Luxury Emojis & Symbols

⠀⠀⠀⢀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣿⣿⣿⣿⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠈⠀⠀⣄⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠸⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⠀⠀⠀⣶⣶⣶⣶⣶⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡿⠆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⣿⠉⣿⠉⠉⠉⡏⢹⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⢸⠀⠀⣿⣀⣿⣀⣀⣠⣧⣸⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠟
⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⠀⠀⠄⠀⠠⠀⠀⠄⠀⠠⠀⠀⠄⠀⠀⠆⠀⠀⠄⠀⠀⠀⠆
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠼⠷⠿⠿⠿⠿⠿⠿⠿⠀⠀⠦⠤⠤⠠⠤⠄⠤⠤⠠⠤⠄⠤⠤⠄⠀⠀⠄

⠀⠴⣶⣦⡀
⠀⠀⠈⠛⠿⣆⠀⢠⣾⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⡀
⠀⠀⠀⠀⠀⠸⡄⢺⡟⢡⠀⣷⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⠀⠸⡇⠀⢀⣴
⠀⠀⠀⠀⠀⠀⢳⣸⡀⠈⣒⡁⣀⠀⢀⡀⠀⠀⠀⠉⢿⣿⣦⠀⠀⣴⣦⣼⣿⡄⠀⠀⠀⣀⣸⣷⣶⣷⣿⡁⠀⣇⣀⠠⠄⠚⠉
⠀⠀⠀⠀⢀⡠⠤⢿⣷⣤⣼⣿⣿⣷⣿⣧⠀⠀⠀⠰⣼⣿⡟⠀⠀⣿⣿⣿⠿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣣⣶⣾⠇⢀⡒⠄
⠀⠀⠀⢰⡁⢤⠀⠰⣾⣿⣿⣿⣿⡛⠉⠉⣷⣶⡄⠀⠈⠙⢧⠀⣿⡿⠋⢀⡀⠠⠶⢦⣭⣭⣽⣿⣼⣿⣿⣿⡇⠀⠀⡑⠤⠃
⠀⠀⠀⠀⣩⣼⣶⣾⣿⣿⣿⣿⣏⣿⡦⢀⣽⣿⣿⣦⡀⠀⣬⣿⣿⣤⣶⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⣧⣼⣿⣧⢀
⠀⠀⠀⢠⣿⣿⣟⣿⣿⣿⣿⣿⣟⠁⠀⠀⠉⠀⠘⠻⠿⢷⣿⡁⢙⣿⣿⣥⣀⡀⠀⠉⠀⠀⠉⠛⠛⠿⣤⣤⣼⣯⣿⣿⡏
⠀⠀⠀⠚⠻⣿⣿⣿⠿⠛⢻⣿⣿⣿⣶⡤⣾⣶⣶⡶⠂⠽⢿⣿⣿⡟⠉⠙⢿⣷⣶⡿⠂⠠⠀⡽⠀⠀⠉⠙⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⠁⠘⢧⠾⠿⢿⣭⠀⠀⠀⢀⣾⣿⠁⢤⡀⠀⠀⠻⣿⣿⠆⠀⣘⣥⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⡷⠻⠁⠀⠀⣠⣶⣿⣿⠉⠀⢸⣿⠢⣄⠀⠀⠈⠛⣝⠻⣅
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠊⠄⡞⣀⠤⡀⠀⠀⢻⣿⢿⣿⠄⠀⢿⣿⣆⠈⠀⠀⡀⢤⠈⡆⠈⠳⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠙⠻⠿⠛⠉⠀⢰⡆⠳⣘⣦⡏⠀⠀⢸⢃⠜⠃⠀⠀⠀⠉⠻⠻⠀⠀⢇⣀⣠⢃⣀⠀⠈⠙⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠀⣽⣧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡍⠠⠈⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⢀⠔⠢⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠤⡐⠊⠐⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⡠⠀⠀⠀⠐⠤⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⠆⠀⠸⠃⠠⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠈⠢⡀⠀⠀⠀⠀⠐⠾⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣑⣀⠈⠢⢀⠄⠀⠈⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⢠⠔⠀⠀⡀⠀⡀⠀⠼⢠⠐⠁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠄⠠⠀⠀⠀⠀⡔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠑⠄⠀⠄⠀⠀⠀⠴⠀⠢⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⣀⠆⠀⠈⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠤⡀⠀⠀⠀⠀⠀⠁⠀⠬⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠚⠁⠘⠡⢀⠀⠀⠀⠠⠀⡆⠠⠀⠀⠀⢀⡠⠚⠀⠉⠒⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠖⠒⠚⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⣠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠂⠀⠀⠁⠀⠀⠠⣀⠀⠀⢀⠉⠀⠀⠐⡀⠄⠀⠀⡀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠈⠂⠀⠀⠐⠂⠀⠀⢀⠁⠢⡄⠁⠀⠂⠀⠀⠀⠈⠤⠆⢈⠀⠀⠀⠐⠀⠀⠈⠳⣱⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠠⠃⠀⣀⠀⠀⠀⠀⠀⠈⠀⠁⠀⢀⠀⠀⠀⠀⢀⠀⠘⠠⠈⠀⠀⠀⠀⡐⠀⡀⠀⡻
⠀⠀⠀⠀⠂⡀⠀⠀⠀⢈⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⢀⠠⠄⠁⠀⠁⠁⠀⠀⣿⠆
⠀⠀⠀⠀⠘⢠⡄⠀⠀⠠⠂⠊⠀⠁⢀⠀⡀⠀⠐⡀⠀⣴⠀⠀⡀⠀⠀⠀⠀⠐⠰⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠈⠂⠈⢀⠀⠄⠀⡠⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⡄⡆⠃
⠀⠀⠀⠀⠀⠀⡈⣁⠀⠀⠀⠀⠀⡱⠊⠂⠀⠀⠀⢀⠀⡀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠀⠀⢉⣃
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠄⠀⠀⠀⠀⢀⠁⡀⠀⠉⠀⠂⢀⡀⠀⠀⠁⠀⠐⠒⠀⠐⢁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⢀⢠⡀⠀⡀⠂⠀⠀⠀⠊⢛⣄⠘⠸⠀⣬⠛⠃⠀⠀⠀⠀⠠⠀⢀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠁⠀⠀⠀⢀⠔⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠐⠀⠈⠢⠀⠀⠀⢠⣀⣀⣀⠀⠀⠀⠈⠀⠀⠐⠢⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠂⠐⠀⡀⠀⠀⠣⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠄⠀⠀⠈⠀⠈⠄⠁⠘⠀⠀⠀⠴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⡨⠉⠀⠔⠀⠎⠀⠐⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢈⠔⢀⠐⠁⠀⠀⠀⠀⠀⢶⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠑⠀⠀⠐⠀⠐⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠒⠡⠄⢠⠀⠀⠀⡠⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠈⠢⠔⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣋⠐⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠀⢈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣮⡾⣿⢣⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣇⣦⡀
⠀⠀⠀⢀⣠⣀⡤⣮⡾⣫⠞⢡⣾⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡏⣾⢏⣿⢠⡀
⠀⢀⣴⣿⣿⡗⣺⢫⡾⠋⣰⡿⣻⣿⣿⣿⣿⣿⣿⣷⣦⣤⣶⡤⣄⣴⣾⣿⣿⣿⣷⣿⣼⣇⣾⡿⡇⡀
⣴⣿⣿⣿⣿⣗⡿⣸⡇⣾⡟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⣷⣧⢻⡆⣤
⣿⣿⣿⣿⣿⣿⡇⣿⢱⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⢷⣝⠷⣝⡻⣿⣿⣿⣿⣿⣏⣾⣿⢻⣮⣻⣿⣛
⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣧⡣⡻⣿⣿⣿⣿⣿⣿⣿⣿⣯⡝⢷⣍⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣙⣿⣿⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢨⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⡀⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠈⠟⠃⠈⠓⠧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⡒⡆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠉⠀⠀⠀⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠓⢤⠀⠀⠀⠀⡀⢠⠀⠀⠜⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⡀⠀⠓⠚⠉⠀⠁⠁⠀⠁⠂⢀⡤⠀⠀⡔⠢⣀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠀⡀⠀⠠⠔⠚⠀⠀⠄⠫⠤⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⢀⠀⠁⠀⠀⠀⠀⣰⠶⠡⠀⠀⠀⠀⠀⡀⠀⠶⠦⠈⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠅⢐⡠⣌⠀⠤⡀⠀⠀⢴⠃⠀⠀⠀⠀⠀⠠⢭⠐⠂⠀⠀⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠊⠁⡌⠙⣆⣆⡤⠞⣓⠒⠢⢄⠀⠀⣀⠀⢠⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣉⢉⢉⣩⠞⠁⢰⠒⣏⠑⡢⣠⣇⡎⣡⠤⠶⣐⠒⡒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠁⡴⣴⠋⡜⣡⢋⢧⠀⠸⡯⠤⠴⣂⠑⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠘⢠⡿⣌⣛⡻⢢⠤⠻⠂⠂⢠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡈⠳⣄⢀⣠⣾⣷⣦⠀⠀⠀⡈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠿⠏⢰⣈⠉⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣤⣬⣁⣤⡈⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⢉⡟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠾⠁⠙⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠑⢿⣀⣮⣤⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠛⠛⢻⠋⣷⣄⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢣⣰⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣄⠙⣿⣦⡄
⠀⠶⣿⣼⣿⣿⣿⠟⢀⣠⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣿⣿⣿⣿⣿⠁
⣶⣶⣿⣿⣿⣿⣿⣆⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠿⠛⢯⣿⣿⣿⣿⣿⣿⣧⠀⣦⣾
⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠏
⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣬⣉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⣤⣀⣀⣻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣽⣿⣿⣿⣶⣶⢈⠉⠉⠙⠛⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠋⠉⣉⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣻⣿⣿⣿⣿⣿⣿⣿⣧⡸⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣾⣿⣿⣿⣿⣿⡿⣿⣿⣿⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠛⠿⢿⣿⣿⣿⡿⠋⢻⣿⣿⣿⣿⣷⡌
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠹⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣴⣿⣿⣿⠏⠀⠀⠈⣿⣿⣿⠈⢿⣿
⠟⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠈⠈⠁⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣿⣿⡿⠃⠀⠀⠀⠀⣿⣿⣿⠀⠀⢿
⠀⣾⣿⣿⠟⣿⣿⡿⢿⣿⣿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⡆⠀⠸
⠸⡿⠻⣿⢰⣿⢰⡇⠈⢿⠀⠌⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⣸⡇⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⢰⣿⣿⣆⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀⠀⢻⡀
⠀⠀⠀⢀⣿⠘⣻⣿⣿⣿⣿⣍⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢦⣾⣿⣿⣥⣄⢸⡇
⠀⠀⠀⢸⡏⢀⢼⣿⢿⣿⡿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣯⡥⢼⡇
⠀⠀⠀⣾⠇⠀⠈⠁⢸⡏⠀⠈⠃⢀⣤⣿⣿⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⡄⠀⠰⡛⠛⢹⢿⡏⠙⠿⠀⣷
⠀⠀⢀⣿⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰⡾⢿⡿⠶⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⠀⠀⢴⢾⣿⣿⡅⠀⠁⠀⠚⠈⠁⠀⠀⠀⣿
⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⣼⣿⡗⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠂⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⣀⣀⣿⡇⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠘⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⡀⣀⠀⣸⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣶⣦⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀
⢤⡀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠈⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠂⠀⢀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣀⣀⠀⢨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠃⠠⢻⡧⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⣀
⠉⠙⠛⠈⠉⢧⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠼⠁⠀⠀⠀⠀⠀⠀⢻⣥⠤⠖⠶⠚⠋⠉⠁
⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⣠⣒⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉
⠖⠒⠒⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⢀⠞⢱⣤⣀⠀⠈⠁⠀⠀⠀⠀⠀⣯⠀⠀⠀⠠⣿⣦⣄⡀
⠀⣀⣀⠠⠤⠀⠂⠈⠁⠀⠀⠀⡰⠁⠘⣿⣿⣿⡆⠀⢀⠃⠀⠀⠀⠀⠙⠀⠀⠀⢀⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣼⣿⣿⣿⣿⡄⠰⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⡿⠹⣿⣷⡂⠀⢸⡄⠀⠀⡀⣀⠀⠀⣿⣿⣿⣻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⢸⣿⣿⣇⠀⣽⡿⠀⠀⠈⡇⠀⠀⢱⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⣿⣿⣿⣾⣿⠇⠀⠀⠀⡇⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⢸⣘⡯⠯⣭⡟⠋⠁⠀⠀⠀⠈⠑⢻⣭⠽⢽⡋⡇⠀⠀⠀⠀⠀⠀⢸⢘⡯⠯⣽⡟⠊⠁⠀⠀⠀⠈⠑⢻⣯⠿⢽⡃⡇
⣀⣀⡀⠈⠽⣧⣽⠦⠉⠀⡀⣠⣿⡀⢀⠀⠉⠴⣯⢼⠿⠁⢀⣀⣀⣀⣀⡀⠈⠿⡧⣽⠦⠉⠀⣀⣠⣟⡄⢀⠀⠉⠴⣯⣼⠯⠁⢀⣀
⡶⡼⠾⡧⡅⠉⠈⠀⣀⣦⡹⡞⣽⢹⢏⣴⣀⠀⠁⠉⢨⣼⠿⢦⢶⡷⡴⠿⣧⡅⠉⠈⠀⣀⣦⡽⡏⣯⢻⢏⣴⣀⠀⠁⠉⢨⢼⠿⢧
⠟⣧⠳⡟⠂⠀⠀⠀⢮⣧⣥⢮⣿⡽⣬⣽⡵⢀⠀⠀⠐⢻⠿⣴⣸⣅⣦⠳⡟⠂⠀⠀⠀⠮⣧⣥⢧⣿⡵⣬⣼⡵⠀⠀⠀⠐⢻⠞⣼
⡿⠉⠃⠀⠤⣼⠮⣿⡅⠉⠓⠒⠚⠒⠚⠉⠨⣼⠷⣧⠄⠀⠘⠑⢟⡻⠈⠃⠀⠠⣼⠾⣧⠅⠉⠓⠒⠛⠒⠚⠉⢨⣼⠵⣧⠤⠀⠘⠋
⠁⠀⠀⣈⣳⠮⡥⣟⢹⠀⠀⠀⠀⠀⠀⠀⡇⣻⢌⠵⣾⣅⠀⠀⠈⠁⠀⠀⣨⣷⠯⡡⣟⢸⠀⠀⠀⠀⠀⠀⠀⡏⣻⢌⠵⣞⣅
⠀⠀⠛⠾⢗⣛⣛⡿⢺⠀⠀⠀⠀⠀⠀⠀⡇⢿⣻⣛⣻⠷⠛⠀⠀⠀⠀⠛⠼⣟⣛⣛⡿⢸⠀⠀⠀⠀⠀⠀⠀⡟⢿⣛⣛⣺⠗⠛
⣆⠀⠀⠐⢉⣵⣂⣿⡞⠀⠀⠀⠀⠀⠀⠀⢳⣿⣰⣮⡙⠃⠀⠀⡰⢆⠀⠀⠘⢋⣵⣆⣿⡞⠀⠀⠀⠀⠀⠀⠀⢳⣿⣰⣾⡙⠃
⡟⣶⢃⡄⠀⠘⠑⠃⣂⣦⣭⣩⣭⣍⣽⣶⣐⠘⠋⠃⠀⢠⣘⣯⢿⡿⣵⢃⡄⠀⠘⠙⠃⣂⣶⣯⣩⣭⣍⣭⣶⣐⠙⠊⠃⠀⢠⡘⣶
⣟⣅⣋⣟⡅⠀⠀⠀⠵⣧⣆⡕⣿⢺⡰⣾⠯⠀⠀⠀⢨⣿⣛⣸⣺⣗⣅⣛⣿⡅⠀⠀⠀⠵⣷⢆⡇⣿⢪⡰⣼⠯⠀⠀⠀⢨⣻⣙⣸
⠭⠬⠟⠃⣂⣶⣴⣀⠀⠁⠞⢿⢾⡞⠷⠈⠀⣀⣦⣶⣐⠘⠻⠥⠭⠯⠬⠟⠃⣂⣶⣴⣀⠀⠁⠞⢱⠷⡿⠳⠈⠀⣀⣦⣶⣐⠘⠻⠥
⠀⠀⠀⢰⠻⣇⢛⣧⢖⠀⠀⠀⠛⠀⠀⠀⡲⣼⡛⣸⠟⡆⠀⠀⠀⠀⠀⠀⢰⠻⣇⢛⣥⢖⠀⠀⠀⠛⠀⠀⠀⣢⣬⡓⣸⠟⡆
⠀⠀⠀⢸⢼⡷⡷⢞⣿⡱⠆⠀⠀⠀⠰⢎⣿⡳⢿⢾⡧⡇⠀⠀⠀⠀⠀⠀⢸⢸⡷⡷⢾⣿⡷⠆⠀⠀⠀⠰⢎⣿⡰⢾⢾⡗⡇
⠀⠀⠀⠸⣴⡏⢬⡟⠮⠀⠀⠀⣤⠀⠀⠀⠵⢻⣥⢹⣦⠇⠀⠀⠀⠀⠀⠀⠸⣴⡏⣬⡟⠯⠀⠀⠀⣤⠀⠀⠀⠴⢻⡥⢹⣦⠇
⣒⢒⣦⡄⠍⠿⠻⠉⠀⡀⢦⣾⣾⢧⡶⢀⠀⠉⠟⠿⠩⢠⣴⡒⣒⣒⢒⣦⡄⠍⠿⠻⠉⠀⡀⢢⡼⡶⣷⡴⢀⠀⠉⠟⠿⠩⢠⣰⡒
⣯⡏⣍⣯⡃⠀⠀⠀⣲⡟⠎⡣⣿⢼⠱⢿⣖⠀⠀⠀⢘⣿⣭⢙⢽⡯⡋⣭⣿⡃⠀⠀⠀⡲⡿⠎⡇⣿⢜⠹⢻⣖⠀⠀⠀⢘⣽⣩⢙
⣷⠿⡌⠃⠀⢠⡠⣄⠍⠟⣛⣙⣛⣋⣻⠻⠩⢠⢄⡄⠀⠘⢩⢿⣾⣷⡻⡍⠃⠀⢠⡠⡄⠍⠿⣛⣙⣛⣋⣻⠿⠭⣠⢄⡄⠀⠘⢡⢿
⠏⠀⠀⢠⣌⡻⠍⣿⢧⠀⠀⠀⠀⠀⠀⠀⡞⣿⠹⢟⣡⡄⠀⠀⠹⠏⠀⠀⢠⣌⡻⠏⣿⢧⠀⠀⠀⠀⠀⠀⠀⡼⣿⠹⢿⣡⡄
⠀⠀⣤⢶⣯⣭⣯⣷⣼⠀⠀⠀⠀⠀⠀⠀⡇⣾⣽⣭⣵⡖⣤⠀⠀⠀⠀⣤⢶⣯⣭⣯⣷⢸⠀⠀⠀⠀⠀⠀⠀⣧⣾⣭⣭⣵⡶⣤
⡀⠀⠀⢉⡽⢞⡑⣯⣸⠀⠀⠀⠀⠀⠀⠀⡇⣽⢊⣲⢿⡋⠀⠀⢀⡀⠀⠀⢉⡿⣖⡑⣯⢸⠀⠀⠀⠀⠀⠀⠀⣇⣽⢊⡳⢿⡋
⣵⣀⡄⠀⠒⢻⢶⡿⡃⣀⡤⠤⣤⠤⢤⣀⢐⢻⡶⡟⠂⠀⠠⡠⣮⣵⢀⡄⠀⠐⢻⢶⡟⡃⣀⡤⠤⡤⠤⢤⣀⢐⢻⡲⡟⠒⠀⢠⣀
⣧⡟⡴⣧⠄⠀⠀⠀⢞⡟⡛⢞⣿⣳⢛⣻⡳⠀⠀⠀⠠⣼⢶⠻⣹⣎⠟⣶⣧⠄⠀⠀⠀⢞⣟⡛⡞⣿⡳⢛⢻⡳⠀⠀⠀⠠⣼⢦⢻
⠷⠳⡶⡟⡃⣀⢀⠀⠉⠟⣱⢧⣻⣸⣞⠻⠉⠀⡀⣀⢘⢺⣶⠞⢾⡷⠳⣶⡟⡃⣀⢀⠀⠉⠟⣳⣏⣿⡽⣎⠻⠉⠀⡀⣀⢘⢺⣶⠞
⠉⠉⠁⢀⣶⡗⣻⠖⣀⠀⠁⠘⣽⠁⠉⠀⣀⠲⣟⢺⣶⡀⠈⠉⠉⠉⠉⠁⢀⣲⡗⣻⠖⣀⠀⠉⠘⣯⠋⠈⠀⣀⠲⣟⢻⣶⡀⠈⠉
⠀⠀⠀⢸⣨⣗⣖⡛⣧⢄⡀⠀⠀⠀⢀⣠⣼⣛⣲⣺⡅⡇⠀⠀⠀⠀⠀⠀⢸⢨⣗⣔⣻⣯⢄⡀⠀⠀⠀⢀⡠⣾⣟⣶⣺⡅⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⢀
⣀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣤⣤⣾⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⣿⡀⠀⠄⠀⠀⠀⠀⠀⠀⠨⢀⣴⣿⣿⣿⣿⠛⣿⣿⣦⠀⢨⡶⣾⣷⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠠⠄⣡⣾⣿⣦⡀
⣿⣿⣿⣷⡀⠐⢀⣴⣶⣦⣤⣶⣿⣿⣿⠟⠻⢿⠀⠈⠻⣿⣿⣿⡇⢈⣿⣦⣀⠀⠢⠀⡀⠀⠀⠀⠀⢀⣀⣾⣿⣷⡄⠈⠀⠴⣿⣿⣿⡿⠃
⣿⣿⣿⣿⣿⣄⠈⠉⠻⣿⣿⡿⢿⣿⣿⣄⠀⣠⣴⣿⣶⣝⠻⣿⣿⣿⣿⢯⠙⢿⣷⣌⠀⠀⢄⣠⣤⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣆⠀⠀⡙⢿⣿⣦⣉⣻⢿⣾⣿⣿⣿⠟⣿⢿⣷⣈⣻⣿⣾⣤⣾⣿⣿⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⢀
⠈⠻⣿⣿⣿⣿⣿⣷⣾⣿⣆⠛⠛⢿⣿⣦⣝⣿⣋⣤⣾⣿⣷⣬⡋⠉⠉⢻⢿⡿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⣠⣦⣤⣾⣿
⣿⣶⣍⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣶⣤⣉⡋⠛⠻⣿⣿⣿⣿⣿⠟⠁⠀⢠⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣦⣩⡿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡙⣿⠟⢁⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣮⣿⢀⡟⡯⢸⠻⢸⠃⡏⠿⠻⠁⠀⠛⠙⠻⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣡⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣄⡒⠁⠄⠂⢁⠀⠠⠀⠀⠀⠀⠁⠸⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⣀⣤⣴⣶⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠴⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⢀⣐⣂⣉⣭⣭⣤⣤⣤⣶⣴⣶⣶⣶⣶⣶⣶⣶⡆⠐⠒⢦⣿⣶⣤⣀⢀⣀⣀⣀⣀
⠀⠀⢠⢖⣿⣶⣶⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠃⠀⢸⡷⠿⠛⡏⠉⠓⠢⢤⣤⠬⠉⠒⢤⣀
⠀⠀⣌⠘⡇⠈⠉⠉⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢘⢺⡇⠀⠀⠙⢄⣀⠀⠀⠈⠉⠒⠦⣄⣈⡙⠒⠤⡀
⠀⠀⣿⣿⠇⠀⠀⢀⡏⣿⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡆⡇⢰⣄⣀⡀⠈⠙⠓⢶⠠⣄⡀⠀⠉⠙⠻⠿⣿⡆
⠀⠀⠉⠉⠀⠀⠀⢸⣵⢸⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠸⠈⣿⣿⠀⠀⠀⠀⣇⠀⠈⠙⡖⠢⢤⣀⠀⠉⠐⠲⡄
⠀⠀⠀⠀⠀⠀⠀⡿⡟⢸⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⡿⣿⠀⠀⠀⠀⢹⠀⠀⠀⢣⠀⠀⢳⠈⠹⡖⢦⣼
⠀⠀⠀⠀⠀⠀⢀⡇⡇⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡄⣿⠠⣄⣀⣀⡘⠀⠀⠀⢸⡀⠀⢸⠀⠀⡇⠀⠛⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠉⠙⠧⠤⠤⠤⠀⠐⠒⠒⠂⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⢸⠀⢸⡇⣿⠀⠀⠀⠀⠀⠉⠉⠉⠒⠁⠀⠀⠀⠀⢁⣀⣾⣇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢀⠤⠐⠒⠂⠚⠉⠉⠁⠀⠀⢁⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣼⠀⠀⣧⢸⢀⠀⠀⠀⠀⠐⠀⠀⠈⢑⣚⣓⣁⣠⣭⣭⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⣾⣶⣶⣶⣶⣶⣒⣲⣾⣓⣈⠉⣩⣿⣏⣩⣭⣭⣭⣽⣿⣿⣿⣿⡆⠀⡟⢸⠀⠵⠶⠖⠒⠒⠒⠒⠛⠉⠉⠉⠉⠉⠉⢙⣿⣿
⠀⠀⠀⠀⠀⠀⢸⠿⠿⠿⠿⠿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠁⠀⠀⢸⡇⠀⣀⣀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⣾⣸⣇⣽
⠀⠀⠀⠀⠀⠀⣤⣌⣥⣄⡀⠀⠨⣭⣤⠀⠀⠀⠀⠀⠀⢰⣾⣿⡿⠂⠀⢀⣤⣤⣤⣤⣤⡤⢤⠀⠀⢸⣧⣾⣿⣿⣇⠀⠀⠀⢀⡀⠤⠄⠒⢰⣿⡇⢀⠴
⠀⠀⠀⠀⠀⠀⢛⣿⣿⣿⣡⡀⠀⠚⡷⠆⠀⠀⣀⠀⠀⢠⣼⠷⠂⠠⢠⣿⣿⣿⣿⣿⣿⡇⢸⠀⢰⠘⣿⣿⣿⣿⣿⠀⠉⠁⠀⠀⠀⣀⣠⣼⣿⠁
⠀⠀⠀⠀⠀⠀⣟⣿⣿⡿⣟⠁⠀⢤⣭⣭⣤⣤⣭⣤⣤⣭⣭⣥⣤⠂⠨⠭⠭⠭⠭⠭⠭⠭⠥⠀⠸⠁⣿⣿⣿⣿⣿⠀⢀⣀⣀⣼⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠬⠷⠶⠶⡶⠒⠒⠒⠒⠒⠒⠖⠂⠀⠈⠉⢛⣟⠉⠩⠁⠀⠀⠈⠁⠀⢀⣿⣿⣿⡿⠁⠀⠭⣻⣿⣿⣿⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠳⣶⣶⣶⣦⡤⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣧⢀⣠⣶⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠟⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⢿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠿⣿⣿⣿⣿⣉⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠔⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣉⣈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠦⢤⣀
⠀⠀⠀⢐⣶⣿⣤⣤⡤⣄⠀⠀⠀⢼⠀⢀⠀⠀⠘⡆
⠀⠀⠀⠛⣿⠋⠀⠀⢱⠈⢳⠀⠀⠈⠲⣎⠀⢸⢂⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢸⡷⣀⢀⡗⠊⠀⠈⡇⠀⠀⠀⣹⡄⣴⣾⣡⣠⡄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠳⣌⠁⠀⠀⠀⠀⠙⠀⠀⠠⠅⡜⠊⣉⣀⣔⡽⠁⡀⠀⢀⡀⠀⠑⠄
⠀⠀⠀⠀⠀⣀⠉⠒⠢⠄⠀⠠⠀⢀⡤⠄⡔⠊⠀⠉⢳⣠⠎⠀⡎⠁⠀⠀⠀⢸⢀⠤⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠉⠉⠀⠀⠀⠀⠀⠀⠜⠑⡆⠀⢇⠀⠀⠀⣀⡞⠁⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⣀⠤⠤⢤⡀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣀⠉⠛⠉⠁⡅⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡞⠀⠀⠀⠀⠘⣠⠎⠀⠀⠐⠦⠉⠀⠤⠶⠤⣤⣖⡻⠯⣄⡋⠰⠀⡈⠠⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⡅⠀⠂⡀⢀⡴⢧⠀⠀⠀⠠⠚⠁⢺⠀⠀⠀⠀⡼⢯⣷⣀⠉⠒⠤⣁⠂⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠑⢄⠀⠀⠀⠐⣿⡷⢲⡀⠀⠀⠀⠈⠀⢀⡠⠞⠁⠐⠈⠂⠀⠀⠓⠂⠤⠤⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠁⠒⠒⠚⣻⠁⠀⠈⠑⣄⠔⠀⠀⠐⠀⢀⠄⣀⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⣀⣀⣠⣂⡀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠋⠁⠀⠈⠢⢀⣀⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⡓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠤⢄⡀⡠⡆⣀⠷⡶⢥⡙⢉⡮⢶⠮⣀⢰⣄⢀⡠⠤⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠻⡑⡞⠙⠠⢹⣜⢤⡲⣝⣫⢖⡤⣣⡏⠄⠋⢣⢊⠟⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠲⠺⠃⢓⣚⣰⢡⡃⡋⠌⢦⢺⡾⢷⡇⡰⠡⢉⢘⡌⣆⣒⡚⠙⠷⠶⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⡛⣬⣠⠒⢾⣗⣹⣷⣏⡄⣨⢻⠛⠛⡟⣅⢠⣹⣾⣟⣺⡷⠒⣄⣥⢛⢸⣇
⠀⠀⠀⠀⠀⠀⠱⡟⣭⡓⢯⢀⢠⠚⡴⠏⠷⡽⣟⢉⠷⣾⠍⣿⡻⢟⣿⠩⣷⠾⡉⣻⢯⠾⠹⣦⠓⡄⡀⡽⢚⣭⢻⠊
⠀⠀⠀⠀⠀⠀⠀⠱⣱⠬⢘⣸⢾⢷⣝⡧⢮⢗⢿⡙⣓⢿⡿⢿⣗⣺⡿⢿⡯⣚⣋⡿⠲⡥⢮⣫⡾⡷⣇⡃⠥⣎⠎
⠀⠀⠀⠀⠀⠀⠠⠾⠓⡀⠡⠲⢿⣾⡟⢙⣟⠳⣾⠟⡧⣲⣯⠋⠿⠿⠙⣭⣖⢼⠻⣷⠞⣻⡋⢻⣷⡿⠖⠬⢀⠚⠵⠄
⠀⠀⠀⠀⠀⠀⢄⡜⣓⢶⡋⠌⠋⠽⣹⣧⣽⣜⢩⣫⡅⣭⣾⣾⣿⣿⣷⣷⣭⢨⣝⡍⣣⣏⣼⣏⠯⠙⠣⢙⡶⣚⢣⡤
⠀⠀⠀⠀⠀⡠⢜⣏⢠⡳⣈⣓⣦⣞⣧⣥⣿⣏⡧⠛⣺⣿⣟⣽⡷⢾⣯⣿⣿⣟⡻⢼⣹⣿⣬⣼⣱⣴⣚⣥⢞⡄⣹⡣⢄
⠀⠀⠀⠀⢾⠰⡇⢈⡷⣹⢾⣋⣿⠀⣿⢎⣹⣹⣿⡇⣿⣿⣹⣏⢾⡿⢹⣟⣿⣿⢸⣯⣏⣏⡱⣿⠀⣿⣙⡷⣏⢾⡁⢼⠆⡷
⠀⠀⠀⠀⠀⠑⢪⣏⠘⡵⢛⡩⠟⢍⡟⡛⣿⣏⡗⣤⢽⣿⣯⣻⡷⢶⣟⣽⣿⡯⣤⣺⣹⣿⢛⢻⡭⠻⢍⡙⢮⠃⣹⡔⠊
⠀⠀⠀⠀⠀⠀⠊⢣⡭⠾⡅⢂⣄⣲⣹⡟⣛⢫⣘⣝⡃⣚⢿⢿⣿⣿⡿⡿⣛⢘⣫⣃⡝⣟⢻⣏⣖⣠⡐⣨⠷⢭⡜⠓
⠀⠀⠀⠀⠀⠀⠐⢶⡤⠁⡒⠴⣾⢿⣧⣨⣯⡴⣿⣦⡗⠽⣛⣄⣶⢶⣠⣻⠯⢺⣴⣿⢦⣽⣅⣼⡿⣷⠦⢒⠈⢤⡲⠂
⠀⠀⠀⠀⠀⠀⠀⡰⡹⢒⢨⢹⢾⡾⣫⡗⠞⡦⣾⣡⡍⢾⣷⣾⡯⢽⣷⣾⣗⢭⣍⣷⠴⡣⢺⣝⢷⡷⡏⡅⡒⢏⢆
⠀⠀⠀⠀⠀⠀⡰⣧⣛⠥⣞⠘⠘⢤⠱⢦⡶⣳⣯⣈⡶⢿⣂⣿⣵⣮⣿⣐⡿⢾⣁⣽⣞⢴⡴⠞⡤⠃⠃⣳⠬⣛⣼⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣥⢛⠙⠤⢾⡯⣽⡿⣏⠃⢑⣼⣤⣤⣧⡊⠘⣹⢿⣯⢽⡷⠤⠋⡛⣬⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠼⢶⣄⡭⠭⠹⡘⡅⣍⢂⠞⢼⢷⡾⡧⠱⡐⣈⢨⢃⠏⠭⢥⣠⡦⠶⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣴⡡⢧⣠⠐⣸⢫⠊⠵⣫⣝⠮⠃⡝⣇⠒⣄⡼⢌⣦⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⠒⠊⠁⠙⠇⠉⡲⠷⡚⣡⣈⢓⠾⢖⠉⠸⠋⠈⠐⠲⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⡥⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠘⢛⢃⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠁⠀⠒⢒⠓⠓⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠈⢉⢁⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠂⠀⠐⠚⠘⠘⣤⡀⠀⠀⠄⢀⣀⠀⠀⠀⠀⢰⠶⢀⣠⡴⠄⠁⠀⠀⣍⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⡤⠆⠲⡄⠀⣌⡓⠀⠤⠔⠒⠒⠉⠉⠉⠀⠈⠉⠉⢉⠙⢒⠤⢤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⡀⠀⢀⠀⣀⣀⣷⠀⠁⠀⠀⠀⠀⠀⢀⡀⠄⠤⠶⢶⡟⠛⠁⢀⣠⣧⡬⢄⣀⠈⠁⠒⠆⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⡀⠀⠀⢰⠀⠀⠀⠀⠨⠁⢀⡾⠀⠀⠀⠀⢀⠀⣈⣒⡤⢓⣂⠄⢐⡀⠄⢈⡀⠠⠥⠀⠠⢉⡉⠥⠄⠌⠐⣀⡠⠽⠧
⠀⠀⠀⠀⠀⡍⠀⠀⠐⠊⣳⢸⠀⠈⡀⠈⠀⠀⢈⢉⠉⢰⠒⣶⣌⣘⠳⡖⡭⠐⠚⠭⠤⣍⡉⠚⢯⠥⢞⡩⠤⣀⠈⠠⣄⣲⣤⠶⠍⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡀⠀⠀⢀⣈⣁⣸⣄⣌⠂⠤⠤⠤⠤⠤⠆⠹⠒⠀⠀⢆⠻⡇⢈⠁⠀⠈⠁⠀⠈⠑⠒⠠⠔⠠⠀⠐⠀⠈⠁⠤⣐⠤⣐⡢⢀⣀⣵⠠⠔
⠀⠀⠀⠀⠈⠁⠒⠒⢺⠙⠉⠐⠒⢜⠂⠀⠀⠀⠀⢀⠁⢾⣣⠄⠀⠰⣒⡟⢛⠾⠛⠴⠖⠢⠾⢷⣶⣦⣴⣄⡀⠇⠴⠖⢒⣉⡁⠥⠔⠒⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⠥⠬⠼⠦⠤⣀⣀⣀⣠⡴⠋⠉⠉⠋⠉⣕⠖⠅⠿⠁⠀⠀⢀⠀⠀⠈⠈⠉⠓⠑⠠⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠉⠀⠀⠈⡍⠁⡟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠐⠈⠉

⡟⠉⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⠿⡿⣿⣿⠿⣿⡿⡿⢿⢿⠿⠿⢷⣿⣿⣿⣿⣏⣿⣹⣻⣿⣿⣿⣿⣯⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⠀⠁⠂⠘⠰⠀⠂⠂⠰⠐⠀⠛⠇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⢋⡀⠀⠀⠀⢀⠀⠄⠢⠀⢀
⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢁⠂⠠⠠⠀⠀⠀⠀⠀⠢⠔⠔⡒⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠅⠪⠚⠀⠀⠀⠀⠀⢀⠀⠈⠒⠄⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠖⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠠⠀⡀⠀⠀⢀⠀⠀⠊⠀⢀⠁⠀⠀⠐⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠚⢀⠀⠀⠐⠒⠀⠀⠀⠂⠠⠀⠴⢁⡡⠀⡄⠘⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠐⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠠⠀⠀⠄
⠀⠀⠀⠀⢀⠀⠀⠀⠘⠀⠀⠀⠈⢀⠀⠈⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣀⡐⠂⠀⠀⣢
⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⡄⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠂⠀⠀⠉⠀⠀⠀⠀⠀⣆⣈⠠⠘
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠦⠤⠄⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠉⠀⠀⠠
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⣠⠞⢉⡞⢁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢿⡞⠡⠁⠘⠛⠉⢉⠉⠉⠙⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠲⣤⡀⠀⠀⢰⠏⠀⠆⠈⠀⠃⡠⠐⠈⠀⠀⣠⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡙⢦⡀⣿⠀⠀⡄⢀⠔⠉⠂⠀⠀⠀⠘⠛⢒⡶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣬⣿⠍⠛⠳⢦⣄⠙⢦⡙⣿⠀⠀⡴⠡⠄⠀⠀⠀⣠⣤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⢾⣯⠁⠀⠠⠀⡀⠀⠉⠂⠈⠳⣄⠹⡌⢧⣈⡤⠶⣤⡴⠶⣄⣈⣩⠿⠋
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠐⠂⢈⠀⠁⠀⠀⡨⠓⢢⡼⢧⡟⠉⠛⣆⠀⡏⠀⠀⢸⡏
⠀⠀⠀⠀⢠⣏⣤⡤⢤⡄⠀⠁⡀⠀⠀⠁⡴⠶⠿⣤⣸⡄⠀⠀⢸⣦⣷⣀⣀⣼⣁
⠀⠀⠀⠀⠈⠀⠀⢰⣯⠤⠶⣺⠏⣀⣠⡼⡇⠀⠀⠈⢻⣷⣦⣤⣿⠁⠀⠙⡿⠁⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠁⠀⣽⢦⣄⣠⣾⠀⠈⢹⣿⣄⣀⣠⣧⡀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⣿⣿⣦⣤⣼⠋⠉⠹⣿⠟⠙⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡶⠃⡴⠛⠛⢿⡀⠀⢀⣟⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⣇⠀⠀⠀⣿⣶⣾⡿⠦⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢳⡶⠖⠛⢻⡏⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⣸⣷⣄⣠⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢛⡟⠉⠩⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠒⠉⠉⠉⠉⠉⠉⠉⠉⠑⠒⠶⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣴⣒⣒⣉⡛⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠒⠒⠶⠦⠤⠤⢤⣀⣀⡀⠀⠀⠀⢀⡤⢶⠯⡉⣾⠁⠀⢀⡔⠙⢦⡀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠈⠉⠑⡞⢁⠔⡁⠀⢹⣿⡀⠀⠊⢀⠄⠀⢱⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⠖⠁⢀⣀⠤⠖⠒⠉⠁⠀⢀⣠⠤⠀⠀⠀⠀⣸⡱⠋⣰⠃⢀⠞⢿⣇⠀⠐⢁⣀⠴⠋⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠈⠀⠀⠀⢠⠤⠒⠊⠁⠀⠀⠀⠀⠀⣰⡿⠁⠀⠁⠠⠋⢀⡸⠝⠒⠚⠉⠀⠀⠀⠀⠀⣀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠉⠓⠲⠤⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢹⠁⣼⣿⣿⣶⠊⠉⠀⠀⠀⠀⠀⠀⠀⢀⠴⢋⣥⠢⣽
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠒⠒⠒⢒⠋⠀⠺⠟⠛⠛⠛⠉⠀⠀⠀⠸⠀⠀⠀⠀⣰⠃⣼⣿⣿⣿⡏
⠀⠀⠀⠀⢠⠖⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠁⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣼⡿⠁⠈⢿⡇
⠀⠀⠀⠀⡇⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⠔⠊⠁⠀⣀⣀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠠⣔⣱⣿⣇⠀⠀⣼⠃
⠀⠀⠀⢰⠧⣄⠁⡈⠀⠀⡀⠤⠄⢀⡀⠀⠀⢀⠔⠈⠁⠀⠀⠈⠀⡴⠊⢉⡠⠤⠬⡑⠦⠀⠀⠀⠀⠀⠀⡀⢄⣰⠷⠟⠋⠉⠻⣿⣿⣦⡶⠋
⠀⠀⠀⠸⣀⠈⣽⣳⣦⣄⣀⣈⠀⠀⠂⠀⢰⠁⠀⠀⠀⣀⠠⢊⡜⠀⣴⣿⣷⣦⣄⢱⢘⠀⡀⢄⣀⠤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⣿⣦⣱⢡⢙⠹⢙⢳⠿⣿⡏⠑⠒⠛⠛⠛⠛⠛⠋⠉⠀⣼⣿⠟⠻⢿⣿⣿⡬⠖⠚⠉
⠀⠀⠀⠀⠈⠻⢭⣛⣦⣤⣌⣌⣸⡿⢡⣶⣿⣷⣶⣶⣶⣿⡗⠀⣼⣿⠁⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠚⠒⠲⠦⠬⠭⠽⢟⣛⣛⣛⡫⠴⠺⣿⣿⣄⠀⣀⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⡿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠀⠀⠀⠄⠀⠾⠉⠉⠓⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠂⠀⣀⠰⣆⢀⠓⠂⠈⠀⠀⠠⠄⠀⠈⠀⠠⠐⠁⠀⠀⡀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⡞⠁⠀⠀⠀⡀⠀⠀⠀⠘⠀⡀⠀⠸⠄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⢂⣀⣤⣧⢹⢀⡀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠃⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠃⣤⠴⠖⠛⠉⠁⢠⡄⠀⠀⠈⠶⠏⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⣰⣷⡀⠀⠀⠀⠤⠀⠀⠀⢸⠷⠃
⠀⠀⠀⠀⠀⠀⠠⠚⠁⢠⡄⠀⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡄⢀⠀⢠⡄⠀⠀⠀⠻⠏⠠⠀⠀⠀⢶⠁⠀⠸⠛⠂
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠛⠋⠉⠀⠀⢠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠃⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟
⠀⠀⠀⠀⠀⣠⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢀⠞⠀⠀⠀⠀⠀⢀⡀⠀⠐⠂⠀⠀⠀⢶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣿⠟⠟⣶⣄⠀⠜⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠀⠀⠐
⠀⠀⠀⠰⢾⣯⣬⣾⡿⡉⢳⡀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢾⡀⠀⢠⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⣴⣦⠀⢀⠀⢠⣾⣿⣿⡄
⠀⠀⠀⠀⢇⠀⣙⢿⣆⣤⣿⣷⠴⣿⡀⠀⠀⠀⠀⢀⣠⣼⣿⠦⣄⢸⡇⠀⠋⠀⠀⠀⠀⣀⣀⡀⠀⠀⢰⣿⡏⠀⠛⡑⠴⣟⠉⢻⡇
⠀⠀⠀⠀⠘⡄⠀⠏⢿⡿⠟⣿⡿⣿⣿⣶⣶⣶⣿⠿⣿⠿⠃⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢁⠶⣶⡄⣿⡄⢸⡆⡭⡄
⠀⠀⠀⠀⠀⠈⢦⡀⠈⢷⣤⣾⠷⡉⠛⠛⠛⠉⠀⠀⠀⠂⠀⣰⡖⣭⠆⠀⠈⡄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠘⠃⡿⠗⠈⠉⠁⠈⠙⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠑⣄⠈⠻⡿⠛⠛⠦⡤⠀⠉⠙⠁⠀⠀⢠⠃⠇⠸⣄⠀⠠⠜⠂⠀⠀⠀⢀⠀⠉⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⠈⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡴⠁⠀⠀⠛⠹⢶⡆⠀⠀⠀⠘⠚⠀⠘⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠒⠀⡀⠀⠔⠲⡆⠀⠀⠀⠁⠀⠀⠐⠀⢘⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⣠⠀⠀⠀⠙⠀⠶⠀⠀⠀⠛⠀⠀⠀⠀⡶⠀⠘⡄⠀⠀⠀⠀⠂⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢉⡄⠀⠘⠋⠀⡀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢦⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠁⠀⠀⠂⠈⠁⡨⠄⠀⠀⠀⢀⠀⡄⠀⠀⠀⢰⡀⢀⣹⣦⣀⠀⠀⠀⠈⠀⠀⠀⠀⠖⢀⣴⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣦⣀⠀⠀⠠⠊⠀⠐⠢⠀⠁⣀⡄⠁⠀⠀⢠⠿⠿⠿⠋⠀⣼⣷⠦⠄⠀⠀⠀⢀⣡⣶⣿⢋⣝⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠻⣿⣷⣦⡇⠀⠀⠘⠟⠀⠀⠀⠈⠠⠀⢀⠂⠀⠀⣀⣤⣾⡿⠋⠀⠀⣹⡿⠛⠋⠉⠉⢈⡥⠍⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢏⠉⣶⠄⠙⠋⠁⠰⠀⠀⢀⣈⣥⣤⣄⠀⠀⢹⣶⣶⣿⡿⠟⠋⠀⣀⣴⡾⠋⠀⠀⠰⡦⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⣜⣋⡒⢃⠀⠀⣴⣶⠖⠚⠛⠉⢉⠀⠙⠋⠀⠀⡸⠉⠩⢤⣤⣴⣶⣿⠟⠁⠀⠀⠏⢣⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠠⢦⣤⣀⣎⠀⠀⠀⠀⠈⠀⢀⡶⢘⠒⠀⠀⢀⡀⠰⠊⠀⠀⠀⠀⡉⢤⠚⣥⣤⣤⣤⠀⢠⠀⡀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠉⠁⠀⠀⢪⠀⢀⠀⠁⠀⠀⠂⠀⠔⠀⠈⠀⠀⠀⠀⠈⠀⡇⢠⣤⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡆⠀⢀⣠⠤⠀⠲⠉⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⢀⣠⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋⠈⠁⠃⠰⠞⢀⡴⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⠉⢀⣀⠀⠈⠑⠂⠁⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⠾⠿⠾⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⢵⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⢠⣧⠤⣤⣦⡽⠃⠀⠀⢶⣀⠀⠀⠀⠐⢠⡐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠐⠒⠻⡾⠀⠀⠀⠀⠈⣋⠀⠐⢦⣾⣯⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠇⠀⠀⠀⠀⣻⡀⠀⠀⠀⢠⠇⠀⠀⢠⡆⠀⠀⠱⡶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠉⢨⡖⠐⢢⡀⠀⠀⢀⠜⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠞⠋⢿⠁⠀⡌⠀⠀⠀⣹⣿⣿⣷⣄⣀⣀⣤⡾⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢆⡀⢀⡀⠀⠰⠒⣉⠛⠉⠈⠉⠛⠉⠙⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠋⠑⠶⠶⠾⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⣿⠀⠀⠀⠀⠀⢀⠿⢶⡞⢃⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⠢⣄⣀⣠⠄⠋⠃⠀⠹⡿⠋⠛⠉⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⡄⢸⣦⡀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡆⠀⠀⠀⢀⡞⠟⠛⠶⡤⣤⡖⠛⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣄⣴⠛⠁⠀⠀⡀⢠⡌⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠛⠁⠙⢄⠠⢀⠰⠀⡼⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⣭⠴⣧⠀⠀⠑⢲⡶⡞⠀⠘⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢛⣧⣤⣠⣬⡌⠑⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠉⠋⠀⠘⠇⢹⣶⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣼⡀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠐⠫⠹⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⢔⡲⠈⠐⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣂⣥⣾⡷⠛⠁⠀⠀⠀⠀⣠⣭⡒⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠊⠙⠲⢿⡛⠻⢥⡀⠀⠀⠀⢀⣤⣮⡥⢋⡴⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢐⣠⣭⣿⣶⣤⣀⠀⠀⠀⠈⠙⠲⢽⣦⣴⣾⠿⣛⣥⣺⢵⣚⣭⣶⡆
⠀⠀⠀⠀⠀⠀⣀⣠⢤⢲⣻⠿⠟⠋⠉⠘⢦⡉⠛⢦⣄⡀⠀⠀⠀⠈⣉⢲⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠈⠐⠠⢉⡛⠋⠁⠀⠀⠀⠀⢀⣠⣤⣿⣷⡶⣞⣻⣷⡆⣾⣿⣿⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢄⡀⠀⠀⣊⠕⡢⢄⡠⢶⣯⡥⢞⣛⣭⣶⣿⣿⣿⣽⡇⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠛⢷⣶⣍⡂⠀⠉⠓⢶⣾⣿⣿⣿⣿⣿⡿⢿⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠑⢤⡀⠨⡛⢿⣷⣤⣀⢸⣿⡿⢿⣻⣭⣷⣿⡿⣟⣯⣷⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠈⢹⣾⣭⣶⣿⣿⠿⣛⣭⣷⣿⡿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠢⣀⠈⡳⢦⣸⡿⢟⣛⣭⢶⣻⣿⣿⣿⡿⣟⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢮⡂⠙⠿⣟⣭⣾⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⢸⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠤⠀⠰⠦⠀⠠⠀⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⡪⠅⠒⠂⠉⠁⠀⢻⠈⠀⠈⠉⠀⠒⠠⢍⡒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣔⠕⠊⠁⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢕⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢌⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢫⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠣⠆⠄⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠄⢢⡀
⠀⠀⠀⠀⠀⢀⠤⠔⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⢄⡀
⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠋⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⢀⠀⢸⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⣛⠁⠀⠀⠀⢀⣼⣿⣿⣿⠟⠛⠛⠛⠻⢿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠿⠛⠉⠉⠙⠻⢿⣿⣿⣦⠀⠀⠀⠀⠀⡏⠙⠄
⠀⠀⢻⠀⠀⠀⠀⣼⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣇⠀⠀⠀⢰⠆⠔
⠀⠀⠈⠓⠀⠤⠄⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠠⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣦⣀⠀⠀⠀⢀⣠⣿⣿⣿⡿⠤⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢿⣿⣿⣷⣤⣀⠀⠀⢀⣠⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⠁






⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢀⣀⠀⠀⠀⠀⣿⣶⣄⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢸⠓⢦⣀⠀⠀⢀⡀
⠛⠿⣿⣿⣷⣶⣿⣿⣿⣿⣶⣾⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢭⡉⠛⠒⠚⠀⠀⣌⠓⠒⠋⠙⡆
⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠉⠓⢤⣸⡄⠀⠀⡀⢿⠀⠀⠀⣠⣤⡀
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣠⣴⣿⣿⣮⡷⠀⠀⠀⠀⠀⠀⠀⢀⡟⠠⠤⠤⢤⣌⣳⡄⠀⡇⢸⢀⡤⠞⠙⢦⣹⠆
⠀⠀⠘⠛⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⣒⣦⡀⠀⠀⠈⠙⣦⠇⣸⠟⠀⢀⣴⣞⡁
⠀⠀⢠⣾⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⡴⠋⠉⠉⢉⡷⠚⠛⠛⠻⣴⠃⢀⡴⢻⡶⣈⡉⢩⣝⢲
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣧⣤⣿⣏⣈⣿⡷⢹⣿⡇⠀⠀⢸⡇⠀⠀⠀⣼⠁⠀⠀⠀⠀⠈⡷⠻⢦⣾⢸⣅⣹⢸⠏⣧
⠀⣰⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⠿⠿⠟⢁⣼⣿⠃⠀⢀⡞⢧⡀⠀⠀⢻⣄⡤⠤⣤⡀⣠⠇⠀⠀⠙⡷⠮⠶⠋⣠⢏
⠀⢿⣿⣿⣿⣿⣿⣿⠛⢻⣿⣿⣿⣿⣧⣷⣴⣿⠀⠀⠀⠘⠛⠁⠀⠀⠸⣇⣠⠭⠿⣶⡟⠁⠀⠀⠀⠙⣧⠀⠀⠀⢀⡇⠀⠀⠀⠛⠋
⠀⣼⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⣸⣤⣤⠴⢿⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢳⣤⣀⣀⣀⡴⣇⠀⠀⠀⠀⣿
⠀⠙⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⣀⣀⣠⡟⠀⠈⠉⠀⠀⢸⡆⠀⠀⣠⠏
⠀⢰⣿⣿⣿⣿⣿⣿⣶⣻⣿⣿⣿⡟⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠉⠉⠀⢷⡀⠀⠀⠀⠀⣸⠓⠒⠛⠁
⠀⠸⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⣹⠶⠶⠶⠚⠁

⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣿⣿⣿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠻⠶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⣩⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣦⣀
⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣛⣿⣦⣄⣀⣀⣀
⠀⠀⢀⡼⠛⠛⠛⠓⠀⠀⠀⠀⠀⠐⠒⠒⠚⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠀⠘⠿⠿⢿⡟⠛⠛⠿⠿⢿⣿⣶⣦⣤⣀
⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣸⣄⣀⣘⣛⣛⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠉⠻⢦⣉⠙⣆
⠀⢸⠃⠀⠀⠀⠀⠒⢉⣩⣭⣭⣭⣍⡉⠁⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣠⣤⣴⣦⣤⣄⠀⠀⠉⠓⢾⣧
⠀⢸⡀⠀⠀⠀⠀⣴⣿⠿⣛⣛⡛⢿⣿⡆⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⣠⣾⡿⢛⠭⢛⠻⣿⣷⡀⠀⠀⠀⣿⡇
⠀⠸⡇⠀⠀⠀⢰⣿⡏⠈⠀⠀⠈⠀⢹⣿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⣿⡟⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⣿⡇
⠀⠀⠉⠓⠦⠤⡌⣿⣇⠀⠀⠀⠀⠀⣼⡿⠀⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠀⢻⣷⡀⠀⠀⠀⠀⣰⣿⣿⣤⣤⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⣦⣤⣴⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣤⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠄⠒⠒⠒⠒⠒⠒⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣤⣴⡶⠶⠶⠶⠶⣦⣴⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠂⠰⣟⠉⣁⣤⣾⣿⠿⠛⠛⠷⢦⣤⣤⣶⣾⠿⠛⢉⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⣿⣿⣿⣿⣭⣀⡀⣀⣤⣶⠿⠟⢋⣹⣿⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⡌⠋⠀⠀⣉⣽⣿⠿⣿⣿⣤⣶⣿⡿⠟⠛⠉⠁⣈⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⡇⠀⣿⣤⣶⣿⣿⣯⣶⣿⠿⠛⠋⠙⠛⣿⣷⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠂⠀⣿⣿⣿⠿⠛⠋⠁⠀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡁⠀⠀⠀⢀⡀⠀⠉⠚⠓⠒⠒⠂⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠁⠀⡠⢒⣬⣭⣽⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⠃⠀⠀⡰⢊⣴⣿⣿⣿⣿⣿⣶⣾⣶⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠈⠑⠛⣛⣛⣻⣛⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣭⣭⣭⣭⣭⣭⣭⣭
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠀⠀⠀⠀⣴⡿⢿⣟⣛⣛⣛⣛⣛⣛⠛⢛⡿⣛⡟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠚⠁⡇⠀⠀⠀⢸⣿⣱⣧⣤⣤⣤⣀⣀⣈⣁⣀⣀⣤⣤⣤⣽⣷⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⠀⠀⡇⢰⡇⣀⢼⡟⣿⣭⡿⢭⡽⢯⣭⣭⣭⣭⣭⣿⣭⡥⣿⡿⠿⠿⠿⠇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣾⡄⠰⡇⠘⠋⣿⠀⠀⣿⣶⣶⣶⣶⣶⣶⣶⣶⣤⣴⣶⣶⣶⣾⡇⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠉
⠀⠀⠀⠀⢀⢹⡇⠀⠇⠀⠀⣿⡾⢄⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠃⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⢸⠀⣿⣠⣶⢀⣄⠘⠻⢾⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣦⣴⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠘⣀⢹⡿⣿⡘⣿⡛⠷⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠦⠤⠤⠴⠶⠶⠶
⠀⠀⠀⢠⡇⠂⠸⣿⠿⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⡇⠀⢸⠋⢿⣿⡌⣿⣿⣿⣿⣿⣿⡟⣷⣿⣷⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⡇⠀⢸⠃⠈⢿⡔⠸⣿⣿⣿⣿⡿⢱⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢁⡀⣿⠀⠀⢀⠻⣄⠈⢿⡏⣴⡷⠿⣿⣿⣿⡿⢛⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣴⣿⣿⣦⣉⠛⢋⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡐⠁
⠀⠀⠀⣀⠀⠀⢤⢈⠲⡀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡰⢊⠀⠄⠀⢀⡀
⠀⠘⠀⠈⠀⢀⡀⠀⠒⠸⣼⠀⠐⠁⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠑⠀⢸⡼⠐⠂⠀⣀⠀⠈⠀⠘
⠐⠂⠠⠽⣄⡣⠀⠪⠰⠄⣹⡀⡐⠀⠀⡁⣠⠰⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⢠⡀⡀⠀⠐⡁⣸⡀⠴⠈⠂⠀⢃⡼⠥⠀⠒
⠴⠀⠀⢂⣈⡉⠙⠮⠥⣤⠌⢳⡁⠀⡐⢀⢅⡀⢰⢀⠅⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠅⢰⠀⣀⢅⠐⡀⠀⣱⠋⢤⡤⠭⠞⠉⣉⡀⠂⠀⠰
⠀⠀⠐⠁⠈⠀⠈⢄⢂⠲⠙⢮⠷⣔⠀⠈⢀⡉⠚⣇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⡟⠊⣁⠈⠀⢐⡴⢯⠞⠱⠂⢂⠌⠀⠈⠀⠑
⠀⠀⠀⠀⠀⠫⠅⢠⣈⠁⢠⣤⡥⠮⠽⣶⢦⣍⣉⡙⣦⠁⠀⠀⠀⠀⠀⠀⠀⠀⢡⡞⣉⣉⣤⢶⡾⠭⠦⣤⣤⠀⢉⣠⠀⠬⠃
⠀⠀⠀⠀⠐⠖⠠⠤⠬⢛⠏⢠⠁⠠⠞⠥⠠⢄⣨⠽⠛⠛⠶⠶⠶⠶⠶⠶⠶⠖⠛⠻⢭⣀⠤⠠⠝⠦⠀⢠⠈⢟⠫⠤⠤⠰⠖
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠁⣐⠀⠀⡇⡦⠠⡤⠞⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠦⡤⠤⡆⡇⠀⢜⡀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠆⠀⡇⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⡇⠀⠦⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣯⣭⣽⣗⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣀⣤⣤⣤⣤⣄⣄⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⣩⡿⠟⠛⠛⠛⠛⢿⣯⡛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⢰⢁⡏⠀⠂⠀⠀⠀⠀⠀⣠⠔⠀⠀⠙⡆⢡
⠀⠀⠀⠀⠀⠀⢰⣿⠀⣿⣿⣿⣟⣛⣛⣛⣻⣿⣿⣿⠰⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠒⣦⡎⠁⣴⣲⠀⠀⣽⠘⣯⡇
⠀⠀⠀⠀⠀⠀⣼⣿⡦⢟⡿⣿⣿⣾⣷⣶⣿⣯⢟⡻⣼⡿⡇⠀⠀⠀⠀⠀⢠⢸⡄⠀⠀⠀⢀⠀⠀⢀⡀⠈⠀⠀⠀⡾⢸⠛⠃
⠀⠀⠀⠀⠀⢠⡗⠋⣪⠗⠋⠁⠀⠀⠀⠀⠀⠉⠑⠪⣗⠝⢿⡀⠀⠀⠀⠀⠈⢎⢷⡄⠀⠀⠈⠁⠈⠉⠉⠀⠀⢀⣾⢇⣿⠇
⠀⠀⠀⠀⣠⠋⣠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠳⣖⣦⠀⠀⠀⠈⢯⣿⣦⣄⡀⠀⠀⠀⠀⣠⣴⣿⣷⠋⠈
⠀⠀⠀⢠⠃⣰⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠹⡄⢹⡿⠂⠀⠀⠀⠘⣿⢻⠿⢿⣿⣿⣿⡿⠿⠟⢻⡏
⠀⠀⠀⡞⢀⡏⠀⠀⠘⠲⠤⣀⡀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⣷⣤⡄⠀⠀⠀⠻⠼⠀⠀⠀⠀⠀⠀⠀⢠⠼⠁
⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⣀⣀⣉⠻⣷⡄⠀⠀⣏⣹⠀⠀⠀⢸⠀⣿⣷⡇
⠀⠀⠀⢧⢸⡇⠀⠀⠀⣾⣿⣿⣿⣧⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡿⠛⠃⠀⠀⠀⣴⢲⠀⠀⠀⠀⠀⠀⠀⠐⢲⡀
⠀⠀⠀⠘⡌⣿⡄⠀⠀⠻⣿⣿⣿⡟⠀⠀⠁⠀⠀⠀⠐⠀⣰⡿⣰⢷⠄⠀⠀⠀⢠⣿⡼⣶⣾⣷⣿⢿⣿⣶⡦⣾⣇
⠀⠀⠀⠀⠙⣜⢿⣄⡀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⡱⠿⠏⠀⠀⠀⢀⡞⢡⠞⠉⠀⠀⠀⠀⠀⠉⠛⢆⠹⡄
⠀⠀⠀⠀⠀⠘⡦⣟⢿⣦⣄⡀⠀⠀⠀⠀⠀⣀⣠⣴⡿⣫⣾⠁⠀⠀⠀⠀⢀⠎⡴⠁⠀⠄⠂⢁⠀⠀⠈⠂⠀⠀⠳⡘⡆
⠀⠀⠀⠀⠀⠀⢻⣿⠗⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢻⣷⡇⠀⠀⠀⠀⠀⠸⢰⠁⠀⠈⠀⠀⠈⡄⠀⠀⠀⠈⠀⠀⢣⢸⣤⡄
⠀⠀⠀⠀⠀⠀⠸⣿⠀⣿⣿⣿⣯⣭⣭⣭⣽⣿⣿⣿⠰⡿⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠻⠃⠐⠐⠀⠂⠀⢸⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠊⠁⠀⠀⠀⠀⠀⠀⠸⡸⣆⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡟⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡻⣦⡀⠉⠂⠀⠀⠀⠐⠀⠀⣠⡾⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣶⣤⣄⣀⣀⣤⣴⣾⣿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣽⣟⣛⣻⡯⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢨⠉⠙⠛⠛⠛⠛⠉⠁⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠀⠀⠀⠀⠀⠀⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠉⠑⠒⠒⠤⠤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡚⠛⠉⠉⠉⠉⠒⠤⣔⣲⠶⣤⡀⠀⠀⠀⠀⠀⢀⣤⠞⣉⣉⣽⡶⠆
⠀⠀⠀⠀⠙⠛⠿⣯⣉⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣽⣗⡢⢄⣀⠀⠠⢀⡀⠈⠉⠓⡓⠤⢤⣀⣴⣾⣯⣷⣿⣿⣿⣷⣶⣿
⠀⠀⠀⠀⠀⣿⠒⠦⣌⡙⠓⠦⣄⣀⠀⠀⠀⠀⠀⠈⢙⣛⣛⣈⣉⣲⣤⣤⠝⠒⠋⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆
⠀⠀⠀⠀⠀⢸⡷⠦⣄⡉⠙⠲⢤⣉⡛⢒⣛⣛⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢰⡀⣻⡿⠿⠿⠿⠿⠯⠅⠀⠐⠒
⠀⠀⠀⠀⠀⠈⣗⠀⠀⠀⠀⠀⠀⠀⠉⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣷⣤⠶⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠠⣟⡒⠤⣀⠀⠀⠀⠀⣹⣟⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⢍⡓⠦⢍⣀⠒⠠⠄⠙⣩⠵⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠈⠁⠀⠉⠓⠢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣍⣙⠒⠒⠖⠋⣱⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠐⠒⠛⢣⣄⡀⠀⠈⠐⠛⠷⠆⠀⠀⠀⣀⣠⣤⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⢟⠂⠤⣀⠀⠀⠀⠀⠀⠈⠙⢋⣉⣐⢤⡶⠒⠋⠀⣻⣿⣿⣿⡿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢤⣤⣤⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠀⠀⠉⠒⠲⠶⣾⣿⣿⣿⣿⣿⣽⡇⢀⣀⣤⣿⣿⣯⣴⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠈⣉⡒⠾⣿⠿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶⠤⣀⠀⠈⢿⣿⣿⣿⠿⠟⢛⣳⣯⠛⠉⢁⣼⣿⣯⡽
⠀⠀⠀⠀⠀⢈⡀⠠⢤⣴⣿⠢⠬⣭⣿⣿⣿⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡠⠿⢤⣀⡈⠀⠒⠛⠉⢀⣴⣾⣿⣿⡿⠒⠚⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠬⢭⣛⣿⣊⣁⡒⠺⣿⡿⣿⡀⠤⢤⣴⣿⡆⠀⠀⠀⠀⢸⡏⠑⠒⠤⢍⣀⡐⠀⠀⢼⣻⣿⣿⣿⡷
⠀⠀⠀⠀⠀⢀⣑⠒⠶⣿⡟⠢⢬⣽⣿⣿⣿⠌⣙⣻⣿⣛⣧⠄⠀⠀⠀⠈⠣⠄⣀⠀⠀⠀⠉⠉⣒⣭⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠬⢉⣛⣿⣛⡁⠒⠲⢿⣯⣿⡓⠲⢾⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠒⠶⠚⠁
⠀⠀⠀⠀⠀⠐⠓⠶⢾⣿⣧⠬⢐⣒⣶⣿⣿⡅⠙⠛⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠲⢾⣿⣿⠿






⡿⢿⡿⠿⣿⡿⢿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣿⣿⣿⣿⣟⣛⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⣀⠀⠤⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢀⡆⠀⠰⠃⢀⠠⠀⠀⠈⠑⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢠⠀⣸⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣜⣰⠟⠁⠀⠀⠀⠐⡀⠀⠀⠀⠀⠁⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣄⣀⠄⡤⠤⣀⣀⢀⣇⡼⠋⠁⢀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⠁⠀⠊⠙⢻⡷⣿⠛⣛⠿⡟⢿⣛⠛⠉⠉⠁⠀⠀⠀⠴⠶⣭⡀⠀⠀⠀⠀⠀⠀⠀⠙⠀⡀
⠀⠀⠀⠀⠀⠀⢻⠁⠀⠀⠂⠀⠀⢘⣘⣇⢿⠟⠁⣰⡇⠀⠹⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⢰⠁⠀⢀⠄⠠⠂⠘⣋⠟⢹⡈⠀⣠⠾⡄⠀⠜⢸⡟⣿⣙⠃⠁⠀⠀⠀⠒⠒⠒⠒⠒⠂⠀⠀⠀⠂⠀⠀⠸
⠀⠀⠀⠀⢰⡇⠀⠈⢁⠀⠀⠒⠈⠉⠐⠈⡇⣾⡻⠀⢴⠀⠀⠐⣇⠸⡟⣷⣤⣀⣀⡀⠀⠀⡀⠀⢀⣀⠀⠀⠀⠊⡘⠃⠐⠚⠦
⠀⠀⠀⠀⢠⢁⠀⠀⠀⢄⣱⠆⠀⠀⠀⢸⠇⠹⡎⢴⠘⢧⠀⠈⠏⢈⡾⣻⠍⠷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠂
⠀⠀⠀⢀⡤⠚⠲⣦⣀⣴⠿⠴⠮⣄⡡⢞⣴⢠⡳⣿⡀⡠⣡⠀⣀⠀⣐⣿⡧⣤⣽⣦⠤⣈⡚⠋⠃⠀⠀⠀⠀⠀⣱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡁⠲⠶⠶⢼⣿⢟⠌⡟⢟⣶⠎⠀⠀⠀⠙⢋⡿⠁⠀⠀⠈⠑⡄⠀⠘⣦⡲⠒⠖⠀⠀⠀⠈⢀
⠀⠀⠀⢇⠀⠀⠀⢀⣜⠃⠀⠀⠀⢸⡏⠛⢶⡇⣎⠸⢂⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⡄⠀⢠⣯⠙⢶⣤⡥⠀⠀⠀⠹⡀
⠠⠂⠉⠙⠳⣦⠎⠁⠀⠈⠳⡀⠠⠜⠀⠀⠀⣿⣁⡽⠗⠒⠠⣠⣾⠭⠦⣀⢀⡀⣠⠴⠶⢾⠉⠸⡇⠀⡛⡳⢤⣐⠀⠒⡟
⣧⠀⠀⠀⢰⣥⠀⠀⠀⠀⠀⡿⢑⣲⢄⣀⣀⣿⣽⣃⠀⠀⠀⠀⡀⠀⠀⠈⢯⢍⡵⠒⠐⠚⠢⡀⢹⡀⠸⣇⠈⠛⣦⢀⠘⢆
⢸⠷⠀⠐⣪⣵⣄⡀⣀⢀⡠⠃⠀⠈⠻⣷⠏⠀⠀⠈⠱⠀⣠⣘⣃⠀⠀⢀⣞⠾⠂⠀⠀⠀⠀⠙⣄⠀⡄⠌⢤⣀⣟⣑⠢⢄⠣
⢸⠀⢀⡞⠁⠀⠈⢻⣿⡛⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣰⡆⠁⠉⠀⠙⢶⠎⠉⠙⣧⡀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⢳⡞⠁⠀⠀⠀⢸⡍⠣⢄⣤⣤⣶⣿⡿⣦⣴⣠⣴⣿⠀⠀⠀⣠⠤⠘⠦⡤⠤⣷⠓⠙⡋⠉
⠀⠀⠛⢄⠀⠀⡠⡎⠀⠐⡞⠁⠀⠀⠉⣿⣿⣿⡿⠿⣿⣷⣧⣼⠁⠀⠀⠀⢸⠀⠈⣆⠔⠁
⠀⠀⠀⠀⠙⣄⠀⣹⣦⣤⡇⠀⠀⠀⢀⣿⣿⣿⡗⠀⣀⠴⠛⠛⠦⡀⠀⢀⣼⠀⣠⡷⢄
⠀⠀⠀⠀⠀⠀⠉⡔⠀⠀⢑⠦⠠⠴⠋⠀⠀⠘⣇⡰⠉⠀⠀⠀⠀⠘⡷⠃⠀⠀⠀⠈⡆⡇
⠀⠀⠀⠀⠀⠀⠀⡥⠀⠀⠀⠑⠠⣆⠀⠀⠀⢀⣼⣧⠁⠀⠀⠀⠀⢀⠗⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⡀⣤⠚⠉⣱⠖⠒⠉⠛⢿⣦⣄⣀⣀⣀⣾⣷⠖⠛⠛⢶⣁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣴⡀⢰⡃⠀⠀⠀⢠⠊⠀⠈⠉⢻⣿⣿⡏⠀⠀⠀⢈⠁⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠚⢷⣄⣀⣀⣼⡀⠀⠀⠀⣸⠟⠛⠻⢦⣤⡴⠎⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⣈⣽⠉⠀⠉⠒⠲⣾⡇⠀⠀⠀⢈⣿⡷⠶⠶⠦⣤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⣰⡿⠷⣄⣀⢀⣼⡿⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⢻⠃⠀⠀⠈⢹⡟⠋⢹⣄⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⢀⡞⠀⠀⠀⠉⡛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠳⣀⣀⢀⠴⠁
⣿⣿⣿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⠤⠖⠒⠒⠒⠒⠒⠚⠉⠉⠉⠉⠓⠢⠤⣄⣀⡀
⠀⠀⠀⠀⠀⣠⠒⠋⠭⠭⠅⠀⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠒⠈⠁⠂⣀⣀⠀⠉⠙⠒⠢⣤⣀⡀
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⢀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡆
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⢀⡠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡑⠀⠀⠀⠀⡐⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠒⠒⠠⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠡⡀⠀⠀⡀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠆⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠉⢢⠀⠀⠀⠀⠀⠀⡀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚
⠀⠀⠀⠀⠀⠀⡤⠃⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠂⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄
⠀⣀⠤⠔⠒⠊⢳⣀⡀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠐⠒⠒⢤
⡎⠁⠀⠀⠀⠄⠀⢀⣉⠉⠉⠉⠉⠉⠓⠋⠀⠀⠘⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠂⠀⠀⣤⣤⡤⢄⣀⠗⠂⠀⢀⡠
⠈⠓⠢⠤⢄⣀⠀⠀⠀⠈⠉⠉⠁⠐⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠐⠒⠂⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⣀⡤⠖⠊⠉
⠀⠀⠀⠀⠀⠉⠛⠒⠶⠶⠤⠤⣤⣤⣤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠠⣄⣀⣀⣒⡤⠦⠤⠭⠖⠒⠚⠉⠁
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⠀⠀⠉⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠒⠉⠀⠀⠀⢦⠾⠃⠀⠉⠁⠠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣴⡀⠀⣀⣀⢀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠻⠋⡏⡏⡏⠍⠋⢀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣤⣤⣤⣤⣤⣤⠤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣂⣀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣟⣚⠓⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠻⠛⠛⠛⠿⠶⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠛⠁⠀⠀⠀⠀⢀⠰⠂⠀⠐⠄⠀⠰⠖⠄⠀⠀⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⣏⣸⡃⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠾⠇⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⡀⠀⠂⣶⡄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢲⡄⠠⠀⡠⠀⠀⠀⢡
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠐⠀⠀⠀⠛⠑⠊⠀⠀⠀⠄⢀⡤⢸⢁⠠⡀⠀⠀⠀⠀⠀⠀⠿⠆⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠈⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠀⠀⠀⣥
⠀⠀⠀⠀⢸⣿⠀⡠⢄⠀⠄⠺⢸⠀⠀⠀⠀⠠⠖⢡⠆⣤⣞⣻⣆⠀⣱⣤⡀⠀⠀⠀⠀⠈⢤⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠸⣿⠀⠁⢀⠀⠀⠀⠀⠐⠒⠚⠛⠙⠀⠂⢀⠀⠈⠉⠀⠸⠷⡽⡧⡀⠀⠀⠀⠁⠈⠀⠀⠀⠁⠀⠀⣿
⠀⠀⠀⠀⠀⣿⡇⢰⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠁⠛⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠃⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠹⣿⡄⠆⠀⠠⠀⠀⣼⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠌⣯⠀⡀⠠⠘⠂⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⢻⣷⡀⢂⠀⠤⠀⠈⠀⠀⠀⠤⠄⠀⠀⠀⠰⡆⠀⠀⠀⠀⠤⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠐⡀⠀⠀⠀⠀⠀⢀⡎⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⡈⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠃⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠙⠢⣄⡀⠀⠀⠀⠀⠈⠀⠀⣄⠀⠀⠀⡀⠤⠚⠁⣠⡴⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠈⠉⠒⠒⠂⠘⠧⠒⠐⠒⠈⠁⣀⣠⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠴⠴⠤⠤⠴⠶⠶⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠔⠁⠀⢀⠤⠀⠀⢀⡠⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠒⠈⠀⠀⠀⠈⠁⠀⠀⠂⠁⠀⠀⠀⠀⠀⠁⠐⣲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈
⠀⠀⠀⠀⠀⠯⡈⠑⡤⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⢀⣀
⠀⠀⠀⠀⠀⠀⠱⡀⢠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣀⠴⠋⠀⠀⠑⠂⠰
⠀⠀⠀⠀⠀⠀⠀⠹⡀⠑⡄⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠡⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⡷⠚⠉⠙⠲⢤⡀⣠⡤⠴⢧⡈⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⢀⡴⠛⠉⠓⠦⣶⡁⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠙⡆⠀⠄⠀⠀⠀⠢⠀⠀⠀⠀⠀⠉⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⣞⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⣷⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⡤⠒⠻⣄⠀⠀⠀⣀⣀⣈⠓⣤⠤⠤⣄⠀⠀⡼⠀⠀⢀⡿⠋⠁⠈⠙⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⠠⠖⠊⠁⠀⠈
⠀⢸⡃⠀⠀⠈⠑⢲⠞⠁⠀⠀⡟⠀⠀⠀⠀⠙⢏⡔⠋⠉⠉⠓⢄⠀⠀⠀⠀⣷⠤⣄⠀⠀⠀⠠⠀⠄⠋⠁⠀⠁
⢀⡤⠛⠉⠉⠉⠉⢿⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣧⣀⣠⠔⠃⠀⠈⢳⠀⣀
⢸⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠈⠓⢤⣀⣀⡤⠞⣄⠀⣀⣀⠀⢀⣿⠤⠤⣄⣀⠀⢀⡼⠀⠀⠆
⠸⣄⠀⠀⢀⡴⠖⠋⠉⠉⠁⠀⣠⠊⠀⠀⠀⠀⠀⡼⠋⠉⠈⠙⢾⡁⠀⠀⠀⠈⠑⢯⡄⠠⠀⠄⠰
⢠⠞⠓⠒⣾⠁⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢰⡅⠀⠀⠀⠀⠈⠧⠀⠀⠀⠀⠀⠀⢻
⡟⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⡸
⢷⡀⠀⠀⠀⣨⠿⠒⠒⠒⠒⠀⠈⠓⠢⠄⠒⠀⠀⠀⠙⠢⠤⢶⡋⠀⠈⠙⠒⠲⡖⠋⠁
⠈⠳⠦⢤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⢀⡇
⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⢸⠶⠤⠴⠚
⠀⠀⠀⠀⢹⣦⣀⣀⣠⡴⠋⠀⠻⣄⡀⢀⣠⠋⣨⠷⢦⣄⣀⣀⣠⡞
⠀⠀⠀⢰⠏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⠇⠀⠀⠀⠈⠁⠀⠘⡆
⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠈⢷⣄⠀⠀⠀⢸⣄⠀⠀⠀⠀⡠⢆⠠⠊⠉⠁⠲⢤⣤⠴⠟
⠀⠀⠀⠀⠀⠈⢩⡟⠋⠉⠈⣱⠒⠊⠛⢦⢰⠃⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣸⠁⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠘⠦⣄⡀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⣹⡀⠀⠀⠀⢀⣎⠀⠀⠀⠙⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠙⣦⠤⠖⠋⠀⠙⡄⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢸⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⣰⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣳⣤⣤⣞⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⢴⡶⢫⢻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣭⡍⢩⡍⡉⠉⡿⠉⡍⢩⣷⣿⣿⣿⠿⡟⢸⣿⠟⣥⣯⣭⣭⣿⣿⣿⣿⣿⣿⣶⠸⣧⠻⠴⠛⠚⢻⢀⣿⡿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣣⠾⠛⠓⠈⠁⢀⠈⠉⠁⠈⠉⣉⠝⣡⡖⢁⡇⡟⣵⠋⠀⢋⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠛⠓⠀⢸⢸⢿⡵⢿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠜⠁⠀⠀⢀⣤⠠⢄⠀⠀⠀⠀⡜⣠⣾⣿⢃⡼⢱⡿⣿⠀⠀⢸⠀⠀⠀⠀⣄⣠⣤⣠⢤⠀⠈⠀⠀⠀⠀⡜⡜⣼⣏⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢀⢎⡴⠁⠠⠄⠀⠀⠸⣌⣉⠴⠀⠀⠀⢸⣵⣿⣟⣡⢃⡆⣼⣼⣿⡄⠀⡼⠀⠀⠀⣸⢻⣿⠿⢯⠏⠀⠀⠀⠀⠀⡰⢱⠡⣽⣾⣏⡝
⠀⠀⠀⠀⠀⠀⡠⢣⠎⠀⠀⠀⢀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠘⡻⣿⡿⢁⢾⡇⣿⣾⡿⠁⡴⠁⠀⠀⣆⠀⢀⠄⣀⡀⣤⣤⣤⣴⡖⣰⢣⢣⡜⣽⣿⡿⠁
⠀⠀⠀⠀⠀⡔⣱⣿⣶⣶⣶⡖⠰⡆⣴⣠⣞⣪⠉⡥⣤⣤⣤⣿⡟⢠⢋⣾⡇⣿⣧⣶⡧⢤⣤⢤⣤⢀⣤⣤⠄⠤⢤⡥⣭⠵⠒⠺⢃⣯⢻⣰⣿⡿⠁
⠀⠀⠀⠀⢾⢤⠴⣶⣶⣶⣶⣶⣶⣶⣶⣶⡦⣤⣤⣭⣉⣉⣉⣹⣡⣏⣤⣼⣄⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⣭⣭⣭⡴⠶⠒⣒⣛⡿⣱⣷⣿⡿⠁
⠀⠀⠀⠀⢘⡿⣿⡿⢗⣒⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡷⠤⢤⣬⣿⣿⣋⠈⠀⡐⠒⠲⠶⠶⠶⢴⣶⣿⣽⣿⠃
⠀⠀⠀⠀⠸⣯⣭⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢷⣽⢿⣯⠿⠿⠿⠿⠿⠿⠦⠶⠶⠿⢿⣾⣿⣿⣿⣿⣿⢿⣻⡿⣿⣿⠇
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠻⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠉⠈⠑⠂⠂⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠖⠢⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⣀⣤⣄⢀⣀⡀⠀⠀⠀⢀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠁⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠻⠿⠿⣿⠿⠃⠀⠀⠈⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠶⣶⣶⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⣄⠀⠀⠉⣩⠟⠛⠻⢅⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⢡⣶⡿⠀⠹⣾⣿⣂⣀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⢀⣀⣄⡀⠀⢸⣿⡿⣶⡄⠙⣿⣿⡋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣠⡶⠷⠒⠀⠀⠀⠀⢽⣿⣿⣿⣿⡇⠀⠙⠛⠛⠁⠀⠈⠻⠿⠷⠟⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠇
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠻⠏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⡇⠀⠀⠀⠈⠿⠿⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢴⠂⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣮⠀⠀⠀⣸⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠠⠀⠀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣯⡀⠀⡀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢠⢤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣤⣀⠀⠀⠙⢟⠛⢋⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⡳⠀⠀⠀⢤⣄⠀⠈⣿⣿⣿⣿⡇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⢀⣀⣿⣿⣧⠀⢿⣿⠏⠙⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡰⠁⢻⣿⣿⣿⠏⠀⠈⠙⠀⠀⠖⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⣀⠤⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣠⠴⠋⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠶⡤⠀⠀⣀⣠⠤⠄⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠇⠀⣀⡀⠤⠐⠂⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠁



⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠴⠦⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⢤⠄⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⣞⠛⣳⠄⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠋⢁⣠⣎⣉⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣀⠍⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⢯⣁⠡⠝⠃⠀⠀⠀⠀⣐⠲⣤⠤⠀⠒⠈⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣩⣤⡤⣀⡀⠀⠀⣀⣀⡀⠤⠤⠐⠒⠂⠉⠉⠈⢁⣀⣀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣋⣡⡼⣇⠈⢊⡾⠋⠉⠀⠀⠀⠀⣠⠤⡄⠀⠀⠀⢠⠞⠻⠮⡺⣝⢷⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁⠀⠀⠀⣯⠀⠀⢘⡩⣲⡟⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢠⣞⣳⠄⠀⡀⠀⠀⠀⠀⢁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣴⡟⢫⡄⠐⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠉⠁⠀⠀⠀⠀⠀⢀⡞⠛⠥⣀⡝⢵⡄⠀⠀⠘⠀⠀⠀⠈⠈⠑⠈⠁⠀⠀⡠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⢆⠦⠀⠘⣧⡀⠀⠀⢊⡼⠁⠀⠀⠀⢰⠒⣦⠀⠀⠀⠀⠀⠀⠀⠓⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠚⠉⠠⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠋⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠻⠇⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠴⠒⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠤⠴⠒⠒⠂⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠒⠒⠃⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⣄⡀⢠⣤⣤⣤⣄⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣄⣀⣀⣀⣀⣁⣀⣀⣀⣀⣠⡤⠖
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠛⠛⠾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⢀⣀⡀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣐⡀⢀⣿⣯⣽⡆⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢻⣿⣿⣿⣿⣿⠇⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣸⡇⡃⠘⣿⣗⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣥⣥⣤⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠇⠤⠭⠿⠧⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠠⠐⠂⠀⠀⠀⠀⠀⠀⠀⠐⢦⣄⡀⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⢀⠔⢈⢠⠀⠀⠀⢀⢸⠀⠀⠀⠀⠀⠙⠻⣦⣄⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⡆⠔⠉⠀⠀⠙⣧⠀⠀⠘⠛⠀⠀⠀⠎⠀⠀⠀⠈⠿⣷⣄⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠋⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣢⡤⠀⠻⣿⣿⡌⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠈⠁⠀⠀⠀⢻⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⢤⡔⠋⠀⠀⠀⠀⢀⡀⣀⡀⢸⣿⣿⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠻⡾⠃⠀⠀⠀⠀⠀⠈⠉⠉⠁⢸⣿⢼⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣧⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣼⣿⣟⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⣟⣆⠀⠾⡫⠋⠀⠀⠀⠀⠂⢰⡇⠀⠠⠀⠀⠀⠙⠻⠿⠆⣰⡿⡿⢃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⣟⢟⣈⠁⠀⠀⣰⡗⠀⢠⢸⡦⠀⠐⠂⢄⠀⠀⠀⢀⣴⣟⡕⠈⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠎⢿⣷⣿⣷⣄⠘⠃⠀⠀⠀⠀⠀⠀⠠⠂⠈⠁⣠⣼⣿⣫⠞⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢼⣿⣿⣯⣳⣖⠂⠀⠀⢀⣉⣀⣠⣤⣶⠿⣛⣥⠞⢁⡴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣶⣦⣬⣭⣭⣽⣷⣶⣿⣫⠵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠛⠻⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠛⠓⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⠚⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⢄⠀⠀⠀⠀⢀⣠⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡀
⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⡀⠉⠲⢄⠑⠀⠐⠚⠛⠛⠛⡃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⣏⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⢀⣀⣀⣤⣼⣧⠤⠤
⠀⠀⠀⠀⠁⠒⠢⢤⣀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠤⠔⠒⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⣲⠶⠤⢤⡀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠛⠉⠉⠀⠀⠰⡅⢃⠴⣣⡼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠲⠭⠙⠛⠫⠷⢦⣄⠀⠀⣠⣾⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡶⠦⣤⣀⣀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⡿⠥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡐⠌⠊⠁⠀⠣⡠⠀⠐⢉⠝⣓⡶⢤⢠⠐⣸⣿⣿⠿⠏⣆⣴⣾⡆
⠀⠀⠀⠀⠀⣠⠐⢊⣥⠞⠁⠛⠀⠀⠀⠀⠀⠈⠉⠉⢉⢥⡪⣾⣿⡀⣳⣯⣿⢍⣡⣾⣿⣿⢏⡇
⠀⠀⠀⠀⠀⣾⢛⠒⠤⣼⣒⠤⣁⠀⡀⠤⣤⠄⢤⣲⣵⣿⣿⣿⡿⢐⣿⡛⠈⠀⡉⣿⡾⣻⢏⡀
⠀⠀⠀⠀⠀⡷⠈⠹⠷⣒⠭⠹⣶⣯⣠⣚⣭⣶⣽⣾⣿⣿⣿⡿⢟⣋⣀⡀⢠⢴⡃⣿⣿⠿⣤⠉⠹⡖⠤⢄⡀
⠀⠀⠀⠀⠀⠑⠢⠤⣐⡢⠭⣝⡂⠝⡓⠮⣿⣯⣷⣿⠿⠋⢉⣁⡠⢴⢻⣻⢮⠷⢻⡿⠖⠚⠹⣧⢤⡐⢋⣵⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣗⡲⢬⣙⡒⢧⣿⣿⡋⠭⠃⠀⠀⠀⡔⣵⣿⣿⠲⣱⡋⠀⢀⠔⠀⠀⠁⢚⠻⠿⡿⢿⡀
⠀⠀⠀⠀⠀⠀⣤⠒⠩⠍⠁⣂⡐⠀⢍⡛⠑⠁⠀⣀⢀⣀⡠⣪⣾⣿⣿⣿⣼⠁⠀⠒⠁⠀⠀⠀⠤⠀⠀⠀⠃⠰⣨⠉⣲⡄
⠀⠀⠀⠀⠀⠠⣗⠂⠈⠩⠥⠐⣂⡈⠓⠲⠤⢔⣉⣐⣊⢔⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⣀⢕⡤⢖⡫⢔⣷
⠀⠀⠀⠀⠀⠀⢿⣊⡠⠬⠍⠁⠐⠒⢒⣀⣤⣬⢐⣶⣂⣭⣾⣿⣿⣧⣤⡠⠤⣤⠄⣀⠀⢀⡀⠠⣐⡮⢝⣪⣵⣾⣿⣾⢿⡃
⠀⠀⠀⠀⠀⠀⠉⣶⢖⠶⡮⢭⠥⡤⡥⣤⢶⢖⡶⡶⣲⢻⣿⡫⡃⠀⠙⠻⣖⠬⣙⣠⣭⣶⣦⣟⣾⣿⣿⡿⣟⣊⣥⣒⣿
⠀⠀⠀⠀⠀⠀⠈⡯⠁⠑⠊⠂⢄⣈⡘⠐⠑⠠⠊⠄⠔⠔⡠⣻⣮⡀⠀⠀⠉⠯⢃⣬⣭⣍⣬⣿⣻⢿⣷⣾⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠈⢻⢦⡉⠭⠽⢓⣾⡿⣿⣿⣿⢿⠶⠄⡀⠀⠰⣷⡶⣶⡷⠶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠀⠀⢯⢍⠑⢾⣻⣷⣿⣿⠟⠋⠁⠀⠀⠀⠑⠒⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⢀⠃⠐⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠢⠰⢀⠄⡠⡀⢀⠀⠀⢐⠅⡨⢀⠀⡠⠀⠔⠁⠀⢀⠔⠡⠊⠀⢀⢀⠑⡡⠂⢀⠔⡁⡠⢀⢔⢕⡔⡰⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠐⠡⠪⠀⠅⠨⠀⠕⡨⢂⠔⠀⡨⠀⠀⡠⠪⢀⢀⠔⡡⠂⠂⠐⠁⠊⠌⠐⠁⠃⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠁⡤⠶⢤⡀⠉⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⠍⠀⠀⠀⠈⠳⠤⠔⠁⠀⠀⠀⠀⢙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠈⢁⡴⠒⢲⣤⠤⠤⠴⠖⠂⠀⠃⠀⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠾⠷⠖⠒⠒⠋⠉⠉⠳⠤⠼⠁⠀⠀⠀⢀⠤⢤⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⡼⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⢠⠖⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢤⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠀⠈⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⡼⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠂⠐⠒⠂⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⡋⠁⠉⢻⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣿⣿⣿⣿⣿⣿⣶⣾⣿⡿⠿⠿⠟⢛⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠿⠿⠿⠟⠁⠀⠀⢩⣭⣭⣴⣶⣶⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣶⣶⣾⣿⣿⣿⣦⣤⣴⣿⣿⣿⣿⣿⠋⠙⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡟⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⠀⠀⠉⠉⠛⠛⠛⠛⠃⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣡⠴⡾⠛⠁⢀⡄⠀⠀⢀⠀⠀⠀⠀⢄⡀⠀⠠⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡷⢊⡵⠂⠀⠀⣴⠋⢀⡴⠀⣾⠀⠀⡇⢠⡀⠹⣦⠘⢦⡹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⣴⠋⠀⠀⣠⡞⠁⢠⡾⠁⡸⠇⣰⠀⣿⠀⢷⡀⠈⢷⡀⠹⣌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⢠⡾⢁⡔⢁⣼⠏⠀⠀⠈⡀⠠⠁⢠⡟⠀⠈⢠⡌⢷⠰⡌⢿⡄⠹⣞⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣾⢡⡿⢁⡞⢀⡞⠁⠀⠀⢀⡼⢡⠇⢠⠿⠃⡤⠀⡼⣿⡘⡆⢿⡸⢷⠀⠻⡆⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⣿⡇⠃⣾⠁⡞⢀⡼⢀⡴⠋⣰⠋⠀⠁⣨⣞⢁⠘⠀⠙⢧⠃⠈⠃⡞⡄⠀⢱⢸⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⣿⠀⠀⠁⠘⣰⣾⠴⠚⠛⠛⠙⠛⠛⠛⠉⠉⠛⠛⠋⠙⠻⡰⠾⣶⢁⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⡀⢀⠂⢰⣯⠴⠷⠶⣤⣄⣀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⡹⢦⢹⣯⠀⢰⠀⢸⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢃⢸⡄⣾⠁⢀⣠⣄⡀⠉⠙⠣⠀⠀⠀⠀⠘⠋⠁⢀⣰⣇⡀⠈⣟⠀⡜⠐⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞⣿⣇⣿⠀⢻⡟⠋⣿⠷⢦⠀⠀⠀⠀⠀⢀⠴⢻⣿⣿⢻⠟⢠⡇⣰⢃⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣇⠈⠳⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠵⠛⠀⣼⣿⣧⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣎⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢟⣥⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⡻⣿⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠚⠛⠁⣿⢿⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣜⠿⡄⠀⠀⠀⣀⣀⡀⠀⠀⠀⠈⣲⣄⡀⠀⠀⠀⢠⣿⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠳⣿⣄⠀⠀⠀⠘⢥⣀⣀⣀⣀⡜⠁⠀⠀⠀⣠⠟⢿⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⡀⣻⣦⡀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢀⡴⠫⣷⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠶⢛⣿⠉⠓⠦⣄⣀⣀⣀⣀⠠⠶⠞⠹⡗⢶⣾⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣛⣿⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠠⣶⡸⢿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣭⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢣⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠜⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡈⠎⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡱⢀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡄⢿⡌⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢠⡟⢠⡄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⣧⠀⢣⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡜⠀⣼⢷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢿⠀⠀⠆⢹⡄⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⡿⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠀⡀⣶⣼⠁⠀⠀⠀⠀⠀⠀⠈⣇⣧⢠⢀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⣷⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣾⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣹⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣏⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁

⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⣰⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤
⠀⠀⠴⢿⣿⡟⠋⠀⠀⠀⠀⠈⠻⣿⣿⠀⠀⠀⠀⠀⠀⢀⣽⣶⡶⠆⠀⠀⠀⠀⠐⠞⣿⣿⠿⠂
⠀⠀⠀⢨⡟⠁⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡁⠀⠀⠀⠀⠀⠀⠀⢈⣿⣄⡀
⠀⣼⣿⣿⣿⣷⡆⠀⠀⠀⢠⣶⣾⣿⣷⣶⡄⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣦⠀⠀⠀⢰⣿⣿⣿⣿
⢠⡟⢿⣿⣿⣿⡇⠀⠀⠀⣾⠛⣿⣿⣿⠙⣧⠀⠀⠀⢀⣿⣿⣿⣿⡟⠀⠙⢷⡄⠀⠈⢿⣿⣿⣿
⢸⡇⠘⣿⣿⣿⡇⠀⠀⠀⣿⠀⣻⣿⢛⡀⣿⠀⠀⠀⣸⡇⢹⣟⣛⠁⣠⡴⠛⠁⠀⢠⣾⣿⡿⢿⡄
⠀⢻⡀⢼⣿⣿⣧⠀⠀⠀⠘⠷⢀⠉⠐⠄⠇⠀⠀⢰⡟⠀⠀⠈⠉⠘⠉⠁⠀⠀⠀⣸⣿⣿⣷⠈⣷⡀
⠀⢸⡇⢨⣿⣿⣿⠀⠀⠀⠠⠂⠀⢄⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⣷⡈⢠⣶⠀⢀⣿⣿⣿⣿⡄⠙⣇
⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠑⡆⠀⠀⠁⠀⠀⠘⠇⠀⠀⠀⠀⠠⣿⣷⣸⠃⠀⣼⣿⣿⣿⣿⣿⠀⠙⠁
⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠛⢻⣿⣿⣿⠿
⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿
⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣼⣿
⠀⠀⣸⠃⡿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⣿⢹⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⣴⠇⣼⡇⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡇⠀⠀⠀⠀⠀⠀⠀⠉⢹⡄
⢸⠟⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣤⣬⣭⣥⣴⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⠿⠟⠋⠉⠉⠉⠉⠉⠛⠻⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣀⣼⣧⣴⣿⣤⣀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠀⢸⣷⣷⣷⣧⣿⣿⣿⡀⠀⣠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣾⣏⣉⣻⣿⣿⣿⣟⣥⣿⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⡿⠿⠛⢛⣛⣛⣛⣛⣛⡛⠉⢿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠿⠛⣉⣥⠶⠞⠛⠋⠉⠉⣉⡉⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⣡⡴⠟⢉⡀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⡀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠁⠀⠀⠀⢠⠐⠁⣀⠀⠙⠿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢀⣾⣿⡟⢁⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠒⠀⠀⠀⠀⢸⠀⣠⠊⠁⡀⠀⠙⢿⣿⣿⣿⣷⡀
⠀⠀⠀⢀⣾⣿⡟⢀⡾⠁⠰⡄⢁⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠞⠀⠀⠀⠀⢂⡠⠈⢻⣿⣿⣿⣷
⠀⠀⠀⣾⣿⡿⢁⡾⠁⠀⡐⠀⠀⠇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⣰⡿⠃⠀⠀⠈⠠⠇⠀⢁⠀⠀⢿⣿⣿⣽⡇
⠀⠀⢠⣿⣿⠇⣸⠃⠀⢰⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠀⠀⠀⠀⠀⠒⠂⠀⠀⠂⠀⠘⣿⣿⣿⣿
⠀⠀⢸⣿⣿⠀⣿⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⡺⠋⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢸⠀⠀⢿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⠀⣿⠀⠰⠆⠘⠇⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣏⡀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⡤⠀⢸⣿⣿⣿⡇
⠀⠀⢸⣿⣿⠀⣿⠀⠀⣇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠈⣿⣿⡄⢹⡄⠀⠸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡤⠔⣶⠾⠿⡋⠛⠟⢦⡀⠀⠀⢀⡀⠀⢀⠄⠀⢀⣿⣿⣿⣽⠁
⠀⠀⠀⢻⣿⣷⣌⣷⠀⠀⢁⡀⡘⢦⠀⠀⠀⠀⠀⣼⡁⠀⠀⢉⠀⠀⢈⣦⠀⠀⠈⠑⠄⠈⡇⠀⠄⠀⠀⣼⣿⣿⣯⡏
⠀⠀⠀⠈⢿⣿⣿⣿⣧⡀⠋⠡⡁⠈⠀⠀⠀⡀⢸⠈⠀⠀⠀⣨⠀⠀⠀⠈⣇⣀⠀⠀⠀⠀⢁⠸⠓⢀⣼⣿⣿⢿⡿
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣄⠀⠈⠀⠀⠀⠀⡩⠻⣀⡀⠀⠀⠉⠀⠀⢀⣀⠏⢠⠀⠀⢀⠐⠃⠀⢠⣾⣿⣿⢫⡿⠁
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠃⠀⠙⠄⡀⠀⣀⠀⢀⡨⠉⠁⠈⣀⠐⠀⠀⣀⣴⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠊⡏⠂⠀⠀⠀⠀⠀⠈⣁⣤⣾⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣀⣈⣁⣀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠰⣶⣶⣶⣤⣤⡤⠆⠄⠀⠀⠛⠛⠛⠛⠛⠛⠛⢿⣿⣟⡛⣛⠛⢛⣛⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣷⣤⠤⠤⠀⢠⠀⠀⠀⠀⠈⠭⢉⣠⣦⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⣿⣿⣿⣾⣾⣾⣶⣶⣶⣶⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⡤⠀⠉⠀⠄⢀⠀⠀⠀⠀⠁⠄⠀⣰⡟⠀⠀⠴⠆⣹⠷⢶⣶⣄⡘⠷⠦⠈⠀⠺⣆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠲⣿⣄⣜⣿⠋⠒⢀⣀⠀⡀⠀⠀⠀⠙⢤⢴⢶⣿⣿⣿⣶⣿⣠⣤⣝⠶⠄⠈⣿⣿⣷⡶⠀⡀⠀⠀⡁⠢⡀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⢀⡜⣻⠙⣼⣡⣤⣶⡉⠀⠀⠀⠀⠀⠒⡇⠘⢻⣰⣿⣿⢿⣿⣿⣿⣝⡋⠐⠀⠀⠀⠉⣿⣷⡤⠀⠠⣳⡌⡀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⠀⠐⠿⣷⣿⡏⢩⠿⢋⣩⠤⢀⣀⢄⡁⢀⣴⣿⡿⣿⠉⣼⣿⣿⣿⠛⠋⠀⠂⠈⠙⠢⠜⢿⣧⡠⠊⢉⠓⠏⠰⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⠓⠹⣶⣿⣿⣾⡁⣀⡉⣀⠐⣎⠉⢹⣷⣿⣿⡟⠷⢃⣴⣿⣿⠿⠁⢶⡀⣀⣉⣀⠀⠌⠀⠀⠹⣷⣾⣥⠀⠀⠰⠆⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢉⣭⠑⢦⣹⣿⡛⣿⡗⠡⠈⠋⠲⡾⠛⢿⣿⣿⣿⡿⠀⣼⣿⢛⣤⣤⣾⣿⣯⣿⣽⣿⣿⣶⣆⡈⢻⣿⣿⣭⡇⠀⠈⠀⣈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣎⠻⣹⣿⣿⡀⠀⠀⠀⠃⢈⣿⣿⣿⣿⣿⠄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⣾⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢶⣽⣦⣿⣿⣧⠀⠀⠄⠁⢠⣿⣿⣿⣿⣿⣡⣼⣿⣭⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠈⢹⣻⣷⣬⡀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣬⣷⣾⣿⣿⣿⣿⡟⠀⠀⢄⣠⣼⣿⣿⣏⣽⣿⣿⣿⣟⠿⣹⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣏⣿⢃⣶⣿⣿⣿⣷⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣬⡿⢿⣿⢣⡄⠀⠀⣼⣿⣿⡿⠉⣹⣿⣿⣽⣿⣷⣾⣿⣥⣼⣿⣿⣿⣿⣿⣿⣿⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣦⡜⠿⣀⣹⣷⠐⣻⣾⣿⡟⣛⢡⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣟⣩⣥⣶⢾⠛⠛⡿⣟⠓⢼⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⣿⢿⣿⣿⠷⣊⣀⣿⢿⡻⠃⠘⢿⣿⣿⣿⣿⣿⠛⠂⢤⣄⣤⡙⠛⠿⠿⡿⠿⠿⡟⠀⠀⠃⢿⣠⢾⣯⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢼⡿⣿⣿⣿⠟⠿⣿⣞⠀⢐⠀⢸⣿⣿⣯⣿⣿⣥⢤⡀⠙⠋⡄⠤⠀⠤⠀⣀⠀⠀⠀⠰⢄⣿⢀⣼⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⣂⠀⢛⣥⡶⠊⠁⢸⣿⣿⣿⣿⣿⣿⡂⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡀⣿⣿⣟⢹⡗
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠈⣿⣿⣿⡿⣇⣰⠞⢃⠀⠠⢳⣾⣿⣿⣿⣿⣿⡛⠿⡦⠄⠀⠀⠀⠀⠀⠀⠀⠀⢴⠛⢠⣼⣿⣿⢋⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠛⣿⣣⠆⡡⠔⣁⠄⠾⢿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠋⣠⢾⣿⡡⢷⠂⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢠⠛⣻⠋⠀⠀⠀⣻⣧⠴⢊⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⢀⠀⠃⠀⠀⣠⣇⣼⣳⣾⠵⢁⠘⢀⠗
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠀⠀⢘⡷⣟⣡⠀⠀⣄⠉⠈⠶⣼⣷⣿⣿⣿⣯⣷⠀⠀⠀⠀⠀⠀⢠⣀⣴⣶⣟⣽⣿⡿⣫⠐⡁⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⢀⡀⠀⠐⣰⣥⡜⢀⣯⠄⠀⣶⡼⣿⣿⣿⣷⣿⣿⡀⠀⠀⠀⠠⠰⣞⠶⠜⠛⢿⣿⢟⡼⢋⠀⠀⠂⠀⠀⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠈⠹⠁⠠⡆⠀⠀⠀⠈⣿⣾⣿⣿⣿⣿⣞⢶⡀⢀⡀⠀⠲⠀⠀⢷⠈⠿⢣⠈⠀⠀⠆⠀⢀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⢀⠀⡀⠀⠀⢉⠠⢠⠄⢀⣽⣿⣿⣿⣿⣿⣿⣿⣦⣷⣸⣧⢀⣾⣿⡃⠀⠀⢰⡄⠀⠠⠀⠋⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠈⠀⢀⡵⠀⠀⠤⠤⠈⠀⡘⢲⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠂⢀⠈⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠐⠚⠀⠀⠀⢀⠀⣤⡤⠶⠒⠀⠰⡦⠉⠀⠁⢀⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⡇⠐⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣀⠀⠄⠒⠐⡎⠂⠀⢀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁⠀⠀⠀⠀⢠⠀⠙⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠾⠂⠉⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⣿⣿⣿⣿⣿⡟⣿⡟⠉⠟⢿⣿⣿⣿⠀⠀⠀⠄⠀⠀⠀⢀⣤⡀⠀⠀⣀
⠀⠀⠀⠀⣀⠀⢂⠀⡄⠀⠀⠀⠐⠀⠀⢀⠀⢰⣰⣦⣵⣿⣿⣿⡿⣻⣥⣿⣿⣷⠀⠘⠡⠆⣈⣿⣿⣿⡄⠊⠀⠀⠀⠀⣼⡟⠉⠁⠀⠀⠁
⠀⠀⠀⢀⣒⣿⣶⣤⢁⢀⠀⠀⠀⣀⣀⣤⣴⣿⠿⢋⢛⣉⣿⣿⣿⣿⣿⡟⠋⢭⠤⠚⠁⠀⠈⠠⢿⣿⣀⡄⠀⢀⣦⡾⢋⠞⠀⠀⠄
⠀⠀⠀⠈⣻⣿⣿⣿⣯⣿⣷⣶⣆⣤⣯⣤⣾⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠒⠲⠀⠀⠀⢴⣾⠿⠿⠄⣰⣾⠋⡅⠀⠀⠀⡀
⠀⠀⠀⢺⡿⠛⠛⢿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣻⠁⠘⣻⣥⣿⣿⡿⠀⠀⠈⠉⠀⠀⠀⠀⠀⢠⣾⣷⢶⣾⣿⠏⠈⠀⠀⠀⡘⠁
⠀⠀⢀⣼⣿⣶⣿⣶⣦⣀⢀⣁⡀⢸⣿⠟⣠⣼⣣⣴⣿⣿⣿⣿⣿⡿⠒⠀⠄⠀⠀⠀⢀⡄⠀⣸⣯⣇⣛⣷⠃⠂⠴⠀⠀⠆
⠀⠀⠀⠀⠟⠛⢉⣿⣿⣿⣶⣉⣋⣠⣷⣌⣩⣾⣿⣿⣿⣿⣿⢿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣸⣟⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠐⠿⠛⠹⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠛⠧⠀⠀⠈⠓⠂⠀⠀⠀⠀⠀⢀⢀⣽⠛⠛⢠⡞⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⣿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠿⠏⠀⠀⠀⠀⠈⠀⠀⠁⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⣿⣿⣿⡏⠁⠀⠀⠀⠀⠲⣤⡀⠀⠀⠀⠀⠀⡠⠀⠈⠛⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣆⠀⠀⠀⠠⢀⠀⠀⠀⠃⠐⠒⠆⠸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠐⠁⠄⠀⠀⠀⠀⠘⠏⠘⠀⠁⠀⠀⠀⠀⠀⠈⠒⠘⣦⡀⠀⠂⠀⠉⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡆⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡾⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠿⠛⠋⠉⠀⠀⠈⣿⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣠⣴⠶⠿⠿⠿⢿⣤⡀⠀⠀⡀⠀⠀⠀⠀⣉⣉⣠⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠾⢿⡿⢋⠀⠰⠄⠀⠀⠒⣈⣉⣤⣤⣴⣶⠶⠿⠟⠛⠋⢿⣇
⠀⠀⠀⠀⠀⠀⣈⣤⣴⣶⠿⠛⠋⠁⠀⢀⣉⣉⣥⣤⣶⡶⠾⠛⠛⠿⢿⣭⡁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠘⠉⣡⣤⣤⣶⡶⠿⠿⠛⠛⠉⣹⡟⡁⠀⢐⣦⣀⠀⠀⠈⢻⣆⠀⠀⡀⢀⣀⡀⡀⣿⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⣩⣥⣤⣤⣤⣤⣤⣬⣭⣬⣤⣤⣭⣭⣬⣥⣤⣤⣥⣭⣤⣤⣥⣭⣭⣥⣥⣭⣭⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⣿⣟⠉⠉⠉⠉⠉⠉⠉⠉⣩⡿⠛⠉⠉⡉⠉⠙⠻⣯⡉⠉⠉⠉⠉⠉⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⡿⠀⠀⢀⡀⠀⠀⠀⣼⠋⠀⠀⣠⠾⡗⣦⠀⠀⠈⢿⡄⠀⠀⠀⣀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⣿⡟⠀⢠⣶⣦⠀⠀⢸⡏⠀⠀⠀⠻⣶⣇⡉⠀⠀⠀⠘⣷⠀⠀⢠⣶⣄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⣿⣷⠀⠈⠛⠋⠀⠀⢸⣧⠀⠀⠀⣤⡌⡟⢻⡆⠀⠀⢠⡿⠀⠀⠈⠛⠋⠄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠙⠲⡷⠛⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣠⣀⣀⣀⣀⣀⣀⣀⣹⣷⣦⣄⣀⣁⣀⣤⣾⣟⣡⣄⣀⣀⣀⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉




⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣄⣺⠗⠁⠀⠀⠠⠀⠉⠁⠉⠉⠀⠷⡿⠀⠁⠀⠁⠀⠀⠉
⢰⠿⠉⠁⠀⣀⠀⠀⠂⠠⠀⠀⠐⠋⠀⠋⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠺⠇⠀⣤
⠀⠀⠀⠀⢈⠻⠈⠀⠀⢀⡄⢰⡄⠐⠂⠀⠀⠀⠀⠀⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠁⠰⡇⢰⡾
⠀⠀⠀⠀⡎⠀⡄⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠂⠀⠀⠐⣃⠠⡆
⠀⠀⠀⠀⠩⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢀⡨⠖⠺⠛⢀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡠⢄⣀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠆⣀⡄⠐⠋⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡿⠿⡇⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠐⠃⢀⣠
⡀⠀⡄⠀⢐⠂⠀⠀⠀⠀⠀⠀⠀⢿⣯⡀⢠⣟⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠄⠐⠋⢀
⠀⠀⠄⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠐⠃⢈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡙⡁⠀⠀⡄⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠒⠃⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⡷⠰⠂⣇⣸⣿⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠢⠀⠀⠀⠀⠤⠆⠐⠃⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣀⠀⠀⢘⣿⣿⣿⣿⡇⠀⣳⠿⣿⣿⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢤⠀⢠⣶⣶⠀⠀⠀⢀⣖⣂⠸⠯⠌⡒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢜⣭⣽⣙⠃⠠⠘⠀⡿⠻⠄⠀⠀⠀⠀⠐⠙⠀⠀⠀⠀⠈⠀⠈⣸⣿⣠⠴⠉⠀⣿⣇⠐⢃⡘⠉⣀⣀
⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠈⠘⠙⣬⠂⡶⠀⣿⡻⠃⠀⠀⠀⠀⠀⠐⠁⠀⠀⡀⠀⠀⠀⠀⢸⣿⣴⡖⢻⡟⣯⡍⢨⣯⢨⣿⡇⠉
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡶⠛⣧⡿⠺⣧⡤⢾⠳⠄⢀⡆⠀⠀⠐⡆⠀⠀⢤⡾⢶⡿⠿⡿⡆⠇⢹⠇⠘⡇⠸⠟⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠊⠣⠀⠻⠇⠀⣸⡗⣾⡧⠒⠀⠐⠀⠀⠐⠐⠒⠀⠈⠀⢈⣡⣤⠐⠢⡄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⠫⡹⣆⠔⡀⠳⡄⠀⠀⠊⠸⠃⠱⠠⡤⣤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣃⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣽⡔⢱⡷⠀⠙⢜⣌⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⠁⣽⣇⠴⠆⠊⡹⠅⡀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢠⣃⠞⣹⠾⠆⠀⠀⢡⣡⣜
⠀⠀⠀⠀⠀⠀⠀⢰⠇⣦⢻⢃⠘⠃⠀⠀⠀⠀⠀⠣⣿⣣⣎⡀
⠀⠀⠀⠀⠀⠀⢠⡿⢚⣀⢃⣾⠄⠀⠀⠀⠀⠀⠀⢰⡀⣳⣵⠰⡄
⠀⠀⠀⠀⠀⠀⠀⡼⢡⡟⣨⣿⠀⠀⠀⠀⠀⠀⠀⠀⣱⢎⡁⢤⡟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠳⡟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⡙⢼⠖⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⡴⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⡀⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢊⠭⠙⠋⠛⠚⠳⠶⢦⣤⣄⣀⣀⠀⠀⢙⣿⣶⡴⣚⠩⠐⠂⠈⠉⠀⠉⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⡡⠊⠁⠀⠀⢀⠤⠀⠠⢄⠀⠀⠉⠉⠙⠛⠛⠃⢚⡙⢷⣿⣷⣄⠀⠀⠀⠀⢀⣀⡀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡰⠋⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠛⢿⡿⣷⣀⡴⢛⣩⣭⣿⣿⣦⣄⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⣰⢃⠎⠀⢀⠔⠂⠒⠒⢄⠑⢤⣀⣠⡴⠏⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⢮⢻⣽⣿⢁⣾⣿⣿⣿⣿⣿⣿⠂⠀⠙⡄
⠀⠀⠀⠀⠀⠀⢰⢁⠃⠀⠀⢊⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣿⠘⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⠃⠄⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠆⡈⠀⠀⠀⠈⠳⠤⣤⠴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣘⣦⢠⡘⣜⢧⡀⢀⡜⠉⠻⠟⣛⣛⡉⠕⠊⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⢠⢀⠀⠀⠀⠀⢀⠤⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⣭⣽⣿⣿⣿⡖⠛⡏⠁⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠠⠃⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠠⣏⢰⣿⣿⣿⣿⣿⣿⡟⡀⠘⡀⠀⠘⢄⠀⢀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠱⢄⣀⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⡠⠁⠀⠱⠀⠀⠈⣮⡴⠖⠒⠀⠈⠳⣄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠛⠛⠭⠚⠁⠀⠀⠀⢳⣴⣾⣿⣆⠀⠀⠀⠀⠀⢹⡆⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢀⠄⣒⣲⡒⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⠀⠀⠀⢠⢎⠃⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠰⢡⡟⠉⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⢀⣀⣠⣀⡀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⣔⠪⠕⠁⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⢃⠘⠀⠀⠀⠀⠀⠀⢀⠽⠀⠀⠀⢠⡞⢉⣴⣶⣿⣿⣷⣄⠀⠀⠀⠈⢿⣿⣿⣿⣷⡄⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠓⣄⣀⠀⣀⢀⠴⡡⠃⠀⠀⠀⣿⣠⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠈⣿⣿⣿⣿⣿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⢡⠃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠯⠭⠅⠒⠁⠀⠀⠀⢀⡠⠖⠙⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠔⠊⠁⠀⠀⠀⠘⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠠⠦⣄⣉⣉⣛⣉⣉⣉⣉⠀⠶⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⣌⣷⣼⡿⠿⠟⠙⠋⠉⠉⠉⠉⠿⢿⣿⣦⣄⠉⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣑⣾⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣄⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢱⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠙⡆⠀⠀⠀⠀⢠⣶⠾⢿⣷⣶⣶⣶⣤⠀⠀⠀⠀⡀⢃⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⡈⠳⡄⠀⢀⣼⡇⠀⢸⣸⣿⣿⣿⣛⢠⡀⣀⠚⢠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣼⣯⣀⣸⣷⣤⣼⣿⢿⡿⢿⣛⣀⣤⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⡿⠶⠶⠶⠶⠶⠾⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠒⣶⡶⣾⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⢈⣿⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠷⣬⣿⣿⣶⠖⠻⠓⠛⠷⠄⣨⡿⠿⠿⠟⠛⠳⣶⣤⣤⣤⣀⢠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣀⠤⠘⠛⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠁⠐⠒⠉⠽⢛⡻⢶⣤⣌⡓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⡋⠁⠊⠁⡀⢠⡀⠢⠈⠉⠒⠀⠄⠀⠠⠤⠠⠤⠔⠒⠋⣉⠤⠖⣀⡤⠘⡻⠶⣹⢿⣶⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡉⠀⠀⢀⡠⣔⠲⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣴⡆⠀⠀⠀⠀⠀⠉⠁⠚⣣⢄⡈⠸⣋⢜⡽⣿⣶⣅⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠑⠋⠀⣠⠶⠓⠉⠀⣀⢶⣄⠀⠀⠀⠀⠀⢀⠀⢠⠀⢠⡄⠀⡀⠀⡀⠀⠀⠀⠀⠀⢀⡉⠙⠫⣦⡉⠺⣿⣽⡽⢿⣷⣄
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⢀⡨⠂⠀⠁⠀⠀⠉⠆⣈⠀⠠⡄⠘⠂⠈⠀⢀⣀⣈⣁⡀⠀⠀⠁⠘⠃⠰⠆⢀⡘⠃⠀⠀⠈⠻⣷⡌⠻⣿⣷⡽⣿⣧
⠀⠀⠀⠀⣰⢋⠄⠀⠀⡰⠋⠠⠀⠀⠀⠀⠀⢠⡀⠙⠃⣀⡀⠀⠀⠀⠀⠀⢻⠈⣿⠀⠀⠀⠀⠀⠀⠀⢀⠙⠃⢀⠀⠀⠀⠀⠈⠻⣷⡘⢻⢿⣞⣿⣧
⠀⠀⢀⡴⢡⠆⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠻⣿⢧⠀⠀⠀⠀⠀⣞⣦⣿⠀⠀⠀⢠⠀⠀⢠⠟⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠻⣦⢫⢻⡾⣻⣧⡄
⠀⠀⣼⠃⠀⠀⣠⡞⠀⠠⣆⡄⠀⢠⠈⠃⠀⠀⠀⠀⠀⠙⣧⡦⠄⠀⠀⠀⠉⠁⠀⠀⠀⢀⡞⠀⢴⠏⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⣀⠙⢷⡋⠾⣣⢿⣎⡄
⠀⣾⡇⠁⠀⢰⡿⠀⠐⠶⠷⠀⣤⡄⢀⡄⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠰⠆⠘⠋⠀⠈⣿⡜⢩⡞⣿⣤
⢠⣿⠀⠀⢀⠻⡄⠀⠀⠀⠀⢤⡀⠀⠾⠷⢦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⣀⡴⢊⡽⠆⠀⠀⠂⠀⠀⠀⢸⣷⠋⣿⢹⣿⡆
⢸⡇⠀⠀⡘⢠⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢙⡿⢂⠀⠀⠀⠀⠀⠀⠀⠻⠞⠉⠀⠀⠀⠀⠒⠂⠀⠀⠀⣿⡞⠛⣿⣿⣧
⣾⠃⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⢸⣷⢸⣿⣿⣿
⣿⠀⠀⠀⢿⠀⣄⣸⠏⠉⠉⠙⣶⣶⢶⣶⣶⣶⣦⣤⣼⣯⣁⣀⣀⣀⣀⠀⢀⣠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⣿⡄⣿⣿⣿
⣿⠀⠀⠀⢳⠀⠐⠲⠆⠙⠃⠀⣿⠭⠭⠷⠆⠀⠁⠀⠸⣏⡍⠙⠻⠿⣿⣿⣯⣶⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⣓⣾⡇⠀⠒⠀⣭⠿⠀⣿⡇⢿⣿⣿
⣟⣹⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⣿⡇⣾⣿⣿
⢻⡏⠀⢀⡿⠀⠀⠀⠀⠈⣁⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣿⢷⣼⣿⣿
⢹⡇⡆⠀⢡⡶⠀⠀⠀⠀⣠⡄⠀⣿⣻⡞⣛⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢆⣤⢀⡀⠀⠐⠂⠀⠀⠀⢸⣇⣸⣿⢿⣿
⠈⣿⠀⠀⠈⣰⠆⠀⠀⢠⡀⣠⠄⠈⣧⡜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠷⣈⣔⡿⠁⠀⠀⠀⠀⠀⢀⣿⣉⣿⡛⣾⡿
⠀⢹⣇⠀⠀⢡⠞⡀⠉⣩⠆⠀⢠⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠸⠷⢀⣛⡦⠀⣼⣏⢻⣞⣿⣿⠃
⠀⠀⢻⣆⠀⡀⠚⡱⠂⠀⠀⠀⠀⢠⡄⠀⠀⠀⢀⣼⠍⣩⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠈⠐⠀⣼⠇⣼⣿⣽⣿⠏
⠀⠀⠀⢻⣦⣉⠄⠰⣫⢆⠀⠀⠀⠀⠰⠆⠀⠀⠻⢮⣰⠋⠀⠀⠀⠀⠀⠀⡦⢠⡀⠀⠀⠀⠀⠀⠘⡷⣀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⢀⣾⢃⣰⣾⣿⣿⠛
⠀⠀⠀⠀⢻⣷⣻⢆⡀⢀⢁⠀⠀⠀⠀⠀⠋⢀⣤⠀⠋⠀⠀⠀⠀⠀⠀⠀⣷⡸⠃⠀⠀⠀⠀⠀⠀⠰⠍⠃⠀⠠⠄⠈⠀⠀⠀⢀⣴⡿⢡⣿⣟⣿⣿⠃
⠀⠀⠀⠀⠀⠹⣷⣾⢦⡄⠲⡝⢦⡀⠀⢸⡅⡠⠀⠀⠆⢀⡀⠀⠀⠀⠀⠀⠷⠥⠶⠀⠀⠀⠀⠀⠀⢀⡀⠸⠆⢀⠀⠀⠀⢀⣤⡾⢋⢔⣵⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠘⢻⣷⣽⣦⡈⠀⣙⠔⣈⡐⠁⠀⠀⠀⠈⠁⠀⠿⠀⢀⡄⠀⣤⠀⢀⠀⠀⡀⠐⠆⠀⠁⠀⠶⣬⠀⢀⣴⠿⠋⠔⣵⣿⣿⢟⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣽⣦⡠⠂⢈⠟⢢⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣁⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⢏⠉⣢⣴⣿⣿⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣮⣕⡣⣎⡳⢮⡽⠻⣶⠶⣀⣤⣀⠀⠀⠈⠉⠓⠀⠀⣀⣠⣤⣴⡶⢿⠛⢏⡡⣜⣶⣿⣯⣾⠛⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠩⣿⣿⣿⣣⡶⣍⡐⠚⠋⠥⠿⠟⠛⠛⠛⠛⠛⠛⠛⠶⢄⣉⣓⣦⣽⣾⣻⣿⣿⡿⠯⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⠟⢿⣿⣷⣿⣿⣶⣶⡾⣿⣿⣿⣭⠷⠶⠿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣩⣵⡶⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⢀⣤⡄
⠀⠀⠀⠀⠀⣼⠃⠀⢀⣀⠀⠈⣷⠀⠀⠀⠀⠀⠀⣞⢷⣤⣾⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠻⣿⣄⢸⡏⣀
⠀⠀⠀⠀⢠⡏⣶⣶⣿⣿⣶⣷⢸⡆⠀⠀⠀⠀⠀⡟⣸⣿⣿⣄⠈⡇⠀⠀⠀⠀⠀⠀⣼⠁⢀⣀⣈⣻⣷⣅⠘⣧
⠀⠀⠀⠀⢸⢱⣿⣿⣿⣿⣿⣿⡆⡇⠀⠀⠀⠀⢠⣧⣿⣿⣿⣿⣦⢷⠀⠀⠀⠀⠀⢸⢇⣄⣘⣛⣻⣿⣿⣿⣧⢸⡆⠀⠀⠀⠀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⢸⡸⣿⣿⣿⣿⣿⣿⢇⡇⠀⠀⠀⢀⡾⣿⣿⣿⣿⣿⣿⡏⣇⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⡇⣶⣤⡌⡇
⠀⠀⠀⠀⠀⢧⡹⣿⣿⣿⣿⢟⡼⠁⠀⠀⠀⠸⡏⣿⣿⣿⣿⣿⣿⠇⡿⠀⠀⠀⠀⠀⢷⡻⣿⣿⣿⣿⣿⣿⢏⡾⠀⠀⠀⠀⢸⢸⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠙⢶⣭⣭⡶⠛⠀⠀⠀⠀⠀⠀⠙⢮⣛⡻⠿⣟⣫⠞⠁⠀⠀⠀⠀⠀⠈⠻⢮⣝⣛⣛⣫⡵⠏⠀⠀⠀⠀⠀⠸⡼⣿⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⣭⣋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇
⠀⠀⠀⠀⠀⢠⣤⣴⣿⣿⣦⣤⡄⠀⠀⠀⠀⠀⢤⣤⣤⣾⣿⣦⣤⣤⠄⠀⠀⠀⠀⠀⠀⣤⣤⣤⣿⣷⣤⣤⡄⠀⠀⠀⠀⠀⠀⢠⣼⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣻⣿⡆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢠⡿⣣⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⣶⠄⠀⠀⠀⠀⠀⠀⠸⣾⣿⣿⣧⡇
⠀⠀⣀⠀⢀⣠⣀⡀⠀⢴⣦⡌⣧⡼⢋⡉⠉⠉⠁⠀⠀⢠⡄⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣬⣜⣿⣥⣾⢃⡀⠀⠀⠀⠀⡆⣿⣿⣿⣿⢠
⠀⠀⠈⠛⢎⢻⣿⣿⣷⣶⣤⣿⢇⡴⠋⠁⠀⠀⠀⠀⠀⣼⢀⣤⣄⠀⠘⡇⠀⠀⠀⠀⡏⣴⣷⣶⣤⣥⣿⣿⣿⣿⣧⢱⠀⠀⠀⠀⡅⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠳⡙⣿⣿⣿⣿⣯⠎⠀⠀⠀⠀⠀⠀⠀⢀⡏⣿⣿⣿⣿⣷⢻⠀⠀⠀⠀⢧⡻⢿⣿⣿⣿⣿⣿⣿⡿⢏⡼⠀⠀⠀⠀⣇⣿⣿⣿⣿⡾
⠀⠀⠀⠀⠀⠀⠙⣎⠿⠟⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⣿⢸⠀⠀⠀⠀⠀⠙⠲⣍⡻⢿⣿⢟⣩⠖⠋⠀⠀⠀⠀⠀⢸⡸⣿⣿⢣⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞⣿⣿⣿⣿⡿⢸⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢳⣮⣵⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢬⣛⣛⣋⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠒⠚⠛⠛⠛⠛⠒⠒⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠒⠒⠛⠛⠛⠛⠒⠒⠀⠀⠀⠀⠀⠀⠒⠚⠛⠛⠓⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠒⢉⡁⢰⣶⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠒⠒⠋⣉⡠⠤⠖⠒⠋⠉⠈⢇⠀⠿⠇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠔⠒⠚⣉⣁⠤⠤⠶⢊⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠘⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠒⠂⣉⣉⡤⠤⠔⢂⠟⠋⠁⢀⣤⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣧
⠀⠀⠀⠀⠀⡔⠒⢈⠉⢠⡤⠤⠒⠊⠉⠁⠀⠀⠀⡔⠁⠀⠀⣴⣾⣿⡿⣷⡆⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⡀
⠀⠀⠀⠀⠀⢱⠀⠘⣢⠀⡇⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢸⣿⡁⢹⣷⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇
⠀⠀⠀⠀⠀⠘⡆⢀⠥⠔⠁⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠈⠻⣿⣾⣿⣦⣤⣀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢹
⠀⠀⠀⠀⠀⠀⢡⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢹⣿⠙⢻⣷⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠀⠏⡇
⠀⠀⠀⠀⠀⠀⠈⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⢠⣿⡇⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠰⡇⠀⢧⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠐⠿⣷⣶⣿⣿⡿⠋⠀⠀⣠⠋⠀⠀⠀⠀⣀⣠⠤⠄⠓⠀⣉⣠⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠈⠛⠁⢀⣠⣞⠥⠤⠒⠒⢉⣁⠤⠤⠖⠒⠈⠉⠀⣳⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⣆⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣒⡲⠶⠶⠒⠊⣉⣁⠤⠔⠒⢺⠋⠁⠀⠀⠀⠀⠀⢀⠞⢁⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢴⠀⠙⡆⠀⣀⣀⠤⠄⠒⠚⢉⣁⠠⠤⠶⣶⣿⣿⠟⠁⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⢀⢸⡄⠈⡳⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠠⠇⠀⢉⣉⡁⠤⠔⠒⠋⠉⠓⢤⣀⠀⠀⠀⠈⠉⢀⣀⣴⣋⡠⠤⠤⠒⠒⠂⢋⣉⣉⡡⠤⠤⠰⡖⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠧⢲⠊⠙⡍⠀⠀⠀⠀⠀⣀⣀⣀⠠⠬⠽⠛⠒⠊⢉⣉⣁⡠⠤⠤⢲⠖⠂⠉⠉⠉⠀⠀⠀⠀⣀⣀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⡷⠀⠻⠒⠒⠊⣉⣁⣤⠤⠤⢤⠒⠒⠊⠉⠻⠟⠉⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⢠⠊⢠⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠦⠤⠖⠲⠶⠬⠭⠥⠤⠤⠤⠤⠤⠤⠭⠶⠦⠤⠤⠤⠤⠶⠮⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠀⠨⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⣶⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣾⣿⣿⣿⣿⣿⣯⣿⣿⣿⡀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⢜⠀⠀⠀⠠⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣷⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠳⠐⠊⠁⠀⢸⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣈⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣏⣽⣟⣿⣿⣿⣿⣿⣿⠻⢻⣾⣿⣟⣡⣤⣀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⠻⠟⠉⠉⠉⠉⠉⠸⣿⣿⣿⣿⣿⣿⣿⣿⣤⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⡀⣾⣿⣿⠈⣻⣿⣋⡙⠿⠟⠁
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⡄⠋⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠟⠛⢛⣿⣿⣿⣉⣿⣷⣤⠹⣿⣿⡗
⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠙⢿⣿⣿⢆⢛⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠁⠀⠀⣷⡄⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⡄⢰⣿⣿⣦⢾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠐⠀⠀⣦⠀⠘⠿⡆⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⡿⣿⣶⣾⢿⣿⣟⢠⣭⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⣻⣤⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠻⣿⠟⢠⣿⣿⣧⣿⣧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠊⠀⠀⠀⠁⠉⣿⡟⠀⠈⠙⡃⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢡⣾⣷⡆⣘⣿⣿⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⡔⠉⢀⣀⡀⠀⠀⠀⠀⠄⠀⠐⠛⠃⡀⠀⢠⡟⠒⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢛⣼⣿⣿⡄⣾⡧⣽⣧
⠀⠀⠀⠀⠀⠀⠘⢀⡞⣁⠠⠃⠀⠀⠈⠀⠠⠘⡇⠀⣠⡟⠀⠀⠸⡄⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⢟⣀⣿⢾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠉⠈⢳⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠆⠀⠀⠀⠀⠀⠈⠄⠠⠋⠀⠀⢰⠁⢰⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡍⢰⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢾⠙⡃⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣶⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⡛⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢾⡏⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠋⢳⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡿⢿⠿⡿⠿⢿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣛⣿⣿⣿⣿⣟⣿⣿⣿⣟⣻⣿

⠀⠀⠀⠀⢀⡤⢖⣛⡲⣤⠀⠀⠀⡠⠔⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⡉⣦⣀
⠀⠀⢀⢤⣾⢰⠋⠀⢹⢘⡇⠀⠘⡄⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠂⢣⠸⡀⠑⢦⡀
⠀⣠⠼⠦⠉⠘⠦⠖⣋⡞⠁⠀⠀⢱⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⡁⠀⠈⡆⢣⠀⠀⠙⢄
⡼⢡⠞⠉⢳⢠⣌⠻⣅⠀⠀⣀⣀⡠⠧⠤⠤⠵⠤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠿⢟⣦⣈⣲⣄⣱⣘⣆⣀⣀⣀⢳⡀
⢧⡸⣄⡤⢎⡼⠁⠑⣬⠟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠒⠲⢤⣀
⠀⠙⠒⠒⠋⠀⠀⣰⠏⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣬⣷
⠀⠀⠀⠀⠀⠀⢠⡯⢬⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢣⣧⣴⣼⣇
⠀⠀⠀⠀⠀⠀⣼⠛⠛⠂⢀⣴⣾⣿⣿⣿⣿⣷⣦⡘⠠⠄⠀⠤⠤⠤⠤⠤⠤⠤⠤⠠⢃⣤⣶⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⢸⡘⠯⠭⠭⣿
⠀⠀⠀⠀⠤⠤⠿⢄⠀⣰⣿⡿⢋⠕⠒⠒⠨⡙⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢟⠥⠒⠒⠂⢝⢿⣿⣷⡀⠀⠀⢨⠍⠉⠉⠹
⠀⠀⠀⠀⡀⠀⠀⣸⣠⣿⣿⢱⠁⠔⠉⠉⢢⠘⡈⣿⣿⣤⣤⣤⡄⠀⠀⠀⠀⠀⠠⣿⣿⠇⠎⡰⠉⠈⠱⠀⢃⢻⣿⡇⣤⡄⠨⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡸⡀⠢⣀⣀⠜⢠⢠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⢆⠱⢄⣀⠴⢀⠆⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣌⡒⠀⠀⣒⣥⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣮⣑⠀⠀⢐⣡⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠶⠒⠒⠒⠒⠒⠒⠒⢶⣶⣒⣒⡲⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⡁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠉⠉⠀⠀⠀⠀⡼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠁⠂⢤⣀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⡏⠀⠀⠀⠀⠈⠉⠒⡤⣄⡀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠁⠂⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠉⠑⠢⢄⡀⠀⠀⠀⠀⠀⠸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣴⡂⠀⠀⠀⠁⠒⢤⡜⢀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠄⢀⣀⠀⠀⠀⠘⠓⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⡇⣿⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢹⠀⠰⣇⣵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⠀⠀⡇⠠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡌⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠁⠀⡇⠀⠸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣧⣶⣷⠀⠀⠀⠀⡀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⡤⢷⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡁⠰⠀⠀⠀⡀⠀⠀⠉⠀⡤⠀⠀⠀⠀⠀⠀⢧⣾⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⢈⠂⢄⠀⠀⠾⠃⠀⠀⠀⠀⠐⠀⡄⠀⠠⡿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣌⡀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠘⠇⡇⠀⠀⢠⠃
⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣤⣤⣀⣀⠀⠈⠙⠓⠤⣐⡠⠐⢶⡄⠀⠀⠀⣣⠀⡠⠃
⠀⠠⣾⣿⣿⡟⣿⢻⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠉⠑⠒⠤⢤⣠⣼⣿⠔⠁
⠀⠀⣿⣿⣿⣏⣿⣣⣃⣕⣩⣼⣿⣿⣿⠇
⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣴⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠘⣿⣿⣇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⠟⣛⡋⠃⠀⠈⢻⣧⠀⠰⣦⣴⢆⡀⣠⣶⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⣀⣠⣶⣴⣿⣿⣿⣿⣿⡏⠓⢄⠀⠀⠉⠁⠀⠈⢻⣦⡟⠉⠛⠓⢦⣀⣀⣠⣿⡿⣧⣄⣀⠀⠀⣠⠿⠿⣦⡀
⠀⠀⠀⠻⡿⢿⡙⢿⣿⠟⢋⣿⣿⣿⡉⠀⠀⠁⠀⠀⠀⣀⣤⢀⠈⠙⠂⠠⠀⡀⠉⠉⠋⠉⠀⠀⣩⣽⣷⠾⡁⠀⠀⠈⠣⠦⣄⣀⣤⣠⣄
⠀⠀⠀⠀⠀⢀⣤⣤⡌⠐⢿⣿⣿⠛⠁⠀⠀⢀⣠⣿⣿⣿⣿⣦⠻⢦⡄⡀⠀⠘⠛⠷⠶⠤⠔⠋⠉⠙⣹⡀⠀⠠⠄⠀⣄⠀⠈⠀⠀⠀⠈⠉⠂
⠀⠀⠀⠀⠀⠈⠋⠉⠙⠤⡀⠈⠻⣤⣤⣤⣴⣿⣿⣿⣿⠿⣿⣿⣄⣤⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⢰⣿⠄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⠏⠿⢁⠎⠀⣽⢿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣀⣴⣿⢿⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⣠⠀⠀⠂
⠀⠀⠀⣀⡀⠀⠀⠀⠁⠀⠄⠀⠀⡀⠀⠀⠈⠛⠂⢀⡈⠀⠀⣡⠟⠋⠁⠘⢀⠀⠀⠀⠀⠈⠙⢿⡿⣦⠀⠀⠀⠀⢰⣿⣿⣯⣤⠴⠀⠖
⠀⠀⠀⣿⣿⣿⣔⢦⣀⣀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠁⠴⠘⠁⡀⣸⣿⣾⣾⣷⡄⠀⠀⠀⠀⠀⠘⠿⢿⡓⠀⠀⣻⣿⣿⣿⣯⡀
⠀⠀⠀⠙⠫⠉⠉⣻⣿⣿⣦⣿⣆⣀⣄⢸⣿⣇⢀⣄⠀⠀⠀⠉⠘⠿⣿⣿⡛⠉⠉⠂⠀⠀⠀⠀⠀⠀⠀⠁⣀⣹⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⣐⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⡿⡇⠀⠀⠀⣴⡆⢀⠈⠙⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣧⣴⣄⠀⣀⡀
⠀⠀⠀⠙⠻⠿⢿⠿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣆⢰⣿⣿⣿⣆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠛⠟⠻⢿⡛⠿⢿⡿⡿⠿⢿⣿⣿⣿⣧⢀⣸⣷⠀⠀⠀⠀⣀⠀⣿⣿⣿⣿⣿⣿⣷⣿⣿⡿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠂⠀⠀⠘⠛⠛⠛⠋⠀⠀⠁⠉⠙⠛⠋

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⢤⣶⣤⣤⣼⣦⣤⣤⣤⣤
⠀⠀⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡆⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣹⠀⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠁⠀⠀⠿⠶⠶⠶⠤⠶⠶⠶⠶⠶⠶⠶⠶⠿
⠈⢅⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠀⠀⡄
⠀⠀⠑⠢⠤⠤⠤⠆⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⠉⢉⣀⠀⠀⠀⠀⠀⠀⢀⡴⢿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⠤⠀⢀⡀⠀⠀⠀⠀⣀⣈⣀⣼⠁⠀⠀⠀⠀⢀⡤⠒⠉⢸⡀⠈⠙⢆
⠀⠀⠀⠀⠀⣇⢠⢄⡠⠀⠘⠀⠀⠈⠦⡄⠀⠀⠈⡇⠀⠀⠉⠀⠀⠀⠀⠀⠜⠁⠀⠀⠘⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠉⢠⠋⢰⣄⠀⢀⡀⠀⠀⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠘⠁⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⢺⠐⠀⠈⠓⠋⠀⠐⠈⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠨⣀⠀⠀⠀⠀⢀⣀⣔⠇
⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠉⠈⠀⢀⠀⠀⠀⠀⠀⠈⠚⠶⢾⢿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠑⠀⠀⣀⡀⠀⠀⠀⠀⠀⣆⢀⡄⠀⠀⠀⠀⠀⠀⠀⠑⠀⠄
⢠⣠⣀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡜⠃⠈⠉⠳⡄⠀⠀⢠⣿⠋⠀⠀⠀⣀⣤⣬⣤⣾⣜⣂⣐⣒⣢⣀
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⣧⣀⠀⠀⠀⠘⣆⢀⣾⡏⠀⠀⠀⠀⠈⠥⢤⣤⣤⣤⣤⣤⣤⣬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⢹⣾⠋⠀⠀⠀⠀⠉⡉⠡⠉⠉⠁⠀⠉⠉⠙⠉
⠀⠀⠀⠀⣀⠄⢂⠄⠀⠀⠀⠀⠀⠀⢴⡈⣿⣷⣄⣠⠾⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠂
⠀⠀⡀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠰⠊⠀⡠⠉⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠁⠈⠁
⠀⢈⠀⠀⠀⠈⠀⡂⠀⠀⠀⠀⡜⠀⠀⢴⠀⠐⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⣀⣂⣀⠂
⠀⠂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡇⠀⠀⢦⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢸⠀⢃
⢠⠆⣶⣔⡐⠚⠒⠐⡆⠀⠀⠀⠁⠌⣀⣤⠶⠗⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠼⢀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢰⠃⣀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡉⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⢈⠁⡐⠆⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⡁⡀⠦⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⢡⠄⠀⠰⠂⠀⠀⠀⠀⠀⠠⡀⠚⠀⠀⠀⠀⠀⠀⢉⡴⠀⠠⠂⠀⠀⠀⠀⠀⠀⣄⠑⠃⠀⠀⠀⠀⠀⠈⣠⠆⠀⠖⠀⠀⠀⠀⠀⠀⢠⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠁
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠚⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠘⠡⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠃⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⡆⠐⠈⠀⡀⠀⠀⠀⠀⠀⠀⠰⢀⠀⠀⠀⠀⠀⠀⡴⠀⠌⠃⡀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⠀⠀⠀⠀⢰⠄⠢⠑⢀⠀⠀⠀⠀⠀⠀⠀⠰⣀
⠀⠀⠀⠀⠴⠋⠀⠈⠀⠀⠀⠀⠀⠀⠐⠆⠁⠀⠀⠀⠀⠀⠀⠠⠚⠀⠈⠁⠀⠀⠀⠀⠀⠀⠢⠈⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠉⠀⠀⠀⠀⠀⠀⠰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠰⠒⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠠⠖⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠈⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⠶⠖⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡄
⠀⠀⠀⢀⡤⣤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣽⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⢈⠀⠀⠀⣀⡀⠂⠀⡀⠀⠀⠀⠀⢀⣬⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠠⠏⣼⠏⠀⠀⣼⣁⡄⠀⠃⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⣿⡇⠀⠀⠀⣽⠆⠀⠐⡁⠄⠈⠀⠀⠈⣼⡇⠀⠀⠀⠁⠀⠀⠀⡉⣰⠼⠚⠀⠀⢀⣈⣿⣄⣀
⠀⠀⢀⣾⡗⠀⢀⡴⢿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠰⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⡷⢲⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠨⠉⠁⠉⢹⡿
⠀⠀⠀⠉⠀⠀⢙⢠⣛⣇⠀⠀⠀⠀⠀⠀⢰⡄⠀⠸⠀⠀⠀⢼⣿⡀⢈⠀⣸⣿⡇⠀⠀⠀⠠⠀⠀⢠⣿⣇⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢠⣾⣟⣿⣿⠀⠐⣦⢀⢰⣶⣶⡆⠀⠀⠄⠀⠀⠈⣿⡇⠈⠀⠉⢹⡇⠀⠴⠀⢰⠀⣠⠻⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣴⠄⠈⣏⡉⢛⣿⠁⠀⠀⠘⠸⢻⠿⠃⠰⠈⠁⠀⠀⠀⢹⡏⠀⠀⠀⢸⡇⠀⠶⣤⠞⠀⣿⢺⣿⣿⣷⠇⠀⠀⠀⠂⠀⠀⢄
⠀⠰⢦⣖⠂⠀⢀⠀⢸⣿⡂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⢀⣄⣼⡇⢀⢀⣀⣸⡇⠀⠀⠁⠀⠀⢸⡄⣿⣿⣿⠀⣀⣠⣦⣄⣀⣀⣈⣳⣄⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠙⠀⠀⢀⡀⣨⣿⠇⣠⠀⠀⠀⢀⠀⠀⠠⣀⠀⢨⠸⣿⣿⣧⠸⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⠇⢿⣿⣿⠀⠻⣯⣽⣭⣽⣿⣿⣿⡿⠃⠀⠀⠀⠀⢸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡆⠀⠉⠙⠒⠦⣄
⠀⠀⠀⠀⠀⠀⣀⣬⣭⣿⣿⣿⣿⣧⣤⣄⣀⠈⠥⢜⢇⣀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣯⣼⣭⣻⣿⣿⣿⣿⠀⠐⣶⠮⠁⠈⠉⠒⠀⠀⠒⠈
⠀⠀⠀⠀⠀⣿⣿⠿⢿⣿⡟⣿⣿⠟⡟⢻⡓⢠⡄⠈⢠⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠙⢡⣾⣿⣿⢹⢟⢣⣶⠈⠈⠇⢸⣟⠳⣾⡆⠀⣿⡀⣰
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣾⠀⣀⣿⣆⠀⣀⣼⣿⢠⣾⠟⠛⢻⡞⠛⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢃⣤⣿⣯⣌⣾⣿⣿⣄⠀⢘⣄⣀⣄⣿⣆⣴
⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⣿⣿⣛⣿⣿⣿⡿⢿⣿⡌⠁⢻⣿⡿⠛⠻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣏⢹⡏⠀⠀⢨⣿⣦⣎⠈⢿⣄⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣿⣿⠿⠾⢷⠋⠀⢸⣿⠷⣿⣋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣤⣤⣼⠷⠒⠻⠃⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⢿⣷⣠⣼⡟⢻⣷⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣤⣾⠋⠉⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣻⡿⣦⣠⠜⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⢦⣀⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠛⠉

⣹⣦
⠈⢳⡀⠉⠲⣄⡀
⠀⢸⡇⣄⡀⠼⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠶⠦⢤⣄⣀
⢀⡏⠸⠀⠁⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠋⠉⠀⠀⠀⠀⣀⣠⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀
⢸⠁⠀⠀⠀⠀⣸⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⢀⣀⣀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠉⡧⠀⠀⠀⠀⢀⡤⠤⢤⡄⠀⠀⠙
⠈⣆⠀⠀⠀⡼⠁⠀⠹⣄⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⡴⠛⠉⠉⠉⢻⡆⠀⠀⠀⠱⢤⣤⣤⠤⠞⠁⠀⠀⠀⡔⠁⠀⠀⠀⡿
⠀⠈⠳⢤⣴⠃⠀⢀⡴⠟⣦⡶⠛⠁⠀⠀⠀⠀⠀⠸⢅⣀⣀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⣀⣀⡤⠞⠁⠀⠀⢠
⠀⠀⠀⠀⠀⢇⢠⠏⠀⠀⠈⢧⣠⠴⠶⠶⣦⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠈⢷⡀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⢠⡿⠳⡀⠀⠀⠀⠀⢳⡔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠲⡆⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠈⣇⠀⠙⣆⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⢀⡴⠃⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠹⢦⣀⠈⢳⡀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠶⠋⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡙⣆⠀⠀⠀⠈⢧⣠⠤⠔⠒⠒⠒⠲⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠳⡀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⡽⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⢀⡟⢦⡀⠀⠘⣧⡀⠀⢀⣠⡴⠋⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠘⢦⣀⣉⣹⣦⣼⡷⠞⠋⠁⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣄⣀⣀⣀⣠⣤⣤⣤⡴⠶⣾⠛⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠘⢷⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⢀⢴⡀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⣀⡠⠤⠔⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠢⢄⡀⠰⠁⠈⠢⡠⠋⠀⠣⡀⠔⠊⢀⣢⠠⠔⠒⠊⠉⠀⠀⠀⠂⢤⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⠀⠀⡄⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⡠⠤⠒⠈⠁⠀⠀⠘⠀⠤⠀⠀⢀⡄⢶⠙⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠀⠀⠈⠂⢠⠀⠀⠠⡆⠻⠭⢀⠐⢄⡁⠈⡁⠐⡄⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠒⠂⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠂⠀⡆⠸⠉⢀⠀⠠⠤⠀⣦⠀⢡⠀⠈⠀⠟⠀⡇⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠠⣀⣁⡀⢤⡀⢱⠀⠤⠀⠈⡀⠃⠀⣁⣀⢰⠊⠀⠀⠀⢘⡀
⠀⠀⠀⠀⢂⠉⠉⢉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢖⡦⠀⠀⠀⠀⡀⠠⠘⠁⡜⠀⠑⠒⠠⠃⠀⠀⣀⣀⠥⠤⠖⠒⠊⠉
⠀⠀⠀⠀⠈⢢⠀⠁⣤⡷⠶⡄⠀⠀⠀⠀⠀⠜⠤⠷⣄⠀⠀⠀⠀⠀⠦⠔⢄⡿⠀⢀⡀⡠⠤⠒⠊⠋⠁⠀⠃⠤⠤⣄
⠀⢀⡠⠤⠒⠊⠀⠀⠈⣀⣤⠿⠶⠶⠶⠶⢦⣄⡀⠀⠈⠳⣄⠀⠀⠀⡀⠄⠀⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠁⠢⡀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠈⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⢄⡀
⠀⠀⠀⢀⡨⢣⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠚⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡳⠂
⠀⠀⣔⠁⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⢳⣝⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠁
⠀⢀⡴⠛⠛⠇⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢻⠟⠒⠀⠀⠀⠀⠉⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⢾⡐⠥⡄⠀⠀⠀⢀⣸⣿⡇⠀⠀⠀⢐⡆⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠜⠃
⠀⠈⣳⡶⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⣠⣴⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢅
⠀⢀⡿⠀⠐⠀⠀⠀⠀⣀⣠⣤⣤⡤⠶⠶⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠑
⠀⢸⣇⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⣤⣴⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⢿⡄⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠼⠄
⠀⠀⠈⠻⣄⡀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠛⠋⠁⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⠻⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠛⠉⠀⠀⠀⠀⠀⢀⡀⠀⢠⠁⠉⠁⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⡟⢉⠝⠋⢂⠀⠀⡴⠓⢄⠀⠀⠎⠈⠢⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠳⠊⠀⠀⠀⠑⠜

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

⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣠⣾⣿⣿⣶⠤⠀⠀⠀⢀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠐⣛⣉⣤⡀
⠀⠀⠀⠀⠋⢻⣿⡿⠀⠀⠀⠀⠀⣾⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠐⠻⣿⣿⠋⠁
⠀⠀⠀⢀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠉⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⣻⣯⣀⣀⡀
⠀⠀⢠⣿⣿⣿⣿⣿⣷⠄⠀⠀⠀⣴⣾⣿⠆⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⡆⠀⠀⢀⣶⣾⣿⣿⣿⣿⣿⣷⡀
⠀⠀⣾⡿⣿⣿⣿⣿⣿⠀⠀⠀⢠⠛⠿⠿⠰⣿⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣿⣿⣿⣿⣿⣧⠀⠀⢸⣿⣿⣿⣿⣿⡿⠀⠙⢿⣦
⠀⢀⣿⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡜⢿⡇⠀⠀⠀⠀⠀⣠⣿⣃⣈⣿⣿⣿⡿⠁⣿⡀⠀⣸⡏⢹⣿⣿⣿⠁⠀⢀⣴⡿
⠀⠈⢿⡄⠈⢛⣿⣯⣿⠄⠀⠀⢀⡴⢀⠘⠄⣸⡇⠀⠀⠀⠀⠐⠛⠛⠉⠙⣿⣿⡿⠁⢀⣼⠇⢠⣿⠁⠸⠛⠷⣖⣤⡶⠛⠁
⠀⠀⠈⣷⠀⢾⣿⣿⣿⡇⠀⠀⠈⠀⠀⠀⠀⣿⠀⠀⠐⠤⣀⠀⠀⠀⣠⣿⣿⣿⣧⡾⠛⠁⢀⡿⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⣿⡄⢀⣿⣿⣿⣿⠀⠀⠘⠘⠀⠀⠀⣿⡀⠄⠀⠀⠈⠙⣷⢰⣿⣿⣿⣿⣿⠇⠀⢀⣾⠁
⠀⠀⠀⠈⠁⠘⣿⣿⣿⡿⠀⠀⠀⠀⠀⢀⠀⠈⠙⠀⠈⠢⣄⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠈⠃⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⡀⢠⣤⣿⡀⠀⣀⡤⠂⠀⠀⠀⠘⣿⣿⣿⠃⠀⠀⠀⢠⣿⡟⠀⢤⣴⣤⠄
⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⠀⢰⣿⣏⠙⠀⠉⠁⠀⠀⠀⠀⠀⠀⢻⣿⡏⠀⠀⠀⠀⠀⠻⠅⠀⢸⣿⣿
⠀⠀⠀⠀⠀⣰⣿⣿⠇⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⢀⣿⣿⡿⠀⠀⠀⠀⠀⣿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⣸⠏⣿⠇⠀⠀⠀⠀⢸⠏⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⡿
⠀⠀⠀⣠⡏⣠⣿⠀⠀⠀⠀⢀⣿⠀⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣿
⠀⠀⣰⡿⠠⢻⡿⠀⠀⠀⠀⢸⡏⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⡄
⠀⠘⠟⠁⠀⠸⣇⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⢿⡿⣿⣿⣶⣶⣦⣤⣤⡄⢠⣄⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣘⣿⣯⣲⣦⠀⠀⠀⠀⢀⠄⠁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣇⠈⣧⠀⠀⢠⣿⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡈⢿⣿⣿⣆⣿⡇⢠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣉⣽⣿⣿⣿⣿⣿⠷⠚⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⣩⣿⣿⣿⣾⣶⡶⢲⣾⡑⢝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣧⣤⣾⣿⣿⣿⣿⠟⣸⠃⠈⠻⡿⣄⠁⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⡀⠋⢠⡄⣤⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠋⠀⢀⣴⣾⣿⣿⣶⣄⡀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⣽⣯⠀⠀⣿⣿⡟⡇⣿⠿⠛⠃⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡗⡀⠀⢿⣿⣿⣧⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⡇⠀⠀⠛⢹⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠒⠀⢠⣤⣿⢸⡟⣿⣿⡄⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⢈⣹⣻⣿⣿⣷⡶⠶⢦⣤⣄⣀⣀⡀⠀⠀⠀⢀⠠⠀⠀⠀⢀⣿⣿⣼⣧⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠉⠙⠻⢿⣿⣷⣤⣮⣿⣿⣿⣿⣿⣿⣷⣂⣀⡤⠖⣾⣿⣏⢻⣿⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠁⢷
⠀⠀⠀⠀⢶⣾⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣽⣿⠿⢛⣁⣀⣀⣿⣿⣷⡬⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠛⠛⣿⣯⣍⣉⣉⣉⡙⠛⠛⠻⠿⡯⠔⠚⠋⠉⠉⣙⡻⣿⣿⣧⢀⡀⢰⣁⠀⠀⠀⡆⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠐⠿⠿⠿⣿⣿⣿⣻⣻⣶⣶⣦⣤⣄⣀⡀⠐⠉⠁⣀⣿⣿⣿⣧⡬⠙⠿⠿⡿⠀⣂⣠⡤⠞⠋⠁
⠀⠀⠀⠀⢸⣶⣦⣤⣀⣀⡀⠊⠉⠉⠛⠛⠻⠿⣿⣯⡤⠴⣚⣉⣽⡿⠿⣿⣿⣿⣿⣿⣻⣟⠁⡿⠟
⠀⠀⠀⠀⠙⠿⠷⠾⣯⣿⣿⣿⣶⣶⣦⣤⣀⣤⠼⠗⠊⢉⣉⡠⠼⠁⠀⠀⣌⠁⠀⠀⢠⡿⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⣿⣧⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⡼⢽⡧⠀⠼⣏⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣻⡗⢀⡚⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠷⢦⣯⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣦⡤⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣒⣒⣢⠉⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠄⠀⠀⢠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⡤⠤⠤⠶⠶⠒⠚⠋⠉⠉⣡⠞⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠤⠴⠶⠶⠶⠒⠚⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠸⡆
⠀⠀⠀⠀⠀⠀⢸⡿⠥⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠈⠉⠛⠲⠦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠿⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⢦⣤⣄⣀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡴⠋⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡇⠀⠀⣾⡷⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⣞⣱⠆⠀⠀⠀⠀⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⣅⣠⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠺⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠲⠶⢤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⠦⣤⣤⣀⣀⣀⣿⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠐⠠⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡏
⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⡇
⠀⠀⢀⡔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⣇
⠀⠀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⢧⡘⣆
⠀⠀⠀⠀⠀⡠⢀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠤⡀⠀⠀⠀⠀⡀⠀⠀⠈⢳⡘⣧
⠀⠀⠀⡀⠊⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢷⣸⡦⠶⠶⠦⣤⡀⠀⣠⡶⠛⠛⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⠶⢦⣀⡾⠋⠀⠀⠀⠀⠈⢻⣤⣿⣤⣀⠀⠀⠀⣨⣧⣀⡀
⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⢈⡏⠀⠀⠉⠻⣦⠟⠉⠀⠈⠙⠳⣄
⠀⠐⠀⠀⠀⠀⠀⠀⠀⡠⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⣿⠀⠀⠀⣠⡶⠋⠉⠉⠛⢦⠀⣠⣤⠤⠼⣧⡀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠹
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠘⢧⣀⣠⡏⠀⠀⠀⠀⠀⠈⡟⠉⠀⠀⠀⠀⠙⢧⡀⠀⢀⡇
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⣧⠀⢀⡴⠋⠀⠘⣇⠀⠀⠀⣀⣠⣄⡁⠀⠀⠀⠀⠀⠀⠈⣧⠴⠾⠷⠦⣄⠀⠀⠀⢠
⠀⠀⠁⠀⢀⠀⠀⠠⠠⠄⠂⠂⠒⠂⠉⠀⣀⠀⠀⠀⠀⠙⢦⣼⠁⠀⠀⠀⠈⠳⣴⠛⠁⠀⠀⠉⠳⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠳⣤⠶⠋
⠴⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⠀⠀⠀⠀⠄⢻⡄⠀⠀⣠⡤⣼⡇⠀⠀⠀⠀⠀⠀⠁⠀⢀⣠⣄⣼⡃⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠂⠀⠀⠀⠀⠙⢳⠞⠁⠀⠘⠃⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠈⠙⢷⡄⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢻⡋⠙⠳⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⡴⠖⠂⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠃⠀⠀⠀⣀⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⣴⠞⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢛⣡⣤⣤⣀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣞⠉⠀⠀⠀⠉⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠈⢻⡄⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢷⣀⠀⣀⡀⠀⠚⠉⠛⠦⠀⠀⠀⠀⣤⣾⣁⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⣠⠞⠉⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣤⣤⠴⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡏⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⢦⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⣤⠶⠛⢉⡿⠒⠛⠁⠀⠀⠀⢤⠀⠰⠞⠳⢤⡀⣀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡿⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⣹⠷⣄⣀⣀⣀⣴⢿⡄⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠛⡟⠀⠀⠉⠉⠉⠀⠀⢹⡶⣴⡶⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⣀⣀⣤⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣶⣶⡦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⣤⣤⣤⣼⣄⣬⣿⣿⣯⣄⣀⣰⣇⣀⡀⠙⢷⡌⢹⣿⣶⣄
⢀⣀⣄⠀⣂⠠⠦⠄⠀⠤⠤⠤⡤⢠⡀⣉⣋⣉⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠋⠁⠐⠚⢛⡈⠉⠉⠉⠉
⠸⣍⠹⣉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⣻⠋⣉⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠨⢡⠻⠁⠛⠉⠙⠫⣍⠛⠛⠛⠛⠓⢸⡥⡌⠻⠒⠀⠀⣿⡿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢻⣿
⠀⠀⠀⣦⣤⡤⣤⣤⣭⣥⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⠆⠋⣴⡾⣿⠀⠐⠛⠛⠛⠛⠋⠉⠁⣈⣿⣗⠛⡿⠂
⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠋⠙⠿⠿⠿⠿⠛⢻⣿⣿⣶⠰⡦⣡⡟⠛⠛⠛⠛⠛⠛⠉⠉⠙⣿⣿⣯⣵⠃

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠂⠀⡄⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⣠⣤⣀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⠂⠀⠀⠀⠹⠉⠛⠋⢁⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡞⠻⣤⣶⣿⣁⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢀⣤⠀⠀⡄⣄⡝⢃⡼⠷⠃⠈⠤
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⣧⠄⠠⠀⠹⣶⠁⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠘⠡⣀⠀⢠⣿⣟⡀⠠⠀⠀⠠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣷⠂⠀⠀⢸⣏⢷⠈⢢⣜⣳⢆⣠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⡀⠀⢰⣿⡇⠀⠀⠈⣿⢸⡇⡄⠙⢣⣴⠋⠑⢆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢤⣾⠿⠃⠀⠀⠀⠈⢸⠇⡇⠀⢈⠁⢀⣉⡋⠓⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠙⢷⣼⣥⠀⠀⠀⠀⠀⠀⣸⠀⡇⢀⣼⠟⢩⣤⣿⠓⠀⠀⠀⠑
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⠌⢻⡀⠀⠀⠀⠀⢀⡇⠀⠁⢹⠿⠉⠉⠛⠋⢦⠀⠀⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⡘⠀⡀⢠⠌⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⣜
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠰⣤⠠⠉⢹⣰⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠘⠧⠤⣀⣀⣀⠴⠇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢰⠆⠀⠄⠀⡤⠤
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⡁⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠀⠸⢇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⠆⠹⠇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⡜
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⢸⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⡵⠋⠀⠀⠀⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢶⡺⠛⠛⠓⠶⣤⠀⠀⢸⡟⠲⠦⠤⠤⠴⢶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣋⠀⠀⠀⠀⣀⣼⡇⠀⣿⣝⠷⠶⠤⠤⠶⣾⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣧⣉⡙⠛⠛⢋⣹⡿⠃⠀⢻⣍⡙⠛⠛⠛⠋⣿⡟⠀⠀⠀⣀⣤⠴⠚⣋⣥⣙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⣥⣍⣉⣛⣫⣽⣿⠀⠀⠀⠀⠈⠙⠛⢛⣛⣫⣥⣤⠶⠚⠋⠉⠀⠀⠉⠁⠀⠉⠓⠌⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⣀⣀⣤⠾⠛⢀⣀⣀⣤⡤⠶⠒⠛⠻⣿⣿⣿⣷⣄⠐⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣟⠋⠉⠉⠀⠀⠀⠀⠀⠀⣷⠈⠻⣿⣿⣿⣷⣄⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⣀⡴⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡶⠟⠳⣄⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠈⠻⣿⣿⣿⣿⣦⡛⠀⠀⠀⢀⣠⡴⢚⣩⡴⣾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⡛⢦⡀⠈⠳⣄⡀⢄⡀⠀⠀⠀⠉⠻⠷⣦⣬⠙⢿⣿⣿⣿⣦⡴⢞⣫⡵⣞⣫⠵⢶⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⡛⢦⡙⢦⣄⠈⠛⢦⡙⠶⣤⣤⠤⠤⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⠿⢛⣋⡭⠶⣚⣻⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡙⢦⡙⢦⡙⠷⣄⠀⠙⠶⠶⠶⣒⣛⣯⠽⣷⣿⡿⢿⣿⣿⣿⠶⣛⣭⠵⠖⠋⢩⡴⣞⠛⠛⠛⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣝⢦⣙⢦⣈⠳⠶⠖⢛⣋⣭⠵⠖⣛⣭⠴⢖⣻⣿⣿⠟⠋⠁⠀⠀⠀⠀⣿⣜⣃⠀⠀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⢦⣝⠷⠶⢚⣋⣥⠴⢞⣫⡭⠶⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣴⢻⣤⣉⠙⠛⢋⣉⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⠷⣚⣫⡥⠖⠛⠉⠁⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀⠀⠀⣿⠦⣤⣍⣉⣩⣭⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢹⠒⠁⠀⠀⠈⠙⣆⠀⠙⠶⣤⣤⣤⣤⡤⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠳⠦⢤⣤⠤⠶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⢤⣤⡤⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⢈⣵⣶⣶⡶⢒⡶⠦⣴⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⣞⡻⠯⢩⠩⠍⠭⠉⠛⠛⠒⠶⠤⣄⡀⠀⠀⠀⢻⣶⣬⡿⢼⣷⣇⣼⡇⢺⠉⣽⣶⣌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⣿⡿⠿⠛⠛⠋⠉⠉⠉⠉⠉⠉⢉⠋⠓⠒⠦⢄⣀⠉⠛⠢⣄⣀⠿⡛⢦⡀⡇⠈⠻⣿⣿⣾⣡⡿⢹⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢪⣿⡷⠛⠁⠀⠀⠀⢀⡀⠐⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠉⠒⢄⡈⠙⢆⢨⣾⣷⣿⣾⣿⢋⣛⣻⣟⡃⣀⡀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠫⢱⡿⠋⠀⠀⠀⠀⠳⣦⢄⠙⢶⣄⡀⠀⠀⠈⠁⠀⠀⠀⠄⠴⠖⠀⠀⠀⠙⢆⢸⣿⣿⣿⠛⠻⠿⠿⠯⣿⣿⣯⠥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⢇⣿⠁⠀⠀⠦⢤⠤⠤⠀⠀⠀⠀⠹⢯⣿⣄⣀⡀⣀⣴⢶⡾⠋⠀⠀⣀⡀⠀⠀⢹⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢸⡀⣇⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠨⣿⣿⣿⡿⣯⠁⠈⠁⠀⠉⠁⠀⠀⠀⠀⠈⣿⡇⣿⣏⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠘⣇⠹⡄⠀⢀⡖⠒⠀⠀⠐⠂⠀⡲⠌⠁⠐⠶⠖⠛⠁⠀⠀⠀⠀⠂⠐⠲⡆⠀⠀⢀⡿⢰⣿⠹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢤⢱⠀⢀⠙⣦⡀⠀⠄⣀⠀⠀⠀⢠⡤⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡐⢦⣀⠀⠀⠀⠀⣠⠞⣱⡿⢧⠘⡌⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡆⢣⡘⢀⢈⠛⢶⣤⣀⠑⠢⢀⣰⠀⠀⠀⢀⡜⠀⠀⠐⣦⠒⡆⠀⠀⠐⢋⡁⠤⠒⣋⣥⡾⠋⡅⢈⡜⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡙⢆⠱⣄⠈⠀⠀⠈⠛⠿⣶⣤⣈⡁⠒⠒⠠⠧⠤⠤⠤⠮⠦⠛⠒⠂⣉⣡⣤⡶⠟⠋⠀⠠⠞⣠⠎⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⠳⢬⡓⢤⣀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠶⠶⢶⣶⣶⣶⠶⠾⠿⠛⠋⠉⠰⢀⡴⠓⢈⡴⢚⣡⠞⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣄⡙⠲⢬⣕⡲⠤⣀⡀⠀⠀⠀⠀⠄⠰⢠⣖⠀⣤⡔⡀⠺⠢⠘⠶⢃⣠⣴⣚⣽⡾⠋⣡⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣀⠀⠉⠙⠷⢿⣭⣍⣙⡓⠒⢶⡶⠶⢶⡶⠶⢶⡶⣶⣶⣾⣿⣿⠿⠟⣋⠁⣠⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣀⠀⠀⠈⠉⠙⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⢋⣉⣡⡴⢖⣫⣵⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠶⣶⣭⣭⣭⣄⣀⣀⣒⣒⣂⣛⣹⣭⣭⣶⣾⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉⠈⠁





⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⡟⣉⣿⠿⠿⠿⠿⠻⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⡤⣤⣤⠄⢠⢤⣤⠀⣤⡤⣤⡤⣤⠀⢤⡤⢠⣤⡤⠤⢤⠠⣤⠠⣤⠄⣤
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣌⡂⢠⡄
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⣉⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡠⠀⠒⠐⠐⠂⢠⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠂⠐⠀⠀⠀⠀⠐⠺⣀⣀⡀⡀⢀⣠⠔⠃⠀⠀⠀⠀⢣⠠⠐⡆⠐⠒⠒⢂⢀⠐⡒⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠄⠀⣀⡄⠀⠀⠀⠀⠀⠀⠆⢠⣼⣧⡼⠇⠉⠁⠀⠀⠀⠆⠀⠀⢋⠦⠀⠇⠀⠀⠀⠀⠀⠠⡄⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠡⠄⠎⢙⣉⡀⠀⠀⠁⠀⣄⣠⠈⠀⠀⠀⠀⠀⠃⠀⠀⠴⡄⠀⢤⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⡙⠉⠁⠀⠀⠀⠀⠠⠞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⡀⠀⠁⠀⠦⢀⠀⠀⠀⠔⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠄⠀⢠⠀⣂⡀⠦⠀⢀⠠⠤⡄⠀⠀⠀⠀⠀⡀⠦⠘⠛⠂⠁⣀⠀⠀⡀⢇⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⡀
⠀⠀⠀⠈⡀⠀⠂⠉⠀⠉⠀⠀⠀⠑⡀⠀⠐⠀⢠⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠰⠀⡃⠸⠀⠀⠀⠀⠂⡬⢁⡤⠂⠘⠓
⠀⠀⠀⠀⠀⠀⡄⠀⠀⢱⡾⠃⠀⠀⡇⠀⠉⠀⠀⠰⠀⠀⠀⠀⠊⠀⠀⠀⡀⠌⢀⣉⡤⠂⠀⠀⠈⢚⠁⢸⡄⣀⠀⠀⠀⡤⢠
⠀⠀⠀⠀⠲⠄⠈⠀⠀⠀⠀⠀⠀⢀⠀⢠⠜⠂⠀⠀⠀⠀⠐⠀⠐⠂⠠⠊⠁⠀⠊⠁⠀⠀⠀⠀⠀⠛⠀⠈⠀⠀⢀⢀⡴⠇⠺⠢
⠀⠀⠀⠀⠀⠤⣄⠀⠁⠒⠐⠒⠀⠀⡠⠐⠠⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡁⠈⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠘⠒⠈⠁⠂⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠰⠶⠲⠆⠐⠗⠶⠂⠲⠖⠲⠖⠶⠀⠶⠶⠰⠶⠲⠚⠲⠀⠖⠒⠆⠳⠆
⠀⠀⠀⢀⠔⠛⠃⠀⠧⠔⢠⠁⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⢢⠀⣴⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡁⠀⠀⠀⠀⠀⠀⠈⠤⢤⡄⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠉⣥⠀⠀⡄⠄
⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠱⣀⡀⠀⠀⢠⠄⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡞
⠀⠀⠀⠙⡇⠲⢲⠒⠆⠀⠒⣠⠛⠄⠀⠀⠀⠀⠀⠀⠉⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠁
⠀⠀⠀⠀⠀⠈⠋⠠⠛⠃⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠆⢆⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢀⠀⡀⠀⣀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣈⠀⠀⢀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠂⢈⠉⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠘⠁⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃⠐⢢⠘⠓
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠀⠈⣈⠀⡜⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠒⠈⠘⠁
⠀⠀⠀⠀⠀⠠⠄⠀⢀⠂⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠂⠀⠉⢈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠑⠀⠉
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣌⠁⠀⠀⠀⢀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠂⠀⠀⠀⠀⠀⠀⡀⠚⠢⡄⠀⠀⠴⠜⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠞⠛⠏⠉⠍⠉⣉⠋⠉⠫⠽⡶⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⠏⠀⣁⣠⠀⠀⠘⠳⢤⠄⠀⠀⠈⢈⡒⠟⣻⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⡟⣿⣿⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠹⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⢏⣿⣏⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢠⣦⣷⣹⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠇⣼⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣿⣸⣿⣾⣿⣿⣧⢿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⠁⠀⠐⠚⢀⠀⠀⠀⠀⠀⠼⠀⠀⢰⡆⣸⣿⣿⣿⣿⣿⣿⣿⣿⡎⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⣱⠀⢰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⣴⣿⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡏⢱⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡞⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡿⢀⣿⣿⣿⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡷⣾⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⠿⠛⠛⣹⣛⣉⣴⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣯⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣰⢏⢴⡿⢿⠉⠀⢠⡆⢺⣿⣾⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣦⡀⠀⣾⣧⣤⣾⠃⠏⠀⠀⠸⠑⣿⣿⢇⣿⣿⣿⣿⣿⣿⡧⣼⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣠⡟⢻⣿⡟⠈⠀⠀⠀⠀⢰⣿⣿⣻⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠟⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣾⣿⣟⢹⢿⣿⡧⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⠋⠀⠀⠀⠀⠀⠀⢰⢿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣼⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣮⡻⣿⣿⣿⣿⡿⠿⣿⣛⡻⣿⣿⣤⡤⡄⠀⠀⠀⠀⠠⣿⣟⣫⠽⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣶⡟⣿⣿⣿⣿⣶⣾⣿⡿⠶⠿⠉⡁⣀⣀⠀⠀⠀⣴⣦⣽⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠻⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠟⠰⢶⠆⢁⣛⠁⢟⠁⢴⣾⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠈⢿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣶⠦⠀⠤⠥⠄⠸⢶⣀⡀⢘⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣦⣶⣶⣶⣦⣀⣉⠁⢠⢩⡽⣿⣿⣿⣿⣿⢻⣿⣷⣿⢹⣿⣿⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢻⠻⠿⠿⠿⠻⠇⠤⠄⠚⢳⢿⣿⣿⣿⡭⢿⡿⣟⣶⣿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠀⠈⠉⠙⠛⠛⠛⠛⠉⠀⢠⣤⣤⣴⣶⣶⣶⣮⣿⣻⣿⠛⠿⠻⠗⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⣿⠿⢿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣷⡼⠂
⠀⠀⠀⠀⠀⠀⠀⣤⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣶⣒⣛⠷⢄⡀
⠀⠀⠀⠀⠀⠀⠸⣿⠿⣦⣄⠂⠄⠴⡤⢄⠀⠀⠀⠀⠀⢀⣾⠋⡀⠠⠀⠄⠉⠑⡍
⠀⠀⠀⠀⠀⠀⢰⡇⣰⣯⢁⢛⣤⣤⣌⢘⣷⠀⠀⠀⠀⣾⡇⢀⠉⢲⡄⠀⣀⡤⠈
⠀⠀⠀⠀⠀⠀⣸⡃⣧⣽⡀⠨⠤⢽⠈⢧⠄⢌⡀⠀⠀⣿⡇⠰⠀⠀⠛⠉⠁⠐⠀⠄
⠀⠀⠀⠀⠀⠀⢹⡇⢻⣧⣀⡠⠀⡠⠤⠞⡀⣼⡇⠀⠀⠘⣷⣄⠘⠠⠄⠠⠈⠀⠈⠀⠀⠀⠀⣀⢀⠀⠀⠀⣼⣙⠢⡀
⠀⠀⠀⠀⠀⢠⡞⠀⠀⠻⣿⣿⣾⡥⣬⣤⣾⡷⠃⠀⠀⠀⠈⠙⠿⢶⣶⣶⠶⠏⠀⠀⣠⠖⠉⢡⠀⡤⠁⠀⣿⠿⣶⡃
⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠈⠙⠛⠿⠿⠟⠋⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢣⣌⠃⠀⡄⠀⠘⠃⢉⢧⠙⣧
⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠖⠛⠋⠉⠉⠙⠛⣦⠀⠀⠀⢀⣄⠀⠀⣯⢈⡁⠀⠀⠷⠄⣀⣈⣈⢸⠀⢸⡆
⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⢀⣠⠴⠋⠉⠀⠀⠀⠀⠀⢀⣀⣤⠟⠀⠀⠀⠸⣿⣿⣄⣿⣎⢡⡀⠀⠀⠀⢠⣄⡹⡾⢀⡞⠁⢀⡴⢦
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⢉⣰⠖⠛⠛⠛⠛⠛⠓⠛⠉⠀⠀⠀⠀⠀⠀⠀⠘⢌⠻⣯⣻⣶⣄⣀⣈⣛⣀⣼⠟⠁⢼⣁⣀⡜⠁⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⢐⡋⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢽⣻⣿⣍⠉⠛⠉⠉⠀⠀⠀⠀⠉⠉⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⠃⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠶⠚⠒⠲⠦⠤⠦⠤⠤⠴⠶⠖⠖⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⣀⣠⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⣀⣼⣍⢉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠉⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣝⣫⣍⣉⣏⣹⣟⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⢿⣿⣿⠿⢿⣿⠟⣣⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⣵⣶⣶⣶⠞⣢⣴⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣡⣾⣿⠿⣋⣵⣾⣿⠟⣋⡛⠿⠿⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣋⣼⡿⢋⣵⣾⣿⠟⣋⣴⣿⣿⣿⣿⣿⣷⡌⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⢛⣥⣾⣿⣿⣯⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⢟⣡⣾⣿⣿⣿⠻⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣱⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢟⣩⣶⣿⡿⢛⣥⣿⣿⠃⣠⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢠⣾⠟⣫⣴⣿⡿⢟⣡⣶⣿⣿⣿⣿⣄⠙⠛⢉⣉⠙⢿⣿⣿⣿⣿⣿⠿⣋⣴⣿⡿⠟⠉
⠀⠀⠀⢀⣴⡿⢱⣿⣿⣿⣿⡐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢨⣿⣿⠿⣋⣵⣾⣿⠟⠉
⠀⠀⠀⣾⣿⣿⡘⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣉⡀⢺⣿⣿⣾⣿⠟⠋
⠀⠀⠀⠻⣿⣿⣿⣌⢿⣿⣧⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣾⣿⠿⠋
⠀⠀⠀⠀⠈⠙⢿⣿⣦⡹⣿⣷⡜⢿⣿⣿⣿⣿⣿⡿⢟⣩⣶⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡜⢿⣿⣮⣥⣤⣭⡛⣩⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣎⠻⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠿⠿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣤⣶⠞⠂⠲⣄⡠⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣠⣾⡁⠀⣹⠏⡰⢻⣿⠃⠀⢀⡴⠋⢀⣀⣀⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡷⣄⠙⣶⠁⣸⠁⡃⣿⡼⢀⣿⣶⣿⣿⡿⣿⣿⣶⣤⣽⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⠿⠛⠂⠀⢨⣧⣿⣆⣿⣿⡇⣸⣿⣿⣏⣀⣠⠤⠾⠿⡿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣁⣀⠀⠀⠀⣈⣿⣿⢿⣿⣿⣿⡿⠿⠛⠛⢿⣟⠒⠐⠀⠈⠺⣿⣯⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠋⢈⣩⣿⣿⣿⣿⠋⠀⠀⠈⠻⠟⠀⠀⠀⠀⠈⠛⢿⣗⡂⠀⠀⠙⢿⣇⣈⣢
⠀⠀⠀⠀⠀⠀⢠⠞⢁⣤⠖⠋⣁⣠⣾⣿⣿⣿⠇⠀⠀⠀⢠⡄⢀⡀⠀⠀⠀⠀⢻⡙⢿⣿⡟⠛⠛⠢⡁⠀⢣
⠀⠀⠀⠀⠀⢠⠃⡤⠋⠀⢠⣾⣿⣿⣷⣿⠟⠉⣀⡀⠀⠀⠀⠙⡟⠀⠀⣀⣠⣄⡀⣧⠘⣿⠙⠷⣦⣄⠈⠢⡈⡃
⠀⠀⠀⠀⠀⡌⠀⠀⠈⣉⣵⣾⣿⢃⣼⢻⠑⢿⣿⠿⣞⡧⠀⠀⠀⢀⠐⣿⣿⠟⠁⠹⣆⣿⠀⠀⠀⠙⠦⡀⠈⣇
⠀⠀⠀⠀⠀⠁⠀⠔⠀⠀⣩⣿⣿⣿⡏⠀⠀⠀⠀⣦⢹⡷⠂⠀⠀⠊⠻⣯⣷⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⢉⠆⢼⡆
⠀⠀⠀⠀⠀⢰⠁⠀⢠⡞⠉⠁⢻⣿⣿⡄⠀⠀⠀⢫⣟⣵⠃⠀⠀⠀⢲⡌⢿⠂⠀⣰⢻⣿⣆⠀⠀⠀⠀⠈⢂⢸⣷
⠀⠀⠀⠀⠀⣿⠊⣰⣾⠁⠀⠆⠘⡟⣧⣑⠀⠀⡶⠀⠻⣿⡟⢛⠛⣹⣿⠗⠈⢱⡄⠁⢀⠟⠙⢆⠀⠀⢦⡀⠈⢧⣻⡄
⠀⠀⠀⠀⠠⣿⡸⢱⡇⠀⠘⠀⠀⢃⠘⢿⣶⣾⣇⡀⣠⡬⠿⢿⡟⢡⣤⣤⣤⡬⣷⣾⡏⢳⡀⠀⠣⠀⠈⢷⡀⠈⢿⠛⠦⠄
⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⣸⡤⠖⣿⣿⡥⠖⠚⠫⠀⣸⣧⣸⣿⣿⣚⣳⣾⡿⠤⢌⣿⠀⠐⠂⠀⢸⣧⠀⢸⠂
⠀⠀⠀⠀⠀⠹⠀⠘⣆⢀⡀⠐⢿⠁⣀⠼⠟⣹⣦⢂⣡⠴⢛⣿⣿⡿⠳⢦⣭⣿⣿⡧⠀⠀⢸⡆⠀⢸⠀⠀⣿⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⠀⠰⡼⡏⠀⣠⣿⣿⣿⡏⡏⢀⡈⠛⠈⠀⠀⣄⢨⡏⠁⠻⣦⡀⠘⠇⢀⣞⠀⠀⣿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢩⣿⠞⢽⣿⡿⠈⡇⣇⡎⠻⣄⠀⢀⡜⢹⠏⠁⠀⠀⠈⢷⣼⣷⡿⠃⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⡾⣿⡇⠀⠀⢸⣄⠀⠈⠀⢀⠀⠀⡆⢸⣆⠀⠀⢸⣿⡿⠁⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠃⢙⠁⠀⣦⡟⠹⡆⠀⠀⡿⡇⠀⠀⠸⠻⣇⠀⢠⣿⠇⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢀⣿⡇⠀⠹⡄⢠⠇⢿⠀⠀⠇⠀⢸⣄⡾⣿⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣴⡀⢸⠀⠃⠀⠀⠹⣾⠀⢸⠀⡜⠀⠀⢸⡟⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣸⡀⠀⠀⠀⠀⠙⡀⢸⡟⠀⠀⠀⠎⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠀⠀⠀⠀⠀⠁⠘

⣿⣏⢹⣟⠻⣿⢛⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣿⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⡠⠊⢸
⠀⠀⠀⠀⠀⠀⠀⣎⡁⣅⣀⡸⠛⠢⡤⠻⡇⠀⢢⠀⠠⠀⠀⠀⠀⠀⢀⠄⢘⠤⠠⢁⠀⠦⣮⠆⠑⢼⡄
⠀⠀⠀⠀⠀⠀⡆⠀⡌⠀⠀⢱⠀⢀⡿⠀⠁⢤⠐⢢⠀⠁⠀⠀⢀⠎⢘⠄⣎⠀⠀⡎⠀⠀⢘⡆⠀⢲⠋⠀⠀⢆
⠀⠀⠀⠀⠀⢄⢸⠁⠀⡤⠤⣞⠀⠈⠻⠄⣠⠞⠀⡜⢢⠀⠀⠀⠀⠁⠺⡀⢸⠁⠀⠈⢠⠤⠎⣇⠀⢀⡇⢀⢀⠎⠀⢱
⠀⠀⠀⠀⠀⡆⢠⠃⠺⣀⠀⣸⠆⠐⢄⣠⠿⣄⠠⣇⠰⠀⠀⠀⠀⠀⠀⠀⢈⠄⡀⠠⡆⠀⠀⢹⠁⢹⡄⢨⡇⠀⣱⠁
⠀⠀⠀⠀⠘⡀⡨⠒⠞⠁⠙⣧⡀⢀⣸⣁⣀⢼⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢣⢰⡁⠀⠑⡄⠄⠃⢠⠈⠀⠣⢱⠐⠁
⠀⠀⠀⠀⠀⠀⡱⢤⠊⠀⣾⠁⠈⡟⠁⠈⡧⠰⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡐⠁⠀⢢⠠⠎⣀⠠⠋⠁
⠀⠀⠀⠀⠀⠀⢇⣀⡦⢴⠁⠀⣏⡩⠄⡈⢹⠖⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠒⢧⠀⠀⠜⠀⠀⡸⠘
⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⡓⠂⠑⣄⢀⡏⢹⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⣰⠋⠈⢹⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⠱⠄⠰⠁⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠈⡢⠀⢻⠄⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡝⠂⠺⣀⢠⡏⠀⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠐⡁⠀⡸⢀⠜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⣸⠉⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡸⠀⢤⠎⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢀⡟⠉⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠓⣦⠀⠀⢀⡤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⡏⢀⠖⠉⢰⠁⠀⢀⣠⠖⠛⠉⢙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢄⡀⠀⠀⠀⠀⠀⡇⠀⢀⣷⠋⠀⠀⢸⢀⡴⠋⠀⠀⠀⣠⠏
⠀⠀⢀⡶⠤⠤⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢳⡀⠀⠀⠀⢰⠃⠀⢸⠃⠀⠀⠀⠈⠉⠀⠀⠀⠀⡰⠃
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠉⠓⠦⣄⠀⠀⠀⡜⠀⠀⠀⢧⠀⠀⠀⣼⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣁⡠⠤⢄⣀⡀
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⢀⠇⠀⠀⠀⢸⠀⢀⡞⡟⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⢸⣤⠋⠀⡇⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠉⠀⠉⠉⠓⠦⣼⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⡤⠤⠤⠤⢄⣀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡁⠀⠀⠀⠀⠉⠓⢴⠋⠀⠀⠀⠀⠀⠀⠈⠑⢦⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠙⣦⡄⠀⢀⣀⣀⡀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⢳⣄⣤⠤⢤⣄⣀⠀⠀⠀⢳⡄
⡴⠖⠒⠚⠛⠋⠉⣡⠖⠋⠉⠉⠉⠉⠙⡇⠀⠀⣠⡤⠴⠶⠦⢤⣀⠀⠀⠀⢸⣆⣀⣀⣀⠀⢠⠖⠋⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⣷
⠘⢦⣀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⢧⡴⠋⠁⠀⠀⠀⠀⠀⠈⠙⢦⡔⠋⠉⠉⠉⠉⠉⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⣿
⠀⠀⠀⠉⠑⡞⠀⠀⠀⠀⠀⠀⢀⣀⣀⣾⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⢷⠃
⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⠀⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⣸⠓⠦⡀
⠀⠀⣠⠞⠉⢧⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠈⢦
⠀⡼⠁⠀⠀⠈⢦⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⣠⠎
⢰⠃⠀⠀⠀⠀⠀⠙⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠤⠤⢤⡤⠖⠋⠁
⢸⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠧⠒⠊⠁⠀⠑⠦⣄⣀⣀⣀⣀⠴⠚⠁⠀⠀⠀⠈⡇
⠀⢧⠀⠀⠀⠀⠀⠀⢀⡼⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡅⠀⠀⠀⠀⠀⠀⠀⠀⣺⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠈⠣⡀⠀⠀⠀⢀⡞⠀⠀⠙⠲⠤⣤⣀⣤⠤⠖⠋⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢀⡷⢤⣀⣀⣀⣀⣠⠴⠋
⠀⠀⠀⠈⠓⠢⠤⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠁⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⠀⠀⠀⠀⢀⣀⠴⠋⠲⠤⣄⣀⣀⣠⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠈⠉⢙⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⣀⠜⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⣴⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠢⠤⣤⠒⠊⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⣄⣀⣀⣀⡠⠴⠊⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⡀⠀⠀⠀⣀⡤⠷⣄⡀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⠉⠁⠀⠀⠀⠉⠓⠒⠒⠒⠚⠉⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠈⠑⠦⣄⣀⣀⣀⣀⣤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⠤⠤⠖⠊⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢄⣀⣀⣀⠤⠖⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⢀⣀⠀⠀⠀⠀⡴⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠺⢍⠀⠀⡄⠀⣀⡈⠃⠀⢀⢿⣦⡆⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢇⣀⠙⠓⠀⠉⠀⠛⠀⣤⡀⡩⡏⠻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⣾⠃⠼⠟⠀⠀⠀⣼⠀⠀⠀⠉⠁⢑⢽⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⢄⠐⣿⡰⣶⠂⠀⠀⠀⠟⠦⣀⡠⠈⠉⢹⢸⡇⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⢿⣧⡱⣿⣧⠁⢔⠀⠀⠀⠀⠀⠈⠉⠺⠗⡌⣸⠀⠀⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⢿⣷⣜⢿⣷⣍⠀⢴⠀⣤⣤⠐⠿⣢⠞⣹⠃⠀⡶⠋⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠻⣿⣦⣻⠷⣵⣢⡤⣭⣍⣐⣬⡴⠛⠁⢀⡿⠁⠀⠀⣴⠏⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⡿⣿⣦⠀⠉⠙⠛⠋⠉⠁⠀⠀⠀⠈⣷⣦⣤⡾⠁⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠻⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⢠⣤⠿⠝⠛⠛⠺⢶⣦⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋
⠀⠀⠀⢠⣿⠁⠀⠀⣖⠛⠓⢦⡀⠉⠉⠙⠓⠿⠿⠷⠧⠶⠶⠶⢶⡶⣶⠶⠽⠟⠛⠋⠁
⠀⠀⠀⠀⠻⠷⣾⣿⠿⣷⣀⣼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⡘⡀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⣤⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢠⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢄⡘⣠⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠠⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢁⡆⠈⠀⠀⣿⡇⠀⠀⢠⣆⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣶⣴⣤⣦⣶⣤⣴⣤⣦⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢤⠈⣡⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⢻⠙⣿⡟⠽⡽⡍⢽⢝⡽⢫⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⣰⡀⠁⠀⠀⠀⠀⠀⣳⠁⠀⠀⠀⠀⠀⠁⣲⡈⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠉⠐⠑⠐⠓⠙⡈⠐⠃⠀⠉⠁
⠀⠀⠀⠀⠀⠀⣒⡈⠀⠀⠀⠐⡡⠀⠀⠢⠤⣀⣀⡔⠋⠱⣄⣀⡠⠄⠂⠈⠈⠑⠀⠀⠀⠀⣐⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠯⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡊⠀⠨⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣯⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣯⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠧⠘⢰⡷⡄⠂⠧⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⡀⠀⠀⢠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡁⠀⠀⠄⢱⡍⠀⠀⠀⢙⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠒⠒⣧⠒⠚⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠑⠂⠉⢻⠛⠉⠐⠋
⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠁⢀⠀⠒⣿⠲⠀⡀⠈⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣠⣤⣤⠀⣡⡇⢀⣤⣤⡄⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠈⠙⣯⢤⠤⣴⣶⠄⢙⣿⡃⢠⣶⣦⢠⡤⣝⠫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⡟⠁⠀⠈⡻⣍⣉⣏⣿⣉⡹⢏⠁⠀⠈⢻⢿⠃
⠀⠀⠀⠀⠀⠀⠀⡄⠐⠀⠈⠁⣤⣿⣻⣷⡴⠸⠰⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠙⠋⠀⠀⠄⠁⠈⠁
⠀⠀⠀⢀⠐⣀⣠⠄⢠⡄⢸⠉⠀⠊⣭⠑⠄⢩⠒⢠⠀⠰⣄⣀⠂⡀⠀⠀⠀⠿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠳
⠀⠀⠀⠀⠨⢿⡧⣀⣼⠀⠘⠃⣼⠀⠾⠀⡆⠘⠃⠈⣧⣀⣼⠿⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠻⡿⠃⠀⣀⠀⠙⠉⠀⣀⠀⠘⢿⠟⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⢀⠠⠱⣾⡿⠶⠃⣰⡀⠀⡂⠀⣰⣆⠸⡶⣿⡷⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡈⢉⠠⢤⠁⠂⢃⡀⠂⢀⡤⠄⡉⡀⠤⢤⡀
⠀⠀⠀⠀⠀⠈⠀⠀⡀⠊⠈⠛⠋⠀⢿⠀⠛⢛⠁⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠈⠀⠐⠛⠛⠁⠀⠁⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠈⠉⠓⠢⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣊⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠚⠃⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠘⠤⠤⠤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠤⠤⠤⠚⡆
⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⣀⣀⣀⣀⠤⠤⠤⠖⠒⠒⠒⠉⠉⡩⠓⠦⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢰⠚⠒⠒⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⡴⠊⠲⡄⠀⠀⠸⡄⠀⢀⡗⠀⠀⠀⠀⢠⠊⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠳⢄⡤⠃⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠈⠉⠢⡀⢰⠁⠀⠀⢳⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⡇⠈⠢⠤⠤⠃⠀⠀⡔⠉⠀⠈⠱⡀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠤⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠶⣏⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⡹⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⡴⠒⠀⠒⢦⡀⠀⣉⣒⣋⣀⠤⠤⠤⠃
⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⢀⣀⣀⣀⠠⠤⠤⠤⠒⠊⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠘⠒⠒⠂⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠛⠳⢤⣀
⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀
⠀⠀⣠⣤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⢤⣄⣀⠀⠉⠛⠶⢤⣀
⠀⠀⡇⠈⠉⠓⠲⢤⣄⣀⡀⠀⠀⠀⠀⠀⠙⠲⠦⢤⣽⠗⠀⠀⠀⠀⠉⠛⠶⣤⣀
⠀⠀⣇⠀⠀⠀⠀⠀⠀⠈⠉⠙⣳⠒⠒⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠉⠛⠶⣄⡀
⠀⠀⡿⢤⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠤⠴⠛⠒⠶⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣄⡀
⠀⠀⣇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⠖⠒⠛⠛⠒⠦⡀⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣀⣀⣀⣽⠒⠒⠒⠲⠶⠦⠤⠤⢽⡇
⠀⠀⡇⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠋⠛⠓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⣸⠁
⠀⠀⡇⠀⠀⠀⢀⡴⠛⠉⠳⡄⠀⠀⠀⠀⠸⣄⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡏
⠀⠀⡇⠀⠀⢰⠋⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⣿⠀⠀⠈⠳⢦⣤⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⡏
⠀⠀⡇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠓⠶⠤⣤⣄⣀⡀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠐⠃⠀⠀⠀⠀⣰⠋⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠿⣅⣀⡀⠈⠉⠳⡄⠀⠀⠀⡀⠀⠀⠙⢦⣄⣀⣠⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠦⣿⣀⡀⠈⠛⠂⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⣰⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤⡀⠀⠀⠀⠉⠙⠛⠲⠦⣤⣄⣀⢸⠀⠀⠀⠯⠭⠽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣠
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠐⠈⠉⠀⠰⠀⢀⠂⠀⢶⡔⠡⠐⡄⠀⠀⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠁⣀⠐⣄⠈⠲⣦⠀⠀⠆⠀⠀⠀⢀⠈⠀⡐⠀⠈⠂⠀⠀⠄⢮⣱
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠊⡀⠀⠀⠀⠘⠂⠀⠈⠁⢀⣀⠀⠀⠄⠡⠀⠀⠀⠁⠀⠐⠀⠀⢀⠈⠱⡃
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⣀⠑⢐⣀⣀⠠⠀⢠⠁⠁⠀⡀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠱⡄⠀⠂⠀⠀⠀⠀⠀⠈⠁⠀⣀⣁⣤⣼⡶⠂⠀⠀⠀⠈⢐⡄⠀⠀⠐⠀⢀⠄
⠀⠀⠀⠀⠀⡀⡀⠀⠈⠀⠀⠀⠀⠄⠀⠈⠉⠘⠿⠋⠊⣀⠀⠘⠀⣄⠀⠁⠀⠀⡀⠀⠀⠀⢠⠐⠀⠤⠀⠀⠀⠀⢐⠀⠀⢀
⠀⠀⠀⠀⢐⠀⠀⠀⡄⠀⠀⠀⠀⠒⠃⠉⠀⠀⠀⠁⠐⠡⠀⠀⠀⠀⠂⠀⠈⠘⡤⢔⡊⠁⠘⠠⠀⠀⠀⠂⠀⠀⠈⠠⡀⠘
⠠⠀⠌⠴⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠛⠆⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠁⢈⣦⠞⠡⠀⠉⢦⡀⠀⠈⠁⠀⠀⠠⠑⠂
⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⢈⣠⠄⠀⠀⠀⢠⠈⣂⠀⠀⠀⠀⠀⠀⠀⣴⡟⠣⠀⠀⠀⠀⢀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠒⠂⣤⣀⣀⣀⠀⠀⠀⠠⠀⠀⠐⠂⠀⠂⠀⠀⠁⠀⢀⣴⣾⡏⠁⠀⠀⠀⠀⠀⠉⠉⠙⢦⠀⠀⢀⠀⡄⠀⠤⢀⠠
⠀⠀⠀⠀⠈⠙⠛⣿⣿⡆⠀⠀⠀⠀⠐⠠⠄⠂⠀⠐⠒⠠⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣰⣶⠶⣶⣶⣦⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠉⠁⠉⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠐⠒⠈⠉⠉⠁⠠⠤⠤⠤⣉⣉⡁⠒⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢦⡈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢠⣤⣤⣤⠤⠶⠋⠁⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⢻⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠙⠻⢿⣿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⠴⠖⠛⠉⢀⡐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⢠⡿⠿⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠶⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⠤⠤⠤⠤⠤⠤⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣠⣴⠶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣤⡀⠀⢰⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠈⠻⣦⡟⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡁⠀⠀⠀⠀⠀⣠⠔⠊⠉⠁⠀⠀⠀⠀⠈⠉⠻⣦⡀⠀⠀⠀⠀⡴⢻⣷⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠙⢷⣄⡀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢸⠛⠂⠀⠀⣿⠇⠀⣀⡴⠋⠀⢸⣿⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠉⠛⠷⢶⣥⡀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠒⢛⣁⠴⠞⠉⠀⠀⢀⣾⢋⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢣⡈⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣠⡴⠟⣱⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠉⠓⠦⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⠴⠖⠋⢉⣤⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⢀⣀⣤⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠿⠻⢯⡉⠉⢉⡽⠿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⡤⢠⡄⠀⠁⢀⡀⢀⢄⠀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢸⣄⠈⢀⣰⠏⠀⠳⢦⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠖⠲⠖⣶⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣠⠴⠦⣄⠘⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⣤⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠀⢀⡠⠞⠙⠶⣤⣤⣤⣤⣀⣤⣤⣤⣤⣶⣶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡆⠀⠀⠀⢰⣀⡀⠀⠀⠀⢰⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⡴⠀⠀⠀⣿⣿⣿⣦⣄⣤⢾⠄⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣀⣠⠔⠁⠀⠀⠀⠈⠪⣟⣻⣛⡯⠋⠀⠀⠀⠀⠙⢿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣠⣧⣀⡀⠀⠀⠀⠀⠀⢀⣀⣼⣄⣀⠀⠀⠀⠀⠀⠀⣀⣠⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣸⣿⣷⣤⣤⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⣠⠎⠀⠀⠀⠀⠀⠻⣻⠿⢿⡷⠁⠀⠀⠀⠀⠘⢿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⡀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇
⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢩⣭⣭⣭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⠂⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⡜⠀⠀⠀⠀⠀⠀⠀⢪⣛⣫⠃⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠒⠂⠀⠐⠒⠀⠀⠀⠀⠀⠀⠒⠚⠛⠒⠂⠀⠀⠀⠀⠀⠐⠒⠛⠓⠒⠀⠀⠀⠀⠀⠀⠒⠚⠓⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⣠⣴⣶⣶⣶⢶⣶⣶⣶⣶⣶⠶⢶⣶⣶⣶⣶⣶⣤⡀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⣼⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠿⠿⣿⣿⠁⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⠀⠀⠀⠸⣿⡿⠿⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⣿⠇⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀⢻⣷⣤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿⣿⣾⠟⠛⣩⣭⣭⣭⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣭⣭⣭⣉⠛⠿⣿⣿⣏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏⠀⠀⣾⡟⠉⠉⠙⣿⣆⠀⠀⢴⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⣼⡿⠉⠉⠙⢿⣧⠀⠈⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⣿⣇⠀⠀⢀⣾⡟⠀⠀⣴⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⢿⣇⠀⠀⠀⣸⡿⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⠻⢿⣶⠿⠛⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠈⠀⠀⠀⠈⠻⠿⣶⠿⠟⠁⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠿⠿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠿⠿⠿⢿⣿⡿⠿⠿⠿⠿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣶⣶⣶⣶⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣶⣶⣶⣶⡿⠏






⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⠿⠻⡿⠛⠿⢿⠿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣟⠁⠲⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⠁⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣆⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣷⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠿⡿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠿⠿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠉⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣶⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⢿⣿⣿⣿⣟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣮⣿⣿⣿⣿⣿⣿⡿⣿⣿⣷⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣋⣱⣾⣿⡿⠏⠙⠛⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣠⣾⡿⢻⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢀⣾⣟⣿⡟⠙⣻⣿⣏⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣭⢠⣿⣿⣿⣟⡏
⠀⠀⠀⠀⢠⣾⣿⣾⣿⢱⣿⣿⣿⣿⣿⣿⠟⠉⣁⣿⣿⣷⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⠏⣿⠉⠉
⠀⠀⠀⠀⠈⠻⣿⣿⣿⡜⣿⣿⣿⣿⣿⣴⡶⣾⡿⣡⣿⣿⡿⠿⢿⡿⠸⣿⣹⣿⢍⣀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⢸⠿⣿⢷⣼⣿⡟⠻⢷⣿⣄⣿⣿⠿⠛⣁⣀⣀⣉⣀⣤⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠻⠄⠀⢠⣿⣿⡿⠶⣾⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡿⠿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣇⠀⠙⢉⣿⠛⢿⠿⢿⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠿⢿⣿⣦⠴⠟⠁⠀⠀⠀⠈