Indie kid Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⠀⠢⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠈⠀⡆⠀⢸⠆⠀⠀⠀⠀⠀⠁⠀⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⡄⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠐⠀⠉⠁⠻⠂
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣈⢠⡀⢰⡄⠄⠀⠀⠈⠁⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠂⠑⠐
⠀⠀⠀⠀⢤⣤⡴⣶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠄⠀⠄⠀⠀⠀⠀⠀⠤⠀⠠⠀⠀⠀⠀⠄⠀⠄⠀⠀⢀⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠄⠀⠄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⡄⠀⠀⠀⢁⣄⠀⣤⣈⠤⠀⠀⢀⡇⢠⠇⠁
⠀⠀⠀⠀⠘⡁⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠈⠉⢀⠀⠀⠀⠀⠀⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠰⠀⠄⠀⠀⠂⠘⠀⠀⠂⠀⠀⢰⠓⠄⠀⠃⠀⠠⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠷⠆⠘⠶⠀⠀⠀⠀⠄⠀⠆⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠒⢀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡈⣀⠀⠀⠀⠁⠰⠀⠘⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⢀⠀⠀⠀⣄⠀⠀⢀⠀⢀⠀⠄⠀⠑⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠛⠂⠀⠈⠁⠀⠀⡀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⣀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⢰⡤⣶⡀
⠀⠀⡀⠀⠐⡀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⡄⠀⠀⠰⠆⠀⠀⠀⠀⠑
⠀⠀⠁⠀⠈⡁⠌⠁⠀⠘⠃⠀⠠⡀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠃⢀⠀⠀⠀⠈⠳⠄⠁⢂⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠠
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠨⠁⠡⠀⠠⠉⠅⠠⠀⠀⠀⠁⠀⠀⠀⠀⢀⣶⡉⠃⠉⢳⣆⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠀⠀⠀⠀⣼⣷⡍⠘⠄⡍⢹
⠀⠀⠂⠐⠀⠒⠀⠁⠀⢀⠀⢀⣠⠤⠴⡇⠉⠉⠒⠀⠀⠐⠒⠂⠀⡀⠀⠀⠘⢿⣅⡴⠀⠈⣈⡄⠀⠀⠶⠀⠀⠗
⠀⠀⠀⠰⠀⡜⣤⠀⠀⠘⠋⠈⠉⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠤⠀⠈⠁⠀⠸⠇⠉⠀⠀⠀⠹⠂⠀⠀⠂⠤⠀⠁⠃
⠀⠀⠀⠉⠙⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠈⠀⠀⠀⠠⠦⠟⢠⡞⠖
⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⠀⠀⡀⠀⠄⠈⠀⠤⠀⠀⠀⣾⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠀⠀⢦⡀⠀⠀⠁⠀⠀⠀⣀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢤⠶⢂⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⡌⠂⠀⠀⠀⠀⠀⠀⠂⠩⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠄⠀⠄⠀⢠⠀⠀⠀⠀⢠
⠀⠀⠐⠀⠀⠀⡀⠀⡄⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠀⠰⢈⠀⠀⠀⠠⠐⠱⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⢃⣠⠌⠖⠀⠀⠐⠁⠀⠈⠁⠘⠀⠘⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠐⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠓⠆⠀⠀⠈⠁⠀⠐

⠀⠀⠀⠀⠀⠀⠀⣠⡔⠲⣤⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⣤⡐⠂⠀⠀⠀⣤⠘⢧⣄⠀⢠
⠀⠀⠀⠀⠀⠀⡀⠙⠆⠀⠛⠁⠀⠀⠘⠛⠛⠛⠛⠛⠛⠻⠿⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠦⣤⡔⠀⠀⠀⠙⢿⣄⠀⡠⢦
⠀⠀⠀⠀⢠⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠄⠀⠀⠁⠈⠛⢶⡾⣀⠀⠙⠷⣀⠨⣶
⠀⠀⣀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠆⠀⠀⠀⠄⠀⠀⠰⠆⠈⡿⠃⣤⡟⢁⡀⣤
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⡀⢀⣠⣤⣆⣀⣤⣤⣆⣀⣤⣠⣆⣀⣠⣇⢀⣤⠈⠉⣻⣿⡇
⠀⠐⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⡄⠀⢀⣄⠀⢶⡄⠉⣼⢻⡇
⠀⠀⠺⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠃⠀⢀⣤⡷⠒⠈⠙⠀⢀⡿⢦⡿⣸⡇
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⢀⣀⡰⠶⠛⠛⢁⠀⠀⠑⠀⢀⡾⠇⣠⡌⠀⢸⡇
⠀⠀⠶⣶⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣤⠐⠛⠉⠉⠠⡀⠀⠀⠈⠃⠀⢀⡾⠯⠄⠀⡾⠁⠀⢻⡇
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⢦⡀⠀⠀⠈⠆⠀⠀⢠⣶⠋⠀⠀⢀⠿⠂⠒⠂⣸⡇
⠀⠐⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠳⠀⠀⠀⠀⢀⡴⠋⠀⠀⠙⢦⡘⢀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⠀⠀⠀⡇⠀⢀⡾⠛⠁⠠⠀⠀⠀⡼⠃⠀⠀⠐⠂⢸⡇
⠀⠀⠰⠾⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢡⡴⢋⡀⠀⠀⠀⠈⢀⡼⠁⣠⣀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠙⠷⠶⠂⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣄⠀⣰⣶⡄⠀⠀⠀⠀⣴⣶⡄⠀⡄⠀⠠⠉⠀⠀⠀⠓⢄⠀⠀⠘⠃⠀⠀⠀⠀⠉⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⡿⣿⡝⢳⣄⠀⠀⠉⣄⠹⠿⠃⠀⣾⡆⠀⠻⠿⠃⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⣿⠇⠀⢹⡆⠀⠀⠘⡇⢀⡀⠀⢀⠀⠀⡀⢀⡜⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠉⠉⠀⣀⢸⣿
⠀⠀⣀⣤⣄⡀⠀⠀⢹⣿⣿⡟⠉⠀⠀⠀⢸⡇⠀⠀⠀⠈⠋⠉⠛⠉⠛⠋⠀⠉⠀⠀⢀⠞⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡟
⠀⠸⣿⣿⡟⣳⠀⠀⠀⠻⣿⣿⣄⡚⢃⣴⠟⠀⠀⠀⠀⣠⠄⠐⠢⡀⠀⠀⠀⠀⠀⠐⢯⡀⣸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇
⠀⠀⣿⣿⠻⣿⡇⠀⠀⠀⠈⠉⠛⠛⠋⠀⠀⠰⠊⢰⡞⠁⣤⠀⢠⣅⡀⠀⠀⢀⡾⣇⠀⢻⡇⠀⠀⣶⣆⡼⢿⣷⣿⢷⣄⠀⠀⠀⢰⣦⣀⣤⠈
⠀⢸⣿⣿⣶⣿⣧⣀⠀⠀⠀⠀⢀⣠⣄⠠⢤⣠⠀⠀⠃⠀⠀⠜⠋⠀⢸⡆⠀⠛⣤⡛⠀⠀⠀⠀⢰⣤⣿⡿⠛⣹⣿⣿⠼⣷⠀⠠⣾⣿⣿⣿⡄
⠀⠀⢉⠙⡿⠟⠉⠿⠟⠀⠀⠰⠉⠁⠈⠒⣄⠀⠀⠀⢀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠈⠀⠀⠀⠀⠀⠸⣿⣙⣧⣼⣤⣿⣿⣷⣯⣀⠀⠀⠻⠏⠉⠁
⠀⠀⠁⠀⡇⠀⠀⡀⢀⠀⠀⢠⠀⠀⡀⠀⠈⢇⠀⠀⡘⠃⠴⠂⠋⠁⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠰⢿⣿⠋⣿⢿⣿⢻⠿⣤⣿⣿⠃⣤⡄
⠀⠀⠀⠀⡃⠀⣿⠀⠀⠙⡄⠘⢠⡀⠘⠂⠀⢸⠃⠚⠀⠀⠀⢀⣤⣶⣤⣄⠀⣰⣿⣿⣿⣷⠀⠀⠀⠀⠙⠛⠧⠤⡿⢛⣶⣿⣿⣉⣿⣧⣴⣄
⠀⠀⠉⠈⠁⠀⠛⡀⠀⠀⠀⡦⠚⠁⠰⣄⡰⠂⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠟⣺⣿⣿⣀⣿⣿⡟⣿⣿
⠀⢰⣦⣴⣿⣿⡆⢃⡀⠀⠀⠁⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣧⣻⡿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⢿⡟⠁
⠀⢸⡿⠛⢻⣿⣷⣤⠙⢦⠀⠠⠔⠊⢁⣀⣠⠄⠶⠀⣾⡆⠀⠀⠀⠉⠙⠻⣿⣷⣼⡿⠁⠀⠀⠀⢀⡠⡤⠀⠂⠠⣄⠈⠙⠛⠋⠙⠷⠎
⠀⢸⣧⣤⣼⣿⡿⠟⠀⠀⠀⠀⠀⠀⠹⢿⠇⠀⠀⠀⠋⠱⣦⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⣀⠀⣠⠎⠁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⢀⡤⣤⡴⠂
⠀⠘⠛⠿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠉⢶⡀⠀⠀⠀⠀⣠⣶⠿⠛⠋⠀⠇⠀⢰⣦⠀⠀⣴⡄⠀⠀⡆⠀⠀⢏
⠀⠀⠀⠀⠀⢀⣶⣀⣤⣀⠀⢤⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠀⢠⣄⣀⡿⠁⠀⠀⠀⠀⡀⢀⠈⠋⠀⠀⠉⠁⣤⠀⡇⠀⠀⠘⠣⣄⣰⠆
⠀⠀⢀⠀⠊⠈⠉⠁⠀⠁⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠈⠛⠓⢀⠀⠀⠀⠀⠁⢎⠱⣀⠀⠀⣀⡰⢂⠖⠀⣠⣶⣶⣦⠈⠉
⠀⢀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠃⠀⠁⡄⠀⠀⠀⠀⠙⠠⠶⠶⠤⠙⠁⣀⠜⢻⣿⣿⣙⡁
⠀⠈⠛⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣶⣿⡟⢙⣿⡇
⠀⠀⠀⢳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢀⣤⣤⣄⠀⠀⠙⠛⠀⠙⡿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢑⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢀⡆⠀⣤⣶⣼⣿⣿⣿⣿⣇⠀⠀⠀⠀⣀⠃⠈⠛⠻⠽⣿⡷
⠀⠀⠀⠀⠀⠉⠑⠢⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣞⠀⠀⠀⢰⣦⣴⣿⣷⣶⣽⢿⣿⣿⣿⣿⣿⣿⠀⠀⢠⡞⠁⠀⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠳⡿⠇⠦⠀⠀⡀⢠⡀⠀⠉⡛⠛⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠈⠣⠄⠠⠆⠀⢠⡄
⠀⠀⠀⢠⡤⠚⠓⡹⠃⠀⠀⠀⣀⣄⡀⠈⠁⢿⡗⠶⠾⠋⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡿⠻⠿⠿⣛⣁⡀⢀⣤⣤⣤⡀⠀⠀⣀⠞⠳⣄
⠀⠀⠀⣤⠀⠀⣠⣧⣤⠀⠀⡰⠋⣀⡙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⡿⠟⠁⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠙⢆⣰⠋⢀⡀
⠀⠀⠀⠟⠀⠀⠀⢠⡍⠀⡜⢃⣠⣿⣷⡏⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠟⠋⠛⣿⣿⣿⣶⣄⠀⠸⠇⢀⣼⠃
⠀⠀⢴⡤⠦⠀⢀⠆⠀⠀⠇⢻⣿⣿⣿⡇⡟⢁⡾⠳⣆⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣄⠀⢀⣼⣿⣿⣿⡿⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⡇⢀⠛⠀⠀⢀⡼⠷⠛⣻⣿⣅⣷⣾⡤⠚⣩⣽⣷⣄⠉⣱⠀⠀⣠⡆⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⢀⣷⠈⠀⡀⠀⣿⢀⣴⣶⡶⢀⣾⣿⠯⣴⣿⣿⣿⣿⠋⢠⠋⠀⢰⣿⠀⠀⠀⢠⠶⣶⣤⠈⠙⠉⠉⠻⢿⣿⡿⠋⠀⠀⠀⣠⠤⣤
⠀⠀⠀⠸⠏⠀⠀⢹⣆⠙⣤⠛⢋⣴⣿⣿⠃⣤⡌⢻⣿⠛⠛⡴⠋⠀⣄⣸⣿⡄⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⡏⠀⠈⡿⠂
⠀⠀⠀⠀⠀⠀⠀⢈⣿⠂⠀⠉⠁⠈⠋⣿⠸⣿⡿⠀⠘⠉⠉⠀⠀⣠⣿⠃⢻⣿⣶⣄⠀⢰⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠸⡆⠀⢀⡀
⠀⠀⠀⠀⠀⢀⣴⣿⡟⢰⣿⠀⠀⠀⠀⠙⠦⠤⠤⠜⠃⠀⠀⢀⣴⣿⡏⠀⠀⢿⣿⣿⣷⡄⠛⠟⠃⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⠀⠀⠹⠛⠋
⠀⠀⠀⠀⣰⣿⣿⡿⠁⠘⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⣸⣿⣿⣇⠀⠀⣸⣿⣿⣿⣧⠀⣠⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟
⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⣿⣿⣿⡀⠀⢀⣄⠀⠀⠀⠀⣿⣷⢿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡇⣾⡏⠀⠀⠀⠀⢀⣾⣿⣿⡟⠀⢴⣄
⠀⠀⣶⠀⣿⣿⣿⣄⢀⣠⣿⣿⣿⣷⣰⡟⠁⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏⢸⣿⡁⠀⢠⡀⠀⣿⣿⣿⣿⡁⠀⢨⣿⣷
⠀⠀⠈⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡇⠀⠀⠀⠀⣸⣿⡟⠀⢻⣿⣿⣿⣿⣿⡟⠁⠀⢸⣿⣷⡀⠈⢿⣆⣿⣿⣿⣿⣷⣶⣿⣿⣿
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡟⠀⣿⣧⠀⠀⢀⣾⣿⣿⡃⠀⠀⢻⣿⣿⣿⣟⠀⠀⠀⣸⣿⣿⣿⣆⢸⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋

⡀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠙⣧⠸⠇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣧⡀⢻⡇⠀⠀⠀⠀⠀⠙⠀⢸⡇
⣷⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⢠⣾⠳⣦⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢀⣀⠀⠀⠈⢿⡘⠁⠀⠀⠀⠀⠀⠀⠀⣸⡇
⣿⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣼⡇⠀⠀⠀⢸⣷⠀⣿⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠠⡾⠃⠀⠀⠀⣿⠙⣷⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢀⣤
⣿⠶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡏⠀⣿⠀⠀⠀⠀⠻⠶⠶⠶⠶⠶⠾⠇⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⢀⣟⠁⠀⠀⠀⣿⠉
⡿⠀⣿⠈⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢘⣟⠀⣿⠛⣷⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢴⣿⠀⣿⣀⡀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⢰⡟⠻⢿⡇⠘⠋⠀⠛⠀⠿⠀⢹⡆⠀⠀⠀⠀⠀⠀⣀⣀⣸⡇⠀⣿⠀⣿⠉⣿⠳⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠺⣿
⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠻⣦⠸⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⣏⠉⢻⡇⠀⠁⠀⠉⠀⠛⠀⢸⡇⠀⠀⠀⠀⠀⢀⣤⣤⣼⡇⠀⡿
⠀⠀⠀⣼⡇⠀⠀⠀⣴⢶⡄⠀⠀⢹⣦⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠘⣷⠘⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢧⡌⢹⡇
⣀⣀⣠⡏⠀⠀⠀⢘⡏⠀⣟⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⣾⠻⡆⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠀⠀⣀⠀⠀⠀⠿⣌⠃
⠉⠉⠉⠁⠀⠀⠀⣈⡏⠀⡏⠀⠀⠀⠀⢿⣦⣤⣤⣤⣤⣴⡿⠀⠀⠀⠀⣷⠀⣿⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⣿⠉⣷⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⢴⡏⢙⡇⠀⡿⠷⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡿⠀⣿⠀⠀⠀⠀⠻⠷⠶⠶⠶⠶⠾⠟⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢹⡄
⠀⠀⢰⡶⢶⣾⠁⠘⠃⠀⠟⠀⡏⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⣏⠀⣿⠛⣿⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣿⠀⣾⣄⣀⠀⠀⠈⠛⠛⠛
⠀⠀⠈⢷⡄⣹⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⡟⠛⢿⡇⠈⠋⠀⠛⠀⠿⠀⢹⡆⠀⠀⠀⠀⠀⠀⣀⣀⣨⡇⠀⣿⠀⣿⠉⢹⠟⢦⡀
⣤⡀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠻⣆⠸⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣯⠉⢻⡇⠀⠁⠀⠈⠀⠘⠀⢸⡇
⠈⡇⠀⠀⠈⢹⡆⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⣴⢶⡄⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠈⢷⡜⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⡇⠀⠀⠀⠀⣷⣄⣀⣀⣀⣀⣤⡯⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⣶⠇⠀⠀⢀⣾⠛⣦⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢀
⠀⣷⣤⣄⣀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⣸⡇⠀⡇⠀⠀⠀⠀⠿⣦⣤⣤⣤⣤⣴⡿⠀⠀⠀⠀⣷⠀⣿⠀⠀⠀⠀⢣⡄⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⣾
⠀⡧⠀⣿⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢹⡇⠀⣿⠿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡯⠀⣿⣀⠀⠀⠀⠘⠷⠺⠗⠺⠶⠾⠛⠀⠀⠀⠀⣿
⠀⠀⠀⠁⠀⣿⠀⠀⠀⠀⠀⠀⢰⡶⠶⣾⡇⠘⠃⠀⠟⠀⣿⠈⢳⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡇⠈⣷⠀⣿⠋⣿⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣿
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠈⢳⡄⢸⠇⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⢰⡟⠛⢿⡇⠈⠉⠀⠛⠀⠿⠀⢸⡇⠀⠀⠀⠀⠀⢀⣀⣀⣸⡇⠀⣿
⠀⠀⠀⠀⣸⡏⠀⠀⠀⣠⣤⡀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠙⣦⠸⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⣯⡉⢻⡇
⣀⠀⠀⢸⡟⠀⠀⠀⢸⡏⠈⣿⠀⠀⠈⢹⡆⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⣴⠶⣤⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠈⢷⡘⠃
⠛⠛⠛⠛⠁⠀⠀⠀⢸⡧⠀⣿⠀⠀⠀⠀⣿⣄⣠⣄⣀⣀⣠⡏⠀⠀⠀⢠⣿⠀⣿⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⣾⠗⠀⠀⠀⣾⠛⣷⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⢠⡞⢻⡇⠀⣿⠴⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⠀⣿⠀⠀⠀⠀⠻⠶⠶⠶⠶⠶⠴⠿⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠈⢳
⠀⠀⠀⢠⣶⣤⣼⡇⠸⡇⠀⣿⠀⣿⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢙⣿⠀⣿⠟⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⠀⣿⣀⡀⠀⠀⠘⠛⠛
⠀⠀⠀⠘⢧⡄⢹⡇⠀⠀⠀⠀⠀⠉⠀⢸⡇⠀⠀⠀⠀⠀⢰⡷⠶⣾⡇⠈⠟⠀⠿⠀⣿⠈⢻⡆⠀⠀⠀⠀⠀⠀⡀⠀⢸⡏⠈⣿⠀⣿⠋⣻⠿⣤
⢀⣀⠀⠀⠀⢿⡌⠁⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠈⠳⣆⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠰⣟⠙⢿⡇⠀⠉⠀⠛⠀⠻⠂⢸⡇
⣿⠉⣷⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⣠⣤⣄⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠘⣧⡸⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⣿⠀⠀⠀⠈⣿⣀⣀⣀⣀⣀⣀⣿⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⣴⠶⣦⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⣿⠀⣿⣤⣀⡀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢼⣧⣤⣤⣤⣤⣤⣿⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⣾⠟
⣿⠀⣿⠀⢿⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⣿⠀⣿⡶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⠀⣿⠀⠀⠀⠀⠸⠷⠶⠶⠶⠶⠶⠿
⠁⠀⠉⠀⠙⠀⢸⡇⠀⠀⠀⠀⠀⢀⣶⢤⣼⡇⠀⠿⠀⢿⠀⣸⡉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⠉⣿⠀⣿⠟⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢷⣄⢹⡇⠀⠀⠀⠀⠀⠈⠀⢸⡇⠀⠀⠀⠀⠀⠀⡾⠷⢾⡇⠀⠛⠀⠻⠀⢸⠇⢹⡆⠀⠀⠀⠀⠀⠀⢀⡀⢘⣯
⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢀⣀⡀⠀⠀⠻⣇⠁⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠹⣦⡈⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠛⠻⣇
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣿⠉⣷⠀⠀⠀⢻⡦⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⣠⣴⣄⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠘⢷⡄⠿
⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⡟⠀⣿⠀⠀⠀⠀⢹⣆⣀⣀⣀⣀⣀⣿⠁⠀⠀⠀⣿⠀⣻⠀⠀⠀⠉⣷⠀⠀⠀⠀⠀⠀⣴⡿⠀⠀⠀⣰⡶⣦⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⣿⠀⣿⣤⣄⡀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢀⣿⠀⢹⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣿⠀⠀⠀⠀⣿⠀⢸⠂⠀⠀⠀⢿⡀
⠀⠀⠀⠀⢀⣤⣄⣸⡇⠀⣯⠀⣿⠀⣿⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⣿⠀⣾⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣽⠀⢸⠂⠀⠀⠀⠸⠷
⠀⠀⠀⠀⠘⣯⠈⢻⡇⠀⠀⠀⠀⠀⠉⠀⢸⡇⠀⠀⠀⠀⠀⢀⣶⠦⣼⡇⠀⠿⠀⠾⠀⢼⠉⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⠉⣿⠀⢸⠟⢳⣶⣄
⠀⠀⣀⠀⠀⠈⢷⡈⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢳⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣾⠓⢾⡇⠀⠛⠀⠘⠃⠸⠄⠈⡇
⠀⡾⠛⣷⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢀⣤⡀⠀⠀⠻⣏⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⣦⡸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢀⡇⠀⣿⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⣿⠉⣿⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⣠⣴⣄⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⣴⡇⠀⣿⣀⡀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢹⣄⣀⣀⣀⣀⣀⣿⠁⠀⠀⠀⣿⠀⣿⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⣰⡿
⠈⡇⠀⣿⠉⣿⠳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠾⣿⠀⣿⣤⣄⡀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⣿⠀⣿⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣾
⠀⠁⠀⠉⠀⠿⠀⢸⡁⠀⠀⠀⠀⠀⢀⣤⣄⣸⡇⠀⡿⠀⣿⠀⣿⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠙⣿⠀⣿⠶⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠘⢯⡀⢹⡇⠀⠀⠀⠀⠀⠙⠀⢸⡇⠀⠀⠀⠀⠀⢠⣶⠶⣼⡇⠀⠿⠀⠿⠀⣾⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⣀⠀⠀⠈⢿⡘⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⢳⣄⢸⡇⠀⠀⠀⠀⠀⠈⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠛⢿⡧
⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⣾⠛⣷⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⢀⣤⡀⠀⠀⠛⣇⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⣧⡸⡇
⠶⠶⠶⠶⠶⠾⠟⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣿⠁⣿⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⣠⡶⣦⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⣿⠀⣿⣀⡀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣿⠁⠀⠀⠀⣿⠀⣹⡇⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⣀⣀⣸⡇⠀⣿⠀⣿⠉⢹⠟⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠻⣿⠀⣻⣦⣄⣀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⣿⠀⢻⡇⠀⠀⠀⢸

⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⡄⢠⡄⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠓⠒⠀⠀⠀⠒⠀⠀⠀⠂⠀⠨⣥⡄⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢀⣤⡶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣅⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⣠⠇⠉⠳⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡤⢅⡀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠾⠋⠀⠀⠀⠀⢾⡶⠀⠀⠄⠒⠉⠐⠐⢀⡁
⠀⠀⠘⠀⠀⠀⠀⢤⠅⠀⡠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⣄⣀⣤⡀⠄⠁⠀⠀⢦⢈⡿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒⠂⠀⠀⠀⠀⠀⠀⠐⠀⠈⠉⠉⠉⠀⠀⠀⠐⠀⠀⠀⠀⠀⡌⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢿⡿⢀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠃⠀⠀⠀⠈⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⠋⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⢤⠿⣄⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⢀⡀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠨⠐⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠶⠤⠤⠔⠃⠀⠀⠀⢰⠶⢶⠀⠀⡀⠂⠑⣀
⠀⠀⠤⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠦⠐⢤⠄⠀⠈⠁⠀⢘⣀⣘⠀⠐⢀⡠⠖⡁
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠠⠀⠂⠀⠒⠰⠿⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⢸⠉⢹⠀⠐⠸⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠘⠀⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣾⠛⡟⠀⠰⠒⠐⠀⣅
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠘⠿⠦⠀⠀⠀⣿⠶⡇⠀⢠⡟⢛⠀⠆
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⣤⣤⠀⠀⢠⣶⣦⠐⠘⠄⠀⢺⣠⣧⠀⠈⠁⠃
⠀⠀⠀⠀⠀⠀⠚⠀⠀⠐⠄⠁⠀⠀⢀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠒⢠⣤⣰⠀⠀⠀⠀⠙⠋⠁⠀⠘⠉⠀⠀⠀⠀⠀⢸⡀⣸
⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠺⠅⠐⢀⣀⣴⠀⠀⡄⠂⠀⡀⠉⠀⠀⠀⠀⢀⢀⡀⠂⢀⠄⠀⠀⠀⠈⠀⠀⣸⣩⠇⠀⠀⣀⡀⢰⡶⠘⠟
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠟⣯⡰⢶⠀⠒⠻⠁⠀⠀⠀⠀⠀⠈⠀⠁⠀⠞⠂⠀⣀⠴⠞⠀⠀⠉⢡⣀⣾⡀⢋⣁⢀⣠⡄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠈⠀⢦⠀⠚⠁⠀⠀⠀⠀⠀⠀⠙⠓⠓⣾⠾⣿⠋⠉⠀⠀⠀⠀⠀⡈⢭⠻⠁⠛⠋⠈⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠋⠀⠀⠀⠀⠀⡉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢹⡥⠊⠀⠀⠀⠀⠀⠀⠀⣿⠈⢂⡒⠂⠈
⠀⠀⠀⠀⠀⠰⡦⠀⠀⠀⠀⠀⣀⠘⠁⠀⠀⠀⠒⠤⠀⡀⠀⠈⠛⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⠂⣀⡀⠀⡤⠄
⠀⠀⠀⠀⠀⢡⣄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠀⠀⠀⠀⣿⡀⠃
⠀⠀⠁⢤⣤⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⠆⠠⠠⠐⠀⠄⠀⠀⠀⠀⠀⠈⠓⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡂⠀⠀⣀⠀⣀⠀⣄⠤⠀⠀⠀⡀⠀⠤⠀⠀⠤⠀⠰⠀⡂⠀⢀⡆⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠃⠈⠁⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠶⠶⠰⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠶⠐⠀⠀⠰⠁⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠄⠀⠀⠀⠀⠀⣀⣀⡴⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⠀⠀⠀⠀⠀⠀⢅⣽⠛⠜⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⡇⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⣴⢶⠿⠓⠈⢇⢘⣿⢁⣴⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣾⢇⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠛⡝⠉⠛⠛⠐⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠆⢱⠐⠀⠀⠀⠀⠀⣰⣾⠏⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⡀⠀⠀⠀⠀⠀⠀⠀⢿⡅⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡇⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠒⠾⠖⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠒⠠⠀⠦⢤⠶⠀⡀⠀⠀⠀⠀⠄⠃

⣶⣿⣶⣿⣿⣿⣶⣉⠀⢀⠀⠠⠌⠀⠀⠐⢀⠀⠠⠀⠀⡀⢠⣖⣦⠠⠒⠆⡀⠀⠀⠋⠛⠛⠆⠀⢀⠀⠀⠁⠀⠀⠀⢄⠀⠀⠖⠈⠀⠀⠤⡄⢢⡤⣬⡳⣧⢸⡐
⣾⣥⡼⠊⠉⠙⠙⣿⣿⠀⠀⠀⠤⠤⠄⠤⣄⠀⠂⠘⠀⠈⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢤⡚⢺⠑⠂⠀⣸⠧⣷⠤⢿⡇⣸⣧
⠆⣙⠀⠜⢀⣀⣀⣿⡏⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢁⢧⠀⠀⠀⠀⠈⣀⠸⢇⣿⢻
⠆⢈⣳⣴⡏⢿⣼⡿⣧⡀⠀⠠⡈⠦⠄⠀⠠⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⢤⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠉⣼⢤⡞⣿⣾
⠁⠲⡇⡁⢀⠘⠁⠀⠀⠋⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠁⠁⠠⢃⠀⠙⣎⣷
⠀⠀⢹⠁⠁⠀⠀⠀⠀⠁⠒⠑⣠⣠⠤⠀⠀⠙⠀⠀⠇⢀⡆⠈⠆⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⢄⠀⠙⢀⠋⠁⠀⠈⢸
⠀⠀⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⡿⠁⠀⠀⠀⠀⠀⠀⢀⠸⠀⠀⡀⠀⡃⠀⠀⠀⠀⠠⠀⡊
⠀⡤⣯⠹⠌⠀⡀⠷⣄⣀⡂⠀⠀⠀⢠⡀⠀⠀⠁⠀⠁⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⡁⢼⠀⠀⠀⠁⠀⠅⣰⠀⠈⠠⠀⢧⠁
⣰⢹⣏⡀⠠⢀⡄⢀⡀⣯⡥⠀⠀⠀⠤⠠⢠⢶⠖⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠠⢨⡄
⢧⣿⡤⣽⣰⢯⣀⣈⡭⢈⡄⠁⠀⠀⠀⠀⣨⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⢠⠀⢠⠀⡂⡁
⣶⣿⣿⣿⣷⣤⡤⣤⢤⠈⠃⠀⠀⠿⠍⠁⠰⡂⠀⠨⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⠋⠉⢃⠀⢀⢸⠒⠅
⣿⣻⣿⣿⣧⡟⢿⡅⢱⢉⡅⠀⠀⠦⠀⡀⠀⠈⠓⢷⣶⠗⢓⠂⠀⣀⠀⠐⠃⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡇⠀⠐⠀⢀⡘⡀
⠛⡿⣟⣿⣍⠗⠉⡠⠊⠉⠀⠀⠀⣄⢄⠁⠈⢀⠀⠀⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⠀⠀⠀⠀⠂⢀⡄⢸⣯⠘⠁⠁⢀⠀⠀⠀⣿⢠
⠁⠀⣹⣿⣿⡆⠰⡇⠀⠀⠀⠀⠀⣵⡎⠀⠄⠊⠀⠴⠘⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡤⠀⠀⠀⢰⠠⠀⡠⣃⠉⢈⣉⡀⠀⢀⣀⡄⣀⣴⠇⠈
⣷⡎⠓⣉⡥⠑⠀⠃⠠⡀⠀⠀⠀⠉⠄⠁⢀⣤⡉⠐⠒⠊⠀⠄⠀⠫⠈⠒⠘⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠃⠀⠀⠈⢠⠀⠉⣇⠀⢸⣿⣷⣼⣮⠉⠙⣨⡿⠀⢐
⣡⠷⠤⡜⠀⠀⠀⠀⠀⢸⠊⠀⠀⠈⠀⠀⠈⠹⣷⡄⠀⠄⠃⡀⠀⠀⠉⢛⠄⠀⠀⠐⠀⠐⠀⠐⢀⠀⠀⠀⠀⠄⠀⡀⠄⠡⠍⠁⡀⢌⠉⢠⡾⠠⣠⣿⢁⢈⣯
⠁⠀⠀⡇⢄⣀⣤⣀⡄⢸⠀⠀⠀⠀⠈⢀⣀⡀⠸⠃⡂⢀⠆⠀⠀⠀⠈⠃⠀⠁⠀⠀⠂⠀⠀⠐⠀⡀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠈⠀⣀⡄⠁⣾⡿⠛⡁⠊
⣀⠀⣠⡇⢇⠬⢤⠤⡄⠈⠄⠀⠀⡀⢈⡂⢍⠠⠄⢁⡀⠘⠑⠀⠀⠀⠀⡇⠀⠄⢢⠀⠀⠀⠠⠴⠦⣾⣕⡀⢅⡀⠀⠀⢀⡦⠠⢂⣄⣶⣞⣿⠰⢌⣣⣮⣁⡌⣰
⠁⠒⠀⢿⡸⠀⠀⠀⢔⢰⠀⠀⠀⠀⠀⠀⣃⣠⢀⡁⠀⠐⠠⠀⠀⠐⣨⡁⠀⠈⠩⡄⠀⡀⠈⠦⠚⠁⡊⡇⠨⠤⣀⠐⢠⣇⠜⠊⣯⣞⣻⠃⡗⢸⣇⠀⠀⠀⢸
⠁⠀⠀⠘⡇⠀⠀⠀⠘⠾⡄⠀⠀⢹⣦⣾⣧⣓⠋⣤⣾⣗⣦⣀⠀⠀⠀⠀⠀⠀⠀⠒⠐⣄⢓⠀⠐⢌⡇⠈⠀⢰⠿⠀⠀⡃⠰⠀⠟⢡⣃⣴⢳⣿⡏⠉⠝⢹⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠖⠐⠆⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠆⠀⠀⠑⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⣀⣀⠀⢩⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⢀⣀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠐⠀⣡
⠀⠀⠂⢁⠄⠁⠀⣠⠄⠉⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢻⡀
⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠠⠴⠂⠀⠘⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢨⠙⢀
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠠⠊⡈⢀⡄⠁⠘⠃⢀⣄⡐⠀⠀⠀⠀⡄⠀⠠⠁⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠘⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡀⠃⢸⡇⠀⠀⢀⡈⠉⠃⠈⠀⠀⠀⠠⣘⢶⠾⠷⢿⠀⠖⠀⠀⠀⠀⠈⠀⢀⡄⠀⠀⠀⣟⠀⢠⠘⡄
⠀⠀⠀⠀⠀⠀⠠⢤⡄⠈⠓⠀⢸⡷⡶⠶⠭⣷⡀⠀⠀⡠⠋⢀⡔⢬⠃⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⠀⠁⠀⠁⠀⢰⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⡇⠀⠀⠀⢈⡡⠛⠀⡀⠠⠋⠀⣴⠶⣤⣄⠀⠀⠀⠘⠀⠀⠀⣠⡾⠋⢄⡶⠀⠀⣀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠉⣧⠞⠣⠀⠀⠀⠀⠀⠀⠀⠈⠛⠖⠀⡀⠄⢀⣾⡏⢠⠀⠈⣏⣠⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⠀⠸⡏⠤⠀⠀⠀⣭⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⣤⣶⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠡⠀⠀⠀⠀⢒⡶⠀⠀⠀⠀⠀⠀⣰⠿⢻⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠁⠀⠀⢻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠀⠀⠀⠀⣀⣀⠴⠞⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠚⠚⣿⠁⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣅⠀⣀⣀⠀⣀⠴⠤⠦⠤⠀⠛⠙⠁⠠⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠚⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠀⢠⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠶⠣⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⣀⠀⠠⠄⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠂⠀⣦⡀⠀⠀⠀⠀⢈⡏⠀⠒⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠐⢴⡀⠀⠈⢟⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢠⣤⣾⡶⣶⣶⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠦⢶⡀⠀⠀⠀⠀⠀⠀⠑⠣⢄⣬⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⢀⣤⠞⠊⠀⠀⠈⠙⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢤⡎⢱⡀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⢄⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠳⢆⣰⣦⡀⠀⠀⠀⢀⠀⠖⣠⠃⠈⠻⠦⣀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣶⣄⠀⠀⠉⠀⠈⣷⠀⠀⠀⠈⠙⠲⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠉⠛⢦⣄⠀⠀⢻⠄⠀⠀⠀⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠓⡄⠀⠙⢻⡤⣯⡘⢢⠀⠀⠀⠻⣄⠀⠀⢀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢠⠀⠀⠈⠀⠀⠀⢻⡀⠈⢢⣶⠀⠀⠀⠈⢷⣏⣩⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⡆⠀⠀⠀⠀⠀⠈⠗⠀⠀⠉⠳⢤⣤⠐⠁⡶⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⣀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⣠⣤⡄⠀⠀⠀⣤⣤⠀⠀⢀⣤⣤⡀⠀⠀⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⢰⣿⣿⠃⠀⠀⣸⣿⠏⠀⢀⣾⣿⡟⠀⠀⢠⣾⣿⠏⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠏⠀⢠⣿⣿⡏⠀⠀⢰⣿⡟⠀⢀⣾⣿⣿⠁⠀⣰⣿⣿⡟⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠁⠀⣠⣿⣿⡟⠀⠀⣠⣿⡟⠀⢀⣾⣿⣿⠃⠀⣰⣿⣿⣿⠁⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⣠⣶⣿⡿⠋⠀⠀⣰⣿⡟⠀⣠⣿⣿⣿⠏⠀⣰⣿⣿⣿⠃⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⠿⠛⠉⠀⠀⣠⣾⣿⠏⢀⣴⣿⣿⣿⠏⠀⢠⣿⣿⣿⠏⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⣿⠟⣡⣴⣿⣿⣿⣿⠋⠀⢀⣿⣿⣿⠏⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⢀⣾⣿⣿⠏⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣠⣿⣿⣿⠋⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⣀⣤⣾⣿⣿⡿⠃⠀⢀⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣰⣿⣿⡿⠋⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⣀⡀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⡿⠟⠁⠀⣠⣾⣿⡟⠉⢀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⠉⠛⠻⠿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⡿⠋⠀⣰⡿⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣄⠀⠈⠙⢿⣿⣷⣶⣤⣀⠀⠀⠀⠙⠛⠛⠛⠛⠁⣰⣾⠿⠋⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⣿⣷⣤⡀⠀⠉⠻⢿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣴⣾⣿⣇⠀⠀⠘⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣈⣻⣿⣿⣦⣄⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡛⠿⢿⣿⣶⣤⣄⣀⣀⣀⠉⠉⠉⠛⠛⠛⠿⠿⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣟⠿⡟⠿⢿⣦⣄⠈⠉⠛⠛⠿⠿⢿⣿⣿⣿⣶⣶⣤⣤⣄⣀⣀⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⢻⡄⠙⣦⡀⠈⢻⣿⠆⠀⣠⣤⡤⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠿⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣷⠀⣿⠏⠀⢸⣏⣴⡾⠟⢋⣠⣤⣶⣶⣶⣶⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⠏⠀⣰⣿⣿⣫⣴⣾⠿⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣾⣿⠋⢀⣴⡟⢩⣿⠟⠋⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣀⣀⡀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣼⠟⠁⣠⣾⠟⠀⠋⣁⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣤⣴⠿⠋⢁⣤⣶⡿⠟⠋⠁⠀⣀⣀⣀⣀⣀⣀⣀⣉⣉⡉⠉⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠘⠛⠛⠉⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⡀⢀⣶⣷⣷⣷⡄⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠈⠀⣀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠙⢄⡈⠀⠀⠀⡂⠀⠈⠉⠔⢂⠁⠀⠀⠀⠀⣠⣴
⠀⠀⠀⠀⢀⠂⠀⠀⠀⣀⣀⣀⠀⠠⠀⠀⠀⠀⢈⠀⠃⠀⠀⠈⠁⠀⠀⠀⠀⠠⠒⠀⠀⠀⠀⠛⡯
⠀⠀⠀⠀⠋⠅⠀⠀⠀⣏⣻⣆⡀
⠀⠀⠀⠊⠡⢦⠀⠀⠀⠀⠀⠀⠀⠁⠲⠀⠀⠀⠈⣔⠀⠀⠀⠈⠁⠀⠀⠀⣄
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠲⡶⢶⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠃⠃
⠀⠀⠀⠀⠈⢹⠀⠀⠀⠇⠀⠢⡀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⢀
⠀⠀⠀⠀⢀⠘⢆⠀⠐⣤⡀⠁⠈⢈⡈⠉⠑⢂⠀⠀⠀⠐⠁⠀⠀⠔⠀⠈⠉⠠⠀⠈⠠⠌⠳⡀
⠀⠀⠀⠀⠒⢰⡄⠐⠀⠀⠛⠂⠈⠉⠉⡀⠀⠀⠐⡄⠠⠄⠀⢤⠠⠄⠀⠡⠄⠀⠀⠀⠀⠐⠄⠙⢂
⠀⠀⠀⠀⠁⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡇⠂⠀⠀⠨⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠀⠀⠀⠀⠘⠂⠀⠀⠀⣧⡃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠊⠈
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠁⠀⠀⠐⠀⠀⠀⠀⢣⡇⠀⠀⠤⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⣇⣓⠀⠀⠑
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠘⠂⠀⠀⡀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⣈⢻⠆
⠀⢠⣾⣿⠀⠀⠁⠀⠀⠀⠀⠀⠌⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡌
⠀⢸⣸⠼⣄⠊⢀⡤⠀⢀⢤⣄⡀⢢⡀⢀⠀⠇⠀⠀⠀⠀⠠⠀⠀⠲⠼⠝⠋⠀⠀⠀⠹⣿⡿⢀
⠀⠈⢛⢭⣲⠀⠈⡛⠀⠀⠀⠈⠄⠈⢩⠓⠀⠀⠁⠠⠀⠠⠂⠀⣄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈
⠀⠀⠀⠙⠈⠁⠨⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠏⠃⠀⠀⠀⠀⠀⠂⠀⠀⠁
⠀⠀⠀⠈⠤⠂⠇⠀⠀⠀⠂⢀⠠⠀⠈⣀⠀⠀⠀⢠⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⡀⠀⠈⠠⠈⠁⠀⠀⠀⠀⠀⠀⣰⡈⠃⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⢀⣠⣴⡇⠀⡠⠀⠀⠠⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⢄⣀⠀⠦⠀⠀⠑⠤⠀⠀⠀⠀⠀⠀⠙⠻⠇⠀⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠈⠀⠐⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣖⠀⠀⣄⠄⠀⠄⠀⡀⠞⠃⠉⣶⠀⠁⠀⠆⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡄⠀⠤⠀⡐⠀⣄⠁⠐⠀⠀⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠂⠰⢅
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠟⠊⠒⠲⠊⠒⠒⠒⢃⣀⠀⢀⠀⠒⠀⠀⠀⠀⢁⣀⠀⡀⢰⠊⢆⠀⡀⠀⠈⠓
⠀⠀⠀⠀⠀⠐⠀⠀⢁⠀⠀⠀⠐⠀⠄⠈⢠⡈⣻⣧⣾⠀⢀⡄⢀⡀⠀⣏⠈⠀⠃⠸⣇⣎⠄⠁⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠀⠀⢇⠙⠴⠀⠈⠙⠟⠟⠓⠀⠀⠀⠀⠀⡠⠈⠁⠉
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠊⢀⠀⠀⠉⠀⠀⠀⠀⠄⠀⠀⠁⠐⠂⢈⠀⢐⠀⠀⠀⡅⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠤⠐⠂⠐⠀⠀⠀⠀⢀⡀⠀⡇⢀⠂⠐⠀⠀⡈⢁⠀⠰⠀⠀⠁⡀⠀⠂⠁⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠄⠘⢻⠁⠀⠤⠄⠀⠀⠃⢠⠀⠀⠀⠰⠤⠦⡀⠀⡄⠄⠀⠂⠀⠉⠀⠸⠟⡅⠠
⠀⠀⠀⠀⠀⠀⠠⡆⠀⠂⠀⢘⠺⠦⠀⠒⠀⠠⠀⠠⠀⠀⠀⠀⡄⠀⠀⡄⠀⠉⢀⠀⣘⠀⢀⡏⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⠀⠚⠁⠉⠹⡉⠁⣫⠀⠁⠐⠂⠑⠆⠑⠂⠐⠀⠀⠁⠱⠃⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⠖⠄⠈⠄⠀⠀⠀⠀⠀⠇⢽⠀⡠⡀⠀⠀⠀⠀⠀⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠊⠠⠀⠀⠀⠀⠀⢀⣀⡁⠾⠀⠀⠄⠀⠀⢀⣀⡀⠀⢰⢎⡹⡄
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⣧⠄⠚⠀⡀⠲⠖⠀⠈⣿⠃⠀⠉⠓⠙⠁
⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠠⠼⠀⢸⡗⠖⠺⠃⠀⠁⠀⠉⠢⠙⠩⠓⢏⠀⠂⢀⠑⠀⠰⢟⡑⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠙⠒⠀⠒⠠⠂⠄⠀⠀⡄⠀⡠⠠⢀⠀⠽⠀⢈⠇⠀⠀⠙⠀⠐⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⡀⠀⠀⢀⠀⠀⢀⠀⠀⢠⠀⠃⠀⠀⠀⠀⠠⠊⠂
⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⣀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⠚⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠬⠀⠀⡀⠈⣻⠖⠒⠚⠢⠀⠀⠀⢀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠘⠂⠔⠋⠀⠀⠁⠀⠈⠂
⠀⠀⠀⠀⠀⠀⣀⠂⠀⠱⡇⡣⡀⢀⣀⡀⠀⠀⠀⠀⢀⠀⠀⢀⠄⠉⠀⠘⠢⠀⠀⠴⠀⠠⠆⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠱⣄⠀⠐⠀⢽⢆⠈⢀⠀⠀⠠⠒⠄⠀⢀⡀⠌⠢⠀⣄⠀⢔⠁⢸⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠓⠀⠀⠈⠀⠀⠀⡌⠀⠀⠀⠘⠀⠀⡀⠀⠀⠦⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣤⣶⣾⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡟⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣮⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⡙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣶⡏⡄⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⢧
⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠩⣍⠛⣷⣤⠾⢧⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⣠⣤⣄⢰⡿⠛⠛⠛⠙⢿⣦⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⣄⢀⣸⠇⠀⠙⠓⠀⠻⡇⢢⡀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⡟⠛⢻⡟⠀⠀⠀⠀⠀⠀⣿⣤⡀⠀⠀⠀⠀⢹⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢉⠈⠘⢙⡳⠀⠀⡠⠶⢄⠘⣷⡀⠀⠀⠀⠀⢰⣿⣿⡏⠀⠀⠀⠀⠀⠀⢠⠀⢠⠵⠤⠀⠀⠞⢋⡗⠁⠘⠃⠀⠀⠀⠀⠸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠀⠀⠈⠁⠀⠀⠀⠘⠂⣇⠸⠃⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠓⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠘⡆⠀⠀⠀⠀⠀⠀⠀⢀⠿⣦⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢼⠁⡾⠛⠻⣆⠀⠈⠓⠀⢀⣴⣯⣄⠙⣧⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠐⠖⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠃⠉⠀⠀⠐⠹⠃⠀⢹⠂⡿⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣲⠄⠀⠠⠖⠉⣂⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡀⠀⠈⣩⡛⠓⢶⣶⠀⡠⠤⣤⣴⣀⡀⠈⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣂⣀⡠⠂⠀⠶⠒⠚⢛⡏⠁⢠⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⣰⡟⠀⠈⠛⠀⠀⠘⣿⡾⠁⠀⠀⠀⢶⡆⠁⡀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⡋⡍⠀⠀⣠⡄⠰⡦⠀⠀⠘⣘⠀⢻⣧⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠈⠁⢸⡇⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢁⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠋⠆⠀⠉⠃⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢠⠛⠀⠘⠐⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣴⣶⣶⣿⣏⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⡶⠶⠿⡿⠿⠟⠛⠉⠉⠉⠈⠉⠨⠍⠚⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⢀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠹⠲⣶⣆⠘⠗⡓⣺⠾⢿⣻⣿⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣉⣙⣁⢈⣛⣈⣛⣈⣛⡋⠇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠘⣯⣟⣀⢿

⠀⠀⠀⠀⠀⠀⢀⣼⣿⡄⠀⠀⣄⠀⠀⠀⠧⢀⠋⢳⣤⡀⠀⠀⠀⣠⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⡦⠀⠀⠀⢀⠉⠉⢠⡀⠀⠈⣕⠢⢴⡀⢸⠆⠀⠙⣀⣈⠶⠁⢠⠀⠀⣆⣰⣶⣄⠀⠀⠀⠀⡄⠀⠀⠀⣀⣄⢀
⠀⡄⢀⠀⠀⠀⡀⠀⠀⠈⢋⡑⠠⡈⣆⠀⠉⠻⠄⠀⠀⠾⠃⡀⠀⢸⠁⣀⣌⣱⣄⠈⠀⠀⠀⠀⠀⠀⢀⣌⢉⣥⣜⠉⢦
⠀⠂⠠⠀⠀⠈⠁⠀⠘⠆⠄⠣⠀⠀⢙⠖⠉⠀⠀⠀⠀⠀⠀⠀⠰⡖⠊⠉⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠘⠿⠛⢁⡌
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣸⣤⣁⣷⡴⠁⠀⠀⠀⠀⠀⠼⢿⡁⠀⠀⠸⣀⡊⠀⠸⠂⡉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠴⠟⠋⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠩⠦⣄⡀⠀⠀⢀⠁⠀⢠⣿⠗⠀⠀⡀⠀⠀⠀⠀⠀⡏⠀⠰⠁⢀⣹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠬⠯⠐⠂⣤⡆⠀⠈⠹⠀⠀⠉⠀⠀⠀⠀⠀⡴⣁⠠⠄⠀⠈⠁⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠁
⠀⠀⠀⠀⣀⠀⠐⠒⢂⠐⠈⣸⠃⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠳⠬⣀⠀⠱⠄⠀⠀⠀⠑⢀⠀⢠⠀⠀⢈⡇⠀⢀⡆⠀⠀⠀⠀⠀⠀⣶⠆
⠀⠀⠀⠀⠈⠁⠠⠀⠀⠈⣱⠆⣀⢠⠺⡇⠰⢦⣄⠠⠄⠶⣌⠀⠀⠘⠆⠒⠚⠂⠀⠀⣞⣱⡆⠈⢳⡌⣷⡖⣿⣥⠀⡼⠃⠀⠀⠀⢀⣠⠴⠟⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠔⠋⡄⠀⣲⡀⢀⡏⢂⠀⢶⠀⠙⣶⣄⣁⠀⠀⠀⢠⠞⣿⣿⣿⡆⣠⣤⣽⡹⣧⡾⣻⡄⠀⡀⠤⠒⠉⠀⠀⠀⠙⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣇⠀⠈⢃⡘⠃⢤⠇⣀⠀⠀⠈⠀⠈⠁⠀⡴⠋⠀⠐⢯⣽⠿⢾⣷⣤⢿⣼⣷⣿⣥⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠉⠀⡀⠀⠀⢠⠤⠄⠀⠀⠀⠀⠀⠀⠀⢻⣞⢳⣮⣽⣾⢷⣯⠤⠤⢤⠄⡀⠤⠤⣤⠂
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⢀⡀⠀⠀⣡⣴⠛⠛⢛⣬⣷⡿⣷⡞⣃⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⠛⠛⠛⠻⠛⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡙⠋⠁⣩⠞⢹⣿⣶⣶⣽⡿⢿⣆⠀⠀⠀⢰⣶⠀⠀⢴⡆⢠
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⠀⠀⠀⠗⣩⠏⠁⢠⡚⢿⣹⢉⣯⣥⢿⣿⠀⠀⡔⠀⠁⢀⣶⣌⠁
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡣⢀⡀⠀⣭⣶⢏⣧⡟⡆⠋⠿⠷⢤⣄⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡥⠠⣄⣀⠿⠞⡛⣷⢿⡟⠛⠂⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠂⠄⣸⣟⢦⡌⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⠐⢄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⢑⣆⠀⠀⠠⠄⠀⠀⢠⡄
⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠘⠧
⡀⠈⠒⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⢰⡿⠄⣦⡄⢰⡄⠸⠊⠃
⠑⠀⠂⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣇⠁⠐⣿⠏⢨⣿
⠆⠐⡂⠀⠀⠀⠤⠄⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⣀⠉⠉⢈⣩⣴⣶⣶⡟⠀⠀⠀⠀⠀⠀⢸
⠀⠚⠀⠐⠀⠂⠀⠀⠀⠃⠈⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢙⢿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠹⣈⢲⡿⣿⣿⡿⠋⠁⣤⣴⠂⢤⡀⠀⠀⠀⠀⠇
⢀⡀⣀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢇⠃⠀⠀⠀⠀⠀⠀⠉⢻⡟⣿⣿⣷⠀⢸⡄⣭⡤⣾⠙⠀⠀⠀⠀⠈⢣
⠈⡽⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⡄⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿⣿⣧⠘⢷⠘⠧⠻⣆⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠤⠌⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠺⠀⠰⣄⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠙⣧⠀⠀⠀⠀⠈⠀⠀⡀⠔
⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠢⡀⠀⠀⠘⣧⣤⣴⣿⠀⠀⠀⠀⠀⠀⠀⡄⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢄⠀⠈⠙⠛⠉⠀⠀⠀⠀⠉⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⣄⠰⠀⠤⢉⠄⠀⠀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⣠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢢⠐⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⠀⡤⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏⠀⣸⣿⣿⣿⣷⣀⣀⡀⠀⠀⣀⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠋⠉⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⡵⠟⣫⣭⣭⡛⠭⣉⠉⠙⠻⢿⣿⣿⡄⢀⡾⠁⢰⣶⣶⣤⣤⣤⣶⠶⠾⣛⠿⢟⣛⡻⡿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⣿⣿⣿⣾⣿⣮⡳⡑⡦⠄⣙⣿⣿⣿⡇⠀⣿⣿⡿⠿⠛⠛⠁⡠⠊⠁⣿⣩⣭⣝⣿⣿⣿⢿⣆
⠀⠀⠀⠀⠀⢀⡿⣴⣲⣶⣶⡟⢿⡿⣿⣿⣿⣇⢱⢱⣤⣤⣿⣿⣿⠏⠀⠻⣆⣠⠒⠒⡲⣰⠁⠀⣀⣻⠛⠛⠙⠽⣿⠿⣦⢿⡄
⠀⠀⠀⠀⠀⣸⡇⣿⣿⣿⣿⠗⣀⠸⣿⣿⣿⣾⢈⢶⣭⣽⣿⣿⣷⠀⠀⣸⡏⠛⣿⣿⡃⡇⣾⢩⣿⣍⢸⡉⡱⠸⣿⡿⣸⢸⣷
⠀⠀⠀⠀⢀⣿⣿⡸⣿⣷⣾⣿⣿⣷⣿⣿⢗⣋⣘⡺⢟⢿⣿⣿⣿⠆⠀⠘⣧⠀⠳⠞⠋⣣⠹⣌⣿⣧⣦⢨⣄⣸⣿⣿⡿⣼⣿⣇
⠀⠀⠀⠀⣸⣿⣿⣿⣜⡻⢿⣿⣿⣯⠟⣵⠟⠁⠀⢨⣷⡹⣹⣿⡏⠀⠀⠀⢸⣧⠀⡔⣫⡶⠒⠚⣭⡻⣿⡻⢿⣛⢿⢟⣽⣿⣿⣿⡄
⠀⠀⠀⢀⣿⣿⡇⠈⠛⠿⣷⣶⣶⣶⡄⣇⠀⣀⣴⡿⢋⡇⣿⣿⣿⠀⠀⠀⠻⣏⢸⢼⠋⠀⠀⣼⡿⣣⢨⣭⣭⣥⣶⠿⢿⣿⣿⣿⢿⡀
⠀⠀⠀⣼⢿⣿⠀⠀⠀⠀⣼⣿⣿⣿⣧⡹⢿⣿⣷⣶⢟⣴⣿⣿⡟⠀⠀⠀⠀⢹⣷⡸⣦⣠⣾⣟⣵⡟⣼⣿⣯⠉⠀⠀⠈⢿⣿⣿⡞⣧
⠀⠀⢠⡟⣿⡇⠀⠀⠀⢰⣿⣿⣿⣿⢿⣟⠲⠦⣴⣶⣿⡿⠁⠈⠀⠀⠀⠀⠀⠸⢿⣷⣬⣛⣛⣛⣩⣾⣿⣿⣿⣇⠀⠀⠀⠈⢿⣿⣿⠹⡆
⠀⠀⢸⠧⣿⠁⠀⠀⠀⣾⣿⢟⡽⠃⠀⠙⠛⠶⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣯⣉⣹⣿⠿⠋⠘⣯⡻⣄⠀⠀⠀⠈⢿⣿⡇⣿
⠀⠀⠸⣧⡙⠀⠀⠀⠴⣟⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⠀⠚⠋⣠⠟
⠀⠀⠀⠈⠛⠓⠶⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡄⠀⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠉⡁⣈⠁⠀⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠠⢌⡣⠥⢠⡌⢱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡃⡞⠀⠊⠈⠳⡄
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠇⠈⠧⣀⢀⢀⡓⠀⠰⠀⡆⠢⡀⠀⣠⠆⠀⠀⠀⣰⢦⠀⢠⢶
⠀⠀⠀⠀⠀⢁⠀⠁⢖⡛⠛⡓⠊⠁⣠⠟⢡⠀⠃⠙⠀⡨⠬⠥⣼⣆⠀⠀⠙⢾⡤⡾⠋⠀⠀⣰⠉⣇⡐⣲⠀⡀
⠀⠀⠀⠀⠀⠂⡖⣀⠀⢙⡆⠀⡀⠈⣈⠕⠂⡎⡤⠒⠢⡀⠠⠤⣄⢉⡄⠀⠀⢈⠀⠇⠀⠀⢠⠤⠀⡄⠀⠙⢠⡞⢀
⠀⠀⠀⠀⠀⠸⡇⣧⠄⠚⢉⣈⠁⢰⠃⠀⠀⠿⠁⠀⠀⠑⣠⠴⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠻⣦⠀⢡⠚⠁
⠀⠀⠀⠀⠀⠡⠙⢀⡴⠋⠉⠁⠀⠚⠂⠀⢀⡴⠋⠉⡥⠋⠁⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠰⠄⠸⡏⠀⠀⣩⠁⢈⡅⠶
⠀⠀⠀⠀⠀⠠⠀⠈⣠⠀⠘⣇⣠⠴⠒⠒⡿⠁⡔⠉⢧⠀⠀⠀⢈⠐⠲⡅⠩⠥⠀⠬⠅⠌⠀⣄⠀⣀⠴⠏⢀⣘⢢⡄
⠀⠀⠀⠀⠀⠥⠶⠄⣏⠀⡀⠈⠃⠀⠀⠀⡇⠀⠃⣠⠄⠂⠀⠀⠀⢱⡀⠘⡇⠻⣿⠟⠀⠀⢠⠏⠈⠁⠀⣤⢌⣧
⠀⠀⠀⠀⠀⠀⢀⠴⠛⢲⡧⠖⠒⠂⠀⠀⠁⢀⡞⠀⡔⠂⠀⠀⢀⠼⠁⠀⡇⠀⠘⡆⠀⢠⠞⠀⣸⠇⠀⠙
⠀⠀⠀⠀⠀⠀⠏⠀⢠⠏⠀⡔⠀⣠⣤⣄⠀⢸⠀⠀⣇⠀⢄⡀⠀⠀⠀⠀⠀⠀⢀⡇⠐⢁⠤⢰
⠀⠀⠀⠀⠐⣴⠒⠂⢾⠀⠸⡅⠈⢀⡠⠤⣄⣘⣆⠀⠘⣆⠀⠉⠛⠂⠀⠀⢀⡠⠊⠀⣰⠁
⠀⠀⠀⠀⠀⢳⠀⣀⠘⢆⠀⠈⠁⡏⠀⠀⠀⠈⠉⠀⠀⠘⡆⠀⢸⠀⣀⡤⠤⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠀⢀⣥⠀⠀⠒⠂⣶⠐⠀⠀⠁⢀⣘⣛⣁⣀⣀⣀⣀⣘⠀⠀⠀⢢⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠉⠀⢠⡀⠀⣿⠠⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⢠⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠐⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠳⠲⠾⢿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠴⠖⠚⣷⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢄⠀⣆⠀⣿⢋⠕⠛⠓⢯⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡷⠼⠀⠀⠀⢈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢰⡇⠀⠀⢀⡠⢎⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣾⡴⣒⣫⣥⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣩⡞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠒⠉⠉⢳⠀⠀⠀⣶⣀⠀⠀⠀⢀⣆⣸⣁⣉⣽⠿⠛⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠰⣀⠀⠀⠉⠁⠈⠣⢤⣄⣙⣒⣒⣪⣭⣭⠷⠛⠈⠁⠈⠉⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢶⣖⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⢖⣲⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠒⠒



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢰⠏⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠘⢷⡄⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠸⠇⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠴⠶⠶⠶⠶⠶⠶⠶⠶⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⡶⠛⠉⠁⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠉⠙⠳⣄⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣴⣶⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣻⣭⣭⠹⣦⠀⠙
⠀⠀⠀⠀⠀⢤⣀⣀⠀⠀⠈⣿⠀⠉⠉⠛⠛⠛⠿⠿⠿⠿⣿⠿⠿⠛⠛⠛⠋⠉⠁⣸⠏⠁⠀⣸⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⣤⠶⣿⢦⣄⠀⠀⠀⠀⠀⢠⣿⣠⡴⠟⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢻⣆⠄⠀⠀⠀⠀⠀⣿⠀⠋⠀⣿⠀⠀⠀⠀⣠⣿⣩⣥⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢀⣙⣧⡀⠀⠀⠀⠀⣿⣀⣀⣀⣿⠀⠀⢀⣴⣟⡉⠁⠀⠀⠀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠋⠉⠉⠙⠷⢦⣤⣤⣬⣥⣭⣭⣤⡤⠶⠋⠉⠉⠙⠛⣦⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⢤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣠⣤⣤⣤⡤⠶⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⡟⠛⠛⠂
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⣠⣄⠀⠀⡾⠁⠀⠀⠀⡇⢀⣤⡄⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⡇⢸⡇⢿⠀⢠⡇⠀⠀⠀⢿⢈⡇⠰⡷⠄⠀⠀⠀⡇⠸⣷⣃⡸⣷⢷
⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠶⠛⠀⠻⠞⠁⠸⡗⠋⠀⠀⠀⠀⠈⠋⠀⠀⣷⠀⠀⠀⠀⠃⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃

⠀⢀⠔⠂⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠢⢄
⠰⠁
⡇
⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣤⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⡇⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡟⠋⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⣿⣿⣿⣿⣿⣿⣷
⡇⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣤⣀⣼⠿⢿⣿⣿⣿⣿⡇
⡇⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⡀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⣀⣿⣿⣇⣀⣨⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⣹⣿⣿⣿⣿⣿⣿⣿⡆
⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠀⠀⠉⠉⢹⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⡇⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⡿
⡇⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛
⡇
⡇
⡇
⠰⡀
⠀⠈⠢⠄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠔⠊

⠀⠀⠀⠀⣀⢄⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠚⠆
⠀⠀⠀⠀⠁⢈⡆⠀⠀⠂⠀⢄⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠠⠀⠄⠀⠂⠤⠀⠀⠀⠀⠀⠀⢀⠀⡀⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂⠆⢉⡀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠁
⠀⠀⠀⠀⢀⡴⠀⡀⡲⠅⠀⠁⠀⠈⡁⠀⠀⠀⠀⠀⠌⠜⠀⠀⠀⠀⠀⠀⣦⠀⠁⣈⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁
⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⢠⣀⣠⣤⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠌
⠀⠀⠀⢔⠒⢂⢀⠀⠀⠤⠤⠌⠀⠀⠀⠀⠄⠀⠀⠠⠄⢀⠈⡀⠁⠀⡇⠀⠀⠈⠐⠀⠀⠀⠀⠀⢀⠀⣠
⠀⠀⠀⠸⠂⠄⢘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠡⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠸⣀
⠀⠀⠀⢀⡐⠘⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢠⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠚⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠨⠈⠣⡼⢆⠂⠀⠂⠀⠀⠀⠀⠀⠇⠀⠀⠀⠂⠀⠀⠚⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠉⠁⠂⠈⠄⠶⠀⡀⠘⢄⠃⣀⠂⠀⡀⠄⠀⠀⠀⠀⡀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⡠⠀⢺⠀⠉⠀⠈⠀⠀⠀⠠⠤⢀⠁⣛⡇⠂⠀⠀⢀
⠀⠀⠘⢀⠀⠀⠀⢧⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠒⢈⠤⡀⠀⠡⠐⠀⠀⢂⠀⠘⠋⢸⠆⠀⠀⠆
⠀⠀⠀⠀⠉⠣⠀⠈⠀⠀⠀⡠⡆⠀⠀⠀⠄⠀⠀⠀⠀⠄⠉⠁⠀⠨⠀⠀⢀⠀⡆⠀⠀⣸⠁⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠈⠛⠂⣀⠀⠀⠦⠀⠀⢀⠀⡀⣀⢀⢄⡁⢀⠔⠒⠁⠁⠀⠈⢣⣬⠂⠰⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈

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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠋⠙⠋⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⣀⣤⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣛⡛⠀⠀⢀⢀⣛⡛⠿⣿⣿⡇⠀⠀⢀⡀⠀⠀⠀⠀⣀⡀⠤⠴⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⣿⣿⣿⣿⣿
⣋⠉⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⢉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠉⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⢀⣀⣀⡀⠀⠊⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢜⠇⠀⠀⠀⠀⠉⠳⣬⣵⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠈⠀⠀⠀⣀⠀⠀⠀⠈⢻⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢿⠃⠀⠀⠀⣤⣨⠷⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣶⠞⡟⠀⠈⡀⠀⠀⠀⠀⠁⣤⣠⠽⠾⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠋⠀⠀⠀⡤⠀⠃⠀⠀⠀⠀⢸⡢⠐⡂⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⢀⠊⢀⣤⣘⡄⢀⣼⡓⠚⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢁⠘⠋⠉⠀⠈⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠧⠂⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠜⠁⠀⠈⠠⣠⣠⡤⠤⠄⠀⠀⢠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⣀⣀⣤⣤⣿⣿⡆⠀⠀⠀⠀⠀⢀⠄⠉⠃⠐⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠃⠀⠒⠈⠀⠀⠠⠈⠛⢷⣄⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣹⣑⠈⠉⠉⠙⠛⠒⠒⠶⠒⠒⠋⠁
⠀⠀⠀⠀⣀⠀⠈⠛⠛⠒⠂⠠⠤⠀⠀⠠⠤⠄⣈⡿⠛⠉⠉⠉⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢤⡢⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⢤⣀⣀⣄⣀⡀⠀⠀⠀⠀⢀⣰⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⣿⣶⡶⠖⠋⠁


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

⠀⠀⠀⠀⠀⣴⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⢀⣀⣀⠀⣸⣿⣿⣿⡄⠀⠀⢠⣶⣶⣾⡂⠀⠀⢀⣤⡀⠀⠀⠀⠰⢾⣿⣿⣿⠆
⠀⢀⣤⡴⠶⣶⣮⡅⠀⠀⠀⣈⡉⣻⡇⠸⠿⠿⠛⠐⢦⡀⠸⣿⣿⣿⡃⠀⠀⠘⠿⠛⠓⠲⠶⠶⢬⠟⣻⣯
⠀⠸⡇⠀⣸⣿⠏⠻⠶⠞⠊⠉⠁⢻⡇⠀⠀⢸⣿⣠⡾⠃⠀⠈⠉⢿⣿⣄⠀⠀⠀⠀⠀⠻⠷⣤⣄⣸⣿⣿⡆
⠀⢰⣧⢀⣿⣟⣀⣀⣀⡀⠀⡴⠀⢸⡇⠀⢀⣼⣿⠋⠀⠀⠀⠀⣀⠀⠻⣿⣿⡛⠛⣿⠂⠀⠀⠀⠈⠉⢸⣿⣷
⠀⠀⠀⢸⣿⠉⠉⠉⢹⡇⠈⠀⠀⢸⣇⣴⠟⣻⣿⠀⠀⢀⣴⣿⣿⣿⡄⠙⣿⡟⠛⠛⠛⠶⣤⣼⠃⠀⣴⡿⢻⡆
⠀⠀⠀⢸⣿⠀⠀⠀⠘⠳⠀⢀⣴⣿⠟⠁⠀⠙⣿⠀⠀⠀⠙⠿⣿⠿⡃⠀⠘⣷⡀⠀⠀⡀⠀⠀⣠⡾⠋⠀⠘⣿
⠀⠀⠀⠸⡿⠀⠀⣤⣶⣶⣴⣿⠟⠁⠀⠀⠀⠀⣿⠀⠀⣤⣦⡄⠀⣿⣿⡀⠀⠘⣷⠀⠀⠇⠀⠀⣿⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠘⠿⠦⠀⣿⣿⣿⡟⠙⢷⣀⣀⡀⠀⣴⡷⠀⠈⠿⠿⠛⠲⢾⣿⣇⠀⠀⠐⣷⡴⠆⠀⢀⣿⠀⠀⠀⠀⠈⣿
⠀⣠⣴⣦⡀⠀⠀⠈⠙⣯⡁⠀⠀⠉⠉⢉⡧⠀⠀⢲⣀⣀⣤⣤⣤⣌⣿⣿⡀⢀⣤⣤⣄⠀⠈⠉⠉⠀⣀⣀⠈⠉⠉
⠐⣿⣿⣿⡿⠂⠀⠀⠀⣌⡗⢀⣠⣾⣿⣦⡀⠀⠀⠈⠉⠉⣁⣀⠀⠉⠉⣿⡇⢾⣿⣿⣿⠷⠀⢀⣀⣼⣿⣿⠆
⠀⠈⣛⣯⡀⠀⢀⣀⡀⠈⠀⠈⠹⣿⣿⡿⠃⢐⣧⡀⠀⣾⣿⣿⣷⡄⠀⣿⡇⠈⢻⣟⣡⠶⠟⠋⠁⠀⢉⣡⣤⣄
⠀⠀⣿⣿⣿⡄⣿⣿⡿⠶⠶⠶⠶⠶⣶⣶⠀⠀⠀⠁⠀⠙⠿⠿⠋⠀⠀⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⠰⢾⣿⣿⣿⡇
⠀⠀⣿⠈⢿⣿⡆⣉⣤⣶⣄⠀⠀⠀⢹⣿⣧⠀⠀⢸⡟⠛⣿⣿⠀⠀⠀⣽⡅⠀⣿⣿⣠⣴⡶⣷⡀⠀⠀⠙⣿⡏
⠀⠀⣿⠀⠈⢻⣿⠋⠁⠀⠙⠷⣄⣰⠎⢿⣿⡆⠀⢸⡇⠀⣿⣿⠀⠐⠛⠛⠃⠀⢿⡟⠉⠀⠀⠈⢻⣄⣀⡀⣿⡇
⠀⠀⠋⠀⠀⠈⢿⡄⢸⡀⠀⠀⠈⢃⣴⠿⠻⣿⠀⠈⠁⠀⣹⣿⣤⣤⣄⠀⠀⠀⢸⡇⠀⠈⢦⠀⠀⠛⠉⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⣠⡾⠛⠁⠀⠀⢻⡇⠀⠀⠀⣿⡏⠉⢩⣿⠀⠀⡶⢟⠁⡀⠀⠀⢀⣀⣤⣤⠶⢹⡇
⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⣾⡇⢸⣿⣿⣿⡌⢻⣄⠀⠀⣿⡇⠀⠨⣿⣤⠘⠟⠋⣄⠹⠿⠘⠋⠉⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⣠⣿⠀⠘⠿⣿⡿⠃⢀⣹⣷⡀⢿⡇⠀⠀⠀⠀⠀⢀⣾⠋⠀⣀⣤⠀⠀⠀⡇⠀⢸⡇
⠀⠀⡈⢧⣀⣼⡿⣦⣄⡀⠀⠀⢀⣶⣶⠶⣶⠀⠀⠀⠀⠀⠘⢛⣦⣀⠀⠀⢠⡾⢁⣴⠾⠋⠀⠀⠀⠀⠁⢀⣼⡇
⠀⠀⣇⣈⣉⣉⣀⣈⣙⣿⣶⣶⣿⡿⠃⠀⣿⠀⠀⢸⠀⠀⢀⣿⣿⣿⣷⢰⣿⣷⣿⣷⣶⣶⣶⣶⣶⣾⢦⣿⡇
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠙⠀⠀⠀⠀⠀⠈⠙⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠈⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠂⠂⠀⠀⠀⠀⠀⠀⠈⢄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠄⠀⢀⠀⠀⣄⠀⠈⠈
⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢄⠀⠀⠀⠀⠈⠀⠀⠀⠰⠄⠀⠀⠂
⠀⠀⠘⠀⠀⢀⣤⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠂
⠀⠀⠠⡀⠀⠈⠿⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢄
⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⡁
⠀⠀⠀⠀⠀⠁⠒⠂⠐⠀⠀⠀⠠⠤⠤⠄⠀⠀⡀⢀⣀⣀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠓⠦⠀⠀⠀⠰⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠀⠀⢀⠁
⠀⠀⠀⠀⣀⣀⠀⠀⠀⠠⠤⠤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠁⠂⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠁⠀⠈⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠁
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠔⠐⠐⠇⠒⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠠⢘⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢅⣀⣴⣀⣀⣀⣐⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢸⡇⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠸⠁⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⢀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠒⠊⠉⠉⠉⠉⠉⠉⠉⠀⠈⠉⠑⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢄⣀⠀⠐⠈⠉⠁⠀⠒⠒⠒⠛⠓⠲⠶⠀⠉⣉⣀⠴⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠀⠀⠐⠐⠒⠒⠛⠋⠉⠉⠉⠁⠀⠀⣸⡥⠔⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠄⢠⡏⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠈⠉⠉⠉⠉⠉⠉⣉⡉⠉⠉⠉⠀⠀⢀⡾⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⡖⠁⠀⠈⠀⢹⡆⠀⠀⠀⢀⡾⠁⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠳⣄⠀⠀⡠⠞⠁⠀⠀⢀⣾⠡⠤⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⣄⣀⣀⣀⣀⣀⡤⠘⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⢠⡿⠁⠀⠀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠘⢷⣄⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠈⢻⡆⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⠟⠁⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⣠⣶⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⢀⣀⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣄⣀⠀⣸⣷⣤⡶⠶⠶⣶⣄⠀⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⣿⡶⠶⠿⣶⡄⢻⡆
⠀⠀⠀⠀⠀⠀⠤⣤⣤⣤⡄⠀⠀⠀⢻⣇⠀⠀⠈⠉⠙⠛⠛⠛⠛⠻⠿⠿⢿⡿⠿⠛⠛⠛⠛⠋⠉⠉⠀⠀⢃⣿⠃⠀⠀⢀⣾⠇⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⢄⣼⡏⣀⣤⡶⠟⣁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠸⡗⠈⣿⡇⠀⠀⠀⠀⠀⠀⣴⡿⠟⠛⣉⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠄⠀⠀⠀⠙⣷⣎⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣼⣿⡷⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣿⣦⡀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣿⡇⠀⠀⢀⣠⣾⣏⣀⡀⠀⠀⠀⠀⠀⠀⠀⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠛⠉⠉⠉⠻⢷⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⡶⠟⠋⠉⠉⠙⠛⠷⣦⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣄⣀⣀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣀⣀⣤⣤⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠿⠿⠿⠷⠶⠶⠶⠾⠿⠿⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⡟⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⡶⠀⠀⠀⠀⢸⡇⠀⢀⣀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⢀⡄⠀⣴⠀⣾⠛⠛⢻⡇⠀⠀⠀⠀⣼⠛⢷⡀⠀⣿⠀⠀⠀⠀⠀⢸⡇⢀⡟⣹⠆⢀⡾⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢀⣀⠈⣧⢀⣿⡀⢻⡄⢀⣼⠃⠀⠀⠀⠀⢸⣆⣼⠃⠘⣿⠛⠀⠀⠀⠀⢸⡇⠈⢿⣯⣤⠌⠷⠟⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢶⡶⠶⠛⠁⠀⠻⠟⠉⠁⠘⣿⠛⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢹⡄⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⢐⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⡐⢀⠁⠀⠀⠀⠀⣨⠑⠒⣂⠐⢀⡚⠒⢉⡀⠄⠀⠀⠀⠀⠛⠛⠀⠀⠀⠄⠀⠀⠀⠀⢠⡠⠀⠀⠀⠀⠀⢄⠠⡀
⠀⠀⢀⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠐⢁⣂⣉⠀⢈⣁⡢⠓⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢠⡔⠇⠐⠂⠂⠀⢄⠠⡄
⠀⠀⠘⠐⠁⠦⠤⠤⠤⠤⠤⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⢈⠋⠁⠀⠀⠈⠁⠀⠀⠀⢸⠀⠀⠀⢤⡤⠀⢀⢠⠁⠀⠀⠀⠀⠀⠀⠤⠁⠀⠈⠠
⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠁⠁⠀⠀⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⢀⠤⠄⠀⢠⠀⠄⠅⠀⠀⠤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠴⠰⠤⠤⠤⠤⠤⡴⠴⠄⡀⠀⠀⠀⠈⠀⠈⠁⠁⠈⠀⠃⠀⠀⢠⠈⠁⡄⠀⢀⠀⠤⠐⢈⡁
⠀⠀⢘⠁⠀⠅⠀⠀⠀⠀⠀⠀⡇⡀⠘⠄⠀⠀⠈⠀⠔⠢⠈⠉⠁⠉⠉⣀⠒⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠒⠀⠒⠈
⠀⠀⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⡇⢀⠈⢘⠀⠀⠄⠑⠄⠐⠀⢁⠀⢀⠠⠀⢒⠀⠀⠀⠀⠀⡀⠀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠘⠠⠂⠂⢀⢀⠀⢀⠀⠀⠁⠺⠗⢈⠀⠀⠈⠀⠀⠒⠀⠉⠁⠉⠁⠐⠂⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⡤⠤⢤
⠀⠀⠄⠀⠐⠀⠉⠈⠁⠈⠈⠁⠐⠀⠀⠄⠀⠀⠀⢀⠤⢄⠀⠀⠀⠀⠀⢀⠤⢄⠀⠀⠀⠀⠘⠄⠀⠀⠀⠠⢀⠀⠀⠀⠠⢊⠠⠀⣤⠀⣤⠀⢴⢵⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠂⢄⠐⠀⡀⡀⠘⡁⠂⠂⡁⠀⠀⠀⢀⠤⡀⠀⠀⠄⠀⠆⠀⢀⠀⠒⠀⢀⣤⣤⣤⡀⠈⠊⠀⡀
⠀⠐⠃⠃⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠳⠀⠀⣂⠐⠀⠐⢡⠉⠁⠁⢄⠁⢄⠄⢁⠀⠀⠀⠐⠤⠈⠀⠉⠄⠁
⠀⠀⠖⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡷⠢⠀⠀⡀⠀⠀⠘⠀⠀⠀⠀⠀⢙⠀⠀⠁⠀⠀⠀⣠⣄⠘⠀⠃⣀⣄⠀⠀⠐⠀⢀⠀⠀⠁⠀⠈⠀⠀⢀⠀⠁
⠀⠀⢀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠄⣀⡀⡀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡁⢈⠂⠀⠀⠀⠀⠖⡁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠢⡀⠀⠀⠀⠀⡤⠒⠒⢄⠀⠀⠀⠐⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠈⠁⠀⠈⠁⠉⠀⠀⠉⠠⠂
⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠌⠄⠉⠀⠀⣄⡀⣠⢠⠀⠜⠃⠈⠀⠀⠸⠿⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⢠⠒⢢⠄⠀⠀⠀⠀⠀⠀⢀⡔⢢⡄⠀⠀⠀⠀⠈⠀⡼⢀⢀⠀⢨⡄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢄⠀⠀⠉⠁⠀⠤⠀⠄
⠀⠀⠀⢑⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⡁⡡⠀⣉⢘⠁⠀⠀⠀⠀⠂⠔⠀⠒⠀⠒⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠛⠃⠘⠃⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠀⠉⠛⢦⡀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣲⣶⠶⠂⠀⠀⠀⠀⠀⢹⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⣰⠋⠠⠙⠃⠀⠀⠀⠀⠀⠛⣅⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠛⢹⠐⠀⠀⠀⠀⠀⢠⡙⠲⢀⡼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⢻⡁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⡴⠉
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠚⢻⡄⠀⠀⢳⡀⠀⠀⢠⠀⠀⠀⠀⢀⠀⢀⡀⠇
⠀⠀⠀⠀⠀⠀⠘⠁⢠⣀⠀⠀⠀⣷⠀⠀⠀⢳⡀⠀⡈⢦⡀⠀⠀⠼⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⢸⡀⠀⠀⠈⣧⠞⢃⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⡟⠙⢣⡈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⣾⣀⣠⠚⠇⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢠⣿⠂⠀⢤⡄⠀⢸⡆⠀⠀⠀⠀⢠⠀⠀⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⣿⠀⠀⡠⣣⠊⠁⠀⠀⢀⣀⣸⣇⠀⠀⣀⠤⠬⠷⢔⣄⡀⠀⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠐⠚⠒⠛⠛⠛⠛⠛⠚⠋⠉⠉⢉⠀⢻⠁⠀⠀⠀⠈⠉⢛⠛⢦⡀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⠁⢸⡇⠀⠀⠀⠀⠀⠤⣍⣲⢽⣆⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣯⣤⣤⠞⠓⠲⠶⠶⢤⡭⠶⠋⠁⠀⠨⠇⢁⣠⠔⠂
⠀⠀⠀⠀⠀⠀⠘⠶⠶⠶⠶⠶⠶⠶⠶⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⣀⣀⣠⣀⠀⠀⢀⣀⣀⢀⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢼⡁⠀⡇⠀⡇⠀⠙⠂⢸⠀⢸⠀⢠⡾⠋⠀⠀⠉⠙⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⣀⠀⠀⠀⣀⠔⡆
⠀⠀⠀⡞⠁⠀⠧⠀⡇⠀⡆⠀⣿⠠⠘⠀⠉⡀⠐⠀⠀⠀⠆⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠄⠀⠐⠒⠉⠁⠀⠀⠋⡖⠒⢲⡏⠂⡁
⠀⠀⠈⠁⠀⠀⠀⠊⠁⠀⠁⠐⠋⠐⠋⠀⠀⠐⠈⠉⠀⠀⠁⠀⠈⠓⠄⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠄⠀⡠⠂⠐⠐⠒⠠⡀⠀⠀⣄⣧⣤⢼⡇⡄
⠀⠀⣀⠂⠀⢐⠂⠋⠘⡒⠀⠀⢰⠈⠉⡆⢀⡄⠀⠐⠖⠀⠠⠲⡂⠀⣀⠀⠀⣠⠄⢠⡤⢠⣄⢠⣶⠠⣴⠌⢰⡀⢠⠄⡀⢰⡀⠀⡄⠉⠀⠀⡀⠈⢁⡄
⠀⢸⡌⠀⣀⢠⠤⠀⠠⡄⣀⠀⠀⠓⠂⠀⠈⠁⠀⠀⠀⢐⡀⠐⠀⠀⠋⠁⠀⠙⠂⠈⠓⠉⠉⠋⠃⠀⢋⠀⠀⠌⠀⠀⠀⠐⠀⢀⡀⠀⠄⢀
⠀⠀⠇⠀⠰⢀⣁⠑⡈⡀⠇⠐⠀⠅⠈⠀⠀⠄⠀⠁⠤⠄⠀⠠⠂⠰⠄⠀⠀⡁⠀⠀⠀⠘⠈⠀⠈⠐⠉⣤⠀⠈⠳⠲⢞⠀⣠⡄⠁⠀⠸⡏⠀⢄⠀⢰
⠀⠀⠄⠀⠀⠅⣙⢒⡋⠰⠀⠀⠀⠁⠠⠀⠠⠀⢀⠀⢰⡆⠀⠿⠀⠀⢠⡠⠆⠠⠄⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠉⠠⠅⠈⠀⢈⡠⡲⠀⠀⠇⠀⠈⠉⠙
⠀⠀⠀⠠⢀⠀⠹⠈⠅⠤⠀⠂⠠⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠀⠉⠀⠉⠀⠀⣀⣀⣈⣀⣂⣀⣁⡀⠀⠀⢄⠀⠠⠀⠠⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠂⢴⠀⠠⠄⢆⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⣿⡁⢀⣀⡀⠀⠀⠐⠉⠉⣉⣉⣉⣉⢉⢉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠡⠈⠀⠀⠀⣀
⠀⠀⠀⡀⠓⠌⠀⠀⠀⠓⠒⠁⠰⠀⠈⠈⠀⠀⢰⡏⠀⠉⠁⣿⠈⠉⠻⠋⠉⢳⡄⠿⡄⢠⣭⣀⢀⣥⡿⠀⢀⣀⠀⢀⠀⠀⠈⠀⠠⠁⠈⠀⠀⢀⡄⣿
⠀⠀⠠⠄⠀⢸⠀⠈⠀⠙⠀⠀⢀⡀⠈⠀⠀⣀⠘⢳⡀⠀⠀⣿⢸⣷⠀⢠⣿⠀⡇⠀⡇⢸⣿⣿⠸⡇⠀⠐⠛⠛⠒⠛⠛⠀⠀⠀⠀⠀⠂⠀⠁⠀⣤⠄
⠀⢀⡀⠀⠆⠈⢀⠀⢀⠈⠀⠀⠐⢰⠢⠤⠄⡞⠀⠈⢪⣀⠀⣿⢸⡿⠀⠸⠟⢰⡇⣀⡇⢸⣿⣿⠠⡇⠠⢠⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀⠀⠀⠀⢀
⠀⢨⣧⠀⡀⠃⢚⠀⢐⠒⢂⡀⢀⣼⠀⣼⢀⡇⠀⠀⠀⣽⡀⢿⡤⠴⠻⣤⡴⠎⠘⡯⠤⠤⠿⣷⣴⡇⠀⠁⢈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⢄
⠀⠀⢀⣈⡁⣤⡁⠀⢀⡀⡀⠁⠀⠉⠉⠉⠀⠀⠀⣀⣀⣬⣥⣄⣀⣀⣀⣋⣀⣀⣀⡀⢀⣀⢀⣀⣛⣀⣀⡀⢀⣀⣈⣀⡈⠈⠈⢀⠀⠀⠀⡀⠀⠀⠅⣀⠬
⠀⠐⠈⠡⡤⣤⡄⢹⡅⠠⡤⠀⠀⡀⠠⠂⢻⡙⣿⠏⣩⠀⣤⡀⠉⢿⣿⠈⠀⣶⡌⠹⢋⣉⠻⢡⣤⣾⡇⣿⢸⡁⠶⢾⡇⠠⠀⠀⠀⠠⠀⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⣶⡀⠁⠄⠀⠈⢷⡘⢀⡏⢸⣿⡇⠀⢸⣿⠀⠀⠛⣡⣴⡸⢿⣶⠘⢛⣿⡇⣿⢸⡇⡶⠺⡇⠀⠄⠀⠀⠄⢸⡄⠲⢖⢾⡄
⠀⠠⠰⠻⢻⠁⠄⠈⠈⠀⠀⠈⠁⠀⠀⠀⠀⢸⡇⢸⣿⠘⠿⠃⡄⢸⣿⠀⠀⣷⡌⣿⢻⡆⠙⢠⣿⣿⠀⣏⣸⡇⣷⠀⡠⢆⡤⡄⠀⠀⠀⢆⢹⣘⣿⡇
⠀⠀⠀⠀⣀⠀⠀⠀⡀⢀⠀⠀⢠⣤⣤⡄⠀⠀⣧⣸⡟⠶⠴⠞⠳⣤⣥⣾⣤⣿⣧⣽⣦⣤⡾⠦⠤⠼⣧⠭⠽⣧⣿⠈⢋⠉⠉⠁⢠⡎⠀⠘⢛⠉⠉
⠀⠀⠈⢶⠃⠀⠀⠀⠄⠠⠀⠀⠸⠿⠿⠇⠀⠀⠀⡉⣁⣀⡤⣠⠀⠉⠀⠈⠉⠙⠀⠀⢸⠇⠀⢴⠒⠠⣀⠀⠀⠁⠀⠀⠘⣆⡇⠀⠉⠀⠀⠀⠀⠀⢢⠶⠂
⠀⠀⢀⡀⠀⠀⠀⠈⠉⠉⠁⢀⣀⢀⡀⠀⠀⠀⠀⠉⠁⠀⠀⠁⠀⠠⡀⠀⠀⠂⠀⢀⠀⢀⠀⡢⣀⡠⠋⢀⣠⣄⣀⣀⠀⠙⠁⠐⣿⢿⣿⡏⠃⠀⢸⡆
⠀⠀⠈⡀⣤⠀⠀⠄⠠⠤⠄⠠⣬⠐⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠰⠂⠁⠆⠀⢠⡈⡾⠁⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠸⠏⡁⠀⠀⠘⠇
⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⡀⠁⠇⣆⠀⠀⠈⠀⢀⣾⠀⠀⡀⠄⠀⠀⠈⠢⠤⠁⠀⠀⣀⠀⡀⠀⢀⡀⠀⠥⠁⠀⠀⢀⠄⠀⠐⠑⠃⢈⠃⠁⠀⠀⢤⡀
⠀⠀⠀⠀⠁⠀⡶⠂⠀⠐⠻⠇⠀⡇⡴⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⠀⠀⠀⢶⡄⣒⡆⠀⠀⠠⠀⠰⡆⠀⠊⠂⠍⠀⠀⡀⣀⡩⠂⠘⠈
⠀⠀⠀⠀⡀⠀⠀⠅⠀⠨⠄⠀⠀⡆⡆⠀⠈⠀⠀⠀⠀⠀⣀⠈⠀⡁⢀⠀⠀⠀⠂⠀⠀⠀⠉⠀⠈⠀⠂⠀⡀⠀⠨⠅⠀⢶⠂⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠸⠀⠇⠀⠀⠀⠨⠐⡀⠀⡄⠃⡧⠀⠀⡡⠁⠀⠀⠂⠈⠐⠀⠄⠂⠀⠀⠀⠂⠀⡀⠂⢄⣐⠄⠀⠀⠀⠀⠀⠈⡅⠀⠉⠁⣠⣤⠀⠀⢠⣤⡀⠀⢹
⠀⠀⢰⠀⢀⡀⡀⠘⠓⠛⢀⣀⡅⠀⠃⠀⠀⡮⠀⠀⠀⠂⠀⠀⠂⠀⠀⠈⢀⢀⠀⠀⢀⠀⠀⡇⠀⣀⢀⡁⠀⣀⠀⡉⢀⣀⡈⡉⠉⠀⠀⠈⣉⣉⣀
⠀⠀⢀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠩⠐⠂⠀⠀⠃⠀⢠⢀⠈⠀⠘⠁⣀⠀⢀⡘⠀⠀⠠⠀⠀⠀⡇⢸⣏⡸⠇⠀⡷⠀⠿⠌⠉⠁⡇⠹⣶⠁⡏⢉⠉⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠉⠀⠀⠀⠀⠁⠀⠁⠀⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⣶⡶⢋⣤⡙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⢿⣿⣿⣿⣶⣮⣹⣿⣦⠀⣵⢄⠀⠀⠀⢀⣠⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⡟⣥⣾⣷⣿⣯⣿⣿⣿⣿⣿⣷⣄⠘⣣⢤⡤⣾⠛⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣽⣿⣿⣿⣿⡿⠟⠉⢻⣿⣿⣿⣿⣏⢂⢻⣷⣳⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠺⣿⣿⡟⠉⠉⠉⢩⡴⠖⠀⠘⠿⣿⣿⣿⣿⣮⣶⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣪⣽⠿⡷⠲⣦⠀⠀⠰⢾⣶⠄⠀⣿⣿⣿⣿⣿⣧⣿⣧⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⡆⠀⠀⣾⣗⣤⡾⣿⡿⡟⢀⠀⠀⠀⠀⠀⠀⠘⡿⣿⢻⣿⣇⣻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠨⣿⣿⣧⡄⠀⠫⣿⣿⣷⣿⡀⢙⣤⣤⡀⠀⠀⠀⠀⠀⡑⢊⣾⣛⣿⣿⣧⣼⢦⡀
⠀⠀⠀⠀⠀⠀⣘⡿⣿⣿⠀⠀⠀⠀⣴⣻⣿⣷⡘⠿⠟⠁⠀⠀⠀⢀⡞⢹⣿⣿⣟⢻⣿⣿⣷⣿⣛⡧
⠀⠀⠀⠀⠀⠀⢨⣛⣿⣝⣆⠀⢀⣾⣿⡿⠛⢿⣷⣄⣀⣀⣤⣴⠾⠋⠀⢸⣿⣿⣿⣿⠟⢩⡟⠛⠛⡧
⠀⠀⠀⠀⠀⠀⠀⣝⣻⣷⣽⢶⣿⠿⢿⣿⣶⣿⣿⣿⣿⣟⠿⡅⠀⠀⠀⣾⣿⣟⣿⡁⢀⡀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣙⣣⡏⢾⣧⡀⡽⣋⡛⠻⢶⣾⠋⠛⠢⠀⠀⠀⢠⣛⣿⣿⠋⣝⣛⣽⣶⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠷⠼⣿⣿⣷⣟⣿⣿⣦⣌⡑⠤⡀⠀⠠⡆⡼⠝⢻⣦⣨⣾⣿⣾⣧⣦⢗
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣳⣼⣶⡿⣿⣿⣿⣷⡄⠀⠉⠻⣷⣮⣷⣄⠁⡇⢀⣼⣫⣷⣿⣿⣿⣿⣿⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢉⢯⣄⣙⠻⣿⠋⢩⣿⣻⡆⠀⠑⠦⣽⣛⣻⣿⣶⣯⡽⠿⣷⣷⣯⠾⠿⠫
⠀⠀⠀⠀⠀⠀⠤⠶⢧⠊⢽⣿⣾⣶⣆⣼⠟⠋⠛⠠⠿⣦⣵⣾⣿⣿⠿⠻⣿⠀⣸⡏⠀⣀⠤⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣽⣿⣿⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠽⠛⠛⣧⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠄⠦⢼⣏⠸⡉⠀⠂⠠⠀⠀⠠⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠱⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠟⢠⠂
⠀⢠⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⣇⠀⡀⠹⢦⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠈⠁⠉⠀⠀⠀⣦⣤⡄⡀
⠀⣏⠀⢱⣶⠀⣧⠀⢸⠁⣄⠀⠀⠀⠲⢿⣷⣶⣾⣿⣿⣛⣛⣛⣻⣛⣿⣿⣷⣶⡾⠟⣃⠸
⠀⠹⣦⡄⢹⣶⡻⠀⡖⣠⡘⢆⠀⢴⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣻⣿⠁⣆⣲⣦⣄⢲⡀
⣀⣀⣈⠳⣄⣹⡇⢰⠃⣿⡇⢸⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢸⠋⠁⠀⡼⣸⠇
⠘⡍⢶⣄⣩⣭⣽⣿⣦⣿⣧⠟⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⠇⣎⣠⠴⢊⣼⠟
⠀⠈⠢⣀⠜⠋⠹⠛⣿⠿⣁⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠛⣸⢋⣤⠾⠛⠁
⠀⠀⠀⠈⠐⠒⠒⠁⠀⠀⠙⠷⠆⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⠏⡰⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⢾⣒⠁⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣼⡿⣫⣞⠓⢤⣤⣐⡢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⠀⠉⠙⠋⣆⠰⢍⡛⠷⠶⠶⠶⠶⠾⡿⢿⣿⠿⠋⠻⢂⠽⠂⢉⣆⡕⠀⠀⢀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣀⣀⣤⡴⣞⡋⠠⡆⠀⢀⠀⢀⠠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣻⣲⣿⣻⣿⣻⣻⣿⣷⣿⣿⣿⠿⣿⠛⠋⠉⠀⠀⠉⠙⠣⡀⠸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣏⠙⠶⣤⣤⠤⠴⠒⠚⣷⠴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣿⡀⠀⠀⠀⢀⣽⣬⠿⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠮⣿⣷⣠⠤⠚⠋⠁
⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣶⣶⣶⣶⣶⣶⣶⣷⣶⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⢏⠁⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⠖⣀⡤⠴⠚⠃⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⠋⡁⠀⠀⢀⡆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠻⣷⣤⠴⠚⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠐⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⢀⡀⠀⠀⠀⠙⠒⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠐⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣦⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⡄⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⢠⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⠟⠉⠉⠙⢛⢿⣿⣿⣿⣿⡄⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠄⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣧⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⣿⣿⠟⠉⠙⢿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⡟⠛⠛⣿⣿⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢸⣿⣿⡀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⣠⣿⣿⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠉⠍⠉⠉⠉⠡⣿⣿⣿⣿⣿⣿⣿⣿⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠸⣷⣾⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣷⣎⠀⠀⠀⠀⠀⠙⣻⣿⣿⣿⣿⣿⠃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢻⣿⣿⣿⣿⡁⠀⢀⣿⣿⣿⣿⣉⣿⣿⣶⣄⢠⣤⣹⣿⣿⣿⣿⡿⠃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠙⣿⣿⣿⣿⣶⣿⣿⣿⣿⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠻⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣸⣿⣿⣿⣿⡿⠛⠉⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠉⠛⠿⢿⣿⣿⣷⣶⣾⡿⠿⠛⠉⢀⡀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢤⣀⣀⠉⠉⣀⣀⡠⠤⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢤⣤⣤⣤⣄⢤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⣆⡤⠖⠚⠛⢳⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠾⠛⠁⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⠋⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠀⢠⣤⣶⣶⣾⡋⠉⠉⢉⡽⣿⣶⣄⠀⠀⠙⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠟⠃⠀⢀⣼⡵⠋⠹⠿⢿⠀⢠⡏⠀⠛⢛⡟⣇⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡾⢹⠁⠀⠀⣠⠞⠀⠸⠷⣤⣠⡞⠀⢹⡄⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣥⡶⠀⠀⢀⡞⠁⠘⠷⠶⠚⠁⠀⠀⠀⠀⠀⠈⢳⡄⠀⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣤⣶⠟⠀⠀⣇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡀⠀⣾⠀⠀⠀⣠⠄⠀⠀⠀⠈⠉⠉⠉⠁⣴⣤⡀⣿⠟⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠛⢲⡄⠀⠈⢻⣄⣿⠀⠀⠚⠉⠳⢦⣀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢡⠖⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⣭⣽⣇⠀⢀⡏⡙⠻⠀⠀⠀⠀⠀⠀⠉⠛⠒⢀⣀⠜⠁⣀⣤⣤⣩⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣠⣿⠀⠀⣧⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣋⣁⣀⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⡤⣶⣿⠀⠀⠘⢷⣄⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⠽⠛⠛⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⣿⢸⡇⢀⣤⠤⠄⠀⠿⠿⠧⣄⡀⠀⠀⠀⠀⠀⠀⢀⣿⡶⠚⠓⠀⣿⠙⠛⠛⠛⠛⠒⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⢸⡇⣸⠀⠀⠀⠀⠀⠀⣀⠠⠿⣦⡄⠀⠀⢀⣶⡛⣯⣤⡶⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇⢻⢸⡇⣿⠀⠀⠀⠀⣠⠞⠁⠀⠸⣿⡁⠀⠀⢸⣿⠛⣾⠁⢀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣸⡇⣿⣠⡴⠖⠒⡯⠀⠀⠀⠀⠈⠛⠒⠚⠛⠋⠀⠹⣶⠟⠁⣸⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣇⢸⣿⡷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠶⠞⠋⠈⢷⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⢻⣿⢸⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠻⣾⡟⠈⣿⠀⢰⠃⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⢳⣄⠀⠀⣿⡇⢀⡿⠀⣿⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣴⡞⢹⡇⠀⠙⣶⣾⢿⣿⡏⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢸⠁
⠐⠛⠛⠓⠲⠶⣾⡏⠸⡇⠀⢷⡀⠀⣸⣿⠷⣿⡿⠶⠿⢤⣤⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣦⣤⣤⣶⡿⠃⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠙⠷⠞⠻⢦⣼⠿⣶⡿⠛⣿⣿⠛⠲⠶⠤⣤⣄⣀⣉⣉⣉⣉⡉⠉⠉⠛⠛⠒⠒⠶⠶⠶⠤⠴⠿⠿⠧⠶⠤⠤⠤⢤⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠁⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⠏
⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠉⠛⠛⠛⠓⠲⠶⠦⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣻⣿⣿⠟⡋⠑⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⣾⣿⣿⣿⣿⣏⠀⠀⠀⠈⠈⠉⠿⢛⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⢀⡠⠀⠀⠀⢈⠔⠋⠙⠻⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠟⣉⣽⣿⣿⣿⣿⣿⣿⣭⡴⣂⠥⠚⠁⠀⠀⠀⠀⢹⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣀⡀⠈⣡⣾⣿⠿⠛⠋⠁⠀⠀⠤⠤⢤⣀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣧⣴⣾⡿⣿⠁⠀⠀⠀⠀⢀⣄⣀⠀⠀⠙⠟⠀⠀⢀⣿⣿⣿⣻⣿
⠀⠀⠀⠀⠀⠀⠴⣪⣵⣿⣿⣿⣿⣿⣿⠉⠃⢻⣌⢧⡀⠀⠀⠀⠈⠻⢷⣶⣿⠄⠀⠀⠀⠛⣿⡿⣿⡟⣸
⠀⠀⠀⠀⠀⢀⣴⡿⢫⣾⣿⣿⣿⣿⣿⡄⠀⠀⠈⠢⡉⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⡾⢋⡴⠁
⠀⠀⠀⠀⢰⣿⣫⣶⣿⣿⣿⣿⣟⠋⣿⣿⠖⠄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⠋⠀⠀⠀⡆
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠗⢡⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⣦⣀⣀⠄⣴
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⠀⠀⠈⣢⡀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠙⢛⣶⣿⡿⠈⠂⠉⠀⣰⠃
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⡿⠀⠀⠀⠈⠛⢶⣤⣀⡀⠀⠀⠀⠀⠈⠙⢫⣿⡟⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡏⠙⢀⡇⠀⠀⠀⠀⠀⠀⠈⠙⣿⣷⣶⣤⣀⣀⣠⣿⠿⠓⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡁⢠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠻⣿⣿⣿⡟⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⡿⠁⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⣼⣻⠅⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠘⠁⠩⡀⠀⡠⠔⠊⣽⡋⣀⠤⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣢⣤⣴⣞⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣙⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡀⠀⠀⠀⢀⠀⡀⢀⡀⠀⠀⢀⢀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⢠⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠸⡆⠀⠀⢀⠀⠀⠀⡾⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠰⠀⢠⠚⠻⣷⡄⠀⣇⠀⠀⢸⠀⠀⠀⠃⠀⠀⠸⠁⠀⠀⣰⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⢸⠀⠀⢸⡇⠀⢸⠀⠀⢸⠀⠀⢠⠀⠀⢀⠀⠀⠀⠰⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠑⣄⠀⠀⢸⡄⠀⠰⣿⠀⠀⣀⣀⡘⠀⠀⡿⠀⢀⡟⢠⠖⠳⣦⠀⢀⡴⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣿⡀⣼⠉⠈⠉⣷⡴⠶⢦⣄⠀⡏⠀⢀⣿⠀⠋⠀⠀⠀⠀⠉⠀⠀⠀⢀⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⣴⠾⠛⠛⠛⠋⠉⢉⣭⣭⣭⣭⣭⣭⡇⠀⠀⠀⣿⠀⠈⠁⣿⣭⣭⣭⣭⣍⠉⠉⠙⠛⠛⠻⢶⡄⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣠⠖⢂⣄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⢀⡀⢸⣿⠀⡀⢰⣿⣿⣿⣿⣿⣿⠀⠀⣠⣶⡒⢦⡀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠀⡾⠰⢡⣄⡸⡟⣀⣀⢸⣿⣿⣿⣿⣿⣿⣿⣇⣠⣿⣿⣏⣀⣿⣿⣿⣿⣿⣿⣿⢠⣤⡝⠛⣡⣤⡱⠸⡆⠴⠒
⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⡇⠀⠘⠛⣩⣥⠛⠛⣸⣿⣟⣩⠴⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⠛⣣⣦⡙⠛⠁⠀⡇⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠒⠀⡇⠀⠀⠀⠙⠛⣰⠟⠁⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣙⠛⠁⠀⠀⠀⡇⠐⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⣿⣇⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣷⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⢹⡀⠀⠀⠀⠉⣿⠀⠀⠀⠛⣿⣿⣿⣿⢿⣿⣿⣿⠿⣿⣿⠿⣿⣿⣿⣿⣿⡿⠀⠈⠉⠁⠀⠀⠀⣸⠁⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠈⢧⣀⡀⢀⠀⢹⡀⠀⠀⠀⢸⣧⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠇⠀⣀⣀⡀
⠀⠀⠀⠐⠒⠛⠋⠉⠉⠀⠀⠀⠀⢉⡛⠛⠛⠛⣧⠀⠀⠀⠈⠉⠉⠙⠛⠿⠛⢿⠿⡿⠿⠿⠿⠿⣿⣿⠛⠛⠛⠛⠛⠛⢛⡋⠁⠀⠀⠀⠈⠉⠉⠛⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠖⠛⠉⠁⠀⣀⡄⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⢀⠈⠙⠦⣄⡀⠀⠉⠀⠀⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠁⠀⣀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢀⠀⠑⠂⠀⠀⠙⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠀⠀⠀⠠⠚⠁⠀⠀⠙⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠈⠁⠀⠀⠀⠢⣄⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠀⠀⠀⣀⠴⠀⠀⠀⣠⠆⠀⠀⢸⡿⠋⠁⠀⠀⠀⠀⠀⠀⠶⣿⡟⠁⠘⠆⠀⠀⢦⡀⠀⠈⠳⠀⠀⠀⠀⠀⠙⠲⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣘⣁⣀⣀⢀⣞⣁⣀⣠⣆⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣧⣀⣀⣀⣦⣀⣀⣙⣆⣀⣀⣀⣙⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢰⠾⠉⣉⣉⣩⣉⣉⠉⠿⠿⠉⣉⣉⣩⣉⣍⠉⠿⠉⣉⣍⠙⠟⢉⣉⣍⠉⢉⣉⣉⣉⣍⣉⣩⡉⠉⢉⣉⣉⣉⣉⣍⡉⠹⣶⣀
⠀⢰⡶⠶⠶⠶⠾⠂⣶⣿⠉⠉⠉⠻⠿⠀⠀⣾⣿⠉⠈⠉⣿⣶⠀⠠⣿⣿⣧⣴⣾⣿⡿⠀⢸⣿⡏⠉⠉⠀⠉⠁⠀⢸⣿⡏⠉⠁⢹⣿⡆⠈⠻⠶⠶⠶⣶⣀
⠀⢸⡇⠸⠿⠿⠂⠀⣿⣿⠀⠀⠺⣿⣿⠀⠀⣿⣿⣤⣤⣤⣿⣿⠀⠈⣿⣿⠘⠟⢸⣿⡇⠀⢸⣿⡿⠾⠿⠿⠂⠀⠀⢸⣿⡿⠿⠷⢿⣯⡀⠀⠺⠿⠿⠀⠈⣿
⠀⠘⢻⣦⣤⣤⣤⡀⠛⣿⣤⣤⣤⠟⣿⠀⠀⣿⣿⠁⠀⠈⣿⣿⠀⢀⣿⣿⠀⠀⢸⣿⣇⠀⢸⣿⣧⣤⣤⣤⣤⡄⠀⢸⣿⡇⠀⠀⢸⣿⡇⠀⣠⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠘⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠛⢉⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡵⣾⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠾⠱⠿⠏⢺⣿⣏⣼⠋⢹⣯⡉⣯⡑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣼⢀⠴⠒⠿⠛⢁⣽⣯⠀⠀⠀⡛⠟⠚⢆⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡼⣶⢁⠀⠄⡤⡸⠈⣻⣦⡀⢄⠹⠆⣀⣀⡹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠿⡀⢻⠀⢶⢋⣽⡿⠯⢭⣟⠿⠖⡺⢿⡝⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠠⣿⣿⣤⠾⣸⠏⠀⠐⡭⠛⠛⠋⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠁⠁⠙⠋⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣔⡐⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠁
⠀⠀⠀⠀⠀⢀⡴⠀⠀⠈⠐⠂⠀⠀⠈⠂⢤
⠀⠀⠀⠀⡴⠡⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⡀
⠀⠀⢀⠎⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⢀⠔⠀⠀⠂
⠀⠀⡎⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠰⠃⠐⠀⠀⠀⠀⣠⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠓⢠⠀⠀⠀⠀⠀⠀⠀⠐⡀⡄⠘⠀⠀⠀⠀⠐⠀⢀⠔⠁
⠀⡆⡠⠀⠀⠀⠀⠀⠀⠠⠼⠀⠈⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠢⠤⠄⠀⠀⠀⠈⢁⠂⠀⠀⠀⠴⢀⡔⠁
⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠀⠠⠁⠀⠀⠀⠀⠀⠄
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄⠂⠠⠘⠂⡀⠁⢦⣀⠴⠤⣀⡀⠀⠀⠈⠀⠀⠀⠀⠨⠡⠀⠀⠀⠀⢀
⡄⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠤⠀⡀⠕⡊⠁⠁⠀⠀⠀⢼⠀⠀⠁⠀⠀⠂⠀⠉⠁⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠊⠃⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠠⢤
⠘⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠠⠊⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠐
⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⡠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠘⠒⠀⠀⠄⠀⠀⡀⢀⡠⠊⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠲⠤⠄⢀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢀⣀⣠⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠂
⠀⠀⠀⠀⠀⠀⠉⠑⠒⠢⠤⢤⣄⡠⠄⠀⡤⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠌

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣽⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣟⣵⣿⡿⠯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⣫⠟⣵⣌⣛⡟⠁⠀⠀⠉⠓⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢫⡀⠀⣠⠞⣡⣾⣿⣿⣿⢠⡞⣠⠆⠀⠀⠀⠀⠀⠉⠓⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⣀⣧⡞⣡⣾⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣀
⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⢀⠞⢃⣴⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⡀
⠀⠀⠀⠀⠀⢠⠏⢠⣄⡴⢃⠴⠋⠀⠘⣜⣆⠀⠀⠀⠀⠀⢀⢄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⠃⣾⣷⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⡞⠀⢀⡏⠐⠣⡀⠀⠀⠀⢈⣨⣤⣶⣶⣶⣿⣿⡼⢟⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⡸⢁⡴⣯⣀⣠⣐⣁⣤⣶⣾⣿⣿⣿⡿⣿⡏⢹⣿⣿⣷⣄⡀⠙⠲⢦⣤⣀⠀⠀⠀⠀⣠⡆⠐⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱
⠀⠀⠀⠀⢣⣞⣵⠟⢹⣿⠛⠛⠻⣿⢿⣿⣫⣿⣾⣿⣷⢘⣿⣿⣿⣿⣿⣷⣶⣦⣤⣭⣶⣦⣴⡾⢻⣿⠇⠀⢸⣿⣿⣿⡁⠟⣿⢹⣿⣿⠸⡸
⠀⠀⠀⣰⣿⠟⠁⠀⠈⠻⢶⡀⠢⡘⣷⠾⢿⣿⡿⠣⠬⣉⠛⢦⣌⠁⡹⣿⣿⣿⡿⣿⡿⠿⣟⢹⣿⡟⠀⠀⣿⣿⣿⣿⣷⣤⡇⣾⣿⡏⠀⡃
⠀⠀⠠⠛⠁⠀⠀⠀⠀⠀⠀⠙⠦⠴⣯⠀⠀⠙⢯⣀⠀⢀⡵⠀⠉⠛⠶⠶⣭⣭⡭⠿⠟⠲⠶⠤⢽⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢄⡀⣸⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⣿⣿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣮⣵⣾⣯⣵⣶⣦⣄⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠤⣄⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠄⠀⠄⠀⠀⢀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠁⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠛⠀⠀⠂⢐⡶⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣀⡀⢀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣧⡀⠀⢀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⢿⣿⣷⣦⡄⠀⠴⠄⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠈⠣⡉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠉⠘⠂⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⡷⠶⣤⡀⠀⠀⠀⠀⠀⣀⣤⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠈⠁⠀⠐⠧⠄⣀⡀⠀⠈⠛⠛⣛⠋⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠈⣀⠋⠈⣩⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⣇
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣆
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⢧⡀
⠀⠀⠀⠀⢸⣇⣀⣀⣀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡹⠆
⠀⠀⠀⠀⠀⣹⠟⠛⠛⠿⢷⣦⣀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⠶⡾⠉
⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⡹⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣿
⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⡀
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠁⣄⠐⠚⠆⠀⠀⠀⠀⠀⠀⠀⢸⠃⠘⣇
⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⠀⠈⠱⠦⣀⠀⠀⠀⠀⠀⢛⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙
⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⠎⠀⠁⠀⠀⠀⢢⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠐⠊⠉⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠈⠁⠉⢹⠀⠀⠀⠀⠀⠒⠒⠒⢶⠚⠃⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠓⠤⢄⠀⠀⠐⠒⡶⢋⠻⠇⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠛⠉⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠒⠒⠒⠛⢻⠛⠛⣿⠛⠛⠛⠒⠲⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣼⣄⣀⣀⣠⣤⡤⠤⠤⠤⠤⠴⠦⠤⠤⠤⢤⣤⣄⣀⡀⣙⣿⠛⠛⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠉⠛⠓⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠳⣶⠒⠒⠛⠋⠉⠁⣿⠶⢦⣄⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⡧⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣯⣄⠀⠀⠀⠀⠀⢰⣇⣀⡤⠔⢁⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢁⣴⡉⣿⠀⠀⠀⢠⠿⠤⠶⠖⠚⠉
⠀⠀⠀⠀⠀⢀⣀⣠⣤⡼⠿⣦⡀⠀⠀⠀⠀⠀⠀⣿⢾⡓⠀⢸⡄⢀⡴⠓⠶⠤⢤⣀⡀
⠀⠀⠀⠀⣾⠋⠉⠀⠀⠀⠀⢨⣿⣶⣤⣀⠀⠀⠀⠻⠾⠶⢚⣫⣴⣿⡄⠀⠀⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠈⠲⣄⡀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣶⣶⣶⣿⣿⠿⠟⠛⠁⠀⠀⢀⣀⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣶⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣶⣶⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠓⠶⠶⠶⠶⠶⠒⠚⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡜⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡈⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡀⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⣠⡄⢠⣿⣿⣿⣿⣇⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣤⡄⣴⣿⣷⣿⣿⡟⢿⣿⣿⣿⣿⣿⣇⢠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⣿⣿⣿⣿⣿⡇⠈⢿⣿⣿⣿⣿⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣿⣿⣿⡟⠈⢻⠇⠀⠘⠋⢻⣿⣿⣿⠞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⣀⣠⡏⣙⠇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣤⣾⡸⣆⣀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⢡⠱⢻⡀⣼⣿⣿⡇⣿⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢿⣿⡇⣿⣿⣧⣀⡟⠎⡌⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢫⡳⣄⠰⢦⠘⣀⣄⣿⣿⣿⣿⡇⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⣿⣿⣿⣿⢰⣀⠃⡴⠂⣠⢞⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⠃⠀⠱⣭⣾⣿⣿⣇⣝⠷⢹⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠟⣻⣿⣿⣷⣭⠎⠀⠘⢡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⣳⣌⢻⣿⣿⢿⣹⣷⣌⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⡿⣿⣿⣟⣡⣞⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⠗⠛⣛⣛⠛⠛⢾⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⣿⣿⡛⠺⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢿⡿⠞⠁⣀⣀⣀⠀⢀⡀⠀⠹⠶⠶⠶⠶⠶⠶⠶⠶⠏⠀⢀⣀⡀⠀⣀⠀⠈⠳⣿⡻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⠀⠀⢀⡯⠟⢹⡀⠈⠉⠀⠀⠀⠀⢀⣀⣈⡀⠀⠀⠀⠠⠋⠉⠀⣞⠉⣹⠀⠀⠈⠻⣟⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡏⡀⢸⣏⡉⠳⣤⠛⢉⣽⡇⠀⢀⣀⣀⠻⠭⠭⠟⣀⣀⡀⠀⢠⡞⠓⡎⠛⢡⠞⠳⡄⠀⢸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⢰⠇⠘⠓⠳⡞⠈⢳⠞⠚⠃⣴⢫⠀⠈⢳⡄⢀⡞⡁⠀⠙⣦⠈⠓⠞⣡⠶⢬⠳⠞⠁⠀⠈⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡏⠸⠀⠀⠀⠀⠷⠷⠾⠀⠀⠀⠻⣜⠳⢂⣼⣿⣿⣧⡛⠦⣠⠟⠀⠀⠀⠻⢤⠞⠀⠀⠀⠀⣆⢹⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠁⢱⠀⠀⠀⠀⠀⠀⠠⠄⣠⣴⠆⣈⣭⣭⣤⣭⣭⣤⣭⣭⣁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⢾⡿⣼⠟⢡⢞⣁⣀⢀⣀⡀⣀⠀⣀⢀⣈⠳⡌⠻⣦⣄⠀⠀⠀⠀⠀⠀⠐⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣷⠀⠀⠀⠀⠀⠀⣠⡾⢃⡴⠋⢸⡇⣤⣸⣭⡗⣿⣾⣿⢸⡧⠀⠙⢦⡘⢿⣷⠀⠀⠀⠀⠀⠀⣠⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢸⣿⠀⠀⠀⠀⠀⠰⠏⣠⠞⠀⠀⠘⠳⠟⠘⠀⠛⠛⠋⠛⠘⠳⠂⠀⠀⠳⣄⠹⠀⠀⠀⡀⠀⠀⣿⡇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⣿⣷⣦⣀⣀⠀⢀⣾⠁⠀⠀⠀⠀⢉⡿⢸⡟⣿⣿⣶⡇⣿⣉⠀⠀⠀⠀⠈⣳⡀⠀⣘⣿⣩⣾⣿⡏⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡙⠛⠟⠛⣉⣴⣻⣦⢳⡀⠀⠀⠀⢼⣧⢸⣧⡿⢿⠹⡇⣿⡤⠀⠀⠀⢀⡞⣵⣟⣦⣈⠛⠻⠛⢋⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠉⢿⣿⣿⣧⣙⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣋⣼⣿⣿⡿⠉⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

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



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⣂⠠⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠵⢻⠆⠁⣸⠇⠀⠀⠀⠀⠀⠀⡀⡀⢀⣀⣬⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡫⠉⢁⡭⡝⡆⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢈⣿⠟⠉⠋⠙⠻⣾⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⡀⠀⠀⠀⣨⠟⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠒⠲⢤⣾⠃⠀⠀⠀⠀⠀⠀⠀⣾⡅⠀⠰⡀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣩⡄⣀⠲⢠⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠭⠖⠁⠀⠀⠀⠀⠰⡞⠉⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣦⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣰⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⠰⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⣠⣾⣿⣴⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣿⣿⣿⣿⣿⣿⣿⡿⠻⢿⣿⡁⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠄⠐⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠾⠛⣻⣿⣿⡿⠿⠋⠋⠀⠀⠀⠀⢻⡉⠀⠀⠀⠀⠀⢿⣹⣇⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠱⡀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠁⠀⠀⡀⠀⠢⡈⠁⠀⢢⠂⠀⠀⠀⠀⠀⢿⡿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡰⣱⡀⠀⠀⠀⠀⠀⢸⠟⢻⠷⡀⠀⠑⠀⠀⠀⠈⠄⠊⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡅⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠤⠂⠀⠠⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠈⠒⠆⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠜⠃⢶⠦⢄⠀⠁
⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠈⠉⠉⠉⠁⢹⣿⣿⠄⠀⠀⠀⠀⠸⣴⡢⢀⣼⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⢀⡀⠀⠀⠀⢸⣿⣿⠀⠀⠘⣅⠁⠃⠘⡆⠀⠀⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⣛⣿⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠀⣿⠎⠈⡀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠁⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠘⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠨⠀⠀⠀⠀⠀⠒⠐⠀⠀⠐⠒⠒⠒⠒⠒⠒⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠠⢄⠀⡀⢲⣾⣻⣍⣯⣿⣦⣾⣤⣌⡓⣒⣠⣤⡤⢤⣀⠀⠀⠀⠀⠀⠀⣰⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣺⡿⠻⠩⠿⣯⡿⠛⠀⠀⠀⠠⠤⢀⣤⣦⣲⠀⠀⠉⠙⠻⢯⠿⠷⠶⠿⢿⣶⣤⡀⡞⡵⠚⠉⠉⢝⠺⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣯⣧⣶⠐⠗⠃⠀⠀⠀⠀⣤⣄⢸⣎⡙⣣⠜⣻⠠⣴⠒⡦⣀⠀⠀⠀⠀⠀⢠⡄⢙⣿⡿⣇⢿⡀⠀⠀⠀⠀⢹⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⡯⡋⡷⢻⣿⣦⠀⠀⠐⣠⣦⡽⣿⣷⡿⢿⣧⠲⣌⠶⢏⠁⠀⠀⠀⠀⠀⢀⣴⢎⡴⢣⣿⡷⡁⢸⣷⠀⠀⠀⠀⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡻⡃⠑⡘⢛⣩⢈⠀⠀⠀⠸⠾⠃⠛⣀⠨⣅⡐⠛⠭⠈⠁⠀⠀⠀⠀⠀⠀⠙⠉⢀⡠⡾⣔⣻⠿⠶⠀⠀⢀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢶⣥⣬⣉⣊⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠆⠀⠀⠀⠀⡠⢤⣲⣖⣠⣥⠶⠿⠋⠀⠖⡦⢧⠀⠧⠄⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠟⠃⢀⠀⠈⠉⠙⠛⠛⠓⠿⠿⠛⠛⠖⠲⠟⠷⠿⠿⠿⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠐⡦⠔⠀⣁⡤⠐⢫⡀
⠀⠀⠀⠀⣠⣮⠀⠀⢀⠰⡇⣀⣂⠀⠄⠀⠀⠀⠄⡀⠀⢠⣄⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⣄⣦⡀⠀⠀⠀⠀⠠⢋⠴⢂⣉⣉⠀⠀⠀⢻⣚⣀
⠀⠀⠀⢠⣿⡇⠀⠀⠋⡐⢛⠉⠼⣣⠀⠀⠀⠀⣼⣇⡀⢈⣁⡜⣠⠤⣶⠚⣯⢣⢄⠀⡀⠀⠛⢾⠏⡀⠀⠀⠀⠀⢀⠊⠈⣋⢬⠏⠉⠀⠀⢀⠉⢹⣷
⠀⠀⠀⢾⡏⡀⠀⠀⠒⣤⣿⡛⣾⣀⢀⠁⠀⠘⠛⠻⠛⢀⣍⣤⢧⣉⡰⢇⡿⡞⠂⡞⣉⡶⠗⠛⠛⠀⠀⠀⠀⢾⣋⠈⠁⠉⠅⠀⠀⣂⢢⢾⣴⢎⡿
⠀⠀⠀⠘⢿⣿⠿⠦⠀⠈⠓⠉⠀⠁⢴⠆⠑⡀⠀⠀⠒⠁⠠⠀⠸⣿⠊⠁⠀⠉⣡⠟⠛⠇⠀⠀⠀⠀⠀⡀⠐⠯⣭⠔⠀⠀⠀⠀⠀⣤⣴⣿⠔⠋
⠀⠀⠀⠀⠈⠙⢶⣢⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠀⢠⢀⡤⠀⠀⣀⣈⣤⠼⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⠾⣦⢶⣤⣀⣀⡀⠀⠀⠀⢫⡑⠂⠤⢤⣤⣠⣤⣤⣔⣶⣴⡬⠴⠖⢋⣝⣀⣤⣬⣭⠬⠖⢛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢄⡀⠉⠉⠓⠒⠒⠛⠒⠒⠶⠦⠴⠶⠒⠒⠒⠒⠒⠒⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠤⢶⣶⣤⣦⣤⣤⣤⣴⣰⣶⠤⠤⠆⣲⠖⠲⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣝⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣻⣥⣝⢆⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⣹⣇⠀⡉⠛⠻⢷⣤⡄⠈⠣⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠄⠐⠉⢴⠿⠛⠻⢿⠀⡀⠀⠀⠈⠉⠳⢤⡙
⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⣀⣠⣾⣄⠀⠀⠀⠀⠛⢷⣆⡆⣀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠃⠀⠀⠀⠀⠀⠙⢿⡓⠦⣄⡀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣾⣟⠀⠀⠀⠱⣦⣄⠀⠀⠀⠈⠓⠺⢿⣦⣀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢈⡙⡛⠿⠿⣆⣠⡁⠙⢻⠶⣷⣂⡀⠀⢨⠟⢻⣟⠁
⠀⠀⠀⢀⣠⣤⣴⣶⣶⣽⣄⡻⣿⣧⣀⠀⠘⠿⣿⣦⣦⣔⣾⣿⣿⣿⣿⡶⠶⠛⢷⣄
⠀⠀⣸⣿⣿⣿⣿⣟⣿⣿⣿⣿⣷⣿⣿⣿⣴⣤⣌⣏⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠈⠑⠚⠓⠚⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣀⣤⣤⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣠⣀
⣿⣯⣤⣥⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣄⠀⠀⠀⠀⣠⣤⣤⣄⣀⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⡄⣠⣾⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⢣⣾⣼⣿⣿⣿⣾⠿⣾⣟⣾⢻⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⠉⠁⣿⡿⢿⣿⣯⣭⣁⡘⢿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣿⣿⣿⢍⠃⣬⡉⢉⣀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣷⣤⣶⣾⢣⡜⣿⣷⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⢿⣿⣿⣿⣿⣏⣻⣿⢸⣿⠈⣱⡆⠀⢀⣼⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡟⠘⣿⣿⣿⣿⣿⣿⣿⡎⢉⣠⣿⡇⠀⢄⣿⠇⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡁⠀⣸⣿⣿⣿⣿⠟⠛⠛⠻⣿⣿⣷⣿⣾⡿⢲⣧⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⢣⣾⣿⣿⣿⣿⣡⣯⣤⣤⣶⣿⣿⣦⣽⣿⣇⣵⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣿⣿⣿⣸⣿⣿⣿⣿⡟⢻⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣰⣿⣿⣿⣿⣿⡄⡈⠐⣲⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢹⡿⢻⣿⣿⣿⣿⣿⣿⣜⢩⢌⣽⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⠀⢠⣼⠏⣸⣿⣿⣿⣿⣿⣟⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⢾⡖⢠⠁⠀⣿⣿⣹⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣈⠈⠃⠁⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠋⠉⣁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⢿⡟⣅⠠⢃⠀⡀⣿⣯⣽⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣟⣼⣂⣸⣿⣿⣿⡿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠃⠘⠛⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣏⣿⣿⣿⡟⠉⢹⡟⠙⢁⣿⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠠⠄⢠⡶⠀⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣵⣿⣿⢀⡴⢋⠀⠲⢻⠋⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⠯⡿⠀⠀⠀⢀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠈⠁⠀⠀⠀⠁⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣀⣋⠻⣶⠃⠰⣾⠇⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠻⣿⡏⢿⣿⡇⠀⣀⡀⠀⠀⠀⠀⠀⣸⣿⣿⣯⣿⣿⣿⣿⣿⣶⣿⣶⣿⣷⣶⢶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣏⣀⣽⣧⡸⣿⡄⢹⣿⡇⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⢎⣽⣿⣿⣧⢸⣷⣾⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣏⣙⣿⣷⣤⡉⠁⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠳⢿⣹⣿⣿⣿⢛⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣟⣁⣀⠁⣤⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣏⡚⢿⣿⣿⣿⣿⠾⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⣼⠟⠛⣿⣶⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣿⣿⣿⣿⠖⠁⢸⡷⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠋⠀⢀⢀⣿⣿⣿⣿⣿⣿⠁⠘⡧⠀⢲⣿⡷⠆⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⡀⠀⡠⠊⠡⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠒⠂⠂⢡⠀⠈⠁⠀⠀⠀⠈⠁⠁⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠒⠈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠤⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠂⠀⠃⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣀⣀⠀⠀⠀⠀⡠⠒⢄⡀⠄⠒⠄⡀⣀⠠⠀⠄⡀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⢀⢴⠃⡠⠊⠀⠀⠀⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠑⢄⠀⠰⠢
⠀⠀⠀⠀⠀⠀⠀⡈⣨⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⣆⠆
⠀⠀⠀⠀⠀⠀⠰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⢀⣀⣀⣀⡀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢀⣀⣀⣀⡀
⠀⠀⠀⠓⠚⠉⠛⠛⠛⠿⠷⠶⠶⢶⣦⣤⣤⣤⣀⣀⣀⡀⢀⣀⣀⣠⣤⣤⣤⣴⣶⠶⠶⠾⠿⠛⠛⠛⠉⠁⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠉⠀⠀⠉⡏⠉
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡐⠈⠐⠃⢁⡼⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡈⠙⠃⠁
⠇⠀⠀⠀⠀⠠⠴⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠦⠄
⠘⢄⠀⠀⠀⠐⣚⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣓⠂⠀⠀⠀⡠
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠂⠀⠠⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠀⠠⠄⠨⠁⣈⣁⡀⠥⠠⠄⠀⠒⠂⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠦⠶⠦⠳⠶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡌⠛⠁⣰⢶⣾⠊⢈⢈⣤⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣧⡀⢛⣉⠉⢨⡁⠛⠚⢺⡏⠉⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣿⠉⠉⠛⠻⢧⠆⣄⠀⡅⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡖⠀⠀⠀⠀⢀⠷⢧⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⢈⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⣠⠞⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⢫⠀⠀⠀⣠⠟⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠉⠛⠲⢤⣀⠀⠀⠀⠀⣰⠏⠀⠀⢂⠈⠓⢲⠎⠀⠀⠀⠈⠑⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⢀⣀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⡾⢫⠀⠀⠄⠈⢀⠀⠠⠁⢀⠀⢈⡀⠉⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠀⣠⣜⣹⠟⡉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠎⠀⠀⠀⠀⠀⠐⢀⠀⠀⡀⠈⠀⠀⠀⢀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⠀⣀⠤⠒⠈⣩⠿⠃⠢⡀⣠⡏⠝⠉⠛⠛⠓⠒⠀⠀⣤⡟⢀⡇⠀⠀⢀⠁⠀⠀⠀⠀⢀⠀⠰⣄⠀⠀⠀⠀⠀⠘⢣⡀
⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⢞⠱⣅⠀⠀⠀⠀⠀⠀⠈⠙⠶⠦⣀⡀⠀⠁⠀⠀⠂⠀⠀⡶⠀⠀⡁⠀⠀⠀⢀⠀⠀⠀⠐⢀⠟⠳⣄⠀⠄⠀⠠⠀⢳
⣀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠹⣆⠉⠓⠶⣀⠄⠈⣀⣼⠃⠀⠀⠀⠄⠂⠀⠀⠄⠀⠈⠀⡜⠀⠀⠈⠳⣄⠀⠀⠄
⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠊⠉⠣⡈⠢⡀⠀⠀⠀⠀⡀⠰⡅⠀⠀⠀⢈⣱⡿⢛⡿⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐
⠀⢸⡟⠻⣦⣄⠀⠀⠀⠀⢀⠀⠀⠀⠳⡀⠀⠙⣆⢹⣦⣠⠖⣉⠤⠚⠁⢀⣤⡶⠛⠁⣠⡞⠁⠉⠀⠀⠈⠐⠀⠈⠀⠀⠤⣴⠀⠀⠀⠀⠀⢀⡟
⠀⢸⡇⠀⠈⠛⢷⣄⡀⠀⢸⠘⡄⠀⠀⠹⣄⠀⠀⣡⠤⠒⠋⠁⣀⣴⠾⠛⠛⠒⠛⠉⠉⠳⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠞⠀⠀⠀⠐⡼
⠀⢸⡇⠀⠀⠀⠀⠉⠻⣦⡀⠁⢱⡄⠀⠀⠈⠲⠋⠀⢀⣠⣴⠟⠋⠁⠀⠀⠀⠀⢀⣤⠖⠀⠀⠙⠲⢦⣄⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣴⠋⠀⠀⠁⣠⠞⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠐⢰⡾⢧⡔⢃⣀⠈⠉⣧⠰⠞⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣠⣴⠾⢻⡇⠀⠀⠀⠀⣈⣢⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠀⠀⢻⠖⠁⠀⡼⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠀⠀⢸⡇⠀⠀⠀⠀⠈⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠒⠋⠁⠀⠀⢀⣀⣴⠛⠦⠤⠚⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣀⠢⣤⡤⠖⢺⣿⠋⠉⠛⠛⠛⠛⣿⠋⠉⣿⡆
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⢻⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡄⠀⣿⣇
⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠸⣧⠀⠀⠀⠀⢀⡿⣆⠀⠀⠀⠀⠀⠈⢧⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡏
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⠁⠹⣆⠀⠀⠀⠀⠀⠘⡇⢸⡿⠀⠀⠀⠀⠀⠀⣿⠃⠀⢸⣷
⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⠀⠀⠹⣆⠀⠀⠀⠀⠀⢻⣼⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣿
⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣿⣀⡀⣀⣀⣼⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⣿⣀⣤⡶⠟⠋⠉⠉⠉⣿⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⣿⠉⠀⠀⢀⣀⣀⣀⣠⡿⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⣿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⢀⣠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠋⠉⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⢀⣀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⢰⡟⠋⠁
⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⣶⡾⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣶⣦⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⡀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠐⠀⠀⠀⠀⢄
⠀⠈⠒⠤⢤⠤⢄⣀⣁⡀⠀⠀⠀⠠⠤⠄⣀⢣⠀⠀⠀⠀⣀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⣀⡀⣑⠀⠀⠀⠘⠆⢧⡄⠀⠀⡱⠊⢍⡉⠉⠐⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠄⠀⠀⠀⠠⡀⡀⡀⠀⣄⢀⠀⠠⣀⠀⡀⠀⠃⠔⢱⢸⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠴⡄⠐⠒⠁⠇⠀⠠⡀⢠⠀⣀⠀⠠⠂⠉⠻⣄⠀⠀⡸⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠘⠚⠀⣬⡀⠀⢀⣱⢀⢀⣀⠞⣧⣤⣤⣴⠏⡷⠊⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⠤⠤⢾⠈⠙⠿⠍⠉⣼⠄⢠⡀⠀⠈⠉⢀⣼⡅⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣼⣦⣤⣠⣤⣶⠗⠛⠻⣌⣐⣓⣫⠽⠷⣄⠠⡀⠀⠀⠀⠀⠀⠈⣇⠄
⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢰⡟⠀⠈⠉⠉⠀⠄⠀⠀⠀⠘⠛⠒⠒⠒⠾⢷⣾⣦⠀⠀⠀⠀⠀⣿⣶
⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡍⢥⠀⠀⠀⠀⢻⠿
⠀⠀⠀⠀⠀⠀⢇⢀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠠⡀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠘⢎⠓⠤⡄⠤⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣈⢉⣹⠧⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠴⠇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⡀⠄⠀⠶⠂⠖⠒⠒⠒⣶⣶⣦⣤⣄⣀⡜⠲⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡤⠀⠀⠀⡀⠠⣍⣀⣀⣤⠈⠉⠉⠂⠛⢛⡛⢿⣷⡆⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠰⣇⠀⠀⠀⠀⣡⠀⢩⡀⠈⠹⠷⣤⣈⣁⣼⡿⢃⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⢤⣒⡐⠀⠀⠀⠀⣠⣶⡀⢼⠭⢽⢟⠉⢰⣏⣹⡷⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠴⠖⠀⠉⠙⠀⠋⡿⣿⠃⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠻⠶⠒⠋
⠀⠀⠀⠀⠀⢀⡔⠟⠛⠉⢛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠿⣶⣤⣀
⠀⠀⠀⠀⠐⠈⠀⠀⠀⣰⠻⡿⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠤⣻⣿⠵⣺⡀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢒⣛⣷⠋⡷⢲⣦⣧⣀⡀⠀⡀⠀⢀⢃⣠⣶⣟⢓⣏⣹⣿⣧⠆⠀⠀⣘⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢋⠥⢠⠈⢉⡛⠛⠛⠿⠛⠿⠛⠻⣿⣯⣯⣯⠋⠁⠀⢀⣠⣸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠛⠁⠀⣀⣀⠀⠀⠀⣀⠀⠈⢀⣤⠄⢰⠿⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠉⠉⠁⠀⠓⠚⠛⠉⠁

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

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

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




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣤⣀⠀⣄⣀⣈⡀⠀⠁⠀⠈⠀⠀⠀⣈⣀⡤⡰⠂⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠈⠳⢦⣤⣭⣍⣛⣢⡄⠀⣐⣙⣥⡍⣴⠚⠁⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⠑⠀⠀⠉⠀⠈⠁⠈⠳⠋⠉⠁⠀⠀⠀⠙⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡀⠀⠁⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠼⠀⠀⠀⠀⠀⢀⡀⣀⠀⢼⡀⠀⡀⠀⠀⠀⠀⠀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡄⠠⠡⡀⠠⡏⠠⢄⡿⠚⢿⠈⡄⠑⡄⢀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠚⠦⣌⠊⠴⢓⢴⣛⠡⣀⠿⢿⣶⠚⡁⠁⢠⢈⡮⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠒⠶⢖⡛⢷⣾⣷⣄⠀⡈⠳⡤⠤⡤⠠⠄⠉⠓⠤⠤⠤⠤⣚⢃⣡⠿⠖⠂⠊⣁⠤⠓⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⣨⠉⠓⠓⠤⠍⣈⠉⠉⠓⠒⣶⣦⡠⠶⣷⠷⠟⢛⡿⣿⠥⠶⠒⢊⡉⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠐⣚⣭⡍⠀⠀⠀⠀⠀⠀⠀⠉⠉⠺⢭⣀⣁⣁⣠⠐⢉⡡⠁⠀⠀⠀⠀⠀⢛⠒⠂⠠⠀⠁
⠀⠀⠀⡔⠀⢐⡩⢴⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢍⡐⠒⠆⠀⠂
⠀⠀⠀⠈⠄⢡⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⠉⠂⠈
⠀⠀⠀⠀⠀⢺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⠤⠴⢒⡟⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠙⢤⡀⠀⠁⠐⠚⢄⡀⣠⠴⠐⠈⠀⠀⣠⠴⠭⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⣭⣮⢐⣦⣄⢀⢀⣀⠀⡀⣀⣀⣤⣾⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠗⠹⠕⠋⠁⠀⠀⠀⠀⠀⠀⠉⠉⠻⢽⡟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣿⠇⠀⢀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠟⠀⠐⠋⠻⠵⢢⣀⢕⣀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢑⣀⣠⢻⠀⠀⠀⠀⠘⠈⠈⠠⠤⠠⡐⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⣢⠓⡽⣶⠂⡀⠀⠀⠠⡞⢉⣀⠠⠂⠀⢀⠀⣷⠀⡏⣠⡇
⠀⠀⠀⠀⠀⠐⠻⣿⣿⣿⣿⣿⣷⢄⣄⣢⠇⠀⢀⠁⠀⠀⠀⠀⠉⡀⠀⠀⢹⡄⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠢⠽⡿⠀⠀⠀⠈⠀⡄⣶⡶⠈⠁⠀⢸⡀⠻⢃⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠄⠀⠀⢸⡦⣄⣀⢸⢘⠀⣿⠃⡅⢠⣄⠸⣇⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡎⠀⠀⠀⠀⣧⡀⠉⠚⠚⢲⠃⠠⡄⠀⢻⣆⣫⣴⡶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢣⡀⠀⠐⢀⣿⣿⣦⣤⣠⣏⣀⣴⣷⠀⠈⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣴⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠐⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⣀⣠⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⠗⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠁⠀⠙⠋⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠴⠆⠀⠀⠀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢹⣿⠿⣿⣿⣿⣿⣿⡷⠀⠀⠠⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⡤⡔⠁⠀⠀⠭⣿⡍⣥⣆⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⡾⡀⢢⠀⠖⠂⢸⡆⠀⠉⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣾⡇⢻⡌⠀⠀⠀⡈⣰⡃⠂⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠊⣧⣷⠈⠁⠀⠀⠀⠐⠊⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡚⣋⢱⢼⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⡶⠔⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠄⠀⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⢻⡋⠛⣛⣛⣒⡢⢤⣀
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⡀
⠀⠀⠀⠀⣜⣀⠊⢗⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢌⡱⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢈⣾⣹⣿⣶⣤⣁⢋⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠇⠈⠑⠢⡤⠤⠀⢀
⠀⠀⣤⠒⠨⡙⠻⢾⣿⢏⠂⡀⢀⠀⠀⠀⢠⠖⠉⠀⠀⠀⠈⠑⢂⠀⠀⣀⢀⠀⢸⡾⠀⠀⠀⠠⠀⠀⠀⣸⡆
⠀⠜⡈⠀⠴⣠⢮⠕⢀⠀⠉⣸⠩⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⣑⠈⢸⡇⠀⠀⠀⡄⠀⠀⢠⣿⠁
⠀⠑⠚⠵⣢⠄⡉⢀⠂⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠛⠉⠙⠛⠓⠳⠶⣾⡇
⠀⠀⠀⠀⠀⠉⠒⠂⠀⠀⠀⠀⠀⠀⠴⠶⡈⢦⠀⠀⠠⠀⢀⡠⠊⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣒⣒⡠⡞⠊⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠸⠈⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡴⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣆⣾⣀⣸⣆⣆⣀⣀⣀⣀⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⢧
⠀⠀⠀⠀⠀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⢾⢾
⠀⠀⠀⠀⠀⠘⡆⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡞
⠀⠀⠀⠀⠀⠀⣟⣠⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⣤⣵⡀
⠀⠀⠀⠀⠀⠀⠘⣄⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⢦⡀⠸⡅⠈⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣆⣬⢽⡀⠀⠀⣠⣤⡴⠦⡿⣅⠈⢸⡇⢠⡇⠀⡇⢀⣴⣶⣾⡿⠿⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢰⠃⠀⠰⣩⣯⡉⠰⣇⣼⣇⠀⢳⣸⣿⣦⡷⠋⢩⣴⣶⣾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣈⣧⣤⣷⣤⣀⣿⣿⣷⠀⠻⢿⣿⣦⣤⠽⣯⣷⣶⣶⡿⣋⣵⢖⠛⣻⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢰⣭⠾⠟⣛⣭⣴⣶⣮⣭⣿⣦⡶⠞⣿⣷⣤⠀⢿⣿⣿⣿⣵⣿⢿⣼⣾⣿⢟⣛⣻⣿⣦
⠀⠀⠀⠀⠀⠀⡞⣀⣰⣿⣻⣿⣟⣛⠿⣿⣿⣿⣷⣦⠈⠻⣿⣷⣦⣿⣿⣿⣿⣴⣼⣾⡿⠿⣿⣶⣿⡟⣿⣧
⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⣯⣭⢻⣿⣿⣮⢿⣿⣿⡿⠿⠾⠿⠋⢻⡉⠻⠿⢿⣾⣿⣿⣿⣿⣼⣿⡿⣱⣿⣿⣧
⠀⠀⠀⠀⠀⢸⣿⣿⠿⢛⣙⠿⣵⣦⡌⣿⣿⣿⣿⣿⣶⣤⣶⣤⡿⠤⢤⣤⣄⣀⡏⠀⠟⢻⣿⣿⣵⣿⣿⣿⡿⣧
⠀⠀⠀⠀⠀⢸⣿⣿⣗⢿⣿⣷⣿⣿⣷⣿⣿⣿⡿⢛⣛⢻⣿⣟⢴⣿⣷⣿⣿⠟⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡙⣆
⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣇⣿⣻⣷⣿⣿⣾⣿⣿⠟⠁⢀⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣭⣭⣭⣭⣾⣿⣿⣿⣿⣿⣿⣯⣿⢿⣿⣿⡯⠀⢀⡎⠀⠀⠀⠙⠮⣽⣿⡿⠿⠿⣿⠟⡼
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡿⠻⣿⣿⠛⠋⠀⣽⣿⢤⣤⣂⡼⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠴⠊
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⡿⢣⢻⢦⣀⣹⣧⣀⠀⢜⣻⠁⣀⣽⠇
⠀⠀⠀⠀⠀⠈⡿⣿⣿⣿⣶⡇⠀⢠⠃⠈⡷⠤⢤⣾⣟⣛⠛⠛⣋⡩⠏
⠀⠀⠀⠀⠀⠀⠙⢬⣛⠿⠿⣁⡴⠋⠀⠀⠉⠋⠉⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣦⣰⣦⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⣤⣤⣽⣿⣿⣿⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⡿⠉⠁
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣼⣷⠟⠑⠂⠀⠀⠀⠀⠀⣤⣀⡀⢀⣀⡀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣤⣎⣁⣠⣤⣤⠠⢬⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⣶⣿⣿⣿⣿⣿⡿⣿⣩⢹⡿⣿⡟⢡⣾⣿⣫⣿⣥⣼⢿⣿⣿⣧⣀⢹⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠈⣿⣿⣿⣿⡟⢻⣇⣿⣦⡿⣼⣿⣿⣾⣿⣿⣿⡿⣿⣛⣿⣽⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⢻⣿⠉⣿⠀⢸⣿⣎⢿⡇⣿⣿⣿⣷⣤⣼⣿⣿⠉⢵⣶⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠘⣿⣧⣿⣄⢠⣿⣿⣷⡇⣿⣿⣏⡻⠉⣿⣿⣿⡇⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟
⠀⠀⠀⠀⢿⣿⣿⣿⣾⣿⣿⣿⣿⣾⣯⣿⣿⠛⢻⣿⣿⣃⣴⣶⣶⣿⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⠉⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⢿⣿⣿⡉⢸⣿⣿⡹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣾⢿⣿⣤⣬⣏⣖⡿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠛⣿⠛⠛⠉⣿⣤⣴⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣿⣿⣷⣿⣶⣶⣿⣿⣽⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣾⣿⡿⠁⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⢀⡀
⠀⠀⠀⢰⣦⡀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡆⣼⣷
⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠈⢻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⡀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣰⡄
⠀⠀⠀⠀⠀⢠⡞⣲⡀⠀⠀⠀⢀⠀⠄⠀⠊⠈⠃⠁⠎⢱⠀⠀⢱⠁⠀⠁⠀⠁⠀⠀⠐⠀⠂⠀⠀⠰⠒⠀⠀⠀⠀⠀⠉⠀⠀⠀⠄⠀⠀⠁⠀⠋⣠
⠀⠀⠀⠀⠀⠁⠀⠀⣣⠀⡆⠐⠐⠆⠠⡆⡀⠠⠀⠄⠀⠈⠀⠠⠈⠀⠀⠠⡀⠀⠀⠀⠀⠃⠸⠄⡰⠶⠀⠘⠅⠴⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠇⢃⠈⠁⠀⠀⡄⠀⡀⢃⠀⠠⠁⠁⠌⠙⠕⠀⠠⠀⢈⠉⠀⠄⠀⠀⠀⠀⠀⠁⠰⠈⡄⠒⠂⡀⠀⠀⠀⠀⢀⠂⠀⡀⠀⠀⢌⠀⠁⠘⢁⡀
⠀⠀⠀⠀⠀⠀⣀⡂⢀⡀⠐⡄⠠⠁⡠⢄⠀⠀⢠⠴⠉⠃⠀⠀⠀⠀⠤⠀⠼⢠⠈⠀⠀⠀⠀⠀⠀⠀⡋⡶⠁⢤⠰⠆⠀⠀⠰⠀⠀⠀⠀⠀⠂⡀⠀⠠⡠
⠀⠀⠀⠀⠈⠀⢨⣞⠡⠠⠀⠭⠗⠜⠀⠀⠂⠀⠀⢠⠀⠀⠀⠂⠀⠀⠀⠳⠤⠊⠀⠐⠀⠀⠀⠒⠀⠀⣤⡇⠀⠀⡠⡠⠁⠀⠠⠀⠄⠈⠂⡀⠀⠈⢔⠆
⠀⠀⠀⠀⠀⢀⠈⠁⠀⠰⠈⠤⠀⠤⠀⢠⠤⢠⢠⡤⣄⠀⠀⡠⢀⠀⠀⠀⡔⠀⠀⢄⠀⠀⠀⠀⠠⠀⢁⡀⠀⠐⠠⠁⠀⢠⠀⠀⠀⠀⠀⡀⠐⠀⠈
⠀⠀⠀⠀⡠⠀⣍⠏⠀⠀⠀⠀⡀⠀⠀⢸⠋⠉⠉⠉⠹⠀⠐⠁⠁⠀⢀⢀⠁⠀⠀⡄⠀⠆⠀⢰⠈⡀⠈⠀⠄⢈⠄⠐⠀⠸⠀⠀⠆⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⣀⡆⠀⠐⢆⡦⠀⠀⠙⠉⠉⢉⠀⠀⢠⠄⠈⠁⡁⢠⡀⠂⠸⠁⠀⠐⡀⢢⠀⢀⠀⡘⠀⠀⠀⠀⢀⠂⠄⠀⠀⠈⠀⠐⠀⠠⠆⣤⠖⠁
⠀⠀⠀⢀⠕⢫⠐⡄⠀⡁⠀⠀⠀⠀⠀⠈⠀⠀⢸⠀⠐⠀⠛⠓⠂⠈⠚⠀⠉⠀⠈⠀⠀⠀⠃⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠒⠀⠀⠁⠤⠀⣻⣀⡄⠠⠄
⠀⠀⠀⠀⠀⠈⠆⠼⠂⠠⠀⠠⠀⠂⢼⠂⡄⠀⠘⠀⢀⠀⠄⠀⠠⠄⢜⠠⡀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢉⠄⡜⠀⡌⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢐⣛⠀⠀⠀⠀⡀⠀⠂⠀⠀⠈⠀⠀⠠⢏⠄⡀⠀⠰⠅⠻⣘⠕⢮⢀⡐⠀⢐⡅⠀⠀⠀⠀⠀⠀⠀⡟⢀⠁⠀⠀⢃⡀⢐⠀⢀⢶⣷⠊⠀⠆
⠀⠀⠀⠀⠐⠚⠃⠐⠀⠂⠐⠀⠀⠀⡀⠁⣀⣀⡀⠀⠀⣴⠀⠀⠅⠀⠈⢶⡁⢣⡈⡈⠀⠦⠀⠀⠀⠀⠀⡀⠀⣋⡊⠀⠀⠀⠃⠀⠀⠐⠀⠀⠀⠠⠀⢀⠄
⠀⠀⠀⠂⡀⠠⠀⠡⠁⠞⡆⠀⠀⢈⠀⠀⠁⠁⠀⠊⠀⠘⠶⠰⡆⠀⣀⠄⠱⣂⠱⡀⠐⠀⠀⢀⡠⠂⠀⠀⠀⠄⠀⠀⡀⠀⠙⠂⡁⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠑⠂⠀⠈⠃⠐⠀⠀⠀⠀⠀⠒⣀⠄⢴⠀⠈⣹⡄⠂⡃⠀⠀⠀⠀⣀⢀⠀⠐⠃⣾⢿⡀⠅⠀⠄⠀⠠⠀⠀⠈
⠀⠀⠀⣠⡀⠀⠀⠀⠀⢠⠀⠀⠀⡀⠈⠀⢠⠀⡃⠐⠀⠠⠀⠀⠀⠈⠛⡀⣀⡼⠁⠀⠀⠀⢤⠑⡀⠀⠈⠀⡈⠉⠀⣦⡶⢹⡎⠀⠘⡀⢀⠂⠒⡆
⠀⠀⠀⠀⢀⣄⡀⢀⢀⠨⡶⠀⢀⠀⠀⠀⢲⠀⠃⡀⠀⢐⠂⡀⠀⠀⠀⠈⠀⠠⡖⢲⠀⠄⠐⡂⠁⠀⠀⡈⢠⠀⠀⠁⠀⣸⡗⠀⠀⠀⠉⠅⢀⠁⠀⢄
⠀⠀⠀⠀⠀⠀⡸⠣⡀⠀⠉⠀⠀⠠⠐⢀⢴⠀⠀⠄⠀⠠⡀⠁⠀⡄⠀⠀⢀⡀⠑⠉⠀⠀⠀⡥⢘⠀⠀⣂⣀⡇⠀⠀⠃⡞⡏⠀⠒⠂⠀⠁⠸
⠀⠀⠀⠓⠀⠀⠀⠲⡇⠀⠀⠀⠀⠐⠀⠹⠿⡦⠤⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⡀⠚⠈⠀⢀⣠⡄⠧⠄⠒⠒⠻⢇⢤⡐⣦⣀⣀⠄⡠⠆
⠀⠀⠀⠀⠈⠀⠀⠅⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠂⠀⠁⠀⠁⠀⢌⠀⠆⠐⠒⠐⠂⠒⠈⠒⠃⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠟⠽⠤⢄⠈⠉⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣷⣭⡄⣤⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠁⠉⢻⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣧⣤⣴⣿⣿⣿⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠛⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠅⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣧⣤⡀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⣠⡀⠀⣤⣤⣀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣷⣿⣿⣿⣿⠿⣿⣿⣿⣿⣦⠀⡜⡿⠛⠛⠉⠉⠋⠉⠀⠀⠀⠸⠿⣦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣟⣼⣿⡿⠛⠟⠗⠀⠀⠙⠿⡿⢿⣮⣀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠹⠿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⢀⣀⡠⠞⠀⠀⠀⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣔⣤⣄⣤⡶⠋⠊⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡄⠈⠢⠀⠀⠀⠀⠀⡀⡀⡠⠂⠀⠘⠁⠀⢀⡴⠤⠤⠦
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠂⠀⠀⠱⡀⠀⠀⠀⠀⢀⠈⠀⠘⡀⠀⠀⢸⠋⠹⠇⠀⠀⠀⠀⢀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠈⢢⣼⣄⡀⠀⠰⠆⠀⢸⡀⠀⡆⠀⠀⠈⠁⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠗⢾⠁⠀⠈⢧⣠⣴⡦⠤⠾⠀⠀⣹⡄⢀⠔⢀⠀⠦⠸⣦⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⣀⣤⣴⣾⡟⠟⢿⠐⠀⠙⠢⣀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⢰⠀⠀⢘⣦⣤⣴⣾⣿⣿⣿⣿⣿⣄⠴⣬⣧⡀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣄⡼⠀⠂⠀⣸⡟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⣄⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡄⠀⠒⠊⠀⣿⣇⣸⣦⣿⣿⣿⣿⣽⠿⠛⠛⠉⠀⠀⠀⢀⣾⣿⣷⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡅⠀⠀⠀⢸⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠙⢿⣿⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠱⡀⢀⠀⢸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡘⠌⢂⠀⢸⣿⣿⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣙⣏⣍⣹⣏⣽⣻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠳⠦
⠀⠀⠀⡴⠉⠀⢴⠄⠀⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠈⠀⠀⠀⠸⡄⠛⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠒⠂⢒⠚⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⣒⣒⣒⣦⣀
⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢀⠀⣀⠈⡇
⠀⠀⠀⠑⢤⡀⠀⠀⠀⡠⠋⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠠⠤⠤⠤⠤⠤⠤⠀⠒⠒⠒⠒⠒⠒⠂⠀⠤⠤⠤⣄⠀⠀⠀⡇⠀⠂⠀⠀⠠⠅⠀⠂
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡠⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠇⢀⣁⣈⠀⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⢦⠀⠀⠸⠀⢀⠀⠐⠀⠀⠀⠀⠀⢠
⠀⠀⣀⡀⣀⣀⣠⣄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠀⠀⠀⢀⢤⡀⡰⣤⠀⣤⢶⢀⣴⠀⠏⢡⠜⡷⠋⠀⠀⢠⠀⠀⡄⠈⠀⠉⠀⠀⠀⡀
⠀⡧⠈⠀⠀⠀⠀⠀⠇⢸⠀⠀⠀⠠⢈⠀⠀⠀⣦⠌⠀⡾⢁⣎⣰⠇⠰⠤⠊⠃⠘⠉⠀⠀⠀⠀⢀⠖⠁⠀⠀⢸⠀⠀⠑⠠⠤⠶⠤⠤⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢰⡂⠀⠈⠀⠳⠠⠊⠑⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠴⠮⠀⠀⠀⠀⢸
⠀⠈⠀⠒⠒⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠡⠀⠀⠀⢸⣀⡀⠤⠀⣄
⠀⠀⠀⠀⠀⠀⢰⢧⢀⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀⠔⠀⠀⠀⢸⠋⡠⠀⡆⠡⡱⠄
⠀⠀⠀⠀⠀⠀⠀⢸⠈⠁⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠶⠆⠠⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠂⠄⠀⢀⠓⠏
⠀⠀⠀⠀⠀⢠⠞⠋⠉⢳⠹⠄⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣈⣀⣀⣈⣴⡄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⡔⠈⠂⠤⠄⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠠⠤⠤⠤⠰⠒⠒⠠⡀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢆⣀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡩⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢀⡀⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡂⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡠⠒⠀⠈⠙⢢⣀⠄⢀⣀⠀⠀⠘⠢⠤⠤⠤⠚⢀⣀⣀
⠀⠀⣠⠞⠉⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⣈⣉⡉⡩⠭⠛⠋⠢⠀⠀⢀⠜⠁⠀⠀⠀⢠⠊⡤⢀⠀⠀⠈⣲⡀⠀⠀⠀⢰⠁⠀⠀⠀⡿⢝
⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⡔⢘⠈⡡⠫⡃⠈⡀⠄⠀⣐⠀⡂⠀⣆⠀⠀⠀⢀⠔⠁⠮⠀⠀⠑⢀⠔⠡⠋⠀⠀⠀⢸⠀⠀⠀⠀⡧⠘
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⡄⠀⡇⣂⠊⢀⠄⠀⢁⡁⠐⠠⠅⠲⠁⢄⠛⠒⠒⠒⢻⢇⡒⠤⢀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠰⠁⠁
⠀⠹⡄⠀⠀⢀⣀⠀⠀⡏⠉⠈⠇⠀⠐⠀⠁⠀⠈⠁⠀⠀⢀⠔⠁⡠⠔⢂⠑⡀⠀⠀⠀⡐⠉⠁⢀⠐⠚⢳⠀⠀⡠⢆⣠⡏⠀⠙⠒⠚⠃
⠀⠀⠈⢦⣰⠉⡀⠱⡀⠘⡈⣀⡄⠀⠀⠀⠀⢀⠤⠒⠀⠀⠙⠄⠈⠔⡨⠐⢀⠐⡄⠀⠀⡑⠠⠀⢀⠕⢀⠎⠀⠌⠀⠀⠉⠁⠊
⠀⠀⠀⠀⠙⢄⠈⢄⢠⠀⠑⠉⠈⠂⠢⡀⠀⢃⠀⠀⠀⠀⠀⠈⢀⠈⠔⠈⢀⡤⠊⠀⠎⠁⠀⠀⠁⡔⠁⠀⠀⢆⡀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⢢⠀⠀⠀⠀⠀⠈⢆⡀⠂⠁⠀⠀⡜⣫⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠡⠀⢀⠠⠐⠀⠉⠀⠀⠀⠀⢀⣡⠏⠐⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⣶⣶⣿⣿⣿⣿⣟⣯⡙⠛⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⠟⠻⠿⣿⣿⣧⢹⠏⢫⡹⡇⠀⠀⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⢰⣿⣿⣿⣿⣿⣿⣎⣧⢀⡘⢙⣯⠀⠀⢸⡃⡄⠀⢀⠀⠘⠺⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡀⣿⣟⣿⣿⣿⠿⡿⣿⣿⣿⣿⣴⣿⣤⣦⣾⣱⣤⣀⣮⣴⠂⠀⣻⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡟⣡⡦⢘⠻⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢟⡻⣿⣫⣩⡷⠎⠀⠀⠀⢹⣿⣟⠻⠭⠉⠙⠛⠛⠿⢿⣿⣿⣌⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⠈⠙⠻⠻⣿⣽⣷⢦⣄⠁⠀⠻⣟⠻⢶⣦⣀⠀⠀⠀⠀⠈⠙⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠘⠻⠷⢶⣦⣌⠐⠈⢢⡌⢠⣿⣷⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢀⣤⡂⠀⠀⣼⣿⠻⣦⣤⡝⣦⡀⠐⣻⣾⡟⠉⠦⣀⠀⠀⠀⠀⠐⠲⣤⣀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣷⡛⠉⡘⣧⠀⠀⠙⢷⣄⣸⣿⣟⣿⡻⠊⣝⡟⠃⠀⠀⠈⠂⠠⣁⠀⠀⠀⠀⠈⠉⠠⡘⣿
⠀⠀⠀⠀⠀⠀⠀⣤⠃⢁⠀⢀⡘⢳⢠⡄⠀⠈⠉⠉⠉⠘⠁⠀⣹⣆⠀⠐⢄⠐⠀⠀⢙⢷⣄⠀⠀⠀⠀⠀⠙⢿
⠀⠀⠀⠀⠀⠀⠸⡻⢷⡈⠶⣤⢤⡛⢶⠀⠀⣾⣿⣿⣦⣀⣾⠿⠻⣿⠀⠀⠈⢳⡀⠀⠈⢻⣿⣿⣦⡀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⢀⡼⡿⠊⠙⠀⠈⠂⢳⣟⢠⣠⢸⣿⣿⣿⠗⢾⣧⡀⠘⣆⠁⠀⠀⠘⠦⠀⠀⢿⡿⣿⡿⠆⠀⠀⠞
⠀⠀⠀⠀⠀⢻⣋⡟⠀⠀⠀⣢⣤⣤⣜⠀⡁⠈⠀⢉⢀⡄⡉⠛⠿⠆⢈⡳⣀⠀⠀⠀⠀⠀⢸⣷⢹⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠙⠢⣤⣀⣸⣿⣿⣿⣿⣷⠀⠀⢀⣠⣾⣿⣿⣶⣶⣿⣿⣿⣏⢷⡀⢀⢀⡌⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢠⢯⡻⣿⣿⣿⢿⢿⣧⣾⣶⣾⣿⣿⣿⣿⣿⣛⡛⣿⢿⡿⣧⣹⣦⢣⣿⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠸⠶⠊⠁⠀⢻⡆⠀⠀⠨⡍⠙⣻⣿⣯⣿⣿⣯⣿⣿⢸⡇⠸⠇⢻⢾⣿⠃⡸⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣤⡄⣿⣷⣘⠄⣐⣠⣲⡿⠟⠉⢁⠿⣹⣿⡧⢸⠇⠀⠀⠀⣸⠏⠀⠃⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣵⣿⣿⣿⣷⢫⢟⡟⡷⠂⡀⠃⠀⠻⠛⢀⡿⣠⠀⠀⢸⡏⠀⠀⠀⣠⡾⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠈⡟⠛⠑⣿⢻⡿⣷⣯⢟⡞⣽⠃⠆⠀⠀⣰⣶⡿⠁⢣⣀⣴⣿⠁⠀⢀⣾⠋
⠀⠀⠀⠘⠇⠀⠀⠀⡰⣧⠀⠀⣿⣄⠁⠹⣷⣏⣼⣱⣿⣤⠀⢠⣿⡟⡱⢃⣾⠉⣸⠇⠀⠀⣼⠃
⠀⠀⠀⠀⠐⣤⣴⣿⣿⡿⡄⠀⣿⡈⡄⠀⠙⠛⣻⣟⣿⣿⡄⠷⢟⠔⣵⣿⡏⢀⡟⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠁⣴⣿⣸⡇⠀⠀⢸⠿⣯⣿⣿⣷⠀⡊⠀⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡣⠁⠀⠀⢸⣿⣿⡙⣿⣿⡄⠁⠸⠏⠿⢷⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢋⠁⢄⡤⠠⠌⠶⣦⡹⣾⣿⣿⡀⠀⠀⠀⠸⣿⠃⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠠⣶⠠⠌⠀⠰⠀⠐⡦⢷⢿⣿⣿⡦⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣟⡡⠈⠴⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠄⢒⢒⠎⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣾⡿⠓⢫⡾⠀⠀⣄⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣀⣺⣾⡇⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣬⣥⣾⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠤⠎⠁⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠴⣶⣄⣀⣀⣀⣠⡄⢀⣠⠞⠁⢴⡄⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠉⠛⠛⠉⠁⠚⠉⠀⠀⠀⠺⣧⠀⠀⠀⢠⣤⡆⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢆⣿⠟⣡⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠀⠀⠀⠀⠨⠋⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠶⢄⠀⠀⠀⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡴⡇⠀⠀⠀⠐⠲⠖⠋⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣧⡀⠀⠀⠀⠀⣰⠁⢐⠢⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⠟⣿⣿⠋⠀⠀⠀⢀⡴⠃⢀⣾⣇⢀⡀⠉⠉⠉⠛⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡎⢧⡀⠀⠀⠀⣀⡴⠋⢀⣴⣿⣿⣿⣿⣗⣴⣦⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣦⣉⠓⠒⠋⠁⠴⠾⠛⠛⠛⠋⠉⠙⣿⣿⣯⣌⡆⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⣀⣠⣤⣴⣶⣷⣾⢿⣿⣿⣿⣿⣶⣆⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡙⠻⡿⠿⢿⣿⡿⠟⠉⢀⣀⣤⣶⣶⣿⠿⠿⠛⠉⡉⠡⠿⢗⣿⣿⣿⣿⣿⡷⠃⣠⠀⠘⢆
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠀⠀⠀⢎⡉⠑⣸⣿⣿⣿⠟⢋⠉⠀⠀⠀⠈⠁⠀⠀⠉⠁⠸⣿⣿⣿⣿⣿⣿⣷⡀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⢻⣽⣿⣿⣛⠛⠛⠛⠛⠋⠙⠈⣿⣿⡟⠘⣿⣿⣯⡿⢷⣾⣯⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠰⠴⠭⠯⣛⣉⡁⠀⠀⠀⠈⢿⣿⣿⡁⢹⣿⣿⣿⣞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣄⠀⠙⣿⣿⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⣀⣠⡼⢖⠀⠀⠀⢸⠏⠻⣇⠠⢿⣿⡿⢛⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⣀⣀⣀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⢀⣠⡯⠗⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣟⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⢁⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠛⠉⠉⠉⠉⠉⠉⠛⠲⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⣶⣿⠿⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠿⣯⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡾⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣹⡁⣠⣾⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⢷⡄⣹⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⢃⡏⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣡⣜⣟⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠇⠀⣎⣔⣼⢯⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⡍⢻⣿⡾⡧⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢈⣶⣿⣿⣿⢿⡄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠁⠀⡆⢸⣿⣿⣿⣫⣖⣽
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⡾⣿⣹⣿⠏⠜⠁⠀⠀⠀⠀⠈⠢⡀⣸⢀⡔⠉⠀⠀⠀⠀⠀⠑⢜⣿⣿⣿⡿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⣼⡿⠃⠘⢊⣀⣀⠀⡀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⢀⣀⣈⠛⠊⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡥⣴⣾⣿⣿⣿⣿⣿⣷⣦⣤⣶⣿⣿⣷⣷⣶⣾⣿⣿⣿⣿⣿⣶⡤⣿⣿⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⢻⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠂⢻⣿⣿⡟⠋⠉⠀⢸⠏⡞⣴⣎⣶⡜⡙⣿⣿⣿⡟⠋⠁⡻⣿⢹⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⡛⠁⠠⠄⠛⢿⣿⡶⠒⠊⢁⡜⣰⣿⣿⣿⣧⡹⡌⠉⠛⠿⣿⡳⠞⠁⠄⠀⠙⢳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⣀⣀⡉⠑⣄⠘⠀⣿⣿⡿⣿⣿⡇⠟⢀⡔⠋⣀⣀⠀⠀⠀⠀⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣤⣤⣭⣽⣟⢷⣌⠆⠳⡈⡻⠃⢻⡋⡴⠃⠊⡴⢛⣭⣭⣤⣤⣤⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⢿⠿⣿⣷⡀⢀⢠⠀⠙⢰⠈⠁⠀⢀⠀⣼⣿⣿⣿⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⣿⡷⣧⡞⡜⣠⠀⣼⠀⢰⠸⢸⣮⡟⣿⡏⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⠀⡿⣿⣬⠉⡇⢻⠟⢿⠞⡾⢸⠋⣇⣿⠋⡇⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡘⠈⣿⠿⡷⣼⣤⣤⣤⣴⠾⡿⢿⠇⡤⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⢘⣣⣆⣻⠀⣬⡀⣾⣠⣴⣋⠜⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠙⢦⡥⣽⢿⡯⠈⡡⠛⠁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠁⠀⢸⠀⠈⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣬⡤⠤⠞⠦⠤⣤⡼⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣻⡟⡛⢛⡛⣻⢿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣄⣀⠀⠀⣤⠤⣴⡟⠉⢙⣶⠾⠛⠲⣄⠀⣾⠟⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢉⣁⣀⣀⡀⠈⠓⢾⣇⣀⣿⠁⠀⢸⡇⠀⠀⠀⠘⡄⣿⣀⣤⠴⠗⠒⠀⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠚⠉⣥⣶⡍⠙⢷⣄⠀⠉⠉⠛⠛⠛⠛⠛⠓⠚⠛⠛⠛⢻⠉⢀⠴⠞⢛⣛⡛⠶⣌⣆
⠀⠀⠀⠀⠀⠀⢀⡟⢀⣤⡀⠙⠿⠃⣀⣄⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⢿⣿⠇⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⣸⠁⠻⣿⠏⠀⠀⠐⣿⣿⠀⡇⠠⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣾⣿⡇⠀⠀⢠⣿⣿⠘⣷
⠀⠀⠀⠀⠀⠀⣿⠀⢦⡀⠀⣾⣿⡆⠀⠀⠘⠀⠀⠀⣿⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠠⡈⠉⠁⣴⣶⡀⠙⠋⠰⢻
⠀⠀⠀⠀⠀⢀⣿⠀⠀⠙⠲⠾⠭⠤⣠⠖⣂⣀⡀⠀⣷⠀⠀⠠⠈⠠⡀⠀⢀⠤⢒⡂⠙⠦⣄⣻⠟⢁⡀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠁⡾⠉⠉⢻⡄⣿⡆⠀⠀⠀⠀⠘⣆⠀⡼⠋⠙⢷⠀⡄⠀⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣠⡾⢡⣏⣿⡄⠀⠀⠀⠔⠉⠳⣷⣀⣀⡾⢡⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣸⠲⠬⠤⠶⠋⠀⠈⠻⣤⡀⠀⠀⠀⠀⠙⢏⣩⣴⣿⡄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⣴⣿⡀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠈⣧⡈⣿⣿⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⢀⡜⠀⢿⣇⠀⠀⠀⠀⠀⠀⣰⣿⢿⡇⠀⠀⠀⠀⠚⠁⣿⠏⢿⣷⡀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠈⠻⢦⡤⠴⠋⠀⠀⠸⣿⡀⠀⠀⠀⠀⢠⣿⠃⠀⢻⡀⠀⠀⠀⠀⠀⣿⠀⠈⠻⣿⣄⡀⢀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠁⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠠⣀⠆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡶⢀⠀⠀⢀⡄⠀⠀⠀⠐⠀⠀⡆⠀⠀⠀⡄⠀⣀⣀⠀⠀⠂
⠀⠀⠀⠀⢠⠄⡄⠀⠁⠀⠀⠄⠀⡂⠈⠰⠃⠂⠠⠀⡌⠁⠀⠀⠀⢺⡠⠀⠁⠀⡀⠐⠂⠈⠀⠡⠄⠁
⢠⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡇⢃⠀⢀⡀⣠⡀⠀⠀⢲⣖⣾⠀⠈⣻⠖⠀⠐⠀⠀⢁⠈⢀⠁⠁⠐
⠈⠀⠀⠀⠀⠀⣤⠀⠀⣀⠀⠀⢸⠘⠐⠀⠀⠉⠀⠀⣠⠟⠛⠛⠀⣄⠉⠀⠘⠃⠀⠀⣤⢀⡄⣤
⠀⠰⠆⠀⠘⠀⠀⠀⠀⠀⠀⢀⠸⠐⠀⢦⠐⠄⠀⡀⠀⠰⠐⡀⠀⢄⠀⡄⠂⠀⠀⠀⠁⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠉⡐⠀⢴⠈⠁⠀⠀⠀⠀⠀⠀⠀⠻⣌⢄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠰⠂⠀⠀⡦⡆⠀⠀⠀⡰⠀⠀⠅⢀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠆⣀⢈⠳⡷
⠀⠒⠀⣀⠉⠁⠀⠀⠀⠁⠀⠶⠀⠀⠄⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠃⠠⠐⠀⠀⠀⡀⠀⠀⠀⠀⠐⠒⠂⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘
⠀⠀⠀⠀⠀⠀⢆⠀⠀⢈⠀⢀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠂
⠀⠀⡀⠀⠀⠀⢀⠀⠐⠀⠠⠄⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠁⠀⡀⠄⠀⠃⢘⠀⠀⠀⠀⠐⠒⠙⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠐⠟⠐⠀⢘⠃⠘⠁⠬⠁⠀⢠⠀⠀⢠⠄⠀⠀⠀⠀⠀⠐⠀⠃⠘⠐⠀⡀⠀⠀⠀⠀⠀⠘
⠀⠘⠀⠸⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠒⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢦⡀⠀⠀⢀⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⡄⠀⠀⠀⢙⣦⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡣⡋⡄⠀⠀⠊⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠅⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠢⠰⠆⠀⠀⠀⠀⠐⠠⠿⠐⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⠀⠀⠒⠀⠀⢀⡀⢀⣀⣒⠀⠀⠀⠤⠀⠐⣀⢤⣤⠷⠟⠍⠂
⠀⠀⠀⠀⠀⠀⠀⡄⠉⠓⠓⠶⢤⣠⠀⢀⠀⠀⠤⣝⡈⠛⠛⠠⣀⣀⣤⡶⠿⠛⠉⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠀⠀⠈⠑⠓⠒⠲⡘⠀⡼⠟⠛⠋⠁⠀⢂⠨⠄⠁⠢⠀⡠⠞⠰
⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⠄⠒⣀⠄⠀⠀⠀⠀⠀⠃⠀⠐⠀⠀⠀⠀⢈⠁⠁⠶⢤⡄⠜⢸⣏⡀
⠀⠀⠀⠀⠀⡬⠐⢧⡀⢠⢊⠰⣢⠀⠀⢠⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⣆⡀⢀⢄⠀⠸⡧⠆
⠀⠀⠀⠀⠀⠡⡈⣰⣿⠀⠀⠀⢂⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠁⠓⠂⠀⠀⠳⡆
⠀⠀⠀⠀⠀⠀⠈⠚⠛⠀⠀⡤⠀⠀⠈⠫⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠄⠛⣙⢰⡆⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⢿⠂⠀⠀⠀⠀⠀⠀⠀⠘⠉⢉⣠⠖⠦⠉⠁⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠈⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠠⠘⠀⠁⠀⠀⢀⠠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⣀⠀⠀⠀⠀⠀⠀⡎⢰⠄⡆⠀⠀⠀⠀⡠⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠒⠢⠤⠄⡹⢍⠄⠠⠖⠀⠉⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠑⣾⣿⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠚⢌⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠒⠈⠀⠁⠀⠀⠄⠀⢀⠔⡠⠀⡴⢿⡟⠁
⠀⠀⠀⠀⠀⠀⡄⡒⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢁⠈⣰⠎⡠⠋
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⡀⠄⠤⠠⠤⠀⠒⠊⠁⠀⠀⠀⠠⠈⠀⢁⡴⢃⠜⠁
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠈⠋⠈⠉⠁⢀⣀⣀⠠⠄⠀⠐⠂⠀⠀⠀⠔⠁⢊⡠⠃⣠⠋
⠀⠀⠀⠀⠀⠀⠙⠀⠐⠀⠀⠒⠂⠁⠁⠀⢀⣀⣀⡀⡀⠤⠒⡰⠀⠔⣠⢟⢀⠞⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠒⠂⠊⠁⠀⠀⠀⠀⠀⢀⠀⠠⢲⡴⣏⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠤⠤⠤⠀⠂⠉⠁⠀⢂⠀⢠⡎⣠⠯⠤⠄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠤⠤⠄⠐⠂⠸⠣⣀⡴⣧⠞⢁⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠠⠤⠀⠐⡆⠀⡠⠗⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢆⠀⠀⠤⠤⠴⠟⠡⠀⠀⠀⠀⢀⢀⣀⣀⠀⠀⢰⠖⠒⠚⠻⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⡀⠀⠀⠀⣀⣀⠤⠔⠒⢛⣋⣉⣩⣥⣥⣤⣶⠾⡿⠽⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⣝⣒⣪⡭⠤⠖⢒⠚⠋⠭⢍⣀⣀⣠⣤⣤⣔⣶⣶⡾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⠛⢉⣐⣉⠠⠎⠁⠚⠋⠏⠋⠀⠂⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠞⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠄⠄⠤⠄⠀⠀⠀⠀⠀⠄⠔⠔⠐⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠄⢀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡹⡆⠀⠀⠀⠀⢠⡀⠀⣠⠄⠶⣇⠤⡨⠂⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⢀⠴⠚⠋⢙⠀⠀⠀⠈⠀⠀⢗⣆⠀⣨⠅⣀⠄⠀⠀⠀⠀⠀⢠⠏⠸⠄⠈⢊⠀⣄⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⢀⠠⠴⠚⠋⠀⠀⠀⠀⠢⠀⠈⠛⠫⠀⢂⠈⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢠⣎⣠⠄⢋⣘⠁⣄
⠀⠀⠀⠀⠀⠀⣀⣠⠄⠴⠋⠀⡀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⣀⠀⠀⡤⢯⡇⠈⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠰⠛⣉⠔⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠦⠰⠚⠁⠸⣏⠲⠀⠀⠀⢘⡇
⠀⠀⠀⠀⣰⣎⣁⠔⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠐⣒⠂⠀⠀⠈⠀⠀⠀⠐⠒⠂⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡔⢢⣤⠛
⠀⠀⠀⠀⠙⢿⡧⠊⠀⠀⠀⠀⠀⠀⠀⢀⡤⣚⠭⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣌⠳⣄⡀⢀⠠⢄⡤⠤⣄⣠⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⣀⣴⡯⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⢀⣀⣀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠘⡆⠑⡶⠉⠑⠂⠛⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠍⠚⡅⢰⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠙⠷⠚
⠀⠀⠀⢠⣍⣤⠞⡙⠠⡤⡌⠄⠀⠄⠀⠀⢱⠘⡧⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⢣⠏⡠⠊⠙⡆⢳⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠉⠚⢣⡼⠲⠔⠼⠀⠀⠀⠀⠀⢃⢧⠈⠑⠒⠤⠤⢄⣀⣀⣀⣀⣀⣀⡀⠤⠤⠒⠊⠉⢀⣮⠏⠉⠁⠀⠜⢁⠌⠀⠀⠀⣠⠘⢃⠀⠙
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣟⡠⠐⣀⠄⠂⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡞⣡⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⣠⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣠⣞⠋⠉⠉⠉⠉⢉⣁⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣟⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢹
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠁⠀⠀⣀⣈⣉⣢⣶⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣫⡄⢸
⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠛⠋⢉⡴⠋⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠈⢐⡇⢸⡆
⠀⠀⠀⠀⢀⡃⣀⣀⣀⡰⠂⠀⢠⠎⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠒⢻⣟⠁⡈⠀⠈⡇⢸
⠀⠀⠀⠀⠈⠁⠀⠀⡼⠁⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣿⣷⠿⣡⣶⡽⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⢠⠈⠁⠱⣈⠳⡴⢁⡎
⠀⠀⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡹⢉⣺⣷⣤⣄⠀⠀⠀⠀⠴⣒⠉⠡⣀⣆⠀⣬⣀⠀⣲⠿⠉⣠⠞
⠀⠀⠀⠀⠀⠀⠀⢸⠅⠒⠀⠓⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⠛⣵⠿⣿⡍⠁⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⣉⣉⣀⣉⣉⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠶⠷⠶⠶⠶⣿⣧⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣶⣮⢔⣩⣵⡶⣚⣛⡳⣶⣽⣷⣾⡇⠻⣷⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣾⡿⢋⣵⠿⣿⣿⣿⣩⣹⣿⣿⣿⣿⣿⣷⣾⣿⠻⣷⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢟⢱⠋⠀⣤⣿⣮⣻⢿⣿⣟⡜⠛⠷⢿⣿⣿⣷⣴⡙⢿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣷⡟⣱⠏⢸⣿⣷⣼⣿⣿⠉⢉⣉⣁⠀⠀⠀⠀⠘⠟⡏⠠⠙⣮⢻⣿⣷
⠀⠀⠀⠀⠀⠀⣴⣻⡟⣰⠃⠀⠸⢿⣯⣹⣿⠁⠀⢁⣤⣬⠛⠁⣰⣯⣝⠀⠉⠀⠀⠈⢧⢻⣿⢦
⠀⠀⠀⠀⠀⠀⣿⣿⢡⡇⠀⢰⢳⣾⣿⡺⡛⠀⠀⠈⠛⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠘⡆⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⡟⢸⠀⠀⣠⣿⠿⢿⣷⣥⡄⠀⠀⠀⠀⠀⠐⠖⠀⣴⣀⠀⠀⠀⠀⠀⣷⢸⣿⡇
⠀⠀⠀⠀⠀⢸⢿⡇⢿⠀⠀⡙⢿⣿⠿⣿⣿⡐⣄⠀⠀⠀⠲⣿⡗⢰⣧⠈⣇⣀⣀⠀⠀⣿⢸⣿⡧
⠀⠀⠀⠀⠀⠘⣿⣷⢸⡀⠀⢸⣟⣤⣴⠿⢿⡇⠈⠳⣤⣀⣀⣀⣰⣿⣿⡿⢿⣽⡿⠀⠀⡟⣸⣿⡇
⠀⠀⠀⠀⠀⠀⢯⣽⡎⣧⠀⠀⢙⢫⣾⣛⡿⠃⠀⠀⠈⢿⡟⣻⣿⠻⣿⣿⣤⣯⣷⠀⢸⢃⣿⡿⠃
⠀⠀⠀⠀⠘⣶⣌⣿⣿⡜⣆⠀⠈⠳⣾⠟⠁⠀⠀⠀⠀⢈⠁⣿⣿⣷⡄⣹⡛⣯⣵⣠⢏⣾⣿⣇⣴⠃
⠀⠀⠀⠀⠀⠘⢿⣿⠭⠿⣿⣷⣽⣾⣗⣉⣉⣉⣉⣤⣄⠈⢣⢿⣾⣿⡿⠿⠿⣻⣿⣿⠟⠯⣽⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠧⣀⢸⣿⢿⣿⣟⠛⠛⠛⠛⠛⠿⠿⠦⠤⠿⠟⠛⠛⠉⣿⣻⡏⣀⡼⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠰⠶⠩⡿⠿⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⡿⠿⠿⠍⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢞⣻⡟⣭⣽⣿⡿⠟⠛⠛⠛⠛⠛⠛⣿⣧⢻⡿⢋⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠂⠀⠺⠷⠀⠐⠂⠈⠁⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣉⣤⣤⣽⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣿⣿⣿⣿⣴⣴⣶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠻⣿⣿⡇⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣿⣿⣷⠒⠀⠀⠈⣻⣿⣿⣿⣿⣿⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣄⣴⣷⣶⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣉⣽⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠱⣦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠶⠀⠼⠁⠈⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢧⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⡄⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⡋⠂⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠜⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⢀⣯⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⡆⠀⠀⣏⣀⡀⠀⠀⠀⣠⡴⠿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣷⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠉⢩⠿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢀⣒⣚⣯⣷⢶⠶⠞⣻⣿⣟⡉⠉⠁⠀⠈⡄
⠀⠀⠀⣀⣤⣠⢄⣀⠀⢳⣴⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣧⡀⠀⣀⣴⣁
⠀⠀⠈⠀⠈⠐⠋⠚⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡋⣙⣰⡿⠋⠉⠁⠀⠀⠙⠉⠓⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉⠀⠀⠀⠀⠉⠉
⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⠀⢀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⢰⣾⢶⣸⡀⢷⢾⣸⢧⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⣴⣿⣿⠇⣿⣿⣷⣤⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⣤⣙⠛⠉⠀⠈⠙⢛⣥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠰⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⢛⣡⣤⣄⠀⢀⣤⣤⡙⠛⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠻⢿⣿⡇⣿⣿⠿⠛⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡄⠀⠔⠂⠀⣸⠀⢀⣑⠠⢀⣀⢏⠍⠉⣙⠒⠦⠎⣉⣈⡵
⠀⠀⠀⠀⠀⡃⠀⡉⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢠⣉⣙⢣⢀⣆⣠⠓⠰⠄⣈⡁⠀⢸⠉⠓⠂⠤⠄⠔
⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⡰⠁⠀⠀⠁⠀⠐⠀⠀⣀⣀⠀⠀⠣⠄⣏⠀⠼⠀⠀⠀⠀⠀⠈⠉⡁
⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣄⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠⠀⠀⡖⠈⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠰⢶⣿⡦⠄⠈⠉⠁⠀⠀⢀⡀⠀⠀⠀⠀⢀⡐⠀⠃⠀⠀⣀⠋⢠⡁⠘⢂⠂⡀⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠈⠻⠿⠟⠂⠀⢀⣀⣤⠀⣰⣶⡟⢀⣾⡇⠀⠀⠀⠀⠄⠐⡄⠀⠀⣀⠀⠀⠀⡀⠠⠠⠐⡄⡂
⠀⠀⠀⠀⠀⢶⣿⣿⣟⣛⣥⠀⣰⣶⡟⢀⣾⣿⣧⣴⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠘⡦⠀⠄⠠⠀⠀⣀⠀⢀⠀⠀⡥⡤⠁
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⡘
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠁⠀⠀⠀⠸⣀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢄⠀⡀⠀⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢠⣾⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⠿⣉
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠉⠄⠀⠀⠀⠀⠀⢀⣴⣶⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣀⡠⢟⠟⠡⣄⠜
⠀⠀⠀⢀⣾⣿⣿⣿⣇⠀⠀⠀⡀⠀⠀⢀⣀⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣀⣀⠈⠿⠿⠿⠟⠁⠀⠀⠀⠀⣀⣤⡔⠁⠀⠂⢀⡷⠁
⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠰⠀⠀⠀⠀⠉⠉⠙⠛⢿⣿⣁⣠⣄⡀⠀⠀⠀⡿⡿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⠇⠀⢀⣴⠟⣡⡄
⠀⠀⠀⠀⠙⠿⠿⡿⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠖⠛⢻⠛⠉⠁⠀⠀⠀⠃⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⡦⢴⣯⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠰⠀⠀⠀⠀⠀⠻⢾⠶⣆⣐⡦⠀⢘⠛⠳⠆⠀⠀⠠⠀⠈⠐⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁