠀⠀⠀⠀⣰⠉⠀⠀⠠⡄⢱⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡼⢮⡹⣆⠀⠀⠀⠀⠀⠀⠀⢧⢠⡤⣸⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠖⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⢰⠁⠸⠁⠀⢀⠁⠀⢳⠀⠀⠀⠀⠀⠀⣰⢻⡄⢳⠀⡟⠋⠀⠀⠀⠀⠀⢰⡋⠚⠁⠀⠙⢋⣷⠀⠀⠀⠀⢰⢏⠀⢧⠀⠀⡼⠁⡹⡆
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠈⠃⢀⡏⠀⠀⠀⠀⠰⣎⠳⣀⢣⣨⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⠀⠰⡏⠀⠀⠀⠀⠀⠸⣄⢹⡚⠓⠚⢑⡟⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⣤⡖⠛⠋⠀⠀⠀⠀⠀⠀⠘⠲⠼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⡴⠲⣄⠙⡆⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠛⠉⠉⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠋⠁⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⣀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⢀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢀⣴⣋⣤⡶⠋⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣴⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡴⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⣀⣸⣇⣀⣤⣤⣾⠟⠋⠀⠀⣀⣀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉⠁
⠀⠀⠀⣠⣤⣶⣾⡿⠟⣿⣧⣹⣿⣿⣿⣿⣿⣷⣼⠟⢷⡿⣯⣼⡧⣽⣿⣿⣿⣿⣿⠿⠿⢿⡿⠿⠿⢿⣶⣦⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠈⠙⠋⢁⣤⣴⣾⣿⣯⣽⣿⣿⣿⣿⣧⣍⡳⠿⣤⢾⣥⡾⠛⠛⠻⢿⠿⠿⣿⣷⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⣸⣿⠟⠋⠉⠀⠀⢠⣿⣿⠛⠛⠛⠛⣻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡀
⠀⠀⠀⢀⣴⣿⡟⠀⠀⠀⣤⣾⣿⡟⠃⠀⠀⠀⣼⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣦⡄
⠀⣠⣾⣿⣿⠟⠁⠀⣴⣿⣿⣿⠏⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣦⡀
⠀⠉⠉⠉⠀⠀⠀⠀⠻⣿⣿⠏⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠁

⠄⢣⣶⢶⡄⠀⠀⣰⣿⣷⡀⠀⠀⠀⠀⠀⠻⠆⢣⣾⠻⣆⠀⠀⣰⡿⣷⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣎
⠀⠈⠻⡾⠃⠀⠀⠙⣻⠟⠀⠀⠀⠀⠀⠀⣾⣿⣧⣛⣚⠃⠀⠀⠈⣛⠛⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠏
⠀⠀⠟⠉⠀⣀⣀⢾⡁⢰⠄⠀⠀⠀⠀⠀⠙⠛⠃⠛⠁⠂⣤⣄⢺⣁⢰⠆⣀⣀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠈⠺⣉⣹⠆⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣽⠃⠉⠁⢺⣏⢹⠂
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠲⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣸⡄⣄⣼⣗⠶⠖⠒⠒⠒⠒⠒⢦⠒⠒⠚⠓⠖
⠀⠀⠀⠘⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣤⣇⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⢀⡆⠈⠁⠀⠀⠈⠉⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⢷⢿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡈
⠀⠀⠀⠀⠀⢈⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠀⠀⠀⠀⣀⠸⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣤⡠⢄⡀⣤⣶
⠀⠀⠀⠀⠀⢿⠶⡄⠀⢘⡃⠀⠀⠀⠀⠀⠈⠉⠒⠤⣄⣀⠉⠀⠈⠛⣀⡄⠀⠀⠀⠀⠲⣿⡵⠂⠀⠀⠀⠀⠀⢈⡿
⠀⠀⠀⠀⠰⠋⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠉⢁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠾⡇⠀⢀⠀⠀⠠⠐⠊⠁
⠀⠀⠀⢠⡇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠈⣄⠀⠀⠸⣿⡂⠀⣀⠀⠀⠀⢰⣾⠚⢗⠀⠀⠀⠀⠀⠀⠀⣴⣂⣽⠒⠀⠆⠀⠀⢀⠀⠀⠀⠀⠀⢰⣿⣷⣰
⠀⠀⠀⠀⡅⠀⠀⠀⠘⠻⠃⢻⠀⠀⠈⠉⠿⠶⠞⠁⠀⠀⠀⠀⠀⠀⠙⠉⠁⠀⠀⠀⠀⠰⣤⢧⠆⠀⠀⠀⡉⢟⣿⣻⢏⠛
⠀⠀⠀⠀⠛⠀⡀⠀⠀⠀⠀⡘⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠛⠞⠀⠀⠀⠀⠀⣼⠈⢑⢕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⡀⠀⠀⠀⠀⠀⡹⣆⢠⢏⠀⠀⠀⠀⠀⢰⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡇⠀⠀⣠⠖⠋⠐⢸⡇⠒⠙⠲⣄⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠹⡦⣤⣤⣸⣇⣤⣤⢴⠏⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⣇⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣸⢇⠇
⠀⠀⠀⠀⠀⠀⠐⢆⠀⠀⠀⠀⠀⠀⠀⠀⡈⠣⡀⢸⠁⠀⠀⠀⠀⠀⠀⠈⡇⢠⠞⢁⠎⠀⠀⠀⠀⠀⠀⣠⡰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠱⠂⠀⠀⠀⠀⠀⠀⠘⢆⠈⠻⠀⠀⠀⠀⠀⠀⠀⠈⠗⠁⡠⠃⠀⠀⠀⠀⠀⠀⡔⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡆⠀⠀⠀⠀⠀⠀⠀⠱⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⠎⠀⠀⠀⠀⠀⠀⠀⢰⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣷⠠⠤⠒⠒⠚⢛⣻⣓⢺⡆⠀⠀⠀⠀⠀⠀⢰⡗⣚⣟⡛⠓⠒⠒⠤⠄⣾⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠛⠉⠉⢁⡼⠃⢠⠆⠀⠀⠀⠀⠰⡄⠘⢧⡈⠉⠉⠛⠒⠒⠒⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠘⣠⠏⠀⠀⠀⠀⠀⠀⠹⣄⠃⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣡⠆⠀⡝⠀⠀⠀⠀⠀⠀⠀⠀⢋⢠⠐⣌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠁⠀⠀⡇⠀⠀⠀⠄⠀⠀⠀⠀⢨⠀⢠⠈⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⡴⡀⠀⢡⠀⡀⠀⠀⠀⠀⢀⠀⡌⠀⢀⢦⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⣴⠰⠀⠀⢀⠐⡀⠁⠈⢀⠀⡐⠀⠀⠆⣦⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠠⠀⣀⠄⠀⢀⡴⣁⠀⠀⣈⠦⡀⠀⠠⣀⠀⡄⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠁⣀⠔⠋⠀⠈⠁⠀⠁⠀⠙⠲⣀⠘⠀⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⣠⡼⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⢧⣀⠀⠁⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⣀⣠⠴⠊⢠⡇⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⡈⠑⠦⣄⣀⣀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⡼⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢧⠀⠀⠀⠈⠉
⣿⡿⢿⣿⢿⣿⡿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣶⣾⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⡀⠤⠤⠤⠒⠒⠂⠀⠐⠒⠠⠤⢄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢹⠆
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⡀⠀⠒⠒⢤⡀⠀⠀⠀⠀⠀⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⡼
⠀⠀⠀⠀⠀⠰⡄⠆⠀⠀⠀⠰⡆⠀⠀⢀⣀⢉⢲⣄⣄⢀⣀⣠⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢰⠃
⠀⠀⠀⠀⠀⢀⠇⠨⠧⠦⠆⠦⠽⠶⠤⠤⠰⠒⢳⣳⣋⡭⠶⠷⢿⠛⠿⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡜
⠀⠀⠀⠀⢰⣅⡰⢀⢠⣀⠴⢬⣀⢀⣠⢀⡴⠖⠉⠈⣃⠀⡀⡄⠈⡆⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣸⠁
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⠱⠚⠓⠊⠉⠓⢳⢹⣔⣃⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢀⡇
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⡃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡼
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢰⠃
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⣠⠶⠒⠒⠛⠶⢦⣀⡋
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⡴⠛⠂⠉⠒⠀⠤⡀⠈⠑⢆
⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣸⡅⠤⣠⣤⡀⠀⠀⠀⠐⠢⣈⡇
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⣼⣁⡀⠀⠀⠀⠀⠉⠙⠢⣀⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⢀⡟⠉⠉⠉⠑⠒⠂⢤⡀⠀⠀⠹⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠶⠴⠶⠒⠐⠋⠁⠀⠀⠀⠀⠀⠀⢸⣀⣀⣤⣤⣀⠀⠀⠀⠈⠒⢴⣀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⠀⠀⠀⠀⠉⠒⠤⡂⠄⢈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠀⠀⠀⠀⠈⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠠⠄⣀⣤⣈⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠳⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠐⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢤⠂⠁⠠⠀⠐⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡇
⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢑⠆⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠑⢦⠀⠀⡀⠀⠈⢀⠀⠰⡆⠁⠀⠀⠀⠀⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠑⠄⠈⠤⠚⠃⠀⠀⠁⠀⡃⠀⠀⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢉⠉⠙⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠐⠀⠀⠀⢄⠹⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠲⢤⡀⠀⢂⠹⡜⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⡎⠙⠢⡀⢢⠱⡘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣀⣷⡀⠀⠈⠢⣳⣣⢡⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠙⠢⣄⠙⢿⣧⡹⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠶⠟⣦⠀⠀⠒⠤⢌⠑⢦⡳⣽⠷⡘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢻⠟⠁⠀⠄⣸⠷⣌⠒⠒⠒⠒⠲⢝⡿⡑⢧⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣠⡿⠾⠶⠦⠴⢼⣧⣄⠈⠻⣦⣒⢄⣠⣀⣙⠮⢪⡳⣌⣧
⠀⠀⠀⢀⣀⣀⣤⠞⠉⠉⠉⠙⢧⡀⠀⠀⢀⠀⠀⠀⢈⠳⣶⣻⣟⠛⡋⠍⣩⡟⠛⠻⠿⢿⣇⡀
⠀⢀⡾⠋⠉⠀⠉⠲⠤⣴⠊⠁⠀⠙⠦⣀⠀⠙⠢⡐⠄⠳⣈⣻⣷⠀⠀⣴⠏⠀⠀⠀⠠⢀⣭⠿⢦⣄
⠀⠘⢷⡶⢤⣀⠀⠀⠀⠀⠙⠒⠦⢤⣄⡈⠙⠲⠤⣌⣢⣶⠃⠈⠉⢻⡋⠁⠀⠀⡀⠀⣰⠏⠁⠀⠠⣹⣷⣶⣦⡀
⠀⠀⠀⠙⠳⠮⣭⣍⣙⡒⠒⠒⠒⠒⠒⠚⠣⠽⠶⣦⣈⣉⠓⠒⠲⢌⠻⣤⠀⠀⠈⣺⠏⠠⠠⢀⣴⠟⠌⢿⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠒⠒⠲⣶⡶⠶⢤⣬⣙⣓⣒⣲⣤⣙⣳⣶⣴⣇⠀⢀⣨⣼⣥⣀⠀⠈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⠿⠀⠀⣠⠾⢋⣉⣩⠝⠋⠉⠙⠷⣿⢶⣿⡿⡫⡪⣪⢿⣅⠁⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣏⡡⠎⠀⠀⠀⠸⡔⢯⠉⠀⠀⠀⠀⠀⠀⢸⠨⣿⣯⡫⡺⡚⣽⣿⠀⣘⢧⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢷⠀⠀⠀⠀⠀⠀⠈⣇⢿⠉⠛⠛⠛⠛⠛⠋⠙⢦⡈⢩⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⢸⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⢀⠔⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⡛⣻⠛⣿⡟⣋⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠼⣾⢻⣷⡿⠼⠔⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⢿⣶⣿⡿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⣿⣼⣿⡷⢤
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣱⠮⠬⠽⣷⡜⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠘⡯⢬⢽⡟⠄⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣚⣿⣿⣿⣜⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⢷⣚⣿⠃⠘⠆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡠⠴⠶⠮⣁⣤⠽⠦⣄
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢦⣤⡴⠒⢻⡄⠀⠀⠀⠳⡀
⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠁⠒⠒⠂⠄⠠⠄⢀⣀⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⣾⣿⡇⠀⢸⠇⠀⠀⠀⠈⡇
⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⠙⠛⠒⠦⡀⠐⠀⠀⢀⡹⣟⠉⠀⣠⡞⠀⠀⠀⠀⠀⡷
⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⡀⠀⠈⣢⠶⠚⠁⠀⠸⡉⠋⠕⠀⠀⠀⠈⠐⢲⠃⠀⠠⡀⢠
⠻⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣡⠞⠁⠀⠄⠀⠀⢀⣹⣆⡐⠆⡀⠀⣀⡴⠋⠀⠀⠠⡀
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠈⠉⠉⠉⠁⠈⠁⠉⠉⠉⠉⠉⠉⠑⠒⠠⠖⠊⣡⠏⠀⠀⠀⠀⠀⣠⡞⠁⠘⠟⠳⡚⠛⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠀⠀⠒⠒⠈⣡⠃⠀⠀⠀⢐⡤⠞⠁⠁⠀⠀⠀⠀⠈⠀⢄⠀⠀⠐⠆⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⠀⠐⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⢀⠞⢧⡀⠀⣠⠶⠏⠝⠃⠀⠀⠀⠀⠀⠠⠆⠈⠁⠒⢁⠀⠀⠀⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⣈⠔⠁⠀⢀⡟⠉⠁⠀⠀⡇⠇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠁⡠⠚⢇⠀⠀⡰⠋⠀⠀⠀⠀⠀⠁⠀⠈⠄⠀⠀⠀⠠⡀⡀⢀⠀⠀⠐⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠄⠄⠐⠊⠁⠀⢀⡴⠊⠀⠀⢀⡹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠣⠑⠔⠆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠘⢦⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠛⠉⠁⠀⠀⢤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⡇⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢂⠤⢀⣀⡀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠙⣦⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢆⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⡀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⢙⡶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⢄⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠢⡼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣚⡫⡽⠕⢫⡙⢿⣗
⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⣤⣾⣉⠀⣉⠁⡀⠠⠀⢰⠲⠀⢹⠀⢀⠤⠂
⠀⠀⠀⠀⣴⢻⠋⣼⠟⠉⠻⠿⣿⣾⠙⠈⠁⠐⠀⠁⠀⠀⢀⣶⠁
⠀⠀⠀⠀⣧⣼⣰⣸⡦⠀⠀⠀⠸⣿⠀⠀⠀⠀⡀⠀⠀⣰⡾⠿⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠈⠁⡮⡇⠹⣦⠀⠀⢴⣿⠆⢰⣾⣷⡻⣖⡛⠋⢠⡆⠂⢀⡐⠀⢱⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠙⣝⠓⣦⣽⣿⣿⣿⣷⢸⠀⢿⣿⣿⡟⠃⢠⣿⠘⠰⠾⠶⠶⠈⡖⠚⠉
⠀⠀⠀⠀⠀⠰⠨⠀⠁⠀⠀⠛⢿⠟⠸⢀⣸⠁⠙⢧⡴⠛⠿⠃⠁⢃⠀⠀⠠⢇⡀
⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⢸⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠀⠀⠀⠉⠢⣤⣀⡀
⠀⠀⠀⢀⡰⠄⣠⣤⠀⠀⠀⠀⠀⢠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠓⠢⢤⣄⣀⡈⠉⠉⠛⠓
⠀⢶⠶⠋⠀⠀⠀⠀⠉⠀⠀⠀⢀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠤⣴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠉⠱⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢀⣧⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣴⢯⣉⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡜⠛⣦⡼⠟⠛⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⠳⠶⢿⠁⠀⠀⠀⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠳⠦⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⣴⠎⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣱⣾⡄⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⢿⣯⡼⠛⣿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠘⣃⡀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⡿⣶⡛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣶⡾⣿⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠈⢧⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢉⣿⣾⠟⠣⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠟⠷⠟⠈⢇⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⢀⣠⠴⠖⠛⣛⣛⣛⣋⣉⣉⣉⣉⣍⠩⠭⠭⠽⠯⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠥⠤⠤⠤⠴⠶⠶⠶⠶⠶⢶⣟⣛⡳⢶⣤⡀
⢠⡞⢻⡷⠖⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⡿⠉⠙⡆
⢻⡄⠈⠳⢦⣤⣤⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣤⣤⡤⠴⠶⠞⠋⠁⠀⠀⠀⢹
⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠲⣄⠀⠀⢀⡤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⢠⡇⠀⠀⠀⠀⣠⠛⢳⡀⠀⣸⠀⠀⢧⣀⣠⠟⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢶⡄⠀⠀⠀⢰⠋⠉⢻
⢸⡇⠀⠀⠀⠀⣿⠆⠈⠳⠞⠃⠀⠀⠈⠉⠁⠀⠀⠀⢨⡇⠀⠀⠀⣠⠞⠋⠛⠳⣄⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⣿⠀⠀⠀⢀⡄⢀⡼
⣾⠀⠀⠀⠀⢸⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠀⠀⢠⡇⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⢸⠇⠀⠀⣀⣹⣦⣤⡴⠾⠛⢻
⡿⠀⠀⠀⠀⣿⠿⠦⠤⣤⠤⠤⠤⠶⠶⠶⠶⠶⠖⠂⣾⠀⠀⠀⠘⡷⠒⠒⠲⠖⠛⠷⢶⡄⠀⠀⣿⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡆
⠱⣄⣀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⣿⡿⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢹⠀⠀⠀⠀⠀⣀⣀⣤⠴⡇
⠀⢹⠶⣮⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣀⣀⣀⠤⠤⠤⠤⢾⡀⠀⠀⠀⣸⠓⠚⠛⠛⠉⠉⠁⠀⠀⢸
⢀⡟⣁⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠁⠈⠁⣈⡉⠉⠉⡁⠀⢠⡀⢀⣀⡀⢀⡀⠛⣷⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⣿⡠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⡆⠀⡟⠉⢹⠀⠘⡛⠳⣼⣇⢸⡦⠀⠀⠙⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡶⠁
⠀⠈⢳⣾⣥⣀⣀⡀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠛⠀⠙⠶⠞⠀⠀⠁⠀⠈⠋⠈⢉⣉⣀⡀⣥⣤⣤⣤⣤⠤⠄⠶⠶⠒⠉⠉⠀⠀⡇
⠀⠀⡿⠂⠀⠈⠉⠉⠉⠛⠋⠙⠛⠛⠒⠋⠘⠀⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠰⣟⡡⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠁
⠀⠀⠈⠙⣿⢤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠤⠤⠤⠤⠤⠤⠤⠄⠒⠛⠋⠉⠹⡆
⠀⠀⠀⠸⣧⠀⠈⠉⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉⠀⠈⠉⠉⠉⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠸⣧⠔⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠃
⠀⠀⠀⠀⠀⠈⢹⡷⢦⣤⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⣠⣤⣤⣤⣤⡤⠤⠶⠶⠶⠖⠊⠉⠉⠸⠁
⠀⠀⠀⠀⠀⠀⠘⣶⡠⢀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣋⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡷⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠖⠒⠒⠚⠛⠛⠙⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡤⢂⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣾⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣮⡟⣩⣉⣐⣀⡜⣁⠀⣠⣀⡜⣠⠆⣠⠆⣤⣤⣼⣷⠷⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⢠⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡄⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠐⢇⠀⠀⠀⠀⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣀⠀⠀⢠⠀⣠⣦⡃⣸⣶⡀⠈⡆⠀⠀⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠙⣆⢹⣿⡿⣿⣿⣇⢠⠏⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⣿⣿⣽⡯⣿⣿⣿⣯⡀⠚⠃⠀⣀⠀⠀⠤
⠁⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⡀⠀⣀⣀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⣐⠐⠒⠂⠶⠒⠐⠀⠀⠀⠀⠀⠐⠂⠀⡀
⠀⠀⠀⠠⠀⢀⣀⠈⠁⠀⠀⠈⠉⠁⠀⠀⠀⠈⠙⣛⣻⣛⡹⠋⠀⠀⠌⠐
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⠿⣿⣿⢿⣷⣾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠹⣿⣿⣁⣿⣏⣸⣿⣿⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⣿⡌⢿⡟⣿⣿⡏⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠈⢿⣷⣾⣿⣿⠟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠩⡸⡿⠋⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣀⢀⠀⠀⢀⣀⠀⠀⣀⠀⣀⠀⣀⠀⠀⣀⠀⡀⠀⠀⢀⡀⠀⢀⡀⠀⡀⠀⢀⣀⠀⠀⠀⣀⠀⢀⡀⠀⢀⡀⠀⣀⠀⢀⡀
⠀⢘⣯⠘⣷⠇⣟⣭⠀⢸⣟⢸⠿⢛⣿⡄⢸⡿⠃⡟⣾⡷⣏⣥⠀⢸⣹⡇⣹⣏⢸⠯⠀⣿⢸⡿⠳⣏⣥⠀⣿⠿⢸⣉⡇⣿⠅
⠀⠈⠁⠀⠈⠀⠈⠉⠀⠈⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠈⠁⠀⠁⠈⠈⠀⠀⠛⠈⠀⠀⠈⠁⠀⠈⠀⠈⠉⠀⠁


⠀⠀⢀⣤⠶⠦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠶⠒⠛⠓⢦
⠀⠀⠰⠐⠂⠈⠈⠉⠉⠙⠳⢦⣄⡀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⢀⡤⠒⠁⠀⠀⠀⠀⣀⣤⣶⢟⣋⡁⠀⠀⠀⠀⣀⠃
⠀⠀⠀⠁⣀⠀⠀⠀⠠⠤⠤⠤⠭⡝⠳⢦⣄⡀⠀⠀⠀⠉⠲⡤⢤⡞⠁⠀⠀⠀⢀⣠⣴⠾⢋⠏⠤⠄⠠⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⢀⡀⠀⠄⠀⠀⠒⠒⠚⠢⠤⠈⣙⠛⠶⣤⡀⢻⠛⠛⢿⢀⣠⡴⠾⣛⡡⠤⠔⠊⠀⠀⠀⠐⠒⠀⠠⡀⠂
⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠈⠀⠠⠄⢀⠀⠀⠀⠈⣉⣉⡛⠏⡚⢹⠛⣿⠛⣊⣩⠍⠁⠀⠀⠀⠀⠀⠀⠀⢈⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣲⠆⠉⠉⠒⢦⡤⢤⡤⠤⠀⢰⡖⣋⢉⣦⡿⠈⠠⠤⢤⠴⠖⠂⠒⠈⢙⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⢐⠨⢀⠀⠋⠋⠉⠡⢊⣿⣉⠈⢿⣎⠀⢌⠉⡋⠑⠄⡀⠁⠐⠠⠈⢐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠤⢀⡠⠊⠀⠀⠀⠀⠾⠀⠀⠀⠀⠇⢢⠀⠑⢌⠂⠀⡄⠠⠄⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠠⠀⠀⠀⠀⡔⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢁⢠⡶⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⢀⢁⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⢂⣴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣤⣤⣤⠋⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣹⡷⠴⣠⣄⠱⣿⣶⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣽⣚⠋⠶⣦⣝⣿⣷⠘⣿⠇⠸⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠞⠉⠐⢿⣦⣈⣿⣷⣻⣧⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠏⡀⠀⠐⠄⠙⡿⣮⣿⠛⢐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠦⡀⣊⣢⠟⡫⠃⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢈⠭⠁⠜⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢤⠤⠄⠠⡇⠀⣤⠀⠤⠤⠤⠤⠤⠄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⣂⣉⣀⣲⡤⢶⠤⠃⢀⠰⣶⠦⠴⠄⠀⣀⣀⡀⠀⠉⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⠴⢏⣙⣇⣸⡠⡧⠼⣦⠀⠈⠐⢦⣤⠤⠤⠤⢤⣄⠀⠙⠢⡀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⣶⢮⣉⣉⠭⠄⣀⠀⠠⠀⣀⣀⠀⠀⠀⣀⡀⠀⠀⠀⢑⣤⠃⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⣽⣂⠈⣉⠃⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠁⠀⡄⠐⣁⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⣝⡻⣟⠀⣴⡂⠀⠀⠀⢠⣶⡄⢰⡯⢦⡀⢂⡀⣇⢈⣀⡥⡞⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢈⢀⡇⠁⠈⠋⢮⡇⠀⠀⠂⢨⠟⡻⢿⠺⠾⠟⠛⠋⠙⠁⠃⠃⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠜⠻⠀⠀⠀⠀⢀⠀⠀⢀⠓⢸⡃⢌⢂⢃⠀⡀⠠⠠⠀⠀⡘⢰⠀⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠃⠀⠀⠀⠀⣠⣆⠀⠤⠅⣼⡷⠶⠶⢄⠀⠀⠁⠆⡀⠀⠃⡈⠀⠁⢈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⢀⣠⡾⠃⠈⠳⠶⠞⣦⣀⠀⡀⢋⣆⠀⠀⠠⠀⡘⢀⠃⠃⠠⢸⢠⠑⠄
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠂⠀⠀⣔⠝⢁⡀⠠⠠⠤⠤⣴⣿⣤⡟⢀⠄⠘⣧⡀⠀⠈⠁⠸⢰⡖⠓⠮⡈⠀⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠉⠠⢄⠀⣠⣶⣯⢁⣿⠃⠀⠀⣀⠘⢟⡄⠀⠀⠃⣼⠀⠀⠀⠡⡆⠀⠐
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠆⠀⠨⡥⣀⡠⢶⣦⣙⡿⣿⠀⢀⡄⠀⠀⡜⡌⠀⠀⢀⠻⠀⣄⠀⠀⠀⠠⢄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡌⣁⠀⠔⢸⠷⣶⣝⣿⢷⠻⡃⠈⣠⠂⡘⡘⠀⠀⢨⡘⢀⠶⣈⡁⠾⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠴⠐⠀⠀⠀⠀⠰⠘⢆⠶⠻⠷⠼⢋⢾⡈⠤⠁⠐⠁⣲⠌⠀⠀⠀⠀⠃⡈⡀⢁⠁⠆⠀⢠⡸⠃
⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠱⠈⠖⢎⢦⡀⠋⠀⠀⠀⠖⢠⡳⠁⠀⠠⢀⠂⠈⠰⢡⠁⠄⡘⠰⠀⠂⡉
⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠚⠭⠴⠶⠖⣂⣀⣀⣰⠏⠀⠀⢀⠂⠌⡀⢀⢃⢃⠆⡠⢁⢃⠂⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦⡀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠠⠀⠀⠐⡐⡀⢀⠎⠎⡰⢁⠎⣀⠎
⠀⠀⠀⠀⠀⠀⠐⠢⠀⠂⠀⠊⠑⣠⣉⠂⠑⢊⠔⡠⠂⠂⠐⠀⠁⡠⢁⠌⠔⠜⠀⠀⠎⠈⠀⣤⠞⣣⠀⡠⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⢂⠠⠀⠄⢉⡰⠛⡰⠦⣤⢀⣀⣀⢀⢀⡀⣀⡀⢀⡤⠀⠄⣐⢞⠕⢁⠈⡡⢊⠐⢁⠔⡡⢂⡰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠡⠒⢁⠔⢉⠔⢊⠤⠂⠠⠀⠀⠀⡁⠀⡠⠂⠔⠠⠂⡠⢁⠄⡡⠊⡄⠁⠔⢁⠘⠁⠊⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠎⠀⠊⠐⠀⠐⠀⠊⠀⠁⠈⠀⠈

⠀⠀⠀⠀⢀⣀⠀⠤⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣩⠶⡀⠰⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠆⠀⠀⠀⠀⡄⠀⠂⠀⠀⠀⠀⠀⡴⠀⠈⠐⣀⣧⢞⠁⢠⠀⠀⢀⠠⠀⠀⠄⠚⢢⡄⠊⠉⠹⡄
⠀⡲⠐⠄⡀⠀⢀⣼⣗⠇⠈⠀⠀⠈⠀⠀⢧⠀⢀⢸⠏⠜⢈⣶⡋⠈⠓⠀⠀⣠⠞⠀⠀⠀⡁⠀⠀⢠⢃⠄
⠀⠸⡂⠬⣟⣴⡯⠦⠯⠋⠀⠀⠀⠀⠀⠂⠈⠿⣿⣴⣮⠜⠉⠜⢁⠠⠀⠀⠀⠘⢄⡢⣀⠢⠁⣀⣀⡜⠁
⠀⠀⠄⠀⡐⡿⠗⠀⠀⠤⠀⠀⠀⠀⠀⢀⡴⠚⢹⠋⠟⣛⣿⢣⡀⠀⠀⠀⠀⠀⠀⣉⡻⠛⢻⠛⠓⡀
⠀⠀⠀⠈⠁⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠜⠀⠀⠀⠈⡆⠀⠉
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⣤⠖⢿⠉⡟⢑⠦⣄⠀⠀⠀⠀⠀⠀⠀⡖⠒⢏⠉⠓⢦⡀⠀⠀⡞⢳⠞⢳⠞⢛⠖⢛⡖⢛⡆
⠀⠀⠀⠀⢀⠾⣀⠳⣸⣼⣡⣋⠴⠚⢧⠀⠀⠀⠀⠀⢠⠟⢦⡀⠑⢄⠀⢳⠀⠀⣇⡏⢀⡎⠀⡞⠀⡞⠀⡼⡇
⠀⠀⠀⠀⢸⠦⠬⢵⣿⣿⣿⣷⣚⣉⣙⡆⠀⠀⠀⠀⠈⣇⠀⠙⢦⡀⠑⡏⠀⠀⣟⣀⣜⣀⣼⣀⣼⣁⣸⣁⡇
⠀⠀⠀⠀⠸⡖⠒⣩⢿⢿⢿⢯⠲⢤⣹⠁⠀⠀⠀⣠⢞⡽⠓⠲⠶⠛⠒⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⡽⣏⣠⠏⡼⠸⡆⢳⡴⠃⠀⠀⣠⢞⡵⠋⢀⣀⠤⣀⣀⠀⠀⠀⠀⣠⠖⠒⠒⠒⠦⣄⠀⠀⠀⢠⠒⡿⠛⠛⠓⣦
⠀⠀⢀⡴⡫⠊⠀⠉⠓⠓⠒⠋⠁⠀⠀⠀⣞⡵⠋⠀⠀⡇⠀⠀⠈⢉⠇⠀⠀⣼⠁⢠⠖⠓⠂⠀⠘⡆⠀⠀⡞⠉⠛⠓⠒⠋⢹
⢀⡴⡫⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⡜⠀⠀⠀⠹⣄⠀⠁⠀⠀⠀⡰⠃⠀⠀⣇⠀⠀⠀⠀⠀⣀
⠻⠚⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣤⣼⡅⠀⠀⠀⠀⠈⠙⠒⠒⠒⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢶⠲⢤⠋⠉⠉⠉⠈⠈⠙⡤⠖⡆⠀⠀⠀⡼⠉⠉⠉⢧⠀⠀⠀⠀⢠⠤⣄⣀⠔⠚⠉⠓⢦⣀⣠⠤⣤
⠀⠀⠀⠀⠀⢯⢀⣸⣄⠀⠀⠀⠀⠀⢠⣇⠀⡝⠀⠀⢰⠁⠀⠀⠀⠈⣇⠀⠀⠀⢞⠀⠀⡏⠀⠀⠀⠀⠀⡯⠀⠀⣣
⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⠉⠉⠉⠉⠁⠀⠉⠁⠀⠀⠺⠤⠤⠤⠤⠤⠿⠀⠀⠀⠾⠴⠒⠛⢦⣀⣀⣀⠴⠋⠒⠮⠇
⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡄⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠰⡅⡇⠀⠀⠀⣀⣠⣄⣀⡀⠀⠀⢸⢨⠆⠀⠀⠀⠀⡏⠀⢠⠃⠀⡼⠁⠀⡼⠁⠀⡼⠀⢈⡷⡄
⠀⠀⠀⠀⠀⠀⢘⡅⡇⠀⠀⡎⠁⠀⠀⠀⢹⠄⠀⢸⠠⡃⠀⠀⠀⠀⠈⠓⠛⠒⠚⠓⠒⠚⠓⠒⠚⠓⢲⠋⠀⢹
⠀⠀⠀⠀⠀⠀⢨⠆⡇⠀⠀⠈⠒⠶⠴⠒⠋⠀⠀⢸⠰⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⠚⠒⠋⡽
⠀⠀⠀⠀⠀⠀⠰⠷⠗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠺⠾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣈⣓⡦⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢤⣄⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⣏⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢦⡘⢶⡿⢈⣷⣫⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣿⣿⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡒⣾⣆⠃⠋
⠀⠀⠀⢠⣴⣶⣶⣶⣾⣶⣦⣸⣿⣿⣿⣀⣀⠀⠀⠀⠀⠀⠀⠙⠶⠚⠀⠀⠀⠀⠀⠀⠀⣀⠛⠶⠟⠀⠀⠀⠀⠀⠀⢲⣤⡀⠀⣥⣭⣿⡆⠺⠟⠁
⠀⠀⠀⢼⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣄⠘⠻⢋⢻⣷⣶⠆
⠀⠀⠀⢾⣿⣸⣿⣿⣿⣿⣃⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣇⢰⣿⣿⣿⠟
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⡀⠙⢿⡿⡾⠿⠛⠁
⠀⠀⢰⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠓⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢿⣿⣿⣆⠀⠘⢀⣐⣒⢢⣄⡀
⠀⠀⠀⠙⠺⣬⣿⣿⣿⠟⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣾⡟⢿⣿⣠⣴⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣦⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣭⣿⡟⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡿⣏⡿⣿⣿⣿⣒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⣿⠃⣴⠀⠀⠀⢀⣴⣤⣄⠀⠀⠀⠀⠀⠘⣿⣿⠋⠀⡜⢿⣿⠟
⠀⠀⠀⠀⠠⣤⡀⠀⢀⢀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠓⠀⢸⡿⠀⠀⠀⠹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⡀⠈⠁
⠀⠀⠀⠀⠀⢿⣿⣷⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣾⠁⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠻⠿⢧⠄⠀⠀⠀⣾⣤⠀⠁⠀⠀⠀⠀⢀⣿⣿⠿⠃⢸⣿⣦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡄
⠀⠀⠀⠀⠀⢤⣤⣶⡚⣠⣶⠀⣠⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠈⠙⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⣾⣿⣿⣿
⠀⠀⠀⢴⣦⡈⢻⣿⣿⡟⠁⠈⠁⢠⡜⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣼⣿⣿⣿⡿⠀⠀⠀⢠⣼
⠀⠀⠀⠸⣿⣿⡄⠈⠛⢿⠟⠃⢤⣜⡁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⡟⣿⣿⣿⣿⠇⢠⢀⣴⠿⡟
⠀⠀⠀⠈⢻⣿⣿⡏⢀⣰⣾⣆⡰⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠈⣿⣿⣿⣿⠟⢀⣼⣿⡿⠞
⠀⠀⠀⣀⣼⣿⣿⣷⡀⠈⠉⠉⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⢄⠹⣌⠛⢃⣴⣤⣤⣤
⠀⠀⠀⠉⠻⢿⣿⣿⣇⠀⢀⣴⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠂⠉⠁⠈⢉⣥⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠠⠼⠿⠂⢸⣿⣿⣿⣿⡿⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⠻⠛⠀⠠⠀⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠿⠿⠟⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣄⠀⠀⣠⡶⠆⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⡉⠉⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠈⠀⠘⢿⡇

⠀⠀⠀⠀⠀⠀⠀⢠⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⠀⠀⠀⠀⠀⠸⣆⢀⣤⣼⠏⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠩⠁⠀⠀⠀⠀⣀⣤⣤⡜⢿⣿⠋⢳⣤⣀⠀⠀⠀⠀⠀⢸⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡎⠀⠀⠀⣠⡾⠙⠈⠻⢷⡾⠍⣰⠋⢆⠙⢷⡄⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⣿⣾⠿⢷⣾⡃⢚⣹⢿⡿⣿⣎⡿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⡀⠀⢬⣿⢯⣿⠿⣿⡟⢻⣿⣾⣿⣿⣿⣧⠀⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢾⣿⣷⣾⣿⢼⣿⡌⣿⠇⠈⢙⣿⡜⢿⡞⣿⣶⣟⣴⡓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣻⠏⢾⡿⠀⣿⠀⠀⢨⣿⠎⣸⣏⣻⣿⡯⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⢸⣏⠄⣿⠀⠀⠸⣿⠅⢹⡷⣻⠇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⡿⢻⢹⣿⠀⣿⡀⠀⣼⣿⡀⢾⣇⣿⣟⣯⠿⣦⣤⡀
⣤⡀⢀⡀⢀⠀⠀⠀⠀⠀⠀⢀⡠⠴⣖⡥⠖⣡⠾⢈⠟⣹⣿⡟⣿⡇⢀⢸⣿⠀⣿⣿⡟⣹⡿⣄⠈⠙⠲⢭⣓⣲⣤⣀⣠⣤⣠⣤⣤⡶⠶
⠻⠉⠙⠛⠻⠟⠳⠗⠶⠖⡻⠷⠖⠋⠁⢀⠞⢁⣧⠏⢀⡟⢿⣿⣼⣿⣶⣾⣿⣾⡿⢋⣠⣌⢣⠈⠳⣄⠀⠀⠀⠙⠯⠉⠉⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⣤⢋⠊⣠⣾⡀⡇⠙⣯⣮⢙⣌⣹⠏⠰⣮⡽⡏⡀⠳⣄⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⡈⡡⢃⢊⠃⡷⣎⠠⡄⢠⣿⣿⣿⣿⠀⢈⢻⠠⣷⠰⣄⡈⢟⠲⢽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡡⠊⡴⢡⠁⡀⠀⣷⢻⣦⢻⣿⣿⣏⠝⣿⠓⡞⠿⣦⣿⣦⠽⣜⢬⡉⡐⠮⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠊⠆⠁⢀⣥⡾⣿⡍⢷⡌⢿⣿⡑⣀⣿⠀⠇⢺⢡⣼⡽⣦⡈⢆⠑⣌⠀⣌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢃⡠⠊⠀⣘⣴⠔⣻⡿⠟⢻⣧⢧⠛⢾⣿⢿⠻⣿⡄⢃⡏⢱⡇⠉⣿⡙⠸⡄⠈⢢⡈⠳⡙⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣞⡕⠁⠀⠀⣼⣻⠗⠋⠛⠀⠀⢘⡟⠓⠆⣼⣏⣂⣐⣙⣧⠀⢲⣋⡇⠀⠈⡟⠢⣿⣄⠀⠹⡄⠘⡈⢧⡀
⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠊⢸⣿⡀⠀⡇⠀⠀⠀⢿⠁⣼⣿⣿⡻⠟⠿⣿⣿⣇⢳⠆⠀⠀⢡⠀⢈⣿⠀⠀⠘⢆⠈⠈⢷⡀
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠈⠙⣧⡐⠀⠀⠀⠀⢀⣾⣿⣽⣯⣧⣀⣈⣬⣤⣾⢧⡀⠀⠀⢸⢠⡿⠉⢧⠀⠀⠈⢣⡀⠀⢧
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠔⠀⠀⠠⠁⠀⠀⣷⠀⠀⢀⡴⠋⠈⠙⢿⣯⣟⡉⠽⢛⣿⠻⠈⠉⠳⣄⣰⠟⠀⠀⠈⢳⡀⠀⠀⠙⠢⢼⡆
⠀⠀⠀⠀⠀⣼⠀⠀⠂⠀⠠⠒⠀⠀⠀⠀⣹⡴⠚⠁⠀⠀⠀⠀⠀⠙⣿⣿⠽⢛⠫⠀⠀⠀⠀⢨⡻⠦⣀⠀⠀⠀⠙⢦⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣏⠀⠠⠀⠀⠀⠀⠀⣀⡴⠋⠉⣽⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠛⣧⠀⠈⠓⢦⣀⡀⠀⠉⠑⠒⢤⡇
⠀⠀⠀⠀⠀⢿⡀⠀⠀⢀⣀⠤⠚⠁⠀⣀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⠈⠉⠛⠶⠶⠶⠛⠁
⠀⠀⠀⠀⠀⠈⠛⠚⠛⠉⠁⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢈⣷⠀⢤⣞⠁⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢀⣾⣇⠂⠀⠀⢸⣿⣆⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢆⠸⣿⣿⣀⢀⠀⣼⡿⣿⠀⠀⢠⢎⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢼⣄⣈⣿⣿⡿⡿⣿⣶⣤⣄⣠⡳⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠁⠀⠀⠈⠀⠀⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣭⠇⠀⠀⠀⠀⠀⠀⠈⣿⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⢝⣍⠄⠀⠀⠀⠀⠀⠀⠀⣼⣆⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢠⣤⠖⣁⠂⣺⣷⣷⣇⣠⣆⢠⣠⣼⣿⣿⡭⢂⠀⠑⢤⣀⣀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠃⡈⠉⠀⠰⢃⣉⣉⣿⠟⠿⠏⣿⡿⠟⣏⣗⣋⠉⢆⠀⠀⠛⠙⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⡄⣮⠷⣿⣀⠀⠁⠀⠀⣠⣤⣾⠛⠲⠭⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠰⡎⣧⠀⠉⠉⠁⠉⠉⠉⠀⠀⠀⠂⢰⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠃⠀⡀⠱⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡆⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡰⣿⠧⠄⠀⡀⠀⢀⡀⠀⠄⠂⠺⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡼⠡⡀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣧⠀⠈⢯⡂⠂⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠿⠻⠄⠀⠀⡀⢀⣠⠬⠘⠇⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣄⡴⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠐⠒⠒⠚⢋⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠾⠛⠳⢦⡀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠈⠻⣄⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠉⠉⠛⠻⠶⣤⣀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⢾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⣠⡾⠋⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⡟⠀⢀⡤⠶⠖⢶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⢀⡼⠓⢛⡭⠿⣿⡲⢯⡀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢄⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⡴⠋⠀⡰⠁⠀⢀⠀⠹⡄⣙⣆
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⠁⠀⠀⡇⠸⣶⣿⠃⢠⠋⢀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡙⠢⣄⡈⠛⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠘⠦⢤⠤⠔⠛⡘⠿⠃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠐⠀⠀⠈⢉⣻⣶⡶⠤⠤⢤⣤⣀⣤⣤⢼⡀⠀⠀⢀⣀⠀⠀⠀⠀⠈⠒⣺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠙⠻⣿⣿⣦⡀⠀⠀⠙⣿⣿⣆⠹⣄⠀⠀⠹⢦⣤⣤⡤⢖⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠀⠀⠀⠈⢿⣿⣷⡄⠀⠀⠈⣿⣿⡆⠈⠙⡶⠦⠤⠤⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⠀⠀⠀⢻⣿⣷⡀⠀⠀⠸⣿⣷⠀⣸⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣦⠀⠀⠈⣿⣿⣇⠀⠀⠀⣿⣿⡴⣿⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡆⠀⠀⢸⣿⣿⠀⠀⢀⡿⠋⢰⡟⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠋⠉⠛⠿⣿⣿⣇⠀⠀⣸⣿⣿⠴⢶⡏⠀⣴⣿⠃⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢉⣼⠏⠀⣾⠃⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠋⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠈⠉



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

⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠎⠀⠴⣾⣿⣿⡇⠈⠃⠉⠛⠂⠀⠀⠂⠀⠀⠀⠰⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⠀⠤⠒⠚⣸⣿⣿⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠈⡇⠀⠀⢀⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡿⠋⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼
⠀⣀⠔⠂⠀⠦⢄⠀⠀⠀⠀⠀⢀⣠⠤⠄⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠠⡀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⡄⢀⣾⡟
⠴⠁⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠈⠙⢂⣀⣀⣀⡀⢀⡠⠐⠒⠦⢄⠀⠀⠀⠑⠇⡀⠀⠀⠀⢀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣤⣾⡿⠋
⠆⠤⠄⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢡⠀⠀⡀⠀⠐⢄⣀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⢤⣶⣶⣼⣷⣷⣤⣾⣿⣿⣿⠖⠈
⠀⠀⠀⠀⠠⠀⠀⠐⠠⠀⠄⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠉⢳⡎⠀⠁⠈⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠉⠉⠹⠛⢛⣟⣿⣿⣿⣷⣶⣖
⣦⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠎⢳⠀⠀⠀⠀⣠⠤⠀⠠⠤⢺⣿⣿⠘⠻⠟⠿
⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠊⠁⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⣆⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠄⠀⠀⠀⠀⠀⠀⡘⠁
⠀⠀⠀⠀⣀⠐⠀⠀⠀⠀⠹⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣻⣷⡀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣤⡄
⠀⠀⠀⠈⠀⣦⢄⡀⣀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣤⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠐
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠤⠱⠶⣦⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⣿⣿⡇⠀⣾⡇⠀⢀⠔⠀⡠⠊⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠢⣀⣀⣀⠀⠉⠙⢿⣿⣿⣇⢰⣿⣷⣾⣿⡵⠄⠀⠀⠀⠀⠀⠀⠀⡚⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠘⠛⠛⠛⢻⣿⣾⣿⡿⢿⡅⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠂⠸⡄⠀⠀⡼⠀⠀⣠⡄
⠀⣠⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠁⠈⠁
⠀⠠⢀⣸⣆⠀⠀⠀⠀⠀⢉⡂⠀⡀⠀⠀⠀⠀⠀⠀⠠⠶⠒⣈⡽⠟⠉⠉⠻⣧⠀⠉⠀⠀⠀⠶⠤⢄⠀⢀⠌⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣠⠂
⠀⠀⠀⠀⠹⠞⠀⢀⠂⠀⠀⠀⠀⠀⠀⠐⠢⠤⠤⠴⢒⢀⠝⠉⠡⠚⠐⠆⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠂⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢠⠀⠀⠀⠀⠀⠀⣠⣄⡀⢸⠀⠀⠀⠀⠀⢀
⠀⣀⠀⠀⠀⠀⠀⠀⠐⠆⢀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠋⠉⠉⠻⢟⢿⣇⣠⡀⡀⠀⠀⠀⠀⠐
⠊⠁⠀⠀⠀⢀⠀⠀⢀⠈⠈⠲⠤⢶⡀⠀⣀⣀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡄⠀⠀⢀⠀⢠⠀⠀⢀⣴
⠀⠀⠀⠀⡤⣈⣀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⢳⣇⠀⢸⣧⣤⣤⣶⡿⠛
⣀⠀⠀⠀⡎⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠀⠀⠀⠹⣆⣧⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⠿⠿⠛⠂
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⡄⠀⡌⠀⠀⠀⢀⣴⣾⡥⠂⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⢿⣍⠀⠀⠀⠐⠊
⠀⠀⠀⣀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠠⠀⠀⠀⠤⣄⡀⠀⠀⠀⣹⣿⣿⣄⣄⡆⠀⣴⣾⣿⠯⢀⡄⠀⠂⠐⠀⠀⠲⠀⠠⠁⠈⠀⠈⠳⣍⡀⠒⠒
⠒⠠⢀⠀⠀⠀⠁⠀⠀⠢⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠀⠙⢿⣿⡏⠃⢠⣿⣿⣯⠖⠁⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⠉⠐
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⣷⣶⡅⣤⠛⠲⣹⣿⣣⠄⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠆⠀⠀⠀⠐⠚⠋⠉⠁⠘⠞⠛⠙⠋⢻⡿⢿⠅⣀⣶⠋⣵⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⣿⣿⣷⡿⡋⠀⠀⠀⠀⠀⠀⢀⣠⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠉⠁⠠
⠀⠀⠀⡀⠻⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠻⠻⣄
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⣿⠁⠀⠀⠀⠀⠀⠀⠐⠢
⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠋⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡷⠀⠻⡇⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈⠓⠦⠤⣌⣳⣏⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠈⢿⡆
⠀⠀⢀⣀⣚⣹⣿⣿⢶⡀⠀⠀⠀⠀⠀⠀⠉⠛⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⡜⠗⠀⠀⠀⠀⠀⠉⠀⠉
⠀⠀⠀⠙⣛⣿⡟⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠐⡾⠋⠉⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀
⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣿⣿⣿⣶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⡼⠛⠉⣹⠂⠀⠀⠀⠀⠙⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠺⠋⠀⠀⠀⠀⠀⠀⠀⠙⠛
⠀⢸⣦⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣀⣸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠤⠄
⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⠛
⣿⡿⠟⠁⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠰⣏⠁
⠀⠀⠀⠀⠀⠀⠻⣟⠲⠤⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣍⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠑⢦⣄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠓⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢿⣷⣿⣿⣦
⠀⠀⠀⠀⠀⣀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣻⣿⣿⣿⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣔⣒⣒⣀⣀⢠⣤⣄⣒⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠾⣟⣋⣉⠉⠉⠁⠀⠀⠈⣀⠈⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠨⣗⡲⠦⢭⣍⡉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣊⢻⣿⣿⣶⡖⠒⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠒⠶⠋⠀⣤⡀⠉⢿⣿⣍⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠟⠀⠰⠶⠶⣾⡋⠁⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣯⡉⠁⠀⡄⠀⠀⠝⠃⠀⠋
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣍⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠲⠤⢤⣀⣀⠀⢀⣿⣿⣿⣿⣿⣿⣯⡅⠀⠀⢀⡀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣏⡛⠻⠿⠿⠿⠯⠷⠢⢒⣣⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠢⠽⠛⠓⠒⠶⠖⠚⠛⠉⠁⠀⠀⠀⡀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡐⡜⢆⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣎⢧⢹⣾⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⡿⠿⠿⠃


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

⠀⠀⠀⠀⠀⠀⠀⣠⡶⠒⢶⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣯⣁⠈⠙⠷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠛⠷⣶⣤⣈⣉⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⢶⣦⣄⡉⠙⠳⠦⡤⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣦⠀⢀⡞⢉⡼⠛⣫⣷⠆⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣿⣀⡟⢁⡞⢁⡴⠋⣺⣎⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣾⡄⣼⡁⣼⣿⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⡴⢆⢀⣀⠀⠀⠀⠀⠀⠀⠉⡛⠿⣿⣷⡿⡿⢃⣏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠢⣼⡈⣻⡇⣸⠃⣸⠤⠄⣀⡴⠖⣋⣩⣥⡷⢤⡉⠧⠆⡼⠀⣉⠉⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠚⠢⣼⣿⣿⣿⣿⣿⡇⢠⠞⣡⣶⣟⣭⠵⠶⣒⣚⣗⣛⣒⡇⠀⠀⠀⠀⠀⣄⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣁⠒⣿⣿⣿⣾⣧⣿⢧⡏⣼⣿⣿⣽⢶⣛⣋⣭⡭⣯⠭⢽⠃⠀⠀⠀⢈⡲⢤⡙⠌⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢤⡬⣝⠛⡿⢿⢿⣻⢸⡇⡜⢿⣿⣿⣻⣭⡷⠶⠖⡟⠲⢺⣷⣶⣤⣄⣲⣽⣶⡝⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠑⣞⣾⠉⢻⣧⡼⣾⡟⣼⣷⣜⠦⣝⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡿⠿⢛⣩⣾⣿⣿⣆
⠀⠀⠀⠀⢠⠖⠦⣄⣇⠀⠘⣇⠀⣸⡏⠀⠸⣼⣿⣿⣿⡷⣤⣉⠀⠀⠀⠉⢉⣉⣉⣩⣭⣥⣴⣶⣾⣿⣿⣿⣿⡿⢿⡆⠀⠀⢀⡀⠀⡀
⠀⠀⠀⣀⣈⣣⣀⠀⢽⣾⣶⣿⣾⣿⣾⣤⢶⣿⣟⠷⣬⡛⠲⢭⣙⠛⠶⠶⠶⠶⠶⠶⠶⠶⢛⣛⣯⣿⣿⣿⣿⡿⣻⣿⠐⢦⡏⠱⡾⢹⣤⢤⡀
⠀⠀⠸⣅⠀⠈⢽⣿⣿⣿⡿⣛⣛⣻⣿⣯⢸⣿⣝⠷⣄⠉⠓⠂⠀⠈⠀⠀⠀⠀⠐⠒⠛⢛⣋⣩⣥⡾⣿⣿⣿⡿⣫⣿⠰⣴⣿⣶⣿⣟⣅⡼⠶⣄
⠀⠀⡴⠚⠛⠓⠺⣿⣿⣯⣾⣿⣿⣿⣿⣿⡞⣿⣟⡳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣉⣩⣥⣶⡾⣿⣿⣿⣿⣫⡿⢬⣿⣿⣽⣯⣿⣿⠦⠶⣅
⠀⠀⢙⡶⠖⠶⠚⢿⣿⣿⣿⣿⣿⣿⣿⣉⠳⢻⣿⣿⢶⣍⡑⠀⠀⠀⠀⠀⠀⣀⣀⣉⣉⣉⣥⣤⠶⢾⣿⣿⣿⣿⣿⣃⡤⠾⣿⣿⣿⣿⣿⠣⣴⠚
⠀⠀⠘⠲⣦⠶⠒⠚⢛⡿⠋⢹⡏⠙⡄⠈⣳⣿⣿⣿⣷⣦⣍⣙⠒⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⢶⣿⣿⣿⣿⢿⣿⣿⣯⣤⡞⠁⡾⢻⡟⢱⡦⠼⠃
⠀⠀⠀⠀⠿⠶⠴⣖⣉⠀⣠⣿⣇⣠⣿⣶⣿⣿⣿⣿⣿⣿⣦⣭⣙⡓⠒⠶⠤⠤⠤⢤⡶⠶⠶⣿⣿⣿⣿⣿⣿⣟⣛⣛⣻⣷⣾⣧⣸⣓⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣷⣯⣝⣛⠳⠶⠶⠶⠶⠶⣾⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠯⠭⠍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠚⠛⠛⠛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣛⣛⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣉⣉⣉⣉⣉⣉⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⠥⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢤⣤⡤⠤⠤⠤⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣖⠻⣽⣿⣿⡗⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠈⠳⡘⣿⢿⣿⣄⡈⢳⣀⡀⠀⠀⠀⡰⣾⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠷⡋⠍⠁⠀⠙⠏⠹⠿⣆⡠⢪⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⣴⡟⠂⠂⢀⠀⠀⠀⠀⠀⢦⡝⣶⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⢣⢠⢾⠁⠀⢀⣀⡠⢐⠒⠓⠻⣿⣗⣍⡢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣼⡿⣯⣔⣆⣦⠿⠉⠀⡀⠀⠠⢀⡘⠿⣷⣍⡛⠦⣕⠢⣶⣶⡶⣦⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠿⠷⠒⢒⡿⢠⡇⣿⠺⣵⠆⡄⠁⢀⠀⠠⠀⠁⣿⢯⡙⢿⣦⣌⠛⢮⡑⢄⡀⠉⠁⠈⠁
⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⣼⡇⢸⡇⣿⠀⢸⠛⣦⡼⣤⣤⠽⠼⠻⢙⡦⡙⢦⠙⣿⣷⡦⣈⠓⢿⣦⣀
⣤⣤⣶⠶⠟⠋⠀⠀⠀⠀⠀⠀⢰⣹⠀⢸⠃⣿⡄⠘⡆⠸⡇⠆⠀⠀⠀⠀⠂⠈⡘⠢⡑⢌⠫⡻⣮⡓⢤⡉⠻⣷⣤⡀
⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠃⠀⢸⠀⣿⣧⠀⢱⠀⡇⠀⠉⠀⠀⡁⣤⠴⠒⠒⠛⢦⠑⢕⢌⠻⣦⡙⠢⡈⠛⢟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣿⠁⠰⠀⢸⠀⣿⢸⡀⠀⣇⠰⢠⡰⠛⠉⠀⠀⠀⠀⠀⠀⠀⠙⠦⣷⡳⡈⠻⡦⡈⠰⣄⠑⢷
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠈⡟⢸⡀⠀⢸⡀⢹⠀⢧⠀⢸⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠙⠲⠬⠮⠦⢬⣷⡊
⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⡇⠀⠀⠀⢸⠁⢸⡆⠘⡆⣸⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡇⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⡀
⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠸⡇⠀⡹⢣⣘⠀⠀⠀⠀⠀⢀⣀⡠⠤⠴⠒⠒⢲⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠋
⠀⠀⢺⠟⠉⠀⠀⠀⠀⠀⠀⡆⡆⠀⠀⢸⠀⠀⢧⠔⠀⠀⠙⢶⣄⣠⠔⠊⠉⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣧⠀⠀⠈⣇⡠⠃⠀⠀⠀⠀⠀⠙⠳⢄⣀⢀⠀⣀⡤⠤⠤⠤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣀⣀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠙⠶⣤⣤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣬⣵⣦⣼⣧⣾⣽⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⣦⠤⣄⣀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⢀⠀⠀⠀⣀⠊⠀⠠⠀⠙⡅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢀⡀⠀⡿⠉⠑⣶⠟⠀⠀⠀⠀⠀⣅⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠀⠀⠀⠁⠈⣾⠇⠀⠀⠀⠀⠀⠢⠐⠀⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢀⡀⢀⠂⠀⠀⢠⡟⣇⡀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠐⠀⠀⠀⠠⣧⡄⠀⣺⣵⣷⠆⠀⠀⢀⢁⠎⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠑⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠙⢯⠁⠀⠀⠘⣿⡏⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⠺⠟⠯⡀⠀⠀⠈⠓⣄⣀⠀⠹⠧⢂⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠓⡄⢸⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠜⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠉⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠤⠄⠀⠤⠀⠀⢀⡀⠀⢀⡀⢠⡔⠋⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠠⡀⡘⠁⠀⠼⣇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠙⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠁⢀⠀⠠⠋⠉⠦⠄⠀⠀⠀⠀⠀⢇⠀⠀⣀⢀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢲⣾⣿⣿⡟⠀⠈⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡟⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠉⠀⠀⢀⠀⠀⠀⠀⢀⣰⡿⠁⢻⡏⠈⠉⢄⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠚⠁⢁⠀⠀⠀⠐⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢚⠃⠀⣀⡠
⠀⠀⠀⡤⠶⠶⣄⠀⣷⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡤⠤⣜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣁⣉⢔⠚⠉⠁
⠀⠀⠀⣧⠀⠀⠈⢷⡸⣆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡘⣖⣒⣾⣂⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣫⣏⠄⠤⠊⠁
⠀⠀⠀⠈⠳⣤⣀⠀⠻⣜⠂⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢚⣩⠃⢠⣯⣉⣉⡽⣧⠀⢫⣉⠲⢤⡀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠋⠝⠟⢩⣱⠤⡀
⠀⠀⠀⠀⠀⠀⢉⣹⠒⣿⡷⠉⢲⣶⡖⠀⠂⠀⠀⢠⣞⡡⡶⠟⣩⠖⠁⠺⢭⣭⣷⠏⠙⢮⡙⠷⡦⣝⣦⠀⠀⠀⠀⠀⠘⡾⡿⠢⣴⠓⢯⣼⣪⡿⣦
⠀⠀⠀⠀⠀⣔⣳⢿⣙⡦⠗⠤⠛⠟⠀⠀⠀⠀⠀⠀⠀⠘⠗⠋⠁⠀⠀⠘⣻⠿⠇⠀⠀⠀⠈⠓⠛⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⣹⠀⠀⣇⠙⢏⣻
⠀⠀⠀⠀⠰⣉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⡏⠉⠲⢄⡀⠀⠀⡢⢄⢠⠀⢀⡠⠔⠉⢹⠀⠀⢀⣀⣀⡀⠀⠀⠀⢂⣠⢶⡄⡄⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⣀⣀⡀⠀⠀⢠⢀⣶⡂⡠⠀⠀⠀⣀⣀⡀
⠀⠀⠘⢦⣤⣤⣬⣱⣾⡟⢻⣧⣶⣭⣤⣤⡴⠃⠀⠀⢅⠀⠈⠩⠙⠛⠒⢸⣯⣉⣽⠗⠒⢟⠋⠉⠀⠀⡜⠀⠐⢏⡀⠀⣉⣉⠩⢤⣿⠿⣿⡶⠬⣉⣉⠀⠀⣹⠂
⠀⠀⠀⠈⠒⠀⠬⢙⢛⠿⠟⡻⡋⠅⠦⠐⠃⠀⠀⠀⠀⠁⠒⠄⠄⠤⠀⣾⣟⣒⣿⣷⠀⠆⠤⠀⠂⠁⠀⠀⠀⠀⠉⠙⠤⠤⡶⠚⣟⠋⣛⢟⠶⠤⠤⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠠⣭⣭⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣷⣶⡞⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠃⢯⠭⣽⠘⠆⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠏

⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄⢀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠌⠉⠉⠳⣦⡀⠁⣀⡀⠀⢀⡴⠛⠉⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢆⡔⠉
⠀⠀⠐⢤⣀⣀⠀⠙⠦⣟⣫⠶⠋⠀⣀⣀⡤⠀⠀⠀⠀⠀⠀⣠⠖⠒⠒⠴⠧⢼⣥⠶⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⡖⠋⠓⠲⠦⣄⣀⡇⢹⣀⡤⠴⠒⠛⠓⡆
⠀⠀⠀⠀⠈⠉⠉⠉⢠⣓⣂⡈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⢀⣠⠶⠋⡀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠙⠲⣤⠤⢀⣀⢬⠛⠫⣥⣀⠠⣤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣟⣒⣒⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠹⣛⡻⢾⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢺⣭⣭⡄⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣭⣵⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠠⢤⣐⠲⢄⠀⢀⡤⠖⣠⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⣄⠀⢀⡤⠖⠂
⠀⠀⠀⠠⡀⠀⠈⠳⣄⣷⣿⡤⠞⠁⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠸⣷⠏⠀⣀⣀⣀
⠀⠀⠀⠠⠉⠙⢛⡶⣾⣿⣿⡷⣖⠛⠋⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣟⣋⣉⣹⡶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⣠⣴⣴⣶⣶⣴⣤⡀⠈⠉⠢
⠀⠀⠀⠐⠤⠖⠋⠀⢭⣶⣾⡅⠈⠛⠦⠔⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⣯⣉⣉⣹⡇⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠐⢤⣖⣯⠟⠋⣶⣶⣶⡌⠛⢯⣗⣦⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⣠⠴⠊⠈⢧⣤⠏⠘⠲⢤⣀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⣶⠆⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠤⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠚⠉⠉⠀⠀⠀⢀⣀⡤⠾⠗⠒⠛⠉⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⢀⣠⠤⠖⠋⠉⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⢀⣀⣠⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⣤⣶⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢀⣠⣤⣶⣿⢟⣁⣠⣤⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣿⣿⣟⣿⣻⣿⣿⣿⣿⣿⣥⣤⣤⣶⣶⣾⣿⣿⣿⣿⠿⠿⠟⠛⠛⠉⠉⠛⠻⢷⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣰⣟⣶⣾⣿⡟⠉⠉⢻⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣟⣿⣿⣿⣧⡀⢀⣰⡟⠛⠻⣿⣿⣄
⠀⠀⠀⠀⠀⣀⣤⡾⠟⠿⣿⣿⠾⠿⢿⣿⣿⠋⠀⠀⠀⠀⠙⠿⣿⣦
⠀⢀⣠⡴⠞⠋⠁⠀⠀⠜⠁⠋⠀⢀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣱⣤⣧⢼⣃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠀⢀⣤⣶⣶⡎⣿⠦⢿⣾⡏⣷⣶⣶⣄⡀⠀⠀⡤⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣷⠀⢰⡿⡈⡺⣿⣷⣿⣗⣻⣿⣇⣿⣿⠖⡝⣷⠀⣸⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡽⠁⠸⣷⢲⣵⣟⣿⣿⡏⣽⣽⣿⣟⡿⣷⢷⡿⠀⠹⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⠸⣿⠿⣻⣥⣼⣿⣿⣿⣥⣝⡻⢿⡿⠀⠀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⣷⡀⠀⣼⣿⣿⣷⣾⣿⣿⣿⣷⣿⣿⣿⡄⠀⣴⣿⣿⡇
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⢓⣿⣼⣿⣿⣧⣟⡟⣼⣾⢣⢻⣿⢿⣿⣷⣾⡷⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⡿⠺⠿⢶⡷⢶⣶⣼⠓⠶⢦⡀⠀⠀⠀⠀⠀⠙⢷⣝⣿⣿⣯⡇⣟⣷⡹⠬⠞⣬⡟⣿⣿⣿⣯⣾⠟⠀⠀⠀⠀⠀⢀⣠⢶⣛⢻⣶⣷⠾⠿⠾⠟⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⣿⠿⡀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣏⡇⣼⣋⣹⣿⣯⣹⣇⣿⣿⣿⡍⠀⠀⠀⠀⠀⠀⠀⡜⢻⡏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡖⣇⡤⢤⣄⣀⠀⠀⠀⢸⣿⣻⢼⣯⡟⢻⡔⡦⢻⣿⢸⣿⣿⠀⠀⠀⢀⣠⡤⠶⣤⣇⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣌⡒⠄⡉⣛⠲⠦⣼⣿⡿⡼⣷⣇⠮⣋⠗⣼⣿⣾⣿⡿⣶⠒⢛⡉⡁⢖⣎⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠷⠶⣿⣿⣟⣥⣿⣿⡿⣾⣿⣷⣶⣾⣿⣿⣽⣿⣧⣬⡻⢾⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢋⣤⠂⠔⠁⢻⣿⣿⣽⣴⣀⣆⢟⣻⣻⠋⡘⢦⠰⣤⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢩⠞⠟⠀⠈⢀⣵⠛⢿⣿⣿⣾⣟⣣⣿⣿⡟⢦⡐⠀⠁⠈⠟⠣⣝⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⢞⢁⡎⠔⠀⢠⣿⠁⠁⡀⠻⣿⣿⣿⣿⡿⠏⡠⠊⢻⣤⡀⠀⡦⣣⠈⡣⡻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⣱⠃⠈⠐⠃⡴⣿⣩⠓⡄⠘⠦⠸⣿⣿⡿⠁⢺⠁⣰⢛⣿⣧⡀⢘⠆⠡⠘⡌⢎⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠌⡶⠃⡾⠆⢺⡏⢰⡟⣛⡆⠩⣮⠀⡞⢸⣿⠡⢹⠈⣴⠃⣼⣻⣇⠙⣼⠗⠺⣧⠙⢦⠠⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠋⣜⢀⡼⣓⣦⡏⠁⣾⠰⢏⣿⠀⣿⡸⢧⠸⣿⢠⡞⢦⣽⠀⡟⠿⣿⠀⠘⢧⣰⣛⢇⠈⢆⢁⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠞⢠⠀⢸⠃⡸⠃⡄⢠⡟⠐⠀⡄⢰⣁⡅⡀⢲⣿⠈⠇⢀⣙⡀⢃⡀⢻⣄⠀⠈⠳⡀⠘⡆⠠⠌⢣⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⣴⣞⣤⡶⠇⡼⠁⠀⠀⣼⣧⠀⢳⣠⣽⡏⣿⠇⣸⣿⡇⢸⡇⣾⣇⣼⡇⢸⠿⡠⠁⠀⠹⡀⠑⢦⣾⣶⢆⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⢀⣽⢿⡇⠁⣧⣀⢀⡴⠁⢻⣆⠘⣧⢿⡳⡹⣰⣿⣿⣿⣜⠏⢹⢏⡿⢀⣼⠂⠱⣄⣀⣠⡇⢀⢸⢻⣧⠀⡀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡌⠈⣿⠀⢠⠀⢇⢘⡅⠠⠖⢛⣿⠁⣏⡏⠀⢸⣿⣿⣿⣿⣿⡃⡌⠉⡕⣸⡏⡙⠲⢄⡯⣠⠇⠀⠘⠀⡅⡁⢥⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣠⡄⠻⣄⢸⠀⡜⣦⢻⡄⠊⢀⠻⣄⢺⠘⣷⣿⣿⣿⣿⣿⣿⣷⣿⢠⢃⡟⠁⡈⠐⣼⢡⡞⢀⠸⠀⡤⠇⢡⣼⢠⡯
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⡙⢣⡂⢼⣀⢶⡁⠚⣆⣹⡔⠻⠿⢝⢶⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⢋⡰⠿⡏⣾⢀⡏⠁⢸⡼⢰⡅⢂⡌⡹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢾⡅⣠⠟⠄⣦⣜⢺⠕⣧⡟⢋⡿⢾⠆⣪⡾⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⢿⡕⠼⠻⣏⠙⣟⡶⢹⢚⣏⣇⠀⠳⣄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⢇⢀⠴⣿⠁⢹⣷⠟⢠⢏⠄⣼⡴⠟⠀⠀⠀⠉⠛⠻⠟⠛⠉⠀⠀⠀⠀⠻⣶⡦⢹⣦⠈⣷⣿⠃⢸⣞⠦⡀⠟⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⠤⠾⡇⢠⡾⠃⣼⠳⣮⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣴⢷⡇⢹⡄⠀⠸⢷⠢⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⢈⡴⣱⣾⠟⢏⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢶⣉⡺⢿⣦⡘⢆⠁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣾⣿⣽⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢿⣿⣦⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⡀⠀⢠⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠀⣰⣞⣋⣙⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠁⡄⠀⠀⠀⠀⣠⡤⠶⠴⠢⢤⣤⠤⠤⠬⢧⣤⡉⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⠶⠶⠖⠒⠒⠛⠻⠀⠀⠠⠄⠀⠀⠀⠄⠀⠀⠀⠢⣹⠀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠖⠒⠒⠒⠀⠀⠀⠘⠒⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡍⠳⣿⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠁⠀⠈⠻⣧
⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⣐⠃⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡤⠤⠀⠹⣇
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡄⠀⡤⠀⢠⠲⡀⠀⠁⠈⠁⠘⠒⠄⠀⠀⢿⡀
⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠃⡅⡌⠉⢠⠋⠀⢱⠸⠙⢦⡃⢰⠖⠑⡞⢀⡱⠞⡔⠢⣀⠔⡆⠀⠀⠀⠸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠐⠃⠀⠈⠈⠙⢸⡗⢠⣄⠑⢪⡄⠀⡇⠡⢄⠀⠘⢆⢠⠊⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⡀⠔⡆⢠⢰⠈⠂⠘⠃⠀⡀⠑⡄⠀⠀⠇⠏⠁⡀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠳⠤⠡⢠⠀⠢⠤⠌⠘⠒⠀⠀⠘⠢⠃⠣⠤⠤⠇⠀⢧⠃⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣄⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣠⣤⣤⣤⣤⠾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣀⣀⠀⠀⣼⠃
⠀⠀⠀⠀⢀⣤⡴⠶⠶⠶⠶⠶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠛⠉⠉⠛⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠶⠶⠶⢦⣄⡀
⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢶⣄⡀⠀⠀⠀⠀⣠⠟⠁⠈⠀⠀⠀⠀⠁⠈⠻⣄⠀⠀⠀⠀⢀⣤⡶⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⡀
⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣀⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⣿⠟⠉⠉⠉⠉⠻⣿⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠈⠙⠷⢦⣤⣄⣀⣀⣀⣀⣠⣤⣤⣴⣶⠾⠟⠛⠉⠀⣴⣯⣤⣤⣤⣤⣤⣤⣽⣆⠀⠉⠛⠻⠷⣶⣦⣤⣤⣄⣀⣀⣀⣀⣀⣤⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣩⠿⠛⠉⠁⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠈⠉⠛⠿⣌⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⢠⣾⣧⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣼⣷⡄⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⣀⡴⠛⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⠛⢦⣀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠙⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠛⠛⠛⠛⠛⠛⠛⠛⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠢⢀⠀⡁⠀⡀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣻⣿⢷⡤⣀⠀⠀⠀⠀⠀⠐⢄⡀⢠⣹⣥⣔⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣤⡶⣦⢳⣱⣿⠿⠿⢯⡀⠀⠀⠀⠀⠀⠁⠀⠀⠏⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡳⣿⣮⡷⠉⠁⠐⢆⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⡃⣿⢿⣿⡿⣃⡈⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠛⢻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⢀⡜⠀⡄⠐⢇⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠐⠁⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⣤⣄⢠⣤⡄⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⣠⣄⠀⣭⡈⠀⣰⡈⢠⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠔⠚⠛⠃
⠀⠀⠀⢀⣀⠀⢈⣀⠁⢉⡀⠉⣁⡉⢉⣩⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢄⡀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠶⠚⠉⠁
⠀⠀⢠⡋⠉⠁⠁⢉⡀⠉⣹⠚⠉⢓⠋⢉⡋⠁⠀⠀⠀⣠⣼⠾⣏⠙⣧⠈⣧⢈⣠⠤⠔⠚⠉⠉
⠀⠀⠈⠇⠙⠈⡃⠉⠛⠐⠊⠑⠘⠛⠃⢘⠋⠀⠀⠀⠀⡏⢹⡆⢹⣆⢽⣶⣾⡿⠃
⠀⠀⠀⠘⠃⡈⠛⠆⠛⠛⠀⠞⠛⢆⠘⠀⠶⠀⠀⠀⠀⠸⣿⣿⣷⣿⣾⣿⠛⠁
⠀⠀⠀⠈⠳⠶⡄⠜⠶⠤⠋⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠳⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠂⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢀⠨⡝⠃⠀⣚⠠⠌⢹⡻⠦⣥⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⡀⠈⢘⠷⢤⣀⣈⣵⡒⠜⢃⡴⠋⣛⣻⣿⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡉⠀⠀⠈⠐⠯⢁⠵⠔⣩⡿⠙⠈⠉⠒⢊
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡟⢶⣴⠀⢳⢉⣶⢻⡇⢠⡴⠖⠚⠩
⠀⠀⠀⠀⠀⠀⠀⠶⣤⣀⣀⠀⠀⠐⢀⣺⣖⢂⣸⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡷⣦⡙⠛⠟⠀⢸⡏⠀⠀⣠⢤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⢉⠛⠳⠛⠛⢻⡯⡹⠁⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⡤⡼⠧⠤⠞⠛⠉⡇
⠀⠀⠀⠀⠀⠤⠀⠐⣷⠀⠁⠀⠒⢄⢚⡄⣿⡝⠯⢱⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣤⣄⠠⢀⡀⠀⠀⠀⠀⠄⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠠⠝⠻⣦⡵⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⡛⠊⠿⢶⣥⠴⡧⠤⠆⠂⠈⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠓⠀⠀⡠⠀⠳⡟⠃⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠅⠀⠀⠈⣀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣦⡇⢀⢓⣶⣾⠿⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢠⡤⠀⡶⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⡀⠀⡤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢤⠀⢀⠀⠀⠀⢀⡶⠛⠗⠢⡀
⠀⠀⠀⠘⢶⡀⠻⢦⣉⣲⣄⢀⠀⣠⣖⣡⡴⠛⢀⠴⠃⠀⠀⠀⠀⠘⠐⠂⠠⣶⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠾⣉⣳⣶⣄
⠀⠀⠀⠀⠀⢳⡀⠀⠀⢻⢿⣿⣿⣿⡏⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠬⠤⡌⢿⡖⠀⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⢀⣀⢤⠄⠐⠓⠛⢛⠁
⠀⠀⠀⠀⠀⠀⠙⢦⠚⣁⢈⣿⣿⡁⡌⠗⡔⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠁⠰
⠀⠀⠀⠀⠀⠀⠀⢸⡈⠷⠏⠸⠓⠇⠻⢁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠕⠒⠋⠕⠫⠑⠒⠪⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⣀⣠⣾⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠶⠄⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠳⣉⠁⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⡀⠀⠀⠀⠀⠿⢿⠿⢛⣉⡁⠈⢱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠟⡀
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⣸⣿⢇⡞⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠈⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⡀⠔⠛⢷⡆⠀⠀⠀⠀⠀⠀⠀⣤⡾⠛⢹⣿⣿⣿⣟⠳⣄
⠀⠀⠀⢠⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠞⢹⣇⠀⠀⠀⠀⠀⠀⢠⠈⢷⣿⡷⠶⠄⢸⡇⠀⠀⠀⠀⠀⠀⠈⠘⢿⡀⢀⣏⠉⠀⠉⠙⣿
⠀⠀⠾⠊⠀⠀⠀⠀⠀⠀⠈⠁⢛⡙⠋⢠⣴⠛⠏⠱⡄⠀⠀⠀⠀⠀⠀⠈⠂⠀⠛⠤⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠙⠙⠰⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠀⠖⠋⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠄

⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⢀⣀⡀⢂⠄
⠀⠀⠀⠀⠀⠑⣤⣔⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠢⢀⣤⡄⡐⠈⠀⢀⣀⡀⠀⠀⠀⠀⠀⠧⣿⣿⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠺⢿⣿⡿⠗⣄
⠀⠀⠀⡠⡖⢹⣿⣿⠉⠢⣄⠀⠀⠀⠘⠠⠀⣀⠀⠈⢹⣿⣟⠁⠀⢀⡀⠤⠊⠀⠀⠀⠐⠒⣻⣿⣿⣟⠒⠂⠀⠀⠀⠀⡠⠊⠉⢀⠞⠨⣿⡇⠱⡄⠈⠑⢄
⠀⢀⠎⠊⠀⢸⣿⣿⠀⠀⠈⢢⠀⠀⠀⠀⠀⠁⡀⠋⠼⡿⠽⡙⢆⡀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣼⣿⢈⠃⠀⠀⠀⠀⠘⠀⠐⠒⠁⠀⢸⣿⡇⠀⠈⠒⠂
⠀⢬⣀⠠⠤⠊⢿⡏⠑⠤⠄⣀⡇⠀⠀⠀⠀⠀⠠⠊⠈⣿⠇⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢸⡇⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⠇

⠀⠀⠀⠈⢢⠀⣀⡀⢠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢸⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠐⢾⣿⣿⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⣰⣿⣆⡼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣚⣛⣛⣓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⢱⣴⣶⣦⣞⠁
⠀⠀⠐⠒⢺⠋⢻⡟⠹⡓⠒⠀⠀⠀⠀⠰⢍⡉⠭⣩⢿⣿⡿⣏⠭⢉⣉⠇⠀⠀⠀⠀⢠⠴⣿⣿⣿⣿⠦⡀⠀⠀⠀⠀⠀⠀⠂⣾⣿⣿⣿⣿⣿⣷⠂
⠀⠀⠀⠠⠃⠀⣸⣇⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⡘⢱⠻⣿⡟⢢⢱⠀⠀⠀⠀⠀⠀⠘⠋⠀⣿⣿⣿⣿⠀⠙⠓⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠱⠔⡏⠙⠉⣷⠢⠂⠀⠀⠀⠀⠀⠀⠊⠁⣸⠀⠉⠁⢦⠀⠉⠂⠀⠀⠀⠀⠀⠀⡎⠹⣿⣿⠏⣱⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⢆
⠀⠀⠀⠀⠀⢡⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⡀⢀⠟⠀⠀⠀⠀⠀⠀⢀⠆⢰⠁⠀⠀⠀⠈⡆⠰⡀
⠀⠀⠀⠀⠱⡄⣿⡟⣠⠃⠀⠀⠀⠘⢅⠈⠁⠒⠼⣦⣠⣶⣎⣠⠮⠔⠊⠉⣀⠕⠀⠀⠙⠲⢽⣾⣷⡯⠖⠋⠀⠀⠀⠀⠀⠀⠑⢌⡣⣀⣀⣀⢜⡡⠊
⠀⠀⠀⢀⡤⢚⣿⣿⠳⢄⡀⠀⠀⠀⠀⠉⡲⠦⠄⠀⢙⣿⡏⠁⠀⠴⢶⡋⠀⠀⠀⣀⡤⠖⣽⣿⣿⣯⠲⢤⣀⠀⠀⠀⠀⠀⢀⠤⡼⢿⣿⡿⢧⠤⡀
⠀⠀⠀⠀⠀⡏⣿⣿⢱⠀⠁⠀⠀⠀⠀⠀⠉⠒⠲⢲⠫⣿⡍⡗⢲⠒⠈⠀⠀⠀⠀⠀⠀⡸⠹⢿⡿⠏⣇⠀⠀⠀⠀⠀⠀⢰⠁⢸⠀⠻⣿⡟⠀⡇⠈⡆
⠀⠀⠀⠀⢠⠃⠙⠃⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢿⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢀⠇


⠀⠀⠀⢠⣀⡀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠑⣶⣶⠊⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣴⠀⣧⠔
⠀⠀⠐⢦⡀⢨⣷⣾⡁⢀⡔⠀⠀⠀⠀⠀⠀⠀⢀⡉⢰⣿⣷⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣿⣿⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢹⣿⡟⠀⡠
⠀⠀⠀⡠⢭⣿⣿⣿⣿⠭⣄⠀⠀⠀⠀⠀⠉⠉⠀⢈⣻⣿⣿⡁⠀⠉⠉⠀⠀⠀⠀⠀⠈⠲⢬⣿⣿⡥⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⣼⣿⣧⠊
⠀⠀⠚⢡⢟⣚⣿⣿⣿⠳⡈⠂⠀⠀⠀⠀⠀⠀⡖⠉⠿⣿⣿⠉⠑⡄⠀⠀⠀⠀⠀⠀⠀⢰⢫⣿⣿⡝⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⢫⢺⣿⡟⡍⠒
⠀⠀⠠⠞⠀⠒⣻⡿⠟⠀⠣⠀⠀⠀⠀⠀⠀⠈⠀⠀⠹⣿⡟⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠜⠈⣿⡿⠁⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠘⣿⠃⢧
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⡔⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢃⠀⢠⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⢀⣠⠖⠂⢉⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠦⠄⠀⣴
⠀⠀⠈⡄⠀⠃⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⡀⡎⠀⠀⠠⢴⠋⠀⣀⠴⠟⠀⠀⠀⢀⡀⠱⠶⠄⠀⣒⡲⠔⠒⠀⠀⠀⠈⠁
⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠴⠀⡀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⡇⠀⠶⢅⡀⠀⣱⠛⠁⠀⢰⠃⠀⠈⠠⠀⠈⠀⢀⠂⠁⠀⠈⠀⠀⢀⠂
⠀⠀⠀⠀⢱⠀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠄⠐⠀⠐⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⠀⠀⠀⠀⠀⠈⠀⠀⠡
⠀⠀⠀⠀⢀⠷⠊⠉⠭⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠄⠀⢠⡶⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢦⠂⢀⠤⠒⠒⠚⢆⠀⠀⠀⠀⠀⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀
⠀⠀⠀⠀⠸⠎⢀⠄⠐⠀⠉⠙⡄⠀⠀⠀⢀⠀⠀⠘⠀⡄⠀⠀⢠⠀⠀⢸⠀⠀⠆⠀⢀⡇⠀⡤⠀⠀⠀⡓⠀⢼⣷⡆⠀⠀⠀⠀⠀⠄⠋
⠀⠀⠀⠀⠀⠻⠁⡠⠒⠊⠉⠁⣇⠀⠀⠀⢸⠀⠀⡇⠀⣇⠀⠀⢸⠀⠀⢸⠀⠸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠂
⠀⠀⠀⠀⠀⠀⠙⢤⢀⠀⡠⠀⠁⠀⠀⠀⠆⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⢀⡀⠀⡇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⠠⢄⡀⠐⠋⠁
⠀⠀⢀⠀⣀⠙⠀⠀⡇⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠤⡀⠀⠀⠀⢀⠀⠀⠀⢀⠀⡀
⠀⠀⢈⢺⡇⠀⠀⠀⠄⠂⠀⠀⡀⠀⠠⣌⠀⢉⢢⣀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣠⠜⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⢋
⠀⠁⠀⠜⠀⠀⠀⠀⠀⠀⠀⣄⣉⣉⠉⠉⠉⣩⡝⢛⢛⠉⠋⢱⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⡐⠐⠉⠀⠈⠀⠘⠀⢄⠉⠀⢌⠀⠈⠁⠀⠀⣁
⠀⠀⠀⢀⠀⠀⠐⠒⠀⠀⠀⠜⠀⠀⠀⠠⠜⠉⠉⠀⠉⢷⠓⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢁⠈⠠⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠘⠆⠀⠀⡀
⠀⣁⠰⠀⣅⠀⠀⡁⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠙⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⡁⠂⠠⠌⠙⠁⢘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠑⢤⠀⢄⠀⠀⡀⢠⠄⠁⠀⠀⠲⠀⢠⠀⠀⠀⠀⢦⣈⠀⠀⠀⠀⠀⠀⠀⡨
⠀⠂⠀⠀⠐⠔⠀⠈⠈⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠑⠈⠒⠘⠂⠀⠀⠠⠀⠠⠈⠀⠈⠀⠀⠀⠀⠀⠛⠥⣴⣀⣀⣀⡩⠔⠁