Mini Emojis & Symbols

⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡄⠀⠀⢀⣤⣤⣀
⠀⠀⠀⢰⣿⣿⣿⠏⠀⠀⠀⠀⠸⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠿⠤⣤⠿⢿⣿⣿⡆
⠀⠀⠀⠈⠻⠿⣿⡟⢙⣿⡟⣿⡙⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡇⢰⣿⣴⣿⡄⠛⠛
⠀⠀⠀⠀⠀⠀⠛⠃⠘⠟⠻⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣁⣈⣀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡶⠶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣭⠉⠫⠌⢠⡦
⠀⠀⠀⠀⠀⠀⠀⠘⠏⢛⣛⠿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠶⢶⣏⣛⣿⣿⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⣟⡛⢧⣤⣼⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠉⠋⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⡄⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠛⠉⠉⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⢰⠀⠈⠉⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⢠⣠⠾⣿⣧⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣥⣤⣄⣠⣤⣴⣯⣽⣿⣿⣿⣿⣿⣿⡋⠸⣷⣼⣿⣿⣠⣤⣧⡀
⠀⠀⠀⠀⢀⣀⠠⠄⣀⣤⣶⠿⠛⠛⠛⠉⠉⠉⠉⢉⣉⡙⠛⠉⠙⠛⠉⠉⠉⢸⠉⠉⠉⠭⠽⢿⣿⣷
⠀⠀⠀⠐⢩⣽⣦⣿⣿⣿⣦⡴⣦⣤⣤⣤⣤⣤⣰⠋⣠⣵⣿⣿⣦⣠⡀⠀⢀⣸⣀⣤⣤⣶⣾⣿⡿⢩⡄
⠀⠀⠀⠐⣤⣬⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⡝⣆⣈⣘⣿⣿⣿⣿⠋⢉⣽⣿⣿⣿⣿⣿⣿⣿⣷⣿⡧
⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣼⡿⠻⣿⣿⣿⠃⢀⣿⠿⣿⣿⣿⣿⣿⠿⣻⣽⣿⣿
⠀⠀⠀⠰⢿⣟⡛⠛⠻⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣼⣷⣷⠸⣿⣛⣍⣀⣀⣘⣿⡿⠇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣯⣭⣭⣭⣭⣤⣼⣾⣿⣿⣿⢇⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⣸⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠖⠚⠛⠛⢿⣯⡉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⡝⣆⠀⠙⠶⣄
⠀⠀⠀⠀⢀⣠⡤⠾⣗⢦⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⡇⠀⠀⠈⠳⣄⣀
⠀⠀⠀⢸⢯⡀⠀⠀⠈⠷⣝⢦⡙⢆⠀⠀⠀⠀⢀⣠⡴⠚⠉⠀⣀⣤⠴⠚⢉⣩⠷⣄
⠀⠀⠀⢸⡀⠙⢦⡀⠀⠀⠈⠓⠷⠬⠶⠶⠖⠛⠉⢀⣠⡤⠖⠛⣉⣤⠴⠚⠉⠀⠀⠈⠳⣄
⠀⠀⠀⠈⠳⣄⠀⠙⠲⣄⠀⠀⠀⠀⣀⣠⠴⠖⠛⣉⣤⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠈⢧⡀⠀⠈⠳⣤⠖⠛⣉⣤⠴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⡏⣷⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡇⡏⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢧⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁⠀⠀⢀⡤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⣄⠀⠀⠈⠳⣄⡀⠀⠀⠀⣀⣤⠴⠚⠋⠁⠀⠀⣀⣤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠙⡶⠒⠋⠁⠀⠀⢀⣠⡴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⢠⡇⠀⣀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⠗⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⡾⠋⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠘⠁⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠙⠦⠄⠀⠘⢦⡀⢦⠀⣿⡄⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⡔⡠⠂⠀⠀⠀⠀⠀⠙⠁⠙⠲⠖⠊⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠎⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢀⠎⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢀⠞⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠈⠀⠀⣰⠋⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⢸⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠱⣄
⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢠⠉⠒⠁⠀⠀⠀⠀⠀⠙⡆⠀⠈⢳⡀
⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠸⠄⠀⠀⠀⠀⢀⣀⣀⣀⠸⠀⠀⠈⠇
⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⠔⠀⠑⠲⠦⠶⠒⠋⠁⠒⢤⣀⣀⡤⠀⢡⠈⠉⠉⠁⠀⠉⢢
⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢬⣉⡀⠀⠼⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠋⣩⠔
⠀⢸⡇⠀⠀⠀⠀⠀⢀⠀⠀⠤⠴⠒⠒⠒⠒⠶⠶⠶⠤⠤⣤⣀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠈
⠀⠸⣇⠀⠀⠀⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠂⠤⠀⠀⠀⠀⠀⠊⢀⣠⠴⠒⢀⠄
⠀⠀⢻⡄⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⣀⣠⠤⠤⠶⠒⠒⠒⠒⠒⠒⠒⠶⠤⠤⠀⡀⠀⠀⠚⠉⠀⣄⠀⠢⣄
⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠳⠀⠐⠊⠉⠀⠀⠀⠀⠙⢦⡀⠀⠀⠉⠢⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠣⣀⠈⠻⣆
⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠈⠳⠄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⠀⠀⠀⠁⠢⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⡰⠁⢠⡏
⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⣰⠃⢀⡞
⠀⠀⢿⡀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⡇⠀⢠⠇⢀⡞
⠀⠀⠈⢷⣄⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢠⠏⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢸⠁⠀⡼⢀⡞
⠀⠀⠀⠀⠉⠓⠤⠀⡘⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣿⠀⢀⡇⡼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠤⠀⣠⢀⣀⣀⣀⣀⣠⡄⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠈⠉⠉⠁⠉⠈⠁⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣠⣤⣠⢄⣀⡤⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠁⠁⠁⠁


⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣆⠀⠀⠀⠀⢀⠀⠀⠀⢠⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣄⡀⠀⠈⠒⠀⠀⢘⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡿⠿⢿⣿⣿⣏⢻⣿⣿⣶⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠈⢿⣿⣾⣿⣟⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢺⣿⣧⡀⠀⣴⣿⡇⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣾⣿⣿⣿⣿⣿⡿⠘⡿⣿⠇⢸⣿⣿⣷⣦⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⣽⢻⣿⣿⣷⡄⡤⢠⣮⢀⣿⣿⣿⣿⣿⣿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣛⠋⢻⣟⢿⣿⠀⠀⠙⣿⣦⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⡋⠀⢰⣛⢽⡻⠀⠀⠀⠈⢙⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣧⠅⠀⢸⣿⣿⡅⠀⠀⠀⢀⣾⣿⣿⣿⣇⡠⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⣷⠀⠀⢸⣿⣿⣧⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⣽⣽⣿⣿⣿⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⡇⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠈⢻⣿⠿⣿⡇⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⣼⣿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢀⣀⣈⣿⣇⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣿⣿⠿⠿⠏⠀⠀⠀⠾⢿⠟⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣧⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣞⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠈⠙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡇⠀⠀⢸⣿⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠠⢿⣿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣰⣿⣯⣿⣿⠀⠠⢼⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣴⡿⠿⠤⡯⣿⣿⣷⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⠿⠿⠿⠿⠿⠀⣿⣿⡏⣿⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠗⢒⣒⠒⠒⠀⡿⣿⣷⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢰⡏⢹⣿⠀⣸⣿⡇⠀⠀⣧⣿⣿⡹⢿⢣⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣧⢀⡀⢀⣾⣷⣸⣿⢀⣿⣿⣿⠀⠀⣀⣿⣿⣇⣲⣸⣷
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣾⡇⢸⢹⣿⣿⣿⣼⣿⠛⣿⣧⣤⣿⡏⣿⣿⣾⠿⡿⠄⢸
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡾⠻⢿⣿⣿⢹⣿⣛⣿⣻⣿⣿⠿⢿⣿⣿⣿⣻⣧⣿⡆
⠀⠀⢀⡀⠀⠀⢸⣭⣿⣿⣿⡿⠛⢿⣿⣿⢹⣿⣿⢿⡟⢿⣿⠶⢾⣿⣿⣠⣾⡿⠿⠛
⠀⠀⠀⡽⢄⠀⠀⠀⢹⡿⠿⠇⠀⠈⢿⢿⣏⣉⣋⣈⣿⡿⠿⠀⠙⣿⣿⣿⡿
⠀⠀⠀⠉⠉⠆⠀⡀⢘⠀⠀⠀⠀⠀⡨⠸⣷⣿⣿⣷⣄⠇⠀⠀⠀⣿⢻⣤⠄
⠀⠀⠀⠀⠈⠃⠱⣧⠞⡄⠀⠀⠀⢸⡇⢰⣿⣿⣿⣿⣿⠶⠀⠀⠀⡏⢈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠋⣆⣄⠀⠀⢘⡃⠸⣿⣿⣿⣿⣿⣀⣠⣄⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠠⠟⠟⠿⣅⣤⡀⠀⠀⠀⠀⡨⣽⡿⠻⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⢀⣀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠏⣠⡗⠻⣍⡱⠀⡀⠀⠀⠀⢠⣧⣆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠
⠀⠀⠀⠀⠘⢈⡶⣦⣾⣗⠟⢉⡝⠀⠀⠎⢻⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠌⠀⢀⣿
⠀⠀⠀⠀⠀⠹⢆⣽⠉⠀⢦⣠⠀⠀⠀⠀⠋⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠄⠁⢀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠈⠻⢶⠀⠈⠻⡄⠠⠁⠆⠀⠀⢸⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⠀⠊⠀⢸⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡼⠂⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠚⣧⠀⢀⣷⣠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⣼⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠉⡇⠙⠻⠜⠉⠻⡄⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⣿⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠹⠀⣀⠀⠀⠀⠀⠀⣿⠀⢸
⠀⠀⠀⣀⠀⢀⠀⣀⡀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠄⠀⠀⠀⣠⠲⠀⠀⠀⠄⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⣿⠀⡀
⠀⣴⣔⡱⣄⠘⣄⡹⢿⡛⠓⠓⠛⠛⡇⠀⢘⡀⢀⢤⣰⠞⠛⢧⠀⠠⠀⠀⡠⠀⢚⠀⠀⡀⠠⠻⠀⠀⠀⢘⠀⠀⠀⣸⡏⠀⡀
⠀⣿⣿⡟⢸⡆⣼⠙⠲⣝⠢⡀⠀⠀⢹⣾⢋⠐⠁⠆⢷⢠⣴⣶⣶⣦⡀⠀⣗⠀⠀⢡⠈⠀⠀⠘⣶⠁⠀⠀⢱⠠⣿⣿⠙⠑⢳
⠀⣿⣿⣧⣌⣀⡏⠀⠀⠈⡷⠇⠀⠀⠘⡿⣧⠀⠀⢸⠈⢺⣿⣿⣿⣿⡇⠀⡿⠀⠀⠀⣶⠻⡀⠀⡿⡋⠀⠀⡙⢸⣻⣿⡦⠄⠀⡇
⠀⢸⡿⠃⡀⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠳⢿⣁⣠⣞⠀⠀⠙⠿⢿⣛⢀⡔⢻⡀⠐⢇⠀⢥⡀⠀⣴⠘⠒⠀⢠⠀⠀⠛⠛⢦⣼
⠀⠀⡧⣤⣭⠿⡇⠀⠀⠀⠀⠀⠀⠀⠸⡭⠭⠭⣭⣽⡇⣼⡁⠀⠀⠀⠈⢧⠀⠉⠀⠈⡇⠀⠀⠈⠙⠀⡀⠀⠉⢰⡆
⠀⠐⠀⠀⢸⡄⢇⠀⠀⠀⠀⠀⠀⠀⢸⢯⣽⣿⡯⣛⣿⣿⠀⠀⠀⠘⣄⡴⠀⠀⠀⠀⣇⠀⠀⠀⠀⠄⠄⠀⠀⢞⡤⢀⠀⠀⠀⡄
⠀⠀⡀⠀⠈⠁⠰⠦⣤⣀⣀⣀⣠⠤⠚⣼⠀⢸⣿⣷⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⠄⡠⠠⠀⣠⣀⣀⣴⠎⣀⠀⢀⠠⣤
⠀⠸⠤⣤⣤⡤⠂⠀⢨⡉⠀⠋⡅⠀⢠⣿⣦⣼⣿⣿⡿⠉⠏⠉⠉⠁⢿⡼⠦⣤⡴⡏⠀⠀⠃⠀⠀⠉⠈⠁⠀⠈⠉
⠀⠀⠀⣻⠀⣇⢀⡴⢾⣧⢰⣤⣇⠀⠈⠙⡟⡟⢻⠋⠁⠀⢁⠀⠇⠀⢰⢀⣀⢠⡤⠃⢀⠀⠀⢀⠀⠀⣀⣀⠀⣀⠀⠀⠀⠁
⠀⠀⠈⠿⠟⠚⠓⠀⠾⢿⣿⣟⠿⠿⠦⠀⣿⣿⣽⣀⠀⠀⢠⠀⡄⠀⢤⣄⡄⠸⢤⡀⢋⣿⣷⣬⣑⣢⠜⠳⠃⠻⠛⠘⠀⠪⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⢿⢊⡛⠚⡳⠄⠛⠖⢿⡤⠚⠉⠀⠀⠉
⠀⢀⠔⠁
⠔⠁
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢀⣴⣶⣤
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣉⡁⠉⠻⣿⣿⣿⠇⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿⣆⢈⡷⡻⢌⢍⣇⡀⠀⡂⣄⡀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠙⠛⠋⣠⣷⡊⣽⡉⢤⠂⠑⡀⢄⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠉⣿⣿⣯⠋⢣⠘⡀⢀⠘⢢⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠌⠀⠈⠢⠀⠀⢠⠄
⠀⢀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠉⢐⠞⠀⠀⢰⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⣀⣀⠀⡠⠀⠀⠀⠀⠁⠀⠀⠀⡈⣠⡞
⠀⠀⠀⢀⠀⠠⠂⠀⠀⡀⠀⠀⠀⠀⠀⢨⠀⠀⠀⡠⠈⠀⠀⠀⠀⠆⠀⠀⠀⢀⠋⠉⢱⢄
⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠁⣑⠀⠔⠀⠀⠂⡚⠀⠀⠀⠀⠘⠀⠀⠂⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠊⠐⠚⠈⠀⠀⠐⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠠⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⣶⣶⠶⠶⠶⠶⠶⢶⣶⣶⣶⣶⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠶⣶
⠀⠀⠀⠀⠀⠀⠀⣿⣭⣤⣀⣀⡀⠀⣾⣿⣿⣿⣿⣿⣿⣎⠛⠿⣿⣿⣿⡿⠿⠲⢦⣤⣿
⠀⠀⠀⠀⠀⠀⠀⣿⢠⠞⠁⠀⣇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣠⣾⣿⠟⠁⠀⢀⣀⠘⠛⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⠀⣠⣼⡶⡟⣻⣿⣿⣿⣿⠟⣿⣿⣿⠃⠀⠀⣰⣿⡟⠀⣤⣿
⠀⠀⠀⠀⠀⠀⠀⣿⢀⣰⣞⠿⡿⠈⠁⠀⠀⠹⣿⣄⠜⠋⠉⠛⠀⠀⢠⣿⡟⢁⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡿⠉⠀⢀⣀⠀⣶⡄⠀⠀⢹⣿⡆⠀⠐⠂⠀⠀⢴⣶⣯⡚⠟⠁⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⡆⢿⣿⠀⠀⢸⡿⢿⠀⠀⠐⣄⠀⠈⠙⠻⠟⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⢻⡷⢘⣛⡀⠀⠀⢀⠀⢱⡀⠀⠿⢷⣦⣤⣀⣀⣀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡧⠀⠀⠀⠾⣿⠿⠃⠀⢀⠀⠀⣸⣿⣆⠂⡈⠈⢻⠟⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⢿⠀⠠⠂⣀⠀⣀⣀⣤⣶⠇⣾⣿⣿⣿⣿⣷⣮⣭⣶⡾⠒⠒⠚⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠈⠣⣀⣀⢄⠙⠋⠉⠁⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣤⡀⢸⡏⠀⣿⣷⣶⡶⢾⣿⣿⣿⡇⠉⢛⣿⣿⣿⠋⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣽⡟⢿⡿⠆⠈⣻⣿⠀⢠⣿⣿⣿⡇⣤⣾⣿⣿⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠿⠿⠷⠶⠿⠶⠾⠶⠿⠶⠾⠿⠿⠿⠿⠿⠿⠿⠾⠶⠶⠶⠶⠶⠶⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⣀⣠⣤⠄⢀⣠⣤⡄⠉⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡶⢄⣀⣀⣀⣀⡀⠈⠋⣀⣠⣬⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⠀⠈⢁⡀⣩⣿⣟⠋⠙⠀⠀⠛⠻⠿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠀⢠⣧⣿⣷⡿⠟⠛⣛⣂⣈⣁⣀⡀⠀⠀⠀⣄⠀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠑⠾⡿⠿⠟⠛⠛⠋⠉⠙⠿⠛⠋⠉⠀⠀⠀⠙⠀⠉⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢁⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣼⣄⠒⣢⠦⠀⢀⣴⡾⠃⠐⢢⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⢋⣽⣿⡶⠀⣀⣉⢁⣄⡞⠿⠟⠁⣀⣶⣤⡀

⠀⠀⠀⠀⠀⣠⣴⣿⣿⡋⠀⠀⢀⣠⣴⣶⣤⣄⠀⢀⢀⣀⣀⣀⡀⠀⠀⠐⠷⣤⡶⠀⠀⠀⢀⣤⣄⡀⠀⣠⣶⣿⣿⣷⣦⡀⠀⢀⠔⡀
⠀⠀⠀⠀⠈⠙⢿⡿⣋⣀⡀⢠⣿⣿⣿⣿⣯⣿⣆⠸⡸⠟⣋⣀⣀⠀⠸⣦⡞⢿⣇⠀⠀⠀⠸⣿⣿⠇⢰⣿⣿⣿⣿⣿⣿⣷⠀⠊⠺⠿⢦⡀
⠀⢠⣾⢻⣟⡛⣮⠑⣿⣿⣿⠸⣿⠻⠿⣿⠯⢻⡿⠀⣰⣾⡟⢉⡻⣿⣦⠙⠓⠟⠛⠀⣀⣤⣤⣤⣄⡀⠘⣿⣿⣙⣉⣿⣿⡏⠀⣎⣤⢒⠀⠉
⠀⣿⣿⣼⡟⣽⣿⡇⢉⣉⣁⡀⠙⠿⣿⣿⡿⠟⠁⠀⣿⣿⣾⣿⣷⣿⣿⡆⢸⠷⢠⣾⣿⠿⣿⡿⢿⣿⣆⠈⠛⠿⠿⢛⡋⠀⠢⠯⣢⣏⠳⢀⣠
⠀⠘⢿⣿⣿⡿⢟⣴⡿⠿⠿⢿⣦⠀⠀⠐⢶⣠⡦⠀⠻⣋⣀⣉⣀⣉⡿⠀⣠⠄⣾⣿⠀⢀⠀⣀⠀⢸⣿⡆⠀⣰⡿⠿⠿⣷⡄⠚⠁⠀⠀⠸⣿
⠀⣠⣴⣦⣄⡀⢸⣷⡄⠀⠀⣰⣾⡇⠸⣦⡞⣿⡁⠀⠀⠈⠛⠛⠛⠉⠀⣿⣿⡇⢹⣿⠀⠈⠀⠈⠀⢸⣿⠁⠀⢿⣆⠀⠀⢸⡟⠀⣠⣶⠛⠓⣦
⣾⡟⠷⠶⢻⣷⠈⢿⣧⣿⣿⣿⡟⠁⠀⠙⠓⠛⠃⠀⠀⣠⣾⡿⣿⡿⣷⣮⡉⠀⠀⠙⠿⣿⣿⣿⡿⠟⣡⣴⠶⢾⣟⠿⠿⢛⣠⠀⣿⣸⠀⠀⣿
⢻⣇⣠⣄⣸⡿⠀⣀⡈⠉⠉⠁⣠⣾⣿⣿⡿⢦⡀⠀⣼⣿⡿⣷⠈⢰⣿⣿⣿⡄⠀⠀⣀⣀⣀⡀⠀⢰⣿⣤⣶⣦⣽⡇⢰⣿⣿⠇⠈⠛⠻⠛⠁
⠀⠉⠛⠛⠉⠀⣶⣿⣿⣶⡀⢰⣿⣿⡛⠉⣰⣿⣷⠀⣿⣿⣷⣌⠃⢈⣻⣿⣿⡇⢠⣾⣿⣿⣿⣿⣦⠈⢿⣁⣉⣈⡹⠃⠋⠛⠯
⠀⠀⠀⠀⠀⠀⠈⠛⢿⠟⠀⠈⢿⣧⣝⣾⣿⣿⠏⡀⠹⣿⣿⠛⠈⠛⣿⣿⡟⠀⢾⡿⣻⣿⣿⣻⣿⠀⢀⣬⣭⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢛⡛⠁⢼⡿⠂⠈⠙⠻⠿⠿⢛⠉⢀⡀⠈⠷⣶⣶⣶⡶⠏⠀⣿⡋⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣧⠀⣠⣶⣾⣷⣶⣄⠀⢄⢸⢞⢣⢤⡀⢠⣤⣍⠁⢀⣰⡆⠙⠳⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⣸⡟⠉⠉⠉⠉⢿⡆⠀⢍⡷⢛⠆⠁⠻⣿⡿⠀⠘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣅⣤⣄⣠⣤⣸⠃⣠⠀⡠⢀⣴⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠁⠀⠀⡄⠀⢸⣿⣎⣾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣧⠀⠈⠙⠿⠟⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠻⠿⣿⣿⣿⣿⠿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⢸⣿⣿⣶⣷⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣼⣄⣈⣉⣋⣉⣀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡷⠶⠟⠙⠶⢾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡀⢉⡉⠀⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡾⠁⢰⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠸⠀⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢀⣤⣄⣠⣶⣤⣀⠀⠀⣰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠟⠉⠉⠉⠁⠀⢽⣟⡛⠙⠻⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠲⣾⣷⠒⠛⠁⠀⠀⠀⠀⣻⣿⣆⠀⠀⢂⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠈⠣⠀⠈⢆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀⠛⠃⠀⠀⢐⡀⢸⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣶⣶⣟⠉⠉⠀⠀⢈⣀⣀⠀⠀⠀⠀⠙⣷⠾⣧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣤⣤⣤⠀⠘⠻⠉⠉⠀⠀⠀⠀⠻⢴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⣠⠇⠀⠂⠐⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⡟⠀⠀⠀⠀⠀⠀⠀⠀⡴⠈⠀⠀⠀⠈⠀⠀⠻⠮⠁⠀⠀⣀⣤⣾⠿⠋⠁⠀⠀⠀⠀⠀⣠⡶⠛⠛⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣽⣦⠀⠰⣾⡦⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣸⠗⠀⠀⢠⡾⠛⠋⠁⠀⠀⠀⠀⠀⠀⢰⣾⠋⠀⠀⠀⠀⣀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠘⠋⡤⠰⠺⠗⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡰⠛⠁⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠠⠤⣠⣤⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠰⣯⣳⣄⣨⣦⢤⣞⠹⠆⠀⠀⠀⠀⠀⠀⡀⢀⣬⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠩⠭⡥⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⠀⠀⠀⠀⠀⠀⠀⡰⠃⢸⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⡀⠀⠀⠀⠀⢠⣆⠀⣹⣿⡟⢿⣷⣦⣀⣀⣀⣀⣤⣤⣾⣿⣿⣿⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⢻⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠈⠳⣿⠹⠀⠰⣿⣿⣿⣿⣿⣿⢭⣿⢛⣿⣿⣿⣧⡀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢿⣤⣄⣴⣿⣿⡉⠁⣁⠀⠀⢉⣱⡿⣿⣿⣿⣿⣦⣄⣻⣷⡀⠀⢀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠈⠛⠿⣿⣿⣿⣷⣴⣦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⡶⠁⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⡴⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⠏⠀⠡⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⣴⠚⠋⠙⠿⣷⣶⣤⡍⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣾⣿⣿⣟⠿⠿⠖⠂⠀⠀⠀⠀⢣⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢰⣿⡿⠛⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣏⣦⠁⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠿⠓⠀⠀⢀⣠⠟⠋⠀⠀⠀⠀⠓⠀⠀⠶⠟⣓⡀⠀⠀⠀⠻⠿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡟⠛⠷
⠀⠀⠀⠀⠀⠀⣠⣿⣿⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣏⠁
⠀⠀⠀⠀⣠⣾⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢫⣿⣿⢟
⠀⠀⠀⣰⠟⠻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠴⢶⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢀⣞⠁⠀⠀⠸⡭⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣀⣤⣿⣿⣿⡿⣿⣿⣧
⣀⡞⠉⠣⠀⠀⠀⠀⠈⠓⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⣒⣽⠟⠉⠋⠉⠠⠛⠂⡙⢷⣄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠋⠉⠳⣄⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⡇⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡷⢤⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⡠⠚⠛⠛⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⠉⠉⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢾⠀⠀⠀⠀⠀⢀⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠘⢿⡿⠃⠀⠀⢀⣀⣼⣄⣀⣀⣧⣀⣀⣸⣇⣀⣀⣼⣀⡀⠀⠀⠀⠈⠻⡿⠃⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⢸⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⡇⠀⠐⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠁⠀⠈⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⡇⠁⠉⠉⠁⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣅⣀⣀⣀⣀⣀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡷⠤⠤⠤⠤⠤⠤⠤⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡧⠤⠤⠤⠤⠤⢴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⣉⣉⣉⣉⣉⣉⣡⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠒⠒⠒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠒⠒⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣯⡀⠀⠀⠀⠀⠙⠿⢿⠿⠿⠿⡿⠿⠿⢿⡿⠿⠿⢿⠿⠟⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⣤⣶⣄⠀⠀⠈⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⣠⠞⠀⠀⣤⣷⣄⠀⢸⡇⠀⠀⠈⠉⠉⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠗⠒⠚⠁⠀⠀⠀⠿⣿⠟⠀⠘⢧⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣟⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⡿⠛⠉⠉⠙⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢻⡇⢀⣄⠀⠀⣰⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⢩⣿⡿⠃⠀⠀⠀⠀⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠛⠃⠀⠀⠈⠀⠜⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⢠⡞⠀⠀⠀⠌⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣿⣀⣀⡠⢻⣿⠂⠀⢠⣄⣀⠘⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⣿⢿⠻⠶⢿⡿⣇⣀⢰⡋⠉⠙⠗⠀⠀⠀⠀⢸⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢻⡆⠀⠐⣿⣿⣿⡿⠉⠳⣄⠀⠀⠀⡀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⡇⠀⠀⢸⣿⣿⢇⡀⠀⠀⠉⠀⠈⢷⠀⠀⢀⠠⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠃⠸⠀⠀⠴⠋⠁⠀⠙⢦⡀⠀⠀⠀⠀⠐⠂⠁⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠂⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠑⠄⠠⡤⠄⠁⠀⠀⠀⠀⠀⢠⢳⡀⠣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠁⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡇⠀⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠘⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣠⣄⠁⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⣿⣄⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⣆⣀⣤⠀⢂⡔⠛⠚⠤⢌⡱⠐⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠑⠠⢤⡆⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⠀⠀⢼⡖⣶⠲⠋⠀⠀⠀⠀⢺⠀⡴⠂⠀⠉⠁⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠆⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⠀⠻⡆⠹⣆⠀⠀⠀⠀⠸⡄⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠉⠆⠀⠀⠀⠀⠀⠸⣿⢻⣇⣧⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣁⠀⠀⠀⠀⠀⠀⠀⠴⣿⠏⠀⠓⢋⠉⣀⠈⠑⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠙⠻⣶⡶⠶⠞⠁⢀⡾⠁⠀⠀⠀⣰⡇⠌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⠂⠀⠀⠉⠠⠤⠖⠋⠀⠀⡠⠤⢐⡵⠏⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡤⠤⠄⠐⠀⠀⠆⠠⡇⠀⠀⠀⢠⣤⡤⠤⠀⠰⠂⠃⠀⠀⢀⡴⠦⣶⡈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠄⠀⠈⢀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢠⡉⠁⡀⠀⠀⣻⠊⠀⠄⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢼⣷⡄⣠⠃⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣀⣠⣜⣷⣷⣶⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉





⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠉⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣄⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠒⠀⡨⠋⢵⣶⣶⣝⠋
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣾⣶⣭⣵⡌⢙⣆⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠞⠛⣿⣿⣿⣿⣿⡁⢰⣮⡅⠀⠛⠁⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠿⠙⠂⠀⠀⠀⠀⠐⢆⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⢿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠐⣿⣭⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣶⣶⣦⣄⡀⠀⠀⠀⠀⠈⠉⢹⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡟⠛⠛⠛⠋⠤⠀⢒⣀⡉⠀⠈⠀⠀⠈⣀⡒⠂⠤⠄⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⡩⠤⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⣁⠒⠀⠠
⠀⠀⠀⠀⠀⠀⣠⣿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⢒⡀⠉⠀⠀⠀⠀⠀⠀⠈⡅⣒⠐⠌⠤
⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡀⠩⠤⠐⠂⡀⠀⠀⠀⠀⠀⡐⢒⣰⣸⣀⣃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡆⠀⠀⠀⠀⣒⣉⠭⠄⠀⠀⠀⠀⠀⠀⠀⠽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠠⠄⠒⣒⠈⠉⠀⠀⠀⠀⠁⠉⣐⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠃⣀⣀⣀⠐⠈⠉⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⢹⣿⣿⣿⣃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⣀⡄⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⢿⣿⠁⠀⠀⠀⠀⠀⢀⣠⣀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠼⣻⣿⣿⣿⣿⣽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣯⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣟⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⣾⣶⣤⣀⡀⠀⠀⠀⣽⣿⣿⣿⣿⣿⡟
⠀⠀⣀⣄⣀⠠⡀⠀⠀⠀⣙⣻⣿⡿⡻⠋⠉⠙⠳⠖⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⣀⠠⠈⢻⣿⣿⣿⡷⠶⢶⣿⡿⠿⡿⠿⠋⠀⠀⠀⠀⢀
⠀⣾⣿⣿⣿⣷⣏⣧⣀⣾⣿⣿⣿⣏⡇⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⣿⠀⠀⠀⠈⣿⠟⠁⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢾⣧⠀⣿⡄⠀⠀⢀⡀⠀⠀⠀⢸⡇⠸⡿⠀⠀⠀⠀⠉⠀⠀⢉⠀⠀⠀⠀⠆⠀⠀⠈⠤
⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠿⢽⡄⠀⠀⠘⢿⡆⠹⠧⠂⢰⣿⣿⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆
⠀⠺⢿⣟⣿⡿⠟⢹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⢀⣤⣀⡠⠒⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⡀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣀⣾⠈⠀⠀⣰⣦⣤⣤⣤⣴⣶⣶⡄⠁
⠀⠀⠠⣴⣶⣶⣶⣶⣶⣿⣦⡀⠀⠏⠙⣿⣶⣤⣤⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⣛⡋⣠⣴⣶⣿⣿⣿⣽⡿⢿⠛⠛⠃⠚⠒
⠀⠀⠀⢹⡿⠿⠛⠛⠙⠛⠛⢿⣿⣮⣀⠀⠀⣛⡛⣵⡄⠀⠀⠀⠀⠀⠠⠀⠒⠀⠀⠀⠀⢀⣠⣴⣿⣿⡿⣳⣿⣿⣿⣿⣿⣿⠟⠀⠀⠱⡄
⠀⠀⢀⠂⠀⠀⡠⠄⠀⠉⣽⣿⣿⣿⣿⣿⣿⣟⣾⣿⣿⣦⣀⢀⠄⠐⠀⣀⠠⢀⣤⣴⣾⣿⣿⣯⠟⢫⠎⠙⠿⠿⠿⠟⠋⠁
⠀⠀⠨⡀⠀⠜⠀⠀⢀⡀⠘⠿⣿⣿⣿⣿⣿⣿⡟⠙⢿⡏⡠⠀⠀⠠⢀⠀⠀⠀⢙⡿⠿⠟⠋
⠀⡀⠀⠀⢰⠀⠀⠐⠁⠀⠀⠀⠀⣉⠉⠉⠉⠁⠱⠀⠈⢧⡀⠀⠀⠐⠁⠊⠀⠀⠀⠀⠄⠀⠆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠡⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠉⢢⠀⠆
⠀⠀⣀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠄⡑⠀⠀⠘⠄⠀⠀⠀⠆⠀⠰⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⣾⣿⣿⡿⠆⡀⠀⡤⢀⡀⠀⠀⠀⠀⠀⣠⣶⣦⡝⠀⠀⠀⢀⣀⣁⠀⠀⠀⠀⠀⠀⠠⢄⠂⠀⣤⣤⣤⣄⣂⣀⣀⠀⠀⠀⣤⣴⣶
⠀⠱⣿⣿⣿⣦⣌⠈⣠⠊⠀⠑⢄⢀⠀⠈⠿⣿⣿⣷⡄⠠⠊⠁⠀⠀⠈⠓⠄⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⡏
⠀⠒⠛⠻⢿⡿⠟⠀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠃⠀⠀⠀⠈⠉⠁⠢⡄⠀⠀⠀⢸⠀⠈⣿⣿⣿⡟⢻⠟⠃⠰⣎⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠹⡿⠟⡡⠂⠀⠈⠰⣿⣿⣿⣿⣧⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠈⢆⠀⠀⠈⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⡀
⠀⣀⣀⣀⣠⡤⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⠀⠀⠀⠀⠀⣀⡀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠀⠀⠀⠀⢀⠁⠀⠙⢿⣿⣿⣿⡷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⢀⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠂⢄⠉⠛⠙⠁⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠤⠀⠰⠃

⠀⢠⡀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⢀⡤⢠⣀⠀⠀⠀⠀⠀⡀⢀⠀⢀⠀⡀⠀⠀⠀⠀⢄⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⣷⡀⠀⠀⠀⠀⠀⡨⠟⠓⢤⣰⣶⠀⢰⠁⠀⠀⠈⢃⠀⠀⠀⢀⠄⠠⡀⢈⠀⠁⠀⠀⠠⠀⠀⠢⡑⡀⠀⠈⢂⢀⠐⡄
⠀⠀⢿⠹⣆⠀⠀⠀⡜⢀⡀⡈⠀⢹⠉⠀⠘⣄⠀⠀⢀⡆⠀⠀⠀⣵⣶⣶⣶⠃⠀⠐⠀⡀⠀⠐⠀⠀⠑⡘⢄⠀⠨⠈⠀⠐⠄
⠀⠀⢸⡄⠘⢿⠀⠀⣹⣄⠀⠀⢀⡜⠀⠀⠀⠈⠉⠉⠁⠹⣷⣄⠀⠙⡿⢿⣯⣿⣿⣿⣾⠀⠘⠀⠀⠀⠀⠈⢄⠒⠁⡄⠀⠀⢈
⠀⠀⠸⠇⠀⠀⠰⠀⠀⠈⠉⠛⠁⠀⠀⠀⠀⡄⢠⡶⠀⠀⠈⠛⠃⠀⠀⠀⠂⠉⠉⠉⢣⣄⠀⠀⠀⠀⠀⠀⠀⠢⡀⠡⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠠⢀⣀⣼⣥⣤⠀⠀⢀⣴⣶⣶⣦⣤⠄⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⢀⣀⠀⠀⠀⠀⠀⣼⠋⠙⡇⠀⠀⠀⠀⢿⣿⣿⣿⣿⠇⠰⢿⣿⣷⣿⣿⡁⠀⡀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠘⢾⣿⣿⣶⣲⠀⠘⠳⢾⡷⠾⢶⣄⠀⠈⠻⠿⠿⠋⠀⠀⢰⣿⣿⣿⠿⠗⠀⠧⠰⡷⠀⠀⠰⠆⠀⠀⠠⠊⠀⠀⠀⠀⠑
⠀⠀⠀⡽⠉⠉⠀⠀⠀⠀⠸⣇⡀⣠⡟⠀⠀⢀⠀⠖⡦⢄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠁
⠀⢠⣴⣧⠀⠀⣤⠴⢶⡀⠀⠉⠛⠉⠀⢠⣾⣦⣤⡀⠀⠐⢳⡀⢀⠠⢀⣸⠀⣆⡀⠤⡀⠀⠀⠀⠠⢢⠀⢃⠀⠀⠀⠂⠢⢀⠆
⠀⠀⠀⠉⠀⠀⠿⣤⡼⢋⡀⠀⠀⠀⠀⠈⣿⣿⣿⡇⢠⣴⡀⡇⠀⠑⢤⣼⣿⣧⣤⠊⠀⠀⠠⠀⠀⠀⠀⠀⠁⠂⠄⠠⠐⠁
⠀⢤⣦⡀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⠀⠈⢄⠙⢿⡄⠀⡿⡱⠁⠀⣊⠀⠼⠓⡧⣤⣑⠄⠀⠁⠀⠸⠁⠀⠀⠀⢀⣴⣤⡀
⠀⠈⠻⡟⠀⠀⠀⢲⣿⣿⡇⠀⠀⠀⢀⠀⠀⠉⠦⢦⡔⠊⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠂⠀⡠⠤⠀⠀⠀⠀⠉⠛⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⢀⠀⠀⠀⠀⠀⠀⠶⠷⠒⠀⠀⠶⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠂⠂⠁⠀⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⢠⣆⠀⢀⡖⣶⠆⣦⡀⠀⠀⠀⠀⠀⠀⣤⠀⣀⣶⣶⡄⠀⠀⠀⠀⠀⠀⢀⠀⠈⠄⠀⠀⠀⠀⠀⠉⠛
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠯⠽⠀⠒⠀⠀⠀⠀⠀⠰⢿⡟⢩⠧⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⡞⠛⠛⠛⠛⠛⠛⠉⠛⣷
⠀⠀⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⡀⠠⠀⢁⡀⠠⡄⠀⠀⠀⠀⠀⠦⣤⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣼⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠘⠀⠀⠈⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢛⣛⣻⣟⣛⠛⠛⠋
⠀⠀⠀⠉⠉⠙⠓⠒⠦⠤⢄⣼⠁⠀⠀⡒⠉⠙⣶⣄⠀⣤⠘⠁⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣌⠉⠉⢉⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣼⣿⡖⠁⠙⠟⢣⠀⠀⠀⣴⠟⠁⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⢀⡴⠋⣹⠆
⠀⠀⠀⠀⡠⠄⠀⢀⡀⢰⠡⠀⠀⢇⡘⠉⠑⣤⣤⡄⡘⠀⢰⠟⠀⠀⠀⠀⠀⠶⠆⠀⠀⣾⣿⡧⠀⠀⠄⠐⣏⣠⠞⡡⣪⣶⣄
⠀⠀⠀⠀⠠⣰⡀⠘⠀⠀⠀⡀⠂⠈⠻⣦⠀⠛⡿⠋⠀⠀⠀⠀⠀⠠⡶⢠⡜⠛⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠁⢻⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢰⡆⠀⠀⢸⣿⢸⣿⠀⠀⠀⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⣿⣴⡆⣿⣿⢸⣿⣿⣶⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠷⣄⣴⣿⣿⣸⣿⣿⡷⠶⠾⢸⣿⣿⣿⣿⣿⣿⣿⣶⣤⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡟⢿⣏⣽⣿⣯⠿⠟⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣰⣿⣼⡟⣿⣩⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣷⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠿⠿⠿⣺⣧⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣹⣿⣻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣯⣷⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠰⣷⡖⣻⣿⡇⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⡇⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢿⡇⣿⣹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⡖⣶⣼⡏⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠹⣿⣶⣯⣿⣙⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡿⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶⣶⢸⣏⣻⣿⣟⣶⣦⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⡶⠛⠸⣿⣿⢹⣿⣿⡷⠶⢶⢸⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⣿⠻⠇⣿⣿⢸⣿⣿⡿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠸⠃⠀⠀⢸⣿⢸⣿⠀⠀⠈⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣋⡿⠓⠢⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠀⠂⠐⠒⠢⠄⠀⠀⠀⠀⠀⣠⣿⣿⣿⣾⣿⣤⣀⣦⡄⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣾⣏⢀⣄⣴⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣾⣿⣿⣯⣄⡀⢨⡀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⢸⠃⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⣻⡟⠁⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠆⠀⢠⠀⠀⡀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠀⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠡⡀⠀⢸⣷⠀⠀⠀⠀⠁⠀⠀⣀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢄⡀⠳⢴⣷⠈⠈⣿⠇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣮⣶⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣀⠠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣡⠞⠙⣄⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⠤⠤⠤⠤⠔⣺⣿⣿⣿⣴⣾⣾⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠂⠠⢤⠒⠉⠉⠁⠒⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠉⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⢿⣿⣿⣿⣿⣿⡇⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠘⡄⢀⢒⣀⡀⠀⠀⠀⢀⠴⠉⠀⠀⠈⠙⠻⣿⣿⣿⣇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠟⢻⣿⣿⣿⣧⣄⡶⠋⠔⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⡏⠉⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⢀⡴⣯⣴⣾⣿⣿⣿⣿⣿⠛⠻⢄⡸⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣴⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⠄⠁⠁⢠⣾⣿⣿⣿⣿⣿⠿⢿⠿⣿⡟⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⢸⣽⣿⣿⣿⡇⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠁⠐⠀⠀⠸⣿⣿⣿⢿⠿⣶⡾⢻⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠋⢁⠤⠵⠶⠖⠒⠶⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠄⡀⠈⠉⣁⠾⠤⠤⠔⠁⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠀⠀⠀⠀⡠⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⣀⠀⠀⠀⣀⠤⠊⠓⠒⠚⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠂⠀⠀⠒⠂⠑⢄⠀⠀⠀⠀⠀⠀⠈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠢⠤⠤⠖⠃⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⢀⣤⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠾⣿⣿⣿⣿⣿⣷⣄⡀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠉⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠈⣇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⢀⢀⡰⠀⣶⠀⠀⠀⢸⡿⠛⠉⠁⠀⠉⠻⣿⡟⢿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡼⠾⠧⠑⠋⠀⠀⠀⠈⠀⠀⢤⠄⠀⠀⠀⢸⠇⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⡅⣰⣖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⣀⠀⠀⠀⢀⣠⣴⣾⠉⠴⣡⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠛⠛⠻⠿⣿⣷⣶⣶⣶⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡁⠀⡠⠊⢳⠀⠀⠹⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠐⠀⠎⠀⢠⠏⠀⠠⣖⠻⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠂⠐⠅⠀⠀⠠⠆⠑⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⡁⠀⣱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠓⢄⡀⢀⠀⠀⠀⠀⠐⢤⡿⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⢣⠀⠣⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⣠⠚⠉⠀⠀⠀⠈⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣤⠤⢤⣤⣤⣶⣍⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⢀⣘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠘⣿⣿⣿⣦⠀⢻⣿⣿⣿⣯⡛⠛⡟⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠁⠀⠀⠀⠉⠑⠲⣎⠉⠻⠿⠿⠏⠲⡝⣿⣿⣿⠋⢩⡇⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠤⠤⠤⠔⠙⢮⢻⠃⡴⠋⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣴⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢢⠀⠀⢀⡜⠁⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⢤⣤⣀⣀⣀⡤⠤⠔⠒⠉⠁⠀⠀⠈⢦⡀⠎⠉⠁⢀⣡⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠀⠂⠀⠀⢺⣿⣿⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠀⠀⠀⠀⠁⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠐⠀⠀⠀⠀⣼⣇⣴⣾⣿⣿⡆⢂⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⣰⣷⡀⢀⡄⢸⣿⣿⣿⣿⣿⡆⢾⡆⠈⡀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠁⠀⠀⣿⣿⣧⣼⣷⣿⣿⣿⣿⣿⣿⣷⣾⣇⢰⣷⠀⡀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⢀⣿⣆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢱
⠀⠀⠀⣠⠀⠀⠐⣧⠀⠀⠐⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣍⡉⠉⠛⣛⣛⣛⣛⡛⣛⣛⣛⡓⠀⠀⠀⠀⡄
⠀⢠⡟⠁⣀⣀⠀⠹⣧⠀⢀⣭⣭⣭⣭⣭⣭⣭⣭⡅⣄⠉⠛⢷⣦⡉⣿⣿⣋⣽⣯⣭⣭⣅⠀⠀⠀⠀⠁
⢀⣿⠀⣾⠋⣿⠀⢠⣿⢠⣾⡉⣽⢽⣭⣽⡯⢉⣩⣵⣿⣤⣤⣄⠈⢿⣎⣍⢉⣬⣭⡯⢍⣍⢰⣆⢤⣴⣦⡄⡤⢀⣤
⠸⣧⡀⣿⡾⠃⣀⣾⢿⣿⢿⣿⣇⣤⣿⡏⠁⠻⣄⣸⣿⠀⠈⠛⢁⣼⡿⣿⢨⣿⣿⣳⣼⡟⣷⣿⣾⣟⣓⢼⣿⣟⣽
⠀⠈⠉⠉⠻⠿⠛⠁⠘⠃⠀⠉⠉⠁⢿⠇⠀⠀⠀⠙⠿⠷⠶⠟⠛⠉⠀⠋⠀⠉⠉⠉⠈⠁⠈⠁⠙⠋⠁⣾⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣙⠆⠀⡟⠀⢰⠊⠙⠂⠀⢻⠛⠀⠘⡇⢸⠀⠀⡟⢱⠀⠀⣿⡙⠀⠰⢏⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠓⠀⠈⠒⠒⠀⠀⠘⠂⠀⠀⠓⠚⠀⠀⠓⠑⠀⠀⠓⠒⠀⠐⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠛⠛⠛⠛⠛⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢴⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠶⠚⠋⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⢿⣿⣿⣶⣦⣤⠀⣠⡴⠋
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⡿⢁⠞⢋⣽⣿⣿⡿⠛⠉
⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⠛⢷⣄⠀⠀⠀⠀⠀⠀⢰⡿⠛⠋⠙⢿⡍⠉⠁⠀⠀⠀⠈⠰⢻⣿⠿⠶⢶⣄
⠀⢠⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠙⠳⠦⣤⣄⣀⣴⠋⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠙⣇
⢠⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⣠⠞⠉⠙⠢⡀⠈⠀⠀⠀⠀⠀⠞⢀⡤⠖⠲⢤⡀⠀⢹⡀
⣼⣿⡿⢻⣿⣿⠃⠀⠀⠀⠀⠀⠀⣷⣤⡴⡾⠁⡼⠃⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⢳⡀⠈⠻⣄⡀
⡿⠋⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⢹⡶⠂⠀⣸⠃⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢷⠀⠀⢈⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣇⡄⠀⡏⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢸⡆⠀⢠⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢸⡇⠀⠀⠀⠀⣴⣿⣦⣸⠀⠀⠀⠀⢸⣇⣤⣤⡀⠀⠀⠀⠀⡇⠀⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⢸⡇⠀⠀⠀⢸⣷⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣧⠀⠀⠀⠀⡇⠀⡿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠓⢦⡘⡇⠀⠀⠀⢸⣿⣿⣿⣿⢀⣀⣀⠀⢸⣿⣿⣿⣿⠀⠀⠀⢰⡇⢈⣛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠻⡷⠙⢷⡀⠀⠀⠈⣿⣿⠿⠛⠉⠉⠉⠉⠛⢿⣿⣿⠇⠀⠀⠀⣼⠖⣩⣤⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⠀⠉⠓⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠖⠛⠁⣠⣾⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠛⠀⠀⠀⠠⠔⠒⠚⠛⠋⠉⠙⠛⠛⠓⠒⠦⠤⠀⠀⠀⠀⠀⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣀⣀⣀⣤⣤⡴⠶⠶⠶⠶⢶⡶⠶⢶⣶⠶⠦⣤⣤⣀⣀⡀⠀⠀⠀⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣧⠘⣇⠀⠀⠀⠀⠈⢷⣀⡞⠁⠀⠀⠀⠀⣸⠏⣹⠟⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢿⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⢠⡏⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡈⠳⣤⣀⣀⣀⣀⣀⣀⣀⣴⠏⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣭⣭⣉⣉⣉⣉⣩⣤⡾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⢿⡿⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢻⠳⢤⣖⠉⠀⠠⡇⠀⠉⣲⠤⠚⡟⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠀⠀⢸⠀⠀⠈⢱⠀⠈⡇⠀⡏⠀⠀⠀⡇⠀⣤⣀⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⡇⠀⣿⣿⣿⣦⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⣇⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠛⠢⣄⡀⢸⠀⢀⡇⠀⡇⠀⣀⠼⠿⠧⡈⠙⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠔⠊⠀⠀⠀⠀⠀⢉⠿⠀⣨⣏⠰⣷⣉⠀⠀⠀⠀⠀⠑⠠⢄⠁⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⢁⠀⠀⠀⠀⢀⡠⠔⠊⠀⣠⠔⠻⠟⠢⣄⠈⠑⠢⣄⡀⠀⠀⠀⠀⡨⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠂⠤⠒⠁⢀⡠⠖⠉⠀⣀⢴⣦⣀⠀⠉⠲⢤⣀⠉⠲⠤⠔⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠄⠊⠁⢀⡠⠔⠈⠀⠐⡇⠈⠙⠢⢄⡀⠈⠑⠦⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⡉⠀⠀⢠⠂⠁⠀⠀⠀⠀⣴⣷⡀⠀⠀⠀⠉⠲⡄⠀⠀⢙⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢼⠀⠀⠀⡠⠔⠊⠉⠉⠙⠢⢄⡀⠀⠀⡧⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⢀⠠⠐⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⠤⣄⣀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢋⠀⠒⠶⠶⠤⣤⡀⠉⠳⣄⠀⢻⣧⠀⠀⠀⣠⠖⢋⣁⣤⠶⠒⢋⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⠀⠉⠷⢦⡘⡆⠀⢿⠀⣠⠞⢁⡴⠛⠉⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣄⣀⣀⣀⣙⣿⡀⣿⢰⢇⡼⠛⠉⠉⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢸⣾⠴⠶⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⠒⠛⠛⠋⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⠛⠓⠲⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣼⣇⣀⣀⣀⣀⠀⠀⠀⠀⢀⣤⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠙⠛⠛⠛⠛⠻⠶⠶⠶⠶⠿⠷⠶⠶⠶⠟⠛⠛⠛⠛⠉⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡏⠓⠲⠶⠶⠤⠤⢤⡤⡤⣤⣤⣤⣤⡤⠤⠶⠶⠒⠚⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢦⣤⣤⣀⣀⣀⣀⣠⡤⠤⠖⠋⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣦⣧⣶⣤⣶⣴⣯⣼⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⣇⠀⠀⠁⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣄⣠⢀⣀⡾⠠⠈⠑⢤⠖⢋⣉⣭⡕⠢⡀⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣛⣿⣿⣿⣀⠀⠀⠈⠒⢄⠀⡀⢻⡇⠀⢻⣇⣠⢧⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠳⡈⡌⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡿⠿⠙⠛⠛⠻⣿⣿⣦⡀⣰⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⢻⣿⠃⠀⢰⡐⡀⠀⠀⠀⠀⠘⠃⠀⢹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠐⠢⡀⠀⠇⢠⣾⣿⡏⠀⠀⠀⠀⠀⢀⡠⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⢀⣴⣾⡏⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢰⠀⠀⠀⠀⠙⠛⠁⣀⣀⠀⠀⣼⣿⡿⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⠙⠋⠁⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣀⣠⣤⣤⡙⠛⠻⠟⠛⢀⣠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣴⣶⣾⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠉⢻⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠙⠛⠿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢀⣿⣿⣿⣿⣷⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡇⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⠀⠀⣠⠊⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣆⣼⠁⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡿⠟⠛⠻⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣦⣤⣴⣶⣿⣿⣿⣿⡆⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⣽⣿⣿⣿⣿⣿⣄⣀⣠⣾⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣬⣭⣖⣛⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠫⠛⠿⠿⠯⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠛⠛⠛⠛⠛⢿⣷⠀⠀⠀⠀⠀⠐⠔⣤⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⢂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠓⠞⠋⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡀⠀⢀⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣅⣈⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⡀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⢉⣉⣉⣉⡁⢀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⠤⠤⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣿⣿⣿⠟⠛⠿⠿⠿⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⣿⣿⣿⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠈⠌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⡀⠀⠀⠀⠰⡆⠀⠀⠀⢀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢏⠀⡗⠀⠀⡇⠀⢸⠂⢸⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠳⡈⢢⠀⡇⡴⠋⡔⠋⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠘⠉⠉⠉⠳⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⡔⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⣀⣀⣀⣀⣀⣿⣿⣟⡍⣭⢹⣭⡍⣭⣩⣩⡝⣿⣿⣀⣀⣀⣀⣀⡤⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣄⣀⣀⣀⣀⣿⣿⣯⣥⣭⣬⣭⣥⣭⣭⣬⡅⣿⣿⣀⣀⣀⣀⣀⠔⠉⠉
⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠉⠉⠉⠉⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⠉⠉⠉⠉⠉⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⠉⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⠔⢁⠜⠀⡇⠳⣄⠣⡀⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣇⠀⡧⠀⠀⡇⠀⢸⠄⢹⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠁⠀⠀⠀⠰⠇⠀⠀⠀⠈⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣾⣿⣿⣿⣿⠿⠦⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢻⡦⠀⠈⠀⠉⠀⠀⠀⠀⠀⠉⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣿⣯⣿⠿⢀⣀⠀⠀⠀⠀⣀⣀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠄
⠀⠘⠀⠀⠀⠲⠂⠀⠀⠀⠀⠄⠉⠀⠁⠔⠀⠀⠀⠀⢸⣿⣿⣾⣉⣹⣷⡆⠐⠺⣭⡀⡄⠀⢻⠇⠀⠀⠀⠐⠄⠁⠉⠁⠄⠀⠀⠀⠀⠀⡔⠀⠀⠐
⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠄⢠⡤⠀⠆⠀⠀⠀⠀⠠⢻⣿⡍⠉⠁⣿⡇⠐⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠆⠠⣤⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠥⠀⠀⠀⠀⠀⠀⠀⠚⣿⣿⡀⡰⠫⢷⡀⠀⠑⢄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠭⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢰⣷⣿⠋⠕⠚⡤⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⣭⣉⣉⡡⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠒⠒⠱⠀⠀⠀⠀⠀⠀⢀⣄⠸⣿⣿⣤⡀⠀⠀⢀⣴⡟⠀⡄⠀⠀⠀⠀⠀⠰⠑⠒⠂⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡀⠙⠿⠿⠿⠿⠛⠉⠈⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⣿⣿⣿⣷⡄⠀⠀⢀⣤⡤⣀⠀⠀⠀⢸⡄⠀⣴⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢿⣾⠃⠛⡃⠀⠀⠀⠴⢿⣷⡄⡾⠓⠀⠀⠀⠳⠞⠛⠋⠁

⠀⠀⢀⣀⡈⠉⠉⠉⠉⠉⡛⠛⢛⣛⣛⣛⣛⣛⣿⣿⣿⡿⠿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣿⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⡧⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣴⠓⣾⣄⣴⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠿⠁⠀⣀⣠⣤⡀⠈⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡀⠀⣻⠿⠿⢿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣀⢿⠇⠀⠀⢰⠀⠀⢹⣻⣿⣿⣿⠁
⣿⣿⣿⣿⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣥⣤⣤⣶⣶⣿⣤⣤⣿⡝⡹⢓⡟
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢠⣶⣾⢿⡟⠟⠫⠍⠉⠉⣹⣿⣿⣿⣿⡒⠻⠁
⣿⣿⣿⣿⣤⣄⣀⠀⠀⢀⠄⠈⠁⠀⠈⢹⠉⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡿⠃⠀⠀⠀⣠⠴⠚⠉⠉⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣵⣴⠀⢤⣐⠒⠻⢿⣶⡆⠀⠀⣀⣴⠿⠟⠋⠀⠀⠀⠀⠀⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠙⠻⠿⣇⠀⠊⢁⣴⣾⠿⣩⣦⠀⠀⠀⠀⢀⣠⡼⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⠀⠂⢁⣠⣾⣿⠟⣡⠞⣿⡇⠙⠒⠛⠛⠋⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣶⣿⡿⢋⡴⠛⠡⠤⠘⠿⠁⠠⠌⣡⣶⣴⣶⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⠟⣫⡶⠋⡀⠀⠀⠀⠀⠀⠀⢀⡗⢿⣿⣿⣿⡧⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⣠⠞⠋⣾⣿⣿⣄⣀⣀⣀⡤⡔⠋⢠⣞⡋⠈⣯⣷⡈⠑⠉⡁

⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⠀⠀⠀⠀⠀⠸⣷⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠀⠀⠀⠀⠀⢸⠀⠀⠘⣧⡀⠀⠐⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⠀⠀⠀⣿⣏⡀⠛⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⡇⣸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠈⠛⢿⣿⣿⣷⠀⠀⠀⠀⠀⢸⡇
⠀⢸⠀⠀⠀⠈⠉⠻⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣷⣼⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⠀⠀⢸⡇
⠀⢸⠀⠀⠀⠀⠀⢀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣿⣿⣿⣧⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⣿⣿⠿⣇⣾⣤⡦⠀⢸⡇
⠀⢸⠀⠀⠀⠀⠀⠀⣽⣿⡿⠀⠀⠀⠀⠀⠀⠀⠠⣶⣾⣿⣿⣿⣿⣿⣿⣷⣧⠀⢸⠀⠀⠀⠀⠀⣿⡿⣿⠀⢻⣿⢧⠀⠀⢸⡇
⠀⢸⠀⠀⢀⣤⣤⣴⣿⣿⣶⣶⡄⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⠿⢿⣷⡿⢿⡇⠀⢸⠀⠀⠀⠀⢀⣿⠀⣿⠀⢸⣿⡇⠀⠀⢸⡇
⠀⢸⠀⠀⠈⠻⠿⠿⠋⠙⠛⠋⠀⠀⠀⠀⠀⠀⠐⠿⠇⠿⠿⠃⠸⠿⠷⠸⠿⠀⢸⠀⠀⠀⠀⠾⠟⠀⠟⠀⠴⠿⡧⠀⠀⢸⡇
⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣉⣉⣉⣉⣁⡀⠀⠀⢀⣀⣠⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠉⠉⠉⠉⠉⠉⠉⢱⣲⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣀⣀⣁⣀⣀⣀⡀⠀⠀⣀⢀⣀⣈⡉⠁⣄⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣷⣤⣀⢠⡇⣿⢸⢸⡇⣏⣿⢹⡟⡿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣯⣟⠁⠸⣷⣿⣾⣾⣧⣿⣼⣼⣇⣷⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠛⠋⠙⠛⠛⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠀⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣠⣀⣠⣤⣮⣴⣧⣿⣿⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠸⠀⠰⠿⠿⢿⠿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⢹⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠩⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠉⠙⢻⣛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⠇⠀⠀⠀⠰⣏⣀⡼⠋⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠯⠬⠤⠈⠀⠀⠀⢠⠏⣭⠇⠀⠀⠈⠛⠦⣄⠤⡐⠒⠛⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁⠚⠀⠀⠀⠀⠀⢸⣿⢛⣿⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠒⠒⠀⢀⣾⡿⠟⠛⢻⣷⣾⣿⠛⠛⠻⢿⣿⣿⠻⡄⠈⠠⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣴⣿⣷⣶⣒⡒⠀⢀⣾⣇⡀⠀⠀⠈⠋⠉⠁⠀⠀⠈⠻⢿⣿⡇⠹⣆⠀⠁⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣞⠻⠿⣿⣛⣃⣰⣶⣦⣾⣛⣛⡁⠀⠀⠄⠀⠀⠀⣀⣴⣤⣤⣿⡏⡏⣿⣏⠀⠀⢀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⠛⢻⣿⡿⢿⣿⡾⠿⠿⠥⠀⠀⠀⠀⠀⠀⣲⣿⣿⣿⣿⡇⡿⡿⡏⣿⣧⠀⠀⢓⡀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠛⢛⣿⣷⣒⡄⠰⣾⣿⣿⣿⣿⣛⠛⠓⠚⣻⣛⣛⡀⠐⠋⠛⡿⣿⣛⣛⣳⣷⣷⣿⢿⠗⠀⠈⠿⠤⠀⢀⣄⠈⢷⡀⠀⠀⢴⡆
⠀⠀⠀⠀⠀⢘⣛⣿⣷⣒⠄⠀⢴⣾⡛⠳⠿⠾⣯⣭⣙⣷⡶⣶⣿⠒⠒⣏⣿⣡⣿⣿⡀⣿⣯⣩⣿⠀⠀⠀⠘⢿⣷⣰⡿⣆⣈⢻⡄⢠⡟⣷⠆
⠀⠀⠀⠀⠘⠛⢻⣿⡿⡿⡄⠀⣠⣤⡟⠃⠀⠀⠀⠀⠈⠉⠛⠿⢿⣖⡂⣖⣗⡛⡭⣿⣁⣿⣽⣩⣿⠀⠀⠀⠀⢀⣽⣿⡟⢨⡟⠂⣹⣿⡀⠭⣽⣗
⠀⠀⠀⠀⠴⠾⠿⣿⡿⠷⢂⣀⠀⢈⣉⣓⣶⠄⠀⠐⠒⠲⠶⣤⣀⣀⠉⠓⠿⢿⣽⣭⠭⠤⠤⠤⢿⠀⠀⠀⠀⠶⣾⣿⣟⠀⠛⣿⣿⠿⠆⠀⣻⣿⣤⠄
⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠳⠜⠛⠀⠀⠐⠒⠓⠶⠦⠄⠀⠀⠉⠁⠀⢀⣤⣈⣉⡙⠓⠶⠦⣼⣧⣀⣀⣐⣾⣿⣿⣖⣀⡀⣼⣿⣿⣧⡤⠉⣳⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠉⠉⠛⠛⠿⠗⠒⠛⠁

⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⢠⣀⡠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⢀⢈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠥⠂⠈⠁⠐⠰⠆⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⢀⣀⡀⠀⠀⢀⣤⡔⠂⢸⠞
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⢠⠀⠈⣿⣷⠆⠀⠈⠻⠟⠁⠘⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⣦⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⣄⠀⠄⣤⠄⠠⣀⠀⠀⠀⠀⢀⠀⡼⡄⠀⠀⠀⠀⠀⢰⠒⣲⡄⠀⠀⠀⠀⠀⡟⢸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⡀⠑⢄⡀⠨⠤⠀⠀⠀⠈⢀⠀⠘⠀⣰⣆⠀⠀⢹⠛⢛⡆⠀⠀⠀⠀⠀⡇⢸⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠫⣁⣄⣠⡀⠀⠀⡀⠀⠀⠀⡰⠧⠼⡄⠀⠀⠉⠉⠀⠀⣀⣀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣰⣷⣬⣦⣝⠩⠒⠀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⣴⠋⠁⠀⠀⢰⡇⢸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠥⣀⣨⠉⠐⠂⢈⠀⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⠤⠄⢸⡇⢸⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠴⠐⠂⢀⠀⠐⠂⠀⣁⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⣖⡀⣀⠀⠒⠀⣈⢀⣐⣢⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡄⠃
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠀⠀⢈⣉⠋⠋⠙⠙⠉⢉⠁⠀⢀⢈⡉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠛⠁⣧
⠀⠀⠂⠀⠁⠀⠃⠀⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⣡⠀⠀⠀⠻⠀⠁⠖⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣶⣄⣀⣀⠀⢀⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢈⡿⣯⡶⠿⢾⣝⢿⠠⣼⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢀
⠀⠀⠀⠀⠀⠀⢰⣾⣿⡇⠰⣷⠀⣿⢾⣷⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢈⣧⡿⢦⣤⡼⣿⣾⠁⡀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠻⣿⠛⠉⣋⡼⣿⡽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢠⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⡃⢶⠄⣄⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢯⣷⣶⣾⡿⠂⢄⠀⠀⠙⣄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠁⠀⠀⠁⠄⠀⠘⢦⠀⠀⠀⢸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠘⡇⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠱⠄⠐⢃⣀⠭⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠔⠚⡈⠁⠀⠐⠢⣶⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⠀⠀⠀⠀⠄⠁⠘⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⢀⡠⣦⣎⣀⠗⠀⠀⠀⠀⠀⠀⢐⢀⢂⢤⣆⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣔⠌⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢇⡼⢿
⠀⠀⠀⠀⠀⠀⡨⢿⣄⡹⠓⢢⠀⠀⠀⠀⠀⠀⠈⠳⣕⣌⡋⠴⠱⢀⠀⠀⠀⠀⠀⠀⠀⢠⢇⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠖
⠀⠀⠀⠀⠀⠘⠤⠜⠉⣀⠜⡡⡰⠀⠀⠀⠀⠀⠀⠀⠑⠬⠯⠷⠿⠩⠆⠀⠀⠀⠀⠀⠀⠘⣊⠁⣙⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠩⠒⠯⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠒


⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⡀⣀⠴⠙⢄⡀⠀⠀⠀⠀⠀⠀⢀⣀⠎⢰⢲⠀⡅⠀⠀⠀⠀⠀⠀⢰⡞⠁⢠⢬⡑⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡀⣫⠄
⠀⠀⠀⠀⠀⠠⢜⢱⠌⠑⡴⠉⠀⠀⠀⠀⠀⠀⠀⠎⠬⡀⢀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠸⣃⠤⡈⠉⢀⠟⠀⠀⠀⠀⠀⠀⢀⡠⠎⠣⣈⠈⠁
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠁⠘⠁⠀⠀⠀⠀⠀⠀⣮⠕⠫⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠤⠴⠟⠀⠀⠀⠀⠀⠀⠀⠀⢱⡠⣀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡠⢄⠀⠠⣍⢆⠀⠀⠀⠀⠀⠀⠰⠆⡠⠔⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢁⡐⢆⡏⠀⠀⠀⠀⠀⠀⢀⠔⢒⣓⡒⣦
⠀⠀⠀⠀⠀⠀⢅⠀⠓⣆⠈⠈⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⢀⣀⣘⡆⠀⠀⠀⠀⠀⠀⠸⡷⡁⠀⠀⣱⠗⠀⠀⠀⠀⠀⠀⠟⠰⢡⣦⡈⡆⢡
⠀⠀⠀⠀⠀⠀⢈⢶⢤⠤⠗⠀⠀⠀⠀⠀⠀⠀⠀⠘⣖⡒⠀⢄⡸⠁⠀⠀⠀⠀⠀⠀⠀⡸⢫⣖⣯⠯⡀⠀⠀⠀⠀⠀⠀⠣⡸⠦⠉⠔⣁⠆
⠀⠀⠀⠀⠀⠀⠟⠻⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠃⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠤⠐⠁⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠊⠉


⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠒⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⢀⠎⢉⡗⠀⣰⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⡜⣍⡅⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢨⣀⣍⡇⠀⠀⠀⠀⠀⠀⢰⡕⠀⢈⠖⢄⡀
⠀⠀⠀⠀⠀⢸⠲⢯⠤⣀⠣⡄⠰⠀⠀⠀⠀⠀⠀⢀⡴⡇⠀⡷⢇⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⣨⣀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠤⠴⠵⠂
⠀⠀⠀⠀⠀⠈⢦⡈⣆⠜⢀⡸⠃⠀⠀⠀⠀⠀⠀⠘⠒⠧⠦⠗⠒⠁⠀⠀⠀⠀⠀⠀⢠⠚⠥⠃⢘⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠖⠑⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣤⡈⠻⠿⣿⣿⣿⣿⡿⠖⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣤⣾⣿⣿⣿⡿⠀⠀⠀⠀⢙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣷⠀⠀⠀⠀⣾⡇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡀⢀⣠⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠛⠋⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⢉⠙⠿⢶⣦⢄⣤⡤⠤⠐⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣦⣀⢀⣈⣵⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⡰⠉⠙⠛⠻⠟⠛⠛⠉⠀⡐⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣠⡇⠀⠀⠀⠀⠀⠀⡴⠀⠀⢧⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⡇⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢦⡀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⢀⣼⣿⡶⠄⡀⣤⠤⣀⣤⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⣠⣴⣿⡿⠋⠁⠒⣾⠃⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠈⠓⢶⣾⣿⡿⠋⠀⠀⢀⡼⠁⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⠀⢀⣴⡋⠠⠊⠀⠿⠿⠃⠰⠀⠒⠒⠒⠦⣀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠰⡇⠀⡀⠔⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢼⡂⠀⠀⠄⠀⢠⠇⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠔⠀⠁⠀⠀⠀⠀⠀⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⠤⠄⠐⠈



⣿⣿⡿⠻⠿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢴⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠝⢻⡍⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡼⠄⠀⠠⠀⣠⡿⠤⠀⡀⠀⠀⠀⠀⠀⠈⢆⠀⠋⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠉⣁⣀⡀⠀⠀⢠⡋⠀⠀⠀⠉⡀⢤⣤⡄⠀⠀⠈⠁⠀⠉⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢴⡏⠀⠉⣦⣾⠁⠀⢹⡆⠀⣼⣿⣿⣿⣷⣶⡗⠘⠛⠁⠀⠒⠒⠂⠀⠀⠀⠙⢷⠄⢷⠇⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠤⠶⠁⠘⠦⠴⠾⠄⠀⢻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠓⢦⣄⠀⠀⣆⢠⠄⢦⢠⠄⠂⠀⠀⣠⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣲⡒⣰⠆⠀⠀⠀⠀⠈⠉⣿⣿⣿⣧⣰⡋⢉⠶⡉⢉⡆⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⠙⣷⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠤⣤⣏⠀⠀⠀⠀⠀⠀⢀⣨⣭⠉⠉⠉⠁⠀⠀⠀⠈⢁⣀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠛⠛⠉⠉⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠠⣤⣤⠹⡆⠀⡈⠛⠁⠀⠀⢿⣻⡆⠀⠀⠀⠀⠀⢀⣶⣾⣅⠀⠀⢹⡆⢠⣴⠀⢀⣴⠟⠉⣠⣾⡁⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠘⢦⣀⣀⠀⠀⢺⣿⣿⣿⣿⣿⠹⣿⣷⠀⠀⠀⠀⠀⠸⣿⣿⡏⠀⠀⠘⣿⠀⠉⣴⡟⠁⠀⠀⠻⠟⠡⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠘⠋⠀⢠⡞⣀⣉⢳⡀⠠⣾⣿⣿⣿⣿⡇⢻⣿⡇⠀⢿⠇⠀⠀⠈⡄⠀⡤⠀⠠⣿⠀⣸⡟⠀⣤⣤⡄⠀⠀⠀⡰⠟⠁⠀⠙⠷
⠀⠀⠀⡀⠀⢠⣤⡄⠀⠈⠀⠀⠀⠈⠁⠀⠉⠉⠉⠉⢉⠁⠈⢉⣁⣀⡀⠀⠀⠰⠆⠀⠀⠀⠶⢸⣿⢀⣿⠃⠀⣿⣿⡇⠀⢰⡖⠀⠀⠀⠀⠀⠀⢰
⠀⢀⡞⠀⠀⠈⣿⠅⡀⢠⡄⢀⠀⠀⠀⠙⢶⡀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⠀⡆⠀⠆⠀⣼⡏⣸⡿⠀⠀⠻⠿⠇⢰⣦⠈⢆⠀⠀⠀⠀⠠⠊
⠀⠙⠛⢻⡶⠀⠠⢄⠙⠀⠁⠈⠠⠆⠀⠀⢸⡇⠀⣘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⢰⣿⢃⣿⡇⠀⠀⠺⣷⣿⣞⣿⡆⠀⠿⠂⠀⠺⠇
⠀⠀⢀⡞⠁⠀⠒⠂⠀⠀⠀⠀⠀⠚⠁⠀⢰⡇⣸⣿⣿⣿⣆⠀⠈⠐⠆⡀⠀⠀⢻⣷⠀⣠⣿⡏⣼⣿⢀⣦⠀⠈⠛⠛⠛⠙⠃⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠈⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⣙⣀⠀⢸⡇⠈⠉⠉⠉⣁⡀⠀⠂⠀⠀⠀⠀⢸⣿⣾⣿⣿⣼⣿⠇⣼⠿⢶⣁⠄⠀⠀⠚⠻⠿⠿⠿⠿⠛⢉⡅
⠀⢠⣶⠿⠀⣬⣐⡀⠀⣿⣾⠇⠀⢡⡞⠁⣾⡇⠰⣶⢿⣿⣿⣿⣿⣶⣤⠀⠀⠀⣸⣿⣿⣿⣿⡿⠋⠀⣀⣀⡼⢣⣆⠀⠀⠐⠒⠂⠀⠤⠐⠊⠁
⠀⡘⠁⠀⠀⠻⣿⡇⠀⠈⠉⠀⣠⡈⠀⠀⢻⣇⠀⠀⠸⢿⣿⡿⠏⠀⠀⠀⠀⣰⣿⣿⣿⡿⠋⠀⠀⠀⢨⣿⣦⠈⠉⡟⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠾⣿⣦⠀⠛⠁⠀⠀⠀⢠⣿⣶⡖⡆⠘⣿⣦⡀⠀⠀⠀⣴⣾⡀⠀⠀⣰⣿⣿⠟⠉⣠⠔⠀⠄⠀⠀⠀⡤⠀⠐⠀⠀⠤⠀⣉⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠈⠀⢠⡟⠀⠀⠈⠻⣿⣷⣤⣄⡙⠛⠀⢀⣼⣿⡿⠁⡀⡀⢀⠀⣀⡀⠀⢰⣿⡇⠀⠀⠀⡀⠐⡀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠉⣿⣯⡀⠀⠀⠀⠂⠀⡙⢿⣿⣿⣷⣤⣾⣿⡟⢀⡄⠀⡰⠀⢀⡔⠁⢠⠀⠉⠀⢤⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠉⠉⠁⡴⠄⠀⠀⢠⠁⠀⠈⢿⣿⣿⣿⣿⡇⠢⠤⢴⡤⠤⠋⠀⠀⣼⡆⠀⠀⣰⣄⠀⠀⢰⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⢠⣿⣿⣷⠀⠀⠀⢸⡄⠀⣠⣾⣿⣿⣿⣿⣿⣦⡀⠚⠓⠀⠀⠀⣠⡿⠀⠶⠁⠀⠀⠑⠖⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠏⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠿⠿⠷⠶⠶⡶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠖⣰⣿⣿⣿⣿⣿⣿⣿⡏⠀⣀⣤⡤⠶⠶⠒⠒⠛⠲⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣄⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢩⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⢠⡞⢁⣀⣤⡤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣤⣤⣀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀⠀⠹⣆
⠀⠀⠀⠀⣠⣿⣞⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠈⠛⠳⢦⣼⢀⣠⣴⢖⣛⣋⣉⣁⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣉⣛⣶⣿⣆
⠀⠀⠀⢠⣿⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠲⠿⠟⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⡄
⠀⠀⠰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽⣿⣻⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠠⣀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠉⠑⠺⢧⣏⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⡰⠀⠀⡠⠔⠊⠉⣡⡿⠷⢦⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢰⠃⡠⠋⠀⠀⠀⠀⠁⠀⠀⣿⠃⡍⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⡞⠀⡤⠦⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰⠀⣿⢠⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⢀⡄⢒⠉⠉⣠⢭⣕⢦⠀⠀⠀⢇⢻⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⣾⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⣰⡙⠁⣀⠌⡆⠳⠭⢞⣀⣱⠀⢠⠊⠆⠋⢠⣾⣇⠀⠀⠀⠀⠀⠀⠀⠈⡿⢽⡆⠈⢸⡇⠀⠀⠀⠀⡆⠀⠀⡇
⠀⠀⢰⠁⡷⡀⠋⣣⡧⠒⠉⣀⣠⠖⠀⠸⣰⠀⢸⣿⣿⡏⢀⠆⠀⠀⠀⠤⠶⣤⡱⡈⢷⡀⠈⣷⠀⠀⠀⢸⠇⡀⢰
⠀⠀⢨⠔⠁⠀⢹⣡⠤⢲⠉⠉⣁⠝⠀⠀⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣤⣼⣿⣿⡕⡌⢳⡀⠘⡆⠀⠀⡞⣰⢃⠃
⠀⠀⢸⠀⠀⠀⠈⠘⠛⢋⠔⠋⠁⠀⠀⠀⢣⠀⠀⠀⠀⠁⠀⠤⠀⠀⠀⠿⢿⠿⠟⠀⠈⠢⡁⠀⠀⠀⠈⠐⢡⠊
⠀⠀⠘⠦⠤⠀⠒⠒⠂⣉⡄⠀⠀⠀⠀⠀⠈⢦⠀⠀⠸⣿⣤⢂⣶⡄⠀⠀⠀⠐⠀⠀⠀⡔⠁⡠⢒⣨⠭⣍⢳
⠀⠀⠀⣇⣀⣒⣂⡈⠥⠐⢣⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠉⣒⠐⠛⠁⠀⠀⠀⠀⠀⠀⠀⣇⠜⠀⠛⢲⠄⠀⡸
⠀⠀⠀⢸⡀⠠⠤⠤⠒⠂⠉⣂⠠⢔⠒⠺⠏⢩⣽⣭⢿⢲⠬⣉⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠒⠤⢬⠖⡎
⠀⠀⠀⠘⣄⣀⡠⡤⢤⢖⡯⡟⡇⠻⠟⢠⣶⣤⠟⠁⠘⡐⢄⡀⢈⡩⠟⡹⠛⢫⣩⡀⠒⠶⣤⡀⡴⠁⠰
⠀⠀⠀⠀⠉⠘⠙⡉⠉⠁⠀⠈⢇⣠⣦⠀⡽⠁⠀⠀⠀⠈⠑⠂⠀⠒⠉⠀⡰⠻⠿⠻⣦⢄⡈⠻⠁⢰⡁
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠘⡌⣡⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠆⢀⡀⠻⠷⣅⠲⣼⠃⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⢘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠻⠟⢀⣤⡈⣷⠈⢘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠤⡼⠁⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢸⠐⠿⠇⠀⣀⠈⠛⢁⡿⢛⢿⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⢀⣀⣙⢦⠑⣤⣄⠀⠀⢸⢠⣦⠀⠘⣿⡣⢾⠋⠀⠀⢱⠮⢷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠈⠒⣬⡙⠃⠘⣬⣀⣀⠼⠞⠛⠛⠿⠅⠀⢸⠀⠀⠀⠈⠢⣰⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠃⠛⠤⢤⡅⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠭⠤⠤⠤⠂⠤⠤⠤⠀⠀⠠⠤⠒⠁

⠀⠀⠀⠀⣠⣤⣤⣄⠀⢀⡀⣀⣠⣀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢠⣿⣟⣿⣷⣿⡇⠘⠓⠛⠛⠚⠐⢠⣺⣷⡄⡀⠀⠀⠀⠰⢆⣂⣂⠈⠀⢀⣀⣠⡶⠋⠀⠀⠠⣀⠀⠀⠀⠀⠳⢎⡸⠀⠀⠀⠀⠄
⠀⠀⠈⠛⠿⠟⢿⠋⠀⠀⠀⣠⠶⢄⠀⠀⠉⠉⠀⠀⠀⠲⢣⣞⠁⠈⢻⣛⣆⠛⠋⠁⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠸⠟⠻⠇⠀⣠⡄
⠀⠀⠀⣠⣴⣤⣌⡀⠀⣀⡤⣷⣤⡾⠃⠀⡀⠀⢠⠀⠀⠀⢾⣿⡄⠀⠈⣿⡏⡇⣰⠆⣠⡦⠀⡴⡄⠀⠀⢠⡶⣦⢠⣶⣦⡸⣧⣸⣿⡀
⠀⠀⢰⠻⣿⣿⡿⠟⠋⠑⠀⡼⠁⡀⠐⡤⡾⣿⡯⠑⢏⠀⠘⢏⡁⣠⣤⣍⡠⠃⠋⠊⠁⠀⠐⣃⡀⢠⣦⠘⢷⡿⢻⣧⣴⣷⠘⠋⠹⣧
⠀⠀⡏⠀⠘⣯⣠⣄⣀⣀⣾⣇⠀⣧⠈⠳⢤⣮⣦⠴⠋⠀⠀⠀⠉⠙⠛⡉⣠⡄⢠⣠⣦⢠⣿⠛⣿⡌⣿⣤⣴⡆⠀⠻⢿⠿⠀⣀⡀⢀⡄
⠀⠀⢿⠀⢤⠁⠉⠻⢮⣿⣿⣿⡇⠈⠷⣶⣆⠀⠀⠀⢤⣰⠚⢣⣤⣄⣺⣧⣸⣷⢻⣿⣿⣎⣿⣦⣿⠇⠻⠿⠛⠋⠀⠀⢰⣾⣿⢿⡇⠈⠇
⠀⠀⠈⢷⣤⡀⠀⠀⠀⣈⡽⠟⠁⠀⠀⠉⠀⣀⣤⣴⡄⠈⣀⣿⡏⠙⠛⣿⡏⢿⡞⡿⠙⠟⠈⡉⠁⠀⠀⠀⣧⠀⠀⠀⣸⣷⣤⣄⣧⣀⡀
⠀⣤⣶⣾⣍⣛⣻⣿⡛⠉⢀⢤⡒⠄⣇⡀⠹⠛⣿⡏⢺⣿⣽⡜⢿⣶⣿⠻⠗⠈⠀⠀⠀⡀⠛⠁⠲⠄⠀⠤⠞⠀⠀⠘⣿⣷⣾⣿⣿⣿⠃
⠀⣿⣟⠛⠛⠛⠛⠛⢻⠃⡜⠀⠀⠀⠀⠂⢀⠀⢹⣿⡘⣿⣷⠶⠀⠉⢁⠀⠀⠀⢹⡀⠀⠸⡄⢠⠤⠒⠀⣀⣠⣤⣤⣀⠀⠙⢿⣿⣿⠃
⠀⣿⠿⡀⠀⠀⠀⢀⡏⠀⢷⠀⠀⠀⠀⢈⡆⠀⠀⠉⠀⣀⡀⠀⠀⢰⣬⣟⣲⠤⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⣟⣿⣿⡟⠀⢠⡀⠙⠁
⠀⠻⣧⣇⣀⣀⣀⡼⠁⡀⠀⠑⠂⠔⠐⠀⢀⣠⠤⠒⠉⠙⣁⣀⣤⣼⣿⣿⡿⢿⣿⣶⡆⠀⡁⠀⠀⠈⠙⢿⣿⣿⣿⠀⠰⡏⠘⠳⠆⣀⡀
⠀⠀⠀⠉⠉⠉⠁⠀⢄⠀⠠⣀⡀⠰⠞⠛⢉⣀⣤⣶⣺⣿⣿⣿⣻⣿⣿⣾⣿⣿⡿⠃⠈⡥⠁⠀⠀⠀⢀⡀⠈⣵⡆⠀⣼⣿⣦⣦⠐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣆⡔⠀⠸⣃⣤⣶⣾⣿⣽⣿⠿⠿⠟⠛⠛⠛⠉⠹⣿⡿⠋⠀⠂⡠⠔⠉⡁⠀⢤⣿⣷⡄⢸⡆⠀⠻⠿⠛⠁⠐⣿⠂
⠀⠀⠀⠃⠁⡔⠀⠀⠉⠉⠀⠈⠛⣛⠋⠉⠉⠀⠀⠀⠀⠀⢈⣶⠃⠀⠀⠉⠈⠆⡆⠀⠀⣀⣀⢿⡿⠄⢉⣩⠥⠔⠧⢄⡀⠀⠀⠀⠬⡁
⠀⠀⠀⠂⠀⠀⠀⠀⠂⣤⡞⢉⣻⣅⣈⠑⣤⠢⠐⠀⠻⢷⣾⠋⠀⠀⠑⠈⠢⣠⠖⣆⠀⡇⠈⡎⠁⣰⠋⣠⣶⣶⣶⣦⣙⡄⠀⠠⣤⣄
⠀⠀⠀⠀⡀⠰⡆⠀⣾⣤⣾⣿⣽⠶⠶⠟⠾⠷⣉⡴⠠⠐⠀⠀⠀⣤⣦⣤⣤⡘⣦⣈⠛⣷⣤⠇⢰⡇⣬⣿⣿⣿⣿⣿⣿⣇⠀⢨⣷⣾
⠀⠀⣠⣾⣷⡀⠉⠘⡿⢋⠉⠀⢻⡀⠀⠀⢤⣤⢸⣷⡀⠐⡅⢰⣟⣛⣿⠻⡟⣿⣎⠙⢻⣿⣿⣆⠈⣧⠸⣿⣿⣿⣿⣿⠏⢹⡀⠘⠿⠟⠂
⠀⠀⠛⣿⡟⠓⠀⢸⣿⡾⠆⢠⣾⣿⣷⣦⣄⣀⠞⠙⣿⡀⢇⢸⣿⣛⣿⣠⣇⣿⣿⠀⠘⣏⡕⡻⣷⣿⡦⠄⣉⣉⣉⡤⠒⠉⢿⡵⣦⡀
⠀⠀⠀⣿⡇⠀⢠⣿⠉⠓⢦⣾⣿⣿⢿⣿⠟⢅⠀⠀⢸⣧⢪⢸⣿⢙⠏⡿⡏⣿⣿⠀⠀⠙⣾⣱⣹⣿⢴⢒⣍⣩⣭⡍⡭⣽⠸⣿⣒⣳
⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠘⢯⣛⣛⠛⣋⡠⠆⠀⠀⣴⣛⢦⡀⣿⡞⡿⣷⣗⢿⣿⡆⢰⠄⠈⠛⠿⣿⡸⣙⠞⠙⢿⠇⢉⡼⠀⣿⣿⣾⡆
⠀⠀⢀⣿⡇⣴⠋⢿⣳⡄⠀⠀⢨⠏⠛⠂⢲⠀⠀⢸⣿⡦⡌⢷⢹⣷⣤⢿⡼⠼⣿⣧⠘⡄⠀⠀⠀⢸⣦⣄⣉⣘⣛⣋⣥⣴⣾⣿⡿⠙
⠀⠀⢸⣿⡇⢹⡀⡏⢿⡷⠀⠀⣸⣀⠀⠀⡇⠀⠀⠈⠛⠓⠘⠉⣿⣿⡆⣾⡿⠾⣿⣿⡙⡄⠀⠀⠀⠈⣿⠿⠿⢿⣿⣿⡿⢟⣹⡞⢷⣰
⠀⠀⣾⣹⣷⠀⠉⢡⣴⣶⠆⠀⠀⢸⠁⠀⢉⠇⢀⣀⣾⣀⡀⠀⣾⡿⣻⢻⡟⢿⢻⣿⡧⡇⠀⠀⠀⢀⣿⣭⣭⣭⣧⢸⣿⣿⣶⡷⣶⣄
⠀⢰⣿⣹⣿⣧⡀⢀⡏⠀⣾⠆⠠⠚⠚⣧⠎⠀⠀⢹⢿⡏⠁⠀⢻⣷⠹⢿⡇⣆⢘⣿⠇⡸⠀⠀⢰⣯⣛⣿⣿⣿⣿⠸⣿⣿⡿⠿⠿⢿⡆
⠀⠈⠉⠉⠉⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠈⠀⠁⠀⠀⠈⠛⠿⠿⠿⠿⠿⠋⠀⠁⠀⠀⠘⠋⠋⠛⠛⠛⠉⠀⠉⠉⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢬⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⡄⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠀⠀⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢀⠠⠴⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠎⢀⡠⠶⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⣇⣸⣤⣶⣶⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣰⢻⣿⢿⢿⣿⡏⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣰⢃⡞⡟⠸⠸⢹⢱⡘⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢁⡞⠘⠀⠃⠀⠘⠈⣇⢹⡿⡟⡿⡓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠊⠀⠀⠀⠀⠀⠀⠀⢻⡀⣷⠁⠀⠙⢆⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢸⠀⠀⠀⠀⠁⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣀⡴⢻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⡀⣠⣀⣤⠴⠖⠛⣉⣤⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠤⢀⣀⡉⠉⠉⠛⠛⠛⠛⠉⡏⠉⠁⠀⣠⠔⠩⠐⡾⡆⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡚⠉⢳⠘⠀⠐⠉⠩⢯⣽⠀⡇⠀⠀⠈⠀⠠⠁⠀⡇⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠘⢰⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⡆⠀⠸⠀⣷⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠚⠉⢼⣤⣀⠀⡘⠀⣸⣄⠈⢀⢰⠀⡇⠀⠀⠀⢧⠳⠿⠻⠏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠃⠀⠀⠈⠉⠘⠚⠁⠀⠙⣒⠻⠥⠞⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⢀⠀⠀⢠⡴⠖⠛⣋⡉⠉⣻⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡦⠶⢶⡶⡶⢲⠖⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠴⠞⠚⡟⠛⠛⣻⠀⠀⢸⡀⠠⡩⠼⠇⠀⡿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⠸⢸⢷⢻⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠈⠸⠁⠀⠁⠈⢀⡏⠀⠀⢸⠀⠀⠀⠀⠀⢀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠀⠘⠘⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⣸⠁⠀⠀⣼⠀⠂⠀⠀⠀⢸⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⣄⣾⣿⣿⣿⡧⠤⠤⠤⠤⠤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠳⠶⠤⢤⡟⠀⠀⠀⠀⠀⣾⠀⠀⢸⠇⠀⠀⠀⡀⣄⣾⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠲⢴⣧⣄⣀⣼⠀⣠⣿⡷⠟⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠴⠇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠠⠀⠀⠀⠀⡆⠀⠀⢸⡄⠀⠀⠀⠀⠀⡄⠀⠀⠀⣌⣉⣉⣉⣉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢸⡸⠀⢸⡁⠀⠀⠀⠀⠀⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠟⠛⠛⠻⠛⠇⠀⠀⠀⠀⠀⠀⡀⠘⠃⠀⢱⡇
⠀⠀⠀⠀⠠⠂⢠⡖⠒⠲⡄⠀⠄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡀⠃⠐⡠⢃⡄⠀⠀⠀⠀⠰⠪⢿⣄⣀⣴⠟
⠀⠀⠀⠀⠠⡀⢀⠻⠻⢟⠃⣀⠄⠀⠀⠀⣧⣀⣀⣀⣠⣄⣀⣀⣀⣠⡇⠀⠀⠀⠛⠛⢏⠔⠩⠬⠤⡄⠀⠀⢀⣤⣤⣤⣭⣥⣤⡀
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⣄⠀⠠⡀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⣶⣶⣶⣀⣀⣨⣶⣶⡆⠀⠀⠀⠀⠁⠀⣤⣶⣶⣦⡀⠂⠀⠀⠀⢀⡎⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠉⠉⠉⠉⢻⣄
⠀⠀⠀⠀⠩⡭⡭⡭⡭⠭⠭⠭⠁⠀⠀⠀⠀⠀⢸⣿⠿⠿⣿⡷⠀⠀⠀⠀⢸⠀⠀⠀⠀⠂⠀⠀⠄⠀⠀⠘⠟⢂⣠⣤⣶⣦⣐⠚⠟
⠀⠀⠀⠀⢀⣃⣓⡓⣓⣒⣒⣒⠀⠀⠀⠀⠀⢀⠈⢻⢸⡟⡼⠃⠀⠀⠀⠀⠀⢣⡀⠀⡐⠀⢀⡠⠀⠀⠀⠀⡏⢌⡟⠛⠛⠛⣿⠩⡃
⠀⠀⠀⠀⠠⠆⠦⠆⠁⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠠⠷⠷⠇⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⠤⠔⠃⠀⠀⠀⠀⣛⣚⠛⣛⣛⣛⢛⣓⡳
⠀⠀⠀⠀⠁⠀⠀⠀⠉⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠒⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠠⠊⠛⠗⠀⠠⠀⠀⠀⠀⠀⠀⡠⠀⡴⢁⠀⠲⠄⢤⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⢰⣾⠖⠚⠛⠓⢾⣶⡆
⠀⠀⠀⠀⠀⢁⠀⠠⠠⠄⢀⠀⠀⠀⠀⠀⠘⣇⠀⡏⠀⢢⡀⠀⢀⠏⠀⠀⠀⠀⠀⠲⣶⣶⠆⠀⠀⠀⠀⠀⢨⠃⠀⠀⣀⡤⠂⢹⡀
⠀⠀⠀⠀⠀⣀⣤⣀⣀⣠⣆⠀⠀⠀⠀⠀⢠⡜⠓⣧⡙⠛⡡⠔⠋⣆⠀⠀⠀⠀⠀⠀⠈⠁⢄⠀⠀⠀⠀⠀⠸⡄⠀⠀⠁⠀⠀⣸⠁
⠀⠀⠀⠀⢠⠒⠛⠛⠛⠛⠓⢐⡄⠀⠀⠀⠀⠙⠒⢻⡉⠉⠓⠃⠜⠁⠀⠀⠀⠀⠀⢠⣶⣶⡄⠀⠀⠀⠀⠀⠀⢘⡦⢤⣤⠤⢾⠁
⠀⠀⠀⠀⠀⠃⠙⠃⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠆⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⣢⣶⠿⠛⠛⠿⣷⣄⠀⠀⠀⢰⠿⠟⠃⠀⠙⠿⢷⡆
⠀⠀⠀⠠⠤⣶⣶⣿⣿⣿⣿⣶⣶⠤⠄⠀⠀⣿⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⡙⢿⣵⢆⡀⠀⠀⠘⣿⡆⠀⠀⢸⠀⣤⣤⣤⡄⠀⢸⡇
⠀⠀⠀⢀⠀⣿⣶⣮⣭⣭⣵⣶⣿⠀⠀⠀⠀⣿⡇⠐⠂⣃⡀⣠⡄⠀⠀⠀⠡⠐⢝⢿⣵⣆⡀⢀⣿⠇⠀⠀⢸⠀⠶⠶⠶⠂⠶⢸⡇
⠀⠀⠀⣸⠀⠻⠿⣿⣿⣿⣿⠿⠟⠀⠀⠀⠀⣿⡇⠀⠀⠿⠧⠿⢧⠀⠀⠀⠀⠈⠢⡁⢝⢿⣯⡾⠋⠀⠀⠀⢸⠀⣏⣉⣉⡁⣋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠥⠙⠟⠁⠀⠀⠀⠸⠤⠤⠤⠤⠤⠤⠼⠃

⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣒⠒⡖⣢⠨⣡⣶⡔⠦⠃⢀⠀⠂⠀⡤⠀⠒⠒⣂⣀⡐⠒⠒⠂
⠀⠀⠀⠀⣀⠤⣺⠁⢠⣤⣤⣤⣶⣶⣶⣾⠀⠀⣾⣦⣴⣿⣿⡧⠀⠨⢎⠌⠜⡼⠠⣖⡻⠷⣔⠃⡎⠀⠀⡀⢰⣷⠀⠁⢻⣿⡇⠀⠿⠿⠿⠿⠗⠐
⠀⠀⠀⠸⢐⠥⠚⡁⢸⣿⣿⣛⣟⣻⣿⣿⠀⠀⠙⢿⣿⣿⡟⠁⠀⣐⣠⣂⣊⣾⡀⠢⠄⠉⣹⡶⣄⠀⢀⠁⠸⠿⠀⢠⠘⠛⣁⣤⣤⣦⣤⣄⠀⠀⡀
⠀⠀⠀⡋⠤⢂⠥⡇⠈⠛⠛⠛⠛⠛⠛⠻⠀⠀⠀⠄⠙⠉⠀⠀⣸⡿⠿⠿⠿⢿⣧⣀⠀⢀⡟⠃⠙⠆⠘⠀⢠⣶⣦⠸⠀⣼⣿⢟⣩⣍⡝⠻⣿⣄⠁
⠀⠀⠀⡇⠊⡡⠔⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⢀⣼⡟⠁⠀⢀⡀⠀⡉⢻⣧⠈⠁⠆⠀⠀⠀⠀⠈⠻⠋⠀⢸⣿⡇⣾⣿⣿⣿⣇⢹⣿
⠀⠀⠀⡇⣈⠔⣊⢸⢰⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠀⠹⣿⣇⠀⠀⢘⡁⠉⠁⣸⡟⠀⠀⡃⠀⠀⠀⠀⠀⠀⢠⠀⠘⣿⣧⡹⠿⣿⠿⣣⣿⡟⡀
⠀⠀⠀⣋⡠⢊⠅⠘⠼⡄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣼⠀⠀⠙⢿⣷⣤⣨⣁⣠⣴⠟⠁⠀⠀⠃⠀⠀⢠⠀⠀⠀⠈⠀⠀⠈⠻⢿⣷⣶⣿⠿⠋⠀⠂
⠀⠀⠀⠧⢈⣤⢊⣠⡀⠀⠀⠀⠀⠀⢀⢼⠀⠀⡒⠄⡁⠠⠐⠀⠀⠈⢻⡻⠻⠟⣿⠀⠀⠀⠀⡆⠀⠀⠀⢄⣀⣀⣀⣠⡄⣠⡠⠠⠤⠤⠤⠤⢠⠲
⠀⠀⠀⠀⡀⠀⠀⠈⠉⠀⠈⠉⠀⠀⠐⠊⠀⠀⠈⠢⣿⠤⠚⠀⠀⠀⠘⠒⠀⠒⠞⢀⡴⣤⠀⠘⠀⠠⡶⣦⠈⠓⠓⠚⠓⠒⠒⠂⠋⠂⠈⠈⠁
⠀⠀⠀⠋⠀⠀⠀⣀⠄⣆⣂⣖⡠⡀⠀⠀⠀⡌⠁⡀⡀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠚⠿⠂
⠀⠀⠀⠀⠀⠠⡎⢰⠊⠉⠙⢿⣿⣮⣢⠀⠈⠀⠁⠠⣿⡴⠂⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣥⣥⣤⣤⡀⢱⡀⠀⡊⠁⠰⠈⠀⣧⢀
⠀⠀⠀⠀⣐⢃⡡⠁⠀⠀⠀⠀⠹⣿⣿⣇⠀⠀⣠⣤⣿⢀⡀⠀⠀⠀⣿⣿⠛⠛⠛⠛⠛⠛⠉⠉⣉⣉⡉⠉⠙⠛⢻⣿⡇⢸⡇⠀⠀⡀⠀⠀⣰⠏
⠀⠀⠀⠀⡉⢀⠃⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⠀⢸⣿⣿⣿⠁⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠿⢷⣆⠀⢸⣿⡇⢸⡇⠀⢀⠀⢠⡴⠃
⠀⠀⠀⢠⡁⣈⠀⠀⠀⠀⠀⠀⠀⡄⣿⣿⣿⠀⠈⠛⠛⠁⠀⠀⢀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⠀⠸⣿⡇⢸⡇⢀⠀⠀⣸⣿⠀⠻⠃
⠀⠀⠀⠀⡁⢌⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣆⠀⠀⢠⣄⣰⡆⢀⠀⣿⣿⠀⠀⠀⠀⠐⠘⢿⣷⣿⣽⣿⡛⠋⠀⠀⣿⡇⢨⡇⢸⠀⠀⣿⣿⡆
⠀⠀⠀⠀⢅⠈⣭⣉⠛⢶⣄⠠⠐⠂⢩⣙⢻⣿⡄⠀⠈⢿⡿⠃⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣿⡇⢠⡇⠸⠂⠀⣿⣼⠇
⠀⠀⠀⠘⠂⠁⠈⢿⣷⣄⠻⣧⠔⢠⣿⣿⡆⣿⣷⠀⠀⠀⠀⠀⠀⢄⠻⠿⠿⠿⠶⠶⠶⠶⠶⢶⣶⣶⣶⣶⣾⠿⠿⠿⢃⣸⠇⠀⠀⠄⠀⡀
⠀⠀⠀⠀⢰⣾⣿⣆⠻⣿⡄⢻⣇⣿⣿⣿⡇⣿⣿⠀⠁⠀⠀⠀⠄⠂⠀⠀⠈⠉⠁⠁⠈⠻⢿⣿⣿⣷⣾⣟⣃⣉⣀⣀⣉⣑⣒⡄⠀⠀⠸⠀⠰⠁
⠀⠀⠀⠀⣿⣿⣿⣿⡆⢹⣷⢸⣿⣿⡿⠏⣰⣿⡟⠀⠀⠠⠀⠀⢷⣶⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠠⠀⠐⠀⠈
⠀⠀⠀⠀⠻⣿⣿⣿⡏⢨⣟⣿⣧⣤⣴⣾⣿⣿⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣠⣲⠄⣀⡲⠄⢠⠰⠤⠀⣀⣀⡀⠀⠀⠀⢀⠠⠐⣒⣒⣒⡀⠄⠁
⠀⠀⠀⠀⠄⠈⠛⠛⠴⠛⠋⠀⢀⠉⠁⠀⠈⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠘⠛⠀⠀⢹⣿⣷⠀⠁⠀⠓⣿⣿⠿⣿⣿⠿⢷⣾⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⢠⠂⠚⠀⠠⡎⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠸⣿⣿⡆⠀⠀⠀⣿⡇⠀⣿⡏⠀⣾⡿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠨⠀⠀⠀⠀⠒⠂⢀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠉⠉⠈⠉⠁⠡⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠘⣿⠀⢹⣷⠀⢸⣇⣼⠏
⠀⠀⠀⠀⠀⠀⠒⣴⣶⣶⢽⣿⢿⡍⠃⣉⣶⣶⣿⡙⠻⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠘⠀⢀⣿⠀⣸⣿⠀⣾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣿⢾⣿⣼⣿⣽⡧⠧⣿⣿⣿⣿⡷⠀⠀⢐⢂⠀⠀⠀⠀⠀⠉⠉⠈⣈⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣴⣾⣧⣀⣿⣧⣸⣿⣷⣦⡀
⠀⠀⠀⠀⢀⠀⠀⠙⠛⠛⡉⢩⣉⣠⡂⠙⠻⠿⠛⠁⢠⠀⠈⠀⠀⠀⡦⠠⠤⠀⠤⠤⠤⠀⢀⠀⠀⠀⢸⣿⣿⡇⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠓⠒⠐⠂⠂⠒⠒⠒⠓⠚⠛⠛⠛⠋⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠄⠀⠀⠀⠒⠂⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠀⣠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⢸⡇
⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠈⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣸⣿⣰⣿⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⠧⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⡀⠀⠀⣸⣿⡧⢿⣿⣿⡇⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⢷⣀⣿⡯⠿⣷⠿⡯⢿⣻⣿⣧⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣮⠉⣿⡇⠈⣍⡉⠁⣾⣿⣿⣿⣠⡄⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⣿⡇⠀⣿⡗⠀⢨⣥⢿⣿⣿⢻⠇⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢠⣄⣸⣿⣛⣛⣛⣛⠁⢠⣿⣿⠀⢸⡇⠈⢿⡍⠀⠀⣬⣽⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠀⢸⣿⡇⣀⠀⠀⠀⠀⢸⣿⣿⣟⣿⣿⣿⣿⡀⣼⣿⣿⡇⢸⣇⢀⣈⣡⣾⣤⣿⣿⣿⠁⠀⠀⠀⢀⣶⡀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⢠⣾⣿⣷⣿⠀⠀⠀⢀⣼⣿⡟⣿⡿⠻⣿⣿⣿⣟⢻⠛⣿⣿⣿⡉⠩⡅⣿⣿⣿⣿⣿⡄⠀⠀⠀⣸⣿⣇⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⣀⣰⣷⣾⣿⣿⣧⣿⣁⣀⣽⣿⣿⣿⣾⣶⣿⣿⣿⣷⣶⣤⣿⣃⣸⣿⣿⣿⣆⣀⣴⣿⣿⣿⠀⠀⢸⡇
⠀⠀⣿⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⠿⢿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⡇
⠀⠀⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⡇
⠀⠀⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣬⡇
⠀⠀⣀⣀⣀⣀⣀
⣾⠟⠛⠛⠛⠛⠛⠻⠿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣦⣄⠀⠀⠀⣾⣫⡾⠛⠉⣸⠇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠙⠻⣷⣄⠀⢻⡿⢷⠶⠟⠁
⠀⠀⠀⠀⠀⢀⣀⣸⣿⣤⣤⣤⣀⡀⠀⠀⠘⢿⣷⡀⠀⣤⠀⢀⣠⣤⣶⣶⣦⡄⠀⣀⡀⠀⣿⡄⠀⢠⣤⣤⣤⣤⡄⠀⣠⠄⠀⣀⣄
⠀⠀⠀⡴⠾⠛⠛⢻⣿⠋⠛⠛⠛⢿⣷⠀⠀⢸⣿⡇⢸⣿⠀⢿⣯⣭⣥⣄⣀⠀⢰⣿⣷⡄⢸⣿⠀⢀⣿⣏⣉⡉⠁⣼⠏⣠⣾⠟⢹
⠀⠀⠀⠱⣄⡀⠀⢸⣿⡆⠀⠀⠀⠛⠋⠀⠀⣸⣿⠇⢸⣿⠀⣴⠚⠾⢿⣏⣹⡇⢸⣿⠘⣿⣾⣿⠀⣼⡟⢉⣉⡁⠰⣿⣾⣿⢃⣠⡿
⠀⠀⠀⠀⠈⠙⠷⣾⣿⣧⣀⣀⣀⣠⣤⣶⣿⠿⠋⠀⠸⠿⠀⠈⠻⠿⠿⠿⠟⠁⠸⠟⠀⠈⠛⠋⠀⢻⣿⣿⠿⠋⢀⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠤⢠⠀⠀⡒⡆⠀⠠
⠐⡆⠀⡄⠀⢀⡠⠄⠀⠀⡀⢠⣾⣿⣛⣛⣛⣛⡛⣿⣷⡄⢀⢁⠠⣀⠔⠀⠁⠐⠀⠊⠀⠀⠀⡁⠀⠀⢀⠃
⠀⠹⡀⠀⠀⠠⠄⢀⡀⠠⠄⢸⣛⡛⡿⠿⣻⠻⠟⣻⢻⡇⠘⡀⠀⢀⠀⢀⠀⠀⠀⠒⠚⠀⠆⠂⠀⠃⠱⠰⠒
⠀⠨⢷⠀⠀⢀⣈⠂⢒⡁⠀⠘⣿⣷⣾⣷⣶⣿⣷⣾⣾⠃⠀⢈⡀⠒⠀⠁⡀⠂⠀⠀⠈⠃
⠀⣄⠂⠁⡀⠀⠸⠒⠐⠣⠀⠀⠈⠙⠛⠛⠿⠛⠛⠋⠁⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠆⠆⠀⠀⠀⢀⢀
⠀⢃⠀⠀⠈⠰⠸⠀⠰⠀⠂⠁⠀⠘⠃⠀⣀⠀⠒⠬⠁⠀⠀⠀⠀⠀⠀⠀⠉⠤⠂⠢⠶⠠⠥⠅⠠⠄⠤⠠⠤
⠀⡤⠁⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠐⠀⠀⡠⡀
⠀⠀⠀⢡⡀⠠⠤⠀⠀⠀⠔⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠉⠀⠀⠀⢀⣤⠀⠀⠀⠀⣋⠀⠴⠰⠤⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠄⠬⠄⠀⠀⢀⠀⠀⠀⠀⠀⢠⣠⠃⢘
⢀⡀⠀⠀⣰⠀⠁⢀⠀⠀⠘⠐⡀⠀⠀⠀⠀⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠸⠀⠀⠀⠤⠔⢨⣸⢰⣈
⠈⠁⠀⠀⠈⠀⠀⢉⢼⠄⠀⠀⠁⠀⠐⠅⠀⠁⢀⡤⠤⠤⠀⠀⢀⣀⡀⠀⠠⠐⠒⠂⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂
⠀⠐⠉⠉⠉⠉⠁⠐⠒⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠁⠀⡀⠀⠀⠀⠀⠀⡀⠀⣠⣴⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠱⡀⠂⠈⠋⠉⠁⢰⡿⠙⠛⢻⡏
⠀⠀⢠⡤⠤⠤⠤⣤⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠔⢀⠀⠀⠰⡀⠀⢣⠀⠀⠂⠀⡀⠀⢆⣼⣅⣸⠐
⠀⠀⡂⠀⠀⠀⠁⠀⠀⠘⠒⠒⠀⠀⠒⠂⠀⠐⠲⠂⠀⠀⠘⠄⠀⠀⢃⠀⠸⢷⡀⠠⡄⠁⢀⡈⠛⠛⡁
⠀⠀⠀⠀⠐⠂⠀⠀⠀⠠⠀⠪⠂⠀⠀⠀⢠⠐⠂⡄⢀⠀⠠⠐⠈⠁⠛⠑⡄⠲⠓⠄⠀⠄⠀⠐⠀⠀⠈⠁⠄
⠀⢀⡁⠐⠀⠀⠀⠀⠁⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠐⢀⡆⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠂⣀⡖⠤⠀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡠⠃⣨⡄⠀⠀⠁⠦⠐⠠⠰⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠿⢖⠋⠀⠀⠁⠁⠠⠀⠀⠀⠀⠀⠀⢀⠱⠴
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠷⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢁⡂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢹⡯⠈⠀⣾⣧⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢠⠇⣨⡝⠂⣸⣿⡏⠀⠀⠀⢋⠁⢤⠤⠒⢂⣷
⠀⠀⠀⢀⡀⠀⠀⠐⠀⢠⠥⠄⢹⣿⣿⣿⣿⣿⡿⠀⠀⠒⣞⠀⠈⡀⢠⣿⣿⠁⠤⢤⣄⡐⢠⢯⢐⣦⣍⡿
⠀⠀⠘⡄⠀⠀⠀⢀⣂⣀⣭⣵⣀⣿⡿⠟⠋⠁⣀⣀⣀⣀⠈⠉⠒⠋⠚⠛⠁⠀⠀⠀⠀⠫⠉⠭⠜⠛⠉
⠀⠀⠀⠙⠛⡛⢛⡻⠛⠉⠀⣹⠛⠛⢋⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠀⠀⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⢠⠉⣐⢎⠐⠁⠀⠀⠀⠀⡇⢠⣿⣿⣷⣶⣶⣶⣶⣦⣤⣬⣤⣤⡀⠀⢿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⢸⢰
⠀⡼⠣⢍⡁⢱⠀⠀⠀⠀⢠⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⡿⠟⠻⣧⣐⣲⣆⡼⣸
⡘⢠⠀⠀⠉⠁⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣨⠛⣙⣤⣶⣿⣿⣿⢟⡠⠋⠔⠁
⠇⢸⡀⠀⠀⠀⠀⢀⣀⡀⣸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⢾⣟⠁⣿⣿⣿⣿⡟⠉
⠈⠀⢱⢤⡒⣈⣭⣤⠄⠀⢀⠗⠘⠛⢛⡿⠿⠿⢿⢿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣶⣿⣿⣿⢤⣤
⠀⡔⠁⠀⠚⠛⠋⣁⡤⠖⠉⠛⢶⣤⣤⣀⣈⣉⠉⠙⢌⠻⣿⣿⣿⣥⠖⣺⣿⣿⣿⢿⣫⣷⣿⣿⠀⣀⡀
⠀⠈⠒⠒⠒⠛⠛⠳⠤⠄⠀⣶⠻⠿⣒⣚⠛⠛⠛⠓⣚⣓⣍⠻⣿⣿⣷⠞⠛⣫⣵⣿⣿⣿⣿⣿⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣐⡒⠫⠉⠩⠉⣙⣓⠒⠓⠾⠛⠿⣶⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠒⠠⠬⠥⢉⣑⣒⠒⠂⠨⠥⠄⠀⣿⣿⣿⣿⣿⣿⠟⠁⠀⡠⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡰⠠⢍⣉⠑⠒⠒⠤⠬⣭⣉⣉⣉⠀⢻⣿⣿⣿⠟⠁⣀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣈⣉⠛⠓⠶⠤⣤⣄⣀⣀⠀⠀⠉⠁⠀⢸⣿⡿⢃⡴⠊⠑⣢⡾
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣀⡂⠭⢉⡓⠒⠦⠤⢤⣉⣉⡙⠛⠻⠶⠶⠾⠋⠠⡏⠑⢢⡾⠋
⠀⠀⠀⠀⠀⠀⠀⣚⠁⠀⠀⠈⢉⠚⠂⠀⠨⠉⠐⠂⠀⠀⠉⣍⣩⢿⠒⠒⢂⡄⠈⠉⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣤⣤⣀⣀⡉⠒⠒⠀⠀⠀⣠⢮⢟⣰⢃⢞⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣷⣶⣦⣤⣽⣷⣶⡖⣡⡾⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣛⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡏⠁⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⣀⣬⣿
⠀⠀⢰⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⣿⡿⠉⡄
⠀⠀⢀⠱⣀⡀⠀⣤⠔⣂⠥⠄⢀⠀⠀⠀⠀⠘⢛⡛⢛⡛⠓⠀⠀⠀⠀⠸⠄⠀⠀⢀⠈⠁⠀⠠⠃⠀⠀⠀⣶⣤⣄⣀⣠⡄⡀
⠀⠈⢄⠣⣒⣒⣐⣒⣫⠔⠊⠁⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢾⣳⢀⡀
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠐⠒⠓⠒⠈⠓⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡻⠴⢻⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡍⣟⣛⣯⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⡄⠠⠄⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠆⠀⠀⠀⠀⠀⠸⠛⠓⠈⠛⠁⣿⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⡇⣀⣤⣶⣤⡀⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠧⢸⣽⣿⢽⠥⠿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⣋⣤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠾⠾⠾⠿⠆⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣾⣷⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣿⣿⡷⢶
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣟⣛⣯⣛⣛⣿⣛⣻⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠇⠘
⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣉⣉⣏⣹⣉⣉⣏⣹⣉⣉⠇⠀⠀⠀⠀⠀⠀⠀⢸⣾⢷⣷⣶⣦⣶⡾⠿⡇
⢀⠀⠀⣀⣾⣿⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠱⣀⣥⣨⣅⣨⣅⣼⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣷⣾⣿⡇
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠵⣷⣾⠮⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⠟⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⣼⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⠀⣠⣤⣤⣶⣶
⠀⢠⣾⣿⣷⣄⠹⡏⠉⢻⡼⠻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⢸⣿⣿⣿⣿⡟
⠀⢺⣿⣿⣿⠋⠉⣷⠶⠾⢦⣀⡜⣋⣠⣤⣄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢴⣿⡿⠟
⠀⠀⠉⢙⣛⡢⢼⡇⠀⠀⠈⣿⠀⢹⣿⣿⡟⠁⠀⠀⠮⣂⠦⠆⠀⠀⠂⠀⠀⢰⣶⣶⣦⣄
⠀⢰⣿⣿⣿⠀⢈⡿⣦⣤⠾⠛⢖⠻⠿⠿⠇⠀⠀⠟⢼⣿⣷⢸⣁⣀⣀⣄⠀⠀⠉⠙⠉⠁
⠀⠸⣿⣿⣿⠟⣹⡁⠀⡸⣄⣠⣾⣷⣄⠀⠀⠀⠀⠺⢎⣭⣥⡿⠋⠙⢻⣿⣷⡀
⠀⠀⠉⠋⠁⣰⣿⣿⣿⡇⡸⣿⣿⣿⣿⠂⠀⠂⠀⠀⠀⠀⠉⣷⣀⠀⠚⢦⣌⡃⠀⠀⠀⠀⠀⠀⠀⠀⢦⢠⣤
⠀⠀⢀⠤⣀⢻⣿⣿⣿⡟⠀⠙⢻⠋⠁⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡇⠀⠲⣌⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠃⠀⠀⢑⠶⣚⣛⠢⣀⡠⢊⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⢀⠀⠹⣆⠘⢿⠑⠲⠤⣤⣤⣬⠭⠁
⠀⠀⠑⠂⣤⡎⣼⣿⣿⣷⠘⠋⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠀⣿⠀⠘⡇
⠀⠀⠊⠀⠀⢱⡘⠿⠿⠟⡸⡦⠤⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠋⠀⠀⠁
⠀⠀⢆⣀⠤⠺⠉⠑⣶⡉⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⢠⣄
⠀⠀⠀⠀⠀⡇⠀⠀⡃⠱⠄⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⡵⣼⣶⡙⠷
⠀⠀⠀⠀⢀⣨⣁⠊⠀⠀⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡁⡟⢹⡇
⠀⠀⠀⣰⣿⡿⢣⣾⣸⣷⣎⠛⠿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢃⠸⣧
⠀⠀⠐⠋⠁⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣾⠄⠿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠐⠂⠀⠐⠰⣿⣿⡷⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⠀⠀⠀⠀⠄⠀⠀⠰⣿⣿⣷⠀⠀⠈⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢀⣿⣿⣏⠀⢀⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⢉⠀⠀⠀⠀⢰⣿⣧⣾⣿⣿⣿⣿⠇⠀⡄⠀⢀⠀⠀⠀⠀⠉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⢀⣾⣧⠀⠀⣤⠀⣿⣿⣿⣿⣿⣿⣿⣇⢸⣷⠀⢀⡄⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠂⠀⠀⣾⣿⣿⣖⣰⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⡀⣼⣧⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢠⣶⡀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣦⣶⡀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⣠⣶⣾⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⢣
⠀⠀⠀⣠⠖⠀⠀⠀⠙⣦⡀⠀⠀⢩⣿⣽⣿⣿⣭⣿⣿⣿⣿⣿⠿⢷⣶⣤⣀⠀⠩⣿⣽⣭⣭⣭⣭⣯⣿⣯⣿⠍⠀⠀⠀⠀⢸
⠀⢀⣾⠃⠀⣠⣄⠀⠈⢸⣷⠀⠀⣠⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⢠⡄⠀⠉⠻⢷⣦⡠⣾⣿⣋⣥⣿⣤⣤⣤⣤⡄
⠀⣾⠇⢀⣾⠃⣿⡇⠀⢸⣿⢀⣴⣏⢉⣿⢭⣍⣩⣿⡍⠉⣉⣁⣾⣧⣤⣤⣀⠀⠙⣿⡌⣩⠉⣁⣭⣭⣭⡍⢉⡁⢲⣄⢠⣤⣤⣤⡀⣠⠄⣀⣀
⠀⣿⠀⢸⣿⣴⠟⠀⢀⣿⣯⣾⣿⣿⣼⡇⢀⣹⣿⠋⠁⢾⡉⠉⣿⡏⠉⠉⢻⠗⠀⣹⣿⣿⡄⣻⣿⣶⢶⣄⣿⢿⣾⣿⣰⣿⠯⠍⣰⣿⣾⢏⣽
⠀⠙⠷⠶⢿⣧⣤⣶⡿⠟⢸⡟⠉⠻⠻⠿⠟⣿⡏⠀⠀⠀⠙⠲⣿⣧⣤⣤⣤⣴⡾⠟⠁⢿⠇⠻⠾⠿⠿⠛⠿⠈⠻⠟⢻⣷⠾⠗⣼⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠙⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣉⡷⠀⠈⡏⠀⢠⡞⠉⠙⠀⠀⠋⣿⠉⠀⠀⡏⠈⡏⠀⠀⢹⠉⡳⠀⠀⣻⣉⠃⠀⠰⣏⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠅⠀⠀⠠⠧⠀⠈⠳⠤⠔⠀⠀⠀⠿⠀⠀⠀⠳⠤⠇⠀⠀⠼⠙⠦⠀⠀⠿⠤⠆⠀⠰⠦⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣿⠟⠉⢀⠉⣀⢸⠸⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣉⣿⣿⣿⣿⢁⠀⢀⡏⢀⡇⢨⣶⡰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣠⢎⡽⣿⣿⣿⣧⡟⢀⡞⣠⠞⣠⢎⣿⣷⠏
⠀⠀⠀⠀⠀⠀⠀⢰⡇⣰⢃⣾⣿⣿⣿⣿⣫⣴⣫⣴⣯⣾⡋⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⣾⢠⡏⣼⣿⢧⣿⣿⣿⡟⠉⠁⠀⣿⡦⠀⠶⠂
⠀⠀⠀⠀⠀⠀⠀⡏⣼⢡⡟⣿⢸⣿⣿⠿⠗⠀⠀⠀⠁⠀⠀⠘⠦⣄
⠀⠀⠀⠀⠀⠀⢸⡇⡇⢸⠃⣿⢸⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⢸⠁⡇⢸⠀⣿⠸⡇⣿⣦⡄⠀⠀⠀⠀⠀⠀⠉⠹⡇⠀⠀⠀⠀⠀⠀⠀⡐⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⢻⠀⡇⠸⡀⢡⠀⠸⡈⢿⣤⡀⠀⣾⣶⣶⡶⠆⢰⠇⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⢸⡆⠱⢆⢣⠈⣷⡀⢣⠀⠈⣿⡇⢹⣿⣿⠁⢠⡞⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣄
⠀⠀⠀⠀⠀⠀⠀⢳⣄⡃⠀⢆⡘⣷⣤⣷⡾⠛⠀⠀⠈⢀⡴⠋⠀⠀⠀⠀⠀⠀⡜⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⢻⡿⠛⠉⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠋⠊⣞⢁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⣀⠈⠙⠾⠀⢠⠆⢀⠀⠀⠀⠀⠀⢠⡟⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠈⠳⣄⠀⠀⠀⠀⠀⣷⣄⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠘⠽⢷⢲⣄⢄⠈⠀⠃⠷⢢⡦⠀⢠⡟⣴⣿⠋
⠀⠀⠀⠀⠀⣀⣤⡾⢫⣽⣷⣄⠀⠉⠒⠤⠤⠔⠁⢸⡛⢶⣦⣄⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠜⠾⡖⠀⠀⠈⠂⢠⠟⣴⣿⠃
⠀⠀⢀⣴⠟⢿⣷⣮⣳⣿⣾⣿⢿⡖⠄⠀⠀⢠⠤⠻⣇⢿⣾⣿⣧⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣼⣿⠏
⠀⢀⡿⠁⣤⠈⠠⠙⣿⣿⣿⣿⢆⢻⣆⠀⠀⠀⠀⠀⢿⣧⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢏⡼⣻⠋
⠀⣸⠇⣮⠁⡟⣷⣽⣹⣿⣿⣿⣜⡆⢻⡄⠀⠀⠀⠀⠸⣏⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢞⣿⠃
⠀⣿⢰⡟⢼⡿⣿⠟⢹⣿⣿⣿⣿⣷⠈⣷⠀⠀⠀⠀⠀⣿⠀⣿⡶⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣄⠀⢠⡿⠱⣱⠇
⠀⣿⣼⣇⡾⢿⣿⣧⡏⢹⣿⣿⣿⣏⢀⢻⡄⠀⠀⠀⠀⣿⣠⡼⠷⠚⣻⡝⠳⢦⣄⡀⠀⠀⣤⡞⢺⡄⢻⡆⢳⡿⠡⢠⠏
⠀⢸⣿⣿⣀⠾⡦⢤⡄⢹⣿⢛⣿⣿⣹⣾⣧⠀⠀⠀⢀⡞⠁⠀⠠⠾⠿⠷⠛⣉⣽⠟⢷⣶⡆⢣⠀⢇⠀⠃⠈⣷⠁⠏
⠀⢸⣿⣹⣿⣿⡛⠈⢁⣿⣿⣿⣿⣿⣻⠏⢿⡀⡠⣄⡜⠀⠀⠀⠀⠠⠤⠒⢊⡡⢾⢶⣬⣙⣿⠘⠃⠈⠀⠀⠀⢸⠇
⠀⣸⠇⣿⣿⢻⡧⠘⠛⣿⣿⣿⣯⣬⣵⡾⠿⠻⣄⠈⢆⠀⠀⠀⠀⠠⠔⠒⣉⣴⣻⣿⡿⠿⡟⠀⠀⠀⠀⠀⠀⡼
⠀⢹⡆⢿⠛⠛⢗⣴⣿⣿⣿⠟⠋⠁⠉⠀⠀⠀⠘⡇⠈⠆⠀⠀⠀⣐⣂⣩⣴⠟⠿⡿⡂⢰⡛⢄⣀⣀⢀⣠⠞⠁
⠀⠀⢻⣾⣶⡰⣾⢿⣿⡿⠋⠀⠀⠀⠀⠀⣀⣀⡀⢹⡀⢸⠞⡛⢋⢹⣿⣷⣿⡇⠖⡓⣬⢦⠳⣤⣬⡭⠋⠁
⠀⠀⢸⡯⣿⣷⣴⣿⣿⣁⣄⣠⣤⣶⢿⣿⣿⣿⣿⢸⣿⣿⡀⠀⠸⠌⣿⡿⣶⣥⣤⣤⣬⡶⠟⠉
⠀⠀⠀⢻⡘⢟⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣯⣭⣴⣿⠋⣿⠃⠂⠂⠀⣿⡇⠀⠈⠉⠉
⠀⠀⠀⠀⠉⠀⠈⠙⠛⠟⠻⠶⠟⠛⠛⠛⠉⠉⠉⠗⠀⠿⠀⠀⠀⠀⢺⡇


⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣴⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⢻⣦⣤⣀⠀⠀⠀⠀⢠⡴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠞⠁⠀⠈⠳⣄⠀⠀⢻⣄⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢸⠂⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡉⢹⡷⢦⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⣠⠞⠉⠉⠉⢷⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢦⣄⠀⠈⠙⠋⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣴⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠸⣧⣤⡟⠀⠀⠀⠀⢀⣸⣀⣀⣀⣸⣀⣀⣀⣸⣇⣀⣀⣸⣃⡀⠀⠀⠀⠀⠘⢷⣤⡿⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠉⣻⠀⠀⠀⡇⠀⠀⠀⠀⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣦⠀⠀⠀⠀⠀⡇⠀⠀⢼⡋⢙⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠋⠀⠀⠀⠓⠒⠒⠒⠚⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠃⠀⠀⠈⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣀⢀⡀⢀⣀⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠒⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣅⣨⡷⠒⠒⠒⠒⠒⠒⠒⠲⣿⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⡘⠃⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠒⠒⣿⣀⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠒⠲⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠀⠀⠀⠀⠻⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠋⠀⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣻⠄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⣀⣀⠀⠀⠀⠀⢿⣄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢸⡀⠀⠀⢸⡇⠀⠀⠸⠧⠤⢴⡏⢉⣷⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠻⣦⣀⣀⣠⠞⠁⠀⠀⣾⠋⢹⡆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠟⠀⠀⠁⠀⠀⠀⠀⠘⠛⠛⠁⠀⠈⠳⣄⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠺⣇⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠀⠐⠒⠒⠒⡒⠒⠒⡒⢀⠉⠉⢉⠉⡉⡉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡰⢁⠃⡘⠀⡇⢸⠀⡇⢸⠀⡇⢸⠀⡇⠸⠀⢃⠸⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⠁⡌⢀⠃⢀⠁⢸⠀⡇⢸⠀⡇⠸⠀⢃⠀⡇⠘⡀⢣⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠇⢠⠁⠸⠀⢸⠀⢸⠀⡇⢸⠀⡇⠘⡇⢸⠀⢃⠀⢣⠈⢆⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠜⠀⠂⠀⠇⣀⣘⠀⠸⠀⡇⠸⠀⠀⠀⠃⠘⢀⣸⣀⣘⣀⣘⣀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡇⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠤⠤⠤⡄⠀⠀⡔⠲⠖⠲⡄⠀⢠⠀⠤⠀⠄⠀⠈⢠⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠗⠒⡇⠀⠀⡇⠀⠓⠚⡇⠀⢸⠀⠘⠂⠃⠀⢠⣟⠓⠂
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡇⠀⠘⠒⠛⠚⠁⠀⠀⠓⠒⠓⠚⠀⠀⠘⠒⠛⠒⠃⠀⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⡄⡇⠀⢀⡀⠀⠀⡄⠀⠀⢠⠤⡤⢤⡀⠀⡤⠤⡤⢤⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢧⡇⠀⢸⠀⠀⠀⢀⠀⠀⡄⠀⡗⠲⡇⠀⡇⠀⠓⢺⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⢸⠠⠀⠀⢸⠀⠀⠑⠒⠓⠒⠃⠀⠓⠒⠒⠚⠀⡀⣧⣯⡒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⢀⣷⣏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣨⣄⣀⣀⣸⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡹⢷⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠝⠉⠁