Hedgewitch Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⡀⠔⠀⠀⠀⠀⡀⠄⠚⠀⠀⠀⠀⠀⠂⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠠⠊⠀⠀⡂⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀⠁⠀⠀⠈⠀⠀⠀⠀⢰⠋⠘⡎⡀⠀⠀⠀⠀⠀⠀⡀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡃⠀⠀⢀⣀⣀⡀⠀⠀⠌⢷⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡸⡆⠀⠚⢉⠉⠉⠛⠷⣦⡀⠀⠀⢠⡴⠾⠛⠛⠛⠲⠤⠀⠀⢣⠀⢸⢇
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠇⡇⠀⠐⠦⣄⣀⣀⠀⠈⡇⠀⠀⡟⠀⡀⠀⠀⣀⡠⠀⠈⠑⣼⠄⣷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠰⣆⢣⣾⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⣽⣶⣆⢰⢘⡿⡐⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⠉⠉⢻⣿⣎⣿⣷⠀⠀⠀⣀⣶⣄⡀⠀⠀⣾⣿⢋⣿⡿⠰⠎⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠠⣰⠈⡏⠀⠀⢀⠙⠻⠿⠟⠀⠀⠀⠀⠉⠁⠀⠀⠀⠻⠷⠿⠋⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠔⠀⠀⠀⠘⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣶⡤⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠈⠄⢀⣠
⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⣙⣛⡋⠁⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⣿
⠀⠀⠀⠀⠀⠠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣄⣀⣿⣹⣿⠀⣀⣠⡄⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠻
⠀⠀⠀⠠⡀⣴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⠿⠟⠉⠀⠀⣨⠄⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠿
⠀⠀⠀⠀⠈⢏⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⣧
⠀⡀⠀⠀⢈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠘⠖⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠈⢇
⠄⡃⡀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣷⠁⠀⠀⠀⠀⠀⠀⠀⢙⠆⠀⠀⠈
⠀⣳⡤⢹⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⢄⡀⠀⠀⠀⢀⣿⠄
⠀⢹⡀⠸⢇⠀⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏
⠀⠘⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠁

⠁









⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣼⣿⣶⣶⣶⣤⣶⣦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠋⢹⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⠈⠁⣾⠇⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⠛⠙⢉⠿⠁⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣄⠀⢀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠶⠶⢦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣀⣤⠀⠐⡾⠶⠋⠀⠈⢿⣿⣿⠶⢶⣦⣤⣄⣴⣴⡆⠀⣀⠀⠀⢰⣿⣷⡄⠈⠉⠉⠉⠉⠉⠛⠛
⣷⣤⣤⣤⣴⣤⣄⣴⣤⣾⣷⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⠟⠁⠀⣴⡛⠀⠀⠀⠀⠀⢿⣿⡀⠀⠻⣿⣿⣿⣿⠃⠀⠿⠀⠀⣿⣿⣿⡇⠀⢸⣷⣶⠀⠀⢠⣤
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⠋⠀⠰⣾⠏⠀⠀⠀⠸⠀⠀⠘⣿⣧⣠⠀⠈⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⡃⠀⠈⠉⠉⠀⠀⠘⠛
⡿⠿⠿⠿⠟⠟⠛⠟⠛⠛⠋⠉⠉⠀⠉⠁⠁⢠⡟⠁⠀⠀⠀⡉⠀⠀⠀⠀⣶⠀⠀⠀⠘⣿⠇⠀⠀⠀⠈⠛⣠⣰⣾⣷⠿⠿⢿⣿⠷⠤⢀⣀⣀⠀⠀⡀
⠶⠒⠶⠶⠴⢦⣤⣤⣤⣄⣀⠀⠀⠀⠀⣠⢴⡆⠀⠀⠀⠀⠀⣿⣶⣶⣶⣿⣿⣶⣤⣴⣴⣿⠀⠀⠀⠀⢶⠿⠟⠛⠁⠀⠀⣴⡿⠛⠀⠀⠀⠀⠀⠀⠈⠉⠉
⡦⢤⣤⣄⣀⡀⠀⠉⠉⠉⠛⠻⠿⠿⠟⠀⠀⣿⡄⠀⡀⢶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢰⣆⣀⡤⠲⠞⠋⠁
⠆⠀⠀⠉⠉⠙⠧⡶⣶⠄⠤⣤⡤⠀⢀⣠⣾⣿⣿⡀⢳⣌⡿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⠀⠀⠀⣾⠁⠁
⣇⠀⢀⣀⡄⠒⠀⠀⠉⠠⠀⠉⢀⣐⣛⠋⣿⣷⠻⣿⣄⣿⡇⣀⡀⠈⠉⢻⠛⠛⠋⠁⢸⣿⠃⡀⠁⣼⣿⣆⠀⠰⠦⠤⢤⣤⡤⠀⣀⠀⡀⠀⠀⢀⠀⠐
⠟⠒⠀⠁⠀⣠⣴⣚⠛⠓⠞⠛⠩⣌⣉⢙⣿⣿⡇⠿⣿⣿⠠⣿⣿⠀⠀⠘⠃⠀⠀⠀⠀⣿⣾⢁⣼⣿⣿⣿⣷⣄⠈⢙⣛⣷⣶⣦⣝⣻⣶⠸⣏⣽⣷⣄⣴
⣦⠴⣶⡟⢿⣦⡉⠛⠿⣛⣩⣽⣛⣛⣿⣻⣿⣿⣷⣰⣹⡟⠀⠻⢻⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣾⡟⣹⣯⣷⣶⣿⣿⣶⣧⣤⣭⣙⣋⣩⣶⣿⣿⣿⣿⣟⣿
⣅⣚⣉⣉⠛⠩⣿⣦⣤⣍⣉⣈⣛⣉⣉⢿⣿⠷⢿⣧⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢹⣿⣿⣯⠿⢿⣯⣭⣿⠦⠿⣷⡿⣷⣼⣿⣿⣷⣾⣿⣿
⠟⢩⣭⣽⣿⣷⣦⡝⠋⣉⣉⣉⡉⠁⠱⣾⣿⠶⢶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣿⡏⠉⠛⠿⢷⣼⣯⣙⠿⢿⣳⢿⣿⣿⣿⣿⣿⣿⣿⠿
⣷⠎⠉⠁⢉⠉⠉⣰⡿⠟⠋⠉⠀⢀⣴⣿⣿⣦⣄⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣿⣷⣤⡀⠀⠀⠙⠛⠛⠛⠾⠿⠿⣶⣽⠿⠟⠋⠙⣷⣄
⠷⠶⠿⠿⠟⠛⠋⠁⠀⣀⣤⣶⣶⡿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢿⣯⣻⣿⣶⣴⣦⣦⣀⡀⡀⠀⠀⣀⢀⣤⣴⣾⠿⢿⣿
⣆⣀⡀⢀⣠⣤⣶⡾⢿⣿⣿⣶⣤⣶⠟⢿⣯⣴⢿⣿⣿⣿⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢸⣿⣷⣶⢾⢿⣿⣮⣽⣛⣿⣿⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣾⣿
⣿⣽⣟⣛⣛⡻⠿⠟⢛⣤⣤⣶⡾⢫⡀⢈⣿⣿⣿⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣟⡟⠙⠛⠿⠛⢥⣬⣥⣶⣮⣭⣭⣍⣉⣛⣛⡛⡻⠿
⣛⣭⣽⣿⡿⢫⣴⣿⣿⠯⢿⣿⢑⠛⠀⣌⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⣷⣤⣄⣀⠀⠈⠉⠛⢻⣿⣤⣿⣿⣿⣿⣿⣿⣿
⣋⣡⣴⣦⣶⣤⣌⠙⠟⠛⢿⡇⠊⠈⠂⡀⣿⣿⣮⣿⣿⣧⠠⠄⢀⠀⢀⠀⠀⣀⡀⢀⡀⢤⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿
⣌⡉⠙⠛⠉⠛⠁⠀⣤⣴⣿⣱⡆⢹⣆⣿⣿⣦⣍⣿⣿⣛⣀⡰⠀⠀⠈⢀⣈⠀⠀⠀⢠⣀⣽⣿⣿⣿⣿⣿⡛⢹⠛⣿⣿⡿⣷⣦⣀⡀⠙⠛⠻⠿⠿⢿⣿
⣿⣿⣷⣦⣾⣶⣷⣾⣿⣿⣿⣿⣷⣾⣧⣿⣿⣿⣿⣯⡟⢸⡟⠀⣿⡟⠉⣹⡿⢿⠏⠙⣿⣭⣿⣿⣿⣿⣽⣿⣹⣿⣿⣿⣿⢷⣿⣿⣿⣶⣦⣄⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠁⢠⣿⣇⣀⣿⣇⣾⣄⢀⣿⣿⣿⣿⣿⢻⣿⡿⣿⣿⡿⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⡿⠙⠛⠻⠻⠋⠙⠛⠙⠛⠉⠉⠉⠈⠉⠋⠉⠛⠿⢟⠻⠿⣿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⣉⡙⠛⣛⣭⣴⣾⣿⡿⠿⠿⠿⠛⠛⢿⡿⠿⢿⣿⣿
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠈⠈⠁⠀⠙⠛⠛⡛⠿⠿⠿⠟⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡞⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣿⠁
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢭

⠀⣠⠐⠶⢶⣶⡶⠒⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠴⢶⠴⠶⠢⣀
⢸⡁⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠂⠀⠀⣹
⢸⡿⢷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⢸⡝⣿
⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠄⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢿
⠘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢢⡁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⢑⣰⡀⣐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⣼
⢈⡭⠓⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣲⠲
⠘⠙⣡⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣍⢿
⠀⢷⡟⠁⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢤⡀⢰⡀⠀⠀⠀⠀⠀⠀⢳⣿
⠐⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⢰⣤⣀⣖⠇⠀⠀⠀⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⠀⠀⠈⠹
⠀⣮⣇⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡾⠴⠖⠛⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠒⠠⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣰
⠀⣇⢾⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠊⠀⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡾⣭
⠐⢷⠠⠄⠄⠁⠀⢈⠄⠀⠄⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠐⠐⠁⠀⠀⠀⠀⠀⢀⡀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠄⠀⠀⠈⠱⠤⠔⠠⠄⠁⠀⠁⠀⠀⠀⠀⠀⠒⠘⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠁⠀⠈⠀⠀⠁⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠄⢠⠀⠠⢤⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⣀⠄
⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠁⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡤⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⡀⣀⣀⣀⡀⠀⣐⢄⢰⣀⣀⣀⢀⠀⠀⠀⠀⠀⠤⢤⣤⢤⢴⣟⠁
⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠑⠂⠀⠀⠚⠉⠉⠈⠉

⣿⣿⣿⣿⣇⠀⠀⠀⠈⠿⠿⠍⠉⠉⠉⠉⠉⠀⠞⠻⠏⠻⠿⠿⠿⠹⠿⠻⠿⠻⠟⠻⠟⠙⠟⠿⠿⠟⠻⠿⠿⠿⠿⠷⣿⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣛⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⣿
⣿⣿⣿⣿⣿⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣏
⣿⣿⣿⣿⡞⢇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠇⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣿⣿⣿⣿
⣿⣿⣿⣿⣁⣾⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿
⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣟⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣭⣞⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣏⡉⣹⡟⣿⡿⢿⣯⣻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⣭⣽⣿
⣿⣿⣿⣿⣿⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⢻⣿⣿⠿⠿⠟⣿⣿⣷⢾⡆⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣭⣶⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣸⣿⣿⣄⣙⣠⣿⣿⢸⠐⠁⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣙⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠲⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠙⣿⢿⣾⠀⠀⣿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣟⣙⣿⣽⣿
⡌⢉⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡇⢠⣿⣿⣿⣆⣲⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠛⢛⣿⣿⣿
⢹⣸⣿⣿⣷⣿⣏⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⣭⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢛⡿⢿⣿⣿
⣿⣟⣿⣿⣿⡿⣧⠇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⣿⣿⣿
⣿⣷⢻⣿⣿⣿⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿
⡛⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⣷⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣬
⣟⢋⣙⠛⠿⢿⣿⣿⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣧⣌⠻
⣿⣿⣿⣿⣷⣶⡞⣿⣭⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣦⣮⣹⣷
⣿⣿⣿⣿⡽⣋⠙⣷⣮⣟⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⣿⣿
⣿⣿⣿⣿⣿⣾⡟⢼⣿⣮⡁⠙⠳⠒⠒⠀⠤⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣟⣻⣋⣻⣿⣀⣀⣀⠀⠀⠀⢈⣂⣀⣀⣀⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣇⣻⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣿⡉⢏⣫⡽⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⢺⣿⡿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣯⣤⡀⠻⢰⣿⠿⠿⠃⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢸⣿⣿⣿⣿⣿⠿⣿⣏⡀⡈⢻⣿⣿⣟⣉⣀⣲⣌⢡⣶⠀⠀⠰⠀⠀⠀⠀⠀⣯⡍⠀⠀⠀⠀⢻⣿⣿⡿⢻⠿⢋⣽⣿⣿⣿⣿⣿⣿⣿⢿⡇
⠀⣿⠉⠿⢿⢿⠀⠙⣿⣿⣿⣦⠈⣿⣿⣿⣿⣿⣿⡇⠉⠀⠀⠀⠀⠀⠀⢈⠁⡇⢁⠀⠀⠀⠀⢸⠟⢁⠐⠀⣠⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⡇
⠁⠀⠀⠂⢰⡆⠁⠀⡘⠛⢿⣿⣮⣽⣛⣿⠟⠉⡅⠠⠀⠀⠘⠀⠀⠀⠀⠀⢀⡤⣦⣀⡤⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡈⠛⢿⣿⣿⡇
⠠⠀⣄⡀⠀⠄⠀⠀⠀⠀⠀⣿⣿⣿⠟⠻⢒⠀⣽⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⣧⠟⠯⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⢣⠀⠀⠙⠻⡇
⠠⠀⠈⠀⢠⡄⠀⠆⢂⠀⠀⠙⣿⡿⠀⠀⠈⠀⠘⠀⠀⡀⠀⠁⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⢀⣤⣴⣾⣶⣴⣾⣿⣿⣿⣇⠸⣶⣅⡀⠀⡄
⡀⠀⠀⠈⢢⡈⠀⠀⠼⠀⠀⠀⢿⣇⠀⠀⠀⠆⠀⠀⠀⠁⠀⠀⠀⢀⢀⠁⠀⢀⠀⠀⢰⣴⠾⡿⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⡷⡈⢿⡷⢚⡇
⠁⣀⠀⠈⠆⠓⠀⠀⡀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⣀⠀⢣⠀⠀⢠⡾⡟⠁⠀⠱⠆⠀⠀⢹⣿⣿⣿⣿⣿⣿⣴⡿⢀⠘⢿⡇
⠀⠀⠑⠀⠀⠀⠈⢦⡆⠀⠀⠈⣿⡏⠀⠀⠀⠐⠀⠀⠀⠀⢠⡴⣿⠃⡴⠟⢇⠀⠀⠙⠷⢀⣤⠖⠀⣀⠐⣾⣿⣿⠿⠟⠻⠿⢿⣷⣞⢡⠈⡃
⡁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠘⣆⠀⠀⠀⠑⢀⡀⠀⠀⠀⢀⠁⠛⠛⠹⣋⠀⢠⠄⠀⣠⠀⠉⠉⠀⠞⡃
⠓⠀⠀⠒⢠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣰⡿⣄⠀⠤⡄⠐⠀⠀⠀⠈⣶⠐⠀⣤⠀⠀⠀⢠⠋⠉⠀⢀⣸⣧⡀⠀⠀⣠⠀⢰
⣶⣶⣦⣀⣸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⢀⡯⡅⠠⡁⠜⡇⠛⠀⠀⠀⢦⣿⠀⠀⠉⠠⠀⠰⣿⣷⣶⣿⣿⣿⣿⣿⣷⡀⢻⣷⣿
⣮⢿⣿⢟⣷⠀⠘⠃⠀⠀⠀⠀⡄⠸⠃⠀⠀⠀⠀⢠⣿⣷⣸⠆⣡⠘⡇⠚⠂⠄⠐⡎⠉⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠹⣿
⢻⠆⠉⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⢠⣿⣾⠁⢏⡸⣠⠃⢄⠠⣤⣼⠃⠀⠀⠀⠀⠀⠀⠀⢻⡿⠿⠿⠛⠿⠿⠿⠿⣿⣿⣷⣾
⢸⡇⠁⠆⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠂⡀⠈⣿⠳⠀⢫⣔⠉⠑⢾⣷⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⠅⠀⠀⠀⠀⠀⠀⠉⠻⢿
⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠈⠃⠹⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⠀⣀⣀⣀⣀⠄⡀⢸
⣶⡄⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⡿⠀⣀⣀⣀⣤⠴⣧⡄⣶⣦⣤⣌⣿⣿⣿⣯⣷⣽⣵⢶⡆⢸
⠄⣀⠉⠈⠈⠁⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⠀⣀⠀⣀⡀⠀⠀⠉⣿⣧⣴⣿⣿⣿⠀⠀⠋⠀⢸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾
⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠬⠴⠟⠉⠻⣋⠁⣄⠀⠀⠘⣿⠟⠻⣿⣿⡆⠀⠀⠀⠘⣿⣷⡦⢨⡛⠛⠿⣿⣿⣿⣿⣿⣿
⠦⢀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠁⢀⠀⣀⠁⠆⠁⢠⠀⠈⠋⠀⠈⠁⠀⡜⠁⣠⣶⣿⣿⣧⠀⠀⠀⠀⢹⣏⣿⠌⠙⠛⡷⢦⣙⣛⠴⠿⣿
⡿⠿⠟⠭⠽⢭⣉⣷⠶⠤⣴⡆⣦⣤⣾⢿⡍⠇⠀⠀⠀⠀⠀⠈⠑⠂⠀⠙⠀⡼⣿⠛⢿⣿⣿⡆⠀⠀⡄⠀⢻⣷⠆⠛⢻⣽⡦⣹⠽⣆⢛⢻
⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⣀⣀⣸⣾⣟⠰⢀⡀⠀⠀⠀⠀⠀⢠⣄⣠⣄⡠⠚⠇⢉⠀⡈⣿⣿⣿⡀⣠⠾⡎⠙⢿⣾⣀⣿⣿⣷⡇⡷⡉⠻⣾
⠤⠀⠀⠀⢀⠄⠲⠄⢑⢿⣿⣿⡞⣿⣿⠷⢮⠃⠠⠀⠀⠠⠀⠀⡉⠻⢹⡄⠀⡄⠸⡀⠠⢻⣿⣿⣿⣧⠀⢠⡀⢠⣿⣿⣿⣿⣯⠙⣷⠉⠀⢾
⢀⠂⠀⠀⣀⢠⣴⣾⣿⣏⣿⡿⣗⣾⣻⡛⣄⡸⠃⠀⠀⠰⢠⣄⠈⡀⠚⡄⢀⡄⠀⠀⣀⠸⣿⣛⣿⣿⠀⣿⣇⣼⣿⣿⣿⣿⣿⣿⣿⡿⠶⢿⠂
⠌⡗⠀⢴⠋⣿⡹⢯⣏⣿⣿⡤⠯⠷⠿⠟⣼⠁⠂⠀⠀⡧⢾⣤⡗⢱⠔⢣⠤⡼⢂⠐⢥⡒⣮⣹⣿⠷⠀⠁⠀⠀⠀⠀⠀⠉⠙⠛⠁⣠⣶⣾
⢀⣇⣰⣾⡶⢿⠿⠳⠈⠁⠉⠀⠠⢠⠀⠀⢀⣞⠃⠀⠀⠀⡀⠁⠞⠋⠚⠙⠂⠳⠎⡶⠰⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘⣿⣿⣿
⡞⢿⠀⣟⠡⠀⠀⠀⠖⢀⣠⣴⣾⢿⣿⡿⠛⠉⠰⣤⣤⣩⣿⢷⣄⠀⠘⢳⡖⠲⢠⠀⠀⠀⠁⠁⠀⢠⡀⢂⣐⠆⠀⢀⠄⣀⠀⠀⠀⠈⠻⢿⡀
⢣⠘⡆⡉⢂⠀⣤⡚⡘⢻⡁⠸⠞⢋⡛⠀⠀⠀⠀⡺⠞⠿⠟⠉⢷⣃⣴⡎⠀⠀⠈⠈⠀⠀⠀⠀⣄⡠⢞⣿⣭⣶⣿⣿⡾⣭⡇⠂⠀⡀⠀⠘⠄
⢱⡀⢤⡔⢻⠐⠈⠣⡈⠀⠉⠲⠎⢋⠀⠀⠀⠀⠀⠐⣦⡀⠀⠀⠀⠈⠌⠀⠀⠀⠂⠀⣀⣠⣤⣿⣯⡴⣿⣿⣿⣿⣿⣿⣿⡿⢁⣄⡀⠀⠀⠸⡆
⡫⢽⡦⢟⣷⡞⣴⢞⢓⣶⠃⠀⢤⣌⢿⡀⠀⢀⣴⣾⠁⠉⢡⣶⡄⢲⣬⡃⣤⣄⣪⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣰⣿⣿⣿⡾⣧⣤⡃
⢤⣼⡇⠈⠯⢃⠘⡞⠁⠉⠓⢦⣤⣀⠀⠘⠛⠻⠉⠉⠁⢴⢦⣤⢶⡈⢽⡁⠈⠀⡈⡟⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠑⣯⡽⣿⣧⠻⣿⣽⡇
⢸⡇⢸⢦⣠⠋⡄⢠⣀⢀⣀⣀⢀⠈⠙⠓⠠⠠⠤⠤⠦⠤⠤⠤⠼⡐⣧⣌⢶⣠⣴⡇⣈⡤⣿⣿⣿⣿⣿⣿⡿⣌⣻⣿⣆⣽⣷⣼⣿⡆⢿⣿⡇
⠀⠈⠉⠻⠣⠽⠴⠟⢋⡈⢥⣝⡀⡈⠁⢀⡀⠀⠀⠀⠀⠀⠠⠈⠐⠁⠀⢘⡦⣌⠳⠟⠀⢀⣀⣉⣉⠙⠛⢿⡷⢟⣻⡟⢈⠿⣿⣽⣿⣿⣦⣹⠃
⣀⠀⠀⠀⠀⠐⠀⠀⣽⢇⣼⣩⠇⠀⠞⣻⡁⠀⣠⡀⠀⢰⣶⣿⣾⢆⣰⣿⡤⠞⠁⢠⡞⢻⣿⣿⣿⣿⣈⣦⡀⢻⣼⣦⣘⣣⢼⡷⣽⣿⣿⣿⡇
⣿⣿⠖⠰⠎⡠⠆⠀⠈⡼⣷⣿⢴⠰⣞⣹⣞⠂⡄⠁⠀⠀⢉⠙⢩⣿⣏⡑⣀⣰⣬⠹⡧⢸⢿⣿⣿⡿⢻⣿⣿⣼⡌⠻⣷⣌⣘⣳⣿⠿⠿⢿⡇
⡿⣆⠀⠀⠀⠀⠀⠀⠀⠄⢀⡞⢰⠀⡅⢰⣿⡔⠁⠄⠀⠌⠀⠀⡞⠸⠛⣿⣿⣿⣿⠘⡥⣿⡚⣻⣏⡵⢘⡥⢿⣿⣿⣎⢶⣤⣉⠉⢵⣾⣿⣿⡇
⡇⢸⠃⠁⠀⠀⢂⠀⢰⠀⠄⡐⣻⢸⣯⠟⢱⡧⠠⠀⠀⠀⠀⡀⠀⠀⠀⠃⠉⢺⠛⢴⠀⠖⢿⣽⣿⣇⢠⣽⣛⣿⣿⠿⣻⣿⣿⣷⣰⣌⣙⢿⡇
⣷⠆⡉⣱⠆⠡⠈⠀⢰⠀⠰⣇⢹⡘⣿⢀⠿⡀⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⡏⠆⠰⢀⠐⣿⣥⣾⣼⡌⢫⣽⣯⣼⣹⣮⣯⣽⣼⡿⢙⣃⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣿⣿⣾⣿⣟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡟⢋⠉⠀⠀⠀⠀⠀⠀⠌⢻⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠃⠀⣀⠀⠀⠀⠀⢀⣀⠀⢸⠋⠉⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠉⠉⠀⠀⠀⠉⠉⢀⠆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠤⠖⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠠⠣⠀⠀⠀⠶⠄⠀⠀⠚⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⡈⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⣦⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣾⣿⣷⣄⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠐⠁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠘⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠃⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⢈⡁⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠴⠁⠀⠀⠰⠤⠄⠀⠀⣸⡿⢿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⡏⠛⠷⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡜⢓⠀⠀⠀⠈⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡧⠀⠀⠄⠀⠀⠀⢶⣖⣿⣿⢘⣿⡿⣿⣿⣿⡛⣿⣿⢸⣿⣹⡷⠀⠀⠀⠀⠀⠀⠓⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠛⠛⠛⠛⠁⠙⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡛⠊⣱⠋⠻⣙⠿⣛⡟⢻⣛⣿⣿⣿⠟⢿⠟⢿⣿⣿⣿⣿⣿⠟⠛⠛⢛⣻⣿⣻⡟⠉⠁⠀⠁⢻⡿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣶⣶⣡⣾⣦⡝⣧⡉⣿⣿⣿⣿⡆⠙⣷⣿⣾⡿⣿⣿⡟⢻⣿⣿⣥⡄⠛⣼⠟⠟⠀⠀⠀⠀⠀⣾⡇⠀⠈⢿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠛⠋⠁⠈⠉⠁⠈⠁⠙⠿⣿⣿⣷⠀⢻⡿⣿⣿⠛⢿⡄⡈⠻⣿⣭⣤⣴⡄⠛⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⡿⠉⠁⣁⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⠀⠈⠉⠀⠀⠹⢿⡀⠸⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠙⠻⣿⡇
⠀⠀⣀⣠⣤⣤⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡛⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠈⠀⠀⠀⠀⠀⠀⢀⣠⣤⣦⠀⣀⠀⠈⠁
⠀⠀⣿⣿⣿⣿⣿⣿⣿⡶⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠓⠻⣿⣿⣿⣿⣿⡧⠘⢷⠦
⠀⠀⣿⣿⠟⠿⠛⣷⠟⠠⠋⣽⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⢶⠏⠁⠀⠀⠀⢈⣿⣿⣿⣿⣦⡟⢀⠘⠇
⠀⠀⣿⢋⡛⠀⢀⠁⠘⠃⠴⣼⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⠀⠀⠀⠀⠀⠀⠛⠀⠊⠁⠀⠀⠀⠀⠀⠀⠤⠾⡿⠋⠉⠉⠙⠻⠆⠀⠂
⠀⠀⣟⣋⢀⣴⣾⣀⠛⠻⣿⢋⣉⣀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠐⠈⠀⣰⣁⠀⠀⠀⢀
⠀⠀⣿⣿⡏⠁⠀⠉⠉⠶⠟⡘⣿⣹⠗⠲⠶⣤⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣴⣶⣾⣿⣿⣿⣷⡄⢷⡎
⠀⠀⡿⠋⠀⠀⠀⠀⠀⠀⠀⢘⣿⡟⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠱⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠿⣿⣿⣿⣿⣤⡁
⠀⠀⡄⠀⠀⣀⣠⡀⠀⡀⠀⠀⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠻⠇
⠀⠀⠆⢠⣾⡿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣀⢀⡀
⠀⠀⣾⣿⡿⢟⣭⣽⣿⠖⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣀⣠⣶⡖⠀⠃⠘⣿⡟⣾⣿⣿⣿⣷⣶⣦⡀
⠀⠀⣿⣿⣝⠿⠿⠉⠠⢴⣿⣧⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠿⠛⢸⡃⠀⠀⠀⢻⣶⠈⣙⡛⠿⣿⡿⢿⡿
⠀⠀⣉⣩⠁⣠⠤⡿⣤⡾⠟⠛⠁⠈⠂⢁⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⢈⣿⡀⠀⠀⠈⣾⣃⠀⠄⠲⢤⠀⠙⠿
⠀⠀⠛⠁⠀⠁⠀⠀⠈⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⣿⣯⠀⢀⡄⠸⣧⣀⣾⣧⡀⠜⠁⠤
⠀⠀⡃⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢻⣿⣷⡄⢀⠀⣾⣿⣿⣷⣤⡆⢰⡄
⠀⠀⣳⣾⣳⡆⠞⠀⠉⢲⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⠛⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣽⠅⠘⠀⠋⠛⠛⠿⠟⠃⣉⡀
⠀⠀⣛⣛⣩⡶⠀⠀⣴⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡇
⠀⠀⠙⣉⣉⠀⠀⠀⠻⣯⡽⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⠊⠛⠁⠀⠀⠀⣠⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠓
⠀⠀⣻⣿⠏⠀⣈⣿⡦⢿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⡸⢿⣿⣿⣷⣶⠀⠀⠀⠀⠠
⠀⠀⠘⢋⡀⠐⠉⠁⠀⠀⠀⠉⠉⠱⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣷⣿⣿⣷⣿⠀⠿⣿⢁⡂⡄
⠀⠀⠐⠀⠀⠀⠀⣀⡄⠀⣠⣄⣰⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⣿⣿⣿⣿⠟⢻⡧⢸⡇⣻⡌⢻⡇
⠀⠀⠃⠀⠀⣴⣟⠛⠃⠀⠛⢻⣿⠗⠀⠀⠀⠀⠈⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣉⠋⠻⠟⡛⠃⠙⢏⡿⣷⣼⡃
⠀⠀⣧⡄⣸⣿⣉⡀⠴⠀⠰⡞⠃⢀⡀⠰⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠈⠀⠀⠠⠀⠈⣿⣿⡗⣦⡀⠓⣦⡀⠰⣛⣿⣿⡇
⠀⠀⣭⣀⣿⡿⣩⠗⢰⣴⠃⠀⣠⣾⣅⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠀⠀⠀⡀⢘⣃⠈⠋⣿⣷⣄⣈⡉⠉⢥⣶⠗
⠀⠀⣿⣿⣉⢲⡄⠀⢀⡄⠀⢰⣿⡿⠋⠠⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⢗⡀⠊⢹⣿⠋⢽⡿⢣⣤⣩⠆
⠀⠀⢧⣀⠰⠙⠇⠀⡿⠀⢠⣿⠀⠑⢻⠀⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠛⢶⣾⡟⠃⠚⡷⢊⠁⣰⢶
⠀⠀⣽⣿⠄⠀⠂⠀⢀⣤⣾⣋⡀⣠⣿⠀⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠄⣠⡔⢾⡟⣶⣿⣿⡏⠀⠉⣀
⠀⠀⣿⡗⠀⣴⣶⡆⢹⣿⣯⣽⣷⣿⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⠙⠈⠛⠟⠋⠀⠀⠁
⠀⠀⡄⠀⠈⠁⠀⠀⠀⠉⠀⠀⠈⠺⠁⠀⠀⢥⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠲⠀⠀⠀⠀⠉⠀⡀⠀⠐⠀⠈⠁
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⡁⠀⠰⠤⢀⠀⠀⠀⠀⠀⠀⠙⠤⠀⠀⠀⠀⠀⢀⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠒⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠼⠃⠀⠀⠀⠆⠸⠄⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠶⣦
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠑⠀⠀⠀⠀⠀⠄⠈⠁⠀⠀⠀⠀⠀⣀⣄⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢀⡤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠖⠀⠀⠀⠀⠀⠀⢸⠉⠇⠘⠉⠋⠀⠃⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠈
⠀⠀⠃⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠃⠀⠀⠀⠀⠀⠀⢠⣄⣼⠂⠆⢈⣤⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠛⠉⠉⠁
⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠡⠊⢀⠄⠀⠀⠀⠀⠀⠀⢀⠐⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢀⠘⠁⠀⠀⢀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠁⠰⠁⠀⠀⡁⠂⠀⠀⠀⠀⠀⢂⡴⠠⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡇⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠠⠐⠯⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢂⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠰⠚⠋⠉⠉⠛⠻⠷⣦⡀⠀⠀⠀⠀⣠⣤⡶⠿⠛⠛⠛⠶⢤⡀⠀⠀⢰⠀⠠⢳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⠇⡄⠀⠀⢤⣈⠁⠀⠈⠀⠀⠈⢿⠀⠀⠀⢰⠟⠁⠀⠀⠀⠀⠐⠀⠀⠀⠐⠦⠸⡄⠠⣦⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢨⠀⢇⢀⣤⣤⣍⡛⠓⠒⠒⠂⠀⠘⠀⠀⠀⡌⠀⠀⠀⠤⠤⠤⠶⣚⠁⠀⣀⢠⣿⡏⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⡸⣇⣸⣿⣿⠻⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡆⠸⠈⡿⢃⣸⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⡛⠈⠁⠙⣿⣿⣆⣻⣿⡇⠀⠀⠀⠄⣠⡛⣆⣠⠀⠀⠀⢼⣿⡿⢁⣽⣿⠇⠴⠏⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡄⠛⡑⠀⠀⢀⠈⠻⢿⣿⣿⠗⠀⠀⠀⠀⠉⠙⠉⠁⠀⠀⠀⢟⣿⣥⣿⠟⠁⠀⠐⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡀⠀⠁⠀⠀⠀⠇⠀⠀⠈⠁⠀⠀⠀⣀⣤⣦⣤⣤⣤⡀⠀⠀⠀⠉⠉⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠋⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣍⡉⠀⠀⠀⠀⠀⠀⡠⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣤⡀⠀⢰⡏⣹⣿⡄⠀⠀⣀⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⢻
⠀⠀⠀⠀⠀⠀⠀⣰⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣶⣿⡿⠟⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⠈⢶
⠀⠀⠀⠀⠀⠠⣄⠏⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⡾⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢁⡜
⠀⠀⠀⠁⠀⢀⠈⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⣿
⠀⢀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠹⠋⠄⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠣
⠀⠛⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢡
⠀⢰⡆⡀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡈⠀⢀
⠀⢠⡿⢉⢺⡄⠀⠀⠀⠀⢀⡀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠀⠀⠀⠒⣵⠍⠀⠸
⠀⠀⣅⠀⢘⢣⠄⠀⠀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡍
⠀⠀⠋⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢠⡄
⠀⠀⠸⠃



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⡆⢰⣶⡄⣤⡄⢠⣤⠠⠄⢀⠠⢤⣠⠠⡆⣀⠀⡤⢀⠀⠀⢀⠀⠀⠀⠄⠄⠀⠀⢀⠀⠀⡠⠀⠀⠀⠐
⠀⠃⠸⢿⡇⠉⡃⠠⣽⢸⡇⢴⣭⡗⣾⠀⣶⠟⠀⣠⠟⡫⢶⣆⣁⠐⡀⣤⠀⠀⢈⠀⡦⢀⠄⣀⣄⢀⡂⠠⣤
⠀⠀⠀⠀⠀⠀⠁⠘⠛⠸⡇⣿⣿⡆⡟⠀⠁⠈⠀⠀⠀⡀⠀⠹⠇⣻⡆⠙⠆⠀⠘⠀⠃⠸⢠⣷⠎⣀⡄⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠛⠀⠷⢠⠀⣄⣶⡆⠤⣿⡇⣴⡇⢹⡄⠀⠀⠀⠀⠀⠚⢡⡿⠁⠀⠘⣿
⠀⢂⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠻⢿⡇⣿⡇⠸⠧⠀⠀⠀⠀⠀⠈⣠⣶⠀⣶⡄⠬
⠀⡿⢿⣿⣿⣿⣿⣿⣿⠶⣂⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣞⡇⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⠠⠿⠏⢈
⠀⡏⣠⣉⣙⣛⠫⠹⡀⢸⢟⢻⠻⣛⡿⣿⠿⢳⣴⣶⣦⣤⣤⣀⡐⠛⢁⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⡇⣿⢃⡀⠛⡇⠀⠁⠘⡇⣶⡶⣤⡍⢺⠀⣿⢛⠛⠻⣿⠏⠶⡛⠑⢛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈
⠀⡇⣇⠀⠙⠀⡄⠁⠀⢰⡇⣿⢃⡀⠃⢰⠀⡁⠘⡇⡟⢰⠃⡈⢉⣂⣼⠷⢂⢁⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⢿
⠀⡇⠉⠀⠀⠀⠀⠗⠀⠀⡇⣿⠘⠀⠀⠜⠀⠁⠀⠀⠀⢐⠈⠁⢷⣿⣏⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠥⠤⠆⠀⠀⠀⠈⠀⠀⣣⣅⣀⡀⣈⠀⠀⠸⣀⡀⠀⡸⠀⣤⣿⣿⠽⠀⠀⡇⠀⠀⠀⠀⢀⣠⣤⣦⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠂⠀⠂⠀⠈⠉⠉⠉⠉⠀⠿⠛⠁⠀⠀⠀⠃⠀⠀⠀⠀⢰⣶⠾⣟⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠈⠙⡁⢸⣿⣿
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠘⠛⠛
⠀⠀⠀⠀⢀⠀⣀⡀⠀⠀⠀⠀⠀⢀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠠⠀⠀⢀⠰⠂⠈⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠸⠒⠙⢣⠖⠀⠀⠐⠓⠒⠒⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠰⡟⠀⠀⠀⡰⠀⢡⡇⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢀⡀⠀⠀⢠⠁⠀⠈⠳⢠⠤⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⡀⠐⢀⡞⠀⠙⠉⢡⠀⠀⠈⡖⢰⣲⢦⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢜⣆⠀⠀⢀
⠀⠀⠀⠀⠀⠠⣠⠎⠀⡠⠐⠃⠈⠁⠐⠊⠀⠘⠁⡜⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠼⠫⢧⡀⠄
⣀⣀⣀⣀⠤⠖⠁⠐⠀⠀⠀⡀⢀⡀⠀⠀⠀⡅⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠐⠉⠲⢄⣀⣀⣀⡀⢀
⠀⠬⡥⠀⠀⠐⠀⠀⠀⠀⠈⠈⠃⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⣁⠀⡛⠤⠁⠀⠀⠈⡉
⠀⢀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠋⠁⠀⣦⠇⠀⠀⠀⠀⢠⡁
⠀⡄⠀⠀⠐⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀⢀⣀⠀⠤⠐⠀⠉⠉⠁⠉⠉⠓⠂⢄⠀⠀⢀⠘⢃⠀⠐⠀⠘⢇
⡰⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡘⠀⠈⠀⡀⢀⣠⡃
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠁⠀⠠⠈⣁⠘
⠀⢠⠀⡀⠂⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣤⡶⢪⣍⡠
⠀⠀⠷⢂⠀⠀⠐⡢⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡉⠻⣦⠕
⠐⢀⠴⠀⠠⢴⠆⢸⡀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⠈⠀⡔
⠀⢀⢂⡑⢦⠹⢥⢀⠹⠚⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣔⠀⢖⠰
⢴⠋⡀⠁⡘⠆⠑⠈⠀⠀⢈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⢸⣦
⠠⠷⡤⢑⡙⠀⠀⠀⡣⢁⡆⣾⠙⢦⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢀⡟⣁
⠢⠂⠀⠀⠀⠀⠦⢠⠈⡄⠠⣰⠀⢸⠀⠀⠀⠀⠰⣅⡀⠀⠀⡀⠀⠄⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡥⡴⠏⣺⣿
⠈⠀⠀⡀⠩⠋⠍⢒⡗⠁⠉⠁⠀⠎⠀⠀⠀⠰⠀⢋⡹⠫⢝⣧⢶⣶⡶⠒⠀⠓⠂⠀⠀⠀⠐⠀⠠⠀⠀⠸⠓⢺⢤⡄⠨
⣤⣄⣀⣀⣀⣥⣤⣭⣥⢦⠎⠀⠀⠀⠀⣠⠋⠠⠶⠄⠁⠄⠉⠲⠶⢿⠅⠠⢀⡀⠀⠤⠤⢤⢤⡄⢢⠀⡌⠐⠤⢤⣤⣤⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⡿⠿⠿⠿⠿⠿⢿⠿⢿⠿⠿⠿⡿⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣟⠀⠛⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⡆⢤⣀⠌⢻⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⢛⡋⠉⠉⢛⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡏⢴⣯⣤⠶⠒⢶⣶⣈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⣿⣧⡸⣿⣿⣐⠆⠘⣿⣿⣿⣾⡿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⣿⣿⣿⣾
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⣻⣿⣗⡈⠉⠉⢀⣜⣿⣿⣿⣿⡇⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⡛⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⣿⣿⣿⣿⣯⣿⣿⡿⠃⢀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣿⣻⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⢿⣿⣿⣿⣟⡉⠀⣀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣉⣽⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡿⢿⣿⣷⡦⢰⠦⠟⢿⣿⣿⣤⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣻⣷⣿⢿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣵⣾⣿⣿⣇⢐⡀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠿⢿⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⢻⣿⣿⣿⣿⣟⣵⣮⣁⡀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣾⡿⣽⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣴⣿⣿⡿⠿⠿⢿⣿⣿⣅⠴⠻⠿⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣼⡯⣶⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣭⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷⣄⢐⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠛⣿⣦⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣏⡀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣧⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⣄⠂⠀⣤⣾⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠓⣬⣿⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢷⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢷
⣿⣿⣿⣿⣿⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣯⣿⢿⣷⣔
⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣀⣴⣶⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⢷⣦⠖⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣇⣴⣿⣿⣿⣿⣿⣷⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⡦⢻⣶⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣼⡷⣲⡶⠀⢹⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⡄⠙⢷⣤⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣤⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢃⣘⣷⡼⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣈⣆⢙⡿⠛⠛⢂⠀⢂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠠⠉⠀⠙⢡⠧⠐⠂⠀⠉⠉⠀⠀⠀⠀⠀⠀⢈⠟⡰⠞⠉⠉⠣⣾⢄⣀⡀
⠀⠀⠀⠀⠀⢀⠀⢰⣦⢸⣇⣯⢕⢤⡗⣴⣗⠁⠄⠄⠀⡠⣦⡤⣦⠠⣤⣀⡤⣿⣎⣱⣾⣗⣷⠔⠳⠋⣿⢅⣢⡀⢘⡀
⠀⠀⠀⠀⠀⣼⣤⡆⢹⡽⠻⣿⡶⣏⣿⣿⠾⠖⠓⠛⣛⣛⣈⣙⣉⣉⣉⣙⣙⡛⠒⠒⠂⠈⠛⠷⢶⣄⠁⠈⡓⢭⡿⣇
⠀⠀⢿⣄⣦⠹⡟⢀⣜⡡⠴⣬⣴⠞⠉⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠈⠑⢤⡙⢦⣇⢫⢀
⠀⠀⠀⠙⢿⡸⣷⡈⠀⣠⠞⠋⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠈⠢⣘⣸⣿
⠀⠀⠀⠰⣮⣿⣽⡷⠊⠁⠀⠀⠀⢀⣾⣿⣿⠛⢋⣉⣉⡉⠉⠉⠉⠉⣠⡤⠉⠉⠉⢹⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠫⣽⠔⠂
⠀⠀⠀⠀⣘⠟⠋⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣁⠀⠀⠨⣙⠀⠀⠀⠈⣱⠂⠉⠀⠦⠸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡏⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠿⢦⣀⡀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡆⠀⠀⠀⢀⡀⣤⢔⣮⠁
⠀⢹⣦⣼⣸⣉⠉⠉⠒⠠⢼⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⢀⡀⣀⡀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⡀⠀⠊⠁⠀⢹⣦⡍
⠀⠀⣍⠁⣏⠛⡄⢀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠁⣀⣠⣄⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣧⠀⠀⠸⣴⠚⡻⠁
⢤⣤⣈⣴⡛⡟⢿⡏⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠺⣿⣟⣟⠋⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠽⠀⣥⣀
⠀⠉⠉⠻⢁⠧⣜⣧⡄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠈⠉⠁⢀⠔⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠪⡿⢸⠍
⠀⠀⠀⠘⠿⣷⣿⢿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠑⠀⠈⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⠚⡉⠍
⠀⠀⠀⠀⠀⠀⢻⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠹⠃
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠈⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠘
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⡠⠁⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⢀⣀⡴⠊⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠠⣀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠠⢤⠀⣴⡄⠀⠀⠀⠀⣤⠀⣀⠀⠀⠁⠀⠀⢀⠀⠸⠁⣀⠀⠀⠐⢿⣶⣤⠀⠀⠀⠀⠀⠠⠾⠷⣴⠊⠀⠀⠀⠀⠀⢀⣷⠀⠤⢀⡶⠶⠶⠻⠚⠁
⠀⠀⠊⠀⠻⠟⠀⠀⠀⠀⠺⣦⣾⡃⠀⠀⠀⠀⠈⠓⠐⡂⠃⠀⠀⠀⠠⣷⣿⡅⠀⠀⢀⠄⣀⠀⠀⢹⣹⡶⢄⠀⠀⠀⠀⠟⣵⢀⣠⠈⠚⡒⠂
⠀⠀⡀⡠⠀⠀⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠐⠂⠈⠁⠀⠨⣆⠀⢀⠈⣿⠀⠀⠀⠈⠘⠋⠀⠁⠬⣿⣿⡛⢧⡀⠀⠀⠀⠀⠀⠘⠓⣾⠉
⠀⠀⡏⠀⠀⠀⠈⠦⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢰⡇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠙⠻⡶⢤⡀⠘⠛⠿⣄⠀⠸⢏⠶⠄
⠀⠀⠀⠀⣠⠀⣄⠀⠀⠀⠀⠘⠷⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⣷⠀⠀⠀⠀⢿⡦⢦⡄⠀⠀⠈⠀⠀⠀⠈⠳⣞⢻⣶
⠀⠀⠀⠀⢙⠀⢸⡆⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠤⢀⡀⠀⠸⠆⢀⣄⠀⠉⠀⣀⠀⢀⣀⡀⠚⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢾⣄⢰⢀⡨⠜⠂⠀⠀⠀⠀⠀⡀⠀⠈⣉⣩⠟⠣⠀⣿⡷⣄⣀⡁⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⣀⡀⠀⠀⠀⠀⠐⢲⡞⡂⠀⠀⠀⠀⠀⠀⢍⠀⠹⠃⡱⠊⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⠀⠿⣧⠀⠉⠙⠆⠀⠀⠤⣀⠀⢀⣀
⠀⠀⠀⢈⣷⠀⠀⠀⣀⣸⡍⠈⠤⠴⠀⠀⠀⠀⢈⡢⡄⠾⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣾⠀⠀⠘⢷⣶⣷⠀⢠⣴⣶⡾⠒⡾⣿⣇⣀⡀
⠀⠀⠐⠁⠈⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⡃⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠈⢻⣿⠀⢀⣉⠿⠿⠛⢿⡛⣻⠿⠿⡤
⠀⠀⠀⠀⢰⡆⠀⣠⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⠀⣠⠟⠂⠁⠀⣉⣀⣙⠀⠀⠁
⠀⠀⠀⠀⠈⠃⢰⠋⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⢿⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠴⣋⣤⣤⡤⠤⠤⠬⠟⠉⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠉⠒⠂⠀⢀⡼⠤⠀⠀⠀⢀⡨⠅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⣠⠴⠟⠒⠒⡂⠋⠈⠁⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣿⣀⣴⠞⠋⠀⠀⠀⠀⠀⠑⠒⠃⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠅⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⣠⣄⣰⠏⠀⠀⠀⠀⠘⣧⣰⠻⡶⣄⠀⠀⠀⠀⠀⠀⠀⣠⠟⢦⡤⢶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣧⢀⣴⣿⠋⠳⡀⠀⠀⠀⢠⡞⢉⣯⠻⡄⠀⠀⣠⠶⠦⠞⠃⠸⣟⠎⠳⢄⠀⠀⠀⠀⣴⠋⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠙⣦⣤⡴⠋⢰⡿⣌⣢⠈⠳⢮⣁⣀⣀⠀⠀⠀⠹⡗⢀⣀⣵⢶⣷⡾⠁⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⣠⠴⣿⠁⠀⠀⠀⠘⢧⣀⡈⠑⠀⠀⠀⠀⠈⠑⠄⠀⠀⠉⠉⠁⠀⠸⣅⠙⠲⠦⢤⡴⢺⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡀⢀⣿⡄⠀⠀⠀⠀⣤⣻⡍⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣹⠇⢸⡄⠀⠀⡞⣰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇⢸⣿⣇⠀⠀⠀⠀⠸⣟⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠀⢠⠘⣆⣿⣇⠀⡼⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⠀⢹⠀⠈⠙⠢⠀⣤⠀⠀⠀⢀⣿⣷⠀⠀⠀⣿⡆⢸⣿⣿⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⢄⠀⠀⣾⡇⠸⡀⠀⠀⠀⢠⣿⡆⠀⠀⢸⣿⣿⡂⠀⢠⣿⡇⢸⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣦⣿⣦⣿⣷⠀⠙⢂⠀⠀⢸⣿⣇⠀⠀⣼⣿⣿⣇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡷⢦⡴⢄⣀⣿⣿⣿⣀⣀⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣷⣾⣧⣴⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠟⠛⠛⠛⠉⠉

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠒⠠⡀⠀⠀⠀⠀⢀⡤⠒⠠⠀⠀⠀⣀⠀⣤⠀⠀⠀⠀⠀⢀⣀⠄⡀
⠀⠀⠀⠀⠈⠀⠀⠀⢰⡆⢀⠀⠀⠀⠀⠀⡀⠠⠀⠈⢀⠀⠈⠆⠀⠠⠀⡞⠠⠀⠈⠀⠀⢉⡉⣠⣅⢠⡄⠀⠀⡔⢎⠀⢸⢖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠅⣾⠀⠀⠀⠀⠀⠀⠍⢠⡌⢡⡄⠀⠀⠀⠻⣬⠑⡯⠃
⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠅⠀⢠⠀⠀⠡⡈⠀⠘⠀⠄⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢁⠉⠀⠀⠀⠈⠀⠀⠀⠒⠊⠀⠀⠀⠀⢀⠐⡀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠃⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠚⡆⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠠⠀⢀⡈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠐⠀⠀⠀⠁⢸⣏⠀⠀⠀⡀⢀⠀⢑⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠆⠐⠤⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⡀⢰⠀⠀⠀⢼⠁⠀⠀⠀⠀⠀⢀⡞⠁⠀⢀⠀⠐⢄⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠆⠀⢀⡁⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠘⠈⠃⠀⡂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠂⠄⠀⠀⠀⠸⠀⠀⠁⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀⠀⠠⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀⡀⠀⠱⣀⢀⣀⣤⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣇⠔⠁
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠐⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡠⠂⠉⣩⠉⠑⠠⡀⠀⠀⠀⠰⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠂⢄⢀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠉⣐⣀⠔⠁⠀⠀⠠⢄⠈⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠔⠉⡑⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⣃⠀⠰⠀⠀⠀⠂⠀⠀⠀⠀⠀⡆⠀⠑⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠃⠃⠂⢀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠊⡀⠀⣀⠈⠳⣄⠀⠰⠌⡤⠂⡐⡄⠀⠀⠛⠀⢠⡷⢾⡇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⠉⠁⠢⠀⡇⠈⠋⠀⠀⠀⠀⠀⠀⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠈⠇⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠁
⠀⠀⠀⢠⡀⠢⡈⠀⠀⠰⠂⠀⠀⠀⠀⠐⢁⡄⠀⠑⡈⠀⡠⠁⠀⠀⠀⠀⠀⢀⠀⢠
⠀⠀⠈⠒⠴⣦⣿⣔⡾⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡷⠈⠀⡌⡆⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠈⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣘⣿⣿⣆⢀⣀⣠⣤⣤⣄⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⡿⣄
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣧⡉⢧⡀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⠁⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢄⠻⣿⣿⣟⣿⣈⣶⡿⣦⣴⣶⠄
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣏⣠⣀⣠⣄⣹⣿⣿⣿⣿⣿⣿⣿⣃⣀⣀⣣⣘⣿⣿⠿⢋⣡⢀⡁⡛⠁
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡘⢻⣿⣿⣽⣿⣿⣿⣿⣿⣷⣾⣇⢀⣀⢠⡷
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢹⠇⣼⣿⣿⣿⣿⣿⣿⡣⢿⡎⠁⢮⠄⣡
⠀⠀⠀⢿⣿⣿⣿⣿⡟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠻⣿⣿⣿⣿⣿⠏⠀⢺⡅⠀⠈⣞⠃
⠀⠀⠀⠸⣿⣿⣿⣿⣧⠀⠘⠿⣿⠿⢿⣿⣿⣿⣿⣿⣿⡿⢄⠈⠿⠁⣾⣗⡆⠀⢀⣀⡀⠄⠈⢢
⠀⠀⠀⠀⠻⣿⣿⣾⣿⣷⣤⣄⣠⣤⡘⠛⢃⣀⡸⠿⢿⣀⣈⢣⡀⢸⣿⠟⠿⣿⠟⣿⣶⢰⠀⠈⡆
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣼⣿⣿⣴⣼⣿⣿⣾⣿⣦⡿⠀⠄⠸⣿⣏⡠⢒⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣥⣀⣀⣠⠞⠁⠀⡌⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⢿⣿⡿⠿⠿⠿⠛⢫⣿⣿⣷⣿⠍⣇⠀⠀⢠⣁⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣷⡄⠿⠀⢠⠋⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠏⠙⢇⣿⣿⣿⣿⣿⣶⢦⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢻⠟⠛⠛⠛⢻⣮⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⡠⢺⠀⠀⠀⠀⡄⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⡠⠊⠀⢸⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸⠀⠀⠀⠀⠀⢃⠀⠀⢀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡼⠀⠀⠀⠀⠀⠀⢣⠀⠀⢣⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⡆⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢪⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⡀⠘⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠁⠀⠀⠀⠀⢠⠀⠀⠘⠦⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠊⠸⠈⠃⠀⠀⠀⠀⢀⠰⠀⠀⠀⠠⡀⠀⠀⠠⠀⠀⡀⠀⡀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⠀⠀⠀⠀⢀⠐⠒⠀⠀⠀⠀⠀⠔⡀⠀⡠⠀⠁⠀⠀⠲⠀⠀⠀⠀⣀⠀⠉⠀⠀⠀⠀⠐⡀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠠⠈⠀⠈⠢⠔⠊⠁⠀⠀⡀⠄⠂⠀⠀⢎⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠄⠀⠀⠀⢀⠇⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠒⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠤⠀⠀⣀⡠⠄⠀⠈⠁⠀⠀⠀⠀⠀⠠⠀⠉⠐⠢⢀⠀⢧
⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠐⠂⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠸⡆
⠀⠀⠀⠀⠀⣎⠠⠂⠁⠀⡀⠀⠀⠀⠀⠈⠁⠀⠡⢄⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⣀⣤⣄⣠⣬⣭⣷
⠀⠀⠀⠀⠀⡏⠡⠀⣀⣀⣀⣤⣄⣀⠀⠀⣠⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣇⠐⠋⠉⠉⠉⠀⠀⠀⣀⣀⠀⠀⠀⣀⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⡿⠿⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡟⠛⠳⢦⣠⣄⡀⠴⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⠀⠀⠀⠀⣤⣾⣦⡐⠛⠉⢁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢷⣦⡀⠀⠈⠁⠀⣀⣤⣾⣷⣦⣤⣄⣀⣀⣤⣦⣴⣶⣾⣷⣷⣶⣶⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠻⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢿⠀⠄⠺⠦⠴⠶⠴⠶⣤⣤⡤⣤⣤⣤⣤⣠⣤⣄⣀⣀⣀⣈⣉⣉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⠛⢻⠖⠊⠈⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣭⣬⣭⣭⢽⠶⠶⠶⠦⢤⠄⢀⣀⠀⠀⠀⠈⠉⠛⠛⢛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⡤⣿⠶⠶⣦⣄⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠠⠅⠀⠀⠀⠀⠀⠀⠉⠙⢩⣙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠆⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⢠⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⡀⠀⠀⡤⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠁⠀⡖⠂⠀⢀⠖⡄⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⠉⠀⠀⠀⠀⠱⡄⠀⠀⠐⠓⣄⠳⠯⡄⡎⠖⢃⡜⠂⠀⠀⢀⡖⠁⠀⠀⠐⠈⢀⠄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠈⠀⠀⠈⠣⣆⠀⣈⣎⠓⠶⠆⠶⠞⢻⡎⠀⣄⡶⠋⢀⠀⠈⢀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀⣄⡼⢦⡼⠋⠀⠠⣤⡤⠀⠈⢻⣴⠾⡀⣄⣁⠤⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢤⣴⠇⠀⠉⠀⠀⠀⠈⠁⠀⣧⣤⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⣄⣊⢙⡆⠀⠤⠀⠙⠀⠠⠄⢀⣟⢉⣆⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠆⠀⠠⠀⠠⠠⠤⠄⠐⠈⠁⠀⡩⣏⣙⠦⣀⣀⣀⣀⣀⡠⢎⣙⠯⡀⠀⠉⠐⠀⠤⠤⠀⠀⠤⠌⠠⠆⢀
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠟⠉⢁⣾⠿⠋⢻⠛⠻⢷⡄⠉⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⢀⠞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠢⣄⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡱⣤
⠀⠀⠀⣠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠁⠀⠀⠐⢦⠀⠀⠀⠀⠁⠀⠀⠀⠀⢤⡀⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣧⡀
⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡊⣢⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠢⡐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠋⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠑⠁⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠘⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀
⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⡀⠀⠀⠀⠀⡀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢬⣀⠀⠀⠀⢀⣩⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣦⡶⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣀⠀⠀⠀⠀⡀⠲⢤⡀⠀⠀⠀⣠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⡾⠉⠀⠀⠸⠂⠹⡗⢦⡙⢦⠀⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿⠀⠹⡆⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢀⡼⠃⣸⠀⣿⠘⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⢦⣄⠀⠀⠀⠀⢠⠞⣁⡴⢋⣠⠌⠁⢠⢟⡄⠀⠀⠐⢾⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⠽⢗⣢⡤⠹⣇⠀⠀⠀⡞⡼⢋⡴⠋⢀⣠⣴⣫⠎⣠⡄⠀⠀⠘
⠀⠀⠀⠀⣀⣠⣴⣶⣻⠿⠗⡋⠩⠖⠛⢩⣴⣾⣷⠿⠶⣤⡀⢷⣧⣟⡴⠚⡉⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠾⠂
⢀⣴⢶⣿⡿⠟⣛⡭⠶⣚⣯⣷⣾⣿⠶⠛⠋⠁⠀⠙⠳⣌⢷⡀⠘⠟⢠⠞⣡⠖⢋⣉⣓⠢⠀⠀⠀⠀⣠⠤⠤⣄
⣿⠁⣬⠴⣚⣭⣵⣾⣿⠷⠟⠋⠁⠀⠀⠀⣀⠀⠀⠀⠀⢹⡜⣧⠀⠀⡏⠀⢡⢞⣩⣤⡈⠳⣄⠀⢠⢎⡴⢒⠒⣦
⢹⣷⣤⣬⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠐⠻⠏⠀⠀⠀⠈⠃⢹⡆⠀⠁⠀⣿⣯⠤⢤⡀⢤⡹⣄⣡⢞⡅⡴⠒⠾⠀⢀⡀
⠀⢿⡦⠤⢿⡆⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣌⢿⡀⠀⠀⠛⡥⠦⡄⢻⡄⠀⠤⠤⠀⣸⣣⠚⠁⠀⠙⠏
⠀⠘⣿⠛⠋⢿⡀⠀⠀⠀⠀⡈⠃⠀⠀⠀⣠⢦⠀⡆⢸⣀⡄⢹⡜⣧⠀⠀⠈⠀⠀⠘⢄⠳⣄⣀⣠⠴⢣⠇⠀⡀
⠀⠀⢹⡆⠀⠸⣧⠀⠀⢾⡄⣿⠀⣼⢻⠀⢯⣾⡟⠙⠋⠉⠀⠀⢷⢻⡆⠀⠀⠀⢤⣶⡶⠦⣄⢠⡀⠐⠋⠀⢀⡿⣤⡄
⠀⠀⠀⢿⡀⠀⢹⡆⠀⢸⣧⣿⣤⠟⠛⠛⠀⣟⡇⠀⠀⠀⠀⠀⠘⣏⣷⠀⠀⠓⡄⠙⣿⣦⠈⣧⠹⡄⠀⠀⠉⠙⡇⠀⠀⠀⢿
⠀⠀⠀⠘⣧⠀⠀⣿⡘⠛⠀⠀⠀⠀⠀⠀⠀⠉⣁⡀⣧⡄⠀⠀⠀⠉⠸⣇⠀⠀⢹⠀⡿⠸⠀⢸⡆⢇⡀
⠀⠀⠀⠀⢻⣟⣫⡽⣧⠀⠀⠀⢰⣆⡀⢰⠶⣾⣇⣹⠻⠛⠆⢀⣄⠀⣦⢻⡄⠀⠉⠀⡇⠐⣇⠀⠳⠤⠵⢿⡆
⠀⠀⠀⠀⠈⣿⡀⠀⢻⡆⠀⠀⠀⣿⣹⠞⠷⠋⠈⠁⠀⠀⠀⠈⠉⠀⣸⠈⣷⠀⠀⠀⠻⣄⠈⠁⠐⠃⣠⠞
⠀⠀⠀⠀⠀⠸⣧⠀⠈⣷⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠴⠋⣀⡾⠃⠀⡀⠀⢮⣙⡓⠚⢋⣡⠞
⠀⠀⠀⠀⠀⠀⢻⡆⠀⢸⣧⠀⠀⠐⣤⠄⠀⠀⢀⣠⡴⠖⣋⣭⡴⠾⠛⠉⠀⠀⣀⣿⣀⡀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣯⢿⡄⠀⠈⠉⠃⠀⣀⣩⡴⠶⠛⠉⠁⠀⠀⠀⠀⠀⠈⠉⢿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠈⣷⣀⣠⣴⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⢀⣤⠖⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡆⠀⠀⣤
⠀⠀⠀⠀⢠⣿⡇⠐⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠿⡿⠃⠀⠀⠀⠀⢀⢠⡀⡄⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠙⠶⠞⠃
⠀⠀⠀⠀⣾⣿⠓⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠾⠁
⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣷
⠀⠀⠀⠀⠛⠛⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠩⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⢀⠀⠀⠀⠀⠀⢈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣸⢀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠅
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠻⣦⣤⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢴⣶⡒⠶⣿⡙⠻⣿⣍⠉⠻⣿⡙⢿⡖⢶⡶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠺⢿⣭⣉⡛⠷⢮⣝⡲⢬⣙⠳⢤⡉⠳⢤⡈⠳⣄⠹⡌⢧⢹⢸⢇⡟⣹⠖⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⢯⣛⡿⠷⠶⢤⠤⠭⡙⠳⠦⣬⣙⠳⢮⣝⡲⠬⡓⠤⣙⠢⠈⠃⠈⠀⠈⠈⠈⠐⠡⢞⡡⠞⣋⣷⢤⡀
⠀⠀⠀⠀⢀⣴⣿⠿⠀⠀⠈⢲⡀⠀⣰⠃⠀⣨⠉⠓⣤⣌⣙⣒⠦⢍⡓⠤⢁⠀⠁⠀⠀⢠⠞⠉⠉⠉⠳⡄⠀⠐⠈⠥⠴⠶⢛⣳⣄
⠀⠀⢀⣴⣫⠔⠢⣅⣠⡞⠉⠳⢄⡀⠉⠀⠠⠉⠁⢀⣼⠷⢻⣿⣷⢶⡌⠉⠂⠀⢀⡴⢲⡇⠀⠀⠀⠀⠀⢸⣀⣠⠞⠛⣯⡩⠭⠭⠭⣷⣄
⠀⢠⣾⣿⣶⣾⣿⣿⣿⣷⣄⠰⠞⣉⡦⠴⠞⣳⠖⠋⠁⠀⠘⣿⡏⠀⠈⠓⣲⡯⣍⡀⣨⢷⠴⠚⠓⢦⣤⣾⣾⡃⠀⣼⡿⠻⢏⣹⢷⠚⠙⢦
⢠⣿⣿⣿⣿⣿⣇⣤⡤⣉⡛⠳⢿⡀⣀⣴⣞⠁⢀⠀⠀⠀⣩⣿⡿⠀⢀⡟⠙⠲⡀⠉⠳⢤⣴⡶⠾⡿⠟⡻⠉⠀⠀⣘⣷⠀⠀⢙⣷⣯⣴⣾⣇
⣼⣿⣿⣿⣿⣷⠟⣽⣻⣯⣿⣷⣶⡉⠛⢿⣿⣿⣶⡄⠀⣴⣿⠓⠀⣠⣿⣤⣤⣠⠞⢀⣀⢤⣬⣻⣮⣄⠈⠀⠀⢀⣠⣿⣿⣤⣾⣿⡿⣿⣟⣿⣿
⣿⣿⡟⣷⡿⣿⢻⡼⣧⣷⣿⣶⣿⣿⣶⣄⣌⡙⠻⢿⣼⣧⣤⣤⣾⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⢬⣿⣿⢿⡟⣿⣾⣿⣿⣻⣯⣿⣿
⣿⣿⢿⣿⣷⣞⣚⣿⣿⣭⣿⣿⣿⢻⣿⣶⡬⠝⠳⣤⣈⣙⠻⢿⣿⣿⣿⣟⡷⠾⠿⠛⢛⣋⣭⣭⣷⣶⣾⣷⣿⣿⠿⣾⣝⣗⣯⣷⣿⣿⢾⣿⣿
⢹⣿⣾⣿⣷⣮⣿⣧⣿⣭⣻⣏⡽⣻⣿⣿⣿⣷⣄⡙⠻⣿⡷⢯⣤⡿⠉⣿⣀⡀⠝⠻⠿⠷⠶⠯⢿⣧⢾⣛⣿⣶⡿⣿⣿⣿⣷⣿⣿⣹⣿⣿⣿
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡟⣿⣆⣤⣍⣳⣯⡳⣄⣈⠙⠛⠻⠶⣬⣵⣾⣿⠿⠷⠿⠯⠿⢶⣿⣟⣿⣞⢻⡿⣽⣿⣿⣿⠃
⠀⠀⠹⣿⡽⢿⣿⣿⣾⣿⣿⣿⣿⣿⣯⣗⣺⠭⣿⡷⣷⣿⣾⣳⣾⣿⣦⣽⡷⠀⠀⠀⠈⠹⣿⣿⣿⣯⠭⠽⢿⣟⣒⣶⣭⣝⣛⣿⣿⣿⡿⠁
⠀⠀⠀⠈⠻⣷⣾⣷⣺⣭⣿⣿⣿⣿⣭⢟⡻⢿⡲⢭⣓⢪⣽⣿⡿⠟⠋⠁⠀⠀⠀⡀⠀⠀⢸⣿⣛⣛⣯⣭⡿⢾⣛⣛⣿⣭⣭⣭⡽⠋
⠀⠀⠀⠀⠀⠈⠛⢿⣞⣛⡻⣍⣯⣭⣝⣿⣮⡓⣭⣷⡾⠟⠋⠁⠀⠀⠤⠀⠀⠀⡀⠀⠀⢀⣼⡿⣟⣻⠽⢗⣛⣭⠭⠴⣶⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣿⣽⣹⣯⣷⡿⣿⣭⠁⠄⠀⠀⠀⠒⡤⠀⠁⠀⠴⠀⣠⡶⣛⠽⣛⣭⠶⣚⡭⢵⣲⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣯⣶⣶⣖⢐⣠⣶⠖⠔⠐⠀⢢⣾⣷⣻⣵⣟⣯⣶⠿⠗⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠤⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⣤⢤⡔⠀⠀⠀⢠⣶⣶⣿⣦⠀⠠⠤⠀⠚⠂⠂⢀⠀⠀⠀⠀⢀⡠⠴⠧⠜⠀⠂⠘⠁⠀⠀⠃⢤⠀⣀
⠀⠀⠀⠀⡶⠀⠀⠀⠀⢸⣷⣄⡻⡿⠦⠠⠀⠈⠊⠁⠀⠀⠀⠀⡜⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠉
⠀⠀⢀⡐⠁⠀⠀⠀⠀⠚⠛⠻⠷⠀⠀⠠⠄⠨⣡⠄⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠸⠀⡀
⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠰⠟⠁⠀⠐⠖⠀⠀⠀⠀⠀⠀⠀⠖⢢⡀
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⠇⠀⠘⠀⠀⢠⡏⠁⢀⣦⣀⣠⣾⣿⡄⠀⠀⠀⠀⠄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠀⣾⣿⣶⠾⣯⡟⠛⠏⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠄⠉⠀⠀⠀⠀⠀⢹⡏⠁⠀⠿⣀⠀⠀⠘⠁
⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⠀⠐⠁⠀⠀⠀⠀⡀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⡀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠃⠆⠀⠚⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⣠⣀⣀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠠
⠀⠀⠃⠀⠀⠢⠀⢀⠀⠀⠀⠀⢀⠀⠀⢀⠐⠶⠚⢿⣿⣷⣾⡿⠟⠶⠶⢀⠀⠀⠀⠀⠀⠀⠀⢐⠒⠠⠒⠉⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠀⠈⠋⠀⠀⠉⠈

⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢿⣿
⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⠿⠿⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢾⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠉⠉⠉


⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⡿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⡿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣯⣽⣿⣿⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠉⠉⠉⠉


⣷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶
⣿⣷⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⣿⣿⣿⣿⣶⡷⢿⣾⣿⡿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣷⡾⠿⠿⠶⡿⠷⣶⡿⠷⢾⠿⠿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣈⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣼⢏⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⡙⢦⡀⡶⢖⠶⡄⣠⠞⣡⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿⣧⠙⢡⣾⡷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣄⣎⡻⢈⠸⢻⣇⡄⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠞⠋⠉⠀⢄⡨⠋⡄⢲⠀⡌⢫⣁⠀⠈⠉⠛⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠁⠀⠀⢀⣀⠄⠀⣅⠂⡡⢬⢄⠁⢈⡂⠀⢄⣀⠀⠀⠀⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⣠⠟⠁⠀⢸⠄⣸⡄⢛⢠⣧⠁⡆⠀⠀⠙⢧⡀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⣏⠺⢵⢿⣇⠕⢫⠇⠀⠀⠀⠈⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⠀⠀⠀⠀⠀⠘⢦⡀⠉⢄⡼⠋⠀⠀⠀⠀⠀⢰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠇⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣟⣿⣻⣻⣛⣟⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣻⣿⣟⣻⣟⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢤⡆⠀⠀⠀⠀⠀⠀⢰⠧⡀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⠀⢰⣌⠍
⠀⠀⠀⠀⠀⣠⡀⢀⠠⠤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠉⠀⠀⡀⠀⠀⠀⢠⠋⠈⠁⠈⠇⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⢀⣤⣬⣿⣄⣸⣧⠘⣇
⠀⠀⠀⠀⠀⠉⠁⠄⠠⠤⠸⢀⠀⠀⠀⠀⠀⠠⡁⢀⣁⣀⣤⡟⠀⠀⠀⠈⠀⢠⣤⢀⠇⠀⠀⠀⠈⠁⠚⠀⠁⠀⡀⠀⠀⠀⢸⣏⠹⣿⡛⠻⣿⡟⠋
⠀⠀⠀⠀⠀⠀⢀⡠⠀⣻⡏⡀⠀⠁⠀⠀⠀⠀⠑⢮⣿⣿⣷⡇⠀⠀⢠⠁⣀⠀⠁⡘⠀⠀⠀⠁⠀⢸⡈⢠⠄⠏⠄⠀⠀⣤⣄⠙⢷⣯⣷⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠞⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠟⡅⠀⠀⠈⠢⠉⠀⠠⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⠁⠈⠳⠾⠏⠈⠉
⠀⠀⠀⠀⠀⠀⣠⠀⠀⠁⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠉⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡄⠂
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢠⠐⡌⠆⠷⠂⢸⠇⠖⣀⡀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⡀⠠⡀⠀⢰⣴⠆⢠⠄⠤⠈⠓
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣘⠦⠉⠀⠀⣶⣄⣀⡀⠀⠉⣁⡤⠀⠀⠀⠀⠤⠀⢄⠀⠀⢀⡤⠤⠤⡈⠀⠁⠀⠁⠘⡀⠁⠂⠀⢸⡇⡀⢠⣄⢀⠔⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢮⠅⠀⣰⢋⣇⠼⢤⠟⣲⠋⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠁⠠⠆⠀⠀⠠⠌⠓⠄⢸⣧⠀⠈⠘⠲⠈
⠀⠀⠀⠀⣠⡄⠀⠀⠀⢚⡃⠀⠛⡯⣤⡃⠀⢈⢎⢩⠀⠀⠤⣤⠀⠀⠀⠐⠒⠴⠀⠀⠀⠈⠀⠀⠠⢰⣆⠀⠉⠀⠒⠂⠐⠀⠀⣿⠀⣴⣾⣿⣿⡆⣀
⠀⠀⠀⠀⠈⢿⣄⠀⠀⠙⢉⠀⠀⠑⢤⢉⡕⠓⡬⣯⠀⢴⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⣤⡤⠤⠀⠚⣛
⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠙⢁⣄⡀⠸⠋⠉⠉⠀⠀⠳⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠘⠺⠟⠋⠉⠈⠁
⠀⠀⠰⢀⠀⢠⡀⠹⣧⡀⠀⠈⠋⠀⠆⡘⠀⣗⠈⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠒⠃⢁⡀⣄⡀⠀⠀⠀⠀⠀⡴⢆
⠀⠀⠈⠀⠀⡈⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠱⠍⢾⣿⢺⡀⠀⠀⠀⠀⠁⠀⠀⢀⢤⠐⠛⢊⠉⠀⠀⠀⠀⠀⣀⡬⠧⠉⠦
⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣤⣀⠀⠛⠸⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⠼⠲⡁⠋⡀⠀⢸⣯⠐⠒⠒⠒⠀⠄
⠀⢀⠀⠂⢰⠀⠀⠀⠀⠀⠀⠈⢳⣄⠄⠀⠀⠀⠀⠀⢸⣿⠏⢿⣿⠇⡀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠰⠀⠀⣠⡶⢶⣶⣟⡢⠀⣰
⠀⠈⠓⠒⠚⠀⠀⣠⣄⣀⠀⠀⠚⢀⡴⢤⡄⣠⠀⠀⣾⣿⣷⣿⡟⢁⠰⠃⠀⠀⠀⠀⠸⠀⠁⠀⠀⠀⠠⠁⠀⠻⠆⠀⠂⠁⣬⣷⣷⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣼⣿⡁⠀⠀⠀⠀⠀⢣⡜⠛⠛⣤⠀⠈⠁⠊⠝⣑⣠⠗⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠲⣿⣿⢿⣿⡿⡿⠿⠿⣿⢇⡤
⠀⠀⠀⠠⣄⢰⣿⣿⣷⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠅⢀⣀⠀⣓⢀⣀⡀⠨⠇
⠀⠀⠐⠂⠈⢘⡟⢁⡴⠀⠀⠀⠀⠀⠀⠶⠄⠀⠀⠈⣩⡄⠀⠀⠀⠀⠠⠭⠭⠭⠥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣿⣶⡏
⠀⠀⠀⣠⣤⣿⣿⣿⣷⣿⣿⣿⠿⠀⠀⠀⠀⠸⠞⠀⠘⠁⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠀⠃⠀⠀⠀⠈⣍⠉⠈⠉⠁⠉⠡
⠀⠀⠀⠈⠉⠈⠉⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠄⠐⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢾⣿⡶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣻⣶⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢋⡉⣸⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡰⢦⣥⣾⡇⠠⢀⠾⠟⠋⡤⢁⢴⠦⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡠⢤⣴⡿⠀⠈⢁⡠⢄⠀⠀⡵⢦⢸⣾⢢⣀⣀⡀⠀⢀⡀⢀⠀⠔⢀⠁⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡁⢹⣏⢀⣚⠿⠗⠂⢰⡍⠀⠺⡃⠀⡇⢈⣿⣻⣿⣿⣿⣶⠃⠈⢀⣀⡠⣠⣾⣿⣿⡀⠀⠀⢀⡔⡋⠙⣿⣿⣀⡿⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠈⠉⡁⠀⠀⠀⠘⣿⠶⢊⡇⠀⣇⣾⣷⣿⣿⡟⣹⣿⣴⣷⣿⣾⣿⣿⣿⣿⣿⠃⣠⡖⣡⢊⣴⠖⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠁⢙⡏⠀⠢⡀⠀⢹⣿⣼⣳⣆⣙⣋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⡁⠈⢿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣷⡄⢀⠤⠫⢀⣀⠉⣈⣶⢺⣦⣟⡩⢷⣷⣶⣬⣙⣿⣿⣿⣿⣿⣿⣿⡿⣛⠁⢾⣭⣶⣶⠏
⠀⠀⠀⠀⠀⠀⣠⡄⡸⡟⠁⠀⠀⠀⠉⢻⣟⣻⣾⣩⣟⣙⣼⣿⣧⣄⣽⣿⣎⢻⣿⣿⣿⡛⢚⡵⣣⡼⠁
⠀⠀⠀⠀⠀⢐⡁⠀⠟⠀⠀⠀⠀⢀⣠⣴⣿⣿⣏⣿⣿⣿⠿⣷⣿⣿⣿⣿⣿⣏⡏⣸⣿⣿⣿⣿⣯⢶⣖⡀
⠀⠀⠀⠀⣠⠈⢾⠗⠀⠀⠀⢀⡠⠾⢛⣿⣧⡟⣛⠛⢿⣿⣷⣿⡏⣿⣜⣿⣯⡿⢋⣌⣽⣿⡿⣿⡃⢿⣍⠿⠓⠄
⠀⢀⣤⣄⣈⠘⠃⠀⠀⠀⣴⡉⠀⠔⣲⣿⣿⣿⣿⣿⣶⣤⣈⠻⣧⣼⣿⣿⠿⡓⣼⢿⣿⡯⣥⠿⠓⠈⠛⠆⣾⣿⢷⡄
⠐⠩⠷⠋⠀⠀⠀⠀⣰⠊⠐⢹⡆⠀⠛⠗⣿⣿⣿⣿⣟⣿⣿⣦⢾⡿⠛⠧⢸⣿⣯⠉⠙⠮⢴⣦⠀⠀⠀⠀⠙⣿⡟⠁
⠈⠁⠀⠀⠀⠀⠀⣮⡀⠀⣴⠟⡠⠾⢿⣷⠈⠭⢾⣳⣿⣤⣽⣿⣿⣻⣿⣷⡌⠙⠛⠦⣄⠀⣤⢾⢀⠀⠀⠀⠀⠈⢣⣾⣣
⠀⠀⠀⠀⠀⣰⢧⢠⣽⠆⠁⢾⣷⠄⠈⠀⠀⠀⠀⣁⡙⢛⣿⠏⠁⠉⢿⡿⠁⠀⠀⠀⠀⢈⣷⣼⣄⡀⠀⠀⠀⠀⢸⠿⠟⠆
⠀⠀⠀⠀⣴⡡⠀⣸⠋⢀⢦⡄⠁⠀⠀⠀⠀⠀⠰⠛⠗⢼⣿⢫⣿⣶⠃⠀⠀⠀⠀⠀⠀⠀⠛⢀⠉⠇⠀⠀⠀⠀⠈⢤⣶⣶⢄
⠀⠀⠀⢠⠃⠁⢸⠃⠀⢈⡛⠀⠀⠀⠀⠀⠀⠀⠱⢴⡆⢨⡟⠈⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⣻⡏⢿⡆⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⡐⢲⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⠁⢸⢁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢹⢸⠃
⠀⠀⢠⠛⠓⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠾⠀⡼⣻⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠙⠺
⠀⠀⣠⡀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⠀⡿⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⢱⡄
⠀⢰⢿⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣱⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠛⠁
⠀⠰⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣈⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣵⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠋⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⡣
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣾⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⣙⣻⣉⣝⣛⣟⢋⣿⢻⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠐⢒⠺⠓⠢⠒⠘⠏⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⣀⡀⠀⠀⡰⣴⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠰⡤⠤⠀⠤⡴⠀⠀⢠⠂⠁⠀⢀⠔⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡛⠚⠒⠐⠢⠐⠚⠁⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠔⠛⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠠⠇⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠛⠀⠀⠷⠀⠀⠀⢀⣠⠖⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠃⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠂⢠⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢳⣶⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠠⡴⠿⠁⠙⠉⠁⠀⠀⠀⠩⠆⠀⠀⠀⠀⠒⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠙⠉⠃
⠀⠀⠀⠀⠀⠀⢐⠀⢓⠀⠀⠀⠀⠀⠘⠘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠜⡁⡐⠲⠂⠀⠀⠀⠈⠇⢀⣄⠀⠀⠀⠀⠀⠀⢰⡶⣶
⠀⠀⠀⠀⠀⠀⠈⢤⠀⠀⠀⠀⠀⠀⡀⣈⠙⠀⠀⢐⣖⣶⡞⠆⠀⠀⠀⠀⢤⠚⣁⡚⣠⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⢀⠌⣄⡀⠀⠈⠉⠈⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠯⣣⠅⠀⠀⠀⠀⠀⠀⠀⣩⠃
⠀⠀⠀⠀⠀⠈⠛⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⡴⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠰⠆⠀⠀⠀⠀⠉⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠄⢺⠠⠄⠀⠀⠀⠁⢀⣦⢤⣤⡤⣤⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠜⠀⠀⣠⠰⠿⠒⠛⠋⡉⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⢤⡘⡛⡀⠜⠁
⠀⠀⠀⠀⠤⠴⠦⣐⡀⠁⣠⠦⠤⢀⣀⡸⠂⠘⠀⠙⠤⠖⠀⠀⠀⠛⠋⠙⠁⠀⠀⠀⢀⡀⠀⣴⠀⠀⠀⠀⢀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢐⡀⠀⠀⠆⣄⠈⠀⣠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⡀⠀⠀⠀⠀⠙⣃⠅⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠐⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠈⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠑⡐⠋⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠈⠄⠀⠀⠀⠀⣤⢘⠃⠀⡀⠀⠀⠀⠀⠈⠁⠀⠂⠀⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡄⠀⠀⠀⠀⠠⣶⢈⠀⣠⢫⠄⠀⠀⠀⠀⠒⠤⢄⣀⠠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣷⡌⠰⠀⠀⠀⠀⠀⠲⢿⡆⣀⡌⠀⠀⠀⠀⠀⠀⠐⠘⠛⡀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠌⣁⢴⡂⠀⠀⠀⠀⠀⢭⣿⠋⠁⠀⠀⠀⠀⠀⢀⡤⠤⣤⠤⡀⠀⠀⠀⠀⠀⠀⠠⢴⣶⠄⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠋⠀⠁⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⣘⠉⠀⣚⠀⠛⣄⠀⠀⠀⠀⠀⠀⢨⡏⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡤⡀⠀⠠⠀⠀⢤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢀⢀⡆⠀⣈⢡⠀⡟⠀⠀⠀⠀⠀⠄⠀⠀⢀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠓⢄⠈⢀⣾⣶⣥⠐⠈⠀⠀⠂⠤⢡⡀⢸⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠅⠎⠁⠂⠈⠀⠀⢶⡟⠀⠀⠀⠘⡢⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠦⡘⡉⠶⢀⡀⠀⠀⠀⠀⠀⠣⡀⠀⠀⢀⡠⢋⡐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠁⠙⠳⠈⡄⠦⣄⡂⠀⠀⡀⠀⠀⠘⣒⣠⠤⠶⢫⣿⠀⠀⠀⡐⡩⠕⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⣤⠒⢺⠀⠀⠀⠀⠀⠀⠉⠉⠛⣉⣀⢠⠤⡶⢮⠻⡍⠀⠀⠀⢫⠂⠈⠀⠈⣡⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠰⠃⠘⠚⠃⠀⠀⣡⠄⣀⡥⠔⠒⠚⣍⠉⠁⠀⠀⠀⡆⠀⠀⡶⢠⠐⣆⣂⡄⠀⠈⢷⠙⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣀⢀⢴⣞⡁⠀⠀⠘⠅⠙⠀⣰⣀⢀⣰⣁⣦⠚⡗⢃⠏⡘⠉⢀⢀⠠⢊⣴⡂⠩⠄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢮⠿⠾⢣⡎⢎⠫⣉⠇⣤⡀⠆⣠⣏⡔⠯⠈⣯⡉⠁⣌⢉⠎⢊⡕⣀⠷⣉⣴⠋⣁⡔⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡁⠈⠢⢿⡿⠿⠃⠀⡇⠡⠈⠻⠀⢯⡇⢀⠹⠎⣛⡫⠥⠖⠻⢝⢻⢖⠽⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣾⡓⠄⠀⠀⠀⢀⠄⠀⢉⢰⣀⠎⠀⣚⣤⠶⠚⠛⠁⠀⠈⠀⠀⠈⠁⠁⡀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡐⡾⣽⣶⡀⠀⠀⠐⡄⠀⡸⣡⠴⠟⠩⢅⠀⠆⣀⠀⠀⠀⠀⠀⢀⣀⡼⢴⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠇⠀⠒⢃⡺⢋⠞⠁⠀⠀⠣⢀⠉⠀⠀⡁⠻⣤⡤⠖⠋⣁⣠⣾⠟⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣆⠠⣀⡀⠠⠀⢈⡐⠻⢮⠵⠶⠀⠀⠀⠐⣷⡴⠶⣾⡷⢿⣿⣾⡿⠿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢪⠜⠂⢭⢁⢃⡎⠀⠀⠁⢄⣀⡀⠄⠀⣈⠥⠒⣋⣽⡾⠟⠋⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠛⠘⣣⠀⠀⢠⡿⠉⠀⢀⣀⣤⡶⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⢤⣿⣧⣶⠶⣿⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠐⠃

⠀⠀⠀⠃⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⣶⣶⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⢸⣣⣼⣦⣿⠀⠀⠀⠀⠀⠀⠀⠈⠀⠿⡿⢿⠟⠿⠆
⠀⠀⠀⠀⠀⢠⣶⣿⠞⢹⢻⣿⡦⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⣠
⡀⠀⠀⠀⢀⣟⠛⠀⠀⢰⠀⠋⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⢂
⢿⠀⠀⠀⠾⣿⡆⠂⣄⣸⠀⠈⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠓
⣿⡆⠀⠀⢸⣿⣃⠀⠙⠛⠀⠀⣽⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣡⣴⠀⠀⠀⠀⠀⣼⣾⣷⣆⢠
⢿⠃⠀⠀⠀⠼⢿⣶⣶⣖⣦⣼⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠸⣿⠟
⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣶⣶⣶⢰⣶⣶⠎
⠀⠀⣀⡀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠃⠀⢻⡟⠀⠀⠀⠀⠀⠀⠠⣄⠠⣶⠖⣀⠂
⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁⠈⢣⣶⣦
⠀⠀⠀⠀⠠⠶⠄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⠀⠠⣽⢹⣿
⠀⠀⠀⠀⠀⠀⠄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣿⡮⣭
⠀⠀⠀⠀⢀⠀⡀⠀⠄⠇⠀⠀⠀⠀⣀⣀⠔⠒⣒⠒⠠⣄⣄⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠛⠋
⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠈⠙⢏⢀⠀⢐⠀⠀⢸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢰⣿⡷⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⣿⣷⣾⡧
⠀⠀⠀⠶⢤⣤⡤⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠛⠛⠛⣿⣷⣿⠆
⠀⠀⠀⠀⠸⢿⡿⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⣰⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⡏⠀⠀⠀⣿⣿⣧⠄
⠀⠀⠀⡖⢢⣤⣴⠐⡇⠀⠀⠀⠀⠀⠀⠀⠓⠒⠉⠉⠖⠚⠀⠀⠀⠀⠀⢨⣿⣿⡇⠀⣶⠄⢿⣿⣷⡄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠤⠤⠤⠤⠴⠶⠶⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⣛⣉⣩⣤⠤⠤⠤⠶⠖⠒⠒⠒⠒⠆⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⡇⠀⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⠈⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⣧⠀⠀⠀⠀⣀⠤⠖⢒⢾⡓⠒⠢⢄⡀⠀⠀⢷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠏⠀⠀⢠⠞⠁⠀⠀⡜⠀⠳⡀⠀⠀⠙⢦⠀⢸⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢷⠀⠀⢰⠃⠀⠀⠀⣸⣁⣀⡠⠵⡴⠒⣒⠞⢣⠸⡆⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⡿⠾⢉⣉⢹⠃⠀⠀⠀⣀⠼⣎⠀⠀⠘⡇⡇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⠀⠀⢧⠀⠀⢨⠏⠒⢦⣤⣊⠁⠀⠈⢆⠀⢠⠇⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡆⠀⠘⣆⢀⡎⡠⠞⠁⠀⠈⠉⠒⠤⣈⣦⡞⠀⢿⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⡇⠀⠀⠈⠺⣋⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢰⠀⡇⠀⠀⠀⠀⠈⠑⠲⠤⠤⠤⠴⠚⠉⠀⠀⠀⠀⢸⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠗⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢰⠎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⠁⢸⠀⠀⠰⠒⠚⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣼⡟⠉⠭⠭⠭⠭⠭⠍⢉⣉⣉⣉⣉⣉⠉⠉⢉⡉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⣀⡀⠈⠙⡍⠉⢏⠩⠥⠤⠴⠶⠒⠒⠒⣂⣀⡉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣇⣀⣀⣈⣉⣣⠀⢸⣉⣉⣉⣉⣉⣉⣭⣭⡥⠤⠤⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣤⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠛


⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠲⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣧⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⣠⣷⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣦⣆⣾⡄⠀⠀⠀⠀⠀⠀⠀⢠⡀⢶⣿⣷⣠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣟⠛⠛⠃⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣓⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠉⣻⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠒⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣾⣿⣤⣀⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣾⣿⣿⣿⣿⡟⠃⠀⠀⠶⠒⠀⠘⣿⠉⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣟⣼⠋⠈⠓⠀⠀⣦⣄⠀⢰⣦⡼⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠿⠿⢿⣿⡀⠉⠧⡀⠀⠀⠀⠘⠆⠀⠀⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠶⠖⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠐⢦⡀⠘⠆⡀⡆⠑
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡎⠁⠀⣀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿⣦⣤⣿⠀⡄⠀⠀⠀⠄⠈⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⡆⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣄⠘⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⢠⠞⠋⣧⣴⠧⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⣦⣀⢻⣦⠀⠀⠂⢀⣤⠖⠉⠀⠀⠀⠀⣤⠈⢹⠁⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⢿⣷⣾⠾⠋⠁⠀⠀⠀⠀⣠⡀⠀⠀⣸⡄⠀⠀⠀⠀⢀⠰⣿⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣶⣿⣷⣤⠀⠖⡀⣿⣷⠞⠀⠀⢢⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⠀⠀⠐⠼⠿⣯⡡⠖⠂⠉⠀⠀⣀⣀⠈⢀⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠁⠀⠀⠀⠀⠀⢀⡀⢠⠐⠂⠈⣿⣿⣿⣧⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠃⠀⠀⠠⠆⠀⠀⠆⠀⠀⠀⡀⠀⠁⢠⠄⣀⣴⣿⣿⣇⠀⠀⠀⠀⣿⡟⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠟⢠⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣧⣤⣾⣷⣾⣿⣿⣿⣿⢿⡆⠀⠀⢀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣰⡞⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⢻⣿⣿⣿⡾⢿⣿⣿⣷⠀⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⢸⡇⠀⠀⠘⠆⠀⠀⠀⠀⢠⠀⠀⠹⣿⣿⣷⠸⣿⣿⣿⡆⢰⡄⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣸⡿⢸⠇⠀⠀⠀⠐⠀⠀⠀⠀⠸⠀⠀⠀⠈⠿⣿⠃⠘⣿⣿⣿⣾⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⣧⣾⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠐⠘⠃⠀⠘⢻⣿⠘⠃⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⠀⠀⠀⠀⡸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⡗⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣾⣿⣷⣿⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⢻⡏⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢹⣿⣼⠇⠀⠀⠀⡟⠀⠀⠀⣠⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡏⢸⡀⠀⠀⢸⠁⠀⠀⢠⣿⠇⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢼⠄
⠀⠀⠀⠀⠀⠤⠀⠀⠘⣿⡇⢸⣧⣼⡇⠀⠀⢸⢳⣶⡄⣼⣿⣆⠀⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢸⣿⢿⡇⠀⢠⡇⢸⣿⢀⣿⣿⡍⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧
⠀⠀⠀⠀⠀⠐⡀⢀⢰⣿⠀⣿⣿⣿⡇⠀⢸⡇⣼⣿⣾⣿⣟⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠥⡤⠈⣙⠋⢸⡟⠀⣿⢿⣿⡃⠀⢸⣿⣿⣿⣿⣿⡿⠃⢸⣿⠁⣀⠀⠀⣀⣀⣀⣠⣤⢸⠀⢠⠀⠀⠀⠠⠔
⠀⠀⠀⠀⠒⠛⣭⡁⣾⡇⢸⣿⢠⣿⡇⠀⠀⠘⣿⣿⣿⣿⣧⣤⣾⣧⣀⣀⣤⣤⣿⣿⣿⣿⣿⡄⠀⠈⠀⠀⠂⠐
⠀⠀⠀⢰⠒⠛⣾⠛⣿⡇⠈⣿⠘⣿⡇⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠘⠀⠡⠈⡀⣀
⠀⠀⠀⢠⣿⣿⣾⣿⣿⣇⠀⣿⠀⣿⣇⣀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⡟⢠⣴⠆⠐⠒⠀⠒⠘⠁⠡⠌⠐⢣
⠀⠀⢳⣾⡿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡻⡿⠿⠿⣿⣷⣬⢻⣀⠀⠂⠀⠀⠀⠀⠁⠀⠒
⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⡢⠀⠀⠉⢋⡘⠻⠿⠓
⠀⠀⠀⠀⠘⠛⠛⠛⠻⠟⠿⠿⠿⠿⠿⠿⠿⠟⠛⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠿⢷⣴⠀⣾⠟

⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢲⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠈⢄⡀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄⠀⠀⠀⡻⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡄⠁⠀⠁⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠤⠤⣾⣿⣟⣶⣃⠀⠊⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠠
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠆⠑⣤⡟⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡿⢿⣿⠁⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⣤⡏⠀⣼⠇⠀⠀⠀⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠁⢸⣦⠀⠀⠀⣀⠴⠃⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⣀⡄⠘
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠘⠋⠀⣴⣿⠃⠀⠀⠀⠁⠀⠀⠀⠘⡿⢽⡢⢄⡀⠀⣀⡆⠸⣿⣷⣷⣼⠋⠀⠀⠀⠀⡄⠀⠀⢣⠀⠸⠇⠤⣿
⠀⠀⠀⠀⠀⠀⠀⡻⣦⡀⠘⣶⠋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⢹⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠁⠀⠀⠘
⠀⠀⠀⠀⠀⣎⢰⡇⢸⣿⡆⢹⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠁⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⢸⣯⠀⡘⠉⣷⡀⢹⡄⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⡀⠄
⠀⠀⠀⠀⠀⠈⠁⠘⠁⠈⠙⠳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠀⠀⠀⠀⡀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⡀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣰⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⠶⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⣷⣴⡿⠇⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⠒⠶⠶⠶⠶⠾⠑⠛⠁⠀⠀⠀⠠⠒⠛⠹⠉⠃⠁⠙⠁⠀⠀⢢⠄⠋⠉
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣹⣄⠀⠼⣿⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣰⠀⠀⠀⠀⡀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⢸⡦⣤⠀⢠⣤⠿⠛⢿⠀⢹⠀⡁⠀⠀⠀⢀⡌⣠⣦⠀⠂⢸⠤⠀⡀⣴⠆⢀⡴⠀⠀⠀⠀⢠⣿⠀⣰⠟⠦⠀⠀⠔⠃
⠀⠀⠀⢠⠊⢀⡾⠀⠈⠀⠀⠀⠀⠀⠈⣆⢘⣴⠃⠀⠀⢀⣼⣴⠋⡰⠃⠀⠀⠀⠀⠀⠀⠀⠎⠇⠀⠀⠀⠀⠘⠁⣼⠁⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⣼⣠⡏⣇⠀⡂⠀⠀⠀⠀⠀⠀⢻⣸⡟⠀⠀⠀⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠸⣁
⠀⠀⠀⢽⣿⡄⠛⠛⠛⠁⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠐⢀⣠⡄⠤⠴⠤⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠔⠁⠀⣠⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠈⠀⠂⠂
⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣂⢄⡀⡀⠀⢄⡀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣰⣾⣍⣹⣿⣿⣆⣱⡈⣦⠀⢉⣳⣤⠀⠀⠄⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠢⢤⠄⠂⢠⣾⣿⣵⠿⠻⣿⣿⣿⠛⣹⣽⣿⣿⣛⣿⣏⣽⣉⡀⢠⣶⢿⣿⠛⢦⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡴⠟⠉⠁⠀⠀⠀⠀⢰⢏⣽⣿⡿⠹⣿⣍⡀⣀⡀⣸⣟⠉⠁⣸⠂⠀⠀⠈⢣⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⣺⣿⠏⠀⠀⠈⠙⠙⠋⠙⢏⢻⣆⠀⠁⠀⠀⠀⠀⠀⠙⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢆⢀⠔⠈⢁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠸⠃⠀⠃⠀⠀⢸⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠢⢶⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠞⠦⣦⡀⠠⣤⠀⣾⡁⣉⣥⠐⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄⠘⠂⡀⠐⠈⡽⢍⠀⠀⠀⠀⠀⠀⡦⠘⠇⠐⠀⠀⠀⠰⠟⠁⠀⠸⠀⠀⢰⠉⠦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢀⠀⠀⠂⠀⠙⡻⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡶⢀⠁⠀⠀⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⡊⡀⠀⠀⠁⠂⢀⠄⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠠⠄⡶⠚⠊⠁⢸⡠⠋⠀⢛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⢘⠁⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⡀⠛⠢⠄⠈⠀⠤⢀⠁⠀⠈⢄⡀⠀⠀⠈⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡦⢠⠤⠈⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠘⠂⠀⠀⠀⠀⣄⠎⠉⠉⠁⠀⠀⠀⠀⠀⠀⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢤⡀⠀⠀⠀⠀⠉⠀⠀⠀⢀⡰⠳⢤⠄⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠻⠀⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡰⠁⠀⠀⠈⠉⠓⠦⣄⠀⣰⠁⠀⠀⠀⠀⠚⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⡰⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠇⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⡛⢄⣀⣀⠀⠀⠒⢭⡀⠁⠀⠈⠹⠂
⠀⠀⠀⠀⠀⠀⠀⢠⠎⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠰⠆⡀⠀⠀⢰⠋⠀⠀⠀⣾⡤⠒⠂⠶
⠀⠀⠀⠀⠀⠀⠀⡎⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⣄⡤⠂⠊⠇⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⡼⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢱⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⡰⢡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠐⠒⠒⢒⠃⠘⡇⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⢰⠃⠐⢀⠶⠒⠒⠠⠤⠤⣄⣀⠀⠀⢠⡁⠀⠁⢀⣀⣀⠠⠤⠤⠀⠈⠄⢳⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⢀⡏⠀⣰⣁⠤⠀⠀⠀⠀⠀⠀⣈⣹⣦⢿⠄⠠⠭⠤⠤⠤⠀⠀⣀⣐⠒⠥⢼⠀⠀⠀⢀⡠⠊
⠀⠀⠀⢸⡀⠀⠘⠓⠂⠂⠀⢀⣀⣀⣀⣀⢠⠀⠀⡕⢀⣒⣓⣈⣉⡡⠤⠤⠖⠒⠒⠛⠒⠐⠈⠁
⠀⠀⠀⠀⠓⠶⠒⠛⠓⠒⠒⠒⠒⠒⠉⠙⠦⣈⣉⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠀⠠⠄⠀⠀⠀⢤⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⣠⠾⠛⠃⠄⣀
⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡴⠟⠉⠀⢀⣾⣿⣿⣿⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⣿⣴⣷⣧⢀⢠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠰⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠉⠉⠃
⣿⣿⣿⣿⣿⣿⣷⣿⣠⡀⣦⣷⢠⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠰⠶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣼⣧⣧⣼⣰⣇⢠⡀⣿⡆⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡆⣠⢸⣷⣿⣦
⣿⣿⣿⣿⢿⢿⡿⠿⠛⠛⠟⠿⠛⠋⠛⠉⠉⠋⠉⠉⠋⠈⠉⠛⠁⠀⠋⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⡀⣸⣧⡆⢠⢀⣿⣾⣷⣿⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⣾⡧⢀⡀⣸⡇⣰⣠⣧⣴⣰⣿⣰⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⢸⣿⣿⣾⣧⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣽⣗⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣸⣷⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡿⠻⣿⣿⢿⡿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠃⠀⠀⠈⠁⠈⠀⠁⠈⠈⠉⠈⠉⠙⠛⠋⠉⠉⠛⠻⠟⠿⢻⡿⣿⠿
⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⢠
⣿⠃⢹⣿⣿⣿⣿⣿⡃⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣸⡇
⡟⠀⢐⣻⣿⣿⣿⣿⣷⣼⣿⣧⡀⢀⣦⠀⠀⠀⢰⣿⠄⠀⠀⠀⠀⠀⢸⣿⣯
⠉⠀⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣸⣿⡄⠀⢰⣾⣿⡇⠀⢰⡆⠀⠀⣿⣿⣿⡁⠀⠀⣾⠄⠀⠀⠀⢰⡧
⠂⠀⠴⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢺⣿⣿⣶⣤⣾⣿⠀⣘⣿⣿⣿⣶⠆⢸⣿⡇⠀⠀⠀⣿⣟
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣷⣘⣿⣿⣿⡟⢻⣿⣿⣦⣽⣿⣿⣿⣿⡥⣿⡿⠿⠀⠀⣽⣿⣿⠇
⠀⠀⠀⠀⣰⣿⣿⣿⣿⠃⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⣶⣤⣿⣿⣿⣷⡤⠀⠀⢰⡇
⠀⠀⠀⣤⣽⣿⣿⣿⡧⠀⠀⢽⣿⣿⣿⡿⢻⣿⣿⠦⠀⠠⢽⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠠⢽⣿⣿⣿⣿⣿⣧⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠤⠼⣿⡟⠓⠀⠀⠙⢿⣿⣿⡃⠀⣿⡟⠃⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⠀⠙⢿⣿⣿⣿⣿⣷⡖⣸⣿⣿⡅⠀⢀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠒⠚⣉⠁⠀⠀⢤⣿⣿⡟⠀⠀⠹⡏⠁⠀⠀⣤⣿⣿⡿⠁⢿⣿⣿⣏⠁⠀⠀⢤⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢁⣾⣇⠀⣸⣿⣟
⠀⠀⠀⠀⠠⠴⠒⠀⠀⠀⠀⠐⠚⠛⠂⠀⠀⠀⠀⠀⠀⠀⠐⠚⠻⠃⠀⠘⣿⣟⠀⠀⠀⠀⠐⢺⠻⣿⣿⠁⢻⣿⣿⡟⢻⣿⣽⣿⣧⣠⣿⣿⣿⣃⣾⠀⠀⣴
⠀⠀⠀⠀⠤⠦⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⢒⣿⠁⠀⢈⣿⣿⠀⠈⣿⠙⣿⣿⣿⠇⢸⣿⣿⣿⡆⢰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠉⣉⠅⠀⠀⠀⠙⠀⠀⠀⠋⠀⣿⣿⡿⠀⠈⣿⣿⣿⣷⣾⣿⣿

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

⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣹⢷⡦⢤⣄⡀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠉⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠃⢠⢹⠆⠀⢸⠏⠀⠀⠀⠀⠀⠐⠒⠂⠉⠉⠀⠀⣀⣠⢴⢾⠀⠀⠈⠳⠄⣀⣀⣀⣀⣀⡤⣤⡔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢰⣏⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠁⢠⠏⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢾⠈⠁⠀⠈⠀⠀⣀⡴⠖⠉⠉⠀⢀⣠⢴⠇⠀⠀⠀⠀⠢⢤⣀⣀⣀⡠⠊⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢸⡆⠀⠀⣻⡷⡚⠁⠀⠀⠀⠒⠉⡉⠠⠁⠀⠀⠀⠀⠀⠀⠀⣈⢝⡟⠀⠀⠀⠈
⠀⠀⠈⠛⡓⠦⢤⣤⣇⣀⣈⣷⣼⣿⣄⣴⠏⠀⠈⠾⡉⠉⠉⠉⢀⡄⠀⠀⠀⠐⠦⣄⡠⠔⠉⠀⠘⠁
⠀⠀⠀⠀⠘⣆⠀⣀⣙⣿⣿⣿⣿⣿⣿⣯⡤⢴⠶⠓⠛⢯⣙⣉⠉⣄⠀⢱⣄⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣠⠾⢛⣿⣿⣿⣿⣿⣿⣿⠉⠁⠘⣗⠲⡶⢧⣤⣯⣭⣶⣪⣍⣀⣄⠀⠀⠰⡄⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠋⠁⠀⠸⣿⣿⣿⣿⣿⡿⠙⣎⡆⠀⡸⠀⢈⠹⣽⠉⣤⢀⡁⢠⣜⠁⠈⠉⠉⠛⠛⠒⠒⠒⠒⠬⠷⠤⣄⣀⣀⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⢸⠃⠀⣸⣷⡀⠀⠀⠈⣷⣁⠄⠈⠋⠀⠀⠀⠀⣠⣼⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⣾⠐⠁⠀⠀⠉⠀⠀⠀⠏⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⢀⡼⠁⣀⡤⠒⠉⣿⠉⠢⠀⠀⠀⠀⠘⣦⣠⠄⠀⠀⢈⡾⠁⠀⠀⠀⠀⠀⠀⠠⠤⣾⣇
⠀⠀⠀⠶⠿⠾⠯⠭⣄⠀⠁⡗⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠒⢤⣀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠇⠀⠀⠀⠘⡄⠀⠀⢀⠠⠤⠤⠤⣄⡀⣀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣌⡓⢄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠋⣰⠆⠀⠀⠀⠀⠀⠈⠙⢿⠃⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢀⣞⣡⠀⠀⠀⠀⠀⠀⠀⣀⠈⢧⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⡿⣿⣿⣷⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠹
⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣠⣿⣿⡟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⡀
⣼⠟⢤⠀⠀⠀⠀⠀⣀⣀⣠⣴⣽⣿⣿⣿⠟⠻⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⡟⢠⣯⡄⠀⠀⠘⠛⠻⠿⠿⣧⣀⣿⣿⣦⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡿⢿⢿⣄
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⣿⣿⣟⡉⢛⣉⡏⡀⣀⣸⣿⣿⡶⠿⠒
⠀⣾⡻⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⠃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣻⣿⣄⡄⠀⠀⠀⠀⠀⢸⡏⣿⣿⠟⢿⣿⠿⢿⣿⣿⣿⡀⢀⣠⣶⡀⣠⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣀⣄
⡟⠀⠙⠃⠀⠀⠀⠀⠀⢸⡿⢯⣿⣤⣀⣻⣆⣠⠿⣿⠟⣿⣯⣨⣿⣹⡟⣹⣿⣿⢿⣋⡀⣄⣽⢳⣻⡿⠁⠀⠀⠀⠀⣼⣇
⠅⠀⢰⡀⠀⠀⠀⠀⠀⠈⢄⣈⣭⣿⣿⣿⣣⡿⠷⣯⣟⣯⣅⠉⣻⣿⣾⣿⡿⠿⣫⡻⣿⠿⢻⣿⣼⣧⠀⠀⠀⠀⠀⠈⢿⣧⡀
⡟⠀⢺⡇⠀⠀⠀⠀⠀⠀⢘⣿⡿⡿⡼⡟⢶⣅⣄⣀⣿⣾⣿⣷⣿⣿⣿⣍⣴⣶⡿⠏⣥⣤⢿⣷⠖⢯⠀⠀⠀⠀⠀⠀⠈⢻⣷
⠁⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢗⡋⢱⣐⣯⣶⣽⣿⠿⠿⣻⣿⣿⣭⣔⡻⠿⢿⣾⣷⣚⣺⣛⣸⣗⡋⣻⡖⠀⠀⠀⠀⠀⠀⠘⢿
⠀⣿⣿⡇⢠⡄⠀⠀⠀⠾⣷⡿⠷⠛⢫⣽⠋⠡⡶⡖⠀⠛⠿⣿⣿⣿⣥⣤⡀⠈⠉⢻⣿⣿⣿⣿⣵⣾⣷⠄⠀⠀⠀⠀⠀⠀⠈
⢸⣿⡿⣰⣿⡇⠀⠀⠀⠈⠛⠿⣿⠿⣿⣯⣤⣤⡍⢵⣶⣿⣿⣿⣿⣿⣿⡿⠟⠙⣛⠋⠛⠋⢻⣾⡿⠟⠛
⣿⣿⣧⣿⣿⠃⠀⠀⠀⠀⢀⠀⠈⠙⢿⣿⣿⣻⣶⣿⣿⣿⣿⡿⠋⠁⠀⣠⣤⣤⣤⡼⣿⣿⡫⢋⣤⣀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠐⢿⠇⡀⠀⢸⣿⢿⣽⣿⣿⣿⣿⢿⣿⣶⡶⣾⠟⡓⣿⣫⣿⣿⠋⢀⣈⠛⠃⠀⠀⠀⠀⠀⠀⢰⣾⡇
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣀⡀⠉⠀⠈⣿⠐⠙⣿⣥⣿⣇⣦⢻⠻⠐⡿⣾⣿⣧⣼⢿⣿⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠁⠀⣈⠀⢿⣰⣿⣿⣿⣿⣿⣷⣼⠀⣰⣿⣿⣿⣿⠀⢘⣿⣤⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣴⣦⡉⠙⣿⣿⣿⢿⣿⣿⠀⣿⣿⡿⠛⠃⠀⢘⣿⠙⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⡏⢀⣶⣦⠀⠀⠀⠀⠈⠀⠀⠻⠟⠁⠀⣿⣿⣿⠀⢻⣿⣶⣿⠏⠀⠀⡀⠀⢸⣿⠀⢀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣝
⣿⣿⣿⣷⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣿⣿⣿⠀⢸⡟⠟⠡⣿⣿⡟⠀⠀⠘⠁⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠛⠿⠿⠀⠘⠇⠀⠀⠉⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢆⡁⠄⠤⠀⠀⠀⠀⠂⠀⠀⠀⠀⠙⠋⠀⠀⢀⣀⣤⣤⣴⣦⣾⡿⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⡶⣾⣿⣿⣿⣿⣶⣶⣋⣽⡿⣾⣿⢷⠀⢀⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⣻⣿⣿⣿⣶⡶⣞⣿⣿⣿⣿⣿⣯⡻⠿⣾⣉⡿⣿⣾⡆
⠀⠀⠀⠀⠀⠀⢀⣼⡿⣿⣧⣪⣾⣝⣻⢻⣿⣻⣿⣿⣿⣽⣿⣿⣿⣯⣿⣮⣻⣿⣿⣿⣿⡉⠂⠀⠘⡄
⠀⠀⠀⠀⠀⠀⣾⡋⠁⢿⣿⣿⣷⣾⣯⡟⣿⣧⡾⣿⢻⣯⢿⡼⣶⡻⣿⢿⣬⣻⣿⣿⣿⣧⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣿⡍⢀⣾⣿⣿⣿⣿⣯⣷⣟⢥⣽⣽⣞⣿⣾⣿⣯⢾⣿⣾⣿⣿⣻⡽⣻⣿⣆
⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣯⣿⣿⡷⣮⢳⣿⡻⣯⣵⣿⣾⡿⣿⣻⣿⣷⣷⢻⣽⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⢿⣿⣯⣿⣿⣼⣿⡷⢿⢿⣟⢿⣾⣇⣿⣷⡿⣾⣿⣿⣿⣇⢟⢹⣷⠀⢰⣡⠄
⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⢀⠖⢿⣿⣿⣿⣿⣿⣆⣾⣧⣿⣿⠟⡛⠙⠿⠷⠋⣻⣿⣿⣿⡼⣿⢿⣇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠰⠁⡘⠚⣿⣿⣧⣿⣿⣿⣿⣿⡏⡉⠀⠀⣀⣀⠠⠜⣿⣿⢿⣿⣷⢿⡽⣿⡇
⠀⠀⠀⠀⠀⠀⠦⠀⠈⢇⠰⢡⣤⡽⣿⣾⣿⣿⣿⣿⣿⣤⡛⠀⠋⣉⠀⡀⠀⣯⠛⢟⣿⣿⣼⣷⣻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢧⠠⠋⢿⣯⣞⣿⢮⣿⡟⢿⣇⠀⠀⠀⠀⣽⣝⢃⡾⣻⣿⣿⣇⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡐⣅⢸⠟⢿⣿⣿⣎⢿⣿⣦⣌⣚⡛⠲⣿⣷⣯⣭⣾⣿⣿⣿⣿⣿⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠘⡅⠠⠂⣿⣿⣿⣦⣛⠿⣿⣿⣷⣿⣷⣯⣻⣿⣿⣿⢟⣼⠿⣦⠡⢽⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠘⡀⠟⣸⣿⣿⣿⣻⣿⣮⣻⣿⣿⣿⣿⣿⣿⣿⡇⣿⡿⣶⣰⣇⣼⠋⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⢦⣠⣾⣿⣿⣿⣿⣯⣮⣭⡿⠿⢿⣿⣟⡿⢯⣷⡟⣯⣭⣿⣿⣿⣿⣴⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣿⣜⣿⠟⢿⣿⡌⠍⢿⣤⣿⣷⣬⣷⣿⣿⣿⣿⣿⣿⠿⡿⠟⠛⣻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢻⣿⠁⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣽⣿⣧⣾⣁⣐⣒⣽⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣋⣻⣿⡿⠿⠿⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠿⠿⣿⠿⢿⡏⡟⠋⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠘⣧⡄⠀⠄⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠞⣶⣢⠀⠀⠀⠀⠀⢰⣢⠔⠒⠒⠒⠒⠒⠲⢖⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢿⠀⠠⠌⠳⣄⠀⠀⠀⠀⢸⠀⠛⢠⣌⡆⠘⠀⠣⣤⠤⢤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⠈⢧⡀⠀⠀⣨⣦⡀⠀⠀⠈⢆⠀⠠⠛⠄⠀⠀⣀⠀⣤⠀⠉⠀⠾⠇⠙⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣶⣿⣿⡉⠳⣄⠹⣦⣴⣿⣾⠇⠀⠀⠀⠀⠙⣶⡖⢒⣌⠀⠉⢀⡀⠀⠛⠃⢠⣄⠛⠃⢸⣦⡀
⠀⠀⠀⠀⢀⣤⠚⠹⡇⠀⢿⠀⠹⡆⠈⠳⡄⠈⠳⡇⠈⠉⠉⠀⠀⠀⠀⠀⠀⢩⡀⣈⠉⢠⡄⠈⢃⡀⠿⠂⣀⡀⢰⡶⠀⡦⣍⣒⠒⣶⣶⡦
⠀⠀⠀⣰⡏⠸⣆⠀⢻⡀⠘⣧⠀⠹⣆⠀⠙⢦⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢱⠉⡷⢠⠊⠉⠉⠉⠉⠑⢻⡁⡎⢑⠸⠃⠀⠀⠈⠉
⠀⠀⢰⠃⡿⣄⡙⢦⡀⢷⡀⠈⣧⣀⠈⠷⣤⢄⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⠀⢰⣼⠀⠀⠀⠀⠀⠀⠀⣷⡇⠸⣶
⠀⠀⠀⢰⣆⢹⢹⡈⡟⠉⠁⠀⠀⠈⢷⡀⡇⠸⢦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣀⣀⣀⣄⣀⣀⡤⣈⣤⡀⢉⡴⠶⢄⡀
⠀⠀⠀⠀⡟⡿⢨⣹⠇⠀⠀⠀⠀⠀⢨⣙⠇⢸⣤⡇⠀⠀⠀⢀⣀⣤⣤⣲⣶⠶⠭⠿⠽⠷⠚⠓⠛⠚⠓⠛⠀⠉⠀⠉⠉⠀⠀⠀⠉⠉⠑⠒⠒⠒⣤
⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⠀⠈⣉⣀⣀⠉⠀⠀⡤⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣋⣹⣉⢭⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠁⠉⠉⠓⢦⡉⠙⠒⠒⠒⠒⠒⠢⠤⣄⡀⠀⣀⠀⢀⠤⠤⠤⠤⠤⠤⠤⣄⠀⣤⡀⢠⠴⠖⠒⠒⠒⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⣀⢀⣀⣀⠀⣀⣀⡀⢀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠱⣼⠉⢦⠏⠀⠀⠀⠀⠀⠀⠀⠘⠴⠁⠣⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠈⢳⠁⢤⢀⠒⠀⣤⠈⡟⠁⢻⠀⠀⠀⢀⡄⢀⡄⠀⠀⠀⠀⠀⠀⠠⡋⠑⠒⠒⠒⠲⢤⡤⢤
⠀⠀⠀⣠⠔⠚⠋⠉⠉⢹⡄⠀⠸⡀⠀⠘⣿⠃⠀⢠⠃⠀⣸⠀⠀⣀⠴⢛⠛⣆⠀⠀⠀⠀⠀⢀⣠⠃⠀⠀⠀⠀⠀⠀⠨⣴⣁⣀⣀
⠀⣰⢿⠁⠀⠀⠀⠀⠀⠀⠳⣀⠀⠑⠦⢄⣀⡡⠴⠋⠀⡰⠃⠀⣞⣉⡀⠀⠀⢸⡿⡄⠀⢠⢊⡉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉⠉⠑⠢⣄⢀⡀
⣰⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⣀⣀⣤⠴⠚⠁⠀⠀⠀⠀⠙⣦⠶⠀⣿⣧⠀⢲⠈⠁⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠏
⠀⠀⢸⢀⡶⡄⢠⡤⠤⠤⠤⠤⠤⣄⠀⠴⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢰⣿⠾⡆⠈⠳⢄⣀⡀⣀⣀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⢸⠀⡇⡿⠀⠀⠀⠀⠀⠀⢹⢸⠀⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⣟⣿⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠘⠞⠀⠳⠃⠀⠀⠀⠀⠀⠀⠘⠞⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢻⠽⡄⠀⠀⠀⢸⠀⠀⣠⡀⠀⢠⠖⠒⠒⠲⡄⠀⢠⡀⠀⢸
⠀⠀⠀⢀⡤⠤⢤⣀⣠⣤⣄⣀⡀⣠⠴⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣴⡼⡗⢶⠀⠀⠀⠸⡀⣀⡇⡇⠀⣸⠀⠀⠀⠀⣇⣀⡾⡇⠀⡸
⠀⠀⣰⠃⠀⢀⠞⠁⠀⠀⠀⠀⠹⡅⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢯⣹⣄⣀⣀⣀⡀⣀⣀⣉⣉⣁⡀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⡇⠀⠀⣜⡀⠀⠀⠀⣀⠀⠀⢸⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⠀⠀⠿⠿⠁⢉⣴⣤⠀⠀⠉⠉⠉⢿⠿⠋⠲⢤
⠀⠀⠹⣄⡀⢫⠁⠀⠀⠈⠉⠁⠀⠘⠤⣀⣀⡠⠎⠉⠛⠲⢤⡀⠀⠀⠀⠀⠀⢸⡿⠃⠀⢠⣤⡀⠈⠛⠋⠀⢰⣿⠆⠀⠀⢰⣿⡆⠈⣇
⠀⠀⢀⣀⠉⠙⣇⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀⠀⠀⠀⠈⣇⠀⠀⠙⠛⠃⠀⢸⣿⠆⠀⠀⠀⢾⡷⠀⠉⠀⡴⢻
⠀⠀⠘⢌⠙⠒⠃⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⣿⢸⢳⠰⡿⠳⠶⠖⠒⠒⠒⠒⠛⠉⢳⢠⢿⢠⠇⢸⡀
⠀⠀⠀⠀⠉⠓⠒⠒⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠇⠀⠀⠀⡇⢸⠘⡇⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢸⢸⢸⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡄⠀⠀⡤⠤⠤⠤⠤⢤⡀⠀⢀⠀⠀⣿⠀⠀⠀⠀⡇⣾⠀⣷⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢻⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⢸⡆⠀⣿⠀⠀⠀⠀⣇⡿⠀⢸⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢹⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡸⢸⡀⢀⠇⠀⠀⠀⠀⠀⢇⠀⡼⡇⠀⡟⠀⠀⠀⠀⢿⡇⠀⠘⣆⡇⠀⠀⠀⠀⠀⠀⠀⠀⢇⡾⢸⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠰⠯⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠁⠀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠁⢤⠤⠀⣼⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣷⣤⣴⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠛⠝⢟⠻⠋⠙⣷⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡄⠀⠈⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠸⠇⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣠⢄⣀⣀⠠⡀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⡿⢿⣄⣽⡄⣸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⢿⣿⣇⣿⣿⠘⣌⣹⢋⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀⠀⠀⠀⠙⢻⣿⣷⣤⣤⣾⣿⠋⠋⠉⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⠀⣤⣠⠀⠉⠛⠛⠛⠋⠁⠀⠀⢀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢤⠈⠁⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠐⠀⠀⠦⠀⠀⠀⠀⠀⢀⡀⠀⠠⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠿⢫⣧⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⣠⡇⠐⠙⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣥⣴⣾⠋⠀⠰⠷⠈⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢾⣋⣭⣵⣾⣯⡐⠷⠀⠀⡀⠀⣿⣿⣿⣿⣦⡀⠀⠘⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠛⠛⠉⣿⣦⣄⠈⠁⠀⠀⠉⠛⠿⣿⣿⣶⣶⢟⣿⠇⠀⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠅⠈⠀⠃⡸⠋⡣⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣏⣀⣠⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⣫⣿⠃⢀⠀⠐⠜⠂⠆⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣽⣿⡿⢛⡛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠁⠀⠠⠀⠤⠜⠀⢀⡠⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⢹⡿⣷⣌⢻⣿⣿⣿⠿⠿⢿⣦⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⡀⠀⠀⠐⠀⠀⠠⠄⠅⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⣾⣿⡟⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠵⡀⢄⣴⣶⣿⣿⠥⢀⠀⠄⢀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣿⣿⡿⣏⣿⡇⠀⠀⣀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⣴⢬⣿⣿⣝⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠀⠀⢰⣿⣿⠀⠻⣾⣿⣦⡀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⠋⠈⢿⣝⡻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣼⣿⣟⢇⠀⠀⠉⠉⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠉⠛⣿⣷⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠰⠿⠟⢻⣮⡓⠦⡤⣶⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠙⠓⠀⠀⠀⠀⠀⠀⠄⠀⠀⣠⣴⡿⠟⠿⣶⡀⠙⠻⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣾⣿⠃⣴⢿⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠹⠦⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⢀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⠀⢠⣾⡿⠙⣷⠀⣀⣤⣤⡴⠶⢦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁⠀⣴⣿⠟⠀⢀⣿⣿⣿⠟⠁⢀⣤⣾⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣝⡲⠶⢶⣶⣶⠿⠟⠋⠀⢀⣠⣾⡿⠋⠀⣠⣾⣿⠟⠁⣠⣶⣿⣿⡿⠛⠁⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⠳⣝⡶⣤⣤⣤⣤⣤⣴⣶⡿⠟⠋⣠⣴⣿⠿⠋⢁⣴⣾⣿⣿⠟⠋⠀⢀⣴⠶⠶⢤⡈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢮⣛⠶⣯⣭⣍⣉⣩⣥⣤⡶⠿⠛⠉⢁⣤⣾⣿⡿⠟⠉⠀⠀⠀⠀⣾⡇⢰⣆⣼⡇⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣍⣉⣉⣁⣀⣤⣤⣶⣾⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⣉⣉⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠀⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠻⡦⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⣰⣷⠀⠀⠀⠀⠀⠀⠀⠀⢱⢡⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡷⡀⠀⠀⠀⢠⢣⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡇⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣱⠀⠀⠀⣾⢾⡃⠀⠀⠀⢀⣀⠀⠀⠀⢀⠏⡹⠀⠀⣼⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⣤⣀⠘⢦⠳⡀⢠⡾⠋⠉⠙⢦⢀⣞⠜⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣌⢷⣹⣿⣠⣤⠤⠤⠬⠿⣎⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⠟⠉⠀⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢋⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⠡⢸⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡈⠛⠢⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⡏⢀⡏⠎⢻⡄⠀⠀⠀⠀⠀⠀⠀⢀⡏⠱⡌⣄⠈⢿⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡼⠀⢸⢻⡀⠀⢻⣆⠀⠀⠀⠀⠀⢀⡞⠀⠀⢻⠹⠂⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠈⣇⣣⠀⠀⠙⢷⣄⠀⠀⣠⠞⠀⠀⠀⢸⣸⡄⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⢯⡣⡀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⡟⡜⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣄⠀⠀⠀⠀⠀⠀⠀⢀⣼⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠀⠀⠚⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⡀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠷⣿⣿⣿⣿⣿⡿⢋⣩⣴⠖⣦⣬⣙⠻⣿⣿⣿⣿⣿⣇⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢁⣀⣤⣾⣿⣿⣿⠋⣴⣿⣿⡏⣴⣘⣿⣿⣷⡜⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⣿⣿⣿⡏⣸⣅⠰⣞⢰⣷⣶⢲⠆⣠⣷⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⡇⢻⣿⣷⢬⣙⡿⢛⡄⢿⣿⡿⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⡌⢿⣏⣈⣥⣶⣯⣁⢸⡿⢡⣾⣿⠿⠿⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠛⢿⣿⣿⣿⣿⣶⣍⣛⠛⠛⠛⣛⣩⣴⣿⣿⡏⠀⢴⡆⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⣶⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠘⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣽⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⢀⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠁⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢟⡛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢿⡇⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣮⣥⣤⣄⣀⣀⣀⣀⣼⡏⠀⠀⢹⣯⣤⣤⣤⣤⣤⣤⣤⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⢹⣇⣀⣀⢸⡏⠉⠉⠉⠉⠉⠉⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠙⠛⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣿⣿⣿⣶⣶⣶⡄⠘⠛⠀⣠⣶⣶⣿⣷⣾⣷⢦⣶⠆⠈⠙⠛⠷⠶⠄
⠀⠀⠀⠀⢸⡿⠊⠉⠉⠙⢻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢰⣦⠾⠿⠛⠃⠀⠀⠀⠀⠐⠓
⠀⠀⠀⠀⣶⡿⠀⠀⠀⢀⣾⠛⠋⠻⣿⣿⣤⣄⣀⣀⡰⠖⠀⠀⣀⣀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢀⣤⣶⣤⣴⣿⠋⠀⠀⠀⠈⢻⣿⣿⣿⣿⡇⠀⠀⠘⢿⣿⣿⣿⠀⠀⠈⠻⣿⣶⣷⡆
⠀⠀⠀⠀⣻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠙⢿⣿⠏⠀⠀⠀⠀⠀⠻⡿⠁⠀⠀⠀⠀⠀⠙⢿⣄⡀⠀⣠⣤
⠀⠀⠀⠀⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿
⠀⠀⠀⠀⠸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠤⢀⡀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠀⠀⠲⣦⠀⠀⠀⠀⠸⠆⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠋⠀⠀⠀⠋

⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠧⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠘⠛⠛⠀⠂⠙⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠂⠀⠾⠐⠀⠆⠀⠀⠀⡠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠄⠠⠀⠀⠀⠀⡠⠐⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠤⠀⠀⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠤⣀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠤⣀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠙⠊⠀⠀⠀⠒⠲⣄⠀⠀⠀⠀⠀⠀⡀⠀⠓⠠⠀⠨⠄⠂
⠀⠀⠀⠀⠑⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠠
⠀⠀⠀⠀⡠⠒⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⡀⠤⠀⠀⠈⠐⠀⠀⠈⠉⠉⠀⣀⡀
⠀⠀⠀⠀⠂⠀⠁⠉⠉⠀⡀⠠⠔⣈⡀⠀⡀⠀⠠⠄⢂⢠⠀⠄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠨⡠⠤⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⢾⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠄⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠨⠤⢄⣤⣬⣥⣤⠄⠀⠀⠀⠀⢀⡀⠀⠀⠠⢔⣽⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠄⣀⠀⠉⠐⠒⠀⠄⠠⠤⠀⠀⠀⠀⠤⠄⠐⠷⢤⣤⣀⣀⣀⣀⣐⣐⣨⡧⢀⠠⣊⠔⠒⠉⠉⠉⠙⢌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠒⠒⠠⠄⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢁⠀⡀⠀⡴⠃⢀⠠⠀⠀⠄⠠⢌⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠄⡀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢰⢧⡔⡥⠚⠉⠉⠉⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣈⠀⠀⠀⠀⠠⠀⠐⡀⠔⡯⠥⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣟⠒⠤⠤⣤⠭⢤⣴⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣌⡀⠉⠉⠙⠛⠛⠛⠛⠛⢑⣂⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠲⠶⠖⠒⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠀⠀⠀⠄⠠⠈⠠⠀⠀⠠⠂⠀⠀⠈

⠀⠀⡀
⠀⡀⡇⠀⠀⡀⣼⡀⠀⢠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐
⢰⡇⣿⣿⠀⣧⣿⡇⢰⢾⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠈⠛⠿⣿⣀⣿⣿⡇⣼⣾⡆⣷⢀⡇⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢿⡧⠇⢄⡀
⠀⠀⠀⠀⠓⠉⠛⠓⠿⠿⢯⣿⣤⣧⣾⣠⡆⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⠇⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠇⠀⠀⢳⠀⢸⠄⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠉⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠑⠄⠀⡬⡆⢀⠂⢠⠈
⠀⠀⠆⠠⠒⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⢀⠀⠀⠒⠣⣀⡤⡀⣸⣭⠆⠠
⠐⡆⢰⣛⣒⡀⣤⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢘⠌⢨⠅⣀⠹⣷⣥⣕⠂⢓

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⣦⠀⠀⠀⠀⠀⢠⡆⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁
⠀⠀⠀⠀⠀⠈⠻⢿⣦⣄⡀⠀⠀⠀⠀⢻⣷⣄⠀⢿⣠⣤⠀⢠⣤⣸⠇⢀⣴⣿⠃⠀⠀⠀⠀⣀⣤⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣤⣄⡀⠀⠙⢿⣷⣼⣿⣿⣿⣿⣿⣿⣴⣿⠟⠁⠀⣀⣤⣴⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⢀⣤⣶⣾⠿⠛⠛⠉⠉⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠈⠉⠙⠛⠿⢿⣶⣦⣄
⠀⠀⠀⣠⣶⡿⠛⠉⠀⠀⠀⠀⠀⠀⣠⣾⠟⢋⣤⣿⣿⣿⣿⣿⣿⣿⣷⣮⣙⢿⣦⡀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣦⡀
⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠿⣛⣫⣭⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⠯⣥⣶⠿⠟⠛⣽⡛⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣛⡯⣙⠶⠽⠓⢛⠫⠥⢬⣖⣚⣩⠽⠴⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⠉⣉⣀⣤⡤⠤⠶⠒⠚⣛⣉⣩⠥⠴⠒⣆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠨⠉⠉⠀⢠⠴⠒⠒⠚⠉⠉⠁⠀⠀⠀⢀⠀⠈⠂⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡼⠀⠠⡄⠰⢠⣤⣠⡄⠆⠰⠃⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⢀⢠⠚⠁⠀⠤⣀⡶⢤⣽⣯⡻⠄⡇⠠⠖⠀⠀⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸⠀⠀⡀⡤⢀⣰⣦⠀⠀⠠⠾⡃⠠⡴⠄⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸⠄⠀⠀⠀⠀⣫⠴⠞⢿⡓⠄⠈⠂⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸⠂⠀⠀⠁⠞⢳⣘⢋⣸⠃⢐⣀⣽⡧⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸⠁⠀⠁⠵⠻⣶⣭⣾⠵⠶⠿⣜⠝⠶⠀⠀⡃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸⠀⠀⠀⠖⠁⠀⣾⣴⣶⢿⡒⠻⡏⠑⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸⠀⠀⠀⠴⢁⡔⣹⠿⣿⣟⡻⡇⠳⠀⢀⡤⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠈⣀⠀⠀⠀⠈⠀⠉⠀⠈⠙⠀⠀⠀⣀⡎⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠙⡆⢀⣀⣀⡤⠤⠖⠒⢒⠛⠉⠉⠀⠐⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⣀⣀⣀⠬⠭⠐⠒⠂⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠀⠿⢿⠑⠻⣧⡀⠀⠀⣀⠺⠄⠉⢸⡏⠉⠉⠉⠰⢁⡤⠁⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠸⣌⣷⡖⢾⡉⠁⠀⠀⠀⠀⠀⢨⡟⠀⠁⠂⠢⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⣬⣧⣀⡁⠀⠀⠀⠀⠀⠈⠉⠓⢀⣄⡰⠁⠀⠀⣀⠜⠀⢠
⠀⠀⠀⠄⠀⠀⠀⠀⠄⠀⠀⠋⠿⠁⠀⠀⠀⠀⠀⠀⠀⠙⠾⠇⠘⠀⡰⠁⠉
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⢀⡦⢀⣀⣴⠀⠀⠀⠄⠀⠐⠦⣄⠀⠇⠀⡤⠀⠀⠀⡄⠈⠀⠀⠠⠄
⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⡠⠁⠀⡸⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠈⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣄⣄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠈⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠠⠳⠀⣅⢴⡀⠀⠀⢀⠀⠀⠀⢀⢠⢄⠦⢄
⠀⠀⠀⠀⠀⠀⣠⣿⢿⠇⠀⠀⠀⢠⣤⠟⠀⠀⠀⠃⠀⠀⠀⠋⠀⠐⢀⠀⠙⡉⠩⠿⢦⣀
⠀⠀⠀⠀⢠⠞⠉⠀⢠⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡬⠶⢃⡀⠀⣀⣀⠌⣷⣆
⠀⠀⠀⠀⣿⣾⡇⠘⠦⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠻⢶⣆⣆⣸⡛⡏⠀⠀⡆⠀⠈⠄
⠀⠀⠀⠀⡟⣿⠇⠀⠀⠛⡀⠀⠰⠀⠀⠀⠀⠀⠀⠇⠀⢈⣀⠀⠀⢀⣹⣾⣿⣷⣧⡀⠀⡄⠀⠀⠘⣄⡀
⠀⠀⡦⢰⡟⠹⠀⠀⠀⠀⡇⠀⠰⠄⠀⠠⠦⠄⠀⠀⠠⠀⠉⠀⠀⣾⣿⣹⣿⣟⡟⠀⠀⣶⠀⣀⠀⠘⣇
⠀⠀⢿⣿⣯⡞⡀⠀⠀⠜⠣⠠⣤⠴⡆⠢⠀⠀⡄⠀⣦⠀⠺⡆⢀⠀⣽⢿⣿⡧⠛⢀⣀⣾⣋⣉⢀⣴⢏⠆
⠀⠀⣿⣻⡻⢍⠓⠰⡷⠋⠀⠀⠈⡠⠚⡅⢸⠁⣷⠀⣫⠓⣈⠚⡃⠙⢋⠀⠉⡄⠔⢉⣿⡇⠈⢙⣋⡵⣼⠠
⠀⠀⣿⣿⣧⢂⠙⠀⠙⣤⣉⠀⠀⡁⠀⠀⠀⠀⢻⡆⠄⠸⣽⣧⠀⣠⠾⠐⢋⡆⠀⣾⡟⣴⠀⢠⡍⡓⠸⡆
⠀⠀⣿⣿⣿⣷⣆⠀⠀⠈⡁⠀⠀⠀⠀⠀⠡⡀⣀⠁⣚⠐⠇⣽⡇⠠⠀⠀⠀⠀⡸⡿⢌⡁⠄⢸⣾⠇⠀⡇
⠀⠀⣿⠿⡇⢃⡙⠂⠀⠀⠀⠐⠂⢀⠀⠀⠀⢹⢿⡆⡿⠂⣼⠟⠁⠀⠈⠀⠀⢰⢃⡇⡸⠀⠀⢸⡏⢩⢹⡇
⠀⠀⣿⠃⡇⢸⠗⠀⠀⠀⠀⠰⠀⡆⠀⠀⢀⠈⠺⣿⠠⡄⢀⣀⡀⢀⡀⠀⠀⡇⢸⡆⡇⠀⠀⠘⠃⢠⡞⣃
⠀⠀⡟⡇⢠⠈⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠨⠻⠶⢇⢀⡟⠸⣿⠿⠄⠀⠀⢰⠋⢸⡇⡇⠀⠀⠀⠀⠀⠁⠟
⠀⠀⠀⣿⠸⠀⠄⠀⠘⠠⡀⠀⠀⠀⠀⠰⡀⣹⠏⠉⠀⠀⠠⣤⡤⢤⠀⢠⡾⠂⣸⡇⣧⡀⠀⠀⠀⠸⣰⢰⡀⡀
⠀⠀⢤⣿⠀⡀⢸⠀⢤⡆⠈⠑⢠⠀⠀⠂⣴⠏⠀⠀⠐⠀⠀⠱⠇⠀⠀⡈⠁⠀⣿⠀⣯⢳⡀⠀⠀⠀⣿⣦⡀⢠
⠀⠀⢸⣿⡆⡇⠀⡇⢸⡇⠀⠀⠀⠀⠀⢈⣿⣿⡆⡄⠠⢄⠠⠀⠆⠀⢰⣧⠂⠀⣿⠀⢸⠀⢇⠀⠀⠀⠀⠈⠳⢻⡄
⠀⠀⢸⣿⠧⣀⠆⠙⣈⠃⠀⠀⠄⠀⠰⢄⣿⡿⠑⡆⠀⠀⠀⣶⠀⢠⠛⠓⠀⠀⠹⠄⠠⡇⢻⠀⠀⠀⠀⢀⣴⡟⠆
⠀⠀⣸⣿⢠⡈⠀⠀⢉⡆⠀⠀⠘⠀⠀⢽⣮⡛⣦⢴⡀⢶⠐⢣⡀⢸⠟⠛⠛⠋⠼⠦⣤⣏⡘⢧⠀⠀⣤⣿⣿⠃
⠀⠀⡿⢻⡎⠡⠄⢃⢸⡇⠀⠀⢀⣱⠀⠎⠝⠻⢿⣻⣿⣏⠰⠼⢧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠾⠛⠪⠸⠟⠋
⠀⢠⡷⡘⠋⠀⠀⡠⢻⡤⢒⠻⣿⡷⠄⠀⢈⡠⠂⠨⣯⠀⠉⠂⢄⡤
⠀⠀⠋⠑⠶⠖⠶⠶⠶⠚⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠖⠃⠀⠀⠀⠀⠟⠟⠋

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠙⠉⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣷⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⡀⠀⣠⣤⣤⣄⡀⠙⠏⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⣼⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⢿⣿⣿⣿⣿⣿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⢀⣸⣧⡀⠙⠛⠛⠛⠁⠈⠋⠀⣀⠀⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠈⢻⡟⠉⠀⠀⠀⠰⠶⠀⠀⠀⠙⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⠇⠀⠃⠀⠈⠁⢀⣤⣶⣷⣦⣄⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⣠⡀⠀⠀⠉⠉⠉⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⡀⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠈⠋⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠈⠁⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠟⠉⠁⠉⠀⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠉⠉⢻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⣤⠀⠈⠋⠀⢰⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠰⣿⣿⠆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣟⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣯⡿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣦⡀⣀⣤⣤⣄⣤⠤⠀⠀⠀⣀⠄⠘⠛⠉⠹⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⡾⠟⠋⠀⠀⠀⠀⠀⠈⠛⠟⠀⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣛⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠻⢿⠛⡿⠿⣿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠡⠀⠀⢀⠀⠀⠀⠀⠀⠀⠔⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀
⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣿⣷⣤⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣮⣩⡄⠀⣥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⢳⠀⠀⠠⠀⡀⠀⠀⣤⣀⠈⢸⠘⢻⠛⠟⠛
⠀⠠⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠐⠀⠀⠱⡄⢸⣿⡿⠟⢻⡆⠖
⢲⣾⣻⠯⢀⣀⡀⢀⣄⠀⠀⢀⠄⠂⠀⢠⠀⠀⠀⠐⡄⢠⡈⠑⠄⣀⠀⢹⢨⣤⣤⡈⢻⠐⣿⣷⣦
⠘⡇⢨⠀⠫⠀⠀⢹⣿⠀⠀⣠⠀⠀⠴⠋⠀⣾⠀⠀⡅⠀⠁⠀⠀⠻⣤⡀⢢⠀⠙⠑⢾⣠⠐⠈⠁
⠠⠶⠾⠶⠴⣤⡄⠉⡉⠀⠘⣿⠗⠀⠈⠀⣤⣤⣤⣦⡥⣤⣤⣤⣥⣴⣴⣶⣶⣷⣶⣶⣼⡷⠶⠛⢻⣯
⢰⣦⢞⣀⣠⡀⡀⣀⣀⣒⣸⣿⣧⣤⣤⡤⠬⣿⠭⢭⠽⠟⢻⣿⣿⣿⣻⣿⣿⣿⡟⡙⣹⡄⢠⡀⣾⣿
⣀⡶⢚⣻⢲⣿⠌⣱⣿⠟⣿⡿⠿⠿⠃⠀⠀⠠⠴⠫⣷⣖⠶⢿⢿⣿⣽⠟⠉⠈⢰⡟⣿⡥⣿⣄⣿⣿⠄
⠡⠺⠯⠖⠁⠀⠀⠿⠷⠏⠁⠀⠐⠂⢚⡁⢀⣉⡉⠉⡉⠙⠛⠿⠓⡬⣁⠉⠃⠀⣷⡀⣿⣗⠥⣸⣿⣿⣦
⠀⠀⢐⣒⣀⠀⠀⠀⠀⠀⠒⠀⠠⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠸⠦⠉⠀⠀⢠⣿⠀⣼⠁⠀⣿⣿⣿⣧
⢴⣾⠏⣽⡿⢿⡶⣴⣏⣿⠦⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⢟⡃⢹⠅⠈⣻⡿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣤⠉⠘⠓⡄
⠀⠀⠀⠀⠀⠀⠀⡀⠠⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈⠀⠈⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠐⢃⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⠀⠠⠴⠐⠆⠀⠀⠁⠰⣀
⠀⠀⠀⠄⠀⠘⠀⠀⠀⠀⠀⠈⠆⠰⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠄⠀⣀⠈⠀⠀⠠⢄⡀⠀⠀⢀⠇
⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⠀⠀⠀⠀⠀⠔⠀⠀⢘⠁⠀⣀⣀⠀⠀⠳⡀⠀⠰⡁
⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠃⠀⠀⠀⠈⠙⠀⠈⠀⠀⠈⡹
⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠞⢳⢇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠙⠉⠉⢢⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⢦⣠⡄⡙⠀⡆⠘⠁⠼⣱⣏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢄⠐⠄⠀⠀⠀⠀⠈⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠰⡮⠁⢲⠃⢠⡄⣾⠀⣤⠘⠿⡛⠀⠀⠀⠀⠀⠀⢠⠁⠁⠈⠀⢀⠈⢢⡄⡠⠊⠀⠈⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⣀⢠⡸⠆⢠⡈⢁⠈⠀⠁⠀⢞⡥⠄⠀⠀⠀⠀⠀⡒⠃⠀⠑⣦⡈⠀⠀⢹⠀⠀⠐⣀⠌⢁⢈⡇
⠀⠀⠀⠀⠀⢦⡻⠃⠀⣦⠸⡇⡼⠀⡼⠀⠀⢨⢖⣫⠔⠀⠀⠀⠀⠘⢆⠀⠀⠠⢹⡀⠈⢢⡆⠁⢠⠅⠄⠀⢰⠁
⠀⠀⠀⠀⠘⠷⠁⠀⠀⠘⠀⠀⠀⠀⠀⠰⠏⠀⢩⡷⠆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠃⠀⠀⠀⠀⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢻⣿⣦⣀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠄⠀⠀⣸⣿⠛⡉⢩⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠒⠂⠠⠤⠮⠂⠀⢀⣀⡾⣫⣿⠀⠁⣈⣰
⠀⠀⠀⠀⠀⠀⢀⠄⠀⣀⢀⠀⠀⠀⣀⠄⠀⠀⢰⣻⡿⣻⡿⡿⠁⢂⣬
⠀⠀⠀⠀⠀⠀⠈⠦⣀⠀⣘⠤⠒⠉⠀⢀⣀⣠⠤⠝⢋⡩⠂⠀⠀⢿⠃⠀⠉⠑⠢⢄
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠠⠔⠒⠋⠉⣀⣠⣴⣶⡿⠁⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠑⢦
⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣧⣤⣶⣿⣧⠀⠈⠐⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⠉⠉⠉⠙⡿⠷⡀⠐⠦⣈⠐⠠⡀⠀⠀⠘⡄
⠀⢀⣿⣿⣿⣿⡿⠁⠀⠈⠻⣿⣿⣿⣿⣿⠟⠉⠀⣼⣶⣤⣤⠞⠀⠀⠙⠦⣀⡈⠙⠒⠬⣐⠄⠀⢸⣠⡴⠖
⠀⢸⡿⠿⡿⠟⠁⠀⠀⠀⠀⠈⠙⠉⠉⠀⠀⢀⣾⣿⣿⣿⡇⠀⠀⠀⣀⣀⣀⡈⠉⠐⢠⡀⡠⠜⠚⡅
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡛⠋⠀⠀⠀⠀⠤⡤⠙⠙⢀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢿⣿⣿⣿⠍⠀⠀⠀⠀⢀⡀⠀⠀⢠⠔⠃⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢉⡻⠿⠒⠒⠚⢋⡝⠁⠀⢀⠔⠁⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢄⣣⡤⠖⠋⠁⠀⠀⠀⠀⢠⠎⠀⣀⠔⠁⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⣀⣀⣠⡴⠖⠚⣋⣠⣬⠷⠿⠀⠀⢄⠀⠀⠀⠠⢾⠿⠃⡠⠊⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⡭⠩⠞⡥⠖⠋⣸⣿⠟⠁⠀⠀⠀⠀⠀⠉⠢⠀⢰⠅⡡⠈⠀⠀⠀⠀⠀⣀⠤⠒⠁
⠀⠀⠴⣚⠯⢀⣶⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠠⢴⠗⢀⠜⠁⠒⠒⠒⠒⠉⠉
⠀⠀⠐⠛⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡯⡎⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢤⡀⠋⠶⠴⠴⣄⡄⠀⠀⠧⢀⠀⡀⠑⢖⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠒⠂⢠⢀⣤⠀⠀⣰⠀⠘⣧⡄⠐⠛⠁⠛⠄⠀⣅⠂⠬⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠠⡔⡖⢻⢀⠠⠒⣄⢼⠝⠏⠋⠀⠀⠶⠋⠑⠒⠚⠷⠂⢀⣀⠀⠂⠰⣌⣆⠀⡀⠈⢹⣄⡄
⠀⠀⠀⠀⠀⠀⢠⠹⣦⣼⣧⣶⡿⠟⠉⠀⢓⡳⠂⠡⣄⠀⠀⠀⠀⠘⣧⠀⠀⢉⡧⢀⠀⢹⡦⣀⠀⠈⢿⡇⠀⡀
⠀⠀⠀⡀⢸⠀⠎⢀⣸⣿⢳⣏⣤⠤⠚⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠲⢇⣆⣷⢀⠁⠘⡧⢸⣦⡀⠎⢷⡴⢻⡆
⠀⠀⠀⣧⠀⡆⠀⡸⢋⡿⣾⣉⣆⢠⢠⡠⣴⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⢿⠳⠷⣧⡄⣣⡑⡄⢼⣷⣾⠁
⠀⠀⢀⠸⡔⡉⢶⡟⣹⣱⡟⠓⠁⠁⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠵⠂⠳⣄⣏⠘⠻⢷⣿⣀⣿⡃⣨
⠀⢳⠈⡦⢻⣦⡟⣿⣧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⢸⣿⡝⣿⡷⡴⠂
⠀⠈⢷⡄⠈⢿⣾⢃⣾⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣌⠳⣿⡇⢻⣇⢽⠄
⣄⠀⣸⢿⣆⣼⣟⣱⣿⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡑⢻⡷⣼⣿⣷
⠘⣧⡽⣎⢿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢟⣸⣿⣼⣿⠏
⠀⠘⣧⣿⢸⣿⣿⢀⣄⠀⢠⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⢹⣿⣿⡏
⠀⢀⢹⣾⢾⣿⣿⣨⠋⠡⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣀⠀⡓⢾⣿⣿⢳
⠐⢾⡤⣿⣤⣿⣿⡄⠀⢸⣿⣫⡀⠀⢠⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣟⡻⡟⣾⣿⣿⢸⠇
⠀⠸⣿⢏⢿⣿⣿⣧⠀⢒⢸⣟⡆⣠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣿⡇⢈⣿⣿⡇⢸⠀⠠
⠀⠀⢹⡥⡂⢻⣿⣿⣧⠀⢖⣿⢩⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣛⣿⠥⣼⣿⣿⣇⡏⠀⠏
⠀⠀⠀⠘⢷⡈⣿⣿⣿⣷⡨⣯⣾⣿⢘⣠⣀⡆⠀⠀⠀⠀⠀⠀⠀⢶⣷⣆⡀⠀⠈⢀⣤⡸⣿⣏⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣾⣿⣛⠛⣿⣻⣿⣶⣄⢀⠀⢀⠀⠀⠘⢝⢯⣇⢆⣇⡞⣿⠩⢿⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠛⠻⣿⣷⣾⠼⠿⠿⠿⠯⣄⣠⣤⡄⢠⠴⠓⢉⣩⣍⠙⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠓⠦⠤⣼⣿⣥⠤⠄⣨⡤⠐⠀⠀⢈⠗⣅⠀⣲⣊⣹⣥⣾⣿⣿⣿⣿⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡏⠉⢻⣿⣟⣩⣹⣷⣶⣊⣠⠶⢻⣯⠟⠉⠀⠀⢤⣤⠼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠙⠿⠏⠀⠊⠁⠐⢤⠞⠂⠀⠐⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠛⠷⣀⠀⠀⢀⠉⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⡜⣻⠀⠀⣹⡆⠀⠐⣀⠘⠃⣐⠃⠀⠀⢠⡆⠀⢱⠒⢛⢳⠦⠐
⠀⠀⠀⠀⠊⠀⠐⢬⣒⢥⡬⣟⠁⠀⠉⠁⠀⠀⠈⠘⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠨⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠊⠑⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠀⠐⠠⠀⠀⠀⠀⠀⠀⢸⣀⠆
⠀⠀⠀⠀⠀⠀⠖⠰⠆⠀⠒⠘⠃⠀⠀⠀⠀⠀⠀⡼⣆⠀⠠⠄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢈⠀⠀⠀⠀⢄⠂⠓⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡴⡒⢃⠈⡐⠀⠀⠀⠠⡄⡄⠀⠁⠀⠀⠀⠀⠀⠄⠎⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⣀⠥⢀⡐⡀⠐⣀⠈⡀⠀⠀⠀⢡⡄⠀⠀⠀⠀⠀⠀⠀⢔⠁⡔⠀⠀⢀
⠀⠀⠀⠀⠀⠒⠒⢀⠃⢐⡈⠀⢩⡀⢡⡄⠀⠀⠀⡆⠀⣤⡆⡀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣀⠀⡘⢃⡄⠈⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠈⡋⠈⠃⠀⡀⠀⠀⠀⠂⠀⠰⠄
⠀⠀⠀⠀⡠⠋⠈⠒⠋⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠐⠙⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠲⠦⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠖⠒⠚⠉⠛⠋⠋⠋⠀⠠⠦⠤⠆⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠠⠤⠄⠂⡀⠒⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠆⠄⠀⠀⠀⠀⠀⢀⠀⣠
⠀⠀⠠⠄⡶⢀⡋⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⠀⠀⠀⠀⠀⠀⡸⢆⠀⠀⣀⠠⠊⠁⡜⠁
⠀⠀⢀⡔⠀⠠⢦⡀⠉⠉⠑⠒⠒⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠉⠉⠀⠀⠀⠀⠄⠈⣩⢁⣠⣴⣤⡎⢀⢂⡀⠀⠱⡀
⠀⠀⠀⣀⠈⠱⡄⠀⠑⠦⣄⡀⠀⡀⠀⠀⠀⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠃⠀⠉⠀⣠⠊⠙⠁⢢⡀⢀⠀⠐⠊⠲⣄
⠀⠀⠀⡀⢁⠀⠈⠢⠀⠀⠀⢉⠠⣤⡀⠀⢈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠠⠁⠀⠈⢾⠀⠁⠀⠀⡴⠊⠁⠈⠉⠂
⠀⠀⠀⡇⠀⠅⠀⠀⠁⢄⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⡔⠀⠢⡱⡅
⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡒⠒⠒⠊⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠣
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠈⢄⠀⠀⡇⠀⠄⡀⢀⢀⡀
⠀⠀⠀⠀⡄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⢴⠒⠮⠀⠀⢊⣀⡈⢆⣠⠀⠠⡤⠒⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡄⠀⢈⣅⠀⠿⡿⣗⠀⢠⣄⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⢀⣤⠥⠚⠉⠁⠀⢠⡶⠞⠲⡁⢀⠀⠀⠀⠲⠦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠡⡌⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠅
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⡉⢀
⠀⠀⠀⠀⠀⠐⣦⡀⠀⠀⠈⢆⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢄⡀⠀⠀⠋⠈⠀⢠
⠀⠀⠀⠀⢀⠀⠘⡄⠀⣀⠀⠀⢡⡢⠀⢀⠼⠁⠀⠁⠤⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢤⡀⢳⢠
⠀⠀⠀⠀⠄⠀⠀⢰⠀⠈⠈⠀⢂⠱⢦⠁⠔⠂⠊⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠤⠒⠒⠂⡈⢸⣗⠘
⠀⠀⠀⠀⠀⠤⠄⣀⠀⠤⠄⠀⠀⠈⠀⠙⠤⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡈⠁⠀⠀⠀⠀⣀⠤⠈⠉⡴⢉⠈⡆
⠀⠀⠀⠀⠀⠀⡀⠀⠑⡄⠀⠀⠔⠁⠀⡀⠤⠀⠓⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠈⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠃⡀⢀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠂⠁⠀⠀⠀⠀⠀⡼⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠠⠠⠎⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠔⠂⠁⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣄⡀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⡀
⠀⠀⠀⠀⢀⡀⠀⠠⠀⠀⠀⠐⡇⡾⣫⠾⣧⢤⡄⠀⢠⢞⠿⣷⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠒⠛⠋⠉⠓⢡⣠⠁
⠀⠀⠀⠀⠈⣷⡀⠀⢳⡀⠀⣀⢿⡟⠡⡎⠀⡄⡇⠀⢸⡘⠆⠈⠀⠀⠀⠀⠀⠤⠀⠀⠀⠄⠀⠀⠀⠉⢀⣤⠞⠿⠃
⠀⣀⠄⠂⢀⣿⣿⡆⠉⢷⣶⡏⢉⢧⠀⢻⠓⢣⠇⠀⣾⣿⠆⠀⠀⠀⠀⠀⢰⣀⠀⠀⢀⢁⡀⢹⣄⠠⡟
⠸⡿⠀⠀⡏⠉⠈⢷⠀⠀⠈⠻⡌⣧⢳⣀⡴⠃⠀⣰⡿⢿⠂⠀⢠⣶⣶⣶⠇⠈⠂⠀⣸⠉⠉⢐⠨⢧⣀
⠀⠀⠀⠀⠀⠠⠀⠈⠇⠀⢀⣴⡷⠸⠋⠁⠀⢀⡼⣯⣟⡞⠀⠀⠀⠀⢸⡟⠁⠀⠀⢸⣇⠀⢰⡏⠀⠀⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠛⠁⠀⠀⠀⣔⣭⣿⣮⣛⠁⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠉⠳⠚⠷⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⢎⠟⡟⡝⣿⡇⠀⠀⢢⠴⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡍⢲⢊⠂⡓⡬⢷⡩⣇⠀⡠⠿⠅⢾⣙⣿⣇⢀⣀⣀⣀⣤⣤⠴⣆⣀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⢇⡍⡦⢚⣣⠋⠠⢽⡇⠘⠿⠋⠁⠿⡿⢻⢳⣋⢟⣿⢴⢫⢗⡭⡛⢒⠏⠑⠢⠴⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢉⢂⡁⣀⡕⠀⠀⡹⠢⠾⡛⣀⠀⠀⠀⠀⠻⠅⢋⡫⣿⢻⣸⠛⠘⣋⣵⡆⠁
⠀⠀⠀⠀⠀⢀⣶⡦⠀⢀⢋⠁⢑⠀⠀⠀⠀⠇⠃⠀⠀⠑⡁⠀⠀⠀⠀⠀⠁⡰⠟⠀⠛⠂⠀⠈⠁⠍⠂
⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠀⠀⡤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠸⠇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀⢀⠠⠀⠀⠀⠀⡀⢀⣧⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠤⠶⠚⢀⡜⢡⣠⣼⣡⣴⡄⣐⣼⣿⣿⣿⠇⠀⠀⢰
⠀⠀⠀⣦⣴⠎⠁⠐⠖⠀⠀⠀⠀⠀⠀⠠⡀⠀⠹⣆⡽⢭⠉⠹⠿⣿⣿⣿⣿⡿⠁⠀⠀⢠⣾⡀⢠⣤⡷⣄
⢰⠀⠀⢸⡿⠀⢀⠀⠀⠐⣧⣆⠀⠀⠐⢰⣷⠀⣦⠹⢄⠨⠀⠀⢠⣿⠿⣿⣿⣿⣦⣤⣴⣿⣿⣧⣿⣿⠇⠘⡇
⠘⣆⠲⣼⠁⠀⢸⣿⣶⣶⣿⣿⣶⣶⣾⣿⣟⣾⢯⡀⠀⠀⠀⢀⠎⣀⡴⠛⢿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣆⠀⠙⠶⠄
⠀⠈⣿⡇⠀⠐⠛⢿⣿⣿⡿⠿⠿⠛⢹⣿⣻⣿⠀⠀⠀⠀⢰⣴⡞⠋⠀⠀⠘⠿⣿⠀⠀⠀⣿⣿⡏⢻⡋
⠀⠀⠈⠁⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⣸⣿⣹⣿⠀⠀⠀⠀⠘⠻⠟⠀⠀⠀⠀⠠⠙⠆⠀⣰⣿⡏⠀⣸⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠁⠈⠙⠛⠁⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⣿⣄⠀⣤⣆⠀⠀⠀⠀⣤⡄⠀⣠⣿⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⣦⠈⠟⢃⠀⠀⠘⠻⠁⣼⣷⠌
⠀⠀⢀⢠⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢈⣁⡇⠸⣟⣀⣀⣻⠇⢨⣌⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠈⠈⠛⠂⠉⠉⠘⠿⢯⣽⣶⣄⡀⢸⣿⣧⣶⣸⣿⣿⣇⣶⣾⣿⠇⢀⣤⣾⣯⠽⠿⠚⠉⠉⠘⠛⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣅⣱⡀⣿⣽⣿⣿⣿⣿⣿⣿⣽⣿⢠⣎⣨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢤⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⠤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠰⣶⡍⠿⠛⠸⠿⢷⢿⡿⢟⣡⣘⠻⢿⣿⡽⠿⠇⠛⠿⢹⣶⠦⣄⡀
⠀⠀⠀⠀⠀⡠⢾⡿⠉⠁⠁⠀⠀⠀⠀⢀⣸⣧⣰⣿⣯⣽⣿⣂⣼⣅⠀⠀⠀⠀⠀⠉⠈⠙⢿⡷⢄
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⠻⠿⡿⣿⣿⣿⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣋⢭⣾⡿⠛⠀⠘⢇⠻⠿⣿⣷⡭⣙⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠃⣾⣿⣷⡇⠀⣾⡇⠀⠘⣿⣿⣷⠘⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡔⠰⣿⣿⠱⠠⠀⠙⣿⠀⠠⠈⣿⣿⠀⢂⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⢿⣯⡦⠀⡀⢀⣿⠀⠀⢰⣿⡿⠀⠘⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠈⢿⣿⣦⡀⠉⠛⢁⣴⣿⡿⠁⠀⠀⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠙⠻⢷⣾⣿⡿⠟⠋⠀⠀⠀⠀⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛

⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⣀⣀⣀⣀⡀⣀⡀⢀⣀⣀⣀⣀⣀⣀⡀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⢀⡀⡀⣀⣀⡀⢀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠠⣤⣼⣢⠞⠀⠀⠘⢦⣸⠃⠀⠀⠀⠹⡄⠀⡄⠀⣠⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣇⣀⣀⣻⡀⠈⢿⡀⠀⢀⣠⡖⢷⡄⢻⠞⠃⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢦⣀⠈⡏⠉⣩⠍⠻⣄⠀⣷⠞⠋⠳⢤⠀⢳⣈⡆⠀⠀⢸⣀⣠⠀⠀⠠⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠓⣷⠚⠁⠀⠀⠙⢾⡇⠀⡀⠀⣄⠀⠈⣿⣀⣤⠶⠛⠙⠀⠁⠀⠀⢀⡤⠚⠻⣝⢩⠻⡏⣏⡙⠟⢦⣀⠀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢼⠄⠀⠀⠀⠀⣿⡤⠞⠁⠀⠘⣇⣸⠋⠁⠖⡄⠀⡃⠀⠀⠀⡴⢋⣀⣀⠀⠈⢻⣴⣬⣤⣧⣷⣧⣽⣦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣸⣇⣤⣦⣶⣠⣏⣀⣀⣠⣶⠀⣸⠏⠀⡄⠡⣧⣤⢶⡤⣄⣼⡇⣯⢉⢹⡇⠀⠀⡇⣶⢶⠲⣆⠀⠀⣷⣤⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣡⣬⣤⣤⣤⣤⣤⣥⣄⣉⣉⠉⠉⠛⠛⠛⠛⠛⢳⠲⠶⠷⠦⠼⣯⣍⣇⣹⡈⠀⡇⠀⠀⡇⠿⠾⠴⠟⠀⠀⣿⣉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠛⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠙⢓⠲⠷⠦⣤⣤⣀⡀⠀⣻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡶⠶⠶⠶⠴⠦⠶⠤⠤⣤⣤⣄⣤⣀⣀⡀⠠⢀⠀⠀⠈⠉⠙⠻⢦⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠶⠤⣤⣄⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡤⠤⠤⠤⢤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣶⡿⢳⡄⠀⠀⠀⠀⠀⠀⠀⠳⣌⠠⠀⠀⠀⢀⠀⠈⠉⠉⠉⠙⠛⡓⠲⠦⠤⢤⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⢀⡞⠳⣬⡛⣛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⢶⣶⣶⠓⠚⡛⠓⠓⠺⠶⡶⢴⣤⣄⡀⠀⢂⠌⡃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⡀⠃⠀⢸⡳⣄⡞⣉⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡌⠁⣀⡀⠁⡀⢀⠈⡀⣵⠟⢳⡄⢍⡓⠦⣄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣞⣯⣹⢦⡀⠈⢇⣸⢉⡽⠋⠀⠀⢀⣀⣠⣤⣤⣤⡄⠀⠠⠀⠙⠲⢤⣄⠠⠀⣀⠀⣀⠙⢳⡟⠂⠠⡸⠁⢨⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⡈⢹⡃⡴⠛⣿⣾⡟⣭⡄⢀⡤⢾⡏⠸⣇⡼⠿⡿⠁⠀⠀⠁⠀⢰⡟⠻⡟⢦⣠⣄⢀⡀⣹⡏⣈⡴⢦⣴⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠉⠛⣟⠁⡴⢻⠍⣿⡿⣠⢾⡇⢸⣧⣞⡉⣩⠟⠁⠀⠀⠀⠀⠀⠈⡿⠛⠁⠈⣧⡶⣄⣤⣼⣏⣉⠷⣴⠛⠖⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⡙⠲⢿⠞⠁⡼⠀⢸⣠⢿⣠⠿⢯⣤⡤⠛⠁⠀⠄⠀⠀⡀⠀⠀⢰⣇⣤⡀⠀⢸⣿⠉⡄⢼⡇⢉⣨⣏⢉⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠙⠲⢬⣇⠼⡁⠘⢿⡿⠿⠛⠛⠉⠁⠀⠀⢐⠀⠀⣻⡟⠛⢦⠀⣸⢿⣬⡿⠀⢸⣿⠤⠴⢾⡇⣤⠼⣯⢠⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢴⣤⣿⣾⠷⣦⣄⣻⣠⣤⢀⣄⡀⠂⠈⡆⠈⠳⢹⣷⣊⡼⠀⢻⠀⣿⠀⠀⢸⢿⠄⠰⢸⡧⠄⠄⡧⠤⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠈⠹⡇⠀⠀⠉⠻⣏⣿⠟⠋⢠⠀⢀⡗⠀⢠⠏⠠⢻⠀⠀⢸⠀⢸⠀⠀⢸⢸⠖⠒⢺⡗⠶⠒⡷⠲⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡑⢢⣷⠀⠀⠀⠀⠀⠀⠀⣠⣦⠀⠀⠀⠀⠀⠀⢠⡏⠒⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢻⡆⠀⠀⠀⠀⠀⣰⡣⢌⣇⠀⠀⠀⠀⠀⣼⠇⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡇⡐⠡⢸⡇⠀⠀⠀⠀⠀⣿⠁⠀⢿⡀⠀⠀⠀⠀⣿⢠⠉⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢧⣡⡀⣼⠃⠀⠀⠀⠀⢸⡇⠰⢂⢸⡇⠀⠀⠀⠀⢹⣄⣄⢡⡏
⠀⠀⠀⠀⠀⠀⣀⣀⣈⣙⣟⣁⣀⣀⡀⠀⠀⠈⣇⢁⠀⣸⠃⠀⠀⣀⣀⣀⣙⣿⣋⣀⣀⣀⡀
⠀⠀⠀⠸⠁⠘⠿⠿⠭⠭⠿⠭⠽⠿⠟⠂⠙⠀⠙⠻⡷⠋⠀⠋⠐⠻⠿⠿⠯⠿⠯⠿⠿⠿⠃⠘⠃
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠸⣿⣿⣟⣻⣿⣛⣿⣿⡷⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠰⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠞⠉⢮⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⣠⠤⡀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣰⠁⠀⠹⡈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⠇⠀⠀⠀⠱⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠾⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⠀⡜⠙⡆⠀⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢸⠀⠀⠸⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣧⢀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣇⢸⡇⠀⣇⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⣿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣿⣦⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠭⣀⣀⠺⣍⣡⣴⣒⣒⣚⣀⣀⣿⣇⣀⣐⣒⣒⣲⣤⣨⣽⢂⣀⡨⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣶⠖⠲⣶⣶⣶⣶⣶⣶⣶⡶⢶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣄⣠⣿⣴⣶⣼⣤⣴⣦⣷⣤⣴⣾⣤⣥⣤⣤⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⣶⣶⣾⣿⣶⣶⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣤⣤⣄⡀⠀⠉⠑⠲⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣻⣿⣇⣨⣿⣶⠀⠀⠀⢹⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢰⡆⠈⣓⣪⡼⡛⣩⣄⠃⢄⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡼⣷⣤⣨⣀⠘⣟⢁⣼⣃⣼⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣄⠀⠀⠀⠈⢻⣿⣿⣿⣿⣦⣿⣼⣿⣿⣷⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⣶⣄⡈⠻⣿⣿⣿⣿⣿⠟⠋⢀⣠⣤⣤⡴⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣿⣿⣿⣿⣿⣷⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⠿⣿⣾⢿⣿⣿⣿⢿⣿⣿⣶⣶⣾⠿⠿⠿⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠋⠀⠀⠀⠀⣴⣿⣿⣿⣴⣾⣏⣽⢿⣶⣶⣶⣶⣄⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉⠀⠀⠀⠀⠀⢸⣿⠋⣿⡝⠛⢿⣏⣼⠂⠀⠀⠈⠉⠛⣷⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉

⠀⠀⠠⠀⠀⠀⠀⠀⠠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⠀⢠⡄⠀⠀⠤
⠀⠘⠻⠇⠀⠀⠂⠂⠀⠀⠀⢠⣶⡄⠀⠀⠀⢠⣠⣠⠀⢠⠀⠉⠟⠓⠀⠀⠀⠀⠀⣀⣀⠀⠀⣠⡀⡀
⠀⠀⠀⠀⣭⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⢠⡈⠉⠹⠂⡄⠀⠀⠀⠀⣀⠀⠀⠀⠈⠋⠉⠀⣀⠉⠀⠉
⠀⠀⠉⠀⠉⠈⠉⢀⠒⠀⠀⠀⠀⠀⡀⠀⠈⠁⠀⠀⠀⡏⠠⠖⠀⡠⠉⠀⠀⠀⠀⠀⢠⠀⠏⠀⠀⠉⠁
⠀⠀⠐⠀⠰⢀⠄⠸⡇⠰⢩⣿⣷⡄⠀⠀⠠⠄⠤⠤⠸⠂⠀⠀⢸⠯⡋⠀⢠⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢠⠁⠘⠃⢠⡈⠻⡿⠃⠊⠅⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⠘
⠀⠀⢀⠀⠀⠀⠠⠂⠲⠀⠈⠈⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠀⠀⠀⠁⠰⠄⠞⠁⠀⠀⠀⠘
⠀⠀⠘⠀⠀⠀⠀⠀⡦⠀⠀⠀⠂⠀⡀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠐⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠂⠀⠈⠀⠀⠈⡛⠛⡉⠃⠈⠉⠈⠉⠁⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀⠀⢀⠀⠈⠁⠀⠀⠀⢸⣶⣾⡇⠀⠀⠀⢈⠀⠀⠀⠀⠀⡇⠀⠐⠒⠒⠂
⠀⠀⠸⡇⠀⠀⠀⠀⠈⣿⣿⣿⠀⠈⠀⠀⡀⠀⠈⢿⠏⠀⠀⠤⠻⠈⠀⠀⣐⠀⠀⠀⠠
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀
⣿⣯⣭⣭⣭⣽⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠈⠀⠀⠀⣹⡀⠀⠀⠀⣷⠀⠀⣴⣀⣿⡿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣄⣠⠶⠛⠉⠁⠀⡀⠀⣿⣷⢠⡿⣿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠁⠸⣇⠀⣿⣿⡾⢡⡏⡿⠈⠙⢄⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠞⡄⠀⢀⠔⠀⠐⠂⠐
⡄⠀⠀⠀⠀⠀⠀⠹⣷⣿⣿⢁⣿⣿⠃⠀⠁⠈⢈⡀⢠⣤⣤⣤⣧⡔⠁⢸⠋⡴⠁⡀⢸⡇
⠂⠀⠀⠀⠀⠄⠀⠀⠈⣿⢾⣼⣻⡏⣠⣴⡆⣪⣶⣗⠉⣉⠛⣯⣝⠳⢄⣾⢉⣁⣠⢀⠏⠈⠐⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⣹⣿⣿⣿⣿⣿⡟⠟⠛⠛⠒⠛⠛⢯⡛⠓⣾⢿⣍⣀⣠⠟⠀⡀⠀⠀⠀⠉⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⡟⠁⢸⡀⠀⠀⠀⠀⠀⠹⢾⣃⠘⠓⡾⠉⠳⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠋⠀⠁⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⢈⡄⠀⠀⠀⠀⠘⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣄⡀⠀⠀⠀⠀⡀⣼⣿⣿⣿⠀⠀⠀⢸⠁
⠛⠒⠀⠘⠀⠼⢹⣿⣿⣿⡇⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢤⠤⠤⠴⠤⠤⠤⠤⠤⠤⠄
⡀⠀⢠⠀⢀⡀⣾⣿⣿⣿⠀⡀⢠⠈⠀⠀⠀⠋⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂
⠙⣶⠞⢣⠸⡇⡸⠋⠉⠙⣨⣡⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣆⠀⠀⢠⠶⠆⡒⣂⠀⢀⠀⣔⢉⠰
⡷⠛⣿⡾⢻⣶⣧⡄⣠⠃⡟⢁⡶⠂⠀⢠⡐⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢉⠚⠀⠌⠘⠛⠉⠀⠀⠀⠀⠈⠉⠀⠉⠀⡐⠋⠀⠀⠉⠘⠻⡟⠛⠂
⣁⣤⣯⣧⣾⣿⣾⡖⣹⣰⢴⠋⠀⠀⠀⠀⢙⣆⠙⣄⣀⣀⣀⣀⢀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⣈⣡⣄⣈⢀⣀⣀⣄⣀⡀⢀⣄⣀⠀⣀
⠉⡁⣿⢹⠗⣿⣷⣧⣽⣿⡏⣉⡔⠁⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠂⠀⠤⢭⣝⣯⡙⠛⣛⠛⠛⠛⠧⠷
⠀⢰⢻⢸⡄⣿⠂⣿⢟⣸⠔⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠘⡇⠸⣿⠃⠈⣯⣌⠃⠖⡾
⠲⣶⣼⣆⢻⢠⣾⡟⣿⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠
⣼⣾⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⡠⠆
⣽⠿⠿⢿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠔⠁⠈
⠀⠀⠀⠘⣋⢩⣿⡛⣯⠈⠙⢢⣴⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⢀⡂⢠⠮⠂⣴⠤⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⡀⠀⠐⢒⠉⢤⣼⣿⣾⣷⣽⣿⡯⠭⢭⠹⣿⣿⡂⢀⣴⣦⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⡜⡩⠋⠀⠀⠿⠤⠄⠠⠀⠘⠀⠀⠀⢸⣻⡅⢀⣀⠺⠄
⡄⠔⠀⠀⠀⡀⢭⣥⣥⣨⣿⣷⣤⡶⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢀⣄⣠⢆⣀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢡
⣄⣸⢀⣶⣭⣀⡨⠙⣻⣿⣯⣿⣇⣠⣼⣶⣾⠋⢻⠿⠿⢟⣿⣻⣿⡿⢿⣿⣿⣿⠋⣩⡿⠉⠀⠀⡀⢀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠁
⣯⢿⣿⣿⣿⣿⣿⣿⣾⠿⠻⠛⠙⣿⣾⣿⡿⣿⣿⠤⣴⢿⣭⣭⣿⣿⣿⣿⣿⣿⠀⠟⢿⣧⣤⠛⠉⠙⠺⠇⠀⠀⢈⣤⣄⡀⠀⠀⠀⠀⠀⢀⡀
⣟⣿⣿⣿⢿⣿⠋⡭⣄⠀⣀⣀⣼⣿⣿⣍⣟⠉⣃⡾⢣⣴⣶⠶⣶⡿⣿⣿⡏⠉⠉⠀⠀⠻⡇⠀⠠⣤⣤⢆⠀⠀⣸⠄⠈⠁⠀⠀⠀⠀⠀⠀⠈⠲⣤⣄⣀⣀
⣿⣿⣿⣷⠿⠁⢀⣀⣙⣿⡟⠋⠉⠙⣿⣿⣿⣿⣿⣾⡿⠛⠛⠛⠻⣿⣿⣿⣠⣤⣤⣤⣀⣀⣼⣶⣾⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠙⠛⠿⠿
⣻⣿⣟⣱⣟⠻⣧⡄⠀⠀⢈⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡶⠒⠒⢛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡀⣠⣤⣤⣤⣤⣴⣶⣟⠋⠉⠀⢄⣤⣄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢶⣿⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⡆⠀⣴⡏
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⢸⣿⠃⠀⡿⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⢠⠿⠟⠉⣿⡜⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⣷⣦⡀
⠀⠀⠀⠀⢀⣤⣶⢷⣄⢻⡇⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠘⠀⢲⣾⣶⣄
⠀⠀⠀⠀⠾⠛⠁⠘⢿⣦⠁⠀⠀⠀⠀⠈⠃⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⡿⢡⣿⣶⣤⡀
⠀⠀⠀⠀⢀⣴⠶⣶⣤⡙⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⢸⡿⠈⣉⠓
⠀⠀⠀⠀⣸⠟⣀⡀⠉⠛⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠘⠃⣸⡟⠻⣷⣄
⠀⠀⠀⠀⠁⢸⣿⠻⢷⣆⠀⠀⠀⠄⠀⠀⠁⠄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠠⠃⠁⠀⠀⠀⠀⠀⣿⠃⣤⡀
⠀⠀⠀⠀⠀⠈⡿⠀⣦⣌⠀⠀⠀⢠⠄⠀⠀⠈⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣾⠟⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠁⣴⣦⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⣾⠟⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡷⢴⣦⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣤⣤⠀⠛⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⣄⣀⠰⢶⣦⣄⡈⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣬⣻⣿⣦⡈⠻⣿⡦⣸⣿⠃⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⢁⣴⣿⠇⣼⡿⠃⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠤⠀⠀⠠⠄
⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⡠⡤⢠⡄⠀⠀⠀⢀⡀⠶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⡤⡄⢀⠁⠀⢠⡄⠀⠀⡂⡀⠀⠀⠀⠀⠀⢀⢡⠀⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⡀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠇⣀⠀⠀⢠⠢⠀⢄⠕⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢀⠀⠀⠀⢀⠀⠀⠀⡈⣠⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠤⠄⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠸⠰⢳⣦⠁⠀⠉⠀⠁⠈⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠠⠌⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡂⠀⡀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠇⢀⡀
⠀⠐⠐⠀⠉⠛⠒⠆⠀⠀⢠⠀⠀⠀⠀⠈⠀⠘⠈⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡄⠀⠀⠠⠄⠐⠛⠀⠀⠐⠒⢀⡄⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠
⠀⠀⠘⢁⠀⠀⠀⠀⠀⠀⠀⠇⢀⠀⠀⠀⠎⠄⠅⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢔⠄⠑⠀⠁⠀⡀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⢠⠀⠉⠁⠀⠀⠀⠀⠀⠒⡀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠠⠄⠂⠀⠐⠠⠀⠀⠠⠤
⠀⠀⠀⠐⠀⢀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠐⠐⠠⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡕⡂⠀⢠⠅⠀⠀⠀⠀⠀⠄⠀⡠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠖⠐⠾⠣
⠀⠀⠀⠐⠂⠀⠑⠀⠄⠀⠀⠁⠀⠋⠃⠀⠀⠘⠉⠀⠀⠀⠐⠀⠀⠋⠀⠀⠀⠀⠐⠀⠀⠀⠠⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⢀⠈⠀⠄
⠀⠀⣤⡄⠡⠄⠀⠀⠀⠀⠀⠀⠂⠀⠐⡐⠂⠀⠀⠀⠈⠀⠐⠀⠀⠀⠒⠲⠒⠐⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠈⠄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⠤⠤⣤⠄
⠀⠀⠀⠀⡄⣅⡀⠀⠀⠠⠦⠄⠀⠀⠀⠰⠀⠀⠀⠈⠥⠖⠀⠀⣔⠠⠀⠀⠐⠡⢐⠄⠀⠀⠀⠀⠂⠠⠀⠁⠀⠀⠂⠀⠀⠒
⠀⠀⣀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠈⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠂⠒⠀⠀⠐⠂
⠀⠀⠁⠀⠂⠂⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠀⠀⠀⠀⠀⠀⠈⠀⠐⡃⠀⠀⠸⠂⠀⠐⠆⢀⡀⢆
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠧⠀⠀⠠⠄⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡈⠀⠉⠈⠀⠱⠎⢠⠃⠈⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠰⣄⡀⠀⠀⠀⢈⠀⠀⠁⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠘
⠀⠀⠀⠀⠐⠀⠂⠀⠀⠒⠀⠀⠄⠀⠀⠀⠂⠀⠔⠀⠀⠈⠉⠀⠀⠀⠂⠄⠰⠀⠀⠀⠀⠀⠀⢈⠈⠀⠉⡀⠸⠀⠀⠀⠀⠨⠀⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠐⠘⠐⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂⠀⠀⠀⠀⠀⠈⠘⠤⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⢦⣼⠀⡌⣿⣿⡆⢻⡰⣴⣃⡶⠋⠹⢲⡼⠀⢠⡸⠀⠀⢸⡄⢴⠥⣰⣷⡞⢻⡈⢸⣿⣿⣧⢸⣿⣀⢼⢿
⣛⣿⣇⡿⣿⣿⣷⠋⣷⠋⢸⠀⣥⠟⡼⠓⠂⢩⡤⣰⠀⠈⢷⡘⠮⢼⡞⣷⠀⢻⣾⣿⣿⣿⣾⣿⣸⣽⣿
⠘⢧⣸⣧⣿⣿⣿⣼⠉⣧⡯⣂⡴⠚⠁⠀⢐⠘⠆⣬⡳⢦⡈⢷⡤⣉⣿⡜⣄⡄⠹⣿⣿⣿⣿⣿⢉⣿⣿
⡦⣹⡟⠈⣿⣿⣿⣇⣾⣿⡞⠉⠀⢬⠌⣠⠀⠤⠄⠰⠭⠵⢿⣿⣷⣌⣐⠿⣎⠻⣄⣿⣿⣿⣿⣿⣼⣿⡷
⣷⣼⣷⣞⣿⣿⣿⠞⢡⡿⢀⣤⣉⣤⣞⠚⠯⠀⠀⠉⠊⠀⠀⠀⠉⠻⢾⣝⠾⣿⡟⣿⣿⣿⣿⣿⣿⣯⣏
⠸⣏⣿⣿⣿⣿⣿⢸⡿⣷⣩⣠⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣷⣿⣿⣿⣿⣿⣿⣦⣾
⠀⢹⣏⣸⣿⣿⣿⣿⣋⡽⠯⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⢹
⢀⡾⠙⣿⣿⣿⣿⣏⠿⣽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡇⢸
⣼⠁⢰⣿⣿⣿⣿⠋⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣿⣿⣿⣿⡇⢸
⣏⢀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡜⣿⣿⣿⣿⣿⣧⢠
⣿⡟⢰⣿⣿⣿⣿⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠤⢻⣿⣿⣿⣿⣿⣿
⣿⠀⢸⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣻⣿⣿⣿⣿⣿⡏
⣿⣇⢸⣿⣿⣿⣿⣧⡀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⡇⠉⣿⣿⣿⣿⣿⣿⡇
⠘⣿⣾⣿⣿⣿⣿⣿⡘⣌⠂⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡇⣴⣿⣿⣿⣿⣿⣿⣇
⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⣰⡿⣝⡞⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢰⣿⣿⣿⣳⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡃⢸⡷⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢹⣸⣿⣿⣿⣿⢿⣿⣿⡏⣿⣿
⢋⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⡕⠀⠀⡀⢀⣀⣀⣠⣤⣤⡤⣶⣷⣿⣿⡟⢻⣿⣿⣆⣿⣿⣧⢻⣿
⣾⠃⣸⣾⢻⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⣦⣴⡷⠛⢋⣭⢿⣉⣴⢿⣿⣿⣿⡿⣠⣾⣧⢻⣇⠟⣿⣞⣷⣻
⡟⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⣷⡟⢣⣾⣱⣷⣟⣿⣿⣿⢇⣿⣿⣿⣾⣿⣷⣻⣿⡟⣿
⣷⠀⣿⡟⢸⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠻⣿⣷⣿⣿⣿⣿⡏⣽⣿⣿⠸⣿⣿⣿⢣⣿⢹⣿⣿⣿⢸

⠀⠀⠀⠀⠀⢀⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣀⠰⠠⡄
⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⡀⠀⠀⠀⡀⣠⣯⣩⣷⣄⡀
⠀⠀⠀⠀⢰⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣿⣷⠀⠀⠀⠀⠀⠀⠀⢶⠇⠀⠀⠠⢂⣼⣿⣿⣷⣿⣿⣿⣷
⠀⠀⠀⠀⣨⣿⣽⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢴⣯⣿⣇⠀⠀⠀⠀⠀⢂⠀⠀⡐⠀⡦⣬⣼⣿⣻⣿⣿⣿⣿⣿⣿⡄
⠀⠀⢠⣾⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢰⣿⣿⣿⡀⠀⠀⠀⠸⡀⠀⠀⢀⣸⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣅
⠀⠀⢮⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⡮⠫⢠⣿⣿⣿⣧⠀⠀⠀⠔⠘⠀⢀⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠂⢠⣾⣿⣿⣿⣿⡄⠀⠀⢠⣴⣗⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣀⡜⢙⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢀⠆⠀⢘⣽⣻⣿⣿⣿⣷⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⣆⣽⣰⣾⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠈⠀⠀⠠⣽⣿⣿⣿⣿⣿⣧⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠸⡛⣮⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⡇⠀⣄⣱⣿⢿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠉⠙⠻⠿⠿⣿⣿⡿⠿⠛⠛⠉
⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠘⠀⠀⠀⣭⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠈⠛⠻⣛⡛⣻⣹⣻⣿⣿⣿⡇⠀⠀⡅⠀⠀⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠠⠀⠀⠀⠀⠀⠠⣹⣿⣿⣿⣿⣿⣿⠀⢀⠀⢀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢆⠀⠀⠈⠉⣷⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⢀⣐⣗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸
⠸⠈⣀⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠸⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⣀⣴⣿⣄

⠀⢐⡄⢠⡤⠞⣣⠖⢓⡖⠀⠀⠀⠀⠀⠀⢀⠾⢷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢦⣻⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠜⠀⠀⠀⠘⠧⠴⠚⠁⠀⠀⠀⠀⠀⠀⠸⣦⣀⣇⡄⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠩⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢈⣁⣶⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠃
⠀⠀⣀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⠀⠁
⠚⠉⢉⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⡟⠀⠀⠀⠀⣀
⡆⠀⠸⠁⠉⠉⢳⡆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣰⠁⠀⠐⣶⣾⠟⠀⠀⠀⠀⠠⡠
⢹⢀⠇⠀⠀⠀⣸⡇⠀⠀⢰⣿⣿⣤⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢰⡞
⠈⣾⠀⠀⠀⢀⣿⠀⠀⠀⣾⡿⣿⣿⣽⡿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⡟⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢴⠋⢳
⢇⢦⣀⡀⠀⣼⡏⠀⠀⠀⠘⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⡰⠂⡜⠁⠉⡠⠀⠀⠀⠀⠀⠀⢳⡀
⢿⣿⠟⠋⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠠⠄⡀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠹
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠐⠏⠀⠀⠀⢠⠀⠀⠈⠒⠒⠁⠀⠀⠀⠸⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰
⢆⠀⠀⠀⠀⠀⠀⠀⢠⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠤⠀⡀⠀⠀⠀⢀⡿⠿⠄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡀
⣤⡄⠀⠀⠁⠀⠀⢀⡀⠁⠀⠶⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠚⢧⣀⣠⡧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠁⠀⠀⠀⠈⠡⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⣶⡄⡉⣁⣤⡄⢀⠀⠀⠀⠀⣠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡈⠉⣰⡇⠻⠽⠃⠄⠀⠀⢠⡿⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⢀⠙⠃⢰⠄⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⣀
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿
⠁⠀⠀⠀⠀⠄⠄⠊⠀⡠⠒⣾⡗⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢸⡟⠁⠉⠀⠀⠀⠀⠀⠦⠀⠀⢀⣀⣳⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣠⡅⡀
⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣹⣏⡛⠦⡜⣦⠀⠀⡀⠀⠀⢤⣅⡤⠖
⠀⣀⣀⣐⢀⠈⣻⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣤⡀⡀⠀⠀⠀⠀⠉⠁⠀⠐⠛⠋⣶⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⣆⣰⡀⣰
⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡀⡀⠀⠀⢀⣼⣿⣦⡀
⠀⠀⠀⠀⢀⣾⣿⣧⠀⠀⠀⣴⣿⣧⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠀⠀⢠⣿⣿⣷⡄⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⢸⣿⣿⣿⠇⠀⢠⣿⣷⡀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⣿⣿⣿⣿⡇⠀⠀⣼⣿⣿⣿⠃⠀⠀⢠⣾⣿⣦
⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣇⣀⣸⣿⣿⡏⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⢰⣾⣿⣿⣿⡏⠉⣹⣿⣿⡟⠛⠿⢿⣿⣿⣇⣀⠀⠀⠀⠀⣀⣿⣿⣿⣀⣀⣸⣿⣿⣿⡀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⣿⣿⣿⠁⠀⠀⢸⣿⣿⡏⠉⠀⠀⠀⠀⠉⢿⣿⣿⠋⠉⢻⣿⣿⣿⠛⠛⠛⣿⣿⣿⡿⠿⠿⠶⣿⣿⣿⣶⠄
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⣿⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢀⣿⣿⣿⠃⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢿⣿⡏⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠸⣿⣿⣿⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠘⠛⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠁⠀⠀⠀⠀⠁⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠛⠛

⠀⠀⠀⠀⠀⠀⢀⡀⢠⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣦⢤⡴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠰⢦⣾⠿⣾⠗⠿⠖⠓⠦⠀⠀⠀⠀⠀⠀⡿⠛⡿⠟⠾⣷⢿⣦⡴⠀⠀⠀⠀⠀⠀⣴⢾⣿⣧⣯⣿⣦⠀⠀⠀⠀⣀⣤⣿⣿⣺⡷⢿⣤⠆
⠀⠀⠀⠀⠀⠛⠡⠀⢀⠆⠔⡠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⡀⠿⠀⡑⠠⠘⠛⠀⠀⠀⠀⢀⡿⠏⠛⠁⠘⠀⠉⠁⢀⠀⠀⠀⠉⠉⠟⠛⢻⡟⠀⠙⡥⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⣴⣮⣷⡾⠷⠦⠀⠀⠀⠐⠏⡃⠀⠀⢧⠀⠀⡀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠂⢈⡌⠀⢀⡧⡂⠀⠀⠀⠀⠁⠐⡠⠀⠀⢄⢢⡥
⠀⠀⠀⠀⠀⣀⠧⣀⠀⠱⠈⢀⠀⠠⠀⠀⠀⠀⠀⠈⣋⠴⠁⠀⡆⠀⢠⡘⣾⠃⠀⠀⠀⠀⠀⢀⡦⡀⠘⠀⢀⠻⡴⠁⠀⠀⠀⠀⠀⠀⡇⢠⠐⠌⠴
⠀⠀⠀⠀⠀⣌⣠⠇⡀⠀⢡⡈⢀⠀⠀⠀⠀⠀⠀⠀⠌⣒⠁⡖⡇⢀⡈⣧⣫⠆⠀⠀⠀⠀⠀⣌⣠⠁⠀⠀⠀⣷⣋⠁⠀⠀⠀⠀⠀⠠⢀⢈⣂⣛⠉
⠀⠀⠀⠀⠀⢨⣅⠚⡡⠀⠈⡰⣮⠜⠀⠀⠀⠀⠀⠀⠠⠤⠼⣀⠠⠰⣙⢾⡀⠀⠀⠀⠀⠀⠀⢬⡰⢃⠀⢀⣸⠿⠃⠀⠀⠀⠀⠀⠀⠰⠈⠙⠀⠁
⠀⠀⠀⠀⠀⠶⢬⣎⠁⡶⢈⢀⡿⣸⠇⠀⠀⠀⠀⠀⢄⡀⠈⠉⠸⠀⣽⣾⡄⠀⠀⠀⠀⠀⠐⢮⣒⠃⡄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠲⠤⠤⢽⣀⢠⠀⡙⣫⠖⠀⠀⠀⠀⠀⠈⠷⣆⡆⠐⢀⡜⠁⠀⠀⠀⠀⠀⠀⠐⢤⣭⡞
⠀⠀⠀⠀⠀⠀⠀⠐⠊⢉⠳⠀⢻⣧⠆⠀⠀⠀⠀⠀⠀⢀⣋⣔⡀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⢆⠀⠀⠀⠀⡀⢀⣦⣸⣆⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣼⠀⠇⣸⠟⠋⣴⠂⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢹⠟⠋⠟⠉⠉⠁⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠨⣌⣡⠄⡸⠁⢿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⣄⣰⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣅⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠠⣏⠵⠀⠡⣈⣬⡭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠛⣿⣍⣿⣽⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠓⢂⡄⢠⠸⠋⠀⠀⠀⠀⠀⣰⢤⢠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠉⡁⢂⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠁⠀⠀⠀⠀⠀⠀⠺⠿⠛⠟⢿⣿⢷⡀⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠀⠀⠠⡿⠾⠿⠟⠛⠃⠀⠀⠀⠀⠀⢁⠔⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠂⠀⠀⠀⠀⠀⢀⣓⠂⠒⠃⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⣝⡸⢋⡡⠄
⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣕⡲⠄⣠⠖⢃
⠀⠀⠀⠀⠀⠀⢠⣿⠀⡀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠐⡆⢀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢀⠖⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣉⡛⢃⡯⠥
⠀⠀⠀⠀⠀⠀⠠⣿⠋⢁⠀⠀⠀⠀⠢⣽⣄⠀⠀⠀⠀⠀⢻⡥⢃⡀⠀⠀⠀⢻⡸⠇⠀⠀⠀⢀⠻⣊⠄⠀⠀⠀⠀⢠⡄⠀⠀⠀⡘⠒⠦⢅⡩⠖⡂
⠀⠀⠀⠀⠀⣄⢷⣿⡷⠞⠁⠀⠀⠈⡓⠶⣛⠀⠀⠀⠀⢀⡚⠇⣹⠂⠀⠀⢘⡎⠙⢠⠆⠀⠀⠀⡣⠤⠀⠀⠀⠀⣀⢃⡜⠀⠀⠀⠉⠛⠲⢤⠴⠊⡁
⠀⠀⠀⠀⠀⢸⣎⣇⣀⣤⡼⠀⠀⠀⠉⣻⠅⣀⠀⠀⠀⠀⠙⢺⢋⡴⠀⠀⢨⠘⠟⢃⡀⠀⠀⣆⠡⢼⡅⠀⠀⠀⣝⠩⠞⠀⠀⠀⠀⣙⠓⠂⢒⡉
⠀⠀⠀⠀⠀⠠⣥⣙⠋⣀⣴⠀⠀⠀⠶⣈⡶⠃⠀⠀⠀⠀⢻⣜⢋⣤⠀⠀⡀⠛⠶⠋⠀⠀⠀⣄⡓⢋⡄⠀⠀⠰⣌⠔⠊⡄⠀⠀⠀⠤⣉⠋⠔
⠀⠀⠀⠀⠀⠀⢠⣉⣷⠟⠁⠀⠀⠀⠲⣿⣷⠀⠀⠀⠀⠀⠀⣼⣯⣡⠀⠀⠘⠳⢄⣰⠀⠀⠠⣌⡛⠉⠀⠀⠀⠤⣌⣱⠚⠀⠀⠀⠀⢉⣀⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠉⣿⠖⠀⠀⠀⠀⠀⠈⠻⡆⠀⠀⠀⠀⣠⡟⠉⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠈⠑⡞⠀⠀⠀⠀⠠⠼⢴⠀⠀⠀⠀⢠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⠂⡸⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠻⡝⠻⢧⣄⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⡀⡏⢠⡎⠹⡆⠀⠀⠋⠒⢤⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠡⡘⢷⣠⡇⡀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣼⠸⣿⡀⠀⠈⡢⣝⠛⠧⢀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⠙⠷⠾⠮⣅⣉⡒⣤⣀⠁⠓⠺⢅⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣤⠞⠟⠓⣲⣦⣤⣄⡀⠈⠙⠺⣟⡲⣦⡂⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠁⠀⢀⣽⡿⠛⠉⠈⠹⣆⠀⠀⠀⠹⡄⠙⢦⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⣸⣟⠂⠀⠀⠀⢠⡿⠀⠀⠀⠀⡇⠈⠈⢷⣁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢠⡋⠉⠙⢻⠒⢤⣾⡟⠁⠀⠀⣠⠞⠀⢀⠀⢸⣿⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠙⢄⡈⢿⡻⠶⠃⠀⠃⢈⢿⠿⢷⣿⣅⡀⠀⠈⢀⣾⠇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠓⢦⢹⡿⢽⣶⣄⣀⠀⠈⠁⠀⠀⠀⠈⠉⠓⠶⢮⣈⡴⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠆⠙⣿⠁⠄⠈⠹⣿⣭⡒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢻⣦⡀⠀⠀⠻⣷⡙⢦⡉⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢤⣀⠀⠀⠀⠀⠀⢈⡄⠀⠈⡆⠉⢳⠢⠄⡘⢷⡼⠟⠋⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢤⣝⢦⠈⠳⢤⣀⠀⠀⠀⢇⡀⠀⢱⣀⣨⠦⠖⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⡀⠀⠀⠈⠓⢄⣀⠼⠖⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡹⠶⠶⠚⠋⠉
⠀⠀⠀⠀⠀⢀⠴⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠄⠀⠀⠀⠀⠀⠀⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⡗⣊⣉⣦⢠⠤⠤⠤⠀⠛⠚⠉⠙⠛⠓⠒⠠⠤⠤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣭⣭⡵⠤⠤⠤⠤⠤⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⠐⠒⠒⠒⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⢹⣧⡀⠀⠀⠀⡀⡀⠀⠀⠀⣠⣿⠃⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠙⢿⣦⣀⣾⣷⣿⣆⣠⣾⠟⠁⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⣶⣶⣮⣿⣿⣿⣿⣿⣿⣿⣧⣶⣶⡶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠉⣩⣽⣿⢳⣿⣿⣿⣿⣿⣿⣿⣿⣭⡉⠙⠛⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢀⣴⡾⠟⠹⣿⢸⣿⣿⣿⣿⣿⣿⣿⡿⠙⠿⣶⣄⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡏⠀⠀⠀⢻⣆⠻⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢿⠀⠀⠀⠀⠀⠙⢷⣦⣭⣭⣿⠟⠁⠀⠀⠀⠀⢸⠆⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠰⠄⣼⠛⢛⢶⠲⠶⠶⠤⠤⢤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢠⡄⢀⡾⠛⢦⡝⡜⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠉⠉⠉⠉⠻⡇⢬⠉⠓⠒⠲⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣀⠀⠰⠆⣠⠶⠋⠀⠀⢨⣿⠇⠀⠀⠀⢀⢠⠀⡀⠀⣚⠃⠀⠀⠀⠀⠀⢻⡄⡀⠀⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣩⡤⠖⠋⠁⠀⠀⠀⠀⠘⠸⣇⠀⠀⠀⠀⡄⢠⠁⣸⠉⠀⠀⠀⠀⠀⠀⠀⠹⣄⣤⠀⠀⠀⣿⣿⣷⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⢰⠀⠀⠀⠀⠀⠀⢀⢤⡄⡿⠒⠂⠀⠛⠷⡮⢀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡀⠤⢰⣿⣾⢹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠆⠀⠀⠀⠀⣠⠞⠁⠀⢀⡧⠀⠀⠀⠀⠀⠀⣨⠉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⠦⣼⣿⡟⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠘⠀⠀⠀⢀⡼⠃⠀⠀⠀⡮⠀⠀⢂⠀⠀⠀⣨⡇⠀⣰⠛⣄⠀⠀⠀⠀⠀⠀⢸⢱⡟⣸⣿⣿⢢⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠄⠀⠀⠀⣜⠀⠀⢀⣀⠚⠃⠀⠀⠀⠱⢆⠀⣍⠀⡴⠃⠀⠈⠄⠀⠀⠀⠀⠀⠈⠸⢣⣿⣿⣿⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⠀⠀⠀⠀⠀⠃⢠⡤⠋⠀⢀⣤⠀⠀⠀⠀⠀⠀⢇⡜⠁⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⡀⣼⣿⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇⠀⠀⣀⠠⠚⠋⢁⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⢐⠎⠑⢄⠀⠀⠀⠐⠀⠀⠀⠀⡐⢠⢠⣿⡿⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠾⠛⠋⠁⠀⠀⢳⠁⠀⠤⠄⠀⢠⡄⠀⠀⣀⣀⣞⡃⠀⠀⠳⡄⠀⠀⠀⢀⣴⡟⠛⠛⡟⣿⣱⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⣄⢀⣄⣠⣏⣸⠉⠁⠀⠒⠚⡶⠀⠀⣸⣿⢠⣤⢠⣷⢀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⡦⠀⠀⠀⠀⠀⢀⡨⠷⡔⠒⠛⠁⢩⠁⢡⠏⠀⣿⠀⠀⠀⣠⠞⠁⠀⠀⣿⡟⣾⡟⢸⡏⡀⡾⢈⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣇⡿⢰⡇⢀⣀⡶⠜⠋⠁⠀⠀⠈⠐⠠⠀⠘⣤⠋⢀⣠⣻⣶⠔⠋⠀⠀⠀⠀⠀⢻⣧⣏⣁⠻⠃⠃⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠓⠻⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠸⣆⠀⠀⠀⠀⠀⣠⣶⣽⣿⢹⣿⣦⡐⢦⡾
⠀⠀⠀⠀⠀⠀⢀⣿⡟⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠂⠀⠀⢿⣄⡠⠄⠀⠚⢹⠋⡇⡄⣾⣿⣿⣿⡾⠇
⠀⠀⠀⠀⠀⠀⢸⣿⣷⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣀⣤⡶⠚⠃⠁⠀⠀⠀⠈⠠⢰⢡⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⡏⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⢴⠒⠋⢹⡉⡏⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⣼⠻⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠉⠹⣆⠀⠀⠆⢀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡏⠈⠁⣈⢧⡀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠹⡀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠃⠂⣀⣤⢤⣼⡟⣿⢿⡇
⠀⠀⠀⠀⢸⠇⠀⠀⠀⢤⢳⡀⠀⠀⠀⠀⣤⡞⠁⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⠘⠓⢸⣾⣇⡏⣿
⠀⠀⠀⠀⣾⣀⣀⠀⠀⠀⢤⢳⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⢸⣿⣸⣿⡇
⠀⠀⠀⠀⠈⠾⣝⡿⣿⣴⣄⣨⣶⣶⣿⣤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⢱⡀⠀⠀⡔⢰⠄⠀⠀⠀⠀⠀⣸⣿⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠺⣿⣿⣿⣿⣟⣛⣛⣛⠛⠛⠛⠿⠿⠿⠿⢿⣿⣿⣷⣶⣾⣜⣓⣶⣶⣶⣦⠤⣤⣿⢿⡏⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠚⠫⠟⠋⠛⠛⠛⠒⠲⠶⠶⠴⠦⢤⣼⣛⣛⣋⣙⠛⠛⠻⠿⠿⠟⣿⣿⣱⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿⠿⠿⠿⠿⠿⠿⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⢒⡈⠵⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠠⠤⠤⠄⢠⠀⠠⠀⠀⣴⣮⡽⠊⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠶⢄⠁⢄⡸⠃⢀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠈⠋⠁
⠀⠀⠀⠀⠀⡀⡀⠁⢠⣀⡄⠈⢠⣼⢿⣄⣸⣒⠉⣉⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠧⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠠⡄⠉⢀⡉⠇⠙⠱⣼⡿⢯⢁⢥⡈⢁⡴⠲⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⡀⠀⠤⠲⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠴⠄⠀⡀⣠⠧⠀⡞⡠⣰⠱⢗⣯⣎⡷⠴⠃⠐⠂⣒⠀⠀⠀⠀⠀⠀⢀⡠⠚⠀⠀⣠⠔
⠀⠀⠀⠀⡐⠤⠀⣧⡇⠪⠣⡗⢃⠙⠮⠽⡖⠓⡀⠴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠋
⠀⠀⠀⠀⠱⠔⣠⡃⠳⣀⠀⢫⣼⢁⠾⠴⣭⠉⠐⢀
⠀⠀⠀⠀⠀⠀⠉⠑⢄⣘⠆⠛⠸⡮⢚⠟⠒⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠀⠒⠀⠀⠠⠐⠂⠀⠲⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⢧⡴⢁⠌⠐⠠⠐⠀⠤⠂⠄⠐⠠⠒⠄⠀⠂⠰⠂⠀⠀⠀⠀⠀⠐⠀⠀⠒⠀⠒⠀⠒⠆⠐⠒⠀⠒⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⣇⡎⠀⠠⠀⠰⠀⠀⠄⠀⠠⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣄⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡜⣿⠿⣿⣿⣯⡽⣻⢳⣿⣄
⠀⠀⠠⠀⠄⠄⠀⠀⠀⠀⠀⢀⣾⣽⣿⣿⣷⣷⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⢠⣶⣯⣿⣟⣾⣬⡼⣽⡾⣵⣿⣷⣽⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⣿⣷⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠿⣻⣿⣿⣷⣿⣻⣯⣟⠿⠋
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢘⣿⣼⣯⣿⣿⣾⡻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⣮⡟⣽⣿⡿⣟⣹⣿⣿⣷⣿⠏
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⣿⣻⣿⣙⣾⣿⣽⣿⣽⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⡟⢻⣿⡞⣻⢡⡟⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⣿⣾⣿⣿⣽⣧⣭⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢏⠈⠋⣿⠇⢸⡥⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢳⣾⡿⣷⢛⡿⣿⣯⣾⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⢹⠀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢿⣿⢿⣿⣿⣿⠻⣻⡿⢷⡼⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣾⣵⣿⣿⣿⣟⣷⣷⣿⣿⡿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡾⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠸⠏⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⢀⠀⢀⠀⠉⠉⢀
⠀⠀⠀⠉⠁⠀⠈⠀⠘⠈⠀⠀⠀⠁⠈⠉⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠈⠀⠀⠀⠁⠉⠁⠈⠁⠀⠀⢀
⠀⠀⠀⠁⠀⠁⠉⠉⠉⠀⠀⠀⠀⠉⠁⠀⠁⠈⠈⠈⠈⠁⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠁⠁⠉⠁⠈⠀⠈⠉⠀⠁⠀⠉⠀⠁⠈⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⡀⠀⠒
⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠙⠛⡷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀⡤
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⣴⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠙⢿⣧⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⣠⣿⠏⠀⠀⠀⠀⣤⠛⠙⣿⣄⠀⠐⣄⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢰⣶⣤⡤⠀⡀⣠⣿⣿⣿⠋⠀⡀⠀⣠⣴⣞⠁⡀⠀⠀⠙⣳⣶⣿⣿⣿⠷⠦⠀⠸⠇⣷⣤⡄
⠀⠀⠀⠤⠈⣋⣾⣿⣯⣿⣿⣿⣿⡟⣿⣿⣶⠿⢛⡿⠟⠁⢤⣾⣿⣿⣿⣿⣯⠱⠶⢶⣄⠀⢀⠈⠁⠀⠄
⠀⠀⠀⠰⣾⣯⣿⣿⣾⣾⣿⣿⣿⣷⣿⣿⣿⣷⣾⣷⣾⣾⡆⠉⢻⣽⣿⣿⠿⠁⠀⢀⣨⣿⣿⣿⣷⣶⣦⣤⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣾⣟⠙⠿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠉⠙⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⠋
⠀⠀⠀⠀⠀⠀⠻⠿⡿⣿⣿⣿⣿⣿⣏⠙⠻⣿⣿⣿⣿⡛⠀⠂⢀⣀⣠⣬⣯⣭⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣶⣶⡀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⣿⠻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠿⠿⠿⠟⠙⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠫⠛⠑⡌⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠛⠡⠐⠁⠀⠀⠀⠈⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢋⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠠⣀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣯⡙⠄⡈⠉⢲⣤⡆⠀⠀⠀⢄⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣵⣄⠢⠈⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⢹⠀⠁⢨⣶⣯⡟⠈⢂⠀⠀⠀⠢⡈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢈⠢⡀⠂⠀⢀⡀⠀⠀⡿⠀⠀⠀⠀⣇⡴⠋⠀⢿⠓⠦⣄⡆⠀⠀⠀⠈⢄⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡟⠀⠀⠁⠘⢄⠀⠘⡇⠀⠀⡇⠀⠀⣠⣶⣿⠶⠖⠺⡟⠂⠀⠾⢿⠶⠤⣀⠀⠀⠑⡈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄⠄⣆⠀⠈⢢⣷⡀⠀⠀⠀⢿⡴⠛⠁⠀⠸⡄⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⢃⠐⣶⣿⣿⣦⠀⠀⠀⠃⡀⠀⠀⠀⡇⠀⡇⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣠⣿⣿⣿⣿⡷⣄⠀⠀⠈⠀⡀⠀⢸⣰⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠃⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡟⢀⣿⣦⡀⠀⠀⠀⠐⠺⣿⠤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⢨⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠁⣼⣿⡿⢱⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠉⠀⢀⣠⡤⢾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣿⣿⢁⣿⣿⣧⡀⠀⠀⠉⠀⠀⠀⠀⢀⣠⠔⠉⠀⠀⠠⣀⡨⢖⣫⠵⠚⠁⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣇⣾⣿⣿⣿⣿⡦⡀⢀⣠⡴⠖⠛⠉⠀⠠⠒⠀⠀⣈⠭⠒⠉⠖⠋⢀⠠⢒⣩⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⠿⡯⢁⣀⠠⠖⠀⠀⣀⣤⣴⣾⠀⠀⢀⡠⠔⣊⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⢸⢸⡋⠉⠁⠀⠔⠲⢿⣿⣿⣿⣿⣔⣊⡥⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⣌⡏⢀⠀⠀⢀⣠⣼⣿⣿⣿⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣝⣤⢔⣨⡵⠛⠙⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠋⠁⠀⠀⠀⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⢰⠆⠳⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⠃⠀⠀⠀⠀⠀⢤⢰⠀⠀⠀⠀⠀⠀⠀⠂⡘
⠀⢀⡎⠠⠀⢸⡁⠀⠀⠀⠀⠀⣄⠀⠺⢢⠚⠃⢀⡄⠀⠀⠀⠲⠀⣀⣀⠐⠃⠀⠀⠀⠠⠴⠀⣀⠄⠀⠀⠤⠀⠒⠂⠀⠀⠀⠈⡟⢻⠀⢐
⠀⠀⢶⣶⠟⡇⡏⠀⠀⠀⠀⠀⠘⡗⣬⣿⡹⡄⡯⠁⠀⢀⣐⠀⠃⢴⣄⠘⡁⢂⡀⠀⣰⣦⠀⣤⠟⠀⠚⡴⠂⠈⠀⠀⠀⠑⢘⣓⣘⡣⠊
⠀⠀⠨⣐⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠿⠗⠉⠀⠀⠀⠀⠀⠠⠧⣙⣁⠴⠑⠂⠀⠈⠘⠁⠀⠀⠉⣀⡘⠁⠀⠀⠀⢠⠀⠀⠀⠏⠹⠄⠀⠀⢀⡀
⠀⠀⠀⢿⡵⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢤⡤⠀⠀⠀⠀⠀⠀⠶⠁⠀⠀⠘⠖⠀⠀⠀⡀⠀⠀⠀⠀⢭⡥⠀⠀⠀⠀⣡⢣⠀⠀⢀⣀⠀⢠⠌⠙⠆
⠀⠀⠀⣘⢠⠀⠀⠀⢀⡷⢯⡀⠀⠀⠀⢀⢀⠀⠀⣀⠤⠄⡄⡀⠀⠸⠗⠀⠀⡀⠠⠀⡁⠠⠀⠀⠀⢰⡆⠀⢀⠀⡘⠔⢥⡏⠀⠨⠍⠀⢰⢚⠀⠈⠀⠀⡀
⠀⠀⠀⢸⡖⠀⠠⣁⠁⣷⣯⡇⠀⠤⠀⠘⢉⠀⠮⢰⡶⠳⣶⠹⡄⠀⡂⠀⠀⠀⠛⠀⡅⠘⠃⠀⠀⠈⣷⠒⠋⠀⠀⣄⣢⠀⠀⢀⣆⠀⠀⠙⠶⣀⡈⠟⠁
⠀⠀⠀⢰⢺⠀⢄⡅⢰⢾⠿⡇⠤⠄⠀⠸⠚⠀⣇⢹⣦⣤⣟⢁⠃⠈⠋⠀⠠⠀⣀⠈⠂⢀⣀⠈⠀⠀⢸⠀⠀⠀⠀⠰⠖⠀⠀⠘⡞⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⢬⡷⣾⠔⠡⠀⠀⢰⣠⠀⠀⠃⠠⡅⠄⠋⠀⠀⠀⠀⠀⠐⠁⠀⡀⠸⠁⠂⠀⠀⡟⠀⠀⠀⠀⠈⡏⠀⠀⢀⢦⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⣴⠄⠀⢐⣸⠀⠐⠀⠀⢠⠀⡀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠰⠇⠀⠀⠀⡏⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⢀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠅⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢐⠀⠀⠀⠘⠿⠀⠀⠀⠀⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠁⠀⠀⠀⠘⠦⠆⠀⠀⠀⠆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠠⡄⡄⠀⠀⠀⡁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠰⡆⠀⠀⢠⠉⠄⠀⠀⠀⠘
⠀⠀⠀⢸⢉⠀⠀⠀⠖⠐⢠⠁⠀⠀⠀⠀⠇⠁⠀⠀⠀⠆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠷⡆⠀⠀⠀⠀⡇⠀⠀⠀⠉⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢦⡆⠀⠀⠀⠂⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢐⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⠀⠀⠀⠀⠈⠀⠀⡀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠴⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⡞⠰⠀⢠⢠⠄⢄⠄⠀⠀⠀⠀⠀⠀⢠⢠⠄⠀⢀⠄
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠄⠰⠒⠚⠀⠀⠀⠀⠀⠞⠀⠐⠀⠰⠠⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠤⣄⣴⣶⣶⣄⢺⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⡄⢰⠃⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⣀⣸⣿⣿⣿⣿⣏⣾⢛⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣾⣜⡁⠀⣀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⢀⡾⠉⠙⣿⣿⣿⣿⣿⡷⠟⠙⠳⡄⠀⠀⢀⣠⠴⣾⣿⣿⣿⣿⣷⣛⣁⠀⠀⠀⠀⠀⣀⣠⢴⣶⡴⡀⡼⠁⠀⠀⢀⣠⠤⠒⠒⠉⠉
⠀⠈⢀⡶⠟⣿⣿⣿⣿⣿⡷⣦⡀⠀⠙⠤⢀⡾⠀⠀⣿⣿⣿⠟⠹⣆⠀⠈⠓⠦⡀⠀⠸⠁⠀⢃⣴⣷⣿⣿⣦⣴⠾⠭⠤⠴⠂⠀⠠⠄
⠀⢀⡞⠀⠀⣽⠉⡟⠋⠛⣇⠈⢿⡄⠀⠀⠘⠁⠀⠀⠀⣠⠟⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠛⢳
⠀⡞⠀⠀⠀⣿⠀⡇⠀⠀⣿⠀⠀⢿⠀⠀⠀⠀⠐⠛⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠛⢹⡆⠀⠳⡀
⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠛⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠠⡆
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢀⣀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢹⡄⠀⠀⢰⠓⡀
⠀⠀⡰⢤⡈⣆⢰⣿⣿⣿⣆⢸⠀⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣿⣿⣧⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠈⡇⠀⠀⡿⠀⢱⡀
⠀⠀⠁⠀⠹⣼⡸⣿⣿⣿⠏⣜⡼⠁⠀⠀⠀⠀⠀⠀⠐⠢⢤⣀⣈⢿⣿⣿⣿⠇⣀⡤⠤⢄⣀⡀⠄⠀⠀⠀⠀⢀⡿⣅⠠⣃⣄⡄⡇⢀⡽⣇
⠀⠀⠀⢀⡴⣾⣿⣿⣿⣿⣾⣿⡴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣻⣿⣿⣷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢈⣻⣿⣿⣿⣿⡋⢀⣼⠂
⠀⢀⠔⠁⠀⡇⢉⠟⠋⠙⢏⠀⡇⠈⠳⡀⠀⠀⠀⠀⠀⣠⠞⢋⡽⢿⣿⣿⠿⡏⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠡⣿⣿⣿⡏⢹⠛⠁
⠀⡜⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⣇⠀⠀⠀⣠⠞⠁⠀⣸⠃⠈⠀⠈⠀⡇⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠻⣿⠟⠀⠘⡄
⠀⠁⠀⠀⠀⠹⡀⠀⠀⠀⠀⢀⠁⠀⠀⠸⠀⠀⠈⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠸⡀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠞⠋⠉⠉⠉⢉⣫⡿⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠂⠀⠀⠀⣀⣤⠞⠉⠁
⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠰⢻⡇
⠀⠀⠈⠻⢤⣌⡉⠹⣟⡓⠶⢯⣀⠀⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⠙⠲⢮⣟⠳⢤⣀⠀⠀⠑⠤⣄⡈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡷⢠⣭⣛⠦⣄⡀⠈⠙⠋⠓⠶⠶⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠁⠐⣴⡮⡙⠚⡟⠛⠷⢦⡦⢤⠦⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠉⠁⠀⠀⢹⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⡇⣰⠟⣷⢁⡀⠀⠀⠀⢸⠀⠀⠀⠀⠙⠦⣤⣤⠤⢴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠁⢀⡾⠛⠛⠋⢀⣴⠏⠀⠀⢠⡀⠐⠲⠬⠍⠛⢯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡋⠀⣀⡴⣾⣿⠃⢀⠀⣢⠞⢷⠾⠶⠒⠒⠓⠶⣝⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢰⣻⠇⣠⠷⠚⠁⠀⠈⢳⡄⠀⠀⠀⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠘⠟⢷⡀⠠⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠈⣷⠀⠐⠀⠀⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣯⠀⠀⢠⡟⠀⠀⠀⢰⡏⠈⠙⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠘⠈⢿⣰⠟⠁⠀⠀⢀⣿⠁⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⢀⣤⠾⢧⡀⠀⠀⣠⡾⠋⠀⠀⠢⣀⣺⠁⢀⠀⠀⠀⠀⠀⠀⠀⠷⢤⣤⣤⣤⡴⡆
⠀⠀⠀⠀⠀⣸⠛⠛⠳⣽⣤⢿⡍⡀⠀⠀⠖⣤⣼⠃⠀⠁⠀⠀⠀⠈⠀⠈⠀⡤⠀⠁⠀⠀⣼⠃
⠀⠀⣏⣉⣉⣳⡞⡛⢛⠛⢳⡆⠙⢦⣱⣄⣤⣾⡃⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠇⢀⠀⠀⣰⠋⠀⠀⠀⣀⠀⠀⠀⢀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠈⠉⢯⣿⣽⣯⣿⣉⡄⢀⢀⣿⡟⠟⠉⠀⠀⠁⠀⠁⠀⠀⠐⢌⡴⠋⠁⠠⠐⢿⡷⠦⠤⠤⢤⡿⢛⣾⡞⠉⢛⡉⠉⠉⠉⠉⠉⠉⠙⠶⠶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⣽⠀⠀⠅⠀⠀⢀⣨⣠⣤⠴⠾⠷⠶⠶⠷⠦⠄⠀⣽⣷⣶⣶⡶⣇⢰⣿⡇⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠀⠀⠄⠁⠝⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣯⡀⠛⠉⠉⠉⠛⢷⣤⣀⣀⣲⣴⣋⣤⡤⠤⠶⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠀⣀⣂⣤⠀⠀⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠛⠚⠛⠛⠛⠯⣯⡉⠉⠁⠐⠀⠅⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠀⠀⢼⠛⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣾⠀⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠴⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠓⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠿⠻⣦⡀⠀⠀⡀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠂⠀⣀
⠀⠀⠀⠈⠙⠛⢻⣿⣦⣄⠀⠀⠀⠀⢸⡴⠚⠛⠀⠠⠷⠦⠤⠄⡀⢰⠀⠀⢸⣿⡌⢀⢁⣿⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠠⠶⠀⠘⣿⣿⠀⠈⢢⣤⡄⠘⢤⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣤⣾⣿⠃⠀⠀⠀⠀⢀⣄⠀⠀⣠⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⣴⠟⠀⠀⠀⠀⠀⠀⠑⠀⠀⢤⣴⠀⠀⠀⠀⠀⠀⠈⡿⠛⠻⠁⠀⠀⠀⠀⠀⠸⠃⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠿⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤
⠀⠀⠀⡨⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⢲⡇
⠀⠀⠀⠐⠀⠐⠃⠀⠀⠀⠀⢀⠀⠲⢠⠜⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣶⣦⡦⠀⠀⡰⡿⠀⠀⢿⢦
⠀⢀⠀⠠⠀⠀⡀⠀⢰⡀⠀⢀⡀⠀⢾⡆⠀⠀⠀⠀⠀⠀⢠⡄⠀⣠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠟⠛⠻⣿⣿⣧⠀⠈⠀⡇⠀⠀⢸⠀⠁
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢺⣿⠀⠀⠀⠀⢰⠈⠁⠀⠘⠻⣆⠀⠈⣧⣤⣤⠆⠀⠀⠀⠀⠀⠀⢀⣿⣿⠏⠀⠀⠀⠃⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢴⠀⠀⠐⠄⠀⠀⠀⢩⣮⢧⣀⠀⠀⢸⣴⠆⠀⠀⢸⠁⠀⠀⠀⣿⡿⠀⠀⠀⠀⢀⣀⣀⣾⠿⠋⠀⠀⠀⠀⠐⠂⠐⠒
⠀⠀⠒⠀⠀⠀⡄⠀⠺⠁⠃⠀⠀⠀⠰⣼⣭⡯⣷⡆⠀⠀⠀⠀⠀⠘⠃⠀⠰⠀⠉⠁⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⣀⠤⣄⣀⠀⠀⢤⡀
⠀⠀⢸⣶⡄⠀⣿⣦⣤⣴⡆⠀⠀⠀⠘⠎⠐⢾⡟⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠚⠉
⠀⠀⠘⣿⣿⠿⠻⣿⣧⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠙⢻⣿⡿⢶⣶⣶⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⡀⠀⣄⡀⢀⡀⠸⠦⣤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡀⠃⠀⠆⠈⠀⠂⠀⠀⠰⠚⢁⡀⠀⠸⠿⣿⣿⡿⠟⢿⣿⠛⠛⠛⠳⠄⠀⠘⠀⠀⠀⠹⠃
⠀⠀⠀⠀⠀⠀⢠⡴⠀⠀⠀⢰⠋⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠸⠟⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦
⠀⠀⠀⠀⠀⠀⢸⡓⠃⠀⠀⠸⠀⠛⢃⢀⡀⢀⠀⠀⠀⠀⢠⠶⢦⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣾⡧⣦⠀⠀⠀⠁⢀⠸⠹⠂⡀⢈⠇⢳⠆⢀⣤⣿⣶⡟⠂⠀⢀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⣿⣿⣿⡿⠊
⠀⠀⠀⠀⣠⣿⣿⣇⢲⡄⠀⠀⢀⣦⠇⠀⠀⠷⢐⠀⠀⠀⢸⡟⠠⣉⣿⣷⠀⠈⠀⠀⠀⢾⣄⡀⠀⣀⣀⣄⠀⢀⡇⠀⠀⠀⠙⢟⢿⠃
⠀⠀⠀⣠⣿⣿⣿⣷⡋⠀⠀⠰⠁⠈⠀⢀⠞⠀⠆⠀⠀⠀⠘⡇⠀⠸⣇⢿⡀⠀⠀⠀⠀⠀⣿⠁⠀⠉⣿⠏⠀⢸⠀⠀⠀⠀⠀⡀
⠀⠘⠋⠉⠀⠛⠛⠛⠉⠀⠀⠀⠀⠂⠀⠘⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠐⠋⠒⠊⠀⠉⠀⠁⠉⠀⠀⠀⠀⠀⠑
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣽⣭⣯⣮⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣾⣿⣿⣾⣿⣷⣿⣿

⠀⠀⠀⣀⣀⡴⠶⠤⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠐⣦⡄
⠀⠀⠀⣿⢿⡇⠒⠒⠢⠤⠤⣉⣉⠉⠛⠒⠲⠦⠤⢤⣄⣀⣀⠀⠀⢀⣀⡤⠤⠔⢒⠒⣄⣉⠉⠀⡄⢐⡀⠈⠁⣿⢻⣿⣿
⠀⠀⠀⡿⠚⡇⠭⢍⣙⣓⣒⠐⠀⠌⠁⠀⠈⠉⠁⠀⠀⠈⠉⢹⣿⣿⠀⢠⠀⢀⡞⠀⣓⡂⠈⢍⣓⣀⣰⠨⠀⣿⢸⣿⣿
⠀⠀⢠⡇⣤⡇⠒⡒⠂⠀⠤⠈⢈⠁⠀⠀⠠⠀⠀⢀⡀⠀⠀⢸⣿⡇⣠⣿⢷⡄⠀⠀⢺⠭⠭⢍⣂⡀⠀⠭⢠⣾⣿⢸⡟
⠀⠀⢸⡧⡟⡇⣂⣢⠤⠀⠉⣀⠂⠀⠀⡀⠀⡈⠡⠀⠀⠀⠀⢸⣿⡇⡹⠟⡟⢉⠡⠞⣬⠭⠭⠥⢒⣒⠋⡁⢸⢸⡇⣿⡇
⠀⠀⢸⣇⢗⠠⠄⠭⠍⠁⢀⣐⣂⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠼⣿⠃⠀⠀⠿⢸⠃⠆⣒⣛⠩⠭⠭⠥⠤⠄⡇⣿⣧⣿⡇
⠀⠀⢸⣿⣸⢀⣒⣒⡒⠒⠒⠂⠶⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⣁⣀⣤⡏⠰⠒⣒⣒⡒⣒⣒⣉⠉⠀⢧⣿⣿⣿⠇
⠀⠀⢸⣿⡟⠀⠀⣈⠉⠉⠀⠉⣍⣀⣒⣒⠐⠒⠒⠠⠤⣤⡀⣿⡇⠦⠍⠿⠹⢷⡌⢨⣉⡁⣀⡀⢠⡀⠀⠀⢿⡿⢹⣿
⠀⠀⢸⡿⡇⠀⠰⢹⠀⠀⠀⣀⠐⠒⠂⠠⠬⠭⠥⠬⠉⠉⢸⣿⡇⢉⣘⣒⣒⡒⠠⠌⠉⠉⠉⠁⠀⢿⠀⢸⢸⡇⣿⡟
⠀⠀⢸⣧⡇⢀⡇⢄⡀⣰⠈⠀⡀⠠⠀⠬⠤⠭⠤⠍⠀⠒⠈⣿⡷⠶⠤⠤⠤⢭⣭⡁⠼⣄⡆⠀⠈⠻⡗⣸⣼⠀⣿⡇
⠀⠀⢸⣿⡇⣬⡇⠈⠀⣾⡇⢸⡁⡀⠀⠭⠉⠭⠉⠥⠈⠁⣴⣿⡧⣖⣳⣒⣒⣒⠒⠶⠀⠨⢵⣄⡠⡀⡇⣿⣿⠈⣿⡇
⠀⠀⢸⣿⡇⢸⣷⡄⢠⣿⣠⣿⠃⠀⠐⣒⠒⣒⡒⣺⠄⠀⡏⠛⣉⣉⡒⠓⠶⠭⢭⣍⣉⣛⠀⠠⣑⠹⠁⣯⡇⣃⣿
⠀⠀⢸⣿⡇⠈⢿⡃⠘⠟⠛⠁⠘⠉⣉⣉⡥⠤⠄⠀⠒⠒⠉⠓⠢⠤⠉⡙⠛⠒⠒⠶⣤⣌⡀⢤⠉⡿⢸⣿⠀⣿⡟
⠀⠀⠸⢿⡧⠤⠴⠶⠒⠒⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂⠠⠄⣉⡁⠁⠇⢸⠏⡘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠚⠈⠉⠛⠁

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