Tofu Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠼⠋⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠉⠁⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠉⠁⠀⠀⠀⠑⠢⣀⠀⠀⠀⠀⢀⣀⡈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢁⠀⠀⠐⠓⠁⠀⠀⠀⠀⠀⢈⡱⠦⡒⠊⠁⠀⠀⠀⠀⠙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠑⠢⢄⡀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠈⠲⢄⠀⠀⠀⠈⠄⠀⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⢀⣠⠴⢾⡇⠔⣊⠄⠀⠈⠓⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠀⠀⠀⠀⠀⣀⠀⠈⣳
⠀⠀⠀⠀⣠⡞⠋⢄⢠⢸⠁⠈⠤⠂⠀⠀⠀⡇⠈⠐⢤⡀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠈⠉⠀⠀⢸⡇
⠀⠀⠀⠀⡏⠙⠳⢤⡈⠈⠳⣄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⠂⠀⢀⡠⠔⠊⠁⠀⠀⡆⠀⠀⠠⡄⠀⢸⣇
⠀⠀⠀⠀⠙⢦⣄⠀⠉⠳⣤⡈⠳⢤⡀⠀⠀⠆⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠏⠻⣄
⠀⠀⠀⠀⠀⠀⢸⠳⣦⡀⠀⠙⢦⡀⠙⠳⣄⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⣁⣤⠤⠶⠒⠛⠉⠀⢀⣨⣷
⠀⠀⠀⠀⠀⠀⣿⣠⣇⣽⠷⣤⡀⠙⢦⣄⠈⠳⢦⡀⠀⠀⢸⠀⠀⢀⣠⡤⠶⠚⠩⠥⠦⣀⣠⣤⣶⣾⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠙⠳⣤⡈⠳⣄⡀⠉⠳⢤⠼⠖⠛⠉⠀⠀⢀⣠⣴⣶⣿⣿⣿⣾⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠦⣄⠙⠦⣄⠀⠀⠀⣀⣤⣶⣾⣿⣿⠿⠛⢯⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢀⣟⣳⣤⡈⢻⡶⣿⣿⢿⡟⢻⣿⣿⠀⠀⠈⠻⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠙⠚⠛⠉⠀⠀⠙⣾⣿⠏

⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠛⠲⠶⠶⠶⠤⠤⣤⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⡄
⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡃
⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠈⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⢻⡄
⠀⠀⢙⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠀⠀⠀⠈⠓⠒⠂⠙⢆⡠⠒⠠⠦⠄⣀⡀⢻⡄
⠀⠀⠀⠘⢧⡀⠀⠀⢀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠈⢳⣶⠚⠋⠀⠀⢀⣤⣤⣄⠀⠀⠀⣀⠀⡀⠀⣠⣀⣠⡀⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⣦⣤⣽⡀⠈⢁⣈⣻⠙⡆⠙⡟⢧⠻⠉⢣⡀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠘⣿⡙⣇⠸⡟⠻⡌⠿⢧⠀⠱⣤⣰⠟⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠘⢷⣬⡄⠹⣆⣳⠘⣶⡇⠀⠹⣅⠀⣿⣀⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⡉⠀⠀⠈⠋⠀⠈⠃⠀⠀⢿⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⢶⡄⠀⠀⠀⠀⣶⠛⠛⠀⢀⡴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣦⠀⠘⣆⠀⠀⠀⠙⣦⠀⠀⠘⠷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠀⠀⠘⣧⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣀⣀⣠
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿
⣿⣿⣿⢁⠈⣁⣀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠐⡂⠀⣤⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⡌⢳⠀⣤⣄⠀⠀⠂⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠠⠄⠀⠀⠒⠊⠀⠀⠀⠉⠀⠀⠘⠄⠀⠈⠀⠰⠂⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠀⠀⠀⠀⠈⠉⡐⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⠐⠟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠈⠆⢀⠀⢀⠀⠀⠁⠀⠒⠀⢀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⢃⠰⠀⡀⠈⣠⡀⢿⡀⠁⠀⢰⠆⠠⡄⣵⡀⠀⠀⠀⠀⠀⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢂⠉⠈⠁⠙⠢⡄⠒⠀⠀⠀⠈⠳⠄⢨⠀⠰⠀⢹⠀⠀⠀⠠⠁⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⠀⠀⡀⠔⠀⠄⠀⠡⡆⠀⠀⠀⠀⠀⠘⣧⢀⠀⠎⢸⡇⠀⠀⠀⠀⠀⢀⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠫⣿⡄⠀⠀⠀⠔⢃⢟⡀⠀⠀⠀⠀⠀⣇⡘⠀⠿⣿⠀⠀⢊⡤⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣟⣦⠀⠂⡀⠀⠙⢿⠃⠀⠀⣀⣀⣠⠇⠈⠑⠂⠍⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠊⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡟⢛⣿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣰⠋⣦⢸⡇⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⠀⣿⢸⡧⢸⡇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠈⠉⢁⣼⠇⠀⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠒⠂⠠⠀⢀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠂⠀⠀⠈⠀⠀⠀⢀⠀⢠⠅⠀⠒⠄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠐⢦⠩⢂⠺⠒⠐⠂⠉⠉⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠈⠀⠂⠠⢀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠁⠈⠁⠀⠀⠀⣠⠈⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠄⠀⠀⠀⡀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠔⠀⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⠄⠀⣴⠀⡀⠀⠀⠀⠁⠀⣄⠀⠂⠀⠀⠀⠰⡆⢀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠤⠥⣈⠀⠂⠀⠸⠂⠻⠤⡀⠀⠸⠉⠘⠆⣽⠄⠀⠀⠀⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠡⠂⠀⠀⠀⠉⠎⡉⠁⠀⠀⠀⠈⢣⡔⠃⠘⣀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡢⢄⣴⠄⠈⠀⠂⠈⡘⣇⠀⠀⠀⠀⠀⠀⡗⢠⠀⣥⣼⠄⠀⣄⠀⢀⣀⠴⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣑⣿⡆⠠⠀⠀⠂⣥⣯⡄⠀⠀⠀⠀⠀⡷⢮⣁⡛⠿⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢵⣄⡁⠀⠀⠀⠹⠥⠤⠀⠒⠒⠒⠁⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣀⣀⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⢨⡟⢀⣄⠀⢀⡄⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⠁⣿⢸⡇⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠻⠤⠟⢸⡏⠸⡧⠼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠰⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣠⡔⣞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⣈⠃⠉
⠀⠀⠀⠀⠀⠀⡟⠰⠖⠦⡄⠀⠀⠀⣀⢀⡀⠀⠠⠤⠀⠀⠀⢀⠀⡀⠀⣀⣄⠄⠆⣀⠀⠀⠀⠁⠷
⠀⠀⠀⠀⠀⠀⠘⠈⢸⡆⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠲⠦⢤⣤⣥⣶⣿⠇⠡⡄⠀⠈⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠰⠆⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣀⠘⠎⠀⣿⡿⠷⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢰⠀⣠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⡿⠳⠆⠀⠀⠀⠐⣇⢀⡀⠁⠀⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⣐⠂⢀⢰⡦⠀⠀⢀⠀⠀⠈⠀⠀⢳⢰⣤⡀⡄⢠⡾⠈⠁⠀⠀⠀⠄⠀⠀⠐⠀⠘⠀⠘⠀⠀⠀⣀⣀⠄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠒⠠⠀⠀⠁⠀⠀⡄⠀⢰⣿⣿⠯⠛⠁⠀⣀⠄⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⢠⠀⢀⣴⣾⣱⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠾⠋⠁⡀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⣤⡾⢻⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠐⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠘⠂⠀⠀⠀⠀⠀⣄⠀⠀⠀⠦⠀⢀⣤⢾⢿⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠠⢤⣤⣴⠾⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠫⣵⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⢹⣟⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⢲⣶⡿⡍⣤⣾⠟⠋⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡯⠀⠀⠀⠐⣦⡐⠂⠀⠀⠀⠐⠒⣠⡄⠀⠀⠀⠀⠈⠓⠢⢼⣿⣿⠟⠋⠀⠀⢸⣤⢉⡀⠀⠀⠀⠙⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠉⠑⠀⣶⡧⢸⡛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠈⣟⣷⡶⠏⠙⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⣿⣃⠈⠁⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡤⠆⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣍⣛⣍⣙⣙⣍⣩⣽⣻⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉⠀⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣺⣷⢶⡄⡴⢒⣶⢖⠛⢉⣉⢭⡓⡆⠀⠀⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⢰⣿⢁⣼⠼⣿⡅⢋⣈⣉⢉⠉⢲⡄⣧⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠀⣠⣴⣺⣟⡟⣋⣏⣠⣿⡟⠹⡇⢸⡎⢃⢈⣤⣸⡄⠀⠀⠀⠀⠀⠲⢌⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⣩⠤⠀⠀⠰⣿⣥⣶⣿⣯⢭⣷⠿⢾⡷⢶⣿⣅⣋⡌⠣⠄⣙⡷⠀⠀⠀⠀⠀⢀⡀⠁⢀⣨⡦
⠀⠀⠀⠀⠀⠀⣠⡞⠉⢤⣈⠁⠀⠀⠀⠀⠀⢻⣳⣭⣯⣿⣷⡒⠛⣏⣁⣉⣹⡄⠈⠓⠛⠁⠀⠀⠀⠀⠀⠚⢉⣠⣶⠋⠀⡇
⠀⠀⠀⠀⠀⠀⢹⠉⢳⡦⣌⡉⠀⠀⠀⠀⠀⠀⠈⣿⣏⣉⣩⢿⠀⠛⠛⠾⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣿⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡆⠀⡇⠀⢹⠳⢤⡀⠀⠀⠀⠀⠈⠛⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⢻⡇⠀⠻⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠷⠀⠸⠀⢸⠈⠓⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉⣇⠀⢸⡇⠀⠀⠀⠿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢻⠀⢤⠀⢠⠀⢸⠀⠀⡇⠈⢳⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡞⠉⢹⠀⠀⢻⠀⠐⠃⠀⠁⠀⢠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢠⠀⢸⠀⠀⡇⠀⠸⠇⠈⢳⣄⡀⠀⢀⡤⣞⠉⠀⡇⠀⢸⠀⠀⠠⠄⠀⠤⠀⣿⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢸⠀⢨⠀⠀⣷⠀⠀⡀⠀⢸⡆⠙⡞⠉⠀⣿⠀⠀⠃⠀⠸⡇⠀⠀⡆⠀⠀⠀⣿⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢸⡇⠈⠀⠀⢈⠀⠀⡅⠀⠈⠃⠀⡇⠀⠀⢸⠀⠀⢈⠀⠀⠅⠀⠀⣇⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⠀⡇⠀⠐⠇⠀⣿⠀⠀⠠⡄⠀⢸⡀⠀⠀⢀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⣀⠀⠀⢿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣇⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢼⡀⠀⠀⠀⢸⡀⠀⠀⢀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢄⣸⣇⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⣤⠀⣴⡿⠛⠛⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣷⣄⠀⠀⢸⡿⣄⠂⠈⠸⣷⣿⢷⡀⢀⣀⣨⣿⡷⣄
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡶⠟⠛⠉⠅⠀⡌⠻⣦⣤⣾⡇⠙⣧⡁⠀⠹⣿⡄⣿⣯⡭⠶⠛⢻⡟⠳⣄
⠀⠀⠀⠀⠀⣾⣿⡉⠡⠀⠀⠀⠂⠀⣀⠀⠀⠀⠙⣿⣿⡇⠀⠘⢿⣌⠀⠙⣿⣿⣿⣦⠀⠀⠀⣹⣦⠈⢳⣄
⠀⠀⠀⠀⠀⣿⡝⢿⣦⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢻⣷⣄⠀⠈⢻⣧⠀⠈⠻⣿⣌⣷⣋⡭⠖⠛⠻⣯⠙⢦
⠀⠀⣀⣠⡴⣿⡟⡫⠻⣷⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⠀⡀⠻⣧⠀⠀⣙⣿⣿⡿⣆⣠⡤⠶⠾⣧⢈⢷⣀
⠀⢸⡿⣔⢊⣿⣷⡋⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⢹⡟⡛⠋⠉⠉⣿⣿⡃⠀⠈⢠⡇⠀⣕⠝⣧⡀
⠀⢸⡇⠹⣯⠐⠙⢿⣆⠀⣈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⡇⢸⡇⠀⠠⠀⢠⣏⣿⣁⣀⣤⢾⣃⡭⢶⠻⢙⡇
⠀⠀⢳⡀⡍⢷⡄⠀⠻⣷⡩⠈⠈⢿⣦⡀⢀⣀⣤⣶⣿⠟⠋⠉⠀⣿⣇⣿⡇⠀⢀⣀⣾⠏⠛⣋⡽⠞⠋⢑⠊⣀⣤⠾
⠀⠀⠀⠹⣆⠀⠙⣶⣕⠼⣿⣄⠀⠀⢻⣿⡟⠋⠉⠙⠀⠀⠠⠀⠀⣿⡿⠿⠷⢞⠫⣋⡤⠖⠋⢁⣠⣴⣾⠷⠛⠉
⠀⠀⠀⠀⠈⢳⣄⠈⠷⣄⠈⢻⣦⡀⢸⣿⡇⠀⠀⠀⠀⢀⣠⣴⠾⢛⡡⣔⣬⢶⡻⠕⢊⣁⣤⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠙⢦⡁⠈⣷⡀⠙⣿⣬⣿⡇⠀⣀⣤⡶⠟⠩⢔⣪⠷⠛⠉⢀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠳⡿⣦⡈⠻⣿⣷⢿⠛⣁⣤⠶⡻⣭⣒⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣇⠈⠻⢮⣦⣬⡷⠛⢉⠤⣚⡼⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⣿⣿⣒⣤⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡿⠞⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠁⡄⠐⠂⠀⠀⠉⠁⠒⠲⠊⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠈⠍⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢍⡀⠈⢬⠉⠀⠁⢃⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠀⠀⣤⡄⠴⠒⢣⡈⠁⣀⠻⠓⠘⠛⠁⠠⠀⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠉⠐⠒⠤⣄⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠘⠁⠋⠁⠀⠀⠀⠀⢀⡌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⠂⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⡤⠒⡏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡀⠁⠀⠀⠰⠀⠇⠃⠠⠈⠀⡀⠀⠈⡑⠒⠂⠤⣀⠀⠀⠀⢀⠠⠖⢁⠄⠀⠈⠀⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠧⢀⠀⠄⠐⠀⠀⠘⠂⠂⠀⠇⠀⡆⠇⡌⢀⠀⢐⠉⠑⣆⠀⠀⠀⠀⠀⠀⠀⠛⠁⢀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⠤⠤⢄⣀⠀⠀⠇⠇⠼⢓⡀⠨⣄⡃⠀⠁⣾⠃⠜⠷⠀⣿⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⡤⠒⣩⡄⠀⠀⠀⢈⠑⢦⠒⠒⠈⠀⠀⠀⠀⠙⢦⣠⡆⠆⠰⠀⡆⡇⡀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⡇
⠀⠀⠀⠀⠀⢾⣁⠀⢀⡁⠀⣠⠔⠀⡀⠀⠜⣞⡢⠀⠀⠤⠤⠄⠀⠀⢹⡄⢐⠘⠀⡇⡴⡇⠀⡀⡀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠘⣆⢍⡲⣶⠁⠀⠀⠀⠐⠀⣴⢘⣼⡄⠀⠀⠀⠀⠀⠀⢸⠁⣸⡄⣇⣇⣗⡇⠀⡃⠁⢀⡠⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠘⣮⢬⣿⡋⠀⡆⡀⠀⢈⣠⣞⣽⡆⡄⠀⠀⠀⠀⠀⢸⠗⠦⣅⡛⠙⡿⠁⣀⡥⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢮⣙⣄⠈⢁⠴⠋⠀⠘⢯⣁⣁⡠⠤⠴⠦⠤⠞⠀⠀⠀⠈⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁





⡶⢶⢶⢷⣿⠷⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⡶⠶⡶⢶⣶⣶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠚⠓⠒⠴⠉⠀⣆⡀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠀⢀⣀⣀⠤⠤⠁⠀⠀⠈⠐⠀⠰⠧⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠉⠀⠄⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠁⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠐⠁⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠁⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡉⠑⠒⢤⣄⣀⣀⣀⡀⣀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⡠⠒⠁
⠀⠀⠀⠀⠀⢀⡠⠔⠊⠀⠀⠈⠁⢢⡎⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⢀⠀⠀⠀⣀⠔⠂⠁⠀⢀⣀⣀⣀
⠀⠀⠀⠀⢰⠁⠀⠀⠀⢀⠐⠁⠀⠀⠁⠀⠒⠀⠀⠐⠀⠀⠀⡇⠤⢬⠤⠖⠋⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠉⣣
⠀⠀⠀⠀⠀⢇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠢⠀⠁⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠠⢄⠅⢹
⠀⠀⠀⠀⠀⠈⢧⠀⠘⡀⠀⠀⠀⣠⠴⣆⠀⠀⠀⠀⣀⣀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠓⠦⣐⡠⠖⠋⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⠀⣠⣄⠀⠀⣤⣤⡄⣀⠀⣤⠀⠈⠉⠉⠑⠒⠒⠒⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠇⢸⡇⢠⡏⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⢀⡇⢸⡷⠶⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⠀⢸⡇⢸⡇⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣇⣼⠃⢸⡇⠀⠀⢿⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠈⠁⠀⠀⠈⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠏⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋⠀⠀⠀⠂⠀⠠⡶⠀⠀⠀⠙⠲⠖⠒⠚⣦⡀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡀⠀⠀⠀⠄⠈⢠⡗⡶⠀⠀⠀⠀⠁⢈⡐⢈⡙⠙⣆⢰
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢻⣵⡆⠀⠀⢀⡄⣀⠀⠀⠀⢐⠖⠮⡁⠁⢀⠀⠀⠹⠈⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⡜⠛⠛⢳⣾⣁⠀⣐⡀⢀⡴⡏⡀⠱⠄⠄⠀⠀⠀⠈⠠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⣽⠂⠀⠀⢀⡘⠥⡂⢀⠢⠞⠁⠉⠣⡀⠀⠀⢀⠟⠀⠀⡇⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⢻⡏⡜⡼⣄⠀⢀⣀⣵⣤⢤⣏⣿⠀⠀⠀⠰⣿⠶⡤⠈⣄⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣞⢎⢜⢮⢣⠟⠉⠑⠾⣓⣷⡄⠀⠀⠐⠊⠻⣶⣛⣊⠆⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡾⡎⡏⡿⡆⠀⠀⠂⢀⢸⡧⢔⠀⡄⠀⠀⢸⡏⡟⢰⢧⠰⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⢿⠙⢳⡘⢶⣦⢄⡈⣤⣧⠣⠠⢔⠂⠀⠨⠚⣡⢎⢸⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⢣⡱⢬⡭⠭⢚⢾⣢⢭⣀⡶⡠⠤⣉⠉⠥⠨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣦⣕⣒⣚⣯⣷⣯⣭⠵⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠿⠷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⢿⣆
⠀⠀⠀⠀⠀⠀⢀⣶⣟⣁⣠⣤⣤⣤⠴⠶⠦⢤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠘⣿
⠀⠀⠀⠀⠀⢠⣿⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠳⠶⠦⣤⣤⠄⢀⣴⠟⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠇
⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡍⠃⠀⠀⠀⠀⠀⠀⠀⠠⡴⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢠⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡟
⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣤⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣾⠟⠁
⠀⠀⠀⠀⣿⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⡿⠁
⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠿⠶⠶⢶⣶⣦⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠙⠉⠛⠙⠋⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢠⣶⣿⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⣾⣭⣛⡿⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⣫⣿⣿⠀⠘⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣷⣾⣽⣛⠿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠋⢀⣀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣽⣛⠿⣶⣤⠀⠀⠀⠀⠀⠀⠀⣰⡟⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠗⠊⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣫⡆
⠀⠀⠀⠀⢀⣾⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣫⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⣿⣯⣽⣿⣛⣛⡿⣫⣷⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⡶⣶⣤⠀⠀⠀⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡿⠋
⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣵⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⠇⠉⠀⠀⠀⢀⣠⣶⡆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣻⣽⣾⣷⣤
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⢟⣽⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠿⢿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡿⣟⣾⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⠶⠿⠿⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣶⠶⠿⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠻⢶⣶⡒⠒⠒⠒⠒⠲⠶⠤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⠶⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣄⡼⠻⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠀⠈⠙⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣍⣉⣙⣻⣦⣄⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⣤⡾⢋⢿⣿⠀⠀⠀⠀⠈⠙⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⠶⠿⢿⡏⠉⠉⠉⠉⠉⠉⠉⠉⢩
⠀⠀⠀⢀⣴⡿⠁⢀⣼⢿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⠶⠾⠟⠛⠛⠉⠉⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣠⡾⢻⠁⣠⡟⢁⢼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣴⡶⠶⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⣰⡟⠀⢸⣼⠋⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⠃
⢰⣿⡃⠀⣾⠃⢀⠔⣿⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣰⣿
⣾⠃⡇⣸⠇⠀⠁⣸⡿⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⢀⡼⠛⣿⡆
⣿⠀⠈⣿⠀⠠⢚⡟⠀⠸⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⣾⠁⠀⣿
⣿⡄⠀⣿⠀⢀⡞⠀⠀⠀⠈⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡗⠀⣠⠞⢡⡟⠀⢠⡿
⠸⣷⣀⠸⣇⠸⣇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣶⡶⠶⠿⠟⠛⢛⣥⡞⠁⢠⡟⢳⠀⣾⠃
⠀⢻⣎⠁⠻⣦⠈⠛⢦⡀⠀⠀⠀⠀⠠⠠⠀⠀⠈⠙⠿⣶⣤⣸⠇⠀⣀⣀⣠⣤⣤⣴⡶⠶⠿⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⢀⣠⠖⠋⠁⢀⣴⠟⠀⣸⣼⠏
⠀⠀⠹⣧⡀⠈⠷⣄⡀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⠛⠛⠉⢉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢚⣉⠄⠀⣀⣴⠟⠁⠀⣠⡿⠃
⠀⠀⠀⠈⠻⣦⡤⠈⠿⢶⣤⣀⠈⠙⠲⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⢖⡻⠝⠋⠉⣀⣤⡴⠟⢋⠇⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠈⠻⢶⣄⡀⠈⠙⠛⠶⢦⣤⣄⣈⣉⠙⠓⠒⠶⠶⠶⠤⠤⠤⠴⠶⠶⠒⠒⠛⠋⣉⣁⣀⣴⣥⡴⠶⡞⠛⠉⠁⠀⣠⣮⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣄⣀⠠⠬⠝⠋⠛⠛⠛⠛⠛⠓⠒⡶⠖⠚⠛⠛⠛⠛⠛⡻⠋⠉⠉⠀⠀⣀⣞⣀⣤⣴⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⢶⣦⣤⣤⣤⣤⣤⣖⣋⣀⣀⣀⣀⣀⣤⣴⣭⣤⣤⣶⠶⠶⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠈⠻⠿⠿⠾⠓⠤⠀⢀⠀⠀⣠⡜⠋⠀⠈⠀⠉⠉⠙⠓⠲⠦⢤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠼⠿⡶⠾⠿⠟⠒⠲⣶⣧⣴⣶⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⢸⠑⠈⠀⠀⠀⠀⠠⠀⠀⠀⠐⡀⠀⠈⠙⠓⠦⣄⣀⡀⠀⠀⠐⠂⠉⠁⠀⠀⠈⠑⠀⣀⣀⣺⣯⢱⡛⠽⣿⢻⢿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣤⡄⠀⠈⠉⠀⡀⠐⠀⠈⠈⠀⡀⠈⠐⠔⣩⣿⣄⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢷⣏⣊⣧⣖⢫⣿⣿⣿⠄
⢀⡀⣀⣤⣴⠎⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣛⠲⠶⠶⠶⣾⢤⠤⣤⣤⣶⣤⠤⡾⠾⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠙⠚⠿⠿⠟⠛⠛⠉
⣿⠿⡁⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠙⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢰⠇⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣠⣴
⣄⣈⣾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠠⣠⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣶⠧⠒⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡴⠞⣡
⡿⠟⠁⠀⠀⠀⢀⣀⠀⢀⠀⠂⢒⣂⣀⡾⢃⢴⡡⠞⠋⠀⠀⠀⢀⡄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⣿⣃⣴⠏⠀⠀⠀⠀⠀⠀⢠⢟⣠⠾⠋⠄⢈⠟⢋
⠶⠂⠀⣠⣴⣿⣿⣿⢷⣾⡿⠋⠉⠉⠛⡻⠛⠛⠓⣠⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢸⡏⢀⠀⠀⠀⠀⠀⠀⣠⡿⠛⡙⠛⣀⢠⣥⣴⡿
⠀⢠⣾⣟⣧⢽⠛⡉⣏⢭⣿⣤⡤⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⠈⣧⢀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⢈⡇⢸⠀⠀⠀⠀⣀⣼⢃⠴⣇⡀⠊⣠⡿⠛⠁
⠀⣿⣿⡿⣿⢫⣴⡌⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⡁⣟⣿⠀⠀⠀⠀⠀⠀⠀⣼⣧⣦⣾⣇⠈⠀⠀⠀⠰⣟⠘⣁⠔⢠⣴⠾⠛⠨⠄
⠀⠈⠙⠛⠳⠻⠶⠞⠛⠁⠀⠀⠐⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡀⠛⢻⣆⠀⠀⠀⢀⣼⡿⢯⠁⣼⣿⣅⠀⠀⠀⠀⣀⠹⠯⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣃⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⣻⡇⣤⣬⠠⣇⠸⠀⠀⢸⣏⣮⠇⣁⢧⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⠋⠁⡌⠩⠙⣦⠀⠀⠀⠀⠀⠀⠀⠙⣿⡁⢘⠛⠄⣿⠀⠀⠀⠸⡿⢛⣼⣿⣻⣿⠇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡴⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠈⣹⠗⠀⠀⠀⡀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣷⢀⠀⡃⣼⡇⠂⠀⠈⣿⣿⣿⣶⡿⠋⠀⠀⠀⢀⣀⡀⠁⠄⠀⠀⠀⢀⣾⣿⢹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠲⣿⢻⠀⠀⠀⠀⠀⠀⠀⣽⡏⠸⠀⢷⢾⡇⣶⠁⠀⠘⠿⠿⠛⠀⠀⢀⣴⣞⠋⠉⠝⠺⡄⠀⠀⠀⢸⣿⡛⡆⢿⣿⣿
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠅⠀⡂⠀⢀⡟⡘⠀⠀⠀⠀⠀⠘⠀⣾⡇⠐⠀⣺⢈⣷⠃⠀⠀⠀⠀⠀⠀⠀⡰⣻⠿⠇⠀⠀⠂⠀⢻⡀⠀⠀⠈⢧⣭⣸⣾⣿⠇
⠀⠀⠀⠀⠀⠀⡟⠀⠀⡐⠘⠠⠁⣸⠁⠁⠀⠀⠀⠀⠀⠀⠀⢸⣧⠈⡓⢺⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢠⡿⠇⠀⠀⠀⡞⠙⢻⡿
⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠁⢠⣰⡏⠀⠀⠀⢀⣶⣶⣦⣀⠀⢸⣧⣤⡆⢰⢸⣧⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠠⠁⠂⠀⢀⡞⠁⠀⠀⠀⠈⠁⢀⡼⢁⠄
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣾⢟⣿⣿⢿⣧⡀⠙⢿⣳⣤⡿⠃⠀⠀⠀⠀⣠⠀⢠⣿⠆⠀⠀⢂⠀⢀⡾⠁⠀⠀⠀⠀⠀⣧⣾⡃⠈
⠀⠀⠀⠀⠄⢽⠀⠠⠁⠀⢠⣿⡀⠀⡇⠀⠀⣿⣮⡜⢿⡧⣿⣿⡄⠀⢩⡀⠀⠀⠀⠀⠀⠈⠀⢀⣾⠁⠀⢀⠀⢀⢀⡞⠀⠀⠀⠀⠀⡄⣸⣟⠈⢿
⠀⠀⠀⠊⠀⣿⠀⠌⡀⢠⡿⠙⡇⠀⠀⠀⠀⣿⣽⠗⢀⣘⢾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣿⠃⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠁⢀⡁⢰⠄
⠁⠀⠀⠀⠀⣿⢠⠀⣤⡿⠁⠠⠀⠀⠀⠀⠀⠘⢿⣧⠆⣉⣏⣽⣃⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⡏⠀⠀⠀⢀⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠾⠀⠀⠈⠉
⠀⠀⠄⠀⠀⢻⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡟⠟⠻⣯⠀⡀⠀⠀⠎⠀⠀⠀⠀⣼⣦⠀⠈⣠⠎⠀⠀⠀⠀⠘⢶⡤⣤⣀⣦
⠀⠀⠆⠠⠀⠈⠛⠃⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⣽⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⡾⠁⠀⠈⠀⠀⠀⠀⣈⣿⡆⠙⠻⣿⠿⢶⣈⡂
⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⡀⠀⢀⡀⠀⠀⠀⠙⢻⣦⣑⡦⣀⣶⣌⠻⣦⣄
⠶⣜⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠌⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠙⠂⠀⠀⠀⠀⠙⠻⢷⣌⠙⢅⡉⠢⢉
⠰⣄⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢎⠒⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠞⠻⢶⣶⣎⠒⠄
⠰⠞⠙⠢⡻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⢷⠘⣇⠀⠀⠀⡄⠀⠀⠤⣤⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣀
⠷⠄⠸⠦⠤⠾⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠰⠦⠘⠄⠙⠷⠀⠐⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⡀⠀⠀⠀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⢠⣤⡀
⠀⠀⢠⡞⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡄⢀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⢸⡇⣀⣀⢸⠉⣷⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠊⠀⠀⠀⠀⠀⠀⠀⠰⢆⣻⣿⣃⠆
⠀⠀⠈⠧⠤⠇⠈⠒⠋⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⣄⢄⡀⢀⣀⢠⢠⠄⠀⠀⠈⠉
⠀⠀⠈⠒⠃⠘⢂⠝⡋⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖
⠀⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡀⠀⠶⠀⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⢈⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣤⣤⣄⠀⣀⠀⢠⣤⣤⡀⠀⠀⠈⠃
⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⣿⡿⠷⢸⣏⣷⡈⣿⡏⠁
⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢀⠀⢀⠀⢨⣧⣶⠀⠀⠀⠀⠀⠀⠰⣌⠿⠷⠀⠀⠀⠻⠿⠗⠻⠏⠿⠇⠻⠇⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⡤⠞⠀⠀⠀⠀⠃⠀⠀⠀⠘⠃⠀⠈⠙⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⢀⡀⠀⢸⡀⢐⠀⠀⠀⠀⢀⠙⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠽⠇⠂⠂⠞⠒⣆⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⠂⠀⠀⠀⠀⠀⠀⠀⠺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠐⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠉⠉⠀⠀⢠⣜⡟⡷⣦
⠀⠀⠀⠀⠈⠁⠀⠀⢀⣀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⠀⢸⢸⠃⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⣠⣦⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⠠⠇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠙⠻⠀⠀⠈⣧⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⡸⠧
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠛⠀⠈⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⠄⠀⠤⠄⠀⠈⠳⡀⠀⠐⢀⣤⢤⡌⢢⠀⠀⡎⠀⠀⣀⠀⠀⠀⢰⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠷⠀⠂⠀⠀⢀⠀⢰⠤⠶⠲⠀⢀⠖⠁⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⢿⡿⠀⢀⠇⠀⠑⡂⠢⡀⠉⢢⡁⣾⠃
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢛⠁⠀⠀⠀⠀⠠⣯⣁⠀⠠⠆⠀⠀⠈⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠔⠃⠀⠀⠀⠈⡀⢠⣤⡀⣷⣿
⠀⠀⠀⠀⠀⣀⣈⣑⠒⠚⠁⢀⠀⢀⡄⠠⢀⠀⠀⠲⣊⠀⠧⠀⠀⠘⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠢⣿⡄
⠀⠀⠀⠐⢉⣤⣀⠈⢣⠀⠀⡏⠁⠛⠹⠆⠀⢁⠀⠀⠉⠈⠀⠀⠀⠀⢄⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠐⠁⠀⠀⠀⠀⠀⠠⠀⢁⠠⡀⠀⠠⠋
⠀⠀⠀⠁⠸⣿⣯⠇⢠⠀⠀⠃⣄⠀⡀⠀⠀⡄⠀⠐⠙⢄⠀⠀⠀⠀⠀⠣⣀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢠⠀⢃⡀
⠀⠀⠀⠈⠀⠀⠀⠴⠋⠀⠀⠀⠹⢉⢀⡠⠚⠀⠀⣠⣸⠦⠗⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⡌⠀⠀⠈⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣼⠍⠁⠀⠀⢰⡁⠀⠀⠒⠀⠂⢱⠀⠊⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠁⠀⠀⠀⠑⠤⠤⠚
⠀⠀⠀⠀⠀⠀⡈⠀⠒⢦⣄⡀⠀⠀⠑⡀⠀⠀⠈⠢⣤⣦⡴⠞⠀⠀⠀⠀⠀⠀⠎⠀⠀⠁⠀⠀⡁⠁⠀⠀⠀⠠⠖⠒⠒⠒⠀⡠⠀⠀⠀⠀⠀⠀⡠⠊⠉⡆
⠀⠀⠀⠀⠐⠀⠈⢀⠀⠀⠀⠉⠓⢶⡤⡀⠢⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡆⢀⢀⠈⠨⠀⠀⠀⠀⢰⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠘⡤⢦⣀⡇
⠀⠀⠀⠀⠀⠘⠲⣤⡄⠐⠶⠶⠤⣤⣀⡍⠉⠛⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠀⠑⠋⠀⠀⠀⠀⠀⠀⠀⢿⢀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⢀⠀⠃⠄
⠀⠀⠀⠈⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣘⣦⠀⠀⠀⠀⡆⠐⠀⢰⠀⢙⠁⠀⠀⡆⠙⠂⠀⡶⠀⡆⠀⠈⠃⠀⠠⣀⣠⡾⠀⢀⠔⠁⠀⠀⠁⢀⠎
⠀⠀⠀⠀⠀⠂⠄⡉⠀⠀⠁⠀⠠⠄⡀⠀⠠⠤⠈⣏⣳⡄⠀⠀⡇⠀⠀⠈⡄⢸⠀⠀⠀⡅⣠⠂⠀⢇⠀⠇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠈⠀⠈⡠⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠈⠉⠀⠀⠀⠀⣢⢀⡀⡇⢸⠀⠀⠀⠁⠈⠄⢰⣸⡀⠄⠀⠀⠀⠀⢀⠔⡁⠔⠀⠀⠀⠀⡰⠎
⠀⠀⢀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠔⠤⠤⠠⠄⠀⠀⠀⠁⠉⠈⠀⠁⠘⠀⠘⠚⠀⠁⠐⠈⠀⠀⠃⠀⠀⠀⠀⠠⠊⠀⠀⠔⠁⢀⡾⠁
⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄⢂⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡏⠀⠐⠉⠂⠎⠉⡆⡋⢢⠀⠀⠀⠀⠀⡀⡰⠁⠀⡠⠂⠀⠀⠁⠀⠀⠀⣠⠂⣢
⠀⠀⠀⠈⠛⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠴⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠂⠀⠀⡀⠀⠁⠀⢸⠀⠀⠀⠀⠀⢣⣇⡠⠊⡀⠀⡀⠀⠀⠀⠀⢐⠃⠀⢻
⠀⠀⠀⠀⠀⠀⠀⡀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣂⣀⠀⠀⠀⠀⡇⠀⢀⠀⡀⡅⠀⠄⠀⠸⠀⠀⠆⠀⠀⠈⠺⠿⠡⠔⠈⠀⣠⠤⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⡠⢀⡀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠈⠡⡀⠀⠀⠁⠀⠈⠉⠀⠈⠊⠀⠉⠁⠀⢠⠀⠀⠀⠀⠀⠀⡀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⡼⠁⠘⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⢰⡀⠀⢰⠂⠀⢀⡀⠉⡆⠀⣀⠀⢀⠌⠀⠀⠀⠀⠐⠁⣠⠚⠁⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠀⠀⠀⠀⠀⠸⠀⠀⠀⠫⠤⠠⠈⠁⠀⠈⣷⠎⣉⠃⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠃⠀⠠⠁
⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠈⠃⠀⡰⠁⠀⠈⢀⠀⠀⠀⠀⠀⠠⠔⠂⠃⠀⠀⠀⠐⠐⠁
⠀⠀⠀⠇⠀⠀⠀⠀⡀⠀⠘⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⠀⠀⢠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⡐⠠
⠀⠀⠀⠀⣀⠀⠀⠠⠃⠀⠀⢀⠄⠀⠀⠀⠠⠀⠀⠴⠊⠀⠀⠀⠈⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠂⠚⣻⣆⠀⢠⡦⢁⠌⠀⡰⠀⠀⠀⢀
⠀⠀⠀⠀⠻⡀⢰⠀⠀⠀⠀⠀⠀⠈⡃⠠⢤⣀⠀⠀⡀⢤⡀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣃⠄⠂⠙⠈⠘⡇⠸⣇⡀⠤⠊⠀⢰⣶⡤⠀⠐⡀
⠀⠀⠀⠀⠀⠘⠘⠂⠀⠀⡀⠀⣄⠀⠈⠑⠀⢈⠐⠋⠀⠠⠁⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⠀⣀⣠⣤⣶⠇⠀⠀⠀⠀⠀⢴⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣉⡁⠀⠙⠃⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠓⠀⠻⠧⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣴⣶⣶⣶⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⠿⢿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡿⠟⠋⠉⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠉⠙⠻⢿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⢹⣿⣿⡏⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⢸⣿⣿⡇⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⣰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⣦⣄⣀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣀⣠⣴⣾⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣷⣶⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠻⠿⠿⠟⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣢⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢄⢠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠟⠀⠀⠀⠀⠀⢀⠠⣂⡜⢣⠀⠣⡀⣕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠡⡀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠄⢣⠴⠀⡘⠈⠂⡠⢤⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠐⡀⠀⠀⢠⠁⠠⠀⠀⠀⠡⠀⢡⠄⠂⣠⡉⠈⠂⠐⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠀⠀⠀⠐⠁⢀⠔⠉⢡⡔⠈⠠⠄⠀⠈⢲⠈⡱⢅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠀⠊⢀⠄⠒⢀⣰⠁⢈⠶⢀⠀⠈⡶⢁⡬⣀⡐⠀⠄⠀⠂⢈⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⢁⠠⠁⠀⡀⡊⠀⡸⢅⠀⠀⢑⠒⢀⡈⠀⠘⠂⠀⠈⡀⠐⠁⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠁⠀⢀⡊⠀⠀⠃⠀⠀⠑⠄⠂⣀⠈⠀⡃⠀⢀⠄⠊⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠁⢈⠖⠈⠒⢄⣀⠔⢄⠠⠃⢄⠔⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⢀⠀⠀⠀⠀⠰⠁⠁⡤⢆⠀⠀⠀⡠⠀⠀⡜⠀⢀⠄⢀⠊⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠊⠀⠀⠈⢦⠂⠀⢀⠁⠀⠑⠁⡠⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⡐⠁⠀⠀⠀⠀⢀⡯⠁⡀⠀⠀⡠⠁⠈⢲⠅⠀⠠⢀⠔⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠊⠀⠀⠈⢢⠺⢀⠀⡠⠂⠀⣨⠔⠁
⠀⠀⠀⠀⠀⠀⠔⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⣄⢀⠀⠀⠐⠉⡈⠌⠉⠁⠐⠒⠃⠀⠎⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠁⠀⠀⠐⠀⠀⠀⠀⣀⠠⠀⠀⠀⠀⢀⠨⠉⠁
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠂⠁⠀⠀⠀⡠⠐
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⡀⠀⠀⠀⡠⠂⠀⠀⠀⠀⢀⠐⠈
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠀⠊⠀⢀⠐⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠈⠄⠀⠀⠀⠐⡀⠠⠋⠀⢀⠀⠂⠁⠀⠀⠀⠀⢀⠐⠁
⠀⠀⠀⠀⠀⠀⢄⠀⠀⢀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠑⠠⠄⠈⠐⠠⠀⠀⠠⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣸⠃⠀⣤⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠸⣧⠀⠀⠀⠀⣿⣤⠤⣄⠀⠀⢀⣤⡀⠀⢀⣤⣴⡇⠀⢠⣿⠀⣾⣡⡼⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢹⡆⠀⠀⢸⣿⣡⠆⠘⣧⣴⣿⡀⡿⢀⡾⢡⣿⠁⢀⣾⠇⠀⣿⠁⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢿⡀⠀⣾⠉⣟⠛⢻⡏⢾⣀⡿⠛⢹⣇⡞⢸⣀⡞⣿⠀⠀⠈⠛⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠘⣷⢀⡿⠀⠻⠶⠛⠀⠈⠋⠁⠀⠀⠉⠀⠈⠉⠘⠷⠶⠶⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠹⣿⠇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⣀⣤⡤⠶⠒⢛⣋⣩⡍⠉⠉⠉⠍⠉⠉⠉⢩⠗⠋⡭⠛⠛⠛⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⠉⠀⢀⣀⡀⠀⠀⠀⢀⣀⣤⣀⣠⠒⡏⢹⠛⣟⢦⡈⢢⡀⠀⠀⠀⠙⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⣰⠋⢉⡉⠉⠓⢲⣯⠔⢶⡞⠉⣦⡑⢌⠲⣌⠳⡙⣄⢹⡀⠀⣀⠀⠀⣧⡙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⣀⣀⣤⣅⠀⠀⠀⢀⣠⠾⠾⠤⣾⢣⡘⢇⠙⢌⠳⣌⢣⡹⣌⢦⠳⠞⠉⠓⡚⢁⠛⢄⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠐⠃⠀⣰⠋⠉⠀⢘⡾⠳⢖⡫⠭⠤⠴⠊⣉⣉⣧⢳⡈⢣⠀⠙⢎⢧⠉⠙⣆⠳⡀⠀⠯⣀⡁⠀⠈⢣⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣀⣀⣸⣠⠴⣻⠿⠶⣒⣋⣠⠴⠋⣉⡭⠤⠤⠜⣧⢹⠀⢙⡷⠛⠦⡓⢄⠸⣗⢭⡢⢄⠀⢹⠀⠀⢀⢧⢀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠐⣇⡔⣋⡥⠤⠤⣴⡾⠕⢒⣊⡵⠒⠒⢒⣫⠥⢖⡒⠂⢁⣹⡆⢧⠈⠓⢦⠀⠈⢢⡱⡌⢆⠙⣦⡱⡌⠑⢒⡡⠊⠘⡇⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠊⠥⠒⠒⣫⡥⠖⠛⠦⣄⠀⣠⠞⣡⠖⣫⠥⠒⢊⡭⠂⣹⡌⢧⡀⠘⠢⠒⠚⣳⣝⡢⣥⣤⡾⠞⠛⠛⠒⠒⢤⡇⠠⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⣷⡩⢤⡟⠁⠀⠀⠀⠀⠈⢳⡏⠐⣅⡀⠓⠤⢤⣈⡳⣄⢣⠘⢦⣉⣓⣒⣈⡭⢶⠎⣩⠋⣼⠀⠀⠀⠄⠀⠀⠀⢳⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠸⡏⢷⣜⠀⣠⠞⠛⢦⡀⠀⠈⣿⠦⣄⣉⡉⠓⢦⡀⠈⢧⠹⡗⠤⠏⢉⢏⣠⠬⠥⢖⣡⠾⠖⠓⠒⢦⡀⠀⠀⢀⡾⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠙⢦⣇⠀⠀⠘⡇⠀⠀⣿⠈⣀⠒⠙⢳⣄⡑⠒⠚⠒⢿⣄⡰⣫⠏⡴⠚⠉⢹⠁⠀⠀⠀⠀⠀⠙⣆⡠⠚⠁⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠙⠷⣤⣀⡇⠀⠀⣿⠀⣒⠛⠛⠋⣊⠽⢛⢝⡛⢦⣉⠙⣁⣾⠴⠒⠒⠺⡄⠀⠀⠀⠁⠀⠀⢻⢀⣠⠞⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠈⠙⠳⢦⣴⣏⠒⠚⠿⠭⠭⢒⡪⢟⡭⠒⠉⠉⠉⡿⠁⠀⠠⠀⠀⢹⡀⠂⠀⠀⣀⣤⠾⠛⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠶⣶⣭⣥⣖⣉⣀⣀⠀⠀⢰⡃⢀⣀⣀⣀⣀⣨⣧⡴⠞⠛⠉⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠠⠄⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠉⠁⠀⠀⠀⢀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢦⡀⠀⣠⡶⠲⡀⠀⣠⠴⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡦⣤⣀⣝⣥⡭⠾⠟⠧⠀⢰⠮⣁⣠⠼⣀⣤⠖⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠂⠄⡀⠀⠀⠀⠀⠀⢸⡃⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⢍⡢⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠑⠀⠹⡟⠛⡫⠀⠈⠀⠀⠀⠀⠀⣒⠒⠄⠀⠀⠀⠀⠀⠀⠨⠢⡰⡒⠒⢛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⢐⣖⠀⠀⡒⠀⠀⠀⠀⠀⠠⠈⠁⠢⠀⠠⠄⠐⠀⠀⢩⠐⢌⢲⣅⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠠⠄⠀⠠⢂⣂⣌⠲⡀⠀⣠⡐⠒⢀⣁⠲⢰⠑⢂⠁⠄⠉⢯⣉⠽⠁⠀⢠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⠀⠀⢣⠀⣜⠀⡥⠤⡃⠸⡅⠈⢸⢟⠈⠃⢸⠀⡇⠁⡄⠀⠘⡰⠀⠀⠀⠀⣄⢠⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⢠⠸⡀⠀⠀⠌⢰⠑⣀⣀⣈⠛⣄⣈⠀⠪⢄⣈⡤⣌⣃⡇⠡⣑⠆⠀⣿⣻⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠁⠂⠘⠓⠷⠤⣀⠀⠋⠀⠈⠈⠠⡀⠀⠀⡇⣸⠔⠁⠀⠀⠀⠀⠀⠀⠀⢀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡀⠀⠐⢍⠒⠀⡠⠋⠀⠈⢢⡀⠀⠀⠀⠀⠀⣸⣞⡟⠞⠀⠀⠀⠐⠀⠀⠀⢠
⠀⠀⠀⠀⡀⠤⠄⠐⠒⠀⠀⠀⠒⢀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠲⡀⠀⠉⠝⠁⠐⠀⠀⠀⠙⣄⠀⠀⠀⢀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠈⠔⠒⠂⠀⠐⠀⠠⢀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠹⢀⠌⠀⠀⠀⠀⠀⠀⠀⠘⡦⣠⢴⣟⠋⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢹⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⡿⠃⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠰⢎⠀⠀⠀⠀⠀⠀⠀⠀⠶⣤⠁⠜⣂⠀⠀⠀⢀⣀⠀⠀⣀⠀⠀⢀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠅⠀⠀⠀⠀⢀⡀⠀⡞⠁⠈⠙⠋⠉⣟⠦⠟⠋⢉⠁⢸⣷⣚⣉⠁⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⡄⢀⣀⠀⠀⣁⠂⠐⠣⠀⠊⡉⠔⠈⠀⠀⠈⣀⠹⢋⢁⣀⣀⡀⠋⠃⠀⢩⢹⣿⡆⠀⠈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⠤⠄⢀⡀⠀⠀⠙⠷⢄⠱⡄⠀⢰⡈⠀⠀⠀⠀⠀⠄⠈⠀⠀⠈⠃⠈⠁⠠⠀⣜⠹⢊⡝⡹⠕⠂⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠀⠀⠉⠐⠒⠠⠶⠌⠠⠭⠤⠤⠀⠂⠁⠀⠀⡀⢚⠄⠀⣀⡄⠀⠀⠀⠑⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠒⢳⡀⠀⠀⠈⠃⠀⠀⠀⠂⠠⠠⠰⠀⠀⠀⠀⠐⠀⡐⢄⠉⠈⠁⢀⣄⢂⢸⡗⠄⣤⣄⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠃⡤⣤⣤⡄⠤⠬⠀⠉⣁⠀⠁⠑⠀⠀⠀⠑⠊⢀⡀⠠⠐⠈⠉⠋⠉⠉⠁⠀⠀⠀⠀⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⣉⠗⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⢒⡂⠭⠉⠙⢉⡉⠉⠉⠉⠉⣉⣉⣉⣉⠉⠉⠑⠒⠲⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣛⣭⠶⠚⠉⠀⡠⢒⣈⠝⢶⣚⠶⣿⣭⢿⣣⣭⣎⣶⠛⠷⣖⢶⠲⢶⣦⢬⣍⠙⢲⢤⡀
⠀⠀⠀⠀⠀⠀⣴⢟⡵⠾⢿⠀⣠⡠⢜⡣⣴⢛⡭⣴⡯⣞⡸⠷⣼⣟⣿⣋⣽⢾⣄⠀⠈⢈⠀⠘⢨⠸⢮⠷⣴⢍⠻⣷⣄
⠀⠀⠀⠀⠀⣼⣿⠁⠁⡤⢚⣏⠻⢌⠹⢿⡯⣽⣿⣭⣤⣌⣽⢿⣦⡿⢫⣗⡾⠓⣿⠛⢿⣿⣷⣮⡽⣾⡧⠔⣿⡌⡇⠛⢫⡇
⠀⠀⠀⠀⠀⣿⣷⡀⢮⣭⠿⣶⣗⣾⣖⣛⣳⣹⣯⣰⠗⡲⣿⠿⣭⡻⣿⠬⣽⠛⣩⡦⢜⣳⣏⡻⢵⣖⣿⣛⣅⣯⡗⣄⢾⡷
⠀⠀⠀⠀⠀⣿⠙⣯⣳⢽⣛⡿⢽⡦⣌⣙⣉⣫⡵⣟⣳⣯⣿⣿⣧⣽⡿⠷⣾⡿⠅⢀⣠⠜⣗⣼⣋⣩⡽⣿⡿⢿⣽⡿⢫⡇
⠀⠀⠀⠀⠀⢹⡆⠀⠉⡛⢲⢭⣍⡘⠲⣟⣳⢀⡞⢳⣶⣾⢷⠿⠟⣾⠿⣿⣶⣻⣾⢏⣻⠆⣽⣒⣾⣯⢿⣟⣿⢿⣻⠃⣼⠃
⠀⠀⠀⠀⠀⠀⢷⠀⢰⠀⢸⠐⠦⣭⣙⡒⠲⠯⠴⢾⣿⣷⣤⣷⣤⣈⣎⣁⣿⣷⣾⣿⣿⣿⠿⠿⠛⠛⠉⠹⠁⢸⡟⢰⡟
⠀⠀⠀⠀⠀⠀⠘⣇⠘⡄⠀⠀⠈⠐⠂⠭⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠒⠒⠉⠉⠀⠀⠀⢀⢢⠏⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⡄⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⠀⠀⠤⠤⠀⠀⣀⣤⠤⠤⠴⠖⠛⣉⡥⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠶⠶⠤⠤⠤⠶⠖⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡈⢦⢸⡰⢁⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣶⣿⣷⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⡾⡽⣷⢿⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣳⢯⣿⣟⣿⣹⣥⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢤⣟⣿⢺⠧⢿⣤⣷⠶⢻⠀⠀⣴⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣄⠀⠀⢠⠓⡾⢤⣏⣹⣗⣻⣗⣛⣖⣩⣧⠞⢡⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⣦⠀⣼⣩⡗⢺⡟⢾⢦⣼⣯⣩⣯⣠⣿⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠈⢳⡗⢼⣍⣹⡇⠈⣧⠾⡦⠴⣷⠤⠾⡆⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣌⢳⡀⢸⣝⣻⠤⢼⠇⠀⢸⣞⣟⠒⢻⡖⢺⣷⡷⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡈⢻⡇⢸⠦⣿⣉⡿⠀⠀⢸⣯⣿⣒⣺⣏⣩⣿⠃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢹⣾⣓⣳⣼⠃⠀⠀⢸⠷⣿⢬⣽⡧⢤⡟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⡬⣿⡏⠀⠀⠀⣾⣉⣹⠒⠚⣟⡾⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠈⢷⢻⠇⠀⠀⠀⣿⢭⢿⡉⢉⡿⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈⣿⠀⠀⠀⠀⣿⠲⢾⡽⠋⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢸⡆⠀⠀⠀⢹⣽⠏⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⢳⡀⠀⠀⠈⣧⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⢳⡀⠀⠀⢸⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣭⣷⠤⣾⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣤⣤⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣈⢍⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠶⠒⠒⠒⠒⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢺⠀⠻⠇⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⠀⠐⠀⠀⠀⠃⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠘⡓⠀⣀⢚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠠⠁⡅⠀⠈⠈⠀⠀⠈⠀⠀⠀⠄⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⠀⠀⠀⡆⠀⢀⠀⢰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀
⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⣠⠐⡄⠀⠃⠀⠐⠀⠞⡏⠀⠣⡀⠠⠛⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⡌⠀⠀⠀⠀⡠⠀⢀⠘⠂⠂⣂⡀⡻⠠⡅⠐⠠⠌⠶⡦⣄⣀⠀⢁⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠉⠉⠉⣔⠋⠊⠔⡂⢀⡸⠄⠠⠀⠀⠈⠉⠉⠠⠤⠄⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⢆⠡⠂⣤⠀⠩⠜⠢⠠⣤⡒⠉⠀⠀⣺⠀⠀⠄⠄⠴⠚⠀⡄⠀⠀⢀⠇
⠀⠀⠀⠀⠘⡄⠀⠘⠈⠉⠈⠀⠔⠐⠀⠂⠁⢈⡍⠓⠒⢾⠳⠄⠀⠀⠀⠀⠀⠈⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠶⡁⠀⡀⠠⠐⠀⠀⠀⡀⠐⠊⠉⠁⢂⠟⢀⡦⠀⠀⠀⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠐⠄⠀⠠⠈⠡⠈⠷⠂⠀⠀⠀⠁⠀⠀⠀⣠⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⡿⡿⠿⡟⢛⣋⣹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣤⣤⣴⣶⣶⣤⣤⣴⣶⣾⠿⠿⠛⣛⣋⣩⣭⣤⣴⣶⣿⡿⠿⠟⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⠿⣿⢿⢛⡏⡟⡇⣿⣿⣿⣿⣶⣶⠿⠿⠿⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⣿⡏⢹⢸⠀⡇⢸⢰⡇⡏⡇⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠿⡿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡇⢾⢸⠀⡇⢸⠸⡆⡇⡇⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⣾⢸⠀⡇⢸⢸⠀⡇⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡏⢸⠀⡇⢸⠸⣤⡇⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡇⢸⠀⡇⢸⢸⣿⡅⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡇⢸⠀⡇⣼⢸⢻⡇⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡇⣾⢸⡇⡟⢸⠀⡇⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⡇⣿⢸⡇⡇⢸⠀⡇⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣨⣿⡇⣇⣿⣸⣇⣧⣸⣀⣇⡇⣿⣧⣤⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣋⠭⢭⣉⣿⣿⡿⠿⠛⠛⠛⠛⠛⠛⠻⠿⢿⣿⣯⣤⠤⠬⣝⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣾⣷⣾⣿⣿⣿⣥⣴⠚⣉⣉⣭⣉⣍⣉⣙⠒⠢⢬⡙⠿⣷⣶⣬⣷⣼⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⣋⡭⠥⠒⠒⠒⠒⠢⠤⢍⣉⠛⢿⣷⣤⠤⠤⠤⠤⢬⣉⡓⢦⣬⣷⡿⢛⣉⣉⣉⣛⠻⢿⣦⣄
⠀⠀⠀⠀⠀⢠⣿⠟⣵⠚⣡⠖⣊⣉⠭⠭⠭⣉⡓⠒⠤⣉⠲⣌⢻⣷⡖⠒⠒⠒⠦⣍⣾⡟⣡⠞⣩⠤⠴⠤⢬⡙⢦⡙⣿⣆
⠀⠀⠀⠀⣠⣼⣿⣴⣧⣾⣵⣾⣷⣶⣶⣶⣶⣶⣿⣷⣶⣿⣷⣾⣷⣿⣿⣿⣿⣿⣶⣾⣿⣶⣿⣼⣧⣶⣿⣿⣦⣽⣤⣷⣼⣿⣄⡀
⠀⠀⠀⠀⣿⣟⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠈⣿⡆⠛⠃⠀⠛⠛⠀⠘⠛⠀⠛⠛⠀⠘⠛⠀⠛⠃⠀⠘⠛⠃⠘⠛⠃⠀⠛⠛⠀⠘⠛⠀⠘⠛⠃⠘⠛⠃⠛⢃⣿⡇
⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣾⡿⠋
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠉⠛⠛⠛⠿⠿⠿⠿⠿⢿⣶⣾⡿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⣿⡇
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⠿⠃
⠀⢰
⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢀⠀⠀⠀⠺⡀⢀⣷⠄⢠⠊⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠉⠙⠻⢦⡄⠀⢿⢋⢿⠂⠀⡇⠀⠰⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠉⠻⣤⡀⠀⢠⢹⣤⠇⣼⡋⠀⠠⠀⢈⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠈⠻⡀⠀⠀⢻⡄⠹⡧⠂⠀⣶⠉⣠⠿⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢤⡀⠀⠀⢹⡆⠀⢸⡇⠘⡇⠀⠀⡇⣼⢃⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠈⣿⡀⠸⣇⢠⡏⠀⠘⢰⡗⠀⠀⠀⠀⠀⠀⠀⠠⠒⡄⠀⣀⠼⠀⠀⢰⠀⠃⡀
⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠸⣇⠀⠻⠸⡏⢰⠃⡼⠇⠀⠀⠀⠀⠀⠀⠊⠠⣼⠏⣾⠁⠀⠀⠀⡴⠀⠀⠷⠈⠂
⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠘⣇⠀⣦⢡⢏⡼⠃⠀⠀⠀⠀⠀⠀⢘⠀⠐⣿⡽⠃⠀⠀⠀⡠⠀⠀⢀⡞⠀⢘
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢹⣼⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⣿⡟⠀⠀⠀⣰⠕⠀⠀⢠⡇⠀⣈⡀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣾⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢿⣯⠀⠀⢀⡟⡀⠀⠀⣸⠀⢠⡟⠁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠹⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣿⣳⣦⡌⣥⡄⣠⠾⢁⣠⠟⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢙⠋⠉⠡⠔⢳⡖⠛⡁⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠀⠐⠁⠐⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣙⡼⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⣷⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢻⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⠀⠀⠀⣀⡀⠈⠁⠀⢨⣳⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠳⡄⠀⠀⡌⠀⢀⠀⣾⠇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠲⠶⠖⣒⣸⠶⢄⠀⠀⠈⢢⣰⡟⠀⢠⣾⣯⠴⠖⠛⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡀⠀⠀⠙⠻⣦⡰⠙⢶⣄⡘⠘⠛⠓⠞⠋⠁⠀⢀⣠⠄⣒⡭⠟⠲⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⡀⠀⠀⠙⢻⣄⡂⠈⠻⣯⡑⠯⠠⠤⠛⠷⠧⠚⠈⠁⡠⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡷⣤⣉⣀⣙⣓⣤⣜⡨⣿⢲⠦⣀⣠⣤⣤⡄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⢀⡀⠤⠟⠋⠀⠉⠻⢊⣁⡀⡀⠀⠀⢀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣿⣿⣿⣿⣿⢟⢹⣿⠀⢸⡇⢻⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠀⠀⣿⠀⢸⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣸⠘⣧⠀⠀⠜⠀⠀⠃⠘⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⡀⠘⡗⣶⣦⠸⣿⣿⠇⠉⢿⣿⣿
⢿⣿⠘⠀⣵⠀⠀⡀⠀⢤⡆⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡿⠿⠟⠋
⠀⢋⠀⠀⠉⠀⠀⠁⠀⠀⠿⠂⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢟⣛⣛⡻⢿⣿⣿⣿⣿⣿⣿⣿
⠃⢸⡂⠀⠀⠀⠄⠇⠀⢠⡄⠀⠀⢰⠀⣿⣿⣿⣿⣿⡿⣡⠞⠉⠉⠇⠉⠳⢝⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⢀
⡀⣘⡯⠐⣿⣷⡀⠠⡄⠺⠿⠀⠀⠸⠀⣿⣿⣿⣿⡏⡜⢁⠐⠀⠀⠒⠐⠒⠀⠊⣻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁
⣧⣿⡦⠏⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣘⡼⠏⡃⠐⠓⡒⠐⡻⠀⢿⣿⣿⣿⣿
⠂⣿⡀⠀⠀⣠⠀⠀⠀⠀⢠⠀⣄⣠⠀⣿⣿⣿⣿⣧⠓⡀⢀⣀⠃⡈⠁⡀⡄⠀⣼⣿⣿⣿⣿⠀⡀
⡿⢻⢟⡀⠹⣿⠺⠷⡞⠛⠛⠙⠿⠿⠀⣿⣿⣿⣿⣿⣅⠄⣈⣀⢀⢀⣀⠁⠁⢰⣿⣿⣿⣿⣿
⠀⠀⠙⠋⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣏⡇⢨⢬⠀⡌⡬⣄⢸⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡷⠿⢸⢾⠁⡇⡿⠊⡷⣺⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣻⣿⣿⣿⣷⡆⠀⠀⠒⠉⠉⢫⣹⢻⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⢞⣾⡿⠿⠛⣻⡿⢿⠄⠀⠀⠀⠀⠘⢻⣯⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⡿⡆⣋⣫⣭⣽⣷⡾⣷⣂⠀⠐⠆⠀⠀⠙⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣮⣿⣽⣿⣟⣿⣿⣟⢳⣾⣿⣷⣶⣶⣶⣶⣶⣾⣿⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣽⣽⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠥⠭⠥⠥⠤⠥⠤⠬⠬⠥⠥⠤⠬⠭⠭⠭⠭⠭⠭⠥⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠀⠾⠺⠈⢁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⠛⠋⣿⣿⠙⠛⢿⣷⣤⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠁⣀⣤⣶⠀⢠⣶⣤⣀⠈⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⣿⣿⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠁⢠⣾⣿⣿⣿⠀⠘⣿⣿⣿⣷⡄⠈⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠁⢀⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⡄⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣠⣿⣿⣄⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⠟⠀⠈⠻⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣷⠀⠀⣠⣾⡿⣿⣿⢿⣷⣄⠀⠀⣾⡿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄⠘⣿⣿⠟⠁⢀⠀⢀⡀⠈⠻⣿⣿⠃⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⢿⣷⣾⡿⠋⠀⣿⣿⠀⠙⢿⣷⣾⡿⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡀⠘⠁⢀⣴⣿⠀⠠⣿⣦⡀⠈⠁⢀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⣄⣿⣿⣠⣤⣾⡿⠟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦⡀⠉⠛⠿⠀⠐⠿⠛⠁⢀⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠝
⠀⠀⠀⠀⠀⠑⢷⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠂⠄⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⣀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠛⠀⠀⠀⠈⠀⠀⢨⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⡀⠀⠀⡀⠠⠄⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠝⠀⣤⠙⡠⠀⠀⠀⠀⠃⠨⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠤⠒⢒⠒⠖⡢⢄⡀⠱⡀⠂⠀⢀⡦⢤⢐⣶⢦⠞⠁⠀⢐⡎⠐⡰⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⢀⡔⠁⠀⠀⢸⣶⣆⡴⠂⠈⠢⡈⠉⠛⠉⠇⠈⠀⠀⢸⠆⠀⠀⡠⢆⡀⠀⠐⠡⠂⣈⠴⠃⣠⣤⡄⠐⠈
⠀⠀⠀⠀⡜⠠⢀⣴⣶⣾⣿⣾⡶⠦⠄⠀⢱⠀⠀⠀⢸⠀⠀⠀⠈⠀⠀⠈⠀⠸⡈⠁⠀⠘⠁⠈⠀⣼⣿⣿⣿⣶⣤
⠀⠀⠀⠀⠅⠀⠈⠁⠀⠘⢿⡀⠀⠁⠀⠀⢈⡆⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⣰⣿⡟⠛⠛⠁
⠀⠀⠀⠀⠈⠳⢄⠀⠐⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡐⠲⢃⣜⡁
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠤⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠃⡌⠑⠁
⠀⠀⠀⠀⠀⢀⠀⢀⠔⠂⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠠⠃⣁⠀⠀⠠⣴⡞⠁
⠀⠀⠀⠀⠔⠁⠁⠊⣠⣦⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡔⢨⠗⣭⡦
⠀⠀⠀⠀⢄⠀⠀⠀⠈⠉⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣮⣎⣿⠓⠃
⠀⠀⠀⠀⠀⠞⢨⣷⠄⠀⠀⠀⢠⠁⠀⠀⠀⢠⠔⠋⠁⠀⠉⢦⣤⣤⠤⠀⣀⠀⠀⠀⠀⠠⠈⠀⠁⠘⡷⠟⠏⠛
⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠐⡁⠀⠙⢆⠵⠀⠀⠐⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠂⠤⠄⠒⠁⠀⠀⣀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⢈⠇⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠔⠁⠀⢀⡠⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⢄⣀⣀⣀⣀⣀⣠⡤⠖⠊⠁⠀⠀⠀⠀⢀⠂⣀⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉

⠀⠀⣰⠖⠺⣄⡂⣋⠀⠀⠀⠘⠰⠃⠀⠀⠀⠀⢰⡙⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠰⠖⠁⠀⠀⣰⠖⢳⡀⡐⡃⠀⠀⠀⠙⠸⠀⠀⠀⠀⠀⢰⠉
⠚⢻⠀⠀⠀⠩⣷⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⠀⣰⡁⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⢠⡼⠂⢀⡤⡴⠚⠛⠀⠀⠀⠹⡷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡠⢈⣉⠲⡀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡴⠊⠀⠀⡔⢩⣑⢆⠀⠀⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻
⢀⠘⣯⠳⡌⠀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠐⠉⠀⠁⠀⠀⡀⢸⣟⢦⡁⢄⡀⠀⠀⠀⠀⠀⢀⣠⣀⡤⡄
⠄⠸⠋⢣⠁⠀⢻⣗⡒⡀⠀⠠⣶⠏⡡⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⠀⠀⣀⠀⣀⠄⠸⠈⣆⠁⠈⢿⣓⢂⡀⠀⢰⣶⠋⡄⠂
⠀⠀⠀⠈⢷⠀⠀⠑⡄⠉⠓⢄⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠈⠡⠈⠉⠲⣀⠟
⠀⠀⠀⠀⠀⠀⠀⡀⢀⢑⡀⢨⡷⠷⠲⠤⠶⣤⠀⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢃⠀⢸⢶⠖⠲⠤⢶⣆⢀
⠀⠀⠀⠀⠀⠀⠘⣹⢸⠀⢈⠹⣅⣀⣀⣤⠟⠛⠉⠙⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢠⣤⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⢱⠘⡌⢻⣅⣀⣠⡼⠛⠋⠉⠑
⠀⠀⢀⠀⠀⠀⠀⢹⡀⡆⠀⣶⠀⠁⡾⡄⠀⠀⢀⠄⢀⡤⠀⢀⣠⡤⠀⠀⠀⢠⣼⢯⠔⣿⠂⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢻⠐⠀⢀⡦⠈⠀⣷⡀⠀⠀⡠⠔
⠀⠀⠤⠀⠀⠀⠀⠘⠇⠁⠀⠛⡀⠀⠀⢰⠆⠰⠡⠿⠿⢾⡋⠁⠈⠀⠀⠀⠀⡾⣶⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠸⠄⠀⠘⠣⡀⠀⠀⢱⠆⠾⠰⠿
⢀⣰⠃⠀⠀⠀⠀⠀⠘⠀⠀⠐⡌⢢⡀⠈⡧⠀⠉⠀⠒⠀⢤⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠈⠀⠀⢠⠉⢂⠀⢘⡆⠀⠑⠐
⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠈⠁⠀⠙⢆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠄⠀⠉⠁⠀⠱⡄
⠀⠀⢀⣀⣲⠦⡀⠀⠜⡷⠒⠀⠀⠀⠐⢹⣄⠈⢧⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣲⠄⡀⢠⠚⠗⠈⠀⠀⠀⠊⣷⡀⠘⣆
⠖⠊⠉⠩⠙⢊⠠⠀⢀⡏⠀⠀⠀⠀⢠⠂⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⢺⣯⣿⣯⠁⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠉⠭⠓⣂⠰⠀⢰⠃⠀⠀⠀⠀⠠⠀⠀⠀⠘
⢀⣀⡀⡀⢊⣎⣀⣠⣌⣠⣤⡄⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠛⠿⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠐⢀⣇⡀⣠⣏⣠⣤⠀⠀⠾
⠔⠉⠉⠀⡏⠘⠼⣧⠈⠋⣉⠙⢖⡀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠔⢠⠎⠉⠉⠰⡏⠘⠸⡆⠈⠡⣉⠻⣅⠀⠀⠀⠀⠀⠀⢠
⠀⠀⢐⡶⢺⡀⡆⢏⠀⠀⠀⠐⠪⠇⠀⠀⠀⠀⢠⡚⠉⢻⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⡔⠁⠀⠀⣰⠶⢷⡀⠰⡃⠀⠀⠀⠐⢸⠀⠀⠀⠀⠀⢰⠋
⠖⠲⠁⠀⠀⠙⣧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣰⡁⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⢠⡼⠆⠀⣠⡤⠒⠾⠁⠀⠈⢹⣷⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠠⢘⣓⠤⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠚⠀⠀⡠⢚⡒⢄⠀⠀⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾
⢀⢼⣿⢦⡐⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠐⠋⠀⠂⠀⠀⠀⢸⡿⢦⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀
⡘⢾⠋⢦⠁⠀⢿⣲⠆⠀⠀⢠⣶⠟⣁⠐⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡼⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠸⠙⣆⠁⠘⣿⣲⠆⠀⠀⢰⣶⠋⡁⠒
⠀⠃⠀⠈⢶⠀⠈⠫⡄⠙⠲⢄⡻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠘⣶⠀⠈⠣⠈⠑⠦⣈⠷
⠀⠀⠀⠀⠀⠀⠀⡠⠁⢒⠀⢈⡧⡶⠶⡄⢤⣄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡰⠀⢂⠀⢹⠴⠶⠦⡤⣤⣠
⠀⠀⠀⠀⠀⠀⠸⣳⢺⠈⢑⢹⣍⣁⣀⣠⡟⠛⠛⠓⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣳⢣⠘⡈⢻⣉⣁⣀⣴⠟⠛⠋⠓
⠀⠆⠀⠀⠀⠀⠀⢹⠀⡀⠀⣢⠈⠉⣿⡀⠀⠀⢀⡠⢀⡥⠀⠀⣠⣄⠀⠀⠀⠠⣜⣟⡭⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠸⠀⢐⡦⠉⢁⣷⡀⠀⠀⣀⠤
⠀⠅⠄⠀⠀⠀⠀⠘⠇⠁⠀⠛⡀⠀⠀⢱⠤⢴⠡⣾⣯⣔⡚⠉⠉⠀⠀⠀⠀⣾⣥⣶⡾⠋⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠸⡄⠀⠘⢇⠀⠀⠀⣣⠄⠴⢡⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢰⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠣
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⡀
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠂⣆⡀⢀⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠀⠠⠤⡀
⠀⠀⠀⢠⢤⣠⡄⠀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠉⠞⢱⠻⡽⠷⠯⠷⠯⠧⠧⠿⠵⠽⠆
⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⡤⠀⣀⣠⠖⠁⠀⣠⣤⣤⣄⣀⢀⣁⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣐⣊⡐⠫⠲⢄⠙⠇⣠⠞⠋⠀⠀⢰⣶⣤⠄⠉⠙⠳⢬⡑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣤⣴⣯⣝⡙⠻⢹⡃⡀⠸⠀⣼⣟⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢉⡤⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣏⠩⠄⠘⣿⠈⠸⠇⠃⣈⠁⣥⡀⣻⣿⡟⢳⠅⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠗⠄⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠙⡄⠀⡀⠀⣿⣧⡄⢠⣁⠻⠧⣻⢵⡈⣿⡿⣥⣄⢠⡁⣠⡶⣿⣶⣄⠀⠀⠀⣰⠆⠂⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⢿⣀⠀⠀⠙⠿⠙⢧⣙⣄⣩⣭⣥⣄⢼⣽⣿⣿⣽⢈⣿⢾⣿⣤⣻⣿⣟⣠⡧⠞⢸⡗⢠⣄⠰⣘⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠻⡏⠁⣀⣽⡆⢠⣬⠟⢩⡿⣿⣿⣴⣿⡏⠈⢻⣿⣿⣿⣿⠟⠋⠉⠉⢉⣀⠔⣾⡅⠀⠛⠶⣡⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣤⣤⠞⠛⣡⣂⡼⢉⣼⣯⢟⠟⣹⡾⠏⢁⡴⣫⣿⣼⣿⢿⡀⡤⠚⠋⠹⡇⣾⢿⣧⣰⢎⠀⠤⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠛⠛⢲⠉⣿⢁⡼⣵⢿⠏⣾⠟⠁⡰⢋⣽⢏⣼⣷⡟⣼⡟⠐⠀⠀⠀⠳⠽⢁⠋⢀⣨⣄⠓⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣅⠖⠾⢷⣿⢸⣷⣿⣹⢰⣿⠀⢰⣧⣽⠋⣼⢻⡏⣰⣿⣧⡀⠀⠀⠀⢀⡴⠛⣢⡤⠥⠠⠅⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⣧⡧⠀⠀⣙⣷⢸⣾⣿⢿⠘⣿⠀⣾⠻⣿⢠⣧⣼⢳⣿⢓⣽⢻⠞⠓⠶⠞⠅⡈⠁⣀⡤⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣶⣶⣿⣿⣦⣟⣻⣾⡄⣿⡀⣷⣴⣇⣺⢷⡏⣸⣏⢸⣁⡴⢶⣭⡀⣤⣴⡷⠛⠛⠠⣄⡀⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢿⣿⣏⣿⡿⢿⠿⠿⠷⣽⣇⡜⣼⣿⣿⣻⣇⣿⣿⣴⣿⡇⠀⠙⡟⢯⡙⢍⠛⠲⢄⠀⠙⠃⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠈⡧⠈⡛⠉⠀⠀⠀⠀⠍⠉⠙⠻⢿⢼⡿⠛⠉⠍⠁⢀⣾⡀⠀⢹⣾⣹⡛⢳⡤⢠⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡘⢰⢴⣾⢰⣶⣿⡍⢸⠇⠀⢾⠏⠁⠀⠨⠁⠀⢹⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⠿⠿⣷⣦⢤⡀⣰⢿⠶⡈⠻⢄⢶⡎⢀⡿⣀⢂⡞⠀⠀⠀⠀⠐⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠙⠻⠻⠋⠻⣷⣯⣦⠴⠃⡀⠀⣀⡉⣉⣭⡵⠞⠀⢀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠋⠒⠒⠀⠐⠚⠛⠀⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⣀⣀⠀⠀⠀⠀⢀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀⢀⡴⠀⣤⠀⣀⠂⠀⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢶⡞⢣⡞⡍⠣⢮⣉⣌⡃⢠⣰⣷⣷⢀⡕⣮⠃⠞⡑⠖⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡶⣽⡥⣾⡀⠓⡆⠉⠁⠘⡴⠀⠀⣻⢋⣵⣌⢶⠯⣸⣯⠀⢑⣂⣽⢦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣿⡀⢰⢅⠳⠺⠴⠇⠀⢆⡴⠄⠀⠀⡀⠈⠈⠙⠚⠀⠈⣌⣹⡴⣓⠀⠨⡝⡀⢹⢆
⠀⠀⠀⠀⠀⠠⣾⣟⠞⣿⢘⣻⠶⡆⠕⠊⠀⠀⠀⠀⡇⠀⡄⠀⠀⠀⠀⠈⠀⠈⠁⠁⠘⣝⠚⠁⠀⠘⠺⣵⣀
⠀⠀⠀⠀⠅⢉⠮⣫⣷⣌⣿⠞⠃⠀⠀⠀⠀⠀⠀⠀⠃⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠒⠠⠆
⠀⠀⠀⠺⢉⠰⣿⠼⡬⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⣠⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢠
⠀⠀⢒⣄⠦⠀⠀⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⠀⠀⠀⣠
⠀⣼⣲⡍⡤⣀⡐⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢡⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡀⠂⠀⠉⡀
⠘⠁⢜⢒⠂⠈⢁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⠠⠡
⠸⢲⠺⠪⡀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠠⠀⠀⢀
⠀⠀⢄⠈⢨⢍⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⢖⠀⠋⠀⠼
⠐⡄⠀⠠⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠰⠉⡀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⠈⡘
⠀⣀⣘⠐⠪⡲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢙⠋⡁⠀⠀⣂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠐⠡⠁⠐
⠀⢉⡀⠐⣰⠺⡀⠈⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠊⠸⠁⠀⠀⠙⠕⡁⠀⠴⠘⢃⡀⠦⠀⠀⢀⡆⠀⠀⠀⠀⡄⢀
⠀⠘⢽⣄⡈⠚⠉⣶⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⢠⣾⠁⠀⠀⠀⢸⠏⠎
⠀⠀⠀⠨⠉⡔⢀⣂⡅⡂⡀⠀⢀⠑⠀⠀⠀⠀⠀⠀⠸⠇⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠝⣿⠃⡆⢀⡄⢀⠂
⠀⠀⠀⠀⢢⠀⠘⢾⡶⠿⠇⠆⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢲⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⠡⣍⠄⠠⣚⢣⠄
⠀⠀⠀⠀⠀⠐⠀⢸⠃⠐⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠑⢾⡖⠀⠀⠀⡀⢤⠀⠚⠀⡀⢦⡅⢎⠘⢲⡗⠉
⠀⠀⠀⠀⠀⠀⠁⠊⠶⠖⠀⠒⣠⣄⠀⠀⢠⡄⣉⣤⠤⢀⣙⡩⠃⠨⢑⠀⠃⢀⠀⢱⡆⢀⡤⣲⡮⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⡴⠀⠹⡢⡇⠄⣈⡽⠈⡳⢄⠘⠞⠃⢀⠂⠀⣂⠀⣈⡁⠀⣤⣀⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢤⣁⠹⣉⣹⡄⠤⢠⢍⠀⠂⢩⠁⢐⡴⠃⠀⠀⠐⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠐⠲⠀⠉⠀⠤⠎⡃⠊⠁⠉
⣿⣿⣿⣿⣟⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣟⣻⣟⣻⣿⣿⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⣟⢽⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣰⠭⣂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠰⡄⠀⢨⡯⠿⢹⡅⠀⢠⠂⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣟⢿⣿⡗⣺⣒⡖⢺⣿⡟⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡀⣚⣃⣛⣘⣃⢠⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡈⣿⠥⠽⠤⡧⢼⡟⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣧⢽⠥⣭⢬⡧⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⢸⡒⣛⠒⡓⢺⠃⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡓⣿⢚⢃⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠈⣇⣭⣉⡭⡼⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡩⣿⢍⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣭⣭⣥⡀⠉⣤⢰⡶⠃⢠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣶⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠙⢢⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡠⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢓⡂⠉⢐⣒⣛⢛⣉⢉⢉⡉⣁⢀⢀⠀⠀⠀⠄⠄⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢦⢊⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⡖⢦⡈⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠲⠉⢘⠇⠀⣖⠀⣈⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡔⣌⠐⠻⠀⢤⡤⡀⣾⠉⡆⠀⠳⠖⢤⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠡⡄⢦⡉⢀⡈⠁⢡⠃⣌⡋⠆⢳⡄⣠⠴⠈⠵⣦⠙⢢⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣔⣴⠂⠠⡆⠐⠛⠙⡍⠩⠐⠂⠨⣤⣠⢖⡃⢰⠙⠃⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢓⠾⣬⣖⡑⠀⠽⢐⠺⢃⢀⣀⠰⡦⠀⢁⣁⣘⣃⣤⠶⢚⡍
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢭⣉⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⢋⣉⡉⠁⠶⠚⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢀⠠⠨⠉⠉⠙⠛⠋⠙⠉⠉⠉⡉⠑⠅⠉⡀⡨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⢀⣟⣶⣷⣤⡿⡭⡷⡤⠀⠀⠐⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠈⡿⣹⣯⣯⣯⣾⣷⣮⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⢙⠉⠙⠉⠉⠉⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⢠⠁⠀⠀⠀⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣧⣴⣶⣶⣦⣶⣀⣦⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠟⠛⢷⣶⣶⣶⣾⣭⣾⣶⣦⣄⠀⠒⢤⣀
⠀⠀⠀⠀⠀⠀⣸⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣶⣶⠶⠶⠾⠿⠿⠿⠿⠶⡾⠿⣿⣿⣟⣛⠷⢶⣭⣭⣿⡷⢦⡈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠢⡙⢧⡉⠻⣆⢻⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠀⣇⣘⣧⣤⢞⡩⠞⠹⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡶⠶⠤⢤⣤⣤⣤⣤⣤⠤⠤⠤⠶⠶⣞⣛⠛⣫⣭⠭⠷⢚⣭⣤⣶⡅
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠈⠉⠛⠿⢿⣿⣿⣿⣿⣿⣄⣀⣀⣙⣛⣛⣛⣛⣛⣛⣛⣛⣭⣭⣭⣤⣶⣶⣶⣿⣿⣿⠿⠋⢻⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢀⡀⠀⢀⣀⣀⡀⠈⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣿⣯⠞⢫⣿⡿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣴⡶⢦⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣣⣿⠉⣀⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⠃⣼⢰⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢰⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡿⣿⣿⣽⣯⠇⣸⢇⣿⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⢻⡿⣱⡿⠃⣸⡿⢉⣿⠉⣿⠛⣻⡿⢁⣟⠛⣿⡟⣱⣾⠙⣎⠈⣿⡿⢰⠏⣾⣷⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠑⣿⠃⣴⠟⢁⣾⠃⣼⠃⣰⡿⢃⣾⡟⣰⡟⣸⣿⢃⡾⢫⠖⡟⠁⡟⠐⣛⣋⣨⠟⢿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣧⣾⣏⣼⣯⣀⡿⢁⣼⣏⡴⢹⡇⣛⣥⡞⢧⣬⠟⣧⠶⠛⠚⠉⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣼⣿⣿⠟⠈⠛⠛⠉⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⢿⣿⣿⣿⣷⡾⢿⣿⣧⣴⣶⣶⣶⣶⣶⡾⣿⡟⡛⣏⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⣿⣿⣾⣿⣿⣷⣿⣙⣿⣿⡏⣿⣳⣿⣤⣿⠷⠶⠿⠋⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣴⣾⣿⣿⣯⣅⠈⠀⠀⣼⣷⣾⣿⣶⣤⣤⣴⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣷⣾⣿⣿⣿⣿⠖⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣤⣼⣎⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢉⡿⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⠟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢻⡿⠃⣴⠟⢶⡄⠀⢠⣤⡀⢀⣤⡄⠀⢠⡤⢴⣀⣶⠒⣺⣦⣾⠷⣽⡄⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⣿⡀⢸⣷⠀⣾⣿⣧⣼⣏⣿⠀⡾⢶⡆⣧⠈⡇⢹⠸⣟⣶⡿⠁⠀⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠀⠀⠘⠿⠾⠃⠀⣿⠟⣿⣿⢿⣿⢰⣧⠖⣦⣼⠌⠧⠜⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠲⣦⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣏⣰⣿⠿⡆⢠⣾⣿⣿⣷⣆⠀⢶⣾⣿⢿⣿⡧⢺⣿⡾⢃⣰⠇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⢠⡿⣟⣿⣿⣆⢸⣿⣿⣟⣿⣿⠀⢸⣿⣿⣸⣿⡇⢸⣿⣿⡋⠁⠀⠀⠀⠀⠀⠀⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣦⡀⠀⠀⠀⠀⠈⠷⠿⢿⡿⠋⠈⠻⣿⣿⣿⠇⠀⠘⢿⣿⣿⡿⠃⠘⠟⠛⠁⠀⠀⠀⠀⠀⣠⣾⠟⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣍⠻⢿⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⠿⢟⣥⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣶⣬⣍⣛⠛⠛⠿⠶⠶⠶⢶⣤⣤⣤⣤⣀⣀⣀⣀⣀⣠⣤⣤⣴⠶⠾⠿⠛⢋⣉⣤⣶⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣷⣶⣶⣶⣶⣤⣤⣤⣤⣥⣍⣉⣉⣭⣭⣭⣤⣤⣶⣶⣶⠶⢾⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠿⠿⠷⠾⣿⣦⣭⣭⣭⣭⣭⣭⣭⣥⣴⡶⠾⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⣿⣛⣻⡿⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⣭⠈⣽⠀⢸⡥⠨⣮⢹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠀⡦⠠⡼⠀⠘⣗⠀⣻⠀⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⡝⢹⠧⠠⣞⠃⠙⣧⠀⣽⡋⣧⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠲⠇⢰⠆⠠⠞⠁⠘⡧⠀⢴⠆⠲⠺⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢺⠆⠲⣄⣀⡜⠃⠚⢇⡀⣴⠂⠸⠇⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡃⠻⠦⢄⠀⠀⠱⠆⠰⠄⠀⠀⠦⠴⠛⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⣠⣀⡈⠁⠈⢣⡀⢠⠎⠁⠙⣄⢠⡌⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠲⣇⠀⠘⠖⠒⢆⡀⢀⡶⠂⠲⡀⢀⡽⠚⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⣿⣷⢤⡏⠉⢳⣄⣠⡌⠉⠉⣡⣄⣤⠋⠉⢳⡤⣿⡇⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠛⣛⣛⣛⠛⠿⣿⣿⣇⢀⠷⠲⢮⠀⠀⡸⠒⠒⢃⠀⢀⠗⠒⠿⣀⣼⣿⡿⠟⢛⣛⣛⣛⠛⠿⢿⣶⣦⣄
⠀⠀⠀⠀⣠⣾⠿⢋⣥⠶⠚⢩⣭⣭⣭⣍⡓⢦⣹⡏⢉⣄⣀⣜⠉⠉⢡⠤⢤⡎⠉⠙⣤⠄⣴⠉⢹⣯⠴⢚⣩⣭⣭⣭⡍⠓⠦⣌⡛⢿⣷⣄
⠀⠀⠀⣼⡿⠡⠖⢛⣋⣛⡛⠲⢬⣙⢿⣿⣿⣶⣽⠗⠚⡀⠀⠘⠒⠒⢆⣀⣀⠐⠂⠚⣀⠀⣈⠓⢻⣧⣾⣿⣿⠟⣫⡴⠖⣛⣋⣛⡛⠢⠙⢿⣷⡀
⠀⠀⣼⣿⣴⡾⠿⠟⣻⣿⣿⣿⣦⣍⢳⡝⢿⣿⣿⡆⢼⠉⠈⢹⡤⠤⡌⠀⠈⣠⠤⢴⠉⠀⠙⠤⢼⣿⣿⡟⣡⠞⣡⣶⣿⣿⣿⡛⠿⠿⣶⣤⣿⣧
⠀⠸⠟⠋⠀⠀⣰⣾⠟⣋⡭⠤⠤⣍⡀⠹⡎⢿⣿⣀⣰⠗⠒⢷⣀⣀⡘⠋⠙⢃⣀⣀⠛⠐⠻⣀⣠⣿⡟⣴⠃⣈⣭⠤⢬⣍⡙⠿⣷⣄⠀⠙⠻⣿
⠀⠀⠀⠀⢀⣾⡟⢡⠞⣡⠖⣛⣓⠶⣍⡳⣄⠘⣿⠁⢉⣤⣤⣜⠁⠈⠱⠦⠴⠋⠁⠉⡦⠤⣴⠁⢈⡿⢁⡴⢚⡥⠖⣛⣓⠶⣝⢷⡘⢿⣧⠀⠀⠈
⠀⠀⠀⠀⣼⡟⢠⢏⣾⣥⠤⢭⣍⡛⢮⡙⣮⠳⣌⠛⠛⣀⣀⡘⠓⠘⢃⣀⣀⡞⠓⠛⣄⣀⣀⠛⠛⣡⢏⡴⢋⡴⢛⣭⡭⠤⣼⣧⢳⡈⢿⣇
⠀⠀⠀⢰⣿⠁⡟⢈⡵⠖⣛⡓⢦⡙⣦⠻⣎⢳⡜⢆⠐⠉⠉⢱⠄⠠⠎⠁⠈⠠⠀⠴⠉⠀⠉⢀⡜⢡⢏⣴⢋⡴⣫⠔⣚⣛⠲⣬⠀⣧⠸⣿⡀
⠀⠀⠀⣼⡏⢸⣧⠏⣰⡿⠟⢿⣦⢹⡌⣦⢻⣧⡹⣎⢧⡀⠐⢣⣄⣀⡌⠀⠘⣠⣀⣠⠋⠀⢠⡞⣴⢣⣾⡏⡾⣱⠋⣾⡿⠻⣷⣌⠳⣼⡀⣿⡇
⠀⠀⠀⣿⡇⢸⠏⣼⡿⠁⠀⠘⣿⡇⢷⢸⡌⣿⣷⡘⣆⢳⡀⢮⠀⠀⡱⠆⠶⠁⠀⠈⠶⢠⠏⡼⢡⣿⣿⢸⢇⡏⣸⡿⠀⠀⠘⣿⣆⢻⡇⢻⡇
⠀⠀⠀⣿⡇⡟⣰⡿⠁⠀⠀⠀⢹⣿⢸⡎⣇⢻⣿⣷⡘⣆⢳⡈⠋⠉⢣⣄⣠⠞⠉⠙⢠⠏⡼⢡⣿⣿⡇⣿⢸⢃⣿⡇⠀⠀⠀⠘⣿⡆⠇⢸⡟
⠀⠀⠀⢹⣇⢠⣿⠃⠀⠀⠀⠀⢸⣿⡄⡇⣿⢸⣿⣿⣷⡘⣎⢣⠀⠰⢆⠀⢀⡰⠄⢀⡏⣼⢡⣿⣿⣿⢃⡇⣿⢸⣿⠀⠀⠀⠀⠀⢹⣿⠀⣿⡇
⠀⠀⠀⠘⣿⣾⡟⠀⠀⠀⠀⠀⠀⣿⡇⣿⢸⡘⣿⣿⣿⣷⠹⡌⣧⠀⡌⠉⠉⢡⠀⡾⣸⢃⣿⣿⣿⣿⢸⡇⡇⣸⣿⠀⠀⠀⠀⠀⠀⣿⣶⡿⠁
⠀⠀⠀⠀⠸⣿⠃⠀⠀⠀⠀⠀⠀⣿⣇⢸⢸⡇⣿⣿⣿⣿⣇⢻⡘⡆⠐⠒⠒⠀⣸⢡⠇⣾⣿⣿⣿⡿⢸⢹⡇⣿⡇⠀⠀⠀⠀⠀⠀⣿⡿⠁
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢹⣿⢸⡎⡇⢿⣿⣿⣿⣿⡎⣧⢹⠀⠤⠤⢠⠇⡞⣸⣿⣿⣿⣿⡇⣿⢸⠁⣿⡇⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⡇⣿⢸⣿⣿⣿⣿⣷⠸⡄⣇⠀⠀⡞⢸⢁⣿⣿⣿⣿⣿⠇⡟⣾⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢿⢹⡌⣿⣿⣿⣿⣿⡇⣧⢹⡀⢰⠇⡏⣼⣿⣿⣿⣿⣿⢸⢇⡏⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠸⡎⣧⢻⣿⣿⣿⣿⣷⢸⡌⡇⣼⢸⢃⣿⣿⣿⣿⣿⠇⡾⣸⢃⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⢻⡜⣧⠻⣿⣿⣿⣿⡈⡇⣷⡇⣼⢸⣿⣿⣿⣿⢏⡼⣱⠏⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡙⣎⠳⣌⡻⢿⣿⡇⠷⠾⠶⠇⣸⣿⡿⢛⡵⢟⡴⢃⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣌⡛⢮⣝⡳⠮⣭⡘⣿⡟⢣⡭⠶⢚⣩⠴⢋⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣬⣉⢻⠠⣇⣙⣃⣼⢰⠋⣩⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⡘⠷⠬⠭⠭⠥⠞⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢷⣶⣶⣶⣶⡾⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦
⠀⠀⠀⢠⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣧
⠀⠀⢀⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣇
⠀⠀⣸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡀
⠀⠀⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇
⠀⠀⢹⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠁
⠀⠀⠈⢿⣿⣿⣿⣧⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠟⠁⢸⣿⣿⣿⣿⠀⠙⢿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢠⣿⣿⣿⣿⠏
⠀⠀⠀⠘⣿⣿⣿⣿⣧⡀⣠⣾⣿⣿⣿⣿⠟⠁⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠙⢿⣿⣿⣿⣿⣦⡀⣠⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⣤⣀⡀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠔⢫⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣒⠟⠋⠁⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⡪⠐⠉⡀⠄⠒⣉⠤⠐⡂⢽⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡒⠕⢊⡁⠄⣊⠩⢐⡊⠥⠒⢉⣡⣤⠨⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠮⢒⣁⡤⠦⣁⠢⠅⠒⢉⣀⠤⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢛⣨⣴⣞⠯⠓⢂⣉⠤⠐⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢵⢾⣿⣽⡻⠐⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣿⣿⣿⣟⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⣯⣷⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠤⠶⠖⠒⠒⠒⠒⢺⣽⣽⣻⢷⠛⠒⠒⠒⠒⠒⠲⠶⠤⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠅⢀⣠⡤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣤⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀⣀⡈⠻⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣛⡻⠿⠿⣯⣿⣺⣿⣿⣿⣿⣿⣿⣻⡿⣝⡻⣟⣿⣟⣿⣿⣿⣻⣿⣿⣿⡿⠿⠾⢛⣋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢐⠈⠉⠐⠒⠰⠄⠠⠤⠭⠭⠭⠉⠩⠉⠉⠩⠍⠉⠉⠭⠉⠉⠠⠀⠖⠐⠒⠈⠉⢐⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⣒⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡱⢊⠄⠒⣂⡀⢑⠕⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⢨⠀⣎⠀⠈⠂⢺⣼⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⢎⢆⠺⢤⣀⠀⡠⠂⡔⡜⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡠⣀⠀⠀⠀⠀⠀⠀⠑⠵⣂⡄⣀⡐⠭⠊⠀⠀⠀⠀⠀⠀⠤⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢥⣀⣒⣒⠀⠠⠤⠤⠤⠤⠤⠤⠤⠄⢐⣒⣂⣀⠥⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠠⠀⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠈⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆
⠀⠀⠀⠀⡀⢀⢲⡄⠀⡆⢠⠚
⠀⠀⠀⠀⠁⠀⠀⢷⣀⠀⠀⠰⣤⡶⠤⣄⡀⠀⠀⠀⢀⡠⢄⠀⠀⠀⡀⠀⠄
⠀⠀⠀⢠⡤⡄⠀⠸⣯⠀⠀⡴⠋⠀⣈⠉⠿⢴⠁⠂⠉⠙⠈⣶⠗⠀⠀⢅⡤⣷⠙⢷⠂⠄
⢀⡀⠋⣁⡠⠛⠀⠀⠇⠀⢠⡞⠀⣤⡠⢤⡔⠸⠀⣬⣉⠄⠀⣵⢶⡿⠀⠐⡇⠠⢤⠀⣄⠀⠠⠂
⠸⣀⡐⠀⠠⠀⠀⠀⠀⢀⡼⢂⠀⢈⢉⢀⠙⠘⣀⠀⢁⡀⠀⡀⢠⡖⠀⢠⡇⡇⢸⠀⡌⠑⢄
⢦⢯⠁⠀⠄⠀⠈⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⡂⢸⠀⢬⡁⢠⠧⠤⡤⡄⠜⠈⠁⡈⠉⠉⠂⠀⠂⠀⡆
⠀⠀⠢⡀⠀⡀⠡⡄⡀⠀⢀⡰⠂⣈⡄⣰⠀⠈⠉⠀⢀⡀⠀⠔⠀⠀⠀⠀⢀⠀⠁⠐⡐⢀⡈⠸⣠⡇
⠀⢇⠀⠘⠦⣠⠀⣄⠀⢄⣀⠘⠛⠆⠈⡉⠀⠀⠀⡀⡀⠀⡀⠁⠀⣠⡈⣀⠀⢀⠔⣠⠁⠀⢀⡴⠼⠁
⠀⠀⠠⡀⠀⠈⠢⠌⠃⠨⠼⠃⠀⠀⠀⠀⠀⣠⠋⡍⠛⠷⠶⠒⠊⠀⠠⠩⠷⠄⠊⠡⠀⡔⠉⠈⠈⠀⠁⠒⠤⣀
⠀⠀⠀⠈⠂⠀⠀⠀⠨⢐⠠⠄⡀⠛⠒⠂⠀⠀⠀⠡⣁⠀⠀⠀⠀⠄⠀⠀⠀⡠⠐⠀⠰⠃⠀⠀⠀⠠⢀⠀⠀⠀⠉⠐⠠⢀⡀
⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠁⠀⠀⠁⠠⠀⢀⠀⠀⠀⠐⡒⡂⢠⠠⠀⠀⠁⠀⠀⠀⠀⠒⠤⣀⠀⠀⠀⠀⠁⠢⢠⣀⠀⠀⠀⠁⠂⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠈⠉⠒⠭⣑⠲⢤⣀⠀⠐⠊⠒⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠂⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠢⢍⡑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⠠⠄⠀⠀⠀⠀⠀⠀⣀⡠⠤⠴⠒⠒⠒⠛⠋⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠈⡇⠈⠻⠀⠀⡀⠀⠀⢀⡠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠂⠀⣀⠂⠀⢈⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⢀⢹⠀⠠⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠛⢀⠃⠱⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⢀⠄⠈⠀⠀⠀⡀⠀⢸⠠⠀⠆⢀⣧⣀⡀⠀⠄⣀⡀⠀⠀⠀⠈⠑⢤
⠀⠀⠀⠀⡰⠁⠀⠀⠀⠐⠃⠘⡄⢀⢐⡀⢅⡌⠡⠀⠈⠲⣙⠁⠐⠀⠀⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠊⢀⣤⣍⠈⢱⢤⠝⠃⠿⠀⡀⠁⠋⠛⢳⡤⠤⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠠⠀⠀⠀⠃⠂⢆⠀⠀⠀⠄⣅⠈⠈⢀⡠⠮⠁⢠⡉⠀⠀⠁⢀⣨⠘⠀⠀⠀⠀⡆
⠀⠀⠀⠀⢇⠀⠀⠘⠣⠬⠴⠃⣀⡉⢈⠥⠙⠷⣤⣄⣀⣼⠅⠐⠁⠁⡏⠀⠠⠂⠀⠀⡰
⠀⠀⠀⠀⠈⢦⠀⠀⠀⢀⡠⠈⠀⠁⢀⠀⠐⠚⠀⢠⡈⠤⠅⡀⣤⠄⡀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠑⠀⣄⠀⠄⠀⠄⠀⢤⠄⠀⠀⠀⠀⠁⠘⠁⠀⠀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣀⡀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⢀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠟⠛⠛⣿⣿⠛⠛⠻⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠉⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠉⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠃⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⠀⠀⢀⣴⣿⣿⠿⣿⣿⠿⣿⣿⣦⡀⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣄⣴⣿⣿⠟⠁⠀⣿⣿⠀⠈⠻⣿⣿⣦⣠⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣟⠁⠀⠀⠀⣿⣿⠀⠀⠀⠈⣻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣦⣤⣤⣿⣿⣤⣤⣴⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⡿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠔⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣤⢤⡀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠆⠹⣆⠀⠀⣠⠟⠁⠀⠀⢰⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣰⠏⠀⠀⠐⢀⡤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡟⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠀⣀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢠⣴⢃⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠔⠚⢳⡈⣧⢘⡇⡈⠉⠉⠉⠉⠓⠲⠖⢀⣴⣋⠁
⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⠑⢭⣳⣷⠐⠂⠀⠀⠉⠀⠀⣴⠏⠈⠙⠫⡀
⠀⠀⠀⠀⠀⠘⡘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⢿⢦⠡⠀⠀⢀⠰⢼⠃⠈⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠹⡶⢥⣀⣀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠀⢀⣠⠾⡾
⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠈⠉⠛⠒⠒⠶⠦⠤⠤⠤⠤⠤⠤⠒⠒⠒⠛⠉⠉⠀⡸⠁
⠀⠀⠀⠀⠀⣞⠛⠁⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠉⠓⢢
⠀⠀⠀⠀⠀⠈⠓⢤⣀⡀⢳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⢀⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠭⣛⠛⠶⠤⢤⣄⣀⣀⣀⣀⣀⣠⣤⠴⠾⢟⡶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠠⠠⠤⠤⠤⠤⠤⠤⠒⠒⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⢚⠛⡦
⠀⠀⠀⠀⠀⠀⠎⠉⠉⠙⠢⢄⣀⣀⣀⣀⠀⠀⠀⠀⢀⠤⠚⠉⠐⠄⠠⠀⢀⡎
⠀⠀⠀⠀⢀⡴⠀⠀⠀⣠⡇⠀⠃⠱⠀⠈⢧⠀⡠⠚⠉⠐⠈⠀⡄⠁⠀⢀⡞
⠀⠀⠀⠀⠏⢀⣠⠞⠉⢩⠃⠀⠀⠀⠀⢀⠼⠋⠀⠀⡀⠀⠑⠊⠀⠀⣐⣏⡠⠤⠒⠊⠙⠦⠒⠒⢄
⠀⠀⠀⢰⡴⠁⠀⠀⢠⠃⠀⠀⠀⢀⡴⠋⡀⢀⠨⠀⠤⠈⠠⠀⢐⡰⠋⠀⠀⠀⡠⣤⠶⠒⠒⠒⠺⠒⣄
⠀⠀⠀⠈⠀⠀⠀⠀⡆⠀⢀⠄⡠⠎⠀⠀⠈⢈⡠⠁⠀⠂⠀⣠⠎⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⣆⠜⠁⠀⠀⢀⡐⠂⠀⠀⢀⣠⣜⡡⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠏⠀⠀⠀⠠⠂⠀⢀⣠⠴⠚⠉⠁⠀⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⢠⠋⠀⡏⠠⠀⠀⠀⣀⡤⠚⠋⣁⣀⣀⣠⣴⣞⣉⡤⠤⠤⢤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⡌⠀⢸⠃⠀⣠⢴⣊⠥⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠒⠒⢒⠦⠴⢄
⠀⠀⠀⢀⣀⣴⣡⣆⣧⡤⠞⢋⣁⣠⠤⠴⠖⠒⠚⠛⠋⠛⠙⠙⠓⠒⠒⠶⠤⠤⠤⠤⠤⢤⣀⣸⠀⣄⣈
⠀⠛⠛⠻⠲⠶⡦⠤⠶⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠚⠉⣠⠖⠒⠛⠉
⠀⠀⠀⠀⠀⠀⢊⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⣀⠤⠒⠋⠁⢀⣀⠴⠚⠁
⠀⠀⠀⢈⣤⣤⡵⠊⠉⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠐⠒⠒⠋⠉⢁⣀⣠⠤⠖⠊⠉
⠀⠀⠀⠀⢀⡤⢭⡹⠋⣠⠤⠖⠛⠓⠒⠒⠒⠶⠶⠖⠒⠒⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣆⡀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀⠁⢾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢂⡀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡈⠻⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣍⢻⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣤⣿⢹⣟⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣸⡏⡾⢸⢿⣟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢋⡾⣡⠏⣼⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡴⣫⡴⣫⣾⣛⣛⠷⣤⡲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡲⢾⣉⡞⣵⢫⡞⠋⢩⠾⠿⣿⢮⣻⣯⠍⣙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣈⠳⣬⣙⠒⢸⡇⣧⡻⢮⣽⢳⡎⠃⣼⣛⠿⢧⣛⠩⠶⢋⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣶⣿⣿⣤⣿⣮⣉⣻⣿⣟⣥⣞⣩⣿⣻⣶⣼⣷⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣦⣤⣀⣉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⢉⣀⣠⣤⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⡀⣠⠖⠓⠒⢒⡦⠒⠒⠒⠒⠚⠉⠉⠳⠤⣀
⠀⠀⠀⠀⠀⣀⣠⠊⠁⢠⠞⠁⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣆
⠀⠀⠀⠀⡼⠉⠁⠀⠀⢧⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⡔⠒⠚⠉⠉⠛⢤⡀
⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠈⢱⡀⠀⠀⠸⡀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⢧⡀
⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⣷⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠓⢢⠀⠀⢰⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠈⡆⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣳⡀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣄⣀⠀⠀⠘⠦⣄⡀⠘⣄⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠲⡄⠀⠉⣶⠚⡄⠀⠀⠀⠀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣤⠤⠟⠀⡇⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠶⠒⠒⠒⠒⠒⠶⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⣠⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠒⠋⣁⣠⣤⡶⠶⠶⠶⠒⠲⠶⠶⠶⢤⣄⣈⠙⠲⢤⣀⠀⠀⢀⠀⠀⠀⣀⣴⣾⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢁⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⡈⡻⠆⠁⢀⣠⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⡾⠛⠁⠀⠀⢀⡴⠒⡛⢛⡭⠟⠽⠿⠉⣻⣶⣤⣤⣀⠀⢀⠀⠉⢀⣠⡾⢿⡏⠁
⠀⠀⠀⠀⠀⢀⡴⠃⣠⡾⠋⠀⢀⣠⣶⣿⡿⡘⢱⣧⣍⡓⣉⣠⡯⠙⣟⣁⠠⡟⠷⠄⢀⣠⠶⠋⠁⠑⠄⠙⢦⡀
⠀⠀⠀⠀⢠⡞⠀⣴⠋⠀⢀⣴⡿⢫⠍⡽⡋⠀⣠⠀⠟⢳⠀⣩⠗⣚⠈⢩⣅⠉⣠⠴⠛⠋⢀⡀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⢠⡟⢀⡾⠁⠀⢠⡾⢩⣤⡔⠛⢳⢿⣿⡟⢀⠰⡶⢊⢡⡎⠀⢘⡈⠩⠇⣿⣴⣄⣸⡞⡛⢆⡀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⡾⠀⣼⠁⠀⢠⡟⡤⣷⣶⡀⣸⠟⣁⣠⠏⣸⡔⠡⢤⡞⠁⢠⣤⣽⠟⣰⡧⠞⠉⣿⣿⡁⠀⣷⠀⠀⠀⠀⠈⣷⡀
⠀⠀⣸⠃⢰⠇⠀⠀⣾⠠⢁⠬⠁⣏⡉⠉⠉⠁⠀⢼⡁⠲⣶⠾⠿⠛⣋⣤⢞⡅⢀⡶⢌⠋⣈⢧⣶⣿⡀⠀⠀⠀⠀⢸⣧
⠀⠀⣿⠀⡾⠀⠀⢰⠓⢰⡞⢳⡄⠙⠃⣄⣀⣈⠛⡞⠳⠦⠬⠴⣶⡿⠋⢴⣿⣀⡘⠓⠀⠛⢯⢸⡏⢙⠌⠀⠀⠀⢠⠀⣿⡀
⠀⠀⡇⠀⡇⠀⠀⠀⠀⠘⣿⣷⣹⣷⣆⠈⠛⠻⣧⡈⢲⠟⠉⣉⣩⡇⡷⡘⢦⣠⣴⣂⡤⠂⣤⠷⢌⢻⣠⡄⠀⠀⠈⠀⣿⡇
⠀⠀⣇⠀⡇⠀⠀⣿⠴⣾⣿⣿⠁⠉⠀⢒⣀⡀⠈⣧⠌⢠⡊⢽⡟⢰⡧⡻⢈⠇⠠⣍⣤⠆⣤⣴⣯⡨⢾⡁⠀⠀⢠⠀⣿⡇
⠀⠀⣿⠀⠃⠀⠀⠈⠠⣿⠁⠙⡀⢲⠀⢀⡿⠿⣷⡇⣐⢚⣏⣆⠀⠀⡱⡡⠊⠁⢀⣶⡤⠞⢷⣇⣿⢳⣬⠁⠀⠀⢸⠀⣿⡇
⠀⠀⢻⡄⠀⠀⠀⠀⣄⣽⡷⢾⣤⣾⣿⣖⠃⣊⣛⣁⠁⡾⠁⢿⡀⢤⡜⢓⣶⣾⡿⠛⢛⣦⠀⡷⠃⣸⠏⠀⠀⠀⡆⢸⣿⠃
⠀⠀⠈⣷⠀⠀⠀⠀⠘⣷⠆⠈⣋⠉⢋⠀⠀⣼⠉⠻⣄⣀⣲⠛⠺⣦⠀⣾⠋⣩⢤⡈⠛⣯⠺⣦⣾⠅⠀⠀⠀⡰⢁⣿⡟
⠀⠀⠀⠸⣧⠀⠀⠀⠀⠈⢧⡄⢉⡀⠉⠐⢀⢿⣷⢾⡿⠋⣸⣿⣷⠘⣷⣄⠃⢃⣾⠏⠀⣢⠆⣿⠏⠀⠀⠀⣰⢃⣾⣿⠁
⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠳⣾⣛⣛⣿⡷⢧⢮⣤⣤⠾⠛⢠⡟⠐⡃⠿⡿⠻⡇⠀⣨⣥⠾⠁⠀⠀⠀⡴⢁⣾⣿⠃
⠀⠀⠀⠀⠀⠘⢷⣄⠀⢄⠀⠀⠈⠻⣴⣟⠛⠠⢴⣎⣠⣤⡬⠡⣘⣎⣉⡷⠔⢂⣰⣆⡵⠃⠀⠀⠀⣠⠞⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠑⢄⡀⠀⠀⠉⠉⠓⠾⢿⣯⣽⣷⣭⣉⣀⣍⠀⠀⠈⠊⠁⠀⠀⠀⡠⠞⣡⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡈⠒⢤⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⢀⡠⠔⣋⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣌⣉⠒⠢⠤⠤⢀⣀⣀⣀⣀⠠⠤⠔⢒⣋⣥⣴⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣶⡖⢲⡶⢶⣶⠶⡶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⢀⣠⠶⠒⣶⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠖⣶⠒⠶⣤⡀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠾⢿⡿⠶⣦⣄
⠀⠀⠀⣰⠋⠀⠀⠀⣿⠀⠀⠀⠹⣄⠀⠀⠀⠀⣠⠟⠁⠀⠀⣿⠀⠀⠀⠙⣦⠀⠀⠀⠀⢠⡟⠁⠀⠀⢸⡇⠀⠀⠙⣷⡀
⠀⠀⢠⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢹⡀⠀⠀⢠⡏⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⣧
⠀⠀⠸⡇⠀⠀⢀⡴⣿⢦⡀⠀⠀⢸⠃⠀⠀⠸⡇⠀⠀⢀⡴⣿⢧⡄⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⢀⣴⣿⣷⣄⠀⠀⢠⡿
⠀⠀⠀⠳⣀⡴⠋⠀⣿⠀⠙⢦⣠⠟⠀⠀⠀⠀⠻⣄⡴⠋⠀⣿⠀⠙⢦⣀⡾⠀⠀⠀⠀⠹⣦⣴⠟⠁⢸⡇⠙⢷⣤⡾⠃
⠀⠀⠀⠀⠙⠲⢤⣀⣿⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠙⠳⢤⣄⣿⣠⣤⠶⠋⠀⠀⠀⠀⠀⠀⠘⠻⣦⣤⣼⣧⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⢀⣴⠟⠋⠉⣿⠉⠙⠻⣦⡀⠀⠀⠀⠀⢀⣴⡿⠛⠉⣿⡏⠛⠻⣷⣄⠀⠀⠀⠀⢀⣴⡿⠟⠋⣿⡏⠛⠻⣷⣄
⠀⠀⢠⣿⠁⠀⠀⠀⣿⠀⠀⠀⠘⣿⡄⠀⠀⢠⣿⠋⠀⠀⠀⣿⡇⠀⠀⠘⣿⡆⠀⠀⢀⣿⡟⠀⠀⠀⣿⡇⠀⠀⠈⢿⣧
⠀⠀⢸⡇⠀⠀⠀⣠⣿⣄⠀⠀⠀⢹⡇⠀⠀⢸⣿⠀⠀⠀⣠⣿⣇⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⢀⣿⣧⡀⠀⠀⢸⣿
⠀⠀⠸⣷⠀⣠⣾⠟⣿⠻⣷⣄⠀⣾⠇⠀⠀⠸⣿⡀⣠⣾⠿⣿⡿⣷⣄⠀⣾⡏⠀⠀⠘⣿⡆⣀⣶⡿⣿⡿⣿⣦⡀⣼⡿
⠀⠀⠀⠹⣿⣟⡁⠀⣿⠀⢈⣻⡿⠋⠀⠀⠀⠀⠹⣿⣿⡁⠀⣿⡇⢈⣻⣿⠟⠀⠀⠀⠀⠙⢿⣿⣏⠀⣿⡇⢈⣻⣿⡿⠁
⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠛⠉

⣿⣿⢟⠿⠛⠛⢿⠟⣻⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠢⡀⡠⠖⠒⠠⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⣽⠖⠋⠑⠫⣍⣆⣸⣤⡠⣌⣣⡀⠀⣀⣀⠄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠒⠉⠉⠀⣀⣀⣀⠀⠈⠀⠀⠀⠘⠉⠀⠀⠁⠀⠈⠐⠄⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠐⠁⡔⡞⠀⡄⠠⠀⠁⢀⣀⠀⠈⠢⠀⠀⠀⢀⠔⠉⠀⠀⠈⠐⢄⠀⠀⠆
⠀⠀⠀⠀⡠⢂⡁⠳⠀⠑⢆⣸⢤⠁⠀⡴⣿⡟⠛⣆⠀⢳⠀⢠⠃⠀⡴⣻⡿⠷⡀⠀⠆⠀⢈⣁⣈⠑⠄
⠀⠀⠀⢠⠀⡁⠇⢀⠗⠠⡴⢃⢸⠀⠀⣿⣿⣷⣶⣿⠀⢸⠀⢸⠀⢰⣿⣿⣷⣴⡇⠀⢸⠸⣮⣍⠙⢆⡘
⠀⠀⠀⠀⢄⠈⢆⠈⠫⣹⠀⡏⠸⣆⠀⠙⢿⣿⡿⠏⢀⠞⠀⠘⣆⠈⢿⣿⣿⠟⠀⢠⠇⠀⣉⠂⠡⢘
⠀⠀⠀⠀⡠⠛⣉⡡⠌⠙⣦⣡⡀⠫⢗⠤⠤⡤⠤⠖⢋⡀⠀⢀⠈⠓⠤⢤⡤⠤⠖⠁⠈⠻⢳⡖⢁⠮⠄⡈⠑⡀
⠀⠀⠀⡰⠀⠠⠁⡠⠄⢄⠈⡖⠻⢒⡀⣙⡀⠀⢶⣄⡈⠟⠛⡛⠀⠀⢀⢀⠤⡴⠀⣧⠀⢶⣄⠈⢿⢊⣩⣭⣉⠓⢤
⠀⠀⡠⠁⣠⣇⣰⠋⣅⠙⠋⡟⢉⣀⠈⢣⡈⡛⡟⠀⡏⠻⠬⠜⣩⣆⠘⠓⠚⢁⠔⢰⠀⢸⡇⣧⠈⠃⠀⠀⠈⢳⣀⡀
⠀⠙⠒⠂⢹⠀⠙⠲⢿⣇⣨⣱⠁⡴⡇⠀⠻⣿⣄⡼⢹⣖⡖⠈⠁⠨⣿⣷⡛⢉⣠⠟⠀⡼⠀⣘⡤⠖⢛⡇⡆⠀⣆⠡
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠉⠙⠚⠓⠾⠦⠤⠤⣥⣤⣞⣉⣉⣀⣩⡖⠤⡉⠉⠉⠉⠳⠚⠛⠉⠁⠀⠄⢸⠏⡁⢴⢳⠀⡆
⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢨⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⡏⠀⠐⢤⠉⠰
⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠸⠠⠔⠁⠀⠀⠀⠀⠀⠀⣸⠁⠘⠂⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠘⣆⠀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠄⠊⠀⡠⠃⢀⠀⠀⠉⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠯⠽⣒⣒⣒⣒⣒⣒⣒⣒⣪⠭⠕⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢀⡽⠛⠒⢄⡖⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢙⢷⡤⠀⣤⠀⠉⠺⠋⠒⢐
⠀⠀⠀⠀⠀⠀⠢⠤⠥⢈⠁⠀⠛⠞⠒⣶⣆⠀⠀⢻⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠏⠀⠓⠲⣿
⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁⠙⠀⠀⢒⠀⠀⠀⠀⠀⢠⢠⠀⣀⡀
⠀⠀⠀⠀⣤⠔⢄⠀⠀⠄⠀⠀⠀⠀⠀⠐⠢⠔⠁⠀⠉⠙⢞⢀⠎⢀⠀⠂⣎⡀⢶⢉⡒⠀⠀⣀⡄
⠀⠀⠀⠀⡇⠀⠘⠳⢤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡳⠛⠀⣐⢋⡉⠀⠛⠠⢁⡀⠈⠉⣀⠤
⠀⠀⠀⠡⡇⠐⠀⠀⠀⠑⠂⠉⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠘⢀⣈⣩⣥⡉⠠⠀⠀⡀⠀⠀⠀⠠⠄⠶⣴⣤⡄
⠀⠀⠀⠀⠈⠀⢀⡀⡀⠀⠀⠀⠀⠀⢶⣿⠀⠂⠀⠀⠀⠀⢀⡤⢹⠓⠘⠃⠀⣠⣬⣍⠉⠄⠀⠁⠀⠬⢿⣿⣧⣀
⠀⠐⠀⠀⠀⠠⣼⢣⡷⠚⠀⠀⠀⡠⠆⠀⠀⠂⠀⠀⠀⠀⠁⠀⣠⣄⠀⢀⣬⡀⠀⠀⢹⠍⢉⠀⠲⠶⠼⠉⠁
⠀⠀⠀⠀⠀⠀⠙⢉⣁⠹⠉⠀⠞⠁⠀⠀⠀⠀⠀⠄⠀⣣⠀⠀⠉⠁⠀⠈⠀⠀⡀⠀⠀⣴⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠉⠁⠀⠂⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠘⠒⠚⠁⠀⠀⠈⠉⠉

⠀⠀⡀
⠀⠀⣸⡍⠢⠴⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡳⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⡈
⠀⠀⢧⡁⢕⣶⣟⣋⠀⠀⠀⠀⠀⠀⢀⡴⠛⠋⠁⠈⠉⢢⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣽⣷⣿⣻⠀⠀⠀⠀⠀⠀⠀⣰⡾⠛⠉⠉⠉⢳⡼⠁
⠀⠀⠀⠉⠙⠛⠛⠻⠶⠀⠀⠀⠀⠀⡈⠃⠀⠀⠀⠀⣠⠏⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠷⠀⠀⠀⠀⠀⢸⣻⠁⠴⠶⠖⣢⣞⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠋⠓⣂⡡⠋⢁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣿⣛⣛⣻⠟⠁⡾⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠍⣱⠏⠉⠀⢀⣾⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣻⡟⠉⡕⣀⣼⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⣮⣷⣶⣿⡛⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣾⣿⣶⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠉⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠉⠠⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⢀⢦⠠⣀⣀⣈⣙⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡡⣤⣄⣉⣛⣷⣤⡀
⠀⠀⠀⠀⠰⠁⠀⠳⢀⠀⠈⠉⠁⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⢻⣦⡈⠉⠉⠉⣉⢿⣄
⠀⠀⠀⠀⠸⡄⠀⢰⠀⠁⢤⡀⠀⠈⢈⢻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⣸⡇⠉⢦⣔⡚⠋⢣⣻⣷⡀
⠀⠀⠀⠀⠀⠙⠷⣾⢄⣀⡀⠈⢱⣤⣤⢤⣭⣿⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣧⣠⣶⣉⢻⣶⣿⣿⣯⣿⣷⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠫⠏⠤⢿⣽⣿⠿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⠿⠿⣿⣿⠿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡠⠤⠤⠖⠒⠒⠒⠒⠐⡛⠒⠢⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠁⠃⠐⣎⡀⢀⠘⠷⠀⠀⠀⡘⠃⢠⡄⠀⠲⠄⢨⠶⠴⠂⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⣶⠀⠠⣄⠀⠘⠃⠀⣶⠀⠐⠇⠀⢀⠤⣄⠠⠋⡡⠖⢴⣧⡘⣒⠂⣼⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠀⢴⡄⡠⢇⠀⡠⢥⣈⠵⢒⠿⠙⡌⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠉⢡⠆⠀⠀⠀⣠⡴⠦⠀⠀⠀⠀⠀⡄⠀⢀⠈⠴⣒⡦⠂⠞⡡⠖⠊⠀⠀⣠⠔⣫⠴⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠐⠇⠀⠀⠉⢀⡘⠃⠀⠀⢀⠤⠷⢼⡬⠈⢀⠟⣩⠟⠉⠀⢀⡠⠖⠋⢀⣀⡑⣌⠓⢌⢢⡀
⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠛⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⡠⠧⠈⠀⣤⣿⣩⣁⠔⠋⠀⢀⡠⠞⠉⠀⠰⡋⠀⠀⠈⠉⠀⠀⠁⠱⣄
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢰⠠⢃⡴⠚⢏⠳⣽⠵⠋⠀⠀⣠⠖⠉⣠⠔⠊⠲⢤⡈⠢⡀⠘⠳⡢⢤⣀⠐⣎⠇
⠀⠀⠀⠀⠀⡈⢠⠄⠀⠀⠠⣲⠀⠀⡀⠀⢡⢇⠜⡊⡍⡎⢣⡤⠒⠁⠀⣀⠴⠋⠁⠀⠸⡅⠀⢺⢢⡀⠉⠲⣜⢮⡢⡉⣢⠈⠉⢁⡤⠒⣄⠀⣀⣀
⠀⠀⠀⠀⠘⠀⠙⠀⠀⡀⢀⡁⡀⡀⡇⠀⡈⠞⡾⠏⣄⠔⠉⠀⢀⡴⠚⠁⢀⡀⢀⣀⠀⠸⡄⠀⠉⣡⠔⢦⡢⣳⡝⠚⢁⡠⠒⢁⢀⣫⣞⣠⣶⡬⠽⠦
⠀⠀⠀⠀⠂⡀⡀⠀⠠⠃⡇⠃⡇⢃⡄⢈⣶⡷⣰⠟⠁⠀⣠⠖⠉⠀⠀⡔⠁⢀⣀⣪⢇⡀⠹⡔⡀⢣⠀⠀⠉⠁⣠⠔⠉⠀⣀⠖⣿⠼⠁⠀⠀⠀⠀⠹⡄
⠀⠀⠀⢸⢱⡰⠘⠘⠺⢷⢁⡠⠴⢋⣀⢘⡴⠊⠀⠀⡠⠚⠁⡠⢦⠀⠀⠁⠀⡋⢠⢟⡉⠘⢦⠱⣘⠼⠀⢀⡴⠊⠁⢀⡼⢊⠵⠺⠈⠀⠀⠀⠠⡄⢢⠱⢸
⠀⠀⠀⠀⡆⠇⣀⡴⡇⠸⣈⡾⡡⢋⠔⠋⠀⢀⡴⠋⢀⠴⠋⠀⣨⠇⠀⢆⠀⠙⠺⠟⠃⠰⡘⠀⢁⡠⠞⠁⠀⡠⢔⣁⡔⠨⠀⠀⠀⠀⠐⢆⢳⠘⠌⠁⡸
⠀⠀⠀⠀⢱⢀⠛⢣⠔⠂⡬⢉⠕⠁⠀⢀⡴⠋⢀⢾⣡⠈⢠⣊⡤⢆⠀⠈⠓⢤⣄⣃⡥⠞⣡⠔⠉⠀⣀⡔⣫⠖⠉⠈⠀⠀⠀⠀⠀⡄⢳⠘⠆⠁⣠⠞⠁
⠀⠀⠀⠀⠈⢋⡐⢒⡤⢘⡔⠁⠀⢊⠔⠁⢠⠔⠉⣧⠳⡀⠀⠉⢔⣫⠋⡳⡄⠀⠀⢀⡴⠊⠁⢀⡤⡞⢉⢇⠀⠀⠀⠀⠀⠀⠀⡄⢣⢸⠀⠃⢀⡜⠁
⠀⠀⠀⠀⢰⢹⡿⣛⡰⢋⡠⢈⠔⠁⠀⠀⢀⠀⠀⠸⡀⠙⣄⠀⠈⢁⡨⠞⠁⡠⠞⠁⠀⡠⠚⠫⠐⠀⢸⢸⠀⠀⠀⠀⠠⡀⠃⢠⠘⠀⠁⣠⠊
⠀⠀⠀⠀⠀⢃⠱⠿⠙⠷⠔⠁⡠⢖⠲⢤⣈⡀⠀⠁⢃⠀⠈⢦⠴⠋⢀⠴⠋⠀⣠⠔⡩⡰⡠⡠⡀⢀⣸⢘⠐⠀⡆⢸⠀⢃⢨⠈⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠈⠎⠀⢰⣦⠀⠈⠓⢬⡓⠂⠈⠁⠀⢐⡘⡄⠀⠀⣠⠜⠁⢀⡴⠛⢱⣹⣤⡥⠥⠥⠏⠘⢛⠈⠃⡃⠙⠈⡃⠙⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠓⠤⣀⠀⠐⡣⠋⣠⠞⠁⣀⠔⣋⡡⠴⠯⠖⠖⢚⠛⢻⠛⣉⡉⢁⢡⡩⡘⠆⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⣠⠞⠁⣀⠞⡅⠉⠁⠠⢀⠀⠀⡇⢸⢰⠈⡄⠁⢃⠈⠌⠆⢁⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠉⡜⢡⢀⡶⢁⣆⠀⢸⠀⠀⠘⠀⠀⢠⠈⡀⡄⢡⠘⠌⠂⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢸⠀⠈⠒⣻⣥⠉⠀⠀⢸⠘⠀⠀⡇⢀⠀⠀⠇⠐⠀⣡⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣠⡇⠀⠌⢸⣃⠠⢠⠀⢀⢸⠀⠆⢇⠰⠘⢀⣁⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠵⠁⠀⠓⠾⠈⠉⠚⠒⠤⠤⠤⠤⠤⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠄⢀⡠⠀⠀⠠⣄⠠⠄⡠⠄⢂⠠⢀⠠⠄⣀⠠⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⡠⢄⡤⢐⡢⠴⣒⠤⠐⣊⠥⣒⡮⢑⡨⠔⣂⠥⠐⣊⠥⢐⡨⢔⡨⠔⣂⠤⢒⠭⢒⡈⠄⢒⡩⠔⣊⠥⠐⣊⠥⢐⡊⠅⢒⡠⢄⡠
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠒⠉⠀⠒⠉⠐⠊⠀⠊⠁⠐⠋⠀⠒⠉⠀⠊⠁⠐⠉⠀⠈⠀⠈⠁⠀⠉⠀⠈⠁⠀⠉⠀⠂⠉⠀⠊⠁⠀⠈⠁⠈⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣺⡟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠉⠀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⡪⠕⠊⠁⣀⠤⠒⠉⣀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣒⠭⠒⢉⡠⠤⠒⣉⠤⠔⣂⠥⠄⠺⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⡪⠕⢊⡠⠄⣂⠥⢔⣂⠥⠐⣊⣁⣤⣴⠦⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣖⠝⠊⣁⣴⠮⢙⡂⠥⠒⠈⣁⡀⠤⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⠮⢛⣩⡴⣞⠫⠕⠂⢉⡡⠤⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣴⣺⣿⣽⢿⠅⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣏⣿⢧⣿⣝⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡷⡽⣟⢯⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣴⣳⢯⣟⡾⡽⢤⣤⣤⣤⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠒⠛⠉⠉⣉⣉⣤⣤⣤⣤⣤⣤⣧⣏⣿⣾⣽⠁⢀⣀⣀⣀⣀⣀⣀⠉⠉⠉⠉⠛⠓⢲⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣉⣠⡴⣞⣿⢽⣷⣶⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡦⢤⣯⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⠂⠭⣙⡛⠛⠾⠿⠿⠶⠶⣿⣿⣷⣶⣿⣭⣟⣶⣝⣳⣽⣿⣿⣿⣿⣿⠭⠿⠷⠾⠟⠛⣛⡋⠥⢐⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠬⠅⠀⠀⠉⠉⠐⠒⠒⠂⠠⠶⠤⠤⠤⠤⠀⠀⠠⠆⠀⠀⠠⠆⠀⠀⠐⠒⠒⠈⠉⠁⠀⠀⠨⢽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡒⠒⣂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠔⢉⠄⠒⡒⡀⠉⠪⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡬⠐⠈⠀⡜⠁⠈⢣⠁⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠹⡀⣼⣀⠈⠁⠀⡤⢀⠌⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠑⢕⠢⢈⣉⣀⡉⠄⡪⠊⠀⠀⠀⠀⠀⠀⢀⡀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡒⠤⠤⣀⣀⣀⠀⠀⠀⠉⠒⠂⠐⠒⠉⠀⠀⣀⣀⠀⠤⠄⢀⣩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠶⠤⠤⢭⣭⣭⣭⣭⣭⣭⣭⡥⠤⠤⠶⠖⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⡠⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣁⡤⠄⠀⢀⣤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⡠⠊⢉⢠⡄⣀⠔⠁⠀⠙⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣵⡶⠿⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡡⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⢠⣤⢴⡾⣿⠶⠚⠛⠶⣶⣤⣄⣀⣤⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢁⡴⢋⠔⠁⠀⣰⠤⠿⠛⠛⠛⠛⠛⠓⠒⠛⠛⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⡴⠟⠡⠃⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⢀⠄⠊
⠀⠀⠠⣊⠔⠁
⠀⠀⠈

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠊⠉⠉⠉⠒⠤⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣊⠭⠥⠤⣀⡠⠔⠒⠈⠉⠒⠲⠤⣀⠀⠈⠑⠒⠒⠒⠦⣀
⠀⠀⠀⠀⣠⠔⠒⠋⠀⡞⠀⠀⠀⠀⠈⠓⠤⣀⣀⠀⠀⠀⠀⠀⠉⠈⠉⠑⠲⢄⠀⠀⠉⠒⠢⢄
⠀⠀⢀⡜⠁⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⢱⡀
⠀⡴⢻⠀⠀⠀⠀⠀⠀⠀⢉⠗⠒⠒⠈⠉⠀⠀⠀⠀⢸⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⣀⣗⠒⠒⠦⡀⢱⢄
⡞⠀⢈⡇⠀⠀⢀⠀⠀⠀⡎⠀⠀⠀⢀⡠⢤⡀⠀⠀⢸⠀⠀⠘⡶⠴⠀⠀⠀⠀⠉⠀⠀⠙⠂⠀⠈⢏⠀⠙⡄
⣧⠚⠉⠘⠦⠤⠞⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⡇⠀⠀⡈⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⡇
⡇⠀⠀⡤⡔⠋⠉⠉⠉⠓⠚⠉⣠⠚⠉⢳⠀⠑⠤⠴⠃⠀⠀⠀⠀⠉⠲⠤⣀⠴⠊⠉⢱⣀⡤⠃⠀⠀⢣⡀⠈⠢⡀
⠘⠦⣀⠀⠸⢠⠖⠒⡶⠒⠒⠊⣡⠴⠒⠁⢀⣀⡀⠤⠤⠀⢀⣀⣀⠠⠤⠚⠁⠀⠀⢀⠞⠀⠀⠀⠀⠀⡄⠘⢆⠀⡇
⠀⠀⠈⢢⡀⠇⠀⠀⠁⠀⡰⠋⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠚⠁⠀⠀⢲⠀⠀⠀⠁⠀⠘⡄⡇
⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠓⠒⢴⠁⠀⢀⠤⠀⠠⠤⠤⠤⠤⠤⠒⢆⠀⠀⠀⡌⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⢠⠃⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⢠⠖⠀⠀⠀⠀⠀⠀⠁⠀⢠⠃⠀⠀⠀⠀⣰⠁⠈⢳⠔⠊⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⠢⠤⠼⠤⢄⣀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠁⠀⠀⠸⠁⠀⢠⠃⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⣀⣀⣀⣀⠀⠀⠀⡱⠀⠀⠀⣠⠔⠒⠉⠓⣖⠒⠒⠉⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠉⠓⠯⣀⡀⢀⡜⠁⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⡇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⠀⠀⠀⠈⠉⠉⠑⠒⠤⣀⠀⠀⠈⠢⢄⣀⡠⠄⢀⡼⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⠑⠒⠤⣀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠒⠒⡖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⢹⠦⣀⣀⣠⠤⠤⠤⠤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡀⠀⠘⠛⠃⠈⠛⠛⠛⠛⠃⠘⠻⠷⠆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠈⠡⠚⡄⠈⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠡⠄⠦⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣨⣴⡶⠦⠀⠀⠀⠀⠀⠤⠤⠄⣀⣤⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣭⣿⣛⣿⣷⣶⣶⣶⣶⣿⣟⣛⣻⣿⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢫⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡿⣻⢻⠟⠛⠛⡿⣛⢻⢫⣜⡋⡿⡽⡿⢼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣾⣇⣏⣼⣼⡹⣱⣫⣞⣻⣦⣹⣧⣶⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡿⣾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣾⣯⣴⡿⠛⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⢻⣿⣿⣿⡶⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⠭⠛⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢰⢳⡄⣶⣰⡆⢰⣶⠘⡟⢸⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠛⠁⠛⠛⠃⠚⠙⠃⠋⢈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣗⢠⡶⣶⡄⣶⠲⡆⢺⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⣀⠀⠀⠛⠟⠉⠿⠾⠃⠹⠿⠃⠘⠁⠀⢀⡠⠖⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⡉⠓⠂⠤⠤⠤⠤⠤⠤⠤⠤⠖⠒⠋⡁⢰⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠐⠢⠤⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠠⣸⣹⠇⠀⠀⠀⠀⠀⢀⣴⡧⠁⠸⣿⣎⠈⣿⡟⣧⠘⣷⡿⣶⣞⠀⠁⡚⣸⢻⣿⣿⣿⣿⡟
⠀⠀⠀⢀⠀⣠⣤⣤⠹⣿⣼⣿⣦⣰⣾⣶⣿⠂⠀⠀⠀⢻⣿⣷⠘⠏⣿⡎⣿⡇⠈⠛⢅⣿⡇⡿⣿⢿⣿⣿⡿⢦⢀⡀⠀⡀⣨
⠀⢀⣠⣾⢼⣿⣿⣿⠑⠜⢿⣿⣻⡙⢻⣿⠎⠂⠀⠀⠀⢸⠘⣿⣧⣤⢻⡷⣿⠂⠀⠀⢠⣿⢫⢷⣿⣿⣿⣿⠁⠀⢻⣷⡿⣫⠟
⣀⢀⢙⣽⣼⣿⣿⣿⡧⢀⣜⣿⡯⢿⣧⡈⠲⣤⣿⣆⠀⠘⣇⠘⢿⣿⢸⣟⣿⠀⠀⢀⠟⣿⣾⣺⡏⣿⣿⡏⠀⢼⠟⢉⠿⣷⣀
⢻⠟⣛⣩⣿⡿⣿⣿⢿⠀⢿⡾⣿⣌⣿⣷⣄⠈⠛⢿⣆⠀⢻⣤⡈⣿⣿⡇⣿⡇⠀⡼⠀⣿⣿⡟⣸⣿⠟⣒⢷⣇⡲⠁⠀⠀⠑
⣴⢺⠟⠉⠀⣿⢿⣸⡌⠃⠀⠀⠈⠙⣏⣿⣿⣷⡄⠀⠉⢳⡈⣿⣧⡈⣿⡇⣿⡇⣰⠃⢰⣿⡿⣱⣿⣿⣿⣾⣽⣿⣯⣐⣠⣀
⠈⠪⠀⢀⠀⣿⡘⣯⢧⠁⠋⣀⠠⡑⡜⣞⣿⣿⣿⡀⠀⠀⠁⠘⣿⣷⣿⣇⣿⣇⡟⠀⣸⡿⢀⣿⢷⣿⣿⣿⢋⣰⠄⣀⠉⠻⣆
⠀⠀⠀⡜⠀⢸⣷⣿⡏⣧⢰⠸⡌⢧⢱⠘⣼⣿⣿⡇⡄⠀⠀⠀⠘⢿⣿⡿⠹⣿⡇⠀⣽⢣⣿⡟⣹⣿⣿⢏⣾⢇⣾⣿⠀⠀⡼
⠀⠀⠀⠀⠀⠘⣿⣿⣷⣿⢘⡀⢫⠉⡌⢧⢹⣞⣿⣧⣿⡒⢠⣶⣶⣤⣿⣿⣿⢿⠇⠀⡟⡼⢏⡸⣿⣿⣏⣼⣏⠌⣿⣿⡀⡰⠁
⠀⠀⠀⠀⠀⠐⠋⣵⣿⣿⣇⢹⡈⡧⠼⡽⣆⢽⡞⣿⣿⣿⣿⡛⠯⠟⢛⣛⡛⠛⠃⠀⢬⣣⣼⣿⣿⣿⣿⡟⡜⠀⡿⠫⢠⠇
⠀⠀⢀⣴⠅⠀⡼⡿⣿⣿⣿⡂⠳⢘⡥⢪⠹⣊⣿⢻⣿⣻⣿⣧⠀⠀⠀⠨⣻⣿⠙⣳⡶⢿⣿⠿⢿⣿⢋⠎⠀⠀⣠⠀⡘
⠀⢰⣯⣿⢀⡾⠀⢻⡍⠟⣿⣷⡈⠄⢻⠄⠣⢰⡸⡿⣿⣟⢿⣿⡆⠀⠀⢈⣿⣿⡀⢺⡏⣾⣿⣷⣷⣵⡁⠀⣠⣾⢯⠀⡇
⠀⢸⣿⡏⣗⡇⠀⠈⣟⡜⠸⣿⣷⢩⠄⣧⠘⡭⢆⢿⢻⣿⣸⣿⣿⡀⠀⠀⣾⣿⡐⡸⣡⣿⣿⡿⠃⠈⢓⣻⣿⣿⣧⢻⠁
⠀⢸⣿⣷⣿⡇⠀⠀⠸⣷⠐⢹⣿⡎⠥⠘⡥⢻⠵⣊⢿⣿⣿⣿⣿⡇⠀⠀⢾⣿⣷⣃⣿⢿⠟⠁⠀⢠⣿⣿⣿⣏⢾⣿⠃
⠀⠐⣿⣿⣿⡇⠀⠀⠀⢹⣷⡀⢿⣷⠩⡏⢷⠈⢧⠵⡊⢿⣿⣿⣿⣿⠀⠀⡬⡿⢛⣫⠔⠁⠀⠀⢀⣾⣿⣿⣿⣟⣾⡏
⠀⠀⢻⣿⣿⣧⠀⠀⠀⠀⢻⣧⠸⣿⣇⠽⠄⢏⠉⡧⢒⡨⣿⣿⣿⣿⡇⠀⠈⣱⣿⠇⠀⠀⠀⠀⠜⡟⣱⡿⠭⡎⣼⣧⣴⠃
⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠈⢿⣵⡽⣿⣆⢯⠽⡇⢹⡄⢗⠹⣿⣿⣿⣷⠀⢀⣿⣿⠀⠀⠀⢀⠆⠀⢠⡏⠀⠀⣰⣿⣿⣿⣶⠞
⠀⠀⠀⣿⣿⢹⠀⠀⠀⠀⠀⠀⢻⣾⡝⣷⡌⢧⠸⡀⢧⡘⡏⣫⣿⣿⣿⣇⣰⣿⡇⠀⠀⠀⠂⠀⢔⠎⠀⠀⣴⡟⠁⢈⠞⠁
⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡭⢧⢻⠈⣇⠹⡺⣿⣿⣿⣿⣼⣿⡇⠀⠀⣰⣴⣿⣿⡣⢀⣼⣿⢇⡴⠃
⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣿⣽⣂⢣⣸⡎⢣⣿⣿⣿⣿⣿⣿⠇⠀⣬⣿⣿⣿⣷⣷⣿⣿⣿⡿⠁
⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣯⣿⣮⡇⢽⣸⢽⣿⣿⣿⣿⣿⠀⢀⣿⣿⣿⣿⣿⣿⣿⡿⡿⠁
⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣝⣧⢻⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⠋⡴⠁
⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣾⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⠙⢹⠁⡼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⡶⠶⠟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⠶⠿⢛⣷⣀⠀⠀⢀⣀⣠⣤⣶⠶⠟⣛⣻⣭⣧⣤⡶⠶⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⡯⡆⠛⢀⡚⠩⡬⢻⣷⣿⣛⣭⣭⡶⠶⠿⠛⣛⣉⣩⣥⣤⣶⠶⠾⠟⣛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣩⡥⠛⠯⢀⡙⠣⠴⠀⣿⣟⣭⣥⡶⠶⠾⣟⣛⣻⣭⣽⣷⠷⠶⠟⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣥⡚⡍⣩⡤⠷⣶⣞⣫⣭⣭⣷⠶⠾⠟⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⠿⠾⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣤⣤⡶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣠⣴⡿⠙⢛⡛⠉⠁⣀⠙⣛⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡿⠟⠛⣋⡡⠆⡀⠉⠩⣤⡄⠙⠁⠉⣡⡝⠿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠷⠶⠿⠉⣀⣤⠆⣉⠀⡈⠋⠀⠶⠶⠤⠋⢋⠀⠐⠷⣤⡌⠛⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⢋⣤⡿⠉⠁⠛⢃⣈⣈⠺⢿⡟⠋⠘⣉⠉⢧⣤⡃⠀⠙⣂⡀⠻⢷⣶⡆
⠀⠀⠀⠀⠀⠀⢰⣿⠛⠡⠟⠉⢀⣐⣛⣁⣀⣈⣉⣀⣐⣒⣤⣀⣛⣋⣀⣀⣈⣁⣀⣀⣉⣁⡀⠳⠈⣻⣆
⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⣧⠉⠋⠀⠛⠃⠀⠚⠂⠒⠒⠀⠒⠒⠀⠲⠶⠀⠲⠖⠀⠶⠶⠀⠰⠶⠀⠐⠒⠂⠀⠛⠃⠘⢻⣿
⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠛⠿⠶⠶⣦⣤⣤⣤⣤⣤⣤⣤⡶⠶⠿⠛⠋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠷⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠛⠁⠀⠀⣀⣤⣴⠶⠾⠟⠛⠛⠛⠶⠶⢦⣤⣀⡀⠀⠈⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢀⣠⠶⠛⠉⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠉⠛⠷⣤⡀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⢀⡴⠛⠁⠀⢀⣠⡴⠞⠛⠉⠉⠉⠉⠉⠉⠛⠒⠶⣄⡀⠀⠈⠙⢦⣀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⣴⠋⠀⠀⣠⡶⠋⠁⠀⠀⣀⣀⣠⣤⢤⣤⣤⣀⠀⠀⠀⠙⠷⣄⠀⠀⠙⢦⡀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⣼⠃⠀⢀⡾⠁⣠⣴⣾⣯⣀⠀⣠⡶⣿⡏⣼⡟⠁⡠⠀⠀⠉⠙⠲⣤⠖⣤⢬⣷⡄⢀⣈⣳⡀⠀⠘⣷
⠀⠀⠀⠀⢰⡏⠀⠀⣾⠁⢸⣿⣿⣿⣿⣿⣿⡁⢡⡈⣷⠜⠓⠚⢳⣾⣿⣷⡀⠀⢸⠀⠃⣜⣹⣿⣿⣿⣿⣷⡀⠀⠘⣧
⠀⠀⠀⠀⣿⠀⠀⣸⠇⢀⣀⣿⡅⠀⢰⠏⠉⠃⣰⡏⠓⢄⠀⠀⣿⠿⠛⠉⠙⣦⣞⡲⣤⣟⢸⣿⢿⡟⠋⠈⣧⠀⠀⢻⡀
⠀⠀⠀⢰⡇⠀⢀⣿⠚⣩⣼⣷⣶⣶⡏⠀⠀⣼⣫⠀⠀⠈⢣⠀⠁⠀⣠⠖⠛⠷⠶⣮⣵⡼⠟⠁⣈⡧⣀⠀⢻⠀⠀⠸⡇
⠀⠀⠀⢸⡇⠀⠀⡏⣼⣿⣿⣿⠋⢸⣧⣴⠖⣾⠃⠀⣄⢢⣠⠇⠄⠴⣁⡀⢰⡆⠀⣠⠙⠇⡆⠘⣇⠀⣸⣄⣼⠁⠀⢐⡇
⠀⠀⠀⢸⡇⠀⠀⣷⠹⣿⣿⠃⠔⠁⡿⠃⢀⠇⠀⠀⣿⠏⠀⠀⠀⠀⠀⠹⡄⠁⠀⠙⠘⠛⠧⠤⠞⠀⣏⣤⢘⠇⣴⠿⠋
⠀⠀⠀⠈⣷⠀⠀⢻⣀⡤⠘⠧⢤⣀⠿⠒⢺⡀⠀⡰⣿⠀⠀⠀⠀⠀⠠⢂⡇⠀⢀⣾⡋⠂⠀⣀⣤⠜⠸⢃⡜⢠⠞⣉⠙⢇⢀
⠀⠀⠀⢀⣽⡊⣠⠜⢿⠀⠀⠀⣰⠃⠀⠀⣾⡷⠛⠛⠻⡶⠛⠓⢄⠀⠀⣸⣥⡀⣼⠀⠻⠉⢭⠀⣸⣄⣠⡍⠙⠃⠾⠋⣠⠇⢸
⠀⠀⠀⠊⠀⡟⠁⠀⣨⠟⠚⠉⠉⠉⢙⣿⡈⣇⠀⠀⠀⠘⣧⡄⡼⠛⢱⡋⠀⠉⠋⠑⠲⣶⡞⢁⡀⠀⠙⡄⠶⠀⡴⠊⠁⠆
⠀⠀⠀⠀⠀⠁⡀⣸⠁⠀⣴⣶⡿⠿⠟⣛⠛⠉⠑⠠⢀⣀⠈⢿⣧⣤⣤⣃⠀⠀⠀⢰⡆⣽⣧⡘⠓⠀⣰⡇⠀⡰⠃
⠀⠀⠀⠀⡠⢸⠀⡏⠀⠀⣿⠁⠘⠓⠀⠉⢸⡆⠀⠈⠀⠉⠑⢄⠐⠀⠀⠈⢳⠤⣄⡈⠀⠙⢿⣿⡦⠚⠛⠶⣞⡥⠖⠲⡶⣄
⠀⠀⠀⠀⠱⠈⠀⣧⡀⠀⢻⡄⠀⢠⣤⣶⣾⣇⠰⠦⠀⠟⢀⢸⡆⠀⢀⠀⡄⠀⠀⣧⣀⣶⣞⡙⢿⠀⣠⡀⠉⠀⣸⣇⣼⡞
⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⠾⢁⣄⡈⠧⣤⣶⣿⣤⣐⣲⣿⣡⡼⢇⠀⣸⢀⡇⢰⢀⣿⠟⠉⣍⡙⢮⠁⢉⣵⣏⠹⠿⠛⡉
⠀⠀⠀⠀⠀⢀⠀⢀⣼⣇⣀⣀⣠⡿⣳⣿⣿⣿⣿⣿⠛⣍⠁⢠⠘⡇⣿⣼⢁⡇⣼⣷⠛⡆⠙⠛⢸⡤⠋⠀⠈⣧⣿⠀⠇
⠀⠀⠀⠀⢀⡏⣰⣿⣿⣿⡟⠉⠀⠙⣾⠟⣿⡏⣤⡜⣧⠗⢒⠢⡄⣿⣿⣿⡿⣿⣿⣇⣰⣷⣦⣤⣼⠦⠈⠀⣠⠟⠁
⠀⠀⠀⠀⠈⠇⢿⣟⣻⠏⣰⣾⠡⣯⣉⣇⠉⠐⣦⣤⠿⠀⠈⠑⣇⠛⢉⣩⡿⢯⠠⠀⣿⣿⣿⣿⣿⣤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢨⠇⣼⣿⣿⠀⠻⡀⠉⠉⢻⠁⠀⣠⡀⠀⢀⡈⣻⢷⣎⠳⡖⣶⠀⡟⠉⠉⢠⣿⣿⠃⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠻⡿⠏⠀⠀⠙⣄⠀⢸⡀⠀⣿⢇⠀⢠⠋⠁⢀⣿⢀⡇⣯⡼⠃⠀⡰⠋⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⣛⠓⢦⣑⠶⠃⠘⠶⠋⠀⠀⡞⡞⢸⠀⢀⣠⡴⢾⣷⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⠀⠐⣧⣀⠀⠈⠙⠓⠲⠤⠤⠤⠼⣇⠣⣌⣻⠿⠛⣉⣩⣍⢹⣜⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⣭⣉⣒⣒⠲⠤⠤⠄⠀⠀⠈⠙⠚⢛⣋⣉⣡⣤⠽⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠠⢈⠓⡄⠀⠀⢠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⢀⠀⢴⠿⣶⡀⣰⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⢠⡏⠼⣿⣿⣳⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⣿⣿⡆⠁⡽⣤⣼⣽⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠇⣿⣿⠃⢘⠃⢀⣐⣗⣲⠺⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠄⠀⠀⠀⠘⣿⡏⣤⠈⡃⢸⠁⠭⠽⣯⣟⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢆⠀⠀⠐⣿⠧⠬⢁⡍⢉⣁⣭⣉⣟⣛⡺⣿⡇⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⠡⡀⠀⠛⠐⠓⠀⠃⠐⠖⠺⡴⠧⣽⢍⣻⡇⠀⠀⠀⢀⣴⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣟⣄⠀⢘⡃⣘⠁⢐⣋⣉⡿⠲⠷⠶⢾⠀⠀⠀⣠⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡇⢨⡅⣨⠀⢈⣁⣀⣋⣈⣛⡛⢻⠀⠀⣰⣿⢿⡿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⢠⡀⢴⠀⢠⡅⢨⣏⣉⣉⣉⣛⠀⣰⣿⡏⣼⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡴⠂⠶⠀⠰⡦⢤⣍⣤⣽⣩⣉⣳⣿⣿⠙⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡇⠙⠀⠐⠂⠶⠦⠤⣼⠤⢯⣿⣿⣏⠀⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⠀⢚⠂⠒⠃⠐⠾⠖⠦⣿⣿⣿⣆⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣙⣉⣉⣏⣙⣺⣛⣟⣿⣿⣿⣿⠃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⠄⢠⣯⣭⣽⣭⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⢴⠀⠰⡶⠶⢶⠶⡿⣿⣿⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡄⠀⠀⢸⡇⠈⣟⢚⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⢸⡇⠠⠧⢤⣷⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⢸⡇⢹⡏⣽⣷⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠈⡇⢸⣃⣸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣷⢸⣷⡾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣷⣿⡇⢿⠛⣿⣿⣿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⠸⣷⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⠻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠘⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣦⣰⣿⣿⣷⣆⣠⣶⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⣼⣿⣿⣿⣿⣿⣯⣿⣽⣯⣿⣿⣿⣿⡆⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣶⣶⣽⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⢿⣿⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⡟⣵⣿⡏⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢀⣶⣿⣟⣿⣿⣿⣿⣿⣿⣿⡿⡟⠇⣿⣿⢸⡿⢻⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣠⣀
⠀⠀⠀⠀⠀⠀⠸⣿⢿⣳⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣿⣿⡜⠑⣣⡾⢋⣥⣶⣶⣶⣶⣾⢿⣿⣿⣿⣿⡿⣿⣿⡇
⠀⠀⠀⠀⠀⡴⣶⣿⣿⣿⡿⢋⣭⡉⠙⣿⣿⣿⡿⠋⢉⣿⣿⣿⠏⡼⠟⠫⢾⣿⡿⠟⣱⣿⣿⣿⣿⣿⣿⣯⣟⣃
⠀⠀⠀⠀⠸⣟⣿⣿⣿⣿⢡⣿⣿⣿⡀⢹⣿⡿⠁⣴⣿⣿⣿⣿⣀⠚⠓⠛⠋⣉⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣷⡄
⠀⠀⠀⠀⢀⣼⢿⣿⣿⣿⠈⠿⠟⢿⡇⢸⡿⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠁
⠀⠀⠀⠀⣿⣿⡷⣿⣿⣿⣷⣦⣴⣿⠀⡼⢁⡾⢋⣭⠉⢻⠟⢉⣥⠉⢩⡿⢋⣭⡍⢉⣿⠋⠹⡋⠉⣿⣿⣿⡿⣫⣶⡄
⠀⠀⠀⠀⠙⢟⣽⣿⣿⣿⣿⣿⣿⣿⠀⠁⣾⠁⠾⢛⣠⠏⢠⣿⡟⢀⡿⠁⣾⡿⠀⣾⠏⢠⣾⠁⣼⢟⣿⣿⣇⣿⣿⡇
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣼⣿⣦⣄⣩⣥⣶⣤⣭⠀⣨⣶⣤⣩⣥⣈⣥⣠⣾⣧⣀⣵⣿⣿⣿⣿⣿⡉
⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡻⠿⣿⣿⡿⢋⡄⢰⣿⠛⢫⣭⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣼⣭⡻⣿⣿⣿⣿⡿⣫⣞⡟⡱⢢⣿⣿⠀⣾⠃⣼⣿⣦⠳⣼⣿⣷⣜⣿⣿⣿⣿⣿⡿⣿⣿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣽⡿⢟⣋⣴⣿⣿⣿⣤⣩⣼⣿⣿⣿⣷⣦⣭⣽⣿⣿⣿⣿⣿⡟⡿⣵⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⣿⣿⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣽⣽⡟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⢻⣿⣿⡿⣿⢟⣿⣿⣻⣟⢿⣿⣿⣿⠻⢿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠻⣿⣿⠟⠈⠳⠿⠿⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠟⠛⠋⠉⠉⢉⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣯⣭⣭⣟⣛⡿⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣫⣵⣶⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣭⣽⣛⣛⣻⠿⣿⣿⣿⣿⣶⣭⡻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣮⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣫⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣾⣭⣟⣛⡻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣛⠛⠋⠉⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣮⣛⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣛⣵⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣶⣶⣾⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣭⣭⣭⣭⣭⣭⣭⣶⣶⣦⣤⣴⣿⣿⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠏⠉⣉⣛⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⣛⡉⠛⢿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣧⠠⡙⠿⣦⣿⣿⠋⠀⣶⣦⠉⠻⣿⣿⡍⠉⢱⣿⣷⡆⣶⣿⣗⠀⢸⣿⡇⠃⣼⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠋⣷⣮⡠⡙⢿⡇⠇⠀⣿⣿⢀⡀⢹⣿⢇⡄⢸⣿⣿⡃⣻⣿⣇⠆⠈⣛⣥⣾⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣄⣙⠻⠟⠁⣸⣷⡀⠀⣿⣿⠀⠀⣼⣿⡇⠀⢸⣿⣿⡇⢿⣿⡿⠀⠸⢿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣭⣭⣤⣾⣿⣿⣷⣤⣈⣉⣭⣶⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣟⢻⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣾⢟⣫⣾⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⠿⠟⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣩⡾⠟⠑⠿⢿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⡁⠀⠀⢀⣠⣤⣶⣶⣶⡿⠿⠿⢿⣿⣿⣿⠿⠶⠶⢾⣭⣤⣀⠀⠀⠀⣹⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣷⣦⣭⣝⣛⣛⣻⣿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣛⣛⣛⣯⣥⣶⣿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢁⣴⣿⣿⣿⣿⠁⠀⠈⣿⣿⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣝⡻⠿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣛⣥⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣶⣯⣭⣭⣽⣟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣻⣯⣭⣭⣽⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠁⠀⠀⠀⢦⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⡀⠠⢀⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢀⡦⠀⠀⢄⡤⠀⣈⡂⠀⠀⣀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞⠛⠀⣀⠗⢊⡤⠀⢀⢌⡆⠈⢹⣦⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠤⠚⠃⠀⣠⠞⢁⡴⠟⠀⢠⣷⠂⣧⡀⠀⠈⠙⢦⠈⢧⡀
⠀⠀⠀⠀⠀⣠⠶⠡⠘⢁⣠⡶⠿⡿⠋⣠⡏⠀⠀⣠⣿⠟⠀⠈⠛⠷⣤⣀⡈⠑⠢⡙⢆
⠀⠀⠀⠀⠞⠑⣁⡰⠾⠋⠀⣠⠊⡱⠋⢀⣤⣶⠾⠋⠁⠀⠀⠀⠀⠀⠉⠛⠻⣦⣄⠘⠠⠽⢦⡀
⠀⠀⠀⠀⠉⠉⠀⠀⠀⢠⠞⠑⠂⣠⣶⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠷⣦⡀⠂⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⢊⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⢡⠀⠠⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠛⢐⢀⢀⠂⠀⣀⣁⣈⣸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⡠⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠒⠒⠉⠉⡙⠈⢩⠀⠄⠉⠀⡤⠋⢁⠀⠈⠉⠑⠒⠒⠤⢄⣀⠀⠁⠀⠉⢀⡤⠊⠁
⠀⠀⠀⠀⠀⢀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⡤⢀⣸⠠⠀⡤⠃⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⡀
⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠈⢁⣀⡀⠐⣔⠘⡄⠀⡇⠀⠐⠀⠀⠅⠀⠀⣀⠀⠉⠀⠀⠀⣀⠀⠈⠈⡄
⠀⠀⠀⠀⠸⣄⡀⠀⠀⠄⠀⠈⣡⠀⢀⠀⠀⠱⠬⠃⣙⠀⢃⡀⠀⠀⠀⠀⠀⠀⠀⠘⢂⡠⠚⠁⠀⠐⣀⠼⠃
⠀⠀⠀⠀⠀⢇⠙⠷⣦⣄⣀⡀⠀⠀⠀⠀⡀⠒⠐⠀⢀⢀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⣃⣀⣤⠴⠚⢁⡘
⠀⠀⠀⠀⠀⠘⣤⡞⡁⠉⡍⠙⡛⠓⠶⠶⡶⢶⣤⣤⣤⣤⢤⢤⢤⠤⠤⠶⡶⠖⢶⢛⠋⣉⠉⠒⠲⣤⢐⠃
⠀⠀⠀⠀⠀⠀⠈⢷⠲⠸⠀⢉⠁⡄⠬⢬⢰⢁⣁⢱⣀⡁⢊⢐⣤⠑⢀⢀⢇⣈⡱⢈⣊⡠⢧⢀⣀⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡬⠁⡈⠙⠿⣧⡸⠘⣶⢅⡊⢤⡱⣾⢸⣰⡎⠀⠈⣺⠈⢁⡾⠋⠉⠃⢭⢁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⡬⠤⣸⡏⢁⢰⡎⠠⠀⢨⣉⣿⠃⠢⠄⡴⠓⢹⡟⣇⣱⠀⠀⠨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣁⠰⠚⠛⢮⣻⡟⢶⢑⠢⡈⢻⡧⢲⠀⣤⢂⣼⣿⠡⠼⣥⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡈⠀⠰⠻⠻⣯⠳⣧⡿⢟⢁⣴⠟⠯⡀⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⣬⣵⣿⢁⣿⠨⣾⣟⣛⣦⠼⠾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠭⠉⠉⠉⠉⠉⠉⠄⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠲⠦⣄⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⢀⡠⠴⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠙⢦⡀⠀⢰⠃⠀⠀⠀⠀⠀⢸⡄⠀⢀⠔⠉⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠑⣄⢸⠀⠀⠀⠀⠀⠀⠈⣇⡰⠃⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠘⢿⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⢀⡤⠒⠒⠒⠲⠤⣄⡀⠹⡄⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⣰⡟⣀⡠⠤⠖⠒⠒⠲⢄
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠑⠿⣄⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⣰⠟⠋⠁⠀⠀⠀⠀⠀⠀⢈⠇
⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⢀⣠⠼⠒⠒⠚⠓⠚⠲⠤⣄⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠖⠉⣠⠴⠒⢳⠀⢰⠛⠒⠤⣀⠙⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣉⣒⣤⣀⠀⠀⠀⠀⠀⡴⠁⣠⡎⠀⠀⠀⢸⠀⢸⠀⠀⠀⠈⠳⡄⠙⣆⠀⠀⠀⢀⣀⣤⣾⣛⠁
⠀⠀⢀⡤⠖⠊⠉⠉⠉⠉⠉⠉⠉⠓⠲⠤⣴⠁⢰⠁⢻⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠘⡄⠘⡖⠋⠉⠉⠉⠉⠈⠉⠉⠉⠓⠢⢄⡀
⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡇⠀⠈⢧⠀⠀⡸⠀⠸⣄⠀⠀⠀⠀⠀⣷⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡇⠀⠀⢈⡷⠊⢀⠀⠀⠈⠳⣄⠀⠀⠀⡿⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡜
⠀⠀⠀⠈⠙⠒⠶⠤⣤⣤⣤⡤⠤⠖⠒⠋⠹⡀⠸⣄⡴⠋⢀⡴⢻⠀⢸⡷⣄⠈⠳⣴⡶⠁⢰⠋⠙⠒⠦⢤⣤⣤⣤⣤⠶⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⡴⠋⠀⢸⠀⢸⠀⠀⠑⣄⠀⠀⣠⠋⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠢⣄⠈⠓⠦⠼⠀⠘⠤⠶⠚⢁⣠⠾⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠈⠙⢲⠤⠤⠤⠤⢤⡔⠒⠉⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢀⣀⣤⢶⡟⠁⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠈⢿⠲⢤⣀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠈⠙⠒⠛⠛⠉⠉⠀⡜⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠈⣧⠀⠈⠉⠉⠓⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣼⠏⢸⠀⠀⠀⠀⠀⠀⢰⡏⠈⢦⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⣠⡴⠛⠁⠀⠘⡇⠀⠀⠀⠀⠀⣸⠇⠀⠀⠙⠦⣀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠖⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣀⣀⣀⣤⣶⣿⣿⣿⣿⣿⡷⣦⣄⣀⣀⣠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⢻⣿⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠻⢿⣿⣿⣾⡟⢻⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣀⣀⡀⠉⠛⠻⠿⣯⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣯⣧⣽⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣻⡿⣿⢹⣿⡿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣞⣷⣾⡷⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⣤⣌⣹⡾⢫⢫⣭⡙⣯⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⢸⣿⠿⠿⣇⠸⣿⣿⡇⣸⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢻⣈⠛⣃⣄⣿⢦⣬⣭⣴⢻⣿⣿⡿⣿⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣽⣟⣞⣿⢿⣿⡿⡿⣿⣿⣍⣿⢹⡿⣎⣷⣿⣿⡿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠙⣿⣻⡷⠿⣾⣮⣽⣿⣏⣯⣿⣿⣿⣿⡿⢿⣿⠿⡇⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⢸⣿⠏⢀⣴⠋⠉⣿⠟⠉⠉⢻⣿⠀⠘⣿⡀⠙⢾⣷⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⠏⢀⡾⢃⣴⣾⡟⢀⣿⣧⣼⣿⠀⣆⠘⣷⡀⢀⠉⠀⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⢠⡿⠁⠀⢹⣿⠁⢸⣧⠼⠿⣿⠀⠛⠀⠘⣧⡈⢷⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⡀⣾⠇⠀⠿⣿⣿⠀⢹⡷⠀⠀⣿⠀⣾⢻⣄⣸⣷⡾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠿⣦⣤⣤⣠⣿⣦⣀⣠⣾⣤⣿⣶⢿⡟⣿⢿⣾⣯⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣞⣸⣻⣷⡿⣿⣼⢿⣿⣿⣻⣿⣿⢻⣷⠿⣿⣵⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣵⣿⣿⣿⡿⢻⣿⣽⣿⣾⣷⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣟⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉

⣏⠉⣹⣛⣛⣟⣛⣛⣟⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣻⣟⣿⣟⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣠⠄⠀⣐⣂⣀⣀⣀⣠⣤⠤⠤⠤⠤⠶⠶⠒⠒⠒⠚⠚⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣛⣚⠀⡄⣇⣩⣉⣩⠩⠅⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠦⠤⠶⠧⠦⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣄⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢚⡡⢭⣍⠂⠘⠒⠒⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢰⡇⠛⠀⠠⠥⠀⢄⣍⡐⡠⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⣄⠄⠘⡇⠀⢤⡤⡄⢴⡄⠈⡆⠄⡈⡴⡂⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⣍⢀⡘⡙⠀⠄⠓⠟⠁⠓⢄⠻⢋⠤⠣⡀⠀⣀⣱⠈⢑⣤⡉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⢀⠭⢄⠁⠂⡀⠉⣐⡂⣮⠤⠶⠞⣠⣀⡀⠃⠉⢉⣀⡄⢍⡸⢼⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢀⡐⠏⡣⡁⠬⢧⡀⠀⣀⡀⠋⢐⠁⣈⡢⠊⠬⠂⠛⠶⡄⡘⢂⠂⣇⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡛⢷⣾⣤⣃⣀⣚⠂⠊⠀⠀⡀⠴⡄⢾⢗⣀⣀⣄⣠⣬⣥⡶⠾⢛⣭⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠳⢦⣭⣉⣙⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢋⣉⣉⣥⡤⠶⠛⠉⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢀⢉⠉⠉⠛⣿⠟⣿⡿⠟⠿⣿⠿⠿⢿⣬⠻⣷⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴⠿⣛⣛⣛⠿⣾⣯⣼⢯⣼⢟⡟⠖⣾⠿⣶⣽⣧⣽⡇⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⢛⣉⣙⠻⢾⡷⡿⡿⣻⢿⢻⢣⠇⠹⣟⣉⣉⡙⢷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⣉⣙⢿⡬⡇⣼⡳⣛⣾⣺⣿⡿⣧⡞⣉⣉⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⢸⣧⣹⣟⢿⣮⣻⡷⠿⢃⣿⢰⡟⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡿⣮⣝⣧⣿⢏⣴⠞⣛⣡⡾⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠿⠿⠿⠾⠿⠾⠟⠛⠋

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

⠀⠀⠀⠀⢀⠄⢄⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⢀⠐⠁⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢄⠀⠀⠀⠠⠄⠀⠀⠀⠁⢰⠅⠀⠀⠀⠀⠪⠀⠀⠀⠀⠃⢀⠀⠀⠈⠰⠨⡄⠐
⠀⢀⢀⠀⠁⠑⡄⠀⢀⠔⠉⢢⠀⠀⠀⠸⠀⠀⢀⠔⠁⠀⠐⠢⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠂⠀⠀⠀⡜⠹⠃⠀⠀⢈⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠄⠔⠩⠀⠈⠣⠀⠀⠜⠀⠀⠀⠀⠀⠀⡀⠑⠠⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⠀⠀⠀⠀⠐⠢⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡰⣆⠀⠀⣠⠈⠁⠀⠐⢁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢬⣕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡠⠃⠀⠀⠩⡌⠀⠀⠀⠊⠄⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠐
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠊⢈⠔⢁⠔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠔⠀⠐⣠⡞⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⠊⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠢⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠐⠁⠀⠀⠀⡀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠐⠉⢱⠆
⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠜⠁⠀⠀⠀⠀⢠⠀⠀⠀⠊
⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠠
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶
⠀⠀⠂⠈⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢓⡀
⠀⠄⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠠⢈⠀⠀⢀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂
⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠐⠀⠀⠲⠃
⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣐⣚⣑⣂⣁⣀
⣿⣿⡋⡛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⢶⣤⡀
⠀⠀⠀⠀⢰⣻⣛⣛⣻⣿⡆⠀⠀⠀⠀⠀⠀⢧⡘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⢧⡀⠀⠀⠀⠀⠀⣀⣤⣤⣿⣿⢿⣿⣿⡇
⠀⠀⠀⠀⢸⢹⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣀⣀⣉⣈⣁⣀⠀⠀⠀⢀⣀⣀⣀⣈⣀⣁⣀⣀⡀⠀⠀⣷⣷⣶⣾⣿⣿⠟⢻⡇
⠀⠀⠀⠀⢸⠘⠻⣶⠙⠛⡇⠀⠀⠀⢸⣯⣤⣤⣤⣤⣤⣬⡷⠀⠀⠻⣼⢹⣿⣿⣿⣿⣿⡧⠇⠀⠀⡟⠿⣿⡿⢿⡇⠖⢺⠇
⠀⠀⠀⠀⢸⠐⠖⠴⠴⠂⡇⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⡇⠄⠬⠠⢼⡗⠋⠉
⠀⠀⠀⠀⠈⠓⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠙⠒⠒⠒⠊


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣼⣧⣤⣤⠀⠀⠀⠀⠀⣤⣤⣴⣧⣤⣤⡀
⠀⠀⠀⣴⡒⠛⠉⠉⠙⢳⣫⡄⠀⠀⢠⣖⠚⠛⠻⠋⡟⣒⣦⠀⠀⢴⣻⣿⣿⢿⡿⣿⣿⣟⡆⠀⢰⣻⣿⣿⣿⣿⣿⣿⣟⡦
⠀⠀⠀⢻⣻⣿⣿⣿⣿⣿⣿⠃⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⢹⣿⣇⠀⠀⣸⣿⡏⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠑⠿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠘⠺⢿⣿⣿⠿⠋⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⠇⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⣴⣊⣉⣉⣉⣉⣑⡦⠀⣠⣶⣛⣿⣛⢦⡄⠀⣤⣄⠀⠀⢀⠲⠚⠊⠑⠑⢣⡀⠀⠀⠀⠀⠀⣤⢀⣄⣠⡀
⠀⠀⣠⠤⠤⠼⠯⠽⠿⡻⣿⣿⠇⢰⣯⣿⣿⣿⣿⣷⣹⡄⢿⡟⠀⢀⡏⠀⠀⠙⠑⠀⠀⢳⠀⠀⠀⠀⠀⣧⠈⠻⠿⠃
⠀⠀⢻⣷⣶⣶⣶⣶⣾⡗⠛⠁⠀⠸⣟⣿⣻⣿⣿⡿⣹⠇⢸⠀⠀⠈⣇⠈⣷⣶⣶⣾⠃⡼⠀⠀⠰⣒⡂⡠⢄⢠⢀⡀⡤⣄
⠀⠀⠈⠻⢿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠙⠾⣽⣿⣭⠾⠋⠀⠘⠀⠀⠀⠈⠳⣬⣉⣉⡥⠞⠁⠀⠀⠐⠶⠃⠓⠚⠘⠞⠣⡗⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⠶⠞⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠻⠶⠶⢶⣤⣤⣀⡀
⠀⠀⠀⢠⢖⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢗⢦
⠀⠀⠀⡇⢇⠀⢀⣠⡤⠴⠶⠒⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠓⠒⠶⠦⣤⣀⡀⢀⠇⡇
⠀⠀⠀⠘⠤⣉⠛⠠⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠘⢋⡡⠞
⠀⠀⠀⠀⠀⠹⡉⠑⠒⠂⠤⠤⠭⠉⢉⣀⣒⣒⣒⣒⣀⣀⣀⣐⣒⣒⣒⣂⣈⠉⠩⠥⠤⠄⠒⠒⠉⡹⠁
⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠂⠀⠤⠤⠄⠤⠄⠀⠀⠒⠒⠀⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠤⠤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀⠤⠤⠀⠒⠒⣒⣂⣉⣉⣡⡤⠤⠤⠤⠔⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣲⠖⣶⣖⣋⣉⣉⡉⠁⠀⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢺⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠀⣄⡀⠁⣁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠈⠩⠠⠤⠀⠀⠠⠦⣠⡀⢄⠀⠀⠤⡠⡀⠉⠑⠒⠢⠤⡀
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⡀⠀⣀⠀⠐⠠⡔⠛⠠⢜⢲⡨⣭⠅⡍⠦⠀⠀⠄⡀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠗⠤⢄⡀⠈⠐⠂⢠⡱⠂⠂⠓⢬⣩⠷⡒⠂⡓⠭⢲⠤⢉⠉⢁⣈⡡⠖⡏
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠈⠁⠒⠚⠒⠐⠰⠮⠲⠽⠴⠠⠔⠲⠀⠂⠓⠊⠉⠁⠈⠁⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡈⠉⠙⠐⠒⠒⠒⠂⠉⠉⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠉⠉⠉⠉⠉⠁



⣿⣿⠛⠿⢻⠿⡿⢻⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠳⠶⣤⡀
⠀⠀⠀⠀⡤⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⣀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢀⡠⠬⢍⣔⠵⢀⠊⢴⡈⠅⠠⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠂⠀⠘⠄⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⣀⢠⣀⠜⠫⠛⠁⠀⣨⡗⠁⡙⡉⠒⠤⣶⡍⠓⣀⡀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⢀⡿⠁⠀⠀⢀⠒⢀⡤⠧⠦⣮⠁⢉⡏⢀⣀⣋⢄⢈⠢⣑⠶⡼⢀⠋⢩⠀⣀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠈⠆⠀⠰⡀⠀⠀⠀⢸⠃⠀⠀⡠⣮⠀⡞⢋⠓⣄⡌⠀⢸⣌⢲⡂⢒⡡⠂⡎⠀⠀⣅⡫⠂⠈⠠⠊⠑⣀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⣿⠀⠀⡜⠀⠀⣸⠁⢰⠂⡉⠀⠀⡁⠀⠉⡋⣱⠄⠀⣩⡀⠀⣀⣀⢠⣶⠊⠔⠃⠀⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢢⠀⠀⢹⣄⠨⡇⣰⠛⠉⢆⢂⠠⠐⠢⠰⢆⠖⡉⢹⢹⡜⡲⢧⣀⡀⢉⠔⠊⢐⠈⣿⡶⠿⠩⠢⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢣⠀⠘⣿⣨⣵⣏⠀⢠⡟⢹⠛⠲⡤⠀⡇⢀⠞⢣⢸⢄⠀⢸⣭⣉⣊⣀⡀⠴⠀⠈⢸⠑⡂⢄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡏⣙⣲⣬⢥⣄⠑⠀⠰⢄⣼⣾⢂⠨⠘⣃⣀⠠⠨⢈⣛⡞⣐⡨⠀⠀⠖⠄⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠆⠀⠈⢷⣆⡎⠈⣄⠁⠬⣡⣄⣠⠉⠈⠳⣄⣦⡦⠛⠉⠍⢉⢘⠀⠉⡔⠣⣀⢈⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠈⠆⠀⠈⢋⠻⣶⣄⢙⠖⠋⣀⣈⡄⠔⠀⣥⠦⢤⣄⡐⢆⠲⠆⢂⠜⡀⢀⣴⠟⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠱⡄⠙⠻⢶⣬⣁⣩⢀⠁⢊⠲⠇⠀⢒⡑⣠⣋⣩⣥⡶⠟⡋⢐⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠸⡀⠀⠀⠈⢦⡀⠀⠀⠉⠙⠛⠻⠿⠿⠷⠿⠿⠿⠛⠛⠉⡑⠂⠅⣪⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠉⠀⠈⠂⠀⠀⠀⠀⣂⡤⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡀⠀⠀⠀⠀⠀⣟⡳⠦⣤⣄⣀⣀⣤⣀⣀⣤⡴⢞⣩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢀⢳⠀⡄⠀⠀⠀⠻⣿⣷⣶⣬⣭⣭⣭⣭⣴⣶⣾⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡜⡆⢻⠹⡄⠀⠀⠀⠈⠑⠪⠭⣭⣭⣭⣭⡭⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠘⡄⢧⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⢸⣿⣿⣿⣿⡇⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⢀⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣧⣠⣶⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⢸⣿⣿⣿⣿⡇⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣼⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⢸⣿⣿⣿⣿⡇⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⠲⡀
⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠈⠻⣧⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣴⠀⣰⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⣇
⠀⠀⠀⢐⠀⠀⢰⠉⠛⡙⠦⡀⠈⣙⣀⣮⠀⡄⠈⠄⠀⠀⠀⠀⠀⠀⠀⡎⣠⡿⢀⢻⠁⢰⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⣦⡆⠀⠈⣼⡀⠀⠈⢢⠱⠆
⠀⠀⠄⠈⢳⣄⡼⣄⠀⠐⠄⠸⣄⣸⡟⢉⢫⣧⠀⠠⢀⠀⠀⠀⠀⠀⠈⠃⣻⡇⡾⣧⢠⡌⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⢐⣾⠄⠀⢸⣧⣄⠀⣿⣶
⠀⠀⠠⠌⠉⠙⠋⡈⠷⣿⡆⣀⠀⠁⠈⠑⣷⣌⢷⡀⠈⠧⠀⠀⠀⠀⣠⠈⣿⣶⣵⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠈⣣⣌⠰⡿⢡⠂
⠀⠀⠀⠀⠲⡀⡈⠀⠀⠈⢹⢿⣣⣘⡀⠀⠈⣋⣄⡳⢄⡀⠄⠀⠀⠀⣿⢸⢠⡻⠉⠄⣴⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠢⠘⡦⠀⠉⡉⠰⠃⠁
⠀⠀⠀⠀⠀⠘⢹⣤⣀⠀⠀⠀⢴⠿⠟⣁⠰⢏⡁⠐⠄⡈⢁⠄⠀⠀⡟⣶⡈⠃⣰⢧⡿⠀⠀⠀⠉⠸⢷⡀⠈⠃⠀⠀⠀⠀⠀⠈⠀⠁⠀⠁
⠀⠀⠀⠀⠐⠒⠤⠬⠟⠋⠈⠁⠋⠁⠀⠀⠀⠀⠈⠁⠀⡈⢦⣦⠂⠀⠘⠌⠻⠜⡷⠋⠀⠀⠀⠀⠣⠀⠛⢯⡀⠀⢰⠀⠀⢠⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⢿⣧⠂⠀⢸⣇⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠘⡄⠀⠀⠙⠇⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣷⠄⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠰⣏⠀⢸⡃⢁⠀⣾⠃⠀⡹⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢉⢀⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣄⠀⡆⠀⠙⢷⣼⠃⢸⡏⠁⠀⠀⡇⣧⠂⠀⠀⠀⠀⠀⠀⠑⠂⠀⢾⠣⠠⠇⠂⠀⠀⠈⣱⡇⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢄⠖⡔⠈⠂
⠀⠀⠀⢰⣇⠀⠾⠀⠿⠀⣾⣝⠀⠂⡠⢡⣵⠀⠀⠀⠀⠀⠈⠀⠀⡠⠀⢠⡀⠀⠀⠀⠠⠤⠀⢈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢦⢀⡼⠃⢆⡤⠄
⠀⠀⢸⠀⣿⠀⠂⢠⡆⢻⣫⠋⣴⠟⢁⡿⡙⠀⠀⠀⠀⠀⠀⢘⡀⠁⠀⡘⠷⠈⣀⣀⣀⠀⣠⣠⣴⡷⢒⠎⠀⠀⠀⠀⠀⢀⢠⣇⣴⠟⠩⢅⡈
⠀⠀⠀⠳⣟⡆⠀⣨⠄⢼⣇⠀⠁⣄⣾⡷⠀⠀⠀⠀⠀⠀⢰⠈⠟⠀⣠⡴⠂⠈⢀⢩⣿⠻⠋⠙⠋⢀⠘⡀⠀⠀⠀⠀⠀⣬⣵⡟⠉⠄⠊⠁
⠀⠀⠀⠀⠸⣧⡤⠀⡠⠀⢈⠀⣴⢿⡟⠁⠀⠀⠀⠀⠀⠀⠘⢠⡙⢴⢫⠁⠀⢰⡏⠀⠀⠀⢀⠀⠀⣸⠦⠀⠀⠀⠀⣰⡾⠋⠈⠁⠀⠰⣄⢀⡀⡄
⠀⠀⠀⠀⠀⣿⠁⣼⣁⠤⡌⠀⡷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢠⡉⣠⣴⣯⡀⠐⣆⠀⠈⠲⣴⡷⠁⠀⠀⢀⡶⠉⠀⠀⠀⠀⠀⠄⠁⠀⢹⠗
⠀⠀⠀⠀⠀⢷⠺⣘⠋⠒⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠂⠒⠲⠽⡟⣛⠡⠾⠛⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠁⠐⣦⣤⠏⠈
⠀⠀⠀⠀⠀⢸⠀⡍⢡⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠤⢸⡼⢖⠸⠀⠄
⠀⠀⠀⠀⠀⠀⠛⠡⠿⠈⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢔⣂⡱⠄⡀⠄⠀⠀⠀⢀⣉⠂⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢏⡅⢨⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠰⠀⢀⣙⡛⠀⣿⢿⣋⡁⣴⠁⡂⠀⠀⠀⠀⠙⣤⢀⣤⠀⠖⠀⠤⣄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢦⠁⣦⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡾⣷⡀⠀⡉⢀⡌⣁⡤⢹⠟⡬⢤⡄⠀⠀⠀⠀⣀⠘⠐⢿⣛⣾⣄⡀⠈⠀⠀⠀⠀⠀⠀⠀⡸⡇⠠⠀⢹⣠⠂⠐
⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⢀⠌⣼⣧⣌⡿⠌⠛⠭⠂⠉⡤⠋⠂⠤⠴⠆⠀⠀⠀⠀⠀⢤⠀⠄⠠⠝⠳⢿⡷⣌⠃⠀⠀⠀⠀⠀⠏⡇⠀⠀⠻⡟⢁⡇
⠀⠀⠀⠀⠀⣨⠁⠀⢠⠾⣋⣀⣡⣬⠒⢀⣾⣌⠙⠃⠉⠁⠲⡈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⡴⣿⣿⠈⠓⢄⠀⠀⠀⠀⠘⣿⡆⣰⢘⡀⡼
⠀⠀⠀⠀⠒⠒⠒⡀⠀⠹⠄⠀⠄⠰⢥⠀⠀⠰⠐⠶⠦⡔⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⣡⠃⠀⠀⠀⠱⡀⠀⠀⠀⠈⠓⠇⠞⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠲⠀⠘⠀⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠝⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡀⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⢀
⠀⠀⠀⢀⡠⠈⠀⠀⠁⠈⠣⢄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⠀⠀⠑⠀⠀⣔⠁⠀⠀⠀⠀⠀⡋⠀⠀⠀⠀⢳⠄⢀⠀⠀⠀⠢⡐⢐⡄⠠⠄
⠀⠀⢠⢊⡠⠤⠄⡀⠀⠀⠀⢀⠵⢄⠀⠀⠀⠂⠨⠆⠀⠀⠀⢀⠤⠐⠂⢀⡀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⢴⠀⠀⠀⡠⠤⠄⢙⣠
⠀⠐⡥⠂⠀⠀⠀⠡⡀⣄⡤⠉⠀⠀⠇⠀⠀⠀⠘⠀⠠⠀⣰⠁⠀⠀⠀⠀⠈⢂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠈⡡⠂⠀⠀⠎
⠀⠀⠉⠀⠀⠀⢀⡴⠀⣾⠀⠀⠀⠀⡌⠀⠀⠀⠈⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠐⠤⠂⠡⠀⠀⠉⠡⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠩⠢⣄⣀⣀⡀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⢀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠃⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠂⢄⠁⠀⠑⢄⠀⠂⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡠⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠐⠌⣄⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⢀⠈⠠⠂⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢠⠄⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠠⠀⠀⣠⡉⠀⠀⠀⠀⠀⠀⡐⠁⠀⠁⠀⠀⠁⢀⢔⠩⠚⣷⠀⠀⠰⡎⢀⠄⠀⠈⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠀⠀⠀⠰⠦⠅⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⡑⡠⢊⠔⠁⠀⠀⠀⠱⡀⠀⠀⠀⠈⠀⠄⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⢠⠒⡂
⠀⠀⠈⠀⠀⠀⠀⠠⠲⠂⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣀⢊⠔⡰⠋⢂⠌⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⢎⠔⡁⢃⠖⡠⠖⢁⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠀⠀⠀⠀⠉⠀⠒
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⠊⡰⢁⠞⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠈⠑⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠄⢁⠎⡰⠋
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣤⠂⠎⡠⠃⠜
⠀⠀⠀⠀⠀⢀⠠⠀⠀⠠⡀⠀⠀⠀⠀⢢⡀⡡⠀⠀⢈⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂
⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠜⠈⢑⡆
⠀⠀⠠⠁⠀⠀⠀⠠⠀⠁⠀⠈⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡠⠊⠀⠀⠂⠀⠀⠀⡀⠂⠀⠀⠴⠁
⠀⠀⠁⠀⠀⠀⠔⠁⠀⠢⠀⠀⡇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠂⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠊⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⡤⠠⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⢀⠁⡀⠀⠀⠄
⠀⢀⠀⠀⠀⢀⡠⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠂⠄⠀⡀
⠀⠀⠁⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠠⠄⢀⡀⠠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢁⠀⣀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⡀
⠀⠀⡠⠒⠈⠁⠀⠰⠊⠀⠀⠀⠈⠂⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠐⢈⠀⠀⢀⡀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠀⠀⠀⠈⠉⠁⠀⠀⠀⠘⢂⠀⠀⠀⠀⠀⠈⠐⠠⠄⠁⠀⠄⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠰⠂⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠄⠈⠠⠀⢈⡀⠀⠀⢉
⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠈⠒⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⡀⠒⠀⢀⠐⠀⣈
⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⢘⠀⠀⠀⠀⠀⠀⠐⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠐⠂⠀⠀⠒⠇
⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠚⠒⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣚⡉⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢃⠰⢤⡐⢠⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⢀⣉⠛⣖⣉⠽⢦⡔⠃⣞⢈⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⡆⡩⠔⢁⠿⢤⠚⣉⠴⣥⠔⠃⣈⠒⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠂⡎⢻⣤⠂⢹⠧⣤⠋⣩⢯⡤⠛⣅⢡⣜⣒⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢻⡕⢃⡼⠈⣇⣤⠋⡹⢤⡎⣁⠼⢤⠔⠃⣄⠒⣀⡲⣔⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠳⢦⡤⠀⠨⠿⠭⠤⠞⣅⣜⣘⣀⡜⢉⣧⣤⢊⡩⠧⡤⠚⣠⠐⢊⣆⠉⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡞⢩⣀⠀⠀⠀⠀⠀⠀⠉⢉⣉⣉⣉⣙⣉⣛⣓⠊⠽⠧⣤⠘⢂⡜⡆⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣄⡙⠢⣠⣀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠯⢔⣀⠈⠁⠐⠻⢴⡀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡋⣹⢉⣓⡸⠀⡭⢴⢀⣆⠽⢓⡦⠥⠾⠒⠑⠂⠲⠤⣀⠀⠉⠒⠤⠀⠀⠉⠂⢤⣀⣄⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠴⣥⡎⢩⠗⢻⠔⣼⣡⠜⢀⠖⣠⢏⡉⢮⠔⠊⠍⠓⠲⢯⣶⣤⣀⡀⠀⠀⠀⠀⠈⠻⣷⡵⡀
⠀⠀⠀⣀⠠⠀⠀⠀⠠⢱⠠⠴⣀⠋⡻⠰⢧⣴⣁⡜⢉⡝⢺⠅⢺⣥⡞⣁⠖⣰⠯⣔⣋⠞⢙⢪⣟⣿⠶⣕⣤⡀⡠⡈⠻⣷
⠀⠀⠈⠉⠉⠉⠒⠢⢄⡀⠛⡔⡉⡕⠣⠴⡁⣼⠁⡙⢛⠖⢦⣴⣃⣰⢁⡜⢺⠧⣎⢤⡎⡩⠗⡤⠚⣷⡳⠄⠁⠻⣖⢌⢷⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡡⠔⡜⢄⠦⢨⡎⠳⠚⠤⣆⢠⠁⡹⠋⡗⢧⢴⣁⡜⢉⠗⢹⠗⢾⠤⣿⡄⠀⠀⠀⡙⢾⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠬⣈⣢⠧⣰⠈⣲⢣⡼⣀⡅⢙⡊⣣⣆⣰⣁⣟⣙⡖⣧⢴⣇⣴⠉⣇⢀⠀⠀⠈⢧⣻⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⢻⣿⣿⣷⠦⡒⠛⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⡒⠦⢄⣠⣸⣿⣧⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠐⣊⠝⡩⠂⣁⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠦⣝⣿⠹⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠡⣢⡶⠟⠒⠐⠒⠒⠓⠒⠂⠒⠠⢌⣑⠂⠄⠦⣄⡂⢕⡢⢀⠠⡀⠀⠀⠀⠀⠀⢹⠇⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡈⠛⠷⣮⣕⣯⣶⣭⣒⣀⣠⣴⣟⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠬⣖⣿⣿⣿⣿⡿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⣶⠛⠛⢦⣄
⠀⠀⠀⢿⡀⠀⠀⠈⠳⣄
⠀⠀⠀⠈⠳⣄⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡾⠋⠻⣦⡤⢤⣤⡤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠈⠳⣄⠀⠀⣀⣀⠀⣼⠋⠉⣷⠶⠶⢾⣇⠘⣰⡷⠦⣴⡏⠠⠈⣷⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠈⢷⣼⡉⠙⢧⣾⠶⢥⡇⠰⣀⣀⣏⣿⠋⣀⠄⠀⣹⠶⠚⠓⠪⢿⠉⠳⣤⡴⠶⠦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⣛⣳⣤⣀⠀⠙⢿⡞⠉⢻⣆⠀⠙⣶⠁⠉⠉⣷⠶⠶⠶⣾⡵⠚⠉⠙⢦⣀⠷⠤⢏⠀⠀⠀⠈⣏⡉⠉⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⣼⠉⠁⢀⣉⣷⣄⣂⣹⠶⡋⠉⢳⣤⣈⣷⣀⡞⠁⠄⠄⢠⠏⠀⠚⠀⣠⡏⠁⡄⢀⠈⣧⡤⠶⠶⣅⠈⣷⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠓⠦⣴⣋⡀⠀⠙⢻⡦⡤⠴⠲⡞⠁⡀⡀⠙⢧⣀⣠⠴⠛⠒⠲⣖⠋⢉⡉⠳⡄⠀⣠⠏⠐⢐⣀⡼⠗⠃⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡇⠀⠀⠀⠉⠉⠒⠚⠳⠦⠤⠼⣧⣄⣀⣁⣀⣀⣯⣁⣀⣠⣦⣤⡼⠧⠤⠤⠾⠷⠚⠛⠚⠋⠉⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠤⣤⣤⣤⣤⣤⣤⣤⠤⠶⠚⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢃⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣠⡟⢡⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡋⠉⠛⠛⠋⠉⠉⠳⣄⠀⠀⣴⠏⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⣉⣤⠶
⠀⠀⠀⠀⠀⠀⠀⢰⠟⠛⠛⠛⠋⠛⠶⠄⠀⠀⠀⠀⠀⠀⠀⠉⢻⡿⢁⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⢋⣡⡴⠟⠉
⠀⠀⠀⢀⣠⣤⢶⣿⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⠀⠀⣰⠟⣰⡟⠉⠙⠛⢶⡄⠀⠀⠀⠀⠀⢀⣠⡶⠛⣉⣤⠾⠋⠁
⠀⢀⡴⠛⣁⣤⠾⣗⠶⠀⠀⣤⠶⠄⠀⠀⠀⠀⠿⠀⠀⢀⡾⢋⣼⠋⠀⠀⠀⠀⠀⡿⠳⣦⣀⣴⠞⢋⣡⡴⠛⠉
⣰⡏⣠⡾⢯⡜⠓⠉⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣡⡿⠁⠀⠀⠀⣀⠀⢘⣻⡶⠟⣉⣴⠞⠋⠁
⣿⠀⣿⣹⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣴⠏⠀⠀⠀⠀⣀⣼⠷⢋⣥⡶⢻⡏⢸⡇
⢻⣦⡟⠷⠀⠀⠀⣠⡤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⣠⡴⠞⣩⡴⠞⠋⠁⠀⢸⡇⢸⡇
⠈⣿⣿⣗⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣥⠶⠛⠉⠀⠀⠀⠀⠀⢈⣹⣿⠁
⠀⠘⣿⡙⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠛⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⣿⠃
⠀⠀⠘⢷⡝⠶⠍⠛⠻⠶⣦⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣶⠶⠟⠛⠉⠀⣼⠏
⠀⠀⠀⠈⠻⣾⠛⠦⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠙⠷⣝⡛⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢯⣭⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡿⠶⣦⣤⣤⣤⣤⣤⣤⣤⣤⡶⠶⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⢦⣤⣤⣤⣤⣤⣤⠶⠶⠟⠋

⠀⠀⠐⠀⠀⠀⢃⡷⢖⠀⣜⡙⣁⡙⢃⣺⠌⠂⠀⡗⠀⠀⠀⠰⠒⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⣃⣷⡦⠘⣦⣴
⡃⠀⠰⠓⠄⢸⢸⡇⢸⢸⢬⣁⡯⠀⢈⡁⠖⠀⠀⡇⠀⠰⠆⠀⣠⣤⣶⢟⣷⣦⣀⠈⠁⢰⠢⣬⣽⣗⠚⢯⣉⣿⠐⢻⣟
⡀⠀⠀⠒⢧⠀⢏⣧⠀⠀⠈⠁⡈⢠⢤⠟⠏⠀⡐⡸⠀⠀⣠⢾⡿⣱⣯⣊⢃⢨⣯⢧⡀⠀⠃⣈⠿⠋⡀⠛⣿⢀⣖⠚⣭
⡀⠀⠀⠐⠺⠀⠀⠈⠳⣄⠙⠲⠤⠀⠀⠈⣠⠔⡡⠁⠀⢠⣿⣾⣿⡿⣿⣿⣿⢻⣿⢞⣷⠀⠀⠀⠠⠞⠗⠨⢽⡎⢹⡞⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⡒⠖⠮⠁⠀⡀⠀⢰⣿⢻⣿⣿⣿⣿⣿⣿⣼⣯⣾⣿⠀⠀⠀⠀⢀⡺⠦⠘⢲⣿⡏⡵
⠀⠀⣠⣤⣤⡀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⢸⣿⣾⣿⠁⡇⣿⠏⣿⣿⣿⣿⠃⠀⠀⢀⠀⠀⠀⠲⠶⠂⠁
⠀⣾⡟⠀⠀⢻⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⢿⣿⣿⣶⣷⣿⣾⣿⣿⣿⡵⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠄
⠀⢻⣿⣶⣶⣶⡿⠀⠀⠀⠀⠀⢀⣔⣴⠂⠁⠀⢠⠀⠀⠸⡿⢻⣿⣿⣿⣛⣿⠟⠋⠀⠀⠀⠀⣤⠀⣲⡖⣀⠀⠀⠀⠀⠶
⠀⠀⠙⠛⠛⠛⠁⣤⡀⠀⠀⢀⠸⠿⠟⠀⠀⠀⠈⠀⠀⣸⡇⠈⠰⢿⡿⠁⠋⠀⣠⠶⡁⠀⠀⠈⠚⠻⠟⢻⠗⠢⠀⠀⡀
⠀⠀⣠⣄⣀⣀⠀⠉⠁⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢰⣾⡇⠀⠀⠀⣃⠀⢨⠶⠁⡀⠂⠲⠄⠈⣤⠤⣷
⠿⣯⠤⢤⣈⡙⠪⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⠃⠀⠀⠀⠠⠁⠉⣦⠼⢏⠛⠂⠀⠉⢣⣤⡿⠀⠰
⠷⢯⡀⠀⠀⠉⠓⠦⣌⣛⢶⣤⣀⡀⠀⠀⢤⠆⠀⠀⠈⢿⣦⠀⣼⠏⠀⠀⠀⠰⣆⢀⣦⣥⣶⡾⣤⣤⣦⣤⣾⠟⠁
⠱⣄⡈⠙⠒⠦⢤⣀⠀⠈⠙⠓⠺⢿⣷⡀⠀⢀⠀⠀⠀⠈⡿⠛⠁⠀⠀⠀⠀⢰⣾⠽⣿⣿⣿⣿⣿⣷⣯⣿⠁⠀⠀⠊
⠀⠀⠉⠓⠲⠦⣤⣬⣉⣳⠦⣤⣄⣠⣼⠂⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠶⣽⣝⣿⣝⢯⡽⠟⠁⢠⣶⣿⣶⣄
⠀⡄⡀⠀⠀⠀⠀⠂⠘⠉⠛⠓⠲⠛⠃⠀⠀⠀⡀⠀⠐⢦⡀⠀⠀⠀⠀⠘⠀⡄⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢻⣿⣿⣿⣿
⠀⡇⠰⠷⣻⣿⣶⡀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠘⠻⢡⠄⠉⢵⠶⡄⡤⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠈⠙⠛⠛⠁
⠀⠘⢄⡑⠃⠐⠇⠋⠠⠀⠀⡀⢀⣦⣶⢻⢿⠃⠀⠉⠀⣤⠤⠦⢤⠀⠀⠔⠀⠀⢩⠗
⠀⠀⣀⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⠉⠡⠾⣠⠀⠈⠁⠋⠉⠓⢤⠄⡀⢀⠀⠀⠠⢬⠀⠀⠀⠀⠉⢦⣀⣀⡀⡀⠄⠀⠐⠶
⠀⠀⢹⡀⠀⠀⠀⠈⠀⠀⣀⡀⠀⠀⠒⠚⠃⣠⠈⠳⢦⠀⠀⠏⠸⡇⠿⠃⠀⠀⠀⠀⠴⠀⠀⠀⠈⢸⡟⠁⠁
⠠⣌⠉⠙⠂⡀⣀⠀⠠⠎⠁⠉⠆⠀⠀⠒⠲⠇⠀⠀⠀⠁⠀⠘⠀⠂⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⢀⣵⣦⠤⠜⠂⠀⠋
⠀⠀⠀⠀⠠⢠⡌⠑⠀⠀⠀⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠁⠐⣿⣿⣿⣧⠀⠀⣀⣈⠀⠀⠀⣤⠤⢤⣄⡀
⢀⢦⠀⠀⠉⢆⠈⢁⢴⡊⠋⠉⠀⠀⠀⠈⠕⡦⠀⠀⠀⣰⣾⣿⣷⣶⡀⠠⠆⠙⢟⣿⣿⠇⠀⢿⣿⣿⡆⠀⠉⠋⠙⠲⠾
⠀⠀⠀⠒⠀⠀⠰⠣⠋⠀⢀⢈⣍⠭⡭⠶⡄⠘⢄⠀⠀⠻⠿⣿⠟⠛⠁⠀⠁⠀⠈⠛⠋⠀⠀⠀⠙⠋⠀⢀⡤⠶⢆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠚⠋⠉⠙⠓⠲⣄
⠀⠀⠀⢀⣤⡶⠤⠤⣀⠀⢠⡴⠶⠒⠒⠦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⣠⡤⠚⠉⠀⠀⠀⠀⠀⠑⠮⠆⠀⠀⠀⢀⢈⣃⣀⣤⣤⡤⠤⠤⡤⣤⣤⣤⣄⣀⣀⡀⠀⠀⢀⡠⠖⠁⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⠚⠁
⠀⠉⠙⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⢽⠋⠁⣀⢩⡤⠤⣤⣧⣤⡀⠀⠀⠀⠀⠉⠉⠙⠋⠄⠀⠀⠀⠀⢀⡠⠔⠊⠁
⠀⠀⠀⠀⡉⠁⠀⠂⠀⠠⠤⠤⠴⠤⡀⠸⣿⠛⠉⠀⠀⠀⠀⠀⣀⠈⠉⠿⢿⣷⡦⠤⣤⣤⣄⡀⠀⠀⢦⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⣸⣷⠶⠂⠀⠀⠈⠀⠀⠀⠈⠆⠠⠀⠀⠀⠀⠀⠛⠛⠻⠶⢬⣈⢢⡀
⠀⠀⠦⣤⣤⣴⣶⣥⣤⡤⠾⠿⠛⠚⠛⠛⠛⠛⠋⠉⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠈⠣⣷
⠀⠀⠀⡄⣿⣿⠻⠭⠁⠀⠀⠁⠈⠁⠉⠉⠁⠈⠁⠈⠉⠙⠻⠿⢍⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠠⠒⠀⠀⢰⣿
⠀⠀⠀⣷⡈⠳⣦⡀⠀⠀⡀⢀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠄⠴⠦⣤⠄⢀⣠⠖⣫⡟
⠀⠀⠀⢹⡹⣦⣈⠉⠛⠶⢤⣬⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠨⠂⠉⠀⢀⣀⣀⣤⠶⠚⢉⣤⠞⢩⡇
⠀⠀⠀⠈⣇⠀⠈⠙⠲⠤⢄⣀⡉⠉⠙⠋⠙⠛⠒⠒⠒⠒⠲⠶⠖⠒⠲⠖⠚⠛⢛⣉⣁⣤⡤⠶⠞⠉⠀⠀⣸
⠀⠀⠀⠀⠘⢦⠄⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⢤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠛⠉⢹⣿⣿⠉⠙⠛⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠈⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⢀⣴⣿⣿⠟⢹⣿⣿⠙⢿⣿⣷⣄⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⠏
⢾⣷⣼⣼⡿⠃⠀⠀⠀⠀⠀⠈⢿⣿⣷⣿⣿⠟⠁⠀⢸⣿⣿⠀⠀⠙⢿⣿⣷⣿⣿⠏⠀⠀⠀⠀⠀⠀⠻⣿⣼⣴⣿
⢀⣈⡉⢇⢠⣶⣦⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⣀⠀⢸⣿⣿⠀⢀⣀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⢠⣶⣦⢀⠏⣉⣀
⠘⠿⠿⠾⣿⠛⣡⣤⣤⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⢠⣤⣤⡙⢻⡿⠾⠿⠟
⠀⣴⣶⣦⣌⣦⣿⡿⠋⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡈⠻⣿⣧⣎⣤⣶⣶⡄
⠀⠉⠛⠛⠛⠋⠫⡀⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠄⡠⠋⠛⠛⠛⠋⠁
⠀⠀⢰⣿⣿⣿⣶⣾⣾⡟⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣾⣶⣿⣿⣿⣷
⠀⠀⠀⠉⠉⠉⠉⠁⠀⠈⠺⣧⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠸⣷⡤⠊⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢋⣉⣹⡇⣠⡶⣤⠀⣠⣤⣤⠀⣴⢶⠄⢻⣾⣃⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠈⠙⠻⠖⢿⣟⣋⣼⣿⣸⣯⣀⣿⠦⠶⠋⠉⠁
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣾⣷⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠁
⠀⠠⠀⠂⠠⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡼⠀⣤⠈
⢠⠀⠀⡀⡀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⢇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢧
⠀⢀⠐⡇⠃⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠁⠀⠂⣀⠐⣀⠐⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⡀
⠀⠈⡄⣐⢁⡼⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠄⡄⢀⡀⠀⠠⠁⠀⠀⠀⠀⠄⠀⢀⢈⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⢀⢀
⠀⠀⢣⢸⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠
⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠣
⠀⠀⣸⣿⢣⠀⠀⠀⠀⠀⠀⠸⠀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠀⠀⠐
⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠂⠈⠃⠀⡀⠀⠀⠄⠢⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢂⠀⡉⠠⠀⠐⠂⠁⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⣜⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡇⢹⡴⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣸⣇⠀⠀⠀⠀⠀⢠⠌⠳⣊⡐⠻⢄⣮⡁⠀⠀⠀⠀⠀⠀⢀⠀⡈⠤⠀⢠
⠀⠁⠀⠀⠀⢤⠀⠀⢠⠀⠀⠀⠀⠀⠸⡿⠘⠀⠀⠀⠀⠀⠀⠹⠁⡁⠈⡎⠀⠈⠁⠃⠀⠀⠀⡀⠀⠀⠀⠁⠀⢠⠈⢠
⠀⠀⠀⠄⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢀⠋⢀⠀⠶⠊⠀⠀⠀⠀⠄⣀⠁⠀⠠⠖⠀⡀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠈⠠⠀⠀⠀⠀⠀⠀⡀⠀⡀⠈⠂⠂⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢐⠀⡇⡇⠀⠀⠀⠀⠀⠠
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⣻⠀⡁⠀⠀⠀⠀⠀⠄⠁
⡌⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠤⠁⠀⠀⠀⠀⠀⠀⢠⠙⡄⡅⠀⠀⠀⠀⠀⠂
⠀⡀⠐⠁⠀⠀⠀⠀⠠⠀⠀⡎⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠃⠁⠀⠀⠀⠀⠀⠄
⡈⢀⠄⠀⠀⠀⠀⠀⠀⠀⡴⢀⠀⣁⡆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣩⠔⢉⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠠⠒⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠝⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠈⠀⠀⠀⣘⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣦⠚⠛⠚⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠒⠒⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⢀⣄⡂⢽⣿⡿⢿⣿⣲⠶⠮⢟⣛⣛⣒⡀⠀⠀⠠⠤⠤⠤⠤⠬⠛⠛⢛⣋⣩⠤⠶⠛⣻⡿⠿⣿⣿⡷⠶⣠⢀
⠀⠀⠀⠀⢠⠏⣾⠗⣫⡶⠛⣋⣥⡶⠖⣂⣨⣭⠶⠶⠖⠚⠛⠛⠋⠙⠋⠛⠛⠒⠲⠶⠶⣤⣬⣉⠑⠲⣬⣙⠳⢤⡁⠒⢼⣷⡝⡄
⠀⠀⠀⠀⠸⣆⢿⡘⠋⠀⢰⠉⠋⢠⢺⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢖⢠⠉⢣⣠⠙⢾⣆⢹⢏⡾
⠀⠀⠀⠀⠀⢹⣷⣮⣑⣂⠸⣆⢳⣄⢘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠸⢃⣊⣼⠤⠞⡋⢥⡟⠁
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣬⣶⣶⣭⣉⣛⠛⠛⠶⠶⠆⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣶⠶⠶⠛⠁⣛⣩⣭⣤⣶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣍⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣩⣭⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣟⣿⡿⠟⣩⣿⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⠀⡾⠿⢛⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⣿⣿⣿⣿⠃⣾⢱⣧⣾⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⡏⢀⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠿⣫⣾⣿⣿⣿⠏⣼⢇⣿⢻⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⠁⣼⣿⡌⣿⠟⣩⡝⢛⣿⡿⠿⠟⢿⡿⠿⣿⣿⠋⠿⠿⣿⣿⠋⡀⠻⣿⠏⣵⣦⡟⣰⡏⣸⡏⢸⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⡄⣿⡿⣸⠇⣼⡟⢀⣾⠟⢀⣶⠂⣴⡆⢠⡿⠃⣴⡶⢀⣾⢀⣾⡿⢀⡿⢆⣹⡿⢠⡿⢰⡿⡿⠸⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⣾⣿⡀⣛⡁⠸⡏⢀⣾⠇⣸⡏⢠⡿⠁⣼⡿⢃⣼⡇⣼⡿⣡⡟⠰⠿⣟⠀⢛⣁⣨⣤⣴⣾⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣶⣿⣤⣼⠃⣤⣭⣴⣾⣿⣧⣤⣾⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⢿⣋⣝⣿⠻⠿⣿⣻⢿⡿⣻⣿⠿⢿⣿⠛⣿⡟⢻⡏⡭⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣶⣿⣭⣾⣧⣤⣿⣷⣶⣿⣭⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣥⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠿⢿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⢿⣷⡆⢠⣴⣦⡀⠀⠀⣀⡀⠀⢀⡀⠀⠀⢀⣀⡀⠀⣠⣤⣤⡀⣰⡾⣷⣆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠁⠙⣿⡄⢸⣿⣷⢀⣿⣿⠀⠀⣾⡿⣿⠀⠛⣿⡏⢀⣿⠀⢸⡿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⠇⠀⢻⣧⣴⣿⠀⣸⡟⣿⣿⡟⣿⠀⢰⣿⠷⣿⡇⠀⣿⡇⠈⢿⣶⠿⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠙⠁⠘⠛⠀⠛⠀⠚⠋⠀⠘⠋⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣤⣦⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⣤⣤⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠘⢿⣿⣯⣟⠃⢰⣿⡿⣿⣿⣷⡄⢺⣿⣿⠛⣿⣿⠀⢹⣿⣏⣿⣿⠇⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⢸⡀⠆⢀⠀⠀⠀⠀⠰⣿⣍⣿⣿⣦⢸⣿⣦⡈⢻⣿⡇⠘⣿⣿⡀⣿⣿⠀⢸⣿⣿⠋⠁⠀⠀⠀⠀⠀⡀⠘⣨⡃
⠀⠀⠀⠀⢀⣼⠛⢦⣌⠀⢠⠄⠀⠀⠙⠛⠻⠛⠃⠈⠻⣿⣿⣿⠿⠃⠀⠿⣿⣿⡿⠇⠀⠿⠛⠛⠀⠀⠀⠀⠀⠆⢀⣥⡾⠛⣇
⠀⠀⠀⠀⠸⣏⣤⡀⠉⠛⠶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⣉⣤⣴⠾⠛⠁⢀⡴⣿⠆
⠀⠀⠀⠀⠀⠙⢿⣙⢦⣤⡀⠀⠉⠉⠛⠻⠶⠶⢤⣤⣤⣤⣀⣀⣀⣀⣀⣈⣀⣤⣤⣤⣤⡶⠶⠾⠛⠛⠉⠁⠀⣀⣤⣶⣿⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣥⣟⠐⠂⠤⠄⢀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⣴⠖⣉⣤⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⠶⢤⣤⣤⣉⣙⠛⣛⠛⠛⠒⠚⠛⢛⣛⣛⣋⣉⣉⣭⣥⠴⠾⠛⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⢋⣉⣩⣤⣤⣤⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣴⠾⢋⣥⡶⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠾⠿⠷⢶⣦⣤⣤⡶⠟⣡⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡋⠳⣤⣀⣀⣀⣁⣤⠼⠃⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠘⠷⣦⣤⣤⣤⣤⣤⣴⠶⠟⠋⠁⠀⠀⠀⣠
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠀⢸⠇
⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⢸⣏⢉⣀⢀⣀⠀⣀⡀⢀⣀⠀⣀⣀⢀⣀⠀⣀⡀⢀⡀⢀⣀⡀⢀⣀⠀⣀⡀⠀⡈⠉⠉⣿
⠀⠀⠀⢿⡌⠁⠀⠀⠀⠈⠀⠈⠁⠀⠉⠁⠈⠉⠀⠉⠁⠈⠁⠈⠉⠁⠈⠉⠀⠉⠁⠈⠉⠉⢱⡟
⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡟⠛⠛⣛⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠢⢀⠄⢂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡀⣠⣶⣿⡆
⠀⠀⠀⠀⡀⠀⣠⠁⠈⡀⡀⠣⣰⡥⠔⢢⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⣽⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⡀⠢⣨⠂⢅⠗⡀⠤⢋⢀⠂⠈⡤⠊⠁⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿
⠀⠀⠀⠀⠡⡀⠘⢀⣠⠋⠐⠀⠃⡾⠂⠀⣄⡃⡇⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡟⢀⠐⠁⠡⡇⠀⣤⣠⠤⣠⠋⢀⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢃⠒⠺⠠⡀⠁⠀⠛⡁⡄⠀⠠⠂⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠑⢄⠳⡈⠰⠁⡀⡰⢂⢊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢨⣦⠈⢢⣠⠟⠶⡵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⡷⢷⣖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⢷⣶⡋
⠀⠀⠀⠀⠀⠀⠀⠠⠁⡏⢨⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠲⠼⠦⠴⠤⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⡜⢫⡞⠁⢗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡘⠢⠄⣈⠧⠀⣘⠃⠯⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣚⠒⢤⠓⠤⠓⠀⣤⠶⢨⡂⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠲⠄⣸⠁⠐⠛⠀⢠⣌⠀⠢⢈⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⣼⠓⠤⠣⢤⠛⠀⢤⡤⠁⣙⡃⢦⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢆⣙⠉⡲⠓⠤⠚⠂⠀⡴⠆⢀⣈⠁⠓⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣑⠦⢀⡘⠓⠢⠞⠢⢄⣀⠓⠂⢴⣁⡙⠲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠓⡶⢀⡈⠉⠲⠾⠂⠀⢀⡜⠁⠲⡆⢀⡛⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢍⠧⢀⡘⠓⠠⣴⠄⠀⠀⣈⠁⠲⠶⢤⣈⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⠤⢄⡠⠆⢀⣠⠄⠀⠀⢨⡇⠁⡒⠃⢤⠌⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡚⠒⠂⢤⠆⠀⢀⡀⠀⠀⢘⡆⠀⣨⠉⢐⡞⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢉⠓⠲⡎⠁⠐⣎⠁⠀⠐⠧⠀⣀⣍⠁⠺⢉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠃⠚⠂⠄⠁⠐⠊⠁⠁⠐⡾⠀⢀⣭⠀⠈⠇⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠉⠛⠃⢶⡁⠀⡒⠁⠀⠠⡴⠀⢀⣨⡀⠀⡛⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢈⠓⠀⠦⠄⣠⠋⠐⠀⣴⡆⠀⠀⠥⢀⣙⠓⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢁⡊⠁⡲⢄⣀⡰⠄⠀⢀⠿⠀⣈⠿⠄⣤⢉⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢤⠄⣨⠤⢀⣨⣀⠀⢀⡟⠀⠠⠜⠠⡄⠂⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣈⡁⣀⡧⢀⣈⡁⠀⠈⣏⠀⠐⣖⠓⢠⠛⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠴⠆⠸⠆⢠⡁⠀⠀⣫⠀⣀⠶⢀⣉⡋⣲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣃⡒⠲⠖⠠⡌⠁⠀⠶⠣⣀⡛⠠⣤⠓⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⡁⡘⠂⢰⠉⠁⣲⠃⠠⣬⡁⠲⠆⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣭⡉⣉⣀⡤⠀⣈⡛⠰⠶⢄⣬⠁⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡾⠴⣇⣰⠏⣁⠉⢠⠒⢤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⣬⣗⡯⢶⣭⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣴⠚⠛⢳⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⢏⠀⠈⠖⠀⠀⠠⡭⠉⠈⠙⠻⡟⠳⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠁⠀⠀⠔⠈⠀⠀⠀⠀⠀⡀⠀⠀⡤⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡛⠉⢀⠀⠀⠁⡀⠀⠆⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠴⠀⠀⡤⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⢀⡄⠀⠀⠀⠀⠀⠀⠁⣿
⠀⠀⠀⠀⠀⠀⢀⣸⡇⣠⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⡀⣿⣄⡀
⠀⠀⠀⠀⢀⡴⠋⣼⠁⠠⡄⠀⠀⠓⠀⠀⠀⡀⠀⠀⠀⠈⠂⠀⠀⡄⠀⠀⠰⠀⠀⠀⠀⠀⠠⢠⣏⠈⣷⡄
⠀⠀⠀⠀⢸⣧⠀⢿⣝⠀⠀⠀⠘⠀⠀⠀⠘⠍⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⢠⠟⡇
⠀⠀⠀⠀⢸⡈⠳⢦⣈⠓⢦⣄⡁⠀⠊⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠛⠀⠀⠀⠲⠀⠀⣀⣤⠞⢉⡴⠋⢰⡇
⠀⠀⠀⠀⠈⣇⠀⠀⠈⠓⠦⣌⣙⠶⠶⠶⢦⣤⣤⣌⣂⣀⣀⣀⣀⣀⣀⣀⡤⠴⠶⠞⣁⡤⠖⠋⠀⠀⡾
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠦⢤⣤⣄⣉⣉⣉⣉⣉⣡⡤⠤⠖⠚⠋⠁⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣈⠉⠓⠲⠶⠤⠤⣤⣤⣤⣄⣤⣤⣤⡤⠤⠖⠛⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠤⣤⣤⣤⣀⣤⣤⣤⣤⣤⠴⠶⠚⠋⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣿⣋⣙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠶⠶⠶⣶⡶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⢶⣶⣶⡾⠿⠿⣦⣶⠿⠻⢿⣶⡿⠻⢿⣭⣭⣉⣉⣩⣿
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡅⠀⠀⣿⠃⠀⠀⢰⣿⠀⠀⣾⣏⣉⣉⣉⣉⠁
⠀⠀⠀⠀⠀⠠⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣻⣇⠀⠀⢿⡇⠀⠀⣸⡇⠀⢠⣿⣋⣉⣉⣉⣹⣷
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡄⠀⢸⣷⠀⠀⣿⡇⠀⢸⣿⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⣿⡇⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⢸⡿⠀⠀⣿⡇⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⣾⡇⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣿⡇⠀⢰⣿⡀⠀⠀⣼⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠛⠉⠁⠀⠀⣿⡇⠀⢸⣿⠻⣷⣴⡿⠁⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⣿⡿⠁⠀⠀⣠⣴⡾⠟⠻⣷⡀⠀⢿⡆⠈⢻⣧⠀⠀⣰⡿⠁⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⢰⡿⠁⠀⢠⣾⠟⠁⠀⣠⣴⣿⡇⠀⠸⣷⠀⠀⢻⣇⣴⣿⡿⠿⠛⠛⠛⠛⠛⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⣠⣾⡇⠀⠀⣿⠃⠀⢀⣾⠏⠁⣼⡇⠀⢀⣿⠀⠀⢸⣿⠋⠁⢀⣀⣤⣤⣤⣀⡀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢰⡿⢿⡇⠀⠀⣿⠀⠀⠸⣿⠀⣰⡿⠀⠀⣼⡟⠀⠀⣼⣧⣴⠿⠛⠋⠉⠉⠉⠙⠻⢷⣄⠀⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠸⣿⡀⠀⠻⣷⣄⣠⣿⡿⠋⠀⢀⣼⠟⠀⠀⣰⡿⠋⠁⣀⣤⣶⠾⠿⣷⡄⠀⢸⣿⠀⠀⠀⠉⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⡿⠶⠶⠿⠷⠶⠿⠿⠶⠶⠾⠿⠿⠿⠶⠶⠶⠿⠿⠶⠶⠾⠿⠷⠶⠾⠿⠿⠶⠶⠶⠿⠷⠶⠿⠿⠶⠶⠶⠶⠾⠿⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠆⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠀⠀⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠄⠀⠀⠀⠀⠀⢀⢠⣴⣶⣶⣶⢶⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣒⣒⣒⢒⠒⣒⣒⣒⣒⣒⣲⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠠⠤⠴⠒⠒⣚⢒⢒⣐⣐⡇⠀⠠⠤⠤⠤⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠉⠉⠉⠉⠉⠀⠐⠒⠒⠒⠒⠒⠂⠠⠬⠭⠭⠭⠭⠍⠉⣹⣷⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠐⠒⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣉⣛⣟⣻⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿